Cellular buckling from mode interaction in I-beams …rspa.royalsocietypublishing.org/content/royprsa/468/2137/...interaction between global lateral-torsional buckling of the beam

Proc. R. Soc. A (2012) 468, 245–268doi:10.1098/rspa.2011.0400

Published online 21 September 2011

Cellular buckling from mode interaction inI-beams under uniform bending

BY M. AHMER WADEE* AND LEROY GARDNER

Department of Civil and Environmental Engineering, Imperial College London,London SW7 2AZ, UK

Beams made from thin-walled elements, while very efficient in terms of the structuralstrength and stiffness to weight ratios, can be susceptible to highly complex instabilityphenomena. A nonlinear analytical formulation based on variational principles for theubiquitous I-beam with thin flanges under uniform bending is presented. The resultingsystem of differential and integral equations are solved using numerical continuationtechniques such that the response far into the post-buckling range can be portrayed. Theinteraction between global lateral-torsional buckling of the beam and local buckling ofthe flange plate is found to oblige the buckling deformation to localize initially at thebeam midspan with subsequent cellular buckling (snaking) being predicted theoreticallyfor the first time. Solutions from the model compare very favourably with a series ofclassic experiments and some newly conducted tests which also exhibit the predictedsequence of localized followed by cellular buckling.

Keywords: nonlinear mechanics; lateral-torsional buckling; plate buckling; thin-walled beams;snaking; numerical continuation

1. Introduction

Beams are possibly the most common type of structural component, but whenmade from thin-metallic plate elements they are well known to suffer from avariety of elastic instability phenomena. There has been a vast amount of researchinto the buckling of thin-walled structural components with much insight gained(Hancock 1978; Schafer 2002; Rasmussen & Wilkinson 2008). In the current work,the well-known problem of a beam made from a linear elastic material with anopen and doubly symmetric cross section—an ‘I-beam’—under uniform bendingabout the strong axis is studied in detail. Under this type of loading, long beamsare primarily susceptible to a global mode of instability namely lateral-torsionalbuckling (LTB), where, as the name suggests, the beam deflects sideways andtwists once a threshold critical moment is reached (Timoshenko & Gere 1961).However, when the individual plate elements of the beam cross section, namelythe flanges and the web, are relatively thin or slender, elastic local buckling ofthese may also occur; if this happens in combination with global instability, thenthe resulting behaviour is usually far more unstable than when the modes aretriggered individually. Other structural components, usually those under axial

*Author for correspondence ([email protected]).

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246 M. Ahmer Wadee and L. Gardner

compression rather than uniform bending, such as I-section struts with thin plateflanges (Becque & Rasmussen 2009), sandwich struts (Hunt & Wadee 1998),stringer-stiffened and corrugated plates (Koiter & Pignataro 1976; Pignataroet al. 2000) and built-up, compound or reticulated columns (Thompson & Hunt1973) are all known to suffer from so-called interactive buckling phenomena, wherethe global and local modes of instability combine nonlinearly.

Experimental evidence suggests that the destabilization from the modeinteraction of LTB and flange local buckling is severe (Cherry 1960; Menkenet al. 1991), the response being highly sensitive to geometrical imperfectionsparticularly when the critical loads coincide (Goltermann & Møllmann 1989;Møllmann & Goltermann 1989). Nevertheless, apart from the aforementionedwork where some successful numerical modelling (both finite strip and finiteelement) and qualitative modelling using rigid links and spring elements werepresented, the formulation of a mathematical model accounting for the interactivebehaviour has not been forthcoming. The current work presents the developmentof a variational model that accounts for the mode interaction between globalLTB and local buckling of a flange such that the perfect elastic post-bucklingresponse of the beam can be evaluated. A system of nonlinear ordinary differentialequations subject to integral constraints is derived, which are solved using thenumerical continuation package AUTO (Doedel & Oldeman 2009). It is indeedfound that the system is highly unstable when interactive buckling is triggered;snap-backs in the response showing sequential destabilization and restabilizationand a progressive spreading of the initial localized buckling mode are alsorevealed. This latter type of response has become known in the literature ascellular buckling (Hunt et al. 2000a) or snaking (Burke & Knobloch 2007), and itis shown to appear naturally in the numerical results of the current model. As faras the authors are aware, this is the first time, this phenomenon has been found inbeams undergoing LTB and local buckling simultaneously. Similar behaviour hasbeen discovered in various other mechanical systems such as in the post-bucklingof cylindrical shells (Hunt et al. 1999) and the sequential folding of geologicallayers (Hunt et al. 2000b).

Experimental results generated for the current study and from the literatureare compared with the results from the presented model; highly encouragingquantitative results emerge in terms of both the mechanical destabilizationexhibited and the nature of the post-buckling deformation with tangible evidenceof cellular buckling being found. This demonstrates that the fundamental physicsof this system is captured by the analytical approach. A brief discussionis presented on whether other structural components made from thin-walledelements may also exhibit cellular buckling when local and global modes ofinstability interact. Conclusions are then drawn.

2. Variational formulation

(a) Classical critical moment derivation via energy

Consider a uniform simply-supported (pinned) doubly symmetric I-beam oflength L made from an isotropic and homogeneous linear elastic material withYoung’s modulus E and Poisson’s ratio n. The overall beam height and flange

Proc. R. Soc. A (2012)



Cellular buckling I-beams 247

b

hM

tw

y

tf

tfx

L

M Mh

(a)

(c)

(b)

(d)

z−f

uw

uw

us

ut

ub

web line

free edge

free edge

b

sbxsby

z

x

−f

M

Mx

My

vulnerableoutstand

Figure 1. (a) An I-section beam under uniform strong axis moment M . (b) Lateral displacementsand torsional rotations owing to lateral-torsional buckling (LTB); a right-hand Cartesian coordinatesystem is employed. After LTB: (c) top flange stress components from strong (sbx ) and weak axis(sby) bending moments; (d) induced bending moment components during LTB, Mx and My , whichforce one of the flange outstands to become more heavily compressed and hence vulnerable to localbuckling.

widths are h and b, respectively, with the web and flange thicknesses being tw andtf , respectively. The beam is under bending about the strong axis with a uniformmoment M , as shown in figure 1a. The second moments of area about the strongand weak axes are defined as Ix and Iy , respectively. When LTB occurs, as notedin the literature (Timoshenko & Gere 1961), the strong axis bending momentand corresponding displacements couple only with the lateral displacements andtorsional rotations at higher orders. For this reason and for the sake of simplicity,the displacements arising from strong axis bending are presently neglected, whichis a common assumption; in future work, however, these coupling effects may beincorporated. Therefore, only two separate lateral displacements are defined fordetermining the respective positions of the local centroids of the web and theflanges: us and uw. The lateral displacements of the top (ut) and bottom flanges(ub), respectively, are thus (figure 1b):

ut = us + uw and ub = us − uw. (2.1)

Moreover, a torsional angle of magnitude f to the vertical is introduced and theapplied moment M is transformed thus into components of strong and weak axismoments, Mx and My , respectively:

Mx = M cos(−f) ≈ M and My = M sin(−f) ≈ −Mf, (2.2)

both expressions being written to the leading order. Figure 1c shows a plan viewof the top flange of the beam and the stress-distribution components from thestrong axis and the induced weak axis bending moments. Therefore, the maximum





strong axis compressive stress occurs when y = h/2 and coupling this with thecoexisting compressive component of the weak axis stress, the most vulnerableflange outstand is defined, as shown in figure 1d. As the important componentof bending under LTB is about the weak axis, the values of the relevant secondmoment of area for the flange (If ) can be approximated to Iy/2 of the wholesection, where Iy = b3tf/6, by assuming that the overall contribution of the web isvery small, which is true for I-beams of practical dimensions. The strain energystored in the beam owing to bending (Ub) is therefore:

Ub = 12EIf

∫L

0(u′′2

t + u′′2b ) dz = 1

2EIy

∫L

0u′′2

s dz + 12EIw

∫L

0f′′2 dz , (2.3)

where primes denote differentiation with respect to the axial coordinate z , uw =hf/2 and Iw = Iyh2/4. The strain energy stored from uniform torsion (UT) is

UT = 12

∫L

0T df = 1

2GJ

∫L

0f′2 dz , (2.4)

where T is the torque, G is the material shear modulus and J is the Saint-Venanttorsion constant defined for an I-section as

J = 13[2bt3

f + (h − 2tf )t3w]. (2.5)

The work done by the external moment, MQ, is given by the induced weak axismoment My multiplied by the average end rotation from bending about the weakaxis; this is given by the following expression (Pi et al. 1992):

MQ =∫L

0Mfu′′

s dz (2.6)

and so the total potential energy is thus:

V = Ub + UT − MQ =∫L

0

[12EIyu′′2

s + 12EIwf′′2 + 1

2GJf′2 − Mfu′′

s

]dz . (2.7)

To find the critical moment Mcr, the calculus of variations could be used toderive the governing differential equation. However, as the solution of the bucklingmode is known to be sinusoidal (Timoshenko & Gere 1961), the same result canbe achieved using a two degree of freedom Rayleigh–Ritz formulation with thefollowing trial functions:

us = qsL sinpzL

and f = −qf sinpzL

, (2.8)

where qs and qf are generalized coordinates for us and f, respectively; owingto the coordinate system used, the negative sign in f ensures that qf > 0when qs > 0. The formulation is in small deflections (linear) and so only acritical equilibrium analysis is possible at this stage. The advantage of using theRayleigh–Ritz method becomes more apparent in the interactive buckling modelmentioned shortly. Substituting us and f into V , performing the integration





and then assembling the Hessian matrix Vij at the critical point C, gives thefollowing condition:

VCij =

∣∣∣∣∣V C

ss V Csf

V Cfs V C

ff

∣∣∣∣∣ =

∣∣∣∣∣∣∣EIy

(p

L

)4 −Mcr

(p

L

)2

−Mcr

(p

L

)2GJ

(p

L

)2 + EIw

(p

L

)4

∣∣∣∣∣∣∣ = 0, (2.9)

where the individual elements of the matrix VCij are thus

V Cab = v2V

vqavqb

∣∣∣∣C

. (2.10)

Solving equation (2.9) gives the classical expression for the critical moment Mcrthat triggers LTB for a beam with a doubly symmetric cross section under uniformbending (Timoshenko & Gere 1961):

Mcr = p

L

√EIyGJ

√1 + p2

L2

EIw

GJ. (2.11)

(b) Interactive buckling model

From §2a, it has been shown that as the displacements and rotations from LTBgrow, the applied moment M can be expressed as a component about the strongaxis (Mx) and an induced component about the weak axis (My) at any point alongz . As a result, the vulnerable outstand of the flange, as identified in figure 1d,may therefore buckle locally as a plate. The critical stress of plate buckling fora uniaxially and uniformly compressed rectangular plate, with one long edge-pinned and the other free, is given by the well-known formula (Timoshenko &Gere 1961):

sClocal = 0.43

p2Db2f tf

, (2.12)

where, in the current case, bf is the width of the vulnerable flange outstand and isgiven by (b − tw)/2 and D is the flexural rigidity of the flange plate that is equalto Et3

f /[12(1 − n2)]. This addresses the case for the flange buckling locally beforeany LTB occurs.

It was shown in Hunt & Wadee (1998) that the intrinsic assumptions inthe Euler–Bernoulli bending theory were insufficient to model any interactionbetween global and local buckling modes. The allowance of shear strainsto develop within the individual elements, however small, being key to theformulation. Figure 2a,b shows a useful way that shear can be introduced; thedisplacement of the top and bottom flanges being decomposed into separate ‘sway’and ‘tilt’ modes (Hunt et al. 1988) after the global instability (LTB) has beentriggered. Each original generalized coordinate qs and qf is defined as a ‘sway’and has a corresponding ‘tilt’ component, with associated generalized coordinatesqt and qt, respectively. This is akin to a Timoshenko beam formulation (Wanget al. 2000); when qs �= qt and qf �= qt, shear strains are developed and allow thepotential for modelling simultaneous LTB and local buckling.





ust

x

qstqsbz

qttqtb

x z

ust = us + uw = qsL + qfh

2sin

pzL

qst ≈∂ust

∂z= ( qs + qfl) p cos

pzL

usb = us – uw = qsL – qfh

2sin

pzL

qsb ≈∂usb

∂z= (qs – qfl) p cos

pzL

utt = − (qt + qtl)px cospzL

qtt ≈ −∂utt

∂x= (qt + qtl)p cos pz

L

utb = − (qt − qtl)px cospzL

qtb ≈ −∂utb

∂x= (qt − qtl )p cos pz

L

usb

x

z

x z

z

web line

b/ 2q

free edge

x

u

u(x, z) = −2x

bu(z ) and q ≈

∂u

∂x= −

2

bu(z )

w(x, z) = −2x

bw(z )

z web line

b/ 2 w

free edge

x

u(0)/ 2average flange end-displacement

outstand vulnerable to local buckling

top flange: plan view

outstand not vulnerable to local buckling

u(L )

web

u(0)

modelled flangeend-displacement u(L )/ 2

MMtop flange

bottom flange

weba0 ≈ u(0) / 2h a L ≈ u(L ) / 2h

beam elevation(x = 0)

(a) (b)

(c)

(e)

(d)

Figure 2. Interactive buckling model. (a,b) Introducing shear deformation: sway (upper row)and tilt (lower row) modes of lateral-torsional buckling in both flanges, where l = h/2L. (c,d)Introducing local buckling: displacement functions u and w defined for the vulnerable part of thetop flange: x = [−b/2, 0] and y = [h/2 − tf , h/2]. (e) Contribution from local buckling in the workdone term, where h is the overall cross-section height.

Previous work on this type of interactive buckling has included experimentalwork combined with the effective width theory (Cherry 1960), somephenomenological modelling using rigid links and springs along with experiments(Menken et al. 1991), some numerical work using a finite strip formulation(Goltermann & Møllmann 1989; Møllmann & Goltermann 1989) and a finite-element formulation (Menken et al. 1997). To model this analytically, however,two displacement functions to account for the extra in-plane displacement uand out-of-plane displacement w from local buckling, see figure 2c,d, need to





be defined. As the strain from the weak axis moment is linear in x and that theboundary condition for the line of the web is pinned out of the plane of the flange,the following linear distribution in x can be assumed for u and w:

u(x , z) = −2xb

u(z) and w(x , z) = −2xb

w(z). (2.13)

It is worth noting that restricting both interactive buckling displacementfunctions u and w to the vulnerable part of the compression flange confines thecurrent model to cases where LTB occurs before local buckling. In the caseswhere local buckling occurs first, the system would be expected to trigger theglobal mode rapidly afterwards (Wadee 2000) and the current model can be usedto indicate the deformation levels where the interaction would occur (as seenlater). However, to obtain an accurate linear eigenvalue solution for pure localbuckling in the current framework, at least another set of in-plane and out-of-plane displacement functions, replicating the role of u and w, respectively, wouldneed to be defined for the non-vulnerable part of the compression flange; thisaddition is left for future work.

(i) Local bending energy

Experimental evidence, presented in Menken et al. (1997), suggests that duringinteractive buckling, the significant local out-of-plane displacements within theflanges are confined to the vulnerable outstand. The component of additionalstrain energy stored in bending Ubl is hence given by

Ubl = D2

∫L

0

∫ 0

−b/2

{(v2wvx2

+ v2wvz2

)2

− 2(1 − n)

[v2wvx2

v2wvz2

−(

v2wvxvz

)2]}

dx dz .

(2.14)Substituting w into Ubl, the following expression is obtained:

Ubl = D2b

∫L

0

[b2

6w ′′2 + 4(1 − n)w ′2

]dz . (2.15)

(ii) Flange energy from axial and shear strains

As the flanges are assumed to behave in the manner of Timoshenko beams, thebending of the flanges when LTB occurs introduce both axial and shear strains,3 and g, respectively. The vulnerable part of the top flange, where x = [−b/2, 0]and y = [h/2 − tf , h/2], also has the possibility of local buckling; the von Kármánplate theory gives a standard expression for the axial strain 3 in the z-directionthat accounts for both LTB and local buckling terms, thus

3t1 = vutt

vz+ vu

vz+ 1

2

(vwvz

)2

= (qt + qtl)p2xL

sinpzL

− 2xb

u′ + 2x2

b2w ′2, (2.16)

where l = h/2L. For the part of the top flange that is not vulnerable to localbuckling, where x = [0, b/2], and the bottom flange, where y = [−h/2, −h/2 + tf ],Proc. R. Soc. A (2012)




the following respective axial strain expressions are obtained:

3t2 = vutt

vz= (qt + qtl)

p2xL

sinpzL

and 3b = vutb

vz= (qt − qtl)

p2xL

sinpzL

.

(2.17)The standard strain energy expression is integrated over the volume of the flanges:

Um = Etf2

∫L

0

[∫ 0

−b/232t1 dx dy +

∫ b/2

032t2 dx dy +

∫ b/2

−b/232b dx dy

]dz , (2.18)

which gives

Um = Ebtf12

∫L

0

[u′2 + 3

20w ′4 + 3

4u′w ′2 − (qt + qtl)p2j sin

pzL

(u′ + 3

8w ′2

)

+(q2t + q2

t l2)p4j2 sin2 pzL

]dz , (2.19)

where j = b/L is a beam aspect ratio parameter. It is worth noting that thetransverse displacement and the strain in the x-direction are omitted from thecurrent formulation. This is a simplification derived from Koiter & Pignataro(1976), where these components were found to have a negligible effect on thepost-buckling stiffness of a uniaxially compressed long plate with one longitudinaledge being pinned and the other being free.

In terms of the shear strain g in the xz plane within the top flange, wherey = [h/2 − tf , h/2], the von Kármán plate theory gives a standard expression,which needs to account for both LTB and local terms for the vulnerable outstandx = [−b/2, 0]:

gt1 = qst − qtt + local buckling terms

gt1 = vust

vz+ vutt

vx+ vu

vx+ vw

vxvwvz

and so gt1 = [(qs − qt) + (qf − qt)l]p cospzL

− 2bu + 4x

b2ww ′, (2.20)

and purely LTB terms for the non-vulnerable part of the top flange, x = [0, b/2],and the bottom flange, where y = [−h/2, −h/2 + tf ], respectively:

gt2 = qst − qtt = vust

vz+ vutt

vx= [(qs − qt) + (qf − qt)l]p cos

pzL

, (2.21)

and

gb = qsb − qtb = vusb

vz+ vutb

vx= [(qs − qt) − (qf − qt)l]p cos

pzL

. (2.22)





The standard shear strain energy expression is integrated over the volume ofthe flanges:

Us = Gtf2

∫L

0

[∫ 0

−b/2g2

t1 dx dy +∫ b/2

0g2

t2 dx dy +∫ b/2

−b/2g2

b dx dy

]dz

Us = Gtfb

∫L

0

{[(qs − qt)2 + (qf − qt)2l2]b2p2 cos2 pz

L+ u2 + 1

3w2w ′2 + uww ′

−[(qs − qt) + (qf − qt)l]bp

2cos

pzL

(2u + ww ′)}

dz . (2.23)

(iii) Work done contribution

An additional work done term from the vulnerable flange’s local in-planedisplacement function u needs to be included. Figure 2e shows that thecompression flange has a distribution of in-plane displacement which is assumedto have an average linear distribution in x . Including this as an average end-rotation, angles a0 and aL are obtained as shown in the diagram. As the endrotation angles can be expressed in terms of the local in-plane displacementfunction u, the expression for the local contribution to the work done is

MQl = Mx(a0 − aL) = −M2h

∫L

0u′ dz . (2.24)

(iv) Total potential energy

The expression for the total potential energy of the interactive buckling modelsystem can be written as a sum of the individual terms of the strain energiesminus the work done terms from §2a,b, with Um and Us replacing Ub to accountfor the change in the bending theory assumptions:

V = Um + UT + Ubl + Us − M (Q + Ql). (2.25)

This new energy function V replaces equation (2.7) and is written in terms ofnon-dimensionalized variables that replace the original ones, thus

u = 2L

u, w = 2L

w and z = 2L

z , (2.26)

giving the full expression for V , where primes henceforth represent derivativeswith respect to z :

V = L2

∫ 2

0

{Db

[j2

3w ′′2 + 2(1 − n)w ′2

]+ GJp2

2L2q2

f cos2 pz2

+ Ebtf12

[u′2 + 3

20w ′4

+34u′w ′2 − (qt + qtl)p2j sin

pz2

(u′ + 3

8w ′2

)+ (q2

t + q2t l2)p4j2 sin2 pz

2

]





+ Gbtf4j2

[u2 + w2w ′2

3+ uww ′ − [(qs − qt) + (qf − qt)l]2pj cos

pz2

(u + ww ′

2

)

+[(qs − qt)2 + (qf − qt)2l2]4p2j2 cos2 pz2

]− M

L

[qsqfp2 sin2 pz

2− u′

4l

]}dz .

(2.27)

(v) Linear eigenvalue analysis

With u and w being zero, along with their derivatives, before any local bucklingoccurs, the Hessian matrix Vij , now including terms associated with the ‘tilt’generalized coordinates, can still be used to find Mcr, the critical moment forLTB. The Hessian matrix is thus

VFij =

⎛⎜⎝

Vss Vst Vsf Vst

Vts Vtt Vtf Vtt

Vfs Vft Vff Vft

Vts Vtt Vtf Vtt

⎞⎟⎠, (2.28)

with the individual terms being

Vss = GLbtfp2,

Vtt = EIyp4j2

2L+ GLbtfp2,

Vff = GJp2

2L+ GLbtfp2,

Vtt = p2l2(

EIyp2

2L+ GLbtf

), (2.29)

Vst = Vts = −GLbtfp2,

Vsf = Vfs = −Mp2

2,

Vft = Vtf = −GLbtfp2l2

and Vst = Vts = Vtt = Vtt = Vtf = Vft = 0,

which—when substituted into Vij with the singular condition at the critical pointC, where M = Mcr—gives

Mcr = p

L

√EIyGJ1 + s

√1 + 1

(1 + s)p2

L2

EIw

GJ. (2.30)

This new expression for Mcr replaces equation (2.11) and is used subsequently. Theterm, s = Ep2j2/12G, accounts for the non-zero shear distortion of both flanges,which tends to zero if G or L become large; this is an entirely logical resultreflecting the difference between Timoshenko and Euler–Bernoulli beam theories(Wang et al. 2000). However, with s > 0, the earlier-mentioned expression givesvalues that are marginally below those given by the classical critical momentgiven in equation (2.11).





(vi) Equilibrium equations

The total potential energy V with the rescaled variables can thus be written as

V =∫ 2

0L(w ′′, w ′, w, u′, u; z) dz . (2.31)

Governing equations are found by finding the first variation of V (or dV ), wheredV is obtained after some standard manipulation (Hunt & Wadee 1998):

dV ={

vLvw ′′ dw

′ +[

vLvw ′ − d

dz

(vLvw ′′

)]dw + vL

vu′ du}2

0

+∫ 2

0

{[d2

dz2

(vLvw ′′

)− d

dz

(vLvw ′

)+ vL

vw

]dw +

[vLvu

− ddz

(vLvu′

)]du

}dz ,

(2.32)

and setting it to zero which gives the condition for stationary potential energythat is necessary and sufficient for equilibrium (Thompson & Hunt 1973). Theintegrand in equation (2.32) has to vanish for all dw and du, which gives twocoupled nonlinear ordinary differential equations:

w ′′′′ − 6(1 − n)j2

w ′′ + 3EL2tf8D

[(qt + qtl)

p2j

4

(sin

pz2

w ′′ + p

2cos

pz2

w ′)

−12(u′′w ′ + u′w ′′) − 3

5w ′2w ′′

]− 3GL2tf

8Dj2w

[u′ + 2

3(w ′2 + ww ′′)

+ [qs − qt + l(qf − qt)]p2j

2sin

pz2

]= 0 (2.33)

and

u′′ + 34w ′w ′′ + p2

4s

{pj cos

pz2

[(qs − qt + l(qf − qt)) − s(qt + qtl)]

−(

u + ww ′

2

)}= 0, (2.34)

subject to the following boundary conditions that arise from minimizing the termsoutside the integral in equation (2.32):

w(0) = w ′′(0) = w(2) = w ′′(2) = 0, (2.35)

u′(0) + 38w ′2(0) + 3M

Ebhtf= 0 (2.36)

and u(1) = w ′(1) = w ′′′(1) = 0, (2.37)

where equation (2.35) refers to pinned boundaries, equation (2.36) refers to theend strain condition and equation (2.37) refers to reflective symmetry of w andantisymmetry of u about the midspan, respectively. The symmetry conditionsare particularly pertinent when LTB occurs simultaneously with or before localbuckling owing to the sinusoidal distribution of f forcing the maximum bendingstress to the beam midspan.





Further equilibrium equations are obtained by minimizing V with respectto the generalized coordinates qs, qt, qt and qf. After some manipulation, thefollowing integral equations are obtained:

∫ 2

0

1pj

[u + ww ′

2

]cos

pz2

dz = 4(qs − qt) − 2MGLbtf

qf (2.38)

∫ 2

0

{6G

Epj2

[u + ww ′

2

]cos

pz2

−[u′ + 3

8w ′2

]sin

pz2

}dz = 2p2j

s[qs − qt(1 + s)]

(2.39)∫ 2

0

{6G

Epj2

[u + ww ′

2

]cos

pz2

−[u′ + 3

8w ′2

]sin

pz2

}dz = 2lp2j

s[qf − qt(1 + s)]

(2.40)

and∫ 2

0

1pj

[u + ww ′

2

]cos

pz2

dz = 4l(qf − qt) + 4JLbhtf

qf − 4MGbhtf

qs. (2.41)

3. Physical experiments

(a) Specimens and procedure

A series of physical experiments were conducted on I-beams fabricated by spot-welding thin-walled channel sections placed back-to-back from cold-formed steel.The key material properties were measured to be thus: Young’s modulus E =205 kN mm−2, Poisson’s ratio n = 0.3 and the yield stress sy = 290 N mm−2. Thechannel sections were 75 × 43 × 2 mm in terms of depth, width and thickness,respectively (figure 3a).

The actual geometry (including corner radii, etc.) was converted into anidealized I-section comprising only flat plate elements based on the mean slopeof the small linear regions of the measured load versus maximum bendingdisplacement curves from the tests. The dimensions of h and b were henceadjusted slightly such that a meaningful comparison with the theory could bemade in terms of LTB (requiring an accurate Iy) and local buckling (requiring anaccurate bf )—figure 3b. Each beam, the idealized properties of which are givenin table 1, had an overall length of 4 m was tested under four-point bendingwith a specified buckling length Le (given in table 2) that was controlled by anadjustable pair of lateral restraints (figure 4a). The critical mode was determinedby comparing the strong axis bending stress when M = Mcr, thus

sCLTB = Mcrymax

Ix= Mcrh

2Ix, (3.1)

with the critical stress of plate buckling being given by equation (2.12).

(b) Testing rig

An idealized representation of the experimental setup is shown in figure 3c witha schematic of the rig shown in figure 4a. The total applied load 2P was splitinto two point loads each of P applied at a distance xL from the end supports.





43 mm

38 mm

2 mm(a) (b)

(c)

2 mm4 mm 71 mm

xL xL

PPlateral restraints

Le

80 mm

2 mm

2 mm

4 mm

38 mm

71 mm

Figure 3. Geometry of steel beams tested in the experimental programme. (a) Actual sectionconstructed from two channel sections. (b) Idealized section geometry derived from experiments.(c) Idealized experimental setup showing loading, restraints and effective length Le.

Table 1. Idealized section properties of the experimental samples.

geometric property value (mm) cross-section property value

b 80 Ix 4.81 × 105 mm4

h 71 Iy 1.71 × 105 mm4

tf 2 Iw 2.16 × 108 mm6

tw 4 J 1.86 × 103 mm4

Table 2. Buckling lengths for each beam test which reflect the symmetric position of the lateralrestraints; these lengths were chosen such that sC

local/sCLTB = [0.75, 1.20] with one beam (test 4)

triggering both modes approximately simultaneously.

test effective length (Le; mm) sClocal/sC

LTB sC (N mm−2) critical mode

1 3200 1.19 186 LTB2 3200 1.19 186 LTB3 3000 1.10 202 LTB4 2750 0.98 221 local5 2500 0.87 221 local6 2250 0.75 221 local

Hence, from simple statics, the uniform moment M between the two loadingpoints was PxL. The loading was displacement-controlled; it was applied witha hand-operated hydraulic jack in conjunction with a gravity load simulator,a mechanism that adjusted the position of the load application relative to thedeflecting beam such that the applied load remained vertical. As the jack washand-operated, the displacement was applied in short controlled increments butit did mean that dynamic loads were inevitable to some extent. At midspan,the beam vertical displacement and lateral displacements of the flanges weremeasured using linear variable differential transformers (LVDTs); the locationsof which are presented in figure 4b.





loadcell

load applicator jack andgravity load simulator

loading beam

array of LVDTs—see (b)

restraining andinstrumentation beam

lateralrestraint

(a)

(b)

Le

tested beam

data logger and PC

loading point loading point xLxL

PP

LVDT: vertical displacementangle to account forbeam vertical displacement

angle to account forbeam vertical displacement

LVDT: top flange

LVDT: bottom flange

plate to account forbeam lateral displacement

Figure 4. (a) Experimental rig designed to test fabricated I-beams under four-point bending.Although each beam was 4 m in length, the distance between the pinned supports was in fact3.8 m. In each test, the lateral restraints were adjusted symmetrically to obtain the desired bucklinglength Le. (b) Beam cross section showing the relative positions of the LVDTs. Angles and plateswere clamped as shown to both flanges such that the overall beam displacement still allowed therelative flange and vertical displacements to be evaluated.

The large displacements and dynamic behaviour of unstable post-buckling,even with displacement control, meant that sometimes the LVDTs measuringthe displacement of the greater displacing top flange—see figure 1b—went out ofrange very quickly after the secondary instability was triggered. However, therewere no such problems associated with the bottom flange measurements and sothese were the primary values used for comparison purposes, as the theoreticalmodel gives both us and f directly.

4. Numerical results and validation

(a) Cellular buckling

The system of nonlinear ordinary differential equations (2.33) and (2.34),subject to boundary conditions from equations (2.35)–(2.37) and integralequations (2.38)–(2.41) are solved using the well-known and tested numerical





0 0.2 0.4 0.6 0.8 1.00.85

0.90

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(us−uw)/b (us−uw)/b

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C1C2C3

C4

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C7C8

C9C10

C11

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/2)/

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C1C2C3

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C11

qs

q t

0 0.5 1.0 1.5 2.0

0.5

1.0

1.5

2.0

C

S

C4

C11

qφ (rad)

q t (r

ad)

S

Figure 5. Numerical equilibrium paths for test specimens 1 and 2. Graphs of (a) normalized momentratio m and (b) local buckling displacement amplitude wmax/tf versus lateral displacement of thebottom flange are shown. Points C and S show the critical and secondary bifurcation points,respectively. Note that the paths, separated by snap-backs that mark the appearance of a newlocal buckling ‘cell’, are denoted as ‘Cn’ in sequence, where n is the cell number. The graphs in(c) and (d) show the generalized coordinates defining lateral-torsional buckling during interactivebuckling; note that the dotted lines represent the relationship between the respective quantitiesassuming Euler–Bernoulli bending, showing that the shear strains are small. (Online version incolour.)

continuation package AUTO (Doedel & Oldeman 2009). For illustrative purposes,figures 5–7 present results from the variational model for test specimens 1 and 2.Figure 5 shows a plot of the (a) normalized moment ratio m, which is defined asthe ratio M/Mcr, and (b) the normalized local buckling displacement amplitude(w(L/2)/tf ) of the vulnerable part of the compression flange versus the normalizedlateral displacement of the bottom flange, (us − uw)/b. The graphs in (c) and(d) show the relationships between the ‘sway’ and ‘tilt’ components of the weakaxis centroidal displacement (qs and qt) and the torsional angle (qf and qt).A dotted line is superimposed on these graphs to show the Euler–Bernoulliassumption, where the sway and tilt amplitudes would be equal; this shows thatthe shear strains developed are small but not zero. Moreover, figure 5a,b showa series of paths separated by a sequence of snap-backs with figure 6 presentingdetailed graphs showing the corresponding numerical solutions beyond individual





0 1000 2000 3000

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0 1000 2000 3000

0 1000 2000 3000

−505

(a)

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u

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u

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00.2

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u

−505

w

−505

w

−505

w

−505

w

−505

w

−505

w

z z

w

z

L L

Figure 6. (a) Numerical solutions of the displacements of local buckling (w and u) for the tip(x = −b/2) of the vulnerable flange of test specimens 1 and 2. Individual solutions on equilibriumpaths C1–C7 (defined in figure 5) are shown in sequence from top to bottom, respectively;note how the local buckling wavelength L drops, which is defined in (b). All dimensions are inmillimetres. (Online version in colour.)

3000

(a) (b) (c) (d)

2000

100050

–500

0100–100 0

x

z

y y

3000

2000

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–500

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z

y

3000

2000

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0100–100 0

x

z

y

3000

2000

100050

–500

0100–100 0

x

z

Figure 7. Numerical solutions of the equilibrium equations visualized on a representation of theactual beam used in tests 1 and 2. All deformation components (w, u, us, f, utb and utt) areincluded with all dimensions in millimetres. The results are shown for individual points on pathsC1, C3, C5 and C7 (figure 5), where m = 0.9998, 0.9900, 0.9718 and 0.9414, respectively. (Onlineversion in colour.)





snap-backs for the local buckling functions. A distinctive pattern is clearly seento emerge where the response passes from one path to the next, i.e. from C1to C2 to C3 and so on, in which each new path reveals a new local bucklingdisplacement peak or trough. A selection of three-dimensional representations ofthe beams using the solutions for the paths C1, C3, C5 and C7 are presentedin figure 7, which include all components of LTB (us, f, utb and utt) and localbuckling (w and u). Note how the local buckling mode develops and how the‘wavelength’ of the local buckling profile within the central portion of the flangein more compression changes as more cells develop.

As the response advances to path C11, which has torsional rotations that arewell beyond the scope of the model in terms of geometrical considerations, thelocal buckle pattern is all but periodic and further loading would restabilize thesystem globally, assuming no permanent deformation has taken place. This globalrestabilization would occur as a result of the boundaries confining the spread ofthe buckling profile any further. Of course, if plasticity were present in the flange,then any restabilization would be less significant and displacements would lockinto plastic hinges.

The phenomenon demonstrated currently and described earlier, where asequence of snap-backs causes a progressive change from an initially localizedpost-buckling mode to periodic, has been termed in the literature as cellularbuckling (Hunt et al. 2000a) or snaking (Burke & Knobloch 2007). It has beenfound to be prevalent in systems where there is progressive destabilizationand subsequent restabilization (Wadee & Bassom 2000; Peletier 2001), suchas in cylindrical shell buckling (Hunt et al. 1999, 2003) and kink banding inconfined layers (Hunt et al. 2000b; Wadee & Edmunds 2005). In the fundamentalstudies concerning the model strut on a nonlinear foundation, the load oscillatesabout the Maxwell load, where the buckling modes progressively transform fromlocalized (homoclinic) profiles to a periodic mode in a heteroclinic connection(Budd et al. 2001; Wadee et al. 2002; Chapman & Kozyreff 2009). The oscillationin the strut model is attributed to the combination of nonlinearities in thefoundation that have softening and hardening properties. In the present context,as in the case for sandwich struts (Hunt et al. 2000a), the destabilizationis derived from the interaction of instability modes with the restabilizationarising from the inherent stretching that occurs during plate buckling owingto large deflections, which accounts for its significant post-buckling stiffness(Koiter & Pignataro 1976). Moreover, as the moment ratio m has a decreasingtrend rather than oscillating about a fixed value, it is suggested that thedestabilization is inherently more severe than the restabilization for thepresent case.

(b) Comparison with existing experiments

Work conducted by Cherry (1960), which focused on the overall bucklingstrength of beams under bending that had locally buckled flanges, presenteda series of test results and proposed a theoretical estimate of the post-bucklingstrength. The theoretical approach was based on the effective width of the locallybuckled flange that originated in von Kármán et al. (1932). However, the modelpresented in Cherry (1960) was limited because of the assumption that bothoutstands of the compression flange behaved symmetrically. Nevertheless, the





Table 3. Cross-section properties of beams tested and results presented in Cherry (1960). Threedifferent lengths of beams were tested under bending for each cross section (A–D) ranging from1 to 2 m.

beam h (mm) b (mm) tw (mm) tf (mm) E (kN mm−2)

A 72.3 76.6 3.68 1.60 66.47B 74.0 76.6 3.78 2.11 64.53C 72.2 76.3 5.13 1.89 64.43D 71.5 75.9 4.65 1.90 66.19

Table 4. Comparisons of buckling wavelengths in millimetres between the tests in Cherry (1960)and the current model.

model wavelength Ltest wavelength

beam (Cherry 1960) minimum maximum error range (%)

A 126.2 127.0 133.4 +0.6 to +5.7B 124.0 153.0 156.7 +23.4 to +26.4C 108.5 103.1 132.6 −4.9 to +22.3D 109.7 102.1 141.2 −7.0 to +28.7

tests that were presented therein provide valuable data for the wavelengths ofthe local buckling mode in the compression flanges that were measured for fourseparate doubly symmetric I-beams, with properties as presented in table 3; thedata are used for comparison purposes in the current study. As the bucklingmode predicted by the current analytical model is not necessarily periodic,but tends to approach this quality in the far post-buckling range, comparisonsbetween the tests in Cherry (1960) and the current model can be made when theprofile for w exhibits periodicity throughout the beam length. Figure 6b showshow the wavelength is defined from the post-buckling mode that has a centralportion that is close to periodic. Table 4 shows the range of wavelengths obtainedfrom the numerical solution of the system of equilibrium equations presentedin §2b(vi). The current model was run for a range of lengths between 1 and2 m, as this was the range for which the vast majority of tests presented inCherry (1960) were conducted. Apart from the beams with cross section B, thecomparisons are very encouraging; the discrepancies between the model and thetests are attributed to boundary effects affecting the results of the analyticalmodel. It has been seen in the cellular buckling results earlier in this section(figures 5–7) that, as each buckling cell develops, the buckling ‘wavelength’ L,figure 6b, drops until the buckling profile eventually tends to true periodicity andthe moment M tends to a constant. For the numerical results from the currentmodel that overestimated the wavelength, lack of convergence became an issueand the local buckling profile w was still showing remnants of the decaying tailsnear the boundaries, which are the signatures of homoclinic behaviour. Hence,those particular comparisons are perhaps not entirely representative of the actualresponse predicted by the model.





(a) (b)

(c) (d)

Figure 8. Selection of photographs from the experimental programme: (a–c) interactive bucklingand (d) four of the beams and their locally buckled flanges that show plastic deformation. (Onlineversion in colour.)

(c) Results from current experiments

For each of the physical experiments performed in the present study (see §3),testing proceeded to failure and all of the beams exhibited an unstable responseonce interactive buckling was triggered. A selection of photographs is presentedin figure 8, which show the beams from a variety of directions while they wereundergoing interactive buckling. In tests 2 and 6, there was visual experimentalevidence of cellular buckling; figure 9 shows a sequence of photographs beforeand after the principal instability showing a new local buckling peak appearingsoon after the initial one. Table 5 presents the results and their comparison withthe individual buckling modes. The maximum applied moment in the experimentsMmax is presented as a ratio of the theoretical critical moment MC calculated fromthe appropriate critical mode given in table 2, whether LTB or local buckling.The local buckling profile was determined by marking (as seen in figure 8d) andmeasuring between adjacent peaks of the local buckling displacement over thelength of the vulnerable part of the compression flange, while the beam was stillloaded but well after the peak moment had been applied. The interactive modewas clearly modulated in each case, with the peak amplitudes from local bucklingdecaying towards the lateral restraints; this was particularly notable in the cases,where LTB was critical since the number of peaks was visibly fewer. In eachtest, two or three peaks of the local buckling mode exhibited significant plasticdeformation almost immediately after the interactive mode was triggered; it wasadjacent to these peaks where the buckling wavelength was, in general, measured





(a)

(b)

(c)

(d)

(e)

( f )

Figure 9. Evidence of cellular buckling. Two sequences of three photographs are shown of tests 2(a–c) and 6 (d–f ), respectively. Photographs (a) and (d), respectively, show the pre-buckling state;(b) and (e), respectively, show the initial post-buckling with one significant peak at midspan; (d)and (f ), respectively, show a newly developed local buckling peak in the top flange. (Online versionin colour.)

Table 5. Results from the experimental programme in terms of the maximum moment and thelocal buckling profile. The final column is a measure of localized nature of the flange buckle—thesmaller the number, the more it was localized.

observations: local bucklinginitial post-buckling drop no. of visible extent

test Mmax/MC in moment (%) buckling peaks (% of Le)

1 1.03 15 5 25.42 0.90 7 7 36.53 0.93 12 7 40.64 1.07 24 8 52.45 1.00 14 9 65.86 1.05 15 9 69.2

to be the smallest values. It is also worth noting that the close correlationdemonstrated experimentally between Mmax and MC, for both LTB and localbuckling, implies that the dimensions of the idealized section shown in figure 3bhave been validated.

Another notable feature shown in table 5 is the immediate proportional drop inthe moment once the interactive mode had been triggered. As would be expectedfrom the literature (Budiansky 1976), the largest drop occurred in test 4, wherethe critical modes had been practically simultaneous. It is also noteworthy thatthe tests with identical buckling lengths (tests 1 and 2) showed very differentpeak moments and moment drops. A rational hypothesis can be devised for thisby postulating that the beam in test 2 contained more geometrical imperfections





0 0.2 0.4 0.6 0.8 1.00.2

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0.8

1.0(a)

(c) (d)

(e)

(b)

(us−uw)/b

(us−uw)/b (us−uw)/b

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0.80.9

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m

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localoC

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LTB

S

S S

Figure 10. Moment ratio m versus bottom flange lateral displacement (us − uw)/b for all theexperiments with the variational model predictions superimposed, denoted as ‘theory’ andevaluated by AUTO. Points C and S refer to the critical and secondary instability points obtainedfrom the model. Note that local buckling was the theoretical critical eigenmode for tests 4–6.(Online version in colour.)

than the beam in test 1. This would account not only for the smaller maximummoment measured in test 2, but also for its smaller relative moment drop andits lower residual moment in the post-buckling range (Thompson & Hunt 1973).Earlier work on a similar problem with local–global mode interaction (Wadee2000) suggests that exactly this response would occur.





Table 6. Comparisons of the experimental results with the variational model in terms of the valuesof the bottom flange displacement at the peak moment point and the local buckling wavelengths.

(us − uw)/b local buckling wavelengths L (mm)

test expt theory expt range expt average theory (minimum)

1 0.169 0.134 185–221 203 2002 0.146 0.134 171–250 195 2003 0.061 0.110 161–242 203 1504 0.068 0.083 161–249 206 1535 0.066 0.052 167–247 206 1636 0.061 0.058 158–225 195 240

(i) Comparisons with variational model and discussion

Figure 10 presents normalized plots of the applied moment m versus themeasured and normalized lateral displacement of the bottom flange, (us − uw)/b.Test 4 clearly gives the best comparison in terms of the correlation between thepost-buckling response of the actual beam and the model prediction. Tests 1and 2 also show good basic agreement with the theory; test 1 showing that thepost-buckling unloading resembles the theory quite well, whereas test 2 shows thatthe instability is triggered at a similar value of the lateral displacement predictedby the theory (table 6). Tests 5 and 6 clearly peak at or marginally above thelocal buckling critical moment, as predicted from linear analysis. However, as intest 2, the instability is triggered at a lateral displacement that correlates wellwith the variational model. For test 6, the variational model yields a lower criticalmoment than the Mcr value for LTB, which triggers a quasi-local buckling mode.However, as stated earlier, a distinct and accurate local buckling mode can bemodelled only with additional displacement functions in the current framework;so this particular result needs to be interpreted with some caution. Test 3 could beconsidered to be an outlier, but the measured response would imply—in a similarway that was discussed above regarding test 2—that the level of geometricalimperfections in this beam was higher than the other tested beams (1, 4, 5and 6). Hence, the measured instability moment and the unloading proportionare less and the response is practically parallel to the model curve, which in factis encouraging.

In terms of the local buckling wavelengths, these are compared with thoseof the buckling profile obtained from the variational model as described in §4b.Even though the theoretical results seemed to be influenced by effects close tothe boundary (particularly in test 6), hence the variability in the predictions, thegeneral correlation between the experiments and theory is good. The apparentconfirmation that the post-buckling behaviour of an I-beam under pure bendingis cellular when global and local instability modes interact nonlinearly poses thefollowing question: is this phenomenon prevalent in other thin-walled structuralcomponents that are known to suffer from overall and local mode interaction?Compressed stringer-stiffened plates (Koiter & Pignataro 1976) and I-sectionstruts (Becque & Rasmussen 2009) are prime examples of other components,where local and global mode interactions are known to occur. Further research isobviously required to determine the answer.





5. Concluding remarks

The current work identifies an interactive form of buckling for an I-beam underuniform bending that couples a global instability with local buckling in one-half ofthe compression flange. In contrast to earlier, more numerical, work (Møllmann &Goltermann 1989; Menken et al. 1997), cellular buckling, the transformationfrom a localized to an effectively periodic mode, is predicted theoretically forthe purely elastic case and evidenced in physical tests. The model compares wellboth qualitatively and quantitatively with the observed collapse of a beam thatundergoes the interaction under discussion that involves global, local, localizedand cellular buckling. The localized buckle pattern first appears at a secondarybifurcation point that immediately destabilizes a portion of the compressionflange; as the deformation grows, the buckle tends to spread in cells untileventually it restabilizes when the localized buckling pattern has become periodicafter a sequence of snap-backs.

Experimentally, the process is unstable and so this sequence occurs rapidlyeven under rigid loading, with the local buckling cells being triggered dynamically.This highlights the practical dangers of the modelled and observed phenomenon;the interaction reduces the load-carrying capacity, and it therefore introduces animperfection sensitivity that would need to be quantified such that robust designrules can be developed to mitigate against such hazardous structural behaviour.

The majority of this work was conducted while M.A.W. was on sabbatical at the School of Civiland Environmental Engineering, University of the Witwatersrand, Johannesburg, South Africafrom April to October 2010. The authors are extremely grateful to Professor Mitchell Gohnert(Head of School), Kenneth Harman (Senior Laboratory Technician) and Spencer Erling (SouthernAfrican Institute of Steel Construction) for facilitating the experimental programme. The work waspartially funded by the UK Engineering and Physical Sciences Research Council through projectgrant no. EP/F022182/1.

References

Becque, J. & Rasmussen, K. J. R. 2009 Experimental investigation of the interaction of localand overall buckling of stainless steel I-columns. J. Struct. Eng. ASCE 135, 1340–1348.(doi:10.1061/(ASCE)ST.1943-541X.0000051)

Budd, C. J., Hunt, G. W. & Kuske, R. 2001 Asymptotics of cellular buckling close to the Maxwellload. Proc. R. Soc. Lond. A 457, 2935–2964. (doi:10.1098/rspa.2001.0843)

Budiansky, B. (ed.) 1976 Buckling of structures. In IUTAM Symp., Cambridge, MA, 17–21 June1974. Berlin, Germany: Springer.

Burke, J. & Knobloch, E. 2007 Homoclinic snaking: structure and stability. Chaos 17, 037102.(doi:10.1063/1.2746816)

Chapman, S. J. & Kozyreff, G. 2009 Exponential asymptotics of localised patterns and snakingbifurcation diagrams. Phys. D 238, 319–354. (doi:10.1016/j.physd.2008.10.005)

Cherry, S. 1960 The stability of beams with buckled compression flanges. Struct. Eng. 38, 277–285.Doedel, E. J. & Oldeman, B. E. 2009 AUTO-07p: continuation and bifurcation software for

ordinary differential equations. Technical Report. Department of Computer Science, ConcordiaUniversity, Montreal, Canada. See http://indy.cs.concordia.ca/auto/.

Goltermann, P. & Møllmann, H. 1989 Interactive buckling in thin walled beams: II. Applications.Int. J. Solids Struct. 25, 729–749. (doi:10.1016/0020-7683(89)90010-3)

Hancock, G. J. 1978 Local, distortional and lateral buckling of I-beams. ASCE J. Struct. Div. 104,1787–1798.



http://dx.doi.org/doi:10.1061/(ASCE)ST.1943-541X.0000051

http://dx.doi.org/doi:10.1098/rspa.2001.0843

http://dx.doi.org/doi:10.1063/1.2746816

http://dx.doi.org/doi:10.1016/j.physd.2008.10.005

http://dx.doi.org/doi:10.1016/0020-7683(89)90010-3



Hunt, G. W. & Wadee, M. A. 1998 Localization and mode interaction in sandwich structures. Proc.R. Soc. Lond. A 454, 1197–1216. (doi:10.1098/rspa.1998.0202)

Hunt, G. W., Da Silva, L. S. & Manzocchi, G. M. E. 1988 Interactive buckling in sandwichstructures. Proc. R. Soc. Lond. A 417, 155–177. (doi:10.1098/rspa.1988.0055)

Hunt, G. W., Lord, G. J. & Champneys, A. R. 1999 Homoclinic and heteroclinic orbits underlyingthe post-buckling of axially-compressed cylindrical shells. Comput. Meth. Appl. Mech. Eng. 170,239–251. (doi:10.1016/S0045-7825(98)00197-2)

Hunt, G. W., Peletier, M. A., Champneys, A. R., Woods, P. D., Wadee, M. A., Budd, C. J. &Lord, G. J. 2000a Cellular buckling in long structures. Nonlinear Dyn. 21, 3–29. (doi:10.1023/A:1008398006403)

Hunt, G. W., Peletier, M. A. & Wadee, M. A. 2000b The Maxwell stability criterion inpseudo-energy models of kink banding. J. Struct. Geol. 22, 669–681. (doi:10.1016/S0191-8141(99)00182-0)

Hunt, G. W., Lord, G. J. & Peletier, M. A. 2003 Cylindrical shell buckling: a characterizationof localization and periodicity. Discrete Contin. Dyn. Syst. Ser. B 3, 505–518. (doi:10.3934/dcdsb.2003.3.505)

Koiter, W. T. & Pignataro, M. 1976. A general theory for the interaction between local and overallbuckling of stiffened panels. Technical Report. WTHD 83, Delft University of Technology, Delft,The Netherlands.

Menken, C. M., Groot, W. J. & Stallenberg, G. A. J. 1991 Interactive buckling of beams in bending.Thin-Walled Struct. 12, 415–434. (doi:10.1016/0263-8231(91)90043-I)

Menken, C. M., Schreppers, G. M. A., Groot, W. J. & Petterson, R. 1997 Analyzing bucklingmode interactions in elastic structures using an asymptotic approach; theory and experiments.Comput. Struct. 64, 473–480. (doi:10.1016/S0045-7949(96)00139-3)

Møllmann, H. & Goltermann, P. 1989 Interactive buckling in thin walled beams. I. Theory. Int. J.Solids Struct. 25, 715–728. (doi:10.1016/0020-7683(89)90009-7)

Peletier, M. A. 2001 Sequential buckling: a variational analysis. SIAM J. Appl. Math. 32, 1142–1168.(doi:10.1137/S0036141099359925)

Pi, Y. L., Trahair, N. S. & Rajasekaran, S. 1992 Energy equation for beam lateral buckling. J.Struct. Eng. ASCE 118, 1462–1479. (doi:10.1061/(ASCE)0733-9445(1992)118:6(1462))

Pignataro, M., Pasca, M. & Franchin, P. 2000 Post-buckling analysis of corrugated panels in thepresence of multiple interacting modes. Thin-Walled Struct. 36, 47–66. (doi:10.1016/S0263-8231(99)00027-0)

Rasmussen, K. J. R. & Wilkinson, T. (eds) 2008 Coupled instabilities in metal structures CIMS2008,vol. 2: Gregory J. Hancock symposium. Sydney, Australia: University Publishing Services,University of Sydney.

Schafer, B. W. 2002 Local, distortional and Euler buckling of thin-walled columns. ASCE J. Struct.Eng. 128, 289–299. (doi:10.1061/(ASCE)0733-9445(2002)128:3(289))

Thompson, J. M. T. & Hunt, G. W. 1973 A general theory of elastic stability. London, UK: Wiley.Timoshenko, S. P. & Gere, J. M. 1961 Theory of elastic stability. New York, NY: McGraw-Hill.von Kármán, T., Sechler, E. E. & Donnell, L. H. 1932 The strength of thin plates in compression.

Trans. ASME 54, 54–55.Wadee, M. A. 2000 Effects of periodic and localized imperfections on struts on nonlinear

foundations and compression sandwich panels. Int. J. Solids Struct. 37, 1191–1209. (doi:10.1016/S0020-7683(98)00280-7)

Wadee, M. K. & Bassom, A. P. 2000 Characterization of limiting homoclinic behaviour in aone-dimensional elastic buckling model. J. Mech. Phys. Solids 48, 2297–2313. (doi:10.1016/S0022-5096(00)00018-1)

Wadee, M. A. & Edmunds, R. 2005 Kink band propagation in layered structures. J. Mech. Phys.Solids 53, 2017–2035. (doi:10.1016/j.jmps.2005.04.005)

Wadee, M. K., Coman, C. D. & Bassom, A. P. 2002 Solitary wave interaction phenomena in astrut buckling model incorporating restabilisation. Phys. D 163, 26–48. (doi:10.1016/S0167-2789(02)00350-0)

Wang, C. M., Reddy, J. N. & Lee, K. H. 2000 Shear deformable beams and plates: Relationshipswith classical solutions. Amsterdam, The Netherlands: Elsevier.





http://dx.doi.org/doi:10.1016/S0045-7825(98)00197-2

http://dx.doi.org/doi:10.1023/A:1008398006403

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