GBT Formulation to Analyse the Buckling Behaviour of Thin-walled

GBT formulation to analyse the buckling behaviour of thin-walled

members with arbitrarily ‘branched’ open cross-sections

P.B. Dinis, D. Camotim *, N. Silvestre

Department of Civil Engineering and Architecture, ICIST/IST, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Received 16 March 2005; received in revised form 19 August 2005; accepted 28 September 2005

Available online 14 November 2005

Abstract

This paper presents the derivation, validates and illustrates the application of a Generalised Beam Theory (GBT) formulation developed to

analyse the buckling behaviour of thin-walled members with arbitrarily ‘branched’ open cross-sections. Following a brief overview of the

conventional GBT, one addresses in great detail the modifications that must be incorporated into its cross-section analysis procedure, in order to be

able to handle the ‘branching’ points — they concern mostly issues related to (i) the choice of the appropriate ‘elementary warping functions’ and

(ii) the determination of the ‘initial flexural shape functions’. The derived formulation is then employed to investigate the local-plate, distortional

and global buckling behaviour of (i) simply supported and fixed asymmetric E-section columns and (ii) simply supported I-section beams with

unequal stiffened flanges. For validation purposes, several GBT-based results are compared with ‘exact’ values, obtained by means of finite strip

or shell finite element analyses.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: Thin-walled members; Generalised beam theory (GBT); ‘Branched’ open cross-sections; Member buckling analysis; Local-plate buckling; Distortional

buckling; Global buckling.

1. Introduction

The Generalised Beam Theory (GBT) was originally

developed by Schardt [1–3] and may be viewed as an extension

of Vlasov’s classical bar theory that incorporates genuine

folded-plate concepts and, thus, is able to take into account in-

plane (local) cross-section deformations. Moreover, the

member deformed configuration or buckling/vibration mode

is expressed as a linear combination of a set of pre-determined

cross-section deformation modes — due to this rather unique

modal nature, the application of GBT is considerably more

versatile and computationally efficient than similar finite strip

or shell finite element analyses. Indeed, it has been recently

shown that GBT provides a rather powerful, elegant and

clarifying tool to investigate a wealth of structural problems

involving thin-walled prismatic members [4,5].

For the last four decades, Schardt and his collaborators, at

the Technical University of Darmstadt, have devoted an

enormous amount of work to the development and application

0263-8231/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.tws.2005.09.005

* Corresponding author. Tel.: C351 21 8418403; fax: C351 21 8497650.

E-mail address: [email protected] (D. Camotim).

of GBT formulations. However, this work was carried out

almost exclusively in the context of the first-order, buckling

and vibration analysis of thin-walled members (i) made of

isotropic elastic materials and (ii) displaying ‘unbranched’

(mostly open) cross-sections.1 Moreover, it was rather poorly

disseminated among the English-speaking scientific and

technical communities — the vast majority of the publications

are available only in German and several of them consist of TU

Darmstadt Reports or Ph.D. Theses. These communities only

became acquainted with GBT in the 1990s, thanks to the work

of Davies and his co-workers [4,6,7], who (i) played a key role

in the dissemination of GBT around the world, (ii) applied it

extensively to investigate the buckling behaviour of cold-

formed steel members (e.g. [8–10]) and (iii) provided strong

evidence that GBT is a valid and often advantageous

alternative to fully numerical finite element or finite strip

analyses.

Quite recently (i.e. in the last 4–5 years), GBT has attracted

the attention of several researchers, which led to the

development of a number of new formulations and appli-

cations. In this regard, Silvestre and Camotim deserve to be

Thin-Walled Structures 44 (2006) 20–38

www.elsevier.com/locate/tws

1 In an unbranched open or closed thin-walled cross-section, no internal

longitudinal edge is shared by more than two walls. Fig. 1(a) and (b) provide

examples of unbranched and branched cross-sections, respectively.

http://www.elsevier.com/locate/tws

Notation

[Bik], [Cik], [Dik] GBT initial matrices

½ ~Bik�, ½ ~Cik�, ½ ~Dik� GBT transformed (modal) matrices

[Xik]j, ½ ~Xik�j GBT initial and transformed (modal) geometric

stiffness matrices

[K(e)], [G(e)] Finite element stiffness and geometric matrices

{d(e)} Finite element displacement vector

Wsi , Wt

i GBT generalised normal and shear stress resultants

l Load parameter

x, s, z Plate coordinate axes

u, v, w Displacement field components

uk(s), vk(s), wk(s) Elementary warping, transverse

membrane and flexural functions

fk(x), ~f kðxÞ Displacement amplitude functions

j (xZx/Le) Cubic Hermitean polynomials

Qi Finite element nodal displacement component

E, G, n Young and shear moduli, Poisson’s ratio

L, Le Member and finite element length

bi, ti, ai Width, thickness and inclination of plate element i

Ki Bending stiffness of plate i

mi, qi Transverse bending moments and rotations at node i

P.B. Dinis et al. / Thin-Walled Structures 44 (2006) 20–38 21

specially credited, as they were the first to depart substantially

from the path set by Schardt’s seminal work. Indeed, these

authors successively developed, validated and illustrated the

application of GBT formulations to analyse the elastic (i)

buckling behaviour of orthotropic members, accounting for the

influence of shear deformation [11–13], (ii) vibration beha-

viour of orthotropic members [14] and (iii) post-buckling

behaviour of isotropic members [15]. Moreover, they also

developed GBT-based analytical formulae to estimate distor-

tional buckling stresses in lipped channel, zed and rack-section

steel members [16–18]. Another very important contribution

was due to Goncalves and Camotim [19,20], who extended the

domain of validity of the GBT buckling analyses to cover

elastic–plastic (aluminium and stainless steel) members with

open and closed cross-sections. Finally, it is still worth

mentioning the works published by Simao and Silva [21] and

Rendek and Balaz [22], dealing respectively with the elastic (i)

buckling behaviour of box and lipped channel columns and (ii)

first-order distortional behaviour of cold-formed steel canti-

levers acted by tip loads.2 However, all the above authors never

challenged the main restrictive feature of Schardt’s original

formulation, namely the exclusion of members with

‘branched’ cross-sections — note that this limitation precludes

the application of GBT to I-section members, by far the most

widely used thin-walled members.3

As far as the authors are aware, the first attempts to perform

GBT analyses of thin-walled members with branched (open)

cross-sections were due to Moller [23] and Morschardt [24], at

the time Schardt’s collaborators at Darmstadt. However, these

attempts were neither (i) adequately reported and validated4

2 Although Simao and Silva [21] derived the fourth-order functional of the

thin-walled member potential energy; most likely with the intention of

performing post-buckling analyses, they did not illustrate or validate its

application.3 It also explains why the GBT-based analyses have been used mostly in the

context of cold-formed steel members — due to their fabrication procedure,

most of these (‘folded-plate’) members display unbranched open cross-

sections.4 Like the vast majority of the work on GBT authored by Schardt and/or his

co-workers, these two specific publications are available only in German.

Moreover, the (few) GBT-based numerical results presented in either of them

are never properly validated, i.e. compared with ‘exact’ values yielded, for

instance, by finite element or finite strip analyses.

nor (ii) complemented by subsequent investigations. Quite

recently, the authors [25] proposed and validated a GBT

formulation to analyse the buckling behaviour of members

with a special class of branched cross-sections: sections that

may be viewed as combining (i) an arbitrary unbranched open

cross-section with (ii) an equally arbitrary number of single-

wall branches, such as the cruciform section depicted in

Fig. 1(b).5 A bit later, some new light was shed on this problem

by Degee and Boissonnade [26], who reported results

concerning the GBT-based first-order analysis of a fixed

beam with a very specific (branched) cross-section shape.6

Therefore, the objective of this paper is to derive, validate

and illustrate the application of a general GBT formulation to

analyse the buckling behaviour of isotropic thin-walled

members displaying arbitrarily branched open cross-sections

— i.e. with any number of branching nodes, branches per node

and walls per branch. Initially, a brief overview of the

conventional GBT is presented,7 which (i) includes a

description of a one-dimensional finite element formulation

enabling the analysis of the buckling behaviour of members

with arbitrary boundary conditions [27] and (ii) concludes with

the identification of the difficulties associated with the

extension to branched cross-sections [25]. Then, one meticu-

lously addresses how the conventional GBT procedure must be

modified in order to be able to handle the presence of branching

nodes — the modifications concern exclusively issues related

to the performance of the GBT cross-section analysis, namely

(i) the choice of the most appropriate ‘elementary warping

functions’ and (ii) the determination of the ‘initial flexural

shape functions’. Finally, the derived GBT formulation is then

employed to investigate the local-plate, distortional and global

5 Although this cross-section class includes the most common I-sections, it

still precludes the application of GBT to, for instance, the I-section with a

stiffened top flange shown in Fig. 1(b) — the ‘single-wall branch’ condition is

violated.6 All the walls sharing the branching node have different orientations — the

methodology proposed by these authors cannot handle aligned walls

converging at a branching node. Moreover, it does not include intermediate

nodes.7 In this context, the designation ‘conventional GBT’ identifies the GBT

formulation derived by Schardt and making it possible to analyse the buckling

behaviour of thin-walled members with unbranched open cross-sections.

(a) (b)

Fig. 1. (a) Unbranched and (b) branched (open and closed) cross-sections.

P.B. Dinis et al. / Thin-Walled Structures 44 (2006) 20–3822

buckling behaviour of (i) asymmetric E-section columns with

both (i1) pinned and free-to-warp and (i2) fixed and warping-

prevented end sections, and (ii) I-section beams with unequal

stiffened flanges and pinned and free-to-warp end sections. For

validation purposes, several GBT-based numerical results

are compared with ‘exact’ values, obtained by means of either

(i) finite strip analyses carried out using the program Cufsm2.6

[28] — members with pinned and free-to-warp end sections —

or (ii) finite element analyses performed in the code Abaqus

[29] and adopting fine shell-element meshes — members with

fixed and warping-prevented end sections.

2. Brief overview of the conventional GBT

The so-called ‘conventional GBT’ is intended to analyse the

buckling (bifurcation) behaviour of linear elastic isotropic

prismatic thin-walled members with unbranched open cross-

sections. Its application involves the performance of two main

tasks, namely (i) a cross-section analysis and (ii) a member

linear stability analysis (e.g. [2,3,11,12]). Next, the main

aspects related to each of them are briefly reviewed — for

illustrative purposes, consider the arbitrary member shown in

Fig. 2, with a cross-section formed by q walls and where (i) x, s

and z are coordinates along the member length, cross-section

mid-line and wall thickness and (ii) u, v and w are the related

displacement components.

In order to obtain a displacement representation compatible

with Vlasov’s classical thin-walled beam theory [30], each

displacement component (u(x, s), v(x, s), w(x, s)) must be

expressed as

uðx; sÞ Z ukðsÞ$fk;xðxÞ vðx; sÞ Z vkðsÞ$fkðxÞ

wðx; sÞ Z wkðsÞ$fkðxÞ;(1)

where (i) ($),xhd($)/dx, (ii) uk(s), vk(s), wk(s) are shape

functions used to approximate the cross-section displacement

field and (iii) fk(x) are their common displacement amplitude

functions — summation convention applicable to subscript k.

Then, after (i) adopting Vlasov’s assumptions of null

x

ds dx s

(a)

x(u)

s(v)

z(w)

dx

ds

t

(b)

Fig. 2. (a) Prismatic thin-walled member with an arbitrary unbranched open

cross-section and (b) infinitesimal wall (plate) element.

membrane shear strains and transverse extensions, (ii)

considering the material constants (Young’s modulus E,

shear modulus G and Poisson’s ratio n) and (iii) employing

either the principle of virtual work or the principle of stationary

potential energy, one is led to the system of equilibrium

equations and boundary conditions (termed GBT equation

system)

ECikfk;xxxx KGDikfk;xx CEBikfk ClWsj:0Xjikfk;xx Z 0 (2)

Wsi dfi;xj

L0 Z 0 ðWt

i CXjiklWsj:0fk;xÞdfij

L0 Z 0; (3)

where (i) Wsj:0 are the pre-buckling internal force/moment

profiles, deemed uniform,8 (ii) l is the load parameter and (iii)

the tensors Cik, Dik, Bik (stiffness), Xjik (geometric) and Wsi , Wt

i

(generalised normal and shear stress resultants) arise from the

cross-section integration of the displacements and their

derivatives — they read

Cik Z C1ik CC2

ik Z

ðS

tuiukds C1

12ð1Kn2Þ

ðS

t3wiwkds

Bik Z1

12ð1Kn2Þ

ðS

t3wi;sswk;ssds

Dik Z D1ik KðD2

ik CD2kiÞ Z

1

3

ðS

t3wi;swk;sds

KnE

12Gð1Kn2Þ

ðS

t3ðwiwk;ss Cwkwi;ssÞds

Xjik Z

ðS

tuj

Cjj

ðvivk CwiwkÞds

Wsi Z ECikfk;xx CGD2

ikfk Wti ZKWs

i;x CGD1ikfk;x:

(4)

Matrices [Cik], [Bik] and [Dik] contain all the relevant

geometric information concerning the cross-section mechan-

ical properties. In (2), (i) the first three terms account for the

member 1st order behaviour, while (ii) the last one concerns

the geometrically non-linear effects, i.e. the interaction

between the cross-section normal stresses (Wsj:0 are their

resultants) and out-of-plane deformations — [Xik]j are

geometric stiffness matrices.

8 If the pre-buckling internal forces and moments vary longitudinally, one

must either (i) change the last term in the l.h.s. of (2) (smooth variation) or (ii)

consider separate beam-segment equilibrium equations (abrupt variation) [31].

q+3

r

r-1

ur=1

dx

x0 x0+dx

x

r+1

(a)

1; q+2

…

2

r-1

r

r+1

q q+1; q+m+1

Natural nodeIntermediate nodeNatural + +Intermediate node

…

q+m

q+p

q+p-1q+p+1

(b)

wq+p=1

q+p q+p-1

q+p+1

(c1)

mp+q

(c2)

Fig. 3. Cross-section (a) discretisation (natural and intermediate nodes), (b) elementary warping (node r) and flexural (node qCp) functions and (c) base system and

redundant mqCp.

9 The distinction between ‘natural’ and ‘intermediate’ nodes, kept here for

‘historical reasons’, is slightly misleading, as the cross-section free end nodes

are both natural and intermediate — see Fig. 3(a). Indeed, it would be more

logical to classify the cross-section nodes according to the nature of the

imposed elementary functions ‘centred’ on them: (i) warping (‘natural’), (ii)

flexural (‘intermediate’) or (iii) warping and flexural (‘natural and

intermediate’).


2.1. Cross-section analysis

The most unique GBT feature consists of the way in which

the member cross-section displacement field is approximated

(cross-section discretisation), i.e. the shape functions uk(s),

vk(s) and wk(s) appearing in (1) are chosen/selected. This

choice/selection, which is not at all straightforward, has been

described in detail by Schardt [2] and by Silvestre and

Camotim [11,12] and involves the following two steps: (i) a

systematic and more or less ‘intuitive’ selection of a set of

initial shape functions and (ii) an elaborate and rational

determination of a set of mechanically meaningful final shape

functions. Concerning the selection of the initial shape

functions, it is worth drawing the attention of the reader to

the following aspects:

(i) The q-walled member cross-section is discretised into

(i1) qC1 natural nodes (wall ends) and (i2) m

intermediate nodes (within the walls), totalling nZqCmC1 — see Fig. 3(a).

(ii) The initial shape functions uk(s), vk(s) and wk(s)

are obtained by sequentially imposing (ii1) elemen-

tary warping functions at each natural node and

(ii2) elementary flexural functions at each intermedi-

ate node,9 all with just one non-null (unit) nodal

value — see Fig. 3(b).

(iii) In view of the null shear strain assumption, each

elementary warping function is associated with a

piecewise constant transverse displacement function

vk(s). In each cross-section wall i, the value of vk(s)

is given by

vkiðsÞ ZKDui

bi

; (5)

where bi and Dui are the width and relative end

node warping displacement of wall i.

(iv) The functions wk(s), stemming from the imposition of

the elementary warping and flexural functions need to be

‘constrained’, in order to ensure that (iv1) the compat-

ibility between the in-plane transverse displacements

10 It is assumed here that the eigenvalue problem may be written in either a

differential or a variational form.11 Recall that these two methods are fully equivalent in conservative

problems. Indeed, provided that the same shape functions are adopted, they

lead to identical solutions (e.g. [35]).


vk(s) and wk(s) and/or (iv2) the continuity of the flexural

rotations wk,s(s) hold at all the cross-section nodes.

Therefore, the determination of each wk(s) requires the

solution of a statically indeterminate folded-plate

problem, a task performed by means of the force

method. The base system is obtained by releasing the

flexural rotations between adjacent wall segments, i.e. by

inserting longitudinal cylindrical rollers at all node

locations, as shown in Fig. 3(c1) — note that the

redundants are transverse bending moments distributed

along each roller (e.g. mqCp in Fig. 3(c2)).

(v) Therefore, all the uk(s), vk(s) and wk(s) can be expressed,

a priori, in terms of the known elementary warping and

flexural functions, the amplitudes of which are the initial

degrees of freedom of the (discretised) cross-section.

Once the initial shape functions are known, the calculation

of matrices [Cik], [Bik], [Dik] and [Xik]j defined in (4) constitutes

a straightforward but time consuming task, which may be

considerably simplified by resorting to symbolic manipulation.

However, all these matrices (i) are fully populated, which

implies that the equilibrium system Eq. (2) is highly coupled,

and (ii) have components with no obvious mechanical

meaning, which obscures the interpretation of the results. In

order to overcome these shortcomings, one further determines

final shape functions, a procedure that is a ‘GBT trademark’

and contributes decisively to its unique modal characteristics.

Although the details concerning the determination of final

shape functions, a key GBT feature, are not presented here

(they can be found in [2,11]), it is important to draw the

reader’s attention to the following aspects and procedures

involved in the performance of this step:

(i) One performs the simultaneous diagonalisation of

matrices [Cik] and [Bik], which strongly reduces the

coupling in system (2). This operation is carried out in

three stages and leads to the identification of n ‘mixed’

(warpingCflexural) eigenvectors f ~akg, which make it

possible to express the final shape functions as a linear

combination of the initial ones — these final shape

functions are termed cross-section deformation modes

and their amplitudes constitute the (discretised) cross-

section final degrees of freedom.

(ii) The cross-section deformation modes have a clear

mechanical meaning and can be divided into three

different categories: (ii1) rigid-body modes (the ones

included in Vlasov’s classical thin-walled bar theory:

axial extension, major and minor axis bending and

torsion), (ii2) distortional modes (combinations of wall

flexural deformation and fold line motions) and (ii3) local-

plate modes (only wall flexural deformation is involved).

(iii) Matrix ½ ~A�, which assembles the various eigenvectors

f ~akg, is then used to determine the ‘transformed’ matrices

½ ~Cik� (diagonal), ½ ~Bik� (diagonal), ½ ~Dik� (approximately

diagonal, as the off-diagonal components are quite small

in comparison with the corresponding diagonal ones) and

½ ~Xik�j (non-diagonal). The components of these matrices

are cross-section modal geometrical properties and most

of them have a clear mechanical meaning — for instance,

as far as the rigid-body modes are concerned, they include

the cross-sectional area, moments of inertia, St Venant

constant and warping constant.

2.2. Member buckling analysis

Before addressing the methods that can be used to carry out

exact or approximate member buckling analyses, the following

remarks are appropriate:

(i) After incorporating (i1) the cross-section geometrical

properties ½ ~Cik� and ½ ~Bik�, (i2) the material constants E

and G, (i3) the quantities related to the applied loads ~Wsk:0

and ½ ~Xik�j and (i4) the member length and end support

conditions into Eqs. (2) and (3), one is led to a one-

dimensional eigenvalue problem defined by a system of

differential equilibrium equations (one per deformation

mode) and boundary conditions that are expressed in

terms of the modal amplitude functions ~fkðxÞ. The

solution of this problem yields the member bifurcation

stress resultants (eigenvalues) and corresponding buck-

ling mode shapes (eigenfunctions).

(ii) A major advantage of the GBT resides in the possibility

of performing buckling analyses involving an arbitrary

set of deformation modes — i.e. one may consider only

the deformation modes that are known to be relevant for

a given problem. Then, by solving the ‘subsystem’ of

equilibrium equations and boundary conditions associ-

ated with those modes, one obtains upper bounds of the

member bifurcation stress resultants and also approxi-

mate buckling mode shapes — if all the relevant modes

are selected, these results are virtually ‘exact’.

The methods that have already been employed to solve the

GBT-based eigenvalue problem are fairly standard in structural

analysis [32]. They include10 (i) the finite difference method

(e.g. [33,34]), (ii) the Galerkin and Rayleigh–Ritz methods11

(e.g. [12,21]) and (iii) the finite element method, using a beam

element that has been specifically developed to perform GBT

analyses [26] — its formulation is briefly outlined in Section

2.2.1. Concerning the suitability of the above methods to solve

a given buckling problem, it is worth pointing out that:

(i) In members with pinned (locally and globally) and free-

to-warp end sections, subjected to uniform applied stress

resultants, the Galerkin and/or Rayleigh — Ritz

techniques are extremely advantageous — indeed,

since the eigenfunctions are known to display pure


sinusoidal shapes, these techniques can be readily used

to obtain exact buckling results.

(ii) In members with other end support and/or loading

conditions, for which there are no exact eigenfunctions

available, the finite element method provides the most

powerful and versatile tool — regardless of the particular

problem under consideration, one always obtains highly

accurate results through analyses that involve only a

modest number of degrees of freedom.

At this point, it should be mentioned that the GBT-based

buckling results presented in this paper have been obtained

through the application of either (i) the Galerkin method

(columns and beams with pinned and free-to-warp end

sections) or (ii) the beam finite element (columns with fixed

and warping-prevented end sections).

2.2.1. Beam finite element

A couple of years ago, Silvestre and Camotim [27]

formulated, implemented and validated an efficient beam finite

element intended to perform GBT-based buckling analyses in

the context of arbitrarily orthotropic thin-walled members. The

most relevant steps involved in this finite element formulation,

specialised for the case of isotropic members, are succinctly

described next:

(i) Rewrite the system of equilibrium equations and

boundary conditions defining the eigenvalue problem

in variational form,ðLe

ðE ~Cik~fk;xxd ~fi;xx CG ~Dik

~fk;xd ~fi;x CE ~Bik~fkd ~fiK

l ~Wsj:0~Xjik

~fk;xd ~fi;xÞdx Z 0;

(6)

where (i1) Le is the finite element length and (i2)

kZ2.qCmC1 — the axial extension mode never

contributes to the member buckling mode shape (this

mode only appears in the pre-buckling equilibrium

paths of axially compressed members).

(ii) Adopt linear combinations of standard cubic Hermitean

polynomials to approximate the deformation mode

amplitude functions ~fkðxÞ. Therefore, one has

~fkðxÞZQ1j1ðxÞCQ2j2ðxÞCQ3j3ðxÞCQ4j4ðxÞ; (7)

where Q1Z ~fk;xð0Þ, Q2Z ~fkð0Þ, Q3Z ~fk;xðLeÞ,

Q4Z ~fkðLeÞ, xZx/Le and

j1 ZLeðx3K2x2 CxÞ j2 Z2x3K3x2 C1

j3 ZLeðx3Kx2Þ j4 ZK2x3 C3x2:

(8)

(iii) Substitute the approximations (7) into (6) and carry out

the integrations, in order to obtain the usual finite

element matrix equation

ð½KðeÞ�Cl½GðeÞ�ÞfdðeÞgZf0g; (9)

where [K(e)], [G(e)] and {d(e)} are the finite element

stiffness matrix, geometric matrix and displacement

vector, which have dimension 4 (qCm) and are of the

form

½KðeÞ�Z

½K22� ½0� ½0� ½K25� .

½K33� ½0� ½K35� .

½K44� ½K45� .

½K55� .

sym: : .

2666666664

3777777775

½GðeÞ�Z

½G22� ½0� ½G24� ½G25� .

½G33� ½G34� ½G35� .

½G44� ½G45� .

½G55� .

sym: .

2666666664

3777777775

(10)

fdðeÞgZffd2gTfd3gTfd4gTfd5gT.gT (11)

The superscripts i, j concern the deformation modes and

the components of each sub-matrix or vector (p, rZ1.4 — finite element degrees of freedom) are obtained

from

Kijpr ZE ~Cij

ðLe

jp;xxjr;xxdxCG ~Dij

ðLe

jp;xjr;xdx

CE ~Bij

ðLe

jpjrdx

(12)

Gijpr ZK ~W

sk:0

~Xkij

ðLe

jp;xjr;xdx djr ZQr: (13)

2.3. Extension to members with branched open cross-sections-

scope and difficulties

First of all, it is convenient to make clear that all the

modifications that must be incorporated into the conventional

GBT procedure, in order to handle the presence of ‘branching’

points, concern the first part of the cross-section analysis,

namely the choice and characterisation of the initial shape

functions. Once this task is completed, both (i) the simul-

taneous diagonalisation of matrices [Cik] and [Bik] (i.e. the

identification of the cross-section deformation modes) and (ii)

the member buckling analysis are absolutely identical to the

ones described in previous sub-sections.

In unbranched open cross-sections, each internal natural

node is shared by only two walls. As far as the GBT


cross-section analysis procedure is concerned, this fact has the

following implications:

(i) At each internal natural node, it is always possible (i1) to

satisfy Vlasov’s assumption of null membrane shear

strains (in the two walls adjacent to that node) and (i2) to

ensure the compatibility between the membrane trans-

verse displacements (in that node), for any combination

of nodal warping values imposed on the natural node

under consideration and the two natural nodes linked to it

— the elementary warping functions defined in Section

2.1 are just convenient particular cases.

(ii) The application of the force method, to determine the

initial flexural shape functions wk(s), involves only one

redundant per internal (natural or intermediate) node —

recall that these redundants are transverse bending

moments acting along the longitudinal edges.

In branched sections, the presence of branching nodes

makes the above assertions no longer valid. Thus, the simple

and ‘intuitive’ extension of the conceptual reasoning pre-

viously used to obtain uk(s), vk(s) and wk(s) requires

considerable modifications to the conventional GBT cross-

section analysis procedure, as now (i) ensuring the compat-

ibility between the membrane transverse displacements at the

branching nodes is not a trivial matter (the warping

displacements are no longer independent12) and (ii) there

exist more than one redundant at a branching node. Moreover,

the very frequent existence of aligned walls emerging from a

branching node (e.g. the I and cruciform sections shown in

Fig. 1(b)) is also a source of additional difficulties.13

In Section 3, one addresses the modifications that need to be

included in the conventional GBT cross-section analysis

procedure in order to handle branching nodes. After reviewing

the previous work dealing with this issue, one presents in detail

a novel methodology that (i) makes it possible to overcome all

the difficulties identified above and, hence, (ii) is applicable to

arbitrarily branched open cross-sections.

14

3. GBT for members with branched open cross-sections

The first attempt to perform GBT analyses of thin-walled

members with branched open cross-sections dates from

1982 and was due to Moller [23]. His main contribution

was to show that, in order to (i) comply with Vlasov’s null

shear strain assumption and (ii) ensure compatibility

between the membrane transverse displacements at a

branching node, the warping displacements at the walls

emerging from that node cannot be chosen independently.

On the basis of the analysis of a branching node associated

12 Due to this lack of independence, the number of elementary warping

functions that must be considered in the cross-section analysis is smaller than

the number of natural nodes — in unbranched sections (conventional GBT),

these two numbers are always equal.13 Recall that, by definition, unbranched sections never exhibit aligned walls

converging at a natural node.

with three non-aligned walls, Moller derived the conditions

that must be satisfied by these warping displacements.

Moreover, he illustrated his approach through the first-order

analysis of a transversally loaded stiffened panel. About

a decade later, Moller’s work was followed by Morschardt

[24], who proposed a systematic procedure to select the

appropriate elementary warping functions and illustrated his

approach by determining the deformation mode shapes of an

unequally flanged I-section.

Quite recently, the authors [25] revisited this problem and

developed a GBT formulation to analyse the buckling

behaviour of members with branched cross-sections. This

formulation is based on a cross-section analysis procedure that

(i) was initially ‘inspired’ by Morschardt’s work14 and (ii)

involves, almost exclusively, conventional GBT

operations and procedures. In spite of the obvious advantages

of the last feature, this approach is only applicable to branched

cross-sections that can be viewed as a combination of (i) an

unbranched open section with (ii) an arbitrary number of

single-wall branches — it cannot handle, for instance, the

stiffened I-section shown depicted in Fig. 1(b). In retrospective,

it becomes clear that this rather severe limitation was

essentially due to the fact that the main goal of the proposed

approach was to employ as many conventional GBT

procedures as possible, namely to ensure that the determination

of the initial flexural shape functions, by means of the force

method, never involved more than one redundant per

branching node. Although this goal was achieved, it led to

unexpected (and insurmountable) difficulties concerning the

choice of the elementary warping functions that must be

imposed in the natural nodes of multiple-wall branches.

Finally, it is still worth mentioning the very recent work of

Degee and Boissonnade [26]: apparently unaware of Moller’s

work, these authors adopted his approach and presented results

concerning the GBT-based first-order analysis of a fixed beam

with a very specific branched cross-section — it has just one

branching node, where converge three non-aligned walls.

At this stage, it is convenient to call the reader’s attention to

a common feature shared by all the previous studies dealing

with the GBT-based analysis of members with branched cross-

sections: like in Schardt’s original work, the initial flexural

shape functions are always determined by means of the force

method. This means that one has to consider mwK1 redundants

at a branching node, where mw is the number of branching

walls, a fact that considerably complicates the cross-section

analysis procedure — most likely, this is the reason why all

the illustrative examples reported involve only branching

nodes with three emerging walls.15

Although there is some (formal and partial) resemblance between the two

approaches, it seems fair to say that they were developed ‘independently’.

Indeed, the fact that Morschardt’s work is (i) insufficiently reported (very few

details given) and (ii) written in German (language not mastered by the authors)

prevented a full grasp of its fundamentals.15 Recall that the approach proposed by the authors [25] only requires the

consideration of one redundant per branching node. However, it cannot be

applied to cross-section with multiple-wall branches.

wq+p=1

mp+q

(a)

= +

θp+q+1

(b)

= +

wq+p=1

q+ p q+p-1

q+p+1

wq+p=1

q+p q+p-1

q+p+1

wq+p=1

θp+q1θp+q

mp+q-1

mp+q+1

.

.

.

.

.

.

.

.

.

.

.

.

Fig. 4. Determination of the initial flexural shape functions in an unbranched cross-section by means of the (a) force and (b) displacement methods.


3.1. Proposed methodology

In this section, one presents in great detail and illustrates the

application of a novel methodology to perform the GBT cross-

section analysis that (i) makes it possible to overcome all the

difficulties and/or limitations outlined above, (ii) is applicable

to a completely arbitrary branched open cross-section16 and

(iii) is computationally as efficient as the conventional

procedure. This methodology concerns the following two

issues, briefly described next and subsequently addressed

individually:

(i) Choice and definition of the elementary warping

functions. First, one provides a systematic and sequential

procedure to choose the most convenient set of

independent natural nodes, i.e. nodes having always

either unit or null warping displacement values (like in

the conventional GBT). Then, one addresses the

evaluation of the warping displacement values at the

dependent natural nodes, based on the compatibility

between the membrane transverse displacements at the

branching nodes.

(ii) Determination of the initial flexural shape functions. One

adopts the displacement method to perform this task — a

radical departure from the conventional procedure

leading to considerable computational savings. This is

due to the fact that, in branched cross-sections, the

degree of kinematical indeterminacy is always lower

than its static counterpart — the difference grows as the

numbers of branching nodes and/or walls increase. In

unbranched cross-sections, on the other hand, these two

16 Cross-sections displaying arbitrary numbers of branching nodes and

branches per node. Moreover, each branch may also include additional

branching nodes, i.e. the cross-section may exhibit multiple branches.

degrees of indeterminacy are equal,17 which means that

the displacement and force methods involve practically

the same computational effort — this makes the use of

the displacement method a viable alternative also for

members with unbranched sections. Fig. 4 provides a

perfect illustration of this statement: instead of having

(ii1) unknown nodal transverse bending moments

determined by means of flexural rotation compatibility

equations (force method), one has (ii2) unknown nodal

flexural rotations yielded by transverse bending moment

equilibrium equations (displacement method).

3.1.1. Choice and definition of the elementary warping

functions

As mentioned earlier, in branched cross-sections it is not a

trivial matter to ensure, simultaneously, (i) the satisfaction of

Vlasov’s assumption of null membrane shear strains and (ii)

the compatibility between the membrane transverse displace-

ments at a branching node. Indeed, this goal can only be

achieved if specific combinations of warping values are

imposed at (i) that branching node and (ii) all the natural

nodes directly linked to it (one per branching wall): aside from

the branching node value, only two additional nodal warping

values can be chosen freely, which means that mwK2 values

(mwO2 is the number of branching walls) are dependent

and must be determined. Therefore, it is possible to identify

three separate tasks, namely (i) to select the set of natural nodes

where the warping values are freely chosen (independent

natural nodes–there are various possibilities, some more

convenient than others), (ii) to choose those values and (iii)

17 Rigorously, the degree of static indeterminacy is always two units lower,

due to the end longitudinal edges (nodes). However, the two degrees can be

readily made equal by using also the stiffness matrix of a pinned-fixed plate

element.

(e)(c)

B2

W11

W10

W9

W13 W12

W8W7

W14 W15

W6W11

W9

W10

W13

W7 W1

B1

W15 W14

W4

B4

W5

W3

W12

W8 W2

B3

W4

W6

W5

W3

W1W2

W15

W3

W13 W12

W8W7

W14

W6

W5

W9W11

W10

W4

W2 W1

Independent natural node

Dependent natural node

(a)

(b) (d)

Fig. 5. Branched section (a) geometry and two most convenient (b) unbranched sub-sections, (c) first-order branches, (d) second-order branches and (e)

independent/dependent natural nodes.


to determine the warping values at the remaining natural nodes

(dependent natural nodes). The second task is carried out in the

obvious way, i.e. following the conventional GBT procedure:

one unit and two null warping values.

Concerning the selection of the most convenient indepen-

dent natural nodes, a general and systematic methodology is

proposed next and illustrated by means of its application to the

branched cross-section depicted in Fig. 5(a) — it has 15 walls

(W1KW15) and 16 natural nodes, four of which are branching

nodes (B1KB4). This methodology comprises the following

sequence of steps:

(i) Choice of an unbranched sub-section that (i1) should

contain as many branching nodes as possible18 and (i2)

must not include aligned walls sharing the same

branching node.19 Fig. 5(b) shows two similarly

convenient unbranched sub-sections for the illustrative

example: each contains three branching nodes and

includes no aligned walls sharing a branching node.

Just for the sake of completion, note that the upper sub-

section would cease to be convenient if the large

vertical wall (W3) was stiffened: the stiffener would

create an additional branching node, shared by two

aligned walls-the two ‘halves’ of W3.

(ii) Definition of a set of first-order branches, which are the

whole or part of the various wall assemblies linked to the

branching nodes belonging to the unbranched sub-

section — note that all the remaining cross-section

branching nodes (if any) are contained in these wall

assemblies. A first-order branch either (ii1) coincides

with a wall assembly, if it contains no branching node, or

(ii2) is a part of a wall assembly, if it contains one or more

branching nodes. In the latter case, the first-order branch

must be chosen similarly to the unbranched sub-section

— i.e. it (ii1) should contain as many wall assembly

18 Although this is not a mandatory requirement, it makes the whole procedure

simpler.19 This ensures that the branching nodes can be treated as natural nodes. If one

fails to do this, it becomes impossible to choose always only unit and null

warping values at the independent natural nodes—recall that the warping

displacement distribution along two aligned walls is linear (and not bi-linear).

branching nodes as possible and (ii2) must not include

aligned walls sharing the same branching node. Fig. 5(c)

displays convenient first-order branches associated with

the unbranched sub-sections showed in Fig. 5(b) — note

that, in both cases, only one of the wall assemblies

contains branching nodes (a single one).

(iii) If necessary, definition of a set of second-order

branches, which (iii1) are the whole or part of the wall

assemblies linked to the branching nodes belonging to

the first-order branches and (iii2) are defined according

to the guidelines given in the previous item. Fig. 5(d)

displays the second-order branches associated with the

first-order branches showed in Fig. 5(c)—only one

second-order branch exists in both cases.

(iv) If necessary, successive definition of sets of higher-

order branches, until all branching nodes are ‘covered’,

i.e. no remaining wall assembly contains a branching

node. For the unbranched sub-sections showed in

Fig. 5(b), there are no branches of order higher than

two.20

(v) Once the unbranched sub-section and the various sets of

branches are completely defined, it is a straightforward

matter to identify the dependent natural nodes: all the

second nodes of the various branches (i.e. the ones

located immediately after the corresponding branching

node). The number of such nodes is equal to S(mwiK2),

where the summation extends to all branching nodes and

mwi is the number of walls emerging from branching

node i — for the cross-section depicted in Fig. 5(a), this

number is 6.

(vi) Therefore, the number of elementary warping functions

is equal to the number of independent natural nodes —

for the cross-section depicted in Fig. 5(a), this number is

10. Each function is characterised by (vi1) a unit warping

value at one independent natural node, (vi2) null values

at all the remaining independent natural nodes and (vi3)

20 If one chose an unbranched sub-section formed by walls W1, W2, W8 and

W7, which contains just one branching node, it would be necessary to define

also third-order branches. Therefore, the choice of such an unbranched sub-

section would just complicate the procedure, without bringing any advantage.

Fig. 6. Branching node r: (a) configuration, (b) determination of the transverse membrane displacement vrj.1 and (c) elementary warping functions associated with

nodes rK1 and r.


warping values at the dependent natural nodes that must

be specifically determined — the determination of these

warping values is addressed in the next paragraphs.

The determination of the warping values at the dependent

natural nodes is based on the fact that (i) Vlasov’s null

membrane shear strain assumption has to be satisfied in all the

walls emerging from a given branching node and (ii) the

compatibility between the membrane transverse displacements

must be ensured at that same branching node. Before

addressing the determination of these values, it is worth

pointing out that:

(i) Each dependent node is linked to a particular branching

node, which is (i1) either independent (most cases) or

dependent21 and (i2) always linked to two and only two

independent nodes.

(ii) The warping value at a given dependent node is

obtained on the basis of (ii1) the warping value at the

associated branching node, (ii2) the warping values at

the two corresponding independent nodes and (ii3) the

widths and inclinations of the three (branching) walls

involved.

(iii) When a dependent node is linked to a branching node

that is also dependent, one must begin by determining

the warping value at the latter. This is always ensured if

one determines the dependent node warping values

following an ascending branch order.

Consider now the arbitrary branching node r depicted in

Fig. 6(a) — without any loss of generality, it is assumed that it

belongs to the unbranched sub-section. From this branching

node emerge kC2 walls, linking it (i) to the 2 independent

nodes rK1 and rC1 and also (ii) to k dependent nodes r1,.,

rk (each contained in a first-order branch). All these walls are

oriented as indicated and have widths and inclinations (relative

21 It depends on the particular unbranched cross-section adopted — this is

illustrated in Fig. 5(e), where the branching nodes B1 and B4 are independent in

one case and dependent in the other.

to the horizontal direction and measured clockwise) designated

as brK1, br, br1, .., brk and arK1, ar, ar1, .., ark. Any

dependent node warping value urj(jZ1, .,k) is a function

of only (i) the three independent warping values urK1, ur and

urC1, and (ii) the widths and inclinations of the three walls

involved (WrK1, Wr and Wrj.1). In order to obtain this value, one

must perform the following operations:

(i) Using Vlasov’s assumption and adopting the conven-

tional GBT procedure, determine the transverse

membrane displacements in the walls WrK1 and Wr,

through the expressions

vrK1 ZKur KurK1

brK1

vr ZKurC1Kur

br

; (14)

where a positive value indicates a displacement

‘following’ the respective wall orientation.

(ii) On the basis of the inclinations of walls WrK1 and Wr

and the values of vrK1 and vr, determine the final

location of the branching node r-identified by a black

circle in Fig. 6(b).

(iii) Based, on the final location of the branching node rand the inclination of wall Wrj.1, evaluate the value of

the transverse membrane displacement vrj.1 required to

ensure that walls WrK1, Wr and Wrj.1 continue to share

node r — see again Fig. 6(b). This can be done by

means of the general expression (valid for vrs022)

vrj:1 Z vr

�cosðarj:1KarÞKsinðarj:1KarÞ!

! KvrK1

vrsinðarK1KarÞC

1

tanðarK1KarÞ

�:

(15)

(iv) Using the values of ur, vrj.1 and brj, evaluate the sought

urj via the expression

22 If one has vrZ0, one just has to ‘switch’ the roles of the walls WrK1 and Wr,

i.e. to reverse their orientations.

Fig. 7. Determination of initial flexural shape functions at a wall and a wall segment: (a) wall flexural displacements due to an elementary warping function, (b) fixed-

end transverse bending moments and deformed configurations and (c) contributions to the initial flexural shape functions.


urj Z ur Kvrj:1brj:1: (16)

After performing the above operations for the k dependent

nodes, it is possible to completely define the elementary

warping functions stemming from the imposition of unit

warping values at nodes rK1, r and rC1 — the first two are

displayed in Fig. 6(c).

23 Recall that the wall membrane transverse displacements can be expressed

in terms of (i) the warping values at nodes rK1, r and rC1 and (ii) the widths

and inclinations of the three walls WrK1, Wr and Wrj.1.

3.1.2. Determination of the initial flexural shape functions

As mentioned earlier, the determination of the initial flexural

shape functions requires the solution of a statically and

kinematically indeterminate ‘plate assembly problem’—in

conventional GBT, which deals exclusively with unbranched

sections, one has a ‘folded-plate problem’ that is viewed only as

statically indeterminate and, therefore, solved by means of the

force method. In branched sections, however, it is much more

advantageous to think in terms of kinematical indeterminacy and

to solve the ‘plate assembly problem’ using the displacement

method: the advantages can be instantly weighed if one realises

that a branching node with k emerging walls is associated with (i)

k-1 static unknowns (transverse bending moments) and (ii) a

single kinematical unknown (transverse flexural rotation).

The initial flexural shape functions that must be determined

stem from the imposition of either (i) elementary warping

functions with a unit value at one independent natural node or

(ii) elementary flexural functions at one intermediate node. The

elementary warping functions lead to flexural displacements at

(i) the natural node with unit warping value and (ii) all the

(dependent or independent) natural nodes directly linked to it.

The elementary flexural functions, on the other hand, involve

merely a unit flexural displacement at one intermediate node.

The fixed-end moments required to apply the displacement

method are evaluated on the basis of these nodal flexural

displacements. The procedure leading to the determination of

any initial flexural shape function is schematically represented

in Fig. 7 and involves the performance of the following

operations:

(i) Evaluation of the nodal flexural displacements associated

with the appropriate elementary function, a step which (i1)

is trivial in the flexural case (unit value at an intermediate

node) and (i2) involves some effort in the warping case.

Fig. 7(a) shows the flexural displacements at a branching

node r stemming from the imposition of an elementary

warping function and concerning the three walls already

depicted in Fig. 6(b) — it is possible to express such

displacements exclusively in terms of the membrane

transverse displacements and inclinations of these walls,23

by means of the expressions (the wrj.1 one is valid again

for vrs0)

wrK1 Zvr

sinðarK1KarÞK

vrK1

tanðarK1KarÞ

wr Zvr

tanðarK1KarÞK

vrK1

sinðarK1KarÞ

wrj:1 Z vr

�sinðarj:1KarÞCcosðarj:1KarÞ

! KvrK1

vrsinðarK1KarÞC

1

tanðarK1KarÞ

�:

(17)

(ii) Determination of the fixed-end distributed (transverse

bending) moments and deformed configurations due to the

imposition of the flexural displacements just evaluated.

Fig. 7(b) illustrates the performance of this step for the

cases of displacements induced (ii1) in a wall by an

elementary warping function and (ii2) in a wall segment by

an elementary flexural function. In the first case, the fixed-

end distributed moments are applied at the longitudinal

edges corresponding to nodes r (mf.r) and rC1 (mf.rC1),

and their values are given by the well-known expressions

mf :r ZKmf :rC1 Z6 Kr

b2r

ðwr KwrC1Þ; (18)

where br is the wall width and Kr ZEt3=12ð1Kn2Þ is

bending stiffness of plate r — E and n are Young’s

Fig. 8. Illustration: cross-section (a) geometry and dimensions and (b) GBT discretisation.


modulus and Poisson’s ratio. In the second case, the two

fixed-end distributed moments are applied at the

longitudinal edges corresponding to the natural or

intermediate nodes24 pK1 (mf.p-1) and pC1 (mf.pC1)

and are determined through

mf :pK1 Z6KpK1

b2pK1

ð0K1Þ mf :pC1 Z6Kp

b2p

ð1K0Þ; (19)

where bp-1, bp are the wall segment widths and Kp-1, Kp are

the associated bending stiffnesses.

(iii) Once all the fixed-end moments are evaluated, it is

possible to obtain the flexural rotations at all the natural

and intermediate nodes, by solving the system of

equations yielded by the application of the displacement

method, which ensures nodal moment equilibrium.

(iv) Finally, the initial flexural shape functions at a wall or

wall segment are obtained by adding (iv1) the

corresponding fixed-end deformed configurations to

(iv2) the deformed configurations associated with the

nodal rotations involved — note that the deformed

configurations due to the unit nodal rotations are

precisely the Hermite polynomials j1(x) and j3(x)

given in (8). Fig. 7(c) shows the various contributions to

the initial flexural shape functions at (iv1) the wall

comprised between nodes r and rC1, and (iv2) the wall

segment including nodes pK1, p and pC1 — their

fixed-end deformed configurations have already been

displayed in Fig. 7(b).

3.2. Illustration

In order to illustrate the proposed GBT cross-section

analysis procedure, described in Section 3.1.2, one determines

(i) the deformation mode shapes and (ii) the modal geometrical

property values of a cross-section with (i) the shape depicted in

Fig. 5(a), (ii) the particular dimensions given in Fig. 8(a) and

(iii) the discretisation shown in Fig. 8(b). This discretisation

involves (i) 16 natural nodes, four of which are branching

nodes, and also (ii) 19 intermediate nodes, eight of which

24 The wall segment under consideration (i) always has an intermediate node

at mid-width and (ii) may be comprised between (ii1) two intermediate nodes,

(ii2) one natural and one intermediate nodes or (ii3) two natural nodes.

correspond to free end longitudinal edges. Since only 10 out of

the 16 natural nodes are independent (as seen earlier), the

initial shape functions adopted to approximate the cross-

section deformation stem from the imposition of 10 elementary

warping functions and 19 elementary flexural functions. Thus,

the performance of the GBT cross-section analysis yields 29

deformation modes, which have the in-plane deformed

configurations depicted in Fig. 9. As far as the nature of

these deformation modes is concerned, one has (i) four rigid-

body modes, which correspond to axial extension (1-not

shown), major axis bending (2), minor axis bending (3) and

torsion (4), (ii) six distortional modes (5–10) and (iii) nineteen

local-plate modes (11–29).

Table 1, on the other hand, gives almost all of the cross-

sectional geometrical properties associated to the above set of

29 deformation modes, namely (i) the diagonal components of

matrices ½ ~Cik�, ½ ~Bik�, and ½ ~Dik� and also (ii) the off-diagonal

component of each matrix ½ ~Dik� line with a higher absolute

value-recall that ½ ~Dik� is only approximately diagonal. It is

worth noting that (i) C11hA (area), (ii) C22hIy (maximum

moment of inertia), (ii) C33hIz (minimum moment of inertia),

(iv) C44hG (warping constant), (v) D44hJ (St-Venant torsion

constant).

4. Buckling analysis of branched section members:

validation and illustration

In order to validate and illustrate the application and

capabilities of the derived GBT formulation, one presents

and discusses a set of numerical results concerning an

investigation on the elastic buckling behaviour of (i)

asymmetric E-section columns with both (i1) pinned and

free-to-warp and (i2) fixed and warping-prevented end

sections (termed henceforth ‘simply supported’ and ‘fixed’

columns), and (ii) simply supported (pinned/free-to-warp

end sections) I-section beams with unequal stiffened

flanges and subjected to uniform positive and negative

major axis bending. The cross-section dimensions and

discretisations considered are given in Fig. 10(a) and (b)

and one adopts the elastic constant values EZ210 GPa

(Young’s modulus) and nZ0.3 (Poisson’s ratio). For

validation purposes, several GBT-based buckling results

are compared with values yielded by either (i) finite strip

analyses carried out in Cufsm2.6 [28] (simply supported

columns and beams) or (ii) shell finite element analyses

Table 1

Modal cross-section geometrical properties

k ~Ckk~Bkk

(!10K4)

~Dkk

(!10K4)

k ~Ckk

(!10K4)

~Bkk

(!10K4)

~Dkk

(!10K4)

i k ~Diik

(!10K4)

i k ~Diik

(!10K4)

1 5.2 0 0 16 4.376 1.350 15.954 1 – 0 16 20 K4.726

2 145.103 0 0 17 6.319 3.169 26.311 2 22 K1.577 17 26 K4.695

3 48.985 0 0 18 8.335 5.459 36.234 3 14 1.169 18 27 4.995

4 924.260 0 173.333 19 2.842 2.235 15.097 4 14 12.620 19 10 K2.868

5 0.4115 0.00214 0.907 20 5.915 4.780 23.301 5 8 0.883 20 28 4.846

6 0.2560 0.00229 0.296 21 2.849 2.856 17.315 6 16 0.690 21 14 K4.586

7 0.0776 0.03093 4.177 22 1.914 2.317 13.720 7 17 K1.556 22 12 1.828

8 0.08779 0.09738 5.872 23 4.650 6.679 26.60 8 18 2.459 23 4 3.280

9 0.08848 0.11150 5.315 24 1.071 1.651 6.846 9 25 1.440 24 4 K9.002

10 0.06972 0.09891 3.151 25 0.988 1.539 6.298 10 29 1.153 25 20 2.311

11 0.000271 0.0372 1.723 26 0.734 2.941 8.598 11 4 6.330 26 4 K1.080

12 0.000205 0.0619 1.566 27 0.781 3.470 8.881 12 22 1.828 27 18 4.995

13 0.000433 0.170 2.823 28 0.781 3.470 8.881 13 23 K2.072 28 18 4.846

14 0.001170 2.695 33.043 29 0.767 3.557 8.747 14 4 12.620 29 19 K4.828

15 0.001025 3.022 31.710 15 27 K2.804

Fig. 9. In-plane deformed configurations of the cross-section deformation modes.


performed in Abaqus [29] and adopting fine S4 element

meshes (fixed columns).

4.1. E-Section columns

In view of the adopted discretisation (8 independent natural

nodes and 14 intermediate nodes), the cross-section analysis

leads to the identification of 22 deformation modes. However,

since the last nine modes (as well as the axial extension mode)

never participate in a column-buckling mode, only the in-plane

deformed configurations (shapes) of the 12 relevant modes are

depicted in Fig. 11.

Fig. 12 shows numerical results concerning the buckling

behaviour of simply supported columns. The curves displayed in

Fig. 12(a) provide the variation of the column bifurcation load Nb

with its length L (in logarithmic scale), under three kinematical

assumptions: (i) single-wave buckling modes (nwZ1) and

including either (i1) all deformation modes (upper lighter solid

curve) or (i2) only a few selected ones (dashed curves — modes

included indicated) and (ii) critical buckling modes (any nw) with

all deformation modes included (lower darker solid curve). For

validation purposes, this figure also provides Nb values yielded

by finite strip analyses performed in Cufsm2.6 [28]. As for the

modal participation diagram presented in Fig. 12(b), it gives

information on how the individual GBT deformations modes

contribute to the column single-wave buckling modes — pi is the

participation of mode i. Finally, typical local-plate, distortional

and global (flexural-torsional) buckling mode shapes are shown

in Fig. 12(c) — they were obtained from shell finite element

analyses performed in Abaqus [29] and correspond to columns

with lengths equal to 6, 54 and 190 cm.

The observation of these buckling results prompts the

following comments:

(i) First of all, there is a virtual coincidence between the

single-wave and critical buckling loads obtained

Fig. 10. Illustrative examples: E and I-section (a) dimensions and (b) GBT discretisations.

Fig. 11. In-plane deformed configurations of the 12 relevant E-section deformation modes: rigid-body (2–4), distortional (5–8) and local-plate (9–13).


through finite strip and GBT-based analyses, which

fully validates the latter.

(ii) The critical buckling curve exhibits three distinct

zones, corresponding to (ii1) 1–4 wave local-plate

buckling, (ii2) 1–3 wave distortional buckling and (ii3)

single-wave global (flexural-torsional) buckling. It only

differs from its single-wave counterpart for (ii1) 9!L!27 cm (2–4 wave local-plate buckling) and (ii2) 65!L!180 cm (2–3 wave distortional buckling).

(iii) Since only 10 deformation modes (2–9, 11 and 13)

participate in column buckling modes, a GBT buckling

analysis including only those modes yields exact

results.

(iv) The single-wave buckling curve exhibits local minima

at Lz6 cm and Lz50 cm, (iv1) the former correspond-

ing to a local-plate buckling mode that combines modes

8, 9 (clearly predominant), 11 and 13, and (iv2) the

latter associated with a distortional buckling mode

combining modes 5, 6 (predominant) and 7.

(v) The final descending branch, common to the single-

wave and critical buckling curves, is associated with

global flexural-torsional buckling modes that always

combines modes 2, 3 and 4 (recall that the cross-section

is asymmetric) — while mode 3 is highly predominant

in the longer columns, modes 2 and 4 are prevalent in

the ‘not so long’ columns.25 Note also that a flexural–

25 Because the cross-section is ‘almost symmetric’, the participation of modes

2 and 4 in the buckling modes of very long columns is negligible — see

Fig. 12(b).

torsional–distortional buckling mode, which includes

small participations of modes 6 and 7, is critical for

200!L!300 cm.

(vi) The modal participation diagram presented in

Fig. 12(b) readily shows that several portions of the

single-wave buckling curve can be very accurately

approximated by means of analyses including only a

few (selected) deformation modes. This statement is

fully backed by the three dashed curves, which (vi1)

involve only modes 8C9C11C13, 5C6C7 or 2C3C4 and (vi2) practically coincide with the ‘exact’

curve for L!12 cm, 50!L!120 cm and LO150 cm.

Next, Fig. 13(a) makes it possible to compare the GBT-

based critical buckling curves (Ncr vs. L) relative to simply

supported and fixed columns. While the former, already

presented in Fig. 12(a), was determined through the application

of Galerkin’s method (sinusoidal shape functions), the latter

was obtained by employing the beam finite element formu-

lation addressed in Section 2.2.1 — the column longitudinal

discretisation involved 4–24 beam elements (depending on the

buckling mode wave number) and up to the 12 cross-section

deformation modes were included in the analyses — the modal

participation diagram presented in Fig. 13(b) shows that only

eight of them are relevant for the fixed columns. As for

Fig. 13(c), it displays the configurations of the relevant modal

amplitude functions concerning the simply supported and fixed

columns with LZ200 cm. In order to validate the GBT-based

results, Fig. 13(a) also includes Ncr values obtained by means

of Abaqus finite element analyses — fine meshes of four-node

Fig. 12. Buckling behaviour of simply supported E-section columns: (a) Nb vs. L curves, (b) modal participation diagram pi vs. L (nwZ1) and (c) typical buckling

mode shapes.

0

20

40

60

80

100Ncr (kN)

L (cm)

(b)

(a) (c)

Fixed

Simply Supported

GBT (all modes, any nw )

FEM (ABAQUS )

GBT (FE formulation and selected modes)

6 1000100

LPM DM

LPM DM FTDM + FTM

0

0.25

0.5

0.75

1.0

10 100 1000

9

6

5

6

5

79

10

3

2

4

67

8

10

50

100

150 200

x (cm)

0.2

0.1

–0.2

–0.3

–0.1

–0.4

0.0

0.4

~

0.3

7

5

6

9

3

2

4

7

60.2

0.1

–0.2

–0.3

–0.1

–0.5

–0.6

–0.4

0.050 100 150

200

x (cm)

k (x)

k (x)

6 L (cm)

~

Fixed

Simply Supported

pi

FTDM + FTM

Fig. 13. Critical buckling behaviour of simply supported and fixed E-section columns: (a) Ncr vs. L curves, (b) fixed column modal participation diagram pi vs. L

(fixed columns) and (c) relevant modal amplitude functions for the columns with LZ200 cm.



isoparametric shell (S4) elements were adopted to discretise

the columns. Concerning these results, it is worth pointing out

that:

(i) Once again, the GBT-based results (critical buckling

loads) are virtually ‘exact’ — indeed, they coincide

with the ones, now obtained by means of Abaqus shell

finite element analyses.

(ii) For 6%L!110 cm and 110%L!370 cm, the fixed

columns buckle in local-plate (9 plus a bit of 5C6C8)

and distortional (5C6C7 plus a bit of 9C10) modes

displaying 1–12 and 2–6 waves, respectively. Unlike its

simply supported counterpart, the fixed column Ncr vs.

L curve exhibits no local minima — it decreases

monotonically and, in the local-plate and distortional

buckling ranges, tends to the simply supported critical

load values Pcr.LPZ52.0 kN and Pcr.DZ43.5 kN. In

particular, notice that no fixed column buckles in a

single-wave distortional mode — the warping restraint

considerably increase the distortional stiffness near the

column supports.

(iii) For 370%L!500 cm and 500%L!2000 cm, buckling

takes place in single-wave flexural–torsional–distor-

tional (2C3C4 plus a bit of 6C7) and flexural–

torsional (2C3C4) modes — recall that, due to the

cross-section ‘slight asymmetry’, the participation of

mode 3 is rather minute.

(iv) The comparison between the two Ncr vs. L curves

clearly shows that, besides the expected critical load

increase, fixing the column end sections leads to a

change in the buckling mode nature and/or longitudinal

configuration (wave number and modal decomposition)

. In order to illustrate these changes, one compares the

buckling behaviours of simply supported and fixed

columns with LZ200 cm. While the former buckles at

PcrZ36.3 kN in a single-wave flexural–torsional–

distortional mode, which combines deformation

Fig. 14. In-plane deformed configurations of all but the first I-section deform

modes 2 (25%), 3 (5%), 4 (45%), 6 (19%) and 7

(5%), the latter exhibits a distortional buckling mode,

occurring for PcrZ46.2 kN and involving modes 5

(41%), 6 (33%), 7 (23%) and 9 (3%). Moreover,

Fig. 13(c) underlines the differences between the

configurations of the corresponding amplitude func-

tions: (iv1) only single-wave sinusoidal functions in the

simply supported column and (iv2) periodic functions

with three ( ~f5 and ~f9) or five ( ~f6 and ~f7) unequal

waves in the fixed column.

4.2. I-Section beams

Taking into account the discretisation shown in Fig. 10(b) (8

independent natural nodes and 11 intermediate nodes), one is

led to 19 deformation modes — with the exception of the first

one (axial extension), their in-plane deformed configurations

are displayed in Fig. 14. It is worth noting that the inclusion of

the flange end stiffeners is responsible for the existence of 4

distortional modes.

Fig. 15 shows numerical results concerning the buckling

behaviour of simply supported beams subjected to uniform

positive and negative major axis bending — all the

conventions adopted in Fig. 12 are retained. The two sets of

curves depicted in Fig. 15(a) provide the variation of the beam

positive and negative bifurcation moment Mb with the length

L, for (i) single-wave (nwZ1) and (ii) critical (any nw)

buckling. In the first case, the GBT analyses included, once

more, either (i) all deformation modes or (ii) just a few

selected ones. This figures also includes several Mb values,

obtained through Cufsm2.6 finite strip analyses and used to

validate the GBT-based results. As for Fig. 15(b) and (c), they

present (i) the two single-wave buckling modal participation

diagrams (MbO0 and Mb!0) and (ii) FEM-based local-plate

(MbO0 and Mb!0) and distortional (MbO0) buckling mode

shapes — they correspond to beams with lengths equal to 17,

ation modes: rigid-body (2–4), distortional (5–8) and local-plate (9–19).

Fig. 15. Buckling behaviour of simply supported I-section beams: (a) Mb vs. L curves and (b) modal participation diagram pi vs. L (nwZ1) for MbO0 and Mb!0 and

(c) typical local-plate (MbO0 and Mb!0) and distortional (MbO0) buckling mode shapes.


24 and 170 cm.26 The observation of these results leads to the

following conclusions:

(i) Once more, a nearly perfect coincidence exists

between the GBT-based and finite strip results.

(ii) The critical buckling curves concerning positive and

negative moments are qualitatively different. While

the MbO0 curve exhibits three distinct zones,

corresponding to 1–6 wave local-plate buckling, 1–

11 distortional buckling wave and single-wave

flexural–torsional buckling, its Mb!0 counterpart

only exhibits two zones, associated with 1–42 wave

local-plate buckling and single-wave flexural–torsional

buckling27 (distortional buckling is never critical).

Moreover, Fig. 15(c) shows that the local-plate

buckling mode is triggered (ii1) by flange buckling

for MbO0 and (ii2) by web buckling for Mb!0 —

26 Note that distortional buckling is never critical for Mb!0 — see Fig. 15(a).27 This beam instability phenomenon is also commonly designated as ‘lateral-

torsional buckling’.

Fig. 15(b) also provides clear evidence of this fact, as

the contribution of deformation mode 9 is considerably

larger in the latter case.

(iii) For MbO0, the single-wave buckling curve exhibits

local minima at Lz17 and Lz170 cm, (iii1) the

former corresponding to a local-plate buckling mode

that combines three dominant modes (9, 10 and 11 —

fairly equal participations) with small contributions

from modes 14 and 16, and (iii2) the latter associated

with a ‘pure’ distortional buckling mode that combines

modes 5 (clearly predominant) and 6.

(iv) For Mb!0, the single-wave buckling curve exhibits

only one local minimum at Lz24 cm, which

corresponds to a local-plate buckling mode that

combines a dominant contribution of mode 9 with

decreasingly important participations of modes 10, 11,

13, 14 and 18. Note also that the negative critical

moment is equal to McrZ14.1 kNm, about 23% below

its positive counterpart (McrZ18.2 kNm) — recall that

the local-plate buckling modes are triggered by

different walls.


(v) Due to the cross-section major axis asymmetry, the

single-wave flexural–torsional buckling modes corre-

sponding to the final descending branches of the two

critical Mb vs. L curves combine modes 3 and 4 in

different ways and torsion is slightly more relevant for

Mb!0 — this asymmetry is also responsible for the

fact that the negative critical moments are always

smaller than the positive ones. At this point, it is worth

noting that the very small participations of mode 4

appearing Fig. 15(b), which are somewhat misleading,

stem from the fact that the method employed to

evaluate the participation factors pi ‘penalises’ the

value of p4 [25].

(vi) As in the case of the E-section columns, several

portions of the single-wave buckling curve can be very

accurately approximated by means of analyses that

include only a few selected deformation modes.

Indeed, Fig. 15(a) shows that analyses involving only

modes 9C10C11, 5C6 or 3C4 yield virtually exact

results for the length ranges associated with local-

plate, distortional and flexural-torsional buckling,

respectively.

5. Conclusion

Following a brief overview of the conventional first and

second-order GBT, which devoted some attention to a recent

one-dimensional finite element numerical implementation, this

paper presented the derivation, validated and illustrated the

application of a more general GBT formulation, in the sense

that it covers isotropic thin-walled members with arbitrarily

branched open cross-sections. With respect to the conventional

GBT, the novel approach involves only modifications in the

cross-section analysis, namely (i) the choice and characteris-

ation of the most appropriate ‘elementary warping functions’

and (ii) the determination of the ‘initial flexural shape

functions’. Concerning these two aspects, the following

methodologies were proposed:

(i) View any branched cross-section as a combination of (i1)

an unbranched sub-section with (i2) an ordered sequence

of branches. This makes it easy to distinguish between

the dependent and independent natural nodes — the

former are the second nodes of all branches and only the

latter are associated with ‘elementary warping functions’

(for each one, it is necessary to calculate the dependent

nodal warping values).

(ii) Determine the ‘initial flexural shape functions’ by means

of the displacement method, taking advantage of the fact

that only a single kinematical unknown is associated

with each branching node, regardless of the number of

branches emerging from it.

In order to illustrate the application of the derived GBT

formulation, several numerical results were presented and

discussed in great detail. These results concerned the local-

plate, distortional and global buckling behaviour of (i) simply

supported and fixed columns with an asymmetric E-section and

(ii) simply supported I-section beams with unequal stiffened

flanges under positive and negative major axis bending. For

validation purposes, a large number of these results were

compared with values obtained through ‘exact’ finite strip or

shell finite element analyses — a virtually perfect coincidence

was found in all cases.

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http://www.ce.jhu.edu/bschafer/cufsm

Documents

GBT Formulation to Analyse the Buckling Behaviour of Thin-walled