First-order, buckling and post-buckling behaviour of GFRP ...

Computers and Structures 89 (2011) 2065–2078

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Computers and Structures

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First-order, buckling and post-buckling behaviour of GFRP pultruded beams. Part2: Numerical simulation

N.M.F. Silva, D. Camotim ⇑, N. Silvestre, J.R. Correia, F.A. BrancoDepartment of Civil Engineering and Architecture, ICIST, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal

a r t i c l e i n f o a b s t r a c t

Article history:Received 6 January 2010Accepted 1 July 2011Available online 3 August 2011

Keywords:GFRP pultruded profilesFlexural behaviourLocal bucklingLateral-torsional bucklingLateral-torsional post-bucklingGeneralised Beam Theory

0045-7949/$ - see front matter � 2011 Civil-Comp Ltdoi:10.1016/j.compstruc.2011.07.006

⇑ Corresponding author. Tel.: +351 21 841 8403; faE-mail address: [email protected] (D. Camo

This work deals with the numerical evaluation of the structural response of simply supported (transver-sally loaded at mid-span) and cantilever (subjected to tip point loads) beams built from a commercialpultruded I-section GFRP profile. In particular, the paper addresses the beam (i) geometrically linearbehaviour in service conditions, (ii) local and lateral-torsional buckling behaviour, and (iii) lateral-torsional post-buckling behaviour, including the effect of the load point of application location. Thenumerical results are obtained by means of (i) novel Generalised Beam Theory (GBT) beam finite elementformulations, able to capture the influence of the load point of application, and (ii) shell finite elementanalyses carried out in the code ABAQUS. These numerical results are compared with (i) the experimentalvalues reported and discussed in the companion paper (Part 1) and (ii) values provided by analyticalformulae available in the literature.

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1. Introduction

Thin-walled pultruded composite profiles made of Glass FibreReinforced Polymers (GFRP) are being increasingly used in the con-struction industry, namely in corrosive environments or in applica-tions requiring fast and light construction [1–7]. However, themechanical properties exhibited by the GFRP material, namelythe high strength-to-stiffness ratio and low shear modulus,render such profiles prone to several specific structural vulnerabil-ities – e.g., brittle rupture, excessive deformations under serviceconditions or high susceptibility to local and global bucklingphenomena. Furthermore, the scarce available specifications and/or guidelines for the design of GFRP members are either tooconservative or incomplete [8–10], thus making the proper use ofthese profiles highly dependent on experimental studies and/orreliable numerical analysis tools.

It is well known that thin-walled beams subjected to major axisbending buckle globally in lateral-torsional modes and the corre-sponding post-buckling behaviour is only marginally stable, whichmeans that the buckling moment/load may be viewed as an upperbound for the elastic strength – in other words, the (elastic) col-lapse of an initially imperfect beam involves large displacementsand occurs for a moment/load close to its critical value. On theother hand, it is also widely recognised that the (vertical) distancefrom the point of load application to the cross-section shear centre

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strongly affects the beam buckling behaviour (buckling loads andmode shapes). Although this effect has been properly quantifiedin the context of steel beams for decades (e.g., [11]), the same isnot true for FRP composite beams – indeed, it was only a few yearsago that Turvey [12] addressed this issue: he conducted an exper-imental, analytical and numerical investigation on the lateral-torsional buckling of FRP pultruded I-section cantilevers acted bypoint loads applied at the end section top flange, bottom flangeand shear centre. It is still worth mentioning the numerical andexperimental work carried out by Samanta and Kumar [13] – how-ever, this work dealt only with distortional buckling. Mottram [14]adapted a formula to calculate critical (lateral-torsional) bucklingmoments in steel beams, included in the ENV (European Pre-Norm)version of Eurocode 3 [15], making it applicable to pultrudedbeams. He used appropriate elastic moduli, as proposed by Bauldand Tzeng [16], and compared the results with values yielded bythe finite difference method. In the context of Generalised BeamTheory (GBT), Gonçalves and Camotim [17] were the first to pro-pose a formulation able to capture the effect of the location ofthe load application point in steel beams. Quite recently, Silvaet al. [18] developed a similar, but more general, approach that isalso capable of capturing this effect in local, distortional and local-ised buckling analyses. The use of GBT to analyse the geometricallylinear and buckling behaviour of composite FRP members is fairlyrecent and due to the research activity of Silvestre and Camotim[19–21] and Silva et al. [22].

Concerning local buckling in FRP members, several researchersconducted numerical and experimental investigations [23–29].Kollár [23] developed simple analytical formulae to estimate the

ved.

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Fig. 1. (a) Arbitrary laminated FRP thin-walled prismatic member and (b) wallelement, together with the corresponding local coordinate system.

2066 N.M.F. Silva et al. / Computers and Structures 89 (2011) 2065–2078

critical local buckling loads of orthotropic beams and columns withI and box cross-sections, which are based on plate solutions andtake into account the rotational restraints due to the walls adjacentto the ‘‘critical wall’’, i.e., that triggers the local buckling. These for-mulae provide rather accurate local critical buckling load esti-mates, when compared with other similar analytical tools [30].

Although there is work available on the use of GBT to analysethe elastic post-buckling behaviour of thin-walled members, al-most all of it was carried out in the context of cold-formed steelmembers, which means that only isotropic materials were dealtwith – aside from the pioneering work of Miosga [31], nowadaysonly of historical interest, it is worth mentioning the studies of(i) Silvestre and Camotim [32], who developed and implementeda beam finite element formulation to analyse geometrically imper-fect members with unbranched open cross-sections,1 (ii) Simão[35], who presented a Rayleigh–Ritz implementation of a formula-tion applicable to geometrically perfect members with arbitrarycross-sections, and (iii) Gonçalves [36], who developed (but didnot implement) a formulation for geometrically imperfect memberswith arbitrary cross-sections. As far as orthotropic thin-walled mem-bers are concerned and to the authors best knowledge, the only GBT-based numerical post-buckling results available were obtained bySilva et al. in the context of FRP composite thin-walled columns withopen [37] and arbitrary [38] cross-sections, based on a beam finiteelement implementation of a geometrically non-linear GBT formula-tion derived a few years ago by Silvestre [39].2

In Part 2 of this two-part paper, the GBT formulations men-tioned in the previous paragraphs are extended in order to be ableto capture the influence of the load application point location onthe local and global buckling and post-buckling behaviours ofFRP composite thin-walled members. Moreover, these formula-tions are based on a novel cross-section analysis approach recentlydeveloped by the authors [41]. Then, GBT-based finite elementmodels are employed to analyse the first order, buckling andpost-buckling behaviour of the GFRP pultruded I-section simplysupported beams and cantilevers whose experimental tests werereported in the Part 1 of this paper [7]. One obtains (i) first order(linear) deformed configurations, (ii) critical moments/load values,together with the corresponding buckling mode shapes, and (iii)non-linear (post-buckling) equilibrium paths and deformed config-urations. While the buckling results concern the (i) local and lat-eral-torsional buckling of a simply supported beam acted by amid-span transverse point load and (ii) lateral-torsional bucklingof a cantilever acted by a tip point load (the lateral-torsional buck-ling results are associated with point loads applied at the beam/cantilever top flange, bottom flange and shear centre), only canti-lever (lateral-torsional) post-buckling analyses are carried out –it is worth pointing out that the latter are still preliminary resultsof an on-going investigation on the development and implementa-tion of a non-linear GBT formulation for FRP composite members(see [42]).3 For validation purposes, the GBT-based first-order,buckling and post-buckling results are compared with (i) theexperimental data presented in the companion paper (Part 1)and also with (ii) values (ii1) yielded by shell finite element analy-

1 Very recently, Basaglia et al. [33,34] have extended this work to cover memberswith arbitrary open cross-sections and also the presence of localised displacementrestraints (e.g., those due to bracing).

2 Some numerical results obtained by means of this formulation were reported bySilvestre and Camotim [40] – however, they were never published with sufficientdetail.

3 The GBT beam post-buckling results presented here are the first ones concerningFRP composite members under non-uniform bending and/or subjected to transverseloads applied off the cross-section shear centre. How to account adequately for theinfluence of these two aspects, which have already been mastered in the context ofbuckling analyses, is a topic of current investigation – this is why the above post-buckling results are termed ‘‘preliminary’’.

ses carried out in the code ABAQUS [43] and (ii2) provided by analyt-ical formulae developed by Mottram [14] (lateral-torsionalbuckling) and Kollár [23] (local buckling).

2. Generalised Beam Theory

2.1. Geometrically non-linear formulation

Consider the supposedly arbitrary laminated FRP thin-walledprismatic member shown in Fig. 1(a), together with its global coor-dinate system X–Y–Z (X is the longitudinal axis). As for Fig. 1(b), itdepicts a wall element and the corresponding local coordinate sys-tem x–s–z (x, s and z are oriented along the member longitudinalaxis, cross-section mid-line and wall thickness) – u, v and w arethe corresponding (local) displacement components.

In a GBT analysis, the displacement components of the membermid-surface are expressed as products of two functions, eachdepending on a single coordinate: either (i) s (the deformationmode shapes uk(s), vk(s) and wk(s)) or (ii) x (the modal amplitudefunctions /k(x)). Then, one has

uðx; sÞ ¼ ukðsÞ/k;xðxÞ vðx; sÞ ¼ vkðsÞ/kðxÞ wðx; sÞ ¼ wkðsÞ/kðxÞ;ð1Þ

where the comma indicates differentiation w.r.t. x. In order to incor-porate the effect of the initial geometrical imperfections into themember post-buckling analysis, it is convenient to express themalso in modal form, i.e.,

�uðx; sÞ ¼ ukðsÞ�/k;xðxÞ �vðx; sÞ ¼ vkðsÞ�/kðxÞ �wðx; sÞ ¼ wkðsÞ�/kðxÞ;ð2Þ

where functions �/kðxÞ provide the variation of the modal imperfec-tion amplitudes along the member length. As in most thin-walledmodels, Kirchhoff’s hypotheses are deemed valid (fibres initiallynormal to the plate mid-plane are inextensible and remain normalto that mid-plane after deformation – cxz = csz = ezz = 0, i.e., negligi-ble wall bending transverse extensions and shear strains), thusmeaning that exx, ess, cxs are the only strain components considered.One adopts here a simplified version of the Green–Lagrange strain–displacement relations, which include only the most relevant non-linear terms – those appearing in the engineering strain definition[37].4 Taking into account these assumptions, the strain componentsread

4 It is again worth noting that the influence of this simplifying assumption on thepost-buckling behaviour of FRP composite members (beams) under non-uniformbending and/or acted by transverse loads applied off the shear centre has never beeninvestigated in the context of GBT-based analyses. As discussed in some detail inSection 3.5.2, it seems that additional non-linear terms will have to be included in thestrain-displacement relations.

N.M.F. Silva et al. / Computers and Structures 89 (2011) 2065–2078 2067

exx ¼ u;x � zw;xx þ12

v2;x þw2

;x

� �� u;x þ z �w;xx �

12

�v2;x þ �w2

;x

� �;

ess ¼ v ;s � zw;ss þ12

u2;s þw2

;s

� �� v ;s þ z �w;ss �

12

�u2;s þ �w2

;s

� �;

cxs ¼ u;s þ v ;x � 2zw;xs þw;xw;s � �u;s � �v ;x þ 2z �w;xs � �w;x �w;s:

ð3Þ

Since most laminated FRP materials employed in structuralapplications (e.g., GFRP or CFRP ones – glass or carbon fibres)exhibit a linear elastic behaviour, one may express the strainenergy directly in terms of the strain and stress components as

U ¼ 12

ZVrijeijdV ¼ U1 þ U2 þ U3 � U1 � U2 � U3; ð4Þ

where Uk and Uk are the strain energy terms associated with the to-tal deformation and the initial geometrical imperfections, respec-tively – they are given by

U1 ¼Z

L

12

Chi/h;xx/i;xx þ12

Bhi/h/i þ12

Dhi/h;x/i;x þ Ehi/h;xx/i

�

þ Hhi/h;xx/i;x þ Fhi/h/i;x

�dx;

U2 ¼Z

L

12

Chij/h;xx/i;x/j;x þ12

Bhij/h/i/j þ Dhij/h;x/i;x/j

�

þ 12

EIhij/h;xx/i/j þ

12

EIIhij/h/i;x/j;x þ HI

hij/h;xx/i;x/j

þ 12

HIIhij/h;x/i;x/j;x þ FI

hij/h/i;x/j þ12

FIIhij/h;x/i/j

�dx;

U3 ¼Z

L

18

Chijk/h;x/i;x/j;x/k;x þ18

Bhijk/h/i/j/k þ12

Dhijk/h;x/i/j;x/k

�

þ 14

Ehijk/h;x/i;x/j/k þ12

Hhijk/h;x/i;x/j;x/k þ12

Fhijk/h/i/j;x/k

�dx;

ð5Þ

U1 ¼Z

LChi/h;xx

�/i;xx þ Bhi/h�/i þ Dhi/h;x

�/i;x þ Ehi/h;xx�/i

�þ Eih/h

�/i;xx þ Hhi/h;xx�/i;x þ Hih/h;x

�/i;xx

þ Fhi/h�/i;x þ Fih/h;x

�/i�dx;

U2 ¼Z

L

12

Chij/h;xx�/i;x

�/j;x þ12

Bhij/h�/i

�/j þ Dhij/h;x�/i;x

�/j

�

þ 12

EIhij/h;xx

�/i�/j þ

12

EIIhij/h

�/i;x�/j;x þ HI

hij/h;xx�/i;x

�/j

þ 12

HIIhij/h;x

�/i;x�/j;x þ FI

hij/h�/i;x

�/j þ12

FIIhij/h;x

�/i�/j

þ 12

Cjhi/h;x/i;x�/j;xx þ

12

Bjhi/h/i�/j þ Djhi/h;x/i

�/j;x

þ 12

EIjhi/h/i

�/j;xx þ12

EIIjhi/h;x/i;x

�/j þ HIjhi/h;x/i

�/j;xx

þ 12

HIIjhi/h;x/i;x

�/j;x þ FIjhi/h;x/i

�/j þ12

FIIjhi/h/i

�/j;x

�dx;

U3 ¼Z

L

14

Chijk/h;x/i;x�/j;x

�/k;x þ14

Bhijk/h/i�/j

�/k þ Dhijk/h;x/i�/j;x

�/k

�

þ 14

Ehijk/h;x/i;x�/j

�/k þ14

Ejkhi/h/i�/j;x

�/k;x

þ 12

Hhijk/h;x/i;x�/j;x

�/k þ12

Hjkhi/h;x/i�/j;x

�/k;x

þ 12

Fhijk/h/i�/j;x

�/k þ12

Fjkhi/h;x/i�/j

�/k

�dx; ð6Þ

where (i) the components of the second, third and fourth order ten-sors C, B and D stem from the longitudinal extensions, transverse

extensions and shear strains, and (ii) those of E, H and F are dueto couplings between (ii1) longitudinal and transverse extensions,(ii2) longitudinal extensions and shear strains and (ii3) transverseextensions and shear strains - these components are obtainedthrough the expressions

Chi ¼Z

b

Zt

Q11ðuh � zwhÞðui � zwiÞdzds;

Bhi ¼Z

b

Zt

Q 22ðvh;s � zwh;ssÞðv i;s � zwi;ssÞdzds;

Dhi ¼Z

b

Zt

Q 66ðuh;s þ vh � 2zwh;sÞðui;s þ v i � 2zwi;sÞdzds;

Ehi ¼Z

b

Zt

Q 12ðuh � zwhÞðv i;s � zwi;ssÞdzds;

Hhi ¼Z

b

Zt

Q 16ðuh � zwhÞðui;s þ v i � 2zwi;sÞdzds;

Fhi ¼Z

b

Zt

Q 26ðvh;s � zwh;ssÞðui;s þ v i � 2zwi;sÞdzds;

ð7Þ

Chij ¼Z

b

Ztðuh � zwhÞðQ 11ðv iv j þwiwjÞ þ Q 12ui;suj;sÞdzds;

Bhij ¼Z

b

Zt

Q 22ðvh;s � zwh;ssÞwi;swj;sdzds;

Dhij ¼Z

b

Zt

Q 66ðuh;s þ vh � 2zwh;sÞwiwj;sdzds;

EIhij ¼

Zb

Zt

Q 12ðuh � zwhÞwi;swj;sdzds;

EIIhij ¼

Zb

Ztðvh;s � zwh;ssÞðQ 12ðv iv j þwiwjÞ þ Q 22ui;suj;sÞdzds;

HIhij ¼

Zb

Zt

Q 16ðuh � zwhÞwiwj;sdzds;

HIIhij ¼

Zb

Ztðuh;s þ vh � 2zwh;sÞðQ 16ðv iv j þwiwjÞ þ Q26ui;suj;sÞdzds;

FIhij ¼

Zb

Zt

Q26ðvh;s � zwh;ssÞwiwj;sdzds;

FIIhij ¼

Zb

Zt

Q26ðuh;s þ vh � 2zwh;sÞwi;swj;sdzds;

ð8Þ

Chijk ¼Z

b

Ztððvhv i þwhwiÞðQ 11ðv jvk þwjwkÞ þ Q 12uj;suk;sÞ

þ uh;sui;sðQ12ðv jvk þwjwkÞ þ Q22uj;suk;sÞÞdzds;

Bhijk ¼Z

b

Zt

Q22wh;swi;swj;swk;sdzds;

Dhijk ¼Z

b

Zt

Q 66whwi;swjwk;sdzds;

Ehijk ¼Z

b

ZtðQ 12ðvhv i þwhwiÞ þ Q 22uh;sui;sÞwj;swk;sdzds;

Hhijk ¼Z

b

ZtðQ 16ðvhv i þwhwiÞ þ Q 26uh;sui;sÞwjwk;sdzds;

Fhijk ¼Z

b

Zt

Q 26wh;swi;swjwk;sdzds;

ð9Þ

where Qij are the (reduced stiffness) components of the ply plane-stress constitutive law, expressed in the x–s–z coordinate systemas (e.g., [44]), given by

rxx

rss

sxs

8><>:

9>=>; ¼

Q11 Q12 Q 16

Q12 Q22 Q 26

Q16 Q26 Q 66

264

375

exx

ess

cxs

8><>:

9>=>;: ð10Þ

Having presented the GBT fundamentals, it is now worth mention-ing that all GBT analyses comprise (i) a cross-section analysis, aimedat determining the deformation mode shapes uk(s), vk(s) and wk(s)


(k = 1, . . ., n, where n is the number of deformation modes consid-ered) and evaluating the associated modal mechanical properties(i.e., the components of matrices C, B, D, E, H and F), and (ii) a mem-ber analysis, in order to solve the first-order, buckling or post-buck-ling problem under consideration. Detailed accounts of the conceptsand procedures involved in the performance of the above analysescan be found in Refs. [32,33,38].5

2.2. Non-linear beam finite element

The member analysis is performed by means of a GBT-basedbeam (one-dimensional) finite element with length Le, whose de-grees of freedom (d.o.f.) are the nodal values and derivatives ofthe deformation mode amplitudes. These deformation modes exhi-bit different natures – indeed, one considers (i) global (axial exten-sion, major/minor axis bending and torsion), (ii) distortional, (iii)local, (iv) shear and (v) transverse extension modes. The longitudinalamplitude functions /k(x) of the deformation modes involvingtransverse displacements v and/or w (i.e., all but the axial exten-sion and shear modes) are approximated by means of linear com-binations of Hermite cubic polynomials wa(n) (a = 1, . . ., 4 � n = x/Le), given by

w1ðnÞ ¼ Leðn3 � 2n2 þ nÞ w2ðnÞ ¼ 2n3 � 3n2 þ 1;

w3ðnÞ ¼ Leðn3 � n2Þ w4ðnÞ ¼ �2n3 þ 3n2;ð11Þ

where, as is usually done, the rotational d.o.f. associated with w1(n)and w3(n) are shared by adjacent finite elements. Therefore, one has

/hðxÞ ¼ waðx=LeÞdha: ð12Þ

As for the axial extension and shear deformation modes, linearcombinations of Hermite cubic polynomials are now used toapproximate the amplitude function derivatives /k,x(x), whichmeans that the corresponding ‘‘rotational’’ (or derivative) d.o.f.are not shared by the adjacent finite elements6 – this approach,which is equivalent to using Lagrange cubic polynomials (as donein [41]) leads to

/h;xðxÞ ¼ waðx=LeÞdha: ð13Þ

In order to perform a member geometrically non-linear analysisby means of an incremental-iterative technique based on Newton–Raphson’s method, one must calculate the tangent stiffness matrixassociated with an arbitrary deformed configuration. Since thefinite element tangent stiffness matrix components are the secondderivatives of the strain energy (see (4)), with respect to the d.o.f.,they can be obtained through the expression

TðeÞhiab ¼ T1hiab þ T2

hiab þ T3hiab � T2

hiab � T3hiab; ð14Þ

where each component is given by

T1hiab ¼ Chik22 þ Bhik00 þ Dhik11 þ phiEhik20 þ phiHhik21 þ phiFhik01;

T2hiab ¼

12

phijChijk211 þ12

phijBhijk000 þ phijDhijk110 þ12

phijEIhijk200

�

þ 12

phijEIIhijk011 þ phijH

Ihijk210 þ

12

phijHIIhijk111

þ phijFIhijk010 þ

12

phijFIIhijk100

�djg;

5 Although the cross-section analysis approach employed is that reported in [41], itis worth mentioning that, for open-section members, it coincides with the one usedby Basaglia et al. [33,34] – indeed, the two approaches only differ for cross-sectionsexhibiting closed cells, which are not dealt with in this work.

6 For these deformation modes, it is not necessary to know the amplitude functionsthemselves, since they always multiply null functions (see (1)).

T3hiab ¼

116

phijkChijkk1111 þ1

16phijkBhijkk0000 þ

14

phijkDhijkk1010

�

þ 18

phijkEhijkk1100 þ14

phijkHhijkk1110 þ14

phijkFhijkk0010

�djgdkl;

ð15Þ

T2hiab ¼ Cjhik112 þ Bjhik000 þ phiDjhik101 þ EI

jhik002 þ EIIjhik110

�

þ phiHIjhik102 þ HII

jhik111 þ phiFIjhik100 þ FII

jhik001

��djg;

T3hiab ¼

12

Chijkk1111 þ12

Bhijkk0000 þ phiDhijkk1010 þ12

Ehijkk1100

�

þ 12

Ejkhik0011 þ12

Hhijkk1110 þ12

Hjkhik1011

þ 12

Fhijkk0010 þ12

Fjkhik1000

��djg

�dkl; ð16Þ

where (i) the components of tensors C, B, D, E, H and F were definedin (7)–(9), (ii) the d and �d components are the finite element d.o.f.associated with the total deformation and initial geometrical imper-fections, (iii) the (h, i, j, k) and (a, b, g, l) subscripts identify thedeformation modes and shape functions, (iv) p is an operatorstanding for the sum of all permutations of the variables includedin (the associated greek variables permute similarly),7 and (v)

kab ¼Z

Le

dawa

dxa

dbwb

dxbdxkabc ¼

ZLe

dawa

dxa

dbwb

dxb

dcwg

dxc dxk20;

kabcd ¼Z

Le

dawa

dxa

dbwb

dxb

dcwg

dxc

ddwl

dxddx ð17Þ

are integrals of products of shape functions wa(x) and/or their deriv-atives. In order to apply the Newton–Raphson method, one mustalso calculate the finite element internal force vector, defined asthe first derivative of the strain energy and given by

f ðeÞha ¼ T1hiabdib þ

12

T2hiabdib þ

13

T3hiabdib � T1

hiab�dib � T2

hiabdib � T3hiabdib

� 12

Chijk211 þ12

Bhijk000 þ Dhijk110 þ12

EIhijk200 þ

12

EIIhijk011

�

þ 12

HIhijk210 þ

12

HIIhijk111 þ

12

FIhijk010 þ

12

FIIhijk100

��dib

�djg: ð18Þ

2.3. First-order analysis

A geometrically linear (or first-order) analysis can be performedby using the linearised expression of the tangent stiffness matrixpresented in (14). The displacement vector d0, due to a geometri-cally linear analysis corresponding to a reference loading f0, is gi-ven by

d0 ¼ ðT1Þ�1f 0; ð19Þ

where T1 � K is the assembled tangent stiffness matrix constantcomponent given in (15).

2.4. Buckling analysis

In order to perform a buckling analysis, one must begin bydetermining the global geometric stiffness matrix. In a GBT-basedbuckling analysis, this matrix may be computed by means of twodistinct approaches, which require either (i) providing the longitu-dinal variations of the pre-buckling modal stress resultants Wk(x)

7 Einstein’s sum convention does not apply to the indices of p .

10

20

1

1

[cm]

123

456

78

9

10

11

1213

E1=32.8 GPaE2=7.5 GPaG12=3.5 GPa

ν12=0.28

Natural

Nodes:

Intermediate

(b)(a)

Fig. 2. I-section GFRP pultruded beam (a) cross-section geometry and materialproperties and (b) GBT cross-section nodal discretisation.

2 3 4 5 6 7


or (ii) performing a preliminary first-order analysis to obtain thestress profile associated with the reference loading f0. The latterapproach was employed in this work, since it is considerably sim-pler in the presence of loadings and/or boundary conditions thatgenerate complex stress resultants. In general, there is a stressresultant associated with each deformation mode k, which variesalong x.

The geometric matrix G is then calculated by means of theexpression

G ¼ �T2ðd0Þ; ð20Þ

where T2(d) is the assembled tangent stiffness matrix linear compo-nent given in (15). Then, one may readily solve the buckling eigen-problem defined by

ðK � kiGÞdi ¼ 0; ð21Þ

where the vector di components provide the shape of bucklingmode i, corresponding to the buckling load kif0.

It is worth mentioning that the imperfections included in theGBT post-buckling analyses are based on the buckling mode shapesobtained by means of the methodology just described – as it is wellknown, the most detrimental imperfection configuration is oftenthe critical buckling mode shape (if not, it can always be expressedas a linear combination of the first few buckling mode shapes).

2.5. Post-buckling analysis – incremental-iterative strategy

In order to determine the beam post-buckling equilibriumpaths, one must perform geometrically non-linear analyses, i.e.,solve systems of non-linear equilibrium equations. In this work,this task is performed by means of an incremental-iterative tech-nique based on the well known Newton–Raphson’s method andemploying an adaptive arc-length control strategy8 – the detailsof this incremental-iterative approach, which is not addressed hereand follows closely the one developed by Silvestre and Camotim[32] for cold-formed steel members, can be found in Silva et al.[37]. However, one must stress that there is one important differ-ence, with respect to the existing GBT non-linear (post-buckling) for-mulations and numerical implementations: it lays in the fact thatthird and fourth order D and B matrices, given in (8) and (9), havebeen first incorporated in a (partially successful) attempt to capturethe effects due to non-uniform bending and/or the transverse loadpoint of application location – the relevance of these effects hasalready been studied in the context of the buckling behaviour ofsteel members [18,48] and also of some GFRP pultruded beams(as reported in this work).

3. Numerical results

In this section one presents several linear (first-order), bucklingand post-buckling numerical results obtained through the simula-tion of the experimental tests described in Part 1 of this paper [7]and concerning I-section pultruded GFRP simply supported andcantilever beams with the cross-section dimensions and materialproperties given in Fig. 2(a). Moreover, comparisons betweennumerical, analytical and experimental values are carried out andpossible explanations are provided for some of the discrepanciesfound. The numerical results are obtained primarily throughimplementations of the GBT beam finite element formulations pre-sented in the previous section. However, results yielded by shell fi-

8 Detailed accounts of the concepts and procedures involved are available in theliterature (e.g., [45,46]). In this work, the arc-length value is adjusted after eachincrement, so that the number of iterations required to achieve convergence in anygiven increment tends to 4. Moreover, the tangent stiffness matrix and internal forcevector are ‘‘continuously’’ updated (i.e., after each iteration).

nite element analyses performed in the code ABAQUS [43] are alsopresented, making it possible to assess the accuracy of the GBT-based counterparts.

After showing the GBT cross-section analysis results (most rel-evant deformation mode shapes), the paper addresses the numer-ical simulation of the following simply supported and cantileverbeam experimental tests (the nomenclature of Part 1 of this paper[7] is adopted):

(i) The first-order and local buckling behaviours of the simplysupported beam V1 – this beam has length L = 400 cm, istransversally braced at 4 in-span cross-sections and is actedby a transverse (vertical) point load applied at the mid-spantop flange mid-point.

(ii) The lateral-torsional buckling behaviour of the simply sup-ported beam V2 – the only difference between this beamand beam V1 is the absence of transverse bracing.

(iii) The first-order behaviour of the simply supported beams V2and V3, in order to assess the relevance of the shear defor-mation contribution to their mid-span vertical displace-ments - beams V3 and V2 only differ in length: L = 144 cmvs. L = 400 cm.

(iv) The lateral-torsional buckling and post-buckling behavioursof cantilevers with various lengths and/or tip point loadlocations – shear centre and top or bottom flange mid-point.

3.1. GBT cross-section analysis

All the beams analysed here have the material properties andcross-section geometry shown in Fig. 2(a) – (i) elastic constants

10 11 12 13 18 19

Fig. 3. Shapes of the most relevant GBT deformation modes.

40 3×40 2×20 8×10 2×20 3×40 40 [cm]

0 1 2 3 4 5 6 14 15 16 17 18 19 20 8 10 12

7 9 11 13

Fig. 4. Beam V1 longitudinal discretisation and support conditions (GBT analysis).


E1 = 32.8 GPa, E2 = 7.5 GPa, G12 = 3.5 GPa and m12 = 0.289, and (ii) I-section profile with width b = 10 cm, height h = 20 cm, thicknesst = 1 cm. Fig. 2(b) shows the cross-section nodal discretisationadopted to perform the GBT analyses – it involves 6 natural and 7intermediate nodes.

The GBT cross-section analysis leads to 39 deformation modes,the most relevant of which have their shapes depicted in Fig. 3 –they are global (1–4), local (5–15), warping shear (16–27) andtransverse extension (28–39) deformation modes.

Table 1Experimental and numerical mid-span vertical displacements(P = 1 kN).

d (P = 1kN) [mm]

Test GBT ABAQUS

1.78 1.91 1.92– +7.3% +7.9%

Table 2Experimental and numerical critical buckling loads and half-wave lengths.

Pcr (kN) Lhw (cm)

Test Anal. [23] GBT ABAQUS Test GBT ABAQUS

60.2 60.1 64.7 63.9 32.0 32.4 32.5– �0.2% 7.5% 6.1% – 1.2% 1.6%

Fig. 5. (a) GBT buckling mode shape and (b) experimental failure modeconfiguration.

0.81

1.2 i

5

3.2. Beam V1 – first-order and local buckling

The first beam analysed is simply-supported, has a total lengthof LT = 480 cm and a span of L = 400 cm, and is acted by a mid-spantransverse point load applied at the upper flange mid-point. Fig. 4depicts the longitudinal discretisation adopted in the GBT analysis:20 beam finite elements with variable length (a finer mesh wasconsidered in the central region, where the local buckling occurs),amounting to a total 1342 d.o.f.10 In order to prevent the occurrenceof global (lateral-torsional) buckling, the beam is laterally braced atnodes 4, 7, 13 and 16 (see Fig. 4) – the displacements along the majoraxis are fully prevented. The GBT-based results obtained are vali-dated through the comparison with values yielded by ABAQUS shell fi-nite element analyses – the beam was discretised into a mesh of 7689-node elements with 5 d.o.f. per node and reduced integration(S9R5 elements) – a total of 16320 d.o.f. were involved. Each beamsupport prevents the vertical displacement of the bottom flangeand the lateral displacement of both flanges – note that thesedisplacements combine contributions from various GBT deformationmodes (nodal displacements expressed in modal form).

The first-order result obtained for this beam concerns a loadP = 1 kN and consists of the vertical displacement of the mid-spanupper flange mid-point (d) – Table 1 shows the experimental andnumerical (GBT and ABAQUS) values. One observes that:

(i) The numerical GBT and ABAQUS values are practically identi-cal (0.6% difference).

(ii) The experimental value is about 7% below the numericalones. This difference is most likely due to elastic constantdiscrepancies and/or variability of the beam material prop-erties - recall, from Part 1 of this paper, that the GFRP mate-rial exhibited some heterogeneity and the standarddeviation of the elastic constants was not negligible.

Next, Table 2 provides experimental, analytical and numericalvalues of the beam (i) critical (local) buckling load Pcr and (ii) the

9 Although the material is deemed homogeneous, different E1 values wereexperimentally measured during the membrane extension (E1.m = 32.8 GPa) andlongitudinal bending (E1.b = 26.9 GPa) tests. This fairly significant discrepancy must bedue to material across-the-thickness heterogeneity. When plate bending is morerelevant than membrane extension (e.g., in local buckling problems), the modulus E1.b

will be used instead of E1.m.10 All the GBT first-order and buckling analyses performed included the full set of 39

deformation modes associated with the cross-section discretisation shown inFig. 2(b). Because these analyses are very fast even when they include all thedeformation modes, it was not felt necessary to select a subset of the most relevantGBT deformation modes. However, it was also found that such a selection is useful inmore complex problems, namely when performing post-buckling analyses like thosepresented in Section 3.5.2 (see also [47]).

top flange half-wave length of the buckling mode (numerical anal-yses) and failure mode (experimental test) – Fig. 5(a) and (b) depictthese mode shapes (the former yielded by the GBT analysis). Theanalytical values are provided by formulae developed by Kollár[23] for the local buckling of I-section pultruded beams, which ac-count for the rotational restraint between the cross-section walls.As for Fig. 6, it shows the amplitude functions /i(x) of the GBTdeformation modes participating in the beam critical bucklingmode shape. The observation of these results prompts the follow-ing remarks:

(i) The GBT and ABAQUS results (critical buckling loads and half-wave lengths) are again almost identical - the Pcr values dif-fer by 1.3% and the Lhw ones virtually coincide.

(ii) The buckling load estimated from the experimental test is7.5% below the GBT value. Once more, this difference maybe attributed to elastic constant discrepancies and/or

-1.2-1

-0.8-0.6-0.4-0.2

00.20.40.6

0 40 80 120 160 200 240 280 320 360 400 440 480

L [cm]

6

7

Fig. 6. GBT modal decomposition of the beam V1 critical buckling mode shape.

-1.2-1

-0.8-0.6-0.4-0.2

00.20.40.60.8

11.2

0 40 80 120 160 200 240 280 320 360 400 440 480

GBT ABAQUS

d2=0.30 d3=4.00

d2=0.31 d3=4.00

(a) (b)

3

4 20 x 6

L [cm]

i

Fig. 7. (a) GBT modal decomposition of the beam V2 critical buckling mode and (b) GBT and ABAQUS buckling mode configurations of the mid-span cross-section.

Fig. 8. (a) GBT buckling mode shape and (b) experimental failure modeconfiguration.

11 It is worth noting that it is now essential to include the beam outstands in thebuckling analysis, since they have a non-negligible effect on the beam lateral-torsional buckling behaviour – in beam V1, which buckled in a local mode, theinfluence of these outstands was virtually null.

12 As reported in Part 1 of this paper, the lowest buckling load, associated with thelowest (but by no means null) beam lateral restraint at mid-span was achieved byapplying the load through a transverse frame formed by two ‘‘dywidag’’ barsconnected to two metal beams, one located above the beam and the other below thelaboratory floor. The other loading system involved placing an hydraulic jack reactingagainst the top beam of a metal frame and applying the load to the GFRP beam topflange through two 8 cm square steel spreading plates – a metal sphere insertedbetween the plates aimed at minimising the transverse load transmission.


variability of the beam material properties. Moreover, it wasalso observed that local buckling (collapse) was accompa-nied by a fair amount of web crushing in the loaded region,which obviously weakened the beam and reduced the failure(buckling) load.

(iii) The experimental and numerical (local) half-wave lengthsare remarkably similar – differences around 1.5%. Note thatwhile the former corresponds to a failure mode configura-tion, the latter is associated with a buckling mode shape.

(iv) Naturally, the major contributions to the GBT buckling modeshape come from the local deformation modes 5 (50.8%), 6(39.8%), 7 (4.0%), 10 (2.8%) and 11 (0.9%). However, thereare also minute participations from the global modes 3(0.5%) and 4 (0.4%) and the shear modes 18 (0.2%) and 19(0.1%) – the presence of such modes stems most likely fromthe slight lateral motion of the beam flanges between bracedcross-sections.

(v) The most relevant GBT modal amplitude functions, shown inFig. 6, provide fresh insight on the beam local bucklingmechanics. In this particular case, they show how the localdeformation varies along the beam length - it is symmetricand mostly restricted to the mid-span region. Moreover, italso becomes clear that the symmetric deformation modes5 and 7 combine with the anti-symmetric mode 6 to gener-ate fair rotation at the mid-span top flange, while keepingthe bottom flange practically in its initial (undeformed) posi-tion – see Fig. 5(a).

(vi) The analytical result is much closer to the experimental onethan the numerical values (GBT and ABAQUS) - indeed there isa virtually perfect match (difference of 0.2%). This suggeststhat Kollár’s formula may become a very practical and effi-cient tool to estimate local buckling loads in pultrudedbeams. Nevertheless, it should be pointed out that theextreme accuracy of this particular analytical estimate mustbe viewed as a coincidence – in fact, the model underlying

the formula takes less physical phenomena into accountthan the two numerical ones – e.g., it does not consider thenon-linear effects stemming from the web transversal com-pressive stresses.

3.3. Beam V2 – lateral-torsional buckling

The second simply supported beam analysed differs from theprevious one only in the fact that it is not laterally braced betweenthe supports - the removal of the bracings leads to the occurrenceof lateral-torsional buckling. The longitudinal discretisationadopted in the GBT analysis consisted of 12 equal length (40 cm)beam finite elements11 involving the full set of 39 deformationmodes – the total amount of d.o.f. is now equal to 830. The GBT-based results are again validated through the comparison with val-ues yielded by ABAQUS shell finite element analyses – the beam dis-cretisation adopted is that already employed to analyse beam V1(16320 d.o.f.). In addition, one also presents the buckling load valueprovided by the formulae proposed by Mottram [25] (based on anadaptation of the EC3-ENV formula [15]).

The numerical results obtained by means of the GBT and ABAQUS

analyses are displayed in Fig. 7(a) and (b): (i) the GBT modaldecomposition of the beam critical buckling mode (the amplitudefunctions shown are the most relevant and concern modes 3, 4and 6 – the last one is magnified 20 times and is only perceptibleat the outstands) and (ii) the buckled configurations of the beammid-span cross-section yielded by both analyses – d2 and d3 arethe bottom and top flange lateral displacements. The experimentaland numerical (GBT) beam critical (lateral-torsional) bucklingmode shapes are shown in Fig. 8(a) and (b). The observation ofthese results leads to the following remarks:

(i) The critical buckling loads yielded by the GBT (Pcr.GBT =5.09 kN) and shell finite element (Pcr:Abaqus = 5.08 kN) analy-ses are virtually identical. However, they fall well belowthe values obtained experimentally: Pcr.exp varies between13.0 kN and 20.7 kN, depending on the loading systemused12. The experimental buckling load values are 2.5–4 times

Table 3Experimental and numerical shear contributions to the mid-span displacement (%).

Beam V2 (L = 400 cm) Beam V3 (L = 144 cm)

Test GBT Test GBT

12.6% 9.1% 41.1% 43.5%

0

2

4

6

8

10

12

14

16

18

20

2 3 4

GBT (TF)GBT (SC)GBT (BF)Exp. (TF)Exp. (SC)Exp. (BF)Shell FE (ABAQUS)

Pcr

[kN]

L [m]

Fig. 9. Variation of the critical buckling load with the cantilever length, for TF, SCand BF loading – experimental and numerical (GBT and ABAQUS) results.

14 The loading system consisted of dead load placed in a bucket hanging from anaxis that could be moved vertically to change the load point of application (see Part 1of this paper). Unlike their simply supported beam counterparts, this loading system


higher than their numerical counterparts, due to the addi-tional stiffness introduced by the loading systems. Indeed,all of them generated considerable lateral restraint, stemmingfrom either (i1) the connection of the loading frame to the lab-oratory floor or (i2) the friction between the metal sphere andsteel plates – i.e., the tested beams were effectively (partially)laterally restrained at mid-span.

(ii) The GBT analysis shows that beam critical buckling modeshape is clearly lateral-torsional, as attested by the highlydominant participation of deformation modes 3 (52.9%)and 4 (45.4%). Nevertheless, there are also minute contribu-tions from the local modes 6 (1.0%), 5 (0.4%), 10 (0.1%) and 7(0.1%), which are mainly due to the occurrence of localiseddeformations at the support and loading regions, as well asto coupling between longitudinal and transversal bending(due to Poisson effects).

(iii) The beam mid-span cross-section buckled shapes providedby the GBT and ABAQUS analyses are practically identical(see Fig. 7(b)). Fig. 8 also shows that the numerical (GBT –the ABAQUS one is identical) beam critical buckling modeshapes compare quite well with the failure mode configura-tion observed during the experimental test.

(iv) The analytical critical buckling load is Pcr = 4.16 kN, consid-erably lower than the value yielded by GBT and ABAQUS. Thismay be due to the fact that the analytical formula does nottake into account the specific support conditions and beamconfiguration. Indeed, it was determined that the presenceof the beam outstands increased the buckling load by asmuch as 20%, a factor that is obviously not taken intoaccount by the analytical formula.

3.4. Beams V2 and V3 – shear deformation

In order to assess the shear deformation contribution to mid-span vertical displacement of beams V2 and V3, their first orderbehaviours were determined by means of GBT first-order analysesincluding and excluding the shear deformation modes – recall thatbeam V3 only differs from beam V2 in the length: 144 cm insteadof 400 cm. Table 3 shows the experimental and numerical (GBT)results concerning the percentage of the beam mid-span verticaldisplacement stemming from shear deformation.13 One observesthat:

(i) There is a reasonable (beam V2) and fairly good (beam V3)correlation between the experimental and numerical (GBT)beam mid-span vertical displacements. Since (i1) the GBTresults may be viewed as ‘‘exact’’ and (ii) the estimation ofthe experimental values is clearly approximate, it seems fairto say that this kind of agreement is quite encouraging, as faras the experimental assessment technique is concerned.

(ii) This numerical investigation highlights the advantages ofthe GBT modal approach. Indeed, the quantitative assess-ment of the shear deformation contribution to the beam

13 The so-called experimental values are, in fact, analytical estimates obtained (i) bymeans of Timoshenko and Euler-Bernoulli beam theories and (ii) on the basis ofelastic constant (EL and GLT) values determined experimentally under the assumptionof an apparent isotropic behaviour – the details of this approach can be found in thePart 1 of this paper [7].

mid-span vertical displacement on the basis of shell finiteelement results would be virtually impossible. Conversely,it is a straightforward matter when using GBT - it merelysuffices to perform analyses that include or exclude theshear deformation modes.

3.5. Cantilevers – lateral-torsional buckling and post-buckling

The last numerical simulations performed concern the lateral-torsional buckling and post-buckling behaviours of cantileversacted by tip transverse point loads applied at the end section topflange (TF), shear centre (SC) and bottom flange (BF). The cantile-vers exhibit three different lengths (L = 200, 300 and 400 cm), arefully fixed at the support and have the warping of their free endsections prevented – this warping restraint was due to the needto connect them to two rigid 8 mm thick plates (attached to eachside of the web and extending 16 cm beyond the profile extremity),required to implement the loading system.14

One addresses separately the buckling and post-bucklingnumerical results: the former in Section 3.5.1 and the latter in Sec-tion 3.5.2. In both cases, the experimental values are comparedwith estimates obtained by means of both GBT (beam finite ele-ment) and ABAQUS (shell finite element) analyses.

3.5.1. Lateral-torsional bucklingThe GBT analyses included the full set of 39 deformation modes

and were based on cantilever discretisations involving 6 finite ele-ments, corresponding to a total of 455 d.o.f. As for the ABAQUS anal-yses, cantilever discretisations into S9R5 shell elements were againadopted – the number of elements considered varied between 320and 640, corresponding to total amounts of 6800 and 13,600 d.o.f..

The curves shown in Fig. 9 stem from GBT buckling analysesand provide the variation of the critical buckling load Pcr withthe cantilever length L, for the three loading cases dealt with(TF, SC and BF). Also shown are the experimental and ABAQUS

values concerning the cantilevers with L = 200, 300 and 400 cm15

– Table 4 provides all the buckling loads for these three cantilevers,

does not generate a significant lateral restraint at the cantilever end section – thisissue will be further addressed when dealing with the cantilever post-bucklingbehaviour in Section 3.5.2.

15 As described in Part 1 of this paper, the experimental critical buckling loads wereassumed equal to the maximum loads attained by the cantilever load-displacementequilibrium paths – since those displacements are bounded, the true critical loadvalues are, most likely, underestimated.

Table 4Experimental and numerical (GBT and ABAQUS) critical buckling loads.

Load point ofapplication

L (cm) Exp.(kN)

GBT(kN)

ABAQUS

(kN)Dif. GBT/exp. (%)

Dif. ABAQUS/GBT (%)

TF 200 5.97 6.93 6.65 +16.1 �4.0300 2.72 2.66 2.70 �2.2 +1.5400 1.38 1.40 1.41 +1.4 +0.7

SC 200 11.68 12.66 12.65 +8.4 �0.1300 4.06 4.36 4.35 +7.4 �0.2400 1.98 2.10 2.09 +6.1 �0.5

BF 200 13.98 18.18 17.94 +30.0 �1.3300 5.33 5.90 5.87 +10.7 �0.5400 2.50 2.70 2.69 +8.0 �0.3

Table 5Modal decomposition of the GBT-based buckling mode shapes.

Load point ofapplication

L (cm) 3 (%) 4 (%) 5 (%) 6 (%) 7 (%) 18 (%) 19 (%)

TF 200 42.5 55.9 0.7 0.3 – 0.2 0.2300 51.0 48.6 0.1 0.1 – 0.1 0.1400 58.0 41.8 0.1 0.1 – – –

SC 200 62.4 36.3 0.8 0.1 0.1 – 0.1300 67.9 31.7 0.2 0.1 – – –400 72.1 27.7 0.1 – – – –

BF 200 77.2 21.2 1.0 0.2 0.1 – 0.1300 79.8 19.7 0.3 0.1 – – –400 81.6 18.2 0.1 – – – –


as well as some of the differences between them. Fig. 10 and Table5 concern the three above cantilever lengths and load applicationpoints: (i) Fig. 10 shows a comparison between the end sectionbuckled shapes yielded by the GBT and ABAQUS analyses, and (ii)Table 5 gives the modal decomposition of the GBT-based bucklingmode shapes. The observation of all these buckling results promptsthe following comments:

(i) Once more, there is a virtually perfect match between theGBT and ABAQUS results. With one single exception, the criti-cal loads never differ by more than 1.5% – the exception isthe L = 200 cm cantilever loaded at the top flange, for whichthe GBT value overestimates its ABAQUS counterpart by about4% difference. Moreover, there is a remarkable similaritybetween the two sets of end section buckled shapes dis-played in Fig. 10.

(ii) Again with one single exception, the experimental criticalloads fall consistently below the numerical ones – the differ-ences with respect to the GBT values may be quite high(30%– one case), reasonable (16.1% – one case) or fairly to verylow (from 10.7% to 1.4% – six cases). The exception is theL = 300 cm cantilever loaded at the top flange, for whichthe GBT value underestimates the experimental one byabout 2%.

TF

SC

BF

2 m 3 m

GBT ABAQUS GBT

d2=-0.47 d3=3.53

d2=-0.61 d3=3.53

d2=0.10 d3=3.90

d2=0.84 d3=3.16

d2=0.78 d3=3.16

d2=1.07 d3=2.93

d2=1.49 d3=2.51

d2=1.50 d3=2.51

d2=1.54 d3=2.46

Fig. 10. Cantilever end section buckled shapes yielded by the GBT and ABAQUS

(iii) The high overestimation mentioned above concerned theL = 200 cm cantilever loaded at the bottom flange and wasdue to the damage of the cantilever support (visible separa-tion of the top flange), which occurred when the loadreached about 12.5 kN – note that this cantilever has, byfar, the highest critical load. In the remaining cases, the dif-ferences can be logically attributed to a combination ofaspects dealing with the performance of the tests, namely(iii1) lack of full fixity of the support, (iii2) lack of full warp-ing restraint at the free end section, and/or (iii3) the inevita-ble underestimation that was addressed in footnote 13.

(iv) Fig. 10 and Table 5 show that the contribution of torsion tothe cantilever critical buckling mode becomes more relevantas the length decreases and the load application point movesupwards (from BF to TF). It is worth noting that the ten-sioned top flange deflects more than the compressed bottomone – this apparently paradoxical behaviour

(v) stems from the influence of the shear stresses associatedwith the non-uniform bending, and was recently investi-gated by Bebiano et al. [48] in the context of I-section steelcantilevers.

(vi) Table 5 clearly shows that all cantilever critical bucklingmodes have almost exclusively contributions from deforma-tion modes 3 (minor axis bending) and 4 (torsion) – they

4 m

ABAQUS GBT ABAQUS

d2=0.03 d3=3.90

d2=0.46 d3=3.44

d2=0.49 d3=3.44

d2=1.03 d3=2.93

d2=1.24 d3=2.76

d2=1.21 d3=2.76

d2=1.55 d3=2.46

d2=1.58 d3=2.42

d2=1.59 d3=2.42

analyses – d2 and d3 are the bottom and top flange lateral displacements.

0

0.4

0.8

1.2

1.6

0 5 10 150

0.4

0.8

1.2

1.6

0 20 40 600

0.4

0.8

1.2

1.6

0 20 40 60

(a) (b) (c)

Exper.

d2 [mm] d3 [mm]

P [kN] P [kN] P [kN]

GBT 39 modes GBT 18 modes ABAQUS

Exper. correct. Exper.

GBT 39 modesGBT 18 modesABAQUS

Exper.


d1 [mm]

Fig. 11. Experimental and numerical post-buckling equilibrium paths (a) P vs. d1, (b) P vs. d2 and (c) P vs. d3 (L = 4.0 m cantilever with top flange loading).

(a) (b) (c)d2 [mm] d3 [mm]

P [kN]

d1 [mm]

0

0.5

1

1.5

2

2.5

0 20 40 60 80 100

Exper.


P [kN]

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50

Exper.


Exper. correct. 0

0.5

1

1.5

2

2.5P [kN]

0 20 40 60 80 100

Exper.


Fig. 12. Experimental and numerical post-buckling equilibrium paths (a) P vs. d1, (b) P vs. d2 and (c) P vs. d3 (L = 4.0 m cantilever with shear centre loading).

18 Note that the cantilever lateral displacements were bounded during the tests –this explains why the experimental curves shown do not extend as far as thenumerical ones.

19 The displacement d1 due to the flexibility of the cantlever support was obtained


always amount to more than 98.4%. However, it is also pos-sible to detect minute contributions (v1) from the localmodes 5, 6 and 7, due to Poisson effects and localised defor-mations near the cantilever ends, and (v2) from the shearmodes 18 and 19, due to flange lateral bending (recall thatGFRP composite exhibit a low shear-to-longitudinal modu-lus ratio). As expected, the shear deformation is more rele-vant in the shorter beam loaded at the top flange.

3.5.2. Lateral-torsional post-bucklingFinally, one presents GBT beam finite element and ABAQUS shell

finite element results concerning the post-buckling behaviour ofsix of the experimentally tested cantilevers, namely those with(i) L = 4.0 m and top-flange, shear centre and bottom-flange load-ing, (ii) L = 3.0 m and shear centre and bottom-flange loading,and (iii) L = 2.0 m and shear centre loading16. Since there are noexperimental measurements concerning the cantilever initial geo-metrical imperfections (shape and amplitude), it was decided (i) toconsider critical-mode initial imperfection shapes and (ii) to selectthe amplitude values that best fit the test results – it was found thatthis is achieved by taking d3.imp = 0.8 cm (initial lateral displacementat the end section top flange).

Two types of GBT post-buckling analysis were performed: (i)one including all the 39 cross-section deformation modes yieldedby the discretisation shown in Fig. 2(b) and involving 12 beam fi-nite elements (1092 d.o.f.), and (ii) another including only the 18most relevant deformation modes17 (the first 8 conventional, 5shear and 5 transverse extension modes) and involving 6 beam finiteelements (252 d.o.f.). The ABAQUS model is the same of the adoptedfor the buckling analysis, which involves between 6800 and 13,600

16 The three beams associated with the lower quality experimental results wereexcluded.

17 Higher-number deformation modes of each type are less likely to contribute tothe member response.

d.o.f. Moreover, the solution of the non-linear equilibrium equationsystem is obtained by means of an incremental-iterative approachbased on Newton–Raphson’s method and employing Riks’s arc-length control strategy.

Figs. 11–16(a)–(c) show the numerical and experimental equi-librium paths P vs. d1 (top flange vertical displacement), P vs. d2

(bottom flange horizontal displacement) and P vs. d3 (top flangehorizontal displacement) concerning the different cantilevers un-der consideration.18 It is worth noting that the experimental d1 val-ues appearing in these figures were corrected, in order to eliminatethe portion due to the major axis bending action stemming fromthe cantilever support rotational flexibility.19 The observation ofthese results leads to the following remarks:

(i) First of all, the experimental load–displacement curves showthe existence of a mild hysteretic behaviour (different load-ing and unloading curves), which is probably due to theoccurrence of friction in the lubricated surface between therod connected to the steel plates and the loading mechanismchain (see Part 1 of this paper [7]20) – it is worth noting thata similar hysteretic behaviour was reported, although notdiscussed, by Turvey [12]. Since it is obvious that this effectcannot be captured by the beam/shell finite elementanalyses carried out, one expects to find the numericalload–displacement curves located somewhere in-between

by subtracting the theoretical value (based on the experimental Young’s and shearmoduli) from the total (measured) value.

20 Note that this hysteretic behaviour is much less pronounced in the P vs. d1

equilibrium paths, since the vertical displacement is only affected by the frictionbetween the rod and the chain for loads close to their critical values, i.e., when thecross-section starts to exhibit rather large lateral displacements.

(a) (b) (c)d2 [mm] d3 [mm]

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Fig. 13. Experimental and numerical post-buckling equilibrium paths (a) P vs. d1, (b) P vs. d2 and (c) P vs. d3 (L = 4.0 m cantilever with bottom flange loading).

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Fig. 15. Experimental and numerical post-buckling equilibrium paths (a) P vs. d1, (b) P vs. d2 and (c) P vs. d3 (L = 3.0 m cantilever with bottom flange loading).

d2 [mm]d1 [mm]

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their experimental loading and unloading counterparts.Moreover, it is observed that the hysteretic loop is more pro-nounced for the shorter beams (those exhibiting higherloads), which provides additional evidence supporting therelevance of the friction effect (the friction forces are propor-tional to the load).

(ii) The assumed initial geometrical imperfections, which arefairly realistic (critical-mode shape with amplitude com-prised between L/500 and L/250) lead to a quite reasonablequalitative agreement between the numerical and experi-mental load–displacement curves, as shown in Figs. 11–16(a) (vertical displacements) and Figs. 11–16(b) and (c)(lateral/horizontal displacements).

(iii) As it would be logical to expect, the GBT analyses includingless deformation modes and/or beam elements lead to stiffercantilever responses. However, there are a few exceptions,which concern the bottom-flange horizontal displacementd2 (e.g., see Figs. 11(b) and 16(b)), and are probably dueto the fact that slightly different buckling mode shapes wereused to define the initial imperfections. For instance, in thecase of Fig. 11(b), the buckling mode (or initial imperfection)shape obtained with 39 modes involves more torsion thanits 18-mode counterpart – since this additional torsion isnot dominant, it just subtracts a higher value from the dis-placement due to minor axis bending, thus leading to alower d2 value.

(iv) Quantitatively speaking, one observes that, in general, thenumerical curves fall more or less inside the experimentalhysteretic loops (recall that some of them were corrected –see footnote 19). However, in some cases the experimentaldisplacements are clearly larger than the numerical ones(e.g., in the L = 4.0 m cantilevers loaded at the shear centreand bottom-flange – see Figs. 12 and 13). This might bedue to an underestimation of the real initial imperfectionamplitude – as mentioned earlier, no initial imperfectionmeasurements were made and an amplitude correspondingto d3 = 0.8 cm was considered for all cantilevers. Naturally,such imperfection amplitude may have varied with (iv1)the cantilever length, (iv2) the cantilever positioning at thesupport, which was probably different for each case (see Part1 of this paper), and (iv3) the load eccentricity caused by the(geometrically imperfect) load application system.

(v) The horizontal plateau observed in most experimentalcurves, in particular those involving higher loads (e.g., seeFig. 15(b) and (c)) is due to a dynamic response that, obvi-ously, cannot be captured in the numerical models – thisoccurs due to a sudden strain energy release taking placewhen the critical buckling load is reached and the loadingsystem friction is no longer capable of restraining the canti-lever from buckling laterally. During the tests, it wasobserved that a higher friction provided by the loading sys-tem led to a more dynamic response (larger strain energyrelease after buckling).

(vi) In the most advanced post-buckling stages, the majority ofthe GBT-based results clearly overestimate the ABAQUS (shellfinite element) ones – the applied load differences mayreach 15–20%.21 A possible explanation for this significantdiscrepancy may have to do with the fact that the rotationvalues are high enough to fall outside the domain of validity

21 It was found that refining the GBT cross-section (number of deformation modes)and longitudinal (number of beam finite elements) discretisations did not improvethe accuracy of the results. Therefore, the discrepancy between the GBT and Abaqusresults presented must be attributed to an insufficiency of the GBT non-linearformulation that is currently implemented. The authors are currently investigatingthis matter.

of the implemented GBT non-linear formulation (based on asmall rotation assumption). Indeed, the cantilever end cross-section may rotate as much as 16�, before it reaches thedevices (side bars) that provide the lateral constraint.22

The consideration of additional non-linear terms in theGreen–Lagrange strain–displacement relations (i.e., morethan those presented in (3)) should lead to more accurateresults. Nevertheless, it is fair to say that, with a few excep-tions, the estimates yielded by the proposed/implementedGBT formulation are reasonably accurate in the early-to-intermediate post-buckling stages, i.e., up until about 80%of the buckling load is reached – within this applied loadrange the differences, w.r.t. to the values provided by theABAQUS shell finite element analyses, are rather small andrarely exceed 10%. Indeed, it is worth mentioning that inclu-sion of the third and fourth order D and B tensors (matrices)makes it possible to capture partially the effects stemmingfrom the non-uniform bending and the location of the loadpoint of application. Note that, in the context of steel mem-bers, these effects are responsible for buckling load changesthat can be as high as 50% [18,48] – therefore, it is logical toexpect a quantitatively similar influence in the correspond-ing post-buckling equilibrium paths.

(vii) The GBT analyses that include 18 deformation modes, whichinvolve about (vii1) one fourth of the number of d.o.f.required to include 39 deformation modes and (vii2) 4–7%of the shell finite element d.o.f., yield very accurate resultsin the early post-buckling stages (they practically coincidewith those obtained with 39 deformation modes). This pos-sibility of reducing very significantly the number of d.o.f.,without sacrificing too much the accuracy, is a trademarkof GBT that explains its high numerical efficiency.

4. Concluding remarks

This work dealt with the numerical simulation of the first-order,buckling and post-buckling behaviour of simply supported and can-tilever beams built from I-section pultruded GFRP profiles. In partic-ular, the paper addressed the beam (i) linear behaviour in serviceconditions, (ii) shear deformation, (iii) local and lateral-torsionalbuckling behaviour and (iv) lateral-torsional post-buckling behav-iour, including the effects of the load point of application location.The numerical results, obtained by means of (i) novel GBT beam fi-nite element formulations and (ii) ABAQUS shell finite element analy-ses, were compared with the test (experimental) values reported inPart 1 of this paper [7], and also between themselves.

Initially, the paper addressed the main concepts and proceduresinvolved in the formulation and numerical implementation of GBT-based beam finite elements intended to perform FRP compositethin-walled member first-order, buckling and post-buckling (geo-metrically non-linear analyses). Then, these finite element modelswere employed to simulate the tests described in Part 1 of this pa-per, i.e., concerning the GFRP pultruded I-section (i) simply sup-ported beams acted by mid-span point loads (first-order, localbuckling and lateral-torsional buckling) and (ii) cantilevers actedby tip point loads applied at different end cross-section locations(lateral-torsional buckling and post-buckling). For comparisonpurposes, most of the GBT-based results were compared with

22 In these cantilevers, the longitudinal normal, transverse normal and shearstresses play equally important roles, as far as the post-buckling behaviour isconcerned – note that the vast majority of beam post-buckling results reported in theliterature only take into account the longitudinal normal stresses. Therefore, acombination of additional non-linear strain-displacement terms and better transverseextension shape functions will be required to achieve more accurate results for awider rotation range – these issues are currently under investigation [42] and,hopefully, new findings should be reported in the near-to-intermediate future.


values yielded by ABAQUS shell finite element analyses – a very goodagreement was found in all first-order and buckling cases and,moreover, the high computational efficiency of the GBT analyseswas clearly demonstrated. The paper also included GBT (prelimin-ary) and shell finite element simulations of the post-buckling (non-linear) behaviour of six cantilevers with different lengths and tipload application points – a fairly reasonable agreement was alsoobtained, even if the limitations of the currently implementedGBT non-linear formulation were exposed and justified.

Concerning the comparison between the numerical and experi-mental results, the main conclusions drawn from this work are thefollowing:

(i) The experimental results dealing with (i1) the simply sup-ported beam first-order and local buckling and (i2) the can-tilever lateral-torsional buckling (including the effect of theload point of application) were numerically replicated withfairly good accuracy by both the GBT and ABAQUS analyses –the very few cases in which some discrepancies were foundcould always be logically explained.

(ii) The experimental results dealing with the simply supportedbeam lateral-torsional buckling could not be properly simu-lated because of the additional stiffness/friction introducedby the loading system, which obviously could not be mod-elled numerically.

(iii) The experimental results dealing with the cantilever lateral-torsional post-buckling were reasonably replicated by boththe ABAQUS (shell) and GBT (beam) finite element analyses,even if they could not capture the hysteretic behaviourstemming from the friction at the mechanical fasteningdevices associated with the loading system. Nevertheless,most numerical load–displacement curves were found to fallwell inside the experimental hysteretic loop, as it would belogical to anticipate.

(iv) The GBT post-buckling analyses could not reproduce accu-rately most of the shell finite element post-buckling curves,particularly in the advanced post-buckling stages. However,they were able to capture, approximately, both the effects ofnon-uniform bending and of the location of the load applica-tion point – this was achieved through the inclusion of thirdand fourth order D and B tensors in the formulation – furtherinvestigations are currently under way, which aim atimproving the accuracy of the GBT post-buckling resultsfor a wider rotation range – they will certainly involve theconsideration of additional non-linear strain-displacementterms and/or the adoption of higher-quality transverseextension shape functions.

Acknowledgements

The first author gratefully acknowledges the financial support ofFCT – Fundação para a Ciência e Tecnologia (Portugal), through schol-arship No. SFRH/BD/21439/2005. All the authors also wish toacknowledge FCT (project ‘‘Generalised Beam Theory (GBT) -Development, Application and Dissemination’’ - reference PTDC/ECM/108146/2008) and ICIST for funding this research and STEP forsupplying the GFRP pultruded profiles used in the experimental tests.

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