ARTICLE IN PRESS

Thin-Walled Structures 47 (2009) 1– 13

Contents lists available at ScienceDirect

Thin-Walled Structures

0263-82

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/tws

Prediction of the elasto-plastic post-buckling strength of uniformlycompressed plates from the fictitious elastic strain at failure

M.C.M. Bakker �, M. Rosmanit, H. Hofmeyer

Structural Design Group, Department of Architecture, Building and Planning, Technische Universiteit Eindhoven (TU/e), Den Dolech 2/VRT 9.33, 5612 AZ Eindhoven, The Netherlands

a r t i c l e i n f o

Article history:

Received 16 November 2007

Received in revised form

9 April 2008

Accepted 10 April 2008Available online 2 June 2008

Keywords:

Plates

Compression

Post-buckling

Elastic

Strength

31/$ - see front matter & 2008 Elsevier Ltd. A

016/j.tws.2008.04.004

esponding author. Tel.: +3140 247 2331; fax:

ail address: M.C.M.Bakker@bwk.tue.nl (M.C.M

a b s t r a c t

This paper discusses the use of the theory of elasticity to determine the post-buckling strength of

uniformly loaded square simply supported thin (steel) plates with sinusoidal-shaped initial

imperfections and longitudinal edges free to wave in plane. Based on the findings from a FEM

parameter study two main types of failure are distinguished: edge failure and center failure. The

parameters determining which failure mode occurs are explained using a simple two-strip model. To

determine the elastic post-buckling behavior of plates a modified far post-buckling solution is proposed,

which is simpler to use than the existing far post-buckling solutions reported in the literature, giving

results with engineering accuracy for loads up to about three times the buckling load. It is shown that

determining the post-buckling strength as the elastic load corresponding to first membrane yield (as is

done in the effective width method) paradoxically gives reasonable results for plates failing by center

failure, but very conservative results for plates failing by edge failure. For plates failing by edge failure a

more accurate strength prediction is obtained by deriving empirical expressions for a fictitious elastic

strain at failure.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

This paper originates in an attempt to develop an effectivewidth method to determine the strength of compression flangesof cold-formed deck sections, with an explicit influence of initialimperfections. It was expected that this could be done by usinganalytical solutions for the elastic initial post-buckling behavior ofplates as given by Rhodes [1] determining the failure load as theelastic load corresponding to first membrane yield. For plates withstress-free longitudinal edges (free to wave in-plane) this load isrelatively easy to calculate, because for these plates firstmembrane yield occurs at the center of the longitudinal edges,where only membrane stresses in the longitudinal direction areworking.

To check the thus predicted failure loads finite elementsimulations were carried out (see Section 2), on square, simplysupported plates with initial imperfections in the shape of thefirst buckling mode, and a reference slenderness l varyingbetween 1 and

ffiffiffi8p

, where

l ¼ffiffiffiffiffiffiffiffiffiffiffiffiffif y=scr

q(1)

ll rights reserved.

+3140 245 0328.

. Bakker).

with fy being the yield stress and scr the buckling stress. This rangewas based on customary slenderness ratios of cold-formed decksections. When the analytically predicted failure loads werecompared to those determined with the finite element simula-tions, it was found that for plates with a reference slenderness lclose to 1, and small initial imperfections, the predicted failureloads are slightly unconservative, while for plates with increasingslenderness, and increasing initial imperfections, they becomemore and more conservative (up to 30%). In an attempt tounderstand why this is so, closer attention was paid to the failuremodes of the plates in the finite element simulations. It was found(see Section 3) that four types of failure can be distinguished:failure by outer fiber yielding at the center of the plate(C-yielding), failure by membrane yielding of the (longitudinal)edges of the plate (E-yielding), failure by outer fiber yielding of thecenter followed by membrane yielding of the edge (CE-yielding),and failure by membrane yielding of the edge followed by outerfiber yielding at the center (EC-yielding). As will be discussed laterthese four types of failure can be reduced to two types of failureby defining edge failure (EF) as the failure mode of plates wheremembrane yielding at the longitudinal edge occurs before orwithout outer fiber yielding at the center, and center failure (CF)as the failure mode of plates where outer fiber yielding at thecenter occurs before or without membrane yielding at thelongitudinal edge. In Section 3.3 it will be explained whichparameters govern the occurrence of either mode, and in

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Notation

a length of plateb width of platebce width of center stripbed width of edge stripbef effective width of platebef;Fu effective width of plate for strengthbef;e effective width of plate for stiffnessbef;sA effective width of plate for maximum membrane

stressfy yield strengtht plate thicknessu in-plane shortening of plateucr in-plane shortening of plate at buckling, equals ecr a

w maximum out-of-plane deflection, occurring at centerof plate

w0 maximum initial out-of-plane deflection, occurring atcenter of plate

wfic;FEM fictitious elastic out-of-plane displacement corre-sponding to Fu;FEM

wfic;u fictitious elastic out-of-plane displacement corre-sponding to efic;u

wfic;fmy fictitious elastic out-of-plane displacement corre-sponding to first membrane yield

A coefficientAu, Bu coefficients in expression to determine u/ucr

AF, BF coefficients in expression to determine F/Fcr

As;A, Bs;A coefficients in expression to determine sA/scr

D flexural rigidity factorE modulus of elasticityE�/E ratio of post-buckling to pre-bucking stiffness of

perfectly flat plate

Fce load carried by center stripFcr buckling load, equals scr bt

Fed load carried by edge stripFu ultimate load of plateFu;efic ultimate load of plate predicted by fictitious elastic

strain method using efic;u

FCFu;�fic ultimate load of plate predicted by fictitious elastic

strain method using �CFfic;u

FEFu;�fic ultimate load of plate predicted by fictitious elastic

strain method using �EFfic;u

Fu;FEM ultimate load of plate determined by FEM simulationFy yield load, equals fy bt

K buckling coefficiente average in-plane strain: u/a

ecr average in-plane strain at buckling, equals scr/E

efic;fmy fictitious elastic strain corresponding to wfic;fmy

efic;u empirically predicted fictitious elastic strain at failure

�CFfic;u efic;u at center failure

�EFfic;u efic;u at edge failure

efic;FEM fictitious elastic strain corresponding to wfic;FEM

ey yield strain, equals fy/E

Z (w/t)2�(w0/t)2

Zfic;fmy (wfic;fmy/t)2�(w0/t)2

Zfic;u (wfic;u/t)2�(w0/t)2

Zfic;FEM (wfic;FEM/t)2�(w0/t)2

l reference slenderness:ffiffiffiffiffiffiffiffiffiffiffiffiffif y=scr

qn Poisson’s coefficients average membrane stress F/(bt)sA membrane stress at point A (in x-direction of plate)scr buckling stresssce stress in center stripsed stress in edge strip

M.C.M. Bakker et al. / Thin-Walled Structures 47 (2009) 1–132

Section 3.4 it will be shown that the different failure modes can beexplained qualitatively from a simple two-strip model originallyproposed by Calladine [2] and modified by Bakker et al. [3].

It is difficult to predict the strength corresponding to thedifferent failure modes by simple, purely analytical formulas. Theelastic determination of first outer fiber yield at the center of theplate is possible (see for instance Mahendran [4]), but requires alot of effort, because it is not only influenced by membranestresses in the longitudinal direction, but also by membranestresses in the transverse direction and by stresses caused bybending moments. Failure by membrane yielding of the edge is anelasto-plastic failure, but no accurate, simple analytical modelsare available yet to determine the elasto-plastic behavior of plates.Elasto-plastic failure loads might be determined as the point ofintersection of elastic loading, and rigid-plastic unloading curvesdetermined from an assumed yield line mechanism [5], but thistype of calculations also requires a lot of calculation effort. Thispaper shows that by deriving empirical expressions for thefictitious elastic strain at failure, the post-buckling strength forboth CF and EF can be determined from relatively simpleanalytical solutions for the elastic plate behavior of plates.Therefore, in Section 4 the elastic theory for the initial and farpost-buckling behavior, available from literature is summarized.Initial post-buckling solutions have engineering accuracy for loadsup to about twice the buckling load. Slender plates may fail atloads higher than twice the buckling load, making it necessary touse far post-buckling solutions for strength predictions. Amodified far post-buckling theory is proposed which is simplerto use than the far post-buckling solutions described in the

literature, giving results with engineering accuracy for loads up toabout three times the buckling load. The results of these theoriesare compared to the results from the finite element simulations.

In Section 5.1, the fictitious elastic strains at failure arecalculated. These are the elastic strains, which in a (purely elastic)modified far post-buckling calculation would result in a load equalto the elasto-plastic failure loads in the FEM simulations. InSection 5.2, empirical expressions are proposed to predict thesefictitious elastic strains at failure, making a distinction betweenedge and center failure. The loads, calculated from the modifiedfar post-buckling theory using the fictitious strains predicted bythese empirical formulas are compared to the failure loads in theFEM simulations. In Section 6, it is explained how the modified farpost-buckling theory and empirical formulas to determinethe fictitious elastic strain at failure can be used in the effectivewidth method. Also it is discussed whether it is necessary toinclude far post-buckling effects. Finally in Section 7 conclusionsare summarized.

2. Finite element simulations

In this paper the post-buckling failure behavior of plates asshown in Fig. 1 is studied. All edges of the plate are simplysupported (uz ¼ 0). The edges loaded by the compression force areforced to remain straight, but free to experience Poisson’scontraction. The other two edges are free to wave in-plane, thusmembrane stresses in the y-direction are equal to zero. Theseboundary conditions correspond to the boundary conditions

ARTICLE IN PRESS

F

w0

b

a= b

A

BA

x

y

z

edge: = 0uzedge:ux u= = 0z

point:= 0uy

edge: = 0 uz

Fig. 1. Schematic view of numerical model: (a) boundary conditions and (b) initial imperfection, load, measures and location of points A and B.

Table 1Considered values for buckling stress and resulting values for parameters depending on buckling stress

scr (N/mm2) l (dimensionless) b (mm) b/t (dimensionless) Fcr (N) ucr (mm)

300 1.000 35.2 50.3 7392.0 5.021�10�2

225 1.155 40.7 58.1 6410.3 4.361�10�2

150 1.414 49.8 71.1 5229.0 3.557�10�2

100 1.732 61.0 87.1 4270.0 2.905�10�2

75 2.000 70.4 100.6 3696.0 2.514�10�2

50 2.449 86.3 123.3 3020.5 2.055�10�2

37.5 2.828 99.6 142.3 2614.5 1.779�10�2

M.C.M. Bakker et al. / Thin-Walled Structures 47 (2009) 1–13 3

usually used for the modeling of compression flanges in thin-walled steel deck sections. The reason to choose these boundaryconditions is that the research described in this paper is part of aresearch project on the strength of cold-formed deck sectionssubjected to the combined action of bending moment andconcentrated load [6]. The concentrated load causes deformationsof the compression flange, which may be quite large. Therefore itwas decided to study the failure behavior of uniformly com-pressed plates with initial imperfections up to two times the platethickness.

When a perfectly flat simply supported plate is subjected touniaxial compression, the stress distribution is uniform over theplate, until the buckling load is reached. After buckling the stressdistribution becomes non-uniform, both over the width b and thelength a of the plate. For plates with initial imperfections thestress distribution is non-uniform from the onset of loading. Inthis paper, it is assumed that the plate has a sinusoidal initialimperfection, with the maximum imperfection w0 occurring atthe center of the plate (see Fig. 1).

In this paper the following results will be discussed asfunctions of the out-of-plane deflection w at the center of theplate, where w is the total out-of-plane deflection at the center ofthe plate, including the initial imperfection w0:

�

the load F or average stress s in x-direction:

s ¼F

bt(2)

�

the axial shortening u or the average strain e in x-direction:

� ¼u

a(3)

�

membrane stresses sA in the x-direction at point A.

These results can be made dimensionless by using the bucklingstress:

scr ¼Kp2D

b2t(4)

from which we can define the buckling strain, ecr:

�cr ¼scr

E(5)

the buckling shortening, ucr:

ucr ¼ �cra ¼scra

E(6)

and the buckling force, Fcr:

Fcr ¼ btscr (7)

where D is the plate flexural rigidity factor:

D ¼Et3

12ð1� n2Þ(8)

where t is the plate thickness, a and b are the length and width ofthe plate (for a square plate a ¼ b), E is the modulus of elasticity, K

is the buckling coefficient (for a square plate K ¼ 4) and n isPoisson’s ratio.

According to Little [7], for two plates having the same values ofn, a/b, b/t,

ffiffiffiffiffiffiffiffiffiffif y=E

q(or fy/scr) and w0/t, but different values of fy, E

and w0/b, numerical analysis will predict precisely the same non-dimensional load-shortening response. In the parameter studysquare plates were studied (a ¼ b) and the plate thickness andthe material properties were kept constant: t ¼ 0.7 mm, fy ¼ 300N/mm2, n ¼ 0.3, and E ¼ 2.1�105 N/mm2. The buckling stress scr

was varied by varying the width b of the plate. Table 1 gives anoverview of the considered values for the buckling stress and thecorresponding values for the reference slenderness l, plate widthb, b/t ratio, buckling load Fcr and buckling shortening ucr. Eachsimulation has been performed once with linear-elastic materialproperties, and once with linear-elastic/ideal plastic materialproperties, using von Mises yield criterion. Thus a total of 7 elasticand 49 elasto-plastic geometrical non-linear simulations with thefinite element program ANSYS 8.1 has been performed (for elasticbehavior a calculation for only one critical stress is needed). In themodel rectangular elements SHELL43 were used. The element hassix degrees of freedom at each node: translations in the nodalx-, y- and z-directions and rotations along the nodal axes.Calculations with this element are based on Mindlin plate theory.The deformation shapes are linear in both in-plane directions. Themesh density for each plate was 40�40 elements. The numericalanalyses were performed for loads up to three times the bucklingload. All boundary conditions, axis convention and the specificpoints on the plate are presented in Fig. 1.

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0.0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3.0

3.3 elastic

F/F

cr

F/F

cr

w/t

- ultimate load

- yield at center

- yield at edge

w0=0,01tw0=0,25t

0.0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3.0

3.3 elastic

u/ucr

- ultimate load

- yield at center

- yield at edge

4.03.02.01.00.0 10.08.06.04.02.00.0

�cr/fy=1/4

�cr/fy=1/2

�cr/fy=1/2

�cr/fy=1

�cr/fy=1/4

�cr/fy=1/8

�cr/fy=1

�cr/fy=1/8

w0=0,01tw0=0,25t

w0=tw0=2tw0=t

w0=2t

Fig. 2. Overview of selected elastic and elasto-plastic curves obtained from the FEM simulations.

M.C.M. Bakker et al. / Thin-Walled Structures 47 (2009) 1–134

3. Failure modes occurring in finite element simulations

3.1. Introduction

Fig. 2 shows selected elastic and elasto-plastic load–deflectioncurves obtained from the performed finite element simulations. Inthis figure the solid dots indicate the ultimate loads, the opensquares indicate first outer fiber yield at the center of the plate(point B), and the open triangles indicate first membrane yield atthe center of the longitudinal edge (point A). These graphs indicatein what order center yielding and membrane yielding occur. Thisorder forms the basis for classifying failure modes as edge orcenter failure modes. Fig. 3 shows the development of yield zonesfor the various failure modes. First yielding and second yieldingindicated in this figure can be either outer fiber yielding at thecenter or membrane yielding at the edges. Outer fiber yielding atthe corners is not included in defining first and second yielding,since it hardly influences the load–deflection behavior.

3.2. Center failure

Plates failing by CF can be distinguished into plates failing bycenter yielding of the plate (C-yielding) and plates failing bycenter yielding followed by edge yielding (EC-yielding). In platesfailing by C-yielding, first yield occurs in the outer fibers of thecenter of the plate (point B), resulting in an immediate deviationof the elasto-plastic load–deformation behavior from the elasticbehavior. Next the outer fibers in an area near the corners of theplate start yielding. First membrane yield (at point A of thelongitudinal edge) occurs after failure, and thus cannot influencethe failure load. Failure by C-yielding results in failure with verylittle elasto-plastic reserve. The failure loads may be smaller thanthe (fictitious) elastic load corresponding to first membrane yieldat the longitudinal edge. Failure by CE-yielding resembles failureby C-yielding, except that the plate has some elasto-plasticreserve after center yielding and fails only when in the long-itudinal edge membrane yielding starts.

3.3. Edge failure

Plates failing by EF can be distinguished in plates failing byedge yielding (E-yielding) and plates failing by edge yielding

followed by center yielding (EC-yielding). In plates failingby E-yielding first yield occurs at the outer fibers of the corner.This outer fiber yield has hardly any influence on the load–deflection behavior of the plate. The elasto-plastic load–deflectioncurve starts to deviate from the elastic load–deflection curveonly after first membrane yield (at point A of the longi-tudinal edge). Failure is not at first membrane yield, but aftersome membrane yielding of the plate edges, at averagestrains larger than the yield strain. The failure loads maybe significantly larger than the (fictitious) elastic load corres-ponding to first membrane yield at the longitudinal edge. Failureby EC-yielding resembles failure by E-yielding, except thatadditional center yielding occurs before failure, resulting insmaller strains at failure and a steeper unloading behavior afterfailure.

3.4. Parameters determining failure modes

Table 2 gives an overview, which failure modes occurred inthe FEM simulations. It shows that the more slender plates(lX2) always fail by E-yielding, regardless of the imperfection.For more stocky plates, (lp1.155) the failure mode changesfrom C-yielding, to CE-yielding, EC-yielding and E-yieldingwith increasing initial imperfections. For plates with a referenceslenderness l ¼ 1.414, the same sequence of failure modesis found, except that the failure by C-yielding does not occur,and that failure by EC-yielding is found for smaller imperfec-tions than in the more slender plates. It is thought that forplates with a very small imperfection (w0/t ¼ 0.01), the failuremode changes from C-yielding, to CE-yielding, EC-yieldingand E-yielding by increasing the slenderness. In Table 2 thefailure by EC-yielding is missing for plates with an imperfectionw0/t ¼ 0.01, but for plates with an imperfection w0/t ¼ 0.25, thisfailure mode is found between the CE-yielding and E-yieldingfailure modes.

The sequence in occurring failure modes suggests that yieldingat the first location is more decisive for the failure mode thanyielding at the second location although one might think that thelatter should be regarded as the final cause of failure. Therefore itseems justified to reduce the four modes to two modes: CFincorporating modes C-yielding and CE-yielding, and EF incorpor-ating modes EC-yielding and E-yielding.

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first yielding second yielding failure of the plate

E

EC

CE

C

Fig. 3. Figures showing yield locations in C-, CE-, EC- and E-yielding failure modes (white—no yielding, gray—outer fiber yielding and black—membrane yielding).

Table 2Failure modes observed in FEM parameter study; the bold values represent center

failure (CF) according to Mahendran’s criterion [4]

C: center yielding; CE: center yielding followed by edge yielding; E: edge yielding;

EC: edge yielding followed by center yielding.

M.C.M. Bakker et al. / Thin-Walled Structures 47 (2009) 1–13 5

According to Mahendran [4], center yielding of the plate willoccur before edge membrane yielding if

w0

tp0:67þ 0:086S� 0:081S2 (9)

where S is a slenderness parameter:

S ¼b

t

ffiffiffiffiffif y

E

s(10)

Using Eqs. (4) and (8), and taking n ¼ 0.3 it can be shown that

S ¼ 1:901l (11)

This formula agreed well with the performed finite elementsimulations. In the few cases where Mahendran’s formula did notagree with the FEM results (see Table 2), edge yielding and centeryielding occurred almost at the same time.

3.5. Explanation by two-strip model

In [3] it was shown that the two-strip model can give anaccurate description of elastic post-buckling behavior. Here it willbe shown that the two-strip model can also be used to give aqualitative explanation of the occurring failure modes. In the two-strip model [2,3,8] there are two edge strips and one center strip.The load carried by the plate is the sum of the loads carried by

ARTICLE IN PRESS

Fig. 4. Explanation of characteristics of failure modes by two-strip model: (a) contribution of center strip, (b) contribution of edge strip and (c) summation of edge and

center strip contributions.

M.C.M. Bakker et al. / Thin-Walled Structures 47 (2009) 1–136

the separate strips:

F ¼ Fed þ Fce ¼ bedtsed þ bcetsce (12)

where Fed and Fce are the loads carried by the edge strips,respectively, the center strip, bed and bce are the widths of the edgestrip, respectively, the center strip and sed and sce are the stressesin the edge strips, respectively, the center strip.

If the plate behavior is elastic then the center strip is assumedto behave like a classical Euler column, that is, the stresses in itcan be calculated as

sce ¼ 1�w0

w

� �scr (13)

where scr is the buckling stress of the plate (not of the strip).The two edge strips always remain straight and togetherconstitute a single element of the system. The stresses in theedge strips are proportional to the in-plane shortening u of theplate:

sed ¼ Eu=a ¼ E� (14)

The only compatibility requirement taken into account is thatthe center strip and edge strips experience the same in-planeshortening u. It can be derived [3] that the relation between thein-plane shortening u and the maximum out-of-plane deflection

w of the modeled plate is given by

u ¼scea

E�

Cwp2ðw2 �w20Þ

4a(15)

with

Cw ¼b

2ðb� bedÞ(16)

The width of the edge strip can be determined from the ratioE�/E of post-buckling to pre-buckling stiffness of the perfectly flatplate, since it can be shown that for the two-strip model:

En

E¼

bed

b(17)

Eqs. (15) and (17) are valid only in the initial post-bucklingrange. They can be modified to be valid also in the far post-buckling range, as explained in [8]. For the qualitative discussionin this section the difference between initial and far post-bucklingformulations (see also Section 4) will be ignored for simplicity.

The occurrence of different failure modes can be understoodfrom the two-strip model by assuming that the center and edgestrip have different failure mechanisms. It is proposed that failureof the center strip can be determined as the point of intersectionof the elastic curve, and a rigid-plastic curve (see Fig. 4a).

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M.C.M. Bakker et al. / Thin-Walled Structures 47 (2009) 1–13 7

The rigid-plastic curve can be determined by assuming a localplastic mechanism with one yield line at the center of the centerstrip, perpendicular to its longitudinal edges. The followingformulas can then be derived:

sce ¼mp;red

wt(18)

where mp;red is the reduced plastic moment capacity per unitwidth of the center strip, which is reduced due to the presence ofa normal force scet per unit width:

mp;red ¼ mp 1�scet

np

� �2 !

(19)

with

mp ¼1

4t2 2ffiffiffi

3p f y (20)

np ¼ t2ffiffiffi3p f y (21)

where mp is the plastic moment capacity per unit width of thecenter strip and np is the plastic normal force capacity per unitwidth of the center strip. The factor 2

. ffiffiffi3p

in Eqs. (20) and (21)arises from the assumption that no strain rates occur in the lengthdirection of the yield line. Inserting Eq. (19) into Eq. (18) andsolving for sce results in

sce ¼�n2

pwþ np

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4m2

p þ n2pw2

q2mp

(22)

It is assumed that relation between u and w is the same forelastic and elasto-plastic plate behavior (see Eq. (15)).

To model failure of the edge strip (see Fig. 4b) it is proposedthat the edge strip remains elastic until the average strain of theplate equals the fictitious elastic average strain efic;fmy correspond-ing to first membrane yield at the longitudinal edge (thedetermination of this strain will be discussed in Section 5.1).After first membrane yield of the edge strip, the stiffness of theedge strip decreases. This is modeled (rather arbitrarily) byassuming that after first membrane yield of the edge the stress inthe edge strip can be calculated as

sed ¼ E�fic;fmy þ 0:5Eð�� �fic;fmyÞ (23)

It is furthermore assumed that failure occurs at an averagestrain corresponding to the yield strain

�y ¼f y

E(24)

Note that many of these assumptions are disputable. They willnot be discussed here since the object goal is not to give anaccurate quantitative description of the post-buckling strength,but only to provide insight in the occurring failure modes.

Fig. 4c shows the resultant behavior if the contributions of theedge strip and center strip are added, for a plate with a referenceslenderness l ¼ 1 and a plate with a reference slenderness l ¼ 2.For the first plate fy ¼ scr and uy ¼ ucr, for the second platefy ¼ 4scr and uy ¼ 4ucr. This figure shows that for plates with areference slenderness close to 1, and very small imperfections,failure of the plate is due to C-yielding. The smaller theimperfection is, the steeper is the reduction of load carryingcapacity after failure. For plates with slightly larger initialimperfections or a slightly larger slenderness, the decrease inload carrying capacity of the center strip may be compensated bythe still increasing load carrying capacity of the edge strip, thusresulting in failure due to CE-yielding. When the imperfections orplate slenderness increase further, failure of the center strip willoccur at strains larger than the elastic strain corresponding to first

membrane yield, but at strains smaller than the yield strain,resulting in failure by EC-yielding. Finally the imperfections orplate slenderness may become so large that the center strip willfail only at strains larger than the yield strain, resulting in failureby E-yielding.

3.6. Discussion

As far as the authors know the described failure modes havenot been distinguished explicitly before. Calladine [2] mentionedthe possibility of different failure modes. He proposed a two-stripmodel (which was modified by the authors to the strip modeldescribed in Section 3.5) to determine the failure load ofuniformly compressed plates with the longitudinal edges keptstraight, assuming that failure occurs at average strains largerthan the yield strain, when outer fiber yield occurs in the center ofthe plate. He noted that failure at strains smaller than the yieldstrain might occur for plates with l ¼ 1 and very small initialimperfections. Calladine [2] furthermore commented that Walkerand Murray [9] had taken a different view on the cause of failure,proposing a design formula based on first yield at the center of thelongitudinal edge, ‘‘with complete indifference to the flexuralstresses at the center of the plate, and to the unstable behaviorwhich occurs in consequence.’’

Calladine [2] was aware of the difference between plates withlongitudinal edges kept straight, where the membrane stresses inx-direction are uniform over the length of the plate, and plateswith stress-free longitudinal edges, where high localized mem-brane stresses in x-direction occur at the center of the longitudinaledge. He noted that these stresses might be responsible for thechoice of final failure mode between the roof mechanism and theflip-disc mechanism, observed experimentally in the collapse ofsteel tubes in compression. This hypothesis was confirmed byMahendran [4], and it is interesting to see that Eq. (9) developedby Mahendran to predict whether the roof or flip-disc mechanismwill occur, can also be used to predict whether the plate will failby EF or CF. Note, however, that the edge failure and center failuredo not have a direct relation to the roof and flip-disc mechanism.First, no yield line mechanism was observed at failure in theperformed finite element simulations. Second, a flip-disc mechan-ism is incompatible with the symmetrical deformation mode atfailure (more or less corresponding to the symmetrical shape ofinitial imperfections).

4. Modeling elastic behavior

4.1. Initial post-buckling solution

The elastic post-buckling behavior of thin plates with initialimperfections is governed by Marguerre’s equations. Approximateanalytical solutions for these equations can be found by postulat-ing a shape for the out-of-plane deflections w. At loads belowabout twice the buckling load the assumption of an unchangingbuckled form gives results of engineering accuracy. According toRhodes [1] the solution based on an unchanging sinusoidaldeflection shape can be described by the following equations:

F

Fcr¼

sscr¼ 1�

w0

w

� �þ AFZ (25)

u

ucr¼

�

�cr¼ 1�

w0

w

� �þ AuZ (26)

sA

scr¼ 1�

w0

w

� �þ As;AZ (27)

ARTICLE IN PRESS

Table 3Table of coefficients A and B

Coefficients Initial post-

buckling

solution

Far post-buckling

solution (in format

modified far post-

buckling solution)

Modified far post-

buckling solution

F/Fcr

AF 0.2356 0.2149 0.2356

BF 0 �0.4283�10�3�0.3137�10�2

u/ucr

Au 0.5775 0.5559 0.5775

Bu 0 0.1257�10�1 0.7799�10�2

sA/scr

As;A 0.9062 0.8429 0.9062

Bs;A 0 0.9572�10�2�0.2608�10�2

M.C.M. Bakker et al. / Thin-Walled Structures 47 (2009) 1–138

with

AF ¼A

K

En

E; Au ¼

A

K; As;A ¼

A

K

En

E

�qsqsA

(28)

where

Z ¼w

t

� �2

�w0

t

� �2

(29)

A is a coefficient, E�/E is the ratio of the pre-buckling to post-buckling stiffness for a perfectly flat plate without initialimperfections and the ratio qs/qsA is a partial variation of theaverage stress s. Rhodes [1] gives values for the coefficients K, A,E�/E and qs/qsA depending on the ratio e ¼ a/b of the buckle halfwidth a and the plate width b of the plate. For the case e ¼ 1(square plates), he gives the following values for a simplysupported plate with stress-free unloaded edges:

K ¼ 4; A ¼ 2:31;En

E¼ 0:408;

qsqsA¼ 0:26 (30)

These values are valid for Poisson’s ratio n ¼ 0.3. The resultingAF, Au and As;A values are given in Table 3.

4.2. Far post-buckling solution

For loads larger than about twice the buckling load the changesin the buckling form must be accounted for. Williams and Walker[10] gave an explicit solution for the elastic far post-bucklinganalysis of compressed plates. The format of their expressions isbased on the perturbation approach, but the value of theconstants in these expressions has been determined fromnumerical simulations (using the finite difference method). Theirsolution takes the following form:

AW&Ww fþ BW&W

w f3¼

ffiffiffiZp

(31)

in which

f ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF

Fcr� 1þ

w0

w

s(32)

u

ucr¼

F

Fcrþ AW&W

u Zþ BW&Wu Z2 (33)

sA

scr¼

F

Fcrþ AW&W

s;A Zþ BW&Ws;A Z2 (34)

For a square plate with simply supported edges, subjected touniaxial compression with the unloaded edges stress free,Williams and Walker [10] give the following values for the

coefficients (valid for n ¼ 0.3):

AW&Ww ¼ 2:157 and BW&W

w ¼ 0:010 (35)

AW&Wu ¼ 0:341 and BW&W

u ¼ 0:013 (36)

AW&WsA ¼ 0:628 and BW&W

sA ¼ 0:010 (37)

4.3. Modified far post-buckling solution

It was found that Eqs. (32)–(34) can be rewritten in a formatsimilar to Eqs. (25)–(27). Therefore first the coefficient f is solvedfrom Eq. (32) (using the Mathematica 5.2 program), resulting in

f ¼�2:71765þ 26:4567C2

C(38)

with

C ¼ 0:0027ffiffiffiZpþ 0:0027

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi148:678þ Z

p� �1=3(39)

From Eqs. (32), (38) and (39) the load F can be solved:

F

Fcr¼�2:71765þ 26:4567C2

C

!2

þ 1�w0

w(40)

Using a power series expansion for F/Fcr about the point Z ¼ 0and leaving out negligible small terms, this equation can befurther simplified to get

F

Fcr¼

sscr¼ 1�

w0

w

� �þ AFZþ BFZ2 (41)

Using Eq. (41), Eqs. (33) and (34) can be written as

u

ucr¼

�

�cr¼ 1�

w0

w

� �þ AuZþ BuZ2 (42)

sA

scr¼ 1�

w0

w

� �þ As;AZþ Bs;AZ

2 (43)

The resulting A and B values, derived from the coefficients AW&W

and BW&W, are given in Table 3, under far post-buckling solution.In the perturbation approach by Williams and Walker [10],

both the coefficients AW&W and BW&W were determined from theresults of numerical solutions. In this paper it is proposed to takethe coefficients A equal to the coefficients determined in the initialpost-buckling solution of the Marguerre equations, and fitting thecoefficient B to the results of numerical simulations, using theformat of Eqs. (41)–(43) instead of (31)–(34). The resultingcoefficients are given in Table 3, under modified far post-bucklingsolution. The coefficients B where fitted for w0 ¼ t and F/Fcr ¼ 3.0,because it was found that these specific values yield the bestresults. For more results see Rosmanit and Bakker [11]. Theadvantage of the modified far post-buckling solution to the farpost-buckling solution given by Williams and Walker [10] is thatthe equations of the modified far post-buckling solution are easierto use. Also the modified far post-buckling solution is moreconsistent, since for small deflections the modified far post-buckling solution will converge to the initial post-bucklingsolution, while the far post-buckling solution will not.

4.4. Comparison of results

Graphical and numerical comparisons of a representativeselection of results are shown in Figs. 5–7. It can be seen thatthe modified far post-buckling solution gives the most accurateresults for the ratios F/Fcr and u/ucr. For small imperfections theinitial post-buckling solution gives results within 5% error forloads up to about twice the buckling load, but for large initial

ARTICLE IN PRESS

0.0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3.0

3.3

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

1,2,3,4

F/F

cr

w/t

21,43

2143

1 - ANSYS

2 - initial post-buckling

3 - far post-buckling

4 - modified far post-buckling

1,2,43

21,43

23

1,4

w0=2tw0=tw0=0.01t

1,2,43

2314

4.5

Fig. 5. Overview of selected elastic curves obtained from the FEM simulations and initial post-buckling, far post-buckling and modified far post-buckling solutions.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

u/u c

r

w/t

w 0=0,01t

1,2,3,4

31,42

1,3,42

w 0= t

1423

1,423

w0=2t

1423

14

2,3

1 - ANSYS

2 - initial post-buckling

3 - far post-buckling

4 - modified far post-buckling

4.5

Fig. 6. Overview of selected elastic curves obtained from the FEM simulations and initial post-buckling, far post-buckling and modified far post-buckling solutions.

M.C.M. Bakker et al. / Thin-Walled Structures 47 (2009) 1–13 9

imperfections the 5% error occurs at lower loads. With respect tothe membrane stresses sA it is surprising to see that the initialpost-buckling solution gives better results than the far post-buckling solution, even for large initial imperfections and/or largedeflections. The modified far post-buckling solution is almostidentical to the initial post-buckling solution, since the coefficientBs;A is very small.

5. Prediction of failure loads

5.1. Fictitious elastic strains at failure and first membrane yield

To check whether it would be possible to use the modified farpost-buckling solution for the prediction of the post-buckling

strength of plates, the fictitious elastic strains efic;FEM correspond-ing to the elasto-plastic failure loads Fu;FEM determined from theFEM simulations as shown in Table 4 were calculated. This can bedone as follows. Eq. (41) describes the non-linear-elastic relation-ship between force and out-of-plane displacement. Now for theforce, the ultimate load Fu;FEM as predicted by the FEM simulationscan be used, and the fictitious displacement wfic;FEM can be solvedfrom

Fu;FEM

Fcr¼ 1�

w0

wfic;FEM

� �þ AFZfic;FEM þ BFZ2

fic;FEM (44)

where

Zfic;FEM ¼wfic;FEM

t

� �2

�w0

t

� �2

(45)

ARTICLE IN PRESS

Fig. 7. Overview of selected elastic curves obtained from the FEM simulations and initial post-buckling, far post-buckling and modified far post-buckling solutions.

Table 4Ratios’s Fu;FEM /Fcr; the bold values represent center failure (CF) according to

Mahendran’s criterion [4]

M.C.M. Bakker et al. / Thin-Walled Structures 47 (2009) 1–1310

Using Eq. (42) the fictitious elastic strain efic;FEM correspondingto Fu;FEM can then be determined as

�fic;FEM

�cr¼ 1�

w0

wfic;FEM

� �þ AuZfic;FEM þ BuZ2

fic;FEM (46)

Note that both the strain efic;FEM and the displacement wfic;FEM

are fictitious, because they are calculated from elastic modified farpost-buckling solutions, disregarding any yielding which mightoccur before the load Fu;FEM is attained. These fictitious strains anddeflections thus do not correspond to the strains and deflectionsin the (elasto-plastic) FEM simulations at failure (see also Fig. 8).

It is interesting to compare the strains efic;FEM to the fictitiouselastic strains efic;fmy, which are the fictitious strains which wouldresult in first membrane yield, if the plate would behaveelastically until this first membrane yield. Thus the influence ofeventual corner yield or center yield is not taken into account. Thefictitious strain efic;fmy can be calculated by first determiningwfic;fmy from:

f y

scr¼ 1�

w0

wfic;fmy

� �þ As;AZfic;fmy þ Bs;AZ

2fic;fmy (47)

where

Zfic;fmy ¼wfic;fmy

t

� �2

�w0

t

� �2

(48)

Eq. (47) is derived from Eq. (43) by requiring sA ¼ fy. Then thefictitious elastic strain efic;fmy corresponding to first membrane

yield can be determined as

�fic;fmy

�cr¼ 1�

w0

wfic;fmy

� �þ AuZfic;fmy þ BuZ2

fic;fmy (49)

5.2. Empirical expressions for fictitious elastic strain at failure and

accuracy of corresponding failure loads

If it is possible to develop empirical formulas to predict thefictitious strain efic;u at failure, then the failure load Fu;efic can becalculated by first determining the out-of-plane deflection wfic;u

from the equation

�fic;u�cr¼ 1�

w0

wfic;u

� �þ AuZfic;u þ BuZ2

fic;u (50)

where efic;u is the ficitious elastic strain at failure predicted by anempirical formula, and

Zfic;u ¼wfic;u

t

� �2

�w0

t

� �2

(51)

The thus determined values for wfic;u and Zfic;u can then be usedto find the failure load, Fu;efic:

Fu;�fic

Fcr¼ 1�

w0

wfic;u

� �þ AFZfic;u þ BFZ2

fic;u (52)

In the ideal case of an ‘‘exact’’ empirical formula the fictitiouselastic strain at failure efic;u will be equal to the fictitious elasticstrain at failure efic;FEM determined from the FEM simulations. Toassist in the development of an empirical formula to determineefic;u, Fig. 9a shows the ratio efic;FEM/efic;fmy as a function of thereference slenderness l and Fig. 9b the ratio efic;FEM/ey.

To find an empirical expression for the prediction of thefictitious strain at failure for plates failing by edge failure, both theratios efic;FEM/efic;fmy and efic;FEM/ey can be used for curve fitting.Since ey is much easier to calculate than efic;fmy the ratio efic;FEM/ey

will be used. It was found that for plates failing by edge failure, fora given imperfection ratio w0/t the relation between the ratio�EF

fic;FEM=�y and the reference slenderness l can be approximatedwith good accuracy by a straight line:

�EFfic;FEM

�y��EF

fic;u

�y¼ clþ d (53)

ARTICLE IN PRESS

Fig. 8. Definition of fictitious elastic strain at failure (shown for scr ¼ fy/4, w0 ¼ 0.01t).

Fig. 9. (a) Ratios efic;FEM/efic;fmy and (b) ratios efic;FEM/ey.

M.C.M. Bakker et al. / Thin-Walled Structures 47 (2009) 1–13 11

where the superscript EF denotes edge failure. After determiningthe coefficients c and d for various values of the imperfectionratio w0/t, and plotting these coefficients as a function of thereference slenderness l, it was found that both the relationbetween c and l, and the relation between d and l can also beapproximated with good accuracy by a straight line, resulting inthe following empirical expression for the fictitious elastic strainat edge failure [12]:

�EFfic;u

�y¼

w0

tð0:139� 0:029lÞ þ 0:844 (54)

Figs. 9a and b show that for plates failing by center failure thereis not such a simple linear relation between the ratio �CF

fic;FEM=�fic;fmy

or �CFfic;FEM=�y and the reference slenderness l, where the superscript

CF denotes center failure. Fortunately the accuracy of theprediction of the ultimate load is not very sensitive to errors inthe assumed fictitious elastic strain at failure. Taking

�CFfic;u ¼ �fic;fmy (55)

gives reasonable results for plates failing by center failure.

Table 5 shows the ratios FCFu;�fic=Fu;FEM and FEF

u;�fic=Fu;FEM , whereFCF

u;�fic and FEFu;�fic are the failure loads calculated with the ficti-

tious strain method, taking �fic;u ¼ �CFfic;u (see Eq. (54)) respec-

tively �fic;u ¼ �EFfic;u (see Eq. (55)).Taking �fic;u ¼ �

CFfic;u ¼ �fic;fmy as

done in the effective width method, paradoxically results inreasonable accurate strength predictions for plates failing bycenter failure, but very conservative predictions for platesfailing by edge failure. For plates failing by edge failure,taking �fic;u¼�

EFfic;u instead of �fic;u¼�

CFfic;u ¼ �fic;fmy, results in a much

improved strength prediction. Surprisingly, �fic;u¼�EFfic;u gives

also very good results for plates failing by center failure,except for the plate with l ¼ 1 and w0/t ¼ 0.01. This suggestsit might be possible to improve the strength predictionsby modifying the criterion to determine whether �CF

fic;u or �EFfic;u

should be used for the strength prediction, for instance bymodifying the coefficients in Mahendran’s formula (Eq. (9)).One can also argue that initial imperfection ratios w0/t ¼ 0.01are so small that they will not occur in practice for plateswith lX1 (see also Section 6.1), and that for all practicalimperfection ratios w0/t40.1, �EF

fic;u can be used for strengthpredictions.

ARTICLE IN PRESS

Fig. 10. Effective width for strength determined from FEM solutions compared to

effective width according to Winter formula.

Table 5

Ratios FCFu;�fic=Fu;FEM and FEF

u;�fic=Fu;FEM; the bold values represent center failure (CF) according to Mahendran’s criterion [4]

M.C.M. Bakker et al. / Thin-Walled Structures 47 (2009) 1–1312

6. Discussion

6.1. Effective width method

As Rhodes [1] noted, the term effective width has been used inthe literature to describe different effects. In this paper, adistinction will be made between an effective width for maximummembrane stress, an effective width for strength, and an effectivewidth for stiffness. Following Rhodes [1] the effective width bef;sA

for maximum membrane stress is defined as that width of fullyeffective (i.e. unbuckled) plate which sustains the same maximummembrane stress sA as the buckled plate under a given load:

bef ;sA ¼ bssA

(56)

The effective width bef;e for stiffness can be defined as thatwidth of unbuckled plate which sustains the same average strain eas the buckled plate for a given load:

bef ;� ¼ bsE�

(57)

If it is assumed that the ultimate load which a plate canwithstand is very close to that which causes first membrane yield,the failure load can be calculated from the effective width formaximum membrane stress bef ;sA

as

Fu ¼ bef ðsA¼fyÞtf y (58)

For that reason Rhodes [1] called the effective width formaximum membrane stress the effective width for strength. Ifone recognizes that a plate does not necessarily fail at a load closeto the load causing first membrane yield, it is more appropriate todefine an effective width bef;sA for strength as

bef ;Fu ¼Fu

tf y

(59)

This definition implies that the assumption that the edges of theplate start yielding at failure is not integral to the effective widthfor strength concept.

The modified far post-buckling formulas and empiricalexpressions for fictitious elastic strains at failure can now beused to determine the above-defined effective widths as explicitfunctions of the initial imperfections, by rewriting Eqs. (56), (57)and (59) as

bef ;sA

b¼

ssA

scr

scr¼

sscr

scr

sA(60)

bef ;�

b¼

sE�

scr

scr¼

sscr

�cr

�(61)

bef ;Fu

b¼

Fu

Fy

scr

scr¼

Fu

Fcr

scr

f y

(62)

where

Fy ¼ btf y (63)

Fig. 10 shows the effective widths ratios bef;Fu/b determinedfrom the finite element simulations taking Fu ¼ Fu;FEM, using Eq.(62), for various values of imperfections w0/t. In Fig. 10 also theWinter formula is shown:

bef

b¼

1

l1�

0:22

l

� �(64)

By determining the points of intersection of the Winter formula,and the effective width curves explicitly depending on initialimperfections, it can be seen what initial imperfections result inthe same strength as the Winter formula (at least for simplysupported square plates with initial imperfections in the shape ofthe first buckling mode). These initial imperfections increase withincreasing reference slenderness l, from imperfection ratios w0/tbetween 0.1 and 0.25 for plates with a reference slenderness lequal to 1, to imperfection ratios w0/t larger than 2 for plates witha reference slenderness l larger than about 2.5.

6.2. Importance of changes in the deflected form

The difference between initial post-buckling and far post-buckling behavior will become significant for plates with areference slenderness l larger than about 2. From Table 4 it can

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M.C.M. Bakker et al. / Thin-Walled Structures 47 (2009) 1–13 13

be seen that these plates fail at ratios F/Fcr of about 1.8 for smallimperfections. For larger imperfections this ratio decreases, butfor larger imperfections the modified far post-buckling solutionstarts to deviate from the initial post-buckling solution at smallerloads. Of course one can also derive expressions for the fictitiousstrain at failure using initial post-buckling equations. This wouldresult in simpler equations to determine the ultimate strength,especially since it would then be possible to derive an explicitsolution for the out-of-plane deflection corresponding to thefictitious elastic strain at failure [12].

7. Conclusions

This paper clarifies the occurrence and character of differentfailure modes in uniformly compressed, square simply supportedplates with stress-free longitudinal edges, depending on theslenderness and initial imperfections of the plate. It has beenshown how relatively simply, elastic modified far post-bucklingsolutions and empirically derived expressions for the fictitiouselastic strain at failure can be used to obtain accurate strengthpredictions. The proposed method can be regarded as a modifica-tion of the effective width method for strength described byRhodes [1], which is based on elastic initial post-bucklingsolutions, and the assumption that efic;u ¼ efic;fmy. The proposedmethod enables the determination of the effective width forstrength with an explicit influence of initial imperfections. It isshown that the effective width for strength may be different fromthe effective width for maximum membrane stress, and that theassumption that the edges of the plates start yielding at failure isnot integral to the effective width for strength concept.

For the prediction of the fictitious elastic strain at failure adistinction has been made between plates failing by edge failureand plates failing by center failure. Which failure mode occurs canbe determined from Eq. (9). Taking �fic;u ¼ �

CFfic;u ¼ �fic;fmy results in

reasonable strength predictions for plates failing by center failurebut very conservative strength predictions for plates failing byedge failure. For plates failing by edge failure a much betterstrength prediction can be obtained by taking �fic;u ¼ �

EFfic;u where

�EFfic;u is given by Eq. (54). Taking �fic;u ¼ �

EFfic;u gives also good results

for plates failing by center yielding, except for plates with areference slenderness l close to 1 and very small imperfections.Based on a comparison of FEM solutions with strength predictionsby Winter’s effective width formula it can be concluded that forplates with lX1 and practical imperfection rations (w0/t40.1),the fictitious elastic strain �EF

fic;u can be used for strength

predictions, regardless whether failure will occur by edge orcenter failure.

Since the study was limited to square simply supported plateswith longitudinal edges free to wave in plane subjected to uniformcompression, further research is necessary to determine whetherthe proposed fictitious elastic strain method is also applicable toplates with other geometries, loading and boundary conditions.

Acknowledgments

This research is part of an Aspasia program and was supportedby the Technology Foundation STW, applied science division ofNWO and the technology program of the Ministry of EconomicAffairs.

References

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[2] Calladine CR. The strength of thin plates in compression. In: Dawe DJ,Horsinton RW, Little AG, editors. Aspects of the analysis of plate structures, avolume in honour of Wittrick. Oxford, England: Clarendon Press; 1985.p. 269–91.

[3] Bakker MCM, Rosmanit M, Hofmeyer H. Elastic post-buckling analysis ofcompressed plates using a two-strip model. Thin-Walled Struct 2007;45:502–16.

[4] Mahendran M. Local plastic mechanisms in thin steel plates under in-planecompression. Thin-Walled Struct 1997;27:245–61.

[5] Murray NW. Introduction to the theory of thin-walled structures. Oxford,England: Clarendon Press; 1986.

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[7] Little GH. The collapse of rectangular steel plates under uniaxial compression.Struct Eng 1980;58B:45–61.

[8] Bakker MCM, Rosmanit M, Hofmeyer H. Elastic post-buckling behavior ofuniformly compressed plates. In: Laboube RA, Yu WW, editors. Recentresearch and developments in cold formed steel design and constructionProceedings of the 18th international specialty conference. Rolla, MI, USA:University of Missouri Rolla; 2006. p. 1–15.

[9] Walker AC, Murray NW. Analysis for stiffened plate panel buckling. Civilengineering research report no. 2/1974. Monash University, 1974.

[10] Williams DG, Walker AC. Explicit solutions for the design of initially deformedplates subjected to compression. Proc Inst Civil Eng 1975(Part 2):763–87.

[11] Rosmanit M, Bakker MCM. Report on elastic post-buckling analysis ofcompressed plates using a two-strip model. Research report O-2007.05,Technische Universiteit Eindhoven, Department of Architecture, Building andPlanning, Structural Design Group, The Netherlands, 2007.

[12] Rosmanit M, Bakker M.C.M. Report on prediction of the elasto-plastic post-buckling strength of uniformly compressed plates from the fictitious elasticstrain at failure. Research report O-2007.23, Technische Universiteit Eindho-ven, Department of Architecture, Building and Planning, Structural DesignGroup, The Netherlands, 2007.