Design of members subject to combined bending and torsion

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

SCI PUBLICATION 057

Design of Members Subject to Combined Bending anu Iorslon m

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m -

D. A. NETHERCOT BSc(Eng) PhD CEng FlStructE University of Nottingham P. R. SALTER BSc(Eng)CEng MlStructE (formerly of) The Steel Construction Institute A. S. Malik BSC MSC The Steel Construction Institute

ISBN 1 870004 44 2

0The Steel Construction Institute 1989 (Reprinted, l99 7)

The Steel Construction Institute Silwood Park Ascot Berkshire SL5 7 0 N Telephone: 0 1 344 23345 Fax: 0 1 344 22944

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

This publication seeks to provide authoritative guidance for the design of steel structures subjected to combined bending and torsion. In most cases, however, it should be possible to avoid the introduction of significant torsion, by paying attention to detail andchoosing a load path forsuch an alternative. This publication is, therefore, concerned with the minority of cases where the loads have to be applied eccentrically with respect to the shear centre. Afterbrief a discussion of the background theory, simple methods of evaluating torsional stresses and deformations are detailed, and worked examples illustrating the use of tables and charts included in the publication are provided.

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The text has been prepared by Professor D A Nethercot of Nottingham University and Messrs. P R Salter and A S Malik of the Steel Construction Institute and reviewed by Mr J C Taylor and Dr R Narayanan of The Steel Construction Institute.

..

11

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

CONTENTS Page SUMMARY NOTATION 1.

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2.

DESIGNING FOR TORSION 1 .l Introduction 1.2 Scope of the Publication 1.3 Choice of Members

1 1 2

BASICTHEORY 2.1 Torsion 2.2 Bending 2.3 Combined Bending and Torsion

3 12 13

3.

LOADAPPLICATION

16

4.

WORKEDEXAMPLES

17

5.

SECTIONPROPERTIES

89

6.

CHARTS TO ASSIST IN EVALUATION 6.1 Standard Cases 6.2 Extreme Cases

98 99 109

REFERENCES APPENDIX A.

Evaluation of Torsional Properties

APPENDIX B. Solution

of Differential Equations

111 118

iii

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Design of Members Subject to Combined Bending and Torsion The causes of torsional loading on structural members are discussed and those situations in which the explicit consideration of torsion needs to form part of the design calculations are identified. The basic theory of the torsion of both open and closed steel sections is presented. Solutions of the resulting equations in terms of both design charts and formulae for a selection of applied torsional loadings and support conditions are provided. A simple method for combining the effects of torsion and bending, consistent with the approach of BS 5950: Part I is presented. The complete design approach for combined bending and torsion is illustrated by means of a number of worked examples. These show that design will frequently be governed by the need to restrict twisting at working load to acceptable levels, rather than by considerations of ultimate strength.

Dimensionnement des Elements Soumis a Flexion et Torsion Combinees

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Rbume Les raisons conduisant a un chargement par torsion des elements structuraux sont discutees et les situations ou une prise en compte explicite dela torsion est nkcessaire dans les calculs de dimensionnement sont identifiees. La thkorie classique de la torsion despiPces en acier a section ouverte ou fermee est prksentke. Des solutions des Pquations qui en resultent sont prksentees sous forme de diagrammes de dimensionnement, d'une part, et sous forme de formules, d'autre part. Elles permettent de sklectionner les charges de torsion a appliquer et les conditions d'appuis. Une mkthode simple pour combiner les effets de la torsion et de la flexion est presentee. Elle est en accord avec la norme BS 5950 : Partie 1. Le mkthode complkte de dimensionnement en flexionet torsion combinkes est illustree au moyen d'exemples. Ils montrent que le dimensionnement est souvent gouverne par la necessitk de restreindre les deformations torsionnelles, sous les charges de service, a des valeurs acceptables, plutbt que par la resistance ultime.

Berechnung von Bauteilen unter Biegung und Torsion

Zusammenfassung Die Ursachen f u r Torsionsbeanspruchung von Bauteilen werden besprochen und die Falle, in denen eine Berucksichtigung der Torsion klarer Bestandteil der statischen Berechnung sein muJ3. Die elementare Theorie der Torsion von offenen und geschlossenen Stahlquerschnitten wird vorgestellt. Die Losungen der sich ergebenden Gleichungen werden fur eine Auswahl von Torsionsbelastungen und Randbedingungen in Form von Bemessungstafeln und Formeln zur Verfugung gestellt. Eine einfache Methode fur kombinierte Beanspruchung aus Biegung und Torsion entsprechend BS 5950, Teil I , wird vorgestellt. Der vollstandige Weg zur Bemessung bei Biegung und Torsion wird anhand einer Reihe von Beispielen aufgezeigt. Die Beispiele zeigen, daJ3 die Bemessung oft von der Notwendigkeit bestimmt wird, die Verdrehung durch die angreifenden Lasten in akzeptablen Grenzen zu halten, weniger von dem Gesichtspunkt der Bruchfestigkeit.

iv

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Progretto di Membrature Soggette a Flessione e Torsione Sommario Vengono discusse le cause dell'impegno torsionale degli elementi strutturali e identificate quelle situazioni nelle qualiP necessario tenere conto esplicitamentedella torsione nei calcoli di pregetto. E presentata la teoria della torsione con riferimento ai profili apertie a quelli a sezione chiusa. Viene quindi riportata la soluzione delle equazioni che reggono il problema, in forma sia di abachi sia di formule, per una gammasignificativa di condizioni di carico e di vincolo. E altresi illustrato un metodo semplice che consente di combinare glieffetti della torsione e della flessione, metodo in accord0 con l'approcciodelle BS.5950: parte I. Una serie di esempi consentela comprensione del'approccio progettuale per elementi soggetti a flessotorsione nella sua completezza. Si mettein luce come la necessita di limitare a livelli accettabili la deformazione torsionale sotto i carichi di esercizio governa il progetto in molti casi, mentre la resistenza ultima riveste minore importanza.

Diseno de Piezas Sometidas a Flexion y Torsion Combinadas

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Resumen Se analizanlas causas que originancargas de torsi6n enlas piezas de unaestructura y se identifican aquellas situaciones en que debe incluirsela consideracion explicita de la torsion en 10s calculos de rrn proyecto. Se presenta la teoria basica de torsion en secciones de acero tanto abiertas como cerradas; t a m b i h se suministran soluciones de las ecuaciones resultantes mediante abacos de disenoy formulas para diferentes tipos cargas de torsoras y condiciones de apoyo. Se incluye, en particular un procedimiento sencillo de combinacion de efectos de flexiony torsion, congruente con el metodo patrocinado porla BS.5950: Parte I . Se desarrollan una serie de ejemplos que ilwtran e l me'todo de analisis completo para flexion y torsion, y demuestran que a menudo el diseno queda controlado por la necesidad de mantener la torsion en tensiones admisibles a niveles aceptables en lugar de por consideraciones deresistencia ultima.

V

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Area enclosed by mean perimeter of closed section Torsion bending constant equal to

[E]

Distance between toe of flange and centre line of web of channel section Beam depth Torsional modulus, constant for closed sections Distance from centre of web to shear centre of channel section Modulus of elasticity of steel (205000 N/mm2) Eccentricity of load with respect to the shear centre

E Shear modulus of elasticity of steel (taken to be 2( 1 +v) ' where Poissons ratio v = 0.3, thus E/G = 2.6 and G has an approximate value of 79000 N/mm2) Warping constant for cross section Depth of open section, centre to centre closed section

of flanges; mean perimeter

of

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Second moments of area of cross section about the major and minor axes Torsional constant for cross section Length of member Bending moment acting on cross section Applied load Statical moment (about the neutral axis of the entire cross section) of the crosssectional area between the free edges of the cross sectionanda plane cutting the cross section across the minimum thickness at the point under examination Value of Q for a point in the flange directly above the vertical face of the web Value of Q for a point at mid-depth of section Plastic modulus about the major and minor axes Warping statical moment at a point

'S'

on cross section

Flange thickness Pure torsional resistance equal to GJ+' Applied torque (torsional moment) at given location Warping torsional resistance equal to EH@'' Thickness generally; web thickness Uniformlydistributedappliedtorque(torsionalmoment)or value of varying applied torque

maximum

Shear acting on cross section Distance from toe of channel section to point on flange where W,, X B' flange is maximum, given by W n o + Wn2 vi

T~

in the

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

W”,

Normalised warping function at a point

xo, Yo

Co-ordinates of the shear centre with respect to the centroid

Y

Perpendicular distance from neutral axis to a point on cross section

Z

Distancefrom left end of member(origin of co-ordinatesystem)to transverse section under examination (Figure 2.3)

‘S’

on cross section

Elastic moduli about the major and minor axes Distance from support to point of applied torsional moment (or to end of uniformly distributed load over a portion of span), divided by the span length (i.e. aL is the distance, (Y is a fraction of L ) Total angle of twist at a transverse section of member, radians First derivative of 4 with respect to z Second derivative of 4 with respect to z

4 with respect to z Fourth derivative of 4 with respect to z

Third derivative of

Combined longitudinal stress Longitudinal stress due to plane bending Warping normal stress, i.e. longitudinal stress at a point on cross section due to restrained warping of the cross section Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

7-

Combined shear stress Shear stress due to plane bending Pure torsional shear stress Warping shear stress at point on cross section of the cross section

due to restrained warping

vii

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

1. 1.l ,Introduction When a member is subject to torsion it will twist about a longitudinal axis which passes through the shear centre of the cross section. However, torsion will not force passes occur if the section is loaded in such a manner that the resultant through the shear centre. In the majority of design situations, the loads are appliedso that the resultant force passes through the centroid. If the section is doubly symmetric, this automatically eliminates torsion because the centroid and the shear centre coincide.

In most cases, the load transfer through the connections of the members applying the loads may be regarded as ensuring that these loads are effectively applied through the shear centre. This is also generally true of loadsfromfloor slabs supported on the top flange of beams, even for channel sections.

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Designing to transferloads by means of torsionshould be avoidedwherever possible, as it is not usually an efficient method of resisting loads. When this is not possible, care should be taken to arrange framing so as to minimise any torsion. Attention to detail, particularly when considering how loads are actually transferred to members,can minimise or even eliminate many potentialdifficulties associated with torsional effects. Where significant torsional eccentricity is unavoidable, consideration should be given to the use of box girders, comprising either a lattice girder fully triangulated on all faces or hollow rolled or plated sections. The assumptions made when using a computer program to analyse a grillage or three-dimensional framework should also be considered. If the members andjoints are assumed to have torsional resistance, then torsional moments will be included in the output. In order to maintain equilibrium with the applied loads, these must then be taken into accountin designing the joints and the members. If, on the other hand, the members and joints are assumed not to have torsional resistance, no torsional moments will arise and the remaining moments and forces will be in equilibrium with the applied loads. In most cases this approach will be the more practical. However, this assumption should not be used for fatigue analysis. The aboveis an exampleof a broader principle, which is valid due to theductility of steelwork. Unless it is necessary to utilise the torsional resistance of a member, it is not necessary to take account of it. As always, the details of the joints must be made consistent with the assumptions made in the analysis.

1.2

Scope of thepublication

This publication is concerned with the minority of cases where the loadis eccentric to the shear centre. Whilst it is important to recognise and deal with such cases when they occur, it is also important not to apply its methods where they are not necessary. When loading is eccentric with respect to the shear centre, the response of the member may conveniently be examined by separating the loading into bending and torsional components. Bending stresses and deflections can be obtainedin the usual manner by assuming that the loadsact through the shear centre and resolving the forces into components parallel to theprincipal axes. Torsional stressesand deformations can be calculated for standard cases using the Tables (Section 5 ) and Graphs (Section 6) in this publication. For non-standard cases and for determination of the torsional effects at other than the critical positions, equations have been provided in Appendix B. The user may then choose between hand or computerised methods of calculation.

1

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

The combined bending and torsional effects (Section 2.3) are then checked by means consistent with the methods used in BS 5950: Part 2.(l3) This publication is principally concerned with providing guidance for the design of hot rolled open sections. However, guidance is also given on the design of hot rolled tubular sections but reference should be made to more detailed literature for the design of Box Girders.(') Memberswhich are curved on plan or which contain particularly slender plate elements, e.g.cold formed sections, are not consideredin this publication. Detailed guidance on these topics is given in References 2 to 5 . Examples in Section 4 have been provided to illustrate the use of the Tables (Section 5) and Graphs (Section 6) for standard cases.

1.3 Choice of member The initial choice of member in design situations not affectedby torsion tends to be governed by the proportionsof axial load to bending moment and the unrestrained length of the section. For members predominantly subject to bending, an I section such as a universal beam will produce an efficient design. Similarly for members subject to axial loading, a universal columnH section is a reasonable choice.When the unrestrained length of the member is high, hollow sections can be advantageous.

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Torsional loading also has a significant influence on the initial choice of section for maximum efficiency. For reasons which are explained later in this publication, I shaped sections are particularly poor at resisting torsion while tubular sections can be very effective. Although H sections are better at resisting torsion than I sections, they are still a poor choice compared with a tubular member. Torsional effects should, therefore, be taken into account early in the design process when the type of member to beused is under consideration and not left to the final stages when perhaps an inappropriate type of member has already been selected. Not only may lighter sections result, but the design time will also be reduced. A distinction is made in this publication between open sections such as I and channel section shapes which are poor at resisting torsion and closed sections such as tubular members which are more effective (Figure 1.1).

OpenClosed sections Figure 1.1 Choice of section

2

sect ions

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

2. When a member is loaded so that the resultant of the applied forces does not pass through the shear centre of the section, the member will be subject to additional stresses due to torsion as well as those due to bending. In the method presented in this publication the effects of torsion and bending are first considered separately and then combined, as explained in Section 2.3.

2.1

Torsion

2.1 .l Shearcentre The shear centre of a cross section lies on the longitudinal axis about which the section would twist if torsion acts on the section. If the resultant force acts through the shear centre, no twist will occur and the torsional stresses will be zero.

The shear centre and the centroid are not necessarily coincident. However, in a rolled I or H section, which is symmetrical about both principal axes, the shear centre, S, coincides with the centroid, c (Figure 2.la). This is also true for sections which are point symmetric such as zed sections (Figure 2. lb).

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For a channel section there is a symmetry about the x-x axis only and, therefore, the shear centre lies on the x-x axis but not the y-y axis (Figure 2 . 1 ~ ) Conversely, . for sections which have symmetry about the y-y axis only, the shear centrelies on the y-y axis but is eccentric to thex-x axis (Figure 2.ld). When the channelsection is asymmetric the shear centre is eccentric to both axes (Figure 2.le). Methods of calculating the position of the shear centre of a cross section are given in Appendix A. Special cases such as angles and tees where the centrelines of the elements intersect at a single point have the shear centre located at that point (Figure 2.lf). 2.1.2 Torsionalresistance The total resistance of a member to torsionalloading is composed of the sum of two components known as ‘uniform torsion’ and ‘warping torsion’.

In some cases only uniform torsion occurs. When warping is included in the torsional resistance, the member is in a state of ‘non-uniform torsion’. Uniform torsion is also referred to as ‘pure’ or ‘St Venant’ torsion. When uniform torsion occurs, the rate of change of the angle of twist is constant along the member (Figure 2.2a).

3

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Y

V

I

X --

IF \

f-X

’U

U

\V

I

I

Y (b) Point symmetric sections

(a) Doubly symmetric sections

Y I

Y

I I

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X+-

I

F-

x

-X

Y

( c ) Sections symmetric about major axis

YI I

(d) Sections symmetric about minor axis

Y

%I

Y

X-

X--

yo

-.

i

Y Asymmetric sections

1 xI 01LY

(f) Sections with a single junction

Figure 2.1 Shear centre ‘S’and centroid ‘c’

4

x 0 4 iY

I S

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Constant torque ends f r e e t o warp

(a) Uniform torsion

Constant torque end warping prevented

(b) Non-uniform torsion

Varyingtorque

-3-

n

(c) Non-uniform torque Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

Figure 2.2

Uniform and non uniform torsion

In this case the applied torque is resisted entirely by shear stresses distributed throughout the cross section. The ratio of the applied torque to the twist per unit length is equal to the torsional rigidity, GJ, of the member, where G is the shear modulus and J is the torsional constant. J is sometimes called the ‘St Venant’ torsion constant. However, when the member is in a state of non-uniform torsion, the rate of change of the angle of twist varies along the length of the member. An example would be a cantilever with an applied torque at the free end and theflanges restrained against warping at the fixed end (Figure 2.2b). Alternatively, for a simply supported beam with an applied torque at the centreof the span, considerations of symmetry about the centreline of the span mean that the cross section must remain plane during twisting and, therefore, the rate of change of angle of twist must vary throughout the span (Figure 2 . 2 ~ ) . In both these cases the warping deflections due to the bending of the flanges vary along the length of the member. Both direct and shear stresses are generated which are additive to those due to bending and pure torsion respectively. The stiffness of the member associatedwith these additional stressesis proportional to the warping rigidity, E H , where E is the modulus of elasticity and H is the warping constant.

When the torsional rigidity, GJ, of the section is very large compared with the warping rigidity, E H , the member will effectively be in a state of uniform torsion. Closed sections, angles and tee sections behave in this manner as do most flat plates and all circular sections. Conversely, if the torsional rigidity of the section is very small compared with the warping rigidity, the member will effectively be in a state of warping torsion. This conditionis closly approximated for very thin walled open sections such as cold formed sections. Between these two extremes, the memberswill be in a stateof non-uniform torsion and the loadingwill therefore be resisted by a combination of uniform and warping torsion. This is the condition which occurs in hot rolled I, H and channel sections. A more detailed explanation of these effects is given by Trahair.(6)

5

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Thus the stresses produced in a member by torsion are:

(1) shear stresses due to pure torsion (2) shear stresses due to warping torsion (3) bending stresses due to warping Thus torsional stresses induced in a member can be identified as pure torsional shear stress, warping shear stress and warping normal stress. Each stress is associated with the angle of twist (4) or its derivatives. Hence, when 4 is determined for different positions along the girder length, the corresponding stresses can be evaluated at each position. In order to determine the direction of these stresses correctly, it is necessary to adopt a standard sign convention as illustrated by Figure 2.3. The longitudinal axis is defined as the z axis. When a member is viewed along the longitudinal axis towards the origin,an anti-clockwise twist is taken to be positive.

Direct ion of

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$rt.icai

torque Positive angle Figure 2.3 General orientation

o f twist

The total torque, therefore, at any cross section may be obtained by resolving the applied loads in two directions parallel to the principal axes (x-x and y-y), multiplying by the relevant eccentricity, and then recombining as shown in Figure 2.4(a) and (b).

Tq

i.e.

=

P,e,+ Pxey

(2-1 1

At any cross section the total torsional resistance is given by:

T,

T, = Tp+

(2.2)

or

T,

(2.3)

=

GJ4’-EHV

For equilibrium, the torsional resistance T, must be equal to the applied torque Tq; Hence,

T GJ

- --

where

a =

4’-a2V

(2.4)

[E]’”

a , is the torsional bending constant. (Values of ‘a’are given in Tables in Section 5).

Tp is the pure torsional resistance and T,,, is the warping torsional resistance. The two sets of shear stresses thus produce torsional moments which together balance the applied torque.

To maintain internal equilibrium within the elements of the member, warping also produces direct bending stresses, but no direct stresses are produced by pure torsion.

6

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

P%

+

ben Ing

I-sections

+ torsion

iPY

px,

f.

bend irrg

torsion

(b) Channel sections Figure 2.4 Bending and torsion

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The values of T p and T , will generally vary along the length of the member, even where T, remains constant. This will be discussed in more detail in Sections 2.1.4 and 2.1.5. 2.1.3 Torsion of closedsections As explained in Section 2.1.2, the torsional rigidity, GJ, of a closed section is very large compared with its warping rigidity, E H , and hence a closed section may reasonably be regarded as subject to pure torsion only.

The total angle of twist 4 is given by: GJ where

Tq

=

the applied torque

z = the length of member subject to Tq

In a closed section the walls are, in general, relatively thin and pure torsion produces a shear flow around the section which is sensibly constant at any point. For a closed thin-walled section, J is given by: J=-

where

4A h* (Slt)

c

Ah = the area enclosed by the mean perimeter of the section

(Figure 2.5) Z((s/t) = the summation around the mean perimeter of the ratio of length along the perimeter to thickness for each element. For a section of uniform thickness:

Z ( s / t ) = h/t where

h = the mean perimeter.

For a closed thin walled section of uniform thickness:

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

t

Mean perimeter h

1

I I -

Figure 2.5 Shear flow in a closed section

S

For thick walled hollow sections, more accurate expressions for J are given in Appendix A.8. Value of J for standard hot-rolled hollow sections are given in Tables 5.5 to 5.7. The shear stress

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Tt

T,

in a thin walled section is given by:

=

Tq 2Aht

(2.8)

For rolled rectangular and circular hollow sections and other closed sections where the walls are relatively thick, the shear stress given by Equation 2.8 is conservative. A more accurate value may be obtained from the theory of thick walled Tt

-

Tq -

(2.9)

c

where C = is the torsional modulus constant (Appendix A.8). Values of C for standard hot-rolled hollow sections are given in Tables 5.5 to 5.7. For large fabricated box sections, reference should be made to more detailed literature.(')

-

2.1.4 Pure torsion open sections If a torque is applied at the ends of the member in such a way that the ends are free to warp, then the member will only develop pure torsion (Figure 2.2a). The resulting shear stresses vary linearly across the thickness of each element (Figure 2.6)* They are maximum at the surfacesof the element, thetwo values being equal but opposite in direction. The stresses are greatest in the thickest element of the section.

Elsewhere there are also small shear stresses orientated perpendicular to the dominant stresses shown. Although they contribute half the resistance to pure torsion in each plane element, due to their much longer lever arms, they are negligible in value and need not be calculated. (For further details see Reference 14 page 376 Figure 5.37.) Figures 2.6(a) and (b) show the stress patterns for I sections and Channels. The total angle of twist

4 is given by:

4 = T Lz GJ

(2.10)

*Strictly speaking this condition will be violated at the junctions between the web and the flanges, particularly in rolled sections wtih radiused root fillets. It is usual to neglect this effect except when determiningthetorsionalconstant, J , for which its inclusionleadstosignificantlylargerThe Tables in Section 5 make due allowance for such fillets.

8

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Figure 2.6 Stress patterns due to pure torsion (Stress diagrams enlarged for clarity)

The maximum shear stress rt =

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2.1.5

T~ in

an element of thickness t is given by: (2.11)

G@'

Warping - open sections

General When a uniform torque is applied to a member of open section restrained against warping, the memberitself will be in non-uniform torsion and the rateof change of angle of twist will vary along the length of the member. The rotationof the section with respect to a restrained end will be accompanied by bending of the flanges in their own plane. The direct and shear stresses generated are shown in Figure 2.7.

Warping stresses are also generated in members of open section when the applied torque varies along the length; even if the ends are free to warp. For an I section member, the action of warping resistance can be visualised as follows. The torque Tq is resisted by a couple comprising forces equal to the shear forces in each flange, and acting at a lever arm equal to the depth between the centroids of the flanges. If each flange is now treated as a beam, the bending

warpingnormal \Stress ow)

d

bending momentin plane o f flange("Bi moment")

- W -- q y r

ressKw) shear flange \/hears

S

1

Y Y

y

Rotation o f cross sect ion Figure 2.7

Warping stresses in open crosssections

9

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

moments produced by the above forces lead to direct stresses,U,, in the flanges as shown in Figure 2.7. For a section with a low value of GJIEH the above provides a reasonable approximation, but in general it over-estimates the direct stresses whilst underestimating the shear stresses, because it neglects the shear stresses due to pure torsion. Also the above treatment is only applicable to I or H sections and cannot readily be applied to a channel section. Forthese reasons the methodsgiven in this publication have been developed. Warping Stresses

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(i) Warping normal stresses These are direct stresses (tension or compression) resulting from the bending of the element due to torsion. In the case of an I beam, the stresses occur in the flanges. They act perpendicular to thesurface of the cross section and are constant across the thickness but vary along the length of an element. Figures 2.8(a) and (b) show the stress patterns in I and channel sections.

(a) I sections

Figure 2.8

(b) Channel sections

Warpingnormalstresses

The magnitude of the warping normal stressat any particular point cross section is given by: U,

where W,,

=

= - EW,,f

‘S’

in the

(2.12)

the normalised warping function at the particular point ‘S’ in the cross section (see Tables 5.1 to 5.7 or formulae given in Appendix A).

(ii) Warping shear stresses These are in-plane shear stresses that are constant across the thickness of the element but vary in magnitude along the length of the element and act in a direction parallel to the edge of the element. Figure 2.9 shows the stress patterns for I sections and channels. The magnitude of the warping shear stress at any point section is given by: r, =

where S,,

10

=

-

E SW&’ t

‘S’

in the cross

(2.13)

the warping statical moment at the particular points ‘S’ in the cross section (see tables 5.1 to 5.7 or formulae given in Appendix A).

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

T (a) l sections Figure 2.9

(b) Channel sections

Warping shear stress

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2.1.6 Endconditions The endconditions of the member influence greatly the torsional stresses along the member. This publication considers the three ideal situations described below. Where the idealconditions do not apply it may be necessary to interpolatebetween conditions or conservatively to assume the worst condition. Torsional fixity must be provided at least at one point in the length of a member, otherwise it will simply twist bodily when a torque is applied. Warpingfixity cannot be provided without also providing torsional fixity. Thus there are3 possible sets of end conditions relevant for torsional calculations: (a) Torsion fixed, warping fixed: This is satisfied when twisting about the of the member longitudinal z-axis and warping of the cross section at the end are prevented. In this situation 4 = 4‘ = 0 at the end. Such a condition may be achieved as shown in Figure 2.10(a). (Note: This torsional end condition is also called ‘Fixed’.) (b) Torsion fixed, warping free: This is satisfied when the cross section at the end of the member is prevented from twisting but is allowed to warp freely. In this situation c) = c)” = 0 at the end. Such a condition may be achieved as shown in Figure 2.10(b). (Note: This torsional end condition is also called ‘Pinned’.) (c) Torsion free, warpingfree:This is achieved when the end is free to warp and twist. The unsupported end of a cantilever illustrates this condition. (Note: This torsional end condition is also called ‘Free’.) Effective warping fixity is not easily provided. A connection providing fixity for bending about both axes is not sufficient. It is also necessary to restrain the flange by means of details such as those shown in Figure 2.10(a), where plates or channel sections are added to provide warping fixity. It is worth considering the fact that provision of warping fixity does not produce such a large reduction in torsional stresses as is obtained from fixity for bending. Thus it may be more practical to assume that the endconditions are ‘warping free’ even when fixity is provided for bending. On the other hand, torsional fixity can be provided relatively simply by standard end connectionsFigure 2.10(b). It should be noted that end conditions for torsion calculations may be quite different from those for bending. A beam may be supported at both ends, but torsionally restrained at one endonly - the torsionalequivalent of a cantilever. On the other hand, torsional restraint (though not normally full fixity) can beprovided at the unsupported end of a cantilever beam.

11

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Schematic

representation

Ideal

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__ ________-_---Plate Feasibleconnection

U -U-

L_-_*

Channel

1

(a) Torsion fixed, warping fixed

I deal

Feasible connection

(b) Torsion fixed, warping free

Figure 2.10 Endconditions

2.2

Bending

Procedures for checking the adequacyof steel members subject to bending fully are documented in the appropriate Sections of BS 5950:Purt I.(13)Thus Clause 4.3 deals with laterally unrestrained members, including allowances for:

(1) The pattern of moments, Clause 4.3.7 (2) End restraint, Clause 4.3.5 (3) Cantilevers, Clause 4.3.6 (4) Destabilising load conditions, Clause .4.3.4 ( 5 ) Angle sections, Clause 4.3.8 Interaction of shear and bending is covered in Clause 4.2, whilst Clause 4.9 deals with moments applied about both principal axes. In the caseof an I or channel bent about its majorprincipal axis and not provided with full lateral restraint, design is likely to be governed by lateral-torsional 12

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

buckling considerations with the design condition being:

M< where

(2.14)

Mb

M

=

equivalent uniform moment

Mb

=

lateral-torsional buckling resistance moment

When determining either Mor Mb advantage may be taken of the beneficial effects of non-uniform moments within the beam segment under consideration according to Clause 4.3.7.6, with the exact procedure to be followed for a particular case being principally dependent o n the nature of the applied loading. If the beam is stocky, e.g. dueto the presenceof closely spaced lateral restraints, or if it is bent about its minor axis or for almost all situations involving the use of closed sections, design will be governed by the moment capacity M , at the most highly stressed cross section. Determination of M , is covered by Clauses 4.2.5 and 4.2.6.

Laterally unrestrained beams must also satisfy the provisions of Clause 4.2. In certainsituationswhereadvantage is taken of afavourablemomentpattern, resulting in an Mvalue in Equation 2.14 becoming less than the maximum moment in the beam, local cross sectional capacity may be the governing condition.

I n addition to direct bending stresses, shear stresses Tb due to plane bending are also present. These shear stresses can be determined from the following:

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Forweb,

Tbw

For flange,Tbf where

=

V QW It

(2.15)

vI Qr T

(2.16)

= -

V = theappliedshearforce I = the second moment of area of the whole section t = the web thickness T = the flange thickness Qw = the statical moment for the web i.e.

plastic modulus of section

(

2

1

Qf = the statical moment for the flange The derivation of Section 5.

2.3

Qw

and Q, is given in Appendix A and values are tabulated il

Combinedbendingandtorsion

The presence of loading which produces simultaneous bending and torsion in member means that some degreeof interaction between the two effectswill occur. This may be regarded as analagousto thesituation in a member subjectto bending and compression, for which the axial load acting through the lateral deflections caused by the bending loads induces additional moments, which in turn amplify the deflections. In the case of bending plus torsion, the angle of twist 4 caused by the torsion is amplified by thebendingmoment, inducingadditionalwarpingmomentsand torsional shears.(") Account must also be taken of the additionalminor-axis moments produced by the major-axis bending effects acting throughthe torsional deformations, including the amplification noted above. Any plasticity is liable to have a disproportionate (and so far unquantified) effect on the torsional deformations. Thedesign criterion is therefore taken asa limit on

13

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

the maximum stress at the most highly stressed cross-section to the design strength p y . This check then effectively becomes the “capacity” check in terms of BS 5950: Part I. Neglecting for the time being the effects of amplification, and assuming that the loads produce bending about the major axis together with torsion, longitudinal direct stresses will arise from three causes as illustrated in Figure 2.11. Assuming elastic behaviour, these may be determined from:

M,

(2.17)

~

2, Obyt

U,

MYt

(2.18)

= -

ZY = EW&“

(2.19)

Determination of u b y t involves calculation of M,,, which depends on the major axis moment M , and the amount of twist 4, thus: = 4Mx

(2.20) Determination of a b , depends directly on M , , torsional deformations having a negligible effect on this quantity. The calculation of U, has already been covered in Section 2.1S . under ‘Warping normal stresses’. Myt

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Whenever lateral-torsional buckling is a design criterion (i.e. when P b is less than p y ) , the values of U, and o b y t will be amplified by the interaction of torsion and lateral-torsional buckling.

Point on cross section at which peak stress occurs

Deformation

OCbx Figure 2.11

14

Deformation and stresses due to combined major axis bending and torsion

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Adopting asimilar form of interaction expressionto thatused in BS 5950: Part for combined axial load and in-plane bending moment, the criteria are:

(i) Buckling Check:

-+

(2.21)

PY

(ii) ‘Capacity’ Check: (2.22) M, is the equivalent uniform moment according to BS 5950: Part I given by: ubyt + u w

Py

M, = mxMx

Equation 2.22 may govern when

M,

< 1 or when M,, > pyZ,

The use of values of M b > pyZ, is justified by the very local nature of the peak stresses and is in general accord with the method of BS 5950: Part I Clause 4.9 for biaxial bending without torsion. If the applied loading also includes minor-axis moment this should be added and the criteria modified to: (i) Buckling Check: M, -+-

M y

+

(2.23)

Mb

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(2.24)

Uby

=

MyfZy

Note that the applied minor axis moment M , is not amplified by the bending-torsion interaction. In Equations 2.23 and 2.24 the various stresses refer to the same point in the cross section at the same point in the length of the member.

in a The torsional shear stresses and warping shear stresses should also be amplified similar manner as follows: 7,1

=

(T,+T,)(~+O.~M,/M~)

(2.25)

and added to the shear stresses due to plane bending. At points of high coincident bending stress and shear stress, a check using BS 5950:Part I Clause 4.2.6 should also be made. The design approach is illustrated by means of a series of worked examples in Section 4. In several cases these show the limit state of acceptable twist at working load as being the governing condition.

15

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

3.

LOADAPPLICATION

In the previous discussion on torsional theory it has been assumed that loads are applied in such a way that they can move freely with the beam as it twists. This is often a conservative assumption because the system applying the load may be attached in such a way to the member under consideration as to reduce the torsional effect. In such cases the stiffness of the loading system should be taken into account in the analysis. For example, consider a load applied to the member through a column in such a way as to cause a torsional loading. Torsional effects can be greatly reduced by ensuring that a moment connectionis provided and taking into account the bending stiffness of the column, see Figure 3.l(a) and Example 10, Section 4. In a similar fashion, consider torsional loading applied to the main beam shown in Figure 3.l(b). In this case, the torsional effects can be reduced almost to zero by taking account of the bending stiffness of secondary beams which frame into the main beam with moment connections. These effects are explained in greater detail by Johnston.('2)

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P

Figure 3.1

16

Torsional effects

-

methods of load application

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

4.

WORKED EXAMPLES

A number of worked examples have been provided to demonstrate the methods proposed and the use of the Tables (Section 5 ) and Graphs (Section 6 ) given in this publication. It should not be assumed that the solutions adopted are the only or even the best method of dealing with the problem. For example, it may be that a tubular section would be a better member to use in some situations than the universal beam or column section used. Alternatively, the loadingcould be applied in such a way as to producenegligible torsional effects. These decisions can only be made by the designer by considering the structure as a whole along with any architectural constraints which have been imposed.

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Examples 1 to 3 have been kept very simple in order to demonstrate the method clearly. As such they are somewhat unrealistic design situations. More complex and realistic problems are dealt with in Examples 4 to 10. It will be seen from Examples 1, 2 and 7 that the shear stresses are relatively insignificant and for normal situations, can be ignored. Only in cases where the span of the member is very short and the rotations arevery high will shear become the governing criterion. In such cases the first and third derivative of 4 (ie 4‘ and @”)will be required to calculate the pure and warping shear stresses. The equations for 4’ and @“ are given in Appendix B. In normal design situations the size of the member will be governed by twist or by the interaction of bending and warping normal stresses. In this case the value of 4 and its second derivative 4‘’ may be obtained from the Graphs provided in Section 6 o r from the equations given in Appendix B. Notes: (i) In Examples 3 , 5 , 6 and 7 it has been assumed that the endsof the beams are ‘fully fixed’ for bending. This has been done to produce simple examples illustrating the treatment of torsion. It should not be taken to imply that bending moments can be determined like this in practice.

In real situationssuch beams are likely to be partof a frame and thevalues of the end moments would depend on the pattern of loading, the relative stiffness of the members joined and the type of connection used. (ii) In Examples 3, 5 and 6 it has been assumed that the ends of the beams are ‘fixed’ against warping. Means of achieving this are illustrated in Section 2.1.6. However, in most practical cases, it is difficult to achieve the condition of ‘warping fixed’. Hence it is usually better to design for the ‘warping free’ condition.

17

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Example The

I

beam shown

bebw

i s unrestml'ned eccentrr'c l o a d ;S applted to the bottom flange a t the centre of the span in s u c h a w a y thht it does not mvrde any L a t e r a c restmint to the me ber. The end condl'trbns are a s s ' u m e d to be simp& supported for bendinq a n d F i x e d against torslbn b o t free for For the factored loads shown , cherk wQrpin% t c l d e q u a c y of the trial sectrbn. abng its

length . An

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R

I

1

I-

L= 4000mm

'It W

18

i

W

W= 100kN

- 1

' i

negative angk of t w k t due to T?

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

The SteelConstruction Institute

---

a &= --

Job No.

PUB rs*

Job Title

Silwood Park Ascot Contract BerksSL5 7QN Telephone: (0990) 23345 Fax:(0990) 22944 Telex: 846843

Client

CALCULATION SHEET

Checked by

Worked

Isheet

2

of

E X . I (Rev.

LxampLe I No.

Made by

Jun. '89 Date Jun. '89 -

Date

m

+

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Loadmq (Factored) Point

load

bistributed

E c c e n trl'ci ty

W Load (sell w t .)

W

9

Q

19

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

IJob No.

PUB

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Try

254x 2S4

-

-,

.@ 89 kg/m UC Grade 43 SteeL

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W,

Add I ti o n a l

proper tie3 from this publlcatiol rablc 5.;

.l3

Itabcc 6.

20

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Client

Contract No.

Made by

m

Checked by

Date Date

Jun. '89 Jun .l89 -

bendinq a ~ dtorston ti> l3uc k linq Check (at U . L . 3 )

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Check combmed

Myr To

=

J

MT.@

coCcuCate

0

21

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

I

Institute o n t r a SL5 ct Silwood Park Ascot CBerks 7QN (0990) 23345 Fax: (0990) 22944 T e l e x : 846843

No.

Client

Telephone:

Made by

Dare

Jun. '89

Date

J'un. '89

Checked by

CALCULATION SHEET

€9.2.18

€9.2.12

t'4. 2.19

To

calcc,late

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Torsional

M'

functlbn

-0".G.J.a

0.4s

Gmph 2

t

Ec+. 2.21

J

22

--

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

The SteelConstruction Institute

e?! --

-

Job No.

Job Title

ISheet 6 of N B &S? Worked Excrmpk I

.I I R e v .

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

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CALCULATION SHEET

. . 93 t IO t 98 = 201 N / i m f ( 265 m

23

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

The SteelConstruction Institute

----

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Job No. Job Title

m8 #S?

[Sheet

7

of

E 1 (Rev.

Wor.ked Example I No.

Contract Client Made by Checked by

FM

Date

Jun. '89

Date

Jon. '89

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I

24

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

The SteelConstruction Institute

4cfl --= --

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

at

O<

at

At

Job No. Job Title

Worked

Lxampk I No.

Contract Client

m

Made by Checked by

Date

Date

J u n . '89 J'on. '89

= 0.6

support

-L= 0

Support

L 25

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

The

=

I

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Client

Made by

-1

Contract

NO.

A

Ey.2. It

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LC+.2.11

E?. 2 .l3

At

ml'dspan

T

26

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Made by Checked by

CALCULATION SHEET

~y

inspection t k

\sheor

stresses 0

Contract No.

Client

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990)22944 Telex: 846843

1

M

maximum

occw-

at

Date

J u n '89

Date

Jun '89

combined

ttx suppart

2

I Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

At

Support

In web

at

3,


=

-21.6N/mrn2

27

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

R e f . 13

(4.2.3)

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These calculations confirm comparativQLy low lwd-,s o f

=

28

1-32'

the

s h e a r str-.

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

I

The SteelConstruction Institute Silwood Park m

.

3

ccAlJob B --

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-

~elepnone:(UYYUJ ~ 5 5 4 3 Telex: 846843

CALCULATION SHEET

I

Sheet

I

of

h.2 IRev*

Worked Example 2 l

No.

Client

m * * * =

Fax: (0990)22944

I

PUB ?S . 7

-

Ascot Contract Berks SL5 7QN

I,.,.,.,.,

No. Job Title

I

Made by Checked by I

m-.

Date

Jun '49

0 I

29

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Contract No.

Client Made by

Date Checked by

W

Check cornbmed bmdlnq

and

Date

Jun '89 Jun '89

torslon

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E?. 2.21

=6S)

Ref. 13 (4 2.3)

-(

30

250 2s0t 150

)x 61.1 =.38*2crn;

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

--

SteelConstruction Institute

-

Silwood Park Ascot BerksSL5 7QN Telephone: (0990)23345 Fax: (0990) 22944 Telex: 846843

136 Urn

Ref. 13

(4.3.7.2)

table

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To

0

caCculate

I

J(

?

U

I

L

100kN

I

I

To

Tf

7

B =

To

E?. 2.5

t

GJ

at

centre

01 span

t = ~=2000rnrn 2

31

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 846843 Telex: 22944 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

No.

C o nCt rl ai ecnt t

lDate Jun - - . . ' 89

M a d e by Checked by

Date

M

lDateJun. '89 '89 I Date

E?. 2.20

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E q . 2 .l8

€ 4.2.22 E?. 2.17

32

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Silwood Park Ascot Ekrks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

Jun '8 un 3 9

Date

CALCULATION SHEET

E?. 2.15 1

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1

2

j

!

€ 9 .2.9

€9.2.2.5 nf

d Kef. 13

l

(4.2.3)

4

33

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Steel Construction Silwood Park Ascot SL5 Berks 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

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CALCULATION SHEET

34

Contract Client

No.

Jun. '89 Date Jun. '89

Date

Checked by

M

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

Contract Client

No.

Made by

Date

Checked by

CALCULATION SHEET

Jun '89

B4

Example 3

3 '

II

I

'

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1

8 1

1 /A

TorsionaI

-I t-e-rsmm

d

Ty lhis

Ty

moment

acts in

a _ a .

=

=

W. e

I~OX?SXIO"=?*S~NITI

q o t i v e sense T? = - ? 6 k N m

35

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

P057: Design of Members Subject to Combined Bending and Torsion

Discuss me ...

L

36

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

ce*

The - -Steel Construction -Institute 1

1

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Job No. Job Title

PUB ?5f

Isheet

3

of

EX

.3IRev.

Morked Example 3 Contract Client

No.

Made by Checked bl

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Chec k

37

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

Contract Client

CALCULATION SHEET

=

To

No.

Made by Checked by

JI,n '89 Date J on '89

Date

M

E . Wno . D"

€9.2.19

I'9. 2.1.5

caCcuCate 0"

Gmpb 4

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!

€9.2.21

Eq.2.22 E?. 2 .If

38

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

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The SteelConstruction Institute

--

---

--

Job No.

PUS ?S;Z Isheet 3

of

Ex.3 IRev.

Job Title

-

Worked Example 3

Silwood Park Ascot Contract Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

Client

CALCULATION SHEET

Checked by

No.

Made by

EM

Date Date

J Un . '8 9 Jun. '89 -

39

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Client

Contract No.

Made by Checked by

-3n '89

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Excrmpk 4

W= IO0 kN

40

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

W =

I kN/m

41

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

L

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Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

CALCULATION SHEET

C o n t r a c t No.

Client Made by Checked by

m

Date

Jun. ' 8 9

Date

Jun . '89

3

t

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42

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L E

=

M

m

m

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m. m

Date

Jun. 'S9

Date

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037

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M = 0.57~'103= 5 9 kNm UsIng

tcrbwhted

inbrrnstl'on from Ref.10

Re!.

IO

43

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

CALCULATION SHEET

Contract Client

No.

Made by Checked by

m

4 859

44

lDate

Date

J u n . ‘89

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

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Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax:(0990) 22944 Telex: 846843

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

CALCULATION SHEET

45

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

CALCULATION SHEET

No.

Contract Client Made by

r Iln. '89

Checked by

I -1' II

7 Sectr'on X-X

46

Sectron

Y- Y

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Date Date

J m . '89

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l

n A-ssurninq

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= 168mm

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II

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CALCULATION SHEET

7QN

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Date

W

n n

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6 T

If

f 2

48

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

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The SteelConstruction Institute

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Contract No.

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Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

CALCULATION SHEET

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E?. 2.19

E?. 2.21 27-3 244

t ( I. I t 6 2 . 4 ) 266

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3.0 kWm

8.1 8 kWm

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P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

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Steel Construction

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Institute

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Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

Job No.

PUB ?5? I Sheet 4 of Ex, Worked Exarnpte 6 No.

Client A

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I Date Date

Jon 3 9 J

l

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Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

CALCULATION SHEET

56

Job No.

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Sheet

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L?. 2.19

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v

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01

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24Sm

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I L

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P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

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Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

CALCULATION SHEET

No.

Contract Client Made by Checked by

EM

Date

Jun '89

Date

J

lo=?-

59

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

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(

60

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BM

Date

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L4.2.21

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:tabLe 3:

F,*

4.3.7.2 E7.2.20

€ 9 .2. I %

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Discuss me ...

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E?.

2.17

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P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

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Contract

No.

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

CALCULATION SHEET

63

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

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Example

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P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

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of

Made by

01

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Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

CALCULATION SHEET

66

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Job No.

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Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

:.

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The SteelConstruction Institute

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Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

CALCULATION SHEET

5

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f U 3 ?5?

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L

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Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

CALCULATION SHEET

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PUB

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Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

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Silwood Park Ascot Berks SLS 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

Contract No.

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CALCULATION SHEET

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

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Date

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Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

€9.2. l?

78

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

P057: Design of Members Subject to Combined Bending and Torsion

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4-

7 mern ber

Bracket 0

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79

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P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

The Steel Construction Institute

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Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

CALCULATION SHEET

r 80

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P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Silwood Park Ascot Berks SLS 7QN Telephone: (0990)23345 Fax: (0990)22944 Telex: 846843

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

CALCULATION SHEET

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Icontract No.

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P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Silwood Park Ascot Berks SL5 7QN Telephone: (0990)23345 Fax: (0990)22944 Telex: 846843

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Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

Date

Checked by

CALCULATION SHEET

bistribut r'on

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Kc

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P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

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CALCULATION SHEET

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J u n '89

Date Date

a

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1.

84

1

L.

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

I

II

~~~~

Silwood Park Ascot Berks SLS 7QN [Contract Client Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

Check

Pecked

No--

~

I

Il'

Column for axial load and bendinq

85

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

I

I I

L

The Steel Construction EEE -Institute ~

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

CALCULATION SHEET

beam

86

wwr

~~

KU Contract No.

Made by

J

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

TO

colluIclte 0

Et+. 2.24

87

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

CALCULATION SHEET

c

88

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

5. SECTION PROPERTIES A list of the basic section properties required for design can be found in Steelwork Design Guide to BS 5950, Volume Tables 5.1 to 5.4 list the additional cross sectional properties that are required in the calculation of angle of twist and torsional stresses for hot rolled I, H, Joist and Channel sections. The torsional constant, J, the torsional bending constant, a, and the warping constant, H , are properties of the entire cross section. The normalised warping function, W,,, and the warping statical moment,S,,, vary at different points on the cross section to which the term applies. The formulae needed to evaluate these torsional properties Appendix A .

are given in

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

Values of the statical moments for the flange, Q,, and the web, Q w , are used to calculate flexural shear stresses in the flange and web respectively. The statical moment for the flange, Q,, is the first moment of area of the portion of the flange between the toe of the flange and the edge of the web about the neutral axis of the whole section. The statical moment for the web,Q,, is the first moment of area of half the cross section about the neutral axis of the whole section. (Qw = plastic modulus/2.) Values of Qf and Qw are also given in Tables 5.1 to 5.4. Tables 5.5 to 5.7 list the additional cross sectional properties for hot rolled structural hollow sections, J and C, described in Section 2.1.3.

89

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Table 5.1 Universal Beams

I

Designation Serial Size mm

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

-r

Torsional

- Constant Mass per metre kg/m

-

J

-

Torsional Properties

Torsional Bending Constant a

Warping Constant

H

Normal ised Warping Warping Statical Moment Function

Statical

Moments

F 1 ange

Web

W,

S,,

01

Q,

cml'

mm

dm6

cm2

cm4

cm3

cm3

914x419

388 343

1730 1190

3640 4050

88.7 75.7

929 920

35700 30800

3280 2860

8830 7740

914x305

289 253 224 201

929 627 421 293

2950 3310 3690 4040

31.2 26.4 22.0 18.4

680 680 674 670

16900 14500 12200 10300

2100 1830 1560 1320

6290 5470 4760 41 80

838x292

226 194 176

51 4 307 222

3120 3590 3900

19.3 15.2 13.0

605 599 595

11900 9500 8150

1560 1260 1090

4580 3820 3400

762x267

197 173 147

405 267 161

2690 3020 3460

11.3 9.38 7.41

499 494 488

8490 7110 5680

1210 1030 835

3580 3100 2590

686x254

170 152 140 125

307 219 169 116

2510 2760 2960 3270

7.41 6.42 5.72 4.79

428 424 422 419

6480 5670 5070 4280

972 86 1 777 662

2810 2500 2280 2000

610x305

238 179 149

788 341 200

2170 2770 3240

14.3 10.1 8.09

469 456 450

11400 8270 6140

1400 l040 867

3730 2760 2290

610x229

140 125 113 l01

217 155 112 77.2

2180 2410 2630 2910

3.99 3.45 2.99 2.51

342 339 337 334

4360 3810 3330 2810

725 640 564 481

2070 1840 1640 1440

533x210

122 109 101 92 82

180 126 102 76.2 51.3

1830 2030 2150 2340 2590

2.32 1.99 1.82 1.60 1.33

277 274 273 271 269

3130 2720 2490 2210 1850

564 496 458 41 1 347

1600 1410 1310 1180 1030

457x191

98 89 82 74 67

121 90.5 69.2 52.0 37.1

1590 1720 1860 2020 2220

1. l 7

1.04 0.923 0.819 0.706

216 214 212 21 1 209

2040 1810 1630 1450 1260

402 362 327 295 259

1120 1010 916 828 736

82 74 67 60 52

89.3 66.6 47.5 33.6 21.3

1290 1390 1530 1730 1950

0.569 0.499 0.429 0.387 0.311

171 170 168 169 167

1240 1100 955 858 694

306 274 241 217 178

900 811 721 642 547

457x152

-

Continues over next page

90

swo?wo

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

. . . Table 5.1 continued Designation

r

Torsional

- Constant

Serial Size mm

Mass per metre kg/m

-

J

cn‘

Torsional Bending Constant a mm

vlarp ing Constant

Normalised Warping Function

rarp i ng Statical Moment

H

W,

SW

dm6

C d

CIb

Statical F 1 ange

Q

Moments Web 0,

cm’

’.C

74 67 60 54

63.0 46 . O 32.9 22.7

1580 1730 1910 2110

0.608 0.533 0.464 0.39

178 177 175 174

1280 1130 994 839

273 244 218 185

752 673 597 524

406x140

46 39

19.2 10.6

1670 1950

0.206 0.155

139 138

554 422

152 118

444 360

356x171

67 57 51 45

55.5 33.1 23.6 15.7

1390 1610 1780 1990

0.413 0.331 0.286 0.238

151 149 147 146

1030 834 727 609

228 188 166 140

606 505 447 387

356x127

39 33

14.9 8.68

1350 1560

0.104 0.081

108 107

362 284

113 90 .O

327 270

305x165

54 46 40

34.5 22.3 14.7

1330 1510 1700

0.234 0.196 0.164

124 122 121

708 599 508

164 141 121

422 361 312

305x127

48 42 37

31.4 21 .o 14.9

913 1020 1120

0.101 0.0842 0.0724

92.8 91.5 90.5

407 344 299

123 106 93.5

353 305 270

305x102

33 28 25

12.1 7.63 4.65

971 1100 1220

0.0441 0.0353 0.0266

77.3 76.4 75.7

213 172 131

254x146

43 37 31

24.1 15.5 8.73

l050 1200 l400

0.103 0 .OR58 0.0662

90.9 89.7 88.7

425 358 280

111 95.1 74 .B

284 243 198

254x102

28 25 22

9.64 6.45 4.31

867 958 1050

0.0279 0.0228 0.0183

63.9 63.3 62.8

163 135 109

61.5 51.2 41.9

177 153 131

203x133

30 25

10.2 6.12

974 1120

0.0373 0.0295

66.0 65.1

212 170

61 .5 49.9

157 130

203x102

23

6.87

760

0.0153

49.3

116

44.6

116

178x102

19

4.37

770

0.00991

43.2

86.6

33.5

85.6

152x89

16

3.61

583

0.0047

32.2

55 .O

24.3

62 .O

13

2.92

422

0.002

22.7

32.9

17.0

42.5

406x178

swo SW0

S WO

S WO

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

S

W 1

127x76

.

79.6 65.5 50 .5

240 204 169

~

91

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Table 5.2

Universal Columns

T

~

Mass per metre kg/m

Torsional Constant

Torsional Bending Constant

-

TorsionalProperties

Warping Constant

Normal ised Warping Warp i ng Statical Function Moment

Statical

Moments

F 1 ange

Web

J

a

H

W"0

S.",

Of

CL

cm"

mm

dm6

cmz

cm4

cm'

cm3

Wno W"0

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

~

~

92

634 551 467 393 340 287 235

13700 9240 5820 3550 2340 1440 812

856 935 1040 1180 1310 1490 1750

38.8 31 .l 24.3 19.0 15.5 12.3 9.54

422 406 390 376 366 356 346

34400 28700 23300 18900 15800 13000 10300

2890 2470 2070 1720 1470 1240 1000

7120 6040 5000 4110 3500 2910 2340

477

5700

l040

23.8

397

22400

1880

4850

202 177 153 129

560 383 25 1 153

1820 2030 2290 2650

7.14 6.07 5.09 4.16

325 320 316 31 1

8220 7100 6040 5010

847 742 638 536

1990 1730 1480 1240

283 240 198 158 137 118 97

2030 1270 734 379 250 160 91 .l

898 1010 1170 1400 1570 1780 2100

6.33 5.01 3.86 2.86 2.38 1.97 1.55

258 250 242 235 231 227 223

9170 7500 5970 4560 3870 3250 2610

1050 883 721 565 486 414 339

2550 2120 1720 1340 1150 976 195

167 132 107 89 73

625 322 173 104 51.3

821 977 1160 1340 1590

1.62

0.716 0.557

170 164 159 155 152

3570 2710 2110 1730 1370

504 394 31 4 262 21 2

1210 937 743 614 494

86 71 60 52 46

138 81.5 46.6 32.0 22.2

772 892 1040 1160 1290

0.317 0.25 0.195 0.166 0.142

105 102 100.0 98.7 97.6

1130 915 728 630 546

205 171 138 121 105

489 401 326 284 249

37 30 23

19.5 10.5 4.87

730 870 1070

0.04 0.0306 0.0214

58.0 56.6 55.5

258 203 144

64.2 51.6 37.2

155 124 92.1

--

1 .l8 0 .894

swo?swo

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Table 5.3 Joists

Torsional Properties

Twno Designation

__

wno

-

Serial Size

Mass per metre

mm

kg/r

__

Torsional Constant

J

cm'

Torsional Bending Constant

warping Constant

a

H

mm

dm6

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

swoTswo 254x203

81.85

254x114

37.20

203x152

Normal ised Warping Statical Warping Function Moment

Statical

Moments

F 1 ange

Web

W,

SW7

0,

OW

cmz

CR4

cm3

cm3

119

1200

228

538

728

0.312

25.5

632

0.0393

68.9

253

52.09

64.9

532

0.0709

71 .l

446

152x127

37.20

34.2

373

0.0183

44.2

185

54.5

127x114

29.76 26.79

20.9 16.9

317 348

0.00807 0.00787

33 .O 33 .O

109 107

35.0 35.5

90.5 85.9

127x76

16.37

284

0.00209

22.4

40.7

20.3

51.8

114x114

26.19

19.0

286

0.00599

29.6

90.3

30.0

75.6

102x102

23.07

14.4

241

0.00321

23.2

60.7

22.2

56.7

191

0.000177

10.6

7.1

6.04

17.6

153

6.69

102x44

7.44

89x89

19.35

11.6

l88

0.00158

17.6

38.7

76x76

14.67 12.65

6.83 4.67

163 182

0.000699 0.000597

13.6 12.9

22.7 20.7

__

1.25

84.3 112

16.0 10.3 10.4

1

230 270 139

41.4 27.1 24.4

93

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Table 5.4

Channels - TorsionalProperties

SW1

I Warp i ng Constanl

Designation Serial Size

Mass per

mm

kg/m

Constant

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

metre

_-

Normalised Warping Functions

Warping Statical Moments

- -

H

W,

W",

SWl

sw2

SW3

dm6

cm1

cm1

cm4

cm"

cm4

- - - -

listance ,tatical Moment! rom entre F 1 ange Web 0, N f web 01 o shear centre eo mm

cm'

'.C ~

132x102 65.54

61 .O

960

0 .217

133

65 .O

720

148

-274

31.3

319

603

381x102 55.10

46.0

930

0 .l53

113

62.7

573

897

-198

34.4

277

466

785 305x102 46.18 720 305x8941.69

35.4 27.6

0. O W 2 0 .0551

87.8

79.4

52.1 42.7

395 296

'56 '10

-128 -105

35.9 29.3

201 161

319 279

!54x89 !54x76

626 35.74 28.29

22.9 12.3

0 .0347

63.0 56.7

38.4 30.9

224 145

41 02

-

70.4 50.9

31.9 25.5

133 93.1

207 159

!29x89 !29x76

32.76 26.06 586

55.4 49.4

35.7 29.3

189 125

11

0 .0151

81 .O

-

55.3 40.5

33.2 27.0

118 85.4

174 135

!03~89 29.78 !03x76 23.82

- 42.1

-

34.3

31.0

28.4

101 75.9

143 113

-

30.8

- 22.6

35.4 29.3

84.7 61.6

115 87.7

641

0 .0194

20.4 11.4

578

0 .0263

17.8 10.4

529 529

0 .0192 0 .0112

48.1 42.5

32.7 27.2

156 105

84.2 62.0

15.1 494 8.13

480

0 .0134 0 .00764

41 .l 36.6

29.3 24.5

125 82.1

61.6 45.2

12.4 5.94

429 461

0 .008a1 0 .00486

34.4 31 .l

25.7 21.3

97.1 60.7

42.9 32.3

- 21.5 - 16.1

36.5 29.7

68.2 46.6

88.8

4.92

315

0 .001a8

21 .o

14.5

34.6

18.1

- 9.04 24.6

31.9

44.7

102x51 10.42

2.55

228

0 .00051?

13.6

16.5

24.4

16x38

l .23

146

178x8926.81

152x8923.84

127x64

94

14.90

6.70

0 .000101

-

7.3,

-

E .89

4.96

14.9

8.49

-

4.24

18.9

5.32

2.92

-

1.46

14.3

- -

8.02

65.0

11.7

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Table 5.5

r

Circular Hollow Sections

Designation

Outside Diam

Thick ness

mm

mm

r;

Mass per metre

orsional onstant

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

orsional lodul us :onstant

C d

ca'

21.3

3.2 ]

1.43

1 .54

1.44

26.9

3.2 ]

1.87

3.41

2.53

33.7

2.6 ] 3.2 ) 4.0 ]

1.99 2.41 2.93

6.19 7.21 8.38

3.67 4.28 4.97

42.4

2.6 ) 3.2 ) 4.0 ]

2.55 3.09 3.79

48.3

3.2 4.0 5.0

60.3

12.9 15.2 18.0

6.10 7.19 8.48

3.56 4.37 5.34

23.2 27.5 32.3

9.59 11.4 13.4

3.2 4.0 5.0

4.51 5.55 6.82

46.9 56.3 67.0

15.6 18.7 22.2

76.1

3.2 4.0 5.0

5.75 7.11 8.77

97.6 118 142

25.6 31 .O 37.3

88.9

3.2 4.0 5.0

6.76 8.38 10.3

158 193 233

35.6 43.3 52.4

114.3

3.6 5.0 6.3

9.83 13.5 16.8

384 514 625

67.2 89.9 109

139.7

5.0 6.3 8.0 10.0

16.6 20.7 26.0 32.0

961 1180 1440 1720

138 169 206 247

168.3

5.0 6.3 8.0 10.0

20.1 25.2 31.6 39.0

1710 2110 2590 3130

203 250 308 372

193.7

5.0+ 6.3 8.0 10.0 12.5 16.09

23.3 29.1 36.6 45.3 55.9 70.1

2640 3260 4030 4880 5870 7110

273 337 416 504 606 735

5.0+

26.4 33.1 41.6 51.6 63.7 80.1 98.2

3860 4770 5920 7200 8690 l0600 12500

352 436 540 657 793 967 140

219.1

6.3 8.0 10.0 12.5 16.09 20.09

.

Torsional Properties

r

-T Mass per

Designation

Outsidc Dian

Thick ness

mm

ma

metre

C

J

kg/m

-

Continues..

..

o r s i ona'

.onstant J

orsional odulus onstant C

cm4

cm3

547 681 830 1010 1230 1470

-

kg/a

6.3 8.0 10.0 12.5 16.0 20. 09

37 .O 46.7 57.8 71 .5 90.2

Ill

6690 8320 10100 12300 15100 17900

273.0

6.3 8.0 10.0 12.5 16.0 20.09 25. 09

41 .4 52.3 64.9 80.3 101 125 153

9390 11700 14300 17400 21 400 25600 30300

688 857 1050 1270 1570 1880 2220

323,9

6.3+ 49.3 8.0 62.3 10.0 77.4 12.5 96.0 16.0 121 20.01. 150 25.01. 184

15900 19800 24300 29700 36800 44300 52800

979 1220 1500 1830 2270 2730 3260

355.6

8.0 10.0 12.5 16.0 20.09 25.09

68.6 85.2 106 134 166 204

26400 32400 39700 49300 59600 71400

1480 1820 2230 2770 3350 401 0

406.4

10.0 12.5 16.0 20. 09 25.09 32.09

97.8 121 154 191 235 295

49000 60100 74900 90900 09000 133000

2410 2960 3690 4470 5380 6540

457 .O

10.0 12.5 16.0 20.09 25. O f 32. 09 40.09

110 137 174 216 266 335 1111

70200 86300 08000 31000 59000 94000 130000

3070 3780 4720 5750 6950 8490 10100

508.0

10.0+ 12.5+ 16.0+ 20. o+l 25.0+4 32.04 40.0+4 50.0d

123 153 194 241 298 376 U62 565

97000 20000 50000 83000 122000 :72000 i24000 182000

3820 4710 5900 7200 8730 10700 12800 15000

244.5

-

+ Sections marked thusarenotincludedin

BS 4848: P a r t 2 ) Sectionsaarkedthusarerolledin grade 43C only 9 Sectionsaarkedthus are r o l l e d i n grade 50C only

95

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Rectangular Hollow Sections - Torsional Properties

Table 5.6 Designation

T

Mass per metre

rors iona 1 :onstant

cm"

Size

Thick ness

mm

mm

kg

50x25

2.5+: 3.0+: 3.2+:

2.72 3.22 3.41

50x30

2.5+ 3.0+ 3.2 4.0+ 5.0+

'ors iona

80x40

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

90x50

00x50

00x60

20x60

20x80

r

Mass per metre

'orsiona :onstant

rors iona lodul us :onstant

Thick ness

cm3

mm

mm

8.41 9.64 10.1

4.62 5.21 5.42

150x100

2.92 3.45 3.66 4.46 5.40

11.7 13.5 14.2 16.6 19.0

5.74 6.52 6.81 7.79 8.71

5.0 6.3 8.0 10.0 12.5+)

18.7 23.3 29.1 35.7 43.6

806 985 1200 1430 1680

127 153 184 215 246

160x80

2.5+ 3.0+ 3.2 4.0 5.0+ 6.3+

3.71 4.39 4.66 5.72 6.97 8.49

25.0 29.2 30.8 36.6 43.0 49.7

9.74 11.2 11.8 13.7 15.8 17.7

5.0 6.3 8.0 10.0 12.5+)

18.0 22.3 27.9 34.2 41.6

599 729 882 1040 1210

106 127 151 175 199

200x100

3 .O+ 3.2 4.0 5.0+ 6.3+ 8 .O+

5.34 5.67 6.97 8.54 10.5 12.8

43.7 46.1 55.1 65.0 75.8 86.3

15.3 16.1 18.9 21.9 24.9 27.6

5.0 6.3 8.0 10.0 12.5 16.0

22.7 28.3 35.4 43.6 53.4 66.4

1200 1470 1800 2150 2540 2990

172 208 251 296 342 393

250x150

6.3 8.0 10.0 12.5 16.0

38.2 48.0 59.3 73.0 91.5

4050 5 0 10 6080 7320 8860

413 506 606 717 85 1

3.0+ 3.6 5.0 6.3+

76.4 89.3 116 138 161

22.4 25.9 32.9 38.2 43.4

300x200

8.0+

6.28 7.46 10.1 12.5 15.3

6.3 8.0 10.0 12.5 16.0

48.1 60.5 75.0 92.6 17

8470 10500 12900 15700 19200

681 840 1020 1220 1470

3.0+ 3.2 4.0 5.0 6.3+ 8.0+

6.75 7.18 8.86 10.9 13.4 16.6

88.3 93.3 113 135 160 187

25.0 26.4 31.4 37.0 43.0 49.1

400x200

8.0+ 10.0 12.5 16.0

73.1 90.7 12 42

15700 19200 23400 28800

1140 1380 1660 2010

450x250 3 .O+ 3.6 5.0 6.3 8 .O+

7.22 8.59 11.7 14.4 17.8

121 142 187 224 266

30.7 35.6 45.9 53.9 62.4

10.0 12.5 16.0

06 32 67

33200 40700 50500

1990 2410 2950

500x300

3.6 5.0 6.3 8.0+

9.72 13.3 16.4 20.4

183 242 290 344

43.3 56.0 66 . O 76.8

10.0+ 12.5+ 16.0+ 20.0+

22 52 92 37

52400 64300 80200 97300

2700 3280 4050 4840

14.8 18.4 22.9 27.9

401 486 586 688

77.9 93 . O 10 26

5.0 6.3 8.0 10.0

Continues

96

Designation Size

J

60x40

lodu1 us :onstant C

.. ..

J

kg

cm"

C

cm3

+ Sections marked thusarenotincludedin BS 4848: P a r t 2 ) Sections marked t h u sa r er o l l e di n grade 43C only

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Square Hollow Sections - Torsional Properties

Table 5.7

r

Designation

Torsional Constant

lorsional lodu 1 us :onstant

Size

Thick ness

mm

mm

20x20

2.0 ) 2.5+)

1.12 1.35

25x25

2 .O+) 2.5+) 3 .O+) 3.2+)

l.43 1.74 2.04 2.15

30x30

2.5+) 3.0+) 3.2 )

2.14 2.51 2.65

40x40

2.5+ 3.0+ 3.2 4.0 5.0+

2.92 3.45 3.66 4.46 5.40

13.6 15.7 16.5 19.5 22.6

2.5+ 3.0+ 3.2 4.0 5.0 6.3+

3.71 4.39 4.6 5.72 6.97 8.49

27.4 32.0 33.8 40.4 47.6 55.3

10.2 11.8 12.4 14.5 16.7 18.9

3.0+ 3.2 4.0 5.0 6.3+ 8.0+

5.34 5.67 6.97 8.54 10.5 12.8

56.9 60.1 72.4 86.3 102.0 119.0

17.7 18.6 22.1 25.8 29.7 33 .5

3.0+ 3.6 5.0 6.3+ 8.0+

6.28 7.46 10.1 12.5 15.3

92.1 108.0 142.0 169.0 200.0

24.8 28.7 36.8 43.0 49.4

3 .O+ 3.6 5.0 6.3 8.0+

7.22 8.59 11.7 14.4 17.8

139.0 164.0 217.0 261 . O 312.0

33.1 38.5 49.8 58.8 68.5

90x90

3.6 5.0 6.3 8 .O+

9.72 13.3 16.4 20.4

237 .O 315.0 381 .O 459.0

00x100

4.0 5.0 6.3 8.0 10.0

12.0 14.8 18.4 22.9 27.9

361 439 533 646 761

50x50

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Mass per metre

60x60

70x70

80x80

Designation

-

C Cm’

I

1.22

1.07 1.21

Thick ness

IR

RB

orsional onstant J

-

kg/m

Cm4

‘orsional lodulus :onstant C cn’

120x120

5.0 6.3 8.0 10.0 12.5+]

18.0 22.3 27.9 34.2 41.6

775 949 1160 1380 1620

122 147 176 206 237

140x140

5 .O+ 5.6+ 6.3+ 7.1+ 8 .O+ 10.0* 12.5+

2 1 .l 23.5 26.3 29.4 32.9 40.4 49.5

1251 1385 1538 1706 1889 2269 2695

170 107 206 227 249 294 342

150x150

5.0 6.3 8.0 10.0 12.5 16.0

22.7 28.3 35.4 43.6 53.4 66.4

1550 1910 2350 2830 3370 4030

197 240 291 345 403 468

180x180

6.3 8.0 10.0 12.5 16.0

34.2 43 .O 53 .O 65.2 81.4

3360 4160 5040 6060 7340

355 434 519 613 725

200x200

6.3 8.0 10.0 12.5 16.0

38.2 48 .O 59.3 73 .O 91.5

4650 5770 7020 8480 10300

444 545 655 779 929

250x250

6.3 8.0 10.0 12.5 16.0

48.1 60.5 75.0 92.6 117

9230 11500 14100 17100 21 100

712 880 1070 1280 1550

300x300

8.0+ 10.0 12.5 16.0

73 .l 90.7 112 142

20200 24800 30300 37600

1290 1580 1900 2330

49.7 64.9 77.1 90.7

350x350

8.0+ 10.0 12.5 16.0

85.7 106 132 167

32400 39800 48900 60900

1790 2190 2660 3260

68.2 81.9 97.9 116 134

400x400

10.0 12.5 16.0+ 20 .o+

122 152 192 237

60000 73800 92300 ,12000

2900 3530 4360 5240

1.41 2.52 2.97 3.36 3.49

1.81 2.09 2.31 2.38

5.40 6.17 6.45

3.22 3.61 3.75

Continues

Size

Mass per metre

6.23 7.11 7.43 8.56 9.65

......

-

Sections narked thus are not included in BS 4848: Part 2 ) Sections marked thus are rolled in grade

+

43C

only

97

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

6.

CHARTS TO ASSIST IN EVALUATION

6.1

Standard cases

The standard cases listed in Table 6.1 have been presented graphically in order to facilitate rapid evaluation of the functions:

4GJ

VGJa T,a T, at a distance z along the member and

~

The graphs 1 to 9 have been obtained by evaluating the functions given for the relevant case number in Appendix B for the appropriatevalues of a L and z , where a L is the distance along the member at which the torque is applied.

Table 6.1 Standard Cases Graph Function Distance Torsional loading App. Case

B. No.

(2) along

Description

member

l

4GJ Tq a

~

CUL

Tq f

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2

4"GJa

4

\r; '

*

c

L(1-U)

UL

(YL

42J

4

t

T+ ,

*

0

=/I

LCldl)

UL

Tq

SA

4GJ

S

ffL

Tq a

YGJa

Concentrated torque on member with torsion fixed, warping fixed at ends. Torque applied varying of a L

applied values

Tq

3

3

Concentrated torque on member with torsion fixed, warping frcc at ends. Torque at varying of a L

*

at values

0.SL

7-4

4

6~

VGJa

Uniform torque on member with ends torsion fixed, warping frcc.

0.SL

Tq

SB

+GJ

0.SL

6B

VGJa

0.SL

+

.T .q T q.

Tq 3

Tq

I

-L-4

3/

3/

4

4

-

L a*

7-4

sc

4GJ

0.5L

Tq

T q T q 2/

2/

SD

4GJ

x

L13

- r +

Y

c

+

-

YGJa

Third point torques on member with torsion fixed, warping free at ends. Sum of case 3 for a=f and

5

7-q

6C

Quarter point torques on member with ends torsion fixed, warping free. Sum of case 3 for a=0.25, 0.5 and 0.75.

L13

Tq

7A

+GJ Tq a

0.SL

7 W L

98

Uniform torque o n

Tq

6

member with ends torsion fixed, warping fixed.

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Table 6.1 continued Graph

7B

4GJ ~

7-q

8B

App. B.

Function Distance Torsional loading Case (2) along member 0.5L

a

&GJa ~

0

il

$ 4 4

c cc

L

L

4

4

E

; a

L

-

L

4

4

Tq

7c

4GJ

0.5L

T,a

7D

4!!J

L13

8C

&GJa -

TQ 2 /

Tq 2/

I '

-

sj

\i

j ,+

Tsa

Description

No.

L 3

Quarter point torques on member with ends torsion fixed, warping fixed. Sum of case 5 for a= 0.25, 0.5 and 0.75 Third point torques on member with ends torsion fixed, warping fixed. Sum of case 5 for a j and f

0

Tq 9A

'?GJ

End torque on member with one end torsion fixed, warping fixed and the other end torsion free, warping free. Use case 7 with a=1.0

L

Tq

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7

Uniform torque on member with one end torsion fixed, warping fixed and the other end torsion free, warping free. Use case 8 with a=1.0

8

6.2

Extreme cases

Cases not included in Table 6.1 can conservatively be evaluated by using the Extreme Cases listed in Table 6.2.

No graphs are provided, but the relevant formulae are Cases l , 2 and 9.

given in Appendix B for

Table 6.2 Extreme Cases.

B

Appendix Torsional loading

Description

Case No.

M

1

Concentrated torques at ends of member, both ends warping free, left end torsion fixed, right end torsion free.

2

Concentrated torques at ends of member, with ends torsion fixed, warping fixed.

9

Uniform torque on member with one end torsion fixed and warping fixed, other end torsion fixed and warping free.

99

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P057: Design of Members Subject to Combined Bending and Torsion

Discuss me ...

0 0 CL

GRAPH

I

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P057: Design of Members Subject to Combined Bending and Torsion

Discuss me ...

GRAPH

2

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P057: Design of Members Subject to Combined Bending and Torsion

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GRAPH

L Q

3

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Discuss me ...

GRAPH

a L

4

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0 P 3

GRAPH

5

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P057: Design of Members Subject to Combined Bending and Torsion

Discuss me ...

GRAPH

n

0-05 0.I

0-2 0.25

-(2/"GJa

'c

6

6

5

4

L -

a 3

2

I

0.3

0.35 0.4

0.45

0.5

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P057: Design of Members Subject to Combined Bending and Torsion

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GRAPH

L 5-0

a 4 -0

3.0

2.0

I -0

0.0 0. I

7

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GRAPH

8

0.35 0.4

0.45

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P057: Design of Members Subject to Combined Bending and Torsion

Discuss me ...

c 0 00

GRAPH

3.0

2.5

2 -0

L a

1-5

I v 0

0.0 0.2

0.4 0.6 0.8

9

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

REFERENCES Structural action in steel box girders CIRIA Guide 3, 1977. VLASOV, V.Z. Thin walled elastic beams. Israel Program for Scientific Translation Ltd. Jerusalem 1961. 3 DABROWSKI, R. Curved thin walled girders: theory and analysis Cement and Concrete Association 1972.

4 KHAN, A.H. and TOTTENHAM, H. The method of bimoment distribution for the analysis of continuous thin walled structures subject to torsion. Proceedings of the Institution of Civil Engineers, Part 2 Volume 63, pp 843-863, December 1977.

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5

TIMOSHENKO, S.P. Theory of torsion and buckling of thin walled members of open cross section. Journal of the Franklin Institute. March, April, May, Volume 239. Philadelphia 1945. TRAHAIR , N. S. The behaviour and design of steel structures. Chapman and Hall. London 1977. TIMOSHENKO, S.P. and GOODIER, J.N. Theory of elasticity. Third edition. McGraw Hill 1970. SHANLEY, F.R. Strength of materials. McGraw Hill 1957. JOHNSTON, B.G. and EL DARWISH, I.A. Torsion of structural shapes. Proceedings of the American Society of Civil Engineers. Journal of the Structural Division, Vol. 91, No. ST1, pp 203-228, February 1965.

10 Steelwork design guide to BS 5950: Part 1 1985, Volume l Section properties and member capacities. The Steel Construction Institute, 1985.

11 PASTOR, T.P. and DE WOLF, J.T. Beams with torsional and flexural loads. Proceedings of the American Society of Civil Engineers. Journal of the Structural Division. Volume 105. No ST3. pp 527-538, March 1979. 12 JOHNSTON, B. G . Design of W shapes for combined bending and torsion. Engineering Journal. American Institute of Steel Construction, pp 65-85, 1982.

109

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

13 BRITISHSTANDARDSINSTITUTION BS 5950: Part 1: 1985. Structural use of steelwork in building part 1: Code of Practice for design in simple and continuous construction: hot rolled sections. BSI, 1985.

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14 McGUIRE, W. Steel Structures Prentice Hall 1968.

110

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Evaluation of torsionalproperties

APPENDIX A A.l

Definitions

In the analysis of cross sections subject to torsion, a number properties are used as follows:

H J SW,

W,,

of special section

warping constant torsional constant warping statical moment normalised warping function

H and J are properties of the entire cross section, while S,, and W,,, apply to specific points on a cross section. The value of S,, at point 1, for example, is denoted SW,.

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Tangent line

Y Figure A1

For a generalised shape (see Figure A l ) these torsional section properties may be defined as follows:

J

=

H =

-1: 1 3

t 3 ds

Iob

W:, t ds

where p . is perpendicular distance to tangent line from shear centre (see Figure A l )

111

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

A.2

Symmetrical I and H sections

I

Y Figure A2

For symmetrical I and H sections (see FigureA2) the following expressions may be used: Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

J

=

$ [ 2 B T 3 + ( D - 2 T ) t 3 ] (but see A4 for rolled sections) hB 4

W",

= -

S,

=

hB2T 16

where Zy is the second momentof area of the section about the minoraxis. For sections with tapered flanges, the flangethickness may be taken as the average value when using the above expressions.

112

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

A.3

Statical moments for I and H sections

In addition the terms Q, and Qw may be calculated thus: Qr

Q,

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where

= Aryr

A

-yW 2 A f = thearea of half the flangeasshown in FigureA3(a) yf = the distance from the neutral axis to the centroidof the area Af as shown in Figure A3(a) A = the total cross sectional area y , = the distance from the neutral axis to the centroid of the area above the neutral axis as shown in Figure A3(b) =

Figure A3

113

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

A.4

Torsional constant for rolled I and H sections

The value of J may be more accurately calculated as follows for sections with parallel flanges (see Figure A 2 ) :

J where

=

rolled 1 and H

t B T 3 + f (D-2T) t3+2a1DI4-0.42T4 t + 0.2204-+ T

a1 = - 0.042

r 0.1355-T

tr t2 0.0865-O.0725--, T2 T

(T+r)*-tt[r+(t/4)] 2r+ T For joist sections with sloping flanges (see Figure A4).

D1

=

+ T2)(TI2+ T22)+
J = (?)(TI

( F + m)2+t[r+ (t/4)] F+r+m

0 2

S =

2(m- TI)

B

V , = 0.10504 Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

~'+~cQD;-~V,T,~

+ 0.1s + 0.08480S2+ 0.06746s' + 0.05153S4

For 8" taper* a2

=

t r tr tL -0.0772+0.2485-+0.1281--0.0815~-0.0837~ T2 T2 T22 T2

B Y

0

--

0

0

1

--

i

,I

0

Figure A4

*Note: Flanges of BS4 joists have a taper of 8". The expressions given abovefor a2 havebeen obtained by linear interpolation between the valuesof the constants for a taperof 16$% and those for parallelflanges.This is asimplification of the more exactprocedure given by Johnstonand El D a r w i ~ h ( but ~ ) is accurate enough for design purposes. The values of J given in the Tables in this publication(Section 5) and in the Steelwork Design Guide,Volume I("'), are basedonthese constants.

114

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

A.5

Channelsections

For channelsectionssymmetrical abouttheirmajor-axis(seeFigure following expressions may be used

J

=

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where

[ 2 B T 3 + ( D - 2 T ) t 3 ](but see A.7 for rolled channelsections)

W",, =

(B-t/2-eo)h 2

SW, =

(B-t/2-eo)2 h T 4

SW2

=

( B- t/2 - 2eo) h( B- t/2) T 4

SW3

=

( B - t / 2 - 2 e o ) h ( B- t/2) T -4 8

H = e, =

A5) the

(B-t/2-3eo) h' (B-t/2)2 T 6

+ e; I ,

(B-t/2)2 T 2( B - t/2)T + ht/3

Figure A5

115

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

A.6

Statical moments for channel sections

In addition the terms Q, and Q, may be calculated thus:

Q,

=

Ar~f

Af = thearea of one flangeasshown in FigureA6(a) y , = the distance from the neutral axis to the centroid of the area A , as shown in Figure A6(a) A = the total cross sectional area y , = the distance from the neutral axis to the centroid of the area above the neutral axis as shown in Figure A6(b).

where

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Centroids of shaded area

Figure A6

A.7

Torsional constant for rolled channels

The value of J may be more accurately calculated as sections (see Figure A5):

J

=

(T)

B-t

follows for rolled channel

( T I+ T2)( T I 2 +T22)+$tT23+

+2a4D44-2VsT14- 0.21 T; where

D4 = 2{ [3r + t + F , ] - [2(2r+ t)(2r+ F l ) ] ” 2 } F1

=

T2- r [ S + 1 - (1 + S ) ” 2 ]

V , = 0.10504+0.1S+0.0848S2+0.06746S3+0.05153S4 CY^ =

t r tr t2 -0.1128+0.2829-+0.1320--0.092~-0.0951~ T2 T2 T2 T,

Note: BS4 Channel sections are rolled with a5“ taper to the flanges. The expression given above for a4 has been obtained by linear interpolation between the valuesof the constants for a taperof 163%

and those for parallel flanges. This is a simplification of the more exact procedure given by Johnston and El Darwish(’) but is accurate enough for design purposes. The values of J given in the Tables in this publication (Section 5 ) and in the Steelwork Design Guide Volume l(1o), are based on these constants.

116

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

A.8

Structural hollow sections

For structural hollow sections the torsional constant J and the torsional modulus constant C may be calculated from the following expressions: For circular hollow sections:

J = 21

c

=

22

For square and rectangular hollow sections:

c=-J t + Klt in which Ah is the area enclosed by the mean perimeter, given by: A h

= ( B - t ) (D-?)-R,2(4-v)

2Ahf K = h and the mean perimeter h is given by:

h = 2[(B-t)+(D-t)]-2RC(4-~)

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where

B D I R,

of the section the depth of the section the second moment of area = the average of the internal and external corner radii t = the thickness of the section Z = the elasticmodulus = thebreadth

= =

117

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

APPENDIX B Solutions of differentialequations N o t e : The l o a d i n g i s i n d i c a t e di nT a b l e s6 . 1 T = Tq iAep. p l i teodr a u e

and6.2

I

CAS NO

1

LIMITS

-

EXPRESSION

'ARAMETEI Tz -

0

GJ

T -

0'

GJ

9''

0

0"'

0

'-

~

9

2

-

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3

o l z l a l

Ta 2a

CJ

T

[tanh

L 2a

-

cosh

0'

G J

L sinh (tanh 2a

0'

-

T CJa

[tanh

0'" 2a

- [tanh -

9

T CJa'

5

L 2a

-

{(I

GJ

-

sinh

+

+

cosh

-

sinh

')

'-

cosh

1'

Z

a

aL

-

-

-

sinh

cosh

p]

sinh

0'

UL

cosh

F]

sinh

f

cosh

[tmh

GJa

sinh

i}

tanh a

--

9''

1'

1'

[-+sinh

U)

-

+ 1

cosh g

L

L

-

tanh

L

-

cosh

F]

f

}

tanh

6'"

sinh

9 tanh

a

9'

GJ

[[ i [ i U +

sinh tanh

UL sinh -

9''

-~ CJa

tanh

aL

9"' L

118

sinh -~ GJa2

tanh

aL

L

a

-

aL

L

-

cosh

f

-

sinh

a

aL

sinh

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Appendix B continued CAS1 No

4

LIMITS

2a

2a

ARMETEI:

EXPRESSION

o l z l a t

[z[ ;- ;;] [z [ ;- ;;] z

2'

L'

1

22

0

GJ

0'

T a -

0''

T a _ [-

L

L

GJ

CJa T

0"' 5

B

L'

L

f

Ta -

0'

T _ _ 1 _ [K1.

T 1 -~

GJa'

L cosh3 '

l]

1'

+ 1)

[K1. K, + K,]

cosh

+

(K1 + 1)

'-

[cosh

sinh

T 1 -~ {[K1. K,

0"'

tanh

L 2a

K, + K,]

+ 1)

GJa (Kl

L sinh

- sinh

tanh

[K1. K, + K,]

GJ

GJ (K1

'-

tanh

L - tanh cosh1 ' 2a

[sinh

0

0''

+ sinh

-z -

1 + cosh

''-

+ cosh

L

z - s i n h -za + a-]

'

-

cosh

-

sinh

']

cosh

']

'-

sinh

K,]

l]

+ l]

~~

Ta -~

0

T -

0' Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

1 2 1 aL (K,+ KS cosh - + [- ( c o s h (1+l/K1) a ~1

GJ

GJ

(l+l/Ki)

T -

0' '

1

+ [z1(cosh

2 1 {K5 cosh - + [ i , ( c o s h

CJa(l+l/Kl)

T 1 -~ {Ks

0"' (l+l/Kl)

z

{K5 s i n h

CJa2

1

aL z 1 s i n h - + [il(cosh

-1

aL

(1

-

cosh

[cosh

-L

+ cosh

+

-

1) + cosh

-1UL

UL

- 1) +

UL

-

-

1) + cosh

-1a L

sinh

aL L

-

1 ) + cosh

-1

cosh

aL

-3

cosh

aL 1 [cosh - L sinh -

z

-

sinh

cosh

a

z - a-]

-l]

]

']

l] + s i n h

Where L

aL

[K,=-

1

sinh

K,

=

[

cosh

L

at.

-

--

- cosh

[cosh

-

aL

--

sinh

a

UL

cosh

cosh

-

L

-

l ] + - (U-l)

L L . L + - sinh a a

cosh

-

sinh

-

sinh

] L

g1 -

- tanh L

s;ih

-

aL -

l] +

-

-+

cosh

aL

-

+

-

tanh L

tanh

L

-

]

119

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Appendix B continued CASE NO.

LIMITS

ARAMETEF

EXPRESSION

0

6

0' L

-

1 + cosh

0''

L a

2CJa

sinh

cosh

-a

2 a

-

-

a L

2

-

sinh

L

Q'''

7

05z5al

0

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

Q'

2CJa'

[

CJ

[

CJ

K s [cosh

i-

1)

-

cosh

-

cosh

-

sinh - + a a

+ 1

Ks s i n h

Q'

Q"'

Q

9'

CJa'

[

Ks sinh

K7

CJ

11

aL [cosh -

-

r

&

[cosh

a-

aL

-

;

l ] [tanh

L

-

6''

L CJa

-

[cosh

a-

aL

+ [cosh -

-

l ] [tanh

a

sinh

'1

L

cosh

i)

+ [cosh

5 1-

+ [cosh

Kc

K,

-

120

sinh

=

tanh

a

+

a

S

aL

-

-

l ] cosh

aL

; - l]

sinh

:I

-

J

r

Where

'

l ] sinh

l

L

0"'

-

l L

1) [ t a n h

r

aL

')

cosh

l

; - l]

[cosh a L

aL

-

L

-

cosh

[tanh L

L

tanh

-

cosh

Z!

-

tanh

+

L

-

-

tanh

sinh

L

UL

J

P057: Design of Members Subject to Combined Bending and Torsion Discuss me ...

Appendix B continued CAS1 NO

a

'ARAMETEI

LIMITS

O l Z l U l

9

la GJ

-

9'

l GJ

-

9''

T a [K8 GJa u L

la

GJ

a UL

l

UL a

9"'

a

cosh

a

[K8 s i n h -

a

-

9' L

g

[K8 s i n h

a UL [ K s

l

-

z

GJa2 u L

GJ

Where

Created on 30 22 March July 2009 2011 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement

a

U4

9

9''

UL

a [K8 [cosh aL

l -a -

9"'

UL

EXPRESSION

GJa uL l -a GJa'

-

-

a

Z a L [a

-

-

-Za U[-a L

z - -)l 2a

'l]

l]

- aUL - cosh 1'

[- a[ s i n ha a UL

[-

UL [sinh a

-

[-

UL [sinh a aa

-

K8

=

ULL tanh a

K9

=

UL L L . tanh - s i n h -

a

-+

- +

sinh

a

UL [ s i n h UL a

(- -

cosh a

aa

z

UL -sinh

l]

- -)

L cosh tanh -

a

ULUL + (sinh a

UL a

-1

L s l. n h z ULUL tanh + [sinh

-1

tanh

-

-

-1

z UL UL L s .i n h tanh + [sinh -

-

--

-

a

UL a

sinh

-1

UL

+ cosh

UL UL cosh a

-

-

cosh g + [ s i n h

-

+ 1 +

1'

UL a

-1

sinh

1'

-1

cosh

1'

U 2 L2

7 2a

9

9

9' -Z

9''

9"'

Where

L

a

GJa

L

-l -

a

GJa'

L

Klo

=

KI,

[-

L

t a n h - cosh g + s i n h

z

-1 a

Klo

[-

tarlh

-L

. slnh

- +

cosh

-1z a

[

-

2: :

1

cosh a + L cosh a

-

1

sinh a + L cosh a

sinh

cosh

tanh

-1a

-1

UL

-

a

a

-

l 1

1'