Thin walled structure

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Thin-Walled Structures 47 (2009) 701–729

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Thin-Walled Structures

0263-82

doi:10.1

� Corr

E-m

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

Calculation of pure distortional elastic buckling loads of members subjectedto compression via the finite element method

Miquel Casafont �, Frederic Marimon, Maria Magdalena Pastor

Departament de Resistencia de Materials i Estructures a l’Enginyeria, E.T.S. d’Enginyeria Industrial de Barcelona, Universitat Politecnica de Catalunya, Av. Diagonal 647, 08028

Barcelona, Spain

a r t i c l e i n f o

Article history:

Received 23 November 2007

Received in revised form

16 October 2008

Accepted 3 December 2008Available online 12 February 2009

Keywords:

Thin-walled members

Cold-formed members

Distortional buckling

Generalised beam theory (GBT)

Constrained finite strip method (cFSM)

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

016/j.tws.2008.12.001

esponding author. Tel.: +34 93 405 43 22; fax

ail addresses: [email protected] (M. C

a b s t r a c t

An analysis procedure is presented which allows to calculate pure distortional elastic buckling loads by

means of the finite element method (FEM). The calculation is carried out using finite element models

constrained according to uncoupled buckling deformation modes. The procedure consists of two steps:

the first one is a generalised beam theory (GBT) analysis of the member cross-section, from which the

constraints to apply to the finite element model are deduced; in the second step, a linear buckling

analysis of the constrained FEM model is performed to determine the pure distortional loads. The

proposed procedure is applied to thin-walled members with open cross-section, similar to those

produced by cold-forming. The distortional loads obtained are rather accurate. They are in agreement

with the loads given by GBT and the constrained finite strip method (cFSM).

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Linear buckling analyses play an important role in the investigation and design of cold-formed members with open cross-sections. Onthe one hand, they help to know the behaviour of a member, predicting the type of instabilities that can experience and the degree ofinteraction between them. On the other hand, the elastic buckling loads deduced from the analysis are used, together withexperimentally determined buckling curves, to calculate the strength of the member.

Nowadays, three different methods are usually applied to perform linear buckling analyses of cold-formed members: the finite stripmethod (FSM), the finite element method (FEM) and the generalised beam theory (GBT). They are suitable for this type of membersbecause they can take into account all relevant buckling modes: local, distortional and global. However, these methods show differentadvantages and limitations that are briefly presented in this introduction.

The generalised beam theory, developed by Schardt [1], may be the most suitable method for carrying out linear buckling analyses. Itsmain feature is that it directly works with uncoupled buckling modes of deformation. These pure modes are obtained in a first step of theanalysis procedure by means of a calculation performed at the cross-section level. Fig. 1 shows the result of the cross-section calculationapplied to a channel profile. Four global and two distortional pure modes of buckling are obtained. In a second step, the solution of theanalysis of the member (a standard linear, a non-linear or a linear buckling analysis) is obtained in the vectorial space defined by thesemodes of deformation.

Since GBT can work with uncoupled modes, it is possible to determine the pure elastic buckling loads independently for each mode ofbuckling. This is very important because, according to the current design standards [2,3], this pure load is one of the input values neededfor the calculation of the buckling strength of members.

GBT presents other advantages: an identical formulation is applied for all modes of buckling; it is easy to investigatethe interaction between modes, because calculations can be easily done for any mode combination; and, since it isformulated as a beam element, the computational cost of the calculations is very low. It should also be pointed out that calculationprocedures have been developed in GBT that allow for the consideration of different types of boundary conditions at member ends. Forexample, GBT-based formulas for the calculation of C and Z members with various end supports were presented by Silvestre andCamotim in [4,5].

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: +34 93 40110 34.

asafont), [email protected] (F. Marimon), [email protected] (M.M. Pastor).

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Transverse displacementsLongitudinal displacements

Fig. 1. GBT pure modes of deformation for a channel section. (a) Axial mode, (b) bending about X, (c) bending about Z, (d) torsion, (e) symmetric distortional, and (f) anti-

symmetric distortional.

M. Casafont et al. / Thin-Walled Structures 47 (2009) 701–729702

GBT is a well known analysis method among the scientific community, mainly due to the articles published by Davies and Schardt inthe nineties [6–9]. Currently, in the field of thin-walled members, the method is used by the Portuguese team led by Silvestre andCamotim [4,5,10,11].

The finite strip method has probably been the most used method of performing linear buckling analysis in investigations on cold-formed members. It has been applied since many years ago [12–14] in some of the most relevant investigations devoted to thedistortional mode of buckling, for example the one carried out by Hancock [15].

The main limitation of the method with respect to GBT is that it does not allow to work with uncoupled modes of buckling. Thebuckling load calculated corresponds to a combination of local, distortional and global buckling. In spite of this problem, goodapproximations to pure distortional buckling loads are obtained for lengths near the critical length, for which there is usually nosignificant interaction with the other modes. The method has had good success and software has been specifically developed for thin-walled members: THIN WALL [16] and CUFSM [17].

FSM helped to the development of theoretical models and formulations for the hand-calculation of distortional (and local) bucklingloads. For example, the formulas proposed by Lau and Hancock [18], Schafer [19] and Serrete and Pekoz [20], were verified and improvedcomparing their results to the results of FSM analyses.

Recently, Adany and Schafer have introduced an improvement in FSM [21,22]. They have developed the constrained finite strip method(cFSM), which is based on the fundamentals of GBT, and can also work with uncoupled modes of buckling. As in GBT, when the cFSM isapplied, the solution of the linear buckling analysis is sought in the vectorial space defined by a specific mode of deformation, for examplesymmetric distortional, or by a chosen combination of modes, for example symmetric and anti-symmetric distortional. This is achievedby properly constraining the nodal displacements of the finite strip model [22].

The last version of CUFSM already includes cFSM. The results obtained with this software are good when compared to the results ofGBT [23]. Furthermore, the calculation time cost is also low. However, nowadays the analyses can only be done for buckling inone-half sine wave, because the semi-analytical finite strip method is applied. The authors believe that in the future this problem will besolved, since there is already a method in FSM that allows for the analysis of members with other boundary conditions, the spline finitestrip method.

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M. Casafont et al. / Thin-Walled Structures 47 (2009) 701–729 703

The FEM has also been used for a long time in investigations on instabilities of cold-formed members. However, there are few of themwhere linear buckling analyses are performed (see for example [24,25]). The problem with FEM is the same as with FSM: buckling modescannot be uncoupled. Furthermore, another important limitation is that the analyses performed via FEM may be too time consuming. Themodel generation and the calculation process take much more time than when GBT or FSM are used.

In spite of these limitations, FEM may be useful under certain circumstances. For example, it can be used to analyse members withperforations. Actually, the origin of the present investigation was a study on members of pallet-rack structures, where the problem ofperforations is relevant [26]. This is the reason why it is believed that linear buckling analyses carried out with FEM may be useful, andthat it makes sense to try to introduce some improvements to the analysis procedure, such as the one shown herein.

The aim of the paper is to present a FEM analysis procedure for the calculation of pure distortional buckling loads. The idea is toconstrain finite element models in such a way that GBT uncoupled distortional deformations are obtained in the analyses (Fig. 1e and f).The constraining method applied is similar to the one used in the cFSM of Adany and Schafer [21,22].

An outline of the paper follows. The fundamentals of GBT needed for the understanding of the presented method are summarised inSections 2 and 4. Those readers who are familiar with GBT can skip these sections, and go directly to Sections 3 and 5, where it isexplained how to constrain finite element models to obtain pure distortional loads. Section 6 includes an illustrative example performedwith commercial finite element software. The accuracy of the procedure is discussed in Section 7 by means of the results of calculationson channel lipped cross-sections subject to pure compression. Finally, the concluding remarks of Section 8 close the paper.

2. Theoretical bases for the calculation of the GBT deformational modes

In the formulation of the generalised beam theory, members are considered to be composed of plates, as shown in Fig. 2. GBT assumesthat these plates behave according to the Kirchoff plate theory: straight lines normal to the mid-plane remain straight, inextensional andnormal to the mid-plane after deformation. Consequently, ezz ¼ gxz ¼ gyz ¼ 0.

Furthermore, two additional simplifying assumptions are considered:

1.
The membrane transverse extensions are zero:
�Mxx ¼ 0. (1)

2.
The membrane shear strains are zero:
gMxy ¼ 0. (2)

These are the two main assumptions of GBT (see [1]).Fig. 2 shows the notation that will be used for the displacements of the plate mid-plane points (the notation given in [21,22] is

followed): u, v, w and y are the displacements expressed in the local plate systems, the x–y–z coordinate systems; and U–V–W and y arethe displacements corresponding to the global coordinate system, the X–Y–Z system. Later on, it will be seen that it is very important tomake the distinction between longitudinal displacements: v and V; and transverse displacements: u, w, U, W and y.

2.1. Strain–displacement relations

It is easy to see from all above considerations that the relevant strain–displacement relations are:

�xx ¼ �Mxx þ �

Bxx ¼ 0� z@2w@x2 ¼ �zw;xx, (3.a)

�yy ¼ �Myy þ �

Byy ¼

@v

@y� z

@2w

@y2¼ v;y � zw;yy, (3.b)

gxy ¼ gMxy þ g

Bxy ¼ 0� 2z

@2w

@x@y¼ �2zw;xy. (3.c)

Y,V

X,U

Z,W

y,v

z,w

x,u

θ

θ

Fig. 2. Local and global degrees of freedom considered in GBT.

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The present subsection summarizes the initial steps of the GBT formulation, which are devoted to express Eqs. (3.a), (3.b), (3.c) in aparticular way.

In GBT, as in FSM, displacements u(x,y), v(x,y) and w(x,y) are expressed as a product of two single-variable functions:

uðx; yÞ ¼ uðxÞcðyÞ, (4.a)

vðx; yÞ ¼ vðxÞc;yðyÞ, (4.b)

wðx; yÞ ¼ wðxÞcðyÞ, (4.c)

where c(y) is usually taken as a sinusoidal function that provides the variation of the u(x), v(x) and w(x) mid-plane displacements alongthe longitudinal direction. It is pointed out that c(y) can also be approximated by polynomials (mostly cubic). In (4.b), v(x,y) depends onthe derivative of c(y) because the member shear strains are considered null (assumption (2)).

Substituting c(y) by a sinusoidal function, the resultant displacement expression is:

uðx; yÞ ¼ uðxÞsinr � p � y

L

� �, (5.a)

vðx; yÞ ¼rpL

vðxÞ cosrpy

L

� �, (5.b)

wðx; yÞ ¼ wðxÞ sinrpy

L

� �, (5.c)

where L is the length of the member and r the number of half-waves.The next step is to express the transverse displacements u(x) and w(x) in terms of the longitudinal displacements v(x). This is one of the

most important steps of GBT, and a direct consequence of assumptions (1) and (2). Firstly, v(x) is expressed in terms of the longitudinal

nodal displacements vi,

vðxÞ ¼Xnþ1

i¼1

viðxÞvi, (6)

where n is the number of elements of the cross-section (Fig. 3a), vi is the longitudinal displacement of node i, and vi(x) is a function of x,that has unit value at node i and zero value at all other nodes. For the moment, the nodes of the section are located in the walllongitudinal edges. These nodes corresponding to the longitudinal edges are called natural nodes.

Subsequently, what is actually done is to express the transverse displacements in terms of the longitudinal nodal displacements:

uðx; yÞ ¼Xnþ1

i¼1

uiðxÞvi sinrpy

L

� �¼Xnþ1

i¼1

uiðxÞfiðyÞ, (7.a)

vðx; yÞ ¼Xnþ1

i¼1

rpL

viðxÞvi cosrpy

L

� �¼Xnþ1

i¼1

viðxÞfi;yðyÞ, (7.b)

wðx; yÞ ¼Xnþ1

i¼1

wiðxÞvi sinrpy

L

� �¼Xnþ1

i¼1

wiðxÞfiðyÞ, (7.c)

where ui(x) and wi(x) are functions of vi(x). These functions can be directly derived from assumptions (1) and (2) through a rather longprocedure presented in [1,10]. Part of this procedure will be shown in Section 4, when the relation between the transverse nodaldisplacements and the longitudinal nodal displacements is derived.

b

Z, W

X, U

z, w

x, u

1

2

n

i

i+1i

n+1

i

i

i

Z, W

X, U

1

2

n

i

i+1

n+sub i

sub 2

sub n

Fig. 3. Nodes and sub-nodes considered in the formulation of GBT.

ARTICLE IN PRESS


At this point, it is important to notice that all mid-plane displacements of the member are expressed in terms of the vi nodaldisplacements (or fi(y) displacements).

Finally, the above equations are introduced in the strain–displacement relations (3.a), (3.b), (3.c):

�xx ¼ �zw;xx ¼ �zXnþ1

i¼1

wi;xxðxÞfiðyÞ, (8.a)

�yy ¼ v;y � zw;yy ¼Xnþ1

i¼1

viðxÞfi;yyðyÞ � zXnþ1

i¼1

wiðxÞfi;yyðyÞ ¼Xnþ1

i¼1

½viðxÞ � z �wiðxÞ�fi;yyðyÞ, (8.b)

gxy ¼ �2zw;xy ¼ �2zXnþ1

i¼1

wi;xðxÞfi;yðyÞ. (8.c)

2.2. Equation of equilibrium

Once the strain–displacement relations are defined, the GBT equilibrium equation is obtained by applying the constitutive law of thematerial and the principle of virtual work.

The constitutive law in GBT is reduced to the following equations:

sxx ¼E

1� u2�xx þ

uE

1� u2�yy ¼

�E

1� u2zw;xx þ

uE

1� u2ðv;y � zw;yyÞ, (9.a)

syy ¼uE

1� u2�xx þ

E

1� u2�yy ¼

�uE

1� u2zw;xx þ

E

1� u2ðv;y � zw;yyÞ, (9.b)

txy ¼E

2ð1þ uÞgxy ¼

�E

ð1þ uÞzw;xy, (9.c)

where E is the Young’s modulus and u is the Poisson’s ratio. These relations are used when the principle of virtual work is appliedZL

Zb

Ztðsxxd�xx þ syyd�yy þ txydgxyÞdz dx dy ¼

ZL

Zbðqxduþ qydvþ qzdvÞdx dy; (10)

where

d�xx ¼ �zXnþ1

i¼1

wi;xxðxÞdfi, (11.a)

d�yy ¼Xnþ1

i¼1

½viðxÞ � zwiðxÞ�dfi;yy, (11.b)

dgxy ¼ �2zXnþ1

i¼1

wi;xðxÞdfi;y, (11.c)

and qx, qy and qz are the three components of the applied load per unit of mid-surface area.The integration of (10) results in the equation of equilibrium. It is not the purpose of this summary on GBT to fully develop this

integration, which is a laborious work that can be consulted in more complete references on the subject [10]. However, the final result isshown, because it is very important for the calculation of the GBT deformation modes:

E ¯Cf;yyyy � G ¯Df;yy þ¯Bf ¼ q. (12)

It should be pointed out that:

1.
¯C, ¯D and ¯B are second order tensors obtained from the integration of (10). They can be calculated from geometric and materialproperties of the member;
2.
f is a vector that contains the longitudinal displacements of the nodes (see (7.a), (7.b), (7.c). These displacements are the unknowns ofthe equilibrium equation.
The equilibrium equation is important because the pure GBT deformation modes are calculated from matrices of tensors ¯C and ¯B. Thelongitudinal components of the buckling modes shown in Fig. 1 are obtained by solving the following eigenvalue problem

ð ¯B� lE ¯CÞv ¼ 0. (13)

This simultaneous diagonalization of matrices ¯Cand ¯B results in n+1 eigenvectors, n�3 of which contain the longitudinal displacementsof the distortional modes. The other 4 eigenvectors correspond to global buckling modes. The calculation of these global eigenvectorsinvolves more work, as it is explained in [1,10].

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3. Constraining the longitudinal displacements of a finite element model

When linear buckling analyses are performed to calculate the elastic buckling loads of a finite element model, an eigenvalue problemis also solved

ð ¯Ke � l ¯KgÞD ¼ 0, (14)

where ¯Ke is the elastic stiffness matrix, ¯Kg is the geometric stiffness matrix, the eigenvalues l are load factors, which allow to determinethe elastic buckling loads, and the eigenvectors D provide the modes of buckling.

As pointed out in Section 1, the aim of the investigation is to verify whether it is possible to obtain good values of pure distortionalbuckling loads by forcing the member to buckle in the GBT pure distortional modes. The way to force the GBT modes is to introduce someconstraints to the components of D.

The members will be modeled by means of plate finite elements with six degrees of freedom per node: U, V, W, yX, yY, y( ¼ yZ). Theconstraints will only be applied to GBT degrees of freedom of D, which are those degrees of freedom used in the GBT formulation: U, V, W

and y.In the present section, it is shown the method followed to constrain only the GBT longitudinal degrees of freedom (V). This case is

explained to make easy to understand the more complex constraining procedure that will be shown in Section 5, where the constraintsare also applied to transverse degrees of freedom.

The first step of the procedure is to solve the GBT eigenvalue problem (13) to obtain the longitudinal displacements corresponding to apure distortional mode m (for example the symmetric or the anti-symmetric distortional modes in Fig. 4)

vm ¼

vm1

vm2

..

.

vmnþ1

8>>>>><>>>>>:

9>>>>>=>>>>>;¼

Vm1

Vm2

..

.

Vmnþ1

8>>>>><>>>>>:

9>>>>>=>>>>>;¼ V

m. (15)

The local and global longitudinal displacements are identical. The global notation is used because the constraining equations will beintroduced in the global coordinate system.

If the member has to buckle according to mode m, the longitudinal components of the constrained nodes should accomplish therelationship given by (15), i.e., the displacement vector of these nodes has to be proportional to (15). This can be accomplished if theselongitudinal degrees of freedom are expressed in terms of only one longitudinal degree of freedom, and the eigenvalue problem (14) is

V11

V12

Fig. 5. Constrained finite element mesh.

V1sd

V2sd

V3sd

V4sd

V5sd

V6sd

V1ad

V2ad

V3ad

V4ad

V5ad

V6ad

Fig. 4. Longitudinal node displacements corresponding to the GBT pure distortional modes of a channel section. (a) Symmetric distortional and (b) anti-symmetric

distortional.

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modified as it will be shown at the end of this section. Now, the next step is to express the longitudinal displacements in terms of V1s,which is chosen to be the unknown longitudinal degree of freedom of the cross-section (Fig. 5):

V s ¼

V1s

V2s

..

.

Vnþ1s

8>>>>><>>>>>:

9>>>>>=>>>>>;¼

Vm1

Vm2

..

.

Vmnþ1

8>>>>><>>>>>:

9>>>>>=>>>>>;

V1s ¼ Vm

V1s, (16)

where s is one of the ns constrained cross-sections, and V1s is the longitudinal displacement of the node located at edge of the constrainedcross-section, see Fig. 5b. This figure also shows that the constraints are not applied to all nodes of the finite element mesh. They areapplied to the nodes which are located in the same place as the GBT natural nodes. Furthermore, it is not necessary to constrain thenatural nodes of all cross-sections of the mesh. In Section 7, it will be shown that good results are obtained when the constraints areapplied to the nodes of some regularly spaced cross-sections (Fig. 5b).

The constraining relations (16) of all ns cross-sections are assembled in one matrix

¯Rm¼

Vm

0 � � � 0

0 Vm� � � 0

0 0 . .. ..

.

0 0 � � � Vm

266664

377775, (17)

which provides the relationship between the constrained degrees of freedom and the unknown degrees of freedom V1s for all constrainedcross-sections

DmCons ¼

Vm

0 � � � 0 � � � 0

0 Vm� � � 0 � � � 0

..

. ... . .

. ... ..

.

0 0 � � � Vm� � � 0

..

. ... ..

. . .. ..

.

0 0 � � � 0 � � � Vm

266666666664

377777777775

V11

V12

..

.

V1s

..

.

V1ns

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;¼ ¯R

mV1. (18)

DConsm is called vector of constrained degrees of freedom of mode m, and V1 is called vector of unknown longitudinal degrees

of freedom.Subsequently, the displacement vector D of (14), which includes all degrees of freedom of the finite element model, is

reduced to Dm, which includes the unknown longitudinal degrees of freedom V1 of the ns constrained sections, plus all degrees offreedom of the model that are not constrained. The relation between these two vectors can be easily formulated by conveniently orderingthe components of D:

D ¼

DNonCons:

��

DmCons

8><>:

9>=>; ¼

¯I ¯0¯0 ¯R

m

" # DNonCons:

��

V1

8><>:

9>=>; ¼ ¯<

mD

m. (19)

Finally, the eigenvalue problem (14) should be changed to the following one (see [23]):

ð ¯<mT

¯Ke¯<

m� l ¯<

mT

¯Kg¯<

mÞD

m:¼ 0. (20)

The constraining procedure presented in this section was the first one used by the authors. This procedure is easy to apply because itonly involves the calculation of the longitudinal GBT displacements in Eq. (13), and the application of the constraints to the finite elementmodel, which is not a difficult task. However, the results of this first procedure were not satisfactory. The main problem is that it does notallow to uncouple the distortional buckling from local buckling (see Section 7).

A second procedure was developed which involved constraining the transverse displacements U, W and y of the nodes located at themid points of the cross-section elements (sub-nodes, sub i in Fig. 3b). The idea was to force these nodes to move according to puredistortional modes, and to reduce the local buckling deformations.

The existing procedures for the calculation of the transverse displacements of the sub-nodes also comprise the calculation of thetransverse displacements of the natural nodes. For this reason, in the end, the transverse displacements of both types of nodes wereconstrained.

These transverse displacements are determined from the longitudinal displacements calculated in (13) following a very involvedprocedure. The next section shows how these transverse–longitudinal displacement relations are derived. The formulation proposed byAdany and Schafer in [21,23] is followed.

A third procedure is presented in Section 7 that is similar to the previous one. The idea is also to constrain the transverse

displacements, but applying constraints derived from FEM buckling analyses. This allows to avoid the complex formulation of thetransverse–longitudinal displacement relations.

4. Relationship between transverse and longitudinal nodal displacements

The transverse–longitudinal displacement relationships, which are needed to constrain the transverse degrees of freedom, aredetermined by following a two step procedure. In the first step, the global transverse displacements Ui and Wi of the internal nodes (nodes

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2 to n in Fig. 3), are expressed in terms of the global longitudinal displacements Vi. This relationship, which is called UWin–V, is derivedfrom the GBT main assumptions (1) and (2), and from geometric compatibility conditions. In the second step, the equilibrium condition ofthe cross-section is applied to determine:

�
UWex–V relationship between V and the U1, Un+1, W1 and Wn+1 transverse displacements of the external nodes (nodes 1 and n+1in Fig. 3); � UWsub–V relationship between V and Usubi and Wsubi transverse displacements of the sub-nodes; and � y–V relationship between V and yi rotations of all nodes and sub-nodes.
4.1. Derivation of the UWin–V relationship

When the generalised beam theory is derived, the variation of u(x) and v(x) in Eqs. (5.a), (5.b) and (5.c) is assumed to be linear withinthe elements of the cross-section [10]. For example, in [21] these displacements are expressed in terms of the nodal degrees of freedom, ui

and vi, and a linear function of x:

uðx; yÞ ¼ 1�x

bi

� �ui þ

x

biuiþ1

� �sin

rpy

L

� �, (21.a)

vðx; yÞ ¼rpL

1�x

bi

� �vi þ

x

biviþ1

� �cos

rpy

L

� �. (21.b)

The relation between ui and vi can be easily deduced from these equations. If the simplifying assumption (1) is taken into account, thefollowing condition is obtained:

�Mxx ¼

@u

@x¼ 0!

uiþ1 � ui

bisin

rpy

L

� �¼ 0. (22)

Since (22) has to be fulfilled in all cross-sections of the member, then:

uiþ1 ¼ ui. (23)

Therefore, the u(x,y) displacement within an element of the cross-section can be expressed as:

uðx; yÞ ¼ uðyÞ ¼ ui sinrpy

L

� �. (24)

Furthermore, if the simplifying assumption (2) is considered, then:

gMxy ¼

@u

@yþ@v

@x¼ 0!

rpL

ui cosrpy

L

� �þ

rpL

viþ1 � vi

bicos

rpy

L

� �¼ 0, (25)

and, consequently, the relation between longitudinal vi and transverse ui displacements is obtained:

ui ¼vi � viþ1

bi. (26)

As vi ¼ Vi, displacement ui can be directly related to the longitudinal global displacement:

ui ¼Vi � Viþ1

bi. (27)

ui

i

21

i

iw

i+1

n+1

n

viui-1

iu

=v /bi-1i

=v /bi i

ui-1

ib

i-1b

dy

i-1w

Fig. 6. Geometric determination of the wi displacement.

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i

i-1

Ui

iWui

i-1u

X

Z

wi

i-1w

αi-1

αi

i

i-1

uii-1u

i+1

Z

wi

i-1u

i+1

i-1w

Δα i

Δαi

ΔαiΔαi

X

Fig. 7. Geometric relationship between transverse displacements.


Once the relationship corresponding to the ui component is known, geometric compatibility allows to express wi in terms of vi. This canbe seen in Fig. 6 where an arbitrary longitudinal vi displacement is applied at node i, considering at the same time that, for the moment,the cross-section elements are connected by means of perfect hinges. Actually, the displacement wi is first expressed in terms of thetransverse displacement ui, taking into account that there must be continuity of the cross-section at nodes. Afterwards, (26) (or (27)) areused to express wi in terms of vi (or Vi).

The relationship between wi and ui can be deduced from the drawing in Fig. 7. This drawing shows an arbitrary displacement of aninternal node of the section. The displacement has been drawn in such a way that the wi–ui relationship is easily derived. However, itshould be pointed out that the final equation is valid for any arbitrary displacement.

This relationship is obtained with the help of the two triangles highlighted in Fig. 7b:

wi ¼ui�1

sin ðDaiÞ�

ui

tan ðDaiÞ, (28)

where

Dai ¼ ai � ai�1. (29)

At this point, the relationships between the transverse ui and wi displacements and the longitudinal vi (or Vi) displacements havealready been obtained (Eqs. (27) and (28)). However, as pointed out in Section 3, the constraints will be applied in the global coordinatesystem. Therefore, the last step is to translate (27) and (28) into global coordinates. This can be done by means of the following equations(see Fig. 7a):

Ui ¼ ui cos ðaiÞ þwi sin ðaiÞ ¼ ui cos ðaiÞ þui�1

sin ðDaiÞ�

ui

tan ðDaiÞ

� �sin ðaiÞ, (30.a)

Wi ¼ ui sin ðaiÞ �wi cos ðaiÞ ¼ ui sin ðaiÞ �ui�1

sin ðDaiÞ�

ui

tan ðDaiÞ

� �cos ðaiÞ. (30.b)

After some trigonometric operations performed taking into account that

sin ðDaiÞ ¼ sin ðaiÞ cos ðai�1Þ � cos ðaiÞ sin ðai�1Þ, (31)

cos ðDaiÞ ¼ cos ðaiÞ cos ðai�1Þ þ sin ðaiÞ sin ðai�1Þ, (32)

the following relations are obtained

Ui ¼sin ðaiÞ

sin ðDaiÞui�1 �

sin ðai�1Þ

sin ðDaiÞui, (33.a)

Wi ¼ �cos ðaiÞ

sin ðDaiÞui�1 þ

cos ðai�1Þ

sin ðDaiÞui. (33.b)

Introducing the relationship between the transverse u displacements and the longitudinal V displacements (Eq. (27)) in the aboveequations, the final relation is obtained:

Ui ¼sin ðaiÞ

sin ðDaiÞ

Vi�1 � Vi

bi�1�

sin ðai�1Þ

sin ðDaiÞ

Vi � Viþ1

bi

¼sin ðaiÞ

sin ðDaiÞbi�1Vi�1 �

sin ðaiÞ

sin ðDaiÞbi�1þ

sin ðai�1Þ

sin ðDaiÞbi

� �Vi þ

sin ðai�1Þ

sin ðDaiÞbiViþ1, (34.a)

ARTICLE IN PRESS


Wi ¼ �cos ðaiÞ

sin ðDaiÞ

Vi�1 � Vi

bi�1þ

cos ðai�1Þ

sin ðDaiÞ

Vi � Viþ1

bi

¼ �cos ðaiÞ

sin ðDaiÞbi�1Vi�1 þ

cos ðaiÞ

sin ðDaiÞbi�1þ

cos ðai�1Þ

sin ðDaiÞbi

� �Vi �

cos ðai�1Þ

sin ðDaiÞbiViþ1. (34.b)

The above equations can be expressed in matrix form

Ui ¼sin ðaiÞ

sin ðDaiÞbi�1

� ��

sin ðaiÞ

sin ðDaiÞbi�1�

sin ðai�1Þ

sin ðDaiÞbi

� �sin ðai�1Þ

sin ðDaiÞbi

� �( ) Vi�1

Vi

Viþ1

8><>:

9>=>;, (35.a)

Wi ¼ �cos ðaiÞ

sin ðDaiÞbi�1

� �cos ðaiÞ

sin ðDaiÞbi�1þ

cos ðai�1Þ

sin ðDaiÞbi

� ��

cos ðai�1Þ

sin ðDaiÞbi

� �( ) Vi�1

Vi

Viþ1

8><>:

9>=>;, (35.b)

and they can be extended to all internal nodes of the cross-section

U2

U3

..

.

Un�1

Un

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;¼

sin ða2Þ

sin ðDa2Þb1

� ��

sin ða2Þ

sin ðDa2Þb1�

sin ða1Þ

sin ðDa2Þb2

� �sin ða1Þ

sin ðDa2Þb2

� �0 0 0 0 0

0sin ða3Þ

sin ðDa3Þb2

� ��

sin ða3Þ

sin ðDa3Þb2�

sin ða2Þ

sin ðDa3Þb3

� �sin ða2Þ

sin ðDa3Þb3

� �0 0 0 0

� � � � � � � �

� � � � � � � �

0 0 0 0 x x x 0

0 0 0 0 0 x x x

26666666666664

37777777777775

V1

V2

..

.

Vn

Vnþ1

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;

(36.a)

W2

W3

..

.

Wn�1

Wn

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;¼ �

cos ða2Þ

sin ðDa2Þb1

� ��

cos ða2Þ

sin ðDa2Þb1�

cos ða1Þ

sin ðDa2Þb2

� �cos ða1Þ

sin ðDa2Þb2

� �0 0 0 0 0

0cos ða3Þ

sin ðDa3Þb2

� ��

cos ða3Þ

sin ðDa3Þb2�

cos ða2Þ

sin ðDa3Þb3

� �cos ða2Þ

sin ðDa3Þb3

� �0 0 0 0

� � � � � � � �

� � � � � � � �

0 0 0 0 x x x 0

0 0 0 0 0 x x x

26666666666664

37777777777775

V1

V2

..

.

Vn

Vnþ1

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;

(36.b)

or,

Uin ¼¯RUinV , (37.a)

Win ¼¯RWinV . (37.b)

4.2. The equilibrium of the cross-section

In the present section, the UWex–V, UWsub–V and y–V relationships are derived. They are determined through the relationbetween the internal displacements Uin and Win and the other transverse displacements of the cross-section (from now on in-exsubyrelationship). Once in-exsuby is known, Eqs. (37.a), (37.b) can be a used to obtain the final relation with the global longitudinal

displacements.The in-exsuby relationship is determined by means of the equivalent 2D beam model shown in Fig. 8. The analogy

between the cross-section and a beam system is applied, assuming that the cross-section deforms so that this 2D system is inequilibrium.

Now, the connection between the cross-section elements of the model is perfectly rigid, and the internal nodes are assumed to besupported by means of external hinges where displacements Uin and Win have been imposed.

ARTICLE IN PRESS

1

2

3

i

i+1

n

n+1

sub i

sub 2

b

t


The following steps should be applied to obtain the in-exsuby relation (similar to the standard 2D beam finite element procedure):

Fig. 8. Cross-section beam model.

1.
For each beam element (Fig. 8a), the equivalent stiffness matrix is calculated in the x–y local coordinate system
¯ki ¼

EAi

bi0 0

�EAi

bi0 0

012EIi

b3i

6EIi

b2i

0 �12EIi

b3i

6EIi

b2i

06EIi

b2i

4EIi

bi0 �

6EIi

b2i

2EIi

bi

�EAi

bi0 0

EAi

bi0 0

0 �12EIi

b3i

�6EIi

b2i

012EIi

b3i

�6EIi

b2i

06EIi

b2i

2EIi

bi0 �

6EIi

b2i

EAi

bi

2666666666666666666666664

3777777777777777777777775

, (38)

where:

Ai ¼ti

ð1� u2Þ, (39)

Ii ¼t3

i

12ð1� n2Þ. (40)

2.
The element stiffness matrices are expressed in the X–Z global coordinate system
¯Ki ¼¯T

T ¯ki¯T , (41)

where ¯Tis the usual coordinate transformation matrix.
3. Assembly of the global stiffness matrix ¯K from ¯Ki. 4. Global equilibrium
¯Kd ¼ ¯K

U1

W1

y1

U2

W2

y2

Usub2

Wsub2

ysub2

..

.

Unþ1

Wnþ1

ynþ1

8>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

¼

0

0

0

FU2

FW2

0

0

0

0

..

.

0

0

0

8>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>>;

¼ q. (42)

ARTICLE IN PRESS


5.
The system of equations of the global equilibrium is reordered to separate the ‘‘known’’ displacements from the ‘‘unknown’’displacements
¯Kkk¯Kku

¯Kuk¯Kuu

" #�

U2

..

.

Un

W2

..

.

Wn

��

U1

Unþ1

W1

Wnþ1

yUsub

Wsub

ysub

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

¼

FU2

..

.

FUn

FW2

..

.

FWn

��

0

0

0

0

0

0

0

0

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

, (43)

or in more compact form according to the notation given by Adany in [23]

¯Kkk¯Kku

¯Kuk¯Kuu

" # dk

��

du

8><>:

9>=>; ¼

qk

��

0

8><>:

9>=>;, (44)

where, dk is the vector of ‘‘known’’ displacements, du is the vector of ‘‘unknown’’ displacements, qk may be seen as the reaction forcesprovoked by the imposed ‘‘known’’ displacements, 0 is a null vector, ¯Kkk, ¯Kuk, ¯Kku and ¯Kuu are constructed from the global stiffnessmatrix ¯K of Eq. (42).

6.
The in-exsuby relationship is calculated from (44)
du ¼ �¯K�1

uu¯Kukdk, (45)

or

U1

Unþ1

W1

Wnþ1

yUsub

Wsub

ysub

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

¼ � ¯K�1

uu¯Kuk

U2

..

.

Un

W2

..

.

Wn

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;

, (46)

and in terms of the longitudinal displacements

U1

Unþ1

W1

Wnþ1

yUsub

Wsub

ysub

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

¼ � ¯K�1

uu¯Kuk

¯RUin

��

¯RWin

2664

3775

V1

V2

..

.

Vnþ1

8>>>>><>>>>>:

9>>>>>=>>>>>;

. (47)

The above equation is finally expressed as follows:

U1

Unþ1

W1

Wnþ1

yUsub

Wsub

ysub

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

¼

RU1

RUnþ1

RW1

RWnþ1

¯Ry

¯RUsub

¯RWsub

¯Rysub

26666666666666664

37777777777777775

V1

V2

..

.

Vnþ1

8>>>>><>>>>>:

9>>>>>=>>>>>;

, (48)

where RU1, RUn+1, RW1, RWn+1, ¯Ry, ¯RUsub, ¯RWsub and ¯Rysub are vectors and matrices derived from Eq. (47).

ARTICLE IN PRESS


will be further illustrated with one example. The present section will be closed with a compact equation that provides the finalrelationship between all transverse displacements and the longitudinal displacements. This equation can be obtained by re-ordering the

Eq. (48) is the last step of the procedure for the determination of the transverse–longitudinal relationships. In Section 6, this procedure

vectors and matrices shown in Eqs. (37.a), (37.b) and (48).

U

V

W

yUsub

Wsub

ysub

8>>>>>>>>>>>><>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;

¼

U1

Uin

Unþ1

V

W1

Win

Wnþ1

yUsub

Wsub

ysub

8>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>;

¼

RU1

¯RUin

RUnþ1

¯I

RW1

¯RWin

RWnþ1

¯Ry

¯RUsub

¯RWsub

¯Rysub

2666666666666666666666664

3777777777777777777777775

V1

V2

..

.

Vnþ1

8>>>>><>>>>>:

9>>>>>=>>>>>;¼

¯RU

¯I¯RW

¯Ry

¯RUsub

¯RWsub

¯Rysub

266666666666664

377777777777775

V1

V2

..

.

Vnþ1

8>>>>><>>>>>:

9>>>>>=>>>>>;

, (49)

where ¯I is the (n+1) identity matrix.

5. Constraining the longitudinal and transverse displacements of a finite element model

It should be noticed that the relationship between ui and vi is not a relationship between nodal displacements, it is actually arelationship between amplitudes of nodal displacements (see Eq. (25)). All other relationships presented in the previous section arederived from this fundamental relationship. As a consequence, Eq. (49) is also a relationship between the amplitude of the longitudinal

displacement and the amplitudes of the transverse displacements of the cross-section.The relationship between longitudinal and transverse displacements is slightly more complex than Eq. (49), due to the fact that U, V, W

and y are function of the longitudinal coordinate y. The relationship that takes into account this variation of the displacements in the y

direction has to be derived to correctly constrain the finite element mesh.The starting point for the calculation of this final relationship is Eqs. (5.a), (5.b) and (5.c), which is slightly modified because now nodal

displacements are considered

UðyÞ ¼ U sinrpy

L

� �, (50.a)

VðyÞ ¼rpL

V cosrpy

L

� �, (50.b)

WðyÞ ¼ W sinrpy

L

� �, (50.c)

yðyÞ ¼ y sinrpy

L

� �. (50.d)

U, V, W and y are the vectors that contain the amplitudes of the nodal displacements. These equations will be used to constrain thenodes corresponding to the longitudinal edges of the finite element mesh (the natural nodes of GBT). For the nodes located at the mid-point of the cross-section elements (the sub-nodes of GBT), similar equations are used

UsubðyÞ ¼ Usub sinrpy

L

� �, (51.a)

WsubðyÞ ¼ Wsub sinrpy

L

� �, (51.b)

ysubðyÞ ¼ ysub sinrpy

L

� �. (51.c)

All vectors of displacement amplitudes of the above equations (U, V, W, y Usub, Wsub and), can be expressed in terms of the vector oflongitudinal displacements V following Eq. (49). Therefore, combining Eqs. (49), (50.a), (50.b), (50.c), (50.d), (51.a), (51.b), (51.c) the final

ARTICLE IN PRESS


relationship is obtained

UðyÞ

VðyÞ

WðyÞ

yðyÞUsubðyÞ

WsubðyÞ

ysubðyÞ

8>>>>>>>>>>>><>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;

¼

¯RU sinrpy

L

� �¯I rp

L cosrpy

L

� �¯RW sin

rpy

L

� �¯Ry sin

rpy

L

� �¯RUsub sin

rpy

L

� �¯RWsub sin

rpy

L

� �¯Rysub sin

rpy

L

� �

26666666666666666666664

37777777777777777777775

V1

V2

..

.

Vnþ1

8>>>>><>>>>>:

9>>>>>=>>>>>;

. (52)

This is the equation used to constrain the finite element mesh. The constraining procedure is similar to the one shown in Section 3. Thefirst step is, again, to solve the GBT eigenvalue problem (13) to obtain the longitudinal displacements corresponding to a pure distortionalmode, Vm. The next step is to choose the ns regularly spaced cross-sections where the constraining equations will be applied.Subsequently, the longitudinal and transverse displacements of the natural nodes and sub-nodes of the cross-section are forced to moveaccording to the m mode of buckling by introducing Vm in (52)

DCons;s ¼

UmðysÞ

VmðysÞ

WmðysÞ

ymðysÞ

UmsubðysÞ

WmsubðysÞ

ym

subðysÞ

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

¼

¯RU sinrpys

L

� �¯I rp

L cosrpys

L

� �¯RW sin

rpys

L

� �¯Ry sin

rpys

L

� �¯RUsub sin

rpys

L

� �¯RWsub sin

rpys

L

� �¯Rysub sin

rpys

L

� �

26666666666666666666664

37777777777777777777775

Vm1

Vm2

..

.

Vmnþ1

8>>>>><>>>>>:

9>>>>>=>>>>>;

V1s ¼¯RsV

mV1s, (53)

where ys is the longitudinal coordinate of one of the ns constrained sections, and V1s is the longitudinal displacement of the edge of theconstrained section, which is chosen to be the unknown degree of freedom. This is the constraining equation of the natural nodes and sub-

nodes located in cross-section s. It is equivalent to Eq. (16) used to constrain the longitudinal degrees of freedoms in Section 3, but nowextended to the transverse displacements of nodes and sub-nodes.

The remaining steps of the constraining procedure are the same as the last steps performed in Section 3 (Eqs. (17)–(20)). Aconstraining matrix is defined that contains the constraining relations of all ns cross-sections

¯Rm¼

¯R1Vm

0 � � � 0 � � � 0

0 ¯R2Vm� � � 0 � � � 0

..

. ... . .

. ... ..

.

0 0 � � � ¯RsVm� � � 0

..

. ... ..

. . .. ..

.

0 0 � � � 0 � � � ¯RnsVm

2666666666664

3777777777775

. (54)

This matrix provides the relationship between the vector of constrained degrees of freedom, and the vector of unknown longitudinal

degrees of freedom

DmCons ¼

¯R1Vm

0 � � � 0 � � � 0

0 ¯R2Vm� � � 0 � � � 0

..

. ... . .

. ... ..

.

0 0 � � � ¯RsVm� � � 0

..

. ... ..

. . .. ..

.

0 0 � � � 0 � � � ¯RnsVm

2666666666664

3777777777775

V11

V12

..

.

V1s

..

.

V1ns

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;¼ ¯R

mV1. (55)

The constraining matrix ¯Rm

is introduced in ¯<m

(Eq. (19)) and, finally, the constrained eigenvalue problem is solved (Eq. (20)) to obtainthe pure GBT buckling load and mode of deformation.

ARTICLE IN PRESS

b =

90

mm

b = 30 mm

b =

5 m

m

t = 1 mm

f

w

s

12

3

45

αs = 90°αf = 180°

αw = 90°

Fig. 9. Geometry of the cross-section S1.


6. Illustrative example

This section presents the calculation of the pure distortional buckling load of a uniformly compressed lipped channel member. Theexample will illustrate the theoretical concepts discussed above.

The cross-section shown in Fig. 9 is chosen for the analysis because it has already been calculated in [4] by means of GBT-basedformulas and, consequently, it will be easy to verify the accuracy of the result obtained herein.

Loads are determined for the buckling of the member in one-half sine wave, considering simple supports withrespect to the distortional mode at both end sections. This was the way pure distortional loads were usually calculatedwhen GBT was derived, and it is also the way the cFSM is applied in CUFSM. Buckling in one-half wave is also considered in thisexample and in Section 7, where the proposed constraining procedure is verified. This will be the first step of the verification of theprocedure, as it was for the other two existing methods. In future investigations, members with more complex boundary conditions willbe analysed.

It is underlined that no specific software has been developed to apply the new procedure. All calculations are performed usingwidespread computer software:

1.
Maple is used for the calculation of the constraint equations. 2. Ansys is used to perform the constrained linear buckling analysis.
The theoretical procedure shown in the previous sections is not strictly followed. Small changes are introduced in the second step,when the finite element model is constrained. The users of Ansys do not directly constrain the finite element stiffness matrices. They onlyhave to introduce the constraining relationships in a particular way, and the constraining operation is automatically performed by Ansys.Section 6.2 will further illustrate this point.

6.1. Calculation of the constraining displacement factors

Fig. 10a shows the nodes and elements that have to be considered for the calculation of the longitudinal components of the pure GBTdeformation modes, Vm; while Fig. 10b shows the sub-nodes and sub-elements that are used in the calculation of the in-exsubyrelationship.

The geometric properties of the cross-section are presented in Fig. 9 and in Table 1. The calculation is performed for a 150 mm longmember, and the material properties considered are: E ¼ 210 000 MPa and n ¼ 0.3.

ARTICLE IN PRESS

Table 1Main dimensions of the cross-section.

Element t (mm) b (mm) a

1 1 5 90

2 1 30 180

3 1 90 270

4 1 30 0

5 1 5 90

t: thickness, b: width, a: element inclination angle with respect to the global X-axis.

1

23

4 5

6

1

2

8

3

4

5

76

sub 2

sub 3

sub 4

1

23

4 5

6

1

2

3

4

5

Fig. 10. GBT discretization for the calculation of (a) longitudinal displacements, and (b) in-exsuby relations.


The following steps should be performed to obtain the constraint equations:

1.
The calculation of matrices ¯C and ¯B of the equilibrium Eq. (12) is performed following the procedure given in Ref. [1].
¯C ¼

5

3

5

60 0 0 0

5

6

35

35 0 0 0

0 5 40 15 0 0

0 0 15 40 5 0

0 0 0 535

3

5

6

0 0 0 05

6

5

3

26666666666666664

37777777777777775

, (56)

¯B ¼

0:0249 �0:0267 0:0038 �0:0038 0:0112 �0:0093

�0:0268 0:0289 �0:0042 0:0042 �0:0133 0:0112

0:0038 �0:0042 0:0008 �0:0008 0:0042 �0:0038

�0:0038 0:0042 �0:0008 0:0008 �0:0042 0:0038

0:0112 �0:0133 0:0042 �0:0042 0:0289 �0:0268

�0:0093 0:0112 �0:0038 0:0038 �0:0268 0:0249

2666666664

3777777775

. (57)

ARTICLE IN PRESS


2.
The eigenvalue problem (13) is solved. The Vm vectors corresponding to the symmetric (sd) and the anti-symmetric (ad) puredistortional buckling modes are obtained
Vsd¼

0:6922

�0:1436

0:0130

0:0130

�0:1436

0:6922

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

(58.a)

Vad¼

0:6868

�0:1630

0:0415

�0:0415

0:1630

�0:6868

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

(58.b)

3.
The UWin–V relations are expressed by means of matrices ¯RUin and ¯RWin of Eqs. (36.a), (36.b) or (37.a), (37.b)
¯RUin ¼

0 �0:0334 0:0334 0 0 0

0 �0:0334 0:0334 0 0 0

0 0 0 0:0334 �0:0334 0

0 0 0 0:0334 �0:0334 0

26664

37775 (59.a)

¯RWin ¼

0:2000 �0:2000 0 0 0 0

0 0 �0:0112 0:0112 0 0

0 0 �0:0112 0:0112 0 0

0 0 0 0 0:2000 �0:2000

26664

37775 (59.b)

4.
The calculation of the in-exsuby relations is performed. The 2D-beam system shown in Fig. 10b should be solved following theprocedure explained in Section 4.2. This procedure consists of several steps that are now summarised.4.1. Calculation of the stiffness matrix of the beam elements in the local coordinate system. The stiffness matrix of the first element
(element number 1 in Fig. 10b) is shown as example

¯k1 ¼

46153:84 0 0 �46153:84 0 0

0 1846:15 4615:38 0 �1846:15 4615:38

0 4615:38 15384:61 0 �4615:38 7692:30

�46153:84 0 0 46153:84 0 0

0 �1846:15 �4615:38 0 1846:15 �4615:38

0 4615:38 7692:30 0 �4615:38 15384:61

2666666664

3777777775

. (60)

4.2. The coordinate transformation matrices ¯T are calculated and the element stiffness matrices are expressed in the global coordinatesystem. The transformation matrix of the first element is

T1 ¼

cos ða1Þ sin ða1Þ 0 0 0 0

� sin ða1Þ cos ða1Þ 0 0 0 0

0 0 1 0 0 0

0 0 0 cos ða1Þ sin ða1Þ 0

0 0 0 � sin ða1Þ cos ða1Þ 0

0 0 0 0 0 1

2666666664

3777777775¼

0 1 0 0 0 0

�1 0 0 0 0 0

0 0 1 0 0 0

0 0 0 0 1 0

0 0 0 �1 0 0

0 0 0 0 0 1

2666666664

3777777775

, (61)

and the resultant stiffness matrix in the global coordinate system

K1 ¼

1846:15 0 �4615:38 �1846:15 0 �4615:38

0 46153:84 0 0 �46153:84 0

�4615:38 0 15384:61 4615:38 0 7692:30

�1846:15 0 4615:38 1846:15 0 4615:38

0 �46153:84 0 0 46153:84 0

�4615:38 0 7692:30 4615:38 0 15384:61

2666666664

3777777775

. (62)

4.3. The element stiffness matrices are assembled to obtain the global stiffness matrix. For space reasons the global matrix of thisexample is not included in the present section. It can be seen in the Appendix (Table A1).

ARTICLE IN PRESS


4.4. The components of the global stiffness matrix are reordered so that the ‘‘known’’ displacements are separated from the‘‘unknown’’ displacements. The global matrix that results from this operation can be decomposed into the four sub-matricesshown in Eq. (43). The first one is

¯Kkk ¼

17230:77 0 0 0 0 0 0 0

0 15387:15 0 0 0 0 0 0

0 0 15387:15 0 0 0 0 0

0 0 0 17230:77 0 0 0 0

0 0 0 0 46222:22 0 0 0

0 0 0 0 0 5196:58 0 0

0 0 0 0 0 0 5196:58 0

0 0 0 0 0 0 0 46222:22

266666666666664

377777777777775

. (63)

The other three sub-matrices, which are rather larger, can be consulted in the Appendix (Tables A2 and A3, ¯Kku ¼¯K

T

uk).4.5. The next step is to determine the in-exsuby relationship applying Eq. (45). See the result in the Appendix (Table A4)4.6. The UWin–V relations are introduced in (47) and, as a consequence, the ‘‘unknown’’ transverse displacements are expressed in

terms of the longitudinal displacements

U1

Unþ1

W1

Wnþ1

y1

y2

y3

y4

y5

y6

Usub2

Usub3

Usub4

Wsub2

Wsub3

Wsub4

ysub2

ysub3

ysub4

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

¼

0:0382 �0:0719 0:0359 �0:0026 0:0022 �0:0018

�0:0018 0:0022 �0:0026 0:0359 �0:0719 0:0382

0:2000 �0:2000 0 0 0 0

0 0 0 0 0:2000 �0:2000

0:00764 �0:00771 0:00052 �0:00052 0:00044 �0:00036

0:00764 �0:00771 0:00052 �0:00052 0:00044 �0:00036

0:00472 �0:00458 0:00007 �0:00007 �0:00088 0:00072

�0:00072 0:00088 0:00007 �0:00007 0:00458 �0:00472

0:00036 �0:00044 0:00052 �0:00052 0:00771 �0:00764

0:00036 �0:00044 0:00052 �0:00052 0:00771 �0:00764

0 �0:0334 0:0334 0 0 0

0:0614 �0:0780 0:0167 0:0167 �0:0780 0:0614

0 0 0 0:0334 �0:0334 0

0:0891 �0:0883 �0:0073 0:0073 �0:0049 0:0041

0 0 �0:0112 0:0112 0 0

�0:0041 0:0049 0:0073 0:0073 0:0883 �0:0891

0:00691 �0:00693 0:00041 �0:00041 0:00011 �0:00009

�0:00100 0:00148 �0:00059 0:00059 �0:00148 0:00100

0:00009 �0:00011 0:00041 �0:00041 0:00693 �0:00691

2666666666666666666666666666666666666666664

3777777777777777777777777777777777777777775

V1

V2

V3

V4

V5

V6

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

(64)

5.
At this point, the constraints presented in Eq. (49) can be obtained from (59.a), (59.b) and (64)
¯RU ¼

3:82 �7:19 3:59 �0:26 0:22 �0:18

0 �3:34 3:34 0 0 0

0 �3:34 3:34 0 0 0

0 0 0 3:34 �3:34 0

0 0 0 3:34 �3:34 0

�0:18 0:22 �0:26 3:59 �7:19 3:82

2666666664

3777777775

10�2, (65.a)

¯RW ¼

20 �20 0 0 0 0

20 �20 0 0 0 0

0 0 �1:12 1:12 0 0

0 0 �1:12 1:12 0 0

0 0 0 0 20 �20

0 0 0 0 20 �20

2666666664

3777777775

10�2, (65.b)

¯Ry ¼

7:64 �7:71 0:52 �0:52 0:44 �0:36

7:64 �7:71 0:52 �0:52 0:44 �0:36

4:72 �4:58 0:07 �0:07 �0:88 0:72

�0:72 0:88 0:07 �0:07 4:58 �4:72

0:36 �0:44 0:52 �0:52 7:71 �7:64

0:36 �0:44 0:52 �0:52 7:71 �7:64

2666666664

3777777775

10�3, (65.c)

ARTICLE IN PRESS

Table 2Constraining displacement factors.

Node Uds Vds Wds yds

1 0.0356 r �p/L � 0.692 0.1671 0.0061

2 0.0052 �r �p/L �0.143 0.1671 0.0061

Sub2 0.0052 – 0.0779 0.0057

3 0.0052 r �p/L � 0.013 0 0.0046

Sub3 0.1078 – 0 0

4 0.0052 r �p/L � 0.013 0 �0.0046

Sub 5 0.0052 – 0.0779 �0.0057

5 0.0052 �r �p/L �0.143 �0.1671 �0.0061

6 0.0356 r �p/L � 0.692 �0U1671 �0.0061


¯RUsub ¼

0 �3:34 3:34 0 0 0

6:14 �7:80 1:67 1:67 �7:80 6:14

0 0 0 3:34 �3:34 0

264

37510�2, (65.d)

¯RWsub ¼

8:91 �8:83 �0:73 0:73 �0:49 0:41

0 0 �1:12 1:12 0 0

�0:41 0:49 0:73 0:73 8:83 �8:91

264

37510�2, (65.e)

¯Rysub ¼

6:91 �6:93 0:41 �0:41 0:11 �0:09

�1 1:48 �0:59 0:59 �1:48 1

0:09 �0:11 0:41 �0:41 6:93 �6:91

264

37510�3. (65.f)

6.
Finally, if the longitudinal displacements Vsd determined in the second step are introduced in (53), the constraint equations for thesymmetric distortional mode are obtained. Table 2 shows the constraining factors which result from (53). These factors will be used inthe next section to constrain the finite element mesh.
From now on the calculations are only performed for the sd mode.

6.2. Constrained linear buckling analysis

Figs. 11 and 12 show the finite element mesh and boundary conditions of the 150 mm long member that is analysed in the example.Only half of the member is included in the model. Symmetry boundary conditions are applied at one end, while at the other end the

simple support with respect to the distortional mode is considered. Fig. 11 shows which degrees of freedom have to be constrained at theend cross-sections. It is pointed out that it is very important to allow the warping of the hinged cross-section, if the one-half sine wavedistortional deformation has to be obtained.

At the hinged cross-section, the nodes located in the web are forced to have identical U displacements (see Fig. 11). Thesedisplacements are coupled because the local buckling of this element of the end cross-section has to be avoided. The same operation isperformed at the nodes of the flange stiffeners, where it is also necessary to avoid local buckling.

The linear buckling analyses are performed in two steps. In the first step, a uniformly distributed normal pressure is appliedat the hinged end of the member, and a linear analysis is carried out. The geometric stiffness matrix is calculated from the results ofthis analysis.

In the second step, the constraints are introduced and the pure elastic distortional loads are determined solving the eigenvalueproblem (20). However, when using Ansys it is not necessary to define matrix ¯R

mand to modify the original elastic stiffness and

geometric matrices. This is done automatically by the program. The user only has to provide the constraining relationships according tothe following procedure:

1.
The constraint relations for the longitudinal displacements Vis are defined. According to Eqs. (53) or (16)
Vis ¼ðrp=LÞVds

i cos ððrp=LÞysÞ

ðrp=LÞVds1 cos ððrp=LÞysÞ

V1s ¼Vds

i

Vds1

V1s ¼Vds

i

0:692V1s, (66)

where Vids is the constraining factor for the longitudinal displacement of node i, that can be obtained from Table 2, and V1s is the

unknown degree of freedom of section s (see equation (53)).
2. The constraint relations for the transverse displacements are set
Uis ¼Uds

i sin ððrp=LÞysÞ

ðrp=LÞVds1 cos ððrp=LÞysÞ

V1s ¼Uds

i

ðrp=LÞVds1

tanrpL

ys

� �V1s ¼

Udsi

0:0145tan ð0:0209ysÞV1s, (67.a)

and in a similar way

Wis ¼Wds

i

0:0145tan ð0:0209ysÞV1s, (67.b)

ARTICLE IN PRESS

Fig. 12. Constrained finite element model.

Symmetry boundaryconditions

Transverse displacement.

Longitudinal displacement.

Coupled displacements.

Rotation.

Distortional hinge

Fig. 11. Boundary conditions applied to the finite element model.


yis ¼yds

i

0:0145tan ð0:0209ysÞV1s. (67.c)

where and Uids , Wi

ds and yids are obtained from Table 2, and ys is the longitudinal coordinate of the cross-section s. If the global

coordinate system of the finite element model is not the same as the coordinate system used when defining the constraining factors, itshould be checked whether the right-hand side of equations (67.a), (67.b) and (67.c) needs to be multiplied by (�1).

These constraining equations should be determined for the ns constrained cross-sections of the member. Eqs. (67.a), (67.b), (67.c) are
valid for natural nodes and sub-nodes.
In the beginning, the idea was to apply this constraining procedure to pallet-rack columns, which show regularly spaced perforations.In Spain, the pattern of perforations of these columns is usually repeated every 50 mm in the longitudinal direction. It was believed thatto constrain two sections of the perforation pattern would lead to accurate values of pure distortional loads. For this reason, theconstraints are applied every 25 mm in the members of this investigation (ys ¼ 25 mm, 50 mm,y in (67.a), (67.b), (67.c)). Later on, seeSection 7, it was verified that most of the times accurate results are obtained if this value is used.

Accuracy problems were only found for members shorter than 200 mm. For this reason, when the length of the member was short, thedistance between constrained cross-sections was reduced to 10 mm. This is the case of the 150 mm long member that is calculated inthis example.

A complete investigation on the sensitivity of the results with respect to the distance between constrained cross-sections should beperformed in the future. For the moment, the chosen values are good for the members analysed herein. It is also pointed out that it has beenverified that the use of shorter distances than 25 mm (or 10 mm for members shorter than 200 mm), do not significantly change the results.

Finally, once the constraints have been applied, the eigenvalue problem (20) is solved and the distortional buckling load correspondingto the pure distortional mode is obtained. The result for the member analysed in this example

ND ¼ 32662 N (68)

The accuracy of this result is discussed in the next section.

ARTICLE IN PRESS

Table 3Mean and standard deviation of the ratio between the cFEM load and the GBTb load for symmetric distortional modes.

Section bf bs as GBT Longitudinal constraints GBT longitudinal constraints +FEM-based transverse

constraints

GBT longitudinal constraints +GBT transverse

constraints

All lengths LoLcrD LXLcrD All lengths LoLcrD LXLcrD All lengths LoLcrD LXLcrD

Mean Dev. Mean Dev. Mean Dev. Mean Dev. Mean Dev. Mean Dev. Mean Dev. Mean Dev. Mean Dev.

1 30 5 90 0.87 0.18 0.61 0.25 0.95 0.02 0.96 0.10 0.80 0.08 1.00 0.03 1.03 0.02 1.00 0.00 1.04 0.01

2 60 5 90 0.91 0.20 0.69 0.27 1.00 0.01 0.99 0.05 0.91 0.03 1.02 0.02 1.01 0.03 0.97 0.02 1.02 0.00

3 90 5 90 0.88 0.21 0.70 0.27 0.99 0.00 0.98 0.05 0.92 0.04 1.01 0.01 0.99 0.03 0.96 0.03 1.01 0.00

4 30 10 90 0.83 0.23 0.48 0.20 0.94 0.03 0.93 0.14 0.73 0.05 1.00 0.07 0.97 0.09 0.86 0.09 1.02 0.01

5 60 10 90 0.85 0.26 0.64 0.31 0.99 0.02 0.96 0.06 0.89 0.03 1.00 0.02 0.98 0.04 0.93 0.03 1.01 0.01

6 90 10 90 0.83 0.28 0.61 0.33 0.98 0.02 0.95 0.08 0.87 0.05 1.00 0.03 0.96 0.05 0.91 0.04 1.00 0.01

7 30 5 �90 0.87 0.19 0.58 0.26 0.95 0.02 0.95 0.11 0.77 0.08 1.01 0.04 1.03 0.03 0.99 0.01 1.05 0.02

8 60 5 �90 0.91 0.20 0.69 0.28 1.00 0.01 0.99 0.05 0.91 0.03 1.02 0.01 1.00 0.03 0.96 0.02 1.02 0.00

9 90 5 �90 0.89 0.20 0.61 0.25 0.99 0.01 0.98 0.05 0.90 0.02 1.00 0.02 0.99 0.03 0.95 0.02 1.01 0.00

10 30 10 �90 0.72 0.35 0.33 0.25 0.95 0.04 0.90 0.19 0.67 0.09 1.02 0.07 0.97 0.09 0.86 0.08 1.02 0.01

11 60 10 �90 0.78 0.34 0.53 0.37 0.99 0.02 0.95 0.09 0.86 0.07 1.01 0.02 0.96 0.07 0.89 0.07 1.01 0.01

12 90 10 �90 0.77 0.34 0.52 0.37 0.98 0.02 0.96 0.07 0.88 0.04 1.00 0.03 0.94 0.10 0.87 0.11 1.00 0.01

All 0.84 0.25 0.58 0.28 0.98 0.03 0.96 0.09 0.85 0.08 1.00 0.03 0.99 0.06 0.92 0.07 1.02 0.02

Fig. 13. Pure distortional buckling mode.


It should be pointed out that the pure distortional buckling mode can be usually found among the first modes given by Ansys, and thatis easy to identify (see Fig. 13).

7. Verification of the calculation procedure

The calculation procedure presented in the previous section is applied to the cross-sections listed in Table 3, which have already beenanalysed by Silvestre and Camotim in [5]. Symmetric and anti-symmetric distortional loads are calculated for different buckling lengths.The results obtained are compared to the loads given by the GBT-based formula proposed by the mentioned authors in [4], and to theloads given by the CUFSM program of Adany and Schafer [21,22].

Actually, the three calculation procedures presented in Section 3 are evaluated. A brief discussion on the accuracy of these proceduresis included, because it gives interesting information on how the different types of constraints affect the uncoupling procedure.

The results are presented by means of an example. The cross-section S1 of Fig. 9, which is the cross-section number 1 in Tables 3 and 4,is again chosen to carry out the example. The results of the calculations on the other cross-sections are also taken into account to fullyevaluate the procedures.

The first step was to determine the distortional buckling loads of S1 by means of the GBT-based formulas (GBTb), and by means ofCUFSM. Fig. 14 shows the results obtained for the symmetric distortional mode of buckling. The third curve included in this graphcorresponds to the elastic buckling loads obtained by means of the finite element method without any constraint, i.e., all modescombined.

As discussed in Section 3, in the beginning, the idea was to obtain the pure distortional deformations by only constraining thelongitudinal displacements Vi of the natural nodes (i.e., by applying only constraining Eqs. (66) to these nodes). The main reasonwhy this decision was taken was to avoid the complex formulation of the longitudinal–transverse displacement relationships presented inSection 4.

ARTICLE IN PRESS

Table 4Mean and standard deviation of the ratio between the cFEM load and the GBTb load for asymmetric distortional modes.

Section bf bs as GBT Longitudinal constraints GBT longitudinal constraints+FEM-based transverse

constraints

GBT longitudinal constraints+GBT transverse

constraints

All lengths LoLcrD LXLcrD All lengths LoLcrD LXLcrD All lengths LoLcrD LXLcrD

Mean Dev. Mean Dev. Mean Dev. Mean Dev. Mean Dev. Mean Dev. Mean Dev. Mean Dev. Mean Dev.

1 30 5 90 0.96 0.05 0.84 - 0.98 0.01 0.98 0.02 0.94 - 0.99 0.02 1.00 0.02 0.97 - 1.00 0.02

2 60 5 90 0.91 0.13 0.78 0.17 0.98 0.01 0.97 0.05 0.91 0.02 1.00 0.02 1.00 0.03 0.95 0.02 1.01 0.02

3 90 5 90 0.88 0.20 0.62 0.24 0.97 0.02 0.97 0.09 0.85 0.03 1.02 0.05 0.99 0.03 0.94 0.02 1.01 0.01

4 30 10 90 0.87 0.21 0.71 0.27 0.98 0.01 0.94 0.08 0.86 0.08 0.99 0.02 0.95 0.08 0.87 0.07 1.00 0.02

5 60 10 90 0.79 0.29 0.57 0.31 0.97 0.02 0.94 0.09 0.84 0.06 1.00 0.04 0.96 0.07 0.88 0.05 1.01 0.02

6 90 10 90 0.74 0.34 0.47 0.35 0.96 0.03 0.95 0.09 0.84 0.04 1.00 0.04 0.97 0.06 0.90 0.04 1.00 0.02

7 30 5 �90 0.95 0.05 0.85 - 0.97 0.05 0.98 0.03 0.94 - 0.99 0.02 1.00 0.02 0.97 - 1.00 0.02

8 60 5 �90 0.91 0.13 0.78 0.17 0.98 0.01 0.97 0.04 0.91 0.02 1.00 0.02 1.00 0.03 0.95 0.02 1.01 0.02

9 90 5 �90 0.88 0.19 0.70 0.25 0.98 0.01 0.97 0.05 0.90 0.03 1.00 0.02 0.99 0.03 0.96 0.03 1.01 0.01

10 30 10 �90 0.86 0.21 0.71 0.28 0.97 0.01 0.94 0.05 0.88 0.02 0.98 0.02 0.95 0.06 0.89 0.03 1.00 0.02

11 60 10 �90 0.81 0.27 0.57 0.31 0.97 0.02 0.94 0.08 0.85 0.05 0.99 0.04 0.96 0.07 0.88 0.04 1.00 0.03

12 90 10 �90 0.76 0.33 0.55 0.36 0.97 0.01 0.92 0.10 0.84 0.10 0.99 0.02 0.95 0.10 0.87 0.10 1.01 0.02

All 0.85 0.23 0.64 0.27 0.97 0.02 0.95 0.07 0.87 0.06 1.00 0.03 0.97 0.06 0.91 0.06 1.01 0.02

0

20000

40000

60000

80000

100000

120000

140000

160000

180000

200000

0L (mm)

N (N

)

GBTbCUFSMFEM

100 200 300 400 500 600 700 800

Fig. 14. Distortional buckling loads of S1 determined via GBT, cFSM and FEM.


All cross-sections of Table 3 were calculated with the longitudinal constraints. Fig. 15 (cFEM curve) shows the result for S1. It can beobserved that distortional loads are rather accurate when the length of the member is higher than the critical buckling length. On thecontrary, for shorter lengths, buckling loads are the same as the previously calculated FEM loads. In this case the distortional mode is stillcoupled with local buckling.

The same occurred to the other cross-sections for both symmetric and anti-symmetric distortional modes. This can be seen in thecolumns titled GBT longitudinal constraints of Tables 3 and 4, which show the mean value and standard deviation of the ratio between thecFEM loads and the GBTb loads for all calculated members. See also Fig. 18.

At this point, it was decided to constrain the transverse deformation of the plates of the cross-section to eliminate the local-distortionalinteraction. For this reason, sub-nodes were placed and constrained at the mid-point of the cross-section elements (Fig. 10b). However, thelongitudinal–transverse displacement relationship was not initially chosen to impose the constraints. Instead, the constraining equations forthe transverse displacements of nodes and sub-nodes were deduced from the just calculated FEM results (GBT longitudinal constraints results).Since good approximations to the distortional buckling loads had been obtained for the critical distortional lengths of the members, thecritical mode was chosen as a pattern of deformation for the other lengths. Constraining factors for the transverse displacements, similar tothe ones shown in Table 2, were deduced from the distortional deformation mode of the critical length. Afterwards, these factors,conveniently modified to take into account the effect of L (rp/L in Table 2), were used for the other buckling lengths.

Applying these FEM-based transverse constraints the results improved. For the members whose length is shorter than the criticallength, the local-distortional buckling interaction is partially removed, as can be seen in Figs. 16 and 18, and Tables 3 and 4 (results incolumns GBT longitudinal constraints+FEM-based transverse constraints). However, the distortional buckling loads are still slightly lowerthan the GBTb loads. For the members with lengths higher than the critical length, the introduction of the FEM-based transverse

constraints results in very good approximations to the distortional loads given by the GBTb formulas.In the end, as the results were still not completely accurate, it was decided to incorporate all longitudinal–transverse relations of

Sections 4 and 5. It seems that this reduced to a very low degree the local-distortional interaction, and good results are obtained for

ARTICLE IN PRESS

0

20000

40000

60000

80000

100000

120000

140000

160000

180000

200000

0L (mm)

N (N

)

100 200 300 400 500 600 700 800

GBTbCUFSMFEMcFEM

Fig. 16. Calculation performed with Lon-GBT + Trans-FEM based constraints.

0

20000

40000

60000

80000

100000

120000

140000

160000

180000

200000

0L (mm)

N (N

)

100 200 300 400 500 600 700 800

GBTbCUFSMFEMcFEM

Fig. 17. Calculation performed with Lon.-GBT + Tran.-GBT constraints.

0

20000

40000

60000

80000

100000

120000

140000

160000

180000

200000

0L (mm)

N (N

)

100 200 300 400 500 600 700 800

GBTbCUFSMFEMcFEM

Fig. 15. Calculation performed with Lon-GBT constraints.


ARTICLE IN PRESS

0

0.2

0.4

0.6

0.8

1

1.2

0L / Lcr d

CFE

M N

d / G

BTb

Nd

Lon.-GBT+Tran.-FEM+Tran.-GBT

1 2 3 4 5 6

Fig. 18. Ratios between the cFEM distortional load and the GBTb distortional load.


members whose length is shorter than the critical length (Figs. 17 and 18 and columns GBT longitudinal constraints+GBT transverse

constraints of Tables 3 and 4) The results for the members with lengths higher than the critical length were also good and similar to theresults obtained when applying the FEM-based transverse constraints.

8. Conclusions

The paper has presented a combined GBT–FEM procedure for the calculation of pure distortional buckling loads of open thin-walledmembers. The bases of the combined procedure are the same as the bases of the constrained finite strip method. In both methods, theidea is to force the member to buckle in a GBT buckling mode by constraining degrees of freedom of the mesh.

The main contribution of the investigation is the translation of the GBT and cFSM concepts into FEM for the distortional mode ofbuckling. This is a small step in the research on GBT, which has consisted in extending the cFSM formulation of the cross-sectionconstraints in the longitudinal direction of the member.

It has been shown that it is necessary to constrain the longitudinal and transverse displacements of natural nodes, together with thetransverse displacements of sub-nodes, to obtain accurate values of distortional buckling loads. It has also been proved that the constraintsdo not have to be applied to all these nodes of the mesh. The constraining equations can be applied at regularly spaced cross-sections.This results in some advantages when the finite element model to constrain has an irregular mesh. It has also been verified that thecomputing time decreases when the number of constrained cross-sections is reduced.

There are several other ways the constraining operation can be performed, some of them are discussed in this section because theymay be subject of future research:

1.
The uncoupling procedure presented was the result of an investigation where, in the beginning, only the longitudinal displacements ofthe natural nodes were constrained. It should be noticed that the longitudinal constraints do not depend on the y coordinate. For thisreason, different cross-sections of the member were independently constrained and, as a consequence, there was one V1s unknowndisplacement per constrained cross-section. Afterwards, it was verified that it is necessary to constrain the transverse degrees offreedom to obtain accurate results. The transverse constraints were introduced together with a sinusoidal function of y that provides arelationship between the constraints of the different cross-sections. Consequently, instead of using one V1s unknown displacement percross-section, it could have been used only one V1 unknown displacement for the whole member.
2.
The transverse constraints could have been applied without the sinusoidal function. This would have avoided the longitudinalrelationship and, consequently, the cross-sections could have been independently constrained. Indeed, some calculations were donewithout this longitudinal relationship, and the results were good for members buckling in one-half sine wave.
3.
An alternative way of determining the transverse constraints has also been presented, which is based on the results of the finiteelement analysis of the critical length of the member. The results obtained with the FEM-based transverse constrains are similar to theGBT results, and the errors of accuracy of these calculations are always on the safe side.
Finally, it is pointed out that more research is currently being done by the authors on the GBT–FEM combined procedure. The first
objective is to extend the constraining procedure to the local and global instabilities. This investigation will also involve a sensitivityanalysis of the effect of the distance between constrained cross-sections and the number of constrained nodes per section (number ofsub-nodes) on the resultant buckling load. Future research will be also focused on the analysis of members subject to bending andmembers with more complex boundary conditions. Finally, it will also be investigated whether the proposed procedure is useful for theanalyses of members with regularly spaced perforations, such as the ones used in pallet-rack structures.
Appendix

Tables A1–A4 show the matrices mentioned in Section 6 that were not included in the main text due to space reasons.

ARTIC

LEIN

PRESS

Table A1Global matrix.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 1846.15 0 �4615.38 �1846.15 0 �46135.38 0 0 0 0 0 0 0 0

2 0 46153.85 0 0 �46153.85 0 0 0 0 0 0 0 0 0

3 �4615.38 0 15384.62 4615.38 0 7692,31 0 0 0 0 0 0 0 0

4 �1846.15 0 4615.38 17230.77 0 46135.38 �15384.62 0 0 0 0 0 0 0

5 0 �46153.85 0 0 46222.22 �512.82 0 �68.38 �512.82 0 0 0 0 0

6 �4615.38 0 7692.31 4615.38 �512.82 20512.82 0 512.82 2564.10 0 0 0 0 0

7 0 0 0 �15384.62 0 0 30769.23 0 0 �15384.62 0 0 0 0

8 0 0 0 0 �68.38 512.82 0 136.75 0 0 �68.38 �512.82 0 0

9 0 0 0 0 �512.82 2564.10 0 0 10256.41 0 512.82 2564.10 0 0

10 0 0 0 0 0 0 �15384.62 0 0 15387.15 0 56.98 �2.53 0

11 0 0 0 0 0 0 0 �68.38 512.82 0 5196.58 512.82 0 �5128.21

12 0 0 0 0 0 0 0 �512.82 2564.10 56.98 512.82 6837.61 �56.98 0

13 0 0 0 0 0 0 0 0 0 �2.53 0 �56.98 5.06 0

14 0 0 0 0 0 0 0 0 0 0 �5128.21 0 0 10256.41

15 0 0 0 0 0 0 0 0 0 56.98 0 854.70 0 0

16 0 0 0 0 0 0 0 0 0 0 0 0 �2.53 0

17 0 0 0 0 0 0 0 0 0 0 0 0 0 �5128.21

18 0 0 0 0 0 0 0 0 0 0 0 0 56.98 0

19 0 0 0 0 0 0 0 0 0 0 0 0 0 0

20 0 0 0 0 0 0 0 0 0 0 0 0 0 0

21 0 0 0 0 0 0 0 0 0 0 0 0 0 0

22 0 0 0 0 0 0 0 0 0 0 0 0 0 0

23 0 0 0 0 0 0 0 0 0 0 0 0 0 0

24 0 0 0 0 0 0 0 0 0 0 0 0 0 0

25 0 0 0 0 0 0 0 0 0 0 0 0 0 0

26 0 0 0 0 0 0 0 0 0 0 0 0 0 0

27 0 0 0 0 0 0 0 0 0 0 0 0 0 0

15 16 17 18 19 20 21 22 23 24 25 26 27

1 0 0 0 0 0 0 0 0 0 0 0 0 0

2 0 0 0 0 0 0 0 0 0 0 0 0 0

3 0 0 0 0 0 0 0 0 0 0 0 0 0

M.

Ca

safo

nt

eta

l./

Th

in-W

alled

Structu

res4

7(2

00

9)

70

1–

72

97

25

ARTIC

LEIN

PRESS

Table A1 (continued )

15 16 17 18 19 20 21 22 23 24 25 26 27

4 0 0 0 0 0 0 0 0 0 0 0 0 0

5 0 0 0 0 0 0 0 0 0 0 0 0 0

6 0 0 0 0 0 0 0 0 0 0 0 0 0

7 0 0 0 0 0 0 0 0 0 0 0 0 0

8 0 0 0 0 0 0 0 0 0 0 0 0 0

9 0 0 0 0 0 0 0 0 0 0 0 0 0

10 56.98 0 0 0 0 0 0 0 0 0 0 0 0

11 0 0 0 0 0 0 0 0 0 0 0 0 0

12 854.70 0 0 0 0 0 0 0 0 0 0 0 0

13 0 �2.53 0 56.98 0 0 0 0 0 0 0 0 0

14 0 0 �5128.21 0 0 0 0 0 0 0 0 0 0

15 3418.80 �56.98 0 854.70 0 0 0 0 0 0 0 0 0

16 �56.98 15387.15 0 �56.98 �15384.62 0 0 0 0 0 0 0 0

17 0 0 5196.58 512.82 0 �68.38 512.82 0 0 0 0 0 0

18 854.70 �56.98 512.82 6837.61 0 �512.82 2564.10 0 0 0 0 0 0

19 0 �15384.62 0 0 30769.23 0 0 �15384.62 0 0 0 0 0

20 0 0 �68.38 �512.82 0 136.75 0 0 �68.38 512.82 0 0 0

21 0 0 512.82 2564.10 0 0 10256.41 0 �512.82 2564.10 0 0 0

22 0 0 0 0 �15384.62 0 0 17230.77 0 �4615.38 �1846.15 0 �4615.38

23 0 0 0 0 0 �68.38 �512.82 0 46222.22 �512.82 0 �46153.85 0

24 0 0 0 0 0 512.82 2564.10 �4615.38 �512.82 20512.82 4615.38 0 7692.31

25 0 0 0 0 0 0 0 �1846.15 0 4615.38 1846.15 0 4615.38

26 0 0 0 0 0 0 0 0 �46153.85 0 0 46153.85 0

27 0 0 0 0 0 0 0 �4615.38 0 7692.31 4615.38 0 15384.62

M.

Ca

safo

nt

eta

l./

Th

in-W

alled

Structu

res4

7(2

00

9)

70

1–

72

97

26

ARTIC

LEIN

PRESS

Table A2

Matrix ¯Kuu:

1 2 3 4 5 6 7 8 9 10

1 1846.15 0 0 0 �4615.38 �4615.38 0 0 0 0

2 0 1846.15 0 0 0 0 0 0 4615.38 4615.38

3 0 0 46153.85 0 0 0 0 0 0 0

4 0 0 0 46153.85 0 0 0 0 0

5 �4615.38 0 0 0 15384.62 7692.31 0 0 0 0

6 �4615.38 0 0 0 7692.31 20513.82 0 0 0 0

7 0 0 0 0 0 0 6837.61 0 0 0

8 0 0 0 0 0 0 0 6837.61 0 0

9 0 4615.38 0 0 0 0 0 0 20512.82 7692.31

10 0 4615.38 0 0 0 0 0 0 7692.31 15384.62

11 0 0 0 0 0 0 0 0 0 0

12 0 0 0 0 0 0 �56.98 56.98 0 0

13 0 0 0 0 0 0 0 0 0 0

14 0 0 0 0 0 512.82 �512.82 0 0 0

15 0 0 0 0 0 0 0 0 0 0

16 0 0 0 0 0 0 0 �512.82 512.82 0

17 0 0 0 0 0 2564.10 2564.10 0 0 0

18 0 0 0 0 0 0 854.70 854.70 0 0

19 0 0 0 0 0 0 0 2564.10 2564.10 0

11 12 13 14 15 16 17 18 19

1 0 0 0 0 0 0 0 0 0

2 0 0 0 0 0 0 0 0 0

3 0 0 0 0 0 0 0 0 0

4 0 0 0 0 0 0 0 0 0

5 0 0 0 0 0 0 0 0 0

6 0 0 0 512.82 0 0 2564.10 0 0

7 0 �56.98 0 �512.82 0 0 2564.10 854.70 0

8 0 56.98 0 0 0 �512.82 0 854.70 2564.10

9 0 0 0 0 0 512.82 0 0 2564.10

10 0 0 0 0 0 0 0 0 0

11 30769.23 0 0 0 0 0 0 0 0

12 0 5.06 0 0 0 0 0 0 0

13 0 0 30769.23 0 0 0 0 0 0

14 0 0 0 136.75 0 0 0 0 0

15 0 0 0 0 10256.41 0 0 0 0

16 0 0 0 0 0 136.75 0 0 0

17 0 0 0 0 0 0 10256.41 0 0

18 0 0 0 0 0 0 0 3418.80 0

19 0 0 0 0 0 0 0 0 10256,41

M.

Ca

safo

nt

eta

l./

Th

in-W

alled

Structu

res4

7(2

00

9)

70

1–

72

97

27

ARTICLE IN PRESS

Table A4

Matrix of the in-exsuby relationship.

1 2 3 4 5 6 7 8

1 1.0000 0.0112 �0.0112 0 0.1909 �0.1909 �0.0091 0.0091

2 0 �0.0112 0.0112 1.0000 0.0091 �0.0091 0.1909 �0.1909

3 0 0 0 0 1.0000 0 0 0

4 0 0 0 0 0 0 0 1.0000

5 0 0.0023 �0.0023 0 0.0382 �0.0382 �0.0018 0.0018

6 0 0.0023 �0.0023 0 0.0382 �0.0382 �0.0018 0.0018

7 0 �0.0045 0.0045 0 0.0236 �0.0236 0.0360 0.0036

8 0 �0.0045 0.0045 0 �0.0036 0.0036 �0.0236 0.0236

9 0 0.0023 �0.0023 0 0.0018 �0.0018 �0.0381 0.0381

10 0.5000 0.0023 �0.0023 0 0.0018 �0.0018 �0.0381 0.0381

11 0 0.5000 0 0 0 0 0 0

12 0 0.5000 0.5000 0 0.3068 �0.3068 0.3068 �0.3068

13 0 0 0.5000 0.5000 0 0 0 0

14 0 �0.0250 0.0250 0 0.4455 0.5545 0.0205 �0.0205

15 0 0 0 0 0 0.5000 0.5000 0

16 0 �0.0250 0.0250 0 �0.0205 0.0205 0.5545 �0.4455

17 0 0.0006 �0.0006 0 0.0345 �0.0345 �0.0005 0.0005

18 0 �0.0145 0.0145 0 �0.5000 0.0005 0.5000 �0.5000

19 0 0.0006 �0.0006 0 0.0005 �0.0005 �0.0345 0.0345

Table A3

Matrix ¯Kku:

1 2 3 4 5 6 7 8 9 10

1 �1846.15 0 0 0 4615.38 4615.38 0 0 0 0

2 0 0 0 0 0 0 56.98 0 0 0

3 0 0 0 0 0 0 0 �56.98 0 0

4 0 �1846.15 0 0 0 0 0 0 �4615.38 �4615.38

5 0 0 �46153.85 0 0 �512.82 0 0 0 0

6 0 0 0 0 0 0 512.82 0 0 0

7 0 0 0 0 0 0 0 512.82 0 0

8 0 0 0 �46153.85 0 0 0 0 �512.82 0

11 12 13 14 15 16 17 18 19

1 �15384.62 0 0 0 0 0 0 0 0

2 �15384.62 �2,53 0 0 0 0 0 56.98 0

3 0 �2.53 �15384.62 0 0 0 0 �56.98 0

4 0 0 �15384.62 0 0 0 0 0 0

5 0 0 0 �68.38 0 0 �512.82 0 0

6 0 0 0 �68.38 �5128.21 0 512.82 0 0

7 0 0 0 0 �5128.21 �68.38 0 0 512.82

8 0 0 0 0 0 �68.38 0 0 �512.82


References

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Standardisation, Brussels, 2004 (Draft).[3] Specification for the Design of Cold-Formed Steel Structural Members. Appendix 1. Design of Cold-Formed Steel Structural Members Using the Direct Strength Method.

American Iron and Steel Institute, 2004.[4] Silvestre N, Camotim D. Distortional buckling formulae for cold-formed steel C and Z-section members. Part I—derivation. Thin-Walled Structures 2004;42:1567–97.[5] Silvestre N, Camotim D. Distortional buckling formulae for cold-formed steel C and Z-section members. Part II—validation and application. Thin-Walled Structures

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[16] Papangelis JP, Hancock GJ. Computer analysis of thin-walled structural members. Computers & Structures 1995;56(1):157–76.[17] Schafer BW. CUFSM Cornell university finite strip method. Ithaca, New York: Cornell University; 2001.[18] Lau SCW, Hancock GJ. Distortional buckling formulas for channel columns. Journal of Structural Engineering, ASCE 1987;113(5):1063–78.[19] Schafer BW. Local, distortional, and Euler buckling of thin-walled columns. Journal of Structural Engineering, ASCE 2002;128(3):289–99.[20] Serrette RL, Pekoz T. Distortional buckling of thin-walled beams/panels. I: theory. Journal of Structural Engineering 1995;121(4):757–66.[21] Adany S, Schafer BW. Buckling mode decomposition of single-branched open cross-section members via finite strip method: derivation. Thin-Walled Structures

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