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Nonlinear elastic response of locally buckled thin-walled beam-columns
Thin-Walled Structures 5 (1987) 211-226
Nonlinear Elastic Response of Locally Buckled Thin-walled Beam-columns
A. J. Davids and G. J. Hancock School of Civil and Mining Engineering, University of Sydney, NSW 2006, Australia (Received 18 September 1986; revised version received 24 November 1986: accepted 3() January 1987)
ABST~A('T A theory is described which combines the finite strip method of nonlinear elastic analysis of locally buckled thin-walled sections with the in)qtwnce co<~t4cient method of attalvsis of beam-columns. The theory combines the advantages of both methods of analysis to produce a computationally e~licient procedure jbr the estimation of the overall nonlinear elastic response of locall3' buckled thin-walled beam-columns. The theory is used to predict the previously reported measured behaviour of l-section test specimens of Bi~laard and Fisher. The theory is also compared with theoretical responses of eccentrically loaded thin-walled beam-cohmtns analvsed by an asymptotic theory. An example oJ a locally huckled thin-walled beam-cohtmn is studied in detail and some observations are made concerning the nonlinear response ~['members of this type.
NOTATION h
{dvl,{dv'l [D] F(k) h
[J]
Flange width. Vectors of small changes of {v} and {v' } vectors respectively. Matrix of finite difference coefficients. Initial uniform loading eccentricity about the minor principal axis. Transverse force at node k. Segment length. Influence coefficient matrix. 211
Thin-Walled Structures 0263-8231/87/$03.50 © Elsevier Applied Science Publishers Ltd, England, 1987. Printed in Great Britain
212
k ( L
L/r M M 1, Mz M ¢f(k)
M~.(k) Mi(k) [M] n
P
[P] [R] t
u ut~ u2 V VO
v(k) {v.} {v'} Wl~ 1422, 1423 at
E E/
{o} ~E ~e ~u
A.J. Davids. G. J. Hancock
Node and segment number. Local buckle half-wavelength. Beam-column length. Column slenderness. Bending m o m e n t of a segment resultant from curvature about the minor principal axis. E n d moments on a beam-column at nodes 1 and n +-J respectively. External m o m e n t at node k due to transverse forces (kth term of vector {M,r}). External moment at node k due to end moments M ,~ M~ tkth term of vector {M,,,}). Internal moment of resistance at node k (kth term of vector {M,}). Matrix of bending moments. Number of local buckle halfwaves. Axial load. Matrix of axial loads. Matrix of finite difference coefficients. Thickness of I-section flange. Axial shortening. Cross-sectional shape of first and second local buckling modes under uniform compression. Beam-column central lateral deflection. Magnitude of initial sinusoidal overall geometric imperfection. Lateral displacement at node k (kth term of vector {v} )~ Initial value of {v} vector. Calculated lateral displacement vector. Local buckling displacements. Maximum amplitude of local geometric imperfection. Flexural rigidity about the minor principal axis of kth segment at constant axial load. Uniform axial compressive strain. Axial strain of equilibrium positions ofjth column of a matrix in the database. Value of ~ at node k (kth term of vector {~}). Nodal rotations vector. Axial load divided by cross-sectional area. Euler buckling stress. Local buckling axial load divided by cross-sectional area. Maximum value of or. Uniform curvature about the minor principal axis.
Nonlinear elastic response of locally buckled thin-walled beam-columns
,/,(k)
213
Curvature of equilibrium positions of ith row of a matrix in the database. Value of ch at node k (kth term of vector {~b}).
1 INTRODUCTION Local buckling of a thin-walled structural member leads to a reduction in its flexural rigidity. This reduction will result in a concentrically loaded thinwalled column buckling in an overall mode at a load less than its classical Euler buckling load, even when the member remains elastic. For eccentrically compressed members, which induce bending from the commencement of loading, the lateral deflections are amplified as a result of the reduced flexural rigidity caused by local buckling. This amplification of displacements leads to a further reduction in the ultimate load attained by the member. The purpose of this paper is to describe and verify an analytical m e t h o d for determining the nonlinear load-defection behaviour, and hence ultimate load, of locally buckled thin-walled beam-columns. The buckling loads for overall buckling of locally buckled thin-walled columns have been estimated by several researchers '-4 by substituting reduced flexurai rigidities into the Euler buckling formulae. 5 The reduced section flexural rigidities were calculated in these papers by various methods, including the Winter effective section method 6 and a finite strip nonlinear analysis. 7.8For eccentrically loaded members, bifurcation into an overall mode does not occur. Consequently, a nonlinear analysis of a beam-column with variable axial and flexural rigidities is required. The variability results from modulation of the local buckling deformations along the length of a thin-walled member. Koiter and Van der Neut 9developed an approximate analytical model based on the concept of shape modulation of short wavelength local buckling modes by the long wavelength overall buckling mode. More recent theoretical approaches by Benito and Sridharan t°-t2 and Sridharan ~3 have employed a combination of the finite strip ~4and mode interaction ~5methods of structural analysis to investigate the interaction of local and Overall buckling in thin-walled beam-columns. Sridharan and Ali 16'17 have modified this approach to include a more satisfactory amplitude modulation technique and more general member end conditions. In this paper, a theory is presented which combines the finite strip method of nonlinear elastic analysis of locally buckled thin-walled sectionsSwith the influence coefficient method of nonlinear analysis of beam-columns developed by Chen. x8.~9The finite strip analysis includes the effects of residual compressive strain in the plate elements forming the section, and
214
A. J. Davids, G. J. Hancock
local geometric imperfections. The nonlinear beam-column analysis includes overall geometric imperfections, general member end conditions and general loading. The theory results in a computationally efficient procedure for the estimation of the nonlinear elastic response of isolated beam-columns with locally buckled thin-walled cross-sections. Of interest also is the rate at which axial load reduces in the post-ultimate collapse state of a thin-walled beam-column, since this may be relevant to the formation of the philosophy of design code rules. The analytical method predicts this unloading behaviour for members which remain elastic.
2 THEORY
2.1 Assumptions The combination of the finite strip nonlinear elastic analysis of locally buckled thin-walled sections and the influence coefficient method for the nonlinear analysis of beam-columns requires certain assumptions. These are:
(i) The beam-column can be replaced by a series of segments of length equal to the half-wavelength of the local buckles. (ii) The m o m e n t and axial force resisted by a segment subjected to defined values of curvature and axial strain, can be determined for each segment in isolation using the finite strip nonlinear analysis. (iii) The average values of moment and axial force along a segment can bc used to describe that segment in the nonlinear analysis of the beamcolumn. (iv) The only interconnection between the segments is by way of equilibrium of the section stress resultants of moment, shear and axial force, as well as compatibility of centreline displacements of adjacent segments. (v) The effect of section shear on the local buckling response of a segment is ignored.
2.2 Finite strip post-local buckling analysis The elastic post-local buckling response of a thin-walled section is modelled using the finite strip nonlinear elastic analysis described in Ref. 8. The thin-walled section is divided into a series o f longitudinal strips which are joined together at nodal lines as shown in Fig. 1. The section length (t) is equal to the half-wavelength for local buckling, which gives a minimum local
Nonlinear elastic response of locally buckled thin-walled beam-columns
///•a
215
g>
Stmps w2
oration=
~-
d)b~
xlat force }ResuLtant ~"~"~ Rotation ~LPrescribe d g moment Forces ~Uniform ~,,, J[3isplacements
Fig. 1. Locallybuckledthin-walledsection. buckling load under the initial conditions of compression, bending and residual strain. This length is calculated using a finite strip analysis as described by Hancock, 2°2~ and is assumed invariant throughout the subsequent nonlinear analysis. The section is assumed to be compressed between rigid frictionless end plattens. The strip displacement functions are identical with those developed in Ref. 8 and described as Version I in Ref. 14. Longitudinal waveform changes of the flexural displacements are not included in the analysis, but this was considered in Ref. 7 to be satisfactory for loads less than approximately 1.8 times the local buckling load. The local geometric imperfections of maximum amplitude equal to c~t are assumed to be of the same shape and half-wavelength as the local buckling mode. Longitudinal membrane residual strains (resulting from such p h e n o m e n a as weld shrinkage after manufacture) have been included in an approximate way, as described in Ref, 22, by superimposing a longitudinal strain distribution in addition to that resulting from compression and bending. The nonlinear analysis involves solution of the equilibrium equations by a Newton-Raphson iteration technique for a prescribed set of nodal line compressive strains. Thin-walled columns in uniform compression v and beams in uniform bending s about the major principal axis have been studied previously. In this paper, the nodal line compressive strains of the beam-column section are prescribed to be a linear combination of the displacements due to uniform compressive strain (E) and curvature (oh) about the minor principal axis of
A.J. Davids, G. J. Hancock
216
the I-section, as shown in Fig. 1. The induced stress resultants of axial load (P) and bending m o m e n t (M) about the minor principal axis are also shown in Fig. 1. 2.3 Influence coefficient method
2.3.1 General The influence coefficient method ([CM) of analysis as described by Chen and Atsutat8 and Han and Chen'9 has been used to model the nonlinear response of the beam-column. In this method, the continuous beam-column is modelled as a series of nodes at which equilibrium is sought by an iterative p r o c e d u r e involving the nodal displacements. The beam-column under Hatf waveteng_~L.j_ N ode 2 Node 1 1
I
i
;(k) / ~Node k
~
Segment k ~ Node n*1
P A
B p
Equitibrium position v(kl u/2
L
~L u/2~ i
Fig. 2. Thin-walledbeam-column. study is shown in its discretised form in Fig. 2. It contains n + 1 nodes which define the centres of n - 1 internal segments and the ends of the two end segments. The internal segments correspond with the lengths of locally buckled sections of length equal to their half-wavelength for local buckling ( t ) . The end segments each represent the state of the beam-column over lengths equal to a quarter-wavelength only. Hence the beam-column contains n local buckle halfwaves between nodes 1 and n - l. The lengths A to 1 and n + t to B in Fig. 2 are assumed to be rigid bodies. Full details of the theory of the influence coefficient method of analysis are described in Ref. 23. A brief s u m m a r y follows.
2.3.2 Equilibrium equations T h e external actions on the beam-column consist of end axial forces P along line A B , external m o m e n t s M 1, M2 applied at nodes 1 and n + 1 respectively and transverse loads F(k) applied at each internal node k as shown in Fig. 2~ M o m e n t equilibrium at node k can be expressed as M~(k) + M.m(k) + M~f(k) - Pv(k) = 0
(1)
Nonlinear elastic response o]'locally buckled thin-walled beam-columns
217
M~(k) is the internal m o m e n t of resistance of the thin-walled section at node k. M~m(k)is the bending m o m e n t at node k resulting from the end moments M ~, M2, which may be applied constants or due to restraint of end rotation or curvature as in the cases of spring and fixed ends, respectively. Mee(k) is the bending m o m e n t at node k resulting from the transverse loads F. Rewriting eqn (1) in matrix notation for all nodes 1, n + 1 gives
{v} = ] [ { M , }
+ {M.~} + {Md}]
(2)
where the vectors {v}, {M,}, {Me,.} and {M,r} are all of length n + 1. The vector {M,,} is independent of the nodal displacements {v} and can be p r e d e t e r m i n e d from the externally applied loading as set out in Ref. 23. The vectors {Mi} and {M.,,} are dependent on the nodal displacements, and must be evaluated at each iteration of the nodal displacement vector {v}. The kth term, M~(k), of the internal m o m e n t of resistance vector {Mi} can be determined from the curvature oh(k) and axial force P at the kth node by reference to matrices (called a database) containing the moments and axial force as a function of curvature and axial strain. The elements of these matrices are predetermined at a number of finite values of curvature and axial strain to give the matrices for axial force [P] and bending m o m e n t [M] shown in Fig. 3. The ith row and jth column of these matrices correspond to discrete values of curvature ~b, and axial strain e j, respectively. Linear interpolation between these discrete values of axial strain and curvature can be used, provided that the matrices are generated at sufficiently small increments of axial strain and curvature. In the case of locally buckled beam-columns, these matrices are predetermined using the finite strip nonlinear analysis. First the axial strain e(k) corresponding to the curvature cb(k) and axial force P is determined from the [P] matrix in Fig. 3(a). Secondly, the m o m e n t M~(k) is determined from the [M] matrix in Fig. 3(b) using the value of e(k) d e t e r m i n e d from Fig. 3(a) along with the value of ~b(k). The vector { M~} can therefore be assembled by repeated reference to the database for the nodes E(k)
(a) Axial Force [P]
E(k)
(b) Bending Moment [M]
Fig. 3. Equilibrium matrices in the database.
218
A. J. Davids, G. .I. Hancock
1 to n + I. Retrieving unique equilibrium states from the database depends upon the assumption that the axial force (P) increases monotonically with axial strain (~) at any fixed value of curvature (&). The nodal curvature vector {4~} and rotation vector {0} are computed from the beam-column displacement vector {v} and initial position vector {v0} using {~} = ~2[D]{v-vo}
i31
and (0}
1
(4)
where [D] and [R] are matrices of finite difference operators. The |ixed segment length (h) was taken to be equal to the half-wavelength of local buckling (t) as shown in Fig. 2. Complete details of the matrices [D] and [RI are given in Ref. 23. Any number of matrices in the database may be formed to contain specilic information about the equilibrium converged position corresponding to the prescribed axial strains and curvatures of the finite strip nonlinear analysis of a section. Examples include the local buckling displacements w,, w, and w, shown on Fig. 1 and the membrane stresses at these locations. These can be used to determine behaviour such as the modulation of the local buckling displacements and stresses along the member and across the sections during the beam-column analysis. However, only the axial force [P] and bending m o m e n t [M] matrices are necessary for the numerical technique to operate.
2.3.3 Newton-Raphson convergence procedure The equations of equilibrium in matrix form (eqn (2)) involve the nodal displacements {v} on both sides of the equation, and so a satisfactory solution may be obtained by iteration of {v} at constant axial load (P). !f a nodal displacement vector is assumed initially to compute {Mt} and {Mo,,}. then a resultant nodal displacement vector {v' } can be computed using eqn (2). This process is iterative until the assumed vector {v} and the calculated vector {v'} converge to within a small predetermined difference. To facilitate the iterative procedure using a Newton-Raphson procedure, it is necessary to know the small change {dr' } of the calculated vector Iv' } that will result from a small change {dv} of the assumed vector {v ~j. To this end it is assumed that {dv'l = [J]{,lv/
Nonlinear elastic response of locally buckled thin-walled beam-columns
219
where [J] is called the influence coefficient matrix. The detailed terms in the matrix [J] and its full derivation are given in Ref. 23. The influence coefficient matrix [J] requires evaluation at each node k of the rate of change a(k) of bending moment M~(k)with curvature ~(k)~ at a prescribed axial load (P). This is achieved by repeating the database retrieval procedure for values of curvature slightly greater and less than the nodal curvature ~(k), and approximating the value of a(k) as a finite difference of the resulting values of bending moment M~(k).
3 C O M P A R I S O N OF NONLINEAR THEORY WITH TESTS OF ALUMINIUM SECTIONS A series of tests on I-section columns was reported by Bijlaard and Fisher. The thin-walled test specimens were extruded from high strength aluminium to ensure a minimum of geometric imperfections, and were tested in a simply supported concentrically loaded condition. The geometry of the K-series and L-series test sections is shown in Figs 4 and 5, respectively.
Number of HaIfwaves S
10
16
14
12
110
°.~u
08 K-seFles
~
O6
Nonkinear theory
603~103 28
Bijtaard and
Fisher tests
0L,
0;
•
ol =2% 6 a=0 001 I
10
I
20
)Io
4lo
~o
6o
L/r Fig. 4. Ultimate load versus slenderness.
70
220
A. I. Davids, G. d. Hancock I
I
Bulaard L-series
15
I
!
1
I
I
117 z, .._.~
and Fisher
L/r=62 OE/Oz=131
730
3 15
~--
3 20
o~ =141.0 • =0 001
O"
ot
/-eo=L/20,000
10 "-eo=L/1 000 05
Nonbnear Theory Test
1, 5
I
10
1
J
I
15
20
25
Lateral
Deflection
v, mrn
1
30
•
I
]5
Fig. 5. Axial load versus lateral deflection
The nonlinear theory requires consideration of an imperfect structure m the local and overall senses for a solution to be obtained, and so a very small value of local imperfection (a) equal to 0-001 and a very small initial loading eccentricity (e0) about the minor principal axis equal to L/20000 were assumed. The ultimate strengths of the K-series test specimens Or,,) nondimensionalised with respect to the local buckling stress ((r¢ '~. 214"6 MPa) are compared in Fig. 4 with the nonlinear theory. The theory accurately predicts the ultimate strengths of the test specimens in the range where the ratio of the Euler buckling stress ((rE) to the local buckling stress ((re) has a value equal to or less than approximately 2-0. The ultimate strengths of the test specimens with a value of the ratio ((rdo-~) greater thm~ approximately 2-0, however, appear to have been influenced by materiat plasticity. The theoretical elastic loading behaviour of a long length locally buckled L-series test specimen ((re = 141.0 MPa, L / r = 62, (rE/O', = 1"31) is shown in Fig. 5. The theory agreed closely with the test measurements, particularly in the region where the column had significant overall deflections. This comparison demonstrates that the nonlinear theory adequately models the elastic unloading behaviour of locally buckled thin-walled beam-columns subjected to significant curvature about the minor principal axis. However, for practical columns, yielding is likely to occur before substantial lateral deflections, with a further reduction in the load capacity.
Nonlinear elastic response of locally buckled thin-walled beam-columns
221
4 COMPARISON OF NONLINEAR THEORY WITH A PERTURBATION THEORY A n example of a concentrically loaded thin-walled beam-column with crosssection shown in Fig. 6 has been studied using the nonlinear theory. The calculated ultimate stress (o-,) nondimensionalised with respect to the local buckling stress (o-~ = 124-2 MPa) is shown in Fig. 6 as a function of column slenderness (L/r). A local geometric imperfection (a) equal to 0.05 and an overall geometric imperfection (v0) about the minor principal axis equal to either L/IO 000, L/2000 or L/IO00 were assumed. The results reported by Sridharan and Ali ~6of the same example, analysed by a perturbation technique, are also shown in Fig. 6. A g r e e m e n t between the two theories appears to be satisfactory at a column length (L) equal to 2000 m m (L/r -- 122), corresponding to a ratio of Euler (O'E) to local (o-~) buckling stress equal to 1.06. The ultimate strength (or,) of the thin-walled column was predicted by both theories to be most sensitive to local and overall imperfections in the column slenderness range where the Euler (o-E) and local (o-~) buckling stresses are approximately equal. At a column length (L) equal to 1200 m m (L/r -- 73), corresponding to a ratio of Euler (O'E) to local (o-~) buckling stress equal to 2.90, the ultimate strength (o-,) predictions of the two theories are in less satisfactory
Number of Halfwaves, n 10 15 20 I ~ I \
5
14
\
25 i
k,/°ler I
12
10
~o:L/?,°°°------"H~\ ~
08
N
du 06
O~
Nonhnear Theory Perturbation Theory
80
J •
02
I
20
I
40
I
I
60
80
I
100
I
120
L/r
Fig. 6. Ultimate load versus slenderness.
140
222
A. J. Davids, G. J. Hancock t.6
1
I
I
I
I
"] 0e/or =2 9 Ln=IS
/ "..."vo=L/?,ooo , -~-vo=L/1,000
10 o
08
~
06
0
_
OZ,
-
O2
-
~L/r=122 "~OEO /, :I.06~,vo=L/2.000 "vo:L/1,000
~^ 80
kn=25 -
...~2
I
10
I
20
l
30
L
40
1
SO
,,
60
Latera~ l]eflection v mm
Fig. 7, Axial load versus lateral deflection.
agreement. Sridharan and Ali lo also report that the imperfection sensitivity remains severe even at column slenclernesses where the ratio (o-E/(r,) far exceeds unity. This observation is not supported by the nonlinear theory since Fig. 6 shows a reduced imperfection sensitivity at high values of (O'E/O-~). In order to further investigate the reasons for the discrepancies between the elastic theories, additional parameters have been plotted, The axial stress (o-), nondimensionalised with respect to the local buckling stress (o-e), has been plotted in Fig. 7 against the central lateral deflection (v) for the short (L/r = 73) and long (L/r = 122) columns. The graphs arc shown for three different levels of overall geometric imperfection (v,,). The elastic ultimate load of the long column occurs (Points A, B) with the column subjected to small overall curvature about the minor principal axis. This behaviour is in contrast with that of the short column, which approaches the elastic ultimate load more gradually with consequently large overall curvature about the minor principal axis (Point C). This behaviour has also been described in a recent paper by Loughlan and Nabavian. :~ The cross-sectional deformed shapes at the centre of the columns near their ultimate loads at points A, B and C in Fig. 7 are shown in Fig. 8. The shapes shown in Fig. 8 have been normalised with respect to the flange tip deflection on the concave side of the column, with the scaling factors in millimetres shown in brackets. The effect of increasing overall column curvature about the minor principal axis has been to distort the crosssectional shape away from that of the local buckling mode under uniform compression (u ,), as shown by the transition in section deformed shape from AtoC.
Nonlinear elastic response of locally buckled thin-walled beam-columns
1.0 08 0.6 OL, 02
Column convex ~ side flange ~
02 0~-
B
C
06 08 1.0
.,~7/
/// ,~A.B.Ul
~ /
'r II~
/
223
Column concave side flange Seating factors Point A (OZ.)
_
~] u~i,
Point C (/.7,
LI?
Ull" ~
Halfweb(syrnmetrfc) I
I
I
02
I
04 06
I
08
I 10
Fig. 8. Cross-sectional distortion.
I
I
I
vo:L/lO 000_
I0 a
f
o,
/
I
('L/r=73
fvo=L/lO.O00
-
/'L.LZ
%,
//
8o-i
02//,/
8o ~E-2 I
05
I
I
I
10 1.5 20 Axial Shortening u, mm
Fig. 9. Axial load versus axial shortening.
The asymptotic theory employed by Sridharan and Ali16 describes the out-of-plane cross-sectional displacements as a linear combination of the first two local buckling modes associated with uniform compression, shown as (u 1) and (u2), respectively, in Fig. 8. Whilst this procedure is satisfactory under nearly uniform strain (Point A, Fig. 7), it may not be satisfactory under conditions of significant cross-sectional distortion (Point C, Fig. 7), since the highly distorted shape of the latter cannot be accurately modelled by a simple linear combination of (u 0 and (u2). In particular, the c o r n -
224
A. J. Davids, G. .L Hancock
bination of (ul) and (u2) required to produce almost zero flange tip deflection on the convex side of the column will result in an underestimate of the deflections in the web by a factor of 0.5. The discrepancy between the elastic theories is likely to be of little consequence in cases where material plasticity due to cross-sectional distortion precipitates failure at a load significantly less than the elastic ultimate strength. The axial stress (o-), nondimensionalised with respect to the local buckling stress (o-e), has been plotted in Fig. 9 against the axial shortening (u) for the short (n = 15, O " E / O " ~ = 2-90) and long (n = 25, crE/o-~ = 1-06) columns. The long column with small imperfections has a highly unstable post-ultimate response following an almost linear rise to the ultimate load whereas the short column displays a gradual and highly nonlinear rise to the ultimate load.
5 CONCLUSIONS A theory has been presented which models the elastic nonlinear response ot locally buckled thin-walled beam-columns, The theory combines the finite strip method of nonlinear analysis of thin-walled sections with the influence coefficient method of nonlinear analysis of beam-columns. The effects ot initial local and overall geometric imperfections, residual strains and general beam-column end conditions and general loading are accounted tot in the estimation of the elastic ultimate load and post-ultimate collapse path of such members. The theory accurately predicts the ultimate strengths of concentrically loaded test specimens which remain elastic and closely models the elastic nonlinear post-ultimate collapse paths of the test specimens. The theory shows that the ultimate strength of long length axially compressed thinwalled beam-columns is well defined and occurs with small overall curvature about the minor principal axis. The ultimate strength is sensitive to initial local and overall geometric imperfections since the post-ultimate collapse response to applied end shortening may be a rapid drop-off in load capacity, The ultimate strength of short length axially compressed thin-walled beamcolumns is approached more gradually, with large overall curvature about the minor principal axis, The ultimate strength is not sensitive to initial local and overall geometric imperfections. The theory also shows that curvature of a locally buckled thin-walled cross-section about the minor principal axis may lead to significant crosssectional distortion, which may not be adequately modelled by present applications of mode interaction theory.
Nonlinear elastic response of locally buckled thin-walled beam-columns
225
ACKNOWLEDGEMENTS This paper forms part of a programme of research into the stability of steel structures being carried out in the School of Civil and Mining Engineering of the University of Sydney. The calculations were performed in the C. A. Hawkins Computing Laboratory at a terminal to a multi-user P R I M E 4(X) minicomputer. Funds to purchase the system were provided by the University and by the Civil Engineering Graduates Association. The first author was supported by a Postgraduate Scholarship from the Civil and Mining Engineering Foundation.
REFERENCES 1. Bijlaard, P. P. and Fisher, G. P., Column strength of H-sections and square tubes in postbuckling range of component plates, TN 2994, NACA, Washington, DC, Aug. 1953. 2. Davids, A. J. and Hancock, G. J., Compression tests of long welded l-section columns, J. Struct. Eng., ASCE, 112(10) (1986) 2281-97. 3. Hancock, G. J., Interaction buckling in I-section columns, J. Struct. Div., A S C E , 107(ST1) Proc. Paper 15978 (1981) 165-79. 4. Wang, S. T. and Pao, H. Y., Torsional-flexurai buckling of locally buckled columns, Int. J. Comp. Struct., 11 (1980) 127-36. 5. Timoshenko, S. P. and Gere, J. M., Theory of Elastic Stability, McGraw-Hill, New York, 1961. 6. Kalyanaraman, V., Pekoz, T. and Winter, G., Unstiffened compression elements, J. Struct. Div., ASCE, 103(ST9) (1977) 1833-48. 7. Hancock, G. J., Nonlinear analysis of thin sections in compression, J. Struct. Div., ASCE, 107(ST3), Proc. Paper 16090 ( 1981) 455-71. 8. Hancock, G. J., Nonlinear analysis of thin-walled I-sections in bending, In Aspects o]' the Analysis of Plate Structures (Dawe, D. J., Little, G. H., Horsington, R. W. and Kamtekar, A. G., Eds), Oxford University Press, 1985, 251-68. 9. Koiter, W. T. and Van der Neut, A., Interaction between local and overall buckling of stiffened compression panels, Proceedings, International Conference on Thin-Walled Structures, University of Strathclyde, 1979. 10. Benito, R. and Sridharan, S., Mode interaction in thin walled structural members, J. Struct. Mech., 12(4) (1984) 517-43. 11. Benito, R. and Sridharan, S., Interactive buckling analysis with finite strips, lnt. J. Numerical Methods in Eng., 21(1) (Jan. 1985) 145--61. 12. Sridharan, S., and Benito, R., Static and dynamic interactive buckling in columns, J. Eng. Mech., ASCE, 110(1) (Jan. 1984) 49-65. 13. Sridharan, S., Doubly symmetric interactive buckling of plate structures, Int. J. Solids and Struct., 19(7) (1983) 625-4l. 14. Sridharan, S. and Graves-Smith, T. R., Postbuckling analyses with finite strips, J. Eng. Mech. Div., ASCE, 107(EM5)(Oct. 1981)869-88.
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15. Budiansky, B., Theory of buckling and post-buckling behavior of elastic structures, Advances in Applied Mechanics, Vol. 14, Academic Press, New York, 1974. 16. Sridharan, S. and Ali, M. A., Response of frames having thin-walled columns, Proceedings, Seventh International Specialty Conference on Cold-Formed Steel Structures, St Louis, Nov. 1984. 17. Sridharan, S., and Ali, M. A., Interactive buckling in thin-walled beamcolumns, J. of Eng. Mech., ASCE, 111(12) (Dec. 1985) 1470-86, 18. Chen, W. F. and Atsuta, T., Theory of beam-columns, Vol. 2, In ,Sp,ce Behaviour and Design, McGraw-Hill, New York, 1977, 337-411. 19. Han, D. J. and Chen, W. F., Buckling and cyclic inelastic analysis ol stccl tubular beam-columns, Eng. Struct., 5 (April 1983) 119-32. 20. Hancock, G. J., Local, distortional and lateral buckling of l-beams, ,I. ~tr, ct. Div., ASCE, 104(STI 1) Proc. Paper 1415.5 (Nov. 1973) 1787-98. 21. Hancock, G. J., Distortional buckling of steel storage rack columns, .1. ,S!rt, t. Eng., ASCE, 111(12) Paper 20240 (Dec. 1985) 2770-83. 22. Davids, A. J. and Hancock, G, J., Compression tests of short weldcd lsections, J. Struct. Eng., ASCE, 112(ST5) Paper No. 20583 (May 1986) 960--7f~. 23. Davids, A. J. and Hancock, G. J., Nonlinear response of locally buckled thin-walled beam-columns, School of Civil and Mining Engineering. University of Sydney, Research Report No. R531, August 1986. 24. Loughlan, J. and Nabavian, M., The behaviour of thin-walled l-section columns after local buckling, Proceedings, Eighth lntermuiomd Siw~::ialty Conference on Cold-Formed Steel Strtwtures. St Louis, Missouri. 1.1SA, November 11-12. 1986.
Nonlinear Elastic Response of Locally Buckled Thin-walled Beam-columns
A. J. Davids and G. J. Hancock School of Civil and Mining Engineering, University of Sydney, NSW 2006, Australia (Received 18 September 1986; revised version received 24 November 1986: accepted 3() January 1987)
ABST~A('T A theory is described which combines the finite strip method of nonlinear elastic analysis of locally buckled thin-walled sections with the in)qtwnce co<~t4cient method of attalvsis of beam-columns. The theory combines the advantages of both methods of analysis to produce a computationally e~licient procedure jbr the estimation of the overall nonlinear elastic response of locall3' buckled thin-walled beam-columns. The theory is used to predict the previously reported measured behaviour of l-section test specimens of Bi~laard and Fisher. The theory is also compared with theoretical responses of eccentrically loaded thin-walled beam-cohmtns analvsed by an asymptotic theory. An example oJ a locally huckled thin-walled beam-cohtmn is studied in detail and some observations are made concerning the nonlinear response ~['members of this type.
NOTATION h
{dvl,{dv'l [D] F(k) h
[J]
Flange width. Vectors of small changes of {v} and {v' } vectors respectively. Matrix of finite difference coefficients. Initial uniform loading eccentricity about the minor principal axis. Transverse force at node k. Segment length. Influence coefficient matrix. 211
Thin-Walled Structures 0263-8231/87/$03.50 © Elsevier Applied Science Publishers Ltd, England, 1987. Printed in Great Britain
212
k ( L
L/r M M 1, Mz M ¢f(k)
M~.(k) Mi(k) [M] n
P
[P] [R] t
u ut~ u2 V VO
v(k) {v.} {v'} Wl~ 1422, 1423 at
E E/
{o} ~E ~e ~u
A.J. Davids. G. J. Hancock
Node and segment number. Local buckle half-wavelength. Beam-column length. Column slenderness. Bending m o m e n t of a segment resultant from curvature about the minor principal axis. E n d moments on a beam-column at nodes 1 and n +-J respectively. External m o m e n t at node k due to transverse forces (kth term of vector {M,r}). External moment at node k due to end moments M ,~ M~ tkth term of vector {M,,,}). Internal moment of resistance at node k (kth term of vector {M,}). Matrix of bending moments. Number of local buckle halfwaves. Axial load. Matrix of axial loads. Matrix of finite difference coefficients. Thickness of I-section flange. Axial shortening. Cross-sectional shape of first and second local buckling modes under uniform compression. Beam-column central lateral deflection. Magnitude of initial sinusoidal overall geometric imperfection. Lateral displacement at node k (kth term of vector {v} )~ Initial value of {v} vector. Calculated lateral displacement vector. Local buckling displacements. Maximum amplitude of local geometric imperfection. Flexural rigidity about the minor principal axis of kth segment at constant axial load. Uniform axial compressive strain. Axial strain of equilibrium positions ofjth column of a matrix in the database. Value of ~ at node k (kth term of vector {~}). Nodal rotations vector. Axial load divided by cross-sectional area. Euler buckling stress. Local buckling axial load divided by cross-sectional area. Maximum value of or. Uniform curvature about the minor principal axis.
Nonlinear elastic response of locally buckled thin-walled beam-columns
,/,(k)
213
Curvature of equilibrium positions of ith row of a matrix in the database. Value of ch at node k (kth term of vector {~b}).
1 INTRODUCTION Local buckling of a thin-walled structural member leads to a reduction in its flexural rigidity. This reduction will result in a concentrically loaded thinwalled column buckling in an overall mode at a load less than its classical Euler buckling load, even when the member remains elastic. For eccentrically compressed members, which induce bending from the commencement of loading, the lateral deflections are amplified as a result of the reduced flexural rigidity caused by local buckling. This amplification of displacements leads to a further reduction in the ultimate load attained by the member. The purpose of this paper is to describe and verify an analytical m e t h o d for determining the nonlinear load-defection behaviour, and hence ultimate load, of locally buckled thin-walled beam-columns. The buckling loads for overall buckling of locally buckled thin-walled columns have been estimated by several researchers '-4 by substituting reduced flexurai rigidities into the Euler buckling formulae. 5 The reduced section flexural rigidities were calculated in these papers by various methods, including the Winter effective section method 6 and a finite strip nonlinear analysis. 7.8For eccentrically loaded members, bifurcation into an overall mode does not occur. Consequently, a nonlinear analysis of a beam-column with variable axial and flexural rigidities is required. The variability results from modulation of the local buckling deformations along the length of a thin-walled member. Koiter and Van der Neut 9developed an approximate analytical model based on the concept of shape modulation of short wavelength local buckling modes by the long wavelength overall buckling mode. More recent theoretical approaches by Benito and Sridharan t°-t2 and Sridharan ~3 have employed a combination of the finite strip ~4and mode interaction ~5methods of structural analysis to investigate the interaction of local and Overall buckling in thin-walled beam-columns. Sridharan and Ali 16'17 have modified this approach to include a more satisfactory amplitude modulation technique and more general member end conditions. In this paper, a theory is presented which combines the finite strip method of nonlinear elastic analysis of locally buckled thin-walled sectionsSwith the influence coefficient method of nonlinear analysis of beam-columns developed by Chen. x8.~9The finite strip analysis includes the effects of residual compressive strain in the plate elements forming the section, and
214
A. J. Davids, G. J. Hancock
local geometric imperfections. The nonlinear beam-column analysis includes overall geometric imperfections, general member end conditions and general loading. The theory results in a computationally efficient procedure for the estimation of the nonlinear elastic response of isolated beam-columns with locally buckled thin-walled cross-sections. Of interest also is the rate at which axial load reduces in the post-ultimate collapse state of a thin-walled beam-column, since this may be relevant to the formation of the philosophy of design code rules. The analytical method predicts this unloading behaviour for members which remain elastic.
2 THEORY
2.1 Assumptions The combination of the finite strip nonlinear elastic analysis of locally buckled thin-walled sections and the influence coefficient method for the nonlinear analysis of beam-columns requires certain assumptions. These are:
(i) The beam-column can be replaced by a series of segments of length equal to the half-wavelength of the local buckles. (ii) The m o m e n t and axial force resisted by a segment subjected to defined values of curvature and axial strain, can be determined for each segment in isolation using the finite strip nonlinear analysis. (iii) The average values of moment and axial force along a segment can bc used to describe that segment in the nonlinear analysis of the beamcolumn. (iv) The only interconnection between the segments is by way of equilibrium of the section stress resultants of moment, shear and axial force, as well as compatibility of centreline displacements of adjacent segments. (v) The effect of section shear on the local buckling response of a segment is ignored.
2.2 Finite strip post-local buckling analysis The elastic post-local buckling response of a thin-walled section is modelled using the finite strip nonlinear elastic analysis described in Ref. 8. The thin-walled section is divided into a series o f longitudinal strips which are joined together at nodal lines as shown in Fig. 1. The section length (t) is equal to the half-wavelength for local buckling, which gives a minimum local
Nonlinear elastic response of locally buckled thin-walled beam-columns
///•a
215
g>
Stmps w2
oration=
~-
d)b~
xlat force }ResuLtant ~"~"~ Rotation ~LPrescribe d g moment Forces ~Uniform ~,,, J[3isplacements
Fig. 1. Locallybuckledthin-walledsection. buckling load under the initial conditions of compression, bending and residual strain. This length is calculated using a finite strip analysis as described by Hancock, 2°2~ and is assumed invariant throughout the subsequent nonlinear analysis. The section is assumed to be compressed between rigid frictionless end plattens. The strip displacement functions are identical with those developed in Ref. 8 and described as Version I in Ref. 14. Longitudinal waveform changes of the flexural displacements are not included in the analysis, but this was considered in Ref. 7 to be satisfactory for loads less than approximately 1.8 times the local buckling load. The local geometric imperfections of maximum amplitude equal to c~t are assumed to be of the same shape and half-wavelength as the local buckling mode. Longitudinal membrane residual strains (resulting from such p h e n o m e n a as weld shrinkage after manufacture) have been included in an approximate way, as described in Ref, 22, by superimposing a longitudinal strain distribution in addition to that resulting from compression and bending. The nonlinear analysis involves solution of the equilibrium equations by a Newton-Raphson iteration technique for a prescribed set of nodal line compressive strains. Thin-walled columns in uniform compression v and beams in uniform bending s about the major principal axis have been studied previously. In this paper, the nodal line compressive strains of the beam-column section are prescribed to be a linear combination of the displacements due to uniform compressive strain (E) and curvature (oh) about the minor principal axis of
A.J. Davids, G. J. Hancock
216
the I-section, as shown in Fig. 1. The induced stress resultants of axial load (P) and bending m o m e n t (M) about the minor principal axis are also shown in Fig. 1. 2.3 Influence coefficient method
2.3.1 General The influence coefficient method ([CM) of analysis as described by Chen and Atsutat8 and Han and Chen'9 has been used to model the nonlinear response of the beam-column. In this method, the continuous beam-column is modelled as a series of nodes at which equilibrium is sought by an iterative p r o c e d u r e involving the nodal displacements. The beam-column under Hatf waveteng_~L.j_ N ode 2 Node 1 1
I
i
;(k) / ~Node k
~
Segment k ~ Node n*1
P A
B p
Equitibrium position v(kl u/2
L
~L u/2~ i
Fig. 2. Thin-walledbeam-column. study is shown in its discretised form in Fig. 2. It contains n + 1 nodes which define the centres of n - 1 internal segments and the ends of the two end segments. The internal segments correspond with the lengths of locally buckled sections of length equal to their half-wavelength for local buckling ( t ) . The end segments each represent the state of the beam-column over lengths equal to a quarter-wavelength only. Hence the beam-column contains n local buckle halfwaves between nodes 1 and n - l. The lengths A to 1 and n + t to B in Fig. 2 are assumed to be rigid bodies. Full details of the theory of the influence coefficient method of analysis are described in Ref. 23. A brief s u m m a r y follows.
2.3.2 Equilibrium equations T h e external actions on the beam-column consist of end axial forces P along line A B , external m o m e n t s M 1, M2 applied at nodes 1 and n + 1 respectively and transverse loads F(k) applied at each internal node k as shown in Fig. 2~ M o m e n t equilibrium at node k can be expressed as M~(k) + M.m(k) + M~f(k) - Pv(k) = 0
(1)
Nonlinear elastic response o]'locally buckled thin-walled beam-columns
217
M~(k) is the internal m o m e n t of resistance of the thin-walled section at node k. M~m(k)is the bending m o m e n t at node k resulting from the end moments M ~, M2, which may be applied constants or due to restraint of end rotation or curvature as in the cases of spring and fixed ends, respectively. Mee(k) is the bending m o m e n t at node k resulting from the transverse loads F. Rewriting eqn (1) in matrix notation for all nodes 1, n + 1 gives
{v} = ] [ { M , }
+ {M.~} + {Md}]
(2)
where the vectors {v}, {M,}, {Me,.} and {M,r} are all of length n + 1. The vector {M,,} is independent of the nodal displacements {v} and can be p r e d e t e r m i n e d from the externally applied loading as set out in Ref. 23. The vectors {Mi} and {M.,,} are dependent on the nodal displacements, and must be evaluated at each iteration of the nodal displacement vector {v}. The kth term, M~(k), of the internal m o m e n t of resistance vector {Mi} can be determined from the curvature oh(k) and axial force P at the kth node by reference to matrices (called a database) containing the moments and axial force as a function of curvature and axial strain. The elements of these matrices are predetermined at a number of finite values of curvature and axial strain to give the matrices for axial force [P] and bending m o m e n t [M] shown in Fig. 3. The ith row and jth column of these matrices correspond to discrete values of curvature ~b, and axial strain e j, respectively. Linear interpolation between these discrete values of axial strain and curvature can be used, provided that the matrices are generated at sufficiently small increments of axial strain and curvature. In the case of locally buckled beam-columns, these matrices are predetermined using the finite strip nonlinear analysis. First the axial strain e(k) corresponding to the curvature cb(k) and axial force P is determined from the [P] matrix in Fig. 3(a). Secondly, the m o m e n t M~(k) is determined from the [M] matrix in Fig. 3(b) using the value of e(k) d e t e r m i n e d from Fig. 3(a) along with the value of ~b(k). The vector { M~} can therefore be assembled by repeated reference to the database for the nodes E(k)
(a) Axial Force [P]
E(k)
(b) Bending Moment [M]
Fig. 3. Equilibrium matrices in the database.
218
A. J. Davids, G. .I. Hancock
1 to n + I. Retrieving unique equilibrium states from the database depends upon the assumption that the axial force (P) increases monotonically with axial strain (~) at any fixed value of curvature (&). The nodal curvature vector {4~} and rotation vector {0} are computed from the beam-column displacement vector {v} and initial position vector {v0} using {~} = ~2[D]{v-vo}
i31
and (0}
1
(4)
where [D] and [R] are matrices of finite difference operators. The |ixed segment length (h) was taken to be equal to the half-wavelength of local buckling (t) as shown in Fig. 2. Complete details of the matrices [D] and [RI are given in Ref. 23. Any number of matrices in the database may be formed to contain specilic information about the equilibrium converged position corresponding to the prescribed axial strains and curvatures of the finite strip nonlinear analysis of a section. Examples include the local buckling displacements w,, w, and w, shown on Fig. 1 and the membrane stresses at these locations. These can be used to determine behaviour such as the modulation of the local buckling displacements and stresses along the member and across the sections during the beam-column analysis. However, only the axial force [P] and bending m o m e n t [M] matrices are necessary for the numerical technique to operate.
2.3.3 Newton-Raphson convergence procedure The equations of equilibrium in matrix form (eqn (2)) involve the nodal displacements {v} on both sides of the equation, and so a satisfactory solution may be obtained by iteration of {v} at constant axial load (P). !f a nodal displacement vector is assumed initially to compute {Mt} and {Mo,,}. then a resultant nodal displacement vector {v' } can be computed using eqn (2). This process is iterative until the assumed vector {v} and the calculated vector {v'} converge to within a small predetermined difference. To facilitate the iterative procedure using a Newton-Raphson procedure, it is necessary to know the small change {dr' } of the calculated vector Iv' } that will result from a small change {dv} of the assumed vector {v ~j. To this end it is assumed that {dv'l = [J]{,lv/
Nonlinear elastic response of locally buckled thin-walled beam-columns
219
where [J] is called the influence coefficient matrix. The detailed terms in the matrix [J] and its full derivation are given in Ref. 23. The influence coefficient matrix [J] requires evaluation at each node k of the rate of change a(k) of bending moment M~(k)with curvature ~(k)~ at a prescribed axial load (P). This is achieved by repeating the database retrieval procedure for values of curvature slightly greater and less than the nodal curvature ~(k), and approximating the value of a(k) as a finite difference of the resulting values of bending moment M~(k).
3 C O M P A R I S O N OF NONLINEAR THEORY WITH TESTS OF ALUMINIUM SECTIONS A series of tests on I-section columns was reported by Bijlaard and Fisher. The thin-walled test specimens were extruded from high strength aluminium to ensure a minimum of geometric imperfections, and were tested in a simply supported concentrically loaded condition. The geometry of the K-series and L-series test sections is shown in Figs 4 and 5, respectively.
Number of HaIfwaves S
10
16
14
12
110
°.~u
08 K-seFles
~
O6
Nonkinear theory
603~103 28
Bijtaard and
Fisher tests
0L,
0;
•
ol =2% 6 a=0 001 I
10
I
20
)Io
4lo
~o
6o
L/r Fig. 4. Ultimate load versus slenderness.
70
220
A. I. Davids, G. d. Hancock I
I
Bulaard L-series
15
I
!
1
I
I
117 z, .._.~
and Fisher
L/r=62 OE/Oz=131
730
3 15
~--
3 20
o~ =141.0 • =0 001
O"
ot
/-eo=L/20,000
10 "-eo=L/1 000 05
Nonbnear Theory Test
1, 5
I
10
1
J
I
15
20
25
Lateral
Deflection
v, mrn
1
30
•
I
]5
Fig. 5. Axial load versus lateral deflection
The nonlinear theory requires consideration of an imperfect structure m the local and overall senses for a solution to be obtained, and so a very small value of local imperfection (a) equal to 0-001 and a very small initial loading eccentricity (e0) about the minor principal axis equal to L/20000 were assumed. The ultimate strengths of the K-series test specimens Or,,) nondimensionalised with respect to the local buckling stress ((r¢ '~. 214"6 MPa) are compared in Fig. 4 with the nonlinear theory. The theory accurately predicts the ultimate strengths of the test specimens in the range where the ratio of the Euler buckling stress ((rE) to the local buckling stress ((re) has a value equal to or less than approximately 2-0. The ultimate strengths of the test specimens with a value of the ratio ((rdo-~) greater thm~ approximately 2-0, however, appear to have been influenced by materiat plasticity. The theoretical elastic loading behaviour of a long length locally buckled L-series test specimen ((re = 141.0 MPa, L / r = 62, (rE/O', = 1"31) is shown in Fig. 5. The theory agreed closely with the test measurements, particularly in the region where the column had significant overall deflections. This comparison demonstrates that the nonlinear theory adequately models the elastic unloading behaviour of locally buckled thin-walled beam-columns subjected to significant curvature about the minor principal axis. However, for practical columns, yielding is likely to occur before substantial lateral deflections, with a further reduction in the load capacity.
Nonlinear elastic response of locally buckled thin-walled beam-columns
221
4 COMPARISON OF NONLINEAR THEORY WITH A PERTURBATION THEORY A n example of a concentrically loaded thin-walled beam-column with crosssection shown in Fig. 6 has been studied using the nonlinear theory. The calculated ultimate stress (o-,) nondimensionalised with respect to the local buckling stress (o-~ = 124-2 MPa) is shown in Fig. 6 as a function of column slenderness (L/r). A local geometric imperfection (a) equal to 0.05 and an overall geometric imperfection (v0) about the minor principal axis equal to either L/IO 000, L/2000 or L/IO00 were assumed. The results reported by Sridharan and Ali ~6of the same example, analysed by a perturbation technique, are also shown in Fig. 6. A g r e e m e n t between the two theories appears to be satisfactory at a column length (L) equal to 2000 m m (L/r -- 122), corresponding to a ratio of Euler (O'E) to local (o-~) buckling stress equal to 1.06. The ultimate strength (or,) of the thin-walled column was predicted by both theories to be most sensitive to local and overall imperfections in the column slenderness range where the Euler (o-E) and local (o-~) buckling stresses are approximately equal. At a column length (L) equal to 1200 m m (L/r -- 73), corresponding to a ratio of Euler (O'E) to local (o-~) buckling stress equal to 2.90, the ultimate strength (o-,) predictions of the two theories are in less satisfactory
Number of Halfwaves, n 10 15 20 I ~ I \
5
14
\
25 i
k,/°ler I
12
10
~o:L/?,°°°------"H~\ ~
08
N
du 06
O~
Nonhnear Theory Perturbation Theory
80
J •
02
I
20
I
40
I
I
60
80
I
100
I
120
L/r
Fig. 6. Ultimate load versus slenderness.
140
222
A. J. Davids, G. J. Hancock t.6
1
I
I
I
I
"] 0e/or =2 9 Ln=IS
/ "..."vo=L/?,ooo , -~-vo=L/1,000
10 o
08
~
06
0
_
OZ,
-
O2
-
~L/r=122 "~OEO /, :I.06~,vo=L/2.000 "vo:L/1,000
~^ 80
kn=25 -
...~2
I
10
I
20
l
30
L
40
1
SO
,,
60
Latera~ l]eflection v mm
Fig. 7, Axial load versus lateral deflection.
agreement. Sridharan and Ali lo also report that the imperfection sensitivity remains severe even at column slenclernesses where the ratio (o-E/(r,) far exceeds unity. This observation is not supported by the nonlinear theory since Fig. 6 shows a reduced imperfection sensitivity at high values of (O'E/O-~). In order to further investigate the reasons for the discrepancies between the elastic theories, additional parameters have been plotted, The axial stress (o-), nondimensionalised with respect to the local buckling stress (o-e), has been plotted in Fig. 7 against the central lateral deflection (v) for the short (L/r = 73) and long (L/r = 122) columns. The graphs arc shown for three different levels of overall geometric imperfection (v,,). The elastic ultimate load of the long column occurs (Points A, B) with the column subjected to small overall curvature about the minor principal axis. This behaviour is in contrast with that of the short column, which approaches the elastic ultimate load more gradually with consequently large overall curvature about the minor principal axis (Point C). This behaviour has also been described in a recent paper by Loughlan and Nabavian. :~ The cross-sectional deformed shapes at the centre of the columns near their ultimate loads at points A, B and C in Fig. 7 are shown in Fig. 8. The shapes shown in Fig. 8 have been normalised with respect to the flange tip deflection on the concave side of the column, with the scaling factors in millimetres shown in brackets. The effect of increasing overall column curvature about the minor principal axis has been to distort the crosssectional shape away from that of the local buckling mode under uniform compression (u ,), as shown by the transition in section deformed shape from AtoC.
Nonlinear elastic response of locally buckled thin-walled beam-columns
1.0 08 0.6 OL, 02
Column convex ~ side flange ~
02 0~-
B
C
06 08 1.0
.,~7/
/// ,~A.B.Ul
~ /
'r II~
/
223
Column concave side flange Seating factors Point A (OZ.)
_
~] u~i,
Point C (/.7,
LI?
Ull" ~
Halfweb(syrnmetrfc) I
I
I
02
I
04 06
I
08
I 10
Fig. 8. Cross-sectional distortion.
I
I
I
vo:L/lO 000_
I0 a
f
o,
/
I
('L/r=73
fvo=L/lO.O00
-
/'L.LZ
%,
//
8o-i
02//,/
8o ~E-2 I
05
I
I
I
10 1.5 20 Axial Shortening u, mm
Fig. 9. Axial load versus axial shortening.
The asymptotic theory employed by Sridharan and Ali16 describes the out-of-plane cross-sectional displacements as a linear combination of the first two local buckling modes associated with uniform compression, shown as (u 1) and (u2), respectively, in Fig. 8. Whilst this procedure is satisfactory under nearly uniform strain (Point A, Fig. 7), it may not be satisfactory under conditions of significant cross-sectional distortion (Point C, Fig. 7), since the highly distorted shape of the latter cannot be accurately modelled by a simple linear combination of (u 0 and (u2). In particular, the c o r n -
224
A. J. Davids, G. .L Hancock
bination of (ul) and (u2) required to produce almost zero flange tip deflection on the convex side of the column will result in an underestimate of the deflections in the web by a factor of 0.5. The discrepancy between the elastic theories is likely to be of little consequence in cases where material plasticity due to cross-sectional distortion precipitates failure at a load significantly less than the elastic ultimate strength. The axial stress (o-), nondimensionalised with respect to the local buckling stress (o-e), has been plotted in Fig. 9 against the axial shortening (u) for the short (n = 15, O " E / O " ~ = 2-90) and long (n = 25, crE/o-~ = 1-06) columns. The long column with small imperfections has a highly unstable post-ultimate response following an almost linear rise to the ultimate load whereas the short column displays a gradual and highly nonlinear rise to the ultimate load.
5 CONCLUSIONS A theory has been presented which models the elastic nonlinear response ot locally buckled thin-walled beam-columns, The theory combines the finite strip method of nonlinear analysis of thin-walled sections with the influence coefficient method of nonlinear analysis of beam-columns. The effects ot initial local and overall geometric imperfections, residual strains and general beam-column end conditions and general loading are accounted tot in the estimation of the elastic ultimate load and post-ultimate collapse path of such members. The theory accurately predicts the ultimate strengths of concentrically loaded test specimens which remain elastic and closely models the elastic nonlinear post-ultimate collapse paths of the test specimens. The theory shows that the ultimate strength of long length axially compressed thinwalled beam-columns is well defined and occurs with small overall curvature about the minor principal axis. The ultimate strength is sensitive to initial local and overall geometric imperfections since the post-ultimate collapse response to applied end shortening may be a rapid drop-off in load capacity, The ultimate strength of short length axially compressed thin-walled beamcolumns is approached more gradually, with large overall curvature about the minor principal axis, The ultimate strength is not sensitive to initial local and overall geometric imperfections. The theory also shows that curvature of a locally buckled thin-walled cross-section about the minor principal axis may lead to significant crosssectional distortion, which may not be adequately modelled by present applications of mode interaction theory.
Nonlinear elastic response of locally buckled thin-walled beam-columns
225
ACKNOWLEDGEMENTS This paper forms part of a programme of research into the stability of steel structures being carried out in the School of Civil and Mining Engineering of the University of Sydney. The calculations were performed in the C. A. Hawkins Computing Laboratory at a terminal to a multi-user P R I M E 4(X) minicomputer. Funds to purchase the system were provided by the University and by the Civil Engineering Graduates Association. The first author was supported by a Postgraduate Scholarship from the Civil and Mining Engineering Foundation.
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