Thin-Walled Structures 71 (2013) 35–45

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Elastic buckling of uniformly compressed thin-walled regularpolygonal tubes

Rodrigo Gonçalves a,n, Dinar Camotim b

a UNIC, Departamento de Engenharia Civil, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugalb ICIST, Departamento de Engenharia Civil e Arquitectura, Instituto Superior Técnico, Technical University Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

a r t i c l e i n f o

Article history:Received 20 January 2013Accepted 29 April 2013Available online 10 June 2013

Keywords:Uniformly compressed thin-walledmembersRegular polygonal cross-sectionsBuckling behaviourGeneralised Beam Theory (GBT)

31/$ - see front matter & 2013 Elsevier Ltd. Ax.doi.org/10.1016/j.tws.2013.04.016

esponding author. Tel.: +351 21 2948580; faxail address: rodrigo.goncalves@fct.unl.pt (R. Gr RCPS with an even number of walls, the sithe exact solution.

a b s t r a c t

This paper investigates the elastic buckling (bifurcation) behaviour of uniformly compressed thin-walledtubular members with single-cell regular polygonal cross-sections (RCPS), such as those employed tobuild transmission line structures, towers, antennas and masts. A specialisation of Generalised BeamTheory (GBT) for RCPS, reported in a recent paper (Gonçalves and Camotim, 2013) [1], is used to obtainboth analytical and numerical results concerning the most relevant buckling modes and provide noveland broad conclusions on the structural behaviour of this type of members. In particular, local, cross-section extensional, distortional and multi-mode (including global flexural) buckling phenomena areaddressed. For validation purposes, the GBT-based results are compared with solutions taken from theliterature and also with numerical values obtained from finite strip analyses.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Thin-walled tubular members with regular (equiangular andequilateral) convex polygonal cross-sections (RCPS—see Fig. 1 forgeometry and notation) are widely employed in many areas of theengineering practice. Due to structural optimisation requirements,these members are often characterised by high wall width-to-thickness ratios and, therefore, are highly susceptible to complexbuckling phenomena, which must be appropriately accounted forwhen assessing their resistance. The widespread use of RCPSmembers has naturally fostered a significant amount of investiga-tion concerning their structural behaviour, but it is also true thatinformation concerning the general elastic buckling (bifurcation)behaviour of these members is relatively scarce.

It may be stated that the local (plate-like) buckling behaviour ofRCPS members is rather well-known. In fact, if the internal anglebetween consecutive walls is well below 1801, critical loads can beaccurately predicted through the analysis of a single wall, simplysupported along the lateral edges1 (e.g., [2]). However, otherbuckling phenomena may be critical and, in that case, thissimplified model may lead to non-negligible errors.

Several theoretical and numerical investigations concerning thebuckling behaviour of RCPS have been carried out in the past. In [3],

ll rights reserved.

: +351 21 2948398.onçalves).mply supported plate model

the local buckling of long polygonal tubes under combined uniformcompression and torsionwas investigated using stability functions andassuming that the line junctions between adjoining walls remainstraight. Equilateral triangle and square sections were studied in detailand, in the first case, it was found that the buckling mode for purecompression is indeterminate, with a longitudinal nodal line appear-ing between the lateral edges of one wall, as first pointed out in [4]. In[5], extensive local buckling solutions for RCPS have been presented inthe form of stability curves, for various cross-section wall numbers n.This study showed that the use of the classical plate buckling formula(E is Young's modulus and ν is Poisson's ratio)

scr ¼kπ2E

12ð1−ν2Þtb

� �2

; ð1Þ

with the buckling coefficient k¼4 (as for simply supported plates)leads to accurate results for most cases, although the triangular andpentagonal members exhibit significantly higher local buckling stres-ses. More recently, the local buckling of triangular (equilateral andisosceles) tubes was examined in [6] and analytical solutions weredeveloped.

When the number of walls n increases, the internal anglebetween walls also increases and buckling may occur with non-null displacements along the wall junctions, as in the case ofcircular tubes. This phenomenon was experimentally investigatedin [7], where it was concluded that, for uniformly compressedtubes, the transition between collapse modes occurs for internalangles greater than 1601 (approximately). Similar conclusionswere obtained in [8], on the basis of the numerical analysis of a

R. Gonçalves, D. Camotim / Thin-Walled Structures 71 (2013) 35–4536

two-plate assembly, and more recently in [9], from experimentalresults.

The local buckling analysis of RCPS members with n¼4–8,under axial compression or bending, was addressed in [10] usingfinite strip analyses. For uniformly compressed members, theresults showed once again that odd n values lead to k44,particularly for low n values. Moreover, for low plate slendernessvalues, a drop in the k value was observed, particularly for n¼5and n¼7. For members under uniform bending, the resultsindicated that the k values increase by about 25%, due to thebeneficial stress distribution around the cross-section. In thisstudy, no displacements along the wall junctions were reported.

The present paper focuses on the elastic buckling (bifurcation)behaviour of RCPS members and aims at providing a general andnovel mechanical insight into the problem, from both analyticaland numerical perspectives. The approach adopted is based on arecently developed Generalised Beam Theory (GBT, e.g., [11–13])specialisation for RCPS [1]. This specialisation (i) stems from thefact that n-sided RCPS exhibit rotational symmetry of order n and(ii) makes it possible to calculate more rationally and efficientlythe cross-section orthogonal deformation mode sets, thus leadingto accurate solutions with just a few modes.

Section 2 presents the GBT fundamental equations for the linearstability analysis of members under uniform compression and sum-marises the conclusions presented in [1] concerning the calculation ofthe various cross-section deformation modes. Section 3 is devoted tothe investigation of single-mode buckling, namely local, cross-sectionextensional and distortional buckling. Moreover, the parameter rangefor which each buckling mode is critical is determined. Section 4addresses multi-mode buckling, which requires the simultaneousconsideration of all GBT deformation modes, including the global(flexural) and shear modes. Finally, the paper closes with someconcluding remarks (Section 5).

Throughout the paper, results of several parametric andnumeric studies are presented and discussed. For validationpurposes, the GBT-based results are compared with availablesolutions and/or values obtained from finite strip analyses per-formed with CUFSM [14].

Concerning the notation, vectors and matrices are representedin bold letters. Partial derivatives are indicated by subscriptsfollowing a comma, e.g., f ;x ¼ ∂f =∂x. A virtual variation is denotedby δ. As previously mentioned, the cross-section geometric para-meters are shown in Fig. 1, which also indicates the GBT cross-section discretisation employed in this work: n natural nodes andan arbitrary number m of equally spaced intermediate nodes ineach wall (the total number of intermediate nodes thus equalsn�m). Moreover, the problems under analysis are generallywritten in terms of the non-dimensional geometric parameters

β1 ¼Lr; β2 ¼

rt; ð2Þ

wall widthwall thicknesscircumradiusapothemnumber of walls (5)number of intermediate nodes per wall (2)natural nodes (5)intermediate nodes (10)

r

bbtr

nm

t

Fig. 1. Geometry and notation for regular convex polygonal sections (RCPS).

where L is the member length. Then, the plate width-to-thicknessratio and plate buckling formula (1) may be written as

bt¼ 2β2 sin

π

n

� �;

scrE

¼ kπ2

48ð1−ν2Þβ22 sin2ðπ=nÞ: ð3Þ

2. GBT buckling analysis for RCPS

2.1. Fundamental equations

Following the usual GBT kinematic description and the nota-tion adopted in [15], with the wall mid-surface local axes (x, y, z)shown in Fig. 2, the displacement field for each wall is given by

Uðx; y; zÞ ¼Ux

Uy

Uz

264

375¼

uðx; yÞ−zw;xðx; yÞvðx; yÞ−zw;yðx; yÞ

wðx; yÞ

264

375; ð4Þ

where u, v, w are the mid-surface displacement components alongx, y and z, respectively, expressed as

uðx; yÞ ¼ ∑D

k ¼ 1ukðyÞϕk;xðxÞ;

vðx; yÞ ¼ ∑D

k ¼ 1vkðyÞϕkðxÞ;

wðx; yÞ ¼ ∑D

k ¼ 1wkðyÞϕkðxÞ; ð5Þ

where uk, vk, wk are the components of the k¼ 1;…;D deforma-tion modes (shape functions) and ϕkðxÞ are their amplitudefunctions along the member length, the problem unknowns.

For uniformly compressed members (sxx ¼ λs, where λ is theload parameter and s is the reference stress value), the associatedGBT equation system for linear stability analyses may be obtainedfrom the linearisation of the virtual work equation, yielding (e.g.,[15,16])

ZL

δϕ

δϕ;x

δϕ;xx

264

375t

B 0 D2

0 D1 0ðD2Þt 0 C

264

375

ϕ

ϕ;x

ϕ;xx

264

375dx

þλ

ZL

δϕ

δϕ;x

δϕ;xx

264

375t 0 0 0

0 Xvþws 0

0 0 Xus

264

375

ϕ

ϕ;x

ϕ;xx

264

375dx¼ 0; ð6Þ

where L is the member length, ϕ is a vector containing thebuckling mode amplitude functions ϕkðxÞ, δϕ is the correspondingvirtual variation and the GBT modal matrices are given by

Bij ¼BMij þ BB

ij ¼ZS

E1−ν2

tvi;yvj;y þt3

12wi;yywj;yy

� �dy;

Cij ¼CMij þ CB

ij ¼ZS

E1−ν2

tuiuj þt3

12wiwj

� �dy;

x

y

x

y

z

z

Fig. 2. Arbitrary thin-walled member local coordinate systems.

R. Gonçalves, D. Camotim / Thin-Walled Structures 71 (2013) 35–45 37

ðD1Þij ¼ ðDM1 Þij þ ðDB

1Þij ¼ZSG tðui;y þ viÞðuj;y þ vjÞ þ

t3

3wi;ywj;y

� �dy;

ðD2Þij ¼ ðDM2 Þij þ ðDB

2Þij ¼ZS

νE1−ν2

tvi;yuj þt3

12wi;yywj

� �dy;

ðXus Þij ¼

ZSstuiuj dy;

ðXvþws Þij ¼

ZSstðvivj þwiwjÞ dy; ð7Þ

where superscripts M and B identify membrane and bendingterms, i; j¼ 1;…;D, S is the cross-section mid-line and G is theshear modulus. These matrices were obtained assuming a planestress state. If sMyy ¼ 0 is further imposed, as customary in GBTanalyses, the term E=ð1−ν2Þ in CM must be replaced by E. It shouldbe mentioned that matrix Xu

s , usually discarded, it is kept inthis work.

Eq. (6) can be solved by approximating the amplitude functionsϕkðxÞ. In some particular cases, however, analytical solutions areavailable. For instance, for simply supported members, ϕkðxÞ ¼ ϕk

sinðaπx=LÞ constitute exact solutions, where ϕk is the amplitude ofthe deformation mode and a is the number of longitudinal half-waves of the buckling mode. Then, using the fundamental lemmaof the calculus of variations, one is led to the simpler eigenvalueproblem

a2π2

L2Cþ Dþ L2

a2π2Bþ scrX

!ϕ ¼ 0; ð8Þ

where D¼D1−D2−Dt2, X¼ ða2π2=L2ÞXu

s þ Xvþws , ϕ is a vector that

contains the amplitudes of each deformation mode (ϕk) and it wasassumed that s ¼ 1. For single-mode buckling (say mode i), onereadily has

scr ¼−1Xii

a2π2

L2Cii þ Dii þ

L2

a2π2Bii

!: ð9Þ

From now on, the minus sign (compression) for critical stresseswill be omitted. If Xu

s is discarded, the minimum value of theprevious expression and the associated half-wavelength are givenby [17]

scr ¼Dii þ 2

ffiffiffiffiffiffiffiffiffiffiffiffiCiiBii

pXii

; ð10Þ

Lcr ¼ πffiffiffiffiffiffiffiffiffiffiffiffiffiffiCii=Bii

4p

: ð11ÞFor non-simply supported members, the former equations may

still be employed (approximately) to assess the distortional orlocal buckling behaviour of long columns, for which the effect ofthe boundary conditions is negligible.

2.2. Deformation modes for RCPS

The calculation of the cross-section deformation modes forRCPS has been thoroughly addressed in [1] and therefore thissection only provides a brief overview. According to the proposedprocedure, using the cross-section discretisation shown in Fig. 1,with n natural nodes and m intermediate nodes in each wall, thedeformation mode set is subdivided into the following sets:

(i)

Natural Vlasov warping modes, corresponding to linear warp-ing functions between natural nodes, complying with Vlasov'sassumption (null membrane shear strains, γMxy ¼ 0) and exhi-biting null membrane transverse extensions (εMyy ¼ 0). This setcontains n modes, where the first three are the axial extens-ion (1) and bending (2,3) modes and the subsequent onesare distortional (combining warping and transverse wallbending).

(ii)

Natural shear modes, which supplement the previous set byallowing for a constant γMxy in each wall. This set also containsn modes, where the first three correspond to torsion (1) andcross-section in-plane translation about orthogonal axes (2,3)and the subsequent modes involve transverse wall bending.

(iii)

Natural transverse extension modes, which allow for constantεMyy in each wall. Again, n modes are obtained, where the firstone corresponds to uniform extension in all cross-sectionwalls.

(iv)

Local u=v=w modes, involving displacements along x=y=z atthe intermediate nodes and null displacements of the naturalnodes. Each set involves n�m modes.

The final deformation modes are obtained by employing aspecial matrix Q , whose columns are the vectors aðlÞ and bðlÞ, withcomponents j¼ 1;…;n given by

aðlÞj ¼ cos2πjln

� �; l¼ 0;…;

n2

h i;

bðlÞj ¼ sin2πjln

� �; l¼ 1;…;

n−12

� �; ð12Þ

where ½k� denotes the largest integer not exceeding k and, there-fore, l¼ n=2 applies for cross-sections with even n values only.For each natural mode set, the GBT matrices are completelydiagonalised by Q tAQ , where A is a GBT matrix. In particular, eachpair ðaðlÞ, bðlÞÞ generates a duplicate diagonal component, a featurewhich is linked to the rotational symmetry of the cross-section.

For the local mode sets, the GBT matrices are first blockdiagonalised by employing matrix Q with each vector componentgiven in (12) multiplied by a m�m identity matrix. Then, theindividual blocks are further diagonalised by solving separateeigenvalue problems. In the particular case of the local w modes,which play a significant role in the present work, the GBT matriceshave the form

½m�m�½2m� 2m�

⋱½2m� 2m�

½m�m�

26666664

37777775; ð13Þ

where (i) the first m�m block is generated by að0Þ (l¼0), (ii) thelast m�m block is generated by aðn=2Þ ðl¼ n=2Þ and does notappear for odd n, and (iii) each intermediate 2m� 2m block isassociated with a pair ðaðlÞ;bðlÞÞ, with l¼ 1;…; ½ðn−1Þ=2�.

3. Single-mode buckling

3.1. Pure local buckling

First, attention is focused on the local buckling behaviour ofuniformly compressed RCPS members. In this case the junctionsbetween adjacent walls remain straight during buckling (thenatural nodes do not move) and it may be assumed thatu ¼ v ¼ 0. Therefore, only local w modes need to be consideredand the accuracy of the solution depends only on the number ofintermediate nodes employed, m.

The only non-null GBT matrices are BB, CB, DB1, D

B2 and Xvþw

s ,which become block diagonal, as (13), upon applying the proce-dure outlined in the previous section. Matrix Xvþw

s was notaddressed in [1], but also becomes block diagonal since it equalsð12ð1−ν2Þ=Et2ÞsCB and, therefore, inherits that feature from CB.It is important to emphasise that, even though mode couplingoccurs, the modes generated by different l values correspond todifferent matrix blocks and, thus, do not couple. Naturally, for the

Square Hexagon Octagon

7 11 15

Fig. 3. Critical local buckling mode shapes for even n and m¼2. The GBTdeformation mode number is indicated in each case.

Table 1Minimum buckling coefficients and associated half-wavelengthsfor even n and β2 ¼ 100, as a function of the number ofintermediate nodes m.

m k=4 Lcr=b

1 1.00426 0.996402 1.00057 0.999463 1.00016 0.99985

R. Gonçalves, D. Camotim / Thin-Walled Structures 71 (2013) 35–4538

particular case of m¼1, all matrices are fully diagonal and there isno mode coupling.

Even-sided RCPS are first analysed. In this case the criticalbuckling mode exhibits the well-known configurations shown inFig. 3, with adjacent walls buckling in opposite directions (out-ward/inward), and the exact solution corresponds to that of asimply supported plate of width b, thickness t and length L.The minimum buckling stress is given by Eq. (1), with k¼4, andthe associated half-wavelength along x equals b.

With the proposed GBT analysis, the critical mode is containedin the last m�m block of (13), since this block is generated byvector aðn=2Þ ¼ ½ð−1Þ 1 ð−1Þ …1�t (see Eq. (12)), which changes signin adjacent walls. By solving the eigenvalue problem ðBB−λCBÞv¼ 0in this block to obtain the final set of deformation modes, asproposed in [1], the first mode (number nm−ðm−1Þ) is uncoupledfrom the remaining ones and corresponds to the critical localbuckling mode. Fig. 3 shows this mode shape and number forn¼ 4;6;8 and m¼2. The equations for single-mode buckling (10)and (11) therefore apply and Table 1 compares the exact and GBT-based values of the minimum k and associated Lcr, for m¼ 1;2;3and β2 ¼ 100. These results show a nearly perfect match with justa single intermediate node per wall.

For odd-sided RCPS, vector aðn=2Þ and the associated (last)m�m block do not exist. The critical mode then corresponds tothe block with the next (smallest) frequency, l¼ ðn−1Þ=2, whosevectors ðaðlÞ;bðlÞÞ change sign in all adjacent walls except one,which is in agreement with [3,5]. In contrast to the case of even n,the matrix block is now generated by a pair of vectorsðaððn−1Þ=2Þ;bððn−1Þ=2ÞÞ and two situations may be identified, depend-ing on the m value (see Fig. 4(a), which shows the deformationmodes and the associated generating vectors for n¼5):

(i)

2 Although not shown in the figure, using GBT with m¼3 leads to k¼4.09 forn¼9 and to k¼4.06 for n¼11. Moreover, the Lcr=b values also gradually approach1 as n increases: 0.881 for n¼3 (as in [3]), 0.948 for n¼5, 0.972 for n¼7, 0.983 forn¼9 and 0.988 for n¼11.

For m¼1 all GBT matrices are fully diagonal with repeateddiagonal entries for each vector pair ðaðlÞ;bðlÞÞ. In this particularcase the deformation modes also appear in pairs and are fullyuncoupled [1]. The l¼ ðn−1Þ=2 mode pair (modes 4 and 5 inthe figure) corresponds to the critical buckling modes, whosebuckling loads are necessarily equal (single eigenvalue withmultiplicity 2).In the figure, modes 4 and 5 are generated by aððn−1Þ=2Þ andbððn−1Þ=2Þ, respectively, and the displacements at the intermediatenodes (numbered in the figure) correspond to the values of the

sinusoidal functions shown in the graph, at the appropriatestations (the values are indicated by the dots). Note that mode4 does not change sign between stations 2 and 3 (the inter-mediate nodes 2 and 3 move inwards), whereas mode 5 has azero value at the station 5 (the intermediate node 5 has nulldisplacements).

(ii)

For m41 the l¼ ðn−1Þ=2 pair generates a 2m� 2m block andthe resulting modes are indicated in the figure as the “initialset”, directly related to vectors a (modes 7 and 9) and b(modes 8 and 10). The procedure proposed in [1] wasfollowed, where each mode shows lateral displacementsat a single intermediate node per wall. In the figure thenode numbering was changed for modes 7/8 and 9/10, tofacilitate a comparison with the sinusoidal function valuesshown in the graph.As for even n, the final deformation modes for this blockmay be obtained by solving the eigenvalue problemðBB−λCBÞv¼ 0, leading to the “final set” of modes shown inthe figure. In this case the buckling modes generally involveparticipation from all modes of this final set, but duplicatebuckling modes are once more obtained. This can be provenby recalling from [1] that, if the GBT matrices are diagona-lised with the Fourier matrix instead of Q , they becomeblock diagonal with m�m Hermitian blocks appearing inpairs (with the blocks in each pair being the transpose ofeach other), thus necessarily leading to real and duplicatebuckling loads.

Fig. 4(b) makes it possible to compare the minimum k valuesand the associated buckling mode shapes obtained with GBT(with m¼ 1;2;3) and finite strip analyses with three intermedi-ate nodes in each wall. These results were calculated forr¼100 mm, t¼1 mm, E¼210 GPa and ν¼ 0:3. The graph showsthat the consideration of only two intermediate nodes alreadyleads to a virtually perfect match between the k values obtainedwith both methods, which also coincide with those reported in[3,5,10]. Note also that the k values decrease as the number ofsides grows and, gradually, approach 4.0.2 The duplicate buck-ling modes are also shown in the figure and, obviously, there isno perfect match between the GBT and finite strip modes, dueto the fact that they are not uniquely defined. Both bucklingmode pairs can be used as a possible base of the same two-dimensional buckling mode space and Fig. 4(c) shows severalpossible configurations for n¼3, as a function of the “rotation”in that space. The rotation leading to an (approximate) modematch is shown in Fig. 4(d).

3.2. Pure cross-section extensional buckling

Cross-section (in-plane) extensional buckling is characterizedby uniform transverse membrane extensions εMyy around the cross-section and small half-wavelengths in the longitudinal direction.This mode is the counterpart of the widely investigated axissymmetric mode occurring in circular cross-sections. To theauthors’ best knowledge, its relevance in RCPS members has neverbeen discussed.

As shown in [1], the first GBT natural transverse exten-sion deformation mode (generated by að0Þ) corresponds to uni-form extension around the cross-section, with the displacement

m = 1

1

23

4

5

1

23

4

5

1

23

4

5

1 2 3 4 5-1

0

1

a b

4(a)

5(b)

m = 2(initial set)

7 8

9 10

m = 2(final set)

Triangle

Pentagon

Heptagon

GBT Finite Strip

4.8

4.6

4.4

4.2

4.01 2 3

k

m

Finite StripGBT

n = 3

n = 5

n = 7

7(a)

9(a)

8(b)

10(b)

0º

45º

90º

135º

180º

225º

270º

315º

1

23

4

51

23

4

5

1

23

4

5

18º288º

GBT

Finite Strip

Fig. 4. (a) Local deformation mode shapes for n¼5 and m¼1,2, with the generating vectors (their components are indicated by the dots at the stations 1;…;5); (b) minimumk values and associated buckling mode shapes for odd n; (c) buckling mode configurations for n¼3. (d) The matching of modes in this case requires a suitable rotation in thebuckling mode space.

R. Gonçalves, D. Camotim / Thin-Walled Structures 71 (2013) 35–45 39

components

u ¼ 0; v ¼−b2þ y; w ¼−

b2tanðπ=nÞ : ð14Þ

This mode is therefore characterised by null warping displace-ments, transverse wall bending ðw ;yy ¼ 0Þ and transverse wallrotations ðw ;y ¼ 0Þ. Fig. 5(a) shows the shapes of this mode forn¼3–7. The non-null GBT matrix diagonal components corre-sponding to this mode can be integrated analytically, yielding

CB ¼ E6ð1−ν2Þnt

3r3 sinπ

n

� �cos2

π

n

� �; ð15Þ

BM ¼ 2E1−ν2

ntr sinπ

n

� �; ð16Þ

DM1 ¼ E

3ð1þ νÞntr3 sin3 π

n

� �; ð17Þ

Xvþws ¼ 2

3ntr3 sin

π

n

� �1þ 2cos2

π

n

� �� �: ð18Þ

With these components, Eqs. (10) and (11) may be readilyemployed to obtain scr and Lcr. However, the use of this singlemode leads to slightly higher buckling loads, which may be

0

Square Pentagon Hexagon HeptagonTriangle

σcrE

n5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40

n

β = 252

0.2

0.15

0.1

0.05

0

β1a

0.25

0.2

0.15

0.1

0.05

0

0.25

0.35

0.3

β = 1002

with warpingwithout warpingν = 0.3

β = 252

β = 1002

Fig. 5. (a) Cross-section uniform extension deformation mode shapes for n¼3–7; (b) minimum buckling stresses and (c) associated half-wavelengths.

R. Gonçalves, D. Camotim / Thin-Walled Structures 71 (2013) 35–4540

lowered by adding the axial deformation mode to the bucklinganalysis. The resulting GBT matrices are diagonal with the excep-tion of DM

2 , and the new components are given by (1—axialextension; 2—cross-section transverse extension)

CM11 ¼

2E1−ν2

ntr sinπ

n

� �; ð19Þ

ðDM2 Þ21 ¼

2Eν1−ν2

ntr sinπ

n

� �; ð20Þ

ðXus Þ11 ¼ 2ntr sin

π

n

� �: ð21Þ

Considering the two modes, the assumption of Xus ¼ 0 in Eq. (8)

leads to the conclusion that the critical buckling mode is char-acterized by the v and w displacements given in (14) and warpingdisplacements

u ¼ νL2

a2π2; ð22Þ

i.e., buckling involves axial shortening of the member. One maytherefore employ a single mode with displacements (14) + (22),leading to the matrix components (15)–(18) and

CM ¼ 2ν2Eπ4ð1−ν2Þ

ntL4r sinðπ=nÞa4

ð23Þ

DM2 ¼ 2Eν2

π2ð1−ν2ÞntL2ra2

sinπ

n

� �ð24Þ

Xus ¼ 2ν2

π4ntL4ra4

sinπ

n

� �: ð25Þ

Note that Eqs. (10) and (11) are not valid for this single mode, evenif Xu

s is discarded, since CM and DM2 depend on L. Moreover, it

should be stressed that this approach is not exactly equivalent toconsidering the two modes independently, since (22) was calcu-lated for Xu

s ¼ 0.The graphs (b) and (c) in Fig. 5 make it possible to compare the

minimum buckling stresses and the associated half-wavelengthsfor β2 ¼ 25;100, n¼3–40 and ν¼ 0;3, obtained by considering asingle mode (I) without warping displacements (using (15)–(18))and (II) with warping displacements (using (15)–(18) and

(23)–(25)). Although not shown here, the authors have alsoobtained results with two independent modes (axial extensionand cross-section extension, (15)–(18) and (19)–(21)), but it wasfound that in all cases the critical stresses fall within 0.03% of thosecalculated with approach (II).

These results show that the buckling stress decreases withincreasing β2 and n. It is also clear that the longitudinal half-wavelengths for this mode are very small ðβ1=ao0:35Þ. Bothapproaches lead to similar results, with maximum differences of4.6% for scr (β2 ¼ 25 and n¼40) and 2.9% for β1=a (β2 ¼ 25 andn¼3).

As n increases, the results tend asymptotically to the solutionsfor circular tubes. With approach (I), one obtains

scrE

¼ 1β2

ffiffiffi3

pð1−ν2Þ

;β1a

¼ffiffiffiffiffiffiffiffiffiffiffiπ4

12β22

4

s; ð26Þ

which, for ν¼ 0:3, yields differences of 4.8% for scr , and 2.3% forβ1=a, with respect to the classical expressions for circular tubes [2]

scrE

¼ 1

β2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð1−ν2Þ

p ;β1a

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

π4

12ð1−ν2Þβ224

s: ð27Þ

For ν¼ 0:5, the differences increase to 15.5% for scr and 6.9% for β1=a.The classical expressions are however retrieved with approach (II),provided that Xu

s is discarded for the calculation of the criticalstresses.

3.3. Pure distortional buckling

Distortional buckling involves cross-section in-plane and out-of-plane (warping) displacements of the natural nodes, causingtransverse wall bending. This buckling mode is primarily capturedusing the inextensional Vlasov ðεMyy ¼ γMxy ¼ 0Þ cross-section defor-mation modes, although membrane shear deformation and localbuckling may also be relevant (see Section 4.2). As shown in [1],these Vlasov distortional deformation modes appear in pairs,except the ½n=2� mode for even n, and therefore duplicate bucklingmodes are generally obtained.

Using only the Vlasov deformation modes, all GBT matricesbecome fully diagonal and the critical loads may be obtained from

Table 2Constants A1-A4 for distortional buckling ðν¼ 0:3Þ.

Modes n 4 5 6 7 8 9 10 11 15 20 40

2 and 3 A1 3.2899 3.7982 4.1123 4.3155 4.4530 4.5500 4.6206 4.6737 4.7926 4.8543 4.9146A2 0.4519 0.4519 0.4519 0.4519 0.4519 0.4519 0.4519 0.4519 0.4519 0.4519 0.4519

4 and 5 A1 1.1072 0.6090 0.4925 0.4681 0.4644 0.4660 0.4689 0.4718 0.4804 0.4858 0.4915A2 0.2955 0.5360 0.6332 0.6689 0.6860 0.6958 0.7020 0.7062 0.7146 0.7184 0.7219A3 0.2959 0.4003 0.4022 0.3958 0.3897 0.3848 0.3810 0.3780 0.3711 0.3675 0.3639A4 0.0749 0.0996 0.0889 0.0822 0.0782 0.0756 0.0738 0.0725 0.0698 0.0685 0.0672

6 and 7 A1 0.3347 0.1950 0.1411 0.1221 0.1144 0.1110 0.1081 0.1083 0.1092A2 0.5360 0.6577 0.7290 0.7605 0.7764 0.7857 0.8010 0.8070 0.8119A3 1.0733 1.2855 1.2891 1.2790 1.2700 1.2627 1.2435 1.2322 1.2203A4 0.5438 0.7031 0.6628 0.6258 0.6025 0.5874 0.5600 0.5482 0.5378

8 and 9 A1 0.1258 0.0797 0.0568 0.0467 0.0374 0.0362 0.0362A2 0.6679 0.7325 0.7798 0.8044 0.8345 0.8430 0.8489A3 2.2833 2.6094 2.6053 2.5755 2.5255 2.5040 2.4809A4 1.9746 2.4831 2.4102 2.2900 2.0838 2.0213 1.9774

R. Gonçalves, D. Camotim / Thin-Walled Structures 71 (2013) 35–45 41

Eq. (9), which requires the computation of the diagonal compo-nents of the six GBT matrices (7).

If Xus is discarded, the closed-form expressions (10) and (11)

may be employed or, instead, the problem may be re-parametrisedand cast in the simpler form

scrE

¼ A1a2

β21þ A2a2

β21β22

þ A3

β22þ A4β

21

β22a2; ð28Þ

where A1–A4 are constants depending on ν, n and the deformationmode number, which can be easily computed from the deforma-tion mode characteristics (see [1]). These constants are given inTable 2, for ν¼ 0:3. Note in the table that, as already mentioned,the buckling modes appear in pairs, except the last mode for evenn, which is alone. Also note that modes 2 and 3, which correspondto global flexural buckling,3 are characterised by A3 ¼ A4 ¼ 0 andthus, Eq. (28) shows that the associated buckling stress approacheszero as β1-∞.

For modes 4 and higher, the minimum buckling stress and theassociated β1 value may be obtained from

scrE

¼A3 þ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðA1β

22 þ A2ÞA4

qβ22

; ð29Þ

β1a

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA1β

22 þ A2

A4

4

s: ð30Þ

Fig. 6(a) plots the variation of the minimum buckling stresswith n, for modes 4–9, β2 ¼ 75 and ν¼ 0:3 (the results arequalitatively similar if β2 is changed). The horizontal lines corre-spond to circular tubes and were obtained using Eq. (9) with theGBT matrix diagonal components for B, C and D derived in [11](see also [1]), together with the components of matrix Xvþw

s from[18] and also Xu

s ¼CM=E. These results show that:

(i)

3

due t

For a given mode pair, the buckling stress decreases withincreasing n, i.e., with decreasing b=t ¼ 2β2 sinðπ=nÞ ratios(recall that β2 is constant).

(ii)

For a given n value, the minimum buckling stress alwayscorresponds to the first distortional cross-section deformationmode pair 4 and 5.

(iii)

As n increases, the buckling stresses asymptotically approachthose obtained for circular tubes. For the first mode pair, the

Although these modes are not “distortional”, they are included in this sectiono the fact that they belong to the Vlasov deformation mode set.

critical stress falls remarkably within 4% of the circular tubevalue for n≥9. For higher mode pairs, this is only achieved athigher n values.

Fig. 6(b) shows the variation of the minimum buckling stresswith n, for modes 4 and 5, by imposing the condition that thecross-section area and thickness are the same for each curve.These results aim at providing help to select the appropriate nnumber for RCPS fabricated from a given metal sheet. Note that, ineach curve, the β2 value drops slightly as n increases (the boundingvalues are indicated).

The results show that the minimum buckling stress decreases(i) if the curve starts with a higher β2 value and (ii) as n increases.However, a comparison between the curve starting with β2 ¼ 75and the curve for modes 4 and 5 in Fig. 6(a) (which corresponds toβ2 ¼ 75) reveals that the decrease of scr with n is less pronouncedin the first case, due to the fact that β2 also decreases with n.Indeed, for n¼40 one has scr ¼ 0:00525E in the first case andscr ¼ 0:004912E in the second.

In order to assess the variation of the distortional bucklingstresses with β1=a, the particular case of n¼20, ν¼ 0:3 and β2 ¼ 75is analysed in Fig. 6(c). For comparison purposes, the graph alsoshows the results for a circular tube with the same parametervalues. Moreover, Fig. 6(d) shows the most relevant (out of the 20)cross-section natural Vlasov warping deformation modes, whichcoincide with the buckling modes: bending (modes 2 and 3) anddistortion with increasing half-wave numbers around the cross-section (modes 4–20). It is concluded that flexural buckling(modes 2 and 3) governs for high β1=a values, whereas distortionalbuckling with duplicate modes having increasing numbers of“lobes” become critical as β1=a decreases. Moreover, as alreadyconcluded from Fig. 6(a), for n¼20 the minimum buckling stressfor modes 4 and 5 virtually coincides with that for circular tubes.As the order of the mode increases, the differences between RCPSand circular tubes also increase, with the lower buckling stressesbeing obtained for circular tubes.

Finally, it is stressed that, as in the case of local buckling, theduplicate distortional buckling modes define a two-dimensionalmode space. Fig. 6(e) shows several possible buckling modeconfigurations for the 6 and 7 mode pair and n¼20, eachassociated with a different “rotation” in that space.

3.4. Critical modes

On the basis of the results obtained in the previous sec-tions, it is now possible to determine the parameter ranges for

50

0.002

0.004

0.006

0.008

0.01

0.012

0.014

σE

10 15 20 25 30 35 40n

RCPSCircular sectionν = 0.3β = 75

modes 4-5

modes 6-7

modes 8-9

0.02

0.015

0.01

0.005

50

10 15 20 25 30 35 40n

σE

β = 25

β = 50

β = 75

β = 100

β = 46.8

β = 70.2

β = 93.6

β = 23.4

modes 4-5ν = 0.3

0º

45º

90º

135º

180º

225º

270º

315º

2 3

4 5

6 7

8 9

10 11

12 13n = 20, single modesCircular section

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

ν = 0.3β = 75

1 10 100

σE

β /a1

modes 2-3

modes 4-5

6-7

8-910-11

12-13

Fig. 6. Pure distortional buckling. Variation of the minimum buckling stress with n for (a) β2 ¼ 75 and modes 4–9, (b) modes 4, 5 and constant cross-section area andthickness. Buckling analysis of a RCPS member with n¼20, β2 ¼ 75 and ν¼ 0:3 (c) variation of scr=E with β1=a (in logarithmic scale), (d) cross-section deformation modes2–13 and (e) buckling mode configurations for the 6 and 7 mode pair and n¼20.

R. Gonçalves, D. Camotim / Thin-Walled Structures 71 (2013) 35–4542

which each pure mode (local/cross-section extensional/distor-tional) is critical. Note that multi-mode buckling is not beingconsidered (it is addressed in Section 4) and, therefore, theresults are necessarily approximate, but nevertheless thisapproach leads to fairly simple solutions that provide a insightinto the buckling behaviour of uniformly compressed RCPSmembers.

Consider first cross-section extensional buckling. If no restric-tions are imposed on β1, this mode is never critical with respect todistortional buckling. This can be demonstrated by simply com-paring (i) Eq. (29) for the mode pair 4 and 5 and n¼4 (whichcorresponds to the highest minimum distortional buckling stress,with A1–A4 taken from Table 2) with (ii) Eq. (27) for circular tubes,which constitutes the lower bound for cross-section extensionalbuckling. It can be shown that, for ν¼ 0:3, the first equation alwaysleads to lower buckling stresses if β2410:4, which holds for thin-walled members.

For illustrative purposes, the graph in Fig. 7(a) makes it possibleto compare the minimum buckling stresses for the distortional andcross-section extensional (with warping) modes, for β2 ¼ 25;100,ν¼ 0:3 and n¼4–40. These results show that distortional bucklingis always critical and, moreover, that the difference between thebuckling stresses of the two modes is higher for lower n and β2values.

Consider now local buckling. Without accounting for the β1values, the boundary between local and distortional buckling maybe expressed in terms of n, ν and β2. The distortional stresses maybe calculated with Eq. (29) and, for ν¼ 0:3, the data in Table 2.As for the local buckling stresses, they may be obtained from the kvalues reported in Section 3.1.

Fig. 7(b) makes it possible to visualise the parameter rangesassociated with critical pure distortional/local buckling for ν¼ 0:3.It is observed that local buckling is critical for higher β2 values,although the transition value increases with n.

Localbuckling

Distortionalbuckling

β2

0

20

40

60

80

100

5 10 15 20

b/t = 30

β = 22.5

cross-section extensionaldistortionalν = 0.3

5

0.01

σE

n n10 15 20 25 30 35 40

β = 252

β = 1002

0

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0

Fig. 7. (a) Comparison of the minimum buckling stresses for the pure cross-section extensional and distortional buckling modes; (b) parameter ranges correspondingto critical local and distortional buckling (ν¼ 0:3).

R. Gonçalves, D. Camotim / Thin-Walled Structures 71 (2013) 35–45 43

In order to extract a few practical considerations for steelsections, the dashed lines in Fig. 7(b) indicate the cross-sectionslenderness values prescribed by Eurocode 3 [19] to separatebetween class 3 and 4 cross-sections4 made of S460 steel (whichleads to the lowest limits): (i) b=t ¼ 30 for internal compressedparts and (ii) β2 ¼ 22:5 for circular tubes. For a given n value, aRCPS with β2 above a given dashed line is classified as class4 according to the corresponding criterion. It is then possible toconclude that:

(i)

4

the c

for the b=t ¼ 30 criterion, class 4 cross-sections with criticaldistortional buckling are only obtained for n419 and β2490.

(ii)

although, strictly, the β2 ¼ 22:5 criterion applies only tocircular tubes, it indicates that RCPS with moderate-to-highn values may be class 4 for n and β2 values lower than thoseobtained with the previous criterion.

No further considerations are made concerning this issue, since itis beyond the scope of the present paper, which deals with elasticbifurcation (and not collapse). Nevertheless, it is recalled that thetransition between local and distortional collapse modes isaddressed in [7–9].

4. Multi-mode buckling

4.1. Local buckling

For local buckling, as reported in [5,10], the k values decreasefor b=t ratios below 50, although the drop is only significant forb=to25 and n¼ 3;5;7. Such cross-section geometries are notthin-walled and, therefore, are hardly affected by local buckling,thus falling outside the scope of this paper. Nevertheless, for thesake of completeness, this effect is addressed for the particularcase of b=t ¼ 20 and n¼3.

For this case, a finite strip analysis using three intermediatenodes in each wall leads to k¼4.595, which is in very closeagreement with the value reported in [5]. Note that this corre-sponds to a 2.5% drop in k with respect to the value obtained forhigh b=t ratios, 4.709 (see also Fig. 4).

Clearly, the GBT approach with local w modes only is not ableto capture the k drop, since the solution is independent of the b=tratio. However, by considering also three intermediate nodes ineach wall (m¼3) and including all the resulting 36 cross-section

For class 4 cross-sections, local buckling occurs before the onset of yielding inross-section.

deformation modes in the analysis (9 natural Vlasov+shear+trans-verse extension modes plus 27 local u þ v þw modes), with allGBT matrices (7), one obtains k¼4.597, which practically coincideswith the finite strip result. The ensuing (duplicate) local bucklingmodes are, however, very similar to those shown in Fig. 4(c),although very small displacements are perceptible at the cross-section natural nodes.

4.2. Distortional buckling

Since it was shown in Section 3.4 that distortional buckling iscritical for relatively low β2 values, it is necessary to investigatewhether shear deformation plays a relevant role.

As proposed in [1], the GBT cross-section natural sheardeformation modes for RCPS may be obtained by retaining thecross-section in-plane displacements of the Vlasov modes, whilediscarding their warping displacements.5 Therefore, the cross-section in-plane configurations of the Vlasov and shear deforma-tion modes are necessarily identical.

Taking into account that the GBT matrices are fully diagonal forthe Vlasov and shear mode sets (individually), the inclusion ofboth of them leads to coupling only between modes correspond-ing to the same in-plane displacements. This makes it possible toconclude that the calculation of the distortional buckling loadswith natural shear deformation modes requires the considerationof only two deformation modes: a given Vlasov distortional modeand its shear mode counterpart.

Fig. 8(a) shows the variation of the minimum distortionalbuckling stress (modes 4 and 5) with n, for ν¼ 0:3 andβ2 ¼ 25;50;75, obtained using either the Vlasov modes (withEq. (29) and Table 2) or the Vlasov+shear modes (with allmatrices (7) and Eq. (8)). For comparison purposes, finite stripresults are provided for n¼ 7;10;20, which also allow formembrane shear deformation. For consistency with the GBTanalyses, no intermediate nodes (m¼0) were considered in thefinite strip analyses.

These results show that the inclusion of the shear modes leadsto slightly lower buckling stresses, particularly for β2 ¼ 25, whereup to 5.4% lower values are obtained. The influence of sheardeformation decreases to 2.7% for β2 ¼ 50 and further to 1.8% forβ2 ¼ 75, with even lower differences being obtained for higher β2values (not shown). The GBT (with shear modes) and finite strip

5 Moreover, the torsional deformation mode must be added to the sheardeformation mode set, since it cannot be retrieved from the cross-section in-planedisplacements of any Vlasov mode.

GBT, Vlasov modesGBT, Vlasov+shear modesGBT, Vlasov+shear+ local w modesFinite Strip, m = 0Finite Strip, m = 3ν = 0.3

50

0.005

0.01

0.015

0.02

σcrE

n10 15 20 25 30 35 40

β = 252

β = 502

β = 752

σcrE

0

0.005

0.01

0.015

0.02

5n

10 15 20

β = 25

β = 502

β = 752

2

Fig. 8. Influence of (a) shear and (b) shear and local w modes on the minimum distortional buckling stress.

R. Gonçalves, D. Camotim / Thin-Walled Structures 71 (2013) 35–4544

results are in excellent agreement, with maximum differences ofabout 4.8% for n¼5, but quickly dropping to 0.6% for n≥7.

Finally, the influence of both shear deformation and local wmodes on distortional buckling is assessed. The graph in Fig. 8(b) supplements the results of Fig. 8(a) with those obtained usingthree intermediate nodes in each wall (m¼3), for both the GBTand finite strip analyses and n¼5–20. Note that, in the GBTapproach, only local w modes are added with respect to theprevious analyses. No such versatility is possible with finite strips,which always involves 4 d.o.f. per new node.

The above results make it possible to conclude that the influence ofthe local modes is more significant for low n and β2 values, but smallerthan that of the shear modes. Moreover, note that the GBT results arein excellent agreement with the finite strip ones throughout thewholen range, although only one local mode set ðwÞ was added in theanalyses, i.e., the local u and v mode sets do not play any role.

4.3. Illustrative example

For illustrative purposes, the particular case of n¼10,r¼100 mm, t¼4 mm ðβ2 ¼ 25Þ and ν¼ 0:3 is analysed in detail,which corresponds to a cross-section with almost coincident purelocal and distortional minimum buckling stresses (see Fig. 7(b)).The graph in Fig. 9 plots the variation of the buckling stresses withβ1=a, obtained with (i) finite strip analyses and (ii) GBT, withvarious combinations of mode sets (natural Vlasov, natural shearand local w) and also with the corresponding single modes.For the finite strip analyses three intermediate nodes in each wallwere considered, while with GBT only two intermediate nodes(m¼2) proved to be sufficient. Also shown in the figure are GBT-based renderings of the buckling mode shapes for half of thecolumns (i.e., 0≤x≤L=2) and corresponding to the three localminima and also to β1=a¼ 17:78, which corresponds to globalflexural buckling.

On the basis of the finite strip results, the following conclusionsconcerning the GBT analyses can be extracted:

(i)

6 See Fig. 6(d) which, although concerns the case of n¼20, makes it possible tocompare the shapes of this mode pair with that shown in Fig. 9 for β =a¼ 7:94.

For β1=ao1:6, local buckling is critical and the single-moderesults are already in excellent agreement with the finite stripones—recall that in Section 4.1 it was discussed that localbuckling is unaffected by other modes for cross-sections withhigh n values.

17 Once again, compare the 6 and 7 mode pair of Fig. 6(d) with the buckling

(ii)

mode for β1=a¼ 2:82 in Fig. 9.

For 1:6oβ1=ao13, distortional buckling is critical. Thelower local minimum corresponds to the 4 and 5 distortional

mode pair6 and, as anticipated in Section 4.2 (see Fig. 8 forβ2 ¼ 25 and n¼10), is influenced by shear deformation.Although with little practical relevance, it should be notedthat the second minimum, corresponding to the 6 and 7distortional mode pair,7 requires the consideration of theshear and local mode sets.

(iii)

For β1=a413, global flexural buckling is critical and sheardeformation is only relevant for a small β1=a range.

(iv)

This example shows that, as predicted in the previoussections, the GBT analyses with only natural Vlasov, naturalshear and local w modes provide extremely accurate results.No such versatility is possible with finite strip (or shell finiteelement) analyses.

5. Concluding remarks

This paper presented an investigation on the buckling (bifurca-tion) behaviour of uniformly compressed thin-walled memberswith regular convex polygonal (RCPS) cross-sections. In particular,a specialisation of Generalised Beam Theory (GBT) for RCPS [1],which makes it possible to calculate more rationally and efficientlythe deformation mode sets, was employed to obtain in-depthknowledge concerning the buckling behaviour of this type oftubular members.

Pure local, cross-section extensional and distortional bucklingmodes were first addressed. Both analytical and numerical resultswere provided and it was shown that, in many cases, the GBTapproach requires only a single mode to provide extremelyaccurate results. Moreover, it was demonstrated that the localand distortional buckling modes are duplicate in some cases.Concerning cross-section extensional and distortional buckling, itwas concluded that, for a given β2 value (circumradius overthickness), the buckling stresses decrease with increasing n(number of sides) and tend to the classical solutions for circulartubes as n-∞. Finally, the parameter ranges for which each puremode is critical were obtained, showing that either local ordistortional buckling is always associated with the lower buckling

0.5

β /a5 50 100

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

σE

GBT, single modeGBT, Vlasov + shear modesGBT, local w + Vlasov modesGBT, local w + Vlasov + shear modesFinite Strip

β /a = 0.60 β /a = 2.82

β /a = 7.94

β /a = 17.78n = 10m = 2β = 25r = 100 mmt = 4 mmν = 0.3

Fig. 9. Illustrative example: variation of the buckling stresses and mode shapes with β1=a.

R. Gonçalves, D. Camotim / Thin-Walled Structures 71 (2013) 35–45 45

stress (if no restrictions are imposed on the longitudinal half-wavelength).

Concerning multi-mode buckling, it was observed that sheardeformation may influence significantly the distortional bucklingstresses. However, with GBT, accurate results are obtained withonly two modes: the relevant distortional mode and the associatedshear mode. It was also shown that local w modes also influencedistortional buckling, but to a smaller extent.

For validation and comparison purposes, solutions taken fromthe literature and finite strip numerical results were considered.An excellent agreement with the GBT results was found inall cases.

Acknowledgement

The financial support provided by Fundação para a Ciência eTecnologia (FCT, Portugal), through project PTDC/ECM/108146/2008 (Generalised Beam Theory (GBT) – Development, Applicationand Dissemination), is gratefully acknowledged.

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