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Inelastic buckling of tapered monosymmetric I-beams
Inelastic buckling of tapered monosymmetric I-beams M. A. Bradford School of Civil Engineering, The University of New South Wales, Kensington, New South Wales 2033, Australia (Received October 1988; revised November 1988)
A finite element method is presented for analysing the inelastic lateral buckling of tapered, monosymmetrlc I-beams with general conditions of loading and restraint. The method incorporates the residual stresses produced by welding the flanges to the web. The lateral buckling solution is based on the stability of a quasi-elastic section, developed by considering the spatial distribution of yielding. The method is verified against independent solutions, and may be used to study the strength of tapered, monosymmetric beams for the lateral buckling limit state. Keywords: buckling, finite elements, inelasticity, monosymmetry, nonuniformity, yielding Tapered and monosymmetric steel girders are most usually made by welding three plates to form an I-section. Although long, slender tapered and monosymmetric beams often buckle laterally in the elastic range, the process of fabrication may reduce the critical load of short or intermediate length beams because the effective stiffnesses are reduced by yielding within the member. Yielding in a stress-relieved beam takes place when the applied moment exceeds the first yield moment, which depends only on the cross-sectional dimensions and the yield stress of the steel. However, the welding of the flanges to the web of a fabricated tapered girder results in significant residual stresses which encourage premature yielding, and consequently reduce the buckling load. It is therefore important to be able to calculate the inelastic buckling load for the strength design of welded tapered and monosymmetriC beams. The elastic lateral buckling of tapered beams has been well documented, with a summary of the earlier work being given by Kitipornchai and Trahair 1 with some references to later studies being cited by Bradford and Cuk =. On the other hand, research into the inelastic lateral buckling of tapered beams has not been thoroughly researched. Shiomi and Kurata 3 have presented a study of the strength of doubly-symmetric web-tapered beamcolumns, but restricted their solutions to members loaded by end moments. They also used the linear residual stress distributions present in rolled sections, which are not appropriate for welded beams4- e. Experimental work on tapered monosymmetric beams has been carried out by Prawel et al. 7, who considered a fillet weld on one side of the web only. The elastic buckling of monosymmetric beams has also been well researched, but limited theoretical work has been done on their inelastic buckling. Recently, a two-stage finite element method for determining the inelastic buckling load of hot-rolled doubly0141-0296/89/02119-08/$03.00 © 1989Butterworth & Co (Publishers)Ltd
symmetric I-section beam-columns in planar steel frames has been presented 8. In the first stage, the pre-buckling in-plane bending was analysed, first by a cross-section analysis for the variations of the yield and strain hardening boundaries with moment and axial force, and then by a nonlinear member analysis to determine the spatial distributions of these boundaries along the member. For the second stage, the out-of-plane bending and torsion of the member was modelled by accounting for the cross-sectional distributions of the elastic, yielded, and strain hardened regions, and by using appropriate tangent moduli of elasticity to calculate effective values of the cross-sectional properties. This second stage adapted a quasi-elastic tapered monosymmetric finite element, for which the partially yielded and strain hardened crosssection at any point along the member was represented by a tapered elastic section, obtained by transforming the thickness according to the strain hardening to elastic modular ratio Est/E. The method has been u s e d 9 t o analyse the test results of Cuk, Rogers and Trahair 1°, and the agreement between the finite element method and experiment was found to be very good. In this paper, the two-stage finite element method described above is extended to include the inelastic lateral buckling of determinate tapered, monosymmetric I-beams fabricated by shearing and welding, with general conditions of loading and with elastic restraints. The in-plane bending analysis is simplified by considering only determinate beams whose moment-curvature relationship is determined apriori. This simplification enables the finite element program to be implemented efficiently on the larger microcomputers. Following development of the theoretical model, its accuracy is demonstrated for elastic and inelastic buckling. The method provides a framework to study the inelastic lateral buckling of monosymmetric and tapered Eng. Struct. 1 989, Vol. 11, April
1 19
Buckling of monosymmetric 1-beams: lid. A. Bradford beams, so that the present limit states design rules can be evaluated and, if necessary, new more accurate design rules can be proposed.
Sfreg.~
Finite element theory
%
I
I
r
General The finite element formulation used in this paper requires a lengthwise subdivision of the beam into a number of one-dimensional elements. The axis system of each element is shown in Figure la. The z-axis is at an arbitrary reference position at the mid-height of the web, and serves as a reference system for the lateral displacement u and twists ~b shown in Figure lb. The idealized trilinear stress-strain relationship for structural steel shown in Figure 2 is used for the analysis, which incorporates a linear elastic region, plastic plateau and a linear strain hardening zone. The initial bending moment distribution within each element is determined prior to the buckling analysis, and this enables the initial moments at each of the four Gauss points along the length of the element to be specified. These moments are increased proportionally by a load factor 2 until buckling occurs. Because the beam is determinate, this increase is proportional, irrespective of the extent of plasticity developed within the tapered beam.
I
I I
I
I
I
Ey
Figure 2
Strain
Est
Idealized stress-strain curve for structural steel
cry
I:'C I ' I c ltOrr Top flange
Bottom flange
__
I~
]Zam cJwb'
Residual stresses The residual stresses assumed for the welded beams are based on the work of the Cambridge University group 4-6, Applications of these models to monosymmettle beams have been presented recently by Kitipornchai and Wong-Chung 11. Tension blocks stressed to ay are assumed to occur at the welds, as shown in Figure 3. The half-widths cr and cB of the tension blocks and compressive stresses in. the flange and web may be calculated from the simplified formulae in Kitipornchai and Wong-Chung's paper. The latter paper presented a comparison of the assumed block distribution with tests, which suggested that there is a finite gradient of residual stress between tension and compression which is not modelled in the assumptions of Kitipornchai and Wong-
Figure 3
Residual stress distributions
Chung. However, welding residual stress distributio~ similar to the block model of Kitipornchai and Wong Chung have resulted in accurate local buckling predictions for welded beams 12'~3 when compared with tests. As the elements are assumed to taper linearly, the residual stress distributions are calculated at each longitudinal node from the cross-sectional geometry at that node. The width of the tension block and compressive residual stress at each Gauss point within the element are then calculated by linear interpolation from the values at the end nodes. This method has been used for tapered members in Reference 3.
vz
/2
l ~-
a
L Elevation
Figure 1 Beam element
120
Eng. Struct. 1989, Vol. 11, April
'-1
b
Section
U
/
Buckling of monosymmetric 1-beams: M. A. Bradford Moment--curvature relationship For a given curvature p in the cross-section at a particular element and Gauss point, a trial position for the neutral axis 9 is assumed, and the applied strains 5, calculated from p, 5' and the residual strains 5, by 5,(x, y, z) = L v - ~)t, + 5,(x, y, z)
O)
The stresses o can then be found from the stress-strain curves for the steel (Figure 2) by e,
o = ~ Etdea Jr Ee r
(2)
M
8r
where E t is the tangent modulus of elasticity. The axial force P in the cross-section corresponding to ~ may then be found by numerical integration over the cross-sectional area A as
and has been shown to result in more accurate solutions. This has its basis on the dislocation theory of yielding. The stability of-the elastic core, derived from the tapered beam, constitutes the crux of the buckling problem. Because of the residual stress distribution and the moment gradient, the elastic core is nonuniform and monosymmetric, and its stability must be handled by theories applicable to beams of this type.
Displacement fields The cross-section of the quasi-elastic elements consists of two unequal flanges connected rigidly at their centrelines to the web. The reference axis is taken arbitrarily and conveniently at the web mid-heights . The lateral deflection u and twist ~ of the reference axis may be represented by cubic polynomials, so that
/"
I~) = | odA
(3)
d A
For pure bending, the condition P(~) = 0
(4)
must be satisfied, and this equation is solved for ~ in equation (1) by a chordal Newton-Raphson technique a`. Finally, the bending moment M subject to equation (4) is found from equations (1) and (2) as f M = I yodA
(5)
,4
where the integration is carried out numerically using a trapezoidal integration technique. The above procedure is repeated, at increments of curvature, until the maximum compressive stress reaches 1.5av. Arrays relating p and ~ to M at each Gauss point for each element are obtained at discrete calculation points, and the relationship is made continuous by linear interpolation between each of these calculation points. Thus, at any value of moment in the beam corresponding to AM, values ofp and are obtained, and the resulting distribution of longitudinal applied strain ~. can then be obtained from equation (1). This strain distribution at each Gauss point in each element is used for the subsequent buckling calculations.
Quasi-elastic element Under the action of external forces, a bending moment distribution AM may be determined in the determinate beam from simple statics. At each Gauss point within the elements, the applied strain e,(x, y, z) may thus be calculated as a function of the load factor 2 from equation 0). The thickness of the elastic cross-section at the Gauss points is then determined from the value of 5= within the section. The thickness is based on the modular ratio E,JE when ~, exceeds the yield strain By, and the elastic core that remains constitutes the quasi-elastic element. The use of the strain hardening modular ratio E,JE in the inelastic region rather than the tangent modulus modular ratio EJE has been adopted by several researcherss'ls,x6,
where [M] is a matrix of cubic interpolation polynomials and < =1,- •., =a > is a vector of displacement coefficients. This vector may be related to the vector {q} of nodal displacements shown in Figure 4 by suitable differentiation and substitution in equation (6), so that
The matrices [M] and [C]-1 have been given explicitly in Reference 2 for the elastic study of tapered beams.
Stiffness matrix for quasi-elastic element The element stiffness matrix [k] contains components due to the flexural and torsional stiffness of the element. The increase in strain energy Ur due to lateral deflection and twist of the quasi-elastic element can be represented as2:
'I{ L
Ur = ~
(EIr + EIjXu") ~ + (h'XEIr + EIBX~') 2
0
h2 + __~(EIr + EisXdp.)2
(8)
+ 2 h ' ( E I r - EIBXu"O') + h ( E I r - E/~Xu"O")
+ hh'(~1T + EGXO'4Y') +
6J(O')e~dz J
2
(q) =
t r ,u~,~z,$z,$1,~2 >T
Nodal displacements
Eng. Struct. 1989, Vol. 11, April
121
B u c k l i n g o f monosymmetric 1-beams: M. A. Bradford
where primes denote differentiation with respect to z. The inegration with respect to z in equation (8) is carried out using four point Gaussian quadrature 14. The top and bottom flange minor axis flexural rigidities E1r and EIn can be obtained numerically as a~/2
The element stiffness matrix may be obtained from equation (8) by suitable differentiation in equation (7), so that the strain energy stored in the quasi-elastic element due to lateral deflection and twist may be written as 1
UT = } {q}r[k]{q}
(17)
P
511` = 2E | x2dA1`
(9)
ot 0
and
The terms in [k] may be determined readily from the elastic stiffness matrix of a tapered 1-beam given in Reference 2.
a,/2 EIR = 2E f x2dAn
(10)
0
and noting that Ar and AB are tapered according to the modular ratio E,JE, and that the applied strain ~= is symmetrical about the web. Note that equations (9) and (10) are equivalent to the alternative expressions Br]2
EIT= 2 f Etx2dAr
(11)
0
and
Restraint stiffness matrix Continuous elastic restraints that may be present along a tapered monosymmetric 1-beam may significantly increase its resistance to inelastic flexural-torsional buckling. These restraints may inhibit the lateral deflection, minor axis rotation, twist or warping of the beam, or a combination of these. The translational restraint of stiffness kt may act at a distance Yt below the reference axis mid-height, and the minor axis rotational restraint of stiffness k~, may act at a distance y, below the reference axis, as shown in Figure 5a. If the torsional restraint has a stiffness k,= and the
B~
EIn = 2 ~ Etx2dAn
(12)
t/ 0
where this time dAr = Trdx and dAe = Ted~ are uniform differential areas, and E t = E when tr < trv and E t = Est when tr/> O'y. For the region of the elastic core where there is no reduction in thickness due to inelasticity, the usual elastic torsional rigidity GJ in equation (8) is used, where E G= - 2(1 + v)
4EEst 4E=(1 + v) + E
j / J - z
I
r
(13)
*r ("-Yt *)~-~
and v is Poisson's ratio. However, in the inelastic zone (where the thickness of the core is reduced by the strain hardening modular ratio), the torsional rigidity GJ is obtained by using the inelastic value G = Gst in the yielded and strain hardened regions, where G,t is the inelastic shear modulus derived by Lay 17 as Gst =
4[
Yr y
(o) Actions applied by restraints on member
(14)
Combining the elastic and inelastic values, if the tapered monosymmetric 1-section is represented as three bimetallic thin rectangular sections, then the approximation of Booker and Kitipornchai 18 can be used, that is
(GS)~ = [G - PR(C,- G,O]JR
(15)
for each rectangle, in which/~R is the ratio of the area of the plastic or strain hardened region to the area of each rectangle, and JR is the torsion constant of each rectangie. The effective torsional rigidity can then be obtained by summation as 3
G./= ~., (GJ)R + C
(16)
l
in which C is a correlation to allow for junction effects as given by El Darwish and Johnston 19 for elastic sections. 122
Eng. Struct. 1989, Vol. 11, April
(b) Actions transformed to reference axis Figure 5 Actions of continuous elastic restraints
Buckling of monosymmetric 1-beams: M. A. Bradford
warping restraint is of stiffness k,,, the restraining actions may he expressed in matrix format as2oj, {f,} = [D,]{~,}
08)
where {f,} is the vector of equivalent reference axis actions per unit length given by {f,} = < f , my, m,, b > T
(19)
as shown in Figure 5b, where b is the bimoment per unit length. The modified strain vector {St} ~ < U, U I, ~ ,
~)'
> T
(20)
can be related to the vector of nodal degrees of freedom {q} of the quasi-elastic element by suitable differentiation of equation (6). The property matrix [Dr] has been given in Reference 21. Finally, the strain energy per unit length stored due to the restraints can be written as dU, I d---f= 2 < e' > {f'}
(21)
so that on substitution of equation (18) suitable differentiation of equation (7) and integration with respect to z produces
1 U, = ~ {q}r[k,]{q}
(22)
where [k,] is the restraint elastic stiffness matrix.
Stability matrix for element The Work Vr done during buckling of the tapered monosymmetric beam is given by'` L
1
[ AM{fl*(~')2 + 2~u" - 2~'~b~b'}dz
(23)
where [0] is the element stability matrix. The effects of concentrated loads as given by equation (25) are accounted for by manipulation of the global stability matrix, and are omitted from the formulation of [01. The terms in the element stability matrix may be obtained readily from the listing given in Reference 2 for elastic tapered monosymmetric beams.
Method of solution By using suitable transformation matrices and an assembly routine'`'`, global stiffness and stability matrices may be assembled that satisfy the minimum potential condition ([K(1)] + [K,] - [G(1)]){Q} = {0}
(27)
where [K] is the global stability matrix assembled from [k], [K,] is the global restraint stiffness matrix assembled from [k J, [G] is the global stability matrix assembled from [01 and {Q} is the global buckling degrees of freedom assembled from {q}. Note that [K] and [G] depend noniinearly on the load factor 1. The load factor i is increased monotonically, and the eigenvalue ici of equation (27) is calculated by the method outlined by Hancock 23. The buckling load factor is that for which A = Aci. When the root is bracketed as i ~ - 1) > 1(")> A~]) after n load factor increments, the method of bisections 14 is used to iterate for the critial load factor. The use of the eigenvalue rather than the determinant was found to produce more manageable numbers, while the bisection method, although a first order scheme, guaranteed convergence always. An eigenvector routine' 3 was invoked at the critical load factor to obtain the buckled shape. The load factor increments need to be kept reasonably small, so that the lowest critical load factor is not missed.
0
where the integration is carried out using four point Gaussian quadrature ~4. The inelastic monosymmetry parameter/]* reflects the Wagner effect of the distribution of stresses a due to the moment AM acting on the monosymmetric section, and is given as 2
'I
/~T = ~-~
~(x'`+ y'`)dA
(24)
A
which is determined by numerical integration over the cross-section using a trapezoidal integration technique. Note that the full cross-section and not the elastic core is used for the determination of ~ . The effect of off-axis vertical load adds to the work done by
1 Va= - ~ ~ W~fiiq~
(25)
where the load W~ is applied at node i which twists ~b~ during buckling, and a~ is the height of application of the load above the reference axis. The terms u", ~ and ~' may be derived by appropriate differentiation of equation (7), so that the work done can be written as VT = {q}r[0]{q}
(26)
Accuracy of solution
Elastic tapered beams In order to obtain verification of his theoretical predictions, Kitipornchai 24 tested a series of tapered aluminium I-beams. All beams had a span of 1524 mm (60 in) and were loaded with a central concentrated load acting just above the top flange (2fi/h = 1.04). The experimental critical loads are shown in Figure 6, along with the material properties and cross-sectional dimensions. The taper constants represent the ratio of the tapered dimension at the smaller end to that at mid-span. The predictions calculated from the theoretical analysis herein using six elements have been increased by between 1.4 and 2.7% to account approximately for the fact that the major axis flexural rigidities El, are not infinitely large25 when compared with other rigidities. This has been done by multiplying them by average values of the factor'`e {(1 - EIy/EI=X1 - GJ/El=) [1 - ~'`EIJ(EI~ - GJ)L "]} 112 which allows for a finite major axis flexural rigidity El, in uniform I-beams with equal end moments. The yield stress ev was made very large in the theoretical analysis to model elastic behaviour.
Eng. Struct. 1989, Vol. 11, April 123
Buckling of monosymmetric 1-beams. M. A. Bradford A4
A,¢
I000
I
,52,
600o
I_ 8O(
/
133
_1
•
/" J
,//~,(
195.2
-,,--5.8
b / 400
/
±
/ E =65160MPo G = 25 6 5 0 MPo
E = 2x 105N/ram 2 G = 8 x 104N/ram 2 (o) Member details and material properties
kt = kry= krz = kw =
200This study
Kitipornchoi tests ~
0 Figure 6
Depth tapered • Width tapered • Thickness tapered • I I I 0.2 0.4 0.6 Taper constant
(b) Restraint stiffnesses
Figure 7 I 0.8
I 1.0
Elastically restrained m o n o s y m m e t r i c beam. (a) M e m b e r details a n d material properties. ( b ) Restraint stiffne~mes
C o m p a r i s o n s t u d y f o r elastic t a p e r e d b e a m
Table I
It can be seen from Figure 6 that the agreement between the theory presented here and the experimental values is very good for width, depth and flange thickness tapered elastic beams. This verifies the finite element model for the prediction of elastic buckling of tapered beams when the load acts away from the shear centre.
Elastically restrained doubly-symmetric beam The elastic lateral buckling of I-beams with elastic restraints has been considered by many researchers, and a listing of these studies has been given by Trahair 2°. In the latter paper, the stability of monosymmetric beams was considered, and is used here as a comparison study for elastic restraints considered in the finite element theory presented in the previous section. For this comparison, the simply supported beam shown in Figure 7 was studied using the finite element method. Six elements were used in the analysis. The finite element solutions are compared in Table 1 with the solutions obtained from the dosed form equation presented in Reference 20. For the restraint stiffnesses considered, it can be seen that the agreement between 'the closed form solutions and the finite element predictions is very dose, with the latter solutions being within 0.3% of those derived from Trahair's formula. This provides verification of the finite element method with elastic restraints. 124
0 . 4 0 6 3 N/mm 2 1482 x 103N 4.1858 x 105N 3 0 5 5 6 x 109Nmm 2
Eng. Struct. 1989, Vol. 11, April
Buckling moments of restrainedbeams (kNm)
Restraint (1) Rotational on tension flange Rotational on compression flange Translational on tension flange Translational on compression flange Torsional Warping
Trahair (1979) This study (2) (3) 28.02 296.2 28.02 136.9 47.27 61.06
28.06 296.8 28.06 137.2 47.31 61.24
Inelastic buckling of uniform beams A final comparison study was undertaken to determine the accuracy of the finite element method when applied to inelastic buckling. The comparison is for doubly symmetric and monosymmetric section beams in uniform bending, as considered by Kitipornehai and WongChung 11. The author could find no inelastic lateral buckling solutions for tapered beams with welding residual stresses. The comparison of the finite dement method with the solutions of Kitipornchai and Wong-Chung is shown in Figure 8. In the finite element model, it was assumed that the strain hardening strain e= equalled 10By, and ~ t the strain hardening modular ratio E J E was 1/33. R~idual stresses were determined in accordance with Kitipomzhai and ;Wong-Chung's recommendations, w i t h the width c = c ~ cs of the tension block being taken as 20 mm. In the th¢orretical analysis herein, a gradient of re,dual stress was also considered between tension and compres-
Buckling of monosymmetric /-beams : M. A. Bradford
1.0
~
0.8
-
0.6
~,.R.. "~
~ "--"-
" ....
M
M
\
~, 0,4 g
Kitipornchoiand Wong_ChurRK)
-E o
g 0,2
--
c= 20 rnrn ~f,~Elostic buckling
Thisstudy
=,,~
T
rt
0
I
I
0.5
I
1.0 1.5 slendernessM~Mp/M~. Comparisonstudyfor inelasticbuckling Modified
Figure 8
sion, as suggested by the test results plotted in Kitipornchai and Wong-Chung's paper. This increases the number of integration points required across the flange, at the expense of obtaining solutions one or two percent lower than those produced from the block assumptions of Figure 3. Figure 8 illustrates that the agreement between the solutions of Kitipornchai and Wong-Chung tl and the finite element method presented herein is satisfactory, with the agreement being particularly good for the monosymmetric beams. Kitipornchai and Wong-Chung validated their solutions against a few tests on welded, monosymmetric beams given by Fukumoto in an unpublished report. Since Kitipornchai and Wong-Chung's study was based on a different theoretical model to that described here, the closeness of the two analyses validates the finite element model presented in this paper. Conclusions A finite element method of analysis has been developed for studying the inelastic lateral buckling of tapered, monosymmetric I-beams. The method allows for the residual stresses produced by welding the shear-cut flanges to the web. A quasi-elastic element that allows for the additional monosymmetry and nonuniformity caused by spatial distribution of inelasticity was used for the buckling model~ The finite element method was shown to model accurately the elastic buckling of tapered beams, the elastic buckling of elastically restrained monosymmetric beams and the inelastic buckling of welded, monosymmetric beams. The method thus forms a basis for the study of the strength of tapered, monosymmetric beams for the lateral buckling limit state, and research is continuing in this area. It would be desirable to have more test data available for welded tapered and monosymmetric beams to validate the inelastic buckling solutions, and to get a better estimate of the level of residual stress that would be encountered in practice. Acknowledgement The author appreciates the comments of Dr M.M. Attard of Monash University, Australia, following a review of the manuscript.
References 1 Kitipornchai, S. and Trahair, N. S. 'Elastic stability of tapered Ibeams', J. Struct. Div. ASCE, 1972, 98 (ST3), 713-728 2 Bradford, M. A. and Cuk, P. E. 'Elastic buckling of tapered monosymmetric I.beams', J. Struct. Eng. ASCE, 1988, 114 (5), 977-996 3 Shiomi, H. and Kurata, M. 'Strength formula for tapered beamcolumns', J. Struct. Eng. ASCE, 1984, 110 (7), 1630-1643 4 Dwight, J. B. 'The effect of residual stresses on structural stability', in Residual stresses and their effects (A. J. A. Farlane, ed.), The Welding Institute, London, 1981, pp. 21-27 5 Dwight, J. B. and Moxham, K. E. 'Welded steel plates in compression', The Structural Engineer, 1969, 47 (4), 49--66 6 Young, B. W. and Schulz, G. W. 'Mechanical properties and residual stresses', Second International Colloquium on Stability of Steel Structures, Liege, Introductory Report, ECCS and IABSE, 1977, pp. 31-46 7 Prawel, S. P., Morreli, M. L. and Lee, G. C. 'Bending and buckling strength of tapered structural members', Weld. J., 1974, 53 (2), 75s-84s 8 Bradford, M. A., Cuk, P. E., Gizejowski, M. A. and Trahair, N. S. 'Inelastic lateral buckling of beam-columns', J. Struct. Eng. ASCE, 1987, 113 (11), 2259-2277 9 Bradford, M. A. and Trahair, N. S. 'Analysis of inelastic buckling tests on beam-columns', J. Struct. Eng. ASCE, 1986, 112 (3), 538-549 10 Cuk, P. E., Rogers, D. F. and Trahalr, N. S. 'Inelastic buckling of continuous steel beam-columns', ,/. Construct. Steel Res., 1986, 6, 21-55 11 Kitipornchai, S. and Wong-Chung, A. D. 'Inelastic buckling of welded monosymmetric I-beams', J. Struct. Eng. ASCE, 1987, 113 (4), 74O-756 12 Bradford, M. A. 'Local buckling analysis of composite beams', Civil Eng. Trans. IE Aust., 1986, CE28 (4), 312-317 13 Bradford, M. A. and Johnson, R. P. 'Inelastic buckling of composite bridge girders near internal supports', Proc. Instn. Cir. Engrs., London, 1987, 83 (2), 143-159 14 Hornbeck, R. W. 'Numerical methods', Quantum Publishers Inc., New York, 1975 15 Nethercot, D. A. 'Lateral buckling of tapered beams', Publications, IABSE, 1973, 33-II, 173-192 16 Trahair, N. S. 'Inelasfic lateral buckfing of beams', in Developments in the stability and strength of structures, (R. Narayanan ed.), Applied Science Publishers, Barking, 1983, vol. 2, ch. 2, pp. 35-69 17 Lay, M. G. 'Flange local buckling in wide-flange shapes', J. Struct. Div. ASCE, 1965, 91 (ST6), 95-116 18 Booker, J. R. and Kitipornchal, S. 'Torsion of multi-layered rectangular sections', J. Eng. Mechs. Div. ASCE, 1971, 97 (EM5), 1451-1468 19 El Darwish, I. A. and Johnston, B. G. 'Torsion of structural shapes', J. Struct. Div. ASCE, 1965, 91 (STI), 327-332 20 Trahalr, N. S. 'Elastic lateral buckling of continuously restrained beam-columns', The Profession of a Civil Engineer, Sydney University Press, Australia, 1979, pp. 61-73 21 Bradford, M. A. 'Lateral stability of tapered beam-columns with elastic restraints', UNI-CIV Report R243, The University of New South Wales, Australia, 1987 22 Rockey, K. C., Evans, H. R., Grifl~ths, D. W. and Nethercot, D. A. 'The finite element method', Granada Publishing, New York, 1979 23 Hancock, G. J. 'Structural buckling and vibration analyses of microcomputers', (:iv. Eng. Trans. IE Aust., 1984,CE26 (4), 327-332 24 Kitipornchai, S. 'Stability of steel structures', PhD Thesis, University of Sydney, Australia, 1973 25 Attard, M. A., Somervaille, I. J. and Kabalia, A. P. 'Lateral buckling of beams including the effects of initial curvature', Proc. Ninth Australasian Conference on the Mechanics of Structures and Materials, Sydney, Australia, 1984, pp. 23-27 26 Trahalr, N. S. and Woolcock, S. T. 'Effect ofmajor axis curvature on I.beam stability', J. Eng. Mechs. Div. ASCE 1973, 99 (EM1), 85-98
Notation The geometry is shown in Figure 1, the material properties in Figure 2, and the restraints in Figure 5. Other principal notation is as below. A a~
area of section height of W~above reference axis Eng. Struct. 1989, Vol. 11, April
125
Buckling of monosymmetric I-beams: M. A. Bradford C
{q}
[:D3
u
correction for junction effects restraint property matrix EI= warping rigidity EIx, Ely major and minor axis flexural rigidities Elf, Eln minor axis flexural rigidities of flanges tangent modulus of elasticity Et vector of restraint actions {fr} element stability matrix M GJ torsional rigidity [k] element stiffness matrix Ik,] element restraint stiffness matrix bending moment M [M] cubic interpolation matrix P axial force
126
Eng. Struct. 1989, Vol. 11, April
N 8= 8r
,l v
P o"
4,
vector of nodal degrees of freedom lateral displacement of reference axis concentrated load coordinate of neutral axis ratio of inelastic area to area of each plate inelastic monosymmetry parameter applied strain vector of restraint strains residual strain load factor elastic Poisson's ratio curvature applied stress twist of reference axis
A finite element method is presented for analysing the inelastic lateral buckling of tapered, monosymmetrlc I-beams with general conditions of loading and restraint. The method incorporates the residual stresses produced by welding the flanges to the web. The lateral buckling solution is based on the stability of a quasi-elastic section, developed by considering the spatial distribution of yielding. The method is verified against independent solutions, and may be used to study the strength of tapered, monosymmetric beams for the lateral buckling limit state. Keywords: buckling, finite elements, inelasticity, monosymmetry, nonuniformity, yielding Tapered and monosymmetric steel girders are most usually made by welding three plates to form an I-section. Although long, slender tapered and monosymmetric beams often buckle laterally in the elastic range, the process of fabrication may reduce the critical load of short or intermediate length beams because the effective stiffnesses are reduced by yielding within the member. Yielding in a stress-relieved beam takes place when the applied moment exceeds the first yield moment, which depends only on the cross-sectional dimensions and the yield stress of the steel. However, the welding of the flanges to the web of a fabricated tapered girder results in significant residual stresses which encourage premature yielding, and consequently reduce the buckling load. It is therefore important to be able to calculate the inelastic buckling load for the strength design of welded tapered and monosymmetriC beams. The elastic lateral buckling of tapered beams has been well documented, with a summary of the earlier work being given by Kitipornchai and Trahair 1 with some references to later studies being cited by Bradford and Cuk =. On the other hand, research into the inelastic lateral buckling of tapered beams has not been thoroughly researched. Shiomi and Kurata 3 have presented a study of the strength of doubly-symmetric web-tapered beamcolumns, but restricted their solutions to members loaded by end moments. They also used the linear residual stress distributions present in rolled sections, which are not appropriate for welded beams4- e. Experimental work on tapered monosymmetric beams has been carried out by Prawel et al. 7, who considered a fillet weld on one side of the web only. The elastic buckling of monosymmetric beams has also been well researched, but limited theoretical work has been done on their inelastic buckling. Recently, a two-stage finite element method for determining the inelastic buckling load of hot-rolled doubly0141-0296/89/02119-08/$03.00 © 1989Butterworth & Co (Publishers)Ltd
symmetric I-section beam-columns in planar steel frames has been presented 8. In the first stage, the pre-buckling in-plane bending was analysed, first by a cross-section analysis for the variations of the yield and strain hardening boundaries with moment and axial force, and then by a nonlinear member analysis to determine the spatial distributions of these boundaries along the member. For the second stage, the out-of-plane bending and torsion of the member was modelled by accounting for the cross-sectional distributions of the elastic, yielded, and strain hardened regions, and by using appropriate tangent moduli of elasticity to calculate effective values of the cross-sectional properties. This second stage adapted a quasi-elastic tapered monosymmetric finite element, for which the partially yielded and strain hardened crosssection at any point along the member was represented by a tapered elastic section, obtained by transforming the thickness according to the strain hardening to elastic modular ratio Est/E. The method has been u s e d 9 t o analyse the test results of Cuk, Rogers and Trahair 1°, and the agreement between the finite element method and experiment was found to be very good. In this paper, the two-stage finite element method described above is extended to include the inelastic lateral buckling of determinate tapered, monosymmetric I-beams fabricated by shearing and welding, with general conditions of loading and with elastic restraints. The in-plane bending analysis is simplified by considering only determinate beams whose moment-curvature relationship is determined apriori. This simplification enables the finite element program to be implemented efficiently on the larger microcomputers. Following development of the theoretical model, its accuracy is demonstrated for elastic and inelastic buckling. The method provides a framework to study the inelastic lateral buckling of monosymmetric and tapered Eng. Struct. 1 989, Vol. 11, April
1 19
Buckling of monosymmetric 1-beams: lid. A. Bradford beams, so that the present limit states design rules can be evaluated and, if necessary, new more accurate design rules can be proposed.
Sfreg.~
Finite element theory
%
I
I
r
General The finite element formulation used in this paper requires a lengthwise subdivision of the beam into a number of one-dimensional elements. The axis system of each element is shown in Figure la. The z-axis is at an arbitrary reference position at the mid-height of the web, and serves as a reference system for the lateral displacement u and twists ~b shown in Figure lb. The idealized trilinear stress-strain relationship for structural steel shown in Figure 2 is used for the analysis, which incorporates a linear elastic region, plastic plateau and a linear strain hardening zone. The initial bending moment distribution within each element is determined prior to the buckling analysis, and this enables the initial moments at each of the four Gauss points along the length of the element to be specified. These moments are increased proportionally by a load factor 2 until buckling occurs. Because the beam is determinate, this increase is proportional, irrespective of the extent of plasticity developed within the tapered beam.
I
I I
I
I
I
Ey
Figure 2
Strain
Est
Idealized stress-strain curve for structural steel
cry
I:'C I ' I c ltOrr Top flange
Bottom flange
__
I~
]Zam cJwb'
Residual stresses The residual stresses assumed for the welded beams are based on the work of the Cambridge University group 4-6, Applications of these models to monosymmettle beams have been presented recently by Kitipornchai and Wong-Chung 11. Tension blocks stressed to ay are assumed to occur at the welds, as shown in Figure 3. The half-widths cr and cB of the tension blocks and compressive stresses in. the flange and web may be calculated from the simplified formulae in Kitipornchai and Wong-Chung's paper. The latter paper presented a comparison of the assumed block distribution with tests, which suggested that there is a finite gradient of residual stress between tension and compression which is not modelled in the assumptions of Kitipornchai and Wong-
Figure 3
Residual stress distributions
Chung. However, welding residual stress distributio~ similar to the block model of Kitipornchai and Wong Chung have resulted in accurate local buckling predictions for welded beams 12'~3 when compared with tests. As the elements are assumed to taper linearly, the residual stress distributions are calculated at each longitudinal node from the cross-sectional geometry at that node. The width of the tension block and compressive residual stress at each Gauss point within the element are then calculated by linear interpolation from the values at the end nodes. This method has been used for tapered members in Reference 3.
vz
/2
l ~-
a
L Elevation
Figure 1 Beam element
120
Eng. Struct. 1989, Vol. 11, April
'-1
b
Section
U
/
Buckling of monosymmetric 1-beams: M. A. Bradford Moment--curvature relationship For a given curvature p in the cross-section at a particular element and Gauss point, a trial position for the neutral axis 9 is assumed, and the applied strains 5, calculated from p, 5' and the residual strains 5, by 5,(x, y, z) = L v - ~)t, + 5,(x, y, z)
O)
The stresses o can then be found from the stress-strain curves for the steel (Figure 2) by e,
o = ~ Etdea Jr Ee r
(2)
M
8r
where E t is the tangent modulus of elasticity. The axial force P in the cross-section corresponding to ~ may then be found by numerical integration over the cross-sectional area A as
and has been shown to result in more accurate solutions. This has its basis on the dislocation theory of yielding. The stability of-the elastic core, derived from the tapered beam, constitutes the crux of the buckling problem. Because of the residual stress distribution and the moment gradient, the elastic core is nonuniform and monosymmetric, and its stability must be handled by theories applicable to beams of this type.
Displacement fields The cross-section of the quasi-elastic elements consists of two unequal flanges connected rigidly at their centrelines to the web. The reference axis is taken arbitrarily and conveniently at the web mid-heights . The lateral deflection u and twist ~ of the reference axis may be represented by cubic polynomials, so that
/"
I~) = | odA
(3)
d A
For pure bending, the condition P(~) = 0
(4)
must be satisfied, and this equation is solved for ~ in equation (1) by a chordal Newton-Raphson technique a`. Finally, the bending moment M subject to equation (4) is found from equations (1) and (2) as f M = I yodA
(5)
,4
where the integration is carried out numerically using a trapezoidal integration technique. The above procedure is repeated, at increments of curvature, until the maximum compressive stress reaches 1.5av. Arrays relating p and ~ to M at each Gauss point for each element are obtained at discrete calculation points, and the relationship is made continuous by linear interpolation between each of these calculation points. Thus, at any value of moment in the beam corresponding to AM, values ofp and are obtained, and the resulting distribution of longitudinal applied strain ~. can then be obtained from equation (1). This strain distribution at each Gauss point in each element is used for the subsequent buckling calculations.
Quasi-elastic element Under the action of external forces, a bending moment distribution AM may be determined in the determinate beam from simple statics. At each Gauss point within the elements, the applied strain e,(x, y, z) may thus be calculated as a function of the load factor 2 from equation 0). The thickness of the elastic cross-section at the Gauss points is then determined from the value of 5= within the section. The thickness is based on the modular ratio E,JE when ~, exceeds the yield strain By, and the elastic core that remains constitutes the quasi-elastic element. The use of the strain hardening modular ratio E,JE in the inelastic region rather than the tangent modulus modular ratio EJE has been adopted by several researcherss'ls,x6,
where [M] is a matrix of cubic interpolation polynomials and < =1,- •., =a > is a vector of displacement coefficients. This vector may be related to the vector {q} of nodal displacements shown in Figure 4 by suitable differentiation and substitution in equation (6), so that
The matrices [M] and [C]-1 have been given explicitly in Reference 2 for the elastic study of tapered beams.
Stiffness matrix for quasi-elastic element The element stiffness matrix [k] contains components due to the flexural and torsional stiffness of the element. The increase in strain energy Ur due to lateral deflection and twist of the quasi-elastic element can be represented as2:
'I{ L
Ur = ~
(EIr + EIjXu") ~ + (h'XEIr + EIBX~') 2
0
h2 + __~(EIr + EisXdp.)2
(8)
+ 2 h ' ( E I r - EIBXu"O') + h ( E I r - E/~Xu"O")
+ hh'(~1T + EGXO'4Y') +
6J(O')e~dz J
2
(q) =
t r ,u~,~z,$z,$1,~2 >T
Nodal displacements
Eng. Struct. 1989, Vol. 11, April
121
B u c k l i n g o f monosymmetric 1-beams: M. A. Bradford
where primes denote differentiation with respect to z. The inegration with respect to z in equation (8) is carried out using four point Gaussian quadrature 14. The top and bottom flange minor axis flexural rigidities E1r and EIn can be obtained numerically as a~/2
The element stiffness matrix may be obtained from equation (8) by suitable differentiation in equation (7), so that the strain energy stored in the quasi-elastic element due to lateral deflection and twist may be written as 1
UT = } {q}r[k]{q}
(17)
P
511` = 2E | x2dA1`
(9)
ot 0
and
The terms in [k] may be determined readily from the elastic stiffness matrix of a tapered 1-beam given in Reference 2.
a,/2 EIR = 2E f x2dAn
(10)
0
and noting that Ar and AB are tapered according to the modular ratio E,JE, and that the applied strain ~= is symmetrical about the web. Note that equations (9) and (10) are equivalent to the alternative expressions Br]2
EIT= 2 f Etx2dAr
(11)
0
and
Restraint stiffness matrix Continuous elastic restraints that may be present along a tapered monosymmetric 1-beam may significantly increase its resistance to inelastic flexural-torsional buckling. These restraints may inhibit the lateral deflection, minor axis rotation, twist or warping of the beam, or a combination of these. The translational restraint of stiffness kt may act at a distance Yt below the reference axis mid-height, and the minor axis rotational restraint of stiffness k~, may act at a distance y, below the reference axis, as shown in Figure 5a. If the torsional restraint has a stiffness k,= and the
B~
EIn = 2 ~ Etx2dAn
(12)
t/ 0
where this time dAr = Trdx and dAe = Ted~ are uniform differential areas, and E t = E when tr < trv and E t = Est when tr/> O'y. For the region of the elastic core where there is no reduction in thickness due to inelasticity, the usual elastic torsional rigidity GJ in equation (8) is used, where E G= - 2(1 + v)
4EEst 4E=(1 + v) + E
j / J - z
I
r
(13)
*r ("-Yt *)~-~
and v is Poisson's ratio. However, in the inelastic zone (where the thickness of the core is reduced by the strain hardening modular ratio), the torsional rigidity GJ is obtained by using the inelastic value G = Gst in the yielded and strain hardened regions, where G,t is the inelastic shear modulus derived by Lay 17 as Gst =
4[
Yr y
(o) Actions applied by restraints on member
(14)
Combining the elastic and inelastic values, if the tapered monosymmetric 1-section is represented as three bimetallic thin rectangular sections, then the approximation of Booker and Kitipornchai 18 can be used, that is
(GS)~ = [G - PR(C,- G,O]JR
(15)
for each rectangle, in which/~R is the ratio of the area of the plastic or strain hardened region to the area of each rectangle, and JR is the torsion constant of each rectangie. The effective torsional rigidity can then be obtained by summation as 3
G./= ~., (GJ)R + C
(16)
l
in which C is a correlation to allow for junction effects as given by El Darwish and Johnston 19 for elastic sections. 122
Eng. Struct. 1989, Vol. 11, April
(b) Actions transformed to reference axis Figure 5 Actions of continuous elastic restraints
Buckling of monosymmetric 1-beams: M. A. Bradford
warping restraint is of stiffness k,,, the restraining actions may he expressed in matrix format as2oj, {f,} = [D,]{~,}
08)
where {f,} is the vector of equivalent reference axis actions per unit length given by {f,} = < f , my, m,, b > T
(19)
as shown in Figure 5b, where b is the bimoment per unit length. The modified strain vector {St} ~ < U, U I, ~ ,
~)'
> T
(20)
can be related to the vector of nodal degrees of freedom {q} of the quasi-elastic element by suitable differentiation of equation (6). The property matrix [Dr] has been given in Reference 21. Finally, the strain energy per unit length stored due to the restraints can be written as dU, I d---f= 2 < e' > {f'}
(21)
so that on substitution of equation (18) suitable differentiation of equation (7) and integration with respect to z produces
1 U, = ~ {q}r[k,]{q}
(22)
where [k,] is the restraint elastic stiffness matrix.
Stability matrix for element The Work Vr done during buckling of the tapered monosymmetric beam is given by'` L
1
[ AM{fl*(~')2 + 2~u" - 2~'~b~b'}dz
(23)
where [0] is the element stability matrix. The effects of concentrated loads as given by equation (25) are accounted for by manipulation of the global stability matrix, and are omitted from the formulation of [01. The terms in the element stability matrix may be obtained readily from the listing given in Reference 2 for elastic tapered monosymmetric beams.
Method of solution By using suitable transformation matrices and an assembly routine'`'`, global stiffness and stability matrices may be assembled that satisfy the minimum potential condition ([K(1)] + [K,] - [G(1)]){Q} = {0}
(27)
where [K] is the global stability matrix assembled from [k], [K,] is the global restraint stiffness matrix assembled from [k J, [G] is the global stability matrix assembled from [01 and {Q} is the global buckling degrees of freedom assembled from {q}. Note that [K] and [G] depend noniinearly on the load factor 1. The load factor i is increased monotonically, and the eigenvalue ici of equation (27) is calculated by the method outlined by Hancock 23. The buckling load factor is that for which A = Aci. When the root is bracketed as i ~ - 1) > 1(")> A~]) after n load factor increments, the method of bisections 14 is used to iterate for the critial load factor. The use of the eigenvalue rather than the determinant was found to produce more manageable numbers, while the bisection method, although a first order scheme, guaranteed convergence always. An eigenvector routine' 3 was invoked at the critical load factor to obtain the buckled shape. The load factor increments need to be kept reasonably small, so that the lowest critical load factor is not missed.
0
where the integration is carried out using four point Gaussian quadrature ~4. The inelastic monosymmetry parameter/]* reflects the Wagner effect of the distribution of stresses a due to the moment AM acting on the monosymmetric section, and is given as 2
'I
/~T = ~-~
~(x'`+ y'`)dA
(24)
A
which is determined by numerical integration over the cross-section using a trapezoidal integration technique. Note that the full cross-section and not the elastic core is used for the determination of ~ . The effect of off-axis vertical load adds to the work done by
1 Va= - ~ ~ W~fiiq~
(25)
where the load W~ is applied at node i which twists ~b~ during buckling, and a~ is the height of application of the load above the reference axis. The terms u", ~ and ~' may be derived by appropriate differentiation of equation (7), so that the work done can be written as VT = {q}r[0]{q}
(26)
Accuracy of solution
Elastic tapered beams In order to obtain verification of his theoretical predictions, Kitipornchai 24 tested a series of tapered aluminium I-beams. All beams had a span of 1524 mm (60 in) and were loaded with a central concentrated load acting just above the top flange (2fi/h = 1.04). The experimental critical loads are shown in Figure 6, along with the material properties and cross-sectional dimensions. The taper constants represent the ratio of the tapered dimension at the smaller end to that at mid-span. The predictions calculated from the theoretical analysis herein using six elements have been increased by between 1.4 and 2.7% to account approximately for the fact that the major axis flexural rigidities El, are not infinitely large25 when compared with other rigidities. This has been done by multiplying them by average values of the factor'`e {(1 - EIy/EI=X1 - GJ/El=) [1 - ~'`EIJ(EI~ - GJ)L "]} 112 which allows for a finite major axis flexural rigidity El, in uniform I-beams with equal end moments. The yield stress ev was made very large in the theoretical analysis to model elastic behaviour.
Eng. Struct. 1989, Vol. 11, April 123
Buckling of monosymmetric 1-beams. M. A. Bradford A4
A,¢
I000
I
,52,
600o
I_ 8O(
/
133
_1
•
/" J
,//~,(
195.2
-,,--5.8
b / 400
/
±
/ E =65160MPo G = 25 6 5 0 MPo
E = 2x 105N/ram 2 G = 8 x 104N/ram 2 (o) Member details and material properties
kt = kry= krz = kw =
200This study
Kitipornchoi tests ~
0 Figure 6
Depth tapered • Width tapered • Thickness tapered • I I I 0.2 0.4 0.6 Taper constant
(b) Restraint stiffnesses
Figure 7 I 0.8
I 1.0
Elastically restrained m o n o s y m m e t r i c beam. (a) M e m b e r details a n d material properties. ( b ) Restraint stiffne~mes
C o m p a r i s o n s t u d y f o r elastic t a p e r e d b e a m
Table I
It can be seen from Figure 6 that the agreement between the theory presented here and the experimental values is very good for width, depth and flange thickness tapered elastic beams. This verifies the finite element model for the prediction of elastic buckling of tapered beams when the load acts away from the shear centre.
Elastically restrained doubly-symmetric beam The elastic lateral buckling of I-beams with elastic restraints has been considered by many researchers, and a listing of these studies has been given by Trahair 2°. In the latter paper, the stability of monosymmetric beams was considered, and is used here as a comparison study for elastic restraints considered in the finite element theory presented in the previous section. For this comparison, the simply supported beam shown in Figure 7 was studied using the finite element method. Six elements were used in the analysis. The finite element solutions are compared in Table 1 with the solutions obtained from the dosed form equation presented in Reference 20. For the restraint stiffnesses considered, it can be seen that the agreement between 'the closed form solutions and the finite element predictions is very dose, with the latter solutions being within 0.3% of those derived from Trahair's formula. This provides verification of the finite element method with elastic restraints. 124
0 . 4 0 6 3 N/mm 2 1482 x 103N 4.1858 x 105N 3 0 5 5 6 x 109Nmm 2
Eng. Struct. 1989, Vol. 11, April
Buckling moments of restrainedbeams (kNm)
Restraint (1) Rotational on tension flange Rotational on compression flange Translational on tension flange Translational on compression flange Torsional Warping
Trahair (1979) This study (2) (3) 28.02 296.2 28.02 136.9 47.27 61.06
28.06 296.8 28.06 137.2 47.31 61.24
Inelastic buckling of uniform beams A final comparison study was undertaken to determine the accuracy of the finite element method when applied to inelastic buckling. The comparison is for doubly symmetric and monosymmetric section beams in uniform bending, as considered by Kitipornehai and WongChung 11. The author could find no inelastic lateral buckling solutions for tapered beams with welding residual stresses. The comparison of the finite dement method with the solutions of Kitipornchai and Wong-Chung is shown in Figure 8. In the finite element model, it was assumed that the strain hardening strain e= equalled 10By, and ~ t the strain hardening modular ratio E J E was 1/33. R~idual stresses were determined in accordance with Kitipomzhai and ;Wong-Chung's recommendations, w i t h the width c = c ~ cs of the tension block being taken as 20 mm. In the th¢orretical analysis herein, a gradient of re,dual stress was also considered between tension and compres-
Buckling of monosymmetric /-beams : M. A. Bradford
1.0
~
0.8
-
0.6
~,.R.. "~
~ "--"-
" ....
M
M
\
~, 0,4 g
Kitipornchoiand Wong_ChurRK)
-E o
g 0,2
--
c= 20 rnrn ~f,~Elostic buckling
Thisstudy
=,,~
T
rt
0
I
I
0.5
I
1.0 1.5 slendernessM~Mp/M~. Comparisonstudyfor inelasticbuckling Modified
Figure 8
sion, as suggested by the test results plotted in Kitipornchai and Wong-Chung's paper. This increases the number of integration points required across the flange, at the expense of obtaining solutions one or two percent lower than those produced from the block assumptions of Figure 3. Figure 8 illustrates that the agreement between the solutions of Kitipornchai and Wong-Chung tl and the finite element method presented herein is satisfactory, with the agreement being particularly good for the monosymmetric beams. Kitipornchai and Wong-Chung validated their solutions against a few tests on welded, monosymmetric beams given by Fukumoto in an unpublished report. Since Kitipornchai and Wong-Chung's study was based on a different theoretical model to that described here, the closeness of the two analyses validates the finite element model presented in this paper. Conclusions A finite element method of analysis has been developed for studying the inelastic lateral buckling of tapered, monosymmetric I-beams. The method allows for the residual stresses produced by welding the shear-cut flanges to the web. A quasi-elastic element that allows for the additional monosymmetry and nonuniformity caused by spatial distribution of inelasticity was used for the buckling model~ The finite element method was shown to model accurately the elastic buckling of tapered beams, the elastic buckling of elastically restrained monosymmetric beams and the inelastic buckling of welded, monosymmetric beams. The method thus forms a basis for the study of the strength of tapered, monosymmetric beams for the lateral buckling limit state, and research is continuing in this area. It would be desirable to have more test data available for welded tapered and monosymmetric beams to validate the inelastic buckling solutions, and to get a better estimate of the level of residual stress that would be encountered in practice. Acknowledgement The author appreciates the comments of Dr M.M. Attard of Monash University, Australia, following a review of the manuscript.
References 1 Kitipornchai, S. and Trahair, N. S. 'Elastic stability of tapered Ibeams', J. Struct. Div. ASCE, 1972, 98 (ST3), 713-728 2 Bradford, M. A. and Cuk, P. E. 'Elastic buckling of tapered monosymmetric I.beams', J. Struct. Eng. ASCE, 1988, 114 (5), 977-996 3 Shiomi, H. and Kurata, M. 'Strength formula for tapered beamcolumns', J. Struct. Eng. ASCE, 1984, 110 (7), 1630-1643 4 Dwight, J. B. 'The effect of residual stresses on structural stability', in Residual stresses and their effects (A. J. A. Farlane, ed.), The Welding Institute, London, 1981, pp. 21-27 5 Dwight, J. B. and Moxham, K. E. 'Welded steel plates in compression', The Structural Engineer, 1969, 47 (4), 49--66 6 Young, B. W. and Schulz, G. W. 'Mechanical properties and residual stresses', Second International Colloquium on Stability of Steel Structures, Liege, Introductory Report, ECCS and IABSE, 1977, pp. 31-46 7 Prawel, S. P., Morreli, M. L. and Lee, G. C. 'Bending and buckling strength of tapered structural members', Weld. J., 1974, 53 (2), 75s-84s 8 Bradford, M. A., Cuk, P. E., Gizejowski, M. A. and Trahair, N. S. 'Inelastic lateral buckling of beam-columns', J. Struct. Eng. ASCE, 1987, 113 (11), 2259-2277 9 Bradford, M. A. and Trahair, N. S. 'Analysis of inelastic buckling tests on beam-columns', J. Struct. Eng. ASCE, 1986, 112 (3), 538-549 10 Cuk, P. E., Rogers, D. F. and Trahalr, N. S. 'Inelastic buckling of continuous steel beam-columns', ,/. Construct. Steel Res., 1986, 6, 21-55 11 Kitipornchai, S. and Wong-Chung, A. D. 'Inelastic buckling of welded monosymmetric I-beams', J. Struct. Eng. ASCE, 1987, 113 (4), 74O-756 12 Bradford, M. A. 'Local buckling analysis of composite beams', Civil Eng. Trans. IE Aust., 1986, CE28 (4), 312-317 13 Bradford, M. A. and Johnson, R. P. 'Inelastic buckling of composite bridge girders near internal supports', Proc. Instn. Cir. Engrs., London, 1987, 83 (2), 143-159 14 Hornbeck, R. W. 'Numerical methods', Quantum Publishers Inc., New York, 1975 15 Nethercot, D. A. 'Lateral buckling of tapered beams', Publications, IABSE, 1973, 33-II, 173-192 16 Trahair, N. S. 'Inelasfic lateral buckfing of beams', in Developments in the stability and strength of structures, (R. Narayanan ed.), Applied Science Publishers, Barking, 1983, vol. 2, ch. 2, pp. 35-69 17 Lay, M. G. 'Flange local buckling in wide-flange shapes', J. Struct. Div. ASCE, 1965, 91 (ST6), 95-116 18 Booker, J. R. and Kitipornchal, S. 'Torsion of multi-layered rectangular sections', J. Eng. Mechs. Div. ASCE, 1971, 97 (EM5), 1451-1468 19 El Darwish, I. A. and Johnston, B. G. 'Torsion of structural shapes', J. Struct. Div. ASCE, 1965, 91 (STI), 327-332 20 Trahalr, N. S. 'Elastic lateral buckling of continuously restrained beam-columns', The Profession of a Civil Engineer, Sydney University Press, Australia, 1979, pp. 61-73 21 Bradford, M. A. 'Lateral stability of tapered beam-columns with elastic restraints', UNI-CIV Report R243, The University of New South Wales, Australia, 1987 22 Rockey, K. C., Evans, H. R., Grifl~ths, D. W. and Nethercot, D. A. 'The finite element method', Granada Publishing, New York, 1979 23 Hancock, G. J. 'Structural buckling and vibration analyses of microcomputers', (:iv. Eng. Trans. IE Aust., 1984,CE26 (4), 327-332 24 Kitipornchai, S. 'Stability of steel structures', PhD Thesis, University of Sydney, Australia, 1973 25 Attard, M. A., Somervaille, I. J. and Kabalia, A. P. 'Lateral buckling of beams including the effects of initial curvature', Proc. Ninth Australasian Conference on the Mechanics of Structures and Materials, Sydney, Australia, 1984, pp. 23-27 26 Trahalr, N. S. and Woolcock, S. T. 'Effect ofmajor axis curvature on I.beam stability', J. Eng. Mechs. Div. ASCE 1973, 99 (EM1), 85-98
Notation The geometry is shown in Figure 1, the material properties in Figure 2, and the restraints in Figure 5. Other principal notation is as below. A a~
area of section height of W~above reference axis Eng. Struct. 1989, Vol. 11, April
125
Buckling of monosymmetric I-beams: M. A. Bradford C
{q}
[:D3
u
correction for junction effects restraint property matrix EI= warping rigidity EIx, Ely major and minor axis flexural rigidities Elf, Eln minor axis flexural rigidities of flanges tangent modulus of elasticity Et vector of restraint actions {fr} element stability matrix M GJ torsional rigidity [k] element stiffness matrix Ik,] element restraint stiffness matrix bending moment M [M] cubic interpolation matrix P axial force
126
Eng. Struct. 1989, Vol. 11, April
N 8= 8r
,l v
P o"
4,
vector of nodal degrees of freedom lateral displacement of reference axis concentrated load coordinate of neutral axis ratio of inelastic area to area of each plate inelastic monosymmetry parameter applied strain vector of restraint strains residual strain load factor elastic Poisson's ratio curvature applied stress twist of reference axis