J. Construct. Steel Research 7 (1987) 317-334

Inelastic Local Buckling of Fabricated I-Beams

M. A. Brad fo rd

.School of Civil Engineering, The University of New South Wales, Kensington, NSW, 2033, Australia

(Received 3 February 1987; revised version received 12 March 1987; accepted 8 April 1987)

SYNOPSIS

A finite strip method of analysis is presented for the inelastic local buckling o f 1-beams fabricated by welding. Stiffness and stability matrices for the section are developed at a monotonically increasing load factor, and the critical moment is that for which the buckling determinant vanishes. Critical moments determined in this way are shown to agree well with test results. The limiting depth-to-thickness ratios for the web which correspond to compact and semi-compact sections are investigated, and it is shown that the values given in BS 5950 : Part I are unconservative for an extensive range of section geometries. Based on a parametric study, alternative and more accurate formulations for the critical web slendernesses are proposed.

NOTATION

The section geometry is defined in Fig. 2 and the material properties are defined in Fig. 3. Other principal notation is as below.

k, g kw q n A B,B* D Et

Stiffness and stability matrices respectively. Web local buckling coefficient. Nodal displacement vector in eqn 7. Displacement vector in eqn 6. Area of section or area of strip. Strain and slope matrices respectively. Property matrix. Tangent modulus.

317 J. Construct. Steel Research 0143-974X/87/$03.50 © Elsevier Applied Science Publishers Ltd, England, 1987. Printed in Great Britain

318 M. A. Bradford

G L M M MI, Mo, Mx, M. Mxy My, Mp P V Ca

~, ¢y, ~/~y E

0, Op h hs, hw hwc, h ,

//

P~,Py,P~y or

Or a

o'~ , o'y , ~'~

Elastic shear modulus. Buckling half-wavelength. Bending moment. Interpolation matrix. Inelastic local buckling moment. Elastic local buckling moment. Infinitesimal buckling moments. Nominal first yield and plastic moments respectively. Axial force. Volume of strip. Applied strain. Membrane buckling strains in eqn 8. Buckling generalised strains in eqn 10. Coordinate of neutral axis. Rotation and rotation at Me respectively. 'Load' factor. Flange and web generalised slendernesses respectively. Web slenderness constants for compact and semi-compact sections respectively. Elastic Poisson's ratio. Infinitesimal buckling curvatures in eqn 8. Generalised stress vector. Applied stress. Infinitesimal buckling stresses. Curvature

1 INTRODUCTION

It is weU-known that slender elastic steel sections can carry bending moments greater than the elastic local buckling moment Mo, because of the postbuckling reserve of strength.' This increase in moment is finally limited by plastification of the component plates of the section, at which point unloading usually occurs. However, stocky sections or sections with sig- nificant residual stresses may buckle inelasticaUy at a moment Mn which is lower than Mol. This inelastic local buckling, which is considered in this paper, will always occur when the elastic critical.moment Mo, is greater than the nominal yield moment Mr. An illustration of the strain hardening buckling, inelastic local buckling and post-local buckling of sections is shown in Fig. 1.

A significant number of experimental studies on the inelastic local buckling of I-sections have been made, mainly with the intention of

Inelastic local buckling of fabricated 1-beams 319

X

X

E

u

%

I

My

MI~ for section with residual stresses ~ ~

\

u

u I

u~

I 1.0

\

Section slenderness /(My/Mo£)

Fig. 1. Local buckling of welded I-section.

ascertaining limiting width-to-thickness ratios of the flanges and webs for which the section is suitable for plastic design. Among the recent studies have been those of Climenhaga and Johnson, 2 Holtz and Kulak, 3'4 Johnson 5 and Kemp. 6 However, relatively little research has been attempted to study the inelastic buckling of sections theoretically.

In principle, the finite element method can be used to investigate the inelastic local buckling of plates and sections. 7 Such a method can lead to the iterative solution of equations with large numbers of degrees of freedom, and is therefore computationally inefficient for many applications. On the other hand, the finite strip method is suitable for analysing prismatic members of arbitrary section such as I-beams, and has been used by Yoshida," and more recently by M a h m o u d 9 and Plank ~° for investigating inelastic local buckling. A pseudo-strip method has also been employed by Dawe and Kulak" for this study. The advantage of the finite strip method is that it is computationally more efficient than the finite element method.

This paper presents a finite strip method of analysis of the elastic and inelastic local buckling of I-beams fabricated by welding along the flange- web junction. The residual stresses induced by such welding are included in

320 M. A. Bradford

the analysis. Although restricted to I-sections herein, it is a general method that may be applied to prismatic sections of arbitrary geometry.

Following development of the finite strip model, its accuracy is demon- strated by a comparison with independent test results. The scope of the analysis is then demonstrated by a study of the limiting depth-to-thickness ratios of the web for compact and semi-compact fabricated sections, and these are compared with the existing provisions of the new BS 5950: Part 1 limit states code.~2

2 FINITE STRIP BUCKLING ANALYSIS

2.1 General

For the analysis in this paper, the I-beam is assumed to be partitioned into finite strips which are connected to one or more other strips along one or both of their longitudinal edges in the normal way.~3 The x, y, z axis system of a typical strip is shown in Fig. 2a, which also indicates the displacements u, v, w in the direction of these axes.

The stress-strain curve assumed for the structural steel is shown in Fig. 3. It is a trilinear idealisation, with a plastic plateau and a constant strain hardening modulus E,t = E/h'.

The finite strip procedure requires a calculation of the distribution of strains applied to the member prior to invoking the bifurcation analysis. This involves:

(a) selection of an appropriate residual stress model for the fabricated 1-beam; and

(b) application of an initial axial strain and curvature which would occur if the member was bent and compressed between rigid plattens.

The above procedure is outlined in Section 2.2. The basic steps in the inelastic bifurcation analysis, presented in the remainder of this section, then involve:

(a) a definition of the displacement functions to describe the membrane and flexural deformations of the plate strips during buckling;

(b) a statement of the strain--displacement relations; (c) selection of an appropriate elastic-plastic plate theory to describe the

membrane and flexural behaviour of the plate strips; (d) application of the principle of virtual displacements to determine the

stiffness and stability matrices; and (e) solution of the buckling equation to determine the lowest load factor

for inelastic buckling.

Inelastic local buckling of fabricated 1-beams 321

Z~W

/ y,v

(a)

Y IL X~U

I T

~ ~ t

d

bf

-I

I

1 n

(b)

Fig. 2. Dimensions and displacements. (a) Strip displacements; (b) cross-section dimensions.

2.2 Applied strains and resulting moment

The residual strains er assumed for the fabricated sections are based on the so-called ' tendon force concept' developed by the Cambridge group as summarised by Kitipornchai and Wong-Chung.14 Tension blocks stressed to o-r are assumed to occur at the welds, accompanied by adjacent compression blocks such that the plate is in longitudinal equilibrium (Fig. 4). Empirical expressions for the size of these blocks depend on the plate thickness, weld size and process efficiency. These expressions, which are set out clearly in Reference 14, are not reproduced in this paper.

Initially, a curvature ~ is applied to the section, and this is increased

322 M. A. Bradford

E

¢y (st

Str l tn

Fig. 3. Stress-strain curve for structural steel.

Est'E/h'

C / / / / f ~

! • /

T av

Fig. 4. Residual stresses in fabricated I-section.

Inelastic local buckling of fabricated 1-beams 323

monotonically by a 'load' factor k until buckling occurs, so that the applied strain ¢, at the point (~, 7?) in the cross-section of the beam may be written as

¢o(~:, r/) = ~.&(~ - , ) + ~,(~:,~) (1)

where ~ is the 77 coordinate of the neutral axis, which in general will be nonzero because of the monosymmetry caused by the presence of the residual strains. The stress or, at (~:,~) can be found from ~, as

I? o'o = E, de, + E~, (2)

where E, is the tangent modulus of elasticity. The value of C-/in eqn I can be determined from

Atr, dA = P = 0 (3)

which is the condition for pure bending. The integration in eqn 3 is carried out numerically over the area A, and the solution of eqns 1, 2 and 3 for ~ is performed by a chordal Newton-Raphson technique. Finally, the resulting moment M in the cross-section corresponding to h~b is determined, subject to eqn 3, by

M = fA tro~dA (4)

which again is integrated numerically over the area A of the cross-section.

2 . 3 B u c k l i n g d i s p l a c e m e n t s a n d s tra ins for a str ip

In accordance with the finite strip method of analysis, the infinitesimal buckling displacements for a strip are combinations of series functions in the longitudinal direction, and shape functions in the transverse direction. Except for a differing axis convention, the displacements for an individual strip are those employed by Hancock.15 These displacements consist of a single harmonic in the longitudinal direction, and a cubic polynomial in the transverse direction, and are appropriate for strips with simple supports for which the moment gradient in the longitudinal direction is zero.

In matrix notation, the displacements may be written as

u = Mq (5)

where

u = [u, v, w] r (6)

324 M. A. Bradford

in which u and v are the membrane displacements, w is the flexural dis- placement (Fig. 2), and q is the vector of nodal line displacements given by

q = [Ul, P1, w1,01,/,/2, P2, w2, 02] T (7)

In cqn 7, the subscripts refer to the nodal lines 1 and 2 of the strip, 0 = Ow/Oy, and M is an interpolation matrix of shape functions.

The generalised strains, which include normal and shear strains as well as bending and twisting curvatures, are obtained by appropriate differentia- tion of the displacement functions with respect to the relevant coordinate variables. The strains and curvatures applicable to a plane stress bending strip are

~x ---- OU/OX, Ey ----- o v l O y , Yxy ---- c)ulOy + OV/OX

Px = -- 0 2 W / o x 2 , Py = -- C92w/OY 2, Pxy = 202W/OXOY (8)

which may be written in matrix format as

= Bq (9)

where

0o)

and B is the strain matrix obtained by the appropriate differentiation of eqn 5.

2.4 Constitutive relationship

The buckling constitutive equations relate the membrane normal and shear stresses to the axial strains and shear strains in the strip, and the bending and twisting moments per unit length of strip to the curvatures and twist of the strip. They may be written as

try = |D21 Dz2 ay

l"xy L 0 0 Da3 "Y~

(11)

Mxy Dss p,,y (12)

Inelastic local buckling of fabricated 1-beams 325

which may be combined into matrix format as

or= De (13)

where D is the property or constitutive matrix. For isotropic elastic buckling (i.e. in regions of the cross-section where c. is less than ev)

Dll = D22 = E/(1- v 2)

D12 = D21 = vDu

D33 = G

(14)

where v is the elastic Poisson's ratio and

E G - 2'l+v------~/t (15)

For inelastic buckling (i.e. in regions where e, is greater than or equal to ¢v), rigidities appropriate for inelastic buckling must be used. In this paper, the rigidities employed by Dawe and Kulak H for buckling of hot-rolled sections have been used, which are based on a derivation from the flow theory of plasticity presented by Haaijer.16 These approximate rigidities are given by

Dll = E~/(I - ulv2)

D22 = 4 E E J ( 3 E , t + E)(1 - u~ v2)

} D~2 = D21 [3Es,+E (2v-1)E~I+E /(1-v~v2) (16)

033 ~--- 4EEs,

4E~,(1 + v) + E

i/lV 2 = {(2v - 1)E,t + E} 2

E(3E,~ + E)

The expression for the shear rigidity D33 differs from that of Haaijer, and was derived by Lay 17 to be applicable for the trilinear stress--strain model shown in Fig. 3. It has been used for tangent modulus lateral buckling, is as well as for local buckling of hot-rolled beam-columns by Dawe and Kulak. u

326 M. A. Bradford

2.5 Buckling solution

Invoking the principle of virtual displacements for a strip gives ~9

(17)

awl ax aaw l ax

where (au/ox, av/Ox, aw/Ox)r is the vector of slopes with respect to the x-axis which may be written as

au/ax

av/ax = B*q (18)

aw/ax

and B* is the matrix obtained by appropriate differentiation of eqn 5. Substituting eqns 9, 13 and 18 into eqn 17, performing the variation and noting that 8q is arbitrary, leads to

I k(k) - g(k) l -- 0 (19)

for a nontrivial q, where

k(k) = n rD(k)ndV (20)

and

g(k) = fv B*r°%(h)B*dV (21)

are the strip stiffness and stability matrices respectively. The strain matrix B and the slope matrix B* have been given by Hancock 2° with permuted axes. In this paper, the integrations with respect to z in eqns 20 and 21 are performed analytically, and are then performed numerically with respect to the area A using four-point Gaussian quadrature.

For a particular 'load' factor h, k and g are determined for each strip, and are then assembled into the global matrices K and G.13 These matrices are calculated at monotonically increasing values of h, and the determinant of their difference is inspected. The procedure is repeated until the deter- minant changes sign, when a suitable interpolation scheme is used to hasten convergence to the load factor for which the determinant is zero.

The critical moment is then determined from the critical value of k using eqn 4. An eigenvector routine ~1 is also invoked at the critical value of h to

Inelastic local buckling of fabricated 1-beams 327

obtain the buckled shape. The increments in h need to be kept reasonably small to ensure that the lowest buckling mode is not missed.

3 E X P E R I M E N T A L V E R I F I C A T I O N

The accuracy of the finite strip method for inelastic buckling was investi- ga ted by comparing the theoretical solutions with tests on fabricated I- beams conducted by Holtz and Kulak. 3,4 These tests were chosen primarily to verify the assumptions made for the material properties and residual stresses in the theoretical model.

T h e theoretical and experimental buckling moments M~, normalised with respect to the plastic moment of resistance Me, are compared in Table 1. For the finite strip analysis, residual stresses were assigned to the sections in accordance with Kitipornchai and Wong-Chung, ~4 while a strain hardening modula r ratio h' of 50 was used. It can be seen that the agreement between theory and exper iment is good, with the mean ratio of theory to experiment being 0.97 with a coefficient of variation of 0.06 for the twelve beams considered. T h e inelastic buckling of beams with low values of Mn/Mp is domina ted by the effects of residual stresses, and the agreement between test and theory for these beams illustrates the accuracy of the residual stress modell ing in the finite strip treatment.

TABLE 1 Comparison of Finite Strip Solutions with Tests by Holtz and Kulak

Specimen Mn (theory) Mn theory Mp Mp (tesO test

WS- 1 0.962 1"023 0-94 WS-2 0.899 0-977 0-92 WS-3 0.920 0"893 1-03 WS-4 0.898 0.863 1"04 WS-6 0.932 0-879 1-06 WS-7-P 0.963 0.993 0.97 WS-8-P 0.944 0-954 0.99 WS-9 1.002 1-077 0.93 WS-10 0.998 1.147 0.87 WS- 11 0.959 1.066 0.90 WS-12-N 0.916 0-944 0.97 WS-13-N 0.919 0.919 1.00

mean = 0.97, c.o.v. = 0.06

328 M. A. Bradford

A non-dimensional moment-rotation plot for specimen WS-6, obtained from the first part of the finite strip analysis, is compared in Fig. 5 with that given by Holtz and Kulak. a The agreement between test and theory is reasonable, particularly for the lower beam rotations. The figure illustrates that the flexural stiffness is reduced by 30% below the linear elastic response due to the presence of residual stresses. It can also be seen that unloading takes place after attainment of the inelastic local buckling moment, and this behaviour was evident with the other tests. 3'4 Because of this, the critical moment Mn, determined by the finite strip method, is a good prediction of the ultimate strength of the section in bending.

Finally, a plot of the elastic and inelastic buckling moments Mol and M~, is given in Fig. 6 as a function of the buckling half wavelength L for specimen WS-13-N. 4 The elastic curve was obtained from the finite strip analysis presented herein by appropriate choice of the material properties. While the elastic critical moment Mot (which is greater than the yield moment My) has the familiar U-shape, the inelastic critical moment Mtt shows a very much flatter curve, with the moment being nearly constant in the range 0.3 < L/ h < 1.0. Also shown in Fig. 6 are the elastic and inelastic eigenmodes at the respective minima. While both plots display a combined flange-web buckle with significant relative web deformations, there is a difference between the elastic and inelastic buckled shapes, particularly with respect to the flange deformations.

4 DEPTH-TO-THICKNESS LIMITS FOR WEB BUCKLING

4.1 General

The finite strip analysis has been used to determine the depth-to-thickness ratios d/t for webs which correspond to a section classification of compact and semi-compact. The limiting d/t value for a compact section is that for which the inelastic buckling moment Mn equals the plastic moment My, while the limiting d/t value for a semi-compact section is that for which Mn equals the nominal yield moment My (Fig. 1).

For the analysis in this section, a strain hardening modular ratio h' of 50 was used, with ~t = 10~v. It was found that the solutions were not greatly sensitive to the value of h'. Residual stresses were assigned in accordance with Reference 14.

4.2 Compact sections

Figure 7 shows plots of the dimensionless buckling moment MII/Mp against the modified web slenderness Xw = (d/t)X/(trv/275) near Mn/Mp = 1,

Inelastic local buckling of fabricated I-beams 329

1.0

0.8

M

Rp

0.6

0.4

0.2

~ L o c a l b u c k l i n g

Linear / / ~ . , ~ e - - - ~ , -e . , , .

/ /pC ~ # / f ~ This study

I I I 1 2 3 B/Bp

Fig. 5. Moment-rotation curveforspecimenWS-6.

z o

z

z

1

\

M0~ \ \ \

\ \

\ \

\

/ /

/ /

etgen~des

0.1 i I I I

0.2 0.5 1,0 2,0

Dimensionless half wavelength L/h (log scale)

Fig. 6. Local buckling behaviour of specimen WS-13-N.

330 M. A. Bradford

1.02

MI£/M P

1.01

1.00

0.99

O. 98

70 \ 80 \ " ¢ < ' - , , "~xloo

- \ \

kf=8 • Ca lcu la ted p0~nts

Fig. 7. Parametric curves for local buckling near Mp.

where o'v is in units of N/ram 2. The curves are for bt/h = 0.12 and 0.24, and for flange slendernesses he = (b¢/73 ~/(o-v/275) of 6.0, 7.0 and 8.0. These flange slendernesses represent a compact classification in BS 5950 : Part 1.12

In te rpo la ted values of the limiting web slenderness (hw),m, for which Mn = Mp, are given in Table 2. The residual stresses were found to have little effect on the values of (h,,)~. It can be seen that the prediction of h,, = 98 given in BS 5950: Part 1 is an unconservative estimate of the critical s lenderness (k,,)m, and does not account for the variation of the critical s lenderness with he and bf/h. This variation was investigated by writing the limiting d/t ratio for the web in the form

(Xw),im = ~/(23"9/kw) (22)

where k,, is the elastic local buckling coefficient for the web (given in Table 2) which can be de te rmined from the charts in Reference 22, while 23.9 is the theoret ical value for k,, for a web in bending with simple supports. l It was found f rom the results in Table 2 that the mean value of h,,c in eqn 22 for the compac t classification was equal to 73, with a coefficient of variation of 2% for the sections considered. This low coefficient of variation means that h,,c may be considered as a near constant, so that eqn 22 can be used to de te rmine the limiting web depth-to-thickness ratios for a compact section w h e n the web buckling coefficient is known. The use of this value of h~,~ implies that the limiting value of k,, = 98 in BS 5950 : Part 1 will be correct for a section with an elastic buckling coefficient k,, = 43.

Inelastic local buckling of fabricated/-beams

TABLE 2 Parametric Results for Compact Sections

331

bf/ h hf (hwh~ k,~ hwc

0-12 6-0 88 35 73 0.12 7-0 81 30 72 0.12 8.0 76 27 72

0-24 6.0 91 37 74 0.24 7.0 90 36 73 0.24 8.0 98 41 75

mean hwc = 73, c.o.v. = 0.02

4.3 Semi-compact sections

The me thod of the previous sub-section was used to investigate the effect of the parameters bf/h and hf on the web slendernesses h~, for which Mn = My. The dimensionless buckling moments Mn/Mv are shown in Fig. 8 in the vicinity of Mn/My = 1 for the same parameter values as those used in the investigation of compact sections.

Interpolated values of the limiting web slenderness (hw)~ for a semi- compact section are given in Table 3. These values of (hw)t~ were found to depend significantly on the welding residual stresses, with the model of Reference 14 being a conservative representation of these stresses and resulting in reasonably low values of (hw)~. The value of hw = 120 in BS 5950 : Part 1 is clearly an unconservative prediction of the limiting web depth- to- thickness ratio for semi-compact sections when the residual stress pat tern of Reference 14 is used.

TABLE 3 Parametric Results for Semi-Compact Sections

bf/ h hf (Xw )tim kw Xw,

0-12 6.0 107 37 86 0.12 7.0 106 36 86 0.12 8.0 102 34 85

0.24 6.0 104 37 84 0-24 7-0 104 37 84 0.24 8-0 108 38 85

mean hws = 86, c.o.v. = 0-01

332 M. A. Bradford

1.06

1.04

MIL/M Y

1.02

I.~

0.~

0.96

0.94

\ \

\

\

U ~ ' ~ ' 8 \ "~ ,L 6 %

% %

\ ).f=7 bf/h - 0.12

0.24 . . . .

• Calculated points

Fig. 8. Parametric curves for local buckling near My.

The variation of (hw),im with the chosen parameters was again investigated by writing

(hw).m = X/(23.9/kw) (23)

where kw is the elastic buckling coefficient for the web which may be determined from Reference 22 and is given in Table 3 for the sections con- sidered herein. The mean value of h,, was found to equal 86, with a coefficient of variation of 1%, and so this value of h,, may be considered as a constant for semi-compact welded sections. The formulation of eqn 23 enables the effects of the geometrical parameters on the limiting depth-to- thickness ratio for a semi-compact section of the web to be included through the presence of the elastic buckling coefficient kw.

The prediction of the limiting modified web slenderness for semi-compact sections of 120 in BS 5950 : Part 1 thus implies an elastic buckling coefficient kw = 46. This is reasonably consistent with the value of kw = 43 implied in BS 5950 : Part 1 for compact sections.

5 CONCLUSIONS

A finite strip method of analysis for the inelastic local buckling of fabricated I-beams in bending has been presented. The method includes nonlinear

Inelastic local buckling o f fabricated l-bearra" 333

material behaviour and the residual stress patterns present in welded sections.

A method for the nonlinear analysis of the cross-section was given, and this enabled the moment of resistance and curvature to be obtained at increasing values of a load factor. The stiffness and stability matrices of the beam were then assembled at each value of this load factor. The critical load factor was that for which the determinant of the difference of the stiffness and stability matrices vanished.

Local buckling moments were calculated in this way for twelve beams tested independently. The good agreement obtained validated the assump- tions made in the finite strip method of analysis.

The limiting web slendernesses corresponding to compact and semi- compact welded I-sections were then obtained parametrically. It was shown that these slenderness limits were proportional to the square root of the elastic web local buckling coefficient for the section for both classifications. Values of the constant of proportionality were proposed for both compact and semi-compact welded I-sections, and it was shown that the limiting web slendernesses given in BS 5950 : Part 1 are unconservative for an extensive range of section geometries.

A C K N O W L E D G E M E N T

The computer program used in this paper was developed while the author was a Visiting Fellow in the Department of Engineering at the University of Warwick, UK. The support provided by the British Steel Corporation for this work is gratefully acknowledged.

REFERENCES

1. Allen, H. G. and Bulson, P. S., Background to buckling, Mc.Graw Hill (UK), 1980.

2. Climenhaga, J. J. and Johnson, R. P., Local buckling in continuous composite beams, The Structural Engineer, 50 (9) (1972) 367-74.

3. Holtz, N. M. and Kulak, G. L., Web slenderness limits for compact beams, Struct. Engng. Report No. 43, Dept. of Civil Engng., Univ. of Alberta, Canada, March 1973.

4. Holtz, N. M. and Kulak, G. L., Web slenderness limits for non-compact beams, Struct. Engng. Report No. 51, Dept. of Civil Engng., Univ. of Alberta, Canada, August 1975.

5. Johnson, D. L., Buckling of beam compression flanges, Research Report, Butler Manufacturing Co., Grandview, Missouri, USA, November 1976.

334 M. A. Bradford

6. Kemp, A. R., Local buckling of webs and flanges in plastic design to SABS 0162 : 1984, Die Siviele Ingenieur in Suid-Afrika, December 1985, 605-11.

7. Owen, D. R. J. and Hinton, E., Finite elements in plasticity theory and practice, Pineridge Press, 1980.

8. Yoshida, H., Coupled strength of local and whole bucklings of H-columns, Proc. Japan Soc. o f Civil Engrs, 243 (November 1975) 19-32 (in Japanese).

9. Mahmoud, N. S., Inelastic stability of plate structures using the finite strip method, PhD Thesis, Univ. of Sheffield, UK, 1981.

10. Plank, R. J., 'A study of the elastic-plastic instability of stiffened panels using the finite strip method', Int. Conf. of Instability and Plastic Collapse of Steel Structures, Manchester, 1983, 399--408.

11. Dawe, J. L. and Kulak, G. L., Plate instability of W shapes, J. Struct. Engng., ASCE, 110 (6) (June 1984) 1278-91.

12. British Standards Institution, BS 5950 : Part 1. Structural use o f steelwork in buildings, London, BSI, 1985.

13. Cheung, Y. K., Finite strip method in structural analysis, Oxford, UK, Pergamon Press, 1976.

14. Kitipornchai, S. and Wong-Chung, A. D., Inelastic buckling of welded mono- symmetric I-beams, J. Struct. Engng., ASCE 113 (1987) 740-56.

15. Hancock, G. J., Local, distortional and lateral buckling of I-beams, J. Struct. Div., ASCE, 1114 (ST11) (1978) 1787-98.

16. Haaijer, G., Plate buckling in the strain hardening range, J. Engng Mechs Div., ASCE, 81 (EM2) (1957) 1-47.

17. Lay, M. G., Flange local buckling in wide-flange shapes, J. Struct. Div., ASCE, 91 (ST6) (1965) 95-116.

18. Bradford, M. A., Cuk, P. E., Gizejowski, M. A. and Trahair, N. S., Inelastic lateral buckling of beam-columns, J. Struct. Engng, ASCE (1987) (in press).

19. Bradford, M. A., Buckling of beams with flexible cross-sections, PhD Thesis, Univ. of Sydney, Australia, 1983.

20. Hancock, G. J., Appendices in Local, distortional and lateral buckling of 1-beams, Research Report R312, School of Civil Engng., Univ. of Sydney, Australia, 1977.

21. Hancock, G. J., Structural buckling and vibration analysis on microcomputers, Civil Engng Trans., lnstn. Engrs Australia, CE26 (4) (1984) 327-32.

22. Hancock, G. J., Bradford, M. A. and Trahair, N. S., Web distortion and flexural-torsional buckling, J. Struct. Div., ASCE, 106 (ST7) (1980) 1557-71.