Machine Design Databook

Source: MACHINE DESIGN DATABOOK CHAPTER 1 PROPERTIES OF ENGINEERING MATERIALS SYMBOLS5;6 a Aj Af A0 Ar Bhn d D E f" f...

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Source: MACHINE DESIGN DATABOOK

CHAPTER

1 PROPERTIES OF ENGINEERING MATERIALS SYMBOLS5;6 a Aj Af A0 Ar Bhn d D E f" f F G HB lf lj l0 Q RB RC

area of cross section, m2 (in2 ) original area of cross section of test specimen, mm2 (in2 ) area of smallest cross section of test specimen under load Fj , m2 (in2 ) minimum area of cross section of test specimen at fracture, m2 (in2 ) original area of cross section of test specimen, m2 (in2 ) percent reduction in area that occurs in standard test specimen Brinell hardness number diameter of indentation, mm diameter of test specimen at necking, m (in) diameter of steel ball, mm modulus of elasticity or Young’s modulus, GPa [Mpsi (Mlb/in2 )] strain fringe (fri) value, mm/fri (min/fri) stress fringe value, kN/m fri (lbf/in fri) load (also with subscripts), kN (lbf) modulus of rigidity or torsional or shear modulus, GPa (Mpsi) Brinell hardness number ﬁnal length of test specimen at fracture, mm (in) gauge length of test specimen corresponding to load Fj , mm (in) original gauge length of test specimen, mm (in) ﬁgure of merit, fri/m (fri/in) Rockwell B hardness number Rockwell C hardness number Poisson’s ratio normal stress, MPa (psi)

The units in parentheses are US Customary units [e.g., fps (foot-pounds-second)].

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PROPERTIES OF ENGINEERING MATERIALS

1.2

CHAPTER ONE

b c s t sf 0sf

transverse bending stress, MPa (psi) compressive stress, MPa (psi) strength, MPa (psi) tensile stress, MPa (psi) endurance limit, MPa (psi) endurance limit of rotating beam specimen or R R Moore endurance limit, MPa (psi) endurance limit for reversed axial loading, MPa (psi) endurance limit for reversed bending, MPa (psi) compressive strength, MPa (psi) tensile strength, MPa (psi) ultimate stress, MPa (psi) ultimate compressive stress, MPa (psi) ultimate tensile stress, MPt (psi) ultimate strength, MPA (psi) ultimate compressive strength, MPa (psi) ultimate tensile strength, MPa (psi) yield stress, MPa (psi) yield compressive stress, MPa (psi) yield tensile stress, MPa (psi) yield compressive strength, MPa (psi) yield tensile strength, MPa (psi) torsional (shear) stress, MPa (psi) shear strength, MPa (psi) ultimate shear stress, MPa (psi) ultimate shear strength, MPa (psi) yield shear stress, MPa (psi) yield shear strength, MPa (psi) torsional endurance limit, MPa (psi)

0sfa 0sfb sc su u uc ut su sb u suc sut y yc yt syc syt s u su y sy sf0

SUFFIXES a b c f s t u y

axial bending compressive endurance strength properties of material tensile ultimate yield

ABBREVIATIONS AISI ASA AMS ASM ASME ASTM BIS BSS DIN ISO

American Iron and Steel Institute American Standards Association Aerospace Materials Speciﬁcations American Society for Metals American Society of Mechanical Engineers American Society for Testing Materials Bureau of Indian Standards British Standard Speciﬁcations Deutsches Institut fu¨r Normung International Standards Organization

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PROPERTIES OF ENGINEERING MATERIALS PROPERTIES OF ENGINEERING MATERIALS

SAE UNS

1.3

Society of Automotive Engineers Uniﬁed Numbering system

Note: and with subscript s designates strength properties of material used in the design which will be used and observed throughout this Machine Design Data Handbook. Other factors in performance or in special aspects are included from time to time in this chapter and, being applicable only in their immediate context, are not given at this stage.

Particular

Formula

For engineering stress-strain diagram for ductile steel, i.e., low carbon steel

Refer to Fig. 1-1

For engineering stress-strain diagram for brittle material such as cast steel or cast iron The nominal unit strain or engineering strain

Refer to Fig. 1-2

The numerical value of strength of a material

"¼

lf l0 l lf A0 Af ¼ ¼ 1¼ l0 l0 l0 A0

ð1-1Þ

where lf ¼ ﬁnal gauge length of tension test specimen, l0 ¼ original gauge length of tension test specimen. F ð1-2Þ s ¼ A where subscript s stands for strength.

Point P is the proportionality limit. Y is the upper yield limit. E is the elastic limit. Y 0 is the lower yield point. U is the ultimate tensile strength point. R is the fracture or rupture strength point. R0 is the true fracture or rupture strength point.

FIGURE 1-1 Stress-strain diagram for ductile material. Subscript s stands for strength.

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PROPERTIES OF ENGINEERING MATERIALS

1.4

CHAPTER ONE

Particular

Formula

¼

The nominal stress or engineering stress

F A0

ð1-3Þ

where F ¼ applied load. F tru ¼ 0 ¼ Af

The true stress

Bridgeman’s equation for actual stress (act ) during r radius necking of a tensile test specimen

where Af ¼ actual area of cross section or instantaneous area of cross-section of specimen under load F at that instant. cal act ¼ ð1-5Þ 4r d ln 1 þ 1þ d 4r "tru ¼ "0 ¼

The true strain

l1 l2 þ l0 l0 þ l1 þ

¼ Integration of Eq. (1-6) yields the expression for true strain From Eq. (1-1) The relation between true strain and engineering strain after taking natural logarithm of both sides of Eq. (1-8)

"tru ¼ ln

l3 þ l0 þ l1 þ l2

ð lf

l0

lf l0

dli li

lf ¼1þ" l0 lf ln ¼ lnð1 þ "Þ or "tru ¼ lnð1 þ "Þ l0 " ¼ e"tru 1

Eq. (1-9) can be written as

ð1-4Þ

There is no necking at fracture for brittle material such as cast iron or low cast steel.

FIGURE 1-2 Stress-strain curve for a brittle material.

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ð1-6aÞ ð1-6bÞ ð1-7Þ ð1-8Þ ð1-9Þ ð1-10Þ

PROPERTIES OF ENGINEERING MATERIALS PROPERTIES OF ENGINEERING MATERIALS

Particular

Percent elongation in a standard tension test specimen Reduction in area that occurs in standard tension test specimen in case of ductile materials Percent reduction in area that occurs in standard tension test specimen in case of ductile materials For standard tensile test specimen subject to various loads

1.5

Formula

lf l0 ð100Þ l0 A0 Af Ar ¼ A0 A0 Af ð100Þ Ar100 ¼ A0 "100 ¼

ð1-11Þ ð1-12Þ ð1-13Þ

Refer to Fig. 1-3.

FIGURE 1-3 A standard tensile specimen subject to various loads.

The standard gauge length of tensile test specimen

pﬃﬃﬃ l0 ¼ 6:56 a

ð1-14Þ d02 df2

ð1-15Þ

lf d ¼ 2 ln 0 l0 df

ð1-16Þ

lf A 0 ¼ ¼ l0 A f

The volume of material of tensile test specimen remains constant during the plastic range which is veriﬁed by experiments and is given by

A0 l0 ¼ Af lf

Therefore the true strain from Eqs. (1-7) and (1-15)

"tru ¼ ln

The true strain at rupture, which is also known as the true fracture strain or ductility

where df ¼ minimum diameter in the gauge length lf of specimen under load at that instant, Ar ¼ minimum area of cross section of specimen under load at that instant. 1 "ftru ¼ ln ð1-17Þ 1 Ar

A0 Af

or

¼ ln

where Af is the area of cross-section of specimen at fracture.

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PROPERTIES OF ENGINEERING MATERIALS

1.6

CHAPTER ONE

Particular

Formula

Refer to Table 1-1A for values of "ftru of steel and aluminum. From Eqs. (1-9) and (1-16) Substituting Eq. (1-18) in Eq. (1-4) and using Eq. (1-3) the true stress From experimental results plotting true-stress versus true-strain, it was found that the equation for plastic stress-strain line, which is also called the strainstrengthening equation, the true stress is given by

A0 ¼1þ" Af

or Af ¼

A0 1þ"

ð1-18Þ

tru ¼ ð1 þ "Þ ¼ e"tru

ð1-19Þ

tru ¼ 0 "ntrup

ð1-20Þ

where 0 ¼ strength coeﬃcient, n ¼ strain hardening or strain strengthening exponent, "trup ¼ true plastic strain. Refer to Table 1-1A for 0 and n values for steels and other materials.

The load at any point along the stress-strain curve (Fig 1-1)

F ¼ s A0

ð1-21Þ

The load-strain relation from Eqs. (1-20) and (1-2)

F ¼ 0 A0 "ntru e"tru

ð1-22Þ

Diﬀerentiating Eq. (1-22) and equating the results to zero yields the true strain equals to the strain hardening exponent which is the instability point

"u ¼ n

ð1-23Þ

The stress on the specimen which causes a given amount of cold work W

w ¼ 0 ð"w Þn ¼

The approximate yield strength of the previously cold-worked specimen

The approximate yield strength since A0w ¼ Aw

Fw Aw

ð1-24Þ

where Aw ¼ actual cross-sectional area of the specimen, Fw ¼ applied load. F ð1-25Þ ðsy Þw ¼ w0 Aw where Aw ¼ A0w ¼ the increased cross-sectional area of specimen because of the elastic recovery that occurs when the load is removed. F ð1-26Þ ðsy Þw ¼ w0 w Aw

By substituting Eq. (1-26) into Eq. (1-24)

ðsy Þw ¼ 0 ð"w Þn

The tensile strength of a cold worked material

ðsu Þw ¼

Fu A0w

ð1-27Þ ð1-28Þ

where Aw ¼ Au , Fu ¼ A0 ðsu Þ0 , su ¼ tensile strength of the original non-cold worked specimen, A0 ¼ original area of the specimen. The percent cold work associated with the deformation of the specimen from A0 to A0w

A0 A0w A A0w ð100Þ or w ¼ 0 A0 A0 W where w ¼ 100

W¼

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ð1-29Þ

PROPERTIES OF ENGINEERING MATERIALS PROPERTIES OF ENGINEERING MATERIALS

Particular

For standard tensile specimen at stages of loading A0w is given by equation

Formula

A0w ¼ A0 ð1 wÞ

Eq. (1-31) can also be expressed as

ðsu Þ0 1w ðsu Þw ¼ ðsu Þ0 e"tru

The modulus of toughness

Valid for Aw Au or "w "u . ð "r Tm ¼ s d"

Expression for ðsu Þw after substituting Eq. (1-28)

1.7

ðsu Þw ¼

0

ð1-30Þ ð1-31Þ ð1-32Þ

ð1-33aÞ

s þ su ð1-34bÞ "r 2 where "r ¼ "u ¼ strain associated with incipient fracture.

HARDNESS The Vicker’s hardness number (HV ) or the diamond pyramid hardness number (Hp )

The Knoop hardness number

The Meyer hardness number, HM

2F sinð=2Þ 1:8544F ¼ ð1-35Þ d2 d2 where F ¼ load applied, kgf, ¼ face angle of the pyramid, 1368, d ¼ diagonal of the indentation, mm, HV in kgf/mm2 . F HK ¼ ð1-36Þ 0:07028d 2 where d ¼ length of long diagonal of the projected area of the indentation, mm, F ¼ load applied, kgf, 0:07028 ¼ a constant which depends on one of angles between the intersections of the four faces of a special rhombic-based pyramid industrial diamond indenter 172.58 and the other angle is 1308, HK in kgf/mm2 . HV ¼

HM ¼

4F d 2 =4

ð1-37Þ

where F ¼ applied load, kgf, d ¼ diameter of indentation, mm, HM in kgf/mm2 . The Brinell hardness number HB

HB ¼

2F pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D½D D2 d 2

ð1-38Þ

where F in kgf, d and D in mm, HB in kgf/mm2 . The Meyer’s strain hardening equation for a given diameter of ball

F ¼ Ad p

ð1-39Þ

where F ¼ applied load on a spherical indenter, kgf, d ¼ diameter of indentation, mm, p ¼ Meyer strain-hardening exponent.

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PROPERTIES OF ENGINEERING MATERIALS

1.8

CHAPTER ONE

Particular

Formula

The relation between the diameter of indentation d and the load F according to Datsko1;2

F ¼ 18:8d 2:53

ð1-40Þ

The relation between Meyer strain-hardening exponent p in Eq. (1-39) and the strain-hardening exponent n in the tensile stress-strain Eq. ¼ 0 "n

p2¼n

ð1-41Þ

The ratio of the tensile strength (su ) of a material to its Brinell hardness number (HB ) as per experimental results conducted by Datsko1;2 For the plot of ratio of (su =HB Þ ¼ KB against the strain-strengthening exponent n (1)

where p ¼ 2.25 for both annealed pure aluminum and annealed 1020 steel, p ¼ 2 for low work hardening materials such as pH stainless steels and all cold rolled metals, p ¼ 2.53 experimentally determined value of 70-30 brass. su KB ¼ ð1-42Þ HB Refer to Fig. 1-4 for KB vs n for various ratios of ðd=DÞ.

FIGURE 1-4 Ratio of ðsu =HB Þ ¼ KB vs strain strengthening exponent n.

The relationship between the Brinell hardness number HB and Rockwell C number RC

RC ¼ 88HB0:162 192

The relationship between the Brinell hardness number HB and Rockwell B number RB

RB ¼

HB 47 0:0074HB þ 0:154

ð1-43Þ ð1-44Þ

Courtesy: Datsko, J., Materials in Design and Manufacture, J. Datsko Consultants, Ann Arbor, Michigan, 1978, and Standard Handbook of Machine Design, McGraw-Hill Book Company, New York, 1996.

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PROPERTIES OF ENGINEERING MATERIALS PROPERTIES OF ENGINEERING MATERIALS

Particular

1.9

Formula

The approximate relationship between ultimate tensile strength and Brinell hardness number of carbon and alloy steels which can be applied to steels with a Brinell hardness number between 200HB and 350HB only1;2

sut ¼ 3:45HB

The relationship between the minimum ultimate strength and the Brinell hardness number for steels as per ASTM

sut ¼ 3:10HB

The relationship between the minimum ultimate strength and the Brinell hardness number for cast iron as per ASTM

sut ¼ 1:58HB 86:2

The relationship between the minimum ultimate strength and the Brinell hardness number as per SAE minimum strength

sut ¼ 2:60HB 110

In case of stochastic results the relation between HB and sut for steel based on Eqs. (1-45a) and (1-45b)

sut ¼ ð3:45; 0:152ÞHB

In case of stochastic results the relation between HB and sut for cast iron based on Eqs. (1-47a) and (1-47b)

sut ¼ 1:58HB 62 þ ð0; 10:3Þ MPa

¼ 500HB

¼ 450HB

SI

ð1-45aÞ

USCS

ð1-45bÞ

SI

ð1-46aÞ

USCS

ð1-46bÞ

SI

ð1-47aÞ

USCS

ð1-47bÞ

SI

ð1-48aÞ

USCS

ð1-48bÞ

SI

ð1-49aÞ

USCS

ð1-49bÞ

MPa psi

MPa psi MPa

¼ 230HB 12500 psi MPa

¼ 237:5HB 16000 psi

¼ ð500; 22ÞHB

MPa

psi

SI

ð1-50aÞ

USCS

ð1-50bÞ

¼ 230HB 9000 þ ð0; 1500Þ psi

Relationships between hardness number and tensile strength of steel in SI and US Customary units [7]

Refer to Fig. 1.5.

The approximate relationship between ultimate shear stress and ultimate tensile strength for various materials

su ¼ 0:82sut

for wrought steel

ð1-51aÞ

su ¼ 0:90sut

for malleable iron

ð1-51bÞ

su ¼ 1:30sut

for cast iron

ð1-51cÞ

su ¼ 0:90sut

for copper and copper alloy ð1-51dÞ

su ¼ 0:65sut

for aluminum and aluminum alloys ð1-51eÞ

The tensile yield strength of stress-relieved (not coldworked) steels according to Datsko1;2

sy ¼ ð0:072sut 205Þ MPa

The equation for tensile yield strength of stressrelieved (not cold-worked) steels in terms of Brinell hardness number HB according to Datsko (2)

sy ¼ ð3:62HB 205Þ MPa

The approximate relationship between shear yield strength ðsy Þ and yield strength (tensile) sy

sy ¼ 0:55sy

¼ 1:05sut 30

kpi

¼ 525HB 30 kpi

SI

ð1-52aÞ

USCS

ð1-52bÞ

SI

ð1-53aÞ

USCS

ð1-53bÞ

for aluminum and aluminum alloys ð1-54aÞ

sy ¼ 0:58sy

for wrought steel

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ð1-54bÞ

PROPERTIES OF ENGINEERING MATERIALS

1.10

CHAPTER ONE

Particular

The approximate relationship between endurance limit (also called fatigue limit) for reversed bending polished specimen based on 50 percent survival rate and ultimate strength for nonferrous and ferrous materials

Formula

For students’ use 0sfb ¼ 0:50sut

for wrought steel having sut < 1380 MPa ð200 kpsiÞ

ð1-55Þ

0sfb ¼ 690 MPa

for wrought steel having sut > 1380 MPa

ð1-56aÞ

0sfb ¼ 100 kpsi

for wrought steel having USCS sut > 200 kpsi

ð1-56bÞ

For practicing engineers’ use 0sfb ¼ 0:35sut

for wrought steel having sut < 1380 MPa ð200 kpsiÞ

ð1-57Þ

0sfb ¼ 550 MPa

for wrought steel having SI sut > 1380 MPa

ð1-58aÞ

for wrought steel having sut > 200 kpsi USCS

ð1-58bÞ

0sfb ¼ 80 kpsi 0sfb ¼ 0:45sut

for cast iron and cast steel when sut 600 MPa ð88 kpsiÞ ð1-59aÞ

0sfb ¼ 275 MPa

for cast iron and cast steel when sut > 600 MPa SI ð1-60aÞ

0sfb ¼ 40 kpsi

FIGURE 1-5 Conversion of hardness number to ultimate tensile strength of steel sut , MPa (kpsi). (Technical Editor Speaks, courtesy of International Nickel Co., Inc., 1943.)

for cast iron and cast steel when USCS ð1-60bÞ sut > 88 kpsi

0sfb ¼ 0:45sut

for copper-based alloys and nickel-based alloys

0sfb ¼ 0:36sut

for wrought aluminum alloys up to a tensile strength of 275 MPa (40 kpsi) based on 5 108 cycle life ð1-62Þ

0sfb ¼ 0:16sut

for cast aluminum alloys up to tensile strength of 300 MPa ð50 kpsiÞ based on 5 108 cycle life

0sfb ¼ 0:38sut

ð1-61Þ

ð1-63Þ

for magnesium casting alloys and magnesium wrought alloys ð1-64Þ based on 106 cyclic life

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PROPERTIES OF ENGINEERING MATERIALS PROPERTIES OF ENGINEERING MATERIALS

Particular

The relationship between the endurance limit for reversed axial loading of a polished, unnotched specimen and the reversed bending for steel specimens The relationship between the torsional endurance limit and the reversed bending for reversed torsional tested polished unnotched specimens for various materials For additional information or data on properties of engineering materials

1.11

Formula

0sfa ¼ 0:850sfb ¼ 0:43sut

ð1-65Þ

sf0 ¼ 0:580sfb ¼ 0:29sut for steel

ð1-66aÞ

sf0

ð1-66bÞ

sf0

0:80sfb

0:480sfb

0:32sut for cast iron 0:22sut for copper

ð1-66cÞ

Refer to Tables 1-1 to 1-48

WOOD Speciﬁc gravity, Gm , of wood at a given moisture condition, m, is given by

Gm ¼

W0 Wm

ð1-67Þ

where W0 ¼ weight of the ovendry wood; N ðlbfÞ; Wm ¼ weight of water displaced by the sample at the given moisture condition, N (lbf ). weight of ovendry wood and the contained water volume of the piece at the same moisture content

The weight density of wood, D (unit weight) at any given moisture content

W¼

Equation for converting of weight density D1 from one moisture condition to another moisture condition D2

D2 ¼ D1

For typical properties of wood of clear material as per ASTM D 143

Refer to Table 1-47.

ð1-68Þ 100 þ M2 100 þ M1 þ 0:0135D1 ðM2 M1 Þ

ð1-69Þ

where D1 ¼ known weight density for same moisture condition M1 , kN/m2 (lbf/ft2 ), D2 ¼ desired weight density at a moisture condition M2 , kN/m2 (lbf/ft2 ). M1 and M2 are expressed in percent.

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PROPERTIES OF ENGINEERING MATERIALS

1.12

CHAPTER ONE

TABLE 1-1 Hardness conversion (approximate) Brinell 29.42 kN (3000 kgf ) load 10 mm ball

Rockwell hardness number

Diameter (mm)

Hardness number

Vickers or Firth hardness number

2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3 55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00 4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40

745 712 682 653 627 601 578 555 534 514 495 477 461 444 429 415 401 388 375 363 352 341 331 321 311 302 293 285 277 269 262 255 248 241 235 229 223 217 212 207 201 197 192 187

840 783 737 697 667 640 615 591 569 547 528 508 491 472 455 440 425 410 396 383 372 360 350 339 328 319 309 301 292 284 276 269 261 253 247 241 234 228 222 218 212 207 202 196

A scale 0.588 kN (60 kgf ) load 84 83 82 81 81 80 79 78 78 77 76 76 75 74 73 73 72 71 71 70 69 69 68 68 67 66 66 65 65 64 64 63 63 62 61 61

B scale 0.98 kN (100 kgf ) load

C scale 1.47 kN (150 kgf ) load

15-N scale 0.147 kN (15 kgf ) load

Shore Tensile strength, sut scleroscope approximate hardness number MPa kpsi

110 109 109 108 108 107 106 106 105 104 103 102 101 100 99 98 97 96 96 95 94 93 92 91

65 64 62 60 59 58 57 55 54 52 51 50 49 47 46 45 43 42 40 39 38 37 36 34 33 32 31 30 29 28 27 25 24 23 22 21 19 18 16 15 14 13 12 10

92 92 91 90 90 89 88 88 87 87 86 85 85 84 83 83 82 81 81 80 79 79 78 77 77 76 76 75 74 74 73 73 72 71 70 70

91 87 84 81 79 77 75 73 71 70 68 66 65 63 61 59 58 56 54 52 51 50 48 47 46 45 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29

2570 2455 2350 2275 2227 2192 2124 2020 1924 1834 1750 1675 1620 1532 1482 1434 1380 1338 12961255 1214 1172 1145 1103 1069 1042 1010 983 955 928 904 875 855 832 810 790 770 748 730 714 690 680 662 645

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373 356 341 330 323 318 309 293 279 266 254 243 235 222 215 208 200 194 188 182 176 170 166 160 155 151 146 142 138 134 131 127 124 120 117 114 111 108 106 103 100 98 96 93

PROPERTIES OF ENGINEERING MATERIALS PROPERTIES OF ENGINEERING MATERIALS

1.13

TABLE 1-1 Hardness conversion (approximate) (Cont.) Brinell 29.42 kN (3000 kgf ) load 10 mm ball

Rockwell hardness number

Diameter (mm)

Hardness number

Vickers or Firth hardness number

4.45 4.50 4.55 4.60 4.65 470 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50 5.60

183 179 174 170 167 163 156 149 143 137 131 126 121 116 111

192 188 182 178 175 171 163 156 150 143 137 132 127 122 117

A scale 0.588 kN (60 kgf ) load

B scale 0.98 kN (100 kgf ) load

C scale 1.47 kN (150 kgf ) load

90 89 88 87 86 85 83 81 79 76 74 72 70 68 65

9 8 7 5 4 3 1

15-N scale 0.147 kN (15 kgf ) load

Shore Tensile strength, sut scleroscope approximate hardness number MPa kpsi 28 27 26 25 24 23 22 21 20 19 18 17

631 617 600 585 576 562 538 514 493 472 451 435 417 400 383

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91 89 87 85 83 81 78 74 71 68 65 63 60 58 55

290 125 90 80 108 225 410 430 260 410 150

Material

RQC-100a 1005-1009 1005-1009 1015 1020d 1045e 1045e 5160 9262 9262 950 931 414 345 414 441 724 1448 1669 924 1565 531 469 476 579

ST, SHg ST and RT ageh ST and AAi

MPa

HRb Plate CDc Sheet HR Sheet Normalized HR Plate Q and Tf Q and T Q and T Annealed Q and T HR Plate

Process/ Condition

68 69 84

135 60 50 60 64 105 210 242 134 227 77

kpsi

Ultimate strength, sut

379 303 469

883 400 262 228 262 634 1365 1531 455 1379 311

MPa

193 122 123 105 103 178 270 280 151 269 145 81 92 108

MPa

1331 841 848 724 710 1227 1862 1931 1041 1855 1000

Steel 128 58 38 33 38 92 198 222 66 200 48 Aluminum: 55 558 44 636 68 745

kpsi

Stress at fracture, f

kpsi

Yield strength, sy

25 35 33

67 64 80 68 62 65 51 42 14 32 72

%

Reduction in area, Af

0.28 0.43 0.41

1.02 1.02 1.60 1.14 0.96 1.04 0.72 0.87 0.16 0.38 1.24

"f

True strain at fracture

0.03 0.20 0.11

0.06 0.05 0.16 0.26 0.19 0.13 0.08 0.06 0.22 0.06 0.19

n

Strain harding exponent

131

903

66 117 120

107 166 302 308 253

738 1145 2082 2124 1744

455 807 827

170 76 77

kpsi

1172 524 531

MPa

Strength coeﬃcient, 0

a Tradename, Bethlehem steel Corp. Rolled quenched and tempered carbon steel. Used in structural, heavy applications machinery. b Hot-rolled. c cold-rolled. d low carbon, common machining steels. e Bar stock, medium carbon high-strength machining steel. f Quenched and tempered. g Solution treated, strain hardened. h Solution treated and RT age. i Solution treated and artiﬁcially aged. Source: SAE j1099, Technical Report on Fatigue properties, 1975.

2024-T351 2024-T4 7075-T6

Brinell hardness HB

TABLE 1-1A Mechanical properties of some metallic materials

PROPERTIES OF ENGINEERING MATERIALS

1.14

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25 mm (1 in) bar 25 mm (1 in) bar

4130 4340

Aged 4828C Aged 4828C Aged 4828C

876

CD 20% + s.r.2h (9008F WQ + (12008F) OQ + (1000 8F) OQ + (800 8F) 1540 1760 1980

814 1262 1531

517 621 805 965 634

586

455 620 710 790 448

Annealed HR CD 20% CD 50% Annealed

(12008F)

CD 0% CD 30% CD 60% CD 80% HR

Condition/Process MPa

225 256 288

118 183 200

127

75 90 117 140 92

85

66 90 102 115 65

kpsi

1480 1630 1920

703 1172 1379

696

352 414 670 855 365

441

275 585 605 660 331

215 237 279

102 170 200

101

51 60 97 124 53

64

40 85 88 96 48

MPa kpsi

724 1310 1517

MPa

105 190 220

kpsi

876 127 1007 146

MPa kpsi

752 855

241

427

269 296 370 410 365

296

240 315 350 365 241

MPa

690 690 760

490 109 669 124 469

35

MPa kpsi

100 100 110

71 97 68

62

d

39 43a 54d 60d 53

43

35d 46d 51d 53d 35d

kpsi

Fatigue limit, sf

207

204

GPa

30.0

29.6

Mpsi

Young’s modulus, E

81

79

83

11.7

11.4

12.0

110 75

100 68

55 62 50

64 52 47

31

57 50 44 25 40

70

70 62 54 26 59

Modulus of Fracture rigidity, G toughness, K IC Reduction in area GPa Mpsi GPa Mpsi A, %

0.80 0.97 0.69

1.02 0.73 0.63

0.37

0.84 0.69 0.58 0.33 0.51

1.20

1.20 0.97 0.78 0.30 0.89

True strain at fracture, "f

c

b

A description of the materials and typical uses follows the table. CD ¼ cold drawn (the percentage reduction in area); HR ¼ hot rolled; OQ ¼ oil quenched; WQ ¼ water quenched (temperature following is the tempering temperature); s:r: ¼ stress relieved. Smooth-specimen rotating-beam results, unless noted A (¼ axial). d 106 cycles. Source: Extracted from Kenneth S. Edwards, Jr, and Robert B. McKee, Fundamentals of Mechanical Component Design, McGraw-Hill, Inc., 1991, which is drawn from the Structural Alloys Handbook, published by the Metals and Ceramics Information Center, Battelle Memorial Institute, Columbus, Ohio, 1985.

a

18% Ni maraging 200 L plate 250 L plate 300 L plate

25 mm (1 in) bar

25 mm (1 in) bar or plate 25 mm (1 in) WQ bar or plate 25 mm (1 in) bar

1050

1040

1030

1020

Steel: 1016

Material Form

Ultimate Yield strength Shear (torsional) strength tensile strength, sut Tensile, syt Compressive, syc Ultimate, su Yield, sy

TABLE 1-1B Mechanical properties of some typical metallic materials

PROPERTIES OF ENGINEERING MATERIALS

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1.15

PROPERTIES OF ENGINEERING MATERIALS

1.16

CHAPTER ONE

TABLE 1-2 Poisson’s ratio ðÞ Material

Material

Aluminium, cast Aluminium, drawn Beryllium copper Brass Brass, 30 Zn Cast steel Chromium Copper Douglas ﬁr Ductile iron Glass Gray cast iron Iron, soft Iron, cast Inconel x Lead Magnesium Malleable cast iron

0.330 0.348 0.285 0.340 0.350 0.265 0.210 0.343 0.330 0.340–0.370 0.245 0.210–0.270 0.293 0.270 0.410 0.431 0.291 0.230

Molybdenum Monel metal Nickel, soft Nickel, hard Rubber Silver Steel, mild Steel, high carbon Steel, tool Steel, stainless (18-8) Tin Titanium Tungsten Vanadium Wrought iron Zinc

0.293 0.320–0.370 0.239 0.306 0.450–0.490 0.367 0.303 0.295 0.287 0.305 0.342 0.357 0.280 0.365 0.278 0.331

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120

121

40

Automotive ASTMA602, SAE J158

ASTM A197 Perlite and martensite: ASTM A220 ANSI G48-2 MIL-1-11444B

Malleable cast iron: Ferrite ASTM A47-52, A338, ANSI G 48-1 FED QQ-1-66e

60

50

111

35

SAE 110

30

Gray cast iron ASTM class 20 25

b

Material, class, speciﬁcation

365

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517

621

M5503e

M7002e

724

517

M5003d

M8501

448

e

345

M4504d

414 448 448 448 483 517 552 552 586 655 724

40010 45008 45006 45010 50005 50007 60004 60003 70003 80002 90001 Grade M3210c

276

345

35018

105

90

75

75

65

50

60 65 65 65 70 75 80 80 85 95 105

40

53

50

62.5

431

Class or grade 32510

52.5

42.5

36.5

31

22 26

kpsi

362

293

252

214

152 179

MPa

Tension, sut

242 242 242 242

1670 1670 1670

220

208

187.5

164

140

124

109

83 97

kpsi

1670

1517

1434

1293

1130

965

855

752

572 669

MPa

Compression, suc

Ultimate strength

689

552

517

338

352

324

496

448

393

338

303

220 255

MPa

100

80

75

49

51

47

72

65

57

49

44

32 37

kpsi

Shear, su

610

503

393

334

276

179 220

MPa

88.5

73

57

48.5

40

26 32

kpsi

Torsional/ shear strength, s

TABLE 1-3 Mechanical properties of typical cast ferrous materialsa

586

483

379

345

310

224

276 310 310 310 345 345 414 414 483 552 621

207

241

220

MPa

85

70

55

50

45

32

40 45 45 45 50 50 60 60 70 80 90

30

35

32

kpsi

Yield strength, sy

276

270

255

220

214

193

169

148

128

110

97

69 79

MPa

40

39

37

32

31

28

24.5

21.5

18.5

16

14

10 11.5

kpsi

Endurance limit in reversed bending, sfb

269–302

229–269

187–241

187–241

163–217

156 max

149–197 156–197 156–207 185 179–229 204 197–241 226 217–269 241–285 269–321

156 max

156 max

156 max

302

262

235

212

210

156 174

186

186

183

180

172

172

141–162

130–157

27

27

26.5

26

25

25

20.4–23.5

18.8–22.8

16.0–20.0

14.5–17.2

13.0–16.4

9.6–14.0 11.5–14.8

Mpsi

Tension, E

110–138

10–119

90–113

66–97 79–102

Brinell hardness, HB GPa

160

160

160

160

172

172

GPa

27

23.2

23.2

23.2

25

25

Mpsi

Compression, E

Modulation of elasticity

Mpsi

54–59 7.8–8.5

50–55 7.2–8.0

44–54 6.4–7.8

40–48 5.8–6.9

36–45 5.2–6.6

27–39 3.9–5.6 32–41 4.6–6.0

GPa

Shear, G

1

2

3

3

4

10

10 8 6 10 5 7 4 3 3 2 1

5

18

10

Elongation in 50 mm (2 in), %

19

19

19

19

22

22

156

108

95

75 75

J

14

14

14

14

16.5

16.5

115

80

70

55 55

ft-lbf

Impact strength (Charpy)

Steering gear housing, mounting brackets Compressor crankshafts and hubs Parts requiring selective hardening, as gears For machinability and improved induction hardening Connecting rods, universal joint yokes Gears with high strength and good wear resistance

General engineering service at normal and elevated temperatures

General purpose at normal and elevated temperature, good machinability, excellent shock resistance. Pipe ﬂanges, valve parts

Soft iron castings Cylinder blocks and heads, housing Flywheels, brake drums and clutch plates Heavy-duty brake drums, clutch plates Cam shafts, cylinder liners Special high-strength castings Special high-strength castings

Typical application

PROPERTIES OF ENGINEERING MATERIALS

1.17

1.18

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g

f

e

d

c

b

a

24–45 30–90 25–45 20–45 34–90 60–100 55–60 58–65 16 35–55

60 65 80 100

141.3

110

81.1

67.3

90–150 100 100–160 70–100

133.5

220.0

56.0

52.5

52.0

kpsi

1240–1380 180–200

620–1040 690 690–1100 480–690

920

1515

386

362

359

MPa

Compression, suc

875

504

475

472

126.9

73.1

68.9

68.5

MPa kpsi

Shear, su MPa kpsi

Torsional/ shear strength, s

40 45 55 70

125.3

72.5

52.5

48.2

47.7

60

40

kpsi

193–241 28–35

276 310 379 483

864

500

362

332

329

414

276

MPa

Yield strength, sy

434

379

345

241

MPa

63

55

50

35

kpsi

Endurance limit in reversed bending, sfb

Source: Compiled from AMS Metals Handbook, American Society for Metals, Metals Park, Ohio, 1988. Minimum values of u in MPa (kpsi) are given by class number. Annealed. Air-quenched and tempered. Liquid-quenched and tempered. Heat-treated and average mechanical properties. Calculated from tensile modulus and Poisson’s ratio in tension.

170–310 210–620 170–310 140–310 235–620 415–690 380–415 400–450 110 241–380

414 448 552 689

974

758

F34800

F36200

559

464

F33100

F33800

461

F32800

66.9

80

552

F34100

kpsi

60

MPa

UNS No. F32800 414

Alloy cast irons Medium-silicon gray iron High chromium gray iron High nickel gray iron Ni-Cr-Si gray iron High-aluminum gray iron Medium-silicon ductile iron High-nickel ductile iron (20Ni) High-nickel ductile iron (23Ni) Durion Mechanite

SAE j 434C

120-9002h D4018 D4512 D5506 D7003

80–55– 06h 100-7003h

Nodular (ductile) cast iron Grade 60-40-18 ASTM A395-76 ASME SA 395 80-60-03 ASTM A476-70(d) SAE AMS5316 ASTM 60-40-18h A536-72 MIL-I-11466 B(MR) 65-45-12h

Material, class, speciﬁcation

Tension, sut

Ultimate strength

TABLE 1-3 Mechanical properties of typical cast ferrous materialsa (Cont.)

170–250 250–500 130–250 110–210 180–350 140–300 140–200 130–170 520 190

170 max 156–217 187–255 241–302

332

257

192

167

167–178

201 min

143–187

158 83

164

162

168

168

169

Brinell hardness, HB GPa

23 12

23.8

23.5

24.4

24.4

24.5

Mpsi

Tension, E

164

165

163

164

GPa

23.8

23.9

23.6

23.8

Mpsi

Compression, E

Typical application

9.0–9.3g 11.2

62–64g

10

18 12 6 3

63.5–64g 9.2–9.3g 1.5

6-10

9.3–9.4g 15

64–65g

3

15–23 20–35 60–150 80–150 5–115 12 28 3

20–31 27–47 80–200 110–200 7–155 16 38 4

Pressurecontaining parts such as valve and pump bodies Machine components subjected to shock and fatigue loads Crankshafts, gears and rollers High-strength gears and machine components Pinions, gears, rollers and slides Steering knuckles Disk brake calipers Crankshafts Gears

ft-lbf

Impact strength (Charpy)

63–65.5g 9.1–9.5g 15

Elongation in 50 mm (2 in), % J

Valves and ﬁttings for steam and chemical plant equipment Paper-mill dryer rollers

Mpsi

Shear, G

18

GPa

Modulation of elasticity

PROPERTIES OF ENGINEERING MATERIALS

37.7 10.6 24.5

43.5 12.2 28.3

50.8 14.2 33.1

58.0 16.2 37.7

FG 260 260 73c 169d

FG 300 300 84c 195d

FG 350 350 98c 228d

FG 400 400 112c 260d

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174.1 32.5 75.4

156.6 28.4 66.0

139.2 24.4 56.6

125.3 21.2 49.0

111.4 17.8 41.5

104.4 16.2 37.7

87.0 12.2 15.2

kpsi

460

403

345

299

253

230

173

MPa

66.7

58.5

50.0

43.4

36.7

33.4

25.1

kpsi

Shear strength, s kpsi 9.9 9.9 13.1 12.6 14.4 13.6 17.0 15.7 19.6 18.4 21.6 18.7 2.0 18.4

MPa 68e 68f 90e 87f 99e 94f 117e 108f 135e 127f 149e 129f 152e 127f

Fatigue limit, sf

145

140

135

128

120

114

100

GPa

21.0

20.3

19.6

18.6

17.4

16.5

14.5

Mpsi

Tension

145

140

135

128

120

114

100

GPa

21.0

20.3

19.6

18.6

17.4

16.5

14.5

Mpsi

Compression

58

56

54

51

48

46

40

GPa

8.4

8.1

7.8

7.4

7.0

6.7

5.8

Mpsi

Modulus of rigidity, G

320a 400b

280a 250b

240a 300b

208a 260b

176a 120b

160a 200b

120a 150b

MPa

46.4 58.0

40.6 50.8

34.8 43.5

30.2 37.7

25.5 32.0

23.2 29.0

17.4 21.8

kpsi

Notched tensile strength, snt

0.28

0.25

0.22

0.20

0.18

0.17

0.15

Elastic strain at failure, % Brinell hardness HB

0.50g

0.50g

0.50g

0.57g

207–270

207–241

180–230

180–230

0.39–0.63g 180–220

0.48–0.67g 160–220

0.6–0.75g 130–180

Total elastic strain at fracture, %

0.26

0.26

0.26

0.26

0.26

0.26

0.26

7300

7300

7250

7200

7150

7100

7050

455.7

455.7

452.6

449.5

446.4

443.3

440.1

11.0

11.0

11.0

11.0

11.0

11.0

11.0

6.1

6.1

6.1

6.1

6.1

6.1

6.1

0.460

0.460

0.460

0.460

0.420

0.375

26.5

0.1089

0.1089

0.1098

0.1098

0.1003

0.0896

0.0640

Speciﬁc heat capacity at 20 to 2008C, c

Poisson’s Density, ratio, kg/m3 lbm /ft3 mm/mK min/in8F kJ/kg K Btu/lbm 8F

Coeﬃcient of the thermal expansion, , 20 to 2008C

44.0

45.7

47.4

48.8

50.1

50.8

52.5

7.75

8.05

8.35

8.59

8.82

8.95

9.25

W/m2 K Btu/ft2 h8F

Thermal conductivity at 1008C, K

Note: The typical properties given in this table are the properties in a 30 mm (1.2 in) diameter separately cast test bar or in a casting section correctly represented by this size of test bar, where the tensile strength does not correspond to that given. Other properties may diﬀer slightly from those given. Source: IS (Indian Standards) 210, 1993.

h

g

f

e

d

c

b

1200 224 520

1080 196 455

960 168 390

864 146 338

768 123 286

720 112 260

600 84 195

MPa

Compressive strength, sc

Modulus of elasticity, E

Circumferential 458 notch-root radius 0.25 mm (0.04 in), notch depth 2.5 mm (0.4 in), root diameter 20 mm (0.8 in), notch depth 3.3 mm (0.132 in), notch diameter 7.6 mm (0.36 in). Circumferential notch radius 9.5 mm (0.38 in), notch depth 2.5 mm (0.4 in), notch diameter 20 mm (0.8 in). 0.01% proof stress. 0.1% proof stress. Unnotched 8.4 mm (0.336 in) diameter. V-notched [circumferential 458 V-notch with 0.25 mm (0.04 in) root radius, diameter at notch 8.4 mm (0.336 in), depth of notch 3.4 mm (0.135 in)]. Values depend on the composition of iron. Poisson’s ratio ¼ 0:26.

32.0 9.0 20.7

FG 220 220 62e 143d

a

29.0 8.1 18.8

FG 200 200 56c 130d

kpsi

21.8 6.0 14.2

MPa

FG 150 150 42c 98d

Grade

Tensile strength, st

TABLE 1-4 Typical mechanical properties of gray cast iron

PROPERTIES OF ENGINEERING MATERIALS

1.19

30–60 61–200 30–60 61–200 30–60 61–200 30–60 61–200 30–60 61–200 30–60 61–200

mm

1.20

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550 362 318 286 272 253 216 216 181

SG 900/2 SG 800/2 SG 700/2 SG 600/2 SG 500/7 SG 450/10 SG 400/15 SG 400/18 SG 350/22

79.8 52.5 46.1 41.5 39.5 36.7 31.3 31.3 31.3

kpsi

MPa

810 720 630 540 45 405 360 360 315

7150 7200 7200 7170 7100 7100 7100 7100 7100

kg/m3

117.5 107.4 91.4 78.3 65.3 58.7 52.2 52.2 45.7

kpsi

Shear strength, sc

1.2–2.4 2.44–8.0 1.2–2.4 2.44–8.0 1.2–2.4 2.44–8.0 1.2–2.4 2.44–8.0 1.2–2.4 2.44–8.0 1.2–2.4 2.44–8.0

in

317 304 280 248 224 210 195 195 180

MPa

46.0 44.1 40.6 35.0 32.5 30.5 28.3 28.3 26.1

kpsi

Fatigue limit, sc

446.4 449.5 449.5 447.6 443.3 443.3 443.3 443.3 443.3

lbm /ft3

Density

0.275 0.275 0.275 0.275 0.275 0.275 0.275 0.275 0.275

67.1 68.6 86.6 67.9 65.9 65.9 65.9 65.9 65.9

GPa

Poisson’s ratio, kpsi

MPa

kpsi

0.2% Proof stress, sy min

9.73 9.95 9.95 9.85 9.56 9.56 9.56 9.86 9.56

Mpsi

Modulus of, Elasticity E

169 169 169 169 169 174 176 176 169

Ten

GPa

169 169 169 169 169 174 176 176 169

Com

24.5 24.5 24.5 24.5 24.5 25.2 25.2 25.2 24.5

Ten

24.5 24.5 24.5 24.5 24.5 25.2 25.2 25.2 24.5

MPsi Com

Modulus of rigidity, G

11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0 11.0

lm/m K

Measured on test pieces from cast-on test samples 700 101.5 400 58.0 2 650 94.3 380 55.1 1 600 87.0 360 52.2 2 550 79.8 340 49.3 1 450 65.3 300 43.5 7 420 61.0 290 42.0 5 390 56.6 250 36.3 15 370 53.7 240 34.8 12 390 56.6 250 36.4 15 370 53.7 240 34.8 12 330 47.9 2231.9 18 320 46.4 210 30.6 15 150

130–180

130–180

170–240

180–270

220–320

280–360 245–335 225–305 190–270 160–240 160–210 130–180 130–180 150

Brinell hardness, HB

14 12b 17b 15b

b

10.3 (8.1) 8.8 (6.6) 12.5 (10.3) 11.1 (8.8)

0.461 0.461 0.461 0.461 0.461 0.461 0.461 0.461 0.461

0.1101 0.1101 0.1101 0.1101 0.1101 0.1101 0.1101 0.1101 0.1101

Btu/lbm 8F

Speciﬁc heat, c at 208 to 2008C kJ/kg K

(11) (9)c (14)c (12)c

c

33.5 31.40 31.40 32.80 35.50 36.5 36.5 36.5 36.5

W/m2 K

5.90 5.53 5.53 5.72 6.25 6.43 6.43 6.43 6.43

Btu/ft2 h8F

Thermal conductivity, at 1008C

Ferrite

Ferrite

Ferrite

Ferrite + pearlite

Ferrite + pearlite

Pearlite

Ferrite

Ferrite

Pearlite Pearlite Ferrite and pearlite Ferrite and pearlite

Predominant structural constituent

Mean value from 3 tests on V-notch test pieces at ambient

6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1

lin/in 8F at 208 to 2008C

b

6.6 (3.2) 12.5 (11.0) 10.3 (8.1) 12.5 (10.3)

ft-lbf

Impact values min (23 58C)

9.0b (4.3)c 17.0b (15.0)c 14.0b (11.0)c 17.0b (14.0)c

J

Thermal coeﬃcient of linear expansion,

Elongationa %, min

Measured on test pieces from separately cast test samples 900 130.5 600 87.0 2 800 116.0 480 69.6 2 700 101.5 420 61.0 2 600 87.0 370 53.7 2 500 72.5 320 46.4 7 450 65.3 310 45.0 10 400 58.0 250 36.3 15 400 58.0 250 36.6 18 350 50.8 220 32.0 22

MPa

Tensile strength, st min

a Elongation is measured on an initial gauge length L ¼ 5d where d is the diameter of the gauge length of the test pieces. c Individual value. temperature. Source: IS 1865, 1991.

MPa

Compression strength, sc

Grade

SG 350/22A

SG 400/18A

SG 400/15A

SG 500/7A

SG 600/3A

SG 700/2A

SG 900/2 SG 800/2 SG 700/2 SG 600/2 SG 500/7 SG 450/10 SG 400/15 SG 400/18 SG 350/22

Grade

Typical casting thickness

TABLE 1-5 Mechanical properties of spheroidal or nodular graphite cast iron

PROPERTIES OF ENGINEERING MATERIALS

3.0 3.0 3.0 3.0 3.0 2.6 2.6 2.6 2.6 2.4 2.4

ASG Ni 13 Mn 7 ASG Ni 20 Cr 2 ASG Ni 20 Cr 3 ASG Ni 20 Si 5 Cr 2 ASG Ni 22 ASG Ni 23 Mn 4 ASG Ni 30 Cr 1 ASG Ni 30 Cr 3 ASG Ni 30 Si 5 Cr 5 ASG Ni 35 ASG Ni 35 Cr 3

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140–150 112–130

112–133 112–133

85–112 120–140 112–130 92–105 91

112–140 112–123

ASG Ni 13 Mn 7 ASG Ni 20 Cr 2

ASG Ni 20 Cr 3 ASG Ni 20 Si 5 Cr 2

ASG Ni 22 ASG Ni 23 Mn 4 ASG Ni 30 Cr 1 ASG Ni 30 Cr 3 ASG Ni 30 Si 5 Cr 5

ASG Ni 35 ASG Ni 35 Cr 3

16.2–20.3 16.2–17.8

12.3–16.2 17.4–20.3 16.2–18.9 13.3–15.2 13.2

16.2–19.3 16.2–19.3

20.3–21.8 16.2–18.9

Mpsi

12.0–14.0 18.0–22.0 18.0–22.0 18.0–22.0 21.0–24.0 22.0–24.0 28.0–32.0 28.0–32.0 28.0–32.0 34.0–36.0 34.0–36.0

Ni

Thermal coeﬃcient of linear expansion,

6.0–7.0 0.5–1.5 0.5–1.5 0.5–1.5 1.5–2.5 4.0–4.5 0.5–1.5 0.5–1.5 0.5–1.5 0.5–1.5 0.5–1.5

Mn

5 5

18.4 14.7 12.6 12.6 14.4

18.7 18.0

18.2 18.7

2.8 2.8

10.2 8.2 7.0 7.0 8.0

10.4 10.0

10.1 10.4

lm/m K lin/in 8F at 20 to 2008C

2.0–3.0 1.5–3.0 1.5–3.0 4.5–5.5 1.0–3.0 1.5–2.5 1.5–3.0 1.5–3.0 5.0–6.0 1.5–3.0 1.5–3.0

Si 0.080 0.080 0.080 0.080 0.080 0.080 0.080 0.080 0.080 0.080 0.080

0.3 1.0–2.5 2.5–3.5 1.0–2.5 <0.5 <0.2 1.0–1.5 2.5–3.5 4.5–5.5 0.2 2.0–3.0

W/m2 K

12.6 12.6

12.6 12.6 12.6 12.6 12.6

12.6 12.6

12.6 12.6

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

Cumax

2.22 2.22

2.22 2.22 2.22 2.22 2.22

2.22 2.22

2.22 2.22

Btu/ft2 h8F

Thermal conductivity, K

Pmax

Cr 455.7 462.0 462.0 462.0 462.0 462.0 462.0 462.0 462.0 474.5 474.5

lbm /ft3 390–460 370–470 390–490 370–430 370–440 440–470 370–440 370–470 390–490 370–410 370–440

MPa 56.6–66.7 53.7–68.2 56.6–71.1 53.7–62.4 53.7–63.8 63.6–68.2 53.7–62.4 53.7–68.2 56.6–70.5 53.7–59.5 53.7–63.8

kpsi

Tensile strength, st min

30.5–37.7 30.5–36.3 30.5–37.7 30.5–37.7 24.7–36.3 30.5–34.8 30.5–39.2 30.5–37.7 34.8–45.0 30.5–34.8 30.5–42.1

kpsi 15–25 7–20 7–15 10–18 20–40 25–45 13–18 7–18 1–4 20–40 7–10

Elongationb %, min

Properties and applications

210–260 210–250 210–260 210–260 170–250 210–240 210–270 210–260 240–310 210–240 210–290

MPa

0.2% proof stress, sy min

130–170 140–200 150–255 180–230 130–170 150–180 130–190 140–200 110–250 130–180 140–190

15.0–27.5 13.5–27.5 12.0 14.9 20.0–33.0 24.0 17.0 8.5 3.9–5.9 20.5 7.0

11.1–20.3 10.0–20.3 8.9 11.0 14.8–24.3 17.7 8.1 6.3 2.9–4.4 15.1 5.2

Brinell Impact valuesd , min hardness, HB J ft-lb

Non-magnetic. Hence used as pressure covers for turbine generator sets, housing for insulators, ﬂanges and switch gears. Corrosion and heat resistance. Used in pumps, valves, compressor exhaust gas manifolds, turbo-supercharger housings and bushings. Good resistance to corrosion. Used in valves, pump components and components subject to high pressure. High value of linear expansion and non-magnetic. Used for pumps, valves, compressor and exhaust gas manifold and turbocharge housings. High impact properties up to 1968C and non-magnetic. Used in castings for refrigerators, etc. Good bearing properties. Used in exhaust manifolds and pumps, valves and turbocharger gas housing. Used in boiler pumps, valves, ﬁlter parts and exhaust gas manifolds. Used in pump components, valves, etc. Power lower linear coeﬃcient of expansion. Used in machine tool parts, scientiﬁc instruments, glass molds, and parts requiring dimensional stability. Possess lower linear thermal expansion. Used in gas turbine housings and glass molds.

7300 7400 7400 7400 7400 7400 7400 7400 7400 7600 7600

kg/m3

Density

b

Unless otherwise speciﬁed, other elements may be present at the discretion of the manufacturer, provided they do not alter the micro-structure substantially, or aﬀect the property adversely. Elongation is measured on an initial gauge length L ¼ 5d where d is the diameter of the gauge length of the test pieces. c Measured on test pieces machined from separately cast test samples. d Mean value from 3 tests on V-notch test pieces at ambient temperature. Source: IS 2749, 1974.

a

GPa

Grade

Modulus of elasticity E

Cmax

Grade

Chemical compositiona , %

TABLE 1-5A Chemical compositiona and mechanical propertiesc of spheroidal graphite austenitic cast iron

PROPERTIES OF ENGINEERING MATERIALS

1.21

1.5–3.0 1.0–2.8 1.0–2.8 1.0–2.8 1.0–2.8 4.5–5.5 1.0–2.0 5.0–6.0 1.0–2.0

3.0 3.0 3.0 3.0 3.0 2.5 2.5 2.5 2.4

AFG Ni 13 Mn 7 AFG Ni 15 Cu 6 Cr 2 AFG Ni 15 Cu 6 Cr 3 AFG Ni 20 Cr 2 AFG Ni 20 Cr 3 AFG Ni 20 Si 5 Cr 3 AFG Ni 30 Cr 3 AFG Ni 30 Si 5 Cr 5 AFG Ni 35

12.4 14.6 5.0

AFG Ni 30 Cr 3 AFG Ni 30 Si 5 Cr 5 AFG Ni 35

6.9 8.1 2.8

460–500 460–500 460–500

460–500 460–500 460–500 460–500 460–500 460–500

0.11–0.12 0.11–0.12 0.11–0.12

0.11–0.12 0.11–0.12 0.11–0.12 0.11–0.12 0.11–0.12 0.11–0.12

Btu/lbm 8F

7300 7300 7300 7300 7300 7300 7300 7300 7300

kg/m3

37.7–41.9 37.7–41.9 37.7–41.9

37.7–41.9 37.7–41.9 37.7–41.9 37.7–41.9 37.7–41.9 37.7–41.9

W/m2 K

6.64–7.38 6.64–7.38 6.64–7.38

6.64–7.38 6.64–7.38 6.64–7.38 6.64–7.38 6.64–7.38 6.64–7.38

Btu/ft2 h8F

Thermal conductivity, K

0.5 5.5–7.5 5.5–7.5 0.5 0.5 0.5 0.5 0.5 0.5

Cu 140–220 170–210 190–240 170–210 190–240 190–280 190–240 170–240 120–180

MPa 20.3–32.0 24.7–30.5 27.6–34.8 24.7–30.5 27.6–34.8 27.6–40.6 27.6–34.8 27.4–34.8 17.4–26.1

kpsi

Tensile strength, st min

– 2 1–2 2–3 1–2 2–3 1–3 – 1–3

Elongation %, min 630–840 700–840 860–1100 700–840 860–1100 860–1100 700–910 560 560–700

MPa 91.4–121.8 101.5–121.8 124.7–159.5 101.5–121.8 124.7–159.5 124.7–159.5 101.5–132.0 81.2 81.2–101.5

kpsi

Ultimate compressive strength, sut

Properties and applications

120–150 140–200 150–250 120–215 160–250 140–250 120–215 150–210 120–140

Brinell hardness, HB

70–90 85–105 98–113 85–105 98–113 110 98–113 105 74

GPa

10.2–13.1 12.3–15.2 14.2–16.4 12.3–15.2 14.2–16.4 16.0 14.2–15.2 15.2 10.7

Mpsi

Modulus of elasticity, E

Non-magnetic. Used in pressure covers for turbine generator sets, housing for switch gears and terminals, and ducts. Resistance to corrosion, erosion, and heat. Good bearing properties. Used for pumps, valves, piston ring covers for pistons, furnace components, bushings. Possess high coeﬃcient of thermal expansion, resistance to corrosion and erosion. Used for pumps handling alkalis. Used in soap, food and plastic industries. Resistance to erosion, corrosion, heat. Used in high temperature application. Not suitable between 500 and 6008C. Resistance to thermal shock and heat, corrosion at high temperature. Used in pumps, pressure vessels, valves, ﬁlters, exhaust gas manifolds, turbine housings. Resistance to erosion, corrosion, and heat. Possess average thermal expansion. Used in components for industrial furnaces, valves, and pump components. Possess low thermal expansion and resistant to thermal shock. Used for scientiﬁc instruments, glass molds and in such other parts where dimensional stability is required

455.7 455.7 455.7 455.7 455.7 455.7 455.7 455.7 455.7

lbm /ft3

Density

1.22

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b

Unless otherwise speciﬁed other elements may be present at the discretion of the manufacturer, provided they do not alter the microstructure substantially, or aﬀect the properties adversely. Measured on test pieces machined from separately cast test pieces/samples. Source: IS 2749, 1974.

a

17.7 18.7 18.7 18.7 18.7 18.0

AFG Ni 13 Mn 7 AFG Ni 15 Cu 6 Cr 2 AFG Ni 15 Cu 6 Cr 3 AFG Ni 20 Cr 2 AFG Ni 20 Cr 3 AFG Ni 20 Si 5 Cr 3

9.3 10.4 10.4 10.4 10.4 10.0

lm/m K lin/in 8F at 20 to 2008C

Grade

0.2 1.0–2.5 2.5–3.5 1.0–2.5 2.5–3.5 1.5–4.5 2.5–3.5 4.5–5.5 0.2

Cr

Speciﬁc hear, c

12.0–14.0 13.5–17.5 13.5–17.5 18.0–22.0 18.0–22.0 18.0–22.0 28.0–32.0 29.0–32.0 34.0–36.0

Ni

J/kg K

6.0–7.0 0.5–1.5 0.5–1.5 0.5–1.5 0.5–1.5 0.5–1.5 0.5–1.5 0.5–1.5 0.5–1.5

Mn

Thermal coeﬃcient of linear expansion,

Si

C

Grade

Chemical composition, %

TABLE 1-5B Chemical compositiona and mechanical propertiesb of ﬂake graphite austenitic cast iron

PROPERTIES OF ENGINEERING MATERIALS

(C 07) (C 10y) (C 14y) (C 15) (C 15 Mn 75) (C 20) (C 25) (C 25 Mn 75+) (C 30+) (C 35) (C 35 Mn 75+) (C 40+) (C 45+) (C 50+) (C 50 Mn 1) (C 55 Mn 75+) (C 60) (C 65)

7C4 10 C 4 14 C 6 15 C 4 15 C 8 20 C 8 25 C 4 25 C 8 30 C 8 35 C 4 35 C 8 40 C 8 45 C 8 50 C 4 50 C 12 55 C 8 60 C 4 65 C 6

0.12 max 0.15 max 0.10–0.18 0.20 max 0.10–0.20 0.15–0.25 0.20–0.30 0.20–0.30 0.25–0.35 0.30–0.40 0.30–0.40 0.35–0.45 0.40–0.50 0.45–0.55 0.45–0.55 0.50–0.60 0.55–0.65 0.60–0.70

%C 0.50 max 0.30–0.60 0.40–0.70 0.30–0.60 0.60–0.90 0.60–0.90 0.30–0.60 0.60–0.90 0.60–0.90 0.30–0.60 0.60–0.90 0.60–0.90 0.60–0.90 0.60–0.90 1.10–1.40 0.60–0.90 0.50–0.80 0.50–0.80

% Mn 320–400 340–420 370–450 370–490 420–500 440–520 440–540 470–570 500–600 520–620 550–650 580–680 630–710 660–780 720 min 720 min 750 min 750 min

MPa 46.5–58.0 49.4–70.0 53.6–65.0 53.6–71.0 61.0–72.5 63.5–75.4 63.5–78.3 68.2–82.7 72.5–87.0 75.4–90.0 79.8–94.3 84.1–98.7 91.4–103.0 95.7–113.1 104.4 min 104.4 min 108.8 min 108.8 min

kpsi

Tensile strength, st

Notes: a , area of cross section; y steel for hardening; + steel for hardening and tempering; Mn 75 ¼ average content of Mn is 0.75%. Source: IS 1570, 1979.

Old

New

Designation

TABLE 1-6 Carbon steels with speciﬁed chemical composition and related mechanical properties

27 26 26 25 25 24 23 22 21 20 20 18 15 13 11 13 11 10

Elongation, % (gauge length pﬃﬃﬃﬃ ﬃ 5.56 a round test piece)

40.6 40.6

40.6 40.6 30.5 30.5

55 55 41.35 41.35

ft-lbf

55 55

J

Izod impact value, min (if speciﬁed)

PROPERTIES OF ENGINEERING MATERIALS

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1.23

(10 S 11) (14 Mn 1 S 14) (25 Mn 1 S 14) (40 S 18) (11 S 25) (40 Mn 2 S 12)

10 14 25 40 11 40

0.15 max 0.10–0.18 0.20–0.30 0.35–0.45 0.08–0.15 0.35–0.45

%C 0.05–0.30 0.05–0.30 0.25 max 0.25 max 0.10 max 0.25 max

% Si 0.60–0.90 1.20–1.50 1.00–1.50 0.80–1.20 0.80–1.20 1.30–1.70

% Mn 0.08–0.13 0.10–0.18 0.10–0.18 0.14–0.22 0.20–0.30 0.08–0.15

%S 0.060 0.060 0.060 0.060 0.060 0.060

370–490* 440–540 500–600* 550–650* 370–490* 600–700*

%P (max) MPa

53.7–71.0 63.8–78.3 72.5–87.0 79.8–94.0 53.7–71.0 87.0–101.5

kpsi

Tensile strength, st

24* 22* 20* 17* 22* 15*

Minimum elongation, % (gauge length pﬃﬃﬃﬃﬃﬃ ﬃ 5.65 a )

40.6

30.2 35.4

55

41 48

100 (4.0)

60 (2.4)

30 (1.2) 30 (1.2)

Izod impact value, Limiting min (if speciﬁed) ruling section, J ft-lbf mm (in)

Notes: a , area of cross section; , steel for case hardening. Minimum values of yield stress may be required in certain speciﬁcations, and in such case a minimum yield stress of 55 percent of minimum tensile strength should be satisfactory. Source: IS 1570, 1979.

C 8 S 10 C 14 S 14 C 12 S 14 C 10 S 18 C 10 S 25 C 15 S 12

Old

New

Designation

TABLE 1-7 Carbon and carbon - manganese free - cutting steels with speciﬁed chemical composition and related mechanical properties

PROPERTIES OF ENGINEERING MATERIALS

1.24

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UNS no.

G10100

G10150

G10200

G10300

G10400

G10500

G10600

G10950

G11170

G11440

G13400

AISIa no.

1010

1015

1020

1030

1040

1050

1060

1095

1117

1144

1340

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Normalized Annealed

As-rolled Normalized Annealed

As-rolled Normalized Annealed

As-rolled Normalized Annealed

As-rolled Normalized Annealed

As-rolled Normalized Annealed

As-rolled Normalized Annealed

As-rolled Normalized Annealed

As-rolled Normalized Annealed

As rolled Normalized Annealed

Hot-rolled Cold-drawn

Treatment

870 800

900 790

900 825

900 790

900 790

900 790

900 790

925 845

870 870

– 925 870

8C

1600 1475

1650 1450

1650 1575

1650 1450

1650 1450

1650 1450

1650 1450

1700 1550

1600 1600

– 1700 1600

8F

Austenitizing temperature

TABLE 1-8 Mechanical properties of selected carbon and alloy steels

836.3 703.3

703.3 667.4 584.7

486.8 467.1 429.5

965.3 1013.5 656.7

813.7 775.7 625.7

723.9 748.1 636.0

620.5 589.5 518.8

551.6 520.0 463.7

448.2 441.3 394.7

420.6 424.0 386.1

320 370

MPa

121.3 102.0

102.0 96.8 84.8

70.6 67.8 62.3

140.0 147.0 95.3

118.0 112.5 90.8

105.0 108.5 92.3

90.0 85.5 75.3

80.0 75.5 67.3

65.0 64.0 57.3

61.0 61.5 56.0

47 53

kpsi

Tensile strength, st

558.5 436.4

420.6 399.9 346.8

305.4 303.4 279.2

572.3 499.9 379.2

482.6 420.6 372.3

413.7 427.5 365.4

413.7 374.0 353.4

344.7 344.7 341.3

330.9 346.5 294.8

313.7 324.1 284.4

180 300

MPa

81.0 63.3

61.0 58.0 50.3

44.3 44.0 40.5

83.0 72.5 55.0

70.0 61.0 54.0

60.0 62.0 53.0

60.0 54.3 51.3

50.0 50.0 49.5

48.0 50.3 42.3

45.5 47.0 41.3

26 44

kpsi

Yield strength, sy

22.0 25.5

21.0 21.0 24.8

33.0 33.5 32.8

9.0 9.5 13.0

17.0 18.0 22.5

20.0 20.0 23.7

25.0 28.0 30.2

32.0 32.0 31.2

36.0 35.8 36.5

39.0 37.0 37.0

28 20

Elongation in 50 mm (2 in), %

62.9 57.3

41.0 40.4 41.3

63.0 63.8 58.0

18.0 13.5 20.6

34.0 37.2 38.2

40.0 39.4 39.9

50.0 54.9 57.2

57.0 60.8 57.9

59.0 67.9 66.0

61.0 69.6 69.7

50 40

Reduction in area, %

248 207

212 197 167

143 137 121

293 293 192

241 229 179

229 217 187

201 170 149

179 149 126

143 131 111

126 121 111

95 105

Brinell hardness, HB

92.5 70.5

52.9 43.4 65.1

81.3 85.1 93.6

4.1 5.4 2.7

17.6 13.2 11.3

31.2 27.1 16.9

48.8 65.1 44.3

74.6 93.6 69.4

86.8 117.7 123.4

110.5 115.5 115.0

J

68.2 52.0

39.0 32.0 48.0

60.0 62.8 69.0

3.0 4.0 2.0

13.0 9.7 8.3

23.0 20.0 12.5

36.0 48.0 32.7

55.0 69.0 51.2

64.0 86.8 91.0

81.5 85.2 84.8

ft-lbf

Izod impact strength

PROPERTIES OF ENGINEERING MATERIALS

1.25

1.26

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G31400

G41300

G41500

G43200

G43400

G46200

G48200

G51500

G61500

G86300

G87400

G92550

G93100

3140

4130

4150

4320

4340

4620

4820

5150

6150

8630

8740

9255

9310

Normalized Annealed

Normalized Annealed

Normalized Annealed

Normalized Annealed

Normalized Annealed

Normalized Annealed

Normalized Annealed

Normalized Annealed

Normalized Annealed

Normalized Annealed

Normalized Annealed

Normalized Annealed

Normalized Annealed

Treatment

890 845

900 845

870 815

870 845

870 815

870 825

860 815

900 855

870 810

895 850

870 815

870 865

870 815

8C

1630 1550

1650 1550

1600 1500

1600 1550

1600 1500

1600 1520

1580 1500

1650 1575

1600 1490

1640 1560

1600 1500

1600 1585

1600 1500

8F

906.7 820.5

932.9 774.3

929.4 695.0

650.2 564.0

939.8 667.4

870.8 675.7

750.0 681.2

574.3 512.3

1279.0 744.6

792.9 579.2

1154.9 729.5

668.8 560.5

891.5 689.5

MPa

131.5 119.0

135.3 112.3

134.8 100.8

94.3 81.8

136.3 96.8

126.3 98.0

109.5 98.8

83.3 74.3

185.5 108.0

115.0 84.0

167.5 105.8

97.0 81.3

129.3 100.0

kpsi

Tensile strength, st

570.9 439.9

579.2 486.1

606.7 415.8

429.5 372.3

615.7 412.3

529.5 357.1

484.7 464.0

366.1 372.3

861.8 472.3

464.0 609.5

734.3 379.2

436.4 360.6

599.8 422.6

MPa

82.8 63.8

84.0 70.5

88.0 60.3

62.3 54.0

89.3 59.8

76.8 51.8

70.3 67.3

53.1 54.0

125.0 68.5

67.1 61.6

106.5 55.0

63.3 52.3

87.0 61.3

kpsi

Yield strength, sy

18.8 17.3

19.7 21.7

16.0 22.2

23.5 29.0

21.8 23.0

20.7 22.0

24.0 22.3

29.0 31.3

12.2 22.0

20.8 29.0

11.7 20.2

25.2 28.2

19.7 24.5

Elongation in 50 mm (2 in), %

58.1 42.1

43.4 41.1

47.9 46.4

53.5 58.9

61.0 48.4

58.7 43.7

59.2 58.8

66.7 60.3

36.3 49.9

50.7 58.4

30.8 40.2

59.5 55.6

57.3 50.8

Reduction in area, %

269 241

269 229

269 201

187 156

269 197

255 197

229 197

174 149

363 217

235 163

321 197

197 156

262 197

Brinell hardness, HB

119.3 78.6

13.6 8.8

17.6 40.0

94.6 95.2

35.5 27.4

31.5 25.1

109.8 92.9

132.9 93.6

15.9 51.1

72.9 109.8

11.5 24.7

86.4 61.7

88.0 58.0

10.0 6.5

13.0 29.5

69.8 70.2

26.2 20.2

23.2 18.5

81.0 68.5

98.0 69.0

11.7 37.7

53.8 81.0

8.5 18.2

63.7 45.5

39.5 34.2

ft-lbf

Izod impact strength

53.6 46.4

J

All grades are ﬁne-grained except for those 1100 series, which are coarse-grained. Heat-treated specimens were oil-quenched unless otherwise indicated. Values tabulated were averaged and obtained from specimen 12.75 mm (0.505 in) in diameter which were machined from 25 mm (1 in); rounded gauge lengths were 50 mm (2 in). Source: ASM Metals Handbook, American Society for Metals. Metals Park, Ohio. 1988

a

UNS no.

AISIa no.

Austenitizing temperature

TABLE 1-8 Mechanical properties of selected carbon and alloy steels (Cont.)

PROPERTIES OF ENGINEERING MATERIALS

PROPERTIES OF ENGINEERING MATERIALS PROPERTIES OF ENGINEERING MATERIALS

1.27

TABLE 1-9 Mechanical properties of standard steels Tensile strength, st

Designation

Yield stress, sy

New

Old

MPa

kpsi

MPa

kpsi

Elongation in 50 mm (gauge pﬃﬃﬃﬃﬃ length 5.65 a )

Fe 290 Fe E 220 Fe 310 Fe E 230 Fe 330 Fe F 250 Fe 360 Fe F 270 Fe 410 Fe E 310 Fe 490 Fe E 370 Fe 540 Fe E 400 Fe 620 Fe E 460 Fe 690 Fe E 520 Fe 770 Fe E 580 Fe 870 Fe E 650

(St 30) – (St 32) – (St 34) – (St 37) – (St 42) – (St 50) – (St 55) – (St 63) – (St 70) – (St 78) – (St 88) –

290 290 310 310 330 330 360 360 410 410 490 490 540 540 620 620 690 690 770 770 870 870

42.0 42.0 45.0 45.0 47.9 47.9 52.2 52.2 59.5 59.5 71.1 71.1 78.3 78.3 90.0 90.0 100.0 100.0 111.7 111.7 126.2 126.2

170 220 180 230 200 250 220 270 250 310 290 370 320 400 380 460 410 520 460 580 520 650

24.7 32.0 26.1 33.4 29.0 36.3 32.0 39.2 36.3 50.0 42.0 53.7 46.4 58.0 55.1 66.7 59.5 75.4 66.7 84.1 75.4 94.3

27 27 26 26 26 26 25 25 23 23 21 21 20 20 15 15 12 12 10 10 8 8

Note: a area of cross-section of test specimen. Source: IS 1570, 1978.

TABLE 1-10 Chemical composition and mechanical properties of carbon steel castings for surface hardening Chemical composition (in ladle analysis) max, % Designation

C

Si

Mn

S

P

Cr

Ni

Mo

Cu

Residual elements

Gr 1 Gr 2

0.4-0.5 0.5-0.6

0.60 0.60

1.0 1.0

0.05 0.05

0.05 0.05

0.25 0.25

0.40 0.40

0.15 0.15

0.30 0.30

0.80 0.80

Yield strength, sy

Tensile strength, st Designation

Mpa

kpsi

Mpa

kpsi

Elongation, % min (gauge length pﬃﬃﬃﬃ ﬃ 5.65 a )

Gr 1 Gr 2

620 700

90.0 101.5

320 370

46.4 53.7

12 8

Brinell hardness HB 460 535

Notes: a area of cross section of test specimen. All castings shall be free from distortion and harmful defects. They shall be well-dressed, fettled, and machinable. Unless agreed upon by the purchaser and the manufacturer, castings shall be supplied in the annealed, or nomalized and tempered condition. Source: IS 2707, 1973.

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0.06–0.24 0.12–0.18 0.14–0.19 0.18–0.22 0.26 max 0.12 max 0.15 max 0.10–0.15 0.12–0.18 0.12–0.18 0.12–0.18 0.12–0.20 0.33–0.40 0.32–0.42 0.30–0.40 0.35–0.45 0.35–0.45 0.30–0.40 0.35–0.45 0.36–0.44 0.20–0.30 0.50–0.60 0.45–0.55 0.18–0.23 0.33–0.41

%C 0.10–0.35 0.10–0.35 0.10–0.35 0.10–0.35 0.10–0.35 0.15–0.60 0.50 max 0.10–0.35 0.10–0.35 0.10–0.35 0.10–0.35 0.10–0.35 1.50–2.00 0.10–0.35 0.10–0.35 0.10–0.35 0.10–0.35 0.10–0.35 0.10–0.35 0.10–0.35 0.10–0.35 1.50–2.00 0.10–0.35 0.20–0.35 1.10–1.40

% Si 1.30–1.70 0.40–0.60 1.00–1.30 1.00–1.40 0.50–0.80 0.40–0.70 0.40–0.70 0.40–0.70 0.40–0.70 0.60–1.00 0.60–1.00 0.40–0.70 0.80–1.00 1.30–1.70 1.30–1.80 1.60–0.90 0.50–0.80 0.60–0.90 0.40–0.70 0.40–0.70 0.40–0.70 0.80–1.00 0.50–0.80 0.70–0.90 1.10–1.40

% Mn

0.20–0.35 0.40–0.70 0.45–0.65

0.15–0.25

0.90–1.20 0.40–0.60

1.00–1.50 1.25–1.75 2.25–2.75 0.30 max

0.40–0.70

0.20–0.35

0.20–0.35

0.80–0.15 0.10–0.20 0.15–0.25

0.90–1.70 0.90–1.20 0.45–0.75 0.90–1.30 0.50–0.80 2.90–3.40

0.15–0.30 0.45–0.65 0.90–1.10

% Mo

0.30 max 0.30 max 3.00–3.50 3.80–4.30 1.00–1.50 1.50–2.00 1.80–2.20

% Cr

0.50–0.80 0.80–1.10 1.00–1.30 0.90–1.20 0.70–1.10 2.00–2.50 0.60–1.00 1.00–1.40 0.75–1.25 0.75–1.25 1.40–1.70

% Ni

0.15–0.30

%V

% Al

Notes: (1) Sulfur and phosphorus can be ordered as per following limits: (i) S and P – 0.30 max; (ii) S – 0.02–0.035 and P – 0.035 max. (2) When the steel is Al killed, total Al contents shall be between 0.02–0.05 percent. Source: IS 4367, 1991.

Mn 2 Cr 65 Mn 1 Cr 95 Mn Cr 1 Cr 1 Mo 28 Cr 90 Mo 55 Cr 2 Mo 1 Ni 3 Cr 80 Ni 4 Cr 1 Ni Cr 1 Mo 12 Ni Cr 1 Mo 15 Ni Cr 2 Mo 20 Si 2 Mn 90 Mn 2 Mn 2 Mo 28 Cr 1 Cr 1 Mo 28 Ni 1 Cr 60 Ni 2 Cr 1 Mo 28 Ni 3 Cr 65 Mo 55 Cr 3 Mo 55 Si 2 Mn 90 Cr 1 V 23

20 15 17 20 21 07 10 13 15 15 15 16 37 37 35 40 40 35 40 40 25 55 50

20 15 16 20 21 07 10 13 15 15 15 16 36 37 35 40 40 35 40 40 25 55 50 20 37

C 15 Cr 3 Mn 5 Cr 4 Mn 5 Cr 5 Cr 4 Mo 2 Cr 4 Mo 6 Cr 9 Mo 10 Ni 13 Cr 3 Ni 16 Cr 5 Ni 5 Cr 4 Mo 1 Ni 7 Cr 4 Mo 2 Ni 8 Cr 6 Mo 2 S 17 C 15 Mn 6 Mo 3 Cr 4 Cr 4 Mo 3 Ni 5 Cr 2 Ni 6 Cr 4 Mo 3 Ni 10 Cr 3 Mo 6 Cr 13 Mo 6 Si 7 Cr 4 V 2 Ni Cr MO 2 Mn 5 Si 5

Old

New

Designation

TABLE 1-11 Chemical composition of alloy steel forgings for general industrial use

PROPERTIES OF ENGINEERING MATERIALS

1.28

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H and T

H and T

R, Q and S.R R, Q and S.R R, Q and S.R H and T

20 C 15

30 C 15

15 16 20 21

R, Q and S.R R, Q and S.R R, Q and S.R R, Q and S.R R, Q and S.R R, Q and S.R H and T H and T H and T H and T

13 15 15 15 16 20 36 37 55 35

H and T

H and T

H and T

40 Cr 4

40 Cr 4 Mo 3

35 Ni 5 Cr 2

Ni 13 Cr 3 Ni 16 Cr 5 Ni 5 Cr 4 Mo 1 Ni 7 Cr 4 Mo 2 Ni 8 Cr 6 Mo 2 Ni Cr Mo 2 Si 7 Mn 5 Si 5 Si 7 Mn 6 Mo 3

N and T N and T

7 Cr 4 Mo 6 10 Cr 9 Mo 10

Cr 3 Mn 5 Cr 4 Mn 5 Cr 5 Cr 4 Mo 2

Condition

Designation 600–750 700–850 600–750 700–850 800–950 600 min 800 min 1000 min 650–800 700–850 800–950 380–550 410–590 520–680 850 min 1350 min 1000 min 1100 min 1350 min 900 min 800–950 780–930 1300–1500 700–850 900–1050 1000–1150 700–850 900–1050 700–850 900–1050 1000–1150 700–850 900–1050

MPa 87.0–108.8 101.5–123.3 87.0–108.8 101.5–123.3 116.0–137.8 87.0 min 116.0 min 145.0 min 94.3–116.0 101.5–123.3 116.0–137.8 55.1–79.8 59.5–85.6 75.4–98.6 123.3 min 195.8 min 145.0 min 159.5 min 195.8 min 130.5 min 116.0–137.8 113.1–134.9 188.6–217.6 101.5–123.3 130.5–152.3 145.0–166.8 101.5–123.3 130.5–152.3 101.5–123.3 130.5–152.3 145.0–166.8 101.5–123.3 130.5–152.3

kpsi

Tensile strength,b st

85.6 78.3 101.5 116.0 78.3 101.5 78.3 101.5 116.0 78.3 101.5

540 700 800 540 700 540 700 800 540 700

61.0 66.7 84.1 32.6 35.5 45.0

58.0 66.7 63.8 78.3 87.0

kpsi

590

420 460 580 225 245 310

400 460 440 540 600

MPa

Yield strength,b sy

TABLE 1-12 Mechanical properties of alloy steel forgings for general industrial use

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220

220

229 241 217 217 229 213 217 217 245 220

170 187

170 207 217 210

220

200

Brinell hardness in soft annealed condition, max, HB

18 15 13 18 15 18 15 13 18 15

14

18 16 18 18 16 13 10 8 16 15 14 19 18 18 12 9 9 9 9 11

Elongation,b %, min (gauge length ﬃ pﬃﬃﬃﬃ 5.65 a )a

55 50 45 55 50 55 50 45 55 50

50 50 50 48 48 48 35 38 60 55 50 60 55 50 48 35 41 35 35 41

J

40.6 36.9 33.2 40.6 36.9 40.6 36.9 33.2 40.6 36.9

36.9 36.9 36.9 35.4 35.4 35.4 25.8 28.0 44.3 40.6 36.9 44.3 40.6 36.9 35.4 25.8 30.2 25.8 25.8 30.2

ft-Ibf

Izod impactb value

100 150 63 30 100 30 150 63 30 150 63

4.00 6.00 2.52 1.20 4.00 1.20 6.00 2.52 1.20 6.00 2.52

4.00

235–280 100 380–440 208–252 268–311 295–341 208–252 266–311 208–252 266–311 295–341 208–252 266–311

2.4 1.20 1.20 1.20 1.20

2.52 1.20 6.00 4.00 2.52 1.20 1.20 1.20 6.00 4.00 1.60 1.60 2.00 60 30 30 30 30

178–221 63 208–252 63 178–221 150 208–252 100 235–280 63 30 30 30 190–235 150 208–252 100 235–280 40 40 50

Brinellb Limiting hardness ruling section number HB mm in

PROPERTIES OF ENGINEERING MATERIALS

1.29

H and T

H and T

40 Ni 10 Cr 3 Mo 6

50 Cr 4 V 2

900–1050 1100–1250 900–1050 1100–1250 1200–1350 1000–1150 1200–1350 1550 min 900–1100 1000–1200

MPa 130.5–152.3 159.5–181.3 130.5–152.3 159.5–181.3 174.0–195.8 145.0–166.8 174.0–195.8 224.8 min 130.5–159.5 145.0–174.0

kpsi 700 880 700 880 1000 800 1000 1300 700 800

MPa 101.5 127.6 101.5 127.6 145.0 116.0 145.0 188.5 101.5 116.0

kpsi

Yield strength,b sy

240

250

230

230

Brinell hardness in soft annealed condition, max, HB 15 12 55 11 10 12 10 8 12 10

Elongation,b %, min (gauge length pﬃﬃﬃﬃ ﬃ 5.65 a Þa 55 41 55 41 30 48 35 15 45 45

J

40.6 30.2 40.6 30.2 22.1 35.4 25.8 11.1 33.2 33.2

ft–Ibf

Izod impactb value

266–311 325–370 266–311 325–370 355–399 295–341 355–399 450 min 266–325 295–355

150 100 150 63 30 150 150 100 100 40

6.00 4.00 6.00 2.52 1.20 6.00 6.00 4.00 4.00 1.60

Brinellb Limiting hardness ruling section number HB mm in

b

a

a , area of cross section. Mechanical properties in heat-treated conditions. Notes: H and T – hardened and tempered; N and T – normalized and tempered; R,Q and S.R – reﬁned quenched and stress-relieved. All properties for guidance only. Other values may be mutually agreed on between the consumers and suppliers. Source: IS 4367, 1991.

H and T

Condition

40 Ni 6 Cr 4 Mo 3

25 Cr 13 Mo 6

Designation

Tensile strength,b st

TABLE 1–12 Mechanical properties of alloy steel forgings for general industrial use (Cont.) PROPERTIES OF ENGINEERING MATERIALS

1.30

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##

Si

Mn

0.32– 0.10– 1.30– 0.42 0.35 1.70

0.30– 0.10– 1.30– 0.40 0.35 1.80

0.30– 0.10– 1.30– 0.40 0.35 1.80

0.35– 0.10– 0.60– 0.45 0.35 0.90

0.35– 0.10– 0.50– 0.45 0.35 0.80

0.10– 0.10– 0.40– 0.30 0.20 0.35 0.70 max

0.20– 0.10– 0.40– 0.30 0.30 0.35 0.70 max

37 C 15 (37 Mn 2)

35 Mn 6 Mo 3 (35 Mn 2 Mo 28)

35 Mn 6 Mo 4 (35 Mn 2 Mo 45)

40 Cr 4 (40 Cr 1)

40 Cr 4 Mo 2 (40 Cr 1 Mo 28)

15 Cr 13 Mo 6 (15 Cr 3 Mo 55)

25 Cr 13 Mo 6 (25 Cr 3 Mo 55)

Ni

0.22– 0.10– 1.30– 0.32 0.35 1.70

0.16– 0.10– 1.30– 0.24 0.35 1.70

C

27 C 15 (27 Mn 2)

20 C 15 (20 Mn 2)

Designation

0.35– 0.55

0.20– 0.35

Mo

2.90– 0.45– 3.40 0.65

2.90– 0.45– 3.40 0.65

0.90– 0.20– 1.20 0.35

0.90– 1.20

Cr

Percent V/Al

TABLE 1–13 Chemical composition and mechanical properties of alloy steels

890–1040 990–1140 1090–1240 1540 min

690–840 790–940

700–850 800–950 900–1050 1000–1150

690–840 790–940 890–1040

790–940 890–1040 990–1140

690–840 790–940 890–1040 990–1140

590–740 690–840 790–940 890–1040

590–740 690–840

590–740 690–840

MPa

129.0–150.8 143.6–165.3 158.1–179.8 223.4 min

100.0–121.8 114.6–136.3

101.5–123.3 116.0–137.8 130.5–152.3 145.0–166.8

100.0–121.8 114.6–136.3 129.0–150.8

114.6–136.3 129.0–150.8 143.6–165.3

100.0–121.8 114.6–136.3 129.0–150.8 143.6–165.3

85.5–107.3 100.0–121.8 114.6–136.3 129.0–150.8

85.5–107.3 100.0–121.8

85.6–107.3 100.0–121.8

kpsi

Tensile strength, st

56.6 71.1 79.9 94.3

71.1 14 79.9 12 650 94.3. 11 750 108.8 10 830 120.4 9 1240 179.8 8

490 550

13 12 11 10

71.1 14 79.8 12 94.3 11 490 71.1 550 79.8 650 94.3 750 108.8

490 550 650

550 79.8 16 650 94.3 15 750 108.9 13

14 12 12 10

18 18 16 15

56.6 18 65.3 16

56.6 18 65.3 16

490 71.1 550 79.8 650 94.3 750 108.8

390 490 550 650

390 450

390 450

40.6 36.8 36.8 35.4

35.4 35.4 35.4 30.2

40.6 36.8 36.8 35.4

50 48 41 14

36.8 35.4 30.2 10.3

55 40.6 50 36.8

55 50 50 48

55 40.6 50 36.8 50 36.8

55 40.6 55 40.6 48 35.4

55 50 50 48

48 48 48 41

48 35.4 48 35.4

48 35.4 48 35.4

255–311 285–341 311–363 444 min

201–248 229–277

201–248 229–277 255–311 285–341

201–248 229–277 255–311

229–277 255–311 285–341

201–248 229–277 255–311 285–341

170–217 201–248 229–277 255–311

170–217 201–248

170–217 201–248

Minimum 0.2% Minimum Izod impact proof stress, elongation value Brinell# min, sy (gauge length pﬃﬃﬃﬃﬃ hardness ¼ 5.65 a )a , MPa kpsi % J ft-lbf HB

150 (6.0) 150 (6.0) 100 (4.0) 63 (2.5)

150 (6.0) 150 (6.0)

150 (6.0) 100 (4.0) 63 (2.5) 30 (1.2)

100 (4.0) 63 (2.5) 30 (1.2)

150 (6.0) 100 (4.0) 63 (2.5)

150 (6.0) 100 (4.0) 63 (2.5) 30 (1.2)

150 (6.0) 100 (4.0) 30 (1.2) 15 (0.6)

100 (4.0) 63 (2.5)

63 (2.5)þ 30 (1.2)

Limiting ruling section, mm (in)þ

PROPERTIES OF ENGINEERING MATERIALS

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1.31

790–940 890–1040 990–1140 1090–1240 1190–1340 1540 min 890–1040 990–1140 1090–1240 1190–1340 1540 min

0.35– 0.10– 0. 50– 3.20– 0.30 0.45 0.35 0.80 3.6 max

0.30– 0.10– 60–90 1.00– 0.45– 0.40 0.35 1.50 0.75

0.26– 0.10– 0.40– 3.90– 1.10– 0.34 0.35 0.70 4.30 1.40

0.35– 0.10– 0.40– 1.20– 0.90– 0.10– 0.45 0.35 0.70 1.60 1.30 0.20

0.35– 0.10– 0.40– 1.25– 0.90– 0.20– 0.45 0.35 0.70 1.75 1.30 0.35

35 Ni 5 Cr 2 (35 Ni 1 Cr 60)

30 Ni 16 Cr 5 (30 Ni 4 Cr 1)

40 Ni 6 Cr 4 Mo 2 (40 Ni Cr 1 Mo 15)

40 Ni 6 Cr 4 Mo 3 (40 Ni 2 Cr 1 Mo 28)

31 Ni 10 Cr 3 Mo 6 0.27– 0.10– 0.40– 2.25– 0.50– 0.40– (31 Ni 3 Cr 65 Mo 55) 0.35 0.35 0.70 2.75 0.80 0.70

1.50– 0.10– Al: 0.90– 1.80. 0.25 1.30

790–940 890–1040 990–1140 1090–1240

1540 min

690–840 790–940 890–1040

790–940 890–1040

690–840 790–940 890–1040

1340 min 1540 min

MPa

40 Ni 14 (40 Ni 31)

V/Al

0.35– 0.10– 0.40– 0.30 0.45 0.45 0.70 max

Mo

3.00– 0.90– V: 0.15– 3.50 1.10 0.25

Cr

40 Cr 7 Al 10 Mo 2 (40 Cr 2 Al 1 Mo 18)

Ni

0.35 – 0.10– 0.40– 0.30 0.45 0.35 0.70 max

Mn

40 Cr 13 Mo 10 V 2 (40 Cr 3 Mo 1 V 20)

Si

C

129.0–150.8 143.6–165.3 158.1–179.8 172.6–194.4 223.4 min

114.6–136.3 129.0–150.8 143.6–165.3 158.1–179.8 172.6–194.4 223.4 min

114.6–136.3 129.0–150.8 143.6–165.3 158.1–179.8

223.4 min

100.0–121.8 114.6–136.3 129.0–150.8

114.6–136.3 129.0–150.8

100.0–121.8 114.6–136.3 129.0–150.8

194.4 min 223.4 min

kpsi

Tensile strength, st

Designation

Percent

TABLE 1–13 Chemical composition and mechanical properties of alloy steels (Cont.)

8 8

179.9

8

71.1 14 79.8 12 94.3 10

79.8 16 94.3 15

71.1 18 79.8 16 94.3 15

152.2 179.8

650 750 830 930 1240

550 650 750 830 930 1240

94.3 108.8 120.4 134.9 179.8

79.8 94.3 108.8 120.4 134.9 179.8

15 12 11 10 8

16 15 11 11 10 6

550 79.8 16 650 94.3 15 750 108.8 13 830 120.4 13

1240

490 550 650

550 650

490 550 650

1050 1240

55 48 41 35 14

55 55 48 41 30 11

55 55 48 41

40.6 35.4 30.3 25.8 10.3

40.6 40.6 36.8 30.3 22.1 8.1

40.6 40.6 35.4 30.3

14 10.3

55 40.6 50 36.8 50 36.8

55 40.6 55 40.6

55 40.6 55 40.6 48 35.4

21 15.5 14 10.3

255–311 285–341 311–363 341–401 444 min

229–277 255–311 285–341 311–363 341–401 444 min

229–277 255–311 285–341 311–363

444 min

201–248 229–277 255–311

229–277 255–311

201–248 229–277 255–311

363 min 444 min

Minimum 0.2% Minimum Izod impact proof stress, elongation value Brinell# min, sy (gauge length pﬃﬃﬃﬃﬃ a hardness ¼ 5.65 a ) , MPa kpsi % J ft-lbf HB

150 (6.0) 150 (6.0) 100 (4.0) 63 (2.5) 63 (2.5)

150 (6.0) 150 (6.0) 100 (4.0) 63 (2.5) 30 (1.2) 30 (1.2)

150 (6.0) 100 (4.0) 63 (2.5) 30 (1.2)

(air-hardened)

150 (6.0)

(air-hardened)

150 (6.0)þ 100 (4.0) 63 (2.5)

100 (4.0) 63 (2.5)

150 (6.0) 100 (4.0) 63 (2.5)

63 (2.5) 30 (1.2)

Limiting ruling section, mm (in)þ

PROPERTIES OF ENGINEERING MATERIALS

1.32

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C

Si

Mn

Ni

Cr

Mo

V/Al 990–1140 1090–1240 1190–1240 1540 min

MPa 143.6–165.3 158.1–179.8 172.6–194.4 223.4 min

kpsi

Tensile strength, st

750 830 930 1240

108.8 12 120.4 11 134.9 10 179.8 8

48 41 35 14

35.4 30.3 25.8 10.3

285–341 311–363 341–401 444 min

Minimum 0.2% Minimum Izod impact proof stress, elongation value Brinell# min, sy (gauge length pﬃﬃﬃﬃﬃ hardness ¼ 5.65 a )a , MPa kpsi % J ft-lbf HB

150 (6.0) 150 (6.0) 150 (6.0) 100 (4.0)

Limiting ruling section, mm (in)þ

Note: a , area of cross section; hardened and tempered condition – oil-hardened unless otherwise stated; # hardness given in this table is for guidance only; x steel designations in parentheses are old designations; þ numerals in parentheses are in inches. Source: IS 1750, 1988.

40 Ni 10 Cr 3 Mo 6 0.36– 0.10– 0.40– 2.25– 0.50– 0.40– (40 Ni 3 Cr 65 Mo 55) (0.44 0.35 0.70 2.75 0.80 0.70

Designation

Percent

TABLE 1–13 Chemical composition and mechanical properties of alloy steels (Cont.)

PROPERTIES OF ENGINEERING MATERIALS

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1.33

PROPERTIES OF ENGINEERING MATERIALS

1.34

CHAPTER ONE

TABLE 1-14 Mechanical properties of case hardening steels in the reﬁned and quenched condition (core properties)

Tensile strength, st Steel designation

MPa

kpsi

Minimum elongation, % (gauge length pﬃﬃﬃﬃﬃ = 5.65 a ) a

Izod impact value, min (if speciﬁed) J

ft-lbf

Limiting ruling section, mm (in)

Brinell hardness number, max, HB 130 143

10 C 4 (C 10) 14 C 4 (C 14)

490 490

71.1 71.1

17 17

54 54

39.8 39.9

10 C 8 S 11 (10 S 11) 14 C 14 S 14 (14 Mn 1 S 14) 11 C 15 (11 Mn 2) 15 Cr 65 17 Mn 1 Cr 95 20 Mn Cr 1 16 Ni 3 Cr 2 (16 Ni 80 Cr 60) 16 Ni 4 Cr 3 (16 Ni 1 Cr 80)

490 588

71.1 85.4

17 17

54 40

39.8 29.7

15 (0.6) >15 (0.6) <30 (1.2) 30 (1.2) 30 (1.2)

588

85.4

17

54

39.8

30 (1.2)

154

588 784 981 686

85.4 113.8 142.3 99.6

13 10 8 15

47 34 37 40

34.7 25.3 27.5 29.7

30 (1.2) 30 (1.2) 30 (1.2) 90 (3.6)

170 207 217 184

834 784 735 834 784 1324 1177 1128 834 686 882 784 735 981 932 1079 932 932 1324 1177 1128

121.0 113.8 106.7 121.0 113.8 192.0 170.7 163.2 121.0 99.6 128.0 113.8 106.7 142.3 135.1 156.5 142.3 135.1 193.0 170.7 163.6

12

40

29.7

217

12

47

34.7

9

34

25.3

12

61

44.8

11

40

29.7

9

40

29.7

9

34

25.3

9

34

25.3

30 (1.2) 60 (2.4) 90 (3.6) 60 (2.4) 100 (4.0) 30 (1.2) 60 (2.4) 90 (3.6) 30 (1.2) 60 (2.4) 30 (1.2) 60 (2.4) 90 (3.6) 30 (1.2) 90 (3.6) 30 (1.2) 60 (2.4) 90 (3.6) 30 (1.2) 60 (2.4) 90 (3.6)

13 Ni 13 Cr 3 (13 Ni 3 Cr 80) 15 Ni 4 Cr 1

20 Ni 2 Mo 25 20 Ni 7 Cr 2 Mo 2 (20 Ni 55 Cr 50 Mo 20) 15 Ni 13 Cr 4 (15 Ni Cr 1 Mo 12) 15 Ni 5 Cr 4 Mo 2 (15 Ni 2 Cr 1 Mo.15) 16 Ni 8 Cr 6 Mo 2 (16 Ni Cr 2 Mo 20)

143 154

229 241

207 213

217 217

229

a area of cross section. Source: IS 4432. 1967.

a

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PROPERTIES OF ENGINEERING MATERIALS PROPERTIES OF ENGINEERING MATERIALS

1.35

TABLE 1-15 Typical mechanical properties of some carburizing steelsa Hardness Ultimate tensile strength, sut

Tensile yield strength, sy

Case

Elongation Core in 50 mm Reduction Brinell, AISI No. MPa kpsi MPa kpsi (2 in), % of area, % H B

Izod impact energy

Thickness Rockwell, RC

mm

in

J

ft-lbf

Machinability

Poor Poor Good Very good Excellent

C1015 C1020 C1022 C1117 C1118

503 517 572 669 779

73 75 83 97 113

317 331 324 407 531

46 48 47 59 77

31 31 27 23 17

71 71 66 53 45

Plain carbon 149 156 163 192 229

62 62 62 65 61

1.22 1.17 1.17 1.14 1.65

0.048 0.046 0.046 0.045 0.065

123 126 110 45 22

91 93 81 33 16

4320b 4620b 8620b

100 793 897

146 115 130

648 531 531

94 77 77

22 22 22

56 62 52

Alloy steels 293 235 262

59 59 61

1.91 1.52 1.78

0.075 65 0.060 106 0.070 89

48 78 66

a

Average properties for 15 mm (1 in) round section treated, 12.625 mm (0.505 in) round section tested. Water-quenched and tempered at 1778C (3508F), except where indicated. Core properties for 14.125 mm (0.565 in) round section treated, 12.625 mm (0.505 in) round section tested. Oil-quenched twice, tempered at 2328C (4508F). Source: Modern Steels and Their Properties, Bethlehem Steel Corp., 4th ed., 1958 and 7th ed., 1972.

b

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PROPERTIES OF ENGINEERING MATERIALS

1.36

CHAPTER ONE

TABLE 1-16 Minimum mechanical properties of some stainless steels Tensile strength, st UNS No. AISI No. MPa

Yield strengtha , sy

kpsi

MPa

Brinell Elongation, hardness, H B %

kpsi

Reduction in area, %

Weldability Machinability Application

Annealed (room temperatures) Austenitic S30200 S30300 S30400 S30500 S30800 S30900 S31000 S31008 S34800 S38400

302 303b 304 305 308 309 310 310 S 348 384

515 585b 515 480 515 515 515 515 515 415–550

75 85b 75 70 75 75 75 75 75 60–80

205 240b 205 170 205 205 205 205 205

30 35b 30 25 30 30 30 30 30

88

40 50b 40 40 40 40 40 40 40

88 88 88 95 95 95 88

55b

Good Poor Good Good

Poor Good Poor

General purpose, springs Bolts, rivets, and nuts Welded structures General purpose

Good Good

Poor Poor

Heat-exchange parts Turbine and furnace Jet engine parts Fasteners and cold-worked parts

Excellent Fair Fair

Fair to good Fair

Screw machine parts, muﬄer Machine parts subjected to hightemperature corrosion

Annealed high-nitrogen Austenitic S20200 202 S21600 216 S30452 304 HN

655 690 620

95 100 90

310 415 345

4560 50

Ferrite S40500 S43000 S44600

405 430 446

415 450 515

60 65 75

170 205 275

25 30 40

88 max 88 max 95 max

22e 20

Martensite S40300 403 S41000 410

485 450

70 65

205 205

30 30

88 max 95 max

25c 22c

795 1450b 1720 1370b

115 210b 250 198b

620 1210b 1480b 1030b

90 175b 215b 149b

b

b

b

b

S41400 S41800d S42000e S43100d S44002 S44003 S44004 S50200

414 418d 420e 431d 440 A 440 B 440 C 502b

725

740 760b 485b

105

b

107 110b 70b

415

b

425 450b 205b

60

b

62 65b 30b

40 40 30

100 100

52RC 95

b

15 18b 8b 16b

b

20

b

18b 14b 30b

96 97b

Bolts, shafts, and machine parts Bolts, springs, cutlery, and machine parts 45 52b 25b 55b

b

High-strength parts used in aircraft and bolts Cutlery, bearing parts, nozzles and ball bearings

70b

a

At 0.2% oﬀset. Typical values. 20% elongation for thickness of 1.3 mm (0.050 in) or less. d Tempered at 2608C (5008F). e Tempered at 2058C (4008F). Source: ASM Metals Handbook, American Society for Metals, Metals Park, Ohio, 1988. b c

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11.5/13.5

Cr

0.20 max 1.0 max 1.5 max

0.03 max 1.0 max 2.0 max 8.0/12.0 17.5/20.0

0.08 max 1.0 max 2.0 max 8.0/10.5 17.5/20.0

0.15 max 1.0 max 2.0 max 8.0/10.0 17.0/19.0

0.08 max 1.0 max 2.0 max 9.0/12.0 17.0/19.0

0.08 max 1.0 max 2.0 max 9.0/12.0 17.0/19.0

0.08 max 1.0 max 2.0 max 10.0/14.0 16.0/18.0 2.0/3.0

0.08 max 1.0 max 2.0 max 10.0/14.0 16.0/18.0 2.0/3.0

0.08 max 1.0 max 2.0 max 10.0/14.0 16.0/18.0 2.0/3.0

0.03 max 1.0 max 2.0 max 11.0/15.0 18.0/20.0 3.0/4.0

0.25 max 2.5 max 2.0 max 18.0/21.0 24.0/26.0

0.12 max 1.0 max 10.0/14.0 3.5/5.5

0.35/0.50 2.5 max 1.0 max 12.0/15.0 12.0/15.0

X 15 Cr 25 N

X 02 Cr 19 Ni 10

X 04 Cr 19 Ni 9

X 07 Cr 18 Ni 9

X 04 Cr 18 Ni 10 Nb

X 04 Cr 18 Ni 10 Ti

X 04 Cr 17 Ni 12 Mo 2

X 02 Cr 17 Ni 12 Mo 2

X 04 Cr 17 Ni Mo 2 Ti 2

X 04 Cr 19 Ni 13 Mo 3

X 20 Cr 25 Ni 20

X 07 Cr 17 Mn 12 Ni 4

X 40 Ni 14 Cr 14 W 3 Si 2

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. 5XC0.80

5XC0.80

Ti

and

P max

0.040

0.040

0.040

0.030 0.045 and N ¼ 0:25 max

0.030

0.030

0.030

Chromium steels 0.030 0.040

S max

0.035 W 2.0/3.0

0.030

0.030

0.030

0.030

0.030

0.030

0.030

0.030

0.030

0.030

0.045

0.045

0.045

0.045

0.045

0.045

0.045

0.045

0.045

0.045

0.045

Chromium–nickel steels 0.030 0.045

10XC1.0

Nb

(785)

515 (517) 515 (517) 515 (517) 485 (483) 515 (517) 515 (517) 515 (490) 550

485 (483) 515 (517) 515

415 (445}# 450 (483) 450 (483) (600 700) 515 (490)

MPa

(113.9)

74.7 (75.0) 74.7 (75.0) 74.7 (75.0) 70.3 (70.0) 74.7 (75.0) 74.7 {75.0) 74.7 (71.1) 79.8

70.3 (70.0) 74.7 (75.0) 74.7

60.2 (64.5) 65.3 (70.0) 65.3 (70.0) (87.0 101.5) 74.7 (71.1)

kpsi

Tensile strength, min, st

(345)

205 (207) 205 (207) 205 (207) 170 (172) 205 (207) 205 (207) 210 (210) 250

170 (172) 205 (207) 205

275 (280)

205 (276)# 205 (276) 205 (276)

MPa

(50.0)

29.7 (30.0) 29.7 (30.0) 29.7 (30.0) 24.7 (25.0) 29.7 (30.0) 29.7 (30.0) 30.5 (30.5) 36.3

24.7 (25.0) 29.7 (30.0) 29.7

39.9 (40.6)

29.7 (40.0) 29.7 (40.0) 29.7 (40.0)

kpsi

0.2% proof stress, min, sy

(269)

217

217

217

217

217

217

183

183

183

183

183

217 (212)

(225)

183

217

183

88

95

95

95

95

95

88

88

88

88

88

–

–

88

95

88

Brinell HB Rockwell RB

Hardness number

(35)

40 (40) 40 (40) 40 (40) 40 (40) 40 (40) 35 (40) 40 (40) 45

40 (40) 40 (40) 40

20 (16)

22 (20)# 20 (20) 22 (20)

(40)

(50)

(50)

(50)

(50)

(50)

(50)

(50)

(50)

(50)

(50)

(45)

(45)

(45)

(45)#

Elongation Reduction in 50 mm of area, (2 in), min, min, % %

Notes: Annealed quenched or solution-treated condition; for free-cutting varieties sulfur and selenium content shall be as agreed between the purchaser and the manufacturer; for electrode steel Nb 10C to 1.0 in place of Ti; # the mechanical properties in parentheses are for bars and ﬂats and the properties without parentheses for plates, sheets, and strips. Source: Compiled from IS 1570 (part 5), 1985.

16.0/18.0

23.0/27.0

0.35/0.45 1.0 max 1.0 max 1.0 max 12.0/14.0

X 40 Cr 13

Mo

0.12 max 1.0 max 1.0 max 1.25/2.50 15.0/17.0

0.80/0.15 1.0 max 1.0 max 1.0 max 11.5/13.5

Ni

X 07 Cr 17

X 12 Cr 12

0.08 max 1.0 max 1.0 max

Mn

X 04 Cr 12

Si

C

Designation of steel

Chemical composition, %

TABLE 1-17 Chemical composition and mechanical properties of some stainless, heat resisting and high alloy steels

PROPERTIES OF ENGINEERING MATERIALS

1.37

1.38

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485 550 620

Grade 60 Grade 70 Grade 80

485

Grade 50

K12249

K02601 K12609 K12700

70 80 90

60

70

65 75 85 70 70 65 95–115

kpsi

415 485 550

345

345

345 415 485 345 345 345 552

310 310

310

345 415 450 345

290

60 70 80

50

50

50 60 70 50 50 50 80

45 45

45

50 60 65 50

42

275–345 42–50

290–345 42–50

290–345 42–50

MPa

Minimum yielda strength,a sy

May vary with product size and mill form. Source: ASM Metal Handbook, American Society for Metals, Metals Park, Ohio, 1988.

a

A715

A690

A656

A618

450 520 590 483 483 448 655–793

65 60

450 410

A607

Grade 50 Grade 60 Grade 70 Grade I Grade II Grade III Grade 1 and 2

65

450

Hot-rolled and annealed or normalized Cold-rolled Grade 45

A606

65 75 80 70

450 520 550 480

Grade 50 Grade 60 Grade 65 Hot-rolled

60

415

Grade 42

415–485 60–70

435–485 63–70

435–480 63–70

kpsi

A572

K12211

A441

K11510

K12810

Type 1

A242

UNS designation MPa

A440

Type, grade, or condition

ASTM speciﬁcation

Minimum tensilea strength,a st

TABLE 1-18 Mechanical properties of high-strength low-alloy steels

18

19 18 18 12

18 16 15

20

18

18

18

In 200 mm (8 in)

20–22 18–20 16–18

22–24

20–22 16–18 14 22 22 20

22 22–25

22

21 18 17 22

24

21

21

21

In 50 mm (2 ln)

Minimum elongation,a %

Truck frames, brackets, crane booms, railcars. and other applications where weight saving is important Dock walls, sea walls, bulkheads, excavations, and similar structures exposed to sea water Structural and miscellaneous applications where high strength, weight savings, improved formability, and good weldability are important

General structural purposes including welded, bolted, or riveted bridges and buildings

Structural and miscellaneous purposes where greater strength or weight saving is important

Structural and miscellaneous purposes where weight saving or added durability is important

Structural members in welded, bolted, or riveted construction Structural members, primarily in bolted or riveted construction Welded, bolted, or riveted structures but primarily welded bridges Welded, bolted, or riveted structures, but used mainly in bolted or riveted bridges and buildings

Intended uses

PROPERTIES OF ENGINEERING MATERIALS

CB-30d C-50d CE-30d CF-8d CH-20d

ACId

ASTM A607 A606 Types 2, 4h A607 715 (sheet)i A656 (plate) A607 A607

138 20 159 23 172 25 214 31 244 34 303 44 255 37 331 48 Cast Stainless Steels

Cast Alloy Steels 24 20 22 20 17 9 14 6

200 172 193 193

29 25 28 28

3 61 14 95 20

81 75 65 54 79 41 61 32 2 45 10 70 15

60 55 48 40 58 30 45 24

255

448 414 483 586

J410Cg J410Cg

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65 60

414 60 448–483 65–70

1758

379 483

345 345

55 70

50 50

20 14

22 24

25 22

245 8 High-Strength Low-Alloy (HSLA) Steels

310 45 310–345 45–50

1689

6.6–12

31

23

17

20–30 15–22

35 40 50 60 85 125 95 145

kpsi

To 2144 To 311 To 1703 To 247

414 448 448 241–276 345

241 276 345 414 586 862 655 1000

MPa

23

95 70–97 87–97 75–85 80–88

65 70 80 90 105 150 120 175

kpsi

Yield strength, sy

60 15 65 18 65 18 35–40 55 50 38 Ultra-High-Strength Steels To 2068 To 300 To 1724 To 250 10

J410CRg

SAE J410Cg

Medium carbon low alloys 4 140 M, 433OV, D 6AC, 4340 Mod. 5 Cr-Mo-V tool steels: H-11 (Mod). H-13 (Mod) Maraging steels (high nickel): 18 Ni (350) Almar 302

448 483 552 620 724 1034 827 1207

Grade LC1a WC4a 80-50a 90-60a 105-85b 150-125b 120-95b 175-145b

ASTM A352-68a A219-6 A148-65 A148-1 A148-65 A148-65 A148-65 A148-65 655 483–669 600–669 517–586 552–607

MPa

Materials classiﬁcation

Tensile strength, st

Fatiguec endurance limit, Elongation Modulus of Impact sj in 50 mm elasticity E Charpy (2 in) MPa kpsi % GPa Mpsi J ft-lbf

TABLE 1-19 Mechanical properties of some cast alloy, cast stainless, high-strength and iron-based super alloy steels

Cb and/or V

Composition Cb and/or V (Proprietary) Cu, Cr, Mn, Ni, P, and other additions Cb and or V (Proprietary) Cb, Ti, Zr, Si, N, V, and others

217 311 262 352

Rupture strength, Brinell 100 h at 5388C hardness, (10008F) temperature, Mpsi 8C (8F), HB GPa

PROPERTIES OF ENGINEERING MATERIALS

1.39

H-11

422

AM 350

Stainless W

16-25-G

17-24 Cu Mo

A-286

610

616

Austenitic 633

635

650

653

660

103–1413 1130 1517–1551 517–552 758–965 621 593–772 448 1007–903

827 531 682–896 758 931–2137 1241 1034–1655 1172

MPa

160–205 160 220–225 75–80 110–140 90 86–112 65 146 131

120 77 125–138 110 135–310 180 150–240 170

kpsi

414–1207 745 1482–2000 255–345 345–689 228 276–620 200 655 607

689 372 655–745 586 689–1655 965 682–1207 869

MPa

kpsi

60–175 482–689 70–100 108 215–290 372–662 54–96 37–50 50–100 33 40–90 29 95 88

12–38 9 1.5 58 20–45 58 30–45 37 25 19

3–17 10 16–19 16

30 20 7

20

19.3

19.5

20.9

20.3

20

21

22

21

c

b

2.88

2.80

2.85

3.02

2.94

2.90

3.05

3.17

3.08

Brinell hardness, temperature, 8C (8F), HB

14

21 538 5–144 4–106 21 538 20 15 21 538 11–35 8–26 21 538 56–81 41-60 21 538

19

70 1000 70 1000 70 1000 70 1000 70 1000

Temperature 8C 8F 21 70 538 1000 21 70 538 1000 14–43 10–32 21 70 538 1000 14–52 10–38 21 70 538 1000

Elongation Modulus of Impact in 50 mm elasticity E Charpy (2 in) % GPa Mpsi J ft-lbf

Iron-Based Superalloys

MPa

100 54 95–108 85 100–240 896 130 140 125–175 621–758 90–100 126

kpsi

Yield strength, sy

Normalized and tempered. Quenched and tempered. Polished specimen. d Corrosion resistance. e Heat resistance. f Heat and corrosion resistance. g Semikilled or killed. h Semikilled or killed-improved corrosion resistance. i Inclusion control-improved formability, killed. Source: Machine Design, 1981 Materials Reference Issue, Penton/IPC, Cleveland, Ohio, Vol. 53, No. 6 (March 19, 1981).

a

Chromalloy

604

Martensitic AISI 601 17-22A

Materials classiﬁcation

Tensile strength, st

Fatiguec endurance limit, sj

TABLE 1-19 Mechanical properties of some cast alloy, cast stainless, high-strength and iron-based super alloy steels (Cont.)

75

49

Mpsi

689

330

538

220

710

400

100

48

78

32

103

58

655–793 95–115

517

338

GPa

Rupture strength, 100 h at 5388C (10008F)

PROPERTIES OF ENGINEERING MATERIALS

1.40

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CS CS CS CS CS

1 2 3 4 5

a a , area of cross section. Source: IS 2644, 1979.

640 700 840 1030 1230

Designation

Grade 640 700 840 1030 1230

MPa 92.8 101.5 121.8 149.4 178.3

kpsi

Tensile strength, min, st

TABLE 1-20 Mechanical properties of high tensile cast steel

390 560 700 850 1000

MPa 56.7 81.2 101.5 123.3 145.1

kpsi

Yield strength (or 0.5% proof stress), min, sy

35 30 28 20 12

Reduction in area, min, %

15 14 12 8 5

Elongation, min, % (gauge length ﬃ pﬃﬃﬃﬃ 5.65 a ) a

190 207 248 305 355

Brinell hardness, min, HB

30 30 29 20

J

22.1 21.1 20.6 14.5

ft-lbf

Izod impact strength, min

PROPERTIES OF ENGINEERING MATERIALS

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1.41

1.30–1.50 1.25–1.40 1.10–1.25 0.65–0.75 0.80–0.90 0.70–0.80 0.60–0.70 2.00–2.30 150–1.70 1.00–1.20 0.90–1.20 0.85–0.95 0.90–1.20 0.90–1.20 0.50–0.60 0.50–0.60 0.45–0.55 0.55–0.65 0.26–0.34 0.50–0.60 0.25–0.40 0.30–0.40 0.30–0.40 0.70–0.80 0.75–0.90 0.50–0.60 0.12–0.20 0.15 max

T 140 W 4 Cr 50 T 133 T 118 T 70 T 85 T 75 T 65 T 215 Cr 12 T 160 Cr 12 T 110 W 2 Cr 1 T 105 W 2 Cr 60 V 25 T 90 Mn 2 W 50 Cr 45 T 105 Cr 1 T 105 Cr 1 Mn 60 T 55 Cr 70 T 55 Si 2 Mn 90 Mo 33 T 50 Cr 2 V 23 T 60 Ni 1 T 30 Ni 4 Cr 1 T 55 Ni 2 Cr 65 Mo 30 T 33 W 9 Cr 3 V 38 T 35 Cr 5 Mo V 1 T 35 Cr 5 Mo W 1 V 30 T 75 W 18 Co 6 Cr 4 V 1 Mo 75 T 83 Mo W 6 Cr 4 V 2 T 55 W 14 Cr 3 V 45 T 16 Ni 85 Cr 60 T 10 Cr 5 Mo 75 V 23

Optional Source: IS 1871, 1965.

a

%C

Steel designation

TABLE 1-21 Chemical composition of tool steels

0.10–0.35 0.10–0.30 0.10–0.30 0.10–0.30 0.10–0.35 0.10–0.35 0.10–0.35 0.10–0.35 0.10–0.35 0.10–0.35 0.10–0.35 0.10–0.35 0.10–0.35 0.10–0.35 0.10–0.35 1.50–2.00 0.10–0.35 0.10–0.65 0.10–0.35 0.10–0.35 0.10–0.35 0.80–1.20 0.80–1.20 0.10–0.35 0.10–0.35 0.20–0.35 0.10–0.35 0.10–0.35

% Si 0.25–0.50 0.20–0.35 0.20–0.35 0.20–0.35 0.50–0.80 0.50–0.80 0.50–0.80 0.25–0.50 0.25–0.50 0.90–1.30 0.25–0.50 1.25–1.75 0.20–0.40 0.40–0.80 0.60–0.80 0.80–1.00 0.50–0.80 0.50–0.80 0.40–0.70 0.50–0.80 0.20–0.40 0.25–0.50 0.25–0.50 0.20–0.40 0.20–0.40 0.20–0.40 0.60–1.00 0.25–0.50

% Mn

0.90–1.20 0.30 max 1.10–1.40 0.50–0.80 2.80–3.30 4.75–5.25 4.75–5.25 4.00–4.50 3.75–4.50 2.80–3.30 0.40–0.80 4.75–5.25

11.0–13.0 11.0–13.0 0.90–1.30 0.40–0.80 0.30–0.60 1.00–1.60 1.00–1.60 0.60–0.80

0.30–0.70

% Cr

0.20–0.30 0.25 max

0.12–0.20a 0.15–0.30

0.25 maxa

0.25–0.40

0.25–0.50 1.00–1.20 0.20–0.40 1.50–1.50 1.75–2.00 0.30–0.60 0.15–0.30

1.20–1.60 1.20–1.60 0.50–1.00 5.50–6.50

0.50–1.00

0.25–0.35

0.80 maxa 0.80 maxa

%V

0.80 maxa 0.80 maxa

%Mo

1.20–1.60 17.50–19.00 5.50–6.50 13.00–15.00

8.0–10.0

1.25–1.75 1.25–1.75 0.40–0.60

3.50–4.20

%W

0.60–1.00

1.00–1.50 3.90–4.30 1.25–1.75

% Ni

5.00–6.00

% Co

PROPERTIES OF ENGINEERING MATERIALS

1.42

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640 2170 710 1827

Annealed 7758C (14258F) Tempered 2058C (4008F)

Annealed 8008C (14758F) Tempered 2058C (4008F)

Annealed 7908C (14508F) Tempered 2058C (4008F)

Annealed 8308C (15258F) Tempered 2058C (4008F)

Annealed 8458C (15508F)b Tempered 5658C (10508F)

P-20

S-1

S-5

S-7

A-8

724 2344

690 2068

103 265

93 315

105 340

100 300

100 270

95 290

103 290

100 295

kpsi

448 1550

380 1448

440 1930

414 1896

517 1413

380 1793

510 1793

365 1724

MPa

65 225

55 210

64 280

60 275

75 205

55 260

74 260

53 250

kpsi

Yield strength, sy

25 9

25 7

25 5

24 4

17 10

25 4

25 5

25 9

97 RB 52 RC

95 RR 58 RC

96 RB 59 RC

96 RB 57.5 RC

97 RB 52 RC

93 RB 54 RC

96 RB 54 RC

96 RB 55 RC

Elongation, % Hardness

1010

940

1850

1725

1600

1700

925 870

1575

855

1550

1575

855 845

1850

8F

1010

8C

Hardening temperature

air

air

oil

oil

oil

oil

oil

air

7

244

206

250

20

12

28

14

Quenched media J

Medium to high

152c

5

Medium

Medium

Medium

184c

180

Medium to high

Medium

High

Medium to high

Machinability

15

9

21

10

ft-lbf

Impact strength Charpy V-notch

b

Single temper, oil-quenched unless otherwise indicated. Double temper, air-quenched. c Charpy impact unnotched tests made on longitudinal specimens of small cross-sectional bar stock. The heat treatments listed were to develop nominal mechanical properties for hardened and tempered materials for test purposes only and may not be suitable for some applications. Source: Machine Design, 1981 Materials Reference Issue, Penton/IPC, Cleveland, Ohio, Vol. 53, No. 6 (March 19, 1981).

a

665 2000

Annealed 7758C (14258F)a Tempered 3158C (6008F)

L-6 690 1860

710 2000

Annealed 7758C (14258F) Tempered 2058C (4008F)

L-2

690 2034

Annealed 8708C (16008F) Tempered 5408C (10008F)

b

MPa

H-11

AISI steel designation Conditiona

Tensile strength, st

TABLE 1-22 Mechanical properties of some tool steels

PROPERTIES OF ENGINEERING MATERIALS

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1.43

90.7–91.5

94.5–95.2

Fine Coarse

Medium

84WC-16Co

72WC-8TiC11.5TaC-8.5Co

Medium 64TiC-28WC2TaC-2Cr3 C2 -4.0Co

6.6

12.6

13.9 13.9

14.6 14.5

15.0 15.0 15.0

0.24

0.45

0.50 0.50

0.53 0.52

0.54 0.54 0.54

Mg/m3 lb/in3

Density

690

1720

3380 2900

3100 2760

1790 2000 2210

MPa

100

250

490 420

450 400

260 290 320

kpsi

4340

5170

4070 3860

5170 4000

5930 5450 5170

MPa

630

750

590 560

750 580

860 790 750

kpsi

Compressive strength, sc

1720

970 700

1590 1170

2550 1930 1450

MPa

250

140 100

230 170

370 280 210

kpsi

Proportional limit compressive strength, sp

558

524 524

620 552

614 648 641

GPa

81

76 76

90 80

89 94 93

Mpsi

Modulus of elasticity, E

1860

1340

1450 1520

MPa

270

195

210 220

kpsi

Tensile strength, st

0.90

3.05 2.83

1.69 2.03

1.02 1.36 1.36

J

8

27 25

15 18

9 12 12

in-lbf

Impact strength

– 50

88

– 112

– 100 121

W/m K

Thermal conductivity

–

5.8

– 5.8

– 5.2

4.3 4.3 4.3

2008C

–

7.0

– 7.0

– –

5.9 5.4 5.6

3.2

– 3.2

– 2.9

2.4 2.4 2.4

10008C 4008F

–

3.8

– 3.8

– –

3.3 3.0 3.0

18008F

Coeﬃcient of linear expansion, lm/m8C at lin/in8F at

Source: Metals Handbook Desk Edition, ASM International 1985, Materials Park, OH 44073-0002 (formerly the American Society for Metals, Metals Park, OH 44073, 1985).

89 86.0–87.5

90.7–91.3 87.4–88.2

Fine Coarse

90WC-10Co

92.5–93.1 91.7–92.2 90.5–91.5

Fine Medium Coarse

Grain size

94WC-6Co

Nominal composition

Brinell Hardness HB

Transverse strength, sb

TABLE 1-23 Properties of representative cobalt-bonded cemented carbides PROPERTIES OF ENGINEERING MATERIALS

1.44

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PROPERTIES OF ENGINEERING MATERIALS PROPERTIES OF ENGINEERING MATERIALS

1.45

TABLE 1-24 Typical uses of tool steel Steel designation

Type

T 140 W 4 Cr 50 T 133 T 118 T 70

Cold-Work Water-Hardening Steels Fast ﬁnishing tool steel Finishing tools with light feeds, marking tools, etc. Carbon tool steels Engraving tools, ﬁles, razors, shaping and wood-working tools, heading and press tools, drills, punches, chisels,shear blades, vice jaws, etc.

T 215 Cr 12 T 160 Cr 12 T 110 W 2 Cr 1 T 105 W 2 Cr 60 V 25 T 90 Mn 2 W 50 Cr 45 T 105 Cr 1 T 105 Cr 1 M 60 T 85 T 75 T 65 T 55 Cr 70 T 55 Si 2 Mn 90 Mo 33 T 50 Cr 1 V 23 T 60 Ni 1 T 30 Ni 4 Cr 1 T 55 Ni 2 Cr 65 Mo 3 T 33, W 9 Cr 3 V 38 T 35 Cr 5 Mo V 1 T 35 Cr 5 Mo W 1 V 30 T 75 W 18 Co 6 Cr 4 V 1 Mo 75 T 83 Mo W 6 Cr 4 V 2 T 55 W 14 Cr 3 V 45a T 16 Ni 80 Cr 60 T 10 Cr 5 bee 75 V 23

Typical uses

Cold-Work Oil and Air-Hardening Steels High-carbon highPress tools, drawing and cutter dies, shear blade thread chromium tool steels rollers. etc. Nondeforming tool steels Engraving tools, press tools, gauge, tape, dies, drills, hard reamers, milling cutters, broaches, cold punches, knives. etc. Carbon-chromium tool steels

Lathe centers, knurling tools, press tools Die blocks, garden and agricultural tools, etc.

Carbon tool steels Shock-resisting tool steels

Pneumatic chisels, rivet shape, shear blades, heavy-duty punches, scarﬁng tools, and other tools under high shock

Nickel-chromemolybdenum tool steels

Cold and heavy duty punches, trimming dies, scarﬁng tools, pneumatic chisels, etc.

Hot-Work and High-Speed Steel Hot-work tool steels Castings dies for light alloys, dies for extrusion, stamping, and forging High-speed tool steels

Drills, reamers, broaches, form cutters, milling cutters, deep-hole drills, slitting saws, high-speed and heavy-cut tools

Low-Carbon Mold Steel Carburizing steels After case hardening for molds for plastic materials

a May also be used as hot-work steel. Source: IS 1871, 1965.

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PROPERTIES OF ENGINEERING MATERIALS

1.46

CHAPTER ONE

TABLE 1-25 Mechanical properties of carbon and alloy steel bars for the production of machine parts Ultimate tensile strength, sut Steel designation

MPa##

kpsi

MPa‡

kpsi

14 C 4 (C 14) 20 C 8 (C 20) 30 C 8 (C 30) 40 C 8 (C 40) 45 C 8 (C 45) 55 C 8 (C 55 Mn 75) 65 C 6 (C 65) 14 C 14 S 14 (14 Mn 1 S 14) 11 C 10 S 25 (13 S 25)

363 432 490 569 618 706 736 432 363

52.6 62.6 71.1 82.5 89.6 102.4 106.7 62.6 52.6

441 510 588 667 696

64.0 74.0 85.3 96.7 101.0

530 481

76.8 69.7

Notes: a , area of cross section; Source: IS 2073, 1970.

##

Minimum (gauge length ﬃ pﬃﬃﬃﬃelongation = 5.65 a ), % 26 24 21 18 15 13 10 22 23

minimum; ‡ maximum; steel designations in parentheses are old designations

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Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. 1473–1123 1473–1123 1473–1123 1473–1123 1473–1123 1473–1123 1473–1123 1473–1123 1473–1123 1473–1123 1473–1123 1473–1123 1473–1123 1473–1123 1473–1123 1473–1123 1473–1123 1473–1123

1473–1123 1473–1123 1473–1123 1473–1123 1473–1123 1473–1123 1473–1123 1373–1123 1373–1123

30 C 8 (C 30) 35 C 8 (C 25 Mn 74) 40 C 8 (C 40) 50 C 8 (C 50) 55 C 8 (C 55 Ma 75) 40 C 10 Si 8 (40 S 18) 40 C 15 Si 2 (40 Mn 2 S 12) 220 C 15 (20 Mn 2) 27 C 15 (27 Mn 2) 37 C 15 (37 Mn 2) 40 Cr 4 (40 Cr 1) 35 Mn 6 Mo 3 (35 Mn 2 Mo 28) 35 Mn 6 Mo 4 (35 Mn 2 Mo 45) 40 Cr 4 Mo 3 (40 Cr 1 Mo 28) 40 Ni 14 (40 Ni 3) 35 Ni Cr 2 Mo (35 Ni Cr Mo 60) 40 Ni 6 Cr 4 Mo 2 (40 Ni Cr Mo 15) 40 Ni 6 Cr 4 Mo 3 (40 Ni 2 Cr 1 Mo 28)

15 Ni Cr 1 Mo 12 (31 Ni 3 Cr 65 Mo 55) 30 Ni 13 Cr 5 (30 Ni 4 Cr 1) 15 Cr 13 Mo 6 (15 Cr 3 Mo 55) 25 Cr 13 Mo 6 (25 Cr 3 Mo 55) 40 Cr 13 Mo 10 V 2 (40 Cr 3 Mo 1 V 20) 40 Cr 7 Al 10 Mo 2 (40 Cr 2 Al 1 Mo 18) 55 Cr 70) 105 Cr 4 (105 Cr 1) 105 Cr 1 Mn 60

a Stabilization 823 K (5508C). Source: IS 1871, 1965.

K

Designation

1200–850 1200–850 1200–850 1200–850 1200–850 1200–850 1200–850 1100–850 1100–850

1200–850 1200–850 1200–850 1200–850 1200–850 1200–850 1200–850 1200–850 1200–850 1200–850 1200–850 1200–850 1200–850 1200–850 1200–850 1200–850 1200–850 1200–850

8C

Hot-working temperature

800–850

850–880 830–860

1123–1153 1103–1133

1073–1123

860–890 850–880 830–860 810–840 810–840 830–860 840–870 860–900 840–880 850–870 850–880

8C

1133–1163 1123–1153 1103–1133 1083–1113 1083–1113 1103–1113 1113–1143 1133–1173 1133–1153 1123–1143 1123–1153

K

Normalizing

TABLE 1-26 Recommended hardening and tempering treatment for carbon and alloy steels

1103–1123 1083–1103 1163–1183 1163–1183 1173–1213 1123–1173 1073–1123 1093–1133 1073–1113

1133–1163 1113–1153 1103–1133 1083–1113 1083–1113 1103–1133 1113–1143 1133–1173 1133–1153 1123–1143 1123–1153 1113–1133 1113–1133 1123–1153 1103–1133 1093–1123 1103–1123 1103–1123

K

830–850 810–820 890–910 890–910 900–940 850–900 800–850 820–860 800–840

860–890 840–880 830–860 810–840 810–840 830–860 840–870 860–900 840–880 850–870 850–880 840–860 840–860 850–880 830–860 820–850 820–850 830–850

8C

Hardening 8C

Oil Air or oil Oil Oil Oil Oil Oil Water or oil Water or oil

Water or oil Water or oil Water or oil Oil Oil Oil Oil Water or oil Water or oil Water or oil Oil Water or oil Oil Oil Oil Water or oil Oil Oil

K

Quenching

823–923 803–1033 823–933 823–933 823–933 823–933 823–933 823–933 823–933 823–933 823–933 823–933 823–933 823–933 823–933 823–933 823–933 823–933 or 423–473 (depending on hardness required) 933 523 823–973a 823–973a 843–923 823–973 773–973 >423 403–453

K

Tempering

660 250 550–700a 550–700a 570–650 550–700 500–700 >150 in oil 130–180

550–660 530–760 550–660 550–660 550–660 550–660 550–660 550–660 550–660 550–660 550–660 550–660 550–660 550–660 550–660 550–660 550–660 550–660 or 150–200

8C

PROPERTIES OF ENGINEERING MATERIALS

1.47

11.2 12.7 12.5

11.6 13.6 13.6 12.6

14.3 14.1 14.1

13.0

5.8 6.3

0.85 1.11 1.28

0.83 1.16 0.93 0.98

0.52 0.75 1.24

0.75

0.90 0.89

0.37 0.6

0.95

1.47 0.99 0.64

0.38 0.60 0.67 0.6

0.57 0.54 0.94

Si

Mo Mo Mo Mo

1.46 Mo 1.20 Mo

3.65 Ni

2.4 Mo 2.0 Mo 3.0 Mo

0.96 1.10 0.96 0.87

Other

Mill liner Plate

Round

Round Round Round

Round Round Plate Plate

Round Round Keel block

Form

100 100

25

25 25 25

25 25 25 50

25 25 100

mm

4 4

1

1 1 1

1 1 1 2

1 1 4

in

Section

340 330a

655

600 745 600

695 560 510 435a

440 450 330a

MPa

49 48a

181

150

3.5 Ni manganese steel 295 43

95

6 Mn-1 Mo alloys 325 47 –

220 183 235

2 Mo manganese steels 370 54 365 53 440 64

87 108 87

163 185 188 –

1 Mo manganese steels 101 345 50 81 400 58 74 365 53 63a – –

kpsi

Brinell hardness, HB – – 245

MPa

Yield strength, sy (0.2% oﬀset)

Plain manganese steels 64 – – 65 360 52 48a – –

kpsi

Tensile strength, st

2 1a

36

15.5 34.5 7.5

30 13 11 4a

14.5 4 1a

Elongation in 50 mm, %

– –

26

13 27 10

29 15 16 –

– – –

Reduction in area, %

b

a

Properties converted from transverse bend tests on 6 13 mm (14 12 in) bars cut from castings and broken by center loading across 25 mm (1 in) span. Charpy V-notch. Source: Metals Handbook Desk Edition, ASM International, 1985, Materials Park, OH 44073-0002 (formerly the American Society for Metals, Metals Park, OH 44073, 1985).

Mn

C

Composition, %

TABLE 1-27 Mechanical properties of some as-cast austenitic manganese steels

ft-lbf

9 –

–

– – –

– – 72 –

7 –

–

– – –

– – 53 –

– – – – 3.4 2.5

J

Impact strength Charpy b

PROPERTIES OF ENGINEERING MATERIALS

1.48

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179 278 331 250

290

303 283

186

-T 43 -T 6 240.0 -F 295.0 -T 4 -T 6 319.0 -F -T 6 C 355.0 -T6 356.0 -T 6

A 390.0 -F -T 6 520.0 -T4 A 535.0 -F

-T 6

C 355.0 -T61 A 356.0 -T 61

-F

-O -H 14 -H 18 -T 3 -T 6 -O -T 451 -T 651 -O -T 451 -O -T 351 -T 3 -T 86

355.0

513.0

1100

2014 -T 4. -T 6. 2017 -T 4. 2024 -T 4.

2011

414 448 235 221 250 186 250 269 228

201.0

90 125 165 380 395 185 425 482 180 425 185 470 485 515

MPa

Alloy no.

13 18 24 55 57 27 62 70 26 62 27 68 70 75

27

44 41

42

26 40 48 36

60 65 34 32 36 27 26 39 33

kpsi

Ultimate tensile strength, sut

35 115 150 295 270 95 290 415 70 275 75 325 345 490

110

234 207

185

179 278 179 124

255 379 200 110 165 124 164 200 164

MPa

5 17 22 43 39 14 42 60 10 40 11 47 50 71

16

34 30

27

26 40 26 18

37 55 29 16 24 18 24 29 24

kpsi

Tensile yield strengthd , syt

117

248 221

185

17

36 32

27

27

25

172

186

56 30 17 25 19 25

kpsi

386 207 117 172 131 172

MPa

Compressive yield strength,d syc

60 75 90 220 235 125 260 290 125 260 125 285 280 310

152

221 193

235

234

9 11 13 32 34 18 38 42 18 38 18 41 40 45

22

32 28

34

34

26

26 31 22 29

179 217 152 200 179

42

kpsi

290

MPa

Shear strength, s

35 50 60 125 125 90 140 125 90 125 90 140 140 125

69

97 90

69

90 55

59

48 52 69 76

MPa

5 7 9 18 18 13 20 18 13 18 13 20 20 18

10

14 13

10

13 8

8.5

7 7.5 10 11

kpsi

Endurance limit in reversed bending, sfb

23 32 44 95 97 45 105 135 45 105 47 120 120 135

60

90 90

90

100 140 75 65

130 90 60 75 70 80 85 70

9.5

11.9

10.5

10.0 10.0 10.7 10.7

10.5

Wrought alloys

72

Permanent mold casting

65

82

72

69 69 74 74

Sand casting alloys

Brinell hardness Modulus of 4.9 kN e (500 kgf) load elasticity, E on 10-mm ball, HB GPa Mpsi

35 9 5 15 17 18 20 13 22 22 20 20 18 6

7.0

3.0 10.0

D B B

D B B B

– C D D

E D D A

1

3 3

3

4 4 1 1

1 1 3 2 2 3 3 3 3

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D D D

A A A D

Machiability Gas

C C

A A A D

1

3 2

3

2 2 1 1

<1.0 <1.0 16 9.0

4.0

4 4 4 3 3 3 3 3 3

Corrosion resistance

17 8 1.0 8.5 5.0 2.0 2.0 5.0 3.5

(2 in), %

Elongation in 50 mm

TABLE 1-28 Mechanical properties, fabrication characteristics,a and typical uses of some aluminum alloysb

D B B C

D B B

A A A D

5

2 2

2

2 2 5 4

2 2 4 2 2 2 2 2 2

Arc

Welding

D B B B

B B B

B A A D

Resistance

Truck wheels, screw-machine products, aircraft structure

Truck frames, aircraft structures

Screw machine products

Sheet metal work, spun holloware, ﬁn stock

Machine-tool parts, aircraft wheels, pump parts, marine hardware, valve bodies Ornamental hardware and architectural ﬁttings

Timing gears, impellers, compressor and aircraft and missile components requiring high strength

Aircraft ﬁttings and components, levers, brackets

Air compressor ﬁtting, crankcase, gear housing Cylinder heads, impellers, timing gears, water jackets, meter parts Automotive engine blocks, pulleys, brake shoes, pumps

Crankcases, spring hangers, housing, wheels

Aircraft structural components

Uses

PROPERTIES OF ENGINEERING MATERIALS

1.49

110

150 200 180 240 285 195

260 290 125

310 90 240 230 570

-O

-H 14 -H 18 -O -H 34 -H 38 -O

-H 34 -H 38 -O

-T 6 -O -T 6 -O -T 6

3003

45 13 35 38 83

38 42 18

22 29 26 35 41 28

16

kpsi

275 50 215 105 505

215 255 55

145 185 70 200 250 90

40

MPa

40 7 31 15 73

31 37 8

21 27 10 29 36 13

6

kpsi

Tensile yield strengthd , syt MPa

kpsi

205 70 150 150 330

145 165 80

95 110 110 125 145 125

75

MPa

30 10 22 22 48

21 24 12

14 16 16 18 21 18

11

kpsi

Shear strength, s

95 55 70 115 160

125 140 60

60 70 95 105 110 110

50

MPa

14 8 10 17 23

18 20 9

9 10 14 15 16 16

7

kpsi

Endurance limit in reversed bending, sfb

95 25 73 60 150

68 77 30

40 55 45 63 77 47

28

ball, HB

Brinell hardness 4.9 kN (500 kgf) load on 10-mm GPa

Mpsi

Modulus of elasticity,e E

12 17 11

12

10 7 25

8 4 20 9 5 25

30

(2 in), %

Elongation in 50 mm

B A A – C

A A B

A A A A A A

A

Corrosion resistance

C D B

C

C C D

D D D C C D

E

A A A D D

A A A

A A B B B A

A

Machiability Gas

A A A C C

A A A

A A A A A A

A

Arc

Welding

A A A B B

A A B

A A B A A B

B

Resistance

Fin stock, cladding alloy Aircraft and other structures

Pipe, railing, furniture, architectural extrusions

Heavy-duty structures requiring good corrosion resistance, truck and marine, railroad car, furniture, pipeline applications

Hydraulic tube, appliances, bus body sheet, sheet metal work, welded structures, boat sheet

Trailer panel sheet, storage tanks, sheet metal works

Pressure vessels, storage tanks, heat-exchanger tubes, chemical equipments, cooking utensils

Uses

b

For ratings of characteristics, 1 is the best and 5 is the poorest of the alloys listed. Ratings A through D are relative ratings in decreasing order of merit. Average of tensile and hardness values determined by tests on standard 12.5-mm (12-in) diameter test specimens. c Endurance limits on 500 million cycles of completely reversed stresses using rotating beam-type machine and specimen. d At 0.2% oﬀset. e Average of tension and compression moduli. Key: Temper designations: F, as cast; O, annealed; Hxx, strain hardened; T1, cooled from an elevated temperature shaping process and naturally aged; T2, cooled from an elevated temperature shaping process, cold-worked and naturally aged; T3, solution heat-treated and cold worked and naturally aged; T4, solution heat-treated and naturally aged; T5, cooled from an elevated temperature shaping process and artiﬁcially aged; T6, solution heat-treated and artiﬁcially aged; T7, solution heat-treated and stabilized; T8, solution heat-treated, cold-worked and artiﬁcially aged; TX 51, stressrelieved by stretching. Source: ASM Metals Handbook, American Society for Metals, Metals Park, Ohio, 1988.

a

7075

6063

6061

5052

3004

MPa

Alloy no.

Ultimate tensile strength, sut

Compressive yield strength,d syc

TABLE 1-28 Mechanical properties, fabrication characteristics,a and typical uses of some aluminum alloysb ðCont:Þ

PROPERTIES OF ENGINEERING MATERIALS

1.50

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0.15–0.35

0.2–0.4

A-4

A-5

A-6

A-8

A-9

A-10 LM10 0.1

A-11 LM11 4.0–5.0

A-12 LM12 9.0–10.5

BS 1490 9.0–11.5 (LM12) A-13 LM 13 0.5–1.3

4223

5230

4600

4250

4635

5500

2280

2585

4685

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A13 (special)

LM9

LM8

LM6

LM5

LM4

LM2

0.1

0.1

0.1

0.1

2.0–4.0

0.7–2.5

0.8–1.5

0.1

9.5–11.0

0.2–0.6

0.3–0.8

0.1

3.0–6.0

0.15

0.3

0.15

A-2

6.0–8.0

4520

LM1

A-1

Mg

2447

Cu

IS old

IS new

BS

Designation

11.0–13.0

2.5

2.0

0.25

0.25

10.0–13.0

3.5 –6.0

10.0–13.0

0.3

4.0–6.0

9.0–11.15

2.0–4.0

Si

0.8

1.0

1.0

0.5

Ni

0.1

0.1

0.1

0.1

0.5

0.6

Ti

0.8

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.5

1.2

0.2 0.2

0.05–0.3 0.05

0.2 0.2 0.3

0.1 0.1

0.2

0.1

0.05

0.2 0.2

0.1

0.2

0.2

0.1

0.1

0.2

0.05

0.2

0.2

0.3

0.3

Pb

0.1

0.2

Ti + Nb

0.2

2.0–4.0 0.2

Zn

2.0–3.0 0.1

0.5

0.5

0.1

0.1

0.3–0.7 0.1

0.5

0.5

0.3–0.7 0.1

0.3–0.7 0.3

0.5

0.6

Mn

0.5–1.5 0.6

0.25

0.35

0.6

0.6

0.6

0.6

0.8

1.0

1.0

Fe

Chemical composition, percent

TABLE 1-29 Chemical composition and mechanical properties of cast aluminum alloy5;6;7

0.10

0.1

0.10

0.05

0.05

0.05

0.05

0.05

0.05

0.05

0.20

0.20

Sn

R

E

D

N

I

A

M

E

R

Al

MPa

WP

Fullly heattreated WP

170 247 170 278 139 201

124 154 As cast 124 147 As cast 139 154 As cast 139 170 As cast 162 185 As cast 124 162 162 232 147 185 232 278 Precipitation- 170 treated 231 Solution278 treated 307 Solution216 treated 263 WP 278 309 Fully heattreated WP 278 As cast

Condition

24.6 35.9 24.6 40.3 20.2 29.2

40.3

18.0 22.3 18.0 21.3 20.2 22.3 20.2 24.6 23.5 26.9 18.0 23.5 23.5 33.6 21.3 26.9 33.6 40.3 24.6 36.4 40.3 44.8 31.3 38.1 40.3 44.8

kpsi

Tensile strength, st

2 1.5 2 8 12 7 13 4 9

2 2 3 5 5 7 2 3 2.5 5 1 2

65 65

100 100

100

Sand-cast Chill-cast Sand-cast Chill-cast Sand-cast Chill-cast

Sand-cast Chill-cast Sand-cast Chill-cast Sand-cast Chill-cast

Sand-cast Chill-cast Sand-cast Chill-cast Sand-cast Chill-cast Sand-cast Chill-cast Sand-cast Chill-cast Sand-cast Chill-cast

Brinell Elongation hardness, Test piece % HB

Mechanical properties

PROPERTIES OF ENGINEERING MATERIALS

1.51

A-18 LM18 0.1

A-22 LM22 2.8–3.8

A-24 LM24 3.0–4.0

4300

4223

4420

0.1

0.05

0.1

0.4–0.6

1.2–1.7

Mg

7.5–9.5

4.0–6.0

4.5–6.0

4.5–5.5

0.6

Si

1.3

0.7

0.6

0.6

0.6

Fe

0.5

Zn

0.1

0.25

0.5

1.0 3.0

0.15

0.1

0.1

1.8–2.3 0.1

Ni

0.3–0.6 0.15

0.5

0.5

0.6

Mn

Chemical composition, percent

0.2

0.2

0.2

0.2

0.2

0.2

0.2

Ti

Ti + Nb

0.3

0.05

0.1

0.05 0.1

0.05

Pb

Notes: IS: Sp-1-1967 Speciﬁcation of Aluminum Alloy Castings and BS 1490 (from LM 1 to LM 24) are same. Refer to both Indian Standards and British Standards; ** refer to British Standards, BS 1490 only. Source IS Sp-1, 1967.

A-16 LM16 1.0–1.5

4225

A-14 (special)

A-14 LM 14 3.5–4.5

Cu

2285

BS

IS old

IS new

Designation

TABLE 1-29 Chemical composition and mechanical properties of cast aluminum alloy (Cont.)

0.1 0.20

0.05

0.05

0.05

0.05

Sn

Al 216 278 185 232

MPa

As cast

177

170 201 232 263 232 278 116 139 Solution (W)- 247 treated WP WP WP WP As cast

Fully heattreated WP Solutiontreated

Condition

25.7

24.6 29.2 33.6 38.1 33.8 40.3 16.8 20.2 35.9

31.3 40.3 26.9 33.6

kpsi

Tensile strength, st

1.5

3 4 8

2 3

100 100 75 75

Chill-cast

Sand-cast Chill-cast Sand-cast Chill-cast Sand-cast Chill-cast Sand-cast Chill-cast Chill-cast

Sand-cast Chill-cast Sand-cast Chill-cast

Brinell Elongation hardness, Test piece % HB

Mechanical properties

PROPERTIES OF ENGINEERING MATERIALS

1.52

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99 min

99.5 min

19000

19500

Remainder

Remainder

Remainder

Remainder

Remainder

Remainder

Remainder

Remainder

24345

24534

43000

46000

52000

53000

54300

63400

19600

Al

Designation

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0.1

0.1

0.1

0.1

0.1

0.1

3.5–4.7

3.8–5.0

0.05

0.1

Cu

0.4–0.9

4.0–4.9

2.8–4.0

1.7–2.6

0.2

0.2

0.4–1.2

0.2–0.8

–

0.2

Mg

0.7

0.7

0.4

0.7

Fe

0.3–0.7

0.4

0.6

0.6

0.6

0.7

0.5

0.5

10.0–13.0 0.6

4.5– 6.00 0.6

0.2–0.7

0.5–1.2

0.3

0.5

Si

0.3

0.5–1.0

0.5

0.5

0.5

0.5

0.4–1.2

0.3–1.2

0.05

0.1

Mn

Chemical composition, %

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.1

0.1

Zn

0.2

0.2

0.2

0.2

–

–

0.3

0.3*

–

–

Ti or others

0.1

0.25

0.25

0.25

–

–

–

0.3*

–

–

Cr

O Ma O Ma O W

Ma O Ma O Ma O Ma

W

Ma O

WP

Ma O Ma O Ma Ma O W

Condition

10 (0.4) 10 (0.4) 75 (3.0) 150 (6.0) – 10 (0.4) 25 (1.0) 75 (3.0) 150 (6.0) – – – 10 (0.4) 75 (3.0) 150 (6.0) – – – – – – – 50 (2.0) – – – All sizes – – 150 (6.0)

Over mm (in)

Size

10 (0.4) 75 (3.0) 150 (6.0) 200 (8.0) 10 (0.4) 25 (1.0) 75 (3.0) 150 (6.0) 200 (8.0) – – 10 (0.4) 75 (3.0) 150 (6.0) 200 (8.0) 15 (0.6) 15 (0.6) 15 (0.6) 15 (0.6) 150 (6.0) 50 (2.0) 150 (6.0) 150 (6.0) 150 (6.0) 150 (6.0) 150 (6.0) – – 150 (6.0) 200 (8.0)

Up to and including mm (in) 2.9 2.6 2.5 13 25.4 32.6 34.1 34.1 32.6 54.4 58.0 60.9 58.7 55.1 13.0 25.0 31.9 34.1 34.1 32.6 – – – – 10.2 14.5 14.5 – 18.9 18.1 – – 11.6 11.6

18 17 90 175 225 235 235 225 375 400 420 405 380 90 175# 220 235 235 225 – – – – 70 100 100 – 130 125 – – 80 80

kpsi

20

MPa

0.2% proof stress, min, sp

TABLE 1-30 Chemical composition and mechanical properties of wrought aluminum and aluminum alloys for general engineering purposes6

65 110# 65 100# 65 150 240# 375 385 385 375 430 460 480 460 430 150 240 375 385 385 375 90 130# 100 150# 160 240# 215 200 260# 275 350 110 130# 140 125

MPa

9.4 16.0 9.4 14.5 9.4 21.6 34.8 54.4 55.8 55.8 54.4 62.4 66.7 69.6 66.7 62.4 21.0 34.8 54.4 55.8 55.8 54.4 13.0 18.9 14.5 21.8 23.2 34.8 31.2 29.0 37.7 40.0 50.8 16.0 18.8 20.3 18.1

kpsi

Tensile strength, min, st

18 25 23 25 23 12 12 10 10 8 8 6 6 6 6 6 12 12 10 10 8 8 18 18 10 12 14 18 14 14 10 11 13 13 18 14 13

Elongation, % (min)

PROPERTIES OF ENGINEERING MATERIALS

1.53

Remainder

Remainder

Remainder

Remainder

64423

64430

65032

74530

1.54

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. 0.4–1.2

0.5–1.3

Mg

0.2

0.4–0.8

0.6–1.3

0.7–1.3

Si

1.0– 1. 5 0.4–0.8

0.15–0.4 0.7–1.2

0.1

0.5–1.0

Cu

0.7

0.7

0.6

0.8

Fe

0.2–0.7

0.2–0.8

0.4–1.0

1.0

Mn

4–5

0.2

1.0

Zn

0.2

0.2

0.2

Ti or others

0.2

0.15–0.35

0.25

Cr

WP

WP

W (Naturally aged for 30 days) WP

WP

Ma O W

All sizes – 6 (0.24) 15 (0.6)

6 (0.24)

– 150 (6.0) – – – – All sizes – – 150 (6.0) – 5 (0.2) 75 (3.0) 150 (6.0) All sizes – – 150 (6.0) – 150 (6.0) – 6 (0.24) 75 (3.0)

P

–

–

Condition

Ma O W WP Ma O W WP

Over mm (in)

Size

6 (0.24) 75 (3.0) 150 (6.0) – 6 (0.24) 75 (3.0) 150 (6.0)

15 (0.6) 150 (6.0) 200 (8.0) 150 (6.0) 200 (8.0) 6 (0.24) 75 (3.0) 150 (6.0)

23 (0.12) 12 (0.5) 150 (6.0) 200 (8.0) – – – – – – 150 (6.0) 200 (8.0) 5 (0.2) 75 (3.0) 150 (6.0) 200 (8.0)

Up to and including mm (in)

245 26245 – 430 455 430

140 110 150 130 – 125# 155 265 80 – 120 100 255 270 270 240 50 115# 115 100 235 200 220 230 220

MPa

35.5 37.7 35.5 – 62.4 66.6 62.4

20.3 16.0 21.8 18.9 – 18.1 22.5 38.4 11.6 – 18.1 14.5 37.0 39.2 39.2 34.8 7.3 16.7 22.5 14.5 34.0 29.0 31.4 33.6 31.4

kpsi

0.2% proof stress, min, sp

285 310 290 290# 500 530 500

170 150 185 150 120 215# 265 330 110 150# 185 170 295 310 295 280 110 150# 185 170 280 245 255 275 265

MPa

41.3 45.0 42.1 42.1 72.5 78.9 72.5

24.7 21.8 26.8 21.8 17.4 31.2 38.4 47.9 16.0 21.8 26.8 24.7 42.8 45.0 42.8 40.6 16 21.8 26.8 24.7 40.6 35.5 37.0 40.0 38.4

kpsi

Tensile strength, min, st

7 7 7 10 6 6 6

7 7 7 6 10 15 13 7 12 16 14 12 7 7 7 6 12 16 14 12 7 6 9 9 9

Elongation, % (min)

Properties in M (as-cast) temper are only typical values and are given for information only. Key: # Maximum, M – as-cast condition; R – stress-relieved only; P – precipitation-treated; W – solution-treated, WP – solution-treated and precipitation treated; WPS – fully heat treated plus stabilization. Source: IS 733, 1983.

a

76528

Al

Designation

Chemical composition, %

TABLE 1-30 Chemical composition and mechanical properties of wrought aluminum and aluminum alloys for general engineering purposes (Cont.)

PROPERTIES OF ENGINEERING MATERIALS

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84 Cu, 10 Sn, 2.5 Pb, 3.5 Ni 324 80 Cu, 10 Sn, 10 Pb 241

81 Cu, 4 Ni, 4 Fe, 11 AI

C 86300

C 87200

C 87500

C 90500

C 92200

C 96300

C 97800

C 99400

C 16200

C 17000

Silicon bronze

Silicon brass

Tin bronze

Leaded tin bronze

Leaded tin nickel bronze C 92900 High-leaded tin bronze C 93700

C 95500

Manganese bronze

Aluminum bronze

Copper-nickel

Nickel-silver

Special alloy

Cadmium copper

Beryllium copper

Leaded beryllium copper C 17300

88 Cu, 6 Sn, 1.5 Pb, 4.5 Zn 276

C 85400

99.5 Cu, 1.9 Be, 0.4 Pb

99.5 Cu, 1.7 Be, 0.20 Co

99.0 Cu, 1.0 Cd

90.4 Cu, 2.2 Ni, 2.0 Fe, 1.2 Al, 1.2 Si, 3.0 Zn

47 35

40

45

67

55

115

34

37

kpsi

179 124

138

152

207

172

572

83

117

MPa

26 18

20

22

30

25

83

12

17

kpsi

517 30

55

25

15

10

12–10

20 20

30

25

21

30

15

25

469-1479 68-200 172-1255 25-182 48-3

483-1310 70-190 221-1172 32-170 45-3

Wrought Alloys 7–69 57–1

234–372 34–54

207

379

241–689 35–100 48–476

455–545 66–79

55

75

Elongation in 50 mm (2 in), %

Cast Alloys 30

Tensile yieldb strength, syt

689–827 100–120 303–469 44–68

310

462

379

793

234

66 Cu, 5 Sn, 2 Pb, 25 Ni, 2 Zn379

79.3 Cu, 20 Ni, 0.7 Fe

88 Cu, 10 Sn, 2 Zn

82 Cu, 14 Zn, 4 Si

89 Cu min, 4 Si

63 Cu, 25 Zn, 3 Fe, 6 Al, 3 Mn

67 Cu, 1 Sn, 3 Pb, 29 Zn

255

Leaded yellow brass

85 Cu, 5 Sn, 5 Pb, 5 Zn

C 83600

Leaded red brass

MPa

UNS no. Composition,a

Alloy name

Ultimate tensile strength, sut

TABLE 1-31 Typical mechanical properties and uses of some copper alloys4

125–170

RB 98 RB 77

50

20

20

50

60

130d d

15

50

192–230d

150

40 80

42

30

50

80 60

65

115 134d 75

40

8

225d

85

80

84

50

60

Brinell, Machin4.9 kN (500-kgf Rockwell,a ability load) HB R ratingc

Hardness

Trolley wire, spring contacts, railbands, high-strength transmission lines, switch gear components, and ware-guide Bellows, diaphragms, fuse clips, fasteners, lock washers, springs, valves, welding equipment, bourdon tubing Bellows, diaphragms, fuse clips, fasteners, lock washers, springs, valves, welding equipment, switch parts, roll pins

Valves, ﬂanges, pipe ﬁttings, pump castings, water pump impellers and housings, small gears, ornamental ﬁttings General-purpose yellow casting alloy, furniture hardware, radiator ﬁttings, ship trimmings, clocks, battery clamps, valves, and ﬁttings Extra-heavy-duty, high-strength alloy, large valve stems, gears, cams, slow heavy-load bearings, screw-down nuts, hydraulic cylinder parts Bearings, bells, impellers, pump and valve components, marine ﬁttings, corrosion-resistant castings Bearings, gears, impeller, rocker arms, valve stems, small boat propellers Bearings, bushings, piston rings, valve components, steam ﬁttings, gears Valves, ﬁttings and pressure-containing parts for use up to 2888C (5508F), bolts, nuts, gears, pump piston, expansion joints Gears, wear plates, cams, guides Bearings for high-speed and heavy-pressure pumps, impellers, pressure-tight castings Valve guides and seats in aircraft engines, bushings, rolling mill bearings, washers, chemical plant equipment, chains, hooks, marine propellers, gears, worms Marine ﬁttings, sleeves and seawater corrosion resistance parts Valves and valve seats, musical instrument components, sanitary and ornamental hardware Valve stems, marine uses, propeller wheels, mining equipment gears

Typical uses

PROPERTIES OF ENGINEERING MATERIALS

1.55

85.0 Cu, 15.0 Zn

C 22000

C 23000

C 26000

C 26800

C 28000

C 34000 C 36000 C 37700 C 44300 C 44400 C 44500 C 46400 to C 46700 C 51000

Commercial bronze (90%) Red brass (85%)

Cartridge brass (70%)

Yellow brass

Muntz metal

Medium leaded brass Free-cutting brass Forging brass Admiralty brass

C 71500

C 77000

Copper-nickel (30%)

Nickel-silver 55-18

1.56

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69–427 10–62

69–400 10–58

MPa

15–60 18–45 20 18–22

172–455 25–66

103–414 124–310 138 124–152

145–379 21–55

138–483 20–70

207–414 30–60

60—145 186–621 27–90

54–75

65–84

56–145 145–483 21–70

47–140 131–552 19–80

55–88

47–88 49–68 52 48–55

54–74

46–128 97–427 14–62

44–130 76–448 11–65

39–105 69–434 10–63

37–72

34–64

kpsi

Tensile yieldb strength, syt

40–2

45–15

33–19

63–3

64–2

50–17

60–7 53–18 45 65–60

52–10

65–3

66–3

55–3

50–3

45–4

Elongation in 50 mm (2 in), %

90-82RB

30

20

30

30

20

30

70 100 80 30

85RF –80RB 40

80RB –64RF 30

82RB –64RF 30

77RB –55RF 30

70RB –53RF 20

64RB –46RF 20

Brinell, Machin4.9 kN (500-kgf Rockwell,a ability load) HB R ratingc

Hardness

Aircraft turn buckle barrels, balls, bolts, nuts, marine hardware, propeller, rivets, shafts, valve stems, welding rods, condenser plate Bellows, bourdon tubing, clutch disks, cotter pins, diaphragms, fasteners, lock washers, chemical hardware, textile machinery Hydraulic pressure liners, anchor screws, bolts, cap screws, machine screws, nuts, rivets, U-bolts, electrical conduits, welding rod Clutch disks, pump rods, shafting, balls, valve stems and bodies Condensers, condenser plates, distiller tubing, evaporator and heat-exchanger tubing, ferrules, salt water piping Optical goods, springs, and resistance wires

Coins, medals, bullet jackets, fuse caps, primers, jewellery base for gold plate Etching bronze, grillwork, screen cloth, lipstick cases, marine hardware, screws, rivets Conduit, sockets, fasteners, ﬁre extinguishers, condenser and heat-exchanger tubing, radiator cores Radiator cores and tanks, ﬂashlight shells, lamp ﬁxtures, fasteners, locks, hinges, ammunition components, rivets Radiator cores and tanks, ﬂashlight shells, lamp ﬁxtures, fasteners, locks, hinges, rivets Architectural, large nuts and bolts, brazing rods, condenser plates, heat-exchanger and condenser tubing, hot forgings Butts, gears, nuts, rivets, screws, dials, engravings Gears, pinions, automatic high-speed screws, machine parts Forgings and pressings of all kinds Ferrules, condenser, evaporator and heat-exchanger tubing, distiller tubing

Typical uses

a b c Nominal composition. unless otherwise noted. All yield strengths are calculated by 0.5 percent oﬀset method. Machinability rating expressed as a percentage of the machinability of d e 29.4 kN (3000 kgf) load. RA , RB , RF , Rockwell numbers in A, B, F scales. C 36000, free-cutting brass, based on 100 percent for C 36000. Note: Values tabulated are average values of test specimens. Source: ASM Metals Handbook, American Society for Metals, Metals Park., Ohio, 1988.

55.0 Cu, 27.0 Zn, 18.0 Ni 414–1000

58.5 Cu, 1.4 Fe, 39.0 Zn, 448– 579 1.0 Sn, 0. 1 Mn 70.0 Cu, 30.0 Ni 372–517

C 67500

Manganese bronze–A

386– 1000

324– 965

379–607

324–607 338–469 359 331–379

372– 510

317–883

303–896

97.0 Cu, 3.0 Si

95.0 Cu. 5.0 Sn, trace P

60.0 Cu, 39.25 Zn, 0.75 Sn

65.0 Cu. 1.0 Pb, 34.0 Zn 61.5 Cu, 3.0 Pb, 35.5 Zn 59.0 Cu, 2.0 Pb, 39.0 Zn 71.0 Cu, 28.0 Zn, 1.0 Sn

60.0 Cu, 41.0 Zn

65.0 Cu, 35.0 Zn

70.0 Cu, 30.0 Zn

269–724

255–496

234–441

MPa

High-silicon bronze -A C 65500

Phosphor bronze (5% A)

Naval brass

90.0 Cu, 10.0 Zn

C 21000

Guilding brass (95%)

95.0 Cu, 5.0 Zn

UNS no. Composition,a

Alloy name

Ultimate tensile strength, sut

TABLE 1-31 Typical mechanical properties and uses of some copper alloys (Cont.)

PROPERTIES OF ENGINEERING MATERIALS

6.0 7.6 9.0

6.0 4.3 9.0

3.0 8.5

3.0

AZ63A–T6 AZ81A–T4 AZ92A–T6 HK3IA–T6 HZ32A–T5 ZE41A–T5 ZH62A–T5 ZK61A–T6

AM60A–F AS41A–Fc AZ91A and B–Fc

AZ31B and C–Fd AZ80A–T5 HM31A–F ZK60A–T5

AZ3IB–H24 HK31A–H24 HM21A–T8 0.6

1.2

0.13 0.35 0.13

0.15 0.13 0.10

3.0 2.0

3.0

0.7

1.8

3.3 3.3

Mn(a) Th

1.0

5.5

1.0 0.5

2.1 4.2 5.7 6.0

3.0 0.7 2.0

Zn

0.6

0.45a

0.7 0.7 0.7 0.7 0.7

Zr

kpsi

MPa kpsi

1.0 Si

290 255 235

260 380 290 365

205 220 230

b

180 160 130

26 23 19

14 35 27 36

Extruded Bars and Shapes 38 200 29 97 55 275 40 240 42 230 33 185 53 305 44 250 Sheets and Plates 42 220 32 33 200 39 34 170 25

17 22 24

115 150 165

Die Castings 30 115 17 32 150 22 33 150 22

19 12 22 15 13 20 25 28

MPa kpsi

Compressive

Bearing

325 285 270

345 405

230

360 305 450 275 255 350 340

47 41 39

50 59

33

52 44 65 40 37 51 49

MPa kpsi

Yield strength, sy

Sand and Permanent Mold Castings 275 40 130 19 130 275 40 83 12 83 275 40 150 22 150 220 32 105 15 105 185 27 90 13 90 1.2 RE 205 30 140 20 140 240 35 170 25 170 310 45 195 28 195

Others MPa

Tensile

Minimum. 4.9-kN (500-kgf) load, 10-mm ball. c A and B are identical except that 0.30% max residual Cu is allowable in AZ91B. d Properties of B and C are identical, but AZ31C contains 0. 15 min Mn, 0.1 max Cu, and 0.03 max Ni. Source: ASM Metals Handbook, American Society for Metals, Metals Park, Ohio, 1988.

a

Al

Alloy

Composition

Tensile strength, st

TABLE 1-32 Nominal compositions and typical room-temperature mechanical properties of some magnesium alloys

15 9 11

15 7 10 11

6 4 3

5 15 3 8 4 3.5 4 10

160 140 125

130 165 150 180

140

145 125 150 145 140 160 165 180

23 20 18

19 24 22 26

20

21 18 22 21 20 23 24 26

73 68

88

49 82

63

73 55 84 55 57 62 70 70

Shear strength, Elongation s Brinellb in 50 mm hardness, (2 in), % MPa kpsi HB

PROPERTIES OF ENGINEERING MATERIALS

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1.57

1.58

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. 1290 830 1620 1460 1280 525

187 120 235 212 185 76

134 204 100

52 107

352 740 924 1410 690

65–110 55–75 95 50 90–120 185 75–90 70–95 145–180 140–170 91 98 185 19.6 162 70 75–100 80–120 140–175 105

kpsi

448–758 379–517 655 345 620–825 1275 517–621 483–655 1000–1240 965–1172 624 672 1276 135 1120 485 512–690 552–827 965–1207 725

MPa

1050 715 1310 1220 795 515

462 965 635

220 365

276–690 103–207 621 110 205–415 910 172–345 205–380 862–1172 724–1034 210 307 910 117 635 455 207–414 241–621 896–1172 260

152 104 190 177 115 75

67 140 92

32 53

40–100 15–30 90 16 30–60 132 25–50 30–50 125–170 105–150 30.4 45 132 17.0 92 66 30–60 35–90 130–170 38

b

10 8 15 14 25 35

52 17 27

14.5 56

35–10 35–40 4 50 55–35 28 60–35 45–25 5–2 30–20 49 46 28 102 24 9 60–30 50–25 5–2 56.0

Tensile yield strength, syt (0.2% oﬀset) Elongation in 50 mm MPa kpsi (2 in), %

Values shown represent usual ranges for common sections. Values tabulated are approximate average ones. Source: ASM Metals Handbook, American Society for Metals, Metals Park, Ohio, 1988.

a

Waspaloy

Unitemp AF2–IDA Rene 95

Hastelloy B Udimet 700

Hastelloy G-3

Hastelloy W

Incoloy 800

Inconel X-750

Inconel 825

Monel K-500 Inconel 600

Monel 400

Durnickel 301

Bar, 218C (708F) 8708C (16008F) Bar, forging 218C (708F) 6508C (12008F) Bar, 218C (708F) 8708C (16008F)

Bar, cold-drawn Annealed Strip, cold-drawn Annealed Bar, cold-drawn, annealed Age-hardened Bar, annealed, 218C (708F) Wire. annealed Spring temper Bar, drawn, age-hardened Rod, annealed As rolled Bar, annealed, 218C (708F) 8718C (16008F) Bar, 218C (708F) 7608C (14008F) Bar, annealed Hot-ﬁnished Wire, spring temper Bar, solution-treated 4258C (8008 F) 9008C (16508F) Sheet, 6.4–19 mm (0.25–0.75 in) thick Bar, cast Bar, 218C (708F) 8708C (16008F)

Nickel 200

Nickel 270

Condition

Name of alloy

Ultimate tensile strength, sut

TABLE 1-33 Mechanical propertiesa of some nickel alloys4

87RB

24RC 75RB 86RB

95RB 35BB 35RC

53

Hardness number J

39

ft-lbf

Impact strength notched Charpy

Jet engines, missiles, turbines where hightemperature strength and corrosion resistance are important

Superalloy, jet engine, turbine, furnace

Springs Corrosion-resistant parts Jet engines, missiles, etc. where corrosion resistance and high strength are required

High strength and hardness, corrosion resistance Corrosion-resistant parts

Corrosion-resistant parts

Typical uses

PROPERTIES OF ENGINEERING MATERIALS

365 255 434

317

207 214

53 37 63

46

30 31

Zinc Foundry Alloys

Die-Casting Alloys

110–120 90–100 110–125

110–125

105–120 105–125

82 91

Tensile yield strength, syt Brinell hardness, MPa kpsi HB

3–6 8–11 1

2

1–3 1–3

10 7 14

Elongation in 50 mm (2 in), %

58 65 54

J

43 48 40

ft-lbf

Impact strength Charpy

Die-cast. Note: Values given are average values. Source: Machine Design, 1981 Materials Reference Issue, Penton/IPC, Cleveland, Ohio, Vol. 53, No. 6 (March 19, 1981); SAE Handbook, pp. 11–123, 1981.

a

58–64 45–47 65

400–440 310–324 448

ZA-27 Sand-cast Sand-cast Die-cast

41 47 47

57

283 324 283

kpsi

393

Z 33520 Z 35531

MPa

40–45 45–50

903 925a 903

UNS

276–310 310–345

AG 40 A AC 41 A

Alloy 3 Alloy 5 Alloy 7

SAE

Ultimate tensile strength, sut

ZA-12 Sand-cast Permanent Mold Die-cast

ASTM

Grade

Designation of alloy

TABLE 1-34 Mechanical properties of some zinc casting alloys8

47 56

MPa

6.8 8.2

kpsi

Fatigue endurance limit, sf , 108 cycles

PROPERTIES OF ENGINEERING MATERIALS

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1.59

550

R 58010

900 1030 1170 1170

Ti-6AI-4Vb Ti-6Al-6V-2Sn Ti-10V-2Fe-3Ala;c Ti-6A1-2Sn-4Zr6Moe Ti-13V-11Cr-3Al Ti-3Al-8V-6Cr4Mo-4Zrb;c 900

790 690 1000

Ti-5Al-2.5Sn Ti-5Al-2.5Sn-ELI Ti-2.25Al-11Sn5Zr-IMo

130

130 150 170 170

115 100 145

50

830

830 970 1100 1100

760 620 900

480

380

280

170

120

120 140 160 160

110 90 130

70

55

40

25

4

10

19

20

22

24

13.5

24.5

35

10

18e

26a

34RC

30RC

40b

22b

30b

40b

MachiHardness nability

Gas turbine engine casting and rings, aerospace structural members, excellent weldability, pressure vessels, excellent corrosive resistance, jet engine blades and wheels, large bulkhead forgings Most widely used alloy, aircraft gas turbine disks and blades, turbine disks and blades, air frame structural components, gas turbine engines, disks and fan blade, components of compressors Missile applications such as solid rocket motor cases, advanced manned and unmanned airborne systems, springs for airframe applications

Resistance to temperature eﬀect of structures, easy to fabricate, excellent corrosive resistance, cyrogenic applications

Uses

b

At 0.2% oﬀset. Mechanical and other properties given for annealed conditions. c Mechanical and other properties given for solution-treated and aged condition. d Based on a rating of 100 for B1112 resulfurized steel. e Approximate values of annealed bars at room temperature. Source: Metals Handbook, Desk Edition, ASM International, Materials Park, Ohio 44073-0002, 1985 (formerly the American Society for Metals, Metals Park, Ohio, 1985).

a

Beta alloy Beta alloy

R 56260

R 56400

80

450

ASTM Grade 3 (Ti-65A) ASTM Grade 4

Alpha-beta alloy

65

340

ASTM Grade 2

Commercially pure titanium Commercially pure titanium Commercially pure titanium Alpha alloy R 54520 Alpha alloy R 54521 Alpha alloy R 54790

35

ASTM Grade 1

Commercially R 50520 pure titanium

240

Designation

Name of alloy UNS no.

Strength Ultimate Tensile yield impact tensile Charpy strength, sut strength, syt Elongation in 50 mm MPa kpsi MPa kpsi (2 in), % J ft-lbf

TABLE 1-35 Mechanical properties of some wrought titanium alloy4

PROPERTIES OF ENGINEERING MATERIALS

1.60

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951 827 920

100 94 100

Forged

100 345 95.5 414 94 427 100 455 100 896 95.5 876 99 min 937 99 1103

Density , % MPa

Blended elemental alloy, cold Blended elemental alloy, forged, preforms or vacuum hot pressed Solution treated and aged Hot isostatically pressed

Forged

Processing

120 133.5

138

50 60 62 66 130 127 136 160

kpsi

738 841

910

344 324 338 365 827 786 862 1013

MPa

107 122

132

50 47 49 53 120 114 125 147

kpsi

62 28 60

60d

414d

kpsi

427 193 414

MPa

Fatigue limit notched, sf

83f

55e 45e 61e

76f

50e 40e 56e

pﬃﬃﬃﬃ pﬃﬃﬃﬃ MPA m kpsi in

Fracture toughness, KIC

114 117 116

103 103

GPa

16.5 17 16.8

15 15

Mpsi

Elastic modulus, E

5 11.5

15

23 10 8 12–18 4.9

5 15

8 25

39

30 20 14 15-40 7.6

35 14

ReducElonga- tion in tion, area, % %

b

0.12% oxygen. 0.2% oxygen. c Consolidated at 811 MPa (58.8 tpsi), 0.5 s dwell in low-carbon steel ﬂuid dies. Preheat temperature was 9408C (17258F) held at temperature 0.75 h. Powder was vacuum ﬁlled into ﬂuid dies following cold static outgassing for 24 h. d Kt ¼ 3. e Ke . f Ktc . Source: Metals Handbook, Desk Edition, ASM International, Materials Park, Ohio 44073-0002, 1985 (formerly the American Society for Metals, Metals Park, Ohio, 1985).

a

Plasma rotating electrode processed Ti-6Al-4V Powder metallurgy Ti-6Al-4Va

Wrought commercial purity titanium Grade II Sponge commercial puritya Powder metallurgy titanium Wrought Ti-6Al-4V (AMS 4298) Powder metallurgy Ti-6Al-4V

Name of alloy

Ultimate tensile Yield strength, sut strength, sy

TABLE 1-35A Mechanical properties of powder metallurgy and wrought titanium and titanium-base alloys

PROPERTIES OF ENGINEERING MATERIALS

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1.61

4350

4930

5367

54520 30 8 MPa at 1008C

54820 34

54915 37

1.62

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7610 2756

55111 52.5 19 MPa at 1008C

33

4790

1450

37

36

32

28

5380

5200

4640

4060

12.5 1810

MPa psi

Shear strength, s

28 10

13

16

14

12

10 30% at 1008C 12

8

30–60 135–200 at 1008C (2128F)

60

130

25

18

10

55

2

20 30 MPa 4358 at at 7 7 2 10 2 10

4–6

30

3.2–4.5

48

1495

624

464

8.1

10.30

4.3

3.2

Fatigue strength at 107 cycles, sf Hardness Elongation number, in 50 mm (2 in), % MPa psi HB

2.9 MPa for 1000 h 0.45 MPa for 1000 h at 1008C (2128F)

2.1 MPa for 1000 h

3.5 MPa for 1000 h 1.1 MPa for 1000 h at 1008C (2128F) 0.790 MPa for 0.01% per day

19.5 MPa-1000 h 7.5 MPa-1000 h at 1008C 28 MPa for 100 h 3% per year at 2.07 MPa

Creep

For general purpose; most popular of all lead alloys

Wiping solder for joining lead pipes and cable sheaths; for automobile radiator cores and heating units

Low melting-point chemical process applications, used as solder for the jobs lead alloyed with tin, bismuth cadmium, indium forms alloys with low melting point. Some of these are fusible alloys, used in automotive devices, ﬁre extinguishers, sprinkler heads.

Uses

Note: Values tabulated are average values obtained from standard test specimens. Source: Metals Handbook, Desk Edition, ASM International, Materials Park, Ohio 44073-0002, 1985 (formerly the American Society for Metals, Metals Park, Ohio, 1985).

4700

55030 32.4

870

4060

54321 28

6 MPa at 1008C

10297

53620 71

10

4002

9570 870–1160

7978

MPa psi

52901 27.6

psi

1740–1885 55 5076 261 at 2128F 10152 66 2320–2755 6–8

MPa

Yield strength, sy

50042 12-13 50132 35 1.8 MPa at 1008C 50750 70 51120 16-19

UNS No.

Ultimate tensile strength, sut

TABLE 1-36 Mechanical properties of some lead alloys

PROPERTIES OF ENGINEERING MATERIALS

Sand-cast (cast-on) Sand-cast (separately cast) Chill-cast

Elongation percent, min

8.0 12.0

5.0

196 (28.4) 216 (31.3)

Class II gun metalc

3.0

186 (27.0)f 206 (29.9)f

Class I phosphora bronze b

4.0

2.0

137 (19.9) 157 (22.8)

Class III leaded

4.0

2.0

157 (22.8) 176 (25.5)

Class IV bronzed

Railway, bronze

12.0

8.0

186 (27.0) 206 (29.9)

Class V leadede gun metal

20.0 20.0

12.0

490 (71.0) 539 (78.2)

Grade II

15.0

647 (93.8) 647 (93.8)

Grade I

20.0

446 (64.7) 196 (28.4)

Grade III

Aluminum bronze

b

Brinell hardness, HB for phosphor bronzes: 60 for sand cast (cast-on) test pieces and 65 for sand-cast (separately cast) test pieces. Used for locomotive side valves, oil-lubricated side rod, pony pivot bushes, steel axle box, oil-lubricated connecting rod. c Used for fusible plugs, relief valves, whistle valve body, stuﬃng box, nonferrous boxes, oil-lubricated connecting rod, large end bearings. d Used for locomotive grease lubricated non-ferrous axle boxes, side rod and motion bushes. e Used for castings for carriage and wagon bearings shells. f sut given in parentheses are the units in US Customary Units (kpsi).

a

Sand-cast (cast-on) Sand-cast (separately cast) Chill-cast

Ultimate strength, min sut , MPa (kpsi)

Property

Mode of casting test pieces

TABLE 1-37 Mechanical properties of bronzes

12.0

216 (31.3) 226 (32.8) 245 (35.5) 8.0

Tin bronze

20.0

309 (44.8)

Silicon bronze

PROPERTIES OF ENGINEERING MATERIALS

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1.63

1.27–3.00 1.30–1.40 1.20–1.30 1.06 1.12–2.00 0.97–1.25 0.98–1.35

Material

Duprene Koroseal (hard) Koroseal (soft) Plioform (plastic) Rubberb (hard) Rubberc (soft) Rubber (linings) 88 758 14 103

MPa

12.8 110.0 2.0 15.1

kpsi 1.4–28 14–62 3.4–17 28–34 7–69 3.5

MPa 0.2–4.0 2.0–9.0 0.6–2.6 4.0–5.0 1.0–10.0 0.6

kpsi

Tensile strength, st

b

Sclerscope. Coeﬃcient of linear expansion from 0 to 333 K (608C ¼ 1408F) is 35 106 . c Coeﬃcient of linear expansion from 0 to 333 K (608C ¼ 1408F) is 36 106 .

a

Speciﬁc gravity

Compressive strength, sc

TABLE 1-38 Mechanical properties of rubber and rubber-like materials

48 62 62 103

MPa

7.0 9.0 9.0 15.1

kpsi

Transverse strength, sb

50a /80

15–95 80–100 30–80

Hardness shore durometer

422 373 361 344–393 328–367 339–367 361

K

149 100 88 71–120 55–71 65–94 88

8C

300 212 190 160–250 130–160 150–200 190

8F

Maximum temperature

Softens Softens Softens

Stiﬀens slightly Softens Softens

Eﬀect of heat

PROPERTIES OF ENGINEERING MATERIALS

1.64

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0.5–2.8

21.0

15.2–47.5 2.2–6.9

8–12

6.5–7.0

55–69

69–138

3–48

55–83

45–48

Cellulosic (cellulose acetate)

Epoxy resin (glass-ﬁber ﬁller)

Fluoroplastic group

Nylon

Phenolic (general purpose)

0.5–7.0

10–20

8–10

60–200

100–300

4.0

40–60

7.6–9.0

1.2–2.9

2.8–3.6

1.5–3.1

Acetal

5.4–10.5 5–50

2.3

37–72

5–20

Acrylics

6

kpsi

41

MPa

0.5–1.6

8.8

J

1.1–1.3

0.18–0.42

3.04

0.4–0.5

1.4–4.5

4.1

2.7–41

Excellent

Good

Available

Available

Heat

Good

Excellent

High

Fair

Fair

0.04-0.13

0.05

With steel

Coeﬃcient of friction,

Chemical With plastic

Resistance to

114–120 R Poor

50-80 D

100–110 M

122 R

80–94 M

92–100 M

103

Hardness, Rockwell

0.30–0.35 70–95 E

1.0–3.3

3

2–30

1.0–7.3

0.4–1.2

6.5

ft-lbf

Izod impact strength

0.065–0.40 1.4–9.9

0.4–0.52

0.22–0.45

0.33

Modulus of Elongation elasticity, E in 50 mm (2 in), % GPa Mpsi

ABS (general purpose

Name of plastic

Tensile strength, sut

TABLE 1-39 Properties of some thermoplastics

Wall plates, industrial switch gears, handles for appliances, housing for vacuum cleaners, automatic transparent rings, housing for thermostats, small motors, small tools, communication instruments, components for aircraft and computers, used as synthetic rubber for tires

Structural, mechanical components such as gears, fan blades, washing-machine agitator, valve, pump, impeller, pistons, and cams

Gears, bearings, tracks, bushings, rollerskate wheels, chute liners

Filament wound structures, aircraft pressure bottles, oil storage tanks and highperformance tubing, reinforced glass-ﬁber composites

Decorative knobs, handles, camera cases, pipes, pipe ﬁttings, eyeglass frames, phone and ﬂashlight cases, helmets, pumps and power tool housings, transparent parts for safety glasses, lens signs, refrigerator shelves and snowmobile windshields, extruded and cast ﬁlm, and sheet for packaging

Mechanical gears, cams, pistons, rollers, valves, fan blades, washing-machine agitators, bushings, bearings, chute liners, wear strips, and structural components

Light-duty mechanical knobs, pipe ﬁttings, automobile-steering wheels, eyeglass frames, tool handles, camera cases, optical and transparent parts for safety glasses, snowmobile windshields, refrigerator shelves

Light-duty mechanical and decorative eyeglass frames, automobile-steering wheels, knobs, handles, camera cases, battery cases, phone and ﬂashlight cases, helmets, housing for power tools, pumps

Application

PROPERTIES OF ENGINEERING MATERIALS

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1.65

55–159

4–38

Polyester

Polyethylene

10.2

Polysulfone

1.66

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10–500

20–1000

1–300

<1

25

0.7–6.2

0.1–1.2

1.9–11.7

2.3-5.9

2-5

0.7–2.6

0.3–23

0.5–1.9

0.25–17

2.7-21.7 2-16

2.7-6.8

ft-lbf

0.36

0.1–0.9

1.8

0.7–3.0

1.3

0.5–2.2

0.014–0.18 0.7–27.1 0.5–20

0.28–1.7

0.34-0.86

0.35-0.93

J

Izod impact strength

120 R

50–110 R

10–65 R

65–100 M

88–120 M

62- 91 M

115-119 R 106-108 L

Hardness, Rockwell

Excellent

Excellent

Excellent

Excellent

Good

Heat

Excellent

Poor

Excellent

Fair

Fair

0.12 0.22

0.52

0.12 0.13

0.39

With steel

Coeﬃcient of friction,

Chemical With plastic

Resistance to

Source: Machine Design, 1981 Materials Reference Issue, Penton/IPC, Cleveland, Ohio. Vol. 53, No. 6 (March 19, 1981).

70

5–14.5

Polypropylene 34–100

0.6–5.5

8–23

3.6–50

25–345

Polyimide

10-125

2.4–6.4

Modulus of Elongation elasticity, E in 50 mm (2 in), % GPa Mpsi

7.0-17.3 4–60

kpsi

8-16

48-123

MPa

Polycarborate 55-110

Phenylene oxide

Name of plastic

Tensile strength, sut

TABLE 1-39 Properties of some thermoplastics (Cont.)

Mechanical cams, pistons, washing machine agitators, fan blades, valves, pump impellers, gears, bushings, chute liners, bearings, tracks, wear strips and other wearresisting parts Pipe ﬁttings, battery cases, knobs, camera and handle cases, trim moldings, eyeglass frames, tool handles, housings for pumps, power tools, phone cases, transparent parts, safety glasses, lenses, snowmobile windshields, signs, refrigerator shelves, and vandal-resistant glazing

Decorative knobs, automobile steering wheels, eyeglass frames, tool handles, camera cases, phone and ﬂashlight cases, housings for pump and power tools

Flashlight and phone cases, housing for pumps, power tools and other appliances, gears, bushings, bearings, tracks, rollerskate wheels and chute liners

Molded polyimides are used in jet-engine vane bushings, high load bearings for business machines and computer printout terminals, gear pump gaskets, hydraulic valve seals, multilayer printed-circuit boards, tubes for oil-well exploration

Mechanical gears, pistons, rollers, pump impellers, fan blades, rotor housing for pumps, power tools, phone cases, transparent parts for safety glasses, lenses and snowmobile windshields, refrigerator shelves, and ﬂashlight cases

Small housing for power tools, pumps, small appliances, hollow shapes for telephones, ﬂashlight cases, helmets, TV cabinets, cable protective cover, bush-bar sleeves, scrubbervane mist eliminators

Application

PROPERTIES OF ENGINEERING MATERIALS

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21–66

28–69

34–69

28–138

34–62

3–45

Alkyd

Allylic

Amino

Epoxy

Phenolics

Silicone

0.4–6.5

5–9

4–20

5–1 0

4–10

3–9.5

kpsi

7–17

2.5–21

9–16

2–21

GPa

1.0–2.5

0.35–3.04

1.3–2.4

0.3–3.0

Mpsi

Modulus of elasticity, E

80–90 M

70–95 E

80–120 M

110–120 M

103–120 M

98E–99 M

Hardness, Rockwell

15

1–10

0.3–0.9

Elongation in 50 mm (2 in), %

0.5–14

0.4–1.5

0.3–41

0.4–24

0.3–16

0.5–14

J

0.3–10

0.26–1.05

0.2–30

0.27–18

0.2–12

0.3–10

ft-lbf

Impact strength Izod

Fair

Chemical

Excellent Excellent

Excellent Good

Excellent Excellent

Excellent Excellent

Excellent Excellent

Good

Heat

Resistance

Source: Machine Design, 1981 Materials Reference Issue, Penton/IPC, Cleveland, Ohio. Vol. 53, No. 6 (March 19, 1981).

MPa

Name

Tensile strength, st

TABLE 1-40 Properties of some thermosets8

Refrigerator equipment, used as a washing, sealant, laminating parts, injection mold silicon rubber

Handles for appliances, automotive power-brake systems and industrial terminal strips, industrial switch gear, housing for vacuum cleaners, handles for pots and pans, automotive transmission rings, and electrical components, thermostat housings, housing for small motors and heavy-duty electrical components, small power tools, electrical components for aircraft and computers, pump housings, synthetic rubber for tires and other mechanical rubber goods, dry ingredients for brake linings, clutch facings and other friction products

Filament wound structures, aircraft pressure bottles, oil storage tank, used with various reinforcements, glass ﬁbers, asbestos, cotton, synthetic ﬁbers, and metallic foils, imprinted circuits, graphite and carbon-ﬁber-reinforced laminates used for radomes, pressure vessels, and aircraft components requiring high modulus and light weight, potting and encapsulating electrical and electronic components ranging from miniature coils and switches to large motors and generators

Electrical wiring devices and switch housings, toaster and other appliance bases, push buttons, knobs, piano keys and camera parts, dinnerware, utensil handles, food-service trays, housings for electric shavers and mixers, metal blocks, connector plugs, automotive and aircraft ignition parts, coil forms, used as baking enamel coatings, particle-board binders, paper and textile treatment materials

Switch gear and TV components, insulators, circuit boards, and housings, tubing and aircraft parts, copper-clad laminate for high-performance printed-circuit boards

Military switch gear, electrical terminal strips, and relay housings and bases, automotive ignition parts, radio and TV components, switch gear, and small-appliance housings

Application

PROPERTIES OF ENGINEERING MATERIALS

1.67

Polycarbonate Marblette [annealed 72 h at 356 K (1818F)]

Bakelite ERL 2774 (50 phthalic anhydride) Hysole 4290: at 296.9 K (758F) at 405.2 K (2708F) Armstrong C-6

Araldite 6020 at 299.7 K (808F) Araldite 6020 at 277.4 K (408F) Araldite B

Methyl methacrylate (unplasticized) Polystyrene Cellulose nitrate Castolite Kriston CR-39 (Columbia resin) Epoxy Resin: Araldite CN-501

38.0–62.0 5.55–9.0 88.2–117.2

3.0

20.6

7.0

48.3

22.4 3.25 at 296.3 K at 748F 34.5 5.0

8.0

55.2

28.3 4.12 at 299 K at 778F

4.0

27.6

82.7 1.38

12.0 0.20

10.0

8.25 7.0–6.0

56.5 48.0–41.4

68.9

7.55

7.0

12.5–17.0

10.0

kpsi

51.7

48.2

69.0

Cataline (61-893)

8.68

60.0

MPa

Glass

kpsi

MPa

Tensile strength, st

Material

Elastic limit, se

3.45 0.014 3.3 at 294 K 2.55 3.79 –4.13

3.2 at 294 K 3.30 0.036 at 433 K

3.10 at 298 K 3.10

2.4 4.3 3.7 1.7–2.6

2.8

4.2–4.3

69.0

GPa 0.20

Poisson’s ratio,

0.38

500.0 2.0 475 at 708F 370.0 550.0 – 600.0

460 at 708F 478.5 5.2 at 3208F

452.0 at 778F 445.0

0.38 0.41

0.34 0.50

0.38

0.362

0.35

350.0 0.33 623.0 0.35 540.0 250.0–380.0 0.42

400.0

615.0–628.0 0.365

10,000

kpsi

Young’s modulus, E

TABLE 1-41 Optical and mechanical properties of photoelastic material6

10.508 0.245 13.222 at 293.6 K 7.355 14.535

10.30 0.435 at 433 K

60.0 1.4 75.5 at 708F 42.0 83.0

58.75 2.5 at 3208F

59.0

60,5

10.595 10.332

61.5 at 778F 58.0

310 245 182 80 83.5—99

54.290 42.907 31.873 14.010 14.623–17.338

10.770 at 298 K 10.157

880

87

1740–2420

5.59 at 293.6 K 4.00 5.33

4.3

4.83

4.57 at 298 K 4.57

25.40 1.00 4.83 11.90–12.50

91.00

4.83

4.83

220 at 708F 157 210

170

191

180 at 778F 180

1000 39 191 468–492

3582

191

191

lin/fri

Strain fringe value, fs

lbf/in fri lm/fri

154,113

304.724– 423.812 15.236

kN/m fri

Stress fringe value, f

347,000 261,000

328,000 57,000 290,000

320,000 83,000

310,000

305,000

290,000

56,000 135,000 264,000 116,000– 150,000

18,200

226,000– 163,000 27,600– 280,000

fri/m

8810 6626

8333 1428 6291

8145 2080

7796

7672

7350

1428 3423 6750 2994–3838

455

7069–7218

5747–4132

fri/in

Figure of merit, Q ¼ ðE=f Þ

4690

1694

4596

5360

2628

1408–1188

643

2500–4070

1970–1415

fri/m

119

43

16

136

67

36.0–30.0

16.3

63.8–100

5–3.5

fri/in

S ¼ e =f

Little optical and mechanical creep Good stress-optical relationship; susceptible for time edge eﬀect

Used for 2-D and 3-D models

Used for 2-D and 3-D models

Used most commonly for 2-D and 3-D models Used for 2-D and 3-D models

Used most commonly for 2-D and 3-D models Used for 2-D and 3-D models

Used for 2-D and 3-D models

Used for 2-D models; free from time edge eﬀect

Low optical sensitivity Free from time edge eﬀect; used for photoplasticity

Used for 2-dimensional (2-D) and 3dimensional (3-D) models; susceptible for time edge eﬀect

Low optical sensitivity; rarely used

Remarks

PROPERTIES OF ENGINEERING MATERIALS

1.68

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2.85 102

46.2

31.0

MPa

6.70

4.50

kpsi

Tensile strength, st

0.003– 0.004 75.8 106

0.003

1.72

1.65

GPa

0.425– 0.625 11 103

0.425

250.0

240.0

kpsi

Young’s modulus, E

0.50

0.46

0.467

0.38

0.40

Poisson’s ratio,

0.025

0.158 0.14

0.9

0.5

10.0 1.7

1.752 0.289

0.084

57.6

57.0

483.00

82.00

40.60

431.80

5.84

7.87

19016

3228

1598

17000

230

310

lin/fri

Strain fringe value, fs

lbf/in fri lm/fri

10.087

9.982

kN/m fri

Stress fringe value, f

19,000– 25,000 3032

35,700

170,000

165,000

fri/m

78

4722–694

850

4340

4210

fri/in

Figure of merit, Q ¼ ðE=f Þ

1076

684

893

fri/m

31.6

17.4

47.4

fri/in

S ¼ e =f

Great optical sensitivity; used for model study of body forces

Used for preparation of models in stress wave propagation and models of dam

Good stress-optical relationship; susceptible for time edge eﬀect

Remarks

Sources: K. Lingaiah, Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986, and K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986.

Gelatin (15% gelatin, 25% glycerine, 60%water)

Hysole 4485

0.17

1.0

6.9

Urethane rubber Hysole 8705 at 26.9 K (758F)

2.70

18.9

Marblette (phenofomaldehyde) Catalin 800 Natural rubber Hard Soft

kpsi

MPa

Material

Elastic limit, se

TABLE 1-41 Optical and mechanical properties of photoelastic material (Cont.)

PROPERTIES OF ENGINEERING MATERIALS

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1.69

W-Ni-Fe alloy 17.0 W-Ni Fe alloy 18.0 W-Ni-Fe alloy 18.5

2 3 4

0.614 895 0.650 925 0.667 795

0.614 785 130 134 115

114 615 655 690

605

MPa

89 95 100

88

kpsi

Yielda strength, spl

260 350 450

205

MPa

38 51 65

30

kpsi

Proportional limit, spl

275 310 345

275

GPa

40 45 50

40

Mpsi

Modulus of elasticity, E

16 6 3

4 27 RC 29 RC 32 RC

27 RC

Elongation in 50 mm (2 in), % Hardness 3.0 3.0 2.9 2.6

5.4 5.3 5.0

lin/in8F

5.4

lm/m8C

Coeﬃcient of thermal expansion,

Virtually nonmagnetic Slightly magnetic Slightly magnetic Slightly magnetic

Magnetic properties

1.70

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. 0.4 0.4 0.4 4 0.2–0.4 5 4 0.5 0.001–0.01 0.2 0.2–0.5 0.8

13 0.025–0.25 5 5–13 20.5

103 in

10.2 10.2 10.2 102.0 5–10 127 102

103 mm

7.64 2.43 2.62 1.11 1.49

2.48 2.43 2.46 2.56 1.43–1.75 1.78 3.40

g/cm3

0.283 0.090 0.097 0.041 0.052

0.092 0.090 0.091 0.095 0.053–0.066 0.066 0.126

lb/in3

Density,

2852–4184 689–2067 1378–2067 827 690

3100 4498 5510 2756 1723–3445 1240 2480

MPa

385–600 100–300 200–300 120 100

450 650 800 400 250–500 180 360

kpsi

Tensile strength, st

200 172 138–413 2.8 4

72.5 85 100 415 241–689 310 414

GPa

29 25 20–60 0.4 0.6

10.5 12.3 14.5 60 35–100 45 60

Mpsi

Modulus of elasticity, E

30 3.7 45–50 45–50

6.5 81–90 81–90

2.8 1.5 6.4 2.2

5.0 2.7 11.5 4.0 54

2.8

lin/in8F 5.0

lm/mK

Coeﬃcient of thermal expansion,

60 87 29

7.5

Btu/(ft2 h8F)/in

381 381

1.7 1.7

22400–38080 100–170

13440 19488 6496

1680

W/(m2 K/m)

Thermal conductivity, K

Class 2, 91–94 Class 3, 94–96 Class 4, 96–98

Class 1, 89–91

Tungsten, % by weight

Courtesy: J. E. Ashton, J. C. Halpin, and P. H. Petit, Primer on Composite Materials—Analysis, Technomic Publishing Co., Inc., 750 Summer Street, Stanford, Conn. 06901, 1969.

E glass S glass 970 S glass Boron on tungsten Graphite Beryllium Silicon carbide on tungsten Stainless steel Asbestos Aluminum Polyamide Polyester

Fiber

Typical ﬁber diameter

TABLE 1-43 Representative properties for ﬁber reinforcement

a 0.2% oﬀset; RC , Rockwell hardness scale C. Source: Metals Handbook Desk Edition, ASM International, Materials Park, OH 44073-0002 (formerly The American Society for Metals, Metals Park, OH 44073), 1985.

W-Ni-Cu alloy 17.0

kpsi

Ultimate tensile strength, sut

Mg/m3 lb/in3 MPa

1

Classiﬁcation Class of alloy

Density,

TABLE 1-42 Typical mechanical properties of commercial machinable tungsten alloys

PROPERTIES OF ENGINEERING MATERIALS

Fe

97.7–100

97.7–100

97.7–100

97.4–99.7

97.4–99.7

97.4–99.7

97.0–99.1

97.0–99.1

97.0–99.1

97.0–99.1

97.0–99.1

97.0–99.1

93.8–98.5

93.8–98.5

93.5–98.2

93.5–98.2

93.5–98.2

93.5–98.2

93.1–97.9

93.1–97.9

93.1–97.9

93.1–97.9

93.1–97.0

93.1–97.9

91.4–95.7

91.0–95.4

91.0–95.4

86.0–93.4

MPIF material designationa

F-0000

F-0000

F-0000

F-0005

F-0005

F-0005

F-0008

F-0008

F-0008

F-0008

F-0008

F-0008

FC-0200

FC-0200

FC-0205

FC-0205

FC-0205

FC-0205

FC-0208

PC-0208

FC-0208

FC-0208

FC-0208

FC-0208

FC-0505

FC-0508

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FC-0508

FC-0808

0.6–1.0

0.6–1.0

0.6–1.0

0.3–0.6

0.6–1.0

0.6–1.0

0.6–1.0

0.6–10

0.6–1.0

0.6–1.0

0.3–0.6

0.3–0.6

0.3–0.6

0.3–0.6

0.3 max

0.3 max

0.6–1.0

0.6–1.0

0.6–1.0

0.6–1.0

0.6–1.0

0.6–1.0

0.3–0.6

0.3–0.6

0.3–0.6

0.3 max

0.3 max

0.3 max

C

6.0–11.0

4.0–6.0

4.0–6.0

4.0–6.0

1.5–3.9

1.5–3.9

1.5–3.9

1.5–3.9

1.5–3.9

1.5–3.9

1.5–3.9

1.5–3.9

1.5–3.9

1.5–3.9

1.5–3.9

1.5–3.9

Cu

Ni

MPIF chemical composition limits and rangesb , %

HT

<6.0

AS AS

6.4–6.8 <6.0

HT

6.0–6.4

AS AS

6.4–6.8 6.0–6.4

AS

HT

AS

HT

6.0–6.4

6.8–7.2

6.8–7.2

6.4–6.8

AS

AS

6.4–6.8

HT

<6.0

AS

HT

AS

6.8–7.2

6.8–7.2

6.4–6.8

6.4–6.8

AS

AS

<6.0 6.8–7.2

HT

AS

HT

AS

HT

AS

HT

AS

AS

AS

6.8–7.2

6.8–7.2

6.4–6.8

6.4–6.8

6.0–6.4

6.0–6.4

6.4–6.8

6.4–6.8

6.0–6.4

7.2–7.6

AS

AS

<6.0 6.8–7.2

Conditionc

MPIF density, , g/cm3

250

515

480

425

455

345

690

550

550

415

295

225

690

425

585

345

255

160

650

395

510

290

400

240

415

220

170

275

205

110

MPa

MPa

kpsi

Yield strength, sy MPa

36

75

70

62

66

50

100

80

80

60

43

33

100

62

85

50

37

23

94

57

74

42

58

35

60

32

25

40

30 26

22

23

–

480

480

395

380

290

655

395

–

330

–

205

655

310

560

260

–

70

70

57

55

42

95

57

–

48

–

30

95

45

81

38

Copper Steel

160

–

195

185

160

170

130

260

210

210

155

110

85

260

160

220

130

95

110 130 130 90 116 90 90 116

30d 30d 38d 19d 25d 24d 27d 29d

–

110

23d

–

70

70 16d

130 13d

90 130

9d 14d

Iron Copper Steel 115 17 60

91

150

38d

130

36d

245

625

40

130

130

22d

275

195

100

24d

110

28d

–

36

150

110

110

14d

250 –

–

110

90

22d

–

19d

90

31d

110

90

155

85 13d

30

57

23

20 23d

205

395

160

140

90

160

15d

110

70 130

6 11d

12d

65

105

80

GPa

–

16

13

13

16

13

19

19

16

16

10.5

10.5

19

19

16

16

19

13

19

19

16

16

13

13

16

16

13

23

19

10.5

7.5

9.5

4.7

4.1

7.5

6.8

3.4

–

–

–

6.1

4.7

6.8

6.1

11

–

–

–

13

–

23

–

–

–

–

6.8

4.7

34

20

4.1

Mpsi J

Modulus of elasticity, E

10d

Steel

180

150

kpsi

Fatigue strength, sf

Iron and Carbon Steel 10 75 11 40

kpsi

Tensile strength, st

TABLE 1-44 Designation, composition and mechanical properties of ferrous powder metallurgy structural steels

5.5

7.0

3.5

3.0

–

–

–

–

–

–

–

4.5

3.5

5.0

4.5

8.0

5.0

2.5

9.5

5.5

17

–

–

–

–

5.0

3.5

25

15

3

ft-lbf

Impact energy

80RH

40RC

<0.5

1.0

55RB

85RB

30RC

<0.5

75RB 65RB

1.0

1.5

60RB

<0.5 1.0

80RB

35RC

70RB

95RB

1.5

<0.5

1.0

<0.5

45RB

35RC <0.5

<0.5

30RC 75RB

3.0

60RB

1.5

30RB

<0.5

7

2.5

30RC

<0.5

25RC 75RB

2.5

65RB

1.5

100RB

50RB

100RB

45RB

20RB

30RB

15RB

10RH

Apparent hardness

<0.5

<0.5

1.0

0.5

2.5

1.5

15

9

2

Elongation in 25 mm (1 in), %

B 426, Grade 3

B 426, Grade 2

B 426. Grade 1

B 310, Class C

B 310, Class B

B 310, Class A

ASTM designation

PROPERTIES OF ENGINEERING MATERIALS

1.71

91.6–98.7

90.2–97.0

FN-0208

FN-0400

1.72

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0.6–0.9

2.0 max

6.0–8.0

87.1–93.4

AS AS

6.8–7.2 7.2–7.6

655

550

HT

FN-0708(e)

620 1160

AS

–

FN-0705

7.2–7.6

705

6.4–6.8 HT

6.0–8.0

370

2.0 max

AS

0.3–0.6

87.4–93.7

585

490

FN-0705

AS AS

6.8–7.2

640

395

1240

510

770

310

400

FN-0700

6.0–8.0

87.7–94.0

FN-0700

2.0 max

AS

0.3 max

AS

3.0–5.5 7.2–7.6

2.0 max

6.4–6.8

0.6–0.9

89.6–96.4

FN-0408

HT

AS

HT

7.2–7.6

6.4–6.8

AS

AS

340

545 1105

HT

690

HT AS

330

AS

760

HT

6.8–7.2. AS

7.2–7.6

6.4–6.8

345

SS

565

HT

7.2–7.6

3.0–5.5

3.0–5.5

3.0–5.5

1.0–3.0

1.0–3.0

6.8–7.2

255

310

195

205

MPa

95

80

168

90

102

54

85

71

93

57

80

74

112

45

58

49

160

79

100

48

110

50

82

37

45

28

30

kpsi

Tensile strength, st

AS

FN-0405

2.0 max

2.0 max

2.0 max

2.5 max

2.5 max

1.0–3.0

6.4–6.8

AS

AS

FN-0405

0.3–0.6

0.3–0.6

0.3 max

0.6–0.9

0.3–0.6

2.5 max

1.0–3.0

7.2–7.6

6.4–6.8

89.9–96.7

91.9–98.7

FN-0208

0.3–0–6

2.5 max

1.0–3.0

1.0–3.0

FN-0405

91.9–98.7

FN-0205

0.3–0.6

2.5 max

2.5 max

AS

<60

6.4–6.8

91.9–98.7

FN-0205

0.3 max

0.3 max

Conditionc

89.9–96.7

92.2–99.0

FN-0200

9.5–10.5

Ni

FN-0405

92.2–99.0

FN-0200

0.3 max

Cu

MPIF density, , g/cm3

7.2–7.6

87.2–90.5

FC-1000

C

MPIF chemical composition limits and rangesb , %

FN-0400

Fe

MPIF material designationa kpsi

–

30

455

380

895

390

57

60

55

130

260

220

500

250

280

150 80

Nickel Steel 240 35 550

240

195 48

330

Iron Nickel 275 40

160

450

205

310

125

255

42

154

43

94

26

160

68

470

290

1060

295

650

180

Nickel Steel

36

140

250

Iron Nickel 205 30

220

275

130

305

140

225

105

415

50

94

30

88

31

65

23

75 125

–

MPa

38

32

65

36

41

22

34

28

37

23

65

30

45

18

23

20

60

32

40

19

44

20

33

15

18

11

–

kpsi

Fatigue strength, sf

155

1070

345

650

205

605

215

450

160

Nickel Steel

205

Iron Nickel 125 18

–

Iron Copper

MPa

Yield strength, sy

160

145

160

160

115

115

160

145

160

115

160

160

115

115

160

145

160

160

115

115

145

145

115

115

160

115

–

GPa

23

21

23

23

17

17

23

21

23

17

23

23

17

17

23

21

23

23

17

17

21

21

17

17

23

17

–

22

16

27

33

11

12

35

28

22

81

19

41

8.1

14

68

47

24

30

8.1

11

22

24

8.1

14

36

19

–

Mpsi J

Modulus of elasticity, E

TABLE 1-44 Designation, composition and mechanical properties of ferrous powder metallurgy structural steels (Cont.)

16

12

20

24

8

9

26

21

16

6

14

30

6

10

50

35

18

22

6

8

16

18

6

10

50

14

–

ft-lbf

Impact energy

3.0

2.5

1.5

5.0

0.5

2.0

6

4

4.5

1.5

1.5

6.0

0.5

3.0

6.5

6

0.5

3.5

0.5

2.0

1.0

3.5

0.5

3.0

11

4

0.5

Elongation in 25 mm (1 in), %

96RB

88RB

40RC

90RB

24RC

69RB

83RB

72RB

95RB

72RB

44RC

80RB

27RC

63RB

67RB

60RB

47RC

87RB

34RC

62RB

42RC

70RB

32RC

50RB

51RB

38RB

70RF

Apparent hardness

B 484, Grade 3, Class C

B 484, Grade 3, Class B

B 484, Grade 3, Class A

B 484, Grade 2, Class C

B 484, Grade 2, Class B

B 484, Grade 2, Class A

B 484, Grade 1, Class C

B 484, Grade 1, Class B

B 484, Grade 1, Class A

B 222, B439, Grade 3

ASTM designation

PROPERTIES OF ENGINEERING MATERIALS

80.5–91.7

80.1–91.4

70.7–85.0

70.4–84.7

70.0–84.4

FX-1005 (e)

FX-1008 (e)

FX-2000 (e)

FX-2005 (e)

FX-2008 (e)

0.6–1.0

0.3–0.7

0.3 max

0.6–1.0

0.3–0.6

C

15.0–25.0

15.0–25.0

15.0–25.0

8.0–14.9

8.0–14.9

Cu

–

–

–

–

–

Ni

MPIF chemical composition limits and rangesb , %

7.2–7.6

–

7.2–7.6

7.2–7.6

7.2–7.6

MPIF density, , g/cm3

895

585 860

HT

790

HT AS

515

AS

450

HT AS

620

830

HT AS

570

MPa

AS

Condition

c

125

85

115

75

65

130

90

120

83

kpsi

Tensile strength, st kpsi

740

515

655

345

–

775

515

740

107

75

95

50

–

105

75

107

Inﬁltered Steel 440 64

MPa

Yield strength, sy

–

–

–

–

–

–

–

–

–

MPa

–

–

–

–

–

–

–

–

–

kpsi

Fatigue strength, sf

125

125

125

125

–

135

135

135

135

GPa

18

18

18

18

–

20

20

20

20

6.8

14

8.1

12.9

20

9.5

16.0

9.5

19

Mpsi J

Modulus of elasticity, E

5.0

10

6.0

9.5

15

7.0

12

7.0

14

ft-lbf

Impact energy

80RB 92RC

1.0

30RC

<0.5 <0.5

75RB

60RB

40RC

80RB

35RC

75RB

Apparent hardness

1.5

1.0

60.5

2.6

1.0

4.0

Elongation in 25 mm (1 in), %

B 303, Class C

B 303, Class B

B 303, Class A

ASTM designation

b

Designation listed are nearest comparable designations, ranges and limits may vary slightly between comparable designations. Metal Powder Industries Federation (MPIF) standards require that the total amount of all other elements be less then 2.0%, except that the total amount of other elements must be less than 4.0% inﬁltered steel. c AS, as sintered; SS, sintered and sized; HT, heat treated, typically austenized at 8708C (16008F), oil quenched and tempered at 2008C (4008F). d Estimated as 38% of tensile strength. e x indicates inﬁltered steel; f Unnotched Charpy test; RB ¼ hardness Rockwell B scale; RC ¼ hardness Rockwell scale C; RF ¼ hardness Rockwell F scale. Source: Metals Handbook Desk Edition, ASM International, 1985, Materials Park, OH 44073-0002 (formerly The American Society for Metals, Metals Park, OH 44073, 1985).

a

Fe

MPIF material designationa

TABLE 1-44 Designation, composition and mechanical properties of ferrous powder metallurgy structural steels (Cont.)

PROPERTIES OF ENGINEERING MATERIALS

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1.73

1.74

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Ti-5Al-3.5Sn Ti-5Al-2.5Sn (EL1) Ti-6Al-4V (EL1)

A 286

Pyromet 538 Nitromic 70 Kromare 5S

347

310S

0.05 1.4

9.5 8.0 9.3

2.0 max

2.0 max

2.0 max

S 30400 0.03 (AISI 304) S 31008 0.03 (AISI 310) S34700 0.03 (AISI 347)

0.5 0.10

304

0.1

0.7

0.5 0.1

15

Ni

2.0

0.4

0.15 3.5 0.05

1.0

1.5

2.6

7.0 8.5 23

9–13

19–22

Austenitic Stainless Steel 1.0 8–12

High Nickel Alloys Rem 1.7 39–44 0.9 Rem 2.5 Rem

Copper and Copper Alloys

0.05

5.5 Rem 18.5 6.8

0.2

Fe

0.07 0.04 0.04 0.04

C

16

Mo

0.35 max 0.4 0.8

Al

0.2 0.1 0.17

N

0.05

0.02 0.07 0.03

0.08 max

0.08 max

0.08 max

1.25

2.2

3.1.

0.20

0.02

2.0 Titanium and Titanium Alloys Omax Sn Hmax 0.20 2.0–3.0 0.02 0.5 max 0.15 max 4.0–6.0 0.07 max 0.12 2–3 0.018 0.2 max 0.08 max 4.7–5.6 0.05 max 3.5–4.5 0.13 0.015 max 0.15 max 0.15 max 0.08 max 5.5–6.5 0.05 max

0.3

20 17 15.5 0.16

17–19

24–26

18–20

15.5 16 18.6 15.0

Smax Aluminum Alloys

0.06

Ti

0.045

3.0 Zn 5.6 Zn

0.18 Zr

Pmax Other

0.30Mmax

0.005 0.008Zr, 0.016B 0.005B

0.045 (10C)(Nb+Tb)

0.045

1.5Co, 4.0W 3.0 (Nb+Ta) 5.0 Nb 0.85 Nb

30 Zn

0.1

V

70.0

0.15 0.20 0.20 0.23

Cr

1.9 Be

4.4 1.0 2.8 2.5

0.5

Mg

98.1

0.28 0.05 0.25 1.6

0.6 0.1

0.8 0.3 0.7

Mn

4.4 6.3

Cu

0.8

Si

No 5500 No 9706 No 7718

UNS No.

Hastelloy C Inconel 706 Inconel 718 Inconel X 750

C 17200 Beryllium copper C 26000 Cartridge brass, 70%

2014 2219 5083 6061 7039 7075

Alloy designation

Nominal composition, %

TABLE 1-45 Nominal composition of some of structural alloys at subzero temperatures (i.e., at cryogenic temperatures)

PROPERTIES OF ENGINEERING MATERIALS

320

423

452 75 320 423 452

253

269 24 196 253 269

24 196 253 6061, T651 temper, L.O. 24 196 269 7039, T6 temper, L.O. 24 196 253 7075, T 651 temper, L.O. 24 196 269 7075, T 7351 temper, L.O. 24 196 269

6061-T6 temper, L.O.

5083-H113, L.O.

75 320 423 75 320 452 75 320 423 75 320 452 75 320 452

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Plate

Sheet

Plate

Sheet

Plate

Plate

Sheet

97.6 48.5 67.1 90.0 85.8

97.6

84.5

320 46.3 425 61.8 495 72.0 310 44.9 400 58.3 485 70.1 455 66.0 575 83.2 665 96.2 580 84.2 705 102 825 120 525 76.2 675 98.2 760 110

675 335 465 620 590

675

585

44.9 58.3 70.1 60.2 66.5 78.8 94.5 67.8

196

2219-T87 temper, L.O.

2219-T62 temper, L.O.

310 400 485 415 460 545 650 465

75 320 452 75 320 423 452 75

24 196 269 24 196 253 269 24

2014-T651 temper, L.O.

Plate

MPa kpsi

Form

8C

Alloy and condition

8F

Tensile strength, st

Temperature

290 340 365 290 335 380 410 495 535 530 650 770 455 570 605

510 235 27.5 305 280

490

460

290 335 380 290 345 370 390 380

42.2 49.6 52.6 42.2 48.9 55.0 59.8 72.0 77.4 77.1 94.4 112 66.2 82.5 88.1

74.2 34.2 39.6 43.9 40.5

71.0

66.6

42.2 48.9 55.0 41.9 50.2 53.6 55.2 56.3

12 18 26 16 23 26 11 15 14 10 7 6 10 11 11

15 15 31 30 29

14

13

16 23 26 10 11 14 23 11

Yield strength, sy Elongation, MPa kpsi % MPa

kpsi

Room temperature yield strength, syr

14 10 9 22 14 12

50 48 42

23 23 31 24 28

21

25

26

403

536

381

289

142*

382

58.5

77.7

55.3

41.9

20.6*

55.4

Aluminum Alloys 50 432 62.7 48 42 382 55.4

Reduction in area, %

35.9 32.1

22.5 27.6

32.3 33.5

32.7 29.2

20.5 25.1

29.4 30.5

26.5 37.9

43.7*

48.0*

29.1 41.6

24.6* 39.5b

36.3 (26.2) 42.4 (31.4) 48.0 (34.0)

21.2 26.1

pﬃﬃﬃﬃ kpsi m

27.0* 43.4b

39.9 (28.8) 46.5 (34.5) 52.5 (37.2)

23.2 28.7

pﬃﬃﬃﬃ MPa m

Fracture toughness KIC (J)

T.S.

T.L.

Bend

Bend

Bend

C.T.

T.L.

T.L.

T.L.

T.L.

Bend, CT T.S.

Bend

Bend

Specimen Orientdesign ation

TABLE 1-46 Typical tensile properties and fracture toughness of structural alloys at sub-zero temperature (i.e. cryogenic temperature)

7039 has good combination of strength and fracture toughness at room and at 196 8C Used for plate of 7075-T6 for the inner tank skirt between the liquid oxygen and liquid hydrogen sections of the external tank of the space shuttle. Improves ductility and notch toughness of 7075 alloy at cryogenic temperature processing it to the T7351 temper.

Fracture toughness values are for 5083O, i.e., KIC (J). 5083-O is not heat treatable and used in annealed (O) condition. Used to build liqueﬁed natural gas (LNG) spherical tanks in ships. Aluminum alloys 6061 in the T6 temper have the same strength and ductility at both room and sub-zero temperatures.

High toughness at room and sub-zero temperatures, used for liquid oxygen and liquid hydrogen tanks for space shuttle. Fracture toughness values in parentheses refer to specimen design C.T.

Possess high strength at room temperature and sub-zero temperature.

Uses and remarks

PROPERTIES OF ENGINEERING MATERIALS

1.75

1.76

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452 75 320 423 452

75 260 320 452

269 GTA weld in Inconel 24 X-750 sheet, X-750 ﬁller 196 metal, aged 20 h at 7008C 253 269

24 162 196 269

24 196 269 24 196

A645 (5Ni Steel), L.O. quenched, tempered, reversion annealed.

304 annealed, L.O.

310 S annealed, T.O.

75 320

24 196

Inconel 718, L.O. [Aged 34 h at 9828C (18008F)]

75 320 452 75 320

75 320 423 75 320 452

24 196 253 Inconel 706 (VIM-VAR) 24 STDA 196 269

Hastelloy C, cold rolled 20%, L.O.

C17200 TD02 temper (Solution treated, cold worked to 12 hard)

75 320 423 75 320 423

Forging

Sheet

Plate

Weldment

Sheet

Bar

Forged billets

Sheet

Sheet

Bar

165 220 252 183 228 243

263 187 224 241

660 95.5 1650 236 1700 247 330 48 670 97

715 104 930 135 1130 164

1810 1290 1540 1660

1410 204 1650 239

1140 1520 1740 1260 1570 1680

95.2 117 132 90 117 137

24 196 253 24 196 253

C26000 03 temper (34 hard) 655 805 910 620 805 945

MPa kpsi

Form

8C

Alloy and condition

8F

Tensile strength, st

Temperature

295 425 570 580 1070

530 570 765

1410 860 945 1020

1170 1340

1000 1280 1380 1050 1200 1250

420 475 505 550 690 750

21 22 50 28

15 21

13 32 33 24 29 30

14 28 32 15 37 45

42.5 55.0 82.5 84 155

75 42 30 40 46

76.8 32 82.9 2.8 111 30

204 125 137 148

170 197

145 186 200 152 174 181

61.0 68.5 73.5 80 100 109

Reduction in area, % MPa

kpsi

Room temperature yield strength, syr

825

1170*

1065

120

170*

154

Austenitic Stainless Steels – – – 76 260 37.9 67

Ferritic Nickel Steels 72 68 535 77.5 62

20

18 20

33 33 33

High-Nickel Alloys

58 63 58

Copper and Copper Alloys (L.O.)

Yield strength, sy Elongation, MPa kpsi %

196 87.1 58.4

112

178 79.3 53.2

134# 176‡

102

87.8* 94

143†

157†

96.4* 103

121†

pﬃﬃﬃﬃ kpsi m

133†

pﬃﬃﬃﬃ MPa m

Fracture toughness KIC (J)

122# 160‡

C.T.

C.T.

C.T.

T.L.

C.R.

Specimen Orientdesign ation

Uses and remarks

Austenitic stainless steels are used extensively for subzero temperature applications to 2698C. † Welding process, SMA, Filler, 310S. b In weld fusion zone, as welded.

(VEB). Used in construction of storage tanks for liqueﬁed hydrocarbon gases, structures and machineries in cold regions.

Fracture toughness values refer to KIC (J). These high nickel alloys exhibit excellent combinations of strength, toughness, and ductility over the entire range of subzero temperatures. Used in energy related equipments such as superconducting motors and generators. * Refer to STDA alloy. The fatigue strengths of high-nickel alloys are higher at cryogenic temperature than at room temperature. # KIJ (J), fusion zone, gas tungsten arc weld (GTA). ‡ KIC (J), vacuum electron beam weld

†

Tensile and fatigue properties of copper alloys increase as the testing temperature decreased. Used in stabilizers, components of the windings in superconducting magnets, solenoids and power cables at super temperatures.

TABLE 1-46 Typical tensile properties and fracture toughness of structural alloys at sub-zero temperature (i.e. cryogenic temperature) (Cont.)

PROPERTIES OF ENGINEERING MATERIALS

75 320 423 75 320 452 75 452 452

A-286 annealed sheet, welded 24 and age hardened, L.O. 196 A-286 STA 253 24 196 269 24 269 269

452

75 320 452 75 320 452

269

452 452 75 320 423 75 320 423 75 320 423 75 320

Nitronic 60, annealed, L.O. 24 196 253 Kromarc 58 annealed 24 plate, tested as welded*, L.O. 196 Kromarc 58 STQ 269

310 S annealed, T.O.

109 218 204 72 124 154 11 16 15

66 60 24 36 46 33

79 66 27 61 41 40

24

74 61

24

370

340

–

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820

125 166 186

58.1 101 125 133 192 209

31

52 47 35 3 29 2 3 10 10 51 48

26

MPa

119

88.2

53.8

49

–

kpsi

Room temperature yield strength, syr

Plate

860 1145 1280

400 695 860 915 1325 1440

239

57 61 63 173 208 226 160 223 259 105 211

160

Reduction in area, %

610

600 87 745 108 870 126

750 1500 1410 495 855 1060

1240 180

1650

255 420 435 1190 1430 1560 1100 1540 1790 725 1455

650 1365 1610 1320 1900 2010 1180 1720 2000 415 1005

94 198 234 191 276 292 171 249 290 60 146

1105

825 120

Yield strength, sy Elongation, MPa kpsi %

Bar

Plate Weldamentc Sheet

Plate

Plate Weldmentc Weldmenta Bar

Plate

Sheet

Sheet

Plate, Weldment† ;a Sheet

Form

MPa kpsi

8F

8C

269 269 347 annealed, L.O. 24 196 253 304 hard cold rolled, L.O. 24 196 253 310, 75% cold reduced, 24 L.O. 196 253 Pyromet 538 annealed 24 Pyromet 538 STQ 196

Alloy and condition

Tensile strength, st

Temperature

125 123 118 161 179 247b

214 155b

181 82b 175b

– 275a

259 116b

pﬃﬃﬃﬃ MPa m

T.L.

Specimen Orientdesign ation

114 112 107 146 163 225b

195 141b

L.T.

T.S.

T.L.

165 at 2698C (4528F) 74b at 2698C (4528F) 159b at 2698C (4528F)

– 250a

236 106b

pﬃﬃﬃﬃ kpsi m

Fracture toughness KIC (J) Uses and remarks

A-286 alloy develops good strength, with good ductility and notch toughness in the cryogenic temperature range.

Welding process: GTA. Filler: Kromrc 58. Kromrc is used for structural appliances. Prototype super-conducting generators

Strength of these steels is increased by rolling or cold drawing at 1968C. Used in liquid hydrogen and liquid oxygen tank construction in Atlas and Centaur rockets. Type 304 stainless steels used in piping, tubing, and valves. Used in transfer of oxygen for storage tanks. Cast steels are used for Bubble Chambers, and for cylindrical magnet tubes for superconducting magnets. ‡ Welding process: GTA. Filler: Pyromet 538. Fracture toughness values refer to Pyromet 538 STQ. a Shielded metal arc weld. b In weld fusion zone, as welded. c Gas tungsten arc weld.

TABLE 1-46 Typical tensile properties and fracture toughness of structural alloys at sub-zero temperature (i.e. cryogenic temperature) (Cont.)

PROPERTIES OF ENGINEERING MATERIALS

1.77

Forging

Sheet/ Plate

Sheet/ Plate

850 1370 1700 800 1300 1570 970 1570 1650

123 199 246 116 188 228 141 227 239

795 1300 1590 740 1210 1450 915 1480 1570

115 188 231 107 175 210 133 214 227

16 14 7 16 16 10 14 11 11

Reduction in area, % MPa

kpsi

Room temperature yield strength, syr

40 31 24

830

120

Titanium and Titanium Alloys 875 127 875 127 875 127 705 102 705 102

Yield strength, sy Elongation, MPa kpsi %

55.5 49.2

54.1

101 81.5

65.4 48.6

pﬃﬃﬃﬃ kpsi m

610

111 89.6

71.8 53.4

pﬃﬃﬃﬃ MPa m

Fracture toughness KIC (J)

C.T. Bend Bend C.T. C.T. Bend C.T.

L.T. L.T. L.T. L.T. L.T. L.T. L.T.

Specimen Orientdesign ation

Ti-5Al-2.5Sn and Ti-6Al-4V, titanium alloys have high strength to weight ratio at cryogenic temperatures and preferred alloys at temperatures of alloys at temperatures of 1968C to 2698C (3208F to 4528F). Used in spherical pressure vessels in Atlas and Centaur Rockets, the Apollo and Saturn launch Boosters and Lunar Modules. Should not be used at cryogenic temperatures for storage or transfer of liquid oxygen, since the condensed oxygen will cause ignition during abrasion.

Uses and remarks

AAM, air arc melted. C.T., compact toughness. GMA, gas metal arc welding process. GTA, gas tungsten arc weld. L.O., longitudinal orientation. S.T., solution treated. STDA, solution treated and double aged. VAR, vacuum arc remelted. VEB, vacuum electron beam weld. VIM, vacuum induction melted. W, weld. Source: Metals Handbook Desk Edition, ASM International, 1985, Materials Park, OH 44073-0002 (formerly The American Society for Metals, Metals Park, OH 44073, 1985).

75 320 423 75 320 423 75 320 423 452

Form

MPa kpsi

8F

8C

Ti-5 Al-2.5 Sn, nominal 24 interstitial annealed, L.O. 196 253 Ti-5 Al-2.5 Sn (EL1) 24 annealed, L.O. 196 253 Ti-6 Al-4 V (EL1) as 24 forged 196 253 269

Alloy and condition

Tensile strength, st

Temperature

TABLE 1-46 Typical tensile properties and fracture toughness of structural alloys at sub-zero temperature (i.e. cryogenic temperature) (Cont.) PROPERTIES OF ENGINEERING MATERIALS

1.78

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0.64 0.67 0.66 0.53 0.43 0.55 0.66 0.55 0.77 0.51 0.68 0.66 0.71 0.55 0.45 0.56

Cedar, western red Cypress Douglas ﬁr, coast region Hemlock, eastern Hemlock, western Larch, western Pine, red Pine, ponderosa Pine, eastern white Pine, western white Redwood Spruce, sitka Spruce, white

Ash, white Beech Birch, yellow Cherry, black Cottonwood, eastern Elm, American Elm, rock Sweetgum Hickory. shagbark Mahogany (swietenia spp) Maple, sugar Oak, red, northern Oak, white Tupelo, black Yellow poplar Walnut, black

6.64 7.11 6.80 5.53 4.42 5.53 7.00 5.69 7.90 5.37 6.95 6.95 7.58 5.53 4.58 6.00

3.62 5.06 5.37 4.42 4.6 6.00 4.90 4.42 3.62 4.27 4.42 4.42 4.42

42 45 43 35 28 35 44 36 50 34 44 44 48 35 29 38

23 32 34 28 29 38 31 28 24 27 28 28 28

kN/m3 lbf/ft3

Densityb , D

4.9 5.1 7.2 3.7 3.9 4.2 4.8 5.4 7.0 3.5 4.9 4.0 5.3 4.4 4.2 5.2

2.4 3.8 4.8 3.0 4.3 4.5 3.8 3.9 2.1 2.6 2.6 4.3 4.7

Rad

7.9 11.0 9.2 7.1 9.2 9.5 8.1 10.2 10.5 4.8 9.5 8.2 9.0 7.7 7.6 7.1

5.0 6.2 7.6 6.8 7.0 9.1 7.2 6.3 6.1 5.3 4.4 7.5 8.2

Tan

Shrinkagec

106.18 102.74 114.46 84.80 58.60 81.36 102.05 86.19 139.28 79.02 108.94 98.60 104.80 66.20 69.64 100.67

51.71 73.09 85.50 61.37 77.11 90.33 75.85 64.81 59.30 66.88 69.00 70.33 67.57

MPa

15.4 14.9 16.6 12.3 8.5 11.8 14.8 12.5 20.2 11.46 15.80 14.30 15.20 9.60 10.10 14.6

7.5 10.6 12.4 8.9 11.3 13.10 11.00 9.4 8.6 9.7 10.0 10.2 9.8

kpsi

Modulus of rupture

12.20 11.86 14.55 10.27 9.45 9.24 10.62 11.31 14.89 10.34 12.62 12.55 12.27 8.27 10.89 11.58

7.65 9.93 13.45 8.28 11.31 12.90 11.24 8.90 8.55 10.10 9.24 10.83 9.24

GPa

kpsi

4.56 6.36 7.24 5.41 7.71 7.64 6.07 5.32 4.80 5.04 6.15 5.61 5.47

7.41 7.30 8.17 7.11 4.91 5.52 7.05 6.32 9.21 6.80 7.83 6.76 7.44 5.52 5.54 7.58

Softwoods 1.11 31.44 1.44 43.85 1.95 49.92 1.20 37.30 1.64 49.02 1.87 52.68 1.63 41.85 1.29 36.68 1.24 33.10 1.46 34.75 1.34 42.40 1.57 38.68 1.34 37.72 Hardwoods 1.77 51.10 1.72 50.33 2.11 56.33 1.49 48.95 1.37 33.85 1.34 38.06 1.54 48.60 1.64 43.58 2.16 63.50 1.50 46.88 1.83 53.98 1.82 46.61 1.78 51.30 1.20 38.06 1.58 38.20 1.68 52.26

Maximumd crushing strength, scr MPa

Mpsi

Modulus of elasticity, E

Static bending

8.00 6.96 6.69 4.76 2.55 4.76 8.48 4.28 12.14 7.58 10.14 7.00 7.38 6.41 3.45 7.00

3.17 5.38 5.52 4.48 3.79 6.76 4.14 4.00 3.03 3.24 4.83 4.00 3.17

MPa

1.16 1.01 0.97 0.69 0.37 0.69 1.23 0.62 1.76 1.10 1.47 1.01 1.07 0.93 0.50 1.01

0.46 0.78 0.80 0.65 0.55 0.98 0.60 0.58 0.44 0.47 0.70 0.58 0.46

kpsi

Compressiona proportionality limit, scp

6.48 7.00 6.34 3.86 4.00 4.55 – 5.24 – 5.17 – 5.52 5.52 3.44 3.72 4.76

1.52 1.86 2.34 – 2.34 3.00 3.17 2.90 2.18 – 1.66 2.55 2.48

MPa

0.94 1.01 0.92 0.56 0.58 0.66 – 0.76 – 0.75 – 0.80 0.80 0.50 0.54 0.69

0.22 0.27 0.34 – 0.34 0.43 0.46 0.42 0.31 – 0.24 0.37 0.36

kpsi

Tensile strengthe;f , st

1092 1041 1397 737 508 990 1422 813 1702 – 483 1092 940 559 610 864

430 864 787 533 660 889 660 483 457 584 483 635 508

mm

43 41 55 29 20 39 56 32 67 – 19 43 37 22 24 34

17 34 31 21 26 35 26 19 18 23 19 25 20

in

Impact bending in a drop of 222 N (50 lbf) hammer

13.45 13.86 13.00 11.72 6.41 10.41 13.24 11.03 16.75 8.48 16.06 12.27 13.80 9.24 8.21 9.45

5.93 6.89 8.00 7.31 6.62 9.38 8.76 7.79 6.21 7.17 6.48 7.93 7.45

MPa

1.95 2.01 1.88 1.70 0.93 1.51 1.92 1.6 2.43 1.23 2.33 1.78 2.00 1.34 1.19 1.37

0.86 1.00 1.16 1.06 1.25 1.36 1.21 1.13 0.90 1.04 0.94 1.15 1.08

kpsi

Sheard strength, s

1320 1300 1260 950 430 830 1320 850 – 800 1450 1290 1360 810 540 1010

350 510 710 500 540 830 560 460 380 420 480 510 480

Hardness average of R and T

f

Seasoned wood at 12% moisture. b Seasoned wood at 12% moisture content. c Percent from green to ovendry condition based on dimensions when green. d Parallel to grain. e Perpendicular to grain. Height of drop of 222 N (50 lbf) hammer for failure, mm (in). Rad, radially. Tan, tangentially. g Tensile strength parallel to grain may be taken as equal to modulus of rupture in bending. Source: Extracted from Wood Handbook and the U.S. Forest Products Laboratory.

0.34 0.48 0.51 0.43 0.44 0.59 0.47 0.42 0.37 0.42 0.42 0.42 0.45

Kind of wood

a

Speciﬁc gravity Gm ovendry volume

TABLE 1-47 Typical properties of wooda of clear material of section 50 mm 50 mm (2 in 2 in), as per ASTM D143

PROPERTIES OF ENGINEERING MATERIALS

1.79

4

24 40 1 40

166 276 7 276 28

60 10 12 7 15

kpsi

414 69 83 48 103

MPa

14

365 310 214 145 152 172 469 290 83 345

GPa

9828C 18008F 218C

2

53 45 31 21 22 25 68 42 12 50 4.0

9.0 8.8 13.5 9.0 9.9 10.1 4.0 4.5 4.7 9.2 2.2

5.0 4.9 7.5 5.0 5.5 5.6 2.2 2.5 2.6 5.1

12320 29120 10528 4480 29120 4480 32480 32480 29120 22400

194880 64960

47040 324800 53760 13888 33600 12992 87360 44800 33600 49280

Speciﬁc heat, c

870

210 1450 240 62 15 58 390 200 150 220 290

55 130 47 20 15 20 145 145 130 100

1.42

1.09 2.09 1.05 0.25 0.59 0.25 0.84 1.51 1.63 0.46

0.34

0.26 0.50 0.25 0.06 0.14 0.06 0.20 0.36 0.39 0.11

2128F 18008F Btu/(ft2 h8F/in) kJ/kg8C Btu/lbm 8F

Thermal conductivity, K at

1008C 9828C lin/in8F W/(m2 8C/m)

Linear coeﬃcientb of expansion,

Mpsi lm/ m8C

708F

Modulus of elasticity, E

3871

2032 2571 2799 3049 2538 2799 2760 2449 2760 2143

8C

7000

3690 4660 5070 5520 4600 5070 5000d 4440 5000 3890

8F

Fusion point

Good

Good Very good Poor Fair Fair Fair Excellent Good Good Good

103

10 0.5 1010 106

>1014 >1014 >1014 >1014 108

Thermal stress 708C resistance (218C)

102

104

107 108 107 105 500

18008F (9828C)

Electrical resistivity, cmc

b

Porosity: 0 to 5%. Between 208C (658F) and 9828C (18008F). c Multiply the values by 0.393 in to obtain electrical resistivity in units of -in. d Stabilized. Courtesy: Extracted from Mark’s Standard Handbook for Mechanical Engineers, 8th edition, McGraw-Hill Book Company, New York, 1978, and Norton Refractories, 3rd edition, Green and Stewart, ASTM Standards on Refractory Materials Handbook (Committee, C-8), 1.

a

3

2.22 0.0802 21

0.143 0.110 0.129 0.361 0.202 0.396 0.116 0.097 0.0813 0.224

100 20 14 12 20 12 24 50 7 100

3.97 3.03 3.58 10.00 5.60 10.96 3.22 2.52 2.25 6.20

Alumina (Al2 O3 ) Beryllium oxide (BeO) Magnesium oxide (MgO) Thoria (ThO2 ) Zirconia (ZrO2 ) Uranium oxide (UO2 ) Silicon carbide (SiC) Boron carbide (BC) Boron nitride (BN) Molybdenum silicide (MoSi2 ) Carbon (C)

lb/in3 MPa kpsi

218C 708F

689 138 97 83 138 83 166 345 48 689

g/m3

Material

Density

Modulus of rupture, sr

TABLE 1-48 Mechanical and physical properties of typical densea pure refractories

PROPERTIES OF ENGINEERING MATERIALS

1.80

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PROPERTIES OF ENGINEERING MATERIALS PROPERTIES OF ENGINEERING MATERIALS

1.81

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

Datsko, J., Material Properties and Manufacturing Process, John Wiley and Sons, New York, 1966. Datsko, J. Material in Design and Manufacturing, Malloy, Ann Arbor, Michigan, 1977. ASM Metals Handbook, American Society for Metals, Metals Park, Ohio, 1988. Machine Design, 1981 Materials Reference Issue, Penton/IPC, Cleveland, Ohio, Vol. 53, No. 6, March 19, 1981. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. Technical Editor Speaks, the International Nickel Company, New York, 1943. Shigley, J. E., Mechanical Engineering Design, Metric Edition, McGraw-Hill Book Company, New York, 1986. Deutschman, A. D., W. J. Michels, and C. E. Wilson, Machine Design—Theory and Practice, Macmillan Publishing Company, New York, 1975. Juvinall, R. C., Fundaments of Machine Components Design, John Wiley and Sons, New York, 1983. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Co-operative Society, Bangalore, India, 1962. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1981 and 1984. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1983. SAE Handbook, 1981. Lessels, J. M., Strength and Resistance of Metals, John Wiley and Sons, New York, 1954. Siegel, M. J., V. L. Maleev, and J. B. Hartman, Mechanical Design of Machines, 4th edition, International Textbook Company, Scranton, Pennsylvania, 1965. Black, P. H., and O. Eugene Adams, Jr., Machine Design, McGraw-Hill Book Company, New York, 1963. Niemann, G., Maschinenelemente, Springer-Verlag, Berlin, Erster Band, 1963. Faires, V. M., Design of Machine Elements, 4th edition, Macmillan Company, New York, 1965. Nortman, C. A., E. S. Ault, and I. F. Zarobsky, Fundamentals of Machine Design, Macmillan Company, New York, 1951. Spotts, M. F., Design of Machine Elements, 5th edition, Prentice-Hall of India Private Ltd., New Delhi, 1978. Vallance, A., and V. L. Doughtie, Design of Machine Members, McGraw-Hill Book Company, New York, 1951. Decker, K.-H., Maschinenelemente, Gestalting und Bereching, Carl Hanser Verlag, Munich, Germany, 1971. Decker, K.-H., and Kabus, B. K., Maschinenelemente-Aufgaben, Carl Hanser Verlag, Munich, Germany, 1970. ISO and BIS standards. Metals Handbook, Desk Edition, ASM International, Materials Park, Ohio, 1985 (formerly the American Society for Metals, Metals Park, Ohio, 1985). Edwards, Jr., K. S., and R. B. McKee, Fundamentals of Mechanical Components Design, McGraw-Hill Book Company, New York, 1991. Shigley, J. E., and C. R. Mischke, Standard Handbook of Machine Design, 2nd edition, McGraw-Hill Book Company, New York, 1996. Structural Alloys Handbook, Metals and Ceramics Information Center, Battelle Memorial Institute, Columbus, Ohio, 1985. Wood Handbook and U. S. Forest Products Laboratory. SAE J1099, Technical Report of Fatigue Properties. Ashton, J. C., I. Halpin, and P. H. Petit, Primer on Composite Materials-Analysis, Technomic Publishing Co., Inc., 750 Summer Street, Stanford, Conn 06901, 1969. Baumeister, T., E. A. Avallone, and T. Baumeister III, Mark’s Standard Handbook for Mechanical Engineers, 8th edition, McGraw-Hill Book Company, New York, 1978. Norton, Refractories, 3rd edition, Green and Stewart, ASTM Standards on Refractory Materials Handbook (Committee C-8).

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PROPERTIES OF ENGINEERING MATERIALS

1.82

CHAPTER ONE

BIBLIOGRAPHY Black, P. H., and O. Eugene Adams, Jr., Machine Design, McGraw-Hill Book Company, New York, 1983. Decker, K.-H., Maschinenelemente, Gestalting und Bereching, Carl Hanser Verlag, Munich, Germany, 1971. Decker, K.-H., and Kabus, B. K., Maschinenelemente-Aufgaben, Carl Hanser Verlag, Munich, Germany, 1970. Deutschman, A. D., W. J. Michels, and C. E. Wilson, Machine Design—Theory and Practice, Macmillan Publishing Company, New York, 1975. Faires, V. M., Design of Machine Elements, 4th edition, McGraw-Hill Book Company, New York, 1965. Honger, O. S. (ed.), (ASME) Handbook for Metals Properties, McGraw-Hill Book Company, New York, 1954. ISO standards. Juvinall, R. C., Fundaments of Machine Components Design, John Wiley and Sons, New York, 1983. Lessels, J. M., Strength and Resistance of Metals, John Wiley and Sons, New York, 1954. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Co-operative Society, Bangalore, India, 1962. Mark’s Standard Handbook for Mechanical Engineers, 8th edition, McGraw-Hill Book Company, New York, 1978. Niemann, G., Maschinenelemente, Springer-Verlag, Berlin, Erster Band, 1963. Norman, C. A., E. S. Ault, and I. E. Zarobsky, Fundamentals of Machine Design, McGraw-Hill Book Company, New York, 1951. SAE Handbook, 1981. Shigley, J. E., Mechanical Engineering Design, Metric Edition, McGraw-Hill Book Company, New York, 1986. Siegel, M. J., V. L. Maleev, and J. B. Hartman, Mechanical Design of Machines, 4th edition, International Textbook Company, Scranton, Pennsylvania, 1965. Spotts, M. F., Design of Machine Elements, 5th edition, Prentice-Hall of India Private Ltd., New Delhi, 1978. Vallance, A., and V. L. Doughtie, Design of Machine Members, McGraw-Hill Book Company, New York, 1951.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

2 STATIC STRESSES IN MACHINE ELEMENTS SYMBOLS3;4;5 A Aw a b c D C1 F Fc Ft F Fcr e et E Ec G Mb Mt i I Ixx , Iyy J k k0 kt

area of cross section, m2 (in2 ) area of web, m2 (in2 ) constant in Rankine’s formula radius of area of contact, m (in) bandwidth of contact, m (in) width of beam, m (in) distance from neutral surface to extreme ﬁber, m (in) diameter of shaft, m (in) constant in straight-line formula load, kN (lbf) compressive force, kN (lbf) tensile force, kN (lbf) shear force, kN (lbf) crushing load, kN (lbf) deformation, total, m (in) eccentricity, as of force equilibrium, m (in) unit volume change or volumetric strain thermal expansion, m (in) modulus of elasticity, direct (tension or compression), GPa (Mpsi) combined or equivalent modulus of elasticity in case of composite bars, GPa (Mpsi) modulus of rigidity, GPa (Mpsi) bending moment, N m (lbf ft) torque, torsional moment, N m (lbf ft) number of turns moment of inertia, area, m4 or cm4 (in4 ) mass moment of inertia, N s2 m (lbf s2 ft) moment of inertia of cross-sectional area around the respective principal axes, m4 or cm4 (in4 ) moment of inertia, polar, m4 or cm4 (in4 ) radius of gyration, m (in) polar radius of gyration, m (in) torsional spring constant, J/rad or N m/rad (lbf in/rad)

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STATIC STRESSES IN MACHINE ELEMENTS

2.2

CHAPTER TWO

l l0 L n n0 l, m, n P T T r q Q v V V Z xy , yz , zx " "T "x , "y , "z

b c sc cr e s t st x , y , z 1 , 2 , 3 y sy u su 0 00

length, m (in) length of rod, m (in) length, m (in) speed, rpm (revolutions per minute) coeﬃcient of end condition speed, rps (revolutions per second) direction cosines (also with subscripts) power, kW (hp) pitch or threads per meter temperature, 8C (8F) temperature diﬀerence, 8C (8F) radius of the rod or bar subjected to torsion, m (in) (Fig. 2-18) shear ﬂow ﬁrst moment of the cross-sectional area outside the section at which the shear ﬂow is required velocity, m/s (ft/min or fpm) volume, m3 (in3 ) shear force, kN (lbf) volume change, m3 (in3 ) section modulus, m3 (in3 ) deformation of contact surfaces, m (in) coeﬃcient of linear expansion, m/m/K or m/m/8C ðin=in=8F) shearing strain, rad/rad shearing strain components in xyz coordinates, rad/rad deformation or elongation, m (in) strain, mm/m (min/in) thermal strain, mm/m (min/in) strains in x, y, and z directions, mm/m (min/in) angular distortion, rad angle, deg angular twist, rad (deg) angle made by normal to plane nn with the x axis, deg bulk modulus of elasticity, GPa (Mpsi) Poisson’s ratio radius of curvature, m (in) stress, direct or normal, tensile or compressive (also with subscripts), MPa (psi) bearing pressure, MPa (psi) bending stress, MPa (psi) compressive stress (also with subscripts), MPa (psi) hydrostatic pressure, MPa (psi) compressive strength, MPa (psi) stress at crushing load, MPa (psi) elastic limit, MPa (psi) strength, MPa (psi) tensile stress, MPa (psi) tensile strength, MPa (psi) stress in x, y, and z directions, MPa (psi) principal stresses, MPa (psi) yield stress, MPa (psi) yield strength, MPa (psi) ultimate stress, MPa (psi) ultimate strength, MPa (psi) principal direct stress, MPa (psi) normal stress which will produce the maximum strain, MPa (psi)

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STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS

s xy , yz , zx !

2.3

normal stress on the plane nn at any angle to x axis, MPa (psi) shear stress (also with subscripts), MPa (psi) shear strength, MPa (psi) shear stresses in xy, yz, and zx planes, respectively, MPa (psi) shear stress on the plane at any angle with x axis, MPa (psi) angular speed, rad/s

Other factors in performance or in special aspects are included from time to time in this chapter and, being applicable only in their immediate context, are not given at this stage. (Note: and with initial subscript s designates strength properties of material used in the design which will be used and observed throughout this Machine Design Data Handbook.)

Particular

Formula

SIMPLE STRESS AND STRAIN Ft ; A

Fc A

The stress in simple tension or compression (Fig. 2-1a, 2-1b)

t ¼

The total elongation of a member of length l (Fig. 2-2a)

¼

Fl AE

ð2-2Þ

"¼

¼ l E

ð2-3Þ

c ¼

ð2-1Þ

FIGURE 2-1

Strain, deformation per unit length

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STATIC STRESSES IN MACHINE ELEMENTS

2.4

CHAPTER TWO

Particular

Formula

FIGURE 2-2

Young’s modulus or modulus of elasticity

E¼

"

ð2-4Þ

The shear stress (Fig. 2-1c)

¼

F A

ð2-5Þ

Shear deformation due to torsion (Fig. 2-18)

¼

L G

ð2-6Þ

Shear strain (Fig. 2-2c)

¼

a ¼ G l

ð2-7Þ

The shear modulus or modulus of rigidity from Eq. (2-7)

G¼

ð2-8Þ

Poisson’s ratio

¼ lateral strain/axial strain ¼

Poisson’s ratio may be computed with suﬃcient accuracy from the relation

¼

E 1 2G

ð2-10Þ

The shear or torsional modulus or modulus of rigidity is also obtained from Eq. (2-10)

G¼

E 2ð1 þ Þ

ð2-11Þ

The bearing stress (Fig. 2-3c)

b ¼

F bd2

ð2-12Þ

¼ x cos2

ð2-13Þ

"t "a

ð2-9Þ

STRESSES Unidirectional stress (Fig. 2-4) The normal stress on the plane at any angle with x axis

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STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS

Particular

2.5

Formula

FIGURE 2-3 Knuckle joint for round rods.

FIGURE 2-4 A bar in uniaxial tension.3;4

The shear stress on the plane at any angle with x axis

¼

x sin 2 2

ð2-14Þ

Principal stresses

1 ¼ x and 2 ¼ 0

ð2-15Þ

Angles at which principal stresses act

1 ¼ 08 and 2 ¼ 908

ð2-16Þ

Maximum shear stress

max ¼

Angles at which maximum shear stresses act

1 ¼ 458 and 2 ¼ 1358

x 2

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ð2-17Þ ð2-18Þ

STATIC STRESSES IN MACHINE ELEMENTS

2.6

CHAPTER TWO

Particular

The normal stress on the plane at an angle þ ð =2Þ (Fig. 2-4d) The shear stress on the plane at an angle þ ð =2Þ (Fig. 2-4d)

Formula

0 ¼ x cos2 þ ð2-19Þ ¼ x cos2 2

cos þ ¼ 12 x sin 2 ð2-20Þ 0 ¼ x sin þ 2 2 ¼ 0 and ¼ 0

ð2-21Þ

The normal stress on the plane at any angle

¼ xy sin 2

ð2-22Þ

The shear stress on the plane at any angle

¼ xy cos 2

ð2-23Þ

The principal stress

1 ¼ xy and 2 ¼ xy

ð2-24Þ

Angles at which principal stresses act

1 ¼ 458 and 2 ¼ 1358

ð2-25Þ

Maximum shear stresses

max ¼ xy ¼

ð2-26Þ

Angles at which maximum shear stress act

1 ¼ 0 and 2 ¼ 908

ð2-27Þ

FIGURE 2-5 An element in pure shear.

FIGURE 2-6 An element in biaxial tension.

Therefore from Eqs. (2-13) and (2-19), (2-14), and (2-20)

PURE SHEAR (FIG. 2-5)

BIAXIAL STRESSES (FIG. 2-6) The normal stress on the plane at any angle

¼

x þ y x y þ cos 2 2 2

ð2-28Þ

The shear stress on the plane at any angle

¼

x y sin 2 2

ð2-29Þ

The shear stress at ¼ 0

¼ 0

ð2-30Þ

The shear stress at ¼ 458

max ¼ ð x y Þ=2

ð2-31Þ

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STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS

Particular

2.7

Formula

BIAXIAL STRESSES COMBINED WITH SHEAR (FIG. 2-7) The normal stress on the plane at any angle

¼

x þ y x y þ cos 2 þ xy sin 2 2 2

ð2-32Þ

The shear stress in the plane at any angle

¼

x y sin 2 xy cos 2 2

ð2-33Þ

The maximum principal stress

x þ y 1 ¼ þ 2

The minimum principal stress

2 ¼

Angles at which principal stresses act

1;2 ¼ 12 arctan

x þ y 2

x y 2 x y 2

1=2

2 þ

2 xy

ð2-34Þ 1=2

2 2 þ xy

2xy x y

ð2-35Þ

ð2-36Þ

where 1 and 2 are 1808 apart

x y 2

Maximum shear stress

max ¼

Angles at which maximum shear stress acts

¼ 12 arctan

The equation for the inclination of the principal planes in terms of the principal stress (Fig. 2-8)

tan ¼

σy

θ x

θ

τxy

¼

1 2 2

x y 2xy

1 x xy

τxy

y

τxy

1=2 2 þ xy

τxy

n

σx

2

σx

θ

σx τxy

σy (a)

n

τxy

σθ

θ τθ σy (b)

FIGURE 2-7 An element in plane state of stress.

FIGURE 2-8

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ð2-37Þ

ð2-38Þ

ð2-39Þ

STATIC STRESSES IN MACHINE ELEMENTS

2.8

CHAPTER TWO

Particular

Formula

MOHR’S CIRCLE Biaxial ﬁeld combined with shear (Fig. 2-9) Maximum principal stress 1

1 is the abscissa of point F

Minimum principal stress 2

2 is the abscissa of point G

Maximum shear stress max

max is the ordinate of point H

FIGURE 2-9 Mohr’s circle for biaxial state of stress.

TRIAXIAL STRESS (Figs. 2-10 and 2-11) The normal stress on a plane nn, whose direction cosines are l, m, n

¼ x l 2 þ y m2 þ z n2

The shear stress on a plane normal nn, whose direction cosines are l, m, n

¼

The principal stresses

1;2;3 ¼ x ; y ; z

The cubic equation for general state of stress in three dimensions from the theory of elasticity

3 ð x þ y þ z Þ 2 þ ð x y þ y z þ z x

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2x l 2 þ 2y m2 þ 2z n2

ð2-40Þ ð2-41Þ ð2-42Þ

2 2 2 xy yz zx Þ 2 2 2 y zx z xy Þ ð x y z þ 2xy yz zx x zy

¼0

The maximum shear stresses on planes parallel to x, y, and z which are designated as

ð2-43Þ

The three roots of this cubic equation give the magnitude of the principal stresses 1 , 2 , and 3 . 3 3 ; ðmax Þ2 ¼ 1 ; ðmax Þ1 ¼ 2 2 2 2 ðmax Þ3 ¼ 1 ð2-44Þ 2

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STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS

Particular

2.9

Formula

MOHR’S CIRCLE Triaxial ﬁeld (Figs. 2-10 and 2-11) Normal stress at point (Fig. 2-11b) on one octahedral plane

t ¼ 13 ð 1 þ 2 þ 3 Þ ¼ 13 ð x þ y þ z Þ

Shear stress at point T (Fig. 2-11b) on an octahedral plane

t ¼ 13 ½ð x y Þ2 þ ð y z Þ2 þ ð z x Þ2

ð2-45Þ

or t is the abscissa of point T ð2-46aÞ

2 2 2 1=2 þ yz þ zx Þ þ 6ðxy qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 13 ½ð 1 2 Þ2 þ ð 2 3 Þ2 þ ð 3 1 Þ2

or t is the ordinate of point T

FIGURE 2-10 An element in triaxial state of stress.

FIGURE 2-11 Mohr’s circle for triaxial octahedral stress state.

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STATIC STRESSES IN MACHINE ELEMENTS

2.10

CHAPTER TWO

Particular

Formula

STRESS-STRAIN RELATIONS Uniaxial ﬁeld 1 ; E

1 ; E

Strain in principal direction 1

"1 ¼

The principal stress

1 ¼ E"1

The unit volume change in uniaxial stress

V ð1 2Þ 1 ¼ "1 ð1 2Þ ¼ E V

"2 ¼

"3 ¼

1 E

ð2-47Þ ð2-47aÞ ð2-48Þ

Biaxial ﬁeld Strain in principal direction 1

"1 ¼

1 ð 2 Þ E 1

ð2-49Þ

Strain in principal direction 2

"2 ¼

1 ð 1 Þ E 2

ð2-50Þ

Strain in principal direction 3

"3 ¼

The principal stresses in terms of principal strains in a biaxial stress ﬁeld

ð þ 2 Þ E 1

ð2-51Þ

1 ¼

E ð"1 þ "2 Þ 1 2

ð2-52Þ

2 ¼

E ð"2 þ "1 Þ 1 2

ð2-53Þ

3 ¼ 0 The unit volume change in biaxial stress

ð2-53aÞ

V ð1 2Þ þ ð 1 þ 2 Þ V E

ð2-54Þ

Triaxial ﬁeld Strain in principal direction 1

"1 ¼

1 ½ ð 2 þ 3 Þ E 1

ð2-55Þ

Strain in principal direction 2

"2 ¼

1 ½ ð 3 þ 1 Þ E 2

ð2-56Þ

Strain in principal direction 3

"3 ¼

1 ½ ð 1 þ 2 Þ E 3

ð2-57Þ

The principal stresses in terms of principal strains in triaxial stress ﬁeld

1 ¼

E ½ð1 Þ"1 þ ð"2 þ "3 Þ ð1 2 2 Þ

ð2-58Þ

2 ¼

E ½ð1 Þ"2 þ ð"3 þ "1 Þ ð1 2 2 Þ

ð2-59Þ

3 ¼

E ½ð1 Þ"3 þ ð"1 þ "2 Þ ð1 2 2 Þ

ð2-60Þ

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STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS

Particular

The unit volume change or volumetric strain in terms of principal stresses for the general case of triaxial stress (Fig. 2-12)

2.11

Formula

dV ð1 2Þ ¼ ð x þ y þ z Þ V E ð1 2Þ ¼ ð 1 þ 2 þ 3 Þ E

e¼

ð2-61aÞ ð2-61bÞ

FIGURE 2-12 Uniform hydrostatic pressure.

The volumetric strain due to uniform hydrostatic pressure c acting on an element (Fig. 2-12)

V 3ð1 2Þ c ¼ ¼ c V E

The bulk modulus of elasticity

¼

E 3ð1 2Þ

ð2-63Þ

The relationship between E, G and K

E¼

9KG ð3K þ GÞ

ð2-63aÞ

ð2-62Þ

STATISTICALLY INDETERMINATE MEMBERS (Fig. 2-13) The reactions at supports of a constant cross-section bar due to load F acting on it as shown in Fig. 2-13

The elongation of left portion La of the bar

Ra ¼

FLb FLb ¼ La þ Lb L

ð2-64aÞ

RB ¼

FLa FLa ¼ La þ Lb L

ð2-64bÞ

a ¼

RA La FLa Lb ¼ AE LAE

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ð2-65Þ

STATIC STRESSES IN MACHINE ELEMENTS

2.12

CHAPTER TWO

Particular

Formula

RA La FL L ¼ a b AE LAE

The shortening of right portion Lb of the bar

b ¼

FIGURE 2-13

FIGURE 2-14

ð2-66Þ

THERMAL STRESS AND STRAIN The normal strain due to free expansion of a bar or machine member when it is heated

"T ¼ ðTÞ

ð2-67Þ

The free linear deformation due to temperature change

¼ LðTÞ

ð2-68Þ

The compressive force Fcb developed in the bar ﬁxed at both ends due to increase in temperature (Fig. 2-14)

Fcb ¼ AEðTÞ

ð2-69Þ

The compressive stress induced in the member due to thermal expansion (Fig. 2-14)

cT ¼

Fcb ¼ EðTÞ A

ð2-70Þ

The relation between the extension of one member to the compression of another member in case of rigidly joined compound bars of the same length L made of diﬀerent materials subjected to same temperature (Fig. 2-15)

s L c L þ ¼ ðc s ÞLðTÞ Es Ec

ð2-71Þ

The forces acting on each member due to temperature change in the compound bar

c Ac ¼ s As

ð2-72Þ

The relation between compression of the tube to the extension of the threaded member due to tightening of the nut on the threaded member (Fig. 2-16)

t L s L þ ¼ [number of turns ðiÞ Et Es ðthreads/meterÞ or pitch ðPÞ ¼ iP

The forces acting on tube and threaded member due to tightening of the nut

s As ¼ t At

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ð2-73Þ ð2-74Þ

STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS

Particular

FIGURE 2-15

2.13

Formula

FIGURE 2-16

COMPOUND BARS X Ei Ai i X Ei Ai ¼ Li Li

The total load in the case of compound bars or columns or wires consisting of i members, each having diﬀerent length and area of cross section and each made of diﬀerent material subjected to an external load as shown in Fig. 2-17

F¼

An expression for common compression of each bar (Fig. 2-17)

F ¼P ðEi Ai =Li Þ

FIGURE 2-17

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ð2-75Þ

ð2-76Þ

STATIC STRESSES IN MACHINE ELEMENTS

2.14

CHAPTER TWO

Particular

Formula

The load on ﬁrst bar (Fig. 2-17)

ðE A =L Þ F1 ¼ P1 1 1 F ðEA=LÞ

The load on ith bar (Fig. 2-17)

Fi ¼

ð2-77Þ

Ei Ai Li

ð2-78Þ

EQUIVALENT OR COMBINED MODULUS OF ELASTICITY OF COMPOUND BARS The equivalent or combined modulus of elasticity of a compound bar consisting of i members, each having a diﬀerent length and area of cross section and each being made of diﬀerent material

The stress in the equivalent bar due to external load F

E1 A1 þ E2 A2 þ E3 A3 þ þ En An A1 þ A2 þ A3 þ þ An P Ei Ai ¼P i ¼ 1;2;:::;n Ai

Ec ¼

The common extension or compression due to external load F

"¼

¼

Ec

Ec

ð2-79bÞ

F

¼P

i ¼ 1;2;3;:::

The strain in the equivalent bar due to external load F

ð2-79aÞ

P

F

P

FL

ð2-80Þ

Ai

i ¼ 1;2;3;:::

Ai

i ¼ 1;2;3;:::;n

L

ð2-81Þ

¼ "L

ð2-82Þ

¼

Ai

POWER The relation between power, torque and speed

P ¼ Mt !

ð2-83Þ

where Mt in N m (lbf ft), ! in rad/s (rad/min), and P in W (hp) ¼

Mt n0 159

SI

ð2-84aÞ

where Mt in kN m, n0 in rps, and P in kW ¼

Mt n 9550

SI

ð2-84bÞ

where Mt in kN m, n in rpm, and P in kW ¼

Mt n 63030

USCS

where Mt in lbf in, n in rpm, and P in hp

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ð2-84cÞ

STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS

Particular

Another expression for power in terms of force F acting at velocity v

2.15

Formula

P¼

F 1000

SI

ð2-85aÞ

where F in newtons (N), in m/s, and P in kW ¼

F 33000

USCS

ð2-85bÞ

where F in lbf, in fpm (feet per minute), and P in hp (horsepower)

TORSION (FIG. 2-18) The general equation for torsion (Fig. 2-18)

Mt G ¼ ¼ J L

Torque

Mt ¼

ð2-86Þ

159P n0

SI

ð2-87aÞ

where Mt in kN m, n0 in rps, and P in kW ¼

9550P n

SI

ð2-87bÞ

where Mt in kN m, n in rpm, and P in kW ¼

63030P n

USCS

ð2-87cÞ

where Mt in lbf in, n in rpm, and P in hp The maximum shear stress at the maximum radius r of the solid shaft (Fig. 2-18) subjected to torque Mt

max ¼

The torsional spring constant

kt ¼

16Mt

D3

ð2-88Þ

Mt GJ ¼ L

ð2-89Þ

FIGURE 2-18 Cylindrical bar subjected to torque.

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STATIC STRESSES IN MACHINE ELEMENTS

2.16

CHAPTER TWO

Particular

Formula

BENDING (FIG. 2-19) The general formula for bending (Fig. 2-19)

M b b E ¼ ¼ I c

ð2-90Þ

FIGURE 2-19 Bending of beam.

Mb c I

The maximum values of tensile and compressive bending stresses

b ¼

The shear stresses developed in bending of a beam (Fig. 2-20)

¼

V Ib

The shear ﬂow

q¼

VQ I

ð2-91Þ

ðc y0

y dA

FIGURE 2-20 Beam subjected to shear stress.

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ð2-92Þ

ð2-93Þ

STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS

Particular

2.17

Formula

ðc

The ﬁrst moment of the cross-sectional area outside the section at which the shear ﬂow is required

Q¼

The maximum shear stress for a rectangular section (Figs. 2-20 and 2-21)

max ¼

3V 2A

ð2-95Þ

For a solid circular section beam, the maximum shear stress

max ¼

4V 3A

ð2-96Þ

For a hollow circular section beam, the expression for maximum shear stress

max ¼

2V A

ð2-97Þ

An appropriate expression for max for structural beams, columns and joists used in structural industries

max ¼

V Aw

ð2-98Þ

y0

y dA

ð2-94Þ

FIGURE 2-21 Element cut out from a beam subjected to shear stress.

where Aw is the area of the web

ECCENTRIC LOADING The maximum and minimum stresses which are induced at points of outer ﬁbers on either side of a machine member loaded eccentrically (Figs. 2-22 and 2-23)

max ¼

F Mb F M and min ¼ b þ Z Z A A

The resultant stress at any point of the cross section of an eccentrically loaded member (Fig. 2-24)

z ¼

F Mbx ey Mby ex Ixx Iyy A

ð2-99Þ

ð2-100Þ

COLUMN FORMULAS (Fig. 2-25) Euler’s formula (Fig. 2-26) for critical load

Fcr ¼

n 2 EA n 2 EI ¼ l2 ðl=kÞ2

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ð2-101Þ

STATIC STRESSES IN MACHINE ELEMENTS

2.18

CHAPTER TWO

Particular

Formula

FIGURE 2-22 Eccentric loading.

FIGURE 2-23 Eccentrically loaded machine member.

FIGURE 2-24

FIGURE 2-25 Column-end conditions. (i) One end is ﬁxed and other is free. (ii) Both ends are rounded and guided or hinged. (iii) One end is ﬁxed and other is rounded and guided or hinged. (iv) Fixed ends.

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STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS

Particular

Johnson’s parabolic formula (Fig. 2-26) for critical load

2.19

Formula

"

y Fcr ¼ A y 1 4n 2 E

2 # l k

ð2-102Þ

FIGURE 2-26 Variation of critical stress with slenderness ratio.

Straight-line formula for critical load

Straight-line formula for short column of brittle material for critical load Ritter’s formula for induced stress

Ritter’s formula for eccentrically loaded column (Fig. 2-23) for combined induced stress

Rankine’s formula for induced stress

The critical unit load from secant formula for a round-ended column

"

2 y Fcr ¼ A y pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 ð y =3nEÞ

# l k

l Fcr ¼ A C1 k " 2 # F e l c ¼ 1þ 2 A n E k " # 2 F e l ce þ 2 c ¼ 1þ 2 A n E k k " 2 # Fcr l 1þa c ¼ A k Fcr ¼ A

y ec l pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 2 sec ðFcr =4AEÞ k k

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ð2-103Þ

ð2-104Þ

ð2-105Þ

ð2-106Þ

ð2-107Þ

ð2-108Þ

STATIC STRESSES IN MACHINE ELEMENTS

2.20

CHAPTER TWO

Particular

Formula

HERTZ CONTACT STRESS Contact of spherical surfaces Sphere on a sphere (Fig. 2-27a) The radius of circular area of contact

2

31=3 1 12 1 22 þ 6 7 6 E E2 7 7 a ¼ 0:7216F 1 4 5 1 1 þ d1 d2

ð2-109Þ

FIGURE 2-27 Hertz contact stress.

2

The maximum compressive stress

cðmaxÞ

31=3 1 1 2 þ 6 7 d1 d2 7 ¼ 0:9186 7 6F 4 1 12 1 22 2 5 þ E1 E2 2

Combined deformation of both bodies in contact along the axis of load

Spherical surface in contact with a spherical socket (Fig. 2-27b) The radius of circular area of contact

6 6 ¼ 1:046F 2 4

2 6 6 a ¼ 0:7216F 4

1 12 1 22 þ E1 E2 d1 d2 d1 þ d2 1 12 1 22 þ E1 E2 1 1 d1 d2

ð2-110Þ

2 31=3 7 7 7 5

ð2-111Þ

31=3 7 7 7 5

ð2-112Þ

2

The maximum compressive stress

cðmaxÞ

31=3 1 1 2 6 7 d1 d2 7 ¼ 0:9186 6F 7 4 1 12 1 22 2 5 þ E1 E2

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ð2-113Þ

STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS

Particular

Combined deformation of both bodies in contact along axis of load

Distribution of pressure over band of width of contact and stresses in contact zone along the line of symmetry of spheres Sphere on a ﬂat surface (Fig. 2-27c) The radius of circular area of contact

The maximum compressive stress

2.21

Formula

2

6 6 ¼ 1:046F 2 4

1 12 1 22 þ E1 E2 d1 d2 d2 d1

2 31=3 7 7 7 5

ð2-114Þ

Refer to Fig. 2-28a. "

a ¼ 0:721 Fd1

1 12 1 22 þ E1 E2

#1=3 ð2-115Þ

31=3 2 F cðmaxÞ ¼ 0:9186 7 4 2 1 12 1 22 2 5 þ d1 E1 E2

ð2-116Þ

where d ¼ d1 (Fig. 2-27c).

Contact of cylindrical surfaces Cylindrical surface on cylindrical surface, axis parallel (Fig. 2-27a and Fig. 2-28b) The width of band of contact

2

6F 6 2b ¼ 1:66 4L

1 12 1 22 þ E1 E2 1 1 þ d1 d2 2

The maximum compressive stress

Cylindrical surface in contact with a circular groove (Fig. 2-27b) The width of band of contact

cðmaxÞ

31=2 7 7 7 5

31=2 1 1 þ 6F 7 d1 d2 7 ¼ 0:7986 7 6 2 2 4L 1 1 1 2 5 þ E1 E2 2

6F 6 2b ¼ 1:66 4L

1 12 1 22 þ E1 E2 1 1 d1 d2 2

cðmaxÞ

Distribution of pressure over band of width of contact and stresses in contact zone along the line of symmetry of cylinders

Refer to Fig. 2-28b.

ð2-118Þ

31=2 7 7 7 5

31=2 1 1 6F 7 d1 d2 7 ¼ 0:7986 7 6 2 2 4L 1 1 1 2 5 þ E1 E2

The maximum compressive stress

ð2-117Þ

ð2-119Þ

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ð2-120Þ

STATIC STRESSES IN MACHINE ELEMENTS

2.22

CHAPTER TWO

Particular

Formula

Cylindrical surface in contact with a ﬂat surface (Fig. 2-27c): The width of band of contact

" Fd1 2b ¼ 1:6 L

1 12 1 22 þ E1 E2

#1=2 ð2-121Þ

31=2 F 1 ¼ 0:7986 ð2-122Þ 7 4Ld1 1 12 1 22 5 þ E1 E2 2

The maximum compressive stress

cðmaxÞ

where d ¼ d1 (Fig. 2-27c). Deformation of cylinder between two plates

d1 ¼

4F L

1 12

E

1 2d þ loge 1 3 b

ð2-123Þ

The maximum shear stress occurs below contact surface for ductile materials For sphere

max ¼ 0:31 cðmaxÞ

ð2-123aÞ

For cylinders

max ¼ 0:295 cðmaxÞ

ð2-123bÞ

The depth from contact surface to the point of the maximum shear

h ¼ 0:786b

ð2-123cÞ

FIGURE 2-28 Distribution of pressure over bandwidth of contact and stresses in contact zone along line of symmetry of spheres and cylinders for ¼ 0:3.

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STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS

Particular

2.23

Formula

DESIGN OF MACHINE ELEMENTS AND STRUCTURES MADE OF COMPOSITE Honeycomb composite For the components of composite materials which give high strength–weight ratio combined with rigidity

Refer to Fig. 2-29.

For sandwich construction of honeycomb structure

Refer to Fig. 2-30.

FIGURE 2-29 Sandwich fabricated panel.

FIGURE 2-30 Honeycomb.

The moment of inertia of sandwich panel, Fig 2-30 Simpliﬁed Eq. (2-124) after neglecting powers of h

The ﬂexural rigidity

3 Bh Hc þ h 2 þ 2Bh I ¼2 2 12 H I ¼ BhHc h þ c 2 D ¼ EI

ð2-124Þ ð2-125Þ ð2-126Þ

where E ¼ modulus of elasticity of the facing metal I is given by Eq. (2-125). D¼

EðH 3 Hc3 Þ 12ð1 2 Þ

ð2-127Þ

The ﬂexural rigidity of sandwich construction for ðHc =hÞ > 5

D¼

EhðH þ Hc Þ2 8ð1 2 Þ

ð2-128Þ

The shear modulus of the core material as per Jones and Hersch

Gcore ¼

The ﬂexural rigidity of sandwich plate/panel

1:5FLc BðH þ Hc Þ2 ð114 82 Þ

ð2-129Þ

where 4 and 2 ¼ deﬂection at quarter-span and midspan respectively F ¼ force over a support span Lc

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STATIC STRESSES IN MACHINE ELEMENTS

2.24

CHAPTER TWO

Particular

Formula

The shear modulus G of isotropic material if the modulus of elasticity E is available

G¼

The modulus of elasticity of the core material (Fig. 2-31)

Ef ¼ Em

E 2ð1 Þ

ð2-130Þ

1 V 2=3 1 V 2=3 þ V

ð2-131Þ

where V ¼ ðHh =HÞ3 , Ef ¼ modulus of elasticity of foam, GPa (psi), Em ¼ modulus of elasticity of basic solid material, GPa (psi). Subscript f stands for foam/ﬁlament, m stands for matrix, and c stands for composite.

FIGURE 2-31 A unit cube foam subject to a tensile load.

The deﬂection for a beam panel according to Castigliano’s theorem

¼

FIGURE 2-32 Phantom load.

FIGURE 2-33

The deﬂection at midspan (Fig. 2-32)

L=2

@U @ ¼ @F @F

ð

@U @ ¼ ¼ @W @W

Mb2 dx þ 2EI

ð

ð

V 2 dx 2GA

Mb2 dx þ 2EI

ð

V 2 dx 2GA

ð2-132Þ

W ¼0

ð2-133aÞ 5FL3 FL þ ð2-133bÞ 349EI 8GA where W is the phantom load (Fig. 2-32). ¼

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STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS

Particular

The deﬂection per unit width for a sandwich panel at midspan (Fig. 2-32) under quarter-point loading

2.25

Formula

5FL3 FL þ 349DB 8Dc B HðH þ Hc Þ where Dc ¼ Gcore 2Hc

L=2 ¼

ð2-134Þ

The deﬂection per unit width for a sandwich panel at quarter panel (Fig. 2-32) under quarter-point loading

L=4 ¼

FL3 FL þ 96DB 8Dc B

ð2-135Þ

The deﬂection/unit width for a sandwich panel at center loading (Fig. 2-33)

L=2 ¼

FL3 FL þ 48DB 4Dc B

ð2-136Þ

The maximum normal stress (Fig. 2-32)

max

F L M FL 2 4 ¼ ¼ ¼ BhHc ðh þ Hc =2Þ Z 8BhHc H=2 ð2-137Þ FL 8BhH

ð2-138Þ

The minimum normal stress

min ¼

The average stress often used in the composite panel design

av ¼

The maximum shear stress in the core

max ¼

V 2V ¼ ½BðH þ Hc Þ=2 BðH þ Hc Þ

ð2-140Þ

The core shear strain

core ¼

max Gcore

ð2-141Þ

FLðL þ Hc Þ FL 16BhHc H 4BhðH þ Hc Þ

ð2-139Þ

FILAMENT REINFORCED STRUCTURES (Fig. 2-34) The strain in the ﬁlament is same as the strain in the matrix of composite material if it has to have strain compatibility

"m ¼ "f

FIGURE 2-34

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ð2-142Þ

STATIC STRESSES IN MACHINE ELEMENTS

2.26

CHAPTER TWO

Particular

Formula

The relation between stress in matrix and stress in ﬁlament

m f ¼ Em Ef

ð2-143Þ

For equilibrium

F ¼ m Am þ f Af ¼ c Ac

ð2-144Þ

f ¼

FEf Af Ef þ Am Em

ð2-145Þ

m ¼

FEm Af Ef þ Am Em

ð2-146Þ

Ec ¼

E f Af ðAm þ Af Þ

ð2-147Þ

The stress in the ﬁlament

The stress in the matrix

The Young’s modulus of composite

The Young’s modulus of chopped-up glass ﬁlaments in resin matrix but still oriented longitudinally with respect to load as proposed by Outerwater

"

ðEc Þchpd-f

Af ¼ Ef Ac

1 4

yf

D2f Lpc

# ð2-148Þ

where ¼ applied tensile stress, MPa (psi) yf ¼ the strength of the ﬁber, MPa (psi) Df ¼ diameter of ﬁber, mm (in) pc ¼ uniform distance of one ﬁber from another on circumference, mm (in) L ¼ length of ﬁber, mm (in) Subscript chpd-f stands for chopped-up ﬁber. The relation between m and f , which has to satisfy Eq. (2-142) at any location on the curves, Fig. 2-35

f m ¼ ðE0 Þm ðE0 Þf

ð2-149Þ

where E0 ¼ secant modulus, GPa (Mpsi) From Eq. (2-144), the expression for c

For structure with all ﬁlament, Am ¼ 0

For structure with no ﬁlament, Af ¼ 0

c ¼

f Ac

¼

ð u Þf Ac

ðE0 Þm Am þ Af ðE0 Þf

ð m Þmax Am þ Af ð u Þf

c ¼

ð u Þf Af ¼ ð u Þf Ac

c ¼

ð u Þf Ac

ð m Þmax Am u

ð2-150Þ ð2-151Þ

ð2-152Þ ¼ ð m Þmax ¼ ð u Þm ð2-153Þ

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STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS

2.27

Particular

Formula

FIGURE 2-35 Stress-strain data for system shown in Fig. 2-34.

FIGURE 2-36 Filament wound cylindrical pressure vessel.

FILAMENT BINDER COMPOSITE (Fig. 2-36) Hoop stress for a closed end vessel/cylinder made of ﬁlaments winding

¼

pd 2h

ð2-154Þ

Longitudinal/axial stress for a closed end ﬁlament wound vessel/cylinder

a ¼

pd 4h

ð2-155Þ

The force carried by a helical ﬁlament wound on a shell of width w subjected to internal pressure p in the -direction

Fa ¼ so wh

The force in helical ﬁlament wound on a shell of width w subjected to internal pressure p in the hoop direction

F ¼ F sin

The hoop stress in the vessel wall due to the pressure p

¼

The stress in the vessel wall in the longitudinal/axialdirection

a ¼ 0 cos2

From Eq. (2-154) to (2-159) the optimum winding angle for closed end cylinders

tan2 ¼

The optimum winding angle for open end cylinders

a cos2 ¼ cot2 ¼ ¼ sin2

ð2-156Þ

where so ¼ strength of the ﬁlaments ð2-157Þ

F ¼ 0 sin2 A

a

ð2-158Þ ð2-159Þ

or 558

ð2-160Þ

or ¼ 908

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ð2-161Þ

STATIC STRESSES IN MACHINE ELEMENTS

2.28

CHAPTER TWO

Particular

The stress in the hoop/circumferential direction for the ﬁlament wound cylinder/vessel consisting windings in longitudinal, hoop and helical directions to satisfy equilibrium condition

Formula

¼

0 h þ h h

ð2-162Þ

where 0 ¼ stress in the circumferential wound layer ¼ circumferential component of stress in the helical layer ht ¼ total thickness ¼ ha þ h þ h ha , h and h are the thicknesses in the preceding layers of ﬁlament windings

The longitudinal stress for the case of winding under Eq. (2-162)

a ¼

0a h0a þ a h h

ð2-163aÞ

so ðh þ h sin2 Þ h so a ¼ ðha þ h cos2 Þ h

where ¼

ð2-163bÞ ð2-163cÞ

so ¼ uniform filament stress a ¼ longitudinal component of stress in helical layer From Eqs. (2-159) and (2-158)

þ a ¼ 0 ðsin2 þ cos2 Þ ¼ 0

From Eqs. (2-154) and (2-155)

h¼

ð2-164Þ

pd 4 a

ð2-165Þ

pd 2h

ð2-154Þ

¼ The sum of stresses and a

þ a ¼ 3 a ¼ 0

For the ideal vessel

h¼

or

a ¼

0 3

3pd 4 0

ha ¼

h h cos2 3

ð2-166Þ ð2-167Þ

ð2-168aÞ

h ¼ 23 h sin2

ð2-168bÞ

2ha h 1 3 cos2

ð2-168cÞ

h ¼

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STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS

Particular

The structural eﬃciency of the wound vessel/cylinder

2.29

Formula

¼

W Ven pi

ð2-169Þ

where W ¼ weight of the vessel, kN (lbf) Ven ¼ enclosed volume, m3 (in3 ) pi ¼ internal pressure, MPa (psi)

FILAMENT-OVERLAY COMPOSITE T uw where T ¼ tension, kN (lbf) uw ¼ area of the element, m2 (in2 )

The stress in the wire which is wound on thin walled shell/cylinder with a wire of the same material (Fig. 2-37)

wr ¼

Under equilibrium condition over the length of shell L, the hoop stress

ð Þsh ¼

T wh

ð2-170Þ

ð2-171Þ

FIGURE 2-37 Shell subjected to an internal pressure.

The tension in the wound wire on the shell under internal pressure

Twr ¼

pd T þ 2ðh þ uÞ wu

ð2-172Þ

The tension in the shell under the above same condition

Tcy ¼

pd T 2ðh þ uÞ wh

ð2-173Þ

The yielding of shell due to internal pressure, i.e., due to plastic ﬂow of material of the shell.

ð 0 Þshy ¼

ð2-174Þ

For the above same winding material under the tension equal to compression yield limits, the stress in the wire.

wr

T ¼ y wh T h ¼ ¼ y uw u

ð2-175Þ

If the vessel material is diﬀerent from the winding material then stress in the wire and vessel

cy ¼ "sh Esh

ð2-176aÞ

wr ¼ "wr Ewr

ð2-176bÞ

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STATIC STRESSES IN MACHINE ELEMENTS

2.30

CHAPTER TWO

Particular

Formula

For uniform distribution of stress in the cylinder/shell and in the wire, strains are proportional to the mean radii

rcy "cy cy Ewr ¼ ¼ rwr "wr wr Ecy

From Eq. (2-177), the stress in the cylinder and the wire

sh ¼ cy ¼ wr

wr ¼ cy

Ecy rcy Ewr rwr

Ewr rwr Ecy rcy

ð2-177Þ

ð2-178aÞ

ð2-178bÞ

where subscripts cy stands for cylinder, sh for shell and wr for winding. rcy and rwr are mean radii of cylinder and winding respectively. The total load on the cylinder and the winding

cy ð2LhÞ þ wr ð2LuÞ ¼ pdL

ð2-179Þ

From Eq. (2-179), the stress in the cylinder ( cy ) and the winding ( wr )

pd cy ¼ Ewr rwr u 2 þh Ecy rcy

ð2-180aÞ

pd wr ¼ Ecy rcy þu 2 Ewr rwr

ð2-180bÞ

The stress in the cylinder is the sum of results of Eqs. (2-180a) and (2-171)

pd T Rcy ¼ Ewr rwr u wh 2 þh Ecy rcy

ð2-181Þ

The resultant stress in the winding is the sum of results of Eq. (2-180b) and (2-170)

pd T þ Rwr ¼ Ecy rcy h wu þu 2 Ewr rwr

ð2-182Þ

For advanced theory using Theory of elasticity and Plasticity construction on composite structures and materials

Refer to advanced books and handbooks on composites, structures, handbooks and design data for reinforced plastics and materials.

For representative properties for ﬁber reinforcement

Refer to Table 2-1

FORMULAS AND DATA FOR VARIOUS CROSS SECTIONS OF MACHINE ELEMENTS For further data on static stresses, properties and torsion of shafts of various cross-sections: shear, moments, and deﬂections of beams, strain rosettes, and singularity functions

Refer to Tables 2-2 to 2-12

For summary of stress and strain formulas under various types of loads

Refer to Table 2-13

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0.4 0.4 0.4 4

0.2–0.4 5 4

0.5 0.001–0.01 0.2 0.2–0.5 0.8

5–10 127 102

13 0.025–0.25 5 5–13 20.5

103 in

10.2 10.2 10.2 102.0

103 mm

7.64 2.43 2.62 1.11 1.49

1.43–1.75 1.78 3.40

2.48 2.43 2.46 2.56

g/cm3 3100 4498 5510 2756

MPa 450 650 800 400

kpsi

Tensile strength, st

0.283 0.090 0.097 0.041 0.052

2852–4184 689–2067 1378–2067 827 690

385–600 100–300 200–300 120 100

0.053–0.066 1723–3445 250–500 0.066 1240 180 0.126 2480 360

0.092 0.090 0.091 0.095

lb/in3

Density,

200 172 138–413 2.8 4

241–689 310 414

72.5 85 100 415

GPa

29 25 20–60 0.4 0.6

35–100 45 60

10.5 12.3 14.5 60

Mpsi

Modulis of elasticity, Eg

30 3.7 45–50 45–50

54 6.5 81–90 81–90

1.5 6.4 2.2

2.8

5.0 2.7 11.5 4.0

2.8

lin/in8F

5.0

lm/m K

Coeﬃcient of thermal expansion,

381 381

22400–38080

13440 19488 6496

1680

W/(m2 K/m)

1.7 1.7

100–170

60 87 29

7.5

Btu/(ft2 h8F)/in)

Thermal conductivity, K

Courtesy: J. E. Ashton, J. C. Halpin, and P. H. Petit, Primer on Composite Materials: Analysis, Technomic Publishing Co., Inc., 750 Summer St., Stanford, Conn. 06901, 1969.

E glass S glass 970 S glass Boron on tungsten Graphite Beryllium Silicon carbide on tungsten Stainless steel Asbestos Aluminum Polyamide Polyester

Fiber

Typical ﬁber diameter

TABLE 2-1 Representative properties for ﬁber reinforcement STATIC STRESSES IN MACHINE ELEMENTS

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2.31

STATIC STRESSES IN MACHINE ELEMENTS

2.32

CHAPTER TWO

TABLE 2-2 Torsion of shafts of various cross sections Polar section modulus, Polar radius of gyration, k0 Cross section Z0 ¼ J=c

D pﬃﬃﬃ ¼ 0:354D 8

D3 16

Angular deﬂection, In terms of torsional moment, Mt In terms of maximum stress, 2l D G

32l Mt

D4 D

at circumference

ðD41 D42 Þ 16D1

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D21 þ D22 ¼ 0:354 D21 þ D22 8

B h

A b

b2 h 16

a 1 4

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ b2 þ h2

2l D1 G

32l Mt

ðD41 D42 Þ G

at outer circumference

16ðb2 þ h2 Þl Mt G

b3 h3

ðb2 þ h2 Þl G bh2

h>b

2b2 h 9

at A b

a

h>b

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ b2 þ h2 12 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:289 b2 þ h2

mðb2 þ h2 Þl Mt G b3 h3 h ¼1 b

nðb2 þ h2 Þl G bh2 at A c

2

4

8

m ¼ 3:56 3:50 3:35 3:21 n ¼ 0:79 0:78 0:74 0:71 b3 20

46:2l Mt b4 G

a

0:289b

2:31l b G at center of side

0:92b3 a

0:645b

0:967l Mt G b4

0:9l b G at center of side

a This value is not true value of Z0 but is the value of Z0 for a circular section of equal strength and may be used for determining the maximum stress by the formula ¼ Mt =Z0 . b At B, shear stress ¼ 10Mt = bh2 . c At B, shear stress ¼ 9Mt =2bh2 . Source: V. L. Maleev and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954.

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STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS

2.33

TABLE 2-3 Shear stress in beams, caused by bending

Section

Shear stress at a distance y from neutral axis, , MPa (psi)

Maximum shear stress, max , MPa (psi)

2 3F 2y 1 2bh h

3F F ¼ 1:5 ðfor y ¼ 0Þ 2bh A

2 4F y 1 r 3 r2

4F F ¼ 1:33 ðfor y ¼ 0Þ A 3 r2

pﬃﬃﬃ pﬃﬃﬃ 2 F 2 y 2 y 4 1þ 2 b b b

1:591

F A

c for y ¼ 4

3F bc2 ðb aÞd 2 ðfor y ¼ 0Þ 3 3 4a bc ðb aÞd

TABLE 2-4 The values of constants a in Eq. (2-107) Yield stress in compression, yc

Value of a for various end-ﬁxity coeﬃcients 1

4

2

n

7

1 750

1 3000

1 1500

1 n 750

549

80

1 1600

1 6400

1 3200

1 n 1600

324

47

1 7500

1 30000

1 15000

1 n 7500

Material

MPa

kpsi

Timber

49

Cast iron Mild steel

TABLE 2-5 End condition coeﬃcient n (Fig. 2-25) Particular

n

TABLE 2-6 End-ﬁxity coeﬃcients for cast iron column to be used in Eq. (2-104)

One end ﬁxed and the other end free Both ends rounded and guided or hinged One end ﬁxed, and the other end rounded and guided or hinged Both ends ﬁxed rigidly Both ends ﬂat

0.25 1 2

End conditions

C1

Maximum, l=k

Round Fixed One ﬁxed, one round

175 88 116

90 160 115

4 1 to 4

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STATIC STRESSES IN MACHINE ELEMENTS

2.34

CHAPTER TWO

TABLE 2-7 Properties of cross sections

Section

Area, A

Moment of inertia, I

Distance to farthest point, c

bh

bh3 12

h 2

bh2 6

0:289h

ðH cÞb

b ðH 3 h3 Þ 12

H 2

bðH 3 h3 Þ 3H

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ H 3 h3 12ðH hÞ

BH bh

BH 3 bh3 12

H 2

BH 3 bh3 6H

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ BH 3 bh3 12ðBH bhÞ

ð6b2 þ 6bb0 þ b20 Þh3 36ð2b þ b0 Þ

ð3b þ 2b0 Þh 3ð2b þ b0 Þ

ð6b2 þ 6bb0 þ b20 Þh2 12ð3b þ b0 Þ

rﬃﬃﬃﬃ I A

D4 64

D 2

D3 32

D 4

2b þ b0 h 2

D2 4

2 ðD D22 Þ 4 1

Section modulus, Z ¼ I=c

Radius of gyration, pﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ k ¼ I=A

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D21 þ D22

ðD4 D42 Þ 64 1

¼ ðR41 R42 Þ 4

D1 ¼ R1 2

ab

ba3 64

a 2

ba2 32

a 4

bh 2

bh3 36

bh2 24

0:236h

ðD41 D42 Þ 32D1

4 ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ R21 þ R22

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2

B to C : Mb ¼ Fðx bÞ

Max MbC ¼ Fa at C

A to B : V ¼ 0

B to C : V ¼ F

4. End supports, center load

3. Cantilever, uniform load

B to C : Mb ¼ þ 12 Fðl xÞ Max MbB ¼ þ14 FL at B

A to B : V ¼ þ 12 F B to C : V ¼ 12 F

Max MbB ¼ 12Wl at B

A to B : Mb ¼ þ 12 Fx

W x l

R1 ¼ þ12 F; R2 ¼ þ 12 F

V¼

1 W 2 x 2 l

A to B : Mb ¼ 0

R2 ¼ þF

Mb ¼

Max MbB ¼ Fl at B

V ¼ F

R2 ¼ þW ¼ wl

Mb ¼ Fx

R2 ¼ þF

1. Cantilever, end load

2. Cantilever, intermediate load

Bending moment Mb , and maximum bending moment

Reactions R1 and R2 , vertical shear V

Loading, support, and reference number

TABLE 2-8 Shear, moment, and deﬂection formulas for beams

ymax ¼

1 F ð3l 2 x 4x3 Þ 48 EI 1 Fl 3 at B 48 EI

A to B: y ¼

1 Wl 3 8 EI

1 W 4 ðx 4l 3 x þ 3l 4 Þ 24 EIl ymax ¼

y¼

1 F ð3a2 l a3 Þ 6 EI

1 F ½ðx bÞ3 3a2 ðx bÞ þ 2a3 6 EI B to C: y ¼

ymax ¼

1 F ða3 þ 3a2 l 3a2 xÞ 6 EI A to B: y ¼

1 Fl 3 at A 3 EI

1 F 3 ðx 3l 2 x þ 2l 3 Þ 6 EI

ymax ¼

y¼

Deﬂection y and maximum deﬂection

STATIC STRESSES IN MACHINE ELEMENTS

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2.35

A O

2.36

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I

W = wl

B M2

8. One end ﬁxed, one end supported, uniform load

7. One end ﬁxed, one end supported, center load

6. End supports, uniform load

5. End supports, intermediate load

Loading, support, and reference number b a ; R2 ¼ þF l l

ab at B l

a ðl xÞ l

b x l

Fbx ½2lðl xÞ b2 ðl xÞ2 6EIl

1 Wx 3 ðl 2lx2 þ x3 Þ 24 EIl

3 Max MbC ¼ 16 Fl at C

B to C: V ¼ 11 16 F

V¼W

3 x 8 l

M2 ¼ 18 Wl

R1 ¼ 38 W; R2 ¼ 58 W

Max MbB ¼ 18 Wl at B

9 Max þMb ¼ 128 Wl at x ¼ 38 l

Mb ¼ Wð38 x 12 x2 Þ

Wl 3 at x ¼ 0:4215l El

1 W ð3lx3 2x4 l 3 xÞ 48 EIl ymax ¼ 0:0054

y¼

ymax ¼ 0:00932

5 Max þMbB ¼ 32 Fl at B

5 F A to B: V ¼ þ 16

Fl 3 at x ¼ 0:4472l EI

1 F ½5x3 16ðx 12 lÞ3 3l 2 x 96 EI

3 M2 ¼ 16 Fl

B to C: y ¼

5 A to B: Mb ¼ 16 Fx

5 F; R2 ¼ 11 R1 ¼ 16 16 F

B to C: Mb ¼ Fð12 l 11 16 xÞ

5 Wl 3 Max y ¼ at x ¼ 12 l 384 EI

y¼

Faðl xÞ ½2lb b2 ðl xÞ2 B to C: y ¼ 6EIl pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Fab ymax ¼ ða þ 2bÞ 3aða þ 2bÞ at 27EIl qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x ¼ 13aða þ 2bÞ when a > b

A to B: y ¼

Deﬂection y and maximum deﬂection

1 F ð5x3 3l 2 xÞ 96 EI

Max Mb ¼ þ 18 Wl at x ¼ 12 l

x2 Mb ¼ 12 W x l

Max MbB ¼ þF

B to C: Mb ¼ þF

A to B: Mb ¼ þF

Bending moment Mb , and maximum bending moment

A to B: y ¼

2x V ¼ 12 W 1 l

R2 ¼ þ 12 W; R2 ¼ þ 12 W

A to B: V ¼ þF

b l a B to C: V ¼ F l

R1 ¼ þF

Reactions R1 and R2 , vertical shear V

TABLE 2-8 Shear, moment, and deﬂection formulas for beams (Cont.)

STATIC STRESSES IN MACHINE ELEMENTS

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1 Max þMb ¼ 24 Wl at x ¼ 12 l

1 1 M1 ¼ 12 Wl; M2 ¼ 12 Wl

2x V ¼ 12 W 1 l

x2 1 Mb ¼ 12 W x l 6 l

R1 ¼ 12 W; R2 ¼ 12 W

Max Mb ¼ M2 when a > b

B to C: V ¼ R1 F ¼ R2

ab2 þ R1 a at B l2

Max Mb ¼ M1 when a < b

Max þMb ¼ F

ab2 þ R1 x Fðx aÞ l2

ab2 þ R1 x l2

A to B: V ¼ R1

ab2 a2 b ; M2 ¼ F 2 l2 l

B to C: Mb ¼ F

Fa2 ð3b þ aÞ l3

R2 ¼

M1 ¼ F

A to B: Mb ¼ F

Fb2 ð3a þ bÞ l3

R1 ¼

Source: J. E. Shigley, Mechanical Engineering Design, 3rd. ed., McGraw-Hill Book Company, New York, 1977.

11. Both ends ﬁxed, uniform load

10. Both ends ﬁxed, intermediate load

Max þMbB ¼ 18 Fl at B

A to B: V ¼ þ 12 F Max MbA;C ¼ 18 Fl at A and C

B to C: Mb ¼ 18 Fð3l 4xÞ

M1 ¼ 18 Fl; M2 ¼ 18 Fl

B to C: V ¼ 12 F

A to B: Mb ¼ 18 Fð4x lÞ

R1 ¼ 12 F; R2 ¼ 12 F

9. Both ends ﬁxed, center load

Bending moment Mb , and maximum bending moment

Reactions R1 and R2 , vertical shear V

Loading, support, and reference number

TABLE 2-8 Shear, moment, and deﬂection formulas for beams (Cont.)

2 F a2 b3 2bl if a < b at x ¼ l 3 EI ð3b þ aÞ2 ð3b þ aÞ

1 Wl 3 at x ¼ 12 l 384 EI

1 Wx2 ð2lx l 2 x2 Þ 24 EIl ymax ¼

y¼

ymax ¼

1 Fa2 ðl xÞ2 ½ð3b þ aÞðl xÞ 3bl 6 EIl 3

1 Fb2 x2 ð3ax þ bx 3alÞ 6 EIl 3

2 F a3 b2 2al if a > b at x ¼ 3 EI ð3a þ bÞ2 ð3a þ bÞ

B to C: y ¼

ymax ¼

1 F ð3lx2 4x3 Þ 48 EI

1 Fl 3 at B 192 EI

A to B: y ¼

ymax ¼

A to B: y ¼

Deﬂection y and maximum deﬂection

STATIC STRESSES IN MACHINE ELEMENTS

2.37

STATIC STRESSES IN MACHINE ELEMENTS

2.38

CHAPTER TWO

TABLE 2-9 Some equations for use with the Castigliano method Type of load

General energy equation U¼

Axial

U¼

Bending Combined axial and bending

U¼

Torsion

U¼

Transverse shear

U¼

Transverse shear (rectangular section)

U¼

Open-coiled helical spring subjected to axial load F

U¼

ðl 0

ðl 0

ðl 0

ðl 0

ðl 0

ðl 0

ðl 0

Energy equation

2

2

2

General deﬂection equation ðl

F ds 2AE

U¼

F l Al ¼ 2AE 2E

¼

Mb2 l ds 2EI

U¼

Mb2 l 2EI

¼

U¼

F 2 l Mb2 l þ 2EA 2EI

Sum of axial and bending load

Mt2 ds 2GJ

U¼

Mt2 l 2GJ

¼

V 2 ds 2GA

U¼

V2l 2 ¼ Al 2GA 2G

¼

3V 2 ds 5GA

U¼

3V 2 l 5GA

¼

F2 ds þ 2AE

Mt2 ds þ 2GJ

ðl 0

ðl 0

Mb2 ds 2EI

Mb2 ds 2EI

Mt2 l Mb2 l þ 2GJ 2EI LFR2 cos2 sin2 ¼ þ GJ EI 2

U¼

0

ðl 0

ðl 0

ðl 0

ðl 0

Fð@F=@QÞ ds AE Mb ð@Mb =@QÞ ds EI

Mt ð@Mt =@QÞ ds GJ Vð@V=@GÞ ds GA 6Vð@V=@QÞ ds 5GA

cos2 sin2 ¼ 2 iFR3 sec þ GJ EI D ¼ mean radius of coil 2 ¼ helix angle of spring i ¼ number of coils or turns

where R ¼

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STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS

2.39

TABLE 2-10 Mechanical and physical constants of some materials1;2 Modulus of elasticity, E

Modulus of rigidity, G

Material

GPa

Mpsi

GPa

Mpsi

Aluminum Aluminum cast Aluminum (all alloys) Beryllium copper Carbon steel Cast iron, gray Malleable cast iron Inconel Magnesium alloy Molybdenum Monet metal Nickel-silver Nickel alloy Nickel steel Phosphor bronze Steel (18-8), stainless Titanium (pure) Titanium alloy Brass Bronze Bronze cast Copper Tungsten Douglas ﬁr Glass Lead Concrete (compression) Wrought iron Zinc alloy

69 70 72 124 206 100 170 214 45 331 179 127 207 207 111 190 103 114 106 96 80 121 345 11 46 36 14–28 190 83

10.0 10.15 10.4 18.0 30.0 14.5 24.6 31.0 6.5 48.0 26.0 18.5 30 30.0 16.0 27.5 15.0 16.5 15.5 14.0 11.6 17.5 50.0 1.6 6.7 5.3 2.0–4.0 27.5 12

26 30 27 48 79 41 90 76 16 117 65 48 79 79 41 73

3.8 4.35 3.9 7.0 11.5 6.0 13.0 11.0 2.4 17.0 9.5 7.0 11.5 11.5 6.0 10.6

43 40 38 35 46 138 4 19 13

6.2 5.8 5.5 5.0 6.6 20.0 0.6 2.7 1.9

70 31

10.2 4.5

Poisson’s ratio,

Density, a, Mg/m3

0.334

2.69

0.320 0.285 0.292 0.211

2.80 8.22 7.81 7.20

0.290 0.350 0.307 0.320 0.332 0.30 0.291 0.349 0.305

8.42 1.80 10.19 8.83 8.75 8.3 7.75 8.17 7.75 4.47 6.6 8.55 8.30

0.33 0.324 0.349 0.326 0.330 0.245 0.431

0.33

8.90 18.82 4.43 2.60 11.38 2.35 6.6

Unit weight, b kfg/m3

kN/m3

lbf/in3

lbf/ft3

2,685 2,650 2,713 8,221 7,806 7,197 7,200 8,418 1,799 10,186 8,830 8,747

26.3 26.0 27.0 80.6 76.6 70.6

0.097 0.096 0.10 0.297 0.282 0.260

167 166 173 513 487 450

83.3 17.6 100.0 86.6 85.80

7,751 8,166 7,750 4,470

76.0 80.1 76.0 43.9

0.307 0.065 0.368 0.319 0.316 0.300 0.280 0.295 0.280 0.16

530 117 636 551 546 518 484 510 484 279

8,553 8,304 8,200 8,913 18,822 443 2,602 11,377 2,353 7,700

83.9 81.4

0.309

534

87,4 184.6 4.3 25.5 111.6 23.1

0.322

556

0.016 0.094 0.411

28 162 710 147

0.24

415

¼ mass density: ¼ weight density; w is also the symbol used for unit weight of materials. Sources: K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, and K. Lingaiah, Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India. 1986. a

b

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E ð" "2 Þ 2ð1 þ Þ 1

0

Maximum shearing stress, max

Angle from gauge 1 axis to maximum normal stress angle, ’p

1 2"2 ð"1 þ "3 Þ tan1 2 "1 "3

E 2ð1 þ Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð"1 "3 Þ2 þ ½2"2 ð"1 þ "3 Þ2

"1 þ "3 1 1þ 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð"1 "3 Þ2 þ ½2"2 ð"1 þ "3 Þ2

"1 þ "3 1 þ 1þ 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð"1 "3 Þ2 þ ½2"2 ð"1 þ "3 Þ2

2

E 1þ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ " þ "2 þ "3 2 "2 "3 2 pﬃﬃﬃ "1 1 þ 3 3

"1 þ "2 þ "3 1 1þ 3ð1 Þ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ "1 þ "2 þ "3 2 "2 "3 2 pﬃﬃﬃ þ "1 3 3

"1 þ "2 þ "3 1 þ 1þ 3ð1 Þ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ " þ "2 þ "3 2 "2 "3 2 pﬃﬃﬃ þ "1 1 3 3

3 1 p ﬃﬃ ﬃ " Þ ð" 2 3 6 7 1 3 7 tan1 6 4 "1 þ "2 þ "3 5 2 "1 3

E

E

Rosette type

Poisson’s ratio. The author has used as symbol for Poisson’s ratio. Source: Perry, C. C., and H. R. Lissner, The Strain Gage Primer, 2nd ed., McGraw-Hill Publishing Company, New York, p. 147, 1962.

E 2

E ð"2 þ "1 Þ 1 2

Minimum normal stress, min

E 2

E ð"1 þ "2 Þ 1 2

Maximum normal stress, max

Required solutions

TABLE 2-11 Relations between strain rosette readings and principal stresses

E 2ð1 þ Þ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 ð"1 "4 Þ2 þ ð"2 "3 Þ2 3 1 2ð" "3 Þ tan1 pﬃﬃﬃ 2 2 3ð"1 "4 Þ

E "1 þ "4 1 2 1 1þ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 ð"1 "4 Þ2 þ ð"2 "3 Þ2 3

E "1 þ "4 1 þ 2 1 1þ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 ð"1 "4 Þ2 þ ð"2 "3 Þ2 3

STATIC STRESSES IN MACHINE ELEMENTS

2.40

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STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS

Table 2-12 Singularity functions Graph of fn ðxÞ

Function Concentrated moment

Meaning hx ai2 ¼ ðx 1

Concentrated force

1

Unit step

Parabolic

x¼a x 6¼ a

1

x¼a

0

x 6¼ a

hx ai1 dx ¼ hx ai0

1

Rump

hx ai0 ¼ ðx

1 0

hx ai2 dx ¼ hx ai1

hx ai1 ¼ ðx

0

x
1

xa

hx ai dx ¼ hx ai1 0

0 x
hx ai2 ¼ ðx

0

x
ðx aÞ2

xa

hx ai3 hx ai2 dx ¼ 3 1

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2.41

Type of loads

Axial load

Bending load

Bending and axial load

Torsion

Figure showing loads

1

2

3

4

sn

Figure showing stress

0

þ

R ¼

b ¼

F A

x

F A

Mb l

Mb c I

0

0

0

0

y

0

0

0

0

z

Applied stresses

Mt r Mt ¼ J Zp

0

0

0

bx ; E

; E

¼

r ¼ G L

"z ¼ "3 ¼ v"1

"y ¼ "2 ¼ v"1 ;

"x ¼ "1 ¼

"z ¼ "3 ¼ "1

"y ¼ "2 ¼ "1 ;

"x ¼ "1 ¼

"z ¼ "3 ¼ "1 ¼ "z ;

"y ¼ "2 ¼ "1 ¼ "x ;

bx

bx

bx 2

bx 2

x max ¼ 2 2

"x ¼ "1 ¼ ¼ x E E

x ¼ max

max

Strain equations/Area /Approach distance max

Maximum stress produced

1t ¼ 1c ¼ at 458 to the shaft axis

1 ¼ E"1 ; 2 ¼ 0; 3 ¼ 0

Principal stresses

Symbols: a ¼ major semi-axis of ellipse of area of contact, mm (in), and also radius of band of contact in case of spheres, mm (in); b ¼ minor semi-axis of ellipse of area of contact, mm (in), and also half-bandwidth of rectangle contact between cylinders with parallel axis, mm (in); d1 , d2 ¼ diameters of small and large spheres respectively, mm (in); E1 , E2 ¼ moduli of elasticities of bodies in contact respectively, GPa (psi); F ¼ load, kN (lbf); F 0 ¼ ðF=‘ Þ ¼ load per unit length, kN/m (lbf/in); k1 , k2 ¼ material constants for small and large solid elastic bodies in contact; L ¼ length of cylinder, m (in); pmax ¼ maximum pressure on surfaces of contact, MPa (psi); c max ¼ maximum contact compressive stress, MPa (psi); ¼ normal stress, also with subscripts, MPa (psi); ¼ shear stress, also with subscripts MPa (psi); 1 , 2 ¼ Poisson’s ratio of materials of small and large elastic bodies in contact respectively; ¼ approach distance along the line of action of the load between two points on the elastic bodies in contact, mm (in); ¼ hoop or circumferential or tangential stress, MPa (psi); a ¼ axial or longitudinal stress, MPa (psi); h ¼ thickness of cylinder/vessel/shell, mm (in). Meaning of other symbols used in this Table are given under Symbols introduced at the beginning of this Chapter. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a2 b2 d d d d 1 12 1 22 b ; k2 ¼ ; ¼ ;e ¼ do ¼ 1 2 ; do0 ¼ 1 2 ; k1 ¼ a a d1 þ d2 d1 d2 E1 E2

TABLE 2-13 Summary of strain and stress equations due to diﬀerent types of loads

STATIC STRESSES IN MACHINE ELEMENTS

2.42

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¼

¼

Thin-walled cylinder under internal pressure with closed ends

Thin-walled cylinder under internal pressure and axial tensile load with closed ends

Thin walled cylinder under internal pressure and compressive load with closed ends.

8

9

10 ¼

a ¼

a ¼

a ¼

pd 2h

pd 2h

0

0

0

y

pd F 4h A

pd F þ 4h A

pd 4h

0

0

0

0

0

0

z

Applied stresses

pd 2h

Mb c F þ l A

Axial, bending and torsion load

Mb c I

F A

x

7

Figure showing stress

Torsion and bending load

Torsion and axial load

Type of loads

6

5

Figure showing loads

TABLE 2-13 Summary of strain and stress equations due to diﬀerent types of loads (Cont.)

0

0

0

Mt r Mt ¼ J Zp

Mt r Mt ¼ J Zp

Mt r Mt ¼ J Zp

h x 2

x 2

x or y whichever is larger

x

1 ð Þ E a 1 " ¼ ð a Þ E

1 ð Þ E a 1 " ¼ ð a Þ E "a ¼

"a ¼

x

þmax

max ¼

r ¼ G L

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ zx þ 4 2 þ

1 2

h x 2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ zx þ 4 2

max ¼

1 2

r ¼ G L

x E

þ

max ¼

General biaxial 1 "a ¼ ð a Þ E 1 " ¼ ð a Þ E

¼

¼

and

x E

r ¼ G L

"x ¼

¼

and

"x ¼

1 ¼ 2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2x þ 4 2

x ; if y > 0 2 x y ; if y < 0 2

max 2

x 2

1 2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2x þ 4 2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2x þ 4 2

max ¼

max

max

1 ¼ 2

Maximum stress produced Strain equations/Area /Approach distance max max

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 2at þ 4 2 2 at

ð"1 "2 Þ 1 2 ð" "1 Þ 2 ¼ E 2 1 2 3 ¼ 0 1 ¼ E

ð"1 "2 Þ 1 2 ð" "1 Þ 2 ¼ E 2 1 2 3 ¼ 0 1 ¼ E

General biaxial: ð" "2 Þ 1 ¼ E 1 1 2 ð" "1 Þ 2 ¼ E 2 1 2 3 ¼ 0

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 2ab þ 4 2 2 ab qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 16 Mb Mb2 þ Mt2 ¼

D3 1 1;2 ¼ ½ at þ ab 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 ð at ab Þ2 þ4 2 2 1 ¼

1;2 ¼

Principal stresses

STATIC STRESSES IN MACHINE ELEMENTS

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2.43

14

h: THICKNESS

Hydraulic pressure

Thick-walled cylinder under internal and external pressure with closed ends

A thin-walled cylinder under internal pressure and torsion with closed ends

12

13

Closed walled spherical shell under internal pressure

11

Mt

Type of loads

Figure showing loads Figure showing stress pd 4h

ðp1 p0 Þd12 d02 4 d02 d12

b¼

p

p1 d12 p0 d02 d02 d12

b r2

a¼

where

¼ a þ

x ¼

¼

x pd 4h

p

y ¼ a

y ¼ a

a ¼

y

b r2

Applied stresses

TABLE 2-13 Summary of strain and stress equations due to diﬀerent types of loads (Cont.)

p

-p

0

0

z

0

Mt r J

0

3pd ð1 Þ 4hE

General triaxial 1 " ¼ ½ ð r þ a Þ E 1 "a ¼ ½ a ð þ r Þ E

1 ð a Þ E 1 "a ¼ ð a Þ E r ¼ ¼ G L " ¼

" ¼

Biaxial hoop stress: Volume strain

0

r 2

max ¼

a 2

Maximum stress produced Strain equations/Area /Approach distance max max

1h ð þ a Þ 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : ð a Þ2 þ4 2

E ½ð1 Þ"1 þ ð"2 þ "3 Þ 1 2 2 E ½ð1 Þ"2 þ ð"3 þ "1 Þ 2 ¼ 1 2 2 E ½ð1 Þ"3 þ ð"2 þ "1 Þ 3 ¼ 1 2 2 1 ¼

12 ¼

Principal stresses

STATIC STRESSES IN MACHINE ELEMENTS

2.44

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3.778 0.408 1.220

6.612 0.319 0.851

p q

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. 2.731 0.493 1.453

30 2.397 0.530 1.550

35

The auxiliary angle deﬁnes the two coeﬃcients p and q and is given by ¼ cos1 ð=Þ. p, q and are obtained from table given below for various values of :

20

General case of contact of two elastic bodies under compressive load

16

10

++General case of loading (Triaxial stress including shear stress)

15

Figure showing stress

, deg

Type of loads

Figure showing loads

1.926 0.604 1.709

45

F

ab

2.136 0.567 1.637

40

av ¼

x

Principal stresses

2 3 1 1 ; ðmax Þ2 ¼ 3 ; ðmax Þ3 ¼ 2 2 2 2

1.611 0.678 1.828

55

1.486 0.717 1.875

60

1.378 0.750 1.912

65

1.286 0.802 1.944

70

1.202 0.845 1.967

75

1.128 0.893 1.985

80

A ¼ ab 3 F ðk1 þ k2 Þ 1=3 3 F ðk1 þ k2 Þ 1=3 a¼p ; b¼q 4 4 1 1 1 1 1 ¼ ðA þ BÞ ¼ þ 0 þ þ 0 2 1 1 2 2 " 1 1 1 2 1 1 2 0 þ 0 þ ¼ ðB AÞ ¼ 2 1 1 2 2 1=2 1 1 1 1 þ cos 2 2 1 01 2 02 " #1=3 9F 2 ¼ ðk þ k2 Þ2 128 1

1.061 0.944 1.996

85

1.000 1.000 2.000

90

c max ¼

3 F 2 ab

b aþb

a aþb

" 1 tan1 e

e2

y¼b

1 tanh1 e 1 e2 e ðÞx¼0 ¼ ð1 2Þ c max

y¼0

3 ¼ ð z Þ ¼ c max : ðÞx¼a ¼ ð1 2Þ c max

ð1 2Þ c max

2 ¼ y ¼ 2 c max

ð1 2Þ c max

½ðmax Þz¼0:63a ¼ 0:34 c max 1 ¼ ð x Þ ¼ 2 c max

For details of general cases of stress, strains and direction cosines refer to Handbook and Theory of Elasticity.

The maximum shear stresses are: ðmax Þ1 ¼

The direction of each principal stress is deﬁned by the cosines of the angles it makes with 0x, 0y, and 0z. The maximum shear stress occurs in each two planes inclined at 458 to the principal stress.

1.754 0.641 1.772

50

z

Maximum stress produced Strain equations/Area /Approach distance max max

The three principal stresses 1 ; 2 and 3 are given by the three roots of the cubic equation in 2 2 2 3 þ y þ x 2 y þ y z þ z x yz zx 2 2 2 x y z þ 2xy yz zx x yz y zx z xy ¼0

y

Applied stresses

TABLE 2-13 Summary of strain and stress equations due to diﬀerent types of loads (Cont.)

STATIC STRESSES IN MACHINE ELEMENTS

2.45

Type of loads

Contact of a solid sphere on a solid plane surface under compressive load

Contact of a solid sphere on solid sphere under compressive load

Contact of a solid sphere with a spherical socket subject to compressive load.

Figure showing loads

17

18

19

x

y F

a2

F

a2

e max ¼

av ¼

c max ¼

av ¼

z

3 F 2 a2

3 F 2 a2

1=3 3 a ¼ Fd0 ðk1 þ k2 Þ 8 " #1=3 9F 2 ðk1 þ k2 Þ2 ¼ 8d0

A ¼ a2

a ¼ 0:721½Fd ðk1 þ k2 Þ1=3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 3 2F ¼ 1:78 ðk1 þ k2 Þ2 d

A ¼ a2

cðmaxÞ ¼

24F þ k2 Þ2 a3 d02 ðk1

"

" z3 ða2 þ z2 Þ 3=2

#

c max

av ¼

F

a2 3 F ¼ 2 a2

1=3 a ¼ 0:721 Fd00 ðk1 þ k2 Þ 1=3 ðk þ k2 Þ ¼ 1:04 F 2 1 d0

The maximum sub-surface shear stress at z ¼ 0:63a is ðmax Þz¼0:63a ¼ 0:34 c max For principal stresses and variation of stresses along the line of action of load refer to Figure 2-28 "

#1=3

#1=3

24 F 1 0

3 d0 2 ðk1 þ k2 Þ2

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 þ 2 7 2

cðmaxÞ

The distance from the surface of contact on the line of action of the load at which the maximum shear stress accurse z ¼ a

3F The maximum Hertz contact stress is ð z max Þz¼0 ¼ pmax ¼ c max ¼ 2 " 2 a 3 # p 1 2 z 3 z pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ The maximum shear stress is 13 ¼ xz ¼ max þ ð1 þ Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 a2 þ z2 a2 þ z2

z ð¼ c Þ ¼ pmax 1

;

#1=3 9 =

0:918

F d 2 ðk1 þ k2 Þ2

"

cðmaxÞ ¼

¼

y max z¼0 ¼

1 þ 2 1 þ 2 pmax ¼ c m 2 2 pmax 1 2 13 max ¼ xz max ¼ 2 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 þ ð1 þ Þ 2ð1 þ Þ 9

z¼0

max

Maximum stress produced Strain equations/Area /Approach distance max max Principal stresses

" 3 # p z z The three principal stresses at the point of contact are: r ¼ ¼ ¼ y ¼ max ð1 þ 2Þ 2ð1 þ Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 2 2 a þz a þz

Figure showing stress

Applied stresses

TABLE 2-13 Summary of strain and stress equations due to diﬀerent types of loads (Cont.)

STATIC STRESSES IN MACHINE ELEMENTS

2.46

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Type of loads

Contact of a solid cylinder on a solid cylinder under compressive load with axes parallel.

Contact of solid cylinder on a solid cylinder under compressive load with axes perpendicular

Contact of a solid sphere with cylindrical groove/socket under compressive load.

Contact of a solid cylinder with a ﬂat surface subject to compressive load

Figure showing loads

20

21

22

23

Figure showing stress

x

y

av ¼ a ¼

av ¼ z ¼

av ¼ z ¼

av ¼ z ¼

z

Applied stresses

TABLE 2-13 Summary of strain and stress equations due to diﬀerent types of loads (Cont.)

F 2Lb

F

ab

F

ab

F 2Lb

Refer to Table Cr for

Refer to Table Cr for

1=2 Fd ð k þ k2 Þ b ¼ 1:6 L 1 F 2d ¼ 4 k1 ln þ 0:41 L b

A ¼ 2Lb

various ratios of 1 1 1 ip ¼ d1 d2 d1

F k1 þ k2 1=2 2b ¼ 1:6 L d00 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2d d1 3 ¼ 1:41Cr 2F 2 2 ðk1 þ k2 Þ2 d1 d2

A ¼ ab

various ratios of 1 1 ip ¼ d2 d1

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2d þ d1 3 ðk1 þ k2 Þ2 ¼ 1:41Cr 2F 2 2 d1 d2

A ¼ ab

b ¼ 1:6

A ¼ 2Lb 1=2 F d ð k þ k2 Þ L 0 1 2F d ¼ k ln 1 þ 0:41 þ b

L 1 d k2 ln 2 þ 0:41 b

Strain equations/Area /Approach distance max

1:5F

ab

1=2 F 1 cðmaxÞ ¼ 0:798 Ld k1 þ k2

1=2 F d00 cðmaxÞ ¼ 0:798 L k2 þ k 2

c max ¼ s ¼

1=2 F 1 cðmaxÞ ¼ 0:798 Ld k1 þ k2

max

Maximum stress produced

ðmax Þz¼0:63a ¼ :034 c max

Principal stresses

STATIC STRESSES IN MACHINE ELEMENTS

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2.47

Contact of a solid sphere on a solid cylinder under compressive load

Cylinder between two ﬂat plates under compressive load

24

25

x

av ¼ z ¼

av ¼ z ¼

z

F 2Lb

F

ab

1=3

c max ¼

max

Refer to Table Cr for

1=2

b ¼ 1:6

Fd ðk þ k2 Þ L 1 F 2d cy ¼ 4 k1 0:41 þ ln L b ! F 1 2

EL ¼4 ln L

E F ð1 2 Þ

A ¼ 2Lb

various ratios of 1 1 1 i ¼ 2 þ d1 d1 d2

a¼

#1=3

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ F Ld ðk1 þ k2 Þ c max ¼ 0:798

2AF 1

3 d04 ðk1 þ k2 Þ2

"

max

Maximum stress produced

3 Fd ðk þ k2 Þ 8 0 1 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð2d2 þ d1 Þ 3 ¼ 1:4Cr 2F 2 ðk1 þ k2 Þ2 d1 d2

A ¼ ab

**TABLE: Crv values for various ratios of i

Approach distance of two points along the line of action of load in two plates is , if E1 ¼ E2 ¼ E and 1 ¼ 2 ¼

Total deformation due to compression of cylinder is cy

y

Strain equations/Area /Approach distance

i 1.00 0.404 0.250 0.160 0.085 0.067 0.044 0.032 0.020 0.015 0.003 Crv 1.00 0.957 0.905 0.845 0.751 0.716 0.655 0.607 0.546 0.510 0.358

Figure showing stress

Applied stresses

Source: Roark, R.J., and W. C. Young, Formulas for Stress and Strain, McGraw-Hill Publishing Company, New York, 1975. + Hertz. H., On the Contact of Elastic Solids, J. Math. (Crelle’s J.) vol. 92, pp 156-171, 1981 Hertz. H., On Gesammelte werke, Vol 1., p 155, Leipzig, 1895.

Type of loads

Figure showing loads

TABLE 2-13 Summary of strain and stress equations due to diﬀerent types of loads (Cont.)

Principal stresses

STATIC STRESSES IN MACHINE ELEMENTS

2.48

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STATIC STRESSES IN MACHINE ELEMENTS STATIC STRESSES IN MACHINE ELEMENTS

2.49

REFERENCES 1. Maleev, V. L. and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954. 2. Shigley, J. E., Mechanical Engineering Design, 3rd edition, McGraw-Hill Book Company, New York, 1977. 3. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. 1 (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 4. Lingaiah, K., Machine Design Data Handbook, Vol II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 5. Lingaiah, K., Machine Design Data Handbook, McGraw-Hill Publishing Company, New York, 1994. 6. Ashton, J. E, J. C. Halpin and P. H. Petit, Primer on Composite Materials-Analysis, Technomic Publishing Co., Inc., 750 Summer St., Stanford, Conn. 06901, 1969. 7. Roark, R. J., and W. C. Young, Formulas for Stress and Strain, McGraw-Hill Publishing Company, New York, 1975. 8. Hertz, H., On the Contact of Elastic Solids, J. Math. (Crelle’s J.) Vol. 92, pp. 156–171, 1981. 9. Hertz, H., On Gesammelte werke, Vol I., p. 155, Leipzig, 1895. 10. Timoshenko, S., and J. N. Goodier, Theory of Elasticity, McGraw-Hill Book Company, New York, 1951.

BIBLIOGRAPHY 1. Black, P. H., and O. Eugene Adams, Jr., Machine Design, McGraw-Hill Book Company, New York, 1965. 2. Lingaiah, K, and B. R. Narayana Iyengar, Machine Design Data Handbook (fps units), Engineering College CoOperative Society, Bangalore, India, 1962. 3. Norman, C. A., E. S. Ault, and I. F. Zarobsky, Fundamentals of Machine Design, The Macmillan Company, New York, 1951. 4. Vallance, A. E., and V. L. Doughtie, Design of Machine Members, McGraw-Hill Book Company, New York, 1951. 5. Timosheko, S., and J. N. Goodier, Theory of Elasticity, McGraw-Hill Book Company, New York, 1951. 6. Timoshenko, S., and J. M. Gere, Mechanics of Materials, Van Nostrand Reinhold Company, New York, 1972. 7. George Lubin, Editor, Handbook of Composites, Van Nostrand Reinhold Company, New York, 1982. 8. John Murphy, Reinforced Plastic Handbook, 2nd edition, Elsevier, Advanced Technology, 1998. 9. Hamcox, N. L., and R. M. Mayer, Design Data for Reinforced Plastics, Chapman and Hall, 1994.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

3 DYNAMIC STRESSES IN MACHINE ELEMENTS2 SYMBOLS2;3 A a, b b c cL cT d E F F Fc Fd Fg Fi Fic Fir Fs g h J k kp K l m M ¼ m=A Mb

area of cross-section, m2 coeﬃcients width of bar or beam, m distance from neutral axis to extreme ﬁbre, m velocity of propagation of plane wave along a thin bar, m/s velocity of propagation of plane longitudinal waves in an inﬁnite plate, m/s velocity of propagation of plane transverse waves in an inﬁnite plate, m/s diameter of bar, m modulus of elasticity, GPa force or load, kN force acting on piston due to steam or gas pressure corrected for inertia eﬀects of the piston and other reciprocating parts, kN centrifugal force per unit volume, kN/m3 the component of F acting along the axis of connecting rod, kN dynamic load, kN gas load, kN inertia force, kN inertia force due to connecting rod, kN inertia force due to reciprocating parts of piston, kN static load, kN acceleration due to gravity, 9.8066 m/s2 depth of bar or beam, m height of fall of weight, m polar moment of inertia, m4 (cm4 ) radius of gyration, m radius of gyration, polar, m kinetic energy, N m length, m mass, kg moving mass, kg ratio of moving mass to area of cross-section of bar bending moment, N m

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DYNAMIC STRESSES IN MACHINE ELEMENTS

3.2

CHAPTER THREE

Mt n n0 n0 ¼ l=r p P r t u u, v, w U Ui Umax Up v V V0 w W Z i s " "x , "y , "z xy , yz , zx

i , i 0 x , y , z

l

n

xy , yz , zx !

torque, m N speed, rpm speed, rps ratio of length of connecting rod to radius of crank pressure power, kW radius of crank, m radius of curvature of the path of motion of mass, m the moment arm of the load, m time, s displacement in x-direction modulus of resilience, N m/m3 displacement components in x, y, and z-directions respectively, m resilience, N m internal elastic energy, N m work done in case of suddenly applied load, N m maximum internal elastic energy, N m potential energy, N m velocity, m/s velocity of particle in the stressed zone of the bar, m/s volume, m3 initial velocity at the time of impact, m/s speciﬁc weight of material, kN/m3 total weight, kN section modulus, m3 (cm3 ) angle between the crank and the centre line of connecting rod, deg unit shear strain, rad/rad weight density, kN/m3 deﬂection/deformation, m (mm) deformation/deﬂection under impact action, m (mm) static deformation/deﬂection under the action of weight, m (mm) unit strain also with subscripts, mm/m strains in x, y, and z-directions, mm/m shearing-strains in rectangular coordinates, rad/rad angle between the crank and the centre line of the cylinder measured from the head-end dead-centre position, deg static angular deﬂection, deg angle of twist, deg angular deﬂection under impact load, deg Lame´’s constants Poisson’s ratio mass density, kg/m3 normal stress (also with subscripts), MPa impact stress (also with subscripts), MPa initial stress at the time of impact and velocity V0 , MPa normal stress components parallel to x, y, and z-axis shearing stress, MPa time of load application, s period of natural frequency, s shearing stress components in rectangular coordinates, MPa angular velocity, rad/s

Note: s and s with ﬁrst subscript s designate strength properties of material used in the design which will be used and followed throughout the book. Other factors in performance or in special aspects which are included from time to time in this book and being applicable only in their immediate context are not given at this stage.

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DYNAMIC STRESSES IN MACHINE ELEMENTS DYNAMIC STRESSES IN MACHINE ELEMENTS

Particular

3.3

Formula

INERTIA FORCE Fv SI ð3-1aÞ 1000 where F is in newtons (N), v in m/s, and P in kW. Fv ¼ US Customary System units ð3-1bÞ 33000 where F is in lbf, v in ft/min, and P in hp.

Power

P¼

Velocity

v¼

2rn US Customary System units 12 where r in in, v in ft/min, and n in rpm.

v¼

2rn SI 60 where r in m, v in m/s, and n in rpm.

(3-2a)

ð3-2aÞ

wv2 rg

ð3-3Þ

The internal elastic energy or work done when a machine member is subjected to a gradually applied load, Fig. 3.1.

U ¼ 12 F

ð3-4Þ

The work done in case of suddenly applied load on an elastic machine member (Fig. 3-2)

Ud ¼ Fd

ð3-5Þ

FIGURE 3-1 Plot of force against deﬂection in case of elastic machine member subject to gradually applied load.

FIGURE 3-2 Plot of force against deﬂection in case of suddenly applied load on a machine member.

The relation between suddenly applied load and gradually applied load on an elastic machine member to produce the same magnitude of deﬂection.

Up ¼ Ud

ð3-6aÞ

Fd ¼ 12 F

ð3-6bÞ

Centrifugal force per unit volume

Fcv ¼

ENERGY METHOD

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DYNAMIC STRESSES IN MACHINE ELEMENTS

3.4

CHAPTER THREE

Particular

The static deformation or deﬂection

Formula

W ð3-7Þ k where k ¼ spring constant of the elastic machine member, kN/m (lbf/in).

st ¼

IMPACT STRESSES Impact from direct load Wv2 2g

Kinetic energy

K¼

Impact energy of a body falling from a height h

K ¼ Wh

The height of fall of a body that would develop the velocity v.

h¼

The maximum stresses produced due to fall of weight W through the height h from rest without taking into account the weight of shaft and collar (Fig. 3-3)

i ¼ max

v2 2g

ð3-8Þ ð3-9Þ ð3-10Þ " rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ# W 2hEA 1þ 1þ ¼ A WL " rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ# 2hEA ¼ st 1 þ 1 þ WL 2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 2h ¼ st 41 þ 1 þ 5 st

ð3-11aÞ ð3-11bÞ

ð3-11cÞ

FIGURE 3-3 Striking impact of an elastic machine member by a body of weight W falling through a height h.

The maximum deﬂection or deformation of shaft due to fall of weight W through the height h from rest neglecting the weight of shaft and collar

"

max

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ# 2hAE ¼ i ¼ st 1 þ 1 þ WL 2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 2h ¼ st 41 þ 1 þ 5 st

ð3-12aÞ

ð3-12bÞ

The stress produced due to suddenly applied load

ðmax Þsud ¼ 2ðmax Þstat

ð3-13Þ

The maximum deﬂection or deformation produced by suddenly applied load

ðmax Þsud ¼ 2st

ð3-14Þ

where subscript stat ¼ st ¼ static and sud ¼ suddenly

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DYNAMIC STRESSES IN MACHINE ELEMENTS DYNAMIC STRESSES IN MACHINE ELEMENTS

Particular

The kinetic energy taking into account the weight of shaft or bar and collar

The relation between , , F and W

3.5

Formula

WVc2 Wb K¼ 1þ 2g 3W

ð3-15Þ

where Vc ¼ velocity of collar and weight W after the load striking the collar, m/s. where Wb ¼ weight of shaft or bar " rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ# F max max 2hAE ð3-16aÞ ¼ ¼ 1þ 1þ ¼ st st W WL 2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 2h ¼ 41 þ 1 þ 5 ð3-16bÞ st 2

The maximum stress due to fall of weight W through the height h from rest taking into account the weight of shaft/bar and collar

i ¼ max

3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ W4 2EAh 1 5 ¼ 1þ 1þ A WL 1 þ ðWb =3WÞ "

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ# W 2EAh ¼ 1þ 1þ A WL 2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 2h5 ¼ st 41 þ 1 þ st where ¼ The maximum deﬂection due to fall of weight W through the height h from rest taking into consideration the weight of shaft/bar and collar

max

ð3-17aÞ ð3-17bÞ

ð3-17cÞ

1 W and ¼ b W 1 þ ð =3Þ

2 3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ WL 4 2hEA 1 5 1þ 1þ ¼ AE WL 1 þ ðWb =3WÞ "

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ# WL 2hAE ¼ 1þ 1þ AE WL 2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 2h5 ¼ st 41 þ 1 þ st

ð3-18aÞ ð3-18bÞ

ð3-18cÞ

Internal elastic energy of weight W whose velocity v is horizontal

U¼

Wv2 2g

ð3-20Þ

Internal elastic energy of weight W whose velocity has random direction

U¼

Wv2 þ W sin 2g

ð3-21Þ

where ¼ angle of velocity, v, to the horizontal plane, deg.

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DYNAMIC STRESSES IN MACHINE ELEMENTS

3.6

CHAPTER THREE

Particular

Formula

Energy

Fig. 3-4a

Fig. 3-4b

Fig. 3-4c

Equation

Up

Wðh þ Þ

W

0

(3-22a)

K

0

Wv2 2g

0

(3-22b)

U

0

0

Wðh þ Þ

(3-22c)

FIGURE 3-4 Impact by a falling body

The equation for energy balance for an impact by a falling body (Fig. 3-4)

ðUp þ K þ UÞa ¼ ðUp þ K þ UÞb 0

Another form of equation for deformation or deﬂection in terms of velocity v at impact Equivalent static force that would produce the same maximum values of deformation or deﬂection due to impact

¼ ðUp þ K þ UÞc

ð3-23Þ

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 v2 A ð3-24Þ max ¼ st @1 þ 1 þ gst 0 0 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 2h v2 A Feq ¼ W @1 þ 1 þ A ¼ W @1 þ 1 þ gst st ð3-25Þ

BENDING STRESS IN BEAMS DUE TO IMPACT Impact stress due to bending

FIGURE 3-5 Impact by a falling body on a cantilever beam

Deﬂection of the end of cantilever beam under impact (Fig. 3-5) The maximum bending stress for a cantilever beam taking into account the total weight of beam

ðb Þmax

" rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ# Wlc 6hEI ¼ bi ¼ 1þ 1þ I Wl 3 2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 Wlc 4 2h ¼ 1þ 1þ 5 I st 0 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 2h ¼ ðb Þst @1 þ 1 þ A st

Wlc Mb c Mb : ¼ ¼ where ðb Þst ¼ Zb I I 0 1 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2h @ max ¼ st 1 þ 1 þ A st 2h ðb Þmax ¼ ðb Þst 1 þ st where ¼

ð3-26aÞ

ð3-26bÞ

ð3-26cÞ

ð3-27Þ ð3-28Þ

mb W b 1 ¼ and ¼ m W 1 þ ð33 =140Þ

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DYNAMIC STRESSES IN MACHINE ELEMENTS DYNAMIC STRESSES IN MACHINE ELEMENTS

Particular

The maximum deﬂection at the end of a cantilever beam due to fall of weight W through the height h from rest taking into consideration the weight of beam The maximum bending stress for a simply supported beam due to fall of a load/weight W from a height h at the midspan of the beam taking into account the total weight of the beam (Fig. 3-6)

3.7

Formula

2

max

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 2h5 ¼ st 41 þ 1 þ st

ð3-28aÞ

2

ðb Þmax

3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2h 1 5 ¼ ðb Þst 41 þ 1 þ st 1 þ ð17 =35Þ ð3-29aÞ

2

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 2h5 ¼ ðb Þst 41 þ 1 þ st where ¼

1 1 þ ð17 =35Þ

and ¼

ð3-29bÞ Wb W

FIGURE 3-6 Simply supported beam

The maximum deﬂection for a simply supported beam due to fall of a weight W from a height h at the midspan of the beam taking into account the weight of beam. (Fig. 3-6)

2

max

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 2h5 ¼ st 41 þ 1 þ st

ð3-30Þ

TORSION OF BEAM/BAR DUE TO IMPACT (Fig. 3-7) The equation for maximum shear stress in the bar due to impact load at a radius r of a falling weight W from a height h neglecting the weight of bar

FIGURE 3-7 Twist of a beam/bar

2

max

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 2h 5 ¼ st 41 þ 1 þ rst

ð3-31Þ

FIGURE 3-8 Displacements due to forces acting on an element of an elastic media.

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DYNAMIC STRESSES IN MACHINE ELEMENTS

3.8

CHAPTER THREE

Particular

The equation for angular deﬂection or angular twist of bar due to impact load W at radius r and falling through a height h neglecting the weight of bar

Formula

2

max

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 2h 5 ¼ st 41 þ 1 þ rst

ð3-32Þ

LONGITUDINAL STRESS-WAVE IN ELASTIC MEDIA (Fig. 3-8) One-dimensional stress-wave equation in elastic media (Fig. 3-8)

2 @2u 2 @ u ¼ c @t2 @x2

ð3-33aÞ

sﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃ Eg E where c ¼ ¼

ð3-33bÞ

¼ velocity of propagation of stress waves, m/s: For velocity of propagation of longitudinal stresswave in elastic media The solution of stress-wave Eq. (3-33a)

The value of circular frequency p

The frequency

Refer to Table 3-1. p p x ¼ A sin x þ B cos x ðc sin pt þ D cos ptÞ ð3-34Þ c c where A, B, C and D are arbitrary constants which can be found from initial or boundary condition of the problem. sﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃ nc n Eg n E p¼ ¼ ¼ ð3-35aÞ l l l where n is an integer ¼ 1; 2; 3; . . . sﬃﬃﬃﬃ p n E c ¼ ¼ f ¼ 2 2l

ð3-35bÞ

where ¼ wave length ¼ 2l=n, c ¼ speed of sound or stress wave velocity, m/s.

LONGITUDINAL IMPACT ON A LONG BAR The velocity of particle in the compression zone The uniform initial compressive stress on the free end of a bar (Fig. 3-9)

The variation of stress at the end of bar at any time t

rﬃﬃﬃﬃﬃﬃﬃ g ¼ pﬃﬃﬃﬃﬃﬃ E E sﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ E ¼ V0 E 0 ¼ V 0 g

V¼

ð3-36Þ

ð3-37Þ

where V0 ¼ initial velocity of the moving weight/ mass at the time of impact, m/s. pﬃﬃﬃﬃﬃﬃ 2l E ¼ 0 exp ð3-38Þ t 0
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DYNAMIC STRESSES IN MACHINE ELEMENTS DYNAMIC STRESSES IN MACHINE ELEMENTS

Particular

The equations of motion in terms of three displacement components assuming that there are no body forces.

3.9

Formula

ð þ GÞ

@" @2u þ Gr2 u ¼ 2 @x @t

ð3-39aÞ

ð þ GÞ

@" @2v þ Gr2 v ¼ 2 @y @t

ð3-39bÞ

ð þ GÞ

@" @2w þ Gr2 w ¼ 2 @z @t

ð3-39cÞ

where " ¼ "x þ "y þ "z

FIGURE 3-9 Prismatic bar subject to suddenly applied uniform compressive stress

r¼

@2 @2 @2 þ 2 þ 2 ¼ the Laplacian operator 2 @x @y @z

¼

E and ð1 þ Þð1 2Þ

¼G¼

E are Lame’s constants 2ð1 þ Þ

Dilatational and distortional waves in isotropic elastic media From the classical theory of elasticity equations for irrotational or dilatational waves

Equations for distortional waves

Equations (3-40) to (3-41) are one-dimensional stress wave equations of the form The velocity of stress wave propagation for the case of no rotation

@ 2 u þ 2G 2 r u ¼ @t2

ð3-40aÞ

@ 2 v þ 2G 2 r v ¼ @t2

ð3-40bÞ

@ 2 w þ 2G 2 ¼ r w @t2

ð3-40cÞ

@2u G 2 ¼ r u @t2

ð3-41aÞ

@2v G 2 ¼ r v @t2

ð3-41bÞ

@2w G 2 ¼ r w @t2

ð3-41cÞ

@2 ¼ a2 r2 @t2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ 2G Eð1 Þ ¼ a ¼ c1 ¼ ð1 þ Þð1 2Þ

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ð3-42Þ ð3-43Þ

DYNAMIC STRESSES IN MACHINE ELEMENTS

3.10

CHAPTER THREE

Particular

The velocity of stress wave propagation for the case of zero volume change The ratio of c1 to c2

Formula

sﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ G E a ¼ c2 ¼ ¼ 2ð1 Þ c1 ¼ c2

ð3-44Þ

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2ð1 Þ pﬃﬃﬃ ¼ 3 ð1 2Þ for Poisson’s ratio of ¼ 0:25 ð3-45Þ

The velocity of stress wave propagation for a transverse stress wave, i.e. distortional wave in an inﬁnite plate

sﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ G E cT ¼ ¼ 2ð1 þ Þ

The velocity of stress wave propagation for plane longitudinal stress wave in case of an inﬁnite plate

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4Gð þ GÞ E ¼ cL ¼ ð þ 2GÞ ð1 2 Þ

ð3-46Þ

ð3-47Þ

TORSIONAL IMPACT ON A BAR Equation of motion for torsional impact on a bar (Fig. 3-10) Torsional wave propagation in a bar subjected to torsion. For velocity of propagation of torsional stress-wave in an elastic bar

FIGURE 3-10 Torsional impact on a uniform bar showing torque on two faces of an element

The angular velocity of the end of a bar subject to torsion relative to the unstressed region

@2 @2 ¼ c2t 2 @t @x2 sﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃ Gg G ct ¼ ¼

ð3-48Þ ð3-49Þ

Refer to Table 3-1.

FIGURE 3-11 Torsional striking impact

2 t pﬃﬃﬃﬃﬃﬃ t 2 t d G ¼ pﬃﬃﬃﬃﬃﬃ t d G

!¼

¼ t

The shear stress from Eq. (3-50)

¼

!d pﬃﬃﬃﬃﬃﬃ G 2

The initial shear stress, if the rotating body strikes the end of the bar with an angular velocity !0

0 ¼

!0 d pﬃﬃﬃﬃﬃﬃ G 2

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ð3-50Þ ð3-51aÞ ð3-51bÞ

DYNAMIC STRESSES IN MACHINE ELEMENTS DYNAMIC STRESSES IN MACHINE ELEMENTS

Particular

The maximum shear stress for the case of a shaft ﬁxed or attached to a very large mass/weight at one end and suddenly applied rotational load at the other end by means of some mechanical device such as a jaw clutch (Fig. 3-11)

3.11

Formula

max

2vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ "rﬃﬃﬃﬃ# 1ﬃ 3 u 0 u1 1 7 ¼

¼ 0 6u 0 4t @ A 5 1þ 3

ð3-52Þ

Ib : I

where ¼

2 d Ib ¼ mass moment of inertia of bar ¼ mb 8 I ¼ mass moment of inertia of striking rotating weight Ib and I correspond to Wb and W of the weight of the bar and the rotating mass or weight respectively.

FIGURE 3-12 A striking rotating weight with massmoment of inertia I rotating at !0 engages with one end of shaft and the other end of shaft ﬁxed to a mass-moment of inertia If

The more accurate equation for the max which is based on stress wave propagation

The initial/maximum ( i ¼ max ) shear stress for the case of a system shown in Fig. 3-12

2

max

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 1 25 ¼ 0 41 þ þ 3

i ¼ max

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 þ Þ ¼ 0 ð1 þ þ Þ

i ¼ max

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 þ Þ ¼ 0 ð1 þ þ Þ

Accurate maximum stress for torsional impact shear stress based on stress-wave propagation as suggested by Prof. Burr

ð3-55Þ

W b mb m ¼ and ¼ mf W m

where ¼ Accurate maximum stress for longitudinal impact stress based on stress wave propagation as suggested by Prof. Burr

ð3-54Þ

Ib I and Ib ¼ Jl: ; ¼ I If

where ¼ A similar equation to Eq. (3-54) for maximum stress for longitudinal impact

ð3-53Þ

2

i ¼ max

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 ð1 þ Þ 5 ¼ 0 41:1 þ ð1 þ þ Þ 2

i ¼ max

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 ð1 þ Þ 5 ¼ 0 41:1 þ ð1 þ þ Þ

ð3-56Þ

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ð3-57Þ

DYNAMIC STRESSES IN MACHINE ELEMENTS

3.12

CHAPTER THREE

Particular

Formula

INERTIA IN COLLISION OF ELASTIC BODIES When a body having weight W strikes another body that has a weight W 0 , impact energy Wh is reduced to nWh, according to law of collision of two perfectly inelastic bodies, the formula for the value of n

n¼

1 þ am ð1 þ bmÞ2 where m ¼

ð3-58Þ W0 ; a and b are taken from Table 3-3 W

RESILIENCE 2 V 1 2 AL ¼ 2 E 2 E

The expression for resilience in compression or tension

U¼

The modulus of resilience

u¼

The area under the stress-strain curve up to yielding point represents the modulus of resilience (Fig. 1.1)

u ¼ 12 " 2 2 k b AL Ub ¼ c 6E 2 2 k b ub ¼ c 6E

The resilience in bending The modulus of resilience in bending

ð3-59Þ

2 2E

ð3-60Þ ð3-61Þ ð3-62Þ ð3-63Þ

where ðk=cÞ2 ¼

1 3

for rectangular cross-section

¼

1 4

for circular section

c ¼ distance from extreme fibre to neutral axis Resilience in direct shear

U ¼

e2 V 2G

ð3-64Þ

The modulus of resilience in direct shear

u ¼

e2 2G

ð3-65Þ

Resilience in torsion

The modulus of resilience in torsion

The equation for strain energy due to shear in bending The modulus of resilience due to shear in bending

2

e2 AL k0 ð3-66Þ c 2G qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where k0 ¼ D21 D20 =8 and c ¼ 12 D0 for hollow shaft. 2

2 k0 ð3-67Þ u ¼ e 2G c ðl k F 2

dx ð3-68Þ U b ¼ 0 2GA U ¼

u b ¼

k e2 2G

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ð3-69Þ

DYNAMIC STRESSES IN MACHINE ELEMENTS

3.13

DYNAMIC STRESSES IN MACHINE ELEMENTS

Particular

Formula

The equation for shear or distortional strain energy per unit volume associated with distortion, without change in volume

U ¼

1 2 ½ þ 22 þ 23 ð1 2 þ 2 3 þ 3 1 Þ 6G 1 ð3-70aÞ

¼

1 ½ð 2 Þ2 þ ð2 3 Þ2 þ ð3 1 Þ2 12G 1 ð3-70bÞ

The equation for dilatational or volumetric strain energy per unit volume without distortion, only a change in volume For maximum resilience per unit volume (i.e., for modulus of resilience), resilience in tension for various engineering materials and coeﬃcients a and b; velocity of propagation c and ct .

Uv ¼

ð1 2Þ ½ð1 þ 2 þ 3 Þ2 6E

ð3-71Þ

Refer to Tables 3-1 to 3-4.

TABLE 3-1 Longitudinal velocity of longitudinal wave c and torsional wave ct propagation in elastic media Density

Modulus of elasticity, E

Modulus of rigidity, G

sﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃ E Eg ¼ c¼

#

sﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃ G Gg # ¼ ct ¼

Material

g/cm3

lbm /in3 kN/m3 GPa

Mpsi

GPa

Mpsi

m/s

ft/s

m/s

ft/s

Aluminum alloy Brass Carbon steel Cast iron, gray Copper Glass Lead Inconel Stainless steel Tungsten

2.71 8.55 7.81 7.20 8.91 2.60 11.38 8.42 7.75 18.82

0.098 0.309 0.282 0.260 0.320 0.094 0.411 0.307 0.280 0.680

10.3 15.4 30.0 14.5 17.7 6.7 5.3 31.0 27.6 50.0

26.2 40.1 79.3 41.4 44.7 18.6 13.1 75.8 73.1 137.9

3.8 5.82 11.5 6.0 6.49 2.7 1.9 11.0 10.6 20.0

5116 3523 5145 3727 3648 4214 1796 5016 4955 4279

16785 11560 16887 12223 12176 13823 5879 16452 15972 14039

3110 2165 3200 2407 2240 2675 1073 2987 3071 2707

10466 7106 10485 7865 7373 8775 3520 9800 10074 8880

26.6 83.9 76.6 70.6 87.4 25.5 111.6 83.3 76.0 184.6

71.0 106.2 206.8 100.0 118.6 46.2 36.5 213.7 190.3 344.7

#

Note: ¼ Mass density, g/cm3 (lbm /in3 ), ¼ weight density (speciﬁc weight), kN/m3 (lbf/in3 ), g ¼ 9:8066 m/s2 in SI units, g ¼ 980 in=s2 ¼ 32:2 ft=s2 in fps units.

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DYNAMIC STRESSES IN MACHINE ELEMENTS

3.14

CHAPTER THREE

TABLE 3-2 Maximum resilience per unit volume (2, 1) Type of loading

Modulus of resilience, J (in lbf) 2e 2E

e2 2G

Tension or compression Shear, simple transverse Bending in beams With simply supported ends: Concentrated center load and rectangular cross-section Concentrated center load and circular cross-section Concentrated center load and I-beam section Uniform load and rectangular section Uniform-strength beam, concentrated load, and rectangular section

2e 18E 2e 24E 32e 32E 42e 45E 2e 6E

Fixed at both ends: 2e 18E 2e 30E

Concentrated load and rectangular cross-section Uniform load and rectangular cross-section Cantilever beam:

2e 18E 2e 30E

End load and rectangular cross-section Uniform load and rectangular cross-section Torsion

e2 4G Di 2 2 1þ Do 4G

Solid round bar Hollow round bar with D0 greater than Di Springs Laminated with ﬂat leaves of uniform strength Flat spiral with rectangular section Helical with round section and axial load Helical with round section and axial twist Helical with rectangular section and axial twist

2e 6E 2e 24E

e2 4G 2e 8E 2e 6E

Sources: K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986; K. Lingaiah, Machine Design Data Handbook, Vol II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986; V. L. Maleev and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954.

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DYNAMIC STRESSES IN MACHINE ELEMENTS DYNAMIC STRESSES IN MACHINE ELEMENTS

3.15

TABLE 3-3 Coeﬃcients in Eq. (3-58) (1) Type of impact

a

b

Longitudinal impact on bar

1 3

1 2

Center impact on single beam

17 35

5 8

Center impact on beam with ﬁxed ends

13 35

1 2

End impact on cantilever beam

4 17

3 8

TABLE 3-4 Resilience in tension Elastic limit, Material

MPa

Cast iron: Class 20 (ordinary) Class 25 Nickel, Grade II Malleable Aluminum alloy, SAF 33 Brass, SAE 40 or SAE 41 Bronze, SAE 43 Monel metal: Hot-rolled Cold-rolled, normalized Steel: SAE 1010 SAE 1030 SAE 1050, annealed SAE 1095, annealed SAE 1095, tempered SAE 2320, annealed SAE 2320, tempered SAE 3250, annealed SAE 3250, tempered SAE 6150, annealed SAE 6150, tempered Rubber

kpsi

Modulus of elasticity, E GPa

Mpsi

Modulus of resilience, u J

in lbf

42.8a 68.9a 117.2a 137.9 48.3 68.9 193.0

62 10.0 17.0 20.0 7.0 10.0 28.0

68.9 89.2 24.5 172.6 66.7 82.4 110.8

10 13 18 25 9.7 12 16

0.22 0.43 0.90 0.90 0.28 0.45 2.77

1.9 3.8 8.0 8.0 2.5 4.0 24.5

206.9 482.6

30.0 70.0

176.5 176.5

25.5 25.5

1.96 10.79

17.6 96

206.9 248.2 330.9 413.7 517.1 310.3 689.5 551.6 1379.0 427.6 1102.3 2.1

30.0 36.5 48.5 60.0 75.0 45.0 100.0 80.0 200.0 62.0 160.0 0.3

206.9 204.8 204.8 204.8 204.8 204.8 213.7 213.7 213.7 213.7 1034 109

30.3 30 29.7 29.7 29.7 29.7 29.7 31 31.0 31 31 150 106

1.69 2.45 4.27 6.77 16.08 3.82 18.83 21.58 72.57 6.96 52.47 33.89

15 22 38 60 94 34 167 193 645 62 466 300

a

Impact strength (Izod no.)

7.9 2.7

120 100

20

52 40 30

Cast iron has no well-deﬁned elastic limit, but the values may be safely used anyway for all practical purposes. Source: Reproduced courtesy of V. L. Maleev and J. B. Hartman, Machine Design, International Textbook Co., Scranton, Pennsylvania, 1954.

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DYNAMIC STRESSES IN MACHINE ELEMENTS

3.16

CHAPTER THREE

REFERENCES 1. Maleev, V. L., and J. B. Hartman, Machine Design, International Textbook Co., Scranton, Pennsylvania, 1954. 2. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 3. Lingaiah, K., Machine Design Data Handbook, Vol II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 4. Lingaiah, K., Machine Design Data Handbook, McGraw-Hill Book Company, New York, 1994. 5. Burr, Arthur H., and John B. Cheatham, Mechanical Analysis and Design, 2nd edition, Prentice Hall, Englewood Cliﬀs, NJ, 1995. 6. Spotts, Merhyle F.,‘‘Impact Stress in Elastic Bodies Calculated by the Energy Method’’, Engineering Data for Product Design, edited by Douglas C. Greenwood, McGraw-Hill Book Company, New York, 1961. 7. Burr, A. H.,‘‘Longitudinal and Torsional Impact in Uniform Bar with a Rigid Body at One End’’, J. Appl. Mech., Vol. 17, No. 2 (June 1950), pp. 209–217; Trans. ASME, Vol. 72 (1950). 8. Timoshenko, S., and J. N. Goodier, Theory of Elasticity, 3rd edition, McGraw-Hill Book Company, New York, pp. 485–513, 1970. 9. Kolsky, H., Stress Waves in Elastic Solids, Dover Publications, New York, 1963. 10. Durellli, A. J., and W. F. Riley, Introduction to Photomechanics, Prentice Hall Inc, Englewood Cliﬀs, NJ, 1965. 11. Arnold, ‘‘Impact Stresses in a Simply Supported Beam’’, Proc. I.M.E., Vol. 137, p. 217. 12. Dohrenwend and Mehaﬀy, ‘‘Dynamic Loading’’, Machine Design, Vol. 15 (1943), p. 99.

BIBLIOGRAPHY Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Co-operative Society, Bangalore, India, 1962. Norman, C. A., E. S. Ault, and E. F. Zarobsky, Fundamentals of Machine Design, The Macmillan Company, New York, 1951.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

4 STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS SYMBOLS6;7;8 a 2a A b b ¼ ðw aÞ 2b B c C d

di do D F

h

diameter of circular hole (cut-out), m (in) semimajor axis of elliptical hole (cut-out), m (in) half of the length of the slot, m (in) length of straight crack, m (in) (Figs. 4-26A and 4-28) area of cross section, m2 (in)2 semiminor axis of elliptical hole (cut-out), m (in) maximum breadth of section of curved bar, m (in) width of notch at the edge, m (in) eﬀective width of plate across the hole, m (in) or net width of plate, m (in) total width of plate with a crack, m (in) (Fig. 4-28) constant in curved bar equation outside diameter of reinforced ring in an asymmetrically reinforced circular hole distance from centroidal axis to extreme ﬁber of beam or inside edge of curved bar, m (in) spring index eﬀective width of plate, m (in) width of U-piece at dangerous section, m (in) diameter of shaft at reduced section, m (in) reduced width of shoulder plate, m (in) diameter of hole (cut-out), m (in) outside diameter of reinforcement, m (in) total diameter of shaft, m (in) total width of plate, m (in) load, kN (lbf) force, kN (lbf) diametrically opposite concentrated loads on ring or hollow roller, kN (lbf) thickness of plate, m (in) thickness of ring or roller, m (in) lever arm from critical section of tooth, m (in)

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STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.2

CHAPTER FOUR

2h h1

length of plate with a crack, m (in) (Fig. 4-28) depth of groove, m (in) (Figs 4-11, 4-12 and 4-13) depth of shoulder, m (in) thickness of reinforcement including plate thickness, m (in) moment of inertia, area, m4 (in4 ) moment of inertia, polar, m4 (in4 ) pﬃﬃﬃﬃ openingpmode ﬃﬃﬃﬃ or mode I of stress intensity factor, MPa m (kpsi in) pﬃﬃﬃﬃ pﬃﬃﬃﬃ mode II of stress intensity factor, MPa m (kpsi in) opening mode or mode I of critical stress intensity factor or pﬃﬃﬃﬃ pﬃﬃﬃﬃ fracture toughness factor, MPa m (kpsi in)

H I J KI KII KIC K ¼ K0 K Kf

Kf Kn Kg Kf Kf

max nom

theoretical stress-concentration factor for normal stress

combined stress-concentration factor (K0 is a theoretical factor) theoretical stress-concentration factor for shear stress (torsion) fatigue stress-concentration factor for normal stress or fatigue notch factor for axial or bending (normal) (Fig. 14-13) or fatigue strength reduction factor f fatigue limit of unnotched specimen (axial or bending) ¼ ¼ nf fatigue limit of notched specimen (axial or bending) stress-concentration factor (normal) based on net area (nominal) of cross section (i.e., net nominal stress) stress-concentration factor (normal) based on gross area of cross section (i.e., gross stress) fatigue stress-concentration factor for shear stress (torsion) or fatigue notch factor (torsion) f fatigue limit of unnotched specimen (torsion) ¼ ¼ nf fatigue limit of notched specimen (torsion) stress-concentration factor for U-grooved plate stress-concentration factor for a V-grooved plate eﬀective stress-concentration factor under a static load, equivalent stress-concentration factor module, mm magniﬁcation factor bending moment, N m (lbf ft) torsional moment, N m (lbf ft) safety factor

Ku Kv Ke m MF Mb Mt n q¼ r r rj rt Ro Ri s VH VR

Kf 1 k 1

index of sensitivity or notch sensitivity factor radius of curvature of groove or notch of curved bar, m (in) minimum radius of curvature of an ellipse, m (in) polar coordinate distance of a point in a plate from the crack tip (Fig. 4-26C), m (in) minimum ﬁllet radius of gear tooth, m (in) cutter tip radius, m (in) outside radius of ring or hollow roller, m (in) inside radius of ring or hollow roller, m (in) thickness of the tooth at critical section, m (in) volume of hole, m3 ðin3 Þ volume of reinforcement, m3 ðin3 Þ

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STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.3

depth of U-piece-arm, m (in) total width of ﬂat plate, m (in) stress components in x, y coordinates, MPa (kpsi) maximum normal stress, MPa (kpsi) nominal normal stress computed from F/A or Mb c=I or from an elementary formula which does not take into account the stress concentration, MPa (kpsi) average stress at the root of gear tooth, MPa (kpsi) maximum shear stress, MPa (kpsi) nominal shear stress computed from Mt r=J, MPa (kpsi) angle of a shallow U-groove, deg angle of V-groove, deg polar coordinate, deg angle made by r the distance of a point from tip of crack with x axis (Fig. 4-26C) Other factors in performance or in special aspects are included from time to time in this chapter, and being applicable only in their immediate context, are not given at this stage

w W x , y , xy max nom 0 max nom

Stress concentration factor theoretical/empirical or otherwise Particular

Extreme value

(a) Keyway (Fig. 4-1 and Tables 4-1 and 4-2: Proﬁle keyway

Formula

Kf ¼ 1:68 Kf ¼ 1:44

Sled-runner keyway

TABLE 4-1 Shear stress-concentration factor for a keyway in a shaft subjected to torsion (by Leven)

TABLE 4-2 Stress-concentration factor Kf for keyways Annealed

r=d K

0.0052 3.92

0.0104 3.16

(b) Curved bar (Fig. 4.1a): For curved bar

0.0208 2.62

0.0417 2.30

0.0833 2.06

Hardened

Type of keyway

Bending

Torsion

Bending

Torsion

Proﬁle Sled runner

1.6 1.3

1.3 1.3

2.0 1.6

1.6 1.6

I 1 1 þ K ¼ 1:00 þ B rc r bc2 ð4-1Þ where B ¼ 1.05 for circular or elliptical cross-section ¼ 0.5 for other cross-section

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STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.4

CHAPTER FOUR

Stress concentration factor theoretical/empirical or otherwise Particular

FIGURE 4-1 Stress-concentration factor for the straight portion of a keyway in a shaft in torsion.

Extreme value

Formula

FIGURE 4-1a Stress distribution in curved bar under bending.

(c) Spur gear tooth (Figs. 4-2 and 4-3, Table 4-3): At root ﬁllet of an involute tooth proﬁle of 14.58 pressure angle

K ¼ 2 to 2:5 K ¼ 0:22 þ

FIGURE 4-2 Stress-concentration factor at root of gear tooth.

FIGURE 4-3 Eﬀect of ﬁllet radius on stress-concentration at root of gear tooth.

rf s

1 0:2 0:4 h s

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ð4-2Þ

STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.5

Stress concentration factor theoretical/empirical or otherwise Particular

Extreme value

Formula

TABLE 4-3 Stress-concentration factors at root ﬁllet of gear m, mm

Kt

Kc

6.24 5 4.25 3.5 3

1.47 1.47 1.42 1.35 1.345

1.61 1.61 1.57 1.50 1.50

At root ﬁllet of a full depth involute tooth proﬁle of 208 pressure angle

K ¼ 0:18 þ

At root ﬁllet of an involute tooth proﬁle of 258 pressure angle

K ¼ 0:14 þ

(d) Circular cut-outs (holes) in plates (Fig. 4-4c): For inﬁnite plate in:

1 0:15 0:45 rf h s s s rf

ð4-3Þ

0:11 0:50 s h

(i) Uniaxial tension (Fig. 4-4c)

K ¼ 3

K ¼

1 a2 3a4 2þ 2 þ 4 ¼ 3 ð4-4Þ 2 r r r¼a

(ii) Biaxial tension

K ¼ 2

K ¼

a2 1þ 2 ¼2 r r¼a

(iii) Pure shear

For stress concentration factor for a semi-inﬁnite plate with a circular hole near the edge under tension.

K ¼ 4

K ¼

3a4 1þ 4 r

ð4-5Þ

r¼a

¼4

K ¼ 3

FIGURE 4-4 Stress distribution around holes (cut-outs) in plate in tension.

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ð4-6Þ

STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.6

CHAPTER FOUR

Stress concentration factor theoretical/empirical or otherwise Particular

Extreme value

Formula

For ﬁnite plate in:

(i) Uniaxial tension (Fig. 4-5)

K ¼ 3

K ¼ 2 þ

(ii) Bending (Fig. 4-6)

K ¼ 2

K ¼ 2

b w

3

b w

ð4-7Þ

ð4-8Þ

(e) Filled circular hole: For ﬁlled circular holes in plate subjected to tension

K ¼ 2:5

(f) Reinforced circular holes: (i) For stress-concentration factor for a symmetrically reinforced circular hole in a ﬂat plate under uniform uniaxial tension (ii) For stress-concentration factor for an asymmetrically reinforced circular hole in a ﬂat plate under uniform uniaxial tension

FIGURE 4-5 Reproduced with permission. Stress-concentration factor for a plate of ﬁnite width with a circular hole (cut-out) in tension. (‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol. 23, Nos. 2 to 7, 1951.)

Refer to Fig. 4-7a, b, and c.

FIGURE 4-6 Reproduced with permission. Stress-concentration factor for a plate of ﬁnite width with transverse circular hole (cut-out) subjected to bending. (‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol. 23, Nos. 2 to 7, 1951.)

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STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.7

Stress concentration factor theoretical/empirical or otherwise Particular

Extreme value

Formula

(g) Noncircular holes (cut-outs) in a plate: (i) An inﬁnite plate with an elliptical hole (cutout) in uniaxial tension (load at right angles to major axis) (Fig. 4-4b)

a b rﬃﬃﬃ a K ¼ 1 þ 2 r

ð4-9aÞ

K ¼ 1 þ 2

(ii) An inﬁnite plate with elliptical hole (cut-out) in uniaxial tension (load parallel to major axis)

K ¼ 1 þ

(iii) An inﬁnite plate with elliptical hole (cut-out) in pure shear

a K ¼ 2 1 þ b

(iv) An inﬁnite plate with an elliptical cut-out in biaxial tension

K ¼

ð4-9bÞ

2b a

ð4-9cÞ

ð4-10Þ

2a b

ð4-11Þ

(v) Transverse bending of a plate containing an elliptical cut-out (or hole)

2a ð1 þ vÞ 3 v þ b K ¼ ð3 þ vÞ

(vi) Slotted plate loaded in tension or bending

K ¼ 1:064 þ 0:788

a b

ð4-12Þ

for v ¼ 0:3 ð4-13aÞ

K ¼ 2 þ For reduction of endurance strength of steel

3 b w

for

a ¼1 r

ð4-13bÞ

Refer to Fig. 4-8.

(h) U-shaped member subjected to bending (Fig. 4-9): (1) At 08 with horizontal axis

KA ¼ 1 þ

d 4r

ð4-14Þ

(2) At 708 with horizontal axis

KB ¼ 1 þ

w 5r

ð4-15Þ

(i) Helical spring: Stress concentration or Wahl’s correction factor for helical spring

K ¼

4c 1 0:615 þ 4c 4 c

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ð4-16Þ

STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.8

CHAPTER FOUR

FIGURE 4-7(a) Stress-concentration factor for an asymmetrically reinforced circular hole (cut-out) in a ﬂat plate subjected to tension. (PhD work of the author.)

FIGURE 4-7(c)

FIGURE 4-7(b) Stress-concentration factor for an asymmetrically reinforced circular hole in a ﬂat plate subjected to uniform unidirectional tensile stress. (PhD work of the author, and R. E. Peterson, Stress Concentration Factors, John Wiley and Sons, Inc., 1974.)

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4.9

Stress concentration factor theoretical/empirical or otherwise Particular

FIGURE 4-8 Reduction of endurance strength of steel, f .

Extreme value

Formula

FIGURE 4-9 U-shaped member. (R. E. Peterson, Stress Concentration Factors, John Wiley and Sons, 1974.)

( j) Ring or hollow roller: For the ring loaded internally

max ½2hðRo Ri Þ 3 2 ðR þ R Þ 1 6 7 o i 5 4 F 1þ3 Ro Ri

K ¼ 2

ð4-17Þ For the ring loaded externally

K ¼

max ½hðRo Ri Þ 3FðRo þ Ri Þ

ð4-18Þ

(k) Shafts with transverse holes (Fig. 4-10): (i) Shaft with a circular hole subjected to transverse bending for a=d ! 0

K ¼ 3:0

(ii) Shaft with a circular hole subjected to torsion for a=d ! 0

K ¼ 2:0

(l) Shafts with grooves: Shaft with U and V circumferential groove in: (i) Tension or bending (Figs. 4-11 to 4-16 and 4-18)

rﬃﬃﬃﬃﬃ h1 r

K ¼ 1 þ 2

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ð4-19Þ

STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.10

CHAPTER FOUR

I d 3 ¼ 32 c nom ¼

D3 32

Mb 2 ðapprox:Þ dD 6

FIGURE 4-10 Reproduced with permission. Stress-concentration factor K for a shaft with transverse circular hole subjected to bending. (R. E. Peterson, ‘‘Stress Concentration Factors,’’ Machine Design, Vol. 23, Nos. 2 to 7, 1951.)

h1

FIGURE 4-11 Types of V-grooves.

FIGURE 4-12 Stress-concentration factor ratio due to notches of various shapes.

FIGURE 4-13 Average notch sensitivity.

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STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.11

Stress concentration factor theoretical/empirical or otherwise Particular

Extreme value

Formula

FIGURE 4-14 Reproduced with permission. Stress-concentration factor K for a grooved shaft in tension. (R. E. Peterson, ‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol. 23, Nos. 2 to 7, 1951.)

FIGURE 4-15 Reproduced with permission. Round shaft in torsion with transverse hole. (R. E. Peterson, ‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol. 23, Nos. 2 to 7, 1951.)

FIGURE 4-16 Reproduced with permission. Stress-concentration factor K for grooved shaft in bending. (R. E. Peterson, ‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol. 23, Nos. 2 to 7, 1951.)

FIGURE 4-17 Plate loaded in tension by a pin through a hole, 0 ¼ F=A, where A ¼ ðw dÞt. When clearance exists, increase Kt 35 to 50 percent. (M. M. Frocht and H. N. Hill, "Stress Concentration Factors around a Central Circular Hole in a Plate Loaded through a Pin in Hole,’’ J. Appl. Mechanics, vol. 7, no. 1, March 1940, p. A-5.)

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STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.12

CHAPTER FOUR

Stress concentration factor theoretical/empirical or otherwise Particular

Extreme value

Formula

rﬃﬃﬃﬃﬃ h1 K ¼ 1 þ r rﬃﬃﬃﬃﬃ h1 K ¼ 1 þ r or

(ii) Torsion (Fig. 4-18)

(iii) Shaft with a small elliptical groove in torsion

ð4-20Þ ð4-21aÞ

h1 b b K ¼ 1 þ r 0:65

ð4-21bÞ

K ¼ 1 þ

(m) Shouldered shaft in torsion (Fig. 4-19):

d K 1 þ S r

where S is some function of 2

3 r 1 þ 2 d 6 6 7 d 7 7 61 K ¼ 1 þ D r 5 12r 4 1þ6 d d For stress-concentration factor and combined factor for stepped-shaft in tension and bending

FIGURE 4-18 Reproduced with permission. Stress-concentration factor K for grooved shaft in tension. (R. E. Peterson, ‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol. 23, Nos. 2 to 7, 1951.)

ð4-21cÞ ð4-22aÞ D d

ð4-22bÞ

Refer to Figs. 4-14, 4-16, and 4-18 to 4-21.

FIGURE 4-19 Reproduced with permission. Stress-concentration factor K for stepped shaft in torsion. (R. E. Peterson, ‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol. 23, Nos. 2–7, 1951.)

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4.13

STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

Stress concentration factor theoretical/empirical or otherwise Particular

Extreme value

Formula

(n) Bar containing grooves: (i) Bar with U, semicircular or shallow grooves symmetrically placed in tension (Figs. 4-11, 4-12, 4-22)

2

3n 1 h1 K ¼ 1 þ 6 7 D r5 4 1:55 1:3 d

ð4-23aÞ

or 3n D 1 6 d7 7 6 d 7 K ¼ 1 þ 6 4 D r5 2 1:55 1:3 d rﬃﬃﬃﬃﬃ D h 1 þ 0:3 1 d r where n ¼ rﬃﬃﬃﬃﬃ D h1 1 þ r d 2

(ii) Bar with deep V-groove in tension for r < 1 (Fig. 4-11a) h1

FIGURE 4-20 Reproduced with permission. Stress-concentration factor K for stepped shaft in tension. (R. E. Peterson, ‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol. 23, Nos. 2 to 7, 1951.)

( Kv ¼ 1 þ ðKv 1Þ 1

180

ﬃ 1 þ 2:4pﬃﬃﬃﬃﬃﬃ r=h1

ð4-23bÞ

) ð4-24Þ

FIGURE 4-21 Reproduced with permission. Stress-concentration factor K for stepped shaft in bending. (R. E. Peterson, ‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol. 23, Nos. 2–7, 1951.)

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STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.14

CHAPTER FOUR

Stress concentration factor theoretical/empirical or otherwise Particular

Extreme value

Formula

FIGURE 4-22 Reproduced with permission. Stress-concentration factor K for notched ﬂat bar in tension. (R. E. Peterson, ‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol. 23, Nos. 2 to 7, 1951.)

(iii) Bar with shallow V-groove in tension for r >1 h1

( Kv ¼ 1 þ ðKv 1Þ 1

180

ﬃ 1 þ 2:4pﬃﬃﬃﬃﬃﬃ r=h1

)

ð4-25Þ (iv) Elliptical groove at the edge of plate in tension

K ¼ 1 þ

2h1 b

rﬃﬃﬃﬃﬃ h K ¼ 1 þ 2 1 r (v) Bar with symmetrical U, semicircular shallow grooves in bending (Fig. 4-23).

2 K ¼ 1 þ 4

3 1 h1 0:85 D r5 4:27 4 d

ð4-26aÞ

ð4-26bÞ

ð4-27aÞ

or 2

30:85 D 1 6 d7 6 7 d 7 K ¼ 1 þ 6 4 D r5 2 4:27 4 d

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ð4-27bÞ

STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.15

Stress concentration factor theoretical/empirical or otherwise Particular

Extreme value

FIGURE 4-23 Reproduced with permission. Stress-concentration factor K for notched ﬂat bar in bending. (R. E. Peterson, ‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol. 23, Nos. 2 to 7, 1951.)

For stress-concentration factors for small grooves in a shaft subjected to torsion. (o) Bar containing shoulders (i) Bar with shoulders in tension (Fig. 4-24)

TABLE 4-4 Stress-concentration factors for relatively grooves in a shaft subject to torsion, K

small

Formula

FIGURE 4-24 Reproduced with permission. Stress-concentration factor K for ﬁlleted ﬂat bar in tension. (R. E. Peterson, ‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol. 23, Nos. 2–7, 1951.)

Refer to Table 4-4.

2

3 1 h1 0:85 K ¼ 1 þ 4 D r5 2:8 2 d or 30:85 2 D 1 7 6 d7 6 d 7 K ¼ 1 þ 6 4 D r5 2 2:8 2 d

h1 r Included angle of V, deg

0.5

1

3

5

2

0 60 90 120

1.85 1.84 1.81 1.66

2.01 2.00 1.95 1.75

2.66 2.54 2.40 1.95

3.23 3.06 2.40 2.00

4.54 3.99 3.12 2.13

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ð4-28aÞ

ð4-28bÞ

STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.16

CHAPTER FOUR

Stress concentration factor theoretical/empirical or otherwise Particular

(ii) Bar with shoulders in bending (Fig. 4-25)

Extreme value

2 K ¼ 1 þ 4 or

2

Formula

3 1 h1 0:85 D r5 5:37 4:8 d

ð4-29aÞ

30:85 D 1 6 d7 6 7 d 7 K ¼ 1 þ 6 4 D r5 2 5:37 4:8 d

ð4-29bÞ

(p) Press-ﬁtted or shrink-ﬁtted members (Table 4-5): (i) Plain member

K ¼ 1:95

(ii) Grooved member

K ¼ 1:34

(iii) Plain member

Kf ¼ 2:00

(iv) Grooved member

Kf ¼ 1:70

(q) Bolts and nuts (Tables 4-6 and 4-7) Bolt and nut of standard proportions

K ¼ 3:85

Bolt and nut having lip

K ¼ 3:00

TABLE 4-5 Stress-concentration factors in shrink-ﬁtted members Particular

K

Kf

Plain Grooved

1.95 1.34

2.00 1.70

FIGURE 4-25 Reproduced with permission. Stress-concentration factor K for stepped bar in bending. (R. E. Peterson, ‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol. 23, Nos. 2 to 7, 1951.)

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4.17

TABLE 4-6 Stress-concentration factors for screw threads Analysis Types of thread

Seely and Smith

Square Sharp V Whitworth US standard Medium Carbon steel National coarse thread Heat-treated Nickel steel

2.0 3.0 2.0

Black (8)

Suggested value

3.35

5 to 6

2.5 2.84 3.85

TABLE 4-7 Stress-concentration factors Kf for screw threads

Type of thread

Peterson (1)

Annealed

Hardened

Rolled Cut

Rolled Cut

Sellers, American National, square thread

2.2

Whitworth rounded roots

1.4

2.8 1.8

3.0 2.6

3.8 3.3

TABLE 4-8 Stress-concentration factors for welds K

Location End or parallel ﬁllet weld

2.7

Reinforced butt

1.2

Tee of transverse ﬁllet weld

1.5

T-butt weld with sharp corners

2.0

TABLE 4-9 Index of sensitivity for repeated stress Average index of sensitivity q

Material

Annealed or soft

Armco iron, 0.02% C Carbon steel 0.10% C 0.20% C (also cast steel) 0.30% C 0.50% C 0.85% C Spring steel, 0.56% C, 2.3 Si, rolled SAE 3140, 0.73 C; 0.6 Cr; 1.3 Ni. Cr–Ni steel 0.8 Cr; 3.5 Ni Stainless steel, 0.3 C; 8.3 Cr, 19.7 Ni Cast iron Copper, electrolitic Duraluminum

0.15–0.20 0.05–0.10 0.10 0.18 0.26

0.25

Heat-treated and drawn at 921 K (6488C)

Heat-treated and drawn at 755 K (4828C)

0.35 0.40 0.45 0.38 0.45 0.25

0.45 0.50 0.57

0.70

0.16 0–0.05 0.07 0.05–0.13

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STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.18

CHAPTER FOUR

Stress concentration factor theoretical/empirical or otherwise Particular

Extreme value

Formula

(r) Crane hook: For crane hook under tensile load

K ¼ 1:56

(s) Rotating disk: For rotating disk with a hole for For thin disk (ring)

Ri !0 Ro

K ¼ 2 K ¼ 1

(t) Eye bar: For eye bar subjected to tensile load

K ¼ 2:8

Stress concentration factors for welds

Refer to Table 4-8.

(u) Notch sensitivity factors (Table 4-9): (i) Notch sensitivity factor for normal stress

For index of sensitivity for repeated stresses. (ii) Fatigue stress concentration factor for normal stress (iii) Notch sensitivity factor for shear stress

(iv) Fatigue stress-concentration factor for shear stress

q ¼

Kf 1 K 1

ð4-30aÞ

q ¼

Kf 1 K0 1

ð4-30bÞ

Refer to Table 4-9. Kf ¼ 1 þ q ðK 1Þ

ð4-31aÞ

Kf ¼ 1 þ q ðK0 1Þ

ð4-31bÞ

q ¼

Kf 1 K 1

Kf ¼ 1 þ q ðK 1Þ

ð4-32Þ ð4-33Þ

STRESS CONCENTRATION IN FLANGED PIPE SUBJECTED TO AXIAL EXTERNAL FORCE The stress in the pipe due to external load F (Fig. 4-25A)

F ð4-33aÞ A where f ¼ depends on the distance x from the ﬂange of the pipe, MPa (psi)

¼ f þ

fm ¼ maximum stress at x ¼ 0, MPa (psi) A ¼ area of the cross section of pipe, m2 (in2 ) F ¼ external load, kN (lbf)

FIGURE 4-25A Pipe and ﬂange under the axial force F

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STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

Particular

Formula

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 3ð1 2 Þ ¼ 10 R2 h2

The value of constant

4.19

ð4-33bÞ

where 2R ¼ 2Ri þ h ¼ mean diameter of pipe, m (in) 2Ro ¼ outer diameter of pipe, m (in) 2Ri ¼ inner diameter of pipe, m (in) h ¼ thickness of pipe, m (in) ¼ Poisson’s ratio of material For plot of the stress ratio

f versus x fm

Refer to Fig. 4-25B.

FIGURE 4-25B Stress concentration region in ﬂanged pipe under axial external force F. Courtesy: Douglas C. Greenwood, Engineering Data for Product Design, McGraw-Hill Publishing Company, New York, 1961.

REDUCTION OR MITIGATION OF STRESS CONCENTRATIONS In designing a machine part, one has to take into consideration the stress concentration occurring in such parts and eliminate or reduce stress concentration. Fig. 4-25C shows various methods used to reduce stress concentration. Stream line ﬂowing analogy in a channel can be applied to force ﬂow lines of a ﬂat plate without any type of ﬂow subject to uniform uniaxial tensile stress as shown in Fig. 4-25C i(a). The stream line ﬂow of water or any ﬂuid is

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STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.20

CHAPTER FOUR

smooth and straight as shown in Fig. 4-25C i(a). If there is any obstruction such as a heavy iron ball or a pipe or stone boulder in the path of ﬂow of water, the ﬂow of water or ﬂuid will not be smooth and straight as shown in Fig. 4-25C i(e). Similarly the force ﬂow lines will not be straight as in case of plate with a circular or elliptical or any shape of holes in a plate as shown in Figs. 4-25C i(b), i(c), i(d) and i( f ). By providing some geometric changes, abrupt change of force-ﬂow lines are smoothened. Fatigue strength of parts with stress raiser can be increased by cold working operation such as shot peening or pressing by balls which creates a nature of stress in thin layers of the part just opposite to the one induced in it. Press ﬁt stress concentration can be reduced by making the gripping portion conical in case of hardening steel parts. Nitriding and plating the parts eliminate the corrosion eﬀect, which combined with stress concentration reduces the fatigue strength of the machine part. (i) Plates:

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(ii) Stepped shafts subject to tensile force:

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4.21

STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.22

CHAPTER FOUR

(iii) Shafts with narrow collar, cylindrical holes and grooves subject to tensile force:

(iv) Shafts subject to bending and torque:

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STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

(v) Screws and nuts under torque:

(vi) Keyways in shafts subject to torque:

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4.23

STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.24

CHAPTER FOUR

(vii) Gears: Maximum Fringe order means maximum stress at Tension side of root point of contact fillet with less stress concentration

Fn

Fr

β

α Fθ

Maximum Fringe order

h ρt

ρt

α

s

b

Forces acting on a Gear Tooth Profile due to Normal Force Fn (a)

Fringe pattern of gear teeth in contact showing stress concentration at root and point of contract (b)

Compression side of root fillet gear tooth with more fringes compare to tension side root fillet stress analysis of gear teeth in contact under load showing photoelastic fringes by using the results of photoelastic experiment (c)

Stress concentration at ﬁllet and at point of contact are shown in photoelastic fringe pattern and also in Fig. c. The stress concentration at ﬁllet can be reduced by providing suitable large ﬁllet radius. Source: From the photoelastic work of K. Lingaiah, Fringe Pattern of Gear-Teeth Showing Stress Concentration at Root and Contact Point, Department of Mechanical Engineering, University Visvesvaraya College of Engineering, Bangalore University, Bangalore, 1973. (viii) Flate plate with and without asymmetrically reinforced circular cutout subjected to uniform uniaxial stress:

FIGURE 4-25C Mitigation of stress concentration in machine members.

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STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.25

STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

Particular

Formula

STRESS INTENSITY FACTOR OR FRACTURE TOUGHNESS FACTOR The energy criterion approach in the fracture mechanics analysis: The energy release rate in case of a crack of length 2a in an inﬁnite plate subject to tensile stress at inﬁnity (Fig. 4-26A). The energy release rate is deﬁned as the rate of change in potential energy with crack area for a linear elastic material.

The critical energy release rate.

G¼

2 a E

ð4-34aÞ

where G ¼ energy release rate E ¼ modulus of elasticity, GPa (psi) a ¼ half crack length, mm (in) Gc ¼

2f ac E

ð4-34bÞ

where f ¼ failure stress, MPa (psi) Gc ¼ material resistance to fracture or critical fracture toughness

FIGURE 4-26A A ﬂat inﬁnite plate with a through thickness crack subject to tensile stress at inﬁnity.

pﬃﬃﬃ pﬃﬃﬃ a

The stress intensity factor for a centrally located straight crack in an inﬁnite plate subjected to uniform uniaxial tensile stress perpendicular to the plane of the crack.

KI ¼

The deﬁnition and unit of critical stress intensity factor KIc .

KIc is the critical stress intensity factor for static loading and plane-strain conditions of maximum constraints and is also referred to as the fracture toughness factor of the material at the onset of rapid pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ fracture dimension of (stress length), i.e. pﬃﬃﬃﬃ pﬃﬃﬃﬃand has MPa m (kpsi in).

The relation between KI and G.

G¼

KI E

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ð4-34cÞ

ð4-34dÞ

STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.26

CHAPTER FOUR

Particular

Formula

Three modes of loading to analyse stress ﬁelds in cracks:

FIGURE 4-26B Three modes of loading for deformation of crack tip.

First mode of loading and stress components at crack tip, Fig. 4-26B (a): The localized stress components at the vicinity of the ‘‘opening mode’’ or ‘‘mode I’’ crack tip in a ﬂat plate subjected to uniform applied stress at inﬁnity from the theory of fracture mechanics (Fig. 4-26C).

KI 3 cos 1 sin sin x ¼ pﬃﬃﬃﬃﬃﬃﬃ 2 2 2 2r KI 3 cos 1 þ sin sin y ¼ pﬃﬃﬃﬃﬃﬃﬃ 2 2 2 2r z ¼ ðx þ y Þ for plane strain ¼0

The crack tip displacement ﬁelds for ‘‘ﬁrst mode’’ (Mode I) in case of linear elastic, isotropic materials.

for plane stress

KI 3 xy ¼ pﬃﬃﬃﬃﬃﬃﬃ cos sin cos 2 2 2 2r rﬃﬃﬃﬃﬃﬃ K r ux ¼ I cos 1 þ 2 sin2 2G 2 2 2 uy ¼

KI 2G

rﬃﬃﬃﬃﬃﬃ r sin þ 1 2 cos2 2 2 2

where G ¼ modulus of shear, GPa (psi) ¼ 3 4 for plane strain ¼ ð3 Þ=ð1 þ Þ for plane stress FIGURE 4-26C State of stress in the vicinity of a crack tip.

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ð4-35aÞ ð4-35bÞ ð4-35cÞ ð4-35dÞ ð4-35eÞ ð4-35f Þ

ð4-35gÞ

STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

Particular

4.27

Formula

Second mode of loading and stress components in the vicinity of crack tip, Fig. 4-26B (b): The localized stress components at the vicinity of the ‘‘second mode’’ or ‘‘sliding mode’’ crack tip in a ﬂat plate subjected to in-plane shear, Fig. 4-26B (b).

KII 3 x ¼ pﬃﬃﬃﬃﬃﬃﬃ sin 2 þ cos cos ð4-35hÞ 2 2 2 2r KII 3 sin cos cos ð4-35iÞ y ¼ pﬃﬃﬃﬃﬃﬃﬃ 2 2 2 2r KII 3 ð4-35jÞ cos 1 sin sin xy ¼ pﬃﬃﬃﬃﬃﬃﬃ 2 2 2 2r z ¼ 0 for plane stress

ð4-35kÞ

z ¼ ðx y Þ for plane strain

ð4-35lÞ

xz ¼ yz ¼ 0 The crack tip displacement ﬁelds for the ‘‘second mode’’ (Mode II) in case of linear elastic, isotropic materials.

ux ¼

KII 2G

uy ¼

rﬃﬃﬃﬃﬃﬃ r sin þ 1 þ 2 cos2 2 2 2

KII 2G

rﬃﬃﬃﬃﬃﬃ r cos 1 2 sin2 2 2 2

ð4-35mÞ ð4-35nÞ ð4-35oÞ

Third mode of loading and stress components in the vicinity of crack tip, Fig. 4-26B (c): The localized stress components at the vicinity of the ‘‘third mode’’ or ‘‘tearing mode III’’ crack tip in a ﬂat plate subjected to out-of-plane shear, Fig. 4-26B (c), in case of linear elastic, isotropic materials.

The crack tip displacement ﬁeld for the ‘‘third mode’’ (Mode III) in case of linear elastic, isotropic materials.

Stress intensity factor:

KIII sin xz ¼ pﬃﬃﬃﬃﬃﬃﬃ 2 2r KIII cos yz ¼ pﬃﬃﬃﬃﬃﬃﬃ 2 2r rﬃﬃﬃﬃﬃﬃ KIII r sin uz ¼ G 2 2

ð4-35pÞ ð4-35qÞ ð4-35rÞ

w¼¼0

The stress intensity factor for a center cracked tension plate (CCT), according to Fedderson (Fig. 4-27a).

pﬃﬃﬃﬃﬃﬃ a KI ¼ a sec 2b

The stress intensity factor for a double edge cracked plate according to Keer and Freedman (Fig. 4-27b).

3 pﬃﬃﬃﬃﬃﬃ a a þ 0:13 Ki ¼ a 1:12 0:61 b b 1=2 a 1 b

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ð4-35sÞ

ð4-35tÞ

STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.28

CHAPTER FOUR

Particular

FIGURE 4-27a

FIGURE 4-27b

The stress intensity factor for the plate with a single edge crack, according to Gross, Srawley and Brown (Fig. 4-27c).

The stress intensity factor for single edged cracked plate/specimen subjected to bending (Mb ) (Fig. 4-27d).

Formula

FIGURE 4-27c

FIGURE 4-27d

2 pﬃﬃﬃﬃﬃﬃ a a KI ¼ a 1:12 0:23 þ 10:6 b b 3 4 a a 21:7 þ 30:4 b b 2 pﬃﬃﬃﬃﬃﬃ a a KI ¼ a 1:112 1:40 þ 7:33 b b 3 4 a a 13:08 þ 14:0 b b

ð4-35uÞ

ð4-35vÞ

Stress intensity factor for the case of angled crack (Fig. 4-27A):

FIGURE 4-27A Through crack in an inﬁnite plate for the general case where the crack plane is inclined at 908 angle from the applied normal stress acting at inﬁnity.

The stress intensity factors for Modes I and II.

KI ¼ KIð0Þ cos2

ð4-36aÞ

KII ¼ KIð0Þ cos sin

ð4-36bÞ

where KIð0Þ is the Mode I stress intensity factor when ¼0

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STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.29

STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

Particular

Equations for stress and displacement components in terms of polar coordinates: The localized stress components at the vicinity of Mode I crack tips in terms of polar coordinates.

The crack tip displacement ﬁelds for ‘‘ﬁrst mode’’ (Mode I) in case of linear elastic, isotropic materials.

Formula

KI 3 r ¼ pﬃﬃﬃﬃﬃﬃﬃ 5 cos cos 2 2 4 2r KI 3 3 cos þ cos ¼ pﬃﬃﬃﬃﬃﬃﬃ 2 2 4 2r KI 3 sin þ sin r ¼ pﬃﬃﬃﬃﬃﬃﬃ 2 2 4 2r

ð4-36dÞ

z ¼ 1 ðr þ Þ

ð4-36f Þ

ð4-36cÞ

ð4-36eÞ

where 1 ¼ 0 for plane stress and 1 is Poisson’s ratio, , for plane strain. These singular ﬁelds only apply as r ! 0. rﬃﬃﬃﬃﬃﬃ K r 3 ð1 þ Þ ð2 1Þ cos cos ð4-36gÞ ur ¼ I 2E 2 2 2 K u ¼ I 2E

rﬃﬃﬃﬃﬃﬃ r 3 ð1 þ Þ ð2 1Þ sin þ sin 2 2 2 ð4-36hÞ

uz ¼

2 z ðr þ 0 Þ E

ð4-36iÞ

where 3 ¼ , 1 ¼ 0, and 2 ¼ for plane stress 1þ ¼ ð3 4Þ, 1 ¼ , and 2 ¼ 0 for plain strain The localized stress components at the vicinity of Mode II crack tip in terms of polar coordinates.

The crack tip displacement ﬁelds for Mode II.

KI is given by Eq. (4-36a). KII sin r ¼ pﬃﬃﬃﬃﬃﬃﬃ 1 3 sin2 2 2 2r

ð4-36jÞ

3KII sin cos2 ð4-36kÞ ¼ pﬃﬃﬃﬃﬃﬃﬃ 2 2 2r rﬃﬃﬃﬃﬃﬃ K r 3 ur ¼ II ð1 þ Þ ð2 1Þ sin þ 3 sin 2E 2 2 2 ð4-36lÞ rﬃﬃﬃﬃﬃﬃ KII r 3 ð1 þ Þ ð2 1Þ cos þ 3 cos u ¼ 2 2 2E 2 ð4-36mÞ

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STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.30

CHAPTER FOUR

Particular

The localized stress components and crack tip displacement ﬁelds for Mode III in terms of polar coordinates.

The critical applied tensile stress necessary for crack extension according to Griﬃth theory for brittle metals.

The modiﬁed Griﬃth’s equation for a small amount of plastic deformation according to Orowan which can be applied to ductile materials at low temperature, high strain rate and localized geometric constraint. The elastic energy release rate for Mode I.

Formula

KIII r ¼ pﬃﬃﬃﬃﬃﬃﬃ sin 2 2r

KIII cos ð4-36pÞ ¼ pﬃﬃﬃﬃﬃﬃﬃ 2 2r rﬃﬃﬃﬃﬃﬃ 2K r sin uz ¼ III ð4-36qÞ G 2 2 rﬃﬃﬃﬃﬃﬃﬃﬃ EU ð4-36rÞ c / a where c ¼ critical applied stress E ¼ Young’s modulus U ¼ surface energy per unit area a ¼ crack length. rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ EðU þ pÞ ¼ ð4-36sÞ a where p ¼ plastic deformation energy per unit area for metallic solid, p U. 1 2 ð4-36tÞ GI ¼ KI2 for plane strain E ¼ KI2 =E

The elastic energy release rate for Mode II. The elastic energy release rate for Mode III. The stress-intensity factor for a centrally located straight crack in an inﬁnite plate subjected to uniform shear stress .

ð4-36nÞ

GII ¼

for plane stress

ð1 2 Þ 2 KII E

ð1 þ Þ 2 KIII E pﬃﬃﬃ pﬃﬃﬃ KI iKII ¼ i a GIII ¼

The stress-intensity magniﬁcation factor for a centrally located straight crack of length 2a in a ﬂat plate whose length 2h and width 2b are very large compared with the crack length subjected to uniform uniaxial tensile stress .

K MF ¼ pﬃﬃﬃ Ipﬃﬃﬃ a

For stress-intensity magniﬁcation factors of plates with straight crack located at various positions in the plate and cylinders subjected to various types of rate of loadings and for various values of a=b, a=d, a=h, a=ðro ri Þ, and other ratios.

Refer to Figs. from 4-28, 4-29 to 4-34.

The factor of safety.

n¼

KIc K

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ð4-36uÞ ð4-36vÞ ð4-36wÞ ð4-37aÞ

ð4-37bÞ

ð4-38Þ

STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.31

FIGURE 4-28 Stress intensity magniﬁcation factor pﬃﬃﬃ pﬃﬃﬃ KI = a for various ratios a=b of a ﬂat plate with a centrally located straight crack under the action of uniform uniaxial tensile stress .

FIGURE 4-29 Stress intensity magniﬁcation factor pﬃﬃﬃ pﬃﬃﬃ KI = a for an oﬀ-center straight crack in a ﬂat plate subjected to uniform unidirectional tensile stress ; solid curves are for the crack tip at A; dashed curves for tip at B.

FIGURE 4-30 Stress intensity magniﬁcation factor pﬃﬃﬃ pﬃﬃﬃ KI = a for an edge straight crack in a ﬂat plate subjected to uniform uniaxial tensile stress for solid curves there are no constraints to bending; the dashed curve was obtained with bending constraints added.

FIGURE 4-31 Stress intensity magniﬁcation factor pﬃﬃﬃ pﬃﬃﬃ KI = a for a rectangular cross-sectional beam subjected to bending Mb .

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STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.32

CHAPTER FOUR

FIGURE 4-32 Stress intensity magniﬁcation factor pﬃﬃﬃ pﬃﬃﬃ KI = a for a ﬂat plate with a centrally located circular hole with two straight cracks under uniform uniaxial tensile stress .

FIGURE 4-33 Stress intensity magniﬁcation factor pﬃﬃﬃ pﬃﬃﬃ KI = a for axially tensile loaded cylinder with a radial crack of a depth extending completely around the circumference of the cylinder.

pﬃﬃﬃ pﬃﬃﬃ FIGURE 4-34 Stress intensity magniﬁcation factor KI = a for a cylinder subjected to internal pressure pi having a radial crack in the longitudinal direction of depth a. Use equation of tangential stress of thick cylinder subjected to internal pressure to calculate the stress at r ¼ ro .

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STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

Particular

4.33

Formula

1

KIc sy =2

2

Critical crack length

ac ¼

For values of critical stress-intensity factor (KIc ) for some engineering materials.

Refer to Table 4-10.

TABLE 4-10 Plane-strain fracture toughness or stress intensity factor KIc for some engineering materials KIc

Material Previous designation Aluminum 2014-T651 2024-T851 7075-T651 7178

UNS designation

A92024-T851 A97075-T651

Titanium Ti-6Al-4V Ti-6Al-4V

R56401 R56401

Steel 4340 4340 H-11 H-11 52100

G43400 G43400 – – G52986

pﬃﬃﬃﬃ MPa m

24.2 26 24 13 115 55 99 60 38.5 27.8 14

Yield strength, xy pﬃﬃﬃﬃ kpsi in

Critical crack length, ac

MPa

kpsi

mm

in

22 24 22 30

455 455 495 490

66 66 72 71

3.6 4.3 3.0 5.8

0.14 0.17 0.12 0.23

105 50

910 1035

132 150

20.5 3.6

0.81 0.14

90 55 35 27 13

860 1515 1790 2070 2070

125 220 260 300 300

16.8 2 <0.6 0.23 <0.06

0.66 0.08 <0.02 0.009 <0.002

Heat treated to a higher strength.

REFERENCES 1. Lingaiah, K., Solution of an Asymmetrically Reinforced Circular Cut-out in a Flat Plate Subjected to Uniform Unidirectional Stress, Ph.D. Thesis, Department of Mechanical Engineering, University of Saskatchewan, Saskatoon, Sask., Canada, 1965. 2. Lingaiah, K., W. P. T. North, and J. B. Mantle, ‘‘Photoelastic Analysis of an Asymmetrically Reinforced Circular Cut-out in a Flat Plate Subjected to Uniform Unidirectional Stress,’’ Proc. SESA, Vol. 23, No. 2 (1966), p. 617. 3. Peterson, R. E., ‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol. 23, No. 27, Pentagon Publishing, Cleveland, Ohio, 1951. 4. Lingaiah, K., ‘‘Eﬀect of Contact Stress on Fatigue Strength of Gears,’’ M.Tech. Thesis, Indian Institute of Technology, Kharagpur, India, 1958. 5. Lingaiah, K., ‘‘Photoelastic Stress Analysis of Gear Teeth Under Load,’’ Department of Mechanical Engineering, University Visveswaraya College of Engineering, Bangalore University, Bangalore, 1980. 6. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 7. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986.

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STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS

4.34

CHAPTER FOUR

8. Lingaiah, K., Machine Design Data Handbook (SI and Customary US Units), McGraw-Hill Publishing Company, New York, 1994. 9. Aidad, T., and Y. Terauchi, ‘‘On the Bending Stress in a Spur Gear,’’ 3 Reports, Bull. JSME, Vol. 5 (1962), p. 161. 10. Dolan, T. J., and E. L. Broghamer, ‘‘A Photoelastic Study of Stresses in Gear Tooth Fillets,’’ Univ. Illinois Exptl. Station. Bull., 335 (1942). 11. Hetenyi, M., The Application of Hardening Resins in Three-Dimensional Photoelastic Studies, J. Appl. Phys., Vol. 10 (1939), p. 295. 12. Shigley, J. E., and L. D. Mitchell, Mechanical Engineering Design, McGraw-Hill Publishing Company, New York, 1983. 13. Greenhood, D. C., Engineering Data for Production Design, McGraw-Hill Publishing Company, New York, 1961. 14. Carlson, R. L., and G. A. Kardomateas, An Introduction to Fatigue in Metals and Composites. 15. Anderson, T. L., Fracture Mechanics—Fundamentals and Application, 2nd edition, CRC Press, New York, 1995. 16. Fedderson, C., ‘‘Discussion’’, in Plane Strain Crank Toughness Testing of High Strength Metallic Materials, ASTM STP410, American Society for Testing Materials, Philadelphia (1967), p. 77. 17. Keer, L. M., and J. M. Freedman, ‘‘Tensile Strip with Edge Cracks,’’ Int. J. Engineering Science, Vol. 11 (1973), pp. 1965–1075. 18. Gross, B., and J. E. Srawley, ‘‘Stress Intensity Factors for Bend and Compact Specimens,’’ Engineering Fracture Mechanics, Vol. 4 (1972), pp. 587–589. 19. Gross, B., J. E. Srawley, and W. E. Brown Jr., Stress Intensity Factors for a Single Edge Notch Tension Specimen by a Boundary Collocation of a Stress Function, NASA Technical Note D-2395, 1964. 20. Damage Tolerant Design Handbook, MICIC-HB-01, Air Force Materials Laboratory, Wright-Patterson Air Force Base, Ohio, December 1972, and supplements.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

5 DESIGN OF MACHINE ELEMENTS FOR STRENGTH SYMBOLS5;6 A b B esz es e0s E F Fm0 G h Ksz Ks K K Kf Kf Mb 0 Mbm Mt 0 Mtm n na nd q qf r t x0 ymax Zb Zt

area of cross-section, m2 (in2 ) a shape factor (b > 0) a constant size coeﬃcient surface coeﬃcient in case of tension and bending surface coeﬃcient in case of torsion Young’s modulus, GPa (Mpsi) normal load (also with suﬃxes and primes), kN (lbf) static equivalent of cyclic load, kN (lbf) modulus of rigidity, GPa (Mpsi) thickness, m (in) size factor surface factor theoretical normal stress-concentration factor theoretical shear stress-concentration factor fatigue normal stress-concentration factor fatigue shear stress-concentration factor bending moment (also with suﬃxes and primes), N m (lbf in) static equivalent of cyclic bending moment, N m (lbf in) twisting moment (also with suﬃxes and primes), N m (lbf in) static equivalent of cyclic twisting moment, N m (lbf in) safety factor a constant actual safety factor (also with suﬃxes) design safety factor (also with suﬃxes) index of sensitivity index of notch sensitivity for alternating stresses notch radius, mm (in) time, h the guaranteed value of x (x0 0) maximum deﬂection ﬂexural section modulus, m3 or cm3 (in3 ) polar section modulus, m3 or cm3 (in3 ) characteristic or scale value ( x0 )

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DESIGN OF MACHINE ELEMENTS FOR STRENGTH

5.2

CHAPTER FIVE

0 su e d x y

00e fb e sy xy nom f " "0 "t "0 "_ v0

normal stress (also with suﬃxes and primes), MPa (psi) initial stress, MPa (psi) ultimate strength, MPa (psi) elastic limit for standard specimen for 12.5 mm (12 in), MPa (psi) design stress (also with suﬃxes), MPa (psi) normal stress in x direction, MPa (psi) yield stress, MPa (psi) normal stress in y direction, MPa (psi) nominal normal stress, MPa (psi) maximum normal stress, MPa (psi) elastic limit for any thickness h between 12.5 mm (12 in) and 75 mm (3 in), MPa (psi) elastic limit for 75 mm (3 in) specimen, MPa (psi) endurance limit in bending, MPa (psi) shear stress (also with suﬃxes and primes), MPa (psi) elastic limit in shear, MPa (psi) yield strength in shear, MPa (psi) shear stress in xy plane, MPa (psi) nominal shear stress, MPa (psi) endurance limit in torsion, MPa (psi) engineering or average strain, mm/m (min/in) true strain, mm/m (min/in) total creep, after a time t, mm/m (min/in) initial creep, mm/m (min/in) creep rate (m/m)/h [(min/in)/h] a constant

Suﬃxes

for

s u y e a b m t max min f o

static strength (u or y ) ultimate strength yield strength elastic limit amplitude bending mean tension maximum minimum endurance limit (also used for reversed cycle) endurance limit repeated cycle

Primes

for

nom max 0e

0 00

(single) (double)

static equivalent combined stress

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DESIGN OF MACHINE ELEMENTS FOR STRENGTH DESIGN OF MACHINE ELEMENTS FOR STRENGTH

Particular

5.3

Formula

STATIC LOADS Inﬂuence of size

FIGURE 5-1 Change of elastic limit with size of section.

FIGURE 5-2 Inﬂuence of size on elastic limits.

The size coeﬃcient (Fig. 5-1, Fig. 5-2, and Table 5-1)

00 esz ¼ 1 0:016 1 e ðh 12:5Þ e

ð5-1Þ

where e ¼ elastic limit for 12.5 mm (0.5 in) 00e ¼ elastic limit for 75 mm (3.0 in) TABLE 5-1 Strength ratios of various materials for use in Eqs. (5-1) and (5-2) Values of 00e =e

Material

Natural state

Annealed

Drawn at 6508C

Drawn at 5358C

Drawn at 4258C

Aluminum, strong, wrought Tobin bronze Monel metal, forged Ductile iron Low-carbon steel, C < 0:20% Medium-carbon steel, 0.30 to 0.50% C Nickel steel, SAE 2340 Cr–Ni steel, SAE 3140 Cast iron, Class no. 20 Cast iron, Class no. 25 Cast iron, Class no. 35 Wrought iron

0.93 0.90 0.80 0.80 0.84 — — — 0.55 0.73 0.60 0.55

— — — 0.98 — 0.85 0.86 0.86 — — — —

— — — — — 0.72 0.80 0.75 — — — —

— — — — — 0.59 0.74 0.70 — — — —

— — — — — 0.53 — 0.65 —

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DESIGN OF MACHINE ELEMENTS FOR STRENGTH

5.4

CHAPTER FIVE

Particular

Formula

250 00 300 4h þ e ð4h 50Þ e

ð5-2Þ

1 ksz

ð5-3Þ

The size factor

Ksz ¼

The relation between size coeﬃcient and size factor

esz ¼

The elastic limit for any thickness h between 12.5 mm and 75 mm can be determined from the relation (Fig. 5-1)

00e ¼ e

ðe 00e Þðh 12:5Þ ð75 12:5Þ

ð5-4Þ

INDEX OF SENSITIVITY Ka 1 K 1

The index of sensitivity

q¼

The actual or real stress-concentration factor

Ka ¼ 1 þ qðK 1Þ

ð5-5Þ ð5-6Þ

SURFACE CONDITION (Fig. 5-3) 1 es

The surface factor for the case of tension and bending

Ks ¼

The surface coeﬃcient in case of torsion

e0s ¼ 0:425 þ 0:575es

FIGURE 5-3 Reciprocals of stress-concentration factors caused by surface conditions.

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ð5-7Þ ð5-8Þ

DESIGN OF MACHINE ELEMENTS FOR STRENGTH DESIGN OF MACHINE ELEMENTS FOR STRENGTH

Particular

SAFETY FACTOR The general equation for design safety factor (Table 5-2)

5.5

Formula

n ¼ k1 k2 k3 k4 . . . km na where k1 k2 k3 k4 .. . na

ð5-9Þ

¼ Ksz ¼ size factor ¼ Ks ¼ surface factor ¼ Kl ¼ load factor (Table 14-3) ¼ material factor ¼ actual safety factor (Table 5-2).

TABLE 5-2 Actual safety factora

Circumstance

Actual factor of safety or reliability factor, na

Strength properties of material well known, load accurately predictable, parts produced with close dimensional control and brought to close tolerance speciﬁcations, and low-weight criteria

1

Load accurately predictable and low-weight and low-cost criteria

1.1–1.5

Load accurately predictable and low-cost criteria (low-weight–no criteria)

1.5–2

Overloads expected, materials ordinary but reliability important

2–3

Strength properties not well deﬁned, loading uncertain, human life at stake if failure occurred, high maintenance and shutdown cost

3

a

These values are recommended for use in design, in the absence of speciﬁc reliability data.

nud ¼ Ksz Ka nua

ð5-10Þ

Tension

0:45sy a 0:60sy

ð5-11Þ

Shear

a ¼ 0:40sy

ð5-12Þ

Bearing

a ¼ 0:90sy

ð5-13Þ

Bending

ð5-14Þ 0:60sy a 0:75sy X X X X Wl þ KFl þ Fw þ Fme F¼ Wd þ

The design safety factor based on ultimate strength The relationships between allowable stress and speciﬁed minimum yield strength using the AISC Code are given here:

The expression for forces or loads used to ﬁnd stresses in machine members or structures as per AISC Code.

ð5-15Þ P where P Wd ¼ sum of dead loads Wl ¼ sum of all stationary or static live loads Fl ¼ impact or dynamic live load Fw ¼ wind load on the structure P Fme ¼ load which accounts for earthquakes, hurricanes, etc. K ¼ service factor obtained from Table 5-3. The value of design normal stress

d a

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ð5-16Þ

DESIGN OF MACHINE ELEMENTS FOR STRENGTH

5.6

CHAPTER FIVE

Particular

Formula

TABLE 5-3 AISC service factor K for use in Eq. (5-15) Particular

K

For support of elevators

2

For cab-operated traveling-crane support girders and their connections

1.25

For pendant-operated traveling-crane support girders and their connections

1.10

For support of light machinery, shaft- or motor-driven

1.20

For supports of reciprocating machinery or power-driven units

1.50

For hangers supporting ﬂoors and balconies

1.33

The value of design shear stress

d a

The design safety factor

nd ¼

ð5-16aÞ

strength ¼ ns nL stress

ð5-17Þ

where ns ¼ safety factor to take into account the uncertainty of strength nL ¼ safety factor to take into account the uncertainty of load. The equation for design safety factor

nd ¼

strength in force units applied force or load

ð5-18Þ

The realized safety factor

nr ¼

s

ð5-19Þ

The design safety factor based on elastic limit

ned ¼ Ksz Ka nea

ð5-20Þ

The design safety factor based on yield strength

nyd ¼ Ksz Ka nya

ð5-21Þ

The design safety factor based on endurance limit on bending

nfd ¼ Ksz Ks Kld nfa

ð5-22Þ

Design stress based on elastic limit

or nr ¼

s

where Kld ¼ load factor ed ¼ e ned

ð5-23Þ

Design stress based on ultimate strength

ud ¼

su nud

ð5-24Þ

Design stress based on yield strength

yd ¼

sy nyd

ð5-25Þ

Design stress based on yield strength in shear

yd ¼

sy nyd

ð5-26Þ

Static design stress

sd ¼

su nud

or

sy nyd

as the case may be

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ð5-27Þ

DESIGN OF MACHINE ELEMENTS FOR STRENGTH DESIGN OF MACHINE ELEMENTS FOR STRENGTH

Particular

5.7

Formula

sf nfd

Design stress based on endurance limit

fd ¼

The corrected fatigue strength or design fatigue strength

sf ¼ Ksr Ksz Kld KR KT Kme 0sf

ð5-28aÞ

se ¼ ksr ksz kld kR kT kme 0se

ð5-28bÞ

The corrected endurance limit or design endurance limit

The size facto ksz for bending or torsion of round bars made of ductile materials according to Juvinall

where

ð5-28Þ

0se 0sf

¼ endurance limit of test specimen ¼ fatigue strength of test specimen Ksr ¼ surface factor Ksz ¼ size factor Kld ¼ load factor KR ¼ reliability factor KT ¼ temperature factor Kme ¼ miscellaneous-eﬀect factor also known as fatigue strength reduction factor 1=Kf (5-28c)

8 1 > > > < 0:9 Ksz ¼ > 0:8 > > : 0:7

d < 10 mm ð0:4 inÞ 10 mm ð0:4 inÞ < d < 50 mm ð2 inÞ 50 mm ð2 inÞ < d < 100 mm ð4 inÞ 100 mm ð4 inÞ < d < 150 mm ð5 inÞ ð5-28dÞ

The size factor for axial force

Ksz ¼ 0:7 to 0:9

The size factor as suggested by the ASME national standard on ‘‘Design of Transmission Shafting’’

Ksz ¼

The surface factor

8 1:00 > > > < 0:90 Ksr ¼ > 0:87 > > : 0:79

d 0:19 1:85d

ð5-28eÞ 2 < d < 10 in

0:19

50 < d < 250 mm

for longitudinal hand polish for hand burnish for smooth mill cut for rough mill cut

ð5-28f Þ

ð5-28gÞ

Also refer to Fig. 5-3 for surface coeﬃcient esr ¼ For a rectangular cross-section in bending

1 Ksr

or Ksr ¼

1 esr

pﬃﬃﬃﬃ d ¼ 0:81 A

ð5-28hÞ

where A ¼ area of the cross section The eﬀective diameter of round-section corresponding to a nonrotating solid or hollow round-section

de ¼ 0:370D

The eﬀective diameter of a rectangular section of dimensions h b which has A0:95cr ¼ 0:05bh

de ¼ 0:808ðhbÞ1=2

ð5-28iÞ

where D ¼ diameter

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ð5-28jÞ

DESIGN OF MACHINE ELEMENTS FOR STRENGTH

5.8

CHAPTER FIVE

Particular

The equivalent diameter rotating-beam specimen for any cross-section according to Shigley and Mitchell

The load factor according to Shigley

Formula

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A95 deq ¼ ð5-28jÞ 0:0766 where A95 is the portion of the cross sectional area of the nonround part that is stressed between 95% and 100% of the maximum stress.

kId

8 0:923 > > > <1 ¼ >1 > > : 0:577

axial loading sut 1520 MPa ð220 kpsiÞ axial loading sut 1520 MPa ð220 kpsiÞ bending torsion and shear

ð5-28kÞ The fatigue stress concentration factor which is used here as the fatigue strength reduction factor at endurance limit 106 cycles

Kf ¼ 1 þ qðkt 1Þ

The fatigue strength reduction factor for lives less 0 than N ¼ 106 cycles is Kf and is given by

0 ¼ aN b Kf

ð5-28lÞ

where Kf , Kt and q have the same meaning as given in Chapter 4. ð5-28mÞ where a ¼

1 Kf

1 1 and b ¼ log 3 Kf

ð5-28nÞ

0 ¼ 1 at 103 cycles. Kf

For reliability factor KR

Refer to Table 5-3A. TABLE 5-3A Reliability correction factor based on a standard deviation equal to 8% or the mean fatigue limit.

The temperature factor as suggested by Shigley and Mitchell

Reliability, %

KR

50 90 99 99.9 99.999

1.000 0.897 0.814 0.743 0.659

8 for T 4508C ð8408FÞ > <1 KT ¼ 1 0:0058 ðT 450Þ for 4508C < T < 5508C > : 1 0:0032 ðT 840Þ for 8408F < T < 10208F

ð5-28pÞ These equations are applicable to steel. These cannot be used for Al, Mg, and Cu alloys. For typical fracture surfaces for laboratory test specimens subjected to range of diﬀerent loading conditions

Refer to Fig. 5-3A.

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DESIGN OF MACHINE ELEMENTS FOR STRENGTH

FIGURE 5.3A Typical fracture surfaces for laboratory test specimens subjected to a range of diﬀerent loading conditions. Courtesy: Reproduced from Metals Handbook, Vol. 10, 8th edition, p. 102, American Society for Metals, Metals Park, Ohio, 1975.

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DESIGN OF MACHINE ELEMENTS FOR STRENGTH

5.10

CHAPTER FIVE

Particular

Formula

THEORIES OF FAILURE

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ðx y Þ2 þ 4xy

The maximum normal stress theory or Rankine’s theory

e ¼ 12 ðx þ y Þ þ

The maximum shear stress theory or Guest’s theory

e ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ðx y Þ2 þ 4xy

ð5-30Þ

The shear-energy theory or constant energy-ofdistortion or Hencky–von Mises theory

e ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ðx y Þ2 þ 3xy

ð5-31Þ

The maximum strain theory or Saint Venant’s theory

e ¼

1 2

ð5-29Þ

ð1 Þðx þ y Þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 þ ð1 þ Þ ðx y Þ2 þ 4xy

ð5-32Þ

The bearing stress which causes failure for no friction at the surface of contact

b ¼ 1:81e

ð5-33Þ

The bearing stress which causes failure for the friction at the surface of contact

b ¼ 2e

ð5-34Þ

The fatigue stress-concentration factor for normal stress

Kf ¼ qf ðK 1Þ þ 1

ð5-35Þ

The fatigue stress-concentration factor for shear stress

Kf ¼ qf ðK 1Þ þ 1

ð5-36Þ

CYCLIC LOADS (Figs. 5-4 and 5-5)

The empirical formula for notch sensitivity for alternating stress of steel

r2u qf ¼ 1 exp 0:904 106

Notch sensitivity curves for steel and aluminum alloys

Refer to Fig. 5-6.

The empirical formula for notch sensitivity for alternating stress for high-strength aluminum alloys having u ¼ 415 to 550 MPa (60 to 80 kpsi)

qf ¼ 1 exp

Endurance strength for ﬁnite life

0f ¼ f

106 N

r 0:01

ð5-38Þ

0:09

where N ¼ required life in cycles. The empirical relation between ultimate strength and endurance limits for various materials

ð5-37Þ

Refer to Tables 5-4 and 5-5.

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ð5-39Þ

DESIGN OF MACHINE ELEMENTS FOR STRENGTH DESIGN OF MACHINE ELEMENTS FOR STRENGTH

5.11

FIGURE 5-4 Types of fatigue stress variations.

1•0

2 Pa (140kgf / mm ) 80M / mm2) f g k 7 0 1 ( σ uf MPa 2) 050 =1 (70kgf / mm σ uf Pa 2) M 0 m m / f 9 g k 2 4 ( =6 a MP σ uf 0 1 =4 σ uf

= 13

Notch sensitivity, q

0•8 0•6 0•4

STEELS

0•2 0

ALUMINUM ALLOY

0

0•5

1•0

1•5 2•0 2•5 Notch radius r, mm

3•0

3•5

4•0

1 kgf/mm2 = 9.8066 N/mm2

FIGURE 5-5 Modiﬁed Goodman diagram.

FIGURE 5-6 Notch-sensitivity curves for steel and aluminum alloys.

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DESIGN OF MACHINE ELEMENTS FOR STRENGTH

5.12

CHAPTER FIVE

TABLE 5-4 Empirical relationship between ultimate strength and endurance limits for various materials (approximate) Tension, compression, and bending (reversed or repeated cycle)a

Torsion (reversed or repeated cycle)a

Gray cast iron

ft ¼ 0:6fb to 0:7fb b ¼ 1:2fb to 1:5fb

¼ 0:75fb to 0.9fb ¼ 1:2f to 1:3f

Carbon steels

ot ¼ 1:6fb ob ¼ 1:5fb

o ¼ 1:8f to 2f

Steels (general)

ft ¼ 0:7fb to 0:8fb ft ¼ 0:36u ; ot ¼ 0:5u fb ¼ 0:46u ; ob ¼ 0:6u

f ¼ 0:55fb to 0:58fb f ¼ 0:22u o ¼ 0:3u

Alloy steels

ft ¼ 0:95fb ot ¼ 1:5ft to 1:6ft ob ¼ 1:6fb

o ¼ 1:8f to 2f

Aluminum alloys

ot ¼ 0:7fb ob ¼ 1:8fb

f ¼ 0:55fb to 0:58fb o ¼ 1:4f to 2f

Material

f ¼ 0:58fb o ¼ 1:4f to 2f 6 0:09 10 0f ¼ f N

Copper alloys

Endurance strength for ﬁnite life a

f —ensurance limit (also for reversed cycle); o—endurance for repeated cycle; t—tension; b—bending; u—ultimate; N—number of cycles

TABLE 5-5 The empirical relation for endurance limit Endurance limit, f Material

Bending

Axial

Torsion

For steel and other ferrous materials [for u < 1374 MPa (199.5 kpsi)] For nonferrous materials

1/2–5/8u 1/4–1/3u

7/20–5/8u 7/40–1/3u

7/80–5/32u 7/160–1/12u

STRESS-STRESS AND STRESS-LOAD RELATIONS Axial load The maximum stress

max ¼

Fmax A

ð5-40Þ

The minimum stress

min ¼

Fmin A

ð5-41Þ

The load amplitude

Fa ¼

Fmax Fmin 2

ð5-42Þ

The mean load

Fm ¼

Fmax þ Fmin 2

ð5-43Þ

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DESIGN OF MACHINE ELEMENTS FOR STRENGTH DESIGN OF MACHINE ELEMENTS FOR STRENGTH

Particular

5.13

Formula

The stress amplitude (Figs. 5-4 and 5-5)

a ¼

Fa A

ð5-44Þ

The mean stress

m ¼

Fm A

ð5-45Þ

The ratio of amplitude stress to mean stress

a F ¼ a m F m

ð5-46Þ

The static equivalent of cyclic load Fm Fa

Fm0 ¼ Fm þ

The static equivalent of mean stress m a

0m ¼

The Gerber parabolic relation (Fig. 5-7)

sd F fd a

Fm0 A a m 2 þ ¼1 fd ud

ð5-47Þ ð5-48Þ ð5-49Þ

FIGURE 5-7 Graphical representation of steady and variable stresses.

The Goodman relation (Figs. 5-5, 5-7, and 5-9)

a þ m ¼1 fd ud

ð5-50Þ

The Soderberg relation (Figs. 5-7 and 5-8)

a þ m ¼1 fd yd

ð5-51Þ

Bending loads The maximum stress

max ¼

MbðmaxÞ Zb

ð5-52Þ

The minimum stress

min ¼

MbðminÞ Zb

ð5-53Þ

The bending moment amplitude

Mba ¼

MbðmaxÞ MbðminÞ 2

ð5-54Þ

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DESIGN OF MACHINE ELEMENTS FOR STRENGTH

5.14

CHAPTER FIVE

Particular

Formula

FIGURE 5-8 Representation of safe limit of mean stress and stress amplitude by Soderberg criterion.

MbðmaxÞ þ MbðminÞ 2

ð5-55Þ

The mean bending moment

Mbm ¼

The bending stress amplitude

ba ¼

Mba Zb

ð5-56Þ

The mean bending stress

bm ¼

Mbm Zb

ð5-57Þ

The ratio of stress amplitude to mean stress

ba M ¼ ba bm Mbm

ð5-58Þ sd Mba fd

The static equivalent of cyclic bending moment Mbm Mba

0 ¼ Mbm þ Mbm

The static equivalent of cyclic stress

0bm ¼

The Gerber parabolic relation (Fig. 5-7)

ba 2bm þ ¼1 fd 2ud

ð5-61Þ

The Goodman straight-line relation (Figs. 5-5, 5-7, and 5-9)

ba bm þ ¼1 fd ud

ð5-62Þ

The Soderberg straight-line relation (Figs. 5-7 and 5-8)

ba bm þ ¼1 fd yd

ð5-63Þ

MtðmaxÞ Zt

ð5-64Þ

0 Mbm Zb

ð5-59Þ ð5-60Þ

Torsional moments The maximum shear stress

max ¼

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DESIGN OF MACHINE ELEMENTS FOR STRENGTH DESIGN OF MACHINE ELEMENTS FOR STRENGTH

Particular

5.15

Formula

FIGURE 5-9 Representation of safe limit of mean stress and stress amplitude by Goodman criterion.

The minimum shear stress

min ¼

MtðminÞ Zt

ð5-65Þ

The load amplitude

Mta ¼

MtðmaxÞ MtðminÞ 2

ð5-66Þ

The mean load

Mtm ¼

MtðmaxÞ þ MtðminÞ 2

ð5-67Þ

The shear stress amplitude

a ¼

Mta Zt

ð5-68Þ

The mean shear stress

m ¼

Mtm Zt

ð5-69Þ

The ratio of stress amplitude to mean stress

a M ¼ ta m Mtm

ð5-70Þ

The static equivalent of cyclic twisting moment Mtm Mta

0 ¼ Mtm þ Mtm

sd Mtd fd

0 Mtm Zt

ð5-71Þ

The static equivalent of cyclic stress

m0 ¼

The Gerber parabolic relation (Fig. 5-7)

a 2 þ 2m ¼ 1 fd ud

ð5-73Þ

The Goodman straight-line relation (Figs. 5-5, 5-7, and 5-9)

a þ m ¼1 fd ud

ð5-74Þ

The Soderberg straight-line relation (Figs. 5-7 and 5-8)

a þ m¼1 fd yd

ð5-75Þ

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ð5-72Þ

DESIGN OF MACHINE ELEMENTS FOR STRENGTH

5.16

CHAPTER FIVE

Particular

Formula

THE COMBINED STRESSES Method 1 sd fd a

ð5-76Þ

sd fd a

ð5-77Þ

The static equivalent of m a

0m ¼ m þ

The static equivalent of m a

m0 ¼ m þ

The maximum normal stress theory or Rankine’s theory

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0 02 e ¼ m þ 02 m þ 4m

The maximum shear theory or Coulomb’s or Tresca criteria or Guest’s theory

1 2

ð5-78Þ

e ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 02 02 m þ 4m

ð5-79Þ

The distortion energy theory or Hencky–von Mises theory

e ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 02 02 m þ 3m

ð5-80Þ

The maximum strain theory or Saint Venant’s theory

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0 1 02 e ¼ 2 ð1 Þm þ ð1 þ Þ 02 m þ 4m

ð5-81Þ

The combined maximum normal stress

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 00max ¼ 12 max þ 2max þ 4max

ð5-82Þ

The combined minimum normal stress

00min

The combined maximum shear stress

00 ¼ 12 max

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2max þ 4max

ð5-84Þ

The combined minimum shear stress

00 ¼ 12 min

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2min þ 4min

ð5-85Þ

Method 2

The combined maximum normal stress according to strain theory

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ¼ min þ 2min þ 4min

ð5-83Þ

1 2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 00max ¼ 12 ð1 Þmax þ ð1 þ Þ 2max þ 4max ð5-86Þ

00min

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ¼ 12 ð1 Þmin þ ð1 þ Þ 2min þ 4min

The combined maximum octahedral shear stress

00 max

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2max þ 3max ¼ 12

ð5-88aÞ

The combined minimum octahedral shear stress

00 min

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2min þ 3min ¼

ð5-88bÞ

The combined mean stress

00m ¼

The combined minimum normal stress according to strain theory

1 2

00max þ 00min 2

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ð5-87Þ

ð5-88cÞ

DESIGN OF MACHINE ELEMENTS FOR STRENGTH DESIGN OF MACHINE ELEMENTS FOR STRENGTH

Particular

The combined stress amplitude The Gerber parabolic relation (Fig. 5-7)

5.17

Formula

00max 00min 2 00 2 00a m þ ¼1 fd ud 00a ¼

ð5-88dÞ ð5-88eÞ

The Goodman straight-line relation (Figs. 5-5, 5-7, and 5-9)

00a 00 þ m ¼1 fd ud

ð5-88f Þ

The Soderberg straight-line relation (Figs. 5-7 and 5-8)

00a 00 þ m ¼1 fd yd

ð5-88gÞ

COMBINED STRESSES IN TERMS OF LOADS Method 1 Maximum shear stress theory

The shear energy theory

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0 0 2 Mbm Fm0 2 Mtm þ þ4 Zb A Zt sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0 2 0 Mbm Fm0 2 e Mtm ¼ þ þ3 ned Zb A Zt e ¼ ned

ð5-89aÞ ð5-89bÞ

where d 3 d 3 and Zt ¼ 32 16 2 d A¼ 4 for solid shafts 2sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 MbðmaxÞ Fmax 2 MtðmaxÞ 2 5 1 1 4 þ þ4 þ Zb A Zt fd d 2sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 MbðminÞ Fmin 2 MtðminÞ 2 5 þ þ4 þ4 Zb A Zt 1 1 ¼2 ð5-90aÞ þ fd d 2sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 MbðmaxÞ Fmax 2 MtðmaxÞ 2 5 1 1 4 þ þ3 þ Zb A Zt fd d 2sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 MbðminÞ Fmin 2 MtðminÞ 2 5 þ þ3 þ4 Zb A Zt Zb ¼

Method 2 Maximum shear stress theory

The shear energy theory

1 1 ¼2 þ fd d

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ð5-90bÞ

DESIGN OF MACHINE ELEMENTS FOR STRENGTH

5.18

CHAPTER FIVE

Particular

Formula

CREEP Creep in tension When the curve for total creep "t is approximated as a straight line its equation is

"t ¼ "0 þ "t

ð5-91aÞ

The creep rate "_ can be approximated by the equation

"_ ¼ B n

ð5-91bÞ

Creep rate "_ , when extrapolated into the region of lower stresses, can be determined with greater accuracy by the hyperbolic sine term

Refer to Table 5-6 for creep constants B and n. ð5-91cÞ "_ ¼ 0 sinh 1

True strain Creep life of aluminum Time for the stress to decrease from an initial value of 0 to a value of

"0 ¼ lnð1 þ "Þ 1 "cr ¼ n "_ n 1 1 0 1 t¼ EBðn 1Þn0 1

ð5-91dÞ ð5-92Þ ð5-93Þ

Creep in bending

The maximum stress at the extreme ﬁbers in case of bending of beam is given by the relation

¼

The maximum deﬂection of a cantilever beam loaded at free end by a load F

ymax ¼

C1 BD

1=n ð5-94Þ

Mb

tF n l n þ 2 Dðn þ 2Þ

ð5-95Þ

2n þ 1 h ð2bÞn 1 2 where D ¼ 1 n B 2þ n Creep constants B and n are taken from Table 5-6. TABLE 5-6 Creep constants for various steels for use in Eqs. (5-91b) to (5-95) Temperature 8C

Steel 0.39% C 0.30% C 0.45% C 2% Ni, 0.8% Cr, 0.4% Mo 2% Ni, 0.3% C, 1.4% Mn 12% Cr, 3% W, 0.4% Mn Ni-Cr-Mo Ni-Cr-Mo 12% Cr

400 400 475 450 450 550 500 500 455

B

n 36

14 10 44 1030 — 10 1019 21 1022 24 1014 12 1016 16 1012 12 1022

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8.6 6.9 6.5 3.2 4.7 1.9 2.7 1.3 4.4

DESIGN OF MACHINE ELEMENTS FOR STRENGTH DESIGN OF MACHINE ELEMENTS FOR STRENGTH

Particular

5.19

Formula

RELIABILITY The probability function or frequency function The cumulative probability function

The sample mean or arithmetic mean of a sample

p ¼ f ðxÞ Fðxj Þ ¼

X

xi xj

ð5-96Þ f ðxi Þ

ð5-97Þ

where f ðxÞ is the probability density x þ x2 þ x3 þ x4 þ þ xn x ¼ 1 n n 1X x ¼ n i¼1 i

ð5-98aÞ ð5-98bÞ

The population mean of a population consisting of n elements

where xi is the ith value of the quantity n is the total number of measurements or elements x þ x2 þ x3 þ x4 þ þ xn ¼ 1 ð5-99aÞ n n 1X x ð5-99bÞ ¼ n i¼1 i

The sample variance

s2x ¼

A suitable equation for variance for use in a calculator

s2x ¼

ðx1 xÞ2 þ ðx2 xÞ2 þ þ ðxn xÞ2 ð5-100aÞ n1 n 1 X ðx xÞ2 ð5-100bÞ ¼ n 1 i¼1 i P

The sample standard deviation (the symbol used for true standard deviation is ^) A suitable equation for standard deviation for use in a calculator

x2 x2 n

"

n 1 X ðx xÞ2 sx ¼ n 1 i¼1 i

8 <X sx ¼ :

#1=2 ð5-102Þ

P 2 91=2 x = x n ; n1 2

The coeﬃcient of variation

xÞ100 c ¼ ðsx =

The normal, or Gaussian, distribution (Fig. 5-10)

2 2 1 f ðxÞ ¼ pﬃﬃﬃﬃﬃﬃ eðxÞ =2^ ^ 2

The normal distribution as deﬁned by parameters, the mean and standard deviation ^ according to the relation for the relative frequency f ðtÞ, which is the ordinate at t

ð5-101Þ

ð5-103Þ ð5-104Þ

1<x<1

2 1 eðt =2Þ f ðtÞ ¼ pﬃﬃ ð2Þ

ð5-105Þ ð5-106Þ

where t ¼ ðx Þ=^ . Refer to Table 5-7 for ordinate f ðtÞ [i.e. y ¼ f ðtÞ] for various values of t.

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DESIGN OF MACHINE ELEMENTS FOR STRENGTH

5.20

CHAPTER FIVE

Particular

FIGURE 5-10 The shapes of normal distribution curves for various and constant .

Formula

FIGURE 5-11 The Gaussian (normal) distribution curve.

Refer to Table 5-8 for area under the standard normal distribution curve. The area under normal distribution curve to the right of t (Fig. 5-11)

Error function or probability integral

BðtÞ ¼ 1 AðtÞ

ð5-107Þ

where AðtÞ is the area to the left of t. The area under the entire normal distribution curve is AðtÞ þ BðtÞ and is equal to unity. The term BðtÞ can be found from Table 5-8 or by integrating the area under the curve. ð 2 x t2 erfðxÞ ¼ pﬃﬃﬃ e dt ð5-108Þ 0 Refer to Table 5-9 for erfðxÞ for various values of x. ¼ s þ

ð5-109Þ

The resultant mean of subtracting the means of two populations

¼ s

ð5-110Þ

The resultant standard deviation for both subtraction and addition of two standard deviations ^s and ^

^ ¼

The resultant mean of adding the means of two populations (Fig. 5-12)

FIGURE 5-12 Distribution curves for two means of populations.

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ^2s þ ^2

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ð5-111Þ

DESIGN OF MACHINE ELEMENTS FOR STRENGTH DESIGN OF MACHINE ELEMENTS FOR STRENGTH

5.21

TABLE 5-7 Standard normal curve ordinates

2 1 y ¼ pﬃﬃﬃﬃﬃﬃ et =2 2p

t

0

1

2

3

4

5

6

7

8

9

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

.3989 .3970 .3910 .3814 .3683 .3521 .3332 .3123 .2897 .2661 .2420 .2179 .1942 .1714 .1497 .1295 .1109 .0940 .0790 .0656 .0540 .0440 .0355 .0283 .0224 .0175 .0136 .0104 .0079 .0060 .0044 .0033 .0024 .0017 .0012 .0009 .0006 .0004 .0003 .0002

.3989 .3965 .3902 .3802 .3668 .3503 .3312 .3101 .2874 .2637 .2396 .2155 .1919 .1691 .1476 .1276 .1092 .0925 .0775 .0644 .0529 .0431 .0347 .0277 .0219 .0171 .0132 .0101 .0077 .0058 .0043 .0032 .0023 .0017 .0012 .0008 .0006 .0004 .0003 .0002

.3989 .3961 .2894 .3790 .3653 .3485 .3292 .3079 .2850 .2613 .2371 .2131 .1895 .1669 .1456 .1257 .1074 .0909 .0761 .0632 .0519 .0422 .0339 .0270 .0213 .0167 .0129 .0099 .0075 .0056 .0042 .0031 .0022 .0016 .0012 .0008 .0006 .0004 .0003 .0002

.3988 .3956 .3885 .3778 .3637 .3467 .3271 .3056 .2827 .2589 .2347 .2107 .1872 .1647 .1435 .1238 .1057 .0893 .0748 .0620 .0508 .0413 .0332 .0264 .0208 .0163 .0126 .0096 .0073 .0055 .0040 .0030 .0022 .0016 .0011 .0008 .0005 .0004 .0003 .0002

.3986 .3951 .3876 .3765 .3621 .3448 .3251 .3034 .2803 .2565 .2323 .2083 .1849 .1626 .1415 .1219 .1040 .0878 .0734 .0608 .0498 .0404 .0325 .0258 .0203 .0158 .0122 .0093 .0071 .0053 .0039 .0029 .0021 .0015 .0011 .0008 .0005 .0004 .0003 .0002

.3984 .3945 .3867 .3752 .3605 .3429 .3230 .3011 .2780 .2541 .2299 .2059 .1826 .1604 .1394 .1200 .1023 .0863 .0721 .0596 .0488 .0396 .0317 .0252 .0198 .0154 .0119 .0091 .0069 .0051 .0038 .0028 .0020 .0015 .0010 .0007 .0005 .0004 .0002 .0002

.3982 .3939 .3857 .3739 .3589 .3410 .3209 .2989 .2756 .2516 .2275 .2036 .1804 .1528 .1374 .1182 .1006 .0848 .0707 .0584 .0487 .0387 .0310 .0246 .0194 .0151 .0116 .0088 .0067 .0050 .0037 .0027 .0020 .0014 .0010 .0007 .0005 .0003 .0002 .0002

.3980 .3932 .3847 .3725 .3572 .3391 .3187 .2966 .2932 .2492 .2251 .2012 .1781 .1561 .1354 .1163 .0989 .0833 .0694 .0573 .0468 .0379 .0303 .0241 .0189 .0147 .0113 .0086 .0065 .0048 .0036 .0026 .0019 .0014 .0010 .0007 .0005 .0003 .0002 .0002

.3977 .3925 .3836 .3712 .3555 .3372 .3166 .2943 .2709 .2468 .2227 .1989 .1758 .1539 .1334 .1145 .0973 .0818 .0681 .0562 .0459 .0371 .0297 .0235 .0184 .0143 .0110 .0084 .0063 .0047 .0035 .0025 .0018 .0013 .0009 .0007 .0005 .0003 .0002 .0001

.3973 .3918 .3815 .3697 .3538 .3352 .3144 .2920 .2685 .2444 .2203 .1965 .1736 .1518 .1315 .1127 .0957 .0804 .0669 .0551 .0449 .0363 .0290 .0229 .0180 .0139 .0107 .0081 .0061 .0046 .0034 .0025 .0018 .0013 .0009 .0006 .0004 .0003 .0002 .0001

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DESIGN OF MACHINE ELEMENTS FOR STRENGTH

5.22

CHAPTER FIVE

TABLE 5-8 Areas under the standard normal distribution curve

AðtÞ ¼

ðt 0

2 1 pﬃﬃﬃﬃﬃﬃ et =2 dt 2p

t

0

1

2

3

4

5

6

7

8

9

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

.0000 .0398 .0793 .1179 .1554 .1915 .2258 .2580 .2881 .3159 .3413 .3643 .3849 .4032 .4192 .4332 .4452 .4554 .4641 .4713 .4772 .4821 .4861 .4893 .4918 .4938 .4953 .4965 .4974 .4981 .4987 .4990 .4993 .4995 .4997 .4998 .4998 .4999 .4999 .5000

.0040 .0438 .0832 .1217 .1591 .1950 .2291 .2612 .2910 .3186 .3438 .3665 .3869 .4049 .4207 .4345 .4463 .4564 .4649 .4719 .4778 .4826 .4864 .4896 .4920 .4940 .4955 .4966 .4975 .4982 .4987 .4991 .4993 .4995 .4997 .4998 .4998 .4999 .4999 .5000

.0080 .0478 .0871 .1255 .1628 .1985 .2324 .2642 .2939 .3212 .3461 .3686 .3888 .4066 .4222 .4357 .4474 .4573 .4656 .4726 .4783 .4830 .4868 .4898 .4922 .4941 .4956 .4967 .4976 .4982 .4987 .4991 .4994 .4995 .4997 .4998 .4999 .4999 .4999 .5000

.0120 .0517 .0910 .1293 .1664 .2019 .2357 .2673 .2967 .3238 .3485 .3708 .3907 .4082 .4236 .4370 .4484 .4582 .4664 .4732 .4788 .4834 .4871 .4901 .4925 .4943 .4957 .4968 .4977 .4983 .4988 .4991 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000

.0160 .0557 .0948 .1331 .1700 .2054 .2389 .2704 .2996 .3264 .3508 .3729 .3925 .4099 .4251 .4382 .4495 .4591 .4671 .4738 .4793 .4838 .4875 .4904 .4927 .4945 .4959 .4969 .4977 .4984 .4988 .4992 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000

.0199 .0596 .0987 .1368 .1736 .2088 .2422 .2734 .3023 .3289 .3531 .3749 .3944 .4115 .4265 .4394 .4506 .4599 .4678 .4744 .4798 .4842 .4878 .4906 .4929 .4946 .4960 .4970 .4978 .4984 .4989 .4992 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000

.0239 .0636 .1026 .1406 .1772 .2123 .2454 .2764 .3051 .3315 .3554 .3770 .3962 .4131 .4279 .4406 .4515 .4608 .4686 .4750 .4803 .4846 .4881 .4909 .4931 .4948 .4961 .4971 .4979 .4985 .4989 .4992 .4994 .4996 .4997 .4998 .4999 .4999 .4999 .5000

.0279 .0675 .1064 .1443 .1808 .2157 .2486 .2794 .3078 .3340 .3577 .3790 .3980 .4147 .4292 .4418 .4525 .4616 .4693 .4756 .4808 .4850 .4884 .4911 .4932 .4949 .4962 .4972 .4979 .4985 .4989 .4992 .4995 .4996 .4997 .4998 .4999 .4999 .4999 .5000

.0319 .0714 .1103 .1480 .1844 .2190 .2518 .2823 .3106 .3365 .3599 .3810 .3997 .4162 .4306 .4429 .4535 .4625 .4699 .4761 .4812 .4854 .4887 .4913 .4934 .4951 .4963 .4973 .4980 .4986 .4990 .4993 .4995 .4996 .4997 .4998 .4999 .4999 .4999 .5000

.0359 .0754 .1141 .1517 .1879 .2224 .2549 .2852 .3133 .3389 .3621 .3830 .4015 .4177 .4319 .4441 .4545 .4633 .4706 .4767 .4817 .4857 .4890 .4916 .4936 .4952 .4964 .4974 .4981 .4986 .4990 .4993 .4995 .4997 .4998 .4998 .4999 .4999 .4999 .5000

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DESIGN OF MACHINE ELEMENTS FOR STRENGTH

5.23

DESIGN OF MACHINE ELEMENTS FOR STRENGTH

TABLE 5-9 Error function or probability integral

2 erfðxÞ ¼ pﬃﬃﬃ p x

0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

.11246 .22270 .32863 .42839 .52050 .60386 .67780 .74210 .79691 .84270 .88021 .91031 .93401 .95229 .96611 .97635 .98379 .98909 .99279 .99532 .99702 .99814 .99886 .99931 .99959 .99976 .99987 .99992 .99996 .99998

ðx 0

2

et dt

1

2

3

4

5

6

7

8

9

.01128 .12362 .23352 .33891 .43797 .52924 .61168 .68467 .74800 .80188 .84681 .88353 .91296 .93606 .95385 .96728 .97721 .98441 .98952 .99309 .99552 .99715 .99822 .99891 .99935 .99961 .99978 .99987 .99993 .99996

.02256 .13476 .24430 .34913 .44747 .53790 .61941 .69143 .75381 .80677 .85084 .88679 .91553 .93807 .95538 .96841 .97804 .98500 .98994 .99338 .99572 .99728 .99831 .99897 .99938 .99963 .99979 .99988 .99993 .99996

.03384 .14587 .25502 .35928 .45689 .54646 .62705 .69810 .75952 .81156 .85478 .88997 .91805 .94002 .95686 .96952 .97884 .98558 .99035 .99366 .99591 .99741 .99839 .99902 .99941 .99965 .99980 .99989 .99994 .99997

.04511 .15695 .26570 .36936 .46623 .55494 .63459 .70468 .76514 .81627 .85865 .89308 .92051 .94191 .95830 .97059 .97962 .98613 .99074 .99392 .99609 .99753 .99846 .99906 .99944 .99967 .99981 .99989 .99994 .99997

.05637 .16800 .27633 .37938 .47548 .56332 .64203 .71116 .77067 .82089 .86244 .89612 .92290 .94376 .95970 .97162 .98038 .98667 .99111 .99418 .99626 .99764 .99854 .99911 .99947 .99969 .99982 .99990 .99994 .99997

.06762 .17901 .28690 .38933 .48466 .57162 .64938 .71754 .77610 .82542 .86614 .89910 .92524 .94556 .96105 .97263 .98110 .98719 .99147 .99443 .99642 .99775 .99861 .99915 .99950 .99971 .99983 .99991 .99995 .99997

.07886 .18999 .29742 .39921 .49375 .57982 .65663 .72382 .78144 .82987 .86977 .90200 .92751 .94731 .96237 .97360 .98181 .98769 .99182 .99466 .99658 .99785 .99867 .99920 .99952 .99972 .99984 .99991 .99995 .99997

.09008 .20094 .30788 .40901 .50275 .58792 .66378 .73001 .78669 .83243 .87333 .90484 .92973 .94902 .96365 .97455 .98249 .98817 .99216 .99489 .99673 .99795 .99874 .99924 .99955 .99974 .99985 .99992 .99995 .99997

.10128 .21184 .31828 .41874 .51167 .59594 .67084 .73610 .79184 .83851 .87680 .90761 .93190 .95067 .96490 .97546 .98315 .98864 .99248 .99511 .99688 .99805 .99880 .99928 .99957 .99975 .99986 .99992 .99996 .99998

The standard variable tR (deviation multiplication factor) in order to determine the probability of failure or the reliability

s ﬃ¼ s tR ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 ^ ^s þ ^

The reliability associated with tR

R ¼ 0:5 þ AðtR Þ

ð5-112Þ

where subscripts s and refer to strength and stress, respectively. ð5-113Þ

where AðtR Þ is the area under a standard normal distribution curve. Refer to Table 5-10 for typical values of R as a function of standardized variable tR .

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DESIGN OF MACHINE ELEMENTS FOR STRENGTH

5.24

CHAPTER FIVE

Particular

Formula

TABLE 5-10 Reliability R as a function of tR Survival rate (R) %

tR

50 90.00 95.00 98.00 99.00 99.90 99.99

0 1.288 1.645 2.050 2.330 3.080 3.700

A safety factor of 1 is taken into account in determining the reliability from Eq. (5-113). The fatigue strength reduction factor based on reliability

CR ¼ 1 0:08ðtR Þ

If a factor of safety n0 is to be speciﬁed together with reliability, then Eq. (5-112) is rewritten to give a new expression for tR

n0 n0 ﬃ¼ s tR ¼ psﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ^ ^2s þ ^2

ð5-115Þ

The expression for safety factor n0 from Eq. (5-115)

n0 ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 s tR ^2s þ ^2

ð5-116aÞ

1 ð tR ^Þ s

ð5-116bÞ

¼ The best-ﬁtting straight line which ﬁts a set of scattered data points as per linear regression

The equations for regression

The correlation coeﬃcient

ð5-114Þ

where tR is also called the deviation multiplication factor (DMF), taken from Table 5-10.

y ¼ mx þ b

ð5-117Þ

where m is the slope and b is the intercept on the y axis P P P x y xy m¼ ð5-118aÞ Pn 2 P 2 x x n P P ym x ð5-118bÞ b¼ n r¼

msx sy

ð5-119Þ

where r lies between 1 and þ1. If r is negative, it indicates that the regression line has a negative slope. If r ¼ 1, there is a perfect correlation, and if r ¼ 0, there is no correlation.

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DESIGN OF MACHINE ELEMENTS FOR STRENGTH DESIGN OF MACHINE ELEMENTS FOR STRENGTH

The equation for frequency or density function according to Weibull

f ðxÞ ¼

b x0

x x0 x0

b 1

5.25

x x0 b exp x0 ð5-120Þ

The cumulative distribution function

FðxÞ ¼

ðx x0

x x0 b f ðxÞ dx ¼ 1 exp x0 ð5-121Þ

b x FðxÞ ¼ 1 exp

Equation (5-121) after simpliﬁcation

ð5-122Þ

REFERENCES 1. Maleev, V. L., and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954. 2. Shigley, J. E., and L. D. Mitchell, Mechanical Engineering Design, McGraw-Hill Book Company, New York, 1983. 3. Faires, V. M., Design of Machine Elements, The Macmillan Company, New York, 1965. 4. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering Co-operative Society, Bangalore, India, Bangalore, India, 1962. 5. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Units), Suma Publishers, Bangalore, India, 1986. 6. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 7. Juvinall, R. C., Fundamentals of Machine Component Design, John Wiley and Sons, New York, 1983. 8. Deutschman, A. D., W. J. Michels, and C. E. Wilson, Machine Design—Theory and Practice, Macmillan Publishing Company, New York, 1975. 9. Edwards, Jr., K. S., and R. B. McKee, Fundamentals of Mechanical Component Design, McGraw-Hill Publishing Company, New York, 1991. 10. Norton, R. L., Machine Design—An Integrated Approach, Prentice Hall International, Inc., Upper Saddle River, New Jersey, 1996. 11. Lingaiah, K. Machine Design Data Handbook, McGraw-Hill Publishing Company, New York, 1994. 12. Metals Handbook, American Society for Metals, Vol. 10, 8th edition, p. 102, Metals Park, Ohio, 1975.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

6 CAMS SYMBOLS3;4 a A Ac b B ao ¼ o þ i d dh Do E1 , E2 f f ðÞ F F Fn Ft h Ki , Ko L n N1 , N2 r Rc Ro Rp Rr R, S S S1 v w

radius of circular area of contact, m (in) acceleration of the follower, m/s2 (in/s2 ) follower overhang, m (in) arc of pitch circle, m (in) half the band of width of contact, m (in) follower bearing length, m (in) distance between centers of rotation, m (in) diameter of shaft, m (in) hub diameter, m (in) minimum diameter of the pitch surface of cam, m (in) moduli of elasticity of the materials which are in contact, GPa (Mpsi) cam factor, dimensionless the desired motion of follower, as a function of cam angle applied load, kN (lbf ) total external load on follower (includes weight, spring force, inertia, friction, etc.), kN (lbf ) force normal to cam proﬁle (Fig. 6-6), kN (lbf ) side thrust, kN (lbf ) depth to the point of maximum shear, m (in) constants for input and output cams, respectively length of cylinder in contact, m (in) total distance through which the follower is to rise, m (in) cam speed, rpm forces normal to follower stem, kN (lbf ) radius of follower, m (in) radius of the circular arc, m (in) minimum radius of the pitch surface of the cam, m (in) pitch circle radius, m (in) radius of the roller, m (in) functions of i and o , in basic spiral contour cams displacement of the follower corresponding to any cam angle , m (in) initial compression spring force with weight w, at zero position, kN (lbf ) velocity of the follower, m/s (in/s) equivalent weight at follower ends, kN (lbf )

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CAMS

6.2

CHAPTER SIX

cartesian coordinates of any point on the cam surface actual lift at follower end, m (in) rise of cam, m (in) radius of curvature of the pitch curve, m (in) radii of curvature of the contact surfaces, m (in) pressure angle, deg maximum pressure angle, deg angle through which cam is to rotate to eﬀect the rise L, rad cam angle corresponding to the follower displacement S, rad angle rotated by the output-driven member, deg angle rotated by the input driver, deg angular velocity of cam, rad/s coeﬃcient of friction between follower stem and its guide bearing Poisson’s ratios for the materials of contact surfaces maximum compressive stress, MPa (kpsi) shear stress, MPa (kpsi)

x, y y yc 1 , 2 m o i ! 1 , 2 c;max

Particular

Cam factor

Formula

Ac L

ð6-1Þ

The length of arc of the pitch circle

Ac ¼ Rp

ð6-2Þ

The pitch circle radius

Rp ¼

f ¼

fL

ð6-3Þ

The displacement of the center of the follower from the center of cam (Fig. 6-1)

R ¼ Ro þ f ðÞ

ð6-4Þ

For pointed cam, the radius of curvature of the pitch curve to roller follower

¼ Rr

ð6-5Þ

For roller follower, the radius of curvature of the pitch curve must always be greater than the roller radius to prevent points or undercuts

> Rr

ð6-6Þ

The radius of curvature for concave pitch curve

¼

RADIUS OF CURVATURE OF DISK CAM WITH ROLLER FOLLOWER

fR2 þ ½ f 0 ðÞ2 g3=2 R þ 2½ f 0 ðÞ2 R½ f 00 ðÞ 2

ð6-7Þ

where R ¼ Ro þ f ðÞ; The minimum radius of curvature

min ¼

dR ¼ f 0 ðÞ; d

d 2R ¼ f 00 ðÞ d2

R2o Ro f 00 ðÞo

where f 00 ðÞo is the acceleration at ¼ 0

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ð6-7aÞ ð6-8Þ

CAMS CAMS

Particular

The minimum radius of curvature of the cam curve c

6.3

Formula

c;min ¼ min Rr

ð6-9Þ

FIGURE 6-1

The radius of curvature for convex pitch curve (Fig. 6-2) The minimum radius of a mushroom cam for harmonic motion The minimum radius of a mushroom cam for uniformly accelerated and retarded motion For cast-iron cam, the hub diameter

fR2 þ ½ f 0 ðÞ2 g3=2 R þ 2½ f 0 ðÞ2 R½ f 00 ðÞ 16200 1 L Ro ¼ 2 13131 1 L Ro ¼ 2 2

ð6-11Þ

dh ¼ 1:75d þ 13:75 mm ð1:75d þ 0:55 inÞ

ð6-13Þ

¼

2

ð6-10Þ

ð6-12Þ

Plate cam design radius of curvature: For cycloidal motion

Refer to Fig. 6-10.

For harmonic motion

Refer to Fig. 6-11.

For eight-power polynomial motion

Refer to Fig. 6-12.

RADIUS OF CURVATURE OF DISK CAM WITH FLAT-FACED FOLLOWER The displacement of the follower from the origin (Fig. 6-2)

R ¼ a þ f ðÞ

The parametric equations of the cam contour (Fig. 6-2)

x ¼ ½a þ f ðÞ cos f 0 ðÞ sin

ð6-15aÞ

y ¼ ½a þ f ðÞ sin þ f 0 ðÞ cos

ð6-15bÞ

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ð6-14Þ

CAMS

6.4

CHAPTER SIX

Particular

Formula

The cam contour given by equations will be free of cusps if

a þ f ðÞ þ f 00 ðÞ > 0

ð6-16Þ

Half of the minimum length of the ﬂat-faced follower or the minimum length of contact of the follower

b ¼ f 0 ðÞ

ð6-17Þ

FIGURE 6-2 (Courtesy of H. H. Mabie and F. W. Ocvivk, Dynamics of Machinery, John Wiley and Sons, 1957.)

PRESSURE ANGLE (Figs. 6-3 and 6-4) The pressure angle for roller follower

¼ tan1

The pressure angle for a plate cam or any cylindrical cam giving uniform velocity to the follower

tan ¼

The pressure angle for a plate cam giving uniformly accelerated and retarded motion to the follower

tan ¼

1 dR R d

360L 360L ¼ 2 Ro Do

360 2L when L > Do ðDo þ LÞ sﬃﬃﬃﬃﬃﬃ 180 2 L when L > Do ¼ Do

ð6-18Þ ð6-19Þ ð6-20aÞ ð6-20bÞ

A precise pressure angle equation for a plate cam giving harmonic motion to the follower or a tangential cam

90L tan ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ R2o þ Ro L

For measuring maximum pressure angle of a parabolic cam with radially moving roller follower

Refer to Fig. 6-3 for nomogram of parabolic cam with radially moving follower

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ð6-21Þ

CAMS CAMS

6.5

FIGURE 6-3 Nomogram for parabolic cam with radially moving follower. Source: Rudolph Gruenberg, ‘‘Nomogram for Parabolic Cam with Radially Moving Follower,’’ in Douglas C. Greenwood, Editor, Engineering Data for Product Design, McGraw-Hill Book Company, New York, 1961.

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CAMS

6.6

CHAPTER SIX

Particular

Formula

FIGURE 6-4 Nomogram to determine maximum pressure angle. (Courtesy of E. C. Varnum, Barber-Coleman Co.) Reproduced with permission from Machine Design, Cleveland, Ohio.

RADIAL CAM-TRANSLATING ROLLERFOLLOWER-FORCE ANALYSIS (Fig. 6-5) The forces normal to follower stem (Fig. 6-5)

FR ¼

lr F sin lg n

lr þ lg Fn sin lg " # 2lr þ lg F ¼ Fn cos sin lg

FL ¼ The total external load

F 2lr þ lg sin cos lg

The force normal to the cam proﬁle

Fn ¼

The maximum pressure angle for locking the follower in its guide

m ¼ tan1

lg ð2lr þ lg Þ

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ð6-22Þ ð6-23Þ ð6-24Þ ð6-25Þ

ð6-26Þ

CAMS CAMS

Particular

6.7

Formula

FIGURE 6-5 Radial cam-translating roller-follower force analysis.

SIDE THRUST (Fig. 6-5) The side thrust produced on the follower bearing

Fi ¼ F tan

ð6-27Þ

ao d 1þ o di do ao di i ¼ d 1þ o di

ð6-28Þ

BASIC SPIRAL CONTOUR CAM The radius to point of contact at angle o (Fig. 6-6) The radius to point of contact at angle i (Fig. 6-6)

o ¼

FIGURE 6-6 Basic spiral contour cam.

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ð6-29Þ

CAMS

6.8

CHAPTER SIX

Particular

Formula

BASIC SPIRAL CONTOUR CAM CONSTANTS The radius to point of contact at angle o

The radius to point of contact at angle i

ao Ko dS 1þ Ki dR K dS ao o Ki dR i ¼ K dS 1þ o Ki dR

o ¼

where R ¼ For characteristic curves of cycloidal, harmonic, and eight-power polynomial motions

ð6-30Þ

ð6-31Þ

i d d ; S ¼ o ; i ¼ ki ; and o ¼ ko : Ki Ko dR dS

Refer to Figs. 6-7 to 6-12

HERTZ CONTACT STRESSES Contact of sphere on sphere The radius of circular area of contact

The maximum compressive stress

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ #ﬃ u " 2 2 u 1 v 1 v 1 2 u3F þ u E1 E2 u a ¼ 3u t 1 1 þ 4 1 2 c;max ¼

3F 2 a2

ð6-32Þ

ð6-33Þ

Contact of cylindrical surface on cylindrical surface Width of band of contact

The maximum compressive stress

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ " u # u 1 v21 1 v21 u16F þ u E1 E2 u 2b ¼ u t 1 1 þ L 1 2 2F bL vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ u u0:35F 1 þ 1 u u 1 2 ¼u t 1 1 þ L E1 E2

ð6-34Þ

c;max ¼

ð6-35Þ

c;max

ð6-36Þ

The maximum compressive stress for 1 ¼ 2 ¼ 0:3

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CAMS CAMS

6.9

FIGURE 6-7 Cycloidal motion characteristics. S ¼ displacement, inches; V ¼ velocity, inches per degree; A ¼ acceleration, inches per degree2 . (From ‘‘Plate Cam Design—with Emphasis on Dynamic Eﬀects,’’ by M. Kloomok and R. V. Muﬄey, Product Eng., February 1955.) Reproduced with permission from Machine Design, Cleveland, Ohio.

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CAMS

6.10

CHAPTER SIX

FIGURE 6-8 Harmonic motion characteristics. S ¼ displacement, inches; V ¼ velocity, inches per degree; A ¼ acceleration, inches per degree2 . (From ‘‘Plate Cam Design—with Emphasis on Dynamic Eﬀects,’’ by M. Kloomok and R. V. Muﬄey, Product Eng., February 1955.) Reproduced with permission from Machine Design, Cleveland, Ohio.

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CAMS CAMS

6.11

FIGURE 6-9 Eighth-power polynomial motion characteristics. S ¼ displacement, inches; V ¼ velocity, inches per degree; A ¼ acceleration, inches per degree2 . (From ‘‘Plate Cam Design—with Emphasis on Dynamic Eﬀects,’’ by M. Kloomok and R. V. Muﬄey, Product Eng., February 1955.) Reproduced with permission from Machine Design, Cleveland, Ohio.

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CAMS

6.12

CHAPTER SIX

FIGURE 6-10 Cycloidal motion. (From ‘‘Plate Cam Design—Radius of Curvature,’’ by M. Kloomok and R. V. Muﬄey, Product Eng., September 1955.) Reproduced with permission from Machine Design, Cleveland, Ohio.

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CAMS CAMS

6.13

FIGURE 6-11 Harmonic motion. (From ‘‘Plate Cam Design—Radius of Curvature,’’ by M. Kloomok and R. V. Muﬄey, Product Eng., September 1955.) Reproduced with permission from Machine Design, Cleveland, Ohio.

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CAMS

6.14

CHAPTER SIX

FIGURE 6-12 Eighth-power polynomial motion. (From ‘‘Plate Cam Design—Radius of Curvature,’’ by M. Kloomok and R. V. Muﬄey, Product Eng., September 1955.) Reproduced with permission from Machine Design, Cleveland, Ohio.

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CAMS CAMS

Particular

6.15

Formula

TABLE 6-1 Cam factors for basic curves Types of motion Pressure angle , deg

Uniform

Modiﬁed uniform

Simple harmonic

Parabolic and cycloidal

10 15 20 25 30 35 40 45

5.67 3.73 2.75 2.14 1.73 1.43 1.19 1.00

5.84 3.99 3.10 2.58 2.27 2.06 1.92 1.82

8.91 5.85 4.32 3.36 2.72 2.24 1.87 1.57

11.34 7.46 5.50 4.28 3.46 2.86 2.38 2.00

The maximum shear stress

max ¼ 0:295c;max

ð6-37Þ

The depth to the point of maximum shear

h ¼ 0:786b

ð6-38Þ

For further data on characteristic equations of basic curves, diﬀerent motion characteristics, cam factors, materials for cams and followers, and displacement ratios

Refer to Tables 6-1 and Figures 6-7, 6-8 and 6-9. For materials of cams refer to Chapter 1 on ‘‘Properties of Engineering Materials.’’

REFERENCES 1. Rothbart, H. A., Cams, John Wiley and Sons, New York, 1956. 2. Marks, L. S., Mechanical Engineers’ Handbook, McGraw-Hill Book Company, New York, 1951. 3. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Co-operative Society, Bangalore, India, 1962. 4. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 5. Rothbart, H. A., Mechanical Design and Systems Handbook, McGraw-Hill Book Company, New York, 1964. 6. Shigley, J. E., Theory of Machines, McGraw-Hill Book Company, New York, 1961. 7. Mabie, H. H., and F. W. Ocvirk, Mechanisms and Dynamics of Machinery, John Wiley and Sons, New York, 1957. 8. Kent, R. T., Mechanical Engineers’ Handbook—Design and Production, Vol. II. John Wiley and Sons, New York, 1961. 9. Klcomok, M., and R. V. Muﬄey, ‘‘Plate Cam Design—with Emphasis on Dynamic Eﬀects,’’ Product Eng., February 1955. 10. Klcomok, M., and R. V. Muﬄey, ‘‘Plate Cam Design—Radius of Curvature,’’ Product Eng., February 1955. 11. Varnum, E. C., ‘‘Circular Nomogram—Theory and Practice Construction Technique,’’ Barber-Coleman Co., Product Eng. 12. Gruenberg, R., ‘‘Nomogram for Parabolic Cam with Radially Moving Follower,’’, in Douglas C. Greenwood, Editor, Engineering Data for Product Design, McGraw-Hill Book Company, New York, 1996.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

7 PIPES, TUBES, AND CYLINDERS SYMBOLS5;6;9 d dc di do e E h or t I K L p pc pcr pi po ri c r rðmaxÞ sa su ðmaxÞ max

diameter of cylinder, m (in) diameter of contact surface in compound cylinder, m (in) inside diameter of cylinder or pipe or tube, m (in) outside diameter of cylinder or pipe or tube, m (in) factor for expanded tube ends modulus of elasticity, GPa (Mpsi) thickness of cylinder or pipe or tube, m (in) moment of inertia, area, m4 or cm4 (in4 ) constant maximum distance between supports or stiﬀening rings, m (in) maximum allowable working pressure, MPa (psi) unit pressure between the compound cylinders, MPa (psi) collapsing pressure, MPa (psi) internal pressure, MPa (psi) external pressure, MPa (psi) inside radius of tube or pipe, m (in) permissible working stress, from Table 7-1, MPa (psi) crushing stress, MPa (psi) radial stress (also with primes), MPa (psi) maximum radial stress, MPa (psi) maximum allowable stress value at design condition, MPa (psi) ultimate strength, MPa (psi) tangential stress (also with primes), MPa (psi) maximum tangential stress, MPa (psi) maximum shear stress, MPa (psi) Poisson’s ratio eﬃciency, from Table 7-4

Note: The initial subscript s, along with , which stands for strength, is used throughout this book.

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PIPES, TUBES, AND CYLINDERS

7.2

CHAPTER SEVEN

Particular

Formula

LONG THIN TUBES WITH INTERNAL PRESSURE The permissible steam pressure in steel and iron pipes (Table 7-1) according to ASME Power Boiler Code

p¼

2sa ðh 1:625 103 Þ 0:9 do

SI ð7-1aÞ

where h, do in m, and p and in MPa. p¼

2sa ðh 0:065Þ 125 do

USCS

ð7-1bÞ

where h, do in in, and p and in psi. For tubes from 6.35 mm (0.25 in) to 127 mm (5 in) nominal diameter p¼

2sa ðh 2:54 103 Þ do

SI ð7-2aÞ

where h, do in m, and p and in MPa. p¼

2sa ðh 0:1Þ do

USCS

ð7-2bÞ

where h, do in in, and p and in psi. For over 127 mm (5 in) diameter The minimum required thickness of ferrous tube up to and including 125 mm (5 in) outside diameter subjected to internal pressure as per ASME Power Boiler Code

The maximum allowable working pressure (MAWP) from Eq. (7-3) as per ASME Power Boiler Code

h¼

pdo þ 0:005do þ e 2sa þ p

ð7-3Þ

where sa is the maximum allowable stress value at design condition and e is the thickness factor for expanded tube ends. Refer to Table 7-1 for sa . Refer to table 7-2 for e. 2h 0:01do 2e p ¼ sa do ðh 0:005do eÞ ¼ sa

2h 0:01do 2e 1:005d0 h þ e

For maximum allowable working pressure

Refer to Table 9-1.

The minimum required thickness of ferrous pipe under internal pressure as per ASME Power Boiler Code

h¼

or

ð7-4Þ

pdo pri þC ¼ þC 2sa þ 2yp sa ð1 yÞp

ð7-5Þ

where ¼ efficiency (refer to Table 7-4 for ) y ¼ temperature coeﬃcient (refer to Table 7-3 for y) C ¼ minimum allowance for the threading and structural stability, mm (in) (refer to Table 7-5 for h values and Table 7-6 for C values).

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2

1

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S31500 S31500

TP410 TP405 TpxM-8 TpxM-8 18Cr-2Mo 18Cr-2Mo TP304L TP304H, TP304 TP304N TP304N TP316L TP316L TP316H XM-15 XM-15 TP316N TP316N XM-29 TP321 TP321H FP347H TP348 TP348H, TP347H S30815

T1 T12 Fp11 T1

Low Alloy Steel: SA-209g SA-213 SA-369 SA-250

(B) High Alloy Steels SA-268 SA-268 SA-268 SA-268 SA-268 SA-268 SA-249. SA-312 SA-213, SA-312 SA-213, SA-312 SA-249, SA-312 SA-213, SA-312 SA-312, SA-688 SA-452 SA-312 SA-213 SA-213 SA-312 SA-312, SA-688 SA-213, SA-312 SA-249, SA-312 SA-430 SA-213 SA-249, SA-312 SA-213, SA-312 SA-789, SA-790 SA-789, SA-790 SA-789, SA-790, SA-669 SA-789, SA-790

C C

SA-210 SA-557b,f

c

(A) Carbon and Low Alloy Steels Carbon Steel: SA-106c A

Grade, alloy designation and temper

Speciﬁcation number

S4 1 000 S40500 S43035 S43035 S44400 S44400 S30403 S30409, S30400 S30451 S30451 S31603 S31603 S31609 S31800 S38100 S31651 S3i651 S24000 S32100 S32109 S34700 S34800 S34809, S34709 S30815 S32550 S32550 S31500 S31500

3

UNS number

13Cr 12Cr-1Al 18Cr-Ti 18Cr-Ti 18Cr-2Mo 18Cr-2Mo 18Cr-8Ni 18Cr-8Ni 18Cr-8Ni-N 18Cr-8Ni-N 16Cr-12Ni-2Mo 16Cr-12Ni-2Mo 16Cr-12Ni-2Mo 18Cr-18Ni-2Si 18Cr-18Ni-2Si 16Cr-12Ni-2Mo-N 16Cr-12Ni-2Mo-N 18Cr-3Ni-12Mn 18Cr-10Ni-Ti 18Cr-10Ni-Ti 18Cr-10Ni-Cb 18Cr-10Ni-Cb 18Cr-10Ni-Cb 21Cr-11Ni-N 25.5Cr-5.5Ni-3.5Mo-Cu 25.5Cr-5.5Ni-3.5Mo-Cu 18Cr-5Ni-3Mo 18Cr-5N-3Mo

1Cr-12Mo 114Cr-12Mo-Si C-12Mo*

C-Mn-Si C-Mn

C-si

4

Nominal composition and size, mm (in)

Smls.Tb Wld.Tbf Wld.Tbd;f Smls.Tbd;f Wld.Tbd;f Smls.Tbd;f Wld.Tbf & Pp Smls.Tbg,h & Pp Smls.Tb & Ppg,h Wld.Th & Ppf,g,h Smls.Tb & Pp Wld.P & Tbf Cast. Ppg Wld.Tbf,g Smls.Tbg Smls.Tbg,h Wld.Ppf,g,h Wld.Ppf & Tb Smls.Tbg,h & Pp Wld.Tb & Ppf Smls.Ppg Smls.Tbg,h Wld.Tb & Ppf,g Smls.Tb & Ppf Smls.Tb & Ppd Wld.Tb & Ppd Smls.Tbd,f Wld.Tbd,f

Smls.Tb Smls.Tb Smls.Pp Wld Pp & Tb

Smls.Tb** Smls.Tb

Smls† .Pp*

5

Product form

TABLE 7-1 Maximum allowable stress values in tension of metals for tubes and pipes, sa

207 207 207 207 276 276 172 207 241 241 172 172 207 207 207 241 241 379 207 207 207 207 207 310 552 552 441 441

207 207 207 207

276 276

207

6

MPa

30 30 30 30 40 40 25 30 35 35 25 25 30 30 30 35 35 55 30 30 30 30 30 45 80 80 64 64

30 30 30 30

40 40

30

7

kpsi

Speciﬁed minimum yield strength, sy

414 414 414 414 414 414 483 517 552 552 483 483 517 517 517 552 552 689 517 517 483 517 517 600 758 758 634 634

379 414 414 379

483 483

331

8

MPa

48

60 60 60 60 60 60 70 75 80 80 70 70 75 75 75 80 80 100 75 75 70 75 75 87 110 110 92 92

55 60 60 55

70 70

9

kpsi

Speciﬁed minimum tensile strength, st

103 88 88 103 88 103 92 130 138 117 108 92 130 110 130 138 117 146 130 110 130 130 110 150 190 161 159 135

10

MPa

15.0 12.8 12.8 15.0 12.8 15.0 13.3 18.8 20.0 17.0 15.7 13.3 18.8 15.9 18.8 20.0 17.0 21.2 18.8 16.0 18.8 18.8 16.0 21.8 27.5 23.4 23.0 19.6

11

kpsi

38 (100)

99 84 84 98 84 99 78 123 138 117 92 78 130 104 122 138 117 143 127 93 123 123 105 149 189 161 153 130

12

MPa

14.3 12.2 12.) 14.3 12.2 14.3 11.4 17.8 131 17.0 13.3 11.3 18.8 15.1 17.7 20.0 17.0 20.8 18.4 13.5 17.9 17.9 15.2 21.6 27.4 23.3 22.2 18.9

13

kpsi

93 (200)

95 81 81 95 81 95 70 115 20.0 111 82 70 127 97 115 132 112 132 119 83 113 113 97 141 177 151 147 125

14

MPa

13.8 11.8 11.8 13.8 11.8 13.8 10.2 16.6 19.0 16.1 11.9 10.1 18.4 14.1 16.6 19.2 16.3 19.2 17.3 12.1. 16.4 16.4 14.0 20.4 25.7 21.9 21.3 18.1

15

kpsi

150 (300)

92 78 78 92 78 92 64 112 126 108 75 63 125 95 111 130 110 119 118 76 107 107 91 135 170 145 146 124

16

MPa

13.3 11.3 11.3 13.3 11.3 13.3 9.3 16.2 18.3 15.6 10.8 9.2 18.1 13.7 16.1 18.8 16.0 17.3 17.1 11.0 15.5 15.3 13.2 19.6 24.7 21.0 21.2 18.0

17

kpsi

205 (400)

Maximum allowable stress, sa

89 75 75 89 75 88 60 110 123 104 69 59 124 93 110 128 109 110 118 70 103 103 88 127 170 145 146 124

18

MPa

12.9 10.9 10.9 12.9 10.9 12.8 8.7 15.9 17.8 15.1 10.0 8.5 18.0 13.5 15.9 18.6 15.8 16.0 17.1 10.2 14.9 14,9 12.7 18.4 24.7 21.0 21.2 18.0

19

kpsi

260 (500)

PIPES, TUBES, AND CYLINDERS

7.3

21

20

22

MPa

23

kpsi

370 (700)

7.4

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21.2 18.0

146 124

146 124

53 105 115 98 59 50 110 89 104 127 108 97 107 63 101 101 86 116

8.0 15.9 17.1 14.6 9.0 7.6 16.3 13.5 15.9 18.6 15.8 14.7 15.8 9.3 14.7 14.7 12.5 17.3

21.2 18.0

65 65

93 99 93 84 77

10.3 10.3

13.8 15.0 14.2 12.8 12.1

64 83 70

24

MPa

(B) High Alloy Steels 73 10.6 71 73 10.6 71 86 12.4 72 10.5 86 12.4 57 8.3 55 110 15.9 110 120 17.4 118 102 14.8 101 51 7.4 62 55 8.0 52 117 17.0 112 93 13.5 93 110 15.9 110 128 18.6 128 109 15.8 109 106 15.4 101 112 16.4 109 67 9.7 64 101 14.7 101 101 14.7 101 86 12.5 86 122 17.7 119

95 103

Low Alloy Steels 13.8 95 15.0 103 98 88 12.8 88 86 12.4 83

(A) Carbon and Low Alloy Steels Carbon Steel: 83 12.0 81 11.7 121 17.5 115 16.6 103 15.0 97 14.1

kpsi

MPa

315 (600)

7.7 15.2 16.6 14.2 8.6 7.3 15.9 12.9 15.1 18.4 15.6 14.1 15.5 9.2 14.7 14.7 12.5 16.8

9.4 9.4

13.5 14.4 13.5 12.2 11.1

9.3 12.0 10.2

25

kpsi

427 (800)

14.7 15.9 13.5

15.5 12.4 14.6 18.1 15.4 15.3 9.0 14.7 14.7 12.5 16.3

107 85 101 125 106 106 62 101 101 86 112

8.2

12.7 11.0 12.5 11.0 9.7

6.5 5.0 5.5

27

kpsi

101 110 93

57

86 76 86 76 67

45 35 38

26

MPa

482 (900)

13.8 8.9 14.4 14.0 12.3 14.9

13.7 17.4 14.8

95 120 102 95 61 99 97 84 103

15.3

13.8 15.0 12.7

3.4

4.8 5.5 6.2 4.1 6.4

2.5 1.5 2.1

29

kpsi

106

95 103 86

23

33 38 48 98 44

17 10 15

28

MPa

538 (1000)

48 52 90 63 76 62

85 72

85

6.9 7.5 13.0 9.1 11.1 9.0

12.4 10.5

12.4

9.8 9.7 8.3

2.9

20

68 67 57

4.0 2.6

31

kpsi

27 18

30

MPa

593 (1100)

24 32 55 30 46 36

51 43

51

42 41 35

7

8 7

32

MPa

3.6 4.6 7.9 4.4 6.7 5,2

7.4 6.3

7.4

6.1 6.0 5.1

1.0

1.2 1.0

33

kpsi

650 (1200)

for metal temperature, 8C (8F), not exceeding

TABLE 7-1 Maximum allowable stress values in tension of metals for tubes and pipes, sa (Cont.)

12 19 30 15 25 21

28

25

34

MPa

1.7 2.7 4.4 2.2 3.7 3.1

4.1

3.7

35

kpsi

704 (1300)

5 11 17 8 15 13

16

16

36

MPa

0.8 1.6 2.5 1.2 2.1 1.9

2.3

2.3

37

kpsi

760 (1400)

2 7 9 5 8 9

9

10

38

MPa

0.3 1.0 1.3 0.8 1.1 1.3

1.3

1.4

39

kpsi

815 (1500)

SA-268 SA-268 SA-268 SA-268 SA-268 SA-249, SA-312 SA-213, SA-312 SA-213, SA-312 SA-249, SA-312 SA-213, SA-312 SA-312, SA-688 SA-452 SA-312 SA-213 SA-213 SA-312 SA-312, SA-688 SA-213, SA-312 SA-249, SA-312 SA-430 SA-213 SA-249, SA-312 SA-312, SA-213 SA-789, SA-790 SA-789, SA-790 SA-789, SA-790, SA-669 SA-789, SA-790

SA-209g SA-213 SA-369 SA-250 SA-268

SA- 106c SA-210c SA-557b,f

40

Speciﬁcation number

PIPES, TUBES, AND CYLINDERS

2

Speciﬁcation number

1

e

SB-234

C700-Ann LCW***

p

C71500 Ann

Nickel and High Nickel Alloys: SB-161 201 Ann SB-163 800H Annk SB-163 825 Annk SB-144 625 Annp SB-468 20 cb.Wld. Annk,p SB-619 C-276 Sol. Annp SB-619 G. Sol. Annk,p

SB-543

SB-467

pp

C71500 Ann

SB-466

p

655. Ann

SB-315

g

192 Ann

SB-1 1 1

Copper and Copper Alloys: SB-111 102, 120, 122, 142i

6061-T6

3003-H25

SB-234

e

3003-H118 5083-H111d,p 1060-H14e

SB-241 SB-241 SB-234

e

6061-T6

SB-210

e

(C) Non-ferrous Metals Aluminum and Aluminum Alloys: SB-210 1060-1114d

Grade, alloy designation and temper

N02201 N08810 N08825 N06625 N08020 N10276 N06007

3

UNS number

Ni Low C Ni-Fe-Cr Ni-Fe-Cr-Mo-Cu Ni-Cr-Mo-Cb Cr-Ni-Fe-Mo-Cu-Cb Ni-Mo-Cr (All sizes) Ni-Cr-Fe-Mo-Cu (All sizes)

(Up to 112.5 incl) (up to 412 incl)

Ann LD‡ HD**

Up to 125 (up to 5.00) 0.250–12.50 (0.010–0.5000) 0.625–6.225 (0.025–0.249)

0.250–12.500 (0.010–0.500) 0.625–12.50 (0.025–0.50) Under 25 (under 1)

4

Nominal composition and size, mm (in)

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Pp & Tb Pp & Tb Pp & Tb

Wld.Cu-Ni-90/10Tb

Smls. Copper, iron alloy condenser Tb Smls. Cu-Si Alloy Pp and Th Smls. Cu-Ni 70/30 Pp & Tb. Wld. Cu-Ni-70/30 Pp

Smls. Copper condenser, Tb.

Smls.Pp Smls. extruded Tb Condenser and heat exchanger Tb Condenser and heat exchanger Tb Condenser and heat exchanger Tb

Smls.Tb

Drawn

5

Product form

69

69 172 241 414 241 283 242

103 241

138

124

103

62 207 276 83

241

131

165 131 69

241

6

MPa

10 25 35 60 35 41 35

15 35

20

18

15

9 30 40 12

35

19

24 19 10

35

10

7

kpsi

Speciﬁed minimum yield strength, sy

TABLE 7-1 Maximum allowable stress values in tension of metals for tubes and pipes, sa (Cont.)

83

345 448 586 827 552 689 620

270 310

345

345

345

207 248 310 262

290

145

186 228 83

290

8

MPa

50 65 85 120 80 100 90

40 45

50

50

50

30 36 45 38

42

21

27 33 12

42

12

9

kpsi

Speciﬁed minimum tensile strength, st

46 112 146 207 117 146 132

59 59

87

83

69

41 62 78 52

72

38

47 57 21

72

21

10

MPa

6.7 16.2 21.2 30.0 17.0 21.2 19.1

8.5 8.5

12.6

12.0

10.0

6.0 9.0 11.3 7.5

10.5

5.5

6.8 8.3 3.0

10.5

3.0

11

kpsi

38 (100)

44 112 146 207 117 146 132

56 56

61

78

69

33 62 78 46

72

38

46 57 21

72

21

12

MPa

6.4 16.2 21.2 30.0 17.0 21.2 19.1

8.1 8.1

8.9

11.3

10.0

4.8 9.0 11.3 6.7

10.5

5.5

6.7 8.3 3.0

10.5

3.0

13

kpsi

93 (200)

43 112 146 207 115 146 132

52 52

61

75

69

32 60 78 42

58

30

37 38 18

58

18

14

MPa

6.3 16.2 21.2 30.0 16.8 21.2 19.1

7.6 7.6

8.8

10.8

10.0

4.7 8.7 11.3 6.1

8.4

4.3

5.4 5.5 2.6

8.4

2.6

15

kpsi

150 (300)

43 112 146 194 110 143 128

50 50

61

71

35

21 57 30

31

17

17 21 8

31

8

16

6.2 16.2 21.2 28.2 15.9 20.7 18.6

7.2 7.2

8.8

10.3

5.0

3.0 8.2 4.3

4.5

2.4

2.5 3.0 1.2

4.5

1.2

17

kpsi

205 (400) MPa

Maximum allowable stress

43 110 146 186 107 140 126

43 43

61

68

18

MPa

6.2 16.0 21.2 27.0 15.5 20.3 18.3

6.3 6.3

8.8

9.9

19

kpsi

260 (500)

PIPES, TUBES, AND CYLINDERS

7.5

21

20

22

MPa

23

kpsi

370 (700)

Nickel and High Nickel 6.2 43 16.0 108 21.2 145 26.4 179 15.1 101 20.0 135 17.9 123

9.6 8.8 4.3 4.3

Alloy 6.2 15.7 21.0 26.0 14.7 19.6 17.8

41 105 143 179 99 134 120

24

MPa

5.9 15.3 20.8 26.0 14.3 19.4 17.4

25

kpsi

427 (800)

4.5 14.8 20.5 26.0 18.9 17.0

130 117

27

kpsi

31 102 141 179

26

MPa

482 (900)

128 111

21 99 36 179

28

MPa

18.5 16.1

3.0 14.4 19.7 26.0

29

kpsi

538 (1000)

12.7

26.0

179 88

2.0 11.6

31

kpsi

14 80

30

MPa

593 (1100)

57

91

8 51

32

MPa

8.3

13.2

1.2 7.4

33

kpsi

650 (1200)

32

34

MPa

4.7

35

kpsi

704 (1300)

21

36

MPa

3.0

37

kpsi

760 (1400)

13

38

MPa

1.9

39

kpsi

815 (1500)

SB-161 SB-163 SB-163 SB-444 SB-468 SB-619 SB-619

SB-111 SB-111 SB-315 SB-466 SB-467 SB-543

SB-210 SB-210 SB-241 SB-241 SB-234 SB-234 SB-234

40

Speciﬁcation number

Source: The American Society of Mechanical Engineers, Boiler and Pressure Vessel Code, Section VIII, Division 1, July 1986. * Pp ¼ pipe; ** Tb ¼ tube; *** LCW ¼ light cold worked; Smls ¼ seamless; Wld ¼ welded; ‡ LD ¼ light drawn; HD ¼ hard drawn; Ann ¼ annealed; Soln Ann ¼ solution annealed. Notes: The superscript letters a, b, c, etc., refer to notes under each category of (A) Carbon and Low Alloy Steels, (B) High Alloy Steels, and (C) Non-ferrous Metals in Tables 8-9, 8-10, and 8-11 in Chapter 8.

43 110 146 182 104 138 123

67 61 30 30

Copper and Copper Alloys:

(C) Non-ferrous Metals Aluminum and Aluminum Alloys:

kpsi

MPa

315 (600)

for metal temperature, 8C (8F), not exceeding

TABLE 7-1 Maximum allowable stress values in tension of metals for tubes and pipes, sa (Cont.)

PIPES, TUBES, AND CYLINDERS

7.6

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PIPES, TUBES, AND CYLINDERS PIPES, TUBES, AND CYLINDERS

7.7

TABLE 7-2 Thickness factor for expanded tube ends e for use in Eqs. (7-3) and (7-4) Particular

Value of e

Over a length at least equal to the length of the seat plus 25 mm (1 in) for tubes expanded into tube seats, except

0.04

For tubes expanded into tube seats provided the thickness of the tube ends over a length of the seat plus 25 mm (1 in) is not less than the following: 2.375 mm (0.095 in) for tubes 31.25 mm (1.25 in) OD 2.625 mm (0.105 in) for tubes >31.25 mm (1.25 in) OD and 50 mm (2 in) OD, including 3.000 mm (0.120 in) for tubes >50 mm (2 in) and 75 mm (3 in) OD, including 3.375 mm (0.135 in) for tubes >75 mm (3 in) OD and 100 mm (4 in) OD, including 3.75 mm (0.150 in) for tubes >100 mm (4 in) and 125 mm (5 in) OD, including

0

For tubes strength-welded to headers and drums

0

Source: ASME Boiler and Pressure Vessel Code, Section 1, 1983.

TABLE 7-3 Temperature coeﬃcient y Temperature, 8C (8F)a

Material

482 (900)a

510 (950)

540 (1000)

565 (1050)

595 (1100)

620 (1150)

Ferrite steels

0.4

0.5

0.7

0.7

0.7

0.7

Austenitic steels

0.4

0.4

0.4

0.4

0.5

0.7

For nonferrous materials

0.4

0.4

0.4

0.4

0.4

0.4

a

Temperatures in parentheses are in Fahrenheit (8F). Values of y between temperatures not listed may be determined by interpolation. Source: ASME Boiler and Pressure Vessel Code, Section 1, 1983.

TABLE 7-4 Eﬃciency of joints, Particular

Eﬃciency,

Longitudinal welded joints or of ligaments between openings, whichever is lower Seamless cylinders

1.00

For welded joints provided all weld reinforcement on the longitudinal joints is removed substantially ﬂush with the surface of the plate

1.00

For welded joints with the reinforcement on the longitudinal joints left in place

0.90

Riveted joints

Refer to Table 13-4 (Chap. 13)

Ligaments between openings

Refer to Eqs. under Ligament (Chap. 8)

Welded joint eﬃciency factor

Refer to Table 8-3 (Chap. 8)

Source: ASME Boiler and Pressure Vessel Code, Section 1, 1983.

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PIPES, TUBES, AND CYLINDERS

7.8

CHAPTER SEVEN

Particular

Formula

TABLE 7-5 The depth of thread h (formula h ¼ 0:8=i ) Number of threads per mm (in), i

Depth of thread, h mm (in)

0.32 (8) 0.46 (11.5)

2.5 (0.100) 1.715 (0.0686)

Source: ASME Boiler and Pressure Vessel Code, Section 1, 1983.

The maximum allowable working pressure from Eq. (7-5) as per ASME Power Boiler Code The minimum required thickness of nonferrous seamless tubes and pipes for outside diameters 12.5 mm (0.5 in) to 150 mm (6 in) inclusive and for wall thickness not less than 1.225 mm (0.049 in) as per ASME Power Boiler Code The maximum allowable working pressure as per ASME Power Boiler Code

p¼

2sa ðh CÞ sa ðh CÞ or p ¼ ð7-6Þ do 2yðh CÞ ri þ ð1 yÞðh CÞ

h¼

pdo þC 2sa

ð7-7Þ

Refer to Table 7-6 for values of C. p¼

2sa ðh CÞ do

ð7-8Þ ð7-9Þ

The minimum required thickness of tubes made of steel or wrought iron subjected to internal pressure which are used in watertube and ﬁretube boilers as per ASME Power Boiler Code

h ¼ 0:0251do

The minimum required thickness of tubes made of nonferrous materials such as copper, red brass, admiralty and copper-nickel alloys used in watertube and ﬁretube boilers with a design pressure over 207 kPa (30 psi) but not greater than 414 kPa (60 psi)

h¼

do þ 0:75 30

SI

ð7-10aÞ

h¼

do þ 0:03 30

USCS

ð7-10bÞ

The minimum required thickness of tubes made of nonferrous materials such as copper, red brass, admiralty and copper-nickel alloys used in steam boilers of less than 103 kPa (15 psi) and water boilers of less than 207 kPa (30 psi)

h¼

do þ 0:75 45

SI

ð7-11aÞ

h¼

do þ 0:03 45

USCS

ð7-11bÞ

The minimum required thickness of tubes when made of nonferrous materials but assembled with ﬁttings, which are based on materials used, and based on whether the pressure is over 207 kPa (30 psi), but not in excess of 1013 kPa (160 psi) or whether the pressure does not exceed 207 kPa (30 psi)

h¼

do þ 0:75 except for copper ¼ 0:027 factor

The formula for permissible pressure in wrought-iron and steel tubes for watertube boilers according to ASME Power Boiler Code

SI ð7-12aÞ do h¼ þ 0:03 USCS ð7-12bÞ factor h 1 103 0:32 SI ð7-13aÞ p ¼ 125 do where h, do in m, and p in MPa. h 0:039 p ¼ 18000 250 USCS ð7-13bÞ do where h, do in in, and p in psi.

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PIPES, TUBES, AND CYLINDERS PIPES, TUBES, AND CYLINDERS

Particular

7.9

Formula

h 1 103 p ¼ 96:5 do

SI

where h, do in m, and p in MPa. h 0:039 USCS p ¼ 14000 do where h, do in in, and p in psi. h 1 103 p ¼ 73 do

SI

where h, do in m, and p in MPa. h 0:039 USCS p ¼ 10600 do

ð7-14aÞ

ð7-14bÞ

ð7-15aÞ

ð7-15bÞ

where h, do in in, and p in psi. Formula (7-13) applies to seamless tubes at all pressures, to welded steel tubes at pressure below 6 MPa (875 psi), and to lap-welded wrought-iron tubes at pressures below 2.5 MPa (358 psi). Formula (7-14) applies to welded steel tubes at pressures of 6 MPa (875 psi) and above. Formula (7-15) applies to lap-welded wrought-iron tubes at pressures of 2.5 MPa (358 psi) and above.

ENGINES AND PRESSURE CYLINDERS The wall thickness of engines and pressure cylinders

h¼

pdi þ 7:5 103 2sta

SI

ð7-16aÞ

where p, st in MPa, and di and h in m. h¼

pdi þ 0:3 2sta

USCS

ð7-16bÞ

where p, t in psi, and di and h in in. sta ¼ 9 MPa (1250 psi) for ordinary grades of cast iron.

OPENINGS IN CYLINDRICAL DRUMS The largest permissible diameter of opening according to D. S. Jacobus

p 3 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ do hð1:0 KÞ

SI

ð7-17aÞ

where do and h in m p 3 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d ¼ 2:75 do hð1:0 KÞ

USCS

ð7:17bÞ

d 0 ¼ 0:81 0

where do and h in in.

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PIPES, TUBES, AND CYLINDERS

7.10

CHAPTER SEVEN

Particular

Formula

K¼

pdo 2h

5 su

USCS

ð7-17bÞ

The maximum diameter of the unreinforced hole should be limited to 0.203 m (8 in) and should not exceed 0:6do .

THIN TUBES WITH EXTERNAL PRESSURE Professor Carman’s formulas for the collapsing pressure for seamless steel tubes

3 h pcr ¼ 346120 do

SI

ð7-18aÞ

where h, do in m, and pcr in MPa. 3 h pcr ¼ 50200000 USCS ð7-18bÞ do h where h, do in in, and pcr in psi when < 0:025. d o h 1:50 SI ð7-19aÞ pcr ¼ 658:5 do where h, do in m, and pcr in MPa h 2090 USCS pcr ¼ 95520 do

Professor Carman’s formula for the collapsing pressure for lap-welded steel tubes

Professor Carman’s formula for the collapsing pressure for lap-welded brass tubes

where h, do in in, and pcr in psi h when > 0:03 do h 0:72 pcr ¼ 574 do

SI

ð7-19bÞ

ð7-20aÞ

where h, do in m, and pcr in MPa h 1025 USCS ð7-20bÞ pcr ¼ 83290 do h where h, do in in, and pcr in psi when > 0:03 d o 3 h SI ð7-21aÞ pcr ¼ 173385 do where h, do in m, and pcr in MPa 3 h USCS ð7-21bÞ pcr ¼ 25150000 do h where h, do in in, and pcr in psi when < 0:025 d o h 1:75 SI ð7-22aÞ pcr ¼ 644 do where h, do in m, and pcr in MPa

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PIPES, TUBES, AND CYLINDERS PIPES, TUBES, AND CYLINDERS

Particular

Formula

h pcr ¼ 93365 2474 do

USCS

where h, do in in, and pcr in psi when

SHORT TUBES WITH EXTERNAL PRESSURES Sir William Fairbairn’s formula for collapsing pressure for length less than six diameters

7.11

ð7-22bÞ

h > 0:03 do

2:19 h pcr ¼ 66580 Ldo

SI

where h, L, do in m, and pcr in MPa 2:19 h pcr ¼ 9657600 USCS Ldo

ð7-23aÞ

ð7-23bÞ

where h, L, do in in, and pcr in psi Thickness of tubes, and pipes when used as tubes under external pressure as per Indian Standards

Refer to Fig. 7-1 to determine the standard thickness of tubes and pipes; see also Table 7-7.

FIGURE 7-1 Thickness of tubes and pipes under external pressure.

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PIPES, TUBES, AND CYLINDERS

7.12

CHAPTER SEVEN

TABLE 7-6 Values of C for use in Eqs. (7-5) to (7-8) Type of pipe

Value of C,b mm (in)

Threaded steel, wrought iron, or nonferrous pipea 19 mm (0.75 in), nominal and smaller 25 mm (1 in), nominal and larger

1.625 (0.065) Depth of thread hc

Plain-end d steel, wrought iron, or nonferrous pipe 87.5 mm (3.5 in), nominal and smaller 100 mm (4 in), nominal and larger

1.625 (0.065) 0

a

Steel, wrought iron, or nonferrous pipe lighter than schedule 40 of the American National Standard for wrought iron and steel pipe, ANSI B36.10-1970, shall not be threaded. b The values of C stipulated above are such that the actual stress due to internal pressure in the wall of the pipe is no greater than the value of S (i.e. sa ) given in Table PG 23.1 of ASME Power Boiler Code as applicable in the formulas. c The depth of thread h in inches may be determined from the formula h ¼ 0:8=i, where i is the number of threads per inch or from Table 7-5. d Plain-end pipe includes pipe joined by ﬂared compression coupling, lap (Van Stone) joints, and by welding, i.e., by any method which does not reduce the wall thickness of pipe at the joint. Source: ASME Boiler and Pressure Vessel Code, Section 1, 1983.

Particular

Formula

LAME´’S EQUATIONS FOR THICK CYLINDERS General equations The tangential stress in the cylinder wall at radius r when subjected to internal and external pressures

¼

pi di2 po do2 di2 do2 ð pi po Þ þ 2 2 do2 di2 4r ðdo di2 Þ

¼aþ The radial stress in the cylinder at radius r when subjected to internal and external pressures

r ¼

b r2

pi di2 po do2 di2 do2 ð pi po Þ 2 2 4r ðdo di2 Þ do2 di2

ð7-24aÞ ð7-24bÞ ð7-25aÞ

b r2

ð7-25bÞ

a¼

pi di2 po do2 do2 di2

ð7-25cÞ

b¼

di2 do2 ð pi po Þ 4ðdo2 di2 Þ

ð7-25dÞ

¼a where

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PIPES, TUBES, AND CYLINDERS PIPES, TUBES, AND CYLINDERS

Particular

7.13

Formula

Cylinder under internal pressure only The tangential stress in the cylinder wall at radius r

The radial stress in the cylinder wall at radius r

pi di2 do2 ¼ 2 1þ 2 do di2 4r

ð7-26Þ

pi di2 do2 1 do2 di2 4r2

ð7-27Þ

pi ðdi2 þ do2 Þ do2 di2

ð7-28Þ

r ¼

The maximum tangential stress at the inner surface of the cylinder at r ¼ di =2

ðmaxÞ ¼

The maximum radial stress

rðmaxÞ ¼ pi

The maximum shear stress at the inner surface of the cylinder under internal pressure

max ¼

pi do2 di2

The radial stress in the cylinder wall at radius r

ð7-30Þ

do2

Cylinder under external pressure only The tangential stress in the cylinder wall at radius r

ð7-29Þ

¼

po do2 di2 1 þ do2 di2 4r2

ð7-31Þ

¼

po do2 di2 1 do2 di2 4r2

ð7-32Þ

DEFORMATION OF A THICK CYLINDER The radial displacement of a point at radius r in the wall of the cylinder subjected to internal and external pressures

u¼

1 pi di2 po do2 r E do2 di2 1 þ di2 do2 ð pi po Þ þ E 4rðdo2 di2 Þ

ð7-33Þ

Cylinder under internal pressure only

The radial displacement at r ¼ di =2 of the inner surface of the cylinder

ui ¼

pi di 2E

The radial displacement at r ¼ do =2 of the outer surface of the cylinder

uo ¼

pi di2 do Eðdo2 di2 Þ

di2 þ do2 þ d02 di2

ð7-34Þ ð7-35Þ

Cylinder under external pressure only The radial displacement at r ¼ di =2 of the inner surface of the cylinder

ui ¼

po di do2 Eðdo2 di2 Þ

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ð7-36Þ

PIPES, TUBES, AND CYLINDERS

7.14

CHAPTER SEVEN

Particular

The radial displacement at r ¼ do =2 of the outer surface of the cylinder

Formula

uo ¼

po do 1 2 E

di2 þ do2 do2 di2

ð7-37Þ

COMPOUND CYLINDERS Birnie’s equation for tangential stress at any radius r for a cylinder open at ends subjected to internal pressure

¼ ð1 Þ

The tangential stress at the inner surface of the inner cylinder in the case of a compound cylinder (Figs. 11-1 and 11-2)

i ¼

The tangential stress at the outer surface of the inner cylinder

ic ¼ pc

The tangential stress at the inner surface of the outer cylinder

oc ¼ pc

The tangential stress at the outer surface of the outer cylinder

o ¼

pi di2 d2d 2 p þ ð1 þ Þ 2 i 2o i 2 2 di 4r ðdo di Þ

ð7-38Þ

do2

2pc dc2 dc2 di2

ð7-39Þ

dc2 þ di2 dc2 di2

do2 þ dc2 þ do2 dc2

ð7-40Þ

ð7-41Þ

2pc dc2 do2 dc2

ð7-42Þ

THERMAL STRESSES IN LONG HOLLOW CYLINDERS The general expressions for the radial r , tangential , and longitudinal z stresses in the cylinder wall at radius r when the temperature distribution is symmetrical with respect to the axis and constant along its length, respectively

2 ð ðr E 4r di2 ro Tr dr Tr dr r ¼ ð1 Þr2 do2 di2 ri ri ¼

ð7-43Þ

2 ð ðr E 4r þ di2 ro 2 Tr dr þ Tr dr Tr ð1 Þr2 do2 di2 ri ri ð7-44Þ

z ¼

ð ro E 8 Tr dr T 1 do2 di2 ri

ð7-45Þ

where do ¼ 2ro and di ¼ 2ri The expressions for radial (r ), tangential ( ), longitudinal (z ) stresses in the cylinder at r when the cylinder is subjected to steady-state temperature distribution, i.e., logarithmic temperature distribution throughout the wall thickness of the cylinder by using equation T ¼ Ti ½ln Ro = ln R

r ¼

¼

ETi 2ð1 Þ lnðRÞ lnðRo Þ

1 ð1 R2o Þ lnðRÞ R2 1

ETi 2ð1 Þ lnðRÞ 1 lnðRo Þ

1 ð1 þ R2o Þ lnðRÞ R2 1

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ð7-46Þ

ð7-47Þ

PIPES, TUBES, AND CYLINDERS

7.15

PIPES, TUBES, AND CYLINDERS

Particular

Formula

z ¼

ETi 2 1 2 lnðRo Þ 2 lnðRÞ 2ð1 Þ lnðRÞ R 1 ð7-48Þ

where R ¼

The expressions for maximum values of tangential (hoop) and longitudinal stresses at inner and outer surfaces of the cylinder under logarithmic temperature distribution. respectively

The simpliﬁed expressions for maximum values of tangential and longitudinal stresses at inner and outer surfaces of the cylinder under logarithmic temperature distribution when the thickness of cylinder is small in comparison with the inner radius of the cylinder, respectively

do ro r d r d ¼ ; Ro ¼ o ¼ o ; Ri ¼ i ¼ i di ri r 2r r 2r

Ti ¼ temperature at inner surface of cylinder, 8C (8F) ETi 2R2 ln R ð7-49Þ i ¼ zi ¼ 1 2 2ð1 Þ ln R R 1 ETi 2 1 2 ln R ð7-50Þ o ¼ zo ¼ 2ð1 Þ ln R R 1 ETi n ð7-51Þ i ¼ zi ¼ 1þ 3 2ð1 Þ o ¼ zo ¼

ETi n 1 2ð1 Þ 3

ð7-52Þ

where do =di ¼ 1 þ n and lnðdo =di Þ ¼ lnð1 þ nÞ The simpliﬁed expressions for maximum tangential and longitudinal stresses for thin cylinders under the logarithmic temperature distribution, respectively

i ¼ zi ¼ o ¼ zo ¼

The expressions for radial (r ), tangential (hoop) ( ), and longitudinal (z ) stresses in a cylinder at radius r subject to linear thermal temperature distribution throughout the wall thickness of the cylinder by using equation T ¼ Ti ðro rÞ=ðro ri Þ when the thickness of the cylinder wall is small in comparison with the outside radius

r ¼

¼

ETi 2ð1 Þ

ETi 2ð1 Þ

ð7-54Þ

2 ETi ðr r2i Þðro þ 2ri Þ 2 6ðro þ ri Þ ð1 Þr 3 3 2ðr ri Þ 3ro ðr2 r2i Þ þ 6ðro ri Þ 2 ETi ðr þ r2i Þðro þ 2ri Þ 2 6ðro þ ri Þ ð1 Þr 2ðr3 r3i Þ 3ro ðr2 r2i Þ ðro rÞr2 ro ri 6ðro ri Þ

The expressions for maximum tangential (hoop), ( ) and longitudinal (z ) stresses at inner and outer surfaces of the cylinder under the linear thermal gradient as per equation T ¼ Ti ðro rÞ=ðro ri Þ

ð7-53Þ

ð7-55Þ

ð7-56Þ

ETi ro þ 2ri ro r z ¼ 1 2ðro þ ri Þ ro ri

ð7-57Þ

ETi 2ro þ ri 1 3ðro þ ri Þ

ð7-58Þ

i ¼ zi ¼ o ¼ zo ¼

ETi ro þ 2ri 1 3ðro þ ri Þ

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ð7-59Þ

PIPES, TUBES, AND CYLINDERS

7.16

CHAPTER SEVEN

Particular

The expressions for maximum tangential and longitudinal stresses at inner and outer wall surfaces of thin cylinder (i.e., ro ri ) under the linear thermal gradient as per equation T ¼ Ti ðro rÞ=ðro ri Þ

The wall thickness of a cylinder made of brittle materials The wall thickness of a cylinder made of ductile materials

Formula

i ¼ zi ¼ o ¼ zo ¼

ETi 2ð1 Þ

ð7-60Þ

ETi 2ð1 Þ

ð7-61Þ

Eqs. (7-60) and (7-61) for the linear thermal gradient are the same as Eqs. (7-53) and (7-54) for a logarithmic thermal gradient. ( ) di þ pi 1=2 h¼ 1 ð7-62Þ 2 pi d h¼ i 2

(

2pi

)

1=2

1

ð7-63Þ

where ¼ permissible working stress in tension, MPa (psi).

CLAVARINO’S EQUATION FOR CLOSED CYLINDERS (Based on the maximum strain energy) The general equation for equivalent tangential stress at any radius r

0 ¼ ð1 2Þa þ

The general equation for equivalent radial stress at any radius r

0r ¼ ð1 2Þa

The wall thickness for cylinders with closed ends

ð1 þ Þb r2

ð7-64Þ

ð1 þ Þb ð7-65Þ r2 where a and b have the same meaning as in Eqs. (7-25c) and (7-25d) " # di 0 þ ð1 2Þpi 1=2 h¼ 1 ð7-66Þ 2 0 ð1 þ Þpi where 0 ¼ permissible working stress in tension, MPa (psi).

BIRNIE’S EQUATIONS FOR OPEN CYLINDERS The equation for equivalent tangential stress at any radius r

0 ¼ ð1 Þa þ ð1 þ Þ

The equation for equivalent radial stress at any radius r

0r ¼ ð1 Þa ð1 þ Þ

b r2

ð7-67Þ

b ð7-68Þ r2 where a and b have the same meaning as in Eqs. (7-25c) and (7-25d)

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PIPES, TUBES, AND CYLINDERS PIPES, TUBES, AND CYLINDERS

Particular

The wall thickness of cylinders with open ends

7.17

Formula

h¼

di 2

0 þ ð1 Þpi 0 ð1 Þpi

1=2

1

ð7-69Þ

BARLOW’S EQUATION The tangential stress in the wall thickness of cylinder

pi do 2h For refer to Table 7-1.

¼

TABLE 7-7 Standard thickness of tubes Diameter, mm (in)

Minimum thickness, mm (in)

25 (1) and over but less than 62.5 (2.5)

2.37 (0.095)

62.5 (2.5) and over but less than 87.5 (3.25)

2.625 (0.105)

87.5 (3.25) and over but less than 100 (4)

3.000 (0.120)

100 (4) and over but less than 125 (5)

3.375 (0.135)

125 (5) and over but less than 150 (6)

3.750 (0.150)

150 (6) and over

h ¼ 0:0251do

Source: ASME Boiler and Pressure Vessel Code, Section 1, 1983.

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ð7-70Þ

PIPES, TUBES, AND CYLINDERS

TABLE 7-8 Comparison of various thick cylinder formulas Symbols: do ¼ 2ro ¼ outside diameter of thick cylinder, in; di ¼ 2ri ¼ inside diameter of thick cylinder, in; h ¼ ðdo di Þ=2 ¼ cylinder wall thickness, in; pi ¼ internal pressure, psi; ¼ Poisson’s statio (for steel ¼ 0:3); ¼ tangential stress, psi; r ¼ radial stress, psi; ð0 Þ po ¼ 0 ¼ tangential stress, r ¼ ri psi R¼

do ro p d 2 po do2 d2d2ð p p Þ p d2 pi di2 do2 ¼ ; a ¼ i i2 ; b ¼ i o 2 i 2 o ; ða0 Þpo ¼ 0 ¼ 2 i i 2 ; ðb0 Þpo ¼ 0 ¼ 2 di ri do di do di 4ðdo2 di2 Þ 4ðdo di Þ

Author

Particular

Formula

Remark

1. Birnie

The equation for an equivalent tangential stress at b 0 ¼ ð1 Þa þ ð1 þ Þ 2 any radius r of a thick cylinder under internal r pressure pi and external pressure po

0 The equation for an equivalent tangential stress ð0 Þp ¼ 0 ¼ ð1 Þa0 þ ð1 þ Þ b o r i r ¼ ri at inner radius ri of a thick cylinder subject to 2 internal pressure pi only when ¼ 0:3 for steel d ð1 Þpi þ ð1 þ Þ o pi di ¼ 2 do 1 di

¼

pi ½ð1 Þ þ ð1 þ ÞR2 R2 1

¼

pi ½0:7 þ 1:3R2 R2 1

2. Clavarino The general equation for an equivalent tangential b 0 ¼ ð1 2Þa þ ð1 þ Þ 2 stress at any radius r of a thick cylinder under r internal pressure pi and external pressure po 2

2 3 do ð1 þ Þ The equation for an equivalent tangential stress 6 ð1 2Þ d 7 7 6 0 ð Þpo ¼ 0 ¼ pi 6 2 þ 2 i 7 at inner radius ri of a thick cylinder subject to 5 4 d d o o r ¼ ri internal pressure pi only when ¼ 0:3 for steel 1 1 di di ð1 2Þ ð1 þ ÞR2 ¼ pi þ 2 2 R 1 R 1 0:4 1:3R2 ¼ pi 2 þ R 1 R2 1 0:4 þ 1:3R2 ¼ pi 2 R 1

3. Barlow

4. Lame´

The tangential stress in the wall thickness of cylinder under internal pressure pi

pi do d0 ¼ pi 2h do di d0 pR d ¼ i ¼ pi i d0 R1 1 di

¼

The tangential stress in the thick cylinder wall at b ¼ a þ 2 any radius r subject to internal pressure pi and r external pressure po

Eqn. (7-67) Used for ductile materials Open ends thick cylinder

Eqn. (7-64) Used for ductile materials Closed ends cylinder

Eqn. (7-70) Open ends cylinder

Eqn. (7-24a) Used for brittle materials

pi di2 d2 0 1 þ o2 Closed ends cylinder The tangential stress in the thick cylinder wall at ð Þpo ¼ 0 ¼ 2 2 do di di r ¼ ri inside radius ri of cylinder subject to internal 2 pressure pi only when ¼ 0:3 for steel pi do 1 þ R2 ¼ 1 þ ¼ p i di R2 1 ðdo =di Þ2 1

Refer to equations in Lingaiah, K., Machine Design Data Handbook, McGraw-Hill Book Company, New York, 1994

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PIPES, TUBES, AND CYLINDERS PIPES, TUBES, AND CYLINDERS

7.19

FIGURE 7-2 Nomogram to ﬁnd the stress in thick cylinder subject to internal pressure using four formulas given in Table 7-8.

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PIPES, TUBES, AND CYLINDERS

7.20

CHAPTER SEVEN

Particular

Formula

PROBLEM A closed end cylinder made of ductile material has inner diameter of 10 in (250 mm) and outside diameter of cylinder is 25 in (625 mm). The pressure inside the cylinder is 5000 psi. Use Clavarino’s equation from Table 7-8 R¼

do 25 ¼ ¼ 2:5 di 10

Mark on scale b at 2.5 Draw a perpendicular from x and this perpendicular meets scale d at y Join y and 5 (5000 psi) on scale e. Produce y–5 to meet scale f at z. y–5–z meets scale f at 8.25 Stress ¼ 8:25 ¼ 8250 psi Stress in SI units ¼ 8250 6:894 103 ¼ 56:88 MPa Check by using Clavarino’s equation from Table 7-8 0:4 þ 1:3R2 0:4 þ 1:3ð2:5Þ2 ¼ 5000 ¼ p1 R2 1 ð2:5Þ2 1 0:4 þ 8:125 4:2625 ¼ 5000 ¼ 104 6:25 1 5:25 ¼ 8120 psi ð56 MPaÞ The stress obtained from nomogram 8250 psi (56.88 MPa) is very close to stress value found from Clavarino’s equation

REFERENCES 1. ‘‘Rules for Construction of Power Boilers,’’ Section I, ASME Boiler and Pressure Vessel Code, American Society of Mechanical Engineers, New York, 1983. 2. ‘‘Rules for Construction of Pressure Vessels,’’ Section VIII, Division 1, ASME Boiler and Pressure Vessel Code, American Society of Mechanical Engineers, New York, July 1, 1986. 3. ‘‘Rules for Construction of Pressure Vessels,’’ Section VIII, Division 2—Alternative Rules, ASME Boiler and Pressure Vessel Code, American Society of Mechanical Engineers, New York, July 1, 1986. 4. Nicholas, R. W., Pressure Vessel Codes and Standards, Elsevier Applied Science Publications, Crown House, Linton Road, Barking, Essex, England. 5. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Co-operative Society, Bangalore, India, 1962. 6. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 7. Courtesy: Durham, H. M., Stress Chart for Thick Cylinders. 8. Greenwood, D. C., Editor, Engineering Data for Product Design, McGraw-Hill Book Company, New York, 1961. 9. Lingaiah, K., Machine Design Data Handbook (SI and U.S. Customary Systems Units), McGraw-Hill Book Company, New York, 1994.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

8 DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS SYMBOLS13;14;15 a

A A A A1 A2 A3 A41 , A42 , A43 A5 Ab Am Am1 ¼ Wm1 =sb Am2 ¼ Wm2 =sa

length of the long side of a rectangular plate, m (in) pitch or distance between stays, m (in) major axis of elliptical plate, m (in) long span of noncircular heads or covers measured at perpendicular distance to short span, m (in) (see Fig. 8-10) factor determined from Fig. 8-3 total cross-sectional area of reinforcement required in the plane under consideration, m2 (in2 ) (see Fig. 8-17) (includes consideration of nozzle area through shell for sna =sva < 1:0) outside diameter of ﬂange or, where slotted holes extend to the outside of the ﬂange, the diameter to the bottom of the slots, m (in) area in excess thickness in the vessel wall available for reinforcement, m2 (in2 ) (see Fig. 8-17) (includes consideration of nozzle area through shell if sna =sva < 1:0) area in excess thickness in the nozzle wall available for reinforcement, m2 (in2 ) (see Fig. 8-17) area available for reinforcement when the nozzle extends inside the vessel wall, m2 (in2 ) (see Fig. 8-17) cross-sectional area of various welds available for reinforcement (see Fig. 8-17), m2 (in2 ) cross-sectional area of material added as reinforcement (see Fig. 8-17), m2 (in2 ) cross-sectional area of the bolts using the root diameter of the thread or least diameter of unthreaded portion, if less, Eq. (8-111), m (in) total required cross-sectional area of bolts taken as the greater of Am1 and Am2 , m2 (in2 ) total cross-sectional area of bolts at root of thread or section of least diameter under stress, required for the operating condition, m2 (in2 ) total cross-sectional area of bolts at root of thread or section of least diameter under stress, required for gasket seating, m2 (in2 )

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.2

CHAPTER EIGHT

length of short side or breadth of a rectangular plate, m (in) short span of noncircular head, m (in) (see Fig. 8-10 and Eq. 8-86a) eﬀective gasket or joint-contact-surface seating width, m (in) basic gasket seating width, m (in) (see Table 8-21 and Fig. 8-13) factor determined from the application material–temperature chart for maximum temperature, psi inside diameter of ﬂange, m (in) corrosion allowance, m (in) basic dimension used for the minimum sizes of welds, mm (in), equal to tn or tx , whichever is less empirical coeﬃcient taking into account the stress in the knuckle [Eq. (8-68)] empirical coeﬃcient depending on the method of attachment to shell [Eqs. (8-82) and (8-85)] empirical coeﬃcients depending on the mode of support [(Eqs. (8-92) to (8-94)] bolt-circle diameter, mm (in) ﬁnished diameter of circular opening or ﬁnished dimension (chord length at midsurface of thickness excluding excess thickness available for reinforcement) of nonradial opening in the plane under consideration in its corroded condition, m (in) (see Fig. 8-17) diameter or short span, m (in) diameter of the largest circle which may be inscribed between the supporting points of the plate (Fig. 8-11), m (in) diameter as shown in Fig. 8-9, m (in) factor, m3 (in3 )

b b bo B B c c c1 c2 c4 , c5 C d

d

d U h g2 V o o U h g2 d¼ VL o o d0 d¼

de di , Di do , Do dk D Dp e

for integral-type ﬂanges for loose-type ﬂanges diameter through the center of gravity of the section of an externally located stiﬀening ring, m (in); inner diameter of the shell in the case of an internally located stiﬀening ring, m (in) [Eq. (8-55)] outside diameter of conical section or end (Fig. 8-8(A)d), m (in) inside diameter of shell, m (in) outside diameter of shell, m (in) inside diameter of conical section or end at the position under consideration (Fig. 8-8(A)d), m (in) inside shell diameter before corrosion allowance is added, m (in) outside diameter of reinforcing element, m (in) (actual size of reinforcing element may exceed the limits of available reinforcement) factor, m1 (in1 )

F ho F e¼ L ho

for integral-type ﬂanges

E Eam

modulus of elasticity at the operating temperature, GPa (Mpsi) modulus of elasticity at the ambient temperature, GPa (Mpsi)

e¼

for loose-type ﬂanges

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

hub stress correction factor for integral ﬂanges from Fig. 8-25 (When greater than one, this is the ratio of the stress in the small end of the hub to the stress in the large end. For values below limit of ﬁgure, use f ¼ 1.) fr strength reduction factor, not greater than 1.0 fr1 sna =sva fr2 (lesser of sna or spa Þ=sva fr3 spa =sva F total load supported, kN (lbf ) total bolt load, kN (lbf ) F correction factor which compensates for the variation in pressure stresses on diﬀerent planes with respect to the axis of a vessel (a value of 1.00 shall be used for all conﬁgurations, except for integrally reinforced openings in cylindrical shells and cones) F factor for integral-type ﬂanges (from Fig. 8-21) FL factor for loose-type ﬂanges (from Fig. 8-23) ga thickness of hub at small end, m (in) thickness of hub at back of ﬂange, m (in) g1 G diameter, m (in), at location of gasket load reaction; except as noted in Fig. 8-13, G is deﬁned as follows (see Table 8-22): When bo 6:3 mm (l/4 in), G ¼ mean diameter of gasket contact face, m (in). When bo > 6:3 mm (1/4 in), G ¼ outside diameter of gasket contact face less 2b, m (in). h distance nozzle projects beyond the inner or outer surface of the vessel wall, before corrosion allowance is added, m (in) (Extension of the nozzle beyond the inside or outside surface of the vessel wall is not limited; however, for reinforcement calculations the dimension shall not exceed the smaller of 2.5t or 2.5tn without a reinforcing element and the smaller of 2.5t or 2.5tn þ te with a reinforcing element or integral compensation.) h hub length, m (in) h, t minimum required thickness of cylindrical or spherical shell or tube or pipe, m (in) thickness of plate, m (in) thickness of dished head or ﬂat head, m (in) ha actual thickness of shell at the time of test including corrosion allowance, m (in) hc thickness for corrosion allowance, m (in) hD radial distance from the bolt circle, to the circle on which HD acts, m (in) hG ¼ ðC GÞ=2 radial distance from gasket load reaction to the bolt circle, m (in) pﬃﬃﬃﬃﬃﬃﬃﬃ ho ¼ Bgo factor, m (in) hT radial distance from the bolt circle to the circle on which HT acts as prescribed, m (in) H ¼ G2 P=4 total hydrostatic end force, kN (lbf ) HD ¼ B2 P=4 hydrostatic end force on area inside of ﬂange, kN (lbf ) HG ¼ W H gasket load (diﬀerence between ﬂange design bolt load and total hydrostatic end force), kN (lbf ) HP ¼ total joint-contact-surface compression load, kN (lbf ) 2b GmP f

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8.3

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.4

CHAPTER EIGHT

HT ¼ H HD Is Is0 I I0 k 1 , k2 , k3 , k4 , k 5 k6 K ¼ A=B K K1 l L

diﬀerence between total hydrostatic end force and the hydrostatic end force on area inside of ﬂange, kN (lbf ) required moment of inertia of the stiﬀening ring cross-section around an axis extending through the center of gravity and parallel to the axis of the shell, m4 or cm4 (in4 ) required moment of inertia of the combined ring-shell crosssection about its neutral axis parallel to the axis of the shell, m4 (in4 ) available moment of inertia of the stiﬀening ring cross-section about its neutral axis parallel to the axis of the shell, m4 (in4 ) available moment of inertia of combined ring shell cross-section about its neutral axis parallel to the axis of the shell, m4 or cm4 (in4 ) coeﬃcients factor for noncircular heads depending on the ratio of short span to long span b=a (Fig. 8-10) ratio of outside diameter of ﬂange to inside diameter of ﬂange (Fig. 8-20) ratio of the elastic modulus E of the material at the design material temperature to the room temperature elastic modulus, Eam , [Eqs. (8-26) to (8-31), (8-55)] spherical radius factor (Table 8-18) length of ﬂange of ﬂanged head, m (in) eﬀective length, m (in) distance from knuckle or junction within which meridional stresses determine the required thickness, m (in) perimeter of noncircular bolted heads measured along the centers of the bolt holes, m (in) distance between centers of any two adjacent openings, m (in) length between the centers of two adjacent stiﬀening rings, m (in) (Fig. 8-1)

te þ 1 t3 factor þ T d m gasket factor, obtained from Table 8-20 m ¼ 1= reciprocal of Poisson’s ratio Mb longitudinal bending moment, N m (lbf in) L¼

FIGURE 8-1 Cylindrical pressure vessels under external pressure.

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

torque about the vessel axis, N m (lbf in) component of moment due to HD , m N (in-lbf ) component of moment due to HG , m N (in-lbf ) total moment acting on the ﬂange, for the operating conditions or gasket seating as may apply, m N (in-lbf ) MT ¼ HT h component of moment due to HT , m N (in-lbf ) N width, m (in), used to determine the basic gasket seating with bo , based on the possible contact width of the gasket (see Table 8-21) pi internal design pressure, MPa (psi) p maximum allowable working pressure or design pressure, MPa (psi) po load per unit area, MPa (psi) external design pressure, MPa (psi) P total pressure on an area bounded by the outside diameter of gasket, kN (lbf ) design pressure (or maximum allowable working pressure for existing vessels), MPa (psi) Pa calculated value of allowable external working pressure for assumed value of t or h, MPa (psi) r radius of circle over which the load is distributed, m (in) ri inner radius of a circular plate, m (in) inside radius of transition knuckle which shall be taken as 0:01dk in the case of conical sections without knuckle transition, m (in) R inner radius of curvature of dished head, m (in) Ri inner radius of shell or pipe, m (in) ro , Ro outer radius of a circular plate, m (in) outer radius of shell, m (in) R ¼ ½ðC BÞ=2 radial distance from bolt circle to point of intersection of hub g1 and back of ﬂange, m (in) (for integral and hub ﬂanges) R inside radius of the shell course under consideration, before corrosion allowance is added, m (in) Rn inside radius of the nozzle under consideration, before corrosion allowance is added, m (in) t or h minimum required thickness of spherical or cylindrical shell, or pipe or tube, m (in) t ﬂange thickness, m (in) t nominal thickness of the vessel wall, less corrosion allowance, m (in) tc weld dimensions thickness or height of reinforcing element, m (in) te tn nominal thickness of shell or nozzle wall to which ﬂange or lap is attached, irrespective of product form less corrosion allowance, m (in) tr required thickness of a seamless shell based on the circumferential stress, or of a formed head, computed by the rules of this chapter for the designated pressure, m (in) trn required thickness of a seamless nozzle wall, m (in) nominal thickness of cylindrical shell or tube exclusive of ts corrosion allowance, m (in) tw weld dimensions tx two times the thickness go , when the design is calculated as an integral ﬂange, m (in), or two times the thickness, m (in), of shell nozzle wall required for internal pressure, when the design is calculated as a loose ﬂange, but not less than 6.3 mm Mt MD ¼ HD hD MG ¼ HG hG Mo

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8.5

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.6

CHAPTER EIGHT

(1/4 in) T U V VL w W W W Wm1

Wm2 y y ymax Y Z , 1 , 2 sy sa e sam sd sa sna sva spa sbat sbd sfd snd

H R 0 r s su z or l

factor involving K (from Fig. 8-20) factor involving K (from Fig. 8-20) factor for integral-type ﬂanges (from Fig. 8-22) factor for loose-type ﬂanges (from Fig. 8-24) width, m (in), used to determine the basic gasket seating width bo , based on the contact width between the ﬂange facing and the gasket (see Table 8-21) weight, kN (lbf ) total load to be carried by attachment welds, kN (lbf ) ﬂange design bolt load, for operating conditions or gasket seating, as may apply, kN (lbf ) minimum required bolt load for the operating conditions, kN (lbf ) (For ﬂange pairs used to contain a tubesheet for a ﬂoating head for a U-tube type of heat exchanger, or for any other similar design, Wm1 shall be the larger of the values as individually calculated for each ﬂange, and that value shall be used for both ﬂanges.) minimum required bolt load for gasket seating, kN (lbf ) gasket or joint-contact-surface unit seating load, MPa (psi) deﬂection of the plate, m (in) maximum deﬂection of the plate, m (in) factor involving K (from Fig. 8-20) factor involving K (from Fig. 8-20) a factor for non-circular heads [Eq. (8-86b)] angles of conical section to the vessel axis, deg (Fig. 8-8(A)d) diﬀerence between angle of slope of two adjoining conical sections, deg (Fig. 8-8(A)d) normal or direct stress, MPa (psi) 0.2 percent proof stress, MPa (psi) maximum allowable stress value, MPa (psi) equivalent stress (based on shear strain energy), MPa (psi) allowable stress at ambient temperature, MPa (psi) design stress value, MPa (psi) allowable stress value as given in Tables 8-9 to 8-12, MPa (psi) allowable stress in nozzle, MPa (psi) allowable stress in vessel, MPa (psi) allowable stress in reinforcing element (plate), MPa (psi) allowable bolt stress at atmospheric temperature, MPa (psi) allowable bolt stress at design temperature, MPa (psi) allowable design stress for material of ﬂange at design temperature (operating condition) or atmospheric temperature (gasket seating), as may apply, MPa (psi) allowable design stress for material of nozzle neck, vessel or pipe wall, at design temperature (operating condition) or atmospheric temperature (gasket seating), as may apply, MPa (psi) calculated longitudinal stress in hub, MPa (psi) calculated radial stress in ﬂange, MPa (psi) calculated tangential stress in ﬂange, MPa (psi) hoop stress, MPa (psi) radial stress, MPa (psi) strength, MPa (psi) ultimate strength, MPa (psi) longitudinal stress, MPa (psi)

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.7

zt tensile longitudinal stress, MPa (psi) zc compressive longitudinal stress, MPa (psi) shear stress (also with subscripts), MPa (psi) Poisson’s ratio joint factor (Table 8-3) or eﬃciency ¼1 (see deﬁnitions for tr and trn ) when an opening is in the solid plate or joint eﬃciency obtained 1 ¼ 1 from Table 8-3 when any part of the opening passes through any other welded joint Note: and with initial subscript s designates strength properties of material used in the design which will be used and observed throughout this Machine Design Data Handbook. Other factors in performance or in special aspect are included from time to time in this chapter and, being applicable only in their immediate context, are not given at this stage.

Particular

PLATES13;14;15

Formula

Refer to Table 8-1

For maximum stresses and deﬂections in ﬂat plates

Plates loaded uniformly The thickness of a plate with a diameter d supported at the circumference and subjected to a pressure p distributed uniformly over the total area The maximum deﬂection

Plates loaded centrally The thickness of a ﬂat cast-iron plate supported freely at the circumference with diameter d and subjected to a load F distributed uniformly over an area (do2 =4) The deﬂection Grashof’s formula for the thickness of a plate rigidly ﬁxed around the circumference with the above given type of loading

h ¼ k1 d

p sd

1=2 ð8-1Þ

Refer to Table 8-2 for values of k1 . p y ¼ k2 d 4 Eh3 Refer to Table 8-2 for values of k2 .

ð8-2Þ

0:67do F 1=2 h ¼ 1:2 1 d sd

ð8-3Þ

0:12d 2 F Eh3 F d 1=2 ln h ¼ 0:65 sd do y¼

y¼

0:055d 2 F Eh3

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ð8-4Þ ð8-5Þ ð8-6Þ

Form of plate

Distributed on circumference of a concentric circle of radius r

8-134

8-133

Edge supported

Edge ﬁxed

8-132

Edge ﬁxed

8-131

Edge supported

Distributed over a concentric circular area of radius r

8-130

Edge ﬁxed

Eq. 8-129

Type of support

Distributed Edge over the entire supported surface

Type of loading

TABLE 8-1 Maximum stresses and deﬂections in ﬂat plates

2rp

2rp

r2 p

r2 p

r2o p

r2o p

Total load, F

3F 4h2

3Fð3m þ 1Þ 8mh2

Center

Edge

Center

Edge

Center

Location of max

8.8

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. r ¼

3F r2 1 2 2 2h ro

All points inside the 2 circle of r r radius r þ ðm þ 1Þ loge o ðm 1Þ 2 r ro 3F Center when r ¼ ¼ ðm þ 1Þ 4mh2 r < 0:31ro Edge when ro r2 r > 0:31ro 2 loge þ 2 1 r ro

r ¼ ¼

3F r ðm þ 1Þ loge ro 2mh2 r2 þ ðm þ 1Þ 2 4ro 3F m 1 r ¼ ¼ 2 2mh2

r ¼ ¼

3F r ðm þ 1Þ loge o 2 r 2mh 2 r ðm 1Þ 2 þ m 4ro 3F r2 r ¼ 1 2 2 2ro 2h

r ¼

r ¼ ¼

Maximum stress, max

ro ð7m þ 3Þ 2 r mþ1 r

3Fðm2 1Þ 2Em2 h3 ð3m þ 1Þðr2o r2 Þ r r2 loge o 2ðm þ 1Þ r 2 3Fðm 1Þ 1 2 ro 2 2 2 ðro r Þ r loge r 2 3 2Em h

3Fðm2 1Þr2o 4Em2 h3

when r is very small (concentrated load)

3Fðm2 1Þ r 4r2o 4r2 loge o 3r2 2 3 r 16Em h

4r2 loge

3Fðm2 1Þr2o 16Em2 h3 3Fðm2 1Þ ð12m þ 4Þr2o mþ1 16Em2 h3

3Fðm 1Þð5m þ 1Þr2o 16Em2 h3

Maximum deﬂection, ymax

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

Form of plate

8-137

8-138

Distributed Outer edge over the entire ﬁxed and surface supported

Distributed Outer edge over the entire ﬁxed and surface supported, inner edge ﬁxed

8-135

8-136

Uniform pressure over entire lower surface

Distributed over a concentric circular area of radius r

Eq.

Distributed Outer edge over the entire supported surface

Type of support

Type of loading

F ¼ ðr2o r2i Þp

F ¼ ðr2o r2i Þp

F ¼ ðr2o r2i Þp

r2 p

Total load, F

TABLE 8-1 Maximum stresses and deﬂections in ﬂat plates (Cont.)

r ¼

¼

¼

ro r

3p ðr2o þ r2 Þ 4h2 4r2 r2 r 2 2 o 2 loge o r ro r

3pðm2 1Þ 4mh2 2 r 3 r2o r4i 12 r2o r2i loge o 4 ri 5 2 ro ðm 1Þ þ r2i ðm þ 1Þ

4ðm þ 1Þr2o r2i loge

þr4i ðm 1Þ 4mr2o r2i

3P r4o ð3m þ 1Þ 4mh2 ðr2o r2i Þ

Maximum stress, max 3F r ðm þ 1Þ loge o r ¼ ¼ 2 r 2mh 2 m1 r 1 2 þ 4 ro

Inner edge

Inner edge

Inner edge

Center

Location of max

3pðm2 1Þ 4 ro þ 3r4i 4r2o r2i 16Em2 h3 r 16r2 r4 r 2 4r2o r2i loge o þ 2 o i2 loge o ri ro ri ri

...

r4i ðm þ 3Þ r2o r2i ð3m þ 1Þ 8ðm þ 1Þ 2ðm þ 1Þ r2o r2i ð3m þ 1Þ r loge o 2ðm 1Þ ri 2 4 2r r ðm þ 1Þ r 2 2 o i2 loge o r ðro ri Þðm 1Þ þ

þ

3Fðm2 1Þ r4o ð5m þ 1Þ 2 3 8ðm þ 1Þ 2Em h

where r is very small (concentrated load) 3Fðm 1Þð7m þ 3Þr2o =16Em2 h3

3Fðm2 1Þ ro 2 2 3m þ 1 log 4r þ 2r e mþ1 r 16Em2 h3 2 2 4 7m þ 3 ðr r Þr r4 þ 2 r2o þ o 2 2 mþ1 r ro ro

Maximum deﬂection, ymax

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

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8.9

8.10

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8-141

8-142

8-143

Uniform over All edges entire surface supported

Uniform over All edges entire surface ﬁxed

Uniform over Short edges entire surface ﬁxed, long edges supported

Uniform over Short edges entire surface supported, long edges ﬁxed

Eq. 8-139

Type of support

Distributed Inner edge over the entire ﬁxed and surface supported

Type of loading

F ¼ abp

F ¼ abp

F ¼ abp

F ¼ abp

F¼

ðr2o

r2i Þp

Total load, F

b ¼

b ¼

b ¼

b ¼

Center of long edge

Center of short edge

Center of long edge

0:5b2 p b6 h2 1 þ 0:623 6 a

0:75b2 p b4 h2 1 þ 0:8 4 a b2 p a4 2h2 1 þ 0:2 4 b

Center

r4o ðm þ 3Þ þ r4o ðm 1Þ þ 4r2o r2i r2o ðm þ 1Þ þ r2i ðm 1Þ

Inner edge

Location of max

0:75b2 p b3 h2 1 þ 1:61 3 a

Maximum stress, max 3p r r ¼ 2 4r4o ðm þ 1Þ loge o r 4h

Note: Positive sign for indicates tension at upper surface and equal compression at lower surface; negative sign indicates reverse condition.

Form of plate

TABLE 8-1 Maximum stresses and deﬂections in ﬂat plates (Cont.)

...

...

0:0284b4 p b5 Eh3 1 þ 1:056 5 a

0:1422b4 p b3 Eh3 1 þ 2:21 3 a

...

Maximum deﬂection, ymax

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.11

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

TABLE 8-2 Coeﬃcients in formulas for cover plates13;14;15 Circular plate

Rectangular plate

Elliptical plate

Material of cover plate

Methods of holding edges

k1

k2

k3

k4

k5

Cast iron

Supported, free Fixed

0.54 0.44

0.038 0.010

0.75 0.62

1.73 1.4; 1.6a

1.5 1.2

Mild steel

Supported, free Fixed

0.42 0.35

... ...

0.60 0.49

1.38 1.12; 1.28

1.2 0.9

a

With gasket.

Particular

Formula

The deﬂection

Rectangular plates UNIFORM LOAD The thickness of a rectangular plate according to Grashof and Bach

h ¼ abk3

abF sd ða2 þ b2 Þ

The thickness of uniformly loaded elliptical plate

1=2 ð8-8Þ

where k4 ¼ coeﬃcient, taken from Table 8-2

Elliptical plate

ð8-7Þ

sd ða2 þ b2 Þ

where k3 ¼ coeﬃcient, taken from Table 8-2

h ¼ k4 CONCENTRATED LOAD The thickness of a rectangular plate on which a concentrated load F acts at the intersection of diagonals

1=2

p

h ¼ abk5

p

1=2 ð8-9Þ

sd ða2 þ b2 Þ

where k5 ¼ coeﬃcient, taken from Table 8-2

SHELLS (UNFIRED PRESSURE VESSEL) Shell under internal pressure—cylindrical shell CIRCUMFERENCE JOINT The minimum thickness of shell exclusive of corrosion allowance as per Bureau of Indian Standards11

h¼

pdi pdo ¼ 2sa p 2sa þ p

ð8-10Þ

Refer to Tables 8-3 and 8-8 for values of and sa , respectively.

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.12

CHAPTER EIGHT

TABLE 8-3 Joint eﬃciency factor ()13;14;15 Requirement

Class 1

Class 2

Class 3

Weld joint 1.00 eﬃciency factor ()

0.85

0.70

0.60

0.50

Shell or end plate thickness

No limitation on thickness

Maximum thickness 38 mm after adding corrosion allowance

Maximum thickness 16 mm before corrosion allowance is added

Maximum thickness 16 mm before corrosion allowance is added

Maximum thickness 16 mm before corrosion allowance is added

Type of joints

Double-welded butt joints with full penetration excluding butt joints with metal backing strips which remain in place Single-welded butt joints with backing strip ¼ 0:9

Double-welded butt joints with full penetration excluding butt joints with metal backing strips which remain in place Single-welded butt joints with backing strip ¼ 0:80

Double-welded butt joints with full penetration excluding butt joints with metal backing strips which remain in place Single-welded butt joints with backing strip ¼ 0:65

Single-welded butt joints with backing strip not over 16 mm thickness or over 600 mm outside diameter

Single full ﬁllet lap joints for circumferential seams only

Single-welded butt joints without backing strip ¼ 0:55

Source: K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962; K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1983; K. Lingaiah, Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986; and IS: 2825-1969.

Particular

Note: A minimum thickness of 1.5 mm is to be provided as corrosive allowance unless a protective lining is employed. The design pressure or maximum allowable working pressure The minimum thickness of shell exclusive of corrosion allowance as per ASME Boiler and Pressure Vessel Code The maximum allowable working pressure as per ASME Boiler and Pressure Vessel Code [from Eq. (8-12)]1;2

Formula

p¼

2sa h 2sa h ¼ di þ h do h

ð8-11Þ

t¼

pRi 2sa þ 0:4p

ð8-12Þ

when the thickness of shell does not exceed one-half the inside radius ðRi Þ p¼

2sa t Ri 0:4t

ð8-13Þ

when the pressure p does not exceed 1:25sa . sa is taken from Tables 8-9, 8-11, and 8-12.

Rules for construction of pressure vessel, section VIII, Division 1, ASME Boiler and Pressure Vessel Code, July 1, 1986.

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

Particular

LONGITUDINAL POINT The minimum thickness of shell exclusive of corrosive allowance as per ASME Boiler and Pressure Vessel Code. [1-10]

8.13

Formula

t¼

pRi pRo ¼ sa 0:6p sa þ 0:4p

ð8-14Þ

when the thickness of shell does not exceed one-half the inside radius Ri sa t sa t ¼ Ri þ 0:6t Ro 0:4t

The maximum allowable working pressure as per ASME Boiler and Pressure Vessel Code [from Eq. 8-14)]

p¼

The design stress for the case of welded cylindrical shell assuming a Poisson ratio of 0.3

d ¼ 0:87

The allowable stress for plastic material taking into consideration the combined eﬀect of longitudinal and tangential stress (Note: The design stress for plastic material is 13.0 percent less compared with the maximum value of the main stress.)

a ¼

The thickness of shell from Eq. (8-17) without taking into account the joint eﬃciency and corrosion allowance

h¼

ð8-15Þ

when the pressure p does not exceed 0.385sa pi ro h

ð8-16Þ

pi do 2:3h

ð8-17Þ

pdo 2:3sa

ð8-18Þ

The design thickness of shell taking into consideration the joint eﬃciency and allowance for corrosion, negative tolerance, and erosion of the shell (hc )

hd ¼

pdo þ hc 2:3sa

ð8-19Þ

The design formula for the thickness of shell according to Azbel and Cheremisineﬀ 10

hd ¼

pdi þ hc 2:3sa p

ð8-20Þ

The factor of safety as per pressure vessel code, which is based on yield stress of material used for shell

n¼

sy a

ð8-21Þ

The factor of safety n should not be less than 4, which is based on yield strength sy of material.

Shell under internal pressure—spherical shell The minimum thickness of shell exclusive of corrosion allowance as per Bureau of Indian Standards

h¼

pdi pdo ¼ 4sa p 4sa þ p

ð8-22Þ

The design pressure as per Bureau of Indian Standards

p¼

4sa h 4sa h ¼ di þ h do h

ð8-23Þ

Rules for construction of pressure vessel, section VIII, Division 1, ASME Boiler and Pressure Vessel Code, July 1, 1986.

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.14

CHAPTER EIGHT

Particular

Formula

The minimum thickness of shell exclusive of corrosion allowance as per ASME Boiler and Pressure Vessel Code

t¼

The design pressure (or maximum allowable working pressure) as per ASME Boiler and Pressure Vessel Code

p¼

Shells under external pressure—cylindrical shell (Fig. 8-1) (a) The minimum thickness of cylindrical shell exclusive of corrosion allowance as per Bureau of Indian Standards

pRi 2sa 0:2p

ð8-24Þ

when thickness of the shell of a wholly spherical vessel does not exceed 0.356Ri 2sa t Ri þ 0:2t

ð8-25Þ

when the maximum allowable working pressure p does not exceed 0.655sa

"

2=3 # 1:15p 4 KL þ 1:1570 10 h ¼ do do SI

ð8-26aÞ

where h, do , and L in m; and p in MPa and h ¼ t ¼ thickness of shell. " 2=3 # 1:15p 6 KL h ¼ do þ 4:19 10 do USCS

The design pressure as per Bureau of Indian Standards

ð8-26bÞ

where h, do , and L in in; and p in psi " 2=3 # h 4 KL p¼ 1:157 10 1:15 do do SI

ð8-27aÞ

where p and in MPa; h, do , and L in m " 2=3 # h 6 KL 4:19 10 p¼ 1:15 do do USCS

ð8-27bÞ

where p and in psi; h, do , and L in in for

L 5:7ð10p=Þ5=2 372:65 103 ðh=do Þ3=2 < or < pK K do SI

ð8-27cÞ

where and p in MPa; do , h, and L in m for

L 5:7ð10p=Þ5=2 5:41 107 ðh=do Þ3=2 or < < do pK K USCS

where and p in psi; L, do and h in in

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ð8-27dÞ

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

Particular

(b) The minimum thickness of cylindrical shell exclusive of corrosion allowance according to Bureau of Indian Standards11

8.15

Formula

h ¼ 2:234 104 do ð pKÞ1=3 but not less than ð3:5=2Þð pdo =Þ

SI

ð8-28aÞ

where do and h in m and p in MPa h ¼ 4:25 103 do ð pKÞ1=3 but not less than ð3:5=2Þð pdo =Þ

USCS

ð8-28bÞ

where do and h in in and p in psi or The design pressure as per Bureau of Indian Standards from Eq. (8-28)

8:97 1010 K

p¼

h do

3 but not greater than SI

2h 3:5do ð8-29aÞ

where p in MPa and h and do in m 13 106 K

p¼

h do

3 but not greater than

2 h 3:5 do

USCS

ð8-29bÞ

where p in psi and h and do in in for

L 97:78 14:6 > or > do ð pKÞ1=6 ð100h=do Þ1=2

for

L 22:4 1:46 > or > do ð pKÞ1=6 ðh=do Þ1=2

or

5:7

0:58

ð10p=Þ5=2 22:4 > pK ð pKÞ1=6

or

372:65 103

USCS

ð10p=Þ5=2 97:78 > pK ð pKÞ1=6

54:1 106 (c) In other cases, the minimum thickness of the shell exclusive of corrosion allowance as per Bureau of Indian Standards

SI

SI

USCS

ðh=do Þ3=2 1:46 > K ðh=do Þ1=2

SI

ðh=do Þ3=2 1:46 > K ðh=do Þ1=2

2=5 L h ¼ 3:576 10 do p K do 5

where h, do , and L in m; p in MPa

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USCS

SI

ð8-30aÞ

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.16

CHAPTER EIGHT

Particular

Formula

2=5 L h ¼ 1:227 103 do p K do

USCS

ð8-30bÞ

SI

ð8-31aÞ

USCS

ð8-31bÞ

where h, L, and do in in; p in psi or The design pressure as per Bureau of Indian Standards

p¼

3:162 1012 ðh=do Þ5=2 LK=do

where h, L, and do in m; p in MPa h¼

189:58 106 ðh=do Þ5=2 LK=do

where h, do , and L in in; p in psi Reference Chart for ASME Boiler and Pressure Vessel Code, Section VIII, Division 112

Refer to Fig. 8-2.

(d) Maximum allowable stress values (1) The maximum allowable stress values in tension for ferrous and nonferrous materials sa The maximum allowable stress values (sa ) for bolt, tube, and pipe materials

Refer to Tables 7-1, 8-8 and 8-13 for sa . Refer to Tables 7-1, 8-8, 8-12 and 8-17.

FIGURE 8-2 Reference chart for ASME Boiler and Pressure Vessel Code, Section VIII, Division 1. (By permission, Robert Chuse, Pressure Vessels—The ASME Code Simpliﬁed, 5th edition, McGraw-Hill, 1977.)12

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.17

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

Particular

(2) The maximum allowable longitudinal compressive stress (ac ) to be used in the design of cylindrical shells or tubes, either seamless or butt-welded subjected to loadings that produce longitudinal compression in shell or tube. shall be as given in either Eq. (a) or (b).

Formula

ac < sa

from Tables 7-1, 8-9 to 8-13 (a)

ac < B

ðbÞ

where B ¼ a factor determined from the applicable material/temperature chart for maximum design temperature, psi, Figs. 8-4, 8-5. [Note: US Customary units (i.e., fps system of units) were used in drawing Figs. 8-3 to 8-5 of ASME Pressure Vessel and Boiler Code, which is now used to ﬁnd the thickness of walls of cylindrical and spherical shells and tubes, unless it is otherwise mentioned to use both SI and US Customary units. Figures 8-3 to 8-5 are in US Customary units. The values from these ﬁgures and others can be used in the appropriate equation to ﬁnd the values or results in SI units, if these values and equations are converted into SI units beforehand.]

(3) The procedure for determining the value of the factor B

The value of factor A

Select the thickness t (¼ h) and outside diameter Do or outside radius Ro of a cylindrical shell or tube in the corroded condition. Then calculate the value of A from Eq. (8-32) A¼

0:125 Ro =t

ð8-32Þ

Using this value of A enter the applicable material/ temperature chart for the material (Figs. 8-4 and 8-5) under consideration to ﬁnd B. In case the value of A falls to the right of the end of the material/temperature line (Figs. 8-4 and 8-5), assume an intersection with the horizontal projection of the upper end of the material/temperature line. From the intersection move horizontally to the right and ﬁnd the value of B. This is the maximum allowable compressive stress for the value of t and Ro assumed. If the value of A falls to the left of the applicable material/temperature line, the value of B, psi, shall be calculated from Eq. (8-33). The expression for value of factor B

AE ð8-33Þ 2 where E ¼ modulus of elasticity of material at design temperature, psi

B¼

Compare the value of B determined from Eq. (8-33) or from the procedure outlined above with the computed longitudinal compressive stress in the cylindrical shell or tube using the selected values of t and Ro . If the value of B is smaller than the computed, compressive stress, a greater value of t must be

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.18

CHAPTER EIGHT

FIGURE 8-3 Geometric chart for cylindrical vessels under external or compressive loadings (for all materials). (Source: American Society of Mechanical Engineers, ASME Boiler and Pressure Vessel Code, Section VIII, Division 1, July 1, 1986.)1;2;3

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.19

FIGURE 8-4 Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of carbon or low-alloy steels (speciﬁed minimum yield strength 24,000 psi to, but not including, 30,000 psi); (1 kpsi ¼6.894757 MPa).1;2;3

FIGURE 8-5 Chart for determining shell thickness of cylindrical and spherical vessels under external pressure when constructed of carbon or low-alloy steels (speciﬁed minimum yield strength 30,000 psi and over except for materials within this range where other speciﬁc charts are referenced) and type 405 and type 410 stainless steels (1 kpsi ¼6.894757 MPa). (Source: American Society of Mechanical Engineers, ASME Boiler and Pressure Vessel Code, Section VIII, Division 1, July 1, 1986.)1;2;3

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.20

CHAPTER EIGHT

Particular

Formula

selected and the procedure outlined above is repeated until a value of B is obtained, which is greater than the compressive stress computed for the loading on the cylindrical shell or tube. (e) Cylindrical shells and tubes. The required thickness of cylindrical shell or tube exclusive of corrosion allowance under external pressure either seamless or with longitudinal butt-welded joint as per ASME Boiler and Pressure Vessel Code can be determined by the following procedure: (1) Cylinders having (Do =t) values 10. Assume the thickness t of shell or tube. Determine Do =t and L=Do . Use Fig. 8-3 to ﬁnd A. Find the value of A from Fig. 8-3 by following the procedure explained in paragraph (d) (3)

In cases where the value of A falls to the right of the end of the material/temperature line, assume an intersection with the horizontal projection of the upper end of the material/temperature line. Using this value of A enter the applicable material/temperature chart for material (Figs. 8-4 and 8-5) under consideration and ﬁnd the value of B. This value of B is the maximum allowable compressive stress for the value of t and Ro assumed, Pa (psi).

The equation for maximum allowable external pressure (Pa ) by using this value of B

Pa ¼

4B 3ðDo =tÞ

ð8-34Þ

The equation for maximum allowable external pressure Pa for values of A falling to the left of the applicable material/temperature line.

Pa ¼

2AE 3ðDo =tÞ

ð8-35Þ

where Pa obtained from Eq. (8-35) is equal to or greater than P. P is the external design pressure, psi. This external allowable pressure is 15 psi (103.4 kPa) or less. The maximum external pressure is 15 psi (103.4 kPa) or 25% more than the maximum possible external pressure, whichever is smaller. (2) Cylinders having (Do =t) values <10. Using the procedure as outlined in section (d)(3), obtain the value of B. For values of (Do =t) less than 4, the value of A can be calculated using Eq. (8-36) The formula to calculate the value of Pa1 The formula to calculate the value of Pa2

A¼

1:1 ðDo =tÞ2

ð8-36Þ

For values of A greater than 0.10, use a value of 0.10 2:167 0:0833 B Do =t 2s 1 ¼ 1 Do =t Do =t

Pa1 ¼

ð8-37Þ

Pa2

ð8-38Þ

where s is the lesser of two times the maximum allowable stress value at design metal temperature, from the applicable Tables 8-9 to 8-12 or 0.9 times the yield strength of the

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

Particular

8.21

Formula

material at design temperature. The yield strength values are twice the B value obtained from the applicable material/ temperature chart. The smaller of the values of Pa1 or Pa2 shall be used for the maximum allowable external pressure Pa . Thus Pa obtained is equal to or greater than the design pressure P.

Shell under external pressure—spherical shell The thickness of a spherical shell as per Bureau of Indian Standards The design pressure as per Indian Standards

h¼

pdo 0:80sa

ð8-39Þ

p¼

0:80sa h do

ð8-40Þ

A¼

0:125 Ro =t

ð8-41Þ

The minimum required thickness of a spherical shell exclusive of corrosion allowance under external pressure, either seamless or of built-up construction with butt joints, shall be determined by the following procedure as per ASME Boiler and Pressure Vessel Code. Select a value for t. Determine Do =L and Do =t. Find the value of A by using Fig. 8-3. The value of the factor A is also calculated from Eq. (8-41). Using this value of A, ﬁnd the value of B from the applicable material/temperature chart as done in case of the cylindrical shell

where Ro is the outside radius of spherical shell in the corroded condition, in

The maximum allowable external pressure Pa for values of A falling to the right of the applicable material/temperature line

Pa ¼

The maximum allowable external pressure Pa for values of A falling to the left of the applicable material/temperature line

Pa ¼

B Ro =t

ð8-42Þ

where Pa is the calculated value of allowable external working pressure for the assumed value of t, Pa (psi), and P is the external design pressure, Pa (psi) 0:0625E ðRo =tÞ2

ð8-43Þ

The smaller value of Pa from Eq. (8-42) or (8-43) shall be used for the maximum allowable external pressure Pa . Pa obtained is equal to or greater than the design pressure P.

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.22

CHAPTER EIGHT

Particular

Formula

For ﬁnding the thickness of a shell in the design of a longitudinal lap joint in a cylindrical or any lap joint in a spherical shell under external pressure

The thickness of the shell shall be determined by the rules already narrated for the longitudinal butt joint of the cylindrical and spherical shell, except that 2P shall be used instead of P in the calculations for the required thickness.

Cylindrical shell under combined loading as per Indian Standards The longitudinal stress

The hoop stress The shear stress

The Huber-Hencky equation for equivalent stress based on the shear strain energy criterion The basic design stress based on distortion energy theory

2 M pd þ W 4 b di 4 i z ¼ hðdi þ hÞ ¼

ð8-44Þ

pðdi þ hÞ 2h

ð8-45Þ

2Mt hdi ðdi þ hÞ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ e ¼ 2 z þ 2z þ 3 2

ð8-46Þ

¼

ð8-47Þ

d ¼ ½2 þ 2z þ 2r 2ð z þ z r þ r Þ1=2 ð8-48Þ

Requirements are: (a) At design conditions

e sa

ð8-49Þ

zt sa

ð8-50Þ

ze 0:125Eðh=do Þ

ð8-51Þ

Refer to Table 8-14 for values of E. (b) At test conditions

e 1:3sam

ð8-52Þ

zt 1:3sam

ð8-53Þ

zc 0:125Esam ðha =do Þ

ð8-54Þ

Stiﬀening rings for cylindrical shells under external pressure The moment of inertia of the stiﬀening rings as per Indian Standards

Is ¼ 0:714 106 pLs d 03 K 4

SI

ð8-55aÞ

0

where Is in m , p in Pa, Ls and d in m Is ¼ 4:29 103 pLs d 03 K 4

USCS 0

where Is in in , p in psi, Ls and d in in

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ð8-55bÞ

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.23

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

Particular

The required moment of inertia of a circumferential stiﬀening ring shall be not less than that determined by one of the formulas given in Eqs. (8-56) and (8-57) as per ASME Boiler and Pressure Vessel Code

Formula

Is ¼

D2o Ls ½t þ ðAs =Ls ÞA 14

USCS

ð8-56Þ

Is0 ¼

D2o Ls ½t þ ðAs =Ls ÞA 10:9

USCS

ð8-57Þ

The expression for factor B

B¼

3 4

For calculating factor A

Use the applicable material/temperature chart to ﬁnd A

Select a member to be used for stiﬀening a ring after knowing Do , Ls , and t of a shell designed already. Then calculate factor B using Eq. (8-58)

For values of B falling below the left end of the material/temperature chart line for the design temperature the value of A can be determined from Eq. (8-59)

A¼

PDo t þ As =Ls

2B E

ð8-58Þ

ð8-59Þ

FORMED HEADS UNDER PRESSURE ON CONCAVE SIDE For domed ends of hemispherical, semiellipsoidal, or dished shape

Refer to Figs. 8-6 for domed end.

The required thickness at the thinnest point after forming of ellipsoidal, torispherical, hemispherical, conical, and toriconical heads under pressure on the concave side of the shell shall be computed by the appropriate formulas The thickness of the ends and/or heads under pressure on concave side (plus heads) as per Indian Standards

h¼

pdo C 2sa

ð8-60Þ

where C is a shape factor taken from Fig. 8-7 The allowable pressure as per Indian Standards

p¼

2hsa do C

FIGURE 8-6 Domed ends.

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ð8-61Þ

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.24

CHAPTER EIGHT

FIGURE 8-7 Shape factor C for domed ends.11

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

Particular

8.25

Formula

Ellipsoidal heads The required thickness of a dished head of semiellipsoidal form. in which half the minor axis (inside depth of the head minus the skirt) equals one-fourth of the inside diameter of the head skirt, shall be determined by Eq. (8-62) as per ASME Boiler and Pressure Vessel Code

t¼

PD 2sa 0:2P

ð8-62Þ

The maximum allowable working pressure or design pressure as per ASME Boiler and Pressure Vessel Code

p¼

2sa t D þ 0:2t

ð8-63Þ

t¼

0:885PL sa 0:1P

ð8-64Þ

Torispherical heads The required thickness of a torispherical head for the case in which the knuckle radius is 6 percent of the inside crown radius and the inside crown radius equals the outside diameter of the skirt, shall be determined by Eq. (8-64) as per ASME Boiler and Pressure Vessel Code The maximum allowable working pressure as per ASME Boiler and Pressure Vessel Code

P¼

sa t 0:885L þ 0:1t

ð8-65Þ

Hemispherical heads The required thickness of a hemispherical head when its thickness does not exceed 0.36L or P does not exceed 0.665sa , shall be determined by Eq. (8-66) as per ASME Boiler and Pressure Vessel Code The design pressure

t¼

PL 2sa 0:2P

ð8-66Þ

P¼

2sa t L þ 0:2t

ð8-67Þ

h¼

pde c1 2sa

ð8-68Þ

Conical ends subject to internal pressure (Fig. 8-8d) as per Indian Standards KNUCKLE OR CONICAL SECTION The thickness of cylinder and conical section (frustrum) within pﬃﬃﬃﬃﬃ the distance L from the junction (L ¼ 0:5 de h=cos ) The thickness of those parts of conical sections not less than a distance L away from the junction with the cylinder or other conical section

Refer to Table 8-4 for values of c1 .

h¼

pdk 2sa p

1 cos

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ð8-69Þ

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.26

CHAPTER EIGHT

FIGURE 8-8(A) (a) Ellipsoidal; (b) spherically dished (torispherical); (c) hemispherical; (d) typical conical shell connection; and (e) toriconical (cone head with knuckle).

Particular

Formula

SHALLOW CONICAL SECTIONS The thickness of conical sections having an angle of inclination to the vessel axis more than 708

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p=sa ð8-70Þ 90 The lower of values given by Eqs. (8-69) and (8-70) shall be used. h ¼ 0:5ðde ri Þ

Conical heads (without transition knuckle) as per ASME Boiler and Pressure Vessel Code The required thickness of conical heads or conical shell sections that have a half-apex angle not greater than 308 shall be determined by Eq. (8-71)

t¼

PD 2 cos ðsa 0:6PÞ

ð8-71Þ

where D ¼ inside diameter ¼ minimum joint eﬃciency, percent

TABLE 8-4 Values of c1 for use in Eq. (18-68) (as function of and ri =de ) ri =de

0.01

0.02

0.03

0.04

0.06

0.08

0.10

0.15

0.20

0.30

0.40

0.50

108 208 308 458 608 758

0.70 1.00 1.35 2.05 3.20 6.80

0.65 0.90 1.20 1.85 2.85 5.85

0.60 0.85 1.10 1.65 2.55 5.35

0.60 0.80 1.00 1.50 2.35 4.75

0.55 0.70 0.90 1.30 2.00 3.85

0.55 0.65 0.85 1.20 1.75 3.50

0.55 0.60 0.80 1.10 1.60 3.15

0.55 0.55 0.70 0.95 1.40 2.70

0.55 0.55 0.55 0.90 1.25 2.40

0.55 0.55 0.55 0.70 1.00 1.55

0.55 0.55 0.55 0.55 0.70 1.00

0.55 0.55 0.55 0.55 0.55 0.55

Source: IS 2825, 1969.

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.27

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

Particular

Formula

P¼

2sa t cos D þ 1:2t cos

ð8-72Þ

The required thickness of the conical portion of a toriconical head, in which the knuckle radius is neither less than 6 percent of the outside diameter of the head skirt nor less than three times the knuckle thickness and pressure shall be determined by Eqs. (8-73) and (8-74)

tc ¼

PDi 2 cos ðsa 0:6PÞ

ð8-73Þ

P¼

2sa t cos Di þ 1:2t cos

ð8-74Þ

The required thickness of the knuckle and pressure shall be determined by Eqs. (8-75) and (8-76)

tk ¼

The design pressure

Toriconical heads

where Di ¼ inside cone diameter at point of tangency to knuckle PLM 2sa 0:2P

ð8-75Þ

or refer to Eqs. (8-66) and (8-67) where M ¼ factor depending on head proportion, L=r L ¼ Di =2 cos

The design pressure

Toriconical heads may be used when the angle 308 P¼

2sa tk ML þ 0:2tk

ð8-76Þ

FORMED HEADS UNDER PRESSURE ON CONVEX SIDE The thickness of heads and ends under pressure on convex side (minus heads) as per Indian Standards (a) Spherically dished ends and heads

The thickness of the spherically dished heads and ends shall be the greater of the following thicknesses: (1) The thickness of an equivalent sphere, having a radius ro or Ro equal to the outside crown radius of the end as determined from Eq. (8-39) (2) The thickness of the end under an internal pressure equal to 1.2 times the external pressure

(b) Ellipsoidal heads

The thickness of ends of a semiellipsoidal shape shall be the greater of the following: (1) The thickness of an equivalent sphere, having a radius ro or Ro calculated from the values of ro =do or Ro =Do in Table 8-5, determined as per Eq. (8-39)

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.28

CHAPTER EIGHT

Particular

Formula

(2) The thickness of the end under an internal pressure equal to 1.2 times the external pressure (c) Conical heads under external pressure: For a conical end or conical section under external pressure, whether the end is of seamless or buttwelded construction as per Indian Standards.

Use Eqs. (8-68), (8-69), and (8-70). Equations (8-68) to (8-70) are applicable, except that the shell thickness shall be no less than as prescribed below: (1) The thickness of a conical end or conical section under external pressure, when the angle of inclination of the conical section to the vessel axis is not more than 708, shall be made equal to the required thickness of cylindrical shell, in which the diameter is ðde = cos Þ and the eﬀective length is equal to the slant height of the cone or conical section, or slant height between the eﬀective stiﬀening rings, whichever is less. (2) The thickness of conical ends having an angle of inclination to the vessel axis of more than 708 shall be determined as a ﬂat cover.

The thickness of formed heads under pressure on convex side (minus heads) as per ASME Boiler and Pressure Vessel Code The required thickness at the thinnest point after forming of ellipsoidal or torispherical heads under pressure on the convex side (minus heads) shall be the greater of the thicknesses given here

(1) The thickness as computed by the procedure given for the heads with the pressure on the concave side of the previous section using a design pressure 1.67 times the design pressure of the convex side, assuming the joint eﬃciency ¼ 1:00 for all cases, or (2) The thickness as determined by the appropriate procedure given in Ellipsoidal heads or Torospherical heads as per ASME Boiler and Pressure Vessel Code

HEMISPHERICAL HEADS The required thickness of a hemispherical head having pressure on the convex side shall be determined by Eqs. (8-41) to (8-43)

A¼

0:125 Ro =t

ð8-41Þ

B Ro =t

ð8-42Þ

Pa ¼

TABLE 8-5 Values of spherical radius factor Ko ¼ Ro =Do for ellipsoidal head with pressure on convex side as a function of ho =Do for use in Eq. (8-41) ho =Do Ko ¼ Ro =Do

0.167 1.36

0.178 1.27

0.192 1.182

0.208 1.08

0.227 0.99

0.25 0.90

0.276 0.81

0.313 0.73

0.357 0.65

0.417 0.57

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0.5 0.5

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

Particular

8.29

Formula

Pa ¼

0:0625 ðRo =tÞ2

ð8-43Þ

where Ro ¼ outside radius in corroded condition The procedure outlined in this section for ﬁnding the thickness of a spherical shell can be used to ﬁnd the thickness of a hemispherical head by using Eqs. (841) to (8-43). ELLIPSOIDAL HEADS The minimum required thickness of ellipsoidal head having pressure on the convex side either seamless or of built-up construction with butt joints shall not be less than that determined by the procedure given here The factor A is given by Eq. (8-41)

TORISPHERICAL HEADS The required minimum thickness of a torispherical head having pressure on the convex side, either seamless or of built-up construction with butt joint

CONICAL HEADS AND SECTIONS The required minimum thickness of a conical head or section under pressure on the convex side, either seamless or the built-up construction with butt joints

Assume a value of t and calculate the value of factor A using the following equation: A¼

0:125 Ro =t

ð8-41Þ

Using the value of A calculated from Eq. (8-41) follow the procedure as that given for spherical shells to ﬁnd the thickness of ellipsoidal heads. where Ro ¼ the equivalent outside spherical radius taken as Ko Do in the corroded condition Ko ¼ factor depending on the ellipsoidal head proportion Ro =Do (Table 8-5) The required thickness shall not be less than that determined by the same design procedure as is used for ellipsoidal heads given in the Ellipsoidal heads section, using appropriate values for Ro . For torispherical head, the outside radius of the crown portion of the head (Ro ) in corroded condition is taken for design purposes. This thickness can be determined following the procedure outlined under cylindrical shells in the Torispherical heads section to ﬁnd factors A and B by assuming a value for te .

FIGURE 8-8(B) Typical conical sections for external pressure

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.30

CHAPTER EIGHT

Particular

The symbols involved in design calculations are

(1) When 608 and for cones having DL =te , values 10: Assume a value of te and determine ratios Le =DL and DL =te . The equation for calculating the maximum allowable external pressure Pa for the case of values of factor A falling to the right of the end of the material/temperature line. Equation for calculating the maximum allowable external pressure Pa for the case of values of factor A falling to the left of the applicable material/temperature line

Formula

te ¼ eﬀective thickness of conical section Le ¼ equivalent length of conical section ¼ ðL=2Þð1 þ Ds =LÞ L ¼ axial length of cone or conical section (Fig. 8-8(B)) Ds ¼ outside diameter at small end of conical section under consideration Pa ¼

4B 3ðDL =te Þ

ð8-76aÞ

Pa ¼

2AE 3ðDL =te Þ

ð8-77Þ

where DL ¼ outside diameter at large end of conical section under consideration (Fig. 8-8(B)) ¼ one-half the apex angle in conical heads and section, deg. (Compare the value of Pa with design pressure P. If Pa < P, then select a new value for te and repeat the design procedure.)

(2) When 608 and for cones having DL =te , values <10: For values of DL =te less than 4, the value of factor A can be calculated by using Eq. (8-78).

A¼

For values of factor A greater than 0.10, use a value of 0.10 The equation for calculating Pa1 using the value of factor B obtained from material/temperature chart The equation for calculating Pa2

1:1 ðDL =te Þ2

2:167 0:0833 B DL =te 2s 1 ¼ 1 DL =te DL =te

ð8-78Þ

Pa1 ¼

ð8-79Þ

Pa2

ð8-80Þ

where s is less than two times the maximum allowable stress value at design metal temperature, from the applicable table or 0.9 times the yield strength of the material at design temperature. The yield strength is twice the value of B determined from the applicable material/temperature chart. The smaller of the values of Pa1 or Pa2 shall be used for the maximum allowable external pressure Pa . Pa is equal to or greater than P. (Design pressure P is obtained from appropriate table for material.) (3) When > 608

The thickness of the cone shall be the same as the required thickness for a ﬂat head under external pressure, the diameter of which equals the largest diameter of the cone.

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

Particular

8.31

Formula

Toriconical head having the pressure on the convex side The required thickness of a toriconical head having pressure on the convex side, either seamless or of built-up construction with butt joints within the head

The length Le (Fig. 8-8B, panel a)

The length Le (Fig. 8-8B, panel b)

The length Le (Fig. 8-8B, panel c)

The thickness shall not be less than that determined using the procedure followed in the case of a cone having DL =te values 10 for conical heads and sections with exception that Le shall be determined using Eqs. (8-81): L DL þ Ds Le ¼ r1 sin 1 þ ð8-81aÞ 2 DLs Le ¼ r2

Dss L sin 2 þ DL 2

Le ¼ r1 sin 1 þ r2

Ds þ DL DL

Dss L sin 2 þ DLs 2

ð8-81bÞ

DL þ Ds DLs

ð8-81cÞ

To ﬁnd the thickness when lap joints are used in formed head construction or for longitudinal joints in a conical header under external pressure

The rules in this section, except the design pressure 2P, shall be used instead of P in the calculations for the design of required thickness.

UNSTAYED FLAT HEADS AND COVERS (Fig. 8-9, Table 8-6) The thickness h of that unstayed circular heads, covers, and blind ﬂanges as per Indian Standards

The minimum required thickness t of unstayed circular heads, covers and blind ﬂanges as per ASME Boiler and Pressure Vessel Code The minimum required thickness t of ﬂat unstayed circular heads, covers, and blind ﬂanges which are attached by bolts, causing an edge moment as per ASME Boiler and Pressure Vessel Code

h ¼ c2 d

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð p=sa Þ

ð8-82Þ

Refer to Table 8-6 for values of c2 . pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ t ¼ d ðCP=sa Þ

ð8-83Þ

Refer to Table 8-6 for values of C. t¼d

CP WhG þ 1:9 sa sa d 3

1=2 ð8-84Þ

where C is taken from Table 8-6 W ¼ flange design bolt load, lbf W ¼ Wm1 ¼ the minimum bolt load for operating condition, lbf t ¼ 0:785D2G P þ ð2b 3:14DG mpÞ

ð8-84aÞ

where W ¼ Wm2 ¼ the minimum required bolt load for gasket seating, lbf t ¼ 3:14bDG y

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ð8-84bÞ

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.32

CHAPTER EIGHT

FIGURE 8-9 Types of unstayed ﬂat heads. (K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962; K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1983; K. Lingaiah, Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986.)

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.33

TABLE 8-6 Coeﬃcients c2 and C for determining head thickness for typical unstayed ﬂat heads (Fig. 8-9) Coeﬃcient, c2 and C

Type of head IS (Fig. 8-9)a

ASME

c2 , IS (Fig. 8-9)

C, ASME

Remarks

AðaÞ

(b-2)

0.50

0.33 m but <0.20 j

Forged circular and noncircular heads integral with or butt-welded to the vessel

AðbÞ AðcÞ B

(b-1) (a)

0.50 0.45

0.35

0.17 0.17

0.10

0.45 0.50 0.1

CðaÞ to CðdÞ

(e), (f ) and (g)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0:7 hr =hs but <0.55 j

D

(h)

0.7

0.33

E

(i)

F

(p)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0:7 hr =hs but <0.55 j 0.42

0.33 m but <0.20 j 0.25

G

(c)

0.45

0.13

0.55 in other cases

0.20

0.33 m but <0.20 j 0.33

0.30

H IðaÞ IðbÞ IðcÞ

(m) (n) (o)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0:31 þ 95ðF=Pd 2 Þ 0.55 0.55 0.55

0.30 0.30 0.30

Flanged circular and noncircular heads forged integral with or butt-welded to the vessel with an inside corner radius not less than three times the required head thickness For circular heads when the pﬃﬃﬃﬃﬃﬃ ﬃ ﬂange length: 1. l ð1:1 0:8h2s =h2 Þ di h; r 2h, d ¼ di r and taper is 1 : 4 (Fig. 8-9) pﬃﬃﬃﬃ 2. l ¼ ½1:1 0:8ððts =th Þ2 td d and taper is 1 : 3 (1) When r 2h, d ¼ di r and taper is 1 : 4 (Fig. 8-9) When d ¼ di and 0:25h r < 2h. For circular heads, when the ﬂange length pﬃﬃﬃﬃ l : l < ½1:1 0:8ðts =thp Þ2 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ th d but the shell pﬃﬃﬃﬃﬃ thickness: ts <j 1:12th 1:1 1= th d; taper is at least 1 : 3 (2) Circular plates welded to inside of the pressure vessel Noncircular plates, welded to the inside of a vessel and otherwise meeting the requirements for the respective types of welded vessels For circular plates welded to the end of the shell when ts is at least 1.25tr For circular plates, if an inside ﬁllet weld with minimum throat thickness of 0.7ts is used For circular and noncircular covers bolted with a full-face gasket, to shells, ﬂanges or side plates Circular heads lap welded or brazed to the shell with corner radius not less than the 3h or 3t and l not less than required by formula (2) Circular and noncircular lap welded or brazed construction as above, but with no special requirement with regard to l Circular ﬂanged plates screwed over the end of the vessel. with inside corner radius not less than 3t or 3h in which the design of threaded joint is based on a safety factor of 4 Autoclave manhole covers d >j 610 mm (24 in) Circular plates inserted in the end of a pressure vessel and held in place by a positive mechanical locking arrangement, and when all possible means of failure are resisted with a safety factor of at least 4; seal welding may be used, if desired

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.34

CHAPTER EIGHT

TABLE 8-6 Coeﬃcients c2 and C for determining head thickness for typical unstayed ﬂat heads (Fig. 8-9) (Cont.) Coeﬃcient, c2 and C

Type of head IS (Fig. 8-9)a

JðaÞ JðbÞ KðaÞ KðbÞ

ASME

( j) (k) (r) (s)

c2 , IS (Fig. 8-9)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0:31 þ 190ðFbhG =Pd 3 Þ 0.7 0.7

C, ASME

0.3 0.3 0.33 0.33

Remarks Circular and noncircular head and covers bolted to the vessel as shown in Fig. 8-9 Circular plates having a dimension d not exceeding 450 mm (18 in) inserted into the vessel as shown in Fig. 8-9 and the end of the vessel shall be crimped over at least 308, but not more than 458; the crimping may be done cold only when this operation will not injure the metal in case of (r); in case of (s) the crimping shall be done when the entire circumference of the cylinder is uniformly heated to the proper forging temperature for the material used

a

Symbols in this column refer to Fig. 8-9. Where F (or W) is load on bolt. Sources: K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962; K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1983; and K. Lingaiah, Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. b

Particular

The minimum thickness of noncircular heads and covers as per Indian Standards

The minimum heads, covers, or blind ﬂanges of square, rectangular, oblong, segmental, or otherwise noncircular shape as per ASME Boiler and Pressure Vessel Code

Formula

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h ¼ c2 k6 b ð pi =sa Þ

ð8-85Þ

Refer to Fig. 8-10 for values of k6 and Table 8-6 for values of c2 . pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ t ¼ d ðZCPÞ=sa ð8-88aÞ where Z ¼ a factor for noncircular heads depending on the ratio of long span to short span a=b Z ¼ 3:4

2:4d D

ð8-86bÞ

Refer to Fig. 8-10 for values of Z (Z ¼ k6 ). Z need not be greater than 2.5. d ¼ diameter or short span as indicated in Fig. 8-9. t, d in in and p and sa in psi

FIGURE 8-10 Coeﬃcient k6 for noncircular ﬂat heads.

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

Particular

The minimum thickness of ﬂat unstayed non-circular heads, covers, or blind ﬂanges attached by bolts causing a bolt load edge moment as per ASME Boiler and Pressure Vessel Code (Note: A stayed ﬂat plate and types of stays are shown in Figs. 8-11 and 8-12.)

8.35

Formula

t¼d

ZCP 6WhG 1=2 þ sa sa Ld 2

ð8-87Þ

where hG ¼ gasket moment arm, equal to the radial distance from the center line of the bolts to the line of the gasket reaction as shown in Fig. 8-13

FIGURE 8-11 Stayed ﬂat plate.

The net cover plate thickness under the groove or between the groove and outer edge (tg ) of the cover plate

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1:9WhG =sa d 3

ð8-88Þ

for circular heads and covers qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ tg <j d 6WhG =sa Ld 2

ð8-89Þ

tg <j d

for noncircular heads and covers The thickness of spherically dished ends and heads secured to the shell through a ﬂange connection by means of bolts as per Indian Standards

h¼

3pdi 100h for R > j 1:3di and > j 10 2sa R

ð8-90Þ

The thickness of a dished head that is riveted or welded to a cylindrical shell according to ASME Boiler and Pressure Vessel Code

h¼

8:33PR 2su

ð8-91Þ

STAYED FLAT AND BRACED PLATES OR SURFACES (Figs. 8-11 and 8-12) The thickness of stayed and braced plate as per Indian Standards

h ¼ c4 d

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð p=sa Þ

ð8-92Þ

Refer to Table 8-7 for values of c4 and Tables 8-8, 8-9, and 8-11 for allowable stress values sa .

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.36

CHAPTER EIGHT

FIGURE 8-12 Types of stays.

FIGURE 8-13 Location of bolt load reaction.

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.37

TABLE 8-7 Coeﬃcients c4 for determining head thickness for stayed and braced plates Type of stay

c4

Remarks

A B C D E F

0.45 0.45 0.55 0.55 0.45 0.40

Flange of a ﬂanged head Welded brace Welded tube stay Expanded and beaded tubular stay Bar stay with washer of diameter not less than 2.5 times the stay diameter Bar stay with washer and reinforcing plate of diameter not less than 0.3d

Particular

The minimum thickness for braced and stayed ﬂat plates with braces or stay bolts of uniform diameter symmetrically spaced as per ASME Boiler and Pressure Vessel Code

The maximum allowable working pressure for braced and stayed ﬂat plates as per ASME Boiler and Pressure Vessel Code

Formula

t ¼ pt

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðP=sa c5 Þ

ð8-93Þ

where t ¼ minimum thickness of plate, exclusive of corrosion allowance, in sa ¼ maximum allowable stress, MPa (psi), taken from Tables 8-9 to 8-13 for shell plates and Table 8-23 for bolts P¼

t2 sa c5 p2t

ð8-94Þ

where c5 ¼ a factor depending on the plate thickness and type of stay taken from Table 8-15

Stayed ﬂat plates with uniformly distributed load Grashof’s formula for maximum stress

The deﬂection

¼ 0:2275

p2t p h2

ð8-95Þ

y ¼ 0:0284

p4t p Eh3

ð8-96Þ

OPENINGS AND REINFORCEMENT For ﬂanged-in and unreinforced openings in cylindrical or conical shell or spherical shell or heads and ends

Refer to Figs. 8-14 and 8-15. Holes cut in domed ends shall be circular, elliptical, or oblong. The radius r of ﬂanged-in openings (Fig. 8-14) shall not be less than 25 mm. Flanged-in and other openings shall be arranged so that the distance from the edge of the end is not less than that shown in Fig. 8-14. In all cases the projected width of the ligament between any two adjacent openings shall be at least equal to the diameter of the smaller openings as shown in Fig. 8-15.

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.38

CHAPTER EIGHT

FIGURE 8-14 Flanged-in unreinforced opening.

FIGURE 8-15 Unreinforced opening.

Particular

The distance between openings spaced apart, Lo , in a cylindrical or conical or spherical shell as per Indian Standards For size of openings in cylinders or conical shells or spherical shells subject to a maximum of 200 mm (8 in) which do not require reinforcement

Formula

Lo <j L ¼

dðha =hr Þ ðha =hr 0:95Þ

ð8-97Þ

Refer to Tables 8-16 and 8-17 for ﬂange bolting. Refer to Fig. 8-16. where h ¼ distance between centers of any two adjacent openings, mm (in) d ¼ diameter or largest opening ¼ mean value of the major and minor axes in case of elliptical or obround openings, mm (in) ha ¼ actual thickness of vessel before corrosion allowance is provided, mm (in) hr ¼ required thickness of vessel putting ¼ 1:0 before corrosion allowance is added, mm (in)

The total cross-sectional area of reinforcement, Ar , as per Indian Standards

Ar <j A ¼ dhr

ð8-98Þ

where d ¼ nominal internal diameter of the branch plus twice the corrosion allowance, mm (in) hr ¼ thickness of an unpierced shell or end calculated from Eq. (8-10)

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

Particular

The expression for K factor

8.39

Formula

K ¼ pDo =1:82a ha

ð8-99Þ

Refer to Fig. 8-16a and b, where K has a value of unity or greater, the maximum size of an unreinforced opening shall be 50 mm (2 in)

Design for internal pressure The total cross-sectional area of reinforcement A required in any given plane through the opening for a shell or formed head under internal pressure shall not be less than as per ASME Boiler and Pressure Vessel code The total cross-sectional area of reinforcement in ﬂat heads that have an opening with a diameter that does not exceed one-half of the head diameter or shortest span, shall not be less than that given by Eq. (8-100) as per ASME Boiler and Pressure Vessel Code

A ¼ 0:5td

For nomenclature and formulas for reinforced openings as per ASME Boiler and Pressure Vessel Code

Refer to Fig. 8-17.

For values of spherical radius factor K1

Refer to Table 8-18.

The length of tapped hole ls to engage a stud

maximum allowable stress value of stud material at design temperature ls ¼ 0:75ds maximum allowable stress value of tapped material at design temperature

ð8-100Þ

where t ¼ minimum required thickness of ﬂat head or cover, exclusive of corrosion allowance, m (in)

ð8-101Þ

and also ls <j ds , where ds ¼ nominal diameter of the stud, except that the thread engagement need not exceed 1:5ds

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471 (68.3) 275 (39.9) 490 (71.1) 294 (42.6) 490 (71.1) 314 (45.5)

432 (62.7) 451 (65.4) 310 (45.0) 414 (60.0) 414 (60.0)

539 (78.2) 461 (66.9) 510 (74.0) 481 (69.8) 510 (74.0) 618 (89.6)

363 (52.6) 412 (59.8) 432 (62.7) 461 (66.9) 491 (71.2) 432 (62.7) 490 (71.1)

Plates 1 2A 2B 20 Mo 6 20 C 15 15 Cr 4 Mo 6 15 C 8

Forgings 20 Mo 6 15 Cr 4 Mo 6 10 Cr 9 Mo 10

Tubes and pipes 1% Cr 12% Mo 20 Mo 6 Fe 170 Fe 240 Fe 290

Castings Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6

Sections, plates, and bars Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade A-N Grade B-N

Plates, bars, forgings, seamless tubes X04 Cr 19 Ni 9 540 (78) X04 Cr 19 Ni 9 Ti 20 540 (78) X04 Cr 19 Ni 9 No 40 540 (78) X05 Cr 18 Ni 11 Mo 3 540 (78) X05 Cr 19 Ni 9 Mo 3 540 (78) Ti 20 Castings Grades 7, 8 461 (66.9)

363 (52.6) 412 (59.8) 510 (74.0) 471 (68.3) 510 (74.0) 490 (71.1) 412 (59.8)

Materials with grade or designation and product

28 28 28 28 28

21

205 (30)

26 25 23 22 21 23 20

17 17 15 17 17 15

950R20 950R20

20 20 20

26 25 20 20 20 20 25

235 (34) 235 (34) 235 (34) 235 (34) 235 (34)

0.55 R20 0.55 R20 0.55 R20 0.55 R20 0.55 R20 235 (34.0) 280 (40.6)

343 (49.8) 245 (35.5) 304 (44.1) 275 (39.9) 304 (44.1) 422 (61.2)

235 (34.1) 245 (35.5) 173 (25.1) 241 (35.0) 290 (42.1)

0.55 R20 a 0.50 R20 0.50 R20 275 (39.9) 294 (42.6) 294 (42.6) 226 (32.8)

Tensile strength, st , min R20 , MPa (kpsi)

Yield stress, sy min, E20 MPa (kpsi)

Mechanical properties

Allowable stress, sa at design temperature, K (8C)

8.40

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. 137 (19.9)

157 (22.8) 157 (22.8) 157 (22.8) 157 (22.8) 157 (22.8)

121 (17.5) 137 (19.9) 143 (20.7) 153 (22.2) 163 (23.6) 143 (20.7) 163 (13.6)

134 (19.4) 115 (16.7) 127 (18.4) 120 (17.4) 127 (18.4) 154 (22.3)

143 (20.7) 150 (21.8) 103 (15.0) 137 (19.9) 137 (19.9)

157 (22.8) 163 (23.6) 163 (23.6)

121 (17.5) 137 (19.8) 170 (24.7) 157 (22.8) 170 (24.7) 163 (23.6) 137 (19.9)

111 (16.1) 126 (18.3) 131 (19.0) 141 (20.5) 150 (21.8) 143 (20.7) 163 (23.6)

134 (19.4) 114 (16.5) 127 (18.4) 120 (17.4) 127 (18.4) 154 (22.3)

143 (20.7) 150 (21.8) 103 (15.0) 137 (19.9) 137 (19.9)

157 (22.8) 163 (23.6) 163 (23.6)

121 (17.5) 126 (18.3) 156 (22.6) 157 (22.8) 170 (24.7) 163 (23.6) 137 (19.9)

127 (18.4)

139 (20.2) 140 (20.3) 140 (20.3) 142 (20.6) 142 (20.6)

102 (14.8) 117 (17.0) 122 (17.7) 130 (18.9) 138 (20.0) 133 (19.2) 158 (23.0)

134 (19.4) 108 (15.7) 127 (18.4) 120 (17.4) 127 (18.4) 154 (22.3)

139 (20.2) 143 (20.7) 97 (14.0) 136 (19.7) 137 (19.9)

157 (22.8) 163 (23.6) 163 (23.6)

113 (16.4) 117 (17.0) 144 (20.8) 157 (22.8) 167 (24.2) 163 (23.6) 127 (18.4)

93 (13.5) 106 (15.4) 111 (16.1) 119 (17.3) 126 (18.3) 121 (17.5) 143 (20.7)

132 (19.1) 100 (14.5) 125 (18.1) 117 (17.0) 127 (18.4) 154 (22.3)

133 (19.3) 133 (19.3) 88 (12.8) 124 (18.0) 137 (19.9)

150 (21.8) 163 (23.6) 163 (23.6)

102 (14.8) 106 (15.4) 130 (18.8) 150 (21.8) 150 (21.9) 163 (23.6) 116 (16.8)

76 (11.0) 88 (12.8) 91 (13.2) 105 (15.2) 109 (15.8) 88 (12.8) 105 (15.2)

110 (16.0) 86 (12.5) 108 (15.7) 104 (15.1) 127 (18.4) 160 (23.2)

119 (17.3) 115 (16.7) 74 (10.7) 103 (15.0) 124 (18.0)

130 (18.9) 149 (21.6) 170 (24.6)

85 (12.3) 88 (12.8) 109 (15.8) 129 (18.7) 126 (18.3) 149 (21.6) 96 (13.9)

70 (10.2) 79 (11.5) 83 (12.0) 94 (13.6) 98 (14.2) 79 (11.5) 94 (13.6)

99 (14.4) 80 (11.6) 100 (14.5) 99 (14.4) 117 (17.0) 152 (22.0)

113 (16.4) 108 (15.7) 66 (9.6) 93 (13.5) 113 (16.4)

121 (17.6) 141 (20.5) 161 (23.4)

77 (11.2) 79 (11.5) 98 (14.2) 121 (17.6) 114 (16.5) 141 (20.5) 87 (17.6)

117 (18.4)

122 (17.7) 124 (18.0) 124 (18.0) 127 (18.4) 127 (18.4)

106 (15.4)

104 (15.0) 106 (15.4) 106 (15.4) 113 (16.4) 113 (16.4)

High-Alloy Steels in Tension

84 (12.2) 96 (13.9) 100 (14.5) 115 (16.7) 119 (17.3) 96 (13.9) 115 (16.7)

120 (17.4) 94 (13.6) 117 (17.0) 110 (17.3) 128 (18.6) 169 (24.5)

127 (18.4) 127 (18.4) 80 (11.6) 113 (16.4) 135 (19.6)

140 (20.3) 157 (22.8) 176 (25.5)

93 (13.5) 96 (13.9) 119 (17.3) 140 (20.3) 137 (19.9) 157 (22.8) 105 (15.2)

Carbon and Low-Alloy Steel in Tension

104 (15.1)

97 (14) 104 (15) 104 (15) 110 (16) 110 (16)

67 (9.7) 76 (11.0) 79 (11.5) 89 (12.9) 93 (13.5)

94 (13.6) 79 (11.5) 98 (14.2) 95 (13.8) 115 (16.7) 146 (21.2)

109 (15.8) 104 (15.1) 63 (9.1) 88 (12.8) 106 (15.4)

107 (15.5) 135 (19.6) 158 (22.9)

74 (10.7) 76 (11.0) 93 (13.5) 117 (17.0) 108 (15.7) 135 (19.6) 82 (11.9)

104 (15.1)

92 (13.3) 104 (15) 104 (15) 110 (16) 110 (16)

64 (9.3) 73 (10.6) 76 (11.0) 81 (11.8) 81 (11.8)

61 (8.8) 76 (11.0) 94 (13.6) 91 (13.2) 116 (16.9) 141 (20.5)

105 (15.2) 101 (14.7) 61 (8.8) 81 (11.8) 81 (11.8)

113 (16.4) 131 (19.0) 155 (22.5)

71 (10.3) 73 (10.6) 81 (11.8) 113 (16.4) 81 (11.8) 128 (18.6) 79 (11.5)

98 (14.2) 94 (13.6) 42 (6.1) 42 (6.1) 42 (6.1)

106 (15.4) 124 (18.0) 146 (21.2)

42 (6.1) 42 (6.1) 42 (6.1) 106 (15.4) 42 (6.1) 124 (18.0) 42 (6.1)

104 (15.1)

85 (12.3) 104 (15.1) 10.4 (15.1) 110 (16.0) 110 (16.0)

58 (8.4) 58 (8.4) 58 (8.4) 58 (8.4) 58 (8.4)

104 (15.1)

79 (11.5) 101 (14.6) 104 (15.1) 109 (15.8) 109 (15.8)

42 (6.1) 42 (6.1) 42 (6.1) 42 (6.1) 42 (6.1)

43 (6.2) 31 (4.5) 74 (10.7) 71 (10.3) 91 (13.2) 82 (11.9) 89 (12.9) 86 (12.5) 109 115.8) 106 (15.4) 137 (19.9) 132 (19.1)

102 (14.8) 98 (14.2) 58 (8.4) 58 (8.4) 58 (8.4)

110 (16.0) 128 (18.6) 150 (21.8)

58 (8.4) 58 (8.4) 58 (8.4) 110 (16.0) 58 (8.4) 128 (18.6) 58 (8.4)

35 (5.0) 35 (5.0) 35 (5.0) 35 (5.0) 35 (5.0)

26 (3.8) 57 (8.8) 57 (8.3) 83 (12.0) 94 (13.6) 66 (9.6)

95 (13.8) 76 (11.0) 35 (5.1) 35 (5.1) 35 (5.1)

76 (11.0) 114 (16.5) 125 (18.1)

35 (5.1) 35 (5.1) 35 (5.1) 76 (11.0) 35 (5.1) 114 (16.5) 35 (5.1)

41 (5.9) 41 (5.9) 64 (9.3) 71 (10.3) 48 (7.0)

84 (12.2) 55 (8.0)

27 (3.9) 27 (3.9) 43 (6.2) 51 (7.5) 34 (4.9)

57 (8.3) 36 (5.2)

56 (8.1) 57 (8.3) 69 (10.0)

57 (8.3)

84 (12.2)

55 (8.0) 84 (12.2) 94 (13.6)

36 (5.2)

55 (8.0)

25 (3,6) 36 (5.2) 25 (3.6)

34 (5.0)

34 (5.0) 48 (7.0)

34 (5.0)

33 (4.8) 16 (2.3)

31 (4.5)

323 K 373 K 423 K 473 K 523 K 573 K 623 K 648 K 673 K 698 K 723 K 748 K 773 K 798 K 823 K 848 K (508C), (1008C), (1508C), (2008C) (2508C), (3008C), (3508C), (3758C), (4008C), (4258C), (4508C), (4758C), (5008C), (5258C), (5508C), (5758C) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi)

TABLE 8-8 Allowable stresses sa for various ferrous and nonferrous materials

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

216 (31.3) 265 (38.4) 265 (38.4) 186 (27.0) 108 (15.7) 294 (42.6) 245 (35.5)

216 (31.3) 108 (15.7) 309 (44.8) 245 (35.5)

275 (40.0) 275 (40.0)

Sheet, strip S1B—12H NS4—14H

Bars, rods, and sections NE5—M NE6—M NE8—0 HE30—W HE30—WP

Drawn tubes HT30—W HT30—WP

Plate sheet and strips Cu Zn 30 Cu Zn 40

284 (41.2) 284 (41.2)

22

45 30

16 7

88+ 18+ 16 18 10

8 8

30 12

69 (10.0) 86 (12.5)

69 (10.0)

69 (10.0) 86 (12.5)

51 (7.4) 72 (10.4)

54 (7.8) 66 (9.6) 69 (10.0) 51 (7.4) 71 (10.3)

21 (3.0) 54 (7.8)

12 (1.7) 43 (6.2)

69 (10.0) 85 (12.3)

69 (10.0)

69 (10.0) 85 (12.3)

50 (7.3) 70 (10.2)

49 (7.1) 70 (10.2)

20 (2.9) 53 (7.7)

13 (1,9) 42 (6.1)

69 (10.0) 81 (11.7)

69 (10.0)

69 (10.0) 81 (11.7)

48 (7.0) 67 (9.7)

47 (6.8) 67 (9.7)

19 (2.8) 52 (7.5)

11 (1.6) 42 (6.1)

69 (10.0) 77 (11.2)

69 (10.0)

69 (10.0) 77 (11.2)

46 (6.7) 65 (9.4)

46 (6.7) 65 (9.4)

18 (2.6) 49 (7.1)

10 (1.5) 41 (5.9)

39 (5.7) 43 (6.2)

39 (6.7) 43 (6.2)

14 (2.0) 37 (5.4)

8 (1.2) 32 (4.6)

28 (4.1) 30 (4.4)

28 (4.1) 30 (4.4)

11 (1.6) 24 (3.5)

7 (1.1) 24 (3.5)

68 (9.9) 71 (10.3)

68 (9.9)

68 (9.9) 71 (10.3)

56 (8.1) 53 (7.7)

56 (8.1)

56 (8.1) 53 (7.7)

38 (5.5) 19 (2.8)

38 (5.5)

38 (5.5) 19 (2.8)

Copper and Copper Alloys

44 (6.38) 55 (8.0)

44 (6.4) 54 (7.8)

16 (2.3) 44 (6.4)

9 (1.3) 37 (5.4)

Aluminum and Aluminum Alloys in Tensions

323 K 373 K 423 K 473 K 523 K 573 K 623 K 648 K 673 K 698 K 723 K 748 K 773 K 798 K 823 K 848 K (508C), (1008C), (1508C), (2008C) (2508C), (3008C), (3508C), (3758C), (4008C), (4258C), (4508C), (4758C), (5008C), (5258C), (5508C), (5758C) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi) MPa (kpsi)

Allowable stress, sa at design temperature, K (8C)

b

These values have been used on a quality factor of 0.75. 0:55R20 ¼ 0:55 363 ¼ 199:7 MPa (29.0 kpsi). Notes: + The elongation values are based on 50.8-mm test piece; a area of cross-section; † tube normalized and tempered. Sources: K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962; K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers. Bangalore. India. 1983; and K. Lingaiah, Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986

a

Tubes Alloy 1 Alloy 2

392 (56.9)

98 (14.2) 196 (28.4)

Plates PIB—M NP4—M

Bars and rods

64 (9.3) 186 (27.0)

Materials with grade or designation and product

Yield stress, sy min, E20 MPa (kpsi)

Tensile strength, st , min R20 , MPa (kpsi)

Mechanical properties

TABLE 8-8 Allowable stresses sa for various ferrous and nonferrous materials (Cont.)

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

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8.41

8.42

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SA 225 SA 302

SA 204

SA 203

Plate SA 202

h

g

A C C C

F

B

A

0.5 Cr-1.25 310 Mn-Si 0.5 Cr-1.25 324 Mn-Si 3.5 Ni, 379 50 mm (2 in.) C-0.5 Mo 255 C-0.5 Mo 296 Mn-0.5 Ni-V 483 Mn-0.5 Mo- 345 0.5 Ni 37 43 70 50

55

47

45

30 40 30 36 22.5 27.5 30.0

207 276 207 248 155 190 207

20 80

36

138 552

C-,Mn C-Mn-SiCb-V

24 40/42 42 32 38 35 38 50

kpsi

248

165 276/290 290 220 262 241 262 345

C C-Mn-Si C-Mn C-Si C-Si C-Mn-Si C-Mn-Si C-Mn-Si

Carbon steel forgings, castings, and bars SA 36b;c bars C-Mn-Si and shapes c SA 216 cast WCA C-Si WCC C-Mn-Si c SA 350 forge LFI C-Mn-Si LF2 C-Mn-Si SA 675b;c bar 45 C 60 C 70 C

Plates and sheets A SA 285b;c;d SA 299c b;c SA 414 F SA 515c 60 70 c 65 SA 516 70 c SA 537 Cl-1 up to 62.5 mm (2.5 in) incl. SA 620 1 and 2 SA 812 80

composition MPa

no.

Grade

Nominal

Speciﬁcation

Speciﬁed minimum yield strength, sy

448 517 724 552

552

586

517

413 483 413 483 310 379 483

400

276 689

310 517 483 413 483 448 483 483

65 75 105 80

80

85

75

60 70 60 70 45 55 70

58

40 100

45 75 70 60 70 65 70 70

MPa kpsi

Speciﬁed minimum tensile strength st

112 130 181 138

138

147

130

103 121 103 121 78 95 121

100

69 147

78 130 121 103 121 115 121

MPa

16.3 18.8 26.3 20.0

20.0

21.3

18.8

15.0 17.5 15.0 17.5 11.3 13.8 17.5

14.5

10.0 21.3

11.3 18.8 17.5 15.0 17.5 16.3 17.5

kpsi

19 to 345 (30 to 650)

11.0 17.7 16.6 14.4 16.6 15.5 16.6

112 130 181 138

136

122

99 115 99 115 76 92 115

56

16.3 18.8 26.3 20.0

19.8

17.7

14.4 16.6 14.4 16.6 11.0 13.3 16.6

13.9

(Fig. 8-5)

76 121 114 99 114 107 114

MPa kpsi

370 (700)

135

112 130

122

108

90 102 90 102 71 83 102

87

71 108 101 90 102 96 102

19.6

16.3 18.8

17.7

15.7

13.0 14.8 13.0 14.8 10.3 12.1 14.8

12.6

10.3 15.7 14.7 13.0 14.8 13.9 14.8

MPa kpsi

400 (750)

10.8 12.0 10.8 12.0 9.0 10.2 12.0

10.5

9.0 12.6 12.0 10.8 12.0 11.4 12.0

60 64 54 54 54 60 64

57

54 66 63 60 64 62 64

130

109 130

83

83

18.8

15.8 18.8

12.0

12.0

123

106 126

54

54

Low-Alloy Steel

75 83 75 83 62 70 83

72

62 87 83 75 83 79 83

455 (850)

17.9

15.3 18.3

7.8

7.8

8.7 9.3 7.8 7.8 7.8 8.4 9.3

8.5

7.8 9.6 9.2 8.7 9.3 9.0 9.3

MPa kpsi

Carbon Steel

MPa kpsi

427 (800)

94

94 94

35

35

45 45 35 35 45 45 45

45 45 45 45 45 45 45

13.7

13.7 13.7

5.0

5.0

6.5 6.5 5.0 5.0 6.5 6.5 6.5

45

6.5 6.5 6.5 6.5 6.5 6.5 6.5

MPa kpsi

482 (900)

56

56 56

21

8.2

8.2 8.2

3.0

3.0

4.5

31

21

4.5 4.5 3.0 3.0

31 31 21 21

6.5

33.0

33.0 33.0

10.0

10.0

17.0

17.0 17.0 10.0 10.0

17.0 17.0 17.0 17.0

31 31 31 31

4.5 4.5 4.5 4.5

17.0

4.8

4.8 4.8

1.5

1.5

2.5

2.5 2.5 1.5 1.5

2.5 2.5 2.5 2.5

2.5

MPa kpsi

538 (1000)

(Fig. 8-5) 31 4.5

MPa kpsi

510 (950)

(Fig. 8-5) (Fig. 8-5)

MPa kpsi

566 (1050)

Maximum allowable stress, sa for metal temperature, 8C (8F), not exceeding

TABLE 8-9 Maximum allowable stress values, sa , in tension for carbon and low-alloy steel

MPa kpsi

593 (1100) MPa kpsi

620 (1150)

MPa kpsi

650 (1200)

SA 302

SA 203 SA 204gb SA 204g SA 225h SA 225h

SA 202

SA 675b;c bar

SA 350c forge

SA 36b;c bars and shapes SA 216c cast

SA 620 SA 812

SA 537c

SA 516c

SA 285b;c;d SA 299c SA 414b;c SA 515c

no.

Speciﬁcation

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

11 Cl.2

SA 387

4N

3

B11

SA 487 cast

SA 541 forge

SA 739 bar

1 Cr-0.5 Mo 1.25 Cr-0.5 Mo-Si C-0.5 Mo 1 Ni-0.5 Cr0.5 MO 9 Cr-1 Mo 1.25 Cr-0.5 Mo-Si 0.5 Ni-0.5 Cr-0.25 Mo V 0.5 Ni-0.5 Mo V 1.25 Cr-0.5 Mo 2.25 Cr- 1 Mo

310

310

345

413

413 276

241 276

276 276

45

45

50

60

60 40

35 40

40 40

30

45

kpsi

517

483

552

620

620 483

448 483

483 483

413

517

75

70

80

90

90 70

65 70

70 70

60

75

MPa kpsi

Speciﬁed minimum tensile strength st

121

138

121

112 121

121 121

130

MPa

17.5

20.0

17.5

16.3 17.5

17.5 17.5

18.8

kpsi

19 to 345 (30 to 650)

121

121

138

141 121

112 121

121 121

95

130

17.5

17.5

20.0

20.5 17.5

16.3 17.5

17.5 17.5

13.7

18.8

MPa kpsi

370 (700)

119

121

138

136 121

112 121

121 121

91

130

17.2

17.5

20.0

19.8 17.5

16.2 17.5

17.5 17.5

13.2

18.8

MPa kpsi

400 (750)

116

121

132

132 121

109 121

121 121

88

130

16.9

17.5

19.1

19.1 17.5

15.8 17.5

17.5 17.5

12.8

18.8

MPa kpsi

427 (800)

113

118

125 118

105 118

118 118

83

126

16.4

17.1

18.2 17.1

15.3 17.1

17.1 17.1

12.1

18.3

MPa kpsi

455 (850)

109

110

114 110

90 103

110 110

75

110

15.8

15.9

10.5 15.9

13.7 15.0

15.9 15.9

10.9

15.9

MPa kpsi

482 (900)

76

76

76 76

56 63

76 76

55

76

11.0

11.0

11.0 11.0

8.2 9.2

11.0 11.0

8.0

11.0

MPa kpsi

510 (950)

52.0

48.0

51.0 48.0

33.0 40.0

45.0 48.0

40.0

48.0

7.6

6.9

7.4 6.9

4.8 5.9

6.6 6.9

5.8

6.9

MPa kpsi

538 (1000)

40

32

35 32

30 32

29

32

5.8

4.6

5.0 4.6

4.3 4.6

4.2

4.6

MPa kpsi

566 (1050)

Maximum allowable stress, sa for metal temperature, 8C (8F), not exceeding

30

19

23 19

18 19

20

19

4.4

2.8

3.3 2.8

2.6 2.8

2.9

2.8

MPa kpsi

593 (1100)

17

15

15

10 15

14

15

2.5

2.1

2.2

1.4 2.1

2.0

2.1

MPa kpsi

620 (1150)

7 8

9

8

10

9

8

1.3

1.2

1.5

1.0 1.2

1.3

1.2

MPa kpsi

650 (1200)

SA 541 forge SA 739 bar

SA 487 cast

SA 336 forge

SA 217 cast

SA 182 forge

SA 387

no.

Speciﬁcation

Notes: a These stress values are one-fourth the speciﬁed minimum strength multiplied by a quality factor of 0.92, except for SA 283, grade D and SA-36. b For service temperature above 4558C (8508), it is recommended that killed steels containing not less than 10% residual silicon be used. c Upon prolonged exposure to temperature above 4268C (8008F) the carbide phase of carbon steel may be converted to graphite. d The material shall not be used in thickness above 50 mm (2 in). e The material shall not be used in thickness above 62 mm (2.5 in). f Only killed steel shall be used above 4558C (8508F). g Upon prolonged exposure to temperature above 4688C (8758F), the carbide phase of carbon molybdenum steel may be converted to graphite. h The maximum nominal plate thickness shall not exceed 14.75 mm (0.58 in). i These stress values apply to normalized and drawn materials only. j For other conditions and speciﬁcations, the reader is referred to the general notes given for Table UCS-23 of ASME Boiler and Pressure Vessel Code, Section VIII, Division 1, July 1, 1986. Source: The American Society of Mechanical Engineers, ASME Boiler and Pressure Vessel Code, Section VIII, Division 1, July 1, 1986.

B22

C12 F11

WC1 WC4

SA 336 forge

SA 217 cast

Forgings, castings, and bars SA 182 forge F12 F11b

5 Cl.1

composition MPa

Grade

no.

1.25 Cr-0.5 310 Mn-Si 5 Cr-0.5 Mo 207

Nominal

Speciﬁcation

Speciﬁed minimum yield strength, sy

TABLE 8-9 Maximum allowable stress values, sa , in tension for carbon and low-alloy steel (Cont.)

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

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8.43

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.44

CHAPTER EIGHT

TABLE 8-10 Maximum allowable stress values, sa in tension for nonferrous metals

Alloy Speciﬁcation no. designation

Temper condition

Sheet and plate SB 209

-H 12

Nominal composition

UNS no.

Size or thickness mm (in)

Speciﬁed minimum tensile strength, st

Maximum allowable stress, sa , for metal temperature 8C (8F), not exceeding

Speciﬁed minimum yield strength, sy

38 (100)

65 (150) kpsi

93 (200)

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

MPa

kpsi

96

14

76

11

24

3.5

24

3.5

24

3.5

110

16

96

14

26

4.0

26

4.0

26

4.0

138

20

117

17

35

5.0

35

5.0

35

5.0

117

17

69

10

30

4.3

30

4.3

30

4.3

193

28

145

21

48

7.0

48

7.0

48

7.0

234

34

179

26

58

8.5

58

8.5

58

8.5

214

31

83

12

54

7.8

54

7.8

54

7.8

248

36

179

26

62

9.0

62

9.0

62

9.0

207

30

110

16

52

7.5

52

7.5

52

7.5

165

24

41

6.0

41

6.0

41

6.0

427

62

290

42

107

15.5

107

15.5

107

15.5

400

58

262

38

100

14.5

100

14.5

100

14.5

241 96 282 290 262 165

35 14 41 42 38 24

96 35 131 179 241

14

8.8 3.4 10.3 10.5 9.5 6.0

61 23 71 72 63 40

8.8 3.4 10.3 10.5 9.2 5.9

23

3.4

19 26 35

61 23 71 72 65 41

62 39

9.0 5.7

379 441 434

55.0 64.0 63.0

207 379 372

30 55 54

95 110

13.8 16.0 15.8

95 110

13.8 16.0 15.8

92 109

13.3 15.9 15.8

262 255 241

38.0 37.0 35.0

241 228 220

35 33 32

65 64 61

9.5 9.3 8.8

65 64 61

9.5 9.3 8.8

65 64 61

9.5 9.3 8.8

50.0 (2.000)

207 172 331

30.0 25.0 48.0

138 124 200

20 18 29

52 43 65

7.5 6.3 9.5

52 43 52

7.5 6.3 7.5

52 43

7.5 6.3

50 mm (2 in) 50 mm (2 in) 12. 5 mm (12 in) 50 mm (2 in)

345 345 496 345

50.0 50.0 72.0 50.0

124 138 220 138

18 20 32 20

83 86 124 86

12.0 12.5 18.0 12.5

83 86 124 86

12.0 12.5 18.0 12.5

82

11.9

124 86

18.0 12.5

>50 mm (2 in) 87.5 mm (3.5 in) 100 mm (4 in) >75(3)–125(5)

310

45.0

103

15

69

10.0

69

10.0

69

10.0

310 345 345 345 310

45.0 50.0 50.0 50.0 45.0

103 138 124 138 124

15 20 18 20 I8

69 86 83 86 78

10.0 12.5 12.0 12.5 11.3

69 86 83 78 70

10.0 12.5 12.0 11.3 10.1

69 86 83 72 65

10.0 12.5 12.0 10.5 9.4

276

40.0

103

15

70

10.1

67

9.7

66

9.5

345 317 482 469

50.0 46.0 70.0 68.0

124 103 172 159

18 15 25 23

83 69 115 105

12.0 10.0 16.7 15.3

78 66 100 93

11.3 9.5 14.5 13.5

75 63 97 90

10.9 9.1 14.0 13.0

358 482 276 379

52 70 40 55

103 262 83 138

15 38 12 20

69 121 55 92

10.0 17.5 8.0 13.3

69 121 55 92

10.0 17.5 8.0 13.3

69 121 55 92

10.0 17.5 8.0 13.3

Aluminum and Aluminum Alloys 1100d

1.275–50.0 (0.051–2.000) 0.225–25.0 (0.009–1.000) 0.15–25.00 (0.006–1.000) 6.25–12.475 (0.250–0.499) 1.275–50.00 (0.051–2.000) 1.275–25.00 (0.051–1.000) 1.275–75.00 (0.051–3.000) 1.275–50.00 (0.051–2.000) 1.275–6.225 (0.051–0.249) 1.275–6.225 (0.051–0.249)

-H 14 SB 209

3003d

-H 14 -H 112

SB 209

3004d

-H 32

SB 209

5052d

-H 34

SB 209

5454d

-O -H 32

SB 209

6061e;f

T4 T 6 Wld

Rods, bars, and shapes SB 221 2024e

SB 221 SB 221 SB 221

5086e 3003d 5456d

SB 308

6061e ’

Die and hand forgings SB 247 2014 Diee

SB 247 SB 247

Castings SB 26 SB 108

-T 4

-H 112 -H 112 -O -H 111 -T 6 -T 6 Wld

100.0 (4.000) 50.0 (2.00) 50.0–100.00 (2.001–4.000) 100.0 (4.00) 100.0 (4.000) 100.025–200.0 (4.001–8.000)

-T 4 -T 6

6061 Diee 6061 Hande

-T 6 -T 6

SG 70 A(356)e

-T 6 -T 71 -T 4

204.0

3.125–12.475 (0.125–0.499) 162.54–200.00 (6.501–8.000) 125.00 (5.000) All 125.00 (5.00) 125.00 (5.00)

5

Copper and Copper Alloys Sheet and plates SB 96 655 SB 169 610 614 SB 171 C 36500, C 36600

Annealed Annealed Annealed Annealed

C 36700, C 36800

Annealed Annealed Annealed Annealed Annealed Annealed

Admiralty Naval brass

SB 171

443, 444, 445 C 46400, C 46500 C 46600, C 46700 715

SB402

706

Annealed

Cu-Ni 90/10

SB 171 SB 171

Die forgings (hot pressed) SB 283h C 37700h

Rods and bars SB 98g SB 98

Cu-Si alloy Al-bronze Al-bronze Lead-Muntz metal

Cu-Ni 70130

As forged

Forging brass

C 64200

As forged

Forgings, Al-Si bronze

655, 661g

Softh Half hardi Soft Half hard

651 j

Cu-Si Cu-Si

62.5 (2.5) 62.5 (2.5) 125(5) incl 62.5 (2.5) 37.5 (1.5) >37.5 (1.5) 37.5 (1.5) >37.5 (1.5)

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.45

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

TABLE 8-10 Maximum allowable stress values, sa in tension for nonferrous metals (Cont.) Maximum allowable stress, sa , for metal temperature, 8C (8F), not exceeding 120 (250) MPa

kpsi

150 (300) MPa

kpsi

176 (350) MPa

kpsi

205 (400) MPa

kpsi

232 (450) MPa

kpsi

260 (500) MPa

kpsi

288 (550) MPa

kpsi

315 (600) MPa

kpsi

343 (650) MPa

kpsi

370 (700) MPa

kpsi

Spec. No.

Sheet and plate SB 209

22 25 34 28 48 58 51 52 51 41

3.2 3.7 4.9 4.0 7.0 8.5 7.4 7.5 7.4 5.9

19 19 30 25 40 43 38 38 48 38

2.8 2.8 4.3 3.6 5.8 6.2 5.5 5.5 6.9 5.5

14 14 21 21 26 32 28 28 43 32

2.0 2.0 3.0 3.0 3.8 4.1 4.1 4.1 6.3 4.6

8 8 16 16 16 16 21 21 31 24

1.2 1.2 2.4 2.4 2.4 2.4 3.0 3.0 4.5 3.5

95 88

13.7 12.8

72 69

10.4 9.7

49 42

6.5 6.1

31 29

4.5 4.2

21

3.0

16

2.4

12

1.8

10

1.4

SB 221 SB 221 SB 221

59 37

8.5 5.4

50 35

7.2 5.0

39 29

5.6 4.2

28 22

4.0 3.2

SB 308

86 102 102 63 61 58

12.5 14.8 14.8 9.1 8.8 8.4

79 79 79 54 53 51

11.5 11.5 11.5 7.9 7.7 7.4

47 47 47 43 43 42

6.8 6.8 6.8 6.3 6.3 6.1

27 27 27 31 31 31

3.9 3.9 3.9 4.5 4.5 4.5

43 42

6.3 6.1

37

5.4

28

4.1

16

2.4

(Fig. 8-9) (Fig. 8-8)

SB 209 SB 209 SB 209 SB 209 SB 209

Rods, bars, and shapes SB 221

Die and hand forgings SB 247 SB 247 SB 247

Castings SB 26 SB 108 Sheet and plates SB 96g SB 169

81

11.7

69

10.0

38

5.0

124 86 69 69

18.0 12.5 10.0 10.0

124 85 69 69

18.0 12.3 10.0 10.0

124 75 69 68

18.0 10.8 10.0 9.8

121 36 36 24

17.5 5.3 5.3 3.5

86 83 72 64 64

12.5 12.0 10.4 9.3 9.3

86 83 72 64 62

12.5 12.0 10.4 9.3 9.0

43 43 72 64 60

6.3 6.3 10.4 9.3 8.7

17 17 72 64 59

2.5 2.5 10.4 9.3 8.5

117

17.0

14

2.0

72 64 57

10.4 9.3 8.2

114

16.5 SB 171 SB 171 SB 171

72 64 55

10.4 9.3 8.0

72 64 48

10.4 9.3 7.0

72 64 41

10.4 9.3 6.0

72 64

10.4 9.3

72 64

10.4 9.3

SB 171 SB 204

Die forgings (hot pressed) 93 96

13.5 12.5

93 86

13.5 12.5

90 83

13.0 12.0

69 121 55 88

10.0 17.5 8.0 12.8

69 121 48 69

10.0 17.5 7.0 10.0

35 69 35 55

5.0 10.0 5.0 8.0

76 76

11.0 11.0

52 52

7.5 7.5

36 36

5.2 5.2

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SB 283h Rods and bars SB 98g SB 98

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.46

CHAPTER EIGHT

TABLE 8-10 Maximum allowable stress values, sa in tension for nonferrous metals (Cont.)

Alloy Speciﬁcation no. designation

Temper condition

Castings SB 61 SB 148 SB 271 SB 584

As Cast As Cast As Cast As Cast

922 954 952 976

Nominal composition

UNS no.

Size or thickness mm (in)

Speciﬁed minimum tensile strength, st MPa

Maximum allowable stress, sa , for metal temperature 8C (8F), not exceeding

Speciﬁed minimum yield strength, sy

kpsi

MPa

kpsi

234 517 448 276

34 75 65 40

110 207 172 117

241

35

345 448

38 (100)

65 (150)

93 (200)

MPa

kpsi

MPa

kpsi

MPa

kpsi

16 30 25 17

59 13 108 52

8.5 18.8 15.7 7.5

59 130 108 52

8.5 18.8 15.7 7.2

59 129 103 48

8.5 18.7 14.9 7.0

172

25

61

8.8

59

8.1

50

7.3

50 65

276 379

40 55

86 112

12.5 16.3

82 107

12.0 15.6

75 99

10.9 14.3

482 345

70 50

345 276

50 40

121 86

17.5 12.5

121 81

17.5 11.7

113 74

16.4 10.7

358

52

207

30

90

13.0

76

11.0

552

80

379

55

138

20.0

114

16.6

Titanium and Titanium Alloys Sheet, strip, plate, bar, billet, and casting SB 265 Grade 1 (F1) SB 381 SB 348

2 (F2) 3 (F3)

SB 367h

12 (F12) Grade C-2

Annealed Annealed

Sheets, strips, plate Forging (F stands for forging) Bar, billet Castingb

Zirconium Flat-rolled products and bars SB 551 Grade R 60702 SB 550

Hot-rolled products Bars

R 60705

Nickel and Nickel Alloys Plate, sheet, and strip SB 127j

400

Ni-Cu

N04400

600 600j B2 825 X

Annealedj Hot-rolled Annealed Hot-rolledj Sol. ann. k Annealed Annealedk

SB 168 SB 168j SB 333k SB 424k SB 435k

Ni-Cr-Fe Ni-Cr-Fe Ni-Mo Ni-Fe-Cr-Mo-Cu Ni-Cr-Mo-Fe

N06600 N06600 N10665 N08825 N06002

SB 435

X

Annealed

Ni-Cr-Mo-Fe

N06002

SB 435k SB 443 SB 463

X 625 20Cb

Annealed Annealed Annealed

N06002 N06625 N08020

SB 575k SB 582 SB 582k SB 709

C22 G G 28

Sol. ann. k Sol. ann. Sol. ann. k Annealed

Ni-Cr-Mo-Fe Ni-Cr-Mo-Cb Cr-Ni-Fe-MoCu-Cb Ni-Mo-Cr Ni-Cr-Fe-Mo-Cu Ni-Cr-Fe-Mo-Cu Ni-Fe-Cr-MoCu Low C

N04400 N06600 N06600 N08825 N08020 N08330l N06625 N06455k N-12 WV CW-12MW

Bars, rods, shapes, and forgings SB 164 400 600k SB 166k SB 166 600 k 825 SB 425 SB 462k 20Cb SB 511l SB 564 SB 574k Castings SA 494h SA 494

Annealed Annealedk Hot ﬁn Annealedk Annealedk

330 625 C-4

Annealed Sol. ann.

Ni-Cu Ni-Cr-Fe Ni-Cr-Fe Ni-Fe-Cr-Mo-Cu Cr-Ni-Fe-MoCu-Cb Ni-Fe-Cr-Si Ni-Cr-Mo-Cb Ni-Mo-Cr

B C

Annealedh Annealed

Ni-Mo Ni-Mo-Cr

N06022 N06007 N06007 N08028

All 0.063 (1/16) 0.188 (3/16) 0.063 (1/16) 0.188 (3/16)k >0.188 (3/16) >100 (4)

19.3 (3/4) >19.3 (3/4) k

All sizes

All sizes

482 517 552 586 758 586 689

70 75 80 85 110 85 100

193 276 241 241 352 241 276

28 40 35 35 51 35 40

388C (1008F) 128 18.6 129 18.7 138 20.0 146 21.2 190 27.5 148 21.5 161 23.3

938C (2008F) 113 16.4 129 18.7 138 20.0 146 21.2 190 27.5 148 21.5 144 20.9

1508C (3008F) 106 15.4 129 18.7 138 20.0 146 21.2 190 27.5 141 20.4 132 19.2

689

100

276

40

161

23.3

161

23.3

116

23.3

655 758 552

95 110 80

241 379 241

35 55 35

161 190 138

23.3 27.5 20.0

144 190 138

20.9 27.5 20.0

132 190 136

19.2 27.5 19.8

689 620 586 503

100 90 85 73

310 241 207 213

45 35 30 31

172 155 138 125

25.0 22.5 20.0 18.2

172 144 138 125

25.0 22.9 20.0 18.2

171 134 138 117

24.8 19.5 20.0 17.0

482 552 586 586 552

70 80 85 85 80

172 241 241 241 141

25 35 35 35 35

114 138 146 146 138

16.6 20.0 21.2 21.2 20.0

101 138 146 146 138

14.6 20.0 21.2 21.2 20.0

94 138 146 141 136

13.6 20.0 21.2 20.4 19.8

482 758 689

70 110 100

207 345 276

30 50 40

121 190 172

17.5 27.5 25.0

121 190 172

17.5 27.5 25.0

112 190 172

16.3 27.5 25.0

524 496

76 72

276 276

40 40

131 124

19.0 18.0

123 118

17.8 17.1

123 112

17.8 16.2

a

The stress values in this table may be interpolated to determine values for intermediate temperatures. Stress values in restricted shear shall be 0.8 times the values in this table. c Stress values in bearing shall be 1.60 times the values in the table. d For weld construction, stress values for this material shall be used. e The stress values given for this material are not applicable when either welding or thermal cutting is employed. f Allowable stress values shown are 90 percent those for the corresponding core material. g Copper-silicon alloys are not always suitable when exposed to certain media and high temperature, particularly steam above 1008C (2128F). h No welding is permitted. i If welded, the allowable stress values for annealed condition shall be used. j For plates only. b

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.47

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

TABLE 8-10 Maximum allowable stress values, sa in tension for nonferrous metals (Cont.) Maximum allowable stress, sa , for metal temperature, 8C (8F), not exceeding 120 (250) MPa

kpsi

150 (300) MPa

kpsi

176 (350) MPa

kpsi

205 (400) MPa

kpsi

232 (450) MPa

260 (500)

288 (550)

315 (600)

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

343 (650) MPa

kpsi

370 (700) MPa

kpsi

Spec. No. Castings

59 129 100 48

8.5 18.7 14.5 6.9

59 129 98 46

8.5 18.7 14.2 6.7

59 125 98

8.5 18.1 14.2

57 120 98

8.3 17.4 14.2

53 110 98

7.7 16.0 14.2

50 96 98

7.2 13.9 14.2

34 76 81

5.0 11.0 11.7

59 51

8.5 7.4

45

6.5

40

5.8

36

5.2

33

4.8

31

4.5

28

4.1

25

3.6

21

3.1

68 90 105 68

9.9 13.0 15.2 9.8

62 81 98 61

9.0 11.7 14.2 8.9

58 72 92 55

8.4 10.4 13.3 8.0

53 64 86 50

7.7 9.3 12.5 7.2

50 57 82

7.2 8.3 11.9

46 52 79

6.6 7.5 11.4

43 46 77

6.2 6.7 11.1

39 41 75

5.7 6.0 10.8

64

9.3

48

7.0

42

6.1

41

6.0

98

14.2

86

12.5

78

11.3

72

10.4

2058C (4008F) 102 14.8 129 18.7 138 20.0 146 21.2 190 27.5 132 19.2 123 17.8 158 22.9 123 17.8

2608C (5008F) 101 14.7 129 18.7 138 20.0 146 21.2 190 27.5 126 18.3 114 16.5 154 22.3 114 16.5

3158C (6008F) 101 14.7 129 18.7 138 20.0 146 21.2 189 27.2 123 17.8 108 15.6 146 21.1 108 15.6

3708C (7008F) 101 14.7 124 1 8.0 135 19.6 145 21.1 187 27.1 119 17.3 103 15.6 140 20.3 103 15.0

4268C (8008F) 98 14.2 98 14.2 132 19.1 141 20.4 137 19.8 118 17.1 101 14.7 136 19.7 101 14.7

4828C (9008F) 55 8.0 28 4.0 110 16.0 135 19.6

5398C (11008F)

5938C (11008F)

48 100

7.0 14.5

21 50

3.0 7.2

116 100 135 100

16.8 14.5 19.6 14.5

115 99 131 99

16.6 14.3 19.3 14.3

98 121 98

185 129

26.8 18.7

180 125

26.1 18.2

175 121

25.4 17.5

172 119

25.0 17.3

170 116

24.6 16.8

165

24.0

163

23.7

166

165 125 138 109

23.9 18.2 20.0 15.8

160 120 138 100

23.2 17.4 20.0 14.5

157 116 134 92

22.7 16.8 19.4 13.3

154 113 131

22.4 16.4 19.0

153 111 128

22.2 16.1 18.6

110 127

16.0 18.4

109 126

15.8 18.3

91 138 146 132 129 105

13.2 20.0 21.2 19.2 18.7 15.3

98 138 146 126 125 101

13.1 20.0 21.2 18.3 18.2 14.6

98 138 146 123 122 94

13.1 20.0 21.2 17.8 17.7 13.7

98 138 146 119 119 92

13.1 20.0 21.1 17.3 17.3 13.4

88 138 141 118 116 89

12.7 20.0 20.4 17.1 16.8 12.9

55 110 134 116

8.0 16.0 19.5 16.8

48 100 114

7.0 14.5 16.6

21 50

3.0 7.2

85

12.3

82

11.9

54

7.8

185 172

26.8 25.0

180 170

26.1 24.7

175 168

25.4 24.4

172 165

25.0 24.0

170 158

24.6 23.0

165

24.0

163

23.7

166

23.4

123 112

17.8 16.2

123 112

17.9 16.2

123 112

17.8 16.2

122 111

17.7 16.1

119 105

17.3 15.2

114 99

16.6 14.4

108 95

15.7 13.8

SB 61 SB 148 SB 271 SB 584 Sheet, strip, plate, bar, billet, and casting SB 265 SB 381 SB 348 Flat-rolled products and bars SB 367h Plate, sheet, and strip 33 4.8 SB 551 68

6488C (12008F) 14 38

2.0 5.5

14.2 17.5 14.2

78 78 78

11.3 11.3 11.3

23.4

91

13.2

4.9

7048C (13008F)

53 53 53

7.7 7.7 7.7

SB 550

7608C (14008F) SB 127j SB 168 SB 168j SB 333k SB 424k 33 4.8 33 4.8 33 4.8 SB 443 SB 463 SB 575k SB 582 SB 582k SB 709

14 38

2.0 5.5

32 91

4.7 13.2

Bars, rods, shapes, and forgings SB 164 SB 166k SB 166 SB 425k 21 3.1 SD 462k 12 1.8

k

SB 564 SB 574k Castings SA 494h SA 494

Nickel alloys have low yield strength. The stress values of these alloys used are slightly on the high side. These higher stress values exceed 2/3 but do not exceed 90 percent of the yield strength at temperature. These stress values are not recommended for the ﬂanges of gasket joints where a slight amount of distortion can cause leakage. Sol. ann. = Solution annealed. l At temperature above 5388C (10008F), these stress values may be used only if the material is annealed at a minimum temperature of 10388C (19008F) and has a carbon content of 0.04% or higher. m These stress values multiplied by a joint eﬃciency factor of 0.85. n A joint eﬃciency factor of 0.85 has been applied in arriving at the maximum allowable stress values in tension for this material. o Alloy NO6225 in the annealed condition is subject to severe loss of impact strength at room temperature after exposure in the range of 5388 to 7608C (10008 to 14008F). p For other conditions and speciﬁcations, it is suggested to refer to the General Notes given for Table UNF-23.1 of ASME Boiler and Pressure Vessel Code, Section VIII, Division 1, July 1, 1986. Source: The American Society of Mechanical Engineers, ASME Boiler and Pressure Vessel Code, Section VIII, Division 1, July 1, 1986.

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.48

CHAPTER EIGHT

TABLE 8-11 Maximum allowable stress values (sa ) in tension for high-alloy steel

Spec. no.

Grade

UNS no.

SA-240, SA-479 SA-240 SA-240 SA-240 SA-240 SA-479 SA-182 SA-217 SA-479 SA-412 SA-182 SA-240, SA-479 SA-351 SA-351 SA-351 SA-336 SA-240, SA-479 SA-182 SA-479 SA-240 SA-351 SA-240 SA-336 SA-240

405 410 S TP 409 18 Cr-2 Mo 430 410 F6 ACI.1 CA 15 430, XM8 201 F 304 L 304 L CF 3 CF 8 CF 8 M Cl-F 304 H 302 F 304 304 H 304 CF 3A 304 N F 304 N 316 L

S 40500 S 41008 S 40900 S 44400 S 43000 S 41000 S 41000 J 91150 S 43000, S 43035 S 20100 S 30403 S 30403 J 92500 J 92600 J 92900 S 30409 S 30200 S 30400 S 30400 S 30400 J 92500 S 30451 S 30451 S 31603

SA-182

F 316 L

S 31603

SA-479

316 L

S 31603

SA-351

CF 8 M

J 92900

SA-182

F 316

S 31600

SA-336

CI-F 316 H S 31609

SA-240

316 Ti

S 31635

SA- 1 82

F 316 H

S 31609

SA-479

316

S 31600

SA-240

317 L

S 31703

SA-240 SA-240

XM-15 316 M

S 38100 S 31651

SA-479, SA-240

XM-29

S 24000

SA-182, SA-336 SA-240, SA-479 SA-182, SA-336 SA-351 SA-240, SA-182 SA-479 SA-351 SA-182, SA-240

F 321 H 321 F 347 CFBC 347,348 F 347, F 348 CG 8 M F 44

S 32100 S 32100 S 34700 J 92710 S 34700 S 34800 S 31254

SA-182, SA-240. F 45 SA-479 SA-240, SA-479

S 30815 S 30815 S 32550

SA-351 SA-351 SA-240 SA-240, SA-182

CH 8 CH 20 309 S, 309Cb 310 Cb

J 93400 J 93402 S 30908, S 30940 S 31040,

SA-479 SA-240 SA-182, SA-336 SA-240, SA-479

CI-F310 310 S TP 329 FXM-27 Cb XM-27

S 31000 S 310 S S 32900 S 44625 S 44627

SA-240 SA-240, SA-479 SA-564 SA-182, SA-336, SA-41Z

XM-33 S 44626 844800 630 H 1100 S 17400 FMX-11, S-21904 NM-11

Speciﬁed minimum

Speciﬁed minimum

yield strength, sy

tensile strength, st

Maximum allowable stress, sa , for metal temperature, 8C (8F), not exceeding 30 to 38 (20 to 100)

93 (200)

150 (300)

205 (400)

Nominal composition

Product form

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

12 Cr-1 Ald 13 Cr 11 Cr-Ti 18 Cr-2 Mod 17 Crd 13 Cr 13 Cr 13 Crd 17 Crd;e ; 18 Cr-Tid;e 17 Cr-4 Ni-6 Mn 18 Cr-8 Ni 18 Cr-8 Ni 18 Cr-S Ni 18 Cr-8 Ni 18 Cr-9 Ni-2 Mo 18 Cr-8 Ni 18 Cr-8 Ni 18 Cr-8 Ni 18 Cr-8 Ni 18 Cr-8 Ni 18 Cr-8 Ni 18 Cr-8 Ni-N 18 Cr-8 Ni-N 16 Cr-12 Ni-2 Mo 16 Cr-12 Ni-2 Mo 16 Cr- 1 2 Ni-2 Mo 16 Cr- 1 2 Ni-2 Mo 16 Cr-12 Ni-2 Mo 16 Cr-12 Ni-2 Mo 16 Cr-12 Ni-2 Mo 16 Cr-12 Ni-2 Mo 16 Cr-12 Ni-2 Mo 18 Cr-13 Ni-3 Mo 18 Cr-18 Ni-2 Si 16 Cr-12 Ni-2 Mo-N 18 Cr-3 Ni-12 Mn 18 Cr-10 Ni-Ti 18 Cr-10 Ni-Ti 18 Cr-10 Ni-Cb 18 Cr-10 Ni-Cb 18 Cr-10 Ni-Cb 18 Cr-10 Ni-Cb 19 Cr-11 Ni-Mo 20 Cr-18 Ni-6 Mo 21 Cr- 1 1 Ni-N 21 Cr-11 Ni-N 25.5 Cr-5.5 Ni3.5 Mo 25 Cr-12 Ni 25 Cr-12 Ni 23 Cr-12 Ni

Plate. bar Plate Plate Plate Plate Bar, forge Bar, forge Cast Bare;g Plate Forgeg Plateg , bare;g Castg Castg;h Castg;h Forgeg Plate, bare;g Forgee;g Barg;e Plate Castg Plateg;h Forge Plateg

172 207 207 276 207 276 276 448 276 310 172 172 207 207 207 207 207 207 207 207 241 241 241 172

25 30 30 40 30 40 40 65 40 45 25 25 30 30 30 30 30 30 30 30 35 35 35 25

414 414 379 414 448 483 483 620 483 655 448 483 483 483 483 483 517 517 517 517 534 552 552 483

60 60 55 60 65 70 70 90 70 95 65 70 70 70 70 70 75 75 75 75 77.5 80 80 70

103 103 95 103 112 111 111 155 121 158 108 108 121 121 121 121 130 130 130 130 134 138 138 108

15.0 15.0 13.8 15.0 16.3 16.2 16.2 22.5 17.5 23.0 15.6 15.7 17.5 17.5 17.5 17.5 18.8 18.8 18.8 18.8 19.4 20.0 20.0 15.7

99 99 90 99 107 106 106 148 114 143 106 108 114 114 121 114 123 123 123 123 125 138 138 108

14.3 14.3 13.1 14.3 15.5 15.4 15.4 21.5 16.6 20.8 15.4 15.7 16.6 16.6 17.5 16.6 17.8 17.8 17.8 17.8 18.2 20.0 20.0 15.7

95 95 97 95 103 103 103 143 111 132 98 105 105 104 118 107 114 114 114 114

13.8 13.8 12.7 13.8 15.0 14.9 14.9 20.7 16.1 19.1 14.2 15.3 15.3 15.1 17.1 15.5 16.6 16.6 16.6 16.6

92 92 84 92 99 99 99 138 107

13.3 13.3 12.2 13.3 14.4 14.4 14.4 20.0 15.5

131 131 108

19.0 19.0 15.7

94 101 104 103 116 104 112 112 112 112 116 126 126 107

13.6 14.7 5.1 15.0 16.8 15.1 16.2 16.2 6.2 16.2 16.9 18.3 18.3 15.5

Forge g

172

25

448

65

108

15.7

108

15.7

108

15.7

107

15.5

Barg;f

172

25

483

70

108

15.7

108

15.7

108

15.7

107

15.5

Cast

207

30

483

70

121

17.5

121

17.5

118

17.1

116

16.8

25 Cr-20 Ni 25 Cr-20 Ni 26 Cr-4 Ni-Mo 27 Cr-Mo 27 Cr-Mo 27 Cr-Mo-Ti 29 Cr-4 Mo-2 Ni 17Cr-4 Ni-4 Cu 20 Cr-6 Ni-9 Mn

Forgeg;h;j

207

30

483

70

121

17.5

121

17.5

118

17.1

116

16.8

Forge

207

30

483

70

121

17.5

111

16.2

100

14.6

92

13.4

Plateg;h;i

207

30

517

75

130

18.8

130

18.8

127

18.4

125

18.1

Forgeg

207

30

517

75

130

18.8

130

18.8

127

18.4

125

18.1

Bare;g;h

207

30

517

75

130

18.8

130

18.8

127

18.4

125

18.1

Plateg

207

30

517

75

130

18.8

112

16.2

98

14.2

92

13.4

Plateg Plateg;h

207 241

30 35

517 552

75 80

130 138

18.8 20.0

122 138

17.7 20.0

114 132

16.6 19.2

111 130

16.1 18.8

Plate, barf;g

379

55

689

100

172

25.0

169

24.5

156

22.6

149

21.6

Forgeg;i Plateg;h , barg;h;e Forgeg;h;i Castg;h Plategg;h , forgeg;h Barg;h;e Castg Forge, plate

207 207 207 207 207 207 241 303

30 30 30 30 30 30 35 44

483 517 483 483 517 517 517 648

70 75 70 70 75 75 75 94

121 130 121 121 130 130 121 162

17.5 18.8 17.5 17.5 18.8 18.8 17.5 23.5

118 127 115 114 123 123 121 162

17.1 18.4 16.7 16.6 17.9 17.9 17.5 23.5

111 119 105 105 113 113 118 147

16.1 17.3 15.3 15.3 16.4 16.4 17.1 21.4

110 118 99 96 107 107 116 137

16.0 17.1 14.4 13.9 15.5 15.5 16.8 19.9

Forge, plate, bar 310 Forge. plate, bar 310 Plate, bar 552

45 45 80

600 600 758

87 87 110

150 150 190

21.8 21.8 27.5

149 149 189

21.6 21.6 27.4

141 141 177

20.4 20.4 25.7

135 135 170

19.6 19.6 24.7

Castg;h Casth Plateg;h;j

193 207 207

28 30 30

448 483 517

65 70 75

112 121 130

16.3 17.5 18.8

103 111 118

14.9 16.1 17.2

98 105 113

14.2 15.3 16.4

95 102 110

13.8 14.8 15.9

Plateg;k;h;j , forgeg;k;h

207

30

517

75

130

18.8

118

17.2

113

16.4

110

15.9

207 483 241 276

30 70 35 40

517 620 414 448

75 90 60 65

130 155 103 112

18.8 22.5 15.0 16.2

118 151 103 112

17.2 21.9 15.0 16.2

113 141 101 110

16.4 20.5 14.6 15.9

109 136 98 110

15.8 19.8 14.2 15.9

310 414 793 345

45 60 115 50

469 552 965 620

68 50 140 90

117 138 241 155

17.0 20.0 35.0 22.5

117 134 241 154

17.0 19.4 35.0 22.4

116 126 241 148

16.8 18.3 35.0 21.4

114 125 235 136

16.6 18.1 34.1 19.7

g;k;h;e

Bar Plated Forged Plated , bar, shaped;e Plated Plated , bard;e Bard;l Forge, plate

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.49

TABLE 8-11 Maximum allowable stress values (sa ) in tension for high-alloy steel (Cont.) Maximum allowable stress, sa , for metal temperature, 8C (8F), not exceeding 260 (500)

315 (600)

370 (700)

427 (800)

482 (900)

538 (1000)

593 (1100)

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

89

12.9

85

12.4

83

12.1

76

11.1

69

9.7

27

4.0

(Fig. 8-5)

89 81 88 96 96 96 133 103

12.9 11.8 12.8 13.9 13.9 13.9 19.3 15.0

85 79 85 93 92 92 129 100

12.4 11.4 12.4 13.5 13.4 13.4 18.7 14.5

83 76

12.1 11.1

76 70

11.1 10.2

69 9.7 (Fig. 8-5)

44

6.4

20

90 90 90 125 97

13.1 13.1 13.1 18.1 14.1

82 82 82 115 89

12.0 12.0 12.0 16.7 12.9

45 44 44 34 45

6.5 6.4 6.4 5.0 6.5

92 99

13.4 14.4

92 96

13.3 14.0

90 93

13.1 13.5

89 90

12.9 13.0

102 102 116 102 110

14.8 14.8 16.8 14.8 15.9

102 102 116 102 110

14.8 14.8 16.8 14.8 15.9

102 102 112 102 110

14.8 14.8 16.3 14.8 15.9

100 100 109 100

14.6 14.6 15.8 14.6

110 110 110 114 123 123 99 99 99 116 116 86 124 124 124 86 110 128 148

15.9 15.9 15.9 16.5 17.8 17.8 14.4 14.4 14.4 16.8 16.8 12.5 18.0 18.0 18.0 12.5 15.9 18.6 21.4

110 110 110 112 120 120 93 93 93 116 116 81 117 117 117 81 110 128 144

15.9 15.9 15.9 16.3 17,4 17.4 13.5 13.5 13.5 16.8 16.8 11.8 17.0 17,0 17.0 11.8 15.9 18.6 20.9

110 110 110 112 118 118 89 89 89 112 112 78 112 112 112 78 110 128 138

15.9 15.9 15.9 16.3 17.1 17.1 12.9 12.9 12.9 16.3 16.3 11.3 16.3 16.3 16.3 11.3 15.9 18.6 20.0

105 105 105 100 114 114 85 85 85 109 110 76 110 110 110 76 104 127 131

110

16.0

110

16.0

109

15.8

118

17.1

113

16.4

109

15.8

96

13.9

94

13.7

94

94 103

13.7 14.9

94 101

13.7 14.7

103 116 128

14.9 16.8 18.5

101 123

127

18.4

127 170

18.4 24.7

93 97 107

72 72 72 76 76

10.5 10.4 10.4 11.0 11.0

kpsi

650 (1200)

704 (1300)

MPa

kpsi

MPa

kpsi

2.9

7

1.0

(Fig. 8-5)

22 3.2 (Fig. 8-5) (Fig. 8-5) 15 2.2

12

1.8

7.0

1.0

760 (1400) MPa

kpsi

815 (1500) MPa

kpsi

Spec. no. SA-240, SA-479 SA-240 SA-240 SA-240 SA-240 SA-479 SA-182 SA-217 SA-479

(Fig. 8-5)

92 107 98

13.4 15.5 14.2

83 103 92

12.0 14.9 13.4

52 61 68

7.5 8.9 9.8

33 37 42

4.8 5.4 6.1

23 23 25

3.3 3.4 3.7

16 16 16

2.3 2.3 2.3

12 11 10

1.7 1.6 1.4

15.2 15.2 15.2 14.6 16.6 16.6 12.4 12.4 12.4 15.8 15.9 11.0 15.9 15.9 15.9 11.0 15.1 18.4 19.0

101 101 101

14.7 14.7 14.7

95 95 95

13.8 13.8 13.8

68 68 68

9.8 9.8 9.8

42 42 42

6.1 6.1 6.1

25 25 25

3.7 3.7 3.7

16 16 16

2.3 2.3 2.3

10 10 10

1.4 1.4 1.4

110 110 83 83 93 107 107 74 103 103 103

15.9 15.9 12.1 12.1 12.1 15.5 15.6 10.8 15.5 15.5 15.5

103 103

15.0 15.0

67 67

9.7 9.7

41 41

6.0 6.0

103 103 73 105 105 105

14.9 15.0 10.6 15.3 15.3 15.3

65 85 71 85 85 85

9.4 12.4 10.3 12.4 12.4 12.4

41 51 51 51 51 51

6.0 7.4 7.4 7.4 7.4 7.4

27 28 28 28 28 28

4.0 4.1 4.1 4.1 4.1 4.1

16 17 16 16 16 16

2.4 2.5 2.3 2.3 2.3 2.3

10 8 9 9 9 9

1.5 1.2 1.3 1.3 1.3 1.3

101 125

14.6 18.1

94 120

13.7 17.4

85

12.4

51

7.4

107

15.5

105

15.3

96

14.0

62

9.0

37

5.4

22

3.2

13

1.9

8

1.1

107

15.5

105

15.3

95

13.8

48

6.9

25

3.6

12

1.7

5

0.8

2

0.3

13.7

94

13.7

94

13.7

91

13.2

63

9.1

30

4.4

15

2.2

8

1.2

5

0.8

94 101

13.7 14.7

94 101

13.7 14.7

94 101

13.7 14.7

91 96

13.2 14.0

72 63

10.5 9.1

34 30

5.0 4.4

19 15

2.7 2.2

11 8

1.6 1.2

7 5

1.0 0.8

14.7

101

14.7

101

14.7

101

14.7

96

14.0

63

9.1

30

4.4

15

2.2

8

1.2

5

0.8

17.9

121

17.5

122

17.7

86

12.5

116

16.8

112

16.3

103

14.9

62

9.0

36

5.2

21

3.1

13

1.9

9

1.3

122

17.7

86

12.5

116

16.8

112

16.3

103

14.9

62

9.0

36

5.2

21

3.1

13

1.9

9

1.3

13.5 14.1 15.5

92 92 105

13.3 13.4 15.3

90 88 104

13.0 12.7 15.1

90 94 103

13.0 12.2 14.9

86 81 96

12.5 11.7 13.9

72 70 72

10.5 10.2 10.5

45 45 45

6.5 6.5 6.5

26 26 26

3.8 3.8 3.8

16 16 16

2.3 2.3 2.3

9 9 9

1.3 1.3 1.3

5 5 5

0.8 0.8 0.8

107

15.5

105

15.3

104

15.1

103

14.9

96

13.9

76

11

59

8.5

41

6.0

24

3.5

11

1.6

5

0.8

107 136 98

15.5 19.8 14.2

105

15.3

104

15.1

103

14.9

95

13.8

76

11

98

14.2

110

15.9

110

15.9

113 125

16.4 18.1

111 125

16.1 18.1

230 123

33.3 17.9

226 117

32.8 17.0

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SA-412 SA-182 SA-240, SA-479 SA-351 SA-351 SA-351 SA-336 SA-240, SA-479 SA-182 SA-479 SA-240 SA-351 SA-240 SA-336 SA-240 SA-182 SA-479 SA-351 SA-182 SA-336 SA-240 SA-182 SA-479 SA-240 SA-240 SA-240 SA-479, SA-240 SA-182, SA-336 SA-240, SA-479 SA-182, SA-336 SA-351 SA-240, SA-182 SA-479 SA-351 SA-182, SA-240 SA-182, SA-240 SA-479 SA-240, SA-479 SA-351 SA-351 SA-240 SA-240, SA-182 SA-479 SA-240 SA-182 SA-336 SA-440, SA-479 SA-240 SA-440, SA-479 SA-564 SA-182, SA-336

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.50

CHAPTER EIGHT

TABLE 8-11 Maximum allowable stress values (sa ) in tension for high-alloy steel (Cont.)

Spec. no.

Grade

SA-351 CG 6 MM SA-240, SA-412, XM-19 SA-479, SA-182

Product form

Speciﬁed minimum

Speciﬁed minimum

yield strength, sy

tensile strength, st

UNS no.

Nominal composition

J 93790 S 20910

22 Cr-13 Ni-5 Mn Cast 241 22 Cr-13 NI-5 Mn Plate, bar, forgef 379

MPa

Maximum allowable stress, sa 30 to 38 (20 to 100)

93 (200)

150 (300)

205 (400)

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

35 55

517 689

75 100

130 172

18.8 25.0

116 172

16.9 24.9

103 163

14.9 23.6

94 156

13.6 22.7

a

The stress value in this table may be interpolated to determine values for intermediate temperatures. Stress values in restricted shear shall be 0.8 times the values in this table. c Stress values in bearing shall be 1.60 times the values in this table. d This steel may be expected to develop embrittlement after service at moderately elevated temperature. e Use of external pressure charts for material in the form of barstock is permitted for stiﬀening rings only. f These stress values are the basic values multiplied by a joint eﬃciency factor of 0.85. b

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.51

TABLE 8-11 Maximum allowable stress values (sa ) in tension for high-alloy steel (Cont.) For metal temperature, 8C (8F), not exceeding 260 (500)

315 (600)

370 (700)

427 (800)

482 (900)

538 (1000)

593 (1100)

650 (1200)

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

90 154

13.0 22.3

87 151

12.6 21.9

85 149

12.3 21.6

83 146

12.0 21.2

81 142

11.8 20.6

79 137

11.4 19.9

131

19.0

57

8.3

704 (1300) MPa

kpsi

760 (1400) MPa

kpsi

815 (1500) MPa

g

kpsi

Spec. no. SA-351 SA-240, SA-412, SA-479, SA-182

Alloy steels have low yield strength. The stress values of these alloy steels used are slightly on the high side. These higher stress values exceed 2/3 but do not exceed 90 percent of the yield strength at temperature. These stress values are not recommended for the ﬂanges of gasket joints where a slight amount of distortion can cause leakage. h At temperature above 5408C (10008F), these stress values apply only when carbon is 0.04% or higher on heat analysis. i These stress values shall be applicable to forging over 125 mm (5 in) in thickness. j For temperature above 5408C (10008F), these stress values may be used only if the material is heat-treated by heating it to a minimum temperature of 10408C (19008F) and quenching in water or rapidly cooling by other means. k These stress values at 5658C (10508F) and above shall be used only when the grain size is ASTM 6 or coarser. l These stress values are established from a consideration of strength only and shall be satisfactory for average service. Source: The American Society of Mechanical Engineers, ASME Boiler and Pressure Vessel Code, Section VIII, Division 1, July 1, 1986.

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Cl 4 Ie,g A 37:6 mm (112 in) E; F 62:5 mm (212 inÞ E; F > 62:5h mm (212 in)

SA-508 SA-522 SA-592

620

90

85 75 100

75 75

85 95 80

85 65 75 70

724

724 690 792

690 690

724 792 758

690 655 655 620

620 690

792

690 792

105

105 100 115

100 100

105 115 110

100 95 95 90

90 100

115

100 115

66 (150)

181

181 163 198

172 146

181 198 190

172 163 164 155

155 172

198

172 198

26.3

26.2 23.7 28.8

25.0 21.3

26.3 28.8 27.5

25.0 23.7 23.8 22.5

22.5 25.0

28.8

25.0 28.8

MPa kpsi

93 (200)

181

181 153 198

161 137

181 199 190

161 163 164 155

155 172

198

161 198

26.3

26.2 22.2 28.8

23.4 19.9

26.3 28.8 27.5

23.4 23.7 23.8 22.5

22.5 25.0

28.8

23.4 28.8

MPa kpsi

181

148 198

156 133

181 198 187

156 161 164 155

155 172

198

157 198

150 (300)

26.3

21.5 28.8

181

198

26.3

28.8

Forgings 181 26.2

181

198

179

181 198 181

Castings 181 26.3 198 28.8 185 26.9

155 172

198

198

163 154

22.5 25.0

28.8

28.8

205 (400)

26.3

28.8

26.0

26.3 28.8 26.3

23.5 22.3

22.5 25.0

28.8

28.8

MPa kpsi

23.8 22.5

164 155

155 172

198

198

Plates

MPa kpsi

Pipes and tubes 22.7 19.3

26.3 28.8 27.2

22.7 23.3 23.8 22.5

22.5 25.0

28.8

22.7 28.8

MPa kpsi

120 (250)

181

198

178

181 198 178

163 154

155 172

198

198

26.3

28.8

25.8

26.3 28.8 25.8

23.5 22.3

22.5 25.0

28.8

28.8

MPa kpsi

260 (500)

181

198

175

181 198 174

163 154

155 172

198

198

26.3

28.8

25.4

26.3 28.8 25.3

23.5 22.3

22.5 25.0

28.8

28.8

MPa kpsi

315 (600)

180

197

173

181 198 172

161 153

155 172

197

197

26.2

28.7

25.1

26.3 28.8 24.9

23.4 22.2

22.5 25.0

28.7

28.7

MPa kpsi

345 (650)

181 198 168

159 151

155 172

26.3 28.8 24.4

23.1 21.9

22.5 25.0

MPa kpsi

370 (700)

Maximum allowable stress values, sa , for metal temperatures, 8C (8F) not exceeding

b

Minimum thickness after forming any section subject to pressure shall be 4.6875 mm (3/16 in). Not welded or welded if the tensile strength of the Section IX reduced section tension test is not less than 600 MPa (100 kpsi). c Welded with the tensile strength of the Section IX reduced tension test less than 690 MPa (100 kpsi) but not less than 655 MPa (95 kpsi). d Grade II of SA-533 shall not he used for minimum allowable temperature below 1708C (2758F). e To these stress values a quality factor as speciﬁed in UG-24 shall be applied for castings. f These stress values are the basic values multiplied by a joint eﬃciency factor of 0.85. g The maximum section thickness shall not exceed 75 mm (3 in) for double normalized and tempered forgings, or 125 mm (5 in) for quenched and tempered forgings. h The maximum thickness of non-heat-treated forgings shall not exceed 93.75 mm (334 in). The maximum thickness as heat treated may be 100 mm (4 in). Source: The American Society of Mechanical Engineers, ASME Boiler and Pressure Vessel Code, Section VIII, Division 1, July 1, 1986.

a

SA-592

517 517

8a,b 8a,c,f

SA-333 SA-334

586 517 690

586 655 551

586 448 517 482

70 83

100

690

482 572

75 100

517 690

MPa kpsi

MPa kpsi

Cl. 4 Qe Cl. 4 QAe Cl. CA 6 NMe

B A, C

A, B, D, J 31 25 mm (1.25 in) 62.5 mm (2.5 in) E 62:5 (212 in) > j 50 mm (6 in) > j 100 mm (4 in) A, B, C, D, Cl 2 B, D, C1 3 > j 62.5 mm (212 in) I, IIa,b,d

Grade and size

SA-487 SA-487 SA-487

SA-553 SA-645a SA-724

SA-533

SA-517

SA-353a,b SA-517

Spec. no.

Speciﬁed minimum tensile strength, st

Speciﬁed minimum yield strength, sy

TABLE 8-12 Maximum allowable stress values, sa , in tension for ferrite steels with properties enhanced by heat treatment

400 (750)

165

152 169

23.9

22.2 24.5

MPa kpsi

427 (800)

161

146

23.3

21.2

MPa kpsi

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.52

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— 20 25 30 35 40 45 50 (Grade 3-2510) 55 60 — 20 25 30 35

SA-667 SA-278 SA-278 SA-278 SA-278 SA-278 SA-278 SA-278 SA-47 SA-278 SA-278 SA-476 SA-748 SA-748 SA-748 SA-748

138 138 172 207 241 276 310 345 345 379 414 552 138 172 207 241

MPa 20 20 25 30 35 40 45 50 50 55 60 80 20 25 30 35

kpsi

Source: ASME Boiler and Pressure Vessel Code, Section VIII, Division 1, July 1, 1986.

Class

Spec. no.

Speciﬁed minimum tensile strength, st

TABLE 8-13 Maximum allowable stress values, sa , in tension for cast iron

13.8 13.8 17.2 20.7 24.1 27.6 31.0 34.5 34.5 37.9 41.4 55.2 13.8 17.2 20.7 24.1

MPa 2.0 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.0 5.5 6.0 8.0 2.0 2.5 3.0 3.5

kpsi

Subzero to 232 (450)

27.6 31.0 34.5 34.5 37.9 41.4

MPa

345 (650)

4.0 4.5 5.0 5.0 5.5 6.0 — — —

kpsi

Maximum allowable stress, sa , for metal temperature, 8C (8F) not exceeding

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

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8.53

Copper—Cu 99.98% Commercial brass— Cu 66%, Zn 34% Leaded tin bronze— Cu 88%, Sn 6%, Pb-1.5%, Zn-4.5% Phosphor bronze— Cu 85.5%. Sn 12.5%, Zn 10% Muntz—Cu 59%, Zn 39% Cupronickel— Cu 80%. Ni 20%

Nickel Nickel-copper alloy— Ni 70%, Cu 30% Nickel-chromium ferrous alloy-Ni 75%, Cr 14%,Fe 10%

1B, N3, N4 H9 H15 A6

Low-carbon steel C 0.03% High-carbon steel C > 0.3% Carbon molybdenum and chrome molybdenum steel up to 3% Cr

Material

273 K (08C)

293 K (208C)

323 K (508C)

348 K (758C)

373 K (1008C)

398 K (1258C)

423 K (1508C)

Design temperature 473 K (2008C)

573 K (3008C)

673 K (4008C)

773 K (5008C)

973 K (6008C)

973 K (7008C)

1023 K (7508C)

77 73 81 87

11.2 10.6 11.7 12.6

73 70 78 84

10.6 10.2 11.3 12.2

13.9 12.9 14.9 101 15.2 100 18.8 128

96 89 103 105 130

95 88

16.0 109

31.0

214

110

30.0 26.3

207 184

8.54

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14.5

14.6

13.8 12.8

15.8

127

96

100

94 87

108

18.4

13.9

14.5

13.6 12.6

15.7

124

89

96

93 85

106

Copper and Its Alloys

Nickel and Nickel Alloy

18.0

12.9

13.9

13.5 12.3

15.4

Aluminum and Aluminum Alloys 69 10.0 68 9.9 67 9.7 66 9.6 65 9.4 64 9.3 64 9.3 63 9.1 73 10.6 72 10.4 71 10.3 70 10.2 79 11.5 78 11.3 77 11.2 76 11.0

203 29.4

206 29.9 206 29.9

69 10.0 65 9.4 73 10.6 79 11.5

203 29.4

206 29.9 206 29.9

70 10.2 67 9.7 74 10.7 80 11.6

Ferrous Materials 191 27.7

192 27.8 192 27.8

169 24.5

17

2.5

83 12.0

87 12.6 85 12.3

99 14.4

203 29.4 197 28.6 172 25.0 157 22.8 128 18.6 118 17.0

200 29.0 184 26.7 162 23.5 137 19.9 115 16.7 107 15.5 176 25.5 173 25.0 166 24.0 159 23.0 152 22.0 147 21.3

197 28.6 190 27.6 181 26.3

122 17.7 116 16.8

81 11.7

93 13.5

89 12.9 82 11.9

104 15.0

65 9.4 59 8.6 67 9.7 75 10.9

26

195 28.3 186 27.0 170 24.7

186 27.0 179

GPa Mpsi GPa Mpsi GPa Mpsi GPa Mpsi GPa Mpsi GPa Mpsi GPa Mpsi GPa Mpsi GPa Mpsi GPa Mpsi GPa Mpsi GPa Mpsi GPa Mpsi GPa Mpsi GPa Mpsi GPa Mpsi

73 K 173 K (2008C) (1008C)

TABLE 8-14 Modulus of elasticity for various materials

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.55

TABLE 8-15 Values of coeﬃcient c5 Coeﬃcient c5

Types of stays

1 2 3

112 120 135

4

150

5

175

Stays screwed through plates 1.1 cm thick, with the ends riveted over Stays screwed through plates >1.1 cm thick, with the ends riveted over Stays screwed through plates and provided with single nuts outside the plate or with inside and outside nuts, but no washers With heads <1.3 j times the stay diameter, screwed through the plates, or made with a taper ﬁt and having heads formed before installing and not riveted over; these heads have a true bearing on the plate Stays with inside and outside nuts and outside washers, when the washer diameter is 0:4a, and the thickness n

TABLE 8-16 Design stresses for bolted ﬂanged beads, d Minimum of speciﬁed range of tensile strength of ﬂange material at room temperature Maximum temperature

3170

3520

3870

4220

4920

Alloy bolt steel

K

8C

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

643 672 696 727 755 783

370 399 423 454 482 510

74.0 63.3 55.8 47.3 38.3 27.9

10.5 8.2 8.0 6.9 5.5 4.0

81.9 73.1 62.8 52.5 41.4 31.1

12.0 10.6 9.0 7.6 6.0 4.5

90.7 77.0 68.6 57.4 45.5 31.2

13.2 11.2 10.0 8.3 6.5 4.5

97.6 87.3 75.5 62.8 50.5 38.3

14.2 12.7 11.0 9.0 7.3 5.5

115.2 102.0 87.3 73.5 58.8 44.2

16.5 14.8 12.0 10.6 8.5 6.4

97.6 87.3 75.5 62.8 50.5 38.3

14.2 12.6 11.0 9.0 7.3 5.5

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.56

CHAPTER EIGHT

FIGURE 8-16(a) Maximum diameter of nonreinforced openings. (Source: IS 2825, 1969.)

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

FIGURE 8-16(b) Maximum diameter of nonreinforced openings. (Source: IS 2825, 1969.)

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8.57

Diameter, mm (in)

63.5 (2.5) >63.5 (2.5) to 102 (4) 63.5 (2.5) >63.5 (2.5) to 102 (4) 63.5 (2.5) >63.5 (2.5) to 102 (4) 102ð4Þ All (1) (2) All (1) (2)

843 min (122.3 min)

431–510 (62.5–74.0) 843 min (122.3) 775 min (112.4) 896 min (130.0) 647 min (93.8) 843 min (122.3) 804 min (116.6) 696 min (101.0 min) 539 min (78.2 min) In softened condition or 863 min (125.2) if cold-drawn

MPa (kpsi)

113 (16.4) 110 (16.0) 195 (28.3)

129 (18.7) 212 (30.8)

187 (27.1) 169 (24.5) 161 (23.4) 109 (15.7) 113 (16.4)

55 (8.0) 181 (26.3) 163 (23.6) 138 (20.0)

MPa (kpsi)

1008C

129 (18.7)

193 (28.0) 174 (25.2) 176 (25.5) 129 (18.7) 129 (18.7)

57 (8.3) 193 (28.0) 174 (25.2) 138 (20.0)

MPa (kpsi)

508C

94 (13.6) 169 (24.5)

100 (14.5)

181 (26.3) 163 (23.6) 141 (20.5) 85 (12.3) 100 (14.5)

53 (7.7) 168 (24.3) 152 (22.0) 138 (20.0)

MPa (kpsi)

2008C

87 (12.6) 160 (23.2)

93 (13.5)

176 (25.5) 159 (23.1) 134 (19.4) 78 (11.3) 93 (13.5)

48 (6.9) 159 (23.0) 145 (21.0) 138 (20.0)

MPa (kpsi)

2508C

83 (12.0) 152 (22.0)

90 (13.0)

170 (24.7) 152 (22.0) 126 (18.3) 76 (11.0) 90 (13.0)

154(22.4) 141(20.5) 138(20.0)

MPa (kpsi)

3008C

79 (11.5) 144 (20.9)

86 (12.5)

165 (23.9) 150 (21.8) 119 (11.3) 73 (10.6) 86 (12.5)

148 (21.5) 134 (19.4) 138 (20.0)

MPa (kpsi)

3508C

Allowable stress, sa , for design metal temperature not exceeding (8C)

1. Austenitic steel bolts for use in pressure joints shall not be less than 10 mm in diameter. 2. For bolts of up to 38 mm diameter use torque spanners. 3. High strength is obtainable in bolting materials by heat treatment of the ferritic and martensitic steels and by cold working of austenitic steels. Values in parentheses are in US Customary units (i.e., fps system of units). Sizes in parentheses are in inches and outside parentheses are in millimeters. Source: IS 2825, 1969.

18/9 Cr Ni Nb All (1) (2) stabilized steel 17/10/212Cr Ni Mo steel All (1) (2) 18/Cr 2 Ni steel 102 (4)

13% Cr Ni steel 18/8 Cr Ni steel 18/8 Cr Ni Ti stabilized steel

1% Cr V steel

5% Cr Mo steel

1% Cr Mo steel

Hot-rolled carbon steel 150 (6)

Material

Speciﬁed tensile strength, st

TABLE 8-17 Allowable stresses (sa ) for ﬂange bolting material

78 (11.3) 127 (18.4)

84 (12.2)

157 (22.8) 143 (20.7) 104 (15.1) 72 (10.4) 84 (12.2)

140 (20.0) 127 (18.4) 138 (20.0)

MPa (kpsi)

4008C

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.58

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.59

FIGURE 8-17 Nomenclature and formulas for reinforced openings. (This ﬁgure illustrates a common-nozzles conﬁguration and is not intended to prohibit other conﬁgurations permitted by the code.) (American Society of Mechanical Engineers, ASME Boiler and Pressure Vessel Code, Section VIII, Division 1, July 1, 1986.)

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.60

CHAPTER EIGHT

TABLE 8-18 Values of spherical radius factor K1 equivalent to spherical radius = K1 D, D=2h = axis ratio D=2h K1

3.0 1.36

2.8 1.27

2.6 1.18

2.4 1.08

2.2 0.99

2 0.90

1.8 0.81

1.6 0.73

Particular

1.4 0.65

1.2 0.57

1.0 0.50

Formula

LIGAMENTS The eﬃciency of the ligament between the tube holes, when the pitch of the tube holes on every row is equal

¼

The eﬃciency of the ligament between the tube holes, when the pitch of tube holes on any one row is unequal (Fig. 8-18)

¼

pd p

ð8-102Þ

where p ¼ longitudinal pitch of tube holes, m (in) d ¼ diameter of tube holes, m (in) p1 nd p1

ð8-103Þ

where p1 ¼ unit length of ligament, m (in) n ¼ number of tube holes in length, p1

FIGURE 8-18 Irregular drilling.

The eﬃciency of the ligament, when bending stress due to weight is negligible and the tube holes are arranged along a diagonal line with respect to the longitudinal axis or to a regular sawtooth pattern as shown in Fig. 8-19a to d

¼

2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A þ B þ ðA BÞ2 þ 4C2

ð8-104Þ

cos2 þ 1 2½1 ðd cos Þ=2a 1 d cos 1 ðsin2 þ 1Þ B¼ 2 a

where A ¼

sin cos C¼ d cos 2 1 a 1 cos ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; 1 þ ðb2 =a2 Þ The smallest value of eﬃciency of all the ligaments (longitudinal, circumferential, and diagonal) in the case of regular staggered spacing of tube holes For minimum number of pipe threads for connections as per ASME Boiler and Pressure Vessel Code

¼

p c PL d ¼ pL PL

or

1 sin ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ a2 =b2

d a

The symbols are as shown in Fig. 8-19d. Refer to Table 8-19.

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ð8-105Þ

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.61

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

FIGURE 8-19(a) A regular staggering of holes.

FIGURE 8-19(c) Regular sawtooth pattern of holes.

FIGURE 8-19(b) Spacing of holes on a diagonal line.

FIGURE 8-19(d)

Particular

Formula

BOLTED FLANGE CONNECTIONS Bolt loads 2 G P þ 2bGmP 4

The required bolt load under operating conditions suﬃcient to contain the hydrostatic end force and simultaneously to maintain adequate compression on the gasket to ensure seating

Wm1 ¼ H þ HP ¼

For additional gasket criteria

Refer to Tables 8-20 and 8-21.

ð8-106Þ

TABLE 8-19 Minimum number of threads for connections Size of pipe connection, mm (in)

12.5 and 18.75 (12 and 34)

25.0, 31.25, and 37.5 (1, 114, and 112)

50.0 (2)

62.5 and 75 (212 and 3)

100–150 (4–6)

200 (8)

250 (10)

300 (12)

Threads engaged

6

7

8

8

10

12

13

14

Minimum plate thickness required, mm (in)

10.75 (0.43)

15.25 (0.62)

17.50 (0.70)

25.0 (1.0)

31.25 (1.25)

37.50 (1.5)

40.5 (1.62)

43.75 (1.75)

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10

10

Dimension N mm (in) (min)

Flat-metal-jacketed, asbestos-ﬁlled

Corrugated metal

Carbon steel, stainless steel or monel metal Soft aluminum Soft copper or brass Iron or soft steel Monel metal or 4–6% chrome steel Stainless steels Soft aluminum Soft copper or brass Iron or soft steel Monel metal or 4–6% chrome steel Stainless steel Soft aluminum Soft copper or brass Iron or soft steel Monel metal or 4–6% chrome steel Stainless steels

68.9 (10.0) 68.9 (10.0) 20.0 (2.9) 25.5 (3.7) 31.0 (4.5) 38.0 (5.5) 44.8 (6.5) 25.5 (3.7) 31.0 (4.5) 38.0 (5.5) 44.0 (6.5) 52.4 (7.6) 38.0 (5.5) 44.0 (6.5) 52.4 (7.6) 55.1 (8.0) 62.1 (9.0)

3.50 2.75 3.00 3.25 3.50 3.75 3.25 3.50 3.75 3.50 3.75

7.55 (1.1)

15.2 (2.2) 20.0 (2.9) 25.5 (3.7)

0 1.37 (0.2) 11.0 (1.6) 25.5 (3.7) 44.8 (6.5) 2.75 (0.40)

Minimum design seating stress, y MPa (kpsi)

2.50 3.00 2.50 2.75 3.00 3.25

1.75

Vegetable ﬁber

Spiral-wound metal, asbestos-ﬁlled Corrugated metal, asbestos inserted or Corrugated metal, jacketed asbestos ﬁlled

2.25 2.50 2.75

0.50 1.00 2.00 2.75 3.50 1.25

Gasket factor, m

Rubber and elastomers ( 3-ply with asbestos fabric 2-ply insertion, with or without 1-ply wire reinforcement

Rubber without fabric or a high percentage of asbestos ﬁber: <70 IRHD* (75A Shore Durometer) 70 IRHD (75A)* or higher ) 3.2 mm (0. 125 in) Asbestos with a suitable binder for the operating 1.6 mm (0.062 In) conditions 0.8 mm (0.031 in) thickness Rubber and elastomers with cotton fabric insertion

Gasket material

TABLE 8-20 Gasket materials and contact facingsa Sketches and and notes

1a, 1b, 1c , 1d 2

1 (a, b, c, d)

1 (a, b)

1 (a, b, c, d), 4, 5

1 (a, b, c, d), 4, 5,

Use facing sketch

II

II

II

II

II

Use column

Refer to Table 8-21

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.62

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Solid ﬂat metal

6

Rubber O-rings: <75 IRHD (75A Shore Dur) 75 (75A) to 85 IRHD (85A) Rubber square section rings: <75 IRHD (75A Shore Dur) 75 (75A) to 85 IRHD (85A) Rubber T-section rings: Below 75 IRHD (75A Shore Dur) Between 75 (75A) and 85 IRHD (85A)

Soft aluminum Soft copper or brass Iron or soft steel Monel metal or 4–6% chrome steel Stainless steels Soft aluminum Soft copper or brass Iron or soft steel Monel metal or 4–6% chrome steel Stainless steels Iron or soft steel Monel metal or 4–6% chrome steel Stainless steels

0.98 (0.14) 2.75 (0.40) 0.98 (0.14) 2.75 (0.40)

4c 9c

179.3 (26.0)

6.50

4c 9c

179.3 (26.0) 124.2 (18.0) 150.3 (21.8)

6.50 5.50 6.00

0.69 (0.10) 1.42 (0.2)

69.6 (10.1) 60.7 (8.8) 89.6 (13.0) 124.2 (18.0) 150.3 (21.8)

4.25 4.00 4.75 5.50 6.00

3c 6c

3.80 (5.5) 44.8 (6.5) 52.4 (7.6) 62.1 (9.0)

Minimum design seating stress, y MPa (kpsi)

3.25 3.50 3.75 3.75

Gasket factor, m

Sketches and and notes

9 only

8 only

7 only

6

1 (a, b, c, d), 2, 3, 4, 5

1 (a, b, c, d), 2, 3

Use facing sketch

II

I

II

Use column

Refer to Table 8-21

c

b

Gasket factors (m) for operating conditions and minimum design seating stress (y). or The surface of a gasket having a lap should not be against the nubbin. These values have been calculated. Note: This table gives a list of many commonly used gasket materials and contact facings with suggested design values of m and y that have generally proved satisfactory in actual service when using eﬀective gasket seating with b given in Table 8-21 and Fig. 8-13. The design values and other details given in this table are suggested only and are not mandatory. Source: IS 2825, 1969.

a

Grooved metal

10

Ring joint

Gasket material

Dimension N mm (in) (min)

TABLE 8-20 Gasket materials and contact facingsa (Cont.)

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

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8.63

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

TABLE 8-21 Eﬀective gasket width Basic gasket seating width, b Facing sketch (exaggerated)

Column I

Column II

1a

N 2

N 2

1ba

1c

w þ 25T ; 3

wþN max 4

w þ 25T ; 3

wþN max 2

1d a

2

wþN 4

wþN ; 4

3

w ; 2

4a

3N 8

7N 16

5

N 4

3N 8

6

w 8

—

7

—

N 2

8

—

N 2

9

—

N 2

a

N min 4

w þ 3N 8

3N min 8

Where serrations do not exceed 0.4 mm depth and 0.8 mm width spacing, sketches 1b and 1d shall be used.

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

Particular

8.65

Formula

Wm2 ¼ bGy

The required initial bolt load to seat the gasket jointcontact surface properly at atmospheric temperature condition without internal pressure

ð8-107Þ

Refer to Table 8-20 for y.

Total required cross-sectional area of bolts at the root of thread

Am > Am1 or Am2

ð8-108Þ

Total cross-sectional area of bolt at root of thread or section of least diameter under stress required for the operating condition

Am1 ¼

Wm1 sbd

ð8-109Þ

Total cross-sectional area of bolt at root of thread or section of least diameter under stress required for gasket seating

Am2 ¼

Refer to Tables 8-17 and 8-23 for sbd . Wm2 sbat

ð8-110Þ

TABLE 8-22 Moment arms for ﬂange loads under operating conditions Type of ﬂange

hD

hT

hG

Integral-type ﬂanges

R þ 0:5g1

Loose-type except lap joint ﬂanges and optional-type ﬂanges

CB 2 CB 2

R þ g1 þ hG 2 hD þ hG 2 CG 2

CG 2 CG 2 CG 2

Lap joint ﬂanges

TABLE 8-23 Maximum allowable stresses in stays and stay bolts, sa Stress For lengths between support not exceeding 120 diameter

For lengths between support exceeding 120 diameter

Type of stay

MPa

MPa

kpsi

(a) Unwelded or ﬂexible stays less than 20 diameter long, screwed through plates with ends riveted over (b) Hollow steel stays less than 20 diameter long, screwed through plates with ends riveted over (c) Unwelded stays and unwelded portions of welded stays, except as speciﬁed in (a) and (b) (d) Steel through stays exceeding 38 mm diameter (e) Welded portions of stays

51

7.5

55

8.0

66

9.5

58

8.5

71 41

10.4 6.0

62 51

9.0 7.5

kpsi

Source: ASME Boiler and Pressure Vessel Code.

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.66

CHAPTER EIGHT

Particular

The actual cross-sectional area of bolts using the root diameter of thread or least diameter of unthreaded portion (if less), to prevent damage to the gasket during bolting up

Formula

Ab ¼

2yGN <j Am sbat

ð8-111Þ

Flange design bolt load W The bolt load in the design of ﬂange for operating condition The bolt load in the design of ﬂange for gasket seating

The relation between bolt load per bolt ðWb Þ, diameter of bolt D and torque Mt

ð8-112Þ

W ¼ Wm1 W¼

Am þ Ab sbat 2

Wb ¼ 0:17DMt

ð8-113Þ

for lubricated bolts USCS

ð8-114aÞ

where Wb in lbf, D in in, Mt in lbf in Wb ¼ 263:5DMt

SI

ð8-114bÞ

USCS

ð8-114cÞ

where Wb in N, D in m, Mt in N m Wb ¼ 0:2DMt

for unlubricated bolts

where Wb in lbf, D in in, Mt in lbf in Wb ¼ 310DMt

SI

ð8-114dÞ

where Wb in N, D in m, Mt in N m

Flange moments The total moment acting on the ﬂange Mo for operating condition

The total ﬂange moment Mo for gasket seating, which is based on the ﬂange design bolt load of Eq. (8-113)

Mo ¼ MD þ Mt þ MG ¼ HD hD þ HT hT þ HG hG

ð8-115aÞ ð8-115bÞ

This is based on the ﬂange design load of Eq. (8-112) with moment arms as given in Table 8-22. Am þ Ab CG Mo ¼ WhG ¼ ð8-116Þ sbat 2 2

Flange stresses The stress in the ﬂange shall be determined for both the gasket seating condition and the operating condition.

The larger of these two controls with the following formulas:

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

Particular

8.67

Formula

INTEGRAL-TYPE FLANGES AND LOOSETYPE FLANGES WITH A HUB There are three types of stress: Longitudinal hub stress

Radial ﬂange stress

Tangential stress

For ﬂange factors values

H ¼

fMo Lg21 B

ð8-117Þ

R ¼

ð1:33te þ 1ÞMo Lt2 B

ð8-118Þ

¼

YMo t2 B

ð8-119Þ

ZR

Refer to Figs. 8-20 to 8-25.

LOOSE-TYPE FLANGES WITHOUT HUB AND LOOSE-TYPE FLANGES WITH HUB WHICH THE DESIGNER CHOOSES TO CALCULATE (a) Stresses without considering the hub YMo t2 B

(1) Tangential stress

¼

(2) The radial and longitudinal stress

H ¼ R ¼ 0

ð8-120Þ ð8-121Þ

(b) Allowable ﬂange design stresses: The ﬂange stresses calculated by Eqs. (8-117) to (8-121) shall not exceed the values of stresses given by Eqs. (8-122) to (8-126). (1) The longitudinal hub stress

H <j sfd j 1:5sfd H >

(i) The longitudinal hub stress for optionaltype ﬂanges designed as integral and also integral type where the neck material constitutes the hub of the ﬂange (ii) The longitudinal hub stress for integraltype ﬂanges with hub welded to the neck, pipe, or vessel wall

j 1:5sfd ðaÞ H >

for cast iron

ð8-122aÞ

for other materials

ð8-122bÞ

or 1:5snd

ð8-123aÞ

The smaller of sfd and snd is to be selected. j 1:5sfd ðbÞ H >

or 2:5snd

ð8-123bÞ

The smaller of sfd and snd is to be selected.

(2) The radial stress

R > j sfd

ð8-124Þ

(3) The tangential stress

j sfd >

ð8-125Þ

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FIGURE 8-20 Values of T, U, Y, and Z for K ¼ ðA=BÞ > 1:5. (Source: IS 2825, 1969.)

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

FIGURE 8-21 Values of F (integral ﬂange factors). (Source: IS 2825, 1969.)

FIGURE 8-22 Values of V (integral ﬂange factors). (Source: IS 2825, 1969.)

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8.69

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.70

CHAPTER EIGHT

FIGURE 8-23 Values of FL (loose hub ﬂange factors). (Source: IS 2825, 1969.)

FIGURE 8-24 Values of VL (loose hub ﬂange factors). (Source: IS 2825, 1969.)

Particular

Formula

(4) The average of H and R , and H and

ðH þ R Þ=2 > j sfd

ð8-126aÞ

j sfd ðH þ Þ=2 >

ð8-126bÞ

Flanges under external pressure The design of ﬂanges for external pressure only shall be based on the formulas given for internal pressure except that for operating conditions.

Mo ¼ HD ðhD hG Þ þ HT ðhT hG Þ

Mo ¼ WhG

for operating conditions

ð8-127aÞ

for gasket seating

ð8-127bÞ

where W ¼ sbat ðAm2 þ Ab Þ=2

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ð8-128Þ

DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.71

FIGURE 8-25 Values of f (hub stress correction factor). (Source: IS 2825, 1969.)

REFERENCES 1. ‘‘Rules for Construction of Pressure Vessels,’’ Section VIII, Division 1, ASME Boiler and Pressure Vessel Code, The American Society of Mechanical Engineers (ASME), New York, 1986 ed., July 1, 1986. 2. ‘‘Rules for Construction of Pressure Vessels,’’ Section VIII, Division 2, Alternative Rules, ASME Boiler and Pressure Vessel Code, ASME, New York, 1986 ed., July 1, 1986. 3. ‘‘Rules for Construction of Power Boiler,’’ Section 1, ASME Boiler and Pressure Vessel Code, ASME, New York, 1983 ed., July 1, 1971. 4. ‘‘Recommended Rules for Care of Power Boilers,’’ Section VII, ASME Boiler and Pressure Vessel Code, ASME, New York, 1983. 5. ‘‘Rules for in Service Inspection of Nuclear Power Plant Components,’’ Section XI, ASME Boiler and Pressure Vessel Code, 1971. 6. ‘‘Heating Boilers,’’ Section IV, ASME Boiler and Pressure Vessel Code, ASME, New York, 1983. 7. ‘‘Recommended Rules for Care and Operation of Heating Boilers,’’ Section VI, ASME Boiler and Pressure Vessel Code, ASME, New York, 1983. 8. ‘‘Part A: Ferrous Materials,’’ Section II, ASME Boiler and Pressure Vessel Code, ASME, New York, 1983. 9. ‘‘Part B: Non-ferrous Materials,’’ Section II, ASME Boiler and Pressure Vessel Code, ASME, New York, 1983. 10. Azbel, D. S., and N. P. Cheremisinoﬀ, Chemical and Process Equipment Design—Vessel Design and Selection, Ann Arbor Science Publishers, Ann Arbor, Michigan, 1982.

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DESIGN OF PRESSURE VESSELS, PLATES, AND SHELLS

8.72

CHAPTER EIGHT

11. Bureau of Indian Standards, ZS 2825-1969 (under revision). 12. Chuse, R., Pressure Vessels—The ASME Code Simpliﬁed, 5th edition, McGraw-Hill Book Company, New York, 1977. 13. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962. 14. Lingaiah, K., and B. R. Narayana lyengar, Machine Design Data Handbook, Vol. I (SI and Customarv Metric Units), Suma Publishers, Bangalore, India, 1983. 15. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

9 DESIGN OF POWER BOILERS

SYMBOLS6;7 C d do Do D.S. E G h or t H l n L P p or P Ri S t (or h) SHS W WHS sa

smoke area consisting of the total internal transverse area of the tube, m2 (ft2 ) diameter of cylinder or shell, in (in) diameter or short span, measured as shown in Fig. 8-9 (Chap. 8) maximum allowable diameter of opening, m (in) outside diameter of cylinder or shell or tube or pipe, m (in) outside diameter of furnace or ﬂue, m (in) disengaging surface or area of water surface through which steam bubbles must be discharged, the water being considered at the middle-gauge cock, m2 (ft2 ) modulus of elasticity, GPa (Mpsi) area of the grate as ﬁnally adopted, m2 (ft2 ) thickness of tube or shell wall, m (in) total heating surface in contact with the ﬁre, m2 (ft2 ) length of the ﬂue sections, m (in) factor of safety to be taken as 5 for usual cases radius to which the head is formed, measured on the concave side of the head, m (in) rated power of boiler maximum allowable working pressure, Pa or MPa (psi) inside radius of cylindrical shell, m (in) volume of steam included between the shell and a horizontal line through the position of the central gauge as ﬁnally determined, m2 (ft2 ) thickness of tube or pipe or cylinder or shell or plate, m (in) total area of superheating surface based on the actual area in contact with the ﬁre, m2 (ft2 ) net water volume in the boiler below the line of the central gauge cock, m2 (ft2 ) total area of water heating surface based on the actual area in contact with the ﬁre, m2 (ft2 ) maximum allowable stress value, MPa (kpsi) from Tables 7-1 (Chapter 7), 8-9 to 8-11, and 8-17 (Chapter 8) eﬃciency of joint

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DESIGN OF POWER BOILERS

9.2

CHAPTER NINE

Other factors in performance or in special aspects are included from time to time in this chapter and, being applicable only in their immediate context, are not given at this stage. Note: and with initial subscript s designates strength properties of material used in the design, which will be used and observed throughout this Machine Design Data Handbook.

Particular

Formula

BOILER TUBES AND PIPES For calculation of the minimum required thickness (t) and maximum allowable working pressure ( p or P) of ferrous and nonferrous tubes and pipes from 12.5 mm (12 in) to 150 mm (6 in) outside diameter used in power boilers as per ASME Boiler and Pressure Vessel Code2;3

Refer to Eqs. (7-1) to (7-15) (Chap. 7).

For eﬃciency of joints (), temperature coeﬃcient (y), minimum allowance for threading, and structural stability (C) as per ASME Boiler and Pressure Vessel Code

Refer to Tables from 7-2 to 7-6 (Chap. 7).

For maximum allowable stress value (sa ) for the materials of tubes and pipes as per ASME Boiler and Pressure Vessel Code3

Refer to Table 7-1.

The maximum allowable working pressure for steel tubes or ﬂues of ﬁre tube boilers for diﬀerent diameters and gauges of tubes as per ASME Power Boiler Code2

p¼

96:5 ðh 1:625 103 Þ do

SI ð9-1aÞ

where p in MPa, h and do in m p¼

14000 ðh 0:065Þ do

USCS

ð9-1bÞ

where p in psi, h and do in in For maximum allowable working pressure and thickness of steel tubes The maximum allowable working pressure for copper tubes for ﬁretube boilers subjected to internal or external pressure as per ASME Power Boiler Code2

Refer to Tables 7-7, 9-1, 9-2 and 9-4 and Fig. 7-1. p¼

83 ðh 1 103 Þ 1:7 do

SI ð9-2aÞ

where p in MPa, do and h in m p¼

12000 ðh 0:039Þ 250 do

USCS

where p in psi, do and h in in For maximum allowable working pressure and thickness of copper tubes

Refer to Tables 9-3 and 9-5.

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ð9-2bÞ

0.055 0.065 0.075 0.085 0.095 0.105 0.120 0.135 0.150 0.165 0.180 0.200 0.220 0.240 0.260 0.280 0.300 0.320 0.340 0.360 0.380 0.400 0.420

1.375 1.625 1.875 2.125 2.375 2.625 3.000 3.375 3.750 4.125 4.500 5.000 5.500 6.000 6.500 7.000 7.500 8.000 8.500 9.000 9.500 10.000 10.500

17 16 15þ 14þ 13 12 11 10þ 9þ 8 7 6 5 4þ 3þ 2

Nearest Bwg no.

3.38 7.52 11.03

MPa

12.5

590 1090 1600

psi

(0.5)

2.41 4.62 6.90 9.24

MPa

19.0

Bwg ¼ Birmingham wire gauge Source: ASME Power Boiler Code, Section I, 1983.

in

mm

Wall thickness

350 670 1000 1340

psi

3.24 5.00 6.62

MPa

(1.75) 25.0

470 720 960

psi

(1.0)

2.42 3.80 5.10 12.13 13.65

MPa

31.25

350 550 740 1760 1980

psi

3.0 4.06 5.24 11.03 12.90

MPa

(1.25) 37.5

430 590 760 1600 1870

psi

(1.5)

3.38 4.34 9.24 10.82 12.34 13.92

MPa

43.75

490 630 1340 1570 1790 2020

psi

2.83 3.65 7.93 9.24 10.62 12.00 13.38

MPa

(1.75) 50.0

410 530 1150 1340 1540 1740 1940

psi

(2.0)

2.76 3.45 7.17 8.20 9.24 10.34 11.45 12.90

MPa

62.5

Tube outside diameter, mm (in)

400 500 1040 1190 1340 1500 1660 1870

psi

(2.5)

2.34 5.80 6.62 7.52 8.34 9.24 10.48 11.65 12.90

MPa

75.0

390 840 960 1090 1210 1340 1520 1690 1870

psi

(3.0)

2.90 5.52 6.27 7.03 7.72 8.76 9.80 10.68 11.86 12.90 13.92

MPa

87.5

420 800 910 1020 1120 1270 1420 1550 1720 1870 2020

psi

(3.5)

4.68 5.38 6.00 6.62 7.52 8.34 9.24 10.14 11.03 12.00 12.90 13.78

MPa

100.1

680 780 870 960 1090 1210 1340 1470 1600 1740 1870 2000

psi

(4.0)

4.62 5.24 5.80 6.55 7.31 8.07 8.90 9.65 10.48 12.24 12.06 12.90 13.72

MPa

112.5

670 760 840 950 1060 1170 1290 1400 1520 1630 1750 1870 1990

psi

(4.5)

4.70 4.62 5.10 7.80 6.48 7.17 7.86 8.55 9.24 10.00 10.68 11.45 12.13 12.90 13.65

MPa

125.0

TABLE 9-1 Maximum allowable working pressures for seamless steel and electric resistance welded steel tubes or nipples for watertube boilers [from Eq. (7-4)]

590 670 740 840 940 1040 1140 1240 1340 1450 1550 1660 1760 1870 1980

psi

(5.0)

DESIGN OF POWER BOILERS

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9.3

0.095 0.105 0.120 0.135 0.150 0.165 0.180 0.200 0.220 0.240

2.375 2.625 3.000 3.375 3.375 4.125 4.500 5.000 5.500 6.000

1.93 2.62 3.58 4.55 5.52 6.48

280 380 520 660 800 940

MPa psi 1.45 1.93 2.69 3.38 4.14 4.83 5.58 6.55 7.52 8.40

210 280 390 490 600 700 810 950 1090 1230

MPa psi

37.50 (1.50) 50.00 (2)

Source: ASME Power Boiler Code, Section I, 1983.

420 560 770 980

13 12 11 10þ 9þ 8 7 6 5 4þ

In

mm

2.90 3.86 5.31 6.76

25.00 (1) Nearest Bwg no. MPa psi

Wall thickness

1.17 1.59 2.14 2.76 3.30 3.86 4.48 5.24 6.00 6.83

170 230 310 400 480 560 650 760 870 990

MPa psi

1.31 1.80 2.28 2.76 3.24 3.72 4.34 5.03 5.65

190 260 330 400 470 540 630 730 820

MPa psi

62.50 (2.50) 75.00 (3)

1.10 1.52 1.93 2.34 2.76 3.17 3.72 4.27 4.83

160 220 280 340 400 460 540 620 700

MPa psi

(4)

1.38 1.72 2.06 2.41 2.83 3.31 3.79 4.28

200 250 300 350 410 480 550 260

MPa psi

87.50 (3.50) 200

Size outside diameter mm (in)

TABLE 9-2 Maximum allowable working pressures for steel tubes or ﬂues for ﬁretube boilers [from Eq. (9-1)]

1.24 1.52 1.86 2.21 2.48 2.90 3.38 4.80

MPa

180 220 270 320 360 420 490 550

psi

1.38 1.65 1.93 1.28 2.62 3.03 3.38

MPa

200 240 280 330 380 440 490

psi

112.50 (4.50) 125.00 (5)

1.52 1.80 2.07 2.41 2.76 3.10

MPa

220 260 300 350 400 450

psi

1.65 1.86 2.21 2.55 2.83

240 270 320 370 410

MPa psi

137.50 (5.50) 150.0 (6)

DESIGN OF POWER BOILERS

9.4

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DESIGN OF POWER BOILERS DESIGN OF POWER BOILERS

Particular

9.5

Formula

The external working pressure, for plain lap-welded or seamless tubes up to and including 150 mm (6 in) external diameter, and if the thickness is greater than the standard one

p¼

1 596h 9:6 n do

where p in Pa, h and d in m 1 86670h 1386 p¼ n do

SI

ð9-3aÞ

USCS

ð9-3bÞ

where p in psi, h and d in in Refer to Table 9-6.

For proportion of standard boiler tubes

TABLE 9-3 Maximum allowable working pressure for copper tubes for ﬁretube boilersa [from Eq. (9-2)] Outside diameter of tube

Gauge, Bwg 12

11

10

9

8

7

6

5

4

MPa psi

MPa psi

MPa psi

MPa psi

MPa psi

MPa psi

MPa psi

MPa psi

1.72 1.72 1.72 1.31

1.72 1.72 1.72 1.59

mm

in

MPa psi

50.00 81.25 100.00 125.00

2 3.25 4 5

1.17

170 1.65

240 1.72 0.76

250 1.72 110 1.03

250 1.72 150 1.52 0.90

250 1.72 220 1.72 130 1.10

250 1.72 250 1.72 160 1.72 1.03

250 250 250 150

250 250 250 190

250 250 250 230

a

For use at pressure not to exceed 1.7 MPa (250 psi) or temperature not to exceed 2088C (4068F). Source: ASME Power Boiler Code, Section I, 1983.

TABLE 9-4 Maximum boiler pressures for use of ANSI B16.5 standard steel pipe ﬂanges and ﬂanged valves and ﬁttings Maximum allowable boiler pressure Primary service pressure rating

Steam service at saturation temperature

Boiler feed and blow-oﬀ line service

Mpa

psi

MPa

psi

MPa

psi

1.14 2.17 2.86 4.23 6.30 10.44 17.33

164.7 314.7 414.7 614.7 914.7 1514.7 2514.7

1.41 4.44 5.75 8.10 11.40 17.23 22.10

204.7 644.7 834.7 1174.7 1654.7 2514.7 3206.0

1.20 3.65 4.68 6.79 10.10 16.13 22.20

174.7 529.7 679.7 984.7 1464.7 2339.7 3220.7

Notes: Adjusted pressure ratings for steam service at saturated temperature corresponding to the pressure, derived from Table 2 to 8 ANSI B 16.5– 1968. Pressures shown include the factor for boiler feed and blow-oﬀ line service required by ASME corrected for saturation temperature corresponding to this pressure. Source: ASME Power Boiler Code, Section I, 1983.

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DESIGN OF POWER BOILERS

9.6

CHAPTER NINE

TABLE 9-5 Maximum external working pressures for use with lap-welded and seamless boiler tubesa Maximum allowable pressure

Nominal diameter, external diameter, mm (in)

Standard thickness, mm

MPa

51 (2) 58 (2.25) 64 (2.5) 70 (2.75) 76 (3) 83 (3.25)

2.4 2.4 2.8 2.8 2.8 3.1

2.84 2.55 2.65 2.45 2.26 2.26

a

Maximum allowable pressure

psi

Nominal diameter, external diameter, mm (in)

Standard thickness, mm

MPa

psi

427 380 392 356 327 327

89 (3.5) 96 (3.75) 102 (4) 115 (4.5) 127 (5) 153 (6)

3.1 3.1 3.4 3.4 3.8 4.2

2.16 1.96 2.06 1.67 1.67 1.37

308 282 303 238 235 199

External diameter 50 to 150 mm (2 to 6 in).

TABLE 9-6 Proportions of standard boiler tubes Nominal diameter, actual external diameter mm (in)

Actual internal diameter, mm

45 (1.76) 51 (2) 58 (2.25) 64 (2.5) 70 (2.75) 76 (3) 83 (3.25) 89 (3.5) 96 (3.75) 102 (4) 115 (4.5) 127 (5) 153 (6)

38 46 50 56 64 71 76 81 89 94 107 120 142

Thickness, mm

External circumference, mm

Internal circumference, mm

External transverse area, mm2

Internal transverse area, mm2

Length of tube m2 of internal heating surface, m

2.4 2.4 2.4 2.8 2.8 2.8 3.0 3.0 3.0 3.3 3.3 3.8 4.2

140 160 181 200 220 240 260 280 300 320 360 400 480

125 144 165 183 200 221 241 260 280 290 340 370 450

1600 2000 2000 3200 3800 4500 5400 6200 7000 8000 10000 12800 18300

1200 1700 2100 2600 3200 3900 4500 5400 6200 6900 9000 11100 16300

7.58 6.58 5.78 5.24 4.74 4.38 3.98 3.71 3.45 3.25 2.86 2.58 2.15

Weight per meter N

lbf

24.5 28.2 32.0 40.7 44.9 49.1 58.5 63.0 68.0 80.8 91.2 112.3 150.0

1.679 1.932 2.186 2.783 3.074 3.365 4.011 4.331 4.652 5.532 6.248 7.669 10.282

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DESIGN OF POWER BOILERS DESIGN OF POWER BOILERS

Particular

9.7

Formula

The external pressure, for plain lap-welded, or seamless tubes or ﬂues over 50 mm (2 in) and not exceeding 150 mm (6 in) external diameter

Refer to Table 9-5.

The minimum required thickness of component when it is of riveted construction or does require staying as per ASME Power Boiler Code2

h¼

pRi 0:8sa 0:6p

ð9-4Þ

The maximum allowable working pressure as per ASME Power Boiler Code

p¼

0:8sa Ri þ 0:6h

ð9-5Þ

h¼

5pL 4:8sa

ð9-6Þ

DISHED HEADS The thickness of a blank unstayed dished head with the pressure on the concave side, when it is a segment of a sphere as per ASME Power Boiler Code

where L ¼ radius to which the head is dished, measured on the concave side of the head, m (in) ¼ eﬃciency of weakest joint used in forming the head. (Refer to Table 8-3 for .) The minimum distance between the centers of any two openings, rivet holes excepted, shall be determined by Eq. (9-7)

AþB 2ð1 KÞ

L¼

ð9-7Þ

where L ¼ distance between the centers of the two openings measured on the surface of the head, m (in) A; B ¼ diameters of two openings, m (in) K ¼ same as deﬁned in Eqs. (9-8a) and (9-8b) The expression for K

K¼

pdo 1:6sa h

ð9-8aÞ

K¼

pdo 1:82sa h

ð9-8bÞ

Equation (9-8a) shall be used with shells and headers designed by using Eqs. (9-4) and (9-5). Equation (9-8b) shall be used with shells and headers designed by using Eqs. (9-9) and (9-10): The minimum required thickness of ferrous drums and headers based on strength of weakest course as per ASME Power Boiler Code

h¼

pdo pRi þ C or þC 2sa þ 2yp sa ð1 yÞp

ð9-9Þ

The maximum allowable working pressure as per ASME Power Boiler Code

p¼

2sa ðh CÞ sa ðh CÞ or do 2yðh CÞ Ri þ ð1 yÞðh CÞ

ð9-10Þ

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DESIGN OF POWER BOILERS

9.8

CHAPTER NINE

Particular

Formula

For values y, C, and sa refer to Tables 7-1, 7-3, and 7-6. The thickness of a blank unstayed full-hemispherical head with the pressure on the concave side

h¼

pL 1:6sa

ð9-11aÞ

h¼

pL ð2sa 0:2pÞ

ð9-11bÞ

Equation (9-11b) may be used for heads exceeding 12.5 mm (0.5 in) in thickness that are to be used with shells or headers designed under Eqs. (9-9) and (9-10) and that are integrally formed on seamless drums or are attached by fusion welding and do not require staying. The formula for the minimum thickness of head when the required thickness of the head given by Eqs. (9-9) and (9-10) exceeds 35 percent of the inside radius

h ¼ Lðy1=3 1Þ

ð9-12Þ

where y¼

2ðsa þ pÞ 2sa p

ð9-12aÞ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Cp=sa

ð9-13Þ

UNSTAYED FLAT HEADS AND COVERS The minimum required thickness of ﬂat unstayed circular heads, covers and blind ﬂanges as per ASME Power Boiler Code

h¼d

where C ¼ a factor depending on the method of attachment of head on the shell, pipe or header (refer to Table 8-6 for C) d ¼ diameter or short span, measured as shown in Fig. 8-9

The minimum required thickness of ﬂat unstayed circular heads, covers or blind ﬂange which is attached by bolts causing edge moment Fig. 8-9( j ) as per ASME Power Boiler Code

h ¼ d½Cp=sa þ 1:78WhG =sa d 3 1=2

ð9-14Þ

where W ¼ total bolt load, kN (lbf ) hG ¼ gasket moment arm, Fig. 8-13 and Table 8-22.

For details of bolt load HG , bolt moments, gasket materials, and eﬀect of gasket width on it

Refer to Tables 8-20 and 8-22 and Fig. 8-13

The minimum required thickness of unstayed heads, covers, or blind ﬂanges of square, rectangular, elliptical, oblong segmental, or otherwise noncircular as per ASME Power Boiler Code

t or h ¼ d

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ZCp=sa

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ð9-15Þ

DESIGN OF POWER BOILERS DESIGN OF POWER BOILERS

Particular

9.9

Formula

where Z ¼ 3:4 2:4d=a

ð9-15aÞ

a ¼ long span of noncircular heads or covers measured perpendicular to short span, m (in) Z need not be greater than 2.5 Equation (9-15) does not apply to noncircular heads, covers, or blind ﬂanges attached by bolts causing bolt edge moment The minimum required thickness of unstayed noncircular heads, covers, or blind ﬂanges which are attached by bolts causing edge moment Fig. 8-9 as per ASME Power Boiler Code

h ¼ d½ZCp=sa þ 6WhG =sa Ld 2 1=2

The required thickness of stayed ﬂat plates (Figs. 8-10 and 8-11) as per ASME Power Boiler Code

h ¼ pt

ð9-16Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ½ p=sa c5

ð9-17Þ

where pt ¼ maximum pitch, m (in), measured between straight lines passing through the centers of the stay bolts in the diﬀerent rows (Refer to Table 9-7 for pitches of stay bolts.) c5 ¼ a factor depending on the plate thickness and type of stay (Refer to Table 8-15 for values of c5 .) For sa refer to Tables 8-8, 8-23, and 8-11 h2 sa c5 p2i

The maximum allowable working pressure for stayed ﬂat plates as per ASME Power Boiler Code

p¼

For all allowable stresses in stay and stay bolts

Refer to Chapter 8

ð9-18Þ

Also for detail design of diﬀerent types of heads, covers, openings and reinforcements, ligaments, and bolted ﬂanged connection

COMBUSTION CHAMBER AND FURNACES Combustion chamber tube sheet The maximum allowable working pressure on tube sheet of a combustion chamber where the crown sheet is suspended from the shell of the boiler as per ASME Power Boiler Code

P ¼ 27000

hðD di Þ wD

USCS

ð9-19aÞ

where h ¼ thickness of tube, in w ¼ distance from the tube sheet to opposite combustion chamber sheet, in

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100 110 120 125 130 140 150 160 170 180 190 200 225 250 300

0.67 0.76 0.83 0.86 0.90 0.96 1.03 1.10 1.17 1.24 1.31 1.38 1.55 1.72 2.07

131.25 125.000 118.750 118.750 115.625 112.500 106.250 103.125 100.000

7.8125

(5.25) (5.000) (4.75) (4.75) (4.625) (4.50) (4.25) (4.125) (4.000)

(0.3125)

159.375 150.000 143.750 140.625 137.500 134.375 128.125 125.000 121.875 118.750 115.625 112.500 106.25 100.000

9.375

Source: ASME Power Boiler Code, Section I, 1983.

psi

MPa

Pressure

(6.375) (6.000) (5.75) (5.625) (5.50) (5.375) (5.125) (5.000) (4.875) (4.75) (4.625) (4.50) (4.25) (4.000)

(0.375)

184.375 175.000 168.750 165.625 162.500 156.250 150.000 146.875 140.625 137.500 134.375 131.25 121.875 115.625 106.250

10.9375

(0.50)

14.0625

(7.375) (7.000) (6.75) (6.625) (6.50) (6.25) (6.000) (5.875) (5.625) (5.50) (5.375) (5.25) (4.875) (4.625) (4.25) 209.375 200.000 193.750 190.625 184.375 178.125 171.875 168.150 162.500 159.375 153.125 146.875 137.50 125.000

(8.375) (8.000) (7.75) (7.625) (7.375) (7.125) (6.875) (6.75) (6.50) (6.375) (6.125) (5.875) (5.50) (5.000) 209.375 200.00 193.750 187.500 184.375 178.125 175.000 162.500 156.250 140.625

Maximum pitch of staybolts, mm (in)

12.500

Thickness of plate, mm (in) (0.4375)

TABLE 9-7 Maximum allowable pitch for screwed staybolts, ends riveted over

(8.375) (8.000) (7.75) (7.500) (7.375) (7.125) (7.000) (6.50) (6.25) (5.625)

(0.5625)

209.375 203.125 196.875 193.750 181.250 171.875 156.250

15.6250

(8.375) (8.125) (7.875) (7.750) (7.25) (6.875) (6.25)

(0.625)

212.500 200.000 175.625 175.00

17.1875

(8.50) (8.00) (7.625) (7.000)

(0.6875)

DESIGN OF POWER BOILERS

9.10

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DESIGN OF POWER BOILERS DESIGN OF POWER BOILERS

Particular

9.11

Formula

D ¼ least horizontal distance between tube centers on a horizontal row, in di ¼ inside diameter of tube, in P ¼ maximum allowable working pressure, psi P ¼ 186

hðD di Þ wD

SI

ð9-19bÞ

where p in MPa; h, D, di , and w in m The vertical distance between the center lines of tubes in adjacent rows where tubes are staggered

Dva ¼ ð2di D þ di2 Þ1=2

ð9-20Þ

where di and D have the same meaning as given under Eq. (9-19) For minimum thickness of shell plates, dome plates, and tube plates and tube sheet for ﬁretube boiler

Refer to Table 9-8

For mechanical properties of steel plates of boiler

Refer to Table 9-9

TABLE 9-8 Minimum thickness of shell plates, dome plates, and tube sheet for ﬁretube boiler Diameter of Shell and dome plates

Minithickness Tube sheet

Shell and dome plates

Tube sheet

m

in

m

in

mm

in

mm

in

0.9 >0.9–1.35 >1.35—1.8 >1.8

36 >36–54 >54–72 >72

1.05 >1.05–1.35 >1.35–1.8 >1.8

42 >42–54 >54–72 >72

6.25 7.81 9.375 12.5

0.25 0.3125 0.375 0.50

9.375 10.94 12.5 14.06

0.375 0.4375 0.500 0.5625

Source: ASME Power Boiler Code, Section I, 1983.

TABLE 9-9 Mechanical properties of steel plates for boilers Tensile strength Grade

MPa

kpsi

Yield stress, percent min of tensile strength

1 2A 2B

333.4–411.9 362.8–480.5 509.9–608.0

48.4–59.7 52.6–69.7 74.0–88.2

55 50 50

Elongation percent gauge length, pﬃﬃﬃﬃ ﬃ 5.65 a a 26 25 20

a area of cross section. Source: IS 2002-1, 1962.

a

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DESIGN OF POWER BOILERS

9.12

CHAPTER NINE

Particular

Formula

Plain circular furnaces FURNACES 300 mm (12 in) TO 450 mm (18 in) OUTSIDE DIAMETER, INCLUSIVE Maximum allowable working pressure for furnaces not more than 412 diameters in length or height where the length does not exceed 120 times the thickness of the plate

p¼

0:36ð18:75T 1:03LÞ D

SI

ð9-21aÞ

USCS

ð9-21bÞ

where p in MPa; T, D, and L in m p¼

51:5ð18:75T 1:03LÞ D

where p in psi D ¼ outside diameter of furnace, in L ¼ total length of furnace between centers of head rivet seams, in T ¼ thickness of furnace walls, sixteenth of an inch The maximum allowable working pressure for furnaces not more than 412 diameter in length of height where the length exceeds 120 times the thickness of the plate

p¼

29:3T 2 LD

SI

ð9-22aÞ

USCS

ð9-22bÞ

SI

ð9-23aÞ

USCS

ð9-23bÞ

SI

ð9-24aÞ

USCS

ð9-24bÞ

where p in MPa; T, L, and D in m p¼

4250T 2 LD

where p in psi; T, L, and D in in

Circular ﬂues The maximum allowable external pressure for riveted ﬂues over 150 mm (6 in) and not exceeding 450 mm (18 in) external diameter, constructed of iron or steel plate not less than 6 mm (0.25 in) thick and put together in sections not less than 600 mm (24 in) in length

p¼

56h d

where p in Pa; h and d in m p¼

8100h d

where p in psi; h and d in in d ¼ external diameter of ﬂue, in The formula for maximum allowable external pressure for riveted, seamless, or lap-welded ﬂues over 450 mm (18 in) and not exceeding 700 mm (28 in) external diameter, riveted together in sections not less than 600 mm (24 in) nor more than 312 times the ﬂue diameter in length, and subjected to external pressure only

p¼

6:7h 0:4l d

where p in Pa; h, l, and d in m p¼

966h 53l d

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DESIGN OF POWER BOILERS DESIGN OF POWER BOILERS

Particular

9.13

Formula

where p in psi and d in in h ¼ thickness of wall in 1.5 mm (0.06 in) l > 600 mm (24 in) and <3 12 d The maximum allowable working pressure for seamless or welded ﬂues over 125 mm (5 in) in diameter and including 450 mm (18 in) (a) Where the thickness of the wall is not greater than 0.023 times the diameter as per ASME Power Boiler Code

p¼

68948h3 D3

SI

ð9-25aÞ

where p in MPa; h and D in m p ¼ maximum allowable working pressure D ¼ outside diameter of ﬂue h ¼ thickness of wall of ﬂue p¼

107 h3 D3

USCS

ð9-25bÞ

SI

ð9-26aÞ

USCS

ð9-26bÞ

SI

ð9-27aÞ

USCS

ð9-27bÞ

where p in psi; h and D in in (b) Where the thickness of the wall is greater than 0.023 times the diameter.

p¼

119h 1:9 D

where p in MPa; h and D in m p¼

17300h 275 D

where p in psi; h and D in in Equations (9-24) and (9-25) may applied to riveted ﬂues of the size speciﬁed provided the section are not over 0.91 m (3 ft) in length and the eﬃciency () of the joint

<j

pD 138h

where p in MPa; D and h in m <j

pD 20000h

where p in psi; D and h in in

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DESIGN OF POWER BOILERS

9.14

CHAPTER NINE

Particular

Formula

THE MAXIMUM ALLOWABLE PRESSURE FOR SPECIAL FURNACES HAVING WALLS REINFORCED BY RIBS, RINGS, AND CORRUGATIONS (a) Furnaces reinforced by Adamson rings

p¼

6:7h 0:4l d

SI

ð9-28aÞ

USCS

ð9-28bÞ

where p in Pa; h and d in m p¼

1080h 59l d

where p in psi h ¼ thickness of wall, 1.5 mm (0.06 in) not to be less 5 than 8 mm (16 in) l ¼ length of ﬂue section, not to be less than 450 mm (18 in) (b) Another expression for the maximum allowable working pressure when plain horizontal ﬂues are made in sections not less than 450 mm 5 (18 in) in length and not less than 8 mm (16 in) in thickness (Adamson-type rings)

p¼

0:4ð300h 1:03LÞ D

SI

ð9-29aÞ

USCS

ð9-29bÞ

SI

ð9-30aÞ

USCS

ð9-30bÞ

where p in MPa; h, L, and D in m p¼

57:6ð300h 1:03LÞ D

where p in psi; h, L, and D in in (c) Corrugated rings

p ¼ 68:5

h d

where p in Pa; h and d in m p ¼ 10000

h d

where p in psi; h and d in in h ¼ thickness of tube wall, mm (in), not to be less than 11 mm (0.44 in) (d) Plain circular ﬂues riveted together in sections

p¼

6:7d 0:4l d

SI

ð9-31aÞ

USCS

ð9-31bÞ

where p in Pa; d and l in m p¼

966h 53l D

where p in psi; l and d in in

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DESIGN OF POWER BOILERS DESIGN OF POWER BOILERS

Particular

9.15

Formula

Ring-reinforced type The required wall thickness of a ring-reinforced furnace of ﬂue shall not be less than that determined by the procedure given here

Assume a value for h (or t) and L. Determine the ratios L=Do and Do =t. Following the procedure explained in Chap. 8, determine B by using Fig. 9-1. Compute the allowable working pressure Pa by the help of Eq. (9-32)

The allowable working pressure (Pa )

Pa ¼

B ðDo =tÞ

ð9-32Þ

where Do ¼ outside diameter of furnace or ﬂue, in Compare Pa with P. If Pa is less than P select greater value of t (or h) or smaller value of L so that Pa is equal to or greater than P, psi The required moment of inertia (Is ) of circumferential stiﬀening ring

A LD2o t þ s A L Is ¼ 14

ð9-33Þ

where Is ¼ required moment of inertia of stiﬀening ring about its neutral axis parallel to the axis of the furnace, in4 As ¼ area of cross section of the stiﬀening ring, in2 A ¼ factor obtained from Fig. 9-1 The required moment of inertia of a stiﬀening ring shall be determined by the procedure given here

The expression for B

Assume the values of Do , L, and t (or h) of furnace. Select a rectangular member to be used for stiﬀening ring and ﬁnd its area As and its moment of inertia I. Then ﬁnd the value of B from Eq. (9-34) B¼

PDo t þ As =L

ð9-34Þ

where P, Do , t, As , and L are as deﬁned under Eq. (9-33) The value of factor A

After computing B from Eq. (9-34), determine the value of factor A by the help of Fig. 9-1 and B. If the required Is is greater than the moment of inertia I, for the section selected above, select a new section with a larger moment of inertia and determine a new value of Is . If the required Is is smaller than the moment of inertia I selected as above, then that section should be satisfactory.

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DESIGN OF POWER BOILERS

9.16

CHAPTER NINE

FIGURE 9-1 Chart for determining wall thicknesses of ring reinforced furnaces when constructed of carbon steel (speciﬁed yield strength, 210 to 262 MPa (30 to 38 kpsi) (1 kpsi ¼ 6.894757 MPa). (Source: ‘‘Rules for Construction of Power Boilers,’’ ASME Boiler and Pressure Vessel Code, Section I, 1983 and ‘‘Rules for Construction of Pressure Vessels,’’ Section VIII, Division 1, ASME Boiler and Pressure Vessel Code, July 1, 1986.)1;2

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DESIGN OF POWER BOILERS DESIGN OF POWER BOILERS

Particular

9.17

Formula

Corrugated furnaces The maximum allowable working pressure (P) on corrugated furnace having plain portion at the ends not exceeding 225 mm (9 in) in length

tC6 D where

P¼

ð9-35Þ

5 in) for t ¼ thickness, in, not less than 7.8 mm (16 Leeds, Morison, Fox and Brown, and not less 7 than 11 mm (16 in) for Purves and other furnaces corrugated by sections not over 450 mm (18 in) long. D ¼ mean diameter, in Values of C6 are taken from Table 9-10

TABLE 9-10 Values of C6 for use in Eq. (9–35) C6 1. 2. 3. 4. 5. 6.

For Leeds furnaces, when corrugations are not more than 200 mm (8 in) from center and not less than 56.25 mm (2.25 in) deep For Morison furnaces, when corrugations are not more than 200 mm (8 in) from center to center and the radius of the outer corrugation is not more than one-half of the suspension curve For Fox furnaces, when corrugations are not more than 200 mm (8 in) from center to center and not less than 37.5 mm (1.5 in) deep For Purves furnaces, when rib projections are not more than 225 mm (9 in) from center to center and not less than 34.375 mm (1.375 in) deep For Brown furnaces, when corrugations are not more than 225 mm (9 in) from center to center and not less than 40.625 mm (1.625 in) deep For furnaces corrugated by sections not more than 450 mm (18 in) from center to center and not less than 37.5 mm (1.5 in) deep, measured from the least inside greatest outside diameter of the corrugations and having the ends ﬁtted into the other and substantially riveted together, provided the plain parts at the ends do not exceed 300 mm (12 in) in length

17,300 15,600 14,000 14,000 14,000

Source: ASME Power Boiler Code, Section I, 1983.

Stayed surfaces The maximum allowable working pressure (P) for a stayed wrapper sheet of a locomotive-type boiler

P¼

11000t P R s sin

USCS

ð9-36aÞ

where t ¼ thickness of wrapper sheet, in R ¼ radius of wrapper sheet, in ¼ minimum eﬃciency of wrapper sheet through joints or stay holes

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DESIGN OF POWER BOILERS

9.18

CHAPTER NINE

Particular

Formula

P

s sin ¼ summated value of transverse spacing (s sin ) for all crown stays considered in one transverse plane and on one side of the vertical axis of the boiler s ¼ transverse spacing of crown stays in the crown sheet, in ¼ angle any crown stay makes with the vertical axis of boiler P¼

The longitudinal pitch between stay bolts or between the nearest row of stay bolts and the row of rivets at the joints between the furnace sheet and the tube sheet or the furnace sheet and the mud ring

R

76t P s sin

SI

where P in MPa; s, t, and R in m 56320t2 2 L¼ USCS PR

ð9-36bÞ

ð9-37aÞ

where t ¼ thickness of furnace sheet, in R ¼ outside radius of furnace, in P ¼ maximum allowable working pressure, psi 2:535 109 t2 2 L¼ SI ð9-37bÞ PR where P in Pa; t, L, and R in m

Cross-sectional area of diagonal stay (A)

A¼

aL l

ð9-38Þ

where a ¼ sectional area of direct stay, m (in) L ¼ length of diagonal stay, m (in) l ¼ length of line drawn at right angles to boiler head or a projection of L on a horizontal surface parallel to boiler drum, m (in) The total cross-sectional area of stay tubes which support the tube plates in multitubular boilers

At ¼

ðA aÞP sa

ð9-39Þ

where A ¼ area of that portion of tuber plate containing the tubes, m (in) a ¼ aggregate area of holes in the tube plate, m2 (in2 ) P ¼ maximum allowable working pressure, Pa (psi) sa ¼ maximum allowable stress value in the tubes, MPa (psi) >48 j MPa (7 kpsi) sa is also taken from Table 8-23 The pitch of stay tubes shall conform to Eqs. (9-17) and (9-18) and using the values of C7 as given in Table 9-11

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DESIGN OF POWER BOILERS DESIGN OF POWER BOILERS

Particular

The pitch from the stay bolt next to the corner to the point of tangency to the corner curve for stays at the upper corners of ﬁre boxes shall be as given in Eq. (9-40)

9.19

Formula

p¼

90½C7 ðT 2 =PÞ1=2 USCS angularity of tangent lines ðÞ

ð9-40aÞ

where T ¼ thickness of plate in sixteenths of an inch P ¼ maximum allowable working pressure, psi C7 ¼ factor for the thickness of plate and type of stay used pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C7 ðT 2 =pÞ pt ¼ 7592 angularity of tangent lines ðÞ SI ð9-40bÞ where pt and T in m, and p in Pa

For various values of C7

Refer to Table 9-11

TABLE 9-11 Values of C7 for determining pitch of stay tubes

Pitch of stay tubes in the bounding rows Where there are two plain tubes between two stay tubes Where there is one plain tube between two stay tubes Where every tube in the bounding rows is a stay tube and each alternate tube has a nut

When tubes have nuts not outside of plates

When tubes are ﬁtted with nuts outside of plates

120 140 —

130 150 170

Source: ASME Power Boiler Code, Section I, 1983.

FINAL RATIOS1 Design of a horizontal return tubular boiler

H ranges from 35 to 45 in firetube boilers; G 37 is a good working value ð9-41aÞ S lines between 12 and W cylindrical boilers C varies from G

1 6

to

1 3

for most types of

1 8

ð9-41bÞ ð9-41cÞ

S ¼ 16:7 103 ð0:6Þ to 19:5 103 ð0:7Þ ð9-41dÞ P H 0:92 to 1:12 m2 (10 to 12 ft2 ) for ¼ P externally fired boiler per hp ¼ 0:74 m2 (8 ft2 ) for Scotch boiler per hp ð9-41eÞ

The units in parentheses are in US Customary System units.

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DESIGN OF POWER BOILERS

9.20

CHAPTER NINE

Particular

Design of a vertical straight shell multitubular boiler

Formula

P ¼ 53 ð5:22Þ G

ð9-41f Þ

DS ¼ 64 103 to 73 103 N

ð9-41gÞ

H ¼ 60 G

ð9-42aÞ

WHS ¼ 45 G

ð9-42bÞ

SHS 1 ¼ WHS 3

ð9-42cÞ

S 1 ¼ W 3

ð9-42dÞ

S ¼ 22:3 103 ð0:80Þ P

ð9-42eÞ

A ¼ Total area of steam segment D ¼ Diameter of shell or drum h ¼ Height of the segment to be occupied by steam

FIGURE 9-2 Disengaging surface in horizontal cylindrical shell. (Source: Reproduced from G. B. Haven and G. W. Swett, The Design of Steam Boilers and Pressure Vessels, John Wiley and Sons, Inc., 1923.)1

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DESIGN OF POWER BOILERS DESIGN OF POWER BOILERS

9.21

FIGURE 9-3 Areas of circular segments. (Reproduced from G. B. Haven and G. W. Swett, The Design of Steam Boilers and Pressure Vessels, John Wiley and Sons, Inc., 1923.)1

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DESIGN OF POWER BOILERS

9.22

CHAPTER NINE

Particular

Formula

Watertube boiler design

For mechanical properties of carbon and carbon manganese steel plates, sections and angles for marine boilers pressure vessels and welded machinery and mechanical properties of steel plates for boilers

C 1 ¼ G 5:5

ð9-42f Þ

H ¼ 1:12 ð12Þ P

ð9-42gÞ

G P ¼ 18:3 103 ð20Þ or ¼ 51 ð5Þ P G

ð9-42hÞ

P ¼ 51 ð5Þ DS

ð9-42iÞ

H ¼ 50 G

ð9-43aÞ

S ¼ 11:2 103 ð0:424Þ p

ð9-43bÞ

H ¼ 0:92 ð10Þ P

ð9-43cÞ

P ¼ 51 ð4:37Þ G

ð9-43dÞ

DS ¼ 27:5 103 ð0:308Þ P Refer to Table 9-12

ð9-43eÞ

For properties of boilers

Refer to Table 9-13

For evaporation of water, average rate of combustion of fuels, and minimum rate of steam produced

Refer to Tables 9-14 to 9-16

TABLE 9-12 Mechanical properties of carbon and carbon manganese steel plates, sections, and angles steel for marine boilers, pressure vessels, and welded machinery Elongation percentage min on gauge length

Tensile strength Grade

MPa

kpsi

pﬃﬃﬃﬃﬃ 5.65 a a

1 2 3 4 5

362.8–441.3 411.9–490.3 431.5–529.6 460.9–559.0 490.3–588.4

52.6–64.0 59.7–71.1 62.6–76.8 66.8–81.0 71.1–85.3

26 25 23 22 21

200 mm

Bond test diameter of former

25 23 21 20 19

2t 2t 3t 3t 312 t

a

Area of cross section. Source: IS 3503, 1966.

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DESIGN OF POWER BOILERS DESIGN OF POWER BOILERS

9.23

TABLE 9-13 Properties of boilers Horizontal return tubular boilers Diameter of shell, mm 910 1070 1220 1370 1520 1680 1830 1980 2130 2290 2440

Diameter of tubes, mm 64 64 64 76 76 76 76 76 76 76 76

76 76 76 89 89 89 89 89 89 89 89

Length of tubes, m 2.44 3.05 3.66 3.97 4.28 4.88 4.88 4.88 4.88 4.88 4.88

102 102 102 102

3.66 3.66 4.28 4.28 4.88 5.18 5.50 5.18 5.50 5.18 5.18

4.28 4.58 4.58 5.50 5.50 6.10 5.50 6.10 5.50 5.50

4.88 4.88

5.5

6.10 6.10 6.10 6.10

Dry-back scotch boilers

Diameter of shell, m

Diameter of tubes, mm

Length of tubes, m

Inside diameter of furnace, mm

Length of grate, m

920 920 970 1150 1270 970 1150 1270

1.22 1.53 1.93 2.03 2.21 1.93 2.03 2.21

970 1040 1140 970 1040 1140

1.93 2.24 2.44 1.93 2.24 2.44

Short Types 1.19 1.99 2.14 2.29 2.44 2.90 3.20 3.50

76 89 89 89 89 89 89 89

2.90 3.50 3.80 3.80 3.97 3.80 3.80 3.89

2.06 2.21 2.36 2.84 3.05 3.28

102 102 102 102 102 102

4.88 4.88 4.88 4.88 4.88 4.88

Long Types

Locomotive-type boilers without dome

Vertical ﬁretube boiler for power plant use

Dimensions of grate Diameter of waist, mm

Length of 7.5 mm tubes, m Width, mm

Length, m

Diameter of tubes, mm

Length of tubes, m

925 1070 1220 1370 1520 1680 1830

2.14 2.44 3.20 3.36 3.97 4.58 4.58

1.22 1.27 1.38 1.53 1.53 1.68 1.83

50 62

3.96 4.27 4.57 4.88

0.76 0.92 1.07 1.22 1.38 1.53 1.68

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DESIGN OF POWER BOILERS

9.24

CHAPTER NINE

Particular

Formula

For permissible strain rates of steam plant equipments

Refer to Table 9-17

For water level requirements of boilers

Refer to Table 9-18

For minimum allowable thickness of plates for boilers

Refer to Table 9-19

For disengaging surface per horsepower

Refer to Table 9-20

For heating boiler eﬃciency

Refer to Table 9-21

TABLE 9-14 Evaporation kg (lb) of water per kg (lb) of fuel reduced to standard condition [from and at 373 K (1008C)] Approximate Type of fuel

kJ/kg

Btu/lb

Evaporation per kg (lb) of fuel, kg (lb)

Anthracite Coke Semibituminous Bituminous Lignite Fuel oil

29,038.3–27842.2 30,228.7 33,703.7 29,098.3 22,106.3 41,868.0

12,500–1 2000 13,000 14,500 12,500 9,500 19,000

9.5–9 9.5 10 9 6 14.5

TABLE 9-15 Average rates of combustion [kg/m2 (lb/ft2 ) of grate surface per hour] draft 12.55 mm (12 in) water column Fuel used

Stationary grate

Anthracite Semibituminous Bituminous Lignite

44–68.5 (9.14) 98 (20) 68.5 (14) 58.5 (12)

TABLE 9-16 Minimum kilograms (pounds) of steam per h per ft2 of surface Firetube boilers Particulars

kg

Boiler heating surface Hand-ﬁred Stoker-ﬁred Oil-, gas-, or powder-ﬁred

11.0 15.4 17.6

Water wall heating surface Hand-ﬁred Stoker-ﬁred Oil-, gas-, or powder-ﬁred

17.6 22.1 30.9

lb

Watertube boilers kg

lb

5 7 8

13.2 17.6 22.1

6 8 10

8 10 14

17.6 26.5 35.3

8 12 16

Source: ASME Power Boiler Code. Section I, 1983.

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DESIGN OF POWER BOILERS DESIGN OF POWER BOILERS

TABLE 9-17 Permissible strain rates for steam plant equipment

Machine part

Strain rate per hour

Turbine disk (pressed on shaft) Bolted ﬂanges, turbine cylinders Steam piping, welded joints, and boiler tubes Superheated tubes

109 108 107 106

TABLE 9-18 Water level requirementsa Horizontal return tubular boilers Distance between gauge cocks, mm

Boiler diameters, mm 910, 1070, 1220 1370, 1520 1680, 1830, 1980, 2130

75 100 125

Vertical ﬁretube boilers

Boiler diameters mm

Distance between gauge cocks, mm

910–1220 1250–1680 1700–2410 2460–3100

100 125 150 175

Dry-back Scotch boilers

Locomotive-type boilers

Low water level 89 mm above surface of tubes for all diameters: distance between gauge cocks may be reduced to a minimum of 75 mm

Low water level must be 75–125 mm above the water surface of the crown sheet; distance between gauge cocks is usually 75 mm for all diameters

a

Low water level 890 mm above surface of tubes.

TABLE 9-19 Minimum allowable thickness of plates for boilers (all dimensions in mm) Power boilers

Minimum thickness 6.5 8.0 9.5 11.0 12.5 14.5

Shell and dome plate diameter 910 >910–1370 >1370–1830 >1830

Tube sheet diameter

1065 >1065–1370 >1370–1830 >1830

9.25

Heating boilers Shell or other plate diameter 1065 >1065–1530 >1530–1980 >1980

Tube sheet or head diameter 1065 >1065–1530 >1530–1980 >1980

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DESIGN OF POWER BOILERS

9.26

CHAPTER NINE

TABLE 9-20 Disengaging surface per horsepower mean water level Disengaging surface Type of boiler Horizontal return tubular Dry-back Scotch Vertical straight shell Vertical (Manning) Locomotive type Sectional water tube

m2 /kW

m2 /hp

0.087–0.10 0.075–0.087

0.065–0.0745 0.056–0.0650

0.020–0.025 0.011–0.013 0.100–0.125

0.0149–0.0186 0.0084–0.0093 0.0745–0.093

0.037–0.0500

0.0279–0.0372

TABLE 9-21 Heating boiler eﬃciency Firing method Hand-Fired Coal Lignite Subbituminous Bituminous Low-volatile bituminous Anthracite Coke Stoker Conversion Bituminous Anthracite Burner Conversion Natural gas Oil Designed for Burner Stoker 45 kg >45 kg Gas Oil Cast-iron boilers Steel boilers Package units

Eﬃciency, %

49 44–63 50–65 44–61 60–75 75–76 55–69 63 69–76 51; 65; 70 60–75 65 70 70–80 70–80 68 70 75

REFERENCES 1. Haven, G. B., and G. W. Swett, The Design of Steam Boilers and Pressure Vessels, John Wiley and Sons, Inc., New York, 1923. 2. ‘‘Rules for Construction of Power Boilers,’’ ASME Boiler and Pressure Vessel Code, Section I, 1983. 3. ‘‘Rules for Construction of Pressure Vessels, ’’ ASME Boiler and Pressure Vessel Code, Section VIII, Division I, July 1, 1986. 4. Code of Unﬁred Pressure Vessels, Bureau of Indian Standards, IS 2825, 1969, New Delhi, India. 5. Nichols, R. W., Pressure Vessel Codes and Standards, Elsevier Applied Science Publishing Ltd., Barking, Essex, England, 1987. 6. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962. 7. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 8. Lingaiah, K., Machine Design Data Handbook, (SI and U.S. Customary Units), McGraw-Hill Publishing Company, New York, 1994.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

10 ROTATING DISKS AND 1 CYLINDERS SYMBOLS1 g r ri ro h h2 r z !

acceleration due to gravity, m/s2 (ft/s2 ) any radius, m (in) inside radius, m (in) outside radius, m (in) thickness of disk at radius r from the center of rotation, m (in) thickness of disk at radius r2 from the center of rotation, m (in) uniform tensile stress in case of a disk of uniform strength, MPa (psi) tangential stress, MPa (psi) radial stress, MPa (psi) axial stress or longitudinal stress, MPa (psi) density of material of the disk, kg/m3 (lbm /in3 ) angular speed of disk, rad/s Poisson’s ratio

Particular

DISK OF UNIFORM STRENGTH ROTATING AT ! rad=s (Fig. 10-1) The thickness of a disk of uniform strength at radius r from center of rotation

Formula

2 ! 2 2 h ¼ h2 exp ðr r Þ 2 2

ð10-1Þ

SOLID DISK ROTATING AT ! rad=s The general expression for the radial stress of a rotating disk of uniform thickness

r ¼

3þ 2 2 ! ðro r2 Þ 8

ð10-2Þ

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ROTATING DISKS AND CYLINDERS

10.2

CHAPTER TEN

Particular

Formula

FIGURE 10-1 High-speed rotating disk of uniform strength.

FIGURE 10-2 Rotating disk of uniform thickness.

The general expression for the tangential stress of a rotating disk of uniform thickness

¼

The maximum values of stresses are at the center, where r ¼ 0, and are equal to each other

rðmaxÞ ¼ ðmaxÞ ¼

3þ 1 þ 3 2 !2 r2o r 8 3þ 3þ !2 r2o 8

ð10-3Þ

ð10-4Þ

HOLLOW DISK ROTATING AT ! rad=s (Fig. 10-2) The general expression for the radial stress of a rotating disk of uniform thickness

3þ 2 2 r2o r2i 2 2 ! ri þ ro 2 r r ¼ 8 r

The general expression for the tangential stress of a rotating disk of uniform thickness

¼

The maximum radial stress occurs at r2 ¼ ro ri

3þ r2 r2 1 þ 3 2 !2 r2i þ r2o þ o 2 i r 8 3þ r

rðmaxÞ ¼

3þ 2 ! ðro ri Þ2 8

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ð10-5Þ

ð10-6Þ

ð10-7Þ

ROTATING DISKS AND CYLINDERS ROTATING DISKS AND CYLINDERS

Particular

The maximum tangential stress occurs at inner boundary where r ¼ ri

10.3

Formula

3þ 2 2 1 2 ! ro þ ri 4 3þ

ð10-8Þ

!2 ½ð3 2Þr2o ð1 þ 2Þr2 8ð1 Þ

ð10-9Þ

ðmaxÞ ¼

SOLID CYLINDER ROTATING AT ! rad=s The tangential stress

¼

The radial stress

!2 r ¼ 8

The maximum stress occurs at the center

The axial strain in the z direction (ends free)

3 2 ðr2o r2 Þ 1

rðmaxÞ ¼ ðmaxÞ ¼

!2 8

ð10-10Þ

3 2 2 ro 1

ð10-10aÞ

"z ¼

!2 r2o 2 E

The axial stress under plane strain condition (ends free)

z ¼

!2 4

ðr2o 2r2 Þ 1

ð10-12aÞ

The axial stress under plane strain condition (ends constrained)

z ¼

!2 1 ð3 2Þr2o 2r2 4ð1 Þ 2

ð10-12bÞ

ð10-11Þ

HOLLOW CYLINDER ROTATING AT ! rad=s The tangential stress at any radius r

!2 ¼ 8

3 2 1

" r2i

þ

r2o

r2 r2 þ i 2o r

# 1 þ 2 2 r 3 2 ð10-13Þ

The radial stress at any radius r

The axial stress (ends free) at any radius r

The axial stress under plane strain conditions (ends constrained) at any radius r

The maximum stress occurs at the inner surface where r ¼ ri

!2 r ¼ 8

z ¼

!2 4

z ¼

!2 4

ðmaxÞ

3 2 1

r2i

þ

r2o

r2 r2 i 2 o r2 r

½r2i þ r2o 2r2 1

3 2 1

!2 ¼ 4

2r2 r2i þ r2o 3 2

3 2 1

ð10-14Þ

ð10-15Þ

ð10-16Þ

" # 1 2 2 2 ð10-17Þ ro þ r 3 2 i

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ROTATING DISKS AND CYLINDERS

10.4

CHAPTER TEN

Particular

The axial strain in the z direction (ends free)

The displacement u at any radius r of a thin hollow rotating disk

Formula

"z ¼ " u¼

!2 2 ðr þ r2o Þ 2E i

!2 r ð3 þ Þð1 Þ E 8

r2o

þ

1 þ r2o r2i 1 þ 2 þ r 1 r2 3þ

r2i

SOLID THIN UNIFORM DISK ROTATING AT ! rad=s UNDER EXTERNAL PRESSURE po (Fig. 10-3)

The radial stress at any radius r

r ¼ po þ !2

The tangential stress at any radius r

¼ po þ !2

The maximum radial stress at r ¼ 0

ð10-18Þ

# ð10-19Þ

3þ ðr2o r2 Þ 8 3þ 8

rðmaxÞ ¼ po þ !2

r2o

1 þ 3 2 r 3þ

ð10-20Þ

3þ 2 ro 8

ð10-21Þ

ð10-22Þ

The maximum radial stress at r ¼ ro

r ¼ po

ð10-23Þ

The maximum tangential stress at r ¼ 0

ðmaxÞ ¼ rðmaxÞ

ð10-24Þ

The displacement u at any radius r

u¼

r !2 ½ð3 þ Þr2o ð1 þ Þr2 ð1 Þ po þ 8 E ð10-25Þ

FIGURE 10-3 Rotating disk of uniform thickness under external pressure.

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ROTATING DISKS AND CYLINDERS ROTATING DISKS AND CYLINDERS

Particular

10.5

Formula

HOLLOW CYLINDER OF UNIFORM THICKNESS ROTATING AT ! rad=s. SUBJECT TO INTERNAL ( pi ) AND EXTERNAL ( po ) PRESSURES (Fig. 10-4) The general expression for the radial stress of a hollow cylinder of uniform thickness rotating at ! rad/s under internal ð pi Þ and external ð po Þ pressure at any radius r

The general expression for the tangential or hoop stress of a hollow cylinder of uniform thickness rotating at ! rad/s under internal ð pi Þ and external ð po Þ pressure at any radius r.

B !2 þ 8 r2

r ¼ A

r2i

3 2 1

r2 r2 i 2 o r2 r

ð10-26Þ

3 2 1 " # r2i r2o 1 þ 2 2 2 2 ri þ ro þ 2 r 3 2 r

¼ A þ

B !2 þ 8 r2

where The tangential or hoop stress in a hollow cylinder rotating at ! rad/s under po and pi at r ¼ ri (Fig. 10-4)

þ

r2o

ð

maxÞr ¼ ri

A¼ ¼

pi r2i po r2o ; r2o r2i

B¼

ð10-27Þ

r2i r2o ð pi po Þ r2o r2i

pi ðr2i þ r2o Þ 2po r2o r2o r2i " # !2 3 2 2 4 2 þ 2r2o þ r 8 1 3 2 i ð10-28aÞ

¼

pi ðr2i

þ r2o Þ r2o r2i

!2 þ 4

2po r2o

3 2 1

" # 1 2 2 2 ro þ r 3 2 i ð10-28bÞ

FIGURE 10-4

FIGURE 10-5

FIGURE 10-6

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ROTATING DISKS AND CYLINDERS

10.6

CHAPTER TEN

Particular

The tangential or hoop stress in a hollow cylinder rotating at ! rad/s under po and pi at r ¼ ro (Fig. 10-4)

Formula

ð

maxÞr ¼ ro

¼

2pi r2i po ðr2o þ r2i Þ r2o r2i !2 3 2 1 2 2 ro þ r2i þ 1 3 2 4 ð10-29Þ

The tangential stress in a cylinder rotating at ! rad/s at any radius r when subjected to internal pressure ð pi Þ only (Fig. 10-5)

ð Þpo ¼ 0 ¼

The tangential stress in a cylinder rotating at ! rad/s at any radius r when subject to external pressure ð po Þ only (Fig. 10-6)

ð Þpi ¼ 0 ¼

pi r2i ðr2o þ r2 Þ !2 3 2 þ 4 1 r2 ðr2o r2i Þ " # r2i r2o 1 þ 2 2 2 2 r ð10-30Þ ri þ ro þ 2 3 2 r po r2o ðr2 þ r2i Þ !2 3 2 þ 1 4 r2 ðr2o r2i Þ " # 2 2 ri ro 1 þ 2 2 2 2 r i þ ro þ 2 r ð10-31Þ 3 2 r

ROTATING THICK DISK AND CYLINDER WITH UNIFORM THICKNESS SUBJECT TO THERMAL STRESSES The hoop or tangential stress in thick disk or cylinder at any radius r rotating at ! rad/s subject to pressure po and pi

The radial stress in thick disk or cylinder at any radius r rotating at ! rad/s subject to pressure po and pi

" # B !2 2 1 þ 3 2 ro ¼ A þ 2 ð3 þ Þ r 3þ 8 r ð E ð10-32Þ ET þ 2 Tr dr r r ¼ A

ð B !2 E 2 2 ð3 þ Þðr Tr dr r Þ o 8 r2 r2 ð10-33Þ

where A and B are Lame´’s constants and can be found from boundary or initial conditions ¼ linear coeﬃcient of thermal expansion, mm/8C (in/8F) T ¼ temperature, 8C or K (8F) ¼ density of rotating cylinder or disk material, kg/m3 (lbm /in3 ) E ¼ modulus of material of disk or cylinder, GPa (Mpsi)

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ROTATING DISKS AND CYLINDERS ROTATING DISKS AND CYLINDERS

Particular

10.7

Formula

ROTATING LONG HOLLOW CYLINDER WITH UNIFORM THICKNESS ROTATING AT ! rad=s SUBJECT TO THERMAL STRESS The general expression for the radial stress in the cylinder wall at any radius r when the temperature distribution is symmetrical with respect to the axis and constant along its length.

r ¼

3 2 r2 r2 r2i þ r2o i 2 o r2 1 r " # ð ð ro E 4r2 di2 ro Tr dr Tr dr þ ð1 Þr2 do2 di2 ri ri

!2 8

ð10-34Þ The general expression for the tangential stress in the cylinder wall at any radius r when the temperature distribution is symmetrical with respect to the axis and constant along its length.

" # 3 2 r2i r2o 1 þ 2 2 2 2 ri þ ro þ 2 r 1 3 2 r 2 ð E 4r þ di2 ro Tr dr þ ð1 Þr2 do2 di2 ri ð ro Tr dr Tr2 ð10-35Þ þ

!2 ¼ 8

ri

The general expression for the axial stress in the cylinder wall at any radius r when the temperature distribution is symmetrical with respect to the axis and constant along its length.

¼

½r2i þ r2o 2r2 1 ð ro E 8 Tr dr T þ 1 do2 di2 ri

!2 4

ð10-36Þ

where do ¼ 2ro and di ¼ 2ri

DEFLECTION OF A ROTATING DISK OF UNIFORM THICKNESS IN RADIAL DIRECTION WITH A CENTRAL CIRCULAR CUTOUT E h

The tangential stress within elastic limit, , in a rotating disk of uniform thickness (Fig. 10-7)

¼

The expression for the inner deﬂection i , of rotating thin uniform thickness disk with centrally located circular cut-out as per Stodalaa (Fig. 10-7)

i ¼ 3:077 106

ð10-37Þ

n 1000

2 ð7:5K 2 þ 5Þ

a

ð10-38Þ

Source: Stodala ‘‘Turbo-blower and compressor’’; Kearton, W. J. and Porter, L. M., Design Engineer, Pratt and Whitney Aircraft; McGraw-Hill Publishing Company, New York, U.S.A. Douglas C. Greenwood, Editor, Engineering Data for Product Design, McGraw-Hill Publishing Company, New York, 1961.

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ROTATING DISKS AND CYLINDERS

10.8

CHAPTER TEN

FIGURE 10-7 Nomogram for radial deﬂection of rotating disks with constant thickness with a centrally located circular hole.

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ROTATING DISKS AND CYLINDERS ROTATING DISKS AND CYLINDERS

Particular

The expression for the outer deﬂection o of rotating thin uniform thickness disk with centrally located circular cut-out as per Stodalaa (Fig. 10-7)

10.9

Formula

o ¼ 3:077 106

n 1000

2 ð1:5K 2 þ 7:5KÞ

ð10-39Þ

where K ¼ ro =ri ¼ tangential stress, psi ¼ i þ o ¼ total deﬂection of disk, in ri ¼ inner radius of disk, in ro ¼ outer radius of disk, in n ¼ speed, rpm The Nomogram can be used for steel, magnesium and aluminum since the modulus of elasticity E ¼ 29 106 psi (200 MPa) for steel and Poisson’s ratio ¼ 1=3. The error involved in using this equation with E and of steel for aluminum is about 0.5% and for magnesium is 2.5%.

REFERENCES 1. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Volume I (SI and Customary Metric Units), Suma Publishers, Bangalore, 1986. 2. Lingaiah, K., Machine Design Data Handbook, McGraw-Hill Publishing Company, New York, 1994. 3. Douglas C. Greenwood, Engineering Data for Product Design, McGraw-Hill Publishing Company, New York, 1961.

a

Source: Stodala ‘‘Turbo-blower and compressor’’; Kearton, W. J. and Proter, L. M., Design Engineer, Pratt and Whitney Aircraft; McGraw-Hill Publishing Company, New York, U.S.A. Douglas C. Greenwood, Editor, Engineering Data for Product Design, McGraw-Hill Publishing Company, New York, 1961.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

11 METAL FITS, TOLERANCES, AND SURFACE TEXTURE SYMBOLS1;2;3 area of cross section, m2 (in2 ) diameter of shaft, m (in) diameter of cylinder, m (in) modulus of elasticity, GPa (Mpsi) modulus of elasticity of cast iron, GPa (Mpsi) modulus of elasticity of steel, GPa (Mpsi) force, kN [lbf or tonf (pound force or tonne force)] length, m (in) length of hub, m (in) eﬀective length of anchor, m (in) original length of slot, m (in) torque or twisting moment, N m (lbf in) pressure, MPa (psi) contact pressure MPa (psi) temperature, 8C (8F) coeﬃcient of linear expansion, (m/m)/8C [(in/in)/8F] total change in diameter (interference), m (in) change in diameter, m (in) Poisson’s ratio stress, MPa (psi) coeﬃcient of friction factor of safety

A d E Ec Es F l L Mt p pc t d n

SUFFIXES a b c d f h i o r

axial bearing surface contact surface, compressive design ﬁnal hub internal, inner original, external, outer radial, rim

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METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.2

CHAPTER ELEVEN

shaft tangential or hoop initial ﬁnal

s 1 2

Particular

Formula

PRESS AND SHRINK FITS Change in cylinder diameter due to contact pressure The change in diameter

d ¼ d"

The change in diameter of the inner member when subjected to contact pressure pc (Fig. 11-1)

di ¼

The change in diameter of the outer member when subjected to contact pressure pc (Fig. 11-1)

do ¼

The original diﬀerence in diameters of the two cylinders when the material of the members is the same

The total change in the diameters of hub and hollow shaft due to contact pressure at their contact surface when the material of the members is the same

ð11-1Þ

pc dc E

pc dc E

dc2 þ di2 dc2 di2

do2 þ dc2 þ do2 dc2

ð11-2Þ

ð11-3Þ

¼ do þ di p d do2 þ dc2 ¼ c c þ E do2 dc2 p d dc2 þ di2 þ c c E dc2 di2 ¼ ds þ dh ¼ ds dh pc ds ds2 þ di2 ¼ s Es ds2 di2 p d do2 þ dh2 þ c h þ exactly h Eh do2 ds2 ¼ pc dc

ð11-4Þ

ð11-5aÞ

dc2 þ di2 d02 þ dc2 þ sþ h 2 2 Es ðdc di Þ Eh ðdo2 dc2 Þ Es Eh

ðapprox:Þ

FIGURE 11-1

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ð11-5bÞ

METAL FITS, TOLERANCES, AND SURFACE TEXTURE METAL FITS, TOLERANCES, AND SURFACE TEXTURE

Particular

The shrinkage stress in the band

11.3

Formula

¼

E dc

ð11-6Þ

The contact pressure between cylinders at the surface of contact when the material of both the cylinders is same (Fig. 11-2)

pc ¼

Eðdc2 di2 Þðdo2 dc2 Þ 2dc3 ðdo2 di2 Þ

ð11-7Þ

The tangential stress at any radius r of outer cylinder (Fig. 11-2a)

o ¼

pc dc2 do2 1 þ do2 dc2 4r2

ð11-8Þ

pc dc2 di2 ¼ 2 1þ 2 do dc2 4r

ð11-9Þ

do2 1 4r2

ð11-10Þ

The tangential stress at any radius r of inner cylinder (Fig. 11-2a)

i

The radial stress at any radius r of outer cylinder (Fig. 11-2a)

r o ¼

pc dc2 2 do dc2

pc dc2 di2 ¼ 2 1 2 dc di2 4r

The radial stress at any radius r of inner cylinder (Fig. 11-2a)

r i

The tangential stress at outside diameter of outer cylinder (Fig. 11-2)

oo ¼

2pc dc2 do2 dc2

ð11-11Þ

ð11-12Þ

The tangential stress at inside diameter of outer cylinder (Fig. 11-2)

oi ¼ pc

do2 þ dc2 do2 dc2

The tangential stress at outside diameter of inner cylinder (Fig. 11-2)

io ¼

pc ðdc2 þ di2 Þ dc2 di2

ð11-14Þ

The tangential stress at inside diameter of inner cylinder (Fig. 11-2)

ii ¼

2pc dc2 dc2 di2

ð11-15Þ

The radial stress at outside diameter of outer cylinder (Fig. 11-2)

r oo ¼ 0

FIGURE 11-2 Distribution of stresses in shrink-ﬁtted assembly.

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ð11-13Þ

ð11-16Þ

METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.4

CHAPTER ELEVEN

Particular

Formula

The radial stress at inside diameter of outer cylinder (Fig. 11-2)

r oi ¼ pc

ð11-17Þ

The radial stress at outside diameter of inner cylinder (Fig. 11-2)

r io ¼ pc

ð11-18Þ

The radial stress at inside diameter of inner cylinder (Fig. 11-2)

r ii ¼ 0

ð11-19Þ

The semiempirical formula for tangential stress for cast-iron hub on steel shaft

¼

Eo dc þ 0:14do

Timoshenko equation for contact pressure in case of steel shaft on cast-iron hub

pc ¼

Ec dc

ð11-20Þ

1 ðdc =do Þ2 1:53 þ 0:47ðdc =do Þ2

for

Es ¼3 Ec ð11-21aÞ

The allowable stress for brittle materials

all ¼

su Ec ½1 þ ðdc =do Þ2 ¼ n dc ½1:53 þ 0:47ðdc =do Þ2

ð11-21bÞ

INTERFERENCE FITS Press The axial force necessary to press shaft into hub under an interface pressure pc

The approximate value of axial force to press steel shaft into cast-iron hub with an interference

Fa ¼ dc lpc

ð11-22aÞ

where ¼ 0:085 to 0.125 for unlubricated surface ¼ 0:05 with special lubricants F ¼ 4137 104

ðdo þ 0:3dc Þl do þ 6:33dc

SI

ð11-23aÞ

where do , dc , l and in m, and F in N F ¼ 6000

ðdo þ 0:3dc Þl do þ 6:33dc

USCS

ð11-23bÞ

where do , dc , l and in in, and F in tonf The approximate value of axial force to press steel shaft in steel hub

F ¼ 28:41 104

ðdo2 dc2 Þl do2

SI

ð11-24aÞ

where do , dc , l and in m, and F in N F ¼ 4120

ðdo2 dc2 Þl do2

USCS

where do , dc , l and in in, and F in tonf

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ð11-24bÞ

METAL FITS, TOLERANCES, AND SURFACE TEXTURE METAL FITS, TOLERANCES, AND SURFACE TEXTURE

Particular

The transmitted torque by a press ﬁt or shrink ﬁt without slipping between the hub and shaft

The temperature t2 in 8C to which the shaft or shrink link must be heated before assembly

11.5

Formula

Mt ¼

dc2 lpc 2

ð11-25Þ

where ¼ 0:10 for press ﬁt ¼ 0:125 for shrink ﬁts 2 þ t1 t2 dc

ð11-26Þ

where t1 ¼ temperature of hub or larger part to which shaft or shrink link to be shrunk on, 8C

Shrink links or anchors (Fig. 11-3) The average compression in the part of rim aﬀected according to C. D. Albert

F c ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ab Ar

ð11-27Þ

FIGURE 11-3 Shrink link.

The tensile stress in link

t ¼

Lf Lo E Lo

ð11-28Þ

The total load on link

F¼

ðLf Lo ÞEA Lo

ð11-29Þ

The compressive stress in rim

c ¼

Lf Lo EA pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Lo Ab Ar

ð11-30Þ

The original length of link

Lo ¼

1þ 1þ

L AE pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E r Ab Ar

r E

d l E

The necessary linear interference for shrink anchors

¼

The force exerted by an anchor

F ¼ abd

ð11-31Þ

ð11-32Þ ð11-33Þ

b ¼ 2 to 3 a d ¼ design stress based on a reliability factor of 1.25

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METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.6

CHAPTER ELEVEN

Particular

Formula

For letter symbols for tolerances, basic size deviation and tolerance, clearance ﬁt, transition ﬁt, interference ﬁt

Refer to Figs. 11-4 to 11-8

For press-ﬁt between steel hub and shaft, cast-iron hub and shaft and tensile stress in cast-iron hub in press-ﬁt allowance

Refer to Figs. 11-9 to 11-11

TOLERANCES AND ALLOWANCES The tolerance size is deﬁned by its value followed by a symbol composed of a letter (in some cases by two letters) and a numerical value as

45 g7

A ﬁt is indicated by the basic size common to both components followed by symbols corresponding to each component, the hole being quoted ﬁrst, as

45H8 g7

For grades 5 to 16 tolerances have been determined in terms of standard tolerance unit i in micrometers (Refer to Table 11-l).

i ¼ 0:45D1=3 þ 0:001D

Values of standard tolerances corresponding to grades 01, 0, and 1 are (values in mm for D in mm)

IT 01 0:3 þ 0:008 D IT 0 0:5 þ 0:012 D IT 1 0:8 þ 0:020 D

or 45H8 g7 or 45

H8 g7 ð11-34Þ

where D is expressed in mm

ð11-35Þ

TABLE 11-1 Relative magnitudes of standard tolerances for grades 5 to 16 in terms of standard tolerance unit ‘‘i ’’ [Eq. (11-34)] Grade

IT 5

IT 6

IT 7

IT 8

IT 9

IT 10

IT 11

IT 12

IT 13

IT 14

IT 15

IT 16

Values

7i

10 i

16 i

25 i

40 i

64 i

100 i

160 i

250 i

400 i

640 i

1000 i

Source: IS 919, 1963.

TABLE 11-1A Coeﬃcient of friction, (for use between conical metallic surfaces) Contacting surface

Nature of surfaces

Coeﬃcient of friction,

Any metal in contact with another metal Any metal in contact with another metal Cast iron on steel Steel on steel Steel on steel Cast iron on steel

Lubricated with oil Greased Shrink-ﬁtted Shrink-ﬁtted Dry Dry

0.15 0.15 0.33 0.13 0.22 0.16

Source: Courtesy J. Bach, ‘‘Kegelreibungsverbindungen,’’ Zeitschrift Verein Deutscher Ingenieure, Vol. 79, 1935.

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METAL FITS, TOLERANCES, AND SURFACE TEXTURE METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.7

TABLE 11-2 Formulas for fundamental shaft deviations (for sizes 500 mm) Upper deviations (es)

Lower deviation (ei)

In lm (for D in mm)

Shaft designation

In lm (for D in mm)

j5–j8 k4–k7 k for grades 3 and 8

No formula pﬃﬃﬃﬃ ¼ þ0:6 3 D ¼0

m n p

¼ þ(IT 7–IT 6) ¼ þ5D0:34 ¼ IT 7 þ 0 to 5

r

c

¼ ð265 þ 1:3DÞ for D 120 ¼ 3:5D for D < 120 l ð140 þ 0:85DÞ for D 160 l 1:8D for D > 160 ¼ 52D0:2 for D 40

d e f g

¼ ð95 þ 0:8DÞ for D > 40 ¼ 16D0:44 ¼ 11D0:41 ¼ 5:5D0:41 ¼ 2:5D0:34

h

¼0

¼ geometric mean of values ei for p and s ¼ þIT 8 þ 1 to 4 for D 50 ¼ þIT 7 þ 0:4D for D > 50 ¼ IT 7 þ 0:63D ¼ þIT 7 þ D ¼ þIT 7 þ 1:25D ¼ þIT 7+1.6D ¼ þIT 7 þ 2D ¼ þIT 7 þ 2:5D ¼ þIT 8 þ 3:15D ¼ þIT 9 þ 4D ¼ þIT 10 þ 5D

Shaft designation

a

For js: The two deviations are equal to

IT 2

s t u v x y z za zb zc

Source: IS 919, 1963.

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METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.8

CHAPTER ELEVEN

TABLE 11-3 Rules for rounding oﬀ values obtained by the use of formulas

Values in lm

Rounded in multiples of

Above Up to

5 45

45 60

60 100

100 200

200 300

300 560

560 600

600 800

800 1000

1000 2000

For standard tolerances for Grades II and ﬁner

1

1

1

5

10

10

For deviations es, from a to g

1

2

5

5

10

10

20

20

20

50

For deviations ei, from k to zc

1

1

1

2

5

5

10

20

50

Source: IS 919, 1963.

FIGURE 11-4 Letter symbols for tolerances.

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2000

1000

METAL FITS, TOLERANCES, AND SURFACE TEXTURE METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.9

TABLE 11-4 Fundamental tolerances of grades 01, 0, and 1 to 16 Values of tolerances in lm (1 lm ¼ 0:001 mm) Diameter steps in mm 3 >3 6 >6 10 >10 18 >18 30 >30 50 >50 80 >80 120 >120 180 >180 250 >250 315 >315 400 >400 500

Tolerance grades 01

0

1

2

3

4

5

0.3

0.5

0.8

1.2

2

3

4

0.4

0.6

1

1.5

2.5

4

0.4

0.6

1

1.5

2.5

0.5

0.8

1.2

2

0.6

1

1.5

0.6

1

0.8

6

7

8

9

10

6

10

14

25

40

5

8

12

18

30

4

6

9

15

22

3

5

8

11

18

2.5

4

6

9

13

1.5

2.5

4

7

11

1.2

2

3

5

8

1

1.5

2.5

4

6

1.2

2

3.5

5

2

3

4.5

2.5

4

3 4

11

14a

15a

16a

12

13

60

100

140

250

400

600

48

75

120

180

300

480

750

36

58

90

150

220

360

580

900

27

43

70

110

180

270

430

700

1100

21

33

52

84

130

210

330

520

840

1300

16

25

39

62

100

160

250

390

620

1000

1600

13

19

30

46

74

120

190

300

460

740

1200

1900

10

15

22

35

54

87

140

220

350

540

870

1400

2200

8

12

18

25

40

63

100

160

250

400

630

1000

1600

2500

7

10

14

20

29

46

72

115

185

290

460

720

1150

1850

2900

6

8

12

16

23

32

52

81

130

210

320

520

810

1300

2100

3200

5

7

9

13

18

25

36

57

89

140

230

360

570

890

1400

2300

3600

6

8

10

15

20

27

40

63

97

155

250

400

630

970

1550

2500

4000

a

Up to 1 mm grades 14 to 16 are not provided. Source: IS 919, 1963.

FIGURE 11-5 Basic size deviation and tolerances.

FIGURE 11-6 Clearance ﬁt.

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METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.10

CHAPTER ELEVEN

TABLE 11-5 Clearance ﬁts (Fig. 11-6) (hole basis) Quality of ﬁt

Large clearance

Combination of shaft and hole H 11 a 9 coarse H 11 b 9

Remarks and uses

Not widely used

H 11 a 11 normal H9a9 fine H8b8 Slack running

Loose running

Easy running

H 11 c 9 coarse H 11 c 11 normal H9c9 H8c8 fine H7c8 H 11 d 11 coarse H9d9 H 8 d 9 normal H8d8 fine H7d8 H8e9 coarse H9e9 H8e8 normal H7e8 H7e7 fine H6e7

Not widely used

Suitable for plummer block bearings and loose pulleys

Recommended for general clearance ﬁts, used for properly lubricated bearings requiring appreciable clearance; ﬁner grades for high speeds, heavily loaded bearings such as turbogenerator and large electric motor bearings

Normal running

H 8 f 8 coarse H 7 f 7 normal H 6 f 6 ﬁne

Widely used as a normal grease lubricated or oil-lubricated bearing having low temperature diﬀerences, gearbox shaft bearings, bearings of small electric motor and pumps, etc.

Close running or sliding

H 8 g 7 coarse H 7 g 6 normal

Expensive to manufacture, small clearance. Used in bearings for accurate link work, and for piston and slide valves; also used for spigot or location ﬁts

H6g6 fine H6g5 Precision sliding

H H H H H

11 h 11 8h7 8h8 7h6 6h5

Widely used for nonrunning parts; also used for ﬁne spigot and location ﬁt

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METAL FITS, TOLERANCES, AND SURFACE TEXTURE

TABLE 11-6 Values of standard tolerances for sizes >500 to 3150 mm IT 6 a

10 I a

IT 7

IT 8

IT 9

IT 10

IT 11

IT 12

IT 13

IT 14

IT 15

IT 16

16 I

25 I

40 I

64 I

100 I

160 I

250 I

400 I

640 I

1000 I

Standard Tolerance Unit I (in mm) 0:004D þ 2:1 for D in mm.

Source: IS: 2101-1962.

FIGURE 11-8 Interference ﬁt.

FIGURE 11-7 Transition ﬁt.

TABLE 11-7 Transition and interference ﬁts (hole basis) Quality of ﬁt

Combination of shaft and hole

Push

H 8 j 7 coarse H 7 j 6 normal H 6 j 5 ﬁne

True transition

H 8 k 7 coarse H 7 k 6 normal H 6 k 5 ﬁne

Fit averaging virtually no clearance-recommended for location ﬁts where a slight interference can be tolerated, with the object of eliminating vibration; used in clutch member keyed to shaft, gudgeon pin in piston bosses, hand wheel, and index disk on shaft

Interference transition

H 8 m 7 coarse H 7 m 6 normal H 6 m 5 ﬁne H8n7 coarse H7n6

Fit averages a slight interference suitable for general tight-keying ﬁts where accurate location and freedom from play are necessary; used for the cam holder, ﬁtting bolt in reciprocating slide

True interference

Remarks and uses

Transition ﬁt (Fig. 11-7) Slight clearance—recommended for ﬁts where slight interference is permissible, coupling spigots and recesses, gear rings clamped to steel hubs

Suitable for tight assembly of mating surfaces

H 6 n 5 ﬁne

Light press ﬁt

H 7 p 6 normal H 6 p 5 ﬁne

Medium drive ﬁt

H 7 r 6 normal H 6 r 5 ﬁne

Heavy drive ﬁt

H8s7 normal H7s6

Force ﬁt

H 6 s 5 ﬁne H8t7 normal H7t6

Heavy force ﬁt or shrink ﬁt

H 6 t 5 ﬁne H8u7 normal H7u6

Interference ﬁt (Fig. 11-8) Light press ﬁt for nonferrous parts which can be dismantled when required; standard press ﬁt for steel, cast iron, or brass-to-steel assemblies, bush on to a gear, split journal bearing Medium drive ﬁt with easy dismantling for ferrous parts and light drive ﬁt with easy dismantling for nonferrous parts assembly; pump impeller on shaft, small-end bush in connecting rod, pressed in bearing bush, sleeves, seating, etc. Used for permanent or semipermanent assemblies of steel and castiron members with considerable gripping force; for light alloys this gives a press ﬁt; used in collars pressed on to shafts, valve seatings, cylinder liner in block, etc. Suitable for the permanent assembly of steel and cast-iron parts; used in valve seat insert in cylinder head, etc.

High interference ﬁt; the method of assembly will be by power press

H 6 u 5 ﬁne

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METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.12

CHAPTER ELEVEN

TABLE 11-8 Preferred basic and design sizes Linear dimensions (in mm) Shaft basis A

B

1.6 2.5 4.0 6.0 10.0 16.0 25.0 40.0 63.0 100.0

5.0 8.0 12.0 14.0 18.0 20.0 22.0 32.0 50.0 80.0

Hole basis Priority 1 1.0 1.6 2.5 4.0 5.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0

Priority 2

22.0 25.0 28.0 32.0 36.0 40.0 45.0 50.0 56.0 63.0 71.0 80.0 90.0 100.0

110.0 125.0 140.0 160.0 180.0 200.0 220.0 250.0 280.0 320.0 360.0 400.0 450.0 500.0

1.2 2.0 3.2 4.5 5.5 7.0 9.0 11.0 13.0 15.0 17.0 19.0 21.0 23.0 26.0 30.0

Priority 3

34.0 38.0 42.0 48.8 53.0 58.0 65.0 75.0 85.0 95.0 105.0 115.0 120.0 130.0 135.0 150.0

170.0 190.0 210.0 230.0 240.0 260.0 270.0 300.0 340.0 380.0 420.0 430.0 470.0 480.0

145.0 155.0 165.0 175.0 185.0 195.0 290.0 310.0 330.0 350.0 370.0 390.0 410.0

440.0 460.0 490.0

Angular dimensions (in deg) Priority 1 2

Preferred angles 1

3 2

6 4

10 5

16 8

30 12

45

60

90

120

20

TABLE 11-9 Formulas for shaft and hole deviations (for sizes >500 to 3150 mm) Shafts d e f (g) h js k m n p r s t u

es es es es es ei ei ei ei ei ei ei ei ei

— — — — — — þ þ þ þ þ þ þ

Formulas for deviations in lm (for D in mm)

Holes

16 D0:44 11 D0:41 5.5 D0:41 2.5 D0:34 0 0.5 ITn 0 0.024 D þ 12:6 0.04 D þ 21 0.072 D þ 37:8 geometric mean between p and s or P and S IT 7 þ 0:4D IT 7 þ 0:63D IT 7 þ D

þ þ þ þ + — — — — — — —

EI EI EI EI EI ES ES ES ES ES ES ES ES ES

a

D E F (G) H JS K M N Pa Ra Sa Ta U

It is assumed that associated shafts and holes are of the same grade contrary to what has been allowed for the dimensions up to 500 mm (see IS 919, 1959). Source: IS 2101, 1962.

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METAL FITS, TOLERANCES, AND SURFACE TEXTURE METAL FITS, TOLERANCES, AND SURFACE TEXTURE

FIGURE 11-9 Press-ﬁt pressures between steel hub and shaft (1 psi ¼ 6894.757 Pa; 1 in ¼ 25.4 mm). (Baumeister, T., Marks’ Standard Handbook for Mechanical Engineers, 8th ed., McGraw-Hill, 1978.)

11.13

FIGURE 11-10 Variation in tensile stress in cast-iron hub in press-ﬁt allowance (1 psi ¼ 6894.757 Pa; 1 in ¼ 25.4 mm). (Baumeister, T., Marks’ Standard Handbook for Mechanical Engineers, 8th ed., McGraw-Hill, 1978.)

FIGURE 11-11 Press-ﬁt pressure between cast-iron hub and shaft (1 psi ¼ 6894.757 Pa; 1 in ¼ 25.4 mm). (Baumeister, T., Marks’ Standard Handbook for Mechanical Engineers, 8th ed., McGraw-Hill, 1978.)

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.

11.14

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.

g5

g4

f8

f7

f6

e9

e8

e7

e6

d10

d9

d8

c11

c9

c8

b9

a9

System of basic shaft

3 6

270 300 140 170 70 88 70 100 70 145 30 48 30 60 30 78 20 28 20 32 20 38 20 50 10 18 10 22 10 28 04 08 04 09

— 3

270 295 140 165 60 74 60 85 60 120 20 34 20 45 20 60 14 20 14 24 14 28 14 39 06 12 06 16 06 20 02 05 02 06

Limits

esb eic es ei es ei es ei es ei es ei es ei es ei es ei es ei es ei es ei es ei es ei es ei es ei es ei

280 316 150 186 80 102 80 116 80 170 40 62 40 76 40 98 25 34 25 40 25 47 25 61 13 22 13 28 13 35 05 09 05 11

6 10 290 333 150 193 95 122 95 138 95 205 50 77 50 93 50 120 32 43 32 50 32 59 32 75 16 27 16 34 16 43 06 11 06 14

10 18

24 30

300 352 160 212 110 143 110 162 110 240 65 98 65 117 65 149 40 53 40 61 40 73 40 92 20 33 20 41 20 53 07 13 07 16

18 24

TABLE 11-10 Tolerancesa for shafts for sizes up to 500 mm

40 50

310 320 372 382 170 180 232 242 120 130 159 169 120 130 182 192 120 130 280 290 80 119 80 142 80 180 50 66 50 75 50 89 50 112 25 41 25 50 25 64 09 16 09 20

30 40

65 80

340 360 414 434 190 200 264 274 140 150 186 196 140 150 214 224 140 150 330 340 100 146 100 174 100 220 60 79 60 90 60 106 60 134 30 49 30 60 30 76 10 18 10 23

50 65

100 120

380 410 467 497 220 240 307 327 170 180 224 234 170 180 257 267 170 180 390 400 120 174 120 207 120 260 72 94 72 107 72 126 72 159 36 58 36 71 36 90 12 22 12 27

80 100 460 560 260 360 200 263 200 300 200 450

120 140 520 620 280 380 210 273 210 310 210 460 145 208 145 245 145 305 85 110 85 125 85 148 85 185 43 68 43 83 43 106 14 26 14 32

140 160 580 680 310 410 230 293 230 330 230 480

160 180

Diameter steps, mm

660 775 340 455 240 312 240 355 240 530

180 200 740 855 380 495 260 332 260 375 260 550 170 242 170 285 170 355 100 129 100 146 100 172 100 215 50 79 50 96 50 122 15 29 15 35

200 225

250 280

280 315

315 355

355 400

400 450

450 500

820 920 1050 1200 1350 1500 1650 935 1050 1180 1340 1490 1655 1805 420 480 540 600 680 760 840 535 610 670 740 820 915 995 280 300 330 360 400 440 480 352 381 411 449 489 537 577 280 300 330 360 400 440 480 395 430 460 500 540 595 635 280 300 330 360 400 440 480 570 620 650 720 760 840 880 190 210 230 271 299 327 190 210 230 320 350 385 190 210 230 400 440 480 110 125 135 142 161 175 110 125 135 162 182 198 110 125 135 191 214 232 110 125 135 240 265 290 56 62 68 88 98 108 56 62 68 108 119 131 56 62 68 137 151 165 17 18 20 33 36 40 17 18 20 40 43 47

225 250

METAL FITS, TOLERANCES, AND SURFACE TEXTURE

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n5

n4

m7

m6

k7

k6

j7

j6

j5

h11

h10

h9

h8

h7

h6

h5

g6

System of basic shaft

es ei es ei es ei es ei es ei es ei es ei esb eic es ei es ei es ei es ei es ei es ei es ei es ei es ei

Limits

3 6

04 12 00 05 00 08 00 12 00 18 00 30 00 48 00 75 þ03 02 þ06 02 þ08 04 þ09 þ01 þ13 þ01 þ12 þ04 þ16 þ04 þ12 þ08 þ13 þ08

— 3

02 08 00 04 00 06 00 10 00 14 00 25 00 40 00 60 þ02 02 þ04 02 þ06 þ04 þ06 þ00 þ10 þ01 þ08 þ02 þ02 — þ07 þ04 þ08 þ04

10 18

05 06 14 17 00 00 06 08 00 00 09 11 00 00 15 18 00 00 22 27 00 00 36 43 00 00 58 70 00 00 90 110 þ04 þ05 02 03 þ07 þ08 02 03 þ10 þ12 05 06 þ10 þ12 þ01 þ01 þ16 þ19 þ01 þ01 þ15 þ18 þ06 þ07 þ21 þ25 þ06 þ07 þ14 þ17 þ10 þ12 þ16 þ20 þ10 þ12

6 10

24 30

07 20 00 09 00 13 00 21 00 33 00 52 00 84 00 130 þ05 04 þ09 04 þ13 08 þ15 þ02 þ23 þ02 þ21 þ08 þ29 þ08 þ21 þ15 þ24 þ15

18 24

30 40 09 25 00 11 00 16 00 25 00 39 00 62 00 100 00 160 þ06 05 þ11 05 þ15 10 þ18 þ02 þ27 þ02 þ25 þ09 þ34 þ09 þ24 þ17 þ28 þ17

40 50

TABLE 11-10 Tolerancesa for shafts for sizes up to 500 mm (Cont.)

50 65 10 29 00 13 00 19 00 30 00 46 00 74 00 120 00 190 þ06 07 þ12 07 þ18 12 þ21 þ02 þ32 þ02 þ30 þ11 þ41 þ11 þ28 þ20 þ33 þ20

65 80

80 100 12 34 00 15 00 22 00 35 00 54 00 87 00 140 00 220 þ06 09 þ13 09 þ20 15 þ25 þ03 þ38 þ03 þ35 þ13 þ48 þ13 þ33 þ23 þ38 þ23

100 120

120 140 14 39 00 18 00 25 00 40 00 63 00 100 00 160 00 250 þ07 11 þ14 11 þ22 18 þ28 þ03 þ43 þ03 þ40 þ15 þ55 þ15 þ39 þ27 þ45 þ27

140 160

160 180

Diameter steps, mm 180 200 15 44 00 20 00 29 00 46 00 72 00 115 00 185 00 290 þ07 13 þ16 13 þ25 21 þ33 þ04 þ50 þ04 þ46 þ17 þ63 þ17 þ45 þ31 þ51 þ31

200 225

225 250

250 280 17 49 00 23 00 32 00 52 00 81 00 130 00 210 00 320 þ07 16 þ16 16 þ26 26 þ36 þ04 þ56 þ04 þ52 þ20 þ72 þ20 þ50 þ34 þ57 þ34

280 315

315 355 18 54 00 25 00 36 00 57 00 89 00 140 00 230 00 360 þ07 18 þ18 18 þ29 28 þ40 þ04 þ61 þ04 þ57 þ21 þ78 þ21 þ55 þ37 þ62 þ37

355 400

400 450

20 60 00 27 00 40 00 63 00 97 00 155 00 250 00 400 þ07 20 þ20 20 þ31 32 þ45 þ05 þ68 þ05 þ63 þ23 þ86 þ23 þ60 þ40 þ67 þ40

450 500

METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.15

es ei es ei es ei

es ei es ei es ei es ei es ei es ei es ei es ei es ei es ei

3 6

6 10

10 18

þ28 þ23 þ41 þ23 — — þ46 þ28 — — þ47 þ35

þ50 þ42 þ62 þ50 þ98 þ80

þ22 þ18 þ32 þ18 — — þ34 þ20 — — þ36 þ26

þ38 þ32 þ50 þ40 þ74 þ60

þ61 þ52 þ82 þ67 þ119 þ97

þ34 þ28 þ50 þ28 — — þ56 þ34 — — þ57 þ42 — — — — — —

þ50 þ60

þ68 þ78

— —

þ40 þ45

þ67 þ72

— þ39

— þ47

11.16

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.

b

a

þ41 þ33 þ60 þ33

þ10 þ17 þ21 þ26 þ06 þ12 þ15 þ18 þ12 þ20 þ24 þ29 þ06 þ12 þ15 þ18 þ16 þ23 þ28 þ34 þ10 þ15 þ19 þ23 — — — —

— 3

Tolerances in micrometers (1 mm ¼ 103 mm). es ¼ upper deviation. c ei ¼ lower deviation.

zc8

zb7

za6

z7

y6

x8

v5

u8

u5

t7

r6

p6

p5

System of basic shaft Limits 24 30

— — — — — —

— — — — — —

þ31 þ22 þ35 þ22 þ41 þ28 þ62 þ41 þ50 þ57 þ41 þ48 þ74 þ81 þ41 þ48 þ56 þ64 þ47 þ55 þ87 þ97 þ54 þ64 þ76 þ88 þ63 þ75 þ94 þ109 þ73 þ88

18 24

40 50

— — — — — —

— — — — — —

þ37 þ26 þ42 þ26 þ50 þ34 þ79 þ54 þ71 þ81 þ60 þ70 þ99 þ109 þ60 þ70 þ79 þ92 þ68 þ81 þ119 þ136 þ80 þ97 þ110 þ130 þ94 þ114 þ137 þ161 þ112 þ136

30 40

TABLE 11-10 Tolerancesa for shafts for sizes up to 500 mm (Cont.)

65 80

— — — — — —

— — — — — —

þ45 þ32 þ51 þ32 þ62 þ43 þ105 þ75 þ100 þ115 þ87 þ102 þ133 þ148 þ87 þ102 þ115 þ133 þ102 þ120 þ168 þ192 þ122 þ146 þ163 þ193 þ144 þ174 þ202 þ240 þ172 þ210

50 65

100 120

— — — — — —

— — — — — —

þ52 þ37 þ59 þ37 þ76 þ54 þ139 þ104 þ139 þ159 þ124 þ144 þ178 þ198 þ124 þ144 þ161 þ187 þ146 þ172 þ232 þ264 þ176 þ210 þ236 þ276 þ214 þ254 þ293 þ345 þ258 þ310

80 100

— — — — — —

þ188 þ170 þ233 þ170 þ220 þ202 þ311 þ248 þ325 þ300 þ405 þ365

120 140

— — — — — —

þ61 þ43 þ68 þ43 þ90 þ65 þ174 þ134 þ208 þ190 þ253 þ190 þ246 þ228 þ343 þ280 þ365 þ340 þ455 þ415

140 160

Diameter steps, mm

— — — — — —

þ228 þ210 þ273 þ210 þ270 þ252 þ373 þ310 þ405 þ380 þ505 þ465

160 180

— — — — — —

þ256 þ236 þ308 þ236 þ304 þ284 þ422 þ350 þ454 þ425 þ566 þ520

180 200

— — — — — —

þ70 þ50 þ79 þ50 þ109 þ80 þ226 þ180 þ278 þ258 þ330 þ258 þ330 þ310 þ457 þ385 þ499 þ470 þ621 þ575

200 225

— — — — — —

þ304 þ284 þ356 þ284 þ360 þ340 þ497 þ425 þ549 þ520 þ686 þ640

225 250

280 315

— — — — — —

— — — — — —

þ79 þ56 þ88 þ56 þ130 þ98 þ292 þ240 þ338 þ373 þ315 þ350 þ396 þ431 þ315 þ350 þ408 þ448 þ385 þ425 þ556 þ606 þ475 þ525 þ612 þ682 þ580 þ650 þ762 þ842 þ710 þ790

250 280

355 400

400 450

450 500

— — — — — —

— — — — — —

— — — — — —

— — — — — —

þ87 þ95 þ62 þ68 þ98 þ108 þ62 þ68 þ150 þ172 þ114 þ132 þ351 þ423 þ294 þ360 þ415 þ460 þ517 þ567 þ390 þ435 þ490 þ540 þ479 þ524 þ587 þ637 þ390 þ435 þ490 þ540 þ500 þ555 þ622 þ687 þ475 þ530 þ595 þ660 þ679 þ749 þ837 þ917 þ590 þ660 þ740 þ820 þ766 þ856 þ960 þ1040 þ730 þ820 þ920 þ1000 þ957 þ1057 þ1163 þ1313 þ900 þ1000 þ1100 þ1250

315 355

METAL FITS, TOLERANCES, AND SURFACE TEXTURE

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J7

H11

H10

H9

H8

H7

H6

H5

G7

F8

F6

E5

D9

D8

C11

C8

B11

B9

A9

ESb EIc ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI

System of basic hole Limits

þ295 þ270 þ165 þ140 þ200 þ140 74 þ60 þ120 þ60 þ34 þ20 þ45 þ20 þ18 þ14 þ12 þ6 þ20 þ6 þ12 þ2 þ4 0 þ6 0 þ10 0 þ14 0 þ25 0 þ40 0 þ60 0 þ4 6

— 3

þ300 þ270 þ170 þ140 þ215 þ140 þ88 þ70 þ145 þ70 þ48 þ30 þ60 þ30 þ25 þ20 þ18 þ10 þ28 þ10 þ16 þ4 þ5 0 þ8 0 þ12 0 þ18 0 þ30 0 þ48 0 þ75 0 þ6 6

3 6

þ316 þ280 þ186 þ150 þ240 þ150 þ102 þ80 þ170 þ80 þ62 þ40 þ76 þ40 þ31 þ25 þ22 þ13 þ35 þ13 þ20 þ5 þ6 0 þ9 0 þ15 0 þ22 0 þ36 0 þ58 0 þ90 0 þ8 7

6 10

14 18

þ333 þ290 þ193 þ150 þ260 þ150 þ122 þ95 þ205 þ95 þ77 þ50 þ93 þ50 þ40 þ32 þ27 þ16 þ43 þ16 þ24 þ6 þ8 0 þ11 0 þ18 0 þ27 0 þ43 0 þ70 0 þ110 0 þ10 8

10 14

24 30

þ352 þ300 þ212 þ160 þ290 þ160 þ143 þ110 þ240 þ110 þ98 þ65 þ117 þ65 þ49 þ40 þ33 þ20 þ53 þ20 þ28 þ7 þ9 0 þ13 0 þ21 0 þ33 0 þ52 0 þ84 0 þ130 0 þ12 9

18 24

TABLE 11-11 Tolerancesa for holes for sizes up to 500 mm

40 50

þ372 þ382 þ310 þ320 þ232 þ242 þ170 þ180 þ330 þ340 þ170 þ180 þ159 þ169 þ120 þ130 þ280 þ290 þ120 þ130 þ119 þ80 þ142 þ80 þ61 þ50 þ41 þ25 þ64 þ25 þ34 þ9 þ11 0 þ16 0 þ25 0 þ39 0 þ62 0 þ100 0 þ160 0 þ14 11

30 40

65 80

þ414 þ434 þ340 þ360 þ264 þ274 þ190 þ200 þ380 þ390 þ190 þ200 þ186 þ196 þ140 þ150 þ330 þ340 þ140 þ150 þ146 þ100 þ174 þ100 þ73 þ60 þ49 þ30 þ76 þ30 þ40 þ10 þ13 0 þ19 0 þ30 0 þ46 0 þ74 0 þ120 0 þ190 0 þ18 12

50 65

100 120

þ467 þ497 þ380 þ410 þ307 þ327 þ220 þ240 þ440 þ460 þ220 þ240 þ224 þ234 þ170 þ180 þ390 þ400 þ170 þ180 þ174 þ120 þ207 þ120 þ87 þ72 þ58 þ36 þ90 þ36 þ47 þ12 þ15 0 þ22 0 þ35 0 þ54 0 þ87 0 þ140 0 þ220 0 þ22 13

80 100 þ560 þ460 þ360 þ260 þ510 þ260 þ263 þ200 þ450 þ200

120 140 þ620 þ520 þ380 þ280 þ530 þ280 þ273 þ210 þ460 þ210 þ208 þ145 þ245 þ145 þ103 þ85 þ68 þ43 þ106 þ43 þ54 þ14 þ18 0 þ25 0 þ40 0 þ63 0 þ100 0 þ160 0 þ250 0 þ26 14

140 160 þ680 þ580 þ410 þ310 þ560 þ310 þ293 þ230 þ480 þ230

160 180

Diameter steps, mm

þ775 þ660 þ455 þ340 þ630 þ340 þ312 þ240 þ530 þ240

180 200 þ855 þ740 þ495 þ380 þ670 þ380 þ332 þ260 þ550 þ260 þ242 þ170 þ285 þ170 þ120 þ100 þ79 þ50 þ122 þ50 þ61 þ15 þ20 0 þ29 0 þ46 0 þ72 0 þ115 0 þ185 0 þ290 0 þ30 16

200 225 þ925 þ820 þ535 þ420 þ710 þ420 þ352 þ280 þ570 þ280

225 250

280 315

315 355

355 400

400 450

450 500

þ1050 +1180 +1340 +1490 +1655 +1805 þ920 +1050 +1200 +1350 +1500 +1650 þ610 +670 +740 +820 +915 +995 þ480 +540 +600 +680 +760 +840 þ800 +860 +960 +1040 +1160 +1240 þ480 +540 +600 +680 +760 +840 þ381 +411 +449 +489 +537 +577 þ300 +330 +360 +400 +440 +480 þ620 +650 +720 +760 +840 +880 þ300 +330 +360 +400 +440 +480 þ271 þ299 þ327 þ190 þ210 þ230 þ320 þ350 þ385 þ190 þ210 þ230 þ133 þ150 þ162 þ110 þ125 þ135 þ88 þ98 þ108 þ56 þ62 þ68 þ137 þ151 þ165 þ56 þ62 þ68 þ69 þ75 þ83 þ17 þ18 þ20 þ23 þ25 þ27 0 0 0 þ32 þ36 þ40 0 0 0 þ52 þ57 þ63 0 0 0 þ81 þ89 þ97 0 0 0 þ130 þ140 þ155 0 0 0 þ210 þ230 þ250 0 0 0 þ320 þ360 þ400 0 0 0 þ36 þ39 þ43 16 18 20

250 280

METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.17

11.18

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.

ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI ES EI

6 10

þ2 7 þ5 10 0 15 4 19 9 24 20 29 17 32 — — 22 37 — — 28 43 — — 42 64 46 61 67 89 97 133

3 6

þ2 6 þ3 9 0 12 4 16 8 20 16 24 15 27 — — 19 31 — — 24 36 — — 35 53 38 50 50 68 80 110

— 3

0 6 0 10 2 12 4 14 6 16 14 20 14 24 — — 18 28 — — 20 30 — — 26 40 32 42 40 54 60 85

b

a

Tolerances in mm; 1 mm ¼ 103 mm ES ¼ upper deviation. c EI ¼ lower deviation.

ZC9

ZB8

ZA7

Z8

Y7

X7

V6

U7

T6

S7

S6

P7

N7

M7

K7

K6

System of basic hole Limits 14 18

þ2 9 þ6 12 0 18 5 23 11 29 25 36 21 39 — — 26 44 — 36 — 47 33 38 51 56 — — — — 50 60 77 87 — — — — — — — — — — — —

10 14

24 30

þ2 11 þ6 15 0 21 7 28 14 35 31 44 27 48 — 37 — 50 33 40 54 61 43 51 56 64 46 56 67 77 55 67 76 88 73 88 106 121 — — — — — — — — — — — —

18 24

40 50

þ3 13 þ7 18 0 25 8 33 17 42 38 54 34 59 43 49 59 65 51 61 76 86 63 76 79 92 71 88 96 113 85 105 110 130 112 136 151 175 — — — — — — — — — — — —

30 40

TABLE 11-11 Tolerancesa for holes for sizes up to 500 mm (Cont.)

65 80

þ4 15 þ9 21 0 30 9 39 21 51 47 53 66 72 42 48 72 78 60 69 79 88 76 91 106 121 96 114 115 133 111 135 141 165 133 163 163 193 172 210 218 256 — — — — — — — — — — — —

50 65

100 120

þ4 18 þ10 25 0 35 10 45 24 59 64 72 86 94 58 66 93 101 84 97 106 119 111 131 146 166 139 165 161 187 165 197 200 232 201 241 236 276 258 310 312 364 — — — — — — — — — — — —

80 100

85 110 77 117 115 140 155 195 195 220 233 273 285 325 365 428 — — — — — —

120 140 þ4 21 þ12 28 0 40 12 52 28 68 93 118 85 125 127 152 175 215 221 246 265 305 325 365 415 478 — — — — — —

140 160

Diameter steps, mm

101 126 93 133 139 164 195 235 245 270 295 335 365 405 465 528 — — — — — —

160 180

113 142 105 151 157 186 219 265 275 304 333 379 408 454 520 592 — — — — — —

180 200 þ5 24 þ13 33 0 46 14 60 33 79 121 150 113 159 171 200 241 287 301 330 368 414 453 499 575 647 — — — — — —

200 225

131 160 123 169 187 216 267 313 331 360 408 454 503 549 640 712 — — — — — —

225 250

149 181 138 190 209 241 295 347 376 408 455 507 560 612 710 791 — — — — — —

250 280 þ5 27 þ16 36 0 52 14 66 36 88 161 193 150 202 231 263 330 382 416 448 505 557 630 682 790 871 — — — — — —

280 315

179 215 169 226 257 293 369 426 464 500 569 626 709 766 900 989 — — — — — —

315 355

þ7 29 þ17 40 0 57 16 73 41 98 197 233 187 244 283 319 414 471 519 555 639 696 799 856 1000 1089 — — — — — —

355 400

450 500 þ8 32 þ18 45 0 63 17 80 45 108 219 239 259 279 209 229 272 292 317 347 357 387 467 517 530 580 582 647 622 687 717 797 780 860 897 977 960 1040 1100 1250 1197 1347 — — — — — — — — — — — —

400 450

METAL FITS, TOLERANCES, AND SURFACE TEXTURE

METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.19

METAL FITS, TOLERANCES, AND SURFACE TEXTURE

TABLE 11-12 Tolerancesa for shafts for sizes 500 to 3150 mm System of basic shaft Limits d10 e8 f9 g6 g7 h6 h7 h8 h9 h10 h11 js9 k6 m6 n6 p6 r7 s7 t7 u7

esb ei c es ei es ei es ei es ei es ei es ei es ei es ei es ei es ei es ei es ei es ei es ei es ei es ei es ei es ei es ei

Diameter steps, mm 500 560

560 630

630 710

710 800

800 900

900 1000

1000 1120

1120 1250

1250 1400

1400 1600

1600 1800

1800 2000

260 540 145 255 76 251 22 66 22 92 0 44 0 70 0 110 0 175 0 280 0 440

290 610 160 285 80 280 24 74 24 103 0 50 0 80 0 125 0 200 0 320 0 500

320 680 170 310 86 316 26 82 26 115 0 56 0 90 0 140 0 230 0 360 0 560

350 770 195 360 98 358 28 94 28 133 0 66 0 105 0 165 0 260 0 420 0 660

390 890 220 415 110 420 30 108 30 155 0 78 0 125 0 195 0 310 0 500 0 780

430 1030 240 470 120 490 32 124 32 182 0 92 0 150 0 230 0 370 0 600 0 920

87.5

100

115

130

155

185

þ66 0 þ106 þ40 þ132 þ66 þ186 þ120 þ355 þ365 þ250 þ260 þ625 þ685 þ520 þ580 þ885 þ945 þ780 þ840 þ1255 þ1405 þ1150 þ1300

þ78 0 þ126 þ48 þ156 þ78 þ218 þ140 þ425 þ455 þ300 þ330 þ765 þ845 þ640 þ720 þ1085 þ1175 þ960 þ1050 þ1575 þ1725 þ1450 þ1600

þ92 0 þ150 þ58 þ184 þ92 þ262 þ170 þ520 þ550 þ370 þ400 þ970 þ1070 þ820 þ920 þ1350 þ1500 þ1200 þ1350 þ2000 þ2150 þ1850 þ2000

þ44 þ50 þ56 0 0 0 þ70 þ80 þ90 þ26 þ30 þ34 þ88 þ100 þ112 þ44 þ50 þ56 þ122 þ139 þ156 þ78 þ88 þ100 þ220 þ225 þ255 þ265 þ300 þ310 þ150 þ155 þ175 þ185 þ210 þ220 þ350 þ380 þ420 þ460 þ520 þ560 þ280 þ310 þ340 þ380 þ430 þ470 þ470 þ520 þ580 þ640 þ710 þ770 þ400 þ450 þ500 þ560 þ620 þ680 þ570 þ730 þ820 þ920 þ1031 þ1140 þ600 þ660 þ740 þ840 þ940 þ1050

2000 2250

2250 2500

480 1180 260 540 130 570 34 140 34 209 0 110 0 175 0 280 0 440 0 700 0 1100

2800 3150

520 1380 290 620 145 685 38 173 38 248 0 135 0 210 0 330 0 540 0 860 0 1350

220 þ110 0 þ178 þ68 þ220 þ110 þ305 þ195 þ615 þ635 þ440 þ460 þ1175 þ1275 þ1000 þ1100 þ1675 þ1825 þ1500 þ1650 þ2475 þ2675 þ2300 þ2500

2500 2800

270 þ135 0 þ211 þ76 þ270 þ135 þ375 þ240 þ760 þ790 þ550 þ580 þ1460 þ1610 þ1250 þ1400 þ2110 þ2310 þ1900 þ2100 þ3110 þ3410 þ2900 þ3200

Tolerances in mm (1 mm ¼ 103 mm). es ¼ upper deviation. c ei ¼ lower deviation. Source: IS 2101, 1962. a

b

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METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.20

CHAPTER ELEVEN

TABLE 11-13 Tolerancesa for holes for sizes 500 to 3150 mm System of basic hole Limits D10 E8 F9 G6 G7 H6 H7 H8 H9 H10 H11 JS9 K6 M6 N6 P6 R7 S7 T7 U7

ESa ESb ES EI ES EI ES EI ES El ES EI ES El ES EI ES El ES EI ES El ES EI ES El ES El ES EI ES EI ES EI ES EI ES EI ES EI

Diameter steps, mm 500 560

560 630

630 710

710 800

800 900

900 1000

1000 1120

1120 1250

1250 1400

1400 1600

1600 1800

1800 2000

2000 2240

2240 2500

2500 2800

2800 3150

þ540 þ260 þ255 þ145 þ251 þ76 þ66 þ22 þ92 þ22 þ40 0 þ70 0 þ110 0 þ175 0 þ280 0 þ440 0

þ610 þ290 þ285 þ160 þ280 þ80 þ74 þ24 þ103 þ24 þ50 0 þ80 0 þ125 0 þ200 0 þ320 0 þ500 0

þ680 þ320 þ310 þ170 þ316 þ86 þ82 þ26 þ115 þ26 þ56 0 þ90 0 þ140 0 þ230 0 þ360 0 þ560 0

þ770 þ350 þ360 þ195 þ358 þ98 þ94 þ28 þ133 þ28 þ66 0 þ105 0 þ165 0 þ260 0 þ420 0 þ660 0

þ890 þ390 þ415 þ220 þ420 þ110 þ108 þ30 þ155 þ30 þ78 0 þ125 0 þ195 0 þ310 0 þ500 0 þ780 0

þ1030 þ430 þ470 þ240 þ490 þ120 þ124 þ32 þ182 þ32 þ92 0 þ150 0 þ230 0 þ370 0 þ600 0 þ920 0

þ1180 þ480 þ540 þ260 þ570 þ130 þ144 þ34 þ209 þ34 þ110 0 þ175 0 þ280 0 þ440 0 þ700 0 þ1100 0

þ1380 þ520 þ620 þ290 þ685 þ145 þ173 þ38 þ248 þ38 þ135 0 þ210 0 þ330 0 þ540 0 þ860 0 þ1350 0

87.5

100

115

130

155

185

220

270

0 66 40 106 66 132 120 186 250 260 355 365 520 580 625 685 780 840 885 945 1150 1300 1255 1405

0 78 48 126 78 156 140 218 300 330 425 455 640 720 765 845 960 1050 1085 1175 1450 1600 1575 1725

0 0 0 44 50 56 26 30 34 70 80 90 44 50 56 88 100 112 78 88 100 122 138 156 150 155 175 185 210 200 220 225 255 265 300 310 280 310 340 380 430 470 350 380 420 460 520 560 400 450 500 560 620 680 470 520 580 640 710 770 600 660 740 840 940 1050 670 730 820 920 1030 1140

0 92 58 150 92 184 170 262 370 400 520 550 820 920 970 1070 1200 1350 1350 1500 1850 2000 2000 2150

0 110 68 178 110 220 195 305 440 460 615 635 1000 1100 1175 1275 1500 1650 1675 1825 2300 2500 2475 2675

0 135 76 211 135 270 240 375 550 580 760 790 1250 1400 1460 1610 1900 2100 2110 2310 2900 3200 3110 3410

Tolerances in mm (1 mm ¼ 103 mm). ES ¼ upper deviation. c EI ¼ lower deviation. Source: IS 2101, 1962. a

b

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H6 h6

Precision location Normal location Loose location Slack assembly

17 10

21 10

12 8

17 8

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Normal

Heavy force H7 u6 Fir or shrink ﬁt

25 12

20 12 29.5 14.5

26.5 14.5

14.5 14.5 19.5 14.5

þ2.5 14.5 3.5 14.5

þ2 12 3 12

þ11 11 þ27 27 þ43 43 þ110 110

þ185 35

þ325 35

þ138 43

þ6.5 14.5

12 12 16 12

18 24

37 17

31 17

18 17 24 17

þ2 17 4 17

þ8 17

þ13 13 þ33 33 þ52 52 þ130 130

þ202.5 42.5

þ342.5 42.5

þ162 52

þ20.5 þ24 14.5 17 þ34 þ41 18 21 þ59 þ73 27 33 þ85 þ107.5 35 42.5

18

10

þ5 12

Tolerance in microns; 1 micron ¼ 103 mm ¼ mm ¼ 106 m

Normal

H7 s6

MHeavy drive ﬁt

Normal

H7 r6

10 10 13 10

—

8 8 11 8

Normal

—

þ3 10

þ2 8

þ9 9 þ22 22 þ36 36 þ90 90

þ179 29

þ159.5 þ164 19.5 24

þ8 8 þ18 18 þ30 30 þ75 75

þ309 29

þ289.5 þ294 19.5 24

þ7 7 þ14 14 þ25 25 þ60 60

þ116 36

þ100 30

þ85 25

þ17 12 þ28 15 þ47 22 þ69 29

þ14 10 þ22 12 þ38 18 þ54 24

Light press ﬁt Medium drive ﬁt

H7 p6

6 10

þ11 8 þ16 9 þ28 14 þ39.5 19.5

2 10

Normal

3 6

1 8

H7 k6

H11 h11

H9 h9

—

3

True H7 h6 Normal transition Interference H7 m6 Normal transition

Push

H8 b9

Position ﬁts

H8 h8

H8 a9

Normal

H8 d9

Position ﬁts

Normal

H8 e8

Normal

Normal

H7 f 7

H9 c9

Normal

H7 g6

Combination of shaft and hole

Slack running

Precision sliding Normal running Easy running Loose running

Quality of ﬁt

44 17

30

24

40 50

50

65 80

80 100

100 120

þ192 62

þ214 74

þ224 74

þ257 74

þ267 87

Clearance Fit (Fig. 11-6) þ34.5 þ40.5 24.5 28.5 þ60 þ71 30 35 þ106 þ126 46 54 þ160 þ190.5 60 70.5

65

140 160

þ300 100

þ310 100

þ46.5 32.5 þ83 40 þ148 63 þ226.5 81.5

140

120

Diameter steps, mm 160

þ330 100

180

180 225

200

þ355 115

þ375 115

þ52.5 37.5 þ96 46 þ172 72 þ263.5 93.5

200

225

þ395 115

250

250

280

þ420 130

þ460 130

315

þ59 42 þ108 52 þ191 81 þ295.5 105.5

280

315

þ500 140

þ64.5 46.5 þ119 57 þ214 89 þ324.5 114.5

355

355

þ540 140

400

55.5 20.5

81.5 24.5

47.5 24.5

64.5 28.5

44.5 28.5

72.5 28.5

47.5 28.5

84.5 32.5

55.5 32.5

60.5 32.5

66.5 37.5

71.5 37.5

75.5 37.5 92.5 100.5 113.5 121.5 131.5 32.5 32.5 37.5 37.5 37.5

57.5 32.5

41.5 37.5

þ4.5 37.5 8.5 37.5

þ21..5 37.5

96.5 117.5 137.5 162.5 182.5 202.5 227.5 249.5 275.5 24.5 28.5 28.5 32.5 32.5 32.5 37.5 37.5 37.5

53.5 24.5

37.5 24.5

35.5 32.5

Interference Fits (Fig. 11-8) 26.5 30.5 24.5 28.5 35.5 24.5

þ4.5 32.5 7.5 32.5

þ3.5 28.5 6.5 28.5

þ3.5 24.5 5.5 24.5

þ18.5 32.5

Transition Fits (Fig. 11-7) þ12.5 þ15.5 24.5 28.5

þ29 29 þ72 72 þ115 115 þ290 290

þ260 þ290.5 þ310.5 þ341.5 þ361.5 þ391.5 þ433.5 þ473.5 þ513.5 60 70.5 70.5 81.5 81.5 81.5 93.5 93.5 93.5 þ25 25 þ63 63 þ100 100 þ250 250

þ19 19 þ46 46 þ74 74 þ190 190

þ250 60 þ22 22 þ54 54 þ87 87 þ220 220

65.5 20.5

38.5 20.5

21.5 20.5 29.5 20.5

þ2.5 20.5 4.5 20.5

þ9.5 20.5

þ16 16 þ39 39 þ62 62 þ160 160

þ220.5 þ230.5 50.5 50.5

305 42

148 42

84 42

46 42

þ6 42 10 42

þ26 42

þ32 32 þ81 81 þ130 130 þ320 320

þ585.5 105.5

340 42

160 42

88 42

þ645.5 105.5

þ794.5 114.5

379.5 46.5

179.5 46.5

97.5 46.5

424.5 46.5

197.5 46.5

103.5 46.5

51.5 46.5

þ6.5 46.5 10.5 46.5

þ28.5 46.5

þ36 36 þ89 89 þ140 140 þ360 360

þ714.5 114.5

Location and Assembly Fit þ360.5 þ370.5 þ400 þ420 þ450.5 þ480.5 þ541.5 þ601.5 þ661.5 þ753.5 þ833.5 þ913.5 þ1025.5 þ1155.5 þ1314.5 þ1454.5 50.5 50.5 60 60 70.5 70.5 81.5 81.5 81.5 93.5 93.5 93.5 105.5 105.5 114.5 114.5

þ182 62

þ29.5 20.5 þ50 25 þ89 39 þ130.5 50.5

40

30

TABLE 11-14 Mean ﬁt and variation about the mean ﬁt for holes for sizes up to 400 mm

METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.21

0–0.12 0.12–0.24 0.24–0.40 0.40–0.72 0.72–1.20 1.20–2.00 2.00–3.20 3.20–4.80 4.80–7.20 7.20–10.00 10.00–12.60 12.60–16.00

0–3 3–6 6–10 10–18 18–30 30–50 50–80 80–120 120–180 180–250 250–315 315–400

0.006 0.008 0.009 0.011 0.013 0.016 0.019 0.022 0.025 0.029 0.032 0.036

mm

IT6

0.0002 0.0003 0.0004 0.0004 0.0005 0.0006 0.0007 0.0009 0.0010 0.0011 0.0013 0.0014

in

Source: Preferred metric limits and ﬁts—BSI 4500.

in

mm

Basic sizes

TABLE 11-15 International tolerance grades

0.010 0.012 0.015 0.018 0.021 0.025 0.030 0.035 0.040 0.040 0.052 0.057

mm

IT7

0.0004 0.0005 0.0006 0.0007 0.0008 0.0010 0.0012 0.0014 0.0016 0.0018 0.0020 0.0022

in 0.014 0.018 0.022 0.027 0.033 0.039 0.046 0.054 0.063 0.072 0.081 0.089

mm

IT8

0.0006 0.0007 0.0009 0.0011 0.0013 0.0015 0.0018 0.0021 0.0025 0.0028 0.0032 0.0035

in 0.025 0.030 0.036 0.043 0.052 0.062 0.074 0.087 0.100 0.115 0.130 0.140

mm

Grades IT9

0.0010 0.0012 0.0014 0.0017 0.0020 0.0024 0.0029 0.0034 0.0039 0.0045 0.0051 0.0055

in

0.040 0.048 0.058 0.070 0.084 0.100 0.120 0.140 0.160 0.185 0.210 0.230

mm

IT10

0.0016 0.0019 0.0023 0.0028 0.0033 0.0039 0.0047 0.0055 0.0063 0.0073 0.0083 0.0091

in

0.060 0.075 0.090 0.110 0.130 0.160 0.190 0.220 0.250 0.290 0.320 0.360

mm

IT11

0.0024 0.0030 0.0035 0.0043 0.0051 0.0063 0.0075 0.0087 0.0098 0.0114 0.0126 0.0142

in

METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.22

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eic

es

ei

es

ei

es

ei

es

ei

es

ei

j

c

k

d

n

f

p

g

s

h

u

0 0 3 0.12

10 0.40 14 0.56

14 0.56 18 0.72

b

a

18 0.72 24 0.96

24 0.96 30 1.20

30 1.20 40 1.60

40 1.60 50 2.0

50 2.00 65 2.60

65 2.60 80 3.20

80 3.20 100 4.00

100 4.00 120 4.80

120 4.80 140 5.60

140 5.60 160 6.40

160 6.40 180 7.20

180 7.20 200 8.00

200 8.00 225 9.00

225 9.00 2.50 10.00

250 10.00 280 11.20

280 11.20 315 12.60

315 12.60 355 14.20

355 14.20 400 16.00

400 16.00 4.50 18.00

450 18.00 500 20.00

0 0 þ18 þ700

20 20 25 25 30 30 36 36 43 43 43 50 50 56 56 62 62 68 68 800 1,000 1,000 1,200 1,200 1,400 1,400 1,700 1,700 1,700 2,000 2,000 2,000 2,200 2200 2,400 2,400 2,680 2,680 þ22 þ26 þ26 þ32 þ32 þ37 þ37 þ43 þ43 þ43 þ.50 þ50 þ50 þ.56 þ.56 þ62 þ62 þ68 þ68 þ900 þ1,000 þ1,000 þ1,300 þ1,300 þ 1,500 þ1,500 þ1,700 þ1,700 þ1,700 þ2,000 þ2,000 þ2,000 þ2,200 þ2,200 þ2,400 þ2,400 þ2,680 þ2,680

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 þ23 þ28 þ33 þ33 þ41 þ48 þ60 þ70 þ87 þ102 þ124 þ144 þ170 þ190 þ210 þ236 þ258 þ284 þ315 þ350 þ390 þ435 þ490 þ540 þ900 þ1,100 þ1.300 þ1,300 þ1,600 þ1,900 þ2,400 þ2,800 þ3,400 þ4,000 þ4,900 þ5,700 þ6,700 þ7,500 þ8,300 þ9,300 þ10,200 þ11,200 þ12,400 þ13,000 þ15,400 þ17,100 þ19,300 þ21,300

5 6 6 7 7 9 9 10 10 12 12 14 14 14 15 15 15 17 17 18 18 20 20 200 200 200 300 300 400 400 400 400 500 500 600 600 600 600 600 600 700 700 700 700 800 800 þ23 þ28 þ28 þ35 þ35 þ43 þ43 þ53 þ59 þ71 þ79 þ92 þ100 þ108 þ122 þ130 þ140 þ158 þ170 þ190 þ208 þ232 þ252 þ900 þ1,100 þ1,100 þ1,400 þ1,400 þ1,700 þ1,700 þ2,100 þ2,300 þ2,800 þ3,100 þ3,600 þ3,900 þ4,300 þ4,800 þ5,100 þ5,500 þ6,200 þ6,700 þ7,500 þ8,200 þ9,100 þ9,100

20 800 þ22 þ900

mm min mm min

4 200 þ19 þ700

16 600 þ18 þ700

2 100 þ14 þ600

16 600 þ18 þ700

mm min mm min

13 500 þ15 þ600

5 200 þ6 þ200

mm min mm min

10 400 þ12 þ500

20 30 40 50 50 65 65 80 80 100 100 120 120 145 145 145 170 170 170 190 190 210 210 230 230 800 1,200 1,600 2,000 2,500 2,600 2,600 3,100 3,100 3,900 3,900 4,700 4,700 5,700 5,700 5,700 6,700 6,700 6,700 7,500 7,500 8.300 8,300 9,100 9,100 þ4 þ8 þ10 þ12 þ12 þ15 þ15 þ17 þ17 þ20 þ20 þ23 þ23 þ27 þ27 þ27 þ31 þ31 þ31 þ34 þ34 þ37 þ37 þ40 þ40 þ200 þ300 þ400 þ500 þ500 þ600 þ600 þ700 þ700 þ800 þ800 þ900 þ900 þ1,100 þ1,100 þ1,100 þ1,200 þ1,200 þ1,200 þ1,300 þ1,300 þ1,500 þ1,500 þ1,600 þ1,600

mm min mm min

280 290 290 300 300 310 320 340 360 380 410 460 520 580 660 740 820 920 1,050 1,200 1,350 1,500 1,650 11,000 11,400 11,400 11,800 11,800 12200 12,600 13,400 14,200 14,900 16,100 18,100 20,500 22,800 26,000 29,100 32,300 36,200 41,300 47,200 53,200 59,000 64,900 2 2 2 3 4 5 5 7 7 9 9 11 11 11 13 13 13 16 16 18 18 18 20 80 80 80 100 160 200 200 280 280 360 360 450 450 450 510 500 500 600 600 700 700 700 800

6 0.24 10 0.40

60 70 80 95 95 110 110 120 130 140 150 170 180 200 210 230 240 260 280 300 330 360 400 440 480 2,400 2,800 3,100 3,700 3,700 4,300 4,300 4,700 5,100 5,500 5,900 6,700 7,100 7,900 8,300 9,100 9,400 10,200 11,000 11,800 13,000 14,200 15,700 17,300 18,900 0 þ1 þ1 þ1 þ1 þ2 þ2 þ2 þ2 þ2 þ2 þ3 þ3 þ3 þ3 þ3 þ4 þ4 þ4 þ4 44 þ4 þ4 þ4 þ5 0 þ40 þ40 þ40 þ40 þ100 þ100 þ100 þ100 þ100 þ100 þ100 þ100 þ100 þ100 þ100 þ160 þ160 þ160 þ160 þ160 þ160 þ160 þ160 þ200

270 10,600 2 80

3 0.12 6 0.24

Diameter steps

mm min mm min

mm 270 min 10,600 mm 2 min 80

mm in mm in

Tolerance in mm (1 mm ¼ 106 m: 1 min ¼ 106 in). es ¼ upper deviations. c ei ¼ lower deviations. Source: Preferred limits and ﬁts—BSI 4500; IS 2101, 1962.

esb

a

System of basic shaft Limits

TABLE 11-16 Fundamental tolerancea (lm and lin) for shafts for sizes up to 400 mm (16 in)

METAL FITS, TOLERANCES, AND SURFACE TEXTURE

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11.23

IT 2

IT 3

IT 1

—

—

Fine turn, ﬁne bore

Cylindrical grind

Fine cylindrical grind

Surface grind

Fine surface grind

b

5

5

2 10 — —

3 105 105

—

—

—

2 10

5 105

5 105

—

4 105

4 105

3 105

104

10 104

—

4

Straightness of cylinders, gaps and tongues

5 105

5 10

5

Flatness of surfaces

Parallelism of cylinders on diameter

2 105

5 105

—

—

5 105

104

10

4

Parallelism squareness

104

3 104

—

—

3 104

3 104

3 10

4

Anyb other angle

Flat surface

2 105

5 105

2 10

5

5 105

5 105

104

10

104

3 104

104

3 104

3 104

3 104

3 104

103

103 4

Anyb other angle

Cylinders, gaps, tongues Parallelism squareness

Angularity

Expressed as mm/mm length of surface or cylinder

Order of tolerance

A roundness tolerance of 0.016 corresponds to a permissible diametrical variation of 0.032 (ovality). The values quoted are for good class of machine tools. Thrice or twice the above values, i.e., tolerance may have to be allowed for worn machine tools.

IT 4

Turn, bore

a

—

Mill, slot, plane

Drill

Machining processes

Roundnessa (circularity) of cylinders

TABLE 11-17 Relation between machine processes and geometry tolerances

METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.24

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Strictly interchangeable

Strictly interchangeable

Strictly interchangeable

Strictly interchangeable

Selective assembly

Selective assembly

Selective assembly

Selective assembly

Loose

Free

Medium

Snug

Wringing

Tight

Medium force

Heavy force or shrink

Class of ﬁt

Method of assembly

0.005 D1=3 0.005 D1=3

0.005 D1=3

0.005 D1=3 0.005 D1=3

0.005 D1=3

0.001 D

0.0005 D

0.00025 D

0.0035 D1=3

0.005 D1=3

0.0000

0.0035 D1=3

0.005 D1=3

0.0000

0.007 D1=3

0.007 D1=3

0.0025 D2=3

0.01 D1=3

0.01 D1=3

0.004 D2=3

0.02 D1

Shaft tolerance

0.02 D1=3

Hole tolerance

0.0075 D2=3

Allowance

Selected average interference of metal

TABLE 11-18 Formulas for recommended allowances and tolerances (all dimensions in mm)

Used for steel external members that have a high yield stress

Suitable for press ﬁts on locomotive wheels, car wheels, generator and motor armature, and crank discs

Slightly negative allowance; suitable for semipermanent assembly and shrink ﬁts

A metal-to-metal contact ﬁt

Closest ﬁt; zero allowance; suitable where no perceptible shake is permissible under load

Accurate automotive parts and machine tools; suitable for running ﬁt

Suitable for running ﬁt; suitable for shafts of motors, generators, engines, and some automotive parts

Suitable for running ﬁt; considerable freedom permissible; used in agricultural, mining, and generalpurpose machinery

Uses

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11.25

METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.26

CHAPTER ELEVEN

TABLE 11-19 Surface ﬁnisha values (CLA) High quality

Normal quality

Coarse quality

Machining process

Tolerance grade

Finish (lm)

Tolerance grade

Finish (lm)

Tolerance grade

Finish (lm)

Drill Mill, slot, plane Turn, bore Ream Commercial grind Fine turn, bore Hone Broach Fine grind Lap

11 9 8 7 7 6 6 6 5 3

1.6–3.2 0.4–0.8 0.4–0.8 0.4–0.8 0.4–0.8 0.2–0.4 0.1–0.2 0.1–0.2 0.1–0.2 0.05–0 1

12 11 9 8 8 7 7 7 6 4

0.8–1.6 0.8–1.6 0.8–1.6 0.8–1.6 0.4–0.8 0.2–0.4 0.2–0.4 0.2–0.4 0.1–0.2

12 11

1.6–3.2 1.6–3.2

9

1.6–3.2

a

The Roughness Number represents the average departure of the surface from perfection over a prescribed ‘‘sampling length’’ normally 0.8 mm, and is expressed in micrometers (mm). The measurements are normally made along a line at right angles to the general directions of tool marks or scratches on the surface.

1 m ¼ 0:001 mm Old machining symbols

Description

Surface roughness

Unmachined surface. cleaned up by sand blasting, brushing, etc.

5–80 m

Surface to be rough machined if found necessary (to prevent fouling) Surface obtained by rough machining under turning, planing, milling etc. Quality coarser than 9

8–25 m

Finish-machined surface obtained by turning, milling etc. Quality 12–7

1.6–8 m

Fine ﬁnish-machined surface obtained by boring, reaming, grinding etc. Quality 9–6

0.25–1.6 m

Super ﬁnish-machined surface obtained by honing, lapping, super ﬁnish grinding. Quality 7–4

0–0.25 m

FIGURE 11-12 Machining symbols.

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METAL FITS, TOLERANCES, AND SURFACE TEXTURE METAL FITS, TOLERANCES, AND SURFACE TEXTURE

TABLE 11-20 Lay symbols

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11.27

METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.28

CHAPTER ELEVEN

FIGURE 11-13 Application and use of surface-texture symbols. (Baumeister, T., Marks’ Standard Handbook for Mechanical Engineers, 8th ed., McGraw-Hill, 1978.)

TABLE 11-21 Preferred series roughness average values (Ra ) (in lm and lin) lm

lin

lm

lin

lm

lin

lm

lin

lm

lin

0.012 0.025 0.050 0.075 0.10

0.5 1 2 3 4

0.125 0.15 0.20 0.25 0.32 0.40

5 6 8 10 13 16

0.50 0.63 0.80 1.00 1.25 1.60

20 25 32 40 50 63

2.00 2.50 3.20 4.0 5.0 6.3

80 100 125 160 200 250

8.0 10.0 12.5 15.0 20.0 25.0

320 400 500 600 800 1000

Source: Reproduced from Baumeister, T., Marks’ Standard Handbook for Mechanical Engineers, 8th ed., with permission from McGraw-Hill Book Company, New York, 1978.

TABLE 11-22 Preferred series maximum waviness height values mm

in

mm

in

mm

in

0.0005 0.0008 0.0012 0.0020 0.0025 0.005

0.00002 0.00003 0.00005 0.00008 0.0001 0.0002

0.008 0.012 0.020 0.025 0.05 0.08

0.0003 0.0005 0.0008 0.001 0.002 0.003

0.12 0.20 0.25 0.38 0.50 0.80

0.005 0.008 0.010 0.015 0.020 0.030

Source: Reproduced from Baumeister, T., Marks’ Standard Handbook for Mechanical Engineers, 8th ed., with permission from McGraw-Hill Book Company, New York, 1979.

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METAL FITS, TOLERANCES, AND SURFACE TEXTURE METAL FITS, TOLERANCES, AND SURFACE TEXTURE

TABLE 11-23 Surface roughness ranges of production processes

Source: Reproduction from Baumeister, T., Marks’ Standard Handbook for Mechanical Engineers, 8th ed., with permission from McGraw-Hill Book Company, New York, 1979.

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11.29

METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.30

CHAPTER ELEVEN

TABLE 11-24 Application of surface texture values to surface symbols (63)

pﬃﬃﬃﬃﬃ 1:6

(63)

1.6

(32)

pﬃﬃﬃﬃﬃ 0:8

(32)

0:05ﬃ pﬃﬃﬃﬃﬃﬃ 0:8

(32)

0:05 100 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0:8

Roughness average rating is placed at the left of the long leg; the speciﬁcation of only one rating shall indicate the maximum value and any lesser value shall be acceptable

(63)

pﬃﬃﬃﬃﬃ 1:6 3:5

The speciﬁcation of maximum value and minimum value roughness average ratings indicates permissible range of value rating

(63)

p ﬃﬃﬃﬃﬃ 1:6

(32)

0:8 pﬃﬃﬃﬃﬃ ?

Maximum waviness height rating is placed above the horizontal extension; any lesser rating shall be acceptable

(32)

pﬃﬃﬃﬃﬃ 0:8 2:5 ð0:100Þ

Maximum waviness spacing rating is placed above the horizontal extension and to the right of the waviness height rating; any lesser rating shall be acceptable

(32)

0:8 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ? 0:5

Machining is required to produce the surface; the basic amount of stock provided for machining is speciﬁed at the left of the short leg of the symbol

Removal of material by machining is prohibited Lay designation is indicated by the lay symbol placed at the right of the long leg

Roughness sampling length or cutoﬀ rating is placed below the horizontal extension; when no value is shown, 0.80 mm is assumed

Where required, maximum roughness spacing shall be placed at the right of the lay symbol; any lesser rating shall be acceptable

Source: Reproduction from Baumeister, T., Marks’ Standard Handbook for Mechanical Engineers, 8th ed., with permission from McGraw-Hill Book Company, New York, 1979.

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METAL FITS, TOLERANCES, AND SURFACE TEXTURE METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.31

TABLE 11-25 Typical surface texture design requirements (250 min) (125 min)

(63 min)

(32 min)

pﬃﬃﬃﬃﬃ 6:3 pﬃﬃﬃﬃﬃ 3:2

pﬃﬃﬃﬃﬃ 1:60

pﬃﬃﬃﬃﬃ 0:80

Clearance surfaces Rough machine parts

(16 min)

pﬃﬃﬃﬃﬃ 0:40

Mating surfaces (static) Chased and cut threads Clutch-disk faces Surfaces for soft gaskets Piston-pin bores Brake drums Cylinder block, top Gear locating faces Gear shafts and bores Ratchet and pawl teeth Milled threads Rolling surfaces Gearbox faces Piston crowns Turbine-blade dovetails Broached holes Bronze journal bearings Gear teeth Slideways and gibs Press-ﬁt parts Piston-rod bushings Antifraction-bearing seats Sealing surfaces for hydraulic tube ﬁttings

(13 min)

(8 min)

(4 min)

(2 min) (1 min)

pﬃﬃﬃﬃﬃ 0:32 pﬃﬃﬃﬃﬃ 0:20

pﬃﬃﬃﬃﬃ 0:10

pﬃﬃﬃﬃﬃ 0:050 pﬃﬃﬃﬃﬃ 0:025

Motor shafts Gear teeth (heavy loads) Spline shafts O-ring grooves (static) Antifraction-bearing bores and faces Camshaft lobes Compressor-blade airfoils Journals for elastomer lip seals Engine cylinder bores Piston outside diameters Crankshaft bearings Jet-engine stator blades Valve-tappet cam faces Hydraulic-cylinder bores Lapped antifriction bearings Ball-bearing races Piston pins Hydraulic piston rods Carbon-seal mating surfaces Shop-gauge faces Comparator anvils Bearing balls Gauges and mirrors Micrometer anvils

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METAL FITS, TOLERANCES, AND SURFACE TEXTURE

11.32

CHAPTER ELEVEN

TABLE 11-26 Range of surface roughnessa Manufacturing process Manual Hack saw cut Chipping Filing Emery polish Casting Sand casting Permanent mold Die casting Forming Forging Extrusion Rolling

With diﬃculty

Normally

Roughing

0.8–1.6 0.1–0.4

6.3–50 3.2–50 1.6–12.5 0.4–1.6

1.6–3.2

0.8–1.6

6.3–12.5 1.6–6.3 0.8–3.2

1.6–3.2 0.4–0.8 0.4–0.8

3.2–25 0.8–6.3 0.8–3.2

3.2–6.3

Machining Drilling Planing and shaping Face milling Turning Boring Reaming Cylindrical grinding Centerless grinding Surface grinding Broaching Superﬁnishing Honing Lapping

0.8–1.6 0.2–1.6 0.2–1.6 0.4–0.8 0.025–0.4 0.05–0.4 0.025–0.4 0.2–0.8 0.025–0.1 0.025–0.1 0.006–0.05

6.3–25 1.6–12.5 1.6–12.5 1.6–6.3 1.6–6.3 0.8–6.3 0.4–3.2 0.4–3.2 0.4–3.2 0.8–3.2 0.1–0.4 0.1–0.4 0.05–0.4

Gear manufacture Milling with form cutter Milling, spiral bevel Hobbing Shaping Shaving Grinding Lapping

1.6–3.2 1.56–3.2 0.8–3.2 0.4–1.6 0.4–0.8 0.1–0.4 0.05–0.2

3.2–12.5 3.2–12.5 3.2–12.5 1.6–12.5 0.8–3.2 0.4–0.8 0.2–0.8

1.6–3.2

3.2–50 0.1–6.3 0.2–0.8 0.05–0.1

Surface process Shot blast Abrasive belt Fiber wheel brushing Cloth buﬃng a

0.1–0.2 0.012–0.05

Surface roughness in mm (1mm ¼ 103 mm ¼ 106 m).

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12.5–25

12.5–50 6.3–50 6.3–50 6.3–12.50 3.2–6.3 3.2–6.3 3.2–6.3

12.5–50 12.5–25 12.5–50 12.5–250

0.8–1

METAL FITS, TOLERANCES, AND SURFACE TEXTURE METAL FITS, TOLERANCES, AND SURFACE TEXTURE

FIGURE 11-14 Symbols for tolerances of form and position.

11.33

FIGURE 11-15 Rivet symbols

REFERENCES 1. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962. 2. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I, Suma Publishers, Bangalore, India, 1986. 3. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 4. Black, P. H., and O. Eugene Adams, Jr., Machine Design, McGraw-Hill Publishing Company, New York. 5. Baumeister, T., Marks’ Standard Handbook for Mechanical Engineers, 8th ed., McGraw-Hill Publishing Company, New York, 1978. 6. Maleev, V. L., and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954. 7. Shigley, J. E., Machine Design, McGraw-Hill Publishing Company, New York, 1956. 8. Vallance, A., and V. L. Doughtie, Design of Machine Members, McGraw-Hill Publishing Company, New York, 1951. 9. British Standard Institution. 10. Bureau of Indian Standards.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

12 DESIGN OF WELDED JOINTS

SYMBOLS2;3;4 A A0 ¼ l! b c cx cy c1 c2 c3 d ex ey h i Ix , Iy , Iz J J! Kf l lt Mb Mt na Na Nb P Px Py

area of ﬂange material held by welds in shear, m2 (in2 ) length of weld when weld is treated as a line, m (in) width of connection, m (in) distance to outer ﬁber (also with suﬃxes), m (in) distance of x axis to face, m (in) distance of y axis to face, m (in) distance of weld edge parallel to x-axis from the center of weld, to left, m (in) distance of weld edge from parallel to x-axis from the center of weld, to right, m (in) distance from farthest weld corner, Q, to the center of gravity of weld, m (in) (Fig. 12-8) depth of connection, m (in) eccentricity of Pz and Py about the center of weld, m (in) eccentricity of Px about the center of weld, m (in) thickness of plate (also with suﬃxes), m (in) number of welds moment of inertia of weld about x, y, and z axes respectively, m4 , cm4 (in4 ) moment of inertia, polar, m4 , cm4 (in4 ) polar moment of inertia of weld, when weld is treated as a line, m3 , cm3 (in3 ) fatigue stress-concentration factor (Table 12-7) eﬀective length of weld, m (in) total length of weld, m (in) bending moment, N m (lbf in) twisting moment, N m (lbf in) actual factor of safety or reliability factor fatigue life (for which sfa is known) for fatigue strength sfa , cycle fatigue life (required) for fatigue strength sfb , cycle load on the joint, kN (lbf ) component of P in x direction, kN (lbf ) component of P in y direction, kN (lbf )

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DESIGN OF WELDED JOINTS

12.2

Pz r R t V w Z Z! 0 sfa sfb d e 0

CHAPTER TWELVE

component of P in z direction, kN (lbf ) distance to outer ﬁber, m (in) ratio of calculated leg size for continuous weld to the actual leg size to be used for intermittent weld throat dimension of weld, m (in) shear load, kN (lbf ) size of weld leg, m (in) section modulus, m3 (in3 ) section modulus of weld, when weld is treated as line (also with suﬃxes, m2 (in2 ) normal stress in the weld (in standard design formula), MPa (psi) force per unit length of weld (in standard design formula) when weld treated as a line, kN/m (lbf/in) fatigue strength (known) for fatigue life Na , MPa (psi) fatigue strength (allowable) for fatigue life Nb , MPa (psi) design stress, MPa (psi) elastic limit, MPa (psi) shear stress in the weld (in standard design formula), MPa (psi) shear force per unit length of weld (in standard design formula) when weld is treated as a line, kN/m (lbf/in) angle, deg eﬃciency of joint

Particular

Formula

FILLET WELD The throat thickness t, for case with equal legs, of weld (Fig. 12-1)

t ¼ w sin 458 ¼ 0:707w

ð12-1aÞ

The allowable load on the weld

P ¼ 0:707 i wl

ð12-1bÞ

FIGURE 12-1 Fillet weld.

FIGURE 12-2 A typical butt-weld joint.

BUTT WELD The average normal stress in a butt weld subjected to tensile or compression loading (Fig. 12-2)

F ð12-2Þ hl where h is the throat dimension. The dimensions of throat (t) are the same as the thickness of plate (h).

¼

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DESIGN OF WELDED JOINTS DESIGN OF WELDED JOINTS

Particular

12.3

Formula

The throat dimension (h) does not include the reinforcement. The average shear stress in butt weld

The allowable load on the weld

F hl

ð12-3Þ

Fa ¼ a hl

ð12-4Þ

¼

TRANSVERSE FILLET WELD The average normal tensile stress

The average normal tensile stress for the case of transverse ﬁllet weld shown in Fig. 12-3.

¼

F F ¼ wl cos 458 0:707wl

ð12-5Þ

¼

F 0:707hl

ð12-6Þ

The stress concentration occurs at A and B on the horizontal leg and at B on the vertical leg in the weld according to photoelastic tests conducted by Norris.1 A double ﬁllet lap weld joint.

Refer to Fig. 12-4.

FIGURE 12-3 A transverse ﬁllet weld.

FIGURE 12-4 A double-ﬁllet lap-weld joint.

PARALLEL FILLET WELD (Fig. 12-5) The average shear stress in the weld

¼

P 0:707wl

ð12-7aÞ

where w ¼ dimension of leg of weld. w can be replaced by h (thickness of plate) when w and h are of same dimension. Either symbol F or P can be used for force or load depending on symbols used in ﬁgures in this chapter. The shear stress in a reinforced ﬁllet weld

¼

P 0:85wl

where throat t is taken as 0.85w

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ð12-7bÞ

DESIGN OF WELDED JOINTS

12.4

CHAPTER TWELVE

Particular

Formula

LENGTH OF WELD The eﬀective length of weld (Fig. 12-5)

l ¼ lt

i 4

ð12-8Þ

where i ¼ total number of free ends The total length of weld (Fig. 12-5)

The relation between the length l1 and l2 (Fig. 12-5)

FIGURE 12-5 Parallel ﬁllet weld.

lt ¼

P where lt ¼ 2ðl1 þ l2 Þ 0:707 wa

l1 l l þ l2 l ¼ 2¼ 1 ¼ t L x x L 2L

ð12-9Þ

ð12-10Þ

FIGURE 12-6

ECCENTRICITY IN A FILLET WELD The bending stress due to ﬁllet weld placed on only one side of the plate (Fig. 12-6)

b ¼ ¼

4Pw 4ð0:707wÞ2 l

¼

2P wl

ð12-11Þ

P 1:414wl

ð12-12Þ

The stress due to tensile load t ¼ The combined normal stress at the root of the weld

n ¼ t þ b ¼

The shear stress ¼ The maximum normal stress The maximum shear stress

4Mb ð0:707wÞ2 l

P 2P þ 1:414wl wl

P 0:707wl

max ¼ 12 ðn þ max ¼ 12

ð12-13Þ ð12-14Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2n þ 4 2 Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2n þ 4 2

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ð12-15Þ ð12-16Þ

DESIGN OF WELDED JOINTS DESIGN OF WELDED JOINTS

Particular

12.5

Formula

ECCENTRIC LOADS Moment acting at right angles to the plane of welded joint (Fig. 12-6) Direct load per unit length of weld

Pd ¼

P l

ð12-17Þ

Load due to bending per unit length of weld

Pn ¼

Pey I

ð12-18Þ

The resultant load or force

PR ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ P2d þ P2n

ð12-19Þ

Per J

ð12-20Þ

Moment acting in the plane of the weld (Fig. 12-7) Load due to twisting moment per unit length of weld

Pn ¼

The resultant load (Fig. 12-7)

PR ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ P2d þ P2n þ 2Pd Pn cos

ð12-21Þ

l2 where cos ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 l1 þ l22

FIGURE 12-7

STRESSES Bending The bending stress

b ¼

Mb wZw

ð12-22aÞ

Mb (treating weld as a line) Zw

ð12-22bÞ

or 0b ¼

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DESIGN OF WELDED JOINTS

12.6

CHAPTER TWELVE

Particular

Formula

Torsion The shear stress due to torsion

Mt r wJw

ð12-23aÞ

0 ¼

Mt r (treating weld as a line) Jw

ð12-23bÞ

0max

1 M ¼ 4 bþ 2 Zw

0 max

1 ¼ 2

¼ or

Combined bending and torsion The resultant or maximum induced normal force per unit throat of weld

The resultant induced torsional force per unit throat of weld

The required leg size of the weld when weld is treated as a line The resultant normal stress induced in the weld

2

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Mb 2 Mt r 2 þ4 Jw Zw

0 actual force 0 or max ¼ max0 permissible force a or a0 2 3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 4 Mb Mb 2 Mt r 2 5 þ þ4 max ¼ wZw 2 wZw wJw

w¼

The resultant shear stress induced in the weld max The required leg size of weld when the weld area is considered

3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Mb 2 Mt r 2 5 þ4 Zw Jw

1 ¼ 2

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Mb 2 Mt r 2 þ4 wJw wZw

ð12-24Þ

ð12-25Þ ð12-26Þ

ð12-27Þ

ð12-28Þ

actual maximum stress induced in the weld permissible stress max or max ¼ a or a

w¼

FATIGUE STRENGTH The fatigue strength related to fatigue life can be expressed by the empirical formula

sfa ¼ sfb or

Na ¼ Nb

Nb Na

sfb sfa

k

ð12-29Þ

1=k

where k ¼ 0:13 for butt welds ¼ 0:18 for plates in bending, axial tension, or compression

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ð12-30Þ

DESIGN OF WELDED JOINTS DESIGN OF WELDED JOINTS

Particular

12.7

Formula

DESIGN STRESS OF WELDS The design stress

d ¼

a na

ð12-31Þ

where

The design stress for completely reversed load

na ¼ actual safety factor or reliability factor ¼ 3 to 4 f fd ¼ ð12-32Þ na Kf

THE STRENGTH ANALYSIS OF A TYPICAL WELD JOINT SUBJECTED TO ECCENTRIC LOADING (Fig. 12-8)2;3;4 Throughout the analysis of a weld joint, the weld is treated as a line Area of cross section of weld

A ¼ ð2b þ dÞw

The distance of weld edge parallel to x axis from the center of weld, to left

c1 ¼

The distance of weld edge parallel to x axis from the center of weld, to right

c2 ¼

The distance from farthest weld corner, Q, to the center of gravity of weld

The moment of inertia of weld about x axis

The moment of inertia of weld about y axis

b2 2b þ d

bðb þ dÞ 2b þ d sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2ﬃ d c3 ¼ c22 þ 2

ð12-33Þ

ð12-34Þ ð12-35Þ

ð12-36Þ

Ix ¼

wd 2 ðd þ 6bÞ 12

ð12-37Þ

Iy ¼

wb3 ð2d þ bÞ 3ðd þ 2bÞ

ð12-38Þ

The moment of inertia of weld about z axis

Iz ¼ I x þ Iy

The section modulus of weld, about x axis

Zwx ¼

Ix wd ¼ ðd þ 6bÞ ðd=2Þ 6

Zwy ¼

Iy wb2 ð2d þ bÞ ¼ c2 3ðb þ dÞ

ð12-40Þ

Zwz ¼

Iz c3

ð12-41Þ

The section modulus of weld, about y axis

The section modulus of weld, about z axis

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ð12-39Þ

DESIGN OF WELDED JOINTS

12.8

CHAPTER TWELVE

Particular

Formula

FIGURE 12-8 A typical weld joint subjected to Eccentric Loading. K. Lingaiah and B. R. Narayana Iyengar, Machine Data Handbook ( fps Units), Engineering College Cooperative Society, Bangalore, India, 1962; K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986; and K. Lingaiah, Machine Design Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986.

Pz component Throughout the analysis of this problem the weld is considered as a line The force per unit length of weld due to direct force Pz

0zd ¼

The force per unit length of weld an account of bending at the farthest weld corner, Q, due to eccentricity ex of load Pz

Pz A0

ð12-42Þ

0zb1 ¼

Pz ex Zwy

ð12-43Þ

The force per unit length of weld an account of bending at the farthest weld corner, Q, due to eccentricity ey of load Pz

0zb2 ¼

Pz ey Zwx

ð12-44Þ

The total force per unit length of weld due to bending

0zb ¼ 0zb1 þ 0zb2

ð12-45Þ

0z ¼ 0zd þ 0zb

ð12-46Þ

The combined force per unit length of weld due to load Pz

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DESIGN OF WELDED JOINTS DESIGN OF WELDED JOINTS

Particular

12.9

Formula

Px component The force per unit length of weld due to direct shear force Px which acts in the horizontal direction (Fig. 12-8)

0 xd ¼

The twisting moment

Mtx ¼ Px ey

ð12-48Þ

The shear force per unit length due to twisting moment Mtx

0 ¼ tx

Mtx c3 Jwz

ð12-49Þ

0 The vertical component of tx

0 The horizontal component of tx

Px A0

ð12-47Þ

0 txv ¼

Mtx c3 cos Jwz

ð12-50Þ

0 ¼ txh

Mtx c3 sin Jwz

ð12-51Þ

where c3 ¼ distance from the center of gravity of the weld to the point being analyzed (i.e., Q) cos

The resultant shear force per unit length of weld in the horizontal direction due to Px only

¼

c2 (Fig. 12-8) c3

0 0 0 ¼ xd ¼ txh txrh

ð12-52Þ

Py component The direct shear force per unit length of weld parallel to y direction due to force Py (Fig. 12-8)

0 ¼ yd

The twisting moment

Mty ¼ Py ex

ð12-54Þ

The shear force per unit length of weld due to twisting moment Mty

ty0 ¼

Mty c3 Jwz

ð12-55Þ

The vertical component of ty0

0 ¼ ty0 cos tyv

ð12-56Þ

The horizontal component of ty0

0 tyh ¼ ty0 sin

ð12-57Þ

0 0 0 tyrv ¼ yd þ tyv

ð12-58Þ

The resultant shear force per unit length of weld in the vertical direction due to Py only

Py A0

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ð12-53Þ

DESIGN OF WELDED JOINTS

12.10

CHAPTER TWELVE

Particular

Formula

COMBINED FORCE DUE TO Px , Py , AND Pz AT POINT Q (Fig. 12-8) From Eqs. (12-46), (12-50), (12-52), (12-57), and (12-58) The total shear force per unit length of weld in the x direction (Fig. 12-8) from Eqs. (12-52) and (12-57) The total shear force per unit length of weld in the y direction (Fig. 12-8) from Eqs. (12-50) and (12-58)

0 0 x0 ¼ tzrh þ tyh

ð12-59Þ

0 0 þ tyrv y0 ¼ txv

ð12-60Þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x02 þ y02

ð12-61Þ

The resultant shear force per unit length of weld at point Q due to Px and Py forces (Fig. 12-8) from Eqs. (12-59) and (12-60)

0 ¼

The resultant actual force per unit length of weld (treating weld as a line) due to components Px , Py , and Pz at point Q from Eqs. (12-46) and (12-61)

0actual ¼

The leg size of the weld

w0 ¼

For the AWS standard location of elements of welding symbol, weld symbols and direction for making weld

Refer to Figs. 12-9 to 12-11.

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 02 02 z þ

0actual 0allowable

FIGURE 12-9 The AWS Standard location of elements of a welding symbol.

FIGURE 12-10 Weld symbols

FIGURE 12-11

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ð12-62Þ

ð12-63Þ

DESIGN OF WELDED JOINTS DESIGN OF WELDED JOINTS

Particular

12.11

Formula

GENERAL For further data on welded joint design

Refer to Tables 12-1 to 12-16.

REFERENCES 1. Norris, C. H., Photoelastic Investigation of Stress Distribution in Transverse Fillet Welds, Welding Journal, Vol. 24, p. 557, 1945. 2. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962. 3. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 4. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 5. Welding Handbook, 3rd ed., American Welding Society, 1950. 6. Bureau of Indian Standards. 7. Lingaiah, K., Machine Design Data Handbook, McGraw-Hill Publishing Company, New York, 1994.

BIBLIOGRAPHY Design of Weldments, The James F. Lincoln Arc Welding Foundation, Cleveland, Ohio, 1968. Design of Welded Structures, The James F. Lincoln Arc Welding Foundation, Cleveland, Ohio, 1966. Maleev, V. L. and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954. Procedure Handbook of Arc Welding Design and Practice, The James F. Lincoln Arc Welding Foundation, Cleveland, Ohio, 1950. Salakian, A. G., and G. E. Claussen, Stress Distribution in Fillet Welds: A Review of the Literature, Welding Journal, Vol. 16, pp. 1–24, May 1937. Shigley, J. E., Machine Design, McGraw-Hill Publishing Company, New York, 1956. Spotts, M. F., Design of Machine Elements, 5th ed., Prentice-Hall of India Private Ltd., New Delhi, 1978. Vallance, A., and V. L. Doughtie, Design of Machine Members, McGraw-Hill Publishing Company, New York, 1951.

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DESIGN OF WELDED JOINTS

12.12

CHAPTER TWELVE

TABLE 12-1 Weld-stress formulas

Source: Welding Handbook, 3rd edition, American Welding Society, 1950.

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DESIGN OF WELDED JOINTS DESIGN OF WELDED JOINTS

12.13

TABLE 12-2 Design formulas used to obtain stress in weld Standard design formula, MPa (psi)

Type of loading

Treating the weld as a line, kN/m (lbf/in)

Primary Welds (transmit entire load) Tension or compression

¼

P A

0 ¼

P Iw

Vertical shear

¼

V A

0 ¼

V Iw

Bending

b ¼

Mb Z

0 ¼

Mb Zw

Twisting

¼

Mb c J

0 ¼

Mc Jw

Secondary Welds (hold section together; low stress) Horizontal shear

¼

VAy Ih

0 ¼

VAy I

Torsional horizontal shear

¼

Mt c J

0 ¼

Mt ch J

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DESIGN OF WELDED JOINTS

TABLE 12-3 Properties of weld—treating weld as line Outline of welded joint b = width, d = depth

Bending (about horizontal axis x–x)

Twisting

Zw ¼

d2 6

Jw ¼

d3 12

Zw ¼

d2 3

Jw ¼

dð3b2 þ d 2 Þ 6

Jw ¼

b3 þ 3bd 2 6

Jw ¼

ðb þ dÞ4 6b2 d 2 12ðb þ dÞ

Jw ¼

ð2b þ dÞ3 b2 ðb þ dÞ2 12 2b þ d

Jw ¼

ðb þ 2dÞ3 d 2 ðb þ dÞ2 12 b þ 2d

Jw ¼

ðb þ dÞ3 6

Jw ¼

ðb þ 2dÞ3 d 2 ðb þ dÞ2 12 b þ 2d

Jw ¼

d 3 ð4b þ dÞ b3 þ 6ðb þ dÞ 6

Jw ¼

b3 þ 3bd 2 þ d 3 6

Jw ¼

2b3 þ 6bd 2 þ d 3 6

Jw ¼

d 3 4

Zw ¼ bd

Zw ¼

4bd þ d 2 d 2 ð4bd þ dÞ ¼ 6 6ð2b þ dÞ top bottom

Zw ¼ bd þ

Zw ¼

2bd þ d 2 d 2 ð2b þ dÞ ¼ 3 3ðb þ dÞ top bottom

Zw ¼ bd þ

Zw ¼

Zw ¼

d2 6

d2 3

2bd þ d 2 d 2 ð2b þ dÞ ¼ 3 2ðb þ dÞ top bottom 4bd þ d 3 4bd 2 þ d 3 ¼ 3 6b þ 3d top bottom

Zw ¼ bd þ

d2 3

Zw ¼ 2bd þ

d2 3

Zw ¼

d 2 4

Zw ¼

d 2 þ D2 2

—

—

Jw ¼

b3 12

Note: Multiply the values Jw by the size of the weld w to obtain polar moment of inertia Jo of the weld.

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DESIGN OF WELDED JOINTS DESIGN OF WELDED JOINTS

12.15

TABLE 12-4 Types of welds and symbols

Form of weld

Sectional representation

Appropriate symbol

Fillet

Sectional representation

Form of weld

Appropriate symbol

Plug or slot

Square butt Backing strip

Single-V butt Double-V butt

Spot

Single-U butt Double-U butt

Seam

Single-bevel butt

Mashed seam

Double-bevel butt Stitch Single-J butt Mashed stitch Double-J butt Stud

Projection

Bead (edge or seal)

Flash

Sealing run

Butt resistance or Pressure (upset)

IS: 696-1960(b) Bureau of Indian Standards.

TABLE 12-5A Properties of common welding rods Melting point

Tensile strength

Rods

8F

8C

MPa

Copper-coated mild steel High-tensile low-alloy steel Cast iron Stainless steel Bronze Ever dur Aluminum White metal Low-temperature brazing rod

2750 2750 2200 2550 1600–1625 1870 1190 715 1170–1185

1510 1510 1204 1399 870–885 1019 643 379 632–640

358.5 52 427.5 62 275.5 40 551.5 80 379.0 55 344.5 50 110.5 16 358.5 52 Varies with parent metal

kpsi

Elongation in 50 mm (2 in), % 23 20 — 30 — 20 25 8

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DESIGN OF WELDED JOINTS

TABLE 12-5 Allowable loads on mild-steel ﬁllet welds Allowable static load per linear cm of weld Bare welding rod Normal weld

Shielding arc

Parallel weld

Normal weld

Parallel weld

Size of weld, mm

N

lbf

N

lbf

N

lbf

N

lbf

23 55 66 88 10 10 12 12 14 14 15 15 18 18 20 20

1667.1 2745.8 3285.2 4373.7 5491.7 6570.4 7659.0 8237.5 9855.6 10944.2

375 617 738.5 983 1235 1477 1722 1852 2216 2460

1323.9 2186.9 2628.2 3501.0 4079.5 5263.3 6129.1 6570.4 7884.5 8757.3

298 491 590 787 983 1182 1378 1477 1772 1968

2059.4 3432.3 4118.8 5491.7 6864.6 8237.5 9581.0 10296.9 12326.9 13680.2

462 772 926 1235 1543 1852 2154 2315 2772 3075

1667.1 2745.8 3285.2 4373.7 5491.7 6570.4 7659.0 8237.5 9855.6 10944.2

375 617 738.5 983 1235 1477 1722 1852 2216 2460

Note: For intermediate sizes interpolate the values. Source: Welding Handbook, American Welding Society, 1950.

TABLE 12-6 Design stresses for welds made with mild-steel electrodes Bare electrodes u ¼ 274.6–380.5 MPa (40–55 kpsi) Type of load Butt Welds Tension Compression Shear Fillet Welds Shear

Covered electrodes u ¼ 416.8–519.7 MPa (60–75 kpsi)

Static loads

Dynamic loads

Static loads

Dynamic loads

MPa kpsi MPa kpsi MPa kpsi

89.70 13.0 103.40 15.0 55.10 8.0

34.30 5.0 34.30 5.0 20.60 3.0

110.30 16.0 124.10 19.5 68.90 10.0

55.10 8.0 55.10 8.0 83.40 12.0

MPa kpsi

78.0 11.5

20.60 3.0

96.50 14.0

34.30 5.0

Source: Welding Handbook, American Welding Society, 1950.

TABLE 12-7 Fatigue stress-concentration factors Kf Type of weld

Stress-concentration factors, Kf

Reinforced butt weld Toe of transverse ﬁllet weld or normal ﬁllet weld End of parallel weld or longitudinal weld T-butt joint with sharp corners

1.2 1.5 2.7 2.0

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DESIGN OF WELDED JOINTS DESIGN OF WELDED JOINTS

12.17

TABLE 12-8 Strength of shielded-arc ﬂush steel welds Limit stress Deposited metal

Type of stress Tension MPa kpsi Compression MPa kpsi Bending MPa kpsi Shear MPa kpsi Shear and tension MPa kpsi

Recommended design stress

Base metal elastic limit, e

Elastic limit, e

Endurance limit, f

Static load

Load varies from O to F

Load varies from +F to –F

220.60 32

275.80 40

151.70 22

110.30 16

100.00 14.5

55.20 8.0

241.20 35.0

303.40 44.0

— —

124.20 10.0

110.30 16.0

55.23 8.0

241.20 35

303.40 44

179.30 26

124.20 18

110.30 16

62.10 9.0

137.90 20

165.40 24

— —

75.80 11

68.90 10

34.50 5

— —

— —

— —

75.80 11

68.90 10

34.50 5

For bare electrode welds, the allowable stress must be multiplied by 0.8 and for gas welds, they should be multiplied by 0.8 to 0.85.

TABLE 12-9 Length and spacing of intermittent welds R, % of continuous weld 75 66 60 57 50 44 43 40 37 33 30 25 20 16

Length of intermittent welds and distance between centers, mm 75–100a 100–150 75–125 50–100

75–150

100–175 100–200 100–225

75–175 50–125 50–160 50–200 50–250 50–300

100–250 75–200 75–225 75–250 75–300

100–300

a

75–100 means a weld 75 mm long with a distance of 100 mm between the centers of two consecutive welds. R in % ¼

calculated leg size (continuous) actual leg size used (intermittent)

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DESIGN OF WELDED JOINTS

12.18

CHAPTER TWELVE

TABLE 12-10 Fatigue data on butt weld joints (average strength values) Endurance strength, f Base metal Material and joint Carbon steel With bead, or welded With bead, tempered 923 K (6508C) Bead machined oﬀ Bead machined oﬀ, tempered 923 K (6508C) Alloy steel As welded Stress-relieved

a

MPa kpsi MPa kpsi MPa kpsi MPa kpsi MPa kpsi MPa kpsi MPa kpsi MPa kpsi

u

y

423 61.3

235 34.0

745.6 108

K ¼ 1 a

K ¼ 0a

K ¼ 0:5 a

No. of cycles 2 106

100.0 14.5 98.0 14 121.6 17.5

152.0 22.0 148.0 21.5 198.0 28.5

155.9 22.5 160.8 23 198.0 28.5

227.5 33 214.7 31 335.3 48.5

253.0 37 264.7 38 304.0 44

114.7 16.5

193.1 28

132.3 19

340.2 49.3

292.2 42.4

400.1 58 456.0 66

539.3 78 593.2 86

368.7 53.5 379.5 55

672.0 97.5

K ¼ þ1 steady; K ¼ 1 complete reversal; K ¼ 0 repeated; K ¼ 12 ﬂuctuating; K ¼

min stress . max stress

Source: Design of Weldments, The James F. Lincoln Arc Welding Foundation, Cleveland, Ohio, 1968.

TABLE 12-11 Stresses as per the AISC Code for weld metal Load type

Weld type

Tension Compression Shear Bending Bending

Butt Butt Butt or ﬁllet Butt Butt

TABLE 12-12 Properties of weld metal

Allowable stress, a 0.60 y 0.60 y 0.40 y 0.90 y 0.60 y –0.66 y

Tensile strength

Yield strength

AWS electrode numbera

Elongation %

MPa

kpsi

MPa

kpsi

E E E E E E

17–25 22 19 14–17 13–16 14

427 483 550 620 690 828

62 70 80 90 100 120

345 393 462 530 600 738

50 57 67 77 87 107

60xx 70xx 80xx 90xx 100xx 120xx

a The American Welding Society (AWS) Speciﬁcation Code numbering system for electrodes.

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DESIGN OF WELDED JOINTS DESIGN OF WELDED JOINTS

12.19

TABLE 12-13 Selection of ﬁllet weld sizes by rule-of-thumb (all dimensions in mm) Designing for rigidity Plate thickness, h mm

Designing for strength, full-strength weld (w ¼ 3=4h)

50% of full-strength weld (w ¼ 3=8h)

33% of full-strength weld (w ¼ 1=4h)

6 8 9.5 11 12.5 14 15.5 19 22 25 28.5 31.5 35 37.5 41 44 50 54 57 60 62.5 66.5 70 75

4.5 6 8 9.5 9.5 11 12.5 14 15.5 19 22 25 25 28.5 31.5 35 37.5 41 44 44 47.5 50 50 56

4.5 4.5 4.5 4.5 4.5 6 6 8 9.5 9.5 11 12.5 12.5 14 15.5 19 19 22 22 25 25 25 25 28.5

4.5 4.5 4.5 4.5 4.5 6 6 6 8 8 8 8 9.5 9.5 11 11 12.5 14 14 15.5 15.5 19 19 19

Source: Welding Handbook, 3rd edition, American Welding Society, 1950.

TABLE 12-14 Equivalent length of ﬁllet weld to replace rivets

Rivet diameter, mm 12.5 15.5 19 22 25 a

Length of ﬁllet weldsa ‘‘Fusion Code’’ (structural) shielded arc welding, mm

Rivet shear value at 100 MPa (10.2 kgf/mm2 ) MPa

kgf/mm2

6-mm ﬁllet

8-mm ﬁllet

9.5-mm ﬁllet

12.5-mm ﬁllet

15.5-mm ﬁllet

20.0 31.5 45.5 61.0 81.2

2.07 3.23 4.66 6.34 8.28

37.5 56 75 105 133

31.5 44.0 61.5 85.5 108.0

28.5 37.5 54 73 92

22 31.5 41 54 70

19 25 35 44 56

6 mm is added to calculated length of bead for starting and stopping the arc.

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DESIGN OF WELDED JOINTS

12.20

CHAPTER TWELVE

TABLE 12-15 Stress concentration factor, K Stress concentration factor, K Weld type and metal Weld metal Butt welds with full penetration End ﬁllet welds Parallel ﬁllet welds Base metal Toe of machined butt weld Toe of unmachined butt weld Toe of machined end ﬁllet weld with leg ratio 1 : 1.5 Toe of unmachined end ﬁllet weld with leg ratio 1 : 1.5 Parallel ﬁllet weld Stiﬀening ribs and partitions welded with end ﬁllet welds having smooth transitions at the toes Butt and T-welded corner plates Butt and T-welded corner plates, but with smooth transitions in the shape of the plates and with machined welds Lap-welded corner plates

Low-carbon steel

Low-alloy steel

1.2 2 3.5

1.4 2.5 4.5

1.2 1.5 2 2.7 3.5

1.4 1.9 2.5 3.3 4.5

1.5 2.7 1.5

1.9 3.3 1.9

2.7

3.3

TABLE 12-16 Allowable stresses for welds under static loads Allowable stresses

Weld type and process

Tension, ta

Compression, ca

Shear, a

Automatic and hand welding with shielded arc and butt welding Hand welding with ordinary quality electrodes Resistance spot welding

t a 0.9t 0.9t

t t t

0.65t 0.6t 0.5t

a

t is the allowable stress in tension of the base metal of the weld.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

13 RIVETED JOINTS SYMBOLS2;3;4 A b c d Di e or l F h hc , h1 , h2 i I J K¼ m Mb p pc pd pt Pf Z a c a

F F0

area of cross-section, m2 (in2 ) the cross-sectional area of rivet shank, m2 (in2 ) breadth of cover plates (also with suﬃxes), m (in) distance from the centroid of the rivet group to the critical rivet, m (in) diameter of rivet, m (in) internal diameter of pressure vessel, m (mm) eccentricity of loading, m (in) force on plate or rivets (also with suﬃxes), kN (lbf) thickness of plate or shell, m (in) thickness of cover plate (butt strap), m (in) number of rivets in a pitch ﬁne (also with suﬃxes 1 and 2, respectively, for single shear and double shear rivets) moment of inertia, area, m4 , cm4 (in4 ) moment of inertia, polar, m4 , cm4 (in4 ) coeﬃcient (Table 13-11) margin, m (in) bending moment, N m (lbf in) pitch on the gauge line or longitudinal pitch, m (in) pitch along the caulking edge, m (in) diagonal pitch, m (in) transverse pitch, m (in) intensity of ﬂuid pressure, MPa (psi) section modulus of the angle section, m3 , cm3 (in3 ) hoop stress in pressure vessel or normal stress in plate, MPa (psi) allowable normal stress, MPa (psi) crushing stress in rivets, MPa (psi) shear stress in rivet, MPa (psi) allowable shear stress, MPa (psi) eﬃciency of the riveted joint angle between a line drawn from the centroid of the rivet group to the critical rivet and the horizontal (Fig. 13-5)

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RIVETED JOINTS

13.2

CHAPTER THIRTEEN

Particular

Formula

PRESSURE VESSELS Thickness of main plates The thickness of plate of the pressure vessel with longitudinal joint

h¼

P f Di 2

ð13-1Þ

For thickness of boiler plates and suggested types of joints

Refer to Tables 13-1 and 13-2.

The thickness of plate of the pressure vessel with circumferential joint

h¼

For allowable stress and eﬃciency of joints

Refer to Tables 13-3, 13-4, 13-5, and 13-6.

P f Di 4

ð13-2Þ

PITCHES Lap joints The diagonal pitch (staggered) (Fig. 13-1) for p, pt , and pd

The distance between rows or transverse pitch or back pitch (staggered)

The rivet diameter

pd ¼

2p þ d 3

ð13-3Þ

Refer to Tables 13-7 and 13-8 for rivets for general purposes and boiler rivets. sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2p þ d 2 p ð13-4Þ pt ¼ 3 2 pﬃﬃﬃ pﬃﬃﬃ d ¼ 0:19 h to 0:2 h

SI

ð13-5aÞ

where h and d in m pﬃﬃﬃ pﬃﬃﬃ d ¼ 1:2 h to 1:4 h

USCS

ð13-5bÞ

where h and d in in pﬃﬃﬃ pﬃﬃﬃ d ¼ 6 h to 6:3 h

CM ð13-5cÞ

where h and d on mm FIGURE 13-1 Pitch relation

TABLE 13-1 Suggested types of joint Diameter of shell, mm (in) Thickness of shell, mm (in)

Type of joint

600–1800 (24–72) 900–2150 (36–84) 1500–2750 (60–108)

Double-riveted Triple-riveted Quadruple-riveted

6–12 (0.25–0.5) 7.5–25 (0.31–1.0) 9.0–44 (0.375–1.75)

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RIVETED JOINTS RIVETED JOINTS

13.3

TABLE 13-2 Minimum thickness of boiler plates Shell plates

Tube sheets of ﬁretube boilers

Diameter of shell, mm (in)

Minimum thickness after ﬂanging, mm (in)

Diameter of tube sheet, mm (in)

Minimum thickness, mm (in)

900 (36) 900–1350 (36–54) 1350–1800 (54–72) 1800 (72)

6.0 (0.25) 8.0 (0.3125) 9.5 (0.375) 12.5 (0.5)

1050 (42) 1050–1350 (42–54) 1350–1800 (54–72) 1800 (72)

9.5 (0.375) 11.5 (0.4375) 12.5 (0.50) 14.0 (0.5625)

TABLE 13-3 Eﬃciency of riveted joints () % Eﬃciency,

Type of joint Lap joints Single-riveted Double-riveted Triple-riveted Butt joints (with two cover plates) Single-riveted Double-riveted Triple-riveted Quadruple-riveted

Normal range

Maximum

50–60 60–72 72–80

63 77 86.6

55–60 76–84 80–88 86–94

63 87 95 98

TABLE 13-4 Allowable stresses in structural riveting (b ) Rivets acting in single shear

Rivets acting in double shear

Load-carrying member

Type of stress

Rivet-driving method

Rolled steel SAE 1020

Tension Shear

Power

124 93

18.0 13.5

124 93

18.0 13.5

Shear Crushing Crushing

Hand Power Hand

68 165 110

10.0 24.0 16.0

68 206 137

10.0 30.0 20.0

Rivets, SAE 1010

MPa

kpsi

MPa

kpsi

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RIVETED JOINTS

13.4

CHAPTER THIRTEEN

TABLE 13-5 Allowable stress for aluminum rivets, a Allowable stressa , a Shear

Bearing

Rivet alloy

Procedure of drawing

MPa

kpsi

MPa

kpsi

2S (pure aluminum) 17S 17S 615–T6 53S

Cold, as received Cold, immediately after quenching Hot, 500–5108C Cold, as received Hot, 515–5278C

20 68 62 55 41

3.0 10.0 9.0 8.0 6.0

48 179 179 103 103

7.0 26.0 26.0 15.0 15.0

a

Actual safety factor or reliability factor is 1.5.

TABLE 13-6 Values of working stressa at elevated temperatures Minimum of the speciﬁed range of tensile strength of the material, MPa (kpsi) Maximum temperatures

(45)

311

(50)

344

(55)

380

(60)

413

(75)

517

8F

8C

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

MPa

kpsi

0–700 750 800 850 900 950

0–371 399 427 455 482 511

61 56 45 37 29 22

9.0 8.22 6.55 5.44 4.33 3.20

68 62 53 41 33 26

10.0 9.11 7.33 6.05 4.83 3.60

76 68 54 46 37 27

11.00 10.00 8.00 6.75 5.50 4.00

82 77 61 51 38 27

12.00 11.20 9.00 7.40 5.60 4.00

103 89 70 57 41 27

15.00 13.00 10.20 8.30 6.00 4.00

a

Design stresses of pressure vessels are based on a safety factor of 5.

TABLE 13-7 Pitch of butt joints Type of joint

Diameter of rivets, d, mm

Pitch, p

Double-riveted— use for h 12:5 mm (0.5 in) Triple-riveted— use for h 25 mm (1 in) Quadruple-riveted— use for h 31:75 mm (1.25 in)

Any

5.5d (approx.)

1.75–23.80 27.00 30.15–36.50 17.50–23.80 27.00 30.15 33.30–36.50

8d–8.5d 7.5d 6.5d–7d 16d–17d 15d (approx.) 14d (approx.) 13d–14d

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RIVETED JOINTS RIVETED JOINTS

13.5

TABLE 13-8 Transverse pitch ( pt ) as per ASME Boiler Code Value of p=d

1

2

3

4

5

6

7

Value of pt

2d

2d

2d

2d

2d

2.2d

2.3d

Particular

Formula

Butt joint pt ¼ 2d to 2:5d qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pt 0:5pd þ 0:25d 2

The transverse pitch

ð13-6aÞ ð13-6bÞ

For rivets, rivet holes, and strap thick

Refer to Tables 13-9, 13-10, and Fig. 13-2.

TABLE 13-9 Rivet hole diameters

TABLE 13-10 Rivet hole diameters and strap thickness

Diameter of rivet, mm 12 14 16 18 20 22 24 27 30 33 36 39 42 48

Rivet hole diameters, mm (min) 13 15 17 19 21 23 25 28.5 31.5 34.5 37.5 41.0 44 50

Plate thickness, h, mm

6.25 7.20 8.00 8.75 9.50 10.30 11.10 12.00 12.50 13.50

Minimum strap thickness, hc mm

6.25

Hole Plate diameter, thickness, d, mm h, mm

8.00

11.10

14.25

11.10

15.90 19.00

12.50

22.25

15.90

25.00 28.50 31.75 83.10

12.50 19.00 22.25 25.00

17.50 20.50

9.50

Minimum strap thickness, hc mm

24.00

Hole diameter, d, mm

27.0 30.15

FIGURE 13-2 Quadruple-riveted double-strap butt joint.

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33.30

36.50 39.70

RIVETED JOINTS

13.6

CHAPTER THIRTEEN

Particular

Minimum transverse pitch as per ASME Boiler Code

Formula

pt ¼ 1:75d

if

p 4 d

pt ¼ 1:75d þ 0:001ð p dÞ

ð13-7aÞ p if > 4 d

SI

ð13-8aÞ

USCS

ð13-8bÞ

where pt , p, and d in m pt ¼ 1:75d þ 0:1ð p dÞ if

p >4 d

where pt , d, and p in in For transverse pitches Haven and Swett formula for permissible pitches along the caulking edge of the outside cover plate

Refer to Table 13-8. sﬃﬃﬃﬃﬃﬃ 3 4 hc pc d ¼ 14 Pf

CM ð13-9aÞ

where pc , d, hc in cm, and Pf in kgf/cm2 sﬃﬃﬃﬃﬃﬃ 3 4 hc pc d ¼ 21:38 USCS Pf where pc , d, hc in in, and Pf in psi sﬃﬃﬃﬃﬃﬃ 3 4 hc pc d ¼ 77:8 Pf

SI

ð13-9bÞ

ð13-9cÞ

where pc , d, hc in m, and Pf in N/m2 Diagonal pitch, pd , is calculated from the relation

2ð pd dÞ ð p dÞ

ð13-10Þ

MARGIN Margin for longitudinal seams of all pressure vessels and girth seams of power boiler having unsupported heads

m ¼ 1:5d to 1:75d

ð13-11aÞ

Margin for girth seams of power boilers having supported heads and all unﬁred pressure vessels

m 1:25d

ð13-11bÞ

COVER PLATES The thickness of cover plate

hc ¼ 0:6h þ 0:0025 if h 0:038 m

SI

ð13-12aÞ

USCS

ð13-12bÞ

SI

ð13-12cÞ

USCS

ð13-12dÞ

where hc and h in m hc ¼ 0:6h þ 0:1 if h 1:5 in where hc and h in in hc ¼ 0:67h

if h > 0:038 m

where hc and h in m hc ¼ 0:67h

if h > 1:5 in

where hc and h in in

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RIVETED JOINTS RIVETED JOINTS

13.7

TABLE 13-11 Rivet groups under eccentric loading value of coeﬃcient K

}

K¼

1 lp 1 þ p21 þ p2 4

K¼

n sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Alcn Alcn 2 þ þ1 2I 2I

n K ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 6l þ1 ðn þ 1Þpt

n K ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 2 lðn 1Þpt lp 1 2 þ 2 1 2 þ p þ 3 ðn 1Þp2t 2 p2 þ 13 ðn2 1Þp2t

n K ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 2 lðn 1Þpt lp 1 2 þ 2 1 2 þ p þ 3 ðn 1Þp2t 3 p2 þ 12 ðn 1Þp2t

n K ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 2 lðn 1Þpt lp 1 2 þ þ p21 þ p2 þ 23 ðn2 1Þp2t 4 p21 þ p2 þ 23 ðn2 1Þp2t

Key: n ¼ total number of rivets in a column F ¼ permissible load, acting with lever arm, l, kN (lbf) F 0 ¼ permissible load on one rivet, kN (lbf) K ¼ F=F 0 , coeﬃcient Source: K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook ( fps Units), Engineering College Cooperative Society, Bangalore, India, 1962; K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1983; and K. Lingaiah, Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986.

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RIVETED JOINTS

13.8

CHAPTER THIRTEEN

Particular

Formula

Thickness of the cover plate according to Indian Boiler Code Thickness of single-butt cover plate

h1 ¼ 1:125h

Thickness of single-butt cover plate omitting alternate rivet in the over rows

h2 ¼ 1:25h

Thickness of double-butt cover plates of equal width

hc ¼ h1 ¼ h2 ¼ 0:625h

Thickness of double-butt cover plates of equal width omitting alternate rivet in the outer rows

hc ¼ h1 ¼ h2 ¼ 0:625h

Thickness of the double-butt cover plates of unequal width

ð13-13Þ

pd p 2d

ð13-14Þ ð13-15Þ pd p 2d

ð13-16Þ

h1 ¼ 0:625h for narrow strap

ð13-17aÞ

h2 ¼ 0:750h for wide strap

ð13-17bÞ

For thickness of cover plates

Refer to Table 13-10.

The width of upper cover plate (narrow strap)

b1 ¼ 4m þ 2pt1

ð13-18Þ

The width of lower cover plate (wide strap)

b2 ¼ b1 þ 2pt2 þ 4m

ð13-19Þ

The tensile strength of the solid plate

F ¼ ph

ð13-20Þ

The tensile strength of the perforated strip along the outer gauge line

F ¼ ð p dÞh

ð13-21Þ

STRENGTH ANALYSIS OF TYPICAL RIVETED JOINT (Fig. 13-2)

The general expression for the resistance to shear of all the rivets in one pitch length

F ¼ ð2i2 þ i1 Þ

The general expression for the resistance to crushing of the rivets

Fc ¼ ði2 h þ i1 h2 Þdc

The resistance against failure of the plate through the second row and simultaneous shearing of the rivets in the ﬁrst row

F1 ¼ ð p 2dÞh þ

d 2 4

ð13-22Þ ð13-23Þ d 2 4

ð13-24Þ

The resistance against failure of the plate through the second row and simultaneous crushing of the rivets in the ﬁrst row

Fc1 þ ð p 2dÞh þ dhc

ð13-25Þ

The resistance against shearing of the rivets in the outer row and simultaneous crushing of the rivets in the two inner rows

Fc ¼

2 d þ idhc 4

ð13-26Þ

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RIVETED JOINTS RIVETED JOINTS

Particular

13.9

Formula

EFFICIENCY OF THE RIVETED JOINT The eﬃciency of plate The eﬃciency of rivet in general case

For eﬃciency of joints The diameter of the rivet in general case

¼

pd p

d 2 ði1 þ 2i2 Þ 4ph h i 2 þ i 1 2 c h ¼ h2 c þ i2 þ i1 h

ð13-27Þ

¼

ð13-28Þ

Refer to Table 13-3. d¼

4hi2 þ i1 h2 c ði1 þ 2i2 Þ

ð13-29Þ

Note: for lap joint i2 ¼ 0 for butt joint i1 ¼ 0 ð2i2 þ i1 Þd 2 þd 4h

The pitch in general case

p¼

For pitch of joint

Refer to Table 13-7.

THE LENGTH OF THE SHANK OF RIVET (Fig. 13-3)

ð13-30Þ

L ¼ h þ h1 þ h2 þ ð1:5 to 1:7ÞD

ð13-31aÞ

L ¼ h þ hc þ ð1:5 to 1:7ÞD

ð13-31bÞ

for butt joint with single cover plate L ¼ 2h þ ð1:5 to 1:7ÞD for lap joint where D ¼ diameter of rivet

FIGURE 13-3

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ð13-31cÞ

RIVETED JOINTS

13.10

CHAPTER THIRTEEN

Particular

Formula

STRUCTURAL JOINT Riveting of an angle to a gusset plate (Fig. 13-4) The resultant normal stress

p

g

a

¼

F

e

i

F Fe þ A Z

a

ð13-32Þ

F

F g

e

(a)

(b)

FIGURE 13-4 Riveting of an angle to a gusset plate.

RIVETED BRACKET (Fig. 13-5) The resultant load on the farthest rivet whose distance is c from the center of gravity of a group of rivets (Fig. 13-5)

" FR ¼

F nn0

2 þ

P

Mb c P x2 þ y2

2

#1=2 F Mb c P P þ2 cos nn0 x2 þ y2

FIGURE 13-5 Riveted bracket. (Bureau of Indian Standards.)

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ð13-33Þ

RIVETED JOINTS RIVETED JOINTS

Particular

13.11

Formula

where n ¼ number of rivets in one column n0 ¼ number of rivets in one row x, y have the meaning as shown in Fig. 13-5 For rivet groups under eccentric loading value of coeﬃcient K

Refer to Table 13-11.

For preferred length and diameter of rivets

Refer to Figs. 13-6 to 13-8 and Tables 13-12 to 13-13.

For collected formulas of riveted joints

Refer to Table 13-14.

REFERENCES 1. Maleev, V. L., and J. B. Hartmen, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954. 2. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook ( fps Units), Engineering College Cooperative Society, Bangalore, India, 1962. 3. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1983. 4. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 5. Bureau of Indian Standards. 6. Lingaiah, K., Machine Design Data Handbook, McGraw-Hill Publishing Company, New York, 1994.

BIBLIOGRAPHY Faires, V. M., Design of Machine Elements, The Macmillan Company, New York, 1965. Norman, C. A., E. S. Ault, and I. F. Zarobsky, Fundamentals of Machine Design, The Macmillan Company, New York, 1951. Vallance, A., and V. L. Doughtie, Design of Machine Members, McGraw-Hill Publishing Company, New York, 1951.

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RIVETED JOINTS

13.12

CHAPTER THIRTEEN

FIGURE 13-6 Rivets for general purposes (less than 12 mm diameter). For preferred length and diameter combination, refer to Table 13-12.

FIGURE 13-7 Rivets for general purposes (12 to 48 mm diameter). For preferred length and diameter combination, refer to Table 13-13.

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RIVETED JOINTS RIVETED JOINTS

13.13

FIGURE 13-8 Boiler rivets (12 to 48 mm diameter). For preferred length and diameter combination, refer to Table 13-13.

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RIVETED JOINTS

13.14

CHAPTER THIRTEEN

TABLE 13-12 Preferred length () and diameter combinations for rivets (Fig. 13-6) Diameter, mm Length, mm

1.6

2

2.5

3

4

5

6

8

10

5 6 7 8 9 10 12 14 16 18 20 22 24 26 28 30 35 40 45 50 55 60 65 70

— — — — — — — — — — — — — — — — — —

— — — — — — — — — — — — — — — — — —

— — — — — — — — — — — — — — — —

— — — — — — — —

— — — — — — — —

— — — — — — — — — —

— — — — — — — — —

— — — — — — —

— — — — — — — — —

Source: Bureau of Indian Standards, IS: 2155, 1962.

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RIVETED JOINTS RIVETED JOINTS

13.15

TABLE 13-13 Preferred lengths () and diameter combinations of rivets (Fig. 13-7) Diameter, mm Length, mm

12

14

16

18

20

22

24

27

30

33

36

39

42

48

28 31.5 35.5 40 45 50 56 63 71 80 85 90 95 100 106 112 118 125 132 140 150 160 180 200 224 250

— — — — — — — — — — — — — — — —

— — — — — — — — — — — — — —

— — — — — — — — — — — —

— — — — — — — — — — — —

— — — — — — — — — — — —

— — — — — — — — — — —

— — — — — — — — — —

— — — — — — — — — —

— — — — — — — — — —

— — — — — — — — — — —

— — — — — — — — — — — — —

— — — — — — — — — — — — — — —

— — — — — — — — — — — — — — — —

— — — — — — — — — — — — — — — — —

Source: Bureau of Indian Standards, IS: 1929, 1961.

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Three rivets per pitch Type d

m

h m

pd p

2 d 3 ph 4

2 d 2 ph 4

pd p

ph

2 d 2 ph 4

Type c

Combined eﬃciency, c LAP JOINT

pd p

d 4

2

Two rivets per pitch Type b

Eﬃciency of rivets, r

pd p

Figure

Eﬃciency of plate, p

One rivet per pitch, Type a

Type of joint

TABLE 13-14 Formulas for riveted joints2;3;4

3:47h þ 40

2:62h þ 40

2:62h þ 40

1:13h þ 40

Longitudinal pitch, p, mm

2d

0:33p þ 0:67d

2d

Transverse pitch, pt , mm

1:5d

1:5d

1.5d

1:5d

Margin, Inner h2 m, mm (wider)

Outer, h1 (narrower)

Thickness of cover plate, mm

RIVETED JOINTS

13.16

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Type g

Four rivets per pitch Type f

Type e

Type of joint

h

Figure

TABLE 13-14 Formulas for riveted joints2;3;4 (Cont.)

pd p

pd p

2 p 2d d 4:14h þ 40 4 p ph 4 2 d þ ph 4

0.2p+1.15d

0:33p þ 0:67d or 2d (whichever is greater)

Transverse pitch, pt , mm

2 p 2d d 4:14h þ 40 4 p ph 4 2 d þ ph 4

Longitudinal pitch, p, mm 0:33p þ 0:67d

2 d 3 ph 4

pd p

Combined eﬃciency, c 3:47h þ 40

Eﬃciency of rivets, r

Eﬃciency of plate, p

1:5d

1:5d

1:5d

Margin, Inner h2 m, mm (wider)

Outer, h1 (narrower)

Thickness of cover plate, mm

RIVETED JOINTS

13.17

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Three rivets per pitch Type d

Type c

Two rivets per pitch Type b

Single butt strap One rivet per pitch Type a

Type of joint

Figure

TABLE 13-14 Formulas for riveted joints2;3;4 (Cont.)

2 d 3 ph 4

pd p

ph

2 d 2 ph 4

pd p

d 2 4

Combined eﬃciency, c

3:06h þ 40

3:06h þ 40

1:53h þ 40

Longitudinal pitch, p, mm

p 2d 4:05h þ 40 p 2 d þ ph 4

BUTT JOINT

2 d 2 ph 4

Eﬃciency of rivets, r

pd p

pd p

Eﬃciency of plate, p

0:33p þ 0:67d or 2d (whichever is greater)

0:33p þ 0:67d

2d

Transverse pitch, pt , mm

1:5d

1:5d

1:5d

1:5d

Margin, m, mm

Inner h2 (wider)

1:125h

1:125d

1:125h

1:125h

pd p 2d

Outer, h1 (narrower)

Thickness of cover plate, mm

RIVETED JOINTS

13.18

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Type h

Two rivets per pitch Type g

Double-butt strap (equal widths) One rivet per pitch Type f

Two rivets per pitch Type e

Type of joint

Figure

TABLE 13-14 Formulas for riveted joints2;3;4 (Cont.)

pd p

pd p

2 d 3:75 4 ph

2 d 3:75 4 ph

2 d 1:875 4 ph

2 d 3 ph 4

pd p

pd p

Eﬃciency of rivets, r

Eﬃciency of plate, p Longitudinal pitch, p, mm

3:5h þ 40

3:5h þ 40

1:75h þ 40

p 2d 4:05h þ 40 p 2 d þ ph 4

Combined eﬃciency, c

0:33p þ 0:67d

2d

0:2p þ 1:15d

Transverse pitch, pt , mm

1:5d

1:5d

1:5d

1:5d

Margin, m, mm

0:625h

0:625h

0:625h

Inner h2 (wider)

0:625h

0:625h

0:625h

1:125h

pd p 2d

Outer, h1 (narrower)

Thickness of cover plate, mm

RIVETED JOINTS

13.19

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Type l

Type k

Type j

Three rivets per pitch Type i

Type of joint

Figure

TABLE 13-14 Formulas for riveted joints2;3;4 (Cont.)

pd p

pd p

pd p

pd p

Eﬃciency of plate, p

2 d 5:625 4 ph

2 d 5:625 4 ph

2 d 5:625 4 ph

2 d 5:625 4 ph

Eﬃciency of rivets, r Longitudinal pitch, p, mm

4:63h þ 40

4:63h þ 40

4:63h þ 40

p 2d 4:63h þ 40 þ 1:875 p 2 d ph 4

Combined eﬃciency, c

0:33p þ 0:67d

0:2p þ 1:15d

2d

0:33p þ 0:67d or 2d (whichever is greater)

Transverse pitch, pt , mm

1:5d

1:5d

1:5d

1:5d

Margin, m, mm

pd p 2d

pd p 2d

0:625h

pd p 2d

0:625h

0:625h

0:625h

0:625h

pd p 2d

0:625h

Outer, h1 (narrower)

0:625h

0:615h

Inner h2 (wider)

Thickness of cover plate, mm

RIVETED JOINTS

13.20

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Type p

Double butt (unequal widths) Two rivets per pitch Type o

Type n

Four rivets per pitch Type m

Type of joint

Figure

TABLE 13-14 Formulas for riveted joints2;3;4 (Cont.)

pd p

pd p

pd p

pd p

Eﬃciency of plate, p

p 2d þ 1:875 d 2 d ph 4

2 d 7:5 4 ph

2 d 2:875 4 ph

2 d 2:875 4 ph

p 2d þ 1:875 d 2 d ph 4

Combined eﬃciency, c

2 d 7:5 4 ph

Eﬃciency of rivets, r

3:5h þ 40

3:5h þ 40

5:52h þ 40

5:52h þ 40

Longitudinal pitch, p, mm

2d

0:33p þ 0:67d

0:2p þ 1:15d

0:33p þ 0:67d or 2d (whichever is greater)

Transverse pitch, pt , mm

1:5d

1:5d

1:5d

1:5d

0:75h

0:75h

0:625h

0:625h

Margin, Inner h2 m, mm (wider)

0:625h

0:625h

0:625h

0:625h

Outer, h1 (narrower)

Thickness of cover plate, mm

RIVETED JOINTS

13.21

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Type s

Type r

Three rivets per pitch Type q

Type of joint

Figure

TABLE 13-14 Formulas for riveted joints2;3;4 (Cont.)

pd p

pd p

pd p

Eﬃciency of plate, p

2 d 4:75 4 ph

2 d 4:75 4 ph

2 d 4:75 4 ph

Eﬃciency of rivets, r

p 2d d 2 d þ ph 4

p 2d d 2 d þ ph 4

Combined eﬃciency, c

4:63h þ 40

4:63h þ 40

4:63h þ 40

Longitudinal pitch, p, mm

0:2p þ 1:15d

2d

0:33p þ 0:67d or 2d (whichever is greater)

Transverse pitch, pt , mm

1:5d

1:5d

1:5d

0:75h

0:75h

0:75h

Margin, Inner h2 m, mm (wider)

0:625h

0:625h

0:625h

Outer, h1 (narrower)

Thickness of cover plate, mm

RIVETED JOINTS

13.22

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Particular

Figure

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Eﬃciency of plate, p

Formula

Combined eﬃciency, c 4:63h þ 40

Longitudinal pitch, p, mm

pﬃﬃﬃ pﬃﬃﬃ ¼ 1:2 h to 1:4 h where d and h in m

Pf Di 2 pﬃﬃﬃ pﬃﬃﬃ d ¼ 0:19 h to 0:2 h where d and h in m h¼

2 d 4:75 4 ph

Eﬃciency of rivets, r 0:33p þ 0:67d

Transverse pitch, pt , mm 1:5d

0:75h

0:625h

Outer, h1 (narrower)

USCS

SI

Margin, Inner h2 m, mm (wider)

Thickness of cover plate, mm

Key: d ¼ diameter of rivet, m (in); h ¼ thickness of main plate, m (in); ¼ hoop stress, MPa (psi); Di ¼ inside diameter of pressure vessel, m (in); Pf ¼ internal ﬂuid pressure, MPa (psi); ¼ efficiency of the riveted joint. Source: K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1983; and K. Lingaiah, Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986.

Unwin’s formula for diameter of rivet

Common Formula: The thickness of the main plate of a longitudinal joint

Type t

Type of joint

TABLE 13-14 Formulas for riveted joints (Cont.)

RIVETED JOINTS

13.23

Source: MACHINE DESIGN DATABOOK

CHAPTER

14 DESIGN OF SHAFTS SYMBOLS1;2;3 width of keyway, m (in) machine cost, $/m ($/in) (US dollars) diameter of shaft (also with subscripts), m (in) inside diameter of hollow shaft, m (in) outside diameter of hollow shaft, m (in) modulus of elasticity, GPa (Mpsi) axial load (tensile or compressive), kN (lbf) the static equivalent of cyclic load, (¼ Fm Fa ), kN (lbf) modulus of rigidity, GPa (Mpsi) depth of keyway, m (in) radius of gyration, m (in) material cost (also with subscripts), $/kg

b c D Di Do E F Fm0 G h k K¼ Kb Kt l Mb Mt 0 Mbm 0 Mtm

P n n0

Di Do

ratio of inner to outer diameter of hollow shaft numerical combined shock and fatigue factor to be applied to computed bending moment numerical combined shock and fatigue factor to be applied to computed twisting moment length, m (in) bending moment, N m (lbf in) twisting moment, N m (lbf in) static equivalent of cyclic bending moment Mbm Mba , N m (lbf in) static equivalent of cyclic twisting moment Mtm Mta , N m (lbf in) power, kW (hp) speed, rpm; safety factor speed, rps speciﬁc weight of material, kN/m3 (lbf/in) stress (tensile or compressive) also with subscripts, MPa (psi) shear stress (also with subscripts), MPa (psi) ratio of maximum intensity of stress to the average value from compressive stress only angular deﬂection, deg

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DESIGN OF SHAFTS

14.2

CHAPTER FOURTEEN

SUFFIXES a b d e h m sc t u y max min f

amplitude bending design elastic limit hollow mean static strength (su or sy ), solid twisting ultimate yield strength maximum minimum endurance

Other factors in performance or in special aspect are included from time to time in this chapter and, being applicable in their immediate context, are not given at this stage. Note: and with the initial subscript s designates strength properties of material used in the design which will be used and observed throughout this handbook. In some books on machine design and in this Machine Design Data Handbook the ratios of design stresses sd =fd and sd =fd ; and design stresses yd , yd 0 , fd , and fd have been used instead of sy =sf , sy =sf ; and yield strengths sy , sy and fatigue strengths, sf , sf in the design equations for shafts [Eqs. (14-1) to (14-65)]. This has to be taken into consideration in the design of shafts while using Eqs. (14-1) to (14-65).

Particular

Formula

SOLID SHAFTS (1) Stationary shafts with static loads The diameter of shaft subjected to simple torsion

The diameter of shaft subjected to simple bending

D¼ D¼

16Mt yd 32Mb yd

1=3 ð14-1Þ 1=3 ð14-2Þ

The diameter of shaft subjected to combined torsion and bending: (a) According to maximum normal stress theory

(b) According to maximum shear stress theory

D¼

16 fMb þ ðMb2 þ Mt2 Þ1=2 g yd

D¼

16 ðMb2 þ Mt2 Þ1=2 yd

(

(c) According to maximum shear energy theory D¼

16 yd

1=3 ð14-3Þ

1=3

)1=3 3 2 1=2 2 Mb þ Mt 4

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ð14-4Þ

ð14-5Þ

DESIGN OF SHAFTS

14.3

DESIGN OF SHAFTS

Particular

Formula

The diameter of shaft subjected to axial load, bending, and torsion:13 "

(a) According to maximum normal theory D¼

(

16 yd

FD Mb þ 8

(

Mb þ

þ

FD 8

2

)1=2 )#1=3 þ Mt2 2

(b) According to maximum shear stress theory

(

FD Mb þ 8

16 D¼4 yd (c) According to maximum shear energy theory

ð14-6Þ

2

(

FD Mb þ 8

16 D¼4 yd

2

)1=2 31=3 5 þ M2

ð14-7Þ

t

2

3 þ Mt2 4

)1=2 31=3 5

ð14-8Þ

(2) Rotating shafts with dynamic loads, taking dynamic eﬀect indirectly into consideration13 For empirical shafting formulas The diameter of shaft subjected to simple torsion

The diameter of shaft subjected to simple bending

Refer to Table 14-1. 1=3 16 ðKt Mt Þ D¼ yd D¼

32 ðK M Þ yd b b

ð14-9Þ

1=3 ð14-10Þ

The diameter of shaft subjected to combined bending and torsion (a) According to maximum normal stress theory

D¼

16 ½K M þ fðKb Mb Þ2 þ ðKt Mt Þ2 g1=2 yd b b

1=3

ð14-11Þ (b) According to maximum shear stress theory

(c) According to maximum shear energy theory

D¼ D¼

16 fðKb Mb Þ2 þ ðKt Mt Þ2 g1=2 yd

1=3

16 3 fðKb Mb Þ2 þ ðKt Mt Þ2 g1=2 yd 4

ð14-12Þ 1=3 ð14-13Þ

The diameter of shaft subjected to axial load, bending, and torsion (

(a) According to maximum normal stress theory D¼

16 yd

Kb Mb þ

" þ

Kb Mb þ

FD 8

FD 8

1=3 #1=2 9 =

2 þ ðKt Mt Þ2

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;

ð14-14Þ

DESIGN OF SHAFTS

14.4

CHAPTER FOURTEEN

Particular

(b) According to maximum shear stress theory

Formula

" 16 D¼ yd

FD Kb Mb þ 8

1=2 #1=3

2 þ ðKt Mt Þ

2

ð14-15Þ (c) According to maximum shear energy theory

" 16 D¼ yd

FD Kb Mb þ 8

2

3 þ ðKt Mt Þ2 4

1=2 #1=3 ð14-16Þ

The diameter of shaft based on torsional rigidity

D¼

584Mt L G

1=4 ð14-17Þ

where Kb and Kt are taken from Table 14-2 (3) Rotating shafts and ﬂuctuating loads, taking fatigue eﬀect directly into consideration13 The diameter of shaft subjected to ﬂuctuating torsion

The diameter of shaft subjected to ﬂuctuating bending

( D¼ ( D¼

16

32

Mtm Mta þ yd fd

)1=3

Mbm Mba þ yd fd

ð14-18Þ )1=3 ð14-19Þ

The diameter of shaft subjected to combined ﬂuctuating torsion and bending: (a) According to maximum normal stress theory

(b) According to maximum shear stress theory

(c) According to maximum shear energy theory

1=3 16 0 02 02 1=2 fMbm þ ðMbm þ Mtm Þ g D¼ yd D¼

16 02 02 1=2 ðMbm þ Mtm Þ yd

( D¼

16 yd

ð14-20Þ

1=3

)1=3 3 02 1=2 02 Mbm þ Mtm 4

ð14-21Þ

ð14-22Þ

where sd M fd ba

ð14-22aÞ

sd M fd ta

ð14-22bÞ

0 ¼ Mbm þ Mbm

0 ¼ Mtm þ Mtm

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DESIGN OF SHAFTS DESIGN OF SHAFTS

Particular

14.5

Formula

The diameter of shaft subjected to combined ﬂuctuating axial load, bending, and torsion (a) According to maximum normal stress theory

( D¼

16 yd

þ (b) According to maximum shear stress theory

" 16 D¼ yd

"

0 Mbm

0 þ Mbm

F 0 D þ m 8

Fm0 D 8

0 Mbm

2

02 þ Mtm

F 0 D þ m 8

2 þ

1=2 #)1=3 ð14-23Þ

02 Mtm

1=2 #1=3 ð14-24Þ

"

(c) According to maximum shear energy theory D¼

16 yd

0 þ Mbm

Fm0 D 8

2

3 02 þ Mtm 4

1=2 #1=3 ð14-25Þ

0 Mbm

0 Mtm

where and have the same meaning as in Eqs. (14-22a) and (14-22b) and Fm0 ¼ Fm þ sd Fa ð14-25aÞ fd

HOLLOW SHAFTS (1) Stationary shafts with static loads

The outside diameter of shaft subjected to simple torsion

Do ¼

The outside diameter of shaft subjected to simple bending

Do ¼

16Mt yd ð1 K 4 Þ 32Mb yd ð1 K 4 Þ

1=3 ð14-26Þ 1=3 ð14-27Þ

The diameter of shaft subjected to combined torsion and bending (a) According to maximum normal stress theory

Do ¼

1=3 16 2 2 1=2 þ ðM þ M Þ g fM t b b yd ð1 K 4 Þ ð14-28Þ

(b) According to maximum shear stress theory

Do ¼ (

(c) According to maximum shear energy theory Do ¼

16 ðMb2 þ Mt2 Þ1=2 yd ð1 K 4 Þ

1=3

)1=3 16 3 2 1=2 2 Mb þ Mt 4 yd ð1 K 4 Þ

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ð14-29Þ

ð14-30Þ

DESIGN OF SHAFTS

14.6

CHAPTER FOURTEEN

Particular

Formula

The outside diameter of shaft subjected to axial load, bending, and torsion (a) According to maximum normal stress theory

( Do ¼

16 yd ð1 K 4 Þ

þ

FDo ð1 þ K 2 Þ Mb þ 8

FDo ð1 þ K 2 Þ Mb þ 8

1=2 !)1=3

2 þ

Mt2 ð14-31Þ

(b) According to maximum shear stress theory

( Do ¼

16 yd ð1 K 4 Þ #1=2 )1=3

" FDo Mb þ ð1 þ K 2 Þ 8

þ Mt2 (c) According to maximum shear energy theory

(

16 Do ¼ yd ð1 K 4 Þ #1=2 )1=3 3 2 þ Mt 4

ð14-32Þ " FDo 2 2 ð1 þ K Þ Mb þ 8 ð14-33Þ

(2) Rotating shafts with dynamic loads, taking dynamic eﬀect indirectly into consideration13

The outside diameter of shaft subjected to simple torsion

Do ¼

The outside diameter of shaft subjected to simple bending

Do ¼

16 Kt M t yd ð1 K 4 Þ

1=3

32 Kb Mb yd ð1 K 4 Þ

ð14-34Þ 1=3 ð14-35Þ

The outside diameter of shaft subjected to combined bending and torsion (a) According to maximum normal stress theory

(b) According to maximum shear stress theory

Do ¼

Do ¼

16 ½Kb Mb þ fðKb Mb Þ2 yd ð1 K 4 Þ 1=3 þ ðKt Mt Þ2 g1=2

ð14-36Þ

1=3

16 fðKb Mb Þ2 þ ðKt Mt Þ2 g1=2 yd ð1 K 4 Þ

ð14-37Þ

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DESIGN OF SHAFTS DESIGN OF SHAFTS

Particular

(c) According to maximum shear energy theory

14.7

Formula

"

1=2 #1=3 16 3 2 2 Do ¼ ðKb Mb Þ þ ðKt Mt Þ 4 yd ð1 K 4 Þ ð14-38Þ

The outside diameter of shaft subjected to axial load, bending and torsion (a) According to maximum normal stress theory

"

( 16 FDo 2 ð1 þ K Þ Do ¼ Kb Mb þ 8 yd ð1 K 4 Þ 2 FDo 2 þ Kb Mb þ ð1 þ K Þ 8 )#1=3 1=2

þ ðKt Mt Þ2 (b) According to maximum shear stress theory

" Do ¼

ð14-39Þ

( 2 16 FDo 2 M þ Þ ð1 þ K K b b 8 yd ð1 K 4 Þ )1=2 #1=3

þ ðKt Mt Þ2 (c) According to maximum shear energy theory

The outside diameter of shaft based on torsional rigidity

ð14-40Þ

(

" 2 16 FDo 2 ð1 þ K M þ Þ Do ¼ K b b 8 yd ð1 K 4 Þ #1=2 )1=3 3 þ ðKt Mt Þ2 ð14-41Þ 4 Do ¼

584Mt L ð1 K 4 ÞG

1=4 ð14-42Þ

(3) Rotating shaft with ﬂuctuating loads, taking fatigue eﬀect directly into consideration The outside diameter of shaft subjected to ﬂuctuating torsion

The outside diameter of shaft subjected to ﬂuctuating bending

"

16 Do ¼ ð1 K 4 Þ "

32 Do ¼ ð1 K 4 Þ

Mtm Mta þ yd fd

#1=3

Mbm Mba þ yd fd

ð14-43Þ #1=3

Please note: If the axial load does not produce column action, the constant need not be used to multiply the term [FDo (1 þ K 2 )/8] throughout this chapter.

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ð14-44Þ

DESIGN OF SHAFTS

14.8

CHAPTER FOURTEEN

Particular

Formula

The outside diameter of shaft subjected to combined ﬂuctuating torsion and bending (a) According to maximum normal stress theory

Do ¼

1=3 16 0 02 02 1=2 fM þ ðM þ M Þ g tm bm bm yd ð1K 4 Þ ð14-45Þ

(b) According to maximum shear stress theory

Do ¼

16 02 02 1=2 þ Mtm Þ ðMbm yd ð1 K 4 Þ

1=3 ð14-46Þ

"

(c) According to maximum shear energy theory

#1=3 16 3 02 1=2 02 Do ¼ Mbm þ Mtm 4 yd ð1 K 4 Þ

ð14-47Þ

0 0 where Mbm , Mtm have the same meaning as in Eqs. (14-22a) and (14-22b)

The outside diameter of shaft subjected to combined ﬂuctuating axial load, bending, and torsion (a) According to maximum normal stress theory

" Do ¼

16 yd ð1 K 4 Þ

þ

0 þ Mbm

(

0 þ Mbm

Fm0 Do ð1 þ K 2 Þ 8

Fm0 Do ð1 þ K 2 Þ 8

2

02 þ Mtm

1=2 )#1=3

ð14-48Þ ( (b) According to maximum shear stress theory

Do ¼

"

16 yd ð1 K 4 Þ

0 Mbm þ

Fm0 Do ð1 þ K 2 Þ 8

2

#1=2 !)1=3 þ ( (c) According to maximum shear energy theory

Do ¼

02 Mtm

ð14-49Þ

16 yd ð1 K 4 Þ

3 02 þ Mtm 4

" Fm0 Do ð1 þ K2 Þ 2 0 Mbm þ 8

#1=2 )1=3 ð14-50Þ

0 0 , Mtm , and Fm0 have the same meaning as where Mbm in Eqs. (14-22a), (14-22b), and (14-25a)

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DESIGN OF SHAFTS DESIGN OF SHAFTS

Particular

14.9

Formula

COMPARISON BETWEEN DIAMETERS OF SOLID AND HOLLOW SHAFTS OF SAME LENGTH For equal strength in bending, torsion, and/or combined bending and torsion, the diameter (a) When materials of both shafts are same

D ¼ Do ð1 K 4 Þ1=3

(b) When materials of shafts are diﬀerent

D ¼ Do

ð14-51Þ

eh ð1 K 4 Þ1=3 es

ð14-52Þ

For torsional rigidity (a) When torsional rigidities are equal (b) When torsional rigidities are diﬀerent

D ¼ Do ð1 K 4 Þ1=4 D ¼ Do

Gh ð1 K 4 Þ Gs

ð14-53Þ 1=4 ð14-54Þ

For equal weight D ¼ Do ð1 K 2 Þ1=2

ð14-55Þ

w 1=2 D ¼ Do ð1 K 2 Þ h ws

ð14-56Þ

(a) For same material and machining cost for both shafts

D ¼ Do ð1 K 2 Þ1=2

ð14-57Þ

(b) For no machining cost for both shafts but with diﬀerent material cost

w k 1=2 D ¼ Do ð1 K 2 Þ h h w s ks

ð14-58Þ

(c) When machining costs are diﬀerent and material cost negligible

D¼

(a) When material of both shafts is same (b) When materials of both shafts are diﬀerent

For equal cost

(d) When machining and material costs are diﬀerent

D¼

ch cs

1=2

8 91=2 <D2o ð1 K 2 Þwh kh þ ch = c : ; ws ks þ s2 D

Note: If the axial load does not produce column action, the constant need not be used to multiply the term [FDo (1 þ K 2 )/8] throughout this chapter

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ð14-59Þ

ð14-60Þ

DESIGN OF SHAFTS

14.10

CHAPTER FOURTEEN

Particular

Formula

STIFFNESS Instead of computing the transverse deﬂection, the maximum distance between the bearings (in meters) may be computed by the empirical formula to limit the transverse deﬂection to 0.8 mm/m of length

1500 cD2=3 n þ 1500 where c is a constant from Table 14-3

L¼

ð14-61Þ

RIGIDITY Moor’s formula for the increase of the angle of twist due to the keyway and applies only to the keyseated length of shaft

K1 ¼ 1 þ

0:4b þ 0:7h D

ð14-62Þ

EFFECT OF KEYWAYS The lowering of the strength of shaft by keyways may be taken into account by introducing a factor similar to a stress-concentration factor (or Moor’s formula for lowering the strength of shaft)

K ¼1þ

0:2b þ 1:1h D

ð14-63Þ

THE BUCKLING FACTOR For short columns or when l=k 115 For long columns or when l=k 115 (Euler’s formula)

1 1 0:0044ðl=kÞ sy l ¼ 2 nE k

¼

ð14-64Þ ð14-65Þ

where n ¼ 1 for hinged ends ¼ 2:25 for fixed ends ¼ 1:6 for both ends pinned or guided and partly restrained ( ¼ 1 for tensile load)

SHAFTS SUBJECTED TO VARIOUS STRESSES (1) Shaft subjected to steady torque and reversed bending moment taking into consideration stress concentration:

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DESIGN OF SHAFTS DESIGN OF SHAFTS

Particular

14.11

Formula

Diameter of solid shaft: (a) According to maximum shear stress failure theory using Soderberg [4] criterion for fatigue strength a m 1 þ ¼ sf sy n

( 32n sy

D¼

2 1=2 )1=3 sy 2 M þ ðKf Mtm Þ Kf sf ba ð14-66Þ

where Kf ¼ fatigue stress-concentration factor due to bending, tension, or compression Kfr ¼ fatigue stress-concentration factor due to torsion

(b) According to maximum shear stress theory of failure using modiﬁed Goodman criterion for fatigue strength a 1 þ m ¼ sf sut n

Kf ¼ Kf ¼ 1 for ductile material under steady state of stress ( 2 1=2 )1=3 32n sut 2 D¼ M þ ðKf Mtm Þ Kf sf ba sut ð14-67Þ

Diameter of hollow shaft: (c) According to distortion-energy theory of failure using modiﬁed Goodman criterion for fatigue strength

(d) According to distortion-energy theory of failure combined with Gerber parabolic relation na nm 2 ¼1 þ sf sut

(e) According to distortion-energy theory of failure using ASME elliptic locus for fatigue strength 2 na nm 2 þ ¼1 sf sy

" Do ¼

16n 2Kf sut Mba sf sut ð1 K 4 Þ #1=3 pﬃﬃﬃ þ 3Kf Mtm

ð14-68Þ

( 16n Kf sut Mba Do ¼ sf sut ð1 K 4 Þ " #1=2 )1=3 2 sut 2 Kf þ M þ 3ðKf Mtm Þ sf ba ( Do ¼

"

sy 16n M 2Kf 4 sf ba sy ð1 K Þ #1=2 )1=3

2

þ 3ðKf Mtm Þ2

Bagci failure locus equation in quartic (fourthdegree) form

i.e.,

and yielding criterion (Langer) equation combined with any theories of failure can be used to predict the fatigue strength of shaft

na þ sf

i.e.,

a þ m 1 ¼ n sy

nm sy

4 ¼1

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ð14-69Þ

ð14-70Þ

DESIGN OF SHAFTS

14.12

CHAPTER FOURTEEN

Particular

Formula

(2) Shaft subjected to ﬂuctuating loads, i.e., reversed bending and reversed torque, taking into consideration stress concentration (a) The diameter of solid shaft according to maximum shear stress theory of failure using Soderberg criterion for fatigue strength

D¼

32n 2 ðMbe þ Mte2 Þ1=2 sy

1=3 ð14-71Þ

where Mbe ¼ static equivalent of cyclic bending moment sy ¼ Kf Mbm þ Kf M sf ba

(b) The diameter of hollow shaft according to distortion-energy theory of failure combined with Soderberg criterion for fatigue strength

Mte ¼ static equivalent of cyclic torsional moment sy M ¼ Kf Mtm þ Kf sf ta 1=3 16n 2 2 1=2 Do ¼ þ 3M Þ ð14-72Þ ð4M te be sy ð1 K 4 Þ where Mbe and Mte have the same meaning as given under Eq. (14-71)

(3) Shaft subjected to constant bending and torsional moments and reversed torsional and bending moments at the same frequency taking into consideration stress concentration (a) The diameter of solid shaft according to maximum distortion energy theory of failure using modiﬁed Goodman criterion for fatigue strength

D¼

16n f½4ðKf Mbm Þ2 þ 3ðKf Mtm Þ2 1=2 g sut 1=3 þ sut f½4ðKf Mba Þ2 þ 3ðKf Mta Þ2 1=2 g sf ð14-73Þ

(b) The diameter of solid shaft according to maximum shear stress theory of failure combined with modiﬁed Goodman criterion for fatigue strength

where Kf ¼ Kf ¼ 1 for constant torsional and bending moments 2 32n D¼ Mbm þ Kf sut Mba sut sf 2 1=2 1=3 sut þ Mtm þ Kf M ð14-74Þ sf ta

(c) The diameter of hollow shaft according to maximum shear stress theory of failure using Soderberg criterion for fatigue strength

Do ¼

32n sy ð1 K 4 Þ

Kf Mbm þ Kf

sy M sf ba

2 1=2 1=3 sy þ Kf Mtm þ Kf Mta sf

2

ð14-75Þ

where Kf ¼ Kf ¼ 1 for constant bending and torsional moments

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DESIGN OF SHAFTS DESIGN OF SHAFTS

Particular

14.13

Formula

(4) Cyclic axial load combined with reversed bending and torsional moments taking into consideration stress concentration as per ASME Code for Design of Transmission Shafting (a) The diameter of solid shaft according to maximum shear stress theory of failure and Soderberg relation for fatigue strength

( D¼

32n sy

1=2 )1=3 Fae D 2 2 þ Mte Mbe þ 8

ð14-76Þ

where Mbe and Mte have the same meaning as given under Eq. (14-71)

(b) The diameter of hollow shaft according to distortion-energy theory of failure combined with modiﬁed Goodman relation for fatigue strength

Fae ¼ static equivalent axial load sy ¼ Kf Fam þ Kf F sf aa " ( 0 32n Fae Do ð1 þ K 2 Þ 2 0 Do ¼ þ M be 8 sut ð1 K 4 Þ )1=2 #1=3 3 þ Mte02 ð14-77Þ 4 where sut M sf ba Mte0 ¼ Kf Mtm þ Kf sut Mta sf 0 Fae ¼ Kf Fam þ Kf sut Faa sf

0 Mbe ¼ Kf Mbm þ Kf

When K ¼ 0, this equation reduces to an equation for a solid shaft

(5) The diameter of solid shaft subjected to axial, bending, and torsional alternating loads according to distortion-energy theory of failure combined with Soderberg relation for fatigue as per ASME Code for Design of Transmission Shafting5

The value of is given by Eq. (14-65) 2 1=2 32n F D 2 3Mta Mba þ a D¼ þ 4 sf 2 )!1=3 ( 2 1=2 32n Fm D 2 3Mtm þ þ Mbm þ 4 sut 2 ð14-78Þ Not explicit in D, use iterative methods to solve

Although ASME has withdrawn the ASME Code for Design of Transmission Shafting, some of the ASME equations given here have historic interest and hence are retained in this book.

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DESIGN OF SHAFTS

14.14

CHAPTER FOURTEEN

Particular

(6) The diameter of shaft made of brittle material, which is subjected to reversed bending and torsional moments taking into consideration stress concentration as per maximum normal stress theory of failure combined with modiﬁed Goodman relation for fatigue strength

Formula

D¼

1=3 16n 0 02 ½Mbe þ ðMbe þ Mte02 Þ1=2 sut

ð14-79Þ

for solid shaft 1=3 16n 0 02 02 1=2 þ ðM þ M Þ ½M Do ¼ te be be sut ð1 K 4 Þ ð14-80Þ 0 and Mte0 have the same for hollow shaft, where Mbe meaning as given under Eq. (14-77)

(7) Shaft subjected to combined axial, bending, and torsional reversed loads taking into consideration stress concentration and shock (a) The diameter of hollow shaft according to distortion-energy theory of failure using Soderberg relation The symbols used in Eqs. (14-80) to (14-85) and Figs. 14-1 and 14-2 are diﬀerent than that of the ANSI/ ASME standard B106. IM-1985 in order to remain consistent with the symbols used in this Handbook.

( 32n Fae Do ð1 þ K 2 Þ 2 Do ¼ þ K M sb be 8 sy ð1 K 4 Þ )1=2 !1=3 3 þ Kst Mte2 ð14-81Þ 4 where Fae , Mbe and Mte have the same meaning as given under Eqs. (14-71) and (14-76) Refer to Table 14-4 for Ksb and Kst

New ASME Code for design of transmission shafting: The diameter of shaft subjected to fully reversed bending i.e., zero mean bending component and torsional ﬂuctuating loads, i.e. alternating loads taking into consideration stress concentration according to distortion energy theory of failure combined with modiﬁed Goodman relation for fatigue as per new ANSI/ASME code for transmission shafting.

The factor of safety, n

" D¼

32n sf

3 ðKf Mba Þ2 þ ðKf Mta Þ2 4

1=2

1=2 #1=3 32n 3 2 2 ðKf Mbm Þ þ ðKf m Mtm Þ þ sut 4 ð14-82Þ when the axial load ¼ Fa is zero " 1=2 1 32 1 3 2 2 M Þ þ M Þ ðK ¼ ðK f ba f ta n D3 sf 4 1=2 # 1 3 2 2 ðKf Mba Þ þ ðKf m Mtm Þ þ ð14-83Þ sut 4

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DESIGN OF SHAFTS DESIGN OF SHAFTS

Particular

The diameter of shaft made of brittle material subjected to reversed bending and torsional moments taking into consideration stress concentration as per maximum normal stress theory of failure combined with modiﬁed Goodman relation for fatigue strength

Formula

D¼

1=3 16n 0 02 fMbe þ ðMbe þ Mte02 Þ1=2 g sut ð14-84Þ

for solid shaft Do ¼

1=3 16n 0 02 02 1=2 fM þ ðM þ M Þ g te be be sut ð1 K 4 Þ ð14-85Þ

for hollow shaft 0 ¼ Kf Mbm þ Kf where, Mbe

Mte0 ¼ Kf Mtm þ For combined fatigue test data for reversed bending combined torsion and combined with reversed torsion on steel specimens.

14.15

sul M sf ba

sul M sf ta

Refer to Fig. 14-1.

FIGURE 14-1 Results of fatigue Tests of steel specimens subjected to Reversed Bending and Torsion. Source: Design of Transmission Shafting, American Society for Mechanical Engineers, New York, ANSI/ASME standard B106IM, 1985. Kececioglu, D. B., and V. R. Lalli, Reliability Approach to Rotating Component Design, Technical Note TND-7846, NASA, 1975. Davies, V. C., H. T. Gough, and H. V. Pollard, Discussion to the Strength of Metals under Combined Alternating stresses, Proc of the Inst. Mech. Eng., 131(3), pp. 66–69, 1935. Loewenthal, S. H., Proposed Design Procedure for Transmission Shafting under Fatigue Loading, Technical Note TM-7802, NASA, 1978. Gough, H. J., and H. V. Pollard, The Strength of Metals under Combined Alternating Stresses, Proc of the Inst. Mech. Eng., 131(3), pp. 3–103, 1935.

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DESIGN OF SHAFTS

14.16

CHAPTER FOURTEEN

Particular

Formula

GENERAL See Tables 14-1 to 14-6 and Fig. 14-2 for further details on shafting design;3 refer to Table 14-4 for shock load factors Ksb and Kst For further design details on shafting

Refer to Tables 14-5 to 14-7.

2 2 5 (0 16 0 ( •4 (0 0•3 16 •2 33 ) 66 ) )

re

6 5 •3

8) 20 6) 0• 16 5( (0• • 12 0 1

ing

mo e ad

a ftw o Fs D SP T R fA o ion s r ve

s du

ifie

en

d mo

e sb

a

is

Th

Fh D P

FIGURE 14-2 Nomogram for determining diameter (d), speed (n), force (F), torque (Mt ), and power (P) in Customary Metric units and System International units. (K. Lingaiah, Machine Design Data Handbook, Vol. II, Suma Publishers, Bangalore, India, 1986.)

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DESIGN OF SHAFTS DESIGN OF SHAFTS

14.17

REFERENCES 1. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative, Bangalore, India, 1962. 2. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI Units and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 3. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI Units and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 4. Soderberg, C. R., ‘‘Working Stresses,’’ J. Appl. Mechanics, Vol. 57, p. A-106, 1935. 5. ASME Code for Design of Transmission Shafting, Standard ANS/ASME B106.1M, 1985. 6. Shigley, J. E., Machine Design, McGraw-Hill Publishing Company, New York, 1956. 7. Kececioglu, D. B., and V. R. Lalli, Reliability Approach to Rotating Component Design, Technical Note TND-7846, NASA, 1975 8. Davies, V. C., H. T. Gough, and H. V. Pollard, Discussion to the Strength of Metals under Combined Alternating stresses, Proc of the Inst. Mech. Eng., 131(3), pp. 66–69, 1935. 9. Loewenthal, S. H., Proposed Design Procedure for Transmission Shafting under Fatigue Loading, Technical Note TM-7802, NASA, 1978. 10. Gough, H. J., and H. V. Pollard, The Strength of Metals under Combined Alternating stresses, Proc of the Inst. Mech. Eng., 131(3), pp. 3–103, 1935.

BIBLIOGRAPHY Berchard, H. A., ‘‘A Comprehensive Method for Designing Shafts to Insure Fatigue Life,’’ Machine Design, April 25, 1963. Black, P. H., and O. Eugene Adams, Jr., Machine Design, McGraw-Hill Publishing Company, New York, 1983. British Standards Institution. Deutschman, A. D., W. J. Michels, and C. E. Wilson, Machine Design—Theory and Practice, Macmillan Publishing Company, New York, 1975. Maleev, V. L., and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954. Marks’ Standard Handbook for Mechanical Engineers, 8th ed., McGraw-Hill Publishing Company, New York, 1978. Vallance, A., and V. L. Doughtie, Design of Machine Members, McGraw-Hill Publishing Company, New York, 1951.

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DESIGN OF SHAFTS

14.18

CHAPTER FOURTEEN

TABLE 14-1 Empirical shafting formulas Power capacity, P

Load factors considered Kind of service

Torsion, Kt

Bending, Kb

kW

hp

Transmission shafts in torsion only Line shafting with limited bending Head or main shafts with heavy bending loads

1.0 1.0 1.0

1.0 1.5 2.5

54,831D3 n0 34,532D3 n0 20,715D3 n0

1:225 106 D3 n 7:715 107 D3 n 4:628 107 D3 n

TABLE 14-2 Shock and endurance factors Nature of loading Stationary shafts Gradually applied load Suddenly applied load Rotating shafts Steady or gradually applied loads Suddenly applied loads, minor shocks only Suddenly applied loads, heavy shocks

TABLE 14-3 Values of constant c Kb

Kt

1.0 1.5–2.0

1.0 1.5–2.0

1.5 1.5–2.0

1.0 1.0–1.5

2.0–3.0

1.5–3.0

Type of shaft loading Shaft heavily loaded, subjected to shock, or reversed under full load Line shafts and countershafts, loaded in bending but not reversed Line shafts or bar with pulleys close to the bearings

MPa

kpsi

0.82

17

2.5

1.1

27

4.0

1.56

44

6.4

TABLE 14-4 Shock load factorsa for use in Eq. (14-81) Nature of load

Ksb , Kst

Gradually applied load Loads applied with minor shocks Loads applied with heavy shocks

1.00 1.0–1.5 1.5–2.0

Allowable stress

Coeﬃcient c in Eq. (14-61)

a

Data from Berchard, H. A., ‘‘A Comprehensive Method for Designing Shafts to Insure Fatigue Life,’’ Machine Design, April 25, 1963.

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DESIGN OF SHAFTS DESIGN OF SHAFTS

14.19

TABLE 14-5 Spacinga for ﬁne shaft bearings

Transmission shaft stressed in torsion only, mm

Line shaft carrying pulleys or gears and subjected to usual bending loads, mm

Diameter of shaft, mm

1–250 rpm

251–400 rpm

1–250 rpm

251–400 rpm

36.5 49.0 62.0 74.5 87.5 100.0 112.5

274.5 305.0 335.5 366.0 396.0 427.0 457.0

244.0 274.5 305.0 335.5 366.0 396.0 427.0

213.5 229.0 244.0 259.0 274.5 289.5 305.0

198.0 213.5 228.5 244.0 259.0 274.5 289.5

a

Center-to-center distance in millimeters.

TABLE 14-6 Sizes of shafts Diameters, mm (in) 4 (0.16) 5 (0.20) 6 (0.24) 7 (0.28) 8 (0.32) 9 (0.36) 10 (0.4)

12 15 17 20 25 30 35

(0.48) (0.60) (0.68) (0.80) (1.0) (1.2) (1.4)

40 (1.6) 45 (1.8) 50 (2.0) 55 (2.2) 60 (2.4) 65 (2.6) 70 (2.8)

75 (3.0) 80 (3.2) 85 (3.4) 90 (3.6) 95 (3.8) 100 (4.0) 105 (4.2)

110 (4.4) 120 (4.8) 130 (5.2) 140 (5.6) 150 (6.0) 160 (6.4) 170 (6.8)

180 (7.2) 190 (7.6) 200 (8.0) 220 (8.8) 240 (9.6) 260 (10.4) 280 (11.2)

TABLE 14-7 Load factors for various machines, kl a Driver

Driven machinery

Factor, kl

Steam turbine

Electric generator, steady load; turbine blower Electric generator, uneven load; centrifugal pump Induced-draft fan; line shaft; gear drive Rolling mill, gear drive Turbine blower; metalworking machinery Centrifugal pump; wood working machinery Line shaft; ship propeller; double acting pump Triplex single-acting pump; elevator; crane Compressor, air or ammonia Rolling mill; rubber mill Values for electric-motor drive multiplied by 1.2–1.5 Values for electric-motor drive multiplied by 1.3–1.6 the factor depending on the coeﬃcient of steadiness of the ﬂywheel

1.00 1.25 1.50 2.00 1.25 1.50 1.75 1.75 1.75 2.50

Electric motor

Steam engine Gas and oil engines a

To be used also in Eqs. (5–9) and (19–79).

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Source: MACHINE DESIGN DATABOOK

CHAPTER

15 FLYWHEELS SYMBOLS1,2 a A b Cf d dh D Do E Fc Fc0 g h i ko I J kt Mtm Mt m n n1 n2 r t T1 T2 v v1 v2 W Z

major axis of ellipse, m (in) negative acceleration or deceleration, m/s2 (ft/s2 ) cross-sectional area of the rim, m2 (in2 ) minor axis of ellipse, m (in) width of rim, m (in) coeﬃcient of ﬂuctuation of rotation diameter of shaft, m (in) hub diameter, m (in) ﬂywheel diameter, m (in) outside diameter of rim, m (in) excess energy, J (ft lbf ) centrifugal force, kN (lbf ) centrifugal force per unit width of rim, kN (lbf ) acceleration due to gravity, 9.8066 m/s2 (32.2 ft/s2 ) depth of rim, m (in) number of arms polar radius of gyration of the rim, m (in) mass moment of inertia, N s2 m (lbf s2 ft) polar second moment of inertia, m4 (in4 ) torsional stiﬀness of shaft, N m/rad (lbf in/rad) mean torque, N m (lbf ft) transmitted torque, N m (lbf ft) coeﬃcient of steadiness mean speed, rpm maximum speed, rpm minimum speed, rpm mean radius of the ﬂywheel, m (in) time, s tension in belt on tight side, kN (lbf ) tension in belt on slack side, kN (lbf ) mean rim velocity, m/s (ft/min) maximum rim velocity, m/s (ft/min) minimum rim velocity, m/s (ft/min) rim weight, kN (lbf ) speciﬁc weight of material or weight density, N/m3 (lbf/in3 ) sectional modulus of the arm cross section at the hub, m3 (in3 )

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FLYWHEELS

15.2 1 , 2 ! !1 , !2

CHAPTER FIFTEEN

stress (also with subscripts), MPa (psi) maximum and minimum angular displacement of ﬂywheel from constant speed deviation, rad (deg) average angular speed, rad/s maximum and minimum angular speed, respectively, rad/s Particular

Formula

The equation of motion of ith rotor of Ii inertia in a multirotor system connected by (i 1) number of shafts of various inertias subjected to external torque

Ii i ¼ Mti Mtði 1Þ

ð15-1Þ

The equation of motion of a ﬂywheel, which is mounted on a shaft between two supports and rotates with an angular velocity and subjected to an input external torque Mti

I ¼ Mti Mto ¼ kt ð2 1 Þ

ð15-2Þ

where Mto ¼ output torque, N m (lbf ft) ¼ angular displacement of ﬂywheel, rad (deg)

KINETIC ENERGY Kinetic energy (Fig. 15-1)

1 Wv2 1 2 K ¼ mv2 ¼ ¼ I! 2g 2 2

For variation of torque with crank angle for twocylinder engine

Refer to Fig. 15-1.

ð15-3Þ

FIGURE 15-1 Torque-crank shaft angle curve for a two-cylinder engine.

The kinetic energy of ﬂywheel at an angular displacement 1 and at angular velocity !1 during one cycle

1 Wv21 K1 ¼ I!21 ¼ 2g 2

ð15-4Þ

The kinetic energy of ﬂywheel at an angular displacement 2 and at angular velocity !2

1 Wv22 K2 ¼ I!22 ¼ 2g 2

ð15-5Þ

The change in kinetic energy or energy ﬂuctuation due to change in angular velocity !1 to !2 in one cycle

1 Wðv22 v21 Þ E ¼ K2 K1 ¼ Ið!22 !21 Þ ¼ 2 2g ¼ 12 Ið!2 !1 Þð!2 þ !1 Þ ¼ Ið!2 !1 Þ! ¼ Wðv2 v1 Þ

v g

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ð15-6Þ

FLYWHEELS FLYWHEELS

Particular

The coeﬃcient of ﬂuctuation of speed or rotation The change in kinetic energy or excess energy

15.3

Formula

Cf ¼

!2 !1 v2 v1 n2 n1 ¼ ¼ ! v n

E ¼ K2 K1 ¼ I!2 Cf ¼

ð15-7Þ

Wv2 Cf g

ð15-8Þ

FLYWHEEL EFFECT OR POLAR MOMENT OF INERTIA

Wk2 ¼

The mean angular velocity

!¼

!2 þ !1 2

ð15-10Þ

The coeﬃcient of steadiness

m¼

1 Cf

ð15-11Þ

182:40gE n21 n22

ð15-9Þ

Refer to table 15-1 for Cf .

STRESSES IN RIM (Figs. 15-2 and 15-3) The component of the centrifugal force normal to any diameter of the ﬂywheel

Fc ¼

2bhr2 !2 g

ð15-12Þ

The tangential force due to hoop stress in the ﬂywheel rim (Fig. 15-3)

F ¼

bhr2 !2 g

ð15-13Þ

The tensile stress created in each cross section of the rim by the centrifugal force

¼ 0:01095

The centrifugal force per unit width of rim (Fig. 15-3)

2 2 r n g

Fc0 ¼ 0:01095

r2 n2 h g

SI

ð15-14Þ

SI

ð15-15Þ

TABLE 15-1 Coeﬃcient of ﬂuctuation of rotation, Cf Driven machine

Type of drive

Cf

AC generators, single or parallel AC generators, single or parallel DC generators, single or parallel DC generators, single or parallel Spinning machinery Compressure, pumps Paper, textiles, and ﬂour mills Woodworking and metalworking machinery Shears and pumps Concrete mixers, excavators, and compressors Crushers, hammers, and punch presses

Direct-coupled Belt Direct-coupled Belt Belt Gears Belt Belt Flexible coupling Belt Belt

0.01 0.0167 0.0143 0.029 0.02–0.015 0.02 0.025–0.02 0.0333 0.05–0.04 0.143–0.1 0.2

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FLYWHEELS

15.4

CHAPTER FIFTEEN

Particular

The bending stress

The combined tensile stress

Formula

b ¼ 0:2146

r3 n2 ghi2

SI

R ¼ 0:75 þ 0:25b

ð15-16Þ ð15-17Þ

STRESSES IN ARMS (Fig. 15-2) The stresses in the arm

1 ¼

Mt ðD dh Þ iZD

ð15-18Þ

FIGURE 15-2 Flywheel.

FIGURE 15-3 Centrifugal force acting on the rim of a ﬂywheel.

When the ﬂywheel is used as a belt pulley, the stresses at the hub

2 ¼

ðT1 T2 ÞðD dh Þ 2iZ

ð15-19Þ

In case of thin-rim ﬂywheel, the stress

02 ¼

ðT1 T2 ÞðD dh Þ iZ

ð15-20Þ

Stress due to centrifugal force

3 ¼ 0:01095

r2 n2 g

The maximum tensile stress in an arm is at hub

max ¼ 1 þ 2 þ 3

The force necessary to stop the ﬂywheel

F¼

SI

Wa g

ð15-21Þ ð15-22Þ ð15-23Þ

RIM DIMENSIONS (Fig. 15-2) The relation between ko in cm and the outside diameter D of the rim in m Cross-sectional area of the rim

k2o ¼ 0:125½D2o þ ðDo 2hÞ2

A¼

W 2k

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ð15-24Þ

ð15-25Þ

FLYWHEELS FLYWHEELS

Particular

15.5

Formula

The relation between depth and width of rim

b ¼ 0:65 to 2 h

ð15-26Þ

The outside diameter of rim

Do ¼ 2ko þ h ðapprox:Þ

ð15-27Þ

The hub diameter in m

dh ¼ 1:75d þ 6:35 103 ¼ 2d

ð15-28Þ

The hub length

l ¼ 2d to 2:5d

ð15-29Þ

rﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 64Z

ð15-30Þ

ARMS (Fig. 15-2) The major axis in case of elliptical section can be computed from the relation

a¼

where z ¼

ba2 32

and

a ¼ 2b

ð15-31Þ

REFERENCES 1. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 2. Lingaiah, K., Machine Design Data Handbook (SI and U.S. Customary Units), McGraw-Hill Publishing Company, New York, 1994.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

16 PACKINGS AND SEALS SYMBOLS1;2 A Ag A1 , A2 b c d d1 d2 da di Dm Dam E Fb F Fo g h hi h i l l1 , l2 (dl) Mt Mti

area of seal in contact with the sliding member, m2 (in2 ) gasket area over which the bolt loads are distributed, m2 (in2 ) area of cross section of unthreaded and threaded portions of bolt, m2 (in2 ) width of U-collar, m (in) gland width or depth of groove, m (in) radial clearance between rod and the bushing, radial deﬂection of the ring, m (in) nominal diameter of the bolt, m (in) diameter of sliding member, m (in) outside diameter of packing material, m (mm) outside diameter of seal ring (Fig. 16-3), m (in) minor diameter of bolt, m (in) actual diameter of wire, m (in) inside diameter of packing material, m (in) estimated mean diameter of conical spring, m (in) actual mean diameter of conical spring, m (in) modulus of elasticity, GPa (psi) bolt load, kN (lbf ) frictional force, kN (lbf ) frictional force of the stuﬃng box when there is no ﬂuid pressure, kN (lbf ) acceleration due to gravity, 9.8066 m/s2 (9806.6 mm/s2 ) (32.2 ft/s2 ) radial ring wall thickness, m (in) uncompressed gasket thickness, m (in) loss of head, m/m (in/in) number of bolts depth of U-collar (Fig. 16-2a), m (in) length of joint, m (in) incremental length in the direction of velocity [Eq. (16-15)], m (in) bolt elongation [Eq. (16-24)], m (in) twisting moment, N m (lbf in) initial bolt torque, N m (lbf in)

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PACKINGS AND SEALS

16.2

CHAPTER SIXTEEN

ﬂuid pressure, MPa (psi) ﬂange pressure on the gasket, MPa (psi) minimum per cent compression to seal pressure diﬀerential in the direction of velocity [Eq. (16-15)], MPa (psi) discharge, m3 /s (cm3 /s, mm3 /s) (in3 /s) equivalent radius, m (in) velocity, m/s (ft/min) nominal packing cross section, m (in) deﬂection of spring, m (in) absolute viscosity of ﬂuid, Pa s (cP) design stress, MPa (psi) coeﬃcient of friction

p pf Ps (dp) Q r v w y d

Particular

Formula

ELASTIC PACKING1–3 Frictional force exerted by a soft packing on the reciprocating rod

F ¼ kpd

ð16-1Þ

where k ¼ 0:005 and p ¼ 0:343 MPa SI k ¼ 0:2 and p ¼ 50 psi USCS

FRICTION Friction resistance

F ¼ Fo þ Ap

ð16-2Þ

where ¼ 0:01 for rubber and soft lubricated leather ¼ 0:15 for hard leather Torsional resistance in a rotary motion friction

F d kd 2 p ¼ 2 2 where k ¼ 0:005 SI k ¼ 0:2 USCS

Mt ¼

ð16-3Þ

c ¼ 0:2d þ 5 mm if d 100 mm pﬃﬃﬃ c ¼ 0:08 d if d > 0:1 mm pﬃﬃﬃ c ¼ 0:5 d if d > 4

ð16-4Þ

METALLIC GASKETS (Fig. 16-1) The empirical relations3

h¼

SI

ð16-5aÞ

USCS

ð16-5bÞ

d þ 12:54 mm or 0:5 in 8

ð16-6Þ

a ¼ d þ 2c

ð16-7Þ

¼ 108 to 158

ð16-8Þ

pﬃ d2 ¼ 0:2ðd þ 0:102Þ= i pﬃ d2 ¼ 0:2ðd þ 4Þ= i

SI

ð16-9aÞ

USCS

ð16-9bÞ

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PACKINGS AND SEALS PACKINGS AND SEALS

Particular

16.3

Formula

FIGURE 16-1 Stuﬃng box with bolted gland. (V. L. Maleev and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954.)

Diameter of bolt is also found by equating the working strength of the bolts to the pressure p exerted by the ﬂuid on the gland and the frictional force F

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðd12 d 2 Þp 4F d2 ¼ þ id id

ð16-10Þ

where d2 ¼ minor diameter of bolt, m (in) d ¼ 68:7 to 83.3 MPa (10 to 12 kpsi)

SELF-SEALING PACKING (Fig. 16-2) Houghton, Welch, and Jenkin’s formula for an approximate thickness of a U-shaped collar for great pressure3

h ¼ 6:36 103 d 0:2

SI

ð16-11aÞ

SI

ð16-11bÞ

USCS

ð16-11cÞ

where h and d in m h ¼ 1:6d 0:2 where h and d in mm h ¼ 0:12d 0:2 where d and d in in

FIGURE 16-2 U-collar.

Width

b ¼ 4h

ð16-12aÞ

Depth

l ¼ 1:2b to 1:8b

ð16-12bÞ

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PACKINGS AND SEALS

16.4

CHAPTER SIXTEEN

Particular

Formula

PACKINGLESS SEALS Leakage of the ﬂuid past a rod can be computed with fair accuracy by the formula

Q¼

c3 d ð p1 p2 Þ 12 l

Q ¼ 1:79ð100cÞ3

ð p1 p2 Þd l

SI

ð16-13aÞ

USCS

ð16-13bÞ

Refer to Table 16-1 for values of . TABLE 16-1 Absolute viscosities Temperature

Absolute viscosity,

Temperature

Absolute viscosity,

Fluid

K

8C

MPa s

cP

K

8C

MPa s

cP

Steam Air Water Water Gasoline Kerosene Fuel oil, 308 Baume´ Fuel oil, 248 Baume´ Spindle oil Machine oil Castor oil

293 293 273 293 293 293 293 293 293 293 293

20 20 0 20 20 20 20 20 20 20 20

0.0097 0.018 1.79 1.0 0.6 2.7 5.0 40 20–35 200–500 1000

0.0097 0.018 1.79 1.0 0.6 2.7 5.0 40 20–35 200–500 1000

539 366 311 333 355 355 355 355 355 372 316

266 93 38 60 82 82 82 82 82 99 43

0.018 0.022 0.69 0.40 0.30 1.30 1.60 4 3–4 1.5–16 200

0.018 0.022 0.69 0.40 0.30 1.30 1.60 4 3–4 5.5–16 200

STRAIGHT-CUT SEALINGS (Fig. 16-3a) The equation for loss of liquid head

h ¼ 64v=2gd12 ðdpÞr 8ðdlÞ

ð16-14Þ

2

Leakage velocity

v¼

Quantity of leakage

Q ¼ vA

ð16-16Þ

Stress in a seal ring

0:4815cE ¼ 2 d h 11 h

ð16-17Þ

For allowable temperatures for materials and surface treatment

Refer to Table 16-2.

FIGURE 16-3(a) Straight-cut seal.

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ð16-15Þ

PACKINGS AND SEALS PACKINGS AND SEALS

Particular

16.5

Formula

V-RING PACKING Single-spring installations The estimated mean diameter of conical spring

The wire size (Table 16-3)

Dm ¼ di þ d¼

3w 2

D2m 139300

ð16-18Þ 1=3 SI

ð16-19aÞ

USCS

ð16-19bÞ

Customary Metric

ð16-19cÞ

where d and Dm in m d¼

D2m 3535

1=3

where d and Dm in in d¼

D2m 193:3

1=3

where d and Dm in mm The actual mean diameter of conical spring The deﬂection of spring

Multiple-spring installations BOLTS AND STRESSES IN FLANGE JOINTS The bolt load in gasket joint The ﬂange pressure developed due to tightening of bolts that hold the gasket joint mechanical assembly together

The load on the bolt when it is tightened

STRESSES IN GROOVED JOINTS The uncompressed gasket thickness that will provide the minimum sealing compression when the ﬂanges are tightened into face-to-face contact

Dam ¼ d1 12 ðw þ da Þ y¼

0:0123D2am da

ð16-20Þ ð16-21Þ

Two standard cylindrical spring sizes are generally used, depending on packing size.

Fb ¼

11mti d

ð16-22Þ

pf ¼

iFb 2iMt ¼ Ag Cu Ag Cu db

ð16-23Þ

where Cu ¼ torque friction coeﬃcient Fb ¼

EðdlÞ ðl1 =A1 Þ þ ðl2 =A2 Þ

ð16-24Þ

hi ¼

100b 100 Ps

ð16-25Þ

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PACKINGS AND SEALS

16.6

CHAPTER SIXTEEN

Particular

Formula

BOLT LOADS IN GASKET JOINT ACCORDING TO ASME BOILER AND PRESSURE VESSEL CODE (Fig. 16-3b)4

FIGURE 16-3(b) Location of gasket load reaction.

The required bolt load under operating condition suﬃcient to contain the hydrostatic end force and simultaneously to maintain adequate compression on the gasket to ensure seating

Wm1 ¼ H þ HP ¼ ð=4G2 PÞ þ 2bGmP

ð16-26Þ

The required initial bolt load to seat the gasket jointcontact surface properly at atmospheric temperature condition without internal pressure

Wm2 ¼ bGy

ð16-27Þ

Total required cross-sectional area of bolts at the root of thread

Am > Am1 or Am2

ð16-28Þ

Total cross-sectional area of bolt at root of thread or section of least diameter under stress required for the operating condition

Am1 ¼

Wm1 sbd

ð16-29Þ

Refer to Tables 8-20 and 8-21 for gasket factor m and minimum design seating stress, y, b, and bo

Refer to Table 8-17 for sbd Total cross-sectional area of bolt at root of thread or section of least diameter under stress required for gasket seating The actual cross-sectional area of bolts using the root diameter of thread or least diameter of unthreaded portion (if less), to prevent damage to the gasket during bolting-up

Am2 ¼

Ab ¼

Wm2 sbat

2yGN <j Am sbat

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ð16-30Þ

ð16-31Þ

PACKINGS AND SEALS PACKINGS AND SEALS

Particular

16.7

Formula

FLANGE DESIGN BOLT LOAD W The bolt load in the design of ﬂange for operating condition The bolt load in the design of ﬂange for gasket seating

W ¼ Wm1 W¼

ð16-32Þ

Am þ Ab sbat 2

ð16-33Þ

The relation between bolt load per bolt (Wb ), diameter of bolt (D) and torque (Mt )

Wb ¼ 0:17DMt for lubricated bolts USCS

(Note: The meanings of symbols given in Eqs. (16-26) to (16-37) are deﬁned in Chap. 8.)

Wb ¼ 263:5DMt

ð16-34Þ

where Wb in lbf, D in in, Mt in lbf in SI

ð16-35Þ

USCS

ð16-36Þ

where Wb in N, D in m, Mt in N m Wb ¼ 0:2DMt for unlubricated bolts where Wb in lbf, D in in, Mt in lbf in Wb ¼ 310DMt

SI

ð16-37Þ

where Wb in N, D in m, Mt in N m For location of gasket load reaction due to tightening of ﬂange bolts

Refer to Fig. 16-3b

The total load on bolts in the gasket joint according to Whalen5

Fb ¼ g Ag

ð16-38Þ

where Ag ¼ contact area of gasket, m2 (in2 ) g ¼ gasket seating stress, MPa (psi), taken from Table 16-35 The load on bolts, which is based on hydrostatic end force

Fb ¼ nPt Am

ð16-39Þ

where Pt ¼ test pressure or internal pressure if no test pressure is available, MPa (psi) Am ¼ hydrostatic area (based on mean diameter of gasket) on which internal pressure acts, m2 (in2 ) n ¼ factor of safety taken from Table 16-36 For more information on design data, selection of packing and seals, properties of sealants and packing materials, dimensions and tolerances of seals, and chamfers on shaft, operating temperatures of various types of seals, data for metallic o-rings, q-rings and oring gaskets, static and dynamic seals, lip seals, and safety factors, etc.

Refer to Tables 16-4 to 16-36

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PACKINGS AND SEALS

16.8

CHAPTER SIXTEEN

Particular

Formula

FIGURE 16-3(c) Plain bush seals. (Panels a and b courtesy of J. M. Neale, Tribology Handbook, Butterworths, London, 1973.)

Leakage through bush seals (Fig. 16-3c): The oil ﬂow (Q) through plain axial bush seal due to leakage under laminar ﬂow condition, Fig. 16-3c, panel a

2aðPs Pa Þ q l where Q in m3 /s (in3 /s)

Q¼

ð16-40Þ

¼ absolute viscosity, Pa s (cP) The symbols used in Eqs. (16-40) to (16-45) have the meaning as deﬁned in Fig. 6-13c, panels a and b. The volumetric ﬂow rate per unit pressure per unit periphery (q) under laminar ﬂow condition for axial bush seal, Fig. 16-3c, panel a

q¼

c3 ð1 þ 1:5"2 Þ a 12

ð16-14Þ

for incompressible ﬂuid where " ¼ c q¼

c3 Ps þ Pa 24 Pa

ð16-42Þ

for compressible ﬂuidb The oil ﬂow (Q) through plain radial bush seal due to leakage under laminar ﬂow condition, Fig. 16-3c, panel b The volumetric ﬂow rate per unit pressure per unit periphery (q) under laminar ﬂow condition for radial bush seal, Fig. 16-3c, panel b

Q¼

2aðPs Pa Þ q ab

c3 a b 12 a log a e b for incompressible ﬂuid q¼

q¼

c3 a b Ps þ Pa 24 a Pa

ð16-43Þ

ð16-44Þ

ð16-45Þ

for compressible ﬂuid

a b

If shaft rotates, onset of Taylor vortices limits validity of formula to ðVc =Þ For Mach number <1.0, i.e., ﬂuid velocity < local velocity of sound.

pﬃﬃﬃﬃﬃﬃﬃ c=a < 41:3 (where ¼ kinematic viscosity).

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PACKINGS AND SEALS PACKINGS AND SEALS

Particular

16.9

Formula

The radial pressure distribution for laminar ﬂow condition between smooth parallel surfaces in case face seal

p p1 ¼

3!2 2 6v r ðr R21 Þ 3 ln 20g R h

ð16-46Þ

where ¼ pressure at radial position r, MPa (lbf/in2 ) ¼ pressure at seal inside radius, MPa (psi) ¼ internal hydraulic pressure MPa (lbf/in2 ) ¼ radial position, m (in) ¼ kinematic viscosity N s/m2 (lbf s/in2 ) ¼ ﬂuid density, lb/in3 (kg/mm3 ) ¼ rotational speed, rad/s ¼ inside radius of rotating member, m (in) ¼ outside radius of rotating member, m (in) ¼ thickness of ﬂuid between members, m (in) h3 3!2 Q¼ ðR22 R21 Þ p2 p1 Þ 6 lnðR2 =R1 Þ 20g p p1 p2 r ! R1 R2 h

The amount of leakage of ﬂuid through face seal

ð16-47Þ where Q ¼ volumetric leakage rate of ﬂuid, m3 /s (in3 /s) The theoretical equation for zero leakage of ﬂuid through face seal

p2 p1 ¼

The power loss or consumed due to leakage of ﬂuid through face seal

P¼

The shape factor (Spf ) for a circular or annular gasket which is the ratio of the area of one load face to the area free to bulge6

3 !2 ðR22 R21 Þ 20

w2 ð16-49Þ ðR4 r41 Þ 13200h 2 where P ¼ power loss, hp D Di Spf ¼ o ð16-50Þ 4h where Do ¼ outside diameter of gasket, m (in) Di ¼ inside diameter of gasket, m (in)

For further design and selection of various types of seals, packings and gaskets, etc.

Refer to Figs. 16-4 to 16-14.

For nomenclature of gasketed joint

Refer to Fig. 16-15.

For packing assembly for a mechanical piston rod For shape factor for various gasket materials

6

For power absorption and starting torque for unbalanced and balanced seals

ð16-48Þ

Refer to Fig. 16-16. Refer to Fig. 16-17. Refer to Fig. 16-18.

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PACKINGS AND SEALS

16.10

CHAPTER SIXTEEN

FIGURE 16-4 Single radial lip seal.

FIGURE 16-5 Exclusion seal.

FIGURE 16-6 Radial exclusion seal. (Produced from ‘‘Packings and Seals’’ Issue, Machine Design, Jan. 20, 1977.)

FIGURE 16-7 Two-piece rod seal. (Produced from ‘‘Packings and Seals’’ Issue, Machine Design, Jan. 20, 1977.)

FIGURE 16-8 Clearance seal idealized labyrinth.

FIGURE 16-9 Face seal.

FIGURE 16-10 Compression packing.

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PACKINGS AND SEALS

TABLE 16-2 Allowable temperatures for materials and surface treatments Temperature

Temperature

Material or surface treatment

8F

8C

Material or surface treatment

Material Low-alloy gray irons Malleable iron Ductile iron Ni-Resist Ductile Ni-Resist 410 Stainless Steel 17-4 PH Stainless Steel Bronze Stellite no. 31 Inconel X Tool steel, Rc 62–65

650 720 720 800 1000 900 900 500 1200 1200 900

343 382 382 427 538 482 482 260 649 649 482

Carbon (high-temperature) K-30 (ﬁlled teﬂons) S-Monel Polymide Surface treatment Chromium plate Tin plate Silver plate Cadmium nickel plate Flame plate LW1 Flame plate LC-1A Flame plate LA-2

8F

8C

950 450–500 950 750

510 232–260 510 399

500 720 600 1000 1000 1600 1600

260 382 315.5 538 538 871 871

TABLE 16-3 Standard wire sizes for V-packing expanders Wire gaugea

Wire diameter, mm

Wire gauge

Wire diameter, mm

19 18 17 16 15 14

1.04 1.20 1.37 1.57 1.83 2.03

13 12 11

2.31 2.67 2.05 3.17 3.31 3.60

a

1 8

10 9

Wire gauge

Wire diameter, mm

5 32

3.82 4.11 4.49 4.77 4.89 5.25

8 7 3 16

6 5

American Wire Gauge (AWG).

TABLE 16-4 Dimensions (in mm) for chamfer on the shaft for mounting the seals d1 h11

d3

d1 h11

d3

d1 h11

d3

d1 h11

d3

d1 h11

d3

6 7 8 9 10 11 12 14 15 16 17 18 20 22

4.8 5.7 6.6 7.5 8.4 9.3 10.2 12.1 13.1 14.0 14.9 15.1 17.7 19.6

24 25 26 28 30 32 35 36 38 40 42 45 48 50

21.5 22.5 23.4 25.3 27.3 29.2 32.0 33.0 34.9 36.8 38.7 41.6 44.5 46.4

52 55 56 58 60 62 63 65 68 70 72 75 78 80

48.3 51.3 52.3 54.2 56.1 58.1 59.1 61.0 63.9 65.8 67.7 70.7 73.6 75.5

85 90 95 100 105 110 115 120 125 130 135 140 146 150

80.4 85.3 90.1 95.0 99.9 104.7 109.6 114.5 119.4 124.3 129.2 133.0 138.0 143.0

160 170 180 190 200 210 220 230 240 250 260 280 300 320

153 163 173 183 193 203 213 223 233 243 252 269 289 309

d1 h11

d3

340 360 380 400 420 440 460 480 500

329 349 369 389 409 429 449 469 489

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PACKINGS AND SEALS

16.12

CHAPTER SIXTEEN

TABLE 16-5 Selection of guide for packing materials

Condition

Leather (natural and synthetic)

Oil Air Water Steam Solvents Acids Alkalis Temperature range Types of metal

Good Good Good Not recommended Not recommended Not recommended Not recommended 558C þ828Ca Ferrous and nonferrous

Metal ﬁnish, rms (max.) Clearances Extrusions or cold ﬂow Friction coeﬃcient Resistance to abrasion Maximum pressure, MPa (kpsi) Concentricity Side loads High shock loads a

Homogeneous

Fabricated

63 Medium Good Low Good 861.7 (125)

Good Good Good Good Good Good Good 558C þ2008Ca Chrome-plated steel and nonferrous alloys with hard, smooth surfaces 16 Very close Poor Medium and high Fair 343.4 (50)

Good Good Good Good Good Good Fair 408C þ2608Ca Chrome-plated steel and nonferrous alloys with hard, smooth surfaces 32 Close Fair Medium Fair 549.4 (80)

Medium Fair Good

Very close Poor Poor to fair

Close Fair Fair

Depending on speciﬁcation or combination of materials.

FIGURE 16-11 Molded packing. Typical U-ring packing.

FIGURE 16-12 Diaphragm seals-rolling diaphragm.

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PACKINGS AND SEALS PACKINGS AND SEALS

16.13

TABLE 16-6 Types of seals and their uses Type

Uses

Radial lip seals

For retaining lubricants in equipments having rotating, reciprocating oscillating shafts, to exclude foreign matter For containing highly viscous materials at low speeds For containing lubricants of lower viscosity at higher speeds in clean atmosphere For excluding contaminants such as dust and dirt For containing lubricant on one side and for excluding ﬂuid on the other For splash system of lubrication For ﬁxed shaft and rotating bore For directing oil ﬂow back into the area to be sealed

Single lip (Fig. 16-4) Single lip—spring-loaded Double lip with one lip spring-loaded Dual lip with both lips spring-loaded Split seal External seal Hydrodynamic seal Exclusion seals (Figs. 16-5 and 16-6) Wipers, scrapers, axial seals, bellows, and boots Clearance seals (Fig. 16-8) Labyrinths, bushing, and ring seals Ring seals—split ring seals Expanding split ring Contracting split ring Straight-cut seal ring (Fig. 16-3a) Step seal ring Circumferential seal

To prevent entry of foreign materials into moving parts of machinery—to avoid contamination of lubricants Dynamic seals-to prevent leakage from a high-pressure station at one end of bushing to a region of low-pressure station at the other end of bushing To seal reciprocating components Used in compressors, pumps, and internal-combustion engines Linear actuators where high-pressure, high-temperature radiation and fatigue are expected Piston seal for low-grade actuators Devices where free-passage leakage is not permissible For rotary applications with low leakage and high performance

Face seals (Fig. 16-9) Stationary, rotating, and metal bellows type

Running seal between two ﬂat precision ﬁnished surfaces, for high-speed applications, stuﬃng boxes, and temperature applications

Compression packing (Fig. 16-10)

For the throat of a stuﬃng box and its gland, dynamic seal

Molded packing (Fig. 16-11)

For automatic-hydraulic or mechanical packings

Lip type Single and multiple spring-loaded packings Squeeze type

Felt radial type

For sealing reciprocating parts Fitted in rectangular grooves machined in hydraulic or pneumatic mechanisms and used as a piston seal in hydraulic actuating cylinder, valve seat, or valve stem packing Used at high speeds from 10 to 20 m/s

Diaphragm seals (Fig. 16-12)

To prevent interchange of a ﬂuid or contaminant between two separated areas, dynamic sealing and force transmitter

Nonmetallic gaskets (Fig. 16-13)

Static sealing

Metallic gaskets (Table 16-7) Corrugated, metal-jacketed, plain or machined (ﬂat metal) round, heavy, or light cross-section (solid metal)

Static sealing, for high pressures and severe conditions, cast iron ﬂanges, ammonia ﬁttings, hydraulic cylinders, gas mains, heat exchangers, boiler openings, vacuum and cryogenic lines, and valve bonnets

Sealants Hardening (rigid or ﬂexible), non-hardening and tapes

To exclude dust, dirt, moisture, and chemicals or contain a liquid or gas-surface coatings to protect against mechanical or chemical attack, to exclude noise, to improve appearance and to perform a joining function, thermal insulating, vibration damping

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PACKINGS AND SEALS

16.14

CHAPTER SIXTEEN

TABLE 16-7 Properties and uses of nonmetallic gasket materials Classiﬁcation

Special characteristics

General uses

Rubber asbestos

Tough and durable, relatively incompressible, good steam and hot water resistance

Heavy duty bolted and threaded joints as in water and steam pipe ﬁttings; temperatures up to 2608C

Cork and rubber

Provides ﬂuid barrier and resilience with compressibility; does not extrude from joint; die cuts well; high coeﬃcient of friction

General-purpose gasketing; enables design of metal-to-metal joints; high friction keeps gasket positioned even where closing pressure is not perpendicular to ﬂange faces

Cork composition

General purpose material compressible; high friction, low cost; excellent oil and solvent resistance; poor resistance to alkalis and corrosive acids

Mating rough or irregular parts; oil sealing at low cost in normal range of temperatures and pressures

Rubber, plastics

Highly adjustable according to compounding, hardness, modulus, fabric reinforcement, etc.; generally impervious, not compressible

Installations involving stretching over projections or where ﬂow of gasket into threads or recesses is desired; for lowest compression set and maximum resistance to ﬂuids such as alkalis, hot water, and certain acids

Low cost, noncorrosive General-purpose material; good oil, gasoline and water resistance

Spacers, dust barriers, splash seals where breathing and wicking not objectionable Machined or reasonably uniform ﬂanges where adequate bolt pressures can be applied

Innumerable modiﬁcations available, depending on materials used and methods of combining

Usually employed for extreme conditions and special purposes

Paper Untreated Treated Combination constructions

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PACKINGS AND SEALS PACKINGS AND SEALS

16.15

TABLE 16-8 Minimum metallic gasket seating stress Minimum seating stressa Type

f

f f f

f

Material

Thickness, mm

MPa

kpsi

Aluminum

3 1.5 and 0.75 3 1.5 and 0.75 3 1.5 and 0.75 3 1.5 and 0.75 3 1.5 and 0.75

109.8 137.3 248.1 309.9 379.0 474.1 448.2 559.9 577.3 646.2

16.0 20.0 36.0 45.0 55.0 69.0 65.0 81.0 84.0 94.0

3b 1.5 b 0.75 b 3b 1.5 b 0.75 b 3b 1.5 b 0.75 b 3b 1.5 b 0.75 b 3b 1.5 b 0.75 b

172.1 206.9 241.2 241.2 275.6 309.9 379.0 413.8 448.2 448.2 482.5 557.6 517.3 557.6 655.1

25.0 30.0 35.0 35.0 40.0 45.0 55.0 60.0 65.0 65.0 70.0 80.0 75.0 80.0 95.0

Copper Soft steel (iron) Monel Stainless steel Aluminum

Copper

Soft steel (iron)

Monel

Stainless steel

Aluminum Copper Soft steel (iron) Monel Stainless steel

3 3 3 3 3

10.3 13.7 27.4 30.9 41.2

1.5 2.0 4.0 4.5 6.0

Aluminum Copper Soft steel (iron) Monel Soft steel

3 3 3 3 3

13.7 17.2 20.6 24.0 27.4

2.0 2.5 3.0 3.5 4.0

Lead Aluminum Copper Soft steel (iron) Monel Stainless steel Inconel Hastelloy c

3 3 3 3 3 3 3 3

3.4 6.9 17.1 24.0 30.9 41.2 51.5 68.6

0.5 1.0 2.5 3.5 4.5 6.0 7.5 10.0

a

Seating stress values shown do not apply to ASME Code. Also they are based on optimum surface ﬁnish and clean ﬂange surface, i.e., no grease, oil or gasket compound. Figures indicated are pitch, and the values of stress apply for all thicknesses.

b

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PACKINGS AND SEALS

16.16

CHAPTER SIXTEEN

TABLE 16-9 Compression packing for various service conditions Service condition Fluid medium

Reciprocating shafts

Rotating shafts

Piston or cylinders

Valve stems

Acids and caustics

Asbestos, metallic, plastic (pliable), semimetallic, TFE ﬂuorocarbon resins and yarns

Asbestos, plastic (pliable), semimetallic, TFE ﬂuorocarbon resins and yarns

TFE ﬂuorocarbon resins

Asbestos, plastic (pliable), semimetallic TFE ﬂuorocarbon resins and yarns

Air, gas

Asbestos, metallic, semimetallic

Asbestos, semimetallic

Leather, metallic

Asbestos, semimetallic

Ammonia, low-pressure Duck and rubber, steam metallic, semimetallic

Asbestos, semimetallic

Duck and rubber

Asbestos, duck and rubber, semimetallic

Cold and hot gasoline and oils

Asbestos, plastic (pliable), semimetallic

Asbestos, plastic (pliable), semimetallic

High-pressure steam

Asbestos, metallic, plastic (pliable), semimetallic

Asbestos, metallic, plastic (pliable), semimetallic

Cold and hot water

Duck and rubber, Asbestos, plastic leather, plastic (pliable), (pliable), semimetallic semimetallic

Asbestos, plastic (pliable), semimetallic Metallic

Asbestos, metallic, plastic (pliable), semimetallic

Duck and rubber

Asbestos, duck and rubber, plastic (pliable), semimetallic

FIGURE 16-13 Common types of gasketed joints.

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PACKINGS AND SEALS

TABLE 16-10 Dimensions of oil seals

Nominala b 0:2, mm Shaft bore diameter Types A diameter d1 , of housing, mm mm and B Type C

cb min, mm

Nominala b 0:2, mm Shaft bore diameter diameter d1 , of housing, Types A mm mm and B Type C

cb min, mm

6

16 22

7

0.3

18

7

0.3

16 22

7

0.3

30 32 35 40

7 8

16 22 24

7

0.3

20

7

0.3

9

22 24 26

7

0.3

30 32 35 40 47

22 7

0.3

— 9

0.3

19 22 24 26

32 35 40 47

7

10

11

22 26

7

0.3

24

7

— 9

0.3

12

22 24 28 30

7

0.3

35 37 40 47

25

7

— 9

0.4

24 28 30 35

7

35 40 42 47 52

26

7

7

— 9

0.4

24 26 30 32 35

37 42 47

28

40 47 52

7

— 9

0.4

16

28 30 32 35

7

0.3

30

7

— 9

0.4

17

28 30 32 35 40

7

0.3

40 42 47 52 62

32

45 47 52

7

—

0.4

14

15

0.3

0.3

9

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PACKINGS AND SEALS

TABLE 16-10 Dimensions for oil seals (Cont.) Nominala b 0:2, mm Shaft bore diameter Types A diameter d1 , of housing, mm mm and B Type C

cb min, mm

35

0.4

47 50 52 62

7

47 50 52 62

7

38

52 55 62

40

36

— 9

Nominala b 0:2, mm Shaft bore diameter diameter d1 , of housing, Types A mm mm and B Type C

cb min, mm

63

85 90

10

12

0.5

65

85 90 100

10

12

0.5

68

90 100

10

12

0.5

70

90 100

10

12

0.5

72

95 100

10

12

0.5

75

95 100

10

12

0.5

78

100 100

10

12

0.5

80

100 110

10

12

0.5

85

110 120

12

15

0.8

90

110 120

12

15

0.8

95

120 125

12

15

0.8

100

120 125 130

12

15

0.8

105

130 140

12

15

0.8

110

130 140

12

15

0.8

115

140 150

12

15

0.8

120

150 160

12

15

0.8

— 9

0.4

7

— 9

0.4

52 55 62 72

8

— 9

0.4

42

55 62 72

8

— 10

0.4

45

60 62 65 72

8

— 10

0.4

48

62 72

8

— 10

0.4

50

65 68 72 80

8

— 10

0.4

68 72

8

70 72 80 85

8

70 72 80 85

8

58

72 80

8

— 10

0.4

125

150 160

12

15

0.8

60

80 85 90

8

— 10

0.4

130

160 170

12

15

0.8

62

85 90

10

12

0.5

135 140 145

170 170 175

11 15 15

15 15 15

0.8 1 1

52 55

56

— 10

0.4

— 10

0.4

— 10

0.4

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PACKINGS AND SEALS

TABLE 16-10 Dimensions for oil seals (Cont.) Nominala b 0:2, mm Shaft bore diameter Types A diameter d1 , of housing, mm mm and B Type C 150 160 170 180 190 200 210 220 230 240 250 260

180 190 200 210 220 230 240 250 260 270 280 300

15 15 15 15 15 15 15 15 15 15 15 20

15 15 15 15 15 15 15 15 15 15 15 20

cb min, mm

Nominala b 0:2, mm Shaft bore diameter diameter d1 , of housing, Types A mm mm and B Type C

cb min, mm

1 1 1 1 1 1 1 1 1 1 1 1

280 300 320 340 360 380 400 420 440 460 480 500

1 1 1 1 1 1 1 1 1 1 1 1

320 340 360 380 400 420 440 460 480 500 520 540

20 20 20 20 20 20 20 20 20 20 20 20

20 20 20 20 20 20 20 20 20 20 20 20

a

For limits of housing, see Tables 16-11 and 16-12. The edges may be chamfered or rounded according to the manufacturer’s discretion. Source: Bureau of Indian Standards: 5129, 1969. b

TABLE 16-11 Press-ﬁt allowances and tolerancesa for type A seals Possible press-ﬁt variation, mm Nominal bore diameter of housing, mm

Housing bore, mm

Outside diameter of seal, mm

High limit

Low limit

High limit

Low limit

Maximum interference

Minimum interference

25 25–55 55–125 125–200 200

þ0.03 þ0.03 þ0.03 þ0.04 þ0.05

0.03 0.03 0.03 0.04 0.05

þ0.20 þ0.25 þ0.30 þ0.38 þ0.48

þ0.10 þ0.15 þ0.20 0.22 0.32

0.23 0.28 0.33 0.42 0.53

0.07 0.12 0.17 0.18 0.27

a

All tolerances are relative to nominal bore diameter of housing. Source: IS 5129, 1969.

TABLE 16-12 Press-ﬁt allowances and tolerances’ for types B and C seals Possible press-ﬁt variation, mm Nominal bore diameter of housing, mm

Housing bore, mm

Outside diameter of seal, mm

High limit

Low limit

High limit

Low limit

Maximum interference

Minimum interference

50 50–90 90–115 115–170 170–215 215–230 230

Nominal Nominal þ0.03 þ0.03 þ0.04 þ0.04 þ0.04

0.03 0.03 0.03 0.03 0.04 0.04 0.04

þ0.12 þ0.14 þ0.18 þ0.20 þ0.23 þ0.25 þ0’30

þ0.04 þ0.06 þ0.08 þ0.10 þ0.13 þ0.15 þ0.20

0.15 0.17 0.21 0.23 0.27 0.29 0.34

0.04 0.06 0.05 0.07 0.09 0.11 0.16

a

All tolerances are relative to nominal bore diameter of housing. Source: IS 5129, 1969.

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PACKINGS AND SEALS

16.20

CHAPTER SIXTEEN

TABLE 16-13 Depth of the housing bore (all dimensions in mm) b

t (0.85b) Min

t2 (b to 0.3) Min

7 8 9 10 12 15 20

5.95 6.80 7.65 8.50 10.30 12.75 17.00

7.3 8.3 9.3 10.3 12.3 15.3 20.3

Source: Indian Standards 5129, 1969.

TABLE 16-14 Types of hollow, metallic O-rings8,9

FIGURE 16-14 Fully conﬁned hollow-metal O-ring: (a) before bolting and (b) after bolting down.

Source: Wes J. Ratelle, ‘‘Seal Selection, Beyond Standard Practice,’’, Machine Design, Jan. 20, 1977 and, ‘‘Packings and Seals’’ Issue, Machine Design, Jan. 1977.

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PACKINGS AND SEALS PACKINGS AND SEALS

16.21

TABLE 16-15 Recommended groove dimensions for metallic Q-ring sealing inside pressure

Nominal O-ring OD Nominal tubing OD mm 0.8

Actual O-ring dimensions

Min B, mm

Incremental increase, I, mm

Tubing OD, O-Ring B, OD, mm mm

6.30

0.8 up to 25

0.74–0.96

1.6

11.0

1.6 thereafter 0.14–0.16

2.4

19

1.6

0.20–0.24

3.2

44

1.6

0.30–0.31

4.0

75

1.6

0.37–0.40

4.8

100

1.6

0.44–0.48

6.3

125

1.6

0.59–0.81

9.5

250

No limit

0.90–0.95

12.7

250

No limit

1.20–1.25

B þ 0:075 0.000 B þ 0:075 0.0006 B þ 0:100 0.000 B þ 0:125 0.000 B þ 0:150 0.000 B þ 0:175 0.000 B þ 0:200 0.000 B þ 0:300 0.000 B þ 0:400 0.000

Open-groove dimensions Maximum ID of closed groove, Y, mm

Maximum, radius of groove corner, R, mm

B 2:160

B 3:000

0.125

B 3:600

B 4:825

0.255

B 5:665

B 6:860

0.510

B 7:495

B 8:635

0.760

B 9:245

B 10:410

0.760

B 11:170

B 12:190

0.760

B 14:730

B 16:000

0.760

B 22:600

B 23:110

0.760

B 30:480

B 30:480

0.760

Depth, C, mm

Groove OD, Minimum X, groove ID, mm Y, mm

0.510– 0.500 1.066– 1.145 1.0650– 1.750 2.290– 2.415 2.920– 3.050 3.685– 3.810 4.955– 5.080 7.495– 7.620 9.910– 10.160

B þ 0:0105 to 0.01525 B þ 0:0105 to 0.01525 B þ 0:0127 to 0.0225 B þ 0:0178 to 0.0305 B þ 0:0203 to 0.0355 B þ 0:0228 to 0.0380 B þ 0:0280 to 0.0480 B þ 0:0355 to 0.0735 B þ 0:0510 to 0.0965

Source: Wes J. Ratelle, ‘‘Seal Selection. Beyond Standard Practice,’’ Machine Design, Jan. 20, 1977, and ‘‘Packings and Seals’’ Issue, Machine Design, Jan. 20, 1977.

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PACKINGS AND SEALS

16.22

CHAPTER SIXTEEN

TABLE 16-16 Rectangular groove dimensions for O-ring gaskets

O-ring nominal cross section, mm

Actual O-ring cross section, mm

Maximum groove depth, V, mm

1.6 1.6 1.6 1.6 1.6 2.4 3.2 4.8 6.4

0.100 0.0075 0.125 0.0075 0.150 0.0075 0.175 0.0075 0.175 0.0075 0.260 0.0075 0.350 0.0100 0.530 0.0125 0.700 0.0150

For Flange Gaskets (Axial) 0.070–0.0050 0.160 0.0050 0.090–0.0050 0.185 0.0075 0.110–0.0050 0.210 0.0075 0.130–0.0100 0.240 0.0075 0.125–0.0100 0.240 0.0075 0.205–0.0105 0.270 0.0125 0.280–0.0200 0.470 0.0125 0.445–0.0250 0.725 0.0125 0.585–0.0250 0.960 0.0125

1.6 1.6 1.6 1.6 1.6 2.4 3.2 4.8 6.4

0.100 0.0075 0.125 0.0075 0.150 0.0075 0.175 0.0075 0.175 0.0075 0.260 0.0075 0.350 0.0100 0.530 0.0125 0.700 0.0150

For Nonﬂange Gaskets (Radial) 0.075–0.0025 0.140 0.0050 0.095–0.0025 0.160 0.0075 0.115–0.0025 0.190 0.0075 0.135–0.0025 0.230 0.0075 0.130–0.0050 0.230 0.0075 0.210–0.0075 0.315 0.0125 0.290–0.0100 0.430 0.0125 0.455–0.0125 0.600 0.0125 0.595–0.0150 0.800 0.0125

TABLE 16-17 Triangular groove dimensions for O-ring ﬂange gaskets

O-ring nominal cross section, mm

Actual O-ring cross section, mm

Width, h, mm

1.6 2.4 3.2 4.8 6.4

0:175 0:0075 0:260 0:0075 0:350 0:0100 0:530 0:0125 0:700 0:0150

0:240 þ 0:0075 0:000 0:345 þ 0:0125 0:000 0:470 þ 0:0175 0:000 0:710 þ 0:0255 0:000 0:950 þ 0:0375 0:000

Groove width, b, mm

Minimum diametral squeeze mm

Bottom radius, R1 , mm

0.025 0.030 0.035 0.040 0.045 0.050 0.060 0.075 0.100

0.0125 0.0200 0.0300 0.0380 0.0380 0.0500 0.0750 0.1250 0.1500

0.0175 0.025 0.030 0.035 0.038 0.043 0.050 0.065 0.090

0.0125 0.0200 0.0300 0.0380 0.0380 0.0500 0.0750 0.1250 0.1500

TABLE 16-18 Packing sizes recommended for various shaft diameters Shaft diameter, mm

Packing size, mm

12.70–15.85 17.45–38.10 39.70–50.80 52.40–63.50 65.10–76.20 77.80–101.60

7.95 9.50 11.10 12.70 14.30 15.85

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PACKINGS AND SEALS PACKINGS AND SEALS

16.23

TABLE 16-19 Temperature limits for gasket materials Maximum sustained temperature Material

K

8C

Asbestos ﬁber and rubber Cellulose-ﬁber and rubber Cork and rubber Synthetic rubber Cork composition

673 423 393 393 393

400 150 120 120 120

TABLE 16-21 Selection of shaft piston seals Type name External-ﬁtted to piston, sealing in bore Internal-ﬁtted in housing, sealing on piston or rod Simple housing design Low wear rate High stability Low friction Resistance to extrusion Availability in small sizes Availability in large sizes Bidirectional sealing

Distributor

U-Ring

Cup

O-Ring

Good Very good Good Fair Good Fair Good

Good Good Fair Fair Good Good Fair Single acting only

Poor Good Very good Fair Good Poor Good

Very good Poor Poor Good Fair Very good Good Eﬀective but usually used in pairs

FIGURE 16-15 Nomenclature of gasketed joint. (J. E. Shigley and C. R. Mischke, Standard Handbook of Machine Design, McGraw-Hill, 1986.)

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0.34–2.94 50–425 27.46–48.05 4000–7000

0.29–0.53

17.16–20.59 2500–3000

0.49–4.90

Acrylic Polyester

Polyurethane-bitumen modiﬁed Butyls—mastic type Butyls—curing type

Polybutene Olcoresin

a

8.33–24.02

Epoxy–modiﬁed

5–20

5–150 650–800

250–400

100–270 3–15

10–20

50–750 250–350 75–125 325 3.0–6.0

150–500

0.17–0.27

0.34–0.69 0.15–0.44 0.34–0.59 0.44–0.69

Elongation %, ASTM D412 MPa

24.5–40

50–100 15–65 50–85.5 65–100

psi

Adhesion in tension, ASA 1161-1960

1.03–1.873

150–270

10.79–23.54 1500–3400

1500–2750

40–100 125–175 1500–3500

0.27–0.69 0.85–1.20 10.29–24.0

10.29–19.1

80–175 240–350 150–200

psi

0.55–1.20 1.72–2.40 1.03–1.37

MPa

Shear strength, ASTM D1002

Compounds built speciﬁcally for plotting and molding, where high strength and abrasion resistance are required.

70–710

42–75

1200–3500

56.5–125 1000–3000 285–780 1000–1500 500–600 1210 4000–13000

0.39–0.86 6.86–20.50 1.96–5.39 6.86–10.29 3.43–4.11 8.33 27.46–89.73

Polysulﬁde Polyurethanea Silicone Ncoprene Hypalon Viton Epoxy

psi

MPa

Tensile strength, ASTM D412

Sealant base test method

TABLE 16-20 Properties of sealants

0.5–5.0 0.25–1.5

0.75–1.50

1.0–5.0 0.25–0.75

0.27–0.50

0.25–1.5 1–3.0 0–1–0.25 0.5–1.5 1.0–5.0 0–3.0 0.04–0.10

Good Good

Good Good

Good

Fair to good Good

Good to excellent

Fair to good Excellent Fair to good Excellent Excellent Fair to good Good to excellent

Moisture resistance, % Abrasion ASTM D570 resistance

–30 to 120 –25 to 95

–20 to 95 –60 to 150

–35 to 95

–25 to 150 –55 to 150

–35 to 150

–50 to 120 –55 to 205 –75 to 370 –40 to 150 –40 to 150 –55 to 230 –35 to 150

Operating temperature 8C

20–40 5–70

5–70 15–75

0–3.0 l5–45

15.0–15.0

0–3.0

5.0–15.0 2.0–10.0

0–3.0 0–3.0

Shore ‘‘D’’ 40–60 Shore ‘‘D’’ 40–100 10–70 Shore ‘‘D’’ 10–45

0–3.0

0–3.0 0–10.0 0–10.0 0–10.0 10–20

Shrinkage 15–60 45–90 25–80 30–80 30–80 40–60 40–100

Shore A hardness ASTM 676

PACKINGS AND SEALS

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PACKINGS AND SEALS PACKINGS AND SEALS

TABLE 16-22 Recommended maximum temperature for materials (supplement to Table 16-2)

TABLE 16-22 Recommended maximum temperature for material (supplement to Table 16-2)

Temperature Material Coil spring material Phosphor-bronze ASTM B159 Silicon bronze ASTM B99 Ni-span C902 Music wire ASTM A228 Hard-drawn spring wire ASTM A227 Oil-tempered wire ASTM A229 Valve spring wire ASTM A230 Beryllium-copper ASTM B197 Chrome-vanadium alloy steel AISI 6150 Silicon-manganese alloy steel AISI 9260 Chrome-silicon alloy steel AISI 9254 Martensite AISI 410 Martensite AISI 420 Austenitic AISI 301 Austenitic AISI 302 17-7 PH Stainless Steel Inconel6OO Nickel-chrome alloy steel A286 Inconel 7l8 Inconel X-750 L-605 S-816 Rene 41 Flat spring material Ni-span C902 Phosphor-bronze ASTM B103 High-carbon AISI 1050 High-carbon AISI 1065 High-carbon AISI 1075 High-carbon AISI 1095 Beryllium-copper ASTM B194 Austenitic AISI 301 Austenitic AISI 302 17-7 PH Stainless Steel Inconel 600 Beryleo-nickel Titanium 6-6-2 Sandvik 11 R51 Duranickel 301 Permanickel Elgiloy Havar Inconel 7l8 Inconel X-750 Rene 41

8F

8C

200 200 200 250 250 300 300 400 425 450 475 500 500 600 600 590 700 950 1200 1300 1400 1400 1400

93 93 93 121 121 149 149 204 218 232 246 260 260 315 315 311 371 510 649 704 760 760 760

200 200 200 200 250 250 400 600 600 700 700 700 750 800 800 800 900 900 1200 1300 1400

93 93 93 93 121 121 204 315 315 371 371 371 399 427 427 427 482 482 649 704 760

16.25

Temperature Material Formed metal bellows materials Brass CDA 240 Phosphor-bronze CDA 510 Beryllium-copper CDA 172 Monel 404 Unstabilized 300 series stainless steel Inconel 600 Inconel X-750 Welded metal bellows materials Ni-span C AM-350 Stainless Steel 410 Stainless Steel Commercially pure titanium Stabilized 300 series Stainless Steel Inconel X-750 Inconel 625 Hastelloy-C Rene 41

8F

8C

300 300 350 450 500 750 800

149 149 177 232 260 399 427

500 800 800 800 1220 1500 1500 1800 1800

260 427 427 427 659 815 815 982 982

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PACKINGS AND SEALS

16.26

CHAPTER SIXTEEN

TABLE 16-23 pv values for seal face material (life of 8000 h) pv Value Unbalanced Product Water Oil Water Oil Water Oil Water Oil Water Water Oil

Combination face material Stainless steel Carbona Lead bronze Carbona Stellite carbona Tungsten carbide Carbonb Solid ceramic Sprayed ceramic

o o

o

Balanced

MPa, m/s

kpsi fpm

MPa, m/s

kpsi fpm

0.9 1.8 1.8 3.5 3.5 9 9 9 15 15 20

25.5 51.0 51.0 100 100

Seldom used Seldom used Seldom used Seldom used 10 70 25 150 Seldom used 90 150

Seldom used Seldom used Seldom used Seldom used 285 2000 710 4280 Seldom used 2570 4280

255 255 430 430 560

a

Metal-impregnated carbon. Retain impregnated carbon. Source: Courtesy M. J. Neale, Tribology Handbook, Butterworths, London, 1973. b

TABLE 16-24 Spring arrangements for various sizes of shaft and speeds Spring arrangement Stationary

Rotary

Shaft diameter, mm

Speed, rpm

Single

Multiple

Single

Multiple

100 >100 75 100 >100

3000 3000 4500 >4500 >4500

Yes No Yes Yes No

Yes Yes Yes Yes Yes

Yes No Yes No No

Yes Yes Yes No No

Source: Courtesy of M. J. Neale, Tribology Handbook, Butterworths, London, 1973.

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PACKINGS AND SEALS PACKINGS AND SEALS

16.27

TABLE 16-25 Types of static and dynamic seals Dynamic seals Clearance seals Static seals

Reciprocating

Fibrous gasket Metallic gasket Elastomeric gasket Plastic gasket Sealant, setting Sealant, nonsetting O-ring Inﬂatable gasket Pipe coupling Bellows

a

Labyrinth (Fig. 16-8) Fixed bushing Floating bushing

Contact seals

Rotary

Reciprocating

Rotary

Labyrinth (Fig. 16-8) Viscoseal Fixed bushing Floating bushing Centrifugal seal

U-ring (Fig. 16-11) O-ring (Table 16-15) Lobed O-ring Rectangular ring Packed gland Piston ring Bellows Diaphragm (Fig. 16-12)

Lip seal (Fig. 16-4) Face seal (Fig. 16-9a) Packed gland (Fig. 10-10) O-ringb (Fig. 16-14) Felt ring

a

Usually for steam or gas. Only for very slow speeds. Source: Courtesy M. J. Neale, Tribology Handbook, Butterworths, London, 1973. b

TABLE 16-26 Operating conditions of lip seals

Particular

TABLE 16-27 Types of seals for reciprocating shafts

Shaft diameter and housing

Remarks

Type of packing

75 mm diameter

60 kPa (8.7 psi)

>75 mm diameter 35 mm diameter

30 kPa (4.35 psi) 8000 rpm

75 mm diameter >75 mm diameter Housing Shaft

4000 rpm 15 m/s Fine-turned Grind and polish to better than 0.5 mm 0.25 mm total indicator reading Depends on speed, 0.25 mm Varies from 208C to 2008C (688F to 2668F)

Remarks

Cups and hats Maximum pressure of ﬂuid Maximum speed

Surface ﬁnish

Eccentricity

Housing Shaft

Temperature

Semiautomatic, leather and rubber/ fabric used U-packing Used for piston rod application up to 10 MPa (1.5 kpsi) (rubber) or 20 MPa (3.0 kpsi) (rubber/fabric) Nylon-supported Used up to 25 MPa (3.6 kpsi) Composite Used with rubber sealing lips, rubber/ fabric supporting portions and nylon wearing portions—used for pressure varying from 15 to 20 MPa (2.2 to 3.0 kpsi) Source: M. J. Neale, Tribology Handbook, Butterworths, London, 1973.

Source: M. J. Neale, Tribology Handbook, Butterworths, London, 1973

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PACKINGS AND SEALS

16.28

CHAPTER SIXTEEN

TABLE 16-28 Materials for lip seals (rubber) Resistance to

Temperature 8F

8C

Type of rubber

Trade names

Mineral oil

Chemical ﬂuids

Acrylate

Thiacril Cyanacryl Viton Fluorel Silastic Silastomer Hycar Polysar

Excellent

Fair

68 to þ266

20 to þ130

Excellent

Excellent

77 to þ392

25 to þ200

Fair

Poor

158 to þ392

70 to þ200

Excellent

Fair

104 to þ212

40 to þ100

Fluoropolymer Polysiloxane Nitrile

Source: Courtesy M. J. Neale, Tribology Handbook, Butterworths, London, 1973.

TABLE 16-29 Seal materials for reciprocating shafts Material

Remarks

Rubber (nitrile) Highest scaling eﬃciency; low cost; easily formed to shape; limited to a pressure of 10 MPa (1.5 kpsi); excellent wear resistance RubberGreat toughness; resistance to extrusion impregnated and cutting; wear resistance inferior to fabric rubber Leather Good wear and extrusion resistance; poor resistance to permanent set; limited shaping capability Nylon Resist extrusion; provide a good bearing surface Source: Courtesy M. J. Neale, Tribology Handbook, Butterworths, London, 1973.

TABLE 16-30 Extrusion clearance for reciprocating shafts—dimensions in mm (in) 10 MPa (1.5 kpsi)

10–20 MPa (1.5–3.0 kpsi)

>20 MPa (3.0 kpsi)

Material

Normal

Short life

Normal

Short life

Normal

Short life

Rubber Rubber/fabric leather Polyurethane Nylon support

0.25 (0.01) 0.40 (0.015) 0.40 (0.015) —

0.50 (0.02) 0.60 (0.025) 0.60 (0.025) —

— 0.25 (0.01) 0.25 (0.01) 0.25 (0.01)

— 0.50 (0.02) 0.50 (0.02) 1.00 (0.04)

— 0.10 (0.005) 0.10 (0.005) 0.10 (0.005)

— 0.25 0.01) 0.25 (0.01) 0.25 (0.01)

Source: Courtesy M. J. Neale, Tribology, Handbook, Butterworths, London, 1973.

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250 300 300

0.700

0.525

7.000 1.750 2.100 2.100

Graphited asbestos with latern ring and jacket cooling arrangement—rotary type

Graphited asbestos with PTFE antiextrusion ring hand surface replaceable sleeve, jacket cooling arrangement—rotary type

Graphited asbestos and PTFE yarn with PTFE antiextrusion ring, jacket cooling arrangement—rotary type Reciprocating, steam-graphited asbestos Reciprocating, water-greased cotton packing Reciprocating, oil-graphited hemp yam

Source: Courtesy M. J. Neale, Tribology Handbook, Butterworths, London, 1973.

1000

0.280

Graphited asbestos with latern ring cooling arrangement—rotary type

75

100

40

15

0.105

Graphited asbestos—rotary type

psi

MPa

Pressure

Type of gland

TABLE 16-31 Operation conditions of packed glands (Fig. 16-1)

500 500 200

545

290

320

240

200

8F

260 260 93

285

143

160

115

93

8C

Temperature

0.75 0.75 0.75 (150)

5.5 (1080)

306 (6100)

17.75 (4000)

17.75 (4000)

17.75 (4000)

Velocity, m/s (fpm)

Steam Water Oil

No latern or jacket ring cooling required Cooling liquid used below 34.5 kPa sealing pressure Latern ring cooling liquid and water to jacket cooler used below sealing pressure of 34.5 kPa Cooling as per type 3; special packing and accurate assembly is required Water to jacket coolant used

Remarks

PACKINGS AND SEALS

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16.29

PACKINGS AND SEALS

16.30

CHAPTER SIXTEEN

TABLE 16-32 Axial stress in packed glands

TABLE 16-33 Selection of number of sealing rings Minimum axial stress required for seal packing

Type of packing

MPa

psi

Teﬂon-impregnated braided asbestos Plastic Braided vegetable ﬁber, lubricated Plaited asbestos, lubricated Braided metallic

1.40

200

1.12 1.75 2.8 3.5

160 255 405 505

Pressure MPa

psi

Number of sets of sealing rings

1.0 1.0–2.0 2.0–5.0 3.5–17.0 7.0–15.0 >15.0

150 (150–250) 250–500 500–1000 1000–2000 above 2000

3 4 5 6 8 9–12

Source: Courtesy M. J. Neale, Tribology Handbook, Butterworths, London, 1973.

Source: Courtesy M. J. Neale, Tribology Handbook, Butterworths, London, 1973.

TABLE 16-34 Selection of packing materials

Material

Hardness of rod, HB

Axial clearance, mm

Lead bronze

250 min

0.08–0.12 (0.003–0.005 in)

Flake graphite gray cast iron White metal (Babbitt)

400 min

0.08–0.12

Filled PTFE

400 min

Reinforced pf resin Carbon-graphite Graphite/metal sinter

0.08–0.12

0.4–0.5 0.25–0.5

400 min

0.030–0.06

250 min

0.08–0.12

Application Optimum material with good lubricated bearing property High thermal conductivity; used where chemical condition exists and suited for pressure up to 300 MPa (50 kpsi) Cheaper suitable up to a pressure of 7 MPa (1.0 kpsi) Used where lead-bronze and ﬂake graphite gray cast iron are not suitable because of chemical condition; used up to a maximum pressure of 35 MPa (5.0 kpsi) and maximum temperature 1208C (2508F) Suitable for unlubricated; very good chemical resistance; suited above 2.5 MPa (400 psi) Used with sour hydrocarbon gases and where lubricant may be thinned by solvents in gas stream Used with carbon-graphite piston rings; must be kept oil free; used up to 3508C (6608F) Alternative to ﬁlled PTFE and carbon-graphite

Source: Courtesy M. J. Neale, Tribology Handbook, Butterworths, London, 1973.

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PACKINGS AND SEALS

TABLE 16-35 Minimum recommended seating stresses for various gasket materials (Supplement to Table 16-8)

Nonmetallic

Metallic

Jacketed metalasbestos

Material, mm (in)

Gasket type

Asbestos ﬁber sheet 3.125 (18 in) thick 1 in) thick 1.563 (16 1 in) thick 0.78 (32

Flat

Asbestos ﬁber sheet 1 in) thick 0.78 (32 Asbestos ﬁber sheet 1 0.78 (32 in) thick Asbestos ﬁber sheet 1 in) thick 0.78 (32 Cellulose ﬁber sheet Cork composition Cork-rubber Fluorocarbon (TFE) 3.125 (18 in) thick 1 in) thick 1.563 (16 1 0.78 (32 in) thick Nonasbestos ﬁber sheets (glass, carbon, aramid, and ceramics) Rubber Rubber with fabric or metal reinforcement Aluminum Copper

Flat with rubber beads

Carbon steel

Flat

Stainless steel

Flat 241–655

Aluminum (soft) Copper (soft) Carbon steel (soft) Stainless steel Aluminum Copper Carbon steel Stainless steel Aluminum Copper Carbon steel Stainless steel Aluminum Copper Carbon steel Stainless steel Stainless steel

Corrugated Corrugated Corrugated Corrugated Proﬁle Proﬁle Proﬁle Proﬁle Plain Plain Plain Plain Corrugated Corrugated Corrugated Corrugated Spiral-wound

Flat with metal grommet Flat with metal grommet and metal wire Flat Flat Flat Flat

Flat

Flat Flat with reinforcement Flat Flat

Minimum seating stress range (psia) MPa

(1400–1600) 9.7–11.0 (3500–3700) 24.1–25.5 (6000–6500) 41.4–44.8 (1000–1500 lb/in) on beads 175–263 kN/m (3000–4000 lb/in) on grommet 525.4–700.5 kN/m (2000–3000 lb/in) on wire 350.2–525.4 kN/m (750–1100) 5.2–7.6 (400–500) 2.8–3.5 (200–300) 1.4–2.1 (1500–1700) 10.3–11.7 (3500–3800) 24.1–26.2 (6200–6500) 42.8–44.8 (1500–3000) depending on composition 10.3–20.7 (100–200) 0.7–1.4 (300–500) 2.1–3.5 (10,000–20,000) 68.9–137.9 (15,000–45,000) 103.4–310.3 depending on hardness (30,000–70,000) 207–483 depending on alloy and hardness (35,000–95,000) 241–655 depending on alloy and hardness (1000–3700) 6.9–25.5 (2500–4500) 17.2–31.0 (3500–5500) 24.1–37.9 (6000–8000) 41.4–55.2 (25,000) 172.4 (35,000) 241.3 (55,000) 379.2 (75,000) 517.1 (2500) 17.2 (4000) 27.6 (6000) 41.4 (10,000) 68.9 (2000) 13.8 (2500) 17.2 (3000) 20.7 (4000) 27.6 (3000–30,000) 20.7–206.8

a

Stresses in pounds per square inch except where otherwise noted. Source: J. E. Shigley and C. R. Mischke, Standard Handbook of Machine Design, McGraw-Hill Book Company, New York, 1986.

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PACKINGS AND SEALS

16.32

CHAPTER SIXTEEN

TABLE 16-36 Safety factors for gasketed joints, n, for use in Eq. (16-39) Safety factor, n

When to apply

1.2 to 1.4

For minimum-weight applications where all installation factors (bolt lubrication, tension, parallel seating, etc.) are carefully controlled; ambient to 2508F (1218C) temperature applications; where adequate proof pressure is applied For most normal designs where weight is not a major factor, vibration is moderate and temperatures do not exceed 7508F (3998C); use high end of range where bolts are not lubricated For cases of extreme ﬂuctuations in pressure, temperature, or vibration; where no test pressure is applied; or where uniform bolt tension is diﬃcult to ensure

1.5 to 2.5 2.6 to 4.0

Source: J. E. Shigley and C. R. Mischke, Standard Handbook of Machine Design, McGraw-Hill Book Company, New York, 1986.

FIGURE 16-16 Packing assembly for a mechanical piston rod. (M. J. Neale, Tribology Handbook, Butterworths, London, 1973.)

FIGURE 16-17 Ratio of retained stress to origins versus shape factor for, various materials: A—asbestos sheet; B— cellulose; C—cork-rubber. (J. E. Shigley and Mischke, Standard Handbook of Machine Design, McGraw-Hill, 1986.)

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PACKINGS AND SEALS PACKINGS AND SEALS

FIGURE 16-18 Power absorption and starting torque for balanced and unbalanced seals. (M. J. Neale, Tribology Handbook, Butterworths, London, 1973.)

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16.33

PACKINGS AND SEALS

16.34

CHAPTER SIXTEEN

REFERENCES 1. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Units), Suma Publishers, Bangalore, India, 1986. 2. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 3. Maleev, V. L., and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954. 4. The American Society of Mechanical Engineers, ASME Boilers and Pressure Vessel Code, Section VIII, Division I, 1986. 5. Whalen, J. J., ‘‘How to Select the Right Gasket Material,’’ Product Engineering, Oct. 1860. 6. Shigley, J. E., and C. R. Mischke, Standard Handbook of Machine Design, McGraw-Hill Book Company, 1986. 7. Neale, M. J., Tribology Handbook, Butterworths, London, 1975. 8. Ratelle, W. J., ‘‘Seal Selection, Beyond Standard Practice,’’ Machine Design, Jan. 20, 1977. 9. ‘‘Packings and Seals’’ Issue, Machine Design, Jan. 1977. 10. Faires, V. M., Design of Machine Elements, Macmillan Book Company, 1955. 11. Bureau of Indian Standards. 12. Rothbart, H. A., Mechanical Design and Systems Handbook, McGraw-Hill Book Company, New York, 1985. 13. Lingaiah, K., Machine Design Data Handbook, McGraw-Hill Book Company, New York, 1994.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

17 KEYS, PINS, COTTERS, AND JOINTS SYMBOLS4;5;6 a A b d d1 d2 d3 d4 dc dpl dm (or dpm ) dnom D F F 0 , F 00 F20 , F200 Ft F h l L lo , so m Mb Mt

addendum for a ﬂat root involute spline proﬁle, m (in) area, m2 (in2 ) breadth of key, m (in) eﬀective length of knuckle pin, m (in) dedendum for a ﬂat root involute spline proﬁle, m (in) diameter, m (in) major diameter of internal spline, m (in) minor diameter of internal spline, m (in) major diameter of external spline, m (in) minor diameter of external spline, m (in) core diameter of threaded portion of the taper rod, m (in) large diameter of taper pin, m (in) mean diameter of taper pin, m (in) nominal diameter of thread portion, m (in) diameter of shaft, m (in) pitch diameter, m (in) force, kN (lbf) force on the cotter joint, kN (lbf) pressure between hub and key, kN (lbf) force applied in the center of plane of a feather keyed shaft which do not change the existing equilibrium but give a couple, kN (lbf) two opposite forces applied on the center plane of a double feather keyed shaft which give two couples, but tending to rotate the hub clockwise, kN (lbf) tangential force, kN (lbf) frictional force, kN (lbf) thickness of key, m (in) minimum height of contact in one tooth, m (in) length of key (also with suﬃxes), m (in) length of couple (also with suﬃxes), m (in) length of sleeve, m (in) length of spline, m (in) space width and tooth thickness of spline, m (in) module, mm, m (in) bending moment, N m (lbf in) twisting moment, N m (lbf in)

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KEYS, PINS, COTTERS, AND JOINTS

17.2

CHAPTER SEVENTEEN

pressure, MPa (psi) tangential pressure per unit length, MPa (psi) maximum pressure where the shaft enters the hub, MPa (psi) pressure at the end of key, MPa (psi) diametral pitch external load, kN (lbf) resistance on the key and on the shaft to be overcome when the hub is shifted lengthwise, kN (lbf) thickness of cotter, m (in) proﬁle displacement, m (in) number of teeth, number of splines stress tensile or compressive (also with suﬃxes), MPa (psi) nominal bearing stress at dangerous point, MPa (psi) shear stress, MPa (psi) angle of cotter slope, deg angle of friction, deg coeﬃcient of friction (also with suﬃxes)

p p1 p2 pd (or P) Q R t xm z b1

SUFFIXES b c d m p s t

bearing compressive design mean pin small end tensile, tangential Particular

Formula

ROUND OR PIN KEYS

pﬃﬃﬃﬃ pﬃﬃﬃﬃ d ¼ 3:035 D to 3:45 D

The large diameter of the pin key

where d and D are in mm pﬃﬃﬃﬃ pﬃﬃﬃﬃ d ¼ 0:6 D to 0:7 D where d and D are in in pﬃﬃﬃﬃ pﬃﬃﬃﬃ d ¼ 0:096 D to 0:11 D

SI

ð17-1aÞ

USCS

ð17-1bÞ

SI

ð17-1cÞ

where d and D are in m

STRENGTH OF KEYS Rectangular ﬁtted key (Fig. 17-1, Table 17-1)

Pressure between key and keyseat

FIGURE 17-1

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Width b Height h

Key cross section

2 2

6 8

6 20

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Source: IS 2048, 1962.

6 36

Keyway radius r2 max

L min L max

0.16

r max r min

Chamfer or radius of key

Length of key

0.25 0.16

t2

þ0.05 0.00 þ0.05 0.00

1.8 1.4

3 3

8 10

Tolerance on keyway depth

t1

Keyway depth In shaft t1 1.2 (nominal) In hub t2 1

Above Up to

For shaft diameters

8 45

2.5 1.8

4 4

10 12 6 6

17 22

10 50

14 71

0.25

0.35 0.25

3.0 3.5 2.3 2.8

5 5

12 17

TABLE 17-1 Dimensions (in mm) of parallel keys and keyways

18 90

22 110

5 3.8

12 8

38 44

28 140

0.40

0.55 0.40

5 3.3

10 8

30 38

þ0.1 0.0 þ0.1 0.0

4.0 3.3

8 7

22 30

36 160

5.5 3.8

14 9

44 50

45 180

6 4.3

16 10

50 58

50 200

7 4.4

18 11

58 65

56 220

7.5 4.9

20 12

65 75

63 250

0.60

0.80 0.60

8.5 5.4

22 14

75 85

71 280

9.0 5.9

25 14

85 110

32 18

110 130

36 20

130 150

40 22

150 170

45 25

170 200

50 28

200 230

56 32

230 260

63 32

260 290

70 36

290 330

80 40

330 380

90 45

380 440

100 50

440 500

80 320

90 360

100 400

110 400

125 400

1.00

1.30 1.00

þ0.15 0.00 þ0.15 0.00

140 400

160 400

180 400

1.60

2.00 1.60

200 400

220 400

250 400

280 400

2.50

2.95 2.50

10 11 12 13 15 17 19 20 22 25 28 31 6.4 7.4 8.4 9.4 10.4 11.4 12.4 13.4 14.4 15.4 17.4 19.5

28 16

95 110

KEYS, PINS, COTTERS, AND JOINTS

17.3

KEYS, PINS, COTTERS, AND JOINTS

17.4

CHAPTER SEVENTEEN

Particular

Formula

Crushing strength The tangential pressure per unit length of the key at any intermediate distance L from the hub edge (Fig. 17-1, Table 17-2)

p ¼ p1 L tan

The torque transmitted by the key (Fig. 17-1)

Mt ¼ 12 p1 DL2 DL22 tan

The general expression for torque transmitted according to practical experience

where tan ¼

p1 p2 p1 ¼ L2 L0

Mt ¼ 14 b1 hDL2

2 1 18 b1 bL2

ð17-3Þ ð17-4Þ

where p2 ¼ 0, when L2 ¼ Lo ¼ 2:25D; tan ¼

For dimensions of tangential keys given here.

ð17-2Þ

p1 h ¼ b1 Lo 4:5D

Refer to Table 17-2.

Shearing strength The torque transmitted by the key (Fig. 17-1)

Mt ¼ 12 1 bDL2 19 1 bL22 where tan ¼

The shear stress at the dangerous point (Fig. 17-1)

1 ¼

ð17-5Þ

p1 b ¼ 1 Lo 2:25D

Mt L2 bð0:5D 0:11L2 Þ

ð17-6Þ

TAPER KEY (Fig. 17-2, Table 17-3) The relation between the circumferential force Ft and the pressure F between the shaft and the hub

F t ¼ 1 F

ð17-7Þ

The pressure or compressive stress between the shaft and the hub

F ¼ blp

ð17-8Þ

The torque

Mt ¼

1 2 1 blpD

ð17-9Þ

where 1 ¼ coeﬃcient of friction between the shaft and the hub ¼ 0:25

FIGURE 17-2

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KEYS, PINS, COTTERS, AND JOINTS KEYS, PINS, COTTERS, AND JOINTS

17.5

TABLE 17-2 Dimensions (in mm) of tangential keys and keyways

Keyway

Keyway

Shaft diameter, D

Height, h

Width, b

Radius, r

Key chamfer, a

100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 320 340 360 380 400 420 440

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 32 34 36 38 40 42 44

30 30 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 95 102 108 114 129 126 132

2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4

3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5

Shaft diameter, D 460 480 500 520 540 560 580 600 620 640 660 680 700 720 740 760 780 800 820 840 860 880 900 920 940 960 980 1000

Height, h 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100

Width, b

Radius, r

Key chamfer, a

138 144 150 156 162 168 174 180 186 192 198 204 210 216 222 228 234 240 246 252 258 264 270 276 282 288 294 300

4 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 8 8 8 8 8 8 8

5 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 9 9 9 9 9 9 9

Notes: (1) The dimensions of the keys are based on the formula: width 0.3 shaft diameter, and thickness ¼ 0.1 shaft diameter; (2) if it is not possible to ﬁx the keys at 1208, they may be ﬁxed at 1808; (3) it is recommended that for an intermediate diameter of shaft, the key section shall be the same as that for the next larger size of the shaft in this table. Source: IS 2291, 1963.

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KEYS, PINS, COTTERS, AND JOINTS

TABLE 17-3 Dimensions (in mm) of taper keys and keyways

Shaft

Key

Above

Up to and including

6 8 10 12 17 22 30 38 44 50 58 65 75 85 95 110 130 150 170 200 230 260 290 330 380 440

8 10 12 17 22 30 38 44 50 58 65 75 85 95 110 130 150 170 200 230 260 290 330 380 440 500

Width, b (h9) 2 3 4 5 6 8 10 12 14 16 18 20 22 25 28 32 36 40 45 50 56 63 70 80 90 100

Height, h 2 3 4 5 6 7 8 8 9 10 11 12 14 14 18 10 25 22 25 28 32 32 36 40 45 50

Keyway in shaft and hub Chamfer or radius r1 , min 0.16 —

0.25 —

0.40 —

0.60 —

1.00 —

1.60 — 2.50

Keyway width, b (D10) 2 3 4 5 6 8 10 12 14 16 18 20 22 25 28 32 36 40 45 50 56 63 70 80 90 100

Depth in shaft, t1

Tolerance on t1

Depth in hub, t2

1.2 1.8 2.5 3.0 3.5 4.0 5.0 5.0 5.5 6.0 7.0 7.5 8.5 9.0 10.0 11.0 12.0 13.0 15.0 17.0 19.0 20.0 22.0 25.0 28.0 31.0

þ0.05 —

0.5 0.9 1.2 1.7 2.1 2.5 2.5 2.5 2.9 3.4 3.3 3.8 4.8 4.3 5.3 6.2 7.2 8.2 9.2 10.1 12.1 11.1 13.1 14.1 16,1 18.1

þ0.10

—

þ0.15

Tolerance on t2

Radius, r2 , max 0.16

þ0.1

0.25 —

— 0.40 —

þ0.2

0.60 —

—

1.00 —

þ0.3

1.60 —

Source: IS 2292, 1963.

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2.50

KEYS, PINS, COTTERS, AND JOINTS KEYS, PINS, COTTERS, AND JOINTS

Particular

17.7

Formula

The necessary length of the key

l¼

The axial force necessary to drive the key home (Fig. 17-2)

Fa ¼ F þ F ¼ 22 F þ F tan

The axial force is also given by the equation

Fa ¼ 0:21pbl

ð17-12Þ

Mt a

ð17-13Þ

2Mt 1 bpD

ð17-10Þ ð17-11Þ

where 2 ¼ 0:10, tan ¼ 0:0104 if the taper is 1 in 100

FRICTION OF FEATHER KEYS (Fig. 17-3) The circumferential force (Fig. 17-3) The resistance to be overcome when a hub connected to a shaft by a feather, Fig. 17-3a and subjected to torque Mt , is moved along the shaft

Ft ¼

R ¼ Ft þ 2 F 0

ð17-14Þ

¼ ð þ 2 ÞFt

ð17-15Þ

0

00

and F ¼ F ¼ Ft ¼ force assumed to be acting at the shaft axis without changing the equilibrium Fig. 17-3a The equation for resistance R, if and 2 are equal

R ¼ 2Ft

ð17-16Þ

The equation for torque if two feather keys are used, Fig. 17-3b

Mt ¼ 2F2 a

ð17-17Þ

The force F2 applied at key when two feather keys are used, Fig. 17-3b

F2 ¼

The resistance to be overcome when the hub connected to the shaft by two feather keys Fig. 17-3b and subjected to torque Mt is moved along the shaft

R2 ¼ 2F2 ¼

For Gib-headed and Woodruﬀ keys and keyways

Refer to Tables 17-4 and 17-5.

Mt Ft þ 2a 2

ð17-18Þ R 2

FIGURE 17-3 Feather key.

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ð17-19Þ

17.8

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12 17 22 30 38 44 50 58 65 75 85 95 110 130 150 170 200 230 260 290 330 380 440 500

Above

10 12 17 22 30 38 44 50 58 65 75 85 95 110 130 150 170 200 230 260 290 330 380 440

Source: IS 2293, 1963.

Up to and including

Shaft diameter, d

4 5 6 8 10 12 14 16 18 20 22 25 28 32 36 40 45 50 56 63 70 80 90 100

Width, b (h9) 4 5 6 7 8 8 9 10 11 12 14 14 16 18 20 22 25 28 32 32 36 40 45 50

Height (nominal) h

þ0.3

—

þ0.2

—

þ0.1

Tolerance on h

Key

TABLE 17-4 Gib-head keys and keyways (all dimensions in mm)

7 8 10 11 12 12 14 16 18 20 22 22 25 28 32 36 40 45 50 56 63 70 75 80

Height of gib-head, h1

2.50

1.60 —

1.00 —

0.60 —

—

0.40

—

0.16 — 0.25

Chamber or radius, r1 (min) 4 5 6 8 10 12 14 16 18 20 22 25 28 32 36 40 45 50 56 63 70 80 90 100

Width of keyway (D10) 2.5 3 3.5 4 5 5 5.5 6 7 7.5 8.5 9 10 11 12 13 15 17 19 20 22 25 28 31

Depth in shaft, t1

þ0.15

—

þ0.1

Tolerance on t1

1.2 1.7 2.1 2.5 2.5 2.5 2.9 3.4 3.5 3.8 4.8 4.3 5.3 6.2 7.2 8.2 9.2 10.1 12.1 11.1 13.1 14.1 16.1 18.1

Depth in hub, t2

Key in shaft and hub

þ0.3

þ0.15

þ0.1

Tolerance on t2

2.50

—

1.60

—

1.00

—

0.60

—

0.4

—

0.25

0.16

Radius at bottom of r2ðmaxÞ keyway

KEYS, PINS, COTTERS, AND JOINTS

Group I

Group II

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1.4 2.6 2.6 3.7 3.7 3.7 5 6.5 5 6.5 7.5 6.5 7.5 9 7.5 9 (10) 11 9 11 13 11 13 16

3 4 6 6 8 8 8 — 10 10 — 12 12 — 17 17 17 — 22 22 — 30 30 —

4 6 8 8 10 10 10 — 12 12 — 17 17 — 22 22 22 — 30 30 — 38 38 —

6 8 10 10 12 12 12 16 17 17 17 22 22 22 30 30 30 30 38 38 38 38 38 38

8 10 12 12 17 17 17 17 22 22 22 30 30 30 38 38 38 38 — — — — — —

4.0 7.0 7.0 10.0 10.0 10.0 13.0 16.0 13.0 16.0 19.0 16.0 19.0 22.0 19.0 22.0 25.0 28.0 22.0 28.0 32.0 28.0 32.0 45.0

Keyslot in shaft

Keyslot in hub

0.2

0.2 0.1

0.1

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 þ 0.2

þ0.1

3.82 6.76 6.76 9.66 9.66 9.66 12.65 15.72 12.65 15.72 18.57 15.72 18.57 21.63 18.57 21.63 24.49 27.35 21.63 27.35 31.43 27.35 31.43 43.08

1.0 2.0 1.8 2.9 2.9 2.5 3.8 5.3 3.5 5.0 6.0 4.5 5.5 7.0 5.1 6.6 7.6 8.6 6.2 8.2 10.2 7.8 9.8 12.8

1.0 2.0 1.8 2.9 2.9 2.8 4.1 5.6 4.1 5.6 6.6 5.4 6.4 7.9 6.0 7.5 8.5 9.5 7.5 9.5 11.5 9.1 11.1 14.1

þ0.2

þ0.1

þ0.2

þ0.1

0.6 0.8 1.0 1.0 1.0 1.4 1.4 1.4 1.7 1.7 1.8 2.2 2.2 2.2 2.6 2.6 2.6 2.6 3.0 3.0 3.0 3.4 3.4 3.4

0.6 0.8 1.0 1.0 1.0 1.1 1.1 1.1 1.1 1.1 1.1 1.3 1.3 1.3 1.7 1.7 1.7 1.7 1.7 1.7 1.7 2.1 2.1 2.1

þ0.1

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4

0.2

0. 1

Chamfer Depth, t Depth t1 Radius, r1 Tolerance or Tolerance on d1 radius, r on r Length L Series A Series B Tolerance Series A Series B Tolerance Nominal Tolerance

Key

Notes: (1) The dimensions d t and d þ t1 may be speciﬁed on workshop drawings; (2) the key size 6 10 is nonpreferred; (3) the key size 2:5 3:7 shall be used in automobile industries only. Source: IS 2294, 1963.

1 1.5 2 2 2.5 3 3 3 4 4 4 5 5 5 6 6 6 6 8 8 8 10 10 10

Diameter of b h Up to and Up to and tolerance (h9) (h12) Over including Over including d1

Key section

Range of shaft dia, d

TABLE 17-5 Woodruﬀ keys and keyways (all dimensions in mm)

KEYS, PINS, COTTERS, AND JOINTS

17.9

KEYS, PINS, COTTERS, AND JOINTS

17.10

CHAPTER SEVENTEEN

Particular

Formula

SPLINES Parallel-sided or straight-sided spline The torque which an integral multispline shaft can transmit (Tables 17-6 to 17-12)

Mt ¼ 12 phliðD hÞ

ð17-20Þ

TABLE 17-6 Proportions of SAE standard parallel side splines Bearing pressure, p Types of spline ﬁttings

Symbols

Proportions

Fit

MPa

kpsi

w h h

w ¼ 0:241D 4A, h ¼ 0:075D 4B, h ¼ 0:125D

A B

20.6 13.7

3.00 2.00

w h h h

w ¼ 0:250D 6A, h ¼ 0:050D 6B, h ¼ 0:075D 6C, h ¼ 0:100D

A B C

20.6 13.7 6.9

3.00 2.00 1.00

w h h h

w ¼ 0:156D 10A, h ¼ 0:045D 10B, h ¼ 0:070D 10C, h ¼ 0:095D

A B C

20.6 13.7 6.9

3.00 2.00 1.00

w h h h

w ¼ 0:098D 16A, h ¼ 0:045D 16B, h ¼ 0:070D 16C, h ¼ 0:095D

A B C

20.6 13.7 6.9

3.00 2.00 1.00

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KEYS, PINS, COTTERS, AND JOINTS KEYS, PINS, COTTERS, AND JOINTS

17.11

TABLE 17-7 Proportions of involute spline proﬁle (American Standard)

Proportions 6 P ¼ 32 through 48 96

Spline characteristics

Symbols

P ¼ 12 through 12 24

Pitch diameter

D

D ¼ zm ¼

Circular pitch

p

p ¼ ð=PÞ

p ¼ ð=PÞ

Tooth thickness

t

m ¼ t¼ 2 2P

t ¼ ðm=2Þ ¼ ð=2PÞ

Diametral pitch

P

P ¼ ð=pÞ

P ¼ ð=pÞ

Addendum

a

a ¼ 0:5m ¼

Dedendum (internal)

b1

b1 ¼ 0:90m ¼

Dedendum

b

b ¼ 0:5m ¼

Dedendum (external)

b1

b1 ¼ 0:9m ¼ 0:900=P

b1 ¼ 1:0m ¼ 1:000=P

Major diameter (internal)

Doi

Doi ¼ ðz þ 1:8Þm ¼ ðz þ 1:8Þ=P

Doi ¼ ðz þ 1:8Þm ¼ ðz þ 1:8Þ=P

Minor diameter (external)

Dme

Dme ¼ ðz 1:8Þm ¼ ðz 1:8Þ=P

Dme ¼ ðz 2:0Þm ¼ ðz 2:0Þ=P

z P

D ¼ zm ¼ z=P

0:500 P 0:900 P

0:500 P

a ¼ 0:5m ¼ 0:500=P b1 ¼ 0:9m ¼ 0:900=P b ¼ 0:5m ¼ 0:500=P

Source: Courtesy H. L. Horton, ed., Machinery’s Handbook, 15th ed., The Industrial Press, New York, 1957.

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KEYS, PINS, COTTERS, AND JOINTS

TABLE 17-8 Straight sided splines (all dimensions in mm)

Nominal size id D

No. of splines, i

Minor diameter, d

Major diameter, D

Width, B

6 23 26 6 26 30 6 28 32 8 32 36 8 36 40 8 42 46 8 46 50 8 52 58 8 56 62 8 62 68 10 72 78 10 82 88 10 92 98 10 102 108 10 112 120

6 6 6 8 8 8 8 8 8 8 10 10 10 10 10

23 26 28 32 36 42 46 52 56 62 72 82 92 102 112

26 30 32 36 40 46 50 58 62 68 78 88 98 108 120

Light-Duty Series 6 22.1 1.25 3.54 6 24.6 1.84 3.85 7 26.7 1.77 4.03 6 30.4 1.89 2.71 7 34.5 1.78 3.46 8 40.4 1.68 5.03 9 44.6 1.61 5.75 10 49.7 2.72 4.89 10 53.6 2.76 6.38 12 59.8 2.48 7.31 12 69.6 2.54 5.45 12 79.3 2.67 8.62 14 89.4 2.36 10.08 16 99.9 2.23 11.49 18 108.8 3.23 10.72

6 11 14 6 13 16 6 16 20 6 18 22 6 21 25 6 23 28 6 26 32 6 28 34 8 32 38 8 36 42 8 42 48 8 46 54 8 52 60 8 56 65 8 62 72 10 72 82 10 82 92 10 92 102 10 102 112 10 112 125

6 6 6 6 6 6 6 6 8 8 8 8 8 8 8 10 10 10 10 10

11 13 16 18 21 23 26 28 32 36 42 46 52 56 62 72 82 92 102 112

14 16 20 22 25 28 32 34 38 42 48 54 60 65 72 82 92 102 112 125

a These values are based on the generating process. Source: IS 2327, 1963.

d1 ,a min

e,a max

Medium-Duty Series 3 9.9 1.55 3.5 12.0 1.50 4 14.5 2.10 5 16.7 1.95 5 19.5 1.98 6 21.3 2.30 6 23.4 2.94 7 25.9 2.94 6 29.4 3.30 7 33.5 3.01 8 39.5 2.91 9 42.7 4.10 10 48.7 4.00 10 52.2 4.74 12 57.8 5.00 12 67.4 5.43 12 77.1 5.40 14 87.3 5.20 16 97.7 4.90 18 106.3 6.40 b

f

a

0.32 0.16 0.45 1.95 1.34 1.65 1.70 0.15 1.02 2.54 0.86 2.44 2.50 2.40 2.70 3.00 4.50 6.30 4.40

g, max

k, mix

r, max

0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.3 0.3 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.3 0.3 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

Centering on

g

g

g

g

Inside centering is not always possible with generating processes.

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Inside diametera

Inside diameter or ﬂanksb

Inside diametera

Inside diameter or ﬂanksb

KEYS, PINS, COTTERS, AND JOINTS KEYS, PINS, COTTERS, AND JOINTS

17.13

TABLE 17-9 Tolerances for straight-sided splines (all dimensions in mm)

Tolerance on Minor diameter of hub, d

Major diameter of hub, D

Soft or hardened

Soft or hardened

Soft or hardened

Shaft sliding or ﬁxed

D9

F10

H7

H11

Shaft sliding inside hub

h8

e8

f7

a11

Shaft ﬁxed in hub Shaft sliding inside hub Shaft ﬁxed in hub

p6 h8 u6

h6 e8 k6

j6 — —

a11 a11 a11

Assembly of splined hub and shaft Splined hub

Width of hub B

For centering on inner diameter or ﬂanks For centering on inner diameter

Splined shaft For centering on ﬂanks

Particular

Formula

Involute-sided spline AMERICAN STANDARD (Table 17-7) The addendum a and dedendum b for a ﬂat root, Table 17-7 The area resisting shear, Table 17-7

The minimum height of contact on one tooth

The corresponding area of contact of all z teeth

a¼b¼m¼ A ¼

ð17-21Þ

DL 2

h ¼ 0:8m ¼ A¼

1 P

ð17-22Þ 0:8 0:8D ¼ P z

0:8D zL ¼ 0:8DL z

DL z

The torque capacity of teeth in shear

Mt ¼

The torque capacity of the spline in bearing with b ¼ 2dc

Mtb ¼ 0:8D2 Ldc

D ¼ 0:7854D2 Ld 2 d

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ð17-23Þ ð17-24Þ

ð17-25Þ ð17-26Þ

KEYS, PINS, COTTERS, AND JOINTS

17.14

CHAPTER SEVENTEEN

TABLE 17-10 Straight-sided splines for machine tools (all dimensions in mm)

4 Splines

6 Splines

Nominal size, ia d D

Minor diameter, d

Major diameter, D

Width, B

4 11 15 4 13 17 4 16 20 4 18 22 4 21 25 4 24 28 4 28 32 4 32 38 4 36 42 4 42 48 4 46 52 4 52 60 4 58 65 4 62 70 4 68 78

11 13 16 18 21 24 28 32 36 42 46 52 58 62 68

15 17 20 22 25 28 32 38 42 48 52 60 65 70 78

3 4 6 6 8 8 10 10 12 12 14 14 16 16 16

a

i ¼ number of splines Source: IS 2610, 1964.

Nominal size, ia d D

Minor diameter, d

Major diameter, D

Width, B

6 21 25 6 23 28 6 26 32 6 28 34 6 32 38 6 36 42 6 42 48 6 46 52 6 52 60 6 58 65 6 62 70 6 68 78 6 72 82 6 78 90 6 82 95 6 88 100 6 92 105 6 98 110 6 105 120 6 115 130 6 130 145

21 23 26 28 32 36 42 46 52 58 62 68 72 78 82 88 92 98 105 115 130

25 28 32 34 38 42 48 52 60 65 70 78 82 90 95 100 105 110 120 130 145

5 6 6 7 8 8 10 12 14 14 16 16 16 16 16 16 20 20 20 20 24

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KEYS, PINS, COTTERS, AND JOINTS

TABLE 17-11 Undercuts, chamfers, and radii for straight-sided splinesa (all dimensions in mm) External splines

Designation, id D

Type A

Type B

Type M

Internal splines

B

d1 , min

g, max

f , min

h

r1 , max

m

n

r2

k, max

r3 , max

Projected tip width of hub

4 11 15 4 13 17 4 16 20 4 18 22 4 21 25 4 24 28 4 28 32 4 32 38 4 36 42 4 42 48 4 46 52 4 52 60 4 56 65 4 62 70 4 68 78

3 4 6 6 8 8 10 10 12 12 14 14 16 16 16

9.6 11.8 15.0 16.9 20.1 23.0 26.8 30.3 34.5 40.2 44.4 49.5 56.2 59.5 64.4

0.2 0.2 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

1.50 2.37 2.87 4.35 5.00 7.30 7.39 9.56 11.03 15.41 16.79 21.63 23.26 23.61 27.57

5.0 5.5 6.7 7.7 8.9 10.4 12.1 14.2 15.9 19.0 20.7 23.7 26.4 28.3 31.2

0.10 0.10 0.15 0.15 0.15 0.15 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25

2.82 3.76 5.64 5.64 7.52 7.52 9.40 9.40 11.28 11.28 13.16 13.16 15.04 15.04 15.04

1.70 1.70 1.70 1.70 1.70 1.70 1.63 2.55 2.55 2.55 2.55 3.40 2.98 3.40 4.25

0.3 0.3 0.3 0.3 0.6 0.6 0.6 0.6 0.6 1.0 1.0 1.0 1.0 1.0 1.0

0.2 0.2 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.15 0.15 0.25 0.25 0.25 0.25 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40

0.5 0.5 0.7 0.7 0.7 0.7 1.0 1.0 1.0 1.0 1.3 1.3 1.6 1.6 1.6

a

Four splines; see Fig. 17-4a. Source: IS 2610, 1964

TABLE 17-12 Undercuts, chamfers, and radii for straight-sided splinesa (all dimensions in mm) External splines

Designation, id D

Type A B

d1 , min

g, max

f , min

h

r1 , max

m

n

r2

k, max

r3 , max

Projected tip width of hub

6 21 25 6 23 28 6 26 32 6 28 34 6 32 38 6 36 42 6 42 48 6 46 52 6 52 60 6 58 65 6 62 70 6 68 78 6 72 82 6 78 90 6 82 95 6 88 100 6 92 105 6 98 110 6 105 120 6 115 130 6 130 145

5 6 6 7 8 8 10 12 14 14 16 16 16 16 16 16 20 20 20 20 24

19.50 21.30 23.40 25.90 29.90 33.70 39.94 44.16 49.50 55.74 59.50 64.40 68.30 73.00 79.60 82.90 87.10 93.40 98.80 108.4 123.9

0.3 0.3 0.4 0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.6

1.95 1.34 1.65 1.70 2.83 4.95 6.02 5.81 5.89 8.29 8.03 9.73 12.67 13.07 13.96 17.84 18.96 19.22 19.25 24.75 29.20

9.7 11.0 11.8 12.9 14.8 16.5 19.3 21.1 23.9 26.7 28.6 31.4 33.4 36.2 38.0 41.3 43.1 46.4 49.2 54.2 61.8

0.15 0.15 0.15 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.30 0.30 0.30 0.30 0.30

4.70 5.64 5.64 6.58 7.52 7.52 9.40 11.28 13.16 13.16 15.04 15.04 15.04 15.04 15.04 15.04 18.80 18.80 18.80 18.80 22.56

1.70 2.13 2.55 2.55 2.55 2.55 2.55 2.55 3.40 3.98 3.40 4.25 4.25 5.10 5.53 5.10 5.53 5.10 6.38 6.38 6.38

0.6 0.6 0.6 0.6 0.6 0.6 1.0 1.0 1.0 1.0 1.0 1.0 1.6 1.6 1.6 1.6 1.6 2.0 2.0 2.5 2.5

0.3 0.3 0.4 0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.6

0.2 0.2 0.3 0.3 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.5

0.7 0.7 1.0 1.0 1.0 1.0 1.0 1.3 1.3 1.6 1.6 1.6 2.0 2.0 2.0 2.0 2.0 2.0 2.4 2.4 2.4

a

Type B

Type M

Internal splines

Six splines see Fig. 17-4b.

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KEYS, PINS, COTTERS, AND JOINTS

17.16

CHAPTER SEVENTEEN

Particular

Formula

The theoretical torque capacity of straight-sided spline with sliding according to SAE

Mt ¼ 6:895 106 i

Dþd hL 4

SI

ð17-26aÞ

where ¼ number of splines ¼ diameter as shown in Table 17-7, m ¼ inside diameter of spline, m ¼ pitch diameter of spline, m ¼ length of spline contact, m ¼ minimum height of contact in one tooth of spline, m Mt in N m Dþd Mt ¼ 1000i hL USCS ð17-26bÞ 4 i D; d d D L h

where Mt in lb in; d, D, L, and h in in D3me ð1 D4i =D4me Þ 4D2 where

Equating the strength of the spline teeth in shear to the shear strength of shaft, the length of spline for a hollow shaft

L¼

ð17-26cÞ

Di ¼ internal diameter of a hollow shaft, m (in) Dme ¼ minor diameter (external), m (in) D3me 4D2

The length of spline for a solid shaft

L¼

The eﬀective length of spline for a hollow shaft used in practice according to the SAE

Le ¼

ð17-26dÞ

D3me ð1 D4i =D4me Þ D2

ð17-26eÞ

For solid shaft Di ¼ 0. For diametrical pitches used in involute splines (SAE and ANSI)

Refer to Table 17-13.

TABLE 17-13 Diametral pitchesa used in involute splines (SAE and ANSI) 2:5 5 a

3 6

4 8

5 10

6 12

8 16

10 20

12 24

16 32

20 40

24 48

32 64

40 80

48 96

Diametral pitches are designated as fractions; the numerator of these fractions is the diametral pitch, P.

INDIAN STANDARD (Figs. 17-4 and 17-5, Tables 17-14 and 17-15) The value of proﬁle displacement (Fig. 17-4)

xm ¼ 12 ðd1 mz 1:1mÞ The value xm varies from 0:05m to þ0:45m

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ð17-27Þ

KEYS, PINS, COTTERS, AND JOINTS KEYS, PINS, COTTERS, AND JOINTS

Particular

17.17

Formula

The number of teeth

z¼

The minor diameter of the internal spline (Fig. 17-4a)

d2 ¼ mz þ 2xm 0:9m ¼ d1 2m

ð17-29Þ

The major diameter of the external spline (Fig. 17-4a)

d3 ¼ mz þ 2xm þ 0:9m ¼ d1 0:2m

ð17-30Þ

The minor diameter of the external spline (Fig. 17-4a)

d4 mx þ 2xm 1:1m ¼ d1 2:2m

ð17-31Þ

1 ðd 2xm 1:1mÞ m 1

ð17-28Þ

FIGURE 17-4(a) Reference proﬁle of an involute-sided spline. (Source: IS 3665, 1966.)

FIGURE 17-4(b) Nomenclature of the involute spline proﬁle.

FIGURE 17-5 Measurement between pins and measurement over pins of an involute-sided spline. (Source: IS 3665, 1966.)

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17.18

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6 7 7 8 9 11 12 14 14 16 17 18 18 20 21 22 22 24 24 26 28 28 30 31 32 34 34 36 38 38 40

15 11 17 13 18 14 20 16 22 18 25 21 28 24 30 26 32 28 35 31 38 33 37 34 40 36 42 38 45 41 47 43 48 44 50 46 ð52 48Þ 55 51 ð58 54Þ 60 56 ð62 58Þ 65 61 ð68 64Þ 70 66 ð72 68Þ 75 71 ð78 74Þ 80 76 ð82 78Þ

12 14 14 16 18 22 24 28 28 32 34 36 36 40 42 44 44 48 48 52 56 56 60 62 64 68 68 72 76 76 80

do

10.392 12.124 12.124 13.856 15.588 19.053 20.785 24.299 24.249 27.713 29.445 31.177 31.177 34.641 36.373 38.105 38.105 41.569 41.569 45.033 48.497 48.497 51.962 53.694 55.426 58.890 58.890 62.354 65.818 65.818 69.283

db 14.6 16.6 17.6 19.6 21.6 24.6 27.6 29.6 31.6 34.6 36.6 37.6 39.6 41.6 44.6 46.6 47.6 49.6 51.6 54.6 57.6 59.6 61.6 64.6 67.6 69.6 71.6 74.6 77.6 79.6 81.6

d3 10.6 12.6 13.6 15.6 17.6 20.6 23.6 25.6 27.6 30.6 32.6 33.6 35.6 37.6 40.6 42.6 43.6 45.6 47.6 50.6 53.6 55.6 57.6 60.6 63.6 65.6 67.6 70.6 73.6 75.6 77.6

d4

Note: Values within parentheses are nonpreferred.

z

Nominal size d1 d 2 14.68 16.68 17.68 19.68 21.68 24.68 27.68 29.69 31.69 34.69 36.69 37.69 39.69 41.69 44.69 46.69 47.69 49.69 51.69 54.70 57.70 59.70 61.70 64.70 67.70 69.70 71.70 74.70 77.70 79.70 81.70

d5 , min

Dimensions (in mm) for involute splines of module 2

TABLE 17-14

10.92 12.92 13.92 15.92 17.92 20.92 23.92 25.91 27.91 30.91 32.91 33.91 35.91 37.91 40.91 42.91 43.91 45.91 47.91 50.90 53.90 55.90 57.90 60.90 63.90 65.90 67.90 70.90 73.90 75.90 77.90

d6 , max þ0.4 þ0.4 þ0.9 þ0.9 þ0.9 þ0.4 þ0.9 0.1 þ0.9 þ0.4 þ0.4 0.1 þ0.9 0.1 þ0.4 þ0.4 þ0.9 0.1 þ0.9 þ0.4 0.1 þ0.9 0.1 þ0.4 þ0.9 0.1 þ0.9 þ0.4 0.1 þ0.9 0.1

xm 3.603 3.603 4.181 4.181 4.181 3.603 4.181 3.326 4.681 3.603 3.603 3.026 4.181 3.026 3.603 3.603 4.181 3.026 4.181 3.603 3.026 4.181 3.026 3.600 4.181 3.026 4.181 3.603 3.026 4.181 3.026

l o ¼ so 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5

Pin diameter, d 7.629 9.324 10.379 12.736 14.460 17.478 20.738 22.484 24.738 27.711 29.571 30.566 32.739 34.589 37.604 39.720 40.740 42.621 44.740 47.724 50.624 52.740 54.650 57.648 60.740 62.663 64.740 67.729 70.672 72.740 74.676

Measurement between pins, Mi

Internal spline

2.42 2.19 1.61 1.66 1.64 1.96 1.68 2.41 1.69 1.88 1.86 2.15 1.70 2.08 1.84 1.84 1.70 2.00 1.71 1.82 1.95 1.71 1.93 1.80 1.71 1.90 1.71 1.79 1.88 1.72 1.87

Deviation factor, fi 5.5 5.0 6.0 6.0 5.5 4.5 5.0 4.0 4.5 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0

Pin diameter d 22.212 22.695 25.588 28.206 28.790 29.898 34.161 34.144 37.016 39.000 40.857 42.181 45.137 46.195 48.938 51.074 51.912 54.218 55.939 59.109 62.235 63.984 66.242 69.058 72.021 74.253 76.036 79.166 82.263 84.063 86.267

Measurement over pins, Ma

1.11 1.13 1.06 1.11 1.13 1.28 1.23 1.46 1.30 1.42 1.42 1.50 1.15 1.52 1.46 1.47 1.43 1.54 1.44 1.50 1.56 1.47 1.57 1.53 1.49 1.59 1.50 1.55 1.60 1.52 1.61

2 2 2 2 — — 3 3 3 3 4 3 4 4 4 4 5 4 5 5 5 6 5 6 6 6 7 7 7 7 7

Deviation factor, fa z0

External spline

9.121 9.214 9.714 9.807 — — 15.621 14.807 15.807 15.493 21.028 15.179 21.621 20.807 21.400 21.493 27.435 21.179 27.621 27.307 26.993 33.435 27.179 33.214 33.807 32.993 39.435 39.121 38.807 39.807 38.993

Tooth thickness deviation factor, 0.866

Tooth thickness over z0 teeth

KEYS, PINS, COTTERS, AND JOINTS

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6 7 8 10 10 11 12 13 14 14 15 16 17 18 18 19 20 22 22 23 24 26 26 27 28 30 30 31 32 34 34 35 36 38 38 40

20 15 22 17 25 20 28 23 30 25 32 27 35 30 37 32 38 33 40 35 42 37 45 40 47 42 48 43 50 45 ð52 47Þ 55 50 ð58 53Þ 60 55 ð62 57Þ 65 60 ð68 63Þ 70 65 ð72 67Þ 75 70 ð78 73Þ 80 75 ð82 77Þ 85 80 ð88 83Þ 90 85 ð92 87Þ 95 90 ð98 93Þ 100 95 105 100

15.0 17.5 20.0 25.0 25.0 27.5 30.0 32.5 35.0 35.0 37.5 40.0 42.5 45.0 43.0 47.5 50.0 55.0 55.0 57.5 60.0 65.0 65.0 67.5 70.0 75.0 75.0 77.5 80.0 85.0 85.0 87.5 90.0 95.0 95.0 100.0

do

d3

12.990 19.5 15.155 21.5 17.321 24.5 21.651 27.5 21.651 29.5 23.816 31.5 25.981 34.5 28.146 36.5 30.311 37.5 30.311 39.5 32.476 41.5 34.641 44.5 36.806 46.5 38.971 47.5 38.971 49.5 41.136 51.5 43.301 54.5 47.631 57.5 47.631 59.5 49.796 61.5 51.962 64.5 56.292 67.5 56.292 69.5 58.457 71.5 60.622 74.5 64.952 77.5 64.952 79.5 67.117 81.5 69.282 84.5 73.612 87.5 73.612 89.5 75.777 91.5 77.942 94.5 82.272 97.5 82.272 99.5 86.603 104.5

db

Note: Values within brackets are nonpreferred.

z

Nominal size d1 d 2 14.5 16.5 19.5 22.5 24.5 26.5 29.5 31.5 32.5 34.5 36.5 39.5 41.5 42.5 44.5 46.5 49.5 52.5 54.5 56.5 59.5 62.5 64.5 66.5 69.5 72.5 74.5 77.5 79.5 82.5 84.5 86.5 89.5 92.5 94.5 99.5

d4 19.58 21.58 24.58 27.58 29.58 31.59 34.59 36.59 37.59 39.59 41.59 44.59 46.59 47.59 49.59 51.59 54.59 57.60 59.60 61.60 64.60 67.60 69.60 71.60 74.60 77.60 79.60 81.60 84.60 87.60 89.60 91.60 94.60 97.60 99.60 104.60

d5 , min 14.92 16.92 19.92 22.92 24.92 26.91 29.91 31.91 32.91 34.91 36.91 39.91 41.91 42.91 44.91 46.91 49.91 52.90 54.90 56.10 56.90 59.90 64.90 66.90 69.90 72.90 74.90 77.90 79.90 82.90 84.90 86.90 89.90 92.90 94.60 99.90

d6 , max

TABLE 17-15 Dimensions (in mm) for involute spline of module 2.5

þ1.125 þ0.875 þ1.125 þ0.125 þ1.125 þ0.875 þ1.125 þ0.875 þ0.125 þ1.125 þ0.875 þ1.125 þ0.875 þ0.125 þ1.125 þ0.875 þ1.125 þ0.125 þ1.125 þ0.875 þ1.125 þ0.125 þ1.125 þ0.875 þ1.125 þ0.125 þ1.125 þ0.875 þ1.125 þ0.125 þ1.125 þ0.875 þ1.125 þ0.125 þ1.125 þ1.125

xm 5.226 4.937 5.226 4.071 5.226 4.937 5.226 4.937 4.071 5.226 4.937 5.226 4.937 4.071 5.226 4.937 5.226 4.071 5.226 4.937 5.226 4.071 5.226 4.937 5.226 4.071 5.226 4.937 5.226 4.071 5.226 4.937 5.226 4.071 5.226 5.226

l o ¼ so 4.6 4.5 4.5 4.55 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5

Pin diameter, d 10.552 12.105 15.552 19.116 20.552 22.265 25.552 27.308 28.316 30.552 32.340 35.552 37.365 38.387 40.552 42.384 45.552 48.424 50.552 52.413 55.552 58.448 60.552 62.434 65.552 62.464 70.552 72.449 75.552 78.476 80.552 82.461 85.552 88.485 90.552 95.552

Measurement between pins, Mi

Internal spline

1.71 1.85 1.72 2.30 1.72 1.81 1.72 1.80 2.26 1.72 1.79 1.73 1.78 2.07 1.73 1.78 1.73 1.99 1.73 1.77 1.73 1.94 1.73 1.77 1.73 1.90 1.73 1.76 1.73 1.88 1.73 1.76 1.73 1.86 1.73 1.73

Deviation factor, fi 9.0 7.0 7.0 5.0 6.5 6.0 6.0 5.5 5.0 6.0 5.5 5.5 5.5 5.0 5.5 5.5 5.5 5.0 5.5 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

Pin diameter d 33.258 30.558 34.113 33.006 38.151 38.835 42.093 42.764 43.096 47.204 47.881 51.035 52.974 53.156 56.100 58.052 61.157 63.198 66.206 66.846 69.924 73.229 74.954 76.920 79.981 83.253 85.004 86.978 90.026 93.273 95.045 97.024 100.063 103.288 105.079 110.094

Measurement over pins, Ma

1.03 1.08 1.13 1.37 1.19 1.23 1.25 1.30 1 43 1.28 1.33 1.33 1.36 1.47 1.36 1.38 1.38 1.51 1.40 1.45 1.44 1.53 1.46 1.48 1.47 1.55 1.48 1.50 1.49 1.57 1.50 1.52 1.51 1.58 1.52 1.53

2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 8

Deviation factor, fa z0

External spline

12.026 11.892 12.252 11.491 19.293 19.160 19.526 19.392 18.759 19.759 19.625 26.793 26.660 26.026 27.026 29.892 27.259 26.491 34.193 34.160 34.526 33.759 34.759 34.625 41.793 41.026 42.026 41.892 42.259 41.491 49.293 49.160 49.526 48.759 49.759 56.793

Tooth thickness deviation factor, 0.866

Tooth thickness over z0 teeth

KEYS, PINS, COTTERS, AND JOINTS

17.19

KEYS, PINS, COTTERS, AND JOINTS

17.20

CHAPTER SEVENTEEN

Particular

The value of tooth thickness and space width of spline

Formula

l o ¼ so ¼ m

þ 2xm tan 2

ð17-32Þ

PINS Taper pins The diameter at small end (Figs. 17-6 and 17-7, Tables 17-16 and 17-17)

dps ¼ dpl 0:0208l

ð17-33Þ

The mean diameter of pin

dm ¼ 0:20D to 0:25D

ð17-34Þ

FIGURE 17-6 Tapered pin.

FIGURE 17-7 Sleeve and tapered pin joint for hollow shafts.

Sleeve and taper pin joint (Fig. 17-7) AXIAL LOAD The axial stress induced in the hollow shaft (Fig. 17-7) due to tensile force F

¼ 4

F ðd22

d12 Þ

2ðd2 d1 Þdm

ð17-35Þ

The bearing stress in the pin due to bearing against the shaft an account of force F

c ¼

F 2ðd2 d1 Þdm

17-36Þ

The bearing stress in the pin due to bearing against the sleeve

c ¼

F 2ðd3 d2 Þdm

ð17-35Þ

The shear stress in pin

¼

2F dm2

ð17-38Þ

The shearing stress due to double shear at the end of hollow shaft

¼

F 2ðd2 d1 Þl2

ð17-39Þ

The shear stress due to double shear at the sleeve end

¼

F 2ðd3 d2 Þl1

ð17-40Þ

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1.60 1.59

1.60 1.54 0.20 1.50 0.30

Max Min

Max Min amax rnom c

dh6

2.00 1.94 0.25 2.00 0.35

2.00 1.99

2.01 2.00

2

1.50 2.44 0.30 2.50 0.40

2.50 2.49

2.51 2.50

2.5

3.00 2.94 0.40 3.00 0.50

3.00 2.99

3.01 3.00

3

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1.50 1.46 0.20

Max Min

Source: IS 549, 1974.

a

dh10

1.5

dnom

2.00 1.96 0.25

2 2.50 2.46 0.30

2.5 3.00 2.94 0.35

3

TABLE 17-17 Dimensions (in mm) for solid and split taper pins

Source: IS 2393, 1980.

dh11

1.61 1.60

Max Min

dm6

1.5

TABLE 17-16 Dimensions (in mm) for cylindrical pins

4.00 3.95 0.40

4

4.00 3.92 0.50 4.00 0.63

4.00 3.98

4.01 4.00

4

5.00 4.95 0.63

5

5.00 4.92 0.63 5.00 0.80

5.00 4.98

5.01 5.00

5

6.00 5.95 0.80

6

6.00 5.92 0.80 6.00

6.00 5.98

6 01 6.00

6

8.00 7.94 1.00

8

8.00 7.91 1.00 8.00 1.60

8.00 7.98

8.02 8.01

8

10.00 9.94 1.20

10

10.00 9.91 1.20 10.00 2.00

10.00 9.98

10.02 10.01

10

12.00 11.93 1.60

12

12.00 11.89 1.60 12.00 2.50

12.00 11.97

12.02 12.01

12

Nominal diameter, dnom , mm

16.00 15.63 2.00

16

16.00 15.89 2.00 16.00 3.00

16.00 15.97

16.02 16.01

16

20.00 19.92 2.50

20

20.00 19.87 2.50 20.00 3.50

20.00 19.97

20.02 20.01

20

25.00 24.92 3.00

25

25.00 24.87 3.00 25.00 4.00

25.00 24.97

25.02 25.01

25

32.00 31.90 4.00

32

32.00 31.84 4.00 32.00 5.00

32.00 31.96

32.02 32.01

32

40.00 39.90 5.00

40

40.00 39.84 5.00 40.00 6.30

40.00 39.96

40.02 40.01

40

50.00 49.90 6.30

50

50.00 49.84 6.30 50.00 8.00

50.00 49.96

50.02 50.01

50

KEYS, PINS, COTTERS, AND JOINTS

17.21

KEYS, PINS, COTTERS, AND JOINTS

17.22

CHAPTER SEVENTEEN

Particular

The axial stress in the sleeve

Formula

¼ 4

TORQUE The shear due to twisting moment applied

For the design of hollow shaft subjected to torsion

F ðd32 d22 Þ 2ðd3 d2 Þdm Mt

¼

2 d d 4 m 2 Refer to Chapter 14.

ð17-41Þ

ð17-42Þ

Taper joint and nut F t ¼ d2 4 c

ð17-43Þ

The bearing resistance oﬀered by the collar

F c ¼ 2 ðd d22 Þ 4 3

ð17-44Þ

The diameter of the taper d2

d2 > dnom

ð17-45Þ

The tensile stress in the threaded portion of the rod (Fig. 17-8) without taking into consideration stress concentration

FIGURE 17-8 Tapered joint and nut.

Provide a taper of 1 in 50 for the length (l l1 Þ

Knuckle joint The tensile stress in the rod (Fig. 17-9) The tensile stress in the net area of the eye

Stress in the eye due to tear of

t ¼

4F d 2

ð17-46Þ

t ¼

F ðd4 d2 Þb

ð17-47Þ

tn ¼

F bðd4 d2 Þ

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ð17-48Þ

KEYS, PINS, COTTERS, AND JOINTS KEYS, PINS, COTTERS, AND JOINTS

Particular

17.23

Formula

FIGURE 17-9 Knuckle joint for round rods.

Tensile stress in the net area of the fork ends

F 2aðd4 d2 Þ

ð17-49Þ

tr ¼

F 2aðd4 d2 Þ

ð17-50Þ

Compressive stress in the eye due to bearing pressure of the pin

e ¼

F d2 b

ð17-51Þ

Compressive stress in the fork due to the bearing pressure of the pin

c ¼

F 2d2 a

ð17-52Þ

Stress in the fork ends due to tear of

Shear stress in the knuckle pin

i ¼

¼

2F d22

The maximum bending moment, Fig. 17-9 (panel b)

Mb ¼

The maximum bending stress in the pin, based on the assumption that the pin is supported and loaded as shown in Fig. 17-9b and that the maximum bending moment Mb occurs at the center of the pin

b ¼

The maximum bending moment on the pin based on the assumption that the pin supported and loaded as shown in Fig. 17-10b, which occurs at the center of the pin

Mb ¼

The maximum bending stress in the pin based on the assumption that the pin is supported and loaded shown in Fig. 17-10b

b ¼

ð17-53Þ

Fb 8

ð17-54Þ

4Fb d23

ð17-55Þ

F 2

b a þ 4 3

ðapprox:Þ

4ð3b þ 4aÞF 3d23

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ð17-56Þ

ð17-57Þ

KEYS, PINS, COTTERS, AND JOINTS

17.24

CHAPTER SEVENTEEN

Particular

Formula

COTTER The initial force set up by the wedge action

F ¼ 1:25Q

ð17-58Þ

The force at the point of contact between cotter and the member perpendicular to the force F

H ¼ F tanð þ Þ

ð17-59Þ

The thickness of cotter

t ¼ 0:4D

ð17-60Þ

The width of the cotter

b ¼ 4t ¼ 1:6D

ð17-61Þ

Cotter joint The axial stress in the rods (Fig. 17-10) Axial stress across the slot of the rod

¼ ¼

4F d 2 d12

ð17-62Þ 4F 4d1 t

ð17-63Þ

Tensile stress across the slot of the socket

¼

The strength of the cotter in shear

F ¼ 2bt

ð17-65Þ

Shear stress, due to the double shear, at the rod end

¼

F 2ad1

ð17-66Þ

¼

F 2cðd4 d1 Þ

ð17-67Þ

4F d12 Þ

ð17-68Þ

Shear stress induced at the socket end The bearing stress in collar

Crushing strength of the cotter or rod

4F ðd32 d12 Þ 4tðd3 d1 Þ

c ¼

ðd22

F ¼ d1 tc

FIGURE 17-10 Cotter joint for round rods.

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ð17-64Þ

ð17-69Þ

KEYS, PINS, COTTERS, AND JOINTS KEYS, PINS, COTTERS, AND JOINTS

Particular

Crushing stress induced in the socket or cotter

17.25

Formula

c ¼

F ðd4 d1 Þt

ð17-70Þ

F¼

ðd22 d12 Þ c 4

ð17-71Þ

Shear stress induced in the collar

¼

F d1 e

ð17-72Þ

Shear stress induced in the socket

¼

F d1 h

ð17-73Þ

The maximum bending stress induced in the cotter assuming that the bearing load on the collar in the rod end is uniformly distributed while the socket end is uniformly varying over the length as shown in Fig. 17-10b

b ¼

Gib and cotter joint (Fig. 17-11)

The width b of both the Gib and Cotter is the same as far as a cotter is used by itself for the same purpose (Fig. 17-11). The design procedure is the same as done in cotter joint Fig. 17-10.

FIGURE 17-11 Gib and cotter joint for round rods.

FIGURE 17-12 Coupler or turn buckle.

The equation for the crushing resistance of the collar

Fðd1 þ 2d4 Þ 4tb2

ð17-74Þ

Threaded joint COUPLER OR TURN BUCKLE Strength of the rods based on core diameter dc , (Fig. 17-12)

2 d 4 c t

ð17-75Þ

The resistance of screwed portion of the coupler at each end against shearing

F ¼ ad

ð17-76Þ

From practical considerations the length a is given by

a ¼ d to 1.25d for steel nuts

ð17-77aÞ ð17-77bÞ

The strength of the outside diameter of the coupler at the nut portion

a ¼ 1:5d to 2d for cast iron F ¼ ðd12 d 2 Þt 4

F¼

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ð17-78Þ

KEYS, PINS, COTTERS, AND JOINTS

17.26

CHAPTER SEVENTEEN

Particular

Formula

2 ðd d22 Þt 4 3

The outside diameter of the turn buckle or coupler at the middle is given by the equation

F¼

The total length of the coupler

l ¼ 6d

ð17-79Þ ð17-80Þ

REFERENCES 1. Maleev, V. L., and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954. 2. Shigley, J. E., and L. D. Mitchell, Mechanical Engineering Design, McGraw-Hill Book Company, New York, 1983. 3. Faires, V. M., Design of Machine Elements, The Macmillan Company, New York, 1965. 4. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962. 5. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 6. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 7. Juvinall, R. C., Fundamentals of Machine Component Design, John Wiley and Sons, New York, 1983. 8. Deutschman, A. D., W. J. Michels, and C. E. Wilson, Machine Design—Theory and Practice, Macmillan Publishing Company, New York, 1975. 9. Bureau of Indian Standards. 10. SAE Handbook, 1981.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

18 THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION SYMBOLS5;6;7 Ab Abr Ac Ag Ar A d d2 d1 dc dm ¼ d2 D D1 D2 Db Di Do e Eb , Eg F Fa Ff Fi Ft h

area of cross section of bolt, m2 (in2 ) area of base of preloaded bracket, m2 (in2 ) core area of thread, m2 (in2 ) loaded area of gasket, m2 (in2 ) stress area, m2 (in2 ) shear area, m2 (in2 ) nominal diameter of screw m (in) major diameter of external thread (bolt), m (in) pitch diameter of external thread (bolt), m (in) minor diameter of external thread (bolt), m (in) mean diameter of thrust collar, m (in) mean diameter of square threaded power screw, m (in) diameter of shaft, m (in) major diameter of internal thread (nut), m (in) minor diameter of internal thread (nut), m (in) pitch diameter of internal thread (nut), m (in) diameter of bolt circle, m (in) inside diameter of a pressure vessel or cylinder, m (in) mean diameter of inside screw of diﬀerential or compound screw, m (in) mean diameter of outside screw of diﬀerential or compound screw, m (in) eccentricity, m (in) moduli of elasticity of bolt and gasket, respectively, GPa (Mpsi) permissible load on bolt, kN (lbf ) tightening load on the nut, kN (lbf ) applied or external load, kN (lbf ) ﬁnal load on the bolt, kN (lbf ) initial load due to tightening, kN (lbf ) preload in each bolt, kN (lbf ) tangential force, kN (lbf ) thickness of a pressure vessel, m (in) thickness of a cylinder, m (in)

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.2

h2 ho

CHAPTER EIGHTEEN

thickness of the ﬂange of the cylindrical pressure vessel, m (in) depth of tapped hole (Fig. 18-1), m (in)

FIGURE 18-1 Flanged bolted joint.

i I K K l

lc lg L Mb Mt n p pc P t t1 W o , i c i , o a b 0b

number of threads in a nut number of bolts moment of inertia of bracket base, area (Fig. 18-6), m4 or cm4 (in4 ) constant (Eq. (18-4a)) stress concentration factor lever arms (with suﬃxes), m (in) distance from the inside edge of the cylinder to the center line of bolt, m (in) lead, m (in) required length of engagement of screw or nut (also with suﬃxes), m (in) gasket thickness, m (in) length of bolt nut to head (Fig. 18-2), m (in) bending moment, N m (lbf in) twisting moment, N m (lbf in) factor of safety pressure, MPa (psi) circular pitch of bolts or studs on the bolt circle of a cylinder cover, m (in) pitch of thread, m (in) thread thickness at major diameter, m (in) thread thickness at minor diameter, m (in) axial load, kN (lbf ) helix angle, deg respective helix angles of outside and inside screws of diﬀerential or compound screws, deg friction angle, deg half apex angle, deg coeﬃcient of friction between nut and screw coeﬃcient of collar friction respective coeﬃcient of friction in case of diﬀerential or compound screw eﬃciency stress (normal), MPa (psi) allowable stress, MPa (psi) bending stress, MPa (psi) bending stress due to eccentric load [Eq. (18-61)] allowable bearing pressure between threads of nut and screw, MPa (psi)

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

c d w a w

18.3

compressive stress, MPa (psi) design stress, MPa (psi) working stress, MPa (psi) applicable shear stress, MPa (psi) allowable shear stress, MPa (psi) permissible working shear stress, MPa (psi)

SUFFIXES vertical horizontal

v h

Particular

Formula

SCREWS The empirical formula for the proper size of a set screw

d¼

The maximum safe holding force of a set screw

F ¼ 54;254d 2:31

D þ 8 mm where D in mm 8

ð18-1Þ SI

ð18-2aÞ

USCS

ð18-2bÞ

where F in kN and d in m F ¼ 2500d 2:31 where F in lbf and d in in Applied torque

Mt ¼ 0:2Fa nominal diameter of bolt)

ð18-3Þ

Ff ¼ KFa þ Fi 2

ð18-4Þ

Gasket joint (Fig. 18-2) Final load on the bolt

3

E b Ab 7 6 L 7 6 7 where K ¼ 6 4Eb Ab Eg Ag 5 þ L lg

Refer also to Table 18-1 for values of K

FIGURE 18-2 Gasket joint.

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ð18-4aÞ

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.4

CHAPTER EIGHTEEN

Particular

Formula

TABLE 18-1 Values of K for use in Eq. (18-4) Type of joint

K

Soft, elastic gasket with studs Soft gasket with through bolts Copper asbestos gasket Soft copper corrugated gasket Lead gasket with studs Narrow copper ring Metal-to-metal joint

1.00 0.90 0.60 0.40 0.10 0.01 0.00

According to Bart, the tightening load for a screw of a steamtight, metal-to-metal joint

F ¼ 2804:69d

SI

ð18-5aÞ

USCS

ð18-5bÞ

SI

ð18-6aÞ

USCS

ð18-6bÞ

SI

ð18-7aÞ

where F in kN and d in m F ¼ 1600d where F in lbf and d in in

Tightening load for screw of a gasket joint

F ¼ 1402:34d where F in kN and d in m F ¼ 8000d where F in lbf and d in in

Cordullo’s equation for the tightening load on the nuts

F ¼ w ð0:55d 2 6:45 103 dÞ

where F in kN, w in MPa, and d in m F ¼ w ð0:55d 2 0:036dÞ

USCS

ð18-7bÞ

where F in lbf, w in psi, and d in in

Bolted joints (Fig. 18-2) The ﬂange thickness of the cylinder or pressure vessel

h2 ¼ 1:25d to 1:5d <j 1:1h to 1:25h

ð18-8Þ

The bolt diameter

d ¼ 0:67h to 0:8h

ð18-9Þ

Circular pitch of the bolts or studs on the cylinder cover to ensure water and steamtight joint

pc ¼ 7d for pressure from 0 to 0.33 MPa (0 to 48 psi) as per American Navy Standards

ð18-10Þ

pc ¼3:5d for pressure from 1.2 MPa (175 psi) to 1.37 MPa (200 psi)

ð18-11Þ

pc ¼ 3d

ð18-12Þ

for tight joint

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

Particular

The average stress for screw for sizes from 12.5 to 75 mm

18.5

Formula

av ¼

490:33 d

SI

ð18-13aÞ

USCS

ð18-13bÞ

where av in MPa and d in m av ¼

2;800;000 d

where av in psi and d in in Unwin’s formula for allowable stresses in bolts of ordinary steel to make a ﬂuidtight joint

d ¼ 17;537:4d 2 þ 11 for rough joint

SI

ð18-14aÞ

USCS

ð18-14bÞ

s ¼ 33;828:9d 2 þ 17:3 for faced joint SI

ð18-14cÞ

where d in MPa and d in m d ¼ 6030d 2 þ 1600 where d in psi and d in in where d in MPa and d in m d ¼ 3070d 2 þ 2500

USCS

ð18-14dÞ

where d in psi and d in in

TENSION BOLTED JOINT UNDER EXTERNAL LOAD Spring constant of clamped materials and bolt (Fig. 18-3A) The spring constant or stiﬀness of the threaded and unthreaded portion of a bolt is equivalent to the stiﬀness of two springs in series. The basic equations for deﬂection (), and spring constant (k) of a tension bar/bolt subject to tension load.

1 1 1 ¼ þ k k1 k 2

ð18-15aÞ

¼

Fl AE

ð18-15bÞ

k¼

F AE ¼ l

ð18-15cÞ

The eﬀective spring constant/total spring rate in case of long bolt consisting of the threaded and unthreaded portion having diﬀerent area of crosssections, the clamped two or more materials of two or more diﬀerent elasticities which act as spring with diﬀerent stiﬀness sections in series.

1 1 1 1 1 ¼ þ þ þ þ kelf k1 k2 k3 kn

ð18-15dÞ

Spring constant of the clamped material

km ¼

Am Em Deff Em ¼ l 4 l

ð18-15eÞ

Spring constant of the threaded fastener

1 l l lt l l ¼ t þ ¼ t þ unt kb At Eb Ab Eb At Eb Ab Eb

2

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ð18-15f Þ

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.6

CHAPTER EIGHTEEN

Particular

Formula

FIGURE 18-3A

FIGURE 18-3B Bolt joint diagram due to external load acting on the joint.

Approximate eﬀective area of clamped material

Am ¼

2 ðD d 2 Þ 4 eff where

Deff d lt lunt

¼ effective diameter, m ¼ round bolt of diameter equal to shank, m ¼ threaded length of bolt, m ¼ unthreaded portion of bolt length, m

PRELOADED BOLT (Fig. 18-3B) The external load

Fa ¼ Fab þ Fam

The bolted joint in Fig 18-3A subjected to external load Fa is such that the common deﬂection is given by

¼

The load shared by bolt The resultant/total bolt load

ð18-15gÞ

Fb Fm ¼ kb km

Fab ¼

ð18-15hÞ

kb F kb þ km a

Fb ¼ Fab þ Fi ¼ Fi þ kb ¼

ð18-15iÞ kb F þ Fi ð18-15jÞ kb þ km a

¼ CFa þ Fi The resultant load on the clamped material

Fm ¼ Fab Fi ¼ Fi km ¼

ð18-15kÞ km F Fi ð18-15lÞ kb þ km a

¼ ð1 CÞFa Fi where C is called the joint constant or stiﬀness parameter Fm ¼ portion of load Fa taken by member/material, kN Ft ¼ preload, kN

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.7

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

Particular

The joint constant or stiﬀness parameter

Formula

kb kb þ km

C¼

or

The preload to prevent joint separation occurs when Fm ¼ 0

Fi ¼ ð1 CÞFao

The external load required to separate joint

Fao ¼

The tensile stress in the bolt

Preload under static and fatigue loading as per the recommendation of R, B and W,a and Bowman

C¼

1 1þ

km kb

ð18-15mÞ

where Fao ¼ external load that cause separation of joint

b ¼

Fi 1C

Fb CFa Fi ¼ þ At At At

ð18-15nÞ ð18-15oÞ

where At ¼ tensile stress area, m2 or mm2 0:75Fp for reused bolt connections Ft ¼ 0:90Fp for permanent bolt connections ð18-15pÞ where Fp is proof load, N

The proof stress load that has to be used in Eq. (18-15p)

Fp ¼ At sp

ð18-15qÞ

where sp ¼ proof strength, taken from tables 18-5c and 18-5d sp 0:85sy The load factor

n¼

Fao Fa

The load factor guarding against joint separation

n¼

Fi Fa ð1 CÞ

or Fao ¼ nFa

ð18-15rÞ ð8-15sÞ

GASKET JOINTS For design of gasket bolted joint

Refer to Chapter 16 under Bolt loads in gasket joints.

PRELOADED BOLTS UNDER DYNAMIC LOADING The mean forces felt by the bolt The alternating forces felt by the bolt

a

Fmn ¼ Fal ¼

Fb þ Fi 2

Fb Fi 2

Russel, Bardsall and Ward Corp., Helpful Hints for Fastener Design and Application, Mentor, Ohio 1976, p. 42.

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ð18-15tÞ ð18-15uÞ

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.8

CHAPTER EIGHTEEN

Particular

The stress due to the preload Fi The fatigue safety factor by using modiﬁed Goodman criterion The alternating component of bolt stress

Formula

i ¼ Kfm

Fi Ai

ð18-15vÞ

nf ¼

se ðsut i Þ se ðm i Þ þ sut a

ð18-15wÞ

a ¼

Fb Fi kb Fa CFa ¼ ¼ 2At kb þ km 2At 2At

ð18-16aÞ

The mean stress

m ¼ a þ

The factor of safety according to the Goodman criterion

n¼

Fi CP Fi ¼ þ At 2At At

sa a

ð18-16cÞ

sa ¼ sm

Fi At

sm ¼ sut 1 sa se Solving of Eqs. (18-16c) and (18-16d) simultaneously sa ¼

sut 1þ

ð18-16bÞ

Fi At

sut se

sy sy ¼ max m þ a

The factor of safety on the basis of yield strength

n¼

For speciﬁcation of SAE, ASTM and ISO standard steel bolts

Refer to Tables 18-5c and 18-5d

The depth of tapped hole (Fig. 18-2)

ho ¼ 1:25d in steel castings

ð18-16dÞ ð18-16eÞ

ð18-16f Þ

ð18-16gÞ

ð18-16hÞ

ho ¼ 1:50d to 1:75d in cast iron

ð18-17Þ

ho ¼ 1:75d to 2d in aluminum

ð18-18Þ

The distance l from the inside edge of the cylinder to the center line of bolts (Fig. 18-2)

l ¼ 1:25d to 1:5d

ð18-19Þ

The diameter of bolt circle

Db ¼ D1 þ 2d

ð18-20Þ

The safe load on each bolt

F 0 ¼ A r d

ð18-21Þ

The number of bolts

i¼

D2b p 4F 0

ð18-22Þ

Another expression for the number of bolts

i¼

Db pc

ð18-23Þ

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

Particular

18.9

Formula

TABLE 18-2 Approximate bolt tension and torque values

TABLE 18-3 Load and working stress for metric coarse threads

Minimum bolt tension

Equivalent torque

Bolt size, mm

kN

lbf

kN m

lbf ft

12.7 15.9 19.6 22.2 25.4 21.6 31.8

51.2 76.9 113.9 139.7 189.1 225.4 286.9

11,500 17,300 25,600 31,400 42,500 50,600 64,500

1.353 2.442 4.835 6.374 9.620 13.013 18.289

1,000 1,800 3,570 4,700 7,090 9,600 13,500

Major diameter, d, mm

Stress Design stress, w area, Ar , MPa psi mm2

kN

16 20 24 30 36 42 48 56

0.016 0.025 0.035 0.056 0.082 0.112 0.147 0.203

2.97 667 5.59 1,260 9.59 2,160 18.04 4,060 30.89 6,940 48.35 10,870 71.10 16,000 111.80 25,130

18.9 22.8 27.2 32.2 37.1 43.1 48.3 55.2

2,740 3,300 3,950 4,670 5,380 6,250 7,000 8,000

Permissible load lbf

Stress in tensile bolt Seaton and Routhwaite formula for working stress for bolt made of steel containing 0.08 to 0.25% carbon and with diameter of 20 mm and over Applied load

w ¼ CðAr Þ0:418

ð18-24Þ

Refer to Table 18-2 for bolt tension and torque values and Table 18-3 for w . Fa ¼ CðAr Þ1:418

ð18-25Þ

where C ¼ 7:8 108 (5000) for carbon steel bolts of u ¼ 414 MPa (60 kpsi) ¼ 23:3 108 (15,000) for alloy–steel bolts ¼ 0:33 108 (1000) for bronze bolts The values of C inside parentheses are for US Customary System units, and values without parentheses are for SI units.

Rotsher’s pressure-cone method for stiﬀness calculation of Fastenera The elongation of frustum of a cone (Fig. 18-3C)

¼

Fa ð2t tan þ D dÞðD þ dÞ ln ð19-25aÞ Ed tan ð2t tan þ D þ dÞðD dÞ

The spring stiﬀness of the frustum

k¼

Fa ¼

Ed tan ð2t tan þ D dÞðD þ dÞ ln ð2t tan þ D þ dÞðD dÞ

a

ð18-25bÞ

Courtesy: Shigley J. E. and C. R. Mischke, Mechanical Engineering Design, 5th Edn., McGraw-Hill Publishing Company, New York, 1989.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.10

CHAPTER EIGHTEEN

FIGURE 18-3C Compression of a member assumed to be conﬁned to the frustum of a hollow cone.

FIGURE 18-3D Forms of threads for power screw.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

Particular

The spring stiﬀness of the frustum when cone angle of frustum ¼ 308 For the members of the joint having same modulus of elasticity E with symmetrical frusta back to back which constitute as two springs in series and using the grip as l ¼ 2t and dw as the diameter of the washer face, the eﬀective spring constant for the system. The eﬀective spring constant for the case of back to back cone frusta with a washer face dw ¼ 1:5d and ¼ 308 from Eq. (18-25d).

18.11

Formula

k¼

0:577 dE ð1:15t þ D dÞðD þ dÞ ln ð1:15t þ D þ dÞD dÞ

ke ¼

ke ¼

ð18-35cÞ

Ed tan ðl tan þ dw dÞðdw þ dÞ 2 ln ðl tan þ dw þ dÞðdw dÞ 0:577 dE 0:577l þ 0:5d 2 ln 5 0:577l þ 2:5d

ð18-25dÞ

ð18-25eÞ

Power screw The helix angle of a V-thread (Fig. 18-3E) The tangential force for a square thread at mean radius of screw

¼ tan1 Ft ¼ W

l d2

ð18-26Þ

tan þ 1 tan

ð18-27Þ

Refer to Table 18-4 for . TABLE 18-4 Coeﬃcient of friction for power screws Lubricant

Coeﬃcient of friction,

Machine oil and graphite Lard oil Heavy machine oil

0.07 0.11 0.14

FIGURE 18-3E Helix angle of a single-start thread.

Torque required to raise the load by a power screw

Mtu ¼ Mtsu þ Mte Wd2 d2 þ l cos d ¼ þ c W c d2 cos l 2 2

ð18-28Þ

where d2 ¼ pitch diameter of thread The tangential force for V-thread or angular thread at mean radius (Fig. 18-4)

The total frictional torque including collar friction torque for square thread

cos Ft ¼ W tan 1 cos " # d2 tan þ dc þ c Mt ¼ W 2 1 tan 2 tan þ

ð18-28aÞ

ð18-29Þ

Refer to Table 18-4 for and Table 18-5a for c .

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.12

CHAPTER EIGHTEEN

FIGURE 18-3G Diﬀerential screw.

FIGURE 18-3F Power screw.

FIGURE 18-4 Forces acting on a triangular thread.

TABLE 18-5a Coeﬃcient of friction on thrust collar, c

TABLE 18-5b Torque factor K for use in Eq. (18-30c)

Material

Coeﬃcient of running friction

Coeﬃcient of starting friction

Soft steel on cast iron Hardened steel on cast iron Soft steel on bronze Hardened steel on bronze

0.121 0.092 0.084 0.063

0.170 0.147 0.101 0.081

Bolt condition

K

Nonplated, black ﬁnish Zinc-plated Lubricated Cadmium-plated With Bowman anti-seize With Bowman-grip nuts

0.30 0.20 0.18 0.16 0.12 0.09

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

Particular

The total frictional torque for V-thread, including collar friction torque

The mean diameter of collar Substituting the value of dc in Eq. (18-30a) and after simplifying

18.13

Formula

3 1 d2 B dc 7 cos C Mt ¼ W 6 4 @ tan A þ c 2 5 2 1 cos 2

0

tan þ

dc ¼ ðd þ 1:5dÞ=2 M t ¼ K Fi d

ð18-30aÞ ð18-30bÞ ð18-30cÞ

where K is the torque factor W ¼ Fi ¼ preload, N (lbf)

The torque factor

0 1 tan þ d cos C þ 0:625 K ¼ 2 B @ A c 2d 1 tan cos

ð18-30dÞ

where d2 ¼ dm Refer to Table 18-5b for K . tan Wl ¼ tanð þ Þ 2 Mt

The eﬃciency of square thread neglecting collar friction

¼

The eﬃciency formula for an angular-type thread with half apex angle and an allowance for nut or end friction on a radius rc

¼

The eﬃciency formula for square thread

d tan ¼ tan þ2 d þ c dc 1 tan 2 ¼

d2 tan tan þ = cos d þ c dc 1 tan = cos 2

l ½d2 tanð þ Þ þ c dc

ð18-31Þ ð18-32Þ

ð18-33Þ

ð18-34Þ

LOADING Lowering the load The tangential force at mean or pitch radius r2 ¼ rm The frictional torque at mean or pitch radius r2 ¼ rm

The condition for overhauling for square threads

Ft ¼ W tanð þ Þ Mt ¼

ð18-35Þ

Wd2 tanð Þ 2

ð18-36Þ

d2 þ c dc d2 c dc

ð18-37Þ

tan

Since the ﬂat faces of hexagonal nut is same as the diameter of washer face which is 1.5 times the nominal diameter d.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.14

CHAPTER EIGHTEEN

Particular

Formula

Diﬀerential screws (Fig. 18-3G) The loading eﬃciency of a diﬀerential screw, not including the collar friction

¼

Do tan o Di tan i tan o þ o tan i þ i Do Di 1 o tan o 1 i tan i

ð18-38Þ

Compound screws The loading eﬃciency of a compound screw, not including collar friction The number of threads necessary in the nut The length of nut

¼

i¼

Do tan o þ Di tan i tan o þ o tan i þ i Do þ Di 1 o tan o 1 i tan i 4W 0b ðd 2 di2 Þ

ln ¼ iP ¼

4WP 0b ðd 2 d12 Þ

ð18-39Þ

ð18-40Þ ð18-41Þ

TABLE 18-5c Metric mechanical-property classes for steel bolts, screws, and studsa

Size range inclusive

Minimum proof strength, sp MPa

4.6

M5–M36

225

400

240

Low or medium carbon

4.8

M1.6–M16

310

420

340

Low or medium carbon

5.8

M5–M24

380

520

420

Low or medium carbon

8.8

M16–M36

600

830

660

Medium carbon, Q and T

9.8

M1.6–M16

650

900

720

Medium carbon, Q and T

10.9

M5–M36

830

1040

940

Low-carbon martensite, Q and T

12.9

M1.6–M36

970

1220

1100

Property class

a

Minimum tensile strength, st MPa

Minimum yield strength, sy MPa

Material

Alloy, Q and T

The thread length for bolts and cap screws is 8 L 125 > < 2d þ 6 LT ¼ 2d þ 12 125 < L 200 > : 2d þ 25 L > 200

where L is the bolt length. The thread length for structural bolts is slightly shorter than given above.

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Head marking

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.15

TABLE 18-5d Grade identiﬁcation marks and mechanical properties of bolts and screws

Identiﬁer

Grade

A

SAE Grade 1

Size range (in)

Min. strength (103 psi) Proof

Tensile

Yield

Material and treatment

to 112

33

60

36

Low or medium carbon

to 112 to 34

33 55

60 74

36 57

Low carbon Low or medium carbon

to 112

33

60

36

SAE Grade 4

1 4 1 4 1 4 7 8 1 4

to 112

65

115

100

SAE Grade 5 and

1 4

to 1

85

120

92

ASTM A449 SAE Grade 5, ASTM A449

118 to 112

74

105

81

ASTM A449

134 to 3

55

90

58

C

SAE Grade 5.2

1 4

to 1

85

120

92

Low-carbon martensite, Q and T

D

ASTM A325, Type 1

1 2

Medium carbon, Q and T

ASTM A307 SAE Grade 2

B

to 1

85

120

92

118 to 112

74

105

81

1 2

Medium carbon, cold drawn Medium carbon, Q and T

E

ASTM A325, Type 2

to 1 118 to 112

85 74

120 105

92 81

Low carbon martensite, Q and T

F

ASTM A325, Type 3

1 2

to 1

85

120

92

Weathering steel, Q and T

118 to 112

74

105

81

G

ASTM A354, Grade BC

to 212

105

125

109

234 to 4

1 4

95

115

99

Alloy-steel, Q and T

SAE Grade 7

1 4

to 112

105

133

115

Medium carbon alloy, Q and T

SAE Grade 8

1 4 1 4

to 112

120

150

130

Medium carbon alloy, Q and T

ASTM A354, Grade BD

to 112

120

150

130

Alloy-steel, Q and T

J

SAE Grade 8.2

1 4

to 1

120

150

130

Low-carbon martensite, Q and T

K

ASTM A490, Type 1

1 2

to 112

120

150

130

Alloy-steel, Q and T

L

ASTM A490, Type 3

1 2

to 112

120

150

130

Weathering steel, Q and T

H I

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.16

CHAPTER EIGHTEEN

Particular

The required length of engagement for adequate shear strength (assuming that the load is distributed over the threads in contact) Neglecting the radial clearance between threads, or allowance at the major and minor diameters and considering the threads as a series of collars the equation for thread engagement The normal length of thread engagement as per Indian standard

Formula

lc ¼

nPF A

leðscrewÞ ¼

leðnutÞ ¼

ð18-42Þ

nPF d1 t1 ðscrewÞ

ð18-43Þ

nPF dtðnutÞ

ð18-44Þ

leNðminÞ ¼ 8:92Pd 0:2

SI

ð18-45aÞ

SI

ð18-45bÞ

SI

ð18-46aÞ

SI

ð18-46bÞ

where leN , P, and d in m leNðminÞ ¼ 2:24Pd 0:2 where leN , P, and d in mm leNðmaxÞ ¼ 26:67Pd 0:2 where leN , P, and d in m leNðmaxÞ ¼ 6:7Pd 0:2 where leN , P, and d in mm Note: If leN has to be between the limits, the length of the thread is said to be normal ðNÞ If leN has to be below the minimum level, length of thread is said to be short ðSÞ If leN has to be above the maximum level, length of thread is said to be long ðLÞ

Eccentric loading The load on bolt 1, Fig. 18-5 (panel a) The general expression for the load carried by ith bolt, Fi The maximum load on the bolt, Fig. 18-5(b)

F1 ¼

Fll1 lða b cos Þ ¼F l12 þ l22 þ l32 þ l42 4a2 þ 2b2

Fi ¼ F Fmax ¼

The maximum load on the bolt, Fig. 18-5(c)

2lða b cos Þ ð2a2 þ b2 Þi 2Flða þ bÞ ð2a2 þ b2 Þi " 2Fl a þ b cos

Fmax ¼

ð18-47Þ ð18-48Þ ð18-49Þ

1808 i

#

ð2a2 þ b2 Þi

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ð18-50Þ

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

Particular

18.17

Formula

FIGURE 18-5 Fastening of a ﬂanged bearing.

Fastening of a bracket Bracket with no preload

Fll1 þ l22 þ l32 Þ

ð18-51Þ

F2 ¼

Fll2 2ðl12 þ l22 þ l32 Þ

ð18-52Þ

F3 ¼

Fll3 2ðl12 þ l22 þ l32 Þ

ð18-53Þ

F1 ¼ Tensile load taken by the bolts, Fig. 18-6(a)

2ðl12

Shear stresses (i) If shear load is taken completely by the lug, shear load on lug is given by

F1 ¼ F

ð18-54Þ

(ii) If shear load is taken completely by the bolt shear load on each bolt is given by

Fb ¼

F i

ð18-55Þ

(iii) If shear load is shared equally between the bolt and the lug

F10 ¼

F 2

ð18-56Þ

Fb0 ¼

F 2i

ð18-57Þ

Shear load due to the eccentricity e, Fig. 18-6(b), in each bolt is given by

Fex Fei0 ¼ P i2 xi

ð18-58Þ

where xi ¼ distance between the center of bolts and the center of the particular bolt Resultant shear load

Fex Fr ¼ Fb ðor Fb0 Þ þ P i2 xi

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ð18-59Þ

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.18

CHAPTER EIGHTEEN

Particular

Formula

FIGURE 18-6 Preloaded bracket.

Preloaded bracket Compression stress in contact area between the bracket base and the wall, Fig. 18-6(c)

c ¼

iFi Ac

ð18-60Þ

Bending stress due to eccentric load, Fig. 18-6(d)

0b ¼

Mb c1 Flc1 ¼ Ic Ic

ð18-61Þ

Resultant compressive stress in the contact area

0c ¼

iFi Mb c1 iFi Flc1 ¼ Ac Ic Ac Ic

ð18-62Þ

Tensile stress in any individual bolt is given by

0b ¼

Fi Mb cb þ Ab Ic

ð18-63Þ

Fi >

Mb c1 Ac iIc

ð18-64Þ

With a 25% margin on the preload to account for overloads, condition to avoid separation of the base and wall

Fi ¼

1:25Mb c1 Ac iIc

ð18-65Þ

Bolt load taking into consideration 25% margin on the preload to account for overloads

Fb ¼

1:25Mb c1 Ac Mb cb þ iIc Ic

ð18-66Þ

Condition to avoid separation of the base and wall

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

Particular

With an additional horizontal load Fh , the preload Fi is given by

18.19

Formula

Fi ¼

1:25Mb c1 Ac Fh iIc i

ð18-67Þ

where (þ) is used when Fh is away from the wall and () when Fh is toward the wall 1:25Mb c1 Ac Fh Mb cb Fh Ab þ iIc i Ic Ac

With the addition of a horizontal load Fh , the bolt load is given by

Fb ¼

Moment on the bracket

Mb ¼ Fl Fh e0

ð18-69Þ

M x Fi ¼ P1 2i xi

ð18-70Þ

ð18-68Þ

Shear loads Shear load due to the eccentricity e in each of the bolts with no horizontal load

where M1 ¼ Fe

Mb c1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a2 þ b2 16Ic P 0:25Mb x0i c1 A2b Ic A c

ð88-70aÞ

where x0i ¼ distance of the center of a particular bolt to the center of the base of the bracket Shear load due to eccentricity e in each of the bolts with a horizontal load, Fh

M x Fi ¼ P1 2i xi

ð18-71Þ

where " 0:25Mb c1 Fh pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a2 þ b2 M1 ¼ Fe Ic Ac 4 Ab Ac

Vertical applied load due to the friction component of the preload

Fv ¼

Condition for the nonexistence of the support for the shearload

F <

0:25Mb c1 Fh Ic Ac

1:25Mb c1 Ac Fh iIc

1:25Mb c1 Ac Fh iIc

X # x0i

ð18-72Þ

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ð18-73Þ

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.20

CHAPTER EIGHTEEN

Particular

Formula

GENERAL See Tables 18-6 to 18-22 and Figs. 18-7 to 18-16 for further particulars on threaded fasteners and screws for power transmission. For British Standard ISO metric precision hexagon bolts, screws and nuts, and machine screws and machine screw nuts.

Refer to Tables 18-23 and 18-24.

For hexagon bolts, ﬁnished hexagon bolts, regular square nuts, hexagon and hexagon jam nuts, ﬁnished hexagon slotted nuts, regular hexagon and hexagon jam nuts, carriage bolts, countersunk, buttonhead and step bolts, machine screw heads, pan, truss and 1008 ﬂat heads, slotted head cap screws, square head setscrews, slotted headless setscrews, etc.

Refer to Tables from 18-25 to 18-42.

For bolts, screws and nuts metric series—American National Standards hexagon cap screws, formed hex screws, heavy hex screws, recommended diameter– length combinations for screws, hexagon bolts, heavy hex bolts, heavy hex structural bolts, hexagon nuts, slotted hex nuts, etc.

Refer to Tables from 18-43 to 18-52.

All dimensions in inches.

TABLE 18-6 Allowable bearing pressure for screws, 0b Material

Safe bearing pressure, 0b

Type

Screw

Nut

MPa

psi

Rubbing velocity, m/s [fpm ¼ (ft/min)]

Hand press Jack screw Jack screw Hoisting screw Hoisting screw Lead screw

Steel Steel Steel Steel Steel Steel

Bronze Cast iron Bronze Cast iron Bronze Bronze

17.2–24.0 12.3–17.2 10.8–17.2 4.4–6.9 5.4–9.8 1.0–1.5

2500–3500 1800–2500 1600–2500 600–1000 800–1400 150–240

Low speed, well lubricated Low speed, not over 0.04 (8) Low speed, not over 0.05 (10) Medium speed, 0.1 to 0.2 (20–40) Medium speed, 0.1 to 0.2 (20–40) High speed, 0.25 and over (50)

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

Particular

18.21

Formula

H ¼ 0:86603 P; H ¼ d 2H1 ¼ d 1:082 P; 2 3 D2 ¼ d2 ¼ d H ¼ d 0:64952 P; 4 H d1 ¼ d2 ¼ d 1:22687 P; 3 D D1 5 H1 ¼ ¼ H ¼ 0:54127 P; 8 2 d d1 17 ¼ H ¼ 0:61343 P; h3 ¼ 24 2 H r ¼ ¼ 0:1443 P; rc ¼ 0:10825 P; 6 d1 þ d2 2 stress area ¼ Ac ¼ 4 2 D1 ¼ d2

FIGURE 18-7 Basic proﬁle ISO metric screw threads.

Designation: A pitch diameter combination of thread size 8 mm and pitch 1 mm shall be designated as M8 1. M8 shall designate pitch diameter combination of thread size 8 mm and pitch 1.25 mm.

In practice the root is rounded and cleared beyond a width of P 8

H1

D D2

Internal threads.

h3 External threads. H H 4 6 P

D1

Internal thread diameters

H 8

r

H 2

H

H 2

d d2

d1 External thread diameters

FIGURE 18-8 ISO metric screw thread design proﬁles of external and internal threads.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.22

CHAPTER EIGHTEEN

TABLE 18-7 Basic dimensions for design proﬁles of ISO metric screw threads Minor diameter, mm Basic diameter, mm 1 2 2.5 3.0 4.0 5.0 6.0 7.0 8.0 10

12

14

16 18

20

22

24

25

Lead angle at basic pitch diameter

Pitch, P, mm

Major diameter, d, mm

Pitch diameter, d2 , mm

External threads, d1

Internal threads, D1

deg

min

Tensile stress area, Ac , mm2

0.25 0.20 0.40 0.25 0.45 0.35 0.50 0.35 0.70 0.50 0.80 0.50 1.00 0.75 1.00 0.75 1.25 1.00 1.50 1.25 1.00 1.75 1.50 1.25 1.00 2.00 1.50 1.25 2.00 1.50 2.50 2.00 1.50 2.50 2.00 1.50 2.50 2.00 1.50 3.00 2.00 1.50 3.00

1.0 1.0 2.0 2.0 2.5 2.5 3.0 3.0 4.0 4.0 5.0 5.0 6.0 6.0 7.0 7.0 8.0 8.0 10.0 10.0 10.0 12 12 12 12 14 14 14 16 16 15 18 18 20 20 20 22 22 22 24 24 24 25

0.837620 0.870096 1.740192 1.837620 2.207716 2.272668 2.675240 2.772668 3.545337 3.675240 4.480385 4.675240 5.350481 5.512861 6.350481 6.512861 7.188101 7.350481 9.025721 9.188101 9.350481 10.863342 11.025721 11.188101 11.350481 12.700962 13.025721 13.188101 14.700962 15.025721 16.376202 16.700962 17.025721 18.376202 18.700962 19.025721 20.376202 20.700962 21.025721 22.051443 22.700962 23.025721 23.051443

0.693283 0.754626 1.509252 1.693283 1.947909 2.070596 2.386565 2.570596 3.141191 3.386565 4.018505 4.386565 4.773131 5.079848 5.773131 6.079848 6.466413 6.773131 8.159696 8.466413 8.773131 9.852979 10.159686 10.466413 10.773131 11.546261 12.159696 12.466413 13.546261 14.159696 14.932827 15.546261 15.159696 16.932827 17.516261 18.159696 18.932827 19.546261 20.159696 20.319392 21.556261 22.159696 21.319392

0.729367 0.783494 1.566987 1.729367 2.012861 2.121114 2.458734 2.621114 3.242228 3.458734 4.133975 4.458734 4.917468 5.188101 5.917408 6.188101 6.646835 6.917468 8.376202 8.646835 8.917468 10.105569 10.376202 10.646835 10.917468 11.834936 12.376202 12.646835 13.834936 14.376202 15.293671 15.834936 16.376202 17.293671 17.834936 18.376202 19.293671 19.834936 20.376202 20.752405 21.834936 22.376202 21.752405

5 4 4 2 3 2 3 2 3 2 3 2 3 2 2 2 3 2 3 2 1 2 2 2 1 2 2 1 2 1 2 2 1 2 1 1 2 1 1 2 1 1 2

27 11 11 29 43 20 24 18 36 29 15 57 24 29 52 6 10 29 2 29 57 56 29 2 36 52 6 44 29 49 47 11 36 29 57 26 14 46 18 49 39 11 36

0.46 0.53 2.07 2.45 3.39 3.70 5.03 5.61 8.78 9.79 14.2 16.1 20.1 22.0 28.9 31.3 36.6 39.2 58.0 61.2 64.5 84.3 88.1 92.1 96.1 115 125 129 157 167 192 204 216 245 258 272 303 318 333 353 384 401 385

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.23

TABLE 18-7 Basic dimensions for design proﬁles of ISO metric screw threads (Cont.) Minor diameter, mm Basic diameter, mm 30

35 42

45

52

60

72

80

90

100

110

Pitch, P, mm

Major diameter, d, mm

3.50 3.00 2.00 1.50 1.50 4.5 4.0 3.0 2.0 1.5 4.5 4.0 3.0 2.0 1.5 5.0 4.0 3.0 2.0 1.5 5.5 4.0 3.0 2.0 1.5 6 4 3 2 6 4 3 2 6 4 3 2 6 4 3 2 6 4 3

30 30 30 30 35 42 42 42 42 42 45 45 45 45 45 52 52 52 52 52 60 60 60 60 60 72 72 72 72 80 80 80 80 90 90 90 90 100 100 100 100 110 110 110

Pitch diameter, d2 , mm 27.726683 28.051443 28.700962 29.025721 34.055721 39.072114 39.401924 40.051443 40.700962 41.025771 42.077164 42.401924 43.051443 43.700962 44.025771 48.752405 49.401924 50.051443 50.700962 51.025721 56.427645 57.401924 58.051443 58.700962 59.025721 68.102886 69.401924 70.051443 70.700962 76.102886 77.401924 78.051443 78.700962 86.102886 87.401924 88.051449 88.700962 96.102886 97.401924 98.051443 98.700962 106.102886 107.401924 108.051443

External threads, d1 25.705957 26.319392 27.546261 28.159696 33.159696 36.479088 37.092523 38.319392 39.546261 40.159696 39.479088 40.092523 41.319392 42.546261 43.159696 45.865653 47.092523 48.319392 49.546261 50.159696 53.252219 55.092523 56.319392 57.546261 58.159696 64.638784 67.092523 68.319392 69.546261 72.638724 75.092523 76.319392 77.546261 82.638784 85.092523 86.319292 87.546261 92.638784 95.092523 96.319392 97.546261 102.638784 105.092523 106.319392

Internal threads, D1 26.211139 26.752405 27.834936 28.376202 33.376202 37.128607 37.669873 38.752405 39.834936 40.376202 40.128607 40.669873 41.752405 42.834936 43.376202 46.587341 47.669873 48.752405 49.834936 50.376202 54.046075 55.669873 56.752405 57.834936 58.376202 66.504809 67.669873 68.752405 69.834936 73.504809 75.669873 76.752405 77.834936 83.504809 85.669873 86.752405 87.834936 93.504809 95.669873 96.752405 97.834936 103.504809 105.669873 106.752405

Lead angle at basic pitch diameter deg

min

Tensile stress area, Ac , mm2

2 1 1 0 0 2 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0

18 57 16 57 48 6 51 22 52 40 57 43 16 50 37 52 29 6 43 32 47 16 57 37 28 36 3 47 31 26 57 42 28 16 50 37 25 8 45 33 22 2 41 30

561 581 621 642 860 1120 1150 1210 1260 1290 1300 1340 1400 1460 1490 1760 1830 1900 1970 2010 2360 2490 2570 2650 2700 3460 3660 3760 3860 4340 4570 4680 4790 5590 5840 5970 6100 7000 7280 7420 7560 8560 8870 9020

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.24

CHAPTER EIGHTEEN

TABLE 18-7 Basic dimensions for design proﬁles of ISO metric screw threads (Cont.) Minor diameter, mm Basic diameter, mm 120

150

160

180

200

250

300

Lead angle at basic pitch diameter

Pitch, P, mm

Major diameter, d, mm

Pitch diameter, d2 , mm

External threads, d1

Internal threads, D1

deg

min

Tensile stress area, Ac , mm2

6 4 3 6 4 3 6 4 3 6 4 3 6 4 3 6 4 3 6 4

120 120 120 150 150 150 160 160 160 180 180 180 200 200 200 250 250 250 300 300

116.102886 117.401924 118.051443 146.102886 147.401924 148.051443 156.102886 157.401924 158.051443 176.102886 177.401924 178.051443 196.102886 197.401924 198.051453 246.102886 247.401924 248.051443 296.102886 297.401924

112.638784 115.092523 116.319392 142.538784 145.092523 146.319392 152.638784 155.092523 156.319392 172.638784 175.092523 176.319392 192.638784 195.092523 196.319392 242.638784 245.092523 246.319392 295.638784 292.092523

113.504819 115.669873 116.752405 143.504809 145.669873 146.752405 153.504809 155.669873 156.752405 173.504809 175.669873 176.752405 193.504809 195.669873 196.752405 243.504809 245.669873 246.752405 293.504809 295.669873

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

57 37 28 45 30 22 42 28 21 37 25 18 33 22 17 27 18 13 22 15

10300 10600 10800 16400 16800 17000 18700 19200 19400 23900 24400 24700 29700 30200 30500 46900 47600 48000 68100 68900

Source: IS: 4218-1967 (Part III).

FIGURE 18-9 Basic proﬁle of square threads.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.25

TABLE 18-8 Basis dimensions (in mm) for square threads

Nominal diameter

Major diameter Bolt, d

Nut, D

Minor diameter, d1

10 14 20 26 30 36 40 44 50 55 60 75 80 85 90 95 90 95 100 110 120 130 140 150 160 170 180 190 200 220 240

10 14 20 26 30 36 40 45 50 55 60 65 70 75 80 85 90 95 100 110 120 130 140 150 160 170 180 190 200 220 240

10.5 14.5 20.5 26.5 30.5 36.5 40.5 44.5 50.5 55.5 60.5 65.5 70.5 75.5 80.5 85.5 90.5 95.5 100.5 110.5 120.5 130.5 140.5 150.5 160.5 170.5 180.5 190.5 200.5 220.5 240.5

8 12 18 23 27 33 37 41 47 52 57 61 66 71 76 84 86 91 96 106 114 124 134 144 154 164 172 182 192 212 232

22 24 26 28 30 36 40 44 50 52 55 60 65 70

22 24 26 28 30 36 40 44 50 52 55 60 65 70

22.5 24.5 26.5 28.5 30.5 36.5 40.5 44.5 50.5 52.5 55.5 60.5 65.5 70.5

17 19 21 23 24 30 33 37 42 44 46 51 55 60

Pitch, P

e

r

h2

b

h1

a

H

2

1

0.12 0.75 0.25 1

0.25 1.25

3

1.5

0.12 1.25 0.25 1.5

0.25 1.75

3

1.5

0.12 1.25 0.25 1.5

0.25 1.75

4

2

0.12 1.75 0.25 2

0.25 2.25

6

3

0.25 2.5

0.5

3

0.25 3.25

8

4

0.25 3.5

0.5

4

0.25 4.25

5

2.5

0.25 2

0.5

2.5

0.25 2.75

6

3

0.25 2.5

0.5

3

0.25 3.25

7

3.5

0.25 3

0.5

3.5

0.25 3.75

8

4

0.25 3.5

0.5

4

0.25 4.25

9

4.5

0.25 4

0.5

4.5

0.25 4.75

5

0.25 4.5

0.5

5

0.25 5.25

Area of core, Ac , mm2 50.3 113 201 415 573 855 1075 1320 1735 2124 2552 2922 3421 3959 4536 5153 5809 5504 7248 8825 10207 12076 14103 16286 18627 21124 23235 26016 28953 35299 42273

Normal Series

10

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227 284 346 415 452 707 855 1075 1385 1521 1662 2043 2376 2827

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.26

CHAPTER EIGHTEEN

TABLE 18-8 Basis dimensions (in mm) for square threads (Cont.)

Nominal diameter

Major diameter Bolt, d

Nut, D

Minor diameter, d1

75 80 85 90 95 100 110 120 130 140 150 160 170 180 190 200 300

75 80 85 90 95 100 110 120 130 140 150 160 170 180 190 200 300

75.5 80.5 85.5 90.5 95.5 100.5 110.5 121 131 141 151 161 171 181 191 201 301

65 70 73 78 83 88 98 106 116 126 134 144 154 162 172 182 274

22 24 26 28 30 36 40 50 60 70 75 80 90 120 150 180 200 300 400

22 24 26 28 30 36 40 50 60 70 75 80 90 120 150 180 200 300 400

22.5 24.5 26.5 28.5 30.5 36.5 40.5 50.5 61 71 76 81 91 121 151 181 201 301 401

14 16 18 20 20 26 28 38 46 54 59 64 72 98 126 152 168 256 352

Pitch, P

e

r

h2

b

h1

a

H

12

6

0.25

5.5 0.5

6

0.25

6.25

14

7

0.5

6

1

7

0.5

7.5

16

8

0.5

7

1

8

0.5

8.5

18

9

0.5

8

1

9

0.5

9.5

26

13

0.5

12

1

13

0.5

13.5

0.5

4

0.25

4.25

Area of core, Ac , mm2 3318 3848 4185 4778 5411 6082 7543 8825 10568 12469 14103 16286 18627 20612 23235 26016 58965

Coarse Series 8

4

0.25 3.5

10

5

0.25

4.5 0.5

5

0.25

5.25

12 14

6 7

0.25 0.5

5.5 0.5 6 1

6 7

0.25 0.5

6.25 7.5

16

8

0.5

7

1

8

0.5

8.5

18 22 24 28 32 44 48

9 11 12 14 16 24 24

0.5 0.5 0.5 0.5 0.5 0.5 0.5

8 10 11 13 15 21 23

1 1 1 1 1 1 1

9 11 12 14 16 22 24

0.5 0.5 0.5 0.5 0.5 0.5 0.5

9.5 11.5 12.5 14.5 16.5 22.5 24.5

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164 201 254 314 314 531 616 1134 1662 2290 2734 3217 4072 8332 12469 18146 22167 51472 97314

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.27

Pitch, mm

Depth of thread, mm

Depth of engagement, mm

e, mm

b, mm

r, mm

2 3 4 5 6 7 8 9 10 12 14 16 18 20 22 24 26 28 32 36 40 44 48

1.736 2.603 3.471 4.339 5.207 6.074 6.942 7.810 8.678 10.413 12.149 13.884 15.620 17.355 19.091 20.826 22.562 24.298 27.769 31.240 34.711 38.182 41.653

1.5 2.25 3 3.75 4.5 5.25 6 6.75 7.5 9 10.5 12 13.5 15 16.5 18 19.5 21 24 27 30 33 36

0.528 0.792 1.055 1.319 1.583 1.847 2.111 2.375 2.638 3.166 3.694 4.221 4.749 5.277 5.804 6.332 6.860 7.388 8.443 9.498 10.554 11.609 12.664

0.236 0.353 0.471 0.589 0.707 0.824 0.942 1.060 1.178 1.413 1.649 1.884 2.120 2.355 2.591 2.826 3.062 3.298 3.769 4.240 4.711 5.182 5.653

0.249 0.373 0.497 0.621 0.746 0.870 0.994 1.118 1.243 1.491 1.740 1.988 2.237 2.485 2.734 2.982 3.231 3.480 3.977 4.474 4.971 5.468 5.965

Designation: A sawtooth thread of nominal diameter 48 mm and pitch 3 mm shall be designated as ST 48 3.

FIGURE 18-10 Basic proﬁle of sawtooth threads. (Source: IS 4696, 1968.)

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.28

CHAPTER EIGHTEEN

TABLE 18-9 Basic dimensions (in mm) for sawtooth threads Bolt

Nut

Nominal diameter

Major diameter, d

Minor diameter, d1

Area of core, mm2

10 12 14 16 20 22 30 36 40 50 55 60 65 70 75 80 85 90 95 100 120 150 180 200

10 12 14 16 20 22 30 36 40 50 55 60 65 70 75 80 85 90 95 100 120 150 180 200

6.528 8.528 10.538 12.528 16.528 16.794 24.794 30.794 34.794 44.794 49.794 54.794 58.058 63.058 68.058 73.058 78.058 83.058 88.058 93.058 109.586 139.586 166.116 186.116

33.5 57.1 87.1 123 215 222 483 745 951 1576 1947 2358 2647 3123 3638 4192 4785 5418 6090 6801 9432 15303 21673 27206

22 24 26 30 36 40 44 50 55 60 70 80 90 100 110 130 150 180 200

22 24 26 30 36 40 44 50 55 60 70 80 90 100 110 130 150 180 200

13.322 15.322 17.322 19.586 25.586 27.852 31.852 36.116 39.380 44.380 52.644 62.644 69.174 79.174 89.174 102.702 122.232 148.760 168.760

139 184 236 301 514 709 797 1024 1218 1547 2177 3082 3758 4923 6246 8775 11734 17381 22368

Pitch diameter, d2

Pitch, P

Major diameter, D

Minor diameter, D1

Fine Series 8.636 10.636 12.636 14.636 18.636 19.954 27.954 32.954 37.954 42.954 57.954 57.954 62.272 67.272 72.272 77.272 82.272 87.272 92.272 97.272 115.909 145.909 174.545 194.545

2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 6 6 8 8

10 12 14 16 20 22 30 36 40 50 55 60 65 70 75 80 85 90 95 100 120 150 180 200

7 9 11 13 17 17.5 25.5 31.5 35.5 45.5 50.5 55.5 59 64 69 74 79 84 89 94 111 141 168 188

Normal Series 18.590 20.590 22.590 25.909 31.909 35.227 39.227 44.545 48.863 53.863 63.181 73.181 81.817 91.817 101.817 120.459 139.089 167.726 187.726

5 5 5 6 6 7 7 8 9 9 10 10 12 12 12 14 16 18 18

22 24 26 30 36 42 44 50 55 60 70 80 90 100 110 130 150 180 200

14.5 16.5 18.5 21 27 31.5 33.5 38 41.5 46.5 55 65 72 82 92 109 126 153 173

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.29

TABLE 18-9 Basic dimensions (in mm) for sawtooth threads (Cont.) Bolt

Nut

Nominal diameter

Major diameter, d

Minor diameter, d1

Area of core, mm2

22 24 26 30 40 50 60 70 80 90 100 150 200

22 24 26 30 40 50 60 70 80 90 100 150 200

8.116 10.116 12.116 12.644 19.174 29.174 35.702 42.232 52.232 58.760 65.290 108.348 144.462

51.4 80.7 115 126 289 668 1001 1401 2143 2712 3348 9220 16391

Pitch diameter, d2

Coarse Series 16.545 18.545 20.545 23.181 31.817 41.817 50.453 59.089 69.089 77.726 86.362 138.634 178.179

Pitch, P

Major diameter, D

Minor diameter, D1

8 8 8 10 12 12 14 16 16 18 20 24 32

22 24 26 30 40 50 60 70 80 90 100 150 200

10 12 14 15 22 32 39 46 56 63 70 114 152

Radii, mm Nominal diameter, d, mm 8–12 14–38 40–100 105–200

Pitch, P, mm

Depth of thread, h1 , mm

Depth of engagement, h2 , mm

Nut Bolt, r

R

R1

2.550 3.175 4.233 6.350

1.270 1.588 2.117 3.175

0.212 0.265 0.353 0.530

0.606 0.757 1.010 1.515

0.650 0.813 1.084 1.625

0.561 0.702 0.936 1.404

Designation: A knuckle thread of nominal diameter 10 mm and pitch of 2.54 mm shall be designated as K10 2:54.

FIGURE 18-11 Basic proﬁle of knuckle threads. (Source: IS 4695: 1968.)

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.30

CHAPTER EIGHTEEN

TABLE 18-10 Basic dimensions (in mm) for knuckle threads Bolt

Nut

Nominal diameter

Major diameter, d

Minor diameter, d1

Area of core, mm2

Pitch diameter, d2

Major diameter, D

Minor diameter, D1

8 9 10 12 14 16 20 24 30 36 40 44 50 55 60 65 70 75 80 85 90 95 100 110 120 130 140 150 160 170 180 190 200

8 9 10 12 14 16 20 24 30 36 40 44 50 55 60 65 70 75 80 85 90 95 100 110 120 130 140 150 160 170 180 190 200

5.460 6.460 7.460 9.460 10.825 12.825 16.825 20.825 26.825 32.825 35.767 39.767 45.767 50.767 55.767 60.767 65.767 70.767 75.767 80.767 85.767 90.767 95.767 103.650 113.650 123.650 133.650 143.650 153.650 163.650 173.650 183.650 193.650

23.4 32.8 43.7 70.3 92.0 129.2 222.3 340.6 565.2 846.3 1005 1242 1645 2024 2443 2900 3397 3933 4509 5123 5777 6471 7203 8438 10145 12008 14029 16207 18542 21034 23683 26489 29453

6.730 7.730 8.730 10.730 12.412 14.412 18.412 22.412 28.412 34.412 37.883 41.883 47.883 52.883 57.883 62.883 67.883 72.883 77.883 82.883 87.883 92.883 97.883 106.825 116.885 126.825 136.825 146.825 156.825 166.825 176.825 186.825 196.825

8.254 9.254 10.254 12.254 14.318 16.318 20.318 24.318 30.318 36.318 40.423 44.423 50.423 55.423 60.423 65.423 70.423 75.423 80.423 85.423 90.423 95.423 100.423 110.635 120.635 130.635 140.635 150.635 160.635 170.635 180.635 190.635 200.635

5.714 6.714 7.714 9.714 11.142 16.142 17.142 21.142 27.142 33.142 36.190 40.190 46.190 51.190 56.190 61.190 66.190 71.190 76.190 81.190 86.190 91.190 96.190 104.285 114.985 124.285 134.285 144.285 154.285 164.285 174.285 184.285 194.285

Source: IS 4695, 1968.

TABLE 18-11 Pitch-diameter combinations for ISO metric threads Pitch, P, mm

Maximum diameter, mm

0.5 0.75 1.00 1.50 2.00 3.00

22 33 80 150 200 300

TABLE 18-12 Tolerance grades 3, 4, 5 for precision; 6 for medium; and 7, 8, and 9 for coarse qualities for bolts and nuts Minor diameter of nut threads Major diameter of bolt threads Pitch diameter of nut threads Pitch diameter of bolt threads

3

4 4 4 4

5 5 5

6 6 6 6

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7 7 7

8 8 8 8

9

41.910 47.803

9.147

40.431 46.324

31.770

25.279

25.793

19.806

15.301

38.952 44.845

30.291

24.117

18.631

14.940

11.445

8.566

Minor, d1

FP 5 FP 6

2.309 2.309

2.309

2.309 FP 4

FP 312

2.309

FP 214

2.309

2.309

FP 3

2.309

FP 214

Pitch, P

FP 2

Designation

138.430 193.830

113.030

100.330

87.884

75.184

62.710

59.614

Major, d2

136.951 162.351

111.551

98.851

86.407

73.705

64.231

58.135

Pitch, P

Basic diameter, internal and external threads

135.472 160.872

110.072

97.372

84.926

72.226

62.752

56.656

Minor, d1

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FIGURE 18-12 Pipe threads for fastening purposes. (Source: IS 2643, 1964.)

Designation: An external pipe thread for fastening purposes of size 2 with class B tolerance shall be designated as Ext-FP 2B, and an internal pipe thread of size 2 shall be designated as Int-FP 2.

2.309 2.309

26.441

1.814

FP 114 FP 112

20.955

1.814 33.249

16.662

1.337

9.728 13.157

Pitch, P

Basic diameter, internal and external threads Major, d2

1.337

0.907

2.309

1 8 1 4 3 8 1 2 3 4

Pitch, P

FP 1

FP

FP

FP

FP

FP

Designation

All dimensions in mm

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.31

5G

S

6G 7G

N

Source: IS 4218 (Part IV), 1967.

Fine Medium Coarse

Tolerance quality

7G 8G

L

Small allowance position G

TABLE 18-14 Preferred tolerance classes for nuts

a

Td in mm; P in mm; Td2 in mm; d in mm. Source: IS 4218 (Part IV), 1967.

Td2 (6) Td2 (6)

Nut

Bolt Nut

Td1 (6)

Bolt

Crest diameter

Pitch diameter

Td (6)

Bolt/nut

Diameter

5H 6H 7H

N 6H 7H 8H

L

—

433P 190P1:22 for P from 0.2 to 0.8 mm 230P0:7 for P from 1 mm and above 90P0:4 d 0:1 90P0:4 d 0:1 0.63 Td2 (6) 0.85 Td2 (6)

—

0.63 Td (6)

4

S

6e

N

7e 6e

L

Large allowance position e

Source: IS 4218 (Part IV), 1967.

Fine Medium Coarse

Tolerance quality

1.25 Td2 (6) 1.7 Td2 (6)

—

—

7

7g 6g

S

6g 8g

N

7g 6g 9g 8g

L

Small allowance position g

Td2 (6) 1.32 Td2 (6)

—

Td (6)

6

Tolerance grades

0.8 Td2 (6) 1.06 Td2 (6)

—

—

5

TABLE 18-15 Preferred tolerance classes for bolts

0.5 Td2 (6) —

—

3

3:15 180P2=3 pﬃﬃﬃﬃ P

Value of tolerance unit

No allowance position H

4H 5H

S

Unit of tolerance

TABLE 18-13 Tolerance for crest and pitch diameters of bolts and nutsa

3h 4h 5h 6h

S

2 Td2 (6) —

—

—

9

4h 6h

N

5h 4h 7h 6h

L

No allowance position h

1.6 Td2 (6) 2.12 Td2 (6)

—

1.6 Td (6)

8

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.32

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.33

TABLE 18-16 Coarse-threaded series—UNC and NC (dimensions in inches)

Sizesa 1 2 3 4 5 6 8 10 12 1 4 5 16 3 8 7 16 1 2 1 2 9 16 5 8 3 4 7 8

1 118 114 138 112 134 2 214 212 234 3 314 312 334 4

UN UN UN UN UN UN UN UN UN UN UN UN UN UN UN UN UN UN UN UN UN UN UN UN

Basic major (nominal) diameter, D

Threads per inch

Basic pitch diameter

Basic minor diameter external thread

0.0730 0.0860 0.0990 0.1120 0.1250 0.1380 0.1640 0.1900 0.2160 0.2500 0.3125 0.3750 0.4375 0.5000 0.5000 0.5625 0.6250 0.7500 0.8750 1.0000 1.1250 1.2500 1.3750 1.5000 1.7500 2.0000 2.2500 2.5000 2.7500 3.0000 3.2500 3.5000 3.7500 4.0000

64 56 48 40 40 32 32 24 24 20 18 16 14 13 12 12 11 10 9 8 7 7 6 6 5 412 412 4 4 4 4 4 4 4

0.0629 0.0744 0.0855 0.0958 0.1088 0.1177 0.1437 0.1629 0.1889 0.2175 0.2764 0.3344 0.3911 0.4500 0.4459 0.5084 0.5660 0.6850 0.8028 0.9188 1.0322 1.1572 1.2667 1.3917 1.6201 1.8557 2.1057 2.3376 2.5876 2.8376 3.0876 3.3376 3.5876 3.8376

0.0538 0.0641 0.0734 0.0813 0.0943 0.0997 0.1257 0.1389 0.1649 0.1887 0.2443 0.2983 0.3499 0.4056 0.3978 0.4603 0.5135 0.6273 0.7387 0.8466 0.9497 1.0747 1.1705 1.2955 1.5046 1.7274 1.9774 2.1933 2.4433 2.6933 2.9433 3.1933 3.4433 3.6933

Root areab in in2 , A 0.0023 0.0032 0.0042 0.0052 0.0070 0.0078 0.0124 0.0152 0.0214 0.0280 0.0469 0.0699 0.0962 0.1292 0.1243 0.1664 0.2071 0.3091 0.4286 0.5629 0.7178 0.9071 1.0760 1.3182 1.7780 2.3436 3.0610 3.7782 4.6886 5.6972 6.8039 8.0088 9.3119 10.7132

Minor diameter internal thread classes 1B, 2B, and 3B for engagement 2 3 3 D to 2 D Minimum

Maximum

0.0585 0.0699 0.0805 0.0894 0.1021 0.1091 0.1346 0.1502 0.1758 0.2013 0.2577 0.3128 0.3659 0.4226 0.4160 0.4783 0.5329 0.6481 0.7614 0.8722 0.9789 1.1039 1.2046 1.3296 1.5455 1.7728 2.0228 2.2444 2.4944 2.7444 2.9944 3.2444 3.4944 3.7444

0.0623 0.0737 0.0845 0.0939 0.1062 0.1140 0.1389 0.1555 0.1807 0.2067 0.2630 0.3182 0.3717 0.4284 0.4223 0.4843 0.5391 0.6545 0.7681 0.8797 0.9875 1.1125 1.2146 1.3396 1.5575 1.7861 2.0361 2.2594 2.5094 2.7594 3.0094 3.2594 3.5094 3.7594

a

Uniﬁed diameter-pitch relationships are marked UN. The actual root area of a screw will be somewhat less than A, but, since the tensile strength of a screw of ductile material is greater than that of a plain specimen of the same material and of a diameter equal to the root diameter of the screw, the tensile strength of a screw may be assumed to correspond to A as given. For complete manufacturing information and tolerances, see ASA Standard B1.1, 1949. b

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.34

CHAPTER EIGHTEEN

TABLE 18-17 Fine-thread series UNF and NF (dimensions in inches)

Basic pitch diameter

Basic minor diameter external thread

Root areab in in2 , A

Minimum

Maximum

80 72 64 56 48

0.0519 0.0640 0.0759 0.0874 0.0985

0.0447 0.0560 0.0668 0.0771 0.0864

0.0016 0.0025 0.0035 0.0047 0.0059

0.0479 0.0602 0.0720 0.0831 0.0931

0.0514 0.0635 0.0753 0.0865 0.0968

0.1250 0.1380 0.1640 0.1900 0.2160

44 40 36 32 28

0.1102 0.1218 0.1460 0.1697 0.1928

0.0971 0.1073 0.1299 0.1517 0.1722

0.0074 0.0090 0.0133 0.0181 0.0233

0.1042 0.1147 0.1358 0.1601 0.1815

0.1079 0.1186 0.1416 0.1641 0.1857

Basic major (nominal) diameter, D

Threads per inch

0 1 2 3 4

0.0600 0.0730 0.0860 0.0990 0.1120

5 6 8 10 12

Sizesa

Minor diameter internal thread classes 1B, 2B, and 3B for engagement 2 3 3 D to 2 D

1 4 5 16 3 8 7 16

UN UN UN UN

0.2500 0.3125 0.3750 0.4375

28 24 24 20

0.2268 0.2854 0.3479 0.4050

0.2062 0.2614 0.3239 0.3762

0.0334 0.0541 0.0824 0.1112

0.2150 0.2714 0.3332 0.3875

0.2190 0.2754 0.3372 0.3916

1 2 9 16 5 8 3 4 7 8

UN UN UN UN UN

0.5000 0.5625 0.6250 0.7500 0.8750

20 18 18 16 14

0.4675 0.5264 0.5889 0.7094 0.8286

0.4387 0.4943 0.5568 0.6733 0.7874

0.1512 0.1919 0.2435 0.3560 0.4869

0.4497 0.5065 0.5690 0.6865 0.8023

0.4537 0.5106 0.5730 0.6908 0.8068

UN UN UN UN UN

1.000 1.1250 1.2500 1.3750 1.5000

12 12 12 12 12

0.9459 1.0709 1.1959 1.3209 1.4459

0.8978 1.0228 1.1478 1.2728 1.3978

0.6331 0.8216 1.0347 1.2724 1.5346

0.9148 1.0398 1.1648 1.2893 1.4148

0.9198 1.0448 1.1698 1.2948 1.4198

1 118 114 138 112 a

Uniﬁed diameter-pitch relationships are marked UN. The actual root area of a screw will be somewhat less than A, but, since the tensile strength of a screw of ductile material is greater than that of a plain specimen of the same material and of a diameter equal to the root diameter of the screw, the tensile strength of a screw may be assumed to correspond to A as given. For complete manufacturing information and tolerances, see ASA Standard B1.1, 1949.

b

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.35

TABLE 18-18 Extra-ﬁne thread series—NEF

Basic pitch diameter, in

Basic minor diameter external thread, in

Root areab in in2 , A

Minimum

Maximum

32 32 32 32 28

0.1957 0.2297 0.2922 0.3547 0.4143

0.1777 0.2117 0.2742 0.3367 0.3937

0.0248 0.0352 0.0591 0.0890 0.1217

0.1855 0.2189 0.2807 0.3429 0.4011

0.1895 0.2229 0.2847 0.3469 0.4051

0.5000 0.5625 0.6250 0.6875 0.7500

28 24 24 24 20

0.4768 0.5354 0.5979 0.6604 0.7175

0.4562 0.5114 0.5739 0.6364 0.6887

0.1635 0.2054 0.2587 0.3181 0.3725

0.4636 0.5204 0.5829 0.6454 0.6997

0.4676 0.5244 0.5869 0.6494 0.7037

0.8125 0.8750 0.9375 1.0000 1.0625

20 20 20 20 18

0.7800 0.8425 0.9050 0.9675 1.0264

0.7512 0.8137 0.8762 0.9387 0.9943

0.4432 0.5200 0.6030 0.6921 0.7765

0.7622 0.8247 0.8872 0.9497 1.0064

0.7662 0.8287 0.8912 0.9537 1.0105

118 3 116 114 5 116 138

1.1250 1.1875 1.2500 1.3125 1.3750

18 18 18 18 18

1.0889 1.1514 1.2139 1.2764 1.3389

1.0568 1.1193 1.1818 1.2443 1.3068

0.8772 0.9840 1.0969 1.2160 1.3413

1.0689 1.1314 1.1939 1.2564 1.3189

1.0730 1.1355 1.1980 1.2605 1.3230

7 116 112 9 116 5 18 111 16

1.4375 1.5000 1.5625 1.6250 1.6875

18 18 18 18 18

1.4014 1.4639 1.5264 1.5889 1.6514

1.3693 1.4318 1.4943 1.5568 1.6193

1.4726 1.6101 1.7538 1.9035 2.0594

1.3814 1.4439 1.5064 1.5689 1.6314

1.3855 1.4480 1.5105 1.5730 1.6355

134 UN 2 UN

1.7500 2.0000

16 16

1.7094 1.9594

1.6733 1.9233

2.1991 2.9053

1.6865 1.9365

1.6908 1.9408

Sizesa 12 1 4 5 16 3 8 7 16

UN

1 2 9 16 5 8 11 16 3 4

UN

13 16 7 8 15 16

UN UN UN UN

1 1 116

UN

Basic major (nominal) diameter, D, in

Threads per inch

0.2160 0.2500 0.3125 0.3750 0.4375

Minor diameter internal thread classes 1B, 2B, and 3B for engagement 2 3 3 D to 2 D, in

a

Uniﬁed diameter-pitch relationships are marked UN. The actual root area of a screw will be somewhat less than A, but, since the tensile strength of a screw of ductile material is greater than that of a plain specimen of the same material and of a diameter equal to the root diameter of the screw, the tensile strength of a screw may be assumed to correspond to A as given. For complete manufacturing information and tolerances, see ASA Standard B1.1, 1949.

b

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.36

CHAPTER EIGHTEEN

TABLE 18-19 8-pitch thread series—8N (dimensions in inches) Minor diameter internal thread classes 1B, 2B, and 3B for engagement 23 D to 32 D

Sizea also basic major (normal) diameter, D

Basic minor diameter external thread

Minimum

1 UN 118 114 138 112 158 134 178 2 218 214 212 234

0.8466 0.9716 1.0966 1.2216 1.3466 1.4716 1.5966 1.7216 1.8466 1.9716 2.0966 2.3466 2.5966

0.8722 0.9972 1.1222 1.2472 1.3722 1.4972 1.6222 1.7472 1.8722 1.9972 2.1222 2.3722 2.6222

Minor diameter internal thread classes 1B, 2B, and 3B for engagement 23 D to 32 D

Maximum

Sizea also basic major (nominal) diameter, D

Basic minor diameter external thread

Minimum

Maximum

0.8797 1.0047 1.1297 1.2547 1.3797 1.5047 1.6297 1.7547 1.8797 2.0047 2.1297 2.3797 2.6297

3 314 312 334 4 414 412 434 5 514 512 534 6

2.8466 3.0966 3.3466 3.5966 3.8466 4.0966 4.3466 4.5966 4.8466 5.0966 5.3466 5.5966 5.8466

2.8722 3.1222 3.3722 3.6222 3.8722 4.1222 4.3722 4.6222 4.8722 5.1222 5.3722 5.6222 5.8722

8.8797 3.1297 3.3797 3.6297 3.8797 4.1297 4.3797 4.6297 4.8797 5.1297 5.3797 5.6297 5.8797

a

Uniﬁed diameter-pitch relationships are marked UN. For complete manufacturing information and tolerances, see ASA Standard B1.1, 1949.

FIGURE 18-14 American Standard screw thread.

FIGURE 18-15 Whitworth screw thread.

FIGURE 18-13 608 uniﬁed and American Standard screw-thread forms.

FIGURE 18-16 British Association screw thread.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.37

TABLE 18-20 12-pitch thread series—12N (dimensions in inches)

Sizea also basic major (normal) diameter, D

Minor diameter internal thread classes 1B, 2B, and 3B for engagement 23 D to 32 D

Minor diameter internal thread classes 1B, 2B, and 3B for engagement 23 D to 32 D

Maximum

Sizea also basic major (nominal) diameter, D

Basic minor diameter external thread

Minimum

Maximum

0.4160 0.4783 0.5405 0.6029

0.4223 0.4843 0.5463 0.6085

2 UN 218 214 UN 238

1.8978 2.0228 2.1478 2.2728

1.9148 2.0398 2.1648 2.2898

1.9198 2.0448 2.1698 2.2948

UN

0.6478 0.7103 0.7728 0.8353

0.6653 0.7276 0.7900 0.8524

0.6707 0.7329 0.7952 0.8575

212 UN 258 234 UN 278

2.3978 2.5228 2.6478 2.7728

2.4148 2.5398 2.6648 2.7898

2.4198 2.5M8 2.6698 2.7948

1 1 UN 116 1 18 3 UN 116

0.8978 0.9603 1.0228 1.0853

0.9148 0.9773 1.0398 1.1023

0.9198 0.9823 1.0448 1.1073

3 UN 318 314 UN 338

2.8978 3.0228 3.1478 3.2728

2.9148 3.0398 3.1648 3.2898

2.9198 3.0448 3.1698 3.2948

114 5 UN 116 138 7 UN 116

1.1478 1.2103 1.2728 1.3353

1.1648 1.2273 1.2898 1.3523

1.1698 1.2323 1.2948 1.3573

312 UN 358 334 UN 378

3.3978 3.5228 3.6478 3.7728

3.4148 3.5398 3.6648 3.7898

3.4198 3.5448 3.6698 3.7948

112 158 134 UN 178

1.3978 1.5228 1.6478 1.7728

1.4148 1.5398 1.6648 1.7898

1.4198 1.5448 1.6698 1.7948

4 414 412 434

UN UN UN UN

3.8978 4.1478 4.3978 4.6478

3.9148 4.1648 4.4148 4.6648

3.9198 4.1698 4.4198 4.6698

5 514 512 534 6

UN UN UN UN UN

4.8978 5.1478 5.3978 5.6478 5.8978

4.9148 5.1648 5.4148 5.6648 5.9148

4.9198 5.1698 5.4198 5.6698 5.9198

Basic minor diameter external thread

Minimum

1 2 9 16 5 8 11 16

0.3978 0.4603 0.0228 0.5853

3 4 13 16 7 8 15 16

a

Uniﬁed diameter-pitch relationships are marked UN. For complete manufacturing information and tolerances, see ASA Standard B1.1, 1949.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.38

CHAPTER EIGHTEEN

TABLE 18-21 16-pitch thread series—16N (dimensions in inches)

Sizea also basic major (nominal) diameter, D

Minor diameter internal thread classes 1B, 2B, and 3B for engagement 23 D to 32 D

Basic minor diameter external thread

Minimum

Minor diameter internal thread classes 1B, 2B, and 3B for engagement 23 D to 32 D

Maximum

Sizea also basic major (nominal) diameter, D

Basic minor diameter external thread

Minimum

Maximum

3 4 13 16 7 8 15 16

UN UN UN

0.6733 0.7358 0.7983 0.8608

0.6865 0.7490 0.8115 0.8740

0.6908 0.7553 0.8158 0.8783

214 UN 5 216 238 7 216

2.1733 2.2358 2.2983 2.3608

2.1865 2.2490 2.3115 2.3740

2.1908 2.2533 2.3158 2.3783

1 1 116 1 18 3 116

UN UN UN UN

0.9233 0.9853 1.0483 1.1108

0.9365 0.9990 1.0615 1.1240

0.9408 1.0033 1.0658 1.1283

212 UN 258 234 UN 278

2.4233 2.5483 2.6733 2.7983

2.4365 2.5615 2.6865 2.8115

2.4408 2.5658 2.6908 2.8158

114 5 116 3 18 7 116

UN UN UN UN

1.1733 1.2358 1.2983 1.3608

1.1865 1.2490 1.3115 1.3740

1.1908 1.2533 1.3158 1.3783

3 UN 318 314 UN 338

2.9233 3.0483 3.1733 3.2983

2.9365 3.0615 3.1865 3.3115

2.9408 3.0658 3.1908 3.3158

112 UN 9 116 158 111 16

1.4233 1.4858 1.5483 1.6108

1.4365 1.4990 1.5615 1.6240

1.4408 1.5033 1.5658 1.6283

312 UN 358 334 UN 378

3.4233 3.5483 3.6733 3.7983

3.4365 3.5615 3.6865 3.8115

3.4408 3.5658 3.6908 3.8158

134 UN 113 16 178 115 16

1.6733 1.7358 1.7983 1.8608

1.6865 1.7490 1.8115 1.8740

1.6908 1.7533 1.8158 1.8783

4 414 412 434

UN UN UN UN

3.9233 4.1733 4.4233 4.6733

3.9365 4.1865 4.4365 4.6865

3.9408 4.1908 4.4408 4.6908

2 UN 1 116 1 28 3 216

1.9233 1.9858 2.0483 2.1108

1.9365 1.9990 2.0615 2.1240

1.9408 2.0033 2.0658 2.1283

5 514 512 534 6

UN UN UN UN UN

4.9233 5.1733 5.4233 5.6733 5.9233

4.9365 5.1865 5.4365 5.6865 5.9365

4.9408 5.1908 5.4408 5.6908 5.9408

a

Uniﬁed diameter-pitch relationships are marked UN. For complete manufacturing information and tolerances, see ASA Standard B1.1, 1949.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.39

TABLE 18-22 Proportions of power threads (dimensions in inches) Square threads Size in 1 4 5 16 3 8 7 16

Threads per inch

Minor diameter

Acme threads Threads per inch

Regular minor diameter

Stub minor diameter

10

0.163

16

0.188

0.213

9 8

0.2153 0.266

14 12

0.241 0.292

0.270 0.325

7

0.3125

12

0.354

0.388

612

0.366

10

0.400

0.440

512

0.466

8

0.500

0.550

5

0.575

6

0.583

0.650

412

0.681

6

0.708

0.775

1

4

0.783

5

0.800

0.880

118

312

0.8750

5

0.925

1.005

114

312

1.000

5

1.050

1.130

138

3

1.0834

4

1.125

1.225

112

3

1.284

4

1.250

1.350

134

212

1.400

4

1.500

1.600

2 214

214 214

1.612 1.862

4 3

1.750 1.917

1.850 2.050

212

2

2.063

3

2.167

2.300

234

2

2.313

3

2.417

2.550

3

134

2.500

2

2.500

2.700

312

158

2.962

2

3.000

3.200

4

112

3.168

1 2 5 8 3 4 7 8

2

3.500

3.700

412

2

4.000

4.200

5

2

4.000

4.700

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.40

CHAPTER EIGHTEEN

TABLE 18-23 British Standard ISO Metric Precision Hexagon Bolts, Screws and Nuts (BS 3692: 1967)

For general dimensions see Tables 2, 3, 4 and 5. Source: Courtesy British Standards Institution, 2 Park Street, London W1A 2BS, 1986.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

TABLE 18-24 British standard machine screws and machine screw nuts—metric series

For dimensions, see Tables 1 through 5. Source: Courtesy British Standards Institution, 2 Park Street, London W1A 2BS, 1986.

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18.41

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2.0000 2.2500 2.5000 2.7500

3.0000 3.2500 3.5000 3.7500 4.0000

2 214 212 234

3 314 312 334 4

3.125 3.438 3.688 3.938 4.188

2.094 2.375 2.625 2.875

1.594 1.719 1.844 1.969 3.0000 3.3750 3.7500 4.1250 4.5000 4.8750 5.2500 5.5250 6.0000

412 478 514 538 6 4.350 4.712 5.075 5.437 5.800

2.900 3.262 3.625 3.988

2.175 2.356 2.538 2.719

1.450 1.631 2.812 1.994

1.5000 1.6875 1.8750 2.0625 2.2500 2.4375 2.6250 2.8125

0.725 0.906 1.088 1.269

0.425 0.484 0.544 0.603

Min

0.7500 0.9375 1.1250 1.3125

0.4375 0.5000 0.5625 0.6250

3 338 334 418

7 16 1 2 9 16 3 8 3 4 15 16 1 38 3 1 16 1 12 1 11 16 1 78 1 2 16 1 24 7 2 16 2 38 2 13 16

Max (basic)

Width across ﬂats F

5.196 5.629 6.062 6.495 6.928

3.464 3.897 4.330 4.763

2.598 2.815 3.031 3.248

1.732 1.949 1.165 2.382

0.866 1.083 1.299 1.516

0.505 0.577 0.650 0.722

Max

4.959 5.372 5.786 6.198 6.612

3.306 3.719 4.133 4.546

2.480 2.686 2.893 3.100

1.653 1.859 2.066 2.273

0.826 1.033 1.240 1.447

0.484 0.552 0.620 0.687

Min

Width across corners G

178 2 218 5 216 212

7 132 138 117 32 111 16

1 3 132 3 132

3 32 13 64 13 64 3 32 3 16 23 64 13 32 35 64 30 64 11 16 23 32 23 32 13 16

Nom

1.935 2.064 2.193 2.385 2.576

1.263 1.423 1.583 1.744

0.974 0.038 1.134 1.198

0.627 0.718 0.813 0.878

0.323 0.403 0.483 0.563

0.163 0.211 0.243 0.291

Max

1.815 1.936 2.057 2.241 2.424

1.175 1.327 1.479 1.632

0.902 0.962 1.054 1.114

0.591 0.658 0.749 0.810

0.302 0.378 0.455 0.531

0.150 0.195 0.226 0.272

Min

Semiﬁnished height H

0.062 0.062 0.062 0.062 0.062

0.062 0.062 0.062 0.062

0.062 0.062 0.062 0.062

0.047 0.062 0.062 0.062

0.031 0.031 0.047 0.047

0.031 0.031 0.031 0.031

Max

0.047 0.047 0.047 0.047 0.047

0.047 0.047 0.047 0.047

0.047 0.047 0.047 0.047

0.031 0.047 0.047 0.047

0.016 0.016 0.031 0.031

0.010 0.010 0.010 0.016

Min

Radius of ﬁllet R

2 3 216 5 216 212 211 16

111 32 112 121 32 113 16

1 1 116 5 132 7 132

11 64 7 32 1 4 19 64 11 32 27 64 1 2 37 64 42 64 3 4 27 32 20 32

Max

2.060 2.251 2.380 2.572 2.764

1.388 1.548 1.708 1.869

1.036 1.100 1.196 1.260

0.700 0.780 0.876 0.940

0.364 0.444 0.524 0.604

0.188 0.235 0.268 0.316

Nom

1.815 1.936 2.057 2.241 2.424

1.175 1.327 1.479 1.632

0.902 0.962 1.054 1.114

0.591 0.658 0.749 0.810

0.302 0.378 0.455 0.531

0.150 0.195 0.226 0.272

Min

Regular height H1

0.188 0.188 0.188 0.188 0.188

0.125 0.188 0.188 0.188

0.125 0.125 0.125 0.125

0.062 0.125 0.125 0.125

0.031 0.062 0.062 0.062

0.031 0.031 0.031 0.031

Max

Radius of ﬁllet R1

Courtesy: Viegas, J. J., ‘‘Standards for Mechanical Elements’’, Horald A. Rothbart, Editor, Mechanical Design and Systems Handbook, McGraw-Hill Publishing company, New York, 1964.

1.5000 1.6250 1.7500 1.8750

112 138 134 178

1.063 1.188 1.313 1.469

0.530 0.675 0.800 0.938

0.5000 0.6250 0.7500 0.8750

1.0000 1.1250 1.2500 1.3750

0.280 0.342 0.405 0.468

0.2500 0.3125 0.3750 0.4375

1 114 114 138

3 4 3 16 3 8 7 16 1 2 5 8 3 4 7 8

Body diam, Max

18.42

Nominal size or basic major diam of thread

TABLE 18.25 Hexagon Bolts21

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

CHAPTER EIGHTEEN

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.43

TABLE 18-26 Regular unﬁnished square bolts21

Width across ﬂats F

Nominal size or basic major diam of thread

Body diam, max

1 4 5 16 3 8 7 16

0.2500 0.3125 0.3750 0.4375

0.280 0.342 0.405 0.468

3 8 1 2 9 16 5 8

1 2 5 8 3 4 7 8

0.5000 0.6250 0.7500 0.8750

0.530 0.675 0.800 0.938

1 118 114 138 112 158

1.0000 1.1250 1.2500 1.3750 1.5000 1.6250

1.063 1.188 1.313 1.469 1.594 1.719

Max (basic)

Width across corners G

Height H Nom

Max

Min

Radius of ﬁllet R, Max

0.498 0.665 0.747 0.828

11 64 13 64 1 4 19 64

0.188 0.220 0.268 0.316

0.156 0.186 0.232 0.278

0.031 0.031 0.031 0.031

1.061 1.326 1.591 1.856

0.995 1.244 1.494 1.742

21 64 27 64 1 2 19 32

0.348 0.444 0.524 0.620

0.308 0.400 0.476 0.568

0.031 0.062 0.062 0.062

2.121 2.386 2.652 2.917 3.182 3.447

1.991 2.239 2.489 2.738 2.986 3.235

21 32 3 4 27 32 29 32

0.684 0.780 0.876 0.940 1.036 1.132

0.628 0.720 0.812 0.872 0.964 1.056

0.062 0.125 0.125 0.125 0.125 0.125

Min

Max

Min

0.3750 0.5000 0.5625 0.6250

0.362 0.484 0.544 0.603

0.530 0.707 0.795 0.884

3 4 15 16 118 115 16

0.7500 0.9375 1.1250 1.3125

0.725 0.906 1.088 1.269

112 111 16 178 1 216 1 24 7 216

1.5000 1.6875 1.8750 2.0625 2.2500 2.4375

1.450 1.631 1.812 1.994 2.175 2.356

1 3 132

Note: Bolt is not ﬁnished on any surface. Minimum thread length shall be twice the diameter plus 14 in for length up to and including 6 in and twice the diameter plus 12 in for lengths over 6 in. Thread shall be coarse thread series class 2A.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.44

CHAPTER EIGHTEEN

TABLE 18-27 Finished hexagon bolts21

Nominal size or basic major diam of thread

Body diam, Width across ﬂats min (max F equal to nominal size) Max (basic) Min

1 4 5 16 3 8 7 16

0.2500 0.3125 0.3750 0.4375

0.2450 0.3065 0.3690 0.4305

7 16 1 2 9 16 5 8

0.4375 0.5000 0.5625 0.6250

1 2 9 16 5 8 3 4 7 8

0.5000 0.5625 0.6250 0.7500 0.8750

0.4930 0.5545 0.6170 0.7410 0.8660

3 4 13 16 15 16 118 5 116

1 118 114 138

1.0000 1.1250 1.2500 1.3750

0.9900 1.1140 1.2390 1.3630

112 158 134 178

1.5000 1.6250 1.7500 1.8750

2 214 212 234 3

2.0000 2.2500 2.5000 2.7500 3.0000

Width across corners G Max

Min

0.428 0.489 0.551 0.612

0.505 0.577 0.650 0.722

0.7500 0.8125 0.9375 1.1250 1.3125

0.736 0.798 0.922 1.100 1.285

112 111 16 178 1 216

1.5000 1.6875 1.8750 2.0625

1.4880 1.6130 1.7380 1.8630

214 1 216 5 28 213 16

1.9880 2.2380 2.4880 2.7380 2.9880

3 338 334 418 412

Height H

Radius of ﬁllet R

Nom

Max

Min

Max

Min

0.488 0.557 0.628 0.698

5 32 13 64 15 64 9 32

0.163 0.211 0.243 0.291

0.150 0.195 0.226 0.272

0.023 0.023 0.023 0.023

0.009 0.009 0.009 0.009

0.866 0.938 1.083 1.299 1.516

0.840 0.910 1.051 1.254 1.465

5 16 23 64 25 64 15 32 35 64

0.323 0.371 0.403 0.483 0.563

0.302 0.348 0.378 0.455 0.531

0.023 0.041 0.041 0.041 0.062

0.009 0.021 0.021 0.021 0.047

1.469 1.631 1.812 1.994

1.732 1.949 1.165 2.382

1.675 1.859 1.066 2.273

39 64 11 16 25 32 27 32

0.627 0.718 0.813 0.878

0.591 0.658 0.749 0.810

0.062 0.125 0.125 0.125

0.047 0.110 0.110 0.110

2.2500 2.4275 2.6250 2.8125

2.175 2.356 2.538 2.719

2.598 2.815 2.031 2.248

2.480 2.686 2.893 3.100

15 16

1 3 132 5 132

0.974 1.038 1.134 1.198

0.902 0.962 1.054 1.114

0.125 0.125 0.125 0.125

0.110 0.110 0.110 0.110

3.0000 3.3750 3.7500 4.1250 4.5000

2.900 3.262 3.625 3.988 4.350

3.464 3.897 4.330 4.763 5.196

3.306 3.719 4.133 4.546 4.959

7 132 138 117 32 111 16 178

1.263 1.423 1.583 1.744 1.935

1.175 1.327 1.479 1.632 1.815

0.125 0.188 0.188 0.188 0.188

0.110 0.173 0.173 0.173 0.173

Note: Bold type indicates uniﬁed thread.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.45

TABLE 18-28 Regular square nuts21

Width across ﬂats F

Nominal size or basic major diam of thread

Max (basic)

Width across corners G Min

Max

Min

Thickness H Nom

Max

Min

1 4 5 16 3 8 7 16

0.2500 0.3125 0.3750 0.4375

7 16 9 16 3 8 3 4

0.4375 0.5625 0.6250 0.7500

0.425 0.547 0.606 0.728

0.619 0.795 0.884 1.061

0.584 0.751 0.832 1.000

7 32 17 64 31 64 3 8

0.235 0.283 0.346 0.394

0.203 0.249 0.310 0.356

1 2 5 8 3 4 7 8

0.5000 0.6350 0.7500 0.8750

13 16

1 118 5 116

0.8125 1.0000 1.1250 1.3125

0.788 0.969 1.088 1.269

1.149 1.414 1.591 1.856

1.082 1.330 1.494 1.742

7 16 35 64 21 32 49 64

0.458 0.569 0.680 0.792

0.418 0.525 0.632 0.740

1 118 114 138 112 158

1.0000 1.1250 1.2500 1.3750 1.5000 1.6250

112 111 16 178 1 216 214 7 216

1.5000 1.6875 1.8750 2.0625 2.2500 2.4375

1.450 1.631 1.812 1.994 2.175 2.356

2.121 2.386 2.652 2.917 3.182 3.447

1.991 2.239 2.489 2.738 2.986 3.235

7 8

0.903 1.030 1.126 1.237 1.348 1.460

0.847 0.970 1.062 1.169 1.276 1.384

1 3 132 113 64 5 116 27 164

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.46

CHAPTER EIGHTEEN

TABLE 18-29 Hexagon and hexagon jam nuts21

Nominal size or basic major diam of thread

Width across ﬂats F Max (basic)

Width across corners G

Min

Max

Min

Finished and regular semi-ﬁnished hexagon nuts thickness H Nom

Max

Min

Finished and regular semi-ﬁnished jam nuts thickness H Nom

Max

Min

1 4 5 16 3 8 7 16

0.2500 0.3125 0.3750 0.4375

7 16 1 2 9 16 11 16

0.4375 0.5000 0.5625 0.6875

0.428 0.489 0.551 0.675

0.505 0.577 0.650 0.794

0.488 0.557 0.628 0.768

7 32 17 64 21 64 3 8

0.226 0.273 0.337 0.385

0.212 0.258 0.320 0.365

5 32 3 16 7 32 1 4

0.163 0.195 0.227 0.260

0.150 0.180 0.210 0.240

1 2 9 16 5 8 3 4 7 8

0.5000 0.5625 0.6250 0.7500 0.8750

3 4 7 8 15 16 118 5 116

0.7500 0.8750 0.9375 1.1250 1.3125

0.736 0.861 0.922 1.088 1.269

0.866 1.010 1.083 1.299 1.516

0.840 0.982 1.051 1.240 1.447

7 16 31 64 35 64 41 64 3 4

0.448 0.496 0.559 0.665 0.776

0.427 0.473 0.535 0.617 0.724

5 16 6 16 3 8 27 64 31 64

0.323 0.324 0.387 0.446 0.510

0.302 0.301 0.363 0.398 0.458

1 118 114 138

1.0000 1.1250 1.2500 1.3750

112 111 16 178 1 216

1.5000 1.6875 1.8750 2.0625

1.459 1.631 1.812 1.994

1.732 1.949 2.165 2.382

1.653 1.859 2.066 2.273

55 64 31 32 1 116 111 64

0.887 0.999 0.094 1.206

0.831 0.939 1.030 1.138

35 64 39 64 23 32 25 32

0.575 0.639 0.751 0.815

0.519 0.579 0.687 0.747

112 158 134 178

1.5000 1.6250 1.7500 1.8750

214 7 216 5 28 213 16

2.2500 2.4375 2.6250 2.8125

2.175 2.356 2.538 2.719

2.598 2.815 3.031 3.248

2.480 2.686 2.893 3.100

9 132 25 164 112 139 64

1.317 1.429 1.540 1.651

1.245 1.353 1.460 1.567

27 32 29 32 31 32 1 132

0.880 0.944 1.009 1.073

0.808 0.868 0.929 0.989

2 214 212 234 3

2.0000 2.2500 2.5000 2.7500 3.0000

3 338 334 418 412

3.0000 3.3750 3.7500 4.1250 4.5000

2.900 3.262 3.625 3.988 4.350

3.464 3.897 4.330 4.763 5.196

3.306 3.719 4.133 4.546 4.959

123 32 159 64 9 264 23 264 237 64

1.763 1.970 2.193 2.415 2.638

1.675 1.874 2.089 2.303 2.518

3 132 113 64 129 64 137 64 145 64

1.138 1.251 1.505 1.634 1.763

1.050 1.155 1.401 1.522 1.643

Note: Bold type indicates uniﬁed threads.

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18.47

TABLE 18-30 Finished hexagon slotted nuts21

Nominal size or basic major diam of thread

Width across ﬂats F Max (basic)

Width across corners G

Min

Max

Min

Thickness H

Slot

Nom

Max

Min

Width, S

Depth, T

1 4 5 16 3 8 7 16

0.2500 0.3125 0.3750 0.4375

7 16 1 2 9 16 11 16

0.4375 0.5000 0.5625 0.6875

0.428 0.489 0.551 0.675

0.505 0.577 0.650 0.794

0.488 0.557 0.628 0.768

7 32 17 64 21 64 3 8

0.226 0.273 0.337 0.385

0.212 0.258 0.320 0.365

0.078 0.094 0.125 0.125

0.094 0.094 0.125 0.156

1 2 9 16 5 8 3 4 7 8

0.5000 0.5625 0.6250 0.7500 0.8750

3 4 7 8 15 16 118 5 116

0.7500 0.8750 0.9375 1.1250 1.3125

0.736 0.861 0.922 1.088 1.269

0.866 1.010 1.083 1.299 1.516

0.840 0.982 1.051 1.240 1.447

7 16 31 64 35 64 41 64 3 4

0.448 0.496 0.559 0.665 0.776

0.427 0.473 0.535 0.617 0.724

0.156 0.156 0.156 0.188 0.188

0.156 0.188 0.219 0.250 0.250

1 118 114 138

1.0000 1.1250 1.2500 1.3750

112 111 16 178 1 216

1.5000 1.6875 1.8750 2.0625

1.450 1.631 1.812 1.994

1.732 1.949 2.165 2.382

1.653 1.859 2.066 2.273

55 64 31 32 1 116 11 164

0.887 0.999 1.094 1.206

0.831 0.939 1.030 1.138

0.250 0.250 0.312 0.312

0.281 0.344 0.375 0.375

112 138 134 178

1.5000 1.6250 1.7500 1.8750

214 7 216 5 28 213 16

2.2500 2.4375 2.6250 2.8125

2.175 2.356 2.538 2.719

2.598 2.815 2.031 3.248

2.480 2.686 2.893 3.100

9 132 125 64 112 139 64

1.317 1.429 1.540 1.651

1.245 1.353 1.460 1.567

0.375 0.375 0.438 0.438

0.438 0.438 0.500 0.562

2 214 212 234 3

2.0000 2.2500 2.5000 2.7500 3.0000

3 338 334 418 412

3.0000 3.3750 3.7500 4.1250 4.5000

2.900 3.262 3.625 3.988 4.350

3.464 3.897 4.330 4.763 5.196

3.306 3.719 4.133 4.546 4.959

123 32 159 64 9 264 223 64 237 64

1.763 1.970 2.193 2.415 2.638

1.675 1.874 2.089 2.303 2.518

0.438 0.438 0.562 0.562 0.625

0.562 0.562 0.688 0.688 0.750

Note: Bold type indicates uniﬁed threads.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.48

CHAPTER EIGHTEEN

TABLE 18-31 Regular hexagon and hexagon jam nuts

Nominal size or basic major diam of thread

Width across ﬂats F Max (basic)

Width across corners G

Min

Max

Min

Thickness regular jam nuts H

Thickness regular nuts H Nom

Max

Min

Nom

Max

Min

1 4 5 16 3 8 7 16

0.2500 0.3125 0.3750 0.4375

7 16 9 16 5 8 3 4

0.4375 0.5625 0.6250 0.7500

0.425 0.547 0.606 0.728

0.505 0.650 0.722 0.866

0.484 0.624 0.691 0.830

7 32 17 64 21 64 3 8

0.235 0.283 0.346 0.394

0.203 0.249 0.310 0.356

5 32 3 16 7 32 1 4

0.172 0.204 0.237 0.260

0.140 0.170 0.201 0.231

1 2 9 16 5 8 3 4 7 8

0.5000 0.5625 0.6250 0.7500 0.8750

13 16 7 8

1 118 158

0.8125 0.8750 1.0000 1.1250 1.3125

0.788 0.847 0.969 1.088 1.269

0.938 1.010 1.155 1.299 1.516

0.898 0.966 1.104 1.240 1.447

7 16 1 2 35 64 21 32 49 64

0.458 0.521 0.569 0.680 0.792

0.418 0.479 0.525 0.632 0.740

5 16 11 32 3 8 7 16 1 2

0.332 0.365 0.397 0.462 0.526

0.292 0.323 0.353 0.414 0.474

1.0000 1.1250 1.2500 1.3750 1.5000

112 111 16 178 1 216 1 24

1.5000 1.6875 1.8750 2.0625 2.2500

1.450 1.631 1.812 1.994 2.175

1.732 1.949 2.165 2.382 2.598

1.653 1.859 2.066 2.273 2.480

7 8

0.903 1.030 1.126 1.237 1.348

0.847 0.970 1.062 1.169 1.276

9 16 5 8 3 4 13 16 7 8

0.590 0.655 0.782 0.846 0.911

0.534 0.595 0.718 0.778 0.839

1 118 114 158 112

1 0 132 13 164 3 116

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.49

TABLE 18-32 Carriage bolts

Countersunk

Nominal diam of bolt D

Diam of head max, A

No. 10

0.520 0.645 0.770 0.895 1.020 1.145 1.400 1.650

1 4 3 16 3 8 7 16 1 2 5 8 3 4

Fin neck

Feed thickness F

Depth of square and countersink max, E

Width of square, max, B

Diam of head, max, A

Height of head, max, H

Depth of ﬁns, max, P

Distance across ﬁns, max, W

Thickness of ﬁns, max, M

0.016 0.016 0.031 0.031 0.031 0.031 0.031 0.047

0.250 0.312 0.375 0.437 0.500 0.562 0.687 0.812

0.199 0.260 0.324 0.388 0.452 0.515 0.642 0.768

0.469 0.594 0.719 0.844 0.969 1.094

0.114 0.145 0.176 0.208 0.239 0.270

0.088 0.104 0.135 0.151 0.182 0.198

0.395 0.458 0.551 0.645 0.739 0.833

0.098 0.114 0.145 0.161 0.192 0.208

Square neck Ribbed neck Nominal diam of bolt D

Diam of head, max, A

Height of head, max, H

Depth of square, max, P

Width of square, max, B

No. 10

0.469 0.594 0.719 0.844 0.969 1.094 1.344 1.594 1.844 2.094

0.114 0.145 0.176 0.208 0.239 0.270 0.344 0.406 0.469 0.531

0.125 0.156 0.187 0.219 0.250 0.281 0.344 0.406 0.469 0.531

0.199 0.260 0.324 0.388 0.452 0.515 0.642 0.768 0.895 1.022

1 4 5 16 3 8 7 16 1 2 5 8 5 8 7 8

1

Ribs below head P

Length of ribs Q

L 78

L1

L 78

L ¼ 1, L ¼ 1 18

L 1 14

Number of ribs

0.031 0.031 0.031 0.031 0.031 0.031 0.094 0.094

0.063 0.063 0.063 0.063 0.063 0.063 0.094 0.094

0.188 0.188 0.188 0.188 0.188 0.188 0.188 0.188

0.313 0.313 0.313 0.313 0.313 0.313 0.313 0.313

0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500

9 10 12 12 14 16 19 22

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.50

CHAPTER EIGHTEEN

TABLE 18-33 Countersunk, Buttonhead, and Step bolts

Countersunk bolt Nominal diam of bolt D

Buttonhead

Step bolt

Head diam, max, A

Head depth H

Head diam, max, A

Head height, max, H

Head diam, max, A

Head height, max, H

Depth of square, max, P

Width of square, max, B

24 20 18 16 14

— 0.493 0.618 0.740 0.803

— 0.140 0.176 0.210 0.210

0.469 0.594 0.719 0.844 0.969

0.114 0.145 0.176 0.208 0.239

0.656 0.844 1.031 1.219 1.406

0.114 0.145 0.176 0.208 0.239

0.125 0.156 0.187 0.219 0.250

0.199 0.260 0.324 0.388 0.452

0.935 1.169 1.402 1.637 1.869

0.250 0.313 0.375 0.438 0.500

1.094 1.344 1.594 1.844 2.094

0.270 0.344 0.406 0.469 0.531

1.594

0.270

0.281

0.515

1

13 11 10 9 8

118 114 138 112

7 7 6 6

2.104 2.337 2.571 2.804

0.563 0.625 0.688 0.750

No. 10 1 4 5 16 3 8 7 16 1 2 5 8 3 4 7 8

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18.51

TABLE 18-34 Machine-screw heads

Flat head

Nominal size No. 0 No. 1 No. 2 No. 3 No. 4 No. 5 No. 6 No. 8 No. 10 No. 12 1 4 5 16 3 8 7 16 1 2 9 16 5 8 3 4

Max diam D

Head diam, max, A

Height of head, max, H

0.060 0.073 0.086 0.099 0.112 0.125 0.138 0.164 0.190 0.216 0.250 0.3125 0.375 0.4375 0.500 0.5625 0.625 0.750

0.119 0.146 0.172 0.199 0.225 0.252 0.279 0.332 0.385 0.438 0.507 0.635 0.762 0.812 0.875 1.000 1.125 1.375

0.035 0.043 0.051 0.059 0.067 0.075 0.083 0.100 0.116 0.132 0.153 0.191 0.230 0.223 0.223 0.260 0.298 0.372

Round head

Width of slot, min, J

Depth of slot, min, T

0.016 0.019 0.023 0.027 0.031 0.035 0.039 0.045 0.050 0.056 0.064 0.072 0.081 0.081 0.091 0.102 0.116 0.131

0.010 0.012 0.015 0.017 0.020 0.022 0.024 0.029 0.034 0.039 0.046 0.058 0.070 0.066 0.065 0.077 0.088 0.111

Total height of head, max, O

Head diam, max, A

Height of head, max, H

Width of slot, min, J

Depth of slot, min, T

0.113 0.138 0.162 0.137 0.211 0.236 0.260 0.309 0.359 0.408 0.472 0.590 0.708 0.750 0.813 0.938 1.000 1.250

0.053 0.061 0.069 0.078 0.086 0.095 0.103 0.120 0.137 0.153 0.175 0.216 0.256 0.328 0.355 0.410 0.438 0.547

0.016 0.019 0.023 0.027 0.031 0.035 0.039 0.045 0.050 0.056 0.064 0.072 0.081 0.081 0.091 0.102 0.116 0.131

0.029 0.033 0.037 0.040 0.044 0.047 0.051 0.058 0.065 0.072 0.082 0.099 0.117 0.148 0.159 0.183 0.195 0.242

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Total height of head, max, O

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.52

CHAPTER EIGHTEEN

TABLE 18-34 Machine-screw heads (Cont.) Oval head

Nominal Size No. 0 No. 1 No. 2 No. 3 No. 4 No. 5 No. 6 No. 8 No. 10 No. 12 1 4 5 16 3 8 7 16 1 2 9 16 5 8 3 4

Max diam D

Head diam, max, A

Height of head, max, H

0.060 0.073 0.086 0.099 0.112 0.125 0.138 0.164 0.190 0.216 0.250 0.3125 0.375 0.4375 0.500 0.5625 0.625 0.750

0.119 0.146 0.172 0.199 0.225 0.252 0.279 0.332 0.385 0.438 0.507 0.635 0.762 0.812 0.875 1.000 1.125 1.375

0.035 0.043 0.051 0.059 0.067 0.075 0.083 0.100 0.116 0.132 0.153 0.191 0.230 0.223 0.223 0.260 0.298 0.372

Fillister head

Width of slot, min, J

Depth of slot, min, T

Total height of head, max, O

0.016 0.019 0.023 0.027 0.031 0.035 0.039 0.045 0.050 0.056 0.064 0.072 0.081 0.081 0.091 0.102 0.116 0.131

0.025 0.031 0.037 0.043 0.049 0.055 0.060 0.072 0.084 0.096 0.112 0.141 0.170 0.174 0.176 0.207 0.235 0.293

0.056 0.068 0.080 0.092 0.104 0.116 0.128 0.152 0.176 0.200 0.232 0.290 0.347 0.345 0.354 0.410 0.467 0.578

Head diam, max, A

Height of head, max, H

Width of slot, min, J

Depth of slot, min, T

Total height of head, max, O

0.098 0.118 0.140 0.161 0.183 0.205 0.226 0.270 0.313 0.357 0.414 0.518 0.622 0.625 0.750 0.812 0.875 1.000

0.045 0.053 0.062 0.070 0.079 0.088 0.096 0.113 0.130 0.148 0.170 0.211 0.253 0.265 0.297 0.336 0.375 0.441

0.016 0.019 0.023 0.027 0.031 0.035 0.039 0.045 0.050 0.056 0.064 0.072 0.081 0.081 0.091 0.102 0.116 0.131

0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.054 0.064 0.074 0.087 0.110 0.133 0.135 0.151 0.172 0.193 0.226

0.059 0.071 0.083 0.095 0.107 0.120 0.132 0.156 0.180 0.205 0.237 0.295 0.355 0.368 0.412 0.466 0.521 0.612

Note: Edges of head on ﬂat- and oval-head machine screws may be rounded. Radius of ﬁllet at base of ﬂat- and oval-head machine screws shall not exceed twice the pitch of the screw thread. Radius of ﬁllet at base of round- and ﬁllister-head machine screws shall not exceed one-half the pitch of the screw thread. All four types of screws in this table may be furnished with cross-recessed heads. Fillister-head machine screws in sizes No. 2 to 38 in, inclusive, may be furnished with a drilled hole through the head along a diameter at right angles to the slot but not breaking through the slot.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.53

TABLE 18-35 Machine screw heads—pan, hexagon, truss, and 1008 Flat heads

Pan head

Nominal size No. 2 No. 3 No. 4 No. 5 No. 6 No. 8 No. 10 No. 12 1 4 5 16 3 8

Max diam D

Head diam, max, A

Height of slotted head, max, H

Width of slot, min, J

Depth of slot, min, T

0.086 0.099 0.112 0.125 0.138 0.164 0.190 0.216 0.250 0.3125 0.375

0.167 0.193 0.219 0.245 0.270 0.322 0.373 0.425 0.492 0.615 0.740

0.053 0.060 0.068 0.075 0.082 0.096 0.110 0.125 0.144 0.178 0.212

0.023 0.027 0.031 0.035 0.039 0.045 0.050 0.056 0.064 0.072 0.081

0.023 0.027 0.030 0.032 0.038 0.043 0.050 0.060 0.070 0.092 0.113

Hexagon head

Radius R

Height of recessed head max, O

Head diam, max, A

Height of head, max, H

Width of slot, min, J

Depth of slot, min, T

0.035 0.037 0.042 0.044 0.046 0.052 0.061 0.078 0.087 0.099 0.143

0.062 0.071 0.080 0.089 0.097 0.115 0.133 0.151 0.175 0.218 0.261

0.125 0.187 0.187 0.187 0.250 0.250 0.312 0.312 0.375 0.500 0.562

0.050 0.055 0.060 0.070 0.080 0.110 0.120 0.155 0.190 0.230 0.295

0.031 0.035 0.039 0.045 0.050 0.056 0.064 0.072 0.081

0.025 0.030 0.033 0.052 0.057 0.077 0.083 0.100 0.131

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.54

CHAPTER EIGHTEEN

TABLE 18-35 Machine screw heads—pan, hexagon, truss, and 1008 Flat heads (Cont.) Truss head

Nominal size No. 2 No. 3 No. 4 No. 5 No. 6 No. 8 No. 10 No. 12 1 4 5 16 3 8 7 16 1 2 9 16 5 8 3 4

1008 ﬂat head

Max diam D

Head diam, max, A

Height of slotted head, max, H

Width of slot, min, J

Depth of slot, min, T

Radius R

0.086 0.099 0.112 0.125 0.138 0.164 0.190 0.216 0.250 0.3125 0.375 0.4375 0.500 0.5625 0.625 0.750

0.194 0.226 0.257 0.289 0.321 0.384 0.448 0.511 0.573 0.698 0.823 0.948 1.073 1.198 1.323 1.573

0.053 0.061 0.069 0.078 0.086 0.102 0.118 0.134 0.150 0.183 0.215 0.248 0.280 0.312 0.345 0.410

0.023 0.027 0.031 0.035 0.039 0.045 0.050 0.056 0.064 0.072 0.081 0.081 0.091 0.102 0.116 0.131

0.022 0.026 0.030 0.034 0.037 0.045 0.053 0.061 0.070 0.085 0.100 0.116 0.131 0.146 0.162 0.182

0.129 0.151 0.169 0.191 0.211 0.254 0.283 0.336 0.375 0.457 0.538 0.619 0.701 0.783 0.863 1.024

Height of recessed head max, O

Head diam, max, A

Height of head, max, H

Width of slot, min, J

Depth of slot, min, T

0.225

0.048

0.031

0.017

0.279 0.332 0.385

0.060 0.072 0.083

0.039 0.045 0.050

0.022 0.027 0.031

0.507 0.635 0.762

0.110 0.138 0.165

0.064 0.072 0.081

0.042 0.053 0.064

Note: Radius of ﬁllet at base of truss- and pan-head machine screws shall not exceed one-half the pitch of the screw thread. Truss-, pan-, and 1008 ﬂat-head machine screws may be furnished with cross-recessed heads. Hexagon-head machine screws are usually not slotted; the slot is optional. Also optional is an upset-head type for hexagon-head machine screws of sizes 4, 5, 8, 12, and 14 in.

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18.55

TABLE 18-36 Machine-screw heads—binding head

Nominal size No. 2 No. 3 No. 4 No. 5 No. 6 No. 8 No. 10 No. 12 1 4 5 16 3 8

Max diam D

Head diam, max, A

Total height of head, max, O

Width of slot, min, J

Depth of slot, min, T

Height of oval, max, F

Diam of undercut,a min, U

Depth of undercut, min, X

0.086 0.099 0.112 0.125 0.138 0.164 0.190 0.216 0.250 0.3125 0.375

0.181 0.208 0.235 0.263 0.290 0.344 0.399 0.454 0.513 0.641 0.769

0.046 0.054 0.063 0.071 0.080 0.097 0.114 0.130 0.153 0.193 0.234

0.023 0.027 0.031 0.035 0.039 0.045 0.050 0.056 0.064 0.072 0.081

0.024 0.029 0.034 0.039 0.044 0.054 0.064 0.074 0.088 0.112 0.136

0.018 0.022 0.025 0.029 0.032 0.039 0.045 0.052 0.061 0.077 0.094

0.124 0.143 0.161 0.180 0.199 0.236 0.274 0.311 0.360 0.450 0.540

0.005 0.006 0.007 0.009 0.010 0.012 0.015 0.018 0.021 0.027 0.034

a

Use of undercut is optional. Note: Binding-head machine screws may be furnished with cross-recessed heads.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.56

CHAPTER EIGHTEEN

TABLE 18-37 Slotted-head cap screws28

Fillister head Nominal size (body diam, max) D

Flat head

Round head

Width of slot min, J

Head diam, max, A

Height of head, max, H

Total height of head, max, O

Depth of slot, min, T

Head diam, max, A

Height of head average, H

Depth of slot, min, T

Head diam, max, A

Height of head, max, H

Depth of slot, min, T

1 4 3 16 3 8 7 16 1 2

0.064 0.072 0.081 0.081 0.091

0.375 0.437 0.562 0.625 0.750

0.172 0.203 0.250 0.297 0.328

0.216 0.253 0.314 0.368 0.412

0.077 0.090 0.113 0.133 0.148

0.500 0.625 0.750 0.8125 0.875

0.140 0.176 0.210 0.210 0.210

0.046 0.057 0.069 0.069 0.069

0.437 0.562 0.625 0.750 0.812

0.191 0.246 0.273 0.328 0.355

0.097 0.126 0.135 0.167 0.179

9 16 5 8 3 4 7 8

0.812 0.875 1.000 1.125 1.312

0.375 0.422 0.500 0.594 0.656

0.466 0.521 0.612 0.720 0.802

0.169 0.190 0.233 0.264 0.292

1.000 1.125 1.375 1.625 1.875

0.245 0.281 0.352 0.423 0.494

0.080 1.092 0.115 0.139 0.162

0.937 1.000 0.125

0.410 0.438 0.547

0.208 0.220 0.227

1

0.102 0.116 0.131 0.147 0.166

118 114 138 112

0.178 0.193 0.208 0.240

— — — —

— — — —

— — — —

— — — —

2.062 2.312 2.562 2.812

0.529 0.600 0.665 0.742

0.173 0.197 0.220 0.244

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18.57

TABLE 18-38 Socket-head cap screws Hexagona Socket width across ﬂats, min, J

Fluted socket

Number of ﬂutes

Socket diam minor, min, K

Socket diam major, min, M

Width of socket land, max, N

Nominal size

Body diam, max, D

Head diam, max, A

Head height max, H

Head side height, max, S

No. 0 No. 1 No. 2 No. 3 No. 4

0.060 0.073 0.0860 0.0990 0.1120

0.096 0.118 0.140 0.161 0.183

0.086 0.099 0.112

0.0803 0.0923 0.1043

1 16 5 64 5 64

6 6 6

0.063 0.080 0.080

0.073 0.097 0.097

0.016 0.022 0.022

No. 5 No. 6 No. 8 No. 10 No. 12

0.1250 0.1380 0.1640 0.1900 0.2160

0.205 0.226 0.270 5 16 11 32

0.125 0.138 0.164 0.190 0.216

0.1163 0.1284 0.1522 0.1765 0.2005

3 32 3 32 1 8 5 32 5 32

6 6 6 6 6

0.096 0.096 0.126 0.161 0.161

0.113 0.113 0.147 0.186 0.186

0.025 0.025 0.032 0.039 0.039

1 4 5 16 3 8 7 16 1 2

0.2500 0.3125 0.3750 0.4375 0.5000

3 8 7 16 9 16 5 8 1 4

1 4 5 16 3 8 7 16 1 2

0.2317 0.2894 0.3469 0.4046 0.4620

3 16 7 32 5 16 5 16 3 8

6 6 6 6 6

0.188 0.219 0.316 0.316 0.383

0.219 0.254 0.377 0.377 0.460

0.050 0.060 0.092 0.092 0.112

9 16 5 8 3 4 7 8

13 16 7 8

1 118 5 116

9 16 5 8 3 4 7 8

1

0.5625 0.6250 0.7500 0.8750 1.0000

1

0.5196 0.5771 0.6920 0.8069 0.9220

3 8 1 2 9 16 9 16 5 8

6 6 6 6 6

0.383 0.506 0.531 0.600 0.681

0.460 0.601 0.627 0.705 0.797

0.112 0.138 0.149 0.168 0.189

118 114 138 112

1.1250 1.2500 1.3750 1.5000

112 134 178 2

118 114 138 112

1.0372 1.1516 1.2675 1.3821

3 4 3 4 3 4

6 6 6 6

0.824 0.824 0.824 1.003

0.966 0.966 0.966 1.271

0.231 0.231 0.231 0.298

1

a

Maximum socket depth T should not exceed three-fourths of minimum head height H. Note: Head chamfer angle E is 28 to 328, the edge between ﬂat and chamfer being slightly rounded. Screw point chamfer angle 35 to 408, the chamfer extending to the bottom of the thread. Edge between ﬂat and chamfer is slightly rounded.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.58

CHAPTER EIGHTEEN

TABLE 18-39 Square-head set screws28

Width across ﬂats F Nominal size No. 10 No. 12 1 4 5 16 3 8 7 16 1 2 9 16 5 8 3 4 7 8

1 118 114 138 112

0.190 0.216 0.250 0.3125 0.3750 0.4375 0.500 0.5625 0.6250 0.750 0.875 1.000 1.125 1.250 1.376 1.500

Width across corners G

Height of head H

Diam of neck relief K

Radius of head X

Radius Width of neck of neck relief relief R U

Max

Min

Min

Nom

Max

Min

Max

Min

Nom

Max

Max

0.1875 0.216 0.250 0.3125 0.375 0.4375 0.500 0.5625 0.625 0.750 0.875 1.000 1.125 1.250 1.375 1.500

0.180 0.208 0.241 0.302 0.362 0.423 0.484 0.545 0.606 0.729 0.852 0.974 1.096 1.219 1.342 1.464

0.247 0.292 0.331 0.415 0.497 0.581 0.665 0.748 0.833 1.001 1.170 1.337 1.505 1.674 1.843 2.010

9 64 5 32 3 16 15 64 7 32 31 64 3 8 27 64 15 32 9 16 21 32 3 4 27 32 15 16 1 132 1 18

0.148 0.163 0.196 0.245 0.293 0.341 0.398 0.437 0.485 0.582 0.678 0.774 0.870 0.966 1.063 1.159

0.134 0.147 0.178 0.224 0.270 0.315 0.361 0.407 0.452 0.544 0.635 0.726 0.817 0.908 1.000 1.091

0.145 0.162 0.185 0.240 0.294 0.345 0.400 0.454 0.507 0.620 0.731 0.838 0.939 1.064 1.159 1.284

0.140 0.156 0.170 0.225 0.279 0.330 0.385 0.439 0.492 0.605 0.716 0.823 0.914 1.039 1.134 1.259

15 32 35 64 5 8 25 32 15 16 3 132 1 14 113 32 9 116 7 18 3 216 212 213 16 318 7 316 3 34

0.027 0.029 0.032 0.036 0.041 0.046 0.050 0.054 0.059 0.065 0.072 0.081 0.092 0.092 0.109 0.109

0.083 0.091 0.100 0.111 0.125 0.143 0.154 0.167 0.182 0.200 0.222 0.250 0.283 0.283 0.333 0.333

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18.59

TABLE 18-40 Square-head setscrew points28

Full-dog, half-dog, and pivot pointa

Nom

Max

Min

Oval (round) point radius J, nom

1 4 5 16 3 8

3 32 7 64 1 8 11 64 13 64

0.102 0.115 0.132 0.172 0.212

0.088 0.101 0.118 0.156 0.194

0.141 0.156 0.188 0.234 0.281

0.127 0.144 0.156 0.203 0.250

0.120 0.137 0.149 0.195 0.241

0.090 0.110 0.125 0.156 0.188

0.045 0.055 0.063 0.078 0.094

7 16 1 2 9 16 5 8 3 4

16 64 9 32 5 16 22 64 7 16

0.252 0.291 0.332 0.371 0.450

0.232 0.270 0.309 0.347 0.425

0.328 0.375 0.422 0.469 0.563

0.297 0.344 0.391 0.469 0.563

0.287 0.334 0.379 0.456 0.549

0.219 0.250 0.281 0.313 0.375

0.109 0.125 0.140 0.156 0.188

7 8

1 118 114 138

33 64 19 32 43 64 3 4 53 64

0.530 0.609 0.689 0.767 0.848

0.502 0.579 0.655 0.733 0.808

0.656 0.750 0.844 0.938 1.031

0.656 0.750 0.844 0.938 1.031

0.642 0.734 0.826 0.920 1.011

0.438 0.500 0.562 0.625 0.688

0.219 0.250 0.281 0.312 0.344

112

29 32

0.926

0.886

1.125

1.125

1.105

0.750

0.375

Diam of cap and ﬂat points C Nominal size No. 10 No. 12

Min

Full dog and pivot Q

Half dog q

Diam P Max

a Pivot points are similar to full-dog point except that the point is rounded by a radius equal to J. Where usable length of thread is less than the nominal diameter, half-dog point shall be used. When length equals nominal diameter or less, Y ¼ 1188 28; when length exceeds nominal diameter, Y ¼ 908 28 Note: All dimensions are given in inches.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.60

CHAPTER EIGHTEEN

TABLE 18-41 Slotted headless setscrews28

Radius of headless crown I

Width of slot J

Depth of slot T

Ovalpoints radius R

Max

Min

Max

Min

Fill Q

Half q

0.125 0.138 0.164 0.190 0.216

0.125 0.138 0.164 0.190 0.216

0.023 0.025 0.029 0.032 0.036

0.031 0.035 0.041 0.048 0.054

0.094 0.109 0.125 0.141 0.156

0.067 0.074 0.087 0.102 0.115

0.057 0.064 0.076 0.088 0.101

0.083 0.092 0.109 0.127 0.144

0.078 0.087 0.103 0.120 0.137

0.060 0.070 0.080 0.090 0.110

0.030 0.035 0.040 0.045 0.055

1 4 5 16 3 8 7 16

0.250 0.3125 0.375 0.4375

0.250 0.313 0.375 0.438

0.045 0.051 0.064 0.072

0.063 0.078 0.094 0.109

0.188 0.234 0.281 0.328

0.132 0.172 0.212 0.252

0.118 0.156 0.194 0.232

0.156 0.203 0.250 0.297

0.149 0.195 0.241 0.287

0.125 0.156 0.188 0.219

0.063 0.078 0.094 0.109

1 2 9 16 5 8 3 4

0.500 0.5625 0.625 0.750

0.500 0.563 0.625 0.750

0.081 0.091 0.102 0.129

0.125 0.141 0.156 0.188

0.375 0.422 0.469 0.563

0.291 0.332 0.371 0.450

0.270 0.309 0.347 0.425

0.344 0.391 0.469 0.563

0.344 0.379 0.456 0.549

0.250 0.281 0.313 0.375

0.135 0.140 0.156 0.188

Nominal size D 5 6 8 10 12

Diam of cup and ﬂat points C

Diam of dog point P

Length of dog pointa

a

Where usable length thread is less than the nominal diameter, half-dog point shall be used. When L (length of screw) equals nominal diameter or less, Y ¼ 1188 28; when L exceeds nominal diameter, Y ¼ 908 28. Point angles ¼ 80 to 908; X ¼ 1188 58; Z ¼ 100 to 1108. Allowable eccentricity of dog point axis with respect to axis of screw shall not exceed 3% of nominal screw diameter with maximum 0.005 in. Note: All dimensions given in inches.

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0.087

0.102

0.115

No. 8

No. 10

No. 12

0.061

No. 4

0.067

0.054

No. 3

0.074

0.047

No. 2

No. 5

0.040

No. 1

No. 6

0.033

No. 0

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0.101

0.088

0.076

0.064

0.057

0.051

0.045

0.039

0.033

0.027

Cup- and Screw ﬂat-point size diameter nominal C diam D Max Min

3 32 7 64 1 8 9 64 5 32

0.084

1 16 5 64

0.055

3 64

Ovalpoint radius R

1 8 1 8 3 16 3 16 3 16

1 16 5 64 3 32 7 64 1 8

118 28 for these lengths and under

3 16 3 16 1 4 1 4 1 4

5 64 3 32 7 64 1 8 5 32

0.103 0.120

0.109 0.127

0.137

0.087

0.144

0.078

0.070

0.075

0.092

0.062

0.066

0.083

0.053

0.045

0.037

0.057

0.049

0.040

0.11

0.09

0.08

0.07

0.06

0.056

0.050

0.043

0.037

0.030

0.055

0.045

0.04

0.035

0.03

0.028

0.025

0.022

0.019

0.015

0.075

0.075

0.062

0.050

0.050

0.040

0.040

0.028

0.028

0.022

90 28 Dog point for these lengths Diam P Key and engagement† over Max Min Full Q Half q min

Cone-point angle Y

Fluted and hexagon socket types

TABLE 18-42 Fluted and hexagon socket-headless setscrews24

0.0947

0.0947

0.0791

0.0635

0.0635

0.051

0.051

0.0355

0.0355

0.0285

Max

1 16 1 16 5 64 3 32 3 32

0.050

0.050

0.035

0.035

0.028

Min

Socket width across ﬂats J

Hexagon type

0.098

0.098

0.082

0.056

0.053

0.051

0.038

0.038

0.026

0.026

Max

0.096

0.096

0.080

0.055

0.052

0.050

0.0375

0.0375

0.0255

0.0255

Min

Socket diam, minor, J

0.115

0.115

0.098

0.079

0.071

0.062

0.050

0.050

0.035

0.035

Max

0.113

0.113

0.097

0.078

0.070

0.061

0.049

0.049

0.034

0.034

Min

Socket diam, major, M

Fluted typea

0.025

0.025

0.022

0.023

0.022

0.014

0.017

0.017

0.012

0.012

Max

0.023

0.023

0.021

0.022

0.021

0.013

0.016

0.016

0.0115

0.0115

Min

Socket land width N

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.61

0.808 0.886

1.039

0.848 0.926

1.086

1.244

2

1.193

112

27 32 15 16 1 132 1 18 5 116

27 64 15 32 9 16 21 32 3 4

3 16 15 64 9 32 21 64 3 8

Ovalpoint radius R

2

114 138 112 134

118

1

9 16 5 8 3 4 7 8

1 4 5 16 3 8 7 16 1 2

118 28 for these lengths and under

214

2

112 158 134

114

118

1

5 8 3 4 7 8

5 16 3 8 7 16 1 2 9 16

112

27 32 15 16 1 132 1 18 5 116

25 64 15 32 9 16 21 32 3 4

5 32 13 64 1 4 19 64 11 32

1.474

1.289

1.011 1.105

0.920

0.826

0.734

0.642

0.456 0.549

0.379

0.334

0.287

0.241

0.149 0.195

1

9 16 5 8 11 16 3 4 7 8

9 32 5 16 3 8 7 16 1 2

1 8 3 32 3 16 7 32 1 4

1 2

9 16 5 16 11 32 3 8 7 16

5 64 3 32 3 16 7 32 1 4

1 16 5 64 3 32 7 64 1 8

0.800

0.800

0.500 0.600

0.500

0.450

0.450

0.400

0.250 0.300

0.200

0.200

0.175

0.150

0.100 0.125

90 28 Dog point for these lengths Diam P Key and engagementb over Max Min Full Q Half q min

1.0040

0.0040

0.6290 0.7540

0.6290

0.5655

0.5655

0.5030

0.3155 0.3780

0.2520

0.2520

0.2207

0.1895

0.1270 0.1582

Max

1

1

9 16 3 8 3 8 3 4

1 4 5 16 3 8 1 2 9 16

1 8 5 32 3 16 7 32 1 4

Min

Socket width across ﬂats J

Hexagon type

1.007

1.007

0.744 0.828

0.685

0.604

0.535

0.509

0.319 0.386

0.254

0.254

0.221

0.190

0.128 0.163

Max

1.003

1.003

0.740 0.824

0.681

0.600

0.531

0.506

0.316 0.383

0.252

0.252

0.219

0.188

0.126 0.161

Min

Socket diam, minor, J

1.275

1.275

0.869 0.970

0.801

0.709

0.631

0.604

0.380 0.463

0.298

0.298

0.256

0.221

0.149 0.188

Max

1.271

1.271

0.865 0.966

0.797

0.705

0.627

0.601

0.377 0.460

0.296

0.296

0.254

0.219

0.147 0.186

Min

Socket diam, major, M

Fluted typea

0.298

0.298

0.207 0.231

0.189

0.168

0.149

0.138

0.092 0.112

0.068

0.068

0.060

0.050

0.032 0.039

Max

0.294

0.294

0.203 0.227

0.185

0.164

0.145

0.134

0.089 0.109

0.066

0.066

0.058

0.048

0.030 0.037

Min

Socket land width N

b

The number of ﬂutes for setscrews Nos. 0, 1, 2, 3, 5, and 6 is four. The number of ﬂutes for Nos. 4, 8, and larger is six. These dimensions apply to cup- and ﬂat-point screws one diameter in length or longer. For screws shorter than one diameter in length, and for other types of points, socket to be as deep as practicable. Note: All dimensions are given in inches. Source: Courtesy: John J. Viegas, Standards for Mechanical Elements, Harold A. Rothbart, Editor-in-Chief; Mechanical Design and Systems Handbook, McGraw-Hill Publishing Company, New York, 1964.

a

0.733

0.767

114 138 112 134

0.655

0.689

118

0.502

0.579

0.530

0.609

0.347 0.425

0.270

0.291

0.371 0.450

0.232

0.252

0.309

0.194

0.212

0.332

0.118 0.156

0.132 0.172

1

9 16 5 8 3 4 7 8

1 4 5 16 3 8 7 16 1 2

Cup- and Screw ﬂat-point size diameter nominal C diam D Max Min

Cone-point angle Y

Fluted and hexagon socket types

TABLE 18-42 Fluted and hexagon socket-headless setscrews24 (Cont.)

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.62

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18.63

TABLE 18-43 American National Standard metric hex cap screws (ANSI B18.2.3.1M-1979, R1989)

Nominal screw diam, D and thread pitch

Body diam, Ds Max

Min

Width across ﬂats, S Max

Min

Width across corners, E Max

Min

Head height, K Max

Min

Wrenching height, K1 Min

Washer face thickness, C Max

Min

M5 0:8 M6 1 M8 1:25

5.00 6.00 8.00

4.82 5.82 7.78

8.00 10.00 13.00

7.78 9.78 12.73

9.24 11.55 15.01

8.79 11.05 14.38

3.65 4.15 5.50

3.35 3.85 5.10

2.4 2.8 3.7

0.5 0.5 0.6

0.2 0.2 0.3

M10 1:5

10.00

9.78

15.00

14.73

17.32

16.64

6.63

6.17

4.5

0.6

0.3

M10 1:5

10.00

9.78

16.00

15.73

18.48

17.77

6.63

6.17

4.5

0.6

0.3

M12 1:75 M14 2 M16 2 M20 2:5 M24 3

12.00 14.00 16.00 20.00 24.00

11.73 13.73 15.73 19.67 23.67

18.00 21.00 24.00 30.00 36.00

17.73 20.67 23.67 29.16 35.00

20.78 24.25 27.71 34.64 41.57

20.03 23.35 26.75 32.95 39.55

7.76 9.09 10.32 12.88 15.44

7.24 8.51 8.68 12.12 14.56

5.2 6.2 7.0 8.8 10.5

0.6 0.6 0.8 0.8 0.8

0.3 0.3 0.4 0.4 0.4

M30 3:5 M36 4 M42 4:5 M48 5 M56 5:5

30.00 36.00 42.00 48.00 56.00

29.67 35.61 41.38 47.38 55.26

46.00 55.00 65.00 75.00 85.00

45.00 53.80 62.90 72.60 82.20

53.12 63.51 75.06 86.60 98.15

50.85 60.79 71.71 83.76 93.71

15.48 23.38 26.97 31.07 36.20

17.92 21.62 25.03 28.93 33.80

13.1 15.8 18.2 21.0 24.5

0.8 0.8 1.0 1.0 1.0

0.4 0.4 0.5 0.5 0.5

M64 6 M72 6 M80 6 M90 6 M100 6

64.00 72.00 80.00 90.00 100.00

63.26 71.26 79.26 89.13 99.13

95.00 105.00 115.00 130.00 145.00

91.80 101.40 111.00 125.50 140.00

109.70 121.24 132.72 150.11 167.43

104.65 115.60 126.54 143.07 159.60

41.32 46.45 51.58 57.74 63.90

38.68 43.55 48.42 54.26 60.10

28.0 31.5 35.0 39.2 43.4

1.0 1.2 1.2 1.2 1.2

0.5 0.6 0.6 0.6 0.6

All dimensions are in millimeters. This size with width across ﬂats of 15 mm is not standard. Unless speciﬁcally ordered hex cap screws with 16 mm width across ﬂats will be furnished. † Transition thread length, X, includes the length of incomplete threads and tolerances gaging length and body length. It is intended for calculation purposes. ‡ Basic thread lengths, B, are the same as given in Table 18-47. For additional manufacturing and acceptance speciﬁcations, reference should be made to Standard. Courtesy: American National Standards Institution, New York, USA. (ANSI B18.2.1M-1979, R1989)

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.64

CHAPTER EIGHTEEN

TABLE 18-44 American National Standard metric formed hex screws (ANSI B18.2.3.2M-1979, R1989)

Nominal screw diam, D, and thread pitch

Max

Min

Max

Min

Max

Min

Max

M5 0:8 M6 1 M8 1:25 M10 1:5 M10 1:5 M12 1:75 M14 2 M16 2 M20 2:5 M24 3

5.00 5.00 8.00 10.00 10.00 12.00 14.00 16.00 20.00 24.00

4.82 5.82 7.78 9.78 9.78 11.73 13.73 15.73 19.67 23.67

8.00 10.00 13.00 15.00 16.00 18.00 21.00 24.00 30.00 36.00

7.64 9.64 12.57 14.57 15.57 17.57 20.16 23.16 29.16 35.00

9.24 11.55 15.01 17.32 18.48 20.78 24.25 27.71 34.64 41.57

8.56 10.80 14.08 16.32 17.43 19.68 22.58 25.94 32.66 39.20

3.65 4.15 5.50 6.63 6.63 7.76 9.09 10.32 12.88 15.44

Body diam, Ds

Width across ﬂats, S

Width across corners, E

Head height, K

Wrenching height, K1

Washer face thickness, C

Washer face diam, Dw

Min

Min

Max

Min

Min

3.35 3.85 5.10 6.17 6.17 7.24 8.51 9.68 12.12 14.56

2.4 2.0 3.7 4.5 4.5 5.2 6.2 7.0 8.8 10.5

0.5 0.5 0.6 0.6 0.6 0.6 0.6 0.8 0.8 0.8

0.2 0.2 0.3 0.3 0.3 0.3 0.3 0.4 0.4 0.4

6.9 8.9 11.6 13.6 14.6 16.6 19.6 22.5 27.7 32.2

All dimensions are in millimeters. This size with width across ﬂats of 15 mm is not standard. Unless speciﬁcally ordered M10 formed hex screws with 16 mm width across ﬂats will be furnished. † Transition thread length, X, includes the length of incomplete threads and tolerances on the grip gaging length and body length. It is intended for calculation purposes. ‡ Basic thread lengths, B are the same as given in Table 18-47. For additional manufacturing and acceptance speciﬁcations, reference should be made to the Standard. Courtesy: American National Standards Institution, New York, USA. (ANSI B18.2.3.2M-1979, R1989)

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.65

TABLE 18-45 American National Standard metric heavy hex screws (ANSI B18.2.3.3M-1979, R1989)

Nominal screw diam, D, and thread pitch

Max

Min

Max

Min

Max

Min

Max

M12 1:75 M14 2 M16 2 M20 2:5 M24 3 M30 3:5 M36 4

12.00 14.00 16.00 20.00 24.00 30.00 36.00

11.73 13.73 15.73 19.67 23.67 29.67 35.61

21.00 24.00 27.00 34.00 41.00 50.00 60.00

20.67 23.67 26.67 33.00 40.00 49.00 58.80

24.25 27.71 31.18 39.26 47.34 57.74 69.28

23.35 26.75 30.14 37.29 45.20 55.37 66.44

7.76 9.09 10.32 12.88 15.44 19.48 23.38

Body diam, Ds

Width across ﬂats, S

Width across corners, E

Head height, K

Wrenching height, K1

Washer face thickness, C

Washer face diam, Dw

Min

Min

Max

Min

Min

7.24 8.51 9.68 12.12 14.56 17.92 21.72

5.2 6.2 7.0 8.8 10.5 13.1 15.8

0.6 0.6 0.8 0.8 0.8 0.8 0.8

0.3 0.3 0.4 0.4 0.4 0.4 0.4

19.0 22.0 25.0 31.0 38.0 46.0 55.0

All dimensions are in millimeters Basic thread lengths, B, are the same as given in Table 18-47. Transition thread length, X, includes the length of incomplete threads and tolerances on the grip gaging length and body length. It is intended for calculation purposes. For additional manufacturing and acceptance speciﬁcations, reference should be made to the Standard.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.66

CHAPTER EIGHTEEN

TABLE 18-46 American National Standard metric hex ﬂange screws (ANSI/ASME B18.2.3.4M-1984)

Nominal screw diam, D, and thread pitch

Max

Min

Max

Min

Max

Min

Max

Min

Min

Max

Min

Max

M5 0:8 M6 1 M8 1:25 M10 1:5 M12 1:75 M14 2 M16 2

5.00 6.00 8.00 10.00 12.00 14.00 16.00

4.82 5.82 7.78 9.78 11.73 13.73 15.73

7.00 8.00 10.00 13.00 15.00 18.00 21.00

6.64 7.64 9.64 12.57 14.57 17.57 20.48

8.08 9.24 11.55 15.01 17.32 20.78 24.25

7.44 8.56 10.80 14.08 16.32 19.68 22.94

11.4 13.6 17.0 20.8 24.7 28.6 32.8

9.4 11.6 14.9 18.7 22.0 25.9 30.1

1.0 1.1 1.2 1.5 1.8 2.1 2.4

5.6 6.8 8.5 9.7 11.9 12.9 15.1

2.30 2.90 3.80 4.30 5.40 5.60 6.70

0.3 0.4 0.5 0.6 0.7 0.8 1.0

Body diam, Ds

Width across ﬂats, S

Width across corners, E

Flange diam, Dc

Bearing circle diam, Dw

Flange edge thickness C

Head height, K

Wrenching height, K1

All dimensions are in millimeters. Basic thread lengths, B, are the same as given in Table 18-47. Transition thread length, X, includes the length of incomplete threads and tolerances on grip gaging length and body length. This dimension is intended for calculation purposes only. For additional manufacturing and acceptance speciﬁcations, reference should be made to the Standard.

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Fillet radius, R

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.67

TABLE 18-47 Received diameter-length combinations for metric hex cap screws, formed hex screws, heavy hex screws, hex ﬂange screws and heavy hex ﬂange screws Diameter—Pitch Nominal Lengtha

M5 0.8

M6 1

M8 1.25

M10 1.5

M12 1.75

M14 2

M16 2

M20 2.5

M24 3

M30 3.5

M36 4

8 10 12 14 16 20

—

— —

— — — b b

— — — — b

— — — — b

— — — — — —

— — — — — —

— — — — — —

— — — — — —

— — — — — —

25 30 35 40 45 50

—

— —

— — —

— — — — —

(55) 60 (65) 70 (75) 80

— — — — — —

— — — —

(85) 90 100 110 120 130

— — — — — —

— — — — — —

— — — — — —

— — —

—

140 150 160 (170) 180 (190)

— — — — — —

— — — — — —

— — — — — —

— — — — — —

— — — — — —

— — — — —

— — —

200 220 240 260 280 300

— — — — — —

— — — — — —

— — — — — —

— — — — — —

— — — — — —

— — — — — —

— — — — — —

— — — — —

— — —

All dimensions are in millimeters. a Lengths in parentheses are not recommended. Recommended lengths of formed Hex Screws, Hex Flange Screws and Heavy Hex Flange Screws do not extend above 150 mm. Recommended lengths of Heavy Hex Screws do not extend below 20 mm. Standard sizes for government use. Recommended diameter-length combinations are indicated by the symbol . Screws with lengths above cross lines are threaded full length. b Does not apply to Hex Flange Screws and Heavy Hex Flange Screws. For available diameters of each type of screw, see respective dimensional table.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.68

CHAPTER EIGHTEEN

TABLE 18-48 American National Standard Metric Hex Bolts (ANSI B18.2.3.5M-1979, R1989)

Nominal bolt diam, D, and thread pitch M5 0:8 M6 1 M8 1:25

For bolt lengths (mm) Body diam, Ds Max

Min

Width across ﬂats, S Max

Min

Widths across corners, E Max

Min

Head height, K Max

Min

Wrenching height, K1 Min

<125

>125 and <200

>200

Basic thread length,b B

5.48 6.19 8.58

4.52 5.52 7.42

8.00 10.00 13.00

7.64 9.64 12.57

9.24 11.55 15.01

8.63 10.89 14.20

3.58 4.18 5.68

3.35 3.55 5.10

2.4 2.8 3.7

16 18 22

22 24 28

35 37 41

10.58

9.42

15.00

14.57

17.32

16.46

6.85

6.17

4.5

26

32

45

M10 1:5

10.58

9.42

16.00

15.57

18.48

17.59

6.85

6.17

4.5

26

32

45

M12 1:75 M14 2 M16 2 M20 2:5 M24 3

12.70 14.70 16.70 20.84 24.84

11.30 13.30 15.30 19.16 23.16

18.00 21.00 24.00 30.00 36.00

17.57 20.16 23.16 29.16 35.00

20.78 24.25 27.71 34.64 41.57

19.85 7.95 22.78 9.25 26.17 10.75 32.95 13.40 39.55 15.90

7.24 8.51 9.68 12.12 14.56

5.2 6.2 7.0 8.8 10.5

30 34 38 46 54

36 49 44 52 60

49 53 57 65 73

M30 3:5 M36 4 M42 4:5 M48 5 M56 5:5

30.84 37.00 43.00 49.00 57.20

29.16 35.00 41.00 47.00 54.80

46.00 55.00 65.00 75.00 85.00

45.00 53.80 62.90 72.60 82.20

53.12 63.51 75.06 86.60 98.15

50.55 60.79 71.71 82.76 93.71

19.75 23.55 27.05 31.07 36.20

17.92 21.72 25.03 28.93 33.80

13.1 15.8 18.2 21.0 24.5

66 78 90 102 —

72 84 96 108 124

85 97 109 121 137

M64 6 M72 6 M80 6 M90 6 M100 6

65.52 73.84 82.16 92.48 102.80

63.80 70.80 78.80 88.60 98.60

95.00 105.00 115.00 130.00 145.00

91.80 101.40 111.00 125.50 140.00

109.70 121.24 132.79 150.11 167.43

104.65 115.60 126.54 143.07 159.60

41.32 46.45 51.58 57.74 63.90

38.68 43.55 48.42 54.26 60.10

28.0 31.5 35.0 39.2 43.4

— — — — —

140 156 172 192 212

153 169 185 205 225

a

M10 1:5

All dimensions are in millimeters. a This size with width across ﬂats of 15 mm is not standard. Unless speciﬁcally ordered, M10 set bolts with 16 mmm width across ﬂats will be furnished. b Basic thread length, B, is a reference dimension. For additional manufacturing and acceptance speciﬁcations, reference should be made to the Standard.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.69

TABLE 18-49 American National Standard Heavy Hex Bolts (ANSI B18.2.3.6M-1979, R1989)

Nominal diam, D and thread pitch

Body diam, Ds

Width across ﬂats, S

Width across corners, E

Head height, K

Max

Min

Max

Min

Max

Min

Max

Min

Min

M12 1:75 M14 2 M16 2 M20 2:5 M24 3 M30 3:5 M36 4

12.70 14.70 16.70 20.84 24.84 30.84 37.00

11.30 13.30 15.30 19.16 23.16 29.16 35.00

21.00 24.00 27.00 34.00 41.00 50.00 60.00

29.16 23.16 26.16 33.00 40.00 49.00 58.80

24.25 27.71 31.18 39.26 47.34 57.74 69.28

22.78 26.17 29.56 37.29 45.20 55.37 66.44

7.95 9.25 10.75 13.40 15.90 19.75 23.55

7.24 8.51 9.68 12.12 14.56 17.92 21.72

5.2 6.2 7.2 8.2 10.5 13.1 15.1

All dimensions are in millimeters. a Basic thread lengths, B, are the same as given in Table 18-47. For additional manufacturing and acceptance speciﬁcations, reference should be made to the Standard.

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Wrenching height, K1

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.70

CHAPTER EIGHTEEN

TABLE 18-50 Recommended clearance holes for metric hex screws and boltsa Clearance hole diam, basic, Dh

Nominal diam, D and thread pitch

Close

Normal, preferred

M5 0:8 M6 1 M8 1:25 M10 1:5 M12 1:75 M14 2 M16 2 M20 2:5 M22 2:5b M24 3 M27 3b

5.3 6.4 8.4 10.5 13.0 15.0 17.0 21.0 23.0 25.0 28.0

5.5 6.6 9.0 11.0 13.5 15.5 17.5 22.0 24.0 26.0 30.0

Clearance hole diam, basic, Dh

Loose

Nominal diam, D and thread pitch

Close

Normal, preferred

Loose

5.8 7.0 10.0 12.0 14.5 16.5 18.5 24.0 26.0 28.0 32.0

M30 3:5 M36 4 M42 4:5 M48 5 M56 5:5 M64 6 M72 6 M80 6 M90 6 M100 6 —

31.0 37.0 43.0 50.0 58.0 66.0 74.0 82.0 93.0 104.0 —

33.0 39.0 45.0 52.0 62.0 70.0 78.0 86.0 96.0 107.0 —

35.0 42.0 48.0 56.0 66.0 74.0 82.0 91.0 101.0 112.0 —

All dimensions are in millimeters. a Does not apply to hex lag screws, hex socket headless screws, or round head square neck bolts. b Applies only to heavy hex structural bolts. Normal Clearance: This is preferred for general purpose applications and should be speciﬁed unless special design considerations dictate the need for either a close or loose clearance hole. Close Clearance: This should be speciﬁed only where conditions such as critical alignment of assembled parts, wall thickness or other limitations necessitate use of a minimum hole. When close clearance holes are speciﬁed, special provision (e.g. countersinking) must be made at the screw used bolt entry side to permit proper seating of the screw or bolt head. Loose Clearance: This should be speciﬁed only for applications where maximum adjustment capability between components being assembled is necessary. Recommended Tolerances: The clearance hole diameters given in this table are minimum. Recommended tolerances are: for screw or bolt diameter M5, þ0:2 mm; for M6 through M8, þ0:3 mm; for M20 through M24, þ0:4 mm; for M48 through M72, þ0:5 mm; and for M80 through M100, þ0:6 mm.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.71

TABLE 18-51 American National Standard metric hex screws and bolts—reduced body diameters

Nominal diam, D, and thread pitch

Shoulder diam,b Ds

Body diam, Dsi

Max

Min

Max

M5 0:8 M6 1 M8 1:25 M10 1:5 M12 1:75

5.00 6.00 8.00 10.00 12.00

4.82 5.82 7.78 9.78 11.73

Metric Formed Hex Screws (ANSI B18.2.3.2M-1979, R1989) 4.46 4.36 3.5 2.5 M14 2 14.00 13.73 12.77 5.39 5.21 4.0 3.0 M16 2 16.00 15.73 14.77 7.26 5.04 5.0 4.0 M20 2:5 20.00 19.67 18.49 9.08 8.86 6.0 5.0 M24 3 24.00 23.67 22.13 10.95 10.68 7.0 6.0 — — — —

M5 0:8 M6 1 M8 1:25 M10 1:5

5.00 6.00 8.00 10.00

4.82 5.82 7.78 9.78

4.46 5.39 7.26 9.08

Metric Hex Flange Screws (ANSI B18.2.3.4M-1984) 4.36 3.5 2.5 M12 1:75 12.00 11.73 5.21 4.0 3.0 M14 2 14.00 13.73 7.04 5.0 4.0 M16 2 16.00 15.73 8.86 6.0 5.0 — — —

M5 0:8 M6 1 M8 1:25 M10 1:5 M12 1:75

5.48 6.48 8.58 10.58 12.70

4.52 5.52 7.42 9.42 11.30

4.46 5.39 7.26 9.08 10.95

Metric Hex Bolts (ANSI B18.2.3.5M-1979, R1989) 4.36 3.5 2.5 M14 2 14.70 13.30 5.21 4.0 3.0 M16 2 16.70 15.30 7.04 5.0 4.0 M20 2:5 20.84 19.16 8.86 6.0 5.0 M24 3 24.84 23.16 10.68 7.0 6.0 — — —

M12 1:75 M14 2 M16 2

12.70 14.70 16.70

M10 1:5 M12 1:75 M14 2

10.00 12.00 14.00

Min

Shoulder length,b Lsh Max

Min

Nominal diam, D, and thread pitch

Shoulder diam,b Ds Max

Min

Body diam, Dsi Max

Shoulder length,b Lsh

Min

Max

Min

12.50 14.50 18.16 21.80 —

8.0 9.0 11.0 13.0 —

7.0 8.0 10.0 12.0 —

10.95 12.77 14.77 —

10.68 12.50 14.50 —

7.0 8.0 9.0 —

6.0 7.0 8.0 —

12.77 14.77 18.49 22.13 —

12.50 14.50 18.16 21.80 —

8.0 9.0 11.0 13.0 —

7.0 8.0 10.0 12.0 —

11.30 13.30 15.30

Metric Heavy Hex Bolts (ANSI B18.2.3.6M-1979, R1989) 10.95 10.68 7.0 6.0 M20 2:5 20.84 19.16 18.49 12.77 12.50 8.0 7.0 M24 3 24.84 23.16 22.13 14.77 14.50 9.0 8.0 — — — —

18.16 21.80 —

11.0 13.0 —

10.0 12.0 —

9.78 11.73 13.73

Metric Heavy Hex Flange Screws (ANSI B18.2.3.9M- 1984) 9.08 8.86 6.0 5.0 M16 2 16.00 15.73 14.77 10.95 10.68 7.0 6.0 M20 2:5 20.00 19.67 18.49 12.77 12.50 8.0 7.0 — — — —

14.50 18.16 —

9.0 11.0 —

8.0 10.0 —

All dimensions are in millimeters. a Shoulder is mandatory for formed hex screws, hex ﬂange screws, and heavy hex ﬂange screws. Shoulder is optional for hex bolts and heavy hex bolts.

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Max

16.70 20.84 22.84 24.84 27.84 30.84 37.00

M16 2 M20 2:5 M22 2:5 M24 3 M27 3 M30 3:5 M36 4

15.30 19.16 21.16 23.16 26.16 29.16 35.00

Min

Body diam, DS

Nominal bolt diam, D, thread pitch 27.00 34.00 36.00 41.00 46.00 50.00 60.00

Max 26.16 33.00 35.00 40.00 45.00 49.00 58.80

Min

Width across ﬂats, S

31.18 39.26 41.57 47.34 53.12 57.74 69.28

Max 29.56 37.29 39.55 45.20 50.85 55.37 66.44

Min

Width across corners, E

10.75 13.40 14.90 15.90 17.90 19.75 23.55

Max 9.26 11.60 13.10 14.10 16.10 17.65 21.45

Min

Head height, K

6.5 8.1 9.2 9.9 11.3 12.4 15.0

Min

Wrenching height, K1

24.9 31.4 33.3 38.0 42.8 46.5 55.9

Min

Washer face diam, Dw

0.8 0.8 0.8 0.8 0.8 0.8 0.8

Max

0.4 0.4 0.4 0.4 0.4 0.4 0.4

Min

Washer face thickness, C

31 36 38 41 44 49 56

Bolt lengths, >100

38 43 45 48 51 56 63

Basic

Bolt lengths, 100

Thread length, Ba

6.0 7.5 7.5 9.0 9.0 10.5 12.0

Max

Transition thread length, Xb

18.72

TABLE 18-52 American National Standard metric heavy hex structural bolts (ANSI B18.2.3.7M-1979, R1989)

THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

CHAPTER EIGHTEEN

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.73

TABLE 18-53 Recommended diameter-length combinations for metric heavy hex structural bolts Nominal length, L

Nominal diameter and thread pitch M16 2

M20 2:5

M22 2:5

M24 3

M27 3

M30 3:5

M36 4

45 50 55 60 65 70 75 80 85

—

— —

— — —

— — — —

— — — — —

— — — — — — —

90 95 100 110 120

130 140 150 160 170

180 190 200 210 220

230 240 250 260 270

280 290 300

All dimensions are in millimeters. Recommended diameter-length combinations are indicated by the symbol . Bolts with lengths above the heavy cross lines are threaded full length.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

TABLE 18-54 American National Standard metric hex nuts, Styles 1 and 2 (ANSI B18.2.4.1M and B18.2.4.2M-1979, R1989)

Nominal nut diam, and thread pitch

Width across ﬂats, S Max

Min

Width across corners, E Max

Min

Thickness, M Max

Min

Bearing face diam, Dw

Washer face thickness C

Min

Max

Min

— — — — — — — — —

— — — — — — — — —

Metric Hex Nuts—Style 1 M1:6 0:35 M2 0:4 M2:5 0:45 M3 0:5 M3:5 0:6 M4 0:7 M5 0:8 M6 1 M8 1:25

3.20 4.00 5.00 5.50 6.00 7.00 8.00 10.00 13.00

3.02 3.82 4.82 5.32 5.82 6.78 7.78 9.78 12.73

3.70 4.62 5.77 6.35 6.93 8.08 9.24 11.55 15.01

3.41 4.32 5.45 6.01 6.58 7.66 8.79 11.05 14.38

1.30 1.60 2.00 2.40 2.80 3.20 4.70 5.20 6.80

1.05 1.35 1.75 2.15 2.55 2.90 4.40 4.90 6.44

2.3 3.1 4.1 4.6 5.1 6.0 7.0 8.9 11.6

a

M10 1:5

15.00

14.73

17.32

16.64

9.1

8.7

13.6

M10 1:5 M12 1:75 M14 2 M16 2 M20 2:5 M24 3 M30 3:5 M36 4

16.00 18.00 21.00 24.00 30.00 36.00 46.00 55.00

15.73 17.73 20.67 23.67 29.16 35.00 45.00 53.80

18.48 20.78 24.25 27.71 34.64 41.57 53.12 63.51

17.77 20.03 23.36 26.75 32.95 39.55 50.85 60.79

8.40 10.80 12.80 14.80 18.00 21.50 25.60 31.00

8.04 10.37 12.10 14.10 16.90 20.20 24.30 29.40

14.6 16.6 19.4 22.4 27.9 32.5 42.5 50.8

— — — — 0.8 0.8 0.8 0.8

— — — — 0.4 0.4 0.4 0.4

2.65 3.00 3.50 4.80 5.40 7.14

4.6 5.1 5.9 6.9 8.9 11.6

— — — — — —

— — — — — —

0.6

0.3

Metric Hex Nuts—Style 2 M3 0:5 M3:5 0:6 M4 0:7 M5 0:8 M6 1 M8 1:25

5.50 6.00 7.00 8.00 10.00 13.00

5.32 5.82 6.78 7.78 9.78 12.73

6.35 6.93 8.08 9.24 11.55 15.01

6.01 6.58 7.66 8.79 11.05 14.38

M10 1:5

15.00

14.73

17.32

16.64

10.0

9.6

13.6

0.6

0.3

M10 1:5 M12 1:75 M14 2 M16 2 M20 2:5 M24 3 M30 3:5 M36 4

16.00 18.00 21.00 24.00 30.00 36.00 46.00 55.00

15.73 17.73 20.67 23.67 29.16 35.00 45.00 53.80

18.48 20.78 24.25 27.71 34.64 41.57 53.12 63.51

17.77 20.03 23.35 26.75 32.95 39.55 50.85 60.79

9.30 12.00 14.10 16.40 20.30 23.90 28.60 34.70

8.94 11.57 13.40 15.70 19.00 22.60 27.30 33.10

14.6 16.6 19.6 22.5 27.7 33.3 42.7 51.1

— — — — 0.8 0.8 0.8 0.8

— — — — 0.4 0.4 0.4 0.4

a

2.90 3.30 3.80 5.10 5.70 7.50

All dimensions are in millimeters. a This size with width across ﬂats of 15 mm is not standard. Unless speciﬁcally ordered, metric hex nuts with 16 mm width across ﬂats will be furnished.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.75

TABLE 18-55 American National Standard metric slotted hex nuts (ANSI B18.2.4.3M-1979, R1989)

Width across Nominal nut ﬂats, diam, D, S and thread pitch Max Min

Width across corners, E Max

Min

M5 0:8 M6 1 M8 1:25

Thickness, M Max

Unslotted thickness, F

Min

Max

Min

4.80 5.40 7.14

6.9 8.9 11.6

3.2 3.5 4.4

Width of slot, N

Washer face thickness, C

Max

Min

Max

Min

2.9 3.2 4.1

2.0 2.4 2.9

1.4 1.8 2.3

— — —

— — —

8.00 10.00 13.00

7.78 9.78 12.73

9.24 11.55 15.01

8.79 11.05 14.38

M10 1:5

15.00

14.73

17.32

16.64

10.0

9.6

13.6

5.7

5.4

3.4

2.8

0.6

0.3

M10 1:5 M12 1:75 M14 2 M16 2 M20 2:5 M24 3 M30 3:5 M36 4

16.00 18.00 21.00 24.00 30.00 36.00 46.00 55.00

15.73 17.73 20.67 23.67 29.16 35.00 45.00 53.80

18.48 20.78 24.25 27.71 34.64 41.57 53.12 63.51

17.77 20.03 23.35 26.75 32.95 39.55 50.85 60.79

9.30 12.00 14.10 16.40 20.30 23.90 28.60 34.70

8.94 11.57 13.40 15.70 19.00 22.60 27.30 33.10

14.6 16.6 19.6 22.5 27.7 33.2 42.7 51.1

5.2 7.3 8.6 9.9 13.3 15.4 18.1 23.7

4.9 6.9 8.0 9.3 12.2 14.3 16.8 22.4

3.4 4.0 4.3 5.3 5.7 6.7 8.5 8.5

2.8 3.2 3.5 4.5 4.5 5.5 7.0 7.0

— — — — 0.8 0.8 0.8 0.8

— — — — 0.4 0.4 0.4 0.4

a

5.10 5.70 7.50

Min

Bearing face diam, Dw

All dimensions are in millimeters. a This size with width across ﬂats of 15 mm is not standard. Unless speciﬁcally ordered, M10 slotted hex nuts with 16 mm width across ﬂats will be furnished.

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THREADED FASTENERS AND SCREWS FOR POWER TRANSMISSION

18.76

CHAPTER EIGHTEEN

REFERENCES 1. Norman, C. A., E. S. Ault, and E. F. Zarobsky, Fundamentals of Machine Design, The Macmillan Company, New York, 1951. 2. Maleev, V. L., and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954. 3. Black, P. H., and O. E. Adams, Jr., Machine Design, McGraw-Hill Publishing Company, New York, 1955. 4. Baumeister, T., ed., Marks’ Standard Handbook for Mechanical Engineers, 8th ed., McGraw-Hill Publishing Company, New York, 1978. 5. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962. 6. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 7. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 8. Bureau of Indian Standards. 9. ISO Standards. 10. John, J. Viegas, Standards for Mechanical Elements, Harold A. Rothbart, Editor-in-Chief, Mechanical Design and Systems Handbook, McGraw-Hill Publishing Company, New York, 1964. 11. Russel, Bardsall and Ward Corp., Helpful Hints for Fastener Design and Application, Mentor, Ohio, 1976, p. 42. 12. Shigley J., E., and C. R. Mischke, Mechanical Engineering Design, 5th ed., McGraw-Hill Publishing Company, New York, 1989. 13. Burr, J. H., and J. B. Cheatham, Mechanical Analysis and Design, 2nd ed., Prentice Hall, Englewood Cliﬀs, New Jersey, 1995. 14. Edwards, K. S., Jr., and R. B. Mckee, Fundamentals of Mechanical Component Design, McGraw-Hill Publishing Company, New York, 1991. 15. Ito, Y., J. Toyoda, and S. Nagata, Interface Pressure Distribution in a Bolt-Flange Assembly, ASME Paper No. 77-WA/DE-11, 1977. 16. Little, R. E., Bolted Joints: How Much Give? Machine Design, Nov. 9, 1967. 17. Osgood, C. C., Saving Weight on Bolted Joints, Machine Design, Oct. 25, 1979. 18. Bowman Distribution-Barnes Group, Fastener Facts, Cleveland, Ohio, 1985, p. 90. 19. American National Standards, ANSI B18.2.3.5M-1979, R1989. 20. British Standards Institution, 2 Park Street, London, 1986. 21. Machinery Handbook, 20th ed., 1999, Industrial Press, U.S.A. 22. Shigley J. E., and C. R. Mischke, Standard Handbook of Machine Design, McGraw-Hill Publishing Company New York, 1996. 23. Lingaiah, K., Machine Design Data Handbook, McGraw-Hill Publishing Company, New York, USA, 1994.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

19 COUPLINGS, CLUTCHES, AND BRAKES SYMBOLS8;9;10 a A Ar Ac b

c c1 c2 d d1 d2 d0 D D1 D2 Di Do Dm e1 , e2 , e3 E

distance between center lines of shafts in Oldham’s coupling, m (in) area, m2 (in2 ) external area, m2 (in2 ) radiating surface required, m2 (in2 ) contact area of friction surface, m2 (in2 ) width of key, m (in) width of shoe, m (in) width of inclined face in grooved rim clutch, m (in) width of spring in centrifugal clutch, m (in) width of wheel, m (in) width of operating lever (Fig. 19-16), m (in) heat transfer coeﬃcient, kJ/m2 K h (kcal/m2 /8C/h) speciﬁc heat of material, kJ/kg K (kcal/kg/8C) radiating factor for brakes, kJ/m2 K s (kcal/m2 /min/8C) diameter of shaft, m (in) diameter of pin, roller pin, m (in) diameter of bolt, m (in) diameter of pin at neck in the ﬂexible coupling, m (in) diameter of hole for bolt, m (in) outside diameter of bush, m (in) diameter of wheel, m (in) diameter of sheave, m (in) outside diameter of ﬂange coupling, m (in) inside diameter of disk of friction material in disk clutches and brakes, m (in) outside diameter of disk of friction material in disk clutches and brakes, m (in) inside diameter of hollow rigid type of coupling, m (in) outside diameter of hollow rigid type of coupling, m (in) mean diameter, m (in) dimensions shown in Fig. 19-16, m (in) energy (also with suﬃxes), N m (lbf in) Young’s modulus of elasticity, GPa (Mpsi)

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COUPLINGS, CLUTCHES, AND BRAKES

19.2

F

F1 F2 Fa0 Fb Fc Fn Fx , Fy F g h

H Hg Hd i

i1 i2 i0 I kl ks l

L Mt Mta n n1 , n2 n P N N p

CHAPTER NINETEEN

operating force on block brakes, kN (lbf ); force at each pin in the ﬂexible bush coupling, kN (lbf ) total pressure, kN (lbf ) force (also with suﬃxes), kN (lbf ) actuating force, kN (lbf ) tension on tight side of band, kN (lbf ) the force acting on disks of one operating lever of the clutch (Fig. 19-16), kN (lbf ) tension on slack side of band, kN (lbf ) total axial force on i number of clutch disks, kN (lbf ) tension load in each bolt, kN (lbf ) centrifugal force, kN (lbf ) total normal force, kN (lbf ) components of actuating force F acting at a distance c from the hinge pin (Figs. 19-25 and 19-26), kN (lbf ) tangential force at rim of brake wheel, kN (lbf ) tangential friction force, kN (lbf ) acceleration due to gravity, 9.8066 m/s2 (9806.6 mm/s2 ) (32.2 ft/s2 ) thickness of key, m (in) thickness of central disk in Oldham’s coupling, m (in) thickness of operating lever (Fig. 19-16), m (in) depth of spring in centrifugal clutch, m (mm) rate of heat to be radiated, J (kcal) heat generated, J (kcal) the rate of dissipation, J (kcal) number of pins, number of bolts, number of rollers, pairs of friction surfaces number of shoes in centrifugal clutch number of times the ﬂuid circulates through the torus in one second number of driving disks number of driven disks number of operating lever of clutch moment of inertia, area, m4 , cm4 (in4 ) load factor or the ratio of the actual brake operating time to the total cycle of operation speed factor length (also with suﬃxes), m (in) length of spring in centrifugal clutch measured along arc, m (in) length of bush, m (in) dimension of operating lever as shown in Fig. 19-16 torque to be transmitted, N m (lbf in) allowable torque, N m (lbf in) speed, rpm speed of the live load before and after the brake is applied, respectively, rpm number of clutching or braking cycles per hour power, kW (hp) normal force (Figs. 19-25 and 19-26), kN (lbf ) frictional force (Figs. 19-25 and 19-26), kN (lbf ) unit pressure, MPa (psi)

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COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES

p

pa pb P P0 r rm rmi rmo R Rc Rd Rr Rx , Ry t Ta Tav T tc v v1 , v2 w W

y b 0b cðmaxÞ db b d1 d2 f s

unit pressure acting upon an element of area of the frictional material located at an angle from the hinge pin (Figs. 19-25 and 19-26), MPa (psi) maximum pressure between the fabric and the inside of the rim, MPa (psi) allowable pressure, MPa (psi) maximum pressure located at an angle a from the hinge pin (Figs. 19-25 and 19-26), MPa (psi) bearing pressure, MPa (psi) total force acting from the side of the bush on operating lever (Fig. 19-16), kN (lbf ) the force acting from the side of the bush on one operating lever, kN (lbf ) radius, m (in) distance from the center of gravity of the shoe from the axis of rotation, m (in) mean radius, m (in) mean radius of inner passage of hydraulic coupling, m (in) mean radius of outer passage in hydraulic coupling, m (in) reaction (also with suﬃxes), kN (lbf ) radius of curvature of the ramp at the point of contact (Fig. 19-21), m (in) radius of the contact surface on the driven member (Fig. 19-21), m (in) radius of the roller (Fig. 19-21), m (in) hinge pin reactions (Figs. 19-25 and 19-26), kN (lbf ) time of single clutching or braking operation (Eq. 19-198), s ambient or initial temperature, 8C (8F) average equilibrium temperature, 8C (8F) rise in temperature of the brake drum, 8C (8F) cooling time, s (min) velocity, m/s speed of the live load before and after the brake is applied, respectively, m/s axial width in cone brake, m (in) width of band, m (in) work done, N m (lbf in) weight of the ﬂuid ﬂowing in the torus, kN (lbf ) weight lowered, kN (lbf ) weight of parts in Eq. (19-136), kN (lbf ) weight of shoe, kN (lbf ) deﬂection, m (in) stress (also with suﬃxes), MPa (psi) allowable or design stress in bolts, MPa (psi) design bearing stress for keys, MPa (psi) maximum compressive stress in Hertz’s formula, MPa (psi) design bending stress, MPa (psi) shear stress, MPa (psi) allowable or design stress in bolts, MPa (psi) design shear stress in sleeve, MPa (psi) design shear stress in key, MPa (psi) design shear stress in ﬂange at the outside hub diameter, MPa (psi) design shear stress in shaft, MPa (psi) one-half the cone angle, deg pressure angle, deg

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19.3

COUPLINGS, CLUTCHES, AND BRAKES

19.4

CHAPTER NINETEEN

friction angle, deg one-half angle of the contact surface of block, deg coeﬃcient of friction factor which takes care of the reduced strength of shaft due to keyway running speed of centrifugal clutch, rad/s speed at which the engagement between the shoe of centrifugal clutch and pulley commences, rad/s

!1 !2

SUFFIXES a d g 1, i 2, o n x y

axial dissipated, design generated inner outer normal x direction y direction tangential friction

Other factors in performance or in special aspects are included from time to time in this chapter and, being applicable only in their immediate context, are not included at this stage.

Particular

Formula

19.1 COUPLINGS COMMON FLANGE COUPLING (Fig. 19-1) i ¼ 20d þ 3 The commonly used formula for approximate number of bolts

SI

ð19-1aÞ

USCS

ð19-1bÞ

where d in m i ¼ 0:5d þ 3 where d in in Mt ¼

d 3 16 s

Mt ¼

1000P !

The torque transmitted by the shaft

The torque transmitted by the coupling

ð19-2Þ SI

ð19-3aÞ

USCS

ð19-3bÞ

where Mt in N m; P in kW; ! in rad/s Mt ¼

63;000P n

where Mt in lbf in; P in hp, n in rpm Mt ¼

9550P n

SI

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ð19-3cÞ

COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES

Particular

19.5

Formula

FIGURE 19-1 Flange coupling.

where Mt in N m; P in kW; n in rpm 159P SI n0 0 where Mt in N m; P in kW; n in rps 2 d1 D Mt ¼ i b 1 4 2 Mt ¼

The torque transmitted through bolts

ð19-3dÞ

ð19-4Þ

The torque capacity which is based on bearing of bolts

Mt ¼ iðd1 l1 Þb

D1 2

ð19-5Þ

The torque capacity which is based on shear of ﬂange at the outside hub diameter

Mt ¼ tðD2 Þf

D2 2

ð19-6Þ

The friction-torque capacity of the ﬂanged coupling which is based on the concept of the friction force acting at the mean radius of the surface

Mt ¼ i Fb rm where rm ¼

ð19-7Þ Dþd ¼ mean radius 2

Fb ¼ tension load in each bolt, kN (kgf ) The preliminary bolt diameter may be determined by the empirical formula The bolt diameter from Eqs. (19-2) and (19-4)

The bolt diameter from Eqs. (19-3) and (19-4)

0:5d d1 ¼ pﬃ i sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d 2 s d1 ¼ 2ib D1 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8000P d1 ¼ i!b D1

ð19-8Þ

ð19-9Þ

SI

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ð19-10aÞ

COUPLINGS, CLUTCHES, AND BRAKES

19.6

CHAPTER NINETEEN

Particular

Formula

where d1 , D1 in m; P in kW; b in Pa; ! in rad/s sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1273P d1 ¼ SI ð19-10bÞ in0 D1 b where d1 , D1 in m; P in kW; b in Pa; n0 in rps sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 76;400P SI ð19-10cÞ d1 ¼ inb D1 where d1 , D1 in m; P in kW; b in Pa; n in rpm sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 50;400P d1 ¼ USCS ð19-10dÞ inD1 b

The diameter of shaft from Eqs. (19-2) and (19-3)

where d1 , D1 in in; P in hp; b in psi; ! in rpm where i ¼ effective number of bolts doing work should be taken as all bolts if they are ﬁtted in reamed holes and only half the total number of bolts if they are not ﬁtted into reamed holes sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 16;000P d¼ SI ð19-11aÞ !s where P in kW; d in m sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 100;800P d¼ ns

USCS

ð19-11bÞ

SI

ð19-11cÞ

SI

ð19-11dÞ

SI

ð19-12aÞ

USCS

ð19-12bÞ

SI

ð19-13aÞ

where D2 in m D2 ¼ 1:5d þ 1

USCS

ð19-13bÞ

D ¼ 2:5d þ 0:075

SI

ð19-14aÞ

USCS

ð19-14bÞ

where P in hp; d in in sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 152;800P d¼ ns where P in kW; d in m sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 2546P d¼ n0 s where P in kW; d in m; n0 in rps The average value of diameter of the bolt circle

D1 ¼ 2d þ 0:05 where D1 in m D1 ¼ 2d þ 2

The hub diameter

The outside diameter of ﬂange

D2 ¼ 1:5d þ 0:025

where D in m D ¼ 2:5d þ 3

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COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES

Particular

The hub length

19.7

Formula

l ¼ 1:25d þ 0:01875

SI

ð19-14cÞ

USCS

ð19-14dÞ

SI

ð19-15aÞ

USCS

ð19-15bÞ

where l in m and d in m l ¼ 1:25d þ 0:75 where l and d in in

MARINE TYPE OF FLANGE COUPLING Solid rigid type [Fig. 19-2(a), Table 19-1] The number of bolts

i ¼ 33d þ 5 where d in in i ¼ 0:85d þ 5

The diameter of bolt

where d in in sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d 3 s d1 ¼ 2iD1 b based on torque capacity of the shaft sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ tD22 f d1 ¼ 4iD1 b

ð19-16aÞ

ð19-16bÞ

based on torque capacity of ﬂange

FIGURE 19-2 Rigid marine coupling.

The thickness of ﬂange

t ¼ 0:25 to 0:28d

ð19-17Þ

The diameter of the bolt circle

D1 ¼ 1:4d to 1:6d

ð19-18Þ

The outside diameter of ﬂange

D ¼ D1 þ 2d to 3d

ð19-19Þ

Taper of bolt

1 in 100

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COUPLINGS, CLUTCHES, AND BRAKES

19.8

CHAPTER NINETEEN

TABLE 19-1 Forged end type rigid couplings (all dimensions in mm) Number coupling

Shaft diameter

Recessed ﬂange

Spigot ﬂange

Flange outside Locating diameter, Flange diameter, Recess depth, c1 Max Min D width, t D2

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16 R17 R18 R19 R20 R21 R22 R23 R24

S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 S19 S20 S21 S22 S23 S24

53 45 55 70 80 90 110 130 150 170 190 210 230 250 270 300 330 360 390 430 470 520 571 620

— 36 46 55 71 81 91 111 131 151 171 191 211 231 251 271 301 331 361 391 431 471 521 571

100 120 140 175 195 225 265 300 335 375 400 445 475 500 560 600 650 730 775 875 900 925 1000 1090

17 22 22 27 32 32 36 46 50 55 55 65 70 70 80 85 90 100 105 110 115 120 125 130

50 60 75 95 95 125 150 150 195 195 240 240 280 280 330 330 400 400 480 480 560 560 640 720

6 6 7 7 7 7 9 9 9 10 10 10 10 10 10 10 10 10 11 11 11 12 12 12

Spigot depth, c2

Pitch circle Bolt Bolt hole diameter, size, diameter, Number D1 d1 d2 H8 of bolts

4 4 5 5 5 5 7 7 7 8 8 8 8 8 8 8 8 8 9 9 9 10 10 10

70 85 100 125 140 160 190 215 240 265 290 315 340 370 400 410 480 520 570 620 670 730 790 850

M10 11 M12 13 M14 15 M16 17 M18 19 20 21 24 25 30 32 33 34 36 38 36 38 42 44 45 46 45 46 52 55 56 60 60 65 68 72 72 76 76 80 80 85 90 95 110 105 110 115

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4 4 4 6 6 6 6 6 8 6 8 8 8 10 10 10 10 10 10 12 12 12 12 12

COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES

Particular

19.9

Formula

Hollow rigid type [Fig. 19-2(b)] i ¼ 50Do

The minimum number of bolts

SI

ð19-20aÞ

USCS

ð19-20bÞ

where Do in m i ¼ 1:25Do where Do in in sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 K 4 ÞD3o s d1 ¼ 2iD1 b

The mean diameter of bolt

where K ¼ The thickness of ﬂange

t¼

ð19-21Þ

Di Do

ð1 K 4 ÞD3o s 8D22 f

ð19-22Þ

The empirical formula for thickness of ﬂange

t ¼ 0:25 to 0:28Do

ð19-23Þ

The diameter of bolt circles

D1 ¼ 1:4Do

ð19-24Þ

For design calculations of other dimensions of marine hollow rigid type of ﬂange coupling

The method of analyzing the stresses and arriving at the dimensions of the various parts of a marine hollow ﬂange coupling is similar to that given for the marine solid rigid type and common ﬂange coupling.

For dimensions of ﬁtted half couplings for power transmission

Refer to Table 19-2.

PULLEY FLANGE COUPLING (Fig. 19-3) The number of bolts

i ¼ 20d þ 3

SI

ð19-25aÞ

USCS

ð19-25bÞ

where d in m i ¼ 0:5d þ 3 where d in in The preliminary bolt diameter

0:5d dt ¼ pﬃ i

FIGURE 19-3 Pulley ﬂange coupling.

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ð19-26Þ

Locating diameter, D4 H8/h7

75 95 125 150 195 240 280 330 400 480 580 720

Nominal diameter, d H7

30 40, 45, 50, 56 63, 71 80, 90 100, 110, 125 140 160, 180 200, 220 250 280, 320 360 400, 450, 500

100 125 160 190 240 290 340 400 480 570 670 850

Pitch circle diameter D1

TABLE 19-2 Fitted half couplings (all dimensions in mm)

4 6 6 6 6 8 8 10 10 10 12 12

No. of bolts 13 17 21 25 25 32 38 44 50 60 68 95

Diameter of hole, d2 H7

Bolt

M12 M16 M20 M24 M24 M30 M36 M32 M48 M56 M64 M90

Bolt size, d1

70 90 120 145 190 230 270 320 380 460 540 690

Hub diameter, D2

80 100 180 155 200 240 285 335 400 480 570 720

Shoulder diameter, D2

125 160 200 240 300 360 420 500 600 710 850 1050

Flange diameter, D

80 110 140 170 210 250 300 350 410 470 550 650

Long

58 82 105 130 165 200 240 280 330 380 450 540

Short

Length of shaft end, l

COUPLINGS, CLUTCHES, AND BRAKES

19.10

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COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES

Particular

The width of ﬂange l1 (Fig. 19-3)

19.11

Formula

l1 ¼ 12 d þ 0:025

SI

ð19-27aÞ

USCS

ð19-27bÞ

SI

ð19-28aÞ

USCS

ð19-28bÞ

SI

ð19-29aÞ

USCS

ð19-29bÞ

SI

ð19-30aÞ

USCS

ð19-30bÞ

SI

ð19-31aÞ

USCS

ð19-31bÞ

SI

ð19-32aÞ

USCS

ð19-32bÞ

where l1 and d in m l1 ¼ 12 d þ 1:0 where d in in The hub length l

l ¼ 1:4d þ 0:0175 where l and d in m l ¼ 1:4d þ 0:7 where l and d in in

The thickness of the ﬂange

t ¼ 0:25d þ 0:007 where t and d in m t ¼ 0:25d þ 0:25 where t and d in in

The hub diameter

D2 ¼ 1:8d þ 0:01 where D2 and d in m D2 ¼ 1:8d þ 0:4 where D2 and d in in

The average value of the diameter of the bolt circle

D1 ¼ 2d þ 0:025 where D1 and d in m D1 ¼ 2d þ 1:0 where D1 and d in in

The outside diameter of ﬂange

D ¼ 2:5d þ 0:075 where D and d in m D ¼ 2d þ 3:0 where D and d in in

PIN OR BUSH TYPE FLEXIBLE COUPLING (Fig. 19-4, Table 19-3) Torque to be transmitted

D1 2 0 D1 Mt ¼ ipb ld 2

Mt ¼ iF

where pb ¼ bearing pressure, MPa (psi) F ¼ force at each pin, kN (lbf ) ¼ pb ld 0 d 0 ¼ outside diameter of the bush, m (in)

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ð19-33aÞ ð19-33bÞ

COUPLINGS, CLUTCHES, AND BRAKES

19.12

CHAPTER NINETEEN

Shear stress in pin

p ¼

F 0:785dp2

ð19-34Þ

where

Bending stress in pin

p ¼ allowable shearing stress, MPa (psi) dp ¼ d1 ¼ diameter of pin at the neck, m (in) l þb F 2 b ¼ ð19-35Þ 3 d 32 p

OLDHAM COUPLING (Fig. 19-5) The total pressure on each side of the coupling

ð19-36Þ

F ¼ 14 pDh

where h ¼ axial dimension of the contact area, m (in) The torque transmitted on each side of the coupling

Mt ¼ 2Fl ¼

pD2 h 6

ð19-37Þ

where l ¼ 13 D ¼ the distance to the pressure area centroid from the center line, m (in) p ¼ allowable pressure >8.3 j MPa (1.2 kpsi) Power transmitted

P¼

pD2 hn 57;277

SI

ð19-38aÞ

USCS

ð19-38bÞ

where P in kW P¼

pD2 hn 378;180

where P in hp; D, h in in; p in psi The diameter of the disk

D ¼ 3d þ a

ð19-39Þ

The diameter of the boss

D2 ¼ 2d

ð19-40Þ

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Type of ﬂexible couplings

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. 30 45 56 75 85 110 130 150

16 22 30 45 56 75 85 110 130

22 30 45 56 75 85 110 130

12 16 22 30 45 56 75 85 110

B3 B4 B5 B6 B7 B8 B9 B10

D1 D2 D3 D4 D5 D6 D7 D8 D9

150

22

16

B2

130

16

12

B1

D10

Max

Min

Coupling number

Bore, d

TABLE 19-3 Cast-iron ﬂexible couplings (all dimensions in mm)

500

400

315

250

200

165

132

110

100

80

500

400

315

250

200

170

132

112

100

280

212

180

140

100

80

280

212

180

140

100

80

D2 min

80

diameter,

D, min

Hub

diameter,

Outside width,

100

90

80

63

56

45

40

32

30

28

100

90

80

63

56

45

40

32

30

28

60

56

50

45

40

35

30

22

20

18

60

56

50

45

40

35

30

22

20

18

l1

length, l, min

Flange

Hub Thickness

55

50, 55

45, 50

40, 45

35, 40

30, 35

25, 30

18, 25

16, 18

15, 16

—

—

—

—

—

—

—

—

—

—

of disk, C

18

18

16

16

12

12

12

10

10

8

18

18

16

16

12

12

12

10

10

8

d1

of bolt,

Diameter

16

16

12

12

8

8

8

6

6

6

8

8

6

6

4

4

4

3

3

3

holes

of bolt

Number

400

315

250

190

150

120

90

73

63

55

400

315

250

190

150

120

90

73

63

53

bolts, D1

28

28

22

22

15

15

15

12

12

10

28

28

22

22

15

15

15

12

12

10

t1

diameter of recess,

Pitch circle Bolt

—

—

—

—

—

—

—

—

—

—

45

45

40

40

30

25

25

22

22

20

diameter

Bush

Nominal gap

—

—

—

—

—

—

—

—

—

—

6

6

5

5

4

4

4

2

2

2

holes, c

coupling

between

Maximum

74.0

52.0

25.0

16.0

6.0

4.0

2.5

0.8

0.6

0.4

74.0

52.0

25.0

16.0

6.0

4.0

2.5

0.8

0.6

0.4

kW

100 rpm,

rating per

COUPLINGS, CLUTCHES, AND BRAKES

19.13

COUPLINGS, CLUTCHES, AND BRAKES

19.14

CHAPTER NINETEEN

Particular

FIGURE 19-5 Oldham’s coupling.

Formula

FIGURE 19-6 Muﬀ or sleeve coupling.

Length of the boss

l ¼ 1:75d

ð19-41Þ

Breadth of groove

D 6 w h1 ¼ 2 w h¼ 2

ð19-42Þ

The thickness of the groove The thickness of central disk The thickness of ﬂange

w¼

ð19-43aÞ ð19-43bÞ ð19-44Þ

t ¼ 34 d

MUFF OR SLEEVE COUPLING (Fig. 19-6) The outside diameter of sleeve

D ¼ 2d þ 0:013

SI

ð19-45aÞ

USCS

ð19-45bÞ

where D, d in m D ¼ 2d þ 0:52 The outside diameter of sleeve is also obtained from equation

where D, d in in sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 16Mt 3 D¼ d1 ð1 K 4 Þ where K ¼

ð19-46Þ

d D

The length of the sleeve (Fig. 19-6)

l ¼ 3:5d

ð19-47Þ

Length of the key (Fig. 19-6)

l ¼ 3:5d sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 16Mt d¼ d

ð19-48Þ

The diameter of shaft

ð19-49Þ

where Mt is torque obtained from Eq. (19-2)

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COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES

Particular

19.15

Formula

The width of the keyway

b¼

2Mt d2 ld

ð19-50Þ

The thickness of the key

h¼

2Mt 0b ld

ð19-51Þ

FAIRBAIRN’S LAP-BOX COUPLING (Fig. 19-7) The outside diameter of sleeve

Use Eqs. (19-45) or (19-46)

The length of the lap

l ¼ 0:9d þ 0:003

SI

ð19-52aÞ

USCS

ð19-52bÞ

SI

ð19-53aÞ

USCS

ð19-53bÞ

where l, d in m l ¼ 0:9d þ 0:12 where l, d in in The length of the sleeve

L ¼ 2:25d þ 0:02 where L, d in m L ¼ 2:25d þ 0:8 where L, d in in

FIGURE 19-7 Fairbairn’s lap-box coupling.

FIGURE 19-8 Split muﬀ coupling.

SPLIT MUFF COUPLING (Fig. 19-8) The outside diameter of the sleeve

D ¼ 2d þ 0:013

SI

ð19-54aÞ

USCS

ð19-54bÞ

SI

ð19-55aÞ

where D, d in m D ¼ 2d þ 0:52 where D, d in in The length of the sleeve (Fig. 19-8)

l ¼ 3:5d or 2:5d þ 0:05 where l, d in m

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COUPLINGS, CLUTCHES, AND BRAKES

19.16

CHAPTER NINETEEN

Particular

Formula

l ¼ 3:5d or 2:5d þ 2:0

USCS

ð19-55bÞ

where l, d in in The torque to be transmitted by the coupling

dc2 t id 16 where

ð19-56Þ

Mt ¼

dc ¼ core diameter of the clamping bolts, m (in) i ¼ number of bolts

SLIP COUPLING (Fig. 19-9) 2 ðD D21 Þp 4 2

The axial force exerted by the springs

Fa ¼

With two pairs of friction surfaces, the tangential force

F ¼ 2Fa

The radius of applications of F with suﬃcient accuracy

rm ¼

The torque

Mt ¼ 0:000385ðD22 D21 ÞðD2 þ D1 Þp SI

ð19-57Þ ð19-58Þ

Dm D2 þ D1 ¼ 2 4

ð19-59Þ ð19-60aÞ

Mt ¼ 0:3927ðD22 D21 ÞðD2 þ D1 Þp USCS

ð19-60bÞ

where the values of and p may be taken from Table 19-4 The relation between D1 and D2

D2 ¼ 1:6 D1

ð19-61Þ

where D1 and D2 are the inner and outer diameters of disk of friction lining

FIGURE 19-9 Slip coupling.

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COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES

Particular

19.17

Formula

SELLERS’ CONE COUPLING (Fig. 19-10) The length of the box

L ¼ 3:65d to 4d

ð19-62Þ

The outside diameter of the conical sleeve

D1 ¼ 1:875d to 2d þ 0:0125

SI

ð19-63aÞ

USCS

ð19-63bÞ

where D, d in m D1 ¼ 1:875d to 2d þ 0:5 where D, d in in Outside diameter of the box

D2 ¼ 3d

ð19-64Þ

The length of the conical sleeve

l ¼ 1:5d

ð19-65Þ

FIGURE 19-11 Hydraulic coupling.

FIGURE 19-10 Sellers, cone coupling.

HYDRAULIC COUPLINGS (Fig. 19-11) Torque transmitted

Mt ¼ Ksn2 Wðr2mo r2mi Þ where K ¼ coefficient ¼

Percent slip between primary and secondary speeds

The mean radius of inner passage (Fig. 19-11)

s¼

ð19-66Þ 1:42 ðapprox:Þ 107

np ns 100 np

ð19-67Þ

where np and ns are the primary and secondary speeds of impeller, respectively, rpm 3 2 r2 r31 ð19-68aÞ rmi ¼ 3 r22 r21

The mean radius of outer passage (Fig. 19-11)

rmo ¼

The number of times the ﬂuid circulates through the torus in one second is given by

i¼

2 3

r34 r33 r24 r23

13;000Mt nWðr2mo r2mi Þ

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ð19-68bÞ ð19-69Þ

Cast iron or steel Cast iron Steel Hard steel Hard steel, chromium plated Hard steel, chromium plated Cast iron or steel Hard steel, chromium plated Cast iron or steel Cast iron or steel Cast iron or steel Cast iron or steel Cast iron or steel

Cast bronze Cast iron Cast iron Hard steel Hard steel

0.08–0.12 0.12 0.05–0.1 0.1–0.15

Hard steel, chromium plated Cast iron or steel Cast iron or steel

Steel Cast iron

Woven asbestos

19.18

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. 0.1

0.1–0.2

0.25 0.25

0.2–0.5 0.32

0.3–0.6

0.2–0.35 0.3–0.5 0.3–0.5 0.22 0.3–0.5

632–811 422

533 533–659

533

444–533

422 363.3 363.3 411 363.3

359–538 149

260 260–386

260

171–260

149 90.3 90.3 138 90.3

538 538

260

149 316 260 260 260

8C

Maximum temperature

2.0682 0.6894

0.3452–1.0346 1.0346

8.2738

0.6894–1.3788

0.03432–0.6894

0.4138–0.6208 0.0686–0.2746 0.0549–0.0981 0.0343–0.0686 0.0686–0.2746

1.0346 2.0682

1.0346

0.5521–0.8277 1.0346–1.7240 0.8277–1.3788 0.6894 1.3788

MPa

0.2109 0.0703

0.0352–0.1055 0.1055

0.8437

0.0703–0.1406

0.0350–0.0703

0.0422–0.0633 0.0070–0.0284 0.0056–0.01 0.0035–0.0070 0.3070–0280

0.1055 0.2109

0.1055

0.0563–0.0844 0.1055–0.1755 0.0844–0.1406 0.0703 0.1406

kgf/mm2

Maximum pressure, p

b

Conservative values should be used to allow for possible glazing of clutch surfaces in service and for adverse operating conditions. Steel, where speciﬁed, should have a carbon content of approximately 0.70%. Surfaces should be ground true and smooth. c For a speciﬁc material within this group, the coeﬃcient usually is maintained within plus or minus 5%. Note: 1 kpsi ¼ 6.894757 MPa or 1 Pa ¼ 145 106 psi or 1 MPa ¼ 145 psi.

a

Carbon graphite Molded phenolic plastic, macerated cloth base

Molded asbestosc Impregnated asbestos

Cast iron or steel

Woven asbestos

0.1–0.2

Cast iron or steel

0.16 0.12–0.15 0.15–0.25 0.18

811 811

0.1–0.4 0.1–0.3

0.05–0.1 0.05–0.1

422 589 533 533 533

K

533

0.15–0.2

Dry

0.03

0.05 0.05 0.06 0.05 0.03

Wet

Friction coeﬃcient,a

Wood Leather Cork Felt Vulcanized ﬁber or paper Woven asbestosc

Hard-drawn phosphor bronze Powder metalc Powder metalc

Opposingb

Wearing

Contact surfaces

TABLE 19-4 Friction materials for clutches

High Low

Very low Moderate

Moderate

Low

Low

Lowest Very low Very low Low Very low

High Very high

High

Low Very low Very low Moderate High

Relative cost

Prolonged slip service ratings given This rating for short infrequent engagements Used in Napier Sabre engine Wide ﬁeld of applications For demanding applications For critical requirements For light special service

Unsuitable at high speed Subject to glazing Cork-insert type preferred Resinent engagement Low speeds, light duty

Good wearing qualities High energy absorption

Good wearing qualities

Subject to seizing Good at low speeds Fair at low speeds Subject to galling Durable combination

Comment

COUPLINGS, CLUTCHES, AND BRAKES

COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES

Particular

Power transmitted by torque converter

19.19

Formula

Mt Kn2 D5

ð19-70Þ

where K ¼ coefficient—varies with the design n ¼ speed of driven shaft, rpm D ¼ outside diameter of vanes, m (in)

19.2 CLUTCHES POSITIVE CLUTCHES (Fig. 19-12) Jaw clutch coupling

a¼ c¼ f ¼ g¼ h¼ i¼ j¼ k¼ l¼

2:2d þ 0:025 m 1:2d þ 0:03 m 1:4d þ 0:0055 m d þ 0:005 m 0:3d þ 0:0125 m 0:4d þ 0:005 m 0:2d þ 0:0375 m 1:2d þ 0:02 m 1:7d þ 0:0584 m

a¼ c¼ f ¼ g¼ h¼ i¼ j¼ k¼ l¼

2:2d þ 1:0 in 1:2d þ 1:2 in 1:4d þ 0:3 in d þ 0:2 in 0:3d þ 0:5 in 0:4d þ 0:2 in 0:2d þ 0:15 in 1:2d þ 0:8 in 1:7d þ 2:3 in (19-71)

The area in shear

The shear stress assuming that only one-half the total number of jaws i is in actual contact

A¼

0:5ða bÞh sin

ð19-72Þ

¼

4F sin iða bÞh cos

ð19-73Þ

¼

2:8F iða bÞh

ð19-74Þ

for tan ¼ 0:7

where ¼ angle made by the shearing plane with the direction of pressure

FIGURE 19-12 Square-jaw clutch.

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COUPLINGS, CLUTCHES, AND BRAKES

19.20

CHAPTER NINETEEN

Particular

Formula

FRICTION CLUTCHES Cone clutch (Fig. 19-13) The axial force in terms of the clutch dimensions

Fa ¼ Dm pb sin

ð19-75Þ

where Dm ¼ 12 ðD1 þ D2 Þ (approx.) ¼ one-half the cone angle, deg ¼ ranges from 158 to 258 for industrial clutches faced with wood ¼ 12.58 for clutches faced with asbestos or leather or cork insert Axial force in terms of normal force (Fig. 19-13)

Fa ¼ Fn sin

The tangential force due to friction

F ¼

Torque transmitted through friction

Mt ¼

Power transmitted

P¼

Fa Dm n 19;100 sin kl

P¼

Fa Dm n 126;000 sin kl

P¼

pD2m bn 19;100kl

ð19-76Þ

Fa sin

ð19-77Þ

Fa Dm 2 sin

ð19-78Þ SI

ð19-79aÞ

USCS

ð19-79bÞ

SI

ð19-79cÞ

FIGURE 19-13 Cone clutch.

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COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES

Particular

19.21

Formula

P¼

pD2m bn 126;000kl

USCS

ð19-79dÞ

where kl ¼ load factor from Table 14-7 Refer to Table 19-4 for p. The force necessary to engage the clutch when one member is rotating The ratio (Dm =b) The value of Dm in commercial clutches

Fa0 ¼ Fn ðsin þ cos Þ

ð19-80Þ

Dm ¼ 4:5 to 8 b sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 Pkl q Dm ¼ 18:2 pn

ð19-81Þ

q¼

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 Pkl q Dm ¼ 34:2 pn

SI

ð19-82aÞ

USCS

ð19-82bÞ

Dm ¼ 5d to 10d

ð19-82cÞ

DISK CLUTCHES (Fig. 19-14) The axial force

Fa ¼ 12 pD1 ðD2 D1 Þ

ð19-83Þ

Refer to Table 19-4 for p. The torque transmitted

Mt ¼ 12 Fa Dm

ð19-84Þ

where Dm ¼

2 ðD32 D31 Þ 3 ðD22 D21 Þ

ð19-85aÞ

for uniform pressure distribution and Dm ¼ 12 ðD2 þ D1 Þ

ð19-85bÞ

for uniform wear

FIGURE 19-14 Multidisk clutch.

Power transmitted

P¼

iFa n 28;650kl

P¼

iFa n 189;000kl

D32 D31 D22 d12

D32 D31 D22 d12

SI

ð19-86aÞ

USCS

ð19-86bÞ

for uniform pressure

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COUPLINGS, CLUTCHES, AND BRAKES

19.22

CHAPTER NINETEEN

Particular

Formula

where Fa ¼ p

D22 D21 4

P¼

ipnD1 ðD22 D21 Þ 76;400kl

SI

ð19-87aÞ

P¼

ipnD1 ðD22 D21 Þ 504;000kl

SI

ð19-87aÞ

for uniform wear The clutch capacity at speed n1

P1 ¼

Pn1 nks

ð19-88Þ

where P ¼ design power at speed, n ks ¼ speed factor obtained from Eq. (19-89) The speed factor

ks ¼ 0:1 þ 0:001n

ð19-89Þ

where n ¼ speed at which the capacity of clutch to be determined, rpm

DIMENSIONS OF DISKS (Fig. 19-15) The maximum diameter of disk

D2 ¼ 2:5 to 3:6D1

ð19-90Þ

The minimum diameter of disk

D1 ¼ 4d

ð19-91Þ

The thickness of disk

h ¼ 1 to 3 mm

ð19-92Þ

The number of friction surfaces

i ¼ i1 þ i2 1

ð19-93Þ

The number of driving disks

i1 ¼

i 2

ð19-94Þ

The number of driven disks

i2 ¼

i þ1 2

ð19-95Þ

FIGURE 19-15 Dimensions of disks.

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COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES

Particular

19.23

Formula

DESIGN OF A TYPICAL CLUTCH OPERATING LEVER (Fig. 19-16) The total axial force on i number of clutch disk or plates

Fa0 ¼ ip0 D1 ðD1 D2 Þ

ð19-96Þ

0

where p ¼ actual pressure between disks Fa0 ¼

4Mta ; iðD1 DÞD2m

MPa ðpsiÞ

Mta ¼ allowable torque, N m (lbf in)

FIGURE 19-16 A typical clutch operating lever.

The force acting on disks of one operating lever of the clutch (Fig. 19-16)

F1 ¼

Fa0 i0

ð19-97Þ

where i0 ¼ number of operating levers The total force acting from the side of the bushing (Fig. 19-16)

P ¼ i0 p1

ð19-98Þ

The force acting from the side of the bushing on one operating lever (Fig. 19-16)

d L cotð þ Þ e1 2 P 1 ¼ F1 d e2 þ e3 þ 2

ð19-99Þ

The thickness of the !ever very close to the pin (Fig. 19-16)

2 31=3 6F 0 e h ¼ 6 a 3 7 4 b 0 5 i db h

ð19-100Þ

where db ¼ design bending stress for the material of the levers, MPa (psi) The diameter of the pin (Fig. 19-16)

Ratio of b=h ¼ 0:75 to 1 sﬃﬃﬃﬃﬃﬃﬃ 2Fr d¼ d

ð19-101Þ

where Fr ¼ resultant force due to F1 and P1 cotð þ Þ on the pin, kN (lbf ) d ¼ design shear stress of the material of the pin, MPa (psi)

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COUPLINGS, CLUTCHES, AND BRAKES

19.24

CHAPTER NINETEEN

Particular

Formula

EXPANDING-RING CLUTCHES (Fig. 19-17) Torque transmitted [Fig. 19-17(a)]

Mt ¼ 2pwr2

ð19-102Þ

where ¼ one half the total arc of contact, rad w ¼ width of ring, m (in)

FIGURE 19-17 Expanding-ring clutch.

The moment of the normal force for each half of the band [Fig. 19-17(a)] The force applied to the ends of the split ring to expand the ring [Fig. 19-17(a)] If the ring is made in one piece (Fig. 19-7(b)] an additional force required to expand the inner ring before contact is made with inner surface of the shell

Mo ¼ pwrL

ð19-103Þ

when rad Fs ¼ pwr Fe ¼

Ewt3 6L

ð19-104Þ

1 1 d1 d

ð19-105Þ

where d1 ¼ original diameter of ring, m (in) d ¼ inner diameter of drum, m (in) w ¼ width of ring, m (in) t ¼ thickness of ring, m (in) F ¼ Fs þ Fe

The total force required to expand the ring and to produce the necessary pressure between the contact surfaces

F ¼ pwr þ

Ewt3 6L

ð19-106Þ

1 1 d1 d

ð19-107Þ

Fn ¼ Fn0 ðsin þ cos Þ

ð19-108Þ

Fn ¼ Fn0 sin

ð19-109Þ

Mt ¼ 12 i1 i2 F D ¼ i1 i2 D2 bp

ð19-110Þ

RIM CLUTCHES (Fig. 19-18) When the grooved rim clutch being engaged, the equation of equilibrium of forces along the vertical axis After the block is pressed on ﬁrmly the equation of equilibrium of forces along the vertical axis Torque transmitted

where i1 ¼ number of grooves in the rim i2 ¼ number of shoes b ¼ inclined face, m (in) 2 ¼ angle of contact, rad

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COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES

Particular

19.25

Formula

FIGURE 19-18 Grooved rim clutch.

D ¼ pitch diameter, m (in) 2 ¼ V-groove angle, deg The width of the inclined face

b ¼ 0:01D þ 0:006 m b ¼ 0:01D þ 0:25 in

Frictional force

SI

USCS ð19-111bÞ

F ¼ Fn0 where Fn0 ¼ 2 Dbp

Torque transmitted in case of a ﬂat rim clutch when i1 ¼ 1 and the number of sides b is only one-half that of a grooved rim

ð19-111aÞ

ð19-112aÞ ð19-112bÞ

Mt ¼

i D2 bp 2

ð19-113Þ

Fc1 ¼

w 2 ! r g 1

ð19-114Þ

Fc2 ¼

w 2 ! r g 2

ð19-115Þ

CENTRIFUGAL CLUTCH (Fig. 19-19) Design of shoe Centrifugal force for speed !1 (rad/s) at which engagement between shoe and pulley commences Centrifugal force for running speed !2 (rad/s) The outward radial force on inside rim of the pulley at speed !2

The centrifugal force for !1 ¼ 0:75!2

Fc ¼ Fc2 Fc1 Fc ¼

w 2 ð! !21 Þr g 2

Fc0 ¼

7w 2 ! r 16g 2

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ð19-116aÞ ð19-116bÞ ð19-117Þ

COUPLINGS, CLUTCHES, AND BRAKES

19.26

CHAPTER NINETEEN

Particular

Formula

FIGURE 19-19 Centrifugal clutch.

Torque required for the maximum power to be transmitted

Mt ¼ 4r0 Fc ¼ 4

w 2 ð! !21 Þrr0 g 2

ð19-118Þ

where r0 ¼ inner radius of the rim The equation to calculate the length of the shoe (Fig. 19-19)

l¼

Fc w ¼ ð!2 !21 Þr bp gbp 2

ð19-119Þ

y¼

1Wl 3 48EI

ð19-120Þ

Spring The central deﬂection of ﬂat spring (Fig. 19-19) which is treated as a beam freely supported at the points where it bears against the shoe and loaded centrally by the adjusting screw The maximum load exerted on the spring at speed !1

W ¼ Fc1 ¼

w 2 ! r g 1

bh3 Wl 3 ¼ 12 48Ey

The cross section of spring can be calculated by the equation

I¼

For other proportionate dimensions of centrifugal clutch

Refer to Fig. 19-19.

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ð19-121Þ

ð19-122Þ

COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES

Particular

19.27

Formula

OVERRUNNING CLUTCHES Roller clutch (Fig. 19-20) FIGURE 19-20 Roller clutch.

The condition for the operation of the clutch

< 2

ð19-123Þ

where ¼ angle of friction, varies from 0.03 to 0.005 The force crushing the roller

For ¼ angle 18430 , the angle < 38260 F ð19-124Þ F¼ tan where F ¼ tangential force necessary to transmit the torque at pitch diameter D

The torque transmitted

Mc ¼ 12 F D

The allowable load on roller

Fa ic k0 ld

ð19-125Þ

where k0 ¼ coefficient of the ﬂattening of the roller 4:64c E for c ¼ allowable crushing stress ¼ 1035.0 MPa (150 kpsi) ¼

The roller diameter

d ¼ 0:1D to 0:15D

The number of roller

i¼

LOGARITHMIC SPIRAL ROLLER CLUTCH (Fig. 19-21) The radius of curvature of the ramp at the point of contact (Fig. 19-21) The radius vector of point C (Fig. 19-21) The radius of the contact surface on the driven member in terms of the roller radius and functions angles and (Fig. 19-21) The tangential force

ðD þ dÞ 2d

Rc ¼ 2ðRd Rr Þ

ð19-126Þ

ð19-126aÞ

sin 2 sin 2

ð19-127Þ

sin 2 R cosð2 þ Þ r cos Rd ¼ Rc 1 þ cosð2 þ Þ

ð19-128Þ

F ¼ F sin

ð19-130Þ

Rv ¼

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ð19-129Þ

COUPLINGS, CLUTCHES, AND BRAKES

19.28

CHAPTER NINETEEN

Particular

Formula

FIGURE 19-21 Logarithmic spiral roller-clutch.

The normal force

Fn ¼

The torque transmitted

Mt ¼

The maximum compressive stress at the surface area of contact between the roller and the cage made of diﬀerent materials

The maximum compressive stress at the surface area of contact between the roller and the cage for vc ¼ vr ¼ 0:3

The maximum compressive stress at the surface area of contact between the roller and the cage made of same material (Ec ¼ Er ¼ E) and vc ¼ vr ¼ 0:3

F ¼ F cos tan

iFn Rd cot where

ð19-130aÞ ð19-130bÞ

2 ¼ angle of wedge, deg (usually varies from 38 to 128) i ¼ number of rollers in the clutch 31=2 2 1 1 þ 6F 7 Rr Rc 7 cðmaxÞ ¼ 0:7986 ð19-131Þ 7 6 2 2 42l 1 vr 1 vc 5 þ Er Ec 2 31=2 1 1 0:35F þ 6 7 6 Rr Rc 7 7 cðmaxÞ ¼ 6 ð19-132Þ 5 4 1 1 l þ Er Ec 2 31=2 1 1 FE þ 4 Rr Rc 5 ð19-133aÞ cðmaxÞ ¼ 0:418 l sﬃﬃﬃﬃﬃﬃﬃ FE if Rc Rr cðmaxÞ ¼ 0:418 ð19-133bÞ lRr rﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2FE ð19-133cÞ cðmaxÞ ¼ 0:418 ld where d ¼ 2Rr ¼ diameter of roller, m (mm) l ¼ length of the roller, m (mm)

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COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES

Particular

19.29

Formula

The design torque transmitted by the clutch

Mtd ¼

ildRd cðmaxÞ tan 0:35E

ð19-134Þ

where 2 varies from 3 to 6 deg. For further design data for clutches

Refer to Tables 19-5, 19-6, 19-7.

TABLE 19-5 Preferred dimensions and deviations for clutch facings (all dimensions in mm) Outside diameter

Deviation

Inside diameter

Deviation

Thickness

Deviation

120, 125, 130 135, 140, 145 150, 155, 150 170, 180, 190 200, 210, 220 230, 240, 250 260, 270, 280 290, 300

0 0.5

80, 85, 90 95, 100, 105 110

þ0.5 0

3, 3.5, 4

0. 1

0 0.8

120, 130, 140 150 175, 203

þ0.8 0 þ1.0 0

0 1.0

325, 350

19.3 BRAKES ENERGY EQUATIONS Case of a hoisting drum lowering a load: The decrease of kinetic energy for a change of speed of the live load from v1 to v2

Ek ¼

Fðv21 v22 Þ 2g

ð19-135aÞ

where v1 ; v2 ¼ speed of the live load before and after the brake is applied respectively, m/s F ¼ load, kN (lbf ) The change of potential energy absorbed by the brake during the time t

Ep ¼

F ðv þ v2 Þt 2 1

The change of kinetic energy of all rotating parts such as the hoist drum and various gears and sheaves which must be absorbed by the brake

Er ¼

X Wk2o ð!21 !22 Þ 2g

ð19-135bÞ ð19-136Þ

where ko ¼ radius of gyration of rotating parts, m (mm) !1 ; !2 ¼ angular velocity of the rotating parts, rad/s

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COUPLINGS, CLUTCHES, AND BRAKES

19.30

CHAPTER NINETEEN

TABLE 19-6 Service factors for clutches

TABLE 19-7 Shear strength for clutch facings Service factor not including starting factor

Type of service Driving machine Electric motor steady load Fluctuating load Gas engine, single cylinder Gas engine, multiple cylinder Diesel engine, high-speed Large, low-speed Driven machine Generator, steady load Fluctuating load Blower Compressor depending on number of cylinders Pumps, centrifugal Pumps, single-acting Pumps, double-acting Line shaft Wood working machinery Hoists, elevators, cranes, shovels Hammer mills, ball mills, crushers Brick machinery Rock crushers

1.0 1.5 1.5 1.0 1.5 2.0

Shear strength Type Facing material

MPa kgf/mm2

A

7.4

0.75

4.9

0.50

Solid woven or plied fabric with or without metallic reinforcement Molded and semimolded compound

B

1.0 1.0 1.0 2.0–2.5 1.0 2.0 1.5 1.5 1.75 2.0 2.0 3.0 3.0 FIGURE 19-22 Single-block brake.

Particular

The work to be done by the tangential force F at the brake sheave surface in t seconds The tangential force at the brake sheave surface

Torque transmitted when the blocks are pressed against ﬂat or conical surface

The operating force on block in radial direction (Fig. 19-22) Torque applied at the braking surface, when the blocks are pressed radially against the outer or inner surface of a cylindrical drum (Fig. 19-22)

Formula

Wk ¼ F ¼

F Dðn1 þ n2 Þt 2 60

ð19-137Þ

38:2ðEk þ Ep þ Er Þ Dðn1 þ n2 Þ

Mt ¼ Fn

ð19-138Þ

Dm 2

ð19-139Þ

where Fn ¼ total normal force, kN (lbf ) F 2 þ sin 2 F¼ ð19-140Þ 4 sin Mt ¼ F

D 2

4 sin 2 þ sin 2

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ð19-141Þ

COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES

Particular

19.31

Formula

FIGURE 19-23 ð4 sin Þ=ð2 þ sin 2Þ plotted against the semiblock angle .

The tangential frictional force on the block (Fig. 19-22)

F ¼ F

4 sin 2 þ sin 2

Refer to Fig. 19-23 for values of Torque applied when is less than 608

Mt ¼ F

ð19-142Þ 4 sin . 2 þ sin 2

D ðapprox:Þ 2

ð19-143Þ

where F ¼ pa ðbrÞ

BRAKE FORMULAS Block brake formulas For block brake formulas

Refer to Table 19-8 for formulas from Eqs. (19-144) to (19-148)

Band brake formulas For band brake formulas

Refer to Table 19-8 for formulas from Eqs. (19-149) to (19-157)

The magnitude of pressure between the band and the brake sheave

p¼

F1 þ F2 Dw

ð19-158Þ

The practical rule for the band thickness

h ¼ 0:005D

ð19-159Þ

Width of band

w¼

F1 hd

ð19-160Þ

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COUPLINGS, CLUTCHES, AND BRAKES

19.32

CHAPTER NINETEEN

TABLE 19-8 Formulas for block, simple, and diﬀerential band brakes Type of brake and rotation

Force at the end of brake handle, kN (kgf ) a ða þ bÞ

Block brake

Rotation in either direction

F ¼ F

Block brake

Clockwise

F¼

F a aþb

Counterclockwise

F¼

F a aþb

Clockwise

F¼

F a aþb

Counterclockwise

F¼

F a aþb

Clockwise

F¼

F b a

Counterclockwise

F¼

F b a

Clockwise

F¼

F b a

Counterclockwise

F¼

F b a

Block brake

Simple band brake

Simple band brake

1 c a

1 c þ a

1 c þ a

1 c a

e e 1

(19-144)

(19-145)

(19-146)

(19-147)

(19-148)

(19-149)

1 e 1

1 e 1

e 1

e

For counterclockwise direction (c=a) must be less than (1=) or brake will be self-locking.

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(19-150)

(19-151)

(19-152)

COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES

19.33

TABLE 19-8 Formulas for block, simple, and diﬀerential band brakes (Cont.) Type of brake and rotation

Force at the end of brake handle, kN (kgf )

Diﬀerential band brake

F¼

F a

Counterclockwise

F¼

F a

F¼

F b a

F¼

F a

}

If b2 ¼ b1 F is the same for rotation in either direction Clockwise

Diﬀerential band brake

Counterclockwise

*

Clockwise

F¼

F a

b2 e þ b1 e 1

b1 e þ b2 e 1

b1 e þ 1 e þ 1

b2 e b1 e 1

b2 b1 e e 1

(19-153)

(19-154)

(19-155)

(19-156)

(19-157)

For the above two cases, if b2 ¼ b1 ¼ b. In this case if b2 b1 e , F will be negative or zero and the brake works automatically or the brake is ‘‘self-locking.’’

**

Particular

Suitable drum diameter according to Hagenbook

Formula

Mt 69

1=3

< 10D <

Mt 54

1=3

where Mt in N m and D in m Mt 1=3 Mt 1=3
SI

ð19-161Þ

USCS

ð19-162Þ

where Mt in lbf and D in in Suitable drum diameter in terms of frictional horsepower

ð79:3PÞ1=3 < 100D < ð105:8PÞ1=3

SI

ð19-163aÞ

where P in kW and D in m ð60PÞ1=3 < D < ð80PÞ1=3

USCS ð19-163bÞ

where P in hp and D in in P is taken as the maximum horsepower to be dissipated in any 15-min period

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COUPLINGS, CLUTCHES, AND BRAKES

19.34

CHAPTER NINETEEN

Particular

Formula

CONE BRAKES (Fig. 19-24) The normal force The radial force The tangential force or braking force

Fn ¼

Fa sin

ð19-164Þ

Fr ¼

Fa tan

ð19-165Þ

F ¼ Fn ¼

The braking torque

Fa sin

Fa D 2 sin where D ¼ mean diameter, m (mm)

Mt ¼

ð19-166Þ ð19-167Þ

CONSIDERING THE LEVER (Fig. 19-24) The axial force

Fa ¼

The relation between the operating force F and the braking force F

F¼

The area of the contact surface using the designation given in Fig. 19-24

A¼

aF h

hF sin a

Dw cos where

ð19-168Þ ð19-169Þ ð19-170Þ

w ¼ axial width, m (mm) ¼ half the cone angle, deg The average pressure between the contact surfaces

Fav ¼

Fn Fa ¼ A Dw tan

Take ¼ from 108 to 188 w ¼ 0:12D to 0:22D

FIGURE 19-24 Cone brake.

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ð19-171Þ

COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES

Particular

19.35

Formula

DISK BRAKES The torque transmitted for i pairs of friction surfaces

Mt ¼

ip1 D1 ðD22 D21 Þ 8

ð19-172Þ

Fa ¼ 12 p1 D1 ðD2 D1 Þ

The axial force transmitted

ð19-173Þ

where p1 ¼ intensity of pressure at the inner radius, MPa (psi) Refer to Table 19-9.

For design values of brake facings TABLE 19-9 Design values for brake facings

Permissible unit pressure

Facing material Cast iron on cast iron Dry Oily Wood on cast iron Leather on cast iron Dry Oily Asbestos fabric on metal Dry Oily Molded asbestos on metal

Design coeﬃcient of friction

1 m/s

10 m/s

MPa

kgf/mm2

MPa

kgf/mm2

0.20 0.07 0.25–0.30

0.5521–0.6824

0.0563–0.0703

0.1383–0.1726

0.0141–0.0176

0.40–0.50 0.15

0.0549–0.1039

0.0056–0.0106

0.35–0.40 0.25 0.30–0.35

0.6209–0.6894

0.0633–0.0703

0.1726–0.2069

0.0176–0.0211

1.0395–1.2062

0.106–0.123

0.2069–0.2756

0.0211–0.0281

Note: 1 kpsi ¼ 6.894757 MPa or 1 MPa ¼ 145 psi.

INTERNAL EXPANDING-RIM BRAKE Forces on Shoe (Fig. 19-25) FOR CLOCKWISE ROTATION The maximum pressure

The moment Mt of the frictional forces

The moment of the normal forces

pa ¼ p

sin a sin

Mt ¼

pa br sin a

Mtn ¼

pa bra sin a

ð19-174Þ ð 2 1

ð 2 1

sin ðr a cos Þ d

sin2 d

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ð19-174aÞ

ð19-175Þ

COUPLINGS, CLUTCHES, AND BRAKES

19.36

CHAPTER NINETEEN

Particular

Formula

FIGURE 19-25 Forces on the shoe. (J. E. Shigley, Mechanical Engineering Design, 1962, courtesy of McGraw-Hill.)

Mtn Mt c

The actuating force

F¼

The torque Mt applied to the drum by the brake shoe

Mt ¼

pa br2 ðcos 1 cos 2 Þ sin a

The hinge-pin horizontal reaction

Rx ¼

pa br sin a

ð19-176Þ

ð

2

1

sin cos d

The hinge-pin vertical reaction

Ry ¼

pa br sin a

ð

2

1

ð19-177Þ

ð 2 1

sin2 d Fx

ð19-178Þ

sin2 d

þ

ð 2 1

sin cos Fy

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ð19-179Þ

COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES

Particular

FOR COUNTERCLOCKWISE ROTATION (Fig. 19-25)

19.37

Formula

Mtn þ Mt c ð 2 p br Rx ¼ a sin cos d sin a 1 ð 2 þ sin2 d Fx F¼

1

Ry ¼

pa br sin a

ð

2

1

ð19-180Þ

ð19-181Þ

sin2 d

ð 2 1

sin cos d Fy

ð19-182Þ

EXTERNAL CONTRACTING-RIM BRAKE Forces on shoe (Fig. 19-26) FOR CLOCKWISE ROTATION The moment Mt of the friction forces Fig. 19-26 The moment of the normal force

Mt ¼

pa br sin a

Mtn ¼

pa br sin a

ð 2 1

ð 2 1

sin ðr a cos Þ d

ð19-183Þ

sin2 d

ð19-184Þ

FIGURE 19-26 Forces and notation for external-contracting shoe. (J. E. Shigley, Mechanical Engineering Design, 1962, courtesy of McGraw-Hill.)

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COUPLINGS, CLUTCHES, AND BRAKES

19.38

CHAPTER NINETEEN

Particular

The actuating force

The horizontal reaction at the hinge-pin

Formula

Mtn þ Mt c ð 2 p br Rx ¼ a sin cos d sin a 1 ð 2 þ sin2 d Fx F¼

ð19-185Þ

ð19-186Þ

1

The vertical reaction at the hinge-pin

Ry ¼

ð 2 pa br sin cos d sin a 1 ð 2 sin2 d þ Fy

ð19-187Þ

1

FOR COUNTERCLOCKWISE ROTATION

Mtn Mt c ð 2 p br Rx ¼ a sin cos d sin a 1 ð 2 sin2 d Fx

ð19-188Þ

F¼

ð19-189Þ

1

Ry ¼

pa br sin a

ð 2

ð 2 1

1

sin cos d

sin2 d þ Fy

ð19-190Þ

HEATING OF BRAKES Heat generated from work of friction

Hg ¼ pAc v J ðjoulesÞ Hg ¼

Heat to be radiated for a brake lowering the load

pAc v 778

H ¼ Wh J ðjoulesÞ

SI

ð19-191aÞ

USCS

ð19-191bÞ

SI

ð19-192aÞ

Wh USCS ð19-192bÞ 778 where h ¼ total height or distance, m (ft)

H¼

The heat generated is also given by the equation

Hg ¼ 754kl P

SI

ð19-193aÞ

USCS

ð19-193bÞ

where P in kW and Hg in J/s Hg ¼ 42:4kl P where P in hp

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COUPLINGS, CLUTCHES, AND BRAKES

19.39

COUPLINGS, CLUTCHES, AND BRAKES

Particular

The rise in temperature in 8C of the brake drum or clutch plates

Formula

T ¼

H mC

ð19-194Þ

where m ¼ mass of brake drum or clutch plates, kg C ¼ specific heat capacity ¼ 500 J/kg 8C for cast iron or steel for cast iron ¼ 0:13 Btu/lbm 8F ¼ 0:116 Btu/lbm 8F for steel The rate of heat dissipation

Hd ¼ C2 TAr

SI

ð19-195aÞ

where Hd in J. Hd ¼ 0:25C2 TAr

Metric ð19-195bÞ

where C2 ¼ radiating factor from Table 19-13 Hd in kcal. The required area of radiating surface

Ar ¼

754kl N C2 T

SI

ð19-196aÞ

where Ar in m2 Ar ¼

0:18kl N C2 T

SI ð19-196bÞ

where Ar in mm2 Approximate time required for the brake to cool

Gagne’s formula for heat generated during a single operation

tc ¼

Wr C2 ln T KAr

ð19-197Þ

where K ¼ a constant varying from 0.4 to 0.8 " # AC nt Nt Hg ¼ ðTav Ta Þ c þ 1:5 1 c nc 3600 3600 ð19-198Þ where ðTav Ta Þ ¼ temperature diﬀerence between the brake surface and the atmosphere, 8C Refer to Table 19-15 for values of C.

For additional design data for brakes

Refer to Tables 19-11 to 19-17.

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COUPLINGS, CLUTCHES, AND BRAKES

19.40

CHAPTER NINETEEN

TABLE 19-10 Working pressure for brake blocks Pressure Rubbing velocity, m/s

Wood blocks

Asbestos fabric

Asbestos blocks

MPa

kgf/mm2

MPa

kgf/mm2

MPa

kgf/mm2

1 2 3 4 5 10

0.5521 0.4482 0.3452 0.2412 0.1726 0.1726

0.0563 0.0457 0.0352 0.0246 0.0176 0.0176

0.6894 0.5521 0.4138 0.2756 0.2069 0.2069

0.0703 0.0563 0.0422 0.0281 0.0211 0.0211

1.1032 1.0346 0.8963 0.6894 0.4825 0.2756

0.1125 0.1055 0.0914 0.0703 0.0492 0.0281

Note: 1 kpsi ¼ 6.894754 MPa or 1 MPa ¼ 145 psi.

TABLE 19-11 Comparison of hoist brakes Block brakes

Band brakes Axial brakes

Brake characteristics

Double block

V-grooved sheave

b a

Average numerical value Relative value

Force ratio

F F

Simple

Both directions of rotation

Cone

Multidisk

b sin a

b aðe 1Þ

bðe þ 1Þ aðe 1Þ

b sin a

b na

0.667

0.282

0.0323

0.165

0.161

0.097

20.6

8.7

1

5.1

5.0

h1 a b sin

h1 a 2b

h1 a 4b

h1 a b sin

a

3.00 b

ih01 a b

Travel at lever end

h1 a b

Average travel, mm (in)

8.0 (0.313)

18.8 (0.74)

74.5 (2.943)

37.36 (1.471)

32.8 (1.292)

5.56 (0.219)

1512.7 (2000)

18.9 (25)

227.0 (300)

75.6 (100)

37.8 (50)

90.8 (120)

Maximum capacity P, kW (hp) a b

h1 ¼ the normal distance between the sheave and the stationary braking surface to prevent dragging. h ¼ b in Fig. 19-21.

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COUPLINGS, CLUTCHES, AND BRAKES

TABLE 19-12 Service factors for typical machines Service factors for prime movers Electric motor steam or water turbine

High-speed steam or gas engine

4 Cyla

4 Cyl

6 Cyl

4 Cyl

Alternators and generators (excluding welding generators), induced-draft fans, printing machinery, rotary pumps, compressors, and exhausters, conveyors

1.5

2.0

2.5

3.0

3.5

5.0

Woodworking machinery, machine tools (cutting) excluding planing machines, calenders, mixers, and elevators

2.0

2.5

3.0

3.5

3.0

5.5

Forced-draft fans, high-speed reciprocating compressors, high speed crushers and pulverizers, machine tools (forming)

2.5

3.0

3.5

4.0

4.5

6.0

Rotary screens, rod mills, tube, cable and wire machinery, vacuum pumps

3.0

3.5

4.0

4.5

5.0

6.5

Low-speed reciprocating compressors, haulage gears, metal planing machines, brick and tile machinery, rubber machinery, tube mills, generators(welding)

3.5

4.0

4.5

5.0

5.5

7.0

Type of driven machine

TABLE 19-13 Radiating factors for brakes

Petrol engine

Oil engine

TABLE 19-14 pv values as recommended by Hutte for brakes

Radiating factor, C2

C2 T

pv

Temperature diﬀerence, T

W/m2 K

cal/m2 s 8C W/m2

cal/m2 s

Service

SI

Metric

55.5 111.5 166.5 222.6

12.26 15.33 16.97 18.40

2.93 3.66 4.05 4.39

162.73 406.83 675.34 976.40

Intermittent operations with long rest periods and poor heat radiation, as with wood blocks Continuous service with short rest periods and with poor radiation Continuous operation with good radiation as with an oil bath

26.97

2.75

13.73

1.40

40.70

4.15

681.36 1703.41 2827.66 4088.19

TABLE 19-15 Values of beat transfer coeﬃcient C for rough block surfaces Heat-transfer coeﬃcient, C

Velocity, v, m/s

W/m2 K

kcal/m2 h 8C

0.0 6.1 12.2 18.3 24.4 30.5

8.5 14.1 18.8 22.5 25.6 29.0

7.31 12.13 16.20 19.30 22.00 24.90

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COUPLINGS, CLUTCHES, AND BRAKES

19.42

CHAPTER NINETEEN

TABLE 19-16 Values of e Leather belt on Wood

Cast iron

Proportion of contact to whole circumference

Steel band on cast iron ¼ 0:18

Slightly greasy ¼ 0:47

Very greasy ¼ 0:12

Slightly greasy ¼ 0:28

0.1 0.2 0.3 0.4 0.425 0.45 0.475 0.500 0.525 0.6 0.7 0.8 0.9 1.0

1.12 1.25 1.40 1.57 1.62 1.66 1.71 1.76 1.81 1.97 2.21 2.47 2.77 3.10

1.34 1.81 2.43 3.26 3.51 3.78 4.07 4.38 4.71 5.88 7.90 10.60 14.30 19.20

1.08 1.16 1.25 1.35 1.38 1.40 1.43 1.46 1.49 1.57 1.66 1.83 1.97 2.12

1.19 1.42 1.69 2.02 2.11 2.21 2.31 2.41 2.52 2.81 3.43 4.09 4.87 5.81

Damp ¼ 0:38 1.27 1.61 2.05 2.60 2.76 2.93 3.11 3.30 3.50 4.19 5.32 6.75 8.57 10.90

TABLE 19-17 Coeﬃcient of friction and permissible variations on dimensions for automotive brakes lining

Type and class of brake lining Type I—rigid molded sets or ﬂexible molded rolls or sets Class A—medium friction Class B—high friction Type II—rigid woven sets or ﬂexible woven rolls or sets Class A—medium friction Class B—high friction

Tolerance on width for sizes, mm

Tolerance on thickness for sizes, mm

Range of coeﬃcient of friction,

Permissible variation in , %

5 mm thickness

>5 mm thickness

5 mm thickness

>5 mm thickness

0.28–0.40 0.36–0.45

þ30, 20 þ30, 20

þ0 0.2

þ0 0.3

þ0 0.8

þ0 0.8

0.33–0.43 0.43–0.53

þ20, 30 þ20, 30

REFERENCES 1. Shigley, J. E., Machine Design, McGraw-Hill Book Company, New York, 1962. 2. Maleev, V. L. and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954. 3. Black, P. H., and O. E. Adams, Jr., Machine Design, McGraw-Hill Book Company, New York, 1968. 4. Norman, C. A., E. S. Ault, and I. F. Zarobsky, Fundamentals oﬀ Machine Design, The Macmillan Company, New York, 1951.

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COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES

19.43

5. Spotts, M. F., Machine Design Analysis, Prentice-Hall, Englewood Cliﬀs, New Jersey, 1964. 6. Spotts, M. F., Design of Machine Elements, Prentice-Hall of India Ltd., New Delhi, 1969. 7. Vallance, A., and V. L. Doughtie, Design of Machine Members, McGraw-Hill Book Company, New York, 1951. 8. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962. 9. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 10. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 11. Lingaiah, K., Machine Design Data Handbook, McGraw-Hill Publishing Company, New York, 1994. 12. Bureau of Indian Standards, New Delhi, India.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

20 SPRINGS SYMBOLS A b b0 c c1 , c2 C1 , C2 d d1 , d2 D D1 D2 esz e0sr E F F Fmax ko Fcr g G h i i0

area of loading, m2 (in2 ) width of rectangular spring, m (in) width of laminated spring, m (in) width of each strip in a laminated spring, m (in) spring index constants taken from Table 20-1 and to be used in Eqs. (20-1) to (20-36) constants to be used in Eqs. (20-20) and (20-21) and taken from Fig. 20-3 diameter of spring wire, m (in) diameter of torsion bar, m (in) diameter of outer and inner wires of concentric spring, m (in) mean or pitch diameter of spring, m (mm) overall diameter of the absorber, m (in) mean or pitch diameter of outer concentric spring, m (in) smallest mean diameter of conical spring, m (in) mean or pitch diameter of inner concentric spring, m (in) largest mean diameter of conical spring, m (in) size coeﬃcient surface inﬂuence coeﬃcient modulus of elasticity, GPa (psi) frequency, cycles per minute, Hz load, kN (lbf ) steady-state load [Eq. (20-84)] maximum force that can be imposed on the housing, kN (lbf ) force to compress the spring one meter (in) N/m (lbf/in) [spring rate, N/m (lbf/in)] critical load, kN (lbf ) acceleration due to gravity, 9.8066 m/s2 9806.06 mm/s2 (32.2 ft/s2 ; 386.4 in/s2 ) modulus of rigidity, GPa (psi) height (thickness) of laminated spring, m (in) axial height of a rectangular spring wire, m (in) total number of strips or leaves in a leaf spring number of coils in a helical spring total number of full-length blunt-ended leaves in a leaf spring

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SPRINGS

20.2

CHAPTER TWENTY

moment of inertia, area, m4 , cm4 (in4 ) stress factor (Wahl factor) correction factor factor depends on the ratio lo =D as shown in Fig. 20-8 reduced stress correction factor or Wahl stress factor or fatigue stress correction factor shear stress correction factor length, m (in) free length of helical spring, m (in) iD length of the coil part of torsion spring, m (in) eﬀective length of bushing, m (in) overall length of the absorber (Fig. 20-15), m (in) constant depends on do =di as indicated in Fig. 20-3 twisting moment, N m (lbf in) factor of safety actual factor of safety or reliability factor resilience, N m (lbf in) energy to be absorbed by a rubber spring, N m (lbf in) volume of spring, m3 , mm3 (in3 ) speciﬁc weight of the spring material, N/m3 (lbf/in3 ) weight of spring, kN (lbf ) weight of eﬀective number of coils i involved in the operation of the spring [Eq. (20-77)], kN (lbf ) deﬂection, m (in) critical deﬂection, m (in) section modulus, m3 , cm3 (in3 ) polar section modulus, m3 , cm3 (in3 ) stress, normal, MPa (psi) shear stress, MPa (psi) constant from Table 20-3 constants from Table 20-3 angular deﬂection, rad Poisson’s ratio

I k, k1 , k2 k4 Kl kr l lf or lo L M Mt n na U V W y ycr Z Zo , 0 , 0

SUFFIXES 1 2 a m max min f o

outside inside amplitude mean maximum minimum endurance lirnit (also used for reversed cycle) endurance limit for repeated cycle

Particular

Formula

LEAF SPRINGS (Table 20-1)1;2;3 The general equation for the maximum stress in springs

¼

c1 Fl bh2

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ð20-1Þ

SPRINGS SPRINGS

Particular

20.3

Formula

The general equation for the maximum deﬂection springs The thickness of spring

y¼

c2 Fl 3 Ebh3

ð20-2Þ

h¼

c2 l 2 c1 yE

ð20-3Þ

Refer to Table 20-1 for values of c1 and c2 . For sizes and tolerances for leaf springs for motor vehicle suspension [4]

Refer to Tables 20-2 to 20-5.

TABLE 20-1 Deﬂection formula for beams of rectangular cross section and constants in beam Eqs. (20-1) to (20-3)

Particular

Maximum deﬂection, ymax

c1 , for the stress

c2 , for the deﬂection

Unit resilience, Nm/m3 (kgf mm/mm3 )

ymax ¼

2F bE

3 1 h

3

2

2 18E

ymax ¼

4F bE

3 1 h

3

4

2 6E

ymax ¼

3F bE

3 1 h

3

3

2 6E

ymax ¼

4F bE

3 1 h

6

4

2 18E

ymax ¼

8F bE

3 1 h

6

8

2 6E

ymax ¼

6F bE

3 1 h

6

6

2 6E

Source: K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962; K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I, Suma Publishers, Bangalore, India, 1986; and K. Lingaiah, Machine Design Data Handbook, Vol. II, Suma Publishers, Bangalore, India, 1986.

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SPRINGS

20.4

CHAPTER TWENTY

Particular

Formula

LAMINATED SPRING (Fig. 20-1)5

FIGURE 20-1 Laminated springs for automobiles.

The load on the spring

ib0 h2 c1 l

F¼

ð20-4Þ

where ib0 ¼ b The maximum deﬂection The maximum deﬂection in case of laminated semielliptical spring for heavy loads The correction factor to be used in Eq. (20-6)

y¼

c2 Fl 3 Eib0 h3

ð20-5Þ

y¼

c2 Fl 3 k4 Eib0 h3

ð20-6Þ

k4 ¼

1 4r þ 2r2 ð1:5 ln rÞ ð1 rÞ3

where r ¼

i0 l

For standard sections of steel plates for laminated springs

Refer to Tables 20-2 to 20-6.

The correction factor k4 can also be obtained from

k4 ¼

Size coeﬃcient

0:73r0:1 1

for 2 < r < 20 for r > 20

0:0025 h where h in mm

esz ¼ 0:8 þ

0:1 h where h in in

esz ¼ 0:8 þ

2:5 h where h in mm

esz ¼ 0:8 þ

ð20-7Þ

ð20-8Þ

SI

ð20-9aÞ

USCS

ð20-9bÞ

SI

ð20-9cÞ

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SPRINGS SPRINGS

Particular

20.5

Formula

TABLE 20-2 Cross section tolerances for leaf springs for motor vehicle suspension—metric bar sizes—SAE J1123a

Width tolerance, mm

Tolerance, mm in thickness (þ)a and in ﬂatness ()b

Maximum diﬀerence in thicknessc

For thickness

For thickness, mm

Width, mm

Minus 0.00

5.00–9.50

10.00–21.20

22.40–37.50

5.00–9.50

10.00–21.20

22.40–37.50

40.0 45.0 50.0 56.0 63.0 75.0 90.0 100.0 125.0 150.0

þ0.75 þ0.75 þ0.75 þ0.75 þ0.75 þ1.15 þ1.15 þ1.15 þ1.65 þ2.30

0.13 0.13 0.13 0.13 0.13 0.15 0.15 0.15 0.18 —

0.15 0.15 0.15 0.15 0.15 0.20 0.20 0.20 0.25 0.30

— — — — — 0.30 0.30 0.30 0.40 0.50

0.05 0.05 0.05 0.05 0.05 0.08 0.08 0.08 0.10 —

0.05 0.05 0.05 0.05 0.05 0.10 0.10 0.10 0.13 0.15

— — — — — 0.15 0.15 0.15 0.20 0.25

a

Thickness measurements shall be taken at the edge of the bar where the ﬂat surfaces intersect the rounded edge. This tolerance represents the maximum amount by which the thickness of the center of the bar may be less than the thickness at the edges. Thickness of the center may never exceed the thickness at the edges. c Maximum diﬀerence in thickness between the two edges of each bar. Source: Reproduced from SAE Handbook, Vol. I, 1981, courtesy SAE. b

Size factor

ksz ¼

1 ¼ 4:66h0:35 esz

SI

ð20-10aÞ

USCS

ð20-10bÞ

SI

ð20-10cÞ

where k in m ksz ¼ 1:27h0:35 where h in in ksz ¼ 0:415h0:35 where h in mm

LAMINATED SPRINGS WITH INITIAL CURVATURE Fg ¼

2ð1 rÞ F 2þr

ð20-11Þ

The load shared by full-length leaves of the spring

Ff ¼

3r F 2þr

ð20-12Þ

The maximum stress in the graduated leaves

g ¼

The load shared by graduated leaves of the spring

The maximum stress in the full-length leaves

Fl lb0 h2 1:5Fl f ¼ 0 2 ib h

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ð20-13Þ ð20-14Þ

SPRINGS

20.6

CHAPTER TWENTY

Particular

Formula

The maximum deﬂection of the leaves (in both graduated and full-length leaves)

y¼

Fl 3 Eib0 h3

ð20-15Þ

The camber to be provided for equalization of stress in both graduated and full-length leaves

c¼

0 Fl 3 ib0 Eh3

ð20-16Þ

The load on the clip bolt to be applied to provide camber

Fb ¼

0 Fl ib0 h2

ð20-17Þ

0 Fl ð20-18Þ ib0 h2 0 0 The values of constants , , and can be obtained from Table 20-7.

The maximum equalized stress

¼

TABLE 20-3 Cross section tolerances for leaf spring for motor vehicle suspension—SAE J510c Tolerance in width

Nominal width Over to and including

in

For thickness

0.0

in

63.5 þ0.030

0.076

63.5 4.00 101.6 þ0.045

+1.14

4.00 101.6 5.00 127.0 þ0.065

+1.65

5.00 127.0 6.00 152.4 þ0.090

+2.90

0.375 9.52 >0.375–0.875 9.52–22.22 0.375 9.52 >0.375–0.875 9.52–22.22 >0.875–1.500 22.22–38.10 0.375 9.52 >0.375–0.875 9.52–22.22 >0.875–1.500 22.22–38.10 >0.375–0.875 9.52–22.22 >0.875–1.500 22.22–38.10

in 0.00 2.50

mm

in

0.0 2.50

mm

0.00

mm

Tolerance in thicknessa

mm

Tolerance in ﬂat surfacesb

in

mm in

0.005 0.006 0.006 0.008 0.012 0.007 0.010 0.016 0.012 0.020

0.13 0.15 0.15 0.20 0.30 0.18 0.25 0.41 0.30 0.51

0.005 0.006 0.006 0.008 0.012 0.007 0.010 0.016 0.012 0.020

Maximum diﬀerence in thicknessc

mm

in

mm

0.13 0.15 0.15 0.20 0.30 0.18 0.25 0.41 0.30 0.51

0.002 0.002 0.003 0.004 0.006 0.004 0.005 0.008 0.006 0.010

0.05 0.05 0.08 0.10 0.15 0.10 0.13 0.20 0.15 0.25

a

Thickness measurements shall be taken at the edge of the bar where the ﬂat surfaces intersect the rounded edge. This tolerance represents the maximum amount by which the thickness of the center of the bar may be less than the thickness at the edges. Thickness at the center may never exceed the thickness at the edges. c Maximum diﬀerence in thickness between the two edges of each bar. Source: Reproduced from SAE Handbook, Vol. I, 1981. b

DISK SPRINGS (BELLEVILLE SPRINGS) The relation between the load F and the axial deﬂection y of each disk is given by the equation (Fig. 20-2)

F¼

" # 4Ey y 3 t þ t ðh yÞ h 2 ð1 v2 ÞMdo2

where t ¼ thickness, m (mm) h ¼ height, m (mm) M is a constant from Fig. 20-3

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ð20-19Þ

SPRINGS SPRINGS

20.7

TABLE 20-4 Width and thickness of leaf springs for motor vehicle suspension—SAE J1123a Width, mm 40.0 45.0 50.0 56.0 63.0 FIGURE 20-2 Disk spring.

The maximum stress induced at the inner edge

The maximum stress induced at the outer edge

75.0 90.0 100.0 125.0 150.0

Thickness mm 5.00 5.30 5.60 6.00 6.30 6.70

7.10 7.50 8.00 8.50 9.00 9.50

10.00 10.60 11.20 11.80 12.50 13.20

14.00 15.00 16.00 17.00 18.00 19.00

20.00 21.20 22.40 23.60 25.00 25.50

28.00 30.00 31.50 33.50 35.50 37.50

Source: Reproduced from SAE Handbook, Vol. I, 1981.

" # 4Ey y C1 h i ¼ þ C2 t 2 ð1 v2 Þdo2

ð20-20Þ

" # 4Ey y o ¼ C1 h C2 t 2 ð1 v2 Þdo2

ð20-21Þ

where C1 and C2 are constants taken from Fig. 20-3 For spring design stresses

FIGURE 20-3 Constants C1 , C2 , and M for a disk spring. (V. L. Maleev and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954.)

Refer to Table 20-8

FIGURE 20-3a Load-deﬂection curves for Belleville springs. Courtesy: Shigley, J. E. and L. D. Mitchell, Mechanical Engineering Design, McGraw-Hill Publishing Company, New York, 1983.

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SPRINGS

20.8

CHAPTER TWENTY

FIGURE 20-3b Load deﬂection characteristics for Belleville washers Courtesy: Jorres, R. L., Springs; Chapter 24 in Shigley, J. E. and C. R. Mischke, eds, Standards Handbook of Machine Design, McGraw-Hill Publishing Company, New York, 1986.

FIGURE 20-4 Helical spring under axial load.

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SPRINGS SPRINGS

Particular

20.9

Formula

HELICAL SPRINGS (Fig. 20-4)5

FIGURE 20-5 Stress factors for a helical spring. (V. L. Maleev and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954.)

The more accurate formula for shear stress

The stress factor or Wahl factor6 k to be used in Eq. (20-2) The spring index The value of stress factor k may be approximated very closely (between c ¼ 2 and 12) by the relation Size factor

8FDk d 3 Refer to Fig. 20-5 for values of k.

ð20-22Þ

¼

4c 1 0:615 þ 4c 4 c D c¼ d 0:25 2 d k ¼ 0:25 ¼ 2 D c

ð20-23Þ

k¼

ksz ¼

1 d 0:25 ¼ esz 0:335

ð20-24Þ ð20-25Þ

SI

ð20-26aÞ

USCS

ð20-26bÞ

SI

ð20-26cÞ

where d in mm d 0:25 0:85 where d in in

ksz ¼

d 0:25 1:89 where d in mm

Ksz ¼

The shear stress taking into consideration k from Eq. (20-25) The angular deﬂection

¼

16FD0:75 5:1Fc0:75 ¼ d 2:75 d2

ð20-27Þ

¼

16FDl d 4 G

ð20-28Þ

The length of the spring wire

l ¼ iD ðapprox:Þ

The axial deﬂection of the whole spring

y¼

8FD3 i iD2 iD2:25 ¼ ¼ kdG 2d 1:25 G d 4G

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ð20-28aÞ ð20-29Þ

SPRINGS

20.10

CHAPTER TWENTY

Particular

Formula

SPRING SCALE (Fig. 20-6)

FIGURE 20-6 Relation between loads and deﬂection in a helical spring.

Force to compress the spring 1 m (mm) (stiﬀness of spring) The total deﬂection (Fig. 20-6)

Fo ¼

F d4G GD Gd ¼ ¼ ¼ y 8iD3 8ic4 8ic3

ð20-30Þ

y2 ¼

F2 y0 F2 ¼ Fo F2 F1

ð20-31Þ

U¼

Fy 4F 2 D3 i 2 d 2 iD 2 ¼ ¼ 2 16k2 G d4G

RESILIENCE The resilience U of a spring is equal to energy absorbed

2 d 1:5 D1:5 i 2 1:55d 2 c1:5 i 2 ¼ 64G G V 2 V 2 D 0:5 Vc0:5 U¼ 2 ¼ ¼ 16G 4k G 16G d U¼

y2 d 4 G 16iD3 V ¼ Dið 14 d 2 Þ ¼

Volume of the spring

ð20-32aÞ ð20-32bÞ

ð20-32cÞ ð20-33Þ

RECTANGULAR SECTION SPRINGS (Fig. 20-7a)5 Shear stress

kFDð1:5h þ 0:9bÞ FD0:75 ð3h þ 1:8bÞ ¼ ð20-34aÞ b2 h2 b1:75 h2 kFDð1:5 þ 0:9mÞ FD0:75 ð3 þ 1:8mÞ 0 ¼ ¼ ð20-34bÞ m2 h3 m1:75 h2 b where m ¼ h 0 ¼

FIGURE 20-7(a) Spring with rectangular section.

The uncorrected shear stress for a rectangular section spring

0 ¼

FD kl bh2

ð20-35Þ

where k1 ¼ factor depending on b=h ¼ m, which is given in Table 20-9 The stress factor

k¼

4c 1 0:615 þ 4c 4 c

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ð20-36Þ

SPRINGS

FIGURE 20-7(b) Minimum tensile strengths of spring wire. (Associated Spring, Barnes Group Inc., Bristol, Connecticut)

FIGURE 20-7(c) Modiﬁed Goodman diagram for Belleville washers; for carbon and alloy steels at 47 to 49 Rc with set removed, but not shotpeened (Associated Spring, Barnes Group Inc., Bristol, Connecticut.)

FIGURE 20-7(d) Deﬂection for helical compression spring at various loads. (Richard M. Phelan, Fundamentals of Mechanical Engineering Design, New Delhi, 1975, courtesy of TataMcGraw-Hill.)

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SPRINGS

20.12

CHAPTER TWENTY

TABLE 20-5 Width of leaf springs for motor vehicle suspension— SAE J510C in

mm

in

mm

in

mm

1.75 2.00 2.25

44.4 50.8 57.2

2.50 3.00 3.50

63.5 76.2 88.9

4.00 5.00 6.00

101.6 127.0 152.4

Source: Reproduced from SAE Handbook, Vol. I, 1981.

TABLE 20-6 Standard sections of steel plates for laminated springs (railway rolling stocks) Width, mm

50

63

75

90

100

115

120

125

140

150

Thickness, mm

10, 13

6, 8, 10, 11, 13

6, 8, 10, 11, 13, 16

6, 8, 10, 11, 16, 19

8, 10, 11, 13, 16, 19

10, 11, 13, 16, 19

16, 19

10, 13, 16

11, 13

11, 13, 16

TABLE 20-7 Constant in Eqs. (20-13) to (20-18) Constant

Cantilever

Simply supported

12 2þr 12 2þr

3 2þr 3 4ð2 þ rÞ

6

1.5

2

1 8

0

0

Particular

The value of stress k may be approximated very closely by the relation The spring index

Formula

2 c0:25 8 D > > < b c¼ > D > : h k¼

ð20-37Þ if b < h

ð20-38Þ

if h < b

where b ¼ breadth of spring wire, m (mm) h ¼ thickness of spring wire, m (mm) The deﬂection

y¼

2:83iFD3 ðb2 þ h2 Þ b3 h3 G

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ð20-39Þ

SPRINGS SPRINGS

Particular

20.13

Formula

The deﬂection for an uncorrected spring of rectangular cross section

y¼

Force required to compress the spring by one meter (millimeter) (i.e., the spring rate)

Fo ¼

m3 h4 G 2:83iD3 ð1 þ m2 Þ

ð20-41Þ

The spring rate for an uncorrected rectangular section spring

Fo ¼

4Gb3 h k1 D3 i

ð20-42Þ

iFD3 k2 bh3 G

for h < b and m > 8

ð20-40Þ

Refer to Table 20-9 for k1 .

SQUARE SECTION SPRING The shear stress, for m ¼ 1

0 ¼

2:4kFD 4:8FD0:75 ¼ h2:75 h3

ð20-43Þ

The deﬂection

y¼

5:66iFD3 h4 G

ð20-44Þ

The approximate equivalent rectangular dimension of a rectangular cross section wire spring to restrict the solid length, which is equivalent to spring of round-wire cross section

h¼

2d 1 þ ðb=hÞ

ð20-44aÞ

The larger dimension of a keystone shape of rectangular wire after coiling

where d ¼ diameter of round wire h1 ¼ h

C þ 0:5 C

ð20-44bÞ

where h ¼ wider end of keystone section h ¼ original, smaller dimension of rectangular section The estimated solid height or length of a uniformly tapered, but not telescoping, spring with squared and ground ends made from round wire

ls ¼ iðd 2 u2 Þ1=2 þ 2d

ð20-44cÞ

where u ¼ outside diameter of large end minus outside diameter of small end divided by 2i

The increase in coil diameter due to compression of a helical spring

Do at solid ¼

D2 þ p2 d 2 2 þ d

The size coeﬃcient for sections above 12.5 mm in section for round wires

esz ¼ 0:86 þ

0:0018 d

1=2 ð20-44dÞ

SI

ð20-45aÞ

SI

ð20-45bÞ

for steel, where d in m esz ¼ 0:986 þ

0:0001 d

for monel metal, where d in m

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SPRINGS

20.14

CHAPTER TWENTY

Particular

Formula

0:07 d for steel, where d in in 0:0043 esz ¼ 0:986 þ d for monel metal, where d in in 1:8 esz ¼ 0:86 þ d for steel, where d in mm 0:1 esz ¼ 0:986 þ d for monel metal, where d in mm esz ¼ 0:86 þ

The general expression for size factor

Wire diameter

USCS

ð20-45cÞ

USCS

ð20-45dÞ

SI

ð20-45eÞ

SI ð20-45fÞ

ksz ¼ 4:66h0:35

where h in m

SI

ð20-46aÞ

ksz ¼ 1:27h0:35

where h in in

USCS

ð20-46bÞ

SI

ð20-46cÞ

ksz ¼ 0:415h0:35 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 8kFD d¼ d esz

where h in mm

ð20-47Þ

SELECTION OF MATERIALS AND STRESSES FOR SPRINGS For materials for springs7

Refer to Tables 20-8 and 20-10 and Figs. 20-7b and 20-7c.

The torsional yield strength

0:35sut sy 0:52sut for steels ð20-47aÞ 8 0:45sut cold-drawn carbon steel > > > > > < 0:50sut hardened and tempered sy ¼ a ¼ carbon and low-alloy steel > > > 0:35sut austenitic stainless steel > > : and nonferrous alloys

The maximum allowable torsional stress for static applications according to Joerres8;9;11

ð20-47bÞ where sy ¼ torsional yield strength, MPa (psi) The maximum allowable torsional stress according to Shigley and Mischke9

sy ¼ a ¼ 0:56sut

ð20-47cÞ

The shear endurance limit according to Zimmerli10

sf ¼ 310 MPa ð45 kpsiÞ

ð20-47dÞ

for unpeened springs sf ¼ 465 MPa ð67:5 kpsiÞ

ð20-47eÞ

for peened springs The torsional modulus of rupture

su ¼ 0:67sut

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ð20-47f Þ

SPRINGS

20.15

SPRINGS

TABLE 20-8 Spring design stress, d , MPa (kpsi) Severe service

Average service

Light

Wire diameter, mm

MPa

kpsi

MPa

kpsi

MPa

kpsi

2.15 2.15–4.70 4.70–8.10 8.10–13.45 13.45–24.65 24.65–38.10

413.8 379.0 331.0 289.3 248.1 220.6

60 55 48 42 36 32

517.3 476.6 413.8 358.4 310.4 275.6

75 69 60 52 45 40

641.4 585.4 510.0 448.2 385.9 344.7

93 85 74 65 56 50

TABLE 20-9 Factors for helical springs with wires of rectangular cross section Ratio b=h ¼ m Factor k Factor k2

1 0.416 0.180

1.2 0.438 0.212

1.5 0.462 0.250

2.0 0.492 0.292

2.5 0.516 0.317

3 0.534 0.335

Particular

The weight of the active coil of a helical spring

For free-length tolerances, coil diameter tolerances, and load tolerances of helical compression springs

5 0.582 0.371

10 0.624 0.398

1 0.666 0.424

Formula

2 d 2 Di ð20-47gÞ 4 where ¼ weight of coil of helical spring per unit volume

W¼

Refer to Tables 20-11 to 20-13.

DESIGN OF HELICAL COMPRESSION SPRINGS Design stress The size factor

ksz ¼

d 0:35 0:355

ksz ¼

d 0:25 0:84

where d in m where d in in

d 0:25 where d in mm 1:89 0:335e ds ¼ e ¼ na ksz na d 0:25

ksz ¼ The design stress

SI

ð20-48aÞ

USCS

ð20-48bÞ

SI

ð20-48cÞ

SI

ð20-49aÞ

USCS

ð20-49bÞ

where e in MPa and d in m ds ¼

e 0:84e ¼ na ksz na d 0:25

where e in psi and d in in

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0.60–0.70 0.60–0.90

0.70–1.00 0.30–0.60

C Mn

C Mn

Chrome-vanadium alloy steel (SAE 6150) AS 32 Silico-manganese alloy steel (SAE 9260) Type 18–8 stainless (Type 302, SAE 30915)

Hot-rolled bars SAE 1095, ASTM A14–42

AS 20

C Mn Cr V C Mn Si C Ni C Mn Si

C Mn

Mn

0.45–0.55 0.50–0.80 0.80–1.10 0.15–0.18 0.55–0.65 0.60–0.90 1.80–2.20 17–20 7–10 0.08–0.15 2 max 0.30–0.75

0.90–1.05 0.25–0.50

0.60–0.70 1034–2068 0.90–1.20

0.85–0.95 0.25–0.60

0.65–0.80 0.50–0.90

C Mn

C Mn

0.90–1.05 0.20–0.50

C Mn

Hard-drawn spring wire (ASTM A227–47) C

High–carbon wire AS 8 Oil-tempered wire (ASTM A229–41) AS10 Music wire (ASTM A228–47) AS 5

Clock spring steel AS 100 SAE 1095 Flat spring steel AS 101 SAE 1074

C Mn

Watch spring steel

1.10–1.19 0.15–0.25

Element %

Material

Analysis

20.16

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2275

1103

1725

1377

1206–1377

150–300

1725–3790

1068–2059

1382–1725

1103–2206

1240–2343

2274–2412

Mpa

1.58

1.24

0.73–0.97

100–200

1.03–2.41

0.83–1.73

1.10–1.45

0.86–1.93

1.03–2.14

2.14–2.28

GPa

180–230

105–140

1 50–350

120–250

160–210

125–280

150–310

310–330

kpsi

Elastic limit

207

196

200

207

200

207

207

207

220

GPa

C42–46

C40–46

28

30

C35–45

C42–48

Alloy and Stainless Spring Materials

28.5

Hot-rolled Special Steel

29

30

29

Not used

1653

828

1206

965

760 965

1515

828

1034 2069

794 1377

1103 1377

Annealed, B70–85 Not used tempered C38–50

C40–52

Carbon Steel Wires 30 C44–48

30

30

Not used

0.90

0.69

0.51 0.76

0.90

0.51

0.62 1.24

0.55 0.90

0.76 1.03

Not used

Not used

Not used

GPa

100–130

75 110

75–130

90–180

80–130

110–150

kpsi

Elastic limit

79

72

79

79 82

79

79

Not used

Not used

Not used

GPa

120–240

0.97

0.31

45–140 69

10

11.5

10.5

11.5

11.5 12.0 depending on size

11. 5

11.5

Mpsi

Modulus in torsion, G

Torsional properties of wire

About the same as chrome vanadium

140–175

110–140

120–220

150–300

115–200

160–200

kpsi

Ultimate strength Rockwell hardness MPa

Flat Cold-rolled Spring Steel 32 C55–55

Mpsi

Modulus of elasticity, E

About the same as chrome vanadium 160–330 0.41 60–260 193 1.79

200–250

175–200

0.69–1.38

250–500

155–300

200–250

160–320

180–340

330–350

kpsi

Ultimate strength

Tensile properties

TABLE 20-10 Chemical composition and mechanical properties of spring materials

Best corrosion resistance, fair temperature resistance

Used as a lower–cost material in place of chrome vanadium

Cold–rolled or drawn: special applications

Hot-rolled heavy coil or ﬂat springs

but lower-quality wire

Same uses as music wire

Miscellaneous small springs of various types— high quality

General spring use

High-grade helical springs or wire forms

Miscellaneous ﬂat springs

Main springs for watches and similar uses Clock and motor springs, miscellaneous ﬂat springs for high stress

Chief uses

SPRINGS

64 26 2.5 2.25 80 14 Balance 66 29 2.75 0.90 98

Ni Cu Mn Fe Ni Cr Fe Ni Cr Al Fe Ni Cu Mn Fe Si Cu Be

98 2

Small amounts

2–3 Small amounts balance

94–96 4–6

7–9

56 25 18

64–74 balance

12–14 0.25–0.40

Si Sn or Mn Cu

Cu Zn Ni Cu Sn or Cu Sn

Cu Zn

Cr C

1103 1377

1583

160–200

180–230

160–180

1103 1241

100–140

140–175

1241

0.76

100–150

0.55 0.76

0.27 0.41

0.90 1.38

0.41

80–110

130–200

103

60–110

110

107

193

0.69 1.03

1.17

0.90

0.79 1.00

0.76 0.93

0.55 0.83

100–150

130–170

115–145

110–135

80–120

110 127

207

179

213

179

28

C42–47

B90–100

B95–100

B90

16–18.5 Subject heat treatment

30

26

31

26

to C35–42

C36–46

C33–40

C30–40

C23–28

Nonferrous Spring Materials

15

16

15

Nonferrous Spring Materials

Properties similar to those of phosphor bronze

691

130–150

100–130

170–250

965 1206

691 964

102

91–93

897 1034

691 897

1171 1725

691 897

1034

828

725 862

651 828

519 760

725

554

588 691

308 622

828 1240

Note: The property values given in this table do not specify the minimum properties. Source: Handbook of Mechanical Spring Design, courtesy Associated Spring, Barnes Group Inc., Bristol, Connecticut.

Beryllium-coppcr AS 45 AS 145

Z–nickel

Inconel AS 40 AS140 K–Monel AS 40 AS 140

Silicon bronze (made under various trade names) AS 46 AS 146 Monel AS 40 AS 140

Phosphor bronze AS 60 AS 160

Nickel silver

Spring brass AS 55 AS 155

Cutlery-type stainless (Type 420)

0.59

0.35

0.41 0.48

0.21 0.41

0.55 0.83

50–85

60–70

30–60

80–120

43

38

38

76

6.25

5.5

5.5

11

100–130

120–150

105–125

95–120

75–110

0.45 0.66

0.68

0.41

0.45 0.58

0.38 0.55

0.31 0.48

65–95

60–90

65–85

55–80

45–70

11

9.5

11

9.5

41 6–7 48 Subject to heat treatment

76

65

76

65

Properties similar to those of phosphor bronze

80–105

85–100

45–90

120–180

Corrosion resistance like copper; high physical properties for electrical work; low hysteresis

Resists corrosion; high stresses to 2888C

Resists corrosion; high stresses to 2328C

Resists corrosion; high stresses to 3438C

Resists corrosion; moderate stresses to 204.58C

Used as substitute for phosphor bronze

Used for corrosion resistance and electrical conductivity

For electrical conductivity at low stresses; for corrosion resistance Used for its color; corrosion resistance

Resists corrosion when polished; good temperature resistance

SPRINGS

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20.17

SPRINGS

20.18

CHAPTER TWENTY

TABLE 20-11 Free-length tolerances of squared and ground helical compression springsa Tolerances: mm/mm (in/in) of free length Spring index (D=d) Number of active coils per mm (in)

4

6

8

10

12

14

16

0.02 (0.5) 0.04 (1) 0.08 (2) 0.2 (4) 0.3 (8) 0.5 (12) 0.6 (16) 0.8 (20)

0.010 0.011 0.013 0.016 0.019 0.021 0.022 0.023

0.011 0.013 0.015 0.018 0.022 0.024 0.026 0.027

0.012 0.015 0.017 0.021 0.024 0.027 0.029 0.031

0.013 0.016 0.019 0.023 0.026 0.030 0.032 0.034

0.015 0.017 0.020 0.024 0.028 0.032 0.034 0.036

0.016 0.018 0.022 0.026 0.030 0.034 0.036 0.038

0.016 0.019 0.023 0.027 0.032 0.036 0.038 0.040

a

For springs less than 12.7 mm (0.500 in) long, use the tolerances for 12.7 mm (0.500 in). For closed ends not ground, multiply above values by 1.7. Source: Associated Spring, Barnes Group Inc., Bristol, Connecticut.

TABLE 20-12 Coil diameter tolerances of helical compression and extension springs Tolerances: mm (in) Spring index ðD=dÞ Wire diameter, mm (in)

4

6

8

10

12

14

16

0.38 (0.015) 0.58 (0.023) 0.89 (0.035) 1.30 (0.051) 1.93 (0.076) 2.90 (0.114) 4.34 (0.171) 6.35 (0.250) 9.53 (0.375) 12.70 (0.500)

0.05 (0.002) 0.05 (0.002) 0.05 (0.002) 0.08 (0.003) 0.10 (0.004) 0.15 (0.006) 0.20 (0.008) 0.28 (0.011) 0.41 (0.016) 0.53 (0.021)

0.05 (0.002) 0.08 (0.003) 0.10 (0.004) 0.13 (0.005) 0.18 (0.007) 0.23 (0.009) 0.30 (0.012) 0.38 (0.015) 0.51 (0.020) 0.76 (0.030)

0.08 (0.003) 0.10 (0.004) 0.15 (0.006) 0.18 (0.007) 0.25 (0.010) 0.33 (0.013) 0.43 (0.017) 0.53 (0.021) 0.66 (0.026) 1.02 (0.040)

0.10 (0.004) 0.15 (0.006) 0.18 (0.007) 0.25 (0.010) 0.33 (0.013) 0.46 (0.018) 0.58 (0.023) 0.71 (0.028) 0.94 (0.037) 1.57 (0.062)

0.13 (0.005) 0.18 (0.007) 0.23 (0.009) 0.30 (0.012) 0.41 (0.016) 0.53 (0.021) 0.71 (0.028) 0.90 (0.035) 1.17 (0.046) 2.03 (0.080)

0.15 (0.006) 0.20 (0.008) 0.28 (0.011) 0.38 (0.015) 0.48 (0.019) 0.64 (0.025) 0.84 (0.033) 1.07 (0.042) 1.37 (0.054) 2.54 (0.100)

0.18 (0.007) 0.25 (0.010) 0.33 (0.013) 0.43 (0.017) 0.53 (0.021) 0.74 (0.029) 0.97 (0.038) 1.24 (0.049) 1.63 (0.064) 3.18 (0.125)

Source: Associated Spring, Barnes Group Inc., Bristol, Connecticut.

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SPRINGS SPRINGS

20.19

TABLE 20-13 Load tolerances of helical compression springs Tolerance: % of load, start with tolerance from Table 20-11 multiplied by LF Deﬂection from free length to load, mm (in) Length tolerance mm (in)

1.27 2.54 3.81 5.08 6.35 7.62 10.2 12.7 19.1 25.4 38.1 50.8 76.2 102 152 (0.050) (0.100) (0.150) (0.200) (0.250) (0.300) (0.400) (0.500) (0.750) (1.00) (1.50) (2.00) (3.00) (4.00) (6.00)

0.13 (0.005) 0.25 (0.010) 0.51 (0.020) 0.76 (0.030) 1.0 (0.040) 1.3 (0.050) 1.5 (0.060) 1.8 (0.070) 2.0 (0.080) 2.3 (0.090) 2.5 (0.100) 5.1 (0.200) 7.6 (0.300) 10.2 (0.400) 12.7 (0.500)

12 — — — — — — — — — — — — — —

7 12 22 — — — — — — — — — — — —

6 8.5 15.5 22 — — — — — — — — — — —

5 7 12 17 22 — — — — — — — — — —

— 6.5 10 14 18 22 25 — — — — — — — —

— 5.5 8.5 12 15.5 19 22 25 — — — — — — —

— 5 7 9.5 12 14.5 17 19.5 22 25 — — — — —

— — 6 8 10 12 14 16 18 20 22 — — — —

— — 5 6 7.5 9 10 11 12.5 14 15.5 — — — —

— — — 5 6 7 8 9 10 11 12 22 — — —

— — — — 5 5.5 6 6.5 7.5 8 8.5 15.5 22 — —

— — — — — — 5 5.5 6 6 7 12 17 21 25

— — — — — — — — 5 5 5.5 8.5 12 15 18.5

— — — — — — — — — — — 7 9.5 12 14.5

— — — — — — — — — — — 5.5 7 8.5 10.5

First load test at not less than 15% of available deﬂection; ﬁnal load test at not more than 85% of available deﬂection. Source: Associated Spring, Barnes Group Inc., Bristol, Connecticut.

TABLE 20-14 Equations for springs with diﬀerent types of ends2,3

Particular Active coils, i Total coils, i

0

i0 lo d p

i0 12 lo p

i0 2 lo 3d p

i0 2 lo 2d þ2 p

Free length, lo or lf

ip þ d

ip

ip þ 3d

ip þ 2d

Pitch, p

lo d i0

lo i0

lo 3d i0

lo 2d i0

Solid height, h

dði0 þ 1Þ

dði0 þ 12Þ

dði0 þ 1Þ

i0 d

Source: K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I, Suma Publishers, Bangalore, India, 1986, and K. Lingaiah, Machine Design Data Handbook, Vol. 11, Suma Publishers, Bangalore, India, 1986.

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SPRINGS

20.20

CHAPTER TWENTY

Particular

Formula

TABLE 20-15 Curvature factor kc c kc

3 1.35

4 1.25

ds ¼ 6 1.15

7 1.13

8 1.11

9 1.1

10 1.09

The actual factor of safety or reliability factor

e 1:89e ¼ na ksz na d 0:25

Metric

ð20-49cÞ

where e in kgf/mm2 and d in mm where na ¼ actual factor of safety or reliability factor na ¼ na ¼

FðcompressedÞ FðworkingÞ

ð20-50aÞ

free length fully compressed length free length working length yþa ¼ ð20-50bÞ y

where y is deﬂection under working load, m (mm), a is the clearance which is to be added when determining the free length of the spring and is made equal to 25% of the working deﬂection The wire diameter for static loading

Generally na is chosen at 1.25. 6na F 0:4 0:3 d ¼ 1:445 D e n F 0:4 0:3 ¼ 2:945 a D e

SI

ð20-51aÞ

where F in N, e in MPa, D in m, and d in m 6na F 0:4 0:3 d ¼ 0:724 D e n F 0:4 0:3 ¼ 1:48 a D Metric ð20-51bÞ e where F in kgf, e in kgf/mm2 , D in mm, and d in mm 6na F 0:4 0:3 d¼ D e n F 0:4 0:3 ¼ 2:05 a D USCS ð20-51cÞ e The wire diameter where there is no space limitation ðD ¼ cdÞ

where F in lbf, e in psi, D in in, and d in in n F 0:57 0:43 d ¼ 4:64 a c SI ð20-51dÞ e where d in m, F in N, e in Pa 6na F 0:57 0:43 c d¼ e

USCS

where d in in, F in lbf, e in psi

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ð20-51eÞ

SPRINGS SPRINGS

Particular

20.21

Formula

n F 0:57 0:43 d ¼ 1:77 a c e

Metric ð20-51fÞ

where d in mm, F in kgf, e in kgf/mm2

Final dimensions (Fig. 20-7d) yd 4 G ydG kydG ¼ ¼ 8FD3 8Fc3 D2

The number of active coils

i¼

The minimum free length of the spring

lf ði þ nÞd þ y þ a

ð20-52Þ ð20-53Þ

where a ¼ clearance, m (mm) n ¼ 2 if ends are bent before grinding ¼ 1 if ends are either ground or bent ¼ 0 if ends are neither ground nor bent Outside diameter of cod of helical spring

Do ¼ D þ d

ð20-53aÞ

Solid length (or height) of helical spring

ls ¼ it d

ð20-53bÞ

Pitch of spring

p¼

ys þd i

ð20-53cÞ

Free length of helical spring lf or lo

lf ls þ ys

ð20-53dÞ

Maximum working length of helical spring

lmax ¼ lf ymax

ð20-53eÞ

Minimum working length of helical spring

lmin ¼ lf ymin

ð20-53fÞ

Springs with diﬀerent types of ends1;2;3

Refer to Table 20-14.

where it ¼ total number of coild in the spring

STABILITY OF HELICAL SPRINGS The critical axial load that can cause buckling

Fcr ¼ Fo Kl lf

ð20-54Þ

where Kl is factor taken from Fig. 20-8

FIGURE 20-8 Buckling factor for helical compression springs. (V. L. Maleev and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954.)

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SPRINGS

20.22

CHAPTER TWENTY

Particular

The equivalent stiﬀness of springs

The critical load on the spring

The critical deﬂection is explicitly given by

Formula

ðEIÞspring ¼

Fcr ¼

ycr lf

Ed 4 l 32iDð2 þ vÞ

ð20-55Þ

2 Ed 4 32ð2 þ vÞiDðlf ycr Þ

2

ycr 2 1 þ v þ lf 2 2þv

D lf

ð20-56Þ 2 ¼0

ð20-57Þ

where l ¼ ðlf ycr Þ

REPEATED LOADING (Fig. 20-9) The variable shear stress amplitude

8D Fmax Fmin 2 d 3 where kw ¼ k kc

a ¼ kw

ð20-58Þ

Refer to Table 20-15 for kc .

FIGURE 20-9 Cyclic stresses in spring. (K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962; K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I, Suma Publishers, 1986; K. Lingaiah, Machine Design Data Handbook, Vol. II, Suma Publishers, Bangalore, India, 1986.)

The mean shear stress

8D Fmax þ Fmin 2 d 3 where k ¼ 1 þ 0:5=c

m ¼ k

ð20-59Þ

Design equations for repeated loadings1;2;3 Method 1 The Gerber parabolic relation

a þ od

m ud

2 ¼1

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ð20-60Þ

SPRINGS SPRINGS

Particular

20.23

Formula

The Goodman straight-line relation

a þ m ¼1 od ud

ð20-61Þ

The Soderberg straight-line relation

a þ m ¼1 od yd

ð20-62Þ

Method 2 The static equivalent of cyclic load Fm Fa

Fm0 ¼ Fm þ

sd F o a

ð20-63aÞ

sd F fd a

ð20-63bÞ

or Fm0 ¼ Fm þ The relation between e and f for brittle material

e ¼ 2f

ð20-64Þ

The static equivalent of cyclic load for brittle material

Fm0 ¼ Fm þ 2Fa

ð20-65Þ

The relation between Fm0 , Fmax and Fmin

Fm0 ¼ 12 ð3Fmax Fmin Þ

ð20-66Þ

The diameter of wire for static equivalent load

3na ð3Fmax Fmin Þ 0:4 0:3 D d ¼ 1:45 e

SI

ð20-67aÞ

where F in N, e in MPa, D in m, and d in m 3na ð3Fmax Fmin Þ 0:4 0:3 D USCS ð20-67bÞ d¼ e where F in lbf, e in psi, D in in, and d in in 3na ð3Fmax Fmin Þ 0:4 0:3 d ¼ 0:724 D e Metric

ð20-67cÞ

where F in kgf, e in kgf/mm , D in mm, and d in mm 3na ð3Fmax Fmin Þ 0:57 0:43 c SI ð20-68aÞ d ¼ 1:67 e 2

The wire diameter when there is no space limitation ðD ¼ cdÞ

where F in N, e in MPa, and d in m 3na ð3Fmax Fmin Þ 0:57 0:43 c USCS d¼ e

ð20-68bÞ

where F in lbf, e in psi, and d in in 3na ð3Fmax Fmin Þ 0:57 0:43 c d ¼ 0:64 e Metric

ð20-68cÞ

where F in kgf, e in kgf/mm , and d in mm 2

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SPRINGS

20.24

CHAPTER TWENTY

Particular

Formula

CONCENTRIC SPRINGS (Fig. 20-10)

The relation between the respective loads shared by each spring, when both the springs are of the same length

F1 ¼ F2

The relation between the respective loads shared by each spring, when both are stressed to the same value

F 1 D2 ¼ F 2 D1

The approximate relation between the sizes of two concentric springs wound from round wire of the same material

F1 ¼ F2

D3 D1

D2 D1

3

d1 d2

d1 d2

3

4

i2 G 1 i1 G 2

ð20-69Þ

k1 k2

0:75

d1 d2

ð20-70Þ 2:5 ð20-71Þ

where suﬃxes 1 and 2 refer, respectively, to springs 1 and 2 (Fig. 20-10)

FIGURE 20-10 Concentric spring.

Total load on concentric springs The total maximum load on the spring The load on the inner spring The load on the outer spring

F ¼ F1 þ F2

ð20-72Þ

F2 ¼ mF1

ð20-73Þ

F ð20-74Þ 1þm where m 1 and F maximum spring load, kN (lbf)

F1 ¼

VIBRATION OF HELICAL SPRINGS The natural frequency of a spring when one end of the spring is at rest

rﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2k0 g k0 ¼ 0:705 fn ¼ W 2 W where

SI

fn ¼ natural frequency, Hz W ¼ weight of vibrating system, N k0 ¼ scale of spring, N/m g ¼ 9:8066 m=s2

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ð20-75Þ

SPRINGS SPRINGS

Particular

20.25

Formula

1=2 k fn ¼ 22:3 0 W

SI

where k0 in N/mm, W in N, fn in Hz, g ¼ 9086:6 mm=s2 1=2 k fn ¼ 4:42 0 USCS W

ð20-75aÞ

ð20-75bÞ

where k0 in lbf/in, W in lbf, fn in Hz, g ¼ 32:2 ft=s2 1=2 k fn ¼ 1:28 0 USCS ð20-75cÞ W

The natural frequency of a spring when both ends are ﬁxed

where k0 in lbf/in, W in lbf, fn in Hz, g ¼ 386:4 in=s2 rﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2k0 g k0 fn ¼ SI ¼ 1:41 W W where k0 in N/m, W in N, fn in Hz, g ¼ 9:0866 mm=s2 1=2 k fn ¼ 44:6 0 SI W where k0 in N/mm, W in N, fn in Hz, g ¼ 9086:6 mm=s2 1=2 k fn ¼ 2:56 0 USCS W

ð20-76Þ

ð20-76aÞ

ð20-76bÞ

where k0 in lb/ft, W in lbf, fn in Hz, g ¼ 32:2 ft=s2 1=2 k fn ¼ 8:84 0 USCS ð20-76cÞ W

The natural frequency for a helical compression spring one end against a ﬂat plate and free at the other end according to Wolford and Smith7 Another form of equation for natural frequency of compression helical spring with both ends ﬁxed without damping eﬀect

where k0 in lbf/in, W in lbf, fn in Hz, g ¼ 386:4 in=s2 k g 1=2 fn ¼ 0:25 0 W fn ¼

1:12ð103 Þd D2 i

Gg

ð20-76dÞ

1=2 SI

ð20-76eÞ

SI

ð20-76f Þ

where G ¼ shear modulus, MPa g ¼ 9:8006 m=s2 d and D in mm, fn in Hz, in g/cm3 fn ¼

3:5ð105 Þd D2 i

for steel

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SPRINGS

20.26

CHAPTER TWENTY

Particular

Formula

fn ¼

0:11d D2 i

Gg

1=2 USCS

ð20-76gÞ

USCS

ð20-76hÞ

where G ¼ modulus of rigidity, psi g ¼ 386:4 in=s2 d and D in in, fn in Hz, in lbf/in2 fn ¼

14ð103 Þd D2 i

for steel

STRESS WAVE PROPAGATION IN CYLINDRICAL SPRINGS UNDER IMPACT LOAD The velocity of torsional stress wave in helical compression springs

Gg 1=2 V ¼ 10:1

SI ð20-76iÞ

where V in m/s, G in MPa, g ¼ 9:8066 m=s2 , in g/cm3 Gg 1=2 V ¼ USCS ð20-76jÞ where V in in/s, G in psi, g ¼ 386:4 in=s2 , in lbf=in3 The velocity of surge wave (Vs ) The impact velocity (Vimp )

(It varies from 50 to 500 m/s.) g 1=2 Vimp ¼ 10:1 2G m=s for steel Vimp ¼ 35:5 g 1=2 Vimp ¼ 2G Vimp ¼

The frequency of vibration of valve spring per minute

131

in=s

SI

SI USCS ð20-76lÞ

for steel

rﬃﬃﬃﬃﬃﬃ k0 fn ¼ 84:627 W where k0 in N/m, W in N rﬃﬃﬃﬃﬃﬃ k0 fn ¼ 2676:12 W where k0 in kgf/mm, W in kgf rﬃﬃﬃﬃﬃﬃ k0 fn ¼ 530 W where k0 in lbf/in, W in lbf

ð20-76kÞ

USCS SI

ð20-77aÞ

Metric

ð20-77bÞ

USCS

ð20-77cÞ

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SPRINGS

20.27

SPRINGS

Particular

Formula

HELICAL EXTENSION SPRINGS (Fig. 20-11 to 20-13) For typical ends of extension helical springs

Refer to Fig. 20-11.

The maximum stress in bending at point A (Fig. 2012)

A ¼

Type

16K1 DF 4F þ 2 d 3 d

ð20-78aÞ

Recommended length min.–max.

Conﬁgurations

Twist loop or hook

0.5–1.7 I.D.

Cross center loop or hook

I.D.

Side loop or hook

0.9–1.0 I.D.

Extended hook

1.1 I.D. and up, as required by design

Special ends

As required by design

FIGURE 20-11 Common-end conﬁguration for helical extension springs. Recommended length is distance from last body coil to inside of end. ID is inside diameter of adjacent coil in spring body. (Associated Spring, Barnes Group, Inc.)

FIGURE 20-12 Location of maximum bending and torsional stresses in twist loops. (Associated Spring, Barnes Group, Inc.)

The constant K1 in Eq. (20-78a)

The constant C1 in Eq. (20-78b)

K1 ¼

4C2 C1 1 4C1 ðC1 1Þ

ð20-78bÞ

C1 ¼

2R1 d

ð20-78cÞ

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SPRINGS

20.28

CHAPTER TWENTY

Particular

Formula

For R1 , refer to Fig. 20-12. The maximum stress in torsion at point B (Fig. 20-12)

B ¼

8DF 4C2 1 d 3 4C2 4

2R2 d For R2 , refer to Fig. 10-12.

The constant C2 in Eq. (20-78d)

C2 ¼

For extension helical spring dimensions

Refer to Fig. 20-13.

ð20-78dÞ ð20-78eÞ

In practice C2 may be taken greater than 4.

FIGURE 20-13 Typical extension-spring dimensions. (Associated Spring, Barnes Group, Inc.)

For design equations of extension helical springs

The design equations of compression springs may be used.

The spring rate

k0 ¼

F Fi Gd 4 ¼ y 8D3 i

ð20-78fÞ

where Fi ¼ initial tension The stress

¼

k8FD d 3

ð20-78gÞ

where k ¼ stress factor for helical springs Refer to Fig. 20-5 for k.

CONICAL SPRINGS [Fig. 20-14(a)] The axial deﬂection y for i coils of round stock may be computed by the relation [Fig. 20-14(a)]

The axial deﬂection of a conical spring made of rectangular stock with radial thickness b and an axial dimension h [Fig. 20-14(c)]

y¼

2iFðD32 þ D22 D1 þ D2 D21 þ D31 Þ d 4G

ð20-79Þ

y¼

iðD32 þ D22 D1 þ D2 D21 þ D31 Þ 4dD2 kG

ð20-80Þ

y¼

0:71iFðb2 þ h2 ÞðD32 þ D22 D1 þ D2 D21 þ D31 Þ b3 h3 G ð20-81Þ

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SPRINGS SPRINGS

Particular

20.29

Formula

FIGURE 20-14 Conical and volute springs.

NONMETALLIC SPRINGS Rectangular rubber spring (Fig. 20-15)

Approximate overall dimension of the shock absorber can be obtained by (Fig. 20-15)

L E ¼ 2 2 D 2F

Spring constant K of an absorber

D2 E L L1 ¼ 0:75L

Dimensions of sleeve and core are found by empirical relations

U ðFmax =FÞ2 1

K¼

ð20-82Þ ð20-83Þ ð20-84Þ

D1 ¼ 0:70D

ð20-85Þ

D2 ¼ 1:12D1

ð20-86Þ

¼

Mt F þ Z A

ð20-87Þ

¼

k0 Mt 2Mt þ Z DA

ð20-88Þ

y¼

Mt LD 2EI

ð20-89Þ

FIGURE 20-15 Rectangular rubber spring.

TORSION SPRINGS (Fig. 20-16)7 The maximum stress in torsion spring The stress in torsion spring taking into consideration the correction factor k0 The deﬂection The stress in round wire spring

8Mt ð4k0 D þ dÞ ð20-89aÞ d 3 D where k0 ¼ k1 can be taken from curve k1 in Fig. 20.5

¼

The torsional moment Mt is numerically equal to bending moment Mb .

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SPRINGS

20.30

CHAPTER TWENTY

Particular

Formula

FIGURE 20-16 Common helical torsion-spring end conﬁgurations. (Associated Spring, Barnes Group, Inc.)

The stress is also given by Eq. (20-90) without taking into consideration the direct stress (F/A)

¼k

The expressions for k for use in Eq. (20-90)

k ¼ ko ¼

4C 2 þ C 1 4CðC þ 1Þ

for outer fiber

ð20-91aÞ

k ¼ ki ¼

4C2 C 1 4CðC 1Þ

for inner fiber

ð20-91bÞ

Mb c I where Mb ¼ Fr

ð20-90Þ

Equation (20-90) for stress becomes

¼ ki

The angular deﬂection in radians

¼

The spring rate of torsion spring

k0 ¼

Mb d4E ¼ 64Di

ð20-94Þ

The spring rate can also be expressed by Eq. (20-95), which gives good results

k00 ¼

d 4E 10:8Di

ð20-95Þ

32Fr d 3

ð20-92Þ

64Mb Di Ed 4

ð20-93Þ

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SPRINGS SPRINGS

Particular

Formula

The allowable tensile stress for torsion springs sy ¼ a ¼

The endurance limit for torsion springs

20.31

8 0:78sut > > > > > 0:87 > sut > < > > > > > 0:61sut > > :

cold-drawn carbon steel hardened and tempered carbon and low-alloy steels stainless steel and nonferrous alloys

sf ¼ 538 MPa (78 kpsi)

Torsion spring of rectangular cross section The stress in rectangular wire spring

6k0 Mt 2Mt ð20-96Þ þ Dbh b2 h where k0 ¼ k2 can be taken from curve k2 in Fig. 20-5 D c¼ ð20-97Þ h

Axial dimension b after keystoning

b1 ¼ b

Another expression for stress for rectangular crosssectional wire torsion spring without taking into consideration the direct stress ( ¼ F=A)

¼

¼

C 0:5 C

6ki Mb bh2

where ki ¼ The spring rate

k0 ¼

ð20-98Þ ð20-99Þ

4C 4C 3

Mb Ebh3 ¼ 66Di

ð20-100Þ

FIGURE 20-17 Torsion bar spring

Torsion bar springs For allowable working stresses for rubber compression springs

Refer to Tables 20-16 and 20-17 and Fig. 20-17. Refer to Table 20-18.

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SPRINGS

20.32

CHAPTER TWENTY

TABLE 20-16 Design formulas for bar springs

TABLE 20-17 Factors for computing rectangular bars in torsion

Cross section of bar

Angular deﬂection, , rad

Maximum shear stress,

Solid circular bar

584Mt l d4G 584Mt l ðd14 d24 G

16Mt d 3 16Mt d1 ðd14 d24 Þ

407Mt l b4 G 57:3Mt l 0 k1 bh3 G

4:81Mt b3 Mt k02 2bh2

Hollow circular bar Square bar Rectangular bar

a

a

a

b=h

k0

k01

k02

1.0 1.2 1.5 2.0 2.5 3.0 4.0 5.0 10.0 1

0.675 0.759 0.848 0.930 0.968 0.985 0.997 0.999 1.000 1.000

0.140 0.166 0.196 0.229 0.249 0.263 0.281 0.291 0.312 0.333

0.208 0.219 0.231 0.246 0.258 0.267 0.282 0.291 0.231 0.333

Values of k01 and k02 can be obtained from Table 20-9.

TABLE 20-18 Suggested allowable working stresses for rubber compression springs Limits of allowable stress Occasional loading

Cont. or freq. loadingb

Durometer hardness

Areaa ratio

MPa

psi

MPa

psi

30 30 30 30 30 50 50 50 50 80 80 80

5 3 2 1 0.5 4 2 1 0.5 2 1 0.5

2.76 2.48 2.24 1.79 1.45 4.82 3.73 2.69 2.07 6.13 4.14 2.90

400 360 325 260 210 700 540 390 300 890 600 420

0.97 0.93 0.86 0.73 0.62 1.86 1.58 1.24 1.03 2.69 2.07 1.65

140 135 125 105 90 270 230 180 150 390 300 240

a

Ratio of load-carrying area available for bulging or lateral expansion

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SPRINGS SPRINGS

20.33

REFERENCES 1. Lingaiah, K. and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962. 2. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 3. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 4. SAE Handbook, Springs, Vol. I, 1981. 5. Maleev, V. L., and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954. 6. Wahl, A. M., Mechanical Springs, McGraw-Hill Book Company, New York, 1963. 7. Associated Spring, Barnes Group Inc., Bristol, CT, USA. 8. Jorres, R. E., Springs; Chap. 24 in J. E. Shigley and C. R. Mischke, eds., Standard Handbook of Machine Design, McGraw-Hill Book Company, New York, 1986. 9. Shigley, J. E., and C. R. Mischke, Mechanical Engineering Design, 5th ed. McGraw-Hill Company, New York, 1989. 10. Zimmerli, F. P., Human Failures in Springs Applications, The Mainspring, No. 17, Associated Spring Corporation, Bristol, Connecticut, Aug.-Sept. 1957. 11. Shigley, J. E., and C. R. Mischke, Standard Handbook of Machine Design, McGraw-Hill Book Company, New York, 1986. 12. Phelan, R. M., Fundamentals of Mechanical Design, Tata-McGraw-Hill Publishing Company Ltd, New Delhi, 1975. 13. Lingaiah, K., Machine Design Data Handbook of Machine Design, 2nd edition, McGraw-Hill Publishing Company, New York, 1996). 14. Shigley, J. E., and C. R. Mischke, Standard Handbook of Machine Design, 2nd edition, McGraw-Hill Publishing Company, New York, 1996.

BIBLIOGRAPHY Baumeister, T., ed., Marks’ Standard Handbook for Mechanical Engineers, McGraw-Hill Book Company, New York, 1978. Black, P. H., and O. Eugene Adams, Jr., Machine Design, McGraw-Hill Book Company, New York, 1968. Bureau of Indian Standards. Chironis, N. P., Spring Design and Application, McGraw-Hill Book Company, 1961. Norman, C. A., E. S. Ault, and I. F. Zarobsky, Fundamentals of Machine Design, The Macmillan Company, New York, 1951. Shigley, J. E., Machine Design, McGraw-Hill Book Company, 1962.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

21 FLEXIBLE MACHINE ELEMENTS SYMBOLS11;12;13 a a1 A ¼ 0:4ðd 2 =4Þ b c C

C1 d

d1 d2 da da1 da2 dc ¼ fp Fb df dp dr D Dr Dd

width of pulley face, m (in) pivot arm length in Rockwood drive, m (in) width of belt, m (in) useful area of cross-section of the wire rope, m2 (in2 ) thickness of arm, m (in) dimension in Rockwood drive (Fig. 21-5), m (in) dimension in Rockwood drive (Fig. 21-5), m (in) center distance between sprockets (also with suﬃxes), m (in) center distance between pulleys, m (in) capacity of conveyor, m3 (ft3 ) constant depends on the rope diameter, sheave diameter, chain, the bearing, and coeﬃcient of friction [Eqs. (21-59) to (21-62) and (21-86) to (21-103)] (also with suﬃxes) tooth width in precision roller and bush chains, m (in) size of chain, m (in) diameter of shaft, m (in) diameter of idler bearing, m (in) diameter of smaller pulley, m (in) diameter of rope, m (in) pitch diameter of sprocket, m (in) diameter of small sprocket, m (in) hub diameter of pulley, m (in) diameter of large sprocket, m (in) tip diameter of sprocket, m (in) tip diameter of small sprocket, m (in) tip diameter of large sprocket, m (in) equivalent pitch diameter, m (in) root diameter of sprocket, m (in) pitch diameter of the V-belt small pulley, m (in) diameter of roller pin, m (in) pitch diameter of sheave, m (in) diameter of large pulley, m (in) wire rope drum diameter, m (in) (Fig. 21-4) diameter of reel barrel, m (in) Eq. (21-76) diameter of the drum in mm as measured over the outermost layer ﬁlling the reel drum

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FLEXIBLE MACHINE ELEMENTS

21.2

CHAPTER TWENTY-ONE

diameter of the sheave pin, m (in) unit elongation of belt corrected elasticity modulus of steel ropes (78.5 GPa ¼ 11.4 Mpsi), GPa (psi) F force, load, kN (lbf ) tension in belt, kN (lbf ) minimum tooth side radius, m (in) Fa correction factor for instructional belt service from Table 21-27 Fc correction factor for belt length from Table 21-26 Fct centrifugal tension, kN (lbf ) Fd correction factor for arc of contact of belt from Table 21-25 tangential force in the belt, required chain pull, kN (lbf ) F Fs tension due to sagging of chain, kN (lbf ) F1 tension in belt on tight side, kN (lbf ) tension in belt on slack side, kN (lbf ) F2 Fc centrifugal force, kN (lbf ) values of coeﬃcient for manila rope, Table 21-32 FR1 the minimum value of tooth ﬂank radius in roller and bush chains, m (in) FR2 the maximum value of tooth ﬂank radius in roller and bush chains, m (in) g acceleration due to gravity, 9.8066 m/s2 (32.2 ft/s2 ) G tooth side relief in bush and roller chain, m (in) h the thickness of wall of rope drum, m (in) crown height, m (in) h1 depth of groove in rope drum, m (in) H ¼ ðDd Dr Þ=2 depth of rope layer in reel drum, m (in) i number of arms in the pulley, number of V-belts, number of strands in a chain, transmission ratio k ¼ ðe 1Þ=e variable in Eqs. (21-2d), (21-4a), (21-6), and (21-123), which depends on ðz1 z2 Þ=Cp kd duty factor kl load factor Kmin center distance constant from Table 21-57 ks service factor coeﬃcient for sag from Table 21-55 ksg l width of chain or length of roller, m (in) minimum length of boss of pulley, m (in) minimum length of bore of pulley, m (in) length of conveyor belt, m (in) length of cast-iron wire rope drum, m (in) outside length of coil link chain, m (in) K1 tooth correction factor for use in Eq. (21-116a) K2 multistrand factor for use in Eq. (21-116a) L length of ﬂat belt, m (in) pitch length of V-belt, m (in) rope capacity of wire rope reel, m (in) Lp length of chain in pitches Mt torque, N m (lbf in) n number of times a rope passes over a sheave, number of turns on the drum for one rope member speed, rpm factor of safety Do e E0

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FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

n1 n2 n0 ¼ nkd P PT p p1 P Pb Ps Pt Pu Pw Q r s s S SA1 SA2 SR1 SR2 t T TDmin TDmax v w W WB wc WI WL z1 z2 1 2 c br c

speed of smaller pulley, rpm or rps speed of smaller sprocket, rpm or rps speed of larger pulley, rpm or rps speed of larger sprocket, rpm or rps stress factor power, kW (hp) power required by tripper, kW (hp) pitch of chain, m (in) pitch of the grooves on the wire rope drum, m (in) distance between the grooves of two-rope pulley, m (in) eﬀort, load, kN (lbf ) bending load, kN (lbf ) service load, kN (lbf ) tangential force due to power transmission, kN (lbf ) ultimate load, kN (lbf ) breaking load, kN (lbf ) working load, kN (lbf ) load, kN (lbf ) radius near rim (with subscripts), m (in) radius, m (in) the amount of shift of the line of action of the load from the center line on the raising load side of sheave, m (in) the average shift of the center line in the load on the eﬀort side of the sheave, m (in) the distance through which the load is raised, m (in) the minimum value of roller or bush seating angle, deg the maximum value of roller or bush seating angle, deg the minimum value of roller or bush seating radius, m (in) the maximum value of roller or bush seating radius, m (in) nominal belt thickness, m (in) thickness of rim, m (in) tension in ropes, chains, kN (lbf ) minimum limit of the tooth top diameter, m (in) maximum limit of the tooth top diameter, m (in) velocity of belt chain, m/s (ft/min) speciﬁc weight of belt, kN/m3 (lbf/in3 ) width between reel drum ﬂanges, m (in) weight of belt, kN/m (lbf/in) weight of chain, kN/m (lbf/in) weight of revolving idler, kN/m (lbf/in) belt load, kN/m (lbf/in) number of teeth on the small sprocket number of teeth on the large sprocket stress, MPa (psi) unit tension in belt on tight side, MPa (psi) unit tension in belt on slack side, MPa (psi) centrifugal force coeﬃcient for leather belt, MPa (psi) breaking stress for hemp rope, MPa (psi) shear stress, MPa (psi) arc of contact, rad angle between tangent to the sprocket pitch circle and the center line, deg coeﬃcient of friction between belt and pulley coeﬃcient of journal friction coeﬃcient of chain friction

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21.3

FLEXIBLE MACHINE ELEMENTS

21.4

CHAPTER TWENTY-ONE

!1 !2

eﬃciency angular speed of small sprocket, rad/s angular speed of large sprocket, rad/s

SUFFIXES bending breaking torque compressive design minimum maximum

b br t c d min max

Other factors in performance or in special aspects of design of ﬂexible machine elements are included from time to time in this chapter and being applicable only in their immediate context, are not given at this stage. Particular

Formula

BELTS Flat belts The ratio of tight side to slack side of belt at low velocities The power transmitted by belt

F1 ¼ e F2 P¼

ð21-1Þ

F v 1000cs

SI

ð21-2aÞ

where F ¼ F1 F2 , P in kW, and v in m/s; F in N P¼

F v 33;000cs

USCS

ð21-2bÞ

where F in lbf; P in hp; v in ft/min P¼

F !r 1000cs

SI

ð21-2cÞ

where F in N, P in kW, r in m, and ! in rad/s Refer to Table 21-1 for cs . Power transmitted per m2 (in2 ) of belt at low velocities

P¼

1 kv 1000

SI

ð21-2dÞ

where k ¼ ðe 1Þ=e , and also from Table 21-2 1 in N/m2 , v in m/s, and P in kW P¼

1 kv 33;000

USCS

where 1 in psi, v in ft/min, and P in hp

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ð21-2eÞ

FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

21.5

TABLE 21-1 Service correction factors, cs Atmospheric condition

Clean, scheduled maintenance on large drives Normal Oily, wet, or dusty Horizontal to 608 from horizontal 608–758 from horizontal 758–908 from horizontal Fiber on motor and small pulleys Cast iron or steel Temporary or infrequent Normal Intermittent or continuous Light, steady load, such as steam engines, steam turbines, diesel engines, and multicylinder gasoline engines Jerky loads, reciprocating machines such as normal-starting-torque squirrelcage motors, shunt-wound, DC motors, and single-cylinder engines Shock and reversing loads, full-voltage start such as squirrel-cage and synchronous motors

Angle of center line

Pulley material Service

Peak loads

1.2 1.0 0.7 1.0 0.9 0.8 1.2 1.0 1.2 1.0 0.8 1.0 0.8 0.6

TABLE 21-2 Values of ðe 1Þ=e ¼ k for various coeﬃcients of frictions and arcs of contact Arc of contact between the belt and pulley (, deg) Value of

90

100

110

120

130

140

150

160

170

180

200

0.28 0.30 0.33 0.35 0.38 0.40 0.43 0.45 0.48 0.50 0.53

0.356 0.376 0.404 0.423 0.449 0.467 0.491 0.507 0.529 0.544 0.565

0.387 0.408 0.438 0.457 0.485 0.502 0.528 0.544 0.567 0.582 0.603

0.416 0.438 0.469 0.489 0.518 0.536 0.562 0.579 0.602 0.617 0.638

0.444 0.467 0.499 0.520 0.549 0.567 0.593 0.610 0.634 0.649 0.670

0.470 0.494 0.527 0.548 0.578 0.597 0.623 0.640 0.663 0.678 0.700

0.496 0.520 0.554 0.575 0.605 0.624 0.650 0.667 0.690 0.705 0.726

0.520 0.544 0.579 0.600 0.630 0.649 0.676 0.692 0.715 0.730 0.750

0.542 0.567 0.602 0.624 0.654 0.673 0.699 0.715 0.738 0.752 0.772

0.564 0.590 0.624 0.646 0.676 0.695 0.721 0.737 0.759 0.773 0.793

0.585 0.610 0.645 0.667 0.697 0.715 0.741 0.757 0.779 0.792 0.811

0.502 0.553 0.684 0.705 0.735 0.753 0.777 0.792 0.813 0.825 0.843

TABLE 21-3 Values of coeﬃcients c for leather belts for use in Eqs. (21-3) and (21-4) Belt velocity, m/s (ft/min) Coeﬃcient, c , kgf/cm2 MPa psi

7.5 (1500) 10.0 (1950) 12.70 (2500) 15.0 (2950) 17.5 (3500) 20.0 (3950) 22.5 (4450) 25.0 (4950) 0.57 1.05 1.63 2.35 3.10 4.07 5.14 6.36 0.0559 0.1030 0.1598 0.2305 0.3040 0.3991 0.5041 0.5237 8.0 15.0 23.2 33.5 45.0 58.0 73.0 76.0

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FLEXIBLE MACHINE ELEMENTS

21.6

CHAPTER TWENTY-ONE

Particular

The ratio of tight to slack side of belt at high velocities

Formula

1 c ¼ e 2 c where c ¼

Power transmitted per m2 (in2 ) of belt at high velocities

P¼

ð21-3aÞ wv2 g

ð1 2 Þkv 1000

ð21-3bÞ SI

ð21-4aÞ

where 1 and c in N/m2 ; v in m/s; P in kW P¼

ð1 c Þkv 33;000

USCS

ð21-4bÞ

where 1 and c in psi; v in ft/min; P in hp Refer to Table 21-3 for values of c . Equation (21-3a) in terms of tension on tight side (F1 ) and slack side of belt (F2 ), and centrifugal force (Fc )

F1 Fc ¼ e F2 Fc

ð21-4cÞ

where F1 ¼ 1 A; F2 ¼ 2 A; Fc ¼ c A; A ¼ a1 t ¼ area of cross section of belt, m2 (in2 ) The relation between the initial tension in the belt (F0 ) and the tension in the belt on the tight side (F1;max ) to obtain maximum tension in the belt

F1;max ¼ 2F0

The power transmitted at maximum tension in belt, i.e., when F1 ¼ 2F0 , from Eq. (21-1)

P¼

F1;max v 2F0 v ¼ 33;000 33;000

P¼

F1;max v 2F0 v ¼ 1000 1000

P¼

2Kp Kv Fa v 33;000Cs

The power transmitted in actual practice taking into consideration pulley correction factor (Kp ), velocity correction factor (Kv ), and service factor (Cs ) at maximum tension in belt.

Stresses in belt (Fig. 21-1c)

ð21-4dÞ

USCS

ð21-4eÞ

SI

ð21-4f Þ

USCS

ð21-4gÞ

2Kp Kv Fa v SI ð21-4hÞ 1000Cs where Fa ¼ allowable tension in belt, N (lbf) v ¼ velocity of belt, m/s (ft/min)

P¼

Tensile stress due to tension on tight side of belt F1 ðS1 Þ

1 ¼

F2 a1 t

ð21-4iÞ

Tensile stress due to tension on slack side of belt F2 ðS2 Þ

2 ¼

F2 a1 t

ð21-4jÞ

¼

F a1 t

ð21-4kÞ

Tensile stress due to tangential force (eﬀective stress)

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FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

Particular

The tensile stress due to belt tension on account of centrifugal force

21.7

Formula

c ¼

Fc v2 ¼ a1 t 9810

ð21-4lÞ

where ¼ specific weight of belt material N/dm3 (lbf/in3 ) The bending stress

b ¼

Fb d

ð21-4mÞ

The maximum belt stress

max ¼ 1 þ c þ b þ tw a

Stress due to twist in belt

tw ¼ E

a1 a

ð21-4nÞ

2 ð21-4pÞ

for crossed belt

¼ 0 for open belt Ea1 D ¼ for half-crossed belt 2a2 where a ¼ distance from centre of bigger pulley diameter to the point of twist of half-crossed belt and crossed belt >2D a ¼ allowable stress in belt, MPa (psi) For distribution of various stresses in belt

Refer to Fig. 21-1C. Refer to Table 21-4B for most commonly used belt materials in practice. The values of Kp and Cs are Table from Tables 21-4C and 21-4D, and Kv from Fig. 21-1B, and also Table 21-4E for minimum pulley sizes. Fa ¼ allowable tension in belt, N (lbf ) v ¼ velocity of belt, m/s (ft/min)

Coeﬃcient of friction ()

¼ 0:54

0:7 2:4 þ v

SI

ð21-5Þ

may also be obtained from Tables 21-4A and 21-5. v ¼ velocity of belt, m/s. ¼ 0:54

140 500 þ v

USCS

where v ¼ velocity of belt, ft/min

For leather belts and belts of similar material c is of importance only if v > 15%.

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ð21-5aÞ

FLEXIBLE MACHINE ELEMENTS

21.8

CHAPTER TWENTY-ONE

TABLE 21-4A Coeﬃcients of frictions of leather belts on iron pulleys depending on velocity of belt Velocity of belt, v, m/s

Coeﬃcient of friction,

Velocity of belt, v, m/s

Coeﬃcient of friction,

Velocity of belt, v, m/s

Coeﬃcient of friction,

0.25 0.50 1.00 1.50 2.00 2.50 3.00 3.50

0.360 0.285 0.307 0.340 0.365 0.384 0.400 0.413 0.423

4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 12.5

0.432 0.440 0.446 0.458 0.456 0.473 0.479 0.494 0.493

15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5

0.500 0.505 0.509 0.512 0.514 0.517 0.519 0.520

TABLE 21-4B Properties of some ﬂat and round materials Minimum pulley diameter, in

Allowable tension per unit width at 600 ft/min, lb/in

Weight, lb/in3

Coeﬃcient of friction

Material

Speciﬁcation

Size, in

Leather

1 ply

t ¼ 11 64

3

30

0.035–0.045

0.4

t ¼ 13 64

3 12

33

0.035–0.045

0.4

t ¼ 18 64

4 12

41

0.035–0.045

0.4

t¼

6a

50

0.035–0.045

0.4

9a

60

0.035–0.045

0.4

10 35 60 60 100 175 275

0.035 0.035 0.051 0.037 0.042 0.039 0.039

0.5 0.5 0.5 0.8 0.8 0.8 0.8

2 ply

t¼

20 64 23 64

Polyamide

F-0 F-1c F-2c A-2c A-3c A-4c A-5c

t ¼ 0:03 t ¼ 0:05 t ¼ 0:07 t ¼ 0:11 t ¼ 0:13 t ¼ 0:20 t ¼ 0:25

0.60 1.0 2.4 2.4 4.3 9.5 13.5

Urethaned

w ¼ 0:50 w ¼ 0:75 w ¼ 0:125 Round

t ¼ 0:062 t ¼ 0:078 t ¼ 0:090 d ¼ 14

See Table 17-4E See

5.2e 9.8e 18.9e 8.3e

0.038–0.045 0.038–0.045 0.038–0.045 0.038–0.045

0.7 0.7 0.7 0.7

d ¼ 38 d ¼ 12

Table 17-4E

18.6e 33.0e

0.038–0.045 0.038–0.045

0.7 0.7

74.3e

0.038–0.045

0.7

b

c

d ¼ 14 a

Add 2 in to pulley size for belts 8 in wide or more. Source: Habasit Engineering Manual, Habasit Belting, Inc., Chamblee (Atlanta), Ga. c Friction cover of acrylonitrile-butadiene rubber on both sides. d Source: Eagle Belting Co., Des Plaines, Ill. e At 6% elongation; 12% is maximum allowable value. Notes: d ¼ diameter, t ¼ thickness, w ¼ width. The values given in this table for the allowable tension are based on a belt speed of 600 ft/min. Take Kv ¼ 1:0 for polyamide and urethane belts. Source: Eagle Belting Co., Des Plaines, Illinois; table reproduced from J. E. Shigley and C. R. Mischke, Mechanical Engineering Design, McGrawHill Book Company, New York, 1989. b

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FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

21.9

TABLE 21-4C Pulley correction factor KP for ﬂat beltsa Small-pulley diameter, in Material

1.6–4

4.5–8

9–12.5

14, 16

18–31.5

>31.5

Leather polyamide, F-0 F-1 F-2 A-2 A-3 A-4 A-5

0.5 0.95 0.70 0.73 0.73 —

0.6 1.0 0.92 0.86 0.86 0.70 —

0.7 1.0 0.95 0.96 0.96 0.87 0.71 —

0.8 1.0 1.0 1.0 1.0 0.94 0.80 0.72

0.9 1.0 1.0 1.0 1.0 0.96 0.85 0.77

1.0 1.0 1.0 1.0 1.0 1.0 0.92 0.91

a

Average values of KP for the given ranges were approximated from curves in the Habasit Engineering Manual, Habasit Belting, Inc., Chamblee (Atlanta), Ga. Source: Eagle Belting Co., Des Plaines, Illinois; table reproduced from J. E. Shigley and C. R. Mischke, Mechanical Engineering Design, McGrawHill Book Company, New York, 1989.

TABLE 21-4D Service factors Cs for V-belt and ﬂat belt drives Power source Driven machinery

Normal torque characteristic

High or nonuniform torque

Uniform Light shock Medium shock Heavy shock

1.0–1.2 1.1–1.3 1.2–1.4 1.3–1.5

1.1–1.3 1.2–1.4 1.4-1.6 1.5–1.8

Source: Eagle Belting Co., Des Plaines, Illinois; table reproduced from J. E. Shigley and C. R. Mischke, Mechanical Engineering Design, McGrawHill Book Company, New York, 1989.

TABLE 21-4E Minimum pulley sizes for ﬂat and round urethane belts (pulley diameters in inches) Ratio of pulley speed to belt length, rev/(ft-min) Belt style

Belt size, in

Up to 250

250 to 499

500 to 1000

Flat

0:50 0:062 0:75 0:078 1:25 0:090

0.38 0.50 0.50

0.44 0.63 0.63

0.50 0.75 0.75

Round

1 4 3 8 1 2 3 4

1.50 2.25 3.00 5.00

1.75 2.62 3.50 6.00

2.00 3.00 4.00 7.00

Source: Eagle Belting Co., Des Plaines, Illinois; table reproduced from J. E. Shigley and C. R. Mischke, Mechanical Engineering Design, McGrawHill Book Company, New York, 1989.

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FLEXIBLE MACHINE ELEMENTS

21.10

CHAPTER TWENTY-ONE

TABLE 21-5 Coeﬃcient of friction for belts depending on materials of pulley and belt Pulley material Cast iron/steel Belt material

Dry

Wet

Greasy

Wood

Compressed paper

Leather face

Rubber face

Leather, oak-tanned Leather, chrome-tanned Canvas, stitched Cotton, woven Camel hair, woven Rubber Balata

0.25 0.35 0.20 0.22 0.35 0.30 0.32

0.20 0.32 0.15 0.15 0.25 0.18 0.20

0.15 0.22 0.12 0.12 0.20 — —

0.30 0.40 0.23 0.25 0.40 0.32 0.35

0.33 0.45 0.25 0.28 0.45 0.35 0.38

0.38 0.48 0.27 0.27 0.45 0.40 0.40

0.40 0.50 0.30 0.30 0.45 0.42 0.42

TABLE 21-6A Thickness and width of leather belts Average thickness, mm Grade

Single

Light

3

Medium Heavy

Double

Width, mm

Triple

Quadruple

Range

6

—

—

12–24 24–102 102–198

3 6 12

4

8

12.5

17.5

200–800 800–1400

25 50

5

10

15

20

800–1400 1500–2100

50 100

TABLE 21-6B Relative strength of belt joints

Type of joint

Relative strength of joint to an equal section of solid leather, eﬃciency, %

Cemented, endless Cemented at factory

90–100

Cemented in shop

80–90

Laced, wire By machine By hand Rawhide, small holes Rawhide, large holes

75–85 70–80 60–70 50–60

Hinged Wire hooks Metal hooks

40 35–40

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Increment

FLEXIBLE MACHINE ELEMENTS

21.11

FLEXIBLE MACHINE ELEMENTS

Particular

Formula

The cross section of the belt is given

1000P a1 t ¼ wv2 v d k g

ð21-6aÞ

SI

where P in kW, v in m/s, g ¼ 9:8066 m/s2 , w in N/m3 , and d in MPa 33;000P a1 t ¼ wv2 4 10 k v d g

ð21-6bÞ

USCS

where P in hp, v in ft/min, g ¼ 386:4 in/s2 ¼ 32:2 ft/s2 , w in lbf/in3 , and d in psi Refer to Tables 21-6A to 21-14. For cross section and properties of belts

TABLE 21-7 Standard widths of transmission belting for diﬀerent plies Standard width, mm Ply

25

32

40

44

50

63

76

90

100

112

125

140

152

180

200

224

250

305

355

400

3 4 5 6 8

pa q — — —

qb q — — —

p p — — —

q q — — —

p p

p p

— —

— —

p p p — —

q p q — —

q p p q —

— p p p —

— p p p —

— q — — —

— p p p —

— — rc p —

— q q p r

— — r — —

— — r r r

— — — — r

— — — — r

— — — — r

p ¼ these sizes are available in Hi-speed and Fort. q ¼ these sizes are available in Hi-speed only. c r ¼ these sizes are available in Fort only. a

b

TABLE 21-8 Widths of friction surface—rubber transmission belting Nominal belt width 103 m

Tolerance 103 m

25, 32, 40, 50, 63 71, 80, 90, 100, 112, 125 140, 160, 180, 200, 224, 250 280, 315, 355, 400, 450, 500

2.0 3.0 4.0 5.0

Source: IS 1370, 1965.

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FLEXIBLE MACHINE ELEMENTS

21.12

CHAPTER TWENTY-ONE

TABLE 21-9 Thickness of friction surface—rubber transmission belting Ply construction

Nominal thickness hard-type fabric 103 m

Tolerance 103 m

3 4 5 6 7 8

3.9 5.1 6.4 7.7 9.1 10.4

0.5 0.7 0.8 0.9 1.0 1.1

Source: IS 1370, 1964.

TABLE 21-10 Properties of leather belting for various purposes Purpose Power transmission

Properties Tensile strength, min

MPa kpsi

Breaking strength, min

N lbf

Temporary elongation, %, max Permanent elongation, %, max Stitch tear resistance thickness, min Grain strength

General

Single belts

Double belts

Splices single and double

20.6 3.0

24.5 3.5

24.5 3.5

20.6 3.0

Round belting for small machine Heavy (5)

Regular (6)

Heavy (7)

441 100

667 150

755 170

6 2 N/m lbf/in

83,356 475 Shall not crack

—

TABLE 21-11 Tensile strength of fabric in ﬁnished rubber transmission belting Tensile strength, N/m (kgf/mm) of width Weight of fabric per square meter

Warp

Weft

Type of fabric

N/m2

kgf/m2

N/m

kgf/mm

N/m

kgf/mm

Soft Hard Soft Hard

8.0 8.8 9.1 3.6

0.815 0.900 0.930 0.975

61,291.3 61,291.3 69,626.9 73,549.7

6.25 6.25 7.10 7.50

29,419.8 35,303.8 32,361.8 44,129.7

3.00 3.60 3.30 4.50

Source: IS 1370, 1965.

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A ¼ warp. b B ¼ weft.

Leather Light Medium Heavy Canvasstitched Balata Rubber

Belt material

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— —

— 14.7 17.7 —

kN/ m

— —

— 1.5 1.8 —

kgf/ mm

— —

16.7 24.5 28.4 —

kN/ m

1.7 2.5 2.9 — — —

B

4.9 7.8

— — 35.3 —

kN/ m

0.5 0.8

— — 3.6 —

kgf/ mm

— 12.1 14.8 17.4 — 118.7 145.1 170.6 — 25.0 31.8 38.6 — 245.2 311.8 378.5 15 8

— 26.4 32.1 37.5 — 258.9 314.8 397.7

A

1AA

kgf/ mm

TABLE 21-13 Allowable tension in width of belt

a

Percentage elongation at break

Tear strength in N for the number of. plies

Bb

— 11.2 13.7 15.9 — 109.8 134.4 155.9 — 20.4 27.2 34.0 — 200.1 266.7 333.4 15 8

— 23.0 28.0 32.7 — 225.6 274.0 320.7

Tensile strength in kgf/mm width for number of plies Tensile strength in N-m 103 width for number of plies Tear strength in kgf for the number of plies

3 4 5 6 3 4 5 6 3 4 5 6 3 4 5 6

Aa

1A

Direction

Belt designation B

6.9 10.8

— — — 6.9

kN/ m

0.7 1.1

— — — 0.7

kgf/ mm

— 14.8 18.0 21.1 — 145.1 176.5 206.9 — 29.5 36.3 43.1 — 289.3 356.0 422.7 15 8

— 32.1 39.3 45.7 — 314.8 385.4 448.2

A

1B B

8.8 12.7

— — — 8.8

kN/ m

B

— 21.3 26.1 — — 209.9 255.0 — — 90.8 113.4 — — 890.4 1112.1 — 17 18

— 39.3 48.0 — — 385.4 470.7 —

A

2A B

— 24.1 29.5 — — 236.3 289.3 — — 104.3 131.4 — — 1022.8 1288.6 — 17 18

— 44.7 54.3 — — 438.4 532.5 —

A

2B

0.9 1.3

— — — 0.9

kgf/ mm

10.8 15.7

— — — 10.8

kN/ m

1.1 1.6

— — — 1.1

kgf/ mm

11.8 18.6

— — — —

kN/ m

1.2 1.9

— — — —

kgf/ mm

13.7 22.6

— — — 11.8

kN/ m

Ply or number of thickness of belt

— 18.6 22.7 26.4 — 182.4 222.6 258.9 — 36.3 45.4 54.4 — 356.0 445.2 533.5 15 8

— 38.6 47.1 55.0 — 378.5 461.9 539.4

A

1C

TABLE 21-12 Properties of ply woven ﬁre-resistant conveyor belting for use in coal mines

B

1.4 2.3

— — — 1.2

kgf/ mm

25.5 25.5

— — — —

kN/ m

21.4 27.9 34.4 — 209.9 273.6 333.4 — 90.8 117.9 149.7 — 890.4 1156.2 1468.0 — 17 18

39.3 51.1 62.2 — 385.4 501.1 610.0 —

A

2C

2.6 2.6

— — — —

kgf/ mm

— 57.2 87.7 — — 560.9 860.0 —

A

3A

28.4 28.4

— — — 13.7

kN/ m

— 21.4 26.1 — — 209.9 256.0 — — — — — — — — — —

B

2.9 2.9

— — — 1.4

kgf/ mm

30.4 30.4

— — — —

3.1 3.1

— — — —

B

33.3 33.3

— — — 15.7

kN/ m

3.4 3.4

— — — 1.6

kgf/ mm

89.3 28.6 116.1 37.2 141.1 45.0 — — 875.7 280.5 1138.5 364.8 1383.7 441.3 — — — — — — — — — — —

A

3C

kgf/ mm

21.4 27.9 34.0 — 209.9 273.6 333.4 — — — — — — — — — —

B

kN/ m

62.5 81.3 99.1 — 612.9 797.3 971.8 —

A

3B

FLEXIBLE MACHINE ELEMENTS

21.13

FLEXIBLE MACHINE ELEMENTS

21.14

CHAPTER TWENTY-ONE

Particular

Formula

BELT LENGTHS AND CONTACT ANGLES FOR OPEN AND CROSSED BELTS (Fig. 21-1A) Length of belt for open drive (Fig. 21-1(A)a) Length of belt for crossed drive (Fig. 21-1(A)b) Length of belt for quarter turn drive For two-pulley open drive the center distance between the two pulleys when the length of the belt is known

L¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4C2 ðD dÞ2 ¼ 12 ðDL þ ds Þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4C2 ðD þ dÞ2 þ ðD þ dÞ 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ L ¼ ðD þ dÞ þ C2 þ D2 þ C2 þ d 2 2 L¼

C¼

l ¼ þ 2 sin1 s ¼ sin1

¼ þ 2 sin1

Dd 2C

Dd 2C

ð21-9Þ

pﬃﬃﬃ e¼ 69;000

where in psi pﬃﬃﬃ e¼ 22 where in kgf/mm pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ 2 F0 ¼ F1 þ F2

ð21-10aÞ

Dþd 2C

where in MPa pﬃﬃﬃ e¼ 21;000

The relation between initial belt tension and ﬁnal belt tension

ð21-8Þ

L 0:393ðD þ dÞ 4 " #1=2 2 L ðD dÞ2 ð21-10Þ 0:393ðD þ dÞ þ 4 8

where

The unit elongation of belt is given by the equation

ð21-7Þ

ð21-10bÞ ð21-10cÞ

SI

ð21-11aÞ

USCS

ð21-11bÞ

Metric

ð21-11cÞ

2

where F0 ¼ initial belt tension, kN (lbf )

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ð21-12Þ

FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

21.15

FIGURE 21-1(A) Open and crossed belts.

FIGURE 21-1(B) Velocity correction factor for Kv for use in Eq. (21-4g) for leather belts.

Belt stresses in open drive: f ¼ c centrifugal stress; 2 slack side stress; 1 tight side stress ¼ 2 þ n ; n eﬀective stress ¼ u ; b1 , b2 bending stresses on pulleys 1 and 2 respectively; G creep angle ( angle over which creep takes place between belt and pulley). Lectrum S2 ¼ slack side F2 ; treibend ¼ driving; Arbeitstrum S1 ¼ tight side F1 ; getrieben ¼ driven FIGURE 21-1(C) Stress distribution in belt. (G. Niemann, Maschinenelemente, Springer International Edition, Allied Publishers Private Ltd., New Delhi, 1978.)

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FLEXIBLE MACHINE ELEMENTS

21.16

CHAPTER TWENTY-ONE

Particular

Formula

PULLEYS (Fig. 21-2 and Fig. 21-3) C. G. Barth’s formula for the width of the pulley face

a ¼ 1:19a1 þ 10 mm for single belt

SI

ð21-13aÞ

a ¼ 1:1a1 þ 5 mm for double belt

SI

ð21-13bÞ

Refer to Table 21-15 for width of pulley. a ¼ 1:1875a1 þ 38 in

USCS

ð21-13cÞ

where a and a1 in in for a single belt 3 in a ¼ 1:09375a1 þ 16

C. G. Barth’s empirical formula for the crown height for wide belts

USCS

ð21-13dÞ

where a and a1 in in for double belt p ﬃﬃﬃﬃﬃ 3 h ¼ 0:00426 a2 SI

ð21-14aÞ

where a in m p ﬃﬃﬃﬃﬃ 3 h ¼ 0:013125 a2

USCS

ð21-14bÞ

Customary Metric Units

ð21-14cÞ

SI

ð21-14dÞ

Customary Metric Units

ð21-14eÞ

where a in in For rubber belts on well-aligned shafts, the crown height

For poorly aligned shafts, the crown height

h¼

a 200

h¼

a 2

h¼

a 120

a SI ð21-14fÞ 0:12 Refer to Tables 21-16, 21-17A, and 21-17B for crown height. pﬃﬃﬃﬃ t ¼ 0:25 D þ 1:5 mm ð21-15aÞ pﬃﬃﬃﬃ t ¼ 0:375 D þ 3:2 mm ð21-15bÞ h¼

The rim thickness at edge for light-duty pulley The rim thickness at edge for heavy-duty pulley for a triple belt The hub diameter of the pulley (Fig. 21-2)

d1 ¼ 1:5d þ 25 mm

ð21-16Þ

Arms The bending moment on each arm The section modulus of the arm at the hub

Mb ¼ Z¼

F D i

F D id

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ð21-17Þ ð21-18Þ

FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

Particular

21.17

Formula

FIGURE 21-2 Cast-iron pulley.

INDIAN STANDARD SPECIFICATION Cast-iron pulley ð21-19Þ

l ¼ 23 a

Minimum length of bore (Fig. 21-2)

It should not exceed a Half of the diﬀerence in diameters d1 and d2 (Fig. 21-2)

p d1 d2 3 ﬃﬃﬃﬃﬃﬃﬃ ¼ 0:412 aD þ 6 mm for a single belt 2 ð21-20Þ p d1 d2 3 ﬃﬃﬃﬃﬃﬃﬃ ¼ 0:529 aD þ 6 mm for a double belt 2 ð21-21Þ

The radius r1 near rim (Fig. 21-2)

r1 ¼ b=2

ð21-22Þ

The radius r2 near rim (Fig. 21-2)

r2 ¼ b=2

ð21-23Þ

TABLE 21-14 Properties of solid woven ﬁre-resistance conveyor belting for use in coal mines Tensile strength/width Belt designation

Direction

kN/m

kgf/mm

Percentage elongation at break

4A

Warp Weft Warp weft Warp Weft

385.4 209.9 525.6 262.8 665.9 262.8

39.3 21.4 53.6 26.8 67.9 26.8

18 19 18 19 18 19

4B 4C

Tear strength kN

kgf

1.3

136.1

1.3

136.1

1.3

136.1

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FLEXIBLE MACHINE ELEMENTS

TABLE 21-15 Width of ﬂat cast-iron and mild steel pulleys

TABLE 21-16 Crown of cast iron and mild steel ﬂat pulleys of diameters up to 355 mm

Width, mm

Tolerance, mm

Nominal diameter, D, mm

Crown, h, mm

20, 25, 32 40, 50, 63, 71 80, 90, 100, 112, 125, 140 160, 180, 200, 224, 250, 280, 315 355, 400, 450, 500, 560, 630

2

40–112 125, 140 160, 180 200, 224 250, 280 315, 355

0.3 0.4 0.5 0.6 0.8 1.0

1.5 2 3

TABLE 21-17A Crown of cast iron and mild steel ﬂat pulleys of diameters 400 to 2000 mma Crown h of pulleys of width Nominal diameter, D, mm

125

140, 160

180, 200

224, 250

280, 315

355

400

400 450 500 560 630 710 800 900 1000 1120 1250 1400 1600 1800 2000

1 1 1 1 1 1 1 1 1 1.2 1.2 1.5 1.5 2.0 2.0

1.2 1.2 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 2 2 2.5 2.5

1.2 1.2 1.5 1.5 2 2 2 2 2 2 2 2.5 2.5 3 3

1.2 1.2 1.5 1.5 2 2 2.5 2.5 2.5 2.5 2.5 3 3 3.5 3.5

1.2 1.2 1.5 1.5 2 2 2.5 2.5 3 3 3 3.5 3.5 4 4

1.2 1.2 1.5 1.5 2 2 2.5 2.5 3 3 3.5 4 4 4.5 4.5

1.2 1.2 1.5 1.5 2 2 2.5 2.5 3 3.5 4 4 5 5 6

a

All dimensions in mm. Source: IS 1691, 1968.

TABLE 21-17B Crown height and ISO pulley diameters for ﬂat belts Crown height, in ISO pulley diameter, in

Crown height, in

ISO pulley diameter, in

w 10 in

w > 10 in

1.6, 2, 2.5 2.8, 3.15 3.55, 4, 4.5 5, 5.6 6.3, 7.1 8, 9 10, 11.2

0.012 0.012 0.012 0.016 0.020 0.024 0.030

12.5, 14 12.5, 14 22.4, 25, 28 31.5, 35.5 40 45, 50, 56 63, 71, 80

0.03 0.04 0.05 0.05 0.05 0.06 0.07

0.03 0.04 0.05 0.06 0.06 0.08 0.10

Crown should be rounded, not angled; maximum roughness is Ro ¼ AA 63 min.

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FLEXIBLE MACHINE ELEMENTS

21.19

FLEXIBLE MACHINE ELEMENTS

Particular

Formula

Arms

Use webs for pulleys up to 200 mm diameter

The number of arms

i¼4

ð21-24aÞ

for pulleys above 200 mm diameter and up to 400 mm diameter i¼6

ð21-24bÞ

for pulleys above 450 mm diameter Use elliptical section rﬃﬃﬃﬃﬃﬃﬃ 3 aD b ¼ 0:294 4i rﬃﬃﬃﬃﬃﬃﬃ 3 aD for single belt b ¼ 1:6 i rﬃﬃﬃﬃﬃﬃﬃ 3 aD b ¼ 0:294 2i rﬃﬃﬃﬃﬃﬃﬃ 3 aD for double belt b ¼ 1:25 i

Cross section of arms Thickness of arm near boss (Fig. 21-2)

SI

ð21-25aÞ

USCS

ð21-25bÞ

SI

ð21-26aÞ

USCS

ð21-26bÞ

The diameter of pulleys and arms in pulleys

Refer to Tables 21-18 to 21-21.

The thickness of arm near rim

b1 —give a taper of 4 mm per 100 mm

The radius of the cross-section of arms

r ¼ 34 b

ð21-27Þ

TABLE 21-18 Minimum pulley diameters for given belt speeds and pliesa Maximum belt speeds, m/s No. of plies

10

15

20

25

30

2 3 4 5 6 7 8 9 10

50 90 140 200 250 355 450 560 630

63 100 160 224 315 400 500 630 710

80 112 180 250 355 450 560 710 800

90 140 200 315 400 500 630 800 900

112 180 250 355 450 560 710 900 1000

a

All dimensions in mm. Source: IS 1370, 1965.

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FLEXIBLE MACHINE ELEMENTS

21.20

CHAPTER TWENTY-ONE

TABLE 21-19 Diameters of ﬂat pulley and tolerances Nominal diameter, mm

Tolerance, mm

Nominal diameter, mm

Tolerance, mm

40 45, 50 56, 63 71, 80 90, 100, 112 125, 140 160, 180, 200 224, 250

0.5 0.6 0.8 1.0 1.2 1.9 2.0 2.5

280, 315, 355 400, 450, 500 560, 630, 710 800, 900, 1000 1120, 1250, 1400 1600, 1800, 2000 — —

3.0 4.0 5.0 6.3 8.0 10.22 — —

Source: IS 1691, 1968.

TABLE 21-20 Minimum pulley diameters for conveyor belting Fabric 28

Fabric 32

Fabric 36

Fabric 42

Fabric 48

No. of plies

A

B

C

A

B

C

A

B

C

A

B

C

A

B

C

>75–100% rated max working tension

2 3 4 5 6 7 8 9 10

205 305 410 510 610 690 765 915 1070

155 255 305 410 460 610 690 690 765

155 205 255 360 410 460 500 610 690

255 360 460 610 690 765 915 1070 1220

205 305 360 460 510 690 765 915 915

155 205 305 360 460 510 610 610 690

305 460 610 690 915 1070 1220 1375 1525

255 36 460 610 690 765 915 1070 1220

205 305 360 460 610 690 690 765 915

305 460 610 765 915 1070 1220 1375 1525

255 360 510 610 765 915 1020 1070 1220

205 305 410 510 610 690 765 915 1070

— 530 710 890 1065 1245 1420 1600 1780

— 460 610 760 915 1065 1220 1370 1525

— 330 510 635 760 890 1015 1145 1245

>50–75% rated max working tension

2 3 4 5 6 7 8 9 10

205 305 360 460 510 610 765 915 915

155 205 305 360 460 510 610 690 765

155 205 255 305 360 410 510 610 610

205 305 410 510 610 690 915 915 1070

155 255 305 410 510 610 690 690 915

155 205 255 360 410 460 610 610 690

255 410 510 690 765 915 1070 1220 1375

205 305 410 510 610 690 915 915 1070

155 255 360 410 510 610 690 765 915

305 460 610 765 915 1070 1220 1375 1525

255 360 460 610 690 915 915 1070 1220

205 305 410 460 610 690 765 915 915

— 430 560 710 865 990 1145 1270 1420

— 355 485 610 735 865 965 1090 1220

— 305 405 510 610 710 815 915 1015

50% rated max working tension

2 3 4 5 6 7 8 9 10

155 255 305 410 510 610 690 765 915

155 205 255 360 410 460 510 610 690

155 155 205 255 360 410 460 510 510

205 305 360 460 510 610 765 915 915

155 205 305 360 460 510 610 690 765

155 205 255 305 360 410 510 610 610

255 360 460 610 690 765 915 1070 1220

205 305 410 460 510 690 705 915 915

155 255 360 410 510 610 690 765 915

255 410 510 690 765 915 1070 1220 1220

205 305 410 510 610 690 765 915 1070

155 255 360 410 510 610 690 765 915

— 380 510 635 735 865 990 1220 1245

— 330 430 530 635 735 865 965 1065

— 280 355 455 535 635 710 815 890

Running

Source: IS 1891 (Part 1), 1968.

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FLEXIBLE MACHINE ELEMENTS

21.21

FLEXIBLE MACHINE ELEMENTS

TABLE 21-21 Number of arms in mild steel pulley Details of spokes Diameter, mm

No.

Of diameter

250–500 560–710 800–1000 1120 1250 1400 1600 1800 2000

6 8 10 12 14 16 18 18 22

19 19 22 22 22 22 22 22 22

Source: IS 1691, 1968.

Particular

Formula

Mild Steel Pulley Minimum length of boss (Fig. 21-3)

l ¼ a=2

ð21-28Þ

<100 j mm for 19-mm-diameter spokes <138 j mm for 22-mm-diameter spokes t ¼ 5 mm for diameters from 400 to 2000 mm t¼

D þ 0:003 m 200

t¼

D þ 0:12 in for single belt 200

t¼

D þ 0:006 m 200

t¼

D þ 0:24 in for double belt 200

SI

ð21-29aÞ

USCS

ð21-29bÞ

SI

ð21-29cÞ

USCS

ð21-29dÞ

FIGURE 21-3 Mild steel pulley.

The crown height

Refer to Tables 21-16, 21-17A, and 21-17B.

Arms for mild steel pulleys

Refer to Table 21-21.

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FLEXIBLE MACHINE ELEMENTS

21.22

CHAPTER TWENTY-ONE

Particular

Formula

V-BELT The formula to obtain the maximum power in kilowatt which the V-belts of sections A, B, C, D, and E can transmit (Table 21-22 and 21-23)

Belt cross section Power rating per strand, kW A

P ¼ v

B

P ¼ v

C

P ¼ v

D

P ¼ v

E

2

0:45 19:62 0:765v de v0:09 104 0:79 51:33 1:31v2 de v0:09 104

1:47 143:27 2:34v2 de v0:09 104

3:16 507:50 4:77v2 de 104 v0:09 4:57 952 7:05v2 P ¼ v 0:09 de v 104

Max. value of dc , mm

SI

125

(21-30)

175

(21-31)

300

(21-32)

425

(21-33)

700

(21-34)

where P ¼ maximum power in kW at 1808 arc of contact for a belt of average length

de ¼ dp Fb

The equivalent pitch diameter

ð21-35Þ

Refer to Table 21-24 for Fb , the small-diameter factor. TABLE 21-22 Classiﬁcation of V-belts Cross-sectional symbol

Nominal top width, W, mm

Nominal thickness, T, mm

A B C D E

13 17 22 30 33

8 11 14 12 23

Source: IS 2494, 1964.

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FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

Particular

The formulas to obtain the maximum horsepower of V-belts of A, B, C, D, and E sections

21.23

Formula

Refer to Eqs. (21-35a) to (21-35e).

Belt section A B C D E

Horsepower rating per strand (equations in USCS) 1:95 3:80 P¼V 0:0136V 2 0:09 kd V 3:43 9:83 0:0234V 2 P¼V 0:09 kd V 6:37 27:0 2 P¼V 0:0416V kd V 0:09 13:6 93:9 2 0:0848V P¼V kd V 0:09 19:9 17:8 0:122V 2 P¼V 0:09 kd V

(21-35a) (21-35b) (21-35c) (21-35d) (21-35e)

where V ¼ belt speed, thousands of ft/min k ¼ small-diameter factor for speed ratio of drive from Fig. 21-4b d ¼ pitch diameter of small sheave, in

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0.13 0.24 0.40 0.58 0.74 0.85 0.94 1.08 1.17 1.32 1.40 1.47 1.54 1.62 1.69 1.77 1.84 1.84 1.84 1.84 1.91 1.91 1.91 1.84 1.84 1.77 1.69 1.62 1.54 1.47 1.32

0.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

21.24

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0.14 0.25 0.43 0.64 0.81 0.95 1.05 1.25 1.35 1.54 1.62 1.69 1.84 1.91 1.99 2.06 2.13 2.21 2.28 2.28 2.35 2.35 2.35 2.35 2.35 2.28 2.28 2.20 2.13 2.06 1.99

0.14 0.27 0.46 0.68 0.88 1.04 1.15 1.39 1.50 1.69 1.77 1.91 2.06 2.13 2.28 2.35 2.43 2.50 2.57 2.65 2.72 2.72 2.72 2.79 2.79 2.79 2.72 2.72 2.65 2.57 2.50

0.15 0.28 0.49 0.72 0.93 1.13 1.25 1.50 1.62 1.77 1.91 2.06 2.21 2.35 2.50 2.57 2.65 2.79 2.87 2.94 3.02 3.02 3.09 3.09 3.16 3.16 3.16 3.09 3.09 3.02 2.94

0.15 0.29 0.51 0.74 0.96 1.18 1.32 1.54 1.69 1.84 1.99 2.13 2.28 2.43 2.50 2.65 2.79 2.87 2.94 3.02 3.00 3.16 3.24 3.24 3.31 3.31 3.31 3.31 3.24 3.24 3.16

0.22 0.37 0.66 0.96 1.18 1.47 1.62 1.91 2.06 2.21 2.35 2.50 2.65 2.79 2.87 2.94 3.02 3.09 3.16 3.16 3.16 3.16 3.16 3.09 3.02 2.94 2.79 2.65 2.50 2.43 2.13

0.22 0.44 0.74 1.03 1.32 1.54 1.84 2.13 2.28 2.43 2.65 2.79 3.02 3.16 3.16 3.38 3.46 3.66 3.60 3.68 3.75 3.75 3.75 3.75 3.68 3.60 3.53 3.45 3.31 3.16 2.94

0.22 0.54 0.81 1.10 1.40 1.69 1.99 2.21 2.43 2.65 2.87 3.09 3.31 3.46 3.60 3.75 3.90 3.97 4.04 4.19 4.19 4.27 4.27 4.27 4.27 4.19 4.19 4.12 3.97 3.82 3.68

0.22 0.44 0.81 1.17 1.47 1.84 2.06 2.35 2.65 2.87 3.09 3.31 3.53 3.75 3.90 4.05 4.19 4.34 4.19 4.46 4.63 4.71 4.78 4.78 4.78 4.78 4.71 4.63 4.56 4.49 4.34

0.29 0.51 0.88 1.25 1.50 1.91 2.21 2.50 2.79 3.02 3.31 3.53 3.75 3.97 4.19 4.34 4.49 4.71 4.78 4.92 5.00 5.07 5.15 5.22 5.22 5.22 5.22 5.15 5.15 5.00 4.92

0.29 0.51 0.88 1.39 1.62 1.99 2.28 2.65 2.94 3.16 3.46 3.95 3.67 4.19 4.56 4.63 4.78 4.92 5.07 5.22 5.44 5.44 5.52 5.55 5.66 5.66 5.66 5.66 5.59 5.52 5.44

0.37 0.64 1.15 1.62 2.06 2.35 2.72 3.02 3.31 3.60 3.82 4.04 4.19 4.34 4.49 4.63 4.71 4.78 4.78 4.78 4.78 4.71 4.56 4.34 4.19 4.15 3.82 3.60 3.24 3.19 2.43

0.44 0.73 1.32 1.84 2.35 2.79 3.16 3.60 3.97 4.27 4.56 4.92 5.15 5.44 5.59 5.81 5.96 6.10 6.25 6.33 6.33 6.33 6.33 6.25 6.18 6.03 5.88 5.66 5.44 5.15 4.18

0.44 0.81 1.47 2.06 2.57 3.09 3.60 4.05 4.49 4.95 5.22 5.59 5.96 6.25 6.55 6.77 7.06 7.21 7.35 7.58 7.65 7.72 7.80 7.80 7.72 7.72 7.58 7.43 7.35 7.06 6.77

0.51 0.88 1.85 2.21 2.76 3.38 3.90 4.41 4.85 4.37 5.81 6.18 6.62 6.91 7.28 7.58 7.87 8.09 8.38 8.53 8.68 8.83 8.90 8.97 9.05 9.05 8.97 8.90 8.75 8.60 8.38

0.51 0.88 1.69 2.35 3.02 3.60 4.19 4.71 5.22 5.81 6.25 6.69 7.13 7.58 7.94 8.31 8.61 8.90 9.19 9.41 9.63 9.86 10.00 10.08 10.15 10.15 10.15 10.15 10.08 9.93 9.71

0.51 0.96 1.77 2.50 3.16 3.75 4.41 5.00 5.59 6.10 6.62 7.13 7.72 8.16 8.53 8.90 9.27 9.56 9.93 10.15 10.44 10.66 10.81 11.03 11.11 11.18 11.18 11.18 11.18 11.03 10.80

280

Cross section, C

0.51 0.96 1.84 2.57 3.31 3.97 4.63 5.22 5.87 6.40 6.99 7.58 8.00 8.46 8.97 9.41 9.78 10.15 10.51 10.81 11.11 11.33 11.62 11.77 11.91 11.99 12.06 12.13 12.13 12.06 11.99

300 0.81 1.47 2.50 3.46 4.34 5.07 5.81 6.47 7.13 7.65 8.23 8.68 9.12 9.49 9.79 10.00 10.30 10.44 10.51 10.51 10.51 10.37 10.22 9.93 9.63 9.19 8.68 8.16 7.51 6.77 5.96

300 0.88 1.54 2.79 3.75 4.78 5.66 6.55 7.28 7.94 8.68 9.34 9.86 10.37 10.86 11.25 11.62 11.91 12.21 12.43 12.50 12.58 13.31 12.58 12.43 12.21 11.91 11.47 11.03 10.51 9.86 9.12

320

Equivalent pitch diameter, de , mm

100 110 120 125 130 140 150 160 170 180 180 200 220 240 260

Source: IS 2494, 1964.

0.13 0.22 0.37 0.51 0.58 0.74 0.81 0.88 0.90 1.03 1.10 1.18 1.25 1.32 1.32 1.32 1.40 1.40 1.40 1.40 1.32 1.32 1.25 1.25 1.18 1.10 1.03 0.88 0.81 0.66 0.51

90

Belt speed, m/s 80

Cross section, A

TABLE 21-23 Ratings for V-belts in kilowatts

0.96 1.69 2.94 4.04 5.15 6.10 7.06 7.94 8.68 9.49 10.22 10.88 11.47 12.06 12.88 13.09 13.46 13.83 14.12 14.27 14.49 14.56 14.56 14.56 14.34 14.27 13.97 13.53 13.09 12.58 11.91

340 0.96 1.77 3.09 4.34 5.44 6.55 7.51 8.46 9.34 10.22 11.03 11.84 12.50 13.16 13.75 14.34 14.78 15.22 15.59 15.89 16.11 16.40 16.47 16.40 16.40 16.25 16.11 15.81 15.44 15.00 14.42

360 1.03 1.84 3.24 4.56 5.74 6.91 7.94 8.97 10.00 10.96 11.84 12.58 13.39 14.05 14.56 15.44 15.96 16.55 16.99 17.36 17.65 17.87 18.02 18.17 18.24 18.17 17.87 17.80 17.43 17.14 16.70

380 1.03 1.91 3.38 4.71 6.03 7.21 8.38 9.41 10.52 11.47 12.43 13.39 14.19 15.00 15.74 16.40 16.99 17.58 18.09 18.53 18.98 19.27 19.49 19.71 19.78 19.78 19.78 19.56 19.34 19.12 18.68

400

Cross section, D

1.10 1.91 3.53 4.92 6.25 7.58 8.75 9.86 10.96 12.06 13.02 14.05 14.93 15.74 16.55 17.28 18.02 18.61 19.20 19.86 20.15 20.52 20.81 21.11 21.26 21.33 21.33 21.26 21.11 20.89 20.52

420 1.10 1.99 3.53 5.00 6.40 7.65 8.90 10.08 11.25 12.28 13.31 14.34 15.30 16.11 16.99 17.72 18.46 19.05 19.71 20.23 20.14 21.11 21.48 21.70 21.99 22.06 21.99 21.99 21.84 21.62 21.33

430 1.40 2.43 4.34 6.03 7.58 9.05 10.44 11.77 13.02 14.12 15.22 16.25 17.21 18.02 18.84 19.56 20.15 20.74 21.18 21.55 21.77 21.99 22.06 21.99 21.92 21.71 21.25 20.74 20.15 19.49 18.61

450 1.47 2.65 4.78 6.69 8.46 10.15 11.69 13.32 14.78 16.11 17.43 18.61 19.78 20.81 21.85 22.80 23.61 24.42 25.08 25.60 26.11 26.48 26.77 27.07 27.07 26.99 26.92 26.62 25.89 25.74 25.08

500

1.54 2.87 5.15 7.21 9.19 11.03 12.80 14.49 16.11 17.72 19.12 20.59 21.92 23.17 24.35 25.45 26.40 27.36 28.24 28.97 29.71 30.23 30.74 31.04 31.33 31.48 31.48 31.41 31.18 30.89 30.45

550

1.62 2.94 5.44 7.65 9.78 11.77 13.68 15.51 17.28 18.98 20.59 22.20 23.68 25.08 26.40 27.65 28.83 29.86 30.89 31.77 32.58 33.17 33.19 34.42 34.79 35.08 35.30 35.38 35.30 35.08 34.79

600

1.69 3.09 5.66 8.02 10.22 12.36 14.49 16.40 18.31 20.15 21.85 23.61 25.15 26.69 28.10 29.49 30.82 31.99 33.10 34.13 35.08 35.97 36.70 37.29 37.88 38.25 38.54 38.69 38.83 39.76 38.54

650

Cross section, E

1.77 3.24 5.88 8.38 12.74 13.02 15.07 17.14 19.05 21.11 22.95 24.71 26.40 28.02 29.57 31.04 32.44 33.76 35.01 36.19 37.22 38.17 38.98 39.72 40.38 40.89 41.46 41.56 41.78 41.78 41.70

700

FLEXIBLE MACHINE ELEMENTS

FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

TABLE 21-24 Small-diameter factor, Fb Speed ratio range

Small diameter factor

1.000–1.019 1.020–1.032 1.033–1.055 1.056–1.081 1.082–1.109 1.110–1.142 1.143–1.178 1.179–1.222 1.223–1.274 1.275–1.340 1.341–1.429 1.430–1.562 1.563–1.814 1.815–2.948 1.949

1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14

Source: IS 2494, 1964.

FIGURE 21-4(a) Factors for power rating of V-belt for use with Eqs. (21-30) to (21-35).

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21.25

FLEXIBLE MACHINE ELEMENTS

21.26

CHAPTER TWENTY-ONE

FIGURE 21-4(b) Factors for horsepower ratings of V-belts for use with Eqs. (21-35a) to (21-35e).

FIGURE 21-4(c) Correction factor K1 for angle of contact.

TABLE 21-25 Correction factors for arc of contact, Fd Correction factor (proportion of 1808 rating)

Arc of contact on smaller pulley, deg

VV

V-ﬂat

180 177 174 171 169 166 163 160 157 154 151 148 145 142 139 136

1.00 0.99 0.99 0.98 0.97 0.97 0.96 0.95 0.94 0.93 0.93 0.92 0.91 0.90 0.89 0.88

0.75 0.76 0.76 0.77 0.78 0.79 0.79 0.80 0.81 0.81 0.82 0.83 0.83 0.84 0.85 0.85

Correction factor (proportion of 1808 rating)

Arc of contact on smaller pulley, deg

VV

V-ﬂat

133 130 127 123 120 117 113 110 106 103 99 95 91 87 83

0.87 0.86 0.85 0.83 0.82 0.81 0.80 0.78 0.77 0.75 0.73 0.72 0.70 0.68 0.65

0.86 0.86 0.85 0.83 0.82 0.81 0.80 0.78 0.77 0.75 0.73 0.72 0.70 0.68 0.65

Source: IS 2494, 1964.

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21.27

TABLE 21-26 Correction factors for belt length, Fc Belt cross-section Nominal inside length, mm

A

B

C

D

610 660 711 787 813 889 914 965 991 1016 1067 1092 1168 1219 1295 1372 1397 1422 1473 1524 1600 1626 1651 1727 1778 1905 1981 2032 2057

0.80 0.81 0.82 0.84 0.85 0.87 0.87 0.88 0.88 0.89 0.90 0.90 0.92 0.93 0.94 — 0.96 0.96 0.97 0.98 0.99 0.99 1.00 1.00 1.01 1.02 1.03 1.04 1.04

— — — — — 0.81 — 0.83 — 0.84 0.85 — 0.87 0.88 0.89 0.90 0.90 0.90 — 0.92 — — 0.94 0.95 9.95 0.97 0.98 — 0.98

— — — — — — — — — — — — — — 0.80 — — — — 0.82 — — — 0.85 — 0.87 — — 0.89

— — — — — — — — — — — — — — — — — — — — — — — — — — — — —

Belt cross-section E

Nominal inside length, mm

A

B

C

D

E

— — — — — — — — — — — — — — — — — — — — — — — — — — — — —

2159 2286 2438 2464 2540 2667 2845 3048 3150 3251 3404 3658 4013 4115 4394 4572 4953 5334 6045 6807 7569 8331 9093 9855 10617 12141 13665 15189 16713

1.05 1.06 1.08 — — 1.10 1.11 1.13 — 1.14 — — — — — — — — — — — — — — — — — — —

0.99 1.00 — 1.02 1.03 1.04 1.05 1.07 — 1.08 — 1.11 1.13 1.14 1.15 1.16 1.18 1.19 — — — — — — — — — — —

0.90 0.91 0.92 — — 0.94 0.95 0.97 0.97 0.98 0.99 1.00 1.02 1.03 1.04 1.05 1.07 1.08 1.11 1.14 1.16 1.19 1.21 1.23 1.24 — — — —

— — — — — — — 0.86 — 0.87 — 0.90 0.92 0.92 0.93 0.94 0.96 0.96 1.00 1.03 1.05 1.07 1.09 1.11 1.12 1.16 1.18 1.20 1.23

— — — — — — — — — — — — — — — — — 0.94 0.96 0.99 1.01 1.03 1.05 1.07 1.09 1.12 1.14 1.17 1.19

Source: IS 2494, 1964.

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FLEXIBLE MACHINE ELEMENTS

21.28

CHAPTER TWENTY-ONE

TABLE 21-27 Correction factors for industrial service, Fa Type of driving unit AC motors; high torque, high slip repulsion induction, single phase, series-wound and slip-ring AC motors; normal torque, squirrel cage, synchronous and split phase DC motors; shunt-wound, multiple cylinder internal combustion engines >600 rpm Severity of service Light-duty

Mediumduty

Heavy-duty

Extraheavy-duty

>16 h and continuous service

DC motors; series-wound and compound wound; single-cylinder internal-combustion engines; multicylinder internal-combustion engines <600 rpm, line shafts, clutches, brakes, direct on-line starting

10 h

>10 to 16 h

>16 h and continuous service

Type of driven machines

10 h

>10 to 16 h

Agitators for liquids, blowers, and exhausters, centrifugal pumps and compressors, fans up to 7.5 kW (10 hp) and light-duty conveyors Belt conveyors for sand, grain, etc; dough mixers; fans over 7.5 kW (10 hp); generators; line shafts; laundry machinery; machine tools; punches, presses and shears; printing machinery; positive-displacement rotary pumps; revolving and vibrating screens Brick machinery, bucket elevators, exciters, piston compressors, conveyors (drag-pan-screw), hammer mills, paper mill beaters, piston pumps, positive displacement blowers, pulverizers, saw mill and woodworking machinery, and textile machinery Crushers (gyratory-jaw-roll), mills (ball-rod-tube), hoists, and rubber (calendersextruders-mills) machinery

1.0

1.1

1.2

1.1

1.2

1.3

1.1

1.2

1.3

1.2

1.3

1.4

1.2

1.3

1.4

1.4

1.5

1.6

1.3

1.4

1.5

1.5

1.6

1.8

Note: This table gives only a few examples of particular machines. If an idler pulley is used, the following values must be added to the service factors: inside: 0:1 inside: 0 Idler pulley on the tight side Idler pulley on the slack side outside: 0:2 outside: 0:1 Source: IS 2494, 1964.

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FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

21.29

TABLE 21-28 Nominal inside length, nominal pitch lengths and permissible length variations for V-belts

Nominal inside length, mm 610 660 711 787 813 889 914 965 991 1016 1067 1092 1168 1219 1295 1372 1397 1422 1473 1524

Nominal pitch length, mm Cross-section A 645 696 747 823 848 925 950 1001 1076 1051 1102 1128 1204 1255 1331 1433 1451 1509 1560

B

C

Pitch length variation D

E

PLLa

MVLb

þ11.4 6.4 þ12.5 7.5

932

2.5 1008 þ14.0 8.9

1059 1110 1212 1262 1339 1415 1440 1466 1567

þ16.0 9.0

1351

1580 5.0

1600 1626 1651 1727 1778 1905 1981 2032 2057 2159 2286 2438 2464 2540 2667 2845 3048 3150 3251 3404 3658 4013 4115 4394 4572

1636 1661 1687 1763 1814 1941 2017 2068 2093 2195 2322 2474

1694 1770 1821 1948 2024 2101 2202 2329

2703 2880 3084

2507 2583 2710 2888 3091

3287

3294

3693

3701 4056 4158 4437 4615

þ17.8 12.5 1783 1991

þ30 16

2113 2215 2342 2494

2723 2901 3104 3205 3307 3459 3713 4069 4171 4450 4628

7.5

þ34 18 3127 3330 3736 4092 4194 4473 4651

10 þ38 21 þ43 24

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12.5

FLEXIBLE MACHINE ELEMENTS

21.30

CHAPTER TWENTY-ONE

TABLE 21-28 Nominal inside length, nominal pitch lengths and permissible length variations for V-belts (Cont.) Nominal pitch length, mm, Cross section Nominal inside length, mm

A

4953 5334 6045 6807 7569 8331 9093 9855 10617 12141

Pitch length variation

B

C

D

E

4996 5377

5009 5390 6101 6863 7625 8387 9149

5032 5413 6124 5886 7648 8410 9172 9934 10696 12220

5426 6137 6899 7661 8423 9185 9947 10709 12233

37 þ76 43

13744 15268 16792

13757 15281 16805

þ89 50 þ105

PLLa

MVLb

þ49 28 þ56 32 þ65

15

17.5 13665 15189 16713

59

a

Pitch length limit. Maximum variation in length within a matched set. Source: IS 2494, 1964. b

TABLE 21-29 Dimensions for standard V-grooved pulleys

Groove section

Pitch width, lp , min

Minimum height of groove above pitch line, bmin , mm

Minimum depth of groove below pitch line, h, min, mm

Center to center distance of grooves, e, mm

A

11

3.3

8.7

15 0:3

B

14

4.2

10.1

19 0:4

C

19

5.7

14.3

25:5 0:5

D

27

8.1

19.9

37 0:6

E

32

9.6

23.4

44:5 0:7

Edge of pulley to ﬁrst groove center, f , mm þ2 1 þ2 12.5 1 þ2 17 þ1 þ3 24 1 þ4 29 1 10

Source: IS: 3142-1965.

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75 80 85 90 95 100 106 112 118 125 132 140 150 160 170 180 190 200 212 224 236 250 265 280 300 315 355

75 80 85 90 95 100 106 112 118 125 132 140 150 160 170 180 190 200 212 224 236 250 265 280 300 315 355

76.3 81.3 86.4 91.4 96.5 101.6 107.7 113.8 119.9 127.0 134.1 142.2 152.4 162.6 172.7 182.9 193.0 203.2 215.4 227.6 239.8 354.0 269.2 284.5 304.8 320.0 360.7

Max, mm

2 2 1 2 1 2 1 3 1 2 1 2 2 1 2

2 1 2 2 1 2

B

3 3 3 1 2 1 2 1 2 1 2 1 2 1 3 1 3 1

A

1 2 1 2 1 2 1 2 1 2

C

1

D

Degree of preferencea for pitch diameters, according to groove section

a Key: 1—ﬁrst preference; 2—second preference; 3—not recommended Source: IS 3142, 1965.

Min, mm

Pitch diameter limits

Nominal value, mm

Series of pitch diameters

TABLE 21-30A Recommended standard pulley pitch diameters

E 375 400 425 450 475 500 530 560 600 630 670 710 750 800 900 1000 1060 1120 1250 1400 1500 1600 1800 1900 2000 2240 2500

Nominal value mm 375 400 425 450 475 500 530 560 600 630 670 710 750 800 900 1000 1060 1120 1250 1400 1500 1600 1800 1900 2000 2240 2500

Min, mm 381.0 406.4 431.8 457.2 482.6 508.8 538.5 569.0 609.6 640.0 680.7 721.4 762.0 812.8 914.4 1016.0 1077.0 1137.9 1270.0 1422.4 1524.0 1625.6 1828.4 1930.4 2032.0 2275.8 2540.0

Max, mm

Pitch diameter limits

Series of pitch diameters

1 3 2 2 1 2 2 1 2 1

1

2 3

1

3

2

2

2

2 1

B

1

A

1

2 1 2

2 2 1 2 1

1 3 2 2 1

2 1 2 2

C

1

2 2 1 2 1 2 2 1 2 2 1 2

1 2 1 3 2 2 1

2 1

D

Degree of preferencea for pitch diameters, according to groove section

2 1 2 2 1 2 2 1 2 1

1 2 1 2 1 2 1 3 1 2 1

E

FLEXIBLE MACHINE ELEMENTS

21.31

FLEXIBLE MACHINE ELEMENTS

21.32

CHAPTER TWENTY-ONE

TABLE 21-30B Standard V-belt sections Minimum sheave diameter, in

hp range, one or more belts

Belt section

Width, a, in

Thickness, b, in

A

1 2 21 32 7 8 1 14 1 12

11 32 7 16 17 32 3 4

9.0 13.0

15–100 50–250

1

21.6

100

B C D E

3.0

1 4–10

5.4

1–25

TABLE 21-30C Inside circumferences of standard V-belts Section

Circumference, in

A B

26, 31, 33, 35, 38, 42, 46, 48, 51, 53, 55, 57, 60, 62, 64, 66, 68, 71, 75, 78, 80, 85, 90, 96, 105, 112, 120, 128 35, 38, 42, 46, 48, 51, 53, 55, 57, 60, 62, 64, 65, 66, 68, 71, 75, 78, 79, 81, 83, 85, 90, 93, 97, 100, 103, 105, 112, 120, 128, 131, 136, 144, 158, 173, 180, 195, 210, 240, 270, 300 51, 60, 68, 75, 81, 85, 90, 96, 105, 112, 120, 128, 136, 144, 158, 162, 173, 180, 195, 210, 240, 270, 300, 330, 360, 390, 420 120, 128, 144, 158, 162, 173, 180, 195, 210, 240, 270, 300, 330, 360, 390, 420, 480, 540, 600, 660 180, 195, 210, 240, 270, 300, 330, 360, 390, 420, 480, 540, 600, 660

C D E

TABLE 21-30D Length conversion dimensionsa

Belt section Quantity to be added a

A 1.3

B 1.8

C 2.9

D 3.3

Add the values given above to the inside circumference to obtain the pitch length in inches.

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E 4.5

FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

21.33

TABLE 21-30E Horsepower rating of standard V-belts Belt speed, ft/min Belt section A

B

C

D

E

Sheave pitch diameter, in 2.6 3.0 3.4 3.8 4.2 4.6 5.0 4.2 4.6 5.0 5.4 5.8 6.2 6.6 7.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0

1000

2000

3000

4000

0.47 0.66 0.81 0.93 1.03 1.11 1.17 1.07 1.27 1.44 1.59 1.72 1.82 1.92 2.01 1.84 2.48 2.96 3.34 3.64 3.88 4.09 4.14 5.00 5.71 6.31 6.82 7.27 7.66 8.01 8.68 9.92 10.9 11.7 12.4 13.0 13.4

0.62 1.01 1.31 1.55 1.74 1.89 2.03 1.58 1.99 2.33 2.62 2.87 3.09 3.29 3.46 2.66 3.94 4.90 5.65 6.25 6.74 7.15 6.13 7.83 9.26 10.5 11.5 12.4 13.2 13.9 14.0 16.7 18.7 20.3 21.6 22.8 23.7

0.53 1.12 1.57 1.92 2.20 2.44 2.64 1.68 2.29 2.80 3.24 3.61 3.94 4.23 4.49 2.72 4.64 6.09 7.21 8.11 8.84 9.46 6.55 9.11 11.2 13.0 14.6 15.9 17.1 18.1 17.5 21.2 24.2 26.6 28.6 30.3 31.8

0.15 0.93 1.53 2.00 2.38 2.69 2.96 1.26 2.08 2.76 3.34 3.85 4.28 4.67 5.01 1.87 4.44 6.36 7.86 9.06 10.0 10.9 5.09 8.50 11.4 13.8 15.8 17.6 19.2 20.6 18.1 23.0 26.9 30.2 32.9 35.1 37.1

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5000

0.38 1.12 1.71 2.19 2.58 2.89 0.22 1.24 2.10 2.82 3.45 4.00 4.48 4.90 3.12 5.52 7.39 8.89 10.1 11.1 1.35 5.62 9.18 12.2 14.8 17.0 19.0 20.7 15.3 21.5 26.4 30.5 33.8 36.7 39.1

FLEXIBLE MACHINE ELEMENTS

21.34

CHAPTER TWENTY-ONE

TABLE 21-30F Belt-length correction factor, K2 a Nominal belt length, in Length factor

A belts

B belts

C belts

D belts

E belts

0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20

35 38–46 48–55 60–75 78–90 96–112 120 and up

46 48–60 62–75 78–97 105–120 128–144 158–180 195 and up

75 81–96 105–120 128–158 162–195 210–240 270–300 330 and up

128 144–162 173–210 240 270–330 360–420 480 540 and up

195 210–240 270–300 330–390 420–480 540–600 660

a

Multiply the rated horsepower per belt by this factor to obtain the corrected horsepower.

Particular

Number of belts

Formula

i¼

PFa P Fc Fd

ð21-36Þ

where P ¼ drive power in kW Obtain Fd , Fc , and Fa from Tables 21-25, 21-26, and 21-27, respectively. dn1 n2

The diameter of larger pulley

D¼

Nominal pitch length of belt

ðD dÞ2 L ¼ 2C þ ðD þ dÞ þ 4C 2

For nominal inside length, nominal pitch lengths and permissible length variations for standard sizes of V-belts

Refer to Table 21-28.

Dimensions for standard V-grooved pulley

Refer to Table 21-29.

For small-diameter factor, for speed ratio and length of belt factor

Refer to Figs. 21-4a and 21-4b.

Recommend standard pitch diameters of pulleys

Refer to Table 21-30A.

For further data for design of V-belts in US Customary system units for use with Eqs (21-35a) to (21-35e)

Refer to Tables 21-30B and 21-30F, and Figs. 21-4b and 21-4c.

Center distance for a given belt length and diameters of pulleys

C¼

L ðD þ dÞ 4 8 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ L ðD þ dÞ 2 ðD dÞ2 þ 8 4 8

ð21-37Þ ð21-38Þ

ð21-39Þ

Maximum center distance

Cmax ¼ 2ðD þ dÞ

ð21-40Þ

Minimum center distance

Cmin ¼ 0:55ðD þ dÞ þ t

ð21-41Þ

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FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

Particular

21.35

Formula

MINIMUM ALLOWANCES FOR ADJUSTMENT OF CENTERS FOR TWO TRANSMISSION PULLEYS Lower limiting value

CL ¼ Cnominal 1:5%L

ð21-42Þ

Higher limiting value

CH ¼ Cnominal þ 3%L

ð21-43Þ

L ¼ 0:5 to 1%L

ð21-44Þ

INITIAL TENSION In order to give the initial tension, the belts may be stretched to Arc of contact angle

For V-belt and pulley dimensions as per SAE J 636C standard

Dd 2C Dd ¼ 1808 608 C ¼ 2 cos1

ð21-45Þ ð21-46Þ

Refer to Table 21-31A and Fig. 21-5A, Tables 21-31B and 21-31C.

SYNCHRONOUS BELT DRIVE ANALYSIS The transmission ratio of synchronous belt drive

i¼

n1 z2 d 02 ¼ ¼ n2 z1 d 01

ð21-46aÞ

where

Datum length of synchronous belt

z1 ; z2 ¼ number of teeth in smaller and larger pulley, respectively d 01 ; d 02 ¼ pitch diameter of smaller and larger pulley, respectively, m (in). p ð21-46bÞ l ¼ 2C sin þ z þ z2 þ ðz z1 Þ 2 2 1 908 2 l 2C

p ðz þ z2 Þ þ 2 1

p 2

2

ðz2 z1 Þ2 l approximate

l pzb where

The minimum number of meshing teeth

¼ angle of contact of belt, deg p ¼ pitch, m (in) zb ¼ number of teeth in belt zb ¼ 6 to 8 teeth

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ð21-46cÞ ð21-46dÞ

FLEXIBLE MACHINE ELEMENTS

21.36

CHAPTER TWENTY-ONE

Particular

Formula

For S1 synchronous belts and pulley dimensions and tolerances

Refer to Figs. 21-5B, 21-5C, 21-5D, 21-5E, 21-5F and Tables 21-31D(a) to 21-31D(i).

For the standard pitch according to ISO 5296 Standard

Refer to Table 21-31D( j).

TABLE 21-31D( j) Standard pitch value Extra light XL

Light L

Heavy H

Extra heavy XH

Double extra heavy XXH

Belt pitch, in

1 4

3 8

1 2

7 8

1 14

Nominal power kW

0.15

1.0

10

40

107

TABLE 21.31B Standard belt center distance tolerances Belt length

Tolerance on center distance

mm

in

mm

in

1270 >1270 to 1524, incl >1524 to 2032, incl >2032 to 2540, incl

50 >50 to 60, incl >60 to 80, incl >80 to 100, incl

3.0 4.1 4.8 5.6

0.12 0.16 0.19 0.22

TABLE 21.31C Maximum center distance for belts in a set SAE size SI units

fps units

mm

in

6A 8A 10A 11A 13A 15A 17A 20A 23A

0.250 0.315 0.380 0.440 0.500 11/16 (0.600) 3/4 (0.660) 7/8 (0.790) 1 (0.910)

0.8 0.8 1.0 1.0 1.0 1.5 1.5 1.5 1.5

0.03 0.03 0.04 0.04 0.04 0.06 0.06 0.06 0.06

Source: V-belts and Pulleys, SAE J 636 C. Reprinted with permission from SAE Handbook, Part I, 1977, Society of Automotive Engineers, Inc.

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0.250 0.315 0.380 0.440 0.500 11/16 (0.600)

3/4 (0.660)

7/8 (0.79)

1 (0.900)

6A 8A 10A 11A 13A 15A

17A

20A

23A

57 57 61 70 76 76 >102 >152 76 >102 >152 89 >114 >152 102 >152 >203

mm 2.25 2.25 2.40 2.75 3.00 3.00 >4.00 >6.00 3.00 >4.00 >6.00 3.50 >4.50 >6.00 4.00 >6.00 >8.00

in 36 36 36 36 36 34 36 38 34 36 38 34 36 38 34 36 38

Groove angle deg. 0.5 A, deg 6.3 8.0 9.7 11.2 12.7 — 15.2 — — 16.8 — — 20.0 — — 23.1 —

mm 0.248 0.315 0.380 0.441 0.500 0.597 — — 0.660 — — 0.785 — — 0.910 — —

in

Eﬀective groove width W

7 9 11 13 14 — 14 — — 15 — — 18 — — 21 —

mm 0.276 0.345 0.433 0.512 0.551 0.551 — — 0.630 — — 0.709 — — 0.827 — —

in

Groove depth minimum D

5.558 7.142 7.938 9.525 11.113 — 12.70 — — 14.288 — — 17.463 — — 20.638 —

0.013 mm 0.2188 0.2812 0.3125 0.3750 0.4375 0.500 — — 0.5625 — — 0.6875 — — 0.8125 — —

0.0005 in

Ball or rod diameter d

4.16 5.63 3.77 5.88 7.99 6.42 7.02 7.56 8.21 8.82 9.38 11.77 12.42 13.02 15.67 16.33 16.94

mm

2K d

0.164 0.222 0.154 0.231 0.314 0.258 0.280 0.302 0.328 0.352 0.374 0.472 0.496 0.520 6.616 0.642 0.666

in

1.0 1.3 1.5 1.8 2.0 — 0 — — 6.5 — — 1.0 — — 1.5 —

mm

2X b

0.04 0.05 0.06 0.07 0.08 0.00 — — 0.02 — — 0.04 — — 0.06 — —

in

8.00 10.49 13.71 15.01 16.79 — 19.76 — — 21.36 — — 24.54 — — 27.71 —

mm

0.315 0.413 0.541 0.591 0.661 0.778 — — 0.84 — — 0.966 — — 1.091 — —

in

Groove spacinga 0:38 S

b

Pulley eﬀective diameters below those recommended should be used with caution, because power transmission and belt life may be reduced. 2X is to be subtracted from the eﬀective diameter to obtain ‘‘pitch diameter’’ for speed ratio calculation. c These values are intended for adjacent grooves of the same eﬀective width ðWÞ. Choice of pulley manufacture or belt design parameter may justify variance from these values. The S dimension shall be the same on all multiple groove pulleys in a drive using matched belts. d 2K dimensions are calculated in millimeters.

a

fps units

SI units

SAE size

Recommended minimum eﬀective diameter

FIGURE 21-5A V-belt pulley dimensions.

1. The sides of the groove are to be 125 min (3.2 mm) A. A. maximum. 2. Radial run-out not to exceed 0.015 in (0.38 mm) full indicator movement (FIM). Axial run-out is not to exceed 0.015 in (0.38 mm) FIM. Run-out in the two directions is measured separately with a ball mounted under spring pressure to follow the groove as the pulley is rotated. Diameter, load, and overhang conditions may require or permit variations in the above speciﬁed run-out limits. 3. Bottom corner radii optional but, if used, it shall be below the depth, D. 4. In pulleys for use with belts in multiple on common centers, the diameters over the ball gages are not to vary from groove to groove in the same pulley more than 0.002 in/in (0.05 mm/25 mm) of diameter, with top limit of 0.012 in (0.30 mm) for diameters 6 in (152 mm) and above. 5. Centerline of groove is to be 90 28 with pulley axis. 6. The X dimension is radial. 2X is to be subtracted from the eﬀective diameter to obtain ‘‘pitch diameter’’ for speed ratio calculation.

Notes:

FLEXIBLE MACHINE ELEMENTS

21.37

FLEXIBLE MACHINE ELEMENTS

21.38

CHAPTER TWENTY-ONE

Particular

Formula

For determining the center distance of synchronous belt pulleys.a

Refer to Fig. 21-5F.a

The distance from belt pitch line to the pulley—tip circle radius (Fig. 21-5C)

a¼

d 0 do 2 2

ð21-46eÞ

The permissible initial tensioning force range FA

Fu FA 1:5Fw

ð21-46f Þ

where Fu ¼ the transmissible peripheral force, kN (lbf ) Fw ¼ the eﬀective shaft tensioning force, kN (lbf ) F1 ﬃ5 F2

The belt side-force ratio

ð21-46gÞ

where F1 ¼ tension belt on tight side of synchronous belt, kN (lbf) F2 ¼ tension belt on slack side of synchronous belt, kN (lbf) a

Courtesy: J. E. Shigley and C. R. Mischke, Standard Handbook of Machine Design, 2nd edition, McGraw-Hill Publishing Company, New York, 1996.

FIGURE 21-5B Pulley generating tool rack form

TABLE 21.31D (a) Pulley generating tool rack form dimensions (mm)

Pulley section

Diameter range (No. of grooves)

Pb Pitch 0.003

0.25 deg

hg þ0.05 0.00

bg þ0.05 0.00

rb 0.03

rt 0.03

2a

ST SU SU STA

10 14–19 >19 19

9.525 12.700 12.700 9.525

40 40 40 40

2.13 2.59 2.59 2.13

3.10 4.24 4.24 3.10

0.86 1.47 1.47 0.86

0.53 1.04 1.42 0.71

0.762 1.372 1.372 1.372

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FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

21.39

The pitch line is situated outside the pulley-tip-circle radius at a distance equaling that of the neutral axis r0 ¼ d 02 ¼ pulley pitch radius do ¼ pulley outside diameter a ¼ distance between the pitch line of belt and the pulley tip circle radius pc ¼ pitch ¼ pitch angle (Fig. 21-5C)

FIGURE 21-5C Pulley dimensions

TABLE 21.31D (b) Pulley tolerance (mm) Pitch to pitch tolerance Outside diameter range

Adjacent grooves

Accumulative over 908

50, incl >50 to 100, incl >100 to 175, incl >175 to 300, incl

0.03 0.03 0.03 0.03

0.09 0.11 0.13 0.15

Outside diameter Up to 50 mm, incl For each additional 25 mm or portion thereof Outside diameter runout Up to 75 mm, incl outside diameter For each additional 25 mm or portion thereof Axial runouta (side wobble) Up to 250 mm, incl outside diameter For each additional 25 mm outside diameter over 220 mm ad 0.01 mm Diametrical taper 0.01 mm per 10 mm of face width Groove helix 0.01 mm per 10 mm of face width a

Tolerance þ0.05 to 0.00 mm þ0.025 to 0.00 mm 0.08 mm (max) 0.01 mm (max) 0.02 mm per 25 mm of diameter add 0.01 mm

Full indicator movement

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FLEXIBLE MACHINE ELEMENTS

21.40

CHAPTER TWENTY-ONE

TABLE 21.31D (d) Belt width tolerances (mm) TABLE 21.31D (c) Nominal belt dimensions (mm) (Fig. 21-5C)

Belt length range

Belt section

Pitch

hb

2 deg

ht

bt

rbb

rbt

ST SU STA

9.525 12.700 9.525

3.6 4.1 4.1

40 40 40

1.9 2.3 1.9

0.5 4.4 3.2

0.5 1.0 0.5

1.0 0.5

Belt width

840, incl

>840 to 1680, incl

40, incl

þ0.6 0.6 þ0.8 0.8

þ0.6 0.6 þ1.0 1.0

>40 to 50, incl

TABLE 21.31D (e) Measuring pulley dimensions, (mm)

Belt section

No. of grooves

Pitch circumference

Outside diam, 0.013

Outside diam, runout FIM,a max

ST SU STA

16 20 20

152.40 254.00 190.50

47.748 79.479 59.266

0.013 0.013 0.013

a b

Axial runout (side wobble) FIM,a max

Min clearanceb

0.025 0.025 0.025

0.33 0.38 0.33

Full indicator movement. See Fig. 21.5.

TABLE 21.31D (f ) Total measuring force (N) Belt width (mm) Belt section

8

10

12

14

16

18

19

20

22

25

28

30

33

35

40

45

50

ST SU STA

55 — —

75 — —

100 245 245

125 300 300

145 370 370

165 420 420

175 445 445

185 475 475

210 530 530

240 610 610

275 700 700

295 750 750

330 840 840

355 900 900

410 1050 1050

470 1200 1200

530 1350 1350

TABLE 21.31D (g) Minimum recommended pulley diameters and ﬂange dimensions (mm) Pulley section

Pitch diam

Min. grooves

Min. pitch diam

Min. outside diam

Min. ﬂange thickness

Min. ﬂange height

ST SU STA

9.525 12.700 9.525

10 14 19

30.32 56.60 57.61

29.56 55.23 56.23

1.3 1.3 1.3

1.6 2.0 2.4

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FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

21.41

TABLE 21.31D (h) Belt length tolerances (mm)

FIGURE 21-5D Belt section

Belt length range

Tolerance on belt pitch length

400, incl >400 to 520, incl >520 to 770, incl >770 to 1020, incl >1020 to 1270, incl >1270 to 1525, incl >1525 to 1780, incl >1780 to 2040, incl >2040 to 2300, incl >2300 to 2560, incl >2560 to 3050, incl

0.46 0.51 0.61 0.66 0.76 0.81 0.86 0.91 0.97 1.02 1.12

TABLE 21.31D (i) Pulley groove tolerances (mm) (Fig. 21-5D) Pulley section

Top curvature band width

Max. top radius tolerance

Flank band width

Bottom curvature band width

Depth band width

Upper reference depth

ST

0.04

0.1 0.0

0.05

0.05

0.05

0.5

SU

0.04

0.1 0.0

0.05

0.05

0.05

0.8

STA

0.04

0.1 0.0

0.05

0.05

0.05

0.5

FIGURE 21-5E Pulley groove proﬁle. Source: Synchronous Belts and Pulleys, SAE J 1313 Oct. 80. Reprinted with permission from SAE Handbook, Part I, Society of Automotive Engineers, Inc., 1997.

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FLEXIBLE MACHINE ELEMENTS

21.42

CHAPTER TWENTY-ONE

FIGURE 21-5F Determination of center distance of synchronous belts. Source: J. E. Shigley and C. R. Mischke, Standard Handbook of Machine Design, 2nd edition, McGraw-Hill Book Company, New York, 1996.

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FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

Particular

21.43

Formula

The power transmitted by synchronous belt

P¼

Ps Cs

ð21-46hÞ

where Ps ¼ standard capacity of the selected belt, kW (hp) Cs ¼ service correction factor

CONVEYOR (Tables 21-12, 21-14, 21-20, and 21-31) The average capacity, C, of conveyor in m3 (in)3 per hour at 0.5 m/s (100 fpm) speed For ﬂat belts

For belts on idlers

For belts on threeto ﬁve-step idlers

when a1 in m

C ¼ 70a21

when a1 in in

C¼

when a1 in mm

C ¼ 0:7

when a1 in m

C ¼ 88a21

when a1 in in

C ¼ 3465a

when a1 in mm

C ¼ 0:88

when a1 in m

C ¼ 132a21 to 154a21

when a1 in in

C¼

when a1 in mm

C ¼ 1:32 105 a21 to 1:54 105 a21

2756a21 105 a21

2

5158a21

105 a21

to

6063a21

SI

ð21-47aÞ

USCS

ð21-47bÞ

SI

ð21-47cÞ

SI

ð21-48aÞ

USCS

ð21-48bÞ

SI

ð21-48cÞ

SI

ð21-49aÞ

USCS

ð21-49bÞ

SI

ð21-49cÞ

TABLE 21-31 Maximum inclination of belt conveyors

Material conveyed

Maximum inclination, deg

Material conveyed

Maximum inclination, deg

Briquets and egg-shaped material Wet-mixed concrete Sized coal Washed and screened gravel Loose cement Crushed and screened coke Sand

12 15 8 18 20 20 20

Glass batch Run-of-mine coal Run-of-bank gravel Crushed ore Crushed stone Tempered foundry sand Wood chips

20 22 22 25 20 25 28

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FLEXIBLE MACHINE ELEMENTS

21.44

CHAPTER TWENTY-ONE

Particular

The power required by a horizontal belt conveyor

Formula

d vL P ¼ ðWI þ 2WB þ WL Þ þ PT D 1000 SI

ð21-50aÞ

where W in N/m, v in m/s, L in m, and P in kW d vL P ¼ ðWI þ 2WB þ WL Þ þ PT D 102 Metric

ð21-50bÞ

where W in kgf/m, v in m/s, L in m, and P in kW FIGURE 21-5 Rockwood pivoted motor base.

d vL P ¼ ðWI þ 2WB þ WL Þ þ PT D 33;000 USCS

ð21-50cÞ

where W in lbf/in, v in ft/min, L in in, and P in hp where ¼ coefficient of friction of idler bearing ¼ 0:15 for roller bearings ¼ 0:35 for grease lubricated idlers

SHORT CENTER DRIVE Rockwood drive (Fig. 21-5) The value of F1

The value of F2

The pivot-arm length for motor of weight W

F1 ¼

aW þ cFn cþb

ð21-51Þ

F2 ¼

aW bFn cþb

ð21-52Þ

F Fn b 1 þ c F 2 a¼ F1 W 1 F2 where Fn ¼ required net pull, kN (lbf ) W ¼ weight of the motor, kN (lbf )

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ð21-53Þ

FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

Particular

21.45

Formula

ROPES Manila rope (Tables 21-32 and 21-34) Pu ¼ 48053d 2

The ultimate load

SI

ð21-54aÞ

USCS

ð21-54bÞ

where d in m and Pu in kN Pu ¼ 7000d 2

where d is diameter of rope in in and Pu in lbf F þ Fc 2 where d in m and F1 in N F F1 ¼ 200d 2 ¼ F þ þ Fc 2 where d in in and F1 in lbf

The maximum tension on the tight side

F1 ¼ 137:5 104 d 2 ¼ F þ

F1 ¼ 0:14d 2

SI

ð21-55aÞ

USCS

ð21-55bÞ

Customary Metric

ð21-55cÞ

where d in mm and F1 in kgf P ¼ vð0:6 6:7 104 Fc Þ

Power transmitted

SI

ð21-56aÞ

where Fc in N, P in kW, and v in m/s 2v ð200 Fc Þ 105 where Fc in lbf and P in hp

P¼

USCS

ð21-56bÞ

Refer to Table 21-32 for Fc ¼ values of coeﬃcients for manila rope

Hemp ropes d 2 4 br where

The load on the hemp rope

ð21-57Þ

F¼

br ¼ breaking stress, MPa (psi) ¼ 9:81 MPa (1.42 kpsi) for white rope ¼ 8:82 MPa (1.28 kpsi) for tarred rope

TABLE 21-32 Value of coeﬃcient Fc for manila rope Velocity, mps Coeﬃcient, Fc

7.50 2.96

10.00 5.40

12.50 8.44

15.00 12.60

17.50 16.10

20.00 21.00

22.50 26.55

25.00 32.89

27.50 39.69

30.00 41.17

32.50 55.34

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35.00 64.40

FLEXIBLE MACHINE ELEMENTS

21.46

CHAPTER TWENTY-ONE

Particular

Formula

The load on the hemp rope in terms of nominal diameter of rope

F ¼ 7:7 106 d 2 for white rope

SI

ð21-58aÞ

USCS

ð21-58bÞ

SI

ð21-58cÞ

USCS

ð21-58dÞ

where d in m and F in N F ¼ 1120d 2 where d in in and F in lbf F ¼ 7 106 d 2 for tarred rope where d in m and F in N F ¼ 1020d 2 where d in in and F in lbf

HOISTING TACKLE The eﬀort on the rope in case of single-sheave pulley (Fig. 21-6)

P¼

D þ d þ 2s Q ¼ CQ D d 2s0

ð21-59Þ

Refer to Table 21-33 for C.

FIGURE 21-6 Rope passing over sheave.

FIGURE 21-7 Load on a hoist.

The eﬀort on the rope in a hoist for raising the load (Fig. 21-7)

P¼

Cn ðC 1Þ Q Cn 1

ð21-60Þ

The pull required on the rope in a hoist for lowering the load

P0 ¼

C1 Q CðCn 1Þ

ð21-61Þ

TABLE 21-33 Value of C Manila rope Wire rope Dry chain Greased chain

1.15 1.07 1.10 1.04

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FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

Particular

Eﬃciency of hoist

21.47

Formula

¼

Cn 1 nCn ðC 1Þ

ð21-62Þ

where n ¼ number of times a rope passes over a sheave

Continuous system Fig. (21-8)

In the continuous system one continuous rope passes around the driving and driven sheaves several times, in addition to making one loop about tension pulley located on a traveling carriage.

FIGURE 21-8 Continuous system.

The relation between ultimate load, bending and service load in wire rope

Pu Pb þ Ps n

The bending load

Pb ¼ kA

Another formula connecting ultimate strength of rope, tensile load on rope (P), dimensions of the rope, wire, and sheave diameter

ð21-63aÞ

dw D where k ¼ 82728:5 MPa (12 Mpsi)

Pu ¼

1 n0

d D

P

dw d

ð21-63bÞ

ð21-63cÞ

E0 u

where D ¼ minimum diameter of sheave or pulley, m (in) n0 ¼ stress factor ¼ nkd n ¼ safety factor kd ¼ duty factor obtainable from Table 21-35 Area of useful cross-section of the rope

The approximate ultimate strength of plow-steel ropes

A¼

u n0

d D

P

dw E0 d

ð21-63dÞ

Pu ¼ 524;000d 2 for 6 7 and 6 19 ropes SI ð21-64aÞ where Pu in kN and d in m Pu ¼ 76d 2

USCS

ð21-64bÞ

SI

ð21-64cÞ

where Pu in lbf and d in in Pu ¼ 517;800d 2 for 6 37 ropes where Pu in kN and d in m

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TABLE 21-34 Manila rope Breaking load Pitch Size designation (C)a mm

Number of yards per strand

25 32 35 38 41 44 51 57 64 70 76 83 89 95 102 108 114 121 127 140 152 165 178 203 229 254 279 305 330 356 381 406 432 457

3 4 5 6 7 8 11 13 17 20 24 28 33 37 43 48 54 60 67 81 96 113 131 171 216 267 323 384 451 523 600 683 771 864

a

Linear density kilotex 53 66 89 107 120 138 191 226 294 346 413 489 569 635 742 831 933 1,090 1,159 1,329 1,661 1,954 2,265 2,958 3,736 4,620 5,583 6,640 7,800 9,044 10,376 11,811 13,335 14,943

Grade 1 2.6C a mm

3.2C a mm

20.7–25.5 26.5–32.6 29.0–36.7 31.5–38.7 34.0–41.8 36.4–44.8 42.2–52.0 47.2–58.1 53.0–65.2 58.0–71.3 62.9–77.4 68.7–M.6 73.7–90.7 78.7–96.8 84.5–104.0 89.4–110.1 94.4–116.2 100.2–123.3 105.2–129.4 116.0–142.7 125.9–154.9 136.6–168.2 147.4–181.4 168.1–206.9 189.6–233.8 210.3–258.8 231.0–284.3 252.5–360.5 273.2–336.3 294.8–362.8 315.5–388.3 336.2–413.8 357.7–440.3 378.4–465.7

kN 5.4 6.9 8.9 10.5 12.3 14.2 19.9 23.9 31.6 37.6 44.8 52.1 59.5 68.0 76.5 85.2 95.4 105.1 116.1 139.0 163.9 190.8 219.7 282.5 353.2 432.9 520.1 616.8 719.9 829.5 953.1 1081.6 1216.1 1362.1

Grade 2

kgf

kN

546 711 902 1,067 1,257 1,448 2,032 2,439 3,226 3,836 4,572 5,309 6,071 6,935 7,798 8,687 9,729 10,719 11,837 14,174 16,714 19,457 22,404 28,805 36,019 44,147 53,038 62,893 73,409 84,586 97,185 10,292 24,009 38,894

4.7 6.2 7.8 9.3 11.0 12.6 17.7 21.2 28.1 33.4 39.9 46.3 53.1 60.5 68.0 75.7 84.7 93.4 103.1 123.6 145.5 169.4 195.3 251.1 313.9 384.6 462.3 548.0 639.7 737.3 846.9 961.8 1081.1 1210.6

kgf 483 635 800 953 1,118 1,283 1,803 2,159 2,870 3,404 4,064 4,725 5,410 5,172 6,935 7,722 8,636 9,525 10,516 12,599 14,834 17,273 19,915 25,604 32,005 39,219 47,145 55,883 65,230 75,188 86,364 98,049 110,241 123,450

Grade 3 kN 4.1 5.5 6.9 8.2 9.6 11.0 15.4 18.4 24.7 29.1 34.9 40.6 46.3 52.8 59.5 66.3 74.2 81.7 95.2 108.1 127.0 148.0 170.9 219.7 274.5 336.3 404.5 479.3 559.5 645.2 740.8 841.5 946.1 1059.2

kgf 419 559 699 838 978 1,118 1,575 1,880 2,515 2,972 3,556 4,140 4,725 5,383 6,071 6,757 7,570 8,332 9,703 11,024 12,955 15,088 17,425 22,404 27,992 34,292 41,252 48,872 57,051 65,789 75,543 85,805 96,474 108,006

C stands for nominal circumference of the rope.

TABLE 21-35 Duty factor and life of mechanism of electric wire rope hoists Duty factor

Average life

Mechanism class

Strength

Wear

Running h/day

Total life h, over

1 2 3 4

1.0 1.2 1.4 1.6

0.4 0.5 0.6 0.7

0.5 0.5 3.0 over 6

2500 9000 20000 40000

Source: IS 3938, 1967.

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FLEXIBLE MACHINE ELEMENTS

21.49

FLEXIBLE MACHINE ELEMENTS

Particular

Formula

Pu ¼ 75d 2

USCS

ð21-64dÞ

where Pu in lbf and d in in The nominal bearing pressure

p¼

2Pt Cu Dr Di

ð21-65Þ

where C ¼ 0:0015 Refer to Table 21-33 for C.

DRUMS Wire rope drum The number of turn on the drum for one rope member (Fig, 21-9) The length of the drum

iS þ2 D 2iS þ 7 p for one rope l¼ D 2iS þ 12 p þ p1 for two ropes l¼ D

n¼

ð21-66Þ ð21-67aÞ ð21-67bÞ

where S ¼ height to which the load is raised, m (in)

FIGURE 21-9 Wire rope drum

The minimum diameter of groove of sheaves and drums (d)

dgs ¼ d þ 0:8 mm to d0 þ 3:2 mm

The thickness of wall of drum made of cast iron

h ¼ 0:02D þ 0:6 to 1:0 cm

ð21-68Þ

The outside diameter of the drum (Fig. 21-9)

Do ¼ ðD þ 6dÞ

ð21-69Þ

The depth of groove in drum or sheave

h1 < 1 1:5d

ð21-70Þ

The outside diameter of sheave (dos )

dos ¼ ds þ 2h1 where ds ¼ minimum diameter of sheave, m

Stresses developed in drum The maximum bending stress

The maximum torque on the drum

8FlD ðD4 D4i Þ Dþd Mt ¼ F 2 b ¼

where d ¼ diameter of rope

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ð21-71Þ ð21-72Þ

FLEXIBLE MACHINE ELEMENTS

21.50

CHAPTER TWENTY-ONE

Particular

The maximum shear stress The crushing stress

The combined stress according to normal stress theory

Formula

¼

16Mt D ðD4 D4i Þ

c ¼

ð21-73Þ

F ph

ð21-74Þ

where p ¼ pitch of the grooves on the drum qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð21-75Þ ¼ 2b þ 2c þ 4 2 d where d ¼ design stress

HOLDING CAPACITY OF WIRE ROPE REELS The rope capacity (L) in meters in any size length may be calculated by the formula

L¼

ðH þ Dr ÞWH 1000d

ð21-76Þ

WIRE ROPE CONSTRUCTION For wire rope strand construction, diameter, weight, breaking load for diﬀerent purposes

Refer to Tables 21-36 to 21-39 and Figs. 21-10 to 21-16.

For wire rope data, factor of safety, values of C, and application

Refer to Tables 21-40 to 21-45.

CHAINS Hoisting chains The working load for the ordinary steel common coil chain

Pw ¼ 84;800d 2

SI

ð21-77aÞ

USCS

ð21-77bÞ

Customary Metric

ð21-77cÞ

where d in m and Pu in kN Pw ¼ 12;300d 2 where d in in and Pu in lbf Pw ¼ 8:65d 2

where d in mm and Pu in kgf The working load for stud chain

Pw ¼ 60;310d 2

SI

ð21-78aÞ

USCS

ð21-78bÞ

where d in m and Pu in kN Pw ¼ 8750d 2 where d in in and Pu in lbf Pw ¼ 6:15d 2

Customary Metric ð21-79Þ

where d in mm and Pu in kgf

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TABLE 21-36 Steel wire ropes (from Indian standards) Nominal breaking strength of rope Tensile strength of wire Diameter of rope mm

Strand construction Group 6 19 6 12/6/1 6 1216 þ 6F/ 1 6 9/9/1 6 10=5 þ 5F/1 (Fig. 21-10)

Group 6 37 6 14/7 and 7/7/1; 6 14=7þ 7F/7/1; 6 1618þ 8F6/1 6 15/15/6/ 1; 6 18=12=6=1; 6 16/8 and 8/1/1 (Fig. 21-11)

1568–1716 MPa (160–175 kgf/mm2)

Approx. weight N/m

kgf/m

kN

tf

1716–1863 MPa (175–190 kgf/mm2) kN

tf

General Engineering Purposes 8 10 12 14 16 18

2.4 4.3 5.3 7.5 9.2 12.3

0.24 0.44 0.54 0.76 0.94 1.25

33.3 64.7 84.3 106.9 131.4 189.3

3.4 6.6 8.6 10.9 13.4 19.3

36.3 70.6 92.2 116.7 144.2 206.9

3.7 7.2 9.4 11.9 14.7 21.1

20 22 24 25 29 32 35 38 41 44 48 51 54 10 12 14 16 18 20 22 24 25 29 32 35 38 41 44 48 51 54 57 64 70

14.4 18.0 20.9 23.6 29.9 36.8 44.6 53.3 62.5 72.4 83.2 94.5 106.8 4.4 5.9 7.3 9.0 12.3 15.5 17.7 20.6 32.2 29.3 36.2 43.9 52.2 61.3 71.0 81.6 92.8 104.7 117.5 145.0 175.3

1.47 1.84 2.13 2.41 3.05 3.75 4.55 5.43 6.37 7.38 8.48 9.64 10.89 0.45 0.60 0.74 0.92 1.32 1.58 1.81 2.10 2.37 2.99 3.69 4.48 5.32 6.25 7.24 8.32 9.46 10.68 11.98 14.79 17.98

221.6 254.0 294.2 333.4 423.6 522.7 623.5 752.2 886.5 1026.8 1175.8 1345.6 1514.1 60.8 79.4 101.0 124.5 179.5 209.9 241.2 278.5 318.7 398.2 493.3 598..2 712.0 836.5 971.8 1116.0 1266.0 1434.1 1604.4 1982.2 2401.6

22.6 25.9 30.0 34.0 43.2 53.3 64.6 76.7 90.4 104.7 119.9 137.2 154.4 6.2 8.1 10.3 12.7 18.3 21.4 24.6 28.4 32.5 40.6 50.3 61.0 72.6 85.3 99.1 113.8 129.1 146.3 163.6 202.2 244.9

241.2 278.5 323.6 368.7 462.9 570.7 692.3 826.7 971.8 1125.8 1295.5 1474.9 1664.2 66.7 87.3 110.8 136.3 196.1 230.5 263.8 304.0 349.1 438.4 543.3 658.0 782.6 916.9 1065.9 1225.8 1394.5 1574.0 1763.2 2172.2 2630.1

24.6 28.4 33.0 37.6 47.2 58.2 70.6 84.3 99.1 114.8 132.1 150.4 269.7 6.8 8.9 11.3 13.9 20.0 23.5 26.9 31.0 35.5 44.7 55.4 67.1 79.8 93.5 108.7 125.0 142.2 160.5 179.8 221.5 288.0

FIGURE 21-10 Round strand group 6 19.

FIGURE 21-11 Round strand group 6 37.

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FLEXIBLE MACHINE ELEMENTS

21.52

CHAPTER TWENTY-ONE

TABLE 21-36 Steel wire ropes (from Indian standards) (Cont.) Nominal breaking strength of rope Tensile strength of wire Diameter of rope mm

Strand construction

1568–1716 MPa (160–175 kgf/mm2)

Approx. weight N/m

kgf/in

1716–1863 MPa (175–190 kgf/mm2)

kN

tf

kN

tf

6 24 Fiber Core (Fig. 21-12)

8 10 12 14 16 18 20 22 24 25 29 32 35 38 41 44 48 51 54

2.1 3.1 5.3 6.6 7.8 11.7 13.8 16.1 18.2 20.4 26.3 31.8 38.8 46.7 54.0 63.1 73.0 82.0 93.4

0.21 0.32 0.54 0.67 0.80 1.19 1.41 1.64 1.86 2.08 2.68 3.24 3.96 4.76 5.51 7.43 7.44 8.36 9.52

29.4 53.9 74.5 92.2 112.8 164.8 196.1 228.5 258.9 289.3 368.7 448.2 548.2 662.9 762.0 891.4 1025.0 1166.0 1315.1

3.0 5.5 7.6 9.4 11.5 16.8 20.0 23.3 26.4 29.5 37.6 45.7 55.9 67.6 77.7 90.9 104.6 118.9 134.1

32.4 59.8 81.4 102.0 123.6 181.4 214.8 249.1 278.5 313.8 403.1 493.3 603.1 722.7 836.5 976.7 1125.8 1274.9 1443.5

3.3 6.1 8.3 10.4 12.6 18.5 21.9 25.4 28.4 32.0 41.1 50.3 61.5 73.7 85.3 99.6 114.8 130.0 147.3

Group 11 F 6 9/12/; 6 10/12/; 6 12/12/; (Fig. 21-14)

14 16 18 20 22 24 25 29 32 35 38 41 44 48 51

8.3 10.2 13.7 16.3 20.1 23.2 26.4 33.2 41.2 49.6 59.0 69.1 81.0 92.7 105.0

0.85 1.04 1.40 1.66 2.05 2.37 2.69 3.39 4.20 5.05 6.02 7.05 8.26 9.45 10.71

112.8 143.2 208.9 246.1 284.4 323.6 363.8 462.9 572.7 692.5 816.9 966.9 1116.0 1275.2 1454.3

11.5 14.6 21.3 25.1 29.0 33.0 37.1 47.2 58.4 69.6 83.3 98.6 113.8 130.1 148.3

121.6 155.9 224.6 263.8 308.9 349.1 393.2 498.2 622.7 737.5 886.5 1036.6 1216.0 1374.9 1574.0

12.4 15.9 22.9 26.9 31.5 35.6 40.1 50.8 53.5 75.2 90.4 105.7 124.0 140.2 160.5

FIGURE 21-12 Round strand group 6 24 ﬁber core.

FIGURE 21-14 Compound ﬂattened strand, group II F.

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21.53

FLEXIBLE MACHINE ELEMENTS

TABLE 21-36 Steel wire ropes (from Indian standards) (Cont.) Nominal breaking strength of rope Tensile strength of wire Diameter of rope mm

N/m

kgf/in

17 7, 18 7 (Fig. 21-15)

8 10 12 14 16 18 20 22 24 25 29 32 35 39

2.5 4.1 5.6 7.8 9.6 12.9 15.2 18.9 21.9 24.8 31.4 38.8 46.8 55.9

34 7 (Fig. 21-13)

15 18 20 22 24 25 29 32 35 38 44 51

10.2 13.4 16.0 19.8 22.8 26.0 32.9 40.6 49.0 58.3 79.5 103.9

Strand construction

1568–1716 MPa (160–175 kgf/mm2)

Approx. weight kN

tf

1716–1863 MPa (175–190 kgf/mm2) kN

tf

0.25 0.42 0.57 0.80 0.98 1.32 1.55 1.93 2.23 2.53 3.20 3.96 4.77 5.70

35.3 68.6 87.3 113.8 142.2 201.0 237.3 268.7 313.8 359.9 443.3 548.2 672.7 802.2

3.6 7.0 8.9 11.6 14.5 20.5 24.2 27.4 32.0 36.6 45.2 55.9 68.6 81.8

1.04 1.37 1.63 2.02 2.32 2.65 3.35 4.14 5.00 5.95 8.21 10.59

134.4 193.2 225.6 263.8 299.1 344.2 433.5 538.4 647.2 771.8 1025.8 1334.7

13.7 19.7 23.0 26.9 30.5 35.1 44.2 54.9 66.0 78.7 104.6 136.1

FIGURE 21-16(a) Metal core. FIGURE 21-13 Multistrand nonrotating ropes 34 7.

FIGURE 21-16(b) Metal core. FIGURE 21-15 Multistrand nonrotating ropes 17 7 and 18 7.

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FLEXIBLE MACHINE ELEMENTS

21.54

CHAPTER TWENTY-ONE

TABLE 21-36 Steel wire ropes (from Indian standards) (Cont.) Nominal breaking strength of rope Tensile strength of wire

Diameter of rope mm

N/m

Group 6 19 6 19 (12/6/1); 6 19 ﬁller wire, 6 19 (9/9/1) Seale

6 8 10 12 14 16 18 20 21 25

1.5 2.5 3.9 5.4 7.4 9.3 12.2 14.2 18.1 22.1

Group 8 19 8 19 ﬁller wire; 8 19 (9/9/1) Seale

8 10 12 14 16 18 20 22 25

2.0 3.4 4.9 6.9 8.3 10.9 13.2 16.7 19.6

0.20 0.35 0.50 0.70 0.85 1.10 1.35 1.70 2.00

10 12 14 16 18 20 22 25

4.4 5.9 8.3 10.3 13.7 16.2 19.6 24.5

0.45 0.60 0.85 1.05 1.40 1.65 2.00 2.50

Strand construction

6 25 ﬂattened strand

Approx. weight kgf/m

1079–1226 MPa (110–125 kgf/mm2) kN

1226–1372 MPa (125–140 kgf/mm2)

tf

kN

tf

1.5 2.3 4.0 5.5 7.7 9.6 12.7 15.0 18.8 23.3

16.7 26.5 44.1 58.8 86.3 107.9 139.5 166.7 207.9 255.0

1.7 2.7 4.5 6.0 8.8 11.4 14.2 17.0 21.2 26.0

21.3 37.6 49.0 68.6 88.3 112.8 137.3 181.4 01.0

2.2 3.8 5.0 7.0 9.0 11.1 14.0 18.5 20.6

24.5 42.2 53.9 79.4 98.1 132.4 152.0 205.9 235.4

2.5 4.3 5.5 8.1 10.0 13.5 15.5 21.0 24.0

42.2 56.9 79.4 102.9 137.3 161.8 203.0 243.2

4.3 5.8 8.1 10.5 14.0 16.5 20.7 24.8

49.0 64.7 90.2 117.7 151.0 184.4 230.5 272.6

5.0 6.6 9.2 12.0 15.4 18.8 23.5 27.8

Lifts and Hoists 0.15 14.7 0.25 22.6 0.40 39.2 0.55 53.9 0.75 75.5 0.95 94.1 1.25 124.5 1.45 147.1 1.85 184.4 2.25 225.6

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TABLE 21-36 Steel wire ropes (from Indian Standard) (Cont.) Nominal breaking strength of rope Tensile strength of wire

Strand construction 67

Diameter Approx. weight of rope, mm N/m kgf/m

1225.8–1373.0 MPa (125–140 kgf/mm2)

1373.0–1520.0 MPa (140–155 kgf/mm2)

1520.0–1667.0 MPa (155–170 kgf/mm2)

1667.0–1814.2 MPa (170–185 kgf/mm2)

kN

kN

kN

tf

kN

tf

19 20 22 24 25 26 27 28 31 35

Winding purposes in mines 166.7 17.0 183.5 18.9 192.2 19.6 211.8 21.6 224.6 22.9 250.1 25.5 254.0 25.9 283.4 28.9 283.3 29.5 325.6 33.2 310.0 31.6 341.3 34.8 332.4 33.9 366.7 37.4 368.7 37.6 410.0 41.8 453.1 46.2 512.8 52.3 553.1 56.4 618.9 63.1

199.1 230.4 268.7 309.1 349.1 399.1 391.9 443.3 548.2 662.9

20.3 23.5 27.4 31.5 35.6 39.9 40.7 45.2 55.9 67.6

213.8 250.1 289.3 333.4 378.6 402.1 430.5 478.6 598.2 717.8

21.8 25.5 29.5 34.0 38.6 41.0 40.9 43.8 71.2 73.2

12.8 15.0 17.7 29.3 23.0 24.6 26.3 29.2 35.9 43.4

1.31 1.53 1.80 2.07 2.35 2.51 2.68 2.98 3.66 4.43

tf

tf

Nominal breaking strength of rope Tensile strength of wire

Strand construction 6 19

1226–1373 MPa (125–140 kgf/mm2)

1373–1520 MPa (140–155 kgf/mm2)

1520–1667 MPa (155–170 kgf/mm2)

1667–1814 MPa (170–185 kgf/mm2)

Diameter Approx. weight of rope, mm N/m kgf/m

kN

tf

kN

tf

kN

tf

kN

tf

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

154.9 179.5 193.2 206.2 222.6 237.3 268.7 291.3 318.7 348.1 372.6 400.1 428.5 447.1 471.7 493.3 518.8 548.2 580.5 611.2 629.6 650.2

15.8 18.3 19.7 21.1 22.7 24.2 27.4 29.7 32.5 35.5 38.0 40.8 43.7 45.6 48.1 50.3 52.9 55.9 59.2 62.4 64.2 66.3

171.6 199.1 213.2 229.5 246.1 263.8 300.1 326.5 352.1 383.4 413.8 443.3 473.7 498.2 522.7 548.2 572.7 608.0 641.3 678.6 696.3 714.9

17.5 20.3 21.8 23.4 25.1 26.9 30.6 33.3 35.9 39.1 42.2 45.2 48.3 50.8 53.3 55.9 58.4 62.0 65.4 69.2 71.0 72.8

189.3 221.6 237.3 254.0 273.6 294.2 334.4 365.8 394.2 423.6 456.0 483.5 522.7 545.2 572.7 608.1 632.5 672.7 707.1 752.2 772.8 792.4

19.3 22.6 24.2 25.9 27.9 30.0 34.1 37.3 40.2 43.2 46.5 49.3 53.3 55.6 58.4 61.5 64.5 68.6 72.1 76.7 78.8 80.8

206.9 243.2 260.8 278.5 301.1 323.6 368.7 399.1 436.4 462.9 502.1 536.4 572.7 603.1 632.5 663.9 692.3 732.5 773.7 826.7 849.3 868.9

21.2 24.8 26.6 28.4 30.7 33.0 37.7 40.7 44.5 47.2 51.2 54.7 58.4 61,5 64.5 67.7 73.6 74.7 78.9 84.3 86.6 88.6

13.2 14.6 16.4 18.0 19.5 20.9 23.6 26.6 28.3 31.3 33.8 35.6 38.2 39.7 41.6 43.1 44.6 47.3 50.2 53.3 55.9 59.2

1.35 1.49 1.67 1.84 1.99 2.13 2.41 2.71 2.89 3.19 3.45 3.63 3.90 4.05 4.24 4.39 4.55 4.82 5.12 5.43 5.70 6.04

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FLEXIBLE MACHINE ELEMENTS

21.56

CHAPTER TWENTY-ONE

TABLE 21-36 Steel wire ropes (from Indian Standard) (Cont.) Nominal breaking strength of rope Tensile strength of wire

Diameter Approx. weight of rope, mm N/m kgf/m

Strand construction

1226–1373 MPa (125–140 kgf/mm2) kN

1373–1520 MPa (140–155 kgf/mm2)

1520–1667 MPa (155–170 kgf/mm2)

1667–1814 MPa (170–185 kgf/mm2)

tf

kN

tf

kN

tf

kN

tf

41 42 44 46 48 51 54

62.5 65.4 72.4 78.1 83.2 94.5 106.8

6.37 6.67 7.38 7.96 8.48 9.64 10.89

726.7 781.6 836.5 893.3 950.3 1100.3 1230.7

74.1 79.7 85.3 91.1 96.9 112.2 125.5

803.2 863.9 926.7 995.4 1057.2 1217.0 1365.1

81.9 88.1 94.5 101.5 107.8 124.1 139.2

886.5 955.2 1025.8 1101.3 1175.8 1345.6 1514.1

90.4 97.4 104.6 112.3 119.9 157.2 155.4

771.8 1048.3 1125.8 1210.1 1225.5 1475.0 1664.2

99.2 106.9 114.8 123.4 192.1 150.4 169.7

19 21 22 24 25 29 22 25 31 41 44 48 51 54 57 64 70

12.9 15.5 17.7 20.6 23.2 29.3 36.2 43.9 52.2 61.3 71.0 81.6 92.8 104.8 117.6 145.0 175.3

1.32 1.58 1.81 2.10 2.37 2.99 3.69 4.48 5.32 6.25 7.24 8.32 9.45 10.68 11.98 14.79 17.88

145.1 170.6 195.8 222.5 260.0 318.7 343.7 478.6 572.7 665.1 676.9 896.3 1006.2 1156.2 1285.6 1624.0 1932.9

14.8 17.4 19.9 23.4 26.4 32.5 35.0 48.8 58.4 67.8 69.0 91.4 102.6 117.9 131.1 165.6 197.3

162.8 190.2 218.7 254.0 289.3 359.0 393.2 548.2 642.3 757.1 857.1 1006.2 1135.6 1295.5 1444.5 1793.6 2172.2

16.6 19.4 22.3 25.9 29.5 36.6 48.1 55.9 65.5 70.2 87.4 102.6 115.8 132.1 147.3 182.9 221.5

179.5 209.8 241.2 278.6 318.4 398.1 493.3 598.2 712.0 836.5 871.8 1116.0 1226.0 1434.7 1604.4 1912.9 2401.6

18.3 21.4 24.6 28.4 32.5 40.6 50.3 61.0 72.6 85.3 99.1 113.8 129.8 146.3 163.6 202.2 244.9

196.1 230.5 263.8 304.0 349.1 438.4 543.3 658.0 782.6 916.9 1066.0 1225.8 1394.0 1574.0 1763.2 2172.2 2630.0

20.0 23.5 26.9 31.0 35.6 44.7 55.4 67.1 79.8 93.5 108.7 125.0 142.2 160.5 179.8 221.5 268.2

67 19 Triangular core 21 22

15.0 17.6 20.1

1.53 1.79 2.05

181.4 205.9 244.2

18.5 21.0 24.9

199.1 228.5 268.7

20.3 23.3 27.4

216.7 249.1 294.2

22.1 25.4 30.0

235.4 272.6 313.7

24.0 27.8 32.6

Group IF 6 7=

24 25 28 31 36

23.24 26.28 33.24 41.19 49.62

2.37 2.68 3.39 4.20 5.06

278.5 313.8 403.1 498.2 598.2

28.4 32.0 41.1 50.8 61.0

306.9 347.1 443.3 553.1 662.9

31.3 35.4 45.2 56.4 67.6

333.4 378.5 483.5 608.0 727.6

34.0 38.6 49.3 62.9 74.2

363.8 413.8 528.6 658.0 792.4

37.1 42.2 53.1 67.1 80.8

Group IIF 6 8/; 6 8/12 Or less/; 6 9/12 Or less/; 6 10/12

19 21 22 24 25 29 32

15.00 17.55 20.10 23.24 26.28 33.24 41.18

1.53 1.79 2.05 2.37 2.68 3.39 4.20

179.5 209.9 234.4 273.6 304.0 393.2 473.6

18.3 21.4 23.9 27.9 31.0 40.1 48.3

194.2 228.5 258.9 299.1 333.4 428.5 522.7

19.8 23.3 26.4 30.5 34.0 43.7 53.3

208.9 246.1 284.4 323.6 363.8 492.9 572.7

21.3 25.1 29.0 33.0 37.1 47.2 58.4

224.6 263.8 308.9 349.1 393.2 498.2 622.7

22.9 26.9 31.5 35.6 40.1 50.8 63.5

6 37

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21.57

FLEXIBLE MACHINE ELEMENTS

TABLE 21-36 Steel wire ropes (from Indian Standard) (Cont.) Nominal breaking strength of rope Tensile strength of wire 1226–1373 MPa (125–140 kgf/mm2)

1373–1520 MPa (140–155 kgf/mm2)

1520–1667 MPa (155–170 kgf/mm2)

1667–1814 MPa (170–185 kgf/mm2)

Diameter Approx. weight of rope, mm N/m kgf/m

kN

tf

kN

tf

kN

tf

kN

tf

Or less/; 6 12/12 Or less/

35 38 41 44 48 51

49.62 5.06 59.03 6.02 69.14 7.05 81.00 8.26 92.67 9.45 105.03 10.71

572.7 677.6 825.2 916.2 1075.8 1216.0

58.4 69.1 84.3 93.6 109.7 124.0

627.6 766.9 896.3 1016.0 1175.8 1334.7

64.0 78.2 91.4 103.6 119.9 136.1

682.5 816.9 966.9 1116.0 1275.8 1454.3

69.6 83.3 98.6 113.8 138.1 140.3

737.5 886.5 1036.6 1216.0 1375.0 1574.0

75.2 90.4 105.7 124.0 140.2 160.5

Group IIIF 6 15/12/A 6 18/12/A

19 21 22 24 25 29 32 35 38 41 44 48 51 54 57 64 70

15.00 17.55 20.10 23.05 26.28 33.24 41.19 49.62 59.04 69.14 81.00 92.67 105.03 118.17 133.57 164.26 198.58

156.9 184.4 208.9 234.4 273.6 354.0 443.3 517.8 627.6 747.3 857.1 986.5 1125.7 1255.2 1448.5 1793.6 2152.5

16.0 18.8 21.3 23.9 27.9 36.1 45.2 52.8 64.0 76.2 87.4 100.6 114.8 128.0 147.3 189.9 219.5

174.6 205.0 234.4 263.8 304.0 388.3 488.4 577.6 682.5 816.9 946.3 1085.6 1235.4 1385.7 1584.2 1954.8 2341.8

17.8 20.9 23.9 26.9 31.0 39.6 49.8 58.9 69.6 83.3 96.5 110.7 126.0 141.3 161.6 199.1 238.8

193.2 226.5 258.9 294.2 333.4 423.6 533.5 537.4 757.1 886.5 1036.6 1185.6 1348.5 1514.1 1724.0 2112.6 2550.7

19.7 23.1 26.4 30.0 34.0 43.2 54.4 65.0 77.2 90.4 105.7 120.9 137.2 154.4 175.8 215.4 260.1

210.8 247.1 284.4 323.6 363.8 458.0 577.6 656.3 821.8 986.1 1125.8 1285.6 1452.8 1643.7 1863.3 2278.2 2740.0

21.5 25.2 29.0 33.0 37.1 46.7 58.9 71.0 83.8 97.5 114.8 131.1 148.3 167.6 190.0 231.7 279.4

Strand construction

1.53 1.79 2.05 2.35 2.68 3.39 4.20 5.06 6.02 7.05 8.26 9.45 10.71 12.05 13.62 16.75 20.25

Minimum break load Approx. weight Strand construction 6 7 ð6 1) Round

For tensile designation

Diameter rope, mm

N/100 m

kgf/100 m

8 9 10 11 12

217.7 275.6 340.3 411.9 490.3

Haulage purposes in mines 22.2 33.3 3400 28.1 42.3 4300 38.7 52.2 3320 42.0 63.1 6430 50.0 75.1 7660

1569.3 MPa

160 kgk/mm2

1765.2 MPa

180 kgf/mm2

37.6 47.5 58.6 71.0 84.4

3830 4840 5980 7240 8610

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FLEXIBLE MACHINE ELEMENTS

21.58

CHAPTER TWENTY-ONE

TABLE 21-36 Steel wire ropes (from Indian Standard) (Cont.) Minimum breaking load of rope For tensile designation

Approx. weight

1569 MPa (160 kgf/mm2)

1765 MPa (180 kgf/mm2)

Diameter of rope, mm

N/100 m

kgf/100 m

kN

kgf

kN

kgf

6 7 (6 1) Round

13 14 16 18 19 20 21 22 24 25 26 27 28 29 31 35

574.7 666.9 870.8 1098.3 1225.8 1363.1 1500.4 1647.5 1961.3 2128.0 2304.6 2481.1 2667.4 2863.5 3275.9 4167.8

58.6 68.0 88.8 112.0 125.0 139.0 153.0 168.0 200.0 217.0 235.0 253.0 272.0 292.0 334.0 425.0

88.1 102 133 169 188 209 229 252 300 326 352 380 409 438 501 638

8980 10400 13600 17200 19200 21300 23400 25700 30600 33200 35900 38700 41700 44700 51100 65100

99 115 150 190 212 234 259 283 337 367 396 428 460 493 564 719

10100 11700 15300 19400 21600 23900 26400 28900 34400 37400 40400 43600 46900 50300 57500 73300

6 19 (9/9/1) Round

13 14 16 18 19 20 21 22 24 25 26 28 29 32 35 36 38

599.2 695.3 908.1 1147.4 1284.7 1422.0 1569.1 1716.2 2039.8 2216.3 2422.2 2785.5 2981.2 3628.4 4344.3 4599.3 5413.2

61.1 70.9 92.6 117 131 145 160 175 208 226 247 284 304 370 443 469 552

87.8 102 133 169 187 208 229 251 299 325 351 407 436 532 636 673 750

8950 10400 13600 17200 19100 21200 23400 25600 30.500 33100 35800 41500 44500 54200 64900 68600 76500

99 115 150 189 211 233 258 282 336 365 395 459 491 598 716 757 843

10100 11700 15300 19300 21500 23800 26300 28800 34300 37200 40300 46700 50100 61000 73000 77200 86000

Strand construction

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21.59

FLEXIBLE MACHINE ELEMENTS

TABLE 21-36 Steel wire ropes (from Indian Standard) (Cont.) Minimum breaking load of rope For tensile designation

Approx. weight

1569 MPa (160 kgf/mm2)

1765 MPa (180 kgf/mm2)

Diameter of rope, mm

N/100 m

kgf/100 m

kN

kgf

kN

kgf

6 8 (7/) Triangular

13 15 16 18 19 20 21 22 24 25 26 28 29 31 35

675.7 783.5 1019.9 1294.5 1441.6 1598.5 1765.2 1931.9 2304.5 2500.1 2696.8 3128.3 3363.7 3844.2 4893.5

68.9 79.9 104 132 147 163 180 197 235 255 275 319 343 392 499

95.9 111 145 183 205 227 250 275 327 354 383 445 478 545 695

9780 11300 14800 18700 20900 23100 25500 28007 33306 36100 39100 45400 48700 55600 70900

106 124 161 204 228 252 278 305 363 393 426 493 530 605 771

10800 12600 16400 20800 23200 25700 28300 31100 37000 40100 43400 50300 54000 61700 78600

6 22 (9/12/) Triangular

13 14 16 18 19 20 21 22 24 25 26 28 29 32 35 38

685.5 794.3 1039.5 1314.1 1461.2 1618,1 1784.8 1961.3 2334.0 2530.1 2736.0 3137.3 3412.7 4148.2 4962.1 5854.5

69.9 81 106 134 149 165 182 200 238 258 279 324 348 423 506 597

93.1 108 141 178 199 221 243 267 317 343 372 431 463 564 675 795

9490 11000 14400 18200 20300 22500 24800 27200 32300 35000 37900 44000 47200 57500 68800 81100

104 120 157 198 222 245 270 296 353 384 414 481 515 629 750 885

10610 12200 16000 20200 22600 25000 27500 30200 36000 39100 42200 49000 52500 64000 76500 90200

Strand construction

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21.60

CHAPTER TWENTY-ONE

TABLE 21-36 Steel wire ropes (from Indian Standard) (Cont.) Maximum breaking load of rope

Approx. weight Strand construction

Diameter of wire, mm

N/m

kgf/m

Small Wire Ropes (Fiber Core) 0.147 0.015 0.324 0.033 0.559 0.057 0.873 0.099 1.255 0.128 1.696 0.172

kN

kgf

2.6 5.9 10.4 16.3 23.5 32.0

260 600 1060 1660 2400 3260

6 7 (6/1)

2 3 4 5 6 7

6 12 (12/ﬁber)

3 4 5 6 7

0.235 0.412 0.637 0.922 1.255

0.024 0.042 0.065 0.094 0.128

3.7 6.5 10.3 14.9 20.3

380 670 1050 1520 2070

6 19 (12/6/1)

3 4 5 6 7

0.314 0.539 0.843 1.206 1.648

0.035 0.052 0.086 0.124 0.168

4.9 8.7 13.5 19.6 26.6

500 890 1880 2000 2710

6 24 (15/9/ﬁber)

4 5 6 7

0.530 0.834 1.206 1.618

0.054 0.085 0.122 0.165

8.6 13.3 19.3 29.3

880 1360 1970 2680

Diameter of wire Strand construction

Max, mm

77 77 7 19 7 19 7 19 7 19 7 10

1.8 2.7 3.5 4.4 5.2 6.0 6.8

Min, mm

Approx. weight, max N/m

kgf/m

Preferred Galvanized Steel Wire Ropes for Aircraft Controls 1.6 0.108 0.011 2.4 0.235 0.024 3.2 0.422 0.043 4.0 0.657 0.067 4.8 0.804 0.082 5.6 1.236 0.126 6.4 1.608 0.164

Minimum breaking load kN

kgf

2.2 4.1 8.9 12.5 18.6 24.9 31.1

220 420 910 1270 1900 2540 3170

Note: kgf ¼ kilogram force; tf ¼ ton force:

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FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

21.61

TABLE 21-37 Round strand galvanized steel wire ropes for shipping purposes Tensile strength of wire, 1373–1569 MPa (140–160 kgf/mm2)

Approx. weight Diameter of wire, mm

N/m

8 9 10 11 12 14 16 18 20 22 24 26 28 32 36 40

2.2 2.8 3.3 4.0 5.1 5.8 8.7 10.9 13.9 16.6 19.5 22.8 27.1 34.7 44.5 54.3

kgf/m kN

67 0.22 0.28 0.34 0.41 0.52 0.69 0.89 1.11 1.42 1.69 1.99 2.32 2.76 3.54 4.54 5.54

6 19 8 9 10 11 12 14 16 18 20 22 24 26 28 32 36 40 44 48 52 60

Breaking strength of rope, min

1.9 0.20 2.8 0.29 3.4 0.35 3.9 0.40 5.1 0.52 6.5 0.66 8.8 0.90 10.6 1.08 13.5 1.38 15.7 1.60 19.2 1.96 23.1 2.36 26.0 2.65 33.7 3.44 42.4 4.32 54.1 5.52 65.1 6.64 78.0 7.95 90.0 9.18 104.0 10.61

kgf Fiber core

31.0 38.8 47.1 56.9 72.6 96.1 123.6 154.0 198.1 236.3 278.5 323.6 385.4 494.3 634.5 773.9

3150 3950 4800 5800 7400 9800 12600 15700 20200 24100 28400 33000 39300 50400 64700 78900

Fiber core 28.0 2850 40.2 4100 47.3 4800 53.9 5500 71.1 7250 90.2 9200 122.6 12500 147.1 15000 188.3 19200 218.7 22300 267.7 27300 321.6 32800 360.9 36800 468.8 47800 588.4 60000 664.9 67800 905.2 92300 1084.6 110600 1251.3 127600 1446.5 147500

Approx. weight N/m

kgf/m kN

16 12 1.5 2.1 2.5 2.8 3.7 4.7 6.4 7.6 9.8 11.4 13.9 16.8 18.7 24.3 30.6 39.0

0.15 0.21 0.25 0.29 0.38 0.48 0.65 0.78 1.00 1.16 1.42 1.70 1.91 2.48 3.12 3.98

6 24 2.2 2.6 3.1 3.7 4.4 5.9 8.4 10.4 12.7 15.0 17.7 22.0 25.1 32.0 41.8 50.5 60.1 73.3 84.7 97.0

Breaking strength of rope, min

0.22 0.27 0.32 0.38 0.45 0.60 0.86 1.06 1.29 1.53 1.80 2.24 2.56 3.26 4.26 5.15 6.13 7.47 8.64 9.90

kgf Fiber core

18.1 26.0 30.4 35.3 46.1 58.4 79.4 95.6 122.1 141.7 173.6 208.9 234.4 304.0 382.5 489.4

1850 2650 3100 3650 4700 5950 8100 9750 12450 14450 17700 21300 23900 31000 39000 49900

Fiber core

Approx. weight N/m

kgf/m kN

6 13

5.1 7.8 9.4 12.4 14.9 18.5 21.0 25.5 30.5 39.4 40.1 61.6

Breaking strength of rope, min

0.52 0.80 0.96 1.26 1.52 1.89 2.14 2.60 3.11 4.02 4.09 6.28

6 37

28.4 2900 2.3 0.23 34.8 3550 2.9 0.30 42.2 4300 3.3 0.34 50.0 5100 4.1 0.42 58.8 6000 5.0 0.51 78.5 8000 7.0 0.71 112.8 11500 9.3 0.95 140.2 14300 12.0 1.22 169.7 17300 14.9 1.52 201.0 20500 18.2 1.86 238.3 24100 20.0 2.04 294.2 30000 23.8 2.43 336.9 34300 28.0 2.85 428.6 43700 37.2 3.79 599.0 57000 47.8 4.87 676.7 69000 56.6 5.77 806.1 82200 69.5 7.09 982.6 100200 83.9 8.55 1136.6 115900 95.2 9.71 1301.3 132700 111.7 11.39

kgf

Fiber core

72.6 109.8 132.4 174.6 209.9 261.8 295.2 361.9 429.5 555.1 703.1 867.9

7400 11200 13500 17800 21400 26700 30100 36900 43900 56600 71700 88500

Fiber core 31.9 41.2 47.1 58.4 70.6 98.1 31.4 168.7 256.9 256.9 282.4 336.4 394.2 524.7 674.7 798.3 981.6 1183.7 1344.5 1577.9

Approx. weight N/m

kgf/m kN

77

3.7 4.4 5.7 7.6 9.7 12.1 15.5 18.5 21.9 25.4 30.2 38.7 49.7

Breaking strength of rope, min

0.38 0.45 0.58 0.77 0.99 1.23 1.58 1.89 2.23 2.59 3.08 3.95 5.07

7 19

Fiber core

52.0 62.8 80.4 106.9 137.3 171.6 219.7 262.8 308.9 359.9 427.6 548.2 704.1

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5300 6400 8200 10900 14000 17500 22400 26800 33500 36700 43600 56000 71800

Wire core

3250 4200 4800 5950 7200 7.3 0.74 101.5 10000 9.9 1.01 138.3 13400 11.9 1.21 165.7 17200 15.2 1.55 211.8 26200 17.7 1.80 245.2 26200 20.8 2.12 289.3 29900 26.0 2.65 361.9 34300 29.1 2.97 406.0 40200 37.9 3.86 526.6 53500 47.6 4.85 662.9 68800 60.8 6.20 847.3 81400 73.1 7.45 1027.7 100100 86.3 8.80 1203.3 120700 101.0 10.30 1407.3 137100 116.6 11.89 1625.0 160900 133.8 13.59 1857.4

Source: IS 2581, 1968.

kgf

10350 14100 16900 21600 25000 29500 36900 41400 53700 67600 86400 104800 122700 143500 165700 189400

FLEXIBLE MACHINE ELEMENTS

21.62

CHAPTER TWENTY-ONE

TABLE 21-38 Dimensions and breaking strength of ﬂat balancing wire ropes Nominal size b s,a mm

Constructions

Doublestitched

Singlestitched

Approximate weight Diameter of the wire, mm

Cross section of the strand, mm2

Double-stitched N/m

kgf/m

Single-stitched N/m

kgf/m

Minimum breaking strength of rope kN

kgf

647

70 17 74 18 78 19 82 20 87 21 91 22 95 23

70 15 74 16 78 17 82 18 87 19 91 20 95 21

1.60 1.70 1.80 1.90 2.00 2.20 2.20

338 381 427 477 528 581 638

34.3 39.2 44.1 49.0 53.9 59.8 65.7

3.5 4.0 4.5 5.0 5.5 6.1 6.7

33.3 37.3 42.2 47.1 52.0 56.9 62.8

3.4 3.8 4.3 4.8 5.3 5.8 6.4

463.9 522.7 585.5 654.1 724.7 797.2 875.7

47300 53300 59700 66700 73900 81300 89300

847

110 20 113 20 116 21 119 21 122 22 125 22 128 23

110 18 113 18 116 19 119 19 122 20 125 20 128 21

1.90 1.95 2.00 2.05 2.10 2.15 2.20

636 670 703 739 775 812 851

65.7 68.7 72.6 76.5 79.4 83.4 87.3

6.7 7.0 7.4 7.8 8.1 8.5 8.9

62.8 65.7 68.7 72.6 76.5 79.4 83.4

6.4 6.7 7.0 7.4 7.8 8.1 8.5

872.8 919.9 956.0 1014.0 1064.0 1116.0 1168.0

89000 93800 98400 103400 108500 113800 119100

6 4 12

112 26 115 26 118 27 121 27 124 28 127 28 130 29

112 23 115 23 118 24 121 24 124 25 127 25 130 26

1.90 1.95 2.00 2.05 2.10 2.15 2.20

818 861 904 950 996 1045 2094

84.3 88.3 98.2 98.1 103.0 107.9 112.8

8.6 9.0 9.5 10.0 10.5 11.0 11.5

80.4 84.3 88.3 93.2 98.1 103.0 106.9

8.2 8.6 9.0 9.5 10.0 10.5 10.9

1122.9 1181.7 1240.5 1304.3 1367.0 1439.6 1483.7

114500 120500 126500 133000 139400 146300 151300

8 4 12

146 26 149 26 154 27 157 27 160 28 165 28 168 29

146 23 149 23 154 24 157 24 160 25 165 25 168 26

1.90 1.95 2.00 2.05 2.10 2.15 2.20

1091 1148 1206 1267 1329 1394 1459

112.8 118.7 124.5 130.4 137.3 143.2 150.0

11.5 12.1 12.7 13.3 14.0 14.6 14.3

106.9 112.8 118.7 124.5 130.4 136.3 143.2

10.9 11.5 12.1 12.7 13.3 13.9 14.6

1497.5 1575.9 1655.4 1738.7 1824.0 1913.3 2002.5

152700 160700 168800 177300 186000 195100 204200

8 4 14

160 27 164 28 168 28 172 29 176 29 180 30 184 30

160 24 164 25 168 25 172 26 176 26 180 27 184 27

1.90 1.95 2.00 2.05 2.10 2.15 2.20

1272 1340 1407 1478 1550 1626 1702

131.4 138.3 145.1 152.0 159.8 167.7 175.5

13.4 14.1 14.8 15.5 16.3 17.1 17.9

124.6 131.4 138.3 145.1 152.0 159.9 166.7

12.7 13.4 14.1 14.8 15.5 16.3 17.0

1745.5 1842.2 1930.9 2029.0 2188.0 2232.0 2335.9

178000 187800 196900 206900 217000 227600 238200

8 4 91

186 31 190 32 194 33

186 28 190 29 194 30

1.90 1.95 2.00

1727 1818 1909

177.5 187.3 191.1

18.1 19.1 20.1

169.7 178.5 187.3

17.3 18.2 19.1

2377.3 2495.8 2620.3

251700 254500 267200

a b ¼ width of rope, s ¼ thickness of rope. Source: IS 5203, 1969.

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21.63

TABLE 21-39 Dimensions and breaking strength of ﬂat hoisting wire ropes

Construction

Nominal size, b s, mm

Weight

Minimum breaking strength of ropea

Nominal wire diameter, mm

Cross section of strand, mm2

N/m

kgf/m

kN

kgf

647

52 10 56 11 60 12 65 14 70 15 74 16 78 16 82 18 87 19 91 20 95 21

1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20

190 223 259 297 338 381 427 477 528 581 638

18.6 21.6 25.5 29.4 33.3 37.3 42.2 47.1 52.0 56.9 62.8

1.9 2.2 2.6 3.0 3.4 3.8 4.3 4.8 5.3 5.8 6.4

298.1 349.1 406.0 465.8 529.6 597.2 669.8 748.2 827.7 911.0 1000.3

30400 35600 41400 47500 54000 60900 68300 76300 84400 92900 102000

847

70 10 75 11 80 12 86 14 92 15 98 16 104 17 110 18 116 19 122 20 128 21

1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20

253 298 345 396 450 508 569 636 703 775 851

24.5 29.4 34.3 39.2 44.1 50.0 55.9 62.8 68.6 76.5 83.4

2.5 3.0 3.5 4.0 4.5 5.1 5.7 6.4 7.0 7.8 8.5

396.2 466.8 541.3 620.8 706.1 796.3 892.4 997.3 1102.3 1216.0 1333.7

40400 47600 55200 63300 72000 81200 91000 101700 112400 124400 136600

a Rope having wires of tensile strength of 1569 MPa (160 kgf/mm2 ). Source: IS 5202, 1269.

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21.64

CHAPTER TWENTY-ONE

TABLE 21-40 Tensile grade Tensile strength range Grade of wire

MPa

kgf/mm2

120 140 160 180 200

1176.8–1471.0 1372.9–1078.7 1569.1–1863.3 1765.2–2059.4 1961.3–2353.6

120–150 140–170 160–190 180–210 200–240

TABLE 21-41 Values of C for wire ropes Rope diameter, mm

C

Rope diameter, mm

C

9.50 11.11 12.70 14.30

1.090 1.083 1.076 1.070

15.90 19.00 22.20 25.40

1.064 1.054 1.046 1.040

TABLE 21-42A Approximate wire rope and sheave data

Rope construction

MN

lbf 103

kN/m

lbf/ft

Wire, diameter dw , mm (in)

6 19 6 37 8 19 67

500:8d 2 473:1d 2 431:3d 2 473:0d 2

72d 2 68d 2 62d 2 68d 2

36:3d 2 35:3d 2 34:3d 2 32:4d 2

1:60d 2 1:55d 2 1:50d 2 1:45d 2

0:063d 0:045d 0:050d 0:106d

Ultimate strength, Fu

Weight

Recommended sheave diameter, mm (in) Area A, mm2 (in2 )

Average

Minimum

0:38d 2 0:38d 2 0:35d 2 0:38d 2

45d 27d 31d 72d

30d 18d 21d 42d

SI units: d ¼ diameter of rope, m. US Customary units: d ¼ diameter of rope, in.

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TABLE 21-42B Wire rope data

Rope

Weight per foot, lb

Minimum sheave Standard diameter, in sizes, d, in Material

6 7 haulage

1:50d 2

42d

1 1 4 –12

6 19 standard hoisting

1:60d 2

26d–34d

1 3 4 –2 4

6 37 special ﬂexible 8 19 extra ﬂexible 7 7 aircraft

1:55d 2

18d

1 1 4 –3 2

1:45d 2

21d–26d

1 1 4 –1 2

1:70d 2

—

1 3 16 – 8

7 9 aircraft

1:75d 2

—

1 3 8 –1 8

19-wire aircraft

2:15d 2

—

1 5 32 – 16

Monitor steel Plow steel Mild plow steel Monitor steel Plow steel Mild plow steel Monitor steel Plow steel Monitor steel Plow steel Corrosion-resistant steel Carbon steel Corrosion-resistant steel Carbon steel Corrosion-resistant steel Carbon steel

Size of outer wires

Modulus of elasticity,a Mpsi

Strength,b kpsi

d/9 d/9 d/9 d/13–d/16 d/13–d/16 d/13–d/16 d/22 d/22 d/15–d/19 d/15–d/19 — — — — — —

14 14 14 12 12 12 11 11 10 10 — — — — — —

100 88 76 106 93 80 100 88 92 80 124 124 135 143 165 165

a

The modulus of elasticity is only approximate: it is aﬀected by the loads on the rope and, in general, increases with the life of the rope. The strength is based on the nominal area of the rope. The ﬁgures given are only approximate and are based on 1-in rope sizes and 14-in aircraftcable sizes. Source: Compiled from American Steel and Wire Company Handbook. b

TABLE 21-43 Common wire rope application Sheave diameter, cm Type of service

Rope construction

Recommended

Minimum

Haulage rope Mine haulage Factory-yard haulage Inclined planes Tramways Power transmission Guy wires Standard hoisting rope (Most commonly used rope) Mine hoists Quarries Ore docks Cargo hoists Car pullers Cranes Derricks Tramways Well drilling Elevators Extraﬂexible hoistings rope Special ﬂexible hoisting rope Steel-mill ladles Cranes High speed elevators

67

72d

42d

6 19

45d 60–100d

30d

20–30d

8 19 6 37

31d 27d

21d 18d

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TABLE 21-44 Recommended safety factors for wire ropes Safety factor 100 or other ﬁgure laid down by the statutory authority Rope application

Class 1

From Indian Standards Mining ropes 3.5 Wire ropes used on the cranes and other hoisting equipment Fixed guys Unreeved rope bridles of jib cranes or ancillary appliances, such as lifting beams Ropes which are straight between terminal ﬁttings Hoisting, luﬃng and reeved bridle systems of inherently ﬂexible crances 4.0 (e.g., mobile crawler tower, guy derrick, stiﬄeg derrick) where jibs are supported by ropes or where equivalent shock absorbing devices are incorporated in jib supports Cranes and hoists in general hoist blocks 4.5

Classes 2, 3

Class 4

4.0

4.5

4.5

5.5

5.0

6.0

From Other Sources Mine Shafts Depths to 152 m 305–610 m 610–915 m >915 m Haulage ropes Small electric and air hoists Hot ladle cranes Slings

8 7 6 5 6 7 8 8

Source: IS 3973, 1967.

TABLE 21-45 Ratio of drum and sheave diameter to rope diameter Minimum, ratioa Purpose

Construction

100

Mining Installation

All

Class 1

Classes 2, 3

Class 4

Cranes and allied hoisting equipment

6 15 8 19 ﬁller wire 8 19 8 19 Warrington 8 19 Seale 34 7 nonrotating 6 24 6 19 ﬁller wire 6 19 6 19 Warrington 17 7 nonrotating 18 7 nonrotating 6 19 Seale

15

17

22

17

18

24

18 18 19

19 20 23

25 23 27

24

28

35

a

The ratio of the sheave diameters speciﬁed are valid for rope speeds up to 50 m/min. For speeds above 50 m/min, the drum or sheave diameter should be increased pro rata by 8% for each additional 50 m/min of rope speed where practicable.

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Particular

The working load for the ordinary steel BB crane chain

21.67

Formula

Pw ¼ 93;750d 2

SI

ð21-79aÞ

USCS

ð21-79bÞ

Customary Metric

ð21-79cÞ

where d in m and Pw in kN Pw ¼ 13;600d 2 where d in in and Pw in lbf Pw ¼ 9:56d 2

where d in mm and Pw in kgf The sheave diameter

D ¼ 20d to 30d

ð21-80Þ

Round steel short link and round steel link chain LENGTH AND WIDTH (Figs. 21-17 and 21-18): The outside dimensions of the links shall fall between the following limits: Outside link length limits (Fig. 21-17)

Maximum outside link width (Fig. 21-18)

Minimum inside link width

l> j 5dn

for uncalibrated chain

ð21-81aÞ

l <j 6dn

for calibrated chain

ð21-81bÞ

Wmax > j 3:5dn

away from weld

ð21-82aÞ

j 1:05 Wmax >

(adjacent width) at weld for noncalibrated chains

ð21-82bÞ

Wmax ¼ 3:25dn

for calibrated chain

ð21-83Þ

Wt <j 1:25dn

except at the weld for noncalibrated chain

ð25-84Þ

Pitch (i.e., inside length)

p ¼ 3dn

Dimensions and lifting capacities and properties of noncalibrated and calibrated chains

Refer to Tables 21-46 to 21-51.

for calibrated chain

FIGURE 21-17 Diameter of material and welded chain.

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ð21-85Þ

1.2 1.4 1.6 1.8 2.0 2.2 2.5 2.8 3.2 3.6 4.0 4.4 5.0 5.6 6.4 7.2 8.0 9.0

þ0.12, 0.36 þ0.14, 0.42 þ0.16, 0.48 þ0.18, 0.54 þ0.20, 0.60 þ0.22, 0.66 þ0.25, 0.75 þ0.28, 0.84 þ0.32, 0.96 þ0.90

þ1.0 þ1.1 1.2 1.4 1.6 1.9 2.0 2.2

Nominal size, dn , mm

6.3 7.1 8.0 9.0 10.0 11.2 12.5 14.0 16.0 18.0

20.0 22.0 25.0 28.0 32.0 36.0 40.0 45.0

Source: IS 2429 (Part I), 1970.

(dw d) max

Diameter tolerance j 16, þ0:02s dn > 0:06s dn 16, 0:05s

7.0 7.7 8.7 9.8 1.1 1.2 1.4 1.6

2.1 2.4 2.8 3.1 3.5 3.9 4.4 4.9 5.6 6.3

(G d) max

Maximum additional weld dimensions

100 110 125 140 160 180 200 225

32 36 40 45 50 56 62 70 80 90

5dn

95 105 120 130 150 170 190 215

30 34 38 43 48 53 59 66 76 86

4:75dn

Outside link length limits

70 77 87 98 110 120 140 160

22 25 28 31 35 39 44 49 56 63

Away from weld, Wmax 3:5dn

3.5 3.9 4.4 4.9 5.5 6.0 7.0 8.0

1.1 1.25 1.4 1.6 1.8 2.0 2.2 2.5 2.8 3.1

Max extra at weld, 0:5Wmax

Maximum outside link width, W

TABLE 21-46 Dimensions and lifting capacities of grade 30 noncalibrated chain (Figs. 21-17 and 21-18)

25 28 31 35 40 45 50 56

7.9 8.9 10 1.1 12 14 16 18 20 22

Minimum inside link width, 1:25dn

189.0 228.0 296.0 372.0 483.0 610.0 757.0 953.0

18.9 23.6 30.2 38.1 47.1 59.2 73.8 93.0 120.0 153.0

32.0 42.0 53.0 67.0 87.0 112.0 136.0 173.0

3.4 4.3 5.5 6.9 8.5 10.7 13.4 16.7 22.0 39.0

Minimum energy Guaranteed absorption minimum factor (energy breaking load absorption stress 30h bar, 0.054 kJ m1 kN mm2 ), kJ/m

49.0 57.0 74.0 93.0 121.0 152.0 189.0 228.0

4.8 5.9 7.5 9.5 11.8 14.8 18.5 23.2 30.0 38.2

Minimum safe working load (stress 7:5h bar), kN

5.0 6.3 8.0 10.0 12.5 16.0 20.0 22.5

0.50 0.63 0.80 1.00 1.25 1.6 2.0 2.5 3.2 4.0

Lifting capacity (stress 7:6h bar), tonnes

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21.68

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0.48 0.56 0.64 0.72 0.80 0.88 1.0 1.1 1.2 1.4 1.6 1.8 2.0 2.2 2.5 2.8 3.2 3.6

þ0.12, 0.36 þ0.14, 0.42 þ0.16, 0.48 þ0.18, 0.54 þ0.20, 0.60 þ0.22, 0.66 þ0.25, 0.75 þ0.28, 0.80 þ0.32, 0.96 0.90 1.0 1.1 1.2 1.4 1.6 1.9 2.0 2.2

Nominal size, dn , mm

6.3 7.1 8.0 9.0 10.0 11.2 12.5 14.0 16.0 18.0 20.0 22.0 25.0 28.0 32.0 36.0 40.0 45.0

Source: IS 2429 (Part II), 1970.

(dw d) max

Diameter tolerance dn > j 16, þ0:02s 0:06s dn 16, 0:05s 2.1 2.4 2.8 3.1 3.5 3.9 4.4 4.9 5.6 6.3 7.0 7.7 8.7 9.8 1.1 12 14 16

(G d) max

Maximum additional weld dimensions

19 21 24 27 30 34 37 42 48 54 60 66 75 84 96 108 120 155

Preferred pitch (inside length), 3dn 0.26 0.30 0.33 0.36 0.40 0.44 0.49 0.55 0.63 0.71 0.79 0.87 0.99 1.1 1.2 1.4 1.6 1.8

Pitch tolerance (one link), 0:00396dn 20 23 26 29 32 36 41 46 52 58 65 73 82 91 100 110 130 150

Preferred outside width, w ¼ 3:25dn 0.45 0.52 0.59 0.67 0.75 0.84 0.93 1.05 1.20 1.35 1.50 1.70 1.90 2.10 2.40 2.70 3.00 3.40

Outside width tolerance away from weld zone þ0:075dn 0

TABLE 21-47 Dimensions and lifting capacities of grade 30 calibrated chain (Figs. 21-17 and 21-18)

0.90 1.0 1.1 1.3 1.5 1.7 1.9 2.1 2.4 2.7 3.0 3.4 3.8 4.2 4.8 5.4 6.0 6.8

At weld zone þ0:15dn 0 18.9 23.6 30.2 38.1 47.1 59.2 73.8 93.0 120 153 189 228 296 372 483 610 757 953

Guaranteed minimum breaking load (stress 30h bar), kN 3.4 4.3 5.5 6.9 8.5 10.7 13.4 16.7 22.0 26.0 39.0 42.0 53.0 67.0 87.0 112.0 136.0 173.0

Minimum energy absorption factor (energy absorption 0.054 kJ m1 mm2 ), kJ/m

4.8 5.9 7.5 9.5 11.8 14.8 18.5 23.2 30.0 38.2 49.0 57.0 74.0 93.0 121.0 152.0 189.0 228.0

Maximum safe working load (stress 7:5h bar), kN

0.50 0.63 0.80 1.00 1.60 1.60 2.0 2.5 3.2 4.0 5.0 6.3 8.0 10.0 12.5 16.0 20.0 22.5

Lifting capacity (stress 7:5h bar), tonnes

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21.69

1 1.4 1.6 1.8 2.0 2.2 2.5 2.8 3.2 3.6 4.0 4.4 5.0 5.6 6.4 7.2 8.0 9.0

0.12, 0.36 0.14, 0.42 0.16, 0.48 0.18, 0.54 0.20, 0.60 0.22, 0.66 0.25, 0.75 0.28, 0.84 0.32, 0.96 þ0.90 þ1.0 þ1.1 1.2 1.4 1.6 1.9 2.0 2.2

Nominal size, dn , mm

6.3 7.1 8.0 9 10 11 12.5 14.0 16.0 18.0 20.0 22.0 25.0 28.0 32.0 36.0 40.0 45.0

Source: IS 3109 (Part I), 1970.

(dw d) max

Diameter tolerance j 16, þ0:02s dn > 0:06s dn 16, 0:05s 2.1 2.4 2.8 3.1 3.5 3.9 4.4 4.9 5.6 6.3 7.0 7.7 8.7 9.8 11 12 14 16

(G d) max

Maximum additional weld dimensions

30 35 40 45 50 55 62 70 80 90 100 110 125 140 160 180 200 225

5dn 28 33 38 43 48 52 59 66 76 86 95 105 120 130 150 170 190 215

4:75dn

Outside link length limits

21 24 28 31 35 39 44 49 56 63 70 77 87 98 110 120 140 160

Away from weld, Wmax 3:5dn 1.0 1.24 1.4 1.6 1.8 2.0 2.2 2.5 2.8 3.1 3.5 3.9 4.4 4.9 5.5 6.0 7.0 8.0

Max extra at weld, 0:05Wmax

Maximum outside link width, W

TABLE 21-48 Dimensions and lifting capacities of grade 40 noncalibrated chain (Figs. 21-17 and 21-18)

7.5 8.8 10 1.1 12 14 16 18 20 22 25 28 31 35 40 45 50 56

Minimum inside link width, 1:25dn 24.9 31.6 40.2 50.9 62.8 79.0 98.4 124.0 161.0 204.0 252.0 304.0 394.0 492.0 644.0 814.0 1010.0 1270.0

Guaranteed minimum breaking load stress 30h bar, kN 4.50 4.70 7.25 9.18 11.30 14.20 17.7 22.2 29.0 37.7 45.3 55.0 70.7 89.0 116.0 147.0 181.0 230.0

Minimum energy absorption factor (energy absorption 0.072 kJ m1 mm2 ), kJ/m

6.2 7.9 10.0 12.7 15.7 19.7 24.5 30.8 40.3 50.5 63.0 76.0 98.5 123.0 161.0 204.0 252.0 318.0

Minimum safe working load (stress 10h bar), kN

0.63 0.80 1.00 1.25 1.6 2.0 2.5 3.2 4.0 5.0 6.3 8.0 10.0 12.5 16.0 20 25 32

Lifting capacity (stress 10h bar), tonnes

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21.70

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0.48 0.56 0.64 0.72 0.80 0.88 1.0 1.1 1.2 1.4 1.6 1.8 2.0 2.2 2.5 2.8 3.2 3.6

0.12, 0.36 0.14, 0.42 0.16, 0.48 0.18, 0.54 0.20, 0.60 0.22, 0.66 0.25, 0.75 0.28, 0.80 0.32, 0.96 0.90 1.0 1.1 1.2 1.4 þ1.6 þ1.9 þ2.0 þ2.2

Nominal size, dn , mm

6.3 7.1 8.0 9.0 10.0 11.2 12.5 14.0 16.0 18.0 20.0 22.0 25.0 28.0 32.0 36.0 40.0 45.0

Source: IS 3102 (Part II), 1970.

(dw d) max

Diameter tolerance dn > j 16, þ0:02s 0:06s dn 16, 0:05s 2.1 2.4 2.8 3.1 3.5 3.9 4.4 4.9 5.6 6.3 7.0 7.7 8.7 9.8 11 12 14 16

(G d) max

Maximum additional weld dimensions

19 21 24 27 30 34 37 42 48 54 60 66 75 84 96 108 120 135

Preferred pitch (inside length), 3dn 0.26 0.30 0.33 0.36 0.40 0.44 0.49 0.55 0.63 0.71 0.79 0.87 0.99 1.1 1.2 1.4 1.6 1.8

Pitch tolerance (one link), 0:0396dn 20 23 26 29 32 36 41 46 52 58 65 73 82 91 100 110 130 150

Preferred outside width, w ¼ 3:25dn 0.45 0.52 0.59 0.57 0.75 0.88 0.93 1.05 1.20 1.35 1.50 1.70 1.90 2.10 2.40 2.70 3.00 3.40

1.1 1.2 1.4 1.9 1.7 1.9 2.1 2.4 2.7 3.0 3.4 3.8 4.3 4.8 5.0 6.1 6.8 7.6

At weld zone þ0:167dn 0

Tolerance on outside width Away from weld zone þ0:075dn 0

TABLE 21-49 Dimensions and lifting capacities of grade 40 calibrated chain (Figs. 21-17 and 21-18)

24.9 31.6 42.2 50.9 62.8 79.0 98.4 124 161 204 252 304 394 492 644 814 1010 1270

Guaranteed minimum breaking load (stress 40h bar), kN

4.50 5.70 7.25 9.18 11.3 14.2 17.7 22.2 29.0 37.7 45.3 55.0 70.7 890 116 147 181 230

Minimum energy absorption factor (energy absorption 0.072 kJ m1 mm2 ), kJ/m

6.20 7.80 10.00 12.7 15.7 19.7 24.5 30.8 40.3 50.5 63.0 76.0 98.5 123 161 204 252 318

Maximum safe working load (stress 10h bar), kN

0.63 0.70 1.0 1.25 1.60 2.00 2.5 3.2 4.0 5.0 6.3 8.0 10.0 12.5 16.0 20.0 25.0 32.0

Lifting capacity (stress 10h bar), tonnes

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21.71

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TABLE 21-50 Requirements of arc welded grade 30 chain for lifting purposes

Proof load based on a stress of 98.1 MPa (10 kgf/mm2 )

Minimum breaking load based on a stress of 294.2 MPa (30 kgf/mm2 )

Minimum energy absorption factor for 1-m gauge length based on an energy absorption of 58.8 MN-m/m2 (6 kgf-m/mm2 )

Maximum safe working load for nominal working condition based on a stress of 49 MPa (5 kgf/mm2 )

Size (nominal diameter), mm

kN

kgf

kN

kgf

Nm

kgf-m

kN

kgf

6 8 9 10 12 14 16 18 20 22 24 27 30 33 36 39

8.6 9.8 12.5 15.4 22.2 30.2 39.4 49.9 61.6 74.5 88.8 102.5 138.7 167.7 192.7 234.4

570 1000 1270 1570 2260 3080 4020 5090 6280 7600 9050 10450 14140 17100 20360 23900

16.7 29.5 37.5 46.2 66.5 90.6 118.3 149.8 184.9 223.7 266.2 336.9 415.9 503.3 598.8 702.9

1700 3010 3820 4710 6780 9140 12060 15270 18850 22810 27140 34350 42410 51320 61070 71680

3.3 5.9 7.5 9.2 13.3 18.1 23.7 30.0 37.0 44.7 53.2 67.4 83.2 100.7 119.8 140.6

340 602 764 942 1356 1848 2412 3054 3770 4562 5428 6870 8482 10264 12214 14336

2.8 4.9 6.2 7.7 11.1 15.1 19.7 25.0 30.8 37.3 44.4 56.14 69.3 83.9 99.8 117.2

285 500 635 785 1130 1540 2010 2545 3140 3800 4525 5725 7070 8550 10180 11950

TABLE 21-51 Requirements for electrically welded steel chain grade 30 chain for lifting purposes

Proof load based on a stress of 157 MPa (16 kgf/mm2 )

Size (nominal diameter), mm

kN

kgf

5 6 7 8 9 9.5 10 11 12 14 16 18 20 22 24 26 28 30 33 36 39 42

6.1 8.9 12.1 15.8 20.0 22.2 24.7 29.8 38.5 48.3 63.1 79.9 98.6 119.3 142.0 166.6 193.2 221.8 268.4 319.4 374.9 434.8

628 904 1232 1608 2036 2268 2514 3042 3928 4926 6434 8144 10054 12164 14476 16990 19704 22620 27370 32572 38228 44334

Minimum breaking load based on a stress of 392.3 MPa (40 kgf/mm2 ) kN 15.4 22.2 30.2 39.4 49.9 55.6 61.6 74.6 96.3 120.8 157.7 199.6 246.5 298.2 354.9 416.5 483.1 554.6 671.0 798.6 937.2 1086.9

kgf 1571 2262 3079 4021 5089 5671 6283 7603 9818 12315 16085 20358 25133 30411 36191 42474 49260 56549 68424 81430 95567 110836

Minimum energy absorption factor for 1-m gauge length based on an energy absorption of 78.5 MN-m/m2 (8 kgf-m/mm2 )

Maximum safe working load for nominal working condition based on a stress of 49 MPa (5 kgf/mm2 )

Nm

kgf-m

kN

kgf

3.1 4.4 6.0 7.9 10.0 11.1 12.3 14.9 19.3 24.2 31.6 39.9 49.2 59.6 71.0 83.3 96.6 110.9 134.2 159.7 187.4 217.4

314 452 616 804 1018 1134 1257 1521 1964 2463 3217 4072 5027 6082 7238 8495 9852 11310 13685 16286 19114 22167

3.1 4.4 6.0 7.9 10.0 11.1 12.3 14.9 19.3 24.2 31.6 39.9 49.3 59.6 71.0 83.3 96.6 110.9 134.2 153.7 187.4 217.4

314 452 616 804 1018 1134 1257 1521 1964 2463 3217 4072 5027 6082 7238 8495 9852 11310 13685 16286 19114 22167

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FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

Particular

21.73

Formula

Chain passing over a sheave (Fig. 21-19) The eﬀort on the chain in case of single-sheave pulley (Fig. 21-19)

P¼

D þ Do þ c d Q ¼ CQ D Do c d

ð21-86Þ

where C ¼ 1:04 for lubricated chains C ¼ 1:10 for chains running dry The eﬃciency of the chain sheave

1 C where ¼ 0:96 for lubricated chains ¼ 0:91 for chain running dry

¼

ð21-87Þ

FIGURE 21-18 Pitch length and width of link.

FIGURE 21-19 Chain passing over sheave.

FIGURE 21-20 Diﬀerential chain block.

Diﬀerential chain block (Fig. 21-20) RAISING THE LOAD Q Q ð1 nÞ 2 d r where n ¼ ¼ D R

The eﬀort required for raising the load without friction

Po ¼

ð21-88Þ

The relation between the tension in the running-oﬀ and running-on chains

T 1 ¼ C1 T 2

ð21-89Þ

The tension in the running-oﬀ chains

T1 ¼

Ct Q 1 þ C1

ð21-90Þ

T2 ¼

Q 1 þ C1

ð21-91Þ

The tension in the running-on chain

where C1 depends on the size of the chain and diameter of the lower sheave

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FLEXIBLE MACHINE ELEMENTS

21.74

CHAPTER TWENTY-ONE

Particular

The relation between eﬀort (P), load (Q), T1 and T2

Formula

PR þ T2 r ¼ C2 T1 R

ð21-92Þ

where C2 depends on the size of the chain and diameter of upper sheave The eﬀort required for raising the load with friction

C2 C2 n Q 1 þ C1

P¼

ð21-93Þ

when C1 and C2 are diﬀerent Or C2 n Q 1þC

P¼

ð21-94Þ

where C is the average value of C1 and C2 The eﬃciency for the diﬀerential chain hoist

1þC C2 n

¼

1n 2

T1 ¼

Q 1þC

ð21-96Þ

T2 ¼

CQ 1þC

ð21-97Þ

ð21-95Þ

Lowering the load The equations for the tension in the running-on running-oﬀ and pull (P0 ) required on the chain so as to prevent running down of the load

T1 R ¼ CP0 R þ CT2 r The pull required on the chain so as to prevent running down of the load

P0 ¼

Q C

The eﬃciency for the reversed motion

0 ¼

2 C

1 nC 2 1þC

ð21-98Þ

1 nC2 ð1 nÞð1 þ CÞ

ð21-99Þ ð21-100Þ

where C varies from 1.054 to 1.09 or obtained from Table 21-33 For mechanical properties of the coil link chain and the strength of hoisting chains in terms of bar from which they are made

Refer to Tables 21-52 and 21-53.

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FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

21.75

TABLE 21-52 Mechanical properties of the coil link chain Requirement Properties

Grade 30

Grade 40

Mean stress at guaranteed minimum breaking load, Fw min, h bar Mean stress at proof load, Fe , h bar Ratio of proof load of guaranteed minimum breaking load Guaranteed minimum elongation at fracture, A min Guaranteed minimum energy absorption factor, Fw A Maximum safe working load mean stress, h bar

30 15 50% 14.4% 0.054 kJ m1 mm2 7.5

40 20 50% 14.4% 0.054 kJ m1 mm2 10

TABLE 21-53 The strength of hoisting chains in terms of the bars from which they are made Particular

% of bar

Standard close link Coil chain BB crane chain Stud chain

138 120 145 165

Particular

Formula

Conditions for self-locking of diﬀerential chain block

1 nC 2 1þC

The condition for self-locking

P0 ¼

For self-locking diﬀerential chain block

n>

1 C2

ð21-102Þ

n¼

1 C2

ð21-103Þ

The initial value of the ratio

r R

Q C

0

ð21-101Þ

Power chains Roller chains !1 n1 d2 z2 ¼ ¼ ¼ !2 n2 d1 z1

The transmission ratio

i

The average speed of chain

v¼

pz1 n1 m=s or 60

v¼

ð21-104Þ pz1 n1 ft=min 12

ð21-105Þ

where z1 ¼ number of teeth on the small sprocket and p in m (in)

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FLEXIBLE MACHINE ELEMENTS

21.76

CHAPTER TWENTY-ONE

Particular

The empirical formula for pitch

Formula

900 2=3 p 0:25 n1

SI

ð21-106aÞ

USCS

ð21-106bÞ

Customary Metric

ð21-106cÞ

where p in m p

900 n1

2=3

where p in in 900 2=3 p 250 n1

where p in mm, n1 ¼ speed of the small sprocket, rpm Bartlett formula relating speed (n1 ) and pitch ( p) based on allowable amount of impact between a roller and a sprocket

1170 n1 ¼ p

sﬃﬃﬃﬃﬃﬃﬃﬃﬃ A wf p

SI

ð21-107aÞ

where n1 in rpm, p in m, wf in N/m, and A in m2 11;800 n1 ¼ p

sﬃﬃﬃﬃﬃﬃﬃﬃﬃ A wf p

Customary Metric

ð21-107bÞ

where n1 in rpm, p in mm, wf in kgf/m, and A in mm2 1920 n1 ¼ p

sﬃﬃﬃﬃﬃﬃﬃﬃﬃ A wf p

USCS

ð21-107cÞ

where n1 in rpm, p in in, wf in lbf/ft, and A in in2 A ¼ ldr ¼ projected area of the roller dr ¼ diameter of rollers l ¼ width of chain or length of roller Maximum allowable chain velocity based on Eq. (21-107)

vmax

sﬃﬃﬃﬃﬃﬃﬃﬃﬃ A 19:48z1 wf p

SI

ð21-108aÞ

where vmax in m/s, A in m2 , p in m, and wf in N/m vmax

sﬃﬃﬃﬃﬃﬃﬃﬃﬃ A 160z1 wf p

USCS

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ð21-108bÞ

FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

Particular

21.77

Formula

where vmax in ft/min, A in in2 , p in in, and wf in lbf/ft sﬃﬃ vmax 0:196z1

A wf p

Customary Metric

ð21-108cÞ

where vmax in m/s, A in mm2 , p in mm, and wf in kgf/m Maximum speed based on the energy of impact per tooth per minute

1437 n p

sﬃﬃﬃﬃﬃﬃ 3 A wf

SI

ð21-109aÞ

where A in m2 , p in m, and wf in N/m 2000 n p

sﬃﬃﬃﬃﬃﬃ 3 A wf

USCS ð21-109bÞ

where A in in2 , p in in, and wf in lbf/ft n

6712 p

sﬃﬃﬃﬃﬃﬃ 3 A wf

Customary Metric

ð21-109cÞ

where A in mm2 , p in mm, and wf in kgf/m Maximum chain velocity based on Eq. (21-109), m/s

vmax

sﬃﬃﬃﬃﬃﬃ 3 A 24z1 wf

SI

ð21-110aÞ

where vmax in m/s, A in m2 , and wf in N/m vmax

sﬃﬃﬃﬃﬃﬃ 3 A 166z1 wf

USCS ð21-110bÞ

where vmax in ft/min, A in in2 , and wf in lbf/ft vmax 0:11z1

sﬃﬃﬃﬃﬃﬃ 3 A wf

Customary Metric

ð21-110cÞ

where vmax in m/s, A in mm2 , and wf in kgf/m Maximum sprocket speed based on the eﬀect of centrifugal force

36350 n p

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A z1 wf

SI

ð21-111aÞ

where p in m, A in m2 , and wf in N/m 9516 n p

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A z1 wf

USCS ð21-111bÞ

where p in in, A in in2 , and wf in lbf/ft

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FLEXIBLE MACHINE ELEMENTS

21.78

CHAPTER TWENTY-ONE

Particular

Formula

sﬃﬃﬃﬃﬃﬃﬃﬃ Az1 600 wf

Maximum velocity based on Eq. (21-111) vmax

SI

ð21-112aÞ

where vmax in m/s, A in m2 , and wf in N/m sﬃﬃﬃﬃﬃﬃﬃﬃ Az1 USCS ð21-112bÞ vmax 793 wf where vmax in ft/min, A in in2 , and wf in lbf/ft sﬃﬃﬃﬃﬃﬃﬃﬃ Az1 Customary Metric ð21-112cÞ vmax 0:2 wf where vmax in m/s, A in mm2 , and wf in kgf/m

Chain pull For preliminary computation, the allowable pull

Fa ¼

Fu no

ð21-113Þ

where Fu ¼ ultimate strength from Tables 21-35B and 21-42 no ¼ working factor, no ¼ 5 for sprocket having over 40 teeth and a speed of 0.5 m/s no ¼ 18 for sprocket having 10 or 11 teeth and a speed of 6 m/s AGMA formula for allowable pull based on velocity factor Cv ¼ 3=ð3 þ vÞ and bearing pressure of 29.4 MPa (4333 psi) for the pin

90 106 ldr v2 wf SI ð21-114aÞ 3þv 9:8 where l and dr in m, v in m/s, and wf in N/m

Fa ¼

Fa ¼

v2 w f ldr 2;600;000 600 þ v 3ð1011 Þ USCS

ð21-114bÞ

where l and dr in in, v in ft/min, and wf in lbf/ft where l ¼ length of roller pins, m (in) v¼

z1 pn1 m=s 60

dr ¼ roller pin diameter, m (in) For dimensions of American Standard Roller Chains—single-strand

Refer to Tables 21-54A.

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21.79

TABLE 21-54A Dimensions of American Standard roller chains—single-strand ANSI chain number 25 35 41 40 50 60 80 100 120 140 160 180 200 240

Pitch, in (mm)

Width, in (mm)

Minimum tensile strength, lb (N)

0.250 (6.35) 0.375 (9.52) 0.500 (12.70) 0.500 (12.70) 0.625 (15.88) 0.750 (19.05) 1.000 (25.40) 1.250 (31.75) 1.500 (38.10) 1.750 (44.45) 2.000 (50.80) 2.250 (57.15) 2.500 (63.50) 3.00 (76.70)

0.125 (3.18) 0.188 (4.76) 0.25 (6.35) 0.312 (7.94) 0.375 (9.52) 0.500 (12.7) 0.625 (15.88) 0.750 (19.05) 1.000 (25.40) 1.000 (25.40) 1.250 (31.75) 1.406 (35.71) 1.500 (38.10) 1.875 (47.63)

780 (3470) 1760 (7830) 1500 (6670) 3130 (13920) 4880 (21700) 7030 (31300) 12500 (55600) 19500 (86700) 28000 (124500) 38000 (169000) 50000 (222000) 63000 (280000) 78000 (347000) 112000 (498000)

Average weight, lb/ft (N/m) 0.09 (1.31) 0.21 (3.06) 0.25 (3.65) 0.42 (6.13) 0.69 (10.1) 1.00 (14.6) 1.71 (25.0) 2.58 (37.7) 3.87 (56.5) 4.95 (72.2) 6.61 (96.5) 9.06 (132.2) 10.96 (159.9) 16.4 (239.0)

Roller diameter, in (mm)

Multiple-strand spacing, in (mm)

0.130 (3.30) 0.200 (5.08) 0.306 (7.77) 0.312 (7.92) 0.400 (10.16) 0.469 (11.91) 0.625 (15.87) 0.750 (19.05) 0.875 (22.22) 1.000 (25.40) 1.125 (28.57) 1.406 (35.71) 1.562 (39.67) 1.875 (47.62)

0.252 (6.40) 0.399 (10.13) — — 0.566 (14.38) 0.713 (18.11) 0.897 (22.78) 1.153 (29.29) 1.409 (35.76) 1.789 (45.44) 1.924 (48.87) 2.305 (58.55) 2.592 (65.84) 2.817 (71.55) 3.458 (87.83)

Source: Compiled from ANSI B29.1-1975.

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FLEXIBLE MACHINE ELEMENTS

21.80

CHAPTER TWENTY-ONE

TABLE 21-54B Rated horsepower capacity of single-strand single-pitch roller chain for a 17-tooth sprocket ANSI chain number Sprocket speed, rpm

25

35

40

41

50

60

50 100 150 200 300 400 500 600 700 800 900 1000 1200 1400 1600 1800 2000 2500 3000

0.05 0.09 0.13a 0.16a 0.23 0.30a 0.37 0.44a 0.50 0.56a 0.62 0.68a 0.81 0.93a 1.05a 1.16 1.27a 1.56 1.84

0.16 0.29 0.41a 0.54a 0.78 1.01a 1.24a 1.46a 1.68 1.89a 2.10 2.31a 2.73 3.13a 3.53a 3.93 4.32a 5.28 5.64

0.37 0.69 0.99a 1.29 1.85 2.40 2.93 3.45a 3.97 4.48a 4.98 5.48 6.45 7.41 8.36 8.96 7.72a 5.51a 4.17

0.20 0.38 0.55a 0.71 1.02 1.32 1.61 1.90a 2.18 2.46a 2.74 3.01 3.29 2.61 2.14 1.79 1.52a 1.10a 0.83

0.72 1.34 1.92a 2.50 3.61 4.67 5.71 6.72a 7.73 8.71a 9.69 10.7 12.6 14.4 12.8 10.7 9.23a 6.58a 4.98

1.24 2.31 3.32 4.30 6.20 8.03 9.81 11.6 13.3 15.0 16.7 18.3 21.6 18.1 14.8 12.4 10.6 7.57 5.76

Type A

Type B

a

Estimated from ANSI tables by linear interpolation. Note: Type A—manual or drip lubrication, type B—bath or disk lubrication; type C—oil-stream lubrication. Source: Compiled from ANSI B29.1-1975 information only section, and from B29.9-1958.

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Type C

FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

21.81

TABLE 21-54C Rated horsepower capacity of single-strand single-pitch roller chain for a 17-tooth sprocket ANSI chain number Sprocket speed, rpm 50 100 150 200 300 400 500 600 700 800 900 1000 1200 1400 1600 1800 2000 2500 3000 Type C

Type B

Type A

80

100

120

140

160

180

200

240

2.88 5.38 7.75 10.0 14.5 18.7 22.9 27.0 31.0 35.0 39.9 37.7 28.7 22.7 18.6 15.6 13.3 9.56 7.25

5.52 10.3 14.8 19.2 27.7 35.9 43.9 51.7 59.4 63.0 52.8 45.0 34.3 27.2 22.3 18.7 15.9 0.40 0

9.33 17.4 25.1 32.5 46.8 60.6 74.1 87.3 89.0 72.8 61.0 52.1 39.6 31.5 25.8 21.6 0

14.4 26.9 38.8 50.3 72.4 93.8 115 127 101 82.4 69.1 59.0 44.9 35.6 0

20.9 39.1 56.3 72.9 105 136 166 141 112 91.7 76.8 65.6 49.0 0

28.9 54.0 77.7 101 145 188 204 155 123 101 84.4 72.1 0

38.4 71.6 103 134 193 249 222 169 0

61.8 115 166 215 310 359 0

Type C0

Note: Type A—manual or drip lubrication; type B—bath or disk lubrication; type C–oil-stream lubrication; type C0 —type C, but this is a galling region; submit design to manufacturer for evaluation. Source: Compiled from ANSI B29.1-1975 information only section, and from B29.9-1958.

TABLE 21-54D Tooth correction factors, K1 Number of teeth on driving sprocket

Tooth correction factor, K1

Number of teeth on driving sprocket

Tooth correction factor, K1

11 12 13 14 15 16 17 18 19 20 21

0.53 0.62 0.70 0.78 0.85 0.92 1.00 1.05 1.11 1.18 1.26

22 23 24 25 30 35 40 45 50 55 60

1.29 1.35 1.41 1.46 1.73 1.95 2.15 2.37 2.51 2.66 2.80

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FLEXIBLE MACHINE ELEMENTS

21.82

CHAPTER TWENTY-ONE

TABLE 21-54E Multistrand factors K2 Number of strands

K2

1 2 3 4

1.0 1.7 2.5 3.3

TABLE 21-54F Service factor for roller chains, ks

Operating characteristics

Intermittent few hours per day, few hours per year

Normal 8 to 10 hours per day 300 days per year

Continuous 24 hours per day

Easy starting, smooth, steady load Light medium shock or vibrating load Medium to heavy shock or vibrating load

0.06–1.00 0.90–1.40 1.20–1.80

0.90–1.50 1.20–1.90 1.50–2.30

0.90–2.00 1.50–2.40 1.80–2.80

Particular

Formula

Power For the rated horsepower capacity of single-strandsingle-pitch roller chains for 17-tooth sprocket and values of K1 and K2

Refer to Tables 21-54B to 21-54E.

Power required

P¼

F v 1000kl ks

SI

ð21-115aÞ

USCS

ð21-115bÞ

Customary Metric

ð21-115cÞ

where F in N and P in kW P¼

F v 33;000kl ks

where F in lbf and P in hp P¼

F v 102kl ks

where F ¼ required chain pull in kgf and P in kW kl ¼ load factor from 1.1 to 1.5 and also obtained from Chap. 14 ks ¼ service factor ¼ 1 for 10 h service per day ¼ 1.2 for 24 h operation and also obtained from Table 21-54F

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FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

Particular

21.83

Formula

The rated horsepower of roller chain per strand

"

# v v1:41 2 90 P¼p 1 þ 5o sin 0:75 3:7 z1 2

Pc ¼ K1 K2 Pr

The corrected horsepower (Pc )

ð21-116Þ ð21-116aÞ

where Pr ¼ rated horsepower and K1 and K2 from Tables 21-54D and 21-54E

CHECK FOR ACTUAL SAFETY FACTOR The actual safety factor checked by the formula

na ¼

Fu F þ Fcs þ Fs

where Fcs ¼

wv2 ; Fs ¼ ksg wC g

ð21-117Þ ð21-117aÞ

33;000P ð21-117bÞ v where F in lbf, P in hp, and v in ft/min

F ¼

1000P v where F in N, P in kW, and v in m/s

F ¼

w ¼ weight per meter of chain, N (lbf ) v ¼ velocity of chain, m/s (ft/min) C ¼ center distance, m (in) ksz ¼ coeﬃcient for sag from Table 21-55 F i¼ Fa

The number of strand in a chain, if F > Fa

ð21-118Þ

Center distance of chain length The proper center distance between sprockets in pitches

Cp ¼ 20p to 30p or Cp ¼ 40 10 pitches

The minimum center distance

Cmin ¼ Kmin C

where pCp ¼ C

where C ¼

da1 þ d2 2

TABLE 21-55 Coeﬃcient for sag, ksg Position of chain drive ksg

Horizontal

<408

>408

Vertical

6

4

2

1

ð21-119Þ

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ð21-120Þ

FLEXIBLE MACHINE ELEMENTS

21.84

CHAPTER TWENTY-ONE

Particular

Formula

TABLE 21-56 Values of k to he used in Eq. (21-123) ðz1 z2 Þ=Cp

0.1

1.0

2.0

3.0

4.0

5.0

6.0

k

0.02533

0.02538

0.02555

0.02584

0.02631

0.02704

0.02828

TABLE 21-57 Minimum center distance constant, Kmin Transmission ratio, i

Minimum center distance constant, Kmin

3 3–4 4–5 5–6 6–7

1 þ ð30–50=c 0 Þ 1.2 1.3 1.4 1.5

da ¼

p þ 0:6p 180 tan z

Refer to Table 21-56 for values of k [used in Eq. (21123)] and Table 21-57 for Kmin . The maximum center distance

The chain length in pitches

Cmax ¼ 80p

ð21-121Þ

where p ¼ pitches of chain, mm z þ z2 z z2 þ 1 (exact) Lp ¼ 2Cp cos þ 1 2 180 ð21-122Þ Lp ¼ 2Cp þ

The chain length, m or in

z1 þ z2 kðz1 z2 Þ2 þ 2 Cp

L ¼ 2C cos þ

ð21-123Þ

z1 pð180 þ 2Þ z2 pð180 2Þ þ 360 360 ð21-124Þ

where z1 ¼ number of teeth on a small sprocket z2 ¼ number of teeth on a large sprocket ¼ angle between tangent to the sprocket pitch circle and the center line d2 d1 ¼ sin1 2C z z2 k ¼ a variable which depends on 1 Cp and obtained from Table 21-56

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FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

Particular

The chain length The pitch diameter of a sprocket

21.85

Formula

L ¼ pLp d¼

ð21-125Þ

p 180 sin z

ð21-126Þ

Roller chain sprocket zmin ¼

4dr þ 5 for pitches of 25 mm p

ð21-127aÞ

zmin ¼

4dr þ 4 for pitches 32 to 58 mm p

ð21-127bÞ

Minimum number of teeth

zmin ¼

4dr þ 6 for pitches to 51 mm p

The root diameter of sprocket

df ¼ d dr

Minimum number of teeth

Silent chain sprocket ð21-128Þ ð21-129Þ

where dr ¼ diameter of roller pin, m (in) The width of sprocket tooth (Fig. 21-22)

C1 ¼ l 0:05p

ð21-130Þ

where l ¼ chain width or roller length Maximum hub diameter

Power per cm of width in hp

180 ðH þ 0:1270Þ z where H ¼ height of link plate, m or in ¼ 0:3p pv v P¼ 1 6:80 2:16ðz1 8Þ Dh ¼ d cos

ð21-131Þ

ð21-132Þ

pz1 n1 ¼ chain speed, m/s; p in m 60 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h ¼ 0:433 S 2 L2 ð21-133aÞ where v ¼

The relationship between depth of sag, and tension due to weight of chain in the catenary (approx.)

F ¼w

S2 h þ 8h 2

ð21-133bÞ

where h ¼ depth of sag, m (in) L ¼ distance between points of support, m (in) S ¼ catenary length of chain, m (in) F ¼ tension or chain pull, kN (lbf ) w ¼ weight of chain, kN/m (lbf/in)

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FLEXIBLE MACHINE ELEMENTS

21.86

CHAPTER TWENTY-ONE

Particular

Formula

Tension chain linkages Allowable load

Fa ¼ 13:1 106 p2

SI

ð21-134aÞ

USCS

ð21-134bÞ

where p in m and Fu in N Fa ¼ 1900p2

where p ¼ pitch of chain, in, and Fu in lbf Allowable load for lightweight chain

Fa ¼ 7 106 p2

SI

ð21-134cÞ

USCS

ð21-134dÞ

where p in m Fa ¼ 1020p2 where p in in, F in lbf

Indian Standards PRECISION ROLLER CHAIN (Figs 21-21 to 21-25, Tables 21-58, 21-59, 21-60) P 180 sin z

ð21-135Þ

Pitch circle diameter (Fig. 21-21)

PCD ¼

Bottom diameter

BD ¼ PCD Dr

FIGURE 21-21 Notation for wheel rim of chain.

FIGURE 21-22 Notation for wheel rim proﬁle of roller chain.

ð21-136Þ

Wheel tooth gap form The minimum value of roller seating radius, mm (Fig. 21-24)

SR1 ¼ 0:505Dr

The maximum value of roller seating radius, mm (Fig. 21-25)

SR2 ¼ 0:505Dr þ 0:069

ð21-137Þ p 3 ﬃﬃﬃﬃﬃﬃ Dr

where Dr ¼ roller diameter, mm

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ð21-138Þ

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25.40 25.40 31.70 31.75 38.10 38.10 50.80 50.80 63.50 63.50 76.20 76.20 88.90 101.60

Pitch, P, mm

7.92 8.51 10.16 10.16 11.91 12.07 15.88 15.88 19.05 19.05 22.23 25.40 27.94 29.21

7.95 7.75 9.53 9.65 12.70 11.68 15.88 17.02 19.05 19.56 25.40 25.40 30.99 30.99

3.96 4.45 5.08 5.08 5.94 5.72 7.92 8.28 9.53 10.19 11.10 14.63 15.90 17.81

Bearing in diameter max, Dp , mm 4.01 4.50 5.13 5.13 5.99 5.77 7.97 8.33 9.58 10.24 11.15 14.68 15.95 17.86

Bush bore, min, db , mm 12.33 12.07 15.35 14.99 18.34 16.39 24.39 21.34 30.48 26.68 36.55 33.73 36.46 42.72

Chain path depth, max, hc , mm 12.07 11.81 15.09 14.73 18.08 16.13 24.13 21.08 30.18 26.42 36.20 33.40 37.08 42.29

Plate depth, H, min, mm 6.9 6.9 8.4 8.4 9.9 9.9 13.0 13.0 16.0 16.0 19.1 19.1 21.3 24.4

Crank linked dimension max, X, mm 11.18 11.30 13.84 13.28 17.75 13.62 22.61 25.45 27.46 29.01 35.46 37.92 46.58 45.57

Width over inner link, min, W , mm

11.31 11.43 13.97 13.41 17.88 15.76 22.74 25.58 27.59 29.14 35.59 38.05 46.71 45.70

Width between outer plates max, mm

17.8 17.0 21.8 19.6 26.9 22.7 33.5 36.1 41.1 43.2 50.8 53.4 65.1 64.7

Width over bearing pin, min, A, mm

3.9 3.9 4.1 4.1 4.6 4.6 5.4 5.4 6.1 6.1 6.6 6.6 7.4 7.9

127.5 127.5 196.1 196.1 284.4 284.4 500.2 500.1 774.7 774.7 1108.2 1108.2 1510.2 2000.6

13 13 20 20 29 29 51 51 79 79 113 113 164 204

Addition width for joint fastener, Measuring load max, B, mm N kgf

13.8 17.9 21.8 22.3 31.2 28.9 55.6 42.3 86.8 64.5 124.5 97.9 129.1 169.1

kN

1410 1820 2220 2270 3180 2950 5670 4310 8850 6580 12700 9980 13160 17240

kgf

Breaking load, min

Notes: (1) The chain path depth Hc is the minimum depth of channel through which the assembled chain will pass; (2) the overall width of chain with joint fastener is A þ B for riveted pin end and fastener on one side; A þ 1:6B for headed pin end and fastener on one side; and A þ 2B for fastener on both sides. The actual dimensions will depend on the type of fastener used, but they should not exceed the dimensions in this column. Source: IS 3542, 1966.

208A 208B 210A 210B 212A 212B 216A 216B 220A 220B 224A 224B 228B 232B

Chain no.

Width Roller between diameter inner plates, max, Dr , W, min, mm mm

TABLE 21-58 Extended pitch transmission roller chain dimensions, measuring loads and breaking loads

FLEXIBLE MACHINE ELEMENTS

21.87

FLEXIBLE MACHINE ELEMENTS

21.88

CHAPTER TWENTY-ONE

Particular

Formula

FIGURE 21-23 Notation for tooth gap form of roller chain.

FIGURE 21-24 Notation for minimum tooth gap form of roller chain.

The minimum value of roller seating angle, deg (Fig. 21-24)

SA1 ¼ 1408

908 z

ð21-139Þ

The maximum value of roller seating angle, deg (Fig. 21-25)

SA2 ¼ 1208

908 z

ð21-140Þ

The minimum value of tooth ﬂank radius, mm (Fig. 21-24)

FR1 ¼ 0:12Dr ðz þ 2Þ

ð21-141Þ

The maximum value of tooth ﬂank radius, mm (Fig. 21-25)

FR2 ¼ 0:008Dr ðz2 þ 180Þ

ð21-142Þ

FIGURE 21-25 Notation for maximum tooth gap form of roller chain.

Tooth heights and top diameters (Fig. 21-23) The maximum limit of the tooth height above the pitch polygon The minimum limit of the tooth height above the pitch polygon The maximum limit of the tooth top diameter, mm The minimum limit of the tooth top diameter, mm

0:8 HTmax ¼ p 0:3125 þ 0:5Dr z 0:6 0:5Dr HTmin ¼ p 0:25 þ z TDmax ¼ PCD þ 0:625p Dr 0:4 TDmin ¼ PCD þ p 0:5 Dr z

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ð21-143Þ ð21-144Þ ð21-145Þ ð21-146Þ

2.0000 2.1519

2.3048

2.4586

2.6131

2.7682

2.9238

3.0798

3.2361 3.3927

3.5494

3.7065

3.8637

4.0211

4.1786

4.3362

4.4940 4.6518

4.8097

4.9677

5.1258

5.2840

512

6 612

7

712

8

812

9

912

10 1012

11

1112

12

1212

13

1312

14 1412

15

1512

16

1612

2812

28

2712

27

26 2612

2512

25

2412

24

2312

23

22 2212

2112

21

2012

20

1912

19

18 1812

1712

17

9.0902

8.9314

8.7726

8.6138

8.2962 8.4550

8.1375

7.9787

7.8200

7.6613

7.5026

7.3439

7.0266 6.1853

6.8681

6.7095

6.5509

6.3925

6.2340

6.0755

5.7588 5.9171

5.6005

5.4422

No. of teeth, Pitch circle z diameter

a The values given are for a unit pitch (e.g., 1 mm). Source: IS 3542, 1966.

1.7013

1.8496

5

No. of teeth, Pitch circle z diameter

4012

40

3912

39

38 3812

3712

37

3612

36

3512

35

34 3412

3312

33

3212

32

3112

31

30 3012

2912

29

12.9045

12.7455

12.5865

12.4275

12.1095 12.2685

11.9506

11.7916

11.6327

11.4737

11.3148

11.1558

10.8380 10.9969

10.6790

10.5201

10.3612

10.2023

10.0434

9.8845

9.5668 9.7256

9.4080

9.2491

No. of teeth, Pitch circle z diameter

TABLE 21-59 Pitch circle diametersa for extended pitch transmission roller chain wheels

5212

52

5112

51

50 5012

4912

49

4812

48

4712

47

46 4612

4512

45

4412

44

4312

43

42 4212

4112

41

16.7212

16.5622

16.4031

16.2441

15.9260 16.0850

15.7669

15.6079

15.4488

15.2898

15.1308

14.9717

14.6537 14.8127

14.4946

14.3356

14.1765

14.0176

13.8585

13.6995

13.3815 13.5405

13.2225

13.0635

No. of teeth, Pitch circle z diameter

6412

64

6312

63

62 6212

6112

61

6012

60

5912

59

58 5812

5712

57

5612

56

5512

55

54 5412

5312

53

20.5393

20.3800

20.2210

20.0619

19.7437 19.6029

19.5847

19.4255

19.2665

19.1073

18.9482

18.7892

18.4710 18.6301

18.3119

18.1529

17.9938

17.8347

17.6756

17.5166

17.1984 17.3575

17.0393

16.8803

No. of teeth, Pitch circle z diameter

20.6982

75

74 7412

7312

73

7212

72

7112

71

70 70122

6912

69

6812

68

23.8802

23.5620 23.7213

23.4031

23.2438

23.0849

22.9256

22.7667

22.6074

22.2892 22.4485

22.1303

21.9710

21.8121

21.6528

21.4939

21.3346

6612 6712

21.0164 21.1757

20.8575 66 6612

6512

65

No. of teeth, Pitch circle z diameter

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21.89

6.35

4310 4960 5540 6070 6500 6940 7290 7590 7840 8050 8230 8380 8480 8560 8610 8780 8200 7580 6820 5950 5010 4020

No. of teeth

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30 35 40 45 50 55 60

2260 2590 2900 3170 3420 3630 3810 3970 4100 4210 4300 4380 4480 4410 4510 4490 4290 3970 3570 3110 2620 2100

9.50 1690 1940 2180 2380 2560 2720 2860 2980 3080 3160 3230 3290 3330 3360 3380 3370 3220 2970 2670 2330 1970 1580

12.70 1220 1400 1570 1720 1850 1969 2060 2150 2220 2280 2330 2370 2400 2420 2440 2430 2320 2140 1930 1680 1420 1140

15.80 920 1050 1110 1290 1390 1480 1550 1610 1670 1720 1750 1780 1800 1820 1830 1830 1740 1610 1450 1270 1070 860

19.05

TABLE 21-60 Maximum speed (rpm), recommended of sprockets for roller chains

580 670 750 820 880 935 985 1020 1060 1090 1110 1130 1150 1160 1100 1160 1110 1020 920 805 675 545

25.40 415 475 535 585 630 670 700 730 755 755 790 805 875 825 830 825 790 730 655 575 410 390

31.75

Pitch

325 375 415 455 490 520 550 750 590 605 620 630 640 645 650 645 615 570 515 450 375 305

38.10 235 270 305 335 360 380 400 415 430 440 450 460 405 470 475 470 450 415 375 325 275 220

44.45 200 230 260 280 305 325 340 355 365 375 385 390 395 400 400 400 380 355 320 275 235 185

50.80

165 190 215 255 255 270 285 295 305 315 320 325 330 300 335 335 320 295 265 230 195 155

57.15

145 165 186 205 220 235 245 255 265 270 280 280 285 290 290 290 275 255 230 200 170 135

63.50

110 125 140 155 165 175 185 195 200 205 210 215 215 220 220 220 210 195 175 150 125 100

76.20

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21.90

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Particular

21.91

Formula

Wheel rim proﬁle (Fig. 21-22) Tooth width

C1 ¼ 0:95W with a tolerance of h=4

ð21-147Þ

The minimum tooth side radius

F ¼ 0:5p

ð21-148Þ

The tooth side relief

G ¼ 0:05p to 0:075p

ð21-149Þ

Absolute maximum shroud diameter

D ¼ p cot

For leaf chain dimension, breaking load, anchor clevises and chain sheaves

Refer to Tables 21-61, 21-62, and 21-63.

1808 1:05H 1:00 2 Koct , mm z ð21-150Þ

Leaf chains PRECISION BUSH CHAINS (Figs. 21-26 to 21-29, Tables 21-64 to 21-68) p 180 sin z

The pitch circle diameter (Fig. 21-21, Table 21-62)

PCD ¼

Bottom diameter

BD ¼ PCD Db

ð21-152Þ

The minimum value of bush seating radius, mm (Fig. 21-28)

SR1 ¼ 0:505Db

ð21-153Þ

The maximum value of bush seating radius, mm (Fig. 21-29)

SR2 ¼ 0:505Db þ 0:0693

The minimum value of bush seating angle, deg (Fig. 21-28)

SA1 ¼ 1408

ð21-151Þ

pﬃﬃﬃﬃﬃﬃ Db

ð21-154Þ

908 z 908 SA2 ¼ 1208 z

ð21-155Þ

The minimum value of tooth ﬂank radius, mm (Fig. 21-28)

FR1 ¼ 0:12Db ðz þ 2Þ

ð21-157Þ

The maximum value of tooth ﬂank radius, mm (Fig. 21-29)

FR2 ¼ 0:008Db ðz2 þ 180Þ

ð21-158Þ

ð21-159Þ

The minimum limit of the tooth top diameter

TDmax ¼ PCD þ 1:25p Db 1:6 Db TDmin ¼ PCD þ p 1 z

The maximum limit of the tooth height above the pitch polygon

HTmax ¼ 0:625p 1:5Db þ

The maximum value of bush seating angle, deg (Fig. 21-29)

TOOTH TOP DIAMETERS HEIGHT (Fig. 21-27)

AND

ð21-156Þ

TOOTH

The maximum limit of the tooth top diameter

0:8p z

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ð21-160Þ ð21-161Þ

TABLE 21-61 Leaf chain dimensions, measuring loads, and breaking loads

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12.70 12.70 12.70 12.70 15.88 15.88 15.88 15.88 19.05 19.05 19.05 19.05 25.40 25.40 25.40 31.75 31.76 31.75 38.10 38.10 38.10 44.45 44.45 44.45 50.80 50.80 50.80

0822 0823 0834 0846 1022 1023 1034 1046 1222 1223 1234 1246 1623 1634 1646 2023 2034 2046 2423 2434 2446 2823 2834 2846 3223 3234 3246

Source: IS: 1072-1967.

Pitch mm

Chain number

Chain width, W1 mm 6.45 8.08 11.30 16.13 7.26 9.09 12.73 18.16 12.50 15.62 21.87 31.24 21.34 29.87 42.67 23.24 32.54 46.68 30.73 43.03 61.47 35.94 50.32 71.88 40.51 56.72 81.03

Lacing

22 23 34 46 22 23 34 46 22 23 34 46 23 34 46 23 34 46 23 34 46 23 34 46 23 34 46 8.69 10.31 13.54 18.36 9.80 11.63 15.27 20.70 15.90 19.02 25.27 34.65 25.48 34.01 46.81 28.35 37.64 51.59 38.05 50.34 68.78 43.89 58.27 79.83 49.43 65.63 89.94

Width over bearing pins, W2 max, mm 4.45 4.45 4.45 4.45 5.08 5.08 5.08 5.08 6.78 6.78 6.78 6.78 8.28 8.28 8.28 10.19 10.19 10.19 14.63 14.63 14.63 15.90 15.90 15.90 17.81 17.81 17.81

Pin body diameter, max, Dp max mm 4.48 4.48 4.48 4.48 5.10 5.10 5.10 5.10 6.80 6.80 6.80 6.80 8.30 8.30 8.30 10.22 10.22 10.22 14.66 14.66 14.66 15.92 15.92 15.92 17.84 17.84 17.84

Articulating plates bore, diameter, min, Dp max mm 11.81 11.81 11.81 11.81 14.73 14.73 14.73 14.73 16.13 16.13 16.13 16.13 21.08 21.08 21.08 26.42 26.42 26.42 33.40 33.40 33.40 37.08 37.08 37.08 42.29 42.29 42.29

Plate depth, min, H mm 1.57 1.57 1.57 1.57 1.78 1.78 1.78 1.78 3.07 3.07 3.07 3.07 4.22 4.22 4.22 4.60 4.60 4.60 6.10 6.10 6.10 7.14 7.14 7.14 8.05 8.05 8.05

Plate thickness, max, T mm 190.0 190.0 280.0 370.0 250.0 250.0 390.0 500.0 450.0 450.0 670.0 890.0 630.0 1020.0 1250.0 980.0 1510.0 1960.0 1600.0 2400.0 3200.0 2400.0 3200.0 4300.0 2800.0 4100.0 5500.0

N 19.10 19.10 28.60 38.10 25.40 25.40 39.90 50.80 45.40 45.40 68.00 90.70 63.50 104.30 127.00 99.80 154.20 199.60 163.30 244.90 326.60 217.70 326.60 435.50 281.20 421.80 562.50

kgf

Measuring load

18.7 18.7 26.3 37.4 24.9 24.9 39.1 49.8 44.5 44.5 66.7 82.0 62.3 102.3 124.5 97.9 151.2 195.7 160.1 240.2 320.3 213.5 320.3 427.1 275.8 413.6 551.9

kN

1910 1910 2860 3810 2540 2540 3990 5080 4540 4540 6800 9070 6350 10430 12700 9980 15420 19960 16330 24490 32660 21770 32660 43550 28120 42180 56280

kgf

Breaking load, min

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21.93

21.94

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12.70 12.70 12.70 12.70 15.88 15.88 15.88 15.88 19.05 19.05 19.05 19.05 25.40 25.40 25.40 31.75 31.75 31.75 38.10 38.10 38.10 44.45 44.45 44.45 50.80 50.80 50.80

0822 0823 0834 0846 1022 1023 1034 1046 1222 1223 1234 1246 1623 1634 1646 2023 2034 2046 2423 2434 2446 2823 2834 2846 3223 3234 3246

Source: IS 1072, 1967.

Pitch P, mm

Chain number

Outside ﬂange thickness, t, min 1.57 1.57 1.57 1.57 1.78 1.78 1.78 1.78 3.07 3.07 3.07 3.07 4.22 4.22 4.22 4.60 4.60 4.60 6.10 6.10 6.10 7.14 7.14 7.14 8.05 8.05 8.05

Lacing

22 23 34 46 22 23 34 46 23 23 34 46 23 34 46 23 34 46 23 34 46 23 34 46 23 34 46 — — 8.18 — — — 9.16 — — — 15.75 — — 21.49 — — 23.42 — — 30.94 — — 36.17 — — 40.77 —

A K þG — — — 13.11 — — — 14.76 — — — 25.25 — — 34.44 — — 37.54 — — 49.58 — — 37.96 — — 65.33

E K1 þ K2 6.35 6.35 6.35 6.35 7.95 7.95 7.95 7.95 9.53 9.53 9.53 9.53 12.70 12.70 12.70 15.88 15.88 15.88 19.05 19.05 19.05 22.23 22.23 22.23 25.40 25.40 25.40

End radius, R, max

TABLE 21-62 Dimensions of anchor clevises for leaf chains (all dimensions in mm)

6.35 6.35 6.35 6.35 7.95 7.95 7.95 7.95 9.53 9.53 9.53 9.53 12.70 12.70 12.70 15.88 15.88 15.88 19.05 19.05 19.05 22.23 22.23 22.23 25.40 25.40 25.40

Slot depth, U, min 0.79 0.79 0.79 0.79 0.79 0.79 0.79 0.79 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59 2.38 2.38 2.38 2.38 2.38 2.38 3.18 3.18 3.18

Fillet radius, B, min — — 4.85 — — — 5.46 — — — 9.37 — — 12.80 — — 13.94 — — 18.44 — — 21.56 — — 24.31 —

K — — — 8.08 — — — 9.09 — — — 15.62 — — 21.34 — — 23.24 — — 30.73 — — 35.94 — — 40.51

K1

Slot pitch

3.33 — 3.33 — 3.73 — 3.73 — 6.38 — 6.38 — — 8.69 — — 9.47 — — 12.50 — — 14.61 — — 16.31 —

G

— 5.03 — 5.03 — 5.66 — 5.66 — 9.63 — 9.63 13.11 — 13.11 14.30 — 14.30 18.85 — 18.85 22.02 — 22.02 24.82 — 24.82

G1

Slot width

þ0:02p þ 0:100 0

þ0:02p þ 0:100 0

Tolerance on A, E G, G1

FLEXIBLE MACHINE ELEMENTS

Source: IS 1072, 1967.

Distance between ﬂanges, L, min 9.12 10.80 14.20 19.28 10.29 12.22 16.03 21.74 16.69 19.96 26.54 36.37 26.75 35.71

Chain number

0822 0823 0834 0846 1022 1028 1034 1046 1222 1223 1234 1246 1623 1634

TABLE 21-63 Dimensions (in mm) for leaf chain sheaves

63.50 63.50 63.50 63.50 79.38 79.38 79.38 79.38 95.25 95.25 95.25 95.25 127.00 127.00

Sheave diameter, SD, min 88.90 88.90 88.90 88.90 104.78 104.78 104.78 104.78 120.65 120.65 120.65 120.65 152.40 152.40

Flange diameter, FD, min 1646 2023 2034 2046 2423 2434 2446 2823 2834 2846 3223 3234 3246

Chain number

49.15 29.77 39.52 51.18 39.95 52.86 72.21 46.08 61.19 83.82 51.89 68.92 94.44

Distance between ﬂanges, L, min

127.00 158.75 158.75 158.75 190.50 190.50 190.50 222.25 222.25 222.25 254.00 254.00 254.00

Sheave diameter, SD, min

152.40 184.15 184.15 184.15 215.90 215.90 215.90 247.65 247.65 247.65 279.40 279.40 279.40

Flange diameter, FD, min

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21.95

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21.96

CHAPTER TWENTY-ONE

TABLE 21-64 Short pitch transmission precision bush chain dimensions, measuring loads, and breaking loads (all dimensions in mm) Chain No. 04C Pitch, p Bush diameter, Db , max Width between inner plates, min, W Bearing pin body diameter, max, Dp Bush bore, db , min Chain path depth, Hd , min Inner plate depth, Hi , max Outer or immediate plate depth, Ho , max Cranked link dimensions X, min Y, min Z, min Transverse pitch, Yp Width over inner link, W1 , max Width between outer plates, W2 , min Width over bearing pins A, max A2 , max A3 , max Additional width for joint fasteners, B, max Measuring load Simplex Duplex Triplex Breaking load, min Simplex Duplex Triplex

06C

6.35 3.30 3.18 2.29 2.34 6.27 6.02 5.21

9.525 5.08 4.77 3.59 3.63 9.30 9.05 7.80

2.64 3.06 0.08 6.40 4.80 4.93

3.96 4.60 0.08 10.13 7.47 7.60

9.10 15.5 21.8 2.5

13.20 23.4 33.5 3.3

0.05 kN 0.10 kN 0.15 kN

5 kgf 10 kgf 15 kgf

3.4 kN 6.9 kN 10. 3 kN

350 kgf 700 kgf 1050 kgf

0.07 kN 0.14 kN 0.20 kN 7.8 kN 15.5 kN 23.2 kN

7 kgf 14 kgf 21 kgf 790 kgf 1580 kgf 2370 kgf

Notes: (1) Dimension C represents clearance between the cranked link plates and the straight plates available during articulation; (2) the chain path depth Hc is the minimum depth of channel through which the assembled chain passes; (3) width over bearing pins for chains wider than triplex ¼ A1 þ Tp (No. of strands in chain—1); (4) cranked links are not recommended for use on chains which are intended for onerous applications. Source: IS 3563, 1966.

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2.9238 3.2361 3.5494 3.8637 4.1786 4.4940 4.8097 5.1258 5.4422 5.7588 6.0755 6.3925 6.7095 7.0266 7.3439 7.6613 7.9787 8.2962 8.6138 8.9314 9.2491 9.5668 9.8845 10.2023

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56

No. of teeth 10.5201 10.8380 11.1558 11.3747 11.7916 12.1096 12.4275 12.7455 13.0635 13.3815 13.6995 14.0176 14.3356 14.6537 14.9717 15.2868 15.6079 15.9260 16.2441 16.5622 16.8803 17.1984 17.5166 17.8347

Pitch circle diameter

a The values given are for a unit pitch length (e.g., 1 mm). Source: IS 3560, 1966.

Pitch circle diameter

No. of teeth 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

No. of teeth 18.1529 18.4710 18.7892 19.1073 19.9255 19.7437 20.0619 20.3800 20.6982 21.0164 21.3246 21.6528 21.9710 22.2892 22.6074 22.9256 23.2438 23.5620 24.8802 24.1985 24.5167 24.8349 25.1513 25.4713

Pitch circle diameter

TABLE 21-65 Pitch circle diametersa for short pitch transmission precision bush chain wheels

81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104

No. of teeth 26.7896 26.1078 26.4260 26.7443 27.0625 27.3807 27.6990 28.0172 28.3355 28.6537 28.9719 29.2902 29.6084 29.9267 30.2449 30.5632 30.8815 31.9097 31.5180 31.8362 32.1545 32.4727 32.7910 33.1093

Pitch circle diameter 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128

No. of teeth 33.4275 33.7458 34.0648 34.3823 34.7006 35.0188 35.3371 35.6554 35.9737 36.2919 36.6102 36.9285 37.2467 37.5650 37.8833 38.2016 38.5198 38.8381 39.1564 39.4776 39.7929 40.1112 40.4295 43.7478

Pitch circle diameter

129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150

No. of teeth

41.0660 41.3843 41.7026 42.0209 42.3291 42.6574 42.9757 43.2940 43.6123 43.9306 44.2488 44.5671 44.8854 45.2037 45.5220 45.8403 46.1585 46.4768 46.7951 47.1134 47.4317 47.7500

Pitch circle diameter

FLEXIBLE MACHINE ELEMENTS

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21.97

FLEXIBLE MACHINE ELEMENTS

21.98

CHAPTER TWENTY-ONE

TABLE 21-66 Recommended design data for silent chains No. of teeth Chain pitch, mm

Speed of small sprocket

Driver

Driven

Min center distance, mm

9.3 12.7 15.8 19.0 22.2 25.4 31.7 38.1 50.8 76.2

2000–4000 1500–2000 1200–1500 1000–1200 900–1000 800–900 650–800 500–650 300–500 300

17–25 17–25 19–25 19–25 19–25 19–25 21–25 25–27 25–27 25–27

21–120 21–130 21–150 23–150 23–150 23–150 25–150 27–150 27–150 27–150

152.4 228.6 304.8 381.0 457.2 533.4 685.8 914.4 1219.2 1676.4

TABLE 21-67A Maximum speed of small sprocket for inverted tooth chains

Pitch, mm

Max width, mm

Number of teeth

9.50 12.70 15.88 19.05 25.40 31.75 38.10 50.80

101.6 177.8 203.2 254.0 355.6 508.0 609.6 762.0

17 19 21 23 25 27 29 31 33 35 37 45 40 50

Speed, rpm 4000 5000 6000 6000 6000 6000 6000 6000 6000 6000 5000 4000 5000 3500

3500 3500 3000 4000 4000 4000 4000 4000 4000 4000 3500 3000 3500 2500

2500 2500 3000 3000 3500 3500 3500 3500 3500 3500 3000 2000 2500 2000

2000 2500 2500 2500 2500 2500 2500 2500 2500 2500 2500 2000 2500 1800

1200 1500 1800 1800 1800 2000 2000 2000 2000 2000 1800 1500 1500 1200

1200 1200 1800 1800 1800 1800 1800 1800 1800 1200 1000 1200 1000

1000 1000 1200 1200 1200 1200 1200 1200 1200 1000 900 900 800

700 700 800 900 900 900 900 900 900 800 700 800 600

TABLE 21-67B Maximum velocity for various types of chains, rpm Number of sprocket teeth Bush roller chain Type of chain

Chain pitch, p, mm

12 15 20 25 30

2300 1900 1350 1150 1000

15

19

23

27

30

Silent chains 17.35

2400 2000 1450 1200 1050

2530 2100 1500 1250 1100

2550 2150 1550 1300 1100

2600 2200 1550 1300 1100

12.7 15.87 19.05 25.40 31.75

3300 2650 2200 1650 1300

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FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

21.99

TABLE 21-68 Safety factor Speed of smaller sprocket, rpm Chains

50

Bush roller chains p ¼ 12, 15 mm p ¼ 20, 25 mm p ¼ 30, 35 mm Silent chains p ¼ 12:7, 15.87 mm p ¼ 19:05, 25.4 mm

7.0 7.0 7.0 20 20

260

7.8 8.2 8.55 22.2 23.4

400

600

800

1000

1200

1600

2000

8.55 9.35 10.2

9.35 10.3 13.2

10.2 10.7 14.8

11.0 12.9 16.3

11.7 14.0 19.5

13.2 16.3

1.48

24.4 26.7

28.7 30.0

29.0 33.4

31.0 36.8

33.4 40.0

37.8 46.5

42.0 53.5

FIGURE 21-26 Notation for wheel rim proﬁles of bush chain.

FIGURE 21-28 Notation for minimum tooth gap form for bush chain.

FIGURE 21-27 Notation for tooth gap form of bush chain.

FIGURE 21-29 Notation for maximum tooth gap form for bush chain.

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FLEXIBLE MACHINE ELEMENTS

21.100

CHAPTER TWENTY-ONE

Particular

Formula

The minimum limit of the tooth height above the pitch polygon

HTmin ¼ 0:5ð p Db Þ

ð21-162Þ

WHEEL RIM PROFILE (Fig. 21-26) The value of tooth width for simple chain wheels (Fig. 21-26)

C1 ¼ 0:93w

ð21-163Þ

The value of tooth width for duplex and triplex chain wheels

C1 ¼ 0:91w

ð21-164Þ

The value of tooth width for quadruplex chain wheels and above

C1 ¼ 0:88w

ð21-165Þ

The value of tolerance shall be h=4. The value of width over tooth

C2 ðor C3 Þ ¼ number of strands 1Tp þ C1 ð21-166Þ with a tolerance value of h=4 where Tp ¼ transmission pitch of strands

The minimum tooth side radius

F ¼p

ð21-167Þ

The tooth side relief

G ¼ 0:1p to 0:15p

ð21-168Þ

Absolute maximum shroud diameter

SD ¼ p cot

For bush chains dimensions, breaking load, pitch circle diameters, etc.

Refer to Tables 21-64 to 21-68.

1808 1:05Hi 1:00 2Kort , mm z ð21-169Þ

REFERENCES 1. Maleev, V. L., and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954. 2. Black, P. H., and O. E. Adams, Jr., Machine Design, McGraw-Hill Book Company, New York, 1968. 3. Norman, C. A., E. S. Ault, and I. F. Zarobsky, Fundamentals of Machine Design, The Macmillan Company, New York, 1951. 4. Shigley, J. E., Machine Design, McGraw-Hill Book Company, New York, 1962. 5. Shigley, J. E., and C. R. Mischke, Mechanical Engineering Design, McGraw-Hill Book Company, New York, 1989. 6. Shigley, J. E., and C. R. Mischke, Standard Handbook of Machine Design, McGraw-Hill Book Company, New York, 1986. 7. Baumeister, T., ed., Marks’ Standard Handbook for Mechanical Engineers, McGraw-Hill Book Company, New York, 1978. 8. Niemann, G., Maschinenelemente, Springer-Verlag, Berlin; Zweiter Band, Munich, 1965. 9. Niemann, G., Machine Elements—Design and Calculations in Mechanical Engineering, Vol. II, Allied Publishers Private Ltd., New Delhi, 1978. 10. Decker, K. H., Maschinenelemente, Gestaltung and Berechnung, Carl Hanser Verlag, Munich, 1971.

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FLEXIBLE MACHINE ELEMENTS FLEXIBLE MACHINE ELEMENTS

21.101

11. Lingaiah, K. and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962. 12. Lingaiah, K. and B. R. Narayana Iyengar, Machine Design Data Handbook, Vol. I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1973. 13. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 14. Bureau of Indian Standards. 15. Albert, C. D., Machine Design Drawing Room Problems, John Wiley and Sons, New York, 1949. 16. V-Belts and Pulleys, SAE J 636C, SAE Handbook, Part I, Society of Automotive Engineers, Inc., 1997. 17. SI Synchronous Belts and Pulleys, SAE J 1278 Oct.80, SAE Handbook, Part I, Society of Automotive Engineers, Inc., 1997. 18. Synchronous Belts and Pulleys, SAE J 1313 Oct.80, SAE Handbook, Part I, Society of Automotive Engineers, Inc., 1997. 19. Wolfram Funk, ‘Belt Drives,’ J. E. Shigley and C. R. Mischke, Standard Handbook of Machine Design, 2nd edition, McGraw-Hill Publishing Company, New York, 1996.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

22 MECHANICAL VIBRATIONS

SYMBOLS a A B C Cc Ct C1 , C2 d D e E f F Fo FT g G h i I J k ke kt K

coeﬃcients with subscripts ﬂexibility acceleration, m/s2 (ft/s2 ) area of cross section, m2 (in2 ) constant constant coeﬃcient of viscous damping, N s/m or N/ (lbf s/in or lbf/) constant critical viscous damping, N s/m (lbf s/in) coeﬃcient of torsional viscous damping, N m s/rad (lbf in s/rad) coeﬃcients constants diameter of shaft, m (in) ﬂexural rigidity ½¼ Eh3 =12ð1 2 Þ displacement of the center of mass of the disk from the shaft axis, m (in) modulus of elasticity, GPa (Mpsi) frequency, Hz exciting force, kN (lbf ) maximum exciting force, kN (lbf ) transmitted force, kN (lbf ) acceleration due to gravity, 9.8066 m/s2 (32.2 ft/s2 or 386.6 in/s2 ) modulus of rigidity, GPa (Mpsi) thickness of plate, m (in) integer (¼ 0, 1, 2, 3, . . .) mass moment of inertia of rotating disk or rotor, N s2 m (lbf s2 in) polar second moment of inertia, m4 or cm4 (in4 ) spring stiﬀness or constant, kN/m (lbf/in) equivalent spring constant, kN/m (lbf/in) torsional or spring stiﬀness of shaft, J/rad or N m/rad (lbf in/rad) kinetic energy, J (lbf/in)

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MECHANICAL VIBRATIONS

22.2

CHAPTER TWENTY-TWO

l m me M Mt p q r R ¼ 1 TR R2 ¼ D2 =2 t T TR U v w W x x1 , x2 xo x_ x€ Xst y ¼

C Cc

length of shaft, m (in) mass, kg (lb) equivalent mass, kg (lb) total mass, kg (lb) torque, N m (lbf ft) circular frequency, rad/s pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ damped circular frequency ð¼ 1 2 Þ radius, m (in) percent reduction in transmissibility radius of the coil, m (in) time (period), s temperature, K or 8C (8F) transmissibility vibrational energy, J or N m (lbf in) potential energy, J (lbf in) velocity, m/s (ft/min) weight per unit volume, kN/m3 (lbf/in3 ) total weight, kN (lbf ) displacement or amplitude from equilibrium position at any instant t, m (in) successive amplitudes, m (in) maximum displacement, m (in) linear velocity, m/s (ft/min) linear acceleration, m/s2 (ft/s2 ) static deﬂection of the system, m (in) deﬂection of the disk center from its rotational axis, m or mm (in) weight density, kN/m3 (lbf/in3 ) damping factor

logarithmic decrement, deﬂection, m (in) static deﬂection, m (in) phase angle, deg wavelength, m (in) Poisson’s ratio mass density, kg/m3 (lb/in3 ) normal stress, MPa (psi) shear stress, MPa (psi) period, s angular deﬂections, rad (deg) angular velocity, rad/s angular acceleration, rad/s2 forced circular frequency, rad/s

st

_

€ !

Particular

Formula

SIMPLE HARMONIC MOTION (Fig. 22-1) The displacement of point P on diameter RS (Fig. 22-1)

x ¼ xo sin pt

ð22-1Þ

The wavelength

¼ 2

ð22-2Þ

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MECHANICAL VIBRATIONS MECHANICAL VIBRATIONS

Particular

22.3

Formula

FIGURE 22-1 Simple harmonic motion.

The periodic time The frequency

¼

2 p

ð22-3Þ

f¼

1 p ¼ 2

ð22-4Þ

The maximum velocity of point Q

vmax ¼ pxo

The maximum acceleration of point Q

amax ¼ v_max ¼ p xo

ð22-5Þ 2

ð22-6Þ

Single-degree-of-freedom system without damping and without external force (Fig. 22-2) Linear system

FIGURE 22-2 Spring-mass system.

The equation of motion

m€ x þ kx ¼ 0

ð22-7Þ

The general solution for displacement

x ¼ A sin pt þ B cos pt

ð22-8Þ

x ¼ C sinð pt Þ

ð22-9Þ

where ¼ phase angle of displacement The equation for displacement of mass for the initial condition x ¼ xo and x_ ¼ 0 at t ¼ 0

x ¼ xo cos pt

The natural circular frequency

rﬃﬃﬃﬃ rﬃﬃﬃﬃﬃ k g ¼ m st rﬃﬃﬃﬃ pn 1 k ¼ fn ¼ 2 2 m rﬃﬃﬃﬃﬃ 1 g fn ¼ 2 st 1=2 3:132 1 1=2 1 0:5 fn ¼ 2 st st pn ¼

The natural frequency of the vibration The natural frequency in terms of static deﬂection st

where st in m and fn in Hz

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ð22-10Þ

ð22-11Þ ð22-12Þ ð22-13Þ ð22-13aÞ

MECHANICAL VIBRATIONS

22.4

CHAPTER TWENTY-TWO

Particular

Formula

fn ¼

99 2

1 st

1=2

1=2 1 15:76 st

SI

ð22-13bÞ

USCS

ð22-13cÞ

USCS

ð22-13dÞ

USCS

ð22-13eÞ

where st in mm and fn in Hz fn ¼

5:67 2

1 st

1=2

1=2 1 0:9 st

where st in ft and fn in Hz fn ¼

19:67 2

1 st

1=2

3:127 ¼ pﬃﬃﬃﬃﬃ st

where st in in and fn in Hz 187:6 fn ¼ pﬃﬃﬃﬃﬃ st

where st in in and fn in cpm (cycles per minute)

FIGURE 22-3 Static deﬂection (st ) vs. natural frequency. (Courtesy of P. H. Black and O. E. Adams, Jr., Machine Design, McGraw-Hill, New York, 1955.)

The plot of natural frequency vs. static deﬂection

Refer to Fig. 22-3.

Simple pendulum The equation of motion for simple pendulum (Fig. 22-4) The angular displacement for ¼ o and _ ¼ 0 at t¼0 The circular frequency for simple pendulum for small oscillation

g g € ¼ sin ¼ € þ ¼ 0 l l rﬃﬃﬃ g ¼ o sin t l rﬃﬃﬃ g p¼ l

ð22-14Þ ð22-15Þ ð22-15aÞ

ENERGY The total energy in the universe is constant according to conservation of energy

K þ U ¼ constant

ð22-16Þ

Kinetic energy

K ¼ 12 mv2 ¼ 12 mx_ 2

ð22-17Þ

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MECHANICAL VIBRATIONS MECHANICAL VIBRATIONS

Particular

22.5

Formula

Potential energy

U ¼ 12 kx2

ð22-18Þ

Maximum kinetic energy is equal to maximum potential energy according to conservation of energy

Kmax ¼ Umax

ð22-19Þ

FIGURE 22-4 Simple pendulum.

FIGURE 22-5 Single rotor system subject to torque.

Torsional system (Fig. 22-5) The equation of motion of torsional system (Fig. 22-5) with torsional damping under external torque Mt sin pt The equation of motion of torsional system without considering the damping and external force on the rotor The equation for angular displacement

The angular displacement for ¼ o and _ ¼ 0 at t¼0 The natural circular frequency The natural circular frequency taking into account the shaft mass The natural frequency

The expression for torsional stiﬀness

I € þ Ct _ þ kt x ¼ Mt sin pt

ð22-20Þ

where Ct ¼ coeﬃcient of torsional viscous damping, N m s/rad € ð22-21Þ I þ kt ¼ 0

¼ A sin pt þ B cos pt

ð22-22aÞ

¼ C sinð pt Þ

ð22-22bÞ

where ¼ phase of displacement pﬃﬃﬃﬃﬃﬃﬃﬃﬃ

¼ o cosð kt =IÞt pn ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ kt =I

" #1=2 Is pn ¼ kt Iþ 3

ð22-23Þ ð22-24Þ ð22-25Þ

fn ¼

pn 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ kt =I 2 2

ð22-26Þ

kt ¼

JG d4 G ¼ l 32 l

ð22-27Þ

where J ¼ d4 =32 ¼ moment of inertia, polar, m4 or cm4

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MECHANICAL VIBRATIONS

22.6

CHAPTER TWENTY-TWO

Particular

Formula

Single-degree-freedom system with damping and without external force (Fig. 22-6) The equation of motion

m€ x þ cx_ ¼ kx ¼ 0

ð22-28Þ

The general solution for displacement

x ¼ C1 es1 t þ C2 es2 t pﬃﬃﬃﬃﬃﬃﬃ ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 x ¼ C1 eð 1Þpn t þ C2 eðþ 1Þpn t

ð22-29Þ

x ¼ Ae pn t sinðqt þ Þ

ð22-31Þ

ð22-30Þ

where C1 , C2 , and A are arbitrary constants of integration. (They can be found from initial conditions.) " #1=2 C C 2 k s1;2 ¼ ð22-32Þ 2m 2m m qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

s1;2 ¼

2 1 pn

ð22-33Þ

C ¼ damping ratio, Cc pﬃﬃﬃﬃﬃﬃﬃ Cc ¼ 2mpn ¼ 2 km

where ¼ FIGURE 22-6 Single-degree-of-freedom spring-mass-dashpot system.

q ¼ frequency of damped oscillation qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 k c2 1=2 ¼ 1 2 pn ¼ ¼ 2 d m 4m ð22-33aÞ

¼ phase angle or phase displacement with respect to the exciting force

For the damped oscillation of the single-degreefreedom system with time for damping factor < 1

Refer to Figs. 22-7 and 22-8.

FIGURE 22-7 Damped motion < 1:0.

FIGURE 22-8 Logarithmic decrement. (Reproduced from Marks’ Standard Handbook for Mechanical Engineers, 8th edition, McGraw-Hill, New York, 1978.)

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MECHANICAL VIBRATIONS MECHANICAL VIBRATIONS

Particular

22.7

Formula

LOGARITHMIC DECREMENT (Fig. 22-8) The equation for logarithmic decrement

¼ ln

xo x U 2 ¼ ln 1 ¼ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 x1 x2 U 1 2

ð22-34Þ

EQUIVALENT SPRING CONSTANTS (Fig. 22-9) The spring constant or stiﬀness

k¼

F x

ð22-35Þ

The ﬂexibility

a¼

x F

ð22-36Þ

The equivalent spring constant for springs in series (Fig. 22-9a)

ke ¼

1 1 1 þ k1 k2

ð22-37Þ

The equivalent spring constant for springs in parallel (Fig. 22-9b)

ke ¼ k 1 þ k 2

ð22-38Þ

For spring constants of diﬀerent types of springs, beams, and plates

Refer to Table 22-1

FIGURE 22-9 Springs in series and parallel.

FIGURE 22-10 Spring-mass-dashpot system subjected to external force.

Single-degree-of-freedom system with damping and external force (Fig. 22-10) The equation of motion

m€ x þ cx_ þ kx ¼ Fo sin !t x€ þ 2 pn x_ þ p2n x ¼

Fo sin !t m

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ð22-39Þ ð22-40Þ

MECHANICAL VIBRATIONS

22.8

CHAPTER TWENTY-TWO

TABLE 22-1 Spring constants or spring stiﬀness of various springs, beams, and plates Formula for spring constant, k

Particular

Figure

Equation

Linear Spring Stiﬀness or Constants [Load per mm (in) Deﬂection] Helical spring subjected to tension with i number of turns

k¼

Gd4 64iR3

(22-41)

Bar under tension

k¼

EA l

(22-42)

Cantilever beam subjected to transverse load at the free k ¼ 3EI l3 end

(22-43)

Cantilever beam subjected to bending at the free end

k¼

2EI l2

(22-44)

Simply supported beam with concentrated load at the center

k¼

48EI l3

(22-45)

Simply supported beam subjected to a concentrated load k ¼ 3EIl a2 b2 not at the center

(22-46)

Beam ﬁxed at both ends subjected to a concentrated load k ¼ 192EI l3 at the center

(22-47)

Beam ﬁxed at one end and simply supported at another k ¼ 768EI 7l3 end subjected to concentrated load at the center

(22-48)

(22-49)

Circular plate clamped along the circumferential edge subjected to concentrated load at the center whose ﬂexural rigidity is D ¼ Eh3 =12ð1 2 Þ, thickness h and Poisson ratio

k¼

16D R2

Circular plate simply supported along the circumferential edge with concentrated load at the center

k¼

16D R2

String ﬁxed at both ends subjected to tension T

1þ 3þ

(22-50)

where ¼ Poisson’s ratio k¼

4T String tension T l

(22-51)

Torsional or Rotational Spring Stiﬀness or Constants (Load per Radian Rotation) Spiral spring whose total length is l and moment of inertia of cross section I

kt ¼

EI l

Helical spring with i turns subjected to twist whose wire diameter is d, the coil

(22-52)

diameter is D

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MECHANICAL VIBRATIONS MECHANICAL VIBRATIONS

kt ¼

22.9

Ed4 64iD

(22-53)

TABLE 22-1 Spring constants or spring stiﬀness of various springs, beams, and plates (Cont.) Formula for spring constant, k

Particular

Figure

Equation

Bending of helical spring of i number of turns

kt ¼

Ed4 1 32iD 1 þ ðE=2GÞ

(22-54)

Twisting of bar of length l

kt ¼

JG l

(22-55)

Twisting of a hollow circular shaft with length l, whose outside diameter is Do , and inside diameter is Di

kt ¼

GIp G D4o D4i ¼ 32 l l

(22-56)

Twisting of cantilever beam

kt ¼

GJ l

(22-57)

Simply supported beam subjected to couple at the center

kt ¼

12EI l

Particular

The complete solution for the displacement

Formula

x ¼ Aepn t sinðqt þ 1 Þ þ Xo sinð!t Þ

ð22-60aÞ

x ¼ Aepn t sinðqt þ 1 Þ þ The steady-state solution for amplitude of vibration

ð22-60bÞ

Fo ﬃ X ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðk m!2 Þ2 þ ðc!Þ2 ¼

The phase angle

ðFo =kÞ sinð!t Þ ½f1 ð!=pn Þ2 g2 þ ð2!=pn Þ2 1=2

Fo =k 2 2

½f1 ð!=pn Þ g þ ð2!=pn Þ2 1=2 " 1

¼ tan

2ð!=pn Þ 1 ð!=pn Þ2

ð22-60cÞ

#

The magniﬁcation factor

Xo 1 ¼ Xst ½f1 ð!=pn Þ2 g2 þ ð2!=pn Þ2 1=2

The plot of magniﬁcation factor ðXo =Xst Þ vs. frequency ratio ð!=pn Þ and phase angle vs. ð!=pn Þ

Refer to Figs. 22-11 and 22-12.

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ð22-61Þ ð22-62Þ

MECHANICAL VIBRATIONS

22.10

CHAPTER TWENTY-TWO

Particular

Formula

FIGURE 22-11 Phase angle vs. frequency ratio ð!=pn Þ.

FIGURE 22-12 Magniﬁcation factor ðXo =Xst Þ vs. frequency ratio ð!=pn Þ.

The amplitude at resonance (i.e. for !=pn ¼ 1)

Fo F X ¼ o ¼ st cpn 2k 2

ð22-63Þ

The equation of motion

M€ x þ cx_ þ kn ¼ ðme!2 Þ sin !t

ð22-64Þ

The steady-state solution for displacement

me!2 X ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðk M!2 Þ2 þ ðc!Þ2

Xres ¼

UNBALANCE DUE TO ROTATING MASS (Fig. 22-13)

X¼

ðm=MÞ eð!=pn Þ2 ½f1 ð!=pn Þ2 g2 þ ð2!=pn Þ2 1=2

ð22-65aÞ

ð22-65bÞ

FIGURE 22-13 External force due to rotating unbalanced mass. (Produced with some modiﬁcation from N. O. Myklestad, Fundamentals of Vibration Analysis, McGraw-Hill, New York, 1956.)

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MECHANICAL VIBRATIONS MECHANICAL VIBRATIONS

Particular

The complete solution for the displacement

22.11

Formula

x ¼ Aepn t sinðqt þ 1 Þ þ

eðm=MÞð!=pn Þ2 ½f1 ð!=pn Þ2 g2 þ ð2!=pn Þ2 1=2

sinð!t Þ ð22-66Þ

Nondimensional form of expression for Eq. (22-65b)

The phase angle

For a schematic representation of Eqs. (22-67) and (22-68) or harmonically disturbing force due to rotating unbalance

M X ð!=pn Þ2 ¼ m e ½f1 ð!=pn Þ2 g2 þ ð2!=pn Þ2 1=2 " # 2 1 2ð!=pn Þ

¼ tan 1 ð!=pn Þ2 Refer to Figs. 22-14 and 22-15

FIGURE 22-14 MX=me vs. frequency ratio ð!=pn Þ.

FIGURE 22-15 Phase angle vs. frequency ratio ð!=pn Þ.

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ð22-67Þ

ð22-68Þ

MECHANICAL VIBRATIONS

22.12

CHAPTER TWENTY-TWO

Particular

Formula

WHIPPING OF ROTATING SHAFT (Fig. 22-16) The equation of motion of shaft due to unbalanced mass

m€ xc þ cx_ c þ kxc ¼ me!2 cos !t

ð22-69aÞ

m€ yc þ c€ yc þ kyc ¼ me!2 sin !t

ð22-69bÞ

where xc and yc are coordinates of position of center of shaft with respect to x and y coordinates The solution

The displacement of the center of the disk from the line joining the centers of bearings

me!2 cosð!t Þ xc ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðk m!2 Þ2 þ ðc!Þ2

ð22-70aÞ

me!2 sinð!t Þ yc ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðk m!2 Þ2 þ ðc!Þ2

ð22-70bÞ

r¼

FIGURE 22-16 Whipping of shaft. (Reproduced from Marks’ Standard Handbook for Mechanical Engineers, 8th edition, McGraw-Hill Book Company, New York, 1978.)

ð22-71aÞ

eð!=pn Þ2

ð22-71bÞ ½f1 ð!=pn Þ2 g2 þ ð2!=pn Þ2 1=2 " # c! 2ð!=pn Þ 1 1 ¼ tan ð22-72Þ

¼ tan k m!2 1 ð!=pn Þ2 r¼

The phase angle

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ me!2 x2c þ y2c ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðk m!2 Þ2 þ ðc!Þ2

FIGURE 22-17 Excitation of a system by motion of support.

EXCITATION OF A SYSTEM BY MOTION OF SUPPORT (Fig. 22-17) The equation of motion The absolute value of the amplitude ratio of x and y

m€ x þ cx_ þ kx ¼ ky þ cy_ #1=2 " 2 X 1 þ ð2!=p Þ n ¼ Y ½1 ð!=pn Þ2 2 þ ð2!=pn Þ2

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ð22-73Þ ð22-74Þ

MECHANICAL VIBRATIONS MECHANICAL VIBRATIONS

Particular

The phase angle

22.13

Formula

" 1

¼ tan

2ð!=pn Þ3 f1 ð!=pn Þ2 g2 þ ð2!=pn Þ2

# ð22-75Þ

Refer to Fig. 22-20 for jX=Yj vs. !=pn . The plot of Eq. (22-55) for motion due to support

INSTRUMENT FOR VIBRATION MEASURING (Fig. 22-18)

The curves are similar.

m€ z þ cz_ þ kz ¼ m€ y ¼ mY!2 sin !t

The equation of motion The steady-state solution for relative displacement Z

The phase angle

Z¼

Yð!=pn Þ2 ½f1 ð!=pn Þ2 g2 þ ð2!=pn Þ2 1=2

¼ tan1

2ð!=pn Þ 1 ð!=pn Þ2

ð22-76Þ ð22-77Þ

ð22-78Þ

Refer to Figs. 22-14 and 22-15. The plot of absolute value of jZ=Yj vs. frequency ratio ð!=pn Þ and the phase angle vs. frequency ratio ð!=pn Þ

FIGURE 22-18 Instrument for vibration measuring. (Reproduced from Marks’ Standard Handbook for Mechanical Engineers, 8th edition, McGraw-Hill, New York, 1978.)

The curves for jZ=Yj vs. !=pn and vs. !=pn are identical.

FIGURE 22-19 External force transmitted to foundation through damper and springs. (Reproduced from Marks’ Standard Handbook for Mechanical Engineers, 8th edition, McGraw-Hill, New York, 1978.)

ISOLATION OF VIBRATION (Fig. 22-19) The force transmitted through the springs and damper

FT ¼ FT ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðkXÞ2 þ ðc!XÞ2 Fo ½1 þ ð2!=pn Þ2 1=2 ½f1 ð!=pn Þ2 g2 þ ð2!=pn Þ2 1=2

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ð22-79Þ ð22-80Þ

MECHANICAL VIBRATIONS

22.14

CHAPTER TWENTY-TWO

Particular

Formula

Transmissibility TR ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ ð2!=pn Þ2

FT ¼ Fo ½f1 ð!=pn Þ2 g2 þ ð2!=pn Þ2 1=2

ð22-81Þ

Refer to Fig. 22-20 for TR and jX=Yj. Comparison of Eqs. (22-81) and (22-85) indicates that the plot of F =Fo is identical to jX=Yj. Transmissibility when damping is negligible

The transmissibility in terms of static deﬂection st

The frequency from Eq. (22-83)

TR ¼

TR ¼

1 ð!=pn Þ2 1

ð22-82Þ

1

ð22-83Þ

ð2fn Þ2 st 1 g

" " #1=2 #1=2 3:132 1 1 1 2R þ1 ¼ 0:5 fn ¼ 2 st TR st 1 R SI

ð22-84aÞ

where fn in Hz and st in m The percent reduction in the transmissibility is deﬁned as R ¼ 1 TR " " #1=2 #1=2 99 1 2 R 1 2R fn ¼ ¼ 15:76 2 st 1 R st 1 R SI

ð22-84bÞ

USCS

ð22-84cÞ

USCS

ð22-84dÞ

where fn in Hz and st in mm " #1=2 19:67 1 2 R fn ¼ 2 st 1 R FIGURE 22-20 Transmissibility (TR ) vs. frequency ratio ð!=pn Þ.

where st in in and fn in Hz "

1 fn ¼ 187:6 st

2R 1R

#1=2

where fn in rpm and st in in For the plot of static deﬂection st vs. R

Refer to Fig. 22-21.

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MECHANICAL VIBRATIONS MECHANICAL VIBRATIONS

FIGURE 22-21 Static deﬂection (st ) vs. disturbing frequency for various percent reduction in transmissibility (TR ) for ¼ 0. (Courtesy of F. S. Tes, I. E. Morse, and R. T. Hinkle, Mechanical Vibration—Theory and Applications, CBS Publishers and Distributors, New Delhi, India, 1983.)

22.15

FIGURE 22-22 Undamped two-degree-of freedom system.

Particular

Formula

UNDAMPED TWO-DEGREE-OF-FREEDOM SYSTEM (Fig. 22-22) WITHOUT EXTERNAL FORCE Equations of motion

The frequency of equation which gives two values for p2

m1 x€1 þ ðk1 þ k3 Þx1 k3 x2 ¼ 0

ð22-85aÞ

m2 x€2 þ ðk2 þ k3 Þx2 k3 x1 ¼ 0 k þ k3 k 2 þ k3 þ p4 p2 1 m1 m2

ð22-85bÞ

þ The amplitude ratio

k1 k2 þ k2 k3 þ k1 k3 ¼0 m1 m2

a1 k3 m p2 k2 k3 ¼ ¼ 2 2 a2 m1 p k1 k3 k3

ð22-86Þ ð22-87Þ

DYNAMIC VIBRATION ABSORBER (Fig. 22-23) Equations of motion

The solution of the forced vibration of the absorber will be of the form

M€ x1 þ ðK þ kÞx1 kx2 ¼ Fo sin !t

ð22-88aÞ

m€ x2 þ kðx2 x1 Þ ¼ 0

ð22-88bÞ

x1 ¼ a1 sin pt

ð22-89aÞ

x2 ¼ a2 sin pt

ð22-89bÞ

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MECHANICAL VIBRATIONS

22.16

CHAPTER TWENTY-TWO

Particular

The ratio of amplitudes a1 and a2 to the static deﬂection of the main system xst

Formula

!2 1 2 a1 pa ¼ 2 xst ! k !2 k 1 2 1þ 2 K pm K pa a2 1 ¼ xst !2 k !2 k 1 2 1þ 2 K pm K pa

ð22-90aÞ

ð22-90bÞ

where xst ¼ Fo =K ¼ static deﬂection of main system p2a ¼ K=m ¼ natural circular frequency of absorber FIGURE 22-23 Dynamic vibration absorber.

FIGURE 22-24 Two-rotor system.

If the main system is in resonance, then considering pa ¼ pm or

k K k m ¼ or ¼ ¼ Rm m M K M

Eqs. (7-90a) and (7-90b) become

The natural frequencies

The mass equivalent for the absorber

p2m ¼ k=M ¼ natural circular frequency of main system Rm ¼

m absorber mass ¼ mass ratio ¼ M main mass

x1 1 ð!=pa Þ2 ¼ sin !t 2 xst ½1 ð!=pa Þ ½1 þ Rm ð!=pa Þ2 Rm ð22-91aÞ x2 1 ¼ sin !t xst ½1 ð!=pa Þ2 ½1 þ Rm ð!=pa Þ2 Rm ð22-91bÞ

! pa

2 ¼

R R2 1=2 1 þ m Rm þ m 2 4

meq 1 ¼ m 1 ð!=pa Þ2

ð22-92Þ ð22-93Þ

where meq ¼ equivalent mass solidly attached to the main mass M

TORSIONAL VIBRATING SYSTEMS Two-rotor system (Fig. 22-24) The torque on rotor A

Mta ¼ Ia p2 a

ð22-94Þ

The total torque on two rotors

Mti ¼ Mta þ Mtb ¼ Ia p2 a þ Ib p2 b ¼ 0

ð22-95Þ

The angular displacement or angle of twist of rotor B

where i ¼ imaginary M I p2 b ¼ a ta ¼ a 1 a kt kt

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ð22-96Þ

MECHANICAL VIBRATIONS MECHANICAL VIBRATIONS

Particular

The frequency equation

The natural circular frequency

The natural frequency

Formula

I I p2 p2 a Ia þ Ib a b ¼0 kt pn ¼ fn ¼

1 2

ðIa þ Ib Þkt Ia Ib

a I l ¼ b¼ a b Ia lb

The relation between Ia , Ib , la , and lb

Ia l a ¼ Ib l b la ¼

ð22-97aÞ

1=2

ðIa þ Ib Þkt Ia Ib

The amplitude ratio

The distance of node point from left end of rotor A

22.17

ð22-97bÞ 1=2 ð22-98Þ ð22-99Þ ð22-100Þ

Ib l Ia þ Ib

ð22-101Þ

Two rotors connected by shaft of varying diameters The length of torsionally equivalent shaft of diameter d whose varying diameters are d1 , d2 , and d3

le ¼ d4

l1 l2 l þ þ 3 d41 d42 d43

ð22-102Þ

Three-rotor torsional system (Fig. 22-25) The algebraic sum of the inertia torques of rotors A, B, and C

Mti ¼ Mta þ Mtb þ Mtc ¼ Ia p2 a þ Ib p2 b þ Ic p2 c ð22-103Þ where a , b , and c are angular displacement or angular twist at rotors, A, B, and C, respectively

FIGURE 22-25 Three-rotor system.

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MECHANICAL VIBRATIONS

22.18

CHAPTER TWENTY-TWO

Particular

The frequency equation

Formula

I I I I I I I I p2 a ðIa þ Ib þ Ic Þ p2 a b þ a c þ a c þ b c kt1 kt1 kt2 kt2 I I I þ p4 a b c ¼0 ð22-104aÞ kt1 kt2 1 kt1 kt2 kt1 þ kt2 þ þ p2 ¼ 2 Ia Ic Ib 1 kt1 kt2 kt1 þ kt2 2 þ þ Ia Ic Ib 2 1=2 k k 4 t1 t2 ðIa þ Ib þ Ic Þ ð22-104bÞ Ia Ib Ic where kt1 and kt2 are torsional stiﬀness of shafts of lengths l1 and l2

The amplitude ratio

The relation between Ia , Ic , la , and lc The relation between Ia , Ib , la , and lc Frequency can also be found from Eqs. (22-108) and (22-109)

b I p2 ¼1 a a kt1

ð22-105aÞ

c Ia I I p4 Ia I ¼ 1 p2 þ c þ b þ kt1 kt2 a kt1 kt2 kt2

ð22-105bÞ

Ia l a ¼ Ic l c 1 1 1 1 ¼ þ Ia la Ib l1 la l2 lc sﬃﬃﬃﬃﬃﬃ 1 ktc fc ¼ Ic 2 GJ2 lc sﬃﬃﬃﬃﬃﬃ k0tb 1 fb ¼ Ib 2

ð22-106Þ ð22-107Þ

ð22-108Þ

where ktc ¼

where k0tb ¼

GJ1 GJ2 þ l1 la l2 lc

For collection of mechanical vibration formulas to calculate natural frequencies

Refer to Table 22-2.

For analogy between diﬀerent wave phenomena

Refer to Table 22-3.

For analogy between mechanical and electrical systems

Refer to Table 22-4.

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ð22-109Þ

MECHANICAL VIBRATIONS MECHANICAL VIBRATIONS

22.19

TABLE 22-2 A collection of formulas Particular

Formula

Natural Frequencies of Simple Systems sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ End mass M, spring mass m, spring k pn ¼ stiﬀness k M þ m=3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ End inertia I, shaft inertia Is , shaft stiﬀness kt p ¼ n kt I þ Is =3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Two disks on a shaft kt ðI1 þ I2 Þ pn ¼ I1 I2 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Cantilever; end mass M, beam mass m, k pn ¼ stiﬀness by formula (22-93) M þ 0:23m rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Simply supported beam central mass M; k pn ¼ beam mass m; stiﬀness by formula (22-95) M þ 0:5m vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u Massless gears, speed of I2 n times as as 1 I1 þ n2 I2 u u p ¼ n speed of I1 t1 1 I1 I2 n2 þ kt1 n2 kt2 p2n

1 ¼ 2

kt1 kt3 kt1 þ kt3 þ þ I1 I3 I2

1 2

ﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ kt1 kt3 kt1 þ kt3 2 kt1 kt3 þ þ 4 ðI þ I2 þ I3 Þ I1 I3 I2 I1 I2 I3 1

Uniform Beams (Longitudinal and Torsional Vibration) sﬃﬃﬃﬃﬃﬃﬃﬃﬃ Longitudinal vibration of cantilever: 1 AE pn ¼ n þ A ¼ cross section, E ¼ modulus of 2 1 l2 elasticity 1 ¼ mass per unit length, n ¼ 0; 1; 2; 3 ¼ number of nodes

For steel and l in inches, this becomes

Organ pipe open at one end, closed at the other

For air at atm. pressure, l in m

f¼

f¼

Longitudinal vibration of beam clamped or free at both ends; n ¼ number of half waves along length

f¼

(22-110)

(22-111)

(22-112)

(22-113) (22-114)

(22-115)

(22-116)

(22-117)

(22-118)

pn 1295 Hz ¼ ð1 þ 2nÞ l 2

pn 84 ¼ ð1 þ 2nÞ Hz l 2

n ¼ 0; 1; 2; 3; . . . Water column in rigid pipe closed at one end (l in m)

Equation

(22-119)

pn 360 Hz ¼ ð1 þ 2nÞ l 2

n ¼ 0; 1; 2; 3; . . . sﬃﬃﬃﬃﬃﬃﬃﬃﬃ AE pn ¼ n 1 l2 n ¼ 1; 2; 3; . . .

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(22-120) (22-121)

MECHANICAL VIBRATIONS

22.20

CHAPTER TWENTY-TWO

TABLE 22-2 A collection of formulas (Cont.) Particular

Formula

Equation

For steel, l in m

pn 2590 Hz ¼ l 2 p 102;000 Hz f¼ n ¼ l 2 p n168 Hz f¼ n ¼ l 2

(22-122a)

For steel, l in inches Organ pipe closed (or open) at both ends (air at 608F, 15.58C)

f¼

n ¼ 1; 2; 3; . . . Water column in rigid pipe closed (or open) at both ends

f¼

(22-123)

n721 Hz l

n ¼ 1; 2; 3; . . . For water columns in nonrigid pipes

(22-122b)

fnonrigid 1 ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ frigid 206D 1þ tEpipe

(22-124) (22-125a)

Epipe ¼ elastic modulus of pipe, MPa D, t ¼ pipe diameter and wall thickness, same units For water columns in nonrigid pipes . . .

fnonrigid 1 ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ frigid 300;000D 1þ tEpipe

(22-125b)

Epipe ¼ elastic modulus of pipe, psi D, t ¼ pipe diameter and wall thickness, same units Torsional vibration of beams . . .

Same as (22-117) and (22-118); replace tensional stiﬀness AE by torsional stiﬀness GIp ; replace 1 by the moment of inertia per unit length i1 ¼ Ibar =l

Uniform Beams (Transverse or Bending Vibrations) The same general formula holds for all the following cases, sﬃﬃﬃﬃﬃﬃﬃﬃﬃ EI pn ¼ an 1 l4

(22-126)

where EI is the bending stiﬀness of the section, l is the length of the beam, 1 is the mass per unit length ¼ W=gl, and an is a numerical constant, diﬀerent for each case and listed below. Cantilever or ‘‘clamped-free’’ beam . . .

Simply supported or ‘‘hinged-hinged’’ beam

a1 a2 a3 a4 a5 a1 a2 a3 a4 a5

¼ 3:52 ¼ 22:0 ¼ 61:7 ¼ 121:0 ¼ 200:0 ¼ 2 ¼ 9:87 ¼ 42 ¼ 39:5 ¼ 92 ¼ 88:9 ¼ 162 ¼ 158 ¼ 252 ¼ 247

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MECHANICAL VIBRATIONS MECHANICAL VIBRATIONS

22.21

TABLE 22-2 A collection of formulas (Cont.) Particular

Formula

‘‘Free-free’’ beam or ﬂoating ship . . .

a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5

‘‘Clamped-clamped’’ beam has same frequencies as ‘‘free-free’’

‘‘Clamped-hinged’’ beam may be considered as half a ‘‘clamped-clamped’’ beam for even a-numbers

‘‘Hinged-free’’ beam or wing of autogyro may be considered as half a ‘‘free-free’’ beam for even a-numbers

Equation

¼ 22:0 ¼ 61:7 ¼ 121:0 ¼ 200:0 ¼ 298:2 ¼ 22:0 ¼ 61:7 ¼ 121:0 ¼ 200:0 ¼ 298:2 ¼ 15:4 ¼ 50:0 ¼ 104 ¼ 178 ¼ 272 ¼0 ¼ 15:4 ¼ 50:0 ¼ 104 ¼ 178

Rings, Membranes, and Plates Extensional vibration of a ring, radius r, weight density sﬃﬃﬃﬃﬃﬃ 1 Eg pn ¼ r

(22-127)

Bending vibrations of ring, radius r, mass per unit length, 1 , in its own plane with n full ‘‘sine waves’’ of disturbance along circumference sﬃﬃﬃﬃﬃﬃﬃﬃﬃ (22-128) nðn2 1Þ EI pn ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 r4 1 þ n2 Circular membrane of tension T, mass per unit area 1 , radius r sﬃﬃﬃﬃﬃﬃﬃﬃﬃ T pn ¼ acd 1 r2

(22-129)

The constant acd is shown below, the subscript c denotes the number of nodal circles, and the subscript d the number of nodal diameters: c d

1

2

3

0 1 2 3

2.40 3.83 5.11 6.38

5.52 7.02 8.42 9.76

8.65 10.17 11.62 13.02

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MECHANICAL VIBRATIONS

22.22

CHAPTER TWENTY-TWO

TABLE 22-2 A collection of formulas (Cont.) Membrane of any shape of area A roughly of equal dimensions in all directions, fundamental mode: sﬃﬃﬃﬃﬃﬃﬃﬃﬃ T pn ¼ const 1 A Circle Square Quarter circle 2 1 rectangle

(22-130)

const ¼ 2:40 ¼ 4:26 const ¼ 4:44 const ¼ 4:55 const ¼ 4:97

Circular plate of radius r, mass per unit area 1 ; the ‘‘plate constant D’’ deﬁned in Eq (22-49) sﬃﬃﬃﬃﬃﬃﬃﬃﬃ D pn ¼ a 1 r4 For free edges, 2 perpendicular nodal diameters For free edges, one nodal circle, no diameters Clamped edges, fundamental mode Free edges, clamped at center, umbrella mode

(22-131)

a ¼ 5:25 a ¼ 9:07 a ¼ 10:21 a ¼ 3:75

Rectangular plate, all edges simply supported, dimensions l1 and l2 : 2 sﬃﬃﬃﬃﬃ n2 D 2 m m ¼ 1; 2; 3; . . . ; n ¼ 1; 2; 3; . . . pn ¼ þ 2 1 l21 l1 Square plate, all edges clamped, length of side l, fundamental mode: sﬃﬃﬃﬃﬃ 36 D pn ¼ 2 1 l

(22-132)

(22-133)

Source: Formulas (Eqs.) (7-110) to (7-133) extracted from J. P. Den Hartog, Mechanical Vibrations, McGraw-Hill Book Company, New York, 1962.

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MECHANICAL VIBRATIONS MECHANICAL VIBRATIONS

22.23

TABLE 22-3 Analogy between diﬀerent wave phenomena Phenomenon

Quantity

String

Transverse wave

Longitudinal wave

Acoustic wave

Torsional wave in bar

Particle velocity

x_

x_

x_

x_

_

c voltage

Mass per unit length

:A

:A

:A

a : A

:J

C capacitance/cm

Inverse spring constant per unit length

1=T

1=G : A

1=E : A

1 pn : k : A

1 J:G

L self-inductance/cm

Elastic force on a mass-element

T? ¼ T :

Velocity of propagation c

sﬃﬃﬃﬃﬃﬃﬃ T pA

sﬃﬃﬃﬃ G p

sﬃﬃﬃﬃ E p

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pn : k pn

sﬃﬃﬃﬃ G p

rﬃﬃﬃﬃﬃﬃﬃ 1 LC

Ratio of force to velocity

sﬃﬃﬃﬃﬃﬃ A x_ ¼ T? : pT

A x_ ¼ pﬃﬃﬃﬃﬃﬃ pG

A x_ ¼ pﬃﬃﬃﬃﬃﬃ pE

pA x_ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pn : pn k

Mt _ ¼ pﬃﬃﬃﬃﬃﬃ pG

i c ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ L=C

Intensity I

ðx_ o Þ2 :p:C 2

ðx_ o Þ2 :p:C 2

ðx_ o Þ2 :p:C 2

ðx_ o Þ2 :p:C 2

energy per sec total ð_o Þ2 :J:p:c 2

energy per sec c2 :C:c 2

Wave impedance

p:c ¼

@x @y

rﬃﬃﬃﬃﬃﬃ pT A

A ¼ G : A :

p:c ¼

@x @y

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ p:G

A ¼ E : A :

p:c ¼

@x @y

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ p:G

pA ¼ pn : k : A :

pn : c ¼

@x @ Mt ¼ J : G : @y @y

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pn : pn k

p:c ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ p:G

Electric cable

i current

inverse wave impedance rﬃﬃﬃﬃ 1 C ¼ Zwave L

Source: Courtesy G. W. van Santen, Introduction to Study of Mechanical Vibration, 3rd edition, Philips Technical Library, 1961. Key: c ¼ capacitance; e ¼ voltage; i ¼ current, A; I ¼ intensity, W/m2 ; J ¼ polar moment of inertia, m4 or cm4 ; k ¼ cp =cv ¼ ratio of speciﬁc heats; L ¼ inductance, H; n ¼ any integer ¼ 1, 2, 3, 4, . . . ; p ¼ pressure of gas, sound pressure, MPa; pn ¼ average pressure of gas, MPa; R ¼ resistance, ; T ¼ tension; T? ¼ component of tension T which returns the string to the position of equilibrium, kN; ¼ speciﬁc mass of the material of string, density of air, kg/m3 ; n ¼ average density of gas, kg/m3 ; ¼ normal stress, MPa; ¼ shear stress, MPa; ¼ wavelength, m. The meaning of other symbols in Table 7-3 are given under symbols at the beginning of this chapter.

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MECHANICAL VIBRATIONS

22.24

CHAPTER TWENTY-TWO

TABLE 22-4 Analogy between mechanical and electrical systems Electrical system Mechanical system

Force—current

Force—voltage

D’Alembert’s principle Force applied Rectilinear system

Kirchhoﬀ’s current law Switch closed Electrical network

Kirchhoﬀ’s voltage law Switch closed Electrical network

i ¼ Cc_, q ¼ C€ c, Energy ¼ 12 C24

e¼L

Torsional system

F ¼ m_ ¼ m€ x, Kinetic energy ¼ 12 m 2

di ¼ L€ q dt

Energy ¼ 12 Li2

F ¼ cx_ , Power ¼ Fx_ ¼ c 2

i¼

c 1 ; q ¼ e_ R R

Power ¼ ci ¼

ð F ¼ kx ¼ k x_ dt

i¼ 1 F20 2 k

1 L

e ¼ Ri ¼ Rq_ , Power ¼ ci ¼ Ri2 ¼ Rq_ 2

c2 R

ð e dt; q ¼

e L

e¼

1 1 q¼ C C

ð i dt

Energy ¼ 12 Li2

Energy ¼ 12 Ce2

(b) Parallel connected electrical elements

(c) Series connected electrical elements

ð mv_ þ c þ c þ k dt ¼ FðtÞ I € ct _ þ kt ¼ Mt ðtÞ

Diﬀerential equation for current ð r 1 e dt ¼ iðtÞ Ce_ ¼ þ R L

m€ x þ cx_ þ kx ¼ FðtÞ

C€ eþ

Diﬀerential equation for voltage ð dl 1 L þ Ri þ i dt ¼ eðtÞ dt C q L€ q þ Rq_ þ ¼ eðtÞ C

Potential energy ¼

ð FðtÞ ¼ kx ¼ x_ dt (a) Spring-mass-dashpot elements

Shaft-rotor-dashpot elements

Diﬀerential equation of motion

1 d e ðtÞ e_ þ ¼ i R L ex

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MECHANICAL VIBRATIONS MECHANICAL VIBRATIONS

22.25

REFERENCES 1. Den Hartog, J. P., Mechanical Vibrations, McGraw-Hill Book Company, New York, 1962. 2. Thomson, W. T., Theory of Vibration with Applications, Prentice-Hall, Englewood Cliﬀs, New Jersey, 1981. 3. Baumeister, T., ed., Marks’ Standard Handbook for Mechanical Engineers, 8th ed., McGraw-Hill Book Company, New York, 1978. 4. Black, P. H., and O. E. Adams, Jr., Machine Design, McGraw-Hill Book Company, New York, 1955. 5. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College CoOperative Society, Bangalore, India, 1962. 6. Myklestad, N. O., Fundamentals of Vibration Analysis, McGraw-Hill Book Company, New York, 1956. 7. Tse, F. S., I. E. Morse, and R. T. Hinkle, Mechanical Vibration—Theory and Applications, CBS Publishers and Distributors, New Delhi, India, 1983.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

23 DESIGN OF BEARINGS AND TRIBOLOGY 23.1 SLIDING CONTACT BEARINGS1,2,11 SYMBOLS distance between bolt centers [Eqs. (23-70) to (23-72)], m (in)

a h a¼ 2 B A ¼ Ld

dimensionless quantity projected area of the journal bearing (Fig. 23-6), m2 (in2) eﬀective area of the bearing, m2 (in2) projected area at full pool pressure in case of hydrostatic journal bearing (Fig. 23-47), m2 (in2) projected area of the region having a linear pressure gradient in case of hydrostatic journal bearing (Fig. 23-47), m2 (in2) width of slider bearing in the direction of motion, m (in) length of journal bearing in the direction of motion, m (in) diametral clearance, m (in) combined coeﬃcient of radiation and convection, W/m2 K (kcal/mm2 s8C) constants in Eq. (23-23)

A0 B c¼Dd C C1 , C2 F F1 CPF1 , CPF2 , CPF3 , CPF4 CPFm , CPFs CF ¼

CPW CQ CS1 to CS7 W W1 C ¼ 1 CP CW ¼

friction leakage factor in Eq. (23-54) constants in Eqs. (23-77b), (23-78b), (23-79b), and (23-80b) friction resistance factor for moving and stationary member, respectively, in pivoted shoe slider bearing in Eqs. (23-96b) and (23-97b) load factor in Eq. (23-95b) ﬂow correction factor from (Fig. 23-42) and Eq. (23-65) constants in Eqs. (23-86b), (23-87b), (23-88b), (23-89b), (23-90b), (23-91b), and (23-92b) load leakage factor in Eqs. (23-52) coeﬃcient of friction factor in Eq. (23-53) coeﬃcient of friction factor in Eqs. (23-98) and Table 23-17

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DESIGN OF BEARINGS AND TRIBOLOGY

23.2

CHAPTER TWENTY-THREE

d di , d2 dc D e ¼ c hmin E Eto F FPFW F F0 Fm Fmp Fs Fsp F1 G h h1 , h2 hc hmin ¼ ho hmax Hd Hg i k k ¼ ðhÞPðmaxÞ PðminÞ

K K1 , K2 , K3 , K4 K5 , K6 KLP1 , KLP2 , KLP3 Klt KPt Kt l1 lc L h m¼ 1 Mt h2 n n0

diameter of journal, m (in) inside and outside diameters of thrust, pivot, and collar bearings, m (in) diameter of capillary in case of hydrostatic journal bearing, m (in) diameter of bearing, m (in) eccentricity, m (in) Young’s modulus, GN/m2 or GPa (Mpsi) Engler, deg force (also with subscripts), kN (lbf ) load factor in Eqs. (23-83) and (23-84) friction force, kN (lbf ) F friction force per unit area of bearing, MPa (Psi) dL friction force on the moving member of bearing (i.e., slider), kN (lbf ) friction force on the moving member of pivoted slider bearing (i.e., slider), kN (lbf ) friction force on the stationary member of bearing (i.e., shoe), kN (lbf ) friction force on the stationary member of pivoted slider bearing (i.e., shoe), kN (lbf ) friction force acting on the moving surface of the same bearing with the same oil-ﬁlm shape but without end leakage, kN (lbf ) ﬂow factor given by Eq. (23-82) oil ﬁlm thickness, m (in) thickness of oil ﬁlm at entrance and exit, respectively, of a slider bearing (Fig. 23-48 and Fig. 23-52), m (in) thickness of bearing cap, m (in) minimum thickness of oil ﬁlm, m (in) maximum thickness of oil ﬁlm, m (in) heat dissipating capacity of bearing, kJ/s (kcal/s) heat generated in bearing. kJ/s (kcal/s) number of collars characteristic number of the given crude oil (’1.4 to 2.8), constant (also with subscripts) heat dissipating coeﬃcient thickness of the oil ﬁlm where the pressure has its maximum or minimum values, m (in) constant for a given grade of oil (varies from 1.000 to 1.004) constants in Eqs. (23-73b), (23-74b), (23-75b), and (23-76b) respectively constants in Eqs. (23-143b) and (23-144b), respectively constants in Eqs. (23-116b), (23-118b), and (23-119b) for parallel surface thrust bearing constant in Eq. (23-121b) for a tilting-pad bearing constant in Eq. (23-120b) for a tilting-pad bearing coeﬃcient of friction factor in Eq. (23-126b) for a tilting-pad bearing length of bearing pressure pad in case of hydrostatic journal bearing (Fig. 23-47), m (in) length of capillary, m (in) axial length of the journal (or of the bearing) normal to the direction of motion, m (in) ratio of the ﬁlm thicknesses at the entrance to exit in the slider bearing torque, N m (lbf in) speed, rpm speed, rps

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

power (also with subscripts), kW (hp) intensity of pressure, MPa (psi)

P P P¼

W Ld

load per projected area of the bearing, MPa (psi)

unit load supported by a parallel surface thrust bearing, MPa (psi) lower pool pressure in hydrostatic journal bearing (Fig. 23-47), MPa (psi) P2 , P4 left and right pool pressure in hydrostatic journal bearing (Fig. 23-47), MPa (psi) P3 upper pool pressure in hydrostatic journal bearing (Fig. 23-47), MPa (psi) P01 ¼ P02 ¼ P03 ¼ the pressure in ﬁrst, second, third and fourth quadrant of the pool, P04 ¼ P0 respectively, when the journal is concentric (e ¼ o) in hydrostatic journal bearing, MPa (psi) Pi inlet pressure, MPa (psi) Po constant manifold pressure, MPa (psi), pressure in the oil ﬁlm in journal bearing at the point when ¼ 0, MPa (psi) h1 q¼ 1 constant used in Eqs. (23-95b) and (23-97b) for a slider bearing h2 Q ﬂow of lubricant through the bearings, m3/s r radius of journal, m (in) r1 , r2 inside and outside radii of thrust bearing, m (in) R number of Redwood seconds in Eqs. (23-15) and (23-16) n0 1 S¼ Sommerfeld number or bearing characteristic number P 2 0 60n 1 bearing characteristic number (Fig. 23-40) S0 ¼ 2 P n bearing modulus (Tables 23-2 and 23-7) S00 ¼ 1 P t running temperature of the bearing, K (8C), number of seconds, Saybolt, in Eqs. (23-7) and (23-8) T ¼ ðtb ta Þ diﬀerence in temperature between bearing housing and surrounding air, K (8C) u average velocity, m/s (ft/min) velocity in the oil ﬁlm at height y (Fig. 23-1), m/s (ft/min) U maximum velocity (Fig. 23-1), m/s (ft/min) v velocity, m/s (ft/min) vm mean velocity, m/s (ft/min) surface speed of journal, m/s (ft/min) V rubbing velocity, m/s (ft/min) W load on the bearing, kN (lbf ) load acting on the journal bearing with end leakage, kN (lbf ) W1 load acting on the journal bearing without end leakage, kN (lbf ) X0 factors used with Eqs. (23-162), (23-165) x the distance of the pivoted point from the lower end of the shoe (Fig. 23-48), i.e., the distance of the pressure center from the origin of the coordinate, m (mm) y distance from the stationary surface (Fig. 23-1), m (in) y0 factors used with Eqs. (23-162) and (23-165) ¼ qa a constant in equation of pivoted-shoe slider bearing [Eqs. (23-86b) and (23-86c)] Pu P1

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23.3

DESIGN OF BEARINGS AND TRIBOLOGY

23.4

CHAPTER TWENTY-THREE

t ¼1" 2e "¼ c h ¼ 1 min d 0 1 2 p o

o ¼

g

attitude or eccentricity ratio or relative eccentricity

absolute viscosity (dynamic viscosity), Pa s absolute viscosity (dynamic viscosity), kgf s/m2 absolute viscosity (dynamic viscosity), cP absolute viscosity (dynamic viscosity), kgf s/cm2 dynamic viscosity of oil above atmospheric pressure P, N s/m2 or Pa s (cP, kgf s/m2 ) dynamic viscosity of oil at atmospheric pressure, i.e., when P ¼ 0, N s/m2 (cP, kgf s/m2 ) the angle measured from the position of minimum of oil ﬁlm to any point of interest in the direction of rotation or the angle from the line of centers to any point of interest in the direction of rotation around the journal, deg coeﬃcient of friction (also with subscripts) viscosity, reyn kinematic viscosity, m2 /s (cSt) density of oil or speciﬁc gravity of oil used, kg/m3 (g/mm3 ) stress (normal), MPa (psi) shear stress in lubricant, MPa (psi) attitude angle or angle of eccentricity, deg

¼ !

angular length of bearing or circumferential length of bearing, deg speciﬁc weight (weight density) at temperature t, 8C, kN/m3 (lbf/in3 ) the minimum ﬁlm thickness variable

c d

diametral clearance ratio or relative clearance angular speed, rad/s

Other factors in performance or in special aspects are included from time to time in this chapter and being applicable only in their immediate context, are not included at this stage.

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.5

Formula

SHEAR STRESS1,2 The shearing stress in the lubricant (Fig. 23-1)

¼

F U u du ¼ ¼ ¼ A h y dy

ð23-1Þ

VISCOSITY The absolute viscosity (dynamic viscosity) in SI units

¼ 103 1

SI

ð23-2aÞ

where in Pa s or (N s/m ) and 1 in cP 2

U

F

y

h

U

FIXED FIGURE 23-1 Shearing stress in lubricant.

The absolute viscosity (dynamic viscosity) in Customary Metric units

¼ 9:80660

ð23-2bÞ

¼ 9:8066 104 2

ð23-2cÞ 0

where in Pa s, in kgf s/m , and 2 in kgf s/cm2 2

104 1:45 o where is Pa s and o in reyn

¼

0 ¼ 0:102 where 0 in

Customary Metric

0 ¼

ð23-3bÞ

kgf s and 1 in cP m2

103 1:422 o

where 0 in

ð23-3aÞ

kgf s and in Pa s m2

0 ¼ 1:02 104 1 where 0 in

ð23-2dÞ

ð23-3cÞ kgf s and o in reyn m2

For absolute viscosity (dynamic viscosity) in centipoise and SI units

Refer to Figs 23-2a and 23-2b

The absolute viscosity (dynamic viscosity) in centipoise

1 ¼ 103

Customary Metric

ð23-4aÞ

where 1 in cP and in Pa s 1 ¼

108 1:02 2

where 1 in cP and 2 in 1 ¼

ð23-4bÞ kgf s cm2

104 0 1:02

where 1 in cP and 0 in

ð23-4cÞ kgf s m2

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DESIGN OF BEARINGS AND TRIBOLOGY

23.6

CHAPTER TWENTY-THREE

Particular

Formula

107 1:45 o where 1 in cP and o in reyn

1 ¼

o ¼ 1:45 104

The viscosity in reyn (lbf s/in2 )

ð23-4dÞ

USCS

ð23-5aÞ

where o in reyn and in Pa s o ¼ 1:45 107 1

ð23-5bÞ

where o in reyn and 1 in cP o ¼ 14:222 where o in reyn and 2 in kgf s/cm 2000 1000 500 400 300 200 150

K

I

H G

F

75

E D C

50 40

B A

100 Absolute viscosoity, η, centipoise

J

ð23-5cÞ 2

30 20 15 10 9 8 7 6 5 4 20

30

40

50

60 70 80 Temperature, C

90

100 110

FIGURE 23-2a Absolute viscosity versus temperature.

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

Formula

o ¼ 1:422 103 0

ð23-5dÞ

where o in reyn and 0 in Kinematic viscosity

¼

g ¼ 2 density

kgf s m2

Customary Metric

104 5 3 2 103 5 3 2 102

SA

Absolute viscosity, mPa s

5

E

70

60

4

50

3 30

2

40

20 10

10

5 4 3

2 10

20

30

40

23.7

50

60 70 80 Temperature, C

90

100 110

120 130 140

FIGURE 23-2b Absolute viscosity versus temperature.

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ð23-6aÞ

DESIGN OF BEARINGS AND TRIBOLOGY

23.8

CHAPTER TWENTY-THREE

Particular

Formula

where v in cm2 =s and 2 in

kgf s ; cm2

g ¼ 980:66 cm=s2 and in Kinematic viscosity

v¼

g 4 10

kgf cm3 SI

ð23-6bÞ

Ns or (Pa s), in N/m3 , and v in m2 /s m2 180 ¼ t 0:22t t where in

Saybolt to centipoises (Fig. 23-3)3 or mPa s

SI=Customary Metric ð23-7Þ where in cP and t in gf/cm3 or N/m3 , t in s Saybolt to reyn

Refer to Table 23-1 for t . "

o ¼ 0:145t 0:22t Kinematic viscosity in centistokes from Saybolt universal seconds (Figs. 23-3 and 23-4)3

Kinematic viscosity

vk ¼

180 t

# USCS

180 0:22t t

ð23-7aÞ

ð23-8aÞ

where vk in cSt and t in s v ¼ 106 vk

SI 2

where v in m /s and vk in cSt

TABLE 23-1 Speciﬁc gravity of oils at 15.58C (608F) No.

Oil characteristics

15:5

A B C D E F G H I J K

Turbine oil, ring-oiled bearing Turbine oil, ring-oiled bearing, SAE 10 All-year automobile oil, SAE 20 Ring-oiled bearing oil, high-speed machinery Automobile oil, SAE 20 Automobile oil, SAE 30 Automobile oil, SAE 40, medium-speed machinery Airplane oil 100, SAE 60 Transmission oil, SAE 110, spur and bevel gears Gear oil, slow-speed worm gears Transmission oil. SAE 60, slow-speed gears

0.8877 0.8894 0.9036 0.9346 0.9254 0.9263 0.9275 0.8927 0.9328 0.9153 0.9365

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ð23-8bÞ

DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.9

Formula

v¼

180 106 0:22t t

ð23-8cÞ

where t in Saybolt seconds and v in m2 /s 141:5 Customary Metric ð23-9aÞ 131:5 þ 8API where 15:5 in gf/ml (gram force/milliliter) 141:5 15:5 ¼ 9807 SI ð23-9bÞ 131:5 þ 8API

Speciﬁc weight at 15.58C

15:5 ¼

where 15:5 in N/m3

10000

500 400 300

H G

1000 750

Viscosity, saybolt universal, seconds

1000

J I

D

300

B 200 A

200 150

F E

100 75

C

50 40

150

30

100 90 80

20 15

70 60 55

Kinematic viscosity, v, centistokes

5000 4000 3000 2000 1500

500

2000

K

10.0 9.0 8.0 7.0

50

6.0

45

5.0 40 30

40

50

60 70 Temperature, C

80

90

100

4.0

FIGURE 23-3 Viscosity Saybolt universal seconds and kinematic viscosity versus temperature.

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DESIGN OF BEARINGS AND TRIBOLOGY

23.10

CHAPTER TWENTY-THREE

Particular

Formula

API ¼ American Petroleum Institute gravity constant t ¼ 15:5 0:000637ðt 15:5Þ

Speciﬁc weight at any temperature

ð23-10Þ

Refer to Table 23-1 for 15:5 t ¼ 60 0:000365ðt 60Þ

USCS

2000

1000

lty

ira

700 500

w ed

R

300

a

l

ro

fu

t ol

yb

Sa

es

re

er

l ng

200

Kinematic viscosity, v, centistokes

d

oo

dm

g de

E

d

o wo

d

Re

100 70

t ol

. No

s

nd

1 er

l ng

o ec

s

E

al

s er

iv

un

yb

50

Sa

30

Ba

20

rb

10

ey

flu

di

ty

7 5 3 2 24 8

1 10

20

30

50 70 100 200 300 Time of eflux, sec

500 700 1000

2000

FIGURE 23-4 Viscosity conversion chart.

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ð23-10aÞ

45

50

60

70

80

100

150

200

300

500 400

1.4

1.5

1.6

1.8

2.0

2.5

3.0

4.0

5

6

10 9 8 7

20

30

40

50

70

10.0 8.0

6.0 5.0

100

170

260

cSt 450

15.0

20.0

30.0

50.0

E

20 30

40

50

60

RE

RA TU

PE

S

NE

LI FO R O S

IL

80 90 100 110 120 Operating temperature, C

EM

/T

TY

SI

70

O

50 C REFERENCE TEMPERATURE FOR VISCOSITY TY PI CA L VI SC

10 000

5000

3000

1500

1000

750

500

300

50

n, rev/min 150

100 200

500 d, mm

DESIGN OF BEARINGS AND TRIBOLOGY

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FIGURE 23-4a Viscosity conversion chart and a guide to suitable oil viscosities for rolling contact bearings (Courtesy: SKF Rolling Bearings).

Example : A bearing having a bore diameter d = 340mm and operating at a speed n = 500 rev/min requires an oil having a viscosity of 13.2 centistokes at the operating temperature. If this operating temperature is assumed to be 70 C an oil having a viscosity of 26 centistokes at 50 C should be selected.

d = bearing bore diameter mm n = rotational speed rev/min An example is given below and shown on the graph by means of the lines of dashes.

In the figure

40

45

50

60

70

80

100

150

200

300

500 400

Viscosity RI SSU

DESIGN OF BEARINGS AND TRIBOLOGY

23.11

DESIGN OF BEARINGS AND TRIBOLOGY

23.12

CHAPTER TWENTY-THREE

Particular

The dynamic viscosity

Formula

¼ ð0:22t 180=tÞ106 where t in 8F where in Pa s, ¼ =g, and in kg/m3

The absolute viscosity (dynamic viscosity) in terms of Engler degree, Et 8

The density ð Þ of oil and its speciﬁc gravity ðÞ relative to water have the same numerical value. 0:635 0 ¼ 106 t 0:737Et 8 Et 8 Customary Metric ð23-11Þ 0

The relation between arbitrary viscosity in Engler degree (V in Et 8) and the absolute viscosity (dynamic viscosity) in kgf s/m2 The change in viscosity 0 depending on temperature is expressed by formula

The relation between viscosity and pressure

where in kgf s/m V ¼ k0

ð23-12Þ

where k ’ 14:9 103 Et 8/(kgf s/m2 ) ¼ proportionality factor 0 ¼

i ð0:1t8Þ3

Customary Metric ð23-13Þ

where i ¼ characteristic number of the given grade of oil i ’ 1:4 to 2.8 0 in kgf s/m2 p ¼ no K P

Kinematic viscosity in centistokes from Redwood No

2

Customary Metric ð23-14Þ

where P ¼ pressure, kgf/cm2 K ¼ constant for the given grade of oil ’ varies from 1.001 to 1.004 for pressure P up to 400 kgf/cm2 (39 MPa). (Changes in oil viscosity due to change in pressure can be neglected.) v ¼ 0:260R

179 when 34 < R < 100 R Customary Metric ð23-15aÞ

where v in cSt and R in number of Redwood seconds

Kinematic viscosity in centistokes from Redwood Admiralty

v ¼ 0:247R

50 when R > 100 R

ð23-15bÞ

2000 Customary Metric ð23-16Þ R where R ¼ the number of Redwood seconds

v ¼ 2:7R

HAGEN-POISEUILLE LAW The rate of laminar ﬂow of lubricant in tubes

Q¼

d 4 dp 128 dz

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ð23-17Þ

DESIGN OF BEARINGS AND TRIBOLOGY

23.13

DESIGN OF BEARINGS AND TRIBOLOGY

Particular

Formula

VERTICAL SHAFT ROTATING IN A GUIDE BEARING (Fig. 23-5) The surface velocity of shaft

U ¼ dn0

The length of bearing in the direction of motion

B ¼ d

The torque (Fig. 23-5)

Petroﬀs equation for coeﬃcient of friction (Fig. 23-5)

Design practice for journal bearing3 The coeﬃcient of friction can also be obtained from expression

ð23-18Þ

8 360

ð23-19Þ

Mt ¼ ðLdÞP

d 2 d 2 Ln0 ¼ 2

ð23-20Þ

Refer to Fig. 23-6 for projected area ðLdÞ. 0 n 1 ¼ 2 2 P

ð23-21Þ

Refer to Table 23-2. 0 n 1 1010 þ ¼ Ka P

ð23-22Þ

where Ka ¼ 5:53 ¼ 1980 for ¼ 3608 Customary Metric 0

where in cP, n in rps, and P in kgf/cm Ka ¼ 1:31 ¼ 473 for ¼ 3608

ð23-22aÞ 2

USCS

ð23-22bÞ

where in cP, n in rpm, and P in psi Ka ¼ 9:23 104 ¼ 0:33 for ¼ 3608 Customary Metric where in cP, n in rpm, and P in kgf/mm d+c d

W L

Projected area ωrad/s FIGURE 23-5 Vertical shaft rotating in a cylindrical bearing.

L

d

FIGURE 23-6 Projected area of a bearing.

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ð23-22cÞ 2

DESIGN OF BEARINGS AND TRIBOLOGY

23.14

CHAPTER TWENTY-THREE

Particular

Formula

Ka ¼ 0:0553 ¼ 19:8 for ¼ 3608 Customary Metric

0.015

0

where in cP, n in rps, and P in kgf/mm Value of

0.010

Ka ¼ 5:4 108 ¼ 1:95 1011 for ¼ 3608 SI ð23-22eÞ where in Pa s, n0 in rps, and P in N/m2

0.005

0

ð23-22dÞ 2

¼ factor to correct for end leakage ¼ 0:002 for L=d ranging from 0.75 to 2.8 0

0.5

1.0 1.5 2.0 Ratio, L/d

2.5

3.0

Refer also to Fig. 23-7 for .

FIGURE 23-7 Correction factor for use in Eq. (23-22).

Louis Illmer equation for coeﬃcient of friction in case of imperfect lubrication

sﬃﬃﬃﬃﬃﬃ 4 P ¼ 0:00012C1 C2 vm

SI

where P in N/m2 and vm in m/s sﬃﬃﬃﬃﬃﬃ 4 P Customary Metric ¼ 0:0066C1 C2 vm where P in kgf/mm2 and vm in m/s sﬃﬃﬃﬃﬃﬃ 4 P USCS ¼ 0:004C1 C2 vm

ð23-23aÞ

ð23-23bÞ

ð23-23cÞ

where P in psi and vm in ft/min Refer to Tables 23-3 and 23-4 for C1 and C2 , respectively. For behaviour of journal at stand still, at start and running in its bearing

Refer to Fig. 23-8.

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DESIGN OF BEARINGS AND TRIBOLOGY

23.15

DESIGN OF BEARINGS AND TRIBOLOGY

TABLE 23-2 Journal bearing design practices Bearing modulus (minimum) Maximum pressure, P

Diameter clearance ratio c ¼ d

L d

Viscosity, 1

Viscosity, S 00 ¼

cP

Pa s 103

1 n P

n0 P SI Units, 109

S 00 ¼

Machinery

Bearing

kgf/mm2

kpsi

MPa

Automobile and aircraft engines

Main Crankpin Wrist pin

0.56–1.19 1.06–2.47 1.62–3.62

0.8–1.7 1.5–3.5 2.3–5.0

5.50–11.70 — 10.40–24.40 15.00–34.80

0.1–1.8 0.7–1.4 1.5–2.2

7 to 8

7 to 8

15 10 8

36.3 24.2 19.3

Gas and oil engines (fourstroke)

Main Crankpin Wrist pin

0.49–0.85 0.90–1.27 1.27–1.55

0.7–1.2 1.4–1.8 1.8–2.2

4.85–8.35 0.001 8.80–12.40 <0.001 12.40–15.20 <0.001

0.6–2.0 0.6–1.5 1.5–2.0

20 to 65

20 to 65

20 10 5

48.4 24.2 12.1

Gas and oil engines (twostroke)

Main Crankpin Wrist pin

0.35–0.56 0.70–1.06 0.85–1.07

0.5–0.8 1.0–1.5 1.2–1.8

3.42–5.50 0.001 6.85–10.40 <0.001 8.35–12.50 <0.001

0.6–2.0 0.6–1.5 1.5–2.0

20 to 65

20 to 65

25 12 10

60.4 29.0 24.2

Marine steam engines

Main Crankpin Wrist pin

0.35 0.42 1.06

0.5 0.6 1.5

3.42 4.14 10.40

<0.001 <0.001 <0.001

0.7–1.5 0.7–1.2 1.2–1.7

30 40 30

30 40 30

20 15 10

48.4 36.3 24.2

Stationary, slow-speed steam engines

Main Crankpin Wrist pin

0.28 1.06 1.27

0.4 1.5 1.8

2.75 10.40 12.50

<0.001 <0.001 <0.001

1.0–2.0 0.9–1.3 1.2–1.5

60 80 60

60 80 60

20 6 5

48.4 14.5 12.1

Stationary, high-speed steam engines

Main Crankpin Wrist pin

0.17 0.42 1.27

0.25 0.6 1.8

1.66 4.14 12.50

<0.001 <0.001 <0.001

1.5–3.0 0.9–1.5 1.3–1.7

15 30 25

15 30 25

25 6 5

60.4 14.5 12.1

Steam locomotives

Driving axle 0.39 Crankpin 1.40 Wrist pin 2.82

0.55 2.0 4.0

3.72 13.70 27.60

0.001 <0.001 <0.001

1.6–1.8 0.7–1.1 0.8–1.3

100 40 30

100 40 30

30 5 5

72.5 12.1 12.1

Reciprocating pumps and compressors

Main Crankpin Wrist pin

0.17 0.42 0.70

0.25 0.6 1.0

1.66 4.14 6.85

<0.001 <0.001 <0.001

1.0–2.2 0.9–1.7 1.5–2.0

30 to 80

30 to 80

30 20 10

72.5 48.4 24.2

Railway cars

Axle

0.35

0.45

3.42

0.001

1.8–2.0

100

100

50

120.9

Steam turbines

Main

0.07–0.19

0.1–0.275

0.69–1.87

0.001

1.0–2.0

2–16

2–16

100

241.8

Generators, motors, centrifugal pumps

Rotor

0.07-0.14

0.1-0.2

0.69-1.37

0.0013

1.0–2.0

25

25

Gyroscope

Rotor

0.60

0.85

5.90

0.0013

—

30

30

55

133.0

Transmission shafting

Light, ﬁxed 0.08 Self-aligning 0.106 Heavy 0.106

0.025 0.15 0.15

0.17 1.04 1.04

0.001 0.001 0.001

2.0–3.0 2.5–4.0 2.0–3.0

25 to 60

25 to 60

100 30 30

241.8 72.5 72.5

Cotton mill

Spindle

0.0007

0.001

0.0069

0.005

—

2

2

10000

24177.5

Machine tools

Main

0.21

0.3

2.06

0.001

1.0–1.4

40

40

40

Punching and shearing machine

Main Crankpin

2.82 5.62

4.0 8.0

27.80 55.60

0.001 0.001

1.0–2.0 1.0–2.0

100 100

100 100

Rolling mills

Main

2.11

3.0

20.60

0.0015

1.1–1.5

50

50

Ratio

USCSU

200

— —

483.5

96.7 — —

10

24.2

Key: ð1 Þ ¼ absolute viscosity, Pa s (cP); n ¼ speed, rpm; n0 ¼ speed, rps; P ¼ pressure, N/m2 or MPa (psi); MPa ¼ megapascal ¼ 106 N/m2 ; Pa ¼ Pascal ¼ 1 N/m2 ; 1 psi ¼ 6894.757 Pa; 1 kpsi ¼ 6.89475 MPa; USCSU ¼ US Customary System units.

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DESIGN OF BEARINGS AND TRIBOLOGY

23.16

CHAPTER TWENTY-THREE

TABLE 23-3 Values of factor C1 in Eq. (23-23) Lubrication

Workmanship

Attendance

Operating condition

Constant C1

Oil bath or ﬂooded

High grade

First class

Clean and protected

1

Oil, free drop (constant feed)

Good

Fairly good

Favorable (ordinary condition)

2

Oil cup or grease (intermittent feed)

Fair

Poor

Exposed to dirt, grit or other unfavorable conditions

4

TABLE 23-4 Values of factor C2 in Eq. (23–23) Type of bearing

Constant C2

Rotating journals, such as rigid bearing and crankpins

1

Oscillating journals, such as rigid wrist pin and Pintle blocks

1

Rotating bearings lacking ample rigidity, such as eccentric and the like

2

Rotating ﬂat surfaces lubricated from the center to the circumference, such as annular step or pivot bearings

2

Sliding ﬂat surfaces wiping over the guide ends, such as reciprocating crossheads; use 2 for relatively long guides and 3 for short guides

2–3

Sliding or wiping surfaces lubricated from the periphery or outer wiping edge, such as marine thrust bearings and worm gears

3–4

Long power-screw nuts and similar wiping parts over which it is diﬃcult to eﬀect a uniform distribution of lubricant or load

4–6

w

r+

c 2

e

A

0’

B

E

0’

0

ω

0’

0

e

0

β α’

D

c 2

r

w

φ

θ

r+

w

h (a) Stand still

(b) At start

(c) Running

FIGURE 23-8 Behaviour of a journal in its bearing.

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.17

Formula

BEARING PRESSURE (Fig. 23-9) General Electric Company’s formula for bearing pressure in the design of motor and generator bearing

Pa ¼ 6:2 105

3 ﬃﬃﬃﬃﬃﬃ p vm

SI

ð23-24aÞ

USCS

ð23-24bÞ

Customary Metric

ð23-24cÞ

2

where Pa in N/m and vm in m/s Pa ¼ 15:5

p 3 ﬃﬃﬃﬃﬃﬃ vm

where Pa in psi and vm in ft/min Pa ¼ 0:0635

p 3 ﬃﬃﬃﬃﬃﬃ vm

where Pa in kgf/mm2 and vm in m/s Victor Tatarinoﬀ’s equation for safe operating load

W¼

1 nd 3 ðL=dÞ2 L 127ð106 Þh 1 þ d

USCS

ð23-25aÞ

where 1 in cP, n in rpm, L, d, h, and c in in, W in lbf W¼

Victor Tatarinoﬀ’s equation for permissible unit pressure

n0 d 3 ðL=dÞ2 L 0:295h 1 þ d

SI

ð22-25bÞ

where in Pa s, n0 in rps, L, d, h, and c in m, W in N 1 n0 L USCS ð23-26aÞ P¼ 3175ð104 Þ 2 L þ d where P in psi, 1 in cP, n in rpm, L and d in in n0 L P ¼ 13:5 2 SI ð23-26bÞ Lþd where P in Pa, in Pa s, n0 in rps, and L and d in m

w

L

Bearing Journal Oil inlet

hmax

d+c

θ

Pmin

o

ho = hmin

n

C E e=

p

c

φ

Line of centers or line joining 0 (centre of bearing) and 0’ (centre of jouurnal)

I

P ma x

Oil film pressure

θ PO Without θPmax groove

With groove

FIGURE 23-9 Oil ﬁlm pressure distribution in the full journal bearing.

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DESIGN OF BEARINGS AND TRIBOLOGY

23.18

CHAPTER TWENTY-THREE

Particular

H. F. Moore’s equation for critical pressure

Formula

pﬃﬃﬃ Pc ¼ 7:23 105 v

SI

ð23-27aÞ

where Pc in N/m and v in m/s pﬃﬃﬃ Pc ¼ 0:0737 v Customary Metric

ð23-27bÞ

where Pc in kgf/mm2 and v in m/s pﬃﬃﬃ USCS Pc ¼ 7:5 v

ð23-27cÞ

2

The critical unit pressure for any given velocity should not exceed according to Louis Illmer

where Pc in psi and v in ft/min rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ vm 3 Pc ¼ 4:6 106 t 288:5

SI

ð23-28aÞ

where Pc in N/m2 , vm in m/s, and t in K rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ vm 3 Customary Metric ð23-28bÞ Pc ¼ 0:47 t 15:5 where Pc in kgf/mm2 , vm in m/s, and t in 8C rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ vm 3 USCS ð23-28cÞ Pc ¼ 140 t 15:5 Stribeck’s equation for the critical pressure when the speed does not exceed 2.5 m/s (500 ft/min)

Stribeck’s equation for the critical pressure when the speed exceeds 2.5 m/s (500 ft/min)

where Pc in psi, vm in ft/min, t in 8F pﬃﬃﬃ Pc ¼ 9:7 105 v SI

ð23-28dÞ

2

where Pc in N/m and v in m/s pﬃﬃﬃ Pc ¼ 10 v

USCS

ð23-28eÞ

where Pc in psi and v in ft/min pﬃﬃﬃ Customary Metric Pc ¼ 0:0986 v

ð23-28f Þ

where Pc in kgf/mm2 and v in m/s pﬃﬃﬃ Pc ¼ 2:9 106 v

SI

ð23-28gÞ

where Pc in N/m and v in m/s pﬃﬃﬃ Pc ¼ 30 v USCS

ð23-28hÞ

2

where Pc in psi and v in ft/min pﬃﬃﬃ Customary Metric ð23-28iÞ Pc ¼ 0:296 v where Pc in kgf/mm2 and v in m/s Refer to Table 23-5 for allowable pressures for reciprocating motion. For permissible Pv values

Refer to Table 23-6.

For values S00 for various combinations of journal bearing materials, abrasion pressure for bearings, allowable bearing pressures for semi-ﬂuid lubricants and diametral clearances in bearing dimensions.

Refer to Tables 23-7 to 23-10.

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

23.19

TABLE 23-5 Allowable bearing pressure, reciprocating motion Pressure, P Type of bearing

Type of machinery

psi

MPa

Steam engine, stationary Steam engine, marine Steam engine, locomotive Gas and oil engines, stationary Compressors and pumps

35–60 55–100 70–90 40–70 50–90

0.24–0.412 0.378–0.688 0.48–0.62 0.275–0.48 0.342–0.62

and oil engines, stationary f Gas Automotive and aircraft engines

20–25 25–40

0.136–0.172 0.172–0.275

f

Crosshead

Trunk pin

TABLE 23-6 Permissible Pv values Values Class of bearing or journal

psi ft/s

N/m s

Mill shafting, with self-aligning cast-iron bearings, grease, or imperfect oil-lubrication, maximum value

12,000

4.2105

Mill shafting, self-aligning ring-oiled babbitt bearings, maximum

24,000

8.45105

Self-aligning ring-oiled bearings, continuous load in one direction

35,000–40,000

Crankshaft journals with bronze bearings

22,000

7.7105

Crankshaft bearings with babbitted bearings, maximum

59,000

20.8105

133,000

46.5105

For excellent radiating condition

12.3105 to 14105

Key: US Customary unit: P ¼ pressure, psi, v ¼ velocity, ft/s; SI unit: P ¼ pressure, N/m2 , v ¼ velocity, m/s

TABLE 23-7 Values S 00 for various combinations of journal bearing materials Bearing-modulus S 00 ¼

1 n P

n0 P SI 109

S 00 ¼

Shaft

Bearing

Metric

Hardened and ground steel

Babbitt

28,500

Machined, soft steel

Babbitt

36,000

61.2

Hardened and ground steel

Plastic bronze

42,700

72.6

Machined, soft steel

Plastic bronze

35,800

60.9

Hardened and ground steel

Rigid bronze

56,900

96.7

Machined, soft steel

Rigid bronze

71,100

120.8

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48.5

DESIGN OF BEARINGS AND TRIBOLOGY

23.20

CHAPTER TWENTY-THREE

TABLE 23-8 Abrasion pressures for bearings Pressure Materials in contact

psi

MPa

Remarks

Hardened tool steel on lumen or phosphor bronze

10,000

68.8

Values applies to rigid, polished and accurately ﬁtted rubbing surface

0.50 C machine steel on lumen or phosphor bronze

8,000

55.0

When not worn to a ﬁt or well lubricated reduce to 4.22 kgf/mm2 (41.4 MPa)

Hardened tool steel on hardened tool steel

7,000

48.0

0.50 C machine steel or wrought iron on genuine hard babbitt

6,000

41.5

Cast iron on cast iron (close grained or chilled)

4,500

31.0

Case-hardened machine steel on case-hardened machine steel

4,000

27.5

0.30 C machine steel on cast iron (close-grained)

3,500

24.0

0.40 C machine steel on soft common babbitt

3,000

20.6

Soft machine steel on machine steel (not casehardened)

2,000

13.8

Machine steel on lignum vitae (water-lubricated)

1,500

10.2

TABLE 23-9 Allowable bearing pressures for semiﬂuid lubrication Allowable pressure, Pa Bearing material

Journal material

psi

MPa

Lumen of phosphor bronze

Hardened tool steel

2500

17.30

Hardened steel

Hardened alloy steel

2000

14.40

Hard babbitt

SAE 1050 steel

1500

10.30

Bronze

Hardened alloy steel

1300

8.90

Cast iron

Cast iron

1100

7.58

Bronze

Alloy steel

850

5.90

Babbit, soft

SAE 1040 steel

750

5.20

Bronze

Mild steel, smooth ﬁnish

540

3.70

Bronze

Mild steel, ordinary ﬁnish

400

2.75

Bronze

Cast iron

400

2.75

Cast iron

Mild steel

350

2.40

Lignum vitae, water lubricated

Mild steel

350

2.40

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DESIGN OF BEARINGS AND TRIBOLOGY

23.21

DESIGN OF BEARINGS AND TRIBOLOGY

TABLE 23-10 Diametral clearance in bearings dimension in micrometers (1 lm ¼ 106 m) Diametral clearances, c in lm d ¼ 12

d ¼ 25

d ¼ 50

d ¼ 100

d ¼ 140

Precision spindle, hardened and ground steel, lapped into bronze bearingvm < 25 m/s; P < 500 psi (3.43 N/m2 ); 0.2–0.4 mm rms

7–19

19–38

38–63

63–88

88–125

Precision spindle, hardened and ground steel, lapped into bronze bearingvm > 25 m/s; P > 500 psi (3.43 N/m2 ); 0.2–0.4 mm rms

13–25

25–50

50–75

75–113

113–163

Electric motors and generators, ground journals in broached or reamed bronze or babbitt bearings; 0.4–0.8 mm rms

13–38

25–50

38–85

50–100

75–150

General machinery, intermittent or continuous motion, turned or cold-rolled journal in reamed and bored bronze or babbitt bearings; 0.8–1.5 mm rms

50–100

63–113

75–125

100–175

125–200

Rough machinery, turned or cold-rolled steel journals in poured babbitt bearings; 1.5–3.8 mm rms

77–150

125–225

200–300

275–400

350–500

38 50 36

63 75 88

Particular about bearing and journal

Automotive crankshaft Babbitt-lined bearing Cadmium silver copper Copper lead

Particular

Formula

IDEALIZED JOURNAL BEARING (Figs. 23-8 and 23-9) The diametral clearance ratio or relative clearance

Attitude or eccentricity ratio or eccentricity coeﬃcient

¼ "¼

c d

ð23-29Þ

2e 2h ¼ 1 min d c

ð23-30Þ

Refer to Fig. 23-10 for ". Oil ﬁlm thickness at any position

c h ¼ ð1 þ " cos Þ 2

For position of minimum oil thickness and max oil ﬁlm pressure

Refer to Figs. 23-11 and 23-11A

Minimum oil ﬁlm thickness

c hmin ¼ ho ¼ ð1 "Þ 2 2h ¼ min ¼ ð1 "Þ c

The minimum oil ﬁlm thickness variable

Refer to Figs. 23-12 to 23-14 and 23-15 for .

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ð23-31Þ

ð23-32Þ ð23-33Þ

DESIGN OF BEARINGS AND TRIBOLOGY 1.0

Attitude, ε

0.8 Lightly loaded bearing

0.6 0.4

Beyond this point 1 S= . 59 36

Reynolds

0.2 0

Petroff 0.10

0

0.20

0.30

ηn Bearing characteristic number, S = P

0.40

1 ψ2

(a) Moderately and lightly loaded bearing

1.0

Attitude, ε

0.8 0.6 0.4 0.2 0

0.010

0

0.020

ηn’ Bearing characteristic number, S = P

0.030 1 ψ2

0.040

(b) Heavily loaded bearing

FIGURE 23-10 Variation of attitude " of full journal bearing with characteristic number S. [Radzimosvksy4 ] 100

Position of Minimum Oil Film Thickness, φ deg

90 80 70

L =∞ d

60

1

50

1 2 1 4

40 30 20 10 0 0

0.01

0.02

0.04 0.06 0.080.1

0.2

0.4 0.6 0.81.0

ηn’ Bearing Characteristic Number, S = P

2

4

6

8 10

1 ψ2

FIGURE 23-11 Position of minimum oil ﬁlm thickness vs. bearing characteristic number S for full journal bearing. (Refer to Fig. 23-9 for deﬁnition of .) [Boyd and Raimondi5 ]

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DESIGN OF BEARINGS AND TRIBOLOGY 1.0

MINIMUM FILM THICKNESS VARIABLE, δ

0.8

0.6

0.4

0.2

0

0

1

2

4

6

8 10

2

4

6

BEARING CHARACTERISTIC NUMBER S’ =

8 102

2

4

6

8 103

60ηn’ 1 P ψ2

FIGURE 23-12 Minimum oil ﬁlm thickness variable based on no side ﬂow. [Boyd and Raimondi5 ] PETROFF

1.0 0.9 0.8 0.7

2nmin c

0.5

δ=

0.6

0.4

REYNOLDS

LIGHTLY LOADED BEARING

0.3 0.2 0.1 0

0

0.10 0.20 0.30 BEARING CHARACTERISTIC NUMBER ηn’ 1 S= P ψ2

FIGURE 23-13 Variation of minimum oil ﬁlm thickness variable of full journal bearing with S.

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DESIGN OF BEARINGS AND TRIBOLOGY

23.24

CHAPTER TWENTY-THREE

Particular

Formula

The safe oil ﬁlm thickness for a bearing in good condition and vm 1 m/s (200 ft/min)

0:2 hmin ¼ ho ¼ 2:37 105 v0:4 m A

SI

ð23-34aÞ

2

where hmin in m, A in m , and vm in m/s 0:2 hmin ¼ 0:0015v0:4 m A

Customary Metric

ð23-34bÞ

2

where hmin in mm, A in mm , and vm in m/s 0:2 hmin ¼ 0:000026v0:4 m A

USCS

ð23-34cÞ

2

where hmin in in, A in in , and vm in in The thickness of oil ﬁlm where the pressure is maximum or minimum

ðhÞPðmaxÞ ¼ k ¼ PðminÞ

2cð1 "2 Þ 2 þ "2

ð23-35Þ

The resultant pressure distribution around a journal bearing excluding Po the oil ﬁlm pressure at the point where ¼ 0 or ¼ 2

Pr ¼ ðP Po Þ 12U "ð2 þ " cos Þ sin ¼ 2 d ð2 þ "2 Þð1 þ " cos Þ2

The pressure at any point (Figs. 23-8 and 23-9)

P ¼ Pr þ Po UL

ð23-36Þ ð23-37Þ

!

W¼

The bearing characteristic number or Sommerfeld number

S¼

For Sommerfeld number S

Refer to Tables 23-10 to 23-12 for Sommerfeld numbers S for full and partial bearings.

MINIMUM FILM THICKNESS VARIABLE, δ

The load carrying capacity of the bearing [Fig. 23-8 (panel c)]

2

2 " pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 "2 ð2 þ "2 Þ

n0 1 P 2

ð23-39Þ

1.0 0.8 0.6 0.4 0.2 0

0

40

80

120

160

200 60 S VALUE OF CL 106

240

280

320

ð23-38Þ

360

FIGURE 23-14 Variation of minimum oil ﬁlm thickness variable with S=CL .

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2hmin c

0.02

0.4

2

4

Optimum Boundary

1

4

6

8 10

0.8

0.7

0.6

0.5

0.2 ηn’ 1 ( 2) ψ P

0.04 0.06 0.08 0.1

Bearing Characteristic Number, S =

0.6 0.8 1.0

2

1.0

0.01

1

0.4

0.3

0

Lo ad

1

0.2

0.1

0.9

0

∞

Ma xim um

L = d

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Friction imum Min

Attitude or Eccentricity Ratio, ε = 2e c

FIGURE 23-15 Variation of minimum oil ﬁlm thickness variable and attitude " of full journal bearing with bearing characteristic number S. [Boyd and Raimondi5 ]

Minimum Oil Film Thickness Variables, δ =

1.0

DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

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23.25

DESIGN OF BEARINGS AND TRIBOLOGY

23.26

CHAPTER TWENTY-THREE

Particular

Formula

n P where in Pa s (cP)

The constant of the bearing or bearing modulus

S 00 ¼

ð23-40Þ

Refer to Table 23-7 for bearing modulus. The calculation of minimum oil ﬁlm thickness from Figs 23-14 and 23-16

The bearing characteristic number or Sommerfeld number as a function of attitude

Hint: S is determined from Eq. (23-39) and CL from Fig. 23-16 for a given ðL=dÞ ratio. Calculate 60S=ðCL 106 Þ. Knowing 60S=ðCL 106 Þ, you can then obtain the minimum ﬁlm thickness variable from Fig. 23-14. From and Eq. (23-33), you can then determine the minimum oil ﬁlm thickness. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð2 þ "2 Þ 1 "2 S¼ ð23-41Þ 12 2 "

The angular positions of points where the maximum or minimum pressure in the oil ﬁlm occur [Fig. 23-8c and Fig. 23-9]

Refer to Fig. 23-10 for " for various values of S. 3" ð23-42Þ ¼ cos1 2 " þ2

For positions of maximum oil ﬁlm pressure and oil ﬁlm termination vs. bearing characteristic number S

Terminating Position of Oil Film, θPo deg

90

25

1 2

1

L = d

1 4

80

20

70 1 4

60

15

50 40

10

1 2

30 20 10 0

5

I

Pmax

L =∞ d

Angular Position of Maximum Oil Film Pressure, θ deg

100

Refer to Fig. 23-15A.

∞ 0

θ Po θ

Pmax

0.01

0.02

0.04 0.06 0.08 0.1

0.2

0.4

0.6 0.8 1.0

Bearing Characteristic Number , S =

2

4

6

0 8 10

η n’ 1 P ψ2

FIGURE 23-15A Position of maximum oil ﬁlm pressure and oil ﬁlm termination versus bearing characteristic number S. [Boyd and Raimondi24 ] (Refer to Fig. 23-9 for deﬁnition of Pmax , and P0 .)

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

23.27

TABLE 23-11 Dimensionless performance parameters for full journal bearings with side ﬂow Values of L=d ratio For maximum load For minimum friction L d

0.25 0.27 0.03

0.5 0.43 0.12

1 0.66 0.6

1.0 0.53 0.3 Qs Q

csp T0 P

P Pmax

3.45 3.76 4.37 4.99 5.60 5.91 6.12 —

0 0.180 0.330 0.567 0.746 0.884 0.945 0.984 1.0

1 1287.0 611.0 245.0 107.6 35.4 14.1 3.73 0

— 0.515 0.489 0.415 0.334 0.240 0.180 0.108 0

1 85.6 40.9 17.0 8.10 3.26 1.60 0.610 0

3.43 3.72 4.29 4.85 5.41 5.69 5.88 —

0 0.173 0.318 0.552 0.730 0.874 0.939 0.980 1.0

1 343.0 164.0 68.6 33.0 13.4 6.66 2.56 0

— 0.523 0.506 0.441 0.365 0.267 0.206 0.126 0

(85) 79.5 74.02 63.10 50.58 36.24 26.45 15.47 0

1 26.4 12.8 5.79 3.22 1.70 1.05 0.514 0

3.37 3.59 3.99 4.33 4.62 4.74 4.82 —

0 0.150 0.280 0.497 0.680 0.842 0.919 0.973 1.0

1 106 52.1 24.3 14.2 8.0 5.16 2.61 0

— 0.540 0.529 0.484 0.415 0.313 0.247 0.152 0

(70.92) 69.10 67.26 61.94 54.31 42.22 31.62 — 0

1 4.80 2.57 1.52 1.20 0.961 0.756 — 0

3.03 2.83 2.26 1.56 0.760 0.411 — 0

0 0 0 0 0 0 0 0 0

1 19.9 11.4 8.47 9.73 15.9 23.1 — 1

— 0.826 0.814 0.764 0.667 0.495 0.358 — 0

"

S

0.25

0 0.1 0.2 0.4 0.6 0.8 0.9 0.97 1.0

1.0 0.9 0.8 0.6 0.4 0.2 0.1 0.03 0

1 16.2 7.57 2.83 1.07 0.261 0.0736 0.0101 0

(89.5) 82.31 75.18 60.86 46.72 31.04 21.85 12.22 0

1 322.0 153.0 61.1 26.7 8.80 3.50 0.922 0

0.5

0 0.1 0.2 0.4 0.6 0.8 0.9 0.97 1.0

1.0 0.9 0.8 0.6 0.4 0.2 0.1 0.03 0

1 4.31 2.03 0.779 0.319 0.0923 0.0313 0.00609 0

(88.5) 81.62 74.94 61.45 48.14 33.31 23.66 13.75 0

1

0 0.1 0.2 0.4 0.6 0.8 0.9 0.97 1.0

1.0 0.9 0.8 0.6 0.4 0.2 0.1 0.03 0

1 1.33 0.631 0.264 0.121 0.0446 0.0188 0.00474 0

1

0 0.1 0.2 0.4 0.6 0.8 0.9 0.97 1.0

1.0 0.9 0.8 0.6 0.4 0.2 0.1 0.03 0

1 0.240 0.123 0.0626 0.0389 0.021 0.0115 — 0

4Q d 2 n0 L

Key: Qs ¼ ﬂow of lubricant with side ﬂow, cm3 /s; ¼ weight per unit volume of lubricant whose speciﬁc gravity is 0.90 ¼ 8.83 kN/m3 (0.0325 lbf/ in3 ); csp ¼ specific heat of the lubricant, kJ/NK (Btu/lbf 8F) ¼ 0.19 kJ/NK (0.42 Btu/lbf 8F); T0 ¼ difference in temperature, 8C. Source: A. A. Raimondi and J. Boyd, ‘‘A Solution for the Finite Journal Bearings and Its Applications to Analysis and Design’’ ASME, J. Lubrication Technol., Vol. 104, pp. 135–148, April 1982.

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DESIGN OF BEARINGS AND TRIBOLOGY

23.28

CHAPTER TWENTY-THREE

TABLE 23-12 Dimensionless performance parameters for 1808 bearing centrally loaded with side ﬂowa Values of L=d ratio For maximum load For minimum friction L d

0.25 0.28 0.03

0.5 0.42 0.23

Qs Q

csp T0 P

P Pmax

3.44 3.71 4.11 4.25 4.07 3.72 3.29 —

0 0.176 0.320 0.534 0.698 0.837 0.905 0.961 1.0

1 653.0 320.0 146.0 79.8 36.5 18.4 6.46 0

— 0.513 0.489 0.417 0.336 0.241 0.180 0.110 0

1 44.0 21.6 9.96 5.41 2.54 1.38 0.581 0

3.41 3.64 3.93 3.93 3.56 3.17 2.62 —

0 0.167 0.302 0.506 0.665 0.806 0.886 0.951 1.0

1 177.0 87.8 42.7 25.9 15.0 9.80 5.30 0

— 0.518 0.499 0.438 0.365 0.273 0.208 0.132 0

90.0 78.50 68.93 58.86 44.67 32.33 24.14 14.57 0

— 14.1 7.15 3.61 2.28 1.39 0.921 0.483 0

3.34 3.46 3.49 3.25 2.63 2.14 1.60 —

0 0.139 0.252 0.425 0.572 0.721 0.818 0.915 1.0

1 57.0 29.7 16.5 12.4 10.4 9.13 6.96 0

— 0.525 0.513 0.466 0.403 0.313 0.244 0.157 0

90.0 72.90 61.32 49.99 43.15 33.35 25.57 15.43 0

1 3.55 2.01 1.29 1.06 0.859 0.681 0.416 0

3.04 2.80 2.20 1.52 0.767 0.380 0.119 0

1 0 0 0 0 0 0 0 0

1 14.7 8.99 7.34 8.71 14.1 22.5 44.0 1

— 0.778 0.759 0.700 0.607 0.459 0.337 0.190 0

"

S

0.25

0 0.1 0.2 0.4 0.6 0.8 0.9 0.97 1.0

1.0 0.9 0.8 0.6 0.4 0.2 0.1 0.03 0

1 16.3 7.60 2.84 1.08 0.263 0.0736 0.0104 0

90.0 81.40 73.70 58.99 44.96 30.43 21.43 12.28 0

1 163.0 79.4 35.1 17.6 6.88 2.99 0.877 0

0.50

0 0.1 0.2 0.4 0.6 0.8 0.9 0.97 1.0

1.0 0.9 0.8 0.6 0.4 0.2 0.1 0.03 0

1 4.38 2.06 0.794 0.321 0.0921 0.0314 0.00635 0

90.0 79.97 72.14 58.01 45.01 31.29 22.80 13.63 0

1

0 0.1 0.2 0.4 0.6 0.8 0.9 0.97 1.0

1.0 0.9 0.8 0.6 0.4 0.2 0.1 0.03 0

1 1.40 0.670 0.278 0.128 0.0463 0.0193 0.00483 0

1

0 0.1 0.2 0.4 0.6 0.8 0.9 0.97 1.0

1.0 0.9 0.8 0.6 0.4 0.2 0.1 0.03 0

1 0.347 0.179 0.898 0.0523 0.0253 0.0128 0.00384 0

a

1 0.64 0.60

1.0 0.52 0.44 4Q d 2 n0 L

See Key and Source under Table 23-11.

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

23.29

TABLE 23-13 Dimensionless performance parameters for 1208 for centrally loaded bearing with side ﬂowa Values of L=d ratio For maximum load For minimum friction L d

0.25 0.26 0.06

0.5 0.38 0.28

Qs Q

csp T0 P

P Pmax

3.34 3.44 3.42 3.20 2.67 2.21 1.69 —

0 0.143 0.260 0.442 0.599 0.753 0.846 0.931 1.0

1 502.0 254.0 125.0 75.8 42.7 25.9 11.6

— 0.456 0.438 0.389 0.321 0.237 0.178 0.112 0

1 36.6 18.1 8.20 4.43 2.17 1.24 0.550 0

3.29 3.32 3.15 2.80 2.18 1.70 1.19 —

0 0.124 0.225 0.386 0.530 0.684 0.787 0.899 1.0

— 149.0 77.2 40.5 27.0 19.0 15.1 10.6 0

— 0.431 0.424 0.389 0.336 0.261 0.203 0.136 0

90.0 72.43 58.25 43.98 35.65 27.42 21.29 13.49 0

1 14.5 7.44 3.60 2.16 1.27 0.855 0.461 0

3.20 3.11 2.75 2.24 1.57 1.11 0.694 —

0 0.0876 0.157 0.272 0.384 0.535 0.657 0.812 1.0

1 59.5 32.6 19.0 15.0 13.9 14.4 14.0 0

— 0.427 0.420 0.396 0.356 0.290 0.233 0.162 0

90.0 66.69 52.60 39.02 32.67 26.80 21.51 13.86 0

1 6.02 3.26 1.78 1.21 0.853 0.653 0.399 0

3.02 2.75 2.13 1.47 0.759 0.388 0.118 0

0 0 0 0 0 0 0 0 0

1 25.1 14.9 10.5 10.3 14.1 21.2 42.3 1

— 0.610 0.599 0.566 0.509 0.405 0.311 0.199 0

"

S

0.25

0 0.10 0.20 0.40 0.6 0.8 0.9 0.97 1.0

1.0 0.9044 0.8011 0.6 0.4 0.2 0.1 0.03 0

1 18.4 8.45 3.04 1.12 0.268 0.0743 0.0105 0

90.0 76.97 65.97 51.23 40.42 28.38 20.55 12.11 0

1 124.0 60.4 26.6 13.5 5.65 2.63 0.852 0

0.50

0 0.1 0.2 0.4 0.6 0.8 0.9 0.97 1.0

1.0 0.9034 0.8003 0.6 0.4 0.2 0.1 0.03 0

1 5.42 2.51 0.914 0.354 0.0973 0.0324 0.00631 0

90.0 74.99 63.38 48.07 38.50 28.02 21.02 13.00 0

1

0 0.1 0.2 0.4 0.6 0.8 0.9 0.97 1.0

1.0 0.9024 0.8 0.6 0.4 0.2 0.1 0.03 0

1 2.14 1.01 0.385 0.162 0.0531 0.0208 0.00498 0

1

0 0.1 0.2 0.4 0.6 0.8 0.9 0.97 1.0

1.0 0.9007 0.8 0.6 0.4 0.2 0.1 0.03 0

1 0.877 0.431 0.181 0.0845 0.0328 0.0147 0.00406 0

a

1 0.53 0.5

1.0 0.46 0.4 4Q d 2 n0 L

See Key and Source under Table 23-11.

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DESIGN OF BEARINGS AND TRIBOLOGY

23.30

CHAPTER TWENTY-THREE

TABLE 23-14 Dimensionless performance parameters for 608 centrally loaded bearing with side ﬂowa Values of L=d ratio For maximum load For minimum friction L d

0.25 0.15 0.10

0.5 0.20 0.16

Qs Q

csp T0 P

P Pmax

3.16 3.04 2.57 1.98 1.30 0.894 0.507 —

0 0.0666 0.131 0.236 0.346 0.496 0.620 0.786 1.0

1 499.0 260.0 136.0 86.1 54.9 41.0 29.1 0

— 0.251 0.249 0.242 0.228 0.195 0.159 0.107 0

1 48.6 24.2 10.3 4.93 2.02 1.08 0.490 0

3.11 2.91 2.38 1.74 1.05 0.664 0.329 —

0 0.0488 0.0883 0.160 0.236 0.350 0.464 0.650 1.0

1 201.0 109.0 59.4 40.3 29.4 26.5 27.8 0

— 0.239 0.239 0.233 0.225 0.201 0.172 0.122 0

90.0 67.92 50.96 33.99 24.56 18.33 15.33 10.88 0

1 29.1 14.8 6.61 3.29 1.42 0.822 0.422 0

3.07 2.82 2.22 1.56 0.883 0.519 0.226 —

0 0.0267 0.0481 0.0849 0.127 0.200 0.287 0.465 1.0

1 121.0 67.4 39.1 28.2 22.5 23.2 30.5 0

— 0.252 0.251 0.247 0.239 0.220 0.192 0.139 0

90.0 65.91 48.91 31.96 23.21 17.39 14.94 10.58 0

1 19.7 10.1 4.67 2.40 1.10 0.667 0.372 0

3.01 2.73 2.07 1.40 0.722 0.372 0.115 0

0 0 0 0 0 0 0 0 0

1 82.3 46.5 28.4 21.4 19.2 22.5 40.7 1

— 0.337 0.336 0.329 0.317 0.287 0.243 0.163 0

"

S

0.25

0 0.1 0.2 0.4 0.6 0.8 0.9 0.97 1.0

1.0 0.9251 0.8242 0.6074 0.4 0.2 0.1 0.03 0

1 35.8 16.0 5.20 1.65 0.333 0.0844 0.0110 0

90.0 71.55 58.51 41.01 30.14 21.70 16.87 10.81 0

1 121.0 58.7 24.5 11.2 4.27 2.01 0.713 0

0.5

0 0.1 0.2 0.4 0.6 0.8 0.9 0.97 1.0

1.0 0.9223 0.8152 0.6039 0.4 0.2 0.1 0.03 0

1 14.2 6.47 2.14 0.695 0.149 0.0422 0.00704 0

90.0 69.00 52.60 37.00 26.98 19.57 15.91 10.85 0

1

0 0.1 0.2 0.4 0.6 0.8 0.9 0.97 1.0

1.0 0.9212 0.8133 0.6010 0.4 0.2 0.1 0.03 0

1 8.52 3.92 1.34 0.450 0.101 0.0309 0.00584 0

1

0 0.1 0.2 0.4 0.6 0.8 0.9 0.97 1.0

1.0 0.9191 0.8109 0.6002 0.4 0.2 0.1 0.03 0

1 5.75 2.66 0.931 0.322 0.0755 0.0241 0.00495 0

a

1 0.25 0.23

1.0 0.23 0.22 4Q d 2 n0 L

See Key and Source under Table 23-11.

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.31

Formula

10

Factor CL

8 6 4 2 0

0

1

2 Ratio, L/d

3

4

The total frictional resistance on an idealized journal bearing surface

F ¼

FIGURE 23-16 Variation of factor CL with L=d ratio.

4 UL

! 1 þ 2"2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð2 þ "2 Þ 1 "2

ð23-43Þ

or F ¼ The total frictional resistance on an idealized lightly loaded journal bearing

F ¼

For the relation between dimensionless quantity n0 1 and Sommerfeld numbers S F0

4 2 n0 Ldð1 þ 2"2 Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð2 þ "2 Þ 1 "2 2 2 n0 Ld

ð23-44Þ

ð23-45Þ

Refer to Fig. 23-17.

0.06 Petroff

0.05

1 ψ

ηn’ Fµ’

0.04 0.03 0.02 0.01 0

0

0.1

1.5

0.2

Bearing characteristic number, S =

0.3 ηn’ P

1 ψ2

FIGURE 23-17 Variation of dimensionless 1 n0 with Sommerfeld number S for quantity F0 an idealized full journal bearing.

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DESIGN OF BEARINGS AND TRIBOLOGY

23.32

CHAPTER TWENTY-THREE

Particular

Formula

The relation between coeﬃcient of friction and bearing characteristic number The relation between the coeﬃcient of friction and attitude "

n0 ¼ 2 P 2

¼

1

1 þ 2"2 3"

ð23-46Þ

ð23-47Þ

For average coeﬃcient of friction at very high pressures

Refer to Table 23-15 for coeﬃcient of friction

The friction coeﬃcient variable

¼

¼

1 þ 2"2 3"

Refer to Figs. 23-20 to 23-24. TABLE 23-15 Average coeﬃcient of friction at very high pressure Angular displacement, deg Material

108

508

Stearic acid

0.022

0.029

Tungsten disulphide

0.032

0.037

Molybdenum disulphide

0.032

0.033

Graphite

0.036

0.058

Silver sulphate

0.055

0.054

Turbine oil plus 1% MoS2

0.060

0.068

Lead iodide

0.061

0.071

Palm oil

0.063

0.075

Castor oil

0.064

0.081

Grease (zinc-oxide base)

0.071

0.080

Lard oil

0.072

0.084

Grease (calcium base)

0.073

0.082

Residual

0.076

0.083

Sperm oil

0.077

0.085

Turbine oil plus 1% graphite

0.081

0.105

Turbine oil plus 1% stearic acid

0.087

0.096

Turbine oil

0.088

0.108

Capric acid

0.089

0.109

Turbine oil plus 1% mica

0.091

0.105

Oleic acid

0.093

0.119

Machine oil

0.099

0.115

Soapstone (powdered)

0.169

0.306

Mica (powdered)

0.257

0.305

Boron (not a lubricant)

0.482

0.710

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ð23-48Þ

DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.33

Formula

INFLUENCE OF MISALIGNMENT OF SHAFT IN BEARING Minimum oil ﬁlm thickness corresponding to the materials factor (km ), the surface roughnesses (Rp ) and amount of misalignment of the journal and bearing (Ma )

Ma L ð23-48aÞ 2 where km ¼ material factor from Table 23-15a L ¼ length of bearing Rpb ¼ surface roughness of bearing from Table 23-15b Ma ¼ x=L amount of misalignment

hmin ¼ km ðRpj þ Rpb Þ þ

Refer to Table 23-15c and Fig. 23-18 for Ma 2Q ð23-48bÞ L dn0 c 3 0 where Q in m /s, L, d, and c in m, n in rps

Dimensionless oil feed rate

Q0 ¼

TABLE 23-15a Material factor, km

TABLE 23-15b Surface ﬁnish, predominant peak height, Rp km

Bearing lining material Phosphor bronze Leaded bronze Tin aluminium White metal (Babbitt) Thermoplastic (bearing grade) Thermosetting plastic

1 0.8 0.8 0.5 0.6 0.7

Courtesy: Neale, M. J., Tribology Handbook, Newnes-Butterworths, 1973

Surface type

Micro-inch cla

Turned or rough 100 ground Ground or ﬁne 20 bored Fine ground 7 Lapped or 1.5 polished

lm RMS

Rp Class

lm lin

2.8

6

12

480

0.6

8

3

120

0.19 0.04

10 12

0.8 32 0.2 9

Courtesy: Neale, M. J., Tribology Handbook, Newnes-Butterworths, 1973

TABLE 23-15c Values of misalignment factor, Ma at two ratios of (hmin =c) (hmin =c)

d

Ma (L=c)

0.1

0.01

0 0.05 0.25 0.50 0.75

100 65 25 12 8

100 33 7 3 1

Courtesy: Neale, M. J., Tribology Handbook, Newnes-Butterworths, 1973

x Ma =

x y

y

L

FIGURE 23-18 Misaligned journal inside the bearing under load

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DESIGN OF BEARINGS AND TRIBOLOGY

23.34

CHAPTER TWENTY-THREE

Particular

Formula

Bearing load capacity number

W0 ¼

W e n0 Ld

ð23-48cÞ

2

where W 0 ðd=LÞ2 ¼ dimensionless load number, W in N, n0 in rps, e in N s/m2 , L, d, and c in m The required grease supply rate per hour for grease lubricated bearing

Qg ¼ kg c dL

The coeﬃcient of friction

¼

The diameter of journal bearing for speeds below 2.5 m/s

The diameter of journal bearing for speeds exceeding 2.5 m/s

ð23-48dÞ

where kg ¼ a factor for grease lubrication at various rotational speeds. Taken from Table 23-15d. ð23-48eÞ

where ¼ = sﬃﬃﬃﬃﬃﬃﬃﬃ 2 5 W d ¼ 3:2 103 2 0 i n

ð23-48f Þ

where W in N, d in m; l=d ¼ i and n0 in rps sﬃﬃﬃﬃﬃﬃﬃﬃ 2 7 W ð23-48gÞ d ¼ 2 103 3 0 i n

POWER LOSS The power loss in the bearing due to viscous friction

P¼

F U 33;000

USCS

ð23-49aÞ

where P in hp, F in lbf, and U in ft/min F U Customary Metric ð23-49bÞ 102 where P in kW, F in kgf, U ¼ dn0 ¼ velocity in m/s, d in m, and n0 in rps F U SI ð23-49cÞ P¼ 1000 where P in kW, F in N, and U in m/s

P¼

TABLE 24-15d Values of factor kg for grease lubrication at various rotational speeds Journal speed, n in rpm

kg

up to

0.1 0.2 0.4 1.0

100 250 500 1000

Courtesy: Neale, M. J., Tribology Handbook, Newnes Butterworth

Lubricant feed rate Q

b

Diametrical clearance Cd d w

n’ Lubricant viscosity ηe

FIGURE 23-19 Journal inside the bearing under Load (W) at speed (n0 ).

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µ Coefficient of friction variable, ψ

0 0

5

10

15

0 0

1

2

3

4

0.10

4. 0 B/L = . 0 3 = B/L 2. 0 B/L = . B/L = 1 0 B/L = 0

1.0

B/L = 0

4. 0 B/L = . 30 = B/L 2. 0 B/L = . B/L = 1 0

1.5 ηn’ 1 Bearing characteristic number, S = P ψ2

0.5

60 Partial journal bearings

0.15 ηn’ 1 Bearing characteristic number, S = P ψ2

0.05

60 Partial journal bearings

2.0

0.20

FIGURE 23-20 Variation of coeﬃcient of friction variable = with S for 608 partial journal bearing.

µ Coefficient of friction variable, ψ

µ Coefficient of friction variable, ψ 0

10

20

0 0

1

2

3

4

0

0.10

1.0

ηn’ Bearing characteristic number, S = P

0.5

B/L = 1.0

120 Partial journal bearings

B/L =

1. 0

.0 =4 B/L .0 =3 /L B .0 2 = B/L

B/L = 0

1.5 1 ψ2

B/L =

0

0 B/L = 2

. B/L = 3. 0

B/L = 4.0

0.15 ηn’ 1 Bearing characteristic number, S = P ψ2 0.05

120 Partial journal bearings

2.0

0.20

FIGURE 23-21 Variation of coeﬃcient of friction variable = with S for 1208 partial journal bearing.

µ Coefficient of friction variable, ψ

5

DESIGN OF BEARINGS AND TRIBOLOGY

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23.35

0

1

2

3

4

23.36

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0

B/L = 2.0

=0

1.0

1.5 ηn’ 1 Bearing characteristic number, S = P ψ2

0.5

B/L = 1.0 B/L = 0

.0 B/L = 3.0 B/L = 4

180 Partial Journal bearings

2.0

0.20

FIGURE 23-22 Variation of coeﬃcient of friction variable = with S for 1808 partial journal bearing.

0

10

20

0.10

B/L

.0 B/L = 1

.0 B/L = 2

.0 =4 B/L 3. 0 = L / B

0.15 ηn’ 1 Bearing characteristic number, S = P ψ2

0.05

180 Partial journal bearings

µ Coefficient of friction variable, ψ 0

1.0

2.0

3.0

4.0

5.0

0

10

20

30

40

45

0

OF

TR

PE

F

. 40

1.0 Bearing characteristic number, S =

0.5

B/L = 3.0 B/L = 2.0 µ 2 PETROFF ψ = 2 π s

B/L = 4.0

360 Journal bearings

0.10

.0 =1 B/L 0 = B/L

2. 0

3. 0

= B/L

= B/L

L=

ηn’ 1 P ψ2

1.5

0.15 ηn’ 1 Bearing characteristic number, S = P ψ2

0.05

B/

360 Journal bearings

2 .0

0.20

FIGURE 23-23 Variation of coeﬃcient of friction variable = with S for 3608 partial journal bearing.

µ Coefficient of friction variable, ψ

5

30

µ Coefficient of friction variable, ψ

µ Coefficient of friction variable, ψ

6.5 6.0

DESIGN OF BEARINGS AND TRIBOLOGY

DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.37

Formula

2

Coefficient of Friction Variable, λµ = µ/ψ

102 5 4 3 2

1 4 L = d

10 5 4 3

1 ∞

2

1 2

1

0

0

0.01

0.02 0.04 0.06 0.08 0.1 0.2

0.4 0.60.81.0 ηn’ 1 Bearing Characteristic Number, S = ( 2) ψ P

2

4

6

8 10

FIGURE 23-24 Variation of the coeﬃcient of friction variable ¼ = with S for 3608 journal bearing. [Boyd and Raimondi5 ]

PARTIAL JOURNAL BEARING (Fig. 23-25) The resultant pressure distribution around the partial journal bearing excluding, Po oil ﬁlm pressure at the point where ¼ 0

Pr ¼ P Po where P Po ¼

12U 2d

ð23-50Þ

ð1 "2 Þ ð2 þ "2 Þðk=cÞ ð1 "2 Þ2=5 ! rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1" arc tan tan 1þ" 2 þ

ðk=2cÞ" sin 2ð1 "2 Þð1 þ " cos Þ2

þ

" sin fð3k=2cÞ 2ð1 "2 Þg 2ð1 "2 Þ2 ð1 þ " cos Þ

where k ¼ h is the thickness of oil ﬁlm at maximum pressure value

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DESIGN OF BEARINGS AND TRIBOLOGY

23.38

CHAPTER TWENTY-THREE

Particular

d

w,

h’max

Formula

+ c d

h’max A

B

θ

φ

Bearing

n

Leading edge

O

e

=

cε

O’

Journal ho = hmin Trailing edge Line of centers

α β

Pmax Pressure distribution

FIGURE 23-25 Partial journal bearing. 90

1.0

80 Attitude angle φ, deg

Attitude, ε

0.8 0.6 0.4 0.2 0

0

0.10

0.20

ηn’ Bearing characteristic number, S = P

0.30 1 ψ2

FIGURE 23-26 Variation of attitude " with S for an idealized oﬀset partial bearing having the maximum load capacity corresponding to a given attitude.

70 60 50 40 30 20 10 0

0

0.1

0.2

0.3

0.4

0.5 0 .6 ηn’ 1 Bearing characteristic number, S = ψ2 P

FIGURE 23-27 Variation of attitude angle with S for an idealized oﬀset partial bearing.

Pressure at any point in a partial journal bearing

P ¼ Po þ Pr

To determine the attitude " and attitude angle for various values of S and for an idealized oﬀset partial bearing having the maximum load capacity corresponding to a given attitude

Refer to Figs. 23-26 and 23-27 respectively.

ð23-51Þ

INFLUENCE OF END LEAKAGE Leakage factors CW , CF , and C

Refer to Fig. 23-28 for CW , CF , and C for various values of B=L ratios

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.39

Formula

Load leakage factor according to Kingsbury6

CW ¼

W W1

ð23-52Þ

Refer to Fig. 23-28 for CW . Load leakage factor CW as a function of B=L ratio for a slider bearing having q ¼ ðh1 =h2 Þ 1 ¼ 1 or h1 ¼ 2h2

Refer to Table 23-16.

Load leakage factor for 1208, centrally loaded partial bearing according to Needs7

Refer to Fig. 23-29 for CW for various attitudes ".

Load correction factor for side ﬂow according to Boyd and Raimondi24

Refer to Fig. 23-30 for CW for various minimum oil ﬁlm thickness variables .

Coeﬃcient of friction leakage factor according to Kingsbury6

C ¼

1

ð23-53Þ

Refer to Fig. 23-28 for C Friction leakage factor according to Kingsbury6

CF ¼

F F 1

ð23-54Þ

Refer to Fig. 23-28 for CF . Refer to Fig. 23-31 for CF for various attitudes ".

Friction leakage factor for 1208, centrally loaded partial bearing according to Needs7 1.0

Leakage factors CW, CF, and Cµ

0.9

CF

0.8 0.7 0.6 0.5 0.4

CW

0.3

Cµ

TABLE 23-16 Load leakage factor CW as a function of B=L ratio for a slider bearing having the quality q equal to unity

0.2 0.1 0

0

0.5

1.0

1.5

2.0

2.5

3.0

2.5

Length in direction of motion Length in direction perpendicular to motion

=

4.0 B L

FIGURE 23-28 Kingsbury’s leakage factors as function of B=L ratios under minimum friction. [Kingsbury6 ]

B=L

CW

B=L

CW

0.00 0.175 0.25 0.50 0.75

1.00 0.92 0.835 0.68 0.55

1.00 1.50 2.00 3.00 4.00

0.44 0.278 0.185 0.090 0.060

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0

1

2

Axial length

Length in direction of motion

Slider

= L

B

3

120 Centrally loaded partial journal bearings

∈=0 .2

∈=0. 6 ∈=0. 4

∈=0. 8

4

FIGURE 23-29 Leakage factors for load for 1208 centrally partial journal bearings for various attitudes. [Needs7 ]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

2 Length in direction of motion

0.2

3 =

L

B

δ = 0. 4 δ = 0. 6 δ = 0. 8

δ=

5

0 .1

0 .0

δ=

δ=

B = 2πr

Length in direction perpendicular to motion

1

r

B

δ=0

L

Load correction factor, CW

FIGURE 23-30 Load correction factor (CW ) for side ﬂow. [Boyd and Raimondi5 ]

0

0.2

0.4

0.6

0.8

1.0

4

23.40

Load factor, CW

1.0

DESIGN OF BEARINGS AND TRIBOLOGY

CHAPTER TWENTY-THREE

DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

Formula

1.0

Friction factor, CF

23.41

∈=0.2 ∈=0.4 ∈=0.6

0.9 120 Centrally loaded partial journal bearings

0.8

SLIDER

∈=0.8 0

1

2

Length in direction of motion Axial length

3 =

4 FIGURE 23-31 Leakage factors for friction force for 1208 centrally loaded partial journal bearings for various attitudes. [Needs7 ]

B L

Refer to Fig. 23-32 for CF for various minimum oil ﬁlm thickness variables and B=L ratios.

The ratios of the maximum pressure in the oil ﬁlm Pmax , and the unit load, P, with B=L ratios for various values of attitude, ", for 1208 central partial journal bearing according to Needs7

Refer to Fig. 23-33 for Pmax =P and Fig. 23-34 for P=Pmax for various values of B=L ratios and attitudes ".

The variation of attitude, ", with bearing characteristic number, S, for various values of B=L ratios for 608, 1208, 1808 partial and full journal bearings

Refer to Figs. 23-35 to 23-38 for " for various values of S and B=L ratios.

The variation of coeﬃcient of friction variable, ¼ = , with bearing characteristic number, S, for various values of B=L ratios for 608, 1208, 1808, partial and full journal bearing

Refer to Figs. 23-20 to 23-24 and 23-39 for ¼ various values of S and B=L ratios.

The friction curves illustrating boundary conditions

Refer to Fig. 23-39.

Friction correction factor, CF

Friction correction factor for side ﬂow according to Boyd and Raimondi5

1.0

δ = 1.0

δ = 0.8 δ = 0.6

0.9

δ = 0.4 δ = 0.2

0.8 0.7

δ = 0. 1 δ = 0. 05

0

1

2 Length in direction of motion

Length in direction perpendicular to motion

3 =

B

4

L

FIGURE 23-32 Friction correction factor for side ﬂow. [Boyd and Raimondi5 ]

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for

DESIGN OF BEARINGS AND TRIBOLOGY

23.42

CHAPTER TWENTY-THREE

5 0. 9

∈=

P

Pressure ratio

Pmax

4

.8 ∈=0

. ∈=0 6 . ∈=0 4

3

. ∈=0 2

2

1

0

0

1

2

3

Length in direction of motion Length in direction perpendicular to motion

=

4

B L

FIGURE 23-33 The ratio of the maximum pressure (Pmax ) and the unit load Pð¼ Pu Þ with B=L ratios for various attitudes for a 1208 central partial bearing. [Needs7 ]

Maximum Oil Film Pressure Ratio,

P Pmax (gauge)

1.0

L =∞ d

0.9 0.8 0.7 0.6 0.5

1

0.4

1 2 1 4

0.3 0.2 0.1 0

0

0.01

0.02

0.04 0.06 0.08 0.1

0.2

0.4

0.6 0.81.0

Bearing Characteristic Number, S =

ηn’ P

2

4

6

8 10

1 ψ2

FIGURE 23-34 Chart for maximum oil ﬁlm pressure ratio P=PmaxðgaugeÞ with bearing characteristic number S for full journal bearing. [Boyd and Raimondi5 ]

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0.6

0.8 0.7

1.0 0.9

0 0 1.0

B/L = 0

B/L = 1.0

B/L = 2.0

B/L = 4.0

0.15 ηn’ 1 2 ψ

B/L = 0

1.5 ηn’ 1 Bearing characteristic number, S = ψ2 P

0.5

B/L = 3.0

ηn’ Bearing characteristic number, S = P

60 Partial journal bearings

0

60 Partial journal bearings 0.05 0.10

B/L = 1.0

2.0

0.20

B/L = 2.0

B/L = 3.0 B/L = 4.0

FIGURE 23-35 Variation of attitude " with S for 608 partial journal bearing.

(b)

0.3 0.2 0.1

0.5 0.4

0.7 0.6

1.0 0.9 0.8

(a)

Attitude, ε

Attitude, ε

0

0.10

B/L = 3.0

1.0

1.5 ηn’ 1 Bearing characteristic number, S = ψ2 P

0.5

B/L = 0

B/L = 1.0

B/L = 2.0

B/L = 3 .0

B/L = 4.0

120 Partial journal bearings

0.15 ηn’ 1 Bearing characteristic number, S = ψ2 P

0.05

B/L = 0

B/L = 1.0

B/L = 4.0 B/L = 2.0

2.0

0.20

FIGURE 23-36 Variation of attitude " with S for 1208 partial journal bearing.

0 0

0.3 0.2 0.1

0.5 0.4

0.7 0.6

0.4 0.3

0.6 120 Partial journal bearings 0.5

0.8 0.7

1.0 0.9 0.8

Attitude, ε Attitude, ε

1.0 0.9

DESIGN OF BEARINGS AND TRIBOLOGY

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23.43

Attitude, ε

23.44

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0

180 Partial journal bearings

0.2 0.1 0

0.4 0.3

0.7 0.6 0.5

1.0 0.9 0.8

0

0 0.10

B/L = 4.0

= 2. 0

= 4. 0 = 3. 0

1.0

1.5 ηn’ 1 Bearing characteristic number, S = P ψ2

0.5

= 1. 0 B/L = 0

B/L

B/L

B/L

B/L

180 Partial journal bearings

0.15 ηn’ 1 Bearing characteristic number, S = P ψ2

0.05

B/L = 0

0.4 0.3

0.2 0.1

B/L = 1 . 0

0.7 0.6 0.5

B/L = 2.0

B/L = 3.0

0.20

2.0

FIGURE 23-37 Variation of attitude " with S for 1808 partial journal bearing.

Attitude, ε

Attitude, ε 0

0.2 0.1

0.4 0.3

0.7 0.6 0.5

1.0 0.9 0.8

0

0

0

0.2 0.1

0.4 0.3

0.7 0.6 0.5

1.0 0.9 0.8

B/L

=0

B/L

0.5

.0

.0

1.0

360 Journal bearings

1.5 ηn’ 1 Bearing characteristic number, S = P ψ2

=2 = 1. 0

B/L

.0

=4 =3

B/L B/L

0.10

0.15 ηn’ 1 Bearing characteristic number, S = P ψ2

0.05

B/L = 0

B/L = 1 . 0

B/L = 2 . 0

B/L = 4 . 0 B/L = 3 . 0

360…Journal bearings

2.0

0.20

FIGURE 23-38 Variation of attitude " with S for 3608 partial journal bearing.

Attitude, ε

1.0 0.9 0.8

DESIGN OF BEARINGS AND TRIBOLOGY

DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.45

Formula

6.0

Coefficient of friction variable,

5.0

Representative experimental Dividing line

4.0

Sommerfeld 3.0

Minimum

Petroff

Boundary Fluid film region 2.0 region

Reynold

1.0

0

0

2

4

6

8

10

12

14 16 60ηn’ 1 Bearing characteristic number, S’ = P ψ2

18

20

FIGURE 23-39 Friction curves illustrating boundary conditions.

FRICTION IN A FULL JOURNAL BEARING WITH END LEAKAGE FROM BEARING The total friction force acting on the surface of a full journal bearing with end ﬂow

2 2 n0 Ld F ¼ 12 W" sin þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 "2

The coeﬃcient of friction variable

" 2 2 S ¼ sin þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1 "2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ " ð1 "2 2 2 S þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 2 1 "2 ¼

ð23-55Þ

ð23-56aÞ ð23-56bÞ

P value Lasche’s equation for P

P ¼ 195;350=t

SI 2

where P in N/m and t in K

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ð23-57aÞ

DESIGN OF BEARINGS AND TRIBOLOGY

23.46

CHAPTER TWENTY-THREE

Particular

Formula

P ¼ 0:02=t

Customary Metric

ð23-57bÞ

where P in kgf/mm and t in 8C 2

P ¼ 51=t

USCS

ð23-57cÞ

SI

ð23-58aÞ

Customary Metric

ð23-58bÞ

where P in psi and t in 8F Lasche’s equation for the coeﬃcient of friction which may be used for bearing subjected to pressure varying from 0.103 MPa (15 psi) to 1.55 MPa (225 psi) and speed varying from 2.5 to 18 m/s and temperature varying from 30 to 1008C

¼

23;126 pﬃﬃ P t

where P in N/m2 and t in K ¼

0:00236 pﬃﬃ P t

where P in kgf/mm2 and t in 8C 4:5 ¼ pﬃﬃ P t

USCS

ð23-58cÞ

SI

ð23-59aÞ

where P in psi and t in 8F The coeﬃcient of friction according to Illmer when bearing is subjected to pressure varying from 0.23 MPa (35 psi) to 0.7 MPa (100 psi) and speed varying from 0.5 m/s to 1.5 m/s (100 ft/min to 300 ft/min)

¼

p 3 ﬃﬃﬃ v pﬃﬃﬃﬃﬃ 0:05 Pt

where P in N/m2 , v in m/s, and t in K ¼

p 3 ﬃﬃﬃ v pﬃﬃﬃﬃﬃ 157:7 Pt

Customary Metric

ð23-59bÞ

where P in kgf/mm2 , v in m/s, and t in 8C ¼

p 3 ﬃﬃﬃ v pﬃﬃﬃﬃﬃ 20 Pt

USCS ð23-59c Þ

where P in psi, v in ft/min, and t in 8F The coeﬃcient of friction according to Tower tests

¼

144;204:5 P

rﬃﬃﬃ v t

SI

ð23-60aÞ

where P in N/m2 , v in m/s, and t in K ¼

0:0147 P

rﬃﬃﬃ v t

Customary Metric

ð23-60bÞ

where P in kgf/mm2 , v in m/s, and t in 8C rﬃﬃﬃ 2 v USCS ð23-60cÞ ¼ P t where P in psi, v in ft/min, and t in 8F

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

The coeﬃcient of friction according to Lasche when the speed exceeds 2.5 m/s (500 ft/min)

23.47

Formula

24:73 ¼ pﬃﬃﬃﬃﬃ Pt

SI

ð23-61aÞ

Customary Metric

ð23-61bÞ

where P in N/m2 and t in K 0:0079 ¼ pﬃﬃﬃﬃﬃ Pt

where P in kgf/mm2 and t in 8C 0:4 ¼ pﬃﬃﬃﬃﬃ Pt

USCS

ð23-61cÞ

where P in psi and t in 8F

OIL FLOW THROUGH JOURNAL BEARING Oil ﬂow through bearing

Q ¼ 0:785cLdn0

SI

ð23-62Þ

0

where c, L, and d in m, n in rps, and Q in m3 /s Q ¼ 785cLdn0

SI

ð23-63aÞ

0

where c, L, and d in m, n in rps, and Q in dm3 /s Q ¼ 7:85 107 cLdn0

SI

ð23-63bÞ

0

where c, L, and d in mm, n in rps, and Q in liters/s or dm3 /s Q ¼ 0:0034cLdn0

USCS

ð23-63cÞ

where c, L, and d in in, n in rpm, and Q in US gallons/min

dc3 Po ð1 þ 1:5"2 Þ 48L

Oil ﬂow through a central groove of bearing from one end

Q¼

Total oil ﬂow through a central groove of bearing from both ends

Qg ¼

Total oil ﬂow through a central groove for lightly loaded bearing [From Eq. (23-65) as " ! 0]

Q’

dc3 Po 24L

ð23-66Þ

Total oil ﬂow through a central groove for heavily loaded bearing [From Eq. (23-65) as " ! 1]

Q’

2:5 dc3 Po 24L

ð23-67Þ

Oil ﬂow through a single hole

Qh ¼

ð23-64Þ

dc3 Po ð1 þ 1:5"2 Þ 24L

c3 Po ð1 þ 1:5"2 Þ tan1 24

ð23-65Þ

d L

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ð23-68Þ

DESIGN OF BEARINGS AND TRIBOLOGY

23.48

CHAPTER TWENTY-THREE

Particular

Formula

The ratio of Qg to Qh in the unloaded region of bearing from Eq. (23-65) and (23-68)

Qg

d ¼ Qh L tan1 ð d=LÞ

ð23-69Þ

where d is the diameter of journal Flow variable (dimensionless)

0Q ¼

4Q 60c2 n0 L

ð26-70Þ

Refer to Figs. 23-40 and 23-41 for ﬂow variable 0Q . Oil ﬂow through a bearing with side leakage

Q¼

600Q d 2 n0 L CQ 4

or

600Q c2 n0 L CQ 4

ð23-71Þ

where CQ ¼ flow correction factor from Fig. 23-42 Oil flow ratio

Qs Q

Refer to Fig. 23-43.

Flow variable,

4Q ψ 60c2n’L

3.0

2.0

1.0

0

0

1

2

4

6

8 10

2

4

6

Bearing characteristic number, S =

8 102

2

4

60ηn’ 1 P ψ2

FIGURE 23-40 Chart for determining oil ﬂow, based on no side ﬂow. [Boyd and Raimondi5 ]

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6

8 103

DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.49

Formula

L/d=1/4

6

L/d=

5

L/d=1

4

=∞

3

2

L/d

ψ Flow Variable, λ’Q = 4Q/d2n’Lψ = 4Q c2 n’L

1/2

1

0

0

0.01 0.02 0.04 0.06 0.1

0.2

0.4 0.6 0.8 1.0 ηn’ 1 Bearing Characteristic Number, S = ψ2 P

2

4

6 8 10

FIGURE 23-41 Chart for oil ﬂow variable Q with bearing characteristic number S for full journal bearing. [Boyd and Raimondi5 ]

THERMAL EQUILIBRIUM OF JOURNAL BEARING The general expression for heat generated in bearing

Hg ¼ Mt ð2 n0 Þ ¼ ¼ ðPLdÞ

d W ð2 n0 Þ 2

d ! ¼ ðPLdÞv SI ðUSCSÞ ð23-72aÞ 2

where Hg in J/s (Btu/s), Mt in N m (lbf in), W in N (lbf ), P in N/m2 (psi), L in m (in), d in m (in), n0 in rps, ! in rad/s, and v in m/s (ft/ min)

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DESIGN OF BEARINGS AND TRIBOLOGY

23.50

CHAPTER TWENTY-THREE

9

δ=

Flow correction factor, CQ

8

05 0.

δ=

6

0.1

δ = 0.2

4

δ = 0.4

2

δ = 0•8 δ = 0.6 δ = 1.0

0

0

1

2

3

Length in direction of motion Length in direction, perpendicular to motion

4 =

B L

FIGURE 23-42 Flow correction factor for side ﬂow. [Boyd and Raimondi5 ] 1.0

L d = 1 4

0.9

Oil Flow Ratio,

Qs Q

0.8

1 2

0.7 0.6 0.5 0.4

1

0.3 0.2

L =∞ d

0.1 0

0

0.01

0.02

0.04

0.08 0.1

0.2

0.4 0.6 0.8 1.0

2

4

6

8 10

ηn’ 1 Bearing Characteristic Number, S = P ψ2 FIGURE 23-43 Oil ﬂow ratio (Qs =Q) versus bearing characteristic number S for full journal bearing. [Boyd and Raimondi5 ]

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DESIGN OF BEARINGS AND TRIBOLOGY

23.51

DESIGN OF BEARINGS AND TRIBOLOGY

Particular

Formula

ðPLdÞv ðPLdÞð dn0 Þ ¼ 778 778 ðPLdÞd! ¼ USCS ð23-72bÞ 1556 where P in psi, L in in, d in in, v in ft/min, ! in rad/s, Hg in Btu/s, and n0 in rps

Hg ¼

The heat generated can also be found by knowing the temperature rise of lubricant oil which is used to carry away heat generated in the bearing

Temperature rise of the lubricant ﬁlm variable T The temperature rise of the lubricant ﬁlm due to heat generated which is to be carried away by the lubricant, can be found from Eq. (23-66c)

Hg ¼ Csp Q T

SI ðUSCSÞ ð23-73Þ

where Hg in J/s (Btu/s) ¼ weight per unit volume of lubricant whose average speciﬁc gravity is 0.90 ¼ 8:83 kN/m3 (0.0325 lbf/in3 ) Csp ¼ speciﬁc heat of lubricant, kJ/N K (Btu/lbf 8F) ¼ 0.19 kJ/N K (0.42 Btu/lbf 8F) Q in m3 /s; T in 8C Refer to Fig. 23-44 for T . T ¼

4 P 427Csp Q

ð23-74aÞ

Customary Metric

where ¼ = ¼ friction coeﬃcient variable Q ¼ flow variable ¼ 4Q=d 2 n0 L P in kgf/m2 , Csp in kcal/kgf 8C, in kgf/m3 , and T in 8C T ¼ 78

P 78ð= ÞP P ¼ ¼ 19:5 Q 4Q=d 2 n0 L Q=d 2 n0 L Customary Metric

ð23-74bÞ

2

where P in kgf/mm , d in mm, L in mm, Q in mm3 /s, n0 in rps, and T in 8C T ¼

8:3 106 ð= ÞP 8:3 106 P ¼ Q=d 2 n0 L Q=d 2 n0 L SI

ð23-74c Þ

where P in N/m , Q in m /s, d in m, L in m, n0 in rps, and T in K or 8C, since T 8C ¼ T K 2

If the end ﬂow is also taken into consideration then the temperature rise of the ﬂow Q Qel due to heat generated, is T and the temperature rise of end leakage is T=2, which is the average of the inlet and the outlet temperatures

T ¼

3

0:103ð= ÞP ½1

1 2 0 2 ðQel =QÞð4Q=d n L

Þ

P ¼ 0:0258 1 ½1 2 ðQel =QÞðQ=d 2 n0 LÞ USCS

ð23-75aÞ

where P in psi, d in in, L in in, Qel and Q in in3 /s, n0 in rps, and T in 8F

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DESIGN OF BEARINGS AND TRIBOLOGY

23.52

CHAPTER TWENTY-THREE

Particular

Formula

103 8 6 4

P

Temperature rise of the lubricant film vanable, λT =

γ Csp∆T

2 For

102 8 6

L =1 d

4 2 10 8 6 4 2

0

0

.01

.02

.04 .06.08.10

.20

.40 .60 .801.0

2.0 4.0 6.0 8.010 ηn’ 1 Sommerfeld number or bearing characteristic number, S = P ψ2

FIGURE 23-44 Variation of temperature rise of the lubrication ﬁlm variable T with Sommerfeld number S. [Boyd and Raimondi5 ]

T ¼ ¼

8:3 106 ð= ÞP ½1 12 ðQel =QÞðQ=d 2 n0 L Þ 8:3 106 P ½1 12 ðQel =QÞðQ=d 2 n0 LÞ

SI

ð23-75bÞ

where P in N/m2 , d and L in m, Qel and Q in m3 /s, n0 in rps, and T in K

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.53

Formula

The temperature rise of the lubricant ﬁlm due to heat generated in pressure fed bearings

T ¼

16 106 ð= ÞSW 2 ð1 þ 1:5"2 ÞPS d 4

SI

ð23-76aÞ

where W in N, PS in N/m2 , d in m, and T in K 9:7ð= ÞSW 2 Customary Metric ð23-76bÞ ð1 þ 1:5"2 ÞPS d 4 where W in kgf, PS in kgf/mm2 , d in mm, and T in 8C ð23-77Þ Hd ¼ CAðtb ta Þ where C ¼ combined coeﬃcient of radiation and convection ¼ 9:6 103 kW/m2 K (1.7 Btu/ft2 h 8F) when the bearing located in still air and oil bath ¼ 11:36 103 kW/m2 K (2 Btu/ft2 h 8F) when the bearing located in air circulation and in oil bath ¼ 15:36 103 kW/m2 K (2.7 Btu/ft2 h 8F) for average design practice ¼ 33:5 103 kW/m2 K (5.9 Btu/ft2 h 8F) when the air velocity over the bearing is 2.5 m/s (500 ft/min) A ¼ eﬀective surface area of bearing housing ¼ (25 104 dL) in m2 for bearing masses of metal as in a ring oil bearing (25 dL in in2 ) ¼ (6 104 dL) in m2 for light construction (6 dL in in2 ) tb ¼ surface temperature of bearing housing, 8C (8F) ta ¼ temperature of surrounding air, 8C (8F) CA Hd ¼ ð23-78Þ ðt ta Þ mþ1 o where m can be assumed as constant which depends on the lubrication system and it is taken from Table 23-17a, to is lubricant ﬁlm temperatures, 8C. C and A are as given under Eq. (23-67a) Hd ¼ kðLdÞðtb ta Þ ¼ kA T Customary Metric ð23-79aÞ where A ¼ Ld ¼ projected area of bearing, cm2 k T ¼ kðtb ta Þ values can be taken from Fig. 23-45(a) and Hd in kcal/min Hd ¼ 697:8kðtb ta ÞðLdÞ SI ð23-79bÞ where kðtb ta Þ in kcal/min cm2 , values are taken from Fig. 23-45(a); ðLdÞ in m2 and Hd in J/s ¼ k < dðtb ta Þ USCS ð23-68cÞ where kðtb ta Þ in ft-lbf/min/in2/8F values can be taken from Fig. 23-45(b) T ¼

Heat dissipated by self-contained bearings

TABLE 23-17a Values of factor, m in Equation (23-67b) Lubrication system

Conditions

Values of m

Oil ring

Moving air Still air Moving air Still air

1-2 1 2-1 1 2-1 1 2 5-5

Oil bath

Another formula for the heat dissipated in bearing in terms of average lubricant oil ﬁlm temperature

The heat dissipating capacity of bearing based on projected area of bearing

Pederson’s equation for heat radiating capacity of bearing due to friction between journal and bearing

Hd ¼

ðT þ 18Þ2 ðLdÞ k

SI ðUSCSÞ ð23-80aÞ

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DESIGN OF BEARINGS AND TRIBOLOGY

23.54

CHAPTER TWENTY-THREE

Particular

Formula

Hd ¼

ðT þ 33Þ2 Ld k

USCS

ð23-80bÞ

where Hd in J/s (ft-lbf/s/in2 ), Ld in in2 m2 (in2), T in 8C (8F) k ¼ 751 (3300) for bearings of light construction located in still air ¼ 7367 Customary Metric ¼ 423 (1860) for bearings of heavy construction and well ventilated ¼ 4152 Customary Metric ¼ 262 (1150) for General Electric Company’s wellventilated bearing ¼ 2567 Customary Metric TABLE 23-17 Quantities CPW , CPFm , CP , and x=B as functions of q for use in Eqs. (23-95) to (23-99) q

CPW

CPFm

CP

0.10 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 2.00 2.20 2.50 2.75 3.00 4.00 5.00 7.00 8.00 9.00

0.007209 0.012585 0.014741 0.016608 0.019618 0.021864 0.023514 0.024714 0.025597 0.026129 0.026481 0.026661 0.026707 0.026645 0.026500 0.026289 0.026026 0.025728 0.025386 0.024653 0.03870 0.022664 0.021668 0.020699 0.017257 0.014528 0.010692 0.009332 0.008225

0.955265 0.919159 0.903630 0.889234 0.864722 0.843721 0.825665 0.809936 0.796076 0.783718 0.772589 0.762470 0.753191 0.744615 0.736633 0.729156 0.722120 0.715441 0.709100 0.697225 0.686283 0.671087 0.659606 0.648392 0.609438 0.580067 0.521586

22.085010 12.172679 10.216742 8.923752 7.346332 6.431585 5.852294 5.462059 5.183394 4.999030 4.862537 4.766450 4.700334 4.657628 4.632912 4.622694 4.624350 4.634646 4.655453 4.713591 4.791810 4.935044 5.073580 5.220786 5.885901 6.654587 8.130471

0.477917

9.684235

x=B

0.533761 0.546881 0.558394 0.563687 0.568688 0.573426 0.577926 0.582209 0.586293 0.590193 0.594111 0.600937 0.607410 0.613416 0.621673 0.633787

Source: F. I. Radzimovsky, Lubrication of Bearings—Theoretical Principles and Designs, The Ronald Press Company, New York, 1959.

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DESIGN OF BEARINGS AND TRIBOLOGY

23.55

DESIGN OF BEARINGS AND TRIBOLOGY

Particular

Formula

900

17

800

13 11

Average

2

19 15

Well ventilated

3

1000

9 7 5

1

0 (a)

Thin shell not attached to radiating mass 0

10

20

30

40

50

60

70

80

90

k(tb ta), ft-ibf/min/in2/ F

4

21

1 - Thin shell not attached to large radiating mass 2 - Average industrial bearing, unventilated 3 - Well ventilated bearing

700 600 500 400

3

300

3

200

1 0

100

(b)

2

0 0

1 20 40 60

80 100 120 140 160

Temperature rise (tb

ta)

FIGURE 23-45 The rate of heat dissipated from a journal bearing.

Hd ¼

ðT þ 18Þ2 ðLdÞ 427k Customary Metric

ð23-80cÞ

where Hd in kcal/s, ðLdÞ in m , T in 8C values of k are as given inside parentheses under Eq. (23-80a) for US customary system units and values of k for customary metric units also given under Eqs. (23-80a) and (23-80b) 2

The diﬀerence in temperature (T) of the bearing and of the cooling medium can be found from the equation

The diﬀerence between the bearing-wall temperature tb and the ambient temperature ta , for three main types of lubrication by oil bath, by an oil ring, and by waste pack or drop feed

BEARING CAP The bearing cap thickness

ðT þ 18Þ2 ¼ K 0 Pv

SI ðMetricÞ ð23-81Þ

where P in N/m (kgf/mm ), v in m/s, and T in K (8C) 2

2

K 0 ¼ 0.475 (4:75 106 ) for bearings of light construction located in still air ¼ 0.273 (2:7 106 ) for bearings of heavy construction and well ventilated ¼ 0.165 (1:65 106 ) for General Electric Company’s well-ventilated bearing t0 tb Refer to Fig. 23-46 for tb ta ’ 2

hc ¼

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3Wa 2L

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ð23-82Þ

DESIGN OF BEARINGS AND TRIBOLOGY

23.56

CHAPTER TWENTY-THREE

Particular

Formula

Drop feed

ga vin ir

Mo

air

Sti

ir

ll a ir

Oil bath

St

40

Mo ving

Stil

l ai

r

50

Oil ring

ill a

60

Moving air

Oil film temperature rise above ambient, to ta, C

70

30

20

10

0

0

10 20 30 Temperature rise of wall above ambient tb ta, C

40

The deﬂection of the cap

The thickness of cap from Eq. (23-71)

FIGURE 23-46 Relation between oil ﬁlm temperature and bearing wall temperature.

y¼

Wa3 4ELh3c

ð23-83Þ

sﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 W hc ¼ 0:63a ELy

ð23-84Þ

where the deﬂection should be limited to 0.025 mm (0.001 in)

EXTERNAL PRESSURIZED BEARING OR HYDROSTATIC BEARING: JOURNAL BEARING (Fig. 23-47) The pressure in the lower pool of quadrant 1 (Fig. 23-47)

P1 ¼ K1 Po

ð23-85aÞ

where K1 ¼

4 1þ

Po 1 P0

1

2 3 þ 2:121" þ 1:93" 0:589" 4 ð23-85bÞ

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.57

Formula

L 3

l1

3

l1

Oil pocket

hmax

b

w 2

d

4

e hmin

t

h e

1

Hydraulic resistance

Constant pressure oil manifold (a)

1

Oil inlet hole (b)

Ps

Oil inlet

l1

(c)

FIGURE 23-47 (a) and (b) schematic diagram of a full cylindrical hydrostatic bearing; (c) oil pressure distribution along the bearing. [Shaw and Macks10 ]

The pressure in the upper pool of quadrant 3 (Fig. 23-47)

P 3 ¼ K3 P o

ð23-86aÞ

where K3 ¼

4 1þ

Po 1 P0

1

þ 2:121" þ 1:93"2 þ 0:589"3 4 ð23-86bÞ

The pressure in the left pool of quadrant 2 (Fig. 23-47)

P 2 ¼ K2 P o where 1 P0 K2 ¼ ð6:283 þ 3:425"2 Þ 8 Po

The pressure in the right pool of quadrant 4 (Fig. 23-47)

The ﬂow of lubricant through the lower quadrant 1 of the bearing from the manifold

P 4 ¼ K4 P o where 1 P0 K4 ¼ ð6:283 þ 3:425"2 Þ 8 Po

ð23-87aÞ

ð23-87bÞ ð23-88aÞ

ð23-88bÞ

3 4

Q1 ¼

d P1 CPF1 96l1

ð23-89aÞ

where CPF1 ¼

2:121" þ 1:93"2 0:589"3 4

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ð23-89bÞ

DESIGN OF BEARINGS AND TRIBOLOGY

23.58

CHAPTER TWENTY-THREE

Particular

The ﬂow of lubricant through the left quadrant 2 of the bearing from the manifold

Formula 3 4

Q2 ¼

d P2 C 768l1 PF2

ð23-90aÞ

where CPF2 ¼ 6:283 þ 3:425"2 The ﬂow of lubricant through the upper quadrant 3 of the bearing from the manifold

ð23-90bÞ

3 4

Q3 ¼

d P3 CPF3 48l1

ð23-91aÞ

where CPF3 ¼ The ﬂow of lubricant through the right quadrant 4 of the bearing from the manifold

þ 2:121" þ 1:93"2 þ 0:589"3 4

ð23-91bÞ

3 4

Q4 ¼

d P4 C 768l1 PF4

ð23-92aÞ

where

The total ﬂow of lubricant through quadrant of the bearing from the manifold assuming P2 ¼ P4 ¼ P0 (good approximation)

CPF4 ¼ CPF2 ¼ 6:283 þ 3:425"2

ð23-92bÞ

Q ¼ Q1 þ Q2 þ Q3 þ Q4

ð23-93aÞ

3 4

Q¼

d Po G 48l1

ð23-93bÞ

where G ¼ flow factor given by Eq. (23-94) The ﬂow factor in Eq. (23-81b)

G ¼ CPF1 K1 þ 18 ðCPF2 K2 þ CPF4 K4 Þ þ CPF3 K3 ¼ CPF1 K1 þ 14 CPF2 K2 þ CPF3 K3

ð23-94Þ

since K2 ¼ K4 and CPF2 ¼ CPF4 The external load on the hydrostatic journal bearing

A0 A0 W ¼ ðP1 P3 Þ A þ ¼ Po A þ FPFW 2 2 ð23-95Þ where FPFW ¼ load factor given by Eq. (23-95)

The load factor

FPFW ¼ K1 K3

ð23-96Þ

The pressure ratio connecting the dimensions of the bearing and its external resistances

3 Po d c lc ¼ 1 þ 6 P0 dc dc l1

ð23-97Þ

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.59

Formula

IDEALIZED SLIDER BEARING (Fig. 23-48) Plane-slider bearing The pressure at any point x w U

zy x 1 x

F x

h

L

2 B

FIGURE 23-48 Plane slider bearing with an angle of inclination.

The load carrying capacity

U C B s1 where

P¼

Cs1 ¼ ¼ W¼

ð23-98aÞ

6x1 ð1 x1 Þ ð 2aÞða þ x1 Þ2 h2 h1 ; B

a¼

h2 ; B

x1 ¼

ð23-98bÞ x B

ð23-98cÞ

6UL Cs2 2

ð23-99aÞ

where Cs2 ¼ ln The resultant shear stress at any point along the slider (Fig. 23-48)

¼

a 2 þ a 2a

U C B s3

ð23-100aÞ

where Bða þ x1 Þ 2y Cs3 ¼ B 3ða þ x1 2ax1 Þ ð 2aÞða þ x2 Þ3 1 þ a þ x1 The shear stress at any point on the surface of the moving member of the bearing (i.e., slider at y ¼ 0) (Fig. 23-48)

U C B s4 where

m ¼

Cs4 ¼

ð23-99bÞ

ð23-100bÞ ð23-101aÞ

4 6aða Þ a þ x1 ð2a Þða þ x1 Þ2 ð23-101bÞ

The shear stress at any point on the surface of the stationary member of the bearing (i.e., shoe at y ¼ h) (Fig. 23-48)

U C B s5 where

s ¼

Cs5 ¼ ¼

ð23-102aÞ

2 6aða Þ a þ x1 ð2a Þða þ x1 Þ2

h2 h1 B

and h ¼ Bða þ x1 Þ

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ð23-102bÞ

DESIGN OF BEARINGS AND TRIBOLOGY

23.60

CHAPTER TWENTY-THREE

Particular

The frictional force on the moving member of the bearing (i.e., slider)

Formula

Fm ¼ ULCs6 where 4 Cs6 ¼ ln

The frictional force on the stationary member of the bearing (i.e., shoe)

The distance of the pressure center from the origin of the coordinates, i.e., from the lower end of the shoe (Fig. 23-48)

a a

6 2a

Fs ¼ ULCs7 where 2 Cs7 ¼ ln

The coeﬃcient of friction

ð23-103aÞ

a 4

ð23-103bÞ ð23-104aÞ

6 ð23-104bÞ 2a a 2ð2a Þ ln 32 Fm a ¼ ð23-105Þ ¼ a W 3ð2a Þ ln þ 6 a 2 3 a 2 2:5 ða Þð3a Þ þ 3a 6 7 6 7 a 7B x ¼ 6 4 5 a 2 2 ð 2aÞ ln a þ

ð23-106Þ

Pivoted-shoe slider bearing (Fig. 23-48 and Fig. 23-52) The load-carrying capacity

W¼

6ULB2 CPW h22

ð23-107aÞ

where CPW ¼

1 2 lnð1 þ qÞ qðq þ 2Þ q2

ð23-107bÞ

Refer to Table 23-17 for CPW . The frictional force on the moving member of the bearing (i.e., slider)

FmP ¼

ULB CPFm h2

ð23-108aÞ

where 4 6 CPFm ¼ lnð1 þ qÞ q 2þq

ð23-108bÞ

Take CPFm from Table 23-17 for various values of q. The frictional force on the stationary member of the bearing (i.e., shoe)

FsP ¼

ULB CPFs h2

ð23-109aÞ

where 2 ð1 þ qÞa 6 CPFs ¼ ln þ q 2 2þq

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ð23-109bÞ

DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

The coeﬃcient of friction

23.61

Formula

¼

FmP h2 ¼ W B

1 CPFm 6 CPW

¼

h2 C B P

ð23-110Þ

where CP ¼ coefficient of friction factor The distance of the pivoted point from the lower end of the shoe (Fig. 23-39), i.e., the distance of the pressure center from the origin of the coordinates

Take CP from Table 23-17 for various values of q. ð1 þ qÞð3 þ qÞ lnð1 þ qÞ qð2:5q þ 3Þ x ¼ B qðq þ 2Þ lnð1 þ qÞ 2q2 ð23-111Þ The ratios x=B are taken from Table 23-17.

DESIGN OF VERTICAL, PIVOT, AND COLLAR BEARING Pivot bearing (Figs. 23-49, 23-50, and 23-53) FLAT PIVOT The total axial load on the ﬂat pivot with extreme diameters of the actual contact d1 and d2

W ¼ p

d12 d22 4

The friction torque based on uniform intensity of pressure with extreme diameters of the actual contact d1 and d2

Mt ¼ 13 W

The friction torque based on uniform wear with extreme diameters of the actual contact d1 and d2

Mt ¼ W

The power absorbed by friction with d as the diameter of ﬂat pivot bearing

P ¼

w

w

ð23-113Þ

d1 þ d2 4

ð23-114Þ

W dn 189;090

USCS ð23-115bÞ

where P in hp, W in lbf, d in in, and n in rpm

Oil Oil groove Radial oil groove

d13 d23 d12 d22

W dn0 SI ð23-115aÞ 478 where P in kW, W in N, d in m, and n0 in rps

P ¼

d (a)

ð23-112Þ

d (b)

FIGURE 23-49 Pivot thrust bearing

CONICAL PIVOT The friction torque based on uniform intensity of pressure with extreme diameters of the actual contact d1 and d2 The friction moment which resists the rotation of the shaft in a conical pivot bearing for uniform wear

1 W d13 d23 3 sin d12 d22 where 2 ¼ cone angle of pivot, deg W d1 þ d2 Mt ¼ 4 sin

The loss of power in vertical bearing

P ¼ 6:2 108

Mt ¼

d 2 Ln02

SI

ð23-116Þ

ð23-117Þ ð23-118aÞ

where P in kW, in Pa s, d and L in m, and n0 in rps

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DESIGN OF BEARINGS AND TRIBOLOGY

23.62

CHAPTER TWENTY-THREE

Particular

Formula

P ¼ 2:35 104

1 d 2 Ln2

Customary Metric ð23-118bÞ where P in hpm, 1 in cP, d and L in cm, and n in rpm d 2 Ln2 P ¼ 2:35 107 1 Customary Metric ð23-118cÞ where P in hpm, 1 in cP, d and L in mm, and n in rpm 0 d 2 Ln2 P ¼ 2:35 106 Customary Metric ð23-118dÞ where P in hpm, 0 in kgf s/m2 , d and L in m, and n in rpm 0 d 2 Ln2 P ¼ 2:35 103

If the journal and the bearing are eccentric and the distance between their axes is ", the power loss is calculated from formula

Customary Metric ð23-118eÞ where P in hpm, 0 in kgf s/m2 , L and d in mm, and n in rpm 3:8 1 d 2 Ln2 P ¼ USCS ð23-118f Þ 3 where P in hp, 1 in cP, d and L in in, and n in rpm 6:2 108 d 2 Ln02 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ P ¼ SI ð23-119aÞ 1 ð2"Þ2 where P in kW, in Pa s, d and L in m, and n0 in rps d 2 Ln2 P ¼ 2:35 107 q1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2"Þ2 Customary Metric ð23-119bÞ where P in hpm, 1 in cP, d and L in mm, and n in rpm 0 d 2 Ln2 P ¼ 2:3 106 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 ð2"Þ2 Customary Metric ð23-119cÞ where P in hpm, 0 in (kgf s/m2 ), d and L in m, and n in rpm 3:8 d 2 Ln2 P ¼ 3 q1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ USCS ð23-119dÞ 10 1 ð2"Þ2 where P in hp, 1 in cP, L and d in in, and n in rpm

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.63

Formula

Collar bearing (Fig. 23-51) The average intensity of pressure with i collars The friction moment for each collar for uniform intensity of pressure The total friction moment for i collars for uniform intensity of pressure The friction moment for each collar for uniform rate of wear The total friction moment for i collars for uniform rate of wear The friction power in collar bearing Oil in

W 0:784ðd12 d22 Þi 1 W d13 d23 Mte ¼ 3 i d12 d22 3 1 d1 d23 Mt ¼ W 3 d12 d22 W d1 þ d2 Mte ¼ 4 i d1 þ d2 Mt ¼ W 4

P¼

P ¼

ð23-120Þ ð23-121Þ ð23-122Þ ð23-123Þ ð23-124Þ

Wðd1 þ d2 Þn0 2;292;296

SI

ð23-125aÞ

where P in kW, W in N, d in m, and n0 in rps w

d

P ¼ FIGURE 23-50a Collar thrust bearing.

Oil in

The coeﬃcient of friction for collar bearing w

The coefficient of friction for collar bearing

Wðd1 þ d2 Þn 252;120

USCSU ð23-125bÞ

where P in hp, W in lbf, d in in, and n0 in rpm ¼ 83:8

v0:5 p0:67

SI

ð23-126aÞ

where v in m/s and P in N/m2 ¼ 0:016

v0:5 p0:67

USCS ð23-126bÞ

where v in ft/min and P in psi

d2 d1

FIGURE 23-50b Plain thrust bearing.

Allowable pressure P may be taken so that Pv value for v ranging from 0.20 to 1 m/s (50 to 200 ft/min)

¼ 1:73 103

v0:5 p0:67

Customary Metric ð23-126cÞ

where v in m/s and P in kgf/mm2 Pv 707;505

SI

ð23-127aÞ

where P in Pa and v in m/s Pv 0:0715

Customary Metric ð23-127bÞ 2

where P in kgf/mm and v in m/s Pv 20;000

USCS

where P in psi and v in ft/min

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ð23-127cÞ

DESIGN OF BEARINGS AND TRIBOLOGY

23.64

CHAPTER TWENTY-THREE

Particular

Formula

PLAIN THRUST BEARING (Fig. 23-50b) W ¼ K1 ðd12 d22 Þ

Recommended maximum load

SI ðUSCSÞ ð23-128Þ

where K1 ¼ 0:3 ð48Þ, W in N (lbf), d1 and d2 in mm (in) d þ d2 0 SI ðUSCSUÞ ð23-129Þ nW P ¼ K2 1 2

Approximate power loss in bearing

where K2 ¼ 70 106 ¼ ð11 106 Þ P in W (hp), n0 in rps, and W in N (lbf) Q ¼ K3 P

Lubrication ﬂow rate to limit lubricant temperature rise to 208C

SI ðUSCSUÞ ð23-130Þ 6

where K3 ¼ 0:03 10 (0.3), Q in m3 /s (q.p.m), and P in W s (hp)

Thrust bearing Parallel-surface thrust bearing (Figs. 23-51 to 23-52)

Refer to Table 23-8 for P and Table 23-6 for Pv values.

The pressure at any point along the bearing

P¼

6UB KLP1 h2 where

KLP1 ¼ 0 ¼

x1 ln½ð 0 1Þx1 þ 1Þ ln 0

2 x ;x ¼ 1 1 B

ω w y x

w y

z

ωr r

1 z

ω

x (a)

U x

h

d

ð23-131aÞ

2 B (b)

FIGURE 23-51 Parallel-surface thrust bearing.

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ð23-131bÞ

DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.65

Formula

ad

be

ari

ng

h22 6ηUB

0.04

h2 6ηUB

Til tin

gp

0.02

ρ2 ρ1

0

y

0.2 x

0.4 0.6 Ratio of x B

0.8

1.0

h2

h

0.96 0.98

c

h1

= 0.92

z Pivot

FIGURE 23-52 Comparison of pressure distribution across tilting-pad and parallel-surface thrust bearing.

2 a ¼ 1 þ ðt2 t1 Þ 1 1

The ratio of the density of the lubricant leaving the bearing to the density of the lubricant entering the bearing

0 ¼

The unit load supported by a parallel-surface thrust bearing

Pu ¼

where a ¼ constant, a= 1 ¼ 0:0004, and t1 and t2 are the temperatures in 8C corresponding to densities 1 and 2 , respectively 6UB KLP2 h2 where

KLP2 ¼ The approximate formula for unit load supported by a parallel-surface thrust bearing

ð23-132Þ

Pu ¼

1 0 1 þ þ 2 1 ln 0

6UB KLP3 h2

where KLP3 ¼ 0:09ð1 0 Þ

ð23-133aÞ

ð23-133bÞ ð23-134aÞ ð23-134bÞ

Refer to Table 23-18 for KLP3 . The pressure distribution along a tilting-pad bearing of inﬁnite width (Figs. 23-48 and 23-52)

P¼

6UB Kpt h21

ð23-135aÞ

where Kpt ¼

ðm 1Þð1 x1 Þx1 ðm þ 1Þðm mx1 þ x1 Þ2

m ¼ h1 =h2 ; x1 ¼ x=B

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ð23-135bÞ

DESIGN OF BEARINGS AND TRIBOLOGY

23.66

CHAPTER TWENTY-THREE

Particular

Formula

TABLE 23-18 Comparison of load capacities of tilting-pad and parallel-surface-type of bearings Temperature rise through bearings, 8C

0

KLP3

Klt (for h0 ¼3)

Relative load capacity, KLP3 =Klt

10 38 93

0.98 0.96 0.92

0.0018 0.0036 0.0072

0.025

14 7 3.5

Source: F. I. Radzimovsky. Lubrication of Bearings—Theoretical Principles and Designs. The Ronald Press Company. New York, 1959.

h

W

(a)

d1

n

(b)

Po oil supply pressure

d2

m Po

n’

m’

FIGURE 23-53 (a) Hydrostatic step bearing; (b) plan view and general character of pressure distribution along diameter of the bearing.

The unit load supported by a tilting-pad bearing of inﬁnite width (Fig. 23-52)

Pu ¼

6UB Klt h22

where 1 Klt ¼ ðm 1Þ2

ð23-136aÞ

2ðm 1Þ ln m mþ1

ð23-136bÞ

OIL FILM THICKNESS The thickness of oil ﬁlm in a parallel-surface thrust bearing

The thickness of minimum oil ﬁlm at location 2 (Figs. 23-48 and 23-52)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h ¼ 6KLP3

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ UB P

Refer to Table 23-18 for KLP3 . sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ UB h2 ¼ 6Klt P Refer to Table 23-18 for Klt .

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ð23-137Þ

ð23-138Þ

DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

For properties of lubricant bearing materials and applications, conversion factors for viscosity, kinematic and Saybolt viscosity equivalents and conversion tables for viscosity equivalent

Formula

Refer to Tables 23-19 to 23-23.

The coeﬃcient of friction in case of a parallel-surface thrust bearing Another formula for coeﬃcient of friction in case of a parallel-surface thrust bearing The coeﬃcient of friction for a tilting-pad bearing of inﬁnite width

1:82 1 0

h B sﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 U ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ P 6KLP3 uB ¼

COEFFICIENT OF FRICTION

23.67

sﬃﬃﬃﬃﬃﬃﬃﬃﬃ U ¼ Kt Pu B where

ð23-139Þ

ð23-140Þ

ð23-141aÞ

31=2 m1 2 4 ln m 6 7 mþ1 7 7 m1 5 6 ln m 12 mþ1

2

6 6 Kt ¼ 6 4

m ¼ h1 =h2 ¼ film thickness ratio

ð23-141bÞ

HYDROSTATIC BEARING: STEP-BEARING (Fig. 23-53) The pressure in the pocket supplied from external source to support the load

Po ¼

The load-carrying capacity

W¼

The rate of ﬂow of lubricant through the bearing

Q¼

Power loss in bearing

8W lnðd2 =d1 Þ

ðd22 d12 Þ

Po ðd22 d12 Þ d2 ln d1

Po h3 2 lnðd2 =d1 Þ

ð23-142aÞ ð23-143Þ

ð23-144Þ

n02 4 ðd d14 Þ SI ð23-145aÞ 16h 2 where P in kW, in Pa s, h, d1 , and d2 in m, and n0 in rps

P ¼ 0:062

P ¼ 8:3 104

n0 4 ðd d14 Þ 16h 2 Customary Metric ð23-145bÞ

where P in hpm, in kgf s/mm, h, d1 , and d2 in mm, and n0 in rps

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0.885

0.895 0.910 0.904

Heavy

Steam Cylinder Oil

0.872

0.858 0.864 0.876 0.884 0.887 0.892 0.892

0.870 0.885 0.891 0.890 0.992 — —

0.900 0.934 0.930 0.937 —

0.877

10W 20W 30 40 50 60 70

75 80 90 140 250

Medium

Turbine-grade oil Light

Aircraft engine oil

Automotive oil

Transmission gear oil

Type and application

23.68

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243

12

14 1.5 15.5

235

12

111 146 215.5 224 232 249 318

65 62 18 18 18 18 7

210

210 227 238 240.5 254.5 —

26 23 20 18 12 — —

18

193 185 232 260 254.5

Flash point, 8C

23 32 23 18 15

Density, Pour g/cm3 , at point, SAE no. 15.58C 8C

TABLE 23-19 Typical properties of lubricants

101 107 103

100

105

109

87 79 106 96 95 95 96

102 96 92 90 90 80 80

121 78 91 82 83

Viscosity index

1800 3750 6470

460

300

150

43 59 350 514 829 1240 1711

190 330 530 800 1250 — —

220 320 1330 3350 5660

130 210 300

62

53

44

33 35 57 64 80 99 120

46 54 64 77 97 115 137

50 52 100 160 220

At 38 8C At 998C

Saybolt seconds, S

kgf s/m2 104

Pa s 103

390 810 1400

99

65

32

5 10 76 111 179 268 369

41 71 114 173 270 — —

47 69 287 725 1220

27 45 64

10.8

8.2

5.4

1.6 2.5 9.3 11.3 15.5 20.1 25.0

6.0 8.5 11.3 14.8 19.7 — —

7.3 7.9 20.4 34 47

8.36

5.51

1.63 2.55 9.49 11.53 15.81 20.50 25.50

6.12 8.67 11.53 15.10 20.10 — —

397.80 27.45 826.20 45.90 1428.00 65.28

100.98 11.02

66.30

32.64

5.10 10.20 77.52 113.22 182.58 273.36 376.38

41.82 72.42 116.28 176.46 275.40 — —

47.94 7.45 70.38 8.06 292.74 20.81 739.50 34.68 1244.40 47.94

390 810 1400

99

65

32

5 10 76 111 179 268 369

41 71 114 173 270 — —

47 69 287 725 1220

27 45 64

10.8

8.2

5.4

1.6 2.5 9.3 11.3 15.5 20.1 25.0

6.0 8.5 11.3 14.8 19.7 — —

7.3 7.9 20.4 34 47

g

g

Railroad stationary steam engines cylindered applications, enclosure gears

Direct-connected turbines electric motors Land-geared turbines electric-motors Marine-propulsion geared turbines

Various reciprocating aircraft engines

Turbojet engines

Automobile, truck and marine reciprocating engines; very-heavy-duty oils used in diesel engines

Combination pinion reduction gear units, enclosed reduction gear sets

At 388C At 998C At 388C At 998C At 388C At 998C Uses

Centipoise, cP

Viscosity

DESIGN OF BEARINGS AND TRIBOLOGY

0.887 0.895 0.901

0.844

0.895 0.898 0.909 0.902

0.881 0.898 0.915 0.915 0.890

Hydraulic oils Light Medium Heavy

Extra-low-temperature

Refrigerating machine oil

Machine tools and general-purpose oil

Type and application

188 207 257

110 146 165.5 182 190.5 177 199 185 199 235

24 45.5 37 29 23 4 4 12 15 9

Flash point, 8C

42 26 12

Density, Pour g/cm3 , at point, SAE no. 15.58C 8C

TABLE 23-19 Typical properties of lubricants (Cont.)

80 80 83 25 80

53 22 34 35

226

64 66 70

Viscosity index

105 205 305 510 930

72 195 235 335

74

150 310 910

39 46 49 59 80

36 43 45 49

43

42 50 74

At 38 8C At 998C

Saybolt seconds, S

kgf s/m2 104

Pa s 103

22 44 66 110 200

14 42 51 72

14

32 67 196

3.9 6.0 7.0 9.9 15.5

2.9 5.1 5.7 7.0

5.2

4.8 7.3 14.0

22.44 44.88 67.32 112.20 204.00

14.28 42.84 52.02 73.44

14.28

3.98 6.12 7.14 10.10 15.81

2.96 5.20 5.81 7.14

5.30

32.64 4.90 68.34 7.45 199.92 14.28

22 44 66 110 200

14 42 51 72

14

32 67 196

3.9 6.0 7.0 9.9 15.5

2.9 5.1 5.7 7.0

5.2

4.8 7.3 14.0

All general-purpose lubrication, machine tools

Ammonia compressor

Aircraft hydraulic systems

Hydraulic ﬂuids for most indoor industrial hydraulic equipments Heavier loads, higher temperature

At 388C At 998C At 388C At 998C At 388C At 998C Uses

Centipoise, cP

Viscosity

DESIGN OF BEARINGS AND TRIBOLOGY

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23.69

Composition, %

23.70

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Sintered iron

Iron base Gray cast iron

Bronze, high-lead Bronze, lead tin Bronze, 80–10–10 Bronze, nickel tin Bronze, aluminum Bronze, zinc

Bronze, high-lead

Copper alloys Clock brass

Cu 7.5, Fe 92.5

C 3.5, Si 2.5, Fe 94.0

Pb 3.0, Zn 35.5, Cu 61.5 Sn 4.0, Pb 14.0, Zn 1.5, Ni 1.0 max, Cu 79.5 Sn 16, Pb 14.0, Cu 70.0 Sn 8, Pb 3.5, Zn 3.5, Cu 85 Sn 10.0, Pb 10.0, Cu 80 Sn 10.0, Ni 3.5, Pb 2.5, Cu 84.0 Al 10.5, Fe 3.5, Cu 86.0 Al 1.0, Si 0.8, Mn 2.5, Zn 37.5, Cn 58.2

Sn 10.0, Sb 15.0, Pb 75.0 Tin base Cu 8.3, Sb 8.3, Sn 83.4 Cadmium base Ni 1.4, Cd 98.6 Aluminum Cu 1.0, Sn 6.5 alloys Ni 1.5, Al 91.0

Babbitts Lead base

Material

0.34 0.33

8.6 2.86

0.15 0.37 0.52 0.39

8.86 — 7.6 8.09

—

0.30

0.37

0.26

8.4

7.2

0.37

0.15

—

—

—

0.28

7.47

8.4

0.34

Dry coeﬃcient of friction,

9.69

Speciﬁc gravity

TABLE 23-20 Journal bearing materials and applications

—

21.10 min

49.22 min

70.30

17.58 min 31.64

21.1 min

—

38.0– 45.0 14.0

— 15.5

7.88

7.03

kgf/mm2

—

207.00 min

482.84 min

689.74

207.00 min 172.46 min 310.04

—

372.48– 445.45 138.00

— 151.76

77.30

68.96

MPa

Ultimate tensile strength, su

—

1.898

1.055

1.125

—

0.773

—

—

1.055

— 0.724

0.534

0.295

kgf/mm2 104

—

186.2

103.5

110.4

—

75.8

—

—

103.5

— 71.0

52.4

28.9

GPa

Modules of elasticity, E

—

180

202

95

65

53

—

45

54–142

— 45

27

45

Brinell

—

—

B 80–92 —

—

—

—

—

—

—

B 40–75

— —

—

—

Rockwell

Hardness numbers

Used in refrigerators, compressors, camshafts, high load at low speed with good lubrication Used with impregnates with oil will give good results; load—3.4 MPa (0.35 kgf mm2 ) and speed 0.67 m/s

General-duty bearing bronze; load up to 20.6 MPa (2.1 kgf/mm2 ); speed—4.5 m/s Used in medium- to heavy-duty application; good strength requirement Used in heavy-duty bearings requiring high strength and good impact resistance Heavy-duty impact loadings; on hardened shaft

Moderately heavy duty

Same as above; can withstand higher loads

Used in poorly lubricated applications with moderately heavy loads

Used for light load

Used in automobiles and electrical equipment Used in automotive and diesel engines, steam turbines and motors Used where lubrication is intermittent Used in high-temperature high-load services and in diesels; requires good lubrication and hardened shaft

Applications

DESIGN OF BEARINGS AND TRIBOLOGY

15.1

C þ Cu þbinder

Tungsten carbide 97.0, Co 3.0

0.18

1.36

P cP kgf s/m2 kg/m s lbf s/ft2 lb/ft s Pa s

cP

100 1 9.80665 103 103 4.788 104 1.4882 103 103

P

1 0.01 98.0665 10 4.788 102 14.882 10

TABLE 23-21 Conversion factors for viscosity

7.03

1.40– 10.55 2.11

7.03

573.00 (compressive)

2.11– 4.22

0.53 1.80

kgf/mm2

0.1 103 9.80665 1 47.88 1.4882

kg/m s

68.96

13.73– 103.50 20.70

68.96

5621 (compressive)

20.7– 41.4

5.17– 17.25

MPa

Ultimate tensile strength, su

0.0102 1.0297 104 1 0.102 4.8824 0.1518 0.102

kgf s/m2

0.17

2.2

Polytetraﬂuoroethylene Textolite 2001 Phenolic, graphite and cotton cloth

Teﬂon

0.25–0.30

0.97–2.00

0.86

0.20

0.17

0.15

Rubber

Plastics and rubber Nylon Polyamide

Wood

1.44

2.9/3.8

C þ binder

Graphite Carbon graphite

Carbon graphite and metal Cemented carbide

1.63–1.86

Composition, %

Material

Speciﬁc gravity

Dry coeﬃcient of friction,

TABLE 23-20 Journal bearing materials and applications (Cont.)

Rockwell

0.0672 6.7197 6.5898 0.6720 32.174 1

2.0886 103 2.0886 105 0.20482 2.0886 102 1 0.0311

M 100

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0.1 103 9.80665

Pa s

Shore scleroscope 50

M 90

C 80

Shore scleroscope 75

Shore scleroscope 75

Brinell

lb/ft s

4.375– 6.35

0.410

2.25

672.5

—

—

GPa

Hardness numbers

lbf s/ft2

0.0443 0.0647

0.0042

0.023

6.885 (compressive)

—

—

kgf/mm2 104

Modules of elasticity, E

Used in many household appliances and other lightly loaded applications; requires little lubrication Marine propellers, pumps, turbine, load 0.54 MPa (0.055 kgf/mm2 ) Useful in corrosive conditions; dairy, textile, and food machinery Used where low wear and good compatibility characteristics are required

Used in high-speed precision grinders which require perfect alignment and good lubrication; can withstand extreme loading and high speeds Used in conveyors; light loads at high speeds under 658C

Particularly suited to high-temperature application (<4558C) where lubrication is diﬃcult; used in electric motors, conveyors Same as above, higher strength

Applications

DESIGN OF BEARINGS AND TRIBOLOGY

23.71

DESIGN OF BEARINGS AND TRIBOLOGY

23.72

CHAPTER TWENTY-THREE

TABLE 23-22 Kinematic and Saybolt viscosity equivalents Kinematic viscosity, Saybolt viscosity, S a

Metric units cSt

cm2 =s 102

SI units m2 /s 106

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 23 25 27 29 30 31 33 35 37 39 40 41 43 45 47 49 50 55 60 65 70 >70

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 23 25 27 29 30 31 33 35 37 39 40 41 43 45 47 49 50 55 60 65 70

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 23 25 27 29 30 31 33 35 37 39 40 41 43 45 47 49 50 55 60 65 70

a

At 388C

At 998C

32.6 36.0 39.1 42.4 45.6 48.8 52.1 55.5 58.9 62.4 66.0 69.8 73.6 77.4 81.3 85.3 89.4 93.6 97.8 102.0 110.7 119.3 128.1 136.9 141.3 145.7 154.7 163.7 172.7 181.8 186.3 190.8 199.8 209.1 218.2 227.5 232.1 255.2 278.3 301.4 324.4 S ¼ cSt 4:635

32.9 36.3 39.4 42.7 45.9 49.1 52.5 55.9 59.3 62.9 66.5 70.3 74.1 77.9 81.3 85.9 90.1 94.2 98.5 102.8 111.4 120.1 129.0 137.9 142.3 146.8 155.8 164.9 173.9 183.0 187.6 192.1 201.2 210.5 219.8 229.1 233.8 257.0 280.2 303.5 326.7 S ¼ cSt 4:667

S ¼ cSt 4:635 at 388C; S ¼ cSt 4:667 at 998C

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

23.73

TABLE 23-23 Conversion table for viscosity equivalents Kinematic viscosity, Metric units

Viscosity

cSt

cm2 =s 102

SI units m2/s 106

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 32 34 36 38

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 32 34 36 38

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10 11 12 13 14 15 16 17 18 19 20 21 24 26 28 30 32 34 36 38

Saybolt, S

Engler 8

Redwood no. 1

32.60 33.40 34.10 34.80 35.40 36.00 36.70 37.30 37.90 38.50 39.1 40.8 42.4 44.0 45.6 47.2 48.8 50.4 52.1 53.8 55.5 57.2 58.9 62.4 66.0 69.7 73.5 77.3 81.2 85.2 89.3 93.4 97.6 106.1 114.7 123.4 132.3 141.1 149.9 158.9 167.9 176.9

1.12 1.14 1.16 1.18 1.20 1.22 1.23 1.25 1.27 1.29 1.31 1.35 1.40 1.44 1.48 1.52 1.56 1.60 1.65 1.70 1.75 1.79 1.84 1.94 2.02 2.12 2.22 2.32 2.43 2.54 2.64 2.75 2.87 3.10 3.33 3.57 3.82 4.07 4.32 4.57 4.82 5.08

30.8 31.3 31.8 32.3 32.8 33.3 33.8 34.3 34.8 35.3 35.8 37.0 38.3 39.6 40.9 42.3 43.6 44.9 46.3 47.7 49.0 50.5 51.9 54.9 58.0 61.2 64.5 67.9 71.3 74.8 78.4 82.0 85.7 93.2 100.8 108.5 116.3 124.2 132.1 140.0 147.9 155.9

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DESIGN OF BEARINGS AND TRIBOLOGY

23.74

CHAPTER TWENTY-THREE

TABLE 23-23 Conversion table for viscosity equivalents (Cont.) Kinematic viscosity, Metric units

Viscosity

cSt

cm2 =s 102

SI units m2/s 106

40 42 44 46 48 50 55 60 65 70 75 80 85 90 95 100 110 120 130 140 150 160 170 180 190 200 220 240 260 280 300 320 340 360 380 400 500 600 700 800 900 1000

40 42 44 46 48 50 55 60 65 70 75 80 85 90 95 100 110 120 130 140 150 160 170 180 190 200 220 240 260 280 300 320 340 360 380 400 500 600 700 800 900 1000

40 42 44 46 48 50 55 60 65 70 75 80 85 90 95 100 110 120 130 140 150 160 170 180 190 200 220 240 260 280 300 320 340 360 380 400 500 600 700 800 900 1000

Saybolt, S

Engler 8

Redwood no. 1

186.0 195.0 204.0 213.0 222.0 232.0 255.0 278.0 301.0 324.0 347.0 370.0 393.0 416.0 439.0 463.0 509.0 555.0 602.0 648.0 694.4 740.0 787.0 833.0 879.0 926.0 1018.0 1111.0 1203.0 1296.0 1388.0 1480.0 1574.0 1666.0 1759.0 1851.0 2314.0 2777.0 3239.0 3702.0 4165.0 4628.0

5.33 5.59 5.84 6.10 6.36 6.62 7.26 7.90 8.55 9.21 9.87 10.53 11.19 11.85 12.51 13.16 14.47 15.80 17.11 18.43 19.75 21.05 22.38 23.70 25.00 26.32 28.95 31.60 34.25 36.85 39.50 42.12 44.75 47.40 50.00 52.65 65.80 79.00 92.20 105.30 118.50 131.60

164.0 172.0 180.0 188.0 196.0 204.0 225.0 245.0 265.0 286.0 306.0 326.0 346.0 367.0 387.0 407.0 448.0 489.0 529.0 570.0 611.0 651.0 692.0 733.0 774.0 815.0 896.0 978.0 1059.0 1140.0 1222.0 1303.0 1385.0 1465.0 1546.0 1628.0 2036.0 2443.0 2850.0 3258.0 3668.0 4074.0

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.75

Formula

SPHERICAL BEARINGS (Fig. 23-54) Wr2 þ 6Wa2 Wr BD

Equivalent bearing pressure (Fig. 23-54)

p¼

Maximum bearing pressure if an average bearing life of 105 number of oscillations is to be expected

po ¼

Bearing life (Fig. 23-54) Wr

φ

D

φ

Bearing type

As a rule ϕ <8

ð23-146bÞ ð23-146cÞ

D

PTFE. FIBRE OR IMPREGNATED METAL

HARDENED STEEL

L ¼ bearing life, i.e. average number of oscillations to failure assuming unidirectional loading f ¼ life-increasing factor depending on periodical relubrication 10–15 for hardened steel on hardened steel 1 for PTFE ﬁber or impregnated metal on hardened steel 5–10 for d > 0:05 m bronze on hardened steel po ¼ maximum allowable bearing pressure, assuming unidirectional dynamic loading and no re-lubriBRONZE cation ¼ 24a MPa (3500 psi) for hardened steel on hardened steel ¼ 97 MPa (14000 psi) for PTFE ﬁber or impregHARDENED nated metal on hardened steel and temperature STEEL up to 2808C (5368 F) ¼ 10a MPa (1450 psi) bronze on hardened steel and temperature up to 1008C (2128 F) Butternl ¼ average number of oscillations to failure ¼ 105 nr ¼ recommended interval between re-lubrication in number of oscillations < 0.3nl for hardened steel on hardened steel < 0.3nl (usually) for bronze on hardened steel Wmax

B HARDENED STEEL

Wmax for n1 ¼ 105 BD 3 po L¼f 105 p

ð23-146aÞ

where B

Wa

provided Wa < Wr

HARDENED STEEL

FIGURE 23-54 Spherical bearings Courtesy: Neale, M. J., Tribology Handbook, Newnes and worths

a

The ﬁgures given above are based on dynamic load conditions. For static load conditions, where the load-carrying capacity of the bearing is based on bearing-surface permanent deformation, not fatigue, the load capacity of steel bearings may reach 10 po and of aluminum bronze 5 po . Ability to carry alternating loading is 1:7 po for metal contact; and is reduced by 0:25 po for DTFE ﬁbre on hardened steel.

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DESIGN OF BEARINGS AND TRIBOLOGY

23.76

CHAPTER TWENTY-THREE

Particular

Load carrying capacity of spherical step bearing

Formula

W¼

Po d22 ðcos 1 cos 2 Þ tanð 2 =2Þ ln tanð 1 =2Þ

ð23-146dÞ

LIFT Inlet pressure

" # 48Q eð4 e2 Þ ð2 þ e2 Þ 1þe þ arctan pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Pi ¼ 3 2 l d 2ð1 e2 Þ2 ð1 e2 Þ5=2 1 e2 ð23-147Þ

Load-carrying capacity of bearing

"

W¼

24Q 2 þ 3e e3 3 2 d ð1 e2 Þ2

#

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ð23-148Þ

DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

23.2 ROLLING CONTACT BEARINGS1 SYMBOLS a1 ; a2 ; a3 b B c C Ca Ca1 , Ca2 ; . . . ; Can Cn Co Coa d db di do dr d1 , d2 D D1 Dw e E f fa fc

fd fk fL fn fs fnt fo foa F

life adjustment factors, Eq. (23-185a), (23-185b) Weibull exponent width of bearing, m (in) permissible increase in diametral clearance, (mm) basic dynamic load rating for radial and angular contact ball or radial roller bearings, kN (lbf ) basic dynamic load rating for single-row, single- and double-direction thrust ball or roller bearings, kN (lbf ) basic load rating per row of a one-direction multi-row thrust ball or roller bearing, each calculated as single-row bearing with Z1 , Z2 ; . . . ; Zn balls or rollers, respectively capacity of the needle bearing, kN (lbf ) basic static load rating for radial ball or roller bearing, kN (lbf ) basic static load rating for thrust ball or roller bearings, kN (lbf ) bearing bore diameter, m (in) diameter of ball, m (in) shaft or outside diameter of inner race used in Eqs. (23-246) and (23-247), m (in) inside diameter of outer race of needle bearing, m (in) roller diameter (mean diameter of tapered roller), m (in) diameter of needle roller, m (in) diameter of spherical balls or cylindrical rollers used in contact stress [Eqs. (23-250) to (23-253)], m (in) outside diameter of bearing, m (in) diameter of revolving race, m (in) diameter of ball, mm bearing constant modulus of elasticity, GPa (psi) a factor use in Eq. (23-155) application factor to compensate for shock continuous duty or inequality of loading a factor which depends on the geometry of the bearing components, the accuracy to which the various bearing parts are made and the material used in Eqs. (23-187), (23-188), and (23-199) to (23-202); a factor which depends on the units used, the exact geometrical shape of the load-carrying surfaces of the roller and rings (or washers in case of thrust bearing), and the accuracy to which the various bearing parts are made and the material, used in Eqs. (23-207), (23-208) a factor for the additional forces emanating from the mechanisms coupled to the gearing used in Eq. (23-154) a factor for the additional forces created in the gearing itself used in Eq. (23-154) index of dynamic stressing speed factor for ball bearings according to Table 23-37 speed factor for roller bearings according to Table 23-38 index of static bearing speed factor used in tapered roller bearing a factor used in Eqs. (23-161) and (23-167) a factor used in Eqs. (23-152) and (23-154) load, kN (lbf )

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23.77

DESIGN OF BEARINGS AND TRIBOLOGY

23.78

Fa Faa Far Fbs Fc Fe Feffg Fna Fnt Fr i k Ka Kh Kt Kn l leff

L LB10 L10 Lh Mt n n0 ne ni nl nm n1 n2 P P Pa Pm Pmax Pmin Po Poa qi R10 X

CHAPTER TWENTY-THREE

theoretical tooth load, kN (lbf ) thrust load, kN (lbf ) applied thrust load, kN (lbf ) thrust component of pure radial load F, due to tapered roller, kN (lbf ) shaft load due to belt drive, kN (lbf ) static load, kN (lbf ) radial equivalent load from combination of radial and thrust loads or eﬀective radial load, kN (lbf ) eﬀective tooth load, kN (lbf ) net thrust load, kN (lbf ) net thrust load on the tapered roller bearing, kN (lbf ) radial load capacity of ball bearing, kN (lbf ) radial bearing load, kN (lbf ) number of rows of balls in any one bearing constant used in Eqs. (23-156), (23-158) to (23-160) application factor, Eq. (23-186) hardness factor used in Eq. (23-247) life load factor taken from the curve in Fig. 23-55 marked ‘‘T-needle’’ and used in Eq. (23-247) a constant used in Eq. (23-152) and Eq. (23-153) length of needle bearing, m (in) the eﬀective length of contact between one roller and that ring (or that washer in case of thrust bearing) where the contact is the shortest (overall roller length minus roller chamfers or minus grinding undercuts), m (in) life of bearing at constant speed, rpm life of bearing at constant speed, h life corresponding to desired reliability, R, used in Eq. (23-194) life factor corresponding to desired B-10 hours of life expectancy used in Eq. (23-195) rating life fatigue life torque, N m (lbf in) speed, rpm speed, rps eﬀective speed, rpm ith speed, rpm limiting speed, rpm mean speed, rpm speed of the inner race, rpm speed of the outer race, rpm power, kW (hp) equivalent dynamic load, kN (lbf ) equivalent dynamic thrust load, kN (lbf ) mean load, kN (lbf ) maximum load, kN (lbf ) minimum load, kN (lbf ) static equivalent load, kN (lbf ) static equivalent load for thrust ball or roller bearings under combined radial and thrust loads, kN (lbf ) percentage time of ith speed 0.90 reliability corresponding to rating life radial factor used in Eqs. (23-177b), (23-182), (23-190), (23-210), and (23-180)

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Xo Y Yo Z

Z1 , Z2 ; . . . ; Zn

!

cðmaxÞ max

radial factor used in Eqs. (23-162), (23-165), (23-173c) and (23-157) Tables (23-37), (23-38), (23-39) thrust factor used in Eqs. (23-163), (23-166), (23-173), (23-178), and (23-180) thrust factor used in Eqs. (23-162), (23-165), and (23-157) number of balls per row number of balls carrying thrust in one direction number of rollers per row number of rollers carrying thrust in single-row one-direction bearing number of needle-rollers number of balls or rollers in respective rows of one-direction multi-row bearings nominal angle of contact, that is, nominal angle between the line of action of the ball load and a plane perpendicular to the bearing axis the angle of contact, that is, the angle between the line of action of the roller resultant load, and a plane perpendicular to the bearing axis angular speed, rad/s coeﬃcient of friction Poisson’s ratio maximum compressive stress, MPa (psi) maximum shear stress, MPa (psi)

Particular

The torque

23.79

Formula

9550P SI ð23-149aÞ n where P in kW, n in rpm, and Mt in N m

Mt ¼

Mt ¼

1000P !

SI ð23-149bÞ

where P in kW, ! in rad/s, and Mt in N m 159:2P SI ð23-149cÞ n0 where P in kW, n0 in rps, and Mt in N m

Mt ¼

Mt ¼

63;000P n

USCS ð23-149dÞ

where P in hp, n in rpm, and Mt in lbf in 716P Customary Metric ð23-149eÞ n where P in hpm, n in rpm, and Mt in kgf m

Mt ¼

937P Customary Metric ð23-149f Þ n where P in kW, n in rpm, and Mt in kgf m

Mt ¼

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DESIGN OF BEARINGS AND TRIBOLOGY

23.80

CHAPTER TWENTY-THREE

Particular

Formula

TABLE 23-32 Coeﬃcient of friction for rolling contact bearings

TABLE 23-33 Safe working values of k for average bearing life

Type of bearing

Material

k 106

Self-aligning bearings Cylindrical roller bearings Thrust ball bearings Angular contact ball bearings Deep groove ball bearings Tapered roller bearings Spherical roller bearings Needle bearings

0.0016–0.0066 0.0012–0.0060 0.0013 0.0018–0.0019 0.0022–0.0042 0.0025–0.0083 0.0029–0.0071 0.0045

For un-hardened steel 3.80 For hardened alloy steel on ﬂat races 6.89 For hardened carbon steel 4.80 For hardened carbon steel on grooved 10.34 races For hardened alloy steel grooved races 13.79 (having radius ¼ 0.67 do )

USCSU* 550 1000 700 1500 2000

* US customary system units.

The equation for friction torque

Mt ¼

Fr d 2

ð23-149gÞ

For values of , refer to Table 23-32.

ROLLING ELEMENTS BEARINGS Deﬁnition, Dimensions and Nomenclature For nomenclature, other details and deﬁnition of a ball bearing

Refer to Fig. 23-49.

For nomenclature, other details and deﬁnition of a taper roller bearing

Refer to Fig. 23-50.

A rule of thumb used for ordinary ball and straight roller bearings

d þD 0 n 8:33 2 where d, D in m and n0 in rps d þD n 500;000 2

ð23-150aÞ

ð23-150bÞ

where d, D in mm and n in rpm

SPEED Eﬀective speed The eﬀective speed which determines the life of the bearing is found from the relation

n ¼ n1 n2

ð23-151Þ

where the plus sign is used when the races rotate in opposite directions and the minus sign is used when the races rotate in the same direction

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Width

23.81

Snap ring groove Corner radius

Seal or shield groove

Shoulders

Seal or shield notch Corner radius Inner ring raceway

Outside diameter

Bore

Inner ring Inner ring land

Separator

Inner ring face

Outer ring raceway Outer ring land

Outer ring face

Outer ring

FIGURE 23-49 Ball bearing nomenclature. (Courtesy: New Departure-Hyatt Bearing Division, General Motors Corporation.)

Cup back face radius, r

Bearing width Cup length

Cup back face

Cone back face rib

Cage

Cone back face

Cone front face rib Cone length

Cone bore

Cup outside diameter

Cup front face radius Cup front face

Cone front face radius Cone

Cone front face

Roller

Cone back face radius

Cup

Standout FIGURE 23-50 Nomenclature of tapered roller bearing.

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DESIGN OF BEARINGS AND TRIBOLOGY

23.82

CHAPTER TWENTY-THREE

Particular

Formula

Limiting bearing speed The limiting bearing speed when the bearing outside diameter is less than 30 mm

nl ¼

The limiting bearing speed when the bearing outside diameter is 30 mm and over

nl ¼

Kn D 10 For values of Kn refer to Table 23-34.

ð23-153Þ

Feffg ¼ fk fd F

ð23-154Þ

3Kn D þ 30

ð23-152Þ

GEAR-TOOTH LOAD The eﬀective tooth load which is used in design of bearings The shaft load due to belt drive which is used in design of bearings

For values of fk and fd refer to Table 23-35. Fbs ¼ fF

ð23-155Þ

For values of f refer to Table 23-35.

TABLE 23-34 Values of Kn to be used in Eqs. (23-l52) and (23-153) Constant, Kn Type of bearing Radial bearings Deep groove bearings Single row Single row with leeds Double row Magneto bearings Angular contact ball bearing Single row Single row paired Double row Self-aligning ball bearings Self-aligning bail bearings with extended inner ring Cylindrical roller bearing Single row Double row Tapered roller bearings Barrel roller bearings Spherical roller bearings Series 213 Thrust bearings Thrust ball bearings Angular contact thrust ball bearings Cylindrical roller thrust bearings Spherical roller thrust bearings

Grease lubrication

Oil lubrication

500,000 360,000 320,000 500,000

630,000 — 400,000 630,000

500,000 400,000 360,000 500,000 250,000

630,000 500,000 450,000 630,000 320,000

500,000 500,000 320,000 220,000 220,000

630,000 630,000 400,000 280,000 280,000

140,000 220,000 90,000 140,000

200,000 320,000 120,000 200,000

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.83

Formula

TABLE 23-35 Value of factors fk , fd and f to be used in Eqs. (23-154) and (23-155) Tooth load Particular

fk

Gear drive Precision gears (errors in pitch and form less than 0.025 mm) Commercial gears (errors in pitch and form 0.025 mm to 0.125 mm)

1.05–1.1 1.1–1.3

Prime movers and driven machines Shock-free rotary machines e.g. electrical machines and turbo-compressors Reciprocating engines, according to the degree of balance Machinery subjected to heavy shock loading, such as rolling mills

fd

f

1.0–1.2 1.2–1.5 1.5–3.0

Belt drive Vee-belts Single leather belts with jockey pulleys Single leather belts, balata belts, rubber belts

2.0–2.5 2.5–3.0 4.0–5.0

Full ball bearing sizes of diﬀerent bearing series

Refer to Fig. 23-51

For summary of types and characteristics of rolling contact bearing9

Refer to Fig. 23-52

Extra light series

Light series

Medium series

Extra light series

Light Medium series series

(100)

(200)

(300)

(100)

(200)

Relative proportions of bearings with same outside diameter

Shaft load

(300)

Relative proportions of bearings with same bore diameter

FIGURE 23-51 Ball bearing size of bearing series. (Courtesy: New Departure-Hyatt Bearing Division, General Motors Corporation.)

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DESIGN OF BEARINGS AND TRIBOLOGY

23.84

CHAPTER TWENTY-THREE

Particular

Formula

STATIC LOADING Stribeck equation for permissible static load

Fc ¼ kdb2

ð23-156Þ

where k ¼ 686:5 106 (100)* for carbon steel balls ¼ 862 106 (125)* for hardened alloy steel balls Use a factor of safety of 10 Fc in N (lbf) and db in m (in) Stribeck equation for permissible static load for ball bearing

Fc ¼

The radial load capacity of ball bearing

Fr ¼

4:37Fr Z

kZdb2 4:37 where db in m and Fr , Fc in kN Refer to Table 23-33 for values of k

kZdb2 5 Refer to Table 23-33 for values of k Fr ¼

Radial load capacity of roller bearing

BASIC STATIC LOAD RATING AS PER INDIAN STANDARDS

ð23-157Þ ð23-158Þ

ð23-159Þ

kZldr ð23-160Þ 5 where k ¼ 48:3 106 (7.0)* for hardened carbon steel ¼ 69 106 (10.0)* for hardened alloy steel ¼ 690 106 (100)* for carbon steel balls l, dr in m (in) and Fr in N (lbf)

Fr ¼

Radial ball bearing The basic static radial load rating for radial ball bearing

Cor ¼ fo iZD2w cos

ð23-161Þ

where Cor in N and Dw in mm Refer to Table 23-36 for fo This formula is applicable to bearings with a crosssectional raceway groove radius not larger than 0:52Dw in radial and angular contact groove ball bearing inner rings and 0:53Dw in radial and angular contact groove ball bearing outer rings and selfaligning ball bearing inner rings.

The static equivalent load for radial ball bearings is greater of the two values given by the formulae

Por ¼ Xo Fr þ Yo Fa

ð23-162Þ

Por ¼ Fr

ð23-163Þ

For values of Xo and Yo refer to Table 23-37. * Values outside the parentheses are in SI units in Pd and inside the parentheses are in US customary system units in kpsi.

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Load Carring Capacity

Yes∗

For Use on Free-end

Yes

One direction only

Excellent

Good

Contact angle 15 , 30 , 40 , Two bearings are usually mounted in opposition Clearance adjustment is necessary.

Yes Yes∗

Yes Yes∗

Duplex Angular Contact Ball Bearings

Fair Two directions

Poor

Yes∗

Yes∗

Yes

Yes

Self Aligning Ball Bearings

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Yes∗ Yes∗

Yes∗

Yes

×

×

Yes

×

×

Yes Yes

Yes

Cylindrical Cylindrical Double Row Cylindrical Roller Cylindrical Roller Roller Bearings Roller Bearings Bearings with Bearings Angle Ring

FIGURE 23-52 Types and characteristics of rolling bearings. (Courtesy: NSK Corp.)

* Can be used as free-end bearings if tap fit allows axial motion.

Legend

Remarks

Tapered Bore of Inner Ring

Yes

For Use on Fixed-end

Ring Separability

Self-Aligning Capability

Angular Misalignment

Rigidity

Low Noise and Torque

High Accuracy

High Speeds

Combined Load

Axial Load

Radial Load

Two bearings are usually mounted in opposition.

Double Row Angular Contact Ball Bearings

Any arrangement of the pair is possible.

Features

Angular Contact Ball Bearings

Yes

Yes

×

×

Needle Roller Bearings

Yes

Tapered Roller Bearings

Two bearings are usually mounted in opposition. Clearance adjustment is necessary.

Magneto Bearings

Yes∗

Yes∗

Yes

Yes

Yes

Spherical Roller Bearings

Yes

Yes

Double Row Tapered Roller Bearings

×

×

Yes

Yes

Yes

×

×

×

Yes

×

×

×

×

×

Angular Contact Thrust Ball Bearings

Thrust Ball Bearings with Aligning Seats

Thrust Ball Bearings

Yes

×

×

×

Cylindrical Roller Thrust Bearings

Needle roller thrust bearings are available.

Deep Groove Ball Bearings

Yes

×

×

×

Tapered Roller Thrust Bearings

Yes

Yes

Spherical Roller Thrust Bearings

This type should have oil lubrication.

Bearing Type

DESIGN OF BEARINGS AND TRIBOLOGY

23.85

DESIGN OF BEARINGS AND TRIBOLOGY

23.86

CHAPTER TWENTY-THREE

Particular

Formula

Radial roller bearing The basic static radial load rating for radial roller bearings The static equivalent radial load for roller bearings with 6¼ 08 is the greater of the two values given by the formulae The static equivalent radial load for radial roller bearings with ¼ 08 and subjected to radial load only, is given by the formula

D Cor ¼ 44 1 we cos iZLwe Dwe cos Dpw

ð23-164Þ

Por ¼ Xo Fr þ Yo Fo

ð23-165Þ

For factors Xo and Yo refer to Table 23-38. Por ¼ Fr

ð23-166Þ

Coa ¼ fo ZD2w sin

ð23-167Þ

THRUST BEARINGS Ball bearings The basic static axial load rating for single- or doubledirection thrust ball bearings

where fo values are taken from Table 23-36 Z ¼ number of balls carrying load in one direction

The static equivalent axial load Poa for thrust ball bearing with contact angle 6¼ 908

The static equivalent axial load for thrust ball bearings with ¼ 908 is given by the equation

Poa ¼ Fa þ 2:3Fr tan

ð23-168Þ

This formula is valid for all ratios of radial load to axial load in the case of double-direction bearings. For single direction bearings it is valid where ðFr =Fa Þ 0:44 cot and gives satisfactory but less conservative values of Poa for ðFr =Fa Þ up to 0:67 cot . Poa ¼ Fa

for ¼ 908

ð23-169Þ

Roller bearings The basic static axial load rating for single- and double-direction thrust roller bearings

D cos ZLwe Dwe sin Coa ¼ 220 1 we DPW

ð23-170Þ

where Z ¼ number of rollers carrying load in one direction The static equivalent axial load for thrust roller bearings with contact angle 6¼ 908

Poa ¼ Fa þ 2:3Fr tan

ð23-171aÞ

This formula is valid for all ratios of radial load to axial load in the case of double-direction bearings. For single-direction bearings, it is valid where ðFr =Fa Þ 0:44 cos and gives satisfactory but less conservative values of Poa for ðFr =Fa Þ up to 0:67 cot .

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

23.87

TABLE 23-36 Values of factor fo for radial ball bearingsa Factor fo Radial ball bearings

Dw cos Dpw

Radial and angular contact groove ball bearings

Self-aligning ball bearings

Thrust ball bearings

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

14.7 14.9 15.1 15.3 15.3 15.7 15.9 16.1 16.3 16 5

1.9 2 2 2.1 2.1 2.1 2.2 2.2 2.3 2.3

61.6 60.8 59.9 59.1 58.3 57.5 56.7 55.9 55.1 54.3

0.10 0.11 0.12 0.13 0.14

16.4 16.1 15.9 15.6 15 4

2.4 2.4 2.4 2.5 2.5

53.5 52.7 51.9 51.2 50.4

0.15 0.16 0.17 0.18 0.19

15.2 14.9 14.7 14.4 14.2

2.6 2.6 2.7 2.7 2.8

49.6 48.8 48.0 47.3 46.5

0.20 0.21 0.22 0.23 0.24

14.0 13.7 13.5 13.2 13.0

2.8 2.8 2.9 2.9 3.0

45.7 45.0 44.2 43.5 42.7

0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35

12.8 12.5 12.3 12.1 11.8 11.6 11.6 11.2 10.9 10.7 10.5

3.0 3.1 3.1 3.2 3.2 3.3 3.3 3.4 3.4 3.5 3.5

41.9 41.2 40.5 39.7 39.0 38.2 37.5 36.8 36.0 35.3 34.6

0.36 0.37 0.38 0.39 0.40

10.3 10.0 9.8 9.5 9.4

3.6 3.6 3.7 3.8 3.8

a The Table 23-36 is based on the Hertz’s point contact formula with a modulus of elasticity ðEÞ ¼ 2:07 105 mPa and a Poisson’s ratio () of 0.3. It is assumed that the load distribution for radial ball bearings results in a maximum ball load of (5Fo =Z cos ), and for thrust ball bearings (Fo =Z sin ). Values of fo for intermediate values of (Dw cos =Dpw ) are obtained by linear interpolation. (IS: 3823-1988, ISO: 76-1987)

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DESIGN OF BEARINGS AND TRIBOLOGY

23.88

CHAPTER TWENTY-THREE

Particular

Formula

TABLE 23-37 Values of factors Xo and Yo for radial ball bearingsa for use in Eqs. (23-162) Single row bearings

Double row bearings

Xo

Yo

Xo

Yo

Angular-contact groove ball bearings

¼ 158 ¼ 208 ¼ 258 ¼ 308 ¼ 358 ¼ 408 ¼ 458

0.6 0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.5 0.46 0.42 0.38 0.33 0.29 0.26 0.22

0.6 1 1 1 1 1 1 1

0.5 0.92 0.84 0.76 0.66 0.58 0.52 0.44

Self-aligning ball bearings

6¼ 908

0.5

0.22 cot

1

0.44 cot

Bearing type Radial contact groove ball bearings

a

Permissible value of Fa =Cor depends on bearing design (internal clearance and raceway groove depth)

TABLE 23-38 Values of factors Xo and Yo for radial roller bearings with 6¼ 08 for use in Eq. (23-165) Bearing type

Xo

Yo

Single-row Double-row

0.5 1.0

0:22 cot 0:44 cot

The static equivalent axial load for thrust roller bearings with ¼ 908 is given by the equation

Poa ¼ Fa

ð23-171bÞ

C o ¼ fs P o

ð23-172Þ

CATALOGUE INFORMATION FROM FAG FOR THE SELECTION OF BEARING The basic static load rating

where fs ¼ index of static dressing ¼ 1:5 to 2.5 for high degree ¼ 1:0 to 1.5 for normal degree ¼ 0:7 to 1.0 for moderate degree The equivalent static load

Po ¼ Fr

for ðFa =Fr Þ 0:8

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ð23-173aÞ

DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.89

Formula

Po ¼ 0:6Fr þ 0:5Fa

for ðFa =Fr Þ > 0:8

P o ¼ Xo Fr þ Yo Fa

ð23-173bÞ ð23-173cÞ

where Co and Po in kN For various values of factors Xo and Yo refer to Table 23-39 and Tables from FAG catalogue.

TABLE 23-39 Calculation of equivalent static and dynamic load Series Bearing type

IS

FAG

SKF

Static, Po

Dynamic, P

For dimensions, C, Ca , nmax , X, e Y, Yo refer to Table

Deep groove ball bearings

02 03 04

62 63 64

62 63 64

Fr when Fa =Fr 0:8 0:6Fr þ 0:5Fa when Fa =Fr > 0:8

Fr when Fa =Fr e 0:56Fr þ YFa when Fa =Fr > e

23-60 23-61 23-62

Self aligning ball bearings

02 03

12 13 22 23

12 13 22 23

Fr þ Yo Fa

XFr þ YFa

02 03

72B 73B

72B 73B

33

33A

Fr when Fa =Fr 1:9 0:50Fr þ 0:26Fa when Fa =Fr >1.9 Fr þ 0:58Fa

Fr when Fa =Fr 1:4 0:35Fr þ 0:57Fa when Fa =Fr > 1:14 Fr þ 0:66Fa when Fa =Fr 0:956 0:6Fr þ 1:07Fa when Fa =Fr > 0:95

23-69

N2 N3 N4 NU22 NU23

N2 N3 N4 NU22 NU23

Fr

Fr

23-70 23-71 23-72 23-73 23-74

322A 22 23

322

02 03

Fr when Fa =Fr 12 Yo 0:5Fr þ Yo Fa when Fa =Fr > 12 Yo

Fr when Fa =Fr e 0:4Fr þ YFa when Fa =Fr > e

11 12 13 14

511 512 513 514

511 512 513 514

Fa

Fa

522

522

Single row angular contact ball bearings Double row angular contact ball bearings

Cylindrical roller bearings

02 03 04

Tapered roller bearings

Single thrust ball bearings

Double thrust ball bearings

Equivalent load

23-63 23-64 23-65 23-66 23-67 23-68

23-75 23-76 23-76B 23-77 23-78 23-79 23-80 23-81 23-82

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DESIGN OF BEARINGS AND TRIBOLOGY

23.90

CHAPTER TWENTY-THREE

Particular

Formula

DYNAMIC LOAD RATING OF BEARINGS The relation between two groups of identical bearings tested under diﬀerent loads F1 and F2 , and length of lives L1 and L2 respectively as per experiments conducted by Palmgern

L1 ¼ L2

F2 F1

m

ð23-174Þ

where m ¼ 3 generally accepted ¼ 3.333 used by Timken ¼ 4 used by New Departures L ¼ life in millions of revolutions ¼ life in hours at constant speed in rpm

For various typical values of bearing life for various applications

Refer to Tables 23-40 and 23-41

LIFE

m

The Antifriction Bearing Manufacturers Association (AFBMA) statistically related formula for the rating life of bearing in millions of revolutions of a bearing subjected to any other load F, which is derived from Eq. (23-174)

For values of C for various types of bearings, refer to Table 23-42.

Equation (23-175) can be written as

C ¼ FL1=m

(The International Organisation for Standardisation (ISO) deﬁnes the rating life of a group of apparently identical rolling elements bearings as that completed or exceeded by 90% of that group before ﬁrst evidence of fatigue)

L¼

C F

ð23-175Þ

ð23-176Þ

where L in millions of revolutions

BASIC DYNAMIC LOAD RATING AS PER FAG CATALOGUEa The load and life of bearings are related statistically as per ISO equation for basic rating life

L10 ¼ L ¼

C F

m or

C 1=m ¼ L10 F

ð23-177aÞ

where L10 ¼ rating life in millions of revolutions (i.e. number of revolutions resulting in 10% failure) C ¼ basic dynamic load rating, N (obtained from manufacturer’s catalogue) F ¼ equivalent dynamic load (also with suﬃx e, i.e. Fe ), N (also symbol P is used in place of F) a

Note: The designer is advised to read carefully the manufacturer’s Catalogue, which explains how load rating and life are established.

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

23.91

TABLE 23-40 Typical values of bearing life for various applications

TABLE 23-41 Life of bearings, Lh

Application

Class of machinery

Life, kh

Instruments and apparatus which are used occasionally

0.5

Aircraft engines

0.5–2

Design life, h

Agricultural equipment Aircraft engines

3000–6000 500–1500

Automobile applications Race car Light motor cycle Heavy motor cycle Light car Heavy car Light truck Heavy truck Bus Boat gearing units Beater mills Briquette presses Domestic appliances Electrical motors (up to 0.5 kW) Electrical motors (up to 4 kW) Electrical motors, medium Electrical motors, large Elevator cables sheaves Small fans Mine ventilation fans

500–800 600–1200 1000–2000 1000–2000 1500–2500 1500–2500 2000–2500 2000–5000 3000–5000 20000–30000 20000–30000 1000–2000 1000–2000 8000–10000 10000–15000 20000–30000 40000–60000 2000–4000 40000–50000

Gearing units Automotive Multi-purpose Machine tool Ship Rail vehicles Heavy rolling mill Grinding spindles Locomotive axle boxes, outer bearings Locomotive axle boxes, inner bearings Machine tools Mining machinery Paper machines

600–5000 8000–15000 20000 20000–30000 15000–25000 50000 and more 1000–2000 20000–25000 30000–40000 10000–30000 4000–15000 50000–80000

Rail vehicle axle boxes Mining cars Motor rail cars Open pit mining cars Street cars Passenger cars Freight cars

5000 16000–20000 20000–25000 20000–25000 26000 35000

Rolling Mills Small cold mills Large multipurpose mills Gear drives Ship gear drives Propeller thrust bearings Propeller shaft bearings

5000–6000 8000–10000 50000 and more 20000–30000 15000–25000 80000 and more

Machines used for period where stoppage of service is of minor importance

4–8

Machine working intermittently whose service is essential

8–14

Machine working for 8 h per day whose service is not fully utilized

14–20

Machine working for 8 h per day whose service is fully utilized

20–30

Machines working continuously for 24 h Machines working continuously for 24 h with high reliability

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50–60 100–200

13800 15010 18894 21119 22367

27557 31115 33006 37788 41336

43561 52459 57339

06 07 08 09 10

11 12 13 14 15

16 17 18 19 20

23.92

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32 34

21 22 24 26 28 30

3842 4067 5557 6323 8134 9957

00 01 02 03 04 05

90679

72451

Light 200 N

SAE No.

137798 148921

80017 88909 101528 113346 122245

46227 58341 56448 68012 72451

18002 21334 27296 34006 38455

6664 7242 9428 9957 14445

113346 122245 137798

56448 68012 72451 88906 101352

27117 31340 35880 44629 48341

Medium 300 Heavy 400 N N

Double-row self-aligning

TABLE 23-42 Values of C for various types of bearings

96011 106683

54223 60907 64896 78233 87122

32674 39122 42895 46227 48902

14445 19110 21830 24451 26009

3332 5067 5557 7115 9604 10309

Light 200 N

144472 166698

92463 99568 108907 117796 135583

53341 60901 68012 75568 82683

21335 25343 30890 40004 46227

6488 8007 8712 10596 14181 16229

76910 84456 92463 115671 124470

33040 42288 48902 58682 64896

28008

Medium 300 Heavy 400 N N

Single-row deep groove

87122 90679 103135 126694 142247

52459 63563 69345 75568 80071

23559 31566 35564 40004 42895

5419 8271 9075 11554 15778 16895

Light 200 N

147921 147921 157809

85789 97794 111132 122245 135583

34672 40004 49794 62495 75568

82104 20002 26676

122245 135583

59564 62230 78233 97794 122245

Medium 300 Heavy 400 N N

Double-row deep groove

184471 191139

106683 122245 137798 157809 173362

56448 63563 73784 78233 87122

32673 39122 44423 48902 50676

14220 18894 22367 25343 26676

10133

Light 200 N

260043

212596

153369 173362

92463 101528 108707 117796 144472

53341 60907 68012 75568 84456

21335 25343 30890 40004 46227

12005 16895

73784 81350 88906 113346 120221

22673 40004 47118 57340 63563

27558

Medium 300 Heavy 400 N N

Single-row angular contact

DESIGN OF BEARINGS AND TRIBOLOGY

1.15 1.24 1.34 1.45 1.56 1.68 1.82 1.96 2.12 2.29 2.47 2.67 2.88 3.11 3.36 3.63 3.91 4.23 4.56 4.93 5.32 6.70

100 500 1,000 1,250 1,600 2,000 2,500 3,200 4,000 5,200 6,300 8,000 10,000 12,500 16,000 20,000 25,000 32,000 40,000 50,000 63,000 80,000 100,000 200,000

1.06 1.15 1.24 1.34 1.45 1.56 1.68 1.82 1.96 2.12 2.29 2.47 2.67 2.88 3.11 3.36 3.63 3.91 4.93

25

Life, Lh hours 10

100

1.06 1.45 1.34 1.82 1.45 1.96 1.56 2.12 1.68 2.29 1.82 2.47 1.96 2.67 2.12 2.88 2.29 3.11 2.47 3.36 2.67 3.63 2.88 3.91 3.11 4.23 3.36 4.56 3.63 4.93 3.91 5.32 4.23 5.75 4.56 6.20 4.93 6.70 5.32 7.23 5.75 7.81 6.20 8.43 7.81 10.6

40

1.56 1.96 2.12 2.29 2.47 2.67 2.88 3.11 3.36 3.63 3.91 4.23 4.56 4.93 5.32 5.75 6.20 6.70 7.23 7.81 8.43 9.11 11.5

125

1.68 1.12 2.29 2.47 2.67 2.88 3.11 3.36 3.63 3.91 4.23 4.56 4.93 5.32 5.75 6.20 6.70 7.23 7.81 8.43 9.11 9.83 12.4

160 1.06 2.82 2.29 2.47 2.67 2.88 3.11 3.36 3.63 3.91 4.23 4.56 4.93 5.32 5.75 6.20 6.70 7.23 7.81 8.43 9.11 9.83 10.6 13.4

200 1.15 1.96 2.47 2.67 2.88 3.11 3.36 3.63 3.91 4.23 4.56 4.93 5.32 5.75 6.20 6.70 7.23 7.81 8.43 9.11 9.83 10.6 11.5 14.5

250

TABLE 23-42a Loading ratio C=P for diﬀerent for ball bearings

1.24 2.12 2.67 2.88 3.11 3.36 3.63 3.91 4.23 4.56 4.93 5.32 5.75 6.20 6.70 7.23 7.81 8.43 9.11 9.83 10.6 11.5 12.4 15.6

320 1.34 2.29 2.88 3.11 3.36 3.63 3.91 4.23 4.56 4.93 5.32 5.75 6.20 6.70 7.23 7.81 8.43 9.11 9.83 10.6 11.5 12.4 13.4 16.8

400 1.45 2.47 3.11 3.36 3.63 3.91 4.23 4.56 4.93 5.32 5.75 6.20 6.70 7.23 7.81 8.43 9.11 9.83 10.6 11.5 12.4 13.4 14.5 18.2

500 1.56 2.67 3.36 3.63 3.91 4.23 4.56 4.93 5.32 5.75 6.20 6.70 7.23 7.81 8.43 9.11 9.83 10.6 11.5 12.4 13.4 14.5 15.6 19.6

630 1.68 2.88 3.63 3.91 4.23 4.56 4.93 5.32 5.75 6.20 6.70 7.23 7.81 8.43 9.11 9.83 10.6 11.5 12.4 13.4 14.5 15.6 16.8 21.2

800 1.82 3.11 3.91 4.23 4.56 4.93 5.32 5.75 6.20 6.70 7.23 7.81 8.43 9.11 9.83 10.6 11.5 12.4 13.4 14.5 15.6 16.8 18.2 22.90

1.96 3.36 4.23 4.56 4.93 5.32 5.75 6.20 6.70 7.23 7.81 8.43 9.11 9.83 10.6 11.5 12.4 13.4 14.5 15.6 16.8 18.2 19.6 24.7

2.12 3.63 4.56 4.93 5.32 5.75 6.20 6.70 7.23 7.81 8.43 9.11 9.83 10.6 11.5 12.4 13.4 14.5 15.6 16-9 18.2 19.6 21.2 26.7

2.29 3.91 4.93 5.32 5.75 6.20 6.70 7.23 7.81 8.43 9.11 9.83 10.6 11.5 12.4 13.4 14.5 15.6 16.9 18.2 19.6 21.2 22.9 28.8

2.47 4.23 5.32 5.75 6.20 6.70 7.23 7.81 8.43 9.11 9.83 10.6 11.5 12.4 13.4 14.5 15.6 16.8 18.2 19.6 21.2 22.9 24.7 31.1

2.67 4.56 5.75 6.20 6.70 7.23 7.81 8.43 9.11 9.83 10.6 11.5 12.4 13.4 14.5 15.6 16.8 18.2 19.6 21.2 22.9 24.7 26.7

2.88 4.93 6.20 6.70 7.23 7.81 8.43 9.11 9.83 10.6 11.5 12.4 13.4 14.5 15.6 16.8 18.2 19-6 21.2 22.9 24.7 26.7 28.8

3.11 5.32 6.70 7.23 7.81 8.43 9.11 9.83 10.6 11.5 12.4 13.4 14.5 15.6 16.8 18.2 19.6 21-2 22.9 24.7 26.7 28.8 31.1

3.36 5.75 7.32 7.81 8.43 9.11 9.83 10.6 11.5 12.4 13.4 14.5 15.6 16.8 18.2 19.6 21.2 22.9 24.7 26.7 28 8 31.1

3.63 6.20 7.81 8.43 9.11 9.83 10.6 11.5 12.4 13.4 14.5 15.6 16.8 .18.2 19.6 21.2 22.9 24.7 26.7 28.8 31.1

3.91 6.70 9.43 9.11 9.83 10.6 11.5 12.4 13.4 14.5 15.6 16.8 18.2 19.6 21-2 22.9 24.7 26.7 28.8 31.1

4.23 7.23 9.11 9.83 10.6 11.5 12.4 13.4 14.5 15.6 16.8 18.1 19.6 21.2 22.9 24.7 26.7 28.8 31.1

4.56 7.81 9.83 10.6 11.5 12.4 13.4 14.5 15.6 16.8 18.1 19.6 21.2 22.9 24.7 26.7 28.8 31.1

1000 1250 1600 2000 2500 3200 4000 5000 6200 8000 10000 12500 16000

Speed, rpm

DESIGN OF BEARINGS AND TRIBOLOGY

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23.93

1.13 1.21 1.30 1.39 1.49 1.60 1.71 1.83 1.97 2.11 2.26 2.42 2.59 2.78 2.97 3.19 3.42 3.66 3.92 4.20 4.50 5.54

100 500 1,000 1,250 1,600 2,000 2,500 3,200 4,000 5,000 6,300 8,000 10,000 12,500 16,000 20,000 25,000 32,000 40,000 50,000 63,000 80,000 100,000 200,000

1.05 1.13 1.21 1.30 1.39 1.46 1.60 1.71 1.83 1.97 2.11 2.26 2.42 2.59 2 78 2.97 3.19 3.42 4.20

25

Life, Lh hours 10

1.05 1.30 1.39 1.49 1.60 1.71 1.83 1.97 2.11 2.26 2.42 2.59 2.78 2.97 3.19 3.42 3.66 3.92 4.20 4.50 4.82 5.17 6.36

40

1.39 1.71 1.83 1.97 2.11 2.26 2.42 2.59 2.78 2.97 3.19 3.42 3.66 3.92 4.20 4.50 4.82 5.17 5.54 5.94 6.36 6.81 8.38

100

1.49 1.83 1.97 2.11 2.26 2.42 2.59 2.78 2.97 3.19 3.42 3.66 3.92 4.20 4.50 4.82 5.17 5.54 5.94 6.36 6.81 7.30 8.98

125

200

1.05 1.60 1.71 1.97 2.11 2.11 2.26 2.26 2.42 2.42 2.59 2.59 2.78 2.78 2.97 2.97 3.19 3.19 3.42 3.42 3.66 3.66 3.92 3.92 4.20 4.20 4.50 4.50 4.82 4.82 5.17 5.17 5.54 5.54 5.94 5.94 6.36 6.36 6.81 6.81 7.30 7.30 7.82 7.82 8.38 9.62 10.3

160 1.13 1.83 2.26 2.42 2.59 2.78 2.97 3.19 3.42 3.66 3.92 4.20 4.50 4.82 5.17 5.54 5.94 6.36 6.81 7.30 7.82 8.38 8.98 11.0

250 1.21 1.97 2.42 2.59 2.78 2.97 3.19 3.42 3.66 3.92 4.20 4.50 4.82 5.17 5.54 5.94 6.36 6.81 7.30 7.82 8.38 8.98 9.62 11.8

320

TABLE 23-42b Loading ratio C=P for diﬀerent lives for roller bearings

1.30 2.11 2.59 2.78 2.97 3.19 3.42 3.66 3.92 4.20 4.50 4.82 5.17 5.54 5.94 6.36 6.81 7.30 7.82 8.38 8.98 9.62 10.3 12.7

400 1.39 2.26 2.78 2.97 3.19 3.42 3.66 3.92 4.20 4.50 4.82 5.17 5.54 5.94 6.36 6.81 7.30 7.82 8.38 8.98 9.62 10.3 11.0 13.6

500 1.49 2.42 2.97 3.19 3.42 3.66 3.92 4.20 4.50 4.82 5.17 5.54 5.94 6.36 6.81 7.30 7.82 8.38 8.98 9.62 10.3 11.0 11.8 14.6

630 1.60 2.59 3.19 3.42 3.66 3.92 4.20 4.50 4.82 5.17 5.54 5.94 6.36 6.81 7.30 7.82 8.38 8.98 9.62 10.3 11.0 11.8 12.7 15.6

800 1.71 2.78 3.42 3.66 3.92 4.20 4.50 4.82 5.17 5.54 5.94 6.36 6.81 7.30 7.82 8.38 8.98 9.62 10.3 11.0 11.8 12.7 13.6 16.7

1.83 2.97 3.66 3.92 4.20 4.50 4.82 5.17 5.54 5.94 6.36 6.81 7.30 7.82 8.38 8.98 9.62 10.3 11.0 11.8 12.7 13.6 14.6 17.9

1.97 3.19 3.92 4.20 4.50 4.82 5.17 5.54 5.94 6.36 6.81 7.30 7.82 8.38 8.98 9.62 10.3 11.0 11.8 12.7 13.6 14.6 15.6 19.2

2.11 3.42 4.20 4.50 4.82 5.17 5.54 5.94 6.36 6.81 7.30 7.82 8.38 8.98 9.62 10.3 11.0 11.8 12.7 13.6 14.6 15.6 16.7 20.6

2.26 3.66 4.50 4.82 5.17 5.54 5.94 6.36 6.81 7.30 7.82 8.38 8.98 9.62 10.3 11.0 11.8 12.7 13.6 14.6 15.6 16.7 17.9

2.42 3.92 4.82 5.17 5.54 5.94 6.36 6.81 7.30 7.87 8.38 8.98 9.62 10.3 11.0 11.8 12.7 13.6 14.6 15.6 16.7 17.9 19.2

2.59 4.20 5.17 5.54 5.94 6.36 6.81 7.30 7.82 8.38 8.98 9.62 10.0 11.0 11.8 12.7 13.6 14.6 15.6 16.7 17.9 19.2 20.6

2.78 4.50 5.54 5.94 6.36 6.81 7.30 7.82 8.38 8.98 9.62 10.3 11.0 11.8 12.7 13.6 14.6 15.6 16.7 17.9 19.2 20.6

2.97 4.82 5.94 6.36 6.81 7.30 7.82 8.38 8.98 9.62 10.3 11.0 11.8 12.7 13.6 14.6 15.6 16.7 17.9 19.2 20.6

3.19 5.17 6.36 6.81 7.30 7.82 8.38 8.98 9.62 10.3 11.0 11.8 12.7 13.6 14.6 15.6 16.7 17.9 19.2 20.6

3.42 5.54 6.81 7.30 7.82 8.38 8.98 9.62 10.3 11.0 11.8 12.7 13.6 14.6 15.6 16.7 17.9 19.2 20.6

3.66 5.94 7.30 7.82 8.38 8.99 9.62 10.3 11.0 11.8 12.7 13.6 14.6 15.6 16.7 17.9 19.2 20.6

3.92 6.36 7.82 8.38 9.98 9.62 10.3 11.0 11.8 12.7 13.6 14.6 15.6 16.7 17.9 19.2 20.6

1000 1250 1600 2000 2500 3200 4000 5000 6200 8000 10000 12500 16000

Speed, rpm

DESIGN OF BEARINGS AND TRIBOLOGY

23.94

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.95

Formula

m ¼ life exponent ¼ 3.0 for ball bearing ¼ 10/3 for roller bearing Load ratio ðC=Pð¼ C=FÞÞ and also taken from Tables 23-42a and 23-42b can be determined from Fig. 23-53. C can be obtained from manufacturer’s catalogue. The equivalent dynamic load used in Eq. (23-177a) is given by

The Eq. (23-177a) in terms of Lh hours of life

Fe ¼ XFr þ YFa

ð23-177bÞ

where Fe ¼ P ¼ equivalent radial load for radial bearings or axial load for axial bearings Fr ¼ radial load, N Fa ¼ axial load, N X ¼ radial factor Y ¼ axial/thrust factor The values of X and Y can be found in tables of the manufacturer’s catalogue (e.g. FAG catalogue). 106 C m Lh ¼ ð23-178aÞ 60n F where Lh ¼ L10h ¼ nominal rating life, h

The relation between fatigue life (Lh ) in hours and the life in millions of revolutions (L)

The Eq. (23-178b) can be written as

106 L ð23-178bÞ 60n Refer to Tables 23-43 and 23-44 for index of dynamic strengthening fL and Lh . L10h ¼ Lh ¼

500ð33 13Þ60L 60n 1 m 33 3 C ¼ Lh ¼ 500 F n

L10h ¼ Lh ¼

ð23-179aÞ

L10h

ð23-179bÞ

Values of L10h in hours as function of speed n and load ratio ðC=F ¼ C=PÞ are given in Fig. 23-53. The simpliﬁed form of Eq. (23-179b)

fL ¼

C f F n

ð23-180aÞ

The basic dynamic load rating from Eq. (23-180a)

C¼

fL f F¼ LP fn fn

ð23-180bÞ

where F ¼P fL ¼ index of dynamic stressing fn ¼ speed factor Refer to Fig. 23-56.

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DESIGN OF BEARINGS AND TRIBOLOGY

23.96

CHAPTER TWENTY-THREE

ROLLER BEARINGS

BALL BEARINGS Speed rev/min 20

Load ratio C/P

Rating life hours Million Lh revolutions 200 L=L10

30 40

1.0

50 60 70 80 90 100

2 1.5

2 150 200 300 400 500 600 700 800 900 1000 1200 1500 1800 2000

3 4 5 6 8 10 20

3

4 5 6 7 8 9 10

3000 20

15000

40

400

40

500

60 70 80 90 100

1500

150

300

4000

400

200

5000

500

6000

600

300 400 500 600 800 1000

7000 8000 9000 10000

700 800 900 1000

30000 40000 50000 60000

1.0

15000

1500

20000

1800 2000

30000

3000

40000

4000

50000

5000

60000 70000 80000 90000 100000

6000 7000 8000 9000 10000

1.5

3 4 6 8 10

200000

20000

30000

300000

30000

500 600 700 800 900 1000

2000

4

30 40 50 60 80 100

5

200

6000

300 400 500 600 800 1000

7000 8000 9000 10000

3000 4000 5000

6 7 8 9 10

15

15000 2000 3000 4000 5000 6000 8000 10000

20000

30000 40000 50000

20

20000 30000 40000 50000 60000 80000 100000

60000 70000 80000 90000 100000

150000

15000

20000

400

1500 20

30 150000

1.0 2

3

1200

Rating life hours Million Lh revolutions 200 L=L10 300

2

3000

3000 4000 5000 6000 8000 10000

Load ratio C/P

50

600 700 800 900 1000

200

20000 30

30

2000

5000 6000 7000 8000 9000 10000

300

30 40 50 60 80 100

2000 15

4000

1.0

Speed rev/min 20

40

200000

200000 300000

FIGURE 23-53 Nomogram chart for determining c/p for ball and roller bearings (ISO-R281) (Note: Information on the calculation of load rating C and equivalent load P can be obtained from ISO Recommendation R281. Values of C for various types of bearing can be obtained from the bearing manufacturers.)

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.97

Formula

rﬃﬃﬃﬃﬃﬃﬃﬃ Lh fL ¼ 500 For a life of 500 h, fL ¼ 1

The value of index of dynamic stressing, fL , can be obtained from Eq. (23-179b)

m

ð23-181aÞ

Refer to Table 23-43 for fL of ball bearings Refer to Table 23-44 for fL of roller bearings and to Table 23-58

TABLE 23-43 Index of dynamic stressing fL for ball bearings for use in Eq. (23-180) Lh hours

fL

Lh hours

fL

100 105 110 115 120

0.585 0.595 0.604 0.613 0.622

300 310 320 330 340

0.843 0.852 0.861 0.870 0.879

125 130 135 140 145

0.631 0.639 0.647 0.654 0.662

350 360 370 380 390

150 155 160 165 170

0.670 0.677 0.684 0.691 0.698

175 180 185 190 195

Lh hours

fL ¼

p 3 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Lh =500

fL

Lh hours

fL

Lh hours

fL

Lh hours

fL

Lh hours

fL

700 720 740 760 780

1.120 1.130 1.140 1.150 1.160

1750 1800 1850 1900 1950

1.520 1.535 1.545 1.560 1.575

4500 4600 4700 4800 4900

2.08 2.10 2.11 2.13 2.14

10000 10500 11000 11500 12000

2.71 2.76 2.80 2.85 2.85

30000 31000 32000 33000 34000

3.91 3.96 4.00 4.04 4.08

0.888 0.896 0.905 0.913 0.921

800 820 840 860 880

1.170 1.180 1.190 1.200 1.205

2000 2100 2200 2300 2400

1.590 1.615 1.640 1.665 1.690

5000 5200 5400 5600 5800

2.15 2.18 2.21 2.24 2.27

12500 13000 13500 14000 14500

2.93 2.96 3.00 3.04 3.07

35000 36000 37000 38000 3900

4.12 4.16 4.20 4.24 4.27

400 410 420 430 440

0.928 0.936 0.944 0.951 0.959

900 920 940 960 980

1.215 1.225 1.235 1.245 1.260

2500 2600 2700 2800 2900

1.710 1.730 1.755 1.775 1.795

6000 6200 6400 6600 6800

2.29 2.32 2.34 2.37 2.39

15000 15500 16000 16500 17000

3.11 3.14 3.18 3.21 3.24

40000 41000 42000 43000 44000

4.31 4.35 4.38 4.42 4.45

0.705 0.712 0.718 0.724 0.731

450 460 470 480 490

0.966 0.973 0.980 0.987 0.994

1000 1050 1100 1150 1200

1.260 1.280 1.300 1.320 1.340

3000 3100 3200 3300 3400

1.815 1.835 1.855 1.875 1.895

7000 7200 7400 7600 7800

2.41 2.43 2.46 2.48 2.50

17500 18000 18500 19000 19500

3.27 3.30 3.33 3.36 3.39

45000 46000 47000 48000 49000

4.48 4.51 4.55 4.58 4.61

200 210 220 230 240

0.737 0.749 0.761 0.772 0.783

500 520 540 560 580

1.000 1.015 1.025 1.040 1.050

1250 1300 1350 1400 1450

1.360 1.375 1.395 1.410 1.425

3500 3600 3700 3800 3900

1.910 1.930 1.950 1.965 1.985

8000 8200 8400 8600 8800

2.52 2.54 2.56 2.58 2.60

20000 21000 22000 23000 24000

3.42 3.48 3.53 3.58 3.63

50000 55000 60000 65000 70000

4.64 4.80 4.94 5.07 5.19

250 260 270 280 290

0.794 0.804 0.814 0.824 0.834

600 620 640 660 680

1.065 1.075 1.085 1.100 1.110

1500 1550 1600 1650 1700

1.445 1.460 1.475 1.490 1.505

4000 4100 4200 4300 4400

2.000 2.020 2.030 2.050 2.070

9000 9200 9400 9600 9800

2.62 2.64 2.66 2.68 2.70

25000 26000 27000 28000 29000

3.68 3.73 3.78 3.82 3.87

75000 80000 85000 90000 100000

5.30 5.43 5.55 5.65 5.85

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DESIGN OF BEARINGS AND TRIBOLOGY

23.98

CHAPTER TWENTY-THREE

Particular

Formula

The value of speed factor, fn , can be obtained from Eq. (23-179b)

sﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃ 33 13 m 100 fn ¼ ¼ n 3n m

ð23-181bÞ

For a speed factor of n ¼ 33 13 min, fn ¼ 1 Refer to Table 23-45 for fn of ball bearings Refer to Table 23-46 for fn of roller bearings The equivalent dynamic load for deep groove ball bearings with increased radial clearance

Fe ¼ Pa ¼ XFr þ YFa

ð23-182Þ

TABLE 23-44 Index of dynamic stressing fL for roller bearings for use in Eq. (23-180) Lh hours

fL

Lh hours

fL

100 105 110 115 120

0.617 0.626 0.635 0.643 0.652

300 310 320 330 340

0.858 0.866 0.875 0.883 0.891

125 130 135 140 145

0.660 0.665 0.675 0.683 0.690

350 360 370 380 390

150 155 160 165 170

0.697 0.704 0.710 0.717 0.723

175 180 185 190 195

Lh hours

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 10=3 Lh =500

fL ¼

fL

Lh hours

fL

Lh hours

fL

Lh hours

fL

Lh hours

700 720 740 760 780

1.105 1.115 1.125 1.135 1.145

1750 1800 1850 1900 1950

1.455 1.470 1.480 1.490 1.505

4500 4600 4700 4800 4900

1.935 1.945 1.960 1.970 1.985

10000 10500 11000 11500 12000

2.46 2.49 2.53 2.56 2.59

30000 31000 32000 33000 34000

3.42 3.45 3.48 3.51 3.55

0.889 0.906 0.914 0.921 0.928

800 820 840 860 880

1.150 1.160 1.170 1.180 1.185

2000 2100 2200 2300 2400

1.515 1.540 1.560 1.580 1.600

5000 5200 5400 5600 5800

2.00 2.02 2.04 2.06 2.09

12500 13000 13500 14000 14500

2.63 2.66 2.69 2.72 2.75

35000 36000 37000 39000 39000

3.58 3.61 3.64 3.67 3.70

400 410 420 430 440

0.935 0.942 0.949 0.956 0.962

900 920 940 960 980

1.190 1.200 1.210 1.215 1.225

2500 2600 2700 2800 2900

1.620 1.640 1.660 1.675 1.695

6000 6200 6400 6600 6800

2.11 2.13 2.15 2.17 2.19

15000 15500 16000 16500 17000

2.77 2.80 2.83 2.85 2.88

40000 41000 42000 43000 44000

3.72 3.75 3.78 3.80 2.83

0.730 0.736 0.742 0.748 0.754

450 460 470 480 490

0.969 0.975 0.982 0.994 0.998

1000 1050 1100 1150 1200

1.230 1.250 1.270 1.285 1.300

3000 3100 3200 3300 3400

1.710 1.730 1.755 1.760 1.775

7000 7200 7400 7600 7800

2.21 2.23 2.24 2.26 2.28

17500 18000 18500 19000 19500

2.91 2.93 2.95 2.98 3.00

45000 46000 47000 48000 49000

3.86 3.88 3.91 3.93 3.96

200 210 220 230 240

0.760 0.771 0.782 0.792 0.802

500 520 540 560 580

1.000 1.010 1.025 1.035 1.045

1250 1300 1350 1400 1450

1.315 1.330 1.345 1.360 1.375

3500 3600 3700 3800 3900

1.795 1.810 1.825 1.840 1.850

8000 8200 8400 8608 8800

2.30 2.31 2.33 2.35 2.36

20000 21000 22000 23000 24000

3.02 3.07 3.11 3.15 3.19

50000 55000 60000 65000 70000

3.98 4.10 4.20 4.30 4.40

250 260 270 280 290

0.812 0.822 0.831 0.840 0.849

600 620 640 660 680

1.055 1.065 1.075 1.085 1.095

1500 1550 1600 1650 1700

1.390 1.405 1.420 1.430 1.445

4000 4100 4200 4300 4400

1.865 1.880 1.895 1.905 1.920

9000 9200 9400 9600 9800

2.38 2.40 2.41 2.43 2.44

25000 26000 27000 28000 29000

3.23 3.27 3.31 3.35 3.38

75000 80000 85000 90000 100000 150000 200000

4.50 4.58 4.66 4.75 4.90 5.54 6.03

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fL

DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

TABLE 23-45 Speed factor fn for ball bearings for use in Eq. (23-180) Speed, n, rpm

Speed factor, fn

10 11 12 13 14

1.494 1.447 1.405 1.369 1.335

15 16 17 18 19

Speed, n, rpm

fn ¼

Speed factor, fn

Speed, n, rpm

Speed factor, fn

Speed, n, rpm

Speed factor, fn

60 62 64 66 68

0.822 0.813 0.805 0.797 0.788

250 260 270 280 290

0.511 0.504 0.498 0.492 0.487

900 920 940 960 980

0.333 0.331 0.329 0.326 0.324

1.305 1.277 1.252 1.225 1.206

70 72 74 76 78

0.781 0.774 0.767 0.760 0.753

300 310 320 330 340

0.481 0.476 0.471 0.466 0.461

1000 1050 1100 1150 1200

20 21 22 23 24

1.186 1.166 1.148 1.132 1.116

80 82 84 86 88

0.747 0.741 0.735 0.729 0.724

350 360 370 380 390

0.457 0.453 0.448 0.444 0.441

25 26 27 28 29

1.100 1.089 1.071 1.060 1.048

90 92 94 96 98

0.718 0.713 0.708 0.703 0.698

400 410 420 430 440

30 31 32 33 34

1.036 1.025 1.014 1.003 0.993

100 105 110 115 120

0.693 0.692 0.672 0.662 0.652

35 36 37 38 39

0.984 0.975 0.966 0.958 0.949

125 130 135 140 145

40 41 42 43 44

0.941 0.933 0.926 0.919 0.912

45 46 47 48 49 50 52 54 56 58

Speed, n, rpm

23.99

q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 33 13 =n

Speed factor, fn

Speed, n, rpm

Speed factor, fn

4000 4100 4200 4300 4400

0.203 0.201 0.199 0.198 0.196

15,000 15,500 16,000 16,500 17,000

0.131 0.129 0.128 0.126 0.125

0.322 0.317 0.312 0.302 0.303

4500 4600 4700 4800 4900

0.195 0.193 0.192 0.191 0.190

17,500 18,000 18,500 19,000 19,500

0.124 0.123 0.122 0.121 0.120

1250 1300 1350 1400 1450

0.299 0.295 0.291 0.288 0.284

5000 5200 5400 5600 5800

0.188 0.186 0.183 0.181 0.179

20,000 21,000 22,000 23:000 24,000

0.119 0.117 0.115 0.113 0.112

0.437 0.433 0.430 0.426 0.423

1500 1550 1600 1650 1700

0.281 0.278 0.275 0.272 0.270

6000 6200 6400 6600 6800

0.177 0.175 0.173 0.172 0.170

25,000 26,000 27,000 28,000 29,000

0.110 0.109 0.107 0.106 0.105

450 460 470 480 490

0.420 0.417 0.414 0.411 0.408

1750 1800 1850 1900 1950

0.267 0.265 0.262 0.260 0.258

7000 7200 7400 7600 7800

0.168 0.167 0.165 0.164 0.162

30,000 32,000 34,000 36,000 38,000

0.104 0.101 0.0993 0.0975 0.0957

0.644 0.635 0.627 0.620 0.613

500 520 540 560 580

0.406 0.400 0.395 0.390 0.386

2000 2100 2200 2300 2400

0.255 0.251 0.247 0.244 0.248

8000 8200 8400 8600 8800

0.611 0.160 0.158 0.157 0.156

40,000 42,000 44,000 46,000 50,000

0.0941 0.0926 0.0912 0.0898 0.0874

150 155 160 165 170

0.606 0.599 0.583 0.586 0.581

600 620 640 660 680

0.382 0.378 0.374 0.370 0.366

2500 2600 2700 2800 2900

0.237 0.234 0.231 0.228 0.226

9000 9200 9400 9600 9800

0.155 0.154 0.153 0.152 0.150

0.905 0.898 0.892 0.585 0.880

175 180 185 190 195

0.575 0.570 0.565 0.560 0.555

700 720 740 760 780

0.363 0.359 0.356 0.353 0.350

3000 3100 3200 3300 3400

0.223 0.221 0.218 0.216 0.214

10,000 10,500 11,000 11,500 12,000

0.149 0.147 0.145 0.143 0.141

0.874 0.863 0.851 0.841 0.831

200 210 220 230 240

0.550 0.541 0.533 0.525 0.518

800 820 840 860 880

0.347 0.344 0.341 0.339 0.336

3500 3600 3700 3800 3900

0.212 0.210 0.208 0.206 0.205

12,500 13,000 13,500 14,000 14,500

0.139 0.137 0.135 0.134 0.132

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DESIGN OF BEARINGS AND TRIBOLOGY

23.100

CHAPTER TWENTY-THREE

TABLE 23.46 Speed factor fn for roller bearings for us in Eq. (23-180) Speed, n, rpm

Speed factor, fn

10 11 12 13 14

1.435 1.395 1.359 1.326 1.297

15 16 17 18 19

Speed, n, rpm

fn ¼

Speed factor, fn

Speed, n, rpm

Speed factor, fn

Speed, n, rpm

Speed factor, fn

60 62 64 66 68

0.838 0.830 0.822 0.815 0.807

250 260 270 280 290

0.546 6.540 0.534 0.528 0.523

900 920 940 960 980

0.372 0.370 0.367 0.365 0.363

1.271 1.246 1.224 1.203 1.184

70 72 74 76 78

0.800 0.794 0.787 0.781 0.775

300 310 320 330 340

0.517 0.512 0.507 0.503 0.498

1000 1050 1100 1150 1200

20 21 22 23 24

1.166 1.149 1.133 1.118 1.104

80 82 84 86 88

0.769 0.763 0.758 0.753 0.747

350 360 370 380 390

0.494 0.490 0.486 0.482 0.478

25 26 27 28 29

1.090 1.077 1.065 1.054 1.0411

90 92 94 96 98

0.742 0.737 0.733 0.728 0.724

400 410 420 430 440

30 31 32 33 34

1.032 1.022 1.012 1.003 0.994

100 105 110 115 120

0.719 0.709 0.699 0.690 0.681

35 36 37 38 39

0.986 0.977 0.969 0.962 0.954

125 130 135 140 145

40 41 42 43 44

0.947 0.940 0.933 0.927 0.920

45 46 47 48 49 50 52 54 56 58

Speed, n, rpm

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 33 13 =n

10=3

Speed factor, fn

Speed, n, rpm

Speed factor, fn

4000 4100 4200 4300 4400

0.238 0.236 0.234 0.233 0.231

15000 15500 16000 16500 17000

0.160 0.158 0.157 0.156 0.154

0.361 0.355 0.350 0.346 0.341

4500 4600 4700 4800 4900

0.230 0.228 0.227 0.225 0.224

17500 18000 18500 19000 19500

0.153 0.152 0.150 0.149 0.148

1250 1300 1350 1400 1450

0.337 0.333 0.329 0.336 0.322

5000 5200 5400 5600 5800

0.222 0.220 0.217 0.215 0.213

20000 21000 22000 23000 24000

0.147 0.145 0.143 0.141 0.139

0.475 0.471 0.467 0.464 0.461

1500 1550 1600 1650 1700

0.319 0.316 0.313 0.310 0.307

6000 6200 6400 6600 6800

0.211 0.209 0.207 0.205 0.203

25000 26000 27000 28000 29000

0.137 0.136 0.134 0.133 0.131

450 460 470 480 490

0.458 0.455 0.452 0.449 0.447

1750 1800 1850 1900 1950

0.305 0.302 0.300 0.297 0.295

7000 7200 7400 7600 7800

0.201 0.199 0.198 0.196 0.195

30000 32000 34000 36000 38000

0.130 0.127 0.125 0.123 0.121

0.673 0.665 0.657 0.650 0.643

500 520 540 560 580

0.444 0.439 0.434 0.429 0.425

2000 2100 2200 2300 2400

0.293 0.289 0.285 0.281 0.274

8000 8200 8400 8600 8800

0.137 0.192 0.190 0.189 0.188

40000 42000 44000 46000 50000

0.119 0.117 0.116 0.114 0.111

150 155 160 165 170

0.637 0.631 0.625 0.619 0.613

600 620 640 660 680

0.420 0.416 0.412 0.408 0.405

2500 2600 2700 2800 2900

0.274 0.271 0.268 0.265 0.262

9000 9200 9400 9600 9800

0.187 0.185 0.184 0.183 0.182

0.914 0.908 0.902 0.896 0.891

175 180 185 190 195

0.608 0.603 0.598 0.593 0.589

700 720 740 760 780

0.401 0.398 0.395 0.391 0.388

3000 3100 3200 3300 3400

0.259 0.257 0.254 0.252 0.250

10000 10500 11000 11500 12000

0.181 0.178 0.176 0.173 0.171

0.896 0.875 0.865 0.856 0.847

200 210 220 230 240

0.584 0.576 0.568 0.560 0.553

800 820 840 860 880

0.385 0.383 0.380 0.377 0.375

3500 3600 3700 3800 3900

0.248 0.246 0.243 0.242 0.240

12500 13000 13500 14000 14500

0.169 0.167 0.165 0.163 0.162

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DESIGN OF BEARINGS AND TRIBOLOGY

23.101

DESIGN OF BEARINGS AND TRIBOLOGY

Particular

Formula

TABLE 23-47a Values of radial factors X and thrust factors Y for deep groove ball bearings with increase In radial clearance Bearing clearance C3a

Normal Bearing Standard clearance

fo

Fa Co

0.3 0.5 0.9 1.6 3.0 6.0

Fa e Fr

Fa >e Fr

Fa e Fr

Bearing clearance C4a

Fa >e Fr

Fa e Fr

Fa >e Fr

e

X

Y

X

Y

e

X

Y

X

Y

e

X

Y

X

Y

0.22 0.24 0.28 0.32 0.36 0.43

1 1 1 1 1 1

0 0 0 0 0 0

0.56 0.56 0.56 0.56 0.56 0.56

2.0 1.8 1.6 1.4 1.2 1.0

0.32 0.35 0.39 0.43 0.48 0.54

1 1 1 1 1 1

0 0 0 0 0 0

0.46 0.46 0.46 0.46 0.46 0.46

1.70 1.56 1.41 1.27 1.14 1.00

0.40 0.43 0.45 0.48 0.52 0.56

1 1 1 1 1 1

0 0 0 0 0 0

0.44 0.44 0.44 0.44 0.44 0.44

1.40 1.31 1.23 1.16 1.08 1.00

a C3 , C4 Standard for a radial clearance that is larger than normal. Values of factor fa are given in Table 23-47b

For values of X and Y, refer to Table 23-39 and also to bearing tables given in FAG catalogue. For values of X and Y of deep groove ball bearings with increased radial clearance, refer to Table 23-47b.

LIFE ADJUSTMENT FACTORS An adjusted fatigue life equation for a reliability of (100-n) percent An approximate equation for adjusted factor a1 , which accounts for reliability, R, is calculated from the Weibull distribution. The adjusted fatigue life for non-conventional materials and operating conditions.

L10a ¼ a2 a3 L10

ð23-183Þ

where a2 ¼ life adjustment factor for materials ¼ 3 for radial ball bearings of good quality as per AFBMA ¼ 0.2 to 2.0 for normal bearing materials ¼ 2.0 for most common material, AISI 52100 a3 ¼ life adjustment factor for application operating conditions ¼ 1 for well and adequately lubricated bearings < 1 for other adverse lubricating conditions and temperatures. The standard does not yet include values of a3

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DESIGN OF BEARINGS AND TRIBOLOGY

23.102

CHAPTER TWENTY-THREE

TABLE 23-47b Factor fo for deep groove ball bearings for use in Table 23-47a Factor fo Bore reference number

Bearing series 160

60

62

63

64

00 01 02 03 04 05 06 07

— — 13.9 14.3 14.9 15.4 15.2 15.6

12.4 13.0 13.9 14.3 13.9 14.5 14.8 14.8

12.1 12.2 13.1 13.1 13.1 13.8 13.8 13.8

11.3 11.1 12.1 12.2 12.1 12.4 13.0 13.1

— — — 10.9 11.0 12.1 12.2 12.1

08 09 10 11 12

15.9 15.9 16.1 16.1 16.3

15.2 15.4 15.6 15.4 15.5

14.0 14.1 14.3 14.3 14.3

13.0 13.0 13.0 12.9 13.1

12.2 12.1 12.2 12.2 12.3

13 14 15 16 17

16.4 16.2 16.4 16.4 16.4

15.7 15.5 15.7 15.6 15.7

14.3 14.4 14.7 14.6 14.7

13.2 13.2 13.2 13.2 13.1

12.3 12.1 12.2 12.3 12.3

18 19 20 21 22

16.3 16.5 16.4 16.3 16.3

15.6 15.7 15.9 15.8 15.6

14.5 14.4 14.4 14.3 14.3

13.9 13.9 13.8 13.7 13.8

12.2

24 26 28 30 32

16.4 16.4 16.4 16.4 16.4

15.9 15.8 16.0 16.0 16.0

14.8 14.5 14.8 15.2 15.2

13.5 13.6 13.6 13.7 13.9

34 36 38 40 44

16.4 16.3 16.4 16.3 16.3

15.7 15.6 15.8 15.6 15.6

15.3 15.3 15.0 15.3 15.2

13.9 13.9 14.0 14.1 14.1

48 52 56 60

16.4 16.4 16.4 16.4

15.8 15.7 15.9 15.7

15.2 15.2 15.3

14.2

64 68 72 76

16.4 16.3 16.4 16.5

15.9 15.8 15.9 15.9

TABLE 23-48 Life adjustment factor for reliability a1 Reliability R, %

Ln

Adjustment factor, a1

90 95 96 97 98 99

L10 L5 L4 L3 L2 L1

1 0.65 0.53 0.44 0.33 0.21

Note: These values of a1 and a2 have to be used judiciously and designers are advised to consult manufacturer’s Catalogue

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.103

Formula

From Eq. (23-177a), an adjusted fatigue life equation taking into consideration adjustment factors a1 , a2 and a3 , and for a failure probability of n as per ISO 281

Lna ¼ a1 a2 a3

C F

m

¼ a1 a2 a3 L

where L in millions of revolutions Lhna ¼ a1 a2 a3 Lh

An adjusted fatigue life equation, which accounts for variation in vibration and shock, speed, environment, etc in addition to adjustment factors a1 , a2 and a3

ð23-184aÞ

ð23-184bÞ

where Lh in h C m L ¼ a1 a2 a3 Ka F

ð23-185Þ

where L in 106 revolutions Ka ¼ application factor taken from Table 23-49

From Eq. (23-186a), the basic equivalent load rating, which accounts for application factor Ka , and adjustment factors a1 , a2 and a3

C ¼ Ka F

L a1 a2 a3

1=m ð23-186Þ

where L in 106 revolutions

Fig. 23-54 shows the plot of relative life vs probability of failure in percent. L10 life (also called as B10 life or minimum life), which indicates a 10% probability of failure, i.e. 90% of the loaded bearings will survive beyond this life, is taken as reference. L50 in Fig. 23-54 indicates median life with a 50% probability of failure, which is also known as average life. From Fig. 23-54 it can be seen that L50 5L10 , which indicates only 50% of the bearings will survive this longer life. For life curves of ball bearings as per SKF and New Departure (ND) and Needle-bearings

Refer to Fig. 23-55

TABLE 23-49 Load application factor Ka for use in Eqs. (23-185) and (23-186) Operating conditions

Smooth operation free from shock Normal operation Operation accompanied by shock and vibration

Applications

Ka

Precision gearing Commercial gearing Applications with poor bearing seals Machinery with no impact: Electric motors, machine tools, air conditioners Machinery with light impact: Air blowers, compressors, elevators, cranes, paper making machines Machinery with moderate impact: Construction machines crushers, vibration screens, rolling mills

1.0–1.1 1.1–1.3 1.2

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1–1.2 1.2–1.5 1.5–3.0

DESIGN OF BEARINGS AND TRIBOLOGY

23.104

CHAPTER TWENTY-THREE

20

Relative fatigue life

15

10

L50

5

L10

1 0

10

20

30 40 50 60 70 Probability of failure, percent

80

90

100

FIGURE 23-54 Typical life distribution in rolling bearings. (Courtesy: SKF Industries, Inc.)

2•5 2•0

Load Factor Kl

1•5 1•0

ND

0•7

T-Ne edle

0•5 0•4

SKF

0•3 0•2 500

1,000

2,000 3,000 5,000 10,000 Life of Bearing, Hours

20,000

50,000

FIGURE 23-55 Life curves of ball and needle bearings.

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.105

Formula

BASIC DYNAMIC LOAD RATING OF BEARINGS AS PER INDIAN STANDARDS Radial Ball Bearing The basic dynamic load rating for radial and angular contact ball bearings.

Cr ¼ fc ði cos Þ0:7 Z2=3 ðDw Þ1:8

ð23-187Þ

for Dw 25:4 mm where Cr in N, Dw in mm Cr ¼ 3:647fc ði cos Þ0:7 Z2=3 ðDw Þ1:4

ð23-188Þ

for Dw > 25:4 mm where Cr in N, Dw in mm For values of factor fc refer to Table 23-50. TABLE 23-50 Values of factor fc for radial ball bearings for use in Eqs. (23-187) and (23-188) Factor, fc Dw cos Dpw

Single row radial contact groove ball bearings and single and double rowangular contact groove ball bearings

Double row radial contact groove ball bearings

Single row and double row selfaligning ball bearings

Single row radial contact separable ball bearings (magneto bearings)

0.05 0.06 0.07

46.7 49.1 51.1

44.2 46.5 48.4

17.3 18.6 19.9

16.2 17.4 18.5

0.08 0.09 0.10

52.8 54.3 55.5

50.0 51.4 52.6

21.1 22.3 33.4

19.5 20.6 21.5

0.12 0.14 0.16

57.5 58.8 59.6

54.5 55.7 56.5

25.6 27.7 29.7

23.4 25.3 27.1

0.18 0.20 0.22

59.9 59.9 59.6

56.8 56.8 56.5

31.7 33.5 35.2

28.8 30.5 32.1

0.24 0.26 0.28

59.0 58.2 57.1

55.9 55.1 54.1

36.8 38.2 39.4

33.7 35.2 36.6

0.30 0.32 0.34

56.0 54.6 53.2

53.0 51.8 50.4

40.3 40.9 41.2

37.8 38.9 39.8

0.36 0.38 0.40

51.7 50.0 48.4

48.9 47.4 45.8

41.3 41.0 40.4

40.4 40.8 40.9

Note: Values of fc for intermediate values of D0w cos =Dpw are obtained by linear interpolation. IS: 3824 (Part 1)–1983

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DESIGN OF BEARINGS AND TRIBOLOGY

23.106

CHAPTER TWENTY-THREE

Particular

The approximate rating life in millions of revolutions for ball bearing The equivalent radial load for radial and contact ball bearings under combined constant radial and axial loads The basic rating life for a radial ball bearing which is based statistically

Formula

Ln ¼

C P

3 ð23-189Þ

Pr ¼ XFr þ Fa

ð23-190Þ

For values of X and Y refer to Table 23-51. L10 ¼

Cr Pr

3 ð23-191Þ

where L10 ¼ basic rating life in millions of revolutions (i.e., the number of revolutions resulting in 10% failure). The values of Cr and Pr shall be calculated in accordance with Eqs. (23-187), (23-188), and (23-190).

Adjusted rating life The adjusted rating life for a reliability of (100-n) percent

Ln ¼ a1 L10

ð23-192Þ

The adjusted rating life for non-conventional materials and operating conditions

L10a ¼ a2 a3 L10

ð23-193Þ

The adjusted rating life for non-conventional materials and operating conditions, and for a reliability of (100-n) percent.

Lna ¼ a1 a2 a3 L10

ð23-194Þ

Refer to Table 23-48 for a1 values

Radial roller bearings The basic dynamic radial load rating of radial roller bearings

Cr ¼ fc ðiLwe cos Þ7=9 Z3=4 D29=27 we

ð23-195Þ

where Cr in N, Lwe and Dwe in mm For values of factor fc refer to Table 23-52.

The equivalent radial load for radial roller bearings with 6¼ 08 under combined constant radial and axial loads

Pr ¼ XFr þ YFa

ð23-196Þ

For values of X and Y refer to Table 23-53.

The equivalent radial load for radial roller bearings with 6¼ 08 and subjected to radial load only

P r ¼ Fr

The basic rating life in millions of revolutions for radial roller bearings

L10 ¼

ð23-197Þ Cr Pr

10=3 ð23-198Þ

The values of Cr and Pr are calculated in accordance with Eqs. (23-195) to (23-197). For adjusted rating life for roller bearings

Refer to Eqs. (23-192) to (23-194) with suitable modiﬁcation.

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DESIGN OF BEARINGS AND TRIBOLOGY

TABLE 23-51 Factors X and Y for radial ball bearings for use in Eq. (23-190) Single-row bearings Fa e Fr ‘Relative axial load’a

Double-row bearings

Fa e Fr

Fa e Fr

Fa e Fr

X

Y

X

Y

X

Y

X

Y

e

1

0

0.56

2.30 1.99 1.71 1.55 1.45 1.31 1.15 1.04 1.00

1

0

0.56

2.30 1.99 1.71 1.55 1.45 1.31 1.15 1.04 1.00

0.19 0.22 0.26 0.28 0.30 0.34 0.38 0.42 0.44

Radial contact groove ball bearings Fa Fa Cor i ZD2w 0.014 0.028 0.056 0.084 0.11 0.17 0.28 0.42 0.56

0.172 0.345 0.689 1.30 1.38 2.07 3.45 5.17 6.89

Angular contact groove ball bearings iFa Fa Cor ZD2w 58

0.014 0.028 0.056 0.085 0.11 0.17 0.28 0.42 0.56

0.172 0.345 0.689 1.03 1.38 2.07 3.45 5.17 6.89

1

0

For this type use the X, Y and e values applicable to single row radial contact groove ball bearings

1

2.78 2.40 2.07 1.87 1.75 1.58 1.39 1.26 1.21

0.78

3.74 3.23 2.78 2.52 2.36 2.13 1.87 1.69 1.63

0.23 0.26 0.30 0.34 0.86 0.40 0.45 0.50 0.52

108

0.014 0.029 0.057 0.086 0.11 0.17 0.29 0.43 0.57

0.172 0.345 0.689 1.03 1.38 2.07 3.45 5.17 6.89

1

0

0.46

1.88 1.71 1.52 1.41 1.34 1.23 1.10 1.01 1.00

1

2.18 1.98 1.76 1.63 1.55 1.42 1.27 1.17 1.16

0.75

3.06 2.78 2.47 2.29 2.18 2.00 1.79 1.64 1.63

0.29 0.32 0.36 0.38 0.40 0.44 0.49 0.54 0.54

158

0.015 0.029 0.058 0.087 0.12 0.17 0.29 0.44 0.58

0.172 0.345 0.689 1.03 1.38 2.07 3.45 5.17 6.89

1

0

0.46

1.47 1.40 1.30 1.23 1.19 1.12 1.02 1.00 1.00

1

1.65 1.57 1.40 1.38 1.34 1.26 1.14 1.12 1.12

0.72

2.39 2.28 2.11 2.00 1.93 1.82 1.66 1.63 1.63

0.38 0.40 0.43 0.46 0.47 0.50 0.55 0.56 0.56

208 258 308 358 408 458

— — — — — —

— — — — — —

1

0

0.43 0.41 0.39 0.37 0.35 0.33

1.00 0.87 0.76 0.66 0.57 0.50

1

1.09 0.92 0.78 0.66 0.55 0.47

0.70 0.67 0.63 0.60 0.57 0.54

1.63 1.41 1.24 1.07 0.93 0.81

0.07 0.68 0.80 0.95 1.14 1.34

1

0

0.40

0.4 cot

1

0.42 cot 0.65

0.65 cot 1.5 tan

1 Single row radial contact separable ball bearings (magnetic bearings)

0

0.5

2.5

—

—

—

Self-aligning ball bearings

—

0.2

Note: Values of X, Y and e for intermediate ‘relative axial loads’ and/or contact angles are obtained by linear interpolation. IS: 3824 (Part 1), 1983. a

Permissible maximum value depends on bearing design (internal clearance and raceway groove depth).

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DESIGN OF BEARINGS AND TRIBOLOGY

23.108

CHAPTER TWENTY-THREE

Particular

TABLE 23-52 Values of fc for radial roller bearings for use in Eq. (23.195) Dw cos Dpw

fc

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30

52.1 60.8 66.5 70.7 74.1 76.9 79.2 81.2 82.8 84.2 86.4 87.7 88.5 88.8 88.7 88.2 87.5 86.4 85.2 83.8

Formula

TABLE 23-53 Values of factors X and Y for radial roller bearings for use in Eq. (23-196) Fa e Fr

Fa >e Fr

Bearing type

X

Y

X

Y

e

Single-row 6¼ 0 Double-row 6¼ 0

1

0

0.4

0:4 cot

1:5 tan

1

0:45 cot

0.67

0:67 cot

1:5 tan

IS: 3824 (Part 2) 1983.

Note: Values of fc for intermediate value of Dw cos =Dw are obtained by linear interpolation. IS: 3824 (Part 2) 1983.

THRUST BEARINGS Ball bearings The basic dynamic axial load rating for a single-row, single- or double-direction thrust ball bearing

ðCa Þ ¼ 908 ¼ fc Z2=3 D1:8 w

ð23-199Þ

for Dw 25:4 mm ðCa Þ 6¼ 908 ¼ fc ðcos Þ0:7 tan Z2=3 D1:8 w

ð23-200Þ

for Dw 25:4 mm ðCa Þ ¼ 908 ¼ 3:647fc Z2=3 D1:4 w

ð23-201Þ

for Dw > 25:4 mm ðCa Þ 6¼ 908 ¼ 3:647fc ðcos Þ0:7 tan Z2=3 D1:4 ð23-202Þ w for Dw > 25:4 mm where Ca in N, Dw in mm For various values of fc refer to Table 23-54. Z ¼ number of balls carrying load in one direction.

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.109

Formula

TABLE 23-54 Values of factor fc for thrust ball bearings for use in Eqs. (23-199) to (23-202) fc

Dw Dpw

fc ¼ 908

Dw cos Dpw

¼ 458

¼ 608

¼ 758

0.01 0.02 0.03

36.7 45.2 51.1

0.01 0.02 0.03

42.1 51.7 58.2

39.2 49.1 54.2

37.3 45.9 51.7

0.04 0.05 0.06

55.7 59.5 62.9

0.04 0.05 0.06

63.3 67.3 70.7

58.9 61.6 65.8

56.1 59.7 62.7

0.07 0.08 0.09

65.8 68.5 71.0

0.07 0.08 0.09

73.5 75.9 78.0

68.4 70.7 72.6

65.2 67.3 69.2

0.10 0.12 0.14

73.3 77.4 81.1

0.10 0.12 0.14

79.7 82.3 84.1

74.2 76.6 78.3

70.7

0.16 0.18 0.20

84.4 87.4 90.2

0.16 0.18 0.20

85.1 85.5 85.4

79.2 79.6 79.5

0.22 0.24 0.26

92.8 95.3 97.6

0.22 0.24 0.26

84.9 84.0 82.8

0.28

99.8

0.28

81.3

0.30

101.9

0.30

79.6

0.32

103.9

0.34

105.8

Note: For thrust bearings > 458, values for ¼ 458 are shown to permit interpolation of values for between 458 and 608. IS: 3824 (Part 3) 1983

Bearings with two or more rows of balls The basic dynamic axial load rating for thrust ball bearings with two or more rows of similar balls carrying load in the same direction

Ca ¼ ðZ1 þ Z2 þ þ Zn Þ " Z1 10=3 Z2 10=3 þ Ca1 Ca2 # 3=10 Zn 10=3 þ þ Can

a

ð23-203Þa

Note: The designers or bearing users are advised to refer to catalogues or standards in this regard or the bearing users should consult the bearing manufacturers regarding the evaluation of equivalent load and life in case where bearing with ¼ 08 are subjected to an axial load. The ability of radial roller bearings with ¼ 08 to support axial loads varies considerably with bearing designer execution.

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DESIGN OF BEARINGS AND TRIBOLOGY

23.110

CHAPTER TWENTY-THREE

Particular

Formula

The load ratings Ca1 ; Ca1 ; . . . ; Can for the rows with Z1 ; Z2 ; . . . ; Zn balls are calculated from appropriate single row bearing formulae from Eqs. (23-199) to (23-202), Values of fc for Dw =Dpw or ðDw cos Þ= DPW and/or contact angle other than shown in Table 23-54 are obtained by linear interpolation or extrapolation.

Dynamic equivalent axial load Pa ¼ XFr þ YFa

The equivalent load for thrust ball bearings with 6¼ 908 under combined constant axial and radial loads

ð23-204Þ

For values of X and Y refer to Table 23-55. Pa ¼ Fa

The equivalent axial load for thrust bearing with ¼ 908 which can support axial loads only

ð23-205Þ

Basic rating life

The basic rating life in millions of revolutions for a thrust ball bearings

L10 ¼

Ca Pa

3 ð23-206Þ

The values of Ca and Pa are calculated in accordance with Eqs. (23-199) to (23-205).

TABLE 23-55 Values of factors X and Y for thrust ball bearings for use in Eq. (23-204) Single direction bearings a

Double direction bearings

Fa >e Fr

Fa e Fr

Fa >e Fr

X

Y

X

Y

X

Y

e

458 508 558 608 658 708 758 808 858

0.66 0.73 0.81 0.92 1.06 1.28 1.66 2.43 4.80

1

1.18 1.37 1.60 1.90 2.30 2.90 3.89 5.86 11.75

0.59 0.57 0.56 0.55 0.54 0.53 0.52 0.52 0.51 10 1 1 sin 13 3

0.66 0.73 0.81 0.92 1.06 1.28 1.66 2.43 4.80

1

1.25 1.49 1.79 2.17 2.68 3.43 4.67 7.09 14.29

1

1:25 tan

6¼ 908

2 1:25 tan 1 sin 3

1

20 1 tan 1 sin 13 3

2 1:25 tan 1 sin 3

Note: For thrust bearings > 458. Values for ¼ 458 are shown to permit interpolation of values for between 458 and 508. a Fa =Fr e is unsuitable for single direction bearings. IS: 3824 (Part 3) 1983.

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DESIGN OF BEARINGS AND TRIBOLOGY

23.111

DESIGN OF BEARINGS AND TRIBOLOGY

Particular

Formula

Adjusted rating life The adjusted rating life of (100-n) percent

Ln ¼ a1 L10

ð23-192Þ

Refer to Table 23-48 for values of factor a1 . For other adjusted rating life with modiﬁcation if required

Refer to Eqs. (23-193) to (23-194).

Roller bearings The basic dynamic axial load rating for single row, single- or double-direction thrust roller bearing

7=9 3=4 29=27 Z Dwe ðCa Þ ¼ 908 ¼ fc Lwe

ð23-207Þ

ð23-208Þ ðCa Þ 6¼ 908 ¼ fc ðLwe cos Þ7=9 tan Z3=4 D29=27 we where Ca in N, Lwe and Dwe in mm For values of factor fc refer to Table 23-56. Z ¼ number of rollers carrying load in one direction. TABLE 23-56 Values of factor fc for thrust roller bearings for use in Eqs. (23-207) and (23-208) Dwc Dpw

fc

Factor fc

¼ 908

Dw cos Dpw

¼ 508

0.01 0.02 0.03

105.4 122.9 134.5

0.01 0.02 0.03

109.7 127.8 139.5

107.1 124.7 136.2

105.6 123.0 134.3

0.04 0.05 0.06

143.4 150.7 156.9

0.04 0.05 0.06

148.3 155.2 160.9

144.7 151.5 157.0

142.8 149.4 154.9

0.07 0.08 0.09

162.4 167.2 171.7

0.07 0.08 0.09

165.6 169.5 172.8

161.6 165.5 168.7

159.4 163.2 166.4

0.10 0.12 0.14

175.7 183.0 189.4

0.10 0.12 0.14

175.5 179.7 182.3

171.4 175.4 177.9

169.0 173.0 175.5

0 16 0.18 0.20

195.1 200.3 205.0

0.16 0 18 0.20

183.7 184.1 183.7

179.3 179.7 179.3

0.22 0.24 0.26

209.4 213.5 217.3

0.22 0.24 0.26

182.6 180.9 178.7

0.28 0.30

220.9 224.3

a

¼ 658

b

a Applicable for 458 < < 608; b Applicable for 608 < < 758; c Applicable for 758 < < 908 Note: Values of fc for intermediate values of Dwc =Dpw or Dw cos =Dpw are obtained by linear interpolation. IS: 3824 (Part 4) 1983.

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¼ 808

c

DESIGN OF BEARINGS AND TRIBOLOGY

23.112

CHAPTER TWENTY-THREE

Particular

Formula

Bearing with two or more rows of rollers The basic dynamic axial load rating for thrust roller bearings with two or more rows of rollers carrying load in the same direction TABLE 23-57 Values of factors X and Y for thrust roller bearings for use in Eqs. (23-210) Fa e Fr Bearings type

X

Single-direction a 6¼ 908 Double-direction 1.5 tan 6¼ 908

Fa >e Fr Y

Ca ¼ ðZ1 Lwe1 þ Z2 Lwe2 þ þ Zn Lwen Þ " Z1 Lwe1 9=2 Z2 Lwe2 9=2 þ Ca1 Ca2 þ þ

Zn Lwen Can

9=2 #2=9 ð23-209Þ

where Ca in N, Lwe and Dwe in mm

Y

X

e

a

tan 1

1.5 tan

0.67 tan 1

1.5 tan

The load ratings Ca1 ; Ca2 ; . . . ; Can for the rows with Z1 ; Z2 ; . . . ; Zn rollers of length Lwe1 ; Lwe2 ; . . . ; Lwen are calculated from the appropriate single row bearing Eqs. (23-207) and (23-208).

* Fa =Fr e is unsuitable for single-direction bearing. IS: 3824 (Part 4) 1983.

The equivalent axial load for thrust roller bearings when 6¼ 908 under combined constant axial and radial load The equivalent axial load for thrust roller bearings with ¼ 908 which can support only axial load The basic rating life in millions of revolutions for thrust roller bearings

Pa ¼ XFr þ YFa

ð23-210Þ

For values of X and Y refer to Table 23-57. Pa ¼ Fa L10 ¼

ð23-211Þ Ca Pa

10=3 ð23-212Þ

The values of Ca and Pa are calculated in accordance with Eqs. (23-207), (23-208), and (23-210).

Adjusted rating life The Eqs. (23-192), (23-193) and (23-194) for adjusted rating life with appropriate modiﬁcation to suit the roller thrust bearings are repeated here

Ln ¼ a1 L10

ð23-192Þ

L10a ¼ a2 a3 L10

ð23-193Þ

Lna ¼ a1 a2 a3 L10

ð23-194Þ

Variable bearing load and speed The mean aﬀective load Fm under varying load and varying speed n1 , n2 , n3 ; . . . ; ni at which the individual loads F1 , F2 , F3 ; . . . ; Fi act.

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ F1m n1 þ F2m n2 þ F3m n3 þ þ Fim ni Fm ¼ n m

rP ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðFi Þm ni m Fm ¼ n

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ð23-213aÞ ð23-213bÞ

DESIGN OF BEARINGS AND TRIBOLOGY

23.113

DESIGN OF BEARINGS AND TRIBOLOGY

Particular

Formula

where F1 ; F2 ; F3 ; . . . ; Fi ¼ constant loads among series of i loads during n1 ; n2 ; n3 ; . . . ; ni revolutions. ni ¼ number of revolutions at which Fi load operates n ¼ total number of revolutions in a complete cycle ¼ n1 þ n2 þ n3 þ . . . þ ni , F1 ; F2 ; F3 ; . . . ; Fi act

during

which

loads

m ¼ exponent mi ¼ 3 for ball bearings mi ¼ 10 3 for roller bearings Fmin þ 2max 3

The mean eﬀective load Fm under linearly varying load from minimum load Fmin to maximum load Fmax at constant speed n.

Fm ¼

The equivalent dynamic load for the varying load which acts in a radial direction only for radial bearings and in a axial direction only for thrust bearing.

P ¼ Fm

ð23-215Þ

In the direction and magnitude of load changes with time then the equivalent loads P1 , P2 , P3 ; . . . ; must be calculated for the individual time periods n1 , n2 , n3 using the general equation.

P ¼ XFr þ YFa

ð23-216Þ

The mean equivalent load Pm by substituting the individual values of P1 , P2 , P3 ; . . . ; obtained from equivalent load’s Eq. (23-119).

ð23-214Þ

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m m Pm 1 n1 þ P2 n2 þ P3 n3 þ n where

Pm ¼

m

ð23-217Þ

m ¼ exponent ¼ 3 for ball bearings

The life of a bearing under variable load and variable speed, taking into consideration life adjustment factors a1 , a2 , a3 and application factor Ka The basic load rating for a required bearing life in case of variable load and variable speed, factor Ka and a1 , a2 , a3

¼ 10 3 for roller bearings m C 1 L ¼ a1 a2 a3 Ka F1m n1 þ F2m n2 þ F3m n3 þ . . . C ¼ Ka ðF1m n1 þ F2m n2 þ F3m n3 þ . . .Þ

L a1 a2 a3

ð23-218Þ 1=m ð23-219Þ

where L is in millions of revolutions; C and F in N; n1 ; n2 ; n3 ; . . . are rotational speeds in rpm under loads F1 ; F2 ; F3 ; . . . m ¼ 3 for ball bearings ¼ 10 3 for roller bearings

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DESIGN OF BEARINGS AND TRIBOLOGY

23.114

CHAPTER TWENTY-THREE

TABLE 23-58 Index fL of dynamic stressing for use in Eq. (23-180) Application

fL

Motor vehicles Motorcycles Light cars Heavy cars Light trucks or lorries Heavy trucks or lorries Buses Tractors Tracked vehicles

1.4–1.9 1.6–2.1 1.7–2.2 1.7–2.2 2.0–2.6 2.0–2.6 1.6–2.2 2.1–2.7

Electric motors For household appliances Small standard motors Medium-sized standard cars Large motors Traction motors

1.5–2.0 2.5–3.5 3.0–4.0 3.5–4.5 3.0–4.0

Railbound vehicles Axle boxes for haulage trolleys Trams Railway coaches Freight cars Overburden removal cars Outer bearings of locomotives Inner bearings of locomotives Gears

3.0–4.0 4.5–5.5 4.0–5.0 3.5–4.0 3.5–4.0 4.0–5.5 4.5–5.5 3.5–4.5

Rolling mills Neck bearings Gears

2.0–2.5 3.0–5.0

Ship building Ship propeller thrust blocks Ship propeller shaft bearings Large marine gears

2.9–3.6 6.0 2.6–4.0

General engineering Small universal gears Medium-sized universal gears Small fans

2.5–3.5 3.0–4.0 2.5–3.5

Application

fL

Medium-sized fans 3.0–4.5 Large fans 4.5–5.5 Centrifugal pumps 2.5–4.5 Centrifuges 3.0–4.0 Winding cable sheaves 4.5–5.0 Belt conveyor idlers 3.0–4.5 Conveyor drums 4.5–5.5 Shovels and reclaimers 6.0 Crushers 3.0–3.5 Beater mills 3.5–4.5 Tube mills 6.0 Vibrating screens 2.5–2-8 Vibrating rolls and large out-of-balance exciters 1.6–2.0 Vibrators 1.0–1.5 Briquette presses 4.5–5.0 Large mechanical stirrers 3.5–4.0 Rotary furnace rollers 4.5–5.0 Flywheels 3.4–4.0 Printing machines 4.0–4.5 Papermaking machines Wet sections Dry sections Reﬁners Calendars

5.0–6.0 5.0–6.0 4.5–4.6 4.0–4.5

Centrifugal casting machines

3.4–4.0

Textile machines

3.6–4.7

Machine tools Lathes, boring and milling machines Grinding, lapping, and polishing machines

2.7–4.5 2.7–4.5

Woodworking machines Milling cutters and cutter shafts Saw mills (con rods)

3.0–4.0 2.8–3.3

Machines for working of wood and plastics

3.0–4.0

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.115

Formula

Reliability The reliability (Ri ) of a group of i bearings

The expression for reliability (R) as per Weibull threeparameter

Ri ¼ ðRÞi

ð23-220Þ

where R ¼ reliability of each bearing " " # # x xo b L=L10 xo b ¼ exp R ¼ exp xo xo ð23-221aÞ

Another Weibull three-parameter equation for reliability (R) for bearings.

" # ðL=L10 Þ 0:02 1:40 R ¼ exp 4:91

The reliability (R) of bearing using Weibull twoparameter for tapered roller bearings.

" # " # x b L=L10 1:5 ¼ exp ð23-222aÞ R ¼ exp 4:48

ð23-221bÞ

where x ¼ life measure xo ¼ guaranteed values of life measure ¼ Weibull characteristic of life measure

Another form of reliability (R) equation for bearing using Weibull two-parameter

b ¼ Weibull exponent/shape parameter " # L b R ¼ exp mL10

ð23-222bÞ

where R ¼ reliability corresponding to life L L10 ¼ rating life ðR ¼ 0:90Þ Weibull two-parameter equation for reliability is obtained from Eq. (23-225a) by putting b ¼ 1.17 and ¼ 6.84.

m ¼ scale constant " 1:17 # L R ¼ exp 6:84L10

Weibull equation for the distribution of bearing rating life based on reliability.

L ¼ L10

The relation between the design or required values and the dynamic load rated or catalog values (Cr ) according to the Timken Engineering is given by

Cr ¼ Fr

lnð1=RÞ lnð1=R10 Þ

"

Ld Lr

ð23-223Þ

1=b

nd nr

ð23-224Þ #1=m ð23-225Þ

where subscripts d and r stand for design and rated values Cr ¼ basic load capacity or dynamic load rating corresponding to Lr hours of L10 life at the speed nr in rpm, kN

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DESIGN OF BEARINGS AND TRIBOLOGY

23.116

CHAPTER TWENTY-THREE

Particular

Formula

Fr ¼ actual radial bearing load carried for Ld hours of L10 life at the speed nd in rpm, kN m ¼ an exponent which varies from 3 to 4 "

The basic dynamic capacity or speciﬁc dynamic capacity of bearing corresponding to any desired life L at the reliability R

Cr ¼ Fr

Ld Lr

nd nr

1 6:84

#1=m

1 ½lnð1=RÞ1=1:17m ð23-226Þ

Another equation connecting catalog radial load rating (Fr ), the design radial load (Fd ) and reliability (R).

Cr ¼ Fd

1=m

Ld nd =Lr nr 0:02 þ 4:439½lnð1=RÞ1=1:483

ð23-227Þ

where Cr ¼ the catalog radial load rating corresponding to Lr hours of life at the rated speed nr in rpm, kN Fd ¼ the design radial load corresponding to the required life of Ld hours at a design speed of nd in rpm, kN R ¼ reliability

Roller bearing, fL 1.00

3

2

0.70

n=

10

1

0.7

.

m r.p.

0.70

20 50 70 100

0.40 Roller bearing, P C

0.50

30

0.50

0.40 0.30

200 300 500 700 0 100

0.30

0.20

000

2

0.20

1.00

0

300

0 500 00 70 00 100

0.10

000

0.10

20 00 300

0.07

0.07

0.05 5

4

3

2 Ball bearing, fL

1

0.7

FIGURE 23-56 Selection of bearing size.

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Ball bearing, P C

4

DESIGN OF BEARINGS AND TRIBOLOGY

THE EQUIVALENT DYNAMIC LOAD FOR ANGULAR CONTACT BALL BEARINGS B (a) Direct Mounting or Fronts of Bearings Fa Facing each others

A

FrA

FrB

B

A

(b) Indirect Mounting or Backto-back Mounting

Fa FrA

FrB

FIGURE 23-57 Angular contact ball bearings mounted on a single shaft. Thrust load to be used in equivalent load calculation Condition of load

Bearing A (Fig. 23-57)

Bearing B (Fig. 23-57)

FrB FrA YB YA

—

Fa þ 0:5

FrA YA

(23-228)

—

Fa þ 0:5

FrA YA

(23-229)

FrB FrA > YB YA F F Fa > 0:5 rB rA YB YA FrB FrA > YB YA F F Fa 0:5 rB rA YB YA

0:5

FrB Fa YB

—

(23-230)

Where thrust factors are: Y ¼ 0:57 for Series 72B (Series 02) and 73B (Series 03); Y ¼ 1:19 for Series LS AC and MS AC; Y ¼ 0:87 for Series 173 and 909; Y ¼ 0:66 for Fa =Fr 0:95 and Y ¼ 1:07 for Fa =Fr > 0:95 for Series 33.

Fraction of basic dynamic capacity, C’ C

1 0.9 0.8 0.7 0.6

ball bearings roller bearings

0.5 0.4 0.3 0.01

0.02 0.04 0.08 0.03 0.06 0.1

0.2 0.30.4 0.6 0.81

2

3 4

6 8 10

Probability of failure, F(percent)

FIGURE 23-58 Reduction in life for reliabilities greater than 90%. (Courtesy: Tedric A Harris, Predicting Bearing Reliability, Machine Design, Vol. 35, No. 1, Jan. 3, 1963, pp. 129–132)

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DESIGN OF BEARINGS AND TRIBOLOGY

23.118

CHAPTER TWENTY-THREE

Particular

Formula

An expression for tapered roller bearings connecting catalog radial load rating (Fr ), the design radial load (Fd ) and reliability

Cr ¼ Fd

The radial equivalent or eﬀective load when the cup rotates in case of tapered roller bearing (Fig. 23-50)

Fr ¼ 1:25Fr

The thrust component of pure radial load (Fr ) due to the tapered roller

Fan ¼

Ld nd =Lr nr

3=10

4:4½lnð1=RÞ1=1:5

ð23-231Þ ð23-232Þ

where Fr is the calculated radial load, kN 0:47Fr K where

K¼

ð23-233Þ

radial rating of bearing thrust rating of bearing

¼ 1:5 for radial bearings ¼ 0:75 for steep-angle bearings The net thrust on the tapered roller bearing when the induced thrust (Far ) is deducted from the applied thrust (Faa )

Fnt ¼ Faa Far

ð23-234Þ

0:47Fr K 0:47Fr Fe ¼ Fr þ K Faa K

ð23-235Þ

Fe ¼ 0:53Fr þ KFnt 0:47Fr Fe ¼ 1:25Fr þ K Faa K

ð23-237Þ

Fe ¼ 0:78Fr þ KFnt

ð23-239Þ

Fnt ¼ Faa

The radial equivalent load when the cup rotates in case of tapered roller bearing (Fig. 23-50)

The radial equivalent load when the cone rotates in case of tapered roller bearing (Fig. 23-50)

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ð23-236Þ

ð23-238Þ

DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.119

Formula

THE EQUIVALENT DYNAMIC LOAD FOR TAPERED ROLLER BEARINGS A (a) Indirect Mounting or Backs of Bearings Fo Facing each other

B

B

A

Fo FrA

FrA

FrB

FrB

(b) Direct Mounting or Fronts of Bearings Facing each other

FIGURE 23-59 Two taper roller bearings mounted on a single shaft.

Thrust load to be used in equivalent load calculation Condition of load

Bearing A (Fig. 23-59)

Bearing B (Fig. 23-59)

FrB FrA YB YA

—

Fa þ 0:5

FrA YA

(23-240)

—

Fa þ 0:5

FrA YA

(23-241)

FrB FrA > YB YA F F Fa > 0:5 rB rA YB YA FrB FrA > YB YA F F Fa 0:5 rB rA YB YA

0:5

FrB Fa YB

—

(23-242)

The thrust factors Y and Ye are taken from Table 23-39 and 23-47a.

The radial equivalent load on bearing A according to Timken Engineering Journal (Fig. 23-59) The radial equivalent load on bearing B according to Timken Engineering Journal (Fig. 23-59)

0:46FrB FaA ¼ 0:4FrA þ KA Fa þ KB FeB ¼ 0:4FrB þ KB

0:47FrA Fa KA

ð23-243Þ

ð23-244Þ

DIMENSIONS, BASIC LOAD RATING CAPACITY, FATIGUE LOAD LIMIT AND MAXIMUM PERMISSIBLE SPEED OF ROLLING CONTACT BEARINGS Deep groove ball bearings—Series 02, Series 03, Series 04

Refer to Tables 23-60, 23-61, and 23-62 respectively.

Self-aligning and deep groove ball bearings—Series 02, Series 03, Series 22 (FAG) and Series 23 (FAG)

Refer to Tables 23-63, 23-64, 23-65 and 23-66 respectively.

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DESIGN OF BEARINGS AND TRIBOLOGY

23.120

CHAPTER TWENTY-THREE

Particular

Formula

Single row angular contact ball bearings—Series 02 and Series 03

Refer to Tables 23-67 and 23-68.

Double row angular contact ball bearings—Series 33 (FAG)

Refer to Table 23-69.

Cylindrical roller bearings—Series 02, Series 03, Series 04, Series NU 22 (FAG), Series NU 23 (FAG)

Refer to Tables 23-70, 23-71, 23-72, 23-73, and 23-74.

Tapered roller bearings—Series 322, Series 02 (22) and Series 03 (23)

Refer to Tables 23-75, 23-76, 23-76A, 23-76B and 2377.

Single thrust ball bearings—Series 11, Series 12, Series 13 and Series 14

Refer to Tables 23-78, 23-79, 23-80, and 23-81.

Double thrust ball bearing-Series 522 (FAG)

Refer to Table 23-82.

Selection of bearing size

Refer to Table 23-83.

NEEDLE BEARING LOAD CAPACITY For various types of needle roller bearings and for some of their characteristics

Refer to Table 23-59.

The capacity of needle bearing at 3000 h average life

Zld Cn ¼ 1:76 107 p 3 ﬃﬃﬃﬃ0 n

ð23-245Þ

where Cn in N, l, and d in m, and n0 in rps The load capacity of needle bearing based on the projected area of the needle-rollers

Cn ¼ 5:33

Lðdi þ dr Þ p 3 ﬃﬃﬃﬃ0 n

ð23-246Þ

where Cn in N, l di ,and dr in m, and n0 in rps The load capacity of needle bearing is also calculated from formula

Cn ¼ Kh Kl pldi

ð23-247Þ

For hardness factors Kh refer to Table 23-83 and for life factor Kl refer to Fig. 23-55.

PRESSURE The pressure for wrist pin rocker arm and similar oscillating mechanism is given by

P ¼ 34:32 MPa

The rotary motion pressure may be computed from the relation

2:86 106 P¼ p 3 ﬃﬃﬃﬃﬃﬃﬃﬃﬃ0ﬃ D1 n

ð23-248Þ

where P in Pa, D1 in m, and n0 in rps Check for total circumferential clearance from formula

c ¼ ðdi þ dr Þ Zdr

For dimensions, design data and sizes for needle bearings.

Refer to Tables 23-84 to 23-88.

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ð23-249Þ

DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.121

Formula

TABLE 23-59 Typical forms of needle roller bearings and some of their important characteristics. Bore size (in.) Type

min

max

Dynamic

State

Limiting speed factor

Drawn cup needle

0.125

7.250

High

Moderate

0.3

Low

Drawn cup needle grease retained

0.156

1.000

High

Moderate

0.3

Low

Drawn cup roller

0.187

2.750

Moderate

Moderate

0.9

Moderate

Heavy duty roller

0.625

9.250

Very high

Moderate

1.0

Moderate

Caged roller

0.500

4.000

Very high

High

1.0

Moderate

Cam follower

0.5000

6.000

Moderate to high

Moderate to high

0.3–0.9

Low

Needle thrust

0.252

4.127

Very high

Very high

0.7

Low

Open end

Open end

Relative load capacity

Misalignment tolerance

Close end

Close end

Courtesy: Machine Design, 1970 Bearings Reference Issue, The Penton Publishing Co., Cleveland, Ohio.

HERTZ-CONTACT PRESSURE Maximum contact pressure between cylinders and spheres of steel ( ¼ 0.3) (i) For cylinders

(ii) For a cylinder and plane

cðmaxÞ

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2FEðd1 þ d2 Þ ¼ 0:418 ld1 d2 rﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2FE ld sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 3 4Fðd1 þ d2 Þ E ¼ 0:388 d12 d22

ð23-250Þ

cðmaxÞ ¼ 0:418

ð23-251Þ

cðmaxÞ

ð23-252Þ

(iii) For two spheres

(iv) For a sphere and plane

cðmaxÞ

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 3 4FE ¼ 0:388 2 d

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ð23-253Þ

d

r

B

r

D

Old

10BC02 12BC02 15BC02 17BC02 20BC02 25BC02 30BC02 35BC02 40BC02 45BC02 50BC02 55BC02 60BC02 65BC02 70BC02 75BC02 80BC02 85BC02 90BC02 95BC02 100BC02 105BC02 110BC02 120BC02

New

10 12 15 17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 120 130 140 150 160 170 180 190 200 220 240 260 280

IS No.

Bearing No.

6200 6201 6202 6203 6204 6205 6206 6207 6208 6209 6210 6211 6212 6213 6214 6215 6216 6217 6218 6219 6220 6221 6222 6224 6226 6228 6230 6232 6234M 6236M 6238M 6240M 6242M 6244M 6246M 6248M

FAG 6200 01 02 03 04 05 6206 07 08 09 10 6211 12 13 14 6215 16 17 18 19 6220 21 22 24 6226 6228 6230 32 34 36 38 40 6244 6248 6252 6256

SKF 10 12 15 17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 120 130 140 150 160 170 180 190 200 220 240 260 280

d 30 32 35 40 47 52 62 72 50 85 90 100 110 120 125 130 140 150 160 170 180 190 200 215 230 250 270 290 310 320 340 360 400 440 480 500

D 9 10 11 12 14 15 16 17 18 19 20 21 22 23 24 25 26 28 30 32 34 36 38 40 40 42 45 48 52 52 55 58 65 72 80 80

B

r 0.6 0.6 0.6 0.6 1.0 1.0 1.0 1.1 1.1 1.1 1.1 1.5 1.5 1.5 1.5 1.5 2.0 2.0 2.0 2.1 2.1 2.1 2.1 2.1 3.0 3.0 3.0 3.0 4.0 4.0 4.0 4.0 4.0 5.0 5.0 6.0

Dimensions, mm

.025 .04 .07 .13 .25 .50

Fa =Co 2.0 1.8 1.6 1.4 1.2 1.0

Y

Factor

TABLE 23-60 Deep groove ball bearings—Diameter series 2 (Series 02) (Indian Standards)

0.22 .24 .27 .31 .37 .44

e 2.60 3.10 3.75 4.75 6.55 7.80 11.20 15.30 18.00 20.40 24.00 29.00 36.00 41.50 44.00 49.00 53.00 64.00 72.00 81.50 93.00 104.00 116.00 122.00 146.00 166.00 170.00 204.00 224.00 245.00 280.00 310.00 355.00 475.00 560.00 600.00

kN

FAG

2360 3100 3750 4750 6550 7800 11200 15300 19000 21600 23200 29000 32500 40500 45000 49000 55000 64000 73500 81500 93000 104000 118000 118000 132000 150000 166000 186000 224000 240000 280000 310000 365000 475000 530000 600000

N

SKF

Static, Co

6.00 6.95 7.80 9.50 12.70 14.00 19.30 25.50 29.00 31.00 36.50 43.00 52.00 60.00 62.00 65.50 72.00 83.00 96.50 108.00 122.00 132.00 143.00 146.00 166.00 176.00 176.00 200.00 212.00 224.00 255.00 270.00 300.00 360.00 405.00 425.00

kN

FAG

5070 6890 7800 9560 12700 14000 19500 25500 30700 33200 35100 43600 47500 55900 60500 66300 70200 83200 95600 10800 124000 133000 143000 146000 156000 165000 174000 186000 212000 229000 255000 270000 296000 358000 390000 423000

N

SKF

Dynamic, C

Basic load rating capacity

100 132 160 200 280 335 475 655 860 915 980 1250 1400 1730 1900 2040 2200 2500 2890 3000 3350 3650 4000 3900 4150 4150 4900 5300 6100 7350 7350 7800 8800 10800 11800 12900

N

Fatigue load limit, Fa SKF/FAG

32000 30000 26000 22000 18000 17000 14000 24000 20000 19000 18000 16000 14000 13000 12000 11000 11000 10000 9000 8500 8000 7500 7000 6700 6300 6000 5600 5600 5300 4800 4300 4000 3600 3400 3200 3000

rpm

Kinematically permissible speed, n SKF/PAG

0.031 0.038 0.044 0.063 0.105 0.128 0.199 0.290 0.372 0.430 0.466 0.616 0.785 1.000 1.080 1.200 1.460 1.870 2.230 2.740 3.300 3.880 4.640 5.630 6.24 8.07 10.30 14.70 18.30 19-00 22.80 27.20 37.90 51.30 68.40 72.90

Mass kg

DESIGN OF BEARINGS AND TRIBOLOGY

23.122

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B

r

D

d

Old

10BC03 12BC03 15BC03 17BC03 20BC03 25BC03 30BC03 35BC03 40BC03 45BC03 50BC03 55BC03 60BC03 65BC03 70BC03 75BC03 80BC03 85BC03 90BC03 95BC03 100BC03 105BC03 110BC03 120BC03

New

10 12 15 17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 120 130 140 150 160 170 180 190 200 220 240

IS No.

Bearing No.

6300 6301 6302 6303 6304 6305 6306 6307 6308 6309 6310 6311 6312 6313 6314 6315 6316 6317 6318 6319 6320 6321 6322 6324 6326M 6328M 6330M 6332M 6334M 6336M 6338M 6340M 6344M 6348M

FAG 6300 6301 6302 6303 04 05 06 07 08 09 6310 11 12 13 14 15 16 17 18 19 20 6321 22 24 26 28 30 32 34 6336 38 40 44

SKF 10 12 15 17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 120 130 140 150 160 170 180 190 200 220 240

d 30 37 42 47 52 62 72 80 90 100 110 120 130 140 150 160 170 180 190 200 215 225 240 260 280 300 320 340 360 380 400 420 460 500

D 11 12 13 14 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 50 55 58 62 65 68 72 75 78 80 88 95

B

r 0.6 1.0 1.0 1.0 1.1 1.1 1.1 1.5 1.5 1.5 2 2 2.1 2.1 2.1 2.1 2.1 3.0 3.0 3.0 3.0 3.0 3.0 3.0 4.0 4.0 4.0 4.0 4.0 4.0 5.0 5.0 5.0 5.0

Dimensions, mm

.025 .04 .07 .13 .25 .05

Fa =Co 2.0 1.8 1.6 1.4 1.2 1.0

Y

Factor

TABLE 23-61 Deep groove hall bearings—Diameter series 3 (Series 03) (Indian Standards)

0.22 .24 .27 .31 .37 .44

e 3.45 4.15 5.40 6.55 7.80 11.40 16.30 19.00 25.00 32.00 38.00 47.00 52.00 60.00 68.00 76.50 86.50 88.00 102.00 112.00 134.00 146.00 166.00 190.00 216.00 245.00 300.00 325.00 365.00 405.00 440.00 465.00 550.00 620.00

kN

FAG

3400 4150 5400 6550 7800 11600 16000 19000 24000 31500 38000 45000 52000 60000 68000 76500 86500 96500 10800 118000 140000 153000 180000 186000 216000 245000 285000 285000 340000 405000 430000 465000 520000

N

SKF

Static, Co

8.15 9.65 11.40 13.40 16.00 22.40 29.00 33.50 42.50 53.00 62.00 76.50 81.50 93.00 104.00 114.00 122.00 125.00 134.00 143.00 163.00 173.00 190.00 212.00 228.00 255.00 285.00 300.00 325.00 355.00 375.00 380.00 430.00 465.00

kN

FAG

8060 9750 11400 13500 15900 22500 28100 33200 41000 52700 61800 71500 81900 92300 104000 114000 124000 133000 143000 153000 174000 182000 203000 208000 290000 251000 276000 276000 312000 351000 371000 377000 410000

N

SKF

Dynamic, C

Basic load rating capacity

143 176 228 275 335 490 670 815 1020 1340 1600 1900 2200 2500 2750 3000 3250 3550 3800 4150 4750 5100 5700 5700 6300 7100 7800 7650 8800 10800 10800 11200 12000

N

Fatigue load limit, Fa SKF/FAG

56000 53000 43000 39000 34000 28000 24000 20000 18000 16000 14000 13000 12000 11000 10000 9500 9000 8000 8000 7500 7000 6700 6300 6000 5600 5300 4800 4300 4000 3800 3600 3400 3200 3000

rpm

Kinematically permissible speed, n SKF/FAG

0.058 0.062 0.087 0.116 0.153 0.237 0.355 0.472 0.639 0.853 1.090 1.400 1.750 2.140 2.610 3.180 3.800 4.350 5.430 6.230 7.670 8.700 10.300 12.800 18.300 22.300 26.700 31.800 37.300 43.600 50.400 56.600 75.000 96.400

Mass kg

DESIGN OF BEARINGS AND TRIBOLOGY

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23.123

B

r

D

6403 6404 6405 6406 6407 6408 6409 6410 6411 6412 6413 6414 6415M 6416M 6417M 6418M

15BC04 17BC04 20BC04 25BC04 30BC04 35BC04 40BC04 45BC04 50BC04 55BC04 60BC04 65BC04 70BC04 75BC04 80BC04 85BC04 90BC04

17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

6403 04 05 06 07 08 09 10 6411 12 13 14 15 16 17 18

SKF 15 17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

d 52 62 72 80 90 100 110 120 130 140 150 160 180 190 200 210 225

D 15 17 19 21 23 25 27 29 31 33 35 37 42 45 48 52 54

B

Dimensions, mm

1.1 1.1 1.1 1.5 1.5 1.5 2 2 2.1 2.1 2.1 2.1 3.0 3.0 3.0 4.0 4.0

r 0.025 .04 .07 .13 .25 .5

Fa =Co 2.0 1.8 1.6 1.4 1.2 1.0

Y

Factor

.22 .24 .27 .31 .37 .44

e

11.0 15.0 19.3 23.2 31.0 36.5 45.0 52.0 62.0 69.5 78.0 104.0 114.0 125.0 137.0 163.0

kN

FAG

10680 15100 18500 22690 29790 36900 42880 48900 57340 64880 75570 99570 106720 117800 128920 142350

N

SKF

Static, Co

23.6 30.5 36.0 42.5 55.0 63.0 76.5 86.5 100.0 110.0 118.0 143.0 153.0 163.0 173.0 196.0

kN

FAG

17350 23570 27540 32690 42240 48950 57330 67590 76880 82680 90700 108880 117800 124460 133280 142250

N

SKF

Dynamic, C

Basic load rating capacity

Source: 1. Extracted with permission from ‘‘FAG Rolling Bearings’’, Catalogue WL 41520 EI, 1995 Edition: FAG Precision Bearings Ltd, Manja, Vadodara, India. 2. Courtesy: Extracted from SKF Rolling Bearings, Catalogue 4000 E, 1989, SKF Rolling Bearings India Ltd., Mumbai, India.

d

r

FAG

Old

New

IS No.

Bearing No.

TABLE 23-62 Deep groove ball bearing—Diameter Series 4 (Series 04) Indian Standards

30000 26000 22000 19000 16000 15000 13000 12000 11000 10000 9500 8500 8000 7500 7000 6700

rpm

Kinematically permissible speed, n FAG

0.275 0.412 0.546 0.746 0.928 1.18 1.51 1.83 2.40 2.90 3.49 4.80 5.64 6.63 9.52 11.6

Mass kg

DESIGN OF BEARINGS AND TRIBOLOGY

23.124

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B

r

d

10B502 12B502 15B502 17B502 20B502 25B502 30B502 35B502 40B502 45B502 50B502 55B502 60B502 65B502 70B502 75B502 80B502 85B502 90B502 95B502 100B502 105B502 110B502 120B502

10 12 15 17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 120

1200TV 1201TV 1202TV 1203TV 1204TV 1205TV 1206TV 1207TV 1208TV 1209TV 1210TV 1211TV 1212TV 1213TV 1214TV 1215TV 1216TV 1217TV 1218TV 1219M 1220M 1221M 1222M 1224M

FAG 1200E 01E 02E 1203E 04E 05E 1206E 07E 08E 1209E 10E 11E 1212E 13E 14 1215 16 17 1218 19. 20 1221 22 24

SKF 10 12 15 71 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 120

d 30 32 35 40 47 52 62 72 80 85 90 100 110 120 125 130 140 150 160 170 180 190 200 215

D 9 10 11 12 14 15 16 17 18 19 20 21 22 23 24 25 26 28 30 32 34 36 38 42

B 0.6 0.6 0.6 0.6 10 1.0 1.1 1.1 1.1 1.1 1.1 1.5 1.5 1.5 1.5 1.5 2.0 2.0 2.0 2.1 2.1 2.1 2.1 2.1

r min

Dimensions, mm

.32 .37 .34 .33 .27 .27 .22 .22 .22 .21 .20 .19 .18 .18 .19 .19 .16 .17 .17 .17 .18 .18 .17 .25

e 2.05 1.77 1.95 2.03 2.34 2.48 2.94 2.65 3.04 3.18 3.32 3.47 3.64 3.74 3.52 3.48 4.08 3.91 3.92 3.91 3.75 3.5 3.78 3.25

Yo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1.95 1.69 1.86 1.93 2.24 2.37 2.53 2.18 2.90 3.04 3.17 3.31 3.47 3.57 3.36 3.32 3.90 3.73 3.74 3.73 3.58 3.68 3.61 3.11

.65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65

3.2 2.62 2.98 2.00 3.46 3.66 3.91 4.34 4.49 4.70 4.90 5.12 5.37 5.52 5.21 5.15 6.03 5.78 5.79 5.78 5.53 5.48 5.58 4.81

Y

X

X

Y

Fn =Fr > e

Fn =Fr e

Factors

1.2 1.27 1.76 2.04 2.65 3.35 4.65 5.20 6.55 7.35 8.15 10.00 11.60 12.50 13.70 15.60 17.00 20.40 23.6 27.0 29.0 32.0 38.0 53.0

kN

FAG kN

FAG

5530 6240 7410 8840 12700 14300 15600 19000 19900 22900 26500 27600 31200 35100 34500 39000 39700 48800 57200 65700 68900 74100 88400 119000

N

SKF

Dynamic, C

1180 5.5 1430 5.6 1760 7.5 3000 8.0 3400 10.0 4000 12.2 4650 15.6 6000 16.0 6950 19.3 7800 22.0 9150 22.8 10600 27.0 12200 30.0 14000 31.0 13700 34.5 15000 39.0 17000 46.0 20900 49.0 23600 57.0 27000 64.0 30000 69.5 32500 75.0 39000 88.0 53000 120.0

N

SKF

Static, Co

Basic load rating capacity

61 72 90 114 176 204 240 305 355 400 475 540 620 720 710 800 830 980 1080 1200 1200 1370 1600 2120

N

Fatigue load limit, Fn SKF

30000 30000 26000 22000 18000 16000 14000 12000 10000 9000 8500 7500 6700 6300 6000 5600 5000 4500 4500 6000 5600 5300 5000 4800

rpm

Kinematically permissible speed, n FAG

0.034 0.041 0.048 0.073 0.119 0.139 0.222 0.322 0.415 0.463 0.525 0.685 0.895 1.16 1.25 1.34 1.66 2.06 2.50 3.40 3.29 4.31 5.67 7.43

kg

Mass FAG

Note: SKF 1984; FAG, 1995,: These values of C and Co of SKF ball bearings refer to old standards in Table 23-62, EK ¼ tapered bore; TV ¼ self-aligning ball bearings with cages of glass ﬁbre reinforced polyamide 66, M ¼ machined brass cage. Source: 1. Extracted with permission from ‘‘FAG Rolling Bearings’’, CatalogueWL 41520EI, 1995 Edition: FAG Precision Bearings Ltd, Maneja, Vadodara, India. 2. Courtesy: Extracted from SKF Rolling Bearings, Catalogue 4000E, 1989, SKF Rolling Bearings. India Ltd., Mumbai, India.

D

r

Old

New

IS No.

Bearing No.

TABLE 23-63 Self-aligning ball bearings—Diameter Series 12 (Series 02, Indian Standards)

DESIGN OF BEARINGS AND TRIBOLOGY

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23.125

r

B

r

d

10B503 12B503 15B503 17B503 20B503 25BS03 30B503 35BS03 40B503 45B503 50B503 55B503 60B503 65B503 70B503 70B503 80B503 85B503 90B503 95B503 100B503 105B503 110B503

10 12 15 17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110

1300 1301 1302 1303TV 1304TV 1305TV 1306TV 1307TV 1308TV 1309TV 1310TV 1311TV 1312TV 1313TV 1314M 1315M 1316M 1317M 1318M 1319M 1320M 1321M 1322M

FAG

d

1300 10 01E 12 02E 15 1303E 17 04E 20 05E 25 1306E 30 07E 35 08E 40 1309E 45 10E 50 11E 55 1312E 60 13E 65 14 70 1315 75 16 80 17 85 1318 90 19 95 20 100 1321 105 22 110

SKF 35 37 42 47 52 62 72 80 90 100 110 120 130 140 150 160 170 180 190 200 215 225 240

D 11 12 13 14 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 50

B 1.0 1.5 1.5 1.0 1.1 1.1 1.1 1.5 1.5 1.5 2.0 2.0 2.1 2.1 2.1 2.1 2.1 3.0 3.0 3.0 3.0 3.0 3.0

r min

Dimensions, mm

.34 .35 .35 .32 .29 .28 .26 .26 .25 .25 .24 .24 .23 .23 .23 .23 .22 .22 .22 .23 .23 .23 .23

e 1.90 1.90 1.90 2.03 2.27 2.40 2.51 2.59 2.64 2.62 2-73 2.79 2.90 2.88 2.93 2.90 3.00 3.02 2.97 2.50 2.81 2.88 2.92

Yo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1.90 1.80 1.80 1.94 2.17 2.29 2.39 2.47 2.52 2.50 2.60 2.66 2.77 2.75 2.79 2.77 2.87 2.88 2.83 2.73 2.68 2.75 2.79

.65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65

2.90 2.80 2.80 3.00 3.50 3.54 3.71 3.82 3.90 3.87 4.03 4.12 4.28 4.26 4.32 4.29 4.44 4.46 4.38 4.23 4.15 4.25 4.32

Y

X

X

Y

Fn =Fr > e

Fn =Fr e

Factors, FAG

3.20 3.35 5.00 6.30 8.00 9.65 12.90 14.30 18.00 20.80 22.80 27.50 30.00 32.50 38.00 43.00 51.00 58.50 65.50 71.00

kN

FAG

7200

2160 2600 3400 4600 5400 6800 8500 11200 13400 14000 18000 22000 25500 27500 30000 33500 38000 44000 51000 57000

N

SKF

Static, Co

9360 10800 12700 17000 19000 22500 26500 33800 39000 43600 50700 58500 65000 74100 79300 88400 97500 117000 133000 143000

N

SKF

12.50 12.50 18.00 21.20 25.00 29.00 38.00 41.50 51.00 57.00 62.00 75.00 80.00 88.00 98.00 108.00 132.00 143.00 156.00 163.00 16300

kN

FAG

Dynamic, C

2750

112 134 176 204 280 355 430 570 695 720 915 1120 1250 1340 1430 1500 1700 1930 2160 2360

N

Fatigue load limit, Fn SKF

18000 16000 14000 11000 9500 8500 7500 6700 6000 5300 5000 7000 6300 6000 5600 5300 5000 4800 4500 4500

rpm

Kinematically FAG

3000

22000 20000 17000 15000 12000 11000 9000 8000 7500 6700 6000 5300 5300 4800 4500 4300 4000 3600 3600 3400

Oil SKF

Permissible speed, n

Source: 1. Extracted with permission from ‘‘FAG Rolling Bearings’’, Catalogue WL 41520EI, 1995 Edition: FAG Precision Bearings Ltd., Maneja, Vadodara, India. 2. Courtesy: Extracted from SKF Rolling Bearings, Catalogue 4000E, 1989, SKF Rolling Bearings India Ltd., Mumbai, India.

D

Old

New

IS No.

Bearing No.

Basic load rating capacity

TABLE 23-64 Self-aligning ball bearings—Diameter Series 03 [Series 03 (Indian Standards)], Dimensions Series 13 FAG and SKF

0.129 0.164 0.262 0.391 0.570 0.711 0.957 1.25 1.59 1.96 1.83 3.42 3.65 4.76 5.19 6.13 6.55 8.70 9.89 11.80

kg

Mass FAG

DESIGN OF BEARINGS AND TRIBOLOGY

23.126

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B

r

r

d

2220M

2200E 01E 02E 2203E 04E 05E 2206E 07E 08E 2209E 10E 11E 2212E 13E 14 2215 16E 17 2218 19 20 2221 22

2200TV 2201TV 2202TV 2203TV 2204TV 2205TV 2206TV 2207TV 2208TV 2209TV 2210TV 2211TV 2212TV 2213TV 2214M 2215TV 2216TV 2217M 2218TV 2219M. 2220TV

10 12 15 17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110

d 30 32 35 40 47 52 62 72 80 85 90 100 110 120 125 130 140 150 160 170 180 190 200

D 14 14 14 16 18 18 20 23 23 23 23 25 28 31 31 31 33 36 40 43 46 50 53

B 0.6 0.6 0.6 0.6 1.0 1.0 1.0 1.1 1.1 1.1 1.1 1.5 1.5 1.5 1.5 1.5 2.0 2.0 2.0 2.1 2.1 2.1 2.1

r min

Dimensions, mm

.28

.58 .58 .46 .46 .44 .35 .30 .30 .26 .26 .24 .22 .23 .23 .27 .26 .25 .26 .27 .27 .27

e

2.33

1.14 1.25 1.44 1.43 1.51 1.86 2.23 2.23 2.54 2.54 2.74 3.06 2.82 2.92 2.45 2.59 2.6 2.58 2.44 2.43 2.44

Yo

1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2.23

1.09 1.20 1.37 1.37 1.45 1.75 2.13 2.13 2.43 2.43 2.61 2.93 2.69 2.78 2.34 2.47 2.48 2.46 2.33 2.32 2.33 .65

.65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 3.45

1.69 1.85 2.13 2.17 2.24 2.75 3.29 3.29 3.76 3.76 4.05 4.53 4.16 4.31 3.62 3.82 3.84 3.81 3.61 3.59 3.61

Y

X

X

Y

Fn =Fr > e

Fn =Fr e

52.00

1.73 1.96 2.08 2.75 3.55 4.40 6.95 9.00 9.50 9.50 9.50 12.70 16.60 19.30 17.00 18.00 20.00 23.60 28.50 34.0 40.50

kN

FAG

1730 1900 2040 2550 4150 4400 6700 8800 10000 10600 11200 13400 17000 20000 17000 18000 25500 23600 28500 34500 40500 45000 52000

N

SKF

Static, Co

N

SKF

8060 8520 8710 10600 16800 16800 23800 30700 31000 32500 33800 39000 48800 57200 44200 44200 65000 58500 70200 83200 97500 108000 125.00 124000

8.3 9.0 9.15 11.40 14.30 17.00 25.50 32.00 31.50 28.00 28.00 39.00 47.50 57.00 44.00 44.00 49.00 58.50 71.00 83.00 98.00

kN

FAG

Dynamic, C

Basic load rating capacity

90 98 104 132 216 228 345 455 510 540 570 695 880 1020 880 900 1250 1120 1320 1530 1760 1900 2120

N

Fatigue load limit, Fn SKF

28000 26000 24000 19000 17000 15000 12000 9500 9000 8500 8000 6700 5300 5300 8500 5300 5000 7000 4300 6000 5600 3000 5000

rpm

Kinematically FAG

28000 26000 22000 20000 17000 14000 12000 10000 9000 8500 7500 7000 6300 6000 5000 5300 4800 4500 4300 4000 3800 3600 3400

Oil SKF

Permissible speed, n

Source: 1. Extracted with permission from ‘‘FAG Rolling Bearings’’, Catalogue WL 41520EI, 1995 Edition: FAG Precision Bearings Ltd., Maneja, Vadodara, India. 2. Courtesy: Extracted from SKF Rolling Bearings, Catalogue 4000E, 1989, SKF Rolling Bearings India Ltd., Mumbai, India.

D

r

r

SKF

FAG

Bearing No.

Factors, FAG

TABLE 23-65 Self-aligning ball bearings—Dimension Series 22—FAG and SKF

0.045 0.050 0.017 0.086 0.136 0.159 0.259 0.404 0.488 0.527 0.567 0.763 1.08 1.36 1.10 1.20 2.10 2 68 3.30 4.10 4.98 6.10 7.10

kg

Mass FAG

DESIGN OF BEARINGS AND TRIBOLOGY

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23.127

B

r

d

2301 02 03 2304 05 06 2307E 08E 09E 2310 11 12 2313 14 15 2316 17 18 2319 20 22

2301TV 2302TV 2303TV 2304TV 2305TV 2306TV 2307TV 2308TV 2309TV 2310TV 2311TV 2312TV 2313M 2314M 2315M 2316M 2317M 2318M 2319M 2320M 2322M

12 15 17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 110

d 37 42 47 52 62 72 80 90 100 110 120 130 140 150 160 170 180 190 200 215 240

D 17 17 19 21 24 27 31 33 36 40 43 46 48 51 55 58 60 64 67 73 80

B

r 1.0 1.0 1.0 1.1 1.1 1.1 1.5 1.5 1.5 2.0 2.0 2.1 2.1 2.1 2.1 2.1 3.0 3.0 3.0 3.0 3.0

Dimensions, mm

.51 .51 .51 .48 .45 .47 .43 .43 .43 .42 .41 .39 .38 .38 .37 .37 .39 .38 .38 .37

e

1.29 1.25 1.29 1.38 1.47 1.42 1.52 1.55 1.44 1.58 1.62 1.70 1.73 1.72 1.78 1.76 1.71 1.74 1.75 1.77

Yo

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

X

1.23 1.19 1.23 1.32 1.4 1.35 1.45 1.48 1.47 1.51 1.55 1.62 1.65 1.64 1.7 1.68 1.68 1.66 1.67 1.69

Y

Fn =Fr e

Factors, FAG

.65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65

X

1.91 1.85 1.9 2.04 2.17 2.1 2.25 2.29 2.27 2.33 2.4 2.51 2.55 2.54 2.62 2.61 2.53 2.57 2.58 2.62

Y

Fn =Fr > e

3.75 3.20 4.65 6.55 8.65 11.2 13.4 16.3 20.0 23.6 28.0 32.5 37.5 42.5 48.0 51.0 57.0 64.0 78.0 95.0

kN

FAG

2700 2900 3550 4750 6550 8800 11200 16000 19300 20000 24000 28500 32500 37500 43000 49000 51000 57000 64000 80000 95000

N

SKF

Static, Co

16.0 13.4 18.0 24.5 31.5 39.0 45.0 54.0 64.0 75.0 86.5 95.00 110.0 122.0 137.0 140.0 153.0 163.0 193.0 216

kN

FAG

11700 11000 14600 18200 24200 31200 39700 54000 63700 63700 76100 87100 95600 111000 124000 135000 140000 153000 165000 190000 216000

N

SKF

Dynamic, C

Basic load rating capacity

140 150 183 240 340 450 585 815 1000 1040 1250 1450 1660 1860 2040 2240 2280 2500 2750 3200 3650

N

Fatigue load limit, Fn SKF

17000 18000 17000 16000 18000 10000 9000 8000 7000 6300 5600 5000 4800 6300 6000 5600 5300 5000 4800 4500 4300

rpm

Kinematically FAG

20000 18000 16000 14500 12000 10000 8500 7500 6700 5500 5600 5300 4800 4500 4000 3800 3600 3400 3200 3000 2300

Oil SKF

Permissible speed, n

Source: 1. Extracted with permission from ‘‘FAG Rolling Bearings’’, Catalogue WL 41520EI, 1995 Edition: FAG Precision Bearings Ltd., Maneja, Vadodara, India. 2. Courtesy: Extracted from SKF Rolling Bearings, Catalogue 4000E, 1989, SKF Rolling Bearings India Ltd., Mumbai, India.

D

r

SKF

FAG

Bearing No.

TABLE 23-66 Self-aligning ball bearings—Dimension Series 23 FAG and SKF

0.095 0.115 0.172 0.226 0.335 0.500 0.675 0.925 1.23 1.60 2.06 2.74 3.33 4.52 5.13 5.50 7.05 8.44 9.86 12.40 16.90

kg

Mass FAG

DESIGN OF BEARINGS AND TRIBOLOGY

23.128

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B

a

r

r1

7200BE 7201BE 7202BE 7203BE 04BE 05BE 06BE 07BE 08BE 09BE 10BE 7211BE 12BE 13BE 14BE 15BE 16BE 17BE 18BE 19BE 20BE 7221BE 22BE 7224BE

7200B 7201B 7202B 7203B 7204B 7205B 7206B 7207B 7208B 7209B 7210B 7211B 7212B 7213B 7214B 7215B 7216B 7217B 7218B 7219B 7220B 7221B 7222B 7222B

15BA02 17BA02 20BA02 25BA02 30BA02 35BA02 40BA02 45BA02 50BA02 55BA02 60BA02 65BA02 70BA02 75BA02 80BA02 85BA02 90BA02 95BA02 100BA02 105BA02 110BA02 120BA02

10 12 15 17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 120

d 30 32 35 40 47 52 62 72 80 85 90 100 110 120 125 130 140 150 160 170 180 190 200 215

D 9 10 11 12 14 15 16 17 18 19 20 21 22 23 24 25 26 28 30 32 34 36 38 40

B 0.6 0.6 0.6 0.6 1.0 1.0 1.0 1.1 1.1 1.1 1.1 1.5 1.5 1.5 1.5 1.5 2 2 2 2.1 2.1 2.1 2.1 2.1

r min

Dimensions, mm

0.3 0.3 0.3 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 1.0 1.0 1.0 1.0 1.0 1.5 1.0 1.1 1.1 1.1 1.1 1.1 1.1

r1 min 13 14 16 18 21 24 27 31 34 37 39 43 47 50 53 56 59 63 67 72 76 80 84 90

a 2.5 3.4 4.3 5.5 7.65 9.30 13.40 18.30 23.20 26.50 28.50 36.00 44.00 53.00 58.50 58.50 69.50 80.00 93.00 100.00 114.00 129.00 143.00 160.00

kN

FAG

3350 3800 4860 6100 83000 10200 15600 20800 26000 28000 30500 38000 45500 54000 60000 64000 73000 83000 96500 108000 122000 137000 153000 163000

N

SKF

Static, Co

5.00 6.95 8.00 10.00 13.40 14.60 20.40 27.00 32.00 36.00 37.50 46.50 56.00 64.00 69.50 68.00 80.00 90.00 106.00 116.00 129.00 143.00 153.00 166.00

kN

FAG

7020 7610 8840 11100 14000 15600 23800 30700 36400 37700 39000 48800 57200 66300 71500 72800 83200 95600 106000 124000 135000 146000 163000 165000

N

SKF

Dynamic, C

Basic load rating capacity

140 160 204 286 355 420 655 880 1100 1200 1290 1630 1930 2280 2500 2650 3000 3250 3650 4000 4400 4800 5200 5300

N

Fatigue load limit, Fuf SKF

32000 28000 24000 20000 18000 16000 13000 11000 9500 8500 8000 7000 6300 6000 5600 5300 5000 4500 4300 4000 3800 3600 3600 3400

FAG rpm

Kinematically

27000 26000 24000 20000 17000 15000 12000 11000 9500 9000 8000 7500 6700 6000 5600 5600 5000 4800 4500 4300 4000 3800 3600 3200

SKF

Oil a

Permissible speed, n

0.028 0.036 0.045 0.07 0.103 0.127 0.207 0.296 0.377 0.430 0.485 0.645 0.779 0.975 1.07 1.19 1.42 1.89 2.22 2.66 3.18 3.19 4.44 5.31

kg

Mass FAG

Use Xe ¼ 1, when Fa =Fr 1:9; Xe ¼ 0:5, Yo ¼ 0:26 when Fa =Fr > 1:9, X ¼ 1 when Fa =Fr 1:4; X ¼ 0:35, Y ¼ 0:57 when Fa =Fr > 1:14. a Oil lubrication, E ¼ cylindrical bore, EK, K ¼ tapered bore, TV ¼ self-aligning bearings with caging of glass-ﬁber reinforced polyamide, M ¼ ball-riding mechanical brass caps. Source: 1. Extracted with permission from ‘‘FAG Rolling Bearings’’, Catalogue WL 41520EI, 1995 Edition: FAG Precision Bearings Ltd., Maneja, Vadodara, India. 2. Courtesy: Extracted from SKF Rolling Bearings, Catalogue 4000E, 1989, SKF Rolling Bearings India Ltd., Mumbai, India.

D d

r

r

SKF

FAG

IS Old No.

Bearing No.

TABLE 23-67 Single row angular contact ball bearings—Dimension Series 02 (Indian Standards)

DESIGN OF BEARINGS AND TRIBOLOGY

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23.129

d

a

r

r1

7303B 7304B 7305B 7306B 7307B 7308B 7309B 7310B 7311B 7312B 7313B 7314B 7315B 7316B 7317B 7318B 7319B 7320B 7321B 7322B

7300B 7301B 7302B 17BA03 20BA03 25BA03 30BA03 35BA03 40BA03 45BA03 50BA03 55BA03 60BA03 65BA03 70BA03 75BA03 80BA03 85BA03 90BA03 95BA03 100BA03 105BA03 110BA03

17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110

7303BE 04BE 05BE 7306BE 07BE 08BE 7309BE 10BE 11BE 7312BE 13BE 14BE 7315BE 16B 17B 7318B 19B 20B 7321B 22B

SKF 10 12 15 17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110

d 35 37 42 47 52 62 72 80 90 100 110 120 130 140 150 160 170 180 190 200 215 225 240

D 11 12 13 14 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 50

B 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.5 1.5 1.5 2.0 2.0 2.1 2.1 2.1 2.1 2.1 3 3 3 3 3 3

r min

Dimensions, mm

.5 .6 .6 .6 .6 .6 .6 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1

r1 min 15 16.0 18.0 20 23 27 31 35 39 43 47 51 55 60 64 68 72 76 80 84 90 94 98

a

5.00 6.55 8.30 10.40 15.00 20.00 25.00 32.50 40.00 47.50 56.00 65.50 75.00 86.50 100.00 114.00 127.00 140.00 153.00 180.00 200.00 224.00

kN

FAG

5000 6700 8300 10400 15600 21200 24500 33500 41000 51000 60000 69500 80000 90000 106000 118000 132000 146000 163000 190000 208000 224000

N

SKF

Static, Co

10.50 12.90 16.00 19.00 26.00 32.50 39.00 50.00 60.00 69.5 78.00 90.00 102.00 114.00 127.00 140.00 150.00 160.00 173.00 193.00 208.00 224.00

kN

FAG

10600 13000 15900 19000 26000 34500 39000 49400 60500 74100 85200 95600 108000 119000 133000 143000 158000 165000 176000 203000 212000 225000

N

SKF

Dynamic, C

Basic load rating capacity

208 280 355 440 655 900 1640 1400 1730 2200 2550 3000 3350 3650 4150 4500 4900 5200 5600 6400 6400 7200

N

Fatigue load limit, Fuf SKF

24000 20000 18000 17000 14000 11000 9500 8500 7500 7000 6300 5600 5300 5000 4500 4300 4000 3800 3800 3600 5300 3400

rpm

FAG

Kinematically

24000 20000 18000 16000 13000 11000 10000 9000 8000 7000 6300 6000 5600 5000 4800 4500 4300 4000 3800 3600 3400 3200

SKF

Oil a

Permissible speed, n

Use Xe ¼ 1, when Fa =Fr 1:9; Xe ¼ 0:5, Yo ¼ 0:26 when Fa =Fr > 1:9, X ¼ 1 when Fa =Fr 1:4; X ¼ 0:35, Y ¼ 0:57 when Fa =Fr > 1:14. a Oil lubrication. Source: 1. Extracted with permission from ‘‘FAG Rolling Bearings’’, Catalogue WL 41520EI, 1995 Edition: FAG Precision Bearings Ltd., Maneja, Vadodara, India. 2. Courtesy: Extracted from SKF Rolling Bearings, Catalogue 4000E, 1989, SKF Rolling Bearings India Ltd., Mumbai, India.

D

r

r

B

FAG

New

Old

IS No.

Bearing No.

TABLE 23-68 Single row angular contact hall bearings—Dimension Series 03 (Indian Standards)

0.059 0.09 0.113 0.147 0.221 0.342 0.447 0.657 0.821 1.050 1.36 1.72 2.10 2.53 3.18 3.75 4.27 5.72 5.99 7.14 9.00 9.74

kg

Mass FAG

DESIGN OF BEARINGS AND TRIBOLOGY

23.130

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a

B

r

d

D

3302A 03A 04A 3305A 06A 07A 3308A 09A 10A 3311A 12A 13A 3314A 15A 16A 3317A 18A 19A 20A

3302 3303 3304 3305 3306 3307 3308 3309 3310 3311 3312 3313 3314 3315 3316 3317 3318 3319 3320

15 17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

d 42 47 52 62 72 80 90 100 110 120 130 140 150 160 170 180 190 200 215

D 19 22.2 22.2 25.4 30.3 34.9 36.5 39.7 44.4 49.2 54 58.7 63.5 68.5 68.3 73 73 77.8 82.6

B

Dimensions, mm

1.5 1.5 2 2 2 2.5 2.5 2.5 3 3 3.5 3.5 3.5 3.5 3.5 4 4 4 4

r 30 34 36 43 51 56 64 72 79 87 96 102 109 117 123 131 136 143 153

a 10,243 14,455 15,092 21,750 29,792 39,200 50,666 63,602 78,353 90,699 106,722 124,460 140,042 157,780 173,361 202,272 228,182 253,379 284,494

N

FAG

9,065 12,642 13,720 19,600 27,146 35,574 44,590 54,390 72,520 78,400 94,570 108,870 126,430 127,940 153,860 173.460 205,800

N

SKF

Static, Co

Basic capacity

12,887 18,032 18,424 25,382 32,814 42,924 52,479 63,602 78,164 88,896 101,332 117,796 128,919 144,520 157,780 177,821 193,608 206,682 224,723

N

FAG

13,720 18,914 18,914 26,008 35,280 43,615 53,410 62,230 85,995 85,840 98,000 115,640 135,730 140,740 157,780 173,460 200,018

N

SKF

Dynamic, C

10,000 8,000 8,000 6,000 6,000 5,000 5,ooo 4,000 4,000 4,000 3,000 3,000 3,000 2,500 2,500 2,500 2,500

Maximum permissible speed, rpm

Note: These bearings are provided with ﬁlling slots on one side; in case of unidirectional thrust loads, the bearings should be so arranged in mounting that the balls on the slot side are relieved from load.Use Xo ¼ 1, Yo ¼ 0:58 and X ¼ 1, Y :66, when Fa =Fr 0:95; X ¼ :6, Y ¼ 1:07 when Fa =Fr > 0:95.

r

SKF

FAG

Bearing No.

TABLE 23-69 Double-row angular contact ball bearings series 33

DESIGN OF BEARINGS AND TRIBOLOGY

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23.131

d

r

D

N203E N204E N205E N206E N207E N208E N209E N210E N211E N212E N213E N214E N215E N216E N217E N218E N219E N220E N221E N223E N224E

10RN02 12RN02 15RN02 17RN02 20RN02 25RN02 30RN02 35RN02 40RN02 45RN02 50RN02 55RN02 60RN02 65RN02 70RN02 75RN02 80RN02 85RN02 90RN02 95RN02 100RN02 105RN02 110RN02 120RN02 N203EC 204EC 205EC N206EC 207EC 208EC N209EC 210EC 211EC N212EC 213EC 214EC N215EC 216EC 217EC N218EC 219EC N220EC 221EC 222EC 224EC

SKF 10 12 15 17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 120

d 30 32 35 40 47 52 62 72 80 85 90 100 110 120 125 130 140 150 160 170 180 190 200 215

D 9 10 11 12 14 15 16 17 18 19 20 21 22 23 24 25 26 28 30 32 34 36 38 40

B 1.0 1.0 1.0 0.6 1.0 1.0 1.0 1.1 1.1 1.1 1.1 1.5 1.5 1.5 1.5 1.5 2.0 2.0 2.0 2.1 2.1 2.1 21 2.1

r min

0.3 0.6 0.6 0.6 0.6 1.1 1.1 1.1 1.1 1.5 1.5 1.5 1.5 2-0 2.0 2.0 2.1 2.1 2.1 2.1 2.1

r1 min

35.1 41.5 46.5 55.6 64.0 71.5 76.5 81.5 90.0 100.0 108.6 113.5 118.5 127.3 136.5 145.0 154.5 163.0 171.5 180.5 195.5

E

14.6 24.5 27.5 37.5 50.0 53.0 63.0 68.0 95.0 104.0 120.0 137.0 156.0 170.0 193.0 216.0 265.0 305.0 320.0 365.0 415.0

kN

FAG

14300 22000 27000 36500 48000 55000 64000 69500 95000 102000 118000 137000 156000 166000 200000 220000 265000 305000 315000 365000 430000

N

SKF

Static, Co

17.6 27.5 29.0 39.0 50.0 53.0 61.0 64.0 83.0 95.0 108.0 120.0 132.0 140.0 163.0 183.0 220.0 250.0 280.0 290.0 335.0

kN

FAG

17200 25100 28600 38000 48000 53000 60500 64000 84200 93500 106000 119000 130000 138000 165000 183000 220000 251000 264000 292000 341000

N

SKF

Dynamic, C

Basic load rating capacity

1730 2700 3350 4550 6100 6700 8150 8800 12200 13400 15600 18000 20400 21200 24500 27000 32500 36500 36500 42500 49000

N

Fatigue load limit, Fuf SKF

18000 16000 15000 12000 10000 9000 8500 8000 7000 6300 6000 5300 5300 4800 4500 4500 3800 4800 5600 3400 3200

FAG rpm

Kinematically

19000 16000 14000 12000 10000 9000 8000 7500 7000 6300 5600 5300 5300 4800 4500 4300 4000 3800 3600 3400 3000

SKF

Oil a

Permissible speed, n

Use Xe ¼ 1, Yo ¼ Y ¼ 0. a Oil lubrication. Source: 1. Extracted with permission from ‘‘FAG Rolling Bearings’’, Catalogue WL 41520EI, 1995 Edition: FAG Precision Bearings Ltd., Maneja, Vadodara, India. 2. Courtesy: Extracted from SKF Rolling Bearings, Catalogue 4000E, 1989, SKF Rolling Bearings India Ltd., Mumbai, India.

E

r1

B

FAG

IS

Bearing No.

Dimensions, mm

TABLE 23-70 Cylindrical roller bearings—Dimension Series 02 (Indian Standards)

0.067 0.107 0.139 0.205 0.300 0.380 0.434 0.493 0.669 0.827 1.040 1.160 1.270 1.550 1.870 2.250 2.750 3.320 4.690 4.840 5.770

kg

Mass FAG

DESIGN OF BEARINGS AND TRIBOLOGY

23.132

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B

r

D

N303 N304 N305E N306E N307E N308E N309E N310E N311E N312E N313E N314E N315E N316E N317EMI N318EMI N319EMI N320 N321 N322EMI N324EMI N326EMI N328EMI N330EMI N332EMI N334M

FAG

N303EC N304EC 305EC 306EC N307EC 308EC 309EC N310EC 311EC 312EC N313EC 314EC 315EC N316EC 317EC 318EC N319EC 320EC 321EC N322EC 324EC 326EC 328EC N330EC 332EC 334EC

SKF 10 12 15 17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 120 130 140 150 160 170

d 35 37 42 47 52 62 72 80 90 100 110 120 130 140 150 160 170 180 190 200 215 225 240 260 280 300 320 340 360

D 11 12 13 14 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 50 55 58 62 65 68 72

B 1.0 1.5 1.5 1.5 2.0 1.1 1.1 1.5 1.5 1.5 2.0 2.0 2.1 2.1 2.1 2.1 2.1 3.0 3.0 3.0 3.0 3.0 3.0 3.0 4.0 4.0 4.0 4.0 4.0

r min

1 2 1.1 1.1 1.1 1.5 1.5 2.0 2.0 2.1 2.1 2.1 2.1 2.1 3.0 3.0 3.0 3 3 3 3 4 4 4 4 4

r1 min

39.1 44.5 54.0 62.5 70.2 80.0 88.5 95.0 106.5 115.0 124.5 133.0 143.0 151.0 160.0 169.5 171.5 191.5 195.0 211.0 230.0 247.0 264.0 283.0 300.0 310.0

E N

SKF

20400 26000 37.5 36500 48.0 49000 63.0 63000 78.0 78000 100.0 10000 114.0 112000 140.0 143000 156.0 160000 190.0 196000 220.0 220000 265.0 265000 275.0 290000 325.0 335000 345.0 360000 380.0 390000 425.0 440000 500.0 500000 510.00 540000 600.00 620000 720.00 750000 800.0 710000 930.0 965000 1060.0 680000 1020.0 1180000

kN

FAG

Static, Co

41.5 51.0 64.0 81.5 98.0 110.0 134.0 150.0 180.0 204.0 240.0 255.0 290.0 315.0 335.0 380.0 410.8 440.0 520.0 610.0 670.0 765.0 865.0 800.0

kN

FAG N

Fatigue load limit, Fuf SKF

20400 2550 30800 3250 40200 4550 51200 6200 64400 8150 80900 10200 99000 12900 110000 15000 138000 18600 151000 20800 183000 25500 205000 29000 242000 33500 260000 36000 297000 41500 319000 43000 341000 46500 391000 51000 440000 57000 468000 61000 539000 69500 627000 81500 594000 75000 781000 100000 501000 72000 952000 110000

N

SKF

Dynamic, C

Basic load rating capacity

4800 4500 4300 3800 3600 3000 3000

12000 10000 9000 8500 6700 6300 5600 5000 4800 4500 4000 3800 5600 5300 5300 5000

FAG rpm

Kinematically

17000 15000 12000 11000 9500 8000 7500 6000 5600 5000 4800 4300 4000 3800 3600 3400 3200 3000 2800 2600 2400 2200 2400 2000 2200 1700

SKF

Oil a

Permissible speed, n

Use Xe ¼ 1, Yo ¼ Y ¼ 0, a Oil lubrication. Source: 1. Extracted with permission from ‘‘FAG Rolling Bearings’’, Catalogue WL 41520EI, 1995 Edition: FAG Precision Bearings Ltd., Maneja, Vadodara, India. 2. Courtesy: Extracted from SKF Rolling Bearings, Catalogue 4000E, 1989, SKF Rolling Bearings India Ltd., Mumbai, India.

E d

r1

10RN03 12RN03 15RN03 17RN03 20RN03 25RN03 30RN03 35RN03 40RN03 45RN03 50RN03 55RN03 60RN03 65RN03 70RN03 75RN03 80RN03 85RN03 90RN03 95RN03 100RN03 110RN03 120RN03

IS

Bearing No.

Dimensions, mm

TABLE 23-71 Cylindrical roller bearings—Dimension Series 03 (Indian Standards)

0.120 0.150 0.234 0.379 0.486 0.649 0.885 1.010 1.400 1.850 2.300 2.730 3.340 3.880 5.220 6.150 7.060 8.750 10.230 11.700 15.200 18.600 22.600 24.000 32.000 36.300

kg

Mass FAG

DESIGN OF BEARINGS AND TRIBOLOGY

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23.133

r

D

N405 N406 N407 N408 N409 N410 N411 N412 N413 N414 N415 N416 N417 N418 N419 N420

15RN04 17RN04 20RN04 25RN04 30RN04 35RN04 40RN04 45RN04 50RN04 55RN04 60RN04 65RN04 70RN04 75RN04 80RN04 85RN04 90RN04 95RN04 100RN04 NU405 NU406 NU407 NU408 NU409 NU410 NU411 NU412 NU413 NU414 NU415 NU416 NU417 NU418 NU419 NU420

SKF 15 17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

d 52 62 72 80 90 100 110 120 130 140 150 160 180 190 200 210 225 240 250

D 15 17 19 21 23 25 27 29 31 33 35 37 42 45 48 52 54 55 58

B 2.0 2.0 2.0 2.5 2.5 2.5 2.5 3.0 3.5 3.5 3.5 3.5 4.0 4.0 4.0 5.0 5.0 5.0 5.0

ra

Dimensions, mm

62.8 73 83 92 100.5 110.8 117.2 127 135.3 152 160.5 170 177 191.5 201.5 211

Ea

23130 32680 42240 51550 63550 81340 81340 99570 110000 143900 157780 164490 213450 235590 253380 283160

N

FAGa

53000 69500 90000 102000 127000 140000 173000 190000 240000 280000 320000 335000 415000 455000 475000

N

SKF

Static, Co

40050 53560 68010 82680 99560 122260 124460 148910 168900 211140 231130 259700 297800 320070 355390 391170

N

FAGa

60500 76500 96000 106000 130000 142000 168000 183000 229000 264000 303000 319000 350000 413000 429000

N

SKF

Dynamic, C

6800 9000 11600 13400 16600 18600 22000 24000 30000 34000 39000 39000 48000 52000 53000

N

Fatigue load Fuf limit SKF

8000 8000 6000 6000 6000 5000 5000 5000 4000 4000 4000 3000 3000 3000 3000 2500

FAGa

Kinematically

7500 6700 6000 5600 5000 4800 4300 4000 3600 3400 3200 3000 2800 2600 2400

rpm

SKF

Greaseb

Permissible speed, n

Use Xe ¼ X ¼ 1, Y ¼ 0, NU Series have two integral ﬂanges on the outer ring and inner without ﬂanges. a Refer to old FAG designation. b Grease lubrication. c Oil lubrication. Source: 1. Extracted with permission from ‘‘FAG Rolling Bearings’’, Catalogue WL 41520EI, 1995 Edition: FAG Precision Bearings Ltd., Maneja, Vadodara, India. 2. Courtesy: Extracted from SKF Rolling Bearings, Catalogue 4000E, 1989, SKF Rolling Bearings India Ltd., Mumbai, India.

Ed

B

FAG

IS Old No.

Bearing No.

Basic load rating capacity

TABLE 23-72 Cylindrical roller bearings—Dimension Series 04 (Indian standards) and series NU 4, SKF

9000 8000 7000 6700 6000 5600 5000 4800 4300 4000 3800 3600 3400 3200 3000

Oilc

0.75 1.00 1.30 1.65 2.00 2.50 3.00 3.60 5.25 6.25 7.30 8.70 10.50 13.6 19.0

kg

Mass SKF

DESIGN OF BEARINGS AND TRIBOLOGY

23.134

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d

B

D

r

NU2203EC NU2204EC 2205EC 2206EC 2207EC 2208EC 2209EC 2210EC NU2211EC 2212EC 2213EC 2214EC 2215EC NU2216EC 2217EC 2218EC 2219EC 2220EC NU2222EC 2224EC 2226EC 2228EC 2230EC 2232EC 2234EC 2236EC 2238EC 2240EC

NU2203E NU2204E NU2205E NU2206E NU2207E NU2208E NU2209E NU2210E NU2211E NU2212E NU2213E NU2214E NU2215E NU2216E NU2217E NU2218E NU2219E NU2220E NU2222E NU2224E NU2226E NU2228E NU2230EMI NU2232EMI NU2234EMI NU2236EMI NU2238EMI NU2240EMI

17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 110 120 130 140 150 160 170 180 190 200

d 40 47 52 62 80 80 85 90 100 110 120 125 130 140 150 160 170 180 200 215 230 250 270 290 310 320 340 360

D 16 18 18 20 21 23 23 23 25 28 31 31 31 33 36 40 43 46 53 58 64 68 73 80 86 86 92 98

B .6 1.0 1.0 1.0 1.5 1.5 1.1 1.1 1.5 1.5 1.5 1.5 1.5 2 2 2 2.1 2.1 2.1 2.1 3 3 3 3 4 4 4 4

r min

Dimensions, mm

0.3 0.6 0.6 0.6 1.1 1.5 1.1 1.1 1.1 1.5 1.5 1.5 1.5 2 2 2 2.1 2.1 2.1 2.1 3 3 3 3 4 4 4 4

r1 min 22.1 26.5 31.5 37.5 46.2 51.0 54.5 59.5 66.0 72.0 78.5 83.5 88.5 95.3 100.05 107.0 112.5 119.0 132.5 143.5 153.5 169.0 182.0 193.0 205.0 215.0 228.0 241.0

F 22.0 31.0 34.5 50.0 63.0 78.0 81.5 88.0 118.0 153.0 183.0 196.0 208.0 215.0 275.0 315.0 375.0 440.0 520.0 610.0 735.0 830.0 980.0 1180.0 1400.0 1500.0 1660.0 1860.0

kN

FAG

21600 27500 34000 49000 63000 75000 81000 88000 118000 153000 180000 193000 208000 245000 280000 315000 375000 450000 520000 630000 735000 830000 930000 1200000 1430000 1500000 1660000 1900000

N

SKF

Static, Co

24.0 32.5 34.5 49.0 64.0 81.5 73.5 78.0 98.0 129.0 150.0 156.0 163.0 186.0 216.0 240.0 285.0 335.0 380.0 450.0 530.0 570.0 655.0 800.0 950.0 1000.0 1100.0 1220.0

kN

FAG

23800 29700 34100 48400 64400 70000 73700 78100 99000 128000 147000 154000 161000 157000 216000 242000 266000 336000 380000 457000 528000 572000 627000 809000 968000 1010000 1100000 1230000

N

SKF

Dynamic, C

Basic load rating capacity

2600 3450 4250 6100 8150 9650 10000 11400 15300 20000 24000 25500 27000 31000 34300 39000 45500 54000 61000 72000 83000 93000 100000 129000 150000 156000 170000 190000

N

Fatigue limit, Fuf SKF

18000 16000 15000 12000 9000 7500 8000 8000 7000 6300 5600 5300 5300 4800 4500 4300 3800 3800 3400 3200 3000 4500 4300 3800 3200 3200 3000 2800

rpm

Kinematically FAG

19000 16000 14000 12000 9500 9000 8000 7000 7000 6300 5600 5300 5300 4800 4500 4300 4000 3800 3400 3000 2800 2600 2460 2200 2200 2000 1900 1800

Oila SKF

Permissible speed, n

Use Xe ¼ X ¼ 1, Yo ¼ Y ¼ 0. EC design series have higher loading capacity for the same boundary dimension than earlier design. a Oil lubrication. Source: 1. Extracted with permission from ‘‘FAG Rolling Bearings’’, Catalogue WL 41520EI, 1995 Edition: FAG Precision Bearings Ltd., Maneja, Vadodara, India. 2. Courtesy: Extracted from SKF Rolling Bearings, Catalogue 4000E, 1989, SKF Rolling Bearings India Ltd., Mumbai, India.

F

r1

SKF

FAG

Bearing No.

TABLE 23-73 Cylindrical roller bearings—Dimension Series NU22

0.092 0.142 0.162 0.359 0.488 0.658 0.530 0.571 0.793 1.08 1.44 1.51 1.60 2.01 2.50 3.18 3.90 4.77 6.73 8.21 10.4 13.2 18.7 23.9 35.7 36.4 36.9 45.1

kg

Mass FAG

DESIGN OF BEARINGS AND TRIBOLOGY

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23.135

B

r

D

NU2304EC NU2305EC 2306EC 2307EC NU2308EC 2309EC 2310EC NU2311EC 2312EC 2313EC NU2314EC 2315 2316 NU2317EC 2318EC 2319EC NU2320EC 2322EC 2324EC NU2326EC NU2328EC 2330EC 2332EC NU2334 NU2336 NU2338EC NU2340EC

NU2304E NU2305E NU2306E NU2307E NU2308E NU2309E NU2310E NU2311E NU2312E NU2313E NU2314E NU2315 NU2316 NU2317 NU2318 NU2319 NU2320 NU2322 NU2324 NU2326MI NU2328MI NU2330MI NU2332MI NU2334M NU2336M NU2338M NU2340

20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 110 120 130 140 150 160 170 180 190 200

d 52 62 72 80 90 100 110 120 130 140 150 160 170 180 190 200 215 240 260 280 300 320 340 360 380 400 420

D 21 24 27 31 33 36 40 43 46 48 51 55 58 60 64 67 73 80 86 93 102 108 114 120 126 132 138

B 1.1 1.1 1.1 1.5 1.5 1.5 2 2 2.1 2.1 2.1 2.1 2.1 3 3 3 3 3 3 4 4 4 4 4 5 5 5

r

Dimensions, mm

0.6 1.1 1.1 1.1 1.5 1.5 2 2 2.1 2.1 2.1 2.1 2.1 3 3 3 3 3 3 4 4 4 4 4 4 5 5

r1 27.5 34.0 40.5 46.2 52.0 58.5 65.0 70.5 77.0 82.5 89.0 95.0 101.0 108.0 113.5 121.5 127.5 143.0 154.0 167.0 180.0 193.0 204.0 220.0 232.0 245.0 260.0

F 39.0 56.0 75.0 98.0 120.0 153.0 186.0 228.0 260.0 285.0 325.0 390.0 425.0 450.0 530.0 585.0 720.0 800.0 1020.0 1220.0 1400.0 1600.0 1830.0 1760.0 2000.0 2200.0 2200.0

kN

FAG

38000 55000 75000 98000 120000 153000 186000 232000 265000 290000 325000 400000 440000 490000 540000 565000 735000 900000 1040000 1250000 1430000 1630000 1860000 1800000 2040000 2550000 2650000

N

SKF

Static, Co

41.5 57.0 73.5 91.5 112.0 137.0 163.0 200.0 224.0 245.0 275.0 325.0 355.0 365.0 430.0 455.0 570.0 630.0 780.0 915.0 1020.0 1160.0 1320.0 1220.0 1370.0 1500.0 1500.0

kN

FAG

41300 56100 73700 91300 112000 136000 161000 201000 224000 251000 275000 330000 358000 396000 440000 468000 583000 682000 792000 935000 1050000 1190000 1320000 1230000 1400000 1830000 2050000

N

SKF

Dynamic, C

Basic load rating capacity

4300 6950 9650 12700 15300 20000 24500 30500 34500 35000 41500 50000 55000 60000 65500 69000 85000 102000 116000 137000 150000 170000 190000 180000 204000 236000 260000

N

Fatigue limit, Fuf SKF

14000 12000 10000 9000 7500 6700 6300 5600 5000 4800 4500 4000 3800 3600 3400 3400 3200 2800 4300 3800 3600 3200 3000 2800 2800 2800 2600

rpm

Kinematically FAG

11000 9000 8000 7000 6300 5600 5000 4600 4300 4000 3600 3400 3200 3000 2800 2600 2400 2000 1900 1800 1800 1700 1500 1400 1300 1200 1200

Oila SKF

Permissible speed, n

Oil lubrication. Source: 1. Extracted with permission from ‘‘FAG Rolling Bearings’’, Catalogue WL 41520EI, 1995 Edition: FAG Precision Bearings Ltd., Maneja, Vadodara, India. 2. Courtesy: Extracted from SKF Rolling Bearings, Catalogue 4000E, 1989, SKF Rolling Bearings India Ltd., Mumbai, India.

a

F

d

r1

SKF

FAG

Bearing No.

TABLE 23-74 Cylindrical roller bearings—Dimensions Series NU23

0.215 0.348 0.530 0.721 0.959 1.300 1.740 2.240 2.780 3.320 4.030 4.93 5.88 6.57 7.84 9.21 12.00 16.80 23.20 26.10 36.50 43.90 46.70 61.00 65.40 83.40 95.70

kg

Mass FAG

DESIGN OF BEARINGS AND TRIBOLOGY

23.136

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a T

B

c

D

r1

r

32206 07 08 32209 10 11 32212 13 14 32215 16 17 32218 19 20 32221 22 24 32226 28 30

32206A 32207A 32208A 32209A 32210A 32211A 32212A 32213A 32214A 32215A 32216A 32217A 32218A 32219A 32220A 32221A 32222A 32224A 32226A 32228A 32230A

30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 120 130 140 150

d 62 72 80 85 90 100 110 120 125 130 140 150 160 170 180 190 200 215 230 250 270

D 20 23 23 23 23 25 28 31 31 31 33 36 40 43 46 50 53 58 64 68 73

B 21.25 24.25 24.75 24.75 24.75 26.75 29.75 32.75 33.25 33.25 35.25 38.5 42.5 45.5 49.0 53.0 56.0 61.5 61.75 71.75 77.0

T 17 19 19 19 19 21 24 27 27 27 28 30 34 37 39 43 46 51 56 58 60

C

Dimensions, mm

1.0 1.5 1.5 1.5 1.5 2.0 2.0 2.0 2.0 2.0 2.5 2.5 2.5 3.0 3.0 3.0 3.0 3.0 4.0 4.0 4.0

r 1.0 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 2.0 2.0 2.0 2.5 2.5 2.5 2.5 2.5 3.0 3.0 3.0

r1 16 18 19 20 21 23 24 26 28 29 31 34 36 39 42 44 46 51 56 60 64

a .37 .37 .37 .40 .42 .40 .40 .40 .42 .44 .42 .42 .42 .42 .42 .42 .42 .44 .44 .44 .44

e 1.6 1.6 1.6 1.48 1.33 1.48 1.48 1.48 1.43 1.38 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.38 1.38 1.38 1.38

Y

Factors

.88 .88 .88 .81 .79 .81 .81 .81 .79 .76 .79 .79 .79 .79 .79 .79 .79 0.79 0.79 0.76 0.76

Yo N

SKF kN

FAG

50100 66000 74800 80900 82500 106000 125000 151000 157000 161000 182000 212000 251000 281000 319000 358000 402000 468000 550000 644000 737000

N

SKF

Dynamic, C

63.0 57000 54.0 85.0 78000 71.0 95.0 86500 80.0 100.0 98000 83.0 110.0 100000 88.0 137.0 129000 110.0 170.0 160000 134.0 200.0 193000 156.0 216.0 208000 163.0 232.0 212000 173.0 265.0 245000 200.0 305.0 285000 228.0 360.0 340000 260.0 415.0 390000 300.0 475.0 440000 335.0 550.0 510000 380.0 600.0 570000 415.0 735.0 695000 490.0 865.0 830000 570.0 1000.0 1000000 655.0 1160.0 1140000 750.0

kN

FAG

Static, Co

Basic load rating capacity

6500 8650 9800 11200 11600 15000 19000 23200 24500 25000 28500 33500 38000 43000 48000 55000 61000 72000 85000 100000 112000

N

Fatigue load limit, SKF

12000 10000 9000 8000 7500 6700 6000 5600 5300 5000 5000 4800 4500 4300 4000 3600 3400 3000 2600 2600 2600

rpm

Kinematically FAG

6000 5300 4000 4500 4300 3800 3400 3000 2800 2600 2400 2200 2000 1900 1800 1800 1700 1600 1500 1400 1200

Greasea SKF

Permissible speed, n

Grease lubrication. Source: 1. Extracted with permission from ‘‘FAG Rolling Bearings’’, Catalogue WL 41520EI, 1995 Edition: FAG Precision Bearings Ltd., Maneja, Vadodara, India. 2. Courtesy: Extracted from SKF Rolling Bearings, Catalogue 4000E, 1989, SKF Rolling Bearings India Ltd., Mumbai, India.

a

d

r

r1

SKF

FAG

Bearing No.

TABLE 23-75 Taped roller bearings—Dimensions Series 322

0.276 0.425 0.555 0.570 0.602 0.872 1.14 1.59 1.69 1 93 2.18 2.76 3.78 4.23 5.67 6.07 7.35 10.1 11.7 14.0 18.5

kg

Mass FAG

DESIGN OF BEARINGS AND TRIBOLOGY

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23.137

DESIGN OF BEARINGS AND TRIBOLOGY

General Plan Boundary Dimensions for Tapered Roller Bearings There are four series in tapered roller bearings. They are: (1) Angle series, (2) diameter series, (3) width series, (4) dimension series. Dimension series is a combination of angle series, diameter series and width series. Dimension series shall be designated by a combination of three symbols, for example 2BD. The ﬁrst symbol is a numeric character which represents a range of contact angles (angle series). The second symbol is an alphabetic character which represents range of numeric values for the outside diameter to bore relationship (diameter series). The third symbol is an alphabetic character which represents a range of numeric values of the width to section relationship (width series).

TABLE 23-76 Series designation

T=ðD dÞ0:95

(D=d 0:77 )

Angle Series

Over

1 2 3 4 5 6 7

Reserved for future use 108 138 520 138 520 158 590 188 550 158 590 188 550 238 238 278 278 308

Up to

Diameter series

Over

A B C D E F G

Reserved for future use 3.40 3.80 3.80 4.40 4.40 4.70 4.70 5.00 5.00 5.60 5.60 7.00

Up to

Width series

Over

A B C D E

Reserved for future use 0.50 0.68 0.68 0.80 0.80 0.88 0.80 1.00

Symbol r

r1 r

α

r1

r r1

r

Dφ E

r1

dφ B

T

d D T B C E r1 r1s min r2 r2s min r3 r3s min r4 r4s min r5

bearing bore diameter, nominal bearing outside diameter, nominal bearing width, nominal cone width, nominal cup width, nominal cup small inside diameter, nominal bearing contact angle, nominal cone back face chamfer height smallest single r1 cone back face chamfer width smallest single r2 cup back face chamfer height smallest single r3 cup back face chamfer width smallest single r4 cone and cup front face chamfer height and width

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

23.139

TABLE 23-77 Dimensions for tapered roller bearings—Contact angle series 2 d

r

r1 r

α

r1

r r1

r

Dφ E

r1

dφ B

T

15 17 17 17 20 20 20 20 22 25 25 25 25 28 28 30 30 30 32 32 35 35 40 40 40 45 45 50 50 50 55 55 55 55 60 60 60 60 65 65 65 65 70 70 70 75 75 75 80 80 80 85 85 85

D

T

B

r1s min r2s min

C

r3s min r4s min

E

Dimension series

42 40 40 47 37 47 47 52 40 42 62 50 52 45 55 47 58 72 52 62 55 68 62 75 90 68 80 72 85 100 80 85 95 120 85 90 100 130 90 100 110 125 100 105 120 105 115 125 110 120 130 120 125 135

14.25 13.25 17.25 20.25 12 15.25 19.25 22.25 12 12 25.25 17 19.25 12 19 12 19 28.75 14 21 14 23 15 24 35.25 15 24 15 24 42.25 17 18 27 45.5 17 18 27 48.5 17 22 31 43 20 22 34 20 25 34 20 25 34 23 25 34

13 12 16 19 12.0 14 18 21 12 12 24 17.5 18.0 12 19.5 12 19.5 27 15 21 14 23 15 24 33 15 24 15 24 40 17 18.5 27 43 17 18.5 27 46 17 22 31 42 20 22 33 20 25 33 20 25 33 23 25 33

1 1 1 1 0.3 1 1 1.5 0.3 0.3 1.5 1.5 1 0.3 1.5 0.3 1.5 1.5 0.6 1.5 0.6 2 0.6 2 2 0.6 2 0.6 2 2.5 1 2 2 2.5 1 2 2 3 1 2 2 2.5 1 2 2 1 2 2.5 1 2 2.5 1.5 2.5 2.5

11 11 14 16 9 12 15 18 9 9 20 13.5 16 9 15.5 9 15.5 23 10 17 11.5 18.5 12 19.5 27 12 19.5 12 19.5 33 14 14 21.5 35 14 14 21.5 37 14 17.5 25 35 16 17.5 27 16 20 27 16 20 27 18 20 28

1 1 1 1 0.3 1 1 1.5 0.3 0.3 1.5 1 1 0.3 1.5 0.3 1.5 1.5 0.6 1.5 0.6 2 0.6 2 1.5 0.6 2 0.6 2 2 1 2 2 2 1 2 2 2.5 1 2 2 2.5 1 2 2 1 2 2 1 2 2 1.5 2 2

108 450 2900 128 570 1000 118 450 108 450 2900 128 128 570 1000 128 2800 118 180 3600 128 128 118 180 3600 138 300 138 300 128 128 100 128 128 500 118 510 3500 128 128 300 118 128 350 108 550 128 070 128 570 1000 128 138 128 500 138 520 128 570 1000 118 390 128 490 128 430 3000 128 570 1000 128 270 138 380 3000 138 270 128 570 1000 138 150 128 100 3000 128 270 128 118 530 128 490 3000 128 220 128 310 128 128 550 138 100 128 330 3000 138 300 128 180 138 70 3000 138 020

33.272 31.408 31.170 36.090 29.621 37.304 35.810 39.518 32.665 34.608 48.637 40.205 41.335 37.639 44.838 39.617 47.309 55.767 44.261 50.554 47.220 55.400 53.388 62.155 69.253 58.852 66.615 62.748 70.969 86.263 69.503 73.586 80.106 94.316 74.185 78.249 84.587 102.939 78.849 87.433 93.090 102.378 88.590 92.004 101.343 93.223 100.414 105.786 97.974 105.003 110.475 106.599 109.650 115.94

2FB 2DB 2DD 2FD 2BD 2DB 2DD 2FD 2BC 2BD 2FD 2CC 2CD 2BD 2CD 2BD 2CD 2FD 2BD 2CD 2BD 2DD 2BC 2CD 2FD 2BC 2CD 2BC 2CD 2FD 2BC 2CC 2CD 2FD 2BC 2CC 2CD 2FD 2BC 2CC 2DD 2FD 2BC 2CC 2DD 2BC 2CC 2DD 2BC 2CC 2BD 2BC 2CC 2DD

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DESIGN OF BEARINGS AND TRIBOLOGY

TABLE 23-77 (Cont.)

r

r1 r

α

r1

r r1

r

Dφ E

r1

dφ B

T

d

D

T

B

r1s min r2s min

C

r3s min r4s min

E

Dimension series

90 90 90 95 95 95 100 100 105 105 105 110 110 110 120 120 130 130 140 140 150 150 160 160 170 180 180 190 190 200 200 220 220 240 240 260 260

125 135 140 130 140 145 140 150 145 155 160 150 160 185 168 180 180 190 190 205 210 215 220 225 235 240 145 255 260 265 270 285 290 305 310 325 330

23 28 34 23 28 34 25 34 25 33 38 25 33 38 29 41 32 41 32 44 38 44 38 44 44 39 44 41 47 41 47 41 47 41 47 41 47

23 27.5 33 23 27.5 33 25 33 25 31.5 37 25 31.5 37 29 40 32 40 32 43 38 43 38 43 43 38 43 40 46 40 46 40 46 40 46 40 46

1.5 2.5 2.5 1.5 2.5 2.5 1.5 2.5 1.5 2.5 3 1.5 2.5 3 1.5 3 2 3 2 3 2.5 3 2.5 3 3 3 3 3 4 3 4 4 4 4 4 4 4

18 23 28 18 23 28 20 28 20 27 31 20 27 31 23 33 25 33 25 36 30 36 30 36 36 31 36 33 38 33 38 33 38 33 38 33 38

1.5 2 2.5 1.5 2.5 2.5 1.5 2.5 1.5 2.5 2.5 1.5 2.5 2.5 1.5 2.5 1.5 2.5 1.5 2.5 2 3 2 3 3 3 3 3 3 3 3 3 3 3 3 4 4

128 510 128 010 3000 128 020 3000 138 250 128 300 128 300 128 230 128 570 3000 128 510 128 170 3000 128 170 3000 138 200 128 420 3000 128 420 3000 138 050 128 080 3000 128 450 128 510 3000 138 300 128 450 128 260 128 370 138 138 140 3000 128 130 3000 128 470 128 460 3000 128 150 128 150 128 450 128 450 128 450 128 128 530 128 520 138 460 138 440 3000

111.282 119.139 121.860 116.082 123.797 126.47 125.717 130.992 130.359 137.045 139.734 135.182 141.607 144.376 148.464 158.233 161.652 167.414 171.032 181.645 187.926 190.810 197.962 200.146 211.346 218.311 220.684 232.395 234.615 241.710 244.043 2U637 265.24 281.653 284.035 300.661 303.004

2BC 2CC 2CD 2BC 2CC 2CD 2CC 2CB 2CC 2CD 2DD 2CC 2CD 2DD 2DD 2CC 2CC 2DD 2CC 2DD 2DC 2DD 2DC 2BD 2DD 2DB 2BC 2DC 2DD 2DC 2DD 2DC 2DD 2DC 2DD 2DC 2DD

All dimensions in mm. Courtesy: Extracted from IS: 7461 (part 1) 1993.

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DESIGN OF BEARINGS AND TRIBOLOGY

TABLE 23-78 Dimensions for tapered roller bearings—Contact angle series 5 d

D

T

B

r1s min r2s min

C

r3s min r4s min

E

Dimension Series

20 25 28 30 32 35 40 40 45 45 50 50 55 55 60 60 65 65 70 70 75 75 80 80 85 85 90 90 95 95 100 100 105

47 52 58 62 65 72 80 50 85 100 90 110 100 120 110 130 115 120 125 130 130 135 135 140 140 145 145 150 150 155 155 160 160

19.25 19.25 20.25 21.25 22 24.25 24.75 27 24.75 38.25 24.75 42.25 30 45.5 34 48.5 34 39 37 42 37 42 37 42 37 42 37 42 37 42 37 42 37

18 18 19 20 21.5 23 23 26.5 23 36 23 40 28.5 43 32 46 32 38 34.5 40 34.5 40 34.5 40 34.5 40 34.5 40 34.5 40 34.5 40 34.5

1 1 1 1 1 1.5 1.5 4 1.5 2 1.5 2.5 4 2.5 4 3 4 4 4 4 4 5 4 5 4 5 4 5 4 5 5 5 5

15 15 16 17 17 19 19 21.5 19 30 18 33 24 35 27 37 27 31 30 34 30 34 30 34 30 34 30 34 30 34 30 34 30

1 1 1 1 1 1.5 1.5 1.5 1.5 1.5 1.5 2.0 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 3 3 3 3 3 3 3 3 3 3

198 218 150 208 340 208 340 208 218 100 208 208 430 3000 218 350 208 218 200 208 208 208 198 3000 208 208 3000 208 280 198 340 198 1100 208 260 208 198 360 208 490 208 240 198 160 198 160 208 208 208 440 208 440 198 200 198 400

33.708 37.555 42.436 46.389 48.523 53.052 61.438 58.963 66.138 71.639 72.169 78.582 77.839 86.300 85.698 94.000 89.829 91.241 98.100 100.186 102.199 104.210 108.128 108.199 112.385 115.106 118.567 119.254 122.832 123.374 127.221 130.033 133.284

5DD 5CD 5DB 5DC 5DC 5DC 5DC 5DD 5DC 5FD 5DC 5FD 5DD 5FD 5DD 5FD 5DD 5ED 5DD 5ED 5DD 5ED 5DD 5ED 5DD 5ED 5DD 5ED 5DD 5ED 5DD 5ED 5DD

All dimensions in mm. Courtesy: Extracted from IS: 7461 (part 1) 1993.

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23.142

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r

r-

r

CD

51100 51101 51102 51103 51104 51105 51106 51107 51108 51109 51110 51111 51112 51113 51114 51115 51116 51117 51118 51120 51122 51124 51126 51128 51130 51132FP 51134FP 51136FP 51138FP 51140FP

10TA11 12TA11 15TA11 17TA11 20TA11 25TA11 30TA11 35TA11 40TA11 45TA11 50TA11 55TA11 60TA11 65TA11 70TA11 75TA11 80TA11 85TA11 90TA11 100TA11 110TA11 120TA11 130TA11 140TA11 150TA11 160TA11 170TA11 180TA11 190TA11 200TA11

51100 01 02 03 04 05 51106 07 08 09 10 11 51112 13 14 15 16 17 51118 20 22 24 26 28 51130 32 34 36 38 40

SKF 10 12 15 17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 100 110 120 130 140 150 160 170 180 190 200

d 11 13 16 18 21 26 32 37 42 47 52 57 62 67 72 77 82 87 92 102 112 122 132 142 152 162 172 183 193 203

C 24 26 28 30 35 42 47 52 60 65 70 78 85 90 95 100 105 110 120 135 145 155 170 178 188 198 213 222 237 247

D 9 9 9 9 10 11 11 12 13 14 14 16 17 18 18 19 19 19 22 25 25 25 30 31 31 31 34 34 37 37

B 24 26 28 30 35 42 47 52 60 65 70 78 85 90 95 100 105 110 120 135 145 155 170 180 190 200 215 225 240 250

E

Dimensions, mm

0.3 0.3 0.3 0.3 0.3 0.6 0.6 0.6 0.6 0.6 0.6 0.6 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.1

r min 0.001 0.001 0.001 0.002 0.004 0.004 0.007 0.009 0.016 0.02 0.024 0.038 0.053 0.06 0.067 0.095 0.1 0.12 0.19 0.36 0.43 0.48 0.75 0.85 0.90 1.0 1.4 1.5 2.4 2.4

Minimum load constant, Mb 14.0 15.3 15.3 15.3 20.8 29.0 33.5 37.5 50.0 57.0 63.0 78.0 93.0 98.0 104.0 137.0 140.0 150.0 190.0 270.0 290.0 310.0 390.0 400.0 400.0 430.0 500.0 530.0 655.0 655.0

kN

FAG

14000 15300 14000 15300 20800 29000 33500 37500 56000 57000 63000 78000 90000 98000 104000 137000 140000 150000 190000 270000 290000 310000 390000 400000 400000 425000 500000 530000 655000 655000

N

SKF

10.0 10.4 10.4 9.65 12.70 15.6 16.6 17.6 23.2 24.5 25.5 31.0 36.5 37.5 37.5 44.0 45.0 45.5 60.0 85.0 86.5 90.0 112.0 112.0 111.0 112.0 132.0 134.0 170.0 170.0

kN

FAG

9950 10400 9360 9750 12700 15900 16800 17400 23400 24200 25500 30700 35800 37100 37700 44200 44900 46200 59200 85200 87100 88400 111000 111000 111000 112000 133000 135000 172000 168000

N

SKF

Dynamic, C

560 620 560 620 850 1160 1340 1530 2040 2280 2550 3100 3600 4000 4150 5500 5700 6000 7500 10000 10200 10800 12900 12900 12500 12900 14300 15000 18000 17600

N

Fatigue load limit, Fuf

9500 9000 8500 8500 7000 6300 5600 5300 4500 4500 4300 3800 3600 3400 3400 3200 3200 3200 2800 2200 2200 2000 1800 1800 1700 1700 1500 1500 1400 1400

b

rpm

Kinematically, FAG

7000 6700 6300 6300 5600 4800 4500 4300 3800 3400 3200 2800 2800 2600 2400 2200 2000 2000 1800 1700 1600 1600 1400 1300 1200 1200 1100 1000 950 950

Greasea , SKF

Permissible speed, n

Grease lubrication. To ﬁnd Fa min ¼ minimum axial load using M refer to table 23-82. Source: 1. Extracted with permission from ‘‘FAG Rolling Bearings’’, Catalogue WL 41520EI, 1995 Edition: FAG Precision Bearings Ltd., Maneja, Vadodara, India. 2. Courtesy: Extracted from SKF Rolling Bearings, Catalogue 4000E, 1989, SKF Rolling Bearings India Ltd., Mumbai, India.

a

r

Ed

B

FAG

IS Old No.

Bearing No.

Static, Co

Basic load rating capacity

TABLE 23-79 Single direction thrust ball bearings—Dimension Series 11 (Indian Standards), FAG and SKF Series 511

0.021 0.023 0.024 0.026 0.041 0.060 0.069 0.087 0.125 0.153 0.169 0.247 9.330 0.359 0.385 0.520 0.557 0.597 0.878 1.300 1.450 1.590 2.370 2.59 2.26 2.39 3.09 3.17 4.08 4.26

kg

Mass FAG

DESIGN OF BEARINGS AND TRIBOLOGY

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r

51200 51201 51202 51203 51204 51205 51206 51207 51208 51209 51210 51211 51212 51213 51214 51215 51216 51217 51218 51220 51222 51224 51226 51228 51230MP 51232MP 51234MP 51236MP 51238MP 51240MP

FAG 51200 01 02 03 04 05 51206 07 08 19 10 11 51212 13 14 15 16 17 51218 20 22 24 26 28 51230 32 34 36 38 40

SKF 10 12 15 17 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 100 110 120 130 140 150 160 170 180 190 200

d 12 14 17 19 22 27 32 37 42 47 52 57 62 67 72 77 82 88 93 103 113 123 133 143 153 163 173 183 194 204

C 26 28 32 35 40 47 52 62 68 73 78 90 95 100 105 110 115 125 135 150 160 170 187 197 212 222 237 260 265 275

D 11 11 12 12 14 15 16 18 19 20 22 25 26 27 27 27 28 31 35 38 38 39 45 46 50 51 55 56 62 62

B 26 28 32 35 40 47 52 62 68 73 78 90 95 100 105 110 115 125 135 150 160 170 190 200 215 225 240 260 270 280

E

Dimensions, mm

0.6 0.6 0.6 0.6 0.6 0.6 0.6 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.1 1.5 1.5 1 1 1 1 2 2

r min 0.002 0.002 0.003 0.004 0.008 0.013 0.014 0.028 0.05 0.043 0.01 0.11 0.12 0.14 0.11 0.18 0.22 0.38 0.53 0.67 0.80 1.0 1.7 1.9 2.8 3.2 4.5 5 7 8

Minimum load constant, Mb 17.0 19.0 25.0 27.5 37.5 50.0 47.5 67.0 98.0 80.0 106.0 134.0 140.0 150.0 160.0 170.0 190.0 250.0 300.0 320.0 360.0 400.0 540.0 570.0 735.0 780.0 930.0 1000.0 1160.0 1220.0

kN

FAG

17000 19000 25000 27500 37500 50000 47500 67000 98000 80000 106000 134000 140000 150000 160000 170000 190000 250000 300000 320000 360000 400000 540000 570000 735000 780000 930000 1000000 1160000 1220000

N

SKF

12.70 13.20 16.6 17.3 22.4 28.0 25.5 35.5 46.5 39.0 50.0 61.0 62.0 64.0 65.5 67.0 75.0 98.0 120.0 122.0 129.0 140.0 183.0 190.0 236.0 245.0 285.0 290.0 335.0 340.0

kN

FAG

12700 13300 16500 17200 22500 27600 25500 35100 46800 39000 49400 61000 62400 63700 65000 67600 76100 97500 119000 124000 130000 140000 186000 190000 238000 242000 286000 296000 332000 338000

N

SKF

Dynamic, C

695 765 1000 1100 1530 2040 1900 2700 4000 3200 4300 5400 5600 6000 6400 6800 7650 8800 11400 11400 12500 13400 15000 17600 22000 22800 26000 27500 31000 31500

N

Fatigue load limit, Fuf

8000 8000 6700 6700 5600 5000 4800 4000 3800 3600 3400 3200 3000 3000 2800 2800 2600 2200 2000 1900 1800 1700 1600 1500 1400 1400 1200 1200 1000 1000

b

rpm

Kinematically, FAG

6000 6000 5300 5000 4500 4000 3600 3000 2800 2600 2400. 1900 1900 1800 1800 1700 1700 1600 1500 1300 1200 1100 950 950 900 850 800 800 750 750

Greasea , SKF

Permissible speed, n

Grease lubrication. To ﬁnd Fa min ¼ minimum axial load using M refer to table 23-82. Source: 1. Extracted with permission from ‘‘FAG Rolling Bearings’’, Catalogue WL 41520EI, 1995 Edition: FAG Precision Bearings Ltd., Maneja, Vadodara, India. 2. Courtesy: Extracted from SKF Rolling Bearings, Catalogue 4000E, 1989, SKF Rolling Bearings India Ltd., Mumbai, India.

a

r

r

Ed

B

10TA12 12TA12 15TA12 17TA12 20TA12 25TA12 30TA12 35TA12 40TA12 45TA12 50TA12 55TA12 60TA12 65TA12 CD 70TA12 75TA12 80TA12 85TA12 90TA12 r 100TA12 110TA12 120TA12 130TA12 140TA12 150TA12 160TA12 170TA12 180TA12 190TA12 200TA12

IS Old No.

Bearing No.

Static, Co

Basic load rating capacity

TABLE 23-80 Single direction thrust ball bearings (with Flat Housing Washer)—Dimension Series 12 (Indian Standards)

0.031 0.034 0.046 0.053 0.083 0.115 0.130 0.215 0.278 0.302 0.371 0.586 0.651 0.737 0.783 0.827 0.908 1.22 1.68 2.22 2.37 2.67 3.99 4.33 6.09 6.56 8.12 8.70 11.70 12.00

kg

Mass FAG

DESIGN OF BEARINGS AND TRIBOLOGY

23.143

b

a

r

r

r

C D

51305 51306 51307 51308 51309 51310 51311 51312 51313 51314 51315 51316 51317 51318 51320 51322MP 51324MP 51326MP 51328MP 51330MP 51332M 51334M 51336M 51338M 51340M

FAGb 51305 06 07 51308 09 10 51311 12 13 51314 15 16 51317 18 20 51322 24 26 51328 30 51332 34 36 38 40

SKFb 25 30 35 40 45 50 55 60 65 70 75 80 85 90 100 110 120 130 140 150 160 170 180 190 200

d 27 32 37 42 47 52 57 62 67 72 77 82 88 93 103 113 123 134 144 154 164 174 184 195 205

C 52 60 68 78 85 95 105 110 115 125 135 140 150 155 170 187 205 220 235 245 265 275 245 315 335

D 18 21 24 26 28 31 35 35 36 40 44 44 49 50 55 63 70 75 80 80 87 87 95 105 110

B 52 60 68 78 85 95 105 110 115 125 135 140 150 155 170 190 210 225 240 250 270 280 300 320 340

E

Dimensions, mm

1.0 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.1 1.1 1.5 1.5 1.5 1.5 1.5 2.0 2.1 2.1 2.1 2.1 3.0 3.0 3.0 4.0 4.0

r min 0.019 0.028 0.05 0.08 0.12 0.18 0.26 0.28 0.32 0.53 0.75 0.8 1.1 1.2 1.8 2.8 4.5 6.0 8.0 9.0 12.0 13.0 18.0 26.0 30.0

Minimum load constant, Mb 55.0 65.5 88.0 112.0 140.0 173.0 208.0 208.0 220.0 300.0 360.0 360.0 425.0 465.0 560.0 720.0 915.0 1060.0 1220.0 1290.0 1500.0 1630.0 1830.0 2200.0 2400.0

kN

FAG kN

FAG

34.5 38.0 50.0 61.0 75.0 88.0 102.0 102.0 106.0 137.0 163.0 160.0 190.0 196.0 232.0 275.0 325.0 360.0 400.0 405.0 455.0 465.0 520.0 600.0 2400000 620.0

55000 65500 88000 112000 140000 173060 208000 208000 220000 300000 360000 360000 425000 465000 560000 720000 915000 1060000 1220000 1290000 1500000 1600000 1830000

N

SKF

2240 2650 3550 4500 5600 6950 8300 8300 8800 11800 14000 13700 16000 16500 19600 24000 28500 32000 35500 36500 41500 43000 47500

N

Fatigue load limit, Fuf

624000 56500

34500 37700 49400 61800 76100 88400 104000 101000 106000 135000 163000 159000 190000 195000 229000 276000 325000 358000 397000 410000 440000 468000 520000

N

SKF

Dynamic, C

4300 4000 3600 3200 3000 2800 2400 2200 2200 1900 1800 1800 1700 1700 1600 1400 1200 1100 1000 950 900 900 800 750 700

rpm

Kinematically, FAG

480

3400 2800 2400 2000 1900 1800 1600 1600 1500 1400 1200 1200 1100 1000 950 850 800 750 700 670 630 600 560

Greasea , SKF

Permissible speed, n

Grease lubrication. Source: 1. Extracted with permission from ‘‘FAG Rolling Bearings’’, Catalogue WL 41520EI, 1995 Edition: FAG Precision Bearings Ltd., Maneja, Vadodara, India. 2. Courtesy: Extracted from SKF Rolling Bearings, Catalogue 4000E, 1989, SKF Rolling Bearings India Ltd., Mumbai, India.

r

Cd

B

25TA13 30TA13 35TA13 40TA13 45TA13 50TA13 55TA13 60TA13 65TA13 70TA13 75TA13 80TA13 85TA13 90TA13 100TA13 110TA13 120TA13 130TA13 140TA13 150TA13

IS Old No.

Bearing No.

Static, Co

Basic load rating capacity

TABLE 23-81 Single Direction thrust ball bearings, Dimension Series 13 (Indian Standards), FAG and SKF series 513

0.170 0.263 0.377 0.533 0.613 0.940 1.300 1.370 1.490 1.910 2.610 2.710 3.530 3.570 4.960 7.700 10.700 13.000 15.700 16.400 21.300 22.500 24.800 31.000 44.300

kg

Mass FAG

DESIGN OF BEARINGS AND TRIBOLOGY

23.144

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b

a

r

r

r

CD

51405 51406 51407 51408 51409 51410 51411 51412FP 51413FP 51414FP 51415FP 51416FP 51417FP 51418FP 51420FP 51422FP 51424FP 51426FP 51428FP 51430FP

FAGb 51405 06 07 51408 09 10 51411 12 13 51414 15 16 51417 18 20 51422 24 26 28 30

SKFb 25 30 35 40 45 50 55 60 65 70 75 80 85 90 100 110 120 130 140 150

d

D

60 70 80 90 100 110 120 130 140 150 160 170 177 187 205 225 245 265 280 154 295

27 32 37 42 47 52 57 62 68 73 78 83 88 93 103 113 123 134

C 24 28 32 36 39 43 48 51 56 60 65 68 72 77 85 95 102 110 112 120

B 1.0 1.0 1.1 1.1 1.1 1.5 1.5 1.5 2.0 2.0 2.0 2.1 2.1 2.1 3.0 3.0 4.0 4.0 4.0

300

r min

60 70 80 90 100 110 120 130 140 150 160 170 180 190 210 230 250 270

E

Dimensions, mm

20.0

0.043 0.08 0.13 0.22 0.32 0.48 0.67 0.85 1.1 1.4 1.8 2.2 2.8 3.4 5.3 7.5 9.0 15.0

Minimum load constant, Mb

1800.0

90.0 125.0 156.0 204.0 245.0 310.0 360.0 400.0 450.0 500.0 560.0 620.0 680.0 750.0 965.0 1140.0 1220.0 1600.0

kN

FAG

90000 125000 156000 204000 240000 310006 360000 400000 450000 500000 560000 620000 680000 750000 965000 114000 122000 160000 160000 180000

N

SKF

560.0

56.00 72.0 86.5 112.0 129.0 156.0 180.0 200.0 216.0 236.0 250.0 270.0 290.0 305.0 365.0 415.0 425.0 520.0

kN

FAG N

Fatigue load limit, Fuf

3600 5100 6200 8300 9600 12500 14300 16000 18000 19300 20800 22400 24000 25500 31500 34500 36000 45000 44000 559000 48000

55300 72800 87100 112000 130000 159000 178000 199000 216000 234000 251000 270000 286000 307000 371000 410000 423000 520000

N

SKF

Dynamic, C

750

3600 3200 3000 2400 2200 2000 1800 1700 1600 1600 1500 1400 1300 1200 1000 950 900 800

rpm

Kinematically, FAG

2600 2600 1800 1700 1600 1500 1300 1100 1100 950 900 850 850 800 700 630 600 560 530 500

Greasea , SKF

Permissible speed, n

Grease Lubrication. Source: 1. Extracted with permission from ‘‘FAG Rolling Bearings’’, Catalogue WL 41520EI, 1995 Edition: FAG Precision Bearings Ltd., Maneja, Vadodara, India. 2. Courtesy: Extracted from SKF Rolling Bearings, Catalogue 4000E, 1989, SKF Rolling Bearings India Ltd., Mumbai, India.

r

Ed

B

25TA14 30TA14 35TA14 40TA14 45TA14 50TA14 55TA14 60TA14 65TA14 70TA14 75TA14 80TA14 85TA14 90TA14 100TA14

IS Old No.

Bearing No.

Static, Co

Basic load rating capacity

TABLE 23-82 Single direction thrust ball bearings—Dimension Series 14 (Indian Standards), FAG and SKF Series 514

0.363 0.576 0.962 1.170 1.600 2.18 2.91 3.70 4.67 5.72 7.06 8.23 9.79 11.60 15.40 20.80 26.50 32.80 34.50 43.10

kg

Mass FAG

DESIGN OF BEARINGS AND TRIBOLOGY

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23.145

r1

d1

d2

r1

r

h

r

52202 04 05 06 07 08 09 52210 11 12 13 14 15 52216 17 18 20 52222 24 26 28 30 32 34

52202 52204 52205 52206 52207 52208 52209 52210 52211 52212 52213 52214 52215 52216 52217 52218 52220 52222 52224 52226 52228 52230MP 52232MP 52234MP

15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 100

d 10 15 20 25 30 30 35 40 45 50 55 55 60 65 70 75 85 95 100 110 120 130 140 150

d1 17 22 27 32 37 42 47 52 57 62 67 72 77 82 88 93 103 113 123 133 143 153 163 173

32 40 47 52 62 68 73 78 90 95 100 105 110 115 125 135 150 160 170 190 200 215 225 240

d2 a D 22 26 28 29 34 36 37 39 45 46 47 47 47 48 55 62 67 67 68 80 81 89 90 97

H 5 6 7 7 8 9 9 9 10 10 10 10 10 10 12 14 15 15 15 18 18 20 20 21

h

Dimensions, mm

0.6 0.6 0.6 0.6 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.1 1.5 1.5 1.5 1.5 1.5

r min 0.3 0.3 0.3 0.3 0.3 0.6 0.6 0.6 0.6 0.6 0.6 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.1 1.l 1.1 1.1 1.1 1.1

r1 min 0.003 0.008 0.013 0.014 0.028 0.05 0.043 0.07 0.11 0.12 0.14 0.16 0.18 0.22 0.38 0.53 0.67 0.81 1.0 1.7 1.9 2.8 3.2 4.5

FAG

Minimum load constant, M

25.0 37.5 50.0 47.5 67.0 98.0 80.0 106.0 134.0 140.0 150.0 160.0 170.0 190.0 250.0 300.0 320.0 360.0 400.0 540.0 570.0 735.0 760.0 930.0

kN

FAG

25000 37500 50000 47500 67000 98000 80000 106000 134000 140000 150000 160000 170000 190000 250000 300000 320000 360000 400000 540000 570000 736000 760000 930000

N

SKF

Static, Co

16.6 22.4 28.0 25.5 36.5 46.5 39.0 50.0 61.0 62.0 64.0 65.5 67.0 75.0 98.0 120.0 122.0 129.0 140.0 183.0 190.0 236.0 245.0 285.0

kN

FAG

16500 22500 27600 25500 35100 46800 32000 49400 61800 62400 63700 65000 67600 76100 97500 119000 124000 130000 140000 186000 190000 238000 242000 286000

N

SKF

Dynamic, C

Basic load rating capacity

1000 1500 2040 1900 2700 4000 3200 4300 5400 5600 6000 6400 6800 7650 9500 11400 11400 11500 13400 17000 17500 22000 22800 26000

N

Fatigue load limit, Fuf SKF

6700 5600 5000 4800 4000 3800 3600 3400 3200 3000 3000 2800 2800 2600 2200 2000 1900 1800 1700 1600 1500 1400 1400 1200

rpm

Kinematically, FAG

5300 4500 4000 3600 3000 2800 2600 2400 1900 1900 1800 1750 1700 1700 1600 1600 1300 1200 1100 950 950 900 850 800

Greaseb , SKF

Permissible speed, n

0.081 0.147 0.215 0.247 0.408 0.553 0.597 0.710 1.100 1.210 1.340 1.470 1.570 1.720 2.390 3.220 4.210 4.63 5.23 7.99 8.66 11.40 12.30 14.00

kg

Mass FAG

b

d2 : Refer to FAG bearings. Grease lubrication. Source: 1. Extracted with permission from ‘‘FAG Rolling Bearings’’, Catalogue WL 41520EI, 1995 Edition: FAG Precision Bearings Ltd., Maneja, Vadodara, India. 2. Courtesy: Extracted from SKF Rolling Bearings, Catalogue 4000E, 1989, SKF Rolling Bearings India Ltd., Mumbai, India. The minimum axial load in case of thrust ball bearings according to FAG Fa min ¼ Mðnmax =1000Þ2 where nmax ¼ maximum operating speed, rpm; M ¼ minimum load constant taken from Tables 23-78 to 23-82 for thrust ball bearings.

a

H

D d r

SKF

IS FAG

Bearing No.

TABLE 23-83 Double direction thrust ball bearings—Dimension Series 522

DESIGN OF BEARINGS AND TRIBOLOGY

23.146

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DESIGN OF BEARINGS AND TRIBOLOGY

TABLE 23-84 Needle bearings—Light Series Bearing No. IS

B

d D

r

D

r

di

B

r

NRB

Basic load rating capacity

Bearing with inner ring

Bearing without inner ring

Bearing with inner ring

Bearing without inner ring

Dimensions, mm d Nom

di Nom

D Nom

B Nom

NA1012 NA1015 NA1017 NA1020 NA1025 NA1030 NA1035 NA1040 NA1045 NA1050 NA1055 NA1060 NA1065 NA1070 NA1075 NA1080

RNA1012 RNA1015 RNA1017 RNA1020 RNA1025 RNA1030 RNA1035 RNA1040 RNA1045 RNA1050 RNA1055 RNA1060 RNA1065 RNA1070 RNA1075 RNA1080

Na1012 Na1015 Na1017 Na1020 Na1025 Na1030 Na1035 Na1040 Na1045 Na1050 Na1055 Na1060 Na1065 Na1070 Na1075 Na1080

Na1012S/Bi Na1015S/Bi Na1017S/Bi Na1020S/Bi Na1025S/Bi Na1030S/Bi Na1035S/Bi Na1040S/Bi Na1045S/Bi Na1050S/Bi Na1055S/Bi Na1060S/Bi Na1065S/Bi Na1070S/Bi Na1075S/Bi Na1080S/Bi

12 15 17 20 25 30 35 40 45 50 55 60 65 70 75 80

17.6 20.8 23.9 28.7 33.5 38.2 44.0 49.7 55.4 62.1 68.8 72.6 78.3 83.1 88 96

28 32 35 42 47 52 58 65 72 80 85 90 95 100 110 115

15 15 15 18 18 18 18 18 18 20 20 20 20 20 24 24

r

Dynamic, C, N

Limiting Static, speed, Co , n, N rpm

.35 .35 .65 .65 .65 .65 .65 .65 .85 .65 .65 .65 .65 .65 .65 .65

11280 12600 12260 19610 21570 23930 26480 28730 31000 33540 36190 37460 41580 43350 64720 68650

9415 10890 12260 18340 21180 23930 27360 30700 34030 37850 41680 43740 49520 52860 80410 86790

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21600 18300 15900 13200 11100 10000 8600 7600 6900 6100 5500 5200 4900 4500 4300 4000

DESIGN OF BEARINGS AND TRIBOLOGY

23.148

CHAPTER TWENTY-THREE

TABLE 23-85 Needle bearings—Medium Series Bearing No. IS

B

di

r

D

NRB

Bearing with inner ring

Bearing without inner ring

Bearing with inner ring

Bearing without inner ring

NA2015 NA2020 NA2025 NA2030 NA2035 NA2040 NA2045 NA2050 NA2055 NA2060 NA2065 NA2070 NA2075 NA2080 NA2085 NA2090 NA2095 NA2100 NA2105 NA2110 NA2115 NA2120 NA2125 NA2130 NA2140 NA2150 NA2160 NA2170 NA2180 NA2190 NA2200 NA2210 NA2220 NA,2230 NA2240 NA2250

RNA2015 RNA2020 RNA2025 RNA2030 RNA2035 RNA2040 RNA2045 RNA2050 RNA2055 RNA2060 RNA2065 RNA2070 RNA2075 RNA2080 RNA2085 RNA2090 RNA2095 RNA2100 RNA2105 RNA2110 RNA2115 RNA2120 RNA2125 RNA2130 RNA2140 RNA2150 RNA2160 RNA2170 RNA2180 RNA2190 RNA2200 RNA2210 RNA2220 RNA2230 RNA2240 RNA2250

Na2015 Na2020 Na2025 Na2030 Na2035 Na2040 Na2045 Na2050 Na2055 Na2060 Na2065 Na2070 Na2075 Na2080 Na2085 Na2090 Na2095 Na2100 Na2105 Na2110 Na2115 Na2120 Na2125 Na2130 Na2140 Na2150 Na2160 Na2170 Na2180 Na2190 Na2200 Na2210 Na2220 Na2230 Na2240 Na2250

Na2015S/Bi Na2020S/Bi Na2025S/Bi Na2030S/Bi Na2035S/Bi Na2040S/Bi Na2045S/Bi Na2050S/Bi Na2055S/Bi Na2060S/Bi Na2065S/Bi Na2070S/Bi Na2075S/Bi Na2080S/Bi Na2085S/Bi Na2090S/Bi Na2095S/Bi Na2100S/Bi Na2105S/Bi Na2110S/Bi Na2115S/Bi Na2120S/Bi Na2125S/Bi Na2130S/Bi Na2140S/Bi Na2150S/Bi Na2160S/Bi Na2170S/Bi Na2180S/Bi Na2190S/Bi Na2200S/Bi Na2210S/Bi Na2220S/Bi Na2230S/Bi Na2240S/Bi Na2250S/Bi

Basic load rating capacity Dimensions, mm D B r d di Nom Nom Nom Nom

Dynamic, C, N

Static, Co , N

Limiting speed, n, rpm

15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 140 150 160 170 180 190 200 210 220 230 240 250

24320 29220 31580 35700 39230 42950 46580 63740 71100 74040 80410 83360 105910 112780 115720 122580 118660 125320 131410 136310 141220 145140 149060 152000 159850 168670 174560 238300 246150 256930 262820 282430 337350 346170 356960 367750

21570 26280 31580 35700 40500 45600 50500 74530 82380 84340 95610 101010 131410 143180 148080 156410 162790 170630 177500 185340 196130 202020 211820 216730 233400 248110 262820 367650 384420 408940 423650 446200 557010 578590 603110 632530

17200 13200 11100 10000 8600 9600 6900 6100 5500 5200 4900 4500 4300 4000 3800 3600 3500 3300 3200 3000 2900 2800 2700 2600 2400 2200 2100 2000 1900 1800 1700 1600 1500 1500 1400 1300

22.1 28.7 33.5 38.2 44 49.7 55.4 62.1 68.8 72.6 78.3 83.1 88 96 99.5 104.7 109.1 114.7 119.2 124.7 132.5 137 143.5 148 158 170.5 179.3 193.8 202.6 216 224.1 236 248.4 258.4 269.6 281.9

35 42 47 52 58 62 72 80 85 90 95 100 110 115 120 125 130 135 140 145 155 160 165 170 180 195 205 220 230 240 255 265 280 290 300 315

22 22 22 22 22 22 22 28 28 28 28 28 32 32 32 32 32 32 32 34 34 34 34 34 36 36 36 42 42 42 42 42 49 49 49 49

.65 .65 .65 .65 .65 .65 .55 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 1.35 1.35 1.35 1.35 1.35 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85

Courtesy: IS: 4215, 1993.

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DESIGN OF BEARINGS AND TRIBOLOGY

TABLE 23-86 Needle bearings—Heavy Series Bearing No. IS

B

r

d

r

D

NRB

Basic load rating capacity

Bearing with inner ring

Bearing without inner ring

Bearing with inner ring

Bearing without inner ring

Dimensions, mm d di D B r Nom Nom Nom Nom

NA3030 NA3035 NA3040 NA3045 NA3050 NA3055 NA3060 NA3065 NA3070 NA3075 NA3080 NA3085 NA3090 NA3095 NA3100 NA3105 NA3110 NA3115 NA3120 NA3125 NA3130 NA3140 NA3150 NA3160 NA3170 NA3180 NA3190 NA3200 NA3210 NA3220 NA3230 NA3240 NA3250

RNA3030 RNA3035 RNA3040 RNA3045 RNA3050 RNA3055 RNA3060 RNA3065 RNA3070 RNA3075 RNA3080 RNA3085 RNA3090 RNA3095 RNA3100 RNA3105 RNA3110 RNA3115 RNA3120 RNA3125 RNA3130 RNA3140 RNA3150 RNA3160 RNA3170 RNA3180 RNA3190 RNA3200 RNA3210 RNA3220 RNA3230 RNA3240 RNA3250

Na3030 Na3035 Na3040 Na3045 Na3050 Na3055 Na3060 Na3065 Na3070 Na3075 Na3680 Na3085 Na3090 Na3095 Na3100 Na3105 Na3110 Na3115 Na3120 Na3125 Na3130 Na3140 Na3150 Na3160 Na3170 Na3180 Na3190 Na3200 Na3210 Na3220 Na3230 Na3240 Na3250

Na3030S/Bi Na3035S/Bi Na3040S/Bi Na3045S/Bi Na3050S/Bi Na3055S/Bi Na3060S/Bi Na3065S/Bi Na3070S/Bi Na3075S/Bi Na3080S/Bi Na3085S/Bi Na3090S/Bi Na3095S/Bi Na3100S/Bi Na3105S/Bi Na3110S/Bi Na3115S/Bi Na3120S/Bi Na3125S/Bi Na3130S/Bi Na3140S/Bi Na3150S/Bi Na3160S/Bi Na3170S/Bi Na3180S/Bi Na3190S/Bi Na3200S/Bi Na3210S/Bi Na3220S/Bi Na3230S/Bi Na3240S/Bi Na3250S/Bi

30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 140 150 160 170 180 190 200 210 220 230 240 250

44 49.7 55.4 62.1 68.8 72.6 78.3 83.1 88 96 99.5 104.7 109.1 114.7 119.2 124.7 132.5 137 143.5 152.8 158 170.5 179.3 193.8 202.6 216 224.1 236 248.4 258.4 269.6 281.5 290.9

62 72 80 85 90 95 100 105 110 120 125 130 135 140 145 150 160 165 170 185 190 205 215 230 245 255 265 280 290 300 315 325 340

30 36 36 38 38 38 38 38 38 38 38 38 43 43 43 45 45 45 45 52 52 52 52 57 57 57 57 57 64 64 64 64 74

0.65 0.65 0.65 0.85 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 1.35 1.35 1.35 1.35 1.35 1.85 1.85 1.85 1.85 1.85 1.85 1.85 1.85

Dynamic, Static, C, Co , N N 35790 91690 102970 106890 115720 119640 126510 131410 137290 140230 149060 153960 189270 196130 201040 207900 215750 221630 228490 273600 280470 294200 307930 368730 380500 397170

68160 98070 107870 121600 134350 141220 151026 160830 169650 184360 190250 198090 249090 262820 270660 283410 300080 311850 323620 397170 409920 441300 463850 568780 593300 632530

421680 490330 504060 517790 514850 666850

691370 813950 847290 882590 921820 11157180

Limiting speed, n, rpm 8600 7600 6900 6100 5500 5200 4900 4500 4300 4000 3800 3600 3500 3300 3200 3000 2900 2800 2700 2500 2400 2200 2100 2000 1900 1800 1700 1600 1500 1500 1400 1300 1300

Courtesy: IS: 4215, 1993.

TABLE 23-87 Hardness factors for needle-roller bearings Rockwell C hardness of raceway

Approximate Brinell hardness (Bhn)

Hardness factor, Kh

63 60 58 56 54 52 50

660 620 595 570 545 515 490

1.00 0.98 0.96 0.92 0.83 0.70 0.50

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DESIGN OF BEARINGS AND TRIBOLOGY

23.150

CHAPTER TWENTY-THREE

Particular

Formula

TABLE 23-88 Torrington needle-roller sizes Diameter, mm

Length, mm

Diameter, mm

Length, mm

1.590 1.590 1.590 1.590 2.380 2.3815 2.3850 2.3850 3.1750 3.1750 3.1750 3.1750

9.40 12.45 15.75 16.95 10.55 19.05 11.745 24.758 9.770 12.750 15.650 19.050

3.175 3.175 3.175 3.175 4.010 4.740 4.765 4.765 4.765 4.765 5.500 6.350

22.25 23.82 25.40 28.575 18.800 13.380 18.900 25.400 30.200 34.950 19.100 31.750

Maximum shear stress occurs below the contact surface for ductile material

Refer to Table 23-91.

(i) For spheres

Refer to Table 23-92.

(ii) For cylinders

SELECTION OF FIT FOR BEARINGS For selection of ﬁt for housing seatings for radial and thrust bearings For selection of ﬁt for shaft (solid) seatings for radial and thrust bearings.

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

23.151

TABLE 23-89 Dimensions for needle bearing without outer ring, type NCS d, mm

do , mm

B, mm

r, mm

30 32 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140

44.2 46.4 50 55.7 61.4 67.1 72.9 80.5 84.3 90.1 96.8 102.4 106.5 111.7 116.1 121.7 126.2 131.7 139.5 144 150.54 155.04 159.8 165.04

18 18 18 18 18 20 20 20 20 20 24 24 32 32 32 32 32 34 34 34 34 34 36 36

1 1 1 1.5 1.5 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

B

r

d

Source: IS 4215, 1967.

TABLE 23-90 Design data for needle-roller bearings Recommended Journal race diameter, mm

Total radial clearance, mm

Needle diameter,

9.50–19.00 19.00–31.75 31.75–50.80 50.80–76.00 76.00–127.00 127.00–177.00

0.0125–0.040 0.0180–0.050 0.0200–0.055 0.0255–0.065 0.0305–0.075 0.0355–0.085

1.55 2.35 3.20 3.20 4.75 4.75

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Do

DESIGN OF BEARINGS AND TRIBOLOGY

23.152

CHAPTER TWENTY-THREE

TABLE 23-91 Selection of ﬁt (a) Housing seatings for radial bearings Conditions

Applications

Tolerance

Solid Housings Rotating outer-ring load Heavy loads on bearings in thin walled housings; heavy shock loads Normal and heavy loads Light and variable loads Direction of loading indeterminate Heavy shock loads Heavy and normal loads; axial mobility of outer ring unnecessary Normal and light loads; axial mobility of outer ring desirable

Roller bearing wheel hubs; big-end bearings

P7

Ball bearing wheel hubs; big-end bearings Conveyor rollers, rope sheaves; belt tension pulleys

N7 M7

Electric traction motors Electric motors, pumps, crankshaft main bearings

M7 K7

Electric motors, pumps; crankshaft main bearings

J7

Split or Solid Housing Stationary outer-ring load Shock loads intermittent All loads Normal and light loads Heat condition through shafts

Railway axle boxes Bearings in general applications Line shafting Drying cylinders; large electric motors

J7 H7 H8 G7

Solid Housings Arrangement of bearing very accurate Accurate running and great rigidity under variable load Accurate running under light loads of indeterminate direction Accurate running; axial movement of outer ring desirable

Roller bearings D > 125 mm For machine-tool D < 125 mm main spindles Ball bearings at work end of grinding spindle; locating bearings in high-speed centrifugal compressors Ball bearings at drive end of grinding spindles; axially free bearings in high-speed centrifugal compressors

N6 M6 K6

J6

(b) Housing seatings for thrust bearings Conditions

Applications

Tolerance

Purely axial load

Thrust ball bearings Spherical roller thrust bearings where another bearing takes care of the radial location Stationary load on housing washer or direction of loading indeterminate Generally Rotating load or housing washer Heavy radial load

H8

Combined (radial and axial) load or spherical roller thrust bearings

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J7 K7 M7

DESIGN OF BEARINGS AND TRIBOLOGY

23.153

DESIGN OF BEARINGS AND TRIBOLOGY

TABLE 23-92 Selection of ﬁt (a) Shaft (solid) seatings for radial bearings Shaft diameter, mm

Conditions

Ball bearings

Application

Cylindrical and tapered roller bearings

Spherical roller bearings

Tolerance

Bearings with cylindrical bore Stationary inner-ring load Easy axial displacement of inner ring on shaft desirable Easy axial displacement of inner ring on shaft unnecessary

Wheels on nonrotating axles

All diameters

g6

Tension pulleys; rope sheaves

All diameters

h6

Rotating inner-ring or direction of loading indeterminate Light and variable loads Electrical apparatus; machine tools; pumps; transport vehicles

18 18–100 100–200

Normal and heavy loads

General application electric motors pumps; turbines; gearing; wood working machines; and internalcombustion engines

18 18–100 100–140 140–200 200–280

Shock and heavy loads

Locomotive axle boxes; traction motors

— — — —

Purely axial load

All kinds of bearing arrangements

Loads of all kinds

Bearing arrangements in general; railway axle boxes Line shafting

— 40 40-140 140-200 — 40 40–100 100–140 140–200 200–400

50–140 140–200 — — All diameters

— 40 49-100 100-200 — 40 40–65 65–100 100–140 140–280 280–500 >500 50–100 100–140 140–200 200–500

h5 j6 k6 m6 j5 k5 m5 m6 n6 p6 r6 r7 n6 p6 r6 r7 j6

Bearings with taper bore and sleeve All diameters

h9

All diameters

h10

(b) Shaft seatings for thrust bearings Conditions

Applications

Tolerance

Purely axial load

Thrust ball bearing, spherical roller thrust bearings

All diameters

j6

Combined (radial and axial) load on spherical thrust bearings

Stationary load on shaft washer Rotating load on shaft washer or direction of loading indeterminate

All diameters d 200 mm d ¼ 200–400 mm d > 400 mm

j6 k6 m6 n6

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DESIGN OF BEARINGS AND TRIBOLOGY

23.154

CHAPTER TWENTY-THREE

23.3 FRICTION AND WEAR1 SYMBOLS a A Aa A0 b c c1 , c2 d E F F Fa Ff Fploughing G h hm H i Qme k 1 , k2 k KE KL KV KW Ksm KsV 0 Ksv L m n Pc PH Pt Ptot P Pa Pm

half the mean diameter of area of contact, Eq. (23-252) real area of contact, m2 (in2) apparent area of contact, m2 (in2) abrasion factor constant used in Eq. (23-222), exponent constant used in Eqs. (23-225) and (23-280) constants as given in Eqs. (23-28lb) and (23-281c) diameter, m (in) Young’s modulus, GPa (psi) force, kN (lbf ) total force of friction, kN (lbf ) adhesive component of friction force or force to shear junctions, kN (lbf ) fatigue resistance is the average number of reversed stress cycles which the surface layer must undergo under given abrasion condition, kN (lbf ) force to plough the asperities on one surface through the other, kN (lbf ) elasticity constant characterizing rubber thickness of layer removed, m (in) eﬀective thickness of the worn-out surface layer, m (in) height of asperities, m (in) hardness of softer material, N/m2 or Pa (psi) number of surface layer which are abraded during a test number of repeated deformation as used in Eqs. (23-256) to (23-258) mechanical equivalent of heat, N m/J (lbf in/Btu or lbf ft/Btu) thermal conductivity of two conducting materials, W/m K (Btu/ft h 8F) constant used in Eq. (23-245) and given in Table 23-77 energetic wear rate or energy index of abrasion linear wear rate volumetric wear rate gravimetric wear rate speciﬁc wear by mass speciﬁc wear by volume modiﬁed speciﬁc wear sliding distance, m (in) mass of wear debris, kg (lb) exponent power used to elongate shred power applied to hysteresis loss which accompanies roll deformation power used to tear shred from surface layer total ﬁctional power yield pressure of soft material (about 5 times the critical shear stress), MPa (psi) apparent pressure over the contact area, MPa (psi) mean pressure over the contact area, MPa (psi) ﬂow pressure of material, MPa (psi)

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

q r R s v v1 V V W Wtb Wtot ¼ Wn2 W z n m a c ploughing

c

n m

friction work done corresponding to a simple stressing cycle which corresponds to a sliding length of , N m (lbf in) radius of curvature, m (in) radius of circular junction (Fig. 23-60), m (in) mean radius of the curvature at the tip of the abrasive particles, m (in) spacing between ridges in the elastomer surface, m (in) velocity, m/s (ft/min) velocity, m/s (ft/min) volume deformed body, m3 (in3) volume of transferred fragment, m3 (in3) volume of layer removed, m3 (in3) applied load at interface, kN (lbf ) the work of adhesion of the contacting metals which can be expressed in terms of their surface energies, N m (lbf in or lbf ft) normal load per unit area, kN (lbf ) weight lost due to abraded layer being removed from the bulk material, kN (lbf ) the average depth of penetration for single sphere, m (in) the absolute approach, m (in) coeﬃcient of hysteresis loss constant depends on the surface treatment taken from Table 23-73 surface tension of the softer sliding member, N/m (lbf/in) abradability as wear index angle of slope of irregularities, deg mean temperature rise at the sliding junction, 8C (8F) coeﬃcient of friction adhesive component of coeﬃcient of friction coeﬃcient of elastic friction coeﬃcient of static friction taken from Table 23-74 ploughing component of coeﬃcient of friction Poisson’s ratio density of the abraded elastomer, kg/m3 (lb/in3) coeﬃcient of abrasion resistance mean wavelength of the surface asperities stress, MPa (psi) contact pressure or pressure over the contours, MPa (psi) tensile strength of elastomer in simple tensions, MPa (psi) shear strength of junction, MPa (psi) mean shear stress, MPa (psi)

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23.155

DESIGN OF BEARINGS AND TRIBOLOGY

23.156

CHAPTER TWENTY-THREE

Particular

Formula

FRICTION The general expression for force of friction

F ¼ Fa þ Fploughing

ð23-256Þ

The total friction force

F ¼ A

ð23-257Þ

The real area of contact

A¼

The general expression for coeﬃcient of friction

¼ a þ ploughing

The total coeﬃcient of friction

F A ¼ ¼ ð23-260Þ W W P " # 1=2 pﬃﬃﬃ Kn K4 hm 2=ð2 þ 1Þ c 2=ð2 þ 1Þ e ¼ r E 2ð þ 1Þ

W P

ð23-258Þ ð23-259Þ

¼

The coeﬃcient of elastic friction when a rigid rough surface is pressed against an elastically deformable second surface

ð23-261Þ where K ’ 1; calculate K4 from Eq. (23-262). The expression for K4 to be used in Eq. (23-261)

K4 ¼

0:75ð1 v2 Þ K2 b

=ð2 þ 1Þ

ð23-262Þ

where K2 ¼ 1, 0.4, 0.12 for ¼ 1, 2, 3 respectively

TABLE 23-93 Constant to he used in Eq. (23-261)

Refer to Table 23-93 for .

Surface treatment

b

Turning, milling Planing Polishing

2 3 3

1–3 4–6 5–10

Greenwood and Tabor’s formula for coeﬃcient of elastic friction

e ¼ n Pm

9 1 v2 64 E

ð23-263Þ

Coeﬃcient of friction under dynamic conditions Franke’s expression for coeﬃcient of friction during rotation

r ¼ o ecv

Stiehl’s formula for coeﬃcient of friction

¼ 0:6

Schutch’s formula for coeﬃcient of friction for leather sliding against slightly lubricated steel plate

¼ 0:5ð1 þ 0:1vÞ

ð23-264Þ

where c ¼ constant taken from Table 23-94 0:6 vþ1

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ð23-265Þ ð23-266Þ

DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.157

Formula

Krumme’s formula for coeﬃcient of friction in textile machinery

¼ 0:38

Formula for coeﬃcient of friction used in design of brakes

¼ 0:6

0:1 0:5 þ v

ð23-267Þ

16P þ 100 100 80P þ 100 3vk þ 100

ð23-268Þ

where P ¼ real pressure on brake shoe, tonne force (tf ) TABLE 23-94 Values of constant c to be used in Eq. (23-264) Sliding combination

State of rubbing surfaces

Coeﬃcient of static friction, o ,

Constant c

Cast iron—steel Forged iron—forged Iron

Dry Dry Slightly moist

0.29 0.29 0.24

1/23 1/50 1/35

Temperature of sliding surface Mean temperature rise at the interface above the material

m ¼

0:25Wv Qme rðk1 þ k2 Þ

ð23-269Þ

where Qme ¼ mechanical equivalent of heat N m/J (lbf in/ Btu or lbf ft/Btu) v ¼ velocity of sliding, cm/s (ft/min) r ¼ radius of the circular junction, cm, m (in) k1 ; k2 ¼ thermal conductivity of the two contacting materials, W/m 8C (Btu/ft h 8F) taken from Table 23-95 TABLE 23-95 Temperature rise per unit sliding velocity

k1

Material combination

dyn/cm

N/m

Steel on steel Lead on steel Bakelite on Bakelite Brass on brass Glass on steel Steel on nylon Brass on nylon Steel on bronze

0.5 0.5 0.3 0.4 0.3 0.3 0.3 0.25

1500 450 100 900 500 120 120 900

1.50 0.45 0.10 0.90 0.50 0.12 0.12 0.90

k2

cal/s cm 8C 0.11 0.08 0.0015 0.26 0.0007 0.11 0.26 0.11

0.11 0.11 0.0015 0.26 0.11 0.0006 0.0006 0.18

k1

k2 W/m 8K

46.055 33.490 0.628 108.856 0.293 46.055 108.856 46.055

46.055 46.055 0.628 108.856 46.055 0.25121 0.25121 75.362

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=, 8C/cm/s 0.75 0.26 2.20 0.15 0.30 0.07 0.03 0.17

DESIGN OF BEARINGS AND TRIBOLOGY

23.158

CHAPTER TWENTY-THREE

Particular

Formula

Simple and crude formula for the mean temperature rise

m ¼ 54:4v ð a factor of 1.67Þ

The radius of a junction (Fig. 23-60)

r ¼ 12;000

The load carried by each junction (Fig. 23-60)

W ¼ r2 P

Mean temperature rise at the interface above the rest of material

m ¼

Material I 2r

9400v Qme ðk1 þ k2 Þ

ð23-271Þ ð23-272Þ ð23-273Þ

where ¼ surface tension of the softer sliding member, N/m (lbf/in) taken from Table 23-95

Load W

Elevation

P

ð23-270Þ

For coeﬃcient of friction refer to Table 23-95.

Material II

Plan

FIGURE 23-60 Assumed junction model.

WEAR AND ABRASION Linear wear rate

KL ¼

Steady state wear rate, depth per unit time

KL ¼ KPVðabcdeÞ

thickness of layer removed h ¼ sliding distance L

ð23-274Þ ð23-275Þ

where K ¼ constant depends on (i) mechanical properties of material and its ability to (ii) smooth the counterface surface and/or (iii) transfer a thin ﬁlm of debris For a, b, c, d, e, refer to Table 23-103. volume of layer removed V ¼ sliding distance apparent area LAa ð23-276Þ

Volumetric wear rate

KV ¼

Energetic wear rate

KE ¼

The energetic and linear wear rate related by equation

KE ¼ KL ðAa =F Þ

volume of layer removed V ¼ work of friction F L

ð23-277Þ ð23-278Þ

where F L is measured in kW h The gravimetric wear rate

KW ¼

W ¼ Kv LAa

where ¼ density of abraded elastomer

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ð23-279Þ

DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.159

Formula

Wear index is given by abradability,

¼

abraded volume V V A0 ¼ ¼ ¼ work of friction F L WL

ð23-280Þ

where A0 ¼ ðV=WLÞ ¼ abrasion factor Energetic wear rate ðKE Þ ¼ abradability ðÞ

The relation between KE and

ð23-281Þ work of friction FL 1 1 ¼ ¼ ¼ ¼ ð23-282Þ abraded volume V A0 KE

The coeﬃcient of abrasion resistance as per work in the former Soviet Union

¼

For surface roughness as obtained by diﬀerent machining processes

Refer to Table 23-96.

Work done during wear

W 0 ¼ Vm

ð23-283Þ

Volume of transferred fragments formed in sliding a distance L

V¼

kWL 300P

ð23-284Þ

For k ¼ coeﬃcient of wear, refer to Table 23-97. For P ¼ hardness of the softer material, Pa (psi), refer to Table 23-99. TABLE 23-96 Surface roughnesses as obtained by machining processes

TABLE 23-97 Wear constant k

Manufacturing process

Surface roughnesses, lm

Sliding combination

Wear constant, k

Turned Coarse ground Fine ground 600 emery Polished Super ﬁnished

1–6 0.4–3 0.2–0.4 0.2 0.05–0.1 0.02–0.05

Zinc on zinc Low-carbon steel on low-carbon steel Copper on copper Stainless steel on stainless steel Copper on low-carbon steel Low-carbon steel on copper Bakelite on Bakelite

0.160 45 32 21 1.5 0.5 0.02

TABLE 23-98 Values of coeﬃcient of wear, k Metal on metal Like

Unlike 105

Condition Clean Poorly lubricated Average lubrication Excellent lubrication

500 20 2 0.2–0.2

20 20 2 0.2–0.2

Metal on non-metal 106 5 5 5 2

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8C

660 630 1400 270 321 838 804 29 1875 1495 1083 1407 1496 827 1312 30 937 1063 2222 1461 156 2454 1534 930 325 180 1652 650 1245 –39 2610 1018 1453 2468 2700 1552 1769 640 64

Metal

Aluminum Antimony Beryllium Bismuth Cadmium Calcium Cerium Cesium Chromium Cobalt Copper Dysprosium Erbium Europium Gadolinium Gallium Germanium Gold Hafnium Holmium Indium Iridium Iron Lanthanum Lead Lithium Lutetium Magnesium Manganese Mercury Molybdenum Neodymium Nickel Niobium Osmium Palladium Platinum Plutonium Potassium

933 903 1673 543 594 1111 1077 302 2148 1778 1356 1680 1769 1100 1585 303 1210 1336 2495 1734 429 2727 1807 1203 598 453 1925 923 1518 234 2883 1291 1726 2741 2973 1825 2042 913 337

K

Melting temperature

0.63 0.80 3.0 0.32 0.56 0.25 0.30 2.6 2.1 1.2 0.63 0.75 0.56 1.56 0.81 0.68 0.11 5.4 2.04 0.39 0.16

0.44

3.0 0.38 2.08 1.05 5.70 1.15 1.50 0.99

2.65 2.14 1.22 0.64 0.78

0.57

1.59 0.83

0.69 0.11 5.50 2.08 0.40 0.16

0.45

3.06 0.39 2.12 1.07 5.81 1.17 1.53 1.01

3.0 0.38 2.08 1.05 5.70 1.15 1.50 0.99

0.44

0.68 0.11 5.4 2.04 0.39 0.16

1.56 0.81

0.56

2.6 2.1 1.2 0.63 0.75

0.63 0.80 3.0 0.32 0.56 0.25 0.30

8.4 1.7 3.2 2.8 3.1 1.6 2.8

3.16 1.63 2.86

1.5 2.5

2.1 2.4 2.2 0.03 6.3 2.5 1.9 0.09

2.7

8.57 1.73 3.26 2.86

1.53 2.55

2.14 2.45 2.24 0.03 6.43 2.55 1.94 0.09

2.76

1.6 7.8 3.2 3.3 2.9

0.72 0.87 1.20

0.73 0.89 1.22 1.63 7.96 3.26 3.37 2.96

1.1 0.11 3.20

1.12 0.11 3.26

23.160

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8.4 1.7 3.2 2.8

1.5 2.5

2.1 2.4 2.2 0.03 6.3 2.5 1.9 0.09

2.7

1.6 7.8 3.2 3.3 2.9

0.72 0.87 1.20

1.1 0.11 3.20

kgf/cm2 103 N/m2 108 MPa 102

kgf/cm2 106 N/m2 1011 MPa 105

0.64 0.82 3.06 0.33 0.57 0.26 0.31

Yield strength, sy

Young’s modulus, E

TABLE 23-99 Properties of metallic elements

240 80 210 160 800 110 100 266 0.04

118 46 3300

58 260 90 0.9 350 82 150 4

125 125 80 117 161 17 97 6.5

27 58 150 7 22 17 48

235.2 78.4 205.8 156.8 784 107.8 98 260.68 0.04

115.64 45.08 3234

56.84 254.80 88.2 0.88 343.0 80.36 147 3.92

122.5 122.5 78.4 115.66 157.78 16.66 95.06 6.37

26.46 56.84 147.0 6.86 21.56 16.66 47.04

235.2 78.4 205.8 156.8 784 107.8 98 260.68 0.04

115.64 45.08 3234

56.84 254.80 88.2 0.88 343.0 80.36 147 3.92

122.5 122.5 78.4 115.66 157.78 16.66 95.06 6.37

26.46 56.84 147.0 6.86 21.56 16.66 47.04

kgf/cm2 102 N/m2 107 MPa 10

Hardness, P

1.80 0.086

1800 86

1.70 2.10 1.19

0.46

460

1700 2100 1190

0.56

0.45 0.40

450 400 560

1.50

0.36 1.12

1.53 1.10

0.900 0.370 1.0 0.39 0.62

N/m

1500

360 1120

1530 1100

900 370 1000 390 620

erg/cm

Surface energy,

0.12

1.10

0.18

0.55 0.19

0.12 0.14

0.33 0.064 0.067 0.56 0.28

m

2300

18

23

0.18

8.1 0.081 13 0.13 1.5 0.015

12

110

18

55 19

12 14

33 6.4 6.7 56.0 28

cm

=P

DESIGN OF BEARINGS AND TRIBOLOGY

8C

919 3180 1966 39 2500 1072 1540 961 98 2996 1356 303 1750 1545 232 1670 3410 1132 1900 824 1495 420 1852

Metal

Praseodymium Rhenium Rhodium Rubidium Ruthenium Samanium Scandium Silver Sodium Tantalum Terbium Thallium Thorium Thulium Tin Titanium Tungsten Uranium Vanadium Ytterbium Yttrium Zinc Zirconium

1192 3453 2293 312 2773 1345 1813 1234 371 3269 1629 576 2023 1818 505 1943 3683 1405 2173 1097 1768 693 2125

K

Melting temperature

0.35 4.70 2.96 4.22 0.35 0.78 1.90 0.58 1.47 0.44 1.13 3.51 1.69 1.34 0.18 0.66 0.91 0.96

0.36 4.79 3.02

4.30 0.36

0.80

1.93 0.59

1.50

0.45 1.15 3.58 1.72 1.36 0.18 0.67 0.93 0.98

0.44 1.13 3.51 1.69 1.34 0.18 0.66 0.91 0.96

1.47

1.90 0.58

0.78

4.22 0.35

0.35 4.70 2.96

kgf/cm2 106 N/m2 1011 MPa 105

Young’s modulus, E

TABLE 23-99 Properties of metallic elements (Cont.)

3.5 0.09 1.5 1.4 1.15 1.4 18.0 2.0 8.4 0.73 1.4 1.3 2.0

0.09 1.53 1.43 1.17 1.43 18.36 2.04 8.57 0.74 1.43 1.33 2.04

2.0

5.5 1.3

2.0 22.0 9.7

3.56

2.04

5.61 1.33

2.04 22.40 9.89

0.09 1.5 1.4 1.15 1.4 18.0 2.0 8.4 0.73 1.4 1.3 2.0

3.5

2.0

5.5 1.3

2.0 22.0 9.7

kgf/cm2 103 N/m2 108 MPa 102

Yield strength, sy

86.24 1.96 36.26 51.94 5.19 64.7 426.3

20.58 36.26 37.24 142.10

21 37 38 145

78.4 0.07

88 2 37 53 53 65 435

80 0.07

382.2 62.72

119.56

122 390 64

74.48

76

20.58 36.26 37.24 142.10

86.24 1.96 36.26 51.94 5.19 64.7 426.3

78.4 0.07

382.2 62.72

119.56

74.48

kgf/cm2 102 N/m2 107 MPa 10

Hardness, P

0.79

2.30

2300

790

0.57

0.40

0.92 0.20

N/m

570

400

920 200

erg/cm

Surface energy,

21

5.3

110

200

11 2800

cm

0.21

0.053

1.10

2.0

0.11 28

m

=P

DESIGN OF BEARINGS AND TRIBOLOGY

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23.161

DESIGN OF BEARINGS AND TRIBOLOGY

23.162

CHAPTER TWENTY-THREE

Particular

Formula

Another formula for volume of transferred fragment formed in sliding a distance

V¼

kAL 3

ð23-285Þ

For k refer to Table 23-98. V W ¼K L Pm

The primary equation of wear according to Archard, Burwell, and Strang

ð23-286Þ

where Pm ¼ flow pressure of material For K, refer to Table 23-100.

Abrasion wear

Wab H

The mean diameter of loose wear particles which are produced at a smooth interface

d ¼ K1

The ratio of half mean diameter of the area of contact to mean radius of the curvature at the tip of the abrasive particle

a W ¼ K2 R GR2

ð23-287Þ ð23-288Þ

where ¼ value of exponent to be determined from experiment

TABLE 23-100 Coeﬃcient of wear Hardness Sliding against hardened tool-steel unless otherwise stated Mild steel on mild steel 60/40 brass Teﬂon 70/30 brass Perspex Bakelite (molded) type 50B Silver steel Beryllium copper Hardened tool steel Stellite Ferritic stainless steel Laminated Bakelite Type 292/16 Molded Bakelite Type 11085/1 Tungsten carbide on mild steel Molded Bakelite Type 547/1 Polythene Tungsten carbide on tungsten carbide

Wear coeﬃcient, K 3

7 10 6 104 2:5 104 1:7 104 0:7 106 7:5 106 6 105 3:7 105 1:3 104 5:5 105 1:7 105 1:5 106 7:5 107 4 106 3 107 1:3 107 1 106

kgf/mm2 18.6 95.0 5.0 68.0 0.0 5.0 320.0 210.0 850.0 690.0 50.0 33.0 30.0 186.0 29.0 1.70 1300.0

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MPa 182.4 931.6 49.0 666.8 196.1 245.2 3138.1 2059.4 8335.6 6776.6 2451.7 323.6 294.2 1824.0 284.4 16.7 12749

DESIGN OF BEARINGS AND TRIBOLOGY

23.163

DESIGN OF BEARINGS AND TRIBOLOGY

Particular

Volumetric wear rate

Volumetric wear rate for ¼ 13 Half the mean diameter of the area of contact for ¼ 13 The spacing s between ridges in the elastomer surface

Formula

KV ¼ K3 n2 R3

Wtot Gn2 R2

3

where n2 ¼ number of abrasive particles per unit area Wtot R KV ¼ K3 ð23-290Þ G a ¼ K1 s’

WR G

1=3 ð23-291Þ

Wtot Rd 2 G

1=3

KV ’ s

ð23-292Þ ð23-293Þ

s ’ d 2=3 The ratio of Kv to s when the abrasive surface consists of closely packed hemisphere so that d ¼ 2R

ð23-289Þ

Wtot G

2=3 ð23-294Þ

Fatigue wear Volume of surface layer removed under fatigue

V ¼ iAh

ð23-295Þ

The required sliding length during abrasion cycle under the given abrasion conditions before failure and separation occurs

L ¼ iFf

ð23-296Þ

The total work of friction

W0 ¼ ðWtot ÞL ¼ iqFf

ð23-297Þ

The coeﬃcient of abrasion resistance

qFf AL 2 3 3 W 2=3 ð1 2 Þ z¼ 4 E 2=3 R1=3 W 2=3 z ¼ 0:683 E 2=3 R1=3

ð23-298Þ

The Hertzian relationship for the average depth of penetration for single spheres

¼

ð23-299aÞ ð23-299bÞ

for rubber ¼ 0:5 where R ¼ asperity tips radius, cm, m (in) E ¼ Young’s modulus for rubber, GPa (psi)

The depth penetration

W ¼ applied load per asperity 2 2 2=3 2 Wtot z ¼ 0:685 A E 2=3 R1=3

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ð23-300Þ

DESIGN OF BEARINGS AND TRIBOLOGY

23.164

CHAPTER TWENTY-THREE

Particular

The number of asperities

The eﬀective thickness of the surface layer of elastomer

Formula

i¼

Wtot A ¼ 2 W

h ¼ k0 z

ð23-301Þ

R2 2

ð23-302Þ

where K 0 ¼ constant The coeﬃcient of abrasion resistance

¼

1=3 Ff 2=3 Wtot E R 2:14K 0 A

The ratio of abrasion resistance to coeﬃcient of sliding friction

1 ¼ A0

The fatigue resistance of rubber taking into consideration tensile strength, geometry of the base surface, and the loading conditions

Ff ¼

ð23-303Þ ð23-304Þ

o K 0 ðW=AÞ1=3 E 2=3 ðR=Þ3 2

ð23-305Þ

Wtot ð1 bÞ=3 ð5 2bÞ=3 ¼ K bo E 2ð1 bÞ=3 A R The ratio of abrasion resistance to coeﬃcient of friction

ð23-306Þ where b ¼ index which is characteristic of the material where K ¼ constant

The relationship between fatigue index b and

b ¼ 13 ð þ 2Þ

ð23-307Þ

Roll formation The coeﬃcient of abrasion resistance

¼

Ptot ðd VÞ=dt

ð23-308Þ

where ðd VÞ=dt ¼ volume abraded per unit time Ptot ¼ total frictional power ¼ Pt þ Pe þ PH The main condition which determines the probable occurrence of roll formation

Ptot o WV

ð23-309Þ

The more general form of the equation for volumetric wear rate which dependence on abrasion by load

KV ¼ CP

ð23-310Þ

where C ¼ constant taken from Table 23-101 P ¼ interfacial pressure, MPa (psi) is obtained from Table 23-101.

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.165

Formula

TABLE 23-101 Wear of rubber on steel, gauze, and abrasive paper Values of constants Rubber

Nature of surface

C 103

A

Steel Gauze Abrasive paper Steel Gauze Abrasive paper Steel Gauze Abrasive paper

1.1 1.5 240 2.7 1.1 305 1.2 5.4 65

1.9 5.3 1.1 1.9 2.0 0.9 3.1 3.0 1.0

B

C

Tread rubber The shearing stress for tread rubber

¼ P

ð23-311Þ

where P ¼ normal pressure, MPa (psi) The critical shearing stress for tread rubber

crit ¼ crit P

For < crit

The fatigue wear predominates.

For > crit

Either wear through roll formation or abrasive wear occurs.

For < crit

The wear is due to surface fatigue.

For > crit

Other forms of wear predominate.

ð23-312Þ

Speciﬁc wear Speciﬁc wear by mass Speciﬁc wear by volume Speciﬁc wear by volume based on the geometry of the aspirities arising out of the surface treatment

Ksm ¼

m Ad

ð23-313Þ

KsV ¼

V Ad

ð23-314Þ

KsV ¼

tan "hm z ¼ ¼ ð þ 1Þ2i ð þ 1Þid ð þ 1Þid

ð23-315Þ

where values of angle of slope of irregularities, , can be obtained from Table 23-102 and the values of from Table 23-93 z ¼ absolute approach " ¼ z=hm

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DESIGN OF BEARINGS AND TRIBOLOGY

23.166

CHAPTER TWENTY-THREE

Particular

Formula

TABLE 23-102 Radii of curvature asperities for diﬀerent methods of surface preparation Angle of slope of irregularities,

Slope radii, micron Treatment

Accuracy class

Transverse

Longitudinal

Transverse

Longitudinal

Shaping Grinding Honing Finishing (lapping)

5–8 5–9 8–11 10–13

20–120 5–20 4–30 15–250

10–25 250–15000 60–160 7000–35000

5–20 7–35 3–13 5–20

5–10 2–10 1–4 2–10

The absolute approach

z¼

6 c K1 2

ð23-316Þ

where K1 K2 ¼ coefficient of rigidity K¼ K1 þ K2 Ki ¼

Ei 2 i ð1 v2i Þ

2 ¼ diameter of contact spot, cm ¼ tangent to the smoothness of the surface equal to the derivative of approach over the contact area ¼ tan An expression for modiﬁed speciﬁc wear Modiﬁed speciﬁc wear formula during microcutting

0 ¼ KsV KsV

0 ¼ KsV

A P ¼ KsV a P Aa

ð23-317Þ

tan Pa ð þ 1Þ2P

ð23-318Þ

Pa P to 0:04 a for tan ¼ 0:1 to 0:2 P P during microcutting ¼ 0:02

Modiﬁed speciﬁc wear formula during plastic contact

0 KSV

¼

hmax rbð1=Þ

5=2

Pa P

ð5 þ 2Þ=2

c "fail

21=2 =8 ð23-319Þ

where hmax ¼

d tan 2"max

"fail ¼ relative elongation corresponding to failure of the specimen c ¼ constant depending on sliding combination taken from Table 23-94

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

Particular

23.167

Formula

Modiﬁed speciﬁc wear formula during elastic contact

0 KSV

" 2=ð2 þ 1Þ ð1 v2 ÞPa K c E ¼ c1 E c2 o ð1 v2 ÞPa ð23-320aÞ

where

pﬃﬃﬃ 3 c1 ¼ 8 K2 ð þ 1Þ c2 ¼

ð23-320bÞ

=ð2 þ 1Þ 1=ð2 þ 1Þ b 0:75 2=ð2 þ 1Þ hmax 2 K2 r

ð23-320cÞ

GENERAL For values of wear rate correction factors; physical and mechanical properties of clutch facings; mechanical properties, performance and allowable operating conditions for various materials; physical and mechanical properties of materials for sliding faces; rubbing bearing materials and applications and allowable working conditions and frictions for various clutch facing materials

Refer to Tables 23-103 to 23-108.

TABLE 23-103 Approximate values of wear rate correction factors Name of factor

Condition

Constant

a. Geometrical factor

Continuous motion þ rotating load Unidirectional load Oscillating motion Metal housing, thin shell, intermittent operation Metal housing, continuous operation Nonmetallic housing, continuous operation PTFE base: 208C 1008C 2008C 208C Carbon graphite thermoset 1008C 2008C Stainless steels, chrome plate Steels Soft, nonferrous metals 0.1–0.2 mm 0.2–0.4 mm 0.4–0.8 mm

0.5 1 2 0.5 1 2 1 2 5 1 3 6 0.5 1 2.5 1 2–5 4–10

b. Heat dissipation factor

c. Temperature factor

d. Counterface factor

c. Surface ﬁnish factor

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DESIGN OF BEARINGS AND TRIBOLOGY

23.168

CHAPTER TWENTY-THREE

TABLE 23-104 Physical and mechanical properties of clutch facings

Thermal conductivity Speciﬁc heat Thermal expansion Speciﬁc gravity Young’s modulus, E Ultimate tensile strength, ut Ultimate shear, stress, u Ultimate compressive strength, uc

Rivet holding capacity

Resin-based material

Sintered metals

0.80 W/m 8C 1.25 kJ/kg 8C 0:50 104 =8C 1.6 for woven 2.8 for molded 352 kgf/mm2 3:45 109 N/m2 3.45 GPa 2.14 kgf/mm2 21 106 N/m2 21 MPa 1.22 kgf/mm2 12 106 N/m2 12 MPa 10.5 kgf/mm2 103 106 N/m2 103 MPa 7.03 kgf/mm2 69 106 N/m2 69 MPa

16 W/m 8C 0.42 kJ/kg 8C 0:13 104 =8C 1488 kgf/mm2 14:5 109 N/m2 14.5 GPa 4.57 kgf/mm2 44:8 106 N/m2 44.8 MPa 3.59 kgf/mm2 35:2 106 N/m2 35.2 MPa 15.6 kgf/mm2 153 106 N/m2 153 MPa

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MPa

106 N=m2

kgf/mm2

MPa

106 N=m2

kgf/mm2

MPa

kgf/mm2

Coeﬃcient of friction,

Speciﬁc gravity

0.32

0.30 4.91 48.2 48.2 7.02 68.9 68.9 10.50 151.6 151.6

6.0

0.92

9.0

9.0 10..54 103.4 103.4

96.5 15.50 152

0.32

96.5

41.3 10.10 103

2.0

9.85

41.3

0.35 1.39 13.8 13.8 1.39 13.8 13.8 10.54 103.4 103.4 17.50 172

8.2

4.21

2.0

8.2

8.2

0.35 1.05 10.3 10.3 0.84

8.2

1.7

8.2 0.84

69 83

0.40 0.84

8.2

kgf/mm2 7.03 8.45

106 N=m2

1.7

1.0 0.50 2.1 20.7 20.7 1.26 12.4 12.4 9.85 96.5 96.5 1.5–2.0 0.4 2.45 24.1 24.1 1.39 13.8 13.8 10.54 103.4 103.4

106 N=m2

Key: 1 psi ¼ 6895 Pa; 1 kpsi ¼ 6:894757 MPa.

Cement

Lining Woven cotton Woven asbestos Molded Light-duty (ﬂexible) Medium (semi-ﬂexible) Heavy-duty Pad Resin-based or asbestos Sintered metals

Materials

MPa 172

152

103

69 83

350 400 500 650

6:1 106 3:1 106 1:8 106 1:2 106

Used at 650 higher temperature 800

150 250

Wear rate at 1008C, mm3 /J 12:2 106 9:2 106

Maximum

Temperature 8C

400

300

300

225

200

175

100 125

Maximum operating

Rivet holding capacity kgf/mm2 103 35.5–107.0

35.5–356.5

35.5–178.5

7.2–71.5

7.2–71.5

7.2–71.5

7.2–71.5 7.2–71.5

Working pressure, Pw

70–700

70–700

70–700

70–700 70–700

103 MPa

Maximum pressure, Pmax

0.390 3.8 3.8

0.296 2.8 2.8

0.214 2.1 2.1

0.152 1.5 1.5 0.214 2.1 2.1

350–1050 350–1050 0.703 6.9 6.9

350–3500 350–3500 0.561 5.5 5.5

350–1750 350–1750 0.561 5.5 5.5

70–700

70–700

70–700

70–700 70–700

103 N/m2

Compressive stress, c

kgf/mm2

Shear stress,

106 N=m2

Tensile stress, t

MPa

TABLE 23-105 Mechanical properties, performance, and allowable operating conditions for various materials

DESIGN OF BEARINGS AND TRIBOLOGY

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23.169

Synthetic carbon 1 Synthetic carbon 2 Carbon 2 Carbon 1 Carbon 3 Carbon 4 Carbon 5 Carbon 6 Graphite 1 Graphite 2 Graphite 3 Graphite 4 Graphite 5 Graphite 6 Hard alloy 1 Hard alloy 2 Hard alloy 3

Hard rubber Synthetic resin 2 PTFE

Phenol resin Synthetic resin 1 Resinimpregnated fabric Acetal resin Bakelite

Nylon

PTFCE

Materials

156.9

164.8

176.6 264.9 245.3 329.6 343.4 230.5 122.6 98.1 122.6 69.7 54.9 137.3 1470 2746.8 1324.4

16.5

18 27 25 33.5 35 23.5 12.5 10 12.5 7.1 5.5 14 150 280 135

98.1– 147

16

10– 15

98.1– 240

10– 24.5

kgf/mm2

215.8 549.4 49.1– 88.3 68.7 207 98.1– 171.7 98.1– 216

MN/m2

20– 56 5– 9 7– 21 10– 17.5 10– 24

MPa

176.6 264.9 245.3 329.6 343.4 230.5 122.6 98.1 122.6 69.7 54.9 137.3 1470 2746.8 1324.4

164.8

156.9

98.1– 147

98.1– 240

215.8 549.4 49.1– 88.3 68.7 207 98.1– 171.7 98.1– 216

2.8 1.83 3.1 3.6 2.1 5.3 1.6 1.55 1.9 1.45 1.4 2.0 3.8 30 53.0

2.3

2.1

27.5 17.2 30.4 35.3 20.6 52.0 15.7 14.7 18.6 14.2 13.7 19.6 372.8 294.3 519.9

22.6

20.1

27.5 49.1 9.81 27.5 14.7– 39.2 40.2

68.7

2.8– 5.0 1.0– 2.8 1.5– 4.0 4.1

7

kgf/mm2 31.4– 39.2 48.1– 73.6 49.154.9 34.3– 48.1 22.6– 61.8

MN/m2

3.2– 4.0– 4.9– 7.5 5.0– 5.6 3.5– 4.9 2.3– 6.3

MPa 27.5 17-2 30.4 35.3 20.6 52.0 15.7 14.7 18.6 14.2 13.7 19.6 372.8 294.3 519.9

22.6

20.1

27.5 49.1 9.81– 27.5 14.7– 39.2 40.2

68.7

31.4– 39.2 48.1– 73.6 49.1– 54.9 34.3– 48.1 22.6– 61.8

kgf/mm2 103 1.4 1.85 1.46 1.60 1.35 2.60 0.7 1.0 1.15 1.30 0.56 1.0 2.4 24.4 23

1.32

0.70– 1.7 0.35 0.10 1.75

0.70– 1.75 0.1

0.34

0.18– 0.28 0.52– 0.70 2.1– 3.5 0.63– 0.90

0.16

GN/m2 1.37 18.15 14.32 15.7 13.24 25.5 6.87 9.81 11.28 12.75 5.49 9.81 235.4 239.4 225.6

12.95

6.87– 17.58 0.343– 0.98 18.06

6.87– 17.16 1.03

3.28

1.77– 2.75 5.1– 6.87 20.7– 34.3 6.2– 8.9

1.55

GPa (0.4)

0.25

0.35

(0.30)

(0.25)

0.25

(0.3)

(0.3)

Poisson’s ratio,

1.37 18.15 14.32 15.7 13.24 15.5 6.87 9.81 11.28 12.75 5.49 9.81 235.4 239.4 225.6

12.95 0.22 0.18 0.2 (0.2) (0.2) 0.2 0.22 0.2 0.18 0.22 0.22 0.22 0.3 (0.3) 0.3

(025)

6.87– (0.25) [17.58 0.343– (0.5) 0.98 18.06 0.2

6.87– 17.16 1.03

3.28

1.77– 2.75 5.1– 6.87 20.7– 34.3 6.2– 8.9

1.55

2.8 1.8 1.82 1.79 2.5 2.4 1.73 1.65 1.85 1.85 1.83 1.66 1.8 8.78 7.77 8.65

85 100 80 75 85 93 65 65 72 60 50 70 58–62 60y 48–50y

1-52– 2.0 1.3– 1.82 1.6– 1.9 2.1– 2.3 2.0

1.425

65

65

55–63

2.1

Hardness, Hh 1.09– 1.14 1.25– 1.3 1.75– 1.25 1.36– 1.43

g/cm3

80

kg/m3 1800 1820 1790 2500 2400 1730 1650 1850 1850 1830 1660 1800 8780 7770 8650

2800

1520 2000 1300– 1820 1600– 1900 2100– 2300 2000

1425

1.09– 1140 1250– 1300 1750– 1250 1360– 1430

2100

Porosity, ", % 0.1 0.4 0.5 2.5 4.0 0.3 14 1.0 0.25 0.3 10.0 7.0

0

0.3

0

0

0

0

8C 180 365 285 280 350 370 540 365 370 180 520 340 1250 1150 1260

170

170

130– 160 280

175– 230 100

100

120– 150 120

135– 150 130

150

453 638 558 553 623 643 813 638 643 453 793 613 1523 1423 1533

443

443

403– 433 553

448– 503 373

373

393– 423 393

408– 423 403

423

K

Density,

5.0 6.1 5.3 6.6 4.82 2.16 4.9 5.25 5.2 3.5 4.5 2.0 11.4 9.9 11.9

20

13.5

70

15–30

54

05–40

81

10–40

19–26

100– 140 25–60

50

312 126 320 (273) (260) 750.0 362 235 260 250 520 780 97 (87) 135

(65)

66

(410)

(75)

180

87

167

(150)

(50)

140

(130)

(320)

Temperature range, T

106 8C1

595 399 593 546 (533) 1023 635 508 533 523 793 1053 370 360 840

338

239

(683)

348

453

360

440

(423)

(323)

(413)

(403)

(593)

K

Tensile strength, t

8C

Expansion coeﬃcient Thermal conductivity

4.0 13 20.0 30.0 34.0 20.0 46.0 90.0 89.0 100.0 60.0 60.0 7.0 9.7 11.0

2.5

2.0

0.4– 1.0 0.2

0.29– 0.58 0.25

0.2

0.12– 0.21 0.1– 0.2 0.36– 0.51 0.14– 0.25

0.052

kcal/m h8C

Maximum temperature, Tmax

4.65 15.12 23.3 34.89 39.54 23.3 53.50 104.67 103.51 116.3 69.78 69.78 8.14 11.28 12.79

2.91

2.33

.465– 1.163 .233

.337– .675 .291

0.233

.140– .244 .116– .233 .419– .593 .163– .291

0.096

W/m K

Modulus of elasticity, E

Thermal stress resistance

1250 1650 6400 (8200) (8800) 15000 16700 21200 23000 25000 26000 47000 680 (850) 1480

(164)

132

(82)

(53)

45.0

38.0

33.5

(30.0)

22.0

(21.5)

(21.5)

(16.6)

kcal/m h

Compressive strength, c

1453.75 1918.95 7443.20 (9536.60) (10234.4) 17445.0 19422.10 24655.60 26749 29075 30238 54461 790.84 (988.55) 1720.24

190.73

153.46

(95.37)

(61.64)

52.34

44.19

39.96

(34.89)

25.59

(25.00)

(25.00)

19.31

W/m

TABLE 23-106 Physical and mechanical properties of materials for sliding face

DESIGN OF BEARINGS AND TRIBOLOGY

23.170

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kgf/mm2

3434

618

1470 687

1648 2746 2070 755 1648 284 1030 2845 3433 4120 4905 3630

3434 2943 3591 2453

63

150 70

168 280 210 77 168 29 105 290 350 420 500 370

350 (300) 366 250

687– 824 206c 279c 981 833 697

MN/m2

350

70 84 21c 28.5 100 85 70

MPa

3434 2943 3591 2453

1648 2746 2070 755 1648 284 1030 2845 3433 4120 4905 3630

1470 687

618

3434

687– 824 206c 279c 981 833 687

49 130 70 7 10 8.4 8.4 11 12.6 17.5 24 14.7 21 17.5 12.5 27 140 120 85 115 14 91 105 56 140

1481 1275.3 687 68.7 98.1 82.4 82.4 107.9 122.6 171.7 235.4 144.2 206 171.7 122.6 264.9 1370 1177.2 833.8 1128.2 137 892.7 1030.1 549 1370

441.5 171.5– 206 834 824 510 235.4 196.2

45 17.50 21 85 84 52 24 20

kgf/mm2

274.6– 343.4 529.7

MN/m2

28– 35 54

MPa 1481 1275.3 687 68.7 98.1 82.4 82.4 107.9 122.6 171.7 235.4 144.2 206 171.7 122.6 264.9 1370 1177.2 833.8 1128.2 137 892.7 1030.1 549 1370

441.5 171.7– 206 834 824 510 235.4 196.2

274.6– 343.4 529.7

15 10.5– 11.3 21.4 20 20.3 21 9.11 11 25 20.6 33 10.5 21.4 14.7 12.5 7.3 22.3 39 35 26 26.6 45.5 48 32 49 56 70 54.5 31.4 41.3 28.7 40 30.4

147.2 103– 111 210 196.2 199 206 89– 108 245.2 202 323.7 103 209.4 144.2 122.6 72.1 218.8 382.6 343.4 255.1 261 446.4 470.8 313.9 480.7 549.4 687 534.6 308 405.2 281.5 392.4 298.2

196.2

20

kgf/mm2 103 171.6

GN/m2

17.5

GPa 147.2 103– 111 210 196.2 199 206 89– 108 245.2 202 323.7 103 209.4 144.2 122.6 72.1 218.8 382.6 343.4 255.1 261 446.4 470.8 313.9 480.7 549.4 687 534.6 308 405.2 281.5 392.4 298.2

196.2

171.6 7.98

[8.0 7.3 9.23 8.94 7.53 8.9 7.25 7.19 7.8 10-2 2.7 3.5 9.69 3.7 2.6 3.4 3.7 3.9 5.9 [6.0 2.51 3.1 7.0 13.0. 14.1 14.8 14.0 [4.9 6.0 6.3 5.8 7.0

155–185f 160f 125–173f 215f 225f 300f 125f 150–220f 180f 64–67e 20–26f 7.5h

(0.3) (0.3) 0.28 0.28 0.25 0.3 0.28 0.324 (0.3) 0.36 0.17 0.35 0.15 0.27 0.31 0.2 0.21 0.26 (0.25) (0.25) 0.26 0.26 0.248 0.216 0.242 0.29 0.25 0.26 (0.25) 0.3

0.3 0.25

0.28

87.5f

8h 800h 9h 9h 9h 37f 50f 2800g 2500g 86.5f 83–64f 86–87f 91.5f 89f 2460g 89f 82.5f

7.7

Hardness, Hh 53–57e

Poisson’s ratio, r

g/cm3

(0.26)

kg/m3 7250 7190 7800 10200 2700 3500 9690 3700 2600 3400 3700 3900 5900 6000 2510 3100 7000 13000 14100 14800 14000 4900 6000 6300 5800 7000

9230 8940 7530 8900

8000 7300

7980

7700

Porosity, ", % 0.1 0.1 0.3 0.1

0.02 0.5 0 0 0

0.02

0

0

0

1400b 1800b 600 550 1000 2800b 3300b 1000 1723b 1400 1550 1750 1800 1700 2500b 2400 1900b 600 600 600 600 3140b 1000 1000 1200b 650

1335b 1285b 1500b 1495b

1425b 1200b

1400b

800

8C 1673b 2073b 873 823 1273 2073b 3573b 1373 1996b 1673 1823 2023 2073 1973 2773b 2673 2173b 873 873 873 873 3413b 1273 1273 1473b 923

1608b 1558b 1773b 1768b

1698b 1473b

1673b

1073

10.0 6.2 14.8 4.8 8.2 13.5 9.2 4.0 0.5 5.5 5.8 6.0 8.0 7.5 4.5 3.9 9.0 9.0 6.8 5.6 7.0 7.4 9.5 10.4 5.7 8.7

10.0 11.3 10.6 12.3

0.9 17.0

16.0

8.5

150 220 305 325 (57) 22 52 108 2550 74 54 92 56 78 (64) (50) 70 240 230 170 225 43 175 260 (185) (370)

(280) (260) 173 67

230 78

121

(157)

Temperature range, T

423 493 578 598 (330) 295 325 381 2823 347 327 365 329 351 (337) (323) 343 513 503 443 498 316 448 543 (462) (643)

553 533 450 340

403 351

394

430

Thermal conductivity

40.0 57.6 45.0 110 2.15 31.0 9.0 4.3 1.37 11.4 16.2 25.0 25.0 29.0 22.3 86.0 (20.0) (30.0) (50.0) (60.0) (40.0) 21.5 26.0 28.0 29.0 45.0

9.7 10.8 19.0 59.5

9.5 34

16.0

12.2

46.5 66.99 52.34 127.90 25.0 36.05 4.65 5.00 1.59 13.37 18.84 29.1 29.1 33.73 25.94 100 (23.3) (34.89) 58.15 (62.78) (46.5) 25.0 30.34 32.56 33.73 52.34

11.28 12.56 22.10 69.20

11.05 39.54

18.61

14.19

Thermal stress resistance

6000 12700 13800 3560 (120) 680 470 465 3500 840 875 2300 1400 2250 1430 (4300) (1400) (7200) (11500) (10000) (9000) 930 4550 7300 (5400) (16700)

(2700) (2800) 3300 4000

2200 2650

1940

1930

6978.0 14770 16049 1440.2 140 790.8 546.6 540.7 4070.5 976.9 1017.6 2674.4 1628.2 2616.7 1663.1 5000.9 (1628.2) 8373.6 13374.5 11630 10467.2 1081.5 5291.6 8489.9 6270.2 (19422.1)

1340.1 (3256.4) 3837.9 (4652.0)

2558.6 3081.9

2253.2

2244.5

Key: a Brinell hardness; b melting point; c Shore hardness; d scleroscope; e Rockwell hardness; ðf Rockwell A; g Knoop hardness; h Mohs hardness; i electric limit; j values in parentheses ( ) are approximate. Preﬁxes: k ¼ 103 ; M ¼ 106 ; G ¼ 109 . Conversion: 1 kgf/mm2 ¼ 9:80665 106 N/m2 ; 1 N/m2 ¼ 1 Pa; 1 kcal/h m 8C¼ 1:163 W/m K; 1 kcal h m ¼ 1:163 W/m¼ 1:163 J/h m; 1 psi ¼ 6894:757 Pa; 1 kpsi ¼ 6:894757 MPa; 1 Mpsi ¼ 6:894757 GPa; 1 Btu=lft2 h 8F ¼ 5:678 W=m2 8C; 1 W=m2 8C ¼ 0:1761 Btu=ft2 h 8F; 1 g=cm3 ¼ 3:6127 102 lb=in3 ¼ 62:428 lb=ft3 . PTFCE ¼ polytetraﬂuorochloroethylene; PTFE ¼ polytetraﬂuoroethylene.

Chrome Steel Molybdenum Steatite Magnesium Thoria Zircon Quartz glass Alumina 1 Alumina 2 Alumina 3 Cement 1 Cement 2 Boron carbide Silicon carbide Chromium carbide Tungsten carbide 1 Tungsten carbide 2 Tungsten carbide 3 Tungsten carbide 4 Titanium carbide 1 Titanium carbide 2 Titanium carbide 3 Titanium carbide 4 Titanium carbide 5

Hastelloy B Hastelloy C Chrome (cast) Cobalt Cast iron

Hardened nickel Stainless steel Steel AISI 316 Invar Niresist (cast)

Materials

K

Density,

106 8C1

K

Tensile strength, t

8C

Expansion coeﬃcient

kcal/m h8C

Maximum temperature, Tmax

W/m K

Modulus of elasticity, E

kcal/m h

Compressive strength, c

W/m

TABLE 23-106 Physical and mechanical properties of materials for sliding face (Cont.)

DESIGN OF BEARINGS AND TRIBOLOGY

23.171

14.28

0.71

14.28

42.84

Thermoplastic with ﬁller bonded to metal back

Filled PTFE

PTFE with ﬁller bonded to steel backing

Woven PTFE reinforced and banded metal backing

3.57

1.03

0.20

Graphite-thermosetting resin Reinforced thermosetting plastic

Thermoplastic with ﬁller or metalbacked

7.14

Graphite-impregnated metal

1.02

35.0

0.31–0.41

Carbon/graphite with metal

Thermoplastic material without ﬁller

2.0

0.14–0.20

Carbon/graphite

420.0

140.0

7.0

140.0

10.14

10.0

70.0

3.0–4.0

1.4–2.0

kgf/mm2

Materials

N/m 106

2

420.0

140.0

7.0

140.0

10.14

10.0

35.0

2.0

70.0

3.0–4.0

1.4–2.0

MPa

Maximum loading, P

TABLE 23-107 Rubbing, bearing materials and applications

0.35

0:35 1:75

1:60

0:0357 0:1785

0:1623

0.035–0.11

0.035

0.35

0.35

1:60

1:75

0:35

0.35

0.035–0.11

0.035

0.35

0.35

0.28–0.35

0.11 0.18 0.145 0.22

0.11 0.18 0.145 0.22 0.28–0.35

MPa m/s

MN/m2 m/s

Pv value

0.0357

0.0036– 0.0112

0.0036

0.0357

0.0357

0.0286– 0.0357

0.0112– 0.0184 0.0148– 0.0224

kgf/mm m/s

2

0.03–0.33, dry

0.05–0.30, dry

0.05–0.35, dry

0.20–0.35, dry

0.15–0.40, dry

0.1–0.45, dry

0.1–0.4, dry, 0.006, waterlubricated

250

280

250

105

100

100

200

250

350–600

130–350

0.10–0.35, dry

0.10–0.15, dry 0.020–0.025, greaselubricated 0.13–0.5, dry

350–500

Maximum temperature, 8C

0.10–0.25, dry

Coeﬃcient of friction,

20 (lining)

60–80

27

80–100

25–80 depending on plane of reinforcement 100

3.5–5.0

12–13 with iron matrix

4.2–5.0

2.5–500

Coeﬃcient of expansion, 106 , 8C

Water-lubricated roll neck bearings in hot rolling mills, rubber bearings, bearings subjected to atomic radiation Textile and food machinery bearings, bushes and thrust washers in automobile, bearing of linkages For more heavily loaded applications, textile and food machinery, automobile and linkage Ball joints, gearbox bushes, kingpin bushes, suspension and steering linkages Bushes, thrust washers, sideways Aircraft controls, linkages, automobile gearboxes, conveyors, bridges and building, expansion bearings, bushes, and steering suspension Aircraft engine controls, automobile suspension, engine mountings, linkages, bridges, and building expansion joint

Conveyors, furnaces, food and textile machinery Bearings immersed in water, acid or alkaline solution, etc. Bearings of foundry plant, coal mining machines, steel plants, etc.

Application

DESIGN OF BEARINGS AND TRIBOLOGY

23.172

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DESIGN OF BEARINGS AND TRIBOLOGY DESIGN OF BEARINGS AND TRIBOLOGY

23.173

TABLE 23-108 Allowable working conditions and friction for various clutch facing materials Temperature Working pressure Working conditions Light-duty Woven Mill board Medium-duty Wound tape yarn Asbestos tape Molded Heavy-duty Sintered Cement Oil immersed Paper Woven Molded Molded (grooved) Sintered Sintered (grooved) Resin/graphite

Coeﬃcient of friction,

Maximum 8C

Continuous 8C

kgf/mm2

N/m2 106

MPa

Power rating, W/mm2

0.35–0.4 0.40

250 250

150 150 200

0.18–0.51 0.18–0.71

1.75–5.00 1.75–7.00

1.75–5.0 1.75–5.0

0.3–0.6 0.3–0.6

0.38 0.40 0.35

350 350 350

200 200

0.18–0.71 0.18–0.71 0.18–0.71

1.75–7.0 1.75–7.0 1.75–7.0

1.75–7.0 1.75–7.0 175–7.0

0.3–0.6 0.6–1.2 0.6–1.2

0.36/0.30 0.40

500

300

0.36–0.29 0.71–1.43

3.5–28 7.0–14

3.5–28.0 7.0–14.0

1.7 4.0

0.11 0.08 0.04 0.06

0.71–1.79 0.71–1.79 0.17–1.79

7.1–17.5 7.1–17.5 7.1–17.5

7.1–17.5 7.1–17.5 7.1–17.5

2.3 1.8 0.6

0.11/0.05 0.11/0.06

0.71–4.28 0.71–4.28

7.0–42 7.0–42

7.0–42.0 7.0–42.0

2.3 2.3

0.10

5.3

REFERENCES 1. Lingaiah, K., Machine Design Data Handbook, Vol. II (SI and Customary Metric Units), Suma Publishers, Bangalore, India. 1986. 2. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962. 3. Maleev, V. L., and J. B. Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954. 4. Radzimovsky, F. I., Lubrication of Bearings—Theoretical Principles and Designs, The Ronald Press Company, New York, 1959. 5. Raimondi, A. A., and J. Boyd, ‘A Solution for the Finite Journal Bearings and Its Application to Analysis and Design’, ASME J. Lubrication Technol., Vol. 104, pp. 135–148, April 1982. 6. Kingsbury, A., ‘Optimum Conditions in Journal Bearing,’ Trans. ASME, Vol. 54, 1932. 7. Needs, S. J., ‘Eﬀect of Side Leakage in 120-degree Centrally Supported Journal Bearings,’ Trans. ASME, Vol. 56, 1934; Vol. 51, 1935. 8. Shigley, J. E., Mechanical Engineering Design, First Metric Edition, McGraw-Hill Book Company, New York, 1986. 9. Edwards, K. S., Jr., and R. B. McKee, Fundamentals of Mechanical Component Design, McGraw-Hill Book company, 1991. 10. Shaw, M. C., and F. Macks, Analysis and Lubrication of Bearings, McGraw-Hill Book Company, New York, 1949. 11. Lingaiah, K., Machine Design Data Handbook, McGraw-Hill Publishing Company, New York, U.S.A., 1994. 12. FAG Rolling bearings, Catalog WL 41520EI, 1995 edition, FAG Precision Bearings Ltd., Maneja, Vadodara, India.

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DESIGN OF BEARINGS AND TRIBOLOGY

23.174

13. 14. 15. 16. 17. 18. 19. 20. 21.

CHAPTER TWENTY-THREE

SKF Rolling Bearings, Catalog 4000E. 1989, SKF Rolling Bearings, India Ltd., Mumbai, India. Bureau of Indian Standards, Manak Bhavan, 9 Bahadur Shah Marg, New Delhi 110 002, India. Neale, M. J., Editor, Tribology Handbook, Butterworth, London, 1973. International Organization for Standards, 1, rue de Varembe, Case Postale 56, CH 1211, Geneve 20, Switzerland. New Departure-Hyatt Bearing Division, General Motor Corporation, USA. NSK Corporation (Corporate), Automotive Products Bearing Division, 3861 Research Park Drive, Ann Arbor, Michigan 48100-1507, USA. The Torrington Company, 59 Field Street, Torrington, Conn 06790, USA. Antifriction Bearing Manufacturers Association, USA. Black, P. H., and O. E. Adams, Jr., Machine Design, McGraw-Hill Publishing Company, New York, 1968.

BIBLIOGRAPHY ASME Standards. Baumeister, T., ed., Marks’ Handbook for Mechanical Engineers, McGraw-Hill Book Company, New York, 1978. Black, P. H., and O. E. Adams, Jr., Machine Design, McGraw-Hill Book Company, New York, 1968. Boswall, R. O., The Theory of Film Lubrication, Longmans, Green and Company, New York, 1928. Bureau of Indian Standards. O’Connor, J. J. ed., Standard Handbook of Lubricating Engineering, McGraw-Hill Book Company, New York, 1968. Fuller, D. P., The Theory and Practice of Lubrication for Engineers, John Wiley and Sons, New York, 1956. Niemann, G., Machine Elements—Design and Calculations in Mechanical Engineering, Vol. II, Springer-Verlag, Berlin, 1950; Student Edition, Allied Publishers Private Ltd. Bangalore, India, 1979. Niemann, G., Maschinenelemente, Springer-Verlag, Berlin, Erster Band, 1963. Niemann, G., Maschinenelemente, Springer-Verlag, Berlin, Zweiter Band, 1965. Hyland, P. H., and J. B. Kommers, Machine Design, McGraw-Hill Book Company, New York, 1943. ISO Standards Lansdown, A. R., Lubrication: A Practical Guide to Lubricant Selection, Pergamon Press, New York, 1982. Leutwiler, O. A., Elements of Machine Design, McGraw-Hill Book Company, New York, 1917. Michell, A. G. M., Lubrication—Its Principles and Practice, Blackie and Son, London, 1950. Neale, M. J., ed., Tribology Handbook, Butterworth, London, 1973. Norman, C. A., E. S. Ault, and I. F. Zarobsky, Fundamentals of Machine Design, The Macmillan Company, New York, 1951. Norton, A. E., Lubrication, McGraw-Hill Book Company, New York, 1942. Slaymaker, R. R., Bearing Lubrication Analysis, John Wiley and Sons, New York, 1955. Rippel, H. C., ‘‘Design of Hydrostatic Bearings,’’ Machine Design, Parts 1 to 16, Aug. 1 to Dec. 5, 1963. SAE Handbook, 1957. Shigley, J. E., Machine Design, McGraw-Hill Book Company, New York, 1962. Shigley, J. E., and C. R. Mischke, Standard Handbook of Machine Design, McGraw-Hill Book Company, New York, 1986. Shigley, J. E., and C. R. Mischke, Mechanical Engineering Design, McGraw-Hill Book Company, New York, 1989. Vallance, A., and V. L. Doughtie, Design of Machine Members, McGraw-Hill Book Company, New York, 1951. Wilcock, D. F., and E. R. Booser, Bearing Design and Application, McGraw-Hill Book Company, New York, 1957.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

24 MISCELLANEOUS MACHINE ELEMENTS2,3 24.1 CRANKSHAFTS2,3 SYMBOLS A b c d de do dm E F Fc Fcomb Fic Fr F G h i0 ¼

lo do

ratio of length to diameter of crank

Di Do

ratio of inner to outer diameter of a hollow shaft

I K¼ Kb Kt l le Mb

area of cross section, m2 (in2 ) width of crank cheek, m (in) distance from the neutral axis of section to outer ﬁber, m (in) diameter (also suﬃxes), m (in) equivalent diameter, m (in) diameter of crankpin, m (in) diameter of main bearing, m (in) modulus of elasticity, GPa (psi) force acting on the piston due to steam or gas pressure corrected for inertia eﬀects of the piston and other reciprocating parts, kN (lbf ) the component of force F acting along the axis of connecting rod, kN (lbf ) combined force, kN (lbf ) magnitude of inertia force due to the weight of connecting rod itself, kN (lbf ) total radial force acting on the crankpin, kN (lbf ) total tangential force acting on the crankpin, kN (lbf ) modulus of rigidity, GPa (psi) thickness of cheek or web (also with suﬃxes), m (in) moment of inertia, m4 , cm4 (in4 )

numerical combined shock and fatigue factor to be applied to the computed bending moment numerical combined shock and fatigue factor to be applied to the computed twisting moment length (also with suﬃxes), m (in) equivalent length, m (in) bending moment, N m (lbf in)

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MISCELLANEOUS MACHINE ELEMENTS

24.2

Mt p r Z

CHAPTER TWENTY-FOUR

twisting moment, N m (lbf in) allowable pressure, MPa (psi) radius, throw of crankshaft, m (in) section modulus, m3 , cm3 (in3 ) normal stress (also with suﬃxes), MPa (psi) shear stress, MPa (psi)

SUFFIXES b c comb e m max r ra rh t s

bending compressive combined elastic main maximum radial resultant in arm resultant in hub torque shaking tangential

Other factors in performance or special aspects which are included from time to time in this chapter and are applicable only in their immediate context are not given at this stage.

Particular

Formula

FORCE ANALYSIS (Fig. 24-1) The radial component of force Fc acting along the axis of connecting rod (Fig. 24-1)

The tangential component of force Fc acting along the axis of connecting rod (Fig. 24-1)

The radial component of force Fic (Fig. 24-1)

F Fc1 ¼ Fc cosð þ Þ ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cosð þ Þ sin 2 1 n0 ð24-1Þ F Fc2 ¼ Fc sinð þ Þ ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sinð þ Þ sin 2 1 n0 ð24-2Þ Fic1 ¼ 23 Fic cos

ð24-3Þ

where ¼ angle between the force Fic and the radial component of Fic The tangential component of force Fic (Fig. 24-1)

Fic2 ¼ 23 Fic sin

ð24-4Þ

The total radial force acting on the crank

Fr ¼ Fic1 Fc1

ð24-5Þ

Fr ¼ 23 Fic cos Fc cosð þ Þ

ð24-6Þ

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

Formula

F ¼ Fic2 Fc2

The total tangential force acting on the crank

¼ 23 Fic sin Fc sinð þ Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Fcomb ¼ F2r þ F2 Fcomb

c1

F

θ

α

r

lo

Fs (b) Shaking force diagram Fc φ

Fic1

Fc

F

1 3l

l

d1

h

c2

x

c1

x

-x

l1

Arm

F

l

ð24-8Þ

b1 = 1.5 d0 b2 = 1.35 d l0 = 1.25 d0

do

Fr = Fic1 – Fc1

Fi

r

Fc2 Fic2 Fic γ

ð24-7Þ

Neutral axis of arm

h1

r1

The resultant force on the crankpin

(θ + φ)

24.3

b1 b2

x dm

d2 = 1.75 d d1 = 2 d0 h1 = 1.4 d0 h2 = 1.4 d

d2

(a)

FIGURE 24-1 (a) Forces acting on crankshaft. (b) Vector sum of F and Fr .

lm

h2

FIGURE 24-2 Overhung built-up crank.

SIDE CRANK Crankpin The maximum bending moment on the crankpin (Fig. 24-2)

MbðmaxÞ ¼ Fcomb

lo t þ 2 2

ð24-9Þ

¼ Fcomb l lo þ c2 ¼ distance from centroidal axis 2 to the application of load (Fig. 24-2), m (in) sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 32lFcomb ð24-10Þ do ¼ b where l ¼

The crankpin diameter with respect to the bending moment

where b ¼ allowable bending stress, MPa (psi) The diameter of crankpin from the consideration of bearing pressure From Eqs. (24-10) and (24-11) neglecting t=2 and eliminating lo the equation for crankpin diameter Empirical relation to determine the length of crankpin

Fcomb lo p sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 4 16Fcomb do ¼ pb

ð24-11Þ

lo ¼ i0 d o

ð24-13Þ

do ¼

where i0 ¼

lo ¼ 1:25 to 1.5 do

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ð24-12Þ

MISCELLANEOUS MACHINE ELEMENTS

24.4

CHAPTER TWENTY-FOUR

Particular

Another relation for the crankpin length/diameter ratio Another relation for the crankpin diameter

Formula

l i ¼ o ¼ do 0

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0:2b p

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Fcomb do ¼ i0 p

ð24-14Þ

ð24-15Þ

HOLLOW CRANKPIN The crankpin length/diameter ratio

l i ¼ o ¼ Do 0

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0:2ð1 K4 Þ p

where K ¼ The crankpin outside diameter

ð24-16Þ

Di Do

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Fcomb Do ¼ i0 p

ð24-17Þ

Crank arm CRANK ON HEAD-END DEAD-CENTER POSITION When the crank is on the head-end dead-center position, the section XX (Fig. 24-2) of the arm is subjected to bending moment

Mb ¼ Fcomb l

ð24-18Þ

The direct compressive stress due to the load Fcomb (i.e., more speciﬁcally by its component Fc )

c ¼

Fcomb A

ð24-19Þ

The resultant stress in the crank arm at XX

ra ¼

Fcomb MbC A I where

ð24-20Þ

A ¼ area of cross section of the arm at XX, m2 (in2 ) c ¼ distance from the neutral axis of section to outer ﬁber of arm, m (in) I ¼ moment of inertia of the section, cm4 (in4 ) CRANK ON CRANK-END DEAD-CENTER POSITION The direct tensile stress in the plane of the hub of crankshaft section passing through the shaft center due to load Fcomb (Fig. 24-2)

t ¼

The bending stress in the section due to bending moment Fcomb a

b ¼

Fcomb h2 ðd2 dÞ

Fcomb a Z where Z ¼ section modulus, cm3 (in3 )

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ð24-21Þ ð24-22Þ

MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.5

Formula

r ¼ t b

ð24-23Þ

The bending moment in the plane of rotation of the crank

Mb ¼ Fcomb l

ð24-24Þ

The bending stress

b ¼

Mb c1 Zb

ð24-25Þ

The torsional moment

Mt ¼ Fcomb r1

ð24-26Þ

The shear stress

¼

Mt c 1 Zt qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ i h max ¼ 12 b þ 2b þ 4 2

ð24-27Þ

The resultant stress in the plane of the hub of crankshaft section passing through the shaft center CRANK PERPENDICULAR TO THE CONNECTING ROD

The maximum normal stress for crank made of cast iron

ð24-28Þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2b þ 4 2

ð24-29Þ

The shaking force on the main bearing from F and Fr (Fig. 24-1b)

Fs ¼ vector sum of F and Fr

ð24-30Þ

The diameter of main bearing taking into consideration the bearing pressure on the projected area of the crankshaft

dm ¼

The maximum shear stress for the crank made of steel

max ¼ 12

DIMENSION OF CRANKSHAFT MAIN BEARING (Fig. 24-2b)

Fs lm p

ð24-31Þ

where lm ¼ length of bearing, m (in) p ¼ allowable bearing pressure, MPa (psi)

The bending movement on the crankshaft

Mb ¼ Fcomb l1

ð24-32Þ

lo l þ h2 þ m 2 2 where

l1 ¼

h2 ¼ hub length, m (in) lo ¼ length of crankpin, m (in) lm ¼ length of bearing on crankshaft, m (in) The torque on the crankshaft

Mt ¼ Fcomb r

The diameter of crankshaft taking into consideration indirectly the fatigue and shock factors

where r ¼ throw of the crank, m (in) ﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 16 dm ¼ Kb Mb þ ðKb Mb Þ2 þ ðKt Mt Þ2 e ð24-34Þ

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ð24-33Þ

MISCELLANEOUS MACHINE ELEMENTS

24.6

CHAPTER TWENTY-FOUR

Particular

Formula

The length of main bearing

lm ¼

Fs dm p

ð24-35Þ

PROPORTIONS OF CRANKSHAFTS For proportions of crankshaft

Refer to Figs. 24-2 to 24-10. 0.8 to 1.1do

0.5 to 0.9do

do

d0

1.4d0

Throw

1.1d1

d

1.8d1 d1

K

2 to 2.5d

FIGURE 24-3 Overhung built-up crank.

1.1 to 1.2d

FIGURE 24-4 Overhung forged crank. Fcomb

L 2

lo

Center of bearing

dm

h

lm 2

12mm c

12mm

lm 2

b

FIGURE 24-6 Center crank (American Bureau of Shipping method).

dc

FIGURE 24-5 Disk crank.

h

Dc

h

r=

d1 d2 d

h1

Center of bearing

dm

h2

t D

do

do

L

do

d3

dj

r

I Dj

dj

Ie e 2

FIGURE 24-7 Equivalent length of crankshaft.

h

c

h

e 2

FIGURE 24-8 Center hollow crank.

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b

MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

Formula

B = Cylinder bore

0.71 to 0.80B

0.5d

R = 0.35B

0.7B + 9.5 = d

0.06 to 0.12B

24.7

0.58B

0.53 31B + 9.5 to 0.6B

0.825B + 9 All dimensions in mm

FIGURE 24-9 Empirical proportion for center crank.

2.35d

1.5d

3.25d

0.580d

3.75d

d

d 2.25d

3.75d

2.15d

1.15d

0.58d

Throw

d

1.5d

2.1d

CENTER CRANK (Fig. 24-6) Crankpin The maximum bending moment treating the crankpin as a simple beam with concentrated load at the center

Fcomb ðlo þ h þ lm Þ 4 where lo ¼ length of crankpin, m (in) lm ¼ length of main bearing, m (in) h ¼ thickness of cheek, m (in) Mbc ¼

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ð24-36Þ

MISCELLANEOUS MACHINE ELEMENTS

24.8

CHAPTER TWENTY-FOUR

Particular

The diameter of the crankpin based on maximum bending moment Mbc

Formula

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 32Mbc do ¼ b

ð24-37Þ

where b ¼ design stress, MPa (psi) The diameter of crankpin based on bearing pressure between pin and the bearing

do ¼

Fcomb lo p

ð24-38Þ

Dimensions of main bearing Fcomb le ð24-39Þ 4 where le ¼ equivalent length of crankshaft, m (in)

The maximum bending moment treating the center crank as a simple beam with load concentrated at the center

Mbb ¼

The twisting moment

Mt ¼ Fcomb r

The diameter of crankshaft at main bearing taking into consideration the fatigue and shock factors

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 16 2 2 Kb Mbb þ ðKb Mbb Þ þ ðKt Mt Þ dm ¼ e

ð24-40Þ

ð24-41Þ The diameter of the crankshaft based on bearing pressure

dm ¼

Fs lm p

ð24-42Þ

American Bureau of Shipping formulas for center crank The thickness h of the cheeks or webs (Fig. 24-6) The diameter of crankpins and journals (Fig. 24-6)

h ¼ 0:4d to 0:6d sﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 Dpc d¼a b

ð24-43Þ ð24-44Þ

where a ¼ coeﬃcient from Table 24-1A D ¼ diameter of cylinder bore, m (in) p ¼ maximum gas pressure, MPa (psi) c ¼ distance over the crank web plus 25 mm (1.0 in) (Fig. 24-6) b ¼ allowable ﬁber stress, MPa (psi) The thickness h and the width b of crank cheeks must satisfy the conditions

bh2 0:4d3

ð24-45aÞ

b2 h d3

ð24-45bÞ

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.9

Formula

EQUIVALENT SHAFTS A portion of a shaft length l and diameter d can be replaced by a portion of length le and diameter de The length he equivalent to crank web

le ¼ l

de d

4 ð24-46Þ

rC B where

ð24-47Þ

he ¼

1 d4e G ¼ torsional rigidity of the crankpin C ¼ 32 1 hb3 E ¼ ﬂexural rigidity of the web B ¼ 12

The equivalent length crankshaft le of Fig. 24-7 varies between

0:95l < le < 1:10l

The equivalent length of commercial crankshaft for solid journal and crankpin according to Carter (Fig. 24-8)

Le ¼

d4e

The equivalent length of commercial crankshaft for hollow journal and crankpin according to Carter (Fig. 24-8)

Le ¼ d4e

The equivalent length of crankshaft for solid journal and crankpin according to Wilson (Fig. 24-8)

Le ¼

The equivalent length of crankshaft for hollow journal and crankpin according to Wilson (Fig. 24-8)

d4e Le ¼ d4e

e þ 0:8a 0:75b 1:5r þ þ 3 D4J D4c ac

ð24-48Þ

e þ 0:8a 0:75b 1:5r þ þ D4J d4J D4c d4c ac3

ð24-49Þ ð24-50Þ

e þ 0:4DJ b þ 0:4Dc r 0:2ðDJ þ Dc Þ þ þ D4J D4c ac3

ð24-51Þ e þ 0:4DJ b þ 0:4Dc r 0:2ðDJ þ Dc Þ þ þ D4J d4J D4c d4c ac3 ð24-52Þ

EMPIRICAL PROPORTIONS For empirical proportions of side crank, built-up crank, and hollow crankshafts

Refer to Figs. 24-2 to 24-10.

The ﬁlm thickness in bearing should not be less than the values given here for satisfactory operating condition: Main bearings

h ¼ 0:0025 mm (0.0001 in) to 0.0042 mm (0.0017 in)

ð24-52aÞ

Big-end bearings

h ¼ 0:002 mm (0.00008 in) to 0.004 mm (0.00015 in)

ð24-52bÞ

The oil ﬂow rate through conventional central circumferential grooved bearings

Q¼

kpc3 d ð1 þ 1:5"2 Þ L

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ð24-52cÞ

MISCELLANEOUS MACHINE ELEMENTS

24.10

CHAPTER TWENTY-FOUR

Particular

Formula

where Q ¼ oil ﬂow rate, m3 /s (gal/min) k ¼ a constant ¼ 0:0327 SI units ¼ 4:86 104 US Customary System Units p ¼ oil feed pressure, Pa (psi) c ¼ D d ¼ diametral clearance, m (in) ¼ absolute viscosity (dynamic viscosity), Pa s (cP) d ¼ bearing bore, m (in) L ¼ land width, m (in) " ¼ attitude or eccentricity ratio For oil ﬂow rate in medium and large diesel engines at 0.35 MPa 0.5 psi

Refer to Table 24-1B.

The velocity of oil in ducts on the delivery side of the pump

v ¼ 1:8 to 3.0 m/s (6 to 10 ft/s)

ð24-52dÞ

The velocity of oil in ducts on the suction side of the pump

v ¼ 1:2 m=sð4 ft=sÞ

ð24-52eÞ

The delivery pressure in modern high-duty engines

p ¼ 0:28 to 0.42 MPa (40--60 psi)

ð24-52f Þ

pmax ¼ 0:56 MPa ð80 psiÞ

ð24-52gÞ

For housing tolerances

Refer to Table 24-1C.

TABLE 24-1A Coeﬃcient a in the American Bureau of Shipping formula [Eq. (24-44)] Ratio of stroke to distance over crank webs ¼ l=c

Number of cylinder Type

Four-stroke

Two-stroke

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

Explosion engines

1, 2, 4 3, 5, 6 8 10, 11, 12 1, 2, 4 3, 5, 6 8 12 16

1, 2 3 8 5, 6 1, 2

1.17 1.17 1.17 1.18 1.17 1.19 1.20 1.22 1.25

1.17 1.17 1.19 1.20 1.19 1.22 1.24 1.25 1.29

1.17 1.17 1.21 1.23 1.22 1.25 1.27 1.29 1.33

1.17 1.17 1.23 1.25 1.25 1.28 1.30 1.32 1.36

1.17 1.19 1.25 1.28 1.28 1.32 1.33 1.36 1.40

1.17 1.20 1.28 1.31 1.31 1.35 1.37 1.39 1.44

1.17 1.22 1.30 1.33 1.34 1.38 1.40 1.42 1.47

1.17 1.24 1.32 1.35 1.36 1.41 1.43 1.45 1.50

Air-injection diesel engines

3 4 5, 6 8

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

TABLE 24-1B Oil ﬂow rate in medium and large diesel engines at 0.35 MPa

24.11

TABLE 24-1C Housing tolerances Parts

Tolerances

Waviness of the surface Run-out of thrust faces Surface ﬁnish Journals

>0.0001d j >0.0003d j

Oil ﬂow rate Diﬀerent parts of engine

liters/min/kW liters/min/hp (gal/h/hp)

Bed plate gallery to mains with piston cooling Mains to big end (with piston cooling) Big ends to pistons (with oil cooling) Total ﬂow of oil with uncooled pistons

0.536

0.4 (5)

0.362

0.27 (3.5)

0.201

0.15 (2)

0.335

0.25 (3)

Gudgeon pins Housing bores Alignment of adjacent housing The ﬁne grinding or honing

0.2–0.25 mm Ra (8–10 in clearance) 0.1–0.16 mm Ra (4–6 in clearance) 0.75–1.6 mm Ra (30–60 in clearance) <1 in 10,000 to 1 in 12,000 0.025–0.05 mm (0.001–0.002 in)

TABLE 24-1D Properties of some steel-backed crankshaft plain bearing materials

Nominal composition, per cent

Lining or overlay thickness, mm

Relative fatigue strength

Guidance peak loading limits, rs1 , MPa

Recommended journal hardness, V.P.N.

Tin-based white metal

Sn 87, Sb 9 Cu 4, Pb 0.35 max

Over 0.1 Up to 0.1

1.0 1.3

12–14 14–17

160 160

Tin-based white metal with cadmium

Sn 89, Sb 7.5 Cu 3, Cd 1

No overlay

1.1

12–15

160

Copper-lead, overlaid with cast white metal

Cu 70, Pb 30

0.2

1.4

15–17

160

Sintered copper-lead, overlay plated with lead-tin

Cu 70 Bb 30

0.05 0.025

1.8 2.4

21–23a 28–31a

230 280

Cast copper-lead, overlay plated with lead-tin or lead-indium

Cu 76, Pb 24

0.025

2.4

31a

Sintered lead-bronze, overlay plated with lead-tin

Cu 74, Pb 22 Sn 4

0.025

2.4

28–31a

400

Tin-aluminum

Al 60, Sn 40

No overlay

1.8

21–23

230

Lining materials

Tin-aluminum

Al 80, Sn 20

No overlay

3.3

28–35

230

Tin-aluminum, overlay plated with lead-tin

Al 92, Sn 6 Cu 1, Ni 1

0.025

2.4

28–31a

400

a

Limit set by overlay fatigue in the case of medium/large diesel engines. Suggested limits are for big-end applications in medium/large diesel engines and are not to be applied to compressors. Maximum design loadings for main bearings will generally be 20% lower. (Courtesy: Extracted from M. J. Neale, ed., Tribology Handbook, Section A11, Newnes-Butterworth, London, 1973)

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MISCELLANEOUS MACHINE ELEMENTS

24.12

CHAPTER TWENTY-FOUR

24.2 CURVED BEAM2,3 SYMBOLS semimajor axis of ellipse, m area of cross section, m2 width of beam, m semiminor axis of ellipse, m distance from the centroidal axis to the inner surface of curved beam, m distance from the centroidal axis to the outer surface of curved beam, m distance from the neutral axis to inner surface of curved beam, m distance from the neutral axis to outer surface of curved beam, m diameter of curved beam of circular cross section, m distance from centroidal axis to neutral axis of the section, m modulus of elasticity, GPa load, kN modulus of rigidity, GPa depth of beam, m moment of inertia, m4 , cm4 stress factor (also with suﬃxes) constant length of straight section between the semicircular ends of chain link, m pure number to be determined for each particular shape of the cross section by performing the integration applied bending moment (also with suﬃxes), N m radius of centroidal axis, m inner radius of curved beam, radius of curvature, m outer radius of curved beam, m radius of neutral axis, m deﬂection, m normal stress (also with suﬃxes), MPa

a A b c1 c2 ci ¼ c1 e co ¼ c2 þ e Hð¼ dÞ e E F G h I k K l m Mb rc ri ro rn y

Please note: The US Customary System units can be used in place of the above SI Units.

SUFFIXES b i h o n max r v x y

bending inner horizontal outer neutral maximum resultant or combined vertical x direction y direction

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.13

Formula

GENERAL Pure bending The general equation for the bending stress in a ﬁber at a distance y from the neutral axis (Figs. 15-11 and 15-12)

b ¼

The maximum compressive stress due to bending at the outer ﬁber (Fig. 24-12)

bo ¼

The maximum tensile stress due to bending at the inner ﬁber (Fig. 24-12)

bi ¼

Mb Ae

y rn þ y

ð24-53Þ

Mb co Aero

ð24-54Þ

Mb ci Aeri

ð24-55Þ

Stress due to direct load ¼

F A

ð24-56Þ

co

axis

b

ral

dA

F

b

d

g

F

ro ri rn rc

F

e

Mb

d

t Neu

Mb ac

troi

dθ

e

Cen

c

cl

F

axis

f

y

o

dy

θ

M

The direct stress due to load F

Mb

rc

rn

b d

O1

FIGURE 24-11 Bending stress in curved beam.

O2 FIGURE 24-12 Analysis of stresses in curved beam.

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MISCELLANEOUS MACHINE ELEMENTS

24.14

CHAPTER TWENTY-FOUR

Particular

Formula

Combined stress due to load F and bending The general expression for combined stress

F Mb A Ae

r ¼

y rn þ y

ð24-57Þ

The combined stress in the outer ﬁber

ro ¼

F Mb co A Aero

ð24-58Þ

The combined stress in the inner ﬁber

ri ¼

F Mb ci þ A Aeri

ð24-59Þ

For values of radius to neutral axis for curved beams

Refer to Table 24-2.

TABLE 24-2 Values of radius to neutral axis for curved beams Radius of neutral surface, rn

Section

ri

ro

b

e

h c

ri

ro

rc rn bi

bo

e

h

ro

b2

bi

bo

e

ao

ri rc rn

d

a2

a1 ro

rn ¼

CL of curvature

rc rn

b

rn ¼

ri

CL of curvature

e

CL of curvature

rc rn

a

CL of curvature

Type

rn ¼

pﬃﬃﬃﬃ pﬃﬃﬃﬃ ð ro þ ri Þ2 4

(24-60)

h r ln o ri

(24-61)

1 þb Þ 2 hðb i o

bi ro bo ri r ln o h ri

(24-62)

ðbi bo Þ

If bo ¼ 0, this section reduces to a triangle rn ¼

A r þ ai r ao ro þ b2 ln o þ bo ln bi ln i ri ri þ ai ro ao

If ao ¼ 0, this section reduces to a ? section; rn is the same for a box section in dotted lines with each side panel 12 b2 thick

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(24-63)

MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

Formula

APPROXIMATE EMPIRICAL EQUATION FOR CURVED BEAMS An approximate empirical equation for the maximum stress in the inner ﬁber

24.15

i ¼ Mb

c1 K þ I bc1

1 1 þ ri ro

ð24-64Þ

where K ¼ 1.05 for circular and elliptical sections ¼ 0:5 for all other sections b ¼ maximum width of the section, m (in) Mb in N m (lbf ft) The stress at inner radius for a curved beam of rectangular cross section The stress at inner radius of circular cross section

The stress at inner radius of elliptical sections according to Bacha

6Mb h 1 þ 0:25 ri bh2 32Mb d 1 þ 0:3 i ¼ ri d3

i ¼

i ¼

32Mb a 1 þ 0:3 ri a2 b

ð24-65Þ ð24-66Þ ð24-67Þ

STRESSES IN RINGS (Fig. 24-13a) Fr ¼ 0:318Fr ð24-68Þ where ve sign refers to tensile load, þ ve sign refers to comprehensive load

Maximum moment for a circular ring at the point of application of the load, A, Fig. 24-13a

MbðmaxÞ ¼

Another maximum momentb for a circular ring at a point B 908 away from the point of application of load

MbðmaxÞ ¼ 0:182Fr

Direct stress for the ring at point B 908 away from the point of application of load

¼

The general expression for bending moment at any cross section DD at an angle with the horizontal (Fig. 24-13b) The stress due to direct load F at any cross section DD at an angle with the horizontal

a b

ð24-69Þ

where ve sign refers to comprehensive load, þ ve sign refers to tensile load F 2A

Mb ¼ MA 12 Frð1 cos Þ

¼

F sin 2A

Courtesy: Bach, Maschinenelemente, 12th ed., p. 43. Moments which tend to decrease the initial curve of the bar are taken as positive.

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ð24-70Þ ð24-71Þ

ð24-72Þ

MISCELLANEOUS MACHINE ELEMENTS

24.16

CHAPTER TWENTY-FOUR

Particular

Formula

F MA A

F A

A A B

B

r

o

r

A

D

A B

D θ B

x

r x (b)

F 2

B MB

B A

F (a) FIGURE 24-13 Bending moments in a ring.

The combined stress at any cross section

MB B

B MB

MA A F FIGURE 24-14 Bending moments in a link.

r ¼

1F M sin b Ae 2A

y rn þ y

ð24-73Þ

DEFLECTION The increase in the vertical diameter of the ring (Fig. 24-13a)

yv ¼ 0:149

Fr3 EIz

ð24-74Þ

The decrease in the horizontal diameter of the ring (Fig. 24-13a)

yh ¼ 0:137

Fr3 EIz

ð24-75Þ

LINK (Fig. 24-14) The moment, MbA , at the point of application of load (Fig. 24-14)

MbA ¼

Frð2r þ lÞ 2ðr þ lÞ

ð24-76Þ

The moment, MbB , at the section 908 away from the point of application of load (Fig. 24-14)

MbB ¼

Frð2r rÞ 2ðr þ lÞ

ð24-77Þ

where l ¼ length of straight section between the semicircular ends.

CRANE HOOK OF CIRCULAR SECTION (Fig. 24-15) The combined stress in any ﬁber of a crane hook subject to a load F The maximum combined stress

F Mb y Fy ¼ A Ae rn þ y Amðrx yÞ F H F rðmaxÞ ¼ ¼ ki A 2mri A r ¼

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ð24-78Þ ð24-79Þ

MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.17

Formula

The minimum combined stress

rðminÞ ¼

F A

H 2mro

¼

F k A o

ð24-80Þ

where ki and ko are stress factors which depend on H=2rc ; ki is the critical one which varies from 13.5 to 15.4 as ratio H=2rc changes from 0.6 to 0.4 For crane hook of trapezoidal section

Refer to Fig. 24-16 and Table 24-3.

Gi

Gi

UNDERCUT

Undercut

G K

B A

U

D

N J

E

D

M B

E

Q

P

F

H

M

N

U K

H

Z F

C

H

I

G

Z

P

C

A

Z D

L

E

F Z

R

H

FIGURE 24-15 Hook of circular section.

M Z

J

F

E H

Z M

FIGURE 24-16 Crane hook of standard trapezoidal section.

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20

25

32

40

50

60

70

85

10.0

12.0

16.0

20.0

25

32

40

50

MS HS MS HS MS HS MS HS MS HS MS HS MS HS MS HS MS HS MS HS MS HS MS HS MS HS MS HS

74 63 105 91 145 126 187 162 233 201 294 256 327 286 369 319 415 360 465 404 520 451 569 481 616 533 680 588

35 38 50 43 69 60 89 71 111 96 140 122 156 136 176 152 198 171 221 193 248 215 271 229 293 254 324 280

27 23 38 33 53 46 68 59 85 73 107 93 119 104 134 116 151 131 169 147 189 164 207 175 224 194 247 214

39 33 55 48 76 66 98 85 122 105 154 134 171 150 193 167 217 189 243 211 272 236 298 252 323 279 356 308

34 29 48 41 66 58 85 74 106 91 134 116 149 130 168 145 189 164 211 184 236 205 259 219 280 242 309 268

27 23 38 33 53 46 68 59 85 73 109 93 119 104 134 116 151 131 169 147 189 164 207 175 224 194 247 214

15 12 20 20 30 25 35 30 45 40 55 50 60 55 70 60 80 70 85 80 100 85 110 100 120 110 130 120

M M M M M M M M M M M M M M M M M M M M M M M M M M M M

14 12 20 18 27 24 33 30 42 30 52 48 60 52 68 60 76 68 80 72 90 80 100 90 110 100 120 110 6 6 6 6 6 6 8 6 6 6 6 6

Course series with graded pitches

c

b

Thread

Ball bearings (per IS 2512, 1963)

25 21 35 31 49 43 63 55 79 68 100 86 111 97 125 108 140 122 157 137 176 153 193 163 208 180 229 199

20 17 28 25 40 34 51 44 64 55 80 70 89 78 100 87 113 98 127 110 142 123 155 131 168 146 187 160

25 21 35 31 49 42 63 54 78 67 98 86 109 96 123 107 139 120 155 135 174 151 191 161 206 178 227 197

MS HS MS HS MS HS MS HS MS HS MS HS MS HS MS HS MS HS MS HS MS HS MS HS MS HS MS HS

19 16 27 23 37 32 48 41 60 51 75 63 83 73 94 81 106 92 118 103 132 115 145 122 157 136 173 150

16 14 23 20 32 28 41 35 51 44 64 56 71 62 80 70 91 79 101 88 113 98 124 105 134 116 148 128

32 28 46 40 64 55 82 71 102 88 128 112 143 125 161 139 181 157 203 176 227 197 248 210 268 233 296 257

14 12 19 16 26 23 34 30 42 36 54 46 60 52 67 58 76 66 84 74 94 82 104 88 112 97 124 107

14 12 19 16 26 23 34 30 42 36 54 46 60 52 67 58 76 66 84 74 94 82 104 88 112 97 124 107

8 7 11 10 16 14 20 18 26 22 32 29 36 31 40 35 45 39 51 44 57 49 62 52 67 58 74 64

3 3 5 4 6 6 8 7 10 9 13 11 14 12 16 14 18 16 20 18 23 20 25 21 27 23 30 26

15 12 20 20 30 25 35 35 45 40 55 50 60 56 70 60 80 70 85 80 100 85 110 100 120 110 130 120

11 11 11 11 12 12 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13

28 26 35 35 52 47 68 60 85 78 105 95 110 105 125 110 140 125 150 140 190 150 210 190 210 190 225 210

9 9 10 10 16 15 24 21 28 26 35 31 35 35 40 35 44 40 49 44 63 49 70 63 70 63 75 70

H J K M N P R U Z Outside (0.93C) (0.75C) (0.92C) L (0.70C) (0.60C) (1.20C) (0.50C) (0.50C) (0.30C) (0.12C) Bore Series diameter Width

Tonne-force, 1 tonne force ¼ 1 tf ¼ 1 Mgf ¼ 1000 kgf ¼ 9086.6 pﬃﬃﬃﬃ N ¼ 9.8066 kN. pﬃﬃﬃﬃ Formula for catching C: for MS (mild steel); C ¼ 26:73 P; for HS (high-tensile steel), C ¼ 23:17 P. Machined shank diameter (min). Source: IS 3815, 1969.

a

16

6.4

3.2

8.0

4

2.0

10

2

1.0

5.0

1

0.5

Safe working Proof load load A D E F Nominal W, tf a P, tf (2.75C) B (3.1C) C b (1.44C) (1.25C) (1.00C) G e size G1 Pitch

TABLE 24-3 Hooks of standard trapezoidal section (Refer to Fig. 24-16)

MISCELLANEOUS MACHINE ELEMENTS

24.18

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

24.3 CONNECTING AND COUPLING ROD2,3 SYMBOLS2,3 area of cross section, m2 (in2 ) Rankine’s constant width, m (in) diameter, m (in) core diameter of bolt, m (in) crankpin diameter, m (in) gudgeon pin diameter, m (in) modulus of elasticity, GPa (psi) force acting on the piston due to steam or gas pressure corrected for inertia eﬀects of the piston and other reciprocating parts, kN (lbf ) the component of F acting along the axis of connecting rod, kN (lbf ) inertia force, kN (lbf ) inertia force due to reciprocating masses, kN (lbf ) crippling or critical force, kN (lbf ) acceleration due to gravity, 9.8066 m/s2 9806.6 mm/s2 (32.2 ft/s2 ) depth of rectangular or other sections, m (in) radius of gyration, m (in) length of connecting rod, m (in) length of crankpin, m (in) equivalent length, m (in) length of gudgeon pin, m (in) bending moment, N m (lbf in) speed of crank, rpm safety factor

A a b d d1 dc dg E F Fc Fi Fir Fcr g h k l lc le lg Mb n n1 n0 ¼ p pf v w W Z !

l r

ratio of connecting rod length to radius of crank allowable pressure, MPa (psi) load due to gas or steam pressure on the piston, MPa (psi) velocity of crank, m/s (fps) speciﬁc weight of material of connecting road, kN/m3 (lbf/in3 ) weight of the reciprocating masses, kN (lbf ) section modulus, m3 , cm3 (in3 ) angular speed of crank, rad/s angle between the crank and the center line of connecting rod, deg angle between the crank and the center line of the cylinder measured from the head-end dead-center position, deg angle between the center line of piston and the connecting rod, deg normal stress (also with suﬃxes), MPa (psi)

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24.19

MISCELLANEOUS MACHINE ELEMENTS

24.20

CHAPTER TWENTY-FOUR

Particular

Formula

The velocity

2rn 60 where r in m

v¼

ð24-81Þ

DESIGN OF CONNECTING ROD (Fig. 24-17) Gas load Load due to gas or steam pressure on the piston

Fg ¼

d2 p 4 f

Fir ¼

Wv2 gr

ð24-82Þ

Inertia load due to reciprocating motion Inertia due to reciprocating parts and piston

cos þ

Fir ¼ 0:01095

Wrn2 g

cos 2 n0

cos þ

ð24-83aÞ cos 2 n0

Wrn2 1 1þ 0 g n

ð24-83bÞ

The maximum value of Fir occurs when ¼ 08 or when the crank is at the head-end dead center

F1irðmaxÞ ¼ 0:01095

At the crank-end dead center, when ¼ 1808, Fir attains the maximum negative value, acting in opposite direction

F2irðmaxÞ ¼ 0:01095

The combined force on the piston

F ¼ Fg Fir

ð24-86Þ

The component of F acting along the axis of connecting rod

F Fc ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sin 2 1 n0

ð24-87Þ

Wrn2 1 1 0 g n

ð24-84Þ ð24-85Þ

The stress induced due to column action on account of load Fc acting along the axis of connecting rod " 2 # Fc l 1 ¼ 1þa e A k

(a) As per Rankine’s formula

ð24-88aÞ

Fi

r

Fc F

α

2l 3

l FIGURE 24-17 Forces acting on a connecting rod.

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

(b) As per Ritter’s formula

(c) As per Johnson’s parabolic formula

24.21

Formula

" 2 # Fc e le 1 ¼ 1þ 2 A n E k 1 ¼

Fc

"

y A 1 4n2 E

2 # le k

ð24-88bÞ ð24-88cÞ

where le ¼ equivalent length, m (in) k ¼ radius of gyration, m (in) n ¼ end-condition coeﬃcient (Table 2-4) a ¼ constant obtained from Table 2-3

Inertia load due to connecting rod The magnitude of inertia force (Fig. 24-17) due to the weight of the rod itself, not including the ends

Fic ¼

Awv2 l sin 2gr

ð24-89aÞ

Fic ¼

Wv2 2gr

ð24-89bÞ

when ¼ 908

where W ¼ Awl ¼ weight of the rod itself, not including the ends, kN (lbf ) The maximum bending moment produced by the inertia force Fic is at a distance (2/3)l from wrist pin

The maximum bending stress developed in the rod due to inertia force Fic

2F l 2Wv2 l MbðmaxÞ ¼ picﬃﬃﬃ ¼ pﬃﬃﬃ sin 9 3 9 3 2gr

ð24-90aÞ

Wv2 l MbðmaxÞ ¼ pﬃﬃﬃ 9 3gr

when ¼ 908

ð24-90bÞ

MbðmaxÞ Wv2 l sin ¼ pﬃﬃﬃ Z 9 3grZ

ð24-91aÞ

bðmaxÞ ¼

Wv2 l bðmaxÞ ¼ pﬃﬃﬃ 9 3grZ The crank angle () at which the maximum bending moment occurs according to B. B. Low

¼ 908

The relation between the moment of inertia in the xx and yy planes in order to have same resistance in either plane

Iyy ¼ 14 Ixx

when ¼ 908

3500 ðn0 þ 7:82Þ2

ð24-91bÞ ð24-92Þ ð24-93Þ

or k2yy ¼ 14 k2xx

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ð24-94Þ

MISCELLANEOUS MACHINE ELEMENTS

24.22

CHAPTER TWENTY-FOUR

Particular

Formula

DESIGN OF SMALL AND BIG ENDS The diameter of crankpin at the big end

dc ¼

F lc p

ð24-95Þ

where p ¼ allowable bearing pressure based on projected area, MPa (psi) ¼ 4.9 to 10.3 MPa (700 to 1500 psi), and lc ¼ 1:25 to 1:5 dc The diameter of the gudgeon pin at the small end

dg ¼

F lg p

ð24-96Þ

where p ¼ 10.3 to 13.73 MPa (1.2 to 2.0 kpsi), and lg ¼ 1:5 to 2 d

DESIGN OF BOLTS FOR BIG-END CAP The diameter of bolts used for ﬁxing the big-end cap

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2F1irðmaxÞ di ¼ d

ð24-97Þ

where F1irðmaxÞ is obtained from Eq. (24-84) d ¼ design stress of bolt material, MPa (psi) The expression for checking load for measuring peripheral length of each thin-walled half-bearing according to J. M. Conway Jonesa

Wc ¼ 6000

Lhb D

SI

ð24-97aÞ

SI

ð24-97bÞ

where Wc ¼ checking load, N L ¼ axial length of bearing, mm hb ¼ wall thickness of bearing, mm D ¼ diameter of housing, mm

The expression for total minimum nip, n

n ¼ 44 106

D2 hb

or 0:12 mm

whichever is larger a

In M. J. Neale, ed., Tribology Handbook, Section A20, Newnes-Butterworth, London, 1973.

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.23

Formula

Note: The ‘‘nip’’ or ‘‘crush’’ is the amount by which the total peripheral length of both halves of bearing under no load exceeds the peripheral length of the housing of the bearing. The compressive load on each bearing joint face to compress nipa

a

b

l h

b

a

SI

ð24-97cÞ

W ¼ Lhsl y 106

SI

ð24-97dÞ

or

whichever is smaller where

α

b

α

D ¼ housing diameter, mm (in) r

a

a

α

hsl ¼ steel thickness þ 12 lining thickness, mm (in) m ¼ sum of maximum circumferential nip on both halves of bearing, mm (in)

(a)

b

ELhsl m ðD hsl Þ106

W¼

b

W ¼ compressive load on each bearing joint face, N (lbf) E ¼ modulus of elasticity of material of backing, Pa (psi)

α

r

¼ 210 GPa (30.45 Mpsi) for steel L ¼ bearing axial length, mm (in)

(b)

FIGURE 24-18 Forces acting on a coupling rod.

y ¼ yield stress of steel backing, Pa (psi) ¼ 350 MPa (50 kpsi) for white-metal-lined bearing ¼ 300 to 400 MPa (43.5 to 58 kpsi) for bearing with copper-based lining ¼ 600 MPa (87 kpsi) for bearing with aluminumbased lining

The bolt load required on each side of bearing to compress nip for extremely rigid housing

Wb ¼ 1:3W

ð24-97eÞ

The bolt load required on each side of bearing to compress nip for normal housing with bolts very close to back of bearing The ratio of connecting rod length (l) to crank radius (r)

Wb ¼ 2W

ð24-97f Þ

l n0 ¼ ¼ 3:4 to 4.4 single-acting engines ð24-98Þ r ¼ 4.6 to 5.4 for double-acting engines ¼ 6.0 or more for steam locomotive engines ¼ 5 to 7 for stationary steam engines ¼ 3.2 to 4 for internal-combustion engines ¼ 1.5 to 2 for aero engines

a

In M. J. Neale, ed., Tribology Handbook, Section A20, Newnes-Butterworth, London, 1973.

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MISCELLANEOUS MACHINE ELEMENTS

24.24

CHAPTER TWENTY-FOUR

Particular

Formula

DESIGN OF COUPLING ROD (Fig. 24-18) The centrifugal force due to the weight of the rod

Fc ¼

wv2 hbl gr

Fc ¼

Wv2 gr

ð24-99Þ ð24-100Þ

wv2 hbl Wv2 cos cos ¼ gr gr

ð24-101Þ

wv2 hbl2 Wv2 l ¼ 8gr 8gr

ð24-102Þ

wv2 Wv2 sin hbl sin ¼ gr gr

ð24-103Þ

The bending component of centrifugal force

Fcb ¼

The maximum bending moment due to the uniformly distributed load of Fcb

MbðmaxÞ ¼

The axial component of the centrifugal force

Fca ¼

For some of the common cross sections of connecting rods

Refer to Fig. 24-19.

For forces acting on a coupling rod

Refer to Fig. 24-18.

For proportions of ends of round and H-section connecting rod

Refer to Fig. 24-20.

For proportions and empirical relations of steam engine common strap end

Refer to Fig. 24-21.

Y x

0.36 to 0.4B

x

d

Y b = 4t

0.6B

t x

t

x

h = 5t

h

t

Y b x

Y (b)

0.02B

Groove in bush 0.02 to 0.04B 0.27B

0.04 to 0.05B

Y (a)

x

Lock washer

Sphere 0.358

Y (c)

FIGURE 24-19 Connecting rod sections.

Rad

Groove in bush B 120 0.06 to 0.13B

Rad

0.26 to 0.33B B = cylinder bore FIGURE 24-20 Two typical end designs for round and H-section connecting rods.

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MISCELLANEOUS MACHINE ELEMENTS

F

C

MISCELLANEOUS MACHINE ELEMENTS

b

D d

F

E

A

B

Taper of cotter 1 in 16

G

H

A

t

b

A = 0.9d B = 0.5d + 10 b=d D = 0.35d + 3

E = 1.15d + 5.5 F = 0.37d + 3 G = 0.1d + 4

d

H = 0.3d + 1.5 C = 0.3d + 1.5 t = 0.2d + 1.5

All dimensions in mm FIGURE 24-21 Steam engine common strap end.

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24.25

MISCELLANEOUS MACHINE ELEMENTS

24.26

CHAPTER TWENTY-FOUR

24.4 PISTON AND PISTON RINGS SYMBOLS2,3 A b B c C d dg D Dr E F F h h1 , h2 , h3 H l g i0 ¼ d g lg L Mb n Pb p r R, Ri , Rh th tr Tc Te w

area of cross section of piston head, m2 (in2 ) width of face of piston, m (in) diameter of bore, m (in) heat-conduction factor, kJ/m2 /m/h/K (Btu/in2 /in/h/8F) higher heat value of fuel used, kJ/kg (Btu/lb) nominal diameter of piston ring, m (in) diameter of piston rod, m (in) diameter of gudgeon pin, m (in) diameter of bore (cylinder), m (in) root diameter of the piston ring groove, m (in) modulus of elasticity, GPa (psi) force, kN (lbf ) diametral load on the piston ring to close the gap which is less than 2.45 N (0.55 lbf ) tangential load on the piston ring to close the gap which is less than 2.45 N (0.55 lbf ) thickness (also with subscripts), m (in) radial thickness of piston ring, m (in) thickness as shown in Fig. 24-24, m (in) heat ﬂowing through the head, kJ/h (Btu/h) length/diameter ratio length of gudgeon pin, m (in) length of piston, m (in) bending moment, N m (lbf in) safety factor brake horsepower (bhp) pressure, MPa (psi) radius, m (in) radius as shown in Fig. 24-22b, m (in) thickness of head, m (in) thickness under ring groove, m (in) temperature at center of head, 8C (8F) temperature at edge of head, 8C (8F) weight of fuel used, kg/bhp/h axial width of ring, m (in) stress (also with subscripts), MPa (psi) angle (Fig. 24-23), deg

TABLE 24-4 Dimensions of cast-iron piston up to 430 mm diameter (Fig. 22-24) (all dimensions in cm) Diameter of cylinder, D

Diameter of piston rod, d

d1

b

a

h

h1

h2

h3

d2

d3

15.0 20.0 25.0 30.0 35.0 40.0

2.8 3.4 4.0 5.0 5.6 5.9

2.5 3.1 3.7 4.7 5.0 5.6

7.5 8.1 9.0 10.0 11.2 11.8

1.2 1.4 1.6 1.7 1.9 1.9

1.2 1.2 1.4 1.6 1.6 1.7

1.4 1.6 1.7 1.7 1.9 2.2

0.8 0.95 0.95 1.2 1.2 1.2

0.8 0.8 0.95 0.95 0.95 0.95

3.1 3.4 4.0 4.6 5.3 5.8

2.5 3.1 3.7 4.7 5.0 5.6

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.27

Formula

STEAM ENGINE PISTONS Piston rods The diameter of piston rod

d¼D

rﬃﬃﬃﬃﬃ p a

ð24-104Þ

where

The diameter of piston rod according to Molesworth

p ¼ unbalanced pressure or diﬀerence between the steam inlet pressure and the exhaust, MPa (psi) u ¼ allowable stress, MPa (psi) a ¼ n Note: a is based on a safety factor of 10 for doubleacting engines and 8 for single-acting engines. (The diameter of piston rod is usually taken as 16 to 17 the diameter of the piston.) pﬃﬃﬃ d ¼ 0:0044D p for cast-iron pistons ð24-105Þ pﬃﬃﬃ d ¼ 0:00338D p for steel pistons ð24-106Þ

Ri R Rh

(a)

θ

(b) FIGURE 24-23 Conical plate piston.

FIGURE 24-22 Plate piston.

h2 h1

d2

a

b

a

h3

d1

h

d3 d D

FIGURE 24-24 Cast-iron piston of diameter 400 mm (16.0 in).

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MISCELLANEOUS MACHINE ELEMENTS

24.28

CHAPTER TWENTY-FOUR

Particular

Formula

PROPORTIONS FOR PRELIMINARY LAYOUT FOR PLATE PISTONS Box type (Figs. 24-22a and 24-24) Width of face

b ¼ 0:3 to 0:5D

ð24-107Þ

Thickness of walls and ribs for low pressure

pﬃﬃﬃ h ¼ ð2R þ 50 cmÞ ð0:003 p þ 0:0275 cmÞ

ð24-108Þ

or

The thickness of walls and ribs for high pressure

h¼

2R þ 10 mm (0.40 in) 60

ð24-109Þ

h¼

2R þ 10 mm (0.40 in) 40

ð24-110Þ

For dimensions of conical plate piston

Refer to Fig. 24-23.

For dimensions of cast-iron piston of 400 mm diameter

Refer to Fig. 24-24 and Table 24-4.

Disk type (Fig. 24-22b) Width of face

b ¼ 0:3 to 0:5D

Thickness of walls and ribs for low pressure

pﬃﬃﬃ h ¼ ð2R þ 12:5 cmÞ ð0:0096 p þ 0:057 cmÞ ð24-112Þ

ð24-111Þ

The hub thickness

h1 ¼ 0:45d

ð24-113Þ

The hub diameter

Dh ¼ 2Rh ¼ 1:6 the piston diameter

ð24-114Þ

Width of piston rings

w ¼ 0:03D to 0:06D

ð24-115Þ

Thickness of piston rings

h ¼ 0:025D to 0:03D

ð24-116Þ

For dimensions of cast-iron piston

Refer to Table 24-4.

STRESSES (a) Distributed load over the plate inside the outer cylindrical wall (i.e., the area R2i ) (1) Stress at the outer edge (Fig. 24-22b)

(2) Stress at the inner edge (Fig. 24-22b)

1 ¼

3p 4R2h Ri 2 2 3R þ ln R i h R2i R2h Rh 4h2

ð24-117Þ

2 ¼

3p 4R2i R2h 2 Ri 2 2 R þ R ln i h Rh R2i R2h 4h2

ð24-118Þ

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

(b) Load on the outer wall, pðR2 R2i Þ distributed around the edge of the plate (1) Stress at the outer edge (Fig. 24-22b) (2) Stress at the inner edge (Fig. 24-22b)

24.29

Formula

3pðR2 R2i Þ 2R2h Ri 3 ¼ ln 1 2 Ri R2h Rh 2h2 3pðR2 R2i Þ 2R2i Ri ln 1 2 4 ¼ Ri R2h Rh 2h2

ð24-119Þ ð24-120Þ

(3) The sum of the stresses at the outer edge

o ¼ 1 þ 3

ð24-121Þ

(4) The sum of the stresses at the inner edge

i ¼ 2 þ 4

ð24-122Þ

(Note: o or i should not be greater than the permissible stress of the material. A safety factor, n, of 8 can be used.)

Dished or conical type (Fig. 24-23) An empirical formula for the thickness of conical piston (Fig. 24-23)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h ¼ 0:288 pD= sin

SI

ð24-123aÞ

where p and in MPa, and D and h in m pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h ¼ 9:12 pD= sin Customary Metric ð24-123bÞ where p and in kgf/mm2 , D and h in mm pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h ¼ 1:825 pD= sin USCS ð24-123cÞ where p and in psi, D and h in in

The height of boss

H ¼ 1:1K

The diameter of boss

Dh ¼ 1:7K for small pistons

ð24-124Þ ð24-125aÞ

Dh ¼ 1:5K for large pistons and light engines ð24-125bÞ The thickness h1 measured on the center line

h1 ¼ Kc

ð24-126Þ

where c ¼ 1 to 0.75 depending on the angle of inclination (Refer to Table 24.5.) ¼ varies from 68 to 358 K ¼ 1 to 4.5 for varying pressure and diameter Also refer to Table 24-6 for values of K. For calculating hub diameter, width of piston rings, and thickness of piston rings

Refer to Eqs. (24-114) to (24-116).

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MISCELLANEOUS MACHINE ELEMENTS

24.30

CHAPTER TWENTY-FOUR

Particular

Formula

PISTONS FOR INTERNAL-COMBUSTION ENGINES Trunk piston (Fig. 24-25) The head thickness of trunk pistons (Fig. 24-25a)

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3PD2 th ¼ 16

ð24-127Þ

where ¼ 39 MPa (5.8 kpsi) for close-grained cast iron ¼ 56.4 MPa (8.2 kpsi) for semisteel or aluminum alloy ¼ 83.4 MPa (12.0 kpsi) for forged steel

COMMONLY USED EMPIRICAL FORMULAS IN THE DESIGN OF TRUNK PISTONS FOR AUTOMOTIVE-TYPE ENGINES th ¼ 0:032D þ 1:5 mm

Thickness of head (Fig. 24-25a)

th ¼ 0:00D þ 0:06 in The head thickness for heat ﬂow

D tr

HD2 0:16cðTc Te ÞA

¼

H 0:194cðTc Te Þ

ð24-128Þ

USCS

ð24-128aÞ

SI

ð24-129aÞ

where

tr

th

th ¼

SI

Tc Te ¼ 2058C (4008F) and Tc ¼ 698K, 4258C (8008F) for cast-iron piston T ¼ Tc Te ¼ 558C (1308F) and Tc ¼ 533K, 2608C (5008F) for aluminum piston

(a)

c ¼ 2.2 for cast iron

Dr

c ¼ 7.7 for aluminum

tr

th ¼

(c) (b)

HD2 H ¼ 16cTA 12:5cT

USCS

ð24-129bÞ

(d)

D

FIGURE 24-25 Trunk piston for small internal-combustion engine. (a) piston laid out for heat transfer; (b) piston modiﬁed for structural eﬃciency; (c and d) alternate pin designs.

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MISCELLANEOUS MACHINE ELEMENTS

24.31

MISCELLANEOUS MACHINE ELEMENTS

Particular

Formula

ð24-130Þ

The thickness of wall under the ring groove

tr ¼ thickness of head ¼ th pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ tr ¼ 12 ðDr D2r 4Dth Þ

The heat ﬂow through the head

H ¼ KCwPb

ð24-132Þ

Thickness of wall under the ring (Fig. 24-25a and b)

ð24-131Þ

where w ¼ weight of fuel used, kJ/kW/h (lbf/bhp/h) K ¼ constant representing that part of heat supplied to the engine which is absorbed by the piston ¼ 0.05 (approx.) Pb ¼ brake horsepower per cylinder The root diameter of ring grooves, allowing for ring clearance

Dr ¼ D ð2w þ 0:006D þ 0:02 inÞ

USCS

Dr ¼ D ð2w þ 0:006D þ 0:5 mmÞ at the compression rings Dr ¼ D ð2w þ 0:006D þ 1:5 mmÞ

SI

ð24-132aÞ

SI ð24-132bÞ

Dr ¼ D ð2w þ 0:00D þ 0:06 inÞ at the oil grooves

USCS

where Dr and D in mm (in) Length L of piston

L ¼ D to 1:5D

For chemical composition and properties of aluminum alloy piston

Refer to Table 24-10B.

ð24-133Þ

Gudgeon pin The diameter of gudgeon pin

sﬃﬃﬃﬃﬃ F dr ¼ i0 p

ð24-134Þ

where F ¼ maximum gas pressure corrected for inertia eﬀect of the piston and other reciprocating parts, kN (lbf ) p ¼ working bearing pressure ¼ 9.81 MPa (1.42 kpsi) to 14.7 MPa (2.13 kpsi) lg ¼ 1:5 to 2 dg

The length/diameter ratio of gudgeon pin

i0 ¼

For gudgeon pin allowable oval deformation

Refer to Fig. 24-28c.

For empirical relations and proportions of pistons

Refer to Figs. 24-26 to 24-28a.

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ð24-135Þ

MISCELLANEOUS MACHINE ELEMENTS CHAPTER TWENTY-FOUR

0.2B

24.32

0.11B

X

0.052B

0. 7D

0.35B=D

Local clearance 0.05D

X

1.25B

0.67B

0.1B

0.4B

0.22B

0.04B

0.0

42B

0.07B

0.37B

Local clearance 0.1B

0.035B

B = cylinder bore

FIGURE 24-26 Proportions of a typical alloy piston.

A

0.2B

0.03 to 0.05B

to

6 0.2 B 0.3

Taper

Offset

0.375 to 0.4B

1.75 to 2.0B

C

0.025 to 0.033B

0.170 to 0.125B

B

1.0B

0.2 to 0.25B

B (Nominal)

0.03 to 0.04B D

FIGURE 24-27 Proportions of an iron piston.

Fig. 24-27

mm (in)

Cylinder 152.5 bore (6) Crown A 16 thickness (5/8) Clearance B 0.760 (0.03) Clearance C 0.125 (0.005) Clearance D 0.125 (0.005)

mm (in)

mm (in)

mm (in)

mm (in)

203.2 (8) 19 (3/4) 0.900 (0.035) 0.225 (0.008) 0.150 (0.006)

254 (10) 32 (114) 1.145 (0.045) 0.230 (0.009) 0.180 (0.007)

305 (12) 41.5 (158) 1.525 (0.06) 0.255 (0.01) 0.200 (0.008)

406.5 (16) 47.5 (178) 2.030 (0.08) 0.255 (0.01) 0.230 (0.009)

Piston weight ¼ 40:715B3 N (approx.) SI where B ¼ cylinder bore, m Piston weight ¼ 0:15B3 lbf USCS where B in in

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Ra d

2.2B B.less 1000

d

0.11B

0.085B

B B.less 1000 B.less 3B 0.045B 4000

0.4B 0.045B

0.425B

1.35 to 1.5B 0.8 to 0.95B 0.22B 0.03B

Ra

Relief in way of pin bosses B = 1000

24.33

0.036B

0.21B

0.03B 3B B.less 4000

650

d

90

600 550

80

A

450

48

400

.4

σ = 0.415P

2 /m

G

2 /m

MN

300 250 200 0.100

0.150

0.200

d t

2

,P=

GAS LOAD AREA A

MN

.5

34

350

70

.2

55

2) n f/i IN lb n2 ) AR 00 f/i ) 2 BE lb 00 n (1 000 lbf/i 2 ) SS BO 0 (9 /in f N/ 00 2) lb PI 8 ( n 0 N 2 f/i 00 EO m b / l 2) (7 DG 0 2 n MN GU f/i 00 /m 69 6 lb 2 ( MN 0 /m 62 00 2 MN (5 /m MN .3

500

0.250

0.300 t GUDGEON PIN d

PR

60 50

ES

SU

RE

’P’

0.350

40

0.400

GUDGEON PIN FATIGUE STRESS,σ. thousands of

× 103 100 t

41

GUDGEON PIN FATIGUE STRESS,σ. MN/m2

700

lbf/in2

FIGURE 24-28(a) Iron piston for small engines (B ¼ cylinder bore).

30 0.450

FIGURE 24-28(b) Fatigue stress in gudgeon pins for various pin and piston geometries. (M. J. Neale, Tribology Handbook, Butterworth-Heinemann, 1973.)

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MISCELLANEOUS MACHINE ELEMENTS

24.34

CHAPTER TWENTY-FOUR

Formula

CYLINDER BORE DIAMETER (NOMINAL), in 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

60 PIN OVAL DEFORMATION δ = (a b) D2 P d m 171 L δ D2 P d ( OR t = 900 in) L δ t=

DEFORMATION, µm

50

L t 40

d

b a D DIA.

30

P MAX. PRESSURE

MN/m2 (lbf/in2)

20

10

2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5

DEFORMATION, THOUSANDTHS OF AN INCH

Particular

100 150 200 250 300 350 400 CYLINDER BORE DIAMETER (NOMINAL), mm

FIGURE 24-28(c) Gudgeon pin allowable oval deformation. (M. J. Neale, Tribology Handbook, Butterworth-Heinemann, 1973.)

For fatigue stress in gudgeon pins

Refer to Fig. 24-28b.

For empirical proportions and values of cylinder cover, cylinder liner, and valves

Refer to Figs. 24-30 to 24-33.

Piston rings Width of rings

For land width or axial width of piston ring (w) required for various groove depths (g) and maximum cylinder pressure, pmax .

w¼

D for concentric rings 20

w¼

D opposite the joint of eccentric rings 27:5 ð24-136bÞ

w¼

D at the joint of eccentric rings 55

Refer to Fig. 24-28d.

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ð24-136aÞ

ð24-136cÞ

MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

24.35

Groove depth g, in 0.200

0.250

0.300

0.350

0.400

0.450

0.500

2400 2200 ALUMINIUM PISTON LAND WIDTH W TO DETERMINE LAND WIDTHS FOR PISTONS IN CAST IRON OR STEEL USE THE FOLLOWING CONSTANTS

16.0

15.0

0.600 1600

13.0

16

1400

0.550

1200

0.500

M

12

13

.8

15

.2

.6

M N/ M m2 N/ . m 4 N/ 2 11 M .0 N/ m 2 9. M m2 65 N/ M m2 N /m 2

STEEL W FROM GRAPH × 0.5

0.650

1800

CAST IRON W FROM GRAPH × 0.700

14.0

2000

W, mm

11.0

3

8.

1000

2

/m

N

M

0.450

2

10.0

m MAX. CYLINDER M PRESSURE, Ibf/in2 0.400 9 6.

9.0

0.350

N/

W, in

12.0

8.0 0.300 7.0

g w

6.0

0.250

5.0

0.200

4.0

0.150

3.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

13.0

Groove depth, mm FIGURE 24-28(d) The land width required for various groove depths and maximum cylinder pressures. (M. J. Neale, Tribology Handbook, Butterworth-Heinemann, 1973.)

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MISCELLANEOUS MACHINE ELEMENTS δf δc Fθ

Fθ

Free ring Compressed ring

r v

r + v+ d

d φ

δc = circumferential clearance gap δf = free piston ring gap h = radial depth or wall thickness of piston ring w = axial width of piston ring d = nominal diameter of piston ring r = radius of neutral axis of ring h w FIGURE 24-28(e) Nomenclature of piston ring and tangential force, F .

d

Variable pv = p (φ)

Constant pc

Fd

Fd φ

pc FIGURE 24-28( f ) Typical variable and constant contact pressure distribution around piston rings for four-stroke engines. (Courtesy: Piston Ring Manual, Goetze AG, D5093 Burscheid, Germany, August 1986.)

FIGURE 24-28(g) Diametrically opposite force (Fd ) applied on piston ring.

Gap clearance Fθ

Fθ

FIGURE 24-29 Tangentially applied force, F , on a piston ring.

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

The modulus of elasticity of piston ring as per Indian Standards

24.37

Formula

3 d 1 F 5:37 h E¼ w

when the ring is diametrically loaded

ð24-137aÞ

3 d 14:14 1 F h E¼ w

when the ring is tangentially loaded

ð24-137bÞ

where E ¼ modulus of elasticity, MPa (psi)

¼ diﬀerence between free gap and gap after applying the load, mm (in) The bending moment produced at any cross section of the ring by the pressure uniformly distributed over the outer surface of the ring at an angle measured from the center line of the gap of the ring (Figs. 24-28e and 24-28f )

Mb ¼ 2pwr2 sin2

2

Mb ¼ pwr2 ð1 þ cos Þ

ð24-138aÞ ð24-138bÞ

where r ¼ radius of neutral axis, mm (in) p ¼ pressure at the neutral axis of the piston ring, Pa (psi)

The bending moment (Mb ) in Eq. (24-138a) in terms of tangential force, F The uniform contact pressure of the piston ring on the wall

Mb ¼ F rð1 þ cos Þ

p¼

En f 7:07dðd=h 1Þ3

ð24-138cÞ

ð24-138dÞ

where d ¼ external piston ring diameter h ¼ radial depth or wall thickness of piston ring En ¼ nominal modulus of elasticity of material of the ring

f ¼ free ring gap The radial distance from a point in piston ring to obtain a uniform pressure distribution (Fig. 24-28e) according to R. Munroa

a

ro ¼ r þ v þ dv

In M. J. Neale, ed., Tribology Handbook, Section A31, Newnes-Butterworth, London, 1973.

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ð24-138eÞ

MISCELLANEOUS MACHINE ELEMENTS

24.38

CHAPTER TWENTY-FOUR

Particular

Formula

where

.067B+4

.04B .05B

.067B + 35.5

.09 to .12B

.065B

All dimensions in mm FIGURE 24-30 Proportion for four-stroke cover, 100- to 450-mm bore (B ¼ cylinder bore).

Fr4 ð1 cos þ 12 sin Þ En I " r Fr3 2 ð 12 cos 12 sin Þ dv ¼ 2 En I # v¼

ð24-138f Þ

ð3 sin þ cos Þ where F ¼ (mean wall pressure ring axial width)

x 0.1 to 0.08 to 0.15B 0.12B

2.5 0.75

0.08 to 0.12B

9.15 .18D to .21D

3 to 4.75

1.25 to 1.35B 1.18 to 1.2B 3.45

0.1 Clear on dia.

0.75 to 1.0B

3.25

r ¼ radius of neutral axis, when the ring is in place inside the cylinder (Fig. 24-28e)

.25 to .30D .25D

0.03B 0.07to 0.08B

18 to 22lbf/sq in of valve area

H J

B 0.0012

.38D

9

.65D

7R

5B 8000 Clear on dia

7.5

G

w 3

0.07B 0.05B

F

B 0.0125 X

Lowest ring on BDC

All dimensions in mm FIGURE 24-31 Empirical rules for average practice in liner design (B ¼ cylinder bore).

D FIGURE 24-32 Valve seated directly in the cylinder head.

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.39

Formula

¼ angle measured from bottom of the vertical line passing through the center of the gap of the ring as shown in Fig. 24-28e I ¼ moment of inertia of the ring The relation between the ratio of ﬁtting stress ft to nominal modulus of elasticity (En ) in terms of h, d, and f

ft 4ð8h f þ 0:00dÞ ¼ En 3hðd=h 1Þ2

The relation between the ratio of working stress (w ) to nominal modulus of elasticity (En ) in terms of h, d, and f

4ð f 0:00dÞ w ¼ En 3hðd=h 1Þ2

The relation between the ratio of the sum of (ft þ w ) to nominal modulus of elasticity (En ) in terms of d and h

ft þ w 32 ¼ En 3ðd=h 1Þ2

where ft ¼ opening stress when ﬁtting the piston ring onto the piston

For preferred number of piston rings

Refer to Table 24-10C.

For properties of typical piston ring materials

Refer to Table 24-10D.

0.27D 3 off

Rib 0.27D D 2

D 2.0

0.16D

D C

E

Two ribs thus (revolved section) Coreplug

0.12D

Cooling water connection N

B

Two bosses thus

1.1D

ð24-138hÞ

where w ¼ working stress when the piston ring is in the cylinder

Equation (24-138i) is independent of f .

Hardened

ð24-138gÞ

Copper joint FIGURE 24-33 Valve with removable cage.

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ð24-138iÞ

MISCELLANEOUS MACHINE ELEMENTS

24.40

CHAPTER TWENTY-FOUR

Particular

The circumferential clearance ( c ) or gap between ends of ring

Formula

c ¼ d p T

ð24-138jÞ

where p ¼ coeﬃcient of expansion of piston ring material T ¼ operating temperature d ¼ cylinder diameter

An expression for pressure acting on ring from Eqs. (24-138b) and (24-138c)

p¼

F rw

ð24-139aÞ

The pressure in the radial outward direction against the cylinder

p¼

2F dw

ð24-139bÞ

For variable and constant radial contact pressure distribution of piston ring

Refer to Fig. 24-28f.

The diametral load which acts at 908 to the gap required to close the ring to its nominal diameter, d (Fig. 24-28g)

Fd ¼ 2:05F

for modulus of elasticity E 150 GPa ð24-140aÞ

Fd ¼ 2:15F

for modulus of elasticity E > 150 GPa ð24-140bÞ

Fd 2:21F

ð24-140cÞ

The maximum bending stress at any cross section which makes an angle measured from the center line of the gap of the ring

12pr2 b ¼ 2 sin2 2 h

ð24-141Þ

12pr2 h2max

ð24-142Þ

The maximum bending stress which occurs at ¼ , i.e., at the cross section opposite to the gap of the ring

max ¼

The bending stress present in the ring of rectangular cross section in terms of free gap ( f ) of the ring, when it is in place in the cylinder

b ¼ 0:424 f

Eh ðd hÞ2

SI

ð24-142aÞ

where b and E in N/mm2 h, d, and f in mm

The bending stress present in the ring of rectangular cross section in terms of tangential force, F (Fig. 2429)

b ¼

The bending stress present in the case of slotted oil control ring of rectangular cross section in terms of free ring gap, f

bso ¼ 0:424

6ðd hÞ F ð24-142bÞ wh2 2 where b in N/mm ; F in N; d, h, and w in mm

f Elco Im ðd hÞ2 Ius

ð24-142cÞ

where bso in N/mm2 and Ius ¼ moment of inertia of the unslotted cross-section ring, mm4 Im ¼

Ius þ Is 2

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.41

Formula

Is ¼ moment of inertia of the slotted cross-section ring, mm4 lco ¼ twice the diameter between center of gravity and outside diameter, mm The bending stress present in the case of slotted oil control ring of rectangular cross section in terms of tangential load, F

The tangential load or force required for opening of a rectangular cross-section piston ringa

bso ¼

6ðd hÞlco Im F wh3 Is

ð24-142dÞ

where bso in N/mm2 ; F in N; lco , d, h, and w in mm; Im and Is in mm4 ;max ¼

hE ð1:26"T 1:84k þ 0:025Þ dh SI

ð24-142eÞ

where dþh 1 dh k ¼ piston ring parameter from Eq. (24-142f ) and (24-142h)

"T ¼

3ðd hÞ2 F E wh3

The piston ring parameter (k) in terms of tangential load F for rectangular cross-section rings

k¼

The tangential load or force required for opening of rectangular cross-section slotted oil control rings

;max T ¼

ð24-142f Þ

lci E I ð1:26"T 1:84k þ 0:025Þ m dh Is ð24-142gÞ

where dþh 1 SI dh lci ¼ twice the distance between center of gravity and inside diameter, mm

"T ¼

k ¼ piston ring parameter from Eq. (24-142e) and (24-142f ) The piston ring parameter (k) in terms of free ring gap ( f ) for rectangular cross-section slotted oil rings for use in Eq. (24-142g)

k¼

2 f 3 d h

ð24-142hÞ

The piston ring parameter (k) in terms of the constant pressure ( p) for rectangular cross-section rings also for use in Eqs. (24-142f ) and (24-142g)

k¼

3 p dðd hÞ2 2E h3

ð24-142iÞ

The radial thickness of the ring at a section which makes an angle measured from the center line of the gap of the ring a

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 3 24pr sin2 h¼ 2 E

Goetze AG, Piston Ring Manual, 3rd ed., Burscheid, Germany, 1987.

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ð24-142jÞ

MISCELLANEOUS MACHINE ELEMENTS

24.42

CHAPTER TWENTY-FOUR

Particular

Formula

The maximum thickness of the ring which occur opposite the gap of the ring (i.e., at ¼ )

hmax

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 24pr4 ¼ E

ð24-142kÞ

For piston ring dimensional deviation, hardness, and minimum wall pressure

Refer to Tables 24-7 to 24-9.

For cylinder bore diameter

Refer to Table 24-10.

TABLE 24-5 Values of c for various inclinations of coned pistons

Cone

Inclination ranges, , deg

c

— Slightly Medium Strong

0–6 6–18 18–28 28–35

1 0.85–0.95 0.75–0.85 0.65–0.75

TABLE 24-6 Values of coeﬃcient K for pistons (admissible pressures, kgf/mm2 absolute) Pressure, kgf/mm2 Diameter of cylinder, mm

0.01 to 0.02

0.02 to 0.04

0.04 to 0.06

0.06 to 0.08

0.08 to 0.10

0.10 to 0.12

0.12 to 0.14

0.14 to 0.16

380–575 575–775 775–975 975–1175 1175–1375 1375–1575 1575–1775 1775–1975 1975–2150 2150–2350 2350–2550 2550–2750

1.000 1.375 1.500 1.750 2.000 2.375 2.500 2.750 3.000 3.000 3.125 3.375

1.125 1.500 1.750 2.000 2.250 2.500 3.000 3.125 3.375 3.500 3.500 3.750

1.375 1.750 2.000 2.375 2.750 3.125 3.375 3.500 3.750 4.000

1.500 2.000 2.500 3.000 3.125 3.500 4.000 4.125 4.375

1.750 2.500 3.125 3.500 3.750 4.000 4.375

2.000 2.750 3.500 4.000 4.125 4.375

2.125 3.000 3.750 4.500

2.500 3.375 4.000

Key: 1 kgf/mm2 ¼ 1:42247 kpsi; 1 kpsi ¼ 6:894757 MPa.

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MISCELLANEOUS MACHINE ELEMENTS

24.43

MISCELLANEOUS MACHINE ELEMENTS

TABLE 24-7 Recommended hardness for piston rings of IC engines Nominal diameter, d, mm <100 100–200 >200

TABLE 24-8 Minimum wall pressure for piston rings of IC engines

Hardness HRD 95–107 93–105 90–102

Compression rings

Petrola Diesel a

Oil rings 2

MPa

kgf/cm

MPa

kgf/cm2

0.059 0.013

0.60 1.05

0.137 0.196

1.40 2.00

Gasoline.

TABLE 24-9 Permissible deviation on the dimensions of piston rings of IC engines Dimensions

Deviations, mm

Axial width, b

0.010 0.022

Radial thickness 80 mm ring diameter >80 mm with 175 mm ring diameter 175 mm ring diameter Parallelism of sides—40% of tolerance on axial width

0.08 0.12 0.15

TABLE 24-10A Preferred cylinder bore diameters for internal-combustion (IC) engines (all dimensions in mm) 30 32 34 (35) 36 38 40 42 44 46 48 50 52 54 56 (57) 58 (59) 60

(62) 65 (68) 70 (72) (73) 74 (76) (78) (79.4) 80 82 85 87.3 88 (88.9) 90 (91.4) (92)

95 98 (98.4) 100 (101.6) (102) (103.2) (104.8) 105 108 110 (111.1) 112 (114.3) 115 (118) 120 (120.6) (122)

125 (127) (128) (128.2) 130 (132) (133.4) 135 (138) (139.7) 140 (142) 142.9 (145) (146) (148) (149.9) 150 (152)

(152.4) 155 (158) (158.8) 160 (162) 165 (165.1) (168) 170 (171.4) (172) 175 (177.8) (178) 180 (182) (184.2) 185

(188) 190 (190.5) (192) 195 (196.8) 198 200 (205) (209.6) 210 (215) (215.9) 220 (225) (228.6) 230 (235) 240

(241.3) (245) (250) (254) (255) 266 (265) 270 (273) (275) 280 (285) 290 (292.1) (295) (298.4) 300 (305) 103

315 (317.5) 320 (325) 330 (335) 340 (343) (345) 350

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34,850 49,582 49,285

2285 4658 4928A 4928B

3.5–4.5 0.8–1.5 0.8–1.5 0.8–1.5

Cu

1.2–1.8 0.8–1.3 0.8–1.3 2.8–1.3

Mg 0.6 11.0–13.0 17.0–19.0 23.0–26.0

Si 0.7 0.8 0.7 0.7

Fe 0.2 0.2 0.2 0.2

Mn 1.1–2.3 1.5 0.8–1.3 0.1–1.3

Nic 0.2 0.35 0.2 0.2

Zn

Chemical composition, %

0.23 0.2 0.2 0.2

Ti 0.05 0.05 0.05 0.05

So 0.05 0.05 0.05 0.05

Pb

c

b

Alloys have been designated in accordance with IS 6051, 1970. Code for designation of aluminum and aluminum alloys. Physical properties are attainable after suitable heat treatment. The purchaser may specify nickel content, if so desired. Source: Bureau of Indian Standards, New Delhi.

a

Forging

Casting

Alloy designationa

0.3–0.6

Cr

MPa 225–275 195–245 175–215 165–205

Hardness, HB 90–130 90–140 90–125 90–125

Al

32.7–39.8 28.5–35.6 25.6–31.3 24.2–29.7

kpsi

49.8–59.7 42.7–52.6 32.7–28.5

kpsi

Forging

345–410 295–365 225–295

MPa

Tensile strength Chill casting

Remainder

Physical propertiesb

TABLE 24-10B Chemical composition of alloys and physical properties of aluminum alloy piston (values in % maximum unless shown otherwise)

23–24 20.5–21.5 18.5–19.5 17–18

mm/mm/ 8C 104

Coeﬃcient of thermal expansion (20 to 2008C)

MISCELLANEOUS MACHINE ELEMENTS

24.44

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

24.45

TABLE 24-10C Preferred number of piston rings Diﬀerential pressure Std. atm. MPa psi Minimum number of rings

0–9 10–14 15–24 25–29 30–49 50–99 100–200 0–0.88 0.98–1.37 1.47–2.35 2.45–2.85 2.94–4.80 4.90–9.71 9.81–19.61 0–128 142–199 213–341 355–412 426–696 710–1406 1422–2844 2 3 4 5 6 7 8

Source: M. J. Neale, Tribology Handbook, Butterworth-Heinemann, London, 1973; reproduced with permission.

TABLE 24-10D Properties of typical piston ring materials

Tensile strength, t Material Metallic: Gray irons Carbide malleable irons Malleable and/or nodular irons Sintered irons Nonmetallic: Carbon-ﬁlled PTFE Graphite/MoS2 -ﬁlled PTFE Resin-bonded PTFE Carbon Resin-bonded carbon Glass-ﬁlled PTFE Bronze-ﬁlled PTFE Resin-bonded fabric

Nominal modulus of elasticity, En

Typical coeﬃcient of expansion,

MPa

kpsi

GPa

Mpsi

Brinell hardness number, HB

Bulk density, g/cm3 106 =8C

230–310 400–580 540–820

33.4–45.0 58.0–84.1 78.3–119.0

83–124 140–160 155–165

12.1–18.0 20.3–23.2 22.5–24.0

210/310 250/320 200/440

Good Excellent Poor

250–390

36.5–56.6

120

17.4

130/150

Good

10.3 19.6

1.49 2.85

2.05 2.20

55 115

29.4 43.4 19.6 16.7 12.8 110.8

4.27 6.30 2.85 2.42 1.85 16.07

1.75 1.8 1.9 2.26 3.90 1.36

30 43 20 80 118 22.5/87.5a

a

Material is anisotropic. Source: M. J. Neale, Tribology Handbook, Butterworth-Heinemann, London, 1973, extracted with permission.

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Wear rating

MISCELLANEOUS MACHINE ELEMENTS

24.46

CHAPTER TWENTY-FOUR

TABLE 24-10E Piston rings and piston ring elements Mechanical properties

Designation

Grade

Hardness

Tensile strength, st , MPa

Modulus of elasticity, E, MPa

Steel GOE 61 GOE 62

Cr steel, 17% Cr min Cr-Si steel

380–450 HV 30 500–600 HV 30

1200 approx. 1900 approx.

230 000 approx. 210 000 approx.

Cr-Si steel Cr steel, 11% Cr min, high C Cr steel, 11% Cr min, low C

450–550 HV 30 300–400 HV 30

1700 approx. 1300 approx.

210 000 approx. 210 000 approx.

270–420 HV 30

1300 approx.

220 000 approx.

GOE 64 GOE 65A GOE 65B

Cast Iron GOE 12 GOE 13 GOE 32

GOE 44 GOE 52 GOE 56

Unalloyed non heattreated gray cast iron Unalloyed non heattreated gray cast iron Alloyed heat-treated gray cast iron with carbides Malleable cast iron Spheroidal graphite cast iron Spheroidal graphite cast iron

Main application

Compression rings Coil spring loaded rings Compression rings Compression rings, nitrided Coil spring loaded rings and segments, nitrided

94–106 HRB

350 min

85 000 typical

97–108 HRB

420 min

95 000–125 000

109–116 HRB

650 min

130 000–160 000

Compression and oil control rings Compression and oil control rings Compression rings

102–111 HRB 104–112 HRB

800 min 1300 min

150 000 min 150 000 min

Compression rings Compression rings

40–46 HRC

1300 min

150 000 min

Compression rings

Source: Goetze Federal Mogul Burscheid GmbH, Piston Ring Manual, 4th ed., January 1995, Burscheid, Germany, reproduced with permission

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

24.5 DESIGN OF SPEED REDUCTION GEARS AND VARIABLE-SPEED DRIVES SYMBOLS2,3 center distance, m (in) number of pinions or planetary pinion (Fig. 24-36) center distance (also with subscripts) (Fig. 24-36) area of reduction gear housing, m2 (in2 ) noncooled, i.e., ribbed, surface of housing of reduction gear drive, m2 (in2 ) cooled surface of reduction gear drive, m2 (in2 ) surface area of contact of teeth when one-fourth of all teeth of wheel in wave-type reduction gears are engaged, m2 (in2 ) width of rim, m (in) diameter of pinion, m (in) diameter of rigid immovable rim with internal teeth of wave-type reduction gears, m (in) diameter of gear, m (in) diameter of ﬂexible movable wheel rim with external teeth of wave-type reduction gear, m (in) maximum diameter of the circumference of the belt arrangement on the V-belt of a variable-speed drive, m (in) minimum diameter of the circumference of the belt arrangement on the V-belt of a variable-speed drive, m (in)

a A An Ac Aw b d1 d2 dmax dmin D¼

dmax dmin

velocity control range for a V-belt drive with only one adjustable pulley velocity control range for a V-belt drive with two adjustable pulleys working height of a V-groove of the pulley, m (in) maximum load acting on the pinion, kN (lbf ) mean load acting on the pinion, kN (lbf ) height of tooth, m (in) coeﬃcient of heat transfer, W/m2 K (Btu/ft2 h 8F) coeﬃcient of heat transfer of noncooled surface, W/m2 K (Btu/ ft2 h 8R) coeﬃcient of heat transfer of cooled surface, W/m2 K (Btu/ft2 h 8R) addendum of tooth, m (in) dedendum of tooth, m (in) transmission or speed ratio

D1 D2 e Fmax Fm h hn hc ha hf i knl ¼ L m Mts n n1 , n2 q

velocity control range for a V-belt drive

Fmax Fm

nonuniform load distribution factor distance between the axes of the pinions (Fig. 24-36d) module, m (in) torque acting on smaller wheel, N m (lbf in) speed, rpm speeds of pinion and gear, respectively, rpm a whole number heat generated, W (Btu/h)

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24.47

MISCELLANEOUS MACHINE ELEMENTS

24.48

CHAPTER TWENTY-FOUR

maximum radius of the circumference of the belt arrangement on the V-belt of a variable-speed drive, m (in) minimum radius of the circumference of the belt arrangement on the V-belt of a variable-speed drive, m (in) temperature of lubricant, 8C (8F) ambient temperature, 8C (8F) number of teeth on sun pinion and planetary pinion of epicyclic gear transmission, respectively, Fig. 24-36 number of teeth on pinion and gear, respectively number of teeth on ring gear 3 (Fig. 24-36a) number of teeth on smaller wheel angular speed of pinion and gear, respectively, rad/s deformation, m (in) clearance between the pinions which should be at least 1 mm (in) half-cone angle of V-belt, deg allowable compressive stress, MPa (psi)

rmax rmin t1 ta z1 , z2 z3 zs !1 , !2

ca

Particular

Formula

!1 n1 d2 z2 ¼ ¼ ¼ !2 n2 d1 z1

Transmission or speed ratio for single reduction gear

i¼

For diﬀerent types of gear reduction drives

Refer to Fig. 24-35 and Table 24-11.

b

dmax

dmin

h

e

Belt

2α Belt (a) V-belt at top position

(b) V-belt at bottom position

FIGURE 24-34 Dimension of V-belt variable-speed drive.

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ð24-143Þ

MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

1. Single-reduction spur gear

5. Single-reduction bevel gear 9. Single-reduction worm gear with worm arranged sideways

H

2

2

H

L 2. Double-reduction coaxial gear

6. Double-reduction bevel spur gear

2

4

H

L 3. Double-reduction spur and hellcal gear I (a)

2

4

(b)

L

H

L 8. Single-reduction worm gear with worm on top

12. Combination wormspur helical reduction gear

I

I 3 4

H

5 6

l 2

11. Double-reduction worm gear

H

4

4. Triple-reduction spur and helical gear

H

7. Single-reduction worm gear with worm underneath

L

I

2

L

L

2

(b)

L

2

3 4

H H

(a)

10. Single-reduction worm gear with worm arranged vertically

H

4 3

H

L

I

H

L

H

L

l 4 4

L I

5 6

H

L H = High Speed

H L

I = Intermediate Speed

L = Low Speed

FIGURE 24-35 Schematic diagrams of various types of spur, helical, herringbone, bevel, and worm reduction gears.

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24.49

MISCELLANEOUS MACHINE ELEMENTS

24.50

CHAPTER TWENTY-FOUR

Particular

Formula

PLANETARY REDUCTION GEARS L ¼ 2A1;2 sin

First condition—mating

¼ z2 m þ 2mð1 þ Þ þ a

ð24-144Þ

where

The sum of the radii of the addendum circles of the mating pinions in planetary reduction gears should be smaller than the distance between their axes (Fig. 24-36d) so that the top of the pinions should not touch each other

a ¼ number of pinions ¼ clearance between the pinions, which should be at least 1 mm A1;2 ¼ center distance as shown in Fig. 24-36

2

1

2π a

A 1,2

A2,3

1

3

2

2 A1,2

A2,3

2 A1,2

1

3

3

1

π a ∆ L

(b)

(a) i=8

i = 15

(c)

(d)

i = 20 to 100

FIGURE 24-36 Planetary reduction gears.

Second condition—coaxiality The center distance of each pair of wheels should be equal (Fig. 24-36)

A12 ¼ A23 ; A12 ¼ A23 ¼ A20 30

The relationship between teeth in corrected or uncorrected gears (Fig. 24-36a)

z1 þ z2 ¼ z3 z2

ð24-145Þ ð24-146aÞ

or z1 þ 2z2 ¼ z3

ð24-146bÞ

The relationship between teeth in corrected or uncorrected gears (Fig. 24-36c) to ratify two conditions (i) First condition

Refer to Eq. (24-146).

(ii) Second condition

m2 ðz3 z2 Þ ¼ m02 ðz03 z02 Þ

ð24-147aÞ

or z3 z2 ¼ z03 z02

since

m2 ¼ m02

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ð24-147bÞ

MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.51

Formula

Third condition—coincidence The condition for the teeth and spaces of the meshed gears should coincide when the pinions are arranged uniformly over the circumference

z1 þ z3 ¼q a where q is a whole number

The moment acting on smaller wheel

Mts ¼

Mt1 knl zs a z1

ð24-148Þ

ð24-149Þ

where zs ¼ z1 or zs ¼ z2 if z1 > z2 knl ¼ 2 maximum value ¼ 1.4 to 1.6 for gears of 7th degree of accuracy ¼ 1.1 to 1.2 when ﬂoating central wheels are used to equalize the load

CONDITIONS OF PROPER ASSEMBLY OF PLANETARY GEAR TRANSMISSION Two planetaries Both the driving pinion (sun pinion) and the planetaries may have either an even or an odd number of teeth.

Three planetaries If z1 (number of teeth on sun pinion) is divisible by 3, then z2 (number of teeth on planetary pinion) must also be divisible by 3. If z2 1 is divisible by 3, then z2 þ 1 must be divisible by 3. If z1 þ 1 is divisible by 3, then z2 1 must be divisible by 3.

Four planetaries If z1 is even, then z2 must be even. If z1 is odd, then z2 must be odd.

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MISCELLANEOUS MACHINE ELEMENTS

24.52

CHAPTER TWENTY-FOUR

Particular

Formula

WAVE-TYPE REDUCTION GEARS Transmission or gear ratio

i¼

z2 d2 ¼ z1 z2 d1 d2

ð24-150Þ

For a double-wave drive, z1 z2 ¼ 2. The necessary deformation

¼ d1 d2 ¼

The condition for obtaining the module for the drive

d1 d2 ¼ ðz1 z2 Þm ¼

ð24-152Þ

¼ 0:5

z1 z2

ð24-153Þ

The module of the drive from Eq. (24-152)

m¼

d2 i

ð24-151Þ

The tooth height

h¼

ð24-154Þ

The tooth addendum

ha ¼ 0:44

ð24-155Þ

The tooth dedendum

hf ¼ 0:56

ð24-156Þ

The rim width

b ¼ 0:1d2 to 0:2d2

ð24-157Þ

The total surface area of contact of teeth when onefourth of all teeth of wheel are engaged

Aw ¼ 0:5h 0:25z2 b

ð24-158Þ

The torque transmitted

Mt ¼ 0:5d2 Aw ca ’ 0:06d22 bz2 ca

ð24-159Þ

where ca ¼ 29:5 MPa (4.28 kpsi) for hardened steel wheels

VARIABLE-SPEED DRIVES (Figs. 24-34 and 24-37, and Table 24-12) For schematic arrangements of various variablespeed drives

Refer to Figs. 24-34 and 24-37.

The velocity control range for V-belt drive with only one adjustable pulley

D1 ¼

The relation between dmax and dmin of V-belt drive

dmax ¼ dmin þ 2ðe hÞ

ð24-161aÞ

dmax ¼ dmin þ b cot 2h

ð24-161bÞ

The velocity control range for V-belt drive from Eqs. (24-160) and (24-161)

D¼

dmax dmin

ð24-160Þ

dmax 2e 2h ¼1þ dmin dmin dmin

D¼1þ

b dmin

cot

2h dmin

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ð24-162aÞ ð24-162bÞ

MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.53

Formula

The velocity control range for V-belt drive when two pulleys are adjustable

D2 ¼ D1

ð24-163Þ

The total range of velocity control of variable-speed drive of two adjustable pulleys of V-belt drive

D ¼ D21

ð24-164Þ

The working height of the V-groove of the pulley

e>

b cot 2

b ’ 1:8h

The width of standard V-belt 6. V-belt

1. Frontal friction L H

L

H

L

L

L H

H

7. Chain

L

LH

LH

4. Ball

L

H

H

H

H

3. Toroidal friction

H

L

L

H 2. Bevel friction

L

H

H

H

L

L

8. Combination

5. Disk

x

H H

x

x

L

x

L

FIGURE 24-37 Variable-speed drives.

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ð24-165Þ ð24-166Þ

MISCELLANEOUS MACHINE ELEMENTS

24.54

CHAPTER TWENTY-FOUR

Particular

Formula

The larger ratio of width to height of specially proﬁled broad V-belts

b ’ 2 to 3 h

ð24-167Þ

The total velocity control range for adjustable pulleys of V-belt drive

D ¼ D41

ð24-168Þ

DISSIPATION OF HEAT IN REDUCTION GEAR DRIVES hðt1 ta Þ

The area of housing required for dissipating heat generated in a closed-type reduction gear drive operating in an oil bath at stable thermal equilibrium condition

A¼

The thermal equilibrium condition of reduction gear drive which has a housing of noncooled surface (ribbed surface) and cooled surface (cooled by blowing of air by fan)

(hn An þ hc Ac ) W (Btu/h)

ð24-170Þ

pﬃﬃﬃ hc ¼ 12 v W/m2 K (Btu/ft2 h 8R)

ð24-171Þ

The expression for coeﬃcient of heat transfer of the housing or reduction gear drive blown over by air The velocity of air which depends on impeller velocity

ð24-169Þ

where h ¼ coefficient of heat transfer, which varies from 8.75 to 17.5 W/m2 K (1.54 to 3.1 Btu/ ft2 h 8R)

where v ¼ velocity of air, m/s (ft/min) v ’ 0:005ni m/s (ft/min) ni ¼ impeller speed, rpm

For minimum weight equations for gear systems

Refer to Table 24-13.

For total weight equations for gear systems

Refer to Table 24-14.

For K factors for preliminary estimate of spur and helical gear size

Refer to Table 24-15.

For comparison of ﬁve gear systems

Refer to Table 24-16.

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24.55

TABLE 24-11 Transmission ratio (i), eﬃciency (), and allowable transmitted power (Pal ) for reduction gears Type of reduction gear

Fig. no.

Single- and triple-spur and helical reduction gear

24-35, serial nos. 3a, 4a 24-35, serial no. 1

Single-spur reduction gear Single worm Helical worm Harmonic drive Planetary reduction gear

24-36a 24-36b 24-36c

Wave-type toothed reduction gear

i

Pal , kW

10–60 8–10

8 15 20–100 100

108 100 100 0.97–0.99 0.97–0.99

1000 100

0.75–0.85

TABLE 24-12 Velocity control range (D), eﬃciency (), and allowable power transmitted (Pal ) for variable-speed drives

Particular

Type of drive

Frontal friction

Single Twin type Single Double Self-locking ring

Bevel friction

Toroidal friction Ball Disk drives

Serial no. in Fig. 24-37 1 2

3 4 5

V-belt drives

Solid disk

6

Chain drives

Grooved disk First type of drive Second type of drive

7

D 3–4 8–10 3–4 4–10 16 4–6 10–12 3 4–5 1.3– 1.7 2 6 7–10

Pal , kW 20 5

0.7–0.8 0.95

10 20

0.8–0.9

800 300 50

0.8–0.9

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30 75

MISCELLANEOUS MACHINE ELEMENTS

24.56

CHAPTER TWENTY-FOUR

MINIMUM AND TOTAL WEIGHT EQUATION FOR GEAR SYSTEMS The following symbols are used in Tables 24-13 to 24-16: a ¼ number of branches in an epicyclic gear: C ¼ ð2Mt =KÞ, m3 ; d ¼ pitch diameter, m (in); i ¼ gear speed ratio; io ¼ overall ratio; is ¼ dp =ds ¼ zp =zs ¼ speed ratio of planet gear to sun gear; j ¼ number of idlers; K ¼ a factor from Table 24-15; Mt ¼ input torque, N m (lbf in); ðio þ 1Þ=io ¼ i0o .

TABLE 24-13 Minimum weight equations for gear systems

TABLE 24-14 Total weight equations for gear systems

Particular

Equation

Particular

Equation

Simple train (oﬀset) Oﬀset with idler

2i3 þ i2 ¼ 1 2i3 þ i2 ¼ i2o þ 1

Oﬀset

1 ðbd2 =CÞ ¼ 1 þ þ i þ i2 i

Oﬀset with two idlers

2i3 þ i2 ¼

i2o þ 1 2

Oﬀset with idler

1 i2 ðbd2 =CÞ ¼ 1 þ þ i þ i2 þ o þ i2o i i

Oﬀset with j idlers

2i3 þ i2 ¼

i2o þ 1 j

Oﬀset with two idlers

ðbd2 =CÞ ¼

Double-reduction

2i3 þ

2i2 i2o þ 1 ¼ 0 io i0o

Doublereduction

1 i 2 i2 ðbd2 =CÞ ¼ 1 þ þ 2i þ i2 þ o þ o þ i2o i io 2i

Double-reduction, double branch

2i3 þ

2i2 i2o þ 1 ¼ 2i0o i0o

Double-reduction, four branch

2i3 þ

2i2 i2o þ 1 ¼ 4i0o i0o

Double-reduction, j branches

2i3 þ

Planetary (theoretical)

2i3s

Star (theoretical)

2i3s

2i2 i2o þ 1 ¼ ji0o i0o

þ

i2s

0:4ðio 1Þ2 þ 1 ¼ a

þ

i2s

0:4i2o þ 1 ¼ a

Doublereduction, double branch Doublereduction, fourbranch Planetary

1 1 i 2 i2 þ þ i þ i2 þ o þ o 2 2i 2i 2

ðbd2 =CÞ ¼

1 1 i2 i2 i 2 þ þ 2i þ i2 þ þ o þ o 2 2i io 2i 2

ðbd2 =CÞ ¼

1 1 i2 i2 i 2 þ þ 2i þ i2 þ þ o þ o io 4i 4 4 4i

ðbd2 =CÞ ¼

1 1 0:4ðio 1Þ2 þ is þ i2s þ þ a ais ais þ

Star

ðbd2 =CÞ ¼

0:4ðio 1Þ2 a

1 1 0:4i2o 0:4i2o þ is þ i2s þ þ þ a ais ais a

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

24.57

TABLE 24-15 K factors for preliminary estimate of spur and helical gear size

Particular Motor driving compressor Engine driving compressor Turbine driving generator Industrial drives

Large industrial gears such as hoists, kilns, and mills Aircraft, single pair Aircraft, planetary Automotive transmission Small commercial vehicles Small gadgets

K factor

Hardness HB ; pinion gear

Pitch line velocity, m/s

225–180 225–180 575–575 225–180 575–575 575–575 350–300 210–180 575–575 300–300 210–180 225–180

>20.5 >20.5

260–210 60RC –60RC 60RC –60RC 60RC –60RC 350; phenolic laminated nylon 200; zinc alloy die casting 200; brass or A1 Brass or A1

kgf/mm2

MN/m2

MPa

0.036 0.032–0.050 0.155–0.320 0.066–0.077 0.280–0.56 0.350–0.703 0.246–0.316 0.120–0.176 0.334–0.527 0.193–0.264 0.088–0.141 0.056–0.070

0.353 0.314–0.49 1.52–3.14 0.65–0.76 2.746–5.50 3.434–6.89 2.234–3.100 1.177–1.726 3.277–5.170 1.893–2.589 0.873–0.138 0.550–0.687

0.353 0.314–0.49 1.52–3.14 0.65–0.76 2.746–5.50 3.434–6.89 2.234–3.10 1.177–1.726 3.277–5.170 1.893–2.589 0.873–0.138 0.550–0.687

<5.1

0.091–0.120 0.703 (at take oﬀ) 0.492 (at take oﬀ) 1.055 0.52 0.035 0.018

0.893–1.177 6.89 4.82 10.35 5.10 0.343 0.176

0.893–1.177 6.89 4.82 10.35 5.10 0.343 0.176

<2.55 <2.55

0.018 0.016

0.176 0.157

0.176 0.157

>20.5 5.1 5.1 5.1 15.3 15.3 15.3 5.1 max

51 15.3–51 <5.1

TABLE 24-16 A comparison of ﬁve gear systems (all systems producing 0.746 kW at 18 rpm) Parameter

Epicyclic

Herringbone

Single worm

Helical worm

Harmonic drive

Speed ratio Safety factor Height, mm Length, mm Width, mm Cubic volume, m3 Weight, kgf Eﬃciency, % Number of gears Number of bearings Tooth-sliding velocity, m/s Pitch line velocity, m/s Tooth contact pressure, kgf/mm2 Tooth contact pressure, GPa Tooth in contact, % Tooth contact Quiet operation Balanced forces

97.4 3 330 381 330 0.0410 111.60 85 13 17 12.75 7.65 35 0.343 7 Line No Yes

96.2 2 356 508 254 0.0458 127.00 85 4 6 12.75 7.65 35 0.343 5 Line Yes No

108 2 580 483 356 0.1000 104.33 40 2 6 7.65 7.65 3.5 0.034 2 Line Yes No

100 2 406 432 254 0.0442 93.00 78 4 6 12.75 7.65 35 0.343 3 Line Yes No

100 36 152 152 152 0.003 13.61 82 2 2 0.143 0.092 0.425 0.0042 50 Surface Yes Yes

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MISCELLANEOUS MACHINE ELEMENTS

24.58

CHAPTER TWENTY-FOUR

24.6 FRICTION GEARING SYMBOLS2,3 a b d1 d2 F Fa Fr FR Ft h i n0 n P p0 vm R 0 !l , !2

1 , 2

center distance, m (in) dimensions as shown in Fig. 24-42 gear face width, m (in) diameter of smaller wheel, m (in) diameter of larger wheel, m (in) pressure on wheels, kN (lbf ) thrust, kN (lbf ) radial force on the grooved spur wheel for each groove, kN (lbf ) normal reaction between two bevel friction gears (Fig. 24-40), kN (lbf ) tangential force, kN (lbf ) depth of groove, m (in) number of grooves, m (in) speed, rps speed, rpm power transmitted, kW (hp) permissible pressure, kN/m (lbf/in) mean circumferential velocity, m/s (ft/min) cone distance, m (in) (Fig. 24-40) half the included angle of the groove, deg ranges from 128 to 188 (should not exceed 208) angle of friction, deg coeﬃcient of friction between wheels coeﬃcient of friction between shaft of wheel and bearings angular speeds of smaller and larger wheels, respectively, rad/s cone center angles of smaller and larger wheels, respectively, deg

Particular

Formula

SPUR FRICTION GEARS Plain spur friction wheels (Fig. 24-38) The radial pressure on the wheels

F ¼ bp0

ð24-172Þ

The tangential force due to radial pressure F

Ft ¼ bp0

ð24-173Þ

The power transmitted

P¼

Ft vm 1000

SI

ð24-174aÞ

where P in kW, Ft in N, and vm in m/s P¼

Ft vm 33,000

USCS

ð24-174bÞ

where P in hp, Ft in lbf, and vm in ft/min

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.59

Formula

Ft

ω2

ω1

d1 d2

FIGURE 24-38 Plain spur friction gears.

Ft vm Customary Metric ð24-174cÞ 75 where P in hpm , Ft in kgf, and vm in m/s

P¼

The gear face width

b¼

1000P p0 dn0

SI

ð24-175aÞ

where P in kW, p0 in N/m, n0 in rps, and b and d in m b¼

102 103 P p0 dn0

Customary Metric ð24-175bÞ

where P in kW, p0 in kgf/mm, n0 in rps, and b and d in mm b¼

33,000P p0 dn

USCS

ð24-175cÞ

where P in hp, p0 in lbf/in, n in rpm, and b in in and d in ft b¼

126,000P p0 dn

USCS ð24-175dÞ

where b and d in in and p0 in lbf/in, n in rpm, and P in hp

Grooved spur friction wheel (Fig. 24-39) The radial force on the wheel for each groove The total tangential force The power transmitted

Fr ¼ 2p0 hðtan þ Þ

ð24-176Þ

Ft ¼ 2ip0 h sec

ð24-177Þ

0

0

2ihp d1 n SI ð24-178aÞ 1000 cos where P in kW, p0 in N/m, n0 in rps, and h and d1 in mm

P¼

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MISCELLANEOUS MACHINE ELEMENTS

24.60

CHAPTER TWENTY-FOUR

Particular

Formula

Fr Fr

B d2

α

α δ

δ Fn C

Fn

h

Fr d1 A

Fr FIGURE 24-39 Grooved spur friction gears.

P¼

ihp0 d1 n 63,000 cos

USCS

ð24-178bÞ

where P in hp, p0 in lbf/in, n0 in rps, and h and d1 in in 2ihp0 d1 n Customary Metric ð24-178cÞ 4500 103 cos where P in hpm , p0 in kgf/mm, n in rpm, and h and d1 in mm

P¼

The empirical relation for the depth of the groove

h ¼ 0:006d1 þ 4 mm (0.15 in)

The recommended value for the mean circumferential velocity

vm 6 þ 0:08d1

ð24-179Þ SI

ð24-180aÞ

USCS

ð24-180bÞ

where vm in m/s and d1 in m vm 1200 þ 4d1 where v in ft/min

BEVEL FRICTION GEARS (Fig. 24-40) Starting The reaction is inclined from the normal by an angle of friction

F0R ¼

F01a F02a ¼ sinð 1 þ Þ cosð 1 þ Þ

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ð24-181Þ

MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.61

Formula

The tangential force transmitted

F0t ¼ F0R cos ¼

1000P vm

F0t ¼ F0R cos ¼

75P vm

SI

ð24-182aÞ

Customary Metric ð24-182bÞ

The least axial thrust on the small wheel F01a ¼

1000Pðsin 1 þ cos 1 Þ vm

F01a ¼

33,000Pðsin 1 þ cos 1 Þ vm

F02a ¼

1000Pðcos 1 sin 1 Þ vm

F02a ¼

33,000Pðcos 1 sin 1 Þ vm

FR ¼

F1a F ¼ 2a sin 1 cos 1

The least axial thrust on the big wheel

SI

ð24-183aÞ

USCS ð24-183bÞ SI

ð24-184aÞ

USCS ð24-184bÞ

Running

b

The reaction in this case is designated by FR bp0 (where p0 is the permissible unit pressure)

ω2

d2 F2a

δ2

FR

R

δ1

ρ

FR

ω1 d1

F1a

FIGURE 24-40 Bevel friction gears.

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ð24-185Þ

MISCELLANEOUS MACHINE ELEMENTS

24.62

CHAPTER TWENTY-FOUR

Particular

The tangential force transmitted

The least axial thrust on the small wheel

Formula

Ft ¼ FR ¼

1000P vm

Ft ¼ FR ¼

33,000P vm

1000P sin 1 vm 2 3 1000P d1 q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 4 5 vm d2 þ d2

SI

ð24-186aÞ

USCS

ð24-186bÞ

SI

ð24-187aÞ

USCS

ð24-187bÞ

SI

ð24-188aÞ

USCS

ð24-188bÞ

SI

ð24-189aÞ

F1a ¼

1

2

33,000P sin 1 vm 2 3 33,000 d1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 4 5P vm d2 þ d2

F1a ¼

1

The least axial thrust on the big wheel

2

1000P cos 1 vm 2 3 1000P d2 ¼ 4qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ5 vm d2 þ d2

F2a ¼

1

2

33,000P cos 1 vm 2 3 33,000 d2 q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 4 5P vm d2 þ d2

F2a ¼

1

2

DISK FRICTION GEARS (Fig. 24-41) The torque on the driving shaft

Mt ¼

1000P !

where Mt in N m, P in kW, and ! in rad/s Mt ¼

9550P n

SI

ð24-189bÞ

where Mt in N m, P in kW, and n in rpm Mt ¼

159P n0

SI

where Mt in N m, P in kW, and n0 in rps

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ð24-189cÞ

MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.63

Formula

Driven spur wheel Driving disc

FIGURE 24-41 Variable-speed disk friction gearing.

Mt ¼

716,000P n

Customary Metric ð24-189dÞ

where Mt in kgf mm, P in hpm , and n in rpm Mt ¼

63,000P n

USCS

ð24-189eÞ

where Mt in lbf in, P in hp, and n in rpm The tangential force acting on the driven wheel for the minimum speed at minimum diameter of driving disk

Ft1 ¼

1000P d1 n0

SI

ð24-190aÞ

where Ft1 in N, P in kW, d1 in m, and n0 in rps Ft1 ¼

33,000P d1 n

USCS ð24-190bÞ

where Ft1 in lbf, P in hp, d1 in ft, and n in rpm Ft1 ¼

102 103 P d1 n0

Customary Metric

ð24-190cÞ

where Ft1 in kgf, P in kW, d1 in mm, and n0 in rps The tangential force acting on the driven wheel for the maximum speed at maximum diameter of driving disk

Ft2 ¼

1000P d2 n0

SI

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ð24-191aÞ

MISCELLANEOUS MACHINE ELEMENTS

24.64

CHAPTER TWENTY-FOUR

Particular

Formula

Ft2 ¼

33,000P d2 n

USCS

ð24-191bÞ

Customary Metric

ð24-191cÞ

SI

ð24-192aÞ

USCS

ð24-192bÞ

where d2 in ft

The minimum thrust to be applied to the disk for the minimum speed

Ft2 ¼

102 103 P d2 n0

Fa1 ¼

Ft1 1000P ¼ d1 n0

Fa1 ¼

33,000P d1 n

where Fa1 ¼ bp0 (b ¼ face width of driven cylindrical wheel) and d2 in ft The maximum thrust to be applied to the disk for maximum speed

F2a ¼

Ft2 1000P ¼ d2 n0

F2a ¼

33,000P d2 n

SI

ð24-193aÞ

USCS

ð24-193bÞ

USCS

ð24-193cÞ

where d2 in ft F2a ¼

126,000P nd2

where d2 in in and F2a in lbf, n in rpm, and P in hp The axial thrust required to shift the driven wheel underload

Fa ¼ F1 ð þ 0 Þ

The eﬃciency

¼

ð24-194Þ

0

where is the coeﬃcient of friction between the shaft of driven wheel and its bearings d dþb

ð24-195Þ

where varies from 0.6 at low speeds when d ¼ d1 to 0.8 at high speeds, when d ¼ d2 The minimum force available on the chain sprocket at minimum speed of driven wheel

F1cs ¼

Ft1 d d3

ð24-196Þ

where d ¼ diameter of driven wheel, m (in) d3 ¼ diameter of chain sprocket, m (in) The maximum force available on the chain sprocket at maximum speed of driven wheel

F2cs ¼

Ft2 d d3

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ð24-197Þ

MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.65

Formula

BEARING LOADS OF FRICTION GEARING (Fig. 24-42, Table 24-17) Driven shaft ðL þ eÞFt ðe þ L þ cÞ

The horizontal force on bearing A due to the tangential force Ft

FhA ¼

The vertical force on bearing A due to thrust Fa and the force on the chain sprocket Fcs

FVA ¼

ð24-199Þ

FRA

ð24-200Þ

The resultant load on bearing A

ðL þ eÞFa þ eFcs ðe þ L þ cÞ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ F2hA þ F2VA

ð24-198Þ

The horizontal force on bearing B due to the tangential force Ft

FhB ¼

The vertical force on bearing B due to the thrust Fa and the force on the chain sprocket Fcs

FVB ¼

ð24-202Þ

FRB

ð24-203Þ

The resultant force on bearing B

cFt ðe þ L þ cÞ

cFa þ ðc þ LÞFcs ðe þ L þ cÞ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ F2hB þ F2VB

Bearing

D Driving shaft a Bearing Fa

C

Driving disc

b

Bearing

Driven spur wheel A

Driven shaft

Chain sprocket B

Bearing

Chain c

L

e

FIGURE 24-42 Bearing loads of disk friction gearing.

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ð24-201Þ

MISCELLANEOUS MACHINE ELEMENTS

24.66

CHAPTER TWENTY-FOUR

Particular

Formula

Driving shaft The horizontal force due to thrust Fa on bearing D

The horizontal force due to the tangential force Ft on the bearing D The resultant force on the bearing D The horizontal force due to thrust Fa on the bearing C The horizontal force due to the tangential force Ft on the bearing C The resultant force on the bearing C

d1 Ft ð24-204Þ 2a where d1 and d2 denote the minimum and maximum diameters of driving disk

FhDa ¼

bFt a qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ F2hDa þ F2hDt

FhDt ¼

ð24-205Þ

FRD

ð24-206Þ

Fhca ¼

d1 Ft 2a

ð24-207Þ

ða þ bÞFt a qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ F2hca þ F2hct

Fhct ¼

ð24-208Þ

FRc

ð24-209Þ

TABLE 24-17 Design data for friction gearing Allowable pressure, p0 Material of driver

kN/m

lbf/in

Coeﬃcient of friction with cast iron

Cast iron Cork composition Paper Rubber Wood

530 8.9 26.5 17.7 26.5

3000 50 150 100 150

0.15 0.21 0.15 0.20 0.15

Allowable pressure, p0 Material of driver

kN/m

Leather Leather ﬁber Straw ﬁber Sulﬁte ﬁber Tarred ﬁber

26.5 42.2 26.5 24.5 44.1

lbf/in

Coeﬃcient of friction with cast iron

Coeﬃcient of friction with aluminum

150 240 150 140 250

0.09 0.18 0.15 0.20 0.28

0.13 0.18 0.16 0.19 0.28

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

24.7 MECHANICS OF VEHICLES SYMBOLS2,3 a A b B c C Dt Dw Ef Esh F Fmax G h i k l l0 l1 l2 l3 l4 l5 l6 l7 l8 m Mt Mtt Mti Mto Mtf Mtr n n0 ni no P r rmi rmo

center distance, m (in) a constant in Eq. (24-216b) frontal projected area of vehicle, m2 (ft2 ) face width of gear, m (in) a constant in Eq. (24-216b) width of bearing, m (in) distance between adjacent rotating parts, m (in) constant (also with suﬃxes) maximum diameter of torus, m (in) diameter of wheel, m (in) ﬂow loss in each member of hydraulic torque converter, N m (lbf in) shock loss in each member of hydraulic torque converter, N m (lbf in) driving force at the tire, kN (lbf ) maximum permissible load on the pitch circle of any particular pair of gears, kN (lbf ) gradient thickness of housing, m (in) gear ratio (total) a constant distance between support bearings on a shaft in gearbox, m (in) distance between bearings of overhanging shaft, m (in) distance of rotating part from the bearing, m (in) distance of bearing from the wall, m (in) cap height from bolt to end, m (in) distance of rotating parts from the bearing cap, m (in) width of boss of rotating parts, m (in) distance of coupling to cap, m (in) distance between gear and shaft, m (in) distance of rotating parts from inner wall of housing, m (in) module, m (in) output torque of the engine, N m (lbf in) torque at the tire surface, N m (lbf in) the input torque, N m (lbf in) the reaction to the output torque, which is opposite in direction to output torque, N m (lbf in) the torque that must be applied to transmission housing to balance the moments of internal friction, oil churning, etc., N m (lbf in) the torque reaction of the transmission housing due to the gear reduction in transmission, N m (lbf in) speed, rpm speed, rps speed of driving shaft, rpm speed of driven shaft, rpm power, kW (hp) radius of the driving wheel, m (in) mean radius of inﬂow to the runner, m (in) mean radius of outﬂow from the runner, m (in)

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24.67

MISCELLANEOUS MACHINE ELEMENTS

24.68

CHAPTER TWENTY-FOUR

air resistance, kN (lbf ) rolling resistance, kN (lbf ) road resistance, kN/tf (lbf/ton) gradient resistance, kN (lbf ) total resistance, kN (lbf ) tonne, t tonne force, tf velocity, m/s (ft/min) speed of vehicle, km/h (ft/s) velocity of ﬂuid relative to the vane, m/s (ft/min) shock velocity, m/s (ft/min) weight of the vehicle, kN (Tonf ) number of teeth angle of inclination of road, deg angle of repose, deg minimum clearance between gears and inner wall of housing, m (in) transmission eﬃciency

Ra Rr R00r Rg Rt t tf v V Vf Vsh W z

SUFFIXES 1 2 b t max min

pinion gear brake tonne maximum minimum

Other factors in performance or in special aspects which are included from time to time in this section and, being applicable only in their immediate context, are not given at this stage. Particular

Formula

CALCULATION OF POWER 1000P SI ð24-210aÞ ! where P in kW, ! in rad/s, and Mt in N m

Torque

Mt ¼

9550P SI ð24-210bÞ n where P in kW, n in rpm, and Mt in N m

φ

Mt ¼ Rg

63,000P USCS ð24-210cÞ n where P in hp, n in rpm, and Mt in lbf in

Mt ¼

+

α

α

+

φ

w FIGURE 24-43 Forces on the vehicle moving up the gradient.

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.69

Formula

Mtt ¼ Mt

Torque at the tire surface

ð24-211Þ

where ¼ 0.90 at top gear ¼ 0.80 at other gears Mt r

The driving force at the tire

F¼

Tractive factor

ftr ¼

Mtt 1000W

ftr ¼

Mtt 2240W

Rg ¼

1000W sinð þ Þ cos

Rg ¼

2240W sinð þ Þ cos

Force required to pull the vehicle of weight W up the slope (Fig. 24-43)

ð24-212Þ SI

ð24-213aÞ

USCS ð24-213bÞ SI

ð24-214aÞ

USCS ð24-214bÞ

Gradient

G¼

W Rg

ð24-215Þ

The air resistance

Ra ¼ kAV2

ð24-216aÞ

where k ¼ constant obtained from Table 24-18 For values of air resistance at diﬀerent speeds of vehicle

Refer to Table 24-19.

The rolling resistance

Rr ¼ ða þ bVÞW

ð24-216bÞ

where a ¼ constant varies from 15 to 600 b ¼ constant varies from 0.1 to 3.5 For rolling or road resistance surfaces

R0r

for various road

The general formula for total resistance or tractive resistance (Fig. 24-44)

Refer to Table 24-20.

Rt ¼ kAV2 þ W

sinð þ Þ cos

þ ða þ bVÞW Another formula for total resistance

Rt ¼ Ra þ Rg þ Rr 1000 þ kAV2 ¼ W R0r þ G

ð24-217aÞ

ð24-217bÞ

where k and R0r are obtained from Tables 24-19 and 24-20 where R0r in N/tf, W in tf, A in m2 , V in m/s

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MISCELLANEOUS MACHINE ELEMENTS

24.70

CHAPTER TWENTY-FOUR

Particular

Formula

Rt ¼ Ra þ Rg þ Rr 2240 0 ¼ W Rr þ þ kAV2 G

USCS

ð24-217cÞ

where R0r in lbf/t, W ¼ weight of vehicle, tonf A ¼ projected frontal area of vehicle, ft2 V ¼ speed of vehicle, ft/s Tractive eﬀort at the tire surface

Ftr ¼

iMt r

ð24-218Þ

where i ¼ gear ratio obtained from Table 24-21 The speed of the vehicle

V ¼ 0:00297

nDw i

USCS

ð24-219aÞ

where V in mph (miles per hour), Dw in in, and n in rpm V ¼ 0:052 Power

AIR RESISTANCE Ra =

kAV2

ROLLING RESISTANCE Rr = (a+bv) W

nDw i

SI

ð24-219bÞ

where V in m/s, Dw in m, and n in rpm P¼

0:002VMtt Dw

SI

ð24-220aÞ

where V in m/s, Mtt in N m, Dw in m, and P in kW W SIN (α+φ) GRADIENT RESISTANCE Rg= COS φ

P¼

USCS

ð24-220bÞ

where V in mph, Mtt in lbf ft, Dw in in, and P in hp P¼

FIGURE 24-44 Various resistances on the moving vehicle.

0:00163VMtt Dw

5:5VMtt Dw

Customary Metric

ð24-220cÞ

where V in km/h, Mtt in kgf m, Dw in mm, and P in kW

TRANSMISSION GEARBOX (Fig. 24-45) The equation for center distance between main and countershafts for the case of three-speed passenger car

a ¼ 0:5

p 3 ﬃﬃﬃﬃﬃﬃﬃ Mt

USCS

ð24-221aÞ

SI

ð24-221bÞ

where a in in and Mt in lbf ft a ¼ 0:0106

p 3 ﬃﬃﬃﬃﬃﬃﬃ Mt

where a in m and Mt in N m

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.71

Formula

F

G

A Clutch I D H II

B

C

E FIGURE 24-45 A typical four-speed gearbox.

The distance between support bearings of shaft

l ¼ 0:0254

p p 3 ﬃﬃﬃﬃﬃﬃﬃ 3 ﬃﬃﬃﬃﬃﬃﬃ Mt to 0:0318 Mt

SI

ð24-222aÞ

where l in m and Mt in N m l ¼ 1:2

p p 3 ﬃﬃﬃﬃﬃﬃﬃ 3 ﬃﬃﬃﬃﬃﬃﬃ Mt to 1:5 Mt

USCS ð24-222bÞ

where l in in and Mt in lbf ft The maximum permissible load at the pitch circle of any pair of gears

Fmax ¼ c1 bm ¼

c1 b Pm

ð24-223Þ

where c1 ¼ constant obtained from Table 24-22 The face width of gear tooth

b¼

The expression for center distance for the case of fourspeed truck transmission

a ¼ 0:017

Fmax Fmax Pn ¼ mc1 c1 p 3 ﬃﬃﬃﬃﬃﬃﬃ Mt

SI

ð24-225aÞ

where a in m and Mt in N m a ¼ 0:8

The distance between support of bearings of shaft

ð24-224Þ

p 3 ﬃﬃﬃﬃﬃﬃﬃ Mt

USCS ð24-225bÞ

where a in in and Mt in lbf ft p p 3 ﬃﬃﬃﬃﬃﬃﬃ 3 ﬃﬃﬃﬃﬃﬃﬃ l ¼ 0:0254 Mt to 0:0318 Mt where l in m, and Mt in N m p p 3 ﬃﬃﬃﬃﬃﬃﬃ 3 ﬃﬃﬃﬃﬃﬃﬃ l ¼ 1:2 Mt to 1:5 Mt

SI

ð24-226aÞ

USCS ð24-226bÞ

where l in in and Mt in lbf ft The face width of gear tooth

b¼

Fmax Fmax Pn ¼ mc1 c1

For values of c1 , refer to Table 24-22.

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ð24-227Þ

MISCELLANEOUS MACHINE ELEMENTS

24.72

CHAPTER TWENTY-FOUR

Particular

The expression for center distance for the case of ﬁve-speed and reverse truck transmission

Formula

a ¼ 0:0170

p 3 ﬃﬃﬃﬃﬃﬃﬃ Mt

where a in m and Mt in N m p 3 ﬃﬃﬃﬃﬃﬃﬃ a ¼ 0:8 Mt The distance between support of bearings of shaft

where a in in and Mt in lbf ft p p 3 ﬃﬃﬃﬃﬃﬃﬃ 3 ﬃﬃﬃﬃﬃﬃﬃ l ¼ 0:0254 Mt to 0:0318 Mt where l in m and Mt in N m p p 3 ﬃﬃﬃﬃﬃﬃﬃ 3 ﬃﬃﬃﬃﬃﬃﬃ l ¼ 1:2 Mt to 1:5 Mt

The face width of gear tooth

The expression for center distance for a farm tractor transmission

Eﬀective face width of gear tooth

The eﬃciency of transmission

SI

ð24-228aÞ

USCS

ð24-228bÞ

SI

ð24-229aÞ

USCS

ð24-229bÞ

where l in in and Mt in lbf ft F F P b ¼ max ¼ max n mc1 c1

ð24-230Þ

For values of c1 , refer to Table 24-22. p 3 ﬃﬃﬃﬃﬃﬃﬃ SI a ¼ 0:021 Mt

ð24-231aÞ

where a in m and Mt in N m p 3 ﬃﬃﬃﬃﬃﬃﬃ a ¼ Mt

ð24-231bÞ

USCS

where a in in and Mt in lbf ft Fmax v SI ð24-232aÞ b¼ 28 105 m where b in m, Fmax in N, m in m, and v in m/s F vP b ¼ max n USCS ð24-232bÞ 8,000,000 where b in in, Fmax in lbf, Pn in in1 , and v in ft/min n M Mto ð24-233Þ ¼ o to ¼ ni Mti ir ½Mto ðMtr Mtf Þ where ir ¼ reduction ratio of transmission ¼ ðni =no Þ

Distance of rotating parts from the inner wall of housing

l8 ¼ 10 to 15 mm or more for high-power and heavyduty operation l8 ¼ 0:4 to 0:6 in

USCS

ð24-234Þ

Distance between adjacent rotating parts

c ¼ 10 to 15 mm (0.4 to 0.6 in)

ð24-235Þ

Minimum clearance between gears and inner wall of housing

1:2h

ð24-236Þ

Distance between bearings of overhanging shaft

l0 ¼ 1:2d to 3d

Distance of bearing from the wall

l2 ¼ 5 to 10 mm (0.2 to 0.4 in)

where h ¼ thickness of housing ð24-237Þ

where d ¼ diameter of shaft ð24-238Þ

Cap height from bolt end

l3 ¼ depends on the design by empirical formula

Distance of rotating parts from the bearing cap

l4 ¼ 15 to 20 mm (0.6 to 0.8 in)

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ð24-239Þ

MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.73

Formula

Width of boss of rotating part

l5 ¼ 1:2d to 1:5d

Distance of coupling to cap

(depends on the type of coupling)

Distance between gear and shaft

l7 20 mm (0.8 in)

ð24-241Þ

Distance of rotating part from the bearing

l1 ¼

B l þ l3 þ l4 þ 5 2 2 Refer to Chapter 24, Section 24.5.

ð24-242Þ

For planetary gear transmission For detail design equations of spur, helical, bevel, crossed-helical and worm gears

ð24-240Þ

Refer to Chapter 23.

HYDRAULIC COUPLING (Fig. 24-46) Torque transmitted by the coupling

Mt ¼ ksn2 Wðr2mo r2mi Þ

ð24-243Þ 7

where k ¼ coefficient ¼ 1:42 10 Percent slip between primary and secondary speeds

r3

where np and ns are primary and secondary speeds of impeller, respectively, rpm r1

C DA

ð24-244Þ

r2

r4

B

(approx.)

ðnp ns Þ 100 s¼ np

FIGURE 24-46 Hydraulic coupling.

The mean radius of the inner passage (Fig. 24-46) The mean radius of the outer passage (Fig. 24-46)

rmi ¼

2 3

rmo ¼

2 3

r32 r31 r22 r21 r34 r33 r24 r23

The expression for number of times the ﬂuid circulates through the torus in one second

if ¼

The torque capacity of hydraulic coupling at a given slip

Mt ¼ Kn2 D5t

ð24-245Þ ð24-246Þ

13,000Mt nWðr2mo r2mi Þ

ð24-247Þ ð24-248Þ

where K ¼ coeﬃcient varying from 0:166 108 to 0:244 108 ¼ 1.56 to 2.28

SI

USCS

Dt ¼ diameter of torus, m (ft) Mt ¼ torque capacity, N m (lbf ft); n in rpm

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MISCELLANEOUS MACHINE ELEMENTS

24.74

CHAPTER TWENTY-FOUR

Particular

Formula

HYDRODYNAMIC TORQUE CONVERTER (Fig. 24-47) Mti ¼ Kn2i D5t

The equation for input torque

ð24-249Þ

where Impeller

Runner

K ¼ coeﬃcient depending on design ni ¼ speed of input shaft, rpm Dt ¼ any linear dimension such as maximum diameter of impeller

Reactor

FIGURE 24-47 Hydrodynamic torque converter.

The equation for the input power

ð24-250Þ

P ¼ Cn3i D5t

where C ¼ coefficient depending on design The expression for ﬂow loss or friction loss in each member of the torque converter under any particular operating conditions in energy unit per kilogram of ﬂuid circulated

Ef ¼

Cf V2f 2g

ð24-251Þ

where Cf ¼ coeﬃcient whose value depends mainly on the Reynolds number and the relative smoothness of the metallic surface ¼ 0.445 to 0.890 SI (where Ef in N m and Vf in m/s) ¼ 0.328 to 0.656 USCS (where Vf in ft/s and Ef in lbf ft)

The expression for shock loss per kg ﬂuid circulated in the impeller of a torque converter

Esh ¼

Csh V2sh 2g

ð24-252Þ

where Csh ¼ coefficient The maximum inside diameter of torus

Dt ¼ 0:00135C

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 Mt =n02

SI

ð24-253aÞ

0

where Dt in m, Mt in N m, and n in rps Dt ¼ 0:00168C

rﬃﬃﬃﬃﬃﬃﬃ 3 Mt n2

USCS

ð24-253bÞ

where Dt in in, Mt in lbf in, and n in rpm C ¼ coeﬃcient ¼ 14 for a ratio of minimum inside diameter to maximum diameter of torus of one-third n ¼ speed in hundreds of rpm

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

Formula

TRACTIVE EFFORT CURVES FOR CARS, TRUCKS, AND CITY BUSES For ﬁnding the diameter of tire of vehicles for a particular wheel speed

Refer to Fig. 24-48.

For tractive eﬀort of a passenger car

Refer to Fig. 24-49.

For tractive eﬀort of trucks, tractors, and city buses

Refer to Fig. 24-50.

850 800 750 700

Pa ss

600

en

i Tra

ge

s,

rc

ler

ars

550

tr u ck

s, an

500

dc ity

Wheel speed, rpm

650

es ss bu

450 400 350 300 250

50

60

70

80 90 100 Tyre diameter, cm

110

120

130

FIGURE 24-48 Wheel speed vs. tire diameter of vehicles.

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24.75

MISCELLANEOUS MACHINE ELEMENTS

24.76

CHAPTER TWENTY-FOUR

Tractive effort, kgf

200

100

50

0

500

1000

1500 Gross vehicle weight, kgf

2000

2500

FIGURE 24-49 Tractive eﬀort curve for passenger cars (1 kgf ¼ 9.8066 N ¼ 2.2046 lbf ). 950 900 850 800 750 700 650

Tractive effort, kgf

600 550 500 450 400 350 300 250 200 150 100

2000 4000

6000

8000 10000 12000 14000 16000 18000 20000 Gross vehicle weight, kgf

FIGURE 24-50 Tractive eﬀort curve for trucks, tractors, and city buses.

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

24.77

TABLE 24-18 Values of k for use in Eq. (24-216) and (24-217) Particular

k in USCSUa

k in SI

Average automobile of modern design Streamlined racing car Truck or omnibus

0.0017

0.20

0.0006 0.0024

0.07 0.28

a

US Customary System units.

TABLE 24-19A Air resistancea

TABLE 24-19B Air resistance in SI units

Speed of vehicle, mph

Velocity of wind, V, ft/s

0.0024V2

0.0017V2

0.0006V2

Speed of vehicle km/h

Velocity of wind V, m/h

0.28V2

0.20V2

0.07V2

10 20 30 40 50 60 90 150

14.67 29.35 44.00 58.60 73.30 88.00 132.00 220.00

0.516 2.060 4.650 8.240 12.900 18.600 — —

0.366 1.460 3.300 5.830 9.130 13.160 29.650 —

— 0.516 1.160 2.060 3.220 4.650 10.450 29.000

10 20 30 40 50 60 70 80 90 100

2.78 5.56 8.34 11.12 13.90 16.68 19.46 22.21 25.02 27.80

2.17 8.68 19.50 34.72 54.25 78.00 106.40 139.00 176.00 217.00

1.55 6.20 13.92 24.80 38.80 57.60 76.00 99.20 125.60 155.00

0.54 2.17 4.88 8.68 13.56 19.50 26.60 34.75 44.00 54.25

a

Values given in this table are in US Customary System units.

TABLE 24-20 Road resistance, R0r Solid Surface

N/tf

Polished marble Concrete Asphalt Stone Good quality Poor quality Vitriﬁed bricks Good macadam (metal road) Good Fair Rough Clay Sand

lbf/ton

Pneumatic %

N/tf

lbf/ton

%

29.3 62.4 67.4

12.12 14.25 15.40

0.541 0.636 0.687

35.3 41.5 448.2

8.08 9.50 10.25

0.360 0.423 0.457

71.8 153.2 85.0

16.40 35.00 19.45

0.732 1.562 0.866

477.6 102.0 56.7

10.92 23.30 12.95

0.487 1.040 0.578

146.2 220.0 307.0 438.4 1314.0

33.50 50.00 70.00 100.00 300.00

1.491 2.240 3.130 4.470 13.400

97.9 186.4 389.3 389.3 874.8

22.20 33.20 46.60 66.60 200.00

0.998 1.900 2.100 3.970 8.920

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SI 106 USCSU

SI 106 USCSU

SI 106

USCSU

III

I

SI 106

USCSU

IV SI 106

USCSU

V

Low speed reduction

Three-speed passenger 124.6–145.2 18,000–21,000 145.2–153.5 21,000–24,000 193.7–221.2 28,000–32,000 car transmission Four-speed truck 76.0–90.0 11,000–13,000 128.0–149.0 18,500–21,500 96.8–110.9 14,000–16,000 175.2–207.5 26,000–30,000 transmission Five-speed reverse 76.0–90.0 11,000–13,000 138.2–152.0 20,000–22,000 105.0–117.7 15,000–17,000 90.0–105.0 13,000–15,000 175.2–207.5 26,000–30,000 truck transmission

No. of speeds and type of transmission

Intermediate speed reduction

Gear wheels belonging to

High speed reduction

TABLE 24-22 Value of coeﬃcient c1 in SI and USCSU

II

4 : 1 to 5 : 1 1:6 : 1 (total 6 or 7 : 1) 2:5 : 1 (total 10 : 1) Same as or higher than low gear ratio <75% above the propeller shaft

Final drive (rear-axle diﬀerential) Second Low

Reverse gear Overdrive

Ratio

Particular

TABLE 24-21 Gear ratios

MISCELLANEOUS MACHINE ELEMENTS

24.78

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

24.79

24.8 INTERMITTENT-MOTION MECHANISMS SYMBOLS2,3 a b d dh e1 , e2 Fn Fnr Ft h m Mb Mt n0 n p p r s1 s2 s02 z Z

¼ tan ¼ ¼ tan1 ¼ ’ b ¼ m b

distance of the pawl pivot point, m (in) face width of ratchet tooth, m (in) diameter (also with suﬃxes), m (in) hub diameter, m (in) dimensions as shown in Fig. 24-52, m (in) normal force through O, Figs. 24-51 and 24-52, kN (lbf ) peripheral force normal to the tooth of ratchet, kN (lbf ) tangential force at diameter, d, kN (lbf ) tooth height or distance from the critical section to the line of action of the load Fnr , m (in) module, m (in) bending moment, N m (lbf in) twisting moment, N m (lbf in) speed, rps speed, rpm tooth pitch, m (in) linear unit pressure, N/m (lbf/ft) radius, m (in) dimension as shown in Fig. 24-53 breadth of tooth land (Fig. 24-53), m (in) thickness of tooth at base (Fig. 24-52), m (in) number of teeth on ratchet wheel section modulus, m3 (cm3 ) (in3 ) pressure angle or angle of the pawl force, deg angle at pawl (¼ 908 8), deg coeﬃcient of friction friction angle, deg ratchet tooth angle or pitch angle, deg varies from 1.5 to 3 stress, MPa (psi) bending stress, MPa (psi) shear stress, MPa (psi)

Particular

Formula

PAWL AND RATCHET The ratchet tooth angle (Fig. 24-51)

The ratchet diameter (Fig. 24-51)

’¼

2 ; rad z

ð24-254aÞ

’¼

360 ; deg z

ð24-254bÞ

pz

ð24-255Þ

d2 ¼ mz ¼

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MISCELLANEOUS MACHINE ELEMENTS

24.80

CHAPTER TWENTY-FOUR

Particular

Formula

The face width of ratchet tooth

b

The allowable unit pressure or force

F ¼

Fn b

ð24-257Þ

Ft ¼

2Mt d

ð24-258Þ

The normal force through O (Fig. 24-51)

Fn ¼

Ft cos

The normal force through O (Fig. 24-52)

Fn ¼ Ft

The bending stress

b ¼

Allowable bending stress (b )

Refer to Table 24-23.

Number of teeth

z ¼ 6 to 30

ð24-261Þ

Module

m > 6 (mostly from 10 to 20)

ð24-262Þ

The ratio of h=m

h=m ¼ 0:6 to 1

ð24-263Þ

For ratchet wheel deﬁnitions and dimensions

Refer to Fig. 24-53.

The ratio of s2 =m

s2 =m ¼ 0:6 to 0.9

ð24-264Þ

The tooth height

h ¼ 5 to 15 for toothed ratchet

ð24-265Þ

The tangential force

Fnr

ð24-256Þ

ð24-259aÞ ð24-259bÞ

Mb 6Ft h ¼ 2 ba Z bs1

ð24-260Þ

e1

µFnr O

s2

Fnr = Ft

e2

µFnr

F nr

s2

α

d

2

p

h2

Fn=Ft α

h2

O

Fn Fn

d2

p

dh

FIGURE 24-51 Ratchet wheel with radial tooth ﬂanks and pawl.

FIGURE 24-52 Ratchet wheel with non-radial tooth ﬂanks and pawl.

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.81

Formula

p

h

1

s2 r1

s1 h

2

2

d

2

r2

FIGURE 24-53 Deﬁnitions and dimensions of ratchet wheel.

For external ratchet

¼ 148 to 178

ð24-266Þ

For internal ratchet

¼ 178 to 308

ð24-267Þ

The ratio of a=d (internal ratchet)

a=d ¼ 0:35 to 0:43 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 Mt m¼2 z ba

ð24-268Þ

The module

ð24-269Þ

The bending moment on pawl

Mb1 ¼ Fn e2

ð24-270Þ

The bending stress

b ¼

6Mb1 Fn þ ba bs1 bs21 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ b 3 Fn þ th d1 ¼ 2:71 2ba 2

ð24-271Þ

The diameter of pawl pin

where th ¼ thickness of hub on pawl TABLE 24-23 F Material

kN/m

Cast iron 49–98 Steel or 98–196 cast steel Hardened 196–392 steel

b

lbf/in 280–560 560–1120 1120–2240

MPa

kpsi

19.5–29.5 2.85–4.27 39–68.5 5.69–10.0 58.8–98

8.54–14.23

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ð24-272Þ

MISCELLANEOUS MACHINE ELEMENTS

24.82

CHAPTER TWENTY-FOUR

24.9 GENEVA MECHANISM SYMBOLS2,3 r1 sin

a¼ F1 F2

F2ðmaxÞ FðmaxÞ FiðmaxÞ

i¼

z2 z

k M1t M2t M2ti M2t n0 n P r1 r2 r02 ra2 rp r2 r1

m

gear ratio

radius ratio total time required for a full revolution of the driver or crank, s time required for indexing Geneva wheel, s time during which Geneva wheel is at rest, s velocity, m/s number of slots on the Geneva wheel crank angle or angle of driver at any instant, deg (Fig. 24-54) angular acceleration, m/s2 (ft/s2 ) angular acceleration of Geneva wheel, m/s2 (ft/s2 ) angular position of the crank or driver radius at which the product ! 2a is maximum, deg angle of the driven wheel or Geneva wheel at any instant, deg (Fig. 24-54)

t ti tr v z 2a

¼

the component of force acting on the crank or the driving shaft due to the torque, M1t , kN (lbf ) (Fig. 24-57) the component of force acting on the driven Geneva wheel shaft due to the torque M2t , kN (lbf ) (Fig. 24-57) maximum force (pressure) at the point of contact between the roller pin and slotted Geneva wheel, kN (lbf ) the component of maximum friction force at the point of contact due to the friction torque M2t , on the driven Geneva wheel shaft, kN (lbf ) the component of maximum inertia force at the point of contact due to the inertia torque on the driven Geneva wheel shaft, kN (lbf )

polar moment of inertia of all the masses of parts attached to Geneva wheel shaft, m4 (in4 ) the working time coeﬃcient of the Geneva wheel total torque on the driver or crank, N m (lbf in) total torque on the driven or Geneva wheel, N m (lbf in) inertia torque on the Geneva wheel, N m (lbf in) friction or resistance torque on Geneva wheel, N m (lbf in) speed, rps speed, rpm power, kW (hp) radius to center of driving pin, m (in) radius of Geneva wheel, m (in) distance of center of semicircular end of slot from the center of Geneva wheel, m (in) outside radius of Geneva wheel, which includes correction for ﬁnite pin diameter, m (in) pin radius, m (in)

J

Rr ¼

center distance, m (in)

r1 a

the ratio of the driver radius to center distance eﬃciency of Geneva mechanism

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

24.83

locking angle of driver or crank, rad or deg ratio of time of motion of Geneva wheel to time for one revolution of driver or crank

¼

360 2z

semi-indexing or Geneva wheel angle, or half the angle subtended by an adjacent slot, deg (Fig. 24-54) crank or driver angle, deg (Fig. 24-54)

2n !¼ 60 !1 , !2

a1

angular velocity of driver or crank (assumed constant), rad/s angular velocities of driver or crank and Geneva wheel, respectively, rad/s φ

β

r2 ω2

a

a2 ω1 α a3

ψ

r1

FIGURE 24-54 Design of Geneva mechanism. Particular

Formula

The angular velocity (constant) of driver or crank

!1 ¼

Gear ratio

i¼

angle moved by crank or driver during rotation angle moved by Geneva wheel during rotation

i¼

z2 z

2n 60

360 2z

The semi-indexing angle or Geneva wheel angle or half the angle subtended by two adjacent slots

¼

The angle through which the Geneva wheel rotates

2 ¼

360 z

ð24-273Þ

ð24-274Þ or or

z

ð24-275Þ

2 z

ð24-276Þ

EXTERNAL GENEVA WHEEL The angle of rotation of driver through which the Geneva wheel is at rest or angle of locking action (Fig. 24-55) The crank or driver angle

¼ 2ð Þ ¼ þ 2 ¼

¼

ðz þ 2Þ z

ðz 2Þ ¼ 2 2z

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ð24-277Þ

ð24-278Þ

MISCELLANEOUS MACHINE ELEMENTS

24.84

CHAPTER TWENTY-FOUR

Particular

ro

2

Formula

r’2

a

φ ω2

ψ

”

90 ω1

r1

FIGURE 24-55 External Geneva mechanism.

DISPLACEMENT r1 sin

The center distance (Fig. 24-55)

a¼

The radius ratio

Rr ¼

The ratio of crank radius to center distance

¼

The relation between crank angle and Geneva wheel angle

r2 ¼ cot r1

r1 ¼ sin ¼ sin a z sin

¼ tan1 1 cos

ð24-279Þ ð24-280Þ ð24-281Þ ð24-282Þ

VELOCITY The angular velocity of the Geneva wheel

!2 ¼

d ðcos Þ ! ¼ dt 1 2 cos þ 2 1

sinð=zÞðcos sin =zÞ !1 1 2 sinð=zÞ cos þ sin2 =z d !2ðmaxÞ ¼ ¼ ! dt max 1 1 1 sin !1 ¼ sin z z

!2 ¼ The maximum angular velocity of Geneva wheel at angle ¼ 0

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ð24-283aÞ ð24-283bÞ

ð24-283cÞ

MISCELLANEOUS MACHINE ELEMENTS

24.85

MISCELLANEOUS MACHINE ELEMENTS

Particular

Formula

ACCELERATION The angular acceleration, a 2a , of Geneva wheel

d2 ð 3 Þ sin ¼ !21 2 dt ð1 þ 2 2 cos Þ2

a

2a ¼

a

2a ¼

sinð=zÞ cos2 ð=zÞ sin !21 1 2 sinð=zÞ cos þ sin2 ð=zÞ

cos ðmaxÞ ¼ þ

The maximum angular acceleration of Geneva wheel which occurs at ¼ ðmaxÞ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 þ 2

ð 2a Þi; f ¼

sinð=zÞ cos3 ð=zÞ !21 ½1 2 sin2 ð=zÞ þ sin2 ð=zÞ

¼ !21 tan ¼ !21 tan =z r ¼ !21 1 r2 Total time required for a full revolution of the crank or driver

α2a 20 ω12

t¼

60 n

ð24-285Þ ð24-286Þ

6

ω2 ω1

4

10

2

0

0 α α2a ω12

ω2 ω1

10 20 30

a

ð24-284cÞ

where 1 1 ¼ þ 4

The angular acceleration of Geneva wheel at start and ﬁnish of indexing

α2a ω12

ð24-284bÞ

Refer to Fig. 24-56.

For angular velocity and angular acceleration curves for three-slot external Geneva wheel with driver velocity, !1 ¼ 1 rad/s

30

ð24-284aÞ

FIGURE 24-56 Angular velocity and angular acceleration curves for three-slot external Geneva wheel.

2a is the symbol used for angular acceleration of Geneva wheel; is the crank or driver angle at any given instant.

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MISCELLANEOUS MACHINE ELEMENTS

24.86

CHAPTER TWENTY-FOUR

Particular

Formula

The ratio of ti to t

ti 2 z2 ¼ ¼ ¼ t 2 2z

ð24-287Þ

The ratio of tr to t

tr 2ð Þ zþ2 ¼ ¼1 ¼ t 2 2z

ð24-288Þ

The sum of angles of ( þ )

þ

ð24-289Þ

The time required for indexing Geneva wheel, in seconds The time during which Geneva wheel is at rest, in seconds The working time coeﬃcient of Geneva wheel Ratio of time of motion of Geneva wheel to time for one revolution of crank or driver The required speed of the driver shaft or crankshaft

¼ 908

z2 z2 t¼ 2z z z þ 2 60 tr ¼ z 2n

ti ¼

60 2n

ð24-290Þ ð24-291Þ

ti z 2 ¼ tr z þ 2 z2 1 v¼ < 2z 2 z þ 2 60 n¼ z 2tr

ð24-292Þ

k¼

ð24-293Þ ð24-294Þ

where n in rpm

SHOCK OR JERK The jerk or shock, J2 , on Geneva wheel

The jerk or shock at ¼ 0 The jerk or shock at start, i.e., ¼

The length of the slot (Fig. 24-54)

The condition to be satisﬁed by diameter on which the driver or crank is mounted

J2 ¼

ð 1Þ½2 cos2 þ ð1 þ 2 Þ cos 4 þ !31 ð1 þ 2 2 cos Þ3 ð24-295Þ

ðJ2 Þ ¼ 0 ¼ ðJ2 Þ ¼ ¼

d3 dt3 d3 dt3

¼0

¼

¼

ð þ 1Þ 3 !1 ð 1Þ3

¼

3 2 !31 1 2

a2 ¼ r1 þ r2 a ¼ a sin þ cos 1 z z d1 < 2a3 ¼ 2ða r2 Þ ¼ 2a 1 cos z

ð24-296Þ ð24-297Þ ð24-298Þ ð24-299Þ

or

The condition to be satisﬁed by the diameter on which Geneva wheel is mounted

d1 < 2 1 cos ¼ 4 sin2 a z 2z d2 ¼ 4 sin2 < 2 1 sin z 4 2z a

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ð24-300Þ ð24-301Þ

MISCELLANEOUS MACHINE ELEMENTS

24.87

MISCELLANEOUS MACHINE ELEMENTS

Particular

Formula

TORQUE ACTING ON SHAFTS OF GENEVA WHEEL AND DRIVER The total torque acting on Geneva wheel shaft

M2t ¼ M2t þ M2ti ¼ M2t þ J 2a

ð24-302Þ

It is assumed that M2t is constant. The torque on the shaft of crank or driver

The eﬃciency of Geneva mechanism

M1t ¼ M2t

!2 1 ! 1 ¼ ðM2t þ J 2a Þ 2 !1 !1

ð24-303Þ

¼ 0.80 to 0.90 when Geneva wheel shaft is mounted on journal bearings (24-304a) ¼ 0.95 when drive shaft is mounted on rolling contact bearings

(24-304b)

¼ 0.75 when the diameter of bearing surface is larger than the outside diameter of Geneva wheel

(24-304c)

INSTANTANEOUS POWER The instantaneous power on the crank or driving shaft

P¼

Mt ! 1000

SI

ð24-305aÞ

where P in kW, Mt in N m, and ! in rad/s Mt ! Customary Metric ð24-305bÞ 102 103 where P in kW, Mt in kgf mm, and ! in rad/s

P¼

Mt ! Customary Metric ð24-305cÞ 75 103 where P in hpm , Mt in kgf mm, and ! in rad/s

P¼

P¼

Mt n 63,000

USCS ð24-305dÞ

where P in hp, Mt in lbf in, and n in rpm

Calculation of average power The average torque MtiðavÞ for complete cycle

MtiðavÞ ¼ 0

The average torque for ﬁrst half-cycle

MtðavÞ ¼ MðavÞ " 2 # 2 zJ 1 M2t þ ¼ !21 z2 2 1

ð24-306Þ

ð24-307Þ

where J ¼ polar moment of inertia, m4 , cm4 (in4 )

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MISCELLANEOUS MACHINE ELEMENTS

24.88

CHAPTER TWENTY-FOUR

Particular

The average power required on the crank or driving shaft

Formula

MtðavÞ ! SI ð24-308aÞ 1000 where Pav in kW, MtðavÞ in N m, and ! in rad/s

Pav ¼

MtðavÞ ! Customary Metric ð24-308bÞ 75 103 where Pav in hpm , MtðavÞ in kgf mm, and ! in rad/s

Pav ¼

Pav ¼

MtðavÞ n 63,000

USCS

ð24-308cÞ

where Pav in hp, MtðavÞ in lbf in, and n in rpm

Calculation of maximum power The maximum torque on the driven shaft of Geneva wheel

M2tðmaxÞ ¼ M2t þ M2tiðmaxÞ where M2t is constant M2tiðmaxÞ ¼ J 2aðmaxÞ ¼

The maximum torque on the driving shaft of the crank ω2

r2

β

a

M1tðmaxÞ ’

ω1

M2t

2n 60

2

ð24-310aÞ

1 1

þ

r1

α

J 1 ð ! Þ !1 2a 2 max

2 ð1 2 Þðcos m Þ sin m ð1 2 cos m þ 2 Þ3 1 J!21 ð24-310bÞ

β+α β+α

F1

J 2aðmaxÞ !21

1 !2ðmaxÞ M1tðmaxÞ ¼ M2t !1 þ

F2

ð24-309Þ

where ¼ m at which Mt1 is maximum

FIGURE 24-57 Forces acting on Geneva wheel.

The maximum power required on the shaft of the crank or driver

M1tðmaxÞ ! SI ð24-311aÞ 1000 where P1ðmaxÞ in kW, M1tðmaxÞ in N m, and ! in rad/s

P1ðmaxÞ ¼

M1tðmaxÞ ! Customary Metric ð24-311bÞ 102 103 where P1ðmaxÞ in kW, M1tðmaxÞ in kgf mm, and ! in rad/s

P1ðmaxÞ ¼

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.89

Formula

P1ðmaxÞ ¼

M1tðmaxÞ n 63,000

USCS

ð24-311cÞ

where P1ðmaxÞ in hp, M1tðmaxÞ in lbf in, and n in rpm

FORCES AT THE POINT OF CONTACT (Fig. 24-57) The maximum force at the point of contact between the roller pin and slotted Geneva wheel

F2ðmaxÞ ¼

M2t r2

ð24-312aÞ

F2ðmaxÞ ¼ FðmaxÞ þ FiðmaxÞ

ð24-312bÞ

where qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r2 ¼ a2 2ar1 cos þ r21 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r ¼ 1 1 2 cos þ 2 The component of maximum friction force at the point of contact due to the friction torque M2t on the driven Geneva wheel shaft

F2ðmaxÞ ¼

M2t M2t ¼ r2ðminÞ r1 1

where r2ðminÞ ¼ a r1 ¼

ð24-313Þ

1 1 r 1

For maximum values of F2i

Refer to Table 24-24.

For design data for external Geneva mechanism

Refer to Table 24-25A.

INTERNAL GENEVA WHEEL The time required for indexing Geneva wheel, s

The time during which Geneva wheel is at rest, s

zþ2 ti ¼ z tr ¼

z2 z

60 2n 60 2n

ð24-314Þ ð24-315Þ

The ti =t ratio

ti z þ 2 ¼ t 2z

ð24-316Þ

The tr =t ratio

tr z 2 ¼ t 2z

ð24-317Þ

The working time coeﬃcient of Geneva wheel

k¼

The relationship between crank or driver angle and Geneva wheel angle

zþ2 >1 z2 sin

¼ tan1 1 þ cos

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ð24-318Þ ð24-319Þ

MISCELLANEOUS MACHINE ELEMENTS

24.90

CHAPTER TWENTY-FOUR

Particular

Formula

The angular velocity of Geneva wheel

!2 ¼

The maximum angular velocity of Geneva wheel

d ¼ dt

!2ðmaxÞ ¼

The angular acceleration, 2a , of Geneva wheel

2a ¼

ðcos þ Þ !1 1 þ 2 cos þ 2

ð24-320Þ

! 1þ 1

ð24-321Þ

d2 ð1 2 Þ sin ¼ !21 2 dt ð1 þ 2 cos þ 2 Þ2

ð24-322Þ

For values of 2a at start and ﬁnish of indexing

Use Eq. (24-285) of external Geneva wheel.

For curves of angular velocity and angular acceleration of internal Geneva wheel

Refer to Fig. 24-58.

The contact forces between the slotted wheel and the pin on the driving crank of the internal Geneva wheel are calculated in a manner similar to that for the external Geneva wheel

0.8 0.6 ω2 ω1 0.4

Materials

1.00

α2a ω12

α 0

3

F2iðmaxÞ

0

α α2a ω12

TABLE 24-24 Maximum F2i values

Jn2 r1

α2a ω12

0.25

0

Chromium steel 40 Cr 1 hardened and tempered to Rc 45 to 55 is used for the sides of slotted Geneva wheel.

0.50

0.2

Chromium steel 15 Cr65 case-hardened to Rc 58 to 65 is used for the roller pin on the driver or crank.

z

0.75

ω2 ω1

4

5

6

8

! 1.966 0.126 0.0318 0.0131 0.00424 FIGURE 24-58 Angular velocity and angular acceleration for four-slot internal Geneva wheel.

TABLE 24-25A Design data for external Geneva mechanism

z

3 4 6 8 10

608 458 308 228300 188

308 458 608 678300 728

i

r1 =a

r2 =a

Rr

r02 =a

!2ðmaxÞ

2aðinitialÞ ¼

ðmaxÞ

Jmax

J ¼ 0

0.5 1 2 3 4

0.886 0.707 0.500 0.383 0.309

0.500 0.707 0.866 0.924 0.951

0.577 1.000 1.732 2.414 3.078

0.134 0.293 0.500 0.617 0.690

3008 2708 2408 2258 2168

0.167 0.250 0.333 0.375 0.400

6.46 2.41 1.00 0.620 0.447

1.732 1.000 0.577 0.414 0.325

48460 118240 228540 318380 388300

31.44 5.41 1.35 0.699 0.465

672 48 6 2.25 1.24

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

24.91

24.10 UNIVERSAL JOINT SYMBOLS2,3 diameter, m (in) shock factor correction factor to be applied to torque to be transmitted correction factor to be applied to power to be transmitted length (also with subscripts), m (in) life, h torque to be transmitted by universal joint, N m (lbf in) design torque, N m (lbf in) speed, rpm speed, rps power to be transmitted by universal joint, kW (hp) design power, kW (hp) angle between two intersecting shafts 1 and 2, deg angle of rotation of the driver shaft 1, deg angle of rotation of the driven shaft 2, deg angular velocities of driver and driven shafts respectively, rad/s

d Ks Kct Kct l L Mt Mtd n n0 P Pd

!1 , !2

Particular

Formula

SINGLE UNIVERSAL JOINT (Figs. 24-59 and 24-61a) The relation between , , and

tan cos

ð24-323Þ

!2 cos ¼ !1 1 sin2 sin2

ð24-324Þ

tan ¼

The relation between the angular velocities of driving shaft 1 or driver (!1 ) to the driven shaft 2 or the follower (!2 )

β R

ω2

r cos β β r P

d1 ω1

1

d2 β O

2

Q 3

S

FIGURE 24-59 A single universal joint.

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MISCELLANEOUS MACHINE ELEMENTS

24.92

CHAPTER TWENTY-FOUR

Particular

The maximum value of !2 =!1

The minimum value of !2 =!1

Formula

!2 !1

¼ max

cos 1 ¼ 1 sin2 cos

ð24-325Þ

when sin ¼ þ1, i.e., ¼ 908, 2708, or =2 or 3=2, etc. !2 ¼ cos ð24-326Þ !1 min when sin ¼ 0, i.e., ¼ 0, , 2, etc.

The angular acceleration of the driven shaft 2, if !1 is constant The value of for which the angular acceleration of the driven shaft is maximum

d 2 d!2 cos sin2 sin 2 2 ¼ !1 ¼ 2 dt dt ð1 sin2 sin2 Þ2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cos 2ðmaxÞ ¼ 2 þ 2

ð24-327Þ ð24-328Þ

where ¼ ð2 sin Þ=2 sin 2

2

The angular acceleration of driven shaft is maximum when is approximately equal to 458, 1358, etc., when the arms of cross are inclined at 458 to the plane containing the axes of the two shafts. The power transmitted by universal joint

P ¼ Mt !=1000

SI

ð24-329aÞ

where P in kW, Mt in N m, and ! in rad/s P ¼ Mt n=63,000

USCS

ð24-329bÞ

where P in hp, Mt in lbf in, and n in rpm The design torque of universal joint

Mtd ¼ Mt Ks Kct

ð24-330Þ

The design power of universal joint

Pd ¼

P KCN

ð24-331Þ

For calculation of torque and power transmitted by universal joint for various angles of inclination

Refer to Figs. 24-62 to 24-65.

For design data of universal joint

Refer to Tables 24-25B and 24-25C.

DOUBLE UNIVERSAL JOINT (Figs. 24-60 and 24-61b) The angular velocities ratio for a double universal joint which will produce a uniform velocity ratio at all times between the input and output ends

!1 ¼1 !2

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ð24-332Þ

MISCELLANEOUS MACHINE ELEMENTS 2

1

ω1 (driver) ω2 (driven)

β

3

β

Intermediate shaft (a)

1

ω1 (driver) 3

β

β

(b)

ω2 (driven)

2

Yoke 1 or fork 1 β1

Yoke 2 or fork 2 1 2

3

β2

(a)

2

β

β

3 β

1

2

3

1

(b)

β

FIGURE 24-60 Double universal joints. 90

L3 2 L4

0.5 × 45 L2

(Refer to Table 25.25B) (b) Double universal joint di = 10 to 50

D × 45 L1

L3

L1

do

L1

L1 =

di

D × 45

do

di

0.5 × 45

L1 =

L3 L2

L2

(a) Single universal joint di = 6 to 50

FIGURE 24-61 Dimensions of universal joints.

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MISCELLANEOUS MACHINE ELEMENTS CHAPTER TWENTY-FOUR

Correction factor, KCN

Correction factor, Kct

24.94

2.5 2.2 2 1.8 1.6 1.4 1.2 1

10

15 20 25 30 35 40 Angle of inclination (β), deg

45

FIGURE 24-62 Angle between two intersecting shafts vs. correction factor (Kct ).

50×90 40×75

20

25×50 32×50

15 10 8 6 5 4

20×40 25×40

3

32×63 40×63

176.5 Nm

98.10 78.45 58.84 49.03 39.23 29.42

16×25 20×32 12×25 16×25

19.61 10×20 12×20

9.81 7.85 5.88 4.90 3.92 2.94

d1×d2

0.2 0.1

1 2 3 5 10 20 50 30

40

45

FIGURE 24-63 Angle between two intersecting shafts vs. correction factor (KCN ).

196.1

2 1 0.8 0.6 0.5 0.4 0.3

20 25 30 35 Angle of inclination (β), deg

981.0 784.5 588.4 490.3 392.3 294.2

120× 104 rpm

Design torque, (Mtd), kgf m

18 kgf m

15

Design torque, (Mtd), N m

100 80 60 50 40 30

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10

1.96 0.98 200 500 2000 5000

4 100 300 1000 3000 10000×10

Speed × life (nL), rpm h

FIGURE 24-64 Design curves for single universal joint with needle bearings for ¼ 108.

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MISCELLANEOUS MACHINE ELEMENTS

kg f

4 5 6 8 10 3 2

40 ×7 5

6× 16

0.1 0.08 0.06 0.05 0.04 0.03

0. 2

1 16 2× 10 ×2 25 1 × 5 8 2× 20 10 ×1 20 ×1 6 6

0.2

0. 1

1 0.8 0.6 0.5 0.4 0.3

0. 0. 0. 0. 0. 1 3 4 5 6 8

32 2 3 5× 40 ×6 25 20 2×5 50 ×63 3 1 × ×4 0 20 6×3 40 0 ×3 2 2

Design power (pd), kW

2

d i ×d o

0.02 0.01

17 .5

50 ×9 0

kW 10 8 5.7 kW 6 5 4 3

10 80 0 60 50 40 30 20

kg f 30 m 0 20 0

m

MISCELLANEOUS MACHINE ELEMENTS

325 rpm

10

200 300 400600 1000 20 30 40 60 100 Speed, (n), rpm

FIGURE 24-65(a) Design curves for single universal joint with plain bearings for ¼ 108.

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24.95

MISCELLANEOUS MACHINE ELEMENTS

TABLE 24-25B Dimensions of universal joint (Fig. 24-61) Maximum allowable rotational play Test torque di H7

do k11

6 8 10

16 20

12 25 16 32 20 40 25 50 32 63 40 50

75 90

L2

L3 þ1

9 11 15 13 18 15 22 19 25 23 32 29 40 36 50 44 54

34 40 52 48 62 56 74 68 86 82 108 105 132 130 166 160 190

L4 1

z

0.196

0.02

45

0.392

0.04

40

0.981

0.10

32

0.667

0.17

28

3.334

0.34

25

5.296

0.54

20

14.710

1.5

18

0.12

21.575 27.458

2.2 2.8

16 14

0.15

Nm

Tolerance on coaxiality of the two bores

74 0.5 88 86 104 124 128 156 160 188 198 238 245 290

0.06

1

m

0.09

USE OF CURVES IN FIGS. 24-62 TO 24-65 do

di

Worked example 1 A single universal joint has to transmit a torque of 10 kgf m at 1500 rpm. The angle between intersecting shafts is 258. The joint is subjected to a minor shock. The shock factor ðKs Þ is 1.5. Design a universal joint with needle bearings for a life of 800 h.

dp

FIGURE 24-65(b) Taper pin joint. The length of the taper pin should conform to diameter do in Table 24-25B.

TABLE 24-25C Dimensions of taper pina (Fig. 24-65b) di

dp

m

6 8 10 12 16 20 25 32 40 50

2 3 4 5 6 8 10 12 14 16

4.5 5 6 7.5 9 11 15 18 22 27

The shear stress of taper pin ¼ ¼ 158:5 to 247.5 MPa (16 to 25 kgf/mm2 ).

a

kgf m

Angular rotational play at an angle of inclination of deg in minutes

SOLUTION From Fig. 24-62 correction factor for

¼ 258 is Kct ¼ 1:2. Design torque ¼ Mtd ¼ Mt Ks Kct ¼ 10 1:5 1:2 ¼ 18 kgf m (176.5 N m). Speed life ¼ nL ¼ 1500 800 ¼ 120 104 rpm h. From Fig. 24-64 for Mtd ¼ 18 kgf m (176.5 N m) and nL ¼ 120 104 rpm h, the size of a single universal joint is ðdi do Þ 40 75 mm.

Worked example 2 Design a single universal joint with plain bearings to transmit 2 kW power at 325 rpm. The angle between two intersecting shafts is 27.58. SOLUTION From Fig. 24-63 correction factor for ¼ 27:58 is KCN ¼ 0:35. Design power ¼ Pd ¼ ðP=KCN Þ ¼ ð2=0:35Þ ¼ 5:7 kW. From Fig. 24-65a the size of a single universal joint for Pd ¼ 5:7 kW and speed ¼ n ¼ 325 rpm is ðdi do Þ 40 75 mm. The permissible torque for this size of joint (Fig. 24-65a) is 17.5 kgf m (171.5 N m).

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

24.11 UNSYMMETRICAL BENDING AND TORSION OF NONCIRCULAR CROSS-SECTION MACHINE ELEMENTS SYMBOLS2,3 semimajor axis of elliptical section, m (in) width of rectangular section, m (in) (in2) A area of cross section, m2 (in) b semi-minor axis of elliptical section, m (in) height of rectangular section, m (in) c distance of the plane from neutral axis, m (in) thickness of narrow rectangular cross section (Fig. 24-68) e the distance from a point in the shear center S (Table 24-26) E Young’s modulus, GPa (MPsi) G modulus of rigidity, GPa (MPsi) I moment of inertia, area (also with suﬃxes), m4 (cm4 ) (in4 ) Iu , Iv moment of inertia of cross-sectional area, respectively, m4 (cm4 ) (in4 ) Jk polar moment of inertia, m4 (cm4 ) (in4 ) k 1 , k2 constants from Table 24-28 for use in Eqs. (24-343) and (24344) L length, m (in) Mb bending moment, N m (lbf ft) Mt twisting moment, N m (lbf ft) Mbu ¼ Mb cos bending moment about the U principal centroidal axis or any axis parallel thereto Mbv ¼ Mb sin bending moment about the V principal centroidal axis or any axis parallel thereto q ¼ t shear ﬂow Q the ﬁrst moment of the section, m4 (cm4 ) (in4 ) S the length of the center of the ring section of the thin tube, m (in) t width of cross section at the plane in which it is desired to ﬁnd the shear stress, m (in) thickness of the wall of the thin-walled section, m u, v coordinates of any point in the section with reference to principal centroidal axes V shear force on the cross section, kN (lbf) Vy resultant shear force acting at the shear center, kN (lbf) x the distance of the section considered from the ﬁxed end (Fig. 24-73) x, y coordinates in x and y directions b bending stress (also with suﬃxes), MPa (psi) shear stress (also with suﬃxes), MPa (psi)

variable thickness of thin tube wall (Fig. 24-70), m (in) angle measured from the V principal centroidal axis, deg angle of twist, deg a

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24.97

MISCELLANEOUS MACHINE ELEMENTS

24.98

CHAPTER TWENTY-FOUR

Particular

Formula

SHEAR CENTER The shear stress at any point in transverse plane or section of a member

¼

The ﬂexural stress in a thin-walled open section

b ¼

Shear ﬂow

q ¼ t ¼

For the equations for locating the shear centers of various thin open sections

Refer to Table 24-26.

VQ It

ð24-333Þ

Mb c I VQ I

ð24-334Þ ð24-335Þ

UNSYMMETRICAL BENDING The ﬂexural stress in case of sections subjected to unsymmetrical bending

b ¼ ¼

Flexural modulus for any cross section on which the stress is desired

ðMb cos Þv ðMb sin Þu þ Iu Iv Mbu v Mbv u þ Iu Iv

Z ¼ Iu Iv =ðvIv cos þ uIu sin Þ

ð24-336Þ ð24-337Þ

TORSION Solid sections ELLIPTICAL CROSS SECTION Shear stress acting in the x direction on the xz plane (Fig. 24-66)

xz ¼

Shear stress acting in the y direction on the yz plane (Fig. 24-66)

yz ¼

Maximum shear stress on the periphery at the extremities of the minor axis (Fig. 24-66 and Table 24-27)

max ¼

2Mt ab2

ð24-340Þ

Minimum shear stress on the periphery at the extremities of the major axis

min ¼

2Mt a2 b

ð24-341Þ

Angle of twist (Fig. 24-66)

¼

2Mt y ab3 2Mt x a3 b

Mt ða2 þ b2 Þ L G a3 b3

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ð24-338Þ ð24-339Þ

ð24-342Þ

MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.99

Formula

lRECTANGULAR CROSS SECTION The maximum shear stress at point A on the boundary, close to the center (Fig. 24-67 and Table 24-27)

A ¼

ð24-343Þ

where k1 depends on ratio a=b (Refer to Table 24-28.) ¼

Angle of twist (Table 24-27)

Mt k1 ab2

Mt L k2 ab3 G

ð24-344Þ

where k2 depends on ratio a=b (Refer to Table 24-28.)

y

Mt

y

τyz

A τxz

y

x

τmax

b

x

x

b

a

a

FIGURE 24-66 FIGURE 24-67

NARROW RECTANGULAR CROSS SECTIONS (Fig. 24-68) Equation for twisting moment (Fig. 24-68)

Mt ¼ 13 Gc3 b

ð24-345Þ

Equation for angle of twist

¼

3Mt Gc3 b

ð24-346Þ

The maximum shear stress

max ¼

3Mt bc2

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ð24-347Þ

MISCELLANEOUS MACHINE ELEMENTS

24.100

CHAPTER TWENTY-FOUR

TABLE 24-26 Location of shear center for various cross sections Location of shear center tf

b t s w

tw 2 x

c

3b2 tf htw þ 6btf

Location of shear center

b2

b1

s

c

e¼

h

h

for b2 < b1

x

e

vy

3ðb21 b22 Þ ðtw =tf Þh þ 6ðb1 þ b2 Þ

tf

e

e¼

Section

tf

Section

tf

y

c

s e

x

where b1 ¼ b t h2 ¼ h t

h1 h 2

y

vy

b

h h 1 2

t

c

e y b

x

e¼

x

e

b c

t2

y

x

vy

t1

c

s

bt1 h31 t1 h31 þ t2 h32

x

e

b b 3 2 b1 b1 e ¼ b1 3 2 pﬃﬃﬃ b b b 2 þ3 3 þ1 b1 b1 b1

b

y

L = Length of Dotted line

2 b þ h1 b bh m ¼ 12 þ 6 þ 12 21 þ6 a a a 2 3 h1 h1 b þ 3 4 a a a

vy y

vy

m n

x h1

a

s

c s h2

1þ

where b1 ¼ b t=2 h2 ¼ h þ t

h1

vy

9 > 1 b1 4 h1 2 > > = 2 h1 3 h2 e ¼ b1 > 1 h2 b1 h 2h > > > > þ þ2 1 1þ 1 > :1 þ ; h2 3h2 6 h1 h1

h1 h 2

s

8 > > > <

t

b

e¼

b 1

h1 h 2

t

b

e

h1

9 > 1 b1 4 h1 2 > > = 2 h1 3 h2 e ¼ b1 > > 1 h b 2h 2h > > 2 > þ 1 1 1 1 > :1 þ ; h2 3h2 6 h1 h1 1þ

t

8 > > > <

b

b t

y

b

vy

h

th s

c

y

e

vy

x tb

e¼

2A h þ Lðth =tb Þ

where A ¼ area

2 b þ h1 h h n ¼ 3 þ 12 3þ 1 þ4 1 a a a a s

c

x α

vy

e

C is at the centroid of triangle e ¼ 0:47a for narrow triangle ð > 128Þ approx.

y

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

24.101

TABLE 24-26 (Cont.) Location of shear center for various cross sections Section φ φ

C e

a

vy

y

Section

Location of shear center b1

2a½ð Þ cos þ sin ½ð Þ þ sin cos 4a For ¼ ; e ¼ 2

e¼

x t

e1 ¼ c

t

s

Location of shear center

½3Ixy ðh cÞ þ Ix ð2b1 3dÞ

t C

e2 ¼ d þ

c

s

x

vx

e1

h

b21 ht 6ðIy Ix I2xy Þ

b21 ht 6ðIy Ix I2xy Þ

t

½Ixy ð2b1 3dÞ þ 3Iy ðh cÞ y

b ex ¼ 2

tw

ey tf

b

s

vx

h 2

þ tf

e2

where

d

vy

y

c¼

b2

h2 þ 2b1 h b21 þ b22 ;d¼ 2hðb1 þ b2 Þ 2hðb1 þ b2 Þ

Ix ; Iy ¼ moment of inertia of section about x and y axes, respectively Ixy ¼ product of moment of inertia

ht3w ht3w

b3

x

y b

e

y

c

vy

tf c h

ey ¼

ht3w ht3w þ tf b3

tw s vy

x

8 9 > < 1 > = 1 3 h tf e ¼ ðtf þ bÞ > > 2 1 þ : ; t3w b

l c

s

vy

e

y

t

ex

h

x

dx

e¼

1 þ 3v 1þv

ð xt3 dx ð t3 dx

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MISCELLANEOUS MACHINE ELEMENTS

24.102

CHAPTER TWENTY-FOUR

TABLE 24-27 Approximate formulas for torsional shearing stress and angle of twist for various cross sections

2b C

2a

C

Angle of twist per unit length , rad/in (rad/m)

Shearing stress, lbf/in2 (N/m2 or MPa)

Cross section

b

b

c ¼

2Mt ab2

¼

¼

2Mt Ab

¼

42 JMt A4 G

¼

20Mt b3

¼

46:2Mt Gb4

c ¼

Mt k1 ab2

¼

Mt k2 ab3 G

¼

Mt 2r2

¼

Mt 2r3 G

¼

3Mt 2r 2

¼

3Mt 2r 3 G

¼

Mt 2ab

¼

Mt

c ¼

Mt 2ab 1

¼

D ¼

Mt 2ab

a

r

Mt ða2 þ b2 Þ Ga3 b3

δ

r

δ C

δ

2b

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2ða2 þ b2 Þ 4a2 b2 G

2a δ1

C

D

b

δ a

Mt ða þ b 1 Þ 2

1 a2 b2 G

A ¼ area of cross section

TABLE 24-28 Variation of k1 and k2 with the ratio a=b for use in Eqs. (24-343) and (24-344) a=b

1

1.2

1.5

2.0

2.5

3.0

4.0

5.0

10.0

1

k1 k2

0.208 0.141

0.219 0.166

0.231 0.196

0.246 0.229

0.258 0.249

0.267 0.263

0.282 0.281

0.391 0.291

0.312 0.312

0.333 0.333

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MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.103

Formula

COMPOSITE SECTIONS Cross Sections Composed of Narrow Rectangles Equation for torque of a narrow rectangular cross section

Mt ¼

k2 bc3 G L

ð24-348Þ

Equation for torque of a narrow rectangular section (b=c ! 1, k2 ¼ 13)

Mt ¼

G bc3 3L

ð24-349Þ

Equation for torque for a cross-section composed of several narrow rectangles (Table 24-27)

Mt ¼

G k2 bc3 L

ð24-350Þ

Angle of twist for a cross section composed of several narrow rectangles

¼

Maximum shear stress

3Mt L Gbc3

max ¼

ð24-351Þ

3Mt cmax k for 2 ¼ 1 k1 bc3

ð24-352Þ

where cmax ¼ maximum thickness of the narrow section For approximate formulas for torsional shearing stress and angle of twist for various cross sections

Refer to Table 24-27.

For variation of stress-concentration factor K with ratio r=c for structural angle (Fig. 24-69)

Refer to Table 24-29.

y

y B

c

A

A B z

0

z

Section on BB

b

z 0 Section on AA

x

x FIGURE 24-68

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MISCELLANEOUS MACHINE ELEMENTS

24.104

CHAPTER TWENTY-FOUR

Particular

Formula

TABLE 24-29 Stress concentration factors for structural angle, K (Fig. 24-69) 0.125 2.550

r=c K

0.250 2.250

0.500 2.000

0.750 1.875

c

1.000 1.800

r c FIGURE 24-69

HOLLOW THIN-WALLED TUBES (Fig. 24-70) The equation for the twisting moment

Mt ¼ qð2AÞ ¼ 2A

ð24-353Þ

where A ¼ area enclosed by the median line of the tubular section Mt S 4A2 G

ð24-354Þ

ds ¼ 2GA

ð24-355Þ

The angle of twist

¼

Þ By membrane analogy the value of ds

þ

The equation for the shear stress

¼

The diﬀerence in level between DC and AB of membrane

h ¼

The equation for twisting moment of thin webbed tubes (or box beams) (Fig. 24-71)

Mt ¼ 2ðA1 h1 þ A2 h2 Þ

ð24-358aÞ

Mt ¼ 2ðA1 1 1 þ A2 2 2 Þ

ð24-358bÞ

The equations for shear stress S 0

x

δ

y z

D 0

ð24-356Þ ð24-357Þ

1 ¼ Mt

3 S2 A1 þ 2 S3 ðA1 þ A2 Þ R

ð24-359aÞ

2 ¼ Mt

3 S1 A2 þ 1 S3 ðA1 þ A2 Þ R

ð24-359bÞ

3 ¼ Mt

1 S3 A1 2 S1 A2 R

ð24-359cÞ

where

C

h A

Mt

2A

B

x

R ¼ 2½ 1 3 S2 A21 þ 2 3 S1 A22 þ 1 2 S3 ðA1 þ A2 Þ2 FIGURE 24-70

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ð24-360Þ

MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.105

Formula

S1

S2 δ2

A2

S3

τ3

δ1

A1

δ3

τ1

τ2 C

B

D

t1

v

h1 δ1

h2

h2

A

E

tw

F

Mt

c

x h 2

h τ3 τ2

v

τ1

b y

FIGURE 24-71

FIGURE 24-72

BENDING STRESSES CAUSED BY TORSION Torsion of I-beam having one section restrained from warping The lateral bending moment in the ﬂanges of an Ibeam subjected to twisting moment at one end, the other end being ﬁxed, Fig. 24-72 The maximum bending moment for long beam

Twisting moment at any section, distance x from the ﬁxed end The angle of twist per unit length

The total angle of twist at the free end Maximum bending moment

Mb ¼

Mt sin hðL xÞ=k k h cos hðL=kÞ

MbðmaxÞ ¼

Mt k h

as

tan h

L ¼1 k

where k ¼ ðh=2Þ½EI=JG 1=2 cos hðL xÞ=k Mtx ¼ Mt 1 cos hðL=kÞ M cos hðL xÞ=k u ¼ t 1 JG cos hðL=kÞ M L ¼ t L k tan h k JG MbðmaxÞ ¼

Mt L k tan h h k

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ð24-361Þ

ð24-362Þ

ð24-363Þ ð24-364Þ ð24-365Þ ð24-366Þ

MISCELLANEOUS MACHINE ELEMENTS

24.106

CHAPTER TWENTY-FOUR

Particular

Formula

The angle of twist at free end if l=k > 2:5

¼

The maximum bending moment if l=k > 2:5

Mt ðL kÞ JG

ð24-367Þ

Mt k h

ð24-368Þ

MbðmaxÞ ¼

The bending stress

b ¼

MbðmaxÞ b 2If

where MbðmaxÞ obtained from Table 24-30 Refer to Table 24-30.

For beams subjected to torsion

TRANSVERSE LOAD ON BEAM OF CHANNEL SECTION NOT THROUGH SHEAR CENTER (Fig. 24-73) σ1 = Mh 2I σ2 = vl b 2If 6Fel = ht b2

b

l

b

l

v t

v

(b)

L

h

v F e

t1

v tw Mt

h 2

y

vy

x

h

Shear centre

s

y b (c)

(a)

FIGURE 24-73

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ð24-369Þ

MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS

Particular

24.107

Formula

The direct stress

t ¼

The bending stress

b ¼

The maximum longitudinal stress (Fig. 24-73b)

¼

For geometrical properties, weight, and nominal dimensions of beams, channels, T-bars, and equal and unequal angles

Refer to Tables 24-31 and 24-32 and Figs. 24-74 to 24-79.

Mh I 2 where M ¼ Fe VIb=2 6Fel ¼ I htb2

Mh 6Fel þ 2I htb2

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ð24-370Þ

ð24-371Þ ð24-372Þ

MISCELLANEOUS MACHINE ELEMENTS

24.108

CHAPTER TWENTY-FOUR

TABLE 24-30 Formulas for maximum lateral bending moment and angle of twist of beams subjected to torsiona Maximum lateral bending moment in ﬂange, lbf in (N m)

Type of loading and support

MbðmaxÞ ¼

Mt

Mt

¼ if

L Mt = wLe w

Mt = wLe w

¼

Mt k h

L L

Mt k h

¼

Mt k h

L1

L2 if

cot h

L 2k 2k L

¼

Mt 2JG

¼

Mt 2JG

if cot h

L k k L

L1 L2 Mt k sin h k sin h k ¼ h L sin h k Mt k ¼ 2h

L1 L and 2 > 2 k k

¼

Mt JG

¼

Mt JG

L k 4

b

L L k tan h 2 2k L k 2

b

L > 2:5 2k

¼

1 Mt 2 JG

¼

1 Mt 2 JG

if

L L k tan h 4 4k

L > 2:5 4k

if

Mt ðL 2kÞb JG

L > 2:5 2k

L is large k

MbðmaxÞ

Mt

L is large 2k

MbðmaxÞ ¼

if

¼ if

Mt k 2h

e

Mt L L 2k tan h 2k JG

Mt k h

MbðmaxÞ ¼

if

¼

L > 2:5 2k

e

L

Mt k L tan h h 2k

Angle of twist of beam of length L, rad

L L k tan h ðapprox:Þ 2 2k

L k 2

b

L > 2:5 2k

error is small a b

Formulas given in Table 24-30 can also be used for Z and channel sections. Error is small for the conditions L=2k > 2:5 and L=4k > 2:5.

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150 175 200 225 75 100 125 150 175 200 225 250 275 300 325 350 400 450 500 550 600 100 125 150 175 200 225 250 300 350 400 450 500 550 600 150 175 200 225

Beam or column section (See Fig. 24-74)

ISJB 150 ISJB 175 ISJB 200 ISJB 225 ISLB 75 ISLB 100 ISLB 125 ISLB 150 ISLB 175 ISLB 200 ISLB 225 ISLB 250 y ISLB 275 b ISLB 300 tf ISLB 325 D ISLB 350 exx !SLB 400 tw b - tw ISLB 450 4 h ISLB 500 x x ISLB 550 ISLB 600 ISMB 100 ISMB 125 r1 r2 ISMB 150 ISMB 175 eyy y ISMB 200 ISMB 225 FIGURE 24-74 ISMB 250 Beam or column ISMB 300 section. ISMB 350 ISMB 400 ISMB 450 ISMB 500 ISMB 550 ISMB 600 ISWB 150 ISWB 175 ISWB 200 ISWB 225

mm

Section 50 50 60 80 50 50 75 80 90 100 100 125 140 150 165 165 165 170 180 190 210 75 75 80 90 100 110 125 140 140 140 150 180 190 210 100 125 140 150

mm

(3)

(1)

(2)

Width of ﬂange, b

Depth of Designation beam, h

3.0 3.2 3.4 3.7 3.7 4.0 4.4 4.8 5.1 5.4 5.8 6.1 6.4 6.7 7.0 7.4 8.0 8.6 9.2 9.9 10.5 4.0 4.4 4.8 5.5 5.7 6.5 6.9 7.5 8.1 8.9 9.4 10.2 11.2 12.0 5.4 5.8 6.1 6.4

mm

(4)

Thickness of web, tw

4.6 4.3 5.0 5.0 5.0 6.4 6.5 6.8 6.9 7.3 8.6 8.2 8.8 9.4 9.8 11.4 12.5 13.4 14.1 15.0 15.5 7.2 7.6 7.6 8.6 10.8 11.8 12.5 12.4 14.2 16.0 17.4 17.2 19.3 20.8 7.0 7.4 9.0 9.9

mm

(5)

Thickness of ﬂange, tf

91.5 91.5 91.5 91.5 91.5 91.5 91.5 91.5 91.5 91.5 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 96 96 96 96

deg

(6)

Slope of ﬂange, D

Dimensions of the section

5.0 5.0 5.0 6.5 6.5 7.0 8.0 9.5 9.5 9.5 12.0 13.0 14.0 15.0 16.0 16.0 16.0 16.0 17.0 18.0 20.0 9.0 9.0 9.0 10.0 11.0 12.0 12.0 14.0 14.0 14.0 15.0 17.0 18.0 20.0 8.0 8.0 9.0 9.0

mm

(7)

Radius at root, r1 ðrr Þ

1.5 1.5 1.5 1.5 2.0 3.0 3.0 3.0 3.0 3.0 6.0 6.5 7.0 7.5 8.0 8.0 8.0 8.0 8.5 9.0 10.0 4.5 4.5 4.5 5.0 5.5 6.0 6.5 7.0 7.0 7.0 7.5 8.5 9.0 10.0 4.0 4.0 4.5 4.5

mm

(8)

Radius at toe, r2 ðrt Þ

cm

(9)

Center of gravity, Cyy

TABLE 24-31 Geometrical properties, weight, and nominal dimensions of beams, channels, and T-bars

9.01 10.28 12.64 16.28 7.71 10.21 15.12 18.08 21.30 25.27 29.92 35.53 42.02 48.08 54.90 63.01 72.43 83.14 95.50 109.97 126.69 14.60 16.60 19.00 24.62 32.33 39.72 47.55 56.26 66.71 78.46 92.27 110.74 132.11 156.21 21.67 28.11 36.71 43.24

cm

2

(10)

Sectional area, A

7.1 8.1 9.9 12.8 6.1 8.0 11.9 14.2 16.7 19.8 23.5 27.9 33.0 37.7 43.1 49.5 56.9 65.3 75.0 86.3 99.5 11.5 12.0 14.9 19.3 25.4 31.2 37.3 44.2 52.4 61.6 72.4 86.9 103.7 122.6 17.0 22.1 28.8 33.9

kg

(11)

Weight per meter, w

322.1 479.3 780.7 1308.5 72.7 168.0 406.8 688.2 1096.2 1696.6 2501.9 3917.8 5375.3 7332.9 9874.6 13158.3 19306.3 27536.1 38579.0 53161.6 72867.6 257.5 449.0 726.4 1272.0 2235.4 3441.8 5131.6 8603.6 13630.3 20450.4 30390.8 45218.3 64893.6 91813.0 839.1 1509.4 2624.5 3920.5

cm

4

(12)

Ixx

9.2 9.7 17.3 40.5 10.0 12.7 43.4 55.2 79.6 115.4 112.7 193.4 287.0 376.2 510.8 631.9 716.4 853.0 1063.9 1335.1 1821.9 40.8 43.7 52.6 85.0 150.0 218.3 334.5 453.9 537.7 622.1 834.0 1369.8 1833.8 2651.0 94.8 188.6 328.8 448.6

cm

4

(13)

Iyy

Moments of inertia

5.98 6.83 7.86 8.97 3.07 4.06 5.19 6.17 7.17 8.19 9.15 10.23 11.31 12.35 13.41 14.45 16.33 18.20 20.10 21.99 23.98 4.20 5.20 6.18 7.19 8.32 9.31 10.39 12.37 14.29 16.15 18.15 20.21 22.16 24.24 6.22 7.33 8.46 9.52

cm

(14)

rxx

1.01 0.97 1.17 1.58 1.14 1.12 1.69 1.75 1.93 2.13 1.94 2.33 2.61 2.80 3.05 3.17 3.15 3.20 3.34 3.48 3.79 1.67 1.62 1.66 1.86 2.15 2.34 2.65 2.84 2.84 2.82 3.01 3.52 3.73 4.12 2.09 2.59 2.99 3.22

cm

(15)

ryy

Radius of gyration

42.9 54.8 78.1 116.3 19.4 33.6 65.1 91.8 125.3 169.7 222.4 297.4 392.4 488.9 607.7 751.9 965.3 1223.8 1543.2 1933.2 2428.9 51.5 71.8 96.9 145.4 223.5 305.9 410.5 573.6 778.9 1022.9 1350.7 1808.7 2359.8 3060.4 111.9 172.5 262.5 348.5

cm

3

(16)

Zxxx

3.7 3.9 5.8 10.1 4.0 5.1 11.6 13.8 17.7 23.1 22.5 30.9 41.0 50.2 61.9 76.6 86.8 100.4 118.2 140.5 173.5 10.9 11.7 13.1 18.9 30.0 39.7 53.5 64.8 76.8 88.9 112.2 152.2 193.0 252.5 19.0 30.2 47.0 59.8

cm3

(17)

Zyy

Section moduli

MISCELLANEOUS MACHINE ELEMENTS

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24.109

h

r1

cyy eyy y

r2

ISJC 100 ISJC 125 ISJC 150 ISJC 175 ISJC 200 ISLC 75 ISLC 100 ISLC 125 ISLC 150 ISLC 175 ISLC 200 ISLC 225 ISLC 250 ISLC 300 ISLC 350 ISLC 400 ISMC 75 ISMC 100 ISMC 125 ISMC 150 ISMC 175 ISMC 200

ISWB 250 ISWB 300 ISWB 350 ISWB 400 ISWB 450 ISWB 500 y b ISWB 550 ISWB 600 tf D ISHB 150 ISHB 200 tw ISHB 225 b - tw ISHB 250 2 x ISHB 300 ISHB 350 ISHB 400 ISHB 450

FIGURE 24-75 Channel section.

x

exx

Section

100 125 150 175 200 75 100 125 150 175 200 225 250 300 350 400 75 100 125 150 175 200

250 300 350 400 450 500 550 600 150 200 225 250 300 350 400 450

mm

45 50 55 60 70 40 50 65 75 75 75 90 100 100 100 100 40 50 65 75 75 75

200 200 200 200 200 250 250 250 150 200 225 250 250 250 250 250

mm

(3)

(1)

(2)

Width of ﬂange, b

Depth of Designation beam, h

3.0 3.0 3.6 3.6 4.1 3.7 4.0 4.4 4.8 5.1 5.5 5.8 6.1 6.7 7.4 8.0 4.4 4.7 5.0 5.4 5.7 6.7

6.7 7.4 8.0 8.6 9.2 9.9 10.5 11.2 5.4 6.1 6.5 6.9 7.6 8.3 9.1 9.8

mm

(4)

Thickness of web, tw

5.1 6.6 6.9 6.9 7.1 6.0 6.4 6.6 7.8 9.5 10.8 10.2 10.7 11.6 12.5 14.0 7.3 7.5 8.1 9.0 10.2 11.4

9.0 10.0 11.4 13.0 15.4 14.7 17.6 21.3 9.0 9.0 9.1 9.7 10.6 11.6 12.7 13.7

mm

(5)

Thickness of ﬂange, tf

91.5 91.5 91.5 91.5 91.5 91.5 91.5 91.5 91.5 91.5 96.0 96.0 96.0 96.0 96.0 96.0 96.0 96.0 96.0 96.0 96.0 96.0

96 96 96 96 96 96 96 96 96 94 94 94 94 94 94 94

deg

(6)

Slope of ﬂange, D

Dimensions of the section

6.0 6.0 7.0 7.0 8.0 6.0 6.0 7.0 8.0 8.0 8.5 11.0 11.0 12.0 13.0 14.0 8.5 9.0 9.5 10.0 10.5 11.0

10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 8.0 9.0 10.0 10.0 11.0 12.0 14.0 15.0

mm

(7) mm

(8)

Radius at toe, r2 ðrt Þ

5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 4.0 4.5 5.0 5.0 5.5 6.0 7.0 7.5 Channel Section 2.0 2.4 2.4 3.0 3.2 2.0 2.0 2.4 2.4 3.2 3.2 3.2 3.2 3.2 4.8 4.8 2.4 2.4 2.4 2.4 3.2 3.2

Radius at root, r1 ðrr Þ

1.40 1.64 1.66 1.75 1.97 1.35 1.62 2.0.4 2.38 2.40 2.35 2.46 2.70 2.55 2.41 2.36 1.31 1.53 1.94 2.22 2.20 2.17

cm

(9)

Center of gravity, Cyy

7.41 10.07 12.65 14.24 17.77 7.26 10.02 13.67 18.36 22.40 26.22 30.53 35.65 42.11 49.47 58.25 8.67 11.70 16.19 20.88 24.38 28.21

52.05 61.33 72.50 85.01 101.15 121.22 143.34 170.38 34.48 47.54 54.94 64.96 74.85 85.91 98.66 111.14

cm

2

(10)

Sectional area, A

TABLE 24-31 Geometrical properties, weight, and nominal dimensions of beams, channels, and T-bars (Cont.)

5.8 7.9 9.9 11.2 13.9 5.7 7.9 10.7 14.4 17.6 20.6 24.0 28.0 33.1 38.8 45.7 6.8 9.2 12.7 16.4 19.1 22.1

40.9 48.1 56.9 66.7 79.4 95.2 112.5 133.7 27.1 37.3 43.1 51.0 58.8 67.4 77.4 87.2

kg

(11)

123.8 270.0 471.1 719.9 1161.2 66.1 164.7 356.8 697.2 1148.4 1725.5 2547.9 3687.5 6047.9 9382.6 13989.5 76.0 186.7 416.4 779.4 1223.3 1819.3

5943.1 9821.6 15521.7 23426.7 35057.6 52290.9 74906.1 106198.5 1455.6 3608.4 5279.5 7736.5 12545.2 19159.2 28083.5 39210.8

cm

4

(12)

14.9 25.7 37.9 50.5 84.2 11.5 24.8 57.2 103.2 126.5 146.9 209.5 298.9 346.0 394.6 460.4 12.6 25.9 59.9 102.3 121.0 140.4

857.5 990.1 1175.9 1388.0 1706.7 2987.8 3740.5 4702.5 431.7 967.1 1353.8 1961.3 2193.6 2451.4 2728.3 2985.2

cm

4

(13)

Weight Moments of inertia per meter, w Ixx Iyy

4.09 5.18 6.10 7.11 8.08 3.02 4.06 5.11 6.19 7.16 8.11 9.14 10.17 11.98 13.72 15.50 2.96 4.0 5.07 6.11 7.08 8.03

10.69 12.66 14.63 16.60 18.63 20.77 22.85 24.97 6.50 8.71 9.80 10.91 12.95 14.93 16.87 18.78

cm

(14)

rxx

1.42 1.60 1.73 1.88 2.18 1.26 1.57 2.05 2.37 2.38 2.37 2.62 2.89 2.97 2.82 2.81 1.21 1.49 1.92 2.21 2.23 2.23

4.06 4.02 4.03 4.04 4.11 4.96 5.11 5.25 3..54 4.51 4.96 5.49 5.41 5.34 5.26 5.18

cm

(15)

ryy

Radius of gyration

24.8 43.2 62.8 82.3 116.1 17.6 32.9 57.1 93.0 131.3 172.6 226.5 295.0 603.2 532.1 699.5 20.3 37.3 66.6 103.9 139.8 181.9

475.4 654.8 887.0 1171.3 1558.1 2091.6 2723.9 3540.0 194.1 360.8 489.3 618.9 836.3 1094.8 1404.2 1742.7

cm

3

(16)

Zxxx

4.8 7.6 9.9 11.9 16.7 4.3 7.3 12.8 20.2 24.8 28.5 32.0 40.9 46.4 52.0 60.2 4.7 7.5 13.1 19.4 22.8 26.3

85.7 99.0 117.6 138.8 17.07 239.0 299.2 376.2 57.6 96.7 120.3 156.9 175.5 196.1 218.3 238.8

cm3

(17)

Zyy

Section moduli

MISCELLANEOUS MACHINE ELEMENTS

24.110

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x

4

rr

eyy = cyy

b - tw

tf

y

b y

tw

x

4

b - tw

eyy = cyy

rt

y

b y

tw

ISMC 225

exx

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100 150 200 250 50 62.5 75 87.5 100 50 62.5 75 87.5 75 100 125 150

ISDT 100 ISDT 150 ISDT 200 ISDT 250 ISMT 50 ISMT 62.5 ISMT 75 ISMT 87.5 ISMT 100 ISMT 50 ISMT 62.5 ISMT 75 ISMT 87.5 ISHT 75 ISHT 100 ISHT 125 ISHT 150

50 75 165 780 75 75 80 90 100 70 70 75 85 150 200 250 250

20 30 40 50 60 75 100 150

80 90 100 100

80

mm

5.8 8.0 8.0 9.2 4.0 4.4 4.8 5.5 5.7 4.5 4.8 5.0 5.8 8.4 7.8 8.8 7.6

4.0 4.0 6.0 6.0 6.0 9.0 10.0 10.0

7.1 7.6 8.1 8.6

6.4

mm

(4)

Thickness of web, tw

10.0 11.6 12.5 14.1 7.2 7.6 7.6 8.6 10.8 7.5 8.0 8.0 9.0 9.0 9.0 9.7 10.6

4.0 4.0 6.0 6.0 6.0 9.0 10.0 10.0

14.1 13.6 13.5 15.3

12.4

mm

(5)

Thickness of ﬂange, tf

98.0 98.0 98.0 98.0 98.0 98.0 98.0 98.0 98.0 98.0 98.0 98.0 98.0 94.0 94.0 94.0 94.0

96.0 96.0 96.0 96.0

96.0

deg

(6)

Slope of ﬂange, D

3.2

mm

(8)

Radius at toe, r2 ðrt Þ

3.2 3.2 4.8 4.8 Normal T-Bar 4.0 3.0 5.0 3.5 5.5 4.0 6.0 4.0 6.5 4.5 8.0 5.5 9.0 6.0 10.0 7.0 Slit and Deep-Legged 8.0 4.0 9.0 4.5 16.0 8.0 17.0 8.5 9.0 4.5 9.0 4.5 9.0 4.5 10.0 5.0 11.0 5.5 9.0 4.5 9.0 4.5 9.0 4.5 10.0 5.0 8.0 4.0 9.0 4.5 10.0 5.0 11.0 5.5

12.0 13.0 14.0 15.0

12.0

mm

(7)

Radius at root, r1 ðrr Þ

0.60 0.32 1.14 1.35 1.56 2.04 2.62 3.61 T-Bar 3.03 4.75 4.78 6.40 0.96 1.30 1.67 1.98 2.13 1.04 1.39 1.73 2.06 1.62 1.91 2.37 2.66

2.30 2.36 2.44 2.42

2.30

cm

(9)

Center of gravity, Cyy

10.37 19.96 36.22 47.75 7.30 8.30 9.50 12.31 16.16 7.35 8.40 9.54 12.43 19.49 25.47 34.85 37.42

1.45 2.26 4.45 5.66 6.85 12.69 18.97 28.88

38.67 45.64 53.66 62.93

33.01

cm

2

(10)

Sectional area, A

8.1 15.7 28.4 37.5 5.7 6.5 7.5 9.7 12.7 5.8 6.6 7.5 9.8 15.3 20.0 27.4 29.4

1.1 1.8 3.5 4.4 5.4 10.0 14.9 22.7

30.4 35.8 41.1 49.4

25.9

kg

(11)

99.0 450.2 1267.8 2774.4 9.7 21.3 40.1 72.6 115.8 10.8 22.8 41.2 75.6 96.2 193.8 415.4 573.7

0.5 0.8 6.1 12.3 21.4 62.0 163.9 541.1

3816.8 6362.6 10008.0 15082.8

2694.6

cm

4

(12)

9.6 37.0 358.2 532.0 20.4 21.9 26.3 42.5 75.0 17.7 19.2 23.4 38.4 230.2 497.3 1005.8 1096.8

0.2 0.8 2.9 5.7 9.7 29.2 76.8 250.3

219.1 310.8 430.6 504.8

187.2

cm

4

(13)

Weight Moments of inertia per meter, w Ixx Iyy

3.09 4.75 5.92 7.62 1.15 1.60 2.05 2.43 2.68 1.21 1.65 2.08 2.47 2.22 2.76 3.45 3.92

0.58 0.89 1.18 1.47 1.77 2.21 2.94 4.33

9.94 11.81 13.66 15.48

9.03

cm

(14)

rxx

0.96 1.36 3.15 3.34 1.67 1.62 1.66 1.86 2.15 1.55 1.51 1.57 1.76 3.44 4.42 5.37 5.41

0.41 0.59 0.81 1.01 1.19 1.52 2.01 2.94

2.38 2.61 2.83 2.83

2.38

cm

(15)

ryy

Radius of gyration

14.2 43.9 83.3 149.2 2.4 4.3 6.9 10.7 14.7 2.7 4.7 7.1 11.3 16.4 24.0 41.0 46.5

0.3 0.8 2.1 3.4 4.8 11.4 22.2 47.5

305.3 424.2 571.9 754.1

239.5

cm

3

(16)

Zxxx

3.8 9.9 43.4 59.1 5.4 5.8 6.6 9.4 15.0 5.0 5.5 6.2 9.0 30.1 49.3 79.9 87.7

0.2 0.5 1.5 2.3 3.2 7.8 15.4 33.4

38.4 46.8 57.0 66.6

32.8

cm3

(17)

Zyy

Section moduli

Key: ISJB—Indian Standard Junior Beams; ISLB—Indian Standard Light-Weight Beams; ISMB—Indian Standard Medium-Weight Beams; ISWB—Indian Standard Wide-Flange Beams; ISHB—Indian Standard Column H-Section Beams; ISJC—Indian Standard Junior Channel; ISLC—Indian Standard Light-Weight Channel; ISMC—Indian Standard Medium-Weight Channel; ISNT—Indian Standard Normal Tee-Bar (T-Bar)., Indian Standard Provisional Slit Medium-Weight Tee-Bars; ISDT—Indian Standard Deep-Legged Tee-Bar; ISLT—Indian Standard Slit Light-Weight Tee-Bar; ISMT—Indian Standard Slit Medium Weight Tee-Bar; ISHT-Indian Standard Slit Tee-Bar from HSection. Source: IS 808, 1964; IS 1173, 1967.

20 30 40 50 60 75 100 150

250 300 350 400

225

mm

ISNT 20 ISNT 30 ISNT 40 ISNT 50 ISNT 60 ISNT 75 ISNT 100 ISNT 150

ISMC 250 ISMC 300 x ISMC 350 ISMC400

cxx

FIGURE 24-77 Slit and deeplegged T-bar. (See Table 24-32.)

h

rr

tf

cxx

exx

FIGURE 24-76 Normal T-bar. (See Table 24-32.)

h

h 2

rt

Section

(3)

(1)

(2)

Width of ﬂange, b

Depth of Designation beam, h

Dimensions of the section

TABLE 24-31 Geometrical properties, weight, and nominal dimensions of beams, channels, and T-bars (Cont.)

MISCELLANEOUS MACHINE ELEMENTS

24.111

90

100

30

35

40

45

50

55

60

65

70

75

80

90

ISA3030

ISA3535

ISA4040

ISA4545

ISA5050

ISA5555

ISA6060

ISA6565

ISA7070

ISA7575

ISA8080

ISA9090

ISA100100 100

80

75

70

65

60

55

50

45

40

35

30

25

25

(3)

20

(2)

20

(1)

Equal Angle ISA2020 Section (see Fig. 24-78) ISA2525

Section

Section dimensions Designation A, mm B, mm

3.0 4.0 3.0 4.0 5.0 3.0 4.0 5.0 3.0 4.0 5.0 3.0 4.0 5.0 3.0 4.0 6.0 3.0 4.0 6.0 5.0 6.0 10.0 5.0 6.0 8.0 5.0 6.0 10.0 5.0 6.0 8.0 5.0 6.0 10.0 6.0 8.0 10.0 6.0 8.0 10.0 6.0 8.0 12.0

(4)

24.112

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. 8.5

8.0

8.0

7.0

7.0

6.5

6.5

6.5

6.0

5.5

5.0

5.0

4.5

4.0

(5)

(6)a 0.59 0.63 0.71 0.75 0.79 0.83 0.87 0.92 0.95 1.00 1.04 1.08 1.12 1.16 1.20 1.25 1.33 1.32 1.37 1.45 1.53 1.57 1.72 1.65 1.69 1.77 1.77 1.81 1.97 1.89 1.94 2.02 2.02 2.06 2.22 2.18 2.27 2.34 2.42 2.51 2.59 2.67 2.76 2.92

(7) 0.59 0.63 0.71 0.75 0.79 0.83 0.87 0.92 0.95 1.00 1.04 1.08 1.12 1.16 1.20 1.25 1.33 1.32 1.37 1.45 1.53 1.57 1.72 1.65 1.69 1.77 1.77 1.81 1.97 1.89 1.94 2.02 2.02 2.06 2.22 2.18 2.27 2.34 2.42 2.51 2.59 2.67 2.76 2.92

(8) 1.12 1.45 1.41 1.84 2.25 1.73 2.26 2.77 2.03 2.66 3.27 2.34 3.07 3.78 2.64 3.47 5.07 2.95 3.88 5.68 5.27 6.26 10.02 5.75 6.94 8.96 6.25 7.44 12.00 6.77 8.06 10.58 7.27 8.66 14.02 9.29 12.21 15.05 10.47 13.79 17.03 11.67 15.39 22.59

(9) 0.9 1.1 1.1 1.4 1.8 1.4 1.8 2.2 1.6 2.1 2.6 1.8 2.4 3.0 2.1 2.7 4.0 2.3 3.0 4.5 4.1 4.9 7.9 4.5 5.4 7.0 4.9 5.8 9.4 5.3 6.3 8.3 5.7 6.8 11.0 7.3 9.6 11.8 8.2 10.8 13.4 9.2 12.1 17.7

(10) 0.4 0.5 0.8 1.0 1.2 1.4 1.8 2.1 2.3 2.9 3.5 3.4 4.5 5.4 5.0 6.5 9.2 6.9 9.1 12.9 14.7 17.3 16.3 19.2 22.6 29.0 24.7 29.1 45.0 31.1 36.8 47.4 38.7 45.7 71.4 56.0 72.5 87.7 80.1 104.2 126.7 111.3 145.1 207.0

(11) 0.4 0.5 0.8 1.0 1.2 1.4 1.8 2.1 2.3 2.9 3.5 3.4 4.5 5.4 5.0 6.5 9.2 6.9 9.1 12.9 14.7 17.3 16.3 19.2 22.6 29.0 24.7 29.1 45.0 31.1 36.8 47.4 38.7 45.7 71.4 56.0 72.5 87.7 80.1 104.2 126.7 111.3 145.1 207.0

(12) 0.6 0.8 1.2 1.6 1.8 2.2 2.8 3.4 3.6 4.7 5.6 5.5 7.1 8.6 8.0 10.4 14.6 11.1 14.5 20.6 23.5 27.5 41.5 30.6 36.0 46.0 39.4 46.5 71.3 49.8 58.8 75.5 61.9 73.1 113.3 89.6 115.6 139.5 128.1 166.4 201.9 178.1 231.8 329.3

(13) 0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.9 0.9 1.2 1.5 1.4 1.8 2.2 2.0 2.6 3.8 2.8 3.6 5.3 6.9 7.0 11.2 7.7 9.1 11.9 9.9 11.7 18.8 12.5 14.8 19.3 15.5 18.4 29.4 22.5 29.4 36.0 32.0 42.0 51.6 44.5 58.4 87.7

(14) 0.58 0.58 0.73 0.73 0.72 0.89 0.89 0.88 1.05 1.05 1.04 1.21 1.21 1.20 1.38 1.37 1.35 1.53 1.53 1.51 1.67 1.66 1.62 1.82 1.82 1.80 1.99 1.98 1.94 2.15 2.14 2.12 2.31 2.30 2.26 2.46 2.44 2.41 2.77 2.75 3.73 3.09 3.07 3.03

(15) 0.58 0.58 0.73 0.73 0.72 0.89 0.89 0.88 1.05 1.05 1.04 1.21 1.21 1.20 1.38 1.37 1.35 1.53 1.53 1.51 1.67 1.66 1.62 1.82 1.82 1.80 1.99 1.98 1.94 2.15 2.14 2.12 2.31 2.30 2.26 2.46 2.44 2.41 2.77 2.75 2.73 3.09 3.07 3.03

(16) 0.73 0.72 0.93 0.91 0.91 1.13 1.12 1.11 1.33 1.32 1.31 1.54 1.53 1.51 1.74 1.73 1.70 1.94 1.93 1.90 2.11 2.10 2.03 2.31 2.29 2.27 2.51 2.50 2.44 2.71 2.70 2.67 2.92 2.91 2.84 3.11 3.08 3.04 3.50 3.47 3.44 3.91 3.88 3.82

(17) 0.37 0.37 0.47 0.47 0.47 0.57 0.57 0.57 0.67 0.67 0.67 0.77 0.77 0.77 0.87 0.87 0.87 0.97 0.97 0.96 1.06 1.06 1.06 1.16 1.15 1.15 1.26 1.26 1.25 1.36 1.36 1.35 1.46 1.46 1.45 1.52 1.55 1.55 1.75 1.75 1.70 1.95 1.95 1.94

(18)

0.3 0.4 0.4 0.6 0.7 0.6 0.8 1.0 0.9 1.2 1.4 1.2 1.6 1.9 1.5 2.0 2.9 1.9 2.5 3.6 3.7 4.4 7.0 4.4 5.2 6.8 5.2 6.2 9.9 6.1 7.3 9.5 7.1 8.4 13.5 9.6 12.6 15.5 12.2 16.0 19.2 15.8 20.0 29.2

(19)

0.3 0.4 0.4 0.6 0.7 0.6 0.8 1.0 0.9 1.2 1.4 1.2 1.6 1.9 1.5 2.0 2.9 1.9 2.8 3.6 3.7 4.4 7.0 4.4 5.2 6.8 5.2 6.2 9.9 6.1 7.3 9.5 7.1 8.4 13.5 9.6 12.6 15.5 12.2 16.0 19.8 15.2 20.0 29.2

(20)

(21)

Moment of inertia Radii of gyration Thick- Radius at Radius at Center of gravity Sectional Weight Moduli of section ness t, root, r1 , toe, r2 , area, A, per meter, Iuu , (max), Ivv , (min), ruu , (max), rvv , (min), mm mm mm Cxx , cm Cyy , cm cm2 w, kg Ixx , cm4 Iyy , cm4 cm4 cm4 rxx , cm ryy , cm cm cm Zxx , cm3 Zyy , cm3 tan

Dimensions of the section

TABLE 24-32 General properties, weight, and nominal dimensions of equal and unequal angles

MISCELLANEOUS MACHINE ELEMENTS

Unequal Angle Section (see Fig. 24-79)

Section

40

200

ISA200200 200

ISA4025

150

ISA150150 150

30

130

ISA130130 130

ISA3020

110

ISA110110 110

A

U

v

cxx

exx

8.0 10.0 16.0 8.0 10.0 12.0 16.0 10.0 12.0 16.0 20.0 12.0 16.0 20.0 25.0

(4)

r1

cyy

90

t

15.0

12.0

10.0

10.0

(5)

y

y

B

eyy

90

(6)a

t v

U

3.00 3.09 3.32 3.50 3.59 2.67 3.82 4.08 4.16 4.31 4.46 5.39 5.56 5.71 5.90

(7)

x r2

3.00 3.09 3.32 3.50 3.59 3.67 3.82 4.08 4.16 4.31 4.46 5.39 5.56 5.71 5.90

(8)

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25

20

3.0 4.0 5.0 2.0 4.0 5.0 6.0 5.0

4.5

0.98 1.02 1.06 1.30 1.35 1.39 1.43

0.49 0.53 0.57 0.57 0.62 0.66 0.69

FIGURE 24-78 Equal-angle section.

x

(3)

(2)

(1)

Section dimensions Designation A, mm B, mm

1.41 1.84 2.25 1.88 2.46 3.02 3.05

17.08 21.12 32.76 20.28 25.12 29.88 39.16 29.21 34.77 45.65 56.21 46.94 61.82 76.38 94.13

(9)

1.1 1.4 1.8 1.5 1.9 2.4 2.8

13.4 16.6 25.7 15.9 19.7 23.5 30.7 22.9 27.3 35.8 44.1 36.9 48.5 60.0 73.9

(10)

A cxx

rn

y

y

cyy

90

t

312.7 381.5 564.3 526.3 644.6 757.1 965.6 1007.4 1186.6 1522.5 1829.6 2905.4 3764.1 4568.6 5501.5

(13)

exx

v

196.8 240.2 357.3 331.0 405.3 476.4 609.1 633.5 746.3 958.9 1153.5 1826.3 2366.2 2875.0 3470.2

(12)

B v

t eyy

90

81.0 98.9 150.0 135.6 166.0 195.6 252.6 259.6 305.9 395.3 481.3 747.2 958.5 1181.4 1438.8

(14)

α

U

3.40 3.37 3.30 4.04 4.02 3.99 3.94 4.66 4.63 4.58 4.53 6.24 6.19 6.14 6.07

(15)

r2

x

3.40 3.37 3.30 4.04 4.02 3.99 3.94 4.66 4.63 4.58 4.53 6.24 6.19 6.14 6.07

(16) 4.28 4.25 4.15 5.10 5.07 5.03 4.97 5.87 5.94 5.77 5.71 7.87 7.80 7.73 7.69

(17)

2.18 2.16 2.14 2.59 2.57 2.56 2.54 2.98 2.97 2.94 2.93 3.99 3.96 3.93 3.91

(18)

1.2 1.5 1.9 3.0 3.8 4.6 5.4

0.4 0.5 0.6 0.9 1.1 1.4 1.6

1.4 1.8 2.1 3.3 4.3 5.1 5.9

0.2 0.3 0.4 0.5 0.7 0.8 1.0

0.92 0.92 0.91 1.25 1.25 1.24 1.23

0.54 0.54 0.53 0.68 0.68 0.67 0.66

0.99 0.98 0.97 1.33 1.32 1.31 1.29

0.41 0.41 0.41 0.52 0.52 0.52 0.52

0.6 0.8 1.0 1.1 1.4 1.8 2.1

24.6 30.4 46.5 34.9 43.1 51.0 66.3 58.0 68.8 89.7 109.7 125.0 163.8 201.2 246.0

(19)

FIGURE 24-79 Unequal-angle section. (See Table 24-32.)

U

x

196.8 240.2 357.3 331.0 405.3 476.4 609.1 633.5 746.3 958.9 1153.5 1826.3 2366.2 2875.0 3470.2

(11)

0.3 0.4 0.4 0.5 0.6 0.7 0.9

24.6 30.4 46.5 34.9 43.1 51.0 66.3 58.0 68.8 89.7 109.7 125.0 163.8 201.2 246.0

(20)

0.43 0.42 0.41 0.38 0.38 0.37 0.37

(21)

Moment of inertia Radii of gyration Thick- Radius at Radius at Center of gravity Sectional Weight Moduli of section ness t, root, r1 , toe, r2 , area, A, per meter, Iuu , (max), Ivv , (min), ruu , (max), rvv , (min), 2 4 4 4 4 w, kg Ixx , cm Iyy , cm cm cm rxx , cm ryy , cm cm cm Zxx , cm3 Zyy , cm3 tan mm mm mm Cxx , cm Cyy , cm cm

Dimensions of the section

TABLE 24-32 General properties, weight, and nominal dimensions of equal and unequal angles (Cont.)

MISCELLANEOUS MACHINE ELEMENTS

24.113

Section

(2)

45

50

60

65

70

75

80

90

100

100

125

125

(1)

ISA4530

ISA5030

ISA6040

ISA6545

ISA7045

ISA7550

ISA8050

ISA9060

ISA10065

ISA10075

ISA12575

ISA12595

24.114

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.

95

75

75

65

60

50

50

45

45

40

30

30

(3)

Section dimensions Designation A, mm B, mm

3.0 4.0 5.0 6.0 3.0 4.0 5.0 6.0 5.0 6.0 8.0 5.0 6.0 8.0 5.0 6.0 8.0 10.0 5.0 6.0 8.0 10.0 5.0 6.0 8.0 10.0 6.0 8.0 10.0 12.0 6.0 8.0 10.0 6.0 8.0 10.0 12.0 6.0 8.0 10.0 6.0 8.0 10.0 12.0

(4)

9.0

9.0

8.5

8.0

7.5

7.0

6.5

6.5

6.0

6.0

5.5

5.0

(5)

4.8

(6)a 1.42 1.47 1.51 1.55 1.63 1.68 1.72 1.76 1.95 1.99 2.07 2.07 2.11 2.19 2.27 2.32 2.40 2.48 2.39 2.44 2.52 2.60 2.60 2.64 2.73 2.81 2.87 2.96 3.04 3.12 3.19 3.28 3.37 3.01 3.10 3.19 3.27 4.05 4.15 4.24 3.72 3.80 3.89 3.97

(7) 0.69 0.73 0.77 0.81 0.65 0.70 0.74 0.78 0.96 1.00 1.08 1.08 1.12 1.20 1.04 1.09 1.16 1.24 1.16 1.20 1.28 1.36 1.12 1.16 1.24 1.32 1.39 1.48 1.55 1.63 1.47 1.55 1.63 1.78 1.87 1.95 2.03 1.59 1.68 1.76 2.24 2.32 2.40 2.48

(8) 2.18 2.86 3.52 4.16 2.34 3.07 3.78 4.47 4.76 5.65 7.37 5.26 6.25 8.17 5.52 6.56 8.58 10.52 6.02 7.16 9.38 11.52 6.27 7.46 9.78 12.02 8.65 11.37 14.01 16.57 9.55 12.57 15.51 10.14 13.36 16.50 19.56 11.66 15.38 19.02 12.92 17.04 21.08 25.04

(9) 1.7 2.2 2.8 3.3 1.8 2.4 3.0 3.5 3.7 4.4 5.8 4.1 4.9 6.4 4.3 5.2 6.7 8.3 4.7 5.6 7.4 9.0 4.9 5.9 7.7 9.4 6.8 8.9 11.0 13.0 7.5 9.9 12.2 8.0 10.5 13.0 15.4 9.2 12.1 14.9 10.1 13.4 16.5 19.7

(10) 4.4 5.7 6.9 8.0 5.9 7.7 9.3 10.9 16.9 19.9 25.4 22.1 26.0 33.2 27.2 32.0 41.0 49.3 34.1 40.3 51.8 62.3 48.6 48.0 61.9 74.7 70.6 91.3 110.9 129.1 96.7 125.9 153.2 100.9 131.6 160.4 187.5 187.8 245.5 300.3 205.5 268.3 328.0 384.8

(11) 1.5 2.0 2.4 2.8 1.6 2.1 2.5 2.9 6.0 7.0 8.8 8.6 10.1 12.8 8.3 10.3 13.1 15.6 12.2 14.3 18.3 21.8 12.3 14.4 18.5 22.1 25.2 32.4 39.1 45.2 32.4 41.9 50.7 48.7 63.3 76.9 89.5 51.6 67.2 81.6 103.6 134.7 164.1 191.8

(12) 5.0 6.5 7.9 9.2 6.5 8.5 10.3 11.9 19.5 22.8 29.0 25.9 30.4 38.7 30.9 36.3 46.3 55.4 39.4 46.4 59.4 71.2 45.7 53.9 69.3 83.3 81.5 105.3 127.3 147.5 110.6 143.6 174.2 124.0 161.3 196.1 228.4 208.9 272.8 332.9 254.0 331.4 404.5 473.7

(13) 0.9 1.1 1.4 1.7 1.0 1.2 1.5 1.8 3.4 4.0 5.2 4.8 5.7 7.4 5.1 6.0 7.8 9.5 6.9 8.2 10.6 12.9 7.2 8.5 11.0 13.5 14.3 18.6 22.8 26.8 18.6 24.2 29.7 25.6 33.6 41.2 48.6 30.5 40.0 49.1 55.1 71.7 87.6 103.0

(14) 1.42 1.41 1.40 1.39 1.59 1.58 1.57 1.56 1.89 1.88 1.86 2.05 2.04 2.02 2.22 2.21 2.19 2.16 2.38 2.37 2.35 2.33 2.55 2.54 2.52 2.49 2.86 2.84 2.81 2.79 3.18 3.16 3.14 3.15 3.14 3.12 3.10 4.01 4.00 3.97 3.99 3.97 3.95 3.92

(15) 0.84 0.84 0.83 0.82 0.82 0.82 0.81 0.80 1.12 1.11 1.10 1.28 1.27 1.25 1.26 1.25 1.24 1.22 1.42 1.41 1.40 1.38 1.40 1.39 1.37 1.36 1.17 1.69 1.67 1.65 1.94 1.83 1.81 2.19 2.18 2.16 2.14 2.10 2.09 2.07 2.83 2.81 2.79 2.77

(16) 1.52 1.51 1.50 1.49 1.67 1.66 1.65 1.64 2.02 2.01 1.98 2.22 2.21 2.18 2.36 2.35 2.32 2.29 2.56 2.55 2.52 2.49 2.70 2.69 2.66 2.63 3.07 3.04 3.01 2.98 3.40 3.38 3.35 3.50 3.48 3.45 3.42 4.23 4.21 4.18 4.43 4.41 4.38 4.35

(17) 0.63 0.63 0.63 0.63 0.65 0.63 0.63 0.63 0.85 0.85 0.84 0.96 0.95 0.95 0.96 0.96 0.95 0.95 1.07 1.07 1.06 1.06 1.07 1.07 1.06 1.06 1.28 1.28 1.27 1.27 1.39 1.39 1.38 1.59 1.59 1.58 1.58 1.62 1.61 1.61 2.07 2.05 2.04 2.03

(18)

1.4 1.9 2.3 2.7 1.7 2.3 2.8 3.4 4.2 5.0 6.5 5.0 5.9 7.7 5.7 6.8 8.9 10.9 6.7 8.0 10.4 12.7 7.5 9.0 11.7 14.4 11.5 15.1 18.6 22.0 14.2 18.7 23.1 14.4 19.1 23.6 27.9 22.2 29.4 36.3 23.4 30.9 38.1 45.1

(19)

0.7 0.9 1.1 1.3 0.7 0.9 1.1 1.3 2.0 2.3 3.0 2.5 3.0 3.9 2.5 3.0 3.9 4.8 3.2 3.8 4.9 6.0 3.2 3.8 4.9 6.0 5.5 7.2 8.8 10.3 6.4 8.5 10.4 8.5 11.2 13.8 16.3 8.7 11.5 44.2 14.3 18.8 23.1 27.3

(20)

0.44 0.43 0.43 0.42 0.36 0.36 0.35 0.35 0.44 0.43 0.42 0.47 0.47 0.46 0.41 0.41 0.40 0.39 0.44 0.44 0.43 0.42 0.39 0.39 0.38 0.38 0.44 0.44 0.43 0.42 0.42 0.42 0.41 0.55 0.55 0.55 0.54 0.37 0.36 0.36 0.57 0.57 0.56 0.56

(21)

Moment of inertia Radii of gyration Thick- Radius at Radius at Center of gravity Sectional Weight Moduli of section ness t, root, r1 , toe, r2 , area, A, per meter, Iuu , (max), Ivv , (min), ruu , (max), rvv , (min), w, kg Ixx , cm4 Iyy , cm4 cm4 cm4 rxx , cm ryy , cm cm cm Zxx , cm3 Zyy , cm3 tan mm mm mm Cxx , cm Cyy , cm cm2

Dimensions of the section

TABLE 24-32 General properties, weight, and nominal dimensions of equal and unequal angles (Cont.)

MISCELLANEOUS MACHINE ELEMENTS

a

75

115

100

150

150

ISA15075

ISA150115 150

ISA200100 200

ISA200150 200

(3)

(2)

(1) 8.0 10.0 12.0 8.0 10.0 12.0 10.0 12.0 16.0 10.0 12.0 16.0

(4)

13.5

12.0

11.0

10.0

(5)

4.8

4.8

4.8

4.8

(6)a 5.24 5.33 5.42 4.48 4.57 4.65 6.98 7.07 7.25 6.02 6.11 6.72

(7) 1.54 1.62 1.70 2.76 2.84 2.92 2.03 2.11 2.27 3.35 3.63 3.79

(8) 17.48 21.62 25.68 20.72 25.66 30.52 29.21 34.77 45.66 34.29 40.85 53.83

(9) 13.7 17.0 20.2 16.3 20.1 24.0 22.9 27.3 35.8 26.9 33.1 42.2

(10)

For the cases for which the radius at toe r2 is not given, the toe should be reasonably square. Source: IS 808, 1964.

Section

Section dimensions Designation A, mm B, mm

410.3 502.2 590.0 474.4 581.8 684.8 1227.8 1449.2 1870.1 1409.2 1655.6 2155.2

(11) 71.1 86.3 100.4 244.4 298.8 350.6 214.7 251.5 319.6 688.9 812.1 1044.9

(12) 435.7 532.7 625.1 589.6 722.6 849.5 1304.9 1539.0 1982.1 1729.6 2043.3 2638.6

(13) 45.7 55.7 65.4 129.2 158.0 185.9 137.6 161.6 207.7 368.5 434.5 561.5

(14) 4.85 4.82 4.79 4.78 4.76 4.74 6.48 6.46 6.40 6.41 6.39 6.33

(15) 2.02 2.00 1.98 3.43 3.41 3.39 2.71 2.69 2.65 4.48 4.46 4.41

(16) 4.99 4.96 4.93 5.33 5.31 5.28 6.68 6.65 6.59 7.10 7.07 7.01

(17) 1.62 1.61 1.60 2.50 2.48 2.47 2.17 2.16 2.13 3.28 3.26 3.23

(18)

42.0 51.9 61.6 45.1 55.8 66.2 94.3 112.1 146.5 100.8 119.9 157.0

(19)

11.9 14.7 17.3 28.0 34.5 40.8 26.9 31.9 41.5 60.2 71.4 93.2

(20)

0.26 0.26 0.26 0.58 0.58 0.57 0.27 0.26 0.26 0.26 0.55 0.55

(21)

Moment of inertia Radii of gyration Thick- Radius at Radius at Center of gravity Sectional Weight Moduli of section ness t, root, r1 , toe, r2 , area, A, per meter, Iuu , (max), Ivv , (min), ruu , (max), rvv , (min), 2 4 4 4 4 mm mm mm Cxx , cm Cyy , cm cm w, kg Ixx , cm Iyy , cm cm cm rxx , cm ryy , cm cm cm Zxx , cm3 Zyy , cm3 tan

Dimensions of the section

TABLE 24-32 General properties, weight, and nominal dimensions of equal and unequal angles (Cont.)

MISCELLANEOUS MACHINE ELEMENTS

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24.115

MISCELLANEOUS MACHINE ELEMENTS

24.116

CHAPTER TWENTY-FOUR

REFERENCES 1. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, fps system, Engineering College Co-operative Society, Bangalore, India, 1962. 2. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Book Handbook, Volume I. (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 3. Lingaiah, K., Machine Design Data Handbook, Volume II (S.I. and Customary Metric Units), Suma Publishers, Bangalore, India, 1986. 4. Lingaiah, K., Machine Design Data Handbook, McGraw-Hill Publishing Company, New York, 1994. 5. Maleev, V. L., and J. B. Hartman, Machine Design, International Text book Company, Scranton, Pennsylvania, 1954. 6. Black, P. H., and O. E. Adams, Jr., Machine Design, McGraw-Hill Book Company Inc., New York, 1968. 7. Neale, M. J., Tribology Handbook, Newnes-Butterworth, London, 1973. 8. Bach, Maschinenelemente, 12th ed., P-43. 9. Heldt, P. M. Torque Converters for Transmissions, Chiltan Company, Philadelphia, 1955. 10. Newton, K., and W. Steeds, The Motor Vehicle, Iliﬀe and Sons Ltd, London, 1950. 11. Steeds, W., Mechanics of Road Vehicle, Iliﬀe and Sons, Ltd., London, 1960. 12. Arkhangelsky, V., et al., Motor Vehicles Engines, Mir Publisher, Moscow, 1971. 13. Heldt, P. M., High Speed Combustion Engines, 6th ed., Chilton Company, Philadelphia, 1955. 14. Thimoshenko, S., and J. N. Goodier, Theory of Elasticity, McGraw-Hill Book Company, Inc., and Kogakkusha Company Ltd., Tokyo, 1951. 15. Timoshenko, S., and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill Book Company, Inc., New York, 1959. 16. Seely, F. B., and J. O. Smith, Advanced Mechanics of Materials, John Wiley and Sons, Inc., 2nd ed., 1959. 17. Timoshenko, S., and J. M. Gere, Mechanics of Materials, Von Nostrand Reinhold Company, New York, 1972. 18. Goetze, A. G. Piston Ring Manual, 3rd ed., Burscheid, Germany, 1987. 19. Bureau of Indian Standards, New Delhi, India.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

25 ELEMENTS OF MACHINE TOOL DESIGN SYMBOLS Ai B c d D Dm Dw E F Fa ¼ Fx Fc Fhc Fmax Fn Fn Fr FR Fs Ff ¼ Fx Ft ¼ Fz ¼ Fc Fy ¼ Fr F F ¼ Fr

cross-sectional area of chip before removal from workpiece, m2 (in2 ) width of V or U die, m (in) chisel edge length, m (in) diameter of the hole, m (in) diameter of the drill, m (in) depth of cut, m (in) blank diameter, m (in) diameter of milling cutter, m (in) shell diameter, m (in) diameter of machined surface, m (in) diameter of workpiece or job, m (in) Young’s modulus, GPa (kpsi) work done in punching or shearing of material, N m (lbf ft) work done per unit volume of removed, J/mm3 force, kN (lbf ) axial component of cutting force, kN (lbf ) cutting force, kN (lbf ) normal cutting force or the resultant force on a single point metal cutting tool in the horizontal plane, kN (lbf ) maximum force at the punch, kN (lbf ) force normal to F , kN (lbf ) force normal to shear force, kN (lbf ) reaction of the cutting force, kN (lbf ) radial component of the cutting force, kN(lbf ) resultant cutting force, kN (lbf ) stripping pressure, kN (lbf ) feed force, kN (lbf ) tangential component of the cutting force, kN(lbf) radial component of the cutting force, kN (lbf ) frictional force on tool face, kN (lbf ) frictional force of the saddle on lathe bed, kN (lbf )

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ELEMENTS OF MACHINE TOOL DESIGN

25.2

CHAPTER TWENTY-FIVE

shear force kN (lbf ) depth of cut, m (in) shell height, m (in) height of the lathe center, m (in) swing over the bed of the lathe, m (in) constant of proportionality in Eq. (25-31) constant coeﬃcient, also with subscripts clearance length of cut or perimeter of the cut, m (in) length of job, m (in) exponents

F h hc hsw K K Kc L Lm m, m1 , m2 , m3 , m4 Mb Mt n n0 p p, q P P Pc Pg Pf Pu ¼ Ps Q r rc rc ¼ ðt1 =t2 Þ rn R Ri s sz t t1 t2 uc us v vc vf V W x, v, z z f n o p tr

bending moment, N m (lbf ft) turning moment N m (ft lbf ) speed, rpm speed, rps pitch of thread, mm (in) exponents periphery, m (in) power, kW (hp) power at cutting tool, kW (hp) gross or motor power, kW (hp) tare power, that is the power required to run the machine at no load, kW (hp) unit power or speciﬁc power, kW (hp) metal removal rate, cm3 /min (in3 /min) radius, m (in) corner radius, m (in) cutting ratio nose radius, m (in) roughness height, m m (m in) inside radius of bend, m (in) feed rate, mm/rev (in/rev) feed per tooth of milling cutter, mm/tooth thickness of material to be punched or sheared, m (in) thickness of chip, m (in) depth of cut, m (in) initial thickness of the chip, m (in) ﬁnal thickness of the chip, m (in) number of chamfered threads number of splines velocity, m/s (ft/min) cutting speed, m/min (ft/min) feed rate, mm/min (in/min) cutting velocity, m/min (ft/min) work done per unit volume, N m/m3 (lbf ft/ft3 ) machine reference co-ordinate axes number of teeth on milling cutter relief or clearance angle, deg side relief or clearance angle, deg normal relief or clearance angle, deg orthogonal clearance angle, deg front or end relief or clearance angle, deg true relief or clearance angle, deg

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

b f n o p "

c

b s c sut y n ð¼ su Þ

o

p

s !

helix angle, deg bevel angle, deg rake angle, also with subscripts, deg side rake angle, deg normal rake angle, deg orthogonal rake angle, deg back rake angle, deg deﬂection, mm (in) gullet angle, deg peripheral pitch angle, deg eﬃciency wedge angle, also with subscripts, deg chip ﬂow angle, deg approach angle in Eq. (25-104b), deg corner angle in Eq. (25-104a) bend angle, deg inclination angle, deg coeﬃcient of friction friction angle, deg stress, also with suﬃces, MPa (kpsi) compressive stress, MPa (kpsi) ultimate tensile strength, MPa (kpsi) yield stress, MPa (kpsi) tool life shear stress, MPa (kpsi) resistance of the material to shearing or the ultimate shearing strength, MPa (kpsi) cutting edge angle, also with subscripts, deg shear angle, deg end cutting edge angle, deg principal cutting edge angle, deg side cutting edge angle, deg engagement angle for milling depth, deg angular speed, rad/s

Note: and with subscript s designates strength properties of material used in the design which will be used and observed throughout this Machine Design Data Handbook. Other factors in performance or in special aspects are included from time to time in this chapter and, being applicable, only in their immediate context, are not given at this stage.

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25.3

ELEMENTS OF MACHINE TOOL DESIGN

25.4

CHAPTER TWENTY-FIVE

Particular

Formula

25.1 METAL CUTTING TOOL DESIGN 25.1.1 Forces on a single-point metal cutting tool Nomenclature of metal cutting tool The normal cutting force or resultant force on a single-point metal cutting tool in horizontal plane (Fig. 25-2) The resultant cutting force on the cutting tool (Fig. 25-2)

Refer to Fig. 25-1. qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Fhc ¼ Ff2 þ Fr2

FR ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 Ft2 þ Fhc

ð25-2Þ

FR ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ff2 þ Fr2 þ Ft2

ð25-3Þ

Shank

Cutting part

Minor cutting edge

Tool axis Base

Minor flank

Major cutting edge

Corner

Face

Major flank

FIGURE 25-1 Nomenclature of metal cutting tool.

FR = resultant cutting force on the cutting tool

z Dw

ω

Ff (=Fx) = feed

force

x

Ft or Fc (=Fx) = tangential cutting force d = depth of cut Fr (=Fy) = radial force y

ð25-1Þ

Dm

Feed motio

n

FIGURE 25-2 Components of cutting force acting on a single point metal cutting tool.

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Particular

25.5

Formula

where Ft or Fc ð¼Fz Þ ¼ tangential cutting force perpendicular to Fr (¼Fy ) and Ff (¼Fx ) in the vertical plane. Fr ð¼Fy Þ ¼ radial force perpendicular to the direction of feed and in the horizontal plane. Ff ð¼Fx Þ ¼ feed force in the horizontal plane against the direction of the feed. x, y and z are machine reference axes along feed force Ff , radial force Fr , and cutting force Ft or Fc directions, respectively.

25.1.2 Merchant’s circle for cutting forces for a single-point metal cutting tool F Fhc þ Fc tan ¼ Fn Fc Fhc tan

The co-eﬃcient of friction in orthogonal cutting (Fig.25-3)

¼ tan ¼

The shear force

F ¼ Fc cos Fhc sin

ð25-5Þ

The friction force

F ¼ Fhc cos þ Fc sin

ð25-6Þ

Mean shear stress

¼

Fc sin cos Fhc sin2 Ai

ð25-7Þ

t2

Chip Fτ Fc

t1

α

Tool

rn

Fhτ = F hc 2φ

Fτ

ρ = tan—1µ

φ

rn

φ

FR

α Fµ

ρ α

FIGURE 25-3 Force acting in orthogonal cutting with a continuous chip. Courtesy: ASTME, Tool Engineers’ Handbook, 2nd Edition, McGraw-Hill Book Company, New York, 1959.

d-rn

ð25-4Þ

d

Feed θ Direction of chip flow FIGURE 25-4 Approximate chip ﬂow direction.

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ELEMENTS OF MACHINE TOOL DESIGN

25.6

CHAPTER TWENTY-FIVE

Particular

Formula

Work done in shearing the material

W ¼ ½cot þ tanð Þ

ð25-8Þ

Work done in overcoming friction

W ¼

F sin Ai cosð Þ

ð25-9Þ

Wt ¼

Fc Ai

The total work done in cutting

ð25-10Þ

rc cos 1 rc sin

The shear angle ( )

tan ¼

The tangential cutting force

Ft ¼ KCs m1 d m2

ð25-11Þ ð25-12Þ

where d ¼ depth of cut, m (mm) m1 ¼ slope of Ft versus s graph (typical values 0.5 to 0.98) m2 ¼ slope of Ft versus d graph (typical values 0.90 to 1.4) K ¼ overall correction coeﬃcient, depends on actual conditions of tool angles and working conditions (varies from 0.9 to 1.0) C ¼ coeﬃcient characterized by material of job, condition of working tool, coolants, etc. (Table 25-1) The values of K in Eq. (25-12) are calculated from equation

K ¼ Km K K Kc

ð25-12aÞ

where Km ¼ material correction coeﬃcient K ¼ correction coeﬃcient, depends on back rack angle Kc ¼ correction coeﬃcient for coolant used K ¼ correction coeﬃcient, depends on top rack angle Values of Km , K , Kc , and K are taken from Tables 25-2 and 25-3.

TABLE 25-1 Values of C and exponents

Type of operation

Material

Turning and boring Facing and parting Turning and boring Parting and facing

Steel Steel Gray cast iron Gray cast iron

Ultimate strength, u , MPa

Hardness, Brinell, HB

C

m1

m2

735 735

215 215 190 190

225 264 98 135

1.0 1.0 1.0 1.0

0.75 1.00 1.75 1.0

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Particular

25.7

Formula

The chip ﬂow angle for zero-degree rake angle (Fig. 25-4)

tan ¼

d rn þ ðd rn Þ tan

ð25-13Þ

where rn ¼ nose radius ¼ side cutting edge angle, deg The equation relating the true rake angle to the corresponding chip ﬂow angle

tan tr ¼ tan sin þ tan cos

The equation for locating the maximum rake angle

tan max ¼

The metal removal rate

Q¼

ð25-14Þ

tan tan

ð25-15Þ

ðD dÞsdn ¼ Vsd 1000 where

ð25-16Þ

s ¼ feed rate, mm/rev Q ¼ metal removal rate, cm3 /min d ¼ depth of cut, mm D ¼ diameter of work piece, mm V ¼ ½ðD dÞn=1000, m/min The approximate relationships between Ft ð¼Fz Þ, Ff ð¼Fx Þ and Fr ð¼Fy Þ

Ff ð¼Fx Þ 0:3 to 0:2 Ft ð¼Fz Þ

ð27-17Þ

Fr ð¼Fx Þ 0:2 to 0:1 Ft ð¼Fz Þ

ð27-18Þ

The turning moment on the work piece due to tangential cutting force

Mtcut ¼ Ft

TABLE 25-2 Material correction coeﬃcient, Km

TABLE 25-3 Values of Kc , K , and K

Material Steel

Cast iron

Ultimate strength , MPa

Km

Coolant

390–490 490–588 686–785 785–880 980–1175 1370–1570 1570–1765 1765–1960 2155–2355 2355–2745

0.76 0.82 1.00 1.10 1.28 0.88 0.94 1.00 1.12 1.17

Dry Soda water Emulsion Mineral oil

D 2

Hard mineral oil

ð25-19Þ

, deg

K

15 10 þ5 0 þ5 þ10 þ15 þ20

1.40 1.30 1.23 1.13 1.06 1.00 0.94 0.89

, deg

K

Kc

30 45 60 75

1.05 1.00 0.96 0.94

1 1.03 1.10 1.15

90

0.92

1.20–1.25

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ELEMENTS OF MACHINE TOOL DESIGN

25.8

CHAPTER TWENTY-FIVE

Particular

The bending moment due to bending of the tool in the vertical plane by tangential cutting force

Formula

Mb ¼ Ft l

ð25-20Þ

where l ¼ cantilever length of the cutting tool, m

25.1.3 Power The total power at the cutting tool, Ptotal

Ptotal ¼ Pc þ Pf þ Pr

ð25-21Þ

where Pc ¼

Ft Vt ¼ power required for turning cut, kW 1000

Pf ¼

Ff Vf ¼ power required to feed in a horizontal 1000 direction, kW. The feed velocity is very low. Power required to feed is approximately 1% of total power. Hence it is neglected.

Pr ¼

Fr Vr ¼ power required to feed in radial direc1000 tion, kW. The radial velocity is zero. Therefore Pr is ignored.

After neglecting Pf and Pr , the power required at the cutting tool, taking Vc for Vt and Ptotal Pc

The gross or motor power

Ft Vc KCs m1 d m2 Vc ¼ ð25-22Þ 1000 1000 where Pc in kW, Ft in N, Vc in m/s, s and d in m

Pc ¼

Pg ¼

Pc þ Pt

ð25-23Þ

where ¼ mechanical eﬃciency of machine tool Pt ¼ tare power, the power required at no-load, kW

25.1.4 Speciﬁc power or unit power consumption The speciﬁc power Pu ð¼Ps Þ, required to cut a material

Pu ¼

Pc (cubic meter or cubic millimeter of material removed by cut per minute)

ð25-24Þ

The speciﬁc power or unit power Pu ð¼Ps Þ, for turning

Pu ¼

Pc Ft C ¼ ¼ Vc sd 1000sd 1000s 1 m1 d 1 m2

ð25-25Þ

where Pc , Pu in kW/m3 /min, Ft in N, s and d in m, and Vc in m/s Another relation connecting speciﬁc powers at diﬀerent cutting feeds and depths of cuts

Pu2 ¼ Pu1

s1 s2

1 m1

d1 d2

1 m2

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ð25-26Þ

ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

25.9

Particular

Formula

TABLE 25-4 Typical values of speciﬁc power consumption Ps or Pu

Refer to Table 25-4 for Pu (m3 /s) or Pu (mm3 /s) for various materials.

Material Plain carbon steel

Alloy steel Free cutting steel Cast iron Aluminum alloy

Brinell hardness number, HB

Speciﬁc power consumption, Ps or Pu ; kW/m3

126 179 262 179 429 229 140 256 55 115

1.6 to 1.8 1.9 to 2.2 2.3 to 2.6 1.5 to 1.86 3.0 to 5.20 1.37 to 1.48 0.60 to 0.90 2.32 to 3.60 0.76 0.46 to 0.57 1.5

Brass

m1 and m2 are taken from Table 25-1.

25.1.5 Tool design For comparison of Orthogonal Rake System (ORS), Normal Rake System (NRS) and American (ASA) tool nomenclature

Refer to Table 25-5.

25.1.6 Tool signatures The tool signature of ASA, ORS and NRS

The tool signature for sintered carbide tipped single point tool

p - f - p - f - o - s - rn

ðASAÞ

s - o - o - 1o

- o - p - rn

ðORSÞ

s - n - n - 1n - o - p - rn

ðNRSÞ

Refer to Fig. 25-5.

TABLE 25-5 Comparison of tool nomenclature system

Particular

Orthogonal rake system (ORS)a

Normal rake system (NRS)a

American Standards Association (ASA)b

Location of cutting edges Orientation of face Orientation of principal ﬂank Orientation of Auxiliary ﬂank Nose radius

p , o o , s o 0o rn

p , o n , s n 0n rn

s , o p , f p , f — rn

a

Tool reference system. b Machine reference system

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ELEMENTS OF MACHINE TOOL DESIGN

25.10

CHAPTER TWENTY-FIVE

Particular

Formula

Tool signature 8 Back rake angle Side rake angle End relief angle End clearance angle

Top view

7

7

9

6 10 8

0 1/64 = 0.4 mm

Side relief angle Side clearance angle End cutting edge angle Side cutting edge angle Nose radius

Carbide insert

8

1/64 in. = 0.4 mm 7

6 10

8

7 9

Right side view

Front view FIGURE 25-5 A straight shank, right cut, sintered carbide tipped, single point tool. Rake angles are negative. Courtesy: American Society of Tool and Manufacture Engineers, Fundamentals of Tool Design, Prentice Hall of India Private Ltd., New Delhi, 1969.

For general recommended various angles for HSS single-point tool

Refer to Table 25-6.

For general recommended various angle for carbide single-point tool

Refer to Table 25-8.

25.1.7 Tool life The relation between the tool life and cutting speed V according to Taylor

Kc ¼ V m

ð25-27Þ

where Kc ¼ constant taken from Table 25-7 or constant equal to the intercept of the tool life and cutting speed curve and the ordinate (V curve) m ¼ slope of the V curve m¼

logðV1 =V2 Þ logð2 =1 Þ

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ð25-28Þ

γp = back rack angle

υp = true wedge angle

φp

F

S

φs

O O1

P

O1

P

O

F

αo

υ s

N

lip angle

γo

υ = wedge angle

υn = true wedge angle

= normal clearance angle

Section N—N

γn = normal rack angle

γn

υn

αn

orthogonal rack angle

λs = inclination angle

φp = principal cutting edge angle φs = side cutting edge angle

N

Section O—O

φo = end cutting edge angle

orthogonal clearance angle

υo

ELEMENTS OF MACHINE TOOL DESIGN

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FIGURE 25-6 Single-point tool geometry (angles of machine reference system ASA, ORS and NRS). Courtesy: Principles of metal cutting—An introduction, Centre for Continuing Education, I.I.T., Madras, November, 1987.5

Section P—P

γp

υp

α1 = α’0

αp = front or end clearance or relief angle.

Section O1 —O1

γ1

υ1

Section F—F

αf = side clearance angle

υf

γf = side rack angle

ELEMENTS OF MACHINE TOOL DESIGN

25.11

∝

λs

n

N

= 10°

n

N

Section O–O

Section N–N

∝ ° ∝ = 6° °

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O

φ = 8° °

φ °

F

P

P

O1

Section F–F

φs

O1

P

φ

O

15'

F φs = 15°

S

∝f = 5°

P

∝ = 22° 6' P

∝

1

∝

Section P–P

P

P

= 2°

Section O1 –O1

1

∝

36'

FIGURE 25-7 Left hand single-point tool geometry (angles of machine reference system ASA, ORS and NRS) Courtesy: Principles of metal cutting—An introduction, Centre for Continuing Education, I.I.T., Madras, November, 1987.5

υ =true wedge angle, deg. ° φ =principal cutting edge angle, deg. P φ =end cutting edge angle, deg. ° φ =side cutting edge angle, deg. s

View S

λs = 0°

= 10°

∝

∝

°

°

∝

∝f

∝

25.12

∝

∝ f

ELEMENTS OF MACHINE TOOL DESIGN

CHAPTER TWENTY-FIVE

ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Particular

25.13

Formula

m ’ 0:1 to 0.15 for high speed steels (HSS) m ’ 0:2 to 0.25 for carbides m ’ 0:6 to 1.0 got ceramics and also taken from Table 25-7. The velocity of the job or bar of diameter D1 at speed n1

V1 ¼

D1 n1 1000 60

ð25-29Þ

TABLE 25-6 Recommended angle for high-speed-steel (HSS) single-point tools Back-rake angle, p , deg

Material High speed, alloy, and high-carbon tool steels and stainless steel Steels: C15 to C40 C45 to C90 14Mn1 S14, 40Mn 2S12 T50 Cr1 V23 40Ni2 Cr1 Mo28 Aluminum Bakelite Brass Commercial bronze Bronze Hard phosphor bronze Gray cast iron Copper Copper alloys, soft Copper alloys, hard Monel and nickel Nickel iron Fiber Formica Micarta Rubber, hard

Front-relief angle, p , deg

Side-rake angle, f deg

Side relief angle, f , deg

5 to 7

6 to 8

8 to 10

7 to 9

10 to 12 10 to 12 12 to 14 6 to 8 6 to 8 30 to 35 0 0 0 0 0 3 to 5 14 to 16 0 to 2 0 8 to 10 6 to 8 0 to 2 14 to 16 14 to 16 0 to 2

8 to 10 8 to 10 7 to 9 7 to 9 7 to 9 8 to 10 8 to 10 8 to 10 8 to 10 8 to 10 6 to 8 6 to 8 12 to 14 8 to 10 6 to 8 12 to 14 10 to 12 12 to 14 10 to 12 10 to 12 14 to 16

10 to 12 10 to 12 12 to 14 8 to 10 8 to 10 14 to 16 0 1 to 3 2 to 4 2 to 4 0 10 to 12 18 to 20 0 0 12 to 14 12 to 14 0 10 to 12 10 to 12 0 to 2

8 to 10 7 to 9 7 to 9 7 to 9 7 to 9 12 to 14 10 to 12 10 to 12 8 to 10 8 to 10 8 to 10 8 to 10 12 to 14 10 to 12 8 to 10 14 to 16 14 to 16 14 to 16 14 to 16 14 to 16 18 to 20

TABLE 25-7 Values of Kc and m Tool material

Kc

m

High-speed steel Carbide Ceramic

60–100 200–330 330–600

0.08–0.15 0.16–0.5 0.40–0.6

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ELEMENTS OF MACHINE TOOL DESIGN

25.14

CHAPTER TWENTY-FIVE

Particular

Formula

D2 n2 1000 60

The velocity of the job or bar of diameter D2 at speed n2

V2 ¼

For standard spindle speeds for machine tools

Refer to Table 23-66.

The relationship between the tool life, speed of cut, feed and depth of cut

K ¼ V m s m3 d m4

ð25-30Þ

ð25-31Þ

where m ¼ exponent taken from Table 25-7 m3 ¼ exponent of feed ’ 0:5 to 0.8 (average values) m4 ¼ exponent of depth of cut ’ 0:2 to 0.4 (average values)

For standard speeds, feeds and etc.

Refer to Tables 23-66 to 23-70

The approximate equation relating tool life to Brinell hardness number (HB )

K ¼ V m s m3 d m4 ðHB Þ

The cutting speed

k V ¼ 0:371 0:77 d s

ð25-32Þ

rﬃﬃﬃﬃﬃ! 6 60 Ccf

ð25-33Þ

TABLE 25-8 Recommended angles for carbide single-point tools

Material Copper, soft Brass and bronze Aluminum alloys Cast Iron: hard chilled malleable Low carbon steels 320 to 470 MPa Carbon steels 620 MPa Alloy steels Free machining steel 700 MPa Stainless steels, austenitic Stainless steels, hardenable High-nickel alloys Titanium alloys

Side cutting End edge, clearance, deg deg

Side clearance, deg

Nose radius rn , mm

6 to 8 6 to 8 10 6 to 8 6 to 8 10 8 to 12 6 to 10 10

10 10 10

2 to 3 2 to 3 2 to 3

2 to 3 2 to 3 2 to 3

0.4 2.00 0.4

þ6 to 7 3 to 6 4 to 8 þ6 to 7

3 to 3 to 3 to 5 to

5 5 5 10

3 to 4 3 to 4 3 to 5 5 to 10

10 10 10 10

10 10 10 10

2 2 2 2

to 3 to 3 to 3 to 3

2 to 2 to 2 to 2 to

3 3 3 3

0.75 0.75 0.75 0.75

þ6 to 7 þ6 to 7 þ6 to 7 þ6 to 7 þ6 to 7 þ6 to þ10 þ6 to 5

5 to 5 to 5 to 4 to 4 to 5 to 5 to

8 10 8 6 6 10 8

5 to 8 5 to 10 5 to 8 4 to 6 4 to 6 5 to 10 5 to 8

10 10 10 10 10 10 10

10 10 10 10 10 10 10

2 2 2 2 2 2 2

to 3 to 3 to 3 to 3 to 3 to 3 to 3

2 to 2 to 2 to 2 to 2 to 2 to 2 to

3 3 3 3 3 3 3

1.00 1.00

Back-rake angle, deg

Side-rake angle, deg

Endrelief, deg

0 to 10 0 to 5 0 to 10

15 to 25 þ8 to 5 10 to 20

0 to 7 0 to 7 0 to 4 0 to 7 0 to 7 0 to 7 0 to 7 0 to 7 0 to 7 0 to 3 0 to 5

Siderelief, deg

End cutting edge, deg

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1.00 1.00 1.00 1.00

ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Particular

Formula

where

1000

Cutting speed, v(m min 1)

25.15

500

Key : 1 HSS 2 Carbide 3 Ceramic

3 2 100

1

50

k1 ¼ an approximate constant taken from Table 25-9 Ccf ¼ correction factor for the tool material (18-4-1 HSS ¼ 100) taken from Table 25-10 rﬃﬃﬃﬃﬃ 6 60 ¼ a factor which will correct the cutting speed from that obtained for a basic 60-minute tool life to the cutting speed for the desired tool life

10

Tool life, t(min)

¼ tool life in minutes

FIGURE 25-8 Tool life () versus cutting speed (v). Courtesy: James Carvill, Mechanical Engineer’s Data Handbook, Butterworth-Heinemann, 1994.

Refer to Fig. 25-8.

For numerical values of d 0:37 and s0:77 for various values of d and s

Refer to Fig. 25-11.

25.2 MACHINE TOOLS For machine tools with a rotary primary cutting motion

Refer to Fig. 25-9.

TABLE 25-9 Numerical value for k1 k1 , for 18-4-1 high-speed-steel tool and tool life of

Material to be cut Light alloys Brass (80–120 HB ) Cast brass Cast steel Carbon steel: SAE1015 SAC1025 SAE1035 SAE1045 SAE1060 Chrome-nickel steel Cast iron: 100 HB 150 HB 200 HB

60 min without cutting ﬂuid, or 480 min with cutting ﬂuid

60 min with cutting ﬂuid

480 min without cutting ﬂuid

25.0 6.7 4.2 1.5

2.1

1.1

3.0 2.4 1.9 1.5 1.0 1.6

4.2 3.3 2.7 2.1 1.4 2.3

2.1 1.7 1.3 1.1 0.7 1.1

2.2 1.4 0.8

3.0 1.9 1.1

1.5 1.0 0.5

Courtesy: Wilson, F. W., Fundamentals of Tool Design, A.S.T.M.E., Prentice Hall of India Private Limited, New Dehli, 1969.

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ELEMENTS OF MACHINE TOOL DESIGN

25.16

CHAPTER TWENTY-FIVE

TABLE 25-10 Correction factors for compositions of tool material Approximate composition, % Type

W Cr V C

Co Mo Ccf

14-4-1 18-4-1 18-4-2 18-4-3 18-4-1 þ 5% Co 18-4-2 þ 10% Co 20-4-2 þ 18% Co Sintered carbide

14 18 18 18 18 18 20 —

— — — — 5 10 18 —

4 4 4 4 4 4 4 —

1 1 2 3 1 2 2 —

0.7–0.8 0.7–0.75 0.8–0.85 0.85–1.1 0.7–0.75 0.8–0.85 0.8–0.85 —

— — 0.75 — 0.5 0.75 1.0 —

0.88 1.00 1.06 1.15 1.18 1.36 1.41 Up to 5

TABLE 25-11 Numerical values for d 0:37 and s0:77 d

d 0:37

d

d 0:37

s

s0:77

s

s0:77

0.25 0.50 1.00 1.50 2.00 2.50 3.50 4.50 5.50

0.60 0.77 1.00 1.16 1.29 1.40 1.60 1.75 1.88

6.25 7.50 8.75 10.00 11.25 12.50 18.75 25.00

1.97 2.11 2.23 2.34 2.45 2.55 2.96 3.29

0.025 0.05 0.10 0.15 0.20 0.25 0.35 0.45 0.55

0.058 0.01 0.17 0.23 0.29 0.34 0.45 0.51 0.63

0.625 0.750 0.875 1.00 1.12 1.25 1.875 2.50

0.70 0.80 0.90 1.00 1.095 1.188 1.623 2.025

V

V

(a) Lathe

(b) Drilling machine

v V

(c) Milling machine

(d) Grinding machine

FIGURE 25-9 Machine tools with a rotary primary cutting motion.

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Particular

25.17

Formula

Vw V

Vr

Vw

(b) Shaping machine

(a) Slotting machine

Vw

Vr

(c) Planing machine FIGURE 25-10 Machine tools with a straight line reciprocating primary cutting motion. Courtesy: N. Acherkan, General Editor, Machine Tool Design, volume 1, Mir Publishers, Moscow, p. 21, 1968.

For machine tools with a straight line reciprocating primary cutting motion

Refer to Fig. 25-10.

25.2.1 Lathe turning (Fig. 25-9a) The tangential cutting force

Ft ¼ ks sd

ð25-34Þ

where ks ¼ speciﬁc cutting resistance or force oﬀered by the workpiece material per unit chip section, MPa ks ’ 3 to 5 u The maximum tangential force is also obtained from equation.

FtðmaxÞ ¼ 98:1 103 hc

The maximum tangential cutting force in terms of swing over the bed of the center lathe

FtðmaxÞ ¼ 49 103 hsw

ð25-35Þ

where hc in m and FtðmaxÞ in N

where hsw in m and FtðmaxÞ in N

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ð25-36Þ

ELEMENTS OF MACHINE TOOL DESIGN

25.18

CHAPTER TWENTY-FIVE

Particular

The maximum torque of the center lathe

Formula

MtðmaxÞ ¼ FtðmaxÞ

hswðmaxÞ 2

ð25-37Þ

where hswðmaxÞ ¼ maximum swing over cross-slide The maximum swing over cross slide for universal lathe

hswðmaxÞ ¼ ð0:55 to 0:7Þhsw

ð25-38Þ

The maximum torque of the lathe by taking hswðmaxÞ ¼ 0:6hsw

MrðmaxÞ ¼ 0:3FrðmaxÞ hsw

ð25-39Þ

The maximum feed force

Ff ¼ FxðmaxÞ þ F ¼ 0:6FcðmaxÞ

ð25-40Þ

where F ¼ Fr ; FxðmaxÞ ¼ 0:3FcðmaxÞ Fr ¼ reaction of the cutting force ¼ 2FcðmaxÞ ¼ coeﬃcient of friction between bed of the lathe and saddle ¼ 0:15 The radial component of cutting force Fy ð¼Fr Þ (Fig. 25-2)

Fy ¼

The deﬂection of the tool taking into consideration the eﬀect of cantilever of tool

t ¼

2:4EIW LW

ð25-41Þ

Ft l 3 3EI

ð25-42Þ

where l ¼ projected length of tool from the tool post, m I ¼ moment of inertia of area of the cross-section of the tool (bh3 =12) b ¼ width of the tool shank, m h ¼ depth of the tool shank, m Ft L3W 0:05 48EIW

The maximum deﬂection of job or work piece in the vertical plane due to cutting force Fc ¼ Ft which should not exceed 0.05 mm and Dw =Lw < 16

j ¼

The diameter of job or work piece

DW ¼ 0:25ds

ð25-44Þ

The length of job or workpiece which is equal to the distance between centers of center lathe

LW ¼ 8DW

ð25-45Þ

The tangential component of cutting force is also calculated from the equation

Ft ¼ 2:5Fy

ð25-46Þ

Another equation for the power due to tangential component of cutting force

Pc ¼

ð25-43Þ

where IW ¼ D4W =64 moment of inertia, m4 (cm4 )

Ft vkw ks sdv ¼ 1000 1000

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ð25-47Þ

ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Particular

25.19

Formula

The cutting speed for carbide tools may be taken as 150 to 170 m/min. The torque

Mt ¼

1000Pc !

ð25-48Þ

159Pc ð25-49Þ n0 where Mt in N m, Pc in kW, n0 in rps and ! in rad=s

Mt ¼

The minimum speed of work piece

nmin ¼

1000vmin Dmax

ð25-50Þ

The maximum speed of work piece

nmax ¼

1000vmax Dmin

ð25-51Þ

where D ¼ diameter of job in mm The bed width of lathe The moment acting on tailstock body in the plane xz1

B ¼ ð1:5 to 2Þhc Mtxz1

Wj ¼ Fz h 2

ð25-52Þ ð25-53Þ

where Fz ¼ Ft The moment acting on tailstock body in the plane xz2 The moment acting on tailstock body in the yz plane

Mtxz2 ¼ Fx hc

ð25-54Þ

Mtyz ¼ Fy hc

ð25-55Þ

where Wj ¼ weight of job, kN h ¼ lever arm of the vertical force, m (mm) Fa ¼ axial force with which tailstock center holds the job, kN

For speeds and feeds for turning of metals and plastics with HSS, carbide and Stellite tools

Refer to Table 25-12.

For cutting speeds and feeds for turning, facing and boring of cast iron, non-ferrous and non-metallic materials with HSS and carbide tools

Refer to Table 25-13.

25.2.2 Drilling machine CALCULATION OF FORCES AND POWER IN DRILLS (Fig. 25-11) For nomenclature of twist drills

Refer to Fig. 25-11.

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ELEMENTS OF MACHINE TOOL DESIGN

TABLE 25-12 Speeds and feeds for turning of metals and plastics with HSS, carbide and stellite tools Turning cutting speed, n (m/min) Depth of cut, d (mm) 0.1–0.5

0.5–2.0

2.0–5.0

6.0–10.0

Feed, s (mm/rev) Work piece material

Tool material

Mild steel

HSS Carbide Stellite HSS Carbide Stellite HSS Carbide Stellite HSS Carbide HSS Carbide Stellite HSS Carbide Stellite HSS Carbide Stellite HSS Carbide Stellite HSS Carbide Stellite HSS Carbide Stellite

Alloy steel

Stainless steel

Free cutting steel Cast, gray

Aluminum alloys Copper alloys

Magnesium alloys Monel metal

Thermosetting plastic

0.05–0.20

0.20–0.30

0.25–0.50

40–80 150–450

30–60 120–200

30–50 80–150

25–35 60–120

20–45 80–180

15–35 60–100

15–26 40–80

10–15 30–65

20–50 50–90

15–30 50–80

15–25 40–70

15–20 40–60

50–120 200–500 110–160 80–120

40–110 150–250 35–45 80–110

40–70 120–180 25–30 70–100

20–40 90–150 20–25 60–90

100–200 150–600

90–120 90–450

70–100 80–180

40–70 60–150

100–200 120–310

90–120 90–180

60–100 60–150

40–60 50–110

100–200 150–160

90–120 90–450

70–100 80–180

40–70 60–150

Speed, V (m/min) and feed s (mm/rev) for stellite tool

v ¼ 70–120; s ¼ 0:125–2.00 v ¼ 20–35; s ¼ 0:125–1.25 v ¼ 30–50; s ¼ 0:075–1.00

v ¼ 60–90; s ¼ 0:20–2.5 v ¼ 120–180; s ¼ 0:075–0.50 v ¼ 70–150; s ¼ 0:125–0.75 v ¼ 85–135 v ¼ 15–20; s ¼ 0:04–0.4; d ¼ 0:6–4 mm v ¼ 50–80; s ¼ 0:04–0.6; d ¼ 0:4–4 mm v ¼ 25–45; s ¼ 0:125–0.325 v ¼ 35–50; s ¼ 0:25; d ¼ 4 mm v ¼ 100–200; s ¼ 0:25; d ¼ 4 mm v ¼ 70–120; s ¼ 0:25; d ¼ 4 mm

Tang

Point angle Taper shank

Tang drive Flutes Shank diameter

0.40–0.60

Axis

Neck Straight shank Shank length

Overall length

Clearance Drill diameter diameter Body dia. clearance Chisel edge Rake Lip relief angle angle or Helix angle Margin

Flute length Body

Lip

Land Web Chisel edge

FIGURE 25-11 Nomenclature of twist drills. Courtesy: MCTI, Metal Cutting Tool Handbook.

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

25.21

Particular

Formula

The equation developed experimentally and analytically by Shaw and Oxford for torque of a twist drill operating in an alloy steel with an hardness of 200HB .

3 2 c 1:8 7 6 1 c d 7 Mt ¼ 37 106 s 0:8 d 1:8 6 0:2 þ 3:2 7 6 d 5 4 c 1þ d 2

ð25-56Þ where Mt in N m, c, s and d in m TABLE 25-13 Cutting speeds and feeds rates for turning, facing and boring of cast iron, non-ferrous and non-metallic materials with HSS and carbide tools. [Speed (at average HB ) for tool life of 1 12 to 2 hours between grinds, m/min] Cast iron Soft Medium hard HB ¼ 160–220 Depth of HB ¼ 160 cut, d Feeds, s, (mm) (mm/rev) HSS Carbide HSS Carbide 0.80

1.6

3.2

6.4

9.6

12.5

0.1 0.2 0.4 0.1 0,2 0.4 0.8 0.1 0.2 0.4 0.8 0.1 0.2 0.4 0.8 1.6 0.1 0.2 0.4 0.8 1.6 2.4 3.2 0.4 0.8 1.6 2.4

Hard HB ¼ 220–360

Aluminum, magnesium

Aluminum alloys, copper, soft bronze, soft brass, ﬁber, and plastic

Hard bronze, hard brass, hard rubber, and marble

HSS Carbide

HSS Carbide HSS

Carbide

HSS Carbide

200 150 110 150 120 71 71 120 100 71 56 110 80 60 45 36 90 70 56 40 25 20 —

400 300 220 300 250 180 140 250 200 150 110 180 140 110 90 45 150 120 90 71 40 32 —

90 71 56 71 63 45 32 63 50 36 25 45 40 32 20 16 40 36 25 20 16 9 —

90 70

160 120

50 40

120 90

32 25

100 71

80 63 50

140 110 100

45 32 25

110 80 63

25 20 16

90 71 56

71 56 45

140 110 90

40 32 25

90 71 63

25 20 16

71 63 50

62 45 40 25

120 90 80 56

32 25 20 16

71 63 45 36

20 16 10 8

63 50 40 25

40 36 25 — 20 40 32 25 16

80 71 50 — 36 80 60 45 25

20 16 10 — 9 20 16 10 8

56 45 32 — — 50 40 12 —

16 10 7 — 5 10 9 7 5

45 36 25 — — 40 12 20 —

500 360 280 400 300 220 180 300 250 200 140 250 180 140 110 71 200 150 120 90 63 45 —

150 120 90 120 100 71 56 95 80 63 45 50 63 50 36 25 71 56 45 32 20 16 —

Key: 1. In case of shock and impact cuts, 70% of above speeds for carbide tools and 80% above speeds for HSS tools are used. 2. The above speeds are for cutting without cutting ﬂuid. 3. A 10% reduction in the above speeds are recommended for soft, medium, hard, hard alloy and malleable irons.

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250 200 150 220 180 140 100 180 140 110 71 140 120 90 63 40 140 110 80 50 32 25 —

ELEMENTS OF MACHINE TOOL DESIGN

25.22

CHAPTER TWENTY-FIVE

Particular

The axial force or thrust acting on a drill

Formula

3 c 1 6 c 0:8 7 d 7 Fa ¼ 2:7 109 s 0:8 d 1:8 6 0:2 þ 2:2 7 6 d 4 5 c 1þ d 2

þ 1:33 108 c2

ð25-57Þ

where Fa in N, c, s and d in m c ¼ chisel edge length 1:15 web thickness for normal sharpening The equation for torque of a drill of regular proportions whose c=d may be set equal to 0.18

Mt ¼ 40 106 s 0:8 d 1:8

The axial thrust for a drill of regular proportions whose c=d is equal to 0.18

Fa ¼ 3:6 109 s 0:8 d 0:8 þ 39 106 d 2

The equation for torque at the spindle of a drill based on Brinell hardness number (HB )

Mt ¼ Cd 2 s 0:8 ðHB Þ0:7

ð25-60Þ

Mt ¼ 2 106 d 2 s 0:8 ðHB Þ0:7

ð25-61Þ

ð25-58Þ

where Mt in N m, s and d in m ð25-59Þ

where Fa in N, c, s and d in m

where Mt in N m, d and s in m The constant C ¼ 2 106 for HSS drill drilling in carbon steel. The equation for axial thrust at the spindle of a drill required for drilling which is based on Brinell hardness number (HB )

Fa ¼ Cds 0:7 ðHB Þ0:8

ð25-62Þ

Fa ¼ 1:9 106 ds 0:7 ðHB Þ0:8

ð25-63Þ

where Fa in N, d and s in m The constant C ¼ 1:9 106 for HSS drill drilling in carbon steel. Another equation for the turning moment on the drill

Mt ¼ Kt d 1:9 s 0:8

ð25-64Þ

Mt ¼ 41:7 106 s 0:8 d 1:9 for steel

ð25-65Þ

Mt ¼ 28:8 106 s 0:8 d 1:9 for cast iron

ð25-66Þ

where Mt in N m, s and d in m Another equation for the axial force acting on the drill

Fa ¼ Ksm d

ð25-67Þ

Fa ¼ 1:1 108 s 0:7 d for steel

ð25-68Þ

Fa ¼ 1:5 108 s 0:8 d for cast iron

ð25-69Þ

where Fa in N, s and d in m

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Particular

25.23

Formula

REAMERS (Fig. 25-12) The equation for torque of a reamer or core drill

The equation for axial thrust for a reamer or core drill

3 d1 2 7 6 1 d 7 Mt ¼ 37 106 Ks 0:8 d 1:8 6 0:2 7 6 5 4 d1 1þ d 2

where Mt in N m, s, d1 and d in m 2 2 3 d1 1 7 6 d 7 9 0:8 1:8 6 Fa ¼ 2:7 10 Ks d 6 0:2 7 5 4 d1 1þ d

ð25-70Þ

ð25-71Þ

where Fa in N, s, d1 and d in m d1 ¼ diameter of hole to be enlarged, m K ¼ a constant depending upon the number of ﬂutes. Refer to Table 25-14. For tapping drill sizes for coarse threads

Refer to Table 25-15.

For cutting speeds and feeds for drills

Refer to Table 25-16.

For drill angles, cutting angles and cutting lubricant for drilling with high speed steel drills.

Refer to Table 25-17.

TABLE 25-14 Values of constant K

TABLE 25-15 Tapping drill sizes for coarse threads

Number of ﬂutes

Constant, K

Number of ﬂutes

Constant, K

1 2 3 4 6

0.87 1.00 1.08 1.15 1.25

8 10 12 16 20

1.32 1.38 1.43 1.51 1.59

Courtesy: Wilson F. W., Fundamentals of Tool Design, A.S.T.M.E., Prentice Hall of India Private Limited, New Dehli, 1969.

Nominal Thread Tap drill Nominal Thread Tap drill diameter, pitch, size, diameter, pitch, size, mm mm mm mm mm mm 1.6 2.0 2.5 3.0 3.5 4.0 5.0 6.0 8.0 10.0 12.0 16.0

0.35 0.40 0.45 0.50 0.60 0.70 0.80 1.00 1.25 1.50 1.75 2.00

1.20 1.60 2.05 2.50 2.90 3.30 4.20 5.30 6.80 8.50 10.20 14.00

20.0 24.0 30.0 36.0 42.0 48.0 56.0 64.0 72.0 80.0 90.0 100.0

2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.00 6.00 6.00 6.00

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17.5 21.0 26.5 32.0 37.5 43.0 50.5 58.0 66.0 74.0 84.0 94.0

a

0.05

0.03

0.08 0.05

0.55

1.10

0.83 0.55

0.05 to 0.075 0.05 to 0.10 0.05 to 0.125 0.05 to 0.10 0.05 to 0.075

mm/rev 0.12

to

to

to

to

to

Feed s, mm

m/s 0.55

4275 5500 4575 6700 1500 1800 9150 1200 3650 1550

Speed n, rpm

0.85 0.55

1.10

0.55

m/s 0.55

2100 to 2800 2300 to 3350 7600 to 9100 4575 to 6100 1800 to 2250

Speed n, rpm

0.13 0.13

0.05

0.12

mm/rev 0.25

0.051 to 0.075 0.05 to 0.10 0.05 to 0.075 0.05 to 0.10 0.05 to 0.075

Feed s, mm

3.0

0.85 0.55

1.10

0.55

m/s 0.55

0 0.15

0

0.10

mm/rev 0.30

0.075 to 0.125 0.10 to 0.15 0.075 to 0.125 0.10 to 0.175 0.075 to 0.10

Feed s, mm

6.0

1050 to 1375 1150 to 1675 3800 to 4600 2300 to 3000 925 to 1150

Speed n, rpm

Extruded, molded or cast. b Extruded or molded. c Cast, molded or ﬁlled.

Plastics: Thermoplastics, polyethylene, polypropylene, TFB ﬂuorocarbona Nylon, acetates, polycarbonate Polystyreneb Thermosetting plastics: Soft gradec Hard tradec

Tool steel, steel castings, stainless steel and monel metal

Bronze, brass

Aluminum

Cast iron

Metals with HSS drills: Mild steel

Material

1.5

TABLE 25-16 Cutting speeds and feeds for drills

— —

—

—

m/s —

700 to 925 750 to 1125 2500 to 3000 1525 to 2025 600 to 750

Speed n, rpm

— —

—

—

mm/rev —

0.125 to 0.175 0.15 to 0.225 0.075 to 0.125 0.171 to 0.25 0.10 to 0.15

Feed s, mm

10.0

0.85 0.55

1.10

0.55

m/s 0.55

525 to 700 575 to 850 1900 to 2280 1150 to 1525 450 to 575

Speed n, rpm

0.20 0.20

0.10

0.20

mm/rev 0.38

0.150 to 0.20 0.20 to 0.30 0.15 to 0.20 0.25 to 0.35 0.15 to 0.225

Feed s, mm

12.0

— —

—

—

m/s —

— —

—

—

mm/rev —

0.25 to 0.35 0.30 to 0.40 0.20 to 0.25 0.35 to 0.45 0.20 to 0.30

Feed s, mm

16.0

425 to 550 450 to 675 1500 to 1800 900 to 1200 350 to 450

Speed n, rpm

Drill sizes, mm

0.85 0.55

1.10

0.55

m/s 0.55

350.0 to 450 375 to 550 1250 to 1500 750 to 1000 300 to 375

Speed n, rpm

0.25 0.25

0.13

0.25

mm/rev 0.46

0 25 to 0.35 0.30 to 0.40 0.20 to 0.25 0.35 to 0.45 0.20 to 0.30

Feed s, mm

20.0

— —

—

—

m/s —

300 to 400 325 to 475 1095 to 1300 650 to 875 260 to 325

Speed n, rpm

— —

—

—

mm/rev —

0.35 to 0.40 0.35 to 0.50 0.25 to 0.325 0.40 to 0.55 0.25 to 0.35

Feed s, mm

22.0

0.85 0.55

1.10

0.55

m/s 0.55

265 to 340 280 to 425 950 to 1125 575 to 750 225 to 280

Speed n, rpm

0.30 0.30

0.15

0.30

mm/rev 0.50

0.35 to 0.40 0.35 to 0.50 0.25 to 0.35 0.40 to 0.55 0.20 to 0.35

Feed s, mm

25.0

0.85 0.55

1.10

0.55

m/s 0.55

Speed n, rpm

0.38 0.38

0.18

0.38

mm/rev 0.64

Feed s, mm

30.0

ELEMENTS OF MACHINE TOOL DESIGN

25.24

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Particular

25.25

Formula

TABLE 25-17 Drill angles and cutting lubricants for drilling with HSS drills Lip relief angle, deg

Chisel edge angle, deg

Helix angle, deg

Cutting lubricants

Workpiece material

Hardness HB

Point angle, deg

Aluminum alloys Magnesium alloys Copper Brass Bronze Cast iron soft medium hard chilled Cast steel Mild steel Medium carbon steel Free machining steels Alloy steels Tool steels Stainless steels Spring steel Titanium alloy Monel metal Pure nickel Manganese steel Duraluminum Wood Bakelite

30–150 40–90 80–85 192–202 166–183

90–140 70–118 100–125 118–130 118

12–15 12–15 12–15 10–12 12–15

125–135 120–135 125–135 125–135 125–135

24–48 30–40 28–40 10–30 10–30

Paraﬃn, lard oil, kerosene Mineral oil Soluble oil Dry, lard oil, paraﬃn mixture Dry, lard oil, paraﬃn mixture

126 196 293–302 402 286–302 225–325 325–425 85–225 423 510 135–325 402 110–402 149–170 187–202 187–217 90–104 — —

90–100 90–100 118 118 118 118 118–135 118 118–135 150 118 150 118–135 118 118 130 118 60 130

8–12 8–12 8–12 8–12 12–15 10–12 8–10 12–15 7–10 7–10 7–10 7–10 7–10 12–15 10–12 7–10 10–12 15–20 7–10

125–135 125–135 125–135 125–135 125–135 125–135 125–135 125–135 125–135 125–135 125–135 125–135 125–135 125–135 125–135 125–135 125–135 135 135

24–32 24–32 24–32 24–32 24–32 24–32 24–32 24–32 24–32 24–32 24–32 24–32 20–32 24–32 24–32 24–32 32–45 24–32 24-32

Dry Dry Dry, soluble oil Dry, lard oil Lard oil, soluble oil Soluble oil, lard oil Sulfur base oil, mineral lard oil Soluble oil, lard oil Soluble oil, mineral oil Soluble oil, lard oil with sulfur Sulfur base oil Lard oil with sulfur Sulfur base oil Lard oil, sulfur base oil Lard oil Mineral oil Mineral oil None None

For elements of metal-cutting reamer

Refer to Fig. 25-12.

For reamer angles and cutting lubricants for reaming with HSS reamers.

Refer to Table 25-18.

25.2.3 Taps and tapping Power, P, at the spindle of tap for tapping of V-thread

1:75p P ¼ 6:3 103 Km Vpuc 0:15 þ uc where Km ¼ material factor taken from Table 25-19 V ¼ cutting velocity, m/min p ¼ pitch of thread, mm uc ¼ number of chamfered threads

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ð25-72Þ

ELEMENTS OF MACHINE TOOL DESIGN

25.26

CHAPTER TWENTY-FIVE

Particular

Formula

For material factor, Km , for use in drilling, reaming tapping

Refer to Table 25-19.

For nomenclature of tap

Refer to Fig. 25-13.

For tip angles and lubricants for tapping with HSS taps

Refer to Table 25-20.

Overall length

Chamfer-relief Chamfer length angle Land width Margin

Shank length Tang

Flute length

Taper shank

Cutter sweep

Chamfer angle

Actual size Chamfer length

Helix angle

Chamfer Radial rake Chamfer angle relief angle Straight flutes shown

Actual size

Straight shank Shank length

Helical flutes right-hand helix shown

(a)

Body

(a) Straight and Taper shank chucking reamer

(b) Machine reamer

FIGURE 25-12 Elements of metal-cutting reamers. Courtesy: MCTI, Metal Cutting Tool Handbook.

TABLE 25-18 Reamer angles and cutting lubricants for reaming with HSS reamers (Fig. 25-12) Chamfer

Workpiece material

Hardness, HB

Radial rack angle, deg

Primary relief angle, deg

Helix angle, deg

Aluminum alloys Magnesium alloys Brass, bronze Cast iron: soft, medium and hard Mild steel Free machining steels Alloy steels Tool steels

30–150 40–90 165–202

5–10 7 0–5

7–8 5–6 5–7

0–10 0.5–1.5 45 0 to 10 0.15–0.30 45 0–12 0.125–0.35 40

126–302 226–325 85–225 423 510

0–10 2–3 2–3 2–3 2–3

5–6 4–5 4–5 2–4 2–4

0–10 0–10 0–10 0–10 0–10

Margin width, mm

0.10–0.65 0.10–0.25 0.15–0.20 0.15–0.20 0.15–0.20

Angle, deg

45 45 45 45 45

Relief angle, deg

Length, mm

Cutting lubricants

10–15 10–15 10–15

1.5 1.5 2.5

Mineral lard oil Mineral oil Soluble oil

7–23 7 7 7 7

1.5 1.5 1.5 1.5 1.5

Dry, soluble oil Mineral lard oil Soluble oil Lard oil Lard oil

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Cutting edge

TABLE 25-19 Material factor, Km , for use in drilling, reaming and tapping calculations

Relieved to cutting edge

No relief

25.27

Heel Size of square Concentric

Eccentric relief

Workpiece material

Length of square

Concentric portion Eccentric relief Flute

Con-eccentric relief

Nagative rake angle

Length of shank Axis

Land

90… Helix angle Positive Root rake angle Crest

Zero rake Hook Tangential measurement

Land Flute

Core diam Hook angle (Crodel measurement)

Aluminum Copper alloys Cast iron

Shank diam

Overall length

Malleable iron Height of thread

Carbon steels

Major diam Pitch diam Minor diam Rear flank Front flank

Flute

Pitch

Thread angle Thread length

Base

Hardness, HB

Alloy steels

Thread half angle

Stainless steel

Chamfer angle

Ultimate tensile strength, ust , MPa

85 175 205 180 220 150 200 165 195 210 240 185 270

278.50 210.00 344.00 540.00 666.50 568.50 588.00 772.50 803.20 633.50 910.00

Material factor, Km 0.55 0.55 0.92 1.40 0.75 0.85 1.50 1.90 1.55 2.04 2.15 2.50 1.55 2.40

Chamfer length Point diam

Internal center External center

TABLE 25-20 Tap angles and cutting lubricants for tapping with HSS taps Rack angle Workpiece material

Hardness, HB

Hook, deg

Positive, deg

Aluminum Copper Bronze Brass Cast iron Cast steel Mild steel Medium carbon steel Alloy steels Tool steels Stainless steels Titanium alloys Phenolic plastics, hard rubber and ﬁbers

30–150 80–85 166–183 192–202 100–300 286–302 225–325 325–425 130–423 300–402 135–325 110–402

10–18 20 4–20 — 0–2

— — — 0 0 10–15

a b

9–12 9–12 10–15 5–10 15–20 5–12

Chamfera relief angle, deg 12 10 10 10 6–12 8 8 8 5–8 10 12

10–15

Number of ﬂutesb

Cutting lubricant

2–3 4 4 4 4 2–3 2–3 2–3 2–3 2–3 2–3 2–3 4

Kerosene and paraﬃn Milk Soluble oil Lard oil, soluble oil Dry or soluble oil Sulfur base oil, lard oil Soluble oil, lard oil Sulfur base oil, lard oil Soluble oil, mineral oil Soluble oil, lard oil with sulfur Sulfur base oil Sulfur base oil Dry

Chamfer length 212 to 3 threads for blind holes; 3 to 4 threads for through holes. For taps of 12 mm or smaller diameter.

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ELEMENTS OF MACHINE TOOL DESIGN

25.28

CHAPTER TWENTY-FIVE

Particular

Formula

25.2.4 Broaching machine BROACHES (Figs. 25-14 and 25-15) AND BROACHING For broach tooth form

Refer to Fig. 25-14.

For nomenclature of round pull broach

Refer to Fig. 25-15.

The allowable pull of internal or hole broach

Fapl ¼

A sut n

The permissible load on push type of round broach (Fig. 25-15) using Euler’s column formula with both ends free but guided

Fpps ¼

2 EI 2 E ¼ 2 nL2 nL

The allowable push in case of push type round broaches when E ¼ 206:8 GPa in Eq. (25-74)

Faps ¼

Note: when (L=D) is greater than 25, a push broach is considered as a long column and strength is based on this. If (L=D) is less than 25, the broach is considered to act as a short column which resist compressive load only.

Land

Rake angle

ð25-73Þ

D4r 64

100;000D4r nL2 where

ð25-74Þ

ð25-75Þ

Fapl ¼ allowable pull, N Faps ¼ allowable push, N A ¼ area of the minimum cross-section of broach which occurs at the root of the ﬁrst roughing tooth or at the pull end, mm2

Back off angle

Pitch

Straight land

Depth Radius FIGURE 25-14 Broach tooth form. Courtesy: American Broach and Machine Division, Sundstrand Machine Tools Company. Pull end

Shank

Front pilot

Semifinish Finishing teeth teeth Broach "length" Round hole broach Burnishing teeth

Roughing teeth

Round hole broach with burnishers

Rear pilot

Rear support

Follow rest grip

FIGURE 25-15 Nomenclature of a typical round pull broach. Courtesy: American Broach and Machine Division, Sundstrand Machine Tools Company.

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Particular

25.29

Formula

n ¼ factor of safety to prevent broach damage because of sudden overloads due to hard spots in material, etc. n ¼ 3 or more dependent on slenderness ratio sut ¼ tensile strength of the broach material, N/mm2 Dr ¼ root diameter of the broach at 1=2L, mm L ¼ length of broach from push end of ﬁrst cutting tooth, mm sa ¼ sut =n

The safe tensile stress for high speed steel

as ¼ 98 MPa for keyway broaches as ¼ 196 MPa for polygon broaches as ¼ 245 MPa for round/circular broaches The number of teeth cutting at a time in case of surface broaching

z¼

lmax þ1 p

ð25-76Þ

where lmax ¼ maximum length of workpiece, mm p ¼ pitch of the broach teeth, mm Sum of the length of all the teeth engaged at any instant in broaching

L ¼ Dz for circular/round broach

ð25-77aÞ

L ¼ bz for spline broach

ð25-77bÞ

L ¼ lz for surface broach

ð25-77cÞ

ks ¼ 4415 þ 3 108 24;515sz

The speciﬁc broaching/cutting force

ð25-78Þ

where ks in N/mm2 Also refer to Table 25-21 for ks

TABLE 25-21 Speciﬁc broaching force, ks Rise per tooth, sz , mm 0.03

0.04

0.05

0.06

0.08

0.1

Speciﬁc broaching force, ks , MPa

Material Mild steel Cast iron: Gray Malleable Alloy steel

4168

3580

3285

3040

2745

2550

3726 3334 5688

3236 2844 4505

2942 2648 4413

2648 2452 4168

2452 2206 3775

2305 2060 3530

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ELEMENTS OF MACHINE TOOL DESIGN

25.30

CHAPTER TWENTY-FIVE

Particular

Formula

¼ tensile strength of workpiece, N/mm2 ¼ rack angle, deg sz ¼ rise per tooth, mm The recommended speeds and feeds for broaching

Refer to Table 25-22

The broaching force

F ¼ kks ðDzÞsz for circular or round broaches ð25-79aÞ F ¼ kks ðbzÞus sz for spline or key broaches ð25-79bÞ F ¼ kks ðlzÞsz for surface broaches

ð25-79cÞ

where b ¼ width of spline or key, mm D ¼ diameter of broached hole, mm l ¼ width to be broached in case of surface broach, mm k ¼ coeﬃcient (may be taken as 1.1 to 1.3) z ¼ number of teeth engaged at a time us ¼ number of spline Another equation for the broaching force in case of key and splines broaching

F ¼ Cszm5 ðbzÞus

TABLE 25-22 Recommended speeds and feeds for broaching

TABLE 25-23 Broach angles for broaching with HSS broaching (Fig. 25-14)

Brinell hardness, HB

Workpiece material Aluminum alloys Copper alloys Cast iron: Gray

Malleable Low alloy steels Carbon steel Free cutting steel

Rise per tooth, mm

Cutting speed, m/min

30–150 40–200

0.15 0.12

10–20 8–10

110–140 190–220 250–320 110–400 85–125 120–375 100–200 275–325 325–375

0.13 0.07 0.05 0.15 0.10 0.08 0.10 0.07 0.05

9 8 4.5 5–30 9 3–8 10–12 6 6

Workpiece material Aluminum/ magnesium Copper alloys Cast iron Lead brass Mild steels Alloy steels Tool steels Stainless steel Titanium

ð25-80Þ

Brinell hardness, HB 30–150 40–200 100–320 — 225–325 130–423 300–402 135–325 110–402

Hook/rake angle, deg

Clearance angle, deg

10–15

1–3

0–10 6–8 5 to þ5 15–20 8–12 8–12 12–18 8–26

1–3 2–3 1–3 2–3 1–3 1–2 2–3 2–8

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Particular

25.31

Formula

where C ¼ coeﬃcient which takes into consideration condition of cutting and characteristic of workpiece. Taken from Table 25-25. sz ¼ feed per tooth, mm (Table 25-25) m5 ¼ exponent taken from Table 25-25 Another equation for the broaching force in case of cylindrical broaching

F ¼ Cszm5 Dz

The velocity of broaching

v¼

Kv m6 szm7

ð25-81Þ

ð25-82Þ

where Kv ¼ velocity coeﬃcient depends on the conditions of metal cutting (Table 25-24) ¼ life of tool, min su ¼ stress of material, N/m2 , from Table 25-24 The power required for broaching by the broaching machine

Fv 1000 where F in N, v in m/s, and P in kW

P¼

ð25-83Þ

25.2.5 Milling machines A knee horizontal-milling machine for plain or slab milling

Refer to Fig. 25-16.

A knee-type vertical milling machine for face milling

Refer to Fig. 25-17.

For nomenclature and tool geometry of milling cutters

Refer to Figs. 25-18 and 25-19a.

For tool angles of millings cutters

Refer to Table 25-26 and Figs. 25-18 and 25-19a. rﬃﬃﬃﬃ z h ð25-84Þ k¼ ¼ " D where

The engagement parameter (Fig. 25-19a)

¼ engagement angle for milling depth, h rﬃﬃﬃﬃ h ¼2 D

ð25-85Þ

" ¼ peripheral pitch angle, deg

2 z

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ð25-86Þ

— — up to 686 686–785 above 785

200 200 up to 200 200–230 above 200 14.0 11.5 16.8 15.5 11.2

Kv 0.50 0.50 0.62 0.62 0.62

m6 0.60 0.60 0.62 0.62 0.62

m7

sz as given in Table 25-25

Circular or round broaching

Alloy steels

Cast steel

Cast iron

Stress su , MPa — — 686 686–785 >785 686 686–785 >735

Brinell hardness, HB

200 >200 200 200–230 >230 200 200–230 >230

Workpiece material

2942 3472 6865 7472 8257 6865 7472 8257

C

6.2 5.1 9.2 8.8 6.3

Kv

0.04–0.08 0.03–0.06 0.02–0.03 0.02–0.05 0.02–0.03 0.02–0.03 0.02–0.04 0.02–0.03

sz 0.73 0.73 0.85 0.85 0.85 0.85 0.85 0.85

0.6 0.6 0.87 0.87 0.87

m6

1128 1344 1735 1980 2452 1735 1980 2452

C

6.2 5.1 7.7 7.0 5.0

Kv 0.6 0.6 0.87 0.87 0.87

m6

0.08–0.15 0.07–0.12 0.04–0.07 0.07–0.12 0.04–0.07 0.03–0.06 0.06–0.10 0.04–0.07

sz

Keyway broaching

0.95 0.95 1.4 1.4 1.4

m7

0.73 0.73 0.85 0.85 0.85 0.85 0.85 0.85

m5

0.95 0.95 1.4 1.4 1.4

m7

sz > 0:07 mm

Keyway broaching sz 0.07 mm

m5

Circular or round broaching

TABLE 25-25 Values of C, sz and m5 for use in Eqs. (25-80) and (25-81)

Steels

Cast iron

Stress su , MPa

Brinell hardness, HB

Workpiece material

1490 2108 2079 2255 2785 2079 2255 2785

C

17.5 14.7 15.5 14.0 10.2

Kv

0.05–0.10 0.04–0.08 0.04–0.06 0.04–0.08 0.03–0.05 0.03–0.05 0.04–0.06 0.03–0.05

sz

Spline broaching

0.5 0.5 0.6 0.6 0.6

m6

0.73 0.73 0.85 0.85 0.85 0.85 0.85 0.85

m5

0.6 0.6 0.75 0.75 0.75

m7

sz as given in Table 25-25

Spline broaching

25.32

TABLE 25-24 Values constant, Kv and exponents m6 and m7 for use in Eq. (25-82)

ELEMENTS OF MACHINE TOOL DESIGN

CHAPTER TWENTY-FIVE

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Particular

25.33

Formula

For up-milling and down-milling processes

Refer to Fig. 25-19.

The minimum number of teeth for satisfactory cutting action (Fig. 20-19a)

2 zmin ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ h=D

ð25-87Þ

where h ¼ depth of milling, mm 1 For h=D ¼ 10 to

The circumferential or circular pitch

pc ¼

1 20

the zmin lies between 20 and 28.

D z

ð25-88Þ

Overarm Arbor Work table

Column

Saddle Knee Base (a) Knee - type horizontal milling machine Y Z Machined surface

Tool

X

Primary motion (C) Continuous feed motion(X’) Workpiece Work surface

(b) Helical milling cutter FIGURE 25-16 Knee-type horizontal milling machine for plane milling. Courtesy: G. Boothroyd, Fundamentals of Metal Machining and Machine Tools, McGraw-Hill Book Company, New York, 1975.9

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ELEMENTS OF MACHINE TOOL DESIGN

25.34

CHAPTER TWENTY-FIVE

Particular

Formula

pc D ¼ tan z tan

The axial pitch

pa ¼

The number of teeth in engagement in case of plain milling cutter whose helix angle is

z b tan þ zs ¼ D

The design equation for the number of teeth on milling cutter

ð25-89Þ

rﬃﬃﬃﬃ! h D

ð25-90Þ

where b ¼ width of cutter, mm pﬃﬃﬃﬃ z¼m D

ð25-91aÞ

where m is a function of helix angle . Table 25-27 gives values of m for various helix angles .

Head

Table

Column

Saddle Knee Z X

Y

Base (a) Knee - type vertical milling machine

Cutting Edge

Primary motion ( C)

Tool

Relief Angle Primary clearance angle (α) Secondary clearance angle (α1)

Lip Angle

Machined surface

Back of Tooth Face of Tooth

Radial Rake Angle = γ1

Land (f) Gash or Chip space

BODY OF CUTTER

(b) Face milling cutter

FIGURE 25-17 Knee-type vertical milling machine for face milling. Courtesy: G. Boothroyd, Fundamentals of Metal Machining and Machine Tools, McGraw-Hill Book Company, New York, 1975.9

Direc

Work surface

tion

Root diameter (Dr)

f Rota

Workpiece

tion o

Continuous feed motion ( X’)

Fillet or Root radius

Tooth depth

Outside diameter (D ) ¡

FIGURE 25-18 Nomenclature and geometry of milling cutter.

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Particular

Formula

R=

γ

α1

ψ

V

D 2 f

∋ h

δ

Feed

α

δ = gullet angle, deg. γ = rake angle α = primary clearance angle α1 = secondary clearance angle h = depth of cut or depth of mmilling, min

Feed

= peripheral pitch angle or angular pitch, deg. = engagement angle for milling depth, h = (ψ / = engagement parameter = land, mm

(∋

∋

ψ k f

25.35

(b) Up-milling

(a) Down-milling

FIGURE 25-19 Horizontal milling process.

The gullet angle (Fig. 25-19a)

¼"þ

if rack angle ¼ 0

ð25-91bÞ

where ¼ wedge angle, deg CHIP FORMATION IN MILLING OPERATION PLAIN MILLING (Fig. 25-21) The maximum undeformed chip thickness in case of plain or slab milling (Fig. 25-21) as per Martellotti10,11 tucðmaxÞ

2 8 91=2 3 D > > > > > > 1 6 < = 7 h 7 cos 6 ¼ 6sz 2 2 7 5 4 > > v v D D > f f > > > 1 : ; 2h V Vh ð25-92Þ

The length of undeformed chip (Fig. 25-21)

l¼

The inherent roughness height

R¼

D 2

1=2 vf D h 1 h V D sz

sz

z

ð25-93Þ ð25-94Þ

where the upper sign (þ) refers to up-milling and the lower sign () refers to down-milling The feed s which is equal to the distance moved by the workpiece during one resolution of tool (Fig. 25-21)

vf n where s in mm/rev

s¼

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ð25-95aÞ

Radial rake γr

(b) End mills

Corner

Axial relief αa

Corner

Axial rake α

Axial rake a or helix angle β

Radial relief αr Axial relief αa

Axial rake λa

Approach angle x

(a) Face mills

Radial relief αr

(c) Side and slot mills

Axial relief αa

Radial relief αr

Radial rake γr

Radial rake γr

Figure 25-20 30–150

Side and slot End

25.36

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6

15 12

Diameter, mm

r1 , deg r2 , deg

13 11

10 13 10

12 12 9

16 10 8

25

85–440

Face

Side 135–425 and slot End

Face

Side and slot End

Face

Side 100–400 and slot End

Face

Brinell hardness, HB

Types of of mills

Note: 1. Use 1 458 or radius for corner. 2. End cutting edge concavity angle: (a) for aluminum, 5 deg. (b) for alloy steels and aluminum, 3 deg. a 3. Radial relief angles (r ) for end mill

Stainless steels

Steels (machineability 100)

Cast iron (machineability 100)

Aluminum alloys

Workpiece material

TABLE 25-26 Tool angles of milling cutters (Figs. 25-18 and 25-19)

HSS Carbide HSS Carbide HSS Carbide HSS Carbide HSS Carbide HSS Carbide HSS Carbide HSS Carbide HSS Carbide HSS Carbide HSS Carbide HSS Carbide

Material of tool

5 to 10 5 to 10 10–15 5 to þ5 10–20 3–5 10–15 0 to 7 5–12 2–4 15 0–3 10–12 0 to 10

10–20 5–15 15–20 5–8 20–35 10–20 10–20 þ5 to 10 12

Radial rake,

r , deg

20–30 5–10 10–15 0 to 5 30–35 15—25 10–15 0 to 7 10–12 5 to þ5 30–35 15–25 10–15 0–10

10–25 10–20 30–45 25 20–35 10–20 10–12 0 to 10 30–35

Axial rake,

a , deg

r2 a 4–8 3–7 3–7 4–8 5–8 — r2 a 8–10

4–7

3–7 5–8 r2 a

10–12

5–11 7–10 r1 a

Radial relief, r , deg

Tool angles

2–4 3–7 3–5 5–7 5–7 3–5 2–4 — 3–7 8–10 3–7

3–7 4–7

2–4 3–5

3–5

5–7 5–7 8–12

Axial relief, a , deg

30

30

30

30–45

Helix angle, , deg

ELEMENTS OF MACHINE TOOL DESIGN

ELEMENTS OF MACHINE TOOL DESIGN

25.37

ELEMENTS OF MACHINE TOOL DESIGN

Particular

Formula

TABLE 25-27

δ

Helix angle, , deg

m

10–20 20–30 30–45

1.25–1.5 0.8–1.25 0.5–0.8

ψ D 2

S

h B

V R A

l

tuc(max)ψ

D 2

h Lw

υf

FIGURE 25-21 Geometry of plain-milling chip.

The engagement angle for milling depth, h The feed per tooth of milling cutter

If V (cutter velocity) vt (feed rate) and trochoidal arcs are replaced by circular arcs, Eqs. (25-92), (25-93) and (25-94) become

FIGURE 25-22 Relative motion between the workpiece and a plain milling cutter.

2h ¼ cos1 1 D vf nz where s in mm/rev, sz in mm/tooth,

sz ¼

tucðmaxÞ ¼ sz sin cos sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 2vf h h tucðmaxÞ ¼ 1þ for ¼ 0 D D zn l¼

D s cos þ z 2 2

R¼ If (h=D) is very small i.e.: when ðh=DÞ 1, Eqs. (25-96), (25-97) and (25-98) become

h (D - h)

s2z 4D

rﬃﬃﬃﬃ! h tucðmaxÞ 2sz cos D rﬃﬃﬃﬃ 2vf h tucðmaxÞ for ¼ 0 zn D pﬃﬃﬃﬃﬃﬃﬃ vf l ¼ Dh 2zn R

s2z 4D

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ð25-95bÞ ð25-95cÞ in rad ð25-96aÞ ð25-96bÞ ð25-97Þ ð25-98Þ ð25-99aÞ ð25-99bÞ ð25-100Þ ð25-101Þ

ELEMENTS OF MACHINE TOOL DESIGN

25.38

CHAPTER TWENTY-FIVE

Particular

Formula

The machining time (Fig. 25-22)

m ¼

Lm þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ hðD hÞ vf

ð25-102Þ

where Lm ¼ length of workpiece, mm The metal removal rate or feed rate which is equal to the product of feed speed and cross-sectional area of the metal removed, measured in the direction of feed motion

hbvf bhsm ¼ 1000 1000 where

Qw ¼

ð25-103Þ

sm ¼ feed ¼ sz nz, mm/min b ¼ back engagement which is equal to the width of the workpiece Qw in cm3 /min FACE MILLING (Fig. 25-23) The maximum chip thickness in case of face-milling

The average value of chip thickness in case of facemilling (Fig. 25-23)

vf cos c ¼ sz cos c ð25-104aÞ nz where c denotes the corner angle, deg " # 57:3 2b1 2ðb b1 Þ tav ¼ þ cos sz sin cos D D tcðmaxÞ ¼

ð25-104bÞ where ¼ approach angle, deg The approximate length of chip

b

l¼

ð25-105Þ

D 2

tc(max)

ψ

D b1

D 2

Lw

D 2

(a)

2

n

vf

FIGURE 25-23 Face milling chip formation.

h (D - h)

Lw

h (D - h) h D 2

(b) FIGURE 25-24 Relative motion between the workpiece and the face milling cutter.

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ELEMENTS OF MACHINE TOOL DESIGN

25.39

ELEMENTS OF MACHINE TOOL DESIGN

Particular

Formula

The angle of engagement with the workpiece for use in Eq. (25-105) (Fig. 25-23) The value of tcðmaxÞ when the corner angle c is zero

¼ sin1

2b1 D

þ sin1

2ðb b1 Þ D

ð25-106Þ

vf ¼ sz nz

ð25-107Þ

The machining time, when the path of tool axis passes over the workpiece, is given by (Lw þ D) [Fig. 25-24(a)]

m ¼ ðLw þ DÞvf

ð25-108Þ

The machine time when the path of the tool axis does not pass over the workpiece [Fig. 25-24(b)]

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Lw þ 2 hðD hÞ m ¼ vf

ð25-109Þ

tcðmaxÞ ¼

where Lw ¼ length of the workpiece, mm END MILLING AND SLOT MILLING The average chip thickness in case of end-milling and slot-milling (Fig. 25-25)

tav ¼

114:6

sz

h D

ð25-110Þ

ω

h

ψ

D

Feed FIGURE 25-25 End-milling and slot-milling.

FORCES AND POWER The empirical equation for the tangential force, Ft , in milling operation as per Kovan12

Ft ¼ Chx syz zbp Dq

ð25-111Þ

where C ¼ constant depends on the material of the tool, and the workpiece taken from Table 25-28 z ¼ number of teeth on milling cutter in simultaneous contact with the workpiece

TABLE 25-28 Values of x, y, p, q and C for use in Eq. 25-111 (approx) Material of workpiece

x

y

p

q

C

Cast iron Steel

0.83–1.14 0.86

0.65–0.70 0.74

1.00–0.90 1.00

0.83 to 1.14 0.86

470–686 392–785

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ELEMENTS OF MACHINE TOOL DESIGN

25.40

CHAPTER TWENTY-FIVE

Particular

Formula

z¼

z 360

b ¼ width of milling cutter or chip ¼ h=sin

x, y, p and q exponents taken from Table 25-28 for steels and cast iron Ft in N The tangential force, Ft , can also be calculated from unit power concept

1000P V where

Ft ¼

ð25-112Þ

P ¼ power at the spindle, kW V ¼ cutting speed, m/s D 2 where Mt in N m, D in m, Ft in N

ð25-113Þ

P¼

Ft V 1000 where P in kW, Ft in N, V in m/s

ð25-114Þ

P ¼ Pu kh kr Q

ð25-115Þ

The torque

Mt ¼ Ft

The power

The power at the spindle from the concept of unit power

where Pu ¼ unit power, kW/cm3 /min or kW/m3 /min as per Table 25-29 kh ¼ correction factor for ﬂank wear as per Table 25-30 kr ¼ correction factor for radial rake angle as per Table 25-31

Another equation for power for peripheral milling

Q ¼ metal removal rate, cm3 /min or m3 /min m9 h ð25-116Þ P ¼ kvzbCszm8 R where k, C, m8 and m9 are taken from Tables 25-32 and 25-33, P in W, v in m/s, b in mm, sz ¼ mm/tooth, h in mm, m8 and m9 are indices, C ¼ constant from Table 25-33, R ¼ D=2 ¼ radius of cutter, mm

The approximate relationships between Fr ð¼Fx Þ, Ff ð¼Fy Þ and Ft ð¼Fz ¼ Fc Þ for diﬀerent milling process

Fr ð¼Fx Þ ¼ 0:5Ft to 0:55Ft ð¼Fz Þ

ð25-117Þ

for symmetrical face-milling Ff ð¼Fy Þ ¼ 0:25Ft to 0:35Ft ð¼Fz Þ for symmetrical face-milling

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ð25-118Þ

ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

25.41

TABLE 25-29 Average unit power Pu , for turning and milling Unit power Pu , 103 kW=cm3 /min

Work material Free machining steels Mild steels Medium carbon steels Alloy steels Tool steels Stainless steelsa

Cast irona Gray, Ductile, Malleable

Aluminum alloys

Copper alloys Magnesium alloys Titanium alloys Ti–Al–Cr Pure Ti Ti–Al–Mn Ti–Al–V Ti–Al–Cr–Mo a

Tensile strength, st MPa

0.025

0.05

0.075

0.1

0.15

0.2

0.3

0.5

0.8

390 490 588 686 785 880 980 1078 1470 1570 1666 1765 1863 1960 1570 1666 1765 1863 1960 2157 2354 2550 2745 98 196 294 392 490 — 98 147 196 245

54 60 66 69 73 78 80 85 80 86 92 99 104 110 30 31 35 36 38 42 46 50 53 13 19 24 28 32 25 9 10 12 13

45 50 55 59 63 65 69 72 71 76 82 90 96 101 26 28 30 31 33 36 40 43 46 11 16 20 23 26 21 7 9 10 11

41 45 50 53 56 59 62 65 66 72 78 84 91 96 24 25 27 29 30 33 36 39 42 9 14 17 21 23 19 6 8 9 10

39 42 47 50 52 56 59 61 61 67 73 80 86 91 22 24 25 27 28 31 34 37 39 9 13 16 19 22 17 6 7 8 9

35 39 42 45 48 50 53 56 57 62 68 75 81 88 21 22 23 24 26 29 31 33 36 8 12 14 17 19 16 5 6 7 8

33 36 39 42 44 47 49 53 52 58 61 69 78 85 19 20 22 23 24 26 29 31 34 7 11 13 16 18 15 5 6 7 7

30 32 35 37 40 42 44 5.1 48 54 56 62 69 78 18 1.9 21 21 22 24 27 29 31 6 10 12 14 16 13 4 6 6 7

26 29 31 33 35 38 39 44 44 50 52 59 64 71 16 17 19 17 20 22 24 26 28 6 8 10 12 14 12 4 5 5 6

23 26 28 30 32 34 35 36 40 46 48 52 58 60 14 15 17 17 18 20 21 23 25 5 7 9 11 12 10 3 4 4 5

1078 — — — —

59 61 67 68 77

51 52 58 59 66

47 48 53 54 60

45 45 50 52 57

41 41 45 47 52

39 38 43 45 49

36 35 39 41 45

32 31 35 37 40

30 28 31 34 36

Average chip thickness, mm

Values in HB .

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ELEMENTS OF MACHINE TOOL DESIGN

25.42

CHAPTER TWENTY-FIVE

TABLE 25-30 Correction factor for ﬂank wear Correction coeﬃcient, kh Hardness of work material Flank wear, mm 0.2

0.4

0.6

0.8

1

Average chip thickness, mm

125

150

200

250

300

350

400

51

56

61

0.1 0.3 0.5 1.0 0.1 0.3 0.5 1.0 0.1 0.3 0.5 1.0 0.1 0.3 0.5 1.0 0.1 0.3 0.5 1.0

1.16 1.06 1.04 1.02 1.50 1.20 1.12 1.06 1.68 1.26 1.17 1.09 1.91 1.35 1.23 1.12 2.18 1.45 1.30 1.15

1.17 1.07 1.05 1.02 1.50 1.20 1.12 1.06 1.71 1.25 1.19 1.10 2.04 1.41 1.28 1.14 2.32 1.50 1.34 1.16

1.18 1.08 1.05 1.03 1.50 1.20 1.14 1.07 1.73 1.29 1.20 1.10 2.10 1.42 1.32 1.15 2.39 1.56 1.39 1.17

1.19 1.08 1.05 1.03 1.53 1.22 1.15 1.07 1.84 1.33 1.23 1.12 2.34 1.52 1.36 1.17 2.54 1.67 1.47 1.20

1.20 1.09 1.05 1.03 1.57 1.23 1.16 1.08 1.94 1.37 1.26 1.14 2.47 1.56 1.38 1.18 2.65 1.70 1.45 1.23

1.21 1.09 1.06 1.30 1.67 1.27 1.19 1.10 2.09 1.44 1.30 1.16 2.54 1.62 1.43 1.23 2.84 1.74 1.51 1.27

1.22 1.09 1.07 1.03 1.78 1.32 1.24 1.12 2.20 1.50 1.37 1.19 2.65 1.70 1.52 1.27 3.15 1.90 1.67 1.35

1.25 1.13 1.08 1.04 1.80 1.36 1.26 1.14 2.43 1.61 1.47 1.25 2.99 1.90 1.66 1.35 3.46 2.16 1.84 1.44

1.33 1.16 1.12 1.06 1.92 1.41 1.30 1.16 2.72 1.78 1.57 1.30 3.26 2.02 1.74 1.40 — — — —

1.38 1.18 1.13 1.07 2.12 1.52 1.38 1.20 2.82 1.85 1.61 1.33 — — — — — — — —

HB

RC

Note: HB ¼ Brinell hardness number, RC ¼ Rockwell hardness scale C

TABLE 25-31 Correction factor for rake angle, kr Rake angle, degrees Correction coeﬃcient, kr

15 10 5

þ5

0

þ10 þ15 þ20

1.35 1.29 1.21 1.13 1.07 1

0.93 0.87

TABLE 25-33 Values of C for use in Eq. (25-116) Material

TABLE 25-32 Values of m8 , m9 , k for use in Eq. (25-116) Material

m8

m9

k

Steel Cast iron

0.85 0.70

0.925 0.85

0.164 0.169

Free machining carbon steel Carbon steels Nickel-chrome steels Nickel-molybdenum and chromemolybdenum steels Chrome-vanadium steels Flake graphite cast iron Nodular cast irons

C 980 (120 HB )

1190 (180 HB )

1620 (125 HB ) 1460 (125 HB ) 1600 (150 HB )

2240 (225 HB ) 220 (270 HB ) 1960 (280 HB )

1820 (170 HB ) 635 (100 HB ) 1110 (annealed)

2380 (190 HB ) 1330 (263 HB ) 1240 (as cast)

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Particular

25.43

Formula

Fr ð¼Fx Þ ¼ 0:5Ft to 0:55Ft ð¼Fz Þ

ð25-119Þ

for asymmetrical face-milling Ff ð¼Fy Þ ¼ 0:30Ft to 0:40Ft ð¼Fz Þ

ð25-120Þ

for asymmetrical face-milling Fr ð¼Fx Þ ¼ 0:15Ft to 0:25Ft ð¼Fz Þ

ð25-121Þ

for end milling (308 helical ﬂute cutters) Ff ð¼Fy Þ ¼ 0:45Ft to 0:55Ft ð¼Fz Þ for end milling (308 helical ﬂute cutters) For types and deﬁnitions of milling cutters

Refer to Table 25-34.

For feed per tooth for milling; cutting speeds for face and end milling; feeds and speeds for hobbing.

Refer to Tables 25-35, 25-36 and 25-37.

For milling cutters selection, dimensions for interchangeability of milling cutters, milling arbors with tenon drive and milling arbor with key drive and diﬀerent types of milling cutters.

Refer to Tables 25-38 to 25-48

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ð25-122Þ

ELEMENTS OF MACHINE TOOL DESIGN

25.44

CHAPTER TWENTY-FIVE

TABLE 25-34 Types and deﬁnitions of milling cutters

Type

Arrangement of teeth

Application

Size Up to 160 160 mm

Cylindrical (slab or rolling)

Helical teeth on periphery

Flat surfaces parallel to cutter axis

Side and face

On periphery and both sides

Steps and slots Up to 200 mm diameter, 32 mm wide

Appearance

Clearance

Straddle ganged

On periphery and both sides

Cutting two steps

Up to 200 mm diameter, 32 mm wide

Side and Teeth on face periphery. Face teeth on staggered tooth alternate sides

Deep slots

Up to 200 mm diameter, 32 mm wide

Single angle Teeth on conical surface and ﬂat face

Angled surfaces 60–858 in 58 steps and chamfers

Feed

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

TABLE 25-34 Types and deﬁnitions of milling cutters (Cont.)

Type

Arrangement of teeth

Double angle

Application

Size

Teeth on two conical faces

Vee slots

458, 608, 908

Rounding

Concave quarter circle and ﬂat face

Corner radius on edge

1.5–20 mm radius

Involute gear cutter

Teeth on two involute curves

Involute gears

Large range

Appearance

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25.45

ELEMENTS OF MACHINE TOOL DESIGN

25.46

CHAPTER TWENTY-FIVE

TABLE 25-34 Types and deﬁnitions of milling cutters (Cont.)

Type End mill

Arrangement of teeth

Application

Helical teeth at one Light work, end and slots, circumferential proﬁling, facing narrow surfaces

Size

Appearance

50 mm

TANGED END

TAPPED END

Parallel Shank

Tee slot

Circumferential and both sides

Tee slots in machine table

Dovetail

On conical surface Dovetail and one end face machine slides

For bolts up to 24 mm diameter

38 mm diameter, 458 and 608

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

TABLE 25-34 Types and deﬁnitions of milling cutters (Cont.)

Type

Arrangement of teeth

Application

Skid end mill

Circumferential and one end

Larger work 40–160 mm than end mill diameter

Cutting Circumferential saw (slot) teeth

Cutting oﬀ or slitting. Screw slotting

Size

Appearance Cutter Arbor

60–400 mm diameter

Clearance

Thick Thin

Concaveconvex

Curved teeth on periphery

Radiusing

1.5–20 mm radius

Concave

Convex

Thread milling cutter

PARALLEL SHANK

TAPER SHANK

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25.47

ELEMENTS OF MACHINE TOOL DESIGN

TABLE 25-35 Suggested feed per tooth for milling various materials, mm

Face mills Materials to be milled Cast iron Soft (up to 160HB ) Medium (160 to 220HB ) Hard (220 to 320HB ) Malleable irona Steel Softa (up to 160HB ) Medium (160 to 220HB ) Harda (220 to 360HB ) Stainlessa Brass and Bronze Soft Medium Hard Copper Monel Aluminuma a

Helical mills

Slotting and side mills

End mills

Form relieved cutters

Circular saws

HSS

Carbide HSS

Carbide HSS

Carbide HSS

Carbide HSS

Carbide HSS

Carbide

0.40 0.32 0.28 0.30

0.50 0.40 0.30 0.35

0.32 0.25 0.20 0.25

0.40 0.32 0.25 0.28

0.22 0.18 0.15 0.18

0.30 0.25 0.18 0.20

0.20 0.18 0.15 0.15

0.25 0.20 0.15 0.18

0.12 0.10 0.08 0.10

0.15 0.12 0.10 0.10

0.10 0.08 0.08 0.08

0.12 0.10 0.08 0.10

0.20 0.15 0.10 0.20

0.35 0.30 0.25 0.30

0.18 0.12 0.08 0.15

0.28 0.25 0.20 0.25

0.12 0.10 0.08 0.12

0.20 0.18 0.15 0.18

0.10 0.08 0.05 0.10

0.18 0.15 0.12 0.15

0.08 0.05 0.05 0.05

0.10 0.10 0.08 0.08

0.05 0.05 0.03 0.05

0.10 0.08 0.08 0.08

0.55 0.35 0.22 0.30 0.20 0.55

0.50 0.30 0.25 0.30 0.25 0.50

0.45 0.28 0.18 0.25 0.18 0.45

0.40 0.25 0.20 0.22 0.20 0.40

0.32 0.20 0.15 0.18 0.12 0.32

0.30 0.18 0.15 0.18 0.15 0.30

0.28 0.18 0.12 0.15 0.10 0.28

0.25 0.15 0.12 0.16 0.12 0.25

0.18 0.10 0.08 0.10 0.08 0.18

0.15 0.10 0.08 0.10 0.08 0.15

0.12 0.08 0.05 0.08 0.05 0.12

0.12 0.08 0.08 0.05 0.08 0.12

Coolant to be used.

TABLE 25-36 Recommended cutting speeds for face and end milling with plain HSS and carbide milling cutters, m/min Depth of cut

Material to be milled Cast iron Soft Medium Hard Malleable Iron Steela : Soft Medium Hard Stainless Brass Average Soft yellow Bronze Copper Monel Aluminuma

Roughing cut, 3 to 5 mm

Semi-ﬁnishing cut, 1.5 to 3 mm

HSS

HSS

Carbide

Carbide

Finishing cut, below 1.5 mm HSS

Carbide

25 15 12 25

68 50 38 68

30 25 16 30

80 68 50 80

36 30 20 36

105 80 68 105

28 22 15 18

120 100 75 50

32 28 20 22

150 120 90 68

40 32 25 28

180 135 105 80

30 60 28 45 18 75

75 120 75 100 50 240

45. 90 36 68 22 105

120 180 100 150 68 300

60 120 45 90 28 150

150 240 128 210 80 450

a

Coolant to be used Note: Cutting speeds for 12% cobalt HSS should be about 25% to 50% higher than those shown for plain HSS. Cutting speeds for cast alloy should be about 100% higher than those shown for plain HSS. Above speeds should be reduced when milling work that has hard spots or when milling castings that are sandy.

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ELEMENTS OF MACHINE TOOL DESIGN

TABLE 25-37 Feeds and speeds for hobbing Feed, mm/rev. of blank

Type of gear

Material

High speed reduction and step up Instrument

Steel Steel Non-ferrous Steel Steel, C.I. Non-ferrous Steel

Aircraft Machine tool and printing press Automotive, including trucks and tractors High quality industrial General industrial Splines

Steel Cast iron Steel Cast iron Steel

Module mm

Roughing (single thread hob)

Roughing (multithread hob)

1.5–8 0.4–1.25 0.4–1.25 2.0–4.0 2.0–6.0 2.0–6.0 1.5–8.10

1–1.5 0.5–1.5 1.0–1.5 1.0–1.5 2.0–3.2 2.0–3.2 2.0–3.2

1–1.5 Up to 3 Up to 3 Up to 3 Up to 2.5 Up to 2.5 Up to 2.5 Up to 2.0 (3 starts)

10.0–25.0 2.5–8.0 10.0–25.0 2.5–8.0

2.0–2.5 1.25–3.2 2.0–2.5 1.25–3.2 1.25–3.0

1.25–1.5

Finishing

Hob speeds, m/min

0.8–1.25 0.5–1.0 0.5–1.0 0.8–1.25 1.0–1.5 1.0–1.5 1.25–2.0

9–25 25–60 25–60 15–45 15–30 25–450 15–45

1.25–2.0

12–30

1.50–2.5

12–30

0.50–1.75

l8–45

TABLE 25-38 Selection of milling cutters Material One-piece construction Two-piece construction Cutting portion Body

High-speed steel High speed steel Carbon steel with tensile strength not less than 700 MPa (190 HN)

Hardness Cutting portion Shank portion Parallel shank Tang of Morse taper shank

760 HV (62 HRC) Min 245 HV (21 HRC) Min 320 HV (32 HRC) Min

Note: The equivalent values within parentheses are approximate.

Recommendations for selection of milling cutters: Tool Type N—For mild steel, soft cast iron and medium hard non-ferrous metals. Tool Type H—For specially hard and tough materials. Tool Type S—For soft and ductile materials. Material to be cut

Tensile strength, MPa

Carbon steel

Up to 500 Above 500 up to 800 Above 800 up to 1000 Above 1000 up to 1300

Steel casting Gray cast iron

Brinell hardness, HB

Up to 180 Over 180

Malleability cast iron Copper alloy Soft Brittle Zinc alloy Aluminum alloy Soft Medium/Hard Aluminum alloy, age hardened Low cutting speed High cutting speed Magnesium alloy Unlaminated a

Tool typea N or (S) N N or (H) H H N H N S or (N) N or (H) S or (N) S N or (S) N S S or (N) N or (S)

Tool types within parentheses are non-preferred. Courtesy: IS 1830, 1971

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ELEMENTS OF MACHINE TOOL DESIGN

25.50

CHAPTER TWENTY-FIVE

TABLE 25-39 Dimensions for interchangeability of milling cutters and arbors with tenon drive A z r1 r

a

a1

dφ A s × 45

dφ

s× 45

b

b1

All dimensions in millimeters Arbor

Cutter

da h6/H7

a h11

b H11

r Max

a1 H11

b1 H13

r1 Max

5 8 10 13 16 19 22 27 32 40 50 60

3 5 6 8 8 10 10 12 14 18 16 20

2.0 3.5 4.0 4.5 5.0 5.6 5.6 6.3 7.0 9.0 8.0 10.0

0.3 0.4 0.5 0.5 0.6 0.6 0.6 0.8 0.8 1.0 1.0 1.0

3.3 5.4 6.4 8.4 8.4 10.4 10.4 12.4 14.4 16.4 18.4 20.5

2.5 4.0 4.5 5.0 5.6 6.3 6.3 8.0 7.0 9.0 10.0 11.2

0.6 0.6 0.8 1.0 1.0 1.2 1.2 1.6 1.2 2.0 2.0 2.0

a

b

s 0.3 0.4 0.5 0.5 0.6 0.6 0.6 0.8 0.8 1.0 1.0 1.0

zb þ 0.1

þ0.2

þ0.3

The tolerance on d is not applicable to gear hobs.

z ¼ maximum permissible deviation between the axial plane of the tenon and the axis of arbor of diameter d.

Courtesy: IS 6285-1971

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0.075 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.125

ELEMENTS OF MACHINE TOOL DESIGN

25.51

ELEMENTS OF MACHINE TOOL DESIGN

TABLE 25-40 Dimensions for interchangeability of milling cutters and milling arbors with key drive r

a

a C

C1

s× 45

r1

C1

C

a

b

KEY SECTION dφ

KEYWAY IN CUTTER

KEYWAY IN ARBOR

All dimensions in millimeters Key da h6/H7

a h9

bb

8 10 13 16 19 22 27 32 40 50 60 70 80 100

2 3 3 4 5 6 7 8 10 12 14 16 18 25

2 3 3 4 5 6 7 7 8 8 9 10 11 14

a

S 0.16

0.25

Keyway Tolerance on S ac þ0.09 0

0þ0.15

9 0.40

0.60

þ0.20 0

2 3 3 4 5 6 7 8 10 12 14 16 18 25

C 6.7 8.2 11.2 13.2 15.6 17.6 22.0 27.0 34.5 44.5 54.0 63.5 73.0 91.0

The tolerance on diameter d is not applicable to gear hobs. Tolerance on thickness b of key: square, h9; rectangular, h11. c Tolerance on keyway width a: light drive ﬁt, N9. For keyway in arbor: running ﬁt, H9; light drive ﬁt, N9. For keyway in cutter: C11

Tolerance on C C1

0 0–0.1

0–0.2

8.9 11.5 14.6 17.7 21.1 24.1 29.8 34.8 43.5 53.5 64.2 75.0 85.5 107.0

Tolerance on C1 r

Tolerance on r r1

Tolerance on r1

0.16

0.6

0–0.1 0 –0.2

0 –0.08

0 –0.3

0.25

1.2

0 –0.09

0–0.5

0.40

0–0.15

0.60

0–0.20 –0.20

0.4 þ0.1 0 1.0

þ0.2 0

1.6 2.0 2.5

IS: 6285, 1971.

b

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ELEMENTS OF MACHINE TOOL DESIGN

25.52

CHAPTER TWENTY-FIVE

TABLE 25-41 American National Standard staggered teeth, T-slot milling cutters with Brown and Sharpe taper and Weldon shanks (ANSI/ASME B94, 19, 1986)

With B. and S. tapera Bolt size 1 4 5 16 3 8

1 5 8 3 4

1

Cutter diam., D

Face width, W

Neck diam., N

9 16 21 32 26 32 13 32 114 115 32 127 32

16 64 17 64 21 64 25 64 31 32 5 8 53 64

17 64 21 64 13 32 17 32 21 32 25 32 1 132

With Weldon shank

Length L

Taper No.

Length L

Diam., S

—

—

219 32

— —

211 16 314 7 316 15 316 7 416 13 416

1 2 1 2 3 4

— — 5

7

614

7

678

9

714

9

1 1 — 114

All dimensions are inches. All cutters are high-speed steel and only right-hand cutters are standard. a For dimensions of Brown and Sharpe taper shanks. See information given in standard Handbook. Tolerances: On D, þ0.000, 0.010 inch; on W, 1 þ0.000, 0.005 inch; on N, þ0.000, 0.005 inch, on L, 16 inch; on S, 0.0001 to 0.0005 inch.

TABLE 25-42 American National Standard 60-degree single-angle milling cutters with Weldon shanks (ANSI/ASME B94, 19, 1985) L

60

w S

D

Diam., D

S

W

L

Diam., D

S

W

L

3 4

3 8 5 8

5 16 9 16

1 216

178

7 8

214

1

13 16 1 116

314

278

138

All dimensions are in inches. All cutters are high-speed steel. Right-hand cutters are standard. 1 inch. Tolerances: On D, 0.015 inch; on S, 0.0001 to 0.0005 inch; on W, 0.015 inch; and on L, 16

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334

ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

25.53

TABLE 25-43 American National Standard multiple ﬂute, helical series end mills with Brown and Sharpe taper shanksa (ANSI/ ASME B94, 19, 1985) L w D

Diam., D

W

L

Taper No.

Diam., D

W

L

Taper No.

–

–

–

–

1

158

558

7

2

714

9

214

712

9

234

8

9

–

–

–

–

1 2 3 4

15 16 114

415 16

7

114 112

514

7

2

All dimensions are in inches. All cutters are high-speed steel. Right-hand cutters with right hand helix are standard. Helix angle is not less than 10 degrees. No. 5 taper is standard without tang: Nos. 7 and 9 are standard with tang only. 1 1 inch; and L, 16 inch. Tolerances: On D, þ0.005 inch; on W, 32 a For dimensions of B. and S. taper shanks, see information given in standard handbook.

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ELEMENTS OF MACHINE TOOL DESIGN

25.54

CHAPTER TWENTY-FIVE

TABLE 25-44 American National Standard form relieved, concave, convex, and corner-rounding arbor-type cuttersa (ANSI/ASME B94, 19, 1985) R w

w

C H

H

D

Concave

Max.

Min.

H

D

Convex

Diameter C or radius R Nom.

w

C

Cutter diam. D b

D Corner - Rounding

Width W :010 c

Diameter of hole H Nom.

Max.

Min.

1

1.00075

1.00000

1 1

1.00075 1.00075

1.00000 1.00000

Concave cuttersc 1 8 1 4 3 8 1 2 3 4

0.1270

0.1240

214

0.2520 0.3770

0.2490 0.3740

212 234

0.5040

0.4980

3

0.7540

0.7480

334

1

0.0040

0.9980

414

1 4 7 16 5 8 13 16 113 16 9 116

1

1.00075

1.00000

114

1.251

1.250

114

1.251

1.250

1

1.00075

1.00000

1

1.00075

1.00000

1

1.00075

1.00000

114

1.251

1.250

114

1.215

1.250

1

1.00075

1.00000

1

1.00075

1.00000

114

1.251

1.250

d

Convex cutters 1 4 3 8 1 2 3 4

0.2520

0.2480

212

0.3770

0.3730

234

0.5020

0.4980

3

0.7520

0.7480

334

1 4 3 8 1 2 3 4

1

1.0020

0.9980

414

1

Corner-rounding cutterse 1 8 1 4 1 2

0.1260

0.1240

212

0.2520

0.2490

3

0.5020

0.4990

4 14

1 4 13 32 3 4

All dimensions in inches. All cutters are high-speed steel and are form relieved. Right-hand corner rounding cutters are standard, but left-hand cutter for 14 inch size is also standard. a For key and keyway dimensions for these cutters, see standard handbook. b c

1 1 Tolerances on cutter diameters are þ 16 , 16 inch for all sizes.

Tolerance does not apply to convex cutters. d Size of cutter is designated by specifying diameter C of circular form. e Size of cutter is designated by specifying radius R of circular form. Source: Courtesy: ANSI/ASME B94, 19, 1985, Erik Oberg Editor Etd., Extracted from Machinery’s Handbook, 25th edition, Industrial Press, N.Y., 1996.

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

25.55

TABLE 25-45 American National Standard roughing and ﬁnishing gear milling cutters for gears with 1412 degree pressure angles (ANSI/ASME B94, 19, 1985)

D

D H

H

ROUGHING Diametral pitch

Diam. of cutter, D

Diam. of hole, H

Diametral pitch

FINISHING Diam. of cutter, D

Diam. of hole. H

Diametral pitch

Diam. of cutter, D

Diam. of hole, H

Roughing gear milling cutters 1

812

2

3

514

112

5

338

1

112

7

134

4

434

134

6

312

114

1

612

134

4

414

114

7

338

114

212

618

134

5

438

134

8

314

114

3

558

134

5

334

114

–

–

–

14

218

1

16

218

114

18

2

112

20

2

1

22

2

7 8 7 8 7 8 7 8 7 8

114 114

24 26

214 134

36

7 8 7 8 7 8

–

Finishing gear milling cutters 812

2

112

7

2

612

212

618

3

558

3 4

514 414

5

438 414 414

134 134 134 134 112 134 134 112 134

1

5 6

6

378

6

318 338 312 278 318

7 8 8 9 10 11 12 14

3 238 278 212

112

7 8

114

40

134 134

1

–

–

1

All dimensions are in inches. All gear milling cutters are high-speed steel and are form relieved. For keyway dimensions refer to standard handbook. 1 1 Tolerances: On outside diameter, þ 16 , 16 inch; on hole diameter, through 1 inch hole diameter, þ0.00075 inch; over 1 inch and through 2 inch hole diameter, þ0.0010 inch. For cutter number relative to number of gear teeth, see standard handbook. Roughing cutters are made with No. 1 cutter form only. Source: Courtesy: ANSI/ASME B94, 19, 1985, Erik Oberg Editor Etd., Extracted from Machinery’s Handbook, 25th edition, Industrial Press, N.Y., 1996.

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ELEMENTS OF MACHINE TOOL DESIGN

25.56

CHAPTER TWENTY-FIVE

TABLE 25-46 American National Standard regular, long and extra length, multiple-ﬂute medium helix single-end end mills with Weldon shanks (ANSI/ASME B94, 19, 1985) L w S

D

AS INDICATED BY THE DIMENSIONS GIVEN BELOW, SHANK DIAMETERS MAY BE LARGER, SMALLER, OR THE SAME AS THE CUTTER DIAMATERS D. Regular mills Cutter diam, D 1a 4 5a 16 3a 8 7 16 1 2 9 16 3 8 11 16 3 4 7 8

1

Long mills

S

W

L

Na

S

W

L

Na

S

W

3 8 3 8 3 8 3 8 3 8 1 2 1 2 1 2 1 2 5 8 5 8 7 8 7 8

5 8 3 4 3 4

7 216

4

3 8 3 8 3 8 1 2 1 2

114

1 316

4

9 316

4

318 314 334

4

2

314

4

4

3 8 1 8 3 8

134

138 112 134

212

414

4

–

–

–

–

4 –

4 –

1 2

–

3 –

5 –

4 –

1 1 138 138 158 158 178 178

212 212 211 16 211 16 338 338 358 358

4

4

6

4 4

4

5 8

212

458

4

5 8

4

4

–

–

–

–

–

–

4

3 4 7 8

3

514

4

4

614

4

312

534

4

3 4 7 8

5

714

4

612 612 612 612 612 612 612 612

4

1

6

812

6

–

–

–

–

6 6

114a –

6 –

812 –

6 –

6

–

–

–

–

6

114

8

1012

6

6

–

–

–

–

8

–

–

–

–

4

2

414

6

1

4

1

2 2

414 414

6 6

1 1

4 4

114

114

2

412

6

114

4

112 134

114 114 114

2

412 412 412

6

114

4

6

114

4

8

114

4

2

2

4

–

1

2

Na

2 –

6

114 112

L

4 4

4

118

Extra long mills

618 –

4 –

6

All dimensions are in inches. All cutters are high-speed steel. Helix angle is greater than 19 degrees but not more than 39 degrees. Right-hand cutters with right-hand helix are standard. 1 1 inch; on L, 16 inch; N ¼ number of ﬂutes. Tolerances: On D, þ0.003 inch; on S, 0.0001 to 0.0005 inch; on W, 32 a In case of regular mill a left-hand cutter with left-hand helix is also standard. Source: ANSI/ASME B94, 19, 1985, Erik Oberg Editor Etd., Extracted from Machinery’s Handbook, 25th edition, Industrial Press, N.Y., 1996.

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

25.57

TABLE 25-47 American National Standard long length single-end and stub-, and regular length, double-end plain- and ball-end, medium helix two-ﬂute end mills with Weldon shanks (ANSI/ASME B94, 19, 1985) L

B

w

S

D L

w

w

C D

S

D C

Single end Long length—plain end

Long length—ball end

Diam., C and D

S

Ba

W

L

S

Ba

W

L

1 4 5 16 3 8 1 2 5 8 1 4

3 8 3 8 3 8 1 2 5 8 3 4

112

5 8 3 4 3 4

1 316

112

5 8 3 4 3 4

1 316

1

4

138

458

158

538

3 8 3 8 3 8 1 2 5 8 3 4

1

1

212

714

1

5

134 134 7 232 23 232 311 32 431 32

5 316 5 316

134 134 214 234 338

5 316 5 316

1

4

138

438

158

538

212

714

Double end Stub length—plain end Diam., C and D

Regular length—plain end

Regular length—ball end

S

W

L

S

W

L

S

5 32 1 4 5 16 3 8 7 16 1 2 5 8 11 16 3 4

3 8 3 8

15 64 3 8

234

318

–

–

–

318

–

–

318 334 334 412

3 8 3 8 3 8 1 2 1 2 5 8

1 2 9 16 9 16 13 16 13 16 118

318

–

5 5

– 3 4

– 5 116

– 5

1

7 16 1 2 9 16 9 16 13 16 13 16 118 5 116 5 116 5 18

578

1

158

578

–

–

–

–

–

–

–

–

–

–

–

–

– –

– –

– –

3 8 3 8 3 8 3 8 1 2 1 2 5 8 3 4 3 4

–

–

–

1

278

318

W

L

318 318 334 334 412

All dimensions are in inches. All cutters are high-speed steel. Right-hand cutters with right hand helix are standard. Helix angle is greater than 19 degrees but not more than 39 degrees. 1 inch; on L, Tolerances: On C and D, þ0.003 inch; for single-end mills, 0.0015 inch for double end mills on S, 0.0001 to 0.0005 on W, 32 a 1 inch. B is the length below the shank. 16 Source: Courtesy: ANSI/ASME B94, 19, 1985, Erik Oberg Editor Etd., Extracted from Machinery’s Handbook, 25th edition, Industrial Press, N.Y., 1996.

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ELEMENTS OF MACHINE TOOL DESIGN

25.58

CHAPTER TWENTY-FIVE

TABLE 25-48 American National Standard Woodruﬀ keyseat cuttersa —shank-type straight teeth and arbor staggered teeth (ANSI/ASME B94, 19, 1985) w L

w D

D

1—DIAM. 2

H

Shank-type cutters Nom. Cutter diam. of No. cutter, D 202 203 403 404 405 505

1 4 3 8 3 8 1 2 5 8 5 8

Width of face, W 1 16 1 16 1 8 1 16 1 8 5 32

Nom. Length Cutter diam. of overall, L No. cutter, D 1 216 1 216 1 28 1 216 1 28 1 232

807

3 4 7 8 7 8 7 8

1008

1

1208

1

506 507 707

Width of face, W 5 32 5 32 7 32 1 4 5 16 3 8

Nom. Length Cutter diam. of overall, L No. cutter, D 5 232 5 232 5 232 1 24 5 216 3 28

809

118

710

114 114 114 112 112

1010 1210 812 1212

Width of face, W

Length overall, L

1 4 7 32 5 16 3 8 1 4 3 8

214

Width of face, W

Diam. of hole, H 1 1 –

9 232 5 216

238 214 238

Arbor-type cutters Nom. Cutter diam. of No. cutter, D 617 817

218 218

1217

218

822

234

Width of face, W 3 16 1 4 3 16 1 4

Nom. Diam. of Cutter diam. of a hole, H No. cutter, D 3 4 3 4 3 4

1

1012 1222

234 234

1262

234

1288

312

Width of face, W 5 16 3 8 1 2 3 8

Nom. Diam. of Cutter diam. of a hole, H No. cutter, D 1 1

1628 1828

312 312

1

2428

312

1 2 9 16 3 4

1

–

–

–

1

All dimensions are given in inches. All cutters are high-speed steel. Shank type cutters are standard with right-hand cut and straight teeth. All sizes have 12 inch diameter straight shank. Arbor type cutters have staggered teeth. For Woodruﬀ key and key-slot dimensions, see standard handbook. 1 3 3 7 5 to 32 inch face þ0.0000, 0.0005: 16 to 32 , 0.002, 0.0007, 14, 0.0003, 0.0008, 16 , 0.0004, 0.0009, Tolerances: Face width W for shank type cutters: 16 3 5 1 8, 0.0005, 0.001, 0.0008, 16, 0.0004, 0.0009, 8 and over, 0.0005, 0.000 inch. Hole size H, þ0.00075, 1:000 inch. Diameter D for shank type cutters; 18, through 14 inch diameter, þ0 016, þ0.015, 78 through 118, þ0.012, þ0.017; 1 114 through 112, þ0.015, þ0.02 inch. These tolerances includes an allowance for sharpening. For arbor type cutters diameter D is furnished 32 inch larger than bore and tolerance of þ0.002 inch applies to the over size diameter. Source: Courtesy: ANSI/ASME B94, 19, 1985, Erik Oberg Editor Etd., Extracted from Machinery’s Handbook, 25th edition, Industrial Press, N.Y., 1996.

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Particular

25.59

Formula

GRINDING The tangential component of grinding force Fz , which constitutes the major value of grinding force Fig. 25-26

Grinding wheel vg

Work piece

Fx Fy = Fr

vw Fz = Ft

FR

FIGURE 25-26 Forces acting on a grinding wheel.

The chip thickness

Ft ¼ Km st

vw vg

ð25-123Þ

where s ¼ feed rate, mm/rev t ¼ thickness of material removed from job or depth of cut, mm vw ¼ peripheral velocity of workpiece/job, m/min vg ¼ peripheral velocity of the grinding wheel, m/min Km ¼ speciﬁc resistance to grinding of the work material, N/m2 (Table 25-51) Fy ¼ Fr ¼ radial component of the force in cylindrical grinding operation, kN Fx ¼ horizontal component of the force against the feed, kN Fz ¼ Ft ¼ vertical component of the force in the cylindrical grinding operation, kN 2pvw t¼ vg

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðdw dg Þs dg dw

ð25-124Þ

where p ¼ pitch of grains, mm dw ¼ diameter of workpiece, mm dg ¼ diameter of grinding wheel, mm þve sign for external grinding wheel, ve for internal grinding wheel The power required by the grinding wheel

P¼

Ft ð¼Fz Þvg 1000

ð25-125Þ

where Ft ¼ Fz ¼ tangential force on wheel, N P ¼ power, W vg ¼ velocity of grinding wheel, mm/s dw ts 1000

Metal removal rate in case of transverse grinding

Q¼

The power at the spindle

P ¼ Pu Q

ð25-126Þ ð16-127Þ 3

where Q in cm /min Refer to Table 25-50 for Pu .

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ELEMENTS OF MACHINE TOOL DESIGN

25.60

CHAPTER TWENTY-FIVE

Particular

Energy per unit volume of material removed

Formula

E¼

P bsvg

ð25-128Þ

where b ¼ width of cut, mm s ¼ feed rate or depth of cut, mm/rev E in J/mm3 Vertical boring: The power required for boring

iFt v 1000 where

P¼

ð25-129Þ

i ¼ number of heads v ¼ cutting speed, mm/s Centerless grinding: The peripheral grinding wheel speed

vg ¼

dw n 1000 60

ð25-130Þ

Through feed rate

st ¼ dr nr sin

ð25-131Þ

where dr ¼ diameter of regulating wheel, mm nr ¼ speed of regulating wheel, rpm ¼ regulating wheel inclination angle, deg Metal removal rate from through feed grinding

dw tst 1000 where Qt in cm3 /min

Qt ¼

ð25-132Þ

st ¼ through feed rate, mm/min Metal removal rate from plunge grinding

dw bsp 1000 where Qp in cm3 /min

Qp ¼

ð25-133Þ

b ¼ width of cut plunge grinding, mm sp ¼ plunge in feed rate per minute ¼ ðsnw Þ, mm/min s ¼ plunge in feed rate per work revolution, mm/rev nw ¼ workpiece revolution per minute For the unit power

Refer to Table 25-50.

Power at the spindle

P ¼ Pu Q

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ð25-134Þ

ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Particular

25.61

Formula

SHAPING (Fig. 25-27) The force of cutting can be found by empirical formula Fz

Fz ¼ Ft ¼ 9:807Cp kd x sy

SI

ð25-135aÞ

where Fz in N Fz ¼ Ft ¼ Cp kd x sy Customary Metric Units ð25-135bÞ where x, y, k and Cp have the same values as in lathe tools; Fz in kgf Equation (25-135) can be also used for the case of planing machine.

The approximate equation 1 expression for cutting force Fz for cast iron

Fz ¼ 1860ds 0:75 K

SI

ð25-136aÞ

Fz ¼ 190ds 0:75 K Customary Metric Units ð25-136bÞ

The power consumption of shaping machine

1 Fz vr ð25-137Þ 1000 60 where vr ¼ the the average velocity of ram in its middle position during its stroke 2rn v1 ¼ ð25-138aÞ 1000 P¼

The velocity of crank pin of r radius The peripheral velocity of the sliding block

v2 ¼ v1 cosð Þ

The peripheral velocity of the driving pin of the rocker arm at point A.

vra ¼ v2

The average velocity of ram at its middle position during its stroke Fig. 25-28

vr av ¼ n

R Ma Rl R þ ðl=2Þ

FR (a)

Fx FIGURE 25-27 Forces acting on a shaping tool.

ð25-138dÞ

vc max vr max vr B C vra

Fz = Ft

ð25-138cÞ

where n in rpm

(b)

Fy

ð25-138bÞ

l

A L

l

v2 a b α−γ

α

c v1 K

γ

d O

R p

M FIGURE 25-28 Ram velocity diagram of a crank shaper.

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ELEMENTS OF MACHINE TOOL DESIGN

25.62

CHAPTER TWENTY-FIVE

Particular

Formula

The approximate/average velocity of ram

vr ¼ vra cos vr ¼

ð25-138eÞ

2nRl cos2 cosð Þ 1000ð2R þ l cos Þ

ð25-138f Þ

R þ ðlmin =2Þ R þ lmin

ð25-139aÞ

2nRl 1000ð2R þ lÞ

ð25-139bÞ

R þ ðlmax =2Þ R lmax

ð25-139cÞ

The maximum speed of ram for the average value of cutting speed vr

nmax ¼ vr

The maximum velocity of ram travel in the cutting stroke when it is at ¼ ¼ 0

vc max ¼

The minimum speed for the average value of cutting speed vr

nmin ¼ vr

vr is a function of l, since and R are constants, i.e. vr ¼ f ðlÞ The maximum velocity of ram travel during the return stroke at ¼ 1808 and ¼ 0

vr max ¼

The average cutting velocity vr av during travel 2l of ram

vr av ¼

2nRl 1000ð2R lÞ

ð25-139dÞ

2ln 1000 where vr av in m/min

ð25-139eÞ

PRESS TOOLS Punching (Figs. 25-29, 25-31 and 25-32): Maximum shearing force or pressure to cut the material

Fmax ¼ pDu t

Work done

W ¼ Fmax x F x

t

Load, F, N

ð25-141Þ

Workpiece

dp

Die 1… 4 to

(b)

for any other contour

Punch

dd

(a)

¼ u tP

F

Work done, N-m (lbf in) x

FIGURE 25-29

Shear punch

112… hτ

Fmax

ð25-140Þ

for round hole

Distance

Workpiece

t

t FIGURE 25-30

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Die

ELEMENTS OF MACHINE TOOL DESIGN

25.63

ELEMENTS OF MACHINE TOOL DESIGN

Particular

Guide pin busing Guide pins

Formula

Punch

Punch holder Punch

Tensile Workpiece

Stripper Die block

Compressive Tensile

Die holder

Die

Bolster plote

FIGURE 25-31 Common components of a simple die. Courtesy: F. W. Wilson, Fundamentals of Tool Design, American Society of Tool and Manufacturing Engineers, Prentice-Hall of India, 1969.

Penetration ratio

FIGURE 25-32 Stresses in die cutting.

c¼

x t

ð25-142Þ

where Fmax ¼ maximum shear force, kN (lbf ) u ¼ ultimate shear stress, taken from Table 25-54 t ¼ material thickness, mm (in) x ¼ penetration, mm (in) P ¼ perimeter of proﬁle, mm (in) Punch Dimensioning: When the diameter of a pierced round hole equals stock thickness, the unit compressive stress on the punch is four times the unit shear stress on the cut area of the stock, from the formula.

4t ¼1 c d

ð25-143Þ

where c ¼ unit compressive stress on the punch, MPa (psi) ¼ unit shear stress on the stock, MPa (psi) t ¼ thickness of stock, mm (in) d ¼ diameter of the punched hole, mm (in) A value for the ratio d=t of 1.1 is recommended.

The maximum allowable length of a punch can be calculated from the formula

d L¼ 8

Ed t

1=2

where d=t ¼ 1:1 or higher value E ¼ modulus of elasticity, GPa (psi) For clearance between punch and die

Refer to Tables 25-52 and Fig. 25-36.

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ð25-144Þ

ELEMENTS OF MACHINE TOOL DESIGN

25.64

CHAPTER TWENTY-FIVE

Particular

Formula

Shearing (Fig. 25-30): Fmax h 1þ x where h is shown in Fig. 25-30

Shearing force

F ¼

The stripper pressure or force

Fstr ¼ 24 106 Pt

ð25-145Þ

SI

ð25-146aÞ

where P ¼ perimeter of cut, m t ¼ thickness of workpiece, m; Fstr in N Fstr ¼ 3500Pt

USCS

ð25-146bÞ

SI

ð25-147aÞ

where Fstr in lbf, t in in, P in in The formula used to compute the force (or pressure) in swaging operation

Fswg ¼ A sut where A ¼ area to be sized in m2

sut ¼ ultimate compressive strength of metal, MPa, and Fswg in N A sut USCS 2000 where A in in2 , sut in psi, Fswg in tonf

Fswg ¼

ð25-147bÞ

SHEET METAL WORK Bending (Figs. 25-33 to 25-36): The bend allowance as per ASTME die design standard (Fig. 25-33)

b ¼

2ri þ Kn t 360

Bend angle = θ

Bend line

Set back

Area under tension Bend allowance Area under compression

ri 0.33 T to 0.5 T

Neutral axis Bend axis

Bend radius

FIGURE 25-33 Bend terms for general angle.

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ð25-148Þ

ELEMENTS OF MACHINE TOOL DESIGN

25.65

ELEMENTS OF MACHINE TOOL DESIGN

Particular

Formula

where

K

O

b ¼ bend allowance (arc length of neutral axis), mm (in)

t T

θ

ri

Flange

¼ bend angle, deg ri ¼ inside radius of bend, mm (in)

We b

t ¼ metal thickness, mm (in) Kn ¼ constant for neutral axis location

FIGURE 25-34

¼ 0:33 when ri is less than 2t Another equation for bending allowance with outside bending angle (Fig 25-34).

¼ 0:50 when ri is more than 2t

t r þ b ¼ ðri þ tÞ tan 2 360 i 2

ð25-149Þ

where in deg

b ¼ 3 tan 0:0218 t 2

ð25-150Þ

when ri ¼ 2t t ri þ 2 2

Initial length of strip of metal (Fig. 25-35)

Li ¼ T t 2ri þ K þ

Bending allowance for right angle bend to take into account reduction of length K and T (Fig. 25-35)

b ¼ ri þ t

t r þ 4 i 2

ð25-151Þ ð25-152aÞ

b ¼ 1:037t

ð25-152bÞ

when ri ¼ 2t The bending force

Fb ¼ Wtu

ð25-153Þ

Planishing force

Fp ¼ WK sy

ð25-154Þ

where K and W are dimensions as shown in Figs. 25-34 and 25-35 sy ¼ yield stress, MPa (psi) Flange

t F T

W

V-Punch

Vee die

Knurled The bending pin punch

ri K

Web

Length of bend

Die

Spring (a) V-bending-clamping a part in a V-die.

FIGURE 25-35

Spring loaded pad

(b) Edge-bending

FIGURE 25-36 Bending methods.

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ELEMENTS OF MACHINE TOOL DESIGN

25.66

CHAPTER TWENTY-FIVE

Particular

Formula

The force/pressure required for V-bending (Fig. 25-36a)

KLt2 sut W where

Fv ¼

ð25-155Þ

F ¼ V-bending force, kN (tonforce) L ¼ length of part, m (in) W ¼ width of V- or U-die, m (in) sut ¼ ultimate tensile strength, MPa (tonf/in2 ) K ¼ die opening factor ¼ 1:20 for a die opening of 16t ¼ 1:33 for a die opening of 8t The force required for U-bending (channel bending)

Fu ¼ 2Fv (approx)

ð25-156aÞ

The force required for edge-bending (Fig.25-36b)

Fed ¼ 12 Fv

ð25-156bÞ

Force required for drawing

F ¼ dt u

ð25-157Þ

Empirical formula for pressure (or force) for a cylinder shell

where u ¼ ultimate tensile stress, MPa (psi) D F ¼ dt sy c ð25-158Þ d

Drawing (Fig. 25-37):

where D ¼ diameter of blank d

T

t

d ¼ diameter of shell h ¼ height of shell r ¼ corner radius T ¼ bottom thickness of shell

t D

t ¼ thickness of wall of shell

FIGURE 25-37

C ¼ constant which takes into account friction and bending

A tentative blank size for an ironed shell can be obtained from equation

¼ 0:6 to 0.7 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ t D ¼ d 2 þ 4dh T pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D ¼ d 2 þ 4dh

ð25-160aÞ

when d=r ¼ 20 or more pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D ¼ d 2 þ 4dh 0:5r

ð25-160bÞ

The blank size taking into consideration the ratio of the shell diameter to the corner (d=r) which aﬀects the blank diameter.

when d=r lies between 15 and 20

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ð25-159Þ

ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Particular

25.67

Formula

D¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d 2 þ 4dh r

ð25-160cÞ

when d=r lies between 10 and 15 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D ¼ ðd 2rÞ2 þ 4dðh rÞ þ 2rðd 0:7rÞ ð25-160dÞ when d=r is less than 10 For die clearance for diﬀerent metals

Refer to Fig. 25-38

For nomograph for determining draw-die radius

Refer to Fig. 25-39

For chart for checking percentage reduction in drawing of cups.

Refer to Fig. 25-40

For clearance between punch and die

Refer to Fig. 25-29 and Table 25-52

For draw clearance

Refer to Table 25-53

For design of speed-change gear box for machine tools, kinematic schemes of machine tools, layout diagrams or structural diagram for gear drives, version of kinematic structures in machine tools, etc.

Refer to subsection ‘‘Designing spur and helical gears for machine tools’’ from pp. 23-109 to 23-138 of Machine Design Data Handbook, McGraw-Hill Publishing Company, New York, 1994.

For ﬁts and tolerances

Refer to Chapter 11 on ‘‘Metal ﬁts, tolerances and surface textures’’, pp. 11.1 to 11.32.

For surface roughness and surface texture

Refer to Chapter 11 on ‘‘Metal ﬁts, tolerances and surface textures’’, pp. 11.26 to 11.32.

For tool steels and die steels

Refer to Chapter 1 on ‘‘Properties of engineering materials’’, Tables 1-31 to 1-36 for tool steels and Tables 1-49 and 1-51 for die steels.

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ELEMENTS OF MACHINE TOOL DESIGN

25.68

CHAPTER TWENTY-FIVE

TABLE 25-49 Metal removal rate in milling operation, Q Material

Metal removal rate, Q, mm3 /kW min

Cast iron, gray Cast steel Mild steel Alloy steel Stainless steel Aluminum Copper Titanium

12600 12600 18900 10500 8400 42000 18900 10500

TABLE 25-50 Average unit power Pu , for grinding Unit power Pu , kW/cm3 /min Depth of grinding, mm per pass Infeed, mm per revolution of work

Work material

Free-machining steels Mild steels Medium carbon steels Alloy steels Tool steels Stainless steels Cast iron: gray, ductile, malleable Aluminum alloys Titanium alloys

0.0125

0.025

0.05

0.075

0.1

0.25

0.5

0.75

1.4

0.88

0.7

0.6

0.51

0.35

0.23

0.18

1.3 1.15 1.4 1.15

0.85 0.82 0.84 0.79

0.68 0.65 0.65 0.65

0.58 0.56 0.58 0.51

0.49 0.46 0.51 0.44

0.34 0.32 0.37 0.3

0.25 0.26 0.29 0.23

0.19 0.21 0.26 0.19

0.58 0.93

0.45 0.79

0.35 0.6

0.33 0.56

0.29 0.51

0.21 0.37

0.17 0.3

0.15 0.25

TABLE 25-51 Values of Km st, mm2

0.3

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.4

Steel 32360 25500 21570 18140 15690 13720 12750 11750 10790 10300 9810 Km N/m2 Cast iron 29910 22550 17650 13730 11770 10780 9810 8825 8430 7845 7350

2.6

2.8

3.0

8830 7355

8830 7355

8330 7110

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

25.69

TABLE 25-52 Clearance between punch and die (Fig. 25-29) Location of the proper clearance determines, either hole or blank size, punch size controls hole size, die size controls blank size. 2C ¼ clearance ¼ dp ddi Clearance between punch and die, mm Sheet thickness, mm

Mild steel

Moderately hard steel

Hard steel

Soft brass

Hard brass

Aluminum

0.25 0.50 0.75 1.0 1.25 1.5 1.75 2.0 2.25 2.5 2.75 3.0 3.3 3.5 3.8 4.0 4.3 4.5 4.8 5.0

0.01 0.025 0.04 0.05 0.06 0.075 0.09 0.10 0.11 0.13 0.14 0.15 0.17 0.18 0.19 0.20 0.22 0.23 0.24 0.25

0.015 0.03 0.045 0.06 0.075 0.09 0.10 0.12 0.14 0.15 0.17 0.18 0.20 0.21 0.23 0.24 0.26 0.27 0.29 0.30

0.02 0.035 0.05 0.07 0.09 0.10 0.12 0.14 0.16 0.18 0.20 0.21 0.23 0.25 0.27 0.28 0.30 0.32 0.34 0.36

0.01 0,025 0.03 0.04 0.05 0.06 0,075 0.08 0.09 0.10 0.12 0.13 0.15 0.16 0.19 0.21 0.23 0.26 0.29 0.33

0.025 0.03 0.04 0.06 0.07 0.08 0.09 0.10 0.11 0.13 0.14 0.16 0.18 0.19 0.22 0.24 0.27 0.30 0.33 0.36

0.02 0.05 0.07 0.10 0.12 0.15 0.17 0.20 0.22 0.25 0.29 0.30 0.33 0.35 0.38 0.40 0.43 0.45 0.48 0.50

TABLE 25-53 Draw clearance, t ¼ thickness of the original blank Blank thickness mm

in

First draws

Redraws

Sizing drawa

Up to 3.81 0.41–1.27 1.30–3.18 3.45 and up

Up to 0.15 0.016–0.050 0.051–0.125 0.136 and up

1.07t–1.09t 1.08t–1.1t 1.1t–1.12t 1.12t–1.14t

1.08t–1.1t 1.09t–1.12t 1.12t–1.14t 1.15t–1.2t

1.04t–1.05t 1.05t–1.06t 1.07t–1.09t 1.08t–1.1t

a

Used for straight-sided shells where diameter or wall thickness is important or where it is necessary to improve the surface ﬁnish in order to reduce ﬁnishing costs.

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ELEMENTS OF MACHINE TOOL DESIGN

25.70

CHAPTER TWENTY-FIVE

TABLE 25-54 Shear strength of various materials Ultimate strength, sut Material

MPa

Ferrous alloys 0.10 carbon steel annealed 0.20 carbon steel annealed 0.30 carbon steel annealed 0.50 carbon steel annealed 1.00 carbon steel annealed Chromium-molybdenum steel: SAE 4130

620 690 862 1035 1240

psi

90,000 100,000 125,000 150,000 180,000

Nickel steel (drawn to 4268C (8008F) and water-quenched): SAE 2320 SAF 2330 SAE 2340 Nickel-chromium steel (drawn) to 4268C (8008F): SAE 3120 SAE 3130 SAE 3140 SAE 3280 SAE 3240 SAE 3250 Nonferrous materials Aluminum and alloys Copper and alloys Magnesium alloys Monel metal K monel Nickel Inconel (nickel chromium iron)

475 745 672 1072 469 831 550 620 689 792 965 1103 1206

69,000 108,000 97,500 155,600 68,000 120,500 80,000 90,000 100,000 115,000 140,000 160,000 175,000

Shear strength, s MPa

psi

240 290 358 550 768

35,000 42,000 52,000 80,000 110,000

380 448 515 620 725

55,000 65,000 75,000 90,000 105,000

675 758 862

98,000 110,000 125,000

655 758 896 930 1035 1138

95,000 110,000 130,000 135,000 150,000 165,000

28–282 150 330 28–145 295–450 450 680 360 520 406 434 455 490 538 580 600

4,000–41,000 22,000 48,000 4,000–21,000 42,900 65,200 65,300 98,700 52,300 75,300 59,000 63,000 66,000 71,000 78,000 84,000 87,000

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

25.71

TABLE 25-54 Shear strength of various materials (Cont.) Ultimate strength, sut Material

MPa

psi

Nonmetallic materials Asbestos board Cellulose acetate Cloth Fiber, hard Hard rubber Leather, tanned Leather, rawhide Mica Papera Bristol board Pressboards Phenol ﬁberb a b

Shear strength, s MPa

34 69 55 124 138 48 90 69 44 33 24 180

psi

5,000 10,000 8,000 18,000 20,000 7,000 13,000 10,000 6,400 4,800 3,500 26,000

For hollow die used one-half value shown for shearing strength. Blank and perforate hot.

0.010 0.009

Die clearance, in

0.008 0.007 0.006 up

0.005

o Gr

0.004

p rou

0.003

G

Gro

up

1

3

m

s

ial

er

at

s, ial

5%

. ,7

+

a

er

av

%

6.0

an

ar

cle

le ec

e

nc

ara

rag

ve

+a

ge

ce

r ver ate +a 2m 5% . 4 , ials ter ma

c age

lea

ran

ce

0.002 0.001

Group 1: 1100S and 5052S aluminum alloys, all tempers. An average clearance of 412 per cent of material thickness is recommended for normal piercing and blanking. Group 2: 2024ST and 6061ST aluminum alloys; brass, all tempers; cold rolled steel, dead soft; stainless steel soft. An average clearance of 6 per cent of material thickness is recommended for normal piercing and blanking. Group 3: Cold rolled steel; half hard; stainless steel, half hard and full hard. An average clearance of 712 per cent is recommended for normal piercing and blanking. Courtesy: Frank W. Wilson, Fundamentals of Tool Design, ASTME, Prentice-Hall of India Private Limited, New Delhi, 1969.

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 Nominal thickness of material which die is designed for, in. Note: 1 in = 24.5 mm

FIGURE 25-38 Die clearances for diﬀerent groups of metals.

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ELEMENTS OF MACHINE TOOL DESIGN

25.72

CHAPTER TWENTY-FIVE

TABLE 25-55 Drawing speeds

TABLE 25-56 Drawing radii Drawing speed, Vd (m/min)

Thickness of stock, mm

Drawing radius, mm

Material

Single action

Double action

Aluminum provide Strong aluminum alloys Brass Copper Steel Steel in carbide dies Stainless steel Zinc

55 — 65 45 18 — — 45

30 10–15 30 25 10–16 20 7–10 13

0.4 0.8 1.25 1.6 2 2.5 3.15

1.6 3.2 4.8 6.3 10 11.2 14

TABLE 25-57 Blank holder force in drawing Thickness of stock, t, mm

Force, N per mm

0.25 0.4 0.5 0.63 0.80 0.9 1.00 1.12 1.25 1.4 1.6 1.8 2.0 2.24 2.5 and over

314 304 294 280 270 260 250 235 225 220 210 196 181 167 157

TABLE 25-58 Recommended die plate thickness (a) For dies with cutting perimeter less than 50 mm Stock thickness 7 mm 0.25 0.5 0.75 Die plate thickness, 2 3 mm/shear stress, kgf/mm2 1 (b) For dies with cutting perimeter greater than 50 mm Cutting perimeter, mm over 50 75 up to 75 150 Factor by which the above tabulated values under (b) should be multiplied 1.25 1.5

1

1.25

1.5

1.75

2

2.25

2.5

4

4.5

5.25

5.75

6.3

6.7

7

150 300

1.75

300 500

2.0

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ELEMENTS OF MACHINE TOOL DESIGN

TABLE 25-59 Chart for bend allowance and correction factor Recommended minimum bend radius ri for sheet metal Aluminum alloys

Magnesium

Steel

Material thickness

24ST and Alclad

24SO and Alclad

2S, 3S and 52S (12 hard)

Cold bend

Hot formed

Stainless Annealed

a 1 2

Up to 0.015 0.016 0.020 0.025 0.032 0.040 0.051 0.064 0.072 0.081 0.091 0.102 0.125 0.156 0.187 0.250 0.375

0.06 0 06 0.09 0.12 0.12 0.12 0.19 0.19 0.25 0.31 0.38 0.44 0.50 0.69 0.81 1.00 —

0.06 0.06 0.09 0.09 0.09 0.09 0.09 0.09 0.12 0.12 0.16 0.19 0.19 0.28 0.38 0.50 —

0.03 0 03 0.03 0.03 0.06 0.09 0.09 0.12 0.16 0.19 0.19 0.19 0.19 0.25 0.38 0.50 —

0.06 0 09 0.12 0.19 0.25 0.31 0.38 0.50 0.56 0.62 0.69 0.75 1.00 1.35 1.50 2.00 3.00

0 06 0.06 0.06 0.06 0.09 0.99 0.12 0.19 0.19 0.19 0.25 0.25 0.31 0.44 0.50 0.62 1.00

0 03 — 0.03 0.03 0.03 0.06 0.06 0.06 0.09 0.09 0.12 0.16 0.19 0.19 0.25 — —

0 03 — 0.06 0.06 0.06 0.09 0.09 0.12 0.12 — — — — — — — —

hard

Low carbon, X-4130 annealed b

— 0.03 0.03 0.03 0.03 0.06 0.06 0.06 0.06 0.09 0.09 0.12 0.12 0.19 0.19 0.25 —

Note: a For bends up to 90 deg. b This applies to 8630 and similar steels. C= t 90 deg angle

t

0.25

90… angle C

C. correction factor

0.20

Closed bevels α’ C

0.15 t

t thickness = 0.125

Open bevels

0.091 0.081

α’

0.10

0.064 0.051 0.040 0.032 0.025 0.020 0.015

0.05

0

80

60

40 Open bevel

20

0

20

α, bevel angle in degrees

40

60 Closed bevel

.80

Courtesy: D. C. Greenwood (ed.), Engineering Data for Product Design, McGraw-Hill Publishing Company, New York, 1961.

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ELEMENTS OF MACHINE TOOL DESIGN

25.74

CHAPTER TWENTY-FIVE

Particular

Formula

FORMING PROCESS: Note: The Symbols, Equations and Examples given in the book entitled ‘‘Mechanical Presses*’’ by Professor Dr. Ing. Heinrich Ma¨kelt and translated by R. Hardbottle, are followed and used in Symbols, Equations and Examples with reference to Figs. 25-41 to 25-49 in this Machine Design Data Handbook. The minimum ram force in mechanical presses

Pmin ¼ 0:5Prat

ð25-161Þ

where Prat ¼ tonnage rating, tonneforce (tf ) The maximum ram force

Pmax ¼ Qkmax ¼ F max

tf

ð25-162Þ

where Q ¼ cross-section kmax ¼ maximum speciﬁc loading, MPa (psi) max ¼ maximum stress, MPa (psi) F ¼ workpiece surface, in2 The press work

A ¼ mPmax h ¼ Pmi h where

ð25-163Þ

m ¼ correction factor taken from Table 25-62 h ¼ work path ¼ 0:5 H H ¼ total maximum stroke setting The volume of the workpiece before and after forming

Q1 h1 ¼ Q2 h2 ¼ constant

The force required to trim the forging

F¼

Pts tf 2000 where F in tonneforce (tf )

ð25-164Þ ð25-165Þ

P ¼ periphery of forging, in t ¼ thickness, in s ¼ shear strength of material, psi For chart for calculating ram path and velocity versus crank angle

Refer to Fig. 25-41

For calculation chart for blanking and piercing with full-edge cutting tool.

Refer to Fig. 25-42

For calculation chart for rectangular bending (a) Vbending on a ﬁxed die, (b) U-bending with back-up

Refer to Fig. 25-43

For calculation chart for deep drawing and redrawing (a) deep drawing with blank holder (b) re-drawing of body

Refer to Fig. 25-44

* Heinrich Ma¨kelt, ‘‘Mechanical Presses’’, translated into English by R. Handbottle, Edward Arnold (Publishers) Ltd., 1968

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Particular

25.75

Formula

For determination of blank-holder force for deep drawing

Refer to Fig. 25-45

For chart for extrusion molding and impact extrusion: a, extrusion molding of hollow bodies in direction of punch travel (forward extrusion); b, impact extrusion (tube extrusion) against direction of punch travel (backward extrusion)

Refer to Fig. 25-46

For determination of multiplication factor for impact extrusion and cold extrusion, and also for stamping and coining

Refer to Fig. 25-47

For chart for calculating stamping and coining

Refer to Fig. 25-48

For chart for calculating hot upsetting and drop forging

Refer to Fig. 25-49

For penetration of sheet thickness before fracture, suggested reductions in diameters for drawing, mean values for m and suggested trimming allowances

Refer to Tables 25-60, 25-61, 25-62 and 25-63

TABLE 25-60 Approximate penetration of sheet thickness before fracture in blanking Work metal

Penetration %

Work metal

Penetration %

Carbon steels 0.10% C annealed 0.10% C cold rolled 0.20% C annealed 0.20% C cold rolled 0.30% C annealed 0.30% C cold rolled

50 38 40 28 33 22

Non-ferrous metal Aluminum alloys Brass Bronze Copper Nickel alloys Zinc alloys

60 50 25 55 55 50

TABLE 25-61 Suggested reductions in diameters for drawing

TABLE 25-62 Mean values for m (standard coeﬃcients)

Material

First draw %

Redraws %

Particulars

Aluminum, soft Aluminum, deep drawing quality Brass Copper Steel Steel, deep drawing quality Steel, stainless Zinc Tin

40 40–50

20–25 20–30

45 40 35–40 40–45 35–40 50 35–45

20–25 15–20 15–20 15–20 15–20 15–20 10–15

Blanking and piercing (full-edge) tough (soft) sheet brittle (hard) sheet Making V- and U-bends with die clash without die clash Deep-drawing and re-drawing Impact extrusion and extrusion forming Stamping Hot, ﬁrst upsetting operations End drop-forging

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m 0.63 0.32 0.32 0.63 0.63 1.00 0.5 0.71 0.36

ELEMENTS OF MACHINE TOOL DESIGN

25.76

CHAPTER TWENTY-FIVE

Particular

Formula

TABLE 25-63 Suggested trimming allowances Allowance per side, mm, for steel with Rockwell B hardness of

First trim or single trim

Second trim or add to ﬁrst trim

Blank thickness, mm

50–66

75–90

90–106

1.20 1.60 2.00 2.36 2.80 3.15 1.20 1.60 2.00 2.36 2.80 3.15

0.063 0.075 0.090 0.106 0.125 0.18 0.03 0.035 0.045 0.050 0.063 0.09

0.075 0.106 0.125 0.150 0.180 0.224 0.035 0.050 0.063 0.075 0.090 0.100

0.106 0.125 0.15–0.180 0.18–0.224 0.224–0.230 0.3–0.355 0.05 0.063 0.075–0.09 0.09–0.100 0.1–0.14 0.15-0.18

MACHINE TOOL STRUCTURES: The optimum ratio l 2 =h for every structure which depends on [], [ ] and E

l 2 6E½ ¼ h ½

ð25-166Þ

where l ¼ length of structure/beam

The natural frequency of an elastic element such as a bar or beam subjected to tension or compression—a case of single degree freedom system

h ¼ distance of outermost ﬁber from the neutral axis in case of bending rﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃ k EA 1 E 1 ¼ ¼ ð25-167Þ f ¼ m l Al l2 where k ¼ stiffness of the system ¼ F=4l

The natural frequency of a simply supported beam subjected to a load at the center of beam – a case of single degree freedom system

¼ mass density of member rﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ k 48E 1 48E bh3 1 E 4bh3 ¼ f ¼ ¼ ¼ 3 3 12 Al m Al 4 l Al l ð25-168Þ where E= is the unit or speciﬁc thickness. It is an important parameter in machine tool structural material and ¼ mass density of material of beam, ¼ specific weight of material or beam. The natural frequency depends on E=.

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

25.77

D t

R

R = 0.8 (D d) t where D = blank diameter, in. Tolerance for R = ± 0.005 in.

d

1.600 0.160 0.140 0.120 0.100 0.080 0.060

1.200 0.800

0.040 0.024 0.020

0.400 0.360 0.320 0.280 0.240

Blank thickness ’t.

Radius R

0.012

0.200 0.160 0.120

16.00

Diameter difference (D d)

12.00

4.80 5.60 6.40 7.20 8.00

2.40 2.80 3.20 3.60 4.00

2.00

1.600

1.200

0.400

0.040

0.480 0.560 0.640 0.720 0.800

0.080

d

h

Example: d = 1.00 h = 0.75 t = 0.020 D = d 2 + 4dh = 2.00 D d=1 R = 0.110 ± 0.005

FIGURE 25-39 Nomograph for determining draw-die radius. Courtesy: American Machine/Metal working Manufacturing Magazine.

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ELEMENTS OF MACHINE TOOL DESIGN

25.78

26

0

CHAPTER TWENTY-FIVE

2

4

6

8

10 12

14 40

24

35

22

30 25 20 15 10 5

18 16 14

Reduction, %

Diameter of blank or cup, in.

20

12 10 8 6 4 2 0

0

2

4

6 8 10 12 Cup diameter, in.

14

FIGURE 25-40 Chart for checking percentage reduction in drawing of cups. The inside diameter is ordinarily used for the cup diameter. Courtesy: From ASM, Metals Handbook, 8th ed., vol. 4, 1969.

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ELEMENTS OF MACHINE TOOL DESIGN

25.79

0 40 5 31 50 2 0

lo

20 60 1 125 00 1 0 8 3 6

ve

50 0 4 .5 31 25

C

m

315 355 40 45 50 56 56

A1

D1

n-

1

0.1 0.08 0.063 E2 0.05 0.04 0.035

0.025

Z

0.02 0.016 0.0125 0.01

0.008

0.0063

∑ α α…

28

mi

0.2

2 .6 1 5 1.2 1 .8 0 .63 0 5 0.

25

n,

F2

20 21

1000 800 600 500 400 315 250 200 160 125 100 80 63 50 D2 40 31.5 25 20 16 12.5 10

E1 0.125

4 15 3. .5 2

18

II

0.16

6.3 5

15

13

A2

es

50 40 .5 31 5 2 0 2 16 2.5 F1 1 10 8

12.5

ok

in

B1

str

/m

r

m

m

v,

pa

ro

m

0.01 0.0125 0.02 0.02 0.025 0.025 0.02 0.05 0.053 0.02 0.1 0.25 0.15 0.235 0.236 0.29 0.355 0.355 0.53 0.11 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.35 0.4 0.45 0.5 0.56 0.63 0.71 0.8 0.9 1 1.12 1.25 1.4 1.6

Crank angle 2, rad

1.61.5 1 4.5 9 5.6 6.3 7.9 8 9 10 11.2 Crank angle o2, degrees 12.5

0.8 0.63 0.5 0.4 0.315 0.25

III

or

g

st

h,

Ra

ty

W

kin

f to

ke

Connecting rod ratio 10 λ = H: 21 = 1: 10 8 6.3 5 Work-path/ stroke h/H 4 0.001 0.002 0.002 0.002 0.005 0.002 0.004 0.005 0.053 0.003

1

ci

20 6 1 .5 12 10

am

4 .5 31 2.5

R

I

0.06 0.07 0.08 0.09 0.1

1000 800 630 500 400 315 250 200 160 125 100 80 63 50 40 31.5 25 20 16 12.5 10 8 6.3 5 4

8 3 6. 5

Total ram stroke H, mm

B2

C

2 6 1. 26 1. 1

1000 800 630 500 400 315 250 200 160 125 100 80 63 53 40 31.5 25 20 16 12.5

0.8.83 0 0.5 .4 0 .15 0 .25 0 .2 0 6 0.7 .15 0 .1 0

Total ram stroke H, mm

ELEMENTS OF MACHINE TOOL DESIGN

8 6.3 5

λ = 1:10

0.925 0.412 0.515 0.58 0.69 0.825 0.875 0.975 0.63 0.463 0.067 0.075 0.085 0.095 0.106 0.118 0.132 0.15 0.17 0.19 0.212 0.236 0.265 0.3 0.335 0.375 0.95 0.75 0.06 0.07 0.08 0.09 3.5

4

4.5

5

0.1

0.11 0.12 0.14 0.16 0.18 0.20 0.22 0.25 0.28 0.32 0.35 0.4

5.6

6.3

7.1

8

9

10 11.2 12.5

14

16

18

20 22.4

0.45 0.5 25

28

0.56 0.63 0.71 0.8 0.9 1 1.12 1.4 1.6 1.25 31.5 35.5 40 45 50 56 63 80 90 71

FIGURE 25-41 Chart for calculating ram path and velocity versus crank angle.

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50

ELEMENTS OF MACHINE TOOL DESIGN

25.80

CHAPTER TWENTY-FIVE

10000

C

2000

800 500 315

630

0.6 3 0.4 0.2 5

80

1600 1000

4 2.5 1.6 1

1250

Sh

400

A

160

16

31.5

25

50

40

125

80

63

k

100

200

160

315

250

E

Hole diametar d, mm

ee

tt hi 6,3 s. ck 4 mm n es 2,5 1,6

1

th

pa

h.

s

0,6 3

m

m

G

4 2.5 1.6

10 6.3

Sh 10

k

ine it l Lim

25 16

ity ac ap kc r o sw es pr r fo 6.3

16

or W

III

500

400 25

Blanking circumferenca U. mm

6,310 16 25 40 63 100 160 250 400 630 1000 1600 2500 4000 6300 10000 16000 25000 40000 100 2 3,2 5 8 12,5 20 32 50 80 125 200320 500 800

0,4

PK d (u)

Fs

δ=hk

0,2 5

F

1

0.63 0.4 0.25

rk th

pa h.

Blanking work Ak, m.kg or tonnes

20

Blanking force p , tonnes

10

6.3

t th s. m ickn m ess

Wo mm

6300 4000 2500 1600 1000 630 400 250 160 100 63 40 25 15 10 6.3 4 2.5 1.5 1

12.5

8

5

E

ee

250

200 125

I

B

10 6.3

20 5 2 .5 31 0 4 0 5 3 6

3150 2500

Limit

4000

C

D

Blanking area, mm2

10 .5 12 16

line s=d ,

II

25

6300 5000

16 10

th ng tre 2 r s mm ea kg/ Sh k ,s

8000

FIGURE 25-42 Calculation chart for blanking and piercing with full-edge cutting tool. Equations and Examples Equations: Area ¼ Fs ¼ Us ¼ ds Section II: The tonnage rating of the press ¼ Pk ¼ Fs ks =1000 tonneforce (tf ). The shear strength of metallic material, ks , is ks ¼ 0:8 B N/mm2 (kgf/mm2 ) The work path is taken as hk ¼ s, where s ¼ thickness of material/sheet. Section III: The cutting work¼ Ak ¼ mPk hk , mm-tonneforce (mm tf) or m kgf where m ¼ correction factor ¼ 0:63 for soft sheet. Example: Blank diameter d ¼ 800 mm (31.5 in) or U ¼ 2500 mm (98 in). Sheet thickness ¼ 1 mm (0.039 in); The blanking area ¼ Fk ¼ 2500 mm2 (3.85 in2 ). Blanking or cutting force Pk ¼ 980 kN (100 tf ). Work¼ 61:78 N m (63 mm-tf or 456 ft-lbf ).

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ELEMENTS OF MACHINE TOOL DESIGN Limit line 1min=2.8 for V-form

Die-span I (only V-form). mm Bending width b.mm 10 12.5 16 20 25 31.5 40 50 63 80 100 125 160 200 250 315 400 500 630 800 1000 0.4 0.5 0.63 0.8 1 1.25 1.6 2 2.5 3.15 4 5 6.3 8 10 12.5 16 20 25 0.06 A1’ A1 A A’ 0.1

nve

rsio

n li

ne

B

for

U-f

B

orm

2 1 50 10 80 63 0 4 2 0 16 5

I

0.16

0.4

C 1 C’ 2.5 6.3

II

0.25

100 160 250 400 630 0 100 D’ 0 160 0 250 0 400

0.63

CC

1.6

4

C’

10

0

(b)

b

a m =r in i

s

hv

ra

250

Die-span I’(only for v-form). mm

400

10 72.5

16

E

8

5

12.

5 3.15 2 1.25

fo

rp

re

ss

F

e 80 lin it 50 m 31.5 Li 20

80

tones

1000 1600 2500 4000 Pv,U/σB E’

125

45

50 63

6300 10000 2.5 2.5 4 6.3 10 16 25 40 63 100160 250 B ’ 3.2 6.3 h vU . th σB pa 8 4 h ork mm t W 5 10 ng tre 12.5 6.3 l s m2 16 8 a i er g/m t 20 10 a k 1 M 12.5 25 12 0 . 5 16 31.5 16 2 2 0 IV 2.0 40 III 31 5 50 40 .5 2.5 5 H G K G 63 0 63 31.5 80 80 40 G’ F’ 50 100 G’ H’ 125 63 80 160 100 200 12.5 250 160 315 I I’ 200 400 500 250 2.5 4 6.3 10 16 25 40 63 100 160 250 400 630 1000 1600 2500 4000 630010000 16000 Bending force Pv,U,tonnes Bending work Av,U’,m.kg or mm.tonnes (+counterforce Pa for U-form) 630

Maximum force Pv max’

160

20 25

.5

Hu

t PG

l

31

b s

lmin

100

100

Pu

ri

D

00

630

(a)

Pv

ri

/σB P v,U

Co

s2//’ for V-form, s/2.5 for U-form

Sheet thickness s.mm

w

or

ki

ng

ca

pa

ci

ty

FIGURE 25.43 Calculation chart for rectangular bending (a) V-bending on a ﬁxed die, (b) U-bending with back-up Courtesy: Heinrich Makelt, Die Mechanischen Pressen, Carl Hanser Verlag, Munich, German Edition, 1961 (Translated by R. Hardbottle, Mechanical Presses, Edward Arnold (Publishers) 1968) Equations and Examples: (Fig. 25-43) Equations: Refer to Eqs. (25-155) and (25-156) for V-bending force and U-bending force. The equation for V-bending force Pv ¼ ½C B bs2 =1000l 0 tonneforce (tf) (kN or lbf ). The bending work Av ¼ mPE hv mm tonneforce (mm tf ). The limit of eﬀective span or width of V-die lmin ¼ 2:8s in mm (in). The work path hv ¼ 0:5l 0 0:4ðs þ ri Þ in mm (in). (a) V-bending Example: s ¼ t ¼ 2:5 mm (0.1 in); b ¼ 630 mm (25 in); B ¼ 618 N/mm2 ¼ 63 kgf /mm2 (90000 lbf/in2 ); Pv ¼ 25 tf [E-F-G ]; PE ¼ 50 tf; h ¼ 0:5l ¼ 5 mm (0.2 in) and m ¼ 0:32. The bending work¼ As ¼ 784 kN m (80 mm tf or 579 ft lbf ) [A-B-C-C-A ], D-D [B-H and G-H-I ]. (b) U-bending: Equations: The equation for U-bending force ¼ Pv ¼ C(2/5) B bs=1000 tf (kN or lbf ). The backing force PG ¼ 25% of PU . The total force PU þ PG ¼ 1:25PU . The bending force ¼ AU ¼ mðPU þ PG ÞhU mm-tf (m N or ft lbf ). The work path ¼ hU ¼ 3s mm (in). The correction factor ¼ m ¼ 0:63. Example: s ¼ 5 mm (0.2 in); s=2:5 ¼ 2 mm (0.08 in) [A0 -B0 -C0 ]. b ¼ 500 mm (19.75 in), PU = B ¼ 1000 [C0 -D0 and A0 -D0 ]. B ¼ 392 N/mm2 (40 kgf/mm2 or 57000 lbf/in2 ). PU ¼ 392 N or 40 tf [D0 -E 0 -F 0 -G0 ]. PU þ PG ¼ 496 N or 50 tf. AU ¼ 4900 mm kN (500 mm tf or 500 m kgf or 3617 ft lbf) for hv ¼ 16 mm (0.70 in) and m ¼ 0:63 [G0 -H 0 -I 0 ]. Prut ¼ 490 kN (50 tf ) [I 0 -K-G ]

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ELEMENTS OF MACHINE TOOL DESIGN

10 12.5 16 20 25 31.5 40 50

2500 2000 1600

63 80

D

C C’

D’

1250 1000 630

800

Body cross-section Sheet thickness s, s2. mm Qz,2. mm2 5

0.4

250

(a)

Pz

II

2.5 5 6.3

20 3.2

25 32 40

6.3 12.5

10 5

E’

s

E

Qz

63 hz

40

500 C’

B1’

25

I’

III

B2’

315

5 0.2

C bo onv dy ers cro ion ss- lin sec e fo tion r Q

I

2

200 125

s

h1

80

(b)

PA d1 (U1)

Q1

d2 (U2)

50

hz

s

Q2 s1 31.5 40 10 16 25 100 160 250 25 63 160 250 100 630 1000 25 63 40 400 10 16 20 80 200 315 31.5 125 50 200 80 1250 2000 3150 50 31.5 500 800 315 500 125 Press tonnage rating Prat. tonnes 2 63 400 250 160 250 100 40 63 400 630 1000 1600 2500 16 100 160 40 25 10 160 500 20 50 80 125 315 800 31.5 12.5 31.5 12.5 200 50 200 20 125 80 8 E’ 12.5 K’ E 3 4 A 16 250

5

50 63

80 0 10

s

d (U)

Drawing force Pz,A. tonnes

4 8

100

D

2 0.3

B

C

400

160

1 0.8 3 0.6 0.5

2 1.6 .25 1

4 .15 2.5 3

G

H

G’

40

0.8 0.63 0.5

20 31.5 G

F’

z,A

50

A2’

40

31.5

400

C

ro s ra s-s tio ec IV q tio n A Drawing ratio βN

F

L’

G’

50 H’ 63 0 80 16 00 50 15 00 00 30 00 000 250 600 000 500 150 000 000 300 000 0000 500 2 2 2 3 4 5 6 8 1 1 1 2 2 3 4 5 6 8 1 1 100 63 re-drawing Work A . 5 12

Drawing Or m.kg or mm.tonnes

A 1

A1’

2.5 2.24 25 20

2

1.8

16

1.6 12.5

Blank height h1. mm

1.4

10

1.2 5 8

M’

630 1000

1600 6.3

Blank cross-section Q1. mm2

Body drawing stress Kz = Z(A) σB. kg/mm2

1 — Mean drawing diameter d, d2, mm; 2 — Mean body circumference U, U2, mm; 3 — Mean drawing diameter d, mm; 4 — Drawing force Pz, tonnes; 5 — Limit line for rational utilisation of press-working capacity; 6 — Work path hz,2 (mm) of cylindrical bodies

FIGURE 25-44 Calculation chart for deep drawing and redrawing (a) Deep drawing with blank holder (b) Re-drawing of body Courtesy: Heinrich Makelt, Die Mechanischen Pressen, Carl Hanser Verlag, Munich, German Edition, 1961 (Translated by R. Hardbottle, Mechanical Presses, Edward Arnold (Publishers) 1968) Equations and Examples: (Fig. 25-44) Force and work requirements for deep drawing and re-drawing: The body cross sectional area Qz ¼ Us ¼ ds mm2 where d ¼ mean diameter of the drawn parts, mm U ¼ mean circumference of the drawn part, mm s ¼ sheet thickness, mm The maximum draw force ¼ Pz ¼ Qz z B =1000 tf where z ¼ drawing factor ¼ y ln N =F ¼ ln N = ln max N ¼ useful drawing ratio ¼ ðD=dÞ max ¼ limiting drawing ratio ¼ Dmax =d D, Dmax ¼ diameters of draw of the blank sheet, mm F ¼ forming eﬃciency For D ¼ 160 mm (6.3 in), d ¼ 100 mm (4 in), N ¼ 1:6 and max ¼ 1:8, the drawing factor is z ¼ 0:8. The strength ratio ¼ y ¼ 1:2 between the deformation strength kfm and the tensile strength B (lines a-b-c and d-e-f ) i.e., y ¼ kfm = B . 2 1Þ mm. The work path for cylindrical drawn part ¼ hz ¼ d=4ðN The drawing work ¼ Az ¼ mPz hz N m or mm tf (m kgf) where m ¼ correction factor ¼ 0:63 Mean body circumference ¼ U ¼ 315 mm (12.5 in) [Left side of Fig. 25-44]. Mean drawing diameter for the case of cylindrical hollow-ware ¼ d ¼ 100 mm (4 in). The body cross section due to drawing stress kz is Qz ¼ 800 mm2 (1.23 in2 ) [Section I top right]. The sheet thickness ¼ s ¼ 2:5 mm (0.1 in) [line A-B-C ]. Section II (top left) gives Pz ¼ 309 kN (31.5 tf) [C-D-E ]. Drawing stress is kz ¼ Z B ¼ 0:8 50 ¼ 392 N/mm2 (40 kgf/mm2 ). Section IV (bottom right) gives hz ¼ 35:5 mm (1.4 in) for BN ¼ 1:6 and d ¼ 100 mm (4 in) [A-F-G ]. The drawing work¼ Az ¼ mPz hz ¼ 0:63 31:5 35:5 ¼ 6908 mm N (705 mm tf or 5136 ft-lbf) [G-H and E-B ]. The press tonnage rating Prat ¼ 63 tf. Force and work requirements for re-drawing: The reduced body cross-section during redrawing ¼ Qz ¼ Uz sz ¼ dz sz mm2 where Uz ¼ mean circumference, mm dz ¼ mean diameter of the ﬁnished product, mm sz ¼ re-drawn wall thickness, mm The reduced body cross-section ¼ Qz ¼ 500 mm2 (0.77 in2 ) undergoing loading by the drawing stress kz [line A01 -B01 -C10 ] (Fig. 25-44). Section II kt ¼ ZA B ¼ 071 45 9:8066 ¼ 312 N/mm2 (31.5 kgf/mm2 or 45000 psi). The re-drawn force ¼ PA ¼ 16 tf [C0 -D0 -E 0 ]. The blank cross section Q1 ¼ Q2 =qA ¼ 800 mm2 (1.23 in2 ) [C0 -B02 -L0 -M 0 ] (Fig. 25-44). The blank height h1 ¼ 31:5 mm (1.25 in). h2 ¼ 50 mm (2 in) [A02 -F 0 -G0 , here F 0 happens to coincide with L0 ] P1 ¼ 16 tf, h2 ¼ 50 mm (2 in), m ¼ 0:63 [E 0 -H 0 and G0 -H 0 ]. Psat ¼ 50 tf, [H 0 -I 0 -K 0 ].

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ELEMENTS OF MACHINE TOOL DESIGN Sum of diameters D + da ¯ d( β + 1). mm

80

25

16

10

6.3

H

4

2.5

50

31.5

20

12.5

5

8

H

3.15

M at σ e B rial .k s g/ tre m n III m 2 gt h 80 63 0 5 0 4 .5

G

31 5 2 0 2 6 1 .5 12 0 1

50 53 58 40 31.5 25 20 16 12.5 10

Material strength σB. kg/mm2

2

1.6

m m

N.

2.2 4 2 1.8 1.6 1.4 1.25

40

Blank-holder pressure P N. kg/cm2

Drawing ratio β

1.25

0.01 0.012 0.016 0.02 0.025 0.032 0.04 0.05 0.063 0.08 0.1 0.125 0.16 F F 0.2 0.25 0.315 0.4 0.5 0.63 0.8 1.0

Dra

win

4

gr

atio

β=

1.2

5

1.4 1.6

IV

E

1.8

Ratio PN / σB

nes

Fl an ge

ton

63

Ratio PN / σB

,

e PN

100

B

forc

K

id th

der

3.15 2 I 1.25 0.8 .5 0 0.32 0.2 0.12 0.08 0.05

Holding-down area FN. cm2

hol

Holding-down area FN. cm2

nk-

II

w

Bla

500 315 200 125 80 50 31.5 20 12.5 8 5

31.5 40 50 63 80 100 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 31500 4000 5000 50 63 80 100 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 500 20000 400 A Mean flange 12500 circumference 315 8000 UN .mm 150 250 5000 125 200 3150 100 160 2000 80 125 1250 63 100 800 50 80 500 40 63 325 31.5 50 200 25 40 I 125 20 31.5 B 16 C 80 C 25 50 12.5 20 D 31.5 BN 10 16 FN 20 8 12.5 DN (UN) 12.5 6.3 s 10 da 8 5 8 5 4 3.25 1

2 2.24

D 5 6.3 8 10 12.5 16 20 25 31.5 40 50 63 80 101 125 100 200 250 325 400 Ratio da / s

FIGURE 25-45 Determination of blank-holder force for deep drawing Courtesy: Heinrich Makelt, Die Mechanischen Pressen, Carl Hanser Verlag, Munich, German Edition, 1961 (Translated by R. Hardbottle, Mechanical Presses, Edward Arnold (Publishers) 1968) Equations and Examples: The blank holder force PN ¼ 10% of the ram drawing force Pz ¼ 0:1Pz . The drawing work ¼ Az ¼ ðmPz þ PN Þhz m N (mm tf or m kgf ). Section I (Fig. 25-45): First holding down area (ﬂange area) under load ¼ FN ¼ UN BN ¼ ð=4ÞðD þ da ÞðD da Þ ¼ ð=4Þdð þ 1Þdð 1Þ cm2 where UN ¼ mean ﬂange width, cm; D ¼blank diameter, cm; da ¼outside diameter after drawing, cm d ¼ mean drawing diameter, cm; ¼ drawing ratio Empirical equation for the ratio PN = B . Section II: PN = B ¼ 0:25 ½ð 1Þ2 þ 0:005 ðda =s)] where PN in N/mm2 (kgf/cm2 ) and in N/mm2 (kgf/mm2 ). Section III: The blank holder force ¼ PN ¼ FN PN =1000 tf. Example: Section I: For the case of cylindrical part from sum of the D and da ðD þ da Þ ¼ 235 mm (9.3 in). For ¼ 1:6 with 458 slope FN ¼ 100 cm2 (15.4 in2 ) [line A-B-C ] (Fig. 25-45). For mean ﬂange circumference ¼ UN ¼ 375 mm (14.8 in). FN ¼ 100 cm2 (15.4 in2 ) [A-B-C ] for BN ¼ 26:5 mm (10 5 in). Section IV: For ðda =sÞ ¼ 40, ¼ 1:6, ðPN = B Þ ¼ 0:14 [D-E-F ] (Fig. 25-45). B ¼ 392 N/mm2 (40 kgf/mm2 , or 57000 psi) in Section III. From this the blank holder pressure ¼ P0N ¼ 55 N/mm2 (5.6 in kgf/cm2 or 80 psi) [F-G-H ] (Fig. 25-45). Finally PN ¼ 5490 N (0.56 tf) [H-I and J-K ] (Fig. 25-45). Section II: FN ¼ 100 cm2 (15.4 in2 ).

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ELEMENTS OF MACHINE TOOL DESIGN CHAPTER TWENTY-FIVE

D’

63 0 8

10 0

50

16 20 0 2 0 31 50 5

5 12

E’

E

5 6.3 8 10 12.5 16 20 25 31.5 40 50 63 80 100 125 150 200 250 325 400 500

Extrusion moulding or impact extrusion force PHF. tonnes

C’

da

C d1

12.5

40

16

50

25

31.5

80

40

100

50

125

3

4

160

L’

6 .1

0

25 40 63

6

10 0 16 0

M’

32 0.

40

25

63

200

F’

F

M N’

80

250

100

315

95 94 92 90 88 84 80 16

5

3.2 4 5 6.3 8 8 10 12.5 16 G 20 G G’ 25 G’ 0 0 5 12 31.5

Forming height

h2. hi. mm A1’ A1 630 500 400 315 250 200 160 125 100 80 63 50 40 31.5 25 20 16 12.5 25

Extrusion moulding or impact extrusion work AH,F. m. kg or mm. tonnes

20

S1

C bl on an ve k rs cr io os n s fin -s e ec for tio n Q A’ H Blank diameter d1.di

4 3.2 8.5

63

0 -1 d1

6.3

1

FN

A 10

31.5

6.3 10 16

50 40

63

III

16 12 20 0 5 0 25 0 31 5 40 0 50 0 63 0 80 0 10 00 12 50 16 00 20 00 25 00 31 50 40 00 50 00 63 00 80 00 100 00

80 0 10

2

.5 12 10 8

L

B 6.3

5

5 63 8 10 12.5 16 20 25 1.5 40 50 63 80 100 125 150 200 250 325 400 500

E’ K’ E Lim it of pline f res or ra sw tio ork na ing l ut cap ilisa aci tion ty I H H’

S1

1

Extrusion moulding or impact extrusion force PHF. tonnes

20 25 .5 31

=

16

B

FBa

75 20 68 25 60 31.5 50 40

50 63 80 100 125 160 N 200 250 7 315 10 8 6.3 400 500 630 800 1000 1250 1600

QZ (mm2)

D

400 630 100 160 0 0 250 0

hZ

25 20

hF s1 s2

0

Qz

d2 (U2)

I

0. .2 25

sZ

QF hi

FBi

di

all l w ss tia ne In ick mm th s 1.

ion ss re m2 mp /m co . kg i fi c ec g K D Sp din loa

QH

(U)

d s3

0. 0. 4 5

hZ~hH

d1 (U1)

b)

0

s

II

0. 0. 05 0 0. 6 0. 08 1 0. 12

PH

4000 3150 2500 2000 1600 1250 1000 C’ 630 500 C 400 325 250 200 150 125 100 200 150 125 100 80

0

a)

Blank cross-section OH. FBl. mm2

25.84

IV

1 — Limit line; 2 — Conversion line for blank cross-section; 3 — Mean blank circumference U1, mm; 4 — Degree of deformation ∈, %; 5 — Cross-section ratio qH, F; 6 — Body cross-section QF, mm; 7 — Body cross-section; 8 — Extrusion moulding or impact extrusion stroke h1, F, mm

FIGURE 25-46 Chart for extrusion molding and impact extrusion: a, extrusion molding of hollow bodies in direction of punch travel (forward extrusion); b, impact extrusion (tube extrusion) against direction of punch travel (backward extrusion). Courtesy: Heinrich Makelt, Die Mechanischen Pressen, Carl Hanser Verlag, Munich, German Edition, 1961 (Translated by R. Hardbottle, Mechanical Presses, Edward Arnold (Publishers) 1968) Extrusion Forming (Fig. 25-46) : (a) Equations: The extrusion molding force ¼ PH ¼ QH kD =1000 tonneforce (tf) (kN or lbf ). The body cross section ¼ QH ¼ U1 s1 ¼ d1 s1 mm2 (in2 ). The reciprocal ratio of cross-section qH ¼ Qz =QH ¼ U2 s2 =U1 s1 , where QH ¼ cross-section of the deformed blank before forming and Qz ¼ cross-section of the blank after forming, Us ¼ mean circumference and s2 ¼ wall thickness of ﬁnished product. The relation between the cross-sectional ratio qH and the degree of deformation " (%) is qH ¼ 1 ð"=100Þ. Example: Blank diameter d ¼ 25 mm (1 in), s1 ¼ 11:2 mm (0.45 in) U1 ¼ 80 mm (3.2 in) Body cross sectional ratio¼ qH ¼ 0:25 corresponding to " ¼ 75%. Speciﬁc compression loading kD ¼ 2196 N/mm2 (224 kgf/mm2 or 319000 psi). The extrusion force ¼ PB ¼ 1960 kN (200 tf) (C-D-E); Qz ¼ 224 mm2 (0.36 in2 ) (B-L-M-N) (Fig. 25-46). Work done due to extrusion ¼ AH ¼ 4000 mm tf (4000 m kgf or 28933 ft/lbf ) [G-H and E-H, H on the limit line]. (b) Equation: The punch force ¼ PF ¼ FB1 kD =1000 tf. The body cross-section ¼ QF ¼ FB1 qF =ðl qF Þ mm2 . The cross-section ratio ¼ qF ¼ QF =FB . The total initial cross-section of the blank disc ¼ FB ¼ QF =qF ¼ QF þ FB mm2 . The wall thickness of the product ¼ sB ¼ ½QF =qF 1=2 d1 =2 mm. The work path ¼ hF ¼ s1 s2 ¼ qF h1 mm. The work for m ¼ 1, AF ¼ PF hF mm tf.

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

25.85

Example: d ¼ 45 mm (1.8 in), kD ¼ 785 N/mm2 (80 kgf/mm2 or 114000 psi) q ¼ 0:2, forming height ¼ h ¼ 12:5mm: B ¼ 196 N/mm2 (20 kgf/mm2 or 28500 psi). Punch force ¼ PF ¼ 1225 kN (125 tf) [C0 -D0 -E 0 ] (Fig. 25-46). Blank area ¼ FB1 ¼ 1600 mm2 (2.46 in2 ) [A0 -B0 -C0 ]. Body cross-section of product ¼ QF ¼ 400 mm2 (0.62 in2 ) [A0 -L0 -M 0 -N 0 ]. The inside body height ¼ h1 ¼ 125 mm (5 in). Work done: AF ¼ 30896 m N (3150 mm tf or 22785 ft-lbf ) [E 0 -H 0 and G0 -H 0 ]. Press rating ¼ Psat ¼ 1960 kN (200 tf) [H 0 -I 0 -K 0 ].

10

6.3

4 3.15 2.5 2

s die bo

5

S Co I ho oli ldd l l o ex w and tru bo de di d, es thi ck wa II lle St dh a oll ow mpi n bo die g s lid So

Multiplication factor, π H,F,P

8

1.6 1.25 1

Degree of forming ∈, % 95 94 92 90 88 84

0.05 0.06 0.08 0.1

80 75 68

0.12 0.16 0.2 0.25 0.32

60 50 37 0.4

0.5

20

0.63 0.8

Cross-section ratio qH,F for extusion moulding and impact extrusion Height ratio S2 / S1 for stamping and cold working FIGURE 25-47 Determination of multiplication factor for impact extrusion and cold extrusion, and also for stamping and coining Courtesy: Heinrich Makelt, Die Mechanischen Pressen, Carl Hanser Verlag, Munich, German Edition, 1961 (Translated by R. Hardbottle, Mechanical Presses, Edward Arnold (Publishers) 1968)

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ELEMENTS OF MACHINE TOOL DESIGN

6 0

II

8 0 10 5

12 60 1 00 2 50 2

5

31

X

n sio es m2 pr m /m co , kg lflc K D ec ng Sp n d i lo

B

50 3

8000 6300 5000 4000 3150 2500 2000 1600 1250 1000 800 A 630 A 500 400 315 250 200 160 125 100

C fo on r d ve ie rsi ar on ea c ur ve

CHAPTER TWENTY-FIVE

X

25.86

Fp

I

D

FP

PP

hP

Vstamp

s1

s2

d

E 20 25 31.5 40 50 63 Blank diemeter d. mm E

80 100

F

Y

3.1 2.5 2 5

1. 1.2 1 6 5

C 80 8 12.5 20 31.5 50 80 125 200 315 500 10 12.5 16 5 6.3 10 16 25 40 63 100 160 250 400 5.3 0.63 20 C 8 0.8 25 L imi .5 10 31 t cu G H G rve 12.5 40 cap for ra 15 50 aci tion ty u al 20 63 tllls pres 25 atio sw 80 o n rkin 0 31.5 III 10 g 5 40 12 0 6 50 1 65

IV

Stamping work Ap, m.kg or mm.tonnes

63 80 10 0 12 5 16 0 20 0 25 0 31 5 40 0

25 31 .5 40 50

10 12 .5 16 20

6. 3 8

20 0 25 0 31 5 40 0 50 0 63 0 80 0 10 00 12 50 16 00 20 00 25 00 31 50 40 00 50 00 63 00 80 00 10 00 12 0 50 0

5

4

Y

he

c la

Stamping volume Vstamp, cm3

X — Projected die area Fp, mm; Y — Stamping stroke hp, mm; Z — Stamping force FIGURE 25-48 Chart for calculating stamping and coining Courtesy: Heinrich Makelt, Die Mechanischen Pressen, Carl Hanser Verlag, Munich, German Edition, 1961 (Translated by R. Hardbottle, Mechanical Presses, Edward Arnold (Publishers) 1968) X- projected die area Fp , mm; Y- stamping stroke hp , mm; Z, stamping force Pp , tonnes.

Key to Fig. 25-49 Equations and Examples: Forging temperature ¼ T ¼ 10008C. Tensile strength of plain carbon steel ¼ B ¼ 588 N/mm2 (60 kgf/mm2 or 86000 lbf/in2 [point B ] (Fig. 25-49). Static deformation resistance ¼ kFg ¼ 49 N/mm2 (5 kgf/mm2 or 7100 lbf/in2 ) [point C of curve]. The deformation rate ¼ w ¼ "r=t(% sec) ¼ 500%/sec [point D ]. The arithmetic proportions of upsetting ¼ "h ¼ 4h=ho ¼ ½1 Fo =F1 100%. The dynamic deformation resistance ¼ kFd ¼ 98 N/mm2 (10 kgf/mm2 or 14200 psi) [point E of the curve] (Fig. 25-49). ¼ 2kFg where kFa ¼ static strength. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ The diameter of non-circular upset or forged component is calculated from d111 ¼ ð4=ÞF1 ¼ 1:13 F1 mm where F1 ¼ crosssection after forming (upsetting surface). The ﬂash ratio ¼ b=s ¼ 4:8 (point F, scale 11). The deformation resistance ¼ kw ¼ 392 N/mm2 (40 kgf/mm2 or 57000 psi) [point G of the curve]. The upsetting force ¼ Ps ¼ 24516 kN (2500 tf) [point I of the curve] A prescribed or theoretical upsetting or die diameter d1 [D ¼ 280 mm (11 in)]. The corresponding upsetting or die area F1 ½Ftot ¼ 63000 mm2 (96 in2 ) [point H ]. The maximum diameter D ¼ d1 þ 2b of forged component The crushed ﬂash or the total cross-sectional area ¼ Ftot ¼ F1 þ Ub where U ¼ periphery of crushed area. The mass ratio ¼ Ls =Bm ¼ 6:3 [point K ]. The maximum upsetting force ¼ Pmax ¼ 30890 kN (3150 tf) [point L of the curve]. The upset path ¼ h ¼ 16 mm (0.65 in) [point M ]. The upsetting work ¼ As ¼ 348134 mm N (35500 mm tf or 256665 ft-lbf) [line N-O ].

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

I Free forging d 1 /h 1m II Drop forging b/s III Form upsetting d 1 /2h 1m

Forging temperature T. …C

ia

ls

te

st

in

g

ac

ne

III

rk

N

40

M

16

6.3

2.5

Upsetting path h to BDC on upsetting. mm 200

80

31.5

12.5

Upsetting path h to BDC on fininshed forging. mm

5

.1 2

0

ter d1 or ttin D. ga mm or nd Ft d i ot . e mm are aF 2 1 250

me dia

25

G 50 100

160

12

100

5

8

80 5 63 3.15 800

20

0×

10 3

80

50

31. 5

H

20

12.

5

se

40

200

0

125

1250

Upsetting force P s . tonnes

2000

I I

3150

5000

Ls

800

1250

e ur

2000

3150

5000

L

t en m m /B

s ea L s m io ie rat D

V K

1 5 2. 3 6. 16

100

L

Max. upsetting work. tonnes

o gw

kg

Pres a t 2 5 s w o r k in g % sp c e e d a p a c it y In c r e ase Pres s a t 1 5 w o r k in g % sp capa eed d e c r c it y ease

ps

in ett

315

15

500

0 ..

Bm

1

1.6

4

2.5

3 10

6.3

16

10

40

25

63

× 100

A

m. s.

50

400

10

250 500

O

VI

500 mm

6.3

150

U

250

G

10

Up

1 1.5 2.1 3 4 5.3 7 9

ig

IV

4

0.8 3 0.6 0.5 0.4 5 0.31 5 0.2 0.2 6 F 0.1 5 0.12 0.1

H

0

2.5

n tio ma for cy De icien eff

1 1.6 2.4 3.4 4.8 6.3 8.5 11.2 14

4

h

~8

E

C

gt

0

Dyn. deformation strength. kg/mm 2

6.3

2.5

2.5

σB

H ig C h-p r- e N rc hs i st ent le N pe ee ag si i s rc n ls e B te en Te C el ta (h s g st ig e ee h- C ls C pe st rs r e te cen els el ta s) g e

n tre

0

4

m

~5

6.3

10

hi

/m

... 5 40

16

m

10

dd ie

X

E

C

1400

2

g .k

ga n

.1

~3

Z Y s er mm ) Ha 5 m/s

ff.

(ue

s s e /s) os 5 m p r .. 0.0 i c 5. u l 0.02 r a f. = y d (uef

er

1250

I

ttin

H

at

1150

A

se

0 00

M

Deformation resistance, kg/mm 2

50

te

1000

900

16

Up

50 160 500 600 000 5 1

25

1 2.5 4 6 8.5 12 16

Static deformation strength K Fs. kg/mm 2

5

800 25

II

ra n io ec t a m /s or % e f w, D D

25.87

X — Explosion deformatuon; Y — Highspeed presses (V eff ~ 0.53); Z — Longitudinal mechanical presses (V eff ~ 0) 125)

FIGURE 25-49 Chart for calculating hot upsetting and drop forging Courtesy: Heinrich Makelt, Die Mechanischen Pressen, Carl Hanser Verlag, Munich, German Edition, 1961 (Translated by R. Hardbottle, Mechanical Presses, Edward Arnold (Publishers) 1968)

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ELEMENTS OF MACHINE TOOL DESIGN

25.88

CHAPTER TWENTY-FIVE

Particular

Formula

Refer to Table 25-64 for unit stiﬀness or speciﬁc stiﬀness E=. The ratio of weights of two bars of same length whose weights are W1 ¼ 1 A1 l and W2 ¼ 2 A2 l

W1 1 A1 l E2 1 E2 =2 ¼ ¼ ¼ W2 2 A2 l E1 2 E1 =1

ð25-169Þ

where E= is the unit stiﬀness or speciﬁc stiﬀness Refer to Table 25-64, which gives E, and E= for some machine tool structural materials The ratio of weights of two bars of same length subjected to tensile load F

W1 nPLð1 = ut1 Þ ut2 =2 ¼ ¼ W2 nPLð2 = ut2 Þ ut1 =1

ð25-170Þ

where ut = is unit strength under tension The ratio of weights of two bars of same length subjected to torque Mt

2=3

W1 ut =2 ¼ W2 ut2=3 =1

ð25-171Þ

2=3

where ut = is an index of the ability of a material to resist torsion and is known as unit strength under torsion The ratio of weights of two bars of same length subjected to bending Mb

2=3

W1 ð1= b1 Þ2=3 1 b2 =2 ¼ ¼ W2 ð1= b2 Þ2=3 2 2=3 =1 b1 2=3

ð25-172Þ

where b = is an index of the ability of a material to resist bending and is known as the unit strength under bending For speciﬁc stiﬀness (in tension)

Refer to Table 25-64.

For comparison of speciﬁc strength and stiﬀness/ rigidity of diﬀerent section having equal cross sectional area

Refer to Table 25-65.

DESIGN OF FRAMES, BEDS, GUIDES AND COLUMNS: For machine frames

Refer to Table 25-66.

For stiﬀening eﬀect of reinforcing ribs

Refer to Fig. 25-50.

For characteristics of bending and torsional rigidities of models of various forms

Refer to Table 25-67.

For variations in relative bending and torsional rigidity for models of various forms

Refer to Table 25-68.

For eﬀect of stiﬀener arrangement on torsional stiﬀness of open structure

Refer to Table 25-69.

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Particular

25.89

Formula

For eﬀect of aperture and cover plate design in static and dynamic stiﬀness of box sections

Refer to Table 25-70.

For typical cross-sections of beds

Refer to Fig. 25-51A, B, C and D.

For classiﬁcation and identiﬁcation of machine tools

Refer to Table 25-72.

For machine tools sliding guides, ball and roller guides made of cast iron, steels and plastics

Refer to Tables and Figures from 25-66 to 25-71. In addition to these, readers are advised to refer to books and handbooks on machine tools. The design of machine tool slideways, guides, beds, frames and columns subjected to external forces are beyond the scope of this Handbook.

For design of spindle units in machine tools

Refer to Chapter 14 on ‘‘Design of shafts’’ in this Handbook.

For design of power screws and lead screws of machine tools

Refer to Chapter 18 on ‘‘Power screws and fasteners’’ in this handbook, and books on power screw design of machine tools.

For vibration and chattering in machine tools

Refer to Chapter 22 on ‘‘Mechanical vibrations’’ in this Handbook.

For variable speed drives and power transmission

Refer to Chapter 23 on ‘‘Gears’’ and Chapter 25 on ‘‘Miscellaneous machine elements’’ in this Handbook.

For lubrication of guides, spindles and other parts of machine tools

Refer to Chapter 24 on ‘‘Design and bearings and Tribology’’ in this Handbook and other books on lubrication.

TOOLING ECONOMICS (Adopted from Tool Engineers Handbook) Symbols: a saving in labor cost per unit C ﬁrst cost of ﬁxture D annual allowance for depreciation, per cent H number of years required for amortization of investment out of earnings I annual allowance for interest on investment, per cent

M N S t T V

annual allowance for repairs, per cent number of pieces manufactured per year yearly cost of setup percentage of overhead applied on labour saved annual allowances for taxes, per cent yearly operating proﬁt over ﬁxed charges

N¼

CðI þ T þ D þ MÞ þ S að1 þ tÞ

ð25-173Þ

C¼

Nað1 þ tÞ S I þT þDþM

ð25-174Þ

Number of years required for a ﬁxture to pay for itself

H¼

C Nað1 þ tÞ CðI þ T þ MÞ S

ð25-175Þ

Proﬁt from improved ﬁxture designs

V ¼ Nað1 þ tÞ CðI þ T þ D þ MÞ S

ð25-176Þ

Number of pieces required to pay for ﬁxture

Economic investment in ﬁxtures for given production

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ELEMENTS OF MACHINE TOOL DESIGN

25.90

CHAPTER TWENTY-FIVE

Particular

Formula

PROCESS—COST COMPARISONS: Symbols: c value of each piece, dollars Cx , Cy total unit cost for methods Y and Z respectively d hourly depreciation rate for the ﬁrst machine (based on machine hours for the base years period) D hourly depreciation rate for the second machine (based on machine hours for the base years period) k annual carrying charge per dollar of inventory, dollar l labor rate for the ﬁrst machine, dollar L lot size, pieces labor rate for the second machine, dollar m monthly consumption, pieces Nt total number of parts to be produced in a single run Number of parts for which the unit costs will be equal for each of two compared methods Y and Z (‘‘breakeven point’’)

number of parts for which the unit costs will be equal for each of two compared methods Y and Z (break-even point) number of pieces produced per hour by the ﬁrst machine number of pieces produced per hour by the second machine unit tool process cost for method Y unit tool process cost for method Z quantity of pieces at break-even point total tool cost for method Y total tool cost for method Z setup hours required on the ﬁrst machine setup hours required on the second machine ratio of machining time piece

Nb p P Py Pz Q Ty Tz s S V

Nb ¼

Ty Tz Pz Py

ð25-177Þ

Cy ¼

Py Nt þ Ty Nt

ð25-178Þ

Total unit cost for method Z

Cz ¼

Pz Nt þ Tz Nt

ð25-179Þ

Quantity of pieces at break-even point

pPðSL þ SD sl sdÞ Pðl þ dÞ pðL þ DÞ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 24mS L¼ kcð1 þ mvÞ

Total unit cost for methods Y

Relatively simple formula for calculation of economic lot size, pieces

Q¼

ð25-180Þ

ð25-181Þ

MACHINING COST: Machining time cost per work piece Non-productive time cost per work piece Tool change time cost per work piece Tool cost per work piece

tm R 60 t R Cn ¼ tL þ s nb 60

Cm ¼

ð25-182Þ ð25-183Þ

Cc ¼

tm tc R 60t1

ð25-184Þ

Ct ¼

Ct1 t t R þ sh m 60t1 1 þ ns

ð25-185Þ

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Particular

25.91

Formula

Total cost of machining

Ctot ¼ Cm þ Cn þ Cc þ Ct

ð25-186Þ

Total tool cost per workpiece

Cn ¼ Cc þ Ct

ð25-187Þ

where tm ¼ machining time per workpiece, min tL ¼ loading and unloading time per workpiece, min ts ¼ setting time per batch, min tt ¼ tool life, min tc ¼ tool charge time, min tsh ¼ tool sharpening time, min R ¼ cost rate per hour nb ¼ number of batch ns ¼ number of resharpening

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ELEMENTS OF MACHINE TOOL DESIGN

25.92

CHAPTER TWENTY-FIVE

TABLE 25–64 Unit stiﬀness/rigidity of some materials Modulus of elasticity, E Material

GPa

Aluminum Aluminum cast Aluminum (all alloys) Beryllium copper Carbon steel Cast iron, gray Malleable cast iron Inconel Magnesium alloy Molybdenum Monel metal Nickel-silver Nickel alloy Nickel steel Phosphor bronze Steel (18-8), stainless Titanium (pure) Titanium alloy Brass Bronze Bronze cast Copper Tungsten Douglas ﬁr Glass Lead Concrete (compression) Wrought iron Zinc alloy Graphite HTS Graphite/5208 epoxy T50 Graphite 2011 Al Boron Boron carbide, BC Silicon carbide, SiC Boron/5505 epoxy Boron/6601 Al Kelvar 49 Kelvar 49/resin Silicon, Si Wood (along ﬁber) Nylon Paper E Glass/1002 epoxy

69 70 72 124 206 100 170 214 45 331 179 127 207 207 111 190 130 114 106 96 80 121 345 11 46 36 14–28 190 83 750 172 160 380 450 560 207 214 130 76 110 11–15.1 4 1–2 39

Mpsi 10.0 10.15 10.4 18.0 30.0 14.5 24.6 31.0 6.5 48.0 26.0 18.5 30.0 30.0 16.0 27.5 15.0 16.5 15.5 14.0 11.6 17.5 50.0 1.6 6.7 5.3 2.0–4.0 27.5 12 108.80 24.95 23.20 55.11 65.28 81.22 30.07 31.03 18.85 11.02 15.95 1.59–2.19 0.58 0.15–0.29 5.65

Modulus of rigidity, G Poisson’s ratio, GPa Mpsi 26 30 27 48 79 41 90 76 16 117 65 48 79 79 41 73

3.8 4.35 3.9 7.0 11.5 6.0 13.0 11.0 2.4 17.0 9.5 7.0 11.5 11.5 6.0 10.6

43 40 38 35 46 138 4 19 13

6.2 5.8 5.5 5.0 6.6 20.0 0.6 2.7 1.9

70 31

10.2 4.5

Density, a

Unit weight, b Unit stiﬀness E=

Mg/m3

kg/m3

kN/m3

lbf/in3 lbf/ft3

0.334

2.69

0.320 0.285 0.292 0.211

2.80 8.22 7.81 7.20

167 166 173 513 487 450

8.42 1.80 10.19 8.83 8.75 8.3 7.75 8.17 7.75 4.47 6.6 8.55 8.30

26.3 26.0 27.0 80.6 76.6 70.6 70.61 83.3 17.6 100.0 86.6 85.80 81.4 76.0 80.1 76.0 43.8

0.097 0.096 0.10 0.297 0.282 0.260

0.290 0.350 0.307 0.320 0.332 0.30 0.291 0.349 0.305

2,685 2,650 2,713 8,221 7,806 7,197 7,200 8,418 1,799 10,186 8,830 8,747 8,304 7,751 8,166 7,750 4,470 6,600 8,553 8,304 8,200 8,913 18,822 443 2,602 11,377 2,353

0.307 0.065 0.368 0.319 0.316 0.300 0.280 0.295 0.280 0.16

530 117 636 551 546 518 484 510 484 279

83.9 81.4 80.0 87.4 184.6 4.3 25.5 111.6 23.1

0.309

534

0.322 1.89 0.016 0.094 0.411

556

2:62 106 2:66 106 2:68 106 1:54 106 2:69 106 1:42 106 2:41 106 2:57 106 2:56 106 3:31 106 2:06 106 1:48 106 2:54 106 2:72 106 1:38 106 2:50 106 2:37 106 2:60 106 1:26 106 1:18 106 1:00 106 1:38 106

28 162 710 147

2:56 106 1:80 106 3:10 106 0:60 106

7,700

76.0 0.24

415

0.33 0.324 0.349 0.326 0.330 0.245 0.431

0.33

8.90 18.82 4.43 2.60 11.38 2.35

6.6 2.25 1.55

22.1 15.2

2.58 2.5 2.4 3.2 1.99 2.60 1.44 1.38 2.30 0.41–0.82 1.1 0.50 1.80

25.3 44.1 22.5 31.4 19.5 25.5 14.1 13.5 22.5 4.0–8.0 10.8 4.9 17.6

2:50 106 1:18 106 34:00 106 11:30 106 6:32 106 11:00 106 19:20 106 17:80 106 8:40 106 8:20 106 9:20 106 5:60 106 4:86 106 2.75–1:86 106 0:37 106 0.20–0:41 106 2:22 106

a

, mass density. , weight density; w is also the symbol used for unit weight of materials. Source: K. Lingaiah and B. R. Narayana Iyengar, Machine Design Data Handbook, Volume I (SI and Customary Metric Units), Suma Publishers, Bangalore, India and K. Lingaiah, Machine Design Data Handbook, Volume II, (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986.

b

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ELEMENTS OF MACHINE TOOL DESIGN

25.93

ELEMENTS OF MACHINE TOOL DESIGN

TABLE 25-65 Comparison of speciﬁc strength and Rigidity/Stiﬀness of diﬀerent sections having equal cross sectional areas (in Flexure) Distance to farthest point, c

Moment of inertia I

Section modulus Z ¼ I=c

i¼

I

Ia

Z

Za

0.14

1

1

0.083

0.166

1.06

1.16

B3r2/6

0.083r

0:166

1.9

1.6

0:05D4 ð1 4 Þ

0:1D3 ð1 4 Þ

0:08

2.1

1.73

B4 ð1 4 Þ 12

B3 ð1 4 Þ 6

1 4 12ð1 2 Þ2

4.6

3.2

9.5

4.6

I A2

Cross-section

Area A

F

0.785D2

D 2

0.05D4

0.1D3

0.08

F

B2

B 2

B4/12

B3/6

F

B2 r ðr ¼ H=BÞ

H 2

B4r3/12

F

0.785D2 (1 2) ð ¼ d=DÞ

D 2

F

B2(1) ð ¼ b=BÞ

B 2

w¼

Z A3=2

D

B pﬃﬃ r

H B

d

1 4 ð1 2 Þ2

0:14

1 4 ð1 2 Þ3=2

D

b

1 4 6ð1 2 Þ3=2

B F b h H B F b

h H

B

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ELEMENTS OF MACHINE TOOL DESIGN

TABLE 25-65 Comparison of speciﬁc strength and Rigidity/Stiﬀness of diﬀerent sections having equal cross sectional areas (in Flexure) (Cont.)

Cross-section

b h H

Area A BHð1 Þ ð ¼ b=B; ¼ h=HÞ

Distance to farthest point, c H 2

Moment of inertia I

Section modulus Z ¼ I=c

i¼

BH 3 12 ð1 3 Þ

BH 2 6 ð1 3 Þ

0:083

I A2 1 3 ð1 Þ2

w¼

Z A3=2

0:166

I

Ia

Z

Za

11

52

1 3 ð1 Þ3=2

B F b/2

b/2

h H

B D3 D4 ; Ia=Moment of Inertia of round solid section= . 32 64 Z/Za and I/Ia for solid and hollow stock having identical cross sectional area in ﬂexure.

* Za=section modulus of round solid section=

TABLE 25-66 Machine Frames Simple frames and beds of horizontal machines Simple frames and beds of vertical machines

Portal frames

Circular frames, housings

Frames of piston machines, banks of cylinders

Frames of conveying machines Crane structures

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

TABLE 25-66 Machine Frames (Cont.) Baseplates Boxes

Pillars, brackets, pedestals, hangers, etc.

Tables, slide blocks, carriages Crossheads, slides, jibs Lids and casings

Source: Courtesy: Dobrovolsky, V., etl., ‘‘Machine Elements’’, Mir Publishers, Moscow, 1974.

TABLE 25-67 Characteristics of Bending and Torsional Rigidities for Models of Various Forms Relative rigidity in bending Sb

Relative rigidity in torsion St

Weight of model G

Sb G

St G

1 (basic)

1.00

1.00

1.00

1.00

1.00

2a

1.10

1.63

1.10

1.00

1.48

2b

1.09

1.39

1.05

1.04

1.32

3

1.08

2.04

1.14

0.95

1.79

4

1.17

2.16

1.38

0.85

1.56

5

1.78

3.69

1.49

1.20

3.07

6

1.55

2.94

1.26

1.23

2.39

Model No.

Model form

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25.95

ELEMENTS OF MACHINE TOOL DESIGN

25.96

CHAPTER TWENTY-FIVE

TABLE 25-28 Variations in Relative Bending and Torsional Rigidity for Models of Various Forms Relative rigidity in bending

Relative rigidity in torsion

Model No.

Relative weight of box-like section

With ribs

With thicker walls

With ribs

With thicker walls

1 (basic) 2a 2b 3 4 5 6

1.00 1.10 1.05 1.14 1.38 1.49 1.26

1.00 1.10 1.09 1.08 1.17 1.78 1.55

1.00 1.15 1.10 1.16 1.29 1.30 1.19

1.00 1.63 1.39 2.04 2.16 3.69 2.94

1.00 1.18 1.10 1.21 1.40 1.46 1.24

Source: Courtesy: Dobrovolsky, V., etl., ‘‘Machine Elements’’, Mir Publishers, Moscow, 1974.

TABLE 25-69 Eﬀect of stiﬀner arrangement on torsional stiﬀness of open structure4

Relative torsional stiﬀness

Relative weight

Relative torsional stiﬀness per unit weight

1

1.0

1.0

1.0

2

1.34

1.34

1.0

3

1.43

1.34

1.07

4

2.48

1.38

1.80

5

3.73

1.66

2.25

Stiﬀener arrangement

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

25.97

TABLE 25-70 Eﬀect of aperture and cover plate design on static and dynamic stiﬀness of box section3

Relative stiﬀness about

Y

Relative natural frequency of vibrations about

Relative damping of vibrations about

X-X

Y-Y

Z-Z

X-X

Y-Y

Z-Z

X-X

Y-Y

Z-Z

100

100

100

100

100

100

100

100

100

85

85

28

90

87

68

75

89

95

89

89

35

95

91

90

112

95

165

91

91

41

97

92

92

112

95

185

X

Y X

Y X

Y X

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ELEMENTS OF MACHINE TOOL DESIGN

25.98

CHAPTER TWENTY-FIVE

Factors Proﬁle

Iben

Itors

A

Iben A

Itors A

1

1

1

1

1

1.17

2.16

1.38

0.85

1.56

1.55

3

1.26

1.23

2.4

1.78

3.7

1.5

1.2

2.45

FIGURE 25-50 Stiﬀening eﬀect of reinforcing ribs.

(a)

(b)

(c)

(d)

FIGURE 25-51A Typical cross-sections of beds.

Male parts 55

55 Female parts

(a)

(b)

(c)

P

P

(d)

FIGURE 25-51B Principal shapes of sliding guides. (a) ﬂat ways; (b) prismatic ways; (c) dovetail ways; (d) cylindrical (bar-type) ways.

(a)

(b)

FIGURE 25-51C Sliding guides. (a) closed type; (b) open type.

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

Px

P

Px

y

2r Z PH

Px P 45”

Z

PH PH

2r

2r (a) P x

P 45”

Z

Px P

y

Z

2r COS45” P PH 45” PH Z 2r

(b)

25.99

y 45”

2r Px P 2r 45”

y

Z

FIGURE 25-51D Rolling (a) open (b)type closed type. (a)guides. open type; (b) type; closed

TABLE 25-71 Traversing Force Calculations – Typical Cases Type of ways

Px P

1

z

2r Px

2

y

P

y

2r

y

Pp

Pp

2r

Pp

Q ¼ Px þ 3T0 þ 1:5 r fr P P ¼ P 2 þ G 1 þ G2

r 1:4

Q ¼ Px þ 4T0 þ 1:4 r fr P

r 1:5

Q ¼ Px þ 2T0 þ 1:5 r fr P

r 2:8

Q ¼ Px þ 4T0 þ 2:8 r fr P P

z Px P

4

r 1:5

45

Px P 2r

Traversing force Q. kgf

45 2r cos 45

z 3

req.cm

45

z

Px

P

45

z 2 r

Px

P

45 z 2r

Pp

y

Pp

y

Q ¼ Px þ 2T0 þ 2:8 r fr P P

Pp

y

Notes: 1. The coeﬃcient of rolling friction fr ¼ 0:001 for ground steel ways and fr ¼ 0:0025 for scraped cast iron ways. The initial friction force, referred to one separator, T 0 ¼ 0:4 kgf: 2. Because of the low value of the friction forces, a simpliﬁed arrangement has been accepted in which the ways are subject only to the feed force Px, vertical component Px of the cutting force, table weight G1 and workpiece weight G2. The tilting moments, force Pp and the components of the traversing force are not taken into account. 3. In the type 4 ways only the feed force Px and the preload force Pp are taken into consideration.

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ELEMENTS OF MACHINE TOOL DESIGN

25.100

CHAPTER TWENTY-FIVE

z Px Pz

ZQ

G fB XC

Qx

Xc XA

Py Pz

Acosα fA O XB

C

(a)

Y

d1 Qz

II

yc

A a

Bcosβ

β

fB

A sinα

α

b

Px

yQ

yG

(C)

β

x

y

(C) (c)

fC

Py

I

G

XQ

Qz

A

yc B

c

I

yp

Zp

C Bcosβ fC

z

Xp

fA

O

α

I (d)

L

(b) FIGURE 25-52 Forces acting on the Slidways of a Lathe – A Typical Case Source: Courtesy: Acherkan, N., ‘‘Machine Tool Design’’, Mir Publishers Moscow, 1968.

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

TABLE 25-72 Classiﬁcation and Identiﬁcation code of Machine Tools – Kinematic Diagram Description

Symbol

Shafts

Description

Symbol

Belt drives: Open ﬂat belts

Shafts coupling: Closed Closed with over-load protection

Crossed ﬂat belts

Flexible Universal V-belts Telescopic Floating Toothed

Chain drive

Parts mounted on shafts: Freely mounted Sliding on feather Engaged with sliding key

Toothed gearing: Spur or helical gears

Fixed Bevel gears Plain bearings: Radial Single-direction thrust

Spiral (crossed helical) gears

Two-direction thrust Worm gearing Antifriction bearings: Radial

Single angular-contact Back-and-pinion gearing Duplex angular contact

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25.101

ELEMENTS OF MACHINE TOOL DESIGN

25.102

CHAPTER TWENTY-FIVE

TABLE 25-72 Classiﬁcation and Identiﬁcation code of Machine Tools – Kinematic Diagram (Cont.) Description

Symbol

Nut on power screw: Solid nuts

Description

Symbol

Single-direction overrunning clutches

Split nuts Two-direction overrunning clutches Clutches: Single-direction jaw clutches Brakes: Cone Spindle noses: Centre type Chuck type

Shoe

Bar type Band Drilling Boring spindles with faceplates

Disk

Milling Two-direction jaw clutches

Grinding

Cone clutches Electric motors: On feet Single disk clutches

Twin disk clutches

Flange-mounted

Built-in

Source: Courtesy: Acherkan, N., et1., ‘‘Machine Tool Design’’, Moscow, 1968.

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ELEMENTS OF MACHINE TOOL DESIGN ELEMENTS OF MACHINE TOOL DESIGN

25.103

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

Lingaiah, K., Machine Design Data Handbook, McGraw-Hill Publishing Company, New York, 1994. Lingaiah, K., Machine Design Data Handbook, Vol. I, Suma Publishers, Bangalore, India, 1986. Merchant, M. E., Trans. Am. Soc. Mech. Engrs., 66, A-168, 1944. Ernst, H., and M. E. Merchant, Chip Formation, Friction and Finish, Cincinneti Milling, Machine Company, USA. American Society of Tool and Manufacturing Engineers (ASTME), Tool Engineers Handbook, 2nd ed., F. W. Wilson, Editor, McGraw-Hill Book Publishing Company, New York, 1959. Cyril Donaldson, George H. Lecain and V.C. Goold, Tool Design, Tata-McGraw-Hill Publishing Company Ltd., New Delhi, India, 1976. Frank W. Wilson, Editor-in-Chief, American Society of Tool and Manufacturing Engineers (ASTME), Fundamentals of Tool Design, Prentice Hall, New Delhi, India, 1969. Kuppuswamy, G., Center for Continuing Education, Department of Mechanical Engineering, Indian Institute of Technology, Madras, India, August 12, 1987. Sen, G. C., and A. B. Bhattacharyya, Principles of Machine Tools, New Central Book Agency, (P) Ltd., Calcutta, India, 1995. Geoﬀrey Boothroyd, Fundamentals of Metal Machining and Machine Tools, McGraw-Hill Publishing Company, New York, 1975. Koenigsberger, F., Design Principles of Metal Cutting Machine Tools, the MacMillan Company, New York, 1964. Shaw, M. C., and C. J. Oxford, Jr., (1) ‘‘On the Drilling Metals’’ (2) ‘‘The Torque and Thrust in Milling’’, Trans. ASME., 97:1, January 1957. Hindustan Machine Tools, Bangalore, Production Technology, Tata-McGraw-Hill Publishing Company Ltd., New Delhi, India, 1980. Central Machine Tool Institute, Machine Tool Design Handbook, Bangalore, India, 1988. Acherkan, A., General Editor, V. Push, N. Ignatyev, A. Kakoilo, V. Khomyakov, Y. U. Mikheyev, N. Lisitsyn, A. Gavryushin, O. Trifonov, A. Kudryashov, A. Fedotyonok, V. Yermakov, V. Kudinov, Machine Tool Design, Vol. 1 to 4, Mir Publishers, Moscow, 1968-69. Milton C. Shaw, Metal Cutting Principles, Clarendon Press, Oxford, 1984. Martelloti, M. E., Trans. Am. Soc. Mech. Engrs., 63, 677, 1941. Kovan, V. M., Technology of Machine Building, Mashgiz, Moscow, 1959. Basu, S. R., and D. K. Pal, Design of Machine Tools, 2nd ed., Oxford and IBH Publishing Company, New Delhi, 1983. Heinrich Makelt, Die Mechanischen Pressen, Carl Hanser Verlag Muchen, 1961 (in German) Translated to English by R. Hardbottle, Mechanical Presses, Edward Arnold (Publishers) Ltd., 1968. Dobrovolsky, K. Zablonsky, S. Mak, Radchik, L. Erlikh, Machine Elements, Mir Publishers, Moscow, 1968. Rivin, E. I., Stiﬀness and Damping in Mechanical Design, Marcel Dekker, Inc., New York, 1999. Machine Tool Design and Numerical Control. Chernov, N., Machine Tools, Translated from Russian to English by Falix Palkin, Mir Publishers, Moscow, 1975. Greenwood, D. C., Engineering Data for Product Design, McGraw-Hill Publishing Company, New York, 1961.

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Source: MACHINE DESIGN DATABOOK

CHAPTER

26 RETAINING RINGS AND CIRCLIPS SYMBOLS a Ch Chmax CF d D f Ftg Fig Frt Fir F 0r F 00r Ftrr Fsg Fsr l n nmax q r rmax t T w ðwaÞg ðwaÞr xo sy saw s

acceleration of retained parts, m/s2 (ft/s2 or in/s2 ) actual chamfer, m (in) listed maximum allowable chamfer, m (in) conversion factor (refer to Table 26-1) depth of groove, m (in) shaft or housing diameter, m (in) frequency of vibration, cps allowable static thrust load on the groove wall, kN (lbf) allowable impact load on groove, kN (lbf) allowable static thrust load of the ring, kN (lbf) allowable impact load on a retaining ring, kN (lbf) listed allowable assembly load with maximum corner radius or chamfer, kN (lbf) allowable assembly load when cornor radius or chamfer is less than the listed, kN (lbf) allowable thrust load exerted by the adjacent part, kN (lbf) allowable sudden load an groove, kN (lbf) allowable sudden load on ring, kN (lbf) distance of the outer groove wall from the end of the shaft or bore as shown in Fig. 26-2, m (in) factor of safety (about 2 to 4 may be assumed) maximum safe speed, rpm reduction factor from Fig. 26-1. actual corner radius or chamfer, m (in) listed maximum allowable corner radius, m (in) ring thickness, m (in) largest section of the ring, m(in) weight of retained parts, kN (lbf) allowable vibratory loading on groove, kN (lbf) allowable vibratory loading on ring, kN (lbf) amplitude of vibration, m (in) tensile yield strength of groove material, Table 26-2, MPa (psi) maximum working stress of ring during expansion or contraction of ring, MPa (psi) shear strength of ring material, MPa (psi) (refer to Table 26-3) coeﬃcient of friction between ring and retained parts whichever is the largest.

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RETAINING RINGS AND CIRCLIPS

26.2

CHAPTER TWENTY-SIX

Note: and with subscript s designates strength properties of material used in the design which will be used and observed throughout this Machine Design Data Handbook. Other factors in performance or in special aspects are included from time to time in this chapter and, being applicable only in their immediate context are not given at this stage. Particular

Formula

RETAINING RINGS AND CIRCLIPS: (Figs. 26-1 to 26-28 and Tables 26-1 to 26-13) Load Capacities of Retaining Rings: Allowable static thrust load on the groove Allowable static thrust load on ring which is subject to shear The allowable thrust load exerted by adjacent part

Allowable assembly load when the corner radius or chamfer is less than the listed (F 00r < F 0r )

Ftg ¼

CF Ddsy nq

ð26-1Þ

Fr ¼

CF Dts n

ð26-2Þ

Ftrr

saw tT 2 18D

ð26-3Þ

F 00r ¼

Fr0 rmax r

Fr00 ¼

Fr0 Chmax Ch

for radius for chamfer

ð26-4Þ ð26-5Þ

Dynamic Loading: Allowable sudden load on ring

Fsr 0:5Fr

ð26-6Þ

Allowable sudden load on groove

Fsg 0:5Ftg

ð26-7Þ

Allowable vibration loading on ring

ðwaÞr 540Fr a

ð26-8Þ

Allowable vibration loading on groove

ðwaÞg 400Ftg

ð26-9Þ

Acceleration of retained parts for harmonic oscillation

a 40xo f 2

ð26-10Þ

Allowable impact loading on groove

Fig ¼ Fr d=2

ð26-11Þ

Allowable impact loading on ring An empirical formula for maximum safe speed with standard types of rings

a

Fir ¼ Fr t=2 nmax ¼ 5000000=D nmax ¼ 20000=D

a

ð26-12Þ where D in mm

ð26-13Þ

where D in inches

ð26-14Þ

Note: Actual tests should be conducted because of repeated or cyclic condition.

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RETAINING RINGS AND CIRCLIPS RETAINING RINGS AND CIRCLIPS

Particular

26.3

Formula

For dimensions of external circlips—Type A—light series

Refer to Table 26-5 and Fig. 26-3.

For dimensions of external circlips—Type A—heavy series

Refer to Table 26-6 and Fig. 26-4. Refer to Table 26-7 and Fig. 26-5.

For dimensions of internal circlips—Type B—light series Refer to Table 26-8 and Fig. 26-6. For dimensions of internal circlips—Type B—Heavy series Refer to Table 26-9 and Fig. 26-7. For dimensions of external circlip—Type C For dimensions, allowable static thrust load, allowable corner radii, chamfers, housing diameter and ring thickness of retaining rings—basic internal, bowed internal, beveled internal, inverted internal, double beveled internal, crescent-shaped, bowed Ering, reinforced, locking prong in grooved housing and on grooved shafts, self locking and triangular self locking ring etc.

Refer to Tables 26-10 to 26-13 and Figs. from 26-1 to 26-28.

Refer to Fig. 26-1. For q reduction factor

TABLE 26-1 Conversion or correction factor CF for calculating Fr and Ftg for use in Eqs. (26-l) and (26-2) Conversion or correction factor CF Ring type

Ring: Fr

Basic, bowed internal Beveled internal Double-beveled internal Inverted internal, external Basic, bowed external Beveled external

1.2 1.2

Groove: Ftg

1.2 1.2 Use d=2 instead of d 2/3 1/2 1 1 1 1 Use d=2 instead of d Crescent-shaped 1/2 1/2 Two-part interlocking 3/4 3/4 E-ring, bowed E-ring 1/3 1/3 Reinforced E-ring 1/4 1/4 Locking-prong ring See manufacturer’s 1.2 speciﬁcations Heavy-duty external 1.3 2 High-strength radial 1/2 1/2 Miniature high-strength See manufacturer’s speciﬁcations Thinner-gage high-strength 1/2 1/2 radial

FIGURE 26-1 Reduction curve

FIGURE 26-2 Edge margin

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RETAINING RINGS AND CIRCLIPS

TABLE 26-2 Tensile yield strength of groove material Tensile yield strength, sy Groove material

MPa

lbf/in2

Cold-rolled steel Hardened steel (Rockwell C40) Hardened steel (Rockwell C50) Aluminum (2024-T4) Brass (naval)

310 1034 1380 276 210

45,000 150,000 200,000 40,000 30,000

TABLE 26-3 Shear strength of ring material for use in Eq. (26-2) Shear strength, s Ring material Carbon spring steel (SAE 1060– 1090)

Ring type

Ring thickness mm (in)

lbf/in2

Basic, bowed, beveled, inverted internal and external rings and crescent-shaped

Up to and including 0.9 (0.035)

827

120,000

Double-beveled internal rings

1.07 (0.042) and over

1034

150,000

Heavy-duty external

0.90 (0.035) and over

1034

150,000

Miniature high-strength

0.510 (0.020) and 0.635 (0.025)

827

120,000

0.9 (0.035) and over

1034

150,000

Two-part interlocking, reinforced E-ring, high-strength radial

All available

1034

150,000

Thinner high-strength radial

All available

1034

150,000

E-ring, bowed E-ring

0.254 (0.010) and 0.380 (0.015)

690

100,000

0.635 (0.025)

827

120,000

1034

150,000

0.9 (0.035) and over

Beryllium copper (CDA 17200)

MPa

Locking-prong

All available

896

130,000

Basic external

0.254 (0.010) and 0.380 (0.015) sizes 12 through 23

758

110,000

Bowed external

0.380 (0.015) sizes 18 through 23

758

110,000

E-ring

0.254 (0.010) (size 4 only)

662

95,000

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RETAINING RINGS AND CIRCLIPS

TABLE 26-4 Maximum working stress of ring during expansion or contraction Maximum allowable working stress, saw Ring material

MPa

lbf/in2

Carbon spring steel (SAE 1075) Stainless steel (PH 15-7 Mo) Beryllium copper (CDA 17200) Aluminum (Alclad 7075-T6)

1724 1724 1380 482

250,000 250,000 200,000 70,000

Courtesy: # 1964, 1965, 1973, 1981 Waldes Kohinoor, Inc., Long Island City, New York, 1985. Edward Killian, ‘‘Retaining Rings’’, Robert O. Parmley, Editor-in-Chief ‘‘Mechanical Components Handbook’’, McGraw-Hill Publishing Company, New York, USA.

TABLE 26-5 Dimensions for external circlips—type A—light series

FIGURE 26-3

All dimensions in millimeters Circlip Shaft Dia d1 8 9

Axial force s h11

a b Max. Approx. d3

Tol. on d4 d5 d3 Expanded Min.

0.8

3.2

þ0.09 0.18 þ0.15

1.5 1.7

10

7.4 8.4 9.3

1.8 1

3.4

15.2 16.4

2

10.2 11 11.9

1:2

d2

Tol. on m1 d2 H13

7.6 8.6

0.9

m2 Min.

n Min.

1.0

N

lbf

1180 1360

265 305

1500

340

2060 2270 2940

460 510 660

0.6 17.6

0.30

3.3 11 12 13

Shaft groove

9.6 1.5

18.6 19.6 20.8

10.5 11.5 12.4

h11

1:1

1:2

0.75

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RETAINING RINGS AND CIRCLIPS

26.6

CHAPTER TWENTY-SIX

TABLE 26-5 Dimensions for external circlips—type A—light series (Contd.) All dimensions in millimeters Circlip Shaft Dia d1 14 15 16 17 18 19 20 21 22 24 25 26 28 29 30 32 34 35 36 38 40 42 45 48 50 52 55 56 58 60 62 63 65 68 70 72 75 78 80 82 85 88 90

Shaft groove Axial force

s h11

a b Max. Approx. d3

Tol. on d4 d5 d3 Expanded Min.

3.5 3.6 3.7 3.8 3:9

þ0:18 0.36

4 4.1 4.2 4.4

1.2

4.5 4.7 4.8 5 5.2 5.4 5.6

1.5

2

5.8 6 6.5 6.7 6.9 6.9 7 7.2 7.3

2.5

7.4 7.5 7.6 7.8 8 8.1 8.2 8.4

1.75

8.6 8.7 8.8 3

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 3 3.1 3.2 3.4 3.5 3.6 3.8 3.9 4 4.2 4.4 4.5 4.7 5 5.1 5.2 5.4 5.5 5.6 5.8 6 6.2 6.3 6.5 6.6 6.8 7 7.3 7.4 7.6 7.8 8 8.2

12.9 13.8 14.7 15.7 16.5 17.5 18.5 19.5 20.5 22.2 23.2 24.2 25.9 26.6 27.9 29.6 31.5 32.2 33.2 35.2 36.5 38.5 41.5 44.5 45.8 47.8 50.8 51.8 53.9 55.8 57.8 58.8 60.8 63.5 65.5 67.5 70.5 73.5 74.5 76.5 79.5 82.5 84.5

þ0.21 0.42

þ0.25 0.25

þ0.39 0.78

þ0.46 0.92

þ0.46 0.92

22 23.2 24.4 25.6 26.8 27.8 29 30.2 31.4 33.8 34.8 36 38.4 39.6 4.1 43.4 45.8 47.2 48.2 50.6 53 56 59.4 62.8 64.8 67 70.4 71.6 73.6 75.8 78 79.2 81.6 85 87.2 89.4 92.2 96.2 98.2 101 104 107 109

1:7

2

2.5

3

d2

Tol. on m1 d2 H13

13.4 14.3 15.2 16.2 17 18 19 20 21 22.9 23.9 24.9 26.6 27.6 28.6 30.3 32.3 33 34 36 h12 37.5 39.5 42.5 45.5 47 49 52 53 55 57 59 60 62 65 67 69 72 75 76.5 h12 78.5 81.5 84.5 86.5

m2 Min.

n Min. 0.9 1.1 1.2

1.5 1.3

1.4 1.7

2.1 1.6

1.7 2.6 3

1.85

2

2.15

2.3

3.8

4.5

4.5 2.65

2.8

3.15

3.3

5.3

N 3190 3920 4809 5100 6770 7110 7550 7900 8300 9900 10400 10790 14710 15300 15890 20590 21770 26180 27070 28540 37360 39230 42170 45110 55900 58350 61780 62760 65210 67665 69625 71100 73550 76880 78940 81395 84336 88260 104930 107870 111795 116700 118660

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lbf 720 880 1080 1150 1520 1600 1700 1780 1860 2230 2335 2425 3310 3440 3570 4630 4890 5890 6085 6415 8400 8820 9480 10140 12565 13120 13890 14110 14660 15210 15650 15985 16535 17285 17748 18300 18960 19840 23590 24250 25130 26236 26675

RETAINING RINGS AND CIRCLIPS RETAINING RINGS AND CIRCLIPS

26.7

TABLE 26-5 Dimensions for external circlips—type A—light series (Contd.) All dimensions in millimeters Circlip Shaft Dia d1 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 210 220 230 240 250 260 270 280 290 300

Shaft groove Axial force

s h11

a b Max. Approx. d3

4

9.4 9.6 9.9 10.1 10.6 11 11.4 11.6 11.8 12 12.2 13 13.3 13.5

8.6 9 9.3 9.6 9.8 10.2 10.4 10.7 11 11.2 11.5 11.8 12 12..2 12.5 12.9 Max 13.5 Max

14 Max

5 16 Max

89.5 94.5 98 103 108 113 118 123 128 133 138 142 146 151 155.5 160.5 165.5 170.5 175.5 180.5 185.5 190.5 198 208 218 228 238 245 255 265 275 285

d5 Tol. on d4 d3 Expanded Min. þ0.54 1.08

þ0.63 1.26

þ0.72 1.44

þ0.81 1.62

115 121 126 132 138 143 149 155 160 165 171 177 182 188 193 197 202 208 213 219 224 229 239 249 259 269 279 293 303 313 323 333

3.5

4

5

d2

Tol. on m1 d2 H13

91.5 96.5 101 106 111 116 121 126 131 136 141 145 150 155 160 165 170 175 180 185 190 h13 195 204 214 224 234 244 252 262 272 282 292

m2 Min.

n Min.

6

4.15

4.3

7.5

9 5.15

5.3

12

N

lbf

125525 132390 158865 166712 174555 181422 189265 197110 204958 212800 220650 283410 294200 304000 313810 322640 331460 341270 331460 328520 320675 312830 478560 502095 524650 518770 493270 533480 514846 498175 491505 463850

28220 29764 35716 37490 39244 40785 42550 44315 46080 47840 49606 63716 66140 69346 70550 72535 74520 76724 74520 73858 72094 70330 107590 112880 117950 116630 110900 119936 115748 112000 108250 104280

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RETAINING RINGS AND CIRCLIPS

TABLE 26-6 Dimensions for external circlips—type A—heavy series

FIGURE 26-4

All dimensions in millimeters Circlip Shaft Dia d1 15 16 17 18 20 22 24 25 28 30 32 34 35 38 40 42 45 48 50 52 55 58 60 65 70 75 80 85 90 100

Shaft groove Axial force

s h11

a b Max. Approx. d3 4.8 5

1.5

1.75

5.1 5.5 6 6.3 6.4

2 6.5

2.5

3

4

6.6 6.7 6.8 7 7.2 7.5 7.8 8 8.2 8.5 8.8 9 9.3 9.5 9.7 9.8 10 10.2 10.5

2.4 2.5 2.6 2.7 3 3.1 3.2 3.4 3.5 4.1 4.2 4.3 4.4 4.5 4.7 5 5.1 5.2 5.4 5.6 5.8 6.3 6.6 7 7.4 7.8 8.2 9

13.8 14.7 15.7 16.5 18.5 20.5 22.2 23.2 25.9 27.9 29.6 31.5 32.2 35.2 36.5 38.5 41.5 44.5 45.8 47.8 50.8 53.8 55.8 60.8 65.5 70.5 74.5 79.5 84.5 94.5

Tol. on d4 d5 d3 Expanded Min. þ0.18 0.36

þ0.21 0.42

þ0.25 0.50 þ0.39 0.78

þ0.46 0.92

þ0.54 1.08

25.5 27.5 28.5 29.5 32.5 35.5 38 39 42.5 44.5 46.5 49 50 53 55.5 58 61.5 65 68 70 73.5 77 79 85 90.5 96 101 106.5 112 124

2

2.5

3

3.5

d2 14.3 15.2 16.2 17 19 21 22.9 23.9 26.6 28.6 30.3 32.3 33 36 37.5 39.5 42.5 45.5 47 49 52 55 57 62 67 72 76.5 81.5 86.5 96.5

Tol. on m1 d2 H13

m2 Min.

n Min. 1.1

h11

1.6

1.7

1.85

2

1.2 1.5

1.7 2.15

2.3

2.1 2.6 3

2.65

2.8 3.8

h12

3.15

3.3 4.5

4.15

4.3 5.3

N 3922 4805 5100 6765 7550 8286 9905 10395 14710 15896 20594 21770 25890 28242 37658 39226 42168 45110 55898 58350 61780 65214 67665 70550 78942 84336 104930 111795 118660 132390

lbf 882 1080 1146 1520 1698 1862 2226 2336 3310 3570 4630 4895 5820 6350 9466 8820 9480 10140 12566 13118 13990 14660 15212 16535 17744 19956 23590 25134 26676 29764

Designation: A circlip of light series in type A for shaft diameter d1 equal to 50 mm shall be designated as: Circlip, Light A 50 IS: 3075, 1965.

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RETAINING RINGS AND CIRCLIPS

TABLE 26-7 Dimensions for internal circlips—type B—light series

FIGURE 26-5

All dimensions in millimeters Circlip Shaft Dia d1 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26 28 30 32 34 35 36 37 38 40 42

Bore groove Axial force

s a b h11 Max. Approx. d3 0.8

2.4 2.5 3.2 3.3 3.4 3.6 3.7 3.8

1.1 1.3 1.4 1.5 1.7 1.8 1.9 2

3.9 4.1

2.1 2.2

1

4.2

1.2

4.4 4.5 4.7 4.8

5.4 1.5 5.5 5.8 5.9

2.3 2.4 2.5 2.6 2.7 2.9 2.9 3.0 3.2 3.3 3.4 3.5 3.6 3.7 3.9 4.1

8.7 9.8 10.8 11.8 13.0 14.1 15.1 16.2 17.3 18.3 19.5 20.5 21.5 22.5 23.5 25.9 26.9 27.9 30.1 32.1 34.4 36.5 37.8 38.8 39.8 40.8 43.5 45.5

Tol. on d4 d5 d3 Compressed Min.

þ0.36 0.18

þ0.42 0.21

þ0.50 0.25

þ0.78 0.39

2.8 3.5 3.1 3.9 4.7 5.3 6 7 7.7 8.4 8.9 9.8 10.6 11.6 12.6 14.2 15 15.6 17.4 19.4 20.2 22.2 23.2 24.2 25 26 27.4 29.2

1 1.2 1.5

1.7

2

d2 8.4 9.4 10.4 11.4 12.5 13.6 14.6 15.7 16.8 17.8 19 20 21 22 23 25.2 26.2 27.2 29.4 31.4 33.7 35.7 37 38 39 40 42.5 44.5

Tol. on m1 d2 H13

m2 Min.

n Min.

0.6

0.75 0.9

H11 1.1

1.2

1.1 1.2

1.5

1.8 1.3

1.4 2.1 2.6

1.6

1.7

3

N 1255 1412 1570 1725 2353 3080 3295 4138 5050 5364 7110 7492 7894 8286 8650 11375 11769 12258 15495 16572 21575 22750 27655 28440 29224 30106 39716 41678

lbf 282 318 352 389 530 692 740 930 1135 1205 1598 1684 1775 1862 1943 2558 2645 2756 3485 3726 4850 5115 6216 6394 6570 6768 8930 9370

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RETAINING RINGS AND CIRCLIPS

TABLE 26-7 Dimensions for internal circlips—type B—light series (Contd.) All dimensions in millimeters Circlip Shaft Dia d1 45 47 48 50 52 55 56 58 60 62 63 65 68 70 72 75 78 80 82 85 88 90 92 95 98 100 102 105 108 110 112 115 120 125 130 135 140 145 150 155 160 165 170 175 180

Bore groove Axial force

s a b h11 Max. Approx. d3 1.75 6.2 6.4 6.5 6.7 6.8 2 6.9 7.3 7.6 7.8 2.5

4.3 4.4 4.5 4.6 4.7 5.0 5.1 5.2 5.4 5.5 5.6 5.8 6.1 6.2 6.4 6.6 6.8

8.5

8.6 3

8.7 8.8 9 9.2 9.5 10.4 10.5

11 11.2 11.4 12 4 13

7.0 7.2 7.4 7.6 7.8 8.1 8.3 8.4 8.5 8.7 9 9 9.1 9.3 9.7 10 10.2 10.5 10.7 10.9 11.2 11.4 11.6 11.8 12.2 Max 12.7 Max 13.2

48.5 50.5 51.5 54.2 56.2 59.2 60.2 62.2 64.2 66.2 67.2 69.2 72.5 74.5 76.5 79.5 82.5 85.5 87.5 90.5 93.5 95.5 97.5 100.5 103.5 105.5 108 112 115 117 119 122 127 132 137 142 147 152 158 164 169 174.5 179.5

d5 Tol. on d4 d3 Compressed Min.

þ0.92 0.46

þ1.08 0.54

þ1.08 0.54

þ1.26 0.63

31.6 33.2 34.6 36 37.6 40.4 41.4 43.2 44.4 46.4 47.4 48.8 51.4 53.4 55.4 58.4 60 62 64 66.8 69.8 71.8 73.6 76.4 79 81 82.6 85.6 88 88.2 90 93 97 102 107 112 117 122 125 130 133 138 145

184.5

149

189.5

153

2.5

3

3.5

4

d2

Tol. on m1 d2 H13

47.5 H12 49.5 50.5 53 55 58 59 61 63 65 66 68 71 73 75 78 81 83.5 85.5 88.5 91.5 93.5 95.5 H12 98.5 101.5 103.5 106 109 112 114 116 119 124 129 134 139 144 149 155 160 165 170 175 180 185

H13

m2 Min.

n Min.

1.85

2

3.8

2.15

2.3

4.5

2.65

2.8

5.3

3.15

3.3

5.3

6

4.15

4.3

N

lbf

44325 46286 47268 59526 61780 65214 66195 68646 71098 73354 74334 76688 80120 82572 84826 88260 91690 109834 112775 116699 120620 123562 126505 130428 134350 137292 159849 164750 169654 172596 175538 180440 188286 195150 202996 210940 219686 226532 226532 294198 312830 322936 332444

9965 10406 10626 13382 13766 14660 14882 15442 15984 16490 16712 17240 18012 18564 19070 19700 20614 24692 25354 26236 27118 27790 28440 29322 30205 30866 35936 37040 38142 38902 39465 40566 42330 43874 45238 47400 49165 50930 50930 66142 70330 72535 74740

341270

76724

338328

76062

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RETAINING RINGS AND CIRCLIPS

TABLE 26-7 Dimensions for internal circlips—type B—light series (Contd.) All dimensions in millimeters Circlip Shaft Dia d1

Axial force s a b h11 Max. Approx. d3 Max 13.7 Max 13.8 Max

185 190 195 200 210 220 230 240 250 260 270 280 290 300

Bore groove

14 Max 5

16 Max

d5 Tol. on d4 d3 Compressed Min.

d2

Tol. on m1 d2 H13

m2 Min.

n Min.

N

lbf

194.5 þ1.44

157

190

343230

77165

199.5 0.72 204.5 209.5 222 232 242 252 262 275 þ1.62 285 0.81 295 305 315

162 167 171 181 191 201 211 221 227 237 247 257 267

195 200 205 216 226 236 246 256 268 278 288 298 308

333424 323618 318715 488368 511904 538392 514846 495232 529556 507980 490330 472678 456006

74960 72755 71652 109795 115096 121040 115748 111338 119055 114205 110236 106268 102520

5

9 5.15

5.3 12

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RETAINING RINGS AND CIRCLIPS

TABLE 26-8 Dimensions for internal circlips—type B—heavy series

FIGURE 26-6

All dimensions in millimeters Circlip Shaft Dia d1

s h11

20 22 24 25 26 28 30 32 34 35 37 38 40 42 45 47 50 52 55 60 62 65 68 70 72 75 80 85 90 95 100

4.5 4.7 4.9 o 1.5 5 5.1 5.3 5.5 5.7 5.9 1.75 6 ) 6.2 6.3 6.5 2 6.7 7 7.2 7.5 2.5 7.7 8 8.5 8.6 8.7 3 8.8 9 9.2 9.3 9.5 9.7 4 10 10.3 10.5

Bore groove Axial force

a Max.

b Approx. d3

Tol. on d3

2.4 2.8 3

þ0.42 0.21

3:1 3.2 3.3 3.4 3.7 3.8 3.9 4.1 4.3 4.4 4.6 4.7 5 5.4 5.5 5.8 6.1 6.2 6.4 6.6 7 7.2 7.6 8.1 8.4

21.5 23.5 25.9 26.9 27.9 30.1 32.1 34.4 36.5 37.8 39.8 40.8 43.5 45.5 48.5 50.5 54.2 56.2 59.2 64.2 66.2 69.2 72.5 74.5 76.5 79.5 85.5 90.5 95.5 100.5 105.5

þ0.50 0.25

þ0.78 0.39

þ0.92 0.46

þ1.08 0.54

d5 d4 Compressed Min.

d2

10 11.6 13.2 14 14.8 16.4 18 19.6 21.2 22 23.6 24.4 26 27.6 30 31.6 34 35.6 38 42 43.8 46.6 49.4 51 52.6 55.5 60 64.6 69 73.4 78

21 23 25.2 26.2 27.2 29.4 31.4 33.7 35.7 37 39 40 42.5 44.5 47.5 49.5 53 55 58 63 65 68 71 73 75 78 83.5 88.5 93.5 98.5 103.5

2

2.5

3

3.5

Tol. on d2

m1 H13

m2 Min.

n Min. 1.5

1.6

1.7

1.8 2.1 2.6

1.85

2 3

2.15

2.3

2.65

2.8

3.8

H12

4.5 3.15

3.3

4.15

4.3

5.3

N

lbf

7895 8650 11375 11768 12259 15495 16572 21575 22750 27655 29224 30106 39716 41678 44325 46286 59526 61782 65214 71098 73354 76688 80120 82570 84926 89260 109834 116698 123564 130429 137292

1775 1945 2558 2646 2755 3484 3726 4950 5115 6218 6590 6768 8930 9370 9965 10406 13382 13890 14660 15984 16490 17240 18017 18564 19070 19942 24692 26236 27780 29322 30966

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RETAINING RINGS AND CIRCLIPS

TABLE 26-9 Dimensions for external circlips—type C

FIGURE 26-7

All dimensions in millimeters Circlip

Shaft Groove

Nominal size d1

d4 Expanded

a H10

s

0.8 1.2 1.5 1.9 2.3 3.2 4 5 6 7 8 9 10 12 15 19 24

2 3 4 4.5 6 7 9 11 12 14 16 18.5 20 23 29 37 44

0.58 1.01 1.28 1.61 1.94 2.70 3.34 4.11 5.26 5.84 6.52 7.63 8.32 10.45 12.61 15.92 21.88

0.2 0.3 0.4 0.5 0.6 0.6 0.7 0.7 0.7 0.9 1.0 1.1 1.2 1.3 1.5 1.75 2.0

Tol. on s

0.02

0.03

To

d3 h11

m

From

d3 1 1.4 2 2.5 3 4 5 6 7 8 9 10 11 13 16 20 25

1.4 2 2.5 3 4 5 7 8 9 11 12 14 15 18 24 31 38

0.8 1.2 1.5 1.9 2.3 3.2 4 5 6 7 8 9 10 12 15 19 24

0.24 0.34 0.44 0.54 0.64 0.64 0.74 0.74 0.74 0.94 1.05 1.15 1.25 1.35 1.55 1.80 2.05

Tol. on m 0:02

0.03

0.06

n Min 0.4 0.6 0.8 1 1 1 1.2 1.2 1.2 1.5 1.8 2 2 2.5 3.0 3.5 4.0

IS: 3075, 1965

REFERENCES 1. Lingaiah K., Machine Design Data Handbook, McGraw-Hill Publishing Company, New York, 1994. 2. Waldes Kohinoor, Inc., Long Island City, New York, U.S.A, 1985. 3. Edward Killian, Retaining Rings, Waldes Kohinoor, Inc., Long Island City, New York, U.S.A, 1985 and Robert O. Parmley, Editor-in-Chief, Mechanical Components Handbook, McGraw-Hill Publishing Company New York, 1985. 4. IS: 3075, 1965, Circlips. 5. Industrial Fasteners Handbook, 2nd Edition, Trade and Technical Press Limited, Morden Surrey, England, 1980. 6. ‘‘General Purpose Uniform Cross-section Spiral Retaining Rings’’, ANSI, B27.6, 1972 (R 1977) 7. ‘‘General Purpose Tapered and Reduced Cross-section Retaining Rings (Metric)’’, ANSI B 27.7, 1977. 8. ‘‘General Purpose Metric Tapered and Reduced Cross-section Retaining Rings’’ ANSI B27.8M, 1978. 9. Joseph E. Shigley, ‘‘Unthreaded Fasteners’’, and Shigley, J. E., and Mischke, C. R., Standard Handbook of Machine Design, McGraw-Hill Publishing Company, New York, 1996.

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From

26.14

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19.050 (0.750) — 20.620 (0.812) 25.40 (1.000) 26.980 (1.062) 38.10 (1.500) 39.680 (1.562) 50.90 (2.000) 52.380 (2.062) 63.50 (2.500) 66.680 (2.625) 76.20 (3.000) 80.160 (3.156) 101.60 (4.000)

Inverted internal in grooved housings (see Fig. 26-11)

0.890 (0.035) 1.070 (0.042) 1.270 (0.050) 1.580 (0.062) 1.980 (0.078) 2.360 (0.093) 2.760 (0.109)

1.350 (0.053) 1.320 (0.052) 1.720 (0.068) 12.080 (0.082)

0.380 (0.015) 0.635 (0.025) 0.890 (0.035) 1.070 (0.042) 1.270 (0.050) 1.580 (0.062) 1.980 (0.078) 2.360 (0.093) 2.760 (0.109) 3.180 (0.125) 3.960 (0.156) 4.750 (0.187)

Nominal ring thickness, in mm (in)

7,340 11,564 18,460 33,805 56,266 95,402 120,096

51,376 57,156 98,292 132,105

1,868 4,670 8,806 20,238 33,138 60,938 101,192 149,898 209,500 412,330 612,440 851,792

N

1,650 2,600 4,150 7,600 12,650 19,200 27,000

11,550 12,850 19,950 29,700

420 1,050 1,990 4,550 7,450 13,700 22,750 33,700 47,100 92,700 137,700 191,500

lbf

From lbf

12,450 14,700 23,900 10,200

— — 14,679 3,300 26,020 5,850 43,368 9,750 68,054 15,300 97,412 21,900 152,122 34,200

55,378 65,385 106,306 45,370

2,358 530 5,694 1,280 13,344 3,000 26,919 6,050 46,926 10,550 77,840 17,500 122,705 27,600 175,696 39,500 342,496 77,000 471,042 105,900 686,326 154,300 1175,162 264,200

N

Through

Groove material having minimum tensile yield strength of 1034 MPa (150,000 lbf/in2 )

2,668 3,114 5,560 11,786 20,016 32,025 47,148

16,012 20,906 29.912 45,370

845 1,556 2,268 7,028 13,566 28,245 48,260 73,392 107,196 266,890 229,596 443,020

N

600 700 1,250 2,650 4,500 7,200 10,600

3,600 4,700 6,500 10,200

190 350 510 1,590 3,050 6,350 10,850 16,500 24,100 60,000 74,100 99,600

lbf

From

— 5,115 11,120 19,126 28,912 42.700 75,170

19,126 27,132 42,700 54,265

1,068 2,046 6,494 13,344 26,688 45,814 69,610 102,970 244,640 305,132 413,364 848.334

N

— 1,150 2,500 4,300 6,500 9,600 16,900

4,300 6,100 9,600 12,200

240 460 1,460 3,000 6,000 10,300 15,650 23,150 55,000 68,600 93,100 190,700

lbf

Through

Groove material having minimum tensile yield strength of 310 MPa (45,000 lbf/in2 )

1.27 (0.050) 1.37 (0.050) 1.95 (0.069) 2.24 (0.088) 3.18 (0.125) 3.81 (0.150) 4.42 (0.174)

1.62 (0.064) 1.62 (0.064) 1.98 (0.078) 2.23 (0.088)

0.28 (0.011) 0.58 (0.023) 0.69 (0.027) 0.89 (0.035) 1.12 (0.044) 1.63 (0.064) 1.93 (0.076) 2.23 (0.088) 2.46 (0.097) 4.26 (0.168) 4.50 (0.177) 5.13 (0.202)

From

— 1.63 (0.064) 2.06 (0.081) 3.00 (0.118) 3.66 (0,144) 4.29 (0.169) 4.42 (0.174)

1.62 (0.064) 1.62 (0.064) 1.98 (0.078) 2.23 (0.088)

0.40 (0.016) 0.68 (0.027) 0.81 (0.032) 1.07 (0.042) 1.22 (0.048) 1.63 (0.064) 1.98 (0.078) 2.33 (0.092) 4.01 (0.158) 4.26 (0.168) 4.98 (0.196) 6.36 (0.270)

Through

Radii

0.79 (0.031) 0.86 (0.034) 1.09 (0.043) 1.40 (0.055) 2.00 (0.078) 2.38 (0.094) 2.77 (0.109)

1.27 (0,050) 1.27 (0.050) 1.58 (0.062) 1.78 (0.070)

0.22 (0.0085) 0.45 (0.019) 0.53 (0.021) 0.71 (0.028) 0.89 (0.035) 1.27 (0.050) 1.55 (0.061) 1.78 (0.070) 1.98 (0.078) 3.40 (0.134) 3.61 (0.142) 4.11 (0.162)

From

— 1.02 (0.040) 1.30 (0,051) 1.88 (0.074) 2.28 (0.090) 2.69 (0.106) 2.77 (0.109)

1.27 (0.050) 1.27 (0.050) 1.58 (0.062) 1.78 (0.070)

0.33 (0.013) 0.53 (0.021) 0.64 (0.025) 0.86 (0.034) 0.97 (0.038) 1.27 (0.050) 1.58 (0.062) 1.88 (0.074) 3.20 (0.126) 3.40 (0.134) 3.61 (0.142) 4.11 (0.162)

Through

Chamfers

3,780 5,560 8,006 12,900 20,460 29,802 40,032

12,676 12,232 20,905 31,136

845 2,358 4,892 7,340 10,675 17,346 27,578 40,032 53,376 66,720 102,304 151,232

N

850 1,250 1,800 2,900 4,600 6,700 9,000

2,850 2,750 4,700 7,000

190 530 1,100 1,650 2,400 3,900 6,200 9,000 12,000 15,000 23,000 34,000

lbf

Allowable thrust load, when rings abut parts with listed corner radii or chamfers, Fr0

FIGURE 26-11 Inverted internal

Maximum allowable corner radii or chamfers of retained parts, mm (in)

FIGURE 26-10 Beveled and double-beveled internal

Allowable static thrust load, when rings abut parts with sharp cornersa

FIGURE 26-9 Bowed internal

c

a

Courtesy: # 1964, 1965, 1973, 1991 Waldes Kohinoor, Inc., Long Island City, New York, 1985. Where rings are of immediate size—or groove materials have intermediate tensile yield strengths—loads may be obtained by interpolation; b Numbers inside the brackets are in inches and numbers outside brackets are in millimeters; Approximate corner radii and chamfers limits for parts with intermediate diameters can be determined by interpolation. Corner radii and chamfers smaller than those listed will increase the thrust load proportionately, approaching but not exceeding allowable static thrust loads of rings abutting parts with sharp corners. Courtesy: Edward Killian, ‘‘Retaining Rings’’, Robert O. Parmley, Editor-in-Chief ‘‘Mechanical Components Handbook’’, McGraw-Hill Publishing Company, New York, USA, 1985.

39.630 (1.562) 142.00 (1.688) 44.450 (1.750) 50.80 (2.000) 52.390 (2.062) 64.28 (2.531) 65.080 (2.562) 71.42 (2.812)

Double beveled internal (see Fig. 26-10)

7.93 (0.312) 11.50 (0.453) 19.05 (0.750) 25.98 (1.023) 38.10 (1.500) 50.80 (2.000) 64.28 (2.531) 76.20 (3.000) 127.00 (5.000) 152.40 (6.000) 177.90 (7.000) 254.00 (10.000)

Through

Housing diameter, mm (in)b

Basic internal

Basic 6.350 (0.250) internal, 9.525 (0.375) Bowed 12.700 (0.500) internal, 19.740 (0.777) Bevelled 26.980 (1.062) internal 39.680 (1.562) in grooved 52.000 (2.047) housings 65.080 (2.562) (see Figs 26-8 77.780 (3.062) and 26-9) 133.350 (5.250) 158.750 (6.250) 184.150 (7.250)

Ring type

FIGURE 26-8

TABLE 26-10 Axially assembled tapered-section internal rings

RETAINING RINGS AND CIRCLIPS

12.700 17.475 26.980 39.675 53.975 69.850 89.900

Inverted external on grooved shafts (see Fig. 26-15)

c

3.962 (0.156) 6.994 (0.236)c 11.912 (0.469) 11.069 (0.672) 25.981 (1.023) 38.100 (1.500) 50.800 (2.000) 68.275 (2.688) 87.325 (3.438) 127.600 (5.000) 152.400 (6.000) 177.800 (7.000) 254.00 (10.000)

Through

(0.500) 17.068 (0.672) (0.688) 25.400 (1.000) (1.062) 38.100 (1.500) (1.562) 50.800 (2.000) (2.125) 66.675 (2.625) (2.750) 85.000 (3.346) (3.500) 101.600 (4.000)

3.175 (0.125) 4.775 (0.188)c 6.350 (0.250) 12.700 (0.500) 17.475 (0.688) 26.980 (1.062) 39.675 (1.562) 52.375 (2.062) 69.850 (2.750) 88.900 (3.500) 133.350 (5.250) 158.750 (6.250) 190.500 (7.500)

Basic external, Bowed external, Beveled external in grooved shafts (see Figs 26-12 to 26-14)

c

From

Ring type

Shaft diameter, mm (in)b

0.890 (0.035) 1.050 (0.042) 1.278 (0.050) 1.570 (0.062) 1.981 (0.078) 2.362 (0.093) 2.768 (0.109)

0.254 (0.010) 0.381 (0.015) 0.635 (0.025) 0.890 (0.035) 1.050 (0.042) 1.270 (0.050) 1.575 (0.062) 1.981 (0.078) 2.362 (0.093) 2.768 (0.109) 3.175 (0.125) 3.962 (0.156) 4.775 (0.188)

Nominal ring thickness, mm (in)b

FIGURE 26-16 Heavy-duty external ring

FIGURE 26-15 Inverted external rings

lbf

N

lbf

Through N

4,993 10,230 18,459 33,805 57,824 89,405 132,995

1,100 6,450 2,300 14,678 4,150 26,020 7,600 43,368 13,000 71,612 20,100 108,976 29,900 152,566

1,450 3,300 5,850 9,750 16,100 24,500 34,300

1,245 280 2,224 500 5,338 1,200 11,565 2,600 20,239 4,550 32,025 7,200 67,152 11,500

35 50 175 550 1,000 2,400 5,200 8,450 14,400 22,800 40,800 58,300 84,800

lbf

From lbf

2,090 4,670 11,120 17,792 29,580 46,704 62,272

470 1,050 2,500 4,000 6,650 10,500 14,000

245 55 534 120 2,002 450 4,225 950 10,008 2,250 22,240 5,000 35,806 8,050 61,605 13,850 97,410 21,900 165,020 37,100 239,302 53,800 323,370 72,700 666,310 149,800

N

Through

Groove material having minimum tensile yield strength of 310 MPa (45,000 lbf/in2 )

489 110 578 130 156 1,068 240 1,378 310 355 2,624 590 4,892 1,100 779 7,340 1,650 9,795 2,200 2,446 5,123 3,400 22,462 5,050 4,448 27,578 6,200 39,142 8,800 10,675 50,707 11,400 64,940 14,600 23,930 84,280 18,950 109,886 24,700 37,596 133,885 30,100 167,690 37,700 64,050 199,715 44,900 285,562 64,200 101,414 343,830 77,300 392,758 88,300 181,478 510,630 114,800 572,012 129,600 259,319 734,810 165,200 979,450 220,200 377,190

N

From

Groove material having minimum tensile yield strength of 1034 MPa (150,000 lbf/in2 )

Allowable static thrust load, when rings abut parts with sharp cornersa

FIGURE 26-13 Bowed external rings

FIGURE 26-12 Basic external rings

TABLE 26-11 Axially assembled tapered-section external retaining rings

1.295 (0.051) 1.696 (0.066) 2.336 (0.092) 2.642 (0.104) 3.378 (0.133) 4.191 (0.165) 5.131 (0.202)

0.254 (0.010) 0.355 (0.014) 0.457 (0.019) 0.864 (0.034) 1.066 (0.042) 1.524 (0.060) 2.082 (0.082) 2.480 (0.098) 2.850 (0.112) 3.29 (0.1295) 4.292 (0.169) 4.750 (0.187) 5.588 (0.220)

From

1.651 (0.065) 2.311 (0.091) 2.540 (0.100) 3.225 (0.127) 4.038 (0.159) 4.928 (0.194) 5.410 (0.213)

0.381 (0.015) 0.419 (0.0165) 0.788 (0.031) 1.016 (0.040) 1.473 (0.058) 2.006 (0.079) 2.438 (0.096) 2.832 (0.1115) 3.276 (0,129) 4.196 (0.165) 4.675 (0.184) 5.283 (0.208) 7.468 (0.294)

Through

Radii

0.813 (0.032) 1.067 (0.042) 1.473 (0.058) 1.676 (0.066) 2.134 (0.094) 2.616 (0.103) 3.226 (0.127)

0.152 (0.006) 0.216 (0.0085) 0.279 (0.011) 0.508 (0.020) 0.635 (0.025) 0.914 (0.036) 1.245 (0.049) 1.498 (0.059) 1.702 (0.067) 1.981 (0.079) 2.565 (0.101) 2.945 (0.112) 3.353 (0.132)

From

N

1.041 (0.041) 1.448 (0.057) 1.600 (0.063) 2.032 (0.090) 2.515 (0.099) 3.073 (0.121) 3.378 (0.133)

3,025 4,448 6,494 10,008 16,680 24,464 34,916

0.228 (0.009) 200 0.254 (0.010) 467 0.457 (0.018) 2,090 0.610 (0.024) 4,048 0.980 (0.035) 5,960 1.194 (0.047) 8,674 1.448 (0.057) 13,344 1.702 (0.067) 22,240 1.956 (0.077) 32,692 2.515 (0.099) 46,704 2.794 (0.110) 60,048 3.175 (0.125) 93,408 4.470 (0.176) 133,440

Through

Chamfers

Maximum allowable corner radii or chamfers of retained parts, mm (in)b

680 1,000 1,460 2,250 3,750 5,500 7,850

45 105 470 910 1,340 1,950 3,000 5,000 7,350 10,500 13,500 21,000 30,000

lbf

Allowable thrust load, when rings abut parts with listed corner radii or chamfers, Fr0

FIGURE 26-17 Miniature high-strength ring

FIGURE 26-14 Beveled external rings

RETAINING RINGS AND CIRCLIPS

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26.15

9.525 (0.375) 12.700 (0.500) 15.975 (0.625)

Average sizes shaft diameter, in 1.270 (0.050) 1.575 (0.062) 1.575 (0.062)

3.400 (0.134) 0.509 (0.020) 5.156 (0.203) 0.635 (0.025) 8.331 (0.328) 0.889 (0.035)

0.890 (0.035) 1.050 (0.042) 1.278 (0.050) 1.981 (0.078) 2.362 (0.093) 2.709 (0.109) 3.175 (0.125)

Nominal ring thickness, mm (in)b

250 490 1,200

2,000 3,000 3,900 9,000 15,000 24,500 37,000

lbf

4,003 5,338 3,451

900 1,200 1,900

CRS/SAE on soft steel shaft

1,112 2,180 5,338

8,896 13,344 17,347 40,032 66,720 108,976 164,576

N

From lbf

330 650 1,800

3,336 5,338 7,116

750 1,200 1,600

Cabra 353 brass

1,468 2,890 8,006

— — — — 23,130 5,200 51,152 11,500 86,736 19,500 129,992 29,000 169,624 38,000

N

Through lbf

60 130 220

2,668 600 4,003 900 4,892 1,100

Cabra 110 copper

266 578 978

3,114 700 4,480 1,000 4,892 1,100 10,760 2,400 21,350 4,800 44,490 10,000 68,054 15,300

N

From lbf

1,334 2,002 2,891

1.194 (0.047) 1.778 (0.070) 1.778 (0.070) 2.200 (0.089) 2.692 (0.106) 3.250 (0.128) 3.886 (0.153)

From — — 1.956 (0.077) 2.548 (0.100) 3.251 (0.128) 3.251 (0.128) 3.886 (0.153)

Through

Radii

0.991 (0.039) 1.473 (0.058) 1.473 (0.059) 1.880 (0.074) 2.235 (0.088) 2.718 (0.107) 3.251 (0.128)

From

— — 1.625 (0.064) 2.108 (0.083) 2.718 (0.107) 2.718 (0.107) 3.251 (0.128)

Through

Chamfers

Maximum allowable corner radii or chamfers of retained parts, mm (in)b

300 450 650

Not applicable

90 0.330 (0.013) 0.355 (0.014) 0.254 (0.010) 0.279 (0.011) 200 0.534 (0.021) 0.584 (0.023) 0.432 (0.017) 0.457 (0.018) 460 0.711 (0.028) 0.965 (0.038) 0.558 (0.022) 0.762 (0.030)

Type 3003 aluminium

400 890 2,046

— — — — 8,451 1,900 17,792 4,000 36,474 8,200 55,155 12,400 75,616 17,000

N

Through

Groove material having minimum tensile yield strength of 310 MPa (45,000 lbf/in2 )

890 1,424 2,668

2,002 2,446 2,890–4,003 11,120 17,792 22,240 26,688

N

Fr0

200 320 600c

450 550 650–900 2,500 4,000 5,000 6,000

lbf

Allowable thrust load, when rings abut parts with listed corner radii or chamfers,

b

a

Courtesy: # 1964, 1965, 1973, 1991 Waldes Kohinoor, Inc., Long Island City, New York, 1985. Where rings are of immediate size—or groove materials have intermediate tensile yield strengths—loads may be obtained by interpolation. Numbers inside the brackets are in inches and numbers outside brackets are in millimeters. c Rings for shafts 3.175 mm (0.125 in) through 6.00 mm (0.236 in) diameter are made of beryllium copper only. Approximate corner radii and chamfers limits for parts with intermediate diameters can be determined by interpolation. Corner radii and chamfers smaller than those listed will increase the thrust load proportionately, approaching but not exceeding allowable static thrust loads of rings abutting parts with sharp corners. Exceptions: for shafts 14.00 mm (0.551 in), 77.125 mm (3.06 in), 89.9 mm (3.500 in), 90.00 mm (3.543 in), 92.00 mm (3.625 in), 101.6 mm (4.000 in), 114.3 mm (4.500 in), 152.4 mm (6.000 in), and 158.75 mm (6.250 in) in diameter, refer to manufacturer’s speciﬁcations for data. Note: Fr0 ¼ 3; 115 N(700 lbf) for ring used with 6.6 mm (0.260 in) in-diameter shaft. Courtesy: Edward Killian, ‘‘Retaining Rings’’, Robert O. Parmley, Editor-in-Chief ‘‘Mechanical Components Handbook’’, McGraw-Hill Publishing Company, New York, USA, 1985

Permanent shoulder on grooved shafts

Through

(0.394) — (0.473) — (0.500) 17.000 (0.669) (0.750) 25.400 (1.000) (1.062) 35.001 (1.378) (1.500) 45.000 (1.772) (1.938) 50.800 (2.000)

2.565 (0.101) 3.962 (0.156) 5.562 (0.219)

10.000 12.014 12.700 19.050 26.980 38.100 49.225

Heavy-duty external on grooved (see Fig. 26-16)

Miniature high-strength external on grooved shafts (see Fig. 26-17)

From

Ring type

mm (in)b

Shaft diameter,

Groove material having minimum tensile yield strength of 1034 MPa (150,000 lbf/in2 )

Allowable static thrust load, when rings abut parts with sharp cornersa

TABLE 26-11 Axially assembled tapered-section external retaining rings (Contd.)

RETAINING RINGS AND CIRCLIPS

26.16

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3.175 (0.125) 5.562 (0.219) 12.700 (0.500) 17.475 (0.688) 28.575 (1.125) 44.450 (1.750)

11.912 16.764 24.994 39.624 50.012 69.850 85.725

3.175 (0.125) 3.556 (0.140) 6.350 (0.250) 9.525 (0.375) 12.700 (0.500)

Crescent-shaped on grooved shafts (see Fig. 26-18)

Two-part interlocking on grooved shafts (see Fig. 26-19)

Bowed E-ring on grooved shafts (see Fig. 26-20)

(0.469) (0.669) (0.994) (1.562) (1.969) (2.750) (3.375)

From

Ring type

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— 5.562 (0.219) 7.925 (0.312) 11.125 (0.438) 15.875 (0.625)

15.815 (0.625) 22.225 (0.875) 38.100 (1.500) 47.625 (1.975) 66.675 (2.625) 82.550 (3.250) —

4.775 (0.188) 11.125 (0.438) 15.875 (0.625) 25.400 (1.000) 38.100 (1.500) 50.800 (2.000)

Through

Shaft diameter, mm (in)b

FIGURE 26-21 E-ring

FIGURE 26-18 Crescent shaped ring

0.254 (0.010) 0.391 (0.015) 0.635 (0.025) 0.890 (0.035) 1.066 (0.042)

0.890 (0.035) 1.066 (0.042) 1.270 (0.050) 1.575 (0.062) 1.981 (0.078) 2.362 (0.093) 2.768 (0.109)

0.381 (0.015) 0.635 (0.025) 0.889 (0.035) 1.066 (0.042) 1.270 (0.050) 1.575 (0.062)

Nominal ring thickness, mm (in)b

TABLE 26-12 Radially assembled external retaining rings

191 334 1,134 3,069 4,937

8,896 14,900 26,020 52,264 81,176 134,232 193,488

378 1,156 3,692 7,562 14,768 23,600

N 578 2,312 4,582 11,031 19,660 32,470

N

43 75 255 690 1,110

512 1,446 3,692 6,316

—

— 115 325 830 1,420

2,650 4,400 9,950 14,100 24,300 35,750 —

130 520 1,030 2,480 4,420 7,300

lbf

Through

2,000 11,797 3,350 19,571 5,850 39,910 11,750 62,716 19,250 108,086 30,200 159,016 43,500 —

85 260 830 1,700 3,320 6,430

lbf

From

Groove material having minimum tensile yield strength of sy , 1034 MPa (150,000 lbf/in2 )

200 266 512 1,401 2,668

2,759 5,560 12,999 25,131 40,032 66,720 91,628

200 445 2,002 3,558 9,786 23,574

N

45 60 115 315 600

620 1,250 2,900 5,650 9,000 15,000 20,600

45 100 450 800 2,200 5,300

lbf

From

— 3,358 1,001 2,135 4,670

3,692 7,116 19,794 30,246 53,376 79,174 —

311 1,556 3,114 8,006 17,792 31,136

N

75 225 480 1,050

—

830 1,600 4,450 6,800 12,000 17,800 —

70 350 700 1,800 4,000 7,000

lbf

Through

Groove material having minimum tensile yield strength of sy , 310 MPa (45,000 lbf/in2 )

Allowable static thrust load, when rings abut parts with sharp cornersa

FIGURE 26-22 Reinforced E-ring

FIGURE 26-19 Two-part interlocking ring

1.016 (0.040) 1.524 (0.060) 1.524 (0.060) 1.650 (0.065) 2.032 (0.080)

1.321 (0.052) 1.650 (0.065) 2.184 (0.086) 2.540 (0.100) 2.890 (0.114) 3.632 (0.143) 4.622 (0.182)

0.355 (0.014) 0.583 (0.021) 0.762 (0.030) 0.864 (0.034) 1.325 (0.052) 2.056 (0.081)

From

— 1.524 (0.060) 1.524 (0.060) 1.650 (0.065) 2.032 (0.080)

1.321 (0.052) 1.650 (0.065) 2.194 (0.086) 2.540 (0.100) 2.895 (0.114) 3.632 (0.143) 4.622 (0.182)

0.533 (0.021) 0.736 (0.029) 0.838 (0.033) 1.168 (0.046) 1.752 (0.069) 2.311 (0.091)

Through

Radii

0.762 (0.030) 1.143 (0.045) 1.143 (0.045) 1.270 (0.050) 1.524 (0.060)

1.016 (0.040) 1.270 (0.050) 1.676 (0.066) 1.956 (0.077) 2.235 (0.088) 2.794 (0.110) 3.556 (0.140)

0.280 (0.011) 0.406 (0.016) 0.584 (0.023) 0.660 (0.026) 1.016 (0.040) 1.575 (0.062)

From

— 1.143 (0.045) 1.143 (0.045) 1.270 (0.050) 1.524 (0.060)

1.016 (0.040) 1.270 (0.050) 1.676 (0.066) 1.956 (0.077) 2.235 (0.088) 2.794 (0.110) —

0.406 (0.016) 0.558 (0.022) 0.635 (0.025) 0.899 (0.035) 1.346 (0.053) 1.778 (0.070)

Through

Chamfers

Maximum allowable corner radii or chamfers of retained parts, mm (in)b

FIGURE 26-23 Locking-prong ring

FIGURE 26.20 Bowed E-ring

lbf

Same values as sharp corner abutment

610 880 1,250 1,900 3,050 4,300 5,950

Same values as sharp corner abutment

2,713 3,914 5,560 8,451 13,566 19,126 26,466

N

Allowable thrust load, when rings abut parts with listed corner radii or chamfers, Fr0

RETAINING RINGS AND CIRCLIPS

26.17

2.336 (0.092) 4.775 (0.188) 9.525 (0.375) 4.775 (0.188) 7.925 (0.312) 11.125 (0.439) 19.050 (0.750) 25.400 (1.000) 4.775 (0.188) 9.525 (0.375) 12.700 (0.500) 19.050 (0.750)

Locking-prong ring on grooved shafts (see Fig. 26-23)

3.962 (0.156) 7.925 (0.312) 11.125 (0.438) 6.350 (0.250) 9.525 (0.375) 15.875 (0.625) — — 7.925 (0.312) 11.125 (0.438) 15.875 (0.625) 25.400 (1.000)

3.175 (0.125) 6.350 (0.250) 11.125 (0.438) 14.275 (0.562)

0.254 (0.010) 0.381 (0.015) 0.508 (0.020) 0.890 (0.035) 1.066 (0.042) 1.270 (0.050) 1.575 (0.062) 1.981 (0.078) 0.635 (0.025) 0.890 (0.035) 1.066 (0.042) 1.270 (0.050)

0.381 (0.015) 0.635 (0.025) 0.890 (0.035) 1.066 (0.042)

0.254 (0.010) 0.381 (0.015) 0.635 (0.025) 0.990 (0.035) 1.066 (0.042) 1.270 (0.050) 1.575 (0.062)

1.270 (0.050) 1.575 (0.062)

Nominal ring thickness, mm (in)b 2,000 3,450 13 45 170 690 1,110 2,000 3,450 50 150 420 820

80 200 550 600 1,300 2,200 4,600 7,500 430 1,300 2,100 3,700

58 200 756 3,069 4,937 8,896 15,346 222 667 1,368 3,647

356 890 2,446 2,668 5,782 9,786 20,460 33,360 1,912 5,782 9,340 16,458

lbf

8,896 15,346

N

From

3,470 8,228 11,120 21,350

534 1,556 3,114 4,003 6,894 13,344

334 1,120 2,669 4,136

89 334 1,446 3,692 6,316 11,787 18,236

11,787 18,236

N

120 350 700 900 1,550 3,000 — — 790 1,850 2,500 4,800

75 250 600 930

20 75 325 830 1,420 2,650 4,100

2,650 4,100

lbf

Through

58 178 600 2,046

27 89 266 1,401 2,669 6,672 6,672

6,672 6,672

156 622 2,002 579 1,120 1,780 7,116 11,565 579 1,334 1,780 7,116

N

35 140 450 130 250 400 1,600 2,600 130 300 400 1,600

13 40 135 460

6 20 60 315 600 1,500 1,500

1,500 1,500

lbf

From

445 1,334 2,668 890 1,334 18,236 — — 1,120 1,780 2,668 11,565

111 334 1,268 4,805

31 200 1,001 2,135 4,670 8,451 10,452

8,451 10,452

N

From

100 300 600 200 300 1,100 — — 250 400 600 2,600

25 75 285 480

7 45 225 480 1,050 1,900 2,350 (0.045) (0.065) (0.070) (0.090)

1.270 (0.050) 1.650 (0.065) 2.032 (0.080) 2.160 (0.085) 2.286 (0.090) 1.270 (0.050) 1.650 (0.065) 2.032 (0.080) 2.286 (0.090)

1.143 1.650 1.778 2.032

0.381 (0.015) 1.016 (0.040) 1.524 (0.060) 1.650 (0.065) 2.032 (0.080) 2.160 (0.085) 2.286 (0.090)

(0.045) (0.065) (0.070) (0.080)

1.270 (0.050) 1.650 (0.065) 2.032 (0.080) — — 1.270 (0.050) 1.650 (0.065) 2.032 (0.080) 2.286 (0.090)

1.143 1.650 1.778 2.032

0.762 (0.030) 1.016 (0.040) 1.524 (0.060) 1.650 (0.065) 2.032 (0.080) 1.956 (0.077) 2.296 (0.090)

1.956 (0.077) 2.296 (0.090)

Through

Radii

1,900 2.160 (0.085) 2,350 2.286 (0.090)

lbf

Through

(0.033) (0.050) (0.055) (0.060)

0.939 (0.033) 1.270 (0.050) 1.396 (0.055) 1.524 (0.060)

0.508 (0.020) 0.765 (0.030) 1.143 (0.045) 1.270 (0.050) 1.524 (0.060) 1.448 (0.057) 1.778 (0.070)

1.448 (0.057) 1.778 (0.070)

Through

1.066 (0.040) 1.270 (0.050) 1.524 (0.060) 1.651 (0.065) 1.778 (0.070) 1.016 (0.040) 1.270 (0.050) 1.524 (0.060) 1.779 (0.070)

1.066 (0.040) 1.270 (0.050) 1.524 (0.060) — — 1.016 (0.040) 1.270 (0.050) 1.524 (0.060) 1.778 (0.070)

Not applicable

0.835 1.270 1.396 1.524

0.254 (0.010) 0.762 (0.030) 1.143 (0.045) 1.270 (0.050) 1.524 (0.060) 1.651 (0.065) 1.778 (0.070)

1.651 (0.065) 1.778 (0.070)

From

Chamfers

Maximum allowable corner radii or chamfers of retained parts, mm (in)b

lbf

250 350 600 1,000 1,800 150 300 400 1,000

Same values as sharp corner abutment

Same values as sharp corner abutment

1112 1556 2669 4448 8006 667 1334 1780 4448

N

Fr0

Allowable thrust load, when rings abut parts with listed corner radii or chamfers,

b

a

Courtesy: # 1964, 1965, 1973, 1991 Waldes Kohinoor, Inc., Long Island City, New York, 1985. Where rings are of immediate size—or groove materials have intermediate tensile yield strengths—loads may be obtained by interpolation. Numbers inside the brackets are in inches and numbers outside brackets are in millimeters. Approximate corner radii and chamfers limits for parts with intermediate diameters can be determined by interpolation. Corner radii and chamfers smaller than those listed will increase the thrust load proportionately, approaching but not exceeding allowable static thrust loads of rings abutting parts with sharp corners. Exeptions: for shafts 14.00 mm (0.551 in), 77.125 mm (3.06 in), 89.9 mm (3.500 in), 90.00 mm (3.543 in), 92.00 mm (3.625 in), 101.6 mm (4.000 in), 114.3 mm (4.500 in), 152.4 mm (6.000 in), and 158.75 mm (6.250 in) in diameter, refer to manufacturer’s speciﬁcations for data. Rings for shafts 3.175 mm (0.125 in) through 6.00 mm (0.236 in) diameter are made of beryllium copper only. Courtesy: Edward Killian, ‘‘Retaining Rings’’, Robert O. Parmley, Editor-in-Chief ‘‘Mechanical Components Handbook’’, McGraw-Hill Publishing Company, New York, USA, 1985

2.388 (0.094) 3.962 (0.156) 7.925 (0.312) 12.700 (0.500)

Reinforced E-ring on grooved shafts (see Fig. 26-22)

1.575 (0.062) 3.556 (0.140) 7.925 (0.312) 11.125 (0.439) 15.975 (0.625) 25.400 (1.000) 34.925 (1.375)

25.400 (1.000) 34.925 (1.375)

19.050 (0.750) 30.175 (1.189)

1.016 (0.040) 2.388 (0.094) 3.556 (0.140) 9.525 (0.375) 12.700 (0.500) 19.050 (0.750) 30.175 (1.189)

Through

From

E-ring on grooved shafts (see Fig. 26-21)

Ring type

mm (in)b

Shaft diameter,

Groove material having minimum tensile yield strength of sy , 310 MPa (45,000 lbf/in2 )

Allowable static thrust load, when rings abut parts with sharp cornersa Groove material having minimum tensile yield strength of sy , 1034 MPa (150,000 lbf/in2 )

TABLE 26-12 Radially assembled external retaining rings (Contd.)

RETAINING RINGS AND CIRCLIPS

26.18

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RETAINING RINGS AND CIRCLIPS RETAINING RINGS AND CIRCLIPS

26.19

TABLE 26-13 Radially assembled external retaining rings

FIGURE 26-26 Internal self-locking ring

FIGURE 26-25 External self-locking ring

FIGURE 26-24 Reinforced external self-locking ring

FIGURE 26-27 Triangular self-locking ring

Allowable static thrust load, when rings abut parts with sharp cornersa

Housing diameter or shaft diameter From

Through

Nominal ring thickness

Groove material having minimum tensile yield strength of 1034 MPa (150,000 lbf/in2 ) From

Through

Ring type

mm

in

mm

in

mm

in

N

lbf

N

lbf

Reinforced self-locking external on shafts, no grooves

2.388 2.388 11.125

0.094 0.094 0.438

9.525 9.525 25.400

0.375 0.375 1.000

0.254 0.380 0.380

0.010 0.015 0.015

— — —

— — —

— — —

— — —

Self-locking external on shafts, no grooves

2.388 11.125

0.094 0.438

9.525 25.400

0.375 1.000

0.254 0.380

0.010b 0.015

— —

— —

— —

Self-locking internal in housing, no grooves

7.925 19.050

0.312 0.750

15.875 50.300

0.625 2.000c

0.254 0.380

0.010 0.015

— —

— —

Triangular retainer on shafts, no grooves

1.575 1.575 4.388 4.388 4.775 9.525 11.231

0.062 0.062 0.094 0.094 0.188 0.375 0.437

— — 3.962 3.962 7.925 —

— — 0.156 0.156 0.312 —

d

d

0.254 0.380 0.254 0.380 0.380 0.509 0.6351

0.010 0.015 0.010 0.015 0.015 0.020 0.025

— — — — — — —

0.381 0.508 0.508 0.635

0.015 0.020 0.020 0.025

622 890 978 978

Triangular nut on threaded, parts

4.761 4.761 6.35–20 6.35–20

6/32 6/32 1/4–20 1/4–20

7.939 7.938 6.35–28 6.35–28

10/32 10/32 1/4–28 1/4–28

Groove material having minimum tensile yield strength of 310 MPa (45,000 lbf/in2 ) From N

N

lbf

120 27 200 45 534 126

289 534 622

65 120 140

— —

58 222

13 50

200 289

45 65

— —

— —

356 334

80 75

200 245

45 55

— — — — — — —

— — — — — — —

— — — — — — —

111 25 178 40 266 60 256 80 622 140 1112 250 1200 270

— — 334 534 890 — —

— — 75 120 200 — —

140 200 220 220

756 978 — —

170 220 — —

645 845 — —

145 190 — —

622 800 978 978

lbf

Through

140 180 220 220

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RETAINING RINGS AND CIRCLIPS

26.20

CHAPTER TWENTY-SIX

TABLE 26-13 Radially assembled external retaining rings (Contd.)

FIGURE 26-28 Radial clamp ring Allowable static thrust loade

Shaft diameter From mm

Shaft without groove

Through

in

mm

in

Nominal ring thickness mm

in

From N

lbf

Shaft with groove, 310 MPa (45,000 lbf/in2 )

Through N

lbf

From

Through

N

lbf

N

lbf

Tapered-section self-locking clamp ring on shafts with or without grooves Inch type

2.388 4.750 7.925 11.100 15.875

0.094 0.187 0.312 0.437 0.625

3.962 6.350 9.525 12.700 19.050

0.156 0.250 0.375 0.500 0.750

0.635 0.890 1.066 1.270 1.575

0.025 0.035 0.042 0.050 0.062

45 111 200 266 378

10 25 45 60 95

98 156 266 289 400

22 35 60 65 90

— — 489 1290 2535

— — 110 290 570

— 400 800 1735 3780

— 90 180 390 850

Millimeter type

2.006 5.004 5.994 8.992 13.538

0.079 0.197 0.236 0.354 0.533

2.947 — 7.010 10.005 14.986

0.118 — 0.276 0.394 0.590

0.610 0.913 0.990 1.194 1.500

0.024 0.032 0.039 0.047 0.059

45 133 155 222 334

10 30 35 50 75

66 — 178 245 356

15 — 40 55 80

— — 178 40 311 70 579 130 1512 340

— — 445 756 1645

— — 100 170 370

Inch type

2.362 4.750 7.925

0.093 0.187 0.312

3.962 6.350 9.525

0.156 0.250 0.375

0.635 0.889 1.066

0.025 0.035 0.042

36 80 142

8 18 32

58 98 187

13 22 42

— — —

— — —

— — —

— — —

Millimeter type

1.981 5.004 7.925

0.078 0.197 0.312

3.962 7.900 9.962

0.156 0.276 0.393

0.660 0.889 1.092

0.024 0.035 0.043

30 85 147

7 19 33

53 102 218

12 23 49

— — —

— — —

— — —

— — —

Radially applied self-locking clamp rings on shafts without grooves

Courtesy: # 1964, 1965, 1973, 1991 Waldes Kohinoor, Inc., Long Island City, New York, 1985. a Where rings are of immediate size—or groove materials have intermediate tensile yield strengths—loads may be obtained by interpolation. b Ring for shaft 6.096 mm (0.240 in) diameter is available only in 0.380 mm (0.015 in) thickness: allowable thrust load ¼ 178 N (40 lbf). c Ring for housing 34.925 mm (1.375 in) diameter is available only as reinforced ring having an allowable thrust load ¼ 667 N (150 lbf). d Round and hexagonal shafts. e Grooved shafts are recommended only for rings used on shafts 0.197 in (5.0 mm) or larger. Courtesy: Edward Killian, ‘‘Retaining Rings’’, Robert O. Parmley, Editor-in-Chief ‘‘Mechanical Components Handbook’’, McGraw-Hill Publishing Company, New York, USA, 1985

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Source: MACHINE DESIGN DATABOOK

CHAPTER

27 APPLIED ELASTICITY SYMBOLS

c C1 , C2

inner radius of cylinder, m (in) inner radius of rotating cylinder, m (in) inner radius of circular plate, m (in) cross-sectional area, m2 (in2 ) outer radius of inner cylinder, m (in) inside radius of outer cylinder, m (in) outer radius of rotating cylinder, m (in) outer radius of circular plate, m (in) outside radius of outer cylinder, m (in) constants of integration, m (in)

D¼

ﬂexural rigidity of a plate or shell, N/m (lbf/in)

a A b

Eh3 12ð1 2 Þ

E G g h I J L l, m, n M Mb Mt Mx , My Mxy

Mn , Mnt Ms , M , Mr n n Nx , Ny Nxy

modulus of elasticity, GPa modulus of rigidity, GPa acceleration due to gravity, 981 cm/s2 thickness of plate, m (in) moment of inertia, cm4 (in4 ) polar moment of inertia, cm4 (in4 ) length, m (in or ft) direction cosines of the outward normal moment (also with subscripts) N m (lbf ft) bending moment, N m (lbf ft) torsional moment, m N (ft lbf ) bending moments per unit length of sections of a plate perpendicular to x and y axes, respectively, N m (lbf ft) twisting moment per unit length of sections of a plate perpendicular to x-axis, N m (lbf ft) bending and twisting moments per unit length of sections of a plate perpendicular to n-direction, N m (lbf ft) radial, tangential and twisting moments in polar co-ordinates normal direction a number, usually but not always, integer normal force per unit length of sections of a plate perpendicular to x and y axis, respectively, N (lbf) shearing force in the direction of y-axes per unit length of section of a plate perpendicular to x axis, N/m (lbf/ft)

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APPLIED ELASTICITY

27.2

CHAPTER TWENTY-SEVEN

Nr , N0 p q Qx , Qy Nr , N r rx , ry r, t T Mtxy u, v, w V W w x, y, z X, Y, Z Z ! x , y , z r , r , , z xy , yz , zx " "x , "y , "z "r , " xy , yz , zx r , z r , z , rz

normal forces per unit length in radial and tangential directions in polar co-ordinates, N (lbf) pressure, MPa (psi) load per unit length, kN/m (lbf/in) shearing forces parallel to z-axis per unit length of sections of a plate perpendicular to x and y axis, N/m (lbf/in) radial and tangential shearing forces, N (lbf ) radius, m (in) radii of curvature of the middle surface of a plate in xz and yz planes polar co-ordinates time, s temperature, 8C tension of a membrane, kN/m (lbf/in) twist of surface components or displacements, m (in) strains energy weight, N (lbf ) displacement, m (in) displacement of a plate in the normal direction, m (in) deﬂection, m (in) rectangular co-ordinates, m (in) body forces in x; y; z directions, N (lbf ) section modulus in bending, cm3 (in3 ) density, kN/m3 (lbf/in3 ) angular speed, rad/s stress, MPa (psi) normal components of stress parallel to x, y, and z axis, MPa (psi) radial and tangential stress, MPa (psi) normal stress components in cylindrical co-ordinates, MPa (psi) shearing stress, MPa (psi) shearing stress components in rectangular co-ordinates, MPa (psi) unit elongation, m/m (in/in) unit elongation in x, y, and z direction, m/m (in/in) radial and tangential unit elongation in polar co-ordinates shearing strain shearing strain components in rectangular co-ordinate shearing strain in polar co-ordinate shearing stress components in cylindrical co-ordinates, MPa (psi) Poisson’s ratio stress function angular deﬂection, deg e ¼ "x þ "y þ "z ¼ "r þ " þ "z e ¼ "x þ "y þ "z ¼ volume expansion shearing components in cylindrical co-ordinates

Note: and with subscript s designates strength properties of material used in the design which will be used and observed throughout this Machine Design Data Handbook

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APPLIED ELASTICITY APPLIED ELASTICITY

Particular

27.3

Formula

STRESS AT A POINT (Fig. 27-1) The stress at a point due to force F acting normal to an area dA (Fig. 27-1b)

Stress ¼ ¼ lim

A ! 0

F A

ð27-1Þ

where F ¼ force acting normal to the area A A ¼ an inﬁnitesimal area of the body under the action of F For stresses acting on the part II of solid body cut out from main body in x, y and z directions, Fig. 27-1b

x ¼ lim

Fx Ax

ð27-2aÞ

xy ¼ lim

Fy Ax

ð27-2bÞ

xz ¼ lim

Fz Ax

ð27-2cÞ

Ax ! 0

Ax ! 0

Ax ! 0

Similarly the stress components in xy and xz planes can be written and the nine stress components at the point O in case of solid body made of homogeneous and isotropic material

x yz

xy y

xz yz

zx

zy

z

ð27-3Þ

Fig. 27-1c shows the stresses acting on the faces of a small cube element cut out from the solid body.

F4

F5

a

F1

Part I

Part II

∆Fy ∆A a N ∆F o

F8

a F6

y

F2

F3

F7

(a) A solid body subject to action of external forces

z

∆Fz a

y

F2

F3

F1

Part II ∆Fx

x F8

F7 (b) An infineticimal area ∆A of Part II of a solid body under the action of force ∆F at 0

σy

τyz τzy dy σ z

o τzx

τyx τxy

τxz

σx

x

dz

dx z (c) Stresses acting on the faces of a small cube element cut out from the solid body

FIGURE 27-1

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APPLIED ELASTICITY

27.4

CHAPTER TWENTY-SEVEN

Particular

Formula

Summing moments about x, y and z axes, it can be proved that the cross shears are equal

xy ¼ yx ;

yz ¼ zy ;

All nine components of stresses can be expressed by a single equation

ij ¼ lim

Fj Ai

Ai ! 0

zx ¼ xz

ð27-4Þ

ð27-5Þ

where i ¼ 1; 2; 3 and j ¼ 1; 2; 3 The FNx , FNy , and FNz unknown components of the resultant stress on the plane KLM of elemental tetrahedron passing through point O (Fig. 27-2)

FNx ¼ x cos N; x þ xy cos N; y þ xz cos N; z FNy ¼ yx cos N; x þ y cos N; y þ yz cos N; z FNz ¼ zx cos N; x þ zy cos N; y þ z cos N; z

The unknown components of resultant stress FNx , FNy and FNz in terms of direction cosines l, m and n (Fig. 27-4) y

τxy

Fz

FNx ¼ x l þ xy m þ zx n FNy ¼ yz l þ y m þ yx n FNz ¼ zx l þ zy m þ z n

Surface area KLM = A TN N (normal to KLM) FNy τzx σz τxz Fx FNz ho’ τzy FNx K x o τyz Fy

l ¼ cos ¼ cos N; x; m ¼ cos ¼ cos N; y, n ¼ cos ¼ cos N; z, l s þ m2 þ n2 ¼ ðlÞ 02 þ ðm0 Þ2 þ ðn0 Þ2 ¼ 1

τyx

y

σy

L

TN = stress vector in N direction Fbx, Fby, Fbz = Body forces in x, y and z - direction

z

σy+

FIGURE 27-2 The state of stress at O of an elemental tetrahedron.

τyz+

y

σx

M Tx’ τx’y’

σz

τxz τzx

L

γ

τxy o τyx z’

β o’ h α τzy

τyz

N

∂τyz ∂y

dy τ zx

τxz

∂τzy

∂σy dy ∂y ∂τyx

τyx+

dy σz ∂τ τxy+ xy dx

∂y

∂x

τzy

∂τxzσx + τzy+ ∂z dz + τ ∂τzy xz ∂x dx τxy o dy τzx+ ∂z dz ∂σz σz+ ∂z dz τ τyz dz yx σy

y’

σx

ð27-7Þ

where the direct cosines are

M

σx

ð27-6Þ

x’

σx’

∂σx ∂x

dx x

dx K

x

τz’x’ σ y

z

FIGURE 27-3 Small cube element removed from a solid body showing stresses acting on all faces of the body in x, y and z directions.

z

FIGURE 27-4 Tx0 , resolved into x0 , x0 y0 and x0 z0 stress components.

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APPLIED ELASTICITY

27.5

APPLIED ELASTICITY

Particular

Formula

cos ¼ l ¼ angle between x axis and Normal N cos ¼ m ¼ angle between y axis and Normal N

The resultant stress FN on the plane KLM

cos ¼ n ¼ angle between z axis and Normal N qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 FN ¼ FNx ð27-8Þ þ FNy þ FNz

The normal stress which acts on the plane under consideration

N ¼ FNx cos þ FNy cos þ FNz cos

The shear stress which acts on the plane under consideration

N ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ FN2 2n

ð27-8aÞ ð27-8bÞ

Equations (27-1), (27-2) and (27-7) to (27-8) can be expressed in terms of resultant stress vector as follows (Fig. 27-2) The resultant stress vector at a point

FN ð27-9aÞ A where TN coincides with the line of action of the resultant force Fn

TN ¼ lim

A ! 0

The resultant stress vector components in x, y and z directions

The resultant stress vector

TNx ¼ x l þ xy m þ xz n

ð27-9bÞ

TNy ¼ xy l þ y m þ zy n

ð27-9cÞ

TNz ¼ zx l þ zy m þ z n

ð27-9dÞ

TN ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 þ T2 þ T2 TNx Ny Nz

ð27-9eÞ

where the direction cosines are cosðTN ; xÞ ¼ TNx =jTN j,

cosðTN ; yÞ ¼ TNy =jTN j,

cosðTN ; zÞ ¼ TNz =jTN j The normal stress which acts on the plane under consideration

N ¼ jTN j cosðTN ; NÞ

ð27-9f Þ

N ¼ TNx cosðN; xÞ þ TNy cosðN; yÞ þ TNz cosðN; zÞ ð27-9gÞ The shear stress which acts on the plane under consideration

N ¼ jTN j sinðTN ; NÞ

ð27-10aÞ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ TN2 2N

ð27-10bÞ

N ¼ The angle between the resultant stress vector TN and the normal to the plane N

cosðTN ; NÞ ¼ cosðTN ; xÞ cosðN; xÞ þ cosðTN ; yÞ cosðN; yÞ þ cosðTN ; zÞ cosðN; zÞ

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ð27-10cÞ

APPLIED ELASTICITY

27.6

CHAPTER TWENTY-SEVEN

Particular

Formula

EQUATIONS OF EQUILIBRIUM The equations of equilibrium in Cartesian coordinates which includes body forces in three dimensions (Fig. 27-3)

@x @xy @xz þ þ þ Fbx ¼ 0 @x @y @z

ð27-11aÞ

@y @yz @yx þ þ þ Fby ¼ 0 @y @z @x

ð27-11bÞ

@z @zx @zy þ þ þ Fbz ¼ 0 @z @x @y

ð27-11cÞ

where Fbx , Fby and Fbz are body forces in x, y and z directions Stress equations of equilibrium in two dimensions

@x @xy þ þ Fbx ¼ 0 @x @y

ð27-11dÞ

@y @yx þ þ Fby ¼ 0 @y @x

ð27-11eÞ

TN ¼ iTNx þ jTNy þ kTNz

ð27-12aÞ

TN 0 ¼ iTN 0 x þ jTN 0 y þ kTN 0 z

ð27-12bÞ

N ¼ il þ jm þ kn

ð27-12cÞ

TRANSFORMATION OF STRESS The vector form of equations for resultant-stress vectors TN and TN0 for two diﬀerent planes and the outer normals N and N 0 in two diﬀerent planes

0

0

0

0

N ¼ il þ jm þ kn

ð27-12dÞ

where i, j and k are unit vectors in x, y and z directions, respectively The projections of the resultant-stress vector TN onto the outer normals N and N 0

Substituting Eqs. (27-9b), (27-9c), (27-9d) and (27-9e) in Eqs. (27-13), equations for TN , N and TN , N 0

TN N ¼ TNx l þ TNy m þ TNz n

ð27-13aÞ

TN N 0 ¼ TNx l 0 þ TNy m0 þ TNz n0

ð27-13bÞ

TN N ¼ x l 2 þ y m2 þ z n2 þ 2xy lm þ 2yz mn þ 2zx nl 0

0

0

ð27-14aÞ 0

0

TN N ¼ x ll þ y mm þ z nn þ xy ½lm þ ml 0 þ yz ½mn0 þ nm0 þ zx ½nl 0 þ ln0 The relation between TN , N 0 and TN0 , N

TN0 N ¼ TN N 0

By coinciding outer normal N with x0 , N with y0 , and N with z0 individually respectively and using Eqs. (27-14a) to (27-14b), x0 , y0 and z0 can be obtained (Fig. 27-4)

x0 ¼ Tx0 x0 ¼ x cos2 ðx0 ; xÞ þ y cos2 ðx0 ; yÞ

ð27-14bÞ ð27-15Þ

þ z cos2 ðx0 ; zÞ þ 2xy cosðx0 ; xÞ cosðx0 ; yÞ þ 2yz cosðx0 ; yÞ cosðx0 ; zÞ þ 2zx cosðx0 ; zÞ cosðx0 ; xÞ

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ð27-15aÞ

APPLIED ELASTICITY APPLIED ELASTICITY

Particular

27.7

Formula

y0 ¼ Ty0 y 0 ¼ y cos2 ðy0 ; yÞ þ z cos2 ðy0 ; zÞ þ x cos2 ðy0 ; xÞ þ 2yz cosðy0 ; yÞ cosðy0 ; zÞ þ 2zx cosðy0 ; zÞ cosðz0 ; xÞ þ 2xy cosðy0 ; xÞ cosðy0 ; yÞ

ð27-15bÞ

z0 ¼ Tz0 z0 ¼ z cos2 ðz0 ; zÞ þ x cos2 ðz0 ; xÞ þ y cos2 ðz0 ; yÞ þ 2zx cosðz0 ; zÞ cosðz0 ; xÞ þ 2xy cosðz0 ; xÞ cosðz0 ; yÞ þ 2yz cosðz0 ; yÞ cosðz0 ; zÞ By selecting a plane having an outer normal N coincident with the x0 and a second plane having an outer normal N 0 coincident with the y0 and utilizing Eq. (27-14b) which was developed for determining the magnitude of the projection of a resultant stress vector on to an arbitrary normal can be used to determine x0 y0 . Following this procedure and by selecting N and N 0 coincident with the y0 and z0 , and z0 and x0 axes, the expression for y0 z0 and z0 x0 can be obtained. The expressions for x0 y0 , y0 z0 and z0 x0 are

ð27-15cÞ

x0 y0 ¼ Tx0 y0 ¼ x cosðx0 ; xÞ cosðy0 ; xÞ þ y cosðx0 ; yÞ cosðy0 ; yÞ þ z cosðx0 ; zÞ cosðy0 ; zÞ þ xy ½cosðx0 ; xÞ cosðy0 ; yÞ þ cosðx0 ; yÞ cosðy0 ; xÞ þ yz ½cosðx0 ; yÞ cosðy0 ; zÞ þ cosðx0 ; zÞ cosðy0 ; yÞ þ zx ½cosðx0 ; zÞ cosðy0 ; xÞ þ cosðx0 ; xÞ cosðy0 ; zÞ ð27-16aÞ y0 z0 ¼ Ty0 z0 ¼ y cosðy0 ; yÞ cosðz0 ; yÞ þ z cosðy0 ; zÞ cosðz0 ; zÞ þ x cosðy0 ; xÞ cosðz0 ; xÞ þ yz ½cosðy0 ; yÞ cosðz0 ; zÞ þ cosðy0 ; zÞ cosðz0 ; yÞ þ zx ½cosðy0 ; zÞ cosðz0 ; xÞ þ cosðy0 ; xÞ cosðz0 ; zÞ þ xy ½cosðy0 ; xÞ cosðz0 ; yÞ þ cosðy0 ; yÞ cosðz0 ; xÞ ð27-16bÞ z0 x0 ¼ Tz0 x0 ¼ z cosðz0 ; zÞ cosðx0 ; zÞ þ x cosðz0 ; xÞ cosðx0 ; xÞ þ y cosðz0 ; yÞ cosðx0 ; yÞ þ zx ½cosðz0 ; zÞ cosðx0 ; xÞ þ cosðz0 ; xÞ cosðx0 ; zÞ þ xy ½cosðz0 ; xÞ cosðx0 ; yÞ þ cosðz0 ; yÞ cosðx0 ; xÞ þ yz ½cosðz0 ; yÞ cosðx0 ; zÞ þ cosðz0 ; zÞ cosðx0 ; yÞ ð27-16cÞ Equations (27-15a) to (27-15c) and Eqs. (27-16a) to (27-16c) can be used to determine the six Cartesian components of stress relative to the Oxyz coordinate system to be transformed into a diﬀerent set of six Cartesian components of stress relative to an Ox0 y0 z0 coordinate system

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APPLIED ELASTICITY

27.8

CHAPTER TWENTY-SEVEN

Particular

For two-dimensional stress ﬁelds, the Eqs. (27-15a) to (27-15c) and (27-16a) to (27-16c) reduce to, since z ¼ zx ¼ yz ¼ 0 z0 coincide with z and is the angle between x and x0 , Eqs. (27-15a) to (27-15c) and Eqs. (27-16a) to (27-16c) y TNy K

TNz M

O

L

x0 ¼ x cos2 þ y sin2 þ 2xy sin cos ¼

x þ y x y þ cos 2 þ xy sin 2 2 2

ð27-17aÞ

y0 ¼ y cos2 þ x sin2 2xy sin cos ¼

N

y þ x y x þ cos 2 xy sin 2 2 2

ð27-17bÞ

x0 y0 ¼ y cos sin x cos sin

TN, σN TNx N

Formula

þ xy ðcos2 sin2 Þ

x

z FIGURE 27-5 The stress vector TN .

¼

y x sin 2 þ xy cos 2 2

z0 ¼ z0 x0 ¼ y0 z0 ¼ 0

ð27-17cÞ ð27-17dÞ

PRINCIPAL STRESSES By referring to Fig. 27-5, where TN coincides with outer normal N, it can be shown that the resultant stress components of TN in x, y and z directions

TNx ¼ N l

Substituting Eqs. (27-9b) to (27-9d) into (27-18), the following equations are obtained

x l þ yx m þ zx n ¼ N l

TNy ¼ N m

ð27-18Þ

TNz ¼ N n

xy l þ y m þ xy n ¼ N m

ð27-19Þ

xz l þ yz m þ z n ¼ N n Eq. (27-19) can be written as

ðx N Þl þ yx m þ zx ¼ 0 xy l þ ðy N Þm þ zy ¼ 0

ð27-20Þ

xz l þ yz m þ ðz N Þn ¼ 0 From Eq. (27-20), direction cosine (N, x) is obtained and putting this in determinant form

Putting the determinator of determinant into zero, the non-trivial solution for direction cosines of the principal plane is

0 yx zx 0 zy y n 0 yz z N cosðN; xÞ ¼ x N yx zx y N zy xy xz yz z N

ð27-21Þ

x N xy xz

ð27-22Þ

yx y N yz

zy ¼ 0 z N zx

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APPLIED ELASTICITY

27.9

APPLIED ELASTICITY

Particular

Expanding the determinant after making use of Eqs. (27-4) which gives three roots. They are principal stresses

Formula

3N ðx þ y þ z Þ2N 2 2 2 þ ðx y þ y z þ z x xy yz zx ÞN 2 2 2 ðx y z x yz y zx z xy þ 2xy yz zx Þ ¼ 0

ð27-23Þ For two-dimensional stress system the coordinating system coinciding with the principal directions, Eq. (27-23) becomes

2 3i ðx þ y Þ2i þ ðx y xy Þi ¼ 0

ð27-23aÞ

where i ¼ 1; 2; 3

The three principal stresses from Eq. (27-23a) are 1;2

1 ¼ ðx þ y Þ 2

ﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x y 2 2 þ xy 2

ð27-23bÞ

3 ¼ 0 The directions of the principal stresses can be found from

2xy sin 2ðN1 ; xÞ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ðx y Þ2 þ 4xy

ð27-23cÞ

x y cos 2ðN1 ; xÞ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ðx y Þ2 þ 4xy

ð27-23dÞ

tan 2ðN1 ; xÞ ¼

2xy x y

ð27-23eÞ

From Eq. (27-15)

TN1 N2 ¼ TN2 N1

From deﬁnition of principal stress

TN1 ¼ 1 N1

ð27-25aÞ

TN2 ¼ 2 N2

ð27-25bÞ

Substituting the values of TN1 and TN2 in Eq. (27-15) and simplifying

ð1 2 ÞN1 N2 ¼ 0

From Eq. (27-20)

N1 N2 ¼ 0

ð27-24Þ

ð27-26Þ

where 1 and 2 are distinct ð27-27Þ

which proves that N1 and N2 are orthogonal. The three invariant of stresses from Eq. (27-23)

I1 ¼ x þ y þ z ¼ x0 þ y0 þ z0

ð27-28aÞ

2 2 2 I2 ¼ x y þ y z þ z x xy yz zx

¼ x0 y0 þ y0 z0 þ z0 x0 x20 y0 y20 z0 z20 x0 ð27-28bÞ 2 2 2 I3 ¼ x y z x yz y zx z xy þ 2xy yz zx

¼ x0 y0 z0 x0 y20 z0 y0 z20 x0 z0 x20 y0 þ 2x0 y0 y0 z0 z0 x0

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ð27-28cÞ

APPLIED ELASTICITY

27.10

CHAPTER TWENTY-SEVEN

Particular

Formula

where I1 ¼ first invariant, I2 ¼ second invariant and I3 ¼ third invariant of stress For the coordinating system coinciding with the principal direction, the expression for invariants from Eq. (27-28)

I 1 ¼ 1 þ 2 þ 3

ð27-29aÞ

I2 ¼ 1 2 þ 2 3 þ 3 1

ð27-29bÞ

I3 ¼ 1 2 3

ð27-29cÞ

x E

ð27-30aÞ

STRAIN (Fig. 27-6) The normal strain or longitudinal strain by Hooke’s law (Fig. 27-6) in x-direction

"x ¼

The lateral strains in y and z-direction

v ¼ v"x E x v "z ¼ x ¼ v"x E y vy "y ¼ ; "x ¼ "z ¼ ¼ v"y E E v "z ¼ z ; "x ¼ "y ¼ z v"z E E "y ¼

The normal strains caused by y and z

ð27-30bÞ ð27-30cÞ ð27-31Þ ð27-32Þ

THREE-DIMENSIONAL STRESS-STRAIN SYSTEM The general stress-strain relationships for a linear, homogeneous and isotropic material when an element subject to x , y and z stresses simultaneously

"x ¼

1 ½ vðy þ z Þ E x

ð27-33aÞ

"y ¼

1 ½ vðz þ x Þ E y

ð27-33bÞ

y

dz

σx

dx’ dx

y

dy

dy’

σx z

dz’

FIGURE 27-6 Uniaxial elongation of an element in the direction of x.

τxy

M N x z

τxy

τxy τ L xy

K

M

τxy

M’ m I

L x τxy L’ τ xy

N

N’ τxy

n

k K’

K

FIGURE 27-7 A cubic element subject to shear stress, xy .

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APPLIED ELASTICITY

27.11

APPLIED ELASTICITY

Particular

The expressions for x , y and z stresses in case of three-dimensional stress system from Eqs (27-33)

Formula

"z ¼

1 ½ vðx þ y Þ E z

x ¼

E ½ð1 vÞ"x þ vð"y þ "z Þ ð1 þ vÞð1 2vÞ

ð27-33cÞ

ð27-34aÞ y ¼

E ½ð1 vÞ"y þ vð"z þ "x Þ ð1 þ vÞð1 2vÞ ð27-34bÞ

z ¼

E ½ð1 vÞ"z þ vð"x þ "y Þ ð1 þ vÞð1 2vÞ

ð27-34cÞ

BIAXIAL STRESS-STRAIN SYSTEM The normal strain equations, when z ¼ 0 from Eq. (27-33)

"x ¼

1 ½ vy E x

ð27-35aÞ

1 ½ vx E y v "z ¼ ½x þ y E "y ¼

The normal stress equation, when z ¼ 0 from Eq. (27.34)

ð27-35bÞ ð27-35cÞ

E ½" þ v"y ¼ J1 þ 2"x 1v x 2 ¼ ð" þ "y Þ þ 2"x

þ 2 x

x ¼

E ½" þ v"x ¼ J1 þ 2"y 1v y 2 ð" þ "x Þ þ 2"y ¼

þ 2 y

ð27-36aÞ

y ¼

z ¼ J1 þ 2"z ¼ 0 xy ¼ xy ;

yz ¼ yz ¼ 0;

ð27-36bÞ ð27-36cÞ

zx ¼ zx ¼ 0 ð27-36dÞ

SHEAR STRAINS For a homogeneous, isotropic material subject to shear force, the shear strain which is related to shear stress as in case of normal strain

xy G yz yz ¼ G zx zx ¼ G

xy ¼

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ð27-37aÞ ð27-37bÞ ð27-37cÞ

APPLIED ELASTICITY

27.12

CHAPTER TWENTY-SEVEN

Particular

Formula

F2

y F3 Unstrained element

P L M dy K w dx N dz

Deformed element due to strain in new position

u

L’

M’

K’

N’

u+

∂v dy ∂y

ν+

v

F4

Y

F1

P’

F5

O

∂u dy ∂y u

L

L’ M

dy

x x

∂u dy M’ ∂y

K’ K

dx

N

u+

N’

∂v dx ∂x

ν

ν+

∂u dx ∂x

z

∂ν dx ∂x

y X

FIGURE 27-8 Deformation of a cube element in a solid body subject to loads.

FIGURE 27-9 Two-dimensional deformation under load.

It has been proved that the shear modulus (G) is related to Young’s modulus (E) and Poisson’s ratio as

G¼

From Eqs. (27-37), shear strain in terms of E and

E 2ð1 þ vÞ

ð27-38Þ

xy ¼

2ð1 þ vÞ xy E

ð27-39aÞ

yz ¼

2ð1 þ vÞ yz E

ð27-39bÞ

zx ¼

2ð1 þ vÞ zx E

ð27-39cÞ

STRAIN AND DISPLACEMENT (Figs. 27-8 and 27-9) The normal strain in x-direction

The normal strain in y and z-directions

The shear strains xy, yz and zx planes

@u change in length dx þ @x dx dx @u ¼ "x ¼ ¼ original length @x dx ð27-40aÞ "y ¼

@v @y

ð27-40bÞ

"z ¼

@w @z

ð27-40cÞ

xy ¼

@v @u þ @x @y

ð27-41aÞ

yz ¼

@w @v þ @y @z

ð27-41bÞ

zx ¼

@u @w þ @z @x

ð27-41cÞ

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APPLIED ELASTICITY APPLIED ELASTICITY

Particular

The amount of counterclockwise rotation of a line segment located at R in xy, yz and zx planes

Formula

xy ¼

1 2

yz ¼

1 2

zx The strain "z and ﬁrst strain invariant J1 in case of plane stress

1 ¼ 2

"z ¼

J1 ¼ The strains components "x0 , "y0 and "z0 , along x0 , y0 and z0 axes line segments with reference to the O0 x0 y0 z0 system

27.13

@v @u @x @y @w @v @y @z @u @w @z @x

ð27-41dÞ ð27-41eÞ

ð" þ "y Þ

þ 2 x

2 ð" þ "y Þ

þ 2 x

ð27-41f Þ

ð27-41gÞ

ð27-41hÞ

"x0 ¼ "x cos2 ðx; x0 Þ þ "y cos2 ðy; x0 Þ þ "z cos2 ðz; x0 Þ þ xy cosðx; x0 Þ cosðy; x0 Þ þ yz cosðy; x0 Þ cosðz; x0 Þ þ zx cosðz; x0 Þ cosðx; x0 Þ ð27-42aÞ "y0 ¼ "y cos2 ðy; y0 Þ þ "z cos2 ðz; y0 Þ þ "x cos2 ðx; y0 Þ þ yx cosðy; y0 Þ cosðz; y0 Þ þ zx cosðz; y0 Þ cosðx; y0 Þ þ xy cosðx; y0 Þ cosðy; y0 Þ ð27-42bÞ "z0 ¼ "z cos2 ðz; z0 Þ þ "x cos2 ðx; z0 Þ þ "y cos2 ðy; z0 Þ þ zx cosðz; z0 Þ cosðx; z0 Þ þ xy cosðx; z0 Þ cosðy; z0 Þ þ yz cosðy; z0 Þ cosðz; z0 Þ ð27-42cÞ

The shearing strain components (due to angular changes) x0 y0 , y0 z0 and z0 x0 with reference to the O0 x0 y0 z0 system

x0 y0 ¼ 2"x cosðx; x0 Þ cosðx; y0 Þ þ 2"y cosðy; x0 Þ cosðy; y0 Þ þ 2"z cosðz; x0 Þ cosðz; y0 Þ þ xy ½cosðx; x0 Þ cosðy; y0 Þ þ cosðx; y0 Þ cosðy; x0 Þ þ yz ½cosðy; x0 Þ cosðz; y0 Þ þ cosðy; y0 Þ cosðz; x0 Þ þ zx ½cosðz; x0 Þ cosðx; y0 Þ þ cosðz; y0 Þ cosðx; x0 Þ ð27-43aÞ

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APPLIED ELASTICITY

27.14

CHAPTER TWENTY-SEVEN

Particular

Formula

y0 z0 ¼ 2"y cosðy; y0 Þ cosðy; z0 Þ þ 2"z cosðz; y0 Þ cosðz; z0 Þ þ 2"x cosðx; y0 Þ cosðx; z0 Þ þ yz ½cosðy; y0 Þ cosðz; z0 Þ þ cosðy; z0 Þ cosðz; y0 Þ þ zx ½cosðz; y0 Þ cosðx; z0 Þ þ cosðz; z0 Þ cosðx; y0 Þ þ xy ½cosðx; y0 Þ cosðy; z0 Þ þ cosðx; z0 Þ cosðy; y0 Þ ð27-43bÞ z0 x0 ¼ 2"z cosðz; z0 Þ cosðz; x0 Þ þ 2"x cosðx; z0 Þ cosðx; x0 Þ þ 2"y cosðy; z0 Þ cosðy; x0 Þ þ zx ½cosðz; z0 Þ cosðx; x0 Þ þ cosðz; x0 Þ cosðx; z0 Þ þ xy ½cosðx; z0 Þ cosðy; x0 Þ þ cosðx; x0 Þ cosðy; z0 Þ þ yz ½cosðy; z0 Þ cosðz; x0 Þ þ cosðy; x0 Þ cosðz; z0 Þ ð27-43cÞ For the case of two-dimensional state of stress when z0 coincides with z and zx ¼ yz ¼ 0, the angle between x and x0 coordinates

"x0 ¼ "x cos2 þ "y sin2 þ zy sin cos ¼

1 2 ½ð"x

ð27-44aÞ

þ "y Þ þ ð"x "y Þ cos 2 þ xy sin 2 ð27-44bÞ

"y0 ¼ "y cos2 þ "x sin2 zy sin cos ¼

1 2 ½ð"y

þ "x Þ þ ð"y "x Þ xy sin 2

ð27-44cÞ ð27-44dÞ

x0 y0 ¼ 2ð"y "x Þ sin cos þ xy ðcos2 sin2 Þ 1 0 0 2 x y

¼ 12 ½ð"x "y Þ sin 2 12 xy cos 2

"z0 ¼ "z ; The cubic equation for principal strains whose three roots give the distinct principal strains associated with three principal directions, is

y0 z0 ¼ z0 x0 ¼ 0

ð27-44eÞ ð27-44f Þ ð27-44gÞ

"3n ð"x þ "y þ "z Þ"2n 2 2 xy yz 2 þ "x "y þ "y "z þ "z "x zx "n 4 4 4 2 2 yz xy xy yz zx 2 "x "y "z "x "y zx "z þ ¼0 4 4 4 4 ð27-45Þ

The three strain invariants analogous to the three stress invariants

J1 ¼ "x þ "y þ "z ¼ first invariant of strain

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ð27-45aÞ

APPLIED ELASTICITY

27.15

APPLIED ELASTICITY

Particular

Formula

J2 ¼ "x "y þ "y "z þ "z "x

2 2 xy yz 2 zx 4 4 4

¼ second invariant of strain J3 ¼ "x "y "z

ð27-45bÞ

2 2 2 "x yz "y zx "z xy xy yz zx þ 4 4 4 4

¼ third invariant of strain

ð27-45cÞ

BOUNDARY CONDITIONS The components of the surface forces Fsfx and Fsfy per unit area of a small triangular prism pqr so that the side qr coincides with the boundary of the plate ds (Fig. 27-10)

Fsfx ¼ lx þ mxy ;

Fsfy ¼ my þ lyx

ð27-46Þ

where l and m are the direction cosines of the normal N to the boundary

O

x

p

q Fsfx

r qr = ds

COMPATIBILITY The six strain equations of compatibility

ds

N

y Fsfy

FIGURE 27-10 Area of a small triangular prism pqr.

@ 2 xy @ 2 "x @ 2 "y ¼ þ 2 @x @y @y2 @x

ð27-47aÞ

@ 2 yz @ 2 "y @ 2 "z ¼ 2 þ 2 @y @z @z @y

ð27-47bÞ

@ 2 zx @ 2 "z @ 2 "x ¼ þ 2 @z @x @x2 @z 2 @yz @zx @xy @ "x @ 2 ¼ þ þ @y @z @x @x @y @z @ 2 "y @ @yz @zx @xy 2 ¼ þ @z @x @y @x @y @z 2 @ "z @ @yz @zx @xy 2 ¼ þ @x @y @z @x @y @z

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ð27-47cÞ ð27-47dÞ ð27-47eÞ ð27-47f Þ

APPLIED ELASTICITY

27.16

CHAPTER TWENTY-SEVEN

Particular

The volume dilatation of rectangular parallelopiped element subject to hydrostatic pressure whose sides are l1 , l2 and l3

Formula

Vf V V ¼ V V where

e¼

ð27-48Þ

Vf ¼ ﬁnal volume after straining of element ¼ l1f l2f l3f V ¼ initial volume of element ¼ l1 l2 l3 The ﬁnal dimensions of the element after straining

Substituting the values of l1f , l2f , l3f , l1 , l2 , l3 in Eq. (27-48) and after neglecting higher order terms of strain

l1f ¼ l1 ð1 þ "1 Þ

ð27-49aÞ

l2f ¼ l2 ð1 þ "2 Þ

ð27-49bÞ

l3f ¼ l3 ð1 þ "3 Þ

ð27-49cÞ

e¼

l1 l2 l3 ð1 þ "1 Þð1 þ "2 Þð1 þ "3 Þ l1 l2 l3 l1 l2 l3

¼ "1 þ "2 þ "3 ¼ J1 1 2v ð1 þ 2 þ 3 Þ ¼ E

ð27-50aÞ ð27-50bÞ

V 3ð1 2vÞ0 ¼ ¼ 0 V E K where K ¼ bulk modulus of elasticity

If hydrostatic pressure (0 ) or uniform compression is applied from all sides of an element such that x ¼ y ¼ z ¼ 0 ¼ 1 ¼ 2 ¼ 3 , xy ¼ yz ¼ zx ¼ 0, Eq. (27-48) becomes

e¼

The bulk modulus of elasticity

K¼

E ¼ 0 e 3ð1 2vÞ

ð27-52Þ

K¼

2ð1 þ vÞG 3ð1 2vÞ

ð27-53Þ

GENERAL HOOKE’S LAW

ð27-51Þ

The general equation for strain in x-direction according to general Hooke’s law in case of anisotropic or non-homogeneous and non-isotropic materials such as laminate, wood and ﬁber-ﬁlled-epoxy materials as a linear function of each stress

"x ¼ S11 x þ S12 y þ S13 z þ S14 xy þ S15 yz

For relationships between the elastic constants

Refer to Table 27-1.

The three-dimensional stress-strain state in anisotropic or non-homogeneous and non-isotropic material such as laminates, ﬁber ﬁlled epoxy material by using generalized Hooke’s law which is useful in designing machine elements made of composite material (Fig. 27-1c) Note: ½S matrix is the compliance matrix which gives the strain-stress relations for the material. The inverse of the compliance matrix is the stiﬀness matrix and the stress-strain relations. If no symmetry is assumed, there are 92 ¼ 81 independent elastic constants present in the compliance matrix [Eq. (27-55)]

þ S16 zx þ S17 xz þ S18 zy þ S19 yz

3 2S 11 "x 6 6 S 7 6 6 "y 7 21 7 6 6 7 6 6 6 "z 7 6 6 7 6 6 6 7 6 xy 7 6 6 7 6 6 7 6 6 yz 7¼6 7 6 6 6 7 6 6 zx 7 6 7 6 6 6 7 6 6 xz 7 6 7 6 6 6 zy 7 6 6 5 4 4 yx S 2

91

ð27-54Þ

3 x 7 76 6 7 S29 7 76 y 7 76 7 6 z 7 7 76 7 76 7 6 7 7 76 xy 7 76 7 6 7 7 76 yz 7 76 7 6 zx 7 7 76 7 76 7 6 7 7 76 xz 7 76 7 7 6 54 zy 7 5

S92

S93

S94

S95

S96

S97

S98

S99

S12

S13

S14

S15

S16

S17

S18

S22

S23

S24

S25

S26

S27

S28

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S19

32

yx

ð27-55Þ

APPLIED ELASTICITY

27.17

APPLIED ELASTICITY

Particular

Formula

TABLE 27.1 Relationships between the elastic constants equals

a equals

E equals

v equals

K equals

, a

–

–

ð3 þ 2Þ

þ

2ð þ Þ

3 þ 2 3

, E

–

b

–

b

, v

–

ð1 2vÞ 2v

ð1 þ vÞð1 2vÞ v

–

ð1 þ vÞ 3v

, K

–

3ðK Þ 2

9KðK Þ 3K

3K

–

, E

2 E E 3

–

–

E 2 2

E 3ð3 EÞ

, v

2v 1 2v

–

2ð1 þ vÞ

–

2ð1 þ vÞ 3ð1 2vÞ

, K

3K 2 3

–

9K 3K þ

3K 2 2ð3K þ Þ

–

E, v

vE ð1 þ vÞð1 2vÞ

E 2ð1 þ vÞ

–

–

E 3ð1 2vÞ

K, E

3Kð3K EÞ 9K E 3Kv 1þv

3EK 9K E

–

3K E 6K

–

3Kð1 2vÞ 2ð1 þ vÞ

3Kð1 2vÞ

–

–

v, K

A þ ðE 3 Þ 4

A ðE þ Þ 4

b

A þ ð3 þ EÞ 6

a

¼G ¼ modulus of rigidity/shear. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A ¼ E 2 þ 2 E þ 9 2 . Courtesy: Dally, J. W. and William F. Riley, Experimental Stress Analysis, McGraw-Hill Publishing Company, New York, 1965. b

Equation (27-55) can be written as given here under Eq. (27-56) with the following use of change of notations and principle of symmetrical matrix in case of stiﬀness matrices xy ¼ yx

"12 ¼ "21

S12 ¼ S21

yz ¼ zy

"23 ¼ "32

S13 ¼ S31

xz ¼ zx

"13 ¼ "31

etc

2

"x

3

2

"1

3

2

S11

6 7 6 7 6 6 "y 7 6 "2 7 6 S21 7 6 7 6 6 7 6 7 6 6 6 " z 7 6 "3 7 6 7 6 7 6 6 7¼6 7¼6 6 6 yz 7 6 "4 7 6 7 6 7 6 6 7 6 7 6 6 6 zx 7 6 "5 7 6 5 4 5 4 4 xy S61 "6

S12

S13

S14

S15

S22

S23

S24

S25

S62

S63

S64

S65

S16

32

x

3

7 76 7 6 S26 7 76 y 7 76 7 7 6 7 76 z 7 76 7 7 6 76 yz 7 7 76 7 6 xz 7 7 54 5 xy S66 ð27-56Þ

and the following changes in Eq. (27-54) x ¼ 1

yz ¼ 4 ¼ 23

"x ¼ "1 2yz ¼ "4 ¼ 23

y ¼ 2

xz ¼ 5 ¼ 13

"y ¼ "2

2xz ¼ "5 ¼ 13

z ¼ 3

xy ¼ 6 ¼ 12

"z ¼ "3

2xy ¼ "6 ¼ 12

a Courtesy: Extracted from Ashton, J. E., J. C. Halpin, and P. H. Petit, Primer on Composite Materials—Analysis, Technomic Publishing Co., Inc., 750 Summer Street, Stamford, Conn. 1969.

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APPLIED ELASTICITY

27.18

CHAPTER TWENTY-SEVEN

Particular

Formula

The general stress-strain equations under linear stress-strain relationship

x ¼ K11 "x þ K12 "y þ K13 "z þ K14 xy þ K15 yz þ K16 zx y ¼ K21 "x þ K22 "y þ K23 "z þ K24 xy þ K25 yz þ K26 zx z ¼ K31 "x þ K32 "y þ K33 "z þ K34 xy þ K35 yz þ K36 zx xy ¼ K41 "x þ K42 "y þ K43 "z þ K44 xy þ K45 yz þ K46 zx yz ¼ K51 "x þ K52 "y þ K53 "z þ K54 xy þ K55 yz þ K56 zx zx ¼ K61 "x þ K62 "y þ K63 "z þ K64 xy þ K65 yz þ K66 zx ð27-57Þ where K11 to K66 are the coeﬃcients of elasticity of the material and are independent of the magnitudes of both the stress and the strain, provided the elastic limit of the material is not exceeded. There are 36 coeﬃcients of elasticity.

The stress-strain relationships for the case of isotropic material

x ¼ ð"x þ "y þ "z Þ þ 2"x

ð27-58aÞ

y ¼ ð"x þ "y þ "z Þ þ 2"y z ¼ ð"x þ "y þ "z Þ þ 2"z xy ¼ xy yz ¼ yz zx ¼ zx where

¼ Lame´’s constant ¼ G ¼ modulus of shear

The strain expressions from Eqs. (27-58a)

"x ¼

þ

ð þ z Þ 3 þ 2 x 2ð3 þ 2Þ y

"y ¼

þ

ð þ z Þ 3 þ 2 y 2ð3 þ 2Þ x

"z ¼

þ

ð þ x Þ 3 þ 2 z 2ð3 þ 2Þ y

xy ¼

1 1 ¼ xy G xy

yz ¼

1 1 ¼ yz G yz

zx ¼

1 1 ¼ zx G zx

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ð27-58bÞ

APPLIED ELASTICITY

27.19

APPLIED ELASTICITY

Particular

Formula

Mb

Element A

Mb A Mb

Mb σ2

τ12

σ1

τ12

FIGURE 27-10A Thin laminae of a composite laminate under bending.

The matrix expression from Eq. (27-55) for orthotropic material in a three-dimensional state of stress

The two-dimensional or a plane stress state matrix expression after putting 3 ¼ 23 ¼ 13 ¼ 0 and 23 ¼ 13 ¼ 0 and "3 ¼ S13 1 þ S23 2 in Eq. (27-59) for orthotropic material The stress-strain relationship for homogenous isotropic laminae of a laminated composite in the matrix form, which is assumed to be in state of plane stress

2

3 2 "1 S11 6 7 6 6 "2 7 6 S12 6 7 6 7 6 6 6 "3 7 6 S13 7 6 6 6 23 7 ¼ 6 0 7 6 6 7 6 6 4 13 5 4 0 0 12

S12 S22 S23 0

S13 S23 S23 0

0 0 0 S44

0 0 0 0

0 0

0 0

0 0

S55 0

3 1 7 7 76 7 6 76 2 7 76 7 76 3 7 76 7 76 23 7 76 7 76 7 0 54 13 5 S66 12 ð27-59Þ 0 0 0 0

32

where there are 9 independent constants in the above compliance matrix which is inverse of stiﬀness matrix 3 2 32 3 2 1 "1 S11 S12 0 7 6 76 7 6 ð27-60Þ 4 "2 5 ¼ 4 S21 S22 0 54 2 5 12 12 0 0 S66 3 2 K11 1 7 6 6 4 2 5 ¼ 4 K21 0 12 n 2

K12 K22 0

3 2 3 0 "1 7 6 7 0 5 4 "2 5 K66 n 12 n

ð27-61Þ

where K is stiﬀness matrix K11 ¼ K12 ¼ E=ð1 v2 Þ K12 ¼ vE=ð1 v2 Þ K66 ¼ E=2ð1 vÞ ¼ G Alternatively Eqs. (27-61) can be written for the nth layer of laminated composite, which is assumed to be in a state of plane stress

1 ¼ ð"1 þ v"2 Þ

E 1 v2

2 ¼ ð"2 þ v"1 Þ

E 1 v2

12 ¼ 12

E 2ð1 vÞ

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ð27-62Þ

APPLIED ELASTICITY

27.20

CHAPTER TWENTY-SEVEN

Particular

Substituting strain-displacement, Eqs. (27-40) and (27-41) into stress-strain Eqs. (27-33) and (27-37) or (27-39), displacement stress equation are obtained with from 15 unknowns to 9 unknowns

Formula

@u 1 ¼ ½ vðy þ z Þ @x E x

ð27-63Þ

@v 1 ¼ ½ vðz þ x Þ @y E y @w 1 ¼ ½ vðx þ y Þ @z E z @u @v 1 þ ¼ @y @x xy @v @w 1 þ ¼ @z @y yz @w @u 1 þ ¼ @x @z zx where ¼ G

Combining stress equation of equilibrium from Eqs. (27-11) with stress displacement Eqs. (27-63) (from 9 to 3 unknowns)

r2 u þ

1 @ 1 2v @x

r2 v þ

1 @ 1 2v @y

1 @ 1 2v @z

@u @v @w þ þ @x @y @z @u @v @w þ þ @x @y @z

þ

1 F ¼0 bx ð27-64Þ

þ

1 F ¼0 by

@u @v @w 1 þ þ þ Fbz ¼ 0 @x @y @z 2 @ @2 @2 þ þ where r2 is the operator @x2 @y2 @z2 1 @ 2 I1 v @Fbx @Fby @Fbz ¼ r 2 x þ þ þ @y @z 1 þ v @x2 1 v @x r2 w þ

Six stress equations of compatibility are obtained by making use of stress strain relations of Eqs. (27-33) and (27-39), the stress equations of equilibrium Eq. (27-11) and the strain compatibility Eq. (27-47) in three dimension in Cartesian system of coordinates

@Fbx ð27-65aÞ @x 1 @ 2 I1 v @Fbx @Fby @Fbz þ þ ¼ r 2 y þ @y @z 1 þ v @y2 1 v @x 2

@Fby ð27-65bÞ @y 1 @ 2 I1 v @Fbx @Fby @Fbz þ þ r 2 z þ ¼ @y @z 1 þ v @z2 1 v @x 2

@Fbz @z 2 1 @ I1 @Fbz @Fby ¼ þ r2 zy þ 1 þ v @x @y @y @x 2

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ð27-65cÞ ð27-65dÞ

APPLIED ELASTICITY

27.21

APPLIED ELASTICITY

Particular

Formula

@Fby @Fbz 1 @ 2 I1 ¼ þ @z @y 1 þ v @y @z 1 @ 2 I1 @Fbx @Fbz ¼ þ r2 zx þ 1 þ v @z @x @z @x r2 yz þ

ð27-65eÞ ð27-65f Þ

AIRY’S STRESS FUNCTION Diﬀerential equations of equilibrium for twodimensional problems taking only gravitational force as body force

@x @xy þ ¼0 @x @y

ð27-66aÞ

@y @yz þ þ g ¼ 0 @y @x r2 ¼ ðx þ y Þ ¼ 0 where r2 ¼

ð27-66bÞ

@2 @2 þ 2 2 @x @y

The stress components in terms of stress function and body force

x ¼

Substituting Eqs. (27-66c) for stress components into Eq. (27-66b) that the stress function must satisfy the equation

@4 @4 @4 þ 2 þ ¼0 @x4 @x2 @y2 @y4

The stress compatibility equation for the case of plane strain

r2 ðx þ y Þ ¼

If components of body forces in plane strain are

Substituting Eqs. (27-68) into Eqs. (27-11d), (27-11e) 2ð þ Þ 1 and Eq. (27-67) and taking ¼

þ 2 1v

By assuming that the stress can be represented by a @2 stress function such that x ¼ 2 þ , @y @2 @2 and substituting y ¼ 2 þ , and xy ¼ @x@y @x these into Eqs. (27-69) and Eq. (27-69c) becomes

@2 gy; @y2

Fbx ¼

@ ; @x

y ¼

@2 gy; @x2

2ð þ Þ

þ 2

Fby ¼

@2 @x @y ð27-66cÞ

xy ¼

ð27-72Þ @Fbx @Fby þ @x @y

@ @y

@x @xy @ þ ¼ @x @y @x @xy @y @ þ ¼ @x @y @y r2 x þ y ¼0 1v r4 ¼

1 2v 2 r 1v

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ð27-67Þ ð27-68Þ ð27-69aÞ ð27-69bÞ ð27-69cÞ ð27-70Þ

APPLIED ELASTICITY

27.22

CHAPTER TWENTY-SEVEN

Particular

Formula

Stresses for plane-stress can be obtained by letting v ! v in Eq. (27-70) and it becomes 1v

r4 ¼ ð1 vÞr2

If body forces are zero or constant then Eq (27-70) becomes

r4 ¼ @

The biharmonic Eq. (27-71a) can be written in expanded form as

ð27-71Þ

ð27-71aÞ

which is a biharmonic equation in and is a stress function @2 @4 @4 þ 2 þ ¼0 @x4 @x2 @y2 @y4

ð27-72Þ

The solution of a two-dimensional problem when the weight of body is the only body force reduces to ﬁnding a solution of Eq. (27-72) which satisﬁes boundary condition Eq. (27-46) of the problem.

CYLINDRICAL COORDINATES SYSTEM General equations of equilibrium in r, and z coordinates (cylindrical coordinates) taking into consideration body force (Figs. 27-13 to 27-15)

@r 1 @r @rz r þ þ þ þ FbR ¼ 0 @r @z r r @

ð27-73aÞ

@rz 1 @z @z rz þ þ þ þ Fbz ¼ 0 @r @z r r @

ð27-73bÞ

@r 1 @ @z 2r þ þ þ þ Fb ¼ 0 ð27-73cÞ @r @z r r @ where FbR , Fb and Fbz are body force components

z

θ r dθ

z ∂τ θz d ∂σz z dz ∂ + ∂z τ θz ∂τ zθ dθ ∂θ + τ zθ ∂σθ σθ + dθ ∂θ F ∂τ τrθ bz τrθ + rθ d θ ∂θ Fbθ τrθ τzr τrz Fb ∂τzr dr n + τ τθ z σ zr ∂r z ∂τθr τzθ dr τ θr + ∂r ∂σr dr dr σr + ∂r ∂τ τrz + rz d z ∂z σz +

σr dz

σθ

FIGURE 27-11 Element showing stresses in r, and in the axial direction.

σθ +

r dθ r

∂σθ dθ ∂θ

σr τθr

∂τ r θ dθ ∂θ ∂τ τθr + θr dr ∂τrz ∂r τrz + ∂z dz ∂σ σr + r dr ∂r F R τ τrθ +

rz

τrθ σθ

dr

FIGURE 27-12 Element showing stresses in r and directions.

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APPLIED ELASTICITY

27.23

APPLIED ELASTICITY

Particular

Formula

Mt

A A1

N τr θ

O

x

θ

τ θz

dθ 0

τrθ

(b)

σr

τrz

σr +

∂σr dr ∂r

∂σr dr ∂r ∂τr θ dr τr θ + ∂r ∂σθ dθ σθ + ∂θ

r

a dr

θ

∂σ σz + z dz ∂z

σθ

σr

τrz

ν r

u

rd

z dz

dθ

σz

a’

A v β

α

σr +

A’ Z

Mt

(a)

FIGURE 27-13

Equations of equilibrium for axial symmetry Eqs. (27-73) reduce to Eqs. (27-74) when there are body forces acting on the body

Equations of equilibrium in two dimension in r and coordinates (polar coordinates) taking into consideration body force components

Equations of equilibrium for an axially symmetrical stress distribution in a solid of revolution when there are no body forces acting on the body (Fig. 27-13), since the stress components are independent of .

∂u ∂θ dθ

y

b’

b u + ∂u dr ∂r v r ∂ν dr ∂r B

ν+ ∂ ν ∂θ dθ B’

FIGURE 27-14 Strain components in polar co-ordinates.

@r @rz r þ þ þ FbR ¼ 0 @r @z r

ð27-74aÞ

1 @ @z þ þ Fb ¼ 0 r @ @z

ð27-74bÞ

@rz 1 @z @z rz þ þ þ þ Fbz ¼ 0 r @ @r @z r

ð27-74cÞ

@r 1 @r 1 þ þ ðr Þ þ FbR ¼ 0 @r r @ r

ð27-75aÞ

1 @ @r 2r þ þ þ Fb ¼ 0 r @ @r r

ð27-75bÞ

@r @rz ðr Þ þ þ ¼0 @r @ r

ð27-76aÞ

@rz @z rz þ þ ¼0 @r @z r

ð27-76bÞ

STRAIN COMPONENTS (Fig. 27-14) The strain components in r, and z coordinates system The strain in the radial direction

"r ¼

@u @r

ð27-77aÞ

The strain in the tangential direction

" ¼

1 @v u þ r @ r

ð27-77bÞ

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APPLIED ELASTICITY

27.24

CHAPTER TWENTY-SEVEN

Particular

Formula

@w @z

The strain in the axial direction

"z ¼

The shear strains

r ¼

1 @u @v v þ r @ @r r

ð27-77dÞ

z ¼

1 @w @v þ r @ @z

ð27-77eÞ

ð27-77cÞ

@u @w þ @z @r 1 @v v 1 @u þ r ¼ 2 @r r r @ 1 1 @w @v z ¼ 2 r @ @z 1 @u @w zr ¼ 2 @z @r

ð27-77f Þ

zr ¼

The rotation of the element in the counter clock-wise direction in the r, z and zr planes

ð27-78aÞ ð27-78bÞ ð27-78cÞ

AIRY’S STRESS FUNCTION IN POLAR COORDINATES When components of body force Fbr and Fb are zero, Eqs. (27-74a) and (27-74b) are satisﬁed by assuming stress function for r , and r

r ¼

1 @ 1 @ 2 þ r @r r2 @2

ð27-79aÞ

¼

@2 @r2

ð27-79bÞ

@ r ¼ @r The stress equation of compatibility Eq. (27-72) in terms of Airy’s stress function referred to Cartesian coordinates x and y, has to be transferred to Airy’s stress function referred to polar coordinates r and system. In this transformation from x and y coordinates transform to r and coordinates

1 @ r @

r2 ¼ x2 þ y2 ;

¼

1 @ 1 @ 2 r2 @ r @r @

¼ tan1

y x

ð27-79cÞ ð27-80Þ

from which @r x ¼ ¼ cos ; @x r

@r y ¼ ¼ sin @y r

@ y sin ¼ 2¼ ; @x r r

@2 @2 þ 2 2 @x @y

@ x cos ¼ ¼ @y r2 r @2 @2 þ @x2 @y2

ð27-81Þ

Eq. (27-72) can be written as

r2 ¼

Using Eqs. (27-79) and (27-80) and transforming Eq. (27-72) into stress equation of compatibility in polar coordinates r and system

@ 2 @ 2 @ 2 1 @ 1 @ 2 þ ¼ 2þ þ r @r r2 @2 @x2 @y2 @r

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ð27-82Þ ð27-83Þ

APPLIED ELASTICITY APPLIED ELASTICITY

Particular

Stress equation of compatibility in terms of Airy’s stress function in polar coordinate r and is obtained by substituting (Eq. (27-83) to Eq. (27-82))

27.25

Formula

@2 1 @ 1 @2 þ þ @r2 r @r r2 @2 2 @ 1 @ 1 @ 2 þ þ @r2 r @r r2 @r2

r4 ¼

ð27-84Þ

SOLUTION OF ELASTICITY PROBLEMS USING AIRY’S STRESS FUNCTION Any Airy’s stress function either in Cartesian coordinates or polar coordinates used in solving any two-dimensional problems must satisfy Eqs. (27-66) and (27-72) in Cartesian coordinates and Eqs. (27-79) and (27-84) in polar coordinates and boundary conditions (27-46)

Cartesian coordinates Solutions of many two-dimensional problems can be found by assuming Airy’s stress function in terms of polynomial and Fourier series, which are

1 ¼ a1 x þ b1 y

first degree polynomial

ð27-85Þ

2 ¼ a2 x2 þ b2 xy þ c2 y2 second degree polynomial

ð27-86Þ

3 ¼ a3 x3 þ b3 x2 y þ c3 xy2 þ d3 y3 third degree polynomial

ð27-87Þ

4 ¼ a4 x4 þ b4 x3 y þ c4 x2 y2 þ d4 xy3 þ e4 y4 fourth degree polynomial

ð27-88Þ

5 ¼ a5 x5 þ b5 x4 y þ c5 x3 y2 þ d5 x2 y3 þ e5 xy4 þ f5 y5 fifth degree polynomial ð27-89Þ ¼ sin

m x f ðyÞ l

Fourier series

ð27-90Þ

where m is an integer ¼

nX ¼1

an cos

n¼0

¼1 n x nX n x bn sin þ l l n¼1

where n is an integer a0 ¼

1 2l

an ¼

1 l

ð 2l 0

ð 2l 0

dx cos

n x dx if n 6¼ 0 l

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ð27-91Þ

APPLIED ELASTICITY

27.26

CHAPTER TWENTY-SEVEN

Particular

Formula

bn ¼

1 l

ð 2l 0

sin

n x dx l

is any periodic function of x, which represents itself at interval of 2l

Polar coordinates

(

)

r4 ¼ 0 is a fourth order biharmonic partial diﬀerential equation. The fourth order diﬀerential equation can be obtained by using a function in r4 ¼ 0 which in term gives four diﬀerent stress functions

cos n n ¼ Rn ðrÞ sin n

One of the stress function for solving many problems in polar coordinates

1 ¼ A ln r þ Br2 ln r þ Cr2 þ D

The second order stress function 2

sin 2 ¼ ðA1 r þ B1 =r þ C1 r þ D1 r ln rÞ cos

ð27-92Þ

ð27-93Þ (

)

3

ð27-94Þ ) sin n n ¼ ðAn rn þ Bm =rn þ Cn r2 þ n þ Dn r2 n Þ cos n (

The third order stress function 3

ð27-95Þ The fourth order stress function

m ¼ Am þ Bm r2 þ Cm r sin þ Dm r cos ð27-96Þ It is sometimes diﬃcult to select a stress function for solving a problem. But it is left to the discretion of the problem solver to select or decide the correct stress function to suit the problem under consideration.

The general expression for the stress function which satisfy boundary conditions and compatibility Eq. (27-84)

¼ A0 þ B0 r2 þ C0 r sin þ D0 r cos þ D00 þ C00 r2 þ B00 r2 ln r þ A00 ln r B þ A1 r þ 1 þ C1 r3 þ D1 r ln r sin r B0 þ A01 r þ 1 þ C10 r3 þ D01 r ln r cos r 1 X Bn Dn n nþ2 An r þ þ Cn r þ n 2 sin n þ rn r n¼2 1 X B0 D0 þ A0n rn þ n þ Cn0 rn þ 2 þ n n2 cos n rn r n¼2 ð27-97Þ

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APPLIED ELASTICITY APPLIED ELASTICITY

Particular

In a general case the loading can be represented by the trigonometric series

Formula

q ¼ A0 þ

1 X m¼1

þ B0 þ

Am cos

1 X m¼1

The stress function can also be represented by

27.27

1 m x X m x A0m cos þ l l m¼1

Bm sin

1 m x X m x B0m cos þ l l m¼1 ð27-98Þ

¼ ðA ey þ B ey þ Cy ey þ Dy ey Þ sin x ð27-99Þ

APPLICATION OF STRESS FUNCTION Thick cylinder Stress function used in this case, Eq. (27-93)

¼ A ln r þ Br2 ln r þ Cr2 þ D

Boundary conditions are

ðr Þr ¼ di =2 ¼ pi

Equation of equilibrium used in this problem

@r r þ ¼0 @r r

and

ð27-93Þ

ðr Þr ¼ d0 =2 ¼ po

ð27-100aÞ ð27-100bÞ

Since it is a case of problem of symmetry with respect to axis of cylinder and no body force acting on it. r ¼

pi di2 po do2 di2 do2 ð pi po Þ 2 2 do2 di2 4r ðdo di2 Þ

ð27-101aÞ

The expression for tangential stress in thickness wall of thick cylinder under pressures po and pi at any radius r

¼

pi di2 po do2 di2 do2 ð pi po Þ þ 2 2 4r ðdo di2 Þ do2 di2

ð27-101bÞ

The shear stress

r ¼ 0

The expression for radial stress in thickness wall of thick cylinder under external pressure (po ) and internal pressure (pi ) at any radius r

Expression for displacement of an element in the thickness wall of cylinder at any radius r in radial direction and tangential direction respectively

u¼

1 d 2d 2ð p p Þ ð1 þ vÞ i o 2 o 2 i E 4rðdo di Þ 1 pi di2 po do2 þ ð1 vÞ r do2 di2

ð27-101cÞ

¼0

ð27-101dÞ ð27-101eÞ

Curved bar under pure bending (Fig. 27-15) Stress function used in this problem Eq. (27-93)

¼ A ln r þ Br2 ln r þ Cr2 þ D

Boundary conditions

ðr Þr ¼ do =2; di =2 ¼ 0

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ð27-93Þ ð27-102aÞ

APPLIED ELASTICITY

27.28

CHAPTER TWENTY-SEVEN

Particular

Formula

ð do =2

y

di =2

l

ð do =2 di =2

Mb

r dr ¼ Mb

ð27-102bÞ

dr ¼ 0

ð27-102cÞ

ðr Þr ¼ do =2; di =2 ¼ 0

ð27-102dÞ

Equation of equilibrium used in this problem of symmetry with respect to the xy-plane perpendicular to axis of the bar

@r r þ ¼0 @r r

ð27-100bÞ

The expression for the radial stress component in the bar at r radius

4Mb r ¼

Mb

r0

d di 0 2 2

O

x

FIGURE 27-15

do2 di2 do do2 2r di2 di ln þ ln þ ln 4 do 4 2r 16r2 di

ð27-103Þ

4Mb do2 di2 do do2 2r di2 di ln þ ln þ ln

4 do 4 2r 16r2 di 1 þ ðdo2 di2 Þ ð27-104Þ 4

The expression for the tangential stress component in the bar at r radius

¼

The expression for shear stress component

r ¼ 0

ð27-105aÞ

where

¼

ðdo2 di2 Þ2 1 2 2 d 2 do di ln o 16 di 4

ð27-105bÞ

STRESS DISTRIBUTION IN A FLAT PLATE WITH HOLES OR CUTOUTS UNDER DIFFERENT TYPES OF LOADS C Ar2 þ Br4 þ 2 þ D cos 2 r

An inﬁnite ﬂat plate with centrally located circular cutout or hole subject to uniform uniaxial tension (Fig. 27-16)

¼

The expression for stress function

ðr Þr ¼ b ¼ cos2 ¼ 12 ð1 þ cos 2Þ

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ð27-106Þ ð27-107aÞ

APPLIED ELASTICITY APPLIED ELASTICITY

Particular

27.29

Formula

ðr Þr ¼ b ¼ 12 sin 2 ð27-107bÞ " # a2 4a2 3a4 r ¼ 1 2 þ 1 2 þ 4 cos 2 2 r r r

Boundary conditions are The radial stress in an inﬁnite plate with a centrally located circular hole (cut-out) subject to uniform uniaxial tension at inﬁnity (Fig. 27-16)

"

The tangential stress in an inﬁnite plate with centrally located circular hole (cutout) under uniform uniaxial tension at inﬁnity

¼

The shear stress in an inﬁnite ﬂat plate with a centrally located circular cutout (hole) subject to uniform uniaxial tension at inﬁnity

r ¼

2

# a2 3a4 1 þ 2 1 þ 4 cos 2 r r

2a2 3a4 1 þ 2 4 sin 2 2 r r

The stress concentration factor

For distribution of tangential stress around circle of hole under uniform uniaxial tension

Refer to Fig. 27-18.

For superposition of stresses in a ﬂat plate with a centrally located circular hole subject to tension, compression and uniform pressure

Refer to Fig. 27-19.

The shear stress around hole at ¼ =2 or 3 =2

ðr Þr ¼ a ¼ 0

ð27-109Þ ð27-110Þ

a2 3a4 ð Þr ¼ a ¼ 2þ 2 þ 4 2 r r 1 a2 3a4 K ¼ ¼ ¼3 2þ 2 þ 4 2 r r r¼a

The tangential stress at hole boundary at ¼ =2 or 3 =2

ð27-108Þ

ð27-111aÞ ð27-111bÞ

ð27-111cÞ

σ

y

σ y

2a

=

P

q =

Q

q

σ

σ FIGURE 27-16 A large ﬂat plate with a centrally located circular hole under uniform uniaxial stress at inﬁnity.

3σ

a S

R

σ h

σ

σ

w

q

=

θ

2a

x

q

σmax

σ

σ

x σ

=

60

σ

σ

σ FIGURE 27-17 A large ﬂat plate with circular hole under pure shear stress.

σ FIGURE 27-18 Distribution of stress around the boundary of circular cutout (hole).

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APPLIED ELASTICITY

27.30

CHAPTER TWENTY-SEVEN

Particular

Formula

σ 2

σ 2

σ

τ=

2a

σ 2

τ=

σ 2

2a =

2a

σ 2

σ σ + 2 2

τ=

σ 2

τ=

σ 2

σ 2

σ 2

σ

σ 2

FIGURE 27-19 Principle of superposition.

Pure shear An inﬁnite ﬂat plate with centrally located circular cutout or hole subject to uniform uniaxial tension and compression (i.e., pure shear) (Fig. 27-17) C Ar2 þ Br4 þ 2 þ D sin 2 r

The expression for stress function

¼

Boundary conditions are

ðr Þ1 ¼ ð Þ1

ð27-113aÞ

r ¼ q sin 2

ð27-113bÞ

r ¼ q cos 2

ð27-113cÞ

ð27-112Þ

where q ¼ shear load

The tangential stress in an inﬁnite plate with a centrally located circular hole subject to uniform tensile and compressive stresses as shown in Fig. 27-17 The radial stress

The shear stress

ðr Þr ¼ a ¼ ðr Þr ¼ a ¼ 0 3a4 ¼ q 1 þ 4 sin 2 r 4a2 3a4 r ¼ q 1 2 þ 4 sin 2 r r 2a2 3a4 r ¼ q 1 þ 2 4 cos 2 r r

The tangential stress

¼ 2 cos 2 ½ 2 cosð2 Þ

The maximum tangential stress at ¼ =2 or 3 =2 i.e., at P and Q (Fig. 27-17)

ð Þ ¼ =2 or 3 =2; r ¼ a ¼ 4

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ð27-113dÞ ð27-114Þ

ð27-115Þ ð27-116Þ ð27-117Þ ð27-118aÞ

APPLIED ELASTICITY APPLIED ELASTICITY

Particular

27.31

Formula

The maximum tangential stress at ¼ 0 or ¼ , i.e., at R and S (Fig. 27-17) The stress concentration factor

ð Þ ¼ 0 or ; r ¼ a ¼ 4

K ¼

max ¼

ð27-118bÞ

3a4 1þ 4 ¼4 r r¼a

ð27-118cÞ

Bi-axial tension (Fig. 27-20) An inﬁnite ﬂat plate with centrally located circular hole (cutout) under biaxial uniform tension (Fig. 27-20) The radial stress at hole boundary

ðr Þr ¼ a ¼ 0

ð27-119aÞ

The shear stress at hole boundary

ðr Þr ¼ a ¼ 0

ð27-119bÞ

The tangential stress in an inﬁnite ﬂat plate with a centrally located circular hole subject to uniform biaxial tensile stress at inﬁnity

¼ 2 cos 2 ½ 2 cosð2 Þ

The stress concentration factor

¼ 2

ð27-120Þ ð27-120aÞ

K ¼

2 ¼2

ð27-120bÞ

K ¼

max nom

ð27-121aÞ

Finite plate (Fig. 27-21) Uniaxial tension (Fig. 27-21)

σ

2a σ

θ

σ

Stress concentration factor, K ’

3.0

F

FIGURE 27-20 Biaxial tension.

d s

F

w

σmax

2.6

2.4

2.2

2.0 σ

q

2.8

h

P

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 Ratio of d/w

FIGURE 27-21 Stress concentration factor for a plate of ﬁnite width with a circular hole (cutout) in tension (Howland).

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APPLIED ELASTICITY

27.32

CHAPTER TWENTY-SEVEN

Particular

The stress distribution in a ﬂat plate of ﬁnite width w with a centrally located small circular hole according to Howland11 , which can be expressed in terms of stress concentration around hole

Formula

K ¼

max d 1 w

ð27-121bÞ

where nom ¼

F ¼ ðw dÞh 1 ðd=wÞ

ð27-121cÞ

¼ F=wh ¼ stress at the end of inﬁnite plate The tangential stress at the points of q and s when w ¼ 2d according to Howland

¼ 4:3

ð27-121dÞ

The tangential stress at the point P according to Howland11

¼ 0:75

ð27-121eÞ

From Eqs. (27-75a) and (27-75b), which can be made use of for a rotating disk of uniform thickness with z-axis perpendicular to the xy-plane and stress components do not depend on . Hence Eqs. (27-75a) taking a body force equal to inertia force i.e., FbR ¼ !2 r, becomes

@r 1 þ ðr Þ þ FbR ¼ 0 @r r

ð27-122aÞ

Equation of force equilibrium from Eqs. (27-122a) after substituting value of FbR becomes

r

@r þ r þ !2 r2 ¼ 0 @r

ð27-122bÞ

It can be seen from Fig. 27-21 that maximum stress which is at the hole boundary, decreases very rapidly and approach the value of average stress at the edge of the inﬁnite plate

ROTATING SOLID DISK WITH UNIFORM THICKNESS (Fig. 27-22)

@ðrr Þ þ !2 r2 ¼ 0 @r where

ð27-122cÞ

FbR ¼ body force per unit volume ¼ !2 r F ¼ rr ¼ stress function ¼ density, kg/m3 ! ¼ angular velocity, rad/s Stress is a function of r only because of symmetry Equation of compatibility

r

Using compatibility equation (27-123) and Hooke’s law after simpliﬁcation, the expression for force equilibrium (27-122b) becomes

r2

@" þ " ¼ "r @r @2F @F þr F ¼ ð3 þ vÞ!2 r3 @r @r2

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ð27-123Þ ð27-124aÞ

APPLIED ELASTICITY APPLIED ELASTICITY

Particular

O

27.33

Formula

x

θ dθ b r

σθ

σr

ω a dr

ρω 2r

σθ y

σ

b ∂σ ∂r r d

r+

r

FIGURE 27-22 Element of a rotating disk.

FIGURE 27-23

The general solution of Eq. (27-124a), when r ¼ e is substituted in it, becomes

F ¼ Ar þ

B !2 r3 ð3 þ vÞ ð27-124bÞ 8 r where A and B are constants of integration to be found by boundary conditions

Boundary conditions (a) The radial stress at outer boundary of rotating disc of radius b

ðr Þr ¼ b ¼ 0

ð27-125aÞ

(b) stress at center of rotating disc

ðr Þr ¼ 6¼ 0

ð27-125bÞ

The expression for radial stress at any radius r

3þv !2 ðb2 r2 Þ 8 3þv 1 þ 3v !2 r2 !2 b2 ¼ 8 8

The expression for tangential stress as any radius r

r ¼

ROTATING DISK WITH A CENTRAL CIRCULAR HOLE OF UNIFORM THICKNESS, Fig. 27-23

ðr Þr ¼ a ¼ 0

Boundary conditions

ðr Þr ¼ b ¼ 0

Using force equilibrium Eq. (27-124b) and boundary conditions Eqs. (27-128) the tangential and radial stresses at any radius r are

ð27-126Þ ð27-127Þ

ð27-128aÞ

ð27-128bÞ # 3þv a2 b2 1 þ 3v 2 2 2 2 ¼ ! a þ b þ 2 r 8 3þv r "

ð27-129Þ r ¼

3þv a2 b2 !2 a2 þ b2 2 r2 8 r

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ð27-130Þ

APPLIED ELASTICITY

27.34

CHAPTER TWENTY-SEVEN

Particular

Formula

3þv !2 ðb aÞ2 8 " # 3þv 1v 2 2 2 ¼ ! b þ a 8 3þv

The expression for maximum radial stresses which pﬃﬃﬃﬃﬃ occurs at r ¼ ab

r max ¼

ð27-131aÞ

The expression for maximum tangential stress which occur at r ¼ a

max

ð27-131bÞ

Rotating disk as a three-dimensional problem The diﬀerential equations of equilibrium from Eqs. (27-76) when body force which is an inertia force (centrifugal force) is included, becomes

After substituting the body forces Fbx ¼ !2 x, Fby ¼ !2 y, Fbz ¼ 0 in Eqs. (27-65) and the last three equations containing shearing stress components remain the same as in Eqs. (27-65), and the ﬁrst three equations in polar coordinates become

@r @rz r þ þ þ !2 r ¼ 0 @r @z r

ð27-132aÞ

@rz @z rz þ þ ¼0 @r @z r

ð27-132bÞ

2 1 @2I 2!2 ð27-133aÞ ðr Þ þ ¼ 2 2 1v 1 þ v @r r

r 2 r r 2 þ

r 2 z þ

2 1 1 @I 2!2 ¼ ð Þ þ r 1v 1 þ v r @r r2 ð27-133bÞ 1 @2I 2v!2 ¼ 2 1v 1 þ v @z

ð27-133cÞ

The equations of shearing stress components in Eqs. (27-65) remain the same without any change even when the body forces are acting. The solution of Eq. (27-132) consists of a particular solution and complementary function. The particular solution call be obtained by assuming

r ¼ Br2 þ Dz2 ;

z ¼ Ar2 ;

¼ Cr2 þ Dz2 ;

rz ¼ 0

The complementary solution is obtained by assuming a stress function, which has to satisfy boundary conditions, compatibility equations and having a form of a polynomial of the ﬁfth degree

¼ a5 ð8z5 40r2 z2 þ 15r4 zÞ

The particular solution

r ¼

þ b5 ð2z5 r2 z3 3r4 zÞ !2 2 !2 ð1 þ 2vÞð1 vÞ 2 r z 3 6vð1 vÞ

dr

r

z ¼ !2 h

1 þ 3v 2 r 6v

ðaÞ

ðbÞ

ðcÞ ðdÞ

b

FIGURE 27-24 Rotating disc of variable thickness.

The complementary function obtained from assuming stress function Eq. (b)

¼

!2 ð1 þ 2vÞð1 þ vÞ 2 z 6vð1 vÞ

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ðeÞ

APPLIED ELASTICITY

27.35

APPLIED ELASTICITY

Particular

Formula

r ¼ !2 ¼ !2

vð1 þ vÞ 2 3 þ v 2 z þ r 2ð1 vÞ 8

1 þ 3v 2 vð1 þ vÞ 2 r þ z 8 2ð1 vÞ

ðfÞ ðgÞ

!2 vð1 þ vÞ c2 ð3 þ vÞa2 þ !2 8 2ð1 vÞ 3

An expression for uniform radial tension on the disk which is superposed on the resultant stresses to express the resultant radial compression along the boundary

Tr ¼

The ﬁnal expressions for stress components

r ¼ !2

ðhÞ

3þv 2 vð1 þ vÞ 2 ða r2 Þ þ ðc 3z2 Þ 8 6ð1 vÞ

ð27-134aÞ

1 þ 3v 2 r 8 vð1 þ vÞ 2 ðc 8z2 Þ þ 6ð1 vÞ

¼ r!2

z ¼ 0;

3þv 2 a 8

rz ¼ 0

ð27-134bÞ ð27-134cÞ

For disk of uniform strength rotating at ! rad/s

Refer to Chapter 10, Eq. (10-1) and Fig. 10-1.

For solid cylinder rotating at ! rad/s, hollow cylinder rotating at ! rad/s and solid thin uniform disk rotating at ! rad/s under external pressure

Refer to Chapter 10, Eqs. (10-9) to (10-25) and Figs. 10-2 and 10-3.

For asymmetrically reinforced circular holes/cutouts in a ﬂat plate subject to uniform uniaxial tensile force/stress

Refer to Chapter 4, Figs. 4-7(a), 4-7(b) and 4-7(c).

ROTATING DISK OF VARIABLE THICKNESS (Fig. 27-24) The equation of force equilibrium in case of rotating disk of variable thickness from Eq. (27-122b)

d ðrhr Þ h þ 2 !rh ¼ 0 dr

Using rhr ¼ F and thickness variation as h ¼ Cr , Hooke’s law, r ¼ e and Eq. (27-123), Eq. (27-135) become

r2

ð27-135Þ

@2F @F þ ð1 þ Þr ð1 þ v ÞF 2 @r @r ¼ ð3 þ vÞ2 !Cr3

ð27-136aÞ

@2F @F ð1 þ v ÞF þ @ @2 ¼ ð3 þ vÞ!2 C eð3 Þ where C and are constants

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ð27-136bÞ

APPLIED ELASTICITY

27.36

CHAPTER TWENTY-SEVEN

Particular

Boundary conditions

Formula

ðr Þr ¼ b ¼ 0

ð27-137aÞ

ðr Þr ¼ 0 6¼ 1

ð27-137bÞ

ð Þr ¼ 6¼ 1 The expression for radial stress

ð27-137bÞ " 2 #

1 þ 1 3þv r r r ¼ !2 b 8 ð3 þ vÞ b b ð27-138Þ

The expression for tangential stress

¼

3þv !2 b2 8 ð3 þ vÞ " # r 1 þ 1 1 þ 3v r 2 1 b 3þv b

where

¼ 2

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 2 ð1 þ v Þ 2

ð27-139aÞ

ð27-139bÞ

1 and 2 are roots of Eq. (27-139b) For a symmetrically reinforced circular cutout in a ﬂat plate under uniform uniaxial tension according to the analysis of Timoshenko

Refer to Fig. 27-25.

NEUTRAL HOLES (MANSFIELD THEORY) Reinforced holes which do not aﬀect the stress distribution in a plate are said to be neutral

Resolving the forces in x-direction, and using stress @2 @2 function for stresses as x ¼ z , y ¼ 2 and @y @x @2 xy ¼ and after integrating, an expression is @x@y obtained as Resolving the forces in y-direction after performing integration etc. as done under Eq. (a), another expression is obtained

There is negligible bonding on the reinforcement since it is thin compared to the radius of the curvature of the hole, and shear across the section of reinforcement is zero. F @ cos ¼ þB ðaÞ t @y

F sin t

From Eqs. (a) and (b) tan

Eq. (c) may be written as

¼

@ þA @x

@ þA dy ¼ @x ¼ @ dx þB @y

@ @ dy þ dx þ B dy þ A dx ¼ 0 @y @x

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ðbÞ

ðcÞ

ðdÞ

APPLIED ELASTICITY APPLIED ELASTICITY

Particular

27.37

Formula

3.0 ANALYSIS OF TIMOSHENKO di = 0.2 w

Stress - concentration Factor, Kσ

σmax σmax di KσB = (1 β) 1 = σnom w σ

2.0

Where Bend cross-section (do-di) (H-h) β= = dih Hole cross-section

KσB H h

X

KσB σ

do di

1.0

σ 1.0

σmax B

0

0

0.1

0.2

0.3

Section X-X

0.4

0.5

X

0.6

B

0.7

0.8

0.9

0 1.0

FIGURE 27-26 Stress-concentration factor KB , for a symmetrically reinforced circular hole (cutout) in a ﬂat plate in tension.

Integrating Eq. (c) an expression for is obtained as

þ By þ Ax þ C ¼ 0

ð27-140Þ

The term Ax þ By þ C can be included or excluded from the stress function without changing the stress distribution. These terms do not aﬀect the shape of the neutral hole but determines the position of the hole. By omitting A; B and C the shape of the neutral hole is given by stress function

¼0

ð27-141Þ

Compatibility Considering the displacements and strain of triangular element klm as shown in Fig. 27-26 x and y directions due to x , y and xy stresses and equating strain in the plate equal to strain in the reinforcement, an expression for area of reinforcement (AR ) for neutral hole is obtained as

Strain in the plate ¼ strain in the reinforcement

" #3=2 F @ 2 @ 2 AR ¼ ¼t þ E"t @x @y "( @ 2 @ 2 @ 2 @ 2 þ 2 @y @x2 @x @y ) ( @ 2 @ @ @ 2 @ 2 v 2 @x @y @x @y @x2 @y )#1 @ 2 @ 2 @ 2 @ @ 2 ð27-142Þ þ 2 @x @x @y @y @x @y

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APPLIED ELASTICITY

27.38

CHAPTER TWENTY-SEVEN

Particular

l σ 2a

m Element k

σ

Reinforcement y

F l

Formula

ψ

σy

τxy

dx

x

(a)

σ

2b

y

σx

dy

σ 2

σ 2

m τxy

2a x

x

2a

(b)

dz

y k

σ y

F + dF

FIGURE 27-26 (a) A reinforced circular hole under tension, (b) element at reinforcement under the action of normal stresses x and y and shear stress xy due to force F acting on the cross-section of the reinforcement.

(a)

(b)

FIGURE 27-27 Thin walled cylinder with a circular hole under stress.

Neutral hole in a thin walled cylinder (Fig. 27-27) The stress function may be taken as

For neutral hole

Eq. (27-144) becomes an ellipse whose minor axispis ﬃﬃﬃ 2r and ratio of major axis (2a) to minor axis (2b) is 2 : 1 Substituting stress function from Eq. (27-144a) in Eq. (27-142), the area of reinforcement

2 y2 þ Ax þ By þ C x þ 2 2 2 y2 ¼ x þ C 2 2 ¼

ð27-143aÞ ð27-143bÞ

if the origin of coordinates is taken at center of hole 2 y2 ð27-144aÞ x þ C ¼0 ¼ 2 2 y2 2r ¼ 0 2 pﬃﬃﬃ x2 3=2 rt 2 1 þ 2 r AR ¼ 3x2 1 2v þ 2 r where t ¼ thickness of plate x2 þ

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ð27-144bÞ

ð27-145Þ

APPLIED ELASTICITY APPLIED ELASTICITY

Particular

27.39

Formula

COMPLEX VARIABLE METHOD APPLIED TO ELASTICITY For equation of equilibrium in three-dimension and two-dimension

Refer to Eqs. (27-11a) to (27-11e).

The combinations of grouping of stress components are

¼ x þ y

ð27-146aÞ

¼ x y þ 2ixy

ð27-146bÞ

¼ xz þ iyz

ð27-146cÞ

z ¼ z

ð27-146dÞ

The body force complex potentials and components of body force complex potentials

Vðx; y; zÞ ¼ Uðz; z; zÞ Fbx ¼

The equations of equilibrium can be reduced to two equations using the stress combinations of Eqs. (27-146) and variable complex number

When the body force is zero Eq. (27-148) can be written as

@V ; @x

Fby ¼

@V ; @y

ð27-147Þ Fbz ¼

@V @z

@ @ @ ð þ 2UÞ þ þ ¼0 @ z @z @z

ð27-148aÞ

where z ¼ x þ iy, z ¼ x iy @ @ @ ð þ UÞ þ þ ¼0 @z z @z @ z @ @ @ þ þ ¼0 @ z @z @z @z @ @ þ þ ¼0 @z @ z @z

ð27-148bÞ ð27-149aÞ ð27-149bÞ

Stress strain relation The complex displacement

Stress-displacement equations by using Eqs. (27-146) and Eqs. (27-58)

D ¼ u þ iv

ð27-150Þ

@w @u @v @w @D @ D þ þ ¼ þ þ ð27-151aÞ @x @y @z @z @ z @z @D @ D @w þ þ 2v ð1 2vÞ ¼ 2G ð27-151bÞ @z @ z @z " # @D @ D @w þ þ ð1 vÞ ð1 2vÞz ¼ 2G v @z @z @z ð27-151cÞ ¼ 4G

@D @ z

@D @w ¼G þ2 @z @ z

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ð27-151dÞ ð27-151eÞ

APPLIED ELASTICITY

27.40

CHAPTER TWENTY-SEVEN

Particular

Formula

STRAIN COMBINATIONS ¼ "x þ "y

The strain combinations are

¼ "x "y þ ixy ¼ xz þ iyz ;

ð27-152Þ

D ¼ u þ iv

@D @ D ¼ þ @z @ z @D ¼2 @ z @D @w ¼ þ2 @ z @ z_ Strain transformation rules (Fig. 27-28)

D ¼ u þ iv;

D0 ¼ u0 þ iv0

D ¼ re ;

0

D ¼ De

i

y y’

0

y v

i

0 ¼

v’ u’

α

¼

x 0

FIGURE 27-28 Strain transformation.

Stress transformation rules

ð being invariantÞ

0 ¼ e2i

x

z

0 @D @ D @D0 @ D þ ¼ þ 0 ¼ 0 @z @ z @z @ z

¼ 0

x’

P (x,y) u

ð27-154Þ

i

z ¼ ze

D

θ

ð27-153Þ

¼

@D @w þ2 @z @z ei

@" @w @"0 ¼ ¼ @z @z @z

ðinvariantÞ

¼ 2ð þ GÞð"x þ "y Þ þ 2 "z ¼ 2ð þ GÞ þ 2 "z ¼ 2ð þ GÞ0 þ 2 "z0 ¼ 0

"

¼ 2G "x "y þ i

@u @v þ @y @x

ð27-155Þ

# ¼ 2G

0 ¼ 2G0 ¼ e2i " # @u @w @v @w ¼G þ þi þ @z @x @z @y ¼ Gðxz þ iyz Þ ¼ G 0 ¼ eia

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APPLIED ELASTICITY APPLIED ELASTICITY

Particular

27.41

Formula

PLANE STRAIN (Figs. 27-29 and 24-30) External forces have to be applied on both top and bottom ﬂat ends of the cylinder to prevent its movement in order to meet the condition that w ¼ 0. The Eq. (27-151b) becomes And Eq. (27-151d) becomes

The expression for F The stress combinations are

@D @ D ð1 2vÞ ¼ 2G þ @z @ z

ð27-156Þ

¼0 ð27-156aÞ @D since U independent of z; ¼ 0 ¼G @z 1 2v 0 ð ð F ¼ 2½ðzÞ þ z w zÞ þ ! zÞ 1v 1 @w 0 ð ¼ 2½0 ðzÞ þ zÞ þ 1 v @z 00 ð 0 ð zÞ þ ! zÞ þ ¼ 2½z

1 2v @w 1 v @ z

ð27-156bÞ ð27-156cÞ ð27-157Þ

0 ð ð zÞ ! zÞ 2GD ¼ ð3 4vÞðzÞ z

The displacement D

þ

1 2v w 2ð1 vÞ

ð27-158Þ

BOUNDARY CONDITIONS Speciﬁed stresses The expression for F

F ¼2

By using Eq. (27-155), Eq. (27-159) becomes

ðs 0

ðn þ ins UÞ

@z ds þ constant @s

ðs

ðn þ ins Þ

@z ds þ constant @s

¼ f1 þ if2 on C

ð27-160Þ

0 ð ðzÞ þ z zÞ þ !ð zÞ ¼

0

where f1 and f2 are functions of z only R y S ( m, l, o)

z (or z)

N (l, m, o)

ds

α

O x

y FIGURE 27-29

ð27-159Þ

x C

FIGURE 27-30

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APPLIED ELASTICITY

27.42

CHAPTER TWENTY-SEVEN

Particular

Formula

Speciﬁed displacement From Eq. (27-158) for D

1 2v 0 ð ð ð3 4vÞðzÞ z zÞ ! zÞ ¼ 2GD w 2ð1 vÞ ð27-161Þ

If body force is absent Eq. (27-161) becomes

0 ð ð ð3 4vÞðzÞ z zÞ ! zÞ ¼ 2Gðg1 þ ig2 Þ on C ð27-162Þ where g1 and g2 are functions of z only

FORCE AND COUPLE RESULTANTS AROUND THE BOUNDARY (Fig. 27-31) The expression for force with components X and Y at point O The expression for couple at O

h iB1 0 ð ð X þ iY ¼ i ðzÞ þ z zÞ þ ! zÞ

ð27-163Þ

A1

h iB1 ð B1 @z þ U ds N ¼ Rl ðzÞ z!ðzÞ z z0 ðzÞ A1 @s A1 ð27-164Þ

GENERALIZED PLANE STRESS The average stress combinations assuming z ¼ 0, a stress free surface, i.e. xz ¼ yz ¼ 0 at the surface and body force potential Uðz; zÞ is independent of z

o ¼ x þ y

ð27-165aÞ

o ¼ x y þ 2ixy

ð27-165bÞ

where

B1

τnb

1 2h

pav ¼

1 2h

τyn

ðh h

ðh

h

σn α

N

X

O

1 2h

ðh h

dz

p dz

O

2h x

y x

FIGURE 27-31

o ¼

z

A1

Y

dz;

τxn

ds

y

o ¼

y

x

FIGURE 27-32

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APPLIED ELASTICITY APPLIED ELASTICITY

Particular

The average complex displacement

The body force Eq.

@ @ @ þ þ ¼ 0 becomes @z @ z @z

Formula

Do ¼ uo þ ivo ¼ 1 2h

ðh h

¼ Taking into consideration the body force, Eq. (27-167) and other expression for F and o become

1 2h

ðh h

ð27-166Þ

D dz

@ @ @ þ þ dz ¼ 0 @z @ z @z

ð27-167aÞ

@o @o þ ¼0 @z @z

@ @o h þ 2Ui þ ¼0 @ z o @z @! v @D @ D þ ¼ 1 v @z @ z @z ¼ 4G

ð27-167bÞ ð27-168aÞ

@D @ z

o ¼ 4G

ð27-168bÞ

@Do @ z

ð27-168cÞ

1v @Do @ D þ o ¼ 2G @z 1þv @ z The equations for generalized plane stress

27.43

0 ð ð zÞ þ ! zÞg þ F ¼ 2fðzÞ þ z 2GD ¼

ð27-168dÞ 1 2K w 1K

ð27-169Þ

3v 1 2K 0 ð ð zÞ ! zÞ ðzÞ z w 1þv 2ð1 KÞ ð27-170Þ

0 ð zÞ ¼ 2 0 ðzÞ þ

1 @w 1 K @z

1 2K @w 00 ð 0 ð zÞ þ ! zÞg ¼ 2fz 1 K @ z

ð27-171Þ ð27-172Þ

CONDITIONS ALONG A STRESS-FREE BOUNDARY, Fig. 27-33 Adding Eqs. (27-169) and (27-170) and putting F ¼ 0 along free boundary, i.e. segment AB, the displacement along AB

D¼

4 ðzÞ E

SOLUTION INVOLVING CIRCULAR BOUNDARIES (Figs. 27-33 and 27-34) From stress strain transformation rules

0 ¼ ¼ r þ

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ð27-173Þ

APPLIED ELASTICITY

27.44

CHAPTER TWENTY-SEVEN

Particular

Formula

y

B

y

A

θ = constant

x r C

FIGURE 27-33

r = constant

θ

x

FIGURE 27-34

z z i i where r ¼ , z ¼ r e , z ¼ r e

0 ¼ F e2i ¼ r þ 2i ¼

1 @w ð27-174Þ 1 K @z 1 2K z @w z 0 ð zÞ þ ! zÞ 0 ¼ 2 z00 ð 1 K z @z z 0 ð 0 ¼ 2f0 ðzÞ þ zÞg

2GD0 ¼ ei The boundary conditions are

F ¼2

ðs 0

3v 0 ð ð zÞ ! zÞ ðzÞ z 1þv

1 2K w 1K

ðr þ ir þ UÞ

ð27-175Þ

ð27-176Þ @z ds þ constant @s ð27-177aÞ

0

ð zÞ þ ! zÞ ¼ f1 þ if2 ðzÞ þ z ð

on C

ð27-177bÞ

APPLICATION OF CONFORMAL TRANSFORMATION (Fig. 27-35) The stress combinations after transformation

Eqs. (27-178) are related stress combinations in rectangular coordinates x and y as

0 ¼ þ

ð27-178aÞ

0 ¼ þ 2i

ð27-178bÞ

0 ¼

ð27-179aÞ

0 ¼ e2i

ð27-179bÞ

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APPLIED ELASTICITY APPLIED ELASTICITY

Particular

Formula

ξ = constant

y

η

η

τξη

ϑ = constant

P

27.45

σξ

η = constant

Q r

ρ

α

x

O

(a) z - plane

ρ = constant ξ = constant

ϑ

O

ξ O

(b) ϕ - plane

(c)

ξ

FIGURE 27-35

An explanation for e2i

where z ¼ zðÞ ¼ f ð; Þ þ igð; Þ ¼ þ i f ð; Þ and gð; Þ are real and imaginary parts of zðÞ e2i ¼ z0 ðÞ=z0 ðÞ ð27-179cÞ or

Using Eqs. (27-179a) and (27-179b), and Eqs. (27-171) and (27-172), when these are no body forces, letting ðzÞ ¼ 1 ðÞ and !ðzÞ ¼ !1 ðÞ The transformation of a given boundary in the zplane into the unit circle in the -plane

Using polar coordinates (, #), the stress components become

d 0 d þ 1 ; ð Þ ð27-180aÞ 0 ¼ 2 01 ðÞ dz d z 0 01 ; ðÞ 1 ðÞ þ 0 ¼ 2 ð27-180bÞ z0 ðÞ z0 ðÞ o 2 n 01 ðÞ þ 0 001 ðÞi þ ! 01 ðÞ zðÞh00 0 ¼ 0 z ðÞ ð27-181aÞ or ( ) 0 2 1 1 ; ð Þ 0 0 1 ; ðÞ ¼ 0 ð27-181bÞ þ! z ðÞ zðÞ z0 ðÞ 00 ¼ þ # ¼ þ # 2i# 00

Using polar coordinates Eqs. (27-180a) and (27-181) in terms of complex potentials become

ð27-182aÞ

00

0

ð27-182bÞ 00

0

2i#

where ¼ and ¼ e " # 0 ðÞ 0 ðÞ 00 þ ¼2 0 z ðÞ z0 ðÞ

¼

0 :

ð27-183Þ

i 2 h 01 ðÞ þ 0 001 ðÞi þ ! 0 ðÞ zðÞh00 0 z ðÞ ð27-184aÞ " # 0 ðÞ 2 0 ðÞ þ! ð27-184bÞ zðÞ 0 00 ¼ 0 z ðÞ z ð Þ 00 ¼

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APPLIED ELASTICITY

27.46

CHAPTER TWENTY-SEVEN

Particular y

Formula

y

T

T

T x

a

T

σy

y

τxy

dy y z

2b

τxy σx

2b

x

Mb

2h

Mb

a

a

FIGURE 27-36

a

FIGURE 27-37

Rectangular plate under all round tension Value of complex potentials ðzÞ and !ðzÞ assumed

ðzÞ ¼ 12 Tz;

From stress combination Eqs. (27-156c) and (27-157)

0 ð zÞg þ ¼ 2f0 ðzÞ þ

!ðxÞ ¼ 0

ð27-185Þ 1 @w 1 v @z

¼ 2½12 T þ 12 T ¼ 2T

ð27-156cÞ

1 2v @w 00 ð 0 ð zÞ þ ! zÞg þ ¼ 2fz 1 v @ z

ð27-157Þ

where ¼ x þ y and ¼ x y þ 2ixy The stress x and y after equating real and imaginary parts

x ¼ T;

The displacement from Eq. (27-158) after equating real and imaginary parts

0 ð ð zÞ ! zÞ 2GD ¼ ð3 4vÞðzÞ z

y ¼ T;

þ

xy ¼ 0

1 2v w 2ð1 vÞ

ð27-158Þ

D¼

T ð1 vÞðx þ iyÞ ¼ u þ i E

u¼

T ð1 vÞx; E

¼

ð27-186Þ

T ð1 vÞy E

ð27-187Þ

Rectangular plate under plane ﬂexure Assume values of complex potentials ðzÞ and !ðzÞ as

ðzÞ ¼ Az2 !ðzÞ ¼ Bz2 Choose A and B, which may be complex, so that edges y ¼ b are stress free.

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APPLIED ELASTICITY APPLIED ELASTICITY

Particular

27.47

Formula

Boundary conditions From stress combinations Eqs. (27-156) and (27-157) boundary conditions

0 ð 0 ðzÞ þ zÞ þ z00 ðzÞ þ !0 ðzÞ ¼ y þ ixy

ð27-156Þ

y ¼ 0, xy ¼ 0 throughout the plate A ¼ iC and B ¼ iC where C is real ¼ 0x þ 0y ¼ 0x ¼ 8Cy

The bending moment

Mb ¼

ðb

x 2hy dy ¼ 8CI

b

ð27-188Þ

where I ¼ moment of inertia about oz C¼

Mb 8I

The values of complex potentials ðzÞ ¼ Az2 and !ðzÞ ¼ Bz2 are

ðzÞ ¼

The displacement from Eq. (27-158)

D¼

iMb 2 z ; 8I

!ðzÞ ¼

iMb 2 z 8I

ð27-188aÞ

i 1 h 0 ð ð zÞ ! zÞ ¼ u þ iv ð3 4vÞðzÞ z 2G

when body forces are zero Substituting the values of ðzÞ and !ðzÞ in the above, u and v can be determined.

Thick cylinder under internal and external pressure Values of complex potentials ðzÞ and !ðzÞ assumed using boundary conditions at r ¼ a or di =2 and r ¼ b or do =2 with no body forces, assuming internal pressure pi , external pressure po , values of A and B in Eq. (27-189), which are real, can be found. From Eqs. (27-174) and (27-175)

The expressions for and r at any radius

ðzÞ ¼ Az

and !ðzÞ ¼

B z

ð27-189aÞ

where A and B are real 0 1 2 ½

0 ð þ 0 ¼ r þ ir ¼ 0 ðzÞ þ zÞ z 00 ðzÞ !0 ð zÞ z z

ð27-189bÞ

The equations for and r are given in Eqs. (27-101b) and (27-101a) respectively.

Rotating solid disk and hollow disk of uniform thickness rotating at ! rad/s Values of complex potentials ðzÞ and !ðzÞ assumed

ðzÞ ¼ Cz

and !ðzÞ ¼

B z

where C and B are real

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ð27-189cÞ

APPLIED ELASTICITY

27.48

CHAPTER TWENTY-SEVEN

Particular

Formula

Using boundary conditions at ðr Þr ¼ b ¼ 0 and ðr Þr ¼ 0 6¼ 0 for solid disk ðr Þr ¼ a ¼ 0 and ðr Þr ¼ b ¼ 0 for hollow disc taking into consideration body forces, values of C and B in Eq. (27-189c) which are real can be found

Refer Eqs. (27-126), (27-127) and (27-128) to (27-131)

The radial displacements at the boundaries

ður Þr ¼ a ¼

!2 a fð1 vÞa2 þ ð3 þ vÞb2 g 4E

ð27-189dÞ

ður Þr ¼ b ¼

!2 b fð1 vÞb2 þ ð3 þ vÞa2 g 4E

ð27-189eÞ

Large plate under uniform uniaxial tension with a centrally located unstressed circular hole Values of complex potentials ðzÞ and !ðzÞ assumed

Using Eq. (27-189b) and above complex potentials Using boundary condition at r ¼ a

Tz A þ 4 z 1 B C !ðzÞ ¼ Tz þ þ 3 2 z z where A, B and C are real

ðzÞ ¼

ð27-190Þ

1 3A 1 z B 3C T 2 þ T þ þ 3 ð27-190aÞ 2 2 z z z z z z 1 B 1 A Tþ 2 þ T þ 2 e2i ðr ir Þr ¼ a ¼ 2 2 a a 3C 3A 2i þ ð27-190bÞ e a4 a2 r ir ¼

A ¼ 12 Ta2 ;

B ¼ 12 Ta2 ;

C ¼ 12 a4

ð27-190cÞ

since hole is stress free The new values of ðzÞ and !ðzÞ

ðzÞ ¼

Using Eqs. (27-174), (27-175) and after equating the real and imaginary parts, the stress components are y A T

A

FIGURE 27-38

θ

a T

x

Tz 1 Ta2 þ 4 2 z

ð27-190dÞ

1 Ta2 a4 ð27-190eÞ þ 3 !ðzÞ ¼ Tz 4 2 2z " # 1 a2 4a2 3a4 r ¼ T 1 2 þ 1 2 þ 4 cos 2 2 r r r

¼

1 T 2

" 1þ

a2 r2

#

3a4 1 þ 4 cos 2 r

1 2a2 3a4 r ¼ T 1 þ 2 4 sin 2 2 r r

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ð27-191Þ ð27-192Þ ð27-193Þ

APPLIED ELASTICITY APPLIED ELASTICITY

Particular

The , r and r at r ¼ a

The maximum tangential stress The stress concentration factor

27.49

Formula

ðr Þr ¼ a ¼ ðr Þr ¼ a ¼ 0

ð27-194aÞ

ð Þr ¼ a ¼ Tð1 2 cos 2Þ

ð27-194bÞ

max ¼ ð Þr ¼ a ¼ 3T

ð27-194cÞ

K ¼

ð Þmax 3T ¼ ¼3 T T

ð27-195Þ

Large plate containing a circular hole under uniform pressure Values of complex potentials ðzÞ and !ðzÞ assumed From Eqs. (27-174) and (27-175) in the absence of body forces

ðzÞ ¼ 0;

!ðzÞ ¼

A z

ð27-196Þ

0 ð zÞg ¼ r þ ¼ 0 0 ¼ 2f0 ðzÞ þ z 0 00 ð ð zÞ þ ! zÞ 0 ¼ 2 z z ¼ r þ 2ir ¼

Boundary conditions are

ðr Þr ¼ a ¼ p ¼

ðaÞ

2A r2

ðbÞ

2A a2

A ¼ pa2 The new complex potentials

ðzÞ ¼ 0;

The stress components are

r ¼

!ðzÞ ¼

pa2 ; r2

¼ r ¼ The displacement from Eq. (27-176)

ðcÞ pa2 z

r ¼ 0 pa2 r2

2GD0 ¼ 2Gður þ iu Þ ¼ ei A ; 2Gr pa ¼ 2G

ður Þ ¼ ður Þr ¼ a

ðdÞ ð27-197Þ

A z

¼

A r

u ¼ 0

ð27-198Þ

Large plate containing a circular hole ﬁlled by an oversize disk 1. Rigid Disk The radius of disk rd

rd ¼ að1 þ "Þ where a ¼ radius of hole

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ðaÞ

APPLIED ELASTICITY

27.50

CHAPTER TWENTY-SEVEN

Particular

From ﬁrst of Eq. (27-198), the radial displacement The stress components

Formula

ur ¼ a" ¼

A 2Ga

r ¼ ¼ 2G"

or A ¼ 2Ga2 "

ðbÞ

a2 r2

ðcÞ

r ¼ 0

ðdÞ

2. Elastic Disk The complex potential for all round pressure on the disk

1 ðzÞ ¼ 12 pz;

The displacement from Eq. (27-176)

2G1 D0 ¼ 2G1 ður1 þ iu1 Þ ¼ ei ¼ ur1 ¼

!1 ðzÞ ¼ 0

ðeÞ

ð1 vÞpz 1þv

pð1 v1 Þpa 1 þ v1

ðfÞ

pð1 v1 Þa E1

ðgÞ

where subscript 1 for disk and 2 for plate The radial displacement of plate The pressure between disc and plate

ur2 ¼ p¼

pa 2G2

ðhÞ

E1 E2 E1 ð1 þ v2 Þ þ E2 ð1 v1 Þ

ð27-198Þ

Elliptical hole in a large plate under tension (Fig. 27-39) The expression for transformation

m ; z¼C þ

m<1

ð27-199Þ

Transforms the outside of an ellipse of semiaxes a and b in the z-plane into the outside of a unit circle r in the -plane, provided ab C ¼ 12 ða þ bÞ; m ¼ ð27-200aÞ aþb z ¼ ei# þ m ei# C ¼ ð1 þ mÞ cos # þ ð1 mÞi sin # ð27-200bÞ aþb 2a 2b z¼ cos # þ i sin # ð27-200cÞ 2 aþb aþb or

x þ iy ¼ a cos # þ ib sin #

# ¼ eccentric angle around the ellipse

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ð27-200dÞ

APPLIED ELASTICITY APPLIED ELASTICITY

Particular

Formula

s

z = x + iy

T

y

27.51

η

η ζ = ξ+iη = eiϑ

ab

b a

x

α

ϑ

ξ

1

T (a) z - Plane

(b) ζ - Plane

FIGURE 27-39

The points at which the transformation ceases to be conformal are

The boundary condition at the stress free ellipse The boundary condition in terms of

Eq. (27-203) on unit circle becomes

z0 ðÞ ¼ 0

m z0 ðÞ ¼ C 1 2 ¼ 0 pﬃﬃﬃﬃ ¼ m; since m < 1 0 ð ð zÞ þ ! zÞ ¼ f1 þ if2 ðzÞ þ z

ð27-201aÞ ð27-201bÞ

ð27-202Þ on the boundary of ellipse ¼ 0 0 ðÞ 1 ðÞ ¼ 0 1 ðÞ þ zðÞ 01 þ ! z ð Þ or 01 ðÞ þ z0 ðÞ þ z0 ðÞ! 1 ðÞ ¼ 0 z0 ðÞ1 ðÞ þ zðÞ ð27-203Þ 01 ð 01 ð z0 ð Þ1 ðÞ þ zðÞ Þ þ z0 ð Þ! Þ ¼ 0 or 1 1 01 1 þ z0 1 ! 1 z0 1 ðÞ þ zðÞ ¼0

The complex potentials for an inﬁnite plate without a hole acted upon by uniaxial tension at an angle to the x-axis in the z-plane and -plane

ð27-204Þ where ¼ ; ¼ ¼ 1 on since ¼ 1 on unit circle ðzÞ ¼ 14 Tz; !ðzÞ ¼ 12 Tz e2i on z plane ð27-205aÞ m z ¼ Cð þ m=Þ ! C and z0 ðÞ ¼ C 1 2 ! C at in -plane

The complex potentials for an inﬁnite plate with stress free elliptic hole subject to tension at an angle to the x-axis in -plane

ð27-205bÞ 1 ðÞ ¼ 14 TC; !1 ðÞ ¼ 12 TC e2i 1 A B C ð27-206aÞ 1 ðÞ ¼ TC þ þ 2 þ 3 þ 4 1 D1 E1 F1 0 2 2i þ 2 þ 3 þ þ z ðÞ!1 ðÞ ¼ TC e 2 ð27-206bÞ

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APPLIED ELASTICITY

27.52

CHAPTER TWENTY-SEVEN

Particular

Using Eqs. (27-206) in Eqs. (27-204), after equating coeﬃcients of powers of or (since E1 ¼ B ¼ C ¼ 0, D1 ¼ 1 þ m2 2Me2i , F1 ¼ e2i , A ¼ 2e2i mÞ

Formula

* + 1 2e2i m 1 ðÞ ¼ TC þ 4

ð27-206cÞ

*

1 !1 ðÞ ¼ TC 2

1 þ m2 2m e2i e2i 3 e2i þ m 1 2

+

The tangential stress on the boundary of elliptical hole from Eq. (27-183) 00 ¼ # þ where ¼ 0 after equating to real part of right hand side of equation and simpliﬁcation (Fig. 27-39)

ð27-206dÞ 1 m2 þ 2m cos 2 2 cos 2ð# Þ # ¼ T 1 þ m2 2m cos 2# ð27-207Þ

The tangential stress on the boundary of elliptical hole for ¼ 0 (Fig. 27-40)

# ¼ T

The maximum tangential stress # max on the contour ab of any elliptical hole for any value of m ¼ and aþb 1 c ¼ ða þ bÞ will be at # ¼ 2

ð# Þmax

1 m2 þ 2m 2 cos 2# 1 þ m2 2n cos 2# 3m 2b ¼T ¼T 1þ 1þm a

ð27-208Þ ð27-209Þ

2

If a ¼ b in Eq. (27-209), then the ellipse becomes a circle

ðv Þmax ¼ 3T

The stress concentration factor

K ¼

By taking ¼ 458 with T ¼ S, and ¼ 458 with T ¼ þS, and on adding these solutions, a solution for pure shear S applied to an inﬁnite ﬂat plate with an elliptical hole at inﬁnity is obtained. The shear will be parallel to the axes of the ellipse with around the elliptical hole is given by

ð27-209aÞ

ðv Þmax ¼3 T 4 sin 2 ¼ S 1 2m cos 2 þ m2

ð27-209bÞ ð27-210Þ

MUSKHELISHVILI’S DIRECT METHOD In this method that a hole L can be transformed conformally into a unit circle in the -plane so that outside of the hole is mopped on the inside of (Fig. 27-41) The form of the conformal transformation will be If the loading of the plate at inﬁnity is given by the complex potential ðÞ, !ðÞ , the full complex potentials which will also satisfy the condition around the hole, can be written as

z¼C

1 þ e1 þ e2 2 þ e3 3 þ þ en n

ðÞ ¼ ðÞ þ o ðÞ

ð27-211Þ ð27-212Þ

!ðÞ ¼ ! ðÞ þ !o ðÞ

ð27-213Þ

where o ðÞ ¼

1 X o

an n ;

!o ðÞ ¼

1 X

bn n

o

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ð27-213aÞ

APPLIED ELASTICITY APPLIED ELASTICITY

Particular

27.53

Formula

S

S 45 b S

45

η

y

θ

η

S

a

o

S FIGURE 27-40

ξ

L

S

S

x

z-Plane

ζ

ζ-Plane

o

FIGURE 27-41

The boundary condition around a stress free hole assuming no body forces is given by (refer to Eqs. (27-203) and (27-204)) Substituting the complex potentials given by Eqs. (27-212), in Eq. (27-214)

zðÞ 0 1 1 ðÞ þ þ ! ¼0 ð27-214Þ 1 z0 zðÞ 0 1 1 o þ! ¼ f1 þ if2 o ðÞ þ o 1 z0 ð27-215aÞ where

2

3 zðÞ 0 1 1 f1 þ if2 ¼ 6 ðÞ þ þ! 7 1 5 4 z0

Using Harnack’s theorem, residue theorem and 1 d Cauchy’s integral, multiplying by and 2 i integrating around Eq. (27-214) can be written as

The complex potential o ðÞ from Eq. (27-216) is

ð27-215bÞ 0o 1 ð ð 1 o ðÞ 1 zðÞ d þ d 2 i 2 i 0 1 z 1 ð ! ð o 1 1 f1 þ if2 d d ¼ þ 2 i 2 i ð27-216Þ 0o 1 ð zðÞ ð 1 1 f1 þ if2 o ðÞ þ d d ¼ 2 i 0 1 2 i ð Þ z ð27-217Þ

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APPLIED ELASTICITY

27.54

CHAPTER TWENTY-SEVEN

Particular

Taking conjugate of Eq. (27-215), remembering that 1 d ¼ 1 and multiplying by and integrating 2 i around

The complex potential !o ðÞ can be found after substituting the value of o ðÞ Eq. (27-218) which can be evaluated from Eq. (27-217)

Formula

1 o 1 ð ð z 1 1 0o ðÞ d þ d 2 i 2 i z0 ðÞ ð ð 1 !o ðÞ 1 f1 if2 d d ¼ þ 2 i 2 i ð27-218Þ 1 ð z ð 1 1 f1 if2 0o ðÞ d ðÞ ¼ d þ ! o 2 i z0 ðÞ 2 i ð27-219Þ y

Stress free square hole in a ﬂat plate under uniform uniaxial tension (Fig. 27-42)

T

T

x d FIGURE 27-42

The form of the conformal transformation will be

The known complex potential in this case

z¼C

1 3 6

ðÞ ¼

ð27-220Þ

1 1 3 TC 4 6

ð27-221Þ

1 1 3 ! ðÞ ¼ TC 2 6

ð27-222Þ

After substituting ðÞ and ! ðÞ from Eqs. (27-221) and (27-222) into Eq. (27-215)

1 2 3 1 f1 þ if2 ¼ TC 2 þ 3 3 3 4

ð27-223Þ

After substituting the value of f1 þ if2 from Eq. (27-223) in Eq. (21-217) and simpliﬁcation

o ðÞ ¼ TC

Substituting the value of o ðÞ from Eq. (27-224) in Eq. (27-219) and after simpliﬁcation

!o ðÞ ¼

3 1 þ 3 7 12

ð27-224Þ

1 1 TC 2 þ 3 4 3 " # 1 1 6 4 3 2 þ TC 3 2 þ 4 7 4

þ

1 TC 14

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ð27-225Þ

APPLIED ELASTICITY APPLIED ELASTICITY

Particular

27.55

Formula

3 1 1 1 3 þ þ 7 4 24 1 91 78 3 !ðÞ ¼ TC þ 2 84ð2 þ 4 Þ

The full complete complex potentials after simpliﬁcation

ðÞ ¼ TC

The tangential stress around the square hole

# ¼ Rl: 4

By adding more terms to the expression for transformation

z¼C

0 ðÞ z0 ðÞ

ð27-227Þ ð27-228Þ

1 1 3 1 þ 7 6 56

ð27-228aÞ

the radius becomes r ¼ 0:025d 1 1 3 1 1 11 þ 7 z¼C 6 56 176

The radius r will be rounded of

ð27-226Þ

ð27-228bÞ

the radius becomes r ¼ 0:014d For graph of # =T versus # in degrees

Refer to Fig. 27-43.

Stress free square hole in a ﬂat plate under pure bending (Fig. 27-44)

The conformal transformation for plate with a square hole such that the diagonals along the coordinate axes as shown in Fig. 27-44

z¼C

The known complex potentials from Eqs. (27-188a)

ðzÞ ¼

8 T 6 4

Square Hole I r = 0.06 d II r = 0.025 d III r = 0.014 d

r

III

I III

20

2

40

iMb 2 z ; 8I

ð27-229Þ

! ðzÞ ¼

iMb 2 z 8I

ð27-188aÞ

T

ϑ

I

0

d

II

2

1 1 3 þ 6

60 ϑ - Degrees

80

II

90 Mb

Mb

4

FIGURE 27-43

FIGURE 27-44 Flat plate with stress-free square hole under pure bending.

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APPLIED ELASTICITY

27.56

CHAPTER TWENTY-SEVEN

Particular

The complete complex potentials in -plane will be of the form

From Eq. (27-217)

From Eq. (27-219)

Formula

iMb C2 1 1 3 2 þ þ o ðÞ 8I 6 iMb C 2 1 1 3 2 þ !o ðÞ !ðÞ ¼ þ 8I 6 iMb C2 4 2 1 4 1 o ðÞ ¼ þ 6 8I 3 3 36 ðÞ ¼

ð27-230aÞ ð27-230bÞ ð27-231Þ

1 1 þ 6 4 0 ðÞ 2 2 4 o iM C2 4 2 1 4 1 37 ¼ b þ 6 8I 3 3 16 18

!o ðÞ

ð27-232Þ The full complex potentials become often simplifying

ðÞ ¼ !ðÞ ¼

iMb C 8I

iMb C 2 8I

2

After knowing full complex potentials, the tangential stress at various angles around the hole/cutout can be calculated For graphs of v =ðMb c=IÞ versus # degree

1 1 2 þ 4 3 2

ð27-233Þ

18 þ 45 2 31 4 þ 36 6 15 8 9 2 ð2 4 Þ ð27-234Þ

Refer to Fig. 27-45.

Large plate containing an elliptical hole subjected to uniform pressure (Fig. 27-46) The expression for transformation

m z¼C þ ;

m<1

ð27-199Þ

where 1 ab C ¼ ða þ bÞ; m ¼ 2 aþb Refer to other details under Eq. (27-199) The complex potential at inﬁnity

ðÞ ¼ ! ðÞ ¼ 0

The required complex potentials

ðÞ ¼

Boundary conditions

n ¼ ¼ p;

1 X an ; n 0

!ðÞ ¼

ð27-235aÞ 1 X bb n 0

ð27-235bÞ

ns ¼ 0 around the hole ð27-236Þ

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APPLIED ELASTICITY APPLIED ELASTICITY

Particular

27.57

Formula

n p

6 5 4 3 2

I

Mb

I - Second moment of inertia ϑ = Angle form z-axis to a point on hole boundary

1

20

0 1

III

II

Mb

40

x

I

I

a p

p

III (a)

II

γ 60

γ 80

III

b

(b)

90

II

A D

B C

R

2 FIGURE 27-46

FIGURE 27-45

From Eq. (27-117b)

0 ð ð ðzÞ þ 2 zÞ þ ! zÞ ¼ ðn þ ns Þ

@z ds @s

¼ pz

Expressing Eq. (27-237) in -plane at all points of

1 @ 2 i [Considering the ﬁrst of these integrals, one has to remember that is now a boundary to the region external to the unit circle. Thus it is necessary to consider an integration around a contour consisting of together with C circle 0 of large radius R joined by two close paths AB and CD, Fig. 27-46] Multiplying Eq. (27-238) by

Using Cauchy’s integral, Harnack’s theorem and residue theorem, Eq. (27-239) gives the expression for ðÞ

at all points on the ellipse ð27-237Þ zðÞ 0 1 1 þ! ¼ pzðÞ ðÞ þ 1 z0 2 þ mðÞ 0 1 1 ðÞ þ þ ! ð1 m2 Þ m ð27-238Þ ¼ pC þ 0 1 ð ð 2 1 ðÞ @ 1 þm þ d 2 i 2 i ð1 m2 Þ 1 ð ! ð þm d 1 pC d ¼ þ 2 i 2 i ð27-239Þ

ðÞ ¼

pCm

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ð27-240Þ

APPLIED ELASTICITY

27.58

CHAPTER TWENTY-SEVEN

Particular

Formula

Taking conjugate of Eq. (27-239) and integrating around , the expression for !ðÞ The stress can be obtained by making use of Eq. (27-183) for 00 and (27-184b) for 00 and equating real parts on both sides of equation

!ðÞ ¼

pC pCm

½ ¼ 1 ¼ p from boundary condition 0 ðÞ ½00 ¼ 1 ¼ 4Rl 0 z ðÞ ¼ 1 ¼ # þ ¼

The tangential stress around by elliptical hole from Eq. (27-243)

1 þ m 2 2 m

# ¼ or

4pmðcos 2# mÞ 1 þ m2 2m cos 2#

4#mðcos 2v mÞ þp 1 þ m2 2m cos 2#

# ¼ p

1 þ 2m cos 2# 3m2 1 2m cos # þ m2

ð27-241Þ ð27-242Þ

ð27-243Þ ð27-244Þ ð27-245Þ

Large ﬂat plate under uniform uniaxial tension with a circular hole whose edge is rigidly ﬁxed (Fig. 27-49) The edge of the hole r ¼ a is held ﬁxed by a rigid circular ring to which the material of the plate adheres at all points The boundary condition is given by T The complex potential form of displacement for generalized plane stress problem from Eqs. (27-170) when there are no body forces

½Dr ¼ a ¼ 0 for all # ð27-246aÞ 3v 0 ð ð ðzÞ z zÞ ! zÞ ¼ 0 ð27-246bÞ 2GD ¼ 1þv or 0 ð ð zÞ ! zÞ ¼ 0 on r ¼ a KðzÞ z

ð27-246cÞ

where 3v K¼ 1þv KðÞ zðÞ

0 ðÞ ðÞ ¼ 0 in terms of ! z0 ðÞ ð27-246dÞ

y a

θ

T

x T

FIGURE 27-47

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APPLIED ELASTICITY APPLIED ELASTICITY

Particular

27.59

Formula

a

The conformal transformation for this problem can be taken as

z¼

The full complex potentials in this case can be taken as

ðÞ ¼

ð27-247Þ 1 Ta þ o ðÞ 4

!ðÞ ¼

1 Ta þ !o ðÞ 2

ð27-248aÞ ð27-248bÞ

where ﬁrst terms in each of the above equations is for stress state at inﬁnity The condition to be satisﬁed on ¼ 1 by o ðÞ and !o ðÞ is

zðÞ 0 1 1 Ko ðÞ ! o o 1 z0 1 a 1 ¼ TðK 1Þ Ta 4 2

1 d and integrating 2 i around , after simpliﬁcation Multiplying Eq. (27-249) by

Multiplying the conjugate of Eq. (27-229) by 1 d and integrating around and after simpliﬁ2 i cation, expression for !o ðÞ The full complex potentials are

o ðÞ ¼

Ta 2K

1 2 3 !o ðÞ ¼ Ta ðK 1Þ 4 K

ðÞ ¼

1 1 2 Ta 4 K

ð27-249Þ ð27-250Þ

ð27-251Þ

ð27-252aÞ

1 a 1 2 !ðÞ ¼ T þ Ta ðK 1Þ 3 ð27-252bÞ 2 4 K From the Eqs. (27-182a) and (27-182b) for 00 and 00 , the following stress components are

¼

1 2 TðK þ 1Þ 1 þ cos 2# 4 K

ð27-253Þ

# ¼

1 2 Tð3 KÞ 1 þ cos 2# 4 K

ð27-254Þ

# ¼

1 K þ1 T sin 2# 2 K

ð27-255Þ

TORSION (Fig. 25-49) The angle of twist , which is proportional to the distance of cross-section from the ﬁxed end

¼ z where ¼ angle of twist per unit length

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ð27-256Þ

APPLIED ELASTICITY

27.60

CHAPTER TWENTY-SEVEN

Particular

Formula

y

y

P’(x+u, y+ν) Mt

O

α Mt

z

P(x, y)

Fixed End

r α

x FIGURE 27-48 Torsion of prismatic bar.

O

v

u

β

x

Pðx; y; zÞ is a point in a section of bar z-distance from ﬁxed end (Fig. 27-48) and it is displaced to a new point P0 ðx þ u; y þ v; z þ wÞ after deformation due to twist such that OP OP0 r

FIGURE 27-49 Shows a cross-section of twisted bar in xyplane.

The displacement of point P in x-direction assuming that is small such that cos ¼ 1 and sin

u ¼ r cosð þ Þ r cos y ¼ zy

ð27-257Þ

The displacement of point P in y-direction

v ¼ r sinð þ Þ r sin x ¼ zx

ð27-258Þ

The warping of bar, which is invariant with z and is deﬁned by a function

w ¼ ðx; yÞ

ð27-259Þ

The component of strains from Eqs. (27-40) and (27-41)

"x ¼ "y ¼ "z ¼ xy ¼ 0 @w @u @ yz ¼ þ ¼ G y @x @z @x @w @v @ þ ¼ G þx xz ¼ @y @z @y

The stress components from Eqs. (27-34) and (27-37)

The equations of equilibrium from Eqs. (27-11)

where ðx; yÞ is a function of x and y only

x ¼ y ¼ z ¼ xy ¼ 0 @ y xz ¼ G @x @ yz ¼ G þx @y

ð27-260aÞ ð27-260bÞ ð27-260cÞ ð27-261aÞ ð27-261bÞ ð27-261cÞ

@x @xy @xz þ þ þ Fbx ¼ 0 @x @y @z

ð27-11aÞ

@y @yz @yx þ þ þ Fby ¼ 0 @y @z @x

ð27-11bÞ

@z @zx @zy þ þ þ Fbz ¼ 0 @z @x @y

ð27-11cÞ

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APPLIED ELASTICITY APPLIED ELASTICITY

Particular

Neglecting body forces in z-direction, Eq. (27-11) yields after substituting the Eqs. (27-261b) and (27-261c) in it

27.61

Formula

2 @ @2 G þ ¼0 @x2 @y2 or

@2 @2 þ 2 2 @x @y

ð27-262aÞ

¼0

ð27-262bÞ

which is true throughout the cross-sectional region of the bar From the equilibrium condition of the surface Eq. (27-7)

FNx ¼ x l þ xy m þ xz n

ð27-7aÞ

FNy ¼ yz l þ y m þ yz n

ð27-7bÞ

FNz ¼ zx l þ zy m þ z n

ð27-7cÞ

When surface forces are absent FNx ¼ FNy ¼ FNz ¼ 0 and cosðNzÞ ¼ n ¼ 0, x ¼ y ¼ z ¼ yz ¼ 0 from Eq. (27-7c)

zx l þ zy m ¼ 0

From the inﬁnitesimal element pqr, if s increasing in the direction from q to r then

l¼

ð27-263Þ

dy ¼ cosðN; xÞ ds

ð27-264aÞ

dx ¼ cosðN; yÞ ds @ dy @ dx y þx ¼0 @x ds @y ds

m¼ Using Eqs. (27-261b), (27-261c), (27-264a) (27-264b) in Eq. (27-263), an expression for boundary condition is obtained (Fig. 27-50)

ð27-264bÞ ð27-265Þ

In torsion problems involving in ﬁnding a function which satisfy Eqs. (27-262) and boundary condition Eq. (27-265)

Stress function From equation of equilibrium

@xy ¼ 0; @z

A function which satisfy the third equation of Eq. (27-266) is

xz ¼

@yz ¼ 0; @z

@xz @yz þ ¼0 @x @y

@ @y

@ @x where is a function of x and y only @ @ ¼ G þx @x @y @ @ ¼ G y @y @x yz ¼

From Eqs. (27-267), (27-268), and Eqs. (27-261), equations involving and are:

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ð27-266Þ ð27-267Þ ð27-268Þ

ð27-269aÞ ð27-269bÞ

APPLIED ELASTICITY

27.62

CHAPTER TWENTY-SEVEN

Particular

Formula

S O

τxz

p dy

Region

y

x

τyz dx

q

b

ds

r qr = ds

a

x N

y FIGURE 27-50 Boundary condition.

FIGURE 27-51 Elliptical cross-section of bar under torsion.

By making use of Eqs. (27-269) and after eliminating from Eqs. (27-269a) and (27-269b) by mathematical method, a diﬀerential equation for stress function is obtained

@2 @2 þ ¼F @x2 @y2

Boundary condition Eq. (27-265) becomes

@ dy @ dx d þ ¼ ¼0 @y ds @x ds ds

ð27-270Þ

where F ¼ 2G

ð27-270aÞ ð27-271Þ

which indicates that the stress function must be constant along the boundary of the cross-section. This constant is taken as zero for a solid bar. The total torque at the ends of the twisted bar due to couple

ðð Mt ¼ 2

dx dy

ð27-272Þ

Torsion of elliptical cross-section bar (Fig. 27-51) The boundary of an elliptical cross-section can be taken as

x2 y2 þ 1¼0 a2 b2

The stress function which satisfy Eq. (27-270) and the boundary condition Eq. (27-271)

¼m

ð27-273Þ

x2 y2 þ 1 a2 b2

ð27-274Þ

where m is a constant Substituting the expression for from Eq. (27-274) in Eq. (27-270) and value of m can be found, and it is

m¼

a2 b2 F 2ða2 þ b2 Þ

Substituting the value of m from Eq. (27-275) into Eq. (27-274) the stress function becomes

¼

a2 b2 F 2ða2 þ b2 Þ

ð27-275Þ x2 y2 þ 1 a2 b2

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ð27-276Þ

APPLIED ELASTICITY APPLIED ELASTICITY

Particular

The torque Mt is obtained after substituting this stress function from Eq. (27-276) into Eq. (27-272) and carrying out integration and simpliﬁcation

Mt Torque

Convex (+ve)

FIGURE 27-52

Formula

Mt ¼

ðð a2 b2 F 1 x2 dx dy a2 þ b2 a2 ðð ðð 1 y2 dx dy þ 2 dx dy b

a3 b3 F 2ða2 þ b2 Þ

Mt ¼

The expression for F from Eq. (27-278)

2Mt ða2 þ b2 Þ a3 b3 M x2 y2 ¼ t þ 1 ab a2 b2

The stress components xz and yz from Eqs. (27-267) and (27-268) after substituting the value of from Eq. (27-280)

The maximum shear stress which occurs at y ¼ b

F ¼

xy ¼

2Mt y ab3

2Mt x a3 b 2Mt ¼ ab2

ð27-278Þ ð27-279Þ ð27-280Þ ð27-281Þ

yz ¼

ð27-282Þ

max

ð27-283Þ

The angle of twist after substituting the value of F from Eq. (27-279) into Eq. (27-270a) and simpliﬁcation

¼ Mt

The torsional rigidity C which is deﬁned as twist per unit length

C¼

For various values of the angle of twist (0 ¼ ) and thereby the values of C for various cross-sections and built up beams

ð27-277Þ

where ðð ba3 x2 dx dy ¼ Iy ¼ 4 ðð ab3 y2 dx dy ¼ Ix ¼ 4 ðð dx dy ¼ A ¼ ab

After substituting the values of Ix , Iy and A into Eq. (27-277) and simpliﬁcation, the expression for Mt

The equation for stress function after substituting the value of F from Eq. (27-279) in Eq. (27-276)

27.63

a2 þ b2 a3 b3 G

a3 b3 G G A4 ¼ a2 þ b2 4 2 Ip

ð27-284Þ ð27-285Þ

where A ¼ ab, Ip ¼ centroidal moment of inertia of the cross-section ¼ ð ab3 Þ=4 þ ð a3 bÞ=4 Refer to Tables 24-27 and 24-30 under Chapter 24.

The expression for warping of elliptical cross-section after substituting Eqs. (27-280), (27-281) and (27-282) into Eqs. (29-260b) and (27-260c) and integrating

w ¼ Mt

For warping of elliptical cross-section

Refer to Fig. 27-52.

Note: The symbol is used for angle of twist here in order to avoid confusion regarding which is used as a stress function

Equations (27-277) to (27-285) are also given in Chapter 24 from Eqs. (24-338) to (24-342), and angle of twist in Chapter 24 in Tables 24-27 and 24-30.

ðb2 a2 Þxy a3 b3 G

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ð27-286Þ

APPLIED ELASTICITY

27.64

CHAPTER TWENTY-SEVEN

Particular

Formula

For torsion of elliptical and rectangular solid sections and other sections (Fig. 24-66 to 24-71)

Refer to Chapter 24 from Eqs. (24-338) to (24-352), Tables 24-27 to 24-30.

Torsion of equilateral triangle bar The expression for stress function

pﬃﬃﬃ pﬃﬃﬃﬃﬃ 2 2 a ¼ x 3y a x þ 3y a x þ A 3 3 3 ð27-287Þ

Substituting Eq. (27-287) in Eqs. (27-267) and (27-268) @2 @2 and 2 can be found. The values the values of 2 @x @y are substituted in Eq. (27-270) to ﬁnd the value A

A ¼ G

ð27-288Þ

The stress function from Eq. (27-287) becomes

1 1 2 ¼ G ðx2 þ y2 Þ ðx3 3xy2 Þ a2 2 2a 27

ð27-289Þ The expression for xz from Eq. (27-267) after using the value of A ¼ G

xz ¼ 0

The expression for yz from Eq. (27-268) after using the value of A ¼ G

yz ¼

The maximum shear stress

ðyz Þx ¼ a=3 max ¼

The shear stress at the center of triangular bar

ðyz Þx ¼ 2a=3

3G 2a

ð27-290aÞ

2ax x2 3

Ga ð27-291aÞ 2 3G 2ax x2 ¼ ¼ 0 ð27-291bÞ 2a 3 x ¼ 2a=3

Ga4 3 pﬃﬃﬃ ¼ GIp 15 3 5

The torque Mt ﬁlter substituting the value from Eq. (27-289) into Eq. (27-272) and carrying out integration and simpliﬁcation

Mt ¼

For shear stress variation along x-axis

Refer to Fig. 27-53.

a

2a 3

o

a 2

ð27-290bÞ

x

y

FIGURE 27-53 Equilateral triangle bar under torsion.

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ð27-292Þ

APPLIED ELASTICITY

27.65

APPLIED ELASTICITY

Particular

Formula

Membrane analogy @2z @2z p þ ¼ T @x2 @y2

Equation of equilibrium of the element klmn

where

Comparing the statement and Eq. (27-293) with Eqs. (27-270) which have been derived for stress function , it can be seen that two problems are identical

p ¼ pressure per unit area of the membrane T ¼ uniform tension per unit length of the membrane z is zero at the edges of the membrane

The quantities which are analogous to each other between torsion and membrane problems are

z is analogous to p=T is analogous to F ¼ 2G

By analogy in terms of stress function and hence in terms of yz and xz from Eq. (27-267) it can be shown that

@ @ @y @ @x ¼ @s @y @s @x @s

The magnitude of the shearing stress at k

¼ yz cosðN; xÞ xz cosðN; yÞ @ dz @ dy d þ ¼ ¼ @x dn @y dn dn

By analogy

¼

T

T dx m dx dy n k T dx l

T dy

b

z dz dy

a o

dz dx

z o’ T

Tdy

z x

dz d2z + dx dx2

(a) z+ dz dx

(b)

z

x

ð27-295Þ

dz dn

ð27-296Þ

Tdx T dz d2z + dy dy2 z+ dz

a

T dy

ð27-294Þ

Maximum slope of the membrane at this point

The resultant shear stress

b

@y @x yz ¼0 @x @s

¼ xz

This proves that the projection of the resultant shear stress at a point k (Fig. 27-56) on the normal N to the contour line is zero

y

ð27-293Þ

T

dy y

(c)

FIGURE 27-54 Membrane subjected to uniform tension at the edges and uniform lateral pressure q.

r z y

O

Tdy T

x

q y

Tdx T

T

O

o

τyz τxz

k

dn dx N dy

FIGURE 27-55

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τ

x

x

APPLIED ELASTICITY

27.66

CHAPTER TWENTY-SEVEN

Particular

Formula

By analogy the slope of the membrane in the direction of the normal is obtained from

This proves that the magnitude of the shearing stress at B is given by the maximum slope of the membrane at this point. @z @z p @n ¼ or ¼ ð27-297Þ p 2G @n 2G t t ð @z ¼ pA ds @n ð ds ¼ 2GA ð27-298Þ

For equation of equilibrium of the portion of the membrane (Fig. 27-55)

where A ¼ horizontal projection of the portion qr of the membran (Fig. 27-55) The membrane analogy can be used to solve problems of build up narrow cross sections, hollow sections, thin tubes, thin webbed tubes, box sections, etc. which are subjected to torsion

Torsion of hollow sections and thin-walled tubes (Fig. 27-57) Equating forces in the two directions acting on an element of hollow section as shown in Fig. 27-56

These conditions can be satisﬁed only if q is constant The torque

@q ds dl q dl ¼ 0 or @s @q qþ dl ds q ds ¼ 0 or @l

qþ

FIGURE 27-56

ð27-300Þ

ð27-301bÞ

where A ¼ area enclosed by the median line of the tubular section. S

qdl ∂q q+ dl ds ∂l

ð27-299bÞ

Mt ¼ q2A

ρ

∂q ds dl ∂s

dq ¼0 @l

ð27-301aÞ

o q+

ð27-299aÞ

q ¼ t ¼ constant ð ð Mt ¼ q ds ¼ q ds

ds qds

@q ¼0 ds

y D

h A

x δ

z o

C B

x

FIGURE 27-57

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APPLIED ELASTICITY APPLIED ELASTICITY

Particular

27.67

Formula

Shear ﬂow in force per unit length

q¼

The shear stress

¼

Mt 2A

ð27-302Þ

q Mt ¼ t 2At where

ð27-303Þ

t ¼ thickness of hollow section or tubular section d ¼ distance from O to mid line or median line of hollow sections or the tubular section (Fig. 27-56)

Torsion of thin-walled tubes (Fig. 27-57) The shear stress at any point is given by the slope of the membrane

h where

¼

ð27-304Þ

h ¼ the diﬀerence in level between DC and AB of membrane ¼ variable thickness of thin-walled tube A ¼ mean area enclosed by the outer and inner boundaries of the cross-section of the tube S ¼ the length of the center line of the ring section of the tube (Fig. 27-57) The expression for torque

Mt ¼ 2Ah ¼ 2A

ð27-305Þ

The shear stress

¼

Mt 2A

ð27-306Þ

The angle of twist for tube with uniform thickness

¼

Mt S 4A2 G

ð27-307Þ

Refer to Fig. 24-71 and Eqs. (24-353a) to (24-360) in Chapter 24.

Torsion of thin-welded tubes (or box beams) (Fig. 27-58) For calculation of each item, assume anticlockwise path for each shell The torque

Mt ¼ 2 volume under membrane 2 ¼ ½2A1 h1 þ A1 h2 ¼ 16 3 a 2

For the section shown in Fig. 27-58, the shear stresses

1 ¼

h1 ;

1 ¼

h1 5Mt ¼ 32a2

2 ¼

h2 ;

3 ¼

ð27-308Þ

h1 þ h2

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ð27-309aÞ

APPLIED ELASTICITY

27.68

CHAPTER TWENTY-SEVEN

Particular

Formula

τ

τ2

τ1 ϕA τ1

δ

τ1

A δ

τ3

h1

O

τ1 A

δ

τ1

a (a)

τ3

τ2

a

u

a δ

ds

+s

τ

ρ

A’ du

A dz

dz

τ1

z

a

(a)

h1

h2

θ

(b)

(b)

FIGURE 27-58

FIGURE 27-59

2 ¼

h2 3 Mt ¼ 16 a2

ð27-309bÞ

3 ¼

h1 h2 Mt ¼ 32a2

ð27-309cÞ

7 Mt 32 a2 G

ð27-310Þ

Mt 32a2 G ¼ 7

ð27-311Þ

The angle of twist of section

¼

The torsional rigidity of section shown

C¼

Warping of a box section (Fig. 27-59) The shearing strain

@u @w sz þ ¼ @z s G where

s ¼

ð27-312Þ

u ¼ displacement in direction s w ¼ displacement in direction z ¼ angle of twist per unit length The expression for incremental displacement in s-direction

du ¼ dz

ð27-313Þ

The slope of the warped cross-section

@w ¼ @s G

ð27-314Þ

TORSION OF CIRCULAR SHAFT OF VARIABLE DIAMETER (Fig. 27-13) For Figure of circular shafts of variable diameter

Refer to Fig. 27-13

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APPLIED ELASTICITY APPLIED ELASTICITY

Particular

For strains and displacements of a solid of revolution twisted by couples applied at ends

Diﬀerential equations of equilibrium when the body force is absent

Since it is a problem of symmetry

27.69

Formula

"¼

@u ; @r

" ¼

u 1 @v þ ; r r @

r ¼

@u @v v þ r d @r r

rz ¼

@w @u þ ; @r @z

"z ¼

@w @z ð27-315aÞ

@w @v þ r d @z

ð27-315bÞ

@r 1 @r @rz r þ þ þ ¼0 @r @z r r @

ð27-316aÞ

@rz 1 @z @z rz þ þ þ ¼0 @r @z r r @

ð27-316bÞ

@r 1 @ @z 2r þ þ þ ¼0 @r @z r r @

ð27-316cÞ

"r ¼ " ¼ "z ¼ rz ¼ 0

ð27-317aÞ

r ¼

@v v ; @r r

z ¼

z ¼

@v @z

ð27-317bÞ

The third of Eqs. (27-316c) can be written as

@ 2 @ 2 ðr r Þ þ ðr z Þ ¼ 0 @r @z

Eq. (27-318) is satisﬁed if the stress function given here used

r2 r ¼

From Eqs (27-317) and (27-319)

@v v @r r @ v 1 @ ¼ Gr ¼ 2 ð27-320Þ @r r r @z @v @ v 1 @ ¼ Gr ð27-321Þ z ¼ Gz ¼ G ¼ 2 @z @z r r @r @ 1 @ @ 1 @ þ ¼0 ð27-322aÞ @r r3 @r @z r3 @z

From the above equation it follows

@ ; @z

r2 z ¼

@ @r

ð27-318Þ ð27-319Þ

r ¼ Gr ¼ G

or

@ 2 3 @ @ 2 ¼0 þ @r2 r @r @z2

ð27-322bÞ

This Eq. (27-322) gives the angle of twist as a function of and z. Boundary conditions

Refer to Fig. 27-50.

The equation which gives the stress function constant value along the boundary of the axial section of the shaft

@ @z @ ds þ ¼0 @z @s @r ds

ð27-324Þ

Equation (27-322) together with the boundary condition (27-324) completely determine the stress function .

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APPLIED ELASTICITY

27.70

CHAPTER TWENTY-SEVEN

Particular

The torque for cross-section ϕ

Formula

ða

Mt ¼

2 r2 z dr

0

r

Mt ¼ 2

ða 0

ð27-325Þ

@ dr ¼ 2 jja0 @r

ð27-326Þ

where a ¼ outer radii of the cross-section

α

z

FIGURE 27-60

Conical shaft (Fig. 27-60) The angle is given by pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ The ratio ðx= r2 þ z2 Þ is constant at the axial section

z cos ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r2 þ z2

The stress function which satisﬁes the boundary condition Eq. (27-324) and equilibrium Eq. (27-316) is taken as

z 1 ¼ c pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r2 þ z2 3

The expression for z

z ¼

Substituting Eq. (27-327) in Eq. (27-325), the value of constant C can be obtained

c¼

The angle of twist

"

ð jÞ

z pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r2 þ z2

3 # ð27-327Þ

where c ¼ constant

¼

1 @ crz ¼ 2 r2 @r ðr þ z2 Þ5=2

ð27-328Þ

Mt 2 ð23 cos þ 13 cos3 Þ

ð27-329Þ

c 3Gðr2 þ z2 Þ3=2

ð27-330Þ

PLATES Diﬀerential equation for cylindrical bending of plates (Figs. 27-61 and 27-62) The unit elongation of a ﬁber at a distance z from the middle plane or surface of plate (Fig. 27-62a)

"x ¼ z

d 2w dx2

ð27-331Þ

where d 2 w=dx2 ¼ the curvature of the deﬂection curve of the plate w ¼ deﬂection of the plate in z direction which is very small compared to the length of plate l The normal bending stress bx

bx ¼

Ez d 2 w 1 v2 dx2

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ð27-332Þ

APPLIED ELASTICITY APPLIED ELASTICITY

Particular

27.71

Formula

σy I

h 2

x

h 2

1 (unit width)

σx

Mb

Mb dz

z

σx

(a) y

w

σx

σy

z

FIGURE 27-61 Bending of a plate due to lateral uniform load acting along the length of plate

σx

(b) Element cut out from the bent plate as shown in Fig (a) FIGURE 27-62

The bending moment in the elemental strip of plate (Fig. 27-63)

Mb ¼

Eh3 d 2w d 2w ¼ D 2 2 12ð1 v Þ dx dx2

ð27-333Þ

where D¼

Eh3 ¼ flexural rigidity of plate ð27-333aÞ 12ð1 v2 Þ

Cylindrical bending of uniformly loaded rectangular plate with simply supported edges, which are restricted from movement (Fig. 27-63) The bending moment at any cross-section of plate at x distance from the left support Substituting Eq. (27-334) in Eq. (27-333) and making use of boundary conditions w ¼ 0 at x ¼ 0 and x ¼ l, the expression for w can be obtained after integrating as

qlx qx2 ð27-334Þ Fa w 2 2 2 3 l cosh u x 7 q 6 qx 2 17 w¼ 4 6 4 5 þ 2 ðl xÞ ul u D 2u D cosh 2 ð27-335Þ

Mb ¼

where u ¼ The expression for axial force

The expression for diﬀerence between deﬂection curve and chord,

The maximum bending moment

Fa D

3 Eh q2 l 5l 5 ul þ tan 2 ð1 v2 Þl D2 24u4 4u6 2u7 l ul þ 6 tanh2 ð27-336Þ 2 4u ð 5ð1 v2 Þl 1 l dw 2 dx ð27-337Þ ¼

¼ Eh 2 0 dx 2 d w ð27-338Þ Mb max ¼ D dx2 x ¼ 2l

Fa ¼

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APPLIED ELASTICITY

27.72

CHAPTER TWENTY-SEVEN

Particular

Formula q intensity of the uniform load per unit length

a q intensity of the uniform load per unit length

Fa ql 2

w

x

l

Fa

Fa Fa ql 2

x

Mbo ql 2

x

a

x

Mbo

w ql 2

l

z z

FIGURE 27-64

FIGURE 27-63

The maximum bending stress

b max ¼

Mb max Z

ð27-338aÞ

where Z ¼ section modulus ¼ The axial stress due to Fa The total stress tot

a ¼

h2 1 6

Fa h1

tot ¼

ð27-338bÞ

Mb max Fa þ Z h

ð27-339Þ

Cylindrical bending of uniformly loaded rectangular plates with built-in edges (Fig. 27-64) Bending moment at any section of plate x-distance from left end

Substituting Eq. (27-340) in Eq. (27-333) and making use of boundary conditions w ¼ 0 at x ¼ 0; dw=dx ¼ 0 for x ¼ 0 and x ¼ l=2, the expression for w can be obtained after integrating and simpliﬁcation

qlx qx2 Fa w þ Mbo ð27-340Þ 2 2 where Mbo ¼ fixed end movement 2 3 2x 4cosh u 1 l 5 ql 4 w¼ 1 cosh u 16u3 D tanh u

Mb ¼

þ

ql 2 ðl xÞx 5u2 D

where u2 ¼ The expression for Mbo Mbo ¼

Fa l 2 D 4

ql 2 ql 2 ql 2 coth u ¼ 2 4u 12 4u

where

ð27-341Þ

1 ðuÞ

¼

1 ðuÞ

3ðu tanh uÞ u2 tanh u

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ð27-342Þ ð27-343aÞ

APPLIED ELASTICITY APPLIED ELASTICITY

Particular

The stresses are

Formula

1 ¼

Eu2 3ð1 v2 Þ

2 h l

6M q 2 ¼ 2bo ¼ 2 h The maximum stress, The maximum deﬂection at x ¼ 12

27.73

ð27-344aÞ

2 l h

1 ðuÞ

ð27-344bÞ

max ¼ 1 þ 2 wmax ¼

ð27-345Þ

ql 4 f ðuÞ 384D 1

24 where f1 ðuÞ ¼ 4 u

ð27-346Þ

u2 u u þ 2 sinh u tanh u

ð27-346aÞ

Cylindrical bending of plate on an elastic foundation (Fig. 27-65) A strip of plate cut out from main plate when treated as a beam on an elastic foundation, it can be shown that

D

@4w @4w k q ¼ q kw or þ w¼ 4 D @x @x4 D

ð27-347Þ

where q ¼ intensity of the uniform load acting on the plate k ¼ reaction of the foundation per unit area for a deﬂection equal to unity

The general solution of Eq. (27-347) consists of complementary function and particular integral which can be written as

w ¼ c1 sin x sinh x þ c2 sin x cosh x þ c3 cos x sinh x þ c4 cos x cosh x q þ ð27-348Þ k where 44 ¼

By taking coordinate as system shown in Fig. 27-65 and making use of symmetry and boundary conditions w ¼ 0 ¼ @ 2 w=@x2 at x ¼ l=2, the ﬁnal equation for deﬂection w can be written as The maximum deﬂection wmax at x ¼ 0

The maximum bending moment of the strip of plate at the middle plane and is obtained by

w¼

k D

q 2 sin sinh 1 sin x sinh x k cos 2 þ cosh 2 2 cos cosh cos x cosh x ð27-349Þ cos 2 þ cosh 2

ðwÞx ¼ 0 ¼ wmax ¼ Mb max ¼ D

q 2 cos cosh 1 ð27-350Þ k cos 2 þ cosh 2

@2w @x2

x¼0

where w is from Eq. (27-349)

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ð27-351Þ

APPLIED ELASTICITY

27.74

CHAPTER TWENTY-SEVEN

Particular

Formula

q intensity of uniform load per unit length O

K

x

L l 2

l

w

x

l

dx

y k

Slope of the plate surface in the y-direction Slope of the surface of plate in n-direction (Fig. 27-66) For y-direction of maximum slope or tan 1

For zero slope,

From Eqs. (d) and (e)

The expression for maximum slope The curvature of the surface of a plate in a plane parallel to xz plane (Fig. 27-67) is The curvature of the surface of a plate in a plane parallel to yz is The twist of the surface of the plate with respect to the x and y axes The curvature of the middle surface of plate in any direction n (Fig. 27-67) is

x b

l

dx

dn

dy t

(a)

y

(b)

m α n

FIGURE 27-66

Pure bending of plates Slope of the plate surface in the x-direction (Fig. 27-66)

dw

x m

z

z

FIGURE 27-65

O

O

Lt

w dw ¼ x dx

ðaÞ

Lt

w dw ¼ y dy

ðbÞ

x ! 0

y ! 0

@w @w dx @w dy @w @w ¼ þ ¼ cos þ sin ðcÞ @n @x dn @y dn @x @y @ @w @w @w ¼0¼ sin þ cos @ @n @x @y @w @y ðsay ¼ 1 Þ ðdÞ or tan 1 ¼ @w @x @w @w @x ðsay ¼ 2 Þ ¼ 0 or tan 2 ¼ ðeÞ @w @n @y tan 1 tan 2 ¼ 1 ðfÞ Therefore the two directions of 1 (maximum slope) and 2 (zero slope) are perpendicular to each other. sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ @w @w 2 @w 2 ¼ þ ðgÞ @n max @x @y 1 @2w ¼ 2 rx @x 1 @2w ¼ 2 ry @y

ðhÞ ðiÞ

1 @2w ¼ ðjÞ rxy @n @y 1 @ @w 1 1 1 ¼ cos2 ¼ sin 2 þ sin2 rn @n @n rx rxy ry ð27-352Þ

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APPLIED ELASTICITY

27.75

APPLIED ELASTICITY

Particular

Formula

The sum of curvature or average curvature of the middle surface of plate at a point

The twist of the surface of plate at b with respect to bm and bt directions Thus twist of the surface of plate

1 1 1 1 þ ¼ þ rn rt rx ry

ð27-353Þ

This shows that the sum of curvature in n and t two perpendicular directions is independent of angle . 1 d dw ¼ ð27-354aÞ rnt dt dn 1 1 1 1 1 ¼ sin 2 þ cos 2 rnt 2 rx ry rxy

ð27-354bÞ

2 rxy tan 2 ¼ 1 1 rx ry

Bending moments and curvature in pure bending of plates (Fig. 27-67)

z ; rx

The unit elongation in x and y directions of an elemental lamina klmO0 at a distance z from the neutral surface (Fig. 27-68)

"x ¼

The corresponding stresses in the lamina klm 00

Ez x ¼ 1 v2 ¼ y ¼

"y ¼

Ez 1 v2

Ez 1 v2

¼

Ez 1 v2

ð27-355Þ

z ry

ð27-356Þ

1 1 þv rx ry

@2w @w2 þ v @x2 @y2

1 1 þv ry rx

ð27-357aÞ

@2w @2w þv 2 @y @x2

ð27-357bÞ dx

dy My

O

n

x

Mx

Mx

n

n z

k

O’

l

O y

My

y

z

FIGURE 27-67

m

h 2 h 2

dz

z

FIGURE 27-68

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x

APPLIED ELASTICITY

27.76

CHAPTER TWENTY-SEVEN

Particular

Formula

The external moments per unit length of element (Fig. 27-68); which are equal to internal couple on the element O

Mby

k

α

Mbx

m y

x

O

l

k

Mbnt

Mx σx α

Mbn (a)

My

h=2

x

σy

τnt

Mbx ¼

ð h=2

l

σn

ð27-358Þ Mby ¼

ð h=2 h=2

m y

1 v x z dy dz ¼ D þ rx ry 2 @ w @2w þ v ¼ D @x2 @y2

1 v þ ry rx 2 @ w @2w þ v ¼ D @y2 @x2

y z dx dz ¼ D

(b)

ð27-359Þ

FIGURE 27-69

Bending moment acting on a section inclined to x and y axes The normal component n and shearing component nt acting on the element klm (Fig. 27-69) in the n and t directions respectively

n ¼ x cos2 þ y sin2

ðaÞ

nt ¼ 12 ðx y Þ sin 2

ðbÞ

The magnitude of the bending moment Mbn per unit length along ml (Fig. 27-69)

Mbn ¼ Mbx cos2 þ Mby sin2

ð27-360Þ

The twisting moment Mbnt per unit length acting on the section ml of the plate

Mbnt ¼ 12 ðMbx Mby Þ sin 2

ð27-361Þ

Substituting in Eq. (27-360), the expressions for Mx and My from Eqs. (27-358) and (27-359), the moment Mbn (Fig. 27-71)

The shearing strain The corresponding shearing stress Figure 27-70(b) shows a section of the middle surface of the plate made by the normal plane through n-axis, ‘ab’ was an element which was initially perpendicular to the xy-plane.

1 1 cos2 þ sin2 rx ry 1 1 þ vD sin2 þ cos2 rx ry 1 1 ¼D þv rn rnt 2 @ w @2w ¼ D þ v @n2 @t2 @u @v nt ¼ þ @t @n @u @v þ nt ¼ G @t @n Mbn ¼ D

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ð27-362Þ ðcÞ ðdÞ

APPLIED ELASTICITY APPLIED ELASTICITY

Particular

Formula

x

O

O

n

p dt

ν + ∂ν dt ∂t

u

s t

s’ u + ∂u dt ∂t

y

(a)

a dn

p’ ν

r

q

27.77

u + ∂u dn ∂n

n q’ ν + ∂v dn ∂n

z

O

∂w ∂n b

φ R

(b)

Mb

Mb

B

A

r’

b

FIGURE 27-70

The displacement in n and t directions

FIGURE 27-71

u ¼ z

@w @n

ðeÞ

v ¼ z

@w @t

ðfÞ

Substituting the values of n and v from Eqs. (e) and (f), Eq. (d) becomes

nt ¼ 2 Gz

The twisting moment, Mbnt

Mbnt ¼

@2w @n @t

ð h=2

h=2

nt z dz ¼

¼ Dð1 vÞ

Strain along the circumference of circular plate acted by uniform edge moments (Fig. 27-71)

ð27-363aÞ Gh3 @ 2 w 6 @n @t

@2w @n @t

3R

The circumferential strain at the edge of the plate

"¼

The maximum bending strain

"maxðbendingÞ ¼

ð27-363bÞ

ð27-364aÞ

h ð27-364bÞ 2R where R ¼ radius as shown in Fig. 27-71

Lagrange’s diﬀerential equation for laterally loaded plates (Figs. 27-72, 27-73, 27-74) Equations of equilibrium for plate

@Qx @Qy þ þ qðx; yÞ ¼ 0 @x @y @Mx @Mxy þ ¼ Qx @x @y @My @Mxy ¼ Qy @y @x

Lagrange’s bending moment equilibrium equation for plate

@ 2 Mxy @ 2 My @ 2 Mx þ 2 ¼ q @x @y @x2 @y2

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ð27-365Þ

ð27-366Þ

APPLIED ELASTICITY

27.78

CHAPTER TWENTY-SEVEN

Particular

z dy

Formula

dx

∂My My+ dy ∂y ∂Myx Myx+ dy ∂y

Mx+

My

∂Mx dx ∂x

Mxy+

∂Mxy dx ∂x

Myx

x ∂Qx Qx+ dx ∂x

y

Myx

Mx

My+

∂Qy Qy+ dy ∂y

∂My dy ∂y

Mxy+ Myx+

∂Myx dy ∂y

FIGURE 27-73

Lagrange’s equilibrium equation for deﬂection surface of plate

@4w @4w @4w q þ2 þ 4 ¼ 4 2 2 D @x @x @y @y

The normal stresses are

xz max ¼

3 Qx 2 h

ð27-369aÞ

yz max ¼

3 Qy 2 h

ð27-369bÞ

xy max ¼

6Mxy h2

ð27-369cÞ

x max ¼

6Mx ; h2

y max ¼

6My h2

ð27-370Þ

Qy

Mxy

Mxy

ð27-367Þ

Qx ¼

The shearing stresses are

x ∂Mxy Mxy+ dx ∂x ∂Mx Mx+ dx + ∂x ∂My My+ dy y ∂y y ∂Mxy Mxy+ dy ∂y (b)

∂Mxy dx ∂x

@Mxy @Mx @ @2w @2w þ ¼ D þ ð27-368aÞ @y @x @x @x2 @y2 @My @Mxy @ @2w @2w ð27-368bÞ ¼ D þ Qy ¼ @y @x @y @x2 @y2

The expression for shearing forces are

My

∂Mx dx ∂x

(a)

FIGURE 27-72

Mx

Mx+

Qx

x q(x,y) Qx+

Qy+

∂Qx dx ∂x

x=a

∂Qy dy ∂y

h

(c)

FIGURE 27-74 Stresses, shearing forces and moments at the middle surface of a plate

FIGURE 27-74A

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APPLIED ELASTICITY APPLIED ELASTICITY

Particular

27.79

Formula

Boundary conditions SIMPLY SUPPORTED EDGE (Fig. 27-74) Boundary conditions for plates at x ¼ a from the simply supported edge

The deflection ðwÞx ¼ a ¼ 0

ð27-371aÞ

The bending moment ðMbx Þ 2 @ w @2w ¼ D þ v ¼0 @x2 @y2 x ¼ a

ð27-371bÞ

or ðwÞx ¼ 0 ¼ 0 BUILT-IN EDGE Boundary conditions for plates at x ¼ a from the built-in edge

FREE EDGE (Fig. 27-75) The total shearing force Qx along the free edge x ¼ a

The deﬂection ðwÞx ¼ a ¼ 0 @w ¼0 The slope @x x ¼ a Vx ¼

Qx

@Mxy @y

ð27-372Þ

x¼a

¼0

ð27-373Þ

FIGURE 27-75

@ 3 wxy @3w þ ð2 vÞ ¼ 0 for all y ð27-374Þ @x @y2 x ¼ a @x3 2 @ w @2w þ v ¼0 Also ðMx Þx ¼ a ¼ D @x2 @y2 x ¼ a ð27-375aÞ or 2 @ w @2w þ v ¼ 0 for all y ð27-375bÞ @x2 @y2 x ¼ a

The magnitude of total reaction which is due to unbalanced force at the corners (Fig. 27-75)

R ¼ 2Mxy ¼ 2Dð1 vÞ

Eq. (27-373) in terms of w Mxy

R = total reaction Mxy

@2w @x @y

x¼a y¼b

ð27-376Þ

ELASTICALLY SUPPORTED EDGE (Fig. 27-76) STRAIGHT BOUNDARY (Fig. 27-76) The pressure transmitted in z-direction at the junction of the plate and beam transmitted from the plate to the supporting beam

Diﬀerential equation of the deﬂection curve of elastic support

@Mxy Vx ¼ Qx @y x ¼ a 2 @ @ w @2w ¼D þ ð2 vÞ ð27-377Þ @x @x2 @y2 x ¼ a 4 @ w @ @2w @2w ¼ D þ ð2 vÞ B @x @x2 @y4 x ¼ a @y2 x ¼ a ð27-378Þ where B ¼ EI of support or beam

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APPLIED ELASTICITY

27.80

CHAPTER TWENTY-SEVEN

Particular

Formula

O

x

a

dy K

x

z

FIGURE 27-76

The rotational equilibrium of the element of the beam

CURVED BOUNDARY (Fig. 27-77) The expressions for bending moment and twist moments at point K

t

y

Mnt

α n

ds y

Qy

Qx

dx

Mn

t Qn

(a)

n (b)

FIGURE 27-77

G

2 @ @ w ¼ ðMx Þx ¼ a @y @x @y x ¼ a 2 @ w @2w ¼D þ @x2 @y2 x ¼ a

Mn ¼

ð27-379Þ

ð h=2 h=2

zn dz

¼ Mx cos2 þ My sin2 2Mxy sin cos ð27-380aÞ ð h=2 znt dz Mnt ¼ h=2

¼ Mxy ðcos2 sin2 Þ þ ðMx My Þ sin cos ð27-380bÞ From equation of equilibrium of an element at K of the curvilinear boundary the shear in the n-direction

Qn ¼ Qx cos þ Qy sin

If the curvilinear edge is a built-in edge then

w ¼ 0;

In case simply supported edge

w ¼ 0;

For the case of free edge of curvilinear plate

Mn ¼ 0;

For the case of free edge of curvilinear plate Eq. (27-380f ) can be represented for boundary condition as

@w ¼0 @n Mn ¼ 0

ð27-380dÞ ð27-380eÞ

V n ¼ Qn

ð27-380cÞ

@Mnt ¼0 @s

@2w @2w þ sin2 2 @x @y2 @2w þ sin 2 ¼0 @x @y

ð27-380f Þ

v w þ ð1 vÞ cos2

ð27-381Þ

@w @w @ þ sin þ ð1 vÞ @x @y @s " 2 # @2w 1 @ w @2w þ sin 2 ¼0 cos 2 @x @y 2 @y2 @x2

cos

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APPLIED ELASTICITY APPLIED ELASTICITY

Particular

27.81

Formula

where w ¼

@2w @2w @x2 @y2

Simply supported rectangular plates under sinusoidal loading (Fig. 27-78) General sinusoidal load is taken as

m x n y sin a b

ð27-382Þ

w ¼ C sin

m x n y sin a b where m and n are integers

ð27-383Þ

From Eq. (26-367)

@4w 2@ 4 w @4w q m x n y þ 2 2 þ 4 ¼ 0 sin sin 4 D a b @x @x @y @y

ð27-384Þ

Substituting Eq. (27-383) in Eq. (27-384) and solving for C

C¼

After substituting value of C in (Eq. 27-383), the expression for w

w¼

The general expression for deﬂection of plate

If m ¼ 1 and n ¼ 1 then Eq. (27-386) becomes

q ¼ q0 sin

D 4 D 4

w¼

D

4

q0 m2 n2 þ a2 b2 q0 m2 n2 þ a2 b2 q0 1 1 þ a2 b2

ð27-385Þ

2

m x n y sin a b

ð27-386Þ

x y sin a b

ð27-387Þ

2 cos

2 cos

x y sin a b

The sinusoidal load is taken as

q ¼ q0 sin

From Eq. (27-367) or Eq. (27-384) becomes

@4w 2@ 4 w @4y q x y þ 2 2 þ 4 ¼ 0 sin sin 4 D a b @x @x @y @y

The boundary conditions

ðwÞx ¼ 0 ¼ 0

ðMx Þx ¼ 0 ¼ 0

ðwÞy ¼ 0 ¼ 0

ðMy Þy ¼ 0 ¼ 0

x¼a

a

o

x sinusoidal loading

b

y¼b

x¼a

y¼b

or ðwÞx ¼ 0 ¼ 0 x¼a

y

z

FIGURE 27-78

ð27-388aÞ

ðwÞy ¼ 0 ¼ 0 y¼b

@2w @x2 @2w @y2

x¼0 x¼a

¼0

y¼0 y¼b

¼0

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ð27-388bÞ ð27-389Þ

APPLIED ELASTICITY

27.82

CHAPTER TWENTY-SEVEN

Particular

Formula

x y sin a b

The expression for deﬂection surface of plate which satisﬁes the boundary conditions Eq. (27-389) and Eq. (27-388b)

w ¼ C sin

Substituting Eq. (27-390) in Eq. (27-388b) and solving for C

C¼

Equation (27-390) for w becomes

q0 x y sin sin ð27-390bÞ 1 1 a b 4 D 2 þ 2 a b q 1 v x y Mx ¼ 0 þ sin sin 2 a b a2 b2 1 1 2 þ a2 b2 ð27-391aÞ q v 1 x y My ¼ 0 þ 2 sin sin 2 2 a b a b 1 1 2 2 þ 2 a b ð27-391bÞ

The expression for Mx , My and Mxy

4 D

1 1 þ a2 b2

q0 ð1 vÞ x y 2 sin sin a b 1 1 2 2 þ 2 ab a b

wmax ¼

4 D

Mx max ¼

q0 1 1 þ a2 b2

4 D My max ¼

2

The maximum deﬂection and bending moments for a square plate The shearing forces from Eqs. (27-368)

wmax ¼

Qx ¼

Qy ¼

ð27-390aÞ

2

w¼

Mxy ¼ The maximum deﬂection and bending moments, which occur at midpoint of plate

q0

ð27-390Þ

q0 a4 ; 4 4 D

1 1 þ a2 b2

1 1 þ a2 b2

ð27-392aÞ

2

q0

q0

ð27-391cÞ

2

2

1 v þ a2 b2

v 1 þ a2 b2

ð27-392bÞ

Mx max ¼ My max ¼

ð27-392cÞ

ð1 þ vÞq0 a2 4 2 ð27-393Þ

q x y cos sin 0 1 1 a b a 2 þ 2 a b q x y sin cos 0 1 1 a b b 2 þ 2 a b

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ð27-394aÞ

ð27-394bÞ

APPLIED ELASTICITY APPLIED ELASTICITY

Particular

The reactive forces at the support edges at x ¼ a and y ¼ b respectively

27.83

Formula

Vx ¼

Qx

@Mxy @y

q0 1 1 2 a 2 þ 2 a b

Vx ¼

ð27-395aÞ

x¼a

1 2v þ 2 a2 b

sin

y b

ð27-395bÞ @Mxy ð27-395cÞ V y ¼ Qy @x y ¼ b q 1 2v x þ sin Vy ¼ 0 2 2 2 a b a 1 1 b 2 þ 2 a b ð27-395dÞ The resultant reaction concentrated at the corners of the plate

The total pressure on all four edges of plate

R ¼ 2ðMxy Þx ¼ a ¼ y¼b

ðb 2 0

vx dy þ 2 ¼

The four corners reactions, which are equal due to symmetry

R ¼

2q0 ð1 vÞ 1 1 2 2 ab 2 þ 2 a b

ð27-396Þ

ða 0

4q0 ab þ 4

vy dx

8q0 ð1 vÞ 1 1 2 ab 2 þ 2 a b

ð27-397Þ

2

8q0 ð1 vÞ 1 1 2 2 ab 2 þ 2 a b

which is the second term on the right hand side of Eq. (27-397) The maximum bending stress if a > b is due to My which is greater than Mx

Using Eq. (27-395d), the expression for maximum shear stress which is at the middle of the longer side of the plate

6My max h2 6q0 v 1 ¼ þ 1 1 2 a2 b2 2 h2 2 þ 2 a b 3q0 1 2v ðyz Þmax ¼ þ 2 a 1 1 2 b2 2 bh 2 þ 2 a b y max ¼

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ð27-398Þ

ð27-399Þ

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