Thin-Walled Structures ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Thin-Walled Structures

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Mode interaction in thin-walled equal-leg angle columns

Enio Mesacasa Jr.a, Pedro Borges Dinis b, Dinar Camotim b,n, Maximiliano Malite a

a Structural Engineering Department, São Carlos School of Engineering, University of São Paulo, Brazil, Av. Trabalhador Sãocarlense,400, 13566-590, São Carlos, SP, Brazilb Department of Civil Engineering and Architecture, ICIST, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais,1049-001 Lisboa, Portugal

a r t i c l e i n f o

Keywords:Cold-formed steel columnsEqual-leg angle columnsGlobal mode interactionImperfection-sensitivityElastic post-buckling behaviour andstrengthElastic–plastic post-buckling behaviour andstrength

31/$ - see front matter & 2013 Elsevier Ltd. Ax.doi.org/10.1016/j.tws.2013.06.021

esponding author. Tel.: +351 21 8418403; fax:ail address: dcamotim@civil.ist.utl.pt (D. Camo

se cite this article as: Mesacasa Jr. E, e//dx.doi.org/10.1016/j.tws.2013.06.02

a b s t r a c t

This paper presents and discusses numerical results, obtained through ANSYS shell finite element analyses,dealing with the post-buckling behaviour (mostly elastic, but also elastic–plastic), ultimate strength andfailure mode nature of fixed-ended and pin-ended thin-walled equal-leg angle steel columns withcoincident critical flexural-torsional and minor-axis flexural buckling loads (i.e., experiencing very strongcoupling effects between these two global instability phenomena) – for comparative purposes, columnsthat are buckling in pure flexural-torsional and flexural modes are also analysed. Since the main aim ofthe work is to investigate the column imperfection-sensitivity, the analyses concern otherwise identicalcolumns containing initial geometrical imperfections with different shapes and amplitudes, combiningthe competing critical buckling modes – particular attention is paid to the sign of the minor-axis flexuralcomponent. The results reported consist of column (i) elastic equilibrium paths and the correspondingpeak loads and deformed configurations and (ii) elastic–plastic collapse loads and mechanisms, making itpossible to assess how they are influenced by the initial geometrical imperfections.

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1. Introduction

Thin-walled columns with all their walls sharing a commonlongitudinal edge (e.g., angle, T-section and cruciform columns)exhibit no primary warping (if the minute rounded corner effectsare neglected, of course) and thus, their warping resistance stemsexclusively from secondary warping – this resistance has beenfound to be quite non-negligible in the presence of end supportwarping fixity [1–3]. Nevertheless, this feature implies a very lowtorsional stiffness, thus rendering the above columns highlysusceptible to instability phenomena involving torsion (torsionalor flexural-torsional buckling). Since this work deals exclusivelywith fixed-ended and pin-ended (singly symmetric) equal-legangle columns with short-to-intermediate lengths, the involve-ment of torsion is felt through flexural-torsional buckling. At thisstage, it should be pointed out the support conditions of the fixed-ended (F) and pin-ended (P) angle columns only differ in theminor-axis flexural rotations, which are fully restrained in the firstcase and completely free in the second case – particularly, it isworth noting that both these support conditions, which arecommonly considered in column tests, fully prevent the endsection warping displacements, torsional rotations and major-axis flexural rotations, which implies that F and P columns exhibit

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+351 21 8497650.tim).

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exactly the same major-axis flexural and torsional behaviour(stiffness and strength).

The structural behaviour of fixed-ended and pin-ended equal-leg angle columns has been investigated, experimentally and/ornumerically, by several authors. Experimental campaigns werereported (i) by Popovic et al. [4], Young [5] and Mesacasa Jr. [6], forF columns, and (ii) by Wilhoite et al. [7], Popovic et al. [4],Chodraui [8] and Maia et al. [9], for P columns. The fairly largeexperimental failure load data obtained by the above researchers,as well as the considerable number of numerical ultimate loadsdetermined by Ellobody and Young [10], Camotim et al. [11],Silvestre et al. [12] and Dinis et al. [13], were used to developand validate several design approaches for equal-leg anglecolumns – those proposed by Young [5] (F columns), Rasmussen[14,15] (P columns) and Silvestre et al. [12] (F and P columns)deserve to be specially mentioned.1

At this stage, it is worth mentioning that the post-bucklingbehaviour and ultimate strength of short-to-intermediate equal-leg angle columns is governed by the interaction between twoglobal buckling modes, (i) one combining major-axis flexure andtorsion corresponding to the almost horizontal plateau of the Pcrvs. L curve shown in Fig. 1(a) (Pcr and L are the column criticalbuckling load and length) and (ii) the other involving exclusively

1 Very recently, Dinis et al. [13] claimed to have found more rational designapproaches for F and P columns, which involve genuine flexural-torsional bucklingconcepts (instead of the “traditional” local buckling ones).

n-walled equal-leg angle columns. Thin-Walled Structures (2013),

Fig. 1. Buckling behaviour of F and P equal-leg angle columns: (a) Pcr vs. L curves, (b) GBT modal participation diagrams, and (c) in-plane shapes of the GBT deformationmodes 2-6.

E. Mesacasa Jr. et al. / Thin-Walled Structures ∎ (∎∎∎∎) ∎∎∎–∎∎∎2

minor-axis flexure corresponding to the descending branch of theaforementioned Pcr vs. L curve [3]. The GBT (Generalised BeamTheory) modal participation diagrams displayed in Fig. 1(b) makeit possible to quantify the relative contributions of the variousdeformation modes, depicted in Fig. 1(c), to the column criticalbuckling modes [1]. Moreover, the observation of the resultspresented in Figs. 1(a)–(c), which concern F and P angle steel(E¼210 GPa and v¼0.3) columns with cross-section dimensions70�70�2.0 mm (rounded corner effects disregarded), show that(i) the short-to-intermediate and intermediate-to-long columnsbuckle in flexural-torsional (2+4)2 and pure flexural (3) modes,respectively and (ii) the “transition length” separating the twobuckling behaviours is naturally much lower for the P columnsthan for their F counterparts (basically due to the 75% minor-axisflexural buckling load drop).

As it would be logical to expect, it was also found [3] that the(detrimental) influence of the coupling between the two globalcompeting buckling modes on the angle column post-bucklingbehaviour and ultimate strength increases considerably as thelength approaches the respective “transition value” (coincidentflexural-torsional and flexural critical buckling loads). Further-more, this coupling is much stronger in the P columns, due tothe absence of minor-axis end moments to counteract the loadeccentricity effects stemming from the effective centroid shiftsoccurring in the advanced post-buckling stages (e.g., [16]).3

Concerning the design of F and P angle columns, even if themost performing available methodologies, namely those devel-oped by Young [5], Rasmussen [14,15] and Silvestre et al. [12],ultimate strength estimates that compare quite well with theavailable experimental and numerical data, it must be recognisedthat such procedures are based on local-global interactive bucklingconcepts (i.e., disregard the fact that short-to-intermediate col-umns buckle in flexural-torsional modes) and have a predominantempirical nature. This implies that they do no seize the realcolumn collapse mechanics, thus reflecting the current lack ofin-depth knowledge about all the aspects involved in the anglecolumn non-linear structural behaviour. Such knowledge is indis-pensable to search for more rational design procedures, combiningaccuracy and safety with a mechanical meaning as solid/realisticas possible. One key aspect requiring proper understanding is the

2 GBT modal participation diagrams in Fig. 1(b) clearly show that the columnbuckling modes (i) are predominantly torsional and (ii) exhibit a (small) major-axisflexural component that grows with the length.

3 It is still worth mentioning that the experimental set-up commonly adoptedto test pin-ended specimens involves “rigid links” connecting the pinned supports(cylindrical hinges) to the rigid plates attached to the column end cross-sections (toensure warping-fixity). Mesacasa Jr. et al. [17] have recently found that these rigidlinks have the net effect of lowering the column minor-axis flexural buckling load,thus rendering it more susceptible to the coupling effects dealt with in this workand contributing to a more pronounced ultimate strength erosion.

iPlease cite this article as: Mesacasa Jr. E, et al. Mode interaction in thihttp://dx.doi.org/10.1016/j.tws.2013.06.021

interaction between flexural-torsional and flexural buckling, atopic seldom investigated in the past – indeed, the existing studieson thin-walled column mode interaction concern almost onlyglobal and local (or, more recently, distortional) buckling. More-over, the investigation on this global interactive behaviour isfurther complicated by the angle (i) deceptive geometrical simpli-city (just two outstands) and (ii) mono-symmetry (no symmetrywith respect to the minor-axis), which greatly affects the columnimperfection-sensitivity.

This work aims at contributing to overcome the shortcomingsmentioned in the previous paragraph, by presenting and discuss-ing numerical results concerning the (i) post-buckling behaviour(mostly elastic, but also elastic–plastic), (ii) ultimate strength and(iii) failure mode nature of F and P thin-walled equal-leg anglesteel columns experiencing different levels of flexural-torsional/flexural (FT–F) interaction: (i) very strong interaction, whichcorresponds to columns with lengths associated with coincidentFT and F critical buckling loads (i.e., the “transition length”separating the “almost horizontal plateau” and the descendingbranch of the Pcr vs L curve illustrated in Fig. 1(a)), and (ii) lessstrong interaction, which corresponds to lengths below (FT criticalbuckling) and above (F critical buckling) the “transition length”just defined.

The numerical results are obtained through ANSYS [19] shell finiteelement analyses (SFEA) based on (i) column discretisations into finefour-node isoparametric element meshes (length-to-width ratio closeto 1), (ii) a steel material behaviour that is either linear elastic orlinear-elastic/perfectly plastic stress–strain curve (residual stresses andcorner effects disregarded) and (iii) column end supports modelled byattaching rigid plates to the end cross-sections, thus ensuring fullwarping and local displacement/rotation restraints. The rigid end-plateare then either (i) clamped (F columns), which prevents all the globaldisplacements/rotations (except the axial displacement, of course), or(ii) placed on cylindrical hinges that allowminor-axis flexural rotations(P columns) – i.e., F and P columns only differ in the minor-axisflexural rotation restraint.

In order to investigate the column imperfection-sensitivityassociated with the FT–F interaction, the results are obtained fromanalyses that concern sets of otherwise identical columns thatcontain initial geometrical imperfections with different shapesand/or amplitudes (various combinations of the normalised com-peting FT and F buckling mode shapes). Particular attention is paidto the sign of the minor-axis flexural component, which isexpected to play a major role in influencing the column response.The results displayed consist of (i) elastic equilibrium paths andassociated peak loads, (ii) deformed configurations and displace-ment profiles, and (iii) elastic–plastic collapse loads, which make itpossible to assess how these equilibrium paths, deformations andloads are influenced by the initial geometrical imperfection shapeand/or amplitude for the columns with the various lengths

n-walled equal-leg angle columns. Thin-Walled Structures (2013),

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selected. Finally, the paper closes with some considerations on theimpact that the findings reported in this work may have on thedesign (ultimate strength prediction) of short-to-intermediate Fand P equal-leg angle columns.

2. Buckling behaviour – column geometry selection

The definition of the column geometries requires performingbuckling analyses to determine (i) cross-section dimensions andlengths ensuring the various structural behaviours under scrutiny,namely (i) very close FT and F critical buckling loads for theintermediate length, which maximises the FT–F interaction, (ii)critical FT buckling and (iii) critical F buckling – the correspondingbuckling mode shapes are subsequently used to define the initialgeometrical imperfections to be considered in the column post-buckling analyses (always expressed as linear combinations of theFT and F buckling modes). Since the SFE buckling and post-buckling analyses adopt exactly the same meshes, it is a straight-forward matter to “transform” the output of the former (bucklingmode shapes) into the input of the latter (column initial configura-tions).

The equal-leg angle F and P columns analysed (i) exhibit cross-section dimensions 70�70�2.0 mm and (ii) have the selectedlengths given in Table 1 and also indicated in Fig. 1(a) (showingtheir locations along the buckling curve – note that the above tablealso provides the column flexural-torsional (Pcr.FT) and flexural(Pcr.F) buckling loads.

2.1. Initial geometrical imperfections

The configuration of the initial geometrical imperfectionsinvariably plays a pivotal role in mode interaction investigations,since its choice may alter considerably the post-buckling behaviourand strength of the structural system under consideration. Thus, itis necessary to determine the post-buckling behaviour of columnsexhibiting various critical-mode initial configurations, combiningarbitrarily the two competing buckling modes: linear combinationsof single half-wave (major-axis) flexural-torsional and minor-axisflexural buckling modes, both normalised to exhibit amplitudescorresponding to L/1000 mid-height values of the (i) leg tip

Table 1Selected F and P 70�70�2.0 mm angle column lengths and corresponding FT andF buckling loads.

P columns F columns

Length (mm) Pcr.FT (kN) Pcr.F (kN) Length (mm) Pcr.FT (kN) Pcr.F (kN)

1500 18.7 49.2 � � �2500 18.2 18.9 5200 16.6 17.53000 18.1 12.4 � � �

Fig. 2. Initial geometrical imperfection representations in the Cd.0�CD.0 plane: (a) “

ments dn and Dn.

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displacements D, due to the torsional rotation (positive counter-clockwise) and minor-axis translation (positive towards the verticalleg), for the FT buckling mode, and (ii) cross-section major-axistranslation d (positive towards the leg tips), for the F bucklingmode – see Fig. 2(c). Therefore, the normalised buckling modeshapes are associated with Dn¼dn¼L/1000. Concerning the linearcombinations of these mode shapes included in the comparativestudy, which are defined by the coefficients CD.0 and Cd.0, twodistinct approaches are followed in this work, namely:

(i)

circle

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To change only the initial imperfection shape, while keepingits initial “overall amplitude” unaltered. This corresponds toenforcing that the linear combination coefficients must satisfythe condition (Cd.0)2+(CD.0)2¼1, which means that the initialimperfection may be defined by an angle θ, measured in theCd.0�CD.0 plane and such that Cd.0¼cosθ and CD.0¼sinθ [20].Fig. 2(a) provides a schematic visualisation of this approach(denoted herein as “circle approach”) and also shows the pureflexural-torsional and flexural initial imperfection shapes(θ¼01, 901 and 1801) – note that (i1) θ¼01 and θ¼1801correspond to minor-axis flexure towards the leg tips andcorner, respectively, and that, due to symmetry, (i2) only01≤θ≤1801 values are considered.

(ii)

To change only one buckling mode shape amplitude,while keeping the other one unaltered (at its normalisedvalue) – naturally, the initial imperfection shape also varies.This corresponds to enforcing that the value of one linearcombination coefficient is always constant. Fig. 2(b) provides aschematic visualisation of this approach (denoted herein as“square approach”) in the Cd.0�CD.0 plane – it concerns thecases in which the absolute value of one linear combinationcoefficient is unitary.

In the first approach, 13 different initial imperfection shapes areconsidered for each column length, corresponding to θ¼151increments (see Fig. 2(a)). In the second approach, on the otherhand, columns with 26 distinct initial imperfection shapes areanalysed, including the 17 ones depicted in Fig. 2(b): combinationsof Cd.0 and CD.0 values in which (i) one of them has a fixed value(�1, 0.25 or 1) and (ii) the other varies between its limit values, in0.25 increments – see the circles and crosses in Fig. 2(b).

3. Elastic post-buckling behaviour

This section deals with the elastic post-buckling behaviour andimperfection-sensitivity of columns with the initial imperfectionshapes described in Section 2.1. Initially, the investigation addressesF and P columns exhibiting strong FT–F interaction, i.e., with thecorresponding “transition lengths” (LT), given in Table 1, illustratedin Fig. 1(a) and ensuring practically coincident FT and F critical

approach”, (b) “square approach” and (c) positive senses of the displace-

alled equal-leg angle columns. Thin-Walled Structures (2013),

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buckling loads – the main objective is to characterise and comparethe imperfection-sensitivities of both columns. Then, in order toprovide a better assessment of the P column imperfection-sensitivity (the most sizeably affected by the FT–F interaction),column with smaller and larger lengths are also analysed – lengthsLFToLT and LF4LT, corresponding to the Pcr vs L curve horizontalplateau and descending branch curve, are considered. Finally, itshould be mentioned that the peak loads of all the columnsanalysed in this work are given in Appendix A.

3.1. Strong FT–T interaction – F and P Columns with length LT

Figs. 3–5(a) and (b) show the upper parts (P48 kN) of theequilibrium paths P vs. d and P vs. D (d and D are mid-span totaldisplacements) concerning F and P columns containing the first 17initial imperfection shapes illustrated in Fig. 2(b), corresponding tothe “square approach” – each equilibrium path is identified by itsimperfection coefficients (Cd.0, CD.0). For clarity purposes, there aremore (twice as many) figures devoted to the P columns – thevarious P vs. d and P vs. D equilibrium paths are first depictedtogether, after which only those concerning Cd.0¼–1 are displayed.The joint observation of all these elastic post-buckling equilibriumpaths leads to the following comments:

(i)

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First of all, it should be noted that all F and P columnequilibrium paths exhibit elastic limit points (peak loads), afeature that clearly indicates the presence of a markedgeometrical non-linearity stemming from strong FT–T inter-action effects.

(ii)

Figs. 3(a) and (b) shows that, if CD.≠0, the F column equilibriumpaths exhibit an almost perfect symmetry with respect to theCd.0 value – i.e., the (Cd.0, CD.0) and (–Cd.0, CD.0) paths aresymmetric with respect to the axes d¼0 (see Fig. 3(a) –minuteasymmetry: lower peak loads for Cd.040) and D¼0 (see Fig. 3

Fig. 3. F column elastic equilibrium paths (a) P vs. d and

Fig. 4. P column elastic P vs. d equilibrium paths: (a) CD.0¼1 or Cd.0

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(b) – D is positive in both sides). For CD.0¼0, the columnexhibits exclusively minor-axis flexural deformations and has ahigher peak load for Cd.0¼1 (less compression in the leg tipregions). Moreover, the variation of CD.0 (≠0) has a minuteimpact on the column peak load, as is clearly shown in Fig. 3(b)– however, it is worth noting that the amount of deformationprior to collapse grows visibly with CD.0.

(iii)

For CD.0¼1, the F column post-buckling strength and peakload decrease significantly as Cd.0 increases. It is also inter-esting to notice that, although the (0, 1) equilibrium pathexhibits positive d values (i.e., minor-axis flexure towards theleg tips), the decrease is practically independent of the Cd.0sign, which reinforces the assertion made in the above item(i). The minor-axis flexure sense does not alter the F columnbehaviour because the load eccentricity effects, associatedwith the effective centroid shifts, are “almost perfectly”counteracted by the column minor-axis end moments.

(iv)

The mechanical explanation for the fact that the (0, 1) Fcolumn develops minor-axis flexure near the peak load canbe found in Stowell's analytical work [18], which wasrecently revisited and confirmed numerically by Dinis et al.[3]: the torsional rotation gradients cause non-linear cross-section mid-line longitudinal stress distributions (“threehalf-wave” pattern) that lead to effective centroid shiftstowards the cross-section corner – these shifts are respon-sible for the positive minor-axis flexural displacementsobtained. This also explains why the (1, 0) column has ahigher peak load than the (–1, 0) one – in the formercolumns, the eccentricities stemming from the initial geo-metrical imperfections and effective centroid shifts haveopposite signs (i.e., counteract each other).

(v)

The comparison between the results concerning the F (Fig. 3(a) and (b)) and P (Figs. 4 and 5(a) and (b)) column providesclear evidence of considerable differences in post-buckling

(b) P vs. D (“square approach”).

¼1, and (b) Cd.0¼�1 (“square approach”).

led equal-leg angle columns. Thin-Walled Structures (2013),

Fig. 5. P column elastic P vs. D equilibrium paths: (a) CD.0¼1 or Cd.0¼1, and (b) Cd.0¼�1 (“square approach”).

E. Mesacasa Jr. et al. / Thin-Walled Structures ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 5

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behaviour and imperfection-sensitivity (even if there are alsosome similarities). The major difference resides in the quali-tative features separating the equilibrium paths of the col-umns associated with positive and “large enough” Cd.0 values(Cd.0≥0.5) from those concerning the remaining columns. Inthe P columns, the former have substantially higher peakloads and, moreover, their P vs. d equilibrium paths exhibitconsiderable ductility prior to failure. Curiously, these peakloads are virtually not influenced by CD.0 (note that the samevalue occurs for all (1, n) columns: P≈17.1 kN, i.e., P/Pcr≈0.94),and the P vs. D equilibrium paths exhibit a D displacementreversal – see Fig. 5(a). This feature is due to the fact that theload eccentricities associated with positive and “largeenough” Cd.0 values successfully “oppose” the effectivecentroid shifts (stemming from the torsional rotation gradi-ents and, most of all, the stress redistribution towards thecorner – recall that there are no reactive bending moments),thus forcing a torsional rotation reversal, responsible for theabove D displacement reversal, and leading to a typical“column flexural post-buckling behaviour”. Moreover, Fig. 5(a) also shows that the peak load increases slightly when Cd.0decreases to 0.75 and 0.5 – this is because the mid-spaneffective centroid location approaches the applied load lineof action. Once that line of action is “crossed” by the mid-span effective centroid, which occurs for a Cd.0 value com-prised between 0.5 and 0.25, the nature of the columnresponse changes. This nature then remains unaltered asCd.0 decreases further and becomes negative – the mid-spaneffective centroid is now always moving away from theapplied load line of action.

(vi)

When the effective centroid shifts dominate (Cd.0 ≤0.5), the Pcolumn peak load is considerably lowered (typical ultimatestrength erosion due to FT–F interaction effects) and thelimit points become sharper in the P vs. d paths than in theirP vs. D counterparts – see Figs. 4 and 5(a) and (b), but notethe scale differences. These figures also show that, naturally,the peak load decreases as (vi1) Cd.0 decreases from 0.25to �1 (for CD.0¼1) and (vi2) CD.0 increases from 0 to 1(for Cd.0¼�1).

(vii)

In order to clarify the mechanical differences between the(0.25, 1.00) and (0.50, 1.00) P column post-buckling beha-viours, Figs. 6(c)-(d) provide the evolutions of the corre-sponding mid-span cross-section deformed configurationsas the applied loading progresses – these deformed config-urations concern the 8 and 9 equilibrium states indicatedalong the two column equilibrium paths depicted in Fig. 6(a).The initial deformed configurations of the mid-spancross-section corner regions (due to the initial geometricalimperfections incorporated in the two columns) are

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displayed in Fig. 6(b) – they only differ in the minor-axisflexure amplitude (one is twice the other). The two cross-section deformed configuration sequences clearly show thatthe impact of the flexural-torsional deformations in the(0.25, 1.00) and (0.50, 1.00) P column responses is verydifferent – it is much more relevant in the former column.A close observation of the various cross-section deformedconfigurations reveals that the “column response diver-gence” occurs between equilibrium states 3 and 5, whichcorrespond to loads P≈8 kN and P≈12 kN, respectively.Indeed, while the (0.25, 1.00) column d value remains almostunaltered (increase of about 16%), its (0.25, 1.00) columncounterpart more than doubles (increase of about 234%).After equilibrium state 5, the (0.25, 1.00) column is governedby minor-axis flexure towards the corner, which stems fromthe effective centroid shift effects (they “successfullyoppose” the small initial imperfection) and grows in parallelwith the FT deformations that are responsible for it – see thedeformed configurations 6–8 in Fig. 6(c). Conversely, the(0.50, 1.00) column exhibits minor-axis flexure towards theleg free ends, which reflects the fact that the initial imper-fection is large enough to preclude the emergence anddevelopment of the above effective centroid shift effects,thus forcing a reversal of the FT deformations and evolvingtowards a pure minor-axis flexural post-buckling behaviour� see the deformed configurations 6–9 in Fig. 6(d).

(viii)

It is still worth noting that, because of the absence of FTinitial imperfections, the (1, 0) and (�1, 0) F and P columnsdo not exhibit D displacements prior to failure (the Pcolumns do it after the peak load is reached) – this confirmsthe predominant role played by minor-axis flexure in thecolumn post-buckling behaviour and strength.

(ix)

In view of what was mentioned in the previous items, it isnot surprising that the lowest peak loads correspond to(i) the (�1, 1) and (1, 1) F columns (the latter is a fractionlower) and (ii) the (�1, 1) P column. As for the highest peakloads, they occur for the (0, 1) F and (0.50, 1) P columns.

In order to provide further insight on the role played by CD.0(amplitude of the initial imperfection flexural-torsional compo-nent), Figs. 7(a) and (b) display the equilibrium paths (P vs. d and Pvs. D) of nine additional P columns with initial imperfection shapesdefined by CD.0¼0.25 and �1≤Cd.0≤1. The comparison betweenthese equilibrium paths and those depicted earlier in Figs. 4 and 5(a), for CD.0¼1, prompts the following remarks:

(i)

First of all, it is worth noting that, in general, the sets ofequilibrium paths concerning CD.0¼0.25 and CD.0¼1 are qua-litatively and quantitatively similar, thus indicating a relatively

led equal-leg angle columns. Thin-Walled Structures (2013),

Fig. 7. P column elastic equilibrium paths (a) P vs. d and (b) P vs. D (“square approach”, and CD.0¼0.25).

Fig. 6. (0.25, 1.00) and (0.50, 1.00) P column (a) elastic equilibrium paths P vs. d, (b) mid-span cross-section initial deformed configurations and (c) and (d) mid-span cross-section deformed configuration evolutions.

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small influence of CD.0 on the column post-buckling behaviour– nevertheless, there are slight differences (recall that a CD.0decrease entails less relevant effective centroid shift effects).

(ii)

The most visible difference resides in the fact that, unlike its(0.25, 1) counterpart, the (0.25, 0.25) column exhibits minor-axis bending towards the leg tips – see Fig. 7(a). This is due tothe fact that CD.0 ¼0.25 is no longer “large enough” to generatea mid-span effective centroid shift capable of successfullyopposing the applied load eccentricity.

Finally, in order to assess the relevance of the initial geometricalimperfection shape, Figs. 8(a) and (b) (F columns) and Figs. 9 and 10(a) and (b) (P columns) show the upper parts (P48 kN) of theequilibrium paths P vs. d and P vs. D of columns with 13 imperfectionshapes, defined according to the “circle approach” illustrated in Fig. 2(a) (all initial imperfections share a common amplitude). Each pathconcerns a θ value such that 01≤θ ≤1801 (Cd.0¼cosθ and CD.0¼sinθ) –the θ¼01, θ¼901 and θ¼1801 columns correspond to the (1, 0), (0, 1)

lease cite this article as: Mesacasa Jr. E, et al. Mode interaction in thitp://dx.doi.org/10.1016/j.tws.2013.06.021

and (�1, 0) previous ones. The observation of all these equilibriumpaths makes it possible to conclude that:

(i)

n-wa

Naturally, the F and P column equilibrium paths depicted inFigs. 8(a)–10(b) follow the same general trends exhibited by theircounterparts shown in Figs. 3(a)–5(b) – there is only a quanti-tative difference, due to the common amplitude restraint.

(ii)

The main novel information extracted from these equilibriumpaths concerns the identification of the least and mostdetrimental initial geometrical imperfection shapes, in thesense that they lead to the highest and lowest column(elastic) peak loads. This information will be subsequentlyused to assess how the initial imperfection shape influencesthe column (elastic–plastic) collapse load.

(iii)

Pure flexural-torsional initial imperfections (θ¼901) are clearlythe least detrimental (i.e., most favourable) for the F columns. Forthe P-columns, on the other hand, the least detrimental initialimperfections, amongst those considered in this work (no furtherrefinement was carried out), correspond to θ¼601, i.e., combine

lled equal-leg angle columns. Thin-Walled Structures (2013),

Fig. 9. P column elastic equilibrium paths P vs. d (“circle approach”): (a) 01≤θ≤1201 and (b) 1201≤θ≤1801.

Fig. 10. P column elastic equilibrium paths P vs. D (“circle approach”): (a) 01≤θ≤1201 and (b) 1201≤θ≤1801.

Fig. 8. F column elastic equilibrium paths (a) P vs. d and (b) P vs. D (“circle approach”).

4

veryexhib

E. Mesacasa Jr. et al. / Thin-Walled Structures ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 7

iPlhtt

a dominant flexural-torsional component (CD.0¼0.87) with arelevant flexural component with curvature towards the leg tips(Cd.0¼0.50) � see Fig. 2(a).4

(iv)

The θ¼1651 F column has the most detrimental initialimperfections (even if the θ¼151 and θ¼1651 columnpeak loads virtually coincide). They combine a highlydominant flexural component, causing minor-axis bendingtowards the cross-section corner (Cd.0¼�0.97), with a muchsmaller flexural-torsional one (CD.0¼0.26) – see Fig. 2(a). Toassess whether any 1651oθo1801 initial imperfection iseven more detrimental, the θ¼1751 column was alsoanalysed – its peak load was found to be 0.5% above theθ¼1651 column one.

Note that a further CD.0 increase (and corresponding Cd.0 decrease) causes alarge peak load drop � the effective centroid shift effects force the column toit minor-axis bending towards the cross-section corner.

ease cite this article as: Mesacasa Jr. E, et al. Mode interaction in thip://dx.doi.org/10.1016/j.tws.2013.06.021

(v)

n-wa

The lowest P column peak load corresponds to θ¼1351, i.e., toinitial imperfections with equal-amplitude flexural and flexural-torsional components (Cd.0¼�0.707 and CD.0¼0.707) – theformer causes bending towards the cross-section corner.

(vi)

The peak load percentage increase associated with switchingfrom the most to the least detrimental initial imperfection isequal to either 22.8% (F columns � 12.09 kN vs.14.85 kN) or46.2% (P columns – 11.93 kN vs. 17.44 kN) – the P columnincrease is much higher (more than twice). Note also thatboth the 14.85 kN and 17.44 kN values are below the corre-sponding critical buckling loads.

3.2. FT and F critical buckling – P Columns with Lengths below andabove LT

In order to assess the variation of the P column interaction effectsand imperfection-sensitivity with the length (with respect to LT),columns with lengths LFT¼1500 mm (critical FT buckling) and

lled equal-leg angle columns. Thin-Walled Structures (2013),

E. Mesacasa Jr. et al. / Thin-Walled Structures ∎ (∎∎∎∎) ∎∎∎–∎∎∎8

LF¼3000 mm (critical F buckling) are analysed in this subsection.Figs. 11(a) and (b) and 12(a) and (b) show the upper parts (P410 kN)of the LFT column equilibrium paths P vs. d and P vs. D containing the17 initial imperfections corresponding to the circles in Fig. 2(b)(“square approach”). The observation of these equilibrium paths, aswell as the comparison with those shown in Figs. 4(a) and (b) and 5(a) and (b) (LT columns), leads to the following conclusions:

(i)

iPht

Regardless of the Cd.0 value, all the LFT columns exhibit minor-axisbending towards the cross-section corner. When Cd.0¼1, there is ad displacement reversal occurring at loads that become progres-sively smaller as CD.0 increases – in the (1, 0) column, this reversalonly takes place after failure. Moreover, except again for the (1, 0)column after failure, the D displacements are always akin to theinitial imperfection FT component, i.e., positive. The above factsprovide solid evidence that the initial imperfection F componentplays a much lesser role in the LFT columns than in their LTcounterparts, i.e., the effective centroid shift effects associated withthe FT post-buckling behaviour clearly prevail.

Fig. 11. LFT P column elastic P vs. d equilibrium paths: (a) CD.0¼

Fig. 12. LFT P column elastic P vs. D equilibrium paths: (a) CD.0¼

Fig. 13. P column elastic P vs. d equilibrium paths: (a) CD.0¼1

lease cite this article as: Mesacasa Jr. E, et al. Mode interaction in thitp://dx.doi.org/10.1016/j.tws.2013.06.021

(ii)

1 or

1 or

or C

n-w

Concerning the influence of the initial imperfection F compo-nent on the LFT column imperfection-sensitivity with respectto the corresponding FT component, it is now completelydifferent in qualitative terms, thus reflecting the lesser roleplayed by minor-axis bending in the column post-bucklingbehaviour. Indeed, the percentage ultimate load drop as CD.0increases from 0 to 1 is (ii1) 20.6%, when Cd.0¼1, and (ii1)12.2%, when Cd.0¼�1 – recall that, for the LT column, therewas virtually no ultimate load change in the former case and amarginally larger drop (13.3%) in the latter. Moreover, forCD.0¼1, the column ultimate load only drops by about 11.5%when Cd.0 varies between 1 and �1, thus providing additionalevidence of the lesser relevance of minor-axis bending – recallthat such drop was about 35% for the LT length columnscontaining the same initial imperfections.

Finally, Figs. 13 and 14(a) and (b) show the upper parts(P46 kN) of the LF column equilibrium paths P vs. d and P vs. Dcontaining the same 17 initial imperfections as before. Their

Cd.0¼1, and (b) Cd.0¼�1 (“square approach”).

Cd.0¼1, and (b) Cd.0¼�1 (“square approach”).

d.0¼1, and (b) Cd.0¼�1 (“square approach”).

alled equal-leg angle columns. Thin-Walled Structures (2013),

Fig. 14. P column elastic P vs. D equilibrium paths: (a) CD.0¼1 or Cd.0¼1, and (b) Cd.0¼�1 (“square approach”).

Table 2LT F column failure loads for four yield stresses and the least or most detrimentalinitial imperfections.

Fixed-ended columns

fy/scr θ ¼901 θ ¼1651 Δ (%)

Pu (kN) Pu /Pue Pu (kN) Pu /Pue

L 250 0.70 9.94 0.67 7.76 0.64 21.91.30 14.44 0.97 11.62 0.96 19.52.50 14.85 1.00 12.09 1.00 18.65.00 14.85 1.00 12.09 1.00 18.6

Table 3LFT, LT, LF P column failure loads for 4 yield stresses and the least or mostdetrimental initial imperfections.

Pin-ended columns

fy/scr θ¼01 θ¼1201 Δ (%)

Pu (kN) Pu /Pue Pu (kN) Pu /Pue

L 150 0.70 12.03 0.56 12.24 0.78 �1.71.30 20.42 0.95 15.19 0.97 25.62.50 21.53 1.00 15.67 0.99 27.25.00 21.53 1.00 15.68 1.00 27.2

L 250 θ ¼601 θ¼13510.70 9.07 0.52 8.36 0.70 7.81.30 13.22 0.76 11.88 0.99 10.12.50 16.52 0.95 11.94 1.00 27.85.00 17.23 0.99 11.94 1.00 30.7

E. Mesacasa Jr. et al. / Thin-Walled Structures ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 9

comparison with the equilibrium paths concerning the LT columns(see Figs. 4 and 5(a) and (b)) prompts the following remarks:

L 300 θ¼801 θ¼15010.70 9.54 0.78 7.01 0.72 26.51.30 11.53 0.94 8.62 0.89 25.2

(i)

2.50 12.02 0.98 9.71 1.00 19.25.00 12.24 0.99 9.71 1.00 20.7

iPlhtt

The column minor-axis bending deformations (d displace-ments) are now in perfect agreement with the initial imper-fection F component – the columns with positive andnegative Cd.0 values exhibit minor-axis bending towards thecross-section leg ends and corner, respectively, regardless ofthe initial imperfection FT component amplitude. This reflectsthe minor role played by the flexural-torsional deformationsand ensuing effective centroid shift effects (no longer able tocause d displacement reversals). The (0, 1) column exhibitsminor-axis bending towards the cross-section corner, due tothe (weak) effective centroid shift effects.

(ii)

Concerning the influence of the initial imperfection FT com-ponent on the LF column imperfection-sensitivity withrespect to the corresponding F component, it is now verysimilar to that exhibited by the LT columns – compare the P vs.d equilibrium paths of the (1, n) and (�1, n) columns inFigs. 4 and 14(a) and (b). The influence of the CD.0 value on thepeak load is (ii1) virtually null for the Cd.0¼1 columns and (ii2)quite moderate for the Cd.0¼�1 columns – the percentageultimate load drop as CD.0 increases from 0 to 1 is 9.1% for theLF columns and 13.2% for the LT ones.

(iii)

Like their LT and LFT counterparts, the LF columns with null noimperfection FT components only exhibit D displacementsafter failure – see the (1, 0) and (�1, 0) column equilibriumpaths in Figs. 5, 12 and 14(a) and (b). This shows that,regardless of the column length (i.e., the ratio between Pcr.FTand Pcr.F), the development of flexural-torsional deformationsonly occurs if CD.0 is not null, which means that they neveremerge (prior to failure) in the context of a pure minor-axisflexural post-buckling behaviour. Nevertheless, the presenceof an initial imperfection FT component has a sizeable impacton the ultimate strength of the LFT and LT columns.

ease cite this article as: Mesacasa Jr. E, et al. Mode interaction in thip://dx.doi.org/10.1016/j.tws.2013.06.021

Summarising, it may be said that, with respect to the LT P columns,either decreasing (LFT) or increasing (LF) the length naturally changesthe column post-buckling mechanics, particularly those associatedwith the minor-axis bending behaviour. Moreover, the correspondingcolumn imperfection-sensitivity is naturally lessened, both qualita-tively and quantitatively – the mitigation is more pronounced for theLFT columns. It also became clear that, as far as the column ultimatestrength erosion is concerned, the initial imperfection F component ismore relevant than its FT counterpart.

4. Ultimate strength erosion due to FT–F interaction

In order to enable the assessment of the potential impact of theinitial geometrical imperfection shape on the column elastic–plasticultimate strength, Tables 2 and 3 provide the failure loads (Pu) of the (i)LT F and (ii) LFT, LT, and LF P columns, respectively – the ratios Pu /Pue,where Pue is the elastic peak load, are also given. The columns analysed(i) exhibit four different yield-to-critical stress ratios (fy/scr≈0.70; 1.30;2.50; 5.00) and (ii) contain initial imperfections with amplitude L/1000and either the least or the most detrimental shape, which wereidentified earlier, in the context of elastic post-buckling analyses (seeFig. 2(a) – “circle approach”). They are (i) θ¼901 and 1651 (LT Fcolumns), (ii) θ¼601 and 1351 (LT P columns), (iii) θ¼01 and 1201 (LFT Pcolumns), and (iv) θ¼801 and 1501 (LF P columns). It is still worthmentioning that the steel material behaviour is assumed to be elastic-perfectly plastic and that residual stress and corner strength effects are

n-walled equal-leg angle columns. Thin-Walled Structures (2013),

E. Mesacasa Jr. et al. / Thin-Walled Structures ∎ (∎∎∎∎) ∎∎∎–∎∎∎10

neglected. Figs. 15–18(a) and (b) show the upper parts of the (a) P vs. dand (b) P vs. D elastic–plastic equilibrium paths of the above columns,providing their post-buckling behaviours and failure loads.

From the analysis of the results concerning the LT F columns(Fig. 15(a) and (b)), it is first noticed that failure only occurs in theelastic–plastic range if the yield-to-critical stress ratio is “smallenough” (in this case, fy/scr≈0.70; 1.30) – in between fy/scr≈1.30and fy/scr≈2.50, there is a value beyond which the column remainselastic up to the peak load, which means that there is no benefit infurther increasing the yield stress. Moreover, there is no significantimpact of the fy/scr value on the percentage difference between thecollapse loads corresponding to the least and most detrimentalinitial imperfection shapes – indeed, this difference (Δ) rangesfrom 21.9% (fy/scr≈0.70) to 18.6% (fy/scr≈2.50; 5.00 – elastic fail-ures), as shown in Table 2.

Next, the attention is focused on the elastic–plastic behavioursof the P columns, which have been found to exhibit a markedasymmetry, with respect to the sign of the initial imperfectionminor-axis flexural component, in the elastic range. The beha-viours of the LFT (Figs. 16(a) and (b)), LT (Figs. 17(a) and (b)) and LF(Figs. 18(a) and (b)) columns are compared, which leads to thefollowing conclusions:

(i)

Fi

iPlhtt

First of all, it is noticeable that the imperfection-sensitivity isclearly different for the LFT, LT and LF columns, thus reflectingthe different deformation natures involved.

(ii)

In the LFT columns, which only exhibit significant non-axialdeformations for θ¼1201 (in the θ¼01 column only a smallamount of minor-axis flexure occurs prior to failure, regardlessof the yield stress level), the percentage difference between thecollapse loads is (ii1) “negative” for fy/scr¼0.70 and (ii2) quiteconsiderable for the remaining three yield stresses. The negativesign means that the most detrimental initial imperfect shapesmay not be the same for the elastic and elastic–plastic columns.

Fig. 15. LT F column (a) P vs. d and (b) P vs. D equilibrium paths for four yield stresse

g. 16. LFT P column (a) P vs. d and (b) P vs. D equilibrium paths for four yield-to-critical

ease cite this article as: Mesacasa Jr. E, et al. Mode interaction in thin-wap://dx.doi.org/10.1016/j.tws.2013.06.021

This slightly surprising fact is closely linked to the low yieldstress (minute non-axial deformation prior to failure in theθ¼1201 column) and has also to do with the difference betweenthe stress distributions associated with the various initialimperfection shapes – obviously, the one associated with anearlier cross-section yielding leads to the lowest failure load. Forfy /scr40.70, the failure load drop is substantial, due to thesignificant flexural deformations developing in the θ¼1201column prior to failure, and remains almost constant, i.e.,exhibits a very small variation with the fy/scr value � this stemsmostly from the fact that failure occurs either very soon after(fy/scr¼1.30) or in (fy/scr41.30) the elastic range.

(iii)

In the LT columns, which exhibit visible flexural-torsional andminor-axis flexural deformations for both θ¼601 and θ¼1351,the percentage difference between the collapse loadsincreases with fy/scr, thus reflecting the growing relevanceof the geometrical non-linearity – it reaches about 30% forfy /scr45.00. However, the rate of increase clearly drops forthe higher yield stresses (Δ only increases 3% betweenfy/scr42.50 and fy/scr45.00), which is due to the fact that(iii1) the θ¼1351 column always fails in the elastic range and(iii2) the θ¼601 column exhibits significant ductility prior tofailure, thus leading to a small variation of Pu with the yieldstress – the elastic peak load (Pu¼17.44 kN) only prevails forfy/scr values a bit larger than 5.00.

(iv)

In the LF columns, which naturally exhibit predominantminor-axis flexural deformations (note the scale differencein Figs. 18(a) and (b)), it is interesting to notice that both theless and the most detrimental initial imperfection shapesinvolve flexural-torsional deformations (i.e., they correspondto θ values that are neither 01 nor 1801). This involvementexplains the fact that the percentage difference between theθ¼801 and θ¼1501 column collapse loads is very high forfy/scr¼0.70 (26.5%) – the θ¼801 column minor-axis flexural

s and the least or most detrimental initial imperfection shapes.

stress ratios and the least or most detrimental initial imperfection shapes.

lled equal-leg angle columns. Thin-Walled Structures (2013),

Fig. 17. LT P column (a) P vs. d and (b) P vs. D equilibrium paths for four yield-to-critical stress ratios and the least or most detrimental initial imperfection shapes.

Fig. 18. LF P column (a) P vs. d and (b) P vs. D equilibrium paths for four yield-to-critical stress ratios and the least or most detrimental initial imperfection shapes.

E. Mesacasa Jr. et al. / Thin-Walled Structures ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 11

iPlhtt

displacements are very small until the applied load reaches avery significant value. The ultimate load difference thendecreases with the yield stress until the θ¼1501 column failsin the elastic range at Pu¼9.71 kN (somewhere betweenfy/scr¼3.00 and fy/scr¼5.00) – for higher fy/scr values, Puincreases slightly, as in the LT columns.

5. Concluding remarks

This paper reported the results of a numerical investigation onhow the initial geometrical imperfections affect the post-buckling,ultimate strength and failure mode nature of fixed-ended and pin-ended equal-leg angle columns experiencing strong of FT–F inter-action – for comparative purposes, columns buckling in pureflexural-torsional and flexural modes were also analysed. In orderto assess the angle column imperfection-sensitivity (the mainfocus of this work), sets of otherwise identical columns containingdifferent initial imperfections were considered � combinations ofcritical-mode flexural-torsional and minor-axis flexural shapeswith different amplitudes. The results displayed were obtainedthrough ANSYS shell finite element analyses and consist of (i) elastic(mostly) and elastic–plastic post-buckling equilibrium paths,including the corresponding peak/failure loads, and also (ii) figuresshowing the evolution of the mid-span cross-section deformedconfiguration along two judiciously selected equilibrium paths.

Amongst the various conclusions drawn from this numericalinvestigation, the following ones deserve to be specially mentioned:

(i)

5 Although the longer LFT columns are also sensitive to the minor-axis flexuralinitial imperfections (particularly the pin-ended ones), this sensitivity is minute

Regardless of the initial imperfection shape, all columnsexhibit elastic peak loads, thus meaning that increasing theyield stress beyond a certain limit does not improve thecolumn load-carrying capacity.

(practically null) for the shorter columns – their response and failure exhibit a pure

(ii) flexural-torsional nature.

Regardless of the column length and end support conditions,the initial imperfection shape has a visible influence on both

ease cite this article as: Mesacasa Jr. E, et al. Mode interaction in thin-wap://dx.doi.org/10.1016/j.tws.2013.06.021

the elastic peak loads and elastic–plastic failure loads. Thisinfluence is more marked in the presence of relevant minor-axis flexural displacements, i.e., for the LT and LF columns.5

(iii)

The sign/sense (mostly) and amplitude of the initial imperfec-tion minor-axis flexural component play a pivotal role in thecolumn imperfection-sensitivity, namely on the peak/failureload erosion – initial displacements towards the cross-sectioncorners. Moreover, the impact of this initial imperfectioncomponent is much more relevant for the pin-ended columnsthan for their fixed-ended counterparts, due to the presence(absence) of the effective centroid shift effects.

(iv)

Finally, one last word to mention that the findings obtained inthe course of this investigation are bound to have far-reachingimplications on the interpretation of the available anglecolumn experimental and numerical results, as well as onthe development of rational and efficient (safe and accurate)design rules for such members.

Acknowledgements

The authors gratefully acknowledge the financial support of (i)Fundação para a Ciência e Tecnologia (FCT – Portugal), throughresearch project PTDC/ECM/108146/2008 (“Generalised BeamTheory (GBT) – Development, Application and Dissemination”)(second and third authors), and (ii) CNPq (Conselho Nacional deDesenvolvimento Científico e Tecnológico – Brazil) (first andfourth authors).

lled equal-leg angle columns. Thin-Walled Structures (2013),

E. Mesacasa Jr. et al. / Thin-Walled Structures ∎ (∎∎∎∎) ∎∎∎–∎∎∎12

Appendix A

Table A.1 gives the elastic peak loads of all the fixed-ended and pin-ended angle columns analysed in this work.

Table A.1Elastic peak loads of the fixed-ended and pin-ended angle columns analysed in this work.

Square approach Circle approach

Combination(Cd.0, CD.0)

Pin-ended Fixed-ended θ(1) Pin-ended Fixed-ended

LFT LT LF LT LT LT(kN) (kN) (kN) (kN) (kN) (kN)

(1.00, 0.00) 21.53 17.13 11.94 12.54 0 17.13 12.54(1.00, 0.25) 19.59 17.13 11.95 12.03 15 17.15 12.09(1.00, 0.50) 18.55 17.14 11.95 11.93 30 17.21 12.20(1.00, 0.75) 17.75 17.14 11.95 11.87 45 17.31 12.48(1.00, 1.00) 17.10 17.14 11.96 11.84 60 17.44 12.95(0.75, 1.00) 16.71 17.28 12.04 12.34 75 13.87 13.65(0.50, 1.00) 16.39 17.44 12.14 12.93 90 12.86 14.85(0.25, 1.00) 16.12 13.71 12.25 13.67 105 12.34 13.73(0.00, 1.00) 15.90 12.86 10.86 14.85 120 12.05 13.00(�0.25, 1.00) 15.71 12.30 10.26 13.76 135 11.93 12.51(�0.50, 1.00) 15.52 11.86 9.82 13.00 150 11.98 12.21(�0.75, 1.00) 15.33 11.50 9.46 12.40 165 12.21 12.09(�1.00, 1.00) 15.14 11.13 9.15 11.89 175 – 12.15(�1.00, 0.75) 15.41 11.42 9.32 11.90 180 12.83 12.28(�1.00, 0.50) 15.76 11.75 9.52 11.94(�1.00, 0.25) 16.25 12.16 9.74 12.03(�1.00, 0.00) 17.24 12.83 10.07 12.28(1.00, 0.25) – 17.13 – –

(0.75, 0.25) – 17.28 – –

(0.50, 0.25) – 17.43 – –

(0.25, 0.25) – 17.61 – –

(0.00, 0.25) – 15.31 – –

(�0.25, 0.25) – 14.05 – –

(�0.50, 0.25) – 13.27 – –

(�0.75, 0.25) – 12.67 – –

(�1.00, 0.25) – 12.16 – –

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