Stability and Ductility of Steel Structures (SDSS'99)

(5)

where the imperfection parameter {r\) is given by Eqn. 2, and A= p ^

{Urf

(6) (7)

In Eqn. 7, L and r are the effective length and radius of gyration respectively. The nondimensional strength (%) is the ratio of ultimate stress (GU) to 0.2% proof stress, X=GJGQ2'

ALUMINIUM COLUMNS On the basis of several hundred tests, the ECCS Recommendations (1978) were prepared for the design of aluminium columns. These included originally three curves, referred to as the a-, b-, and c-curves, and a fairly complex selection system for determining which curve to use. The selection system depended on the type of cross-section (open or closed), the symmetry of the cross-section (symmetric or asymmetric) and the type of alloy (heat-treated or non-heat-treated). In the final ECCS Recommendations, the selection system was simplified to depend only on the type of alloy, and only the a- and b-curves were adopted. The effect of asymmetry was dealt with through a reduction factor which depended on the degree of asymmetry. The a- and b-curves were nominated for heat-treated and nonheat-treated alloys respectively.

104

It is shown in Rasmussen and Rondal (1998) that the general column curve formulation (Eqns 2, 4-5) can represent closely the ECCS a-, b- and c- curves, using the values of a, P, Xo and Xi shown in Table 1. The comparison is shown in Fig. 1.

VALUES OF

TABLE 1 a, p, XO AND Xi FOR THE ECCS a-, b- AND C-COLUMN CURVES FOR ALUMINIUM COLUMNS . Column curve a-curve b-curve c-curve L new a-curve

3

a 0.4 0.7 0.95 0.3

Ao 0.55 0.55 0.35 0.55

0.2 0.15 0.25 0.2

Xi

1

0.2

0.2

1

0.2 0.2

1.2 Proposed curves ECCS curves

1.0 Euler(Eo)

0.8 0.6 0.4

0.2

J 0.5

L

I

1.0

I

1.5

\

L 2.0

2.5

X

Figure 1: Comparison between the ECCS a-, b- and c-column curves and their approximations using the generalised column curve formulation. On the basis of a statistical analysis using more than 400 tests, it is also shown in Rasmussen and Rondal (1998) that better agreement with tests can be achieved by raising slightly the a-curve using oc=0.3, rather than a=0.4, as shown in Table 1. It was suggested that this improved a-curve and the bcurve defined in Table 1 be adopted in Eurocode9 for heat-treated and non-heat-treated alloys respectively. Figure 2 shows a comparison between the proposed column curves and tests on heat-treated and nonheat-treated alloys (Arnault 1967, Bernard et al. 1973, Djalaly & Sfintesco 1972, Kloppel & Barsch 1973). The ECCS a- and b- curves, the column curves of the ISO Recommendations for aluminium structures (ISO, 1992), and the column curve of the pre-standard Eurocode9 (1998) are also shown in the figure. The proposed curve for heat-treated alloys is the slighdy raised a-curve based on (x=0.3. Eurocode9 and the ISO Recommendations use the Perry curve and the linear imperfection parameter given in Eqn. 3. For heat-treated alloys, the imperfection parameter is defined by (a, ?io)=(0.2, 0.1) and (a, ^o)=(0.2, 0.3) in Eurocode9 and the ISO Recommendations respectively. For non-heat-treated alloys, the values (a, Xo)=(0.32, 0) and (a, Xo)=(OA, 0.3) apply for Eurocode9 and the ISO Recommendations respectively. It is clear from Figure 2 that the proposed column curves and the colunm curves of the ECCS Recommendations follow very closely the variation of the test results for both heat-treated and non-heat-

105

treated alloys. The columns curves of the ISO Recommendations and the draft Eurocode9 are in reasonable agreement with the tests for heat-treated alloys. However, for non-heat-treated alloys, the ISO column curve is unconservative at short and intermediate lengths, and the draft Eurocode9 column curve is too conservative at intermediate and long lengths. The main problem with the ISO and Eurocode9 curves is that being based in the linear imperfection parameter, they do not take the real material behaviour into account. As a result, the gradients of the curves are too small at short lengths, particularly for non-heat-treated alloys. 1.2 1

,

^

^

,

1

1

1

r

. ISO, h-t /y ISO, n-h-t

Tests, h-t 0 Tests, n-h-t +

1.0 ><^.^^<^^

0.8

EC9, n-h-t/

U 0.6 h

\

6 \

/

1 -J

Euler(Eo)

^S4N^-->^

Proposed a-curve, h-t

f

(a = 0.3)

/

Proposed b-curve, n-h-t

>?4 ^ * ' j \ .

v

^^•*'V^

N«tl^vStfvO

-^

0.4 0.2 h-t, heat-treated n-h-t, non-heat-treated 1

1

1

1

0.5

1

1

1.0

1

'

1

'

2.0

1.5

1 2.5

Figure 2: Design curves and tests of heat-treated and non-heat-treated aluminium alloys.

STAINLESS STEEL COLUMNS Proposed Column Curves The current draft of EurocodeS, Part 1.4, (1996) only applies to austenitic and austenitic-ferritic (duplex) alloys, and not to ferritic alloys. The latter alloys can, however, be designed using the informative Annex D of EurocodeS, Part 1.4, but as shown in Rasmussen and Rondal (1999), the design approach described in Annex D of EurocodeS, Part 1.4, is very conservative for compression members. TABLE 2 a, p, Xo AND X\ VALUES FOR PROPOSED COLUMN CURVES , Eo=200 GPa Column curve a-curve b-curve c-curve

Alloy austenitic, duplex austenitic, duplex ferritic, 3Crl2

"" >r

Strength class/grade

C?0.2

n

a

P

Xo

S350, S480, C70G, C850

(MPa) 480

4

1.31

0.18

0.67

0.37

S220, S240, S290

240

5

1.31

0.18

0.57

0.30

250

7.5

0.93

0.15

0.56

0.26

The draft Eurocode3, Part 1.4, specifies two strength curves for columns failing by flexural buckling. It uses a Perry-Robertson curve with a linear imperfection parameter, as given by Eqn. 3. For cold-formed open and rolled tubular sections, the imperfection parameter is defined by a = 0.49 and Ao = 0.4. For

106 welded sections, the values of a = 0.76 and XQ = 0.2 apply. In specifying these values, no regard is made to the fact that the mechanical properties of the alloys covered by EurocodeS, Part 1.4, are different and consequently, the corresponding strength curves are different. The differences arise partly because the chemical compositions are different and partly because different degrees of cold-working may be used in the forming process. The effects of chemical compositions and cold-working are recognised by use of strength classes (S220, S240, S290, S350, S480) and cold-worked strength grades (C700, C850, etc) respectively. The numerals refer to the 0.2% proof stress and tensile strength for the Sxxx and Cxxx strength classes/grades respectively. However, despite the differences in 0.2% proof stress of the alloys pertaining to these strength classes and cold-worked grades, a single column curve is specified in the current draft of Eurocode3, Part 1.4, for cold-formed open and rolled tubular sections. As shown in Rasmussen and Rondal (1999), there are significant differences in the strength curves of the various strength classes and cold-worked grades, and consequently, several column curves should be used. It was proposed in Rasmussen and Rondal (1999) to use two column curves for cold-formed austenitic and duplex alloys and a third curve for cold-formed ferritic alloys and 12% Chromium weldable structural steels (3Crl2). The proposed classification is set out in Table 2 which also lists representative values of ao.2 and n for each column curve. The n-values are suitable averages of those given in the EURO INOX Design Manual (EURO INOX 1994), those found from tests on cold-formed hollow sections, and those given in the ASCE Specification (ASCE 1991) and the South African specification (SABS 1997) for stainless steel structures. Using the values of ao.2 and n, and assuming EQ = 200 GPa, the values of a, P, XK) and X\ shown in Table 2 were obtained using the expressions given in Rasmussen and Rondal (1997a). The column curves defined by the values of a, P, Xo and Xi are shown in Fig. 3. The figure also contains the current Eurocode3, Part 1.4, column curve for cold-formed sections, which is closest to the proposed a-curve. 1.4

1

r

1.2 h 1.0 0.8 Euler (EQ) 0.6 0.4 0.2 J.

J-

0

0.25

0.5

0.75

1.0 X

1.25

1.5

1.75

2.0

Figure 3: Proposed column curves for stainless steel columns

Discussion Figure 4 shows a comparison between tests (Johnson and Winter, 1966) and the current column curve o Eurocode3, Part 1.4, as well as the proposed b-curve for annealed and skin passed AJSI 304 (1.4301 ii

107 EN 10088 (1995)) stainless steel alloy. The lower set of design curves is based on the properties of the flat material (£0=204 GPa, ao.2=238 MPa, n=4.1); see SCI (1990) for the n-value. It appears that the EurocodeS, Part 1.4, column curve based on ao.2=238 MPa provides a better fit to the tests than the proposed b-curve based on ao.2=238 MPa. In fact, the calibration (EURO INOX, 1994) of the current column curve of EurocodeS, Part 1.4, was such that the curve provided a suitable lower bound to tests (Johnson and Winter 1966, Hammer and Petersen 1955, Coetzee et al. 1990) on cold-formed columns. Thus, it may not seem appropriate to adopt a new column curve which is lower than the current curve of EurocodeS, Part 1.4. However, the following important points need to be observed: • All tests referred to above were conducted using the American column testing procedure of the 1950s and 1960s which required the column to be positioned on the end platens so as to reduce the effect of geometric imperfections. The columns were shifted on the platens until overall bending under initial loading was eliminated. Loading eccentricities were thus introduced to counter-balance the effect of overall geometric imperfections and achieve bifurcation behaviour. Consequently, the test results represent closely the strengths of geometrically perfect columns. It follows that the current column curve makes no allowance for overall geometric imperfections. • In the calibration described in EURO INOX (1994), SCI (1990), the test strengths were nondimensionalised with respect to the 0.2% of the proof stress of the flat portion of the crosssections. The mechanical properties of the comers were significantly enhanced beyond those of the flat portions because of cold-working. As a result, if the proof stress had been determined on the basis of a stub column test or as an area-weighted mean of the proof stresses of the flat portions and comers, as allowed for in Eurocode3, Part 1.3, (Sections 3.1.2 and 9) and in the South African Standard for Cold-formed Stainless Steel Structures (SABS, 1997), it would have been higher than the proof stress of the flat portions. This would have affected the column curve calibration, as will be demonstrated below. Because of the trend in national standards towards allowing the enhanced comer properties to be utilised in design, the column curve calibration should allow for situations where this procedure is followed. 100

1

1 1 I-section concentric (Johnson & Winter, 1966) • I-section eccentric — - Design — Finite element strength ECS, 1.4 (Oo.2=302MPa) o

80

ECS, 1.4 (ao.2 = 2S8MPa)

60

40

b(ao.2=2S8MPa)

r uiooo 20

O0 2=238MPa^ L/1500 I L/10000 ao.2 = 302MPa , L/10000

25

50

75

_L

1

1

I

100 L/r

125

150

175

Figure 4: Influence of geometric imperfections and proof stress

200

108

Figure 4 demonstrates the effects of changes in overall geometric imperfections and proof stress. The figure shows several analytic strength curves (shown dashed) obtained using the advanced finite element program described by Clarke (1994). Three curves are shown in Fig. 4 based on the properties of the flats (£o=204 GPa, Oo.2=238 MPa, n=4.1). The curves were obtained using the magnitudes of overall geometric imperfection of L/10000, L/1500 and L/1000. A fourth curve based on the mechanical properties obtained from a stub column test is shown using a magnitude of overall geometric imperfection of L/10000. The stub column properties, referred to as "average" properties, were derived in Appendix A of Rasmussen and Rondal (1999) as £0=192 GPa, ao.2=302 MPa and n=3.7. The increase in proof stress from 238 MPa to 302 MPa is a result of the cold-working of the comers. It is clear from Fig. 4 that geometric imperfections and the proof stress value influence the column strength. The finite element strength curve based on the average properties obtained from a stub column test is in fairly good agreement with the test strengths (shown with open circular markers), although slightly higher suggesting that the proof stress derived in Appendix A of Rasmussen and Rondal (1999) may be slightly too high. As a means of accounting for geometric imperfections, the test strength at each slendemess value has been reduced by the ratio of the finite element strength for a geometric imperfection of L/1500 to the finite element strength for a geometric imperfection of L/10000 based on ao.2=238 MPa. The reduced test strengths are shown with solid circular markers. It follows from Fig. 4 that the Eurocode3, Part 1.4, design curve based on the average proof stress of ao.2=302 MPa is higher than the test strengths and so would lead to unconservative design values. In conclusion, the current Eurocode3, Part 1.4, strength curve for cold-formed sections does not allow for overall geometric imperfections and is unconservative if the design yield stress is based on the average mechanical properties allowing for the enhanced properties of the comers. At present, Section 2 Materials of Eurocode3, Part 1.4, does not mle out the use of enhanced average properties. Furthermore, the testing provisions of Eurocode3, Part 1.4, refer to the testing provisions of Eurocode3, Parts 1.1, and 1.3. Section 9 Design Assisted by Testing of Eurocode3, Part 1.3, describes procedures for stub column tests and Section 3.1.2 Average Yield Strength of Eurocode3, Part 1.3, explicitly allows enhanced properties to be utilised in the design of cold-formed carbon steel structures. Thus, the intent of Eurocode3 is to allow enhanced average properties to be utilised in design. In view of this, the current column curve of Eurocode3, Part 1.4, for cold-formed sections is too high when applied to strength grades S220, S240 and S290. The proposed b-curve based on the average properties is in reasonable agreement with the solid markers shown in Fig.4 representing tests on geometric imperfect columns.

CONCLUSIONS The paper summarises recent research on applications of the "general column curve formulation" (Rasmussen and Rondal 1997a) to aluminium and stainless steel columns. The formulation uses the imperfection parameter given by Eqn. 2 in conjunction with a Perry-curve, as defined by Eqns 4-5. Equation 2 is a simple modification of the linear imperfection parameter currently used in the Eurocodes for carbon steel, stainless steel and aluminium colunms. For bi-linear materials with distinct yield plateaus (such as carbon steel), the parameters P and ^i attain the values of unity and zero respectively so that Eqn. 2 simplifies to the linear form for these materials. It is shown that the imperfection parameter defined by Eqn. 2 can produce column curves that approximate closely the ECCS a-, b- and c-curves for aluminium alloy columns. The a- and b-curves obtained using the general formulation have been shown to be in better agreement with tests than the current column curves of the pre-standard Eurocode9 and the ISO Recommendations for aluminium stmctures. A new slightly raised a-curve has been proposed for heat-treated alloys.

109 In the application to stainless steel alloy columns, two strength curves are proposed for the strength classes (S220, S240, S290, S350, S480) and cold-worked strength grades (C700, C850) included in EurocodeS, Part 1.4. These strength classes and grades apply to austenitic and austenitic-ferritic alloys. One further curve is proposed for ferritic alloys and 12% Chromium weldable structural steels (3Crl2). It is shown that the current EurocodeS, Part 1.4, strength curve for cold-formed austenitic and austeniticferritic alloys of strength classes S220, S240 and S290 is too high when the design incorporates the enhanced properties of the cold-worked comers. This is partly because the strength curve was calibrated against tests on nominally geometric perfect columns and partly because the measured proof stress of the flat portions of the sections was used in the calibration, as opposed to the average proof stress based on a stub column test. The general column curve formulation offers a generalisation of the imperfection parameter expression currently used in the Eurocodes. The expression is versatile and capable of producing a wide range of strength curves, including curves with triple curvature as is required for materials with n-values of 8-10. The general formulation is capable of unifying the European rules for carbon steel, stainless steel and aluminium columns while maintaining high accuracy,

REFERENCES Arnault, P., (1967), "Recherche sur le Flambement des Profiles en Alliages Legers", Centre technique Industriel de la Construction Metallique (CTICM), Paris. ASCE (1991), Specification for the Design of Cold-formed Stainless Steel Structural Members, American Society of Civil Engineers, ANSI/ASCE-8-90, New York. Bernard, A., Frey, F., Janns, J. & Massonet, C, (1995), "Reserches sur le Comportement au Flambement de Barres en Aluminium", lABSE Memoires, 33-1, Zurich, 1-32. Clarke, M.J., (1994), Advanced Analysis of Steel Frames: Theory, Software and Applications, Chapter 6: Plastic-zone Analysis of Frames, eds W.F. Chen & S. Toma, CRC Press, London. Coetzee, J.S., van den Berg, G.J., and van der Merwe, P., (1990), "The Behaviour of Stainless Steel Lipped Channel Axially Loaded Compression Members", Report MD'55, Faculty of Engineering, Rand Afrikaans University, 1990. Djalaly, H. & Sfintesco, D., (1972), "Reserches sur Flambement de Barres Aluminium", Reports of the working commissions, lABSE, 23, International Colloqium on Column Strength, Paris. ECCS, (1978), European Recommendations for Aluminium Alloy Structures, European Convention for Constructional Steelwork, 1st ed., Brussels. EN 10088, (1995), Stainless Steels - Part 1: List of Stainless Steels, EN 10088-1, European Committee for Standardisation (CEN), Brussels. EurocodeS (1992), EurocodeS: Design of Steel Structures, Part 1.1: General Rules and Rules for Buildings, ENV 1993-1.1, European Committee for Standardisation (CEN), Brussels. Eurocode3 (1996), Design of Steel Structures, Part 1.4: General Rules - Supplementary Rules for Stainless Steels, ENV 1993-1.4, European Committee for Standardisation (CEN), Brussels.

110 Eurocode9 (1998), Design of Aluminium Structures - Part 1-1: General Rules and Rules for Buildings, ENV 1999-1-1, European Committee for Standardisation (CEN), Brussels. EURO INOX (1994), Design Manual for Structural Stainless Steel, European Stainless Steel Development Information Group (EURO INOX), Nickel Development Institute, Toronto, Canada. Hammer, E.W. Jr., & Petersen, R.E., (1955), "Column Curves for Type 301 Stainless Steel", Aeronautical Engineering Review, 14, Part 12, 33-39 and 48. ISO (1992), Aluminium Structures: Material and Design, Part 1: Ultimate Unit State - Static Loading, Technical Report, Doc. No. 188, International Standards Organisation Conmiittee TC 167/SC3. Johnson, A.L. & Winter, G., (1966), "Behaviour of Stainless steel Columns and Beams", Journal of the Structural Division, American Society of Civil Engineers, 92(ST5), 97-118. Kloppel, K. & Barsch, W., (1973), "Versuche zum Kapitel "Stabilitatsfalle" der Neufassung von DIN 4113", Aluminium, 10, 690-699. Rasmussen, K.J.R. & Hancock, G.J., (1993), "Design of Cold-formed Stainless Steel Tubular Members. I: Columns", Journal of Structural Engineering, American Society of Civil Engineers, 119(8), 23492367. Rasmussen, K.J.R. & Rondal, J., (1997a), "Strength Curves for Metal Columns", Journal of Structural Engineering, American Society of Civil Engineers, 123(6), 721-728. Rasmussen, K.J.R. & Rondal, J., (1997b), "Explicit Approach to Design of Stainless Steel Columns", Journal of Structural Engineering, American Society of Civil Engineers, 123(7), 857-863. Rasmussen, K.J.R. & Rondal, J., (1998), "Strength Curves for Aluminium Alloy Columns", Research Report, Department of Civil Engineering, University of Sydney. Rasmussen, K.J.R. & Rondal, J., (1999), "Column Curves for Stainless Steel Alloys", submitted to the Journal of Constructional Steel Research. SABS (1997), Structural Use of Steel, Part4: The design of Cold-formed Stainless Steel Structural Members, SABS 0162-4, South African Bureau of Standards, Pretoria, 1997. SCI, (1990), "Flexural Buckling of Stainless Steel Columns", Document No. SCl-RT-141, The Steel Construction Institute, London. Talja, A., & Salmi, P., (1995), "Design of Stainless Steel RHS Beams, Columns and Beam-columns", VTT Research Notes No. 1619, Technical Research Centre of Finland (VTT), Espoo.

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

111

ELASTO-PLASTIC STABILITY OF COLUMNS WITH AN UNSYMMETRICALLY STRENGTHENED I-CROSS SECTION G. Salzgeber University Assistant, Institute of Steel, Timber and Shell Structures, Technical University Graz, A-8010 Graz, Lessingstr. 25, AUSTRIA

ABSTRACT A parametric case study is performed concerning the effect of unsymmetrically and partially strengthening of an I-column which is loaded by an eccentric compression force at its upper end and centrically supported at its lower end. The main effects investigated are, firstly, the length of the additional flange plate which is welded onto the compression flange and, secondly, the boundary condition at the top of the column where the external load is applied (hinged versus built-in condition). The study is mainly oriented on the in-plane buckling case. A reduction of the load carrying capacity occurred in a wide range with different boundary conditions. For comparison the flexural buckling and lateral torsional buckling of a laterally unrestrained column is enclosed. KEYWORDS unsymmetrically and partially strengthened column, eccentric loading, limit load analysis, parametric study, in-plane flexural buckling INTRODUCTION The supporting colunms of a silo for the storage of limestone powder with a length of L = 1300cm are eccentrically loaded (Guggenberger 1998). This is because of the necessity of aflatoutside of the whole silo (Figure la). After a damage - caused by a production failure - in the connection between the silo and the columns the outer flange had to be strengthened by an additional plate welded onto the flange. The strengthening was done in a fairly unstressed situation. The cross-section of the column is a HEB360 profile and the dimension of the additional flange plate is 280/30 mm (see Figure la). In a first step in-plane flexural buckling about the major axis was considered by the assumption that lateral buckling failure is excluded by adequate bracing. Thereby state-of-the-art Finite Element procedures were applied taking into account geometrical and material nonlinearities as well as representative geometrical and material imperfections (NLBEAM3D, Salzgeber 1998). The used nonlinear beam elements (including warping effects) are established by a consistently quadratic approximation of the finite strains (moderate rotation assumptions) including the normal hypothesis. A fully automatic path following algorithm (Guggenberger 1992), arbitrary geometrical imperfections and arbitrary one-dimensional

112 material behaviour are essential parts of the used finite element code. In order to get insight into the specific stability behaviour the geometric imperfections were varied with respect to direction (sign), shape and amplitude. The main effects to be investigated on the one hand, the length of the additional flange plate which was welded onto the compression flange and, on the other hand, the boundary condition at the top of the column where the external load is applied and the boundary condition at the bottom (hinged versus built-in condition). In a second step the study was extended by including the 3D lateral torsional buckling failure. As conclusion, the effectiveness of eccentric strengthening is discussed by comparing the different results. IN-PLANE BUCKLING OF AN ECCENTRICALLY LOADED COLUMN System and in-plane buckling behaviour (system II) In Figure l(b and c) the imperfect and deformed column of an appropriate idealized system (hinged at the bottom and clamped at the top) is shown. The displacement plots are magnified. The plastic cross section interaction between the axial force and the bending moment about the major axis is given in Figure 2. The interaction diagrams for HEB 360 (CSO) and the strengthened section (CSl) are established for axial forces acting in the centroid of CSO. The strengthened section (CSl) has approximately the same capacity if an axial force acts in the centre of CSO only, however under pure axial force (centre of CSl) the capacity is 48% higher (under pure bending 23.5%) than the resistance of CSO. For a specific relation between the acting bending moment and the axial force the limiting bending moment is 1.633 higher than for the unstrengthened section. 0.860

0.844

A = 0.0

0.790

0.763

Fref = 3662[KN] oi: W "i h (D | i i ^ \

•B " \

^O I "U

' N ^v^

p.

a1 Co \fh nzlff 3'i w ,f ^- II

S I

^o !i

,

^

^

1

1

PH.

i

HEB 360

a)

PH...plastic hinge

Figure 1: Effect of imperfections on the load capacity (^ = 0.5), imperfect system and deformed structure under loading (magnified) a) connection detail, b) imperfection outwards (negative), c) imperfection inwards (positive)

113 The material has an young's modulus of 21000kN/cm^ and the yield stress is fy = 21 .OkN/cm^. The initial stresses are idealized as a triangular shaped pattern (Figure 2) in the HEB 360 section. Initial stresses due to the welding process in the section and in the plate had been neglected, because the throat area is very small. The plastic resistance of the HEB 360 has been chosen as reference load. Dependent on the imperfection direction the limit loads reach different values as it is shown for a relative strengthening length I, = 0.5. If the geometrical imperfection has an amplitude to the outside of the silo (negative direction) the load maximum reaches a factor of 0.86Npi Q. In the opposite case the limit load is about 10% lower and larger deflections are resulting. The plastic hinges (zones) are located below the strengthening plate, whereby the location of the hinges are different in both cases. In this zone the weak section and the bending moment due to the discontinuity of the centroid fibre are in common. 2 1.5 1

T"

1.6331 Np,,o = 3662[kN] Mp, 0 = 543.5 [kNm]T7 ! ^^?T235 O -..sS-rr^,

iN + Nsd = Nsd • YM Nsd = Nk- Tf. TM

0.5 "Q.

0

+0.5L

•0.5

-1

-0.5 f

'1.5

.5

-0.5

0

0.5

1

1.5

N/N,pl.O

initial stresses

Figure 2: Plastic cross section interaction (N + M) and initial stresses Figure 3a presents the load deflection curves (lateral displacement U) for both cases. For a positive imperfection the initial stiffness is lower than in the opposite case. Varying the relative strengthening length ^ (Figure 3b) it can be stated that the resulting load maxima decrease with larger ^ (for positive amplitude of the geometrical imperfections) by an amount of 10 percent. For ^ greater than 0.55 the resulting limit load factor increases again. Approximately 80 percent strengthening length is necessary to reach the load carrying capacity of the unrestrained double synmietric member. The bearing capacity of a fiilly strengthened member is only 10 percent higher than of an unrestrained member. In this considered case an unsynmietric strengthening is very uneconomic. This behaviour seems strange, because normally it is stated, that additional material cannot decrease the bearing capacity. In this case it is different because due to the unsymmetric strengthening a discontinuity of the centre line occur, which leads to a local bending moment reducing the load carrying capacity in a certain amount. If the geometrical imperfection is chosen outwards (negative sign) an interesting shape of the limit load curve results. Up to ^ = 0.2 a steady increase (10%) results; using longer strengthening the capacity decreases and then growth for t, greater than 0.55 up to 0.80. The maximum limit load can be stated with 0.98Npi 0- If a longer plate is used the capacity is getting lower. This relationship of the strengthening length is caused because in same regions the primary deflection is in the opposite direction as the postbuckling deflection (increasing parts of the curve). The signs in Figure 3b determine the primary (first sign) and postbuckling (second sign) deformation direction. The necessary resistance for the present silo (Ns^ / Npi Q = 0.811) is marked as a dashed line. In a region from ^ = 0.30 to 0.70 this capacity is not available.

114

0.95

Z

z z

z

0

a)

1

2

3

4

displacement U [cm] at the location ^ = 0.5

Figure 3: a) Load-deflection curves, b) Load carrying capacity with varying strengthening length Variation of the boundary conditions and imperfections The behaviour of different systems are compared in the following. In the parametric case study the variation of the strengthening length, boundary conditions, different types of strengthening (unsymmetric and symmetric) and different geometrical imperfections were considered. A classical eigenvalue calculation for the in-plane buckling failure (Figure 4a) shows in principal a monotonic increase of the critical load for each system. However the increase is unsteady and depends on the considered case and the relative strengthening length.

Figure 4: Classical eigenvalue calculation for varying strengthening length and different systems a) in-plane buckling, b) out of plane buckling (same constraints in and out of plane) Four different systems (boundary conditions) with unsynmietric strengthening is investigated by performing a limit load analysis using initial stresses and different geometrical imperfections - two parabolic and two cubical shapes (Figure 5a-d). The amplitude of the geometrical imperfection is given in each case with 8o = L/10(X). In case I (simple beam) and case III (clamped at the bottom) a steady increase of the bearing capacity can be stated, however in a first range (a short plate is welded onto the flange) by a very small amount only. A fully strengthened column has a higher bearing capacity by a

115 factor two than the unstrengthened I-section. The system 11 (clamped at the top) and system IV (both end fixed) a certain reduction in respect to the pure I~section goes in hand with increasing strengthening length in a wide range. The cubical imperfection (T| = 0.75 - the maximum amplitude is located at three quarter from the top of the column) leads to a further reduction of two percent in comparison to a parabolic shape. In the last case (system IV) an unsteady behaviour occurs at the transition to a full strengthening. The curves tend to a factor from 0.96 to 0.98. Full strengthening gives for all imperfections a factor 1.373 for OMNIA (geometrical and material nonlinear imperfect analysis) and 1.482 for MNA (material nonlinear analysis). This example shows, that a small region with less capacity and discontinuities can reduce the limit load by a certain amount. This problem arises directly the question, whether a built-in boundary condition (foundation) will act in reality in the limit state as a built-in condition with full capacity. In thefiguresthe limit loads neglecting stability effects (MNA) are given. Especially for system I a large amount of the reductions occurs by the plastification of the unstrengthened cross section due to the axial compression force and the existing bending moment at the location of the discontinuity. 1.2

1.05

! stability ^S1)I cross |section(CS1)

0.95 h

0.85 R

0.75 I

0.2

b) ' ' I • '

' 1 '

0.6

5 MNA(CSO)

'

0.4

0.8

1

GMNIA 1.373 i MNA 1.482

' ' I ' ' '' •' ' . J e ^

: i%d i

s-^^^^}r^

J

0.95

0.6

z z

•

\^^^

1

' y ; : : ^ ^ ^ ^ \ 0.4 h

^

i 0.487! i

i

Z

f :

5-

• 6+

^^ Z

0.85

;

0.2 U

: 1 1 ! 1m 0.75 r , , , 0.2i , < ,0.4 i , , ,0.6 i . , < 0.8 1 .. 0 0.2 0.4 0.6 c) I d) ^ Figure 5: In-plane buckling behaviour of unsymmetrically strengthened columns, a) system I - simple beam, b) system n - built-in at the top, c) system III - built in at the bottom, d) system IV - fixed at both ends

0.8

1

116 For the systems I and II the different failure modes are pointed out (Figure 5) in the diagrams using parabolic imperfections. In case I nearly in the whole region the unstiffened part is limiting the capacity. The stability failure mode of the stiffened part is limiting in the last zone only. In case II the stability of the strengthened part is the dominating failure mode for ^ larger than 0.8. Due to interaction effects between both zones small increase of the limit load in respect to the independent failure mode of the unrestrained part occur. If the I-section is nearly full strengthened all boundary conditions result in approximately the same limit load. For comparison a synmietric strengthening with two plates (280/30) has been investigated (Figure 6). The main difference is that in each case of considered boundary conditions a strengthening leads to an increase of the calculated limit load. However in the simple beam case (system 5, Figure 6a) nearly the same capacity results as in system I, so the symmetric strengthening has no benefit in this loading case. In system VI (Figure 6b, clamped at the top) strengthening give a smart increase of the limit load first, in the middle region (^ = 0.4 to 0.7) the capacity is nearly a constant, a remarkable increase follows.

0.95

0.85

5+(Ti=0.50) 5- (Ti= 0.50) —.<^—S+(Ti= 0.25) --6-.S+(T|=0.75) I MNA 5 = 0

|W

0.8

0.75

0

0.2

a)

.--• J0;855i

0.4

0.6

0.2

0.8

0.4

0.6

0.8

b)

Figure 6: In-plane buckling behaviour of symmetrically strengthened columns, a) system V - simple beam column, b) system VI - built-in at the top Sensitivity on imperfections of a simple beam column In the study before, the column was eccentrically loaded (systems I and III) with a special ratio between the acting bending moment and the axial compression force. The load carrying behaviour depending on an unsymmetric strengthening of a centrically loaded simple beam column is studied. Centrically loaded in this sense means when the axial force acts in the centroid of the I-beam. Performing the parametric case study (Figure 7a) with increasing strengthening length - using the same geometrical imperfection shapes and initial stresses as above - results in a monotonic reduction of the calculated limit load. The I-section has a limit load factor of 0.725Npi Q. In contrary full strengthening gives a limit load factor of 0.665 which is 9% lower than without any strengthening, although the slendemess is nearly the same in both cases. In order to clarify this unexpected reduction a sensitivity study of full strengthened member had been carried out (Figure 7b). The effects of an eccentrical loading with a constant bending moment had been studied. The eccentricity is measured from the centroid of the I-section. In this situation a general reduction due to stability effects are about 30% in respect to the plastic cross section resistance. If the axial force acts in the centre (e/h = 0) the centroid of the I-section, the bending moment in the strengthened section leads to additional geometrical imperfections and to a reduction of the bearing capacity. If the axial force acts closer to the centroid of the strengthened section the limit load factor increases. The maximum factor results when the axial force acts in between the centroid of CSl and the strengthened flange. The question arises, where the hinge has to be located best, because the choice of the location of the hinge has a direct influence on the calculated limit load. Possibly, only connection elements including a realistic nonlinear behaviour lead to a save and economic limit load.

117

0.75

0.665 0.6

—o--5+(T|=0.50) — c ^ - 5 - ( 1 1 = 0.50) ....<>.... 5+ (n = 0.25) - - A - - 5+ (Ti = 0.75)

0.5

-J

0.2

0.4

- 6+ (Tl= 0.25) -S+(11= 0.75) -MNA 6 = 0

0.25

.

0.6

1

> i_

0.8

1

b)

0.75

0.25

0.5

0

-0.25

-0.5

e/h

OUT OF PLANE BUCKLING BEHAVIOUR OF AN UNRESTRAINED COLU^LN

i CS1 i 1.017:

MNAJCSO j

-Jf^s^

[_ L

.•''^''

J i ^ 1

i

^..^r*—o"

;

..^.^...-A^-r.—fWfl^

[0620]

i1 m-m

t i l l c) ^

"-^

--^ ^ " -

-

'

d)

1

•

1

i

-

i 8-I-;

^"

1

\p.8S0

1

0.2

Figure 8: Flexural buckling behaviour a) system I - d) system IV

1

1

i

1

0.4

1

1

^ j

0.6

.

.

.

i

m

[

1

0.8

1

118 The 3D flexural buckling behaviour of an unrestrained member is described in the following overview. The boundary conditions are the same about the major axis and the weak axis. In Figure 4b the classical eigenvalue calculations are given for all investigated systems. Figure 8 presents the results of the limit load case study for theflexuralbuckling failure of the systems I to IV. As a general result it can be stated that no decrease of the calculated limit load occur when the strengthening length increases. However the increase is limited on a small amount in a wide range. A full strengthened member (HEB 360 and plate 280/30) reaches a limit load factor which is about 50 to 100% higher than for the I-section. For the parabolic imperfection the independent failure modes for each part of the column are shown in the figures. The grey areas sign the decrease of the limit loads due to interaction effects between parts of the colunm. In case IV in the transition to full strengthening a discontinuity can be stated. An increase of 15% of the limit load occur. Performing the study, oriented on the lateral torsional buckling failure, the question of a correct geometrical coupling between both sections arises, because the unit warping functions are not compatible in this case (Salzgeber 1999). The results for lateral torsional buckling are in this special situation nearly the same as for flexural buckling only. In case in a relevant reduced limit load factor results if a short strengthening length is given. SUMMARY A parametric case study on the effect of partially unsynmietric strengthening of an eccentrically loaded colunm was investigated. As a general result it may be stated that for hinged boundary conditions at the point of the load application at the top of the colunm a monotonic increase of the ultimate limit loads (in-plane buckling) is obtained with increasing length of the additional plate. By contrast, for built-in boundary conditions at the top of the colunm reduces the load carrying capacity by about 10 percent, if the additional flange plate extends over less than three quarter of the length of the colunm. Fully strengthening leads to an capacity increase of 10%. Therefore, eccentric strengthening of an I-column does not seem to be effective for built-in conditions. This result seems strange since strengthening is normally supposed to effect an increase in load bearing capacity. However, for the weaker hinged end conditions the load bearing capacity is increased by a remarkable amount (about a factor of two) compared to the unstrengthened situation which is in accordance with our expectation. In the flexural buckling case of a laterally unrestrained column a monotonic increase of the limit load results by varying the (unsynmietric) strengthening length. Full strengthening of the compression flange leads to an increase of about 50% to 100%. REFERENCES Guggenberger W. (1992). Incremental Iterative Solution of Nonlinear Equation Systems in Structural Mechanics - Application of a New Consistent Automatic Variable-Parameter Control, Institute of Steel, Timber & Shell Structures, Technical University Graz, Austria. Guggenber W. (1998). Schadensfall, Schadensanalyse und Schadensbehebung eines Silos auf acht Einzelstutzen, Stahlbau 67 (1998), Heft 6 Salzgeber G. (1998). Nichtlineare Berechnung von raumlichen Stabtragwerken, Dissertation, Technical University Graz, Austria. Salzgeber G. (1999). Modellierung von Konstruktionsdetails in raumlichen Stabberechnungen mittels geometrischer Zwangsgleichungen, Baustatik-Baupraxis 7, Aachen.

Technical papers on STABILITY OF PLATES AND PLATED STRUCTURES

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Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

119

PROPOSALS FOR A MODIFIED BEAM FINITE ELEMENT EMBEDDING THE DEFORMABILITY OF THE CROSS SECTION H. Degee MSM Department, Institute of Civil Engineering, University of Liege, Belgium

ABSTRACT The correct modelling of some phenomena occurring in steel profiles requires a due account for the deformability of the cross section of the profile. Such phenomena are for example local buckling, distortional buckling, lateral buckling, distortional torsion...Different numerical methods are available to study these behaviours (classical finite strips, spline finite strips, plate finite elements, generalised beam theory, special beam elements...). A first part of the paper consists in a qualitative comparative analysis of these methods. In a second part, a new method is proposed, that can be used essentially for the analysis of structures made of members subject to local buckling, or any other phenomenon including a distortion of the cross section. It consists in the integration of the deformability of the cross section directly in a beam element, by enriching the displacement field. Some results related to local buckling, obtained with this modified beam element, are presented. KEYWORDS Numerical methods, finite elements, thin-walled structures, structural analysis, local buckling, coupled instabilities, distortion. INTRODUCTION In the current steel constructions, the use of thin-walled members has become an everyday reality, through the use of welded profiles on one side and cold-formed profiles on the other side. Another tendency is the use of high strength steel. These two tendencies, associated with a wish of limiting operations such as the welding of stiffeners, force the designers, even for structural elements having a beam aspect, to account for a wide range of phenomena, neglected in the traditional steel constructions, associated with the deformability of the cross sections of the used profiles. So phenomena such as local buckling, distortional buckling, lateral buckling of the flanges, or torsion resistance by distortion of the profile cannot be ignored anymore.

120 During the last years, a great number of tests and numerical simulations have been carried out, and all these phenomena are better known. But most of these tests and simulations have been carried out on single members. The main aim was indeed to develop practical rules for the design or verification of the structural elements. At the point of view of the numerical models, a lot of methods have been used and have produced very good results. It remains that the following question raises : what about their use for the analysis of complete structures such as multi-storeys buildings, or bridges ? So, is it possible, for example, to represent the effect of a loss of rigidity of a member subjected to local buckling, on the overall stability of a frame. Present paper aims at answering this question. REVIEW OF DIFFERENT EXISTING METHODS Three main classes of methods do exist that enable the study of beams with deformable cross section. In ref. [2], a short description of each method is presented. In the following paragraphs, only the basic advantages and disadvantages of each class of methods are pointed out. Finite strip method This method is very efficient to study a single member. In contrast, it is practically unusable to study complete structures, essentially because of the support conditions. In addition the use of finite strips is restricted to simply supported prismatic members only. Plate finite elements This method allows a great freedom for what regards the geometry of the profiles, and the support conditions. Good results with such a kind of model require a rather fine meshing. For example, even with very efficient elements, it is necessary to use 6 to 8 elements over the heigth of an I-profile's web, that are square, or ahnost square, finite elements. When applied to the modelling of complete structures (frames, bridges...), such a way to proceed can lead to the necessity of very important computation time and storage space. Moreover practical difficulties related to the meshing and to the analysis of a lot of results must be overcome. Modified beam finite elements This class gathers all the methods for which either the displacement field, or the constitutive law or the resistant section of a beam finite element is adapted so as to account for phenomena such as local buckling. Coiiq)ared to plate finite elements models these methods aim at reducing the total number of degrees of freedom of the structural system. They also facilitate the reaUsation of the meshing: the topology of a beam finite element is still defined by two nodes (an axis) only and the orientation of the cross section around its longitudinal axis.

121 These elements present however a number of disadvantages: • • •

•

They are usually developed to study one phenomenon (generally local buckling), and sometimes only one kind of loading (pure compression, pure bending...)In examples where the local behaviour is accounted for through a complementary displacement field, the shape of this local field is very often chosen at the beginning of the computation and kept such during the whole loading range. Almost all these elements require a preliminary computation (critical analysis, or even non linear analysis), using finite strip method or plate finite elements, in order to determine a shape of local deformation or a local constitutive law. In the example quoted as ref [1], the element works well if the half-wavelength of the local phenomenon is significantly inferior to the length of the beam finite element. Therefore a refinement of the meshing can lead to a deterioration of computation results.

Conclusion There is clearly a duality between plate finite elements on one side and modified beam elements on the other side. The elements of the first kind suit for any type of member geometry, and are likely to model a large range of phenomena with a same discretisation. Unfortunately they are very costly in terms of computation time and storage space. The second kind of elements are less costly, but have a more limited use. The future of the plate modelling is essentially linked with the increase of the computer's performances, while the future of beam modelling depends on the generalisation of the capacities of the beam elements, in such a manner that a same element can exhibit a wide range of behaviours. PROPOSALS FOR A MODIFIED BEAM ELEMENT Regarding the conclusion of the first part of the paper, the main ideas holding up the proposals are the following: • • • •

•

•

Keep the fecilities of a beam meshing. An element should be represented by its axis (two nodes), and the orientation of the cross-section principal axes. Limit the increase of the number of degrees of fi-eedom of the element, compared to a finite element with undeformable cross section. Allow for different kinds of support conditions and member-to-member connections. Be able to represent an as great as possible number of phenomena related to the deformability of the cross section. So the local displacement field of the element should be able to represent any shape of deformation of the cross section. Avoid any preliminary computation, before FE computation itself. So the element should be able to determine by itself the shape of the local deformation and the wavelength of the local phenomenon (even if both should be varying over the length of the structural member). Try to obtain results going better, or at worst staying stable, when refining the longitudinal meshing of a structural member.

122 Based on these ideas, a special beam element is currently being developed by the author of present paper (see ref. [3]). BASIC ASSUMPTIONS OF THE ELEMENT Constitutive law In order to model correctly the local behaviour of the wall elements composing the member, a biaxial constitutive law is used (plane stress state). So for example a wall with its normal axis oriented in the z-direction will be subject to axial ax stresses, as for usual beam elements, but also to additional ax, ay and Xxy stresses that are associated with the local behaviour. Strain field The total strains are obtained by adding the classical beam strains and the strains associated with the local deformation of the wall elements of the section (see Eqn. 1.1, 1.2 and 1.3). Since a biaxial law is used, the transverse strain associated with the beam behaviour must be explicitly expressed, in order to set a usual beam behaviour in the absence of local deformation (see Eqn. 2.2). The strains associated with the beam displacements are obtained by a Bernoulli formulation, while the strains associated with the local plate displacements obey a KirchhofiF formulation. The general formulation of linear strains for a wall element with its normal axis oriented in the zdirection is presented here below (Eqn. 1 and 2):

^xx " ^xx ^yy

''" ^xx

(1.1)

"•" ^yy

(1.2)

^^yy

s^=s^+s^

(1.3) (2.1)

s';;=-ys^=-

5M *" -V

<=o

(2.3) (2.4)

8x dv'" < - ^

^'-'^

£ ^ = ^ + ^r-

'^

(2.2)

dx

dy

dx

(2.6)

123 Local displacement field Similarly to the general formulation of strains, only the case of a wall element with its normal axis oriented in the z-direction is here considered. The local displacement is assumed to be perpendicular to the reference plane of the wall element. The secondary local displacements, such as described in ref. [1], are neglected. In order to be able to represent any shape of local deformation, a superposition of elementary local transverse modes will be used (see Eqn. 3) :

>v,(x,>^) = c o s ^ * / ; ( x ) h

(3.2)

In Eqn. 3.2, h is the width of the wall element perpendicularly to the longitudinal axis of the beam, and fi(x) represents the magnitude modulation of the elementary / mode over the length of the element. This kind of local displacement, associated with the possibility of a rigid rotation of the cross section around the longitudinal axis of the beam, allows the representation of almost any shape of local deformation of the cross section, provided a sufficient number of elementary modes be included in (3.1). Modulation function fi(x) The use of one modulation function for each separate elementary local mode (see Eqn. 3) was preferred to the use of a single modulation fimction for a local transversal mode obtained by a preliminary superposition of local elementary modes (see Eqn. 4). Though it leads to a slight increase of the number of degrees of freedom of the element. w(x,;;) = / ( x ) * X 4 c o s ^

(4)

Indeed only the first formulation allows to represent a deformation shape of the cross section that varies continuously over the length of a member. That is especially necessary to model correctly shear buckling. Modulation cubic functions are chosen, similarly to the interpolation functions of classical beam finite elements. The four parameters of each ftinction that enable the connection of an element with another one are both the magnitude and slope of the local displacement at both nodes of the element.

124 SOME RESULTS Some very simple tests have been carried out in order to validate the model. Local buckling of a rectangular plate in compression The buckling coefficient ko (=acr/cTEuier) of a rectangular plate that is considered as a beam having a thin rectangular cross section is first computed. The dimensions of the cross section of the beam elements are h (width of the plate) and b (thickness of the plate). The results are given in Table 1, for different kinds of longitudinal meshing, and different values of the local buckling halfwavelength (A,/2). TABLE 1 LOCAL BUCKLING COEFFICIENT FOR A SIMPLY SUPPORTED 5000 x

1250 MM (ASPECT RATIO a=4)

PLATE S U B J E C T TO PURE COMPRESSION

|Nbr. of half-waves XI2 (mm) 4 1250 1000 5 1667 3 6 833 714

1

!__

ka,th

4.00 4.20 4.34 4.70 5.39

16 ELM. 4.00 4.20 4.34 4.70 5.40

10 ELM. 4.00 4.21 4.34 4.73 5.46

8 ELM. 4.01 4.23 4.34 4.77 5.55

6 ELM. 4.02 4.27 4.34 5.18 6.00

4 ELM. 4.20 4.60 4.33 5.68 6.25

It can be observed that the use of a succession of cubic functions over the length of the plate, instead of the theoretical sine waves, provide quite good results, even when rough meshing. With 1 EF over a local buckling half-wavelength, the error is only 5 %. Also the non coincidence of nodal lines of the local phenomenon with nodes of the meshing has no consequence on the computation results. Last a refining of the meshing gives better results. Local buckling of a longitudinally stiff ened plate In order to represent the behaviour of a plate stiffened by a longitudinal stiffener at mid-width of the plate, only the second elementary mode is activated. The results are given in table 2. TABLE 2 LOCAL BUCKLING COEFFICIENT FOR A SIMPLY SUPPORTED 5000

x 1250 MM (ASPECT RATIO a=4)

PLATE IN PURE COMPRESSION STIFFENED BY ONE LONGITUDINAL STIFFENER AT MID-WIDTH

1 Nbr. of half-waves X/2 (mm) 8 625 556 9

ko,th

16.00 16.22

16 ELM. 16.03 16.28

10 ELM. 16.14 16.47

'8EiJVf. 1 16.81 17.23 1

|

125 The results are going bad faster than for the unstiffened plate. The local half-wavelength is here shorter, so that more elements are required to get a same accuracy of results. The main interest of those computations is to show how much easily the presence of a longitudinal stififener can be represented, only by preventing the &st elementary local mode at each node of the meshing. Local buckling of a transversally stiffened plate The effect of an effective transverse stififener is obtained by preventing any local transverse displacement at the node corresponding to the line where the stififener is located. Results are given in table 3. TABLES LOCAL BUCKLING COEFFICIENT FOR A SIMPLY SUPPORTED 6250 x 1250 MM (ASPECT RATIO a=5) PLATE IN PURE COMPRESSION STIFFENED BY ONE TRANSVERSE STIFFENER AT MID-LENGTH

1 Nbr. of half-waves A-/2 (mm) 1041.7 6 1562.5 1 ..._.. ._4

k
4.13 4.20

20 ELM. 4.14 4.20

16 ELM. 4.14 4.20

10 ELM. 4.16 4.20

6 ELM. | 4.44 4.28 J

The stififener forces the plate to buckle in non square local waves. For the vmstiffened plate, the buckling coefificient would be k^ = 4 (with a 5 half-wavelength buckling shape). Other examples Critical analyses have also been carried out on plates subject to pure bending, or compression and bending, or shear, (see ref [3]). The fiormulation presented above has also been extended to Iprofiles with thin web, and could easily be extended to other kinds ofi cross sections (channels, boxes...) CONCLUSION The future ofi modified beam elements is lying in their ability in representing correctly an as great as possible number ofi behaviours. In this perspective, a kind ofi "inverted finite strip"fiormulationis proposed. Displacements are defined by polynomial functions along the longitudinal axis and sinusoidal functions in the transverse direction. This new fiormulation is fiound quite efficient fior critical analysis ofi members. It is now being extended to post-critical behaviour (that means non linear analysis).

REFERENCES [1] Ali, M.A. & Sridharan, S., (1989), "A special beam fmite element fior the analysis ofi thinwalled structural components". International Journal for Numerical Methods in Engineering, Vol. 28, pp 1733-1747. [2] Degee, H., (1999), Internal Research Report n° 236, MSM Department, University ofi Liege. [3] Degee, H., (1999), Internal Research Report n° 237, MSM Department, University ofiLiege.

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Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

j 27

TOWARDS A BETTER KNOWLEDGE ABOUT THE WEB BREATHING PHENOMENON Y. Duchene\ R. Maquoi\ and M. Skaloud ^ MSM Department, University of Liege, Liege (Belgium) ^ Institute of Theoretical and Applied Mechanics, Czech Academy of Sciences, Prague (Czech Republic)

ABSTRACT The slender webs of steel plate and box girder bridges are by nature imperfect and are subjected to repeated in-plane loads. That results in so-called "web breathing", i.e. repeated out-of-plane buckling, and in very significant secondary bending stresses. The latter can induce cumulative fatigue damage and possible fatigue cracks. Based on lessons drawn from experimental and numerical investigations and from analytical studies, two approaches are presented such as to enable to account for web breathing in design.

KEYWORDS Civil Engineering, Plate Girders, Fatigue, Web Breathing.

INTRODUCTION Recently built steel and composite bridges of medium and long spans have webs that are substantially more slender than in the past. These webs always exhibit an initial out-of-flatness that, for a given depth b^ of the web, tends to increase when the web thickness t^ decreases. This imperfection causes a second order behaviour when the girder is subjected to repeated in-plane loading. The imperfect plate deflects out-of-its plane according to the load spectrum. The change in curvature generates significant bending stresses in the web, especially at the boundary of the web plate, i.e. where flexural restraints are exerted by the flanges and stiffeners. These bending stresses are usually termed "secondary" - though they are not at all "secondary" in magnitude - because they can only be assessed based on a second order theory. In service conditions, the web buckles repeatedly; this phenomenon is usually termed "web breathing". The pronounced cumulative damage that results from the repeated variation of the secondary bending

128 stresses (Figure 1) is likely to initiate fatigue cracks and thereby substantially to influence the limit state of the girder. Fatigue cracks can occur at several places (Figure 2) of the breathing web, according to the type of loading and the type of weld detailing. When the web is exposed to predominant bending, a so-called type 1 crack is prone to appear at the toe of the fillet weld that connects the web with the compression flange. A so-called type 4 crack may develop in a web subjected to predominant shear, in the region where the tension band associated with the postcritical behaviour anchors into the boundary frame of the web panel; such a crack, once initiated, propagates along both the related flange and the transverse stiffener. Furthermore, a crack may occur in the tensile region of the girder, either in the tension flange (type 3 crack) or in the close vicinity of the stiffener end (type 2 crack). Type 2 and type 3 cracks can be controlled by referring to the existing fatigue design rules for the relevant details. In contrast, there is a lack of a reliable and suitable model for the onset of type 1 and/or type 4 cracks. Compression flange /Transverse stiffener

Secondary bending stresses

Fatigue crack

Out-of-plane deflection due to web breathing Figure 1 - Secondary bending stresses induced by web breathing Type 4

Type 3

D

Tension band

B

Type 4

Fatigue crack Figure 2 - Cracks induced by cumulative damage In an attempt to solve the problem, it was in the past proposed to limit either the b^ty^ ratio of the web or the maximum load acting on the web plate in order to make the cumulative damage disregardable. Such rough procedures cannot provide a satisfactory solution to the problem and are just a lower bound. Therefore it is desirable to develop a more economical and physically based approach. The aim of the present paper is twofold : • To draw the main conclusions from experiments in the field; • To present two approaches to the fatigue assessment of breathing webs.

129 LESSONS DRAWN FROM EXPERIMENTS The process of cumulative damage in breathing webs was very little known till recently. A first step towards a better understanding of the phenomenon was reached based on tests conducted on suitably designed and instrumented experiments. The latter were aimed respectively at: • •

Updating the fatigue resistance of the relevant detail category; Performing breathing tests on complete girder specimens.

Updated evaluation of the fatigue resistance Type 1 and 4 cracks initiate and propagate as a result of secondary bending stresses induced by the breathing of the web. They are characterized by the same fatigue resistance. The latter was established based on several kinds of fatigue tests : • So-called "plate model" and "beam model" tests [9], which aimed at investigating respectively the crack propagation along the fillet weld and across (but till half the width of) the plate thickness; • Tests on closed section models with either one or both-sided welds and either good or poor fitting [10]; only results for both-sided welds (current practice in bridge fabrication) are considered here; • Tests on large scale specimens subjected to repeated shear [3] with measurements of secondary bending stresses (on the surface of the web in the direction perpendicular to the weld and therefore termed maximum perpendicular surface stress) and of the number of cycles corresponding to a 100 mm crack length (criterion taken as the onset of the ultimate limit state). The characteristic fatigue resistance at 2.10^ cycles was obtained from a statistical evaluation of these tests (see Table 1). TABLE 1 CHARACTERISTIC FATIGUE RESISTANCE

Fatigue resistance at 2.10^ cycles Characteristic value

MAEDA's tests [9] Plate Beam 116MPa 137 MPa (*) (*)

DOWLING's tests (both-sided welds) [10]

DAVIES'tests

[3]

1

114 MPa 109 up to 114 MPa (little effect of the fitting) (*) Evaluations conducted in [1] and [2] in accordance with the procedure described in [11]. Considering the set of the aforementioned test results (except MAEDA's beam model tests - because of a different criterion for fatigue resistance) and using the evaluation method described in [11], DUCHENE [2] obtained the characteristic fatigue resistance at 2.10^ cycles Aa= 110 MPa. Should reference be made to the method described in [12], then this resistance increases up to 115 MPa. Therefore the detail category 112 would be appropriate to control fatigue cracks of type 1 and 4 by referring to the maximum perpendicular surface stress. Supplementary information provided by ENV 1993-1 [12] for welds experiencing shear flow indicates that the detail category 80 as appropriate for fillet welds and 100 for full penetration welds.

130 Tests on girder specimens An extensive investigation [5]-[8] was conducted in the Czech Republic on (to date) 132 carefully instrumented two web panel girders with a web depth-to-thickness ratio ranging from 175 up to 320. These girders were simply supported and loaded by a single concentrated load F applied upright the transverse stiffener at mid-span. The loading cycle ranged repeatedly from a minimum load of 10 kN up to a maximum load which varied from test to test. Both panels were thus subjected to combined bending and predominant shear. Li the present paper, it is conmiented only on the results that are relevant for the cumulative damage in breathing webs and constitute therefore the background for an assessment of the fatigue resistance of such webs. In all of the test specimens, a fatigue crack initiates at the toe of the filled welds assembling the web plate with its boundary elements, where the principal surface stress range is maximum. The crack propagates along the weld with an increase in the number of load cycles and frequently goes diagonally across the web comers, where the tension band in the buckling web is anchored. The number of load cycles at failure is a measure of the fatigue life of the test girder; it depends mainly on the stress range and may vary very significantly with its magnitude. Crack propagation is basically a continuous process. A sudden increase in the rate of crack growth has never been observed. The collapse of the test girders occurs : • Either via a shear failure mode, with pronounced plastic buckles developing along the tension diagonal in the web and plastic zones occurring in the flanges till the final stage when the web behaviour looks like one of a web with an opening (the latter being due to a crack cutting the tension band); • Or via a progressive separation of the compression flange from the web plate due to crack propagation, followed by flange buckling in the web plane. It was also observed that the initiation and propagation of cracks, and consequently the fatigue life of the test specimens, was significantly influenced by the quality level of the relevant welds and by both the pattern and magnitude of the initial out-of-flatness of the web plate. FATIGUE RESISTANCE OF BREATHING WEBS Two methods for the fatigue assessment of breathing webs were separately developed in Liege and Prague. Though basically different, each of them constitutes a suitable basis for the fatigue analysis of slender webs.

Cardiff-Prague approach This approach is based on specific S-N fatigue curves that relate a nominal/geometric stress range S to the number A^ of loading cycles. As the crack-prone areas of breathing webs belong to unclassified detail categories, it was thought preferable to base the fatigue assessment on the geometric stress range, i.e. on the maximum principal surface stress range AcJp in the close vicinity of the weld concerned. Based on a large number of experimental results, it was found appropriate to propose two S-N curves in order to account, on the one hand, for the paramount influence of the quahty level of the welds and of the shape and magnitude of the initial curvature of the web plate on the other :

131 •

For girders with good quality automatic and MIG welds and small initial curvatures, the detail category 125 fatigue strength curve provides a satisfactory basis for fatigue assessment: log N 125 = 12.601 - 3 log A(7p log Ni25 = 16.536 - 5 log AOp

•

(N<5.10^) (N>5.10^)

(La) (Lb)

For rod welded girders and/or girders with relatively large initial curvatures, the detail category 100 fatigue strength curve seems more appropriate : logNioo = 12.301 - 3 log A(7p log A^ioo= 16.036-5 log zlcT^

(N<5.10^) (A^>5.10^)

(2.a) (2.b)

A reliable determination of the principal surface stress range AOp preconditions a successful use of the above fatigue strength curves. In waiting for the results of extensive calculations that are currently under way in both Liege and Prague with the view to provide reliable formulae for Aap, a set of formulae derived by T.M. Roberts [4] via a simple analytical solution can be used : •

The shear buckling coefficient kf for a web plate with fixed boundaries, width bw, depth dw and thickness t^: kf=2yfAB

(3)

where : A = 12.69[(d^ fbj'

+ ( b , / d j ] + 7.256(d^ /b J

B = 0 . 8 2 3 4 [ ( d ^ / b , ) ^ + ( b , / d J ] + 0.8062(d,/b^) •

(4) (5)

The elastic critical shear buckling stress for the same plate : T,^=kf[7r'E/l2(l-v')YK/dj'

(6)

with Young's modulus E and Poisson's ratio v. •

The maximum secondary bending stress an in the direction perpendicular to the web boundary^ and the nominal membrane stress Tnt: c7„=[E/180(l-v')](t^/h)(11.29 + 2.24kf/B)C Kt^P^dj^ with: h : the lesser of the web plate dimensions Z?^ and dy,; P : in-plane shear load; C = 1.5a + 0.25a'

' Established based on an initial curvature affme to the shear buckling mode and having an amplitude of hf ISO.

(7) (8)

132 •

The maximum principal surface stress Cp at the plate boundary, from Mohr's circle : dp =0.5(l + v)cT„ + V o , 2 5 ( l - v ) V > r „ ' ,

•

(9)

The principal surface stress range ACp, from the difference in stresses CTp at the extremes of the applied load range: Ao =0

-o

(10)

As the above procedure was only validated for aspect ratios h^l d^ < 2, this ratio shall conservatively be limited to 2 when using (5) and (6) for larger aspect ratios. Liege approach The Liege approach is based on extensive numerical simulations, analytical developments and updated assessment of the fatigue resistance to web breathing [1], [2], [13]. A large number of non-linear simulations were respectively conducted on complete girders, on girder panels and on individual plates with idealised flexural and membrane boundary conditions. They aimed at examining the influence of the mechanical and geometrical parameters on the magnitude of the secondary bending stresses. They showed that it was sufficient to analyse individual plates only, which speeded up the calculations and stimulated the development of analytical solutions. The effects of the shape and magnitude of the initial curvature of the web plate on the magnitude of the secondary bending stresses were highlighted. It was also observed that a small increase in loading, when the latter approached the actual buckling load of the web panel, was likely to induce, due to a snap-through phenomenon, a large secondary stress range. Therefore, while an initial out-of-flatness affine to the first buckling mode produces the maximum secondary bending stresses in the pre-critical range, other shapes of this initial deflection need to be considered when the web panel reaches the post-critical range. Using the energy methods and solving the Von Karman-Marguerre plate equations by the Galerkin method provided the format of analytical expressions for the secondary bending stresses Ct induced by web breathing in plates subjected to compression, bending or shear. For any type of loading, this stress, normalized with respect to the so-called reference Euler stress : (T^=[;r'E + l2(\'-v')](t^/dj\

(11)

writes : a, /a, =c,(e/t^ )^c,(e/t^ f +c,(ee^ /tl )

(12)

Co designates the magnitude of the initial deflection while e is that of the additional deflection. The latter is related to the loading magnitude through a complex cubic equation, which establishes a link between the secondary bending stress and the relevant loading. The coefficients c in equation (12) can be expressed in terms of the aspect ratio of the web plate, by fitting them to the results of the numerous non-linear numerical simulations which were performed. It is henceforth possible, though lengthy, to perform a fatigue resistance check according to the socalled Procedure 1. The variation Aab induced by a variation of the applied stress cr ov r is first

133

computed and then compared with the relevant fatigue resistance of the constructional detail. However, it is more convenient to base the check of the fatigue resistance on a control of the applied stress than on one referring to the secondary bending stresses. Therefore the above procedure, requiring the solution of non-linear equations, was run in an extensive parametric study in order to establish a set of curves which could be used as design aids. These curves provide the maximum value of the applied stress that, producing a variation of 110 MPa (see Section 2) in the secondary bending stress, should not be exceeded with a view to prevent cracks of type 1 or 4 from occurring prior to 2.10^ cycles (so-called Procedure 2). They are plotted versus the main parameters, i.e. the aspect ratio iK' /tw) of the web panel, the slendemess fi = dw /tw, the relative magnitude (eo /t) of the initial imperfection and the extreme stress ratio R. Though user friendly, these curves cover only a limited number of plate geometries. Therefore a further attempt was made towards the formulation of simple criteria to be fulfilled to prevent the initiation of type 1 or 4 cracks. These criteria involve only directly computable parameters of the web plate and require only a minimum of calculations. They were derived via a careful analysis of the available set of test results. These simple rules are presented below in the format of possible normative clauses. Girder subjected to any distribution of repeated longitudinal direct stresses The maximum longitudinal direct stress shall not exceed the smaller of the following values : • The elastic critical stress of the web plate panel fitted with simple supports along the transverse (loaded) edges and fully clamped supports at the longitudinal (unloaded) edges; • The fatigue resistance to type 2 crack (i.e. the one occurring in the tension zone of the web at its junction with the transverse stiffener), which anyway has to be controlled independently. Girder subjected to repeated shear The maximum shear stress shall not exceed the value drawn from the following equation :

(l-R)(r^

/r:r ){A+(rT / Ax\ j ; = l

where : min '

max

A = 1.27R + 0.34 zlr; = 68 MPfl Girder subjected to combined longitudinal direct and shear stresses The applied maximum direct and shear stresses shall fulfil respectively the relevant of the above criteria.

CONCLUSIONS The slender webs of the large plate and box girders of steel bridges are prone to breathing under the many times repeated loading to which they are subjected. That can result in cumulative damage, which influences significantly the limit state of the structure. Though further research work in the field is still desirable - and there are new investigations in progress - before a complete satisfactory solution can be found, two first approaches are described in the present paper, such as to enable the designer to account for the breathing phenomenon in the design of steel and composite bridges.

134 ACKNOWLEDGEMENTS The authors gratefully acknowledge the financial support of their research on web breathing and the fatigue limit state of thin-walled steel girders, provided by : • The Belgian National Foundation for Scientific Research; • The Grant Agency of the Czech Republic. Thanks are also addressed to all those (researchers, technicians, collaborators) who have successfully contributed to the authors'research efforts in the field. REFERENCES Only the basic bibliography to which reference is made in the paper is listed below. A much more complete one on the general subject of fatigue resistance to web breathing may be found in references [1], [2] and [6]-[8]. 1. Remadi A. (June 1995). Etude theorique et experimentale du risque de fatigue a la fissuration amesemelle de poutres metalhques a ame elancee. Ph.D. Thesis, Institut National des Sciences Appliquees, Rennes. 2. Duchene Y. (September 1998). Etude par voie analytique et numerique des effets de la respiration des ames elancees sur la resistance ultime des poutres metalliques a ame pleine ou en caisson. Ph. D. Thesis, Universite de Liege. 3. Davies A.W., Roberts T.M., Evans H.R. and Bennet J.H. Fatigue of slender plates subjected to combined membrane and secondary stresses. Journal of Constructional Steel Research, 30, 1994. 4. Roberts T.M. (1996) Analysis of geometric fatigue stresses in slender web plates. Journal of Constructional Steel Research, 37, 33-45. 5. Skaloud M., Zomerova M. (September 1997) Experimental research on the breathing of slender steel webs. Proc. of the Int. Conf Experimental Model Research and Testing of Thin-Walled Structures. Prague, 275-278. 6. Skaloud, M., Zornerova, M. and Roberts, T.M. (May 1997) Fatigue behaviour of the breathing webs of steel plate girders. In : Proc. of the 18^^ Czech-Slovak Int. Conf. on Steel Structures and Bridges, Brno. 7. Skaloud, M. and Roberts T.M. (May 1998) Fatigue crack initiation and propagation in slender webs breathing under repeated loading. In : Proc. of the 2"^ World Conference on Steel in Construction, San Sebastian. 8. Skaloud, M. (1999) Ermtidungsverhalten von Stahltragem mit schlanken Stegblechen unter wechselnder Beanspruchung. Stahlbau, 68:1, 3-8. 9. Maeda Y. (September 1978) Fatigue cracks in deep thin-walled plate girder. Bridge Engineering, Transportation Research Journal, National Academy of Sciences, USA, 1, 120-128. 10. Ghavami K. and Dowling P.J. (1986) Fatigue performance of welded joints in closed section structures. Journal of Pipelines, 5, 289-299. ll.Brozzetti J., Hirt M., Sedlacek G., Smith I. (December 1989) Background documentations Chapter 9 - Document 9.01 : Background informations on fatigue design rules. Statistical evaluation. Prepared for the Commission of the European Communities, First draft, Eurocode 3, Editorial Group. 12. Comite Europeen de Normahsation (July 1992) Eurocode 3 - Design of steel structures - Part 1.1 - General rules and rules for buildings. 13. Duchene Y., Maquoi R. (1998) Fatigue resistance to web breathing of slender plate girders subjected to shear. Proceedings of the T^ World Conference on Steel in Construction, edited by G.W. Owens, Elsevier Science Limited.

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

135

INTERACTION OF STIFFENER-END-GAP AND STIFFENER SIZE IN THE ULTIMATE STRENGTH OF THIN-WALLED GIRDERS L. Dunai, J. Nezo Department of Steel Structures, Technical University of Budapest Budapest, H-1521 P.O.B. 91, Hungary

ABSTRACT In an ongoing joint research project the effect of stiffener-end-gaps on the ultimate behaviour of thinwalled plate girders are studied. Papers have been presented on the results of experimental and numerical studies (Okura et. al. 1997, Dunai et. al. 1998). In this paper a parametric study is presented with the aim to show the interaction between the sizes of the stiffener-end-gaps and the sizes of the horizontal stiffeners. The details of the geometrically and physically nonlinear finite element (FE) model and its verification are discussed. The parametric study is designed to investigate a wide range of stiffener-end-gaps and horizontal stififener sizes. The nonlinear behaviour is determined by the FE analysis and the results are summarised with a focus on the ultimate strength.

KEYWORDS Thin-walled girder, robotics welding, ultimate strength, stiffener-end-gap, horizontal stiffener, FEM analysis, parametric study.

INTRODUCTION Background Advanced fabrication methods use the advantages of CAD and CAM systems, which provide more effective and economical fabrication of steel plate and box girders. In an ongoing joint research of the Department of Steel Structures, Technical University of Budapest, and the Department of Civil Engineering of Osaka University the effect of advanced fabrication methods and ultimate behaviour of plate girders are studied. The main focus of the research is on the structural details of horizontal

136 stifFeners due to the requirements of robotics welding. The studied part is the gap between the horizontal and vertical stifFeners, which allows the application of simpler welding robots in the automated process. Previous Studies Parallel experimental and numerical studies have been completed to investigate the ultimate behaviour under pure bending of four specimens (Okura et. al. 1997, Dunai et. al. 1998). The results of the nonlinear finite element analyses and experimental tests were in good agreement, so the developed FE models could be used for additional numerical tests. Based on these experiences numerical parametric studies were designed to determine the effects of different parameters on the uhimate behaviour. In this paper first the characteristics of the developed FE models and nonlinear solution techniques are discussed. The second part of the paper concentrates on the ultimate strength of plate girders in the interaction of stiffener-end-gaps and horizontal stiffener sizes. In a separate paper the reduction of ultimate strength due to wide range of stiffener-end-gaps are presented (Okura et. al. 1999).

PARAMETRIC STUDY Test Girders In the parametric study the finite element models were built up according to the four different test girders, as shown in Fig. 1 (note, that more details about the experimental test can be found in Okura et. al. 1999). 2301 300 "I

F I-

2150

4 ^ 250

'2200

2150 250 'B30A\

1176

1886.6

12 1500 750 |500| 2501

1000 |500|

li 750 feOoT 230 250

=^ 110

No welding Girder I

No welding

1^ Girder II and III

"B Girder IV

Figure 1: Test girders In the parametric study seven different horizontal stiffeners were applied on the four girders. Altogether twenty-eight cases were investigated. The horizontal stiffeners were plates with a range of relative area (8), relative stiffness (y) and b/t ratio, as detailed in Table 1. Stiffener no. 1 is calculated according to the relevant Japanese Standard while stiffeners specified by MSZ are related to the Hungarian Standard. The larger b/t ratios of stiffeners nos. 4 and 5 are designed to study the effects of local buckling of the horizontal stiffener.

137 TABLE 1 CHARACTERISTICS OF HORIZONTAL STIFFENERS [No. Spec.

1 "l" JSHB 2 3 4 MSZ [5 r"" 7

MSZ

5 0.048 0.05 0.05 0.05 0.05 0.10 0.10

y 23.108 32 40 48 67.784 48 82.052

b,,[inni]

t^fmm]

bb^U

55 63.246 70.711 77.46 92.048 54.772 71.612

4.8 4.372 3.91 3.569 3.004 10.096 7.722

11.458 14.468 18.084 21.701 30.642 5.425 9.274

1

cross-sectional area of the stiflfener ' bt

t .j ^ ' Db

thickness of the web.

where

D=

Et'

b height of the web plate stiffness

12[l-v') moment of inertia of stiffeners 3

Finite Element Model The finite element models are developed and analysed by COSMOS/M (1993) program by applying four-noded plane shell elements (SHELL4T). The general view of FE mesh is shown in Fig. 2. The mesh density is increased in the region of the stifiFener-end-gaps. The applied mesh is developed by an accuracy/efficiency study. The measured geometrical and material properties are icluded in the FE model (for details see Okura et. al. 1999). In the presented parametric study, however, not the measured initial imperfection but a Active one is used, as shown in Fig. 2, in order to make the computed results comparable. The support is modelled according to the test setup. In the load introduction zone Active supports are used instead of the web stiffeners, in order to avoid the failure of these web fields. The load is applied as it is in the laboratory test: concentrated forces over the second stiffeners, increasing by displacement control. 1152mm

b) Mesh division

4.60mm

a) Applied initial imperfection

Figure 2: Finite element model Nonlinear Solution Technique The nonlinear solution technique is optimised in order to perform an accurate and eflScient geometrically and physically nonlinear parametric study. The load increment technique with Newton-

138 Raphson method is used, with the convergence check of the unbalanced residual forces (the convergence factor is constant 0.001). The AUTOSTEP incremental procedure of COSMOS compared to a uniform, very fine-step load increments, what is assumed as "accurate". The comparison is done in the cases of the four test girders. Figure 3 and Table 2 shows the results of comparison on Girder II. On the bases of the results it is concluded, that the AUTOSTEP method is not accurate enough and an "optimal" load increment is determined, to be applied in the parametric study.

Figure 3 : Comparison of different load increments on the nonlinear solution TABLE 2 RESULTS OF DIFFERENT INCREMENTAL TECHNIQUES

ultimate load [kN] deflection [mm] nmnber of steps calculation time [h] diff. to accurate result f%]

AUTO STEP 758.12 18.63

"accurate" 754.28 18.40

11

92

-0.81 0.51

~4.96

-

"optimal" 754.51 18.40 33 -1.34 0.03

1

ANALYSES AND RESULTS The nonlinear analyses are completed on four different gap sizes and seven different stiffener sizes; altogether 28 specimens are investigated in the numerical tests. The results are studied on the bases of the nonlinear load - displacement relationships. Figures 4 to 7 show the limit point region of load deflection curves of Girders I to IV, horizontal stiffeners 1 to 7. The curves illustrate the features of the ultimate behaviour. It is a complex behaviour by an interaction of different local and global phenomena. On the bases of these results the ultimate behaviour is determined and classified, as presented in Nezo et. al. (1999).

139

Figure 5 : Load - defl. curves - Girder 11

Figure 4 : Load - defl. curves - Girder I 1 nfin 1.050-

*>'***C*'

1.040-

x4li^^^w\.

y^v^v^x

1.030-

1*" 1.0201.0101.0000.990

^•-ll

/

/

A

-^2

^

-*-4

^

-H-6

/ /

-^71

15

19

17

21

U[mm]

Figure 6 : Load - defl. curves - Girder III

Figure 7 : Load - defl. curves - Girder IV

ULTIMATE STRENGTH Ultimate Load of Specimens The ultimate strength of the test specimens are expressed by the ultimate load (Fu) of the pure bending loading condition. These load intensities are compared to the yield load (Py), which is calculated from the yield moment of the plate girder on the bases of measured yield stress of the flange. The theoretical plastic load of the plate girder (Ppi) is also determined for comparison purposes: Ppi / Py = 1.136. The calculated ultimate load to yield load relationships of the 28 specimens are given in Table 3. TABLE 3 ULTIMATE LOAD TO YIELD LOAD RATIO Stiffener • 1

1

2 3 4 5 6 7

GIRDER 1 1.060 1.071 1.077 1.082 1 1.087 1.083 1.095

GIRDER II 1.060 1.070 1.076 1.079 1.078 1.083 1.092

GIRDER III 1.045 1.046 1.046 1.046 1.045 1.052 1.052

GIRDER IV 1.001 1.002 1.002 1.002 1.003 1.003 1.003 1

140 The ultimate load data are analysed in order to obtain the • gap size effect, • stifFener size effect, and • stiffener b/t ratio effect. Gap Size Effect Figure 8 shows the effect of the gap size on the ultimate load. On the horizontal axis the horizontal stififeners are marked by their codes, as given in Table 1. The different lines are related to the test girders with given gap sizes, as shown in Fig. 1. From the results the following conclusions can be drawn: • Girders I and II have almost the same ultimate load, which means that the applied gap size of Girder II (g=35 mm, g/a=0.035) practically does not reduce the strength of the girder. • The gap size of test Girders III and IV (g=55 and 100 mm, g/a=0.055 and 0.1) reduces the strength by 2-5 % and 6-9 %, respectively, depending on the applied stiffeners. • The results of the more detailed study on the gap size effect can be found in Okura et. al. (1997) and Okura et. al. (1999).

-•-GIRDER I -*-GIRDERII -A-GIRDERIII -M-GIRDER IV

3

4

5

Stiffeners

Figure 8 : Gap size effect on the ultimate load Stiffener Size Effect Figure 9 shows the effect of the horizontal stiffener size on the uhimate load. On the horizontal axis the different girders are marked by their codes, as given in Fig. 1. The different lines are related to the different horizontal stiffeners, as detailed in Table 1. From the results the conclusions are as follows: • The investigated stiffener sizes result in max. 4% differences in the ultimate strength of Girders I and II. • The strength of Girder I and II are almost the same, however, the features of the ultimate behaviour are different, as shown in Figs. 4 and 5. • The stiffener sizes have negligible effect (max. 0.5%) on the ultimate strength of Girder III; it can be noted, however, that the behaviour of stiffeners #1-5 with 5=0.05 and stiffeners #5-6 with 5=0.10 have different behaviour, as shown in Fig.6.

141 •

The stifFener sizes have practically no effect on the ultimate strength of Girder IV; in this case the uhimate behaviour is dependent only on the bigger gap size, as shown in Fig. 7.

1.100 n 1.080-

k

-*-2

>. 1.060

U<-4 ^

1.040-

-•-6

1.0201.000 GIRDER 1

GIRDER il

GIRDER ill

GIRDER IV

Girder types

Figure 9 : Stiffener size effect on the ultimate load Stiffener b /1 Ratio Effect Figure 10 shows the effect of the b/t ratio of horizontal stiffener on the ultimate load. The different lines of the figure show the strength as a function of the b/t ratio of horizontal stififeners. The effect of the stiffener size is more significant for Girders I and 11, that is why the effect of b/t ratio is detailed in these cases as follows: • The different stififeners relative stifi&iess (y) of the same relative area (5) resuh in bigger b/t ratio; by increasing the stiffness the strength is increased, as shown between 12 and 22 b/t ratios. • For larger b/t ratios the plate buckling of the stiffener becomes dominant in the ultimate behaviour and the strength of the girders will increase in a smaller scale.

1.100 1.080 ^

1.060

tt. 1.040

•

•

•

•

•

1.020 1.000 10.00

15.00

20.00

25.00

30.00

b^/tb ratio of stiffeners

Figure 10 : Stiffener b / t effect on the ultimate load

GIRDER GIRDER GIRDER GIRDER

I II III IV

142

CONCLUSION In this paper the ultimate strength of plate girders under bending are discussed. In a parametric study four different stiffener-end-gaps and seven different horizontal stififener sizes are investigated. The ultimate behaviour is determined by geometrically and physically nonlinear finite element analyses. The applied FE model and computational technique is developed and verified by the results of experimental tests.

The conclusions on the interaction of stififener-end-gap and horizontal stiffener sizes can be summarised as follows: • The bigger stiffiiesses of the horizontal stiffeners increase the ultimate strength of the girders with smaller gap sizes (g < 35 mm, g/a < 0.035); these strength increments, however, are not higher than 4% in the investigated cases. • The ultimate strength of the girders with higher gap sizes (g = 55 and 100 mm, g/a = 0.055 and 0.1) are practically independent fi'om the sizes of the horizontal stiffeners; the roles of the gaps become dominant in the ultimate behaviour.

Acknowledgement This research is part of the Hungarian-Japanese Scientific and Technological Co-operation; it was supported by the project JAP/R-4/96 of OMFB, Hungary and STA, Japan; and OTKA project T 023378.

References COSMOS/M (1993). Finite Element Analysis System. Version 1.70. Structural Research and .^alysis Corporation. Santa Monica, CA, USA. Dunai L., Okura I., Kesmarky Z., Nagy A. and Ivanyi M. (1998). Effects of Stiffener-End-Gaps on Ultimate Behaviour of Thin-Walled Girders. Journal of Constructional Steel Research, Vol. 46, No. 13, 428-429. (fiill paper on CD-ROM). Nezo J., Dunai L., Okura I. and Ivanyi M. (1999). Interaction of Local and Global Phenomena in the Ultimate Behaviour of Plate Girders. 2"^^ European Conference on Steel Structures, Proceedings Paper No. 144, Praha. Okura I., Kazashi A., Kesmarky Z. and Nagy A. (1997). Effects of Stiflfener-End-Gaps on Ultimate Strength of Girders under Bending. Technology Reports of Osaka University, 47:2294, 225-235. Okura I., Kawarabayashi M. and Kazashi A. (1999). Reduction of Ultimate Strength of Girders due to Stiffener-End-Gaps. 2"'' Intl Conf on Stability and Ductility of Steel Structures, Proceedings, Timisoara.

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

143

INFLUENCE OF LOCAL BUCKLING OF FLANGES ON THE ULTIMATE LOAD OF I-SECTIONS J. Lindner ^ and A. Rusch * ^ Technische Universitat Berlin, Fachgebiet Stahlbau, Sekr. Bl, Hardenbergstr. 40 A, D-10623 Berlin

ABSTRACT Thin-walled open cross sections always contain unstiffened plate elements and - provided there is a sufficient local slendemess - the load carrying capacity is governed by local plate buckling. If the unstiffened elements are the weakest part of the cross-section, restraintsfromthe stronger elements are still present. In particular, this applies for I-sections with wide flanges. At the Technical University of Berlin in a current DFG-research project the behaviour of short columns of welded I-sections with very thin-walled flanges is investigated. The buckling pattern observed in tests and its comparison to finite strip calculations cover the interaction between the flanges and the web. Formulae will be presented, which allow to calculate the critical buckling stress and the half-wave length of the whole I-section in a direct way. As previously reported the comparison of test results with different design formulae shows that in codes the cross-section check is not sufficient enough. Therefore, a new buckling curve for the threesided supported plate has been developed. When applying the new buckling formula, the concepts of the European as well as the American code reflect the test ultimate loads much more accurately. KEYWORDS Columns, Thin-Walled Structures, Elastic Buckling Theory, Effective Widths, Post-critical Behaviour, Load Carrying Capacity, Coupled Instabilities. TEST PROGRAMME Tests on thin-walled I-sections were performed as reported in Lindner and Rusch (1999). The crosssections were designed such a way that the major influence on the load carrying capacity came from the three-sided simply supported plates. Although using relatively short columns, the test results differ from stub colunm tests due to the boundary conditions and the slightly longer column length. The loads have been applied centrally by end plates welded to the columns. The bearings led to pin-ended support conditions in order to allow rotations about major and minor axes, but to avoid lateral

144

displacements and twist rotations. The test results reported in this study (Table 1) extend analogous, previously published (Lindner and Rusch (1999)) tests. TABLE 1 EXPERIMENTAL DATA

Specimen

b [mm] 100 100

Xp

H

hw [mm] 80 80

t

fy [N/mm^] 222' 186'

[mm] 1.5 1.9

1.360 1066 1068 0.983 1114 364 1.307 2.0 100 80 1115 1116 364 80 2.0 1,879 150 1117 364 2.448 1118 200 80 2.0 1112 364 1 1.958 150 120 2.0 1113 1089 161' 1 1.772 100 200 1.5 1 ^ cold rolled material: yield stress fy defined with 0.2 % offset method

L

Nu,te8t 1

[mm] 1200 1200

im

1200 1200 1200 1200

1

1200

55.5 1 76.4 127.5 122.0 149.9 152.3 168.2 155.3 157.2 46.2 1 _J

VERIFICATION CONCEPTS The particular combination of column length L and local slendemess Xp (Eqn. 2) of the single plates (Table 1) causes coupled instabilities for centrally loaded thin-walled I-sections. For overall stability, the influence of local buckling on the global slendemess must be determined. Several international codes differ with respect to their treatment of the reduced stiffness (for details, see: Lindner and Rusch (1999)). The European standard EC 3 Part 1.3 (1996) uses the Q-factor approach, whereas in the American specification AISI (1986) the Unified Approach is used. In a completely different approach, the DIN 18 800-2 (1990) demands an iterative calculation using second order theory. For reasons of comparability the European buckling curves have been applied throughout this investigation (ECCScurves b and c for buckling about the y- and z-axes, respectively).

t

a C

^1T .4i bf

^ b f ^

^^.f

tj J--bf

i

(^ •^>w

fi •«P,f

tr J--bf

\

Figure 1:1-section considered as single plates Table 2 shows the results calculated for the sections of Table 1 including the cross-section capacity. The effects of local plate buckling are taken into account by the established method of effective widths with the "Winter Formula" Eqn. 1.

145

(1) The cross-section capacity in Table 2 is determined in two different ways, (i) The sum of the individual load carrying capacities is calculated on the basis of the critical buckling stresses Gcrco of simply supported flanges and web. (ii) £qn. 1 is evaluated with the critical buckling stress OctCn) of the whole member, which has also been used in the three mentioned design concepts. Especially for Isections with wide flanges, the design concepts clearly underestimate the test results Nu,test. As discussed in Lindner and Rusch (1999), the cross-section capacity calculated with the Winter-Formula (Eqn. 1) in most cases is also significantly lower than the experimental ultimate load N^test. TABLE 2 ULTIMATE LOADS ACCORDING TO DIFFERENT DESIGN CONCEPTS

Specimen

Nu,test

1066 1068 1114 1115 1116 1117 1118 1112 1113

rkNi 55.5 76.4 127.5 122.0 149.9 152.3 168.2 155.3 157.2 46.2

1 1089

DIN 18 800-2 AISI EC 3 Part 1.3 (1986) (1990) (1996) with ECCS curves FkNl FkN] rkNi 51.7 35.1 51.6 70.9 54.2 69.5

cross-section capacity (i) [kNl 55.5 74.8

(ii)

1

FkNl

57.5 1 78.1

107.9

109.2

72.0

125.1

129.7

124.5

121.7

83.3

128.9

130.0

127.3

123.9

91.7

130.8

129.9

132.1

128.9

87.6

134.5

138.6

43.8

43.0

28.1

47.8

49.7 J

ELASTIC CRITICAL BUCKLING STRESSES FOR I-SECTIONS The non-dimensional plate buckling slendemess X,p (Eqn. 2) has a strong influence on the load carrying capacity. As mentioned above, the elastic critical buckling stress Gcr can be determined for the single plates acr(i) or for the whole member CcrCu). The buckling stress acr(u) of the whole members in Table 1 has been evaluated numerically by the finite strip method (FSM) (Thierauf (1986), Schafer (1998)).

;ip=.

[fyy

_

/.

(2)

12(l-//^)UJ Furthermore, an analytical solution can be derived as follows. Considering an I-section consisting of single plates connected by rotational springs (Figure 1), the stiffness ratio of the single plates determines which plates are restrained by the others. For example, the wide flange I-sections studied here have flanges that are restrained (positive spring Cp,f) by the web (negative spring c
146 plates, respectively (Lindner et al. (1974)). In these equations, L is the half-wave length and r represents the non-dimensional restraint degree defined by Eqn. 4.

[

120.- UJ "T^UJ ^-T2^f l^^UJ 4;r'

120 ;r-

^T;^UJ "

^-^f

"2[UJ "UJ J"^

2

(3a)

^.=-

10 6 U J

2 42 U J ^ ' K J 1

5 _

100n2j 23lUJ '^^'[bfj

53 '450

(3b)

5 _2

r52'>^^'> _12(l-AiO/»,

.

- ^12(l-;i0^/ >./

(4)

With the transition conditions defined by Eqn. 5 and 6, the critical stress Cc^a) of an I-section of length L with the same thickness t for flanges and web can be determined iteratively. (5)

(6) The critical buckling stress Ccr is a function of the length-to-width ratio of the plate which buckles. For example, the four-sided simply supported plate reaches the first minimum Ccr(i) at Li/h = 1 with a kw-value of 4.0, while the three-sided simply supported plate has its minimum at Ll^f=oo with kf = 0.425. The decisive length Li corresponds to the half-wave length of the buckling pattern. Thus, a four-sided simply supported plate with a length L twice the decisive length Li buckles in two halfwaves. The same applies to whole I-sections, where the critical buckling stress Ga^u) of the whole member reaches its first minimum at a half-wave length Li. As shown in Figure 2 for tests listed in Table 1 and those reported in Lindner and Rusch (1998), a column buckles in n half-waves Li, if the column is approximately n half-waves long. The half-wave length Li is independent of the plate thickness t. This is also observed in the tests. In analogy to single plates, Eqn. 7 is valid for the critical stress Oci^ii) of whole I-sections. ^cr(ii)i^l)

=

-r^cr{ii){0

(7)

With a known half-wave length Li and critical stress Ccr(u), r can be recalculated by Eqn. 3 (Figure 3a). The restraint degree r has a value of zero at a ratio bf^h of flange width to web height, where the single, simply supported plates (web and flanges) have the same critical buckling stresses_aci
h

V 4.0 V * ,

(8)

147

-I . [N/mmT 120 -I- - '^-l N

PCII 90

-1

1

I

>

y '

2 *

\

* \

3^4

•half-waves in the test 1 oo

Li

"*

* \

^ 5 \

\

\

6 \

7\

J}

V

8half-waves 150

Z3o

A

., ^U T

1

0

200

400

600

800

1000

1200 L [mm]

Figure 2: Critical stress Caiii) and half-wave length: numerical results in comparison to test results FSM calculations for several I-sections with constant t and varying V h (Figure 3b) show that the halfwave lengths Li can be described by Eqn. 9. The section unwinding (4 bf + h) equals 2 Li. Li is limited to h for I-sections with bf < Vi h, which corresponds to the half-wave length of the web, when considered as an four-sided simply supported plate. 4b^ +h 2 Ly = m a x ^ h

L i / h ,I 43-

-0.326 / \

(9)

i

M/h(Eqn.9)^

-4 -3

' • FSM results^^^.."''''^'^''''^

-2

21 •

Df ^^--^

0>V

t :, >!t

1 l.^r'"'^^'^^^^*^'^^ ^f (flanges) " 1 ^>^^i^t 1 1 • « 0.25!V).50 0.75 1.00 1.25 1.50 . \ ^ r^ (web) s^

bf/h 1

1.75 1

1

>

web

/

^^ flanges

* will be restrained

M

^

-1 - n u

r"'' h-2 *- -3

Figure 3: a) half-wave-length Li and b) elastic restraint r depending on the bf/h ratio Therefore, replacement of h by (4 bf + h) in Eqn. 2 and some simple transformations lead to Eqn. 10, which defines a k-value k(ii) for the whole I-section with a constant plate thickness t. Figure 4 shows the dependence of the k-value k(ii) on the hflh ratio. The critical buckling stress aci
148

«•(//)

*^(«) ~

(10)

TT^ E

1 2 ( 1 - / / ' ) Abf

+ h

6 0 - ^ + 4.0 n^E <^cr(U}—.-..

..2N

12(l-/i^) ^4*,+/,^

mm

:^i

' hi)

1

25-

(11)

12.81^1 ' + 6 . 8

Eqn. lOwithEqn. 11

20-

^-^^^<.^_\__

bf

15 1050.0

• FSM results

. - 0.326 0.5

1

1

•

1.0

1.5

bf/h

±

Figure 4: Dependence of the buckling value Ic(ii)of the whole I-sectionon the bf/h ratio PLATE BUCKLING CURVE FOR THREE-SmED SUPPORTED PLATES The Winter-Formula (Eqn. 1) is derived from stub column tests for the four-sided supported plate. Further stub-column tests (Kalyanaraman et al. (1977)) have proved that the three-sided supported plate has a higher relative post-critical load-bearing capacity (Eqn. 12 and Figure 5) than the four-sided supported plate (Rhodes 1982).

(12) Measurements in the post-critical range (Lindner and Rusch (1999)) have shown that tensile stresses occur at the free flange ends. Anchoring these stresses at the ends of the member (for example in stiff end plates), the ultimate load is significantly higher than compared to an analogous situation where no anchorage is possible. In usual stub colunrn tests, the tensile stress can not be anchored because of the test rig. This is the reason for thefindingthat the cross-section capacity determined with Eqn. 1 (Table 2) or Eqn. 12 (Lindner and Rusch 1998) is higher than the test member capacity including overall stability influence (for details, see: Lindner and Rusch (1999)). The anchorage of tensile stresses can accounted for by the non-linear deformation theory of Stowell (1950).This theory and comparisons with test results led to Eqn. 13. As discussed in Lindner and Rusch (1999), the FEM parameter study of Fischer et. al (1996) with a loading caused by an edge displacement ("geometric loading") also shows the tensile anchorage (Figure 5).

X. = l i > ^ l 9 x:

(13)

149 Eqn. 13 Stowell(1950)

• bf' / bf 1.0 0.8 0.6

Eqn. 1 Winter-Formula

0.4

Eqn. 12 Kalyanaraman et al. (1977)

0.2

X FEM results Fischer et al. (1996) 0.5

1

2.5

1.5

Figure 5: Comparison of different equations for the effective widths of three-sided supported plates On the basis of these results, the Winter-Formula (Eqn. 1) can be modified into Eqn. 14 as a new buckling formula for the three-sided supported plate (Figure 5). Table 3 shows the evaluation of the design concepts with the new Eqn. 14 for the sections in Table 1 and for the tests of Lindner and Rusch (1999). Clearly the Q-factor as well as the Unified approach now lead to a much more satisfying agreement of calculated and test results. As reported previously (Rusch and Lindner (1999)), this remains also valid for thin walled I-sections subjected to bending about the z-axis. 0.22 + 0.05Ap

(14)

TABLES ULTIMATE LOADS ACCORDING TO DIFFERENT DESIGN CONCEPTS WITH NEW BUCKLING CURVE EQN. 14

cross- 1 cross- 1 Q-factor Unified Q-factor Unified Specisection section 1 Nu,test approach approach approach approach men capacity capacity with ECCS curves with ECCS curves (ii) (ii) [IcN] ricNi [kN] fkNl [kNl fkNl 68.9 81.8 1 72.7 82.1 1112 155.3 1072 160.0 150.7 140.6 87.7 85.9 83.4 1060 100.9 1113 157.2 88.0 91.0 93.2 62.0 1066 1065 101.1 52.0 55.2 55.5 44.9 42.4 49.0 81.6 70.3 49.5 1068 72.2 1070 76.4 50.1 47.8 1025 53.5 1114 127.5 51.6 105.0 1 139.3 114.3 55.2 53.3 55.6 1115 122.0 1063 60.2 54.5 57.0 56.5 1116 149.9 1013 55.7 134.0 150.5 142.5 56.9 1117 152.3 58.1 1009 61.6 58.1 93.8 1057 110.3 90.4 93.0 1118 168.2 151.9 1 165.5 160.2 66.0 63.0 67.2 1 1089 43.3 71.4 47.2 52.0 1 1059 46.2 56.6 56.6 1055 64.7 55.1 62.1 1028 61.2 67.2 62.1 1.075 1.129 Statistical evaluation Nu,test / Nu,approach (22 tests): Imean value 0.053 [standard deviation 0.055 Specimen

Nu,test

\m

\m

150 CONCLUSION Tests on centrally loaded thin-walled I-sections with constant plate thickness have been performed. In the post-critical range the I-sections buckle in several half-waves. In agreement with additional FSM calculations, the half-wave length can be described by a simple formula containing only the section geometry. Furthermore a buckling value k can be defined for the whole section. With this novel parameter, the critical buckling stress of centrally loaded I-sections with a constant plate thickness can well be approximated. Therefore the critical buckling stress of the whole section can be calculated in a direct way. We intend to generalise this concept for other types of sections and stress distributions. In extension of the work reported in Lindner and Rusch (1999), a new buckling curve for the threesided supported plate is presented, which includes the positive effect of the anchorage of tensile stresses on the load carrying capacity. Replacing the Winter-Formula in the design concepts by the new buckling curve, the ultimate loadsfromtests are reflected much more accurately than before. ACKNOWLEDGEMENTS The authors thank the Deutsche Forschungsgemeinschafl DFG for funding. We are gratefiil to the Salzgitter AG for the generous gift of steel plates. REFERENCES AISI (1986). Specificationfor the Design of Cold-formed Steel Struct Members. Washington, USA DIN 18 800-2(1990). Stahlbauten - Stabilitdtsfdlle, Knicken von Staben und Stabwerken. BeuthVerlag, Berlin, Germany EC 3 Part 1.3 (1996), ENF 1993'l-3 General Rules, Supplementary Rules for Cold-formed Thin Gauge Members and Sheeting, CEN, Brussels, Belgium Fischer M., Zhu J., Priebe J. (1996). ne Method of Effective Width used for Bars with Thin-walled Cross-sections - Remarks on Insufficiencies and Improvements. Proceedings of the 2^ Intern. ConC of Coupled Instabilities in Metal Structures, Imperial College Press, London, UK Kalyanaraman V., Pek6z T., Winter G. (1977). Unstiffened Compression Elements. Journal of Structural Division, ASCE 103:9, 1833-1848. Lindner W., Loffler, V., Strathmann, U. (1974). Beulen von StabstOhlen mit offenen und geschlossenem Querschnitt unter Normalldngsspannung, Ph.D.-Thesis, Leipzig, Germany Lindner J. and Rusch A. (1998). Stiitzen mit diinnwandigen Querschnittsteilen im Bereich geringer Schlankheiten; L Zwischenbericht DFG-Forschungsvorhaben Li-351-15/1, Fachgebiet Stahlbau, TU Berlin, Germany Lindner J. and Rusch A (1999). Load Carrying Capacity of Thin-walled Short Columns, Proceedings of 4^ International Conference on Steel and Aluminium Structures, Espoo, Finland Rhodes J. (1982). Effective Widths in Plate Buckling, in; Developments in Thin-walled Structures - 7, edited by Rhodes J., Walker A C , Applied Science Publishers, London, UK Rusch A. and Lindner J. (1999). Tragfahigkeit von beulgefahrdeten I-Profilen bei Biegung um die z-Achse, submitted to Stahlbau Schafer B.W. (1998). CU-FSM, Finite Strip Analysis of Thin-walled Members, Users Manual vl.Od, Cornell University, USA Stowell E. (1950). Compressive strength of flanges, Technical Note 2020, National Advisory Committee for Aeronautics NACA, Washington, USA Thierauf G. (1986). PLATE. Finite Strip Analysis of Thin-walled Columns for Elastic Stability and Local Buckling, UGH Essen, Germany

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Iv^yi, editors © 1999 Elsevier Science Ltd. All rights reserved

151

UNDULATING WEBS UNDER PATCH LOADING Rostislav Novak ^ and Prof. Josef Machacek ^ ^ Department of Steel Structures, Czech Technical University, Prague, Thakurova 7, 166 29, Czech Republic

ABSTRACT The use of trapezoidal or undulating shaped profiles has increased considerably in recent years. The profiled panels are widely used for girders, roofing, decking, wall cladding and in offshore structures. This paper concerns the patch load capacity of steel girders with thin undulating web. Experimental strengths and comparison with finite-element method are presented. Beams were tested to the failure under knife patch loading acting through the crane rail. The influence of transverse force eccentricity on collapse web resistance was investigated. The magnitude of eccentricity varied between zero and 40 mm. The failure was sudden and due to web crippling or buckling.

KEYWORDS Undulating web, patch loading, eccentricity, finite-element model, geometric and material nonlinearity.

INTRODUCTION The experimental series included 18 tests of 1000 mm long specimens WTA 625 - 250 x 20, Siokola (1996). It means that the girders have web depth 625 nmi, 2 mm web thickness and cross-section of flanges 250x20 mm. The geometry of corrugation is shown in Figure 1.

Figure 1: The shape of corrugation (one wave is 155 mm long)

152 The web is joined to the flanges by a fillet weld from one side only. Besides investigation of the ultimate strength, the experiments were also focused on the weld resistance. After each test a possible crack in the weld was checked. The results confirmed that at the collapse point never appeared any crack in welding, and the collapse was due to web crippling/buckling only.

EXPERIMENTAL TESTING The test series includes a total of 18 individual tests Al ~ A18. The specimens, shown in Figure 2, were reinforced by two end vertical stiffeners 250x10 mm. All specimens were simply supported on two cylindrical bearings in the distance of 900 mm, except series of two tests A17 - A18, in which the girder was supported on a plate at the centre of bottom flange (see Figure 3). undulating web i^ 2mm

lateral restrain

end vertical stiffeners 250x10

cytindrical bearing

^^

=-^ ilateral restrain

Figure 2: Typical test set-up and definition of parameters Three transducers (PO, PI, P2) were installed at all beams for measuring of vertical deflections. Some specimens were equipped with the set of strain gauges for finding of strain distribution. The testing was made in a HAPZ machine with an ultimate capacity of 600 kN. For all tests the load from the hydraulic jack was applied at the upper flange through a 300 mm long crane rail with rectangular cross-section 50x30mm. The loading was controlled by magnitude of force. The load step 21 kN was applied and afler each step followed by 2 minutes break before reading strains and deflections.

153

undulating web t / 2 m m crane rail 50x30x300

20

ir*

lateral restrain

M

N

78 78! 78 781

end vertical stiffeners

250x10 support X plate

2 transducers .rigid frame

-i*-flanges 250x20

" ^ lateral restrain

Figure 3: Test set-up with steel support plate at the centre of bottom flange Tests with eccentricity e = 0 mm First of all, 4 pilot tests Al - A4 with e=0 mm were performed. These tests should inform about basic behaviour of the girders subjected to local loading: i.e. about collapse mode, collapse load, force deflection relation, stress distribution along the web, etc. Two positions of force with respect to the shape of corrugation were considered as shown in Figure 4. (a)

(b)

Figure 4: Position of force with respect to the shape of corrugation (e = 0 mm) (a) (b)

Tests Al and A3 Tests A2 and A4

All four beams collapsed by web crippling in vicinity of crane rail about 50 mm below the upper flange. Buckling area affected approximately 7 half-waves of corrugation. The longitudinal position of the force had no influence on ultimate strength. Average ultimate load was 230 kN. Tests with eccentricity e = 20 mm and typical boundary conditions The beams from Figure 2 were simply supported on two cylindrical bearings (tests A5 - A8, A l l , A13 and A15).

154 Three positions of force with respect to the shape of corrugation were considered as shown in Figure 5. (a)

(c)

(b)

Figure 5: Position of force with respect to the shape of corrugation (e = 20 mm) (a) (b) (c)

TestsA5,A6andA15 TestsA7andA8 TestsA12andA13

The results proved that both collapse load and damaged area of the web practically did not change in relation with zero eccentricity of loading. The position of force onto corrugation had negligible influence. Average ultimate load was 228 kN. Tests with eccentricity e = 20 mm and different boundary conditions The beams from Figure 3 were supported on a steel plate at the centre of bottom flange. The plate width Ss was 200mm and 50 mm for test A17 and A18, respectively. In both cases the loading was located in vertex of wave as can be seen in Figure 5 (b). The failure mode of specimens A17 and A18 was different to that described above. In test A17 {ss = 200 mm) the collapse arose due to web buckling about 200 mm below the upper flange, in test A18 (ss = 50 mm) the web buckling/crippling appeared near the plate support and with lower level of uhimate loading. Damaged area included 7 half-waves of corrugation. It indicates that the web collapse was governed by the length of loading or support Ss and moved accordingly along depth of the web. Also ultimate capacity was decreasing (Fuit = 216 kN for test A17 and Fuh = 198 kN for test A18). Tests with eccentricity e = 30 mm Additional increasing of eccentricity was applied in tests A14 and A16. There were used two positions of force according to Figure 6. (b)

(o)

Figure 6: Position of force with respect to the shape of corrugation (e = 30 mm) (a) (b)

TestAN TestA16

Crippling developed 50 mm below the upper flange. Deformed zone rapidly expanded - the failure of the web could be observed in 12 half-waves of corrugation. In both cases the maximum load of 210 kN was reached. Tests with eccentricity e-40 mm Extreme value of eccentricity, determined as e = 40 mm, was used in three tests A9 - A l l (see Figure 7).

155 (b)

(a)

Figure 7: Position of force with respect to the shape of corrugation (e = 40 mm) (a) test A9, All (b) testAlO Application of such eccentricity on a girder with planar web is quite impossible and would probably lead to immediate collapse due to cross-section distortion. Undulating web crippled in the major part of girder length about 50 mm below the upper flange. Visible vertical deformations appeared at the upper flange due this eccentric load. Average ultimate force was 208 kN. Example of deformed web is presented in Figure 8.

Figure 8: Typical deformed shape, test A13, e = 20 mm, Fuit = 231 kN On next page a simple list of experimental results and recapitulation of all tests are presented (see TABLE 1). Average values of ultimate resistance depending on load eccentricity are also given. They do not include results from test A17 and A18 with different boundary conditions.

156 TABLE 1 SUMMARISED DATA OF EXPERIMENTAL RESEARCH

• ir^?iimhiMar?inimmid

..^i.^

M£I^

Al

e=0

with strain gauges

231

A2

e^b

with strain gauges

231

A3

e^O

with strain gauges

228

A4

e^O

with strain gauges

11^

A5

e=20

upside down girder A4

226

A6

e=20

with strain gauges

224

A7

e=20

upside down girder A6

219

A8

e=20

A9

e=^0

upside down girder A8

210

AW

e=40

upside down girder A3

210

All

e=40

AI2

e=20

upside down girder All

230

A13

e=20

upside down girder AI

231

A14

e=30

upside down girder A2

210

A15

er-20

strain gauges, flanges are welded only in 40 mm length part

231

A16

e=30

upside down girder Al5

210

A17

e=20

A18

e=20

e e e e

= 0 mm = 20mm = 30mm = 40mm

\flanges are welded to the end vertical\ 231 stiffeners only within undulating

\flanges are welded to the end vertical] 205 stiffeners only within undulating

strain gauges, support plate at the centre of lower flange, Ss = 200 mm 216 strain gauges, support plate at the 198 centre of lower flange, Ss = 50 mm

Favg= 230 kN Favg=228kN Favg=210kN Favg= 208 kN

100,0 % 99,0% 91,3% 90,4 %

157 FINITE-ELEMENT ANALYSIS WITH ANSYS ANSYS is multi-purpose finite-element program and among other things, it contains tools for buckling analysis. In this case, non-linear buckling analysis was used (including material and geometric non-linearity). As shown in Figure 9, a 3D model was chosen for the analysis. Because of symmetry, only half of the beam was taken into account. AMWS

NOV 16 1998 11:40:03 ELEMENTS REAL NUM

{a2

Figure 9: 3D finite-element model used in analysis

ANSYS

NOV 16 1998 11:43:55 NODAL SOLUTION TIME=210 USUM TOP RSYS=0 DMX =1.255 SMX =1.255 S S S

{a2

Figure 10: Deformed shape of the girder

.13942 .278839 .418259 .557678 .697098 .836518 .975937 1.115 1.255

158 In all calculations a four node shell element, supporting plasticity, was used (SHELL63) for modelling of the web, flanges and vertical stiffener. The crane rail is made from eight node brick elements, called SOLID45, connected to the upper flange by special non-linear contact elements CONTAC52. Finite-element boundary conditions and loads are corresponding with those in experiments. The material properties are according to the tensile tests of specimens taken from the web and flange. Steel used for flanges is a standard steel with fy = 289 MPa and elasto-plastic bilinear stress-strain relationship is assumed. Steel used for the web is cold formed and it has not visible yield point. Therefore good approximation seems to be trilinear stress-strain relationship with conventional yield stress fo,2 = 256 MPa. The results from FE analysis of the simulated girder show similar behaviour as the tested girders. In Figure 10 can be seen web crippling under the upper flange. Ultimate resistance calculated by ANSYS is 210 kN for girders with eccentricity e=0 mm. SUMMARY AND CONCLUSION Experimental strengths of a series of girders with undulating webs under patch loading are presented. The load was applied with transverse eccentricity 0 - 4 0 mm. Under compressive patch loads, a girder with undulating web failed due to web crippling. When a short girder was simply supported by two cylindrical bearings, the undulating web collapsed close to the crane rail below the upper flange. In the case of support by a plate at the middle of bottom flange the deformed area moved down from the upper flange to the bottom flange. The support plate width Ss decided about the position of the crippling/buckling. If the width was 200 mm, the web buckled about 200 mm below the upper flange. If the width was only 50 mm, the web collapsed in vicinity of the bottom flange. The eccentricity of applied force had a small eflfect on ultimate capacity. Because of undulating, this type of girder was not too sensitive to eccentric loads. If the eccentricity was smaller than or equal to 20 mm (i.e. within undulating), the resistance was the same. If the eccentricity was outside the undulating (e = 30 - 40 mm), the patch load capacity dropped down about 10%. The patch loading resistance cannot be described by formulas given for planar webs, Lagerqvist (1994); the corrugation provides continuous stiffening and its buckling resistance is much greater than that of a planar web. Statistical and theoretical evaluations are in progress. The test specimens were analysed using finite-element models, and the correlation between the analytical and experimental results seems to be reasonably good.

REFERENCES Lagerqvist O. (1994). Patch loading. Resistance of steel girders subjected to concentrated forces, Lulea University of Technology, Sweden Siokola W. (1996). Wellstegtrdger - Technische Dokumentation, Zeman+Co. GmbH, Wien, Austria

ACKNOWLEDGMENTS The work presented in this paper is part of a study carried out under Grant No. 103/98/0062 granted by the National Grant Agency GACR. The support provided by Central Laboratories of CTU in Prague and company Kovove profily is gratefully acknowledged.

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

159

REDUCTION OF ULTIMATE STRENGTH OF GIRDERS DUE TO STIFFENER-END-GAPS I. Okura\ M. Kawarabayashi^ and A. Kazashi^ ^Department of Civil Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0871, Japan ^Bridge Design Department, Ishikawajima-Harima Heavy Industries Co., Ltd., 1, Shin-nakahara-cho, Isogo-ku,Yokohama 235-8501, Japan

ABSTRACT The modified structural details of horizontal stiffeners to meet the requirement of robotics welding are needed in the plate girder fabrication in Japan. The larger stiffener-end-gaps are desirable to improve the movability of welding robots. In this research, the effects of horizontal-stiffener-end-gaps on the ultimate strength of girders under bending are investigated. Static loading tests are carried out for girders with different stififener-end-gaps. Elasto-plastic large-deflection analysis is performed by FEM to obtain the relationship between the reduction of the ultimate strength and the size of stiffenerend-gaps.

KEYWORDS ultimate strength, girder, stififener-end-gap, robotics welding, bending, static loading test, elasto-plastic large-deflection analysis, FEM

INTRODUCTION It is rather difficult to apply robotics welding to the connections of a horizontal stiffener with a vertical

160 stifFener and to those of a vertical stiffener with a flange in the plate girder fabrication in Japan. Placing a large gap between a horizontal-stififener-end and a vertical stiffener and between a verticalstiffener-end and a flange are desirable to improve the movability of welding robots. In this research, the effects of horizontal-stiffener-end-gaps on the ultimate strength of girders under bending are investigated. Static loading tests under bending are carried out for girders with different stiffenerend-gaps. Elasto-plastic large-deflection analysis is performed by FEM to obtain the relationship between the reduction of the ultimate strength and the size of stiffener-end-gaps.

TESTS Test Girders As shown in Figure 1, four test girders were fabricated, which had different sizes of gap between a horizontal-stiffener-end and a vertical stifFener (Okura et al., 1997). The two web panels between the points A and C of the girders were tested under bending by applying the loads of equal magnitude to the points D and E. The rolling supports were placed at both the ends of the girders. The steels used are JIS SM400 with the guaranteed yielding stress 235.3 MPa and the guaranteed tensile strength 400 MPa. Table 1 presents their material properties obtained from tensile tests. The aspect ratio a (= ajb ) of each web panel is 0.87, and the web slendemess ratio yff (= b/t^ ) is 256. Here, a is the interval between vertical stiffeners, b is the web height, and t^ is the web thickness, yff =256 is the limiting value specified in the Japanese Specifications for Highway Bridges, JSHB, Japan Road Association (1996) for the case one horizontal stiffener is provided on a web of SM400. The horizontal stiffeners were located at one fifth of the web height from the compression flange. The moment-of-inertia 7^, of the horizontal stiffeners satisfies the following equation specified in JSHB: 230 I 300 2150

AX 250

'2200

21^0

250 I3

F

>

'\

D '

1176

LA

9^ 1

50

c

E

1230:4^

\ h 4.5

1 500 750^ 500 500 1000

886. 6

^1000 ^500 500 750 500

* — > •

r

230 '250

1152

'35

g 35

Girder

n

5.'5

m

4.5j

m

100

^ • ^ 1 - =

No welding

UJL No welding

Girder I

Ul

Girder II and III Figure 1: Test girders

'_fe If Girder IV

161 h=^r,

(1)

r,=30a

(2)

in which

The moment-of-inertia I^ of the vertical stiffeners satisfies the JSHB formulas /^ >\btl/llJiS/a^), b^ >b/30 + 50 (in mm), and t^ >b^/l3, where b^ and t^ are the width and thickness of a vertical stiffener, respectively. The upper end of the vertical stiffeners were welded to the top flange. The lower end of the vertical stiffeners were cut short of the upper surface of the bottom flange by 35 mm. The cross section of the flanges has such dimensions that the top flange can not locally buckle before it reaches its yielding stress. To avoid the lateral-torsional buckling of the girder before the girder reaches its yielding load, supporting frames were equipped in the lateral direction on the top flange at an interval of 1000 mm. In Girder I, the ends of the horizontal stiffeners were tightly fitted to the vertical stiffeners where welding was not used. The radius of the scallops at the stiffener-ends is 35 mm. In Girders II, in and IV, the ends of the horizontal stiffeners were cut short of the vertical stiffener. In Girder 11, the gap size at the stiffener-ends is 35 mm, and the ends are cut at 45 degrees. In Girder III, the gap size is 55 mm, and the ends are also cut at 45 degrees. Cutting the stiffener-ends at 45 degrees improves the movability of welding robots. In Girder IV, the gap size is 100 mm, and the ends are cut at 90 degrees. Test Results Figure 2 shows the initial out-of-plane deflection of the web in Girder II. Figure 3 shows the residual vertical deflection of the flange and the residual out-of-plane deflection of the web when the test has been finished. These figures are for the observations from the side where the horizontal stiffeners are provided. In the out-of-plane deflection of the web, plus quantities express the deflection in the direction of coming to us (in the positive direction), and minus ones express the deflection in the direction of going away from us (in the negative direction). It is seen from Figure 3(b) that while the left panel has the residual deflection in the positive direction, the right panel has that in the negative direction. Figure 4 presents the relation between the load and the additional out-of-plane deflection at the middle of the horizontal stiffener in each of the left and right panels. At small loads, each panel deforms in the positive direction. However, when the load reaches to a certain quantity, the right panel suddenly deforms in the negative direction. A snap-back buckling occurs in the right panel. The left panel deforms greatly in the positive direction, and the top flange on the panel has a local buckling, as shown in Figure 3(a). The girder had the maximum load when the snap-back buckling

162 TABLE 1 MATERIAL PROPERTIES Horizontal stiffener and web SM400 0.48 301.6 0.28 195000

1 Material 1 Measured thickness(cm) 1 Yielding stress(MPa) 1 Poisson's ratio 1 Young's modulus(MPa)

Flange SM400 1.24 296.3 0.28 203000

20

40

60

80

too

120

|

140

tSO

180

200 (cm)

(a) Residual vertical deflection of flange

Left panel Left panel

Ri^t panel

Figure 3: Residual deflection of flange and web

(kN) 1000

(kN) 800r-

-5 0 5

15

25 (mm) (a) Left panel

<^>

(b) Residual out-of-piane deflection of web

Figure 2: Initial out-of-plane deflection of web

(kN) 800|

Right panel

600

800

400

600

200

400

<5o

1 YLeJ^ in^los

(mm) (b) Right panel

-J

Figure 4: Relation between load and additional out-of-plane deflection

1 I

n m

1 ^

734.2 733.8 703.1 682.2

!

Girder I Girder II Girder III Girder IV

i

20

25 (mm)

Figure 5: Relation between load and vertical deflection

TABLE 2 MAXIMUM LOAD Test(kN)

,^jj_

O • D •

-10

1 Girder

d

FEM(kN) FEM/Test | 782.5 783.5 752.4 729.0

1.066 1.068 1.070 1.069

163 occurred.

The above phenomena were observed in other test girders.

The relation between the load and the vertical deflection at the span center is shown in Figure 5 for all the test girders. The maximum load is listed in Table 2. The yielding load Py of the test girders is 729.1 kN. The cross-sectional area of the horizontal stiffener is not included in Py. The maximum load of Girder I is the same as that of Girder II, and both are just a little larger than the yielding load. The larger gap reduces the maximum load. The maximum load of Girder IV is 7 % smaller than that of Girder I.

ANALYSIS FE Models To examine the reduction of the ultimate strength due to stiffener-end-gaps, elasto-plastic largedeflection analysis was performed with MARC (1994), a commercial program for FEM. Figure 6 shows the general view of the mesh division for the test girders. The finite elements used are an eight-node thick shell element (Element type 22 in MARC). The out-of-plane deflection of the webs between the supporting and loading points in the right and left sides are restrained. In Girder I, the nodes at the ends of the right and left horizontal stiffeners are on the vertical stiffener, but they are separated from the vertical stiffener so that they can move freely. The measured plate thickness and the material constants in Table 1 were used. The steels were assumed to be a linearly elasticperfectly plastic material. The load was imposed under load-controlled conditions. Figure 7 shows the out-of-plane deflection of the web after the maximum load for Girder II. The outof-plane deflection has the same phenomena as observed in the test. The right panel has a snap-back buckling at the maximum load. The top flange on the left panel has a local buckling, which occurs after the load reaches its maximum. Figure 8 shows the comparison of the FEM and test results for the relation between the load and the vertical deflection at the span center of Girder II. The maximum load given by the FEM is 6.8 % higher than the test one. Although to find this reason, we calculated for the FE model with residual stresses introduced into the top and bottom flanges, the maximum load was almost the same. The maximum loads computed by the FEM are listed in Table 2\ for all the test girders. The ratios of the FEM value to the test one are between 1.066 and 1.070. The variation of the ratios are less than 0.4 % for the different sizes of the stiffener-end-gaps. Therefore, we can use the FEM values to know the relative reduction of the ultimate strength due to stiffener-end-gaps.

164

Figure 6: FE model for test girder

Figure 7: Out-of-plane deflection of web after snap-back buckling

(kN)

1000

i i

Lrt

1

-O- Test - ^ FEM 1 10

! 15

i 20

-2 5 (mm)

Figure 8: Comparison of FEM and test results

Effects of Stiffener-End'Gaps The effects of the moment-of-inertia of horizontal stiffeners on the reduction of the ultimate strength due to stiffener-end-gaps were investigated by the FEM, The FEM analysis was carried out for the girders with the dimensions in Table 3. The general shape of the girders is identical to that of the test girders in Figure 1. The considered steel is JIS SM490Y, which has the guaranteed yielding stress 353.0 MPa and the guaranteed tensile strength 490 MPa. This steel is usually used in bridge design in Japan. In the FEM analysis, 353.0 MPa, 2.06x 10^ MPa and 0.3 were used for the yielding stress. Young's modulus and Poisson's ratio, respectively. The aspect ratio a of the two middle web panels is 1.0 and the web slendemess ratio P is 209. yf = 209 is the limiting value for SM490Y steels specified in JSHB. The cross-sectional dimensions of the horizontal stiffeners meet the value of /;, for each of r^ =30, 45 and 60 in Eqn. 1. From Eqn. 2, r^ =30 is the same value as specified in JSHB. The initial out-of-plane deflection in each panel is one half sine wave in the horizontal and vertical directions with the maximum value 7.52 mm according to the JSHB formula w^ =b/250. For the shape of the initial out-of-plane deflection in the horizontal direction. Case (5) in Figure 9 was assumed, because it gave the smallest ultimate strength among the five cases (Okura et al, 1997).

165 TABLE 3 CROSS-SECTIONAL DIMENSIONS OF EACH COMPONENT OF ANALYZED GIRDERS P 1

Flange (mm) Web (mm)

Horizontal stiffener (mm)

1 Vertical stiffener (mm)

437x20 1 1881x9 104xl0(a=1.0, rh=30) 104xl5(a=L0, rh=45) 104x20(a=1.0, rh=60) 85.5x9(a=0.5, rh=15) 115.2xll(a=1.5, rh=45) | 113x9 1

(2)-

Figure 9: Shape of initial out-of-plane deflection

Figure 10 shows the relation between P^JPY and gla. Here, P„ is the ultimate strength, and g is the stiffener-end-gap. For g/a<0.02, Pu/Py becomes larger with the increase of r^,. For g/a>0.02, however, P^/Py does not differ, indepentent of the magnitudes of /-;,, and it decreases linearly with the increase of g/a. After g/a is larger than 0.10, P^/Py is nearly equal to 0.95, which is the value for the case without the horizontal stiffeners. For g/a<0.02, P^/Py for the stiffener-ends cutted at 45 degrees of ^^,=30 is almost equal to that for the stiffener-ends cutted at 90 degrees. For 0.02 < g/a < 0.10, however, the former is lower than the latter. Figure 11 shows the relation between P^/Py and g/a for a=0.5, 1.0 and 1.5.

The cross-sectional

dimensions of the horizontal stiffeners were changed so that they could satisfy the moment-of-inertia /^ given by Eqns. 1 and 2. The following equation gives the limiting values of g/a corresponding to the decreasing rate TJ of the ultimate strength, 77 = 1 - (P^ /Py )/[p^ /P^ for g/a = 0.0 and cuttmg angle 90°): g

•• k^ +

where

This equation is effective for 0.5 < a < 1.5 .

rh

["•

0.04

0.06

0.08

defined

as

(3)

D A 0

0.02

is

k^a

where k^, k^= coefficients shown in Table 4.

0.94 0.00

//

0.10

60 45 30 30

0.12 g / a

Figure 10: Relation between P^/Py and g/a

Cutting angle 1 90° 45°

166 Pu/Pl

PU/PY

1.08

1

1.U5

1 1 1 1

I

a

} 9 ^ i 6 0 A ^ ^ k 6 I

D 0 A

_ •'

0.5 1.0 1.5

1 nrJ

d

1 1 1 1 f

• • •

' i 4 i 1 ni'

^ ^ •

a 0.5 1.0 1.5

H

1

1

^ 9

II

1.02

A

jii^

/^

1.00

—

6-

0.98

^S

0.96

1 i

! ! 1 i i ! •

1

1/1

• 0.00

0.02

0.04

0.06

0.08

0.10 g/a

(a) Cutting angle 90°

0.00

0.02

0.04

0.06

0.08

0.10 g/a

(b) Cutting angle 45°

Figure 11: Relation between P^/Py and g/a TABLE 4 VALUES OF k, AND k^

1 Cutting angle 90° 45°

k

_

1

k2

1% 0.0277 -0.0005

2% 0.0370 -0.0015

3% 0.0480 -0.0035

4% 0.0593 -0.0055

-0.0070 1

k2

0.0183 0.0020

0.0310 -0.0010

0.0433 -0.0040

0.0557 -0.0070

-0.0095 1

kj

x

5% 1 0.0700 0.0677

CONCLUSION In this paper, the effects of horizontal-stiffener-end-gaps on the ultimate strength of girders under bending were investigated. Static loading tests were carried out for girders with different stiffenerend-gaps. The relationship between the reduction of the ultimate strength and the size of stiffenerend-gaps was examined by the FEM. The limiting size of the stiffener-end-gap corresponding to the decreasing rate of the ultimate strength was given.

REFERENCES Japan Road Association. (1996). Japanese Specifications for Highway Bridges, Part 2: Steel Bridges (in Japanese) MARC. (1994). KdARC Version K6, MARC Analysis Research Corporation, USA Okura L, Kazashi A., Kesmarky Z. and Nagy A. (1997). Effects of Stiffener-End-Gaps on Ultimate Strength of Girders under Bending. Technology Reports of Osaka University 47:2294, 225-235.

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

167

ADVANCED FINITE ELEMENT BUCKLING ANALYSIS IN ENGINEERING PRACTISE Peter Osterrieder\ Frank Werner^, Matthias Friedrich^ and Olaf Ortlepp^, ^Department of Civil Engineering, Brandenburgische Technische Universitat Cottbus D-03044 Cottbus, FRG, ^Department of Civil Engineering, Bauhaus Universitat Weimar, D-99423 Weimar, FRG

ABSTRACT Buckling design of stiffened plates and webs according to national and international standards requires the evaluation of lowest buckling loads due to action of one or more stress resultants. Applying as conunon in the past buckling tables to determine buckling slendemess leads in general to mostly conservative structural system and loading idealizations and limitations. The present study is concerned with the investigation of the effect of those simplifications by applying straight forward finite element buckling analysis. Lowest buckling loads are calculated within this analysis considering the total, arbitrarily stiffened plate with any type of loading. Stiffeners are idealized within thefiniteelement procedure as beam elements with or without axial and torsional rigidity. As a result of this study it is obvious, that applying PC-based finite element buckling analysis in engineering practise with closely code-related software, plate buckling design can be far more effective, economic and safe.

KEYWORDS Plate buckling, nonlinear finite elements, advanced design methods, interaction buckling, bifurcation problem

1

INTRODUCTION

Stability design with respect to the German Steel Code DIN 18800 part 2 [1] or EC 3 [2] basically allows for two different approaches. For checking flexural and flexural torsional beam buckling stability either equivalent slendemess concept or alternatively theory second order analysis may be applied. Within the first approach determination of buckling load or buckling length respectively is required to calculate the buckling slendemess. Depending on the latter and the cross section shape reduction factors on the plastic capacity are to be determined from associated code specifications. Reduced section capacity has then to be checked against maximum internal forces obtained from a theory first order analysis. The second approach is based on internal forces and stresses derived from a theory second order analysis taking into

168 account appropriate geometric imperfections. However for checking plate buckling stability codes only allow for the equivalent slendemess concept. Specifications in [ 3] are closely oriented towards the application of available buckling tables and buckling curves respectively. Buckling tables like in [ 7] contain buckling loads for specific load and boundary conditions. Thus any given stress situation is decomposed into maximum three load cases. Also multiple stiffened plates must be investigated considering up to three different systems - the total buckling panel stiffened by longitudinal and transverse stiffeners, part of the total panel limited by two transverse stiffeners and extending over the total height and each single unstiffened subpanel. Since plate buckling curves with reduction factors depending on the buckling slendemess are only available for specific boundary conditions the procedure is limited to plates with simple boundary conditions. Therefore especially for multiple stiffened panels this design procedure is time consuming and prone to errors. Thus a closely practice oriented finite element based buckling analysis not only will save time, but also may be useful to both economically and safely idealize load and boundary conditions.

2

ELEMENTS OF ADVANCED METHODS FOR PLATE BUCKLING DESIGN

Nonlinear finite element analysis has reached a level of practical applicability which allows to use the method not only for the solution of linear problems but also for stability design of any structure in daily engineering practise. However it has has been realized at the moment only for flexural beam buckling design [ 4]. For flexural torsional buckling of open thin-walled steel sections applications based on a first yield criterion according to [ 1] were presented in [ 5] and [ 6] and are widely used in engineering practise. Using advanced nonlinear finite elements for solution of plate buckling problems according to present codes the following requirements should to be met. 2.1 Simple, user-friendly interfaces to create structural system (High Torsional Restraint) This includes automatic generation of the finite element mesh, code related loads and load cases and geometric boundary conditions. Specifications in [ 3] are restricted to rectangular plates free, fixed or elastically restrained boundaries with respect to lateral displacement three load cases Therefore any loading situation has to be decomposed in maximum three load cases shown in figure 1.

load case 1

Figure 1

load case 2

load case 3

Load cases for plate buckling design

Edge loads o^ for load case 1 are assumed to be constant in x-direction and Oy-loads in load case 2 do not create essential a^-bending stresses. Decomposing any given load situation into load cases 1 to 3 as show in fig. 1 indicates that the real behaviour may be superimposed from these three cases. However having

169 in mind that the plate under consideration is part of a girder or column o^-, Oy- and T-stresses are not independent and will interact in buckling. It is especially difficult to separate local stresses from load case 3 from beam-stresses to be treated in load case 1. 2.2

Investigation of total buckling panel with arbitrary numbers of longitudinal and transverse stiffeners of arbitrary shape concentric flat stiffener

T-stiffener

excentric flat stiffener

L-stiffener

trapezoidal stiffener

center line

J center line

center line bq

Figure 2

Buckling Stiffeners

Stiffeners shown in fig. 2 are commonly used to prevent plate buckling. However to allow for any other type of stiffeners also user-defined stiffeners should be supported via explicit input of axial and bending stiffnesses. Finite element buckling analysis with discrete stiffeners will include buckling of each single unstiffened subpanel, buckling of the overall stiffened panel and also flexural buckling of stiffeners. It is therefore sufficient to check only the overall panel for plate buckling. This leads to a more straight forward and easier to judge design. 2.3 Closely code related analysis and check procedure With respect to [ 3] this requires the following steps for each of the maximum 3 load cases determination of membrane stresses eigenvalue analyses calculation of buckling slendemess and reduction factor buckling check If more than one load component is present in addition buckling due to simultaneous action of all loads must be checked. This will be accomplished in [ 3] by applying an interaction equation which summarizes the effect of all stress components on the overall buckling capacity. This procedure is based on buckling capacity curves in [ 3] which include post buckling strength for 3 different load cases and various boundary conditions. In cases where the problem is governed by flexural beam buckling rather then by plate buckling a correction factor taking into account reduced post buckling capacity must be determined and applied to the resistance. 2.4 Automatic, effective and stable algorithms for eigenvalue analysis for lowest eigenvalue This seems to be the most crucial requirement for practical application. Applying geometric nonlinear finite element formulation leads to a general eigenvalue problem for determination of the lowest eigenvalue T^Ki from [ K

^ Ti K 1 = 0

In this equation K^ is the linear elastic stiffness matrix of the overall unstiffened or stiffened panel and Kg the geometric or initial stress matrix. K depends on the membrane stress distribution and the geometric

170

dimensions. In cases of strongly varying membrane stresses, very localized loading or complex eigenmodes for plates with several stiffeners the convergence behaviour during eigenvalue iteration can become rather poor. Since this is also true for very small membrane stresses it is advisable in these cases to magnify the applied loads by a constant factor. It has been observed that in most cases inverse vector iteration using random starting vectors and continuous shift till the lowest positive eigenvalue was found will be highly effective. In only few cases when the previous method failed an automatically initiated eigenvalue analysis using subspace method with QZ-algorithm was able to find a solution. Though there are at present no buckling curves available for combined loading allowing for postbuckling strength bifurcation factors not only for single components but also for combined action are found to be very useful for judgement of buckling behaviour. 2.5 Assessment of results through graphics output Graphics representation of calculated results is essential for the following reasons input verification plate geometry, boundary conditions, finite element mesh, dimensions, stiffeners and applied loading in-plane stress distribution For subsequent eigenvalue analysis the distribution of membrane stress components for load cases under consideration must be calculated via a linear in-plane stress analysis. This could in general be omitted for load cases 1 and 3, but is necessary in case of partial loading like in load case 2 (see fig. 3) and also if load redistribution into stiffeners is to be considered. Though there is at present no code based design check available based on von-Mises-stresses visualization of its distribution can be very useful for better understanding of load carrying behaviour.

a) Figure 3

a^-stress contours

b)

buckling shape

Unstiffened plate with partial loading of upper edge

Design of shape and location of buckling stiffeners Buckling stiffeners should be arranged effectively at locations where buckling modes have maximum displacements. As shown in fig. 3b a 3d-graphic visualization of the buckling shape will greatly support this assignment.

171 3

APPLICATION TO STANDARD BUCKLING PROBLEMS

To find out about differences and possible advantages of employing a finite element based plate buckling program instead of doing traditional hand calculations using buckling tables selected practical problems were solved and results compared. Subsequently two selected problems for one unstiffened and one longitudinally stiffened plate are presented. Example 1 - see [ 8], Steel St52, r, = 15,4 kN/cm, ry - 14,7 kN/cm I

b=800

t=7

0.2r

B X B'l LJ 0.2r

I

Tlcr,x-1.78

Figure 4

Tlcr,y=4.43

Biaxially loaded, unstiffened plate

The unstiffened plate in fig. 4 is laterally supported at all boundaries and loaded in both directions. Also shown in fig. 4 are lowest buckling modes and associated bifurcation factors calculated by the finite element program. From buckling stresses nondimensional slendemesses are determined with respect to [ 3] as Ap^ = 0.96 and Ap = 0.62. Using buckling curves in [ 3] the program finds K^ = 0.93 and K = 1.0. As it is obvious from buckling mode shape shown in fig. 4 for r^-loading a large portion of the panel buckles in beam mode. This is considered in [ 3] via a further reduction of the factor K^ which accounts for reduced post buckling capacity. Evaluation of interaction equation (14) in [ 3] at the centre of each finite element results in a value 0.67 at the governing location with coordinates X = 10.0 cm and Y = 78.6 cm. Linear eigenvalue analysis with combined loads leads to a bifurcation factor of r]^^^ = 1,543 and an eigenmode, which is dominated by r^-loading. Example 2 - see [ 9], r, = 27.72 kN/cm, i|; = -1, ry = 13.08 kN/cm, r,y = 9.0 kN/cm, Steel St 52 330

1 1 .. 11 i l ' '

ii

iY

r

120x80x10

1

r

120x80x10

1

X

t=12

^ r^

a=3300

^

Figure 5 Stiffened plate with boundary loading

172 Buckling analysis of the overall panel in fig. 5 with discrete longitudinal stiffeners and laterally supported boundaries leads to buckling modes and bifurcation load factors given in figure 6.

Ti,,,=2.38

^

Ti,,y:«2.82

^

Ti,,,=177

Tlcr,v=1.56

Figure 6 Buckling due to single loads and combined loading According to [ 3] buckling slendemesses are calculated as Ap^ = 0.81, Ap 1.08 and X^^ = 1.25. From appropriate buckling curves in [ 3] reduction factors are determined as K^ = 1.0, K = 0.74 and K^ = 0.67. Within the checking procedure the program further evaluates interaction equation (14) in [ 3] at the centre of each element to find the governing location at X « 1,58m and Y = 2,13m with a value of 1.18 > 0! Exceeding the allowable value of 1.0 requires reinforcement of the panel by increasing either panel thickness or stiffness of stiffeners. From buckling modes in fig. 6 it is obvious, that buckling occurs basically in the upper subpanel and therefore strengthening of the upper stiffener will be sufficient. Moreover the associated buckling mode in fig. 6 due to combined loading clearly indicates that stability behaviour is dominated by load case 3 - partial loading of the upper edge. While a finite element program will find also for this case the lowest bifurcation load factor, buckling curves taking into account post buckling strength are not available. Though afiniteelement based buckling design for total loading would be both safer and more economic missing code specification do not allow for it. However the finite element procedure with discrete stiffeners allows to deal only with the total, stiffened panel which includes eventually buckling of a single subpanel. Also different from values in [ 7], bifurcation loads from finite element analyses may account for axial and torsion stiffnesses of stiffeners. This can result in considerable increase in buckling capacity depending on the location and type of the stiffeners.

4.

EXTENDED APPLICATION

4.1 Rigidity of Longitudinal Stiffeners For buckling analysis proper definition of boundary conditions is crucial. However since buckling specifications in codes are limited to single stiffened or unstiffened panels it is up to the designers understanding to define boundary conditions.

Figure 7

Buckling shape of I-beam

173 This requires quite often enormous engineering knowledge about the behaviour and the stiffness of adjacent structural members. Especially for buckling design of beams with slender webs like in fig. 7 the designer has to decide whether flanges will provide enough stiffness to prevent lateral displacements and rotations respectively at the longitudinal edges of the plate. The following calculations with flanges idealized as longitudinal stiffeners will demonstrate the effectivity and usefulness of a finite element based buckling design by supporting the designers job of structural idealization. In subsequent studies the following parameters for an unstiffened I-beam like infig.7 were considered, geometry length a == 2000 mm, width b = 800 mm, a = 2.5, web thickness t variable transverse boundaries all edges are laterally supported and free to rotate longitudinal boundaries LBCl edges laterally supported and free to rotate (no flanges) LBC2 edges laterally supported and fixed against rotation (no flanges) LBC3 edges are free with flange 300x15 - I-^s = 33.8cm'^ (Low Torsional Restraint) LBC4 edges are free with flange 300x30 - I^s = 270.0cm^ (High Torsional Restraint) constant o^ - stress, \|;^ = 1 (pure compression) constant shear stress T Fig. 8 shows buckling stresses T^^ for pure shear loading (fig. 8a) and o^ ^r ^^^ P^^^ compression (fig. 8b) depending on b/t-ratio of the web and the boundary conditions in longitudinal direction. i iTcr [^^/^f^^]

62.5-

1

b)

^^

75.0 -

LBC2

\

^

i f..cr

LBC4

50.037.5-

[kN/cm^]

50.0\ \

LBC3

LBC4

37.5-

V \

25.0-

LBC2

LBCl

LBC3

25.0^ . ^

12.5-

LBCl

12.5i 50

1

j

1

100

150

200

Figure 8

H^" 250

(T).

—I

H

\

i

1

50

100

150

200

250

•

Buckling Stresses for Unstiffened I-Beam

From fig. 8 it becomes obvious, that even comparatively small flanges (LBC3) provide enough lateral bending stiffness to justify lateral support (LCBl) of the upper edge of the web. Furthermore it is to observe, that moderate increase of flange thickness (LBC4) and therefore torsional rigidity makes the buckling solution rapidly converge towards the case of total fixed boundaries (LCB2). Additional calculations lead to the conclusion, that these findings are nearly independent of differing a - and i|; ratios. Results from the finite element analysis were also found to agree well with those given in [10] including torsional rigidity of flanges. 4.2 Interaction of Local and Global Buckling Buckling design of problems outlined infig.9 with respect to current code specifications without use of an appropriate design program is a very time consuming process and requires detailed knowledge of stability theory. As shown infig.9a stability failure of I-beams with slender webs will be in a coupled mode of local web buckling and global lateral torsional buckling. In fig. 9b the buckling load q^r is plotted versus the web slendemess b/t for 4 cases. Beam buckling curves infig.9b are calculated assuming that the cross section remains undistorted and load q^ is applied either at centroid (upper curve) or at top flange. The finite plate buckling model consists of plate elements for the web laterally supported only at

174 the beam ends. Flanges are idealized as longitudinal stiffeners providing axial stiffness, lateral stiffness and in the second plate model also torsional stiffness. Beam span is kept constant to 6,0m. From fig. 9b it follows, that for slender webs local buckling reduces global stability considerably, even when torsional rigidity of stiffeners is included. For stocky webs results of the plate model converge to those of the beam model. Qcr [ ' < N / c m ]

b)

10

beam buckling "(load at centre)

Qzzszzzsa -

beann buckling (load at top flange)

1000

vhi/iift2zzi •

k^oo^

plate buckling (with torsion rigidity) plate buckli ^ (no torsion rigidity) -H—\—I—I—I—\—\—I—I—\—I—I—h100 50

Figure 9

Interaction buckling

For the problem indicated in fig. 9a with b/t«100 the finite element analysis finds the critical load to be q„ « 3.31 kN/cm. A semiempirical solution for interaction buckling is provided by eqn. (99) in [ 1]. Applying this eqn. the critical global buckling load may be determined taking into account reduced web stiffness caused by local web buckling. However the critical plate buckling load - required in eqn. (99) for the web under transverse loading depends heavily on the structural model. Based on bending stiffness distribution the web will carry 21% of the total load. Scaling thus the critical web buckling load of 1.53 kN/cm with a factor 1/0.21 leads to q^^^p = 7.29. With the critical load for lateral beam buckling q^r^B = 4.49 the reduced overall buckling load with respect to eqn. (99) in [ 1] then becomes red q^r * 3.82 kN/cm.

REFERENCES [ 1] [ 2] [ 3] [ 4] [ 5] [ 6] [ 7] [8] [ 9] [10]

DIN 18800 Part 2 - Stahlbauten, Stabilitatsfalle, Knicken von Staben und Stabwerken, 11/1990 DIN V ENV 1993, Eurocode 3: Bemessung und Konstruktion von Stahlbauten. Marz 1994 DIN 18800 Part 3 - Stahlbauten, Stabilitatsfalle Plattenbeulen, 11/1990 Bridge,R.Q., Clarke,M.J., Osterrieder, P., Pi, YL., Trahair, N.S., Design by Advanced Analysis, Journal of Constructional Steel Research, Vol. 46, No. 1-3, Paper No. 144,1998 Osterrieder,P., Voigt,M., Saal,H., Vergleichende Betrachtungen zum Biegedrillknicknachweis nach DIN 18800 Teil 2, Stahlbau 58 (1989), Heft 11, 341 - 347 Osterrieder,P., Wemer,F., Kretzschmar,!., Biegedrillknicknachweis Elastisch-Plastisch fiir gewalzte I-Querschnitte, Stahlbau 67,1998, 794 - 801. Kloppel,K., Scheer,J. und K16ppel,K.,M611er,H., Beulwerte ausgesteifter Rechteckplatten, Band 1 und 2, Ernst und Sohn, Berlin, 1960 und 1968 Peil,U., Kompaktseminar DIN 18800, Universitat Karlsuhe, Lehrstuhl fiir Stahl-und Leichtmetallbau, 1994 Lindner,;., Scheer,J., Schmidt,H., (Hrsg.): Stahlbauten. Erlauterungen zu DIN 18800 Teil 1 bis Teil 4 (Beuth Kommentare). Beuth, Berlin, Koln; Ernst & Sohn, Berlin 1993. Kutzelnigg,E., Beulwertdiagramme fiir Stegbleche ohne Zwischensteifen nach der linearen Beultheorie bei Beriicksichtigung von in Tragerlangsrichtung veranderlichen Spannungen und der Torsionssteifigkeit der Tragergurte, Der Stahlbau 47,1978

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

175

INFLUENCE OF WEB IMPERFECTIONS IN WELDED I-BEAMS WITH SLENDER WEBS N. Rangelov Department of Steel Structures, University of Architecture and Civil Engineering, Boul. «Hristo Smimenski» 1, BG-1421 Sofia, Bulgaria

ABSTRACT The Study is devoted to the influence of apparent initial out-of-flatness together with residual stresses in the typical case of welded I-beams in bending. The theoretical model is a portion of an I-girder, analysed by means of a nonlinear FEM package. A simple approach is used to account for the postcritical response of the slender web to the initial compression due to welding. A parametric study is carried out on the combined influence of both the initial geometrical imperfections and residual stresses on the "stub-beam" bending capacity. Some contradictory trends in the effects of the two factors are found to interact, thus leading to a favourable balance. The results demonstrate that the design procedure of ECS - due to its empirical origin - covers a high level of imperfections. Therefore even large out-of-flatness has no practical influence and thus no theoretical justification may be found for keeping more severe and restrictive tolerances in practice.

KEYWORDS I-girders, slender webs, imperfections, out-of-flatness, residual stresses, nonlinear analysis INTRODUCTION Built-up girders made of thin sheared plates are rather usual in the modem practice of steel construction since they can be proportioned so as to achieve the best balance between internal forces and resistance. It is well known that the most effective beam is the one having a slender web but relatively thick flanges. However, the contrast between the thinness of the materials used and the "brutality" of the fabrication processes results in introducing considerable initial geometrical imperfections (out-of-flatness) together with residual stresses. In general, these imperfections are presumed unfavourable but their deleterious effect has not always been demonstrated. The tolerances that limit the initial out-of-flatness have often been established on basis having no relation to the resistance, but to aesthetic or other considerations, and thus appear undue conservative and restrictive. Modem codes base the design of slender cross-sections on the effective width concept, which has an experimental background. For example, ECS design model, using the empirical Winter formula,

176 undoubtedly covers certain level of imperfections, which may be much higher than the relevant tolerance. In other words, if it can be shown that the actual design procedure implicitly accounts for large imperfections, and/or their influence on the beam capacity is relatively small, then those imperfections should also be allowed in practice. An attempt for a systematic research in this direction has been made by Rangelov (1992), Braham et al. (1995) and Rangelov & Braham (1995). The last contribution is especially noteworthy, in which a parametric study based on a nonlinear FEM model is reported and on that basis conclusions are drawn that the influence of large imperfections is small and that the design procedure of EC3 covers a variable but high level of imperfections. However, despite that some residual stresses are accounted for, the attention in that paper is mostly focused on the effect of geometrical imperfections. On the other hand, a study on the influence of post-welding residual stresses was recently published by Pastemak et al. (1996), underlining a redistribution of the initial stresses due to the post-critical response of the slender web. A surprising conclusion is made there for a strong (-30%) effect of the residual stresses in a thin-webbed beam. However, the considered failure mode is lateral-torsional buckling and it is well known that all the overall buckling modes are sensitive to residual stresses, but the relevant buckling factors (x, Xvr) account for that effect. Moreover, only one example has been solved. Therefore, the conclusions of this report can hardly be generalized. The present study is aimed at further investigating the effect of the initial out-of-flatness together with the residual stresses on the local ("stub-beam") behaviour of girders with slender webs. Based on a nonlinear FEM model, a parametric study is carried out on the interactive influence of both types of imperfections on the bending capacity, emphasizing the specific aspects regarding the post-welding residual stresses in the case of slender cross-sections as suggested by Pastemak et al. (1996).

GENERAL ASSUMPTIONS. MODEL OF STUDY The structural behaviour of members with slender cross-sections is rather complicated, being controlled by interactive buckling (local buckling/post-buckling — overall stability). According to the design approach, the local behaviour is accounted for by the effective width concept and is "transferred" to the global behaviour via the member slendemess. In the case of beams in bending, which is the case of consideration herein, the latter can be written as ^LT = ^ M s t u b / M c r ,

(1)

Mstub being the resistance of the "short coupon" ("stub beam") with account taken of the local postbuckling behaviour, while Mcr is the linear buckling moment independent on the cross-section class. In this context, the study is directed to the local behaviour of a girder, i.e. to Mstub. Thus it is believed that the local effect of imperfections on the overall lateral-torsional stability can be accounted for through the relevantly defined beam slendemess >ILT . On the other hand, an assumption is made that the global effect of imperfections is to be taken into consideration by the lateral-torsional buckling factor ;fLT. As only the local behaviour is studied, the model is a portion of I-girder, the length of which corresponds to the half-wave length of the web undulations, a. Additionally the symmetry is taken into account (Figure 1). The typical metal plasticity law is used to represent the material nonlinearity. Since the loaded cross-section corresponds to a nodal line of the out-of-plane deflection pattern and therefore is "perfect", Bemoulli's hypothesis is to be kept valid. This is modelled by constraint equations leaving only two independent nodes in the cross-section. The external bending moment is applied directly as a couple offerees at those nodes. It is assumed that the shear force is favourable because of

177 the stabilizing role of the tension field action (ECCS-TWG8.3, 1986), therefore it is not included in the analysis.

<

Figure 1: Model of study and FE discretisation The initial deflection shape is generally unpredictable, therefore it is derived from the basic buckling mode of the model (the most imfavourable case after some theoretical suggestions), scaled by the outof-flatness amplitude, wom- The model of the residual stresses and their treatment is discussed in the next section. The nonlinear FEM code ABAQUS is employed with S4R5 shell elements both for the web and flanges. The bending capacity is referred to the ultimate theoretical load, which is obtained using Riks' method for automatic loading strategy. MODEL OF RESIDUAL STRESSES The main cause for the residual stresses (and out-of-flatness as well) in the steel members is their uneven heating and cooling during the fabrication process. In the specific case considered here, these are mostly due to the welding of the component plates together and obviously can hardly be predicted. A theoretical approach is applied by Dwight & Moxham (1969) to welded steel plates and a model for their distribution is proposed. The work of Plumier (1981«) is especially noteworthy, where an analytical method is used to assess the post-welding stresses depending on the welding parameters. The approach is directed to different welded cross-sections (Plumier, 1981Z?) and relevant models are suggested. As such models are complicated and sometimes based on questionable assumptions, a simple stress distribution is widely used nowadays as an approximation (ECCS-TWG8.2,1984), which in this paper is referred to as "classical" distribution (Figure 2). These models, however, are applicable to compact cross-sections where the component plates are stable enough to carry the residual compression. As theoretically shown by Pasternak et al. (1996), in case of Class 4 sections the picture is different, since a post-critical redistribution of stresses takes place even during welding. This

178 results in unloading of the buckled part of the slender web and shading of the compressive stresses into the stiffer flanges. The precise approach of Pasternak et al. (1986), using FE thermal analysis to simulate the heat input during welding and, based on the resulting temperature fields, to calculate the post-welding stresses, appears rather time-consuming. Therefore, another approach is adopted in the present study to obtain a more realistic picture of the residual stresses in case of welded girders with slender webs.

0,25/y (in both flanges)

Figure 2: "Classical" distribution of residual stresses The starting point is the "classical" distribution, applied however to the geometrically imperfect model. Thus defined, the residual stresses are no longer in self-equilibrium as far as web out-offlatness is considered. Therefore, before applying any load, an initial nonlinear step is performed to reach equilibrium, and the result of this step is regarded as new modified residual stress pattern. This is illustrated for one typical case in Figure 3. In this manner, a similar to that reported by Pasternak et al. (1996) effect is simply obtained; especially, the compression in flanges is up to 50% higher compared to the "classical" diagram. This is accompanied by additional web out-of-plane deflections tending to a new more symmetrical equilibrium shape as a consequence of the symmetrical initial stresses applied.

Figure 3: Modified residual stress distribution after the initial step Of course, this approach is based on some arbitrary assumptions. For instance, the greatest part of the initial deflections is a consequence of the post-welding stresses, while herein both the initial out-offlatness and the residual stresses ("classical") are applied simultaneously to the model to interact as initial conditions. However, the post-critical response of the web to the initial compression is anyhow accounted for, and the modified picture obtained seems very similar to the theoretical as well as measured post-welding stresses (Pasternak et al., 1996).

179 PARAMETRIC ANALYSIS Scope of the Study The following parameters are considered to account for different cross-section geometry: •

The ratio Ax,/A. ^w being the web area and A - the whole area. Two cases are considered: A^ I A = 0-50 (a typical 'conventional' case) and ^w I A = 0*375 (a more representative case for a beam with slender web). Flange dimensions are determined assuming bf/tf= const ~ 20, thus ensuring full effectiveness (at least Class 3).

•

The web reference slendemess X^, which is the basic parameter governing the post-buckling behaviour. The study is directed to slender webs, therefore the range of 0'8
•

The initial out-of-flatness, measured by its relative amplitude womfh^, within the range 1/1000 ("perfect") to 1/50.

•

The residual stresses are treated as described above and in fact two cases are taken into consideration: without and with residual stresses.

•

As shown by Brahametal. (1995) and Rangelov & Braham (1995), the influence of the aspect ratio and the steel grade is negligible, therefore constant value a/hyf = 0-6 is adopted and only steel grade S 355 is considered.

Remarks on the Observed Behaviour The behaviour observed is qualitatively similar in all cases despite the web slendemess and imperfection level. Due to the web buckling in the compression zone, the neutral axis moves towards

{a)

(P)

Figure 4: Typical picture of the middle surface von Mises stresses at ultimate load without (a) and with (b) residual stresses

180 the tension flange. Thus the compression flange appears more or less (depending on Aw) overloaded and therefore critical for the ultimate capacity. From this point of view it can be expected that the higher compressive residual stress after the redistribution (Section 3) would be rather unfavourable. However, it turns out that the strips with initial tension stresses compensate this effect to a great extent. As illustrated in Figure 4, there is a "perturbation" of the stress flow if residual stresses are present, and, due to the initial tension, some parts of the compression zone remain elastic up to collapse, which is favourable. The load-shortening curves show similar behaviour both at relatively low and high web slendemess. It is characterized by an earlier loss of stiffiiess in case of residual stresses, which is obviously due to the initial compression. On the other hand, the behaviour is "smoother" and the collapse — less sudden, probably because of the tension strips that remain elastic. Influence of Imperfections on the ^*Stub-Beam" Capacity In this study the attention is exclusively focused on the ultimate bending resistance. To assess the relative effect of imperfections, the case of wom/hy,= 1/1000 without residual stresses is adopted as 'perfect', and all results are referred to its capacity taken as 100%. In this manner, for both the sets — without and with residual stresses — the same "benchmarks" are used, therefore it may be thought that the interactive influence of the two types of imperfections is estimated. The results in this format are given in Figure 5. —•— -1/500 —1 1/250 —

— A —

-1/150 — — A —

-1/100 — 1/72 — 1/50 —

— « . . -1/500 + . . - A - . -1/250 + . - - • - - -1/150 + - - - a - - -1/100 + -•-A--

-1/72 +

0-- •

1/50 +

- EC3

2 0.8

1

1.2

1.4

1.6

1.8

2

Figure 5: Relative "stub-beam" capacity for different imperfection levels without (-) and with (+) residual stresses Firsty, regarding the influence of initial out-of-flatness (solid lines), the conclusions of Braham et al. (1995) and Rangelov & Braham (1995) are confirmed, which may be summarized as follows: •

the influence of web imperfections is higher in case of larger Ay, I A, which is obvious, because the contribution of the web in such a case is greater;

•

the effect of undulations rapidly decreases with increasing the slendemess, tending to vanish a high Aw.

•

the small effect of initial out-of-flatness is confirmed by the fact that, for example, an increase o: the amplitude wom from 1/150 to 1/50 (3 times) results in "loss" of capacity within only 2-4%.

181 The effect of residual stresses can be estimated by the difference between the "welded" and "nonwelded" case. In this direction the following observations are made, more clearly seen in the case of Ay,/A = 0-50: •

there is a tendency that the effect of residual stresses when compared to the corresponding "nonwelded" case becomes smaller in case of larger geometrical imperfections;

•

the influence of residual stresses appears higher in case of more slender webs. This is surprising and can be explained with the more pronounced post-critical redistribution of the initial compression in the cross-section.

The latter observation, however, is only valid for the model, where as a starting point the "classical" distribution is always taken. In reality, in case of slender webs, the leg sizes of the welds would be smaller, thus introducing less heat and therefore less post-welding stresses. Besides that, some favourable balance between the effects of both imperfection types can be found in the following. In case of high slendemess on one hand the actual out-of-flatness is quite larger however its influence nearly vanishes; on the other hand, in such a case the influence of residual stresses appears slightly higher. As a result, the combined influence of both, represented by the dotted lines, appears nearly constant with the slendemess. When comparing to the resistance of the "perfect" case (100%), this combined effect does not seem so negligible. However, the curves representing different amplitudes wom /h^^ are very close to each other, therefore an increase of WQ^ several times leads to a difference of only a small percentage. Thus, no justification can be found to keep more severe tolerances. On the other hand, as outlined by Braham et al. (1995) and Rangelov & Braham (1995), referring the results to the "perfect" case is mostly of theoretical interest to assess the "pure" relative influence of imperfections. However, from a practical point of view it is substantial to estimate what level of imperfections is implicitly covered by the empirically based design approach. For this reason the bending capacity according to ECS, based on the effective width concept with the Winter formula, is included in Figure 5, normalized by the resistance of the corresponding "perfect" beam. The results now can be interpreted as follows. Where a curve representing certain imperfection level lies above that of ECS, the design model covers "automatically" the influence of those imperfections. In this context the results seem much more optimistic. In the case of ^w I A = 0-375 almost all the data points are situated above the ECS curve, therefore even a level wom/^w= 1/50 is acceptable because the "unsafety" is only about 1% in the worst case. Similarly, the difference is maximum 2% in the case of A^ I A = 0-50. Consequently, more liberal tolerance limits may be allowed for the webs of welded girders, provided their capacity is calculated according to ECS design method.

CONCLUSIONS The present study is focused on the very typical case of welded I-girders in bending when apparent initial imperfections are unavoidable. To study their influence, together with that of the post-welding residual stresses, a geometrically and physically nonlinear model is developed using ABAQUS FEM package. A simple approach is used to account for the nonlinear redistribution of the residual stresses in case of slender cross-sections. A parametric analysis is carried out on some 110 cases to investigate the combined effect of both the initial imperfections and residual stresses, and relevant conclusions are drawn. The results demonstrate that the design procedure of ECS - due to its empirical origin - covers a high level of imperfections, therefore even large imperfections have no practical influence. In this context no theoretical justification can be found for keeping more severe and restrictive tolerances in practice.

182 ACKNOWLEDGEMENTS The main part of the present study has been carried out at Lehrstuhl fur Stahlbau, BTU Cottbus, Germany. Thanks are therefore due to all the colleagues there and especially to Professor Hartmut Pasternak. The financial support of DAAD is also gratefully acknowledged.

REFERENCES ABAQUS - GQiiQrsil purpose nonlinear finite element program. User's Manual, Version 5.5, Hibbitt, Karlsson & Sorensen, Inc., 1995. Braham M., Maquoi R., Rangelov N. and Richard Ch. (1995). L'influence des defauts de planeite de Tame des profiles reconstitues sondes sur leur resistance en flexion et compression. Construction metallique :1, 3-28. Dwight J.B. and Moxham K.E. (1969). Welded steel plates in compression. Structural Engineer 47:2, 49-66. ECCS-TWG8.2 (1984). Ultimate limit state calculation of sway frames with rigid joints. ECCS Publication JVTo 33.

ECCS-TWG8.3 (1986). Behaviour and Design of Steel Plated Structures (ed. by P. Dubas & E. Gehri). ECCS Publication X044. PasternakH., Schillings, and EngstW. (1996). How welding does influence the ultimate load capacity of thin-walled structural members - an example. European Workshop ''Thin-Walled Steel Structures'", Krzyzowa (Kreisau), Poland, 135-142. Plumier A. (1981a). Determination analytique de la distribution des contraintes r^siduelles de soudage dans \me plaque mince. Revue de la soudure 37:1, 29-45. Plumier A. (1981Z>). Determination de la distribution des contraintes r^siduelles de soudage dans les plaques soudees d'epaisseur quelconque, dans les assemblages de plaques et apres soudages multipasses. Revue de la soudure 37:2, 91-112. Rangelov N. (1992). A theoretical approach to the limiting of initial imperfections in steel plates. Stahlbau 61:5, \5\-\56. Rangelov N. and Braham M. (1995). Are the web imperfections in built-up I-beams of any importance? International Colloquium ''Stability of Steel Structures'", European Session, Budapest, Hungary. Preliminary Report, vol. I, 263-270.

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

183

Prediction of Buckling Strength of Stiffened Plates by Use of the Optimum Eccentricity Method Gunnar Solland and Morten Rotheim Det Norske Veritas, H0vik, Norway

ABSTRACT The buckling capacity of axially loaded stiffened plates, as found in bridges and marine structures, are often analysed by so-called strut models, ref e.g. Galambos (1987). The strut model is traditionally taken as a single span column with hinged end connections. However, the stiffeners in a real structure are continuos over several spans or to some degree rotational fixed at the ends. Therefore, a new strut model is developed. This new concept is called the optimum eccentricity method. This paper presents the background for the method of analysis and the capacity formulations. Furthermore, the present method is compared with numerical calculations made with non-linear finite element methods as well as full-scale tests. The new formulation shows improved predictability in most cases compared to current codes. The method is valid for stiffened plates loaded with axial compression stress both longitudinal as well as transverse to the stiffener combined with in plane shear and lateral load on the plate. The validation of the method presented in this paper is limited to longitudinal compression stress combined with lateral pressure acting both on the stiffener as well as on the plate side. The new formulation for calculation of buckling strength of stiffened panels is implemented in the Norsok standard N-004 for Design of Steel Structures, which is used for design of offshore structures in Norway. Whilst the capacity checks are developed for stiffened plates the same method may be utilised for check of beam-columns loaded by axial force and lateral load with buckling about the same axis as the moment caused by the lateral load.

KEYWORDS Stiffened plates, buckling, combination of axial load and lateral load, continuous over several spans, strut model, marine structures, design codes.

184 PROBLEM FORMULATION Normally buckling of stiffened plates is checked by a stmt analogy as shown in Figure 1. In this paper buckling due to axial and lateral load will be treated. Presence of normal stress transverse to the direction of the stiffener and shear stress may be included in the formulation (see Carlsen (1977) or Norsok (1998)), but these effects are not deaJt with in this paper.

Nx.Sd=Nx.Sd (O'x.Sd. 7 s d )

.qsd^q(Psd. Po)

Nx.Sd

Psd/

STIFFENED PLATE

BEAM COLUMN

Figure 1 The strut model In a laboratory test of an axial loaded single span stiffened plate panel, the working point of the axial load will be fixed by the test set up. The position of the working point for the axial load in a stiffened panel being part of a real structure will, however, in the general case be indeterminate. When analysing the ultimate load carrying resistance of plated structures, the best estimate of the global strength is found when the position of the working point for the strut is selected so that the maximum resistance of the stiffened panel is obtained. By transferring the stiffened plate-buckling problem to an equivalent strut, it is possible to use ordinary beam column formulas for checking of the capacity. However, traditional design codes do not give explicit guidance on how to calculate the resistance of a beam-colunui with axial and lateral load and having only one axis of symmetry. In general the idealised strut or beam-column to be analysed is therefore a continuous beam-column with different section modulus for the plate and stiffener side of the strut.

DESCRIPTION OF THE METHOD The design formulas for continuous beam column with moment about one axis and buckling about the same axis are developed on the basis of traditional beam-column interaction formulas using linear elastic cross-section properties as well as elastic moment distributions. The cross-section may be nonsymmetric or single symmetric as long as the buckling is about the same axis as the moment caused by the lateral load. Furthermore it is assumed that the beam-column interaction formulas need to be

185 satisfied both at the supports as well as at mid-span and at both the plate and the extreme fibre of the stiffener flange. From this assumption it follows that the interaction of axial force and moment need to be made at four positions in the equivalent beam-column, see Figure 2. Consequently there will be four design equations to be checked.

• HniHHJHI

%d

Figure 2 Check points for interaction equations The check at the four design points is dependent upon the location of the working point of the axial force. The distance from the centroid of the effective cross section to the assumed working point for the axial force is denoted z*, see Figure 3. As described the value for z may be optimised for continuous stiffeners such that the maximum capacity is obtained. This means that the calculations need to be done repeatedly for all four design equations with varying values for z* and selecting the z* value giving the maximum capacity. If only a conservative assessment for the capacity is needed a z value equal to zero may be assumed. For calculation of the forces and moments in the total structure, of which the stiffened panel is a part, the working point for the stiffened panel should correspond to the assumed value of z*. In most cases, the influence of variations in z* on global forces and moments will be negligible.

Nsd

HiiHHiitH

%d J^sd

" *

N|

Neutral axis Figure 3 Definition of z

186 DESIGN EQUATIONS The following design equations are developed from traditional formulas for buckling of beamcolumns. Separate formulas are developed dependent upon whether the lateral load acts on the stiffener or the plate side. Lateral pressure on plate side: Stress point 1 Interaction equation number. See Figure 2 Nsd

Equation 1 number

, M^-Ns,-z

^^ (1)

1

Nsd 2

Nkp,Rd

Ns, ^ M ^ - N s , - z * NRJ

(I

^^ (2)

^sd 1

(3)

3

Nsd

1 M,+Ns,-z'

^^ (4)

4

Lateral pressure on stiffener side: Nsd

Ns, ^ M , + N s , - z

N

N

^^ ks,Rd

^^ Rd

Ms,Rd

N

Nsd , M , + N s , - z N kp.Rd M p.Rd l_Nsd

^^ '

^^ (6)

N. N.

M , - N o . -z / M.

N, N

M , - N ^Sd H-Z* N. -<1 N -+' ^ Rd M p,Rci NE J

^^ kp.Rd

V

(5)

<1

(7)

(8)

187 For a continuous stiffener the buckling length (k) is assumed to be a function of the lateral load. With no lateral load one need to assume a buckling length of 1.0 corresponding to a half sinus wave in each span. With increasing lateral load the buckling mode will tend to approach a mode with rotational fixed supports. Hence the following expression for the buckling length is assumed:

K=i\ 1-0.5

Ml, Sd

=

qsd

^sf 12

(9)

for continuous stiffeners with equal spans and equal lateral pressure in all spans

absolute value of the actual largest support moment for continuous stiffeners with unequal spans and/or unequal lateral pressure in adjacent spans M2,Sd

=

qsd^^

24

for continuous stiffeners with equal spans and equal lateral pressure in all spans

absolute value of the actual largest field moment for continuous stiffeners with unequal spans and/or unequal lateral pressure in adjacent spans

qsd

=

design lateral pressure

qf

=

the lateral pressure giving first yield in extreme fibre at support

W

=

the smaller of the section modulus for plate or stiffener side

/

=

span length

NE

=

Euler buckling strength

Nsd

=

design axial force

Nks,Rd =

design stiffener induced axial buckling resistance

Nkp,Rd =

design plate induced axial buckling resistance

Ms,Rd =

design bending moment resistance on stiffener side

Mp,Rd =

design bending moment resistance on plate side

z

the distance from the neutral axis of the effective section to the working point of the axial force, z may be varied so that the capacity from equations(l) to (4) or (5) to (8) is maximised. The value of z* is taken positive towards the plate. The simplification z* = 0 is always allowed.

=

COMPARISON WITH NUMERICAL CALCULATIONS In order to compare the design equations with non-linear numerical calculations, the design equations were compared with results from analysis from the general non-linear software Abaqus and the inhouse developed colunm-buckling program Colbuck. The result of the comparison is shown in Figure 5 and Figure 6 for a stiffened plate with a T-stiffener with reduced slendemess of 0.25 and 1.5 respectively.

188 The Colbuck analyses are made with two different material models denoted A and B, see Figure 4. Model A is used in order to achieve a conservative representation of post peak behaviour for stiffeners involving plate and torsional buckling. For some load combinations numerical instability was experienced and therefore also results with material model B are presented.

0.24E

Abaqus

Colbuck A

Colbuck B

Figure 4 Material models used for the numerical analyses. The imperfections used in the Abaqus analyses are in compliance with the tolerances in Norsok, but no residual stresses are included. The runs with Colbuck include both the tolerances in Norsok as well as residual stresses according to the modified Column buckling curve b according to Eurocode 3 (ENV 1993-1-1). In all cases the capacity equations are producing results to the safe side.

fqanao

V

r

^

•

X

^.%, t

•

\r \

" ^ - • - C u r r e n t paper xr^r-

A

ABAQUS

o

Colbuck A

X

Colbuck B

r

[~

1

e-1

-0.8

0 q/qf

Figure 5 Capacities for stiffened plate with reduced slendemess 0.25

k

-1

-0.8

-0.6

-0.4

Figure 6 Capacities for stiffened plate with reduced slendemess 1.5

COMPARISON WITH TESTS The design equations were compared with results from test of stiffened plates with and without lateral pressure (Smith (1975)). The results are presented in Figure 7. 300

Figure 7 Comparison of tests results with design recommendations

190 In the calculations the measured yield strength of the plate is used. In some tests the yield strength of the stiffener was significant lower than the plate and then the calculations were also made using this yield strength, (denoted minimum yield). The design equations produce results on the safe side compared with the tests except for test 3b. It should be noted that the out of straightness tolerances for this test was 2.2 times greater than the requirement in the Norsok N-004 code. The same tests are also calculated according to the DNV Classification Note 30.1 1992 edition. The proposed formulations give either similar or better correspondence with test than this DNV recommendation.

CONCLUSIONS A new formulation for calculation of buckling capacity of continuous stiffeners loaded by axial and lateral load developed. Comparisons with tests and numerical calculations show that the capacity predictions correspond well and the deviations are to the safe side. The new formulations show also improved predictions of the capacity compare with existing recommendations. REFERENCES Andreassen, E., 1996, DNV Report No. 96-0181, Users Guide to Computer Program for Buckling of Continuous Columns ver. 1. (Colbuck 1). C.A. Carlsen, 1977, Simplified Collapse Analysis of Stiffened Plates, Norwegian Maritime Research No. 4 1977. 20-36. DNV Classification Note 30.1. ,1995, Buckling Strength Analysis. Dubas P.and Gehri E. (editors.), 1986, ECCS publication No 44 Behaviour and Design of Steel Plated Structures. Eurocode 3, ENV 1993 1-1, Design of steel structures. Part 1.1: General rules and rules for buildings. 1992 Hibbitt, Karlsson, Sorensen, 1994, Abaqus Users Manuals, Version 5.4. Galambos,T.H. (editor) 1987, Guide to stability Design Criteria for Metal Structures, Fourth Edition, John Wiley & Sons, USA. Norsok Standard N-004,1998, Design of steel structures, Norwegian Technology Standards Institution, Norway. Smith C. S, Compressive Strength of Welded Steel Ship Grillages, The Royal Institution of Naval Architects, 1975. Steen, E, 1998, DNV Report No. 95-0445, Buckling of Stiffened Plates Under Combined Loads Abaqus Analyses.

Technical p a p e r s on CONNECTIONS

This Page Intentionally Left Blank

Stability and Ductility of Steel Structures (SDSS '99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

191

BEHAVIOUR OF UNSYMMETRIC BOLTED CONNECTIONS SUBJECTED TO DYNAMIC LOADING D. Beg^ C. Remec^ and P. Skuber^ ^Faculty of Civil and Geodetic Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia ^Institute for Metal Structures, 1000 Ljubljana, Slovenia

ABSTRACT The unsymmetric endplate bolted moment resistant connections which are primarily suitable for the transmission of bending moments arising from gravity loadings v^ere in the past sometimes used in Slovenia also for sway frames to resist wind and seismic loadings. Because of questionable resistance and ductility of such connections under cyclic loading, we decided to perform two full-scale dynamic tests on the typical T-shape beam-column assemblies containing unsymmetric bolted connections. The tests were accompanied by nonlinear numerical simulation where simple one and two-storey frames were subjected to different seismic loadings. The results show that at stronger seismic excitations connections are damaged and bolts at the weaker side of the connection can fail in tension but in spite of bolt failure all the analysed frames survived.

KEYWORDS bolted connections, imsymmetric connections, dynamic loading, seismic loading, tests, numerical analysis, seismic resistance INTRODUCTION The unsymmetric endplate bolted connections with two rows of bolts at the upper flange and one row of bolts at the bottom flange (see Figure 1) are suitable to resist gravity loading in non-sway frames. They can be successftilly used also in sway frames in the case of moderate wind loading in non-seismic regions. They have been used mainly in portal frame design for instance in United Kingdom and in Germany. Unfortunately this type of cormections was adopted in Slovenia mainly from the German practice, usually overlooking the fact that Slovenia is a seismic region and Germany not. Old Slovenian codes for seismic resistance of structures (which are still in force together with Eurocode 8) have implicitly incorporated behaviour factor 5 or even more for steel structures. Such behaviour factors require fully ductile design. Before Northridge and Kobe earthquakes steel structures were recognized as ductile more or less automatically. The consequence of such approach in Slovenia is the fact that

192

certain number of sway frames were designed using unsymmetric end plate connections whose ductility and resistance characteristics are questionable. In order to analyse the behaviour of unsymmetric endplate connections we performed two full-scale dynamic tests on unsymmetric moment connections. On the basis of the test results we also performed nonlinear dynamic numerical analysis on simple frames containing respective connections.

M20 (10.9)

•m m. Actuator — Displacement Transducer

^ 'HEB200 ^

I®

(L)

HEB200

|7-T|:f J-

i^ I

n

m-

^

395 ^,

600

Figure 1 : Unsymmetric endplate bolted connection

600

Figure 2 : Test set-up and instrumentation

TESTS Typical beam-column assembly containing unsymmetric bolted endplate connection was tested under dynamic loading. Two nominally equal full-scale test specimens with beam section made of IPE 300 hot rolled profile and column section made of HEB 200 profile were prepared. The details of the bolted connection are shown in Figure 1. M20 bolts of grade 10.9 were used and half of the full preloading was applied. The basic material was nominally of grade S235, but actual mechanical properties for beam section were as high as 315.5 MPa for yield strength fy and 460.1 MPa for ultimate tensile strength fu. Endplate was welded to the beam by full penetration single V butt welds and MAG welding procedure was used. The test set-up and instrumentation are presented in Figure 2. The dynamic load was introduced by hydraulic actuator with the stroke ±200 mm and capacity ±250 KN. The introduced force was measured by an electronic load cell and the displacements and relative beam-to-column rotations by displacement transducers. During the tests the displacement control was used. To simulate the effect of the gravity loading on the beam in the frame the displacement causing bending moment of the magnitude 0.3 Mp (plastic moment of the beam) was introduced first. This bending moment certainly caused the tension at the stronger part of the connection. Then the displacements following the sinusoidal pattern with constant amplitude of approximately two times the displacement of first yield Vy and frequency of 0.5 Hz were applied. The displacement Vy was estimated to 8-10 mm. The first specimen was subjected to the displacement amplitude of ±18 mm and the second one to ±20 mm. Unfortunately this increase caused tensile rupture

193 of the two bolts at the weaker side of the connection at the beginning of the second test. New bolts were installed and the second test was run at the displacement amplitude of ±16 mm. The test results in the form of moment-rotation diagrams are shown in Fig. 3 for the first test (the results of the second test are very similar to those in Fig. 3). In Figure 3.a relation moment - total relative rotation for the connection is plotted. Strong pinching effect mainly due to plastic extension of bolts producing larger deformability and decrease of absorbed hysteretic energy is evident. In Fig. 3.b moment-rotation diagram only for web panel is plotted. Hysteretic loops are more regular and the influence of rather weak panel is important. 300

300 -, M (kN/m)

M (kN/m)

200 H

O(Rad*1000) -40

20

-20

40

3.b Web panel

3.a Connection -300

Figure 3 : Moment-rotation diagrams from the first test In both tests the rupture caused by low cycle fatigue occurred in the beam flange at the stronger part of the connection at the edge of the weld (HAZ). Due to preloading in those flanges plastic deformations were larger than at the weaker side of the connection. The first test specimen survived 195 loading cycles and the second one 124. The number of cycles is quite high, because the extent of plasticity was not so high. Lower number of cycles in the second test can be explained by higher preloading introduced unintentionally after the rupture of the bolts in the second test. This preloading caused higher tension deformations in the beam flange at the stronger part of the connection. It is worth mentioning that the strain rate in the beam flanges at the connection reached the magnitude of 0.01/s (measurement by strain gauges), which is less than in stronger earthquakes (about 0.05/s or even more), but at least the order of magnitude is the same. NUMERICAL MODELLING OF THE CONNECTION Our next step was directed to the assessment of the behaviour of unsymmetric bolted connection in simple frames. From our test results it is evident that numerical modelling of the connection must include unsymmetric hysteretic behaviour with strong pinching effect and low ductility on the weak side of the connection due to the possibility of tension rupture of the bolts. After some study of the capabilities of the computer programs for nonlinear dynamic analysis CANNY and DRAIN 2DX, we decided to use the finite element developed by R. Zamic & S. Gostic (1997) and implemented into the program DRAIN 2DX. This element with rotational springs at the ends was

194 originally developed for the analysis of masonry infilled RC frames, but can be used also for steel structures (with some precaution). The behaviour of the connection was described by the hysteretic envelope and the pinching effect shown in Figure 4. Envelope

M Mpi Mpi = 172.70 kNm Ko(R)= 1.463 105 kNm Ko(SR) = 3.9 104kNm Oui = 0.05 Rad

/ 1 /

i^Typ »ical h]

; /

1 / // /

1

^

/

u2 ^ ^ -

1

/—I

/

f

r

m

Oul

O

Mp2= 122.50 kNm Ou2(R) = 0.01Rad Ou2(SR) = 0.015 Rad

Mp2

Figure 4 : Modelling of the connection spring Although simplified, the numerical model represents well all the important features of the behaviour ol the unsymmetric connections. It is important that the numerical model can act like the real connection also in the case of the rupture of the two bolts on the weaker side of the connection. In this case the connection behaves like the moment connection when the bending moment causes compression in the part where the bolts failed and acts like pinned connection when the bending moments is opposite and opens the gap at the position of the failed bolts. FRAMES WITH UNSYMMETRIC CONNECTIONS g,q

n u u u n n \n u

i

*

IPE300

g^

n

V

g>q

n u u i n 1 w u 111

n i n i n i n n t /*i

'\ ^

IPE300

''I

g JS235

g = 16.10 kN/m q = 15.00 kN/m

6.00 m

-^

seismic loading combination: g + 0.30 q normal mass (m): g + 0.3 q double mass (2 m): 2 (g + 0.30 q) (only portal frame)

Figure 5 : Frames used in the numerical analysis

195 For our numerical analysis we have chosen single and double storey frame. The geometry and loading are shown in Figure 5. All beam-to-column connections were modelled by nonlinear springs shown in Figure 4 using both rigid (R) and semi-rigid (SR) option. As said before, the computer program DRAIN 2DX was used for the dynamic analysis and 5% damping was considered. A single storey frame with normal mass was designed to satisfy all ultimate and serviceability limit state requirements according to EUROCODE 8, taking into account 0.3 g for the design ground acceleration. In all other cases only the inter-storey drift requirement was violated. Frames were subjected to two accelerograms: Kobe - 1995 and Ulcinj - 1979, which were scaled by different factors from 0.5 to 3.5 (see Figure 6). 0.8 0.4 (^) 30

40

50

M0'^^v^

0

60

10

20

^— 30

(s) 40

50

60

-0.4 -0.8-J

Kobe 1995

-0.8

Ulcinj 1979

Figure 6 : Accelerograms from Kobe and Ulcinj (Montenegro) used in the numerical simulation The characteristics of Kobe accelerogram are few very large peaks that reach acceleration of 0.9 g. On the other hand the spectrum of Ulcinj accelerogram exhibits very wide plateau with the peak ground acceleration of 0.24 g. The results of the parametric study are presented in Table 1 for the single storey frame and Table 2 for the two-storey frame. As the key parameters we have chosen maximum horizontal displacement at the top of the first storey and the description of damage caused by the earthquake. Regarding the damage elastic means that all connections remained elastic throughout the earthquake. Small means that rotations less than 0.01 rad occurred in one or more connections and large means that rotations greater than 0.01 rad occurred. Bolt rupture means that in at least one connection bolt at the weaker side of the connection failed in tension. It is interesting that in all numerical simulations frames survived, although some of the seismic loadings were quite severe. Regarding single storey frames and Ulcinj accelerogram in almost all cases frames remained elastic. Only in the case of double mass and scaling factors 2 and 3 the damage is small and large, respectively. Stronger Kobe accelerogram caused large damage at scaling factor 1 and double mass when semi-rigid initial stiffness was considered (frame No. 14). The change to rigid initial stiffness caused bolt rupture in the left cormection (frame No. 12). To see the influence of the connection preloading due to gravity loadings in the last numerical analysis the preloading was removed (frame No. 15) and the damage in the left connection was increased from large to bolt rupture. It is evident that the preloading, which is usually present in normal structures, acts favourably.

196 TABLE 1 NUMERICAL SIMULATION (SINGLE STOREY FRAMES)

Frame No. 1 2 3 4 5 6 7 i 8

Scaling factor 1 1 2 1 1 2 3 1 0.5 1 1.25

Mass

10 11

Accelerogram Ulcinj Ulcinj Ulcinj Ulcinj Ulcinj Ulcinj Ulcinj Kobe Kobe Kobe Kobe

12

Kobe

13 14 15

1 9

m m m 2m 2m 2m 2m m m m m

Initial stiffness R SR SR R SR SR SR R SR SR SR

Max. horizontal displ. (cm) 2.0 2.2 4.3 3.7 4.3 8.4 11.7 7.2 3.8 7.5 9.5

1

2m

R

15.5

Kobe Kobe

0.5 1

2m 2m

SR SR

8.2 15.4

Kobe

1*

2m

SR

15.4

Damage Elastic Elastic Elastic Elastic Elastic Small Large Small Elastic Small Small Bolt rupture, left conn. Small Large Bolt rupture, left conn.

Preloading of connections due to gravity loading was not considered TABLE 2 NUMERICAL SIMULATION (TWO- STOREY FRAMES)

Frame No. 1 2 3

Accelerogram Ulcinj Ulcinj Ulcinj

Scaling factor

4

1.0 2.0 3.0

Initial stiffness R R R

Max. horizontal displ. (cm) 4.2 8.4 12.9

Ulcinj

3.5

R

13.4

5

Kobe

1.0

R

14.1

6

Kobe

1.5

R

18.9

Damage Elastic Small Large Bolt-rupture right conn., L storey Large Bolt rupture, both conn., L storey

The numerical analysis of two-storey frames shows (Table 2) that Ulcinj accelerogram at scaling factor 1 caused no damage, at 2 small and at 3 large damage. The factor of 3.5 was needed to produce holt failure in the right connection of the first storey. Stronger Kobe earthquake needed scaling factor 1 for large damage and 1.5 for bolt rupture in both connections of the first storey. As expected, there is more damage in the case of two-storey frames than in the case of portal frames. The last case where the bolts failed in two connections is very interesting. The moment-rotation relations are plotted in Figure 7, moment-time diagrams in Figure 8 and displacement-time diagram for the first storey in Figure 9.

197 M(kN/m)

M(kN/m) Right conn.

Left conn.

0(Rad) -0.03

PO

-0.02i

0.01

0.02

-0.03

-0.02 \ - 0 . 0 1 1

/0.|)0

0.01

0.02

0.03

OOl

-200 J

-200

Figure 7 : Moment-rotation diagram for the connections in the first floor of the two-storey frame No.6 20

200 1 M(kN/m)

u(cm)

M(leftcorai) t(s)

30

40

50

60

(ri^t cona)

Figure 8 : Moment-time response for the connections in the firs floor (two-storey frame No. 6)

Figure 9 : Horizontal displacement (first floor)-time diagram (two-storey frame No. 6)

^ after bolt failure

'^^^^ Before bolt failure

Figure 10 : Behaviour of the frame after the failure of the bolts in both beam-to-column connections

After the first few strong impacts almost at the same time rupture of bolts occurred in both connections. Afterwards these connections acted alternately as rigid and as pinned when the frame moved in both directions. This behaviour is illustrated in Figure 10 and can be seen also in Figure 8 from the typical moment response.

198 CONCLUSIONS From the test and complementary numerical simulations it is evident that the behaviour of unsymmetric bolted endplate connection under consideration is better than it was expected regarding low energy dissipation and possible failure of the bolts at the weaker part of the connection. In our opinion the most important reasons are as follows: -

with the rupture of the bolts on the weaker side the rotation capacity of the connection is not exhausted. On the contrary, the rotation capacity is increased for rotations in one direction, because the connection behaves as pinned. because of the pinching effect and particularly after the bolt failure the structure becomes more flexible. This results in substantially larger vibration periods and lower seismic loadings.

It is important to note that bolted connection must be designed with adequate overstrength for bending moments causing tension on the stronger side with four bolts and also for shear forces. Namely, the rupture of these bolts would cause an instant failure of the structure. There is also a question of low cycle fatigue which is very important and was not handled into detail. Our tests show that before bolt failure the sensibility to low cycle fatigue is not much different than in other regular connections. After the bolt failure one beam flange is relieved, but some other elements can be stressed more (endplate, remaining bolts). The fi-ames we analysed were low and not loaded very heavily and it is not possible to draw very general conclusions regarding the application of unsymmetric bolted connections in swayfi-ames.But our analysis shows that the behaviour of these connections in simple frames can be satisfactory even in severe seismic conditions.

REFERENCES Berruzzi C, Calado L. and Castiglioni C.A. (1997). Steel Beam-to-Column Joints: Failure Criteria and Accumulative Damage Models. STESSA 97: Behaviour of Steel Structures, Edizioni 10/17, Salermo, Italy, pp. 538-545. CEN (1994). ENV 1998: Design Provisions for Earthquake Resistance of Structures. Zamic R. and Gostic S. (1997). Masonry infilled frames as an effective structural sub-assemblage. Seismic Design Methodologies for the Next Generation of Codes. Proceedings of the International Workshop, Bled, Slovenia, pp. 335-346.

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

199

BEHAVIOUR OF BEAM-TO-COLUMN JOINTS IN MOMENT-RESISTING STEEL FRAMES Claudio BERNUZZI, Carlo A. CASTIGLIONI, Stefano VAJNA de PAVA ^ Department of Structural Engineering, Technical University of Milan, Milan -1.

ABSTRACT This paper presents a study on welded beam-to-column joints in moment-resisting steel frames, which comprises both an experimental and a numerical phase. Main features of two series of joint specimens are summarised, which have been tested to identify key parameters influencing joint response as well as low-cycle fatigue endurance. Moreover, a criterion to predict the type of failure and an approach to appraise the fatigue endurance are presented. Finally, the preliminary results of an on-going finite element study on the stress distribution in the nodal zone are summarised and discussed.

KEYWORDS Earthquake, Steel, Welded Connections, Fatigue, Frames, Ductility, Collapse, Tests, Finite Elements Method.

INTRODUCTION Moment-resisting (MR) steel frames are traditionally designed assuming that the structural system has to provide sufficient strength, ductility and energy dissipation capabilities to resist severe earthquake, even if it is severely damaged (Mazzolani & Piluso, 1996). As pointed out by Zandonini et al. (1997), several researches carried out in the last decades on nodes in moment-resisting (MR) frames permitted to develop a satisfactory knowledge on joint cyclic behaviour (Enghelard & Husain, 1993, Tsai et al., 1995, Plumier, 1994). As a consequence, modem seismic provisions require that dissipation occurs at the beam ends and, eventually, at the base section of the columns. Nodal zones should hence embody sufficient strength and rotational stiffness so as to allow yielding as well as strain hardening in the dissipative zones, without any brittle fracture of the key structural components. Therefore, during Northridge (1994) and Kobe (1995) earthquakes, a lot of MR frames suffered local damages in several beam-to-column joints (Bertero et al., 1997, Tremblay et al, 1995, Kuwamura & Yamamoto, 1997). Unprecedented combined phenomena (i.e., notch effect due to backing bars, lack of preheating for thick flange plates and inadequate workmanship and inspection) appear as key factors responsible of the severe failure modes of joints. Moreover, high strain rate effects, which are associated with ground motion, generated material overstrength and, as a consequence, joint ductility resulted remarkably reduced, in comparison with the one expected on the basis of previous experimental studies. Several studies (Zandonini et al., 1997) were hence carried out to

200

explain both fracture locations and failure modes, which were observed during the aforementioned earthquakes. A research project is currently in progress between some European Universities (Athens, Liege, Lisbon, Milan and Trento), with the aim of defining suitable criteria for the seismic design of steel frames with rigid and semi-rigid joints. This paper is focused on the behaviour of welded beam-to-column joints in MR steel frames. The experimental phase of the study, which comprised two series of cyclic tests, is summarised and key features of joint response are presented. A suitable definition of a threshold value of displacement is proposed to predict the type of failure (i.e., brittle, ductile or mixed failure). Moreover, an approach to assess fatigue endurance life is discussed, which depends on the threshold value, too. Finally, the preliminary results of an on-going finite element study on the nodal zones are outlined, mainly with reference to the local strains in the vicinity of the weld toes, where unexpected brittle fractures could occur, as observed during tests as well as recent earthquakes.

EXPERIMENTAL PROGRAMME A muhi-specimen testing programme (Ballio et al., 1997, Bemuzzi et al., 1997, Ballio & Castiglioni, 1994, Plumier et al., 1998, Calado et al., 1998) on several types of rigid joints, carried out at the Universities of Lisbon and Milan, is herein shortly presented. Table 1 shows the general layout of all the considered tests on rigid joints (fig. 1). In particular, details are reported about the loading history, the total number of cycles performed during the tests, Ntot, and the number of cycles to conventional failure, N, which has been evaluated in accordance with the approach presented in the following. Moreover, the type of failure is reported, too. As a general remark, it should be mentioned that the spread of plasticity was observed only in the nodal zone, while beam and column remained in the elastic range in the otlier parts of the specimen. As a consequence, joint behaviour is herein presented with reference to the global response of the specimen, i.e., by considering the relationship between the force applied at the beam free end, F, and the associated displacement, v.

iMi

yifh

-|pM-f

h~

BCC6

1 Figure 1: Test specimen details

i m

201 TABLE 1 LISBON AND MILAN JOINT TEST CHARACTERISTICS

1 JOINT 11

BCC5

BCC6

CI

C2

C4

SPECIMEN J1 LOADING HISTORY | A Sat 100 mm B Sat 150 mm S at 76 mm D M E A Sat 100 mm Sat 150 mm B S at 76 mm D E M A30 S at 60 mm S at 60 mm C30 B50 Sat 100 mm Sat 100 mm C50A C50B Sat 100 mm D50 Sat 100 mm B75 Sat 150 mm S at 200 mim BlOO S at 60 mm B30 Sat 100 mm B50 B75 Sat 150 mm Sat 150 mm C75 S at 200 mm BlOO S at 250 mm B125 S at 60 mm B30 B50A Sat 100 mm B50B Sat 100 mm B75 Sat 150 mm S at 200 mm BlOO S at 250 mm B125 S at 250 mm C125

LOADING HISTORY: S=symmetrical constant amplitude M=monotonic

20 8 27

N 15 4 21

19 15 22

14 10 16

D D B

16 16 19

8 12 17

51 1 20 14 82 36 4 33 15 8 52 8 10 28 15 12 7

48 1 18 12 75 36 3 32 14 7 48 6 8 26 13 8 7

B2 B2 B2 B2 D Bl M D B2 Bl Bl M D D Bl Bl Bl M D D

Ntot

FAILURE MODE | B B B

D

1

FAILURE MODE: D=ductile Bl=brittle with crack formed at the centre of the weld, between beam and cohimn flanges B2=brittle with crack formed at the edges centre of the weld, between beam and column flanges M=mixed

The Lisbon tests A series of 29 tests was executed on 5 different types of beam-to-column joint specimens, with the aim of investigating main parameters affecting cyclic response of the rigid as well as semi-rigid joints (Calado & Castiglioni, 1996, Bernuzzi et al., 1997). The tested specimens were representative of a node to external columns (i.e., a stub beam attached to a column by means of the connection details). For each typology of joint, several specimens were realised and tested, under both constant and variable

202

amplitude loading histories. As to rigid joints, two different series of welded joint specimens were tested: BCC5 and BCC6 (in fig. la) and lb), respectively), differing in the type of profile (i.e., HEB160 and HEB200, respectively) used for the columns. Rigid joint with the weaker column (BCC5) exhibited a very stable behaviour, in terms of F-v hysteresis loops (fig. 2), with a very limited deterioration of stiffness, strength and energy absorption capabiUties. Collapse was due to the fracture of the beam flange in the vicinity of the welded connections. Also in case of joint with stronger column (BCC6), the behaviour resulted very stable. The column profile influenced remarkably the failure mode. A plastic hinge formed in the zone of beam flanges interested by local buckling and failure was associated with cracking of these ones in correspondence of the weld with the column. A comparison between the responses of the two type of specimens shows no remarkable differences in terms of hysteric behaviour, as it can clearly seen from fig. 3, which presents two cycles selected at different level of displacement, while the column type affects moderately the peak values of the force. 250

300

1 50

-5 0

1 50

-250

100

15 0

Figure 2 : Hysteresis loops for BCC5A and BCC5E joint specimens

Figure 3: Hysteresis loops for BCC5 and BCC6 joint specimens

The Milan tests In the past, several experimental studies were carried out at the Technical University of Milan on the cyclic behaviour of both beams and beam-columns in order to identify key parameters affecting member response (BaUio & Castiglioni, 1994). Recently, attention has been paid also to beam-tocolumn joints for steel and steel-concrete composite frames. As far as joint tests in MR frames are concerned, the tested specimens, very similar to the ones tested in Lisbon, consisted in a beam (IPE450A profile) attached to a column (HE300B) by means of rigid welded connections, with butt welds connecting beam flanges to the column flange. Tliree different welding procedures (Plumier et al., 1998, Calado et al., 1998) which represent solutions commonly used to weld beam-to-column joints in accordance with European and North American steel construction practice, were considered. In particular, the following types of connections were tested (fig. Ic): • connection CI, typically adopted in the US practice. All welds are executed in a flat position with single bevel V groove with a steel backing bar underneath, usually not removed after welding; • connection C2, similar to the CI one but with a copper backing bar easily removable after welding. One additional weld pass is executed from underneath to melt down eventual defects and flows and to remove the eventual presence of weld slag almost unavoidable in the CI specimens; • connection C4, realised by means of butt welds, with K grooves. It normally corresponds to shop

203

welding, and is typical for the Italian (and in general European) steel constructional practice. It has been decided to analyse the behaviour of some specimens with column stiffeners, in order to investigate the influence of the stiffness of the panel zone on both resistance and ductility of the connection. Moreover, some specimens were modified to assess also the influence of the web access hole in the beam web in order to have a continuous weld along the whole (lower) flange and to allow continuous backing bar to be positioned also on the upper flange. With reference to table 1, indexes A e B are related to specimens with the web access holes and with or without column stiffeners, respectively. Specimens without the web access holes are labelled as C, if with column stiffeners, and D, if without column stiffeners. The label of each specimen is composed by a first letter (A, B, C or D) indicating the geometry of the welded node, as previously mentioned, the value of half displacement range AV under which constant amplitude test has been executed and, eventually, a letter (A or B) related to the number of test replications (under the same displacement amplitude). Five constant amplitude loading histories, each of them characterised by a different value of the imposed displacement (Tab. 1), have been selected and considered for the experimental analysis. As a general remark, it can be said that the behaviour of these types of welded joints results generally characterised by stable hysteresis loops and by a satisfactory energy dissipation capabilities. Moreover, independently on the welding procedure, three types of collapse were observed. F(kN)

400

400

200

200

-200

-200

C 4 B 50 ^^-1 0 0 - 5 0

0

^ 50

^^^ -40 0 1 00

Figure 4: Hysteresis loops for C4B50 joint specimen

-400

-100

-50

50

100

Figure 5: Hysteresis loops for C4B100 joint specimen

A re-analysis of test results shows that the collapse modes occurred for different values of the ductility range Av/Vy, defined as the ratio between the imposed displacement range (Av) and the conventional yield displacement (v^), evaluated in accordance to the ECCS procedure (ECCS, 1986). In particular, the observed collapse modes are: • brittle failure mode, which was usually achieved under cycles corresponding to small values of Av/Vy. Specimen response (see, as an example, fig. 4) was not affected by remarkable deterioration of strength, stiffness and for energy absorption capability. Collapse was sudden, without warning signs, owing to a crack either at the weld toes or in the base material; • ductile failure mode, generally associated with a large ductility range Av/v . Joint performance was

204

significantly affected (fig. 5 can be considered an example) by a remarkable and progressive deterioration of the key behavioural parameters, with the formation of a plastic hinge associated with local buckling of the beam flanges. Collapse was caused by the gradual propagation of a crack initiating at surface striations forming, due to attainment of the maximum tensile strain of the material, at the buckles at the plastic hinge location. mixed failure mode, which appears as a combination of the two previously described failure modes. It is characterised by a progressive deterioration of the key behavioural parameters, such as stiffness, strength and dissipated energy. This failure mode was usually associated with both plastic hinge formation and local buckling phenomena in the beam flange. However, collapse was generally due to a crack at the weld toes. Furthermore, in case of brittle collapse, two different failure modes were observed, conventionally identified as Bl and B2. The first one, which was due to a crack formed in the centre of the weld between the beam flange and the column and propagated toward the edges, was observed in all the type of considered rigid joints. Otherwise, failure mode B2, observed only in some CI and C2 specimens, was caused by through cracks starting at the beam flange edges and propagating toward flange centre. It must be noticed that the previous definitions of "brittle" or "ductile" failure modes are not correct, in terms of fracture mechanics or metallurgy. However, it is intention of the authors to give an "engineering" definition associated with the "global" behaviour of the joints, and not with "local" parameters, such as those usually adopted in the metallurgic and/or fracture mechanics approach. As far as these ones are considered, all the observed fracture modes should be considered as ductile, as large "local" plastic deformations of the material microstructures occurred.

THE THRESHOLD VALUE OF THE DISPLACEMENT Re-analysis of the constant amplitude test data confirms that the previously mentioned types of failure mode depend on the imposed displacement. For each set of similar specimens, joint responses were compared also in terms of F-v envelopes, in order to single out differences related to the observed failure modes (Castiglioni et al., 1997). As a result, a suitable threshold displacement, Av-^,^, was identified. It has been defined as the displacement associated with the maximum strength obtained from a monotonic test, suitably amplified in order to take into account the cycling loading history. Moreover, a criterion to predict the type of failure and an approach to assess fatigue endurance of the tested components have been developed, both strictly depending on the value of Avy,^. In absv^nce of experimental data, an approach to estimate the value of Av^,^ by means of a simple equatio has been developed and validated, with reference to the aforementioned joint tests as well as to other «.omponents tested under cyclic loading. In particular, it has been assumed that Av^,^ depends on the following parameters: • the conventional elastic displacement, v^, which in the following has been identified in accordance with the ECCS procedure (ECCS, 1986); • the beam web slenderness ratio, X^=d/\^, defined as the ratio between the depth of the profile, d, and the web thickness, t^; • the beam flange slenderness rafio, Xf=c/t^, where c represents half width of the flange and t^- is its thickness; • the weld quality and/or the severity of the detail, globally accounted for by the introduction of the numerical coefficient r|. This term ranges from r|=0.5 for poor quality welds, to r|=1.0 for good quality (or no) welds.

205 The threshold value, Avy,^, has been hence defined as:

As to the non-dimensional coefficient y, on the basis of the available experimental data, a value of 2000 ± 15% (i.e., in the range 1700 - 2300) was suggested, independently on the considered component. The value of Av^,^ related to each tested specimen has been directly obtained also from the envelopes of the cyclic response. The scatter between these values for each set of similar specimens and the ones obtained via eq. (1) is quite limited, confirming the validity of the proposed approach. In this paper, the threshold value is presented with reference to the displacement, which represents the parameter governing the considered tests. However, it should be noted that the threshold value can be proposed in term of generalised displacement, i.e., if the control parameter is the rotation, the associated value of the threshold rotation can be obtained from eq. (1) by substituting Vy with the conventional yielding rotation Oy. Prediction of the type of collapse As previously mentioned, the value of the threshold displacement, Av^j^, can be used to identify the type of failure mode. In case of constant amplitude loading history, on the basis of the considered displacement range Av, the following collapse modes are expected: • brittle failure mode, if Av is lower than Av^,^^, defined as 0.85 Av^,^; • ductile failure mode, if Av is greater than AvTh"^, defined as 1.15 AVTH; • mixed failure mode, if Av falls in the range [Av^,^^ ^ ^^Th^ ]• It should be noted that the approach has been proposed with reference to constant amplitude tests. However, some preliminary results indicate its validity also in case of variable amplitude tests for which term Avhas been considered as an equivalent displacement range (Calado et al., 1998). Failure prediction Calado and Castiglioni proposed a criterion to assess the fatigue endurance of steel components (Calado & Castiglioni, 1996, Castiglioni & Calado, 1996) under cyclic loading. In case of constant amplitude tests, failure is achieved in correspondence of the first cycle that satisfies the following condition: ^
(2)

F

where E^.^- is the absorbed energy at the considered cycle (i.e., the cycle during which failure occurs), E^Q represents the absorbed energy at the first cycle in the plastic range, and a^^ is a constant value, for which the value of 0.5 was recommended. Ballio et al. (1997) already proposed for the term a^- the value of 0.5, which was conservative only for ductile fracture phenomena; hence a new value of a^ was defined on the basis of the cycle displacement amplitude Av by the same authors. In the present study, a more refined criterion is proposed, which takes into account the failure mode by

206 considering the dependency of a^ from Av/Av^,^. Figure 6 shows the value of a^, (i.e. the value of the ratio EU/EQ, with reference to Eu, the dissipated energy determined at the complete failure of the specimen) plotted versus, Av/Av^,^. An appraisal of the fatigue life can be obtained in the region where the experimental values of a^, are non contained. For the sake of simplicity, the value of at governing eq. (2) which bounds this "safe" domain can be defined by the following simplified multi-linear relationship:

af=1.65-(Av/Av^,J af=0.5

ifAv/Av^,^<0. if0.85 1.15

(3)

It can be noted that for values of Av/Av^^,^ greater than 1.15 (i.e. for Av>Av^,^^) a ductile failure mode is expected, and a constant value of aj=0.5 is recommended, in accordance with the original Calado and Castiglioni proposal. If Av/Av-^,^ falls in the range [0.85-1.15], i.e., Av e [Av^,^^ , Av^,^^]^ where a mixed failure mode is expected, a strong dependence of a^- on the cycle amplitude Av is accounted for by means of a slope equal to unity. For Av/Av-^^,^ lower than 0.85, i.e., Av < Avy,^^^ a brittle failure mode is expected, and a^- is assumed moderately decreasing with the increasing of Av.

Av-Av^,^

Av=Av^th

Figure 6: Criterion to predict the type of failure

NUMERICAL ANALYSIS The influence of the key joint components can be singled out from experimental analyses, wliich result nevertheless a costly approach and are however essential to establish the fundamental background for the validation of ail theoretical approaches, but have, by their very nature, a limited scope. Hence, the necessary parametric investigations appear possible only by means of finite element (FE) simulations, which enable for an exhaustive understanding of both global joint behaviour and local transfer force

207

mechanisms between frame members through joints, over a sufficiently extensive range of both mechanical and geometrical parameters. In the framework of the current study, an extensive numerical FE analysis has been carried out on the tested joints. The main scope is to analyse local strains near the weld toes, where unexpected brittle fractures were observed after recent earthquakes as well as during the tests previously described. The first results of some numerical simulations, which were executed using the FE non-linear code ABAQUS, (1996) are in the following briefly outlined. A three-dimensional (3D) model of the complete specimen (specimen model) was at first developed, which encompassed of 1465 nodes and of 1554 shell elements (fig. 7a). In the nodal zone, i.e., in the part of beam and column in the vicinity of the joint, the mesh was refined to allow a more accurate appraisal of the transfer force mechanisms. In order to simulate local buckling effects, an initial out of straightness was considered by means of geometrical imperfections, which were taken into account via a linear combination of the first two buckling modes with a sinusoidal shape characterised by an amplitude of 0.02mm (i.e, approximately L/10000, where L is the beam length in the model). The FE model simulated the complete tested specimens. As a consequence, the load was applied normally to the beam at its free edge; only the transversal displacements at the top of the beam were restrained, consistently with the actual testing conditions. Moreover, to obtain more accurate information about the local behaviour in the nodal zone, reference was made to the substructuring technique, and a second 3D model (fig. 7b) was considered to simulate the response of a part of the nodal zone (node model), by means of 1200 nodal points and 1150 shell elements. Two models of the node were considered, respectively with and without the web access hole. As to the constitutive law of the materials, on the basis of the tensile coupon tests, both a bilinear model with strain hardening and a Ramberg-Osgood model (Ramberg & Osgood, 1943), which was approximated by means of a multi-linear relationship, were considered. These uni-axial stress-strain relationships, for which a kinematic hardening was considered, have been correlated to the more complex and representative state of loading via the von Mises yielding criterion with associated plastic flow.

a)

b) Figure 7: Specimen (a) and node (b) F.E. model

The accuracy of the specimen models was evaluated on the basis of the experimental data, by comparing not only the shape of the hysteresis loops, but also the absorbed energy (both per cycle and cumulated) and the peak values of the force in each cycle as well as the stiffness and strength degradation. An agreement more than satisfactory between numerical and experimental data was observed for all the considered numerical cases. Moreover, an accurate analysis of the local behaviour in term of stresses and strains has been carried

208

out on the models of the node (fig.7b). Five different cyclic loading conditions were considered, four constant amplitude and one variable amplitude loading history. Numerical simulation of nodal zone pointed out a strong dependency of the trend of the principal strains on the loading history. The part of the beam in the vicinity of the welds results subjected to a severe state of stress, when small displacement cycles (i.e., with Av < AVTH) are imposed to the specimen. In these cases, the amplitude of the strains near the welds increases remarkably during the loading phase with a moderate decreasing during the unloading phase. The final results is that the strains however increase «in mean» with the number of the cycle. As an example, in case of a cycle amplitude Av = ±50mm, the strains assume approximately a value up to 0.3 in the first cycles, independently on the constitutive law adopted for the material and, in correspondence of the 5^^-6^^ cycle reach the maximum tensile strain of the material. It should be noticed that the strains in the beam flange, at the expected plastic hinge location, are negligible. No local buckling effects are evident and there is a shakedown phenomenon, i.e., the stresses in the beam flanges are in the elastic range (Fig. 8). On the contrary, when the imposed displacements are greater than the value of Avjh, the principal strains near the weld decrease after the first cycles and the associate stresses are contained in elastic range, so that the weld zone seems not to be interested by the phenomenon of crack propagation. At the same time, however, the strains in the beam flange increase with the cycle number (fig. 9) and cause the formation of a plastic hinge in the beam, accompanied by evident local buckling of both the web and the flanges (fig. 10) up to the complete failure of the joint, in accordance with experimental features. In case of small amplitude cycles, the presence of the web access hole affected remarkably the values of the local strains in the weld area, which result greater than those associated with the model without access holes. Strain values indicate an earlier collapse due to weld failure or cracking of the base material in the weld area. On the contrary, if large amplitude cycles are considered, when collapse is achieved at the plastic hinge location, the presence of the web access hole does not influence the failure conditions. O.Si

Av=± 50mm

10

A\F:±100mm

N

15

Figure 8-.Strain trend vs cycle number for cycle amplitude of Av=±50 mm

10

N

15

Figure 9: Strain trend vs cycle number for cycle amplitude of Av=± 100 mm

209 Despite the fact that numerical analyses are currently on going, it should be remarked that these preliminary results show a very good qualitative agreement with the experimental study. In particular, it can be noted that failure modes observed in tests can be associated with different limit conditions of material. If AV>AVTh^, numerical results seem to indicate a local state of strain causing collapse for lowcycle fatigue. In case of AV
(IVRa)

• C1-a2« C1-D 1 .gci-aii]C2-si

i maM:> nCXBI 0 CM) i^OlM A C2M A C^m 1001 1

Figure 10: Defomied shape of the joint at collapse

,.

^_ 10

N

TO

Figure 11: Fatigue resistance lines for tested joints

CONCLUDING REMARKS A study on the behaviour on joints in moment-resisting steel frames, which comprises both experimental and numerical phases, is briefly presented in the paper. As to the experimental part of the research, which was carried out on 3 different types of rigid joints for a total of 31 tests, it has been pointed out that different failure modes can occur on the basis on the applied loading history. With reference to constant amplitude loading history, failure modes have been correlated to the cycle amplitude Av by a suitable threshold value of the displacement range Av^,^. As a consequence, ductile, brittle or mixed failure mode can be predicted with a satisfactory degree of accuracy. It should be noted that, by using the unified approach proposed by Ballio and Castiglioni (1995) to assess low-cycle fatigue resistance of steel components, different resistance lines are associated with the identified failure modes. It can also be noted that the connection detail has a limited influence on the fatigue life. As an example, figure 11 plots in the logAa*-logN scale the experimental points related to failure and the fatigue resistance lines defined on the basis of the experimental data. It should be noted that the limit lines related to the ductile failure mode (Cl-D, C2-D and C4-D) are the highest, while the ones associated with brittle failure (Cl-Bl, C1-B2, C2-B1 and C4-B1) are the lowest. The numerical phase of the study, which encompasses numerical finite element analysis of the nodal zone, is currently in progress. First results indicate that the local strains in the vicinity of the node strictly depend on the loading history and, as a consequence, different failure modes can be expected. Moreover, numerical results indicate that, owing to low-cycle fatigue or ratcheting of the base material, the presence of the web access hole leads to earlier collapse in the weld area in the case of small amplitude cycles while, in the case of large amplitude cycles, does not influence the failure conditions.

210 A ckno wledgements The research was supported by grants from the ItaUan Ministry of the University and Scientific and Technological Research (M.U.R.S.T.), cofinanziamento 1997. References Ballio G., Calado L., Castiglioni C. A., 1997, Low cycle fatigue behaviour of structural steel members and connections. Fatigue & Fracture of Engineering Materials & Structures, 20: 8 Ballio G., Castiglioni C.A., 1994, Seismic Behaviour of Steel Sections, Journal of Constructional Steel Research, 29, 21-54. Ballio G., Castiglioni C.A., 1995, A unified approach for the design of steel structures under low and high cycle fatigue, J. of Constructional Steel Research, 34 Bernuzzi C, Calado L. Castiglioni C.A., 1997, Experimental Research on Cyclic Behaviour of Steel Beam-to-Column Connections, 5th Int. Colloquium on Stability and Ductility of Steel Structures, Nagoya, Japan. Bertero V., Anderson J.C, Krawinkler H, 1997, Performance of Steel Building Structures during the Northridge Earthquake, EERc, University of California, Berkeley. Calado L., Castiglioni C.A., 1996, Steel Beam-to-Column Connections Under Low-Cycle Fatigue Experimental and Numerical Research, Proc. of the XI World Conf. on Earthquake Engineering, Acapulco Calado, L., Castiglioni, C. A., Barbaglia, P., Bernuzzi, C, 1998, Seismic design criteria based on cumulative damage concepts, Proc. of 11th European Conf. on Earthquake Engineering, Paris. Calado, L., Castiglioni, C. A., Barbaglia, P., Bernuzzi, C, 1998, Procedure for the assessment of lowcycle fatigue resistance for steel connections, Liege Conf. on Control of the Semi-Rigid Behaviour of Civil Engineering Structural Connections, Liege. Castiglioni C. A., Calado L., 1996, Seismic damage assessment and failure criteria for steel members and connections. Int. Conf. on Advances in Steel Structures, Hong Kong. Castiglioni C.A., Bernuzzi C. Calado L., Agatino M.R., 1997, Experimental study on steel beam-tocolumns joints under cyclic reversal loading SAC-CUREe, USA Enghelard M.D., Husain A.S., 1993, Cyclic-Loading Performance of Welded Flange-Bolted Web Connections, J. of Struct. Eng. ASCE, 119:12, 3537-3549. European Convention for Constructional Steelworks (ECCS) 1986, Recommended testing procedure for assessing the behaviour of structural elements under cyclic loads. Technical Committee 1, TWG 1.3 - Seismic Design, ECCS Publication No. 45. Hibbitt, Karlsson & Sorensen Inc., 1996, "ABAQUS User's Manual" Version 5.5 Kuwamura H., Yamamoto K., 1997, Ductile Crack as Trigger of Brittle Fracture in Steel, J. of Struct. Eng. ASCE, 123:6, 729-735. Mazzolani P.M., Piluso V., 1996, Theory and Design of Seismic Resistant Steel Frames, Chapmann and Hall, E&FN Spon. Plumier A., 1994 Behaviour of Connections, Journal of Constructional Steel Research, 29 95-119. Plumier, A., Agatino, M. R., Castellani, A., Castiglioni, C. A., Chesi, C, 1998, Resistance of steel connections to low-cycle fatigue, Proc. of 11th European Conf. on Earthquake Engineering, Paris. Ramberg W., Osgood, W., 1943 Description of stress-strain curves by three parameters, Technical Note n^ 902, NACA Tremblay R., Timler P., Bruneau M., Filiatrault A., 1995, Performance of Steel Structures during the Northridge Earthquake, Canadian Journal of Civil Engineering, 2,4,Vol.22. Tsai K.C., Shun W., Popov E., 1995, Experimental Performance of Seismic Steel Beam-Column Moment Joints, J. of Struct. Eng. ASCE, 121:6, 925-931. Zandonini R., Bernuzzi C, Bursi O., 1997, Steel and Steel-Concrete Composite Joints Subjected to Seismic Actions, General report of the STESSA'97 Conference, Kyoto-Japan.

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

211

CYCLIC BEHAVIOUR OF STEEL BEAM-TO-COLUMN CONNECTIONS: INTERPRETATION OF EXPERIMENTAL RESULTS L. Calado ^ G. De Matteis ^ R. Landolfo ^ F.M. Mazzolani ' ^ Dept. of Civil Engineering Institute Superior Tecnico of Lisbon, PORTUGAL ^ Dept. of Structural Analysis and Design University of Naples Federico II, Naples, ITALY ^ D.S.S.A.R., University of Chieti G. D'Annunzio, ITALY

ABSTRACT The present paper is focused on the structural behaviour of beam-to-column steel joints. In particular, full-scale bolted cleat angle connections subjected to cyclic flexural moment loading are considered. Aiming at assessing the influence of column size on the global response of the joint, experimental tests have been preliminarily carried out on different joint configurations. Both fatigue constant amplitude deformation tests and variable amplitude deformation tests are performed. Obtained results show that the global joint behaviour is mainly influenced by connecting elements, independently from the applied deformation history. Experimental outputs are then used for the calibration of an available cyclic model, which shows its capability to correctly interpret the response of the joint. Comparison between experimental and numerical results, in terms of both energy dissipation and moment capacity degradation all over the deformation history, are finally provided.

KEYWORDS Beam-to-column joints. Semi-rigid connections. Experimental tests. Low-cycle fatigue. Cyclic model, Energy dissipation. Strength degradation.

INTRODUCTION Since many years, the international scientific community is aware of the remarkable influence of beamto-column connection behaviour on the global seismic response of steel structures (Mazzolani & Piluso, 1996). It has been, in fact, reckoned that the accurate identification of the connection features overlooks the correct assessment of collapsing state of moment resisting steel frames, where beam-tocolumn joints are often regarded as the major dissipative source of the structure, suffering extended damages. In particular, connection ductility supply represents the key parameter to be detected and

212 correctly made use of in performing accurate seismic analyses and design (Krawinkler, 1997; Faella et al, 1993). Notwithstanding, under repeated actions, connection rotation capacity is strongly affected by damage phenomena, which are related to low-cycle fatigue. The above remarks demand for a faithful prediction of the joint behaviour by setting up reliable models, which account for the strength and stiffness degradation arising as the number of plastic excursions increases. In particular, the problem to assess the behaviour of semi-rigid joints subjected to cyclic loading seems to be still far to be completely solved and require further research efforts. Several Authors have been already studied through experimental analyses the cyclic behaviour of semi-rigid beam-to-column angle connections. For instance, Ballio et al (1987), among several joint typologies, examined bolted web and flange cleat angle connections with different stiffening details, subjected to a loading history according to the testing procedure provided by ECCS Recommendations (1986). Astaneh et al. (1989) investigated the response of double angle bolted connections usually adopted for steel frames, by considering several angle, bolt and column types with reference to an increasing level of cyclic deformation up to the failure of specimens. Calado & Ferreira (1994) conducted a wide experimental research on the influence of several connection details (i.e. number of bolts, thickness of the angle), by adopting as deformation history only the testing procedure suggested by the ECCS (1986). Madner et al (1994) analysed top-and-seat angle connections under several constant amplitude deformation histories, using both symmetric and asymmetric displacement conditions. Bemuzzi et al (1996) investigated a different geometry of top-and-seat angle connection specimen, by considering four different loading histories with different sequences of increasing deformation amplitude. Calado et al (1997) tested several joint typologies where, among other, a bolted web and flanges with cleat connection specimen was also included. A multi-specimen testing program, consisting of different cyclic loading histories with both constant and variable amplitudes, was considered. In the framing of the Copernicus European project RECOS 'Reliability of moment resistant connections of steel building frames in seismic areas' (Mazzolani, 1999), a co-operation between University of Naples (Italy) and Instituto Superior Tecnico of Lisbon (Portugal) has been stated. The activity developed at the Instituto Superior Tecnico of Lisbon is mainly addressed at the experimental behaviour of beam-to-column connections usually adopted in steel frames. Conversely, the activity developed at University of Naples is focused on the development of suitable analytical models, able to well interpret the behaviour of beam-to-colimm joints as it is experimentally observed. In particular, such a model account for all main features of connections, such as slip phenomena, stiffness and resistance degradation, softening behaviour due to instability occurrences. In this paper, experimental test results are firstly presented. They are dealing with steel bolted beam-tocolumn connections, where angle cleats are used for both flange and web connection. Details of the joint are different from the ones above mentioned with reference to previous experimental research programs. Besides, different joints configuration, where column size is considered as the main parameter and several flexural deformation histories are considered. Then, obtained results are used for the calibration of the above analytical model, showing an accurate degree of approximation in both the prediction of energy dissipation capacity and strength degradation due to repeated actions. The final aim of the research is to investigate, by means of suitable analytical models properly interpreting the hysteretic behaviour of beam-to-column joints, on the non-linear dynamic response of moment resisting (rigid and semi-rigid) steel frames.

EXPERIMENTAL PROGRAM AND TESTING PROCEDURE Full scale experimental tests on bolted beam-to-column joints, where both web and flange cleat angles are used as connection elements, have been carried out at Laboratory of Material and Structures of

213 Institute Superior Tecnico de Lisbon. Both monotonic and cyclic tests have been performed. Several deformation histories have been accounted for, aiming at studying the influence of the column size on the cyclic behaviour of the joint as well as the effect of amplitude deformation on plastic fatigue behaviour. The testing apparatus is the same previously described by Calado & Ferreira (1994), Specimen geometry is depicted in Figure 1. Forces (F) are applied through an actuator acting at the top of the beam, at a distance h to the external upper flange of the column where Figure 1: General specimen lay-out horizontal displacements (v) are measured too. Several electrical transducers are also used to monitor displacements of the colunm supporting plates, vertical displacements of the beam, deformation of the panel zone and beam-to-colimm relative rotations. Tested specimens consist of a beam made of IPE300 section attached to several column sizes (HEB160, HEB200 and HEB240) by bolted web and flanges with cleats. As connecting angles, L section 120x120x10 has been considered. As far as material 1 ? ^ ^ L I20xi20xt0 characteristics are concerned, beam, colxunns and angles are L 180x120x10 in steel grade S 235, while 8.8-M 16 bolts, pre-tensioned by a pre-loading equal to 88 kN, have been made use of. Several specimens were fabricated and tested according to a multi-specimen testing program, consisting of different cyclic loading histories with both constant and variable deformation amplitudes. In Figure 2 details of all considered joint typologies are shown. [BCC91

As far as loadmg history is concerned, in order to follow the cyclic behaviour of the specimen up to the complete failure with increasing amplitude deformations, the so called Complete Testing Procedure provided by ECCS Recommendations (1986) has been fu^tly applied. In addition, since the testing program is mainly focused at the determination of beam-to-column joint response under arbitrary loading histories, as suggested by ATC Guidelines (1992), three different inelastic constant amplitude deformation levels have been also considered. This allows damage phenomena due to cumulated plastic excursions to be properly evaluated. In particular, displacement amplitude (Vmax) equal to ± 37.5 mm, ± 50.0 mm and ± 75.0 mm have been considered. In conclusion, the following cyclic sequence has been executed: a) for all the specimens, four cycles in elastic range, with a semi-amplitude equal to: V*, Vz,

IPE300 I igOxigQxtO

^40

Ir

L 120x180x10

L 180x180x10

Figure 2: Tested specimens

214 VA, 1 Vy, where Vy is the elastic displacement which has been conventionally and preventively assumed equal to 10mm; then: (b) in case of constant amplitude tests, repeated cycles at Vmax, until the complete failure of the specimen; (c) in case of ECCS procedure, three cycles at each amplitude i-Vy (with 7=2,....) up to the complete failure of the specimen. In Table 1, the adopted nomenclature is reported for each analysed specimen. In particular, size of the column, distance h relative to the point of application of loading actuator and displacement history type are reported. TABLE 1 ADOPTED NOMENCLATURE FOR TESTED SPECIMENS

Diplacement history Vmax =± 37.5 mm ^max =± 50.0 mm Vmax =± 75.0 mm ECCS * Beam Section IPE 300

Column section* (Actuator position h Fmml) HEB240 HEB200 BCClO-B (/;=765) BCC7-D (h=7S0) BCC10-A(/i=765) BCC7-A (/z=780) BCC7-B (/7=780) BCC10-D(/7=765) BCC10-C(/i=760) BCC7-C (/z=770)

HEB160 BCC9-C (/i=793) BCC9-A ih=192) BCC9-B (h=190) BCC9-D (h=793)

THE OBTAINED EXPERIMENTAL RESULTS Test results have been represented in terms of moment-rotation curves (M-cj)). Conventionally, flexural moments have been evaluated respect to the column axis (therefore M = F'(h + dy2), being dc the column depth). Similarly, rotations have been evaluated as <|) = v/(h + dy2), thus representing a sort of rotation equivalent to the measured displacement, the latter incorporating both relative beam-to-column rotation due to connection deformation and elastic deformations of beam and column members. In Figure 3, M-<j) curves are depicted. In case of specimen typology BCC7, the one with intermediate column size, results are shown for all considered loading histories, while for other specimen typologies only tests relative to ECCS procedure and the one with constant amplitude Vfftax =± 30.0 mm are reported herein. Other results can be found in D'Avino (1999). In all cases, the response of the specimen is characterised by large slippage, emphasising the pinching type behaviour, which is typical for this kind of connections. Then, it is to be noticed that the qiialitative behaviour of the specimens seems to be only slightly influenced by the column size, even if specimen BCC9, corresponding to the smallest column section, present lower resistance values for both ECCS and constant amplitude deformation histories. Nevertheless, in all cases the connecting elements always represent the weakest component of the joint (Figure 4), completely ruling the whole response of the joint at collapse. In fact, failure is due to cleat in tension, which, generally, exhibits cracks at the internal edge of leg root. First cracks appear after few plastic cycles, quickly propagating as far as the number of cycles increases up to the complete detachment of the specimen (Figure 5). Contemporary, bolt holes of flange and web cleats suffer large ovalizations. The above phenomena induce strength and energy dissipation degradation, which is growing up with the number of plastic excursions. Besides, it is important to observe that, actually, first plastic deformation appears for displacement values v equal to about 5 mm rather than for v=Vy=10 mm, as conventionally assimied in defining deformation history to be applied. Obviously, the loading history strongly conditions the number of cycles causing the collapse of the specimen (Nj). In Table 2, Nf is reported for each tested specimen, where collapse has been intended as the complete rupture of the connection. Results confirm the similarity among specimens with different column member, as well as the remarkable influence of the deformation amplitude on the low-cycle

215 fatigue failure.

150

150

M (kNm)

M (kNm)

BCC7-D

100

100

50

50

0

0

-50

-50

ly

-100

'

BCC7-A

^^

-100

^ (mrad) -150

(j> (mrad) -150

-100

-80

-60

-40

-20

0

20

40

60

80

100

-100

-80

-60

-40

-20

0

20

40

60

80

100

150

M (kNm)

M (kNm)

BCC7-B

100

BCC7-C

100

50

-

50

0

/»

'

'

nil u)

'

'

_L^-^—"^i^s^v '

^^^;3/]l

0

-50 -50

-100

nn^^^'

-100

-150

«|)(mfBd)


-100

-80

-60

-40

-20

0

20

40

60

80

100

-100

-80

-60

-40

-20

0

20

40

60

80

100

150

M (kNm)

M (kNm)

BCC9-A

100

BCCIO-A

100

-

50

0

'

M

'

50

0

' /ni %v__^2—

1^^^^^-^"^^^

-50

_ '

luX-

-50

-100

-100

^ (mrad)

^ (mrad) -150

-100

-80

-60

-40

-20

0

20

40

60

80

100

-100

-80

-60

-40

-20

0

20

40

60

80

100

150

150

M (kNm)

M (kNm)

BCC9-D

BCCIO-C 1

100

100

-

50

50

0

0

-50

-50

i

1

-100

-100

<|) (mrad)

(|) (mrad)

-150 -100

-80

-60

-40

-20

0

20

40

60

80

100

-100

-80

-60

-40

-20

Figure 3: Obtained moment-rotation curves (M-<())

0

20

40

60

80

100

216

Figure 4: Typical observed behaviour

Figure 5: Typical collapse mode TABLE 2

NUMBER OF CYCLES TO FAILURE (N/)

Column section HEB240 HEB200 HEB 160

Vmax =± 37.5 mm 22 22 21

Displacement history Vmax =± 50.0 mm Vmax =± 75.0 mm 13 7 13 8 13 8

ECCS 19 20 19

THE CALIBRATION OF THE ANALYTICAL MODEL Brief description of the model The cyclic response of steel beam-to-column joints evidences several behavioural aspects, hinting at a very difficult modelling of the corresponding moment-rotation relationship. A general cyclic model interpreting the hysteretic behaviour of steel joints has been previously presented (De Matteis et aL, \99Sa,b; Delia Corte, 1998). It may be applied to both welded and bolted connections, contemplating several types of dissipative behaviour, including pinching phenomena as well. In particular, in order to account for damaging of mechanical features due to low-cycle fatigue, the model makes allowance for stiffness degradation and strength deterioration, by considering a damage index, progressively varying along the loading history as a function of both the number of reversals and amplitude of plastic excursions. With reference to generalised force (F) and generalised displacement (v), the non-linear model is based on the anal)^ical expression proposed by Richard and Abbott (1975). It is defined on the basis of four independent parameters characteristic of the monotonic curve: the initial stiffiiess ko, the hardening of the system kh, a reference force Fo, which is assumed as the intersection of the hardening asymptotic line with the load axis, and a shape factor n, modelling the sharpness of the curve. Once the monotonic curve has been defined, both unloading and reloading branches are defined by combining linear function and Richard-Abbott (R-A) functions. As shown in Figure 6a, firom the reversal point of the previous load-deformation step (point A), a linear function defines the unloading branch up to the intersection of the asymptotic line of virgin curve translated at the origin (B). Subsequent unloading and then the reloading branches - up reaching a new reversal point (C)- are again modelled by means of Richard-Abbott function. In order to account for stiffness and strength degradation phenomenon as well as for isotropic hardening, stiffness ko, hardening kh, reference force FQ are varied as a function of experienced deformation history.

217 Upper bound cufve Transition curve — J

Hardening line

—

Y^)

X X

... (A)^

V / Lower bound curve

(C)

Displacement

Displacement

a) general definition

b) schematisation of pinching behaviour

Figure 6: The adopted cyclic model For the sake of simplicity, deterioration functions due to low-cycle fatigue could be based upon same parameters usually adopted to define damage index assessing the structural collapse under repeated actions. Similarly to the criterion proposed by Park et al (1985), for defining the potential of damage in structural components, the degradation of F-v relationship could be assessed as afimctionof maximum experienced plastic excursion v combined with hysteretic energy dissipated during the loading history (Kunnath et al, 1990). In particular, the current damage index may be simply defined as 5 = v/vaw^ , where Vavw is a sort of equivalent cyclic available ductility (Fajfar, 1992), reducing the monotonic one as a function of loading history through the relationship: '

E

^

Vavlb = Vult

(1)

being Fy a conventional yielding strength evaluated on the virgin curve, Vw/zthe available plastic deformation on the virgin curve and EH the dissipated hysteretic energy through the whole experienced deformations. The coefficient P is an empirical parameter, which controls the deterioration rate due to each reversal. Theoretically, it should be a function of deformation amplitude, but it is usually assumed to be constant. Then, in a simplified way, deterioration function for mechanical parameters may be assumed to be dependent on the same above defined damage index d. In particular, both reference force FQ and post-limit stiffness kh, at each new plastic excursion, are reduced as in (2). 0.red

= a-Fn

ku ^oA = d' ku ^h,red

(2)

When pinching phenomena are accounted for, in order to represent the typical shape of reloading branches, one should consider that once got through zero deflection point (PR), the moving point approaches the envelope curve at deformation amplitude corresponding to the previous reversal. A transition curve can be therefore introduced, regulating the shifting between a lower bound curve and an upper bound curve (Figure 6b). The latter can be assumed to be the curve without any damage due to pinching, eventually reduced to make allowance for deterioration during loading history. The lower bound curve is instead characterised by a similar curve but with different fundamental parameters (Fo,p, koj>, kh,p, np), which are based on experimental evidence. Then, it can be assumed that the generic point along the transition curve describing the pinching is given by a curve expression, which is determined by setting intermediate parameters between the ones related to upper and lower bound curves. In particular, three transition parameters (^i, ti, X) are used, ruling the real deformation path between lower and upper boimd curves.

218

The can^mrisim wHk experimeniai i^suiti Ih^ ^h&m luiikiyticid model has been calibmled on tlie basis of previously pitesemed e}!Cperimenlal resiulfs. bi patticulur,, m order to assess the niodel reliability to well inteipiiet the oonncclion i^ponse in caise of dififeireat defbrmaiioEi histories, comparisons have been carried out for both conslaiit amplitude deJOormation and ECCS histories, mtti feta:ence to speeimen typology with intesmediate eolunm Mze (BCC7). In ofder to geit the best approximation^ model behavioural parameters faa\'e beeo set up differently on the ba^s of each test Tikis is due to the feet that experimeniai test$ show slight difO^tenoes between e^h othef, not allowing for a unique set of parnmeters. Hie avail^le plaslie defbrmation on the virgin curve v^ has been assumed equal to the ultimate detbrmation obtained ftom mcmolonic cxpciimental curve. Parameter P^, ruling the deterioration iate» has been slightly varied, being incr^ing with defbrmaiion amplitude. MOtMn)

M(P(Nn4

BCC7-D

»CC7^

m

•50 •IQO

^m^^

i^tf^m) -150 ']
-BO

-60

-40

20

-21^

4U

«0

n

100

-100 -so

-«a

-40

-30

0

20

40

60

10

iOO

190

M<M«m)

MflkNm}

BCC7^B

BCCTvC

100 50 0

$0

fff

U

'

/^—"^—

' IXVWJ

-—-Oi"4l^

4) -50

^^

-IHI

^•OiiriiD

4(innK0 .130 ^JOO ^§0

.tiO

•^O

-20

0

30

40

«0

W

too

'100

•«)

.40

<«)

.30

0

20

40

60

SO

100

Figure 7: Qualitative comparison betM^'een experiniental and numerical results Qualitative comparison betvi'een experimental (light-line) and numerical (dark line) resuhs are; shovm in Figure 7, while adopted values for all model parameters are reported in Table 3. The model appears to be adequate to account for all the eharacteri^c phenomenological aspects of tested eonnections. In &ct, virgin curve^ pinching behaviour and mechanical damage ai« satisfactor}' reproduced. In Pipire 8, in case of ECCS procedure (BCC7*C) as well as in one case of constant amplitude deformation history (BCC7-A), the comparison is shown in terms of both peak moment and energy dissipation per cyck. In these gmphs, cycle n. 1 is referred to first cycle corresponding to a deformation amplitude i'=v> in case of ECCS piooedure^ while it represents the first cycle at v^v^m in cuse of constant amplitudie tests. This allows to emphasise the model capability to v^'ell interpret the actual connection response in terms of the main mechanical features for each cycle and all ox'er the deformation history. In fact, scatters w^ always in a reasonable limit (about than 10%). allowing for a correct energy capability prediction up to the collapse of the system. Finally, it should be observed that numerical simulations have been carried out by assuming symmetry in positive and negative ranges, while exp^imental results show bending moment resistance higher in negative range than in positive one.

219 TABLES ADOPTED VALUES FOR MODEL FUNDAMENTAL PARAMETERS Specimen BCC7-A BCC7-B BCC7-C BCC7-D Specimen BCC7-A BCC7-B BCC7-C BCC7-D

Fo(kNm) 90 150 85 95 ti

50 50 50 50

Upper bound curve kh (kNm) n 750 0.6 380 0.6 750 0.6 750 0.6 Transition curve t2 X 0.50 1 0.50 1 0.50 1 0.50 1

Fo,(kNm) ko (kNm) 25000 55 25000 85 25000 45 25000 55 Damage parameters P

Lower bound curve khn(kNm) nn 250 0.6 100 0.6 250 0.6 0.6 250

kop(kNm) 25000 25000 25000 25000

Vult

0.08 0.10 0.025 0.065

0.1 0.1 0.1 0.1 BCC7-A typology

BCC7-C typology 150 •Analytical ^ Experlmenta1

100

1 III! IllIII

I 50 I 0 2

VbTTH | k | k

j-30

fltttt'II

III

•I

-100 -150

6" 6 4 2 0

lilllPPPP 11 • li li 11 li • a I

1

2

3

4

5 Cycle

6

Figure 8: Energy and peak moment per cycle comparison CONCLUDING REMARKS The current study is part of a general research project devoted to analyse the influence of connection behaviour on the seismic response of steel frame structures. In particular, full-scale tests on currently utilised bolted beam-to-column joint typology have been preliminarily performed. Analysed cleat angle connections subjected to cyclic flexural moment loading have behaved as semi-rigid and partial strength, showing an appreciable energy dissipative capability. The experimental outputs have been used to calibrate the proposed cyclic model, which is based upon a damage index procedure to account for mechanical degradation of the joint all over the deformation history. The comparison with experimental results has shown a deserving agreement in terms of both strength and energy dissipation, ensuring for the model capability to predict correctly the connection response up to the collapse. The subsequent phase of the study will be twofold. In fact, on one side, experimental results will be reelaborated in order to provide major information on the influence of column size on both monotonic

220 and cyclic behaviour of the connection as well as to draw out relative low-cycle fatigue curves. On the other hand, the proposed analytical model will be used to analyse the dynamic behaviour of whole frame structures, pointing out the effect due to semi-rigid and partial strength connections, when their actual dissipative behaviour is accounted for the evaluation of the global structural response. ACKNOWLEDGEMENTS

The current study has been carried out in the framework of the INCO-COPERNICUS Joint Research Project '^Reliability of Moment Resistant Connections of Steel Building Frames in Seismic Areas^' (RECOS). The financia support of the EC is gratefully acknowledged. Thanks are also due to students Michele D'A vino and Roberto Cappuccio who developed their graduation thesis at Instituto Superior Tecnico of Lisbon within the European mobility project Erasmus/Socrates.

REFERENCES

Astaneh, A., Nader, M. and Malik, L. (1989). Cyclic behaviour of double web angle connections. Journal of Structural Engineering, ASCE, 115(5), 1101-1118. ATC, (1992). Guidelines for cyclic seismic testing of components of steel structures. Applied Technology Council, Publication N. 24. Ballio, G., Calado, L., De Martino, A., Faella, C. and Mazzolani, P.M. (1987), Cyclic behaviour of steel beam-tocolumn joints experimental research. Costruzioni Metalliche, 2,69-88. Bemuzzi, C, Zandonini, R. and Zanon, P. (1996). Experimental analysis and modelling of semi-rigid steel joints under cyclic reversal loading. J. Constructional Steel Research, Elsveier, 38(2), 95-123. Calado, L. and Ferreira, J. (1994). Cyclic behaviour of steel beam-to-column connections - An experimental research. In Proc. of Behaviour of Steel Structures in Seismic Areas (STESSA '94), Eds. P.M. Mazzolani, and V. Gioncu, E & FN Spon. Calado, L., Bemuzzi C , Castiglioni, C..A. (1997). Behaviour of steel beam-to-column joints under cycling reversal loading: an experimental study. In Proc. of the 5*^ International Colloquium on Stability and Ductility of Steel Structures, Nagoya, Japan, 29-31 July. D'Avino, M. (1999). Cyclic behaviour of beam-to-column bolted connections in steel seismic resistant structures (in Italian). Graduation thesis. University of Naples Federico II, Department of Structural Analysis and Design. Delia Corte, G. (1998). Modelling of beam-to-column joints in steel moment resisting frames (in Italian). Graduation thesis. University of Naples Federico II, Department of Structural Analysis and Design. De Matteis, G., Landolfo, R. and Mazzolani F. M. (1998). Hysteretic behaviour of steel connections: the influence of behavioural parameters. RECOS Copernicus project document, Sofia, November. De Matteis, G., Landolfo, R. and Mazzolani F. M. (1998). Cyclic models for shear diaphragms with different cormecting system. COST CI Int. Conference on "Control of the semi-rigid behaviour of civil engineering structural connections", Liege. ECCS, (1986). Recommended Testing Procedure for Assessing the Behaviour of Structural Steel Elements under Cyclic Loads. European Convention for Constructional Steelwork, Publication No. 45, Brussels. Faella, C, Piluso, V. and Rizzano, G. (1993). Connection deformability influence on the inelastic response of seismicresistant frames. Giomate Italiane della Cosatruzione in Acciaio (XIV Congresso CTA), Viareggio, Italy. Fajfar, P., (1992). Equivalent ductility factor taking into account Low-Cycle Fatigue. Earthquake Engineering and Structural Dynamics, Vol. 21, 837-848. Krawinkler, H. (1997). System Behaviour of structural steel frames subjected to earthquake ground motion. Background Reports SAC 95-09, FEMA-288 'Program to reduce the earthquake hazards of steel moment frame structures. Sacramento, California. Kunnath, S.K., Rinhom, A.M., Park, Y.J., (1990). Analytical modelling of inelastic seismic response of R/C structures". Journal of Structural Division, ASCE, 116 (4), 996-1017,1990. Madner, J.B, Chen, S.S and Pekcan, G. (1994). Low-cycle fatigue behaviour of semi-rigid top-and-seat angle connections. AISC, Engineering Journal (Third quarter), 111-122. Mazzolani, P.M. and Piluso, V (1996). Theory and design ofseismic resistant steelframes, Chapman & Hall, London, UK. Mazzolani, P.M. (1999). Reliability of moment resistant connections of steel building frames in seismic areas: the first year of activity of the RECOS project. In Proc. of T*^ European Conference on Steel Structures, Eurosteel 1999, Praha, Czech Republic. May 26-29. Richard, R.M. and Abbott, B.J. (1975). Versatile elastic-plastic stress-strain formulas. J. Engineering. Meek Div., ASCE, 101(4), 511-515. Park, Y.J., and Ang, A.H-S. (1985). Mechanistic Seismic Damage Model for Reinforced Concrete. Journal of Structural Engineering, ASCE, 111 (4), 723-739.

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

221

NUMERICAL AND EXPERIMENTAL STUDY OF BEHAVIOUR OF COLUMN-BASE CONNECTIONS UNDER BENDING Miklos Ivanyi^ Jeno Balogh^ and Laszlo Hegediis^ ^ Department of Steel Structures, Technical University of Budapest, P.O.Box 91, Budapest, H-1521 Hungary 'Inter-CAD Ltd., P.O.Box 317, Budapest, H-1461 Hungary

ABSTRACT Numerical modelling and experimental analysis of base plated column-base connections is the subject of this paper. The actual semi-rigidity of these connections could have a significant influence on the structure's response. A nominally pinned and a fixed column-base connection was studied by numerical simulations, in the framework of a parametric study, completed and verified with experimental tests. To model the moment-(relative)rotation characteristic of the connections, the Richard function is integrated in the finite element model of a beam element.

KEYWORDS Column-base connection, semi-rigid connection, non-linear finite element INTRODUCTION The simple and in-expensive base plated solution is often applied to the pinned and fixed column bases, in the design practice. In the following, the base plated column-base connections will be referred to as column-base connections. To take into account the action of the column-base connections in the steel frame design, is basic to be able to approximate their behaviour. The experimental investigations show that this behaviour is semi-rigid. The research of the semi-rigid connections in the case of beam-to-column connections is in an advanced stage and the results have already been incorporated into design codes (e.g., Eurocode 3). On the other hand, in the case of column-base connections the experimental and modelling research has come into the foreground of interest only in the last few years. A great problem is the small number of experimental tests.

222

In this study the results of numerical simulations, on a three-dimensional finite element model, are presented and completed with the results of experimental tests, which at the same time validate the numerical model. The results could be used to calibrate simphfied approximate models for the design of the connection, or directly, in the global analysis of frames with this connection type.

NUMERICAL STUDY To point out the influence of the base plate thickness, of the bolt diameter and of the model of the concrete on the moment-relative rotation behaviour of a pinned and of a fixed column-base connection, a parametric study using finite element numerical analysis had been performed. In the following, the moment-relative rotation characteristics will be referred to simply as momentrotation characteristics. Geometry of the Connections The analysed pinned column-base connections consist of a 190x180 and 270x180 steel base plate for the pinned and fixed connection case respectively, welded to the end of a HEA 180 steel column stub fixed on a special concrete square foundation. The base plate is bolted to the foundation by two HSFQ high strength fiiction bolts of d diameter. The bolts are embedded in the concrete. The detailed geometry of the connections are given in Fig. 1 and Fig. 2. 190 *

45

90

45

1 1

J{ '1

+ 1 + 1

270

180

K - ^ - ^ K - -^^-M Figure 1: Geometry of the pinned connection

Figure 2: Geometry of the fixed connection

The Numerical Model The non-linear module of the COSMOS/M Explorer program [1] was appHed to the analysis. In the analysis different sources of non-linearity (material, geometrical, due to contact) were taken into account. The base plate and the column were modelled by 6-sided solid isoparametric finite elements [1] with three degree offi-eedomper node. The anchor bolts were modelled using a system of truss [1] elements. The equivalent bolt lengths (Z) were obtained [2] as L = \0'd-^t + tg, where d is the diameter of the anchor bolt, / is the thickness of the base plate and t^ is the thickness of the grout. For the steel material model the Ruber von Mises - Hencky yield criterion was used with a bilinear stress-strain curve. Table 1. Large deflections non-linear analysis in updated Lagrangian formulation (small strain assumption is made) was performed.

223

TABLE 1 MATERIAL CHARACTERISTICS E [N/mm'] 210000 210000

Material of steel parts bolts Concrete (r) (e)

k [N/mm^/mm] -

V

0.3 -

[N/mm'l 275 1000

[N/mm'l 2100 2100

00

75

20

27400

(P)

274

The steel-to-concrete contact was modelled by 2-node gap elements [1]. No friction between contact surfaces had been considered. The concrete block was modelled by non-linear spring elements [1] with stiffness k connected to the gap elements. According to the concrete material model, these springs were rigid (r) or linear-elastic (e) or elastic-plastic (p) ones, Table 1. A unit moment (1 kNm) was applied using a system of forces acting on the top of the stub only in the flanges. To control the solution process a displacement control method (monotone loading) has been used with a regular Newton-Raphson scheme for equilibrium iterations. The Parametric Study During the numerical study the values of the parameters were: thickness of the base plate {t) 6, 10, 15 mm; diameter of the bolts {d) 12, 16, 20 mm; model of concrete material {r) rigid, {e) linearelastic, (p) elastic-plastic.

EXPERIMENTAL STUDY The experimental investigations were carried out in the Laboratory of the Department of Steel Structures, Technical University of Budapest. A test program on semi-rigid frames was performed within the framework of COST research project. Due to the fact, that in the full-scale tests the frames had nominally pinned column-base connections, the test results are available only for this type. In the tests COST-10, and COST-11, the geometry of the specimens were as in Fig. 1. The diameter of the bolts was d = \6 nmi, and the base plate thickness was t = \5 mm in COST-10, and ^ = 6 mm in COST-11. p; t-6; d=16

ir^

'Trn

Un Um 0

5

10

15

p; t=15; d=16

^ -Test: COST-11 - Finite Element

^

20

rotation {mrad]

25

30

35

Kfrr

15

I

-TestCOST-IO -Finite Element

5

0

5

10

15

20

25

30

rotation [mrad]

Figure 3: Comparison of numerical and test Figure 4: Comparison of numerical and test results of COST-11 results of COST-10

224

RESULTS The main result is the load, corresponding to the applied unit moment, obtained in the function of the nodal deflection that was used to control the solution procedure. The variation of the load represents the moment-rotation curve of the connection. In the case of the pinned column-base connection, the following results are presented: comparison of the results of numerical simulation to the test results Fig. 3 and Fig. 4; values of the initial stiffiiess *S^„,v in the case of plastic {p) modelling of concrete, in the function of bolt diameter d and base plate thickness t, Fig. 5; values of bending moment Af at a column base rotation of (|) = 30 mrad in case of plastic (p) modelling of the concrete, in function of boh diameter d and base plate thickness t. Fig. 6; moment-rotation characteristics corresponding to different concrete models, Fig. 7. The extents of contact zone in these cases are as in the Fig. 8. e;p p; (t>=30 mrad

B22S0-2S00 •2000-2250 • 1750-2000 1500-1750 iQl2SO-1500 01000-1250 D750-1000 •500-750 11250-500 ao-250

•35-40 30-35 D25-30 020-25 IEI15-20 10-15 05-10 IDO-5

connection

connection

£20

I 15

s

/

raV ]£.

.<^ * r^'

r-AcAal

UJA #*•

• • ^ 10

15

20

25

rotation fmrad]

Figure 7: Comparison of different concrete models t=6; cl=16;

t=15; d=16;

Figure 8: Contact zones of the pinned connection

225 In the case of the fixed column-base connection, the following results are presented: momentrotation characteristics corresponding to the bolt diameter d=20 mm, for different base plate thickness t values, Fig. 9; values of the initial stiffiiess Si^^^ in case of plastic (p) modelling of the concrete in function of base plate thickness t, Fig. 10; values of bending moment M at a column base rotation of (t)=7 mrad in case of plastic (p) modelling of the concrete, in function of base plate thickness t, Fig. 11. It should be noted that the results refer to the pure bending case and therefore additional studies are required to clarify the effect of the normal forces on the initial stiffiiess and on the moment resistance of the connections. p;

ds16

-t=6 -t=10 -t=15

Figure 9: Moment-rotation characteristics of the fixed connection

p;

ds16

10000-,

^ ^ ^ ^

80006000 4000

p; d=16;4>=12mrad

^ ^ ^ - • ^ " " ^

-Sinitj

^^-"^^^^

20000-

1

1

1

1

1

1

10 11

1

1

12 13

1

1

14 15

16

tfrran] 5

6

7

8

9

10 11 12 13 14 15

16

t[mm]

Figure 10: Initial stiffiiess of the fixed connection

Figure 11: Moment resistance of the fixed connection

MODELLING THE BEHAVIOUR OF THE CONNECTION To include the behaviour of the connections in the global analysis of semi-rigid fi-ames, it is necessary to model their moment-rotation characteristic. In the global analysis the non-linear moment-rotation characteristics of the connections are often modelled by fiinctions (power fimction, Ramberg-Osgood fimction, exponential fimction, Richard function). These fiinctions use two or three main parameters to describe the characteristic, and one or two empirically determined shape factors to control the transition fi-om elastic to non-elastic behaviour. The main parameters of the connection moment-rotation characteristic would be established fi-om the results of the numerical

226 simulations and these are as follows: Mj^^ the moment resistance; Si^u initial (elastic) stiffiiess; Spi the strain-hardening stiffness. MA

Figure 12: Richard function The Richard function [3], which expresses the moment M in terms of the rotation (|), and takes strain-hardening or strain-softening behaviour into account, gives a good representation of experimentally or analytically determined moment-rotation curves of semi-rigid connections of steel frames. (5,„,v-5,,)-*

-ln2

^ + 5^, • <|), with n =

M((|)) =

,\>A

1+

K.,-5,,)-

In *J/n,V "" *J

M,

p/y

where, Mj, M2 are reference moments as in Fig. 12. The particularity of column-base connections, as semi-rigid connections, is that its elements are of two substantially different materials. The Richard function was applied to the moment-rotation characteristics of the analysed column-base connections, in order to study the apphcability to this kind of connections. The results show a good modelling accuracy [4]. The Richard function was used in conjunctions with a beam finite element capable to taking into account the non-linear behavior of the connections in global analysis of steel frames. Below a model that was implemented in the Inter-CAD's Axis-3D software for line and surface finite elements is presented. The finite elements are connected to nodes by springs that have to model the non-linear connection behavior. The secant stiffness S^ and the tangent stiffness S^ of a spring are expressed as Sp-Sp ^ c

=

1+

respectively.

(^o-^J-* M,

—rr + ^n \l/n

P

and S, =

Sp-S,

dM

NM)/«

f

1+

('^o-^ph M,

^^P

227 Denote [r]=(rj

r^ ... r^„) the diagonal flexibility matrix of the springs, where r. =^—

and i

z = 1,2,...,6« with w the number of the finite element nodes. If {w„}= {w. displacement vector at nodes, and {we}={wg, u^. ... u^^f

u.

... u^f

is

the

is the vector of the element

displacements, the displacements of the springs are:

The element equilibrium equation is

[K]-k}={Pe}+{Ps] where {p^} is the element load vector and {p^} is the internal force in the springs.

[kMk}-[r]-{Ps}h{pM{Ps}

([KnrMirAK]-k]=
l + rrk„ mathematical transformation [6], to the above equations, with {e,} being a unit vector in the direction /,

1 + r.-A:..

'3,.+A:..

are obtained, and

[k\-k}-^}={Ps} The new secant or tangent element stiffness matrix is obtained depending on the expression used for the determination of Sj. The equation {u^} = [^J * {Ps } = i'^]^ ' {Ps} is not adequate for determination of {u^} due to possible singularities of [s]~^. Recalling the equation [k^]- {u^]=:[pj+ {p^} the components u^. are expressed successively.

228

The equation corresponding to the component u^. is:

from which

SUMMARY The behaviour of the connection is determined by the interaction of the behaviour of its component elements. In any case, the deformabihty of the concrete should be taken into account in the analysis because the assumption that the concrete is rigid [4] leads to large contact forces, and overestimates the initial stiffness and the moment resistance of the connection. Fig. 7. The basis of the determination of main parameters of moment-rotation characteristics of the connections could be the results of numerical simulations completed with those from experimental tests [5]. Once the parameters of the characteristics have been determined, using the Richard function to model the moment-rotation behaviours of the connections, a global analysis can be carried out with software for frame structures. ACKNOWLEDGEMENT This work was partially supported by Hungarian Science Foundation (OTKA) resarch grants T 020358 and F 020355. References [1] [2] [3] [4] [5] [6]

COSMOS/M (1993) The Advanced Modules, Structural Research & Analysis Corporation, Santa Monica, USA Sokol, Z., Wald, F., Dunai, L. (1994) ne Column-Base Stiffness Prediction, COST CI Workshop, Prague, October Ahnusallam, T.H., Richard, R.M. (1993) Steel Frame Analysis with Flexible Joints Exhibiting a Strain-Softening Behavior, Computers & Structures Vol. 46, No. 1 J. Balogh, M. Ivanyi (1996) Analysis of Steel Frames with Semi-Rigid Column-Base Connections, Third International Conference on Computational Structures Technology, Budapest, August Penserini, P., Colson, A. (1992) Caracterisation des Liaisons Structure MetaliqueFondation: Application au Flambement des Poteaux, Construction Metallique, n° 2 Zs. Caspar (1993) Tartok Statikdja III - Rudszerkezetek, Muegyetemi Kiado, Budapest

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

229

REMARKS ON THE USE OF EC3-ANNEX J FOR THE PREDICTION OF ALUMINIUM JOINT BEHAVIOUR G. De Matteis , A. Mandara , F. M. Mazzolani * Department of Structural Analysis and Design University of Naples Federico II, Naples, ITALY ^ Department of Civil Engineering Second University of Naples, Aversa (CE), ITALY

ABSTRACT A critical examination of the possibility to apply EC3-Annex J component method for the prediction of the ultimate resistance of aluminium T-stub joints is presented in this paper. Based upon numerical results, the need to develop alternative simplified procedures is firstly stated. This is essentially due to the fact that in case of aluminium alloy structural components the material strain hardening and ductility may strongly influence the behaviour of the system, especially at the collapse limit state. Therefore, the codified approach for steel appears to be inadequate to be simply extended to aluminium joints, being unable to predict the collapse mechanism type, as well as to account for the beneficial effect of hardening. On the basis of a wide parametric analysis, a correction procedure is proposed in this paper. It allows such effects to be properly accounted for at least when the collapse mechanism type 1 occurs - namely in case of strong bolt-weak T-stub flange combination - through a balancing factor to be used in the same simplified relationship proposed by EC3-Annex J. Finally, the variation of this factor with both material strain hardening and ductility is provided. KEYWORDS Joints, Annex J, T-stub, Aluminium alloys. Component method, Strain hardening. Ductility INTRODUCTION A wide research work is presently being carried out on aluminiimi alloy joints in the framework of EC9 activity. One of the main concerns of this research is the possible application of the existing regulations holding for steel joints to aluminium alloy joints. This would be, in fact, rather convenient from the practical point of view, a vsdde number of design rules having been already assessed in the field of steel structures. In particular, the EC3 - Annex J (ENV 1993, 1997) covers a large amount of joint typologies, providing a great deal of calculation rules in both elastic and plastic range. The suggested procedure is based upon the well known component method approach. An extension to aluminium

230

alloy joints of such an effective method could be very advantageous, allowing most of the codified rules, well familiar to researchers and designers, to be preserved in EC9 (ENV1999, 1998) as well. Nevertheless, it is not possible to extend roughly the provisions of Annex J to these materials, but some particular adaptations should be fitted. This is a consequence of the mechanical feature of these materials which, contrary to steel, exhibit a general a a-8 relationship of round-house type, which can not be interpreted through the classic elastic-perfectly plastic idealisation (Mazzolani, 1995). hi particular, both the strain hardening and the relatively low ductility of some alloys have to be properly accounted for. The problem is rather difficult to be completely solved, being collapse mechanisms of basic joint components strongly influenced by these parameters which are peculiar for each alloy. Besides, it should be considered that in case of aluminium structural components the need to determine exactly the full non-linear curve representing their mechanical response is generally felt more than in steel structures. In fact, when instability phenomena are avoided, collapse mechanisms are practically always ruled by the available ductility of the alloy. When the structural component is made of a number of separate sub-components, as generally occurs in the case of joints, sub-components interact between each other owing to the hardening. This results in a collapse which is not necessarily ruled by the subcomponent exhibiting the minimum conventional resistance at the elastic limit state. Some recent investigations on the plastic analysis of aluminium alloy structures have been already developed by the authors, which proposed a concentrated plasticity model with non-linear plastic hinges aiming at representing the structural behaviour of the joints by means of simplified analyses of monodimensional aluminium alloy structural components (De Matteis & Mandara, 1997; De Matteis et al, 1999). Still referring to T-stub components, the possibility to introduce slight modifications into Annex J is examined in this paper, aiming at taking into account all the relevant aspects of alimiinium alloys. In particular, the prediction of the ultimate joint moment capacity is focused, this being the most important objective to be investigated. The goal is pursued by means of a wide numerical analysis, allowing a suitable analytical formulation to be set up. BASIC CONCEPTS FOR THE EXTENSION OF THE COMPONENT METHOD TO ALUMINIUM ALLOY CONNECTIONS In order to assess the design moment-rotation characteristic of a joint, this may be taken apart into basic components, namely, in case of beam-to-colimm joints, column web panel in shear, column web in compression and in tension, column flange in bending, end plate in bending, flange cleat in bending, bolts, etc. The design moment-rotation characteristic of a joint is built up from the individual properties of the basic components. Therefore, for the sake of the determination of the overall behavioural properties, all components contributing to stiffness, strength and ductility have to be preventively recognised. Besides, according to EC3-Annex J procedure, simplified models to compute the force versus displacement relationship must be provided for each joint component. The overall deformability of the joint is then obtained by combining the single component deformability contributions, whereas the joint resistance, as well as its deformation capacity, is assumed equal to that of the weakest component. The above concerns are strictly valid only in case of elastic-perfectly plastic behaviour, when the weakest component surely governs both resistance and ductility of the joint. When the strain hardening characterising the response of basic components is high, or when there exist some brittle-type components, the approach could be unsafe. This may happen in case of aluminium joints. In fact, on one side, strain hardening of the material has to be considered in plastic design, since it may have a nol negligible effect. On the other side, since some alloys are characterised by limited values of ultimate elongation, if section instability phenomena are avoided, the reduced deformation capacity of the joini

231 component may be caused by mechanical factors. As an example, the negative effect of the brittle behaviour of a joint component (which can be not necessarily the less rigid) is shown in Fig. 1 in case of a joint made of just two components supposed to be arranged in series. Other influencing factors to be considered are due to the fact that aluminiimi alloys may be strongly different between each other due to chemical composition of the material, fabrication process (extrusion and successive straightening) and heat treatments. Besides, it has not to be forgotten that the effect of welding is more important than in steel, compelling the designer to account for the correct behavioural properties of affected zones. As a consequence, the analytical computation of the joint response suffers this drastic increment in the number of variables, also by considering the possibility to combine different aluminium alloys for basic components and bolts.

Figure 1 Interaction between components With reference to the generalised load-displacement curve of one of the basic components, the essential parameters to be determined are the ultimate resistance, the initial stiffness and the deformation capacity. The analysis must be conducted taking into account the actual stress-strain relationship of the base material, which, contrary to the common elastic-perfectly plastic idealisation adopted for mild steels, has a continuous trend and, depending on the alloy, a relatively low limit of deformation (st). The stress-strain relationship is usually described by means of the well known Ramberg-Osgood law, where an exponent n is used to describe the strain hardening of the material. As an effective alternative the following exponential model (Hopperstad, 1993) may be conveniently used for describing the postelastic branch: For

a > CTn

G = ao+a[l-exp(-ys^)]

(1)

where: GQ is the stress at proportionality limit in an uniaxial test, which can be conventionally assumed equal to the elastic limit strength^oiJ £p is the accumulated plastic strain; the material parameters a and P determine the magnitude of the strain hardening and the shape of the hardening curve, respectively. According to experimental tests, a suitable value for y is 10, whereas the factor a varies depending on the alloy. In particular, a values equal to 80 N/mm^ (low hardening), 140 N/mm^ (medium hardening), 200 N/mm^ (high hardening), 300 N/mm^ (very high hardening) can be practically be assumed for aluminium alloys. For evaluating the deformability of a general basic component of the joint, in general the same formulas as adopted for steel connections could be used. Particular attention must be paid to the assumption of the Young's modulus of the basic material, owing to the effect of welding, as well as to the plate-to-bolt stiffness ratio, by including the possibility to use steel bolts.

232

As far as the ultimate resistance is concerned, the conceptual approach of ECS may be still used. For each basic component, a possible collapse mechanism must be therefore established. Nevertheless, it is important to observe that, while for steel structures the collapse load corresponding to the limit analysis developed with plastic hinge method leads generally to a satisfactory degree of accuracy in the prediction of the ultimate resistance, this ultimate load for roimd-house type materials could be less significant. In order to account for the post-elastic behaviour of structures made of such materials is possible to define suitable correction factors of the conventional yield stress assessing the effects due to hardening and reduced ductility. The correction factor r\ considered in Eurocode 9 (ENV 1999, 1998) represents a first attempt to apply such a procedure, where r| was determined on the basis of an equivalence in terms of load bearing capacity corresponding to a given ductility level between the concentrated plasticity model and the spread plasticity one (Mandara & Mazzolani, 1995). A different approach, which is somewhat complementary to that proposed in EC9, is presented herein for the evaluation of the ultimate resistance of T-stub joint components, allowing the same EC3-Annex J formulas to be applied by introducing a correction factor (k), which accounts for all material peculiar mechanical features.

FRAMING OF THE STUDY Many connection typologies, such as end plate and angle cleat joints, can be represented by simple tensioned equivalent T-stub components, whose behaviour is ductile in case of steel, provided that boh resistance is sufficient to avoid their premature failure. According to EC3-Annex J (ENV 1993, 1997), three possible failure modes for steel equivalent T-stub components are recognised. The complete yielding of the flange takes place in the type-1 mechanism, with the onset of four plastic hinges at ultimate limit state, two of which are located at bolt axes. The type-2 mechanism occurs when bolt failure takes place together with yielding of the flange at the sections corresponding to flange-to-web connection. The type-3 mechanism is due to the sole bolt failure with the overall up-lift of the flange. In this case, neither prying forces nor flange yielding take place. The examination of this component in case of aluminium joint is particularly interesting due to the fact that collapse mechanisms can be deeply different from the ones related to steel, being the possibility to develop completely the collapse mechanisms strongly influenced by the mechanical features of the alloy. In particular, previous studies (De Matteis et al, 1998) pointed out that the type-2 mechanism, i.e. failure caused by attainment of plastic deformation in both flange and bolt material, can not be so clearly defined as for steel. When the T-stub at the flange-to-web connection is in plastic range and while bolts are developing their axial deformation, the collapse may be attained in either flange or bolt depending on their ultimate deformation capacity as well as on their deformation gradient. Therefore, at least three different situations can take place, depending on which element rules the collapse. The first case (type-2a mechanism) is related to a flange plastic deformation higher than that arising in the bolt; the third case (type-2c mechanism) reflects the bolt failure with limited plastic deformation occurred in the flange, whereas an ideal boundary of the behavioural response may be assumed to be correspondent to a situation where both flange and bolts govern the problems together (type-2b mechanism). 150

.30. 45

o1 o1

1 ^ 1 ^'

Figure 2 Geometry of analysed T-stub model

Several numerical analyses have been performed, emphasising all these possible collapse mechanisms. To this purpose, a detailed threedimensional finite element model developed by means of the explicit non-linear code Abaqus has been set up. The model has been calibrated on the basis of experimental results available for steel (Bursi & Jaspart, 1997), showing an

233

excellent accuracy of the FEM model in predicting the joint response up to failure. The typical double T-stub symmetrical arrangement shown in Figure 2 has been considered in the study, where M12 bolts are used to connect the 10.7mm thick flange. In order to analyse cases with different flange-to-bolt resistance ratios, several alloys, including steel for bolts, have been considered. Details on the assumed material behaviour are given in De Matteis et al (1998), where the obtained results are also outlined. The corresponding force-displacement relationships for T-stub joints under consideration have allowed the ultimate resistance of the structural component to be determined. Collapse loads may be now evaluated as a function of the conventional ultimate elongation of either T-tub flange or bolt materials. For aluminium alloys, reference to uniform strain (s„) instead of ultimate strain (s/) should be made. Uniform strain, which corresponds of the attainment of maximum stress in engineering stress-strain experimental curves, may been approximately assumed equal to half the ultimate elongation. Since the ultimate load is determined by the true collapse of the material, i.e. when the conventional ultimate strain is attained at least in one point over the section, this assumption makes the method proposed safer for design purposes.

- i ^ - A W 6082-T6 - • — A W 6082-14 - X - A W 6061-T6

Steel 6.8

AW 6062-T6

Steel 4.6

AW 7075-T6

AW 5063

Bolt type

Figure3: Nimierical and codified (Annex J) ultimate resistance (F„//) values The numerical assessment of collapse loads for examined T-stubs are compared in Figure 3 with the design plastic resistance suggested by EC3-Annex J (1997). The latter has been therefore determined by considering an equivalent four-support beam, whose effective width (beff) can be evaluated according to code provisions, as a function of the geometry of the actual scheme. In the analysed case, for each bolt row, ^^^has been therewith assimied equal to 40 mm, which represent the geometrical width (b) as well. Codified formula has been applied to aluminium material by replacing the steel yielding stress Jj, with aluminiimi conventional limit elastic stress ^.2- It is clearly evident that values suggested by Annex J are largely conservative, leading to an underestimation of collapse load up to 50%. So high scatters are mainly due to the fact that the codified procedure does not include strain hardening of the material (this can be accepted for steel only, but not for alimiiniimi as well). In fact, errors are greater for T-stubs flange materials characterised by stronger hardening values, i.e for AW 6061-T6 and AW 6082-T4 alloys. Besides, the codified procedure is generally inadequate for predicting T-stub mechanism types (De Matteis et al, 1998), confirming the need to establish different rules in predicting the failure mode and the corresponding ultimate load for aluminium joints. THE PARAMETRICAL STUDY The examined cases In order to investigate to a greater extent the effect of hardening and ductility of aluminium alloys on the ultimate resistance of T-stub joints, several numerical analyses have been carried out. Since the

234

effect of hardening is especially exploited in case of collapse mechanism type 1, where plastic hinges forming in T-stub flanges rule the development of post-elastic behaviour, the analysis has been firstly focused on strong bolt-to-weak flanges cases. Parameters considered in the study are the ones mostly influencing T-sub collapsing behaviour, i.e. hardening of the alloy (a), uniform elongation (SM), flange-to-bolt resistance ratio and T-stub geometry. The influence of both hardening and ductility of T-stub flange material has been inspected by analysing conventional alloys. Therefore, with reference to some fixed value of elastic stress, namely y&.2=l 20 and 240 N/mm^, values of a=0, 80,140, 200 and 300 N/mm^ and values of e„ = 3, 4, 6, 8 % have been considered, thus practically covering all realistic cases. For a conventional yielding stress yb.2=240 N/mm^, stress-strain curves of flange material are depicted in Figure 4. The case a=0, corresponding to an ideal elastic-plastic material, has been also included for the sake of comparison. As far as bolts are concerning, bolt steel grade 8.8 have been considered for all flange materials, these flange-to-bolt combinations allowing for mechanism types 1 to be almost invariably involved. Three T-stub geometry has been considered. The first one is related to the geometry depicted in Figure 1. In the following it will be labelled as T-stub type 1. The second one (T-stub type 2) practically coincides with the previous one, but with a different thickness for flange, namely 14 mm instead of 10.7 mm a=0 [N/mm2) (Figure 5a). This allows the influence of a=80[N/mm2l flange thickness on the T-stub collapse a=140 [N/mmri a=200 [N/mm2]| behaviour to be investigated. Because of the a=300 (N/mm2]| close relative distance between bolt rows, ^ 0 according to Annex J both the above cases 0 2 4 6 8 10 12 correspond to T-stub collapse in the situation Figure 4 Analysed conventional alloys of combined bolt patterns, i.e. collapse mechanism which yielding lines involve both bolt rows. In order to cover the opposite situation, i.e. T-stub collapse in the situation of separate bolt pattern, the geometry depicted in Figure 5b has been considered as well. In the following it will be labelled as T-stub type 3. a [kN/mm^

The obtained results The results obtained from FEM analysis are shown in Figure 6 for all T-stub typologies, where in dashed lines resistance values corresponding to assumed ductility values are indicated. The curves well emphasise the role of strain hardening of the flange alloy, which strongly influences the ultimate resistance of the joint component. This is particularly evident for the we^er flange material (/J).2=120 N/mm^), which always involves a failure mechanism type-1, where bolt is completely excluded by the collapse of the system and the flanges develop their maximum plastic deformations. In the same figure, for each curve, the corresponding number indicates the type of collapse mechanism as it has been deduced by observing specimen yielding patterns and bolt plastic engagement. For the sake of comparison with the codified procedure, it should be observed that for all the cases EC3 annex J suggests to consider a mechanism type 1. The numerical and codified values corresponding to the ultimate resistance of T-stubs under consideration are summarised in Table 1. Scatters are remarkable in all cases, but they increase with the hardening, especially for more ductile alloys. Besides, the importance of the flange ductility should be underlined, which only in case of high hardening, i.e. for a > 140 N/mm^, produces a considerable reduction of T-stub deformation capacity for low 8„ values. In particular, in case of T-stub types 1 and 2, the ultimate resistance values proposed by the code are strongly conservative, while they are unsafe for T-stub type 3, which is characterised by bolts resisting to tensioned forces individually.

235

.39^ 4?.,

n^

Figure 4: Geometry of T-stub-type 2 and 3

CRITICAL EXAMINATION OF RESULTS The codified approach A careful examinations of results demands for other considerations. In fact, one should note that the error due to codified approach is not negligible even in case of flange without any hardening, which practically corresponds to the idealised steel material. Obviously, this is a consequence of the approximated procedure, which is based upon the equivalence between the actual three-dimensional behaviour of the examined joint component and the one related to a monodimensional scheme. The design strength, in fact, is evaluated by means of equilibrium conditions based on a concentrated plastic hinge approach. The conventional resistance is therefore a function of the assumed failure mechanism as well as of the nominal geometrical and mechanical features of the structural component. In case of mechanism type 1, the governing parameter is the bending plastic resistance of the flange rectangular section (M^), and the following relationship is suggested: ^M,l

(2)

m where m = d-0.^r ,dbeing the distance between the bolt axis and the T-stub web and r the root radius of the flange-to-web connection. In particular, Mu is the fully plastic bending moment evaluated with respect to the conventional yield stress^.2 and to the effective width beff, which is defined in such a way that the design tensile resistance of the equivalent T-stub is approximately equal to the actual one. Its determination represents, therefore, the critical point of the procedure. In both cases, when the bolt-row is considered individually and when the bolt row is considered as part of a bolt-group, EC3-Annex J allows the effective width to be determined by assuming the smallest value between the one related to the circular yielding pattern and the one related to other non-circular patterns. In particular, for bolts as a part of a group, ^e^is limited to the T-stub geometric width b. Besides, yielding patterns observedfi*omnumerical results show a substantial difference between the shape and therefore the effective length of the inner plastic hinge, located close to the root of flange-toweb connection, and the outer one, which is close to the inner edge of bolt hole (Figure 7). Yielding lines are, in fact, almost straight in the first position, while are mainly circular in the second one. Consequently, the effective v^dth used in relation (2) accoimts for this difference by assuming an average value for both hinges. Finally, really speaking, another factors influencing results could be due to membrane actions developed by T-stub flanges, which confers a sort of geometrical hardening on joint behaviour. Such complementary actions beneficially affect the actual specimen response, but it is not taken into account by the simplified procedure suggested by the code.

236 T'Stub type 1 Steel bolt grade 8.8

steel bolt grade 8.8

6%

' • a * 8 0 IN/mm^] " * " a » 1 4 0 IN/mm ] — • " a « 2 0 0 [N/mml —•— a = 3 0 0 [N/mml Displacement [mm]

10

12

14

16

18

20

T'Stub type 2 steel bolt grade 8.8

Steel bolt grade 8.8

Displacement [mm]

'1-2a

6%

8% ' a « 0 IN/mm ]

- • - a « 0 {NAnm 1 • "•

a « 8 0 |N/mm^l

-*"

a « 1 4 0 fN/mm^J

a « 8 0 (N/mm^J a«140 [N/mm^j • a « 2 0 0 JN/mm^l 2 • a « 3 0 0 IN/mm J

- * - a « 2 0 0 |N/mm^] - * — a « 3 0 0 JN/mm^l Dispaeemcnt [mm]

Aluminiumflangef 0,2=240 N/mm^

Displacement [mm]

Aluminiumflangefo.2=120 N/mm^

Figure 6: Numerical F-A relationship for analysed cases The evaluation of effective width In order to provide a correction procedure based on the same simplified relationships suggested by EC3-Annex J, accounting for the effects due to peculiar aspects of aluminium, i.e. strain hardening and alloy available ductility, the effective width should be preUminarily determined. This allows the main

237

imprecision due to the approximate method suggested by ECS to be skipped, hampering the remaining difference to the effect of the hardening and ductihty. According to the basic conceptual definition, the effective width beff may assumed as the one which, in case of not hardening material, is able to reproduce through the relation (2) the same ultimate resistance Fuu as predicted by numerical simulation. For each T-stub geometry, the numerical results for a=0 lead to effective width values summarised in Table 2. Conventionally, the ultimate resistance has been evaluated as the one corresponding to the attainment of a global strain in the most developed plastic hinge equal to 8%. This allows to assume that the plastic mechanism is almost completely developed. TABLE 1 NUMERICAL AND CODIFIED ULTIMATE RESISTANCE FUU [kN]

3

Material Hardening g fN/mm^l

8=3 % 89.3

80

93.9

140

97.1

100.5

Numerical 8=6 %

8=8 %

Codified (EC3-Annex J)

90.0

90.6

90.9

62.7

96.2

99.2

101.6

62.7

105.1

109.6

62.7

8=4 %

200

101.6

105.6

112.4

117.7

62.7

300

107.2

113.0

122.8

130.6

62.7

46.1

46.4

46.8

47.2

31.4

80

51.5

53.1

56.0

58.2

31.4

140

55.3

58.0

62.6

66.3

31.4

200

59.0

62.7

68.9

74.2

31.4

300

64.8

70.4

79.5

87.1

31.4

140.6

142.5

145.6

148.1

107.4

80

146.8

151.1

157.9

163.6

107.4

w):z;

140

148.6

153.7

162.3

169.8

107.4

is

200

154.8

161.8

174.1

184.3

107.4

300

158.3

167.3

183.3

197.4

107.4

71.9

72.9

74.4

75.7

53.0

80

78.4

81.4

86.4

90.4

53.0

140

83.3

87.8

95.4

101.6

53.0

200

88.2

94.2

104.5

112.9

53.0

300

96.3

104.8

119.3

128.0

53.0

203.2

206.2

209.1

210.4

256.0

213.2

219.3

226.7

231.6

256.0

80

JS II

l§ II

140

219.5

227.8

238.4

245.1

256.0

200

225.2

235.5

249.4

258.9

256.0

300

234.1

247.9

266.3

279.9

256.0

107.0

108.0

109.0

110.0

128.0

80

117.8

121.2

126.2

130.1

128.0

140

124.2

129.6

137.7

143.7

128.0

200

130.7

137.4

148.3

157.4

128.0

140.4

149.8

165.8

179.5

128.0

300

238

hinge line

hinge line

Figure 7: Observed yielding lines for plastic hinges (T-stubs type 1 and 2) It can be seen that effective vsddths are slightly dependent on the flange-to-bolt resistance ratio, as basically assumed by EC3-Annex J. Evidently, in case of aluminium, the actual values of beff will be also dependent on the hardening of the material, which should allow for a major spreading of plasticity within the structural component. Nevertheless, exact determinations are quite difficult and may be disregarded in a first tentative approach. TABLE 2 EFFECTIVE WIDTH VALUES [nim]

T-stub type

Flange type /o 2=240 N/mm ^ 2=120 N/mm /o.2=240 N/mm ^ 2=120 N/mm /o.2=240 N/nmi /h2=120N/mm

1 TYPE 1

1 TYPE 2 TYPE 3

Effective width be/f [mm] Numerical value Codified value 57 40 40 59 52 40 54 40 109 138 138 115

The proposed correction factor The comparison between numerical results and the ones obtained by applying the codified procedure showed that concentrated plasticity model proposed by Annex J for interpreting T-stub ultimate resistance is inadequate essentially due to the fact that the system hardening of the material is completely disregarded. The values of ultimate resistance deduced according to the effective widths corrected as previously mentioned, allow the effect of hardening and ductility to be easily evaluated. As a first attempt, in order to incorporate such effects, relationship (2) could be applied by considering the ultimate resistance fu instead of the conventional elastic stress yb.2- In turn, fu may be determined through the material c - e relationship (1), as a fiinction of the assumed limit strain. In fact, it should be intended as a conventional ultimate stress, corresponding to the attainment at least in one point of the section of the conventional ultimate elongation, previously assumed as the uniform strain (s„). In such a way, the T-stub conventional ultimate resistance would be overestimated due to the variation of normal stress through the section. A balancing factor k may be therefore introduced in order to reduce the section modulus respect to the one corresponding to fiilly plastic behaviour (Figure 8). The following corrected relationships may be therefore assumed:

\Kfft M u,mod = fuT (3) k 4 m From the theoretical point of view, the Mactor should be related to the possibility to develop uniform stress distribution through the section depth, being essentially dependent on the hardening of the alloy 4M.u,mod

u,\,mod

239 (a) and the conventional ultimate elongation (Su). In particular, it ranges from 1, in case of nothardening material and unlimited ductility, to 1.5, for an indefinitely elastic material. On the other side, A:-factor also accounts for other aspects that are strictly connected to the adopted simplified model based upon a monodimensional scheme, such as the different behaviour of plastic hinges at bolt location and root of flange-to-web connection. In the present study, ^-factors have been estimated by applying relationship (3), backward, where T-stub resistance values F„ have been evaluated by means of the numerical results referring to collapse mechanism type 1. e«

fy

Figure 8: Schematised plastic hinge behaviour Results are depicted in Figure 9, where the variation of ^-factor is diagrammed as a function of both material hardening and ductility supply. It is shown that for all the examined T-stubs typologies, which should cover the majority of realistic cases, the correction factor k ranges between 1 and L45, increasing with both a and s„ values. In particular, ^-factor seems to be slightly influenced by ductility alloy (with an influence not higher than 15%), while it is more remarkably dependent on the hardening, whose influence is up to 40%. Going deeply into details, it is interesting to observe that in case of no hardening material (a=0), ^-factor decreases as far as available ductility increases, but this is due to the geometrical hardening effect which, owing to the spreading of plastic zones, produces an an increase of the actual effective width. This trend is opposite for a>0, showing that the effect of stress redistribution over the flange thickness is predominant as respect to the increase of the effective width. Likewise, it should be noticed that the A:-factor tendencies are also slightly influenced by the T-stub geometry and flange-to-bolt resistance ratio. The latter concern is particularly important, proving that the T-stub behaviour is also affected by other factors, rather than the ones considered by the codified approach. This is especially due to the actual effective width evolution with the increase of plastic deformations. Finally, it is to be observed that these differences seem to be due separately to both the effects of flange-to-bolt resistance ratio and to the thickness of the flange.

CONCLUSIVE REMARKS Within a general research project devoted to the codification of rational methods to be applied to the prediction of aluminium joint structural behaviour, the current study has been focused on the possibility to profitably use the existing approach for steel joints. In particular, in case of simple T-stub joints characterised by strong bolt-weak T-stub flange ratio, a wide numerical analysis has been carried out, aiming at the calibration of a simplified procedure that could allow the ultimate resistance of T-stub aluminium joints to be simply evaluated. In order to account for the effects due to the material strain hardening as a fimction of tiie available ductility, a balancing ^-factor has been therefore introduced in the same simplified formulation provided by EC3-Annex J. The results obtained have shown that the actual behaviour of T-stub joints is rather complex, being influenced by several factors, which are not all covered by the codified method. In particular, effective width values suggested by EC3-Annex J appear to be inadequate; besides, in case of hardening materials, the evolution of the effective width during the loading process, plays an important role, compelling somewhat variation of ^-factor with Tstub geometry and flange-to-bolt resistance ratios. Anyway, the first results are encouraging, demanding for succeeding steps of the study. Further analyses should be therefore addressed to the definitive assessment of the procedure as well as to its extension to different collapse mechanisms.

240 T-stub flange materialfo.2=240 Wmnf

T-stub flange material f0.2^120 N/mm^ 1.50

T-stub type 1

T-stub type 2

T-stub type 3

T-stub type 1 ^

T-stub type 2

3 ^

1.30

A

tS 1.20

Io

O

1.10

^

j

^

A j^P^

\^f^'^, , ^^, 0

80 140 200

300 80 140 200

300

80 140 200

300

^ 1.30 c o •5 1.20 o

O

1.10

y^

- ^

^ ^

M-—1—1—1 a—1—1—1 0 80 140 200 300 80 140 200

/ / x

///^

L^==^/^

Hardening a [N/mm^J Eu=3%

T-stub type 3

1.40

^

%—1—)—1 300 80 140 200

1 300

i

Hardening a [N/mm^] eu=4%

8u=6%

eu=8%

Figure 9: ^-factor variation for examined cases (collapse mechanism type 1)

ACKNOWLEDGEMENTS This work has been developed in the framework of activity of the research project "Methods of Behaviour Prediction for Aluminium Alloy Joints", sponsored by the Italian Ministry of University and Scientific and Technological Research (MURST). The helpful co-operation of the graduate student Marco Sciarra is also gratefully acknowledged.

REFERENCES Bursi, O. S. and Jaspart, J.P. (1997). Benchmarks for Finite Element Modelling of Bolted Steel Connections. Journal of Constructional Steel Research, 43, (Issue: 1-3), pp. 17-42. De Matteis, G. and Mandara, A. (1997). A Concentrated Plasticity Model for Hardening Material Structures. XVI Congresso C T.A., Ancona, 281 -92. De Matteis, G. Mandara, A. and Mazzolani, F.M. (1998). Numerical Analysis for T-stub Aluminium Joints. 4^^ Int Conf. on "ComputationalStructures Technology", CST'98, Edinburgh. De Matteis, G. Mandara, A. and Mazzolani, F.M. (1999). Interpretative Models for Aluminium Alloy Connections. International Conference on Steel and Aluminium Structures (ICSAS'99), Espoo (Finland) June 20^^-23"^. ENV 1993-1.1 - Eurocode 3 - Annex J, (1997), Joints in Building Frames, CEN/TC250/SC3-PT9. ENV 1999-1.1 - Eurocode 9, (1998), Design ofAluminium Alloy Structures, CEN/TC250/SC9. Hopperstad, O.S.,(1993), Modelling of Cyclic Plasticity with Application to Steel and Aluminium Structures, Dr. Ing. Dissertation, Division of Structural Mechanics, The Norvegian Institute of Technology, Trondheim, Norway. Mandara, A and Mazzolani, F.M. (1995). Behavioural Aspects and Ductility Demand of Aluminium Alloy Structures. International Conference on Steel and Aluminium Structures (ICSAS'95), Istanbul. Mazzolani F.M. (1995), Aluminium Alloy Structures, E & FN SPON, London. Mazzolani, F.M., De Matteis, G. and Mandara, A. (1996). Classification System for Aluminium Alloy Connections. Z4M^M Colloq. on "Semi-RigidStructural Connections", Istanbul

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

241

IMPROVED BEAM-TO-COLUMN JOINTS FOR MOMENT-RESISTING FRAMES AN EXPERIMENTAL STUDY P. Sotirov, N. Rangelov, O. Ganchev, Tz. Georgiev, Z.B. Petkov and J. Milev Faculty of Civil Engineering, UACEG, Boul. Hristo Smimenski 1, 1421 Sofia, Bulgaria

ABSTRACT The good performance of steel moment-resisting frames under seismic motions depends strongly on the behaviour of their beam-to-column connections. The Northridge earthquake generated an urgent need to increase the ductility of the connections normally used in practice. This paper presents the study of beam-to-column joints in column-tree configuration of moment-resisting frames. Tapered flanges are used to improve the seismic behaviour of beams. Two types of joints are proposed and tested: type A, considered as an improved pre-Northridge joint, and type B, as a post-Northridge detail. Experimental investigation on six (3+3) full-scale specimens was carried out. The tests were performed in the Steel Structures Research Laboratory of the University of Architecture, Civil Engineering and Geodesy (UACEG), Sofia. Both specimen types exhibited stable hysteretic behaviour with adequate ductility and energy dissipation capacity. Though both variants are quantitatively comparable, the most significant advantage of type B is found in its predictable behaviour.

KEYWORDS Beam-to-column connections, moment-resisting frames, column-tree configuration, ductility, cyclic hysteretic response, plastic rotation, energy dissipation, dog bone, experimental study. INTRODUCTION Steel moment-resisting frames have been popular in many regions of high seismicity not only for their architectural versatility, but also for their good seismic performance as highly ductile systems. However, the seismic response of a ductile moment frame will be satisfactory only if the connections between the framing members have sufficient strength, stiffness and plastic deformation capacity. The traditional strong-column — weak-beam design concept implies that the input energy is absorbed and dissipated primarily by plastic hinges formed at the beam-to-column connections. Experimental

242

work carried out by Popov et al. (1986), demonstrated a lack of deformation capacity of the welded flange — bolted web connections, normally used in practice. The Northridge earthquake definitely proved that conclusion and generated an urgent need for improved detailing. Two key strategies have been developed: (i) strengthening the connection, and (ii) weakening the beam that frame into the connection (Bruneau et aL, 1998). Within the framework of (i) cover plates, upstanding ribs, side plates, and haimches are implemented. The weakening strategies (ii) are based on the idea of shaving beam flanges to intentionally weaken the beam at a predetermined location, originally proposed by Plumier (1990). Besides the classical "dog bone" profile and circular cuts, Chen et al. (1996) proposed to taper flanges according to a linear profile so as to approximately follow the varying bending moment diagram. Popov et al. (1996) suggested a combination between the two strategies, namely a cover plates connection with circular cuts in the beam flanges. Both strategies aim at effectively moving the plastic hinge away from the column face, thus avoiding the problem of poor ductile behaviour and potential fragility of the welds. All the proposed and tested connections have some merits and disadvantages. It seems that no detail can be assumed to be perfect and new solutions are still worth exploring. In Bulgarian practice, there are two features regarding the steel moment-resisting frames: built-up members which have the advantage to be proportioned more flexibly to attain a better balance between internal forces and resistance, and the column-tree configuration witib fully shop-welded rigid beam-tocolumn connections and field-welded erection splices in the beams. High ductility can be achieved by several tj'pes of connections, however fully welded rigid connections are still reckoned to possess the highest dissipation capacity (Mazzolani & Piluso, 1996). Two different beam-to-column joints are considered in the present study. In both of them tapered flanges are used for improvement of the seismic behaviour of the beams. An experimental programme is reported, carried out on six full-scale specimens. Shop-weided connection

i«aa»m^KP««|KP»l4**^«wBcsag&sjk9i;-«4V9»K«o«i|j»jK«an^

\y provided bending capacity - A ^provided bending c a p a c i t y - 6 T h e Idea

^ P'l bending moment diagram (demand) ,!v!||'

Figure 1 : Beam-to-colounm joints considered

Q

I

243

BEAM-TO-COLUMN JOINTS UNDER CONSIDERATION The two types of joints are shown in Figure 1. In type A the flanges are tapered to follow approximately the varying bending moment diagram. The solution is more economical, and, under an earthquake load, the yielded zones in flanges will be larger, thus dissipating more energy than in the conventional non-tapered flanges. However, the expected dissipative zone is mostly at the column face. Therefore type A is to be regarded as an improved pre-Northridge joint. Detail B is of post-Northridge type. The shifting of the plastic hinge from the column flange is achieved by a specific flange profile, following the idea illustrated in Figure 1. Thus both the abovementioned strategies are incorporated in one detail. On the one hand, the connection is strengthened without additional cover plates, ribs or haunches, keeping a constant beam depth, and, on the other hand, the beam is weakened at a predetermined location to provoke the development of a plastic hinge. TEST SPECIMENS, EXPERIMENTAL SETUP AND LOADING HISTORY The test beam-to-column assemblies were fiill-scale, "extracted"fi-oman especially designed two-bay four-storey regular moment-resisting fi-ame. The specimens were fabricated by a Bulgarian industrial manufacturer. The column sections were produced by automatic welding under flux, whereas the beam sections, small in size for the welding equipment, were built-up by semiautomatic welding under CO2 shield. ThefiiUpenetration welds between beam and column flanges were also made under gas shield. The erection splices were hand-welded with covered electrodes in the laboratory, thus simulating the field conditions. The steel grade used corresponds to S235. A total of 6 specimens were tested: 3 identical ones (labeled A-1, A-2 and A-3) of type A, and 3 identical specimens (B-1, B-2 and B-3) of type B. The layout and main dimensions are shown in Figure 2.

Beam sections: Flanges'. 12x120 to 12x220 for A 12x120 to 12x280 for B Web: 8x350 next to column 6x350 in the span Column section: Flanges'. 16x300 Web'. 10x300

hydraulic

Figure 2: Test specimens and experimental setup

244

The test setup was designed to accommodate specimens in a horizontal position as shown in the figure. The load was applied to the cantilever beam end by a hydraulic actuator with a displacement range of ±200mm and a capacity of ±200kN. No axial load was applied to the column. To prevent out of plane motion of the beam, lateral bracing was provided near the beam end and near the splice. A total of 14 inductive displacement transducers were installed (Figure 2) to measure the global behaviour of the specimens: beam end displacement, joint rotation, panel zone shear deformation, and possible movement of the supports. Strain gauges were used to investigate the local response of the beam flanges including the splice plates. All the instruments were connected to a PC-based data acquisition system with software developed by the laboratory staff. A Keithly DAC-02 card was used to send the displacement command signal to the actuator servovalve. The testing programme was based on the recommendations of ECCS (1986). The specimens were tested under displacement control, following a loading history consisting of stepwise increasing deformation cycles. Initially, four cycles in elastic range were applied with amplitudes of ±0.25vy, ±0.50vy, ±0.75vy, and ±Vy, where Vy is the expected first-yield displacement of the beam end. Eventually, correction of Vy was made. Then the testing continued in the plastic range with three full cycles at each amplitude level ±2vy, ±3vy, and so on (instead of the recommended by ECCS (1986) ±(2+2«)vy; the latter was only applied to specimen A-3). For specimens A-3 and B-3 an initial displacement corresponding to approximately 30% of the plastic bending capacity was applied to simulate the effect of the gravity loading. A typical loading history and the corresponding actuator force response are shown in Figure 3. 200

I 150 ^ 100

l-^

,J

iiiii'i

V

B

50

W

.S

CL w

0

io -50

4

I-100 I -150 4 < -200

(«)

-----

\

il

I-I

200 400 600 800 Number of reading {'time')

-200

1000

(b)

200 400 600 800 Number of reading ('time')

1000

Figure 3: Typical loading history (a) and force response (b)

EXPERIMENTAL RESULTS AND DISCUSSION In type A specimens plastic deformations started to develop first in the panel zone and later in the flanges near the column. Satisfactory stable hysteretic behaviour was observed. The first tested specimen, A-1, failed by lateral-torsional buckling due to improper bracing. In specimen A-2 flange local buckling occurred, followed by web buckling. Shear deformations of the panel zone were apparent, demonstrating its very significant contribution. This was more pronounced in A-3, where no local buckling appeared. Both A-2 and A-3 failed due tofiractureof the upper flange splice plate. The datafi*omthe strain gauges showed that considerable plastic deformations had developed there. Due to some lack of overstrength and different areas of the two splice plates, the top one became overloaded. This fact, together with the stress concentration, led to low-cycle fatigue. However, no other cracks

245 450

450

300

300

S

7:

;z; ^ 150

S o

s 150 -300

-450 -0.075 -0.05 -0.025

0

0.025

0.05

0.075

150

I

0

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!iii 1",

1 1 1 ' III

II i l l 1 I I

if i

ii|"f] 0

0.025

0.05

0.075

0.05

0.075

0.05

0.075

Plastic rotation [rad] 450

450

^

fi:

-450 -0.075 -0.05 -0.025

Plastic rotation [rad]

300

i

flilu 1 \1 1

o

-300

-l±iJ

-EO

71

/i^Mi--J ill if 1

11 mmm 1 ill //7

iLL^^^^**^ ;

-450 -0.075 -0.05 -0.025

; -450

0

0.025

0.05

0.075

-0.075 -0.05 -0.025

Plastic rotation [rad]

0

0.025

Plastic rotation [rad] 450

450

-0.075 -0.05 -0.025

0

0.025

Plastic rotation [rad]

0.05

0.075

-450 -0.075 -0.05 -0.025

0

0.025

Plastic rotation [rad]

Figure 4: Hysteretic behaviour in terms of moment versus total ( ) and beam (—) plastic rotation (moment and rotation determined for the reference cross-section: at the column face for type A, and at the theoretically expected location of the plastic hinge for type B)

246 were observed elsewhere. The good behaviour of the shop-welded full penetration welds was clearly proven. All type B specimens exhibited a very stable cyclic hysteretic response, and behaved exactly as predicted. Plastic zones developed at the designed location, accompanied by flange local buckling in the three specimens. Finally, the resulting secondary bending stresses at the crest of the buckles led to a low-cycle fatigue rupture of the buckled flange. No other cracks were detected either in the web-toflangefilletwelds in that region, or elsewhere. Both in A-3 and B-3, the effect of initial gravity loading was negligible and vanished well within the plastic range. The cyclic response of the joints is illustrated in Figure 4 in terms of moment versus plastic rotation. The latter is obtained using the elastic stiffness determined from the experimental curve by a linear fit. The plastic rotation in the beam is separated to assess the relative contribution of the plastic zone in the beam and the panel zone. Excluding A-1, all joints exhibit adequate ductility. In both types A and B, total plastic rotations of 0,05 rad were reached without a brittle fracture, which is well above the demand of 0.03 rad adopted recently (Bruneau et al., 1998). However, due to the different contribution of the panel zone, beam plastic rotations reached in type A were 0.02-0.025 rad, whereas in type B they were up to 0.03-0.04rad. A very important parameter characterizing the structural behaviour of a joint under seismic loading is its energy dissipation capacity. Herein, it is determined as the plastic work based on the hysteretic area of the experimental loops (Figure 4), and is plotted in Figure 5 versus the relevant cumulative beam end displacement. Regarding type A specimens, one can see that the contribution of the plastic zones in the beams is only about 30%. However, the check of the local response showed that theflangeshad

g400

1 iJ-i i;

300 - ^

;.. Total

I

-^X^-

J<'

j ^ ^

--\---i

^2 ^'

1

B'i y

.« 200 Beam contribution

S 100 0

- ^ • ^ ^ ^ ^ ^ 1 ^ i— 1000 2000 3000

1 4000

j-

j

j

5000

6000

7000

Cumulative beam end displacement [mm] Figure 5: Dissipated energy

8000

247

yielded within the whole region between the column and the splice, thus ensuring higher energy dissipation than in the case of constant cross-section, where the yielded zone would be more localized and therefore the plastic volume would be smaller. The higher gradient of the curves representing A-3 is due to lack of local buckling and probably to the different loading history. For type B, the contribution of the beam plastic zones is about 50% and only for B-1 is -60%, due to earlier local buckling. Note the very close agreement of the curves for the total dissipated energy, demonstrating similar behaviour of the three specimens. To compare the dissipation capacity of the different specimens, the nondimensional dissipated energy is defined as £* = ^pl

(1) ^tot

where E is the dissipated energy (plastic hysteretic work), Fpi is the force at the beam end corresponding to the theoretical plastic resistance moment at the reference cross-section (see the caption for Figure 4), and Vtot is the total cumulative beam end displacement. Thus, referring the dissipated energy to that of a relevant perfectly plastic system, it is believed that not only the different bending resistance of the two types of specimens, but also some variations in the loading history may be accounted for. 0.8 B column & panel zone contribution

0.7 0.6 :2fe3

0.5 0.4 0.3

• beam contribution

0.2 e o Z.

0.1 0 A'1

A-2

A'3 B-1 Test specimen

B-2

B-3

Figure 6: Relative dissipation capacity The comparative results are shown in Figure 6. In this format, firstly, the different contributions of the two components - beams and panel zones - are well seen. Secondly, the relative dissipation capacity of type B specimens is on average some 20% higher. Moreover, the portion of the energy dissipated only by the beam plastic zones in specimens B is almost twice higher, which undoubtedly is an additional advantage.

CONCLUDING REMARKS In this paper, an experimental study is reported on two types of full-scale beam-to-column assemblies. Type A can be regarded as an improved pre-Northridge joint, while type B is of post-Northridge type. Cyclic tests were carried out following the ECCS (1986) Recommendations. Both specimen types exhibited stable hysteretic behaviour with adequate ductility. The plastic rotation capacity of type A is largely due to the flexible panel zone. The energy dissipation capacity of both types is high, type B being some 20% better. Though the two variants are quantitatively comparable, the most significant advantage of type B lies in its predictable behaviour. The lack of a structural fuse in type A leads to some variable behaviour and imposes higher demands to the panel zone.

248

It should be concluded that both joint typologies considered exhibited satisfactory cyclic behaviour and high dissipation capacity. However, type B, having behaved exactly as predicted, possesses some definite advantages, therefore should be preferred in the design practice.

ACKNOWLEDGEMENTS The present study has been carried out in the framework of the INCO-COPERNICUS Joint Research Project '^Reliability of Moment Resistant Connections of Steel Building Frames in Seismic Areas'' (RECOS). Thanks are therefore due to all the partners for their comments and advices, and especially to Professor F. M. Mazzolani, the coordinator of the Project. The financial support of the EU is also gratefully acknowledged.

REFERENCES Bruneau M., Uang C.-M. and Whittaker A., (1998). Ductile Design of Steel Structures, McGraw-Hill, New York, USA. ChenS.-J., YehC.H. and ChuJ.M. (1996). Ductile steel beam-to-column connections for seismic resistance. ASCE, Journal of Structural Engineering 122:11,1292-1299. ECCS (1986), TWG1.3. Recommended Testing Procedures for Assessing the Behaviour of Structural Elements under Cyclic Loads. ECCS Publication JV245. Mazzolani P.M. and Piluso V. (1986). Theory and Design of Seismic Resistant Steel Frames, E & FN Spon, London, UK. Plumier A. (1990). New Idea for Safe Structures in Seismic Zones, University of Liege, lABSE Symposium, Brussels, Belgium. Popov E.P., AminN.R. and Louie J.C. (1986). Cyclic behaviour of large beam-column assemblies. Engineering Journal 23:1, 9-23. Popov E.P., BlondetM. and StepanovL. (1996). Application of "Dog Bones" for Improvement of Seismic Behaviour of Steel Connections, Report No.UCB/EERC 96/05, University of California, Berkeley, USA.

Technical p a p e r s on L O C A L DUCTILITY

This Page Intentionally Left Blank

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

249

DUCTILITY OF IPE AND HEA BEAMS AND BEAM-COLUMNS A.Anastasiadis\ V.Gioncu^ ^ Politehnica University Timisoara, 1900 Timisoara, str. Traian Lalescu 2, Romania

ABSTRACT The paper deals with the determining of available local ductility of steel beams and beam-colimms made by hot-rolled IPE and HEA profiles. Using the local plastic mechanism the influence of junction between flange and web was investigated. Based on the analysis of a great number of nimierical tests, using DUCTROT'96 computer program, simplified design relationships are proposed for direct checking of rotational capacity of beams and beam-columns. A member behavioral classes for IPE and HEA also are presented. KEYWORDS Local ductility. Local plastic mechanism. Available ductility. Hot-rolled sections, DUCTROT computer program, simplified design relationships.

INTRODUCTION An efficient seismic design requires to use the method of capacity design, which is based on the principle to obtain a global plastic mechanism, ensuring the formation of sufficient number of plastic hinges only in the beam ends and not in the columns. But assure of a good structural behaviour requires also the verification of rotation capacity of these plastic hinges, which must be greater than the necessary rotational demands (Gioncu & Petcu, 1997). So, the ductility capacity of a structure strongly depends on available ductility of individual structural members that constitute the structure, and first of all, these ductility capacities should be quantified. Current steel and aseismic codes as EC3, EC8, includes only qualitative specifications without a clear definition of 'sufficient ductility' and how these ductility must be assured, not only by constructional details but with a direct checking providedfi*oma practical method. The classification of cross section ductility classes, as given in ECS, contains many shortcomings presented by Gioncu and Mazzolani (1995) and must be reconsidered in order to evaluate the real inelastic deformation capacity of structural members, taking into account the member span. The present study deals with the assessment of available ductility of IPE and HEA sections, widely used in practice as beams and beam-columns. In the first part of the paper the rotation capacity, based on the local plastic mechanism and the influence of junction on double T sections is presented and

250

investigated. In the second part, using a great number of numerical tests, simplified design relationships for available rotation capacity of beams and beam-columns is proposed. AVAILABLE ROTATION CAPACITY An adequate measure for determining local ductility of steel members is the rotation capacity of plastic hinges which can be calculated by different ways: (i) using FEM methods, (ii) integration of the moment-curvature relationship, (iii) using a collapse mechanism coming from experimental evidence, (iv) determining the effective width, (v) approximate formulae, (vi) simple formulae obtained from statistical analysis of experimental results. Among these methods, the use of plastic collapse mechanism method seems to be the most efficiently for design purposes (Gioncu & Petcu 1997, Gioncu & Mazzolani, 1998). The formula to calculate this rotation capacity, R, determined in the lowering postbuckling curve at the intersection with the theoretical full plastic moment, taking into account both stable and unstable part of curve, is given by (Fig.l):

where: 0u- ultimate rotation obtained with the intersection of theoretical full plastic moment 0p- elastic rotation at the level of fiill plastic moment However, for a complete definition of available local ductility, it is essential to be defined also the fracture rotation capacity, Rf, for which the first crack occurs in the buckled flange. This fracture could be developed in the stable or imstable part of curve, depending on geometrical confonnation of member, given by (Fig. 1): Rf=7^"l

(2)

where: 0r fracture rotation of a buckled flange 0p- elastic rotation at the level of full plastic moment This fracture ductility, strongly depends on the yield ratio, buckled length and ultimate strain of the used steel (Gioncu & Mazzolani, 1999, Anastasiadis, 1999). It is clearly that for a good behaviour of plastic hinges must be assured that: R„.p^Rf where: Rav.p- plastic available rotation capacity of a member Rf- fracture rotation capacity

(3)

The available rotation capacity of a plastic hinge must be determined taking into account that the member belongs to a structure with a complex behaviour. This complex behaviour is studied using of a simple substitute member with very similar behaviour (Gioncu & Petcu, 1997). In Fig, 2 is presented the determination manner of 'standard beam' span for beams and beam-columns, for seismic loads. Fig. 2a, and both for gravitational and seismic loads, Fig.2b. For columns, the influencing factors is the ratio between upper and lower column bending moment.

251 I M/Mp

fracture by tension

"WDT fracture by compression

Ou/ep ef/ep -e/ep

Figure 1: Definition of rotation capacity

t

•—r—i~-T-—] "-T-T—t

•'1

a)

1

3:

+ ^1

t—^

A

4

*

- 4

1

1

1-/2

t

I-/2

.t

1

1

21

m

a

&

^

B—t

a

X

Lsb«2(L-xf)

*

« *

i

1

i

*

4

*

•]

31

t- -

4.

L

i,

Figure 2: Determination of 'standard beam' for beams and beam-columns A computer program DUCTROT'96 for welded sections was elaborated at INCERC Timisoara on the basis of the local mechanism of the above mentioned standard beam (Gioncu & Petcu, 1996). The values obtained using this computer program, relies a good correspondence with experimental data, giving confidence in the theoretical results (Gioncu & Petcu, 1997, Gioncu & Mazzolani, 1998, Anastasiadis, 1999). A problem which appears in design practice is relies to the hot-rolled profiles, namely if the results obtained with DUCTROT can be used for these sections, when the presence of junction reduces the flange slendemess.

252

INFLUENCE OF JUNCTION ON PLASTIC ROTATION CAPACITY One of the main fectors influencing section ductility, clearly affecting the superior level, member available ductility, is the mode of fabrication. Hot-rolled profiles widely used in structural design provide different ductility capacity than welded sections (Gioncu & Mazzolani, 1999, Anastasiadis, 1999). In the plastic mechanism method for hot-rolled sections, the ultimate plastic rotation, Ou, of plastic hinge is determined, in the same way as for welded sections, using a local plastic quasimechanism composed by plastic zones and yield lines, taking into account the influence ofrigidzone created by the junction of flange and web, Fig.3 (Anastasiadis, 1999, Piluso, 1995). The comparison between theoretical and experimental values, for improved proposed new shape mechanism is presented in Fig.4 showing a good correspondence.

Figure 3: Local plastic mechanism In Figs.5 and 6 the influence of jimction for different IPE and HEA beams is plotted. One can see the important increasing of plastic rotation capacity of hot-rolled sections as compared with the same sections in which the influence of a rigid zone is neglected. For IPE beams the increasing is about 64%, while for HEA beams it is about 38-48%. In the aim to use the results obtained using the DUCTROT'96 computer program a simplified coefficient of correction, Cr, is proposed;

where:

b , b c =(-)^=( ' ^c^ %~0.5t„-0.8r b- half width flange c- width between half widthflangeand junction r- junction of section between web and flange

(4)

253 35 1 in -

^^

1 Cv = 0.306

25 -

4^'-'

'^n 1^ ij

J^mi ,.r'

7f^ r

10 -

• Baeraeve etal, 1993 AFEM

A-

z

[Jit' ()

• Spaiighemaher,1991 1

.•^ • ! , ^ A ^-l*

"TB

c_

•

^

p = 0.932

;

10

15

30

25

20

35

Rexp

Figure 4: Correlation between theoretical and experimental data 17,5 .

;

;

1

;

15,5 -

jr

13,5 -

1

i^-*<^^ "~~

\

11,5 9,5-

—

;

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7,5 - - - J r - * rrHlrrtTTT^....: 5,5 3,5 -

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•:

L^SOOOmm (Fe360)

^m •

J...,_,.!^.L-

j^»'r* i

• - - — •• — 1—*" ;

"•"" Junction

;

;

IPE 400

IPE 500

D^E 6 00

Proffl

IPE 240

IPE 300

IPE 360

Figure 5: Influence ofjunction on IPE profiles

at

HE200A

HE240A

HE300A HE360A Profii

1

- 4 1 — Without junction 1

;

1,5 IPE 200

jr

HE400A

HE500A

Figure 6: Influence of junction on HEA profiles

254

The correlation between the exact values and the corrected values obtained using DUCTROT '96 and relation (4) is presented in Fig.7, showing that the simple methodology allows to determine the improved values for rotation capacity of hot rolled sections. 30 25

-^^

t 20

1 15 ^ 10 5

BPE, HEA ICv =0.144 p=0.936

.• A# ^

J^

iJ^

o¥10

15 Rcalc

20

25

30

Figure 7: Correlation between exact and corrected values SIMPLIFIED DESIGN RELATIONSHIPS FOR ROTATION CAPACITY Rotation capacity of beams Design relationship, based on the very high number of numerical tests using DUCTROT computer program is proposed for beams under monotonic loads (Gioncu & Mazzolani, 1998): R...O. =3X10^C,-^E[O.8+0.2^J ; e=235 / f^

(5)

where: Cr-coefiRcient taking into account the influence ofjunction tr thickness of flange b- half width of flange Lsb- span of standard beam fyw, fyT yield tensile strength for web and flange In relationship (5), in comparison with the original equation, a new coefficient, c^, is introduced for considering the effect of flange and web junction. Rotation capacity of beam-columns In the same manner as for beam, an approximate relationship for beam-column rotation capacity considering the effect of axial force and moment diagrams is proposed (Anastasiadis, 1999): K.^ -im.3clxy^^^]

R»m«, =5099 av.mon

where: b/ tr flange slendemess

•^'i^%^T''

np=0.10

(6a)

np= 0.40

(6b)

255 X - non dimensional slenderaess >i =

Tc'ET

fy- yield tensile strength of section Lgb- span of standard beam Over 160 numerical tests were performed to obtain an adequate statistical parameters. Fig. 8. The resuhs from the relationships (6a,b) cover the practical domain for HE 100A-HE600A considering the ratio of moments, M^^ I Minf=1...0 as well as the influence of axial force, np. 1,5

#

1,4

•

1,3

^

I 0,8

.

1

• np = 0.10 (DuctRot)

r* -

t

i

Cv=0.11

t|Hrrvr:::::

ii==160

_; p=0,967

^ 0,7 0,6 0,5 0,4 J

i

(

1

R

1,3 1,2 1 1 1,1 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 -1

^k

At

^

^ ^ I

j ^^T9^f^

#-

• n p = 0.40 (DuctRot)

Cv« 0.106 ii»160

p = 0.978

\

i

R

10

20

30

Figure 8: Correlation between the relation 6a,b and D U C T R O T The above relationships take into accoimt only the basic factors affecting the monotonic rotation capacity, Rooon- However, other factors influencing local ductility is given elsewhere (Gioncu &Mazzolani, 1999, Anastasiadis, 1999). For seismic available ductility, Rseism. a simple design formulae was proposed by Gioncu & Petcu (1997): Rseism""- h ( b / t f , ttp) Rmon

(7)

M £ M B £ R BEHAVIOUHAL CLASSES FOR EPE AND HEA PROFILES It was demonstrated that the concept of cross-section behavioural classes does not correspond with the real inelastic deformation capacity. As a consequence this concept should be substituted by the member behavioural classes approach (Gioncu & Petcu, 1997, Mazzolani &Piluso, 1995). A new classification according to the member concept is proposed for hot-rolled sections, based on the classification criteria (High ductility, H, R >7.5, Medium ductility, M, 4.5
256 TABLE 1 MEMBER CLASSIFICATION FOR BENDING BEAMS

CONCLUSIONS Available local ductility of hot-rolled beam and beam column was investigated by approximate methods using local plastic mechanism and DUCTROT computer program. It was found that the influence of flange and web junction is very high, the rotation capacity showing an important increasing for hot-rolled sections. Simplified relationships for beams and columns are proposed for design purposes. For IPE and HEA members, a classification infimctionof member classes also was proposed. It must be underlined that due to the results obtained in the last decade on the field of local and global ductility it is the time to introduce in ECS an annex containing proposals for directly checking of structure ductility, as the same level as for strength and rigidity.

257 TABEL 2 MEMBER CLASSIFICATION FOR ELEMENTS UNDER COMBINED COMPRESSING AND BENDING

Profile

J3L

M-N Msup /Minf

iMii Mi

HP

iii

H=3000

H=5000

R BM

iSiiiiti

WSm^ V^Um. iii^^M^

iliilli

•••I

liiiliii

fliiiili

I^iliP " "lifiliill

iiMilil

•aiiili

liiiMt

Fe430 H^OOO

H=5000

IMii

iiill iliii

^^

i«^|

HI

liiiiiiiK]

^^H •••'•"• • l a i '

HE340A

H=3000

liiiiiiii

iill

HE240A

Fe360 H«4000

,._.ilBii:"' ^ i»

^

liiiMiili WM ^^m

•«iiil

v^i^^^ll^S^gm

REFERENCES Anastasiadis A. (1999). Ductility Problems of Steel Moment Resisting Frames, Ph. D dissertation, Politehnica University Timisoara, Romania (in romanian) Gioncu V. and Mazzolani F.M. (1995). Alternative methods for assessing local ductility, in Behaviour of Steel Structures in Seismic Areas, E&FN Spon, U.K, 182-190. Gioncu V. and Mazzolani F.M (1998). Simplified approch for evaluating the rotation capacity of double T steel sections. In Behaviour of Steel Structures in Seismic Areas, E&FN Spon, U.K, Gioncu V. and Mazzolani F.M(1999): Ductility of Seismic Resistant Steel Structures, accepted for publication, E& FN Spon, U.K Gioncu V. and Petcu D. (1996). DUCTROT'96: Plastic rotation of steel beams and beam-columns. Guide for Users, INCERC Timisoara, Romania Gioncu V. and Petcu. D (1997). Available rotation capacity of wide-flange beams and beam-columns: Part IiTheoretical approches. J. Construct. Steel Research 43:1-3,161-217. Gioncu V. and Petcu. D (1997). Available rotation capacity of wide-flange beams and beam-columns: Part II: Experimental and numerical tests, J. Construct. Steel Research 43:1-3,219-244. Mazzolani F.M and Piluso V.(1993): Member behavioural classes of steel beams and beam-columns. XIV Congreso, CTA Italy, 405-416 Piluso V. (1995). Post-local buckling behaviour of rolled steel beams subjected to nonuniform bending. XV Congresso CTA, Italy, 542-562

This Page Intentionally Left Blank

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

259

RELIABILITY OF JOINT SYSTEMS FOR IMPROVING THE DUCTILITY OF MR-FRAMES

A.Anastasiadis\ G. Mateescu^, V.Gioncu , F.M. Mazzolani ^Politehnica University Timisoara, 1900 Timisoara, Str T.Lalescu 2, Romania ^ INCERC Timisoara, 1900 Timisoara, Str. T. Lalescu 2, Romania 3iUniversity of Naples 'Federico 11', Italy

ABSTRACT Two important issues regarding steel moment resisting frames (MRFs) are investigated in this paper. Firstly, a parametrical study was conducted on local ductility of new joint types resulted by the beam section reduction or by the beam-to-column joint strengthening, using DUCTROT'96 computer program. The second part deals with the inelastic global performance of different MRFs types subjected to some scaled ground motions. Usmg Drain-2D computer program, ductility requirements were determined. The numerical results indicate that the proposed modified MRFs could be an effective solution for controlling structural response, but in some specific cases the new solutions must be used with caution.

KEYWORDS Moment resisting frames (MRFs), Reliability of joint details, Plastic rotation capacity, Lielastic global analysis, Earthquake, Ductility.

INTRODUCTION Steel moment resisting frames are the most popular structural system in many high seismicity areas for several reasons. Firstly, moment resisting frames (MRFs) are considered as highly dissipative systems. Secondly, moment resisting frames are widely used because of their simplicity in execution, as well as due to their architectural versatility. During the last severe earthquakes (Michoacan, 1985, Loma Prieta, 1989, Northridge, 1994, Kobe, 1995), many modem steel moment resisting frames were seriously damaged, challenging the assumption of high ductile systems. However, evidence of global structural collapses has been reported in few occasions only (for instance Pino Suarez Building, Mexico, 1985).The main lessons leamed from the recent earthquake events (Northridge and Kobe) are related to the important differences between the characteristics of earthquakes recorded at different distances from the source (near field vs. far field), as well as the incomplete understanding of the inelastic behaviour of beam-to-column joints in moment resisting frames (Gioncu & Mazzolani, 1999).

260 Unprecedented cases of brittle and unpredicted failure of welded-flanges-bolted-web connections were identified in rigid frames, designed as special (ductile) moment-resisting frames. In all cases there was little, or no evidence, that plastic hinges had developed in the beams prior to weld fractures. As a consequence, a variety of ideas for improving the jomt inelastic behaviour by reducing the beam section or by strengthening the joint, were proposed (Plumier, 1996, Chen et al, 1997, SAC, 1995). hi the case of frames, recent investigations confirmed that the beam-to-column joints have fundamental importance in seismic response, because dissipative zones have to be located at the beam ends, and that their rotational ductility supply is strictly dependent to the detailing of connections (Mazzolani, 1998). hi this way, inelastic behaviour, dynamic characteristics, natural period and influence of superior modes, must be evaluated related to different site ground motions, for ensuring the reliability of the chosen joint details. The subject of the present paper is focused on two main aspects: the local performance of the joint details, resulted by weakening the beam section or strengthening the joint, and the effectiveness of the overall seismic behavior of a modified moment-resisting frame, as compared with the special and ordinary frames, for different characteristic earthquakes. Using the structural analysis package DrainID, global ductility requirements were evaluated, while for determining the available local ductility, the DUCTROT'96 computer program was used.

LOCAL PERFORMANCE OF THE MODIFIED MOMENT RESISTING JOINTS For detennining the local available plastic rotation, 0p av, a proper methodology based on the local plastic mechanism and the 'standard beam concept' was used (Gioncu & Petcu, 1997). A computer program DUCTROT'96 has been elaborated at INCERC Timisoara (Gioncu & Petcu,1996) for evaluation of the ductility of local plastic mechanism (Fig. I). Based on the high number of theoretical and experimental data, the available plastic rotation is proposed to be determined by the following relation which corrects the values obtained by DUCTROT'96, taking into account some factors affecting local ductility (Anastasiadis, 1999): ep..=c,^e,.„

(1)

where: 0p av - available plastic rotation capacity; Gp.u - ultimate plastic rotation capacity determined using DUCTROT'96; Cr - coefficient taking into account the influence ofjunction between flange-web in case of hot rolled sections (1.63...1.76 for IPE profiles and 1.50. ..1.69 for HE profiles); r^ - coefficient taking into account the influence of incomplete mechanism, ( rw « 0.75.. .0.50); rs - coefficient taking into account the cyclic seismic action as a function of flange slendemess, 0.80. Local ductility of reduced-beam section (Dog-bone solution) During severe earthquakes, great moment capacity demands are developed at the face of column, producing high stress concentrations in this region. When a cross section of the beam near to the beamto-column interface is reduced at a selected location, having a smaller moment capacity than required, the first plastic hinge would form at that location, away fi-om the column face, protecting the joint wedings. A reduced beam section illustrating the main geometrical parameters, as well as the concept of sizing such sections is presented in Figure 2a. The influence of geometrical characteristics on the beam ductility is plotted in Figure 3.

261

„1P

Local buckling

Local plastic mechanism

Standard beam concept

Figure 1: Determination of available plastic rotation capacity Dog-bone solution

/T''*>n

p^ Hzb

Reinforced solution

\

I

.-JL

yh4_^i3_4

Ui-41^ Provided moment capacity

Provided moment capacity

Mp.StT'l

Required moment capacity

Required moment capacity

MC

^p.red

"p,

-""^

a)

b)

Figure 2: Analyzed moment joints

262 IPE300(Fe360) L=5000inin, Lni=165mm Mp /qL^=O.30

0,089 0,082 0,075 0,068 0,061 0,054 0,047 f:;^ihSl--:ti

;4^

0,04

0,297 0,9

0,377 0,83

0,42 0,8

Li/Lt bi/b

Figure 3: Influence of geometrical parameters High plastic rotation capacities can be achieved in the case of 'dog-bone' solution. When beam flange reduction is about 40%, an increasing with 37% of rotation capacity can be obtained , as compared with the normal unreduced beam section. It is very important to consider in analysis the influence of gravitational loads, otherwise the above mentioned effect of weakening could be questionable (Fig.4). A more comprehensive parametrical study about the factors influencing the local ductility of dog-bone solution is given by Anastasiadis & Gioncu (1998).

IPS 300(Fe 360) L"5000mm, Lin-165mm| bl / b»0.83

Ms- Moment from seismic loading MS-KJ -Momentfromseismic and gravitational loading

0,7

0,8

0,9

1

•»2/b

Figure 4: Influence of gravitational loads Ductility of reinforced beam section A second solution for improving ductility of moment-resisting joints is the application of different joint strengthening schemes, using ribs, haunches, cover plates, side plates, etc. (SAC, 1995). The dimensions of strengthened beam with ribs are presented in Figure 2b. The influence of ribs length and the plastic hinges position on the plastic available rotation capacity is plotted in Figure 5. One can see an increasing of this capacity of about 8%, comparing with the unreinforced beam section. Experimental evidence shows that for these type of strengthening the plastic hinge occurs at the distance d / 3 from the edge of the reinforced connection. From Figure 5 it can be observed that relocating the plastic hinge away from the ribs end, an additional rotation capacity can be achieved, of about 4%, as compared with the situation when plastic hinge is formed exactly at the edge of the rib. One can see that the reinforced beam section is less effective than the dog-bone solution.

263

IPE 300, Lr ^lOOmm L=5000iiim, Mp/qL^=0.30

0,069

-Lr

0,067 0,065 0,063 0,061 0,059 0,057 0,055 0

0,01

0,02

0,03

Lr/L

Figure 5: Influence of rib length

GLOBAL PERFORMANCE OF DIFFERENT MRFs TYPES Considerations of the parametrical study The parametrical study is developed for 3 stories - 2 bay frames, as a function of the inelastic behaviour demands due to the different conformation of the frames. Special moment-resisting frames, SMRF, ordinary moment-resisting frames, OMRF, as well as modified moment-resistmg frames, the dog-bone frame, DBF, and reinforced frames, RF, were investigated. The geometry of frames and the gravitational loads are presented in Figure 6a. SMRF is sized to form a global mechanism, having HE280B columns and IPE 300 beams, while OMRF is dimensioned to develop a storey mechanism, having HE-240B columns and IPE 300 beams (Fig.6b,c). The DBF and RF are based on the OMRFs general dimensions, intending to transform them in special moment-resisting frames (Fig.6d,e), using dog-bone or remforced solutions. All the frames were subjected to main Romanian earthquakes, Vrancea (1977) and Banat (1991) and also to Kobe earthquake (1995). Each of these ground motions introduce different aspects concerning the structural response. The Vrancea earthquake is a typical far-field earthquake, while Kobe and Banat earthquakes are specific near-field ones, with velocity impulse characteristics. The difference between them is that the Kobe earthquake has more velocity pulses, while the Banat earthquake has a single velocity pulse only. The effectiveness and the reliability of the above solutions were studied, normalizmg the proposed accelerograms at 0.35g, 0.25g, 0.15g, considered as high, moderate and low seismic accelerations (Mazzolani & Piluso, 1996) (Table 1). The numerical modeling of the modified frames, DBF, RF, was made by using the beam-to-column element from the Drain-2D computer program, introducing rigid or weakened short elements and fictive bearings as showed in Figure 6f,g.

Analysis of the results Studying the results of the parametrical analysis, the following main findings should be emphasized:

264 TABLE 1 NORMALIZED GROUND ACCELERATIONS

Vrancea INCERC 1977 1 Banat Long 1991 Kobe 1994

0.199

1.60

0.03

11.6

0.296

1.18

n

nzm

MM

0.15

5.0

0.506

0.85

IPE300

IPE300

IPE300

IPE300

s

20KN/m i

0.25

8.33

0.35

20KN/m ill U H i i M M

i20

0.682

1.14

M U U i 20KN/ni

JM l U

Weak earthquake Normalized Scaling acceleration factor X (g)

1 Moderate earthquake Strong earthquake Normalized Scaling Normalized Scaling acceleration factor acceleration factor, X (g) (g)

P.G.A ao (g)

Ground motion

3 CM

1PE300

LU

IPE300

X

X

4,00

^4

S,0 ^

M ir

5,0

^r

m

»

^

j

OMRF

SMRF ,.IPE300. ^^IPE300 ^^

JPE 300^

'

M^

^lUieooq, M

pn DBF

I

4470

300.

f-7.

'

hPE Imed

d) P

^

LlPE300 . LlPE300 ^ RF

7r

"2

>m^

P^

4800

llstr

LiPLm.

,^0^ Istr ^

lStr = 1.45 IlPE

JT^e)

g)

Figure 6: Geometry and modeling of examined types of frames

^

c)

265 • For low seismic ground motions, 0.15g, the inelastic behaviour requirements were not so significant, all the frame types sustain the ductility demands, while for Banat earthquake the structures does not enter in plastic domain. • Li case of far-field ground motion (Vrancea), Fig.7,8, medium ductility requirements for both strong and moderate seismic actions were marked. Modified moment-resisting frames, DBF, RF, as well as SMRF undergo these inelastic requkements, formmg plastic hinges only at the beam ends, while OMRF develops plastic hinges both at beams and columns, having a prone inelastic behaviour for life safety. • hi case of near-field ground motion (Kobe), Fig.8,9, high ductility requkements were marked. One can see that OMRF and SMRF do not satisfy the ductility requirements for high and moderate earthquakes, the first one forming a storey mechanism as expected. In exchange the modified moment-resisting frames, DBF, RF, concentrating the plastic hinge away from the column face, have sufficient available plastic rotation capacities, forming a global mechanism. • In case of local Banat earthquake (Fig.8.10), some new specific characteristics were observed. For high and moderate accelerations, 0.35g, 0,25g, respectively, the ductility demand was minor, because other earthquake parameters than peak ground acceleration, have influence on the structure behaviour. Completely different inelastic behaviour was find as compared with the Kobe earthquake due to the presence of a single velocity pulse. While SMRF has a good global performance, modified moment-resisting frames can be ineffective for this earthquake type (Fig. 9,10). Scalded ace:Vrancea INCERC, X=l. 16,^0.35g|

Scalded acc.Vrance INCERC, X=1.14, a=0.25g

8nec (rad) 0

0,03

0,06

0,09

0,12

0,15

Onec (rad)

0 0

0,03

0,06

0,09

0,12

0,15

Figure 7: Ductility demands for the normalized Vrancea earthquake From the local point of view, the moment time history demands of the beam-to-column interface, at the first storey of SMRFs, DBFs and RFs, are shown in Figure 11. One can see the small requirements of moment at the case of DBF, RF, avoiding stress concentration at the weldments. From the global point of view, taking into account the shear index, Vm. / Vp ( ratio between shear force obtamed for 0.35g peak acceleration to storey shear at the first plastic yield), it is clearly showed the superior overall seismic performance of the modified frames. RFs have a little bit better global performance as compared with the DBF, due to smaller inelastic deformations, causing smaller rotation demands. In some specific cases (see Banat earthquake) the use of such joint details could be unreHable. It is concluded from the parametrical study that, in many cases, the modified moment-resisting frames using dog-bone or reinforced solutions represent attractive solutions to satisfy ductility demand, as well as to mitigate the brittle failures observed in recent earthquakes. In the same time, the case of Banat

266 earthquake shows that these solutions are not universally valuable, depending to the seismicity site characteristics. KOBE, a= 0.35g

p

Y

ti

U

BU

J

T

"Y

n

JL

JL

li

•! I >

m

m

n Mi

•!

»

i n

m

M

•»•

-H

!M

i

>d

JL

VRANCEA, a= 0.35g

» »

• •

L

•

^

ii

J

•

aU

•I

JL

•!

"P

•!

.|^,,n.

J

— ^ ^^^'i

1^

-* 1

JL

J

Ml,.

L

""F

KW-

BANAT, a= 0.35g i|

\

^

SMRF

OMRF

DBF

RF

Figure 8: Plastic collapse mechanisms IScalded ace: Kobe, X=1.18, a=0.35g

[Scaled acciKobe, X=0.85, a=0.25g

1 6nec(rad) Onec (rad) 0

0,03

0,06

0,09

0,12

0,15

0,03

0,06

0,09

0,12

Figure 9: Ductility demands for the normalized Kobe earthquake

0.15

267 jScaled ace: Banat Long, X=l 1.6, a=0.35g Scaled ace: Banat Long, A,=8.33, a=0.25g ^^^•^"T!^?;

3 i

}^m —•—SMRF 1

^i^M.]kr''^ |lf:

-•—OMRFl

5|l^^

—A-DBF 1 -B—RF 1

jifiy -ij^Pj^T-!^?,,

pB

J

r ': " /'' % ^i 'M %i^ : ,- '- >;'f\Cf--

l^^e^m'

'Ifi^^^i^^^^

'v^-M

—»-OMRF —A—DBF

."->:•.:':;'-"

"- <' ;

ys^ y-y:,;./ Onec (rad) OB' 0.006

0.003

..

'''" ' '''

0.009

0.002

': . -',.'. , '-' J

0.004

Gnec (rs

0.006

Figure 10: Ductility demands for normalized Banat earthquake 1,5

Scalded ace: Kobe, X=1.18, a=0.35g

r¥':';1

0,5

-' % - \ . ~, -'y

>;. '--J

^- "\;-':;>:, "-y^yy-'r//; ''rz^^-i/>r'i:^: iy^;ry;;vj''--;

W-:'^-A''f&^:

i;Sf%^i;: rllflvSiSl^ti

y"=-'v-

. =>j5?^^ n5^W^^^

0

H-'^\/

tvtf|i;C-4/^ ;21^H

^

-0,5

\

\

;/;"-rv--rr-^\. '^'^^"~/ ~ '•'-••'*"/''"V'"''V-:*^-^'-':]

1

.' y-^l^-:^':;:'-:'^^ ^r'"-\^.~ l[:J-':'

rK,:'^-;:'

'\

T(sec)

Figure 11: Time histor>^ moment demands at beam-to-coioumn interface of the first storey IKobe,X=1.18,a=0.35g i Vrancea, X=1.6, a =0.35g B Banat Long, X=l 1.66, a=0.35g

SMRF(T=0,909sec)

OMRF (T=1.05sec)

DBF (T=1.14sec)

T- Natural period of the structure

Figure 12: Shear base index of effectiveness

RF (T=0.92sec)

268 CONCLUSIONS Analytical investigations were performed on both local and global performance of different momentresisting jframe typologies, using DUCTROT'96 and Drain-2D computer programs. The study concludes that the modified moment-resisting firames, DBF, RF, can help to control the seismic response avoiding the formation of undesirable collapse mechanisms, as storey mechanism. In the same time, these solutions transform OMRFs, in ductile moment-resisting firames, obtaining a predetermined failure mode and the ductility control through concentrated rotation requirements only at the beams, far from joints. Li some cases this effect seems to be unreliable; in this way it is need to consider the specific seismicity of the site. Further research work are planned to lighten these aspects.

REFERENCES Anastasiadis A. (1999). Ductility problems of steel MR frames. Ph. D Thesis, Politehnica University Timisoara, Romania Anastasiadis A. and Gioncu V. (1998). Influence of joint details on the local ductility of steel momentresisting frames, Greek National Conference on Steel Structures, Thessaloniki, Greece (in print) Chen S.J., Chu J.M. and Chou Z.L. (1997). Dynamic behavior of steel frames with beam flanges saved around connection. J. Construct. Steel Research 42:1, 49-70. Gioncu V. and Mazzolani F.M. (1999). Ductility of Seismic Resistant Steel Structures, manuscript, E& FN Spon, U.K Gioncu V. and Petcu D. (1996). DUCTROT'96: Plastic rotation of steel beams and beam-columns, Guide for Users, INCERC Timisoara, Romania Gioncu V. and Petcu D. (1997). Available rotation capacity of wide-flange beams and beam-columns: Part I:Theoretical approaches. J. Construct. Steel Research 43:1-3, 161-217. Gioncu V. and Petcu D. (1997). Available rotation capacity of wide-flange beams and beam-columns: Part n: Experimental and numerical tests. J. Construct. Steel Research 43:1-3, 219-244. Mazzolani F.M. and Piluso V. (1993). Design of Steel Structures in Seismic Areas, ECCS Document Mazzolani F.M. (1998). Design of steel structures in seismic regions: The paramount influence of connections, COST 98, Liege Conference, 11-24. Plumier A. (1996). Reduced beam section; a safety concept for structures in seismic areas, Buletin Stiintific, Ser. Constructii, Arhitectura, Tom 41(55), Fasc. 2, 46-60. SAC (1995). Interim Guidelines: Evaluation, Repair, Modification and Design of Welded Steel Moment Frame Structures, Report FEMA 267/SAC-95-02, SAC Joint Venture, California, U.S.A.

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Iv^nyi, editors © 1999 Elsevier Science Ltd. All rights reserved

269

FACTORS INFLUENCING DUCTILITY IN HIGH PERFORMANCE STEEL I-SHAPED BEAMS C. J. Earls^ Department of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261-2294, USA

ABSTRACT In general, compactness and bracing provisions associated with ultimate strength design of steel beams are formulated so as to ensure that the resulting beam designs exhibit adequate structural ductility. The specification of such compactness and bracing requirements oftentimes involves assumptions about the constitutive nature of the structural steel being used in construction. Historically, mild carbon steel has been the most common type of steel used in buildings and thus the characteristic constitutive properties of this steel are usually assumed to be the norm. While such material property assumptions form a good basis for compactness and bracing provisions of mild carbon steel structural elements, the same is not true when they are applied to structural members made from newer high performance steels. These new high performance steels have constitutive properties that differ significantly from those of mild carbon steel. It appears from the research reported herein that separate bracing and compactness requirements may not be valid when used to evaluate the ductility of I-shaped beams made from some new high performance steel grades. Results from this same research point to the fact that I-shaped beams made from high performance steel, and subjected to moment gradient loading, display inelastic mode shapes which do not lend themselves to a notional de-coupling of so called local buckling and lateral-torsional buckling phenomena. Rather, the inelastic mode shapes of these high performance steel beams display two distinct types of geometry at failure; both of which possess localized and global buckling components. The structural ductility of the beams is very much dependent upon which of the two inelastic mode shapes govern at failure. Cross sectional proportions, bracing configuration, and geometric imperfections all play a role in influencing which mode governs in the beam at failure. Currently held views as to the impact of cross sectional compactness and bracing on structural ductility often do not apply to I-shaped beams made from these high performance steels. Hence current design specifications will not be able to predict the response of these high performance steel flexural members. This paper will present results from numerical studies of high strength steel I-shaped beams conducted using the nonlinear finite element analysis technique. These studies will focus on the impacts which cross sectional proportions have on the manifestation of the inelastic mode shape and

270

subsequent structural ductility under a moment gradient type loading. In addition, the validity of two methods for predicting flexural ductility in I-shaped members, as obtained from the literature, will be compared with current research results.

KEYWORDS High Performance Steel, Flexural Ductility, Local Buckling, Lateral Torsional Buckling, Non-Linear Finite Element Method, Inelastic Buckling, Unbraced length, Rotation Capacity, Compactness, Bracing Stiffness, HSLA80 Steel.

INTRODUCTION The American (AISC 1994) and European (ECS 1992) steel design specifications simplify the flexural design of I-shaped members such that cross sectional proportions are considered independently from unbraced length when computing design capacity, thus assuming that cross sectional instabilities of constituent plate elements may be considered independently from global instabilities such as lateral-torsional buckling. It is clear that this simplification is made out of necessity due to the substantial complexity of treating the true interactive nature of the local and global modes as outlined by the plastic design commentary of the American Society of Civil Engineers (ASCE 1971): "Even though local and lateral-torsional buckling in the inelastic range are manifestations of the same phenomena, namely, the development of large crosssectional distortions at large strains, they have been treated as independent problems in the literature dealing with these subjects. This is mainly due to the complexity of the problem." Researchers such as Climenhaga et. al (1972) and Gioncu et. al (1996) have addressed the complicated coupled flexural instabilities of I-shaped beams by employing the energy method in conjunction with a yield line mechanism simplification. In this technique, after a predetermined cross-sectional rotation capacity is achieved, the descending portion of the moment rotation curve is traced by plotting the flexural response emanating from two different types of buckling geometry; one dominated by a so called "in-plane mechanism", and the other dominated by an "out-of-plane mechanism". The descending portion of the moment-rotation curve resulting from the mechanism which produces the least capacity is assumed to correspond to the governing failure mode. Gioncu et. al (1996) have also considered interactions that arise between the in-plane and out-of-plane modes. These yield line mechanism methods have produced satisfactory results as compared with experimental data reported in certain studies (Gioncu et. al 1996, Kuhlmann 1989). Other researchers have treated the complicated coupled flexural response of I-shaped beams in terms of simplifications involving cross-sectional geometry alone (Climenhaga et. al 1972, White et. al 1997). Kemp (Kemp et. al 1991, Kemp 1996) addresses the complicated global and local buckling interaction through the consideration of both cross-sectional geometry and unbraced length of the member. Methods for estimating rotation capacity of I-shaped beams have been given by Kemp (Kemp et. al 1991, Kemp 1996) as well as White (White et. al 1997, White et. al 1998). These predictive methods can perform adequately in certain restricted ranges of applicability as is shown later in this paper.

271 VALIDATION OF MODELING TECHNIQUES The commercial multipurpose finite element software package ABAQUS (ABAQUS 1998) is employed in this research. The finite element models described herein consider both geometric and material nonlinearities. These nonlinearities tend to create formidable computing requirements since dense meshes of shell finite elements must be used to properly model localized instabilities that occur in conjunction with global instabilities, both of which are inelastic in nature. Incremental solution strategies are necessary to properly trace the nonlinear equilibrium path of the inelastic Ishaped beams. The ABAQUS modified Riks-Wempner strategy is used. All analyses reported here are carried out in parallel using sixteen processors of a CRAY T90 supercomputer. The ABAQUS S9R5 shell finite elements, used to model the beams, are oriented along the planes of the middle surfaces corresponding to the constituent plate components of the members. A uniaxial representation of the constitutive law given in terms of true stress and logarithmic strain is recorded in the ABAQUS input deck. ABAQUS then uses the von Mises yield criterion to extrapolate a yield surface in three-dimensional principal stress space from the uniaxial material response given in the input deck. The corresponding ABAQUS metal plasticity model is characterized as an associated flow plasticity model with isotropic hardening being used as the default hardening rule. Three finite element models were constructed so as to compare finite element results with the experimental results obtained by Lay and Galambos (Lay et al. 1965). The subjects of the comparison study are B8xl3 wide flange beams of varying length and subjected to three point bending resulting in a moment gradient loading. Both the experimental specimens and the finite element models used fiill depth stiffeners on both sides of the beam web, at the mid-span loading point and supports, so as to control cross sectional distortions under the action of the load. Similarly, out-of-plane translation and rotation were constrained at the supports and load point of the physical specimens and the finite element models. The experimental tests were carried out at Lehigh University in 1965 as part of a study aimed at extending the applicability of plastic analysis and design methods to steel with up to a 345 MPa yield stress. The experimental specimens were made from ASTM A441 steel. A piece-wise linear uniaxial representation of the material model used in the finite element comparison studies has a yield stress of 354.4 MPa. The ratios f^ / fy = 1.49, s„ / 8y = 72.4, and s^^ / Sy = 9.66 fiirther characterize the material model used in the finite element comparison studies. A relatively good agreement can be observed in the comparison plots of the moment-rotation responses obtained from the experiments and finite element models (see Figures 1,2, and 3). In all three cases, the finite element models displayed slightly higher ultimate moment capacities as compared with the experiments. This could be due to necessary differences in the material properties of the experiments and finite element models. In the report of Lay and Galambos (Lay et al. 1965), only the yield strength, ultimate strength, and percent elongation were given. Assumptions as to the strain hardening strain, ultimate strain, and strain hardening modulus had to be made in the finite element modeling. The finite element models did not exhibit the same degree of "roundness" in the portion of the moment-rotation response corresponding to the transition from elastic to inelastic behavior. This discrepancy is most likely due to the fact that residual stresses were not incorporated into the finite element models and thus the onset of first yield was delayed slightly in comparison to the experimental tests which undoubtedly possessed residual stresses.

272

INFLUENCE OF SLENDERNESS ON DUCTILTY Finite element studies of HSALSO beams subjected to a moment gradient loading have been conducted (Earls 1999) so as to develop a notional understanding of potential underlying mechanisms associated with the ultimate response of I-shaped beams. The results of these early studies are seen to run counter to expectations concerning the influence which cross-sectional proportioning has on structural ductility as quantified by plastic hinge rotation capacity. The definition of rotation capacity, used below, is consistent with that given by ASCE (ASCE 1971) in which R = {(Gu / Op) -1} where 0^ is the rotation when the moment capacity drops below Mp on the unloading branch of the M-0 plot and 0p is the theoretical rotation at which the full plastic capacity is achieved based on elastic beam stiffness. The influence of flange compactness on overall rotation capacity is addressed by Table 1. The results presented in this table seem to contradict current practical notions regarding the role of flange compactness in wide flange ultimate response. It is seen from Table 1 that an increase in flange slendemess increases overall rotation capacity for a bf / 2tf increasing from 3 to 6. However a further increase in bf / 2tf from 6 to 7 results in a significant decrease in overall section rotation capacity. The influence of brace spacing on beam rotation capacity is also addressed in this earlier study. Tables 2 provides a summary of results relevant to a discussion of the impact that unbraced length has on such beams. The results outlined in Table 2 contradict practical notions concerning the role of beam slendemess on the rotation capacity of a flexural element governed by lateral torsional buckling. It is seen from these results that both increasing and decreasing the unbraced length can lead to substantial improvements in the rotation capacity exhibited by a beam.

DOMINANT FAILURE MODES Based on the Author's examination of the inelastic mode shapes accompanying the finite element studies of HSLA80 I-shaped beams subjected to a moment gradient loading (as reported in Earls 1999) two distinct failure modes are identified. These two modes will be referred to respectively as Mode 1 and Mode 2. Figure 4 displays a typical Mode 1 geometry, while Figure 5 shows the Mode 2 geometry. Mode 1 is characterized by a local instability of the flange, either with or without substantial web participation, which occurs in close proximity to the mid-span stiffener. The plastic hinge region of Mode 1 is well defined and proximal to the mid-span stiffener region Mode 2 is characterized as a highly asynmietrical inelastic mode whereby local and global buckling are highly coupled. As can be seen in Figure 5, the flange buckling components, or flange-web buckling components, occur at a substantial distance from the mid-span. This distance from the stiffener to the center of the flange buckling wave is in general different for each half-span, but on average this distance is roughly equal to d/2. Similarly, the degree to which the flange-web buckling component of Mode 2 manifests itself varies significantly between the half-spans. Generally speaking, substantial out-of-plane deflections between brace points occur in the Mode 2 failures. The Mode 2 "plastic hinge" is better described as being a "zone of plastification" thereby not implying the usual connotation of a tightly formed concentrated region of plastification. On the contrary, the zone of plasticity in Mode 2 is very ill-defined and quite distributed in nature. Another characteristic feature of the Mode 2 inelastic mode shape is the formation of a mechanism in the compression flange of the beam. An example of this type of compression flange mechanism can be

273

seen in Figure 6 which displays a typical top view of the compression side of an I-shaped beam. From this figure, it can be seen that the compression flange behaves somewhat like a three-barlinkage; the kinematics of which are driven by the location of the mid-span stiffener and the linkage articulations. These articulations coincide with the locations of the flange buckling component (or flange-web buckling components) within the overall Mode 2 manifestation. Beyond the pronounced geometric differences between the inelastic mode shapes of Mode 1 and Mode 2, there are other more quantifiable differences in response. Mode 1 failures achieve a higher ultimate moment capacity and exhibit larger cross-sectional rotation capacities as compared with Mode 2 failures. It has also been found (Earls 1999) that bracing location, bracing stiffness, and the presence of cross-sectional imperfections all play a role in determining whether a Mode 1 or Mode 2 manifestation exists at failure. Climenhaga (Climenhaga et. al 1972) and Gioncu (Gioncu et. al 1996) have also observed that two distinct failure mode manifestations exist in I-shaped flexural members. These researchers categorize them as "in-plane" and "out-of-plane" mechanisms which respectively correspond to Mode 1 and Mode 2 as described previously. Clear examples of Mode 2 failures have also been reported in the experimental work of Schilling (Schilling 1988) and Azizinamini (Azizinamini 1998) as well as in the finite element studies of White (White 1994). Kemp and Dekker have made similar distinctions between mode shapes (Kemp et. al 1991). The categorization of Mode 1 and Mode 2, as defined in their paper, is opposite to that outlined above and given elsewhere (Earls 1999). For clarity of discussion, the definitions of Mode 1 and Mode 2 given earlier in this section of the current paper will be used in the subsequent discussion. Despite their two distinct failure mode categorizations, Kemp and Dekker believe that a single cause results in a reduction of observed moment capacity, or load shedding. This cause is a phenomenon which they term "strain weakening". They describe "strain weakening" as the displacement of the compression flange, due to lateral buckling, inducing a transverse strain gradient across the width of the flange. Kemp and Dekker conclude that "such lateral displacement of the member will lead to loss of moment resistance unless considerable strain hardening occurs on the opposite edge of the flange" (Kemp et. al 1991). Kemp and Dekker also describe their version of the Mode 2 failure as being predominantly elastic in nature with only a short segment adjacent to the hinge region yielding. This characterization is not supported by the Author's own research which shows very clearly that the lateral motion of the compression flange only occurs after very significant yielding is achieved along large portions of the beam longitudinal axis. Schilling (Schilling 1998) also observes that any lateral motion of the compression flange, as observed in his experimental tests, follows fi-om extensive yielding of the beam adjacent to the hinge region.

COMPARISON OF ROTATION CAPACITY PREDICTIVE EQUATIONS This portion of the paper is focused on comparing the rotation capacities observed in several test beams, modeled with validated finite element techniques, with those obtainedfirompredictive equations found in the literature (Kemp 1996, Kemp et. al 1991, White et. al 1997, White et. al 1998). These predictive equations have various geometrical limitations concerning their proposed range of validity. These equations also include scaling parameters to address the differences associated with different steel grades. It is interesting to note however that these scaling parameters consider yield stress only.

274

Kemp's predictive equation (Kemp 1996, Kemp et. al 1991) is based on the use of several basic parameters. The first is a "yield stress factor" for the flange or web given as

7 =

250

for F in MPa. The second parameter is a "slendemess ratio in lateral-torsional buckling" given as 'L^ 7f in which Lj is the length from the section of maximum moment to the adjacent point of inflection, rye is the radius of gyration of the portion of the elastic section in compression. The "flange slendemess factor in local buckling" is the third factor and is of the form 'b^ ry K. _Vj_2 subject to 0.7 < Kf < 1.5 as the range of applicability. Similarly, the "web slendemess factor in local buckling" is

y.

K =

70

subject to 0.7 < K^ < 1.5 as the range of applicability. Kemp also defines a distortional restraint factor to account for the effects of a concrete slab in negative moment regions of a beam. However, since the current comparison study is only applied to bare steel beams this value is taken to be unity as per Kemp (Kemp 1996). The above parameters are then combined to form the "effective lateral slendemess ratio" of the form ' L ^ e

J

^

\^y^ J

valid in the range 25 < X^ < 140. Kemp goes on to define an empirical expression for the relationship between the effective lateral slendemess ratio and the plastic length of the member at maximum moment as

= 0.067

r6o_

i.

The rotation capacity is then predicted with the following equation

if

=45

275 It is noted that all rotation capacities, both from predictions and finite element models, are given with respect to the same definition as outlined by ASCE (ASCE 1971) and described earlier. In the case of White's predictive model (White et. al 1997, White et. al 1998), a deterministic equation, fit to finite element data, for the plastic rotation at which the descending portion of the moment rotation curve passes below the fiill plastic capacity is given as

Oj^ =0.128-0.0119

•0.0216

'D^

+ 0.002

K^fcJ

White recommends that the following restrictions apply to the use of the above equation

D

<4.25,

^ <

It,

-<0.4

2Z).,

-^<6.77

^<0.75, D

V — <0.6,

0.124 yc J

It is noted that the last restrictive equation has been specialized for the moment gradient case which is being studied in the current paper. It is interesting to note that White's predictive equations consider only cross-sectional proportions directly while Kemp's predictive equation considers unbraced length in addition to cross-sectional geometry directly. Using White's predictive equation for GRL it is possible to compute cross sectional rotation capacities for comparison with Kemp's predictions and the results from the current finite element studies. Such a comparison is presented in Table 3. In this table, the finite element results are tabulated in three columns. The first column corresponds to finite element analyses conducted on meshes with no geometric imperfections, while the other two columns correspond to those results obtained with meshes possessing cross sectional imperfections associated with a canting of the compression flange by 3mm, representing Imperfection A, and a canting of both flanges by 3mm toward the same side characterizing. Imperfection B. From the results outlined in the table, it is observed that Kemp's predictions agree quite well with the finite element results of three beams whose proportions are given as b/2tf = 6, h/t^ = 25, hjx^ = 55, and b/2tf = 6, h/t^ = 25, hJXy = 57, and finally b/2tf = 6, h/t^ = 25, Lb/ry = 65. The remaining beams' rotation capacities were over predicted by Kemp's equation by quite a large margin as can be observed from Table 3. White's predictions generally agreed less well with the finite element results (two notable exceptions to this are the perfect geometry case of a beam with b/2tf = 3, h/t^ = 25, L^/ry = 65, and a beam with b/2tf = 6, h/t^ = 25, L^/ry = 65 with Imperfection B) than Kemp's. However, it is important to realize that White's equations were developed from finite element studies of plate girders. Similarly, strictly speaking all of the test cases considered were outside of White's pubUshed limits of applicability for his predictive equations based on the parameter L/ry^; though this exceedence was only slight. In addition, it is noted that one of the test cases fell slightly outside the range of applicability stipulated by Kemp (Kemp 1996) for use with his predictive equation. Any violation of published guidelines are clearly marked in Table 3.

276 CONCLUSION Based on experimental studies contained in the literature, and the finite element studies conducted as part of the current research, there appears to be two dominant failure modes which occur in I-shaped beams. These modes display the complicated inter-relationship that exists between localized buckling phenomena and more global manifestations. This paper defines these dominant modes as either Mode 1 or Mode 2. The Mode 1 failure results in larger moment capacities and greater ductility as compared with the Mode 2 failure. It is concluded that the two predictive methods for computing available rotation capacities in Ishaped beams, while useful in certain very restricted ranges of application, are not in a general sense valid for predicting I-beam ductility. Based on on-going research, it is believed that in formulating effective methods for identifying the transition between Mode 1 and Mode 2, or in predicting beam rotation capacity, it is necessary to consider features of the constitutive relationship used to model the specific steel grade of the beam (specifically yield stress, ratio of yield strength to ultimate strength, and whether or not a yield plateau is present in stress-strain response of the steel). Such constitutive parameters would then need to be complimented with geometric parameters relating both to unbraced beam length, and cross sectional slendemess.

REFERENCES ABAQUS Theory Manual, Hibbitt, Karlsson & Sorensen, Inc., Pawtucket, Rhode Island, USA, 1998. American Institute of Steel Construction, Load and Resistance Factor Design, American Institute of Steel Construction Inc., Chicago, Illinois, 1994. American Society of Civil Engineers, Plastic Design in Steel, A Guide and Commentary, American Society of Civil Engineers, New York, New York, p.80, 1971. Azizinamini, A., Yakel, A., Mans, P., Inelastic Behavior of Steel Plate Girders with 70 ksi Steel, Draft Report, University of Nebraska-Lincoln, Lincoln, Nebraska, USA, 1998. Climenhaga, J.J., Johnson, R.P., Moment-Rotation Curves for Locally Buckled Beams, Journal of the Structural Division, Proceedings of the American Society of Civil Engineers, Vol. 98, No. ST 6, pp. 12391254, June, 1972. European Committee for Standardization (CEN), Design of Steel Structures. Part 1: general rules and rules for building; Eurocode 3, Brussels, Belgium, 1992. Earls, C.J., On the Inelastic Failure of High Strength Steel I-Shaped Beams, Journal of Constructional Steel Research, Elsevier Science Ltd., Great Britain, Vol. 49, No. 1, pp. 1-24,1999. Gioncu, v., Tirca, L., Petcu, D., Interaction Between In-Plane and Out-of-Plane Plastic Buckling of WideFl2inge Section Members, Proceedings of the Coupled Instabilities in Metal Structures Symposium, Liege, Belgium, Imperial College Press, London, pp. 273-282, 1996. Kemp, A.R., Inelastic Local and Lateral Buckling in Design Codes, Journal of Structural Engineering, American Society of Civil Engineers, Vol. 122, No. 4, pp. 374-382, April, 1996.

277 Kemp, A.R., Dekker, N.W., Available Rotation Capacity in Steel and Composite Beams, The Structural Engineer, Vol. 69, No. 5, pp. 88-97, March 1991. Kuhlmann, U., Definition of Flange Slendemess Limits on the Basis of Rotation Capacity Values, Journal of Constructional Steel Research, Elsevier Science Ltd., Great Britain, Vol. 14, pp. 21-40, 1989. Lay, M.G., Galambos, T.V., Experiments on High Strength Steel Members, Welding Research Council Bulletin, Bulletin No. 110, pp. 1-16, November, 1965. Schilling, C.G., Moment-Rotation Tests of Steel Bridge Girders, Journal of Structural Engineering, Vol. 114, No. 1, pp. 134-149, January, 1988. White, D.W., Analysis of the Inelastic Moment-Rotation Behavior of Non-Compact Steel Bridge Girders, Report CE-STR-94-1, School of Civil Engineering, Purdue University, West Lafayette, Indiana, February, 1994. White, D.W., Ramirez, J.A., Barth, K.E., Moment-Rotation Relationships for Unified Autostress Designs of Continuous-Span Bridge Beams and Girders, Report FHWA/IN/JTRP-97/8, U.S. Department of Transportation Federal Highway Administration, 1997. White, D.W., Barth, K.E., Strength and Ductility of Compact-Flange I-Girders in Negative Bending, Journal of Constructional Steel Research, Elsevier Science Ltd., Great Britain, Vol. 45, No. 3, pp. 241-280, 1998.

TABLE 1 Summary of the Influence of Flange Slendemess on HSLA80 Flexural Ductility Steel HSLA-80

1 HSLA-80

b/2tf

3

6

h/t. 25

25

L/r, 65

65

R 3.0

3.4

Notes

1

Failure Mode: Combined LTB & LB (Flange)

1

Highly unsymmetric mode shape with flange local buckling occurring away from stiffener. Highly unsymmetric hinge region.

\

Failure Mode: LB (Flange & Web) & Slight LTB Flange local buckling occurs close to stiffener in one half- span only. Slightly unsymmetric hinge region.

HSLA-80

7

25

65

2.1

Failure Mode: LB (Flange & Web) & LTB Highly unsymmetric mode shape with flange local buckling occurring away from stiffener. Highly unsymmetric hinge region.

TABLE 2 Summary of the Influence of Unbraced Length on HSLA80 Flexural Ductility Steel HSLA-80

b/2tf 3

h/t. 25

L/r, 65

R 3.0

Notes Failure Mode: Combined LTB & LB (Flange) Highly unsymmetric mode shape with flange local buckling occurring away from stiffener. Highly unsymmetric hinge region.

1 HSLA.80

3

25

55

»3

HSLA-80

3

25

75

4.5

Rotation capacity greater than 3; Failure Mode: Slight LTB & LB Symmetric flange local buckling. Symmetric hinge region.

Failure Mode: LTB & very localized LB Symmetric flange local buckling and LTB. Symmetric hinge region.

278

TABLE 3 Comparison of Rotation Capacities Obtained from Predictive Equations in the Literature and Those Obtained from the Current Finite Element Study. Rotation Capacity White

| FEM 1 Perfect | Imp A ImpB 1 4.2 NA 1 NA

Specimen Geometry

Kemp

1 b/2tf-3,h/t^-25 Wry = 55 b/2tf-3,h/t^-25 Wry = 57 b/2tf=3,h/t^ = 25 Wry = 65 b/2tf=6,h/t^ = 25 Wr, = 55 b/2tf=6,h/t^ = 25 Wry = 57 b/2tf=6,h/t^ = 25 Wr, = 65 b/2tf=6,h/t^ = 29.3 Wry = 27 b/2tf=3,h/t, = 70

12.3

3.1^

11.7

3.1^

4.1

3.8

3.9

9.6

3.1^

3.0

NA

NA

4.3

1.8^

»3

NA

NA

4.1

1.8^

3.9

2.7

2.7

3.4

1.8^

3.4

3.5

2.2

10.0^

3.5^

4.1

NA

NA

2.5^

1.3

NA

NA

1

Wr, = 65

2.0

1

1

^ L/ryc value exceeds White's Prescribed limit of (L/ryJ^a^ = 42.3 * Kf and K^ are slightly outside Kemp's prescribed limits. NA indicates that FE analyses were not carried out for these cases

279 B8X13 - Lehigh Test Specimen HT52

-Experiment - Finite Element

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 (rads)

Figure 1: Plot of Normalized Moment vs. Normalized Rotation, L / ry = 72.3

B8x13 - Lehigh Test Specimen HT28 1.4

-Experiment -Finite Element

4 0/0P

Figure 2: Plot of Normalized Moment vs. Normalized Rotation, L / r = 35.0

280 B8X13 - Lehigh Test Specimen HT43

-Experiment -Finite Element

0.005

0.01

0.015

0.02

0.025

0.03

© (rads)

Figure 3: Plot of Normalized Moment vs. Normalized Rotation, L / ry = 22.9

Figure 4: Mode 1 loelastic Buckling

281 wmmmmmmm^mmm

Mgure 5; Mode 2 Inelastic Buckling mm^^&mm^mmm

Figure 6: Compression Flange Mechanism of Mode 2 Failure

This Page Intentionally Left Blank

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

283

DUCTILITY OF PLATE GIRDER PANELS UNDER CYCLIC SHEAR

Y. Fukumoto\ M. Uenoya\ M. Nakamura^ and T. Takaku^ ^Department of Civil and Environmental Engineering, Fukuyama University, Fukuyama, 729-0292, Japan ^Engineering Department, NKK Corporation, Yokohama, 230-0045, Japan

ABSTRACT This paper presents an experimental investigation on the hysteretic behavior of plate girder panels under shear. Six specimens with two square panels of 800 mm side are made of low-yield steel LY 100 (nominal 0.2% offset yield stress is 120 MPa) or mild steel SM 400 (nominal yield stress 235 MPa) in webs. All the flanges are made of SM400. Hysteretic curves in shear for SM and LY series were compared for three different web slendemess ratios of 114, 160 and 200. The shear strength was almost uniform up to the large shear displacement, (20-28) times the elastic limit value. Energy dissipation increases in proportion to the shear displacement. Energy dissipation by the low-yield steel is larger than the energy dissipation by the mild steel, and the thicker web is more effective from the energy dissipative point of view.

KEYWORDS Experiments, Plate Girder, Histeretic Curves, Cyclic Shear Strength, Low-Yield Steel, Energy Dissipation, Ductility, Diagonal Tension Field

284 INTRODUCTION The 1995 Kobe Earthquake left serious structural damages in buildings and bridges, but at the same time provided researchers with an invaluable set of buckling data on a large scale. Shear buckling of web plates under seismic loading was often observed at the mid portion of the cross beams in a single and two story portal framed piers which carry the urban expressway. Diagonally cross-shaped patterns in the web panels indicate that the efficient energy dissipation can be obtained in the process of the formation of the alternating tension field. The damages were locally restricted in the several panel zones and survived the other parts of the structures. This paper aims to reproduce the failure patterns by the laboratory test and to obtain the cyclic behavior of girder webs under shear. The development of the energy dissipative device is also examined.

TEST SPECIMENS AND TEST SETUP The plate girder specimen is shown in Fig. 1. The specimen consists of two web panels made of a mild (SM 400, nominal yield stress is 235 MPa) or a low-yield steel (LY 100, nominal 0.2 % offset yield stress 120 MPa) and upper and lower flanges and three vertical stiffeners made of SM 400. The length and height of the specimens are 2 240 mm and 824 mm, respectively. The web is composed of two square panels of 800 mm side with the thickness being one of 4, 5, and 7 mm. The 4 and 5 mm webs were scraped from the 7 mm web plate. The full width and thickness of the flanges are 300 mm and 12 mm, respectively. Fig. 2 shows the loading setup employed in the tests. The test specimen is installed between the two loading beams and fastened to them by high-tension bolts. Equal and opposite loads were applied quasistatically by two jacks at the both ends of the loading beams as shown in Fig. 2. The loading procedure would minimize bending deformation of the test specimens as shown in Fig. 3.

LOADING PROGRAM AND MEASUREMENT The specimens were loaded cyclically according to the history shown in Fig. 4, in which the ordinate indicates the amount of the shear displacement defined by a multiple of the yield shear displacement 6y . The shear displacement is defined as the diagonal displacement caused by the shear deformation as shown in Fig. 5 and 8 = | 6i | +1 62 |, and by is the yield displacement when the web yields in pure shear. The diagonal displacement of the two web panels were monitored by digital displacement transducers which can detect the prescribed diagonal displacement in order to control the loading jacks. Also, 25

285

digital displacement transducers for measuring the deflection of the web panels and flanges were installed to the test specimen. 39 triaxial and 56 uniaxial strain gauges were placed on the web panels, the flanges and the vertical stiffeners. 1120 72=216 800 ol 1 mt of

1120 800 3@72 216

inr

in^

V4@16-26h0le5

2240 320

300 800

800

iJTOo

320

T A-Panel 5

ids

!nr

B-Panel

X2@i4-26holes

2

D85=170

Loading Beam

9iAi

Fig.l Dimensions of test specimens

f IkN

i 3m

-IkN

^ j J.5m J

Shear t Displacement (5 3^y|-

s^l.SkN

2ay 1 ay 0 -l(5y -2(5y

e

-IkN

0 IkN

IkN

Fig.2 Test setup

IkN

5 kN-nN

IkN

Plate Girder

2@85 = 170

©

SFD IkN

Fig.3 Bending and shear force diagrams in loading system

-3 ay

Fig. 4 Displacement history in test

TEST RESULTS AND DISCUSSION The tensile coupon test results of SM 400 and LY 100 are summarized in Table 1 and the stress-strain relationships obtained from the tests are shown in Fig. 6. Figs. 7 (a) and (b), respectively, show the cyclic curves of the shear force (Q) versus the shear displacement (6) for SM 1 (t^=4mm) and SMS (t^=7mm), and Figs. 8 (a) and (b) for LYl (t^=4mm) and LY2 (t^=7mm), respectively. The specimens made of the mild steel web reach to Q^^^ at the early stage of loading cycles and then gradually decrease to the constant level after the cyclic formation of the diagonal tension field. Cyclic behavior for t^=7mm is significantly better than for t^=4mm because of

286 the prevention of shear buckling. Only one panel of the web collapsed where the initial deflection was larger than another one as shown in Figs.7. The cyclic shear capacities were kept constant up to 6=54mm(28 6^ ), 54mm(28 6^ ) and 35mm (20 6y ), for SM1,SM2 and SM3,respectively. TABLE 1 DIMENSION AND MATERIAL PROPERTIES OF TEST Web specimen

SMI SM2 SM3 LYl LY2

1 LY3

Flange

Yield stress Oyy,

Width

Height

Thickness

K

MPa

mm

h mm

mm

274 274 295 91 91 91

800 800 800 800 800 800

800 800 800 800 800 800

4.0 5.1 6.7 4.3 4.7 7.2

Slendemess ratio h

200 158 119 185 170 112

Stiffener

Yield stress

Width

Thickness

MPa

mm

mm

268 268 273 268 273 268

300 300 300 300 300 300

11.9 12.1 11.9 12.0 12.0 12.1

Yield stress

Width

Thickness

MPa

mm

mm

288 288 288 268 273 268

100 100 100 100 100 100

7.8 7.9 7.8 8.2 8.1 7.8

'/

450

— SM400 -LYIOO

^400

I 350

hr

^300 h Sg250 u 5 200 150 100 50 0 -5

Fig.5 Shear displacement 6

-A-Panel B-Panel

•/'

5

10

15

20

25 30 35 Strain (%)

40

45

Fig. 6 Stress-strain curves of SM and LY steels

Shear Force Q(kN)

Shear Force Q(kN)

-90 -80 -76 - ^ - 5 b . . U a ^ b ; ^

0

/ i

'•W^O-lsb 70 80 90 Shear Displacement d

60 70 80 90 lear Displacement (5 (mm)

(a) SM 1 (b) SM 3 Figs. 7 (a) and (b) Shear force( Q) versus shear displacement (6) relationships

287 Shear Force Q(kN)

500 J- Shear Force Q(kN) 400 30

60 70 80 90 Shear Displacement 6 (mm)

(a) LY 1 Figs. 8 (a) and (b) Q versus 6 relationships The specimens made of the low-yield steel web gradually reach to the constant shear level at 6=28mm(36 5y ) and continue to 6=57mm at the completion of the tests. There was no sign of decrease in the constant shear load capacity. Due to inset of strain- hardening without any yield plateau in the low-yield tensile stress-strain relationships, the two panels are almost simultaneously into the tension field range. The envelope Q^^^ -6 curves of each panel for SM and LY series are obtained from the cyclic curves as shown in Figs. 9 (a) and (b). The different Qn^^x -S curves can be seen for material properties and web slendemess ratios as test parameters. Table 2 summarizes the test results of the six specimens with the reference shear strengths.

900 r Shear Force Q(kN)

-^SMl-A

•

900 - Shear Force Q(kN)

-^LYl-A

o LYl-B

800 [

-*-SM2-A

*• SM2-B

800

--r-LY2-A

-^ LY2-B

-*-SM3-A

-81^3-8

700

— LY3-A

o LY3-B

700 [ 600 f

f^

600

p

500 f 400

SMl-B

500

ntrir^^-^--. ••

••••••-...,.,.•.«-•#•••••

300

300 T B ^ 200

200

•

^ ^ ^ ^ ^ #»-"•'«n

100

100 0 •—' 0

*-~«i

400

'—' 10

' 20

"30

40

50 60 70 80 Shear Displacement 5 (mm)

90

0 < 0

10

20

30

40

50 60 70 80 90 Shear Displacement 6 (mm)

(a) SM series (b) LY series Figs. 9 (a) and (b) Envelope curves of Q versus 8 relationships Energy dissipation behavior is the important factor in evaluating the seismic performance of the shear links as energy dissipators. Dissipated energy was calculated for each cycle of the histeretic curves. Fig. 10 for SM series plots the cumulative dissipated energy against the shear displacement for each panel A and B, and the shear displacement for the combined two panels is taken as the larger value in the panel A

288

and B. The thicker webs are more effective in the shear capacity and the energy dissipation. However, only one panel of the webs can contribute to the energy dissipation in MS series. TABLE 2 SUMMARY OF TEST RESULTS WITH REFERENCE LOADS

Specimen

SMI SM2 SM3 LYl LY2 LY3

Slendemess ratio

Elastic shear buckling

_h_

Qcr^th kN 138 288 653 173 226 811

200 158 119 185 170 112

Plastic shear force

Ultimate shear strength

Maximum shear force

Constant shear force

PRE*

Easier**

kN

QuU,th kN

Quit,th kN

kN

Qconst,ex kN

504 642 915 181 198 303

389 524 907 201 198 303

363 505 812 178 198 303

400 536 818 236 279 436

375 482 678 236 279 436

^const,ex ^mz\,ex

0.94 0.90 0.83 1 1 1

*Porter, Rockey and Evans(1975) **Basler(1961)

Energy Dissipation Capacity (kN-m)

90

-SMl-A+B -SM2-A+B -SM3-A+B

80 70

•SMl-A • SM2-A •SM3-A

60 h 50 h 40 \ 30 \ 20 h 10 0 0

10

20

30

40 50 60 70 Shear Displacement (5 (mm)

80

90

Fig. 10 Energy dissipation capacity of SM series Fig. 11 for LY series which is similar to Fig. 10 shows that the thicker webs are more effective in the shear capacity and the energy dissipation. Both panels of the LY web can contribute effectively to the energy dissipation. The low-yield steel is more effective than the mild steel from the energy dissipative point of view.

289 90 p Energy Dissipation Capacity (kN-m) 80 70

LYl-A+B LY2-A+B LY3-A+B

LYl-B LY2-B LY3-B

-LYl-A -LY2-A -LY3-A

60

10

20

30

40

50 60 70 80 Shear Displacement d (mm)

90

Fig. 11 Energy dissipation capacity of LY series CONCLUSIONS FoUowings are the main results from the cyclic shear tests: The SM specimens reach to the maximum shear strength at early stage of loading cycles and then the shear capacity gradually decrease to the constant level. Failure of the specimens occurs on only one side of the two web panels. The LY specimens gradually reach to the constant shear level with the cycles due to the strain-hardening effect of the material. Web failure occurs on both sides of the two web panels. The constant shear capacity starts to decrease when the large flange deformation occurs due to a lateral loading from the tension field. The energy dissipation increases in proportion to the shear displacement. The thicker webs are more effective in the cyclic shear capacity and the energy dissipation. The low-yield steel is more effective to use in webs in the energy dissipative point of view.

ACKNOWLEDGMENT Financial support by Grant-in-Aid for Scientific Research (B)(2) No.09450179 from the Ministry of

290 Education of Japan is greatly appreciated.

REFERENCES Easier K (1961). Strength of Plate Girders in Shear. Journal of the Structural Division ASCE 87:8X7,151-180. Kasai K. and Popov E.P. (1986). Cyclic Web Buckling Control for Shear Link Beams. Journal of the Structural Engineering ASCE 112:3,505-523. Nakashima M. (1995). Strain-Hardening Behavior of Shear Panels Made of Low-Yield Steel. I : Test. Journal of the Structural Engineering ASCE 121:12,1742-1749. Porter D.M., Rockey K.C., and Evans H.R.(1975). The Collapse Behaviour of Plate Girders Loaded in Shear. Structural Engineer 8:53,313-325.

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

291

A LIMIT-STATES CRITERION FOR DUCTILITY OF CLASS 1 AND 2 COMPOSITE AND STEEL BEAMS A. R. Kemp Department of Civil Engineering, University of the Witwatersrand, P. O. WrrS, 2050 Johannesburg, South Africa.

ABSTRACT A limit-States criterion of ductility is presented for continuous composite and steel beams in terms of a comparison of required and available inelastic rotations at notional plastic hinges. Calculation of the required inelastic rotation is illustrated for a three-span beam using the Principle of Virtual Work and is expressed as a simple function of the proportion of moment distribution and the simply-supported end rotations due to load in each span. The available inelastic rotation at each notional hinge is determined from experimental or theoretical components of rotation due to yielding of the steel beam, distortion of the end connection and cracking of the concrete. The Principle of Virtual Work is used to demonstrate the applicability of double-cantilever test results and idealised moment-curvature relationships to the calculation of available inelastic rotation at notional plastic hinges.

KEYWORDS Composite, connection, ductility, hinge, inelastic, limit-state, rotation, steel, virtual work, yielding

INTRODUCTION TO DUCTILITY LIMIT STATE Increasing attention is being given to the use of continuous composite beams in buildings for spans exceeding 10m or for heavily loaded members. Relative to steel beams, the redistribution of moments and therefore the ductility required for an efficient design is particularly demanding for the following reasons: 1. Elastic distributions of moment generally give higher moments at intemal supports than midspan regions, whereas resistances are greater in positive midspan regions. 2. The shape factor for composite beams is high (1.25 to 1.35) and this benefit is only utilised if stress block analysis is used to determine the resistance. Designers are given the opportunity to analyse ultimate load effects either by plastic analysis or elastic analysis with or without moment redistribution, and to determine the ultimate moment resistance either by using rigid-plastic stress blocks or elastic stress limits. To clarify these options a classification has been

292 introduced into American, European and Canadian codes as shown in Table 1. TABLE 1 CLASSMCATION OF FLEXURAL DUCTILITY

II

Method of calculating moment»resistance

AISCLRFD (1986)

Eurocode 3 (1992)

CSA16.1 (1989) 1

Plastic analysis, or elastic analysis with moment redistribution

Yielded stress blocks

Compact

Class 1

Class 1 Plastic

Elastic analysis with limited redistribution

Yielded stress blocks

Class 2

Class 2 Compact

Elastic analysis with very little redistribution

Elastic stress distribution limited to yield stress

Non compact

Class 3

Class 3 Non compact 1

Elastic analysis with no redistribution

Elastic stress limit below yield

Slender

Class 4

Method of analysing 1 load effects

Class 4 Slender

Predicting inelastic behaviour of Class 1 and 2 members therefore involves not only assessing the ultimate resistance but also the expected level of ductility. Due to the considerable interaction between the different modes of failure in the support regions of composite and steel beams Kemp and Dekker (1991) have expressed this requirement in terms of a single-limit states criterion which requires the available inelastic rotation O^ (resistance) to exceed the required inelastic rotation 9^ (action effect) as follows:

(ea/Ymd)>e,

(1)

in which Ymd is a partial material factor for ductility probably in the range 1.3 (ductile failure) to 3.0 (brittle modes) which allows for the uncertainties in the assessment of 0^ and 8^. Modelling approaches are described in this paper for quantifying the required and available inelastic rotations in this equation. Specific attention is given to the following important concepts: 1. Notional plastic hinges concentrated at sections of maximum moment are used to compare the available and required components of inelastic rotation on a consistent basis. 2. These notional hinges also enable the inelastic rotations due to yielding of the steel section, rotation within the end connection and cracking of the concrete to be combined in a logical way.

REQUIRED INELASTIC ROTATION The required inelastic rotation (0^ in Eqn.l) to achieve a plastic collapse mechanism or a defined level of moment redistribution has been investigated widely (Johnson and Hope-Gill (1976), Rotter and Ansourian (1979), Li et al (1995) and Gioncu and Peteu (1997)). A comprehensive analysis has been undertaken of many common cases by Nethercot et al (1995). On the basis of their work and investigations by the author, the three-span beam shown in Figure 1 a will be used for illustrating the concept of notional hinge rotations using the following simplifying assumptions which are not inherent in the approach:

293 1. 2.

3. 4. 5. 6.

The symmetrical beam arrangement is based on the conclusion by Nethercot et al (1995) that this represents a limiting condition for assessing the required inelastic rotation in composite beams. For reasons given at the start of the paper the largest inelastic rotations occur in arrangements requiring redistribution of moment from the two internal supports to the adjacent midspan regions: redistribution from midspan to the support is unusual and is of limited magnitude. Columns which form part of the support structure can be represented by increasing the effective stiffness of the outside spans. A uniformly distributed load is adopted as representative of general structural loading although there are unusual arrangements of concentrated loads requiring greater rotation capacities. Inelastic curvatures and rotations are confined to the central span, conservatively reflecting the influence of columns at the internal supports. A fully-plastic mechanism is developed in the central span in which inelastic behaviour during the formation of the midspan hinge increases the required inelastic rotations at the supports.

The inelastic rotations 0^ concentrated at notional plastic hinges are determined using the Principle of Virtual Work in the following form:

P*9r = er=IJM*6

(2)

in which P* is a pair of unit virtual moments in the directions of the change in slope on either side of the notional plastic hinge, M* is the resulting distribution of virtual moments and 6 is the elastic distribution of curvature due to the applied loads on the structure. In this expression the * reflects the virtual force system and the moments are taken as positive in sagging bending. The integration of M* 6 applies over the length of each member and is undertaken simply using integral tables. The summation applies over each of the members in the structure. For the three-span beam of Figure la, the distributions of elastic moments M and curvatures 6 due to the applied loads are shown in Figures lb and Ic respectively. To simplify the analysis a uniform flexural rigidity is adopted within each span and cracking of the concrete in composite beams is considered separately as a component of available inelastic rotation.

nnirmnnrinnintrtintntnTmTmnnmiinnrrnrrrrnniiniinmminiiiirim AA

Figure I: Three-span model of a continuous beam (a) beam arrangement; (b) bending moment distribution; (c) curvature distribution; (d) and (e) virtual moments.

The elastic value of the internal support moments M'e is determined conventionally using Eqn. 2 and equating the change in slope at the hinge at B or D to zero: M'e = O'BA + Q'BD) / (Lo/3EI, + LJ2EI,)

(3)

294 in which Q'^^=:v/jJj24El^ and Q'^Q=V/JJJ24EI^ are the end rotations due to the apphed loads if the spans AB and BD are simply-supported and the other properties are defined in Figure la. The' symbol reflects properties in the negative moment region adjacent to the internal supports and the subscripts o and c refer to the outside and central spans respectively. The dashed lines in Figures lb and Ic represent the distributions of moment and elastic curvature after a proportion m=(M'e - M'p) / M'e of the elastic negative moment M'^ is redistributed from the internal supports to the positive moment regions of the adjacent spans. For this proportion of moment redistribution and provided the outer supports are pinned, the virtual work Eqn. 2 is used again to determine simple, generally applicable expressions for the required inelastic rotation 6^3 of the notional hinge at the internal support B from the virtual moments in Figure Id (or 6^ at C from Figure le): 6.B = ni(e'BA + 0'BD)

(4)

AVAILABLE INELASTIC ROTATION The total available inelastic rotation 6j,of the notional plastic hinge at each internal support in composite or steel beams is made up of the sum of the effective components due to yielding of the steel section in the negative moment region S'^y and in the positive moment region O^y, rotation of the end connection 6acon and cracking of the concrete slab 6acr in composite beams: ea=(0'ay-9ay+eacon + eacr)

(5)

Each of these four components of inelastic rotation will be considered separately before showing how they are combined on a consistent basis using the same virtual work approach. Yielding of the Steel Section Specimens in negative bending (inelastic rotationff^y^and curvature ^'^ J The contributions to the available inelastic rotation of a notional plastic hinge in the negative moment region are usually measured in tests on doublecantilever specimens, as shown in Figure 2. Each half-span represents approximately the moment gradient over the length Lj between the section of maximum negative moment and the adjacent point of inflection. Failure is assumed to occur when the negative moment resistance falls below the ultimate design resistance M'p in Figure 3.

i jop of slab i i ^ ^ , . r - - i , , . i ~ - - , ,^^ i —,-;-'^r--'—S—"--^^^^-^— '^^^y^ " "-C^os \.^' , i \>* ^ ^ •

Figure 2: Double-cantilever test arrangement

Bradford and Kemp (1999) have reviewed briefly the results of 47 of these double-cantilever tests as well as 22 tests on continuous composite beams. The dominant modes of failure observed are local flange, local web and lateral-distortional buckling, although vertical shear failure occurred in a limited number of tests. Kemp et al (1995) and Kemp (1996) have described the interaction between local flange, local web and lateral buckling and have emphasised the importance of inelastic lateral buckling in inducing strain-weakening behaviour of both composite and steel beams. Inelastic lateral distortional buckling is less likely in fully loaded composite beams using hot-rolled steel sections than in plain steel beams due to the restraint from the slab and shorter buckling lengths, although the required inelastic rotation is greater. Kemp and Dekker (1991) have shown that in composite beams large areas of longitudinal

295 reinforcement in the slab relative to the area of the steel section contribute to a significant reduction in the available inelastic rotation for the following reasons: 1. Increased depth of the web in compression induces earlier web buckling and reduces strainhardening in the reinforcement. 2. Reduced curvature, caused by the increased height of the neutral axis when the critical strain is reached in the compression flange, reduces the inelastic rotation at maximum moment. Kemp and Dekker (1991) have analysed all of these aspects and concluded that in a double-cantilever test (Figure 2), if the proportion of web depth in compression is the same, a particular steel section in a composite beam can be assumed to achieve at least the same inelastic rotation as a plain steel section prior to failure. Kemp (1996) has proposed and Couchman and Lebet (1996) have independently evaluated the following empirical expression for this available inelastic rotation 6'ayt in such tests: e',yt = 1.5(0.5 M, Li / El3)(60 / A,)' /a

(6)

in which 0.5 M^ L^ / EI^ = elastic rotation of steel section in double-cantilever specimen at M^ Ms = ultimate moment resistance of the steel section Lj = length between sections of zero and maximum negative moment in Figure la. EIs= flexural rigidity of steel section a = ratio of web depth in compression to total web depth Ag = effective lateral slendemess ratio K^K^K^iL^ /izcV^r Kf = (b/tf)/20ef is the flange factor for width b and thickness t^ K^ = (ad/t^)/35€^ is the web factor for depth d and thickness t^ Kd = distortional restraint factor (1 for steel section, 0.71 for composite section) i^ = radius of gyration of the elastic portion of the web and flange in compression € = JlSOJf , and fy = yield stress of the flange or web. The observed momentcurvature curve in negative bending (solid line) may be represented by the dotted bilinear relationship in Figure 3 based on an effective yield moment M'y shown in this figure. The corresponding distribution of inelastic negative curvature in the yielded length L'y adjacent to the supports in the central span of the beam is shown in Figure 4a. The yield moment M'y in Figure 3 is selected to give the same inelastic rotation as the test value predicted by Eqn.6 for a linear moment gradient:

actual design idealised (bi-linear)

Curvature

Figure 3: Moment-curvature relationship (j);^ = 2 0;^/LV

(7)

Specimens in positive bending {inelastic rotation Q and curvature (|)„„)

296 Inelastic curvatures in positive (sagging) regions of continuous beams which occur during the development of the ultimate moment resistance, increase the required inelastic rotations in the negative moment regions for a particular level of moment redistribution. If the moment-curvature curve is represented in the same way as for negative bending by the idealised bi-linear relationship in Figure 3, a parabolic distribution of inelastic positive curvature is obtained as shown in Figure 4b. The curvature at the ultimate moment ^^y is calculated from the change in strain over a depth d^ between the maximum compressive strain on the upper surface of the concrete of 0.0035 and the plastic neutral axis, as follows: (t),y = ( 0.0035/d„) -(Mp/EI)

(8)

Rotations in End Connection, d^^^ Increasingly, semi-rigid connections have been investigated with a moment resistance similar to or less than that of the adjacent composite beam in negative bending for the purpose of providing significant ductility in the connection rather than in the adjacent member. As these rotations are located at the notional plastic hinge, the available inelastic rotation Q^^^ in Eqn.5 is equated to the rotation in the connection at the ultimate moment M'p. A review of these composite connections has recently been provided by Leon (1998) in which a range of appropriate details are described. If the moment resistance of the connection is less than the adjacent yield moment in the composite beam, the available inelastic rotation due to yielding will be negligible. Nethercot et al (1995) have evaluated the required rotations in such cases.

itnf»»niMiiMriiiiiiiiinmnt»inimiiiiinniriiiinmnnmiiiimiiimirimm AS BS C S"5

I

(a)

(c)

i /

^

Figure 4: Inelastic curvatures (a) and (b) yielding in negative and positive bending; (c) cracking in composite beams

If a partial height, end-plate connection is adopted which does not provide for the transfer of tension force in the upper region of the steel section, the ultimate moment is obtained from the couple formed by the tensile force in the slab reinforcement and the compressive force in the bottom flange and adjacent web of the steel section. Kemp et al (1995) have demonstrated how local and distortional buckling is limited by reducing the proportion of the steel section in compression in this way. Excellent ductility has been achieved in tests in which the rotation capacity was increased more than sixfold over the normal rigid endplate connection. As yielding of the reinforcement is common to the behaviour of both the connection and the adjacent beam, it is proposed that the combined inelastic distortion be considered as part of the yielding of the adjacent beam in accordance with Eqns. 6 and 7. Cracking of the concrete, 0acr in composite beams If the uncracked and cracked elastic flexural rigidities of the composite beam in negative bending are EI and EI' respectively and the moment at which cracking of the concrete effectively occurs is M'^, the associated distribution of inelastic curvature in composite beams is shown in Figure 4c.

297 Calculation of available inelastic rotation The preceding evaluation of the four components of available rotation defined in Eqn. 5 is based on measurements made in a wide range of double-cantilever tests. To assess the limit state of ductility (Eqn. 1) for the representative three-span structure in Figure 1, it is necessary to know whether these test results are applicable to the notional plastic hinge model and on what basis the different components of available rotation should be added together as shown in Eqn.5. The virtual work approach of Eqn.2 may be extended as follows to assess the available rotation Q^ at the notional plastic hinges at B and D in the three-span structure of Figure 1, based on the inelastic curvature distributions shown in Figures 4a to 4c and the connection rotations:

p*ea = e.=LJM*6 + i:M*„„e_

(9)

For this purpose P* is a pair of unit virtual moments in the directions of the change in slope on either side of any notional plastic hinge, M* is the resulting distribution of virtual moments in Figures Id and le, 6 is the distribution of available inelastic curvature in the central span in Figures 4a to 4c, M^^on is the virtual moment at each end connection and Q^^^^^ is the concentrated rotation in the end connection. If the distribution of curvature in Figures 4a and 4b is simplified by replacing the curved relationship by the linear function shown by the dashed line, the following expression is obtained for 6^^ or d^ from Eqn.9: 6aB = 9aD = ^'ay " ^ay + 9acon+ ^acr

= 0.5 (j)',y,LV - 0.333 (|),yLy + 6,,,„ + 0.5 L'„ [(l/EI) - (1/ED]( M^ + M'„)

(10)

in which all the terms are defined in the preceding descriptions of available rotation or in Figures 4a to 4c. The overestimate of O^B caused by replacing the curved distributions of curvature (shown by the solid lines in Figures 4a and 4b) by the dashed linear distributions is less than 2% over a wide range of practical parameters. Due to the symmetrical distribution of inelastic curvatures in the central span, the integral JM* 5 is directly related to the area of each curvature distribution irrespective of the length over which the curvatures are distributed. The distribution of total curvature in the central span is given by the elastic distribution in Figure Ic from which the notional required inelastic rotations are derived, plus the inelastic distributions in Figures 4a to 4c from which the notional available inelastic rotations are determined. These two rotations form the basis of the limits-state criterion of ductility in Eqn. 1 and are compatible as they both refer to the rotation of notional hinges at the supports. Non-Dimensional Form The limit-states criterion in Eqn.l can be expressed in non-dimensional form by considering the ratio of available to required rotation Q^/Qr obtained from Eqns. 4 and 5: ra = Qa/Q'e = (QjQXV&e) = [2Ymd m{ 1 + 0.667(W L,) / (Ely Ey}] / [ ( l - m ) ( l - l / y r ^ ) ]

(11)

in which r^^sO^^ /O'^ is the non-dimensional available rotation capacity obtained by dividing the total available rotation 6^ in Eqn. 10 by the elastic rotation 6'^= 0.5 Mp Lj / EI^. in the test specimen in Figure 2 at moment Mp; Lj = 0.5Lc(l - l/\/l +n) is the length between the section of maximum negative moment Mp and the adjacent point of inflection; EI^, is the flexural rigidity in the central span; n is the ratio of moment resistances Mp / Mp and m is the proportion of moment redistribution defined in Eqn. 4. This equation identifies the factors influencing the amount of rotation capacity which should be available

298 to satisfy the limit-states Eqn. 1 for a specified partial material factor Ymd- Thus, for example, for m=0.3 representing 30% redistribution, n=0.7 and constant ratio of length to flexural rigidity (L/EI) in all spans, a partial material factor of 1.3 is achieved by providing an available rotation capacity of ra=8.0. Conclusions A consistent limit-states equation for ductility in continuous composite and steel beams is satisfied if the available inelastic rotation at each notional plastic hinge (the resistance) sufficiently exceeds the required inelastic rotation (action effect) at the same hinge due to moment redistribution from the elastic condition. The Principle of Virtual Work provides a consistent basis for determining both the required and available rotations at the notional plastic hinges. This principle also enables the components of rotation due to yielding of the steel section, rotation of the end connection and cracking of the concrete to be appropriately combined at each notional hinge on a rational basis. For symmetrical arrangements the rotation at each notional hinge is equal to the area under the inelastic curvature distribution and is therefore compatible with rotations measured in double-cantilever tests. References Bradford M.A. and Kemp A.R. (1999). Buckling in Continuous Composite Beams. Accepted for publication in Progress in Structural Engineering and Materials. Couchman G. and Lebet J.-P. (1996). A New Design Method for Continuous Composite Beams. Structural Engineering International 6:2, 96-101. Gioncu V. and Peteu D. (1997). Available Rotation Capacity of Wide-Flange Beams and Beam-Colunms. Journal of Constructional Steel Research 43:1-3, 161-218 and 219-244. Johnson R.P. and Hope-Gill M.C. (1976). Applicability of Simple Plastic Theory to Continuous Composite Beams. Proceedings of the Institution of Civil Engineers, London Part 2 61:3, 127-143. Kemp A.R. and Dekker N.W. (1991). Available Rotation Capacity in Steel and Composite Beams. The Structural Engineer 69:5, 88-97. Kemp A.R., Trinchero P. and Dekker N.W. (1995). Ductility Effects of End Details in Composite Beams. Journal of Constructional Steelwork 34:2, 187-206. Kemp A.R.. (1996). Inelastic Local and Lateral Buckling in Design Codes. Journal of Structural Engineering, ASCE 122:4, 374-382. Leon R. (1998). Composite Connections. Progress in Structural Engineering andMaterials 111, 159-169. Li T.Q., Choo B.S. and Nethercot D. A. (1995). Determination of Rotation Capacity Requirements for Steel and Composite Beams. Journal of Constructional Steel Research 32:3, 303-332. Nethercot D.A., Li T.Q. and Choo B.S. (1995). Required Rotations and Moment Redistribution for Composite Frames and Continuous Beams. Journal of Constructional Steel Research 35:2, 121-164. Rotter J.M. and Ansourian P. (1979). Cross-Sectional Behaviour and Ductility in Composite Beams. Proceedings of the Institution of Civil Engineers, London. Part 2. 67:3, 111-\ A3.

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

299

DUCTILITY OF THIN-WALLED MEMBERS A. Moldovan \ D. Petcu ^ and V. Gioncu ^ * Building Research Institute, 1900 Timisoara, T. Lalescu 2, Romania West University, 1900 Timi§oara, V. Parvan 4, Romania ^ Polytechnical University, 1900 Timi§oara, T. Lalescu 2, Romania

ABSTRACT The study of ductility for thin-walled members finally reduces to the analysis of local plastic mechanisms for steel plates subjected to in-plane compression. The type of mechanism determines the rigid-plastic curve shape of the plate, which is important for estimating the ductility of thin-walled members. In evaluating the plastic rotation capacity three types of possible mechanisms are studied: pyramid, roof and flip-disc shapes mechanisms. A computer program named DUCTROT TWM was elaborated by the authors for rapidly determine the ultimate rotation and the ductility of a thin-walled section. Comparison with the experimental results showed that the obtained theoretical rigid-plastic curves can be used to draw the framework aroimd the actual behaviour of thin-walled members. A rotation capacity of about 1.7 can be used for seismic design.

KEYWORDS Thin-walled members, plastic mechanism, rigid-plastic curve, ductility, reduced plastic moment, computer program. INTRODUCTION Thin-walled steel structural members may lead to a more economic design as a result of ease of construction and of their superior strength to weight ratios. In seismic areas the use of cold-formed profiles has not attained the same importance as hot-rolled ones because their bending behaviour is strongly affected by the local buckling of the compressed part. The evaluation of the complete loaddeformation history pointed out that the exclusion of thin-walled profiles in dissipative zones of seismic resistant structures could be revised. For one or two levels structures and for low or medium earthquake intensities, seismic loads have a lower influence and thin-walled members can be used. The paper has the aim to provide useful data on the use of these profiles in seismic resistant structures. The

300

ductility of thin-walled sections in the plastic range is studied by evaluating the plastic moment, the ultimate moment and the rotation capacity of thin-walled members. PLASTIC MECHANISMS FOR THIN-WALLED MEMBERS Thin-walled steel members often develop local buckling and local plastic mechanisms, both of which are essential features determining their behaviour. The study of plastic mechanisms for thin-walled members is governed by the local plastic mechanisms for thin-walled steel plates subjected to in-plane compression. These plates can develop three types of local plastic mechanism, namely the pyramidshaped, the roof-shaped and the flip-disc mechanism. For doubly supported plates they are observed in Fig.l (a - pyramid-shaped, b - roof-shaped, c - flip-disc).

u '

a)

Figure 1: Types of plastic mechanisms for doubly supported thin-walled plates The type of mechanism determines the rigid-plastic curve shape of the plate. The intersection of elastic and rigid-plastic curves can be used to estimate the ultimate load and the framework around the actual behaviour of thin-walled members can be drawn. For the idealised roof mechanism, the average axial stress a is given by eqn.(l), derived by Mahendran and Murray (1991):

(l-r(fj-l

~(l^r)^K (1)

c —s b

4(l + r)YA

+1

2(1+ r) A k t

-In 2(l + r)A

4(l + r ) V A

+1

2(1+ r) A

where k=cosec^ a + cosec^ p, A is the deflection of the flange and fy the yield stress. The mechanism was considered in two parts, the inner region of width b-2c and the two identical edge regions each of width c. In the above mentioned paper, the basic geometric parameters a, c and r were assumed to be 30°, 0.2b and 0.6, respectively, to obtain the rigid-plastic curve. For the idealised flip-disc mechanism of parabolic shape, the average axial stress a was obtained by eqn.(2). Post-plastic behaviour for the pyramid mechanism was studied by Feldman (1994), who obtained its characteristic equation (3). It was concluded that the value given by eqn.(3) remains practically unchanged for a = 45°.. .55°, so a = 45° may be considered.

301

— = 1.

^1.1

-+4

+1

.2\

1 + 4-

— = 1. /."2

2A

+1

2A

/:/ ,

+—\ni kt 2A

(2)

1+4 b'

2A|

(3)

kt

where k=l+cosec a. The eqns.(l)...(3) determined for doubly supported plates, can be used also for simply supported plates if half of the mechanism shape is used, as Gioncu and Mazzolani (1999) have mentioned. So these mechanism types can describe the collapse shapes of beam flanges. Theoretical and experimental research developed by Batista (1989) showed that for simply supported thin-walled plates, flip-disc mechanisms are formed. The authors of the present paper observed that for the flip-disc mechanism, the variation of ratio a/b from 0.1 to 0.5 gives small differences for the value of plastic moment, so the ratio a/b was assumed to be 0.1. In evaluating the ductility of thin-walled members, a type of section commonly used for beams and columns was considered: I section formed of two U profiles. Since the web of this section has the thickness of 2t, where t is the thickness of the U profile, the local buckling will only affect the compressed flanges. The three types of plastic mechanisms discussed above were taken into account. The local buckling of the compressed flange causes the decrease of its capacity, which is expressed by: C(0) = C,

(4)

fy where Cp is the load at which flange is fiilly plasticised. a/fy depends on the type of the mechanism. The reduction in the width of the compressed flange because of local buckling causes neutral axes to shift from the midpoint of the web. Distance z from the compressed flange to the neutral axes is determined from internal forces equilibrium (Fig.2) and is given by eqn.(5):

C(0 )

T ^ •

2

m i

\

rr^/ ' . — Figure 2: Intemal forces equilibrium for I section

1

J

--^1

302

h z = —1 +- 1 - ^ 2h V 2 Jy)

(5)

and the rotation 9 from eqn (6): (9 = 2

A t h t

1

(6) a

2hy

where b is the width of the flange and h is the depth of web for the I section. Plastic moments become: M = Cz +I(h-z)+fytz^ +fyt(h-z)^

(7)

where C, I and z are functions of rotation 9. Hence, the rigid-plastic curve can be drawn. The ductility of thin-walled beams can be established by intersection of the rigid-plastic curve and the reduced plastic moment curve affected by the safety factor YM. Reduced plastic moment is the plastic bending moment for the effective section of the member. DUCTROT TWM COMPUTER PROGRAM Since the method presented above for evaluating the plastic rotation capacity is complicated enough, a computer program named DUCTROT TWM ( DUCTilitv RQJation of Thin-Walled Members) was elaborated by the authors of the present paper. The ultimate rotation 0r and the ductility of a section can be determined with its contribution. The program consists of 8 sheets. Some of the most important of them are presented in Fig.3. Sheet 1. Types of sections considered: built-up U section, built-up C section, cold-rolled box section, welded I section. Sheet 2. Material characteristics. The user introduces: yield limit, uhimate tensile strength, yield strain, hardening strain, ultimate strain, modulus of elasticity, tangent modulus. Sheet 3. Geometrical characteristics. The user introduces: half-width of the flange, depth of the web, thickness of the section, radius of comers. DUCTROT TWM gives: cross section area, effective cross section area for bending and for compression, position of centroid, gross and effective second moment of area, elastic and plastic section modulus, effective elastic and plastic section modulus. Sheet 4. Loading system. Standard beam may be acted by one or two equal concentrated loads, with or without axial load. Type of mechanism can be in-plane (symmetric or asymmetric) and out of plane. Sheet 5. Mechanical sectional characteristics. The program gives: plastic moment and reduced plastic moment, plastic and reduced plastic axial forces. Sheet 6. Geometrical member characteristics. DUCTROT TWM gives: slendemess of the element, plastic rotation and ultimate rotation.

303 Ahooi DUCTROT r\i/M 98

DUOTROT-TWM

•0®f»:-^Tar, yki^^^^.Mim&imsmm

IDUCTHOT 37

g<|.f|p|#|,i;^4fl

.

P*feM<sfSfedii8«eife«^jipK S t ^ H ^ i l s e e ^

• ^^^^ tmii^mMmt?(WM rw^

mcnom

mm

^•^ff^^^^A

^^^

Figure 3. Sheets of DUCTROT TWM computer program

304

. ^

j

75.Q0

flWfc

I ^

1 300 00

9m

,1 u

^ i 1 I t

i

^^;j 1

^^

imT \

^^ ppSTT

'^oo

«».^^;i

n^l

_.2

3 ^14 '^^ 'Iw^MJM^Mi

S^ ^

T^ |S|

f 1/2

|.

1/2

^

1^^^

7 7

1 | : (l-a)/2 i a .[ (l-a|/2 ^

J

^ ~^!«?ttiaft.rA'-r-

?l

.

I iM^Xi

{ I 4000 00

9«t

1

^1 r

: r| •GHBteue I

Figure 3 (continued)

^tii

305

DUCTROT 97 j^?)^^ ^ N P

^^IlIsM^^

|f^igSsdSS>Si^^^^N^^^^^ S;tem&&rCfi!9i*i^^ ?J^«<»&<::^pA<% I

<J»|

»d:

8

pof^iim

132^32

\

VditȤ})^Oe{A-

; DUCTROT 97

BiMe

^1

L6j)i»fLM 5000 00 Wji^-teil^i;i^«

Figure 3 (continued)

taj^faHrf j 6000 GO

VSafefeott5l<» j 250 00

306 Sheet 7. Rotation capacity. The program determines: rotation corresponding to plastic hinge and rotation capacity and draws the moment-rotation curve. Sheet 8. Parametric variation. The user can demand the draw of variation of ultimate rotation and rotation capacity in function of the following parameters: width of the flange, depth of the web, thickness of the section, axial load, yield limit and length of the beam. COMPARISON OF THEORETICAL RESULTS WITH EXPERIMENTS Though some thin-walled beams were tested, there are only a few complete moment-rotation (or loaddeflection) curves pubhshed till now. De Martino et al. (1992) have tested 5 beams with built-up U section and loaded with two equal forces. Fig.4 presents for those five tests, moment-rotation experimental curves in comparison with elastic and rigid-plastic theoretical curves. It is mentioned that pyramid-shaped mechanism gives a very close curve with roof-shaped mechanism, so only roof mechanism and flip-disc mechanism curves were represented. The reduced plastic moment horizontal line was also figured. Fig.4 shows that for the analysed sections, flip-disc mechanism curve gives smaller moments for the same plastic rotation and is closer to the experimental curves. It can be seen that the obtained theoretical curves can be used to draw the fi-amework around the actual behaviour of thin-walled members. The differences for high values of plastic rotations are due to the high non-linear plastic deformations, which are not considered in analysis. For determining the ductility, which is obtained for the first part of the rigid-plastic curve, the accuracy is adequate. There is an exception with the beam noted P6, whose experimental moment-rotation curve is not typical for built-up U sections, as the authors of tests have mentioned. For the others tested beams, a ductility of 1.7... 1.75 was obtained. CONCLUSIONS The comparison between theoretical plastic curves of a thin-walled beam and the experimental curve established that flip-disc mechanism is characteristic for this type of sections and decides its plastic behaviour. More experimental data are necessary to emphasize this conclusion. On the other hand, the influence of imperfections on the theoretic plastic curves must be studied and the behaviour of other types of sections used at thin-walled members must be analysed. Even if for thin-walled structures only the elastic design is allowed by seismic codes, the theoretical and experimental results show that a reduction factor for seismic loads q=1.7 can be appUed.

REFERENCES Batista M. (1989). Etude de la stabilite des profils a parois minces et section ouverte de types U et C. These de doctorat, Universite de Liege. De Martino A., Ghersi A., Landolfo R., Mazzolani F.M. (1992). Calibration of a bending model for cold-formed sections. Xf International Specialty Conference on Cold-Formed Steel Structures, St. Louis, p.503-511. Feldman M. (1994). Zur Rotazionskapazitat von I-Profilen statisch und dynamisch belasteter Trager. Ph. Doctoral Thesis, Aachen Universitat.

307

/roof

MpNr/^

rMpNr/!fM

/flip-disc

/nx)f

/flip-disc

701

\ /F.ri L B ^ 71

SO-

^

1

.

!

.

1 1;

SO E 40-

1

/I

i'30

J 30

/

150

0200x10'^ 0

20

40

60

80

_ i

100

i

120"

140 X10"^

ecradl

e(rad]

yTOof

i

/

LI

100

1

/

/ i

50

1 1

/flip-disc

r3

150x10r3

lOOx-IO

)[rad]

[ P10

^pNr^^M

.roof

.flip-disc

Figure 4. Comparison between theoretical and experimental moment-rotation curves Gioncu v., Mazzolani F,M. (1999). Ductility of Seismic Resistant Steel Structures, E & FN Spon (manuscript). . Mahendran M., Murray N.W. (1991). Effect of initial imperfections on local plastic mechanisms in thin steel plates with in-plane compression. International Conference on Steel and Aluminium Structures, Singapore, p.491-500.

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T e c h n i c a l p a p e r s on

FRAMED STRUCTURES: GLOBAL PERFORMANCES STA TIC AND STABILITY BEHA VIO UR

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Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

309

NUMERICAL STUDIES IN SEMIRIGID METALLIC FRAME STRUCTURES P. Alexa^ and A. Catarig^ ^Faculty of Civil Engineering, Technical University, 3400 Cluj-Napoca, RO

ABSTRACT The study presents several aspects involved and/or emphasised in the semirigid behaviour of steel frames. Out of the abundence of analytical bending moment - relative rotation models, two models have been associated to several connection types. As the quasitotahty of proposed analytical models M- 6r depend on the geometric, mechanic and elastic properties of the connecting elements, a data - base has been worked out for rolled steel profiles produced in Romania. The recomended M - Or multilinear models of EUROCODE 3 are also presented. Using this adapted M - Or models and currently designed beam - column connection of skeletal structures, the study focuses on stability behaviour, the instability being generated by large lateral displacements. Several numerical studies have been carried out in a sensitivity analysis manner. The results have been focused an P - J curve aiming at emphasising the degree of non-linearity. The analysis has been extended up to either strength limit state or stability loss. Using flexible and rigid beam - column connection types and several analytical M- Or models, it has been proved the prevalence of the analytical model over that of the connection type, i.e. associating an arbitrary analytical model to a given connection type, the structural behaviour is marked stronger by the model than by the connection type. A rather rigid beam - column connection type as it is that with top and seat and, also double web angles may behave quite flexible if a "soft" M- Or, as the three parameter model, is associated to it. The study is part of larger investigations on semirigidity including spatial fi-ames and post-elastic behaviour.

KEYWORDS Semirigid, metallicfi-ames,stability, parametrical study, M- Or data base. INTRODUCTION The dihotomy "rigid - semirigid" has followed a classical evolution, also common to several other problems of Structural Mechanics. At the beginning -fi-oma cronological point of view - it was the period of awamess of the problem of semirigidity by investigation and later by the design engineers, Rathburn (1936).

310 During this first stage, the elastic connectivity (semirigidity) of mainly beam - columns joints of the metaUic frame structures has been signalled by the investigations as a simple real fact and has been viewed rather as an imperfection of the connection. The first investigations carried out aimed at developing a technique able to correct the results obtained under the "rigid connexion" hypothesis, as well as, to recognize, in the engineering practice, a given connection as being rigid or semirigid. The possibiUty of taking into account, in structural analysis and design, the semirigidity as "another type of connectivity", rather than an imperfection, lead naturally to the necessity of modelling the behaviour of semirigid connections, Livesley (1964), Frye & Morris (1975). The specific weight of the influence of semirigid behaviour on state stress of frame structures, as well as the real fact that semirigidy is emphasised mainly in the rotation of the connecting cross sections, directed the theoretical and experimental investigations towards the behaviour of semirigid connections from the point of view of bending moment relative rotation (M- 6r), Jones et al. (1983). The experimental investigations reported in the literature in an overwhelming amount emphasized several factors governing the bending moment - relative rotation behaviour. - connection type, - initial stiffiiess of the connection, - stress and strain state of the connection. The influence of these factors is quantified using mechanical, geometrical and elastic parameters of the connecting elements. Several aspects have to be underlined in order to have a clear picture of the connection behaviour along bending moment - relative rotation direction: - the connection's behaviour does not depend on the mechanical, elastic or geometric parameters of the connected elements (beams, columns, etc.), - the connection's behaviour is nonhnear (the nonlinearity is due to mainly the "loading level" of the connection). This fact is easily proved by the shapes of M - Or experimental curves. The great number of experimental investigations reported in the literature, Chen & Kishi (1989), can be explained, on one hand, by the dependence of the connection behaviour on the connection type (leading to the necessity of conducting investigations on every connection type used in the engineering practice) and, on the other hand, by the necessity of associating M - Or analytical models as loyal as possible to the experimental M- Or curves. The loyalty of the analytical M- Or models to their experimental curves comes from the ingeniosity of processing the mechanical, elastic and geometrical parameters of each connection type and, also, from the use of some shape nondimensional parameters taking values in a relativelly small domain, Barakat (1991). By suitable choosing of shape parameters, the analytical M - Or curves may by closely brought to their experimental counterparts, along the entire loading process of the connection, up to its limit state, Chen (1977). In the case of planarframeswith semirigid beam column connections, several analytical M - Or models, Chen & Lui (1991), imposed themselvs through their accuracy (Table 1).

311 TABLE 1 M' Or ANALYTICAL MODELS

Analytical Analytical expression Remarks model Kj, K2, K3, K4 ' parameters I Rathbun model M- '~j-^i{^2^A + ^ 5 ^ 5 / — y r — of beam relative and (1936) connection stiffness Ang & Morris model (1984) I Polynomial model (1977) I Frye & Morris

I Richard and Abbott model (1975) I Exponential model (1983) Colson & Louveau Exponential model (1990) I BCishi & Chen

'Or

m

KM 7+ M

0^,M,n,K depend on connection type

e, = C^CMy +Q(CM)' +C,{CMy Ci, C2, C3 depend on the connection type K depends on the geometry of connecting elements 0O - reference plastic R. M= rotation n depends on connection 7+ type Mu - ultimate moment of the M connecting section M_ n depends on connection type Mu - ultimate moment of the M ^.= connecting section M n depends on connection M,. type

I Exponential model (1990) Wu & Chen

M = «ln 7 + —^ M„

I Exponential model (1993) King & Chen Differential model (1993) King & Chen

dM = dO.

Mu - ultimate moment of the connecting section 60 - reference plastic rotation n depends on connection

M5 R\1-

dM dM = /?,, dd.

M

M-M„ M,. - M. M<M^

c depends on connection type Mo is the upper value of M>M\ bending moment in Unear behaviour

312

As a synthesis of the curvihnear M - Or analytical models, the EUROCODE 3 recommends, EUROCODE 3 (1993), two models for unbraced and braced planar frames (Table 2). TABLE 2 RECOMENDED A/- Or MODELS BY EUROCODE 3

Analytical model EUROCODE 3 model 1 for frames: * unbraced

Analytical expression

Remarks

|

_ M m= S _ 256^^4 2 _ ,^ /w = '- , -<m<1.0 7 3

* braced

M - current value of bending moment Mo - ultimate moment of the connecting section m = 80^, m<3 E - Young modulus 1 20(9+5 2 _ ,^ h - inertia moment of m= , -<m<1.0 connected beam 7 3 ^-rotation corresponding toM Lb - span of beam

Based on these models the research group of the Structural Mechanics Chair of Civil Engineering Faculty of the Technical University of Cluj-Napoca, developed an M - ft data - base associated to rolled steel profiles manufactured in Romania, Catarig et al. (1993). The theoretical (numerical) investigations carried out at the Structural Mechanics Chair, Catarig et al. (1991), have been extended over space metallic frames and over the bending moment - axial stress resultant (M- N) interaction, Alexa et al. (1994). The necessity of incorporated the axial stress resultant - axial deformation {N - Ur) became necessary, and this, in its turn, emphasised the dependence of the semirigid behaviour on the mechanical, elastic and geometrical parameters of the connected elements (beams and columns mainly), Catarig et al. (1988). Using the analytical M - ft models and the Romanian rolled steel profiles data - base, the authors carried out numerical parametrical studies specific to stability loss by large displacements of planar frames, Moldovan (1997).

NUMERICAL RESULTS A ten - story one bay steel frame (Fig.l) with its beams semirigidly connected to continuous columns has been analysed from the point of view of its stability. The stability behaviour is emphasised using the P - ^ curve, where P is the loading (unique) parameter and A is the lateral displacement of one end of the top beam. Several types of beam - column connections have been considered associated to two M - ft analytical models, Catarig et al. (1998).

313

A sensitivity technique has been apphed by changing the associated analytical model, the connection type and the loading in turn. A""

-^

h

m.e>L n i l h

\...../A... -"^

h

-^

h

iiijfij^iij. l l l l X l l l l l

h

.q J,

X X%1?NL \lr 4 . ^

d

[^^^^.^q^^^

h h

y q X AXija

j ^ m

h

/q

q=^qo q^r 40daN/cm OdG P = 2000C1QN

Element A, (cm2) type

® (D

190 150

(cm^) 105100

(cm3) 3890

78050 2890

columns - t y p e ® beams - type ® I = 900 cm h = 360 cm

^ •sLX'JJX X 4 / X 4/

1 '''

h h

ITT

4

777 "77

f

'—^ Figure 1: Ten story frame

Figure 2 shows the numerical results of theframewith a two web angles beam - column connection associated to a three parameters M - Or model under a constant lateral force P and a steadily increasing gravitational force q.

100 DISPLACEMENT (c#n]

Figure 2: P- A curves for several connection types and analytical models

314 The increasing gravitational load q took several "starting" values in order to emphasise de degree of non-linearity in ihtP - A curve. The same frame has been subjected to several values of the lateral force P and steadily increasing gravitational load q (Fig.3).

1 ' r 20 25 30 HORIZONTAL DISPLACEMENT I cm]

Figure 'i. P - A curves under variable vertical load Three different types of connection, each one associated in turn to two analytical M - Or models have been used under constant lateral force P and increasing gravitational force q (Fig.4).

1 ^ 120 HORiZOKTAL DfSaACE^€NT (cm]

Figure A.P - A curves under variable horizontal disturbing force

315 CONCLUDING REMARKS The two web angle connection is a flexible connection and, except, for small values of the vertical load q, the general behaviour of the structure is strongly non-linear from the very beginning (Fig. 2) even for small lateral forces P. Nevertheless the loss of stability is reached only under large values of gravitational loads, while the strength limit state is reached rather early. Increasing the lateral force P and keeping the gravitational force q in the medium range of the strength state, the P - A curve emphasise more and more the pattern of loss of stability by large displacements (Fig.3). Associating a "rigid" M - Or polynomial model to a "flexible" connection (two web angles) emphasises the prevalence of the model over the connection type as it can be seen in the first four P ' A curves of figure 4, which are almost linear. The prevalence of the M - Or model over the connection type is proved fiirther, when a flexible analytical model (three parameters model) is associated to a "rigid" connection (top and seat and two web angles), as it can be seen in figure 4. The results prove the importance of connection type - analytical model association in structural behaviour study.

REFERENCES Rathbun J.C. (1936).Elastic Properties of Riveted Connections. Transactions of American Society of Civil Engineers 101, 524-563. Livesley R.K. (1964). Matrix Methods of Structural Analysis, Pergamon Press, Oxford, UK. Frye M.J. and Morris G.A. (1975). Analysis of Flexible Connected Steel Frames. Canadian Journal of Civil Engineers 2:3, 280-291. Jones S.W. et al. (1983). The Analysis of Frames with Semi-Rigid Connections. Journal of Constructional Steel Research 3:2, 1-13. Chen W.F. and Kishi N. (1989). Simi-Rigid Steel Beam-to-Column Connections: Date Base and Modelling. The Journal of Structural Engineering WZil, 105-119. Barakat M. and Chen W.F. (1991). Design Analysis of Semi-Rigid Frames: Evaluation and Implementation. Engineering Journal, AISC, Second Quarter, 55-64. Chen W.F. (1977). Theory of Beam - Columns - The State of the Art Review. Reprinted from the Proceedings of the International Colloquium on Stability and Dynamic Loads, SSRC/ASCE, Washington, 631-648. Chen W.F. and Lui EM. (1991). Stability Design of Steel Frames, CRC Press, Boca Raton, Florida, USA. EUROCODE 3 (1993). Design of Steel Structures, Technical University, Timisoara, RO.

316 Catarig A. et al. (1993). Problems of Structural Analysis Versus Connections Behaviour. Proceegings ofInternational Symposium of Cluj-Napoca 1, 293-302. Catarig A. et al. (1991). Elastic and Mechanic Properties of the Strainght Bar EUastically Connected into Finite Rigid Zones. The Scientific Journal of the Polytechnical Institute of ClujNapoca 34, 53-62. Alexa P. et al. (1994). Modem Techniques in Structural Engineering, Research Report, Ministry of Education, Bucharest, RO. Catarig A. et al. (1988). Stability and Nonlinear Analysis of Semirigid Sfeletal Structures with Finite Dimension Joints. Construction Journal 11/12, 30-34. Moldovan Corina (1997). Mechanical and Analytical Models in Analysis of Semirigid Metallic Structures, Ph.D. Thesis, Technical University, Cluj-Napoca, RO. Catarig A. et al. (1998). Plane and Spatial Semirigid Steel Structures, Research Report, Ministry of Education, Bucharest, RO.

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

317

Influence of residual stress on the carrying-capacity of steel framed structures. Numerical investigation G. M. Barsan and C. G. Chiorean Faculty of Civil Engineering, Technical University of Cluj, 3400 Cluj-Napoca, Romania

ABSTRACT In order to assess explicitly the influence of residual stresses on the inelastic behavior of the steel frames a new computer program NILDFA (Non-linear Inelastic Large Deflection Frame Analysis) was elaborated by the authors. The states of strain, stress and yield stress are monitored explicitly during each step of the analysis and the effects of residual stresses are accurately included in the analysis. Three residual stress patterns, constant, linear and parabolic distribution has been considered for three types of calibration frames: a portal frame, a six-story two-bay frame and a two-story frame. The numerical tests and the comparisons made proved the effectiveness of the proposed analysis method. KEYWORDS Residual stress, material imperfection, advanced analysis, steel frames, inelastic analysis, semi-rigid frames, large-deflection analysis. INTRODUCTION An upcoming technology for limit-states design, termed advanced analysis, is defined as any method of analysis that sufficiently represents the strength and stability behavior, such that specification member capacity is not required ( Chen & Toma, 1994). The reason for this is that, since advanced analysis can directly assess the strength and stability of the overall structural system as well as interdependence of member and system strength and stability, separate member capacity checks are not required. A plasticzone analysis that includes the spread of plasticity, residual stresses, initial geometric imperfections, and any other second-order behavioral effects, would certainly be suitable to be classified as advanced inelastic analysis in which the checking of beam-column interaction is not required. An efficient object-oriented Turbo-Pascal computer program NEFCAD for large deflection elastoplastic analysis of semi-rigid steel frames, in the plastic-zone approach has been developed recently, by the authors (Barsan & Chiorean, 1999). In this approach, the gradual plastification of the cross-section of each member are accounted for by smooth moment-rotation curves of Ramberg-Osgood type, experimentally calibrated, and the residual stresses are only implicitly considered. In order to assess explicitly the influence of residual stresses on the behavior of the structure, and in particular, on the carrying-capacity of the frames, and to compare this to results obtained by NEFCAD program, a new computer program NILDFA (Non-linear Inelastic Large Deflection Frame Analysis) was elaborated by the authors. This time the states of strain, stress and yield stress are monitored explicitly during each step of the analysis and the effects of residual stresses can be accurately included in the analysis. Three

318 residual stress patterns, constant, linear and parabolic distribution has been considered for calibration frames, as it will be shown in the following. In the same purpose three types of frames were selected as well: a portal frame first analyzed by El-Zanaty (1980) followed by others researchers, a six-story twobay frame (Vogel, 1985) and a two-story frame that has been proposed by Ziemian (1992).

MATHEMATICAL FORMULATION The magnitude and distribution of residual stresses in hot-rolled members depend on the type of cross section and manufacturing processes and different patterns are proposed. In the US, the residual stress is considered constant in the web although, when the depth of a wide flange section is large, it varies more or less parabolically. Another possible residual stress pattern in the web is the one simplified by a linear variation as used in European calibration frames (Vogel, 1985).These three residual stress patterns for the web, i.e., linear, constant, or parabolic distribution, were chosen for calibration frames (Fig. La, b, andc).

^ aot

TT

otH

•^tH

""^"^i

I

-lO^t

I

k If h/lol.a a =0.3 a.

bt+w(h-2t> Ac

If h/b1.2i«=0.3

10^

c.

b.

Figure 1: Residual stress patterns

Figure 2: Stress-strain relation

Consider the cross-section of a beam-column subjected to the action of the bending moment Af axial force isF and residual stresses oj., as shown in Figure 3. The strain £ij in an arbitrary point of the section can be expressed as follows:

Figure 3: Cross-section characteristics

319

£ij=

W+ (l>.yi^

€r,ij

(1)

in which u is the strain produced be axial force, (/> is the curvature in the section and £r,ij is the strain produced by residual stresses. The equations of static equilibrium in the section are: ^'Nint

(U,(t>)=0

Nf-Mint (u,(j>)^0

(2)

where A^/„/ (u, (j)) is the internal axial force and Mint (u, (j>) is the internal bending moment in the section. They can be expressed in the following form: N,,,{uJ)^\(j{u,(j>)dA A

(3)

Mi„,(«,(*)=Ja(»,^)yrf/(

The curvature cj) and the axial strain u are the solutions of Eqs. (2) that may be rewritten as follows: (4)

M{u,(l>)=M,,,{u,(l>)-M'=(^

The Eqs. 4 are solved numerically using the modified Newton-Raphson method, and results in two recurrence relationships to obtain the unknowns u and (p. The stress-strain relation of the material is assumed as elastic-perfectly plastic, and a tri-linear type of curve such as shown in Figure 2 is used in the analysis. The elastic modulus E = 29000 ksi and fy = 32.4 ksi. Two types of residual stress pattern was considered: type 1, European linear distribution (Fig.l.a) and type 2, US constant distribution (Fig. l.b). Using these data and the recurrence relationships obtained from Eqs.4 the M-N-O curves of a W8x31 section were computed for different N/Np ratio considering, both compression and tension, with and without residual stresses, as they are shown in Figure 4.a and b. A detailed layout of the procedure used to consider the development of plastification, second-order effects and non-linear behavior of the connections, is presented by Barsan and Chiorean (1999). These curves compare favorably to those computed by Kanchanalai (1977) presented by Chen and Toma (1994). b. Compression

a. Tension

0

0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 Curvature

0

0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004

Figure 4: M-N-O relationship of section W8x31

CurvMure

320

NUMERICAL EXAMPLES Based on the mathematical formulation above described an object-oriented Turbo-Pascal program, NILDFA (Non-linear Inelastic Large Deflection Frame Analysis) has been developed to study the influence of residual stresses on the behavior of steel-framed structures. It combines the structural analysis routine with a graphic routine to display the results, as deformed shape of the structure, M, T, N diagrams, percentages of section areas yielded at ultimate stage, in compression or tension, loaddeflection response curves and the development of plastic zones in the cross-sections and along the members of the structure. This program was used to study the influence of residual stresses on the behavior of steel frames. As a standard for comparison three types of frames were selected: a portal frame first analyzed by El-Zanaty (1980), a six-story two-bay frame (Vogel , 1985) and a two-story frame that has been proposed by Ziemian (1992). Three residual stress patterns, constant, linear and parabolic distribution in the web has been considered for calibration frames. The differences between the linear and parabolic distribution was irrelevant so only the linear (type 1) and constant (type 2) distribution have been represented. Example 1: Portal frame The El-Zanaty (1980) portal frame, member and material properties, and applied loading are shown in Figure 5. The plastic zone developments are represented in the Figure 6. In Figure 7 the lateral-load displacement curves are represented for type 1, type 2 distributions of the residual stress, and without residual stress. Example 2: Six-story frame The Vogel's six-story frame, member and material properties, and the applied loading are shown in Figure 8. In this example the influence of residual stress on the inelastic behavior of the structure and a comparison of computer time necessary to carry out the analysis using NEFCAD and NILDFA programs are presented. Comparable lateral load-displacement curves for the sixth-stories are shown in Figure 9. The NEFCAD program (a^l and n=35) results are in close agreement with NILDFA program where the effects of residual stresses are explicitly taken. Computing time for NEFCAD program was 100 times shorter than the time taken to complete the same analysis with NILDFA program. Percentages of section-areas yielded (Fig. lO.a) and spread of plastic zones in the characteristic cross-sections (Fig. lO.b and Fig. 1 l.a,b,c) are represented. Example 3: Two-story frame The Ziemian's two-story frame, member and material properties, and applied loading are shown in Figure 12. The percentages of section-areas yielded are represented in Figure 13 for all three cases considered and the lateral load-displacement curves, for the same three cases, are shown in Figure 14. CONCLUSIONS In order to assess explicitly the influence of residual stresses on the inelastic behavior of the structures, and in particular, on the carrying-capacity of the steel frames, and to compare this to results obtained

321 by NEFCAD program, a new computer program NILDFA (Non-linear Inelastic Large Deflection Frame Analysis) was elaborated by the authors.

ih r-i

HEA340

With residual stress, type2, Rim=1.03 With residual stress, type 1. Rim=1.03 Without residual stress, Rim=1.06

%=i/4UU

nfn

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

D(m)

Figure 5: Example 1. Portalframedescription

W

45.6 (38.3) [0.0]

51.2 (48.8) [45.6]

Figure 7: Lateral load-displacement curves for portal frame

52.0 (45.2) [41.1]

62.6 (57.6) [57.5]

a. Percentages of section areas yielded at ultimate for the portal frame Type 1 distribution. (Type 2 distribution.) [Without residual stress]

T T b. Type 1

c. Type 2

Figure 6: Spread of plastic zones for portal frame

d. Without residual stress

322

23 .-K

1M j M i ii M 1 1 i i MM iPE240

.4

Kh.

lilillMI

o

IPE300

H,

1 i

i

IMIMIM

oj

MIMMII

1 1 1 ! 1 1 IPE300 I

x|

! 1 1 M!1 M

H,

Ml

c

IFE330

-,

: M :MM

H,

i

1 ; 1 1 i

1 1 1

X

ol

1

l|

1 1 i 1 1 i! ! 1 i ! i i l 1 IPE400

U - c':

o

/

¥

Figure 9: Lateral load-displacement curves for sixstory frame Figure 8: Example 2. Six-story frame description 89.6

[90.2|

,-1

yflTV

96.6

'^-^J ^

t

75.7 (74.8) (77.01

-1 yf[)

86.2 (86.6) |86.6|

Value indicates total % of section area yielded, Type 1 distribution (Type 2 distribution) [Without residual stress]

r Al

- Tension

Cross section 1-1

T

Cross section 2-2

T

Compression

Cross section 3-3

3 3 Figure lO.a: Percentages of section-areas yielded

Figure lO.b: Spread of plastic-zones

323

a. Type 1 distribution

b. Type 2 distribution

c. Without residual stress

} \^mc n I he spread of pla<5ltc /CHICS oi th-j AB hcani o-hjp ikogc, |3"latcral view of the beam, f-bolfoin llange

_ J

t

!

;

f

;

<

<

^

?

!

!

i

i

!

?

<

f

I

T

!

}

?

>

?

f

t

?

Figure 12: Example 3. Two-story frame description

Bi

I Figure 13: Percentages of section areas yielded at ultimate for the two-story frame

I

324

FlinrFl.01 -Wflhreekici staseB,^1,

Q6'

J2

F«nrFl.01 -VVKhalreasid 8bQBses.HJrTFl.(E|

-QQ25

-Q02

-Q015


-Q01

-QOOB

0

Figure 14: Lateral load-displacement curves for two-story frame

The states of strain, stress and yield stress are monitored explicitly during each step of the analysis and the effects of residual stresses are accurately included in the analysis. Three residual stress patterns, constant, linear and parabolic distribution has been considered for calibration frames. Three frame types were selected: a portal frame, a six-story two-bay frame and a two-story frame. From the numerical tests it may be concluded that the influence of the residual stress on the carryingcapacity of the steel frames analysed vary within 1-3 % but the influence on inelastic behavior during the loading process is more important and must be considered in the analysis. Also, the NEFCAD program (a=l and n=35) results are in close agreement with NILDFA program where the effects of residual stresses are explicitly taken. That means that NEFCAD program conectly calibrated can implicitly capture accurately the influence of residual stress. Computing time for NEFCAD program was 100 times shorter than the time taken to complete the same analysis with NILDFA program.

REFERENCES Barsan G.M. and Chiorean C.G. (1999). Computer program for large deflection elasto-plastic analysis of semi-rigid steel frameworks. Computers &. Structures, (to be printed in 1999), 1-13. Chen W.F, and Toma S. (1994). Advanced Analysis of Steel Frames, CRC Press, U.S. El-Zanaty M.H., Murray D.W. and Bjorhodve R. (1980). Inelastic Behavior of Multystory Steel Frames, Structural Engineering Report, No. 83, The University of Alberta, Edmonton, Canada, April, 248 pp. Kanchanalai T. (1977). The Design and Behavior of Beam-Column in Unbraced Steel Frames. AISI Project No. 189, Report No. 2, Civil Engineering/Structures Research Laboratory, University of Texas, Austin, 300pp. Vogel U. (1985). Calibration frames. Stahlbau, 10,1-7. Ziemian R.D. (1992). A Verification Study for Methods of Second-Order Inelastic Analysis, Proceedings, Annul Technical Session, Structural Stability Research Council, Pittsburgh, PA, 315-326.

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

325

AN ADVANCED ANALYSIS FOR STEEL FRAME DESIGN: COMPARISON WITH TEST RESULTS A.M. Barszcz and M. A. Gizejowski ^ Department of Metal Structures, Faculty of Civil Engineering, Warsaw University of Technology, 00-637 Warsaw, POLAND

ABSTRACT The second phase of a research project carried out at the Warsaw University of Technology is summarized in this paper. The project aims at developing the theoretical and experimental basis for steel frame design using advanced analysis. The phase reported herein deals with the evaluation of test results. The influence of different arrangements of beam-to-column joints in laterally restrained frames is investigated. The main parameters of the joint M-cp characteristic are identified from the examination of the structural system behaviour as reproduced from test results. Using the predicted joint characteristics, the behaviour of laterally unrestrained frames is modelled and the member out-of-plane stiffiiess degradation model for advanced analysis is verified. A simple prediction model is then suggested for practical applications which enables satisfactory approximation of theframeresponse for use in the refined advanced analysis developed.

KEYWORDS Experimental investigations, semi-rigid joints, member stiffness degradation model, advanced analysis. INTRODUCTION In the fiiture, limit states design codes will to a greater extent utilize the system reliability based assessment. Such an assessment requires an advanced analysis of planar frames which should be able to represent the strength and stability behaviour of the whole structural system (continuous or semi-continuous, sway or nonsway, laterally restrained or laterally unrestrained) with the degree of accuracy acceptable from the practical design point of view. Present methods of advanced analysis concentrate on the in-plane loaddeflection response. The assumptions are made that all the frame member sections and joints are of class 1 (plastic) and frame structural components are suflSciently restrained in the out-of-plane direction in order to ensure that the unrestricted plastic redistribution of moments between the members and joints may take place, Chen & Kim (1997). A research study at the Warsaw University of Technology is underway which aims at developing the refined advanced analysis having an ability to capture the effects of nonplastic sections and frame out-of-plane stability on the structure ultimate strength, Gizejowski et al. (1998). Although the phase of theoretical modelling is well advanced, the available experimental data cannot yet be regarded as sufficient to create the background information required for the verification purposes. There are a few full-scale tests

326 conducted on the behaviour of frame specimens. Full-scale frame tests on the behaviour of steel semicontinuousframeswere reported by Ivanyi (1996), among others. The experimental verification phase of the research project presented herein is devoted to the quasi-static response offramespecimens with some types of semi-rigid connections. The connection response is assessed on the basis of connection component deformations which were monitored during the full scale tests carried out for laterally restrained portal frame specimens (in-plane behaviour). The connection strength, initial stififiiess and shape parameter of M-cp curves are evaluatedfromtest results and then used for the validation of the theoretical model of the refined advanced analysis developed in the first phase of the research project. Finally, a simple member stifihess degradation rule is suggested for the prediction of system out-of-plane stability in the refined advanced analysis.

EXPERIMENTAL SETUP A total of 15 portal frame specimens were prepared for testing in two series, the first aiming at behaviour investigations of the 9frames,beams of which were laterally restrained, and the second - 6 frames, beams of which were laterally unrestrained. The portalframespecimens were tested upside-down, i.e. the beam-tocolumn connections were placed at the bottom while the column free end points were heading up. One column end was simply supported and held in position. The other one was moveable and loaded with a horizontal force. The point of load application was allowed to rotate freely in the frame plane. The bracing points offrameswith laterally restrained beams were located at the distances of 1/3 of the beam length and at the column ends, below the beam-to-column connections. Frames with laterally unrestrained beams were braced only at the column ends, below the beam-to-column connections. The numbering of tested frame specimens is given in Table 1. Models being tested were difierentiated according to the connection detail, type of lateral restraints and the direction of applied load. All specimens consist of the beam member of an IPE 120 section, attached through connections to the column members of an IPE 240 section. Four types of connections were examined - one welded connection with no stiffeners in the columns, and three unstiffened bolted connections with different arrangements of bolts in the tension zone. Steel rolled sections and connection plate elements of 6 mm in thickness selected for frame specimens were approximately of grade Fe 360. The 10.9 grade high strength nonpreloaded bolts Ml2 were used for all the bohed connections. The loading programme was chosen in such a way that it was able to reproduce an average connection behaviour in building steel frameworks. All the specimens were loaded in such a way that the applied horizontal force was increasing step-by-step with the increment of 0.2 kN in the connection elastic region and of 0.1 kN - in the inelastic region. After each incremental load step, a sequence of load cycles was applied with the amplitude of 0.4 times the load magnitude of the particular load increment. Load cycles were then repeated up to the point at which the specimen deflections were adapted to the incremental load bounds, and within the tolerance limit prescribed. A tolerance limit of 0.01 times the deflection of the first cycle at each incremental step was adopted. Usually 2-4 cycles were needed for the deflections to stabilize. The quasi-static nature of the test procedure was ensured by a very low rate of load application. The duration of each test ranged from approximately 1 to 2 hours, depending upon the ultimate strength being reached by the particular frame specimen. More time was needed for testing laterally restrainedframeswith relatively rigid connections while less time - for laterally unrestrained frames with semi-rigid connections. The experiments were on-line guided by the PC. The PC was equipped with the GENIE software developed by Advantech. The results recorded during experiments were stored on the computer hard disc in the format compatible with the Microsoft Office of WINDOWS 95.

327

TABLE 1 NUMBERING OF TESTED FRAME SPECIMENS Type of beam-to-column connections Frame specimens i H M

3,4

3,6

7,8

10

12

11

13

14

15

16

17

#

Each two series of tests had the same loading conditions - thefirsttwo were characterized by the beam being in tension, while the second two - by the beam being in compression. The tests on laterally restrained frames were designed to investigate thefi-amein-plane response and the connection behaviour as monitored during the flill-scale tests of portalfi"amespecimens. The goal was to single out the key parameters enabling the full description of the connection response. The measurement setup was devised in such a way that the behaviour of main connection components (end plate, bolt shank, column section flange and web) could be monitored, as well as the overall response of the connection. The arrangement of measurement setup was similar to that developed by Bemuzzi et al. (1996). The sensitivity offi^amesystems to lateral buckling was investigated for laterally unrestrained frames. A special instrumentation was developed to monitor the components of space movement of the midspan beam section during the tests of laterally unrestrained frame specimens.

328

All the experimental results are used to verify the model of refined advanced analysis of steel frames developed by Gizejowski et al. (1998). Using the model parameters evaluatedfi"omthe first two series, the second two series with limited number of specimens are considered in order to investigate the effect of beam compression force on the overallfi*amebehaviour. MODEL PARAMETERS FOR IN-PLANE BEHAVIOUR The first set of experimental tests aimed in understanding of the main behavioural characteristics of different beam-to-column connections deducedfi-omthe behaviour ofjoint components monitored during the full scale testing offi-amespecimens. The connection standardized model II-l of Gizejowski et al. (1998) is used herein and the model parameters are verified on the basis of experimental resultsfi-omtests described above. In each run of the verification procedure, the advanced analysis is performed with different values of connection strength and initial stififtiess, and with a constant shape factor. The initial values of the above connection model parameters are taken in such a way that they corresponded to the model parameters adopted in the Annex J of EC3. These parameters are calculated using the software developed by Kozlowski & Pisarek (1998). The frame response modelling is carried out for each step of the verification procedure in order to get the loaddeflection charaaeristic best fitted to the experimental one. When the best fitted fi-ame load-deflection characteristic was predicted, a new value of the connection shape factor is selected and the advanced analysis repeated for different values of two other parameters. The target is to define a single connection shape factor which is independent from the type of the connection and to specify the other two connection model parameters in such a way that the advanced analysis is bestfittedto the experimental behaviour. 16-T

16-

144 12 10 8 6 4 2

X laT -1

\

0 5 10 15 Joint rotation [rad]

1

20

0

- 1

0

5

\

\

1

10

15

20

1

1

\

25 30 35 Joint rotation [rad]

S,^=5000kNm ' Muit=12.6kNm . —^ jx 10"^ 1 1 — 40 45 50 55 60

Experiment J

-fi-amespeciment no. 3, 5 and 7

"

- frame speciment no. 4, 6 and 8 - FE modelling Annex J of ECS

0

5 10 15 20 Joint rotation [rad]

0

5 10 15 20 Joint rotation [rad]

Figure 1: The verification of connection model parameters for in-plane behaviour

329 As a result, the shape factor of 0.7 is selected to model the behaviour of all the connection types. The other two connection parameters are found to best reproduce theframeexperimental behaviour. Thefinalvalues of connection parameters are then used to compare the connection model developed herein with that reproduced from the measurement ofjoint deformations during the tests. Figure 1 shows the connection moment-rotation responses. The black triangles and squares represent the moment upper bound while the blank ones - the moment lower bound, each corresponding to the load cycles applied at each incremental stage of the loading programme. The joint characteristic M-cp of the present study corresponds well to the upper bound results both in the elastic and thefinalinelastic ranges. In the intermediate inelastic range, differences are greater but the overall performance is estimated on the safe side. Such a discrepancy may be associated with a larger gap created between the columnflangeand the beam end plate (in the tension zone) under alternate loading. This effect is more pronounced for connections subjected to cyclic loads since the gap is developed here less gradually than under monotonic loads and starts at lower values of the applied moment. The analytical solution to the Annex J of ECS is much lower than that predicted herein, even less than the results corresponding to the moment lower bound under alternate loading. It can be concluded that all the connection characteristics of the present study correlate reasonably well with test results in contrast to the prediction according to the Annex J of EC3 which tends to underestimate the real connection behaviour. Since the Annex J model was verified with use of the SERICON data bank which collects the results of experimental investigations carried out for isolated joints, the additional safety margin identified herein could not be displayed. It proves that the reliable verification of connection response can be obtained only if the joint behaviour is monitored through the experimental investigations of the behaviour of the whole structural system. ^

Experiment frame speciment no. 3, 5, 7 and 9 frame speciment no. 4, 6 and 8

10 20 30 40 50 60 70 80 90 Displacement [mm] 35 T FE modelling — - 4 FEs / member - - - 1 FE / member

10 20 30 40 50 60 70 80 90 100 110 120 Displacement [mm]

FE modelling - 4FEs / member rigid joint frame model semi-rigid jointframemodel 0 10 20 30 40 50 Displacement [mm]

10 20 30 40 50 60 70 80 90 100 110 120 130 140 Displacement [mm]

Figure 2: The verification of frame behaviour (frame beams under tension and bending) The connection moment-rotation characteristics evaluated herein are used for the final comparisons of the load-deflection response of tested frames. Figure 2 shows the overall response of frame specimens. The

330

advanced analysis using 4 FEs per member is represented by the solid line. It correlates well with the set of points representing the experimental equilibrium path. Experimental results describe the upper bound of the applied load envelope to which theframespecimens adapt at each increment of the loading programme. The results for theframewith the discretization scheme based on 1 FE per member are also plotted (dashed line) for comparison. It is obvious that only the discretization based on 4 FEs is able to correctly represent the frame response. It can be concluded that all the frame load-deflection characteristics of the present study correlate reasonably well with test resuhs.

MODEL PARAMETERS FOR OUT-OF-PLANE BEHAVIOUR In advanced analyses proposed so far, such as the one based on the PU model, the in-plane modes of failure were not integrated with the lateral buckling of the structural system, Chen & Kim (1997). In the refined advanced analysis proposed by Gizejowski et al. (1998), the out-of-plane stiflSiess degradation model is based on the member tangent stifi&iess matrix which is approximated by using a similar concept to that used for the in-plane stiflGhess degradation concept adopted in the PU model. The member stiflSiess degradation effect associated with the out-of-plane stability is described however by a difierent fiinction, denoted by ^lo in contrast to the n-fiinction used for in-plane bending. An effort is made herein to calibrate the no-fiinction in such a way that it is related to the expression of ri. After some calibrations, a simple relationship of rio = r] is recommended for practical applications in the assessment of system out-of-plane stability. This expression is used together with connection model parameters established earlier in order to verify the behaviour of laterally restrained frame specimens tested. The frame responses of specimens no. 10, 11, 12 and 13 are presented in Figure 3.

10

10 20 30 40 50 Displacement [mm]

20

30

40 50 60 70 80 Displacement [mm]

90 100 110 120

- experiment FE modelling - 4 FEs / member - 1 FE / member - out-of-plane buckli 10 20 30 40 50 Displacement [mm]

60

10 20 30 40 50 Displacement [mm]

60

Figure 3: The verification of frame model parameters for out-of-plane behaviour

331

The beam-to-column minor axis joint stifihesses are set to be of a large value, corresponding to the nominally rigid connection according to ECS. The warping stifihesses are calculated according to Lindner (1995). Only the frame specimens no. 10 and no. 13 are sensitive to lateral buckling. Satisfactory estimates are obtained in both cases considered. For the other two cases, i.e.fi"amespecimens no. 11 and no. 12, the analysis cannot detect any mode of out-of-plane instability. This results from the fact that the connection in-plane strength is too low for such a moment transfer from theframecolumns to the beam, which could cause the members to laterally buckle and twist. As a result, an excessive yielding of the end-plate is the local mode responsible for the structural failure, instead of the global mode of lateral buckling.

MODEL PARAMETERS FOR FRAME MEMBERS EV BENDING AND COMPRESSION The frame structures members of which are subjected to bending and axial force may respond differently on the applied loads, depending on the amount of tension/compression transferred by structural members. In general, frames composed of members subjected to tension and bending are less sensitive to stability effects than frames composed of members which are under bending and compression. The frame specimens no. 14 and no. 15 were designed to experimentally examine the behaviour of laterally restrained beams subjected to bending and compression. The advanced analysis is carried out with use of the elasticity modulus E for the calculation of beam rigidity in the initial stage of loading. The results are given in Figure 4. Theframefailure mode is identified as the system out-of-plane instability in both cases. The set of points representing the experimental equilibrium path is below the frame load-deflection curve predicted by the refined advanced analysis. This proves that the compression member initial rigidity has to be reduced in order to achieve a better correlation between test results and FE modelling. The above effect results from the sensitivity of compression members to geometrical imperfections which tend to reduce the member effective flexural stiffiiess. 35 30 25-[

S 20 tin

10 5-

10

20 30 40 50 Displacement [mm]

60

70

20 30 40 50 Displacement [mm]

70

30 - experiment

25

FE modelling

g^20 ^ 154

I 104

- 4 FEs / member - 1 FE / member

5 - out-of-plane buckling

0 10 20 30 40 Displacement [mm]

0 10 20 30 Displacement [mm]

Figure 4: The verification of frame behaviour (frame beams under compression and bending)

332

The behaviour offramespecimens with the laterally unrestrained beam subjected to bending and compression is also investigated - specimens no. 16 and no. 17. The results of verification are presented in Figure 4. The dependence of theframebehaviour on compression beam geometrical imperfections is here even greater than in the case of laterally restrained frames. The bifurcation points associated with system lateral buckling are found to be located much lower on the in-plane equilibrium path than those for restrained beam specimens. The overestimation of the load-deflection characteristic by the refined advanced analysis is also more pronounced in this case. It proves that the advanced analysis has to consider the reduction of compression member initial rigidity in order to correctly account for the influence of geometrical imperfections. It can be shown that by applying a factor of 0.85, as suggested by Chen & Kim (1997), an accurate modelling can be achieved.

CONCLUSIONS The method of refined advanced analysis developed by Gizejowski et a. (1998) is verified by experimental investigations of the behaviour of 15 portalframespecimens of different beam-to-column connection types. In particular, the effects of joint characteristic and the members stifl5iess degradation model on the structure mode of failure are examined. The following can be concluded. - a single value of the semi-rigid joint model shape parameter equal to 0.7 can be selected to describe accurately the behaviour of all types of beam-to-columns connections used in frame models tested experimentally, - the joint initial stiflfoess and ultimate strength parameters assessedfromtest results of theframebehaviour are found to be higher than those obtainedfromtests on isolated joints, - the member out-of-plane stiflSiess degradation model can be characterized by the rio-fijnction which is taken as the squared in-plane stifihess degradationfianctionri in the PU model, - the stiffhess degradation model of flexural members in compression has to be based on the reduced modulus of elasticity Eo instead of its initial value E and the value 0.85 E is verified to be adequate. The verification exercise presented herein is limited to design situations in which member plastic sections are used. Further tests are needed to verify the method in the case of frame systems composed of nonplastic section members. This couldfinalizethe verification procedure of all the model parameters specified for the refined advanced analysis as presented by Gizejowski et al. (1998) and lead to its recommendation for the fiiture generation of limit states design codes.

REFERENCES Bemuzzi C, Zandonini R. and Zanon P. (1996). Experimental Analysis and Modelling of Semi-Rigid Steel Joints under Cyclic Reversal Loading. J. Construct Steel Res. 38:2, 95-123. Chen W.F. and Kim S.E. (1997). LRFD Steel Design Using Advanced Analysis. CRC Press, Boca RatonNew York, USA. Gizejowski M.A., Papangelis J.P. and Parameswar H.C. (1998). Stability Design of Semi-Continuous Steel Frame Structures. J. Construct Steel Res. 46:1-3, 99-101. Ivanyi M. (1996). Large-Scale Tests on Steel Frames with Semi-Rigid Connections. Effect of Beam-toColumn Connections and Column-Bases. In: Connections in Steel Structures III - Behaviour, Strength and Design. Eds. R. Bjorhovde, A. Colson and R. Zandonini, 309-320, Pergamon, UK-USA-Japan. Kozlowski A. and Pisarek Z. (1998). The computer EXCEL based program for WINDOWS 95: calculation of steel semi-rigid connection characteristics to the Annex J of ECS. Rzeszow University of Technology, Poland. Lindner J. (1995). Influence of constructional detailing on the lateral torsional buckling capacity of beamcolumns. In: Structural Stability and Design. Eds. S. Kitipomchai, G.J. Hancock and M.A. Bradfod, 61-66, A.A. BALKEMA, Rotterdam, Brookfield, Netherlands.

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

333

DESIGN OF COMPOSITE SWAY FRAMES: A CONCISE GUIDE J S Hensman^ and D A Nethercot^ ^ Caunton Engineering Ltd., Moorgreen Industrial Park, Nottingham, NG16 3QU, U.K. ^ School of Civil Engineering, University of Nottingham, University Park, Nottingham NG7 2RD, U.K.

ABSTRACT The findings from a number of recent studies of the ability of composite connections to deliver the levels of performance required for the adoption of the semicontinuous approach to overall frame design have been combined with recently developed improved understanding of the justification for the Wind Moment Method of design for unbraced steel frames to extend the concept to cover composite construction. Numerical parametric studies have formed the basis for defining a range of composite frames to which the Wind Moment Method may safely be applied.

KEYWORDS Connections Frames Joints Steel Structures Structural Analysis Structural Design INTRODUCTION The Wind Moment Method (WMM) has long been used for the design of unbraced steel frames. Its main characteristic is that it simplifies the analysis and subsequent design of a bare steel frame that has to resist both gravity and lateral loading by frame action i.e. no bracing is provided. Hughes (1999) has recently discussed the origins, development and use of the method from the point of its attractiveness to designers. In addition to pointing out how it strives for overall frame economy by recognising the benefits of less than full strength and full stiffness connections, he has set the method's use into the context of modem thinking on connection design, BCSA/SCI (1995). Given the ready availability and heavy reliance on computer based methods for overall

334

frame analysis nowadays, Hughes, quite rightly, notes that the original main attraction of the method not requiring a full frame analysis is no longer such an important factor. It was not until methods of frame analysis had developed to incorporate realistic representations of joint behaviour that detailed justifications for the WMM could be undertaken. Previously, a number of authors had used pragmatic arguments based on the concept of shake-down under repeated cycles of applied loading to demonstrate that the method led to safe designs. Ackroyd and Gerstle (1982) were the first investigators to make detailed studies for a particular range of frames of the actual load factors against collapse and working load deflections obtained for frames designed by the WMM. Subject to certain limitations, Ackroyd (1987) subsequently confirmed the method's viability for structures up to about nine storeys. In the UK, Reading (1989) and Kavianpour (1990) conducted similar studies, that were subsequently used as the basis for the publication of a full design method, Anderson et al (1991). Since then, developments in connection design have demonstrated that rather than those considered in the original SCI Guide simpler joints are likely to be more cost-effective; Brown and Anderson (1999) have recently extended their studies to consider the effect of using these more flexible, standard endplate connections. All of the foregoing relates to bare steel frames; no attention was given to the utilisation of composite action between the floors and the beams or within the beam to column connections. However, over the past two decades considerable study of the ability of composite connections to deliver the sort of performance in terms of moment capacity, rotational stiffness and ductility necessary for the adoption of a semicontinuous approach to fraime design has taken place. To date, this work has concentrated on non-sway frames, leading to the development of a full design approach, Nethercot (1995). The possibility of utilising the available improved body of knowledge on the performance of composite connections, in conjunction with recently developed understanding of the WMM, as a way of extending the scope of the WMM to cover composite frames has been the subject of a recent study in Nottingham, Hensman (1998). [Initial findings, Hensman and Nethercot (1997) suggested that because connections might not exhibit reversal of load and rotation during realistic loading sequences, it would not be necessary to investigate additional connection M-N characteristics]. Further study has shown this to be a valid assumption. By utilising the results of an extensive numerical parametric study for a series of representative frames, it has been possible to develop a new set of design recommendations (which are presented in a similar manner to the original SCI guidelines) that enable a wide range of composite frames to be designed using the WMM. Details of this study are available elsewhere, Hensman (1998); the present paper concentrates on presenting the main features of the adopted design approach. PRINCIPLES OF THE WIND MOMENT METHOD The WMM separates the treatment of gravity and wind loads into the pair of frame analyses illustrated in Fig. 1 : • Gravity load analysis: horizontal beams are assumed simply supported at their ends, and designed accordingly. • Wind load analysis: beam to column connections are assumed rigid and internal moments and forces are determined by assuming points of contraflecture at the mid-span of the beams and mid-height of the columns.

335

Q-

p

zrr (a) Connections assumed to act as pins for gravity loading (b) Connections assumed to act rigidly for wind loading

Figure 1: Basic premise of Wind Moment Design Approach The principle of superposition is then used to combine results from the two separate treatments to obtain the ultimate design loads for which the columns and connections are designed. The simplicity of the method is inherently attractive, since by employing the portal method for the wind load analysis, the frame is rendered statically determinate and all calculations are therefore kept brief and simple. The two main concerns with this apparent lack of rigor are: • That frames designed using this approach will possess adequate factors of safety against collapse. • That sway deflections under serviceability load conditions will be within acceptable limits. Since extensive physical testing is clearly not feasible on economic grounds, the use of modem numerical simulations as a basis for comparing the simplified treatment with better approximations of real behaviour is the obvious approach to adopt when seeking to extend and validate the WMM. PARAMETRIC STUDY The numerical study was based on a modified version of the set of frames used for the SCI Guide, Anderson et al (1991). This was, however, modified somewhat as a result of discussions with practitioners, consideration of the different layouts likely to be adopted when using composite beams and recent changes to loading requirements. Table 1 and Fig.2 identify the boundaries of the set of frames considered. Each frame was initially designed using the proposed form of the WMM and then analysed using a finite element modelling technique that had been developed specifically for this study, Hensman (1998). In addition to overall frame behaviour, attention was paid to the rotations required at the connections since composite joints have consistently demonstrated that increased moment capacity and rotational stiffness (as compared with bare steel equivalents) is obtained at the expense of ductility.

336 -2 bay

2 storey -4 bay

4 storey - 4 bay

2 bay - 2, storey

Open plan top storey

Uneven bay widths

Figure 2: Frame Layouts considered in the Parametric Study

337

TABLE 1 LIMITATIONS OF PARAMETRIC STUDY Maximum | Minimum 4 1 Number of storeys 2 1 Number of bays 2 4*' 1 Bay width (m) 12.0 6.0 1 Bottom storey height (m) 6.0 4.5 1 Storey height elsewhere (m) 3.5 5.0 1 Dead load on floors (kN/m'') 2.50 5.00 1 Imposed load on floors (kN/m^) 2.50 7.50 1 Dead load on roof (kN/m^) 3.75 3.75 1 Imposed load on roof (kN/m^) 1.50 1.50 [wind loads (kN) 10*" 40*' J * frames can have more than 4 bays, but a core of 4 bays is the maximum that can be considered to resist the applied wind load. *^ Wind loads = concentrated point load on plane frame at each floor level

All previous studies of the WMM have assumed fixed column bases (in accordance with the design assumptions of the WMM). It was found from discussions with practitioners that fully rigid foundations are offputting because of their increased fabrication costs and design complexity. [The parametric study therefore included investigations of the extent to which the inherent stiffness of practical equivalents of the "notionally pinned" column base might be sufficient for them to have a worthwhile influence in limiting frame deformations]. Initial studies had shown that analyses assuming actual pin bases generally resulted in sway deflections at working load levels well in excess of acceptable limits. By making an assessment of the benefits likely to be achievable from practical baseplate arrangements, it was possible to demonstrate how sway deflections could be kept to acceptable levels, without the need for complicated and costly connection details. FRAME LIMITATIONS Because the scope of the parametric study was restricted, the proposed design approach should only be used when the following conditions are met: • The frame consists principally of horizontal beams and vertical columns • Only composite beams with insitu slabs and metal decking are used • The frame does not exceed four storeys • The number of "active bays" does not exceed four (further bays may be 'hung' from the WMM frame) • The width of each bay is constant over the height of thefi-ame,except that the columns may terminate at the top floor to allow an open-plan top storey • Frames are effectively braced against out of plane sway at each floor level and at the roof level • Beam grids must be consistent with one of the layouts shown in Fig. 3 • Steel sections may be either S275 or S355 • Universal Beams should be used for the composite beams • Universal Columns should be used for the vertical columns, with a minimum section size of 203 x 203 x 60 • Sections should be oriented in such a way that loads in the plane of the frame tend to cause bending about the major axis

338

Internal beam to column connections should be flush endplate composite connections External connections should be designed as bare steel, or composite if adequate rebar anchorage can be achieved Column bases should employ a minimum of four bolts, with a minimum base plate thickness of 25mm and the centreline of the bolt rows should be located at least 50mm outside the external face of the column flanges The total unfactored dead and imposed load on any floor should not exceed 12.5kN/m^ Characteristic wind loads on the wind-moment frame at each floor level should not be taken as less than a lOkN point load The wind load should not be such that it controls the design of any beam

M

M

iH H

HFigure 3: Beam layouts at each floor level

DESIGN FOR ULTIMATE LIMIT STATE Global analysis should consider the three load combinations of dead plus imposed plus notional horizontal forces, dead plus imposed plus wind and dead plus wind. An end restraint moment of 10% of the maximum sagging moment in the beam is assumed at the connections. When analysing the frame under horizontal loading, the portal method should be used. Composite beams should be Class 1 plastic with the sagging moment capacity limited to 90% of the plastic moment of resistance. Column effective lengths should be taken as 1.5L for in-plane behaviour and l.OL for out of plane behaviour. Connections should be designed for moments equal to the sum of the end restraint moment due to partial fixity plus the moment due to horizontal loading. The connection shear force should be taken as the sum of the beam end shear due to gravity load and the shear force in the beam due to horizontal loading. Once the frame has been designed for the ultimate limit state, it should be analysed as an elastic rigid-jointed frame to determine the sway displacements. Since properties of the composite beams will not be constant along their length, use of an equivalent beam model is required. Comparisons against the rigorous analyses have confirmed that

339 assuming the hogging lengths as L/5 and taking the representative beam stiffness from Hensman(1998)as:

in which Ig In

== =

second moment ofareaoftheuncracked "sagging" composite beam secondmomentof area of the cracked "hogging" composite beam

when using the graphical procedure proposed by Wood & Roberts (1975) leads to acceptable predictions providing the calculated sway deflections are increased by: 1.35 1.55

when external connections are assumed to act as composite when external cormections are assumed to act as non-composite.

In addition to checking that overall frame sway does not exceed h/300, it is important to check that a similar limit is observed for each individual inter-storey sway. Studies have shown that it is often the first storey that is the most critical. Should the deflections be found to be excessive, then additional frame stiffness must be provided, normally by increasing member sizes. Since all the steps required by this process may readily be brought together into a logical design procedure for which all the necessary mathematics is available in explicit form, implementation by means of spreadsheets is particularly appropriate. CONCLUSIONS An extension of the Wind Moment Method to cover composite frames has been described. This has been justified by comparison against results from an extensive parametric study limited to a specific range of structures. Providing these limitations are observed, use of the WMM represents an attractive approach for the economic design of unbraced composite frames. ACKNOWLEDGEMENTS The work reported herein was undertaken on behalf of the Steel Construction Institute, who provided a studentship to the first author. The assistance of Dr G Couchman in many aspects of the study is gratefully acknowledged.

REFERENCES Ackroyd, M H and Gerstle K H (1982), "Behaviour of Type II Steel Frames" Journal of the Structural Division, ASCE Vol. 108, No. 7, pp 1541-1556. Ackroyd, M H, (1987), "Design of Flexibly Connected Unbraced Steel Building Frames", Journal of Constructional Steel Research, No 8, pp 261-286.

340

Anderson, D, Reading, S J and Kavianpour K, (1991),"Wind Moment Design for Unbraced Frames" The Steel Construction Institute, Publication No. 082. BCSA/SCI, (1995), "Joints in Steel Construction: Moment connections". Publication no. 207/95 The Steel Construction Institute. Brown, M D and Anderson, D, (1999) "Wind-Moment Steel Frames with Standard Ductile Connections" submitted for publication Hensman, J S, (1998) "Investigation of the Wind-Moment Method for Unbraced Composite Frames", MPhil thesis, University of Nottingham. Hughes, A F, (1999), "A fresh look at the wind-moment method", submitted for publication. Kavianpour, K, (1990), "Design and Analysis of Unbraced Steel Frames" PhD thesis. University of Warwick. Nethercot, D A, (1995), "Semi-Rigid Joint Action and the Design of Non-Sway Composite Frames" Engineering Structures, Volume 17, No. 8, pp 554-567. Nethercot, D A and Hensman, J S (1997), "Design Considerations for Composite Sway Frames", Structures in the New Millenium ed PKK Lee, Balkema, pp 503-510 Reading, S J, (1989), "Investigation of the Wind Connection Method", MSc thesis, University of Warwick. Wood, R H and Roberts E H, (1975), "A Graphical Method of Predicting Side Sway in the Design of Multi-Storey Buildings", Proc. Institution of Civil Engineers, Part 2, Vol 59, June, pp 353-372.

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

341

THE SPATIAL BEHAVIOUR OF THE SCAFFOLD USED FOR THE COAL CONVEYANCE A. Ivan\ M. Ivan^ and D. Cozma^ ^'^'^ Department of Steel Structures and Structural Mechanics, "Politehnica" University, Timisoara, Str. Stadion nr.l, Romania

ABSTRACT The belts used for coal conveyance in the bunkers of thermoelectrical power plant are sustained by spatial steel-framed structures. The spatial structures are sustained by steel columns and by adjacent buildings. Present paper describes the loads that were taken into account, the finite element modelling of structure, the computation of stresses and displacements and the safety checking of the structure. KEYWORDS Scaffold, Spatial steel-framed structure. Linear analysis, Stresses in a scaffold. Displacements in a scaffold, The stability of a scaffold. GENERAL PRESENTATION The conveyance of the coal from the crushing plant to the bunkers is provided by conveyor belts made of rubber. The conveyor belts lie on cylindrical rolls, each one sustained by a steel structure. The weight of the coal, the belts and the sustaining framework is taken over by a spatial steel-framed system. The spatial structure consists of two vertical inclined lattices and of framed bracings, disposed at the superior and the inferior flanges of the frame girder (Figure 1). The main dimensions of this structure are: Li = 26.45 m, L2 = L3 = 38.90 m, L4 = 37.70 m and H = 44.50 m. On the inferior flange of the frame girder lies the circulation flooring, made of ribbed steel sheet, which allows the inspection and the checking of the conveyor belts working. The roof frames, which are sustaining the covering, are tied with the rest of the structure in the joints of the superior flange. The inclined framed structure lies at the ends on reinforced concrete structures and on the length of the span on steel framed columns. The leaning of trusses is achieved using closed frames, made of rolled steel shapes. The members are made of Romanian steel OL - 52 with a load carrying capacity of fy = 350 N/mml

342

§)

343

THE MODELLING OF STRUCTURE In order to study the state of stresses and displacements produced by the loads that occur in the exploitation process, the steel-framed structure was divided in TRUSS and BEAM finite elements [HabibuUah A., Wilson E. (1991)]. The inclined truss's members, the bracings, the roof frames and the wedge pieces were considered pin-ended members, which are spatially loaded. The column bases were modeled using BEAM 3D finite elements, while the diagonals and the posts of the column were modeled using TRUSS 3D finite elements. The stresses and the displacements in the joints of the steel-framed structure were determined using specialised software and a PCS86 computer. The loads that were taken into account are listed below: • the self weight; • the load due to the coal weight on the conveyor belt; • the service loads on the circulation flooring; • the snow loading; • the load due to the wind pressure; • the load due to the temperature variation; • the seismic loading. Using these loads 15 hypothesis of loading and 59 load combinations were established, which contain the overloading and the grouping coefficients for the considered loads. The spatial steel-framework structure was computed considering two phases: the assembly stage and the exploitation stage. THE RESPONSE OF THE STRUCTURE The stresses and the displacements were computed for each load separately and were then combined in 59 load combinations. The members of the structure were designed based on the computed results and then the serviceability condition was checked. The stability of the compressed members (the members under compression) of the steel-framed structure was checked using the relationships from the Romanian standard STAS 10108/0-78: N (1) a =- ^ < R (p-A where N is the axial compression force cp is the minimum value of the buckling coefficient, which depends on the steel quality and the slendemess ratio X A is the gross cross-section of the member R is the yield strength for steel The condition of stability of members, under bending and axial force in accordance with STAS 10108/0-78 is fiilfilled if the maximum stress a satisfies the criterion:

344

where

N is the axial compression force Mx is the maximum bending moment in the member A is the cross-section of the member Wx is the resistance moment with respect to the x-x axis (p is the minimum value for the buckling coefficient Cx is a coefficient which depends on the distribution of the bending moment along the member Gg is the Euler's buckling load for with respect to x-x axis (p g is a coefficient depending on the transformed slendemess ratio:

K = r^

(3)

in which y depends on

.- 1 / 1 is the length of the member Ir is the torsional moment of inertia h is the height of the cross-section ly the second moment of area with respect to the y-y axis N a =— A The relationships for checking the stability of members in accordance with EUROCODE 3 are as follows: The stability of members subjected to axial compression shall be checked using the following relationship: N^^X-^^

(4)

YMI

where

% is the reduction factor for the relevant buckling mode A is the cross-section of the member fy is the yield strength for steel Y j^i is the partial safety factor

The reduction factor x is a function of X:

(5)

1

where

\ =^J~

(^)

For the flexural buckling, the slendemess X shall be taken as:

^4

(7) 1

where

1 is the buckling length of a compression member i is the radius of gyration for the relevant axis

345 In accordance with ECS, members in bending and axial compression shall be checked as described bellow. In case of compression and bending we shall apply the rules for buckling and lateral torsional buckling of members. Without lateral-torsional buckling phenomena: k,-M^s, , k , - M , s , MyRd

M,R,

•^l-n„»

(8)

1.50-M^3, ^ 1.50-M,3, ^^ ^ M,.Kd

(9)

M,

where the plastic resistance is: M,Ka = W,,, i - ; M , , , = W , , , ^ YMI

(10)

YMI

the elastic resistance is:

t

I

TMI

TMI

M-^y.Rd , R a- = ""y W,^;M,R, = W , ^

(11)

n„ax=max|n^=-^;n,=-^

(12)

^zY/2 lateral-torsional buckling phenomena:

1.00M,,^1.50.M.,a^^. M M where the plastic resistance is: M,.Ra = XLT • Wp,, - ^ ; M,,, = Wp„ ^ V /Ml

(15)

V /Ml

the elastic resistance is: M,.Ra = XLT " W, - ^ ; M,.,, = W, - ^ 'Ml

(16)

1u\

XLT=f(\T)

The result of these checkings is that these cross-sections of the members satisfied the condition of stability imposed by the mentioned Design Codes. Table 1 shows the maximum intemal efforts and stresses in the most challenged members of the structure, stresses which results from the strength or the stability condition. It is obvious that in all the members the effective stresses are lower then the values of the yield strength.

346 TABLE 1 MAXIMUM INTERNAL EFFORTS AND STRESS IN THE MEMBERS OF THE STRUCTURE Typeofme«*er Column SI Column S3 Column S3 1 Diagonal Inferior flange Diagonal Superior flange Fl 1 Superior flange Fl

Section

Section 1 Section 1 Section 2 Section 4 Section 4

Crossing section [niinl 2x(20x300)+12x600 2x(20x400)+l 5x900 2x(20x300)+12x900 2L 120x120x10 2L 120x120x10 2L 80x80x8 2L 60x60x6 2L 60x60x6

N[kK] -26.42 -46.45 -6.45 10.55 -13.09 7.84 0.3104 0.3275

MxfNm] 1120 1460 1760 10.4 -6.4 3.1 0 0

M^pSfmJ 0[N/mtn^]] 323.7 0 0 294.6 0.01 321.5 0 241.7 291.1 0 324.9 0 5.08 290.9 223.5 3.78

Using the cross-sections of the members that were determined as described before, the natural periods for the spatial structure were calculated. The numerical values of this periods are: Ti = 1.372 s; T2 = 1.221 s; T3 = 0.721 s; T4 = 0.697 s; T5 = 0.681 s; Te = 0.665 s; T7 = 0.625 s. In figures 2 and 3 are shown the first and the third mode shapes, which are the most unfavourable ones.

Figure 2

347

Figure 3 These periods were used to evaluate the seismic loadings and the response of the structure to earthquake. Fortunately the response of this structure to the seismic loadings is good enough. But some forced vibrations affect the stability of the columns and of the inclined trusses. These vibrations are induced by the eccentricity of cylindrical rolls, by tensioning device of the rubber belts. An important problem was the study of the influence of these vibrations on the general behaviour of the scaffold. These vibrations were measured on the real structure in different positions of the coal on the rubber belts. A linear buckling analysis was then performed with the same loading combinations that were used previously for the stability checking (Ivan M. et al (1997)) A significant decreasing of the buckling load (10%) was met just on the diagonals of the second and third column, in the presence of lateral wind. It was not necessary to resize those members. The safety coefficient for those decreased from 1.9 to 1.71.

348 CONCLUSIONS As a result of the studies performed, we can draw a few important conclusions: • The stresses and the displacements have to be determined by taking into account the spatiality of the structure. • In case of inclined trusses with large spans, the resulting stresses and displacements in the structure are verified using also the geometric non-linear analysis. • The safety of structure during the exploitation process grows if the truss members are protected using fire resistant paint. • The guiding cylindrical rolls for the conveyor belt must be of high quality and they must be endowed with reliable bearings. • The vibrations induced in the spatial steel-framed structure depend on the tear-and-wear degree of the conveyor rubber belt, of the leaning rolls and of the bearings.

REFERENCES Ashraf HabibuUah, Edward L. Wilson (1991). SAP90^^ Users Manual, Computers and Structures, hic, Berkeley, California, USA **** EUROCODE 3: Design of steel structures Ivan M., Vulpe A., Banut V. (1982). Statica, stabilitatea si dinamica constructiilor, Editura Didactica si Pedagogica, Bucuresti, RO Ivan M., Botici A., Dogaru E., Ivan A. (1997). Statica, stabilitatea si dinamica constructiilor - Teorie siprobleme, Editura Tehnica, Bucuresti, RO **** STAS 10108/0 - 78: Calculul elementelor din otel

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

349

INFLUENCE OF SEMI-RIGID JOINTS ON THE STABILITY OF SWAY-FRAMES H.H. Snijder^'^ and CM. Steenhuis^ ^ Eindhoven University of Technology, Faculty of Architecture Building and Plaiming, P.O. Box 513, 5600 MB Eindhoven, The Netherlands ^ Holland Railconsult, P.O. Box 2855, 3500 GW Utrecht, The Netherlands ^ TNO Building and Construction Research, P.O. Box 49, 2600 AA Delft, The Netherlands

ABSTRACT To calculate the force distribution in a sway-frame with a first-order elastic analysis, the non-linear behaviour of semi-rigid joints may be taken into account using an iterative method. The secant stiffiiess of a beam-to-column joint decreases with increasing moment on the joint. Therefore, the secant stiffiiess to be used has to be determined in an iterative way, taking second-order effects into account. Then, appropriate checks can be carried out. Columns are checked, taking the flexibility of the joint into account through an adapted buckling length. To avoid an iterative method when calculating the force distribution, EC3 Annex J (1998) suggests adopting a secant stiffiiess of the joint in the frame analysis being equal to half of the initial stiffiiess. Then, again appropriate checks can be carried out. In the paper both methods are compared for a one-bay two-storey sway frame with semi-rigid bolted end plate beam-to-column joints. Both ways to obtain the force distribution are illustrated and the appropriate checks are presented accordingly. Although the iterative method to obtain the force distribution is in fact a more precise method, the other method using a secant stiffiiess is preferred for reasons of simplicity.

KEYWORDS Joints, Connections, Semi-Rigid Joints, Stability, Frames, Sway-Frames, Unbraced Frames, Eurocodes, Beam-to-Column Joints, Joint Stiffiiess, Frame Analysis, Second-Order Analysis

INTRODUCTION Traditionally, unbraced steel sway-frames are designed with rigid beam-to-column joints. In that case, the joints have to be stiffened resulting in a complex detailing of the joints (figure la). This will

350

9~> b. unstiffened

a. stiffened

c. moment-rotation characteristic

Figure 1: Examples of beam-to-column joints and their properties increase the cost of steel frames. Leaving out the stiffeners (figure lb) makes the joints semi-rigid leading to semi-rigid sway-frame design. Anderson et al. (1997) show that semi-rigid design reduces the complexity of the joints, resulting in significant savings in the cost of fabrication. A comprehensive report on the effect of semi-rigid joints on frame behaviour is given in ECCS (1992). In this paper, elastic global analysis is used to calculate the force distribution in a multi-storey sway-frame. This is necessary in case of class 2 to 4 sections or in case of insufficient support at plastic hinge locations. Moreover, in practice this method is most frequently used because it is easy to apply. The simplest way is to use first-order theory with indirect allowance for second-order effects. In case of semi-rigid joints, the joint stiffiiess has to be taken into account exphcitly. The joint stiffiiess can be taken from a moment-rotation characteristic of the joint (figure Ic). The moment-rotation characteristic is nonlinear. Usually a secant stiffiiess is used. The secant stiffness of a beam-to-column joint decreases with increasing moment on the joint. Therefore, the secant stiffness to be used has to be determined in an iterative way, taking second-order effects into account. In this paper, this method is called the "iterative secant stiffiiess method". Alternatively, to avoid an iterative method when calculating the force distribution, ECS Annex J (1998) suggests to adopt a fixed value for the secant stiffiiess of the joint in the frame analysis being equal to half of the initial stiffiiess, see Jaspart (1997). Using half of the initial joint stiffiiess has been verified against second-order elasto-plastic frame analyses Jaspart (1991). In this paper, this method is called the "half initial stiffness method". After calculating the force distribution with either one of these two methods, ultimate limit state stability member checks and cross-section verifications have to be carried out as well as serviceability limit state deflection checks. Whether or not both methods yield comparable results is the subject of this paper. Therefore, as an example, a one-bay two-storey sway frame with semi-rigid boUed end plate beam-to-column joints is analysed.

EXAMPLE Both the iterative secant stiffiiess method and the half initial stiffness method are illustrated and TABLE 1 LOADS ON THE FRAME

1

1

Load qi (kN/m) q2 (kN/m) qs (kN/m) q4 (kN/m)

Serviceability limit state 21,6 46,8 4,2 2,1

Ultimate limit state 29,7 67,5 6,3 3,2

|

351 qi 1 1 1 ^ 1 1 1 1 1 1 ^ 11 j ;

IPE400

Si

1111

S3 4,5 m

q3

q2

'S2

q4

IPE550

S4 4,5 m

HE360B

iL

HE360B

9m

4

Figure 2: Example of a one-bay two-storey sway frame Bolts M24 8.8

105 105 J

1. — / - .

/ Bolts M24 8.8

t=14

90 90

IPE550

^

IPE400

HE 360 B

HE 360 B

a. Knee-joint

b. T-joint

Figure 3: Semi-rigid bolted end plate beam-to-column joints compared for a one-bay two-storey sway frame with semi-rigid bolted end plate beam-to-column joints. The frame is shown in figure 2. The columns are HE360B sections, the lower floor beam is an IPE550 section and the upper roof beam is an IPE400 section. The sections are (assumed to be) class 1. The steel grade used is S235. The joints used in this frame are given in figure 3. The load case considered consists of dead load, roof and floor loads and wind load as shown in figure 2 and given in table 1. In this paper, only in plane behaviour is considered, assuming sufficient lateral supports.

ITERATIVE SECANT STIFFNESS METHOD Clause J.2.1.2 (2) of EC3 Annex J (1998) states that the secant rotational stiffiiess Sj to be used in the global elastic analysis should be taken as equal to the value corresponding to the actual bending moment Mj,sd according to clause J.4.1 (4): 8.= - ^ where: Sj,ini is the initial rotational stiffiiess of the joint;

(1)

352

^ = 1 for:Mj,sd< 2/3 Mj,Rd; ^ = (1.5Mj^s,/Mj.Rj''for: 2/3 Mj,Rd<Mj,sd < Mj,Rd; Mj,Rd is the moment resistance of the joint. The initial rotational stiffiiess of the joints given in figure 3 are Sj,ini = 42266 kNm/rad for the kneejoint and Sj,ini = 113667 kNm/rad for the T-joint respectively. Their moment resistances are 179.1 kNm for the knee-joint and 342.9 kNm for the T-joint respectively. Now, using Eqn. 1, the moment-rotation characteristics can be determined as represented by tiie non-linear curves in figure 4.

10 12 9 (10-^ rad) —n

9 (10-^ rad)

a. Knee-joint

b. T-joint

Figure 4: Moment-rotation characteristics of joints considered (figure 3) The response of the sway-frame is determined using first-order elastic global analysis. The joints are modelled by rotational springs with a certain rotational stiffiiess. For reasons of simplicity, the moment-rotation curves of figure 4 have been used for positive as well as negative moments on the joints. The secant rotational stiffiiess Sj is used which varies depending on the actual bending moment Mj,sd (Eqn. 1). Therefore, an iterative method has to be used. This is illustrated schematically in the joint moment-rotation characteristic of figure 5. First, a secant stiffiiess So belonging to a joint moment Mo is used in the elastic global analysis. This analysis yields a joint moment Mi being either greater or lower than MQ. If MI is greater than M©, the joint behaves too stiff and a lower secant stiffness has to be used in the next iteration; if Mi is lower than Mo, a greater secant stiffiiess has to be used. With the new secant stiffiiess, again a global elastic frame analysis is carried out. This iterative procedure repeats until the assumed and calculated moment (M© and Mi respectively) show less than 1% difference. Of course, in every iteration step, the secant stiffnesses of all joints have to be adapted separately. Also, in every step of the iteration procedure, second-order effects have to be taken into account. This can be done using EC3 (1992) clause 5.2.6.2(3) and (6). However, here a sHghtly different approach is used. The sway frame has a P- A -effect, which can be taken into account using an ampHfication factor:

n-1

(2)

where: n is the quotient of the Euler buckling load of the frame and the factored vertical load on the frame. The value of n is approximated (Home (1975)) by placing the vertical storey loads horizontally on the frame as point loads at each storey: see figure 6. With this load case, the horizontal displacement A of the lower floor beam is calculated and the value of n is found as:

353

^^, M„ ^

14'

cp->

M„ < M, decrease S

9 ">

Figure 5: Iteration procedure

Figure 6: Approximation of n TABLE 2 ITERATION RESULTS

Sin kNm/rad Si S2 S3 S4

1 n

Serviceability limit state (SLS) Iteraticm step 1 2 3 14089 42266 42266 37889 113667 113667 14089 21933 25836 37889 50548 52700 13.1 16.9 17.1

4 42266 113667 27966 53853 17.1

Sin kNm/rad Si S2 S3 S4 1 n

h n«-A

Ultimate limit state (ULS) Iteratic)n step 1 3 2 14089 37045 42266 37889 113667 113667 14089 16243 12800 37889 19018 19542 9.2 9.9 9.9

| 4 42266 113667 12584 19542

9.9 1 (3)

where: h is the height of the lower storey, in this case 4.5 m. The iterative secant stiffiiess method as described above has been apphed for both load cases of table 1. The iteration results are summarised in table 2: per step the joint secant stiffiiesses of each joint and the value of n are shown. From the n values it can be concluded that the P-A effect is limited. For the serviceability limit state load case the frame can even be classified as non-sway (n>10). For both load cases, 4 iteration steps are required. As first approximation, the secant stifJBiess belonging to the moment resistance of the joint is used (Sj,ini/3). For both load cases and for both lefl side joints, these stiffiiesses change iteratively into Sj,ini. For the ultimate limit state load case and for both right side joints, a lower secant stiffiiess than Sj,ini/3 is reached. This means that the horizontal branches of the

354 moment-rotation characteristics are used, indicated by the moment resistances of the joints. Therefore, the joints need to have sufficient rotation capacity. Though the joints fail by column web in compression, it is likely that they have sufficient rotation capacity being a-symmetrical joints (Zoetemeijer (1981) and Jaspart (1991)), The final results are summarised in figure 7.

480.2

210.1

ULS NinkN

Figure 7: Results for the iterative secant stiffiiess method So far, EC 3 (1992) has been followed literally with respect to moment-rotation characteristic and rotational stiffness, see clauses 6.9.2(4) and 6.9.4(3). However, a remark is appropriate here. For the ultimate limit state load case and for both right side joints, very low secant stiffiiesses are found, corresponding to the horizontal branches of the moment-rotation characteristics. In fact, plastic hinges are formed in the right side joints and though EC 3 (1992) does not indicate this, it seems appropriate to model these joints by hinges. When this is done, the n value changes inton = 6.6 and the left side joints are not strong enough: the frame can not carry the loads applied. It seems appropriate to indicate in EC 3 (1992) whether the low secant stif&iess or the hinge approach should be used. The hinge approach will lead to safe results. When it is necessary to use the hinge approach, EC3 (1992) clause 6.9.2(4) is questionable and should be at least hmited with respect to joint resistance. More research is required. In the continuation of this paper, the former low secant stiffiiess approach is used. With displacements and force distribution known, joint, cross-section, member and displacement checks have to be carried out. The iterative secant stiffness method implies that strength and stiffiiess of joints are satisfactory. Since it is likely that the joints have sufficient rotation capacity, the joints fulfil all requirements. The most relevant checks according to EC3 (1992) are summarised in table 3. The horizontal deflections do not satisfy the criteria: stiffer joints and/or heavier sections should be used. The deflections are calculated making due allowance for second-order effects using the amphfication factor of Eqn. 2 including the effect of semi-rigid joints as described above, EC3 (1992) clause 4.2.1(5). The cross-section resistances are sufficient, even when second-order effects are taken into account. The member resistances are sufficient also. The checks are carried out on the basis of first-order forces and moments calculating the sway-mode buckling length of the column, taking semirigid joints into account, EC3 (1992) clause 5.2.2.1(7). This is done by translating the effect of joint stiffness into the beam length (TGB (1991)) used when determining restraint parameters at the ends of the column. This results in a quite substantial buckling length.

HALF INITIAL STIFFNESS METHOD Clause J.2.1.2 (4) of EC3 Annex J (1998) states that as a simpUfication, the secant rotational stiffiiess for bolted end-plate beam-to-column joints to be used in the global elastic analysis may be taken as: Sj — Sj,ini / 2

(4)

Thus, the secant rotational stiffness having a constant value, an iterative method is avoided. The straight lines in figure 4 represent Eqn. 4. Again the response of the sway frame is determined using first-order elastic global analysis with rotational springs representing the joints. These springs now

355 have the secant stiffness of Eqn. 4. Analyses are straightforward: second-order effects are taken into account as previously described. The values of n are now n = 15.5 and n = 10.9 for the serviceability and ultimate limit state load cases respectively. The results for the half-initial stiffiiess method are shown in figure 8. 33.0

20.3

480.2

251.2

ULS N i n k N

Figure 8: Results for the half initial stiffiiess method Again, joint, cross-section, member and displacement checks have to be carried out. Looking at the bending moments in figure 8, it is clear that the strength of the right T-joint is insufficient (410.4 x n/(n-l) = 451.9 kNm > Mj,Rd; However, EC3 (1992) clause 6.9.2(4) suggests to use the intersection point of the curved and straight line in figure 4b. This should be made consistent with EC3 Annex J clause J.2.1.2(3)). The same holds for the right knee-joint. The right side joints should be strengthened. Altematively, a lower secant stiffness should be used leading to an iterative analysis as described in the previous section. However, it was the intention to avoid this. Again, the most relevant checks according to EC3 (1992) are summarised in table 3. The same general remarks and conclusions hold as for the iterative secant stiffiiess method. COMPARISON OF RESULTS The results for the iterative secant stiffiiess method and the half initial stiffiiess method can be compared on the basis of the figures 7 and 8 and table 3. However, this has to be done with care because of the plastic hinges formed in the right side joints when the iterative secant stiffiiess method is used for the ultimate limit state load case and because of insufficient joint strength of these joints in the half initial stiffiiess method. The horizontal deflections are somewhat greater for the iterative secant stiffiiess method than for the half initial stiffiiess method. For the iterative secant stiffiiess method for ULS, the beams are heavier loaded and the buckling length of the column is greater resulting in a more utilised column despite a lower bending moment at the top of the column, when compared to the half initial stiffiiess method. This is due to the lower joint stiffiiess. Differences in results of the two methods are however relatively small. TABLE 3 SUMMARY OF MOST RELEVANT CHECKS 1 Criterion in EC3 (1992) SLS deflection Clause 4.2.2

1 ULS cross-section resistance Clause 5.4.8.1 1 ULS member resistance 1 Clause 5.5.4 (1)

Location Roof beam - vertical deflection (mm) Floor beam - vertical deflection (mm) First storey - horizontal deflection (mm) Structure - horizontal deflection (mm) Right lower column - unity check Roof beam - unity check Floor beam - imity check Right lower column - imity check Buckling length (mm) - TGB (1991)

Iterative secant stiffness method 19.2 < 45 17.6 < 36 24.5 > 15 38.8 > 18 0.41 < 1 0.77 < 1 0.97 < 1 0.86 < 1 23850

Half initial stiffness method | 21.7 <45 18.K36 22.6 > 15 35.5 > 18 0.48 < 1 0.69 < 1 0.80 < 1 0.72 < 1 17100 J

356 CONCLUSIONS In this paper two ways to obtain the global elastic force distribution of a sway-frame with semi-rigid joints are illustrated and subsequently the necessary checks are carried out. The methods used are the iterative secant stiffness method and the half initial stiffiiess method. The iterative secant stiffiiess method to obtain the force distribution is in fact a more precise method than the half initial stiffiiess method. The half initial stiffiiess method showed to be easier to apply than the iterative secant stiffiiess method. When using the iterative secant stiffiiess method for the frame considered, low secant stiffiiesses corresponding to the horizontal branch of joint moment-rotation characteristics were found. This means that these joints need to have sufficient rotation capacity. In fact, plastic hinges are formed and it seems appropriate to indicate in EC 3 (1992) whether low secant stiffiiesses may be used or hinges should be used in this case. The hinge approach will lead to safe results. When this approach has to be used, EC3 (1992) clause 6.9.2(4) becomes questionable. More research on this subject is required. When using the half initial stiffiiess method for the frame considered, joint strength showed to be insufficient and these joints should be strengthened. Alternatively, a lower secant stiffiiess may be used. However, this again leads to the iterative secant stiffiiess method. It is difficult to compare both methods for the example considered due to the considerations mentioned above. Differences in results do occur in deflections, force distribution and appropriate checks, but are relatively small. More research is necessary and a comparison for a semi-rigid frame that fiilfils all requirements seems appropriate.

ACKNOWLEDGEMENT The calculations of the beam-to-column joints according to EC3 Annex J (1998) have been carried out using the computer program CoP (Connection Optimisation Program) made available by ECCS bv, P.O. Box 212, 2130 AE Hoofddorp, The Netherlands. The authors are indebted to Cezary Ceglarek and Erik van Baars for carrying out the (frame) calculations. REFERENCES Anderson D., Colson A. and Jaspart J.P. (1997). Connections and Frame Design for Economy, Publication No 77, ECCS-CECM-EKS, Brussels, Belgium EC3 Annex J (1998). ENV 1993-1-1:1992/A2:1998, Eurocode 3 Annex J, Joints in building frames, CEN, Brussels, Belgium EC3 (1992). ENV 1993-1-1:1992, Eurocode 3: Design of steel structures, Part 1.1: General rules and rules for buildings, CEN, Brussels, Belgium ECCS (1992). Analysis and Design of Steel Frames with Semi-Rigid Joints, Technical Committee 8 Structural Stability WG 8.1 / 8.2, Publication No 67, ECCS-CECM-EKS, Brussels, Belgium Home M.R. (1975). An Approximate Method for Calculating the Elastic Critical Loads of MultiStorey Plane Frames. The Structural Engineer 53:6,242-248. Jaspart J.P. (1991). Etude de la semi-rigidite des assemblages poutre-colonne et son influence sur la resistance et la stabilite des ossatures en acier. Thesis, Department MSM, University of Liege, Liege, Belgium Jaspart J.P. (1997). Recent Advances in the Field of Steel Joints, Professor's Thesis, Department MSM, University of Liege, Liege, Belgium TGB (1991). TGB 1990 Steel Structures - Basic requirements and basic rules for calculation of predominantly statically loaded structures (in Dutch), NEN 6770, NNI, Delft, The Netherlands Zoetemeijer P. (1981). Summary of Research on Bolted Beam-to-column Connections, Report 6-85-7, Technical University Delft, The Netherlands

Technical papers on FRAMED STRUCTURES: GLOBAL PERFORMANCES DUCTILITY AND SEISMIC RESPONSE

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Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

357

DYNAMIC BEHAVIOUR CONTROL OF STEEL FRAMES IN SEISMIC AREAS BY EQUIVALENT STATIC APPROACHES J. M. Aribert^ and D. Grecea^ ^ Laboratory of Structures and Applied Mechanics, INS A Rennes, 20, av. des Buttes de Coesmes, 35043 Rennes, France ^ Department of Steel Structures and Structural Mechanics, "Politehnica" University, 1, Stadion, 1900 Timisoara, Romania

ABSTRACT This paper intends to control the dynamic behaviour of steel frames in seismic areas using equivalent static approaches with a new concept of q-factor related to the maximum inelastic shear force at the structure base. This new concept appears only consistent with rigid-plastic or elastic-plastic static global analyses, as illustrated by two numerical examples.

KEYWORDS Steel frames, Seismic behaviour, q-Factor, Ground accelerogram, Elastic response spectra. Equivalent static analysis, Rigid-plastic analysis. Elastic-plastic analysis, P-A effects.

INTRODUCTION Generally codes require only an elastic static analysis at geometrical first order for designing steel frames subject to seismic actions. Advantage of the very significant dissipative phenomena in steel structures is taken by means of a q-factor reducing the seismic forces which would be obtained assuming a perfectly elastic behaviour. Simulating frame responses by non-hnear dynamic analyses, it was realized that most of the definitions of this q-factor in the relevant literature are optimistic and conventional, leading consequently to an unsafe determination of the internal forces and moments in the structures and also of the forces applied to the foundations. In fact it is understandable easily that the first order elastic static analysis may not be well adapted to the dynamic behaviour of steel structures. Indeed, this type of analysis appears too elementary to express suitably the real dissipative behaviour of steel structures due to the occurrence of a very large number of plastic hinges associated with inevitable important P-A effects.

358

NEW METHOD FOR EQUIVALENT STATIC ANALYSIS OF STEEL FRAMES

Preliminary Use ofNon-Linear Dynamic Analysis It is assumed the time-history response of a structure to be simulated by means of an appropriate nonhnear dynamic analysis which can be performed using a specific computer code, for instance DRAIN2DX (1993). Under some ground motion defined by an accelerogram a(t), this analysis provides stepby-step on the time t the internal forces and moments in members, the displacements of nodes, the location of the plastic hinges, the cumulated plastic rotations, etc., all these resuUs including P-A effects if non negligible. The new method presented hereafter deals with regular 2D steel frames subject to seismic horizontal motion acting in their frame plane. For the steel elements (beams and columns), cyclic behaviour of elastic-perfectly plastic type (without strain-hardening) is considered as well as for the partial resistant beam-to-column joints. For the columns with high compression, interaction of axial force and plastic resistance moment is taken into account.

Definition of a Global Behaviour q-Factor In previous papers, the authors have introduced a new definition of the q-factor which is appHcable to any type of steel frame and any type of ground accelerogram. This definition is related essentially to the maximum inelastic shear force at the structure base, extracted from the time-history response. Usual definition A series of dynamic non-linear analyses is performed increasing step-by-step the accelerogram X,ag(t) where X means some positive multiplier. For each value of X, the maximum displacement 6 is extracted from the time-history response, for example the top sway displacement of the frame. So, Figure 1 shows the inelastic real behaviour of the frame resulting from the series of dynamic analyses, in terms of maximum top displacement 6 as a function of accelerogram multipher X. Also in the figure, the so-obtained non-linear curve (6, X) is compared with the straight hne corresponding to the indefinitely elastic response which can be evaluated by means of a first order elastic analysis. For multi-storey and multi-bay frames, the ultimate value of multiplier X^ corresponds to the attainment of rotation capacity of one particular plastic hinge in the elements (depending on the crosssection class) or possibly of rotation capacity of one partial resistant joint when this type of joint is employed. The usual definition of the q-factor is given by :

As aheady mentioned by Aribert and Grecea (1997,1998), this definition of q-factor is opened to criticism : on the one hand, it is not connected efficiently to the intemal forces and moments in the frame ; on the other hand, it keeps up some confusion of the extemal ground acceleration with the inelastic spectral response in acceleration of the frame, whose consequence may be the more significant as the structure is governed by several degrees of freedom. Complementary to Figure 1, the maximum base shear force of the frame as a function of the accelerogram multiplier X is represented in Figure 2. It is clear that using the definition (1) of q through a first order elastic analysis of the frame, the real inelastic state of the frame (marked as point U in the figure) is led down to the elastic limit state (poini

359 E). But, instead of the real inelastic values for base shear force and displacement (V^'"^'^, 5^'°^'^), the elastic values (V^^\ 6^*^^) are obtained. Generally V^^ is clearly less than V^'"^^^, so definition (1) leads to underestimate the internal forces and moments in the frame and consequently the forces applied to the foundation. Displacement

V

§(e,thUg(inel) g(e,th)„g(inel)*

y(e,th)

u*

v(e,th)'

u

u

Shear force

V -*~ Inelastic shear force -it-Elastic shear force

-*- Inelastic displacements -•-Perfect clastic displacements

y(inel) y(uiel)^

5(e)

Y.y^l^.-^^' "

^./"'^^--'^""^"^ Multiplier

K

K•

K

•

Multiplier

O

Figure 1 : Maximum displacement of the top storey

\

-k:

K '

Figure 2 : Maximum base shear force

Ne'w definition of behaviour q-factor Basing essentially on Figure 2, the authors have proposed another definition of the q-factor corresponding to the ratio between the elastic theoretical base shear force V^^*^ and the inelastic one yCmei) (determined both at multiplier X^, namely : yCcth) 4 ~ yCinel)

^'''K

\/(™')i

(2)

As main advantage of this new definition, it is worth mentioning the exact determination of the reduced forces applied to the foundation, which in principle allows to expect a suitable evaluation of the internal forces and moments in the structure provided that an appropriate global analysis is used including the dissipative effects. As another advantage, the new definition avoids knowing an accurate value of the elastic limit multipHer X^ whose part played predominantly seems debatable and whose determination is often ambiguous in dynamic response. Indeed, relationship (2) needs to know the ratio V^^VA-g instead of X^ alone. Following up the previous definition, three complementary aspects should be underlined : (i) The q-factor considered above is in direct correspondence with the ultimate value of the multipUer \ ; therefore, it should be associated with a specific value of nominal ground acceleration, for example a^^ = X^ • ag where ag is the maximum ground acceleration (ag=max|a(t)|). (ii) A value called q* with its associated nominal acceleration a^^* may be adopted as reference, corresponding to the criterion of equality of displacements (point U* in Figure 2, where §(e,th) _ g(mei)* y r^^^ ^^^^^ ^£ dissipation is very interesting because of the equality of first order elastic displacements with the second order inelastic ones. For levels of nominal accelerations different from a^j^^*, due to another seismic intensity of the site or another rotation capacity of the elements or joints, the q-factor will be adjusted moving apart this reference factor q*, as explained in the next paragraph.

360 (iii) With definition (2), it is possible to determine the q-factor in any case, even when the d(k) curve has not a regular shape and when there is no intersection between the inelastic curve and the elastic one.

Use ofq-Factor with Associated Elastic Response Spectrum In practice, the q-factor defined above will be nothing else than a global tool applied to the elastic response spectrum associated to the ground accelerogram in order to determine easily the total base shear force of the structure V^"'^'^ In fact, several similar accelerograms should be used to deduce a suitable response spectrum, but it is not the matter of this paper. Adjustment of q-factor When the nominal acceleration a^ ^ is different from the reference one aj^"^*, the q-factor has to be adjusted. A large parametrical investigation considering several types of frame and ground acceleration (not presented here; see Grecea (1999)) demonstrated that the hereafter linear interpolation gives always results very often on the safe side : q=q;(^

(3)

provided that a^^ ^ lies within the following interval: ^52if' <^'^' <\5^!^f

(4)

Of course, the upper bound of (4) is reached seldom because it implies high dynamic stability of the structure and large rotation capacity of its elements. An illustration of relationship (3) in a particular case will be shown farther. Evaluation of base shear force and distribution of equivalent static forces at different storeys From the elastic response spectrum, it is possible to evaluate the elastic base shear force V^^'**'^ corresponding to the nominal acceleration a^ ^, as given by relationship (5) where Re(Ti) is the normalized elastic response, Tj the fundamental period and M the total mass of the structure. When the accelerogram is very representative of the normalized spectrum, the so-obtained value V^^*^ is generally close to the elastic one deduced from the dynamic analysis. Let be : V^^'*^=MR,(T,)a^"^

(5)

Using the adjusted q-factor of relationship (3) and the elastic response spectrum, the reduced inelastic base shear force V^™^^' corresponding to the nominal acceleration a^^ ^ is given by :

V(-^)=M.R,(T,)-al,"Vq

(6)

It should be noted that relationship (6) may be slightly approximative because the inelastic response spectrum is never perfectly deduced by anamorphosis from the elastic one, specially when the latter is very irregular.

361 The distribution of the seismic horizontal forces is given by relationship (7) where Xji represents the displacement of storey j in the fundamental mode of vibration : m-x • p(inel) _

J J^

Y(inel)

/y\

j

This relation is true only for regular structures, which means that the participant mass Mj of the fundamental mode is close to the total mass M of the structure. If it is not the case, it should be taken account of several vibration modes in such a manner that the sum of participant modal masses exceeds 90% of the total mass of the structure ( J ] M^ > 0.90M). i

Evaluation of inelastic displacements Inelastic displacements are simply given by the familiar elastic relationship (8) when a^^ is strictly equal to a^"^* : T^

X M

5(ineO ^ 5 ( e , . ) ^ _ k _ _ J l

^(u)^^^^,)

(8)

k=l

In the general case where a^ ^ ^ a^^ ^*' ^ linear variation may be proposed again : ;;(inel) _ 5(e,th) **N "j ~"j «(u)*

^QX y^^

Indeed, the parametrical investigation already mentioned for relationship (3) demonstrated the inelastic displacements 6^'"^^^ to be quasi-proportional to the effective value of factor q, hence to a^^ • An illustration of relationship (9) in a particular case will be shown farther. The evaluation of 6^'"^'^ is useful to control the interstorey drift at the « serviceability limit state (SLS) » under seismic action, often transformed in equivalent ultimate limit state (ULS) by most of the seismic codes. Also, this evaluation is considered hereafter at the ultimate limit state to know if the second order effects on the global rigid-plastic analysis or the elastic-plastic one are negligible or not. Evaluation ofP-A effects P-A effects can be evaluated by the interstorey drift sensitivity coefficient 9j defined as follows : A6j W.

where Ad- is the horizontal displacement at the top of the storey j , relative to the bottom of the storey, hj is the storey height, Hj is the total horizontal shear force at the bottom of the storey and Wj is the total vertical axial force acting at the bottom of the storey. For values of 0j < 0.1, second-order effects can be neglected.

362 Checking of the Structural Load Capacity Equivalent static global analysis a. Case of fully dissipative structure When the condition a^^^ > a^jj^* is satisfied and the rotation capacity of dissipative members is large (for example, beam cross-sections in Class 1), the structure can be considered as fully dissipative. In this case, three criteria have to be checked : al. Development of global mechanism (l^^ principle) This principle requires that the global mechanism can be developed with plastic hinges at the ends of beams and at the base of the first storey columns. Using a rigid plastic analysis for the global mechanism and taking account of second order effects if 9j>0.10, a plastic multiplier ap of the horizontal forces associated to the modal response : F<'"=a,-mjXj,

(11)

can be determined. Then, it should be checked that: ^P|P)

< yCinel)

(J2)

a2. Prevention of local storey mechanisms (2^^ principle) The second principle consists in making sure of no development of premature local storey mechanisms for a value of base shear force less than V^'"*''^ Explicitly, a local storey mechanism means a plastic hinge mechanism which is concentrated on one storey or a few ones including possible plastic hinges at column ends. a3. Checking of element stability (3^^ principle) In addition to the above prevention of local mechanisms, any risk of column buckling and beam lateral-torsional buckling should be checked carefully. For column buckling, French code PS92 specifies to satisfy condition (13) where X is the column nondimensional slendemess : N — ^ + l,35?i < 1 if •Npl,Rd

X < 1,10

if

> 0,15

(13a)

Nsa < 0,15 -^^^^

(13b)

^:^

M ^^ pl.Rd

^ pl,Rd

For beam lateral-torsional buckling, EC3 mentions that no allowance is necessary when the nondimensional slendemess A.LT does not exceed 0.4. b. Case of partially dissipative structure This case may correspond to the criterion aj^"^ < aj^"^* or to a limited rotation capacity of elements or joints (for instance when partial resistant joints are used). A first or a second order elastic-plastic analysis (depending again on coefficient 6j) should be substituted for the rigid-plastic one proposed above (check al). This analysis is performed step-by-step

363 up to the horizontal forces F|'"^^^ distributed from the base shear force V^'"^^^ in accordance with the modal response. Then, it is necessary to check only that the rotation capacity must match or exceed the required rotation to attain the base shear force V^'"^'l Of course, checks a2 and a3 have to be kept though generally not predominant. n

As a remark, it is possible to meet a situation where V^'"^*^ > 2^Fj^^^, in particular if the level of j=i

acceleration a^^ is rather close to the reference value a{^"^ ; in this situation, the elastic-plastic analysis could not be performed up to the inelastic base shear force V^™^^\ To avoid any discontinuity between cases a and b, a possible solution is to strengthen the members, especially the beams, in order n

to obtain a sufficient value for ^ F|^^.

EXAMPLE OF 6 STOREY 3 BAY FRAME SUBJECT TO VRANCEA EARTHQUAKE The 6 storey-3 bay frame presented in Figure 3 is analysed in both previous cases a and b. The seismic action is characterized by the accelerogram of Vrancea (Romania) recorded to Building Research Institute INCERC Bucharest, the 4* of March 1977, and illustrated with its associated elastic response spectrum in Figure 4. All the cross-sections of members are Class 1. IPE330 3.00

m, = 47250kg T,=l,33 sec. 1,0001 HEB300 0,896

HEB260

IPE330

3.00

IPE330

3.00

HEB300

IPE330

3.00

IPE330

3.00

4.50

4.50

0,300 0,101j HEB360 Steel grade : S 275 Mp,,b = 221,1 kNm(IPE 330) 737,0 kNm (HEB 360) HEB360 Mpi^c 513,7 kNm(HEB 300) 352,6 kNm(HEB 260) HEB300

IPE330

3.00

0,7381

{x,} = 0,5321

4.50

Figure 3 : 6 storey-3 bay frame 0,7 . Sa/g

S,(Ti)=0,63

£=5%

0,6-

0,5 0,4

Time [sec] 0,3 0,2-

0,1 0-

;Ti=l,33 \

1

T[sec]

1 — 1

Figure 4 : Accelerogram and elastic response spectrum of Vrancea earthquake (04/03/1977).

364 Using rigid and full strength joints, the frame can be fully dissipative and the value of q* deduced from non-linear dynamic analyses is 3.4 with the associated nominal acceleration a^^^"^* = 4.8m/s^. For the same frame, but for other accelerograms, it may be mentioned that the q-factor would be q* = 6.7 with a^N^^* =7.9m/s^ for the accelerogram of Kobe (1995) and q* = 4.0 with a^"^* = 12.0m/s' for the accelerogram of El Centro (1940). In order to illustrate relationships (3) and (9), the variation of the q-factor and of the inelastic top sway displacement are represented in Figure 5 and 6 respectively, as a function of a^ ^, considering the three accelerograms of Vrancea, Kobe and El Centro. 1,-i - " g(inel)^g(e,th)

1 0,8 -^ Vrancea 0,6 -

-Kobe -*- El Centro

0,4 0,2-

« (u).

Acceleration a^ ^ [m/s ] 1

1

1

10

Figure 5 : Variation of q-factor

0 - -^^—

\

{

0,2

0,4

i

1

\

0,6

0,8

(u)*

\

1

1,2

Figure 6 : Variation of inelastic displacement

Results from the inelastic dynamic analyses are presented in Figure 7 at the first yielding stage and at the ultimate stage corresponding to the attainment of the rotation capacity in members (namely the bottom columns with rotation capacity d)^^^ = 0.070rad) for a^^ = 5m/s^. So, the ultimate state of the structure can be classified as fully dissipative. The associated base shear force is V^™*^^ = 1144.5kN and the top sway displacement 56=0.950m. Ultimate stage

Yielding stage Fj(kN)

Sj(m)

1

—

I

—

*\

•

5j(m)

Fj(kN)

T

(

•

•.

•

'^ *'

*

•

•!•

•

,•

•!•

•,

1

I 1

^^

imft

V('=^ = 510,3kN

Vmi

m m V(i»«=i)=n44 5kN

Figure 7 : Numerical results extracted from the inelastic dynamic analysis a. For acceleration a^^"^ = 5.0m/s^ the q-factor is equal to 3.5 accordingly to relationship (3). Also, relationships (6) and (9) give V'"'*^=1138kN for the inelastic base shear force and b^""'^^ = 1.060m for the top inelastic displacement respectively. Evaluating the interstorey drift sensitivity coefficient accordingly to (10), it is obtained 6^'"' = 0.153, so that all plastic analyses will be at second order. Calculating the plastic multipHer ap of the horizontal forces defined in relationship (11) by a second

365 order rigid-plastic analysis, it is obtained ^ F j ^ ^ =675.9kN, so that the 1'^ principle al is well j=i

satisfied. Checking all the local mechanisms with limited number of active storeys as illustrated partially in Figure 8, by second order rigid-plastic analyses, it can be demonstrated that the corresponding base shear forces V*''^ are always greater than V^™^'\ so that the 2"^ principle a2 is well satisfied. As for the third principle related to the element stabihty, only relationship (13) for the columns is considered here ; however, it is mentioned that condition X < 0.4 requires several lateral bracings to be used for the beams. For external columns where Ns/Np,^ = 0.11 < 0.15, it is checked that: X = 0.46 < 1.1 ; for

internal

columns where

Ng/Np, ^ = 0.22 > 0.15, it is checked

that:

Ns/Npi,R + U5X = 0.22 +1.35 x 0.46 = 0.84 < 1.0.

0,716m H 0,468m H

0,468m

. 0,223m

\

mM

W W

MM

m iLk X& m

m m

M

V(p.i)=1609kN

V*'2> = 3606kN

V*''>=1292kN

V(P'4) = 4242kN

Figure 8 : Checking of base shear force corresponding to various local plastic mechanisms (1) b. To illustrate also the procedure in the case of partially dissipative structure, the same structure is considered with partial resistant joints whose characteristics are rotational stiffiiess Sj = 137300kNm, resistant moment Mj^ =0.8Mpj^ = 177kNm and rotation capacity <^f^ =0.030rad. The q-factor deduced from non-linear dynamic analyses for the accelerogram of Vrancea is q = 2.1 with associated acceleration a^^ = 2.1 m/s^ (for information, it would be q = 1.8 for a rotation capacity of 0.015rad, and q = 2.9 for 0.045rad). Results from the inelastic dynamic analyses are illustrated in Figure 9, noting that the ultimate stage corresponds to the attainment of the joint rotation capacity 0.030rad. The associated base shear force is ycinei) ^ 773 3 ] ^ g^d thc top sway displacement b^ = 0.252m. Yielding stage

Fj(kN)

K

ssssbs

5j(m)

Fj(kN)

Ultimate stage (0.030rad)

M

WM

mm

yiincl) ^ 445 J ^^

WM y(inel) ^ 7-73^3 j ^

Figure 9 : Numerical results extracted from the inelastic dynamic analysis

5j(m)

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Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

367

GLOBAL PERFORMANCE OF STEEL MOMENT RESISTING FRAMES WITH SEMI-RIGID JOINTS D. Dubina\ A. Ciutina^ A. Stratan\ & F. Dinu^ ^Department of Steel Structures and Structural Mechanics, The "Politehnica" University, Timisoara, Romania ^Centre of Advanced and Fundamental Technical Studies, Romanian Academy of Science, Timisoara, Romania

ABSTRACT Beam-to-column joints have a fundamental importance in case of seismic moment resistant frames, because dissipative zones must be located at the beam-ends, so that their rotational ductility supply is strictly related to the detailing of connections. Based on a large numerical study, the present paper investigates the seismic performance of two series of moment resisting frames with beam-tocolumn joints with different configurations. Performance of the analysed structures is expressed in terms of characteristic failure acceleration, ductility demand, damage indices and the q factor.

KEYWORDS moment resisting frames, beam-to-column joints, steel frames, semi-rigid joints, rotation capacity, ductility, damage, earthquake. 1. INTRODUCTION The use of semi-rigid joints in frames subjected to seismic loads is a matter of controversy. None of the existing design codes include provisions for their use and, in zones characterised by a high seismicity, the use of rigid full-strength joints is mandatory. To ensure that a beam-to-column joint is ductile enough and has the capacity to provide the required rotations, the related connection must be capable to develop adequate plastic hinge while sustaining its yield moment capacity. Due to their higher flexibility, semi-rigid steel frames are prone to increases in inter-storey drifts. The inter-storey drift condition is related to the serviceability limit state which corresponds to minor frequent earthquake. The design objective, when building serviceability is checked, is that the building, including both structural and non-structural components, should suffer no damage and discomfort of the inhabitants should be minimal. The first requirement, which leads to the avoidance of damage, is satisfied by ensuring that the structure behaviour during the earthquake remains in the elastic range. In order to fiilfil the second requirement - both non-structural components damage and discomfort of inhabitants is avoided- it is necessary to provide sufficient stiffness to prevent significant deformations. The ultimate limit state of a building in seismic circumstances could be regarded either as damage

368 limit state or failure limit state. The damage limit state allows some minor damages to non-structural components due to large local deformation in certain zones. The failure limit state pertains to very infrequent severe earthquake ground motions in which both structural and non-structural damages are expected, but safety of the inhabitants must be guaranteed. Furthermore, the structure must be able to absorb and dissipate large amounts of energy. In case of MR frames with semi-rigid partial-resistant beam-to-column connections, the connection design properties can be expressed in rotational stiffness, moment capacity and plastic rotation supply. In case of strong seismic motion, the plastic rotation demand for beam-to-column connection could be greater than the plastic rotation supply, which leads to its damage (partial or total). This was the situation of most beam-to-column welded connections in muhi-storey MR frames during the Northridge earthquake (Kato B. et al, 1997). The problem in such cases is whether damaged connections can be recuperated by repairing works or not? The SAC joint venture project suggests several reparatory design detailing for seismic damaged welded connections. It is obviously that reparatory detailing can be also proposed for bolted connections. It is recommendable, of course, to have an initial design of beam-to-column connections such that both damage and reparatory works to be easy controlled. Authors' opinion is that based on the Performance Design Philosophy (Bertero V., 1997), semirigid steel frames could satisfy the required strength and stiffness conditions for "Operational and Functional" or "Life Safety" performance levels. In terms of rotation capacity of connections, it means that, if a certain percentage of damage in frame connections can be accepted, a larger rotation capacity of the connections (related to design) can be considered as a supplementary ductility supply for the global behaviour of the structure. A parametrical study was developed by the authors in order to analyse the effects of beam-to-column connection properties on the global performance of two series of MR frames. The present paper shows the results and the conclusions of this study.

2. ANALYSED FRAMES The parametrical study is developed on two frames: a 6 storey - 3 bay frame (C36) made out of Fe430 steel and a 3 storey - 3 bay frame (C33) made out of Fe360 steel, shown in Figure 1. The member cross-sectional dimensions and joint properties were obtained by means of an equivalent static design procedure, using an elasto-plastic analysis. Different frame typologies have been considered in the study. Member characteristics for the C36 and C33 frames are given in Table 1.

HEB260

IPE300 3.00

HEB260

IPE300

HEB300

3.00

IPE300

q IPE330

3.00

IPE300

5|

3.00

IPE330

IPE300

q

4.50

4.50

4.50

4.0

Figure 1. Geometry of the analysed frames

4,0

4.0

369 The rigid and semi-rigid joints have been designed according to Annex J of Eurocode 3 [8]. Joint Table 1. Design values of the section characteristics. Frame

Column

C36

HEB360 HEB300 HEB260 HEB240

1 C33

Beam

Mpl,b.Rd

Mpi,c.Rd

MpLcJRd/

xlO^ Nxmm

xlO^ Nxmm 670.8 467.3 320.7 225.0

MDl,b.Rd

IPE300

157.1

IPE330

171.8

J

4.27 2.97 2.04 1.31

material Fe430

Fe360 1

configurations are given in Figure 2 and the design values of moment capacity and stiffness in Table 2. Distinction is made between single-sided and double-sided joint, subjected to unbalanced moments, as is the case under seismic horizontal forces.

CI

C4

C3

C2

C5

C6

r

10

Figure 2. Detailing of semi-rigid joints. Table 2. Joints Characteristics Joint

1 Type 1 C36.1A C36.1B C36.1C C36.2A C36.2B C36.2C C36.3A C36.3B C36.3C C36.4A C36.4B C36.4C C36.5A C36.5B C36.5C C36.6A C36.6B

1 C36.6C C33.1 C33.2 C33.3 C33.4 C33.5

'"

Mj,Rd

xlO^Nxmm 153.4 153.4 153.4 133.8 109.3 91.4 153.4 153.4 153.4 153.4 153.4 153.4 219.3 280.8 172.3 280.8 280.8 243.4 130.9 74.63 155.8 152.5 140.9 210.4

•••

'*

•

-

'

m (Mi,Rd/Mpi.b.Rd)

I 1 1 !

0.98 0.98 0.98 0.85 0.70 0.58 0.98 0.98 0.98 0.98 0.98 0.98 1.40 1.79 1.10 1.79 1.79 1.55 0.76 0.43 0.91 0.89 0.82 1.22

f 1

"irn

c.. . xW^ Nxmm/rad 0.628 0.561 0.493 0.421 0.340 0.305 1.235 1.274 1.292 1.235 1.274 1.292 1.186 1.509 0.936

xlO" Nxmm/rad 0.314 0.281 0.246 0.211 0.127 0.152 0.618 0.640 0.646 0.618 0.640 0.646 0.977 1.509 0.468

00

00

00

oo

00

00

00

00

00

0.501 0.307 1.362 1.362 0.907

0.250 0.153 0.681 0.681 0.454

4.05 2.48 11.02 11.02 7.34 ! 00 1

1

8.05 7.21 6.31 5.40 3.26 3.90 15.85 16.41 16.57 15.85 16.41 16.57 25.05 38.70 12.00

1 C336 00 00 A - connections at levels 1-2; B - connections at levels 3-5; C - connections at the roof level

370 If the web panel of the C2 joints is considered as unstiffened to shear, compared to C4, the drop of stiffness is very important in the case of unbalanced bending moments. This penalisation is probably too severe. The topology of the frames, including the joint distribution, is shown in Figure 3. The elasto-plastic dynamic analysis has been performed using the DRAESf2DX computer program and the set of three accelerograms described in Chapter 3. The main aim of the analysis was to investigate the influence of the different ground motions on the seismic response of the frames. For this study, a value of 3% inelastic inter-storey drift and two values of plastic rotation capacities i.e. 0.02 rad. for welded joints and 0.03 rad. for bolted joints were accepted. The monotonic plastic rotation capacity of members (9p) was computed by DUCTROT computer program (Gioncu et al, 1997). Plastic rotation capacity of members for cyclic behaviour conditions (Qp^^) has been computed by adjusting the monotonic values by coefficients depending on the member cross-sectional slenderness and the axial force. A material partial safety factor Y=1 5 was used (Gioncu, 1997). Computed values of plastic rotation supply of members are given in Table 3. Table 3 Plastic rotation supply of members Frame type

member

%

Op''"

€36

IPE300 HEB360 HEB300 HEB260 IPE330 HEB240

0.0946 0.1085 0.0926 0.1267 0.0934 0.0963

0.054 0.049 0.042 0.065 0.053

€33

0044

1

In order to assess the structural response to the different ground motions, local and global damage indexes are computed as follows:

e„

I. =-

lo.-

0. ''p.\m

Zh 2^

Du

where. IDL is the local damage index, $p is the plastic rotation demand and 6pMm is the minimum guaranteed plastic rotation supply and log is the global damage index C33RIG B

C-B

C33SR1

C33SR2

a

a

^

rf—T—T—\

B[1

i

i J

T

T

T

\

Ji

(j)

J

L

*[

I 1

r

T

T

1

C-p

0"

4

D

rr

T

r — \

BL|

i

Jj

H

U

k

J)

H

k AU)L

i J (j

1) \ J) [ rfJ

Q

- joint type CI

•

-joint type C2

0

-joint type C3

0

- jointtypeC4

D

- joint type CS

0^

- joint type C6

O

- pined Joint

u

1——( 1

g

D

t—4

Figure 3. Frame typologies

1

(

11

(

371 The q-factor is evaluated as the ratio between the ground acceleration leading to failure (^u) and that corresponding to the first yielding (kc):

K

'-r. The ultimate design limit state multiplier Xud may be evaluated as: X^ =min(i^,^,A^) where: XA is the accelerogram multiplier corresponding to the attainment of 3% inter-storey drift limit; XQ is the accelerogram multiplier corresponding to the plastic rotation capacity either in joint or members; Xm is the accelerogram multiplier corresponding to the plastic mechanism.

3. SELECTION AND SCALING OF RECORDS Three different earthquakes have been chosen to study the dynamic response of the considered structures. The ground records have been chosen so as to assure a large variety of Control Periods Tc [7] (considered to be a very significant parameter for an earthquake if it is reported to a structure with a given eigenperiod): SV T^^lK

2-L

Table 4 Ground motions characteristics PGA(g)

Record Incerc-Bucharest NS, March 04,1977 Kobe NS, 17 Jan 1995 IBrienza NS, November 23, 1980

0.199 0.836 0.21

Tc (sec) 1.335 0.622 0.19

EPA (g) 0.240 0.704 0.22

scaling factor 1.042 0.355 1.155

scaled"! J^V 1 PGA(g) (cm/sec) | 52.18 1 0.207 97.17 0.297 0.248 9.06 1

Table 4. gives the values of the most important parameters which characterise the considered earthquakes (INCERC Bucharest - Romania, 4 March 1977, Kobe NS - Japan, 17 Jan. 1995 and Brienza NS - Italy, 23 Nov. 1980), and in Fig. 4 are given the spectral accelerations of the same eatrhquakes with a damping factor of 5%. For comparison, in Tab. 5 are given the fundamental periods of the resulted structures. ELASTIC ACCELERATION RESPONSE SPECTRA 5% DAMPING I N C E ° C BUCSARES^ NS MARCH (M 1977 T c " ! 33Ss — K O B E NS Tc=0 622s

Fig.4 Acceleration response spectra of records. The scaling of records for structural analysis was done so as to correspond to the same Effective Peak Acceleration (EPA).

372

Table 5 Fundamental periods of vibration of the analysed frames. C36RIG C36DU4 C36SR2 C36DU1 C36DU2 C36DU5 C36SR1 |C36DU3| 1 Frame 1.47 1.61 1.63 1.35 1.52 1 Fund. Period 1.33 1.69 1 1.86 1 C33RIG C33DU4 C33SR2 C33DU1 C33SR1 C33DU5 C33DU3 1 Frame 1.01 1.09 1.20 [Fund. Period 0.95 1.04 1.15 1.31 1

4. NUMERICAL RESULTS As it was described in Chapter 2 ~ Analysed Frames, the parameters that have been monitored were the Damage Index - ID and the q-Factor. A different parameter that has been taken into account for results is the Effective Peak Acceleration (EPA) in the failure situation. EPA VALUES FOR INCERC 1977 EARTHQUAKE (C36 SERIES) Isms EPA ««t

EPA VALUES FOR INCERC 1977 EARTHQUAKE (C33 SERIES)

1

BBSaiPA rotellon • • • EPA KMchanisni MMnVaLDrM. Mtan val. roMion. ^ " M # m VBL M9C.

I^SSSEPAAWt eezzaCPA rotation • m BPA nMchanism MMnVal.DriR. Mean Vai. rotation. — • M o a n Val. Mec.

2.5-

«2

I

s,

ui 1

0.50 C36raG

C36DU4 C36SR2 C3SDU1 CMDU2 C36DUS CMSR1 C36DU3

C33RIO

lii '^wmmm

C330U4

C33SR2

''•iiiii''i

C33DU1

03391(1

1 ^ ^ H

1 .^^iffllW 1

C33DU5

C33DU3

Fig. 5 EPA values for C36 frame series INCERC 1977 earthquake

Fig. 8 EPA values for C36 frame series INCERC 1977 earthquake

EPA VALUES FOR KOBE NS EARTHQUAKE (036 SERIES)

EPA VALUES FOR KOBE NS EARTHQUAKE ( C M SERIES)

C36raO

C36DU4 C36SR2 C3SDU1 C36DU2 C3eDU5 C36SR1 C3C0U3

Fig 6 EPA values for C36 frame series - Kobe NS earthquake EPA VALUES FOR BRIEN2A EARTHQUAKE (C36 SERIES)

Fig. 9 EPA values for C33 frame series NS earthquake EPA VALUES FOR BRIENZA EARTHQUAKE (C33 SERIES) GSsePAdrMt CZZaepArMation moB EPA medianlsm Mean Val. D m t MeanVaLrataHon. MeanVaLMec.

C36RIG

C36DU4 C36SR2 C36DU1 C36DU2 C36DUS C38SR1 C36DU3

Fig. 7 EPA values for C36 frame series Brienza earthquake

C33RIG

C33DU4

C33SR2

C33DU1

C33SR1

C330U5

C33DU3

Fig. 10 EPA values for C33 frame series Brienza earthquake

373 This parameter has been introduced in order to see the maximum values of EPA, as frame response, for each earthquake. These values are computed for each failure criterion type (drift - 3%, rotation capacity - 0.02 or 0.03 rad and mechanism). In Fig. 5-7 are given the charts with the obtained values for C36 frame series (for the three earthquakes considered) and the similar values for the C33 frame series are given in Fig. 8-10. In the charts are also given the mean values for each criterion. The Global Damage Indices for each earthquake are given in Fig. 11 and Fig. 12 for frame series C36 and C33 respectively. Global Damage Indices for C33 frame series

Global Damage Indices for C36 frame series

B1977

C36RI6

C36DU4

Fig. 11 Global Damage Indices C36 frame series

Fig. 12 Global Damage Indices C33 frame series

For a better identification of the global and partial Damage Indices. These values are given in the Table 6. It is to be noted that the Global Damage Index is not a linear sum of partial indices. Also, the damage index in joints is computed for the ductility criteria stated in Chapter 2. Table 6. The Global and Partial (Columns, Beams and Joints) Damage Indices. Frame Global Damage Indices ID.COI I iD.beam 1 iD.ioint 1 ID,G INCERC-BUCHAREST NS, MARCH 04, 1977 ROMANIA 0.384 0.168 0.397 0 C36RIG C33RIG 0.162 0.398 0 0.386 C36DU4 C33DU4 0.710 C33SR2 0.125 0 0.696 C36SR2 0.698 0.651 ' 0.122 0 C33DU1 C36DU1 0.047 0 0.656 0.651 C33SR1 C36DU2 0.018 0.383 0.348 0.206 C36DU5 C33DU5 0.049 0 0.661 0.656 C33DU3 C36SR1 0.024 0 C36DU3 0.642 0.639 KOBENS JAPAN 0.027 0.269 0 0.267 C36RIG C33RIG 0.037 0.284 0 C36DU4 C33DU4 0.281 0.043 0 0.488 C33SR2 C36SR2 0.483 0.064 0 0.506 0.499 C36DU1 C33DU1 0.564 0.564 C36DU2 0.011 0 C33SR1 C36DU5 0.014 0.351 0.207 0.320 C33DU5 0.021 0 0.540 0.538 C33DU3 C36SR1 C36DU3 0.055 0 0.577 0.568 BRIENZA, 24 NOV. 1980, N-S 0 0.018 0.314 0.311 C33RIG C36RIG C36DU4 0.018 0.304 0 0.301 C33DU4 C36SR2 0.021 0 0.758 0.755 C33SR2 0.020 0 0.709 0.706 C33DU1 C36DU1 0.019 0 C36DU2 0.706 0.703 C33SR1 0.257 J 1 C33DU5 C36DU5 0.020 0.275 0.211 0 0.022 0.314 0.311 1 C33DU3 C36SR1 0.655 ! 0.650 1 C36DU3 0.027 0 Frame

Global Damage Indices ID.COI I lD,beam 1 lo.ioint

1 h),G

| 1 |

1 0.442 0.352 0.459 0.428 0.451 0.266 0.327

0.256 0.192 0 0 0 0 0

0 0.762 0.488 0.505 0.649 0.758 0.763

0.360 1

0.228 0.318 0.263 0.270 0.055 0.086 0.166

0.281 0.272 0 0 0 0 0

0 0.860 0.646 0.583 0.513 0.470 0.743

0.258 0.543 0.561 0.503 0.501 0.440 0.700

0.325 0.319 0.361 0.378 0.118 0.089 0.076

0.243 0.206 0 0 0 0 0

0 0.681 0.473 0.547 0.601 0.796 0.215

0.286 1

0.494 0.478 0.479 0.613 0.680 0.690 | 1

| 0.439 0.437 0.495 0.576 0.759 0.199 j

374

q-Factor values, may be regarded as a measure of global ductility, are given in Fig. 13 and Fig. 14 (frame series C36 and C33 respectively), comprising also the mean values for each earthquake. They have been computed according to the three failure criteria, but only the minimum values are represented in the Fig. 13 and Fig. 14. ..flMCTP.?.FOR CM F

•••g.PACTQR.FOR.C33JFR^^ ^SSINCERC1977

C3anO

Fig. 13 q factors for C33 frame series

C33DU4

C33Slt2

C330U1

C33SR1

C33DU5

C33DU3

Fig. 14 q factors for C33frameseries

5. CONCLUSIONS Previous chapter presents the results obtained by the numerical analysis for the considered frame series, for three main parameters: failure value of EPA, global damage index and the q-factor. In order to draw the final conclusions. Table 7 sumarises the qualitative description of the structural behaviour. Table 7. Global structural behaviour. Output Results 1 iFrame series C33/ Types of joints EPA-high iRigid Frame ID - low (theoretical) q-fct - high RIG EPA - med-liigh iReal Rigid Frame ID - high (rigid joints) q-fct - high pU4

iFrame series C36/ Types of joints iRigid Frame (theoretical) RIG iReal Rigid Frame (rigid joints) pU4

Output Results

iFrames with semirigid joints: (semi-rigid joints) SR1-SR2

EPA - SRl-medium - SR2 - med-high ID - SRI - low -SR2-high q-fct - SRI - med -low -SR2-high EPA-DUl-med-low - DU4 - med-high ID - D U l - h i g h -DU4-high q-fct - DUl - medium - DU4 - high EPA - DU3-medium - DU5 - medium ID - DU3 - medium -DU5-high q-fct - DU3 - low -DU5-high 1

JDual frames without pimied joints (rigid and SR joints) p U l - DU4

iDual frames with pimied joints (rigid and SR joints) p U 3 - DU5

EPA-high ID - low q-fct - high EPA-high ID - high q-fct - high

iFrames with semirigid joints: (semi-rigid joints) SR1-SR2

iDual frames without pinned joints (rigid and SR joints) p U l - DU4

iDual frames with pinned joints (rigid and SR joints) p U 3 - DU5

EPA-SRI-medium - SR2 - med-high ID - SRI - high - SR2 - medium q-fct - SRI - med -high - SR2 - medium EPA-DUI-high -DU4-high ID -DUl-med-high - DU4 - med-high q-fct-DUl -medium - DU4 - medium EPA-DU3-10W - DU5 - low ID -DU3-high -DU5-high q-fct - DU3 - low - DU5 - low

Comments

1

Ideal behaviour, but 1 it remains theoretical High values of 1 damages. This is the consequence of real rigid joints. Frame behaviour is 1 influenced by the joint propreties

Frame behaviour is 1 influenced by the joint propreties

Pinned joints have a 1 bad influence on frame behaviour

375

iHorizontal Dual Frames (semi-rigid joints)

Tab. 7. (Continued) Global structural behaviour. EPA - med-low 1 ID - lugh q-fct - low

pU2

Intermediar results, 1 between the two types of SR frames. No conclusions for just a frame |

Concluding remarks: - The perfect rigid frames resulted to have a very good behaviour, but they still remain unrealisable. The real rigid frames have big damages, damages due to Joint degradation mainly. This may be related to the relatively low value of ductility (plastic rotation capacity of 0.02 rad. and of 0.03 rad. respectively) that have been consideredfor computing the damage index in joints. It is expected to improve theframeperformances if the joints rotation capacity could be increased. - The frames having semi-rigid joints have a good general behaviour. The problem is to find the right joint properties that 'fit'' betterfor the given frame. Anyway, the semi-rigid frames seem to be an alternative to the classical ones. The only problem is to establish the limits of their use. ' Dual frames with rigid / semi-rigid joints represent an open problem: it seems possibly to find configurations with improved performances, butfurther studies are necessary. - Dual frames with pinnedjoints in middle span have generally a bad behaviour and they should be avoided. They have a bad behaviour of all the three parameters monitored. - The main conclusion of this study is that global performances of MR frames can be controlled by beam-to column joint properties. We can either improve or get worse frame performances. This problem is important enough to focus further researches. 6. REFERENCES Bertero V. (1997): General Report on Codification, Design and Applications, STESSA '97 Behaviour of Steel Structures in Seismic Areas, Proceedings of the Second International Conference 3-8 Aug. 1997, Kyoto, Japan. Gioncu et al. (1997): Simplified Approach for Evaluating the Rotation Capacity of Double T Steel Sections, STESSA '97 Behaviour of Steel Structures in Seismic Areas, Proceedings of the Second International Conference 3-8 Aug. 1997, Kyoto, Japan. Kato. B. et al (1997): Seismic Damage of Steel Beam-to-Column Rigid Connections in the Hyogoken-Nanbu Earthquake, STESSA '97 Behaviour of Steel Structures in Seismic Areas, Proceedings of the Second International Conference 3-8 Aug. 1997, Kyoto, Japan. Lungu D. et al (1997): Effective Peak Ground Acceleration (EPA) Versus Peak Ground Acceleration (PGA) and Effective Peak Ground Velocity (EPV) Versus Peak Ground Velocity (PGV) for Romanian Seismic Records. Eurocode 8 - Worked examples Mazzolani F. M. & Piluso V. (1996): Theory and Design of Seismic Resistant Steel Frames. London: E& FN Spon. ENV 1993-1-1 (1993) EUROCODE 3: Design of Steel Structures. Part 1.1. General Rules and Ridesfor Buildings. Brussels: CEN, European Committee for Standardisation. ENV 1998-1-1, (1993) - second draft - EUROCODE 8: Earthquake Resistant Design of Structures. Part L: General Rules and Rules for Buildings - Seismic Actions and General Requirements for Structures. Brussels: CEN, European Committee for Standardisation. SAC Joint Venture (1996). Connection Test Summaries. SAC 96-02. Sacramento, California.

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Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

377

A SIMPLE PROCEDURE TO DESIGN STEEL FRAMES TO FAIL IN GLOBAL MODE A. Ghersi, E. Marino and F. Neri Institute "Scienza delle Costruzioni", University of Catania, 95125 Catania, Italy

ABSTRACT In the paper a new method to design moment resisting steel frames able to fail in global mode is presented. Starting from the study of the inelastic behaviour of such type of structures subjected to increasing horizontal forces, the authors have defined a design procedure based on the plastic analysis; the beam crosssections are assigned such in way to resist to vertical loads, while the column cross-sections are determined by an application of limit design, taking also in account the necessity of minimising the structural cost. The application of the proposed design method is shown by means of an example; the inelastic behaviour of the structure thus dimensioned, compared to that of structures designed according to criteria proposed by seismic codes and by other researchers, confirm the effectiveness of the proposal. KEYWORDS Seismic behaviour, steelfi*ame,global ductility, local ductility, plastic design, capacity design. INTRODUCTION As it is well known, every seismic code allows to design a structure basing on internal actions quite smaller than those the structure could experience during strong earthquakes. Such reduction is possible because of the capability of the structure to undergo large deformations in plastic range without a substantial reduction of strength and to dissipate in this way the input energy of the seismic action. However, the inelastic behaviour is strongly conditioned by number and location of the developed plastic hinges. One of the main goals of the seismic design is therefore to avoid partial mechanisms of collapse (like the storey mechanism, in which the plastification concerns all top and bottom sections of the columns of a storey) and to lead the scheme to fail in global mode. A number of criterions aiming at achieving this have been included in seismic codes or proposed by researchers. In the paper some of these are recalled and discussed, pointing out their limits. Finally, a new procedure is proposed; the simplicity of this one is shown with some examples and its effectiveness is tested by means of push-over and dynamic response analyses. CRITERIONS DEFINED IN SEISMIC CODES The idea which underlies all code provisions is conceptually very simple. The designer should firstly define the best mechanism of collapse for the structure and then grant to the sections which have to remain elastic

378

a strength greater than the maximum resistant capacity of the contiguous plastic sections (capacity design criterion). The procedures suggested by all recent seismic codes are therefore arranged in three steps: 1. evaluation of internal actions produced by vertical loads and by horizontal forces smaller than the inertial actions which could be produced by a strong seismic event if the scheme would remain elastic; 2. definition of strength and executive details of the critical sections, i.e. of those which are expected to undergo inelastic deformations (plastic hinges); 3. definition of strength of all other sections, which have to remain elastic. The opportunity of taking into account horizontal forces in step 1 is usually explained with the necessity of avoiding structural damages under small seismic events, which have a short return period; another goal to be achieved in this case, the limitation of damage in non structural elements, leads to restrictions of the inter-storey drift which might strongly condition the design procedure (see Ghersi et al. 1999). Step 3 is the critical point of the procedure. In the case of moment resisting frames, for which the desired collapse mechanism presents hinges at both ends of the beams and at the bottom section of the first order columns, all seismic codes prescribe to evaluate the design value of bending moment in the columns basing on the equilibrium of each node, by proportionally increasing either the global results of the structural analysis, vertical loads plus horizontal forces, (GNDT, 1984) or the effect of horizontal actions only (ECCS, 1988). These criterions, based on node equilibrium, could be effective only if the plastic hinges would develop all together, at the same instant, according to the foreseen distribution of intemal actions. This never happens in actual structures, mainly because of two reasons: a) the intemal actions vary, during the seismic event, in a non proportional way. Even when modal analysis is used as a design tool, it provides only a statistical esteem of the maximum values of bending moments, which are reached in different instants at each section; b) also in the case of a proportional increase of intemal actions (like in push-over analyses) a simultaneous plastification is not possible because of the unavoidable (and different each other) overstrength of the sections. This is often caused by technological reasons, like the necessity of using commercial sections in steel structures (this aspect is thus usually less important in reinforced concrete structures, where strength may be more accurately calibrated by adding single reinforcing bars); however, it is much more prominent the overstrength caused by the necessity of defining the design internal actions as the maximum among the results of many load conditions (vertical loads plus horizontal forces acting along two opposite directions, vertical loads increased by partial safety factors without any horizontal force). A clear example is shown in figure 1 (Neri, 1999), which compares the bending moment distribution in an elastic scheme to the distribution of incremental bending moments just before failure, when plastic hinges developed only in the beams of the lower storeys; it is apparent that in this case the bending moment may largely increase in both columns connected to a node, without violating node equilibrium.

(a)

r-

=

r r

TWJ^

3!!

1

1

r

\

r

1/

(b)

u ri \r.rf\ J1

r

•/

•[ j

•

Figure 1: Bending moment distribution in an elastic scheme (a) and distribution of incremental bending moments before failure (b)

379 CRITERIONS PROPOSED BY RESEARCHERS An improvement of the code design procedure has been proposed by Lee (1996), basing on the observation of the bending moments evolution in a scheme subjected to constant vertical loads and horizontal forces increased up to the structural failure. He emphasised that, while during the elastic phase the points of contraflexure are generally located close to midheight of column, when collapse is incipient the moment distribution is much different and "the ratios of the moment of the lower column to that of the upper column are generally about 3". He therefore suggested a three-quarter rule, i.e. to assume for each column M^j > 0.75 p ^^Mj^j , being (3=1.1 to take into account the steel hardening in beams. beams

A completely different approach has been followed by Mazzolani and Piluso (1997), starting from the kinematic theorem of plastic collapse. In a defined structural scheme, with assigned values of vertical load and with a given distribution of horizontal forces, the cinematic multiplier corresponding to the actual mechanism of failure is the smallest one among the cinematic multipliers of all possible mechanisms. The proposed procedure is therefore arranged in the following steps: 1. definition of the cross-section for all beams, so as to grant enough strength to bear design vertical loads (or to comply any other requirement prescribed); 2. definition of a trial value for the sum of the plastic moments at the first order columns; 3. evaluation of the sum of the plastic moments at each other storey, by satisfying a set of conditions which impose that the collapse multiplier for the global mechanism be less than the multipliers associated to partial and storey mechanisms; 4. iteration of step 2 and 3, so as to comply the technological condition that the strength of the columns at each storey be not smaller than the one of the columns of the upper storey; 5. definition of the cross-section of the columns, so as to grant the previously evaluated total storey strength, taking into account also the effect of axial forces (effect of vertical loads plus the beam shear corresponding to the plastic moments). The procedure does not require the study of the scheme subjected to seismic forces and therefore it gives no information about the internal actions which will arise in the structure during a strong or weak earthquake. However it grants a global failure, provided that the actual distribution of forces during the seismic event coincide to the one used in the numerical analysis. PROPOSED PROCEDURE A numerical way to define which strength has to be provided to columns in order to obtain a global failure of the structure might be based on dynamic response analysis or on push-over analysis. After having defined the cross-section for all beams, like in the Mazzolani-Piluso method, and having chosen a tentative value for the columns, the above mentioned analyses can be carried on, by assuming that all column sections, but those at the base of the scheme, will remain elastic. The maximum bending moment in these sections, given by the analysis, shows therefore the strength required to obtain the desired collapse mechanism. For both kinds of analysis (but mainly for the dynamic one) the results are conditioned by the stiffness properties of each member. The procedure should therefore be iterative, especially in the case of steel structures, in which strength and stiffness of the members are strictly related each other. Anyway, basing on the consideration that in push-over analyses the maximum bending moment within each column is always achieved at the collapse (Lee, 1996; Neri, 1999), although in some sections of the column it may reach its maximum value before collapse, a much simpler approach may be based on limit analysis. After having defined the limit bending moment of the sections where the plastification has to occur {Mb,ik at each end of the / * beam at the A:* storey; Mcj\ at the bottom end of they ^^ column at the first storey) and the distribution of horizontal forces a Fk, the collapse multiplier ac may be immediately evaluated by equating the energy dissipated by the plastic hinges to the work of horizontal forces owed to a rigid displacement according to the global mechanism (fig. 2)

380

aF/

acFk

storey r

«v ric rih

Mc,2r

number of storeys number of columns number of beams

Mc,3r

// length of/^ beam

Figure 2: Global collapse mechanism

Figure 3: Equilibrium of the upper part of the scheme

7=1

k=l

/=1

a. =

(1)

The value ac is independent of the entity of vertical loads, because they do no work in consequence of the displacements corresponding to the mechanism if second order effects are neglected. Knowing a^, it is possible to evaluate the sum of the bending moments Mcjr at the top or bottom end of the columns at a storey r, by imposing the rotation equilibrium of the upper part of the frame (fig. 3) and remembering that the axial forces correspond to the shear forces of the beams and thus to their limit moments,

t,Kjr=^c E^* (^.-^)-ZZ2M,,,

(2)

7=1

being h= hr for the top section and h = hr.\ for the bottom section of the columns. Equilibrium conditions do not allow to evaluate the bending moment at each column. Push-over analyses show that when all columns at a storey have the same section the bending moments in proximity of the collapse are quite similar for all columns (slightly larger values are obtained in the internal columns, but the differences are smaller than 20 %). It seems therefore acceptable to assume for each column, as a design value of the bending moment, the average of the results of the limit analysis

^.v.=-i:^c.> ''c

(3)

7=1

It is important to point out that, differently from dynamic response analysis or push-over analysis, the limit analysis does not requires a preliminary definition of the stiffness of all members, but only an assumption of the strength of the sections in which plastic hinges have to arise. It is not economically convenient to provide beams with a larger strength, because it would require an analogous increase of column strength, although it could be in some cases necessary in order to limit the interstorey drift during low return period seismic events; in this case it is questionable whether a proportional increase of strength at all storeys is more convenient than a larger increase at some storeys.

381 2.4 2.2

1

'

2.0 1.8

1 __

• exact value — approximate value

1.6

1

I

T

1

1.4 1.2 1.0 2000

J

^,y^

1

T

1

ZAr

1 ^x-^

'

^"^ '

,

,

2000

-1000

^

M[kNiiil

1

T

1

1

1

^

5

r 1

i'

6

10

INs

Figure 4: Bending moment in the columns, for dif- Figure 5: Exact and approximate multipliers of the ferent values of the strength of the base sections first storey sum of beam plastic moment The influence of the value of the plastic moment at the base of the first order columns is more remarkable (fig. 5). When it is too small (c), the maximum moment will be attained at the upper storeys and technological reasons will compel to use in most columns sections larger than what strictly necessary; on the contrary, when it is too large (a) the columns will be severely stressed at all lowerfloors;a better distribution is obtained by imposing that the maximum positive and negative moments along the height be coincident (b). The design could therefore be based on an iterative procedure. However, a wide set of numerical analyses performed by the authors showed that the optimum value of the global strength of the base sections is related to the total strength of first order beams (fig. 6) by a relation linearly dependent on the number of storeys f;M,,,,= (0.7 + 0.15 ^ , ) X ^ M

(4)

The proposed design procedure is therefore fully defined and it may be resumed in the following steps. 1. definition of the cross-section for all beams, so as to grant enough strength to bear design vertical loads (or to comply any other requirement prescribed); 2. definition of the sum of the plastic moments at the first order columns, according to eq. 4; 3. evaluation of the collapse multiplier by means of eq. 1; 4. evaluation of the plastic moments of the columns at each storey, according to eq. 2 and 3; 5. definition of the cross-section of the columns, so as to grant the previously evaluated strength, taking into account also the effect of axial forces (effect of vertical loads plus the beam shear corresponding to the plastic moments). APPLICATION OF THE PROPOSED PROCEDURE AND COMPARISON WITH OTHER METHODS The proposed procedure has been applied to a six-storey frame, the geometrical scheme and the loads of which are shown in figure 6. The design bending moment for the beams has been evaluated as the maximum between the values given by two limit conditions: resistance for vertical loads only, evaluated as ^gSk'^lq Qk) /^/lO (Mj^71.4 kNm); maximum positive moment in seismic conditions achieved at the end of the beam (in order to obtain the desired mechanism), i.e. resistant moment evaluated as {gk + \|/ qk) P /4 (Msd=^\.\ kNm). Using a Fe 430 steel grade, an IPE 300 cross-section (MRO^ISI kNm) has therefor been assigned at every storey. Eq. 4 gives a design bending moment Msdr'^l'hn kNm at the bottom end of the first order columns which, together with the axial force caused by vertical loads and beam limit moments, makes necessary the use of a HEB300 cross-section (M/?^422 kNm). Eq. 1 gives a collapse multiplier ac=565, while eq. 2 and 3 give the design bending moments reported in tab. 1 together with the values of the axial forces. Basing on these values, the cross-sections have been assigned such in a way to verify the M,'Nlimit diagram (fig, 7).

382 -HE320B -••HE300B •HE280B —HE260B HE240B - -HE220B

300

+

G;t=15kN/m

300

a=10kN/m

300

w=24.3 t

300

+

+ + 300

300

•f-

450

450 I 450

1000

Figure 6: Geometrical scheme, vertical loads and masses of the example frame

2000

4000

3000

Figure 7: Column M-N limit domain and design values at different storeys TABLE 1

DESIGN ACTIONS AND COLUMN CROSS-SECTIONS

Storey 6 5

1 ^3 2 1

Fc(kN) 161.3 134.4 107.5 80.6 53.7 26.8

M(kNm) 235.5 350.0 363.6 296.7 169.2 1.4

Mt (kNm) 114.5 128.2 61.2 -66.2 -234.0 -422.0

600 i

A^e./(kN) 110 221 331 441 551 662

iV,„,(kN) 81 162 243 324 405 486

35

38

34

31

1h9 23 32

26 33

22

21 15

0

20

40 60 80 top sway displacement [cm]

100

120

Figure 8: Load-displacement curve of the frame designed according to the proposed and other methods

HEB 240 1 HEB 280 ' HEB 280 HEB 280 HEB 280 HEB 300

40

400

200

C0lint=C0lext

^i^41

12 24

10

•J 17

6

16

4

11 25 5

13

2

14

3

15

1

18

8

20

9

19

7

I3O

27

28

29

Figure 9: Order of yielding of the frame designed according to the proposed method

The same frame has been designed according to the ECS provisions (with ^^=0.35 g, soil C and ^=5) and the procedures proposed by Lee and Mazzolani-Piluso. In order to make easier the comparison, the beam cross-sections previously selected have been used for all the cases, being the maximum bending moment obtained according to EC8 (98.3 kNm) close to the values formerly evaluated; the structures differ only in the column cross-sections (tab. 2). It may be observed that the total weight of columns is minimum (55 kN) when ECS is used and it is +7% and +58% more when Lee and Mazzolani-Piluso procedure are used; the proposed design procedure gives intermediate weight (+33%).

383

beams: IPE 300 internal columns: HEB 240 external columns: HEB 220

17

22

23

fe"

" •< f-

1 13

12

^28

16

il^

18

•(^?32

^i 31 7 24

9 25 •

•<

beams: IPE 300 internal columns: HEB 260 external columns: HEB 220

^27 •"

3 14

*29

4 20

6 19

JO

Al

26 22

8

35

28

20 29

21 31

19

22

10 24

12 23

7

13

5 17

6 18

3

3

14

2 15

4 16

1

7

25

9 26

11 27

8

^4

j30

^2

17

'f31 8^|24

9

5

18

4 19

5 20

425

^5

^6

ECS capacity design criterion

37 40

14 28

11

15

36 39

•^32

2

30

38

10 27 6 23 ^29" 1 12

••A

po

beams: IPE 300 1-2 storeys: HEB 360 3-4-5 storeys: HEB 300 6 storey: HEB 260

'33

Lee three-quarter rule

21

33

Mazzolani-Piiuso plastic design method

Figure 10: Pattern of yielding of the frame designed according to the different method In order to check the effectiveness of the proposed design procedure, all the schemes have been firstly tested by means of push-over analyses up to the reaching of the mechanism. The F-5 diagrams (fig. 8) give information about the global strength of the frames and their ductility, while fig. 9 and 10 show the plastic hinges sequence and the collapse mechanism achieved. In these figures the dashed lines and the white circles are referred to plastic rotation of the base cross section of the columns greater than the ultimate rotation (assumed to be equal to 0.04). The frames designed according to the proposed procedure and to the Mazzolani-Piluso method reach the goal of obtaining a global mechanism of collapse and show an high global ductility, while those designed according to ECS and Lee criterion fail in partial mechanisms (which involve 3 or 4 storeys respectively) with a lower ductility. All the schemes have been successively tested by means of an inelastic dynamic analysis, using a set of ten artificial accelerograms compatible with the ECS provisions for a subsoil of class C and a strong motion phase duration of 22.5 s (Falsone and Neri, 1999). These accelerograms have been amplified in order to obtain three different values of peak ground acceleration (respectively PGA=0.25, 0.35 and 0.50 g). In all the cases, an elastic-perfectly plastic behaviour has been assumed for the end of the beams and the base cross-section of the first order columns, while other cross-sections of the columns have been assumed to remain in the elastic range. It is thus possible to judge the effectiveness of the design criteria simply by checking whether the maximum moment is smaller than the resisting capacity of the cross-section or not. The values of the ratio RM of maximum bending moment over resisting moment (fig. 11) show that at the bottom storeys the columns of the frame designed according to ECS provisions are not able to remain in the elastic range for high value of PGA, while all other methods give acceptable bending moments.

CONCLUSIONS The proposed design approach, based on limit analysis, allows a very easy definition of the cross-sections of the members. The beams may be designed so as to resist to vertical loads only or to comply with seismic code provisions. The cross-sections of the columns are immediately given by simple formulations, in function of the beam cross-sections strength. The weight of the structure thus designed is larger than that obtained following ECS provisions, but the increase is smaller than that suggested by other researchers (Mazzolani and Piluso). The effectiveness of the proposed design approach has been tested by means of pushover analyses and dynamic inelastic response analyses, which proved its capability in obtaining a global mechanism of collapse and a large dissipative capacity. The inelastic behaviour of frames designed ac-

384 cording to the proposal is significantly better than that shown by frames designed according to the ECS provisions and comparable to that of frames according to the more complex and costly procedure suggested by Mazzolani and Piluso. A wider set of parametric analyses, reported by Neri (1999), confirm the above conclusions. Although some aspects of the proposal need further detailing, the easiness of use and the effectiveness of the proposed design approach made it a powerful tool for seismic design.

^v^—\—^(^

1

/y^y'.,'

4

lyjl/

|3 2

'

•

\

0^ ' Q
'

^

1

^^^'^^'^^^

0 0.2

0.4

0.6

0.8

1.2

0

0.2

0.4

0.6

0.8

1.2

RM

•e-0.25g

-^0.35 g

-^0.50g

Figure 11: i?^ values in the internal columns of the frame designed according to the ECS capacity design criterion (a) and the proposed method (b) for different PGA

REFERENCES Falsone G. and Neri F. (1999). Stochastic modelling of earthquake excitation following the ECS: power spectrum and filtering equations. Accepted for the publication on European Earthquake Engineering. Ghersi A., Neri F. and P.P. Rossi. (1999). A Global Approach to the Design of Steel Frames. Presented to 6* International Colloquium on Stability & Ductility of Steel Structures, Timisoara, Romania. GNDT. (1984). Norme tecniche per le costruzioni in zone sismiche. Gruppo Nazionale per la Difesa dai Terremoti. Italy ECCS. (1988). European Recommendations for Steel Structures in Seismic Zones. European Convention for Constructional Steelwork, Brussels. EUROCODE 8. (1996). Design Provisions for Earthquake Resistance of Structures. Part 1.3, General rules, specific rules for various materials. CEN, European Committee of standardisation, Brussels. Lee H.S. (1996). Revised Rule for Concept of Strong-Column Weak-Girder Design. Journal of Structural Engineering 122, 359-364. Mazzolani F.M. and Piluso V. (1997). Plastic Design of Seismic Resistant Steel Frames. Earthquake Engineering and Structural Dynamics 26, 167-191. Neri F. (1999). Comportamento Sismico di Telai in Acciaio a Nodi Rigidi. Doctorate Thesis Universita di Catania, Catania Italy Powell G.H. (1993). DRAIN-2DX, Program Description and Users Guide. Department of Civil Engineering University of California, UCB/SEMM-93/17, Berkeley, US

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

385

A GLOBAL APPROACH TO THE DESIGN OF STEEL FRAMES A. Ghersi, F. Neri and P.P. Rossi Institute "Scienza delle Costruzioni", University of Catania 95125 Catania, Italy

ABSTRACT The design of steel structures requires, as general rule, a double check. The first one involves the ability of the structure to withstand strong seismic events in the inelastic range, experiencing large deformations consistent with the material mechanical properties (ultimate limit state). The second one involves the response of structures to low intensity seismic events, which have to be withstood without inelastic deformations of structural elements and without relevant damage of non-structural elements (serviceability limit state). A design approach, described in a companion paper and aiming at the achievement of a global collapse mechanism, is herein deeply examined by means of inelastic static and dynamic analyses in order to analyse the process of formation of plastic hinges within the structures in occurrence of strong seismic events; furthermore elastic analyses according to Eurocode 8 have been carried out so as verify the limits imposed by the code to the internal forces and displacements in the ultimate and serviceability limit states respectively.

KEYWORDS Steel frames, design, dual-level approach, ultimate limit state, serviceability limit state, ECS. INTRODUCTION In order to design structures adequately able to withstand seismic events, different levels of peak ground acceleration have been associated to structural performances which, compatibly with the intensity of the ground motion, assure different safety levels both of human beings and structures (Bertero 1997). On the basis of this concept quite everywhere seismic codes impose, for the design of structures in seismic areas, two elastic analyses corresponding to different levels of the peak ground acceleration. The first one allows to verify the structural behaviour in occurrence of strong ground motions: it is carried out with reference to design horizontal actions reduced respect to those obtained from the elastic response spectrum scaled at high-level design peak ground accelerations, specified by each country by economic and safety considerations. In occurrence of such actions the check of the structural performance turns into a check of the internal forces and implies specific values of the available local and global ductility. The second elastic analysis simulates the behaviour of structures subjected to low-intensity seismic events: for its application Eurocode 8 does not impose the use of a design elastic response spectrum scaled at a low level of the design peak ground acceleration but

386 derives the design forces by scaling the design inelastic response spectrum corresponding to the occurrence of high-level intensity earthquakes. In occurrence of strong ground motions most design procedures aim at improving the seismic structural performance by favouring a global collapse mechanism characterised by plastic hinges at the ends of beams and at the base of the columns of the first storey (Mazzolani & Piluso,1997 - Lee, 1996). Such a distribution of plastic hinges, global ductility being equal, improves the inelastic response of structures in occurrence of high-level intensity seismic events because it increases the amount of energy dissipated by hysteresis and decreases the values required to the local ductility. In the present paper a design procedure (Ghersi et al., 1999) aiming at the achievement of a global collapse mechanism, is deeply examined by means of inelastic and elastic analyses with reference to earthquakes having both long and short return period. The design procedure, based on the application of the limit design theory, defines strength distributions which are independent of the frequency content of the expected earthquake and leads to global stiffness of the frames which are consequence of the adoption of commercial sections able to satisfy the design strength requirements only. NUMERICAL TESTS With reference to moment resisting steel frames designed according to the proposed procedure a large number of numerical tests has been carried out in order to analyse their structural performance in occurrence of both low and strong seismic events. The different stages of the whole checking process of the structural behaviour have involved both static and dynamic analyses so as to investigate different aspects of the structural response. In particular push-over and dynamic non-linear analyses have been carried out to evaluate the collapse mechanism mode and the ductility and stress demands while elastic analyses have further allowed to check, according to Eurocode 8, the displacement and stress state in the serviceability and ultimate limit states respectively. In order to get results of general validity the numerical tests have been carried out with reference to a large nimiber of frames (108 different systems) characterised by some geometrical, mechanical and loading parameters varying in a wide range of values. The frames are characterised by a different number of both storeys, having height equal to 3.0 m, and columns so as to represent different actual steel structures. For each geometrical scheme three different types of IPE sections have been assigned having strength constant at all storeys. Depending on the beam section, two values of the intensity of the vertical loads have been examined, called later on as high and low intensity vertical loads. The first one is defined as the value of the vertical load which in the non-seismic load combination involves the plastification of the ending cross-sections of the beams while the second one has been considered one half of the first one; each frame is recalled in the paper by means of a label which allows to know all the properties of the selected structures. The first character of the label individuates the number of columns and beams of the geometrical scheme, the second one the beam section, the third one the intensity of the vertical loading and the last one the span length (Table 1).

PUSH-OVER ANALYSES Push-over analyses have been carried out in order to check that the strength conferred to the columns by means of the proposed procedure be able to grant a global mechanism mode. These analyses have remarked the influence of the approximations and limits of the proposed procedure which lead to a performance of the steel frames which slightly moves away from the target (Marino et al., 1999). In a first time, in order to evaluate the difference between demanded bending moments and designed strength, the frames are characterised by infinite strength at the ending cross-sections of the columns, where the global mechanism mode imposes no plastification (all columns cross-sections with the exception of the bottom end of the first order columns). The strength of the other cross-sections has

387 TABLE 1 EXAMINED FRAMES AND RELATIVE LABELS

GEOMETRY

|

Type n.°

1

2

3

4

5

6

7

8

9

storeys

3

3

3

6

6

6

9

9

9 1

columns

2

4

6

2

4

6

2

4

6

VERTICAL LOADING

BEAM SECTION

1 Type

Ll

A

B

IPE 220

IPE 240

C

IPE 300

(m)

1

E

F

D

H

L

IPE 330

high

low

4.5

5.5 1

D

IPE 270

SPAN LENGTH

1

been assigned according to the proposed procedure described in the companion paper (Ghersi et al, 1999). The results of the numerical analyses have shown that in some cases the value of strength demanded in the columns is greater than that conferred by the proposed design procedure to the same elements. The bending moment of the external columns remains always lower than the value of strength while that of the internal columns, close to the collapse mechanism condition, in some cases overcomes the strength of the element causing its plastification. In particular in Fig. 1 -a is shown the process of the bending moment during the push-over analysis at the top cross-sections of the internal and external columns of the fifth order of the frame labelled as 5BFH; the mean value of the bending moment at that storey and the strength really conferred by the design are reported in order to compare each other the demanded and available strength values. Owing to the change in the structural configuration the maximum value of the bending moment is not always reached in correspondence of the collapse mechanism (Fig.l-b). In order to have parameters which can synthesise the abovementioned phenomenon, with reference to the cross-sections of the columns for which no plastification is expected, the following ratios have been evaluated:

c,

C,=

M,.

(1)

where Mi is the maximum value of the bending moment experienced at a cross-section during the push-over analysis, Mc is the bending moment corresponding to the collapse mechanism. Mm is the mean value of the bending moment corresponding to the collapse configuration at a storey level. The maximum demanded value of the bending moment at the cross-sections of the columns, where no plastification is expected, represents the minimum value of strength which allows to assure the global mechanism mode and can be obtained by the product of the two described values; it establishes how much the bending moment obtained by the limit analysis has to be increased to match the correct design value: M,=M„

c,c,

(2)

The analysis of the previous parameters have led to the following considerations: C/ mainly depends on the number of storeys and on the location of the columns within the frame (internal or external): it is equal to unity in the three storey steel frames while assumes different values in the six and nine storey frames. The numerical tests have shown that the maximum bending moment is reached before the occurrence of the collapse mechanism at the lower storeys of the frames only and precisely at the second storey of six storey frames and at the second and third storey of nine storey

388

(b) M

(-) M (kNm)

y—Resisting

bending moment at the storey

150 100

1

I

^'''—z^

1 1 Mean value-^^V,,.-^ ^.-''Xi'^s^X^^ 150 T Extemall columrpp^^^T^ ,,r''' ^ / ^ i V * \ ^

Internal cx)iumn—

1

(kNm) 200 1 Internal column~T~\ M j ^ ^ - ^ K ^ ' \ ^

200

I Mean valueExternal column-

100

V^j^^^^

50 Y""^

50

J 0.9

Vu/V

0.5

1

s

\ 0.6

T

1

1 0.7

r

\

1 0.8

1

\

\

•

\

0.9 VJV

Figure 1. Exemplifying bending moment process: internal and external columns at top cross-sections of the system 5BFH (a) and at bottom cross-sections at the third order of the system 8CFL (b)

2BFL 9Rnw

•N

LoxJW

2BDL 2CDH 2CDL

>•

o—o ] 0—q 0

^

1i

T

oto—ok>

\

5BFH

\ 5BDH

O

i\— - i

ok)—c>k> o(o—ok>

i

Figure 2. Collapse mechanisms frames. The greater values are obtained with reference to the external columns which are generally characterised by low values of the bending moment compared to the design values; C2 reaches values greater or lower than unity for the internal and external columns respectively. The principal parameter which influences the different performance of internal and external columns is represented by the number of columns: indeed the difference of structural behaviour decreases on increasing the number of the columns. In a second time, in order to evaluate the influence of the above mentioned approximations on the actual collapse mechanism, the same set of steel frames designed according to the proposed procedure have been analysed by push-over analysis with reference to distributions of the column strength obtained by means of frie proposed design method. The frames have highlighted collapse me<:hanisms characterised by plastic hinges in all beam ending cross-sections and by plastic hinges in th^ bottom cross-sections of the columns of the first storey so that a large amount of energy can be assumed to be dissipated by the inelastic structural behaviour during the seismic event. In particular frames having two columns have shown the desired global mechanism mode while all the other frames have highlighted the presence of plastic hinges at the top cross-section of the internal columns of the upper storey. Just in few cases only the presence of plastic hinges has been noticed at the top cross-sections of internal columns in intermediate storeys (Fig.2); anyway such behaviour does not seriously affect the seismic response of the steel frames which experience a mechanism where the external columns avoid the formation of storey mechanisms.

389 DYNAMIC ANALYSES Dynamic analyses have been carried out with reference to the same set of steel frames previously defined with columns having infinite strength with the exception of the bottom cross-sections of the first order columns. The structural systems have been subjected to ten artificially generated accelerograms (Falsone and Neri, 1999) matching the elastic response spectrum proposed by Eurocode 8 for subsoil of class C and characterised by a damping factor ^ equal to 0.05. The accelerometric signals have a strong motion phase of 22.5 s and a global duration of 35 s. The choice of the spectral shape corresponding to the subsoil of class C is justified by the greater amplification of the structural response imposed by such spectrum in the range of frequency characteristic of the examined steel fi-ames. The accelerograms have been scaled at different values of the peak ground acceleration equal to 0.20, 0.35, 0.50 g so as to reproduce the effects of moderate and strong seismic events. For all the cross-sections which are expected to yield a check of the damage level has been carried out by means of the normalised cumulative ductility demand which corresponds to the ratio of the maximum plastic rotation recorded during the dynamic analysis over the ultimate value of the same parameter calculated according to Mazzolani and Piluso (1992). The damage is quite always below the failure threshold: on increasing the value of the peak ground acceleration the damage level grows up uniformly (Fig.3-a) with the exception of the bottom cross-sections of the first order columns where sometimes, in presence of peak ground acceleration equal to 0.50 g, the damage index denotes a remarkable increase. The normalised cumulative ductility demand presents slight increases at the intermediate storeys of the frames having a large number of storeys respect to the values of frames having lower number of storeys and decreases at the upper and lower storeys. At the cross-sections where the formation of plastic hinges is not expected the following parameters have been evaluated: Ri=max|^|

R^=max{Rj

R„={2:R./na}

i = U.,n,

(3)

where rriiit) is the value of the bending moment at a time t, Mi{t) is the value of the plastic resisting bending moment reduced because of the presence of the axial force N{i) at the same time, while ria is the number of the accelerograms. Values of i?/ minor than unity points out cross-sections which behave elastically while values greater than unity individuate the overcoming of the available strength. The evaluation of the parameter Ri has allowed, with reference to each cross-section and all the accelerograms, the knowledge of the maximum RM and the mean Rm values. The dynamic analyses, so as the push-over analyses do, have shown a mean increase of the bending moment in the internal columns respect to the external ones up to 20%. With reference to 0.20 g and 0.35 g values of the peak ground acceleration, the columns present values of RM always lower than unity with the exception of the base cross-sections where the formation of plastic hinges frequently takes place (Fig.3-b). For a 0.50 g value of the peak ground acceleration values of RM greater than unity sometimes occur at the top cross-sections of the columns of the second order in frames with three storeys and at the bottom cross-section of the columns of the second order in frames with six and nine storeys. Such results do not deeply affect the general results because they occur in correspondence of very high values of the peak ground acceleration.

LIMIT STATE CHECKS ACCORDING TO EUROCODE 8 Commonly serviceability limit state checks are recognised conditioning for the design of steel structures due to the high deformability of such structures. Displacement limits are severe for beams subjected to vertical loads but much emphasis acquires the problem if the serviceability limit state check is extended to the horizontal displacement experienced by steel structures, particularly by moment resisting steel structures, due to wind or seismic lateral forces.

390

1.0 -e- Pga=0.20 g

-e- Pga=0.35 g

RM

- ^ Pga=0.50 g

Figure 3: Nonnalised cumulative ductility demand D (a) and values of RM (b) obtained for different values of PGA at different storeys of the system 5CFL In Eurocode 8 (1996) the limits for the design interstorey drift dr are specified, evaluated as difference of the average lateral displacements ds at the top and bottom cross-sections of the storey under consideration. For buildings having elements attached to the structure respectively the limits are: ^ / v < [0.004-0.006]/2

(4)

where h is the interstorey height and v is the reduction factor to take into accoimt the lower return period of the seismic event associated with the serviceability limit state. The displacement ds induced by the design seismic action is to be evaluated on the basis of the elastic deformation of the system {ds = qa de yi; where qd is the displacement behaviour factor, de is the displacement of a point of the structural system induced by the design seismic action, yi is the importance factor). The ratio dr/v is therefore evaluated by an elastic analysis of the system by means of the design spectrum Sd multiplied by the displacement behaviour factor qd and by the importance factor /; and divided by the reduction factor V. 5,(T) = SA^)^ci

Yi

(5)

With reference to the design of ordinary buildings (characterised by yj equal to unity and v equal to 2.0), the comparison between the elastic response spectrum in terms of acceleration and the design response spectrum to be used for the serviceability limit state check is shown in Fig. 4 for subsoil of class A and C. The above-described elastic analyses which use the spectrum proposed for subsoil class A evidently show the different influence of the parameters which characterise the response of the tested steel frames on the maximum interstorey drift (Fig.5). Structures having different intensity of the vertical loads provide different values of tiie evaluated interstorey drift; the increase of the intensity of the vertical loads affects the structural deformability because it requires greater sections of the columns and induces higher inertial forces. The difference of values of the interstorey drifts corresponding to low and high intensity vertical loads grows up on increasing the number of storeys, particularly if the frames are endowed with long beams. On the contrary the number of bays only slightly influences the elastic displacement state of the frames, being equal the other structural examined parameters. The values of the displacement parameter of interest are quite never minor than the limit value imposed by ECS for buildings having non-structural elements of brittle materials and only sometimes minor than that one imposed for buildings having non-structural elements fixed in a way as not to

391 Sa/g SUBSOIlJ CLASS A

1.00

—Elastic spectrum Design spectrum 0.60 - (serviceability limit state) k^ 1 Design spectrum 0.40 t ^ ^ \ / - /^---^_l__(ultimate limit state)

0.80

0.20

y

0.00 0.00

1.00

1

1

\

\

2.00

3.00

~-0.00

T(s)

0.00

1.00

2.00

3.00

T(s)

Figure 4 : Comparison between the elastic response spectrum and the design response spectrum to be used for the serviceability limit state check interfere with structural deformations. In particular this last value is verified only by three-storey frames subjected to low intensity vertical loads; the values of the maximum interstorey drift are indeed generally high reaching values of about two-three times the limit value proposed by ECS in the case of nine-storey frames with long beams. Nevertheless it has to be noted that the design spectrum proposed by ECS for the serviceability limit state checks seems to be extremely conservative particularly for structures having a long fundamental period of vibration. Such consideration partially justifies the results obtained for the examined steel frames not always in accordance with the limits imposed by the Eurocode S. The frames already examined in the previous sections have been then subjected to the combination of loads imposed by Eurocode 8 for the ultimate limit state. The steel frames have been analysed with reference to the design spectrum proposed by ECS for subsoil A and C, scaled by a value of the behaviour factor q equal to 5. In the columns the bending moments do not ever exceed the resisting bending moments corresponding to the maximum axial force; the ratio R of the last two terms evaluated for each storey is always smaller than 0.4 and presents a mean value ranging from 0.25 to 0.05 quite equal at all storeys. In the beams only few times the bending moments exceed, slightly, member strength; the mean value of the ratio R ranges from 0.25 to 0.8 showing a remarkable scattering of the values depending on the influence of the intensity of the vertical loads and of the geometrical characteristics of the frame (Neri, 1999).

CONCLUSIONS The inelastic numerical analyses have shown that the proposed design procedure lead in occurrence of strong ground motions to global mechanism modes characterised by plastification of the ending crosssections of all beams and of the bottom cross-sections of the first order columns. The limits and approximations of the procedure do not significantly affect the development of such behaviour. At the same time the maximum plastic rotation of the yielding sections is always lower than the ultimate values. The elastic analyses referred to the serviceability limit state according to Eurocode 8 have underlined great values of the maximum horizontal displacements, the reason of which has probably to be found in the high values of the accelerations imposed by Eurocode 8 to structures having long fundamental periods of vibration. The maximum values of the internal forces evaluated by the design spectnmi method in the ultimate limit state rarely exceed the ultimate values showing safety factors which are very high for columns and lower for beams but anyway quite always greater than unity.

392 SPAN LENGTH (m) 4.5 2.00 1

1 bay

3 bays

SPAN LENGTH (m) 5.5

1 5 bays

2.00

i

1 bay

3 bays

i

5 bays

o o

o

o

3B

3C

1.60 1.20

*

0.80

L

JL

i /— high load /—l-low load

>

t~--A

1.20 f 0.80 t # -

0.40 i 0 _ _ O - - 0 - - G - - Q - - 0 - i O - - o - - Q 0.00 IB

IC

2A

2B

2C

3A

3B

3C

2.00

2.00 1.60 4-

T

1

T

1.20 -p

T

1

1.20

1

0.00

lA

IB

IC

o

2A

2B

2C

3A

]

1

1

1

r

'"

^ ~

i

"

r

T

1

11 •

_^r,

r

r

r

'

•

-^,; •^^1 T

-

0.80

|o--o--or^-o--6^^-o--50.40

o 1 o

1.60 4"

1 '

0.80

CO

•

o

0.00 lA

g

^

•

o

T

i

'

0.40

'

0.00 7C

7D

7E

8C

8D

8E

9C

9D

9E

'

T

1

'

J

! 7C

7D

7E

8C

8D

Figure 5. Maximum interstorey drifts evaluated for the frames designed according to the proposed design procedure (subsoil class A). REFERENCES Bertero Vitelmo V. (1997). Performance-based seismic engineering: A critical review of proposed guidelines. Seismic Design Methodologies for the Next Generation of Codes, eds. Fajfar & Krawinkler, 1-31. EUROCODE 8. (1996). Design Provisions for Earthquake Resistance of Structures. CEN, European Committee of standardisation, Brussels. Falsone G. and Neri F. (1999). Stochastic modelling of earthquake excitation following the ECS: power spectrum and filtering equations. Accepted for publication on European Earthquake Engineering. Lee H.S. (1996). Revised Rule for Concept of Strong-Column Weak-Girder Design. Journal of Structural Engineering 111, 359-364. Marino E., Neri F. and P.P. Rossi. (1999). A simple procedure to design steel frames to fail in global mod. Presented to 6* International Colloquium on Stability & Ductility of Steel Structures, Timisoara. Mazzolani F.M. and Piluso V. (1992). Evaluation of the Rotation Capacity of Steel Beams and Beam-Columns. l^^Cost CI Workshop, Strasbourg . Mazzolani F.M. and Piluso V. (1997). Plastic Design of Seismic Resistant Steel Frames. Earthquake Engineering and Structural Dynamics 26, 167-191. Neri F. (1999). Compotamento sismico di telai in acciaio a nodi rigidi: proposta di una metologia per il controUo del meccanismo di coUasso. Ph.D. Thesis, Catania

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

393

A NEW SERIES OF FULL-SCALE TESTS ON STEEL FRAMES WITH SEMI-RIGID CONNECTIONS M, Ivanyi and G. Varga Department of Steel Structures, Technical University of Budapest Budapest, H-1111, Hungary

ABSTRACT The paper presents a test series consisting of six single-bay, two-storey frames, five of them tested under cyclic quasi-static loading, and one, under proportionally increasing loading. The connections between beams and columns are semi-rigid and partial strength, and have been designed to exhibit the performance of a plastic hinge with sufficiently large rotation capacity for a plastic mechanism to occur. The purpose of the tests was (i) to examine the behaviour of the frame as a whole, and (ii) to examine the behaviour of such connections when they form part of a frame. The paper puts an emphasis on the methods used during testing, including load histories, loading system and measuring devices. Of particular importance is a specifically designed device by which absolute rotations of cross-sections have been measured. The paper also presents some generic results concerning the frame behaviour under the loads applied.

KEYWORDS Testing, cyclic loading, semi-continuous frame, connection behaviour, testing methods, rotation measurement INTRODUCTION Semi-continuous framing, i.e. framing in which partial strength and semi-rigid joints according to the Eurocode 3 definitions (CEN, 1992) are applied, is gaining importance in building structures due to the economy offered and the possibilities for modelling and calculation provided by a new set of design codes (Davies, 1996). The connection models presented in these design standards and in the technical literature in general are based on the enormous number of tests, experimental and numerical, which were performed in the last decade or two (see e.g. Mazzolani, 1992), most of them examining joints isolated from the whole structure in which they are to be incorporated. However, it is not evident how these joints behave when they become part of a real frame; it is recognised that this problem needs experimental investigations performed on frill scale frames and realistic solutions for connection behaviour.

394 The first author directed a series of tests in 1993/94 consisting of four, two-storey and two-bay frames (Ivanyi et al., 1998), in which partial strength and semi-rigid joints were applied for column bases as well as beam-to-column connections. However, these joints were not designed to accommodate plastic hinges, therefore the failure of the frames in all instances was due either to loss of member stability or to connection failure in a brittle manner. The new test series presented in this paper consists of frames prepared with semi-rigid beam-tocolumn joints where special attention was devoted to the detailing of these joints in order to ensure a rotation capacity sufficient for a plastic mechanism to occur within the frame. Furthermore, the tests concentrated mostly on the behaviour of these frames under quasi-static cyclic loading with different types of load histories. This way two objectives were set at the start of the tests: (i) to examine the behaviour of thefi*ameitself under the circumstances covered, and (ii) to look at the behaviour of the partial strength and semi-rigid connections within these frames. The purpose of this paper is to present the testing methods and the testing apparatus including the specimen itself and the measuring systems, and to present some illustrative results. It is not intended to discuss the results in detail or to draw delicate conclusions; because there still needs to be some work done to reach that objective, further publications will follow with that subject.

TESTING Structure An overall view of the testing arrangement is shown in Figure 1. Thefi-amesexamined are two-storey single-bay ones. Both the columns and the beams are welded I sections, see Table I. Pairs of test frames are identical. Columns are connected to a rigid steel base element by two bolts through an end plate (layout generally regarded as pinned joint in the practice). Beams and columns are connected with flush end plate joints (Figure 2). In frames 1 to 5, the connections are strengthened with single-sided additional web plates. These were found to be necessary on the basis of an analysis of joint behaviour according to Revised Annex J of Eurocode 3 (CEN, 1992). According to the predictions, the joints were to fail in their tension zone, by Mode 1 bending of the end plate, a failure mode generally recognised as one ensuring sufficient rotation capacity to accommodate a plastic hinge within the connection. These predictions also showed that omitting the additional web plates would have caused column web buckling to become the governing failure mode, which is regarded to be ineffective in terms of plastic hinge behaviour. Despite these predictions, joints in frame 6, in which these plates were indeed omitted, showed similarly ductile behaviour as those in frames 1 to 5. In addition, these joints showed a type of asymmetric behaviour which, according to the best knowledge of the authors, is not treated in the technical literature, and which is not covered by design code regulations. This behaviour, characterised by excessive yielding of the column flange at one side and similarly excessive yielding of the end plate at the other, is due to the asymmetry of the joint because of the single-sided additional web plate, and suggests that these web plates play a more sophisticated role in the joint behaviour than as implied in current European regulations. This issue is discussed in more detail elsewhere (Ivanyi and Varga, 1999). In order to avoid lateral-torsional buckling, lateral restraints are applied to the frame at the beam-tocolumn joint locations and at the mid-spans of the beams, see Figure 1.

395

/

I*

(a)

/

/

(b)

Figure 1: The tested frames: (a) main geometry and (b) lateral restrains

00

20

160

^

L ^

J

P^r\

seas:

:

•^TTi" 4

CO

welding mm length)

,15 130

^

^

fillet weld of section column section additional web plate (218x400x8 mm)

^ ^ 5 ^ 7P ^ ^ 5 ^ «0

Figure 2: Beam-to-column connection and the detail of the additional web plate

TABLE 1 BEAM AND COLUMN SECTIONS APPLIED IN THE TESTS

Frame

Beam (flange/web)

Column (flange/web)

1,2

130-8/260-8

200-12/260-8

3,4

160-10/260-8

200-12/260-8

5,6

130-8/260-8

160-10/260-8

396 Loading The frame is loaded by two vertical concentrated loads at the mid-spans of the beams, and two horizontal loads applied at one side of the frame in the levels of the beams (Figure 1). The two vertical loads are increased and decreased proportionally using three hydraulic jacks (one larger to the lower beam and two smaller and identical to the upper) connected into one oil circuit. Because of the slight difference between the pressure surfaces of the larger jack on one hand and the two smaller jacks on the other, the lower beam was loaded by a concentrated load 89% in magnitude of the load on the upper beam. The vertical loads are applied through so-called gravity load simulators (Halasz and Ivanyi, 1979), devices which ensure the verticality of the loads within certain limits of lateral displacements of the points of application of the loads. The horizontal loads are applied using one hydraulic jack through a simply supported vertical beam, which ensures the applied load to be equally distributed between the two beam levels. The direction of these horizontal loads is reversible. The loading histories under which test frames 1 to 6 were examined are shown in Table 2 and Figure 3. Frame 1 was loaded by proportionally increasing loads; frames 2 to 6 were loaded by increasing vertical loading and cyclic horizontal loading at the levels of the vertical loads. In the case of frames 2, 3, 5 and 6, the amplitude of the horizontal loads was set as a ratio of the vertical loads, while in the case of frame 4, this amplitude was set to a constant value. In the case of all frames except frame 4, the ratio of the vertical loads and horizontal loads was set to 1:6 (value of one horizontal force as compared to the vertical load acting on the upper beam); for frame 4, the amplitude of the horizontal load cycles was set to a constant value of 36 kN (value of one horizontal force).

TABLE 2 LOADING PROGRAMMES

Programme description

Frame

amplitude of load cycles

Highest vertical load (kN)

vertical load

horizontal load

1

monotonic

monotonic

2

monotonic

cyclic

fixed ratio

208

3

monotonic

cyclic

fixed ratio

240

4

monotonic

cyclic

fixed value

240

5

monotonic

cyclic

fixed ratio

196

6

monotonic

cyclic

fixed ratio

200

193,4

397 280 -r— 240-

200 160 4 120 4 80 4 40 41

in frame 2

frame 4

frame 3

frames

frame 5

vertical ft)rce (kN) 200 ^

120 j 80 J 40 i (b) 1 -40

—h-30

\ -20

1

-10

9- 1

j

1

10

20

horizontal force (kN) 1 1 30

40

Figure 3: Loading programmes: (a) general programme in terms of vertical forces for frames 2 to 6 (columns represent cycles of horizontal loads); (b) vertical force versus horizontal force for frame 2 as measured

MEASUREMENTS During the tests we measured beam mid-span deflections and lateral storey displacements by using inductive transducers, forces by pressure transducers built into the oil circuit of the hydraulic system, strains by strain gauges and cross-section rotations by purpose-designed devices (Figure 4). The strain measurement was taken at the end cross-sections of beams and columns, by using ten gauges for each cross-section. The arrangement we applied was first used by Gibbons, Kirby and Nethercot (1991), and makes possible to derive the four cross-sectional internal forces (the axial force, the two bending moments and the bimoment) even in the case of plasticity occurring within a part of the cross-section (Figure 5). We did not, however, expect excessive plasticity in these cross-sections due to the fact that the beams and columns were connected by partial strength connections. We applied a special purpose-designed device referred to as rotation indicator for the measurement of the absolute rotations of cross-sections (Figure 6). This device consists of a spring steel connected rigidly on its top to the cross-section being examined. While the cross-section and the top of the spring element rotates, a weight connected to the bottom of this spring element tries to remain in

398

Figure 4: Locations of measurements: (a) measurement of displacements; (b) cross-sections where rotations and strains were measured 160 = 20 + 3x40 + 20

f

^ i?i

?i I

ii

y

260-10/160-8

Figure 5: The arrangement of strain gauges within a cross-section, after Gibbons, Kirby and Nethercot (1991)

htmM

strain gauges spring steel element -

0\O

-hole

©

weight box wall -

fastening plate

34

weight spring steel elennent box wail

^5=^ Figure 6: The rotation indicator. The "weight" element is a (t)50x67 solid element; the "spring" element is a 0,5x34x40 steel element, with a crosssection weakened to 7 mm^ at the strain gauges

399 place, which causes bending within the spring steel element. The deformations depend on the rotation to be measured, and are assessed by two strain gauges on the two surfaces of the spring steel element. The device is suitable for the measurement of rotations within the range of ±10 degrees. The device was calibrated for this range and was found to behave linearly; its coefficient, i.e. the ratio between the rotation and the difference of strains as measured by the two strain gauges, was in the range of 7.8 to 9.0 X 10'^ degrees per microstrains (the results come from the calibration of the fifteen rotation indicators constructed). Material properties of the applied steel was determined on the basis of tension tests, see Table 3. TABLE 3 MATERIAL PROPERTIES

Plate element

Yield strength (MPa)

Ultimate strength (MPa)

Thickness

Purpose

8 mm

web, flange

300

429

10mm

flange

276

384

10mm

end plate

320

444

12 mm

flange

274

382

ILLUSTRATIVE RESULTS It is not the purpose of this paper to analyse the results of the testing programme. It is worth mentioning, however, that the ultimate behaviour of the frames was characterised by excessive yielding within the joint zones (plastic mechanism), and in the beams below the vertical loads, and the failure was in nearly all instances due to the buckling of the compression flanges of the top beams below the vertical loads, and then by the initiation of lateral-torsional buckling in the top beams. For the sake of completeness. Figure 7 presents two diagrams related to frame 2. Further results will be presented elsewhere. Horizontal 40 load (kN) 30

Figure 7: Representative results of the experiments: (a) horizontal load versus top storey drift; (b) horizontal load versus top beam end rotation - both for fi'ame2

400

CONCLUSIONS The paper sets the objective to present the testing methods apphed during a test series consisting of six frames with similar geometry and, in five cases, under quasi-static cyclic loading. The testing methods presented in the paper prove to be suitable for the given purpose. Special attention is paid to a device, a new development, referred to as rotation indicator, which has been constructed to measure the rotations of cross-sections. Further work related to these tests will concentrate on the detailed analysis of the results collected. The focus will be to look at the connection behaviour, which in one sense is peculiar due to its asymmetry relative to the plane of the frame, and which will also be interesting from the point of view of the information available about behaviour of connections tested off-frame. Another main direction of the fiirther assessment of results will concentrate on the behaviour of the frame as a whole.

ACKNOWLEDGEMENTS This work is supported by the National Scientific Research Fund of the Republic of Hungary (OTKA) under grant No. T020358. The test arrangement and the device for the measurement of rotations was designed by Mr. Laszlo Kaltenbach, chief engineer at the Structural Testing Laboratory of the Technical University of Budapest. The measurement system was installed and operated by Dr. Miklos Kallo. The valuable contribution of both these persons, as well as other staff members of the Structural Testing Laboratory and the Department of Steel Structures, is gratefrilly acknowledged.

REFERENCES GEN. (1992). European Prestandard ENV 1993-1-1:1992 Eurocode 3: Design of Steel Structures. General Rules. General Rules and Rules for Buildings, European Conmiittee for Standardisation, Brussels, Belgium, April 1992 Davies J.M. (1996). The Stability of Steel Frames. General Report, Stability of Steel Structures, 1995, Budapest, ed. M. Ivanyi, Akademiai Kiado, Budapest, Hungary, Vol. 1, 377-392 Gibbons C., Kirby P.A. and Nethercot D.A. (1991). The Experimental Assessment of the Force Components within Structural Steel 'H' Sections. Journal of Strain Analysis, 26:1, 31-37. Halasz O., Ivanyi M. (1979). Tests with Simple Elastic-Plastic Frames. Periodica Polytechnica Civil Engineering (Budapest), 23:3-4, 151-182. Ivanyi M., Hegedus L., Ivanyi M. Jr. and Varga G. (1998). Failure Tests of Two-Storey Steel Frames, Journal of Constructional Steel Research, 46:1-3, 86. Ivanyi M. and Varga G. (1999). Full Scale Tests of Steel Frames under Quasi-Static Cyclic Loading. Paper submitted to the Eurosteel 99 Conference, Prague, Czech Republic, 27-29 May 1999 Mazzolani F.M. (1992). Stability of Steel Frames with Semi-Rigid Joints, Stability Problems of Steel Structures, CISM Courses and Lectures No. 323, ed. M. Ivanyi and M. Skaloud, Springer-Verlag, Vienna, Austria and New York, U.S.A., 357-415

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

401

INFLUENCE OF ASYMMETRY ON SEISMIC RESPONSE OF MOMENT-RESISTING STEEL FRAMES Damjan MARUSIC and Peter FAJFAR Faculty of Civil and Geodetic Engineering, University of Ljubljana, Jamova 2, SI-1000 Ljubljana, Slovenia

ABSTRACT In the paper results of a study of inelastic seismic response of five-storey steel moment-resisting frame buildings are presented. Two different torsionally stiff structural systems were investigated. For one structural system, a torsionally flexible variant was also studied. The symmetric structures were designed according to Eurocode 8. Asymmetry was introduced in plan by taking into account 5% accidental eccentricity, as proposed by Eurocode 8. In addition, larger mass eccentricity was also applied. Seismic response was determined by nonlinear dynamic analyses of spatial structural models subjected to six recorded ground motions. Excitation was simultaneous in two horizontal directions. For comparison, uni-directional excitation was also applied. Four different levels of ground motion intensity were used. In addition, nonlinear static ("pushover") analyses were performed. In the paper, the main results have been summarised and several conclusions have been drawn.

KEYWORDS Inelastic seismic response, Nonlinar dynamic analysis. Asymmetric building. Torsion, Steel frame. Moment-resisting frame. Pushover analysis, Eurocode 8. INTRODUCTION Inelastic seismic response of asymmetric structures is an important and popular research topic. Nevertheless, due to a very large number of parameters, which influence the response, the progress has been rather slow. In this paper, a small part of the research performed at the University of Ljubljana, aimed to contribute to a better understanding of inelastic torsional response, is presented. The effects of the structural system, the magnitude of mass eccentricity, and the intensity of ground motion on the maximum response of moment-resisting steel frame buildings are studied. In addition, the difference of the response of structures subjected to uni- versus bi-directional horizontal excitation is investigated. The objective is to contribute to the solution of a problem that is very important for the future development of practical design procedures: How to perform a simplified nonlinear analysis of asymmetric building structures? Such analyses, which are rapidly gaining popularity, combine a

402

nonlinear static (pushover) analysis with response spectrum approach. Simplified nonlinear methods have been developed for planar structures (e.g. Fajfar and Gaspersic, 1996) and their applicability has to be extended to asymmetric structures that experience torsional rotations. Here the problems arise where to apply the static horizontal loading and how to combine the influence of the two horizontal directions of loading. Answers to these questions are among the objectives of the research at the University of Ljubljana.

TEST STRUCTURES AND GROUND MOTION Three five-storey steel buildings are used as test structure. The symmetric structures were designed by Mazzolani and Piluso (1997) according to Eurocodes 3 and 8. The design spectrum for stiff soil normalised to peak ground acceleration 0.35g was used. The behaviour (reduction) factor q = 6 was applied. A conservative estimate of the natural period was made, which resulted in desigA base shear of about 10% of the total weight of the building. The first structure is a space moment- resisting frame (S, Figure 1) in which all the beam-to-column connections are moment-resistant. In the second structure, there are only few moment-resisting bays at the perimeter (Fl, Figure 1). In this structure, the cost of connections is reduced. On the other hand, the cross-sections of the columns carrying the lateral loads have to be increased. As a result, the displacement shape of Fl building resembles to a cantilever beam and not to a frame (Figure 5). According to Mazzolani & Piluso (1997), the expected seismic behaviour of S-building is superior to that of Fl-building. The third structure (F2) consists of the same moment-resisting frames as the second structure. However, the layout in the plan is different. In the new structure, the moment-resisting frames are located in the interior of the plan (Figure 1). Thus, the torsional stiffness and torsional resistance are greatly reduced. In such a way, a torsionally flexible structure was created. The fundamental period of torsional vibration amounts to 2.13 s, compared to 0.73 s in the original structure. Masses and initial translational stiffnesses of all structures are practically the same. Accordingly, the fundamental translational periods are almost equal: 1.25 s for S structure, and 1.26 s for F structures. Storey heights are 4 and 3.5 m for the first storey and other storeys, respectively. Beams are hot-rolled I-profiles. The sections of all columns in S structure, and of some columns in F structures are composed of two hot-rolled I or H-profiles welded along the longitudinal axis to form X-shaped section. Other columns are hot-rolled H profiles. Masses amount to 331 tons at the top storey and to 315 tons at all other storeys. Asymmetry is introduced in plan by taking into account 5% accidental eccentricity, as proposed by Eurocode 8. The influence of larger mass eccentricity is also investigated. Six different ground motion records (with 2 horizontal components each) obtained during the 1979 Montenegro earthquake (3 records), the 1994 Northridge earthquake (2 records) and the 1995 Kobe earthquake (1 record) are used. The components with the higher peak ground velocity are applied in Ydirection and the others in X-direction. Peak ground velocities in Y-direction (Vg) are scaled to the same value. Acceleration spectra are shown in Figure 2.

Fl

+ 't'

F2

I

i

't' V"

+ *v—*v—4*

*^

I

•J '•'

I

'•'

I

'•'

I

Figure 1: Plans of investigated buildings

J 4-

403

X direction

Kobe-JMA Newhall Sylmar Ulcinj 1 Ulcinj 2 Bar MMMMean ••••£C8

22-| 20181614-

Y direction '\\

121086420-

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Period [s]

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Period [s]

Figure 2: Acceleration spectra for 5% damp, (spectra for different ground motions, stronger comp. in Y-dir. scaled to Vg = 70 cm/s; mean spectrum; Eurocode 8 design spectrum without reduction) MATHEMATICAL MODELLING A pseudo three-dimensional global mathematical model, usually applied for linear analyses of multistorey structures, is used. The model is composed of planar frames connected with rigid diaphragms. The compatibility of axial deformations in columns, belonging to two frames (in two directions) is not considered. For structural elements, a relatively simple mathematical model, typical for steel frame structures, is applied. All beams and columns are modelled as planar elastic beam elements with two nonlinear rotational springs at the two ends. The moment - rotation relationship for each spring is assimied to be elasto-plastic with zero post-yield stiffness. Torsional stifnesses and strengths of all elements are neglected. An initial study (Marusic and Fajfar, 1999) has shown that influences of axial force - bending moment interaction in columns and of the second order theory are small. Thus, they have not been taken into account. Initial moments due to vertical loading were considered. Secondary beams are oriented in Y-direction. Consequently, the initial moments influence the frames in Xdirection. The damping matrix is proportional to mass matrix and initial stiffness matrix. The target damping is 5% in the first two modes.

PARAMETRIC STUDY A parametric study of dynamic behaviour of three different structural systems was performed with the CANNY program (Li 1996). The dynamic response to simultaneous ground excitations in two orthogonal horizontal directions was studied. The parameters, varied in the study, include eccentricity of the mass centre and the intensity of ground motion. In addition to the symmetric structures (no eccentricity), eccentric structures with 5, 10, and 15% eccentricity (the values are defined as the eccentricity divided by the dimension of the building in the plan and multiplied by 100) were analysed. The same eccentricity applies to both directions. Different intensities, defined by peak ground velocity Vg, were used. All accelerograms applied in Y-direction were scaled to the same Vg. Each accelerogram in X-direction was scaled with the same factor as the corresponding accelerogram in Y-direction. Vg amounted to 10, 40, 70, and 100 cm/s. In the first case (Vg = 10 cm/s), the structures remain in the elastic range. The extent of plasticity increases with increasing velocity. For all investigated cases (3 structural systems x 4 eccentricities x 4 intensities = 48 cases) dynamic analyses were performed for 6 input ground motions. Maximum values for each ground motion were determined. Finally, mean values and standard deviations of selected response quantities were computed. The same procedure was repeated for uni-directional ground excitation in X- and in Y-direction. In addition, pushover

404

TABLE 1 MEAN VALUES OF TOP DISPLACEMENTS FOR SYMMETRIC STRUCTURES [cm] Building

Vg= 10 cm/s X - dir. Y - dir. 0.040 0.053 0.042 0.056

S F1,F2

Vg = 40 cm/s X - dir. Y - dir. 0.152 0.196 0.155 0.220

Vg = 70 cm/s X - dir. Y - dir. 0.226 0.303 0.230 0.346

Vg= 100 cm/s X - dir. Y - dir. 0.298 0.379 0.318 0.438

X Design base shear Symmetric Asymmetric

10

20 30 40 50 60 Top displacement [cm]

10

20 30 40 50 60 Top displacement [cm]

70

Figure 3: Base shear - top displacement relationships for S and Fl symmetric and asymmetric Structures (10% eccentricity). analyses were performed. Horizontal loads with inverted triangular distribution were applied in mass centres, separately in X- and in Y-direction. Target displacement of the mass centre at the top of each building was equal to the top displacement in the symmetric structure. RESULTS Selected results are shown in Figures 3-6 and in Tables 1 and 2. The base shear - top displacement (at geometrical centre) relationships, determined by pushover analyses (Figure 3), show that the overstrength factor, i.e. the ratio between the actual and design strength, is large (about 2.5 to 3). Practically, there is no influence of eccentricity on stiffness and strength. In the case of S structures, there is some influence of initial moments due to vertical loading (compare X- and Y-direction, initial moments are larger in frames in X-direction). Maximum displacements at the top of symmetric structures (mean values) are given in Table 1. Asymmetric and corresponding symmetric structures are compared in Figure 4. The results correspond to four edges and to geometrical centre. In the legend, the designation of the building (e.g. S) is followed by the value of the peak ground velocity (e.g. 10). The values of the coefficient of variation, indicating the scatter of results obtained for 6 time-histories, vary from about 0.1 to about 0.5 with an average of about 0.3. Mean values of envelopes of displacements and storey drifts for two cases (structures S and Fl, 10 % eccentricity, ground motion scaled to Vg = 70 cm/s) are shown in Figure 5. In the same figure, the results obtained by pushover analyses are also shown. For the same structures, rotations at element ends in frames at the flexible edge in Y-direction are also presented (Figure 6). They correspond to Kobe-JMA ground motion, stronger component scaled to Vg = 70 cm/s. Only values larger than yield rotation are shown. Comparisons of results obtained by uni- and bi-directional input are summarised in Table 2. The uni-directional results in the upper part of Table 2 correspond to the larger of the two values obtained by performing two separate analyses with uni-directional ground motion in X- and Y-direction, respectively. In the bottom part of Table 2 both values, obtained by separate analyses with uni-directional input, are combined according to the SRSS rule, i.e. D=V(DX^+DY^). DX and Dy are mean values of top displacements obtained by applying ground

405

excitation only in X- and only in Y-direction, respectively. DXY corresponds to bi-directional input. Xc, Xstiff and Xflex correspond to displacements in X-direction in geometrical centre and at the stiff and flexible edge, respectively. Yc, Ystiff and Yflex are the corresponding values in Y-direction. TABLE 2 RATIO OF TOP DISPLACEMENTS DUE TO UNI- AND BI-DIRECTIONAL INPUT Max(Dx, DY)/DXY Xc ^g [cm/s] Xstiff 10 Xflex Yc Ystiff Yflex Xc Xstiff 40 Xflex Yc Ystiff Yflex Xc Xstiff 70 Xflex Yc Ystiff Yflex Xc Xstiff 100 Xflex Yc Ystiff Yflex

805 0.97 0.89 0.97 0.99 0.96 0.97 0.99 0.90 0.98 0.99 1.00 0.96 0.98 0.84 1.04 1.01 1.03 0.98 0.98 0.81 1.06 1.00 0.96 0.96

SIO 0.87 0.81 0.88 0.95 0.91 0.97 0.98 0.79 0.92 1.00 0.97 0.97 1.00 0.73 1.00 1.05 1.01 0.99 0.95 0.68 1.05 0.99 0.93 0.96

S15 0.73 0.85 0.75 0.94 0.83 0.94 0.90 0.77 0.86 1.00 0.86 0.99 0.93 0.74 0.90 1.02 0.84 1.00 0.92 0.63 0.96 0.97 0.85 0.93

F105 0.99 0.98 0.99 1.00 0.99 1.00 1.01 1.02 0.97 1.01 1.04 0.97 1.00 1.01 0.99 1.02 1.07 0.97 1.02 1.01 1.01 1.01 1.04 0.98

FllO 0.96 0.94 0.96 0.99 0.98 0.98 1.02 1.00 0.94 1.02 1.07 0.96 1.04 0.97 1.00 1.04 1.17 0.97 1.04 0.94 1.05 1.01 1.08 0.97

F115 0.92 0.84 0.90 0.97 0.97 0.97 1.04 0.96 0.92 1.03 1.05 0.97 1.08 0.88 1.02 1.02 1.10 0.98 1.08 0.88 1.01 0.99 1.08 0.96

F205 1.00 0.90 0.89 1.01 0.97 0.94 1.02 0.90 0.86 1.02 0.95 0.98 0.99 0.79 1.07 1.00 0.97 1.03 0.83 0.72 0.96 1.00 1.00 0.99

F210 0.89 0.81 0.78 1.00 0.93 0.85 0.97 0.89 0.86 1.00 0.93 0.87 0.94 0.80 0.97 1.01 0.98 0.96 0.92 0.79 1.04 1.02 1.00 0.99

F215 0.81 0.75 0.85 1.00 0.91 0.92 0.86 0.78 0.96 1.03 0.94 0.88 0.87 0.75 1.03 1.04 0.98 0.94 0.84 0.77 1.04 1.00 0.96 0.93

V(DX'+DY^ )/DxY

S05 0.98 0.98 1.01 0.99 1.00 0.98 1.01 0.99 1.03 0.99 1.04 0.98 1.00 0.91 1.09 1.02 1.08 1.00 0.99 0.86 1.11 1.00 1.00 0.98

SIO 0.99 1.06 1.03 0.98 1.04 1.00 1.08 1.02 1.07 1.04 1.12 1.01 1,08 0.90 1.17 1.09 1.16 1.03 1.01 0.82 1.18 1.01 1.05 1.00

S15 1

F105 0.99 1.00 1.00 1.00 1.00 1.00 1.01 1.04 0.98 1.01 1.05 0.97 1.01 1.02 1.00 1.02 1.08 0.97 1.02 1.03 1.02 1.01 1.05 0.98

FllO 0.99 1.01 1.00 0.99 1.00 0.99 1.04 1.08 0.98 1.02 1.09 0.97 1.05 1.04 1.03 1.05 1.20 0.98 1.05 0.99 1.08 1.01 1.11 0.98

F115 1

0.95 1.10 0.99 1.02 1.05 1.01 1.15 1.01 1.11 1.08 1.08 1.05 I.IO 0.96 1.15 1.09 1.08 1.06 1.03 0.82 1.16 1.03 1.06 0.99

F205 1.02 1.00 1.11 1.01 0.99 1.01 1.03 0.97 1.02 1.03 0.97 1.04 0.99 0.86 1.24 1.01 0.99 1.08 0.84 0.80 1.10 1.07 1.09 1.09

F210 0.99 0.98 0.99 1.06 1.02 1.04 1.04 1.01 1.13 1.03 1.00 1.07 0.99 0.93 1.29 1.03 1.05 1.12 0.95 0.93 1.31 1.03 1.06 1.11

F215 1.00 0.95 1.04 1.15 1.06 1.21 1.05 0.98 1.19 1.13 1.08 1.16 0.99 0.91 1.34 1.10 1.08 1.18 0.93 0.94 1.34 1.04 1.05 1.13

Vg

[cm/s] 10

40

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!

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Figure 6: Maximum rotations at element ends (in lO""^ rad) of symmetric and asymmetric structures (10% ecc.) for Kobe-JMA ground motion (bi-directional excitation, Vg = 70 cm/s) OBSERVATIONS AND CONCLUSIONS Based on the results of the study, several conclusions can be drawn. The numerical values given in the conclusions apply only to the investigated structures. The qualitative observations, however, are considered to be generally valid. They mostly confirm the conclusions obtained within our research on single-storey and other multi-storey buildings. Similar conclusions have been obtained also by some other researchers. Once again, it has been confirmed that asymmetry may substantially increase the seismic demand at edges. The global behaviour of the structures, designed according to Eurocode 8 (without taking into account asymmetry, but with substantial overstrength), suggests that the appropriately detailed buildings would in most cases survive ground motions, which are more severe than design ground motion. Some more detailed observations and conclusions are as follows. Asymmetric versus symmetric structures The displacements at the centre of asymmetric buildings are in the range from 80 to 105% of the corresponding displacements of symmetric buildings. The difference increases with the increase in eccentricity. It is interesting to note that, in the case of inelastic behaviour, the displacements of asymmetric buildings are almost always smaller than the displacements of symmetric buildings. The behaviour of torsionally stiff and torsionally flexible buildings is qualitatively different. In the case of torsionally stiff buildings, the influence of torsion generally increases with an increase in eccentricity. However, the relation is not linear. In general, the influence of torsion decreases with an increase of the intensity of ground motion. The influence is the largest in the elastic range. As regards to structural systems, the largest influence of torsion can be observed in the torsionally flexible system. In this

408

system, the increase of top displacements due to torsion amounts to almost 100% (in the most critical case). Note that displacements are measured at the perimeter. At the location of structural elements the increase due to torsion is much lower. Within the group of torsionally stiff buildings, larger influence of torsion can be observed in S (space moment-resisting) structure. In this structure, the maximum increase of top displacements due to torsion amounts to about 23, 40, and 50% for 5, 10, and 15% eccentricity. Uni- versus bi-directional excitation Table 2 demonstrates that, in the majority of cases, a fair (mostly slightly underestimated) estimate of displacements can be obtained from uni-directional ground motion. In general, the estimates based on uni-directional ground motions can be improved if the SRSS rule is applied. This rule has been widely used for elastic structures. In our study it provides in most cases slightly overestimated results. Pushover versus dynamic analysis In the case of torsionally stiff structures, pushover analysis yields qualitatively correct results. If the loading is applied in mass centre (separately in each horizontal direction), pushover analysis usually underestimates the torsional effects at the most critical edge (see Figure 5). This error can be eliminated or at least reduced by increasing the eccentricity of the static lateral loading (as is usual in the case of linear analyses according to codes). Pushover analysis does not take into account the higher mode effect and thus underestimates the seismic response in the upper part of the structures. In the case of torsionally flexible buildings, the dynamic and static responses are qualitatively different. The study has not been concluded yet. Influence of semi-rigid connections and influence of stiffness and strength eccentricity are still under investigation. ACKNOWLEDGEMENTS The results presented in this paper are based on the work supported by the Ministry of Science and Technology of the Republic of Slovenia and by the INCO-COPERNICUS project RECOS (coordinated by F.Mazzolani).

REFERENCES Fajfar P. and Gaspersic P (1996). The N2 method for the seismic damage analysis of RC buildings. Earthquake Engineering and Structural Dynamics 25, 31-46. Li, K.N. 1996. Three-dimensional nonlinear dynamic structural analysis computer program package CANNY-E. CANNY Consultants Pte Ltd, Singapore. Marusic D. and Fajfar P. (1999) Comparative study of seismic response of steel frame structures. Proceedings EURODYN'99, Balkema, Rotterdam. Mazzolani F.M. and Piluso V. (1997). The influence of the design configuration on the seismic response of moment-resisting frames. Behaviour of steel structures in seismic areas: STESSA '97 (F.M.Mazzolani and H.Akiyama, eds.), Edizioni, Salerno.

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivdnyi, editors © 1999 Elsevier Science Ltd. All rights reserved

409

SEISMIC RESPONSE OF MR STEEL FRAMES WITH DIFFERENT CONNECTION BEHAVIOURS G. De Matteis ^ R. Landolfo ^, F. M. Mazzolani * & L. A. Fulop ^ D. Dubina ^ ^ Dept. of Structural Analysis and Design University of Naples Federico II, Naples, ITALY ^ D.S.S.A.R., University of Chieti G. D'Annunzio, ITALY ^ Dept. of Steel Structures and Structural Mechanics Politehnica University of Timisoara, ROMANIA

ABSTRACT The present study aims at investigating the influence of the connection behaviour on the seismic response of moment-resisting frames. In particular, with reference to both dual joint and homogeneous structural configurations, the influence of the building structural layout is pointed out through several time-history dynamic analyses, which are referred to six different natural records. Both space moment resistant frames and perimeter moment resistant frames are considered. In both cases, parametrical analyses are concerning with moment resistant frames where members are assembled with different combinations of rigid, semi-rigid and pinned joints. Besides, in every case, several joint rigidity and resistance levels are accounted for. The results, summarised in terms of ductility demand as a function of the peak ground acceleration, allow the effect of the joint semi-rigidity on the global behaviour of the structure to be emphasised. Finally, the variation of the q-factor as well as the distribution of the structural damage within the structure for all examined schemes are outlined.

KEYWORDS Moment-resisting frames. Semi-rigid joints. Structural typology. Seismic behaviour. Dynamic analyses, Ductility demand, q-factor.

INTRODUCTION Moment-resisting frames are more and more used in the field of steel structure buildings as a suitable and convenient solution resisting to both vertical and horizontal actions (Mazzolani & Piluso, 1996). They may be preferred to alternative structural schemes where pin-jointed steel frames are stabilised against lateral forces by means of purposely designed bracing systems (usually, steel diagonal bracing.

410 concrete core and concrete walls). Framed structures allow, in fact, the optimisation of architectural space, also providing higher dissipative features in case of earthquake loading. Middlemost these two solutions, dual structures, where horizontal loading are resisted in part by moment resisting frames and in part by bracing systems acting in the same plane, must be mentioned. In seismic applications, in order to maximise lateral stif&iess and system energy dissipation capability, traditional space moment-resisting frames present all the beam-to-colimm connections which are moment-resisting full-strength type. In fact, this solution should provide the greatest nimiber of dissipative zones which, according to the strong column - weak beam design philosophy, have to be located at the beam ends (ENV 1998, 1994). Obviously, since moment-resisting connections are more expensive than shear connections, a more economical design can be achieved by reducing the number of such kind of connections. The study of different design criteria is therefore now in progress, in order to verify the possibility to adopt even better solutions. Aiming at optimising the structural configuration, as a first attempt, building layouts where some frames, interior and/or perimeter, have a reduced number of bays with moment-resisting connections could be considered. In particular, because of their economical benefits, in remaining bays either simple connections or semi-rigid joints could be employed. This corresponds to an alternative category of dual system, where, for each frame, duality must be intended in terms of joint mechanical properties. This structural configuration has became almost popular in United States and Japan (Shen, 1996, Kishi et ai, 1996) and has been frequently used as seismic-resistant system especially in Southern Califomia. Another way would be the employment of semirigid joints everywhere in the structure. On the other side, the possibilities of combining economy and seismic performance through the above mentioned alternative design philosophy, which is essentially based upon the use of semirigid joints, have not yet been completely established. In fact, the presence of not fUlly restoring joints could affect the global performance of the structure under seismic loading. For instance, one of the most important concerns is the difficulty to yield the serviceability conditions imposed by technical codes, as much as the number of rigid connections is reduced. Two main effects can be identified: the reduction of lateral stiffness, which reduces the performance of the structure under horizontal loading, and the increase of the period which is beneficial due to the reduction of the effect of the earthquake on the structure. The objective is, to try to understand the combined effects of these two influences, contemporary exploring for the possible convenient key of economy and seismic performance under both serviceability and ultimate conditions.

EXAMINED CASES The influence of the structural layout on the inelastic response of moment resisting frames have been examined through inelastic dynamic analyses, which have been carried out by means of the general-purpose computer program DRAIN 2D-X (Prakash et al, 1993). The symmetric 3-bay 5-storey building designed by Mazzolani & Piluso (1997) has been considered, by analysing the response of the single frame (Figure 1). Member sizes, which are have been carried out according to ECS (ENV 1998, 1994), by considering stiff soil conditions, a peak ground acceleration equal to a^0.35g and a design behavioural factor qd6. Member hierarchy criterion has been therefore applied. As far as the serviceability conditions are

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411 concerned, the design value of the inter-storey drift has been limited to /2/250 (where Joint Stiffness Joint Moment Capacity STRUCTURAL h is the inter-storey height). CONFIGURATION Despite this less restrictive Pinned Semirigid Semirigid Pinned limitation, the design has (S**) (P) (P) rs**) been governed by Dualjoints 1.2 (S*l) 1 (SI*), 1 (S*2), 0 0.8 (S2*) serviceability conditions 0.8 (S*3) 0.6 (S3*) rather than by member 0.6 (S*4) strength requirements. Rigid Semirigid Semirigid Rigid With reference to the above (R) (R) rr**) m frame structure, in order to Homogeneous joints 1 (Tl*), 1.2 (T*l) 00 00 1 (T*2) investigate the effect of joint 0.8 (T2*) 0.8 (T*3) 0.6 (T3*) behaviour on the global 0.6 (T*4) structural response, several (Symbols in brackets indicate nomenclature usedfor contracted names) behavioural parameters for beam-to-column connections have been considered, by using both semirigid and pinned joints as alternative to rigid joints. As a first step, semirigid joints have been located in the middle bay only, obtaining a dual joint structural configuration (Figure la). In the following, these configurations are labelled vAih the letter '5". For the sake of comparison, configuration where central bay is characterised by pinned joints T * has been considered too. In the second step semirigid joints are present everywhere in the structure, this corresponding to a homogeneous joint structural configuration (Figure IV), In the following, these configurations are labelled as T . For the sake of comparison, the configuration where all joint are fiilly rigid and total strength 'R' has been considered too. Therefore, the seismic performance of four structural schemes have been determined and compared each other. TABLE 1

EXAMINED CASES

In all cases, several joint rigidity and resistance levels have been considered. With reference to the stiffness, joint rigidity values {kj) equal to kum, 0.8 kum and 0.6 kum have been assumed, where kim is the joint rigidity value corresponding to the boundary between rigid and semirigid joint behaviour as suggested by the above code for unbraced frames (ENV1993, 1998). In the following, the semirigidity level is indicated by the first number of contracted names. Therefore, SI* and Tl* refer to ^ = 1 kum, S2* and T2* refer to kj = 0.8 kn^n and S3* and T3* refer to kj = 0.6 kn^. At the same way, several moment capacity levels (Mj) have been considered, by analysing both partial strength andftiUstrength joints. Values have been referred to moment capacity of the connected beam (Mb). M/ Mb ratios equal to 1.2, 1, 0.8 and 0.6 have been considered. In contracted names, moment capacity level is associated to the second number, by using S/T*l, S/T*2, S/T*3, S/T*4, in a decreasing order for resistance level. All examined cases in terms of both joints stiffness and moment capacity level are summarised in Table 1.

a) dual Joint structural configuration (S-P)

b) homogeneous joint structural configurations (T-R)

Figure 2: Analysed frame configurations

412 ANALYSIS METHODOLOGY General The performed parametrical study has been carried out by means of 6240 time-history analyses. The structural performance of 26 structural typologies under 6 earthquake records has been examined, by considering progressively increasing peak-ground acceleration values, starting from O.OSg to 2.0g. The value O.OSg has been chosen because of, according to the design procedure, the first yield of the structure theoretically should appear at 0.35/^^0.058g, so probably at this step the structure should behave elastically. The value of 2.0g has been chosen after a preliminary analysis, by checking that at this earthquake level the structure has reached a conventional collapsing state, as stated below. Integration step equal to 0.01s has been adopted, after checking that unbalanced forces produce numerical errors less than 10% in term of the plastic rotation demand to member sections. Selection of records In order to take allowance for the seismic input, six records corresponding to different historical earthquakes have been considered. Accelerograms have been selected in such a way the corresponding elastic spectra was similar to the one assumed by Eurocode 8 (ENV 1998, 1994) for subsoil class A. In order to eliminate the differences in terms of peak ground acceleration, all the records have been scaled at the same acceleration value. In Figure 3, in case of viscous damping coefficient equal to of 5%, normalised elastic response spectra of the examined earthquakes are depicted together with the design one (EC8-class A). The figure allows the shape and the frequency content of analysed spectra to be compared each other. It is clearly shown as the average shape of analysed earthquakes is very similar to the one assumed by the code, especially for vibration periods higher than 0.5s. Dispersion factor of spectral amplifications of each record respect to the design one, has been evaluated in the period range 1.4s-4.0s, which should be representative of the structural behaviour all over the seismic event, being the ftmdamental period of vibration of all examined frame configurations ranging between 1.4s and 1.5s. Percentage values of standard deviation for examined records are reported in Table 2 together with the one corresponding to the average spectrum. Dispersion among earthquakes seems to be acceptable. In fact, standard deviation value is about 10% for the average. Nevertheless, one should mention that for some earthquakes, namely Shandon and Tolmezzo, it is higher than 20%. TABLE 2 STANDARD DEVIATION (%) OF ELASTIC RESPONSE SPECTRA

El Centra Kobe 6.98 13.60

Korinthos Shandon 12.91 23.11

Tolmezzo Vrancea 25.98 17.13

Average 10.19

Adopted collapse criteria In order to compare the ultimate structural performance of analysed structures, collapse criteria should be preventively defined allowing ultimate peak ground acceleration values of considered records to be determined. A local criterion based on the comparison between plastic rotation supply and plastic rotation demand in either the most engaged member section or beam-to-column connection has been applied. In order to account for the structural damage due to repeated actions, two different simplified approaches have been adopted: the first one is simply based upon the maximum plastic deformation demand, the second one is an energy-type approach concemed with the cumulated plastic deformation computed along the time-history analysis. Conventionally and conservatively, in both cases, in order to asses the collapsing state, plastic rotation demand may be then compared with plastic rotation supply deduced on the basis of either experimental results or analytical methods.

413 each structure and for each record, jleration multiplier versus maximum tic rotation curves have been efore plotted. They have been rred to the most engaged plastic IQ among those formed in beam ions, column sections and lections, by considering both :imum plastic excursion and lulated plastic excursion. These ^es can be simply used for jrmining the maximum peak ground jleration that the structure is able to 4.0 istand when the available plastic rotation capacity for beam sections, TIs] column sections is fixed. Consequently, Figure 3: Elastic spectra of examined earthquakes actual q-factor of the structure may be promptly defined as the ratio between the acceleration level corresponding to the attainment of conventional collapse (Ouit) and the yielding acceleration) corresponding to the appearance of the first plastic hinge in the structure (o^). In the current study, in order to determine the ductility supply, the stable part of available plastic rotation capacity (Rst) for member sections have been assessed as the average of values suggested by the theoretical method proposed by Mazzolani & Piluso (1996) and the empirical one proposed by Mitani & Makino (1980). Besides, in order to account for the unstable part in the evaluation of total of rotation capacity (R), the simpUfied method proposed by Gioncu & Pectu (1997) has been considered too. Corresponding ultimate plastic rotation values 9^/ and Qput are summarised in Table 3, where rotation supply has been computed for each member section. Then, as final values of available plastic rotations, minimum values, which correspond to section IPE 450 for beam and to section 2xIPE 550 for columns, have been considered. TABLE 3 MEMBER PLASTIC ROTATION SUPPLY ( R A D )

Member Section IPE 450 IPE 400 2XIPE 550 2XIPE450

Mazzolani'Piluso method 0.0610 0.0663 0.0379 0.0618

Stable part (^uiS Mitcmi'Makino method 0.0513 0.0529 0.0353 0.0372

Average

Total (0^/) Gioncu *s method

0.0562 0.0596 0.0366 0.0495

0.0837 0.0850 0.0415 0.0747

The problem of determining available plastic rotation capacity for beam-to-column joints is more complicated due to the lack of both satisfactory theoretical approaches and comprehensive experimental results. Usually, plastic rotation supply rangingfi-om0.02 to 0.03 rad are considered to be realistic values (Tsai & Popov, 1988) and they are generally assumed for performing seismic analyses by checking that they are not exceeded by maximum plastic excursions experienced all over the deformation history. On the other side, further studies suggest to relate the coimection rotation capacity to the beam depth, also providing in some cases higher values than the ones above mentioned (Roeder & Fountch, 1996). Anyway, once this value has been established, the collapse state of plastic hinge involving the connection may be assessed through plotted 0^/ versus PGA curves. Finally, in order to account for a global parameter also in estimating the structure capability to

414 withstand increasing intensity earthquakes, interstorey drift criterion has been applied as well, by assuming a ultimate value equal to 0.03/?, where h is the interstorey height. THE OBTAINED RESULTS The influence of joint resistance The performed numerical analysis has produced thousands results, allowing all selected parameters to be investigated in terms of influence on the seismic structural performance of the building. Herein, only the main results are shown and discussed, while a more comprehensive examination is reported by Fulop (1998). In Figure 4, the influence of joint strength is shown in case of dual joint structural configuration and rigid beam-to-column connections {k/kum-l). Plotted curves represent the average of rotation demand for all examined earthquakes. It can be seen the increasing influence of the connection as the moment capacity decreases, while no significant modification of the plastic rotation requirement in plastic hinges on the beams and columns can be observed. Obtained results show that generally beam sections present the highest plastic rotations, unless partial strength connection where Mj=0.6Mb are considered. This is due to the effect of dual joint configuration, which induce a concentration of plastic rotation within members of outer bays rather than in the central one. Really speaking, this does not mean that the collapse of the fi-ame is always due to the beam (*beam controlled collapse'), because of the different ductility supply of beam sections, column sections and connections. For instance, collapse of frame types SI 1 and S12 seems to be ascribed to the column sections rather than beams, being rotation requirements slightly higher for beams than for columns and, concurrently, rotation capacity supply FRAME r y P E S11

FRAME T / P E S12

Accelerogram multiplier

Accelerogram multiplier

FRAME TYPE S14

FRAME TVPE S13 0.10 ^ 0.09 1

9-

0.08

£

0.07

O

0.06

^

0.06

1 "•'" |g

0.03-

E

0.02 0.01 0.00

Accelerogram multiplier Cumulated rotation I Beam section A Column section O Connection

•frW'^^T^^

1—1—\—1—1—1—1—1—1—1—1—\—

8 ? R ^ § 8 8 S g 8 8 S a 8 § S 8 S g Accelerogram multiplier Maximum rotation I Beam section A Column section # Connection

Figure 4: The influence of joint resistance for dual joint structural configurations

415 FRAME TYPE T13

FRAME TYPE T12

•g 0.08-

2

^ 0.07 c ;2 0.06 f

0.05-

O 0.04 (0 0.03 ^ 0.02 0.01 • 0 00-

• >tTir ^ ^ § Accelerogram multiplier Cumulated rotation

D Beam section A Column section O Connection

8 8 o o o o

S S P § 8 8 S S 8 § g ai

O

O

O

O

•r^

r^

1-=

-r^

•»-• 1-'

Accelerogram multiplier Maximum rotation I Beam section A Column section > Connection

Figure 5: The influence of joint resistance for homogeneous joint structural configurations substantially lower for the latter. Similarly, in case of frame type S13, the collapse is ruled by connections if they have a limited rotation capacity respect to beam and column sections. Structural performances are clearly worst in case of homogeneous joint configuration, especially for partial strength connections. In Figure 5, plots for Mf=lMb and Mj=0.8M, are represented. It is clear that, in both cases, connections overrule the collapse of the structure. In figure 6, in case of dual 0.10 the joint configurations, Shewvbfv Koritr^hi>i' Vtamecu Kiobe^ 0,09 elCe^tforequired rotation capacity for 0.08 connections and column 0,07 sections providing a collapse 0.06-1 ruled by beam sections is 0.05-1 plotted. It allows the minimum 0,04 connection and column 0.03 rotation capacity supply in 0,02 0,01 order to have collapse due to 0.00 the rotation capacity Umitation of the beams. The above figure is referred to the case of I Beam section D Column section i Connection structures with joint stiffness Figure 6:Required plastic rotations for 'beam controlled collapse' kj= him and joint moment capacity Mj=lMb, O.SMfe, and 0.6Mb, respectively, while beam rotation capacity has been Hmited to the stable part. It is possible to verify that only in case of partial strength connections, namely Mj-O.^Mb and 0.6Mh, required plastic rotation for connections are higher than 0.03 rad. Besides, generally, connection rotation capacity demand decreases as far as the joint stiffness decreases, emphasising that semirigid connections could be particularly effective in dual joint configurations. Finally, it is to be observed the influence of the seismic input on the collapse mechanism, being the required plastic rotation in columns very high in case of Korinthos, Shandon and Vrancea records. The comparison between dual and homogeneous joint structural configurations [n Figure 7, the structural performance is synthesised for all examined structures and earthquakes. In ?act, for each adopted collapse criterion (exhausting of available stable rotation capacity, exhausting of ivailable total rotation capacity, interstorey drift higher than 3%), the corresponding values of ultimate iccelerogram multipMer are reported. Conventionally, for all the connections, available plastic rotation

416 capacity values have been fixed equal to 0,045 rad. Depicted graphs allows the influence of all main behavioural parameters to be easily recognised with respect to both the accelerogram type and the collapse criterion. Obtained results seem to be quite homogeneous for all examined collapse criteria, leading to slightly increasing resistance values going from stable rotation capacity to interstorey drift limitation. On the contrary, record typology has a strong influence on the evaluation of structural performance, resulting a the ratio up to three between ultimate acceleration of best behaving earthquake (Shandon) and the worst (Vrancea). Anyway, in all cases, analysed structures are able to withstand all examined earthquakes up to peak ground accelerations much higher than the one assumed for building design (^^=0.35 g). Collapse criterion: stable part of available rotation capacity

ipiiiiiiliS

uim

R S11 S21 S31 S12 S22 S32 S13 S23 S33 S14 S24 S34 P

T11 T21 T31 T12 T22 T32 T13 T23 T33 T14 T24 T34

Frame type

Frame type

Collapse criterion: total available rotation capacity

iiiiimm

R S11 S21 S31 S12 S22 S32 S13 S23 S33 S14 S24 S34 P

Til T21 T31 T12 T22 T32 T13 T23 T33 T14 T24 T34

Frame type

Frame type

Collapse criterion: interstorey drift limit (3%)

1^7^^-^-KXI V

'~ai • ^

/ \

c o

UUpljtJfl/'lI P l

•fl)

u o 2

o.aoo

ra

.1

I|H|H|H|H|W|H|W|BH|M|H

0.000

R S11 S21 S31 S12 S22 S32 S13 S23 S33 S14 S24 S34 P

H

1 H 1 M 1 B 1 W 1 i

Frame type

lAverage

-Tolmezzo

L-H'

Til T21 T31 T12 T22 T32 T13 T23 T33 T14 T24 T34 Frame type

-Vrancea

a) dualjoint configurations

-El Centre

-Kobe

-Korjnthos

-Shandon

b) homogeneous joint configurations

Figure 7: Intensity earthquake structural resistance of examined frames

417 The comparison between dual and homogeneous configuration shows that generally structural performance is better in the former case. This is particularly evident if reference to interstorey drift limitation criterion is made. Corresponding graphs emphasise lateral stiffness reduction effect due to semirigid joints. Besides, obtained results confirm the important rule of connection moment capacity, especially in case of homogeneous joint configuration, generally providing undermining of seismic response for weaker connections. As a consequence, in case of partial strength connections, in homogeneous joint configurations, connection ductility became the key parameter controlling the collapse of the structure, while in dual joint configurations beam sections of outer bays firstly exhaust their available rotation capacity. Truly, for a connection plastic rotation capacity equal to 0.045 rad, there is no significant drop in the performance for partial strength joints, if reference stable part of rotation capacity is made. This is because the plastic rotation capacity of the joint is comparable with the one of the beam, 0.045 rad against 0.0562 rad. If total rotation capacity is considered, the differences are much more significant due to the bigger difference between the two values, 0.045 rad and 0.0837 rad. Nevertheless, results are sometimes contradictory, showing an increasing of structural performance in case of moment capacity Mj=O.SMb, but this outcome is strongly affected by the individual earthquake record. On the contrary, within a structural typology, for both dual and homogeneous joint configurations, the reduction of connection stiffness does not negatively affect frame structural performance. Conversely, dual joint configurations show a slightly increase of ultimate accelerogram multiplier as connection rigidity decreases, but result dispersion due to record type in not negligible. This is due to two main beneficial effects: firstly, the shifting of the structure toward higher vibration periods; secondly, the moving of damage localisation from the central bay to the outer ones, where connection are rigid and full strength and the collapse is ruled by beam element sections. The first of this effect is also significant in case of homogeneous joint configuration when interstorey drift criterion is adopted. In Figure 8, in order to examine the different localisation of damage within frame members and connections, for both dual and homogeneous joint configurations, global damage index distribution is depicted. It is referred to Vrancea earthquake at peak ground acceleration equal to O.Sg. For each element (member sections and connections), local damage index IDJ. has been evaluated as the ratio between the attained plastic rotation 0^ and the stable part of plastic rotation supply 0^/,^^. Then, according to Park et al (1985), a global damage index ID,G within the whole structure and in each bay ID.GJ has been computed as the weighted average of the local damage indexes, by using the following relationship type:

Z^l.M

/z.,a=^7^

(1)

It can be clearly noticed as in homogeneous joint configurations global damage indexes strongly increase as far as connection resistance decreases, while they slightly decrease as far as the connection stiffness decreases. With reference to dual joint configurations, the connection behaviour influences damage distribution within the structure only in case of very poor connections, i.e A^=0.6A/6. In fact, in all the other cases, damage is concentrated in outer bays only, where connections are rigid and full strength. Consequently, global damage index is substantially less than the corresponding one in homogeneous joint configurations. The effect of structural overstrength Finally, a meaningfiil aspect to be mentioned is the effect due to overstrength. In Figure 9, q-factor values determined through the ratio between yielding acceleration and ultimate accelerogram multiplier evaluated with reference to stable part of plastic rotation supply are depicted. Over and above the influence of the record, it can be observed that higher q-factor values correspond to frames with partial

418

strength connections, even if they provide a worst structural response in terms of maximum intensity earthquake that the structure is able to withstand (see Figure 7).

( O ( o c o c o a > ( o c o ( 0 0 } ( 0 ( o c o

a) dualjoint configurations

b) homogeneous joint configurations

Figure 8: Damage distribution among bays This contradictory result evidences that q factor, as it is usually defined, is not enough to describe by itself the seismic performance of the structure, another fundamental parameter to be considering being the structural overstrength p. It can be defined as the ratio between the actual yielding acceleration ay and the theoretical one a^y = a^ /q^ = O.OSSg. For the sake of example, obtained values for dual joint structures are presented in Figure 10a, where an important overstrength can be observed in each fi-ame typology. In particular, results are remarkably influenced by the examined record, varyingfi-om7 to 1.8 in case of rigid full-strength joints.

•'*>

0

o-""*

•«

~

Ck.

—A ^•"""-1^

A

A

6

A

*

^

^

T11 T21 T31 T12 T22 T32 T13 T23 T33 T14 T24 T34

R S11 S21 S31 S12S22S32S13S23S33S14S24S34 P Frame type

I Ave rage

-Tolmezzo

-Vrancea

a) dualjoint configurations

Frame type -El Centre

.Kobe

-Korinthos

-Shandon

b) homogeneous joint configurations

Figure 9: q-factor values for examined fi'ames Structural overstrength is basically caused by two effects: the fu-st is due to differences between the design spectra and the actual one corresponding to the examined record; the second one is an intrinsic factor of the structure, being imputable to the design procedure. In particular, this intrinsic factor is remarkable for the analysed structure, due to the following aspects: (1) the effect of seismic forces on the structure have been evaluated by means of a simplified analysis, while numerical time-history analysis account for actual mass modal participation; (2) in design evaluation of base shear force, reference to the approximate fundamental vibration period suggested by EC8-Part 1.2-Appendix C has been made. Since the effect due to earthquake response spectrum may be easily calculated as ratio of

419 spectral ordinate at the fundamental period of the stmcture for the considered record and the design value, in Figure lOZ^, overstrength rate due structural design is presented. Even if results are much less scattered, one can see that intrinsic overstrength of the structure due to the design remains remarkable and tends to 1 only in case of very partial strength connections. Similar conclusions may be drawn up for homogeneous joint structural configurations.

1 .^ Ifejtnft-w-'

1 Ba 1BB 1 IB 1 BB1 IW|M|W|M|m|m|M|M|m|ml R S11 S21 S31 S12 S22 S32 S13 S23 S33 S14 S24 S34

R

P

S11 S21 S31 S 1 2 S 2 2 S 3 2 S 1 3 S 2 3 S 3 3 S 1 4 S 2 4 S 3 4

lAverage

-Tolmezzo

P

F r a m e type

F r a m e type

- E l Centre

-Vrancea

-Kobe

-Korinthos

-Shandon

a) total overstrength b) intrinsic structural overstrength Figure 10: Overstrength for dual joint structural configurations Actually, once the overstrength has been computed, the actual structural seismic performance may be quantified through the coefficient ^'= p • cty/a^it, which may be intended as a modified q-factor value. Corresponding values are reported in Figure 11. It can be observed that now partial strength connection structures provide the worst performance, confirming that this factor is able to correctly quantify the ultimate resistance of the structure. It is also important to observe that results show that analysed structures are able to withstand earthquake whose peak ground acceleration is up to 25 times higher than the one that is expected to produce the first yielding. Only in case of the most damaging earthquake (Vrancea), such modified q-factor is similar to the one which has been hypothesised in design (qd=6), being in this case total overstrength value almost equal to one. This result emphasises the high ductility capability of moment-resisting structures independent on the joint behavioural parameters, provided that member and connections guarantee significant local ductility supply.

»|BM|iW(WI|BH|«M|im|Mjm|IW|BHI|BW|lW| R S11 S21 S31 S12 S22 S32 S13 S23 S33 S I 4 S24 S34

Frame type ^^lAverage

0 |iiW|Wi|™|™|Waiw«|HW|iwi|W«i|i— P

i w i i

T11 T21 T31 T12 T22 T32 T13 T23 T33 T14 T24 T34

Frame type

—D—Tolmezzo —A—Vrancea —>t—El Centre —o—Kobe —•—Korinthos

—A—Shandon

a) dualjoint configurations b) homogeneous joint configurations Figure 11: modified q-factor values for examined frames

420

CONCLUSIONS A wide parametrical analysis has been performed in this study. It has allowed important outcomes on the influence of connection behaviour on the seismic response of moment-resisting frames to be drawn. (1) Generally, when semirigid of partial strength connections are employed, dual joint structural configuration behave better that analogous homogeneous joint configurations. (2) The effect of semirigd joint on the global structural behaviour is not strongly influencing. (3) Connection moment capacity is able to remarkably undermine the seismic performance of the structure in homogeneos joint configurations. In such a case, joint ductility is the key parameter, having to be defined in order to be consistent with plastic rotation demand, which is a ftinction of the structure itself as well as of the seismic records; on the contrary, dual joint configurations, allowing the major damage to be concentrated in frame bays where rigid, full strength connections are employed, may lead to a Hmited penalisation of the frame behaviour due to the employment of poor connections. (4) The structural overstrength for usual frame structures designed according to recent code provisions is considerable and should not be disregarded in analysing results coming from numerical results; besides, it covers the essence of many other effects, which should be investigate on properly designed structures. ACKNOWLEDGEMENTS The current study has been carried out in the framework of the INCO-COPERNICUS Joint Research Project ''Reliability of Moment Resistant Connections of Steel Building Frames in Seismic Areas'' (RECOS), financed by EC, which is also gratefully acknowledged for supporting the L. Fulop's stage in Naples, within Tempus Project JEP 011297 ''Experimental Analysis^'. REFERENCES ENV 1993, Eurocode 3, (1994). Joints in Building Frames, CEN/TC250/SC3-PT9. ENV 1998, Eurocode 8, (1994). Design Provisions for Earthquake Resistance Structures, Commission of the European Communities. Fulop, L. A., (1998). Seismic Performance of Moment-Resisting Frames with Semirigid Joints, Diploma project. University of Naples Federico II, Univesitatea Politehnica Timisoara, Supervisors: Prof F. M. Mazzolani and Prof D., Dubina, Co-ordinator: Dr. G. De Matteis. Gioncu, V. and Pectu, D., (1997). Available rotation capacity of wide flange beam-columns. J. Constructural Steel Research, Vol. 43 (1 -3), 161 -217. Kishi, N., Chen, W.F., Goto, Y. and Hasan, R., (1996). Behaviour of tall buildings with mixed use of rigid and semi-rigid connections. Computer & Structures, No. 6, Elsvier Science Ltd, 1193-1206. Mazzolani, F.M. and Piluso, V (1996). Theory and design of seismic resistant steel frames, Chapman & Hall, London, UK. Mazzolani, F.M. and Piluso, V. (1997). The influence of the design configuration on the seismic response of moment-resisting frames. In Proc. of Behaviour of steel Structures in Seismic Areas STESSA '97, Kyoto, Japan, 444-453. Mitani, I. and Makino, M., (1980). Post local Buckling Behaviour and Plastic Rotation Capacity of Steel Beam-Columns. In Proc. f^ World Conference on Earthquake Engineering, Istanbul. Park, YJ., Ang, A.H.S. and Wen, Y.W., (1985), Seismic damage Analysis of Reinforced Concrete Buildings, Journal of Structural Engineering, ASCE, 111. Prakash, V., Powell, G.H. and Campbell, S. (1993). Drain 2DX Base program description and user guide. Version 1.10. Roeder, C. W. and Fountch, D.A. (1996), Experimental results for seismic resistant steel frame connections. Journal of Structural Engineering, ASCE, Vol. 122:6. Shen, J., (1996). A new dual system for seismic design of steel buildings. In Proc. oi Advances in Steel Structures, ICASS'96, Hong-Kong, Vol 2, Edited by S. L. Chan and J. G. Teng, Pergamon, Elsevier Science ltd, 1027-1033. Tsai, K. and Popov, E.P., (1988). Steel beam-column joint is seismic moment resisting frames. Report no. UCB/EERC-88/19, University of California, Berkeley.

Stability and Ductility of Steel Structures (SDSS '99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

421

ENERGY BASED METHODS AND TIME HYSTORY RESPONSE ANALYSIS FOR BEHAVIOR FACTOR EVALUATION OF MOMENT-RESISTING STEEL FRAMES J. Milev, Z. B. Petkov, P. Sotirov, N. Rangelov, & Tz. Georgiev Faculty of Civil Engineering, UACEG, Boul. Hristo Smimenski 1, 1421 Sofia, Bulgaria

ABSTRACT Parametric studies are carried out to evaluate the behavior factors of some typical for the Bulgarian construction practice steel moment resisting frames. Several frames are designed according to current Bulgarian codes and the capacity design procedure of ECS. The q-factors are evaluated by the modified version of the Akiyama-Kato energy based method. Static pushover analysis with inverted triangular load distribution is made to obtain story stiffiiess and yield shear forces. Time history response analysis for several acceleration records is performed. DRAIN-2DX software is used for both type of analysis. Some other methods as N2 method and Balio-Seti method are used for the behavior factor evaluation. The acceleration time histories are scaled until the structure is lead to the ultimate state. The latter is defined by various criteria. Parameters that influence q-factors are discussed and some conclusions are drawn.

KEYWORDS Energy method, behavior factor, moment resisting frames, time history analysis INTRODUCTION Sufficient lateral stiffness is required in building systems in order to limit interstory drifts to levels which imply an acceptable level of structural and non-structural damage. This stiffness is provided by restricting the flexibility of beams, columns and connections. Capacity design procedures require the resistance of columns and connections to exceed that of beams to which they are connected. This consideration implies that a significant contribution of lateral stiffness will be provided by all elements of the structure. In many codes including Eurocode 8, a maximum interstory drift is implicitly stipulated at design load level. The interstory drift control is likely to be the most significant design criterion in case of

422

moment-resisting sway frames. However the reliable drift determination usually requires non-linear analysis. Considering this fact, it is essential for a safe and efficient design that reliable force reduction (or behavior factors) to be employed. Some practical considerations complicate the experimental assessment of these behavior (or q-) factors for all but a limited range of structures. The analytical methods should be used to obtain the desired information. This may be achieved if the analytical tools employed are capable of accurately capturing the essential response features upon which the behavior factor of a particular structure depends.

STUDIED FRAMES A total number of 12 moment resisting steel frames are designed. The frame design is carried out according to the current Bulgarian codes. Provisions for the capacity design methodology are taken into account. Some of the important features of the frames are the column-tree configuration (erection splices located within the beam spans) and the flange tapering of beams. Mild steel is considered for a grade equivalent to the grade S235 vnihfy = 235 N/mm^. The materied safety factor is assumed as y^ = 1.1. The frames have 1 to 3 bays and 2 to 6 stories. They are refereed as RFnmh , where n is the number of bays (n varies from 1 to 3), m is the number of stories (m varies from 1 to 6) and A is a key letter for the flange tapering type (see Fig. 1) Erection splice /h=1000

J:"

/h

^\ i

i [ill i 1

'l

T] r"t L)

'•

^ ^

1000

1000

[ "A-type" 1

"B-type"

J

Figure 1: Tapered flange type The adopted dimensions are: all bays = 6.0m, all story heights = 3.6 m, distance between frames in longitudinal direction = 6.0m. The results for the frames RF23B, RF34B and RP36B only are presented herein. They are shown on the Fig. 2-4. Both columns and beams are designed with welded built-up I-sections. The beam-to-column connections are fully shop-welded with continuity plates (stiffeners) in the columns, and therefore they may be classified as rigid full-strength connections. The colimm bases are considered fixed. The erection splices are made with splice plates (single in the flanges and double in the web), site-welded by fillet welds. Such a detail may also be regarded as a rigid connection, therefore no semi-rigidity is actually introduced.

EARTHQUAKE RECORDS Several earthquake records are preliminary selected and only two of them (El-Centro 1940, NS component and Vrancea 1977 NS component) are finally used for the inelastic time history analysis. The first record is suitable for simulation of local source earthquakes. The second one is typical for the region. Some of their parameters are presented in Table 1 and they are refereed as El Centro and Vrancea.

423

Figure 2: Frame RF23B layout

Figure 3: Frame RF'34B layout

424

Figure 4: Frame RF36B layout TABLE 1 PARAMETERS OF THE SELECTED EARTHQUAKE RECORDS

Name of the event El Centro Vrancea

Record Step 0.02 0.033

PGA/g 0.348 0.201

Time at max. acceleration 2.12 1.45

Number Record Length [sec] of Steps 2691 53.8 1496 49.4

BEHAVIOR FACTOR EVALUATION BASED ON ENERGY APPROACH The modified version of Akiyama-Kato energy method is used (Milev 1998). The original KatoAkiyama method (Akiyama 1985 and Mazzolani&Piluso 1996) does not require any non-linear analysis. However the evaluation of each story capacity needs several assumption concerning the simplified expression of the inelastic strain energy adsorbed by the entire structure and the optimum distribution of this energy between different storeys. The modification of Akiyama-Kato method does not require the simplified expression of the inelastic strain energy adsorbed by the entire structure and the optimum distribution of this work between different storeys. However it needs both non-linear static pushover and dynamic time history analysis. The software DRAIN-2DX and El5 of its library is used for such analysis. The procedure is as follows:

425

(1) The story stiffness Kt and the story yield shear force Qyi are calculated by performing the nonlinear static push-over analysis with inverted triangular load distribution. The story yield shear force coefficient ai is calculated (1)

a, =

(2) The limit value of the cumulated story ductility ratio Tium =10 is assumed; (3) The cumulative inelastic strain energy Wp for the entire structure and the story inelastic strain energy Wpt for each story are calculated by performing the dynamic non-linear time history analysis with the scaled acceleration records. The program RESPONSE developed in BRJ - Tsukuba under supervision of Prof Akiyama is used. The simplified "shear" model is used in this software; (4) The damage distribution coefficient yi is calculated W /

(2)

r. = yw^ ; (5) The cumulated story ductility ratio rii is calculated by applying. W ./ ^^= 7Q.5. where:

(^)

h^^ = -7^: elastic story deformation under Qyi;

If the maximum r^i is equal (with some tolerance) to the assumed rium then go to step 6. Otherwise the acceleration time history is scaled again and steps (3)-(5) are repeated; (6) The lower bound of the behaviour factor is derived by the assumption that the collapse in the i-th story of the building is prevented The q-factor of the structure is calculated OC:

q i = J l + 2cjiTiiHa,

(4)

q = min(qi,q2,...,qN)

where:

4;r' T'^K,

c, =-

IL^^i y=i

The results for the frame RE64B from the static pushover analysis and from the time history response analysis applying El Centro record scaled by die factor 2.4 are presented in Fig. 5. The relative interstory drifts in the critical stories of all frames are around 2%. The behavior factors obtained by the modified Akiyama-Kato method are summarized in Table 2. TABLE 2 BRHAVIOR FACTORS OBTAINED BY MODIFIED AKIYAMA-KATO METHOD

Frame RE23B RE34B RE36B

q-factor 4.29 4.41 4.70

El Centro Critical story 2 2 2

IDI[%] 2.10 2.05 2.05

q-factor 3.87 3.93 4.18

Vrancea Critical story 1 1 1

iDir%i 2.00 2.10 2.05

426

0.03

0.04

0.05

Storey Drift [cm] El Centro*2.4

Time [sec]

Figure 5: Frame RF36B - pushover and time history response analysis OTHER METHODS The resuhs of the evaluation of the behaviour factor of some of the designed studied frames by means of time history analysis with lumped plasticity model and by some of the known approximate methods are presented. The ECS response spectrum is used for an approximate analysis with an equivalent SDOF system. The sameframes RF23B, RF34B and RF36P are analysed. Three types of models are used for a numerical study of the seismic behaviour of the frames and evaluation of the q-factor. These models, applied to each one of the frames, are as follows:

427

E02-E04 model. The DRAn>I-2DX is used to study the frames, subjected to static and dynamic loads. The element E02 is used for modelling of beams and columns. The points for development of plastic hinges are selected at corresponding distances - 4 (see Fig. 1). The joints at those points are modelled by connection hinge element E04. Shear model. The story stifftiess is modelled by springs for horizontal displacements. Bi-linear force-displacement relationships are assumed for the springs. The equivalent stiffness parameters for each story spring are determined by a pushover analysis. Equivalent SDOF system. When a structure undergo elastic behaviour its response is predominantly as to a SDOF. Some authors (Fajfar 1998) developed the idea that structural behaviour can be identified by the response of SDOF. The results for the q-factor, obtained by the Ballio-Setti method (Mazolani & Piluso 1996) are presented in Table 3. Nonlinear dynamic analysis is performed, using both shear type and E02-E04 models. It is assumed that the ultimate state is reached when the story drift exceeds 2%. Then the value of the q-factor is calculated as a ratio between the scaling factors Xy and A„, characterising the state of initiation of inelastic deformations and the ultimate limit state, respectively. Thus the q-factor is defined as q=^

(5)

TABLES BALIO-SETI METHOD

FRAME

Acceleration record

Model type

El-Centro

E02-E04 Shear E02-E04 Shear E02-E04 Shear E02-E04 Shear E02-E04 Shear E02-E04 Shear

RF36P Vrancea El-Centro RF34P Vrancea El-Centro RF23P \

Vrancea

Scale factors Ay

Au

0.36 0.30 0.11 0.15 0.35 0.29 0.18 0.19 0.28 0.25

2.45 0.90 0.90 0.76 1.72 1,18 1.22 0.93 1.76 1.43 1.10 0.97

1 0.32 0.35

q-factor

8-top [m]

6.80 3.0 8.18 5.07 4.91 4.07 6.78 4.90 6.28 5.72 3.44 2.77

0.38 0.22 0.32 0.21 0.27 0.20 0.38 0.22 0.19 0.16 0.17 0.12

TABLE 3 N 2 METHOD

FRAME Ru q

1

5*fm] S[ml

RF23P 2.108 4.216 0.128 0.161

RF34P 2.378 4.756 0.197 0.251

RF36P 2.85 5.70 0.214 0.275

428 The N2 method (Fajfar 1998) is applied to establish the value of the q-factor. The calculations are performed for an equivalent SDOF model. The ECS design response spectrum is applied for the analysis. The overstrength factor Rs is assumed to be equal to 2.0. The behaviour factor is defined as q^RsRfi

(6)

where: jR^ is the ductility dependent coefficient, calculated by equivalent SDOF model. The results for q and J?^ are presented in Table 3. The values of the maximum displacement S* of the equivalent SDOF system and the maximum top displacement <^ of thefi-amesare also listed in Table 3. CONCLUSIONS The following conclusions can be drawnfi*omthe presented resuhs of the parametric studies: - The values of the behavior factors, calculated by the modified Akiyama-Kato method, are more conservative than the values obtained by the other methods and accepted by both Bulgarian code and Eurocode 8. Probably more experimental and analytical study is necessary for the evaluation of critical value of the cumulated ductility ratio rium. Moreover, the energy method gives the smallest scatter of the behavior factor for different acceleration histories and different frames; - The results in Table 4, derived byfiromthe time history response analysis show that the values for the q-factor are sensitive to the used computational models. The largest scatter of the results is observed in the 6-story frame. The scatter between different mechanical models (shear model and E02-E04) decrease when the number of stories decreases. The possible reason for this tendency could be that the shear model is no longer adequate to model the story stif&iess with bi-linear model. The shear stiffness could be modelled by multi-linear model. - The values of q-factors obtained by shear models are more conservative than ones obtained by E02-E04 models; ACKNOWLEDGEMENTS The present study has been carried out in thefirameworkof the INCO-COPERNICUS Joint Research Project "Reliability of Moment Resistant Connections of Steel Building Frames in Seismic Areas'' (RECOS). Thanks are therefore due to all the partners for their comments and advices, and especially to Professor F. M. Mazzolani, the coordinator of the Project. The financial support of the EU is also gratefiiUy acknowledged.

REFERENCES Akiyama H. (1985). Earthquake-Resistant Limit State Design for Buildings, University of Tokyo Press, Japan Fajfar P. (1998). Towards a nonlinear Methods for the Future Seismic Codes, in : Seismic Design Practice into the Next Century, Booth (ed.), Balkema Mazzolani F. M. and Piluso V. (1996) The Theory and Design of Seismic Resistant Steel Frames, E&FN Spon Milev J. (1998) Modelling of Shear Walls for Seismic Analysis of Frame-Wall Structures, PhD Thesis, (in Bulgarian)

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

429

DUCTILITY DEMANDS FOR MRFs AND LL-EBFs FOR DIFFERENT EARTHQUAKE TYPES L.Tirca ^ and V. Gioncu^ ^Department of Architecture, Civil Engineering and Architecture Faculty, "Polytehnica" University of Timisoara, str. T.Lalescu nr. 2, 1900 Timisoara, Romania,

ABSTRACT The eccentrically braced frames with long links, LL-EBFs, is used in practice to safety the architectural demands to obtain free spaces in facades and to reduce lateral displacements. The paper shows that this solution must be used with the precaution, because, in some cases, the braced frames may behave badly than the unbraced frames.

KEYWORDS Long link beams, top displacement, rotation capacity, base shear forces, ductility demands.

mTRODUCTION In the design of high-rise buildings in seismic areas there is an increasing demands for reliable structural systems with predictable behavior. The modem approach demands that the strength, stiffness and ductility triad must be considered in design. Moment resisting frames, MRFs, provide a satisfactory strength and possess an excellent ductility, but tend to behave too flexible, and so, especially for highrise buildings, MRF become uneconomical in developing the designed stiffness required by the drift condition. Aiming to solve this problem, in the last period, braced systems have been developed. Concentrically braced frames, CBFs, are capable of easily ensuring both the required strength and stifftiess, but the ductility demands are unsatisfied. Eccentrically braced frames, EBFs, combine the good strength and the stiffness of CBFs with the very good ductility of MRF and have fined a very good position in seismic design of high-rise buildings F.M. Mazzolani, D. Georgescu, A. Astaneh-Asl (1994). They possess good stiffness under moderate lateral loads and provide stable mechanism for dissipating

430

large amounts of energy when subjected to severe seismic load. The link beam between the braced parts of the frame is a ductile element if the system is designed to prevent buckling of the braces. But for CBFs, and partially the EBFs, the presence of bracing systems Umits the possibility to have free spaces for answering the architectural demands. So, new forms of bracing systems are studied, between these being the most suitable the EBFs with long links, LL-EBFs, satisfying both structural and architectural requirements. The behavior of MRFs is very well known, the main problem is to obtain a good collapse mechanism, which is the global one (Fig, la). In exchange, the behavior of EBFs is less known. During a severe seismic load, the link beams may be subjected to very large deformations (Fig. lb). By introducing braces, the frame strength and stiflftiess rapidly increases with the decreasing of link length, but in the some time, the rotation requirements increases. For the long link beams the moment ductility is dominant, while for short link beams the system ductility is given by the shear deformations.

b)

a) Fig.lMRFandLL-EBF

So, the choosing of the link length in an EBFs is crucial in design decision. In a practical way, using the rolled wide flange beam, according to the provisions of European and USA codes the link elements are presented in the following classification: - short links (SL):

MP /<1.6Vp

(la)

intermediate links (IL):

l,6^^l<3^P Vp

(lb) Vp

long links(LL): \p where: 1 - the length of the link; Mp, Vp - the plastic bending moment and plastic shear strength of the link.

(Ic)

431 The long links in a EBFs have a behavior like a beam in a MRFs, and many resuhs concerning local ductility of beams may be used for link design. In the paper the ductility demands for LL-EBFs are compared with them of MRFs for different earthquake types characterized with different acceleration amplitudes and natural periods.

SELECTED FRAMES AND PARAMETRES The typical structures selected for the numerical analysis are presented in Table 1; three MRFs and four EBFs with 6 stories and one bay. For beams IPE profiles are used, while for columns and braces are variable in two steps on thefi-amehigh. On each part the sections of elements are the same. TABLE 1 CHARACTERISTICS OF ANALYSED FRAMES Columns Frame type

Beams

Braces

Name Tl

T2

Tl

T2

Tl

T2

Collapse mechanism

HE240B HE200B IPE360

IPE300

local

MRF2 HE260B HE220B IPE360

IPE300

global

MRF3 HE450B HE400B IPE360

IPE300

global

EBFl

HE240B HE200B IPE360

IPE300 HE140B HE120B

local

EBF2

HE240B HE200B IPE360

IPE300 HE240B HE200B

global

EBF3

HE400B HE360B IPE360 IPE300 HE260B HE240B

global

EBF4

HE450B HE400B IPE360 IPE300 HE280B HE260B

global

MRFl CD

6J0O

^

i

KM .

6QQ

.

The MRFl is an ordinary moment resisting frame, sized so that story mechanism occurs. The MRF2 and MRF3 are special moment resisting frames at which global mechanism must be the collapse type. The first has column sections more reduced than the second. The EBFl is obtained from MRFl by introducing weak braces. One can see that the mechanism type is not changed, the collapse mode being the story mechanism, as for the ordinary MRFl. So, a special attention must be paid for the sizing the bracing system in the aim to make this an efficient one. The

432

EBF2 is obtained from MRFl using strongest bracing, so global mechanism results. The EBF4 are obtained from MRF3 using the types of bracing for obtain global mechanism. The EBF3 also form global mechanism, but is not so strong that EBF4. The analysis of these frames is time-history and has been performed using Drain-2D computer program. For study were used the following accelerograms: Vrancea (1997); Kobe (1995); Banat (1991) and Loma Prieta (1989) records. The earthquakes are selected to cover the far field (Vrancea); intermediate field (Loma Prieta) and near field (Banat and Kobe) ground motions types. For all these earthquakes the influence of local soil conditions are analyzed. In Fig.2 the selected earthquakes are superposed in function on the natural period. In each case, the ground motions are characterized by short natural periods of 0.20-0.30 sec, which are the periods corresponding to the base excitation. These short characteristic periods are followed by a graduated increasing of periods, due to the effect of local conditions. One can see that the most affected by these local soil conditions is the Vrancea earthquake, recorded at Bucharest, where the period is about l.Ssec. The second case is the Kobe earthquake, with maximum periods of about O.Tsec, followed by Banat and Loma Prieta earthquakes, with maximum periods of 0.3-0.4 sec.

-Vrancea

-Kobe

-Banat-

-Loma Pfieta i

Fig.2 Variation of natural periods All the accelerogram use, have been normalized for three level of accelerations: 0.15g; 0.25g and 0.3 5g for low, medium and high earthquakes.

NUMERICAL TESTS

Fundamental periods Fig. 3 shows the first three fundamental periods for the analyzed frames. One can see the great influence of bracing system in the reducing of fundamental periods. If compares the natural periods of used earthquakes and these fundamental periods of frames, on can conclude which mode of vibration is amplified due to the resonance. For instance, it is very clearly that for Vrancea earthquake the first vibration is amplified, while for the other earthquakes, the second vibration modes is affected.

433

2 ^-g^™ 1.8 1.6 1.4

1.72

E3T1

j^T2 Jc3T3

1.35 1.08

— 1 H 0.8 Tid:65 tj (ter-t 0.6 0.4 0.2 0

089 0.86 0.82 39"

43_Joba Ct26-i fe14 13 @

Ilj mj MRF1 MRF2 MRF3

El

EBF1 EBF2 EBF3 EBF4

Fig.3 Structure fundamental periods Base shear forces The variation of base shear forces in function of earthquake accelerations, frame types and level of peak accelerations are presented in Fig.4. The interrupted diagrams show that the frame collapse is produced before the reaching the corresponding seismic loads. One must observe that the bracing of MRFs has the effects very contradictory, in some cases the shear forces increase, while for other cases, these forces decrease, showing that is not valuable a very simple rule. These differences can be explained for elastic behavior by the position in spectrum curve of the structure fundamental period related with the natural period of the ground motion. The possible situation is presented in Fig. 5. In the case of plastic deformations, the behavior is more complicated, the shear forces can increase or decrease in ftmction of the position on the increasing or decreasing part of spectrum curve. One can see that MRFl frame has an inadequate behavior only for Vrancea earthquake, due to correspondence of the frame first vibration mode and ground motion periods. In exchange, the EBFl, obtained from MRFl has a very bad behavior for all the considered earthquakes due to insufficient rigidity of bracing system. The increasing of dimensions of braces, as in the case of EBF2, has effect only for Banat and Loma Prieta earthquakes, due to the differences between the two natural periods. The bracings of MRF2 and MRF3 have a good effect in the case of Vrancea, Banat and Loma Prieta, resulting reduced shear forces. In exchange, for Kobe earthquake, the increasing of shear forces for EBF3 is important.

Top displacements For buildings, EC3 and ECS recommend that the horizontal deflections at the structure as a whole does not exceed 0.2% H (H being the height of the structure). Considering this recommendation as a limit for operational level of seismic loads, the life safe level is given by the 0.4%H. In Fig.6 the top displacement for mentioned earthquakes and frames are plotted. Considering the 0.15g as the accelerations for operational level and 0.35g as the acceleration for life safe level, one can see that these limit displacements are exceeded by all the selected earthquakes, especially for Vrancea earthquake. Only for Loma Prieta earthquakes (MRFl and MRF'2) these conditions are satisfied. This observations give rise the question of the validity of these code requirements. An other very important observation is concerning the effectiveness of the long link bracing, because the reducing of the top displacement is not

434

400 350 300 250H

200J 15oJ 100 50 0

^^ / ^ ^^"^ a) Vrancea S[KN] 400

K
.^<^

b) Kobe Opga=0.15g Bpga-0.25g Dpga=0.35g

S[KN] 400

Ppga=0.15g Bpga=0.25g Dpga=0.35g

d) Loma Prieta Collapse before reaching the corresponding seismic loads

Fig.4 Base shear forces Sa

UD

unbraced frame

Ts/Tg

braced frame

Fig. 5 Influence of reducing offtmdamentalperiod due to bracing

435 so important as supposed. For operational level, when the structure works almost elastically, one can see in some cases on contrary of the presumed behavior, an increasing of top displacement for braced structure in comparison with the unbraced structures, due to the increasing of seismic forces, as the results of reducing the fundamental periods. Thus, the balance of the increasing the lateral rigidity due to bracing and the reduction of top displacements, plays a primordial role in the design process. This is not a simple problem, because this balance depends on the earthquakes type. The strengthening of a frame with a long link bracing system must be performed with caution, because in some cases the behavior of braced frame may be badly than the unbraced one.

Ductility demands The available ductilities for beams are determined from the relationship: Oa = Cr X Cc X 9p/y„,

(2)

De/H % 1.2i

/

/

De/H% 1.2i

/

b) Kobe

a) Vrancea Qpga=0.15g ©pga=0.25g apga=0.35g

d) Loma Prieta c) Banat Collapse before reaching the corresponding seismic loads limit for life safe level limit for operational level Fig.6 Top displacements

npga=0.15g M pga=0.25g npga=0.35g

436 where 9p is the plastic rotation determined with the Ductrot computer program (Gioncu and Petcu, 1997), Cr, coefficient which considers the effect of root radius (Anastasiadis and Gioncu, 1999) Cr=(b-tf-0.8r)^

(3)

where b is the half of flange width, tf, the web tickness and r, root radius. In relationship (2), Cr is a coefficient, which consider the effect of cyclic loads. In absence of more exact values, it is proposed to consider Cc =0.8; ym is a partial safety factor, ym = 13. Using the relationship (2), the following values are obtained: for EPE 300 . three stories) and for EPE 360 =:> 63= 7.82% ( for first three stories). The ductility condition is given by relationship:

Oa = 7.34% ( for ultimate

e.>Qr

(4)

where 0r is the required ductility determined fi^om the global behavior of structures. The required ductilities for the considered frames and earthquakes (for the three levels of peak accelerations) are presented in Fig. 7. If some cases lack from the plotted curves, then the frame behavior is elastically, the nQ.„Qf_stQr|es_

no. of stc!rie$^^

el3 " ^ 5| 4

DEBF4 eEBF3 nEBF2 DMRF3 • MRF2 • MRF1

S

a,

'eai 1 I •

1

-^

•

1

1 —

,

2

3

4

5

6

"|-

7

•

"

-

—

'

"

0EBF4

•

'" •'

mEBF3

sfefc 4f^

3

5

—

'

"

'

^...™...,.™-

! 6a •

y

: • :

\ \

1 ,,.„-..™-.i.-

2

3

4

5

6

4 " ^ ;

nMRF3

3h\:--.

• MRF2

E1EBF4 1

m •'-••

^

i

6

7

rotation % c) Banat earthquake (pga = 0.25g)

•

•

•

•

•

I

•

•

0

.:;-•.••

1

^

)

• MRF1

" ^ L '.

2P

^M

'

DEBF2 •j':'---'-'

__:'^;-^.-^.-.-.^^. 4

'

•

. • \

\'

.

2

•

\

I 1

1

..„

• EBF4 0EBF3 nEBF2 nMRF3 • MRF2 DMRF1

liEBF3

• l

:

—-.,.,

5 f ""

2]

'•

—

no. of stories

OMRFI

••

-

6 ^ ^ ^

• MRF2

1M:,

'

Te;

aEBF2 Q.MRF3.

3pli

•

rotation % b) Loma Prieta earthquake (pga -• Q.35g)

a) Banat earthquake (pga = 0.35g) '

'

1

1

rotation %

np,,Qf,§<:QTies^

"

p_

^A,

'

jl.

"

F

r 1

•

" |_ '

-:-:

- L ^

r 1; 1:

L^

\

p.,^\

:

2 1

hjggggggmggf^':

l:-.^

• : • : • • • ' ' ^ : , i

2

3

4

5

6

rotation %

d) Loma Prieta earthquake (pga = Q.25g)

Fig. 7 Ductility requirements

437

plastic rotations being not present. One can see that for Vrancea earthquake, the required ductilities is more that the other types of earthquakes. With the exception of Banat earthquake, for the other types of accelerogram records, the ductility demands for braced frames are higher than for the unbraced frames, due to the fact that the link beam is shorter than the beam. no...j3t stories-,^

\—r

^....

r -",-

n^^

1 1. 1 i

ji '

1 1 1

H I

•

r-l

>

==ui

'

h 1 1 k^ 1 1

I I I " [ 1 1

^4^'^^! 7

'

1

p^^l 3

4 5 rotation %

©EBF3

j^sssa^sMssBa "

.

^

^

~

-

nEBF2 •

DMRF3

l,iT-iwii?iTwrifflTirilfrmrimirn
U

:

,

' • • • - . : ^

3]^^

-..„—^..-J.

2

3

4

5

rotation %

•

;

;

1: :

no. of stories

p7T:"i

2

•

7

h) Kobe earthquake (pga = Q.25g)

^

•

„.„.„„•-J....,:

6

rotation %

no. of stories

0

.

...

g) Vrancea earthquake (pga = Q.25g)

r

•

V- . : - ^

1

5

1 . :

eMRFI

1 B 4

r*

0EBF4

6 ^ ^

2^^-.

3

1

'

rimrzxn

1,

no. otstories

-Q.

aEBF4 ^EBF3 nEBF2 DMRFS • MRF2 DMRFI

'

1

\

rotation % f) Kobe earthquake (pga = 0.35g)

e) Vrancea earthquake (pga = Q.35g) rro^of-slpries

r

DEBF4 ^EBF3 nEBF2 DNARFa • IVrRF2 • MRF1

1 6a-

•<^--M "

t

•

;

•

•

•

4

ilEBF4 eEBF3

5

^

'••'

• MRF3

3

• MRF2 HMRFI

-.-.'i--i

2

i ^ . i •;;••:

6

7

1 a-;:/,,,.-

8

3

rotation %

i) Vrancea earthquake (pga = 0.15g)

I

•

- 1 - • . . • • • ; J - - . ; ; .

4 5 rotation %

j) Kobe earthquake (pga = 0.15g)

Fig. 7 (continued)

. " • :

•

•.•.;

•

•

•

• ' • : • : : • • - •

• - ' : • ; ' { • •

^:--^"^-^:^^ 5

-eai

eEBF2

4

;^;.v':;:::|::;=

-^

Wir:

•

j:^:-:-;-!---^

^-

6

•

•

;

A

I

438 CONCLUSIONS The LL-EBFs are used for seismic resistant structures aiming to reduce the lateral displacements of MRFs and in the same time, to answer the architectural demands for maximum free spaces. The comparison of MRFs and LL-EBFs behaviors emphasized the following conclusions: All the record accelerations used for analysis start with natural periods of 0.2-0.3sec. and in fiinction of site conditions, this period can be modified or not. In this way the Vrancea earthquake shows the most important increasing of ground period; The shear forces of braced frames may be smaller or greater than the corresponding unbraced frames in functions of the ratio between frame and ground period. This is a very complex phenomenon, which can not be described with a simplified approach as the use of a design spectrum; The top displacement exceeds the limits for both operational and life safe levels in the majority of analysed frames. So, it is suggested that the code requirements must be revised because seems to be too severe. The ductility demands varies in function of earthquake and structure types. The ductility checking, shows that for earthquake with long natural periods (Vrancea), this condition is more severe that for earthquakes with shorts natural periods (Banat, Loma Prieta); The LL-EBFs must be used with great precaution because, in some cases, its behavior may be badly than the unbraced corresponding frame.

References Anastasiadis A., Gioncu W . (1999): Ductility of IPE and HE A beams and beam-column. Stability and Ductility of Steel Structures ,9-11 September 1999, Timisoara (in this volume). Gioncu v., Petcu D. (1997). Available rotation capacity of wide-flange beams and beam-column J. Construct, Steel Research, voU3, no 1-3: Part 1, 161-217; Part 2, 219-244. Mazzolani F. M., Georgescu D., Astaneh-Ash A.(1994). Remarks on behavior of concentrically and eccentrically braced steel frames. Behaviour ofSsteel Structures in Seismic Areas STESSA ' 94, F & FN Spon, 310-322.

Stability and Ductility of Steel Structures (SDSS'99) D. Dubina and M. Ivanyi, editors © 1999 Elsevier Science Ltd. All rights reserved

439

DUCTILITY AND OVERSTRENGTH OF MOMENT FRAMES I. Vayas^ and A. Spiliopoulos* ^Department of Civil Engineering, National Technical University of Athens, 106 82 Athens, Greece

ABSTRACT A method of the evaluation of the global ductility and overstrength of moment frames is proposed. The method is based on an energy approach, where the elastic energy stored and the plastic energy dissipated in the structure are determined. Various definitions for the yield state and the ultimate state are discussed and a relevant proposal is made. The method is applied to various frames, where the frame geometry, the flexibility of beam-to-column joints, the level of vertical loading and the level of local ductility were varied. Conclusions on the influence of the various parameters on the ductility and the overstrength were drawn.

KEYWORDS Ductility, overstrength, rotation capacity, plastic hinge, energy method, yield state, ultimate state INTRODUCTION As well known, part of the energy imparted in a building frame during earthquakes is dissipated through inelastic deformations. Consequently, the elastic strain energy, and accordingly the developing seismic forces, may be reduced leading in most cases to a more economic design. Li order to secure a stable hysteretic behaviour under reversed loading, the structure must be designed for ductility in addition to stiffiiess and strength. However, the structural response beyond yielding is not only influenced by its deformation capacity, as determined for bilinear elastoplastic systems. It has been recognized, that steel frames possess additional strength, an overstrength, beyond the yield load. This additional strength, in combination with ductility, constitute important parameters for a structure to resist seismic induced actions. Moment frames resist horizontal seismic loads through bending action on their structural members. A key element for this type of frames are their beam-to-column joints that shall transfer moments in addition to forces. Therefore, the joint characteristics in respect to stif&iess and strength largely influence the behaviour of the overall frame. This paper presents an energy method for the determination of the ductility and the overstrength of moment frames. The method is based on a pushover analysis of the frame and calculation of the elastic and plastic energies stored and dissipated by its members. Subsequently a parametric study is performed, in order to study the influence of various parameters, such as the level of local ductility.

440

the level of vertical forces, the rigidity of the joints, the geometric properties etc., on the above characteristics. LOCAL DUCTILITY As referred to above, parts of the structure are expected to exhibit inelastic behaviour during a seismic event. This corresponds for moment frames to the formation of plastic hinges, preferably in the beams, where plasticity is supposed to concentrate. The relationship between the rotation capacity of the members and the overall ductility of the system is therefore straightforward. Local ductility is usually determined from three point bending tests. Local ductility in terms of rotation is given by eq. (1): A/.=^

(1)

where: (Py ^„ = yield rotation, correspondingly ultimate rotation The rotation capacity may be determined from eq. (2): (2)

(Pp = ( p u - 9 y = ( l ^ c p - l ) ( p y

Calculating the end rotation at yielding for the three-point test and considering that the rotation at midspan is twice as large, the rotation capacity is finally given by: IM, where EI is the stiffiiess and / the length of the beam and Mp the plastic moment of its cross section. The rotation capacity of a member that is part of an overall frame, is determined by transformation to an equivalent single span, three point loaded beam, Spangemacher et al. Starting from the moment distribution at the ultimate state, a member is isolated from the overall frame, and according to its moment diagramme is transformed to a single span beam as above by introduction of an equivalent span length. If a member has two plastic hinges at its ends, it will be transformed to two equivalent single span beams (Figure 1). The rotation capacity at each plastic hinge is then found by application of eq. (3). In the present study, following the provisions of Eurocode 8, three levels of ductility H(high), M(edium) and L(ow) are introduced. The corresponding values of jLi