Journal of Constructional Steel Research 73 (2012) 12–30

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Journal of Constructional Steel Research

Post-buckling behaviour and direct strength design of lipped channel columnsexperiencing local/distortional interaction

Nuno Silvestre, Dinar Camotim ⁎, Pedro B. DinisCivil Engineering Department, ICIST, Instituto Superior Técnico, Technical University of Lisbon, Portugal

⁎ Corresponding author. Fax: +351 21 8418403.E-mail address: dcamotim@civil.ist.utl.pt (D. Camoti

0143-974X/$ – see front matter © 2012 Elsevier Ltd. Aldoi:10.1016/j.jcsr.2012.01.005

a b s t r a c t

a r t i c l e i n f o

Article history:Received 30 September 2011Accepted 13 January 2012Available online 17 February 2012

Keywords:Direct Strength Method (DSM)Cold-formed steelLipped channel columnsUltimate strengthLocal bucklingDistortional bucklingInteractive buckling

This paper reports an investigation aimed at developing a Direct Strength Method (DSM) approach to estimatethe ultimate strength of lipped channel columns affected by local/distortional bucklingmode interaction. Follow-ing a brief presentation of a few relevant aspects concerning the shell finite element analysis of the geometricallyandmaterially non-linear behaviour of thin-walledmembers, one illustrates themethodology adopted to obtaina lipped channel column ultimate load “data bank” intended to be used in the development and assessment of aDSM design approach. Next, the current DSM expressions to predict the load-carrying capacity of columnsfailing in local and distortional modes are briefly reviewed, devoting special attention to an approach thattakes into account the above mode interaction. Then, the results of a parametric study, carried out by means ofthe code ABAQUS, are presented and discussed — this study involves the evaluation of the “exact” ultimate loadsof 276 lipped channel columnswith various geometries and twoboundary conditions (pinned andfixed end sup-ports), all exhibiting local/distortional interaction. Finally, these ultimate strength data are compared with theestimates provided by the available DSM expressions and, on the basis of this comparison, one identifies severalfeatures that a DSM approach successfully accounting for local/distortional interaction must incorporate.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Most cold-formed steel members display very slender thin-walledopen cross-sections, making them highly susceptible to several insta-bility phenomena, namely local, distortional and global (flexural orflexural–torsional) buckling (see Fig. 1(a)–(d)) — depending on themember length and cross-section shape and dimensions, any ofthese buckling modes may be critical. Moreover, since several com-monly used members exhibit geometries associated with rather sim-ilar local and distortional buckling stresses, their overall structuralbehaviours are likely to be affected by the occurrence of mode inter-action phenomena involving these two buckling modes.

It is well known that thin-walled members exhibit stable local andglobal post-buckling behaviours with high and low post-criticalstrength reserves, respectively. On the other hand, recent studiesshowed that the distortional post-buckling behaviour fits in betweenthe previous two and exhibits a non-negligible asymmetry with re-spect to the flange-lip motion (outward or inward) — e.g., [1,2].

Until the beginning of this century, the classical “Effective WidthMethod” (EWM) provided the only approach to predict the ultimatestrength of thin-walled members, namely those built from cold-formed steel profiles. This widely employed method is able to take

m).

l rights reserved.

into account both the local post-buckling strength reserve and the in-teraction between local and global (flexural or flexural–torsional)buckling. However, the advent of more complex cross-section shapes,exhibiting several walls and including additional lips and/or interme-diate stiffeners, made the application of the EWM excessively cum-bersome and time-consuming, thus paving the way for thedevelopment of the “Direct StrengthMethod” (DSM), viewed as capa-ble of overcoming the above problem.

The DSM was originally proposed by Schafer and Peköz [3], abouttwelve years ago, and has been continuously improved since, mainlydue to the research activity carried out and/or fostered by Schafer[4,5]. Moreover, it is worth mentioning that the DSM has already beenincluded in the current versions of the Australian/New Zealander [6]and North American [7] specifications for the design of cold-formedsteel structures. The method has been shown to provide an efficientand general approach to estimate the ultimate strength of cold-formed steel columns and beams (i) exhibiting global (flexural, torsion-al or flexural–torsional), distortional or local failuremodes or (ii) failingin collapse mechanisms involving interaction between local and globalbuckling modes. Indeed, the most recent DSM version stipulates theneed to perform three independent safety checks, regardless of themember critical buckling mode nature: (i) one against distortional fail-ure, (ii) another against global failure and (iii) a third one against a localor a combined local/global collapse. In the latter case, the DSM providesan efficient alternative to themore traditional (and conservative) EWM.However, as repeatedly pointed out by Schafer [4,5,8,9], further re-search is needed before the DSM approach can be successfully applied

Fig. 1. Cross-section deformed shapes of lipped-channel columns experiencing (a) local,(b) distortional, (c) flexural–torsional and (d) flexural buckling.

13N. Silvestre et al. / Journal of Constructional Steel Research 73 (2012) 12–30

tomembers (i) under compression and bending or (ii) influenced by in-teraction phenomena involving distortional buckling modes. Indeed,considerable research has been carried out on the application of theDSM to beam-columns [10,11] and, most of all, to cold-formed steellipped channel columns affected by local/distortional mode interaction[12–20]— note that it was shown by the authors [21,22] that the post-buckling and ultimate strength behaviours of columns exhibiting vari-ous commonly used lipped channel cross-section shapes can be strong-ly affected by coupling between local and distortional buckling modes.Concerning this last topic, it would be obviously very convenient tohave the local/distortional mode interaction phenomenon also coveredby the DSM — the purpose of this work is precisely to contribute to-wards achieving this goal.

Yang and Hancock [12] and Kwon et al. [20] carried out tests onhigh strength steel lipped channel columns, with and without weband/or flange intermediate stiffeners, which provided strong experi-mental evidence of the occurrence of local/distortional interaction.These authors also compared their test results (ultimate strengths)with the estimates provided by the current DSM design curves andconcluded that the latter considerably overestimate the former, thusconfirming that the current DSM cannot handle adequately the ulti-mate strength erosion stemming from local/distortional interaction.Recently, Yap and Hancock [23,24] performed additional tests onhigh strength steel lipped channel columns with intermediately stiff-ened webs and concluded again that (i) the columns failed prema-turely due to visible local/distortional interaction, (ii) the DSMdistortional strength curve was inadequate to account for the interac-tion effects — moreover, a modified DSM design curve was proposedto predict more accurately these column ultimate strengths.

As mentioned before, the objective of the present work is to con-tribute towards the extension of the domain of application of the cur-rent DSM, by making it able to estimate the ultimate strength oflipped channel steel columns affected by interaction involving local(L) and distortional (D) buckling modes. In order to achieve thisgoal, one begins by illustrating the numerical (shell finite element)analysis methodology adopted to (i) select the column geometries(buckling analysis), (ii) identify the most detrimental (critical) im-perfection shapes and (iii) estimate the column collapse loads(post-buckling analysis). All the analyses were carried out in thecode ABAQUS [25], adopting 4-node isoparametric shell finite elementsto discretise the columns. Then, one reports an extensive parametricstudy involving the determination of the elastic–plastic failure loadsof simply supported and fully fixed lipped channel columns with dis-tinct cross-section dimensions, lengths and yield stresses (a total of276 columns), and containing critical-mode small-amplitude initialgeometrical imperfections — the columns were carefully selected, inorder to exhibit strong L/D interaction. The ultimate strength valuesobtained by means of the second-order elastic–plastic ABAQUS shell fi-nite element analyses (SFEA) then provide a “data bank” that makes itpossible to propose and validate preliminary recommendations onthe use of a novel DSM approach to estimate column collapse loadsunder L/D mode interaction. As noted earlier, the results reportedhere concern an ongoing investigation aimed at proposing modifica-tions to the existing DSM expressions, thus making them applicableto design lipped channel columns against the interaction phenome-non under consideration.

2. Local/distortional interaction — numerical analysis

In order to assess the performance of the Direct Strength Method(DSM) to estimate the ultimate strength of lipped channel columnsaffected by L/D interaction, it is indispensable to possess a significantamount of reliable experimental and/or numerical ultimate strengthvalues concerning columns with close (almost coincident) local anddistortional critical buckling stresses. Therefore, it was decided tocarry out an extensive parametric study, intended to acquire a fairlylarge data bank of column ultimate strengths on which to base the as-sessment and/or improvement of the current DSM design approach—

these ultimate strengths were yielded by ABAQUS SFEA, as briefly de-scribed next.

Following a brief discussion of relevant aspects dealing with the fi-nite element modelling of the geometrically and materially non-linear behaviour of thin-walled members, the methodology adoptedto obtain the column ultimate strength data bank is illustrated. It in-volves two types of analyses, namely (i) elastic buckling analyses, toselect the most appropriate column geometries (cross-section dimen-sions and lengths) to be considered in the post-buckling analyses, aswell as to obtain the corresponding buckling mode shapes, requiredto define the column initial geometrical imperfections, and (ii) elas-tic–plastic post-buckling analyses, to obtain “exact” ultimate strengthvalues (collapse loads). This methodology is illustrated by showingsome results obtained in two recent investigations on the elastic–plastic post-buckling behaviour and ultimate strength of simply sup-ported and fixed lipped-channel columns strongly affected by L/D in-teraction [22,26].

2.1. Finite element modelling

This section deals with a few relevant issues concerning the use ofthe code ABAQUS to perform shell finite element (i) elastic bucklingand (ii) elastic and elastic–plastic post-buckling analyses of thin-walled members. Concerning the performance of these SFEA, the fol-lowing aspects deserve to be mentioned here [22,27,28]:

(i) Discretisation. In order to analyse both the local and distortion-al behaviour of a given thin-walled member, a two-dimensional modelling must be adopted (i.e., its mid-surfacemust be discretised — the round corners have been neglectedin this work), a task adequately performed by means of iso-parametric shell finite elements. The column mid-surfaceswere discretised into S4 finite elements (ABAQUS nomenclature:4-node shell elements with the shear stiffness yielded by a fullintegration rule), which were found to provide accurate results.20–30 elements were considered along the cross-section mid-line (width of about 5–10 mm) and previous convergence/accuracy studies showed that the element length-to-widthratio should be comprised between 1 and 2.

(ii) Support conditions. The columns dealt with in this work exhibittwo end support conditions, namely (ii1) simply supported(pinned and free-to-warp) or (ii2) fixed and warping-prevented end sections— the torsional rotations are preventedin both cases. The pinned condition was modelled by imposingnull transverse displacements at all end section nodes and thefixed condition by means of rigid end-plates attached to thecross-section centroid (i.e., the column longitudinal axis). Inorder to preclude numerical difficulties, the longitudinal dis-placements were prevented in either (ii1) one single mid-section point (pinned columns) or (ii2) an entire end section(fixed columns). Finally, note also that, due to Poisson's effects,the above support condition modelling generates non-negligible stress concentrations near the column end sections— however, it has been shown that this phenomenon only af-fects the stability behaviour of “very short” columns [27,28].

14 N. Silvestre et al. / Journal of Constructional Steel Research 73 (2012) 12–30

(iii) Loading. Compressive forces, statically equivalent to a uniformnormal stress distribution, are applied at the nodes of the col-umn axially free end-sections — both end-sections in the sim-ply supported columns and only one in the fixed columns.Since the reference value of the load parameter p is t N/mm (t— wall thickness), which corresponds to a 1MPa uniformstress distribution, the value of p obtained is equal to the aver-age stress acting on the column, expressed in MPa.

(iv) Material modelling. The column (carbon steel) material behav-iour, deemed isotropic and homogeneous, was modelled bymeans of (iv1) linear elastic (buckling analysis) and (iv2) elas-tic/perfectly-plastic (post-buckling analysis) stress–strainlaws. In the latter case, the well-known Prandtl–Reuss model(J2-flow theory), combining Von Mises's yield criterion withthe associated flow rule, was adopted. These stress–strainlaws are readily available in the ABAQUS material behaviour li-brary and one just needs to provide the values of E, v and fy.

(v) Initial imperfections. Initial geometrical imperfections can beincorporated automatically in an ABAQUS column post-bucklinganalysis through a linear combination of buckling modeshapes, obtained from a preliminary buckling analysis. In thiswork, the initial geometrical imperfections exhibit critical(mostly distortional) mode shapes. Residual stress effects areneglected, since they have been shown to have little impacton the ultimate strength of cold-formed steel columns [29].

2.2. Methodology adopted

The methodology adopted to obtain a fairly large data bank of nu-merical column ultimate strengths involved the following main steps:

(i) Performance of sequences of “trial-and-error” buckling ana-lyses, in order (i1) to select column geometries (cross-sectiondimensions and lengths) associated with strong L/D interac-tion (almost coincident local and distortional critical bucklingstresses) and (i2) to obtain the corresponding local and distor-tional buckling mode shapes, required to define the columninitial geometrical imperfections (in the subsequent post-buckling analyses). Fig. 2(a1)–(a2) illustrates the end productof this selection procedure for simply supported and fixed

Simply Supported Columnsfcr(MPa)

L(cm)

+

200

150

100

50

0

(mm)

100

505

1

LL/D=27cm

(fcr=100.5MPa)

1 10 100 1000

(a1)

(b1) (b2)

Fig. 2. Selection of simply supported and fixed lipped channel columns exhibiting strong L/D(simply supported) and LL/D=250 cm (fixed) columns.

lipped channel columns [22,26] — the curves shown providethe variation of the critical stress fcr with the length L (logarith-mic scale). As for Fig. 2(b1)–(b2), they show the “mixed” criti-cal buckling mode shapes of (i1) the simply supported columnwith LL/D=27 cm, combining a single distortional half-wavewith three local ones (triggered by the web), and (i2) thefixed column with LL/D=250 cm, combining 27 local half-waves with 3 distortional ones. The whole set of columncross-section dimensions and lengths selected, as well as thecorresponding critical local (fcrl) and distortional (fcrd) bucklingstresses, are given in tabular form in Annex A (simply sup-ported columns) and Annex B (fixed columns).

(ii) Definition of sets of critical-mode initial geometrical imperfec-tions than span the whole shape spectrum, while sharing thesame amplitude — 10% of the wall thickness (0.1 t) was thevalue considered. Fig. 3(a) provides a good visualisation and“feel” of these initial imperfection sets: each point on the unitradius circle drawn in the CL.0–CD.0 plane, associated with anangle θ measured counter clockwise from the horizontal axis,corresponds to a particular critical-mode initial imperfectionshape, combining uniquely the normalised shapes of the “coin-cident” local and distortional critical buckling modes —

each combination is defined by the ratio CD.0/CL.0 (CD.0=cosθand CL.0=senθ) [22]. Fig. 3(b1)–(b2) concerns the simply sup-ported and fixed columns identified in Fig. 2(a) [22,26] andshows their initial imperfection shapes associated with θ =0,180° (pure distortional) and θ =90, 270° (pure local) —

while θ=0° and θ=180° involve inward and outward flange-lip motions, θ=90° and θ=270° exhibit outward and inwardmid-span web bending.

(iii) Performance of elastic and elastic–plastic post-buckling ana-lyses of columns containing the initial geometrical imperfec-tions defined in the previous item, in order (iii1) to acquireknowledge about the L/D mode interaction mechanics and(iii2) to identify the most detrimental initial imperfectionshapes, in the sense that they are those leading to higher col-umn strength erosions.

(iv) Performance of parametric studies involving the various col-umns whose geometries were selected in step (i), all contain-ing the most detrimental imperfection shapes identified in

200

150

100

50

0

1 10 100 1000

L(cm)

Fixed Columns

+

fcr(MPa)

LL/D=250cm

(fcr=102.7MPa)

95

(mm)10

1

80

(a2)

interaction: (a) fcr vs. L curves and (b) “mixed” critical mode shapes of the LL/D=27 cm

0

CD.0

CL.0

r =1

1

1

−1

−1

wL.0=−0.1t

vL.0

vD.0= 0.1t

wD.0

wD.0

vD.0= -0.1t

wL.0=0.1t

vL.0

(a)

0°°

90°°

270°°

180°°

270°°

0°°

180°°

90°°

(b1) (b2)

Fig. 3. (a) Initial imperfection shapes considered in the column post-buckling analyses, represented in the CD.0–CL.0 plane, and (b) pure distortional (θ=0°, 180°) and pure local(θ=90°, 270°) initial imperfection shapes for the (b1) simply supported and (b2) fixed columns identified in Fig. 2(a).

15N. Silvestre et al. / Journal of Constructional Steel Research 73 (2012) 12–30

step (iii) and exhibiting several yield stress values, chosen tocover wide slenderness ranges, in order to obtain a “databank” of column numerical ultimate strengths. This ultimatestrength data bank will be subsequently used to develop/vali-date a DSM-based design approach that accounts for L/Dmode interaction effects.

fy /fcr=5.5

P/Pcr

θ6090

120150180

θ45300

1.2

1.4

fy /fcr=3.5

fy /fcr=5.5

fy /fcr=3.5

Elastic Elastic

Simply Supported Columns(a) 1800 θ

3. Column post-buckling behaviour and ultimate strength

The influence of L/D mode interaction on the post-buckling behav-iour and strength of thin-walled columns has attracted the attentionof several researchers in the recent past (e.g., [12,22,26,30–32]).This section briefly reports the main results obtained in two recentnumerical investigations concerning simply supported [22] andfixed [26] lipped channel columns. These results concern columnswith the geometries given in Fig. 2(a1)–(a2) and deal with (i) theidentification of the most detrimental initial imperfection shapes,(ii) the characterisation of the most common column failure mecha-nisms and (iii) the assessment of the strength erosion due to the L/D mode interaction.

1

15

fy /fcr=2

fy /fcr=2

10 5 10 155

0.8

> 0 < 0

0

(b) P/Pcr

300270240210180

3153300

15

1.2

1.4

1

10 5 10 155

Elastic Elastic

0.8

<0

0

|v|/t

Simply Supported Columns 360=0180 θ

θθ

fy /fcr=5.5

fy /fcr=3.5

fy /fcr=2fy /fcr=2

fy /fcr=3.5

fy /fcr=5.5

ν 0ν

|v|/t

ν

>

ν

Fig. 4. Simply supported column elastic and elastic–plastic (fy/fcr≈2, 3.5, 5.5) post-buckling equilibrium paths P/Pcr vs. v/t: (a) 0°≤θ≤180° and (b) 180°≤θ≤360°.

3.1. Simply supported columns

Fig. 4(a)–(b) shows the upper portions (P/Pcr>0.8) of the elastic andelastic–plastic equilibrium paths P/Pcr vs. v/t (v is the mid-span flange-lip corner vertical displacement) of columns (i) containing 14 initial im-perfection shapes (θ=0, 30, 45, 60, 90, 120, 150, 180, 210, 240, 270, 300,315, 330°−0≤θ≤180° in Fig. 4(a) and 180≤θ≤360° in Fig. 4(b)),all with the same overall amplitude of 0.1 t, and (ii) exhibiting 3 yield-to-critical stress ratios fy/fcr≈2, 3.5, 5.5 (fcrl=fcrd=100.5MPa,corresponding to Pcr=21.1 kN)− fy/fcr=∞ identifies the elasticequilibrium paths. On the other hand, Fig. 5(a)–(b) shows the elastic(i) web deformation evolution and (ii) deformed configurations, atthe advanced post-buckling stages, of the θ=0° and θ=180° columns.Finally, Fig. 6(a)–(b) provides information about the evolution ofthe plastic strains in the two columns defined by θ=0°+fy/fcr≈3.5and θ=90°+fy/fcr≈3.5 — in each case, 4 plastic strain diagramsare presented, corresponding to different equilibrium states locatedalong the post-buckling equilibrium path (as indicated in Fig. 6(a)).Note that (i) the deformed configurations corresponding to thepoints I are amplified 10 times with respect to the remaining ones,and that (ii) points IV always correspond to equilibrium states immedi-ately after the column collapse — this means that the associated de-formed configurations provide information about the column failuremodes.

The observation of the post-buckling results presented in Figs. 4 to6 prompts the following remarks:

(i) There is a visible elastic post-buckling asymmetry concerningthe columns with pure distortional imperfections: the θ=0°(inward) column exhibits slightly higher post-bucklingstrengths than its θ=180° (outward) counterpart.

(ii) All elastic equilibrium paths merge into one of the two curvesconcerning the θ=0° or θ=180° columns. At first sight, the

16 N. Silvestre et al. / Journal of Constructional Steel Research 73 (2012) 12–30

deformed configurations of all those columns, at the advancedpost-buckling stages, appear to be purely distortional (seeFig. 5(b)). However, the gradual emergence of a (small) localcomponent is perceptible — Fig. 5(a) compares the evolutionof the θ=0° and θ=180° column web deformed configura-tions with an exact sinusoidal shape, thus revealing the pres-ence of the local 3 half-wave (sinusoidal) component.

(iii) All elastic equilibrium paths associated with predominantlylocal imperfections (namely those concerning the θ=90° andθ=270° columns) exhibit a distinct “irregular” behaviour: re-gardless of the CL.0 sign, they (iii1) are less “smooth” than theθ=0° and θ=180° column paths, which always evolve in amonotonic fashion, (iii2) exhibit limit points, associated with“snap-through” phenomena,1 and, for P/Pcr>1.1, (iii3) mergewith either the θ=180° column (most of them) or θ=0° col-umn paths. This means that the columns evolve towards pre-dominantly distortional single half-wave deformedconfigurations — these evolutions often include significantweb bending reversals, which explain the occurrence of thelimit points appearing in the (elastic) equilibrium pathsshown in Fig. 4(a)–(b).

(iv) The various column elastic–plastic equilibrium paths alsomerge with those corresponding to θ=0° or θ=180°. Depend-ing on the fy/fcr value, this merging may occur after orbefore the column has reached its ultimate strength – seeFig. 4(a)–(b).

(v) The onset of yielding always takes place inside 0.8bP/Pcrb1.3(obviously, the exact value varies with the yield-to-criticalstress ratio and initial imperfection shape) and defines thepoint of separation between the elastic and elastic–plasticequilibrium paths. For a given fy/fcr value, this separationtakes place for higher load levels in columns with pure localimperfections (θ=90° or θ=270°) than in columns withpure distortional ones (θ=0° or θ=180°) — the differencemay reach about 20% (e.g., θ=180° vs. θ=270° for fy/fcr≈2).

(vi) Plasticity first appears at the mid-span lip free end regions, asshown in Fig. 6(b1)–(b2). However, note that, for fy/fcr≥3.5,the onset of yielding does not precipitate failure, even whenassociated with a “snap-through” phenomenon (e.g., fy/fcr≈3.5 and any imperfection shape with a local component).Indeed, the column still exhibits a certain amount of post-buckling strength reserve, which is more substantial for fy/fcr≈5.5 and outward flange-lip motion (see Fig. 4(a)). Notealso that failure occurs after the various equilibrium pathshave already merged together, whichmeans that the initial im-perfection shape has little influence on the column load-carrying capacity and failure mechanism.

(vii) For fy/fcr≥3.5, collapse only occurs after the full yielding of thecolumn mid-span zones located around the web-flange cor-ners, as shown in the right side of Fig. 6(b1)–(b2). However,the collapse mechanism is different in columns with outwardand inward flange-lip motions: (vii1) a “three hinge flangemechanism” in the former (see the right side of Fig. 6(b2))and (vii2) plastic deformations restricted to the close vicinityof the mid-span cross-section (see the right side of Fig. 6(b1)).

(viii) The outward pure distortional initial imperfection shape is themost detrimental one, since it leads to the lowest column load-carrying capacities. Therefore, all the column ultimatestrengths included in the data bank used to develop/validatethe DSM-based design approach accounting for L/D mode in-teraction effects are obtained with outward pure distortionalinitial imperfections (see Section 5).

1 The “snap-through” becomes more “abrupt” as the amplitude of the initial imper-fection local component grows. When the imperfections are pure or “almost pure”(e.g., the θ=90° column), one even observes a minor “snap-back” phenomenon.

3.2. Fixed columns

Fig. 7(a) shows the upper portions (P/Pcr>0.8) of the elastic andelastic–plastic equilibrium paths P/Pcr vs. v/t (v is again the mid-spanflange-lip corner vertical displacement) of columns (i) containing 8 ini-tial imperfection shapes (θ=0, 45, 90, 135, 180, 225, 270, 315, 315°), allhaving once more amplitude of 0.1 t, and (ii) with yield-to-criticalstress ratios equal to fy/fcr≈2, 3.5, 5.5 (now fcrl=fcrd=102.7MPa,corresponding to Pcr=28.2kN). Fig. 7(b) shows the θ=0 and 180°column deformed configurations at the advanced post-buckling stages.As for Fig. 8(a)–(b), they display the plastic strain evolution in the col-umns associated with θ=0°+fy/fcr≈3.5 and θ=180°+fy/fcr≈3.5— ineach case, plastic strain diagrams corresponding to three equilibriumstates are shown in Fig. 8(b) (their locations are indicated inFig. 8(a)). The deformed configurations corresponding to the points Iand 1 are again amplified 10 times, while those associated with pointsIII and 3 always concern the column collapse modes. The observation ofthe results presented in these two figures leads to the followingremarks:

(i) There is a close resemblance between the elastic post-bucklingbehaviours of the fixed and simply supported columns affectedby L/D interaction, notably (i1) the post-buckling asymmetryconcerning the columns with pure distortional imperfections(the θ=180° column is slightly more flexible), (i2) the “irregu-lar” behaviour (limit points and “snap-through” phenomena)of the column equilibrium paths associated with predominant-ly local imperfections and (i3) the occurrence of deformed con-figurations in advanced post-buckling stages with eitherinward or outward mid-span flange-lip motions.

(ii) However, there are also some differences between the two col-umn post-buckling behaviours, namely the fact that the fixedcolumns exhibit (ii1) more than one “common curve” forboth inward and outward mid-span flange-lip motions and(ii2) more visible local deformations (see Fig. 5(b)).

(iii) For fy/fcr≥3.5, the onset of yielding (iii1) never precipitates thecolumn collapse and (iii2) often triggers a mild “snap-through”, followed by a further strength increase before failureis reached. Note also that, for the 90≤θb270° columns, thestrength increase following the onset of yielding is slightlyhigher (e.g., for fy/fcr≈3.5, 0.06Pcr for θ=0° and 0.09Pcr forθ=180°).

(iv) For fy/fcr≈3.5, yielding (iv1) starts at the most outward cross-section lip free end regions (see the plastic strain diagrams Iand 1 in Fig. 8(b)), then (iv2) spreads along the whole widthof those “outward” lip regions and only after (iv3) begins atthe most inward cross-section lip free end regions (see theplastic strain diagrams II and 2 in Fig. 8(b)). Finally, the columncollapse is precipitated by the full yielding of the mid-spanweb-flange corner regions, which correspond to eitheroutward (θ=180°) or inward (θ=0°) cross-sections — seethe plastic strain diagrams III and 3 in Fig. 8(b).

(v) Pure distortional initial imperfections involving predominantlyoutward flange-lip motions (θ=0° in this case — but note thatthe central half-wave is inward) are the most detrimental ones,as it leads to the lowest column strengths. Thus, they areadopted to obtain the column ultimate strength data bankreported in Section 5.

4. Direct Strength Method (DSM)

When compared with the traditional “effective width approach”,the DSM exhibits three major innovative features, all stemmingfrom the fact that the cross-section is viewed as a whole: (i) wall-restraint effects are automatically taken into account, (ii) no effectivewidth calculations are needed and (iii) it is possible to provide

P/Pcr

1.50 1.30 1.14 1.04 0.86

(a)

9 18 27

(b)

distortional

w / t θ=0

0

2

4

0

P/Pcr

0.84 0.96 1.09 1.17 1.43

w / t

9 18 27

distortional

θ=180

-4

-2

00

θ=0 θ=180

Fig. 5. Elastic (a) web deformed configuration evolution and (b) deformed configurations, at the advanced post-buckling stages, of the θ=0° and θ=180° columns.

17N. Silvestre et al. / Journal of Constructional Steel Research 73 (2012) 12–30

strength estimates for members failing in distortional modes. More-over, the DSM provides a rational and systematic framework for thedesign of thin-walled members with arbitrary cross-section shapes,loadings or failure modes— naturally, a given application needs prop-er calibration and validation (i.e., comparison with a fair number ofexperimental and/or numerical results). Finally, note that both theDSM and effective width approaches share the same basic assump-tion: the member ultimate strength can be accurately predicted solelyon the basis of its elastic buckling and yield stresses.

The available DSM applications adopt “Winter-type” designcurves, which have been calibrated against a large number of experi-mental and/or numerical results [4,5]. It was shown that, whenever agiven column fails in pure local or distortional modes, it is possible toobtain safe and accurate ultimate strength estimates on the basis of

0

fy/fcr=3.5

θ=0

I

1.2

0.8

IV II III

P/Pcr(a1)

15

I II

(b1)

(b2)

I II

v / t||

Fig. 6. (a) Equilibrium states on the P/Pcr vs. v/t post-buckling equilibrium paths and (b) assoand (b2) θ=90° (fy/fcr≈3.5).

the elastic buckling and yield stress values. Thus, the DSM prescribesthat the column nominal strengths against local and distortional fail-ures (fnl and fnd) are given by the expressions [4,5,33]

f nl ¼ f y if λl≤0:776; ð1aÞ

f nl ¼ f yf crlf y

!0:4

1−0:15f crlf y

!0:4" #if λl > 0:776; ð1bÞ

f nd ¼ f y if λd≤0:561; ð2aÞ

f nd ¼ f yf crdf y

!0:6

1−0:25f crdf y

!0:6" #if λd > 0:561; ð2bÞ

v / t

15 0

fy/fcr=3.5

θ=90

1.2

0.8I

IV

II III

P/Pcr(a2)

III IV

III IV

||

ciated plastic strain diagrams and failure modes for the columns defined by (b1) θ=0°

(a) ⏐v⏐/ t

P/PcrFixed Columns

(b2)

30 20 10 20 30100

0

1.2

1.4

1.0

Elastic Elastic

0.8

v > 0 v < 0 180

27090

45 315

135225

fy /fcr=5.5

fy /fcr=3.5

fy /fcr=2

fy /fcr=5.5

fy /fcr=3.5

fy /fcr=2

(b1)

θ θ

θ = θ =

θ =θ =

Fig. 7. Fixed column (a) elastic and elastic–plastic (fy/fcr≈2, 3.5, 5.5) post-buckling equilibrium paths PPcr vs. v/t and (b) deformed configurations, at the advanced post-bucklingstages, of the (b1) θ=0° and (b2) θ=180° columns.

18 N. Silvestre et al. / Journal of Constructional Steel Research 73 (2012) 12–30

where (i) λl=(fy/fcrl)0.5 and λd=(fy/fcrd)0.5, (ii) fy is the yield stress and(iii) fcrl and fcrd are the local and distortional critical buckling stresses. Inorder to capture the local/global interaction, the DSM approach pro-poses the replacement of fy by fne in Eqs. (2a)–(2b), where fne is the col-umn buckling strength associated with global (overall) failure. Veryrecently, Kwon et al. [34] investigated the applicability of the DSM to es-timate the ultimate strength of welded columns undergoing local/

fy /fcr =3.5

P/Pc

1

1.

Elastic

0v < 0

I II

III

(a)

0

(b1)

20

I

II

III

IV

IV

10

Fig. 8. (a) Equilibrium states on the P/Pcr vs. v/t post-buckling paths and (b) associated plastθ=0° (fy/fcr≈3.5).

global buckling interaction. At this point, it is worth mentioning thatan accurate prediction of the column distortional failure load has con-siderable practical relevance, since (i) the distortional post-criticalstrength reserve is considerably lower and more imperfection-sensitive than its local counterpart and (ii) there exists clear (numeri-cal) evidence that the collapse of columns buckling in local modes isoften associated with a distortional failure mechanism [35].

r

.2

0

Elastic

.8

v > 0

v / t

fy /fcr=3.5 1 2

3

20

(b2)

1

2

3

4

4

10

||

ic strain diagrams and failure modes for the columns defined by (b1) θ=180° and (b2)

19N. Silvestre et al. / Journal of Constructional Steel Research 73 (2012) 12–30

4.1. DSM for local/distortional interaction

Following a strategy similar to that used to develop safety-checkingrules accounting for local/global interaction effects, it is possible to pro-pose expressions to estimate the ultimate strength of columnsexperiencing L/D interaction. This was first done by Schafer [36,37],who proposed two distinct approaches: (i) replace fy by fnd inEqs. (1a)–(1b) (NLD approach) or (ii) replace fy by fnl inEqs. (2a)–(2b) (NDL approach)— fnl and fnd are the local and distortion-al buckling strengths, obtained from Eqs. (1a)–(1b) and Eqs. (2a)–(2b).Yang and Hancock [12] and Kwon et al. [20] recently adopted the NLDapproach. After comparing the ultimate strength estimates providedby the NLD approach with the results of experimental tests performedon lipped channel columns (plain or with “v-shape”web and/or flangeintermediate stiffeners), which provided evidence of adverse local/dis-tortional interaction, they concluded that (i) the coefficient 0.15 inEq. (1b) should be replaced by 0.2 in order to obtain safe and reasonablyaccurate DSM estimates, and also that (ii) further investigation is re-quired on the design of columnswith nearly coincident local and distor-tional buckling stresses. Recently, Silvestre et al. [19] assessed theperformance of these two approaches and concluded that they yieldvery similar results. In the present work, the NDL approach is used,which means that the column strength is given by

f ndl ¼ f nl if λdl≤0:561; ð3aÞ

f ndl ¼ f nlf crdf nl

� �0:61−0:25

f crdf nl

� �0:6� �if λdl > 0:561; ð3bÞ

where λdl=(fnl/fcrd)0.5 is the “distortional–local slenderness” and fnl iscalculated by means of Eqs. (1a)–(2b). Finally, note that the use ofEqs. (1), (2) and (3) requires de evaluation of accurate local and distor-tional buckling stresses (fcrl, fcrd), which may be obtained through ana-lyses employing (i) shell finite element models, (ii) the (spline orsemi-analytic) finite strip method or (iii) Generalised Beam Theory(GBT) — in this work, this task was performed by means of GBT-basedbuckling analyses [38–40], which is able to calculate accurately bothvalues for members with different support conditions.

5. Parametric study — column ultimate strength data

In order to carry out a fairly large parametric study involving the ul-timate strength of lipped channel columns affected by L/D interaction, itwas necessary to begin by selecting carefully their geometries (cross-section dimensions and lengths), always controlling the ratio betweenthe local (fcrl) and distortional (fcrd) critical buckling stresses. This wasachieved by adopting the following strategy:

(i) A trial-and-error approach was initially employed to find six-teen “basic” columns, exhibiting commonly used cross-section dimensions and lengths that ensure coincident fcrland fcrd values. This search led to (i1) eight stocky (ST) cross-sections (λdb1.5), defined by

(i1) bw=180, bf=100, bs=20 and t=3.4mm (ST1).(i2) bw=110, bf=78, bs=30 and t=2.8mm (ST2).(i3) bw=100, bf=100, bs=26 and t=2.0mm (ST3).(i4) bw=100, bf=100, bs=25 and t=2.4mm (ST4).(i5) bw=110, bf=100, bs=17.5 and t=1.9mm (ST5).(i6) bw=110, bf=85, bs=15 and t=2.4mm (ST6).(i7) bw=100, bf=50, bs=10 and t=2.4mm (ST7).(i8) bw=115, bf=80, bs=10 and t=1.5mm (ST8).and also to the eight slender (SL) columns (λd≥1.5)(i9) bw=100, bf=50, bs=5 and t=1.0mm (SL1).

(i10) bw=120, bf=80, bs=10 and t=1.3mm (SL2).(i11) bw=95, bf=80, bs=10 and t=0.95mm (SL3).(i12) bw=95, bf=80, bs=10 and t=1.0mm (SL4).(i13) bw=150, bf=140, bs=10 and t=1.2mm (SL5).(i14) bw=200, bf=190, bs=12.5 and t=1.5mm (SL6).(i15) bw=235, bf=160, bs=12.5 and t=1.9mm (SL7).(i16) bw=180, bf=120, bs=10 and t=1.9mm (SL8).

It should be mentioned that all these columns satisfy the cross-section dimension requirements adopted in the DSM approach(i.e., they are all “pre-qualified columns”).

(ii) Concerning the boundary conditions, both (ii1) simply sup-ported and (ii2) fixed columns were considered. The lengths(L) of the simply supported columns always correspond to asingle distortional half-wave and were determined, togetherwith the associated buckling stress fcrd, by means of GBT-based buckling analyses. Choosing the fixed column lengthsconstitutes a more difficult task, since they must be largeenough to (ii1) ensure the occurrence of L/D interaction (mul-tiple distortional half-waves) and (ii2) stay away from flexur-al–torsional buckling. The lengths of all the columnsconsidered in this study are given in the tables included inAnnex A (simply supported columns) and Annex B (fixedcolumns).

(iii) Subsequently, the ratio between fcrl and fcrd was slightly variedfrom the unit value, by changing only one cross-section dimen-sion (flange width bf, web width bw or stiffener width bs). Thisprocedure led to the identification of 31 simply supported and61 fixed columns (iii1) having cross-section dimensions gener-ated from the sixteen basic shapes and (iii2) exhibiting close(but not necessarily equal) fcrl and fcrd values — they are suchthat one has always 0.90≤ fcrl/fcrd≤1.10.

As mentioned earlier, the column (steel) material behaviour ischaracterised by E=210 GPa (Young's modulus), ν=0.3 (Poisson'sratio) and fy=250–350-550MPa (yield stress — these three valuesmeet the limit stress stipulated for pre-qualified columns), whichmeans that a total of 276 columns were analysed. Regardless of thecritical stress ratio fcrl/fcrd value, all columns contained initial geomet-rical imperfections having the distortional buckling mode shapes in-volving predominantly outward flange-lip motions (found to be themost detrimental shapes) and small amplitudes (maximum flange-lip corner displacement equal to 10% of the wall thickness t). Natural-ly, higher (more realistic) amplitudes would have further adverseconsequences on the column ultimate strength erosion.

The parametric study involved 48 slender simply supported col-umns, 45 stocky simply supported columns, 93 slender fixed columnsand 90 stocky fixed columns (a total of 276 columns), correspondingto various combinations of slender and stocky cross-section shapesand 3 yield stress values, were analysed. All cross-section dimensions(bw, bf, bs, t — web, flange, lip widths and thickness), lengths (L), crit-ical stresses (fcrl, fcrd) and yield stresses (fy) considered are given inAnnex A (simply supported columns) and Annex B (fixed columns)— see Tables A.1 (stocky simply supported columns), A.2 (slendersimply supported columns), B.1 (stocky fixed columns) and B.2 (slen-der fixed columns). Finally, recall that the “exact” (numerical) columnultimate strengths (fu) presented in those tables and used to assessthe merits of the fndl approach, were obtained by means of ABAQUS

shell finite element analyses.

6. Assessment of the DSM estimates

6.1. Simply supported columns

The numerical and DSM results presented in Annex A make it pos-sible to compare the “exact” fu values with their DSM estimates fnl, fnd

Table 1Average and standard deviation of the simply supported column ultimate strengthestimates.

Stocky columns (λd≥1.5) Slender columns (λd≥1.5)

fnl/fu fnd/fu fndl/fu fnl/fu fnd/fu fndl/fu

Average 1.08 0.96 0.88 1.93 1.44 1.06Standard deviation 0.12 0.07 0.06 0.39 0.22 0.12

20 N. Silvestre et al. / Journal of Constructional Steel Research 73 (2012) 12–30

and fndl for the simply supported columns analysed. Table A.1 con-cerns the 45 stocky columns (geometries ST1–ST3) and Table A.2the 48 slender ones (geometries SL1–SL3). Fig. 9(a)–(c) shows thevariations of fnl/fu, fnd/fu and fndl/fu with the distortional slendernessλd and their averages and standard deviations are given in Table 1.The observation of these results prompts the following remarks:

(i) The fnl values clearly overestimate the column ultimatestrengths, particularly those concerning the slender ones. In-deed, the averages/standard deviations of the fnl/fu values areequal to 1.08/0.12 (stocky columns) and 1.93/0.39 (slender col-umns)— even if the former values appear to be almost “accept-able”, it should be mentioned that only 8 out of the 45 fnl/fulvalues fall below 1.0.

(ii) The fnd values (ii1) estimate quite accurately the stocky columnultimate strengths (average/standard deviation of fnd/fu equalto 0.96/0.07) and (ii2) evidently overestimate those concerningthe slender columns (average/standard deviation of fnd/fu equalto 1.44/0.22). In the former case, out of the 45 fnd estimates, 26are safe and accurate (0.90≤ fnd/fu≤1.00), 11 are too conserva-tive (0.80≤ fnd/fub0.90), 6 are accurate but slightly unsafe(1.00b fndl/fub1.10) and 2 are too unsafe (1.10b fndl/fu≤1.12).

(iii) The fndl values (iii1) clearly underestimate, with a fairly lowscatter, the stocky column ultimate strengths (average/stan-dard deviation of fndl/fu equal to 0.88/0.06) and (iii2) estimatereasonably accurately those dealing with the slender columns(average/standard deviation of fndl/fu equal to 1.06/0.12). Inthe former case, out of the 45 fndl estimates, 19 are safe and ac-curate (0.90≤ fndl/fub1.00), and the remaining 26 are overlyconservative (fndl/fub0.90). In the latter case, out of the 48 fndlestimates, 17 are safe and accurate (0.90≤ fndl/fu≤1.00), 2 aretoo conservative (0.80≤ fndl/fub0.90), 17 are accurate butslightly unsafe (1.00b fndl/fub1.10) and 14 are too unsafe(1.10b fndl/fu≤1.26).

(iv) The contents of the previous items show that the adequateprediction of the ultimate strengths of simply supported lippedchannel columns affected by L/D interaction depends heavilyon the distortional slenderness: while (iv1) the stocky column(λdb1.5) ultimate strengths are fairly accurately estimated bythe current DSM design curve against distortional failure(Eqs. (2a)–(2b)), which indicates that the L/D interactiondoes not play a significant role, (iv2) the prediction of the slen-der column (λd≥1.5) ultimate strengths must take into ac-count the L/D interaction effects — adopting the DSM NDL

fnl / fu fnd / fu

D buckling governs

L/D Interactive buckling governs

(a)

D buckling governs

(b)

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00

0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5

λd

Fig. 9. Variation of (a) fnl/fu, (b) fnd/fu and (c) f

approach (combination of Eqs. (1a)–(1b) and (3a)–(3b)) isclearly a good option.

(v) In other words, the erosion of the column distortional failureloads, due to the presence of local buckling, increases with thestress ratio fy/fcrd and must be taken into account for columnswith moderate-to-high and high distortional slenderness λd.

(vi) As clearly shown in Fig. 9(c), the DSM “combined” D-NDL ap-proach provides accurate ultimate strength estimates withinthe whole slenderness range, i.e., for both the stocky and slen-der columns. Indeed, the corresponding overall average/stan-dard deviations are equal to 1.03/0.12.

Finally, Fig. 10 shows the variation of the fu/fy (white dots) and“combined” (fnd or fndl)/fy (black dots) stress ratios with the distor-tional slenderness λd (≡λl), for all the columns analysed (three yieldstresses). Also included are the DSM design curves providing the col-umn ultimate strengths associated with local, distortional and local/distortional failures, defined by Eqs. (1a)–(1b), (2a)–(2b) and(3a)–(3b), respectively. The close observation of this figure promptsthe following comments:

(i) The proposed predictions (i1) lie between the L andD curves forλd≤1.1, (i2) are located around the D curve for 1.1bλd≤1.4, (i3)lie between the D and L/D curves for 1.4bλd≤1.5and (i4) are lo-cated around the L/D curve for λd>1.5. This confirms that, atleast for the critical stress ratio range considered in this work(0.90≤ fcrd/fcrd≤1.10), the L/D interaction only causes signifi-cant ultimate strength erosion (with respect to the predicteddistortional failure loads) for the most slender columns.

(ii) Changing the “transition” between stocky and slender columnsfrom λd=1.5 to λd=1.4 improves slightly the “quality” of theproposed estimates. Indeed, the overall average/standard devi-ations change from 1.03/0.12 to 1.01/0.11.

fndl / fuL/D Interactive

buckling governsD buckling

governsL/D Interactive

buckling governs

(c)

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00

2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0

λd λd

ndl/fu with the distortional slenderness λd.

λd

fnl /fy (L curve)

fnd /fy (D curve)

Exact results - fu /fyDSM results - (fnd or fndl)/fy

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0

fndl /fy (L/D curve) (λd=λl)

Fig. 10. Variation of fu/fy and (fndl or fndl)/fy with the column distortional slenderness λd.

Table 2Average and standard deviation of the fixed column ultimate strength estimates.

Stocky columns (λdb1.5) Slender columns (λd≥1.5)

fnl/fu fnd/fu fndl/fu fnl/fu fnd/fu fndl/fu

Average 1.06 0.95 0.87 1.51 1.11 0.81Standard deviation 0.13 0.07 0.07 0.19 0.12 0.10

21N. Silvestre et al. / Journal of Constructional Steel Research 73 (2012) 12–30

6.2. Fixed columns

The numerical and DSM results concerning the 90 stocky (geome-tries ST1–ST8) and 93 slender (geometries SL1–SL8) fixed columnsanalysed in this work are presented in Tables B.1 and B.2, includedin Annex B. Fig. 11(a)–(c) shows the variations of the ratios fnl/fu,fnd/fu and fndl/fu with λd and their averages and standard deviationsare given in Table 2. After observing these results, it is possible to con-clude that:

(i) The fnl values (i1) slightly overestimate the stocky column ulti-mate strengths (average/standard deviation of fnl/fu equal to1.06/0.13) and (i2) clearly overestimate the slender columnones (average/standard deviation of fnl/fu equal to 1.51/0.19).

(ii) The fnd values (ii1) estimate quite accurately, and with a fairlylow scatter, the stocky column ultimate strengths (average/standard deviation of fnd/fu equal to 0.95/0.07) and (ii2) moder-ately overestimate those concerning the slender columns (av-erage/standard deviation of fnd/fu equal to 1.11/0.12). In theformer case, out of the 90 fnd estimates, 57 are safe and accu-rate (0.90≤ fnd/fu≤1.00), 20 are too conservative (0.80≤ fnd/fub0.90), 10 are accurate but slightly unsafe (1.00b fndl/fub1.10) and 3 are too unsafe (1.10b fndl/fu≤1.24).

(iii) The fndl values (iii1) clearly underestimate, with a quite lowscatter, the stocky column ultimate strengths (average/stan-dard deviation of fndl/fu equal to 0.87/0.07) and (iii2) stronglyunderestimate, with a reasonable scatter, the slender columnultimate strengths (average/standard deviation of fndl/fu equalto 0.81/0.10). In the former case, out of the 90 fndl estimates,25 are safe and accurate (0.90≤ fndl/fub1.00), and the

fnl /fuD buckling governs

D or L/D Interactive buckling governs

fnd /fuD buckling

governsD or L/D

bucklin

0.50

0.75

1.00

1.25

1.50

1.75

0.5 1.0 1.5 2.0 2.5 3.00.50

0.75

1.00

1.25

1.50

1.75

0.5 1.0 1.5 2.0

2.00 2.00

3.5 4.0

λd

(a) (b)

Fig. 11. Variation of (a) fnl/fu, (b) fnd/fu and (c)

remaining 46 are overly conservative (fndl/fub0.90). In the lat-ter case, out of the 93 fndl estimates, 21 are safe and accurate(0.90≤ fndl/fu≤1.00) and 73 are too conservative (fndl/fub0.90).

(iv) In the case of the slender columns, the “quality” of the fnd andfndl column ultimate strength estimates varies considerablyamong the various column “families” SL1–SL8. Indeed, one has:(iv.1) SL1, SL7 and SL8: while the fndl estimates are too conser-vative, the fnd ones are very accurate — out of the 33 fnd esti-mates, 13 are safe and accurate (0.90b fnd/fu≤1.00), 19 areslightly unsafe but accurate (1.00b fnd/fu≤1.10) and only 1 istoo unsafe (fnd/fu>1.10).(iv.2) SL2, SL5 and SL6: while the fnd estimates are unsafe, thefndl ones are very conservative — out of the 36 fndl estimates, 1is safe and accurate (0.90b fndl/fu≤1.00) and all the remaining35 are overly conservative (fndl/fu>0.90).(iv.3) SL3 and SL4: while the fnd values are very unsafe, the fndlones are very accurate— out of the 24 fndl estimates, 17 are safeand accurate (0.90b fnd/fu≤1.00), and the remaining 7 are a bittoo conservative (0.80b fnd/fu≤0.90).

(v) A distinctive feature setting the various column “families”apart is the relation between their local and distortional half-wave lengths, which can be reasonably well estimated by theratio between the corresponding simply supported columnminima (obtained from the cross-section signature curve).This assertion is confirmed by comparing the Lcrd/Lcrl ratios(simply supported columns) and nl/nd (nd and nl are the fixedcolumn distortional and local half-wave numbers):

Interactivg governs

2.5 3

fndl/fu wi

(v1) Lcrd/Lcrl=270/80=3.4 vs. nl/nd=16/5=3.2 (SL1),(v2) Lcrd/Lcrl=540/95=5.7 vs. nl/nd=23/4=5.8 (SL2),(v3) Lcrd/Lcrl=610/85=7.2 vs. nl/nd=28/4=7.0 (SL3),(v4) Lcrd/Lcrl=595/85=7.0 vs. nl/nd=28/4=7.0 (SL4),(v5) Lcrd/Lcrl=830/150=5.5 vs. nl/nd=9/2=4.5 (SL5),(v6) Lcrd/Lcrl=1105/200=5.5 vs. nl/nd=8/2=4.0 (SL6),(v7) Lcrd/Lcrl=905/200=4.5 vs. nl/nd=9/4=4.5 (SL7) and(v8) Lcrd/Lcrl=620/155=4.0 vs. nl/nd=6/2=3.0 (SL8).

(vi) It seems that the value of the ratio Lcrd/Lcrl (≈nl/nd) plays an impor-tant role in the ultimate strength of slender fixed columns affected

e

.0 3.5 4.0

fndl / fuD buckling

governsD or L/D Interactive

buckling governs

0.50

0.75

1.00

1.25

1.50

1.75

0.5 1.0 1.5 2.0 2.5 3.0

2.00

3.5 4.0

λd λd

(c)

th the distortional slenderness λd.

λd=λl

fnl /fy (L curve)

fnd /fy (D curve) ≡ f * ndl /fy (Lcrd/Lcrl≤4)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

fndl /fy (L/D curve) ≡ f * ndl/fy (Lcrd/Lcrl≥8)

D buckling governs L/D Interactive buckling or D buckling governs

fu /fy and f * ndl /fy

Fig. 12. Novel DSM proposal: variation of fndl* /fy with the column slenderness λd≡λl

(fcrl=fcrd).

Table 3Average and standard deviation of the fixed column ultimate strength estimates.

Stocky (λdb1.5) Slender (λd≥1.5)

fnl/fu fnd/fu fndl/fu fndl* /fu fnl/fu fnd/fu fndl/fu fndl* /fu

Average 1.06 0.95 0.87 0.94 1.51 1.11 0.81 1.00Standard deviation 0.13 0.07 0.07 0.06 0.19 0.12 0.10 0.06

22 N. Silvestre et al. / Journal of Constructional Steel Research 73 (2012) 12–30

by L/D interaction: as Lcrd/Lcrl increases, (vi1) the column ultimatestrength decreases (the strength erosion due to interaction effectsismore relevant) and (vi2) the fndl estimates become progressivelymore accurate. Conversely, as Lcrd/Lcrl decreases the relevance ofthe interaction effects diminishes, which implies that the columnultimate strength becomes gradually “more distortional” and,therefore, the accuracy of the fnd estimates improves.

(vii) In the stocky columns, there are no visible “quality” differencesbetween the ultimate strength estimates concerning the vari-ous column “families” ST1–ST8, even if they exhibit a widerange of Lcrd/Lcrl values: 670/145=4.6, 765/90=8.5, 850/100=8.5, 845/100=8.5, 755/105=7.2, 540/95=5.7, 285/80=3.6 and 500/95=5.2, respectively.

(viii) The contents of the previous two items show that the Lcrd/Lcrlratio (viii1) has virtually no impact on the ultimate strength ofstocky fixed columns affected by L/D interaction (distortionalfailure always governs) and (viii2) must necessarily be includedin a DSM-based approach aimed at predicting accurately the ul-timate strength of slender fixed columns affected by L/D interac-tion (governed by either distortional or L/D interactive failures).

(ix) As clearly shown in Fig. 11(b), the DSM distortional design curveprovides quite accurate ultimate strength estimates for the stockyfixed columns— λdb1.5. If this “limit” value decreases to λd=1.4,the fnd estimates become a bitmore conservative— the overall av-erage/standard deviations change from 0.95/0.07 to 0.94/0.05.

(x) Fig. 11(c) provides clear evidence that, unlike in the slender sim-ply supported columns, the DSMNDL approach does not yield ac-curate ultimate strength estimate for the slender fixed columns.Indeed, the fndl values excessively underestimate the “exact” col-umn ultimate strengths. This means that (x1) the DSM “com-bined” D-NDL approach is not adequate to predict the ultimatestrength of columns affected by L/D interaction and (x2) anotherdesign approach is required, including the Lcrd/Lcrl ratio for theslender columns — such an approach is proposed next.

6.2.1. Novel DSM approach to account for L/D interactionIn order to achieve accurate column ultimate strength predictions

within the whole slenderness range, i.e., for both stocky and slenderfixed columns, a modified DSM “combined” D-NDL approach is pro-posed. The basic modification consists of replacing the column localstrength fnl (see Eqs. (1a)–(1b)) by fnl* , defined as

f �nl ¼ f y ifLcrdLcrl

≤4; ð4aÞ

f �nl ¼ f y þ 1� 0:25LcrdLcrl

� �f y � f nl� �

if 4 bLcrdLcrl

b8; ð4bÞ

f �nl ¼ f nl ifLcrdLcrl

≥8: ð4cÞ

It depends on fy, fnl (obviously) and also on the critical half-wavelength ratio Lcrd/Lcrl. Then, modified ultimate strength estimates fndl*

for columns affected by L/D interaction are obtained by replacing fnlby fnl* in Eqs. (3a)–(3b), thus defining a “modified NDL approach”.Note that Eqs. (4a)–(4c) lead to the previous D (fnd) and NDL (fndl) ulti-mate strength estimates for Lcrd/Lcrl≤4 and Lcrd/Lcrl≥8, respectively —

only for 4bLcrd/Lcrlb8 occur novel column ultimate strength estimates.Recalling that (i) it was found that L/D interaction effects play no

relevant role in stocky columns (λdb1.5 or λdb1.4) and (ii) the Lcrd/Lcrl limit values appearing in Eqs. (4a)–(4c) are based solely on the ul-timate strength data reported in this work, which concern exclusivelyfixed columns with critical stress ratios fcrl/fcrd comprised between0.90 and 1.10. Concerning this last issue, it is possible that differentLcrd/Lcrl limit values apply for other cross-section shapes. For instance,the stiffened and “plain” lipped channel columns analysed by Yang

and Hancock [12] have similar Lcrd values, but quite different Lcrlones (the former is about half of the latter) — this difference shouldbe reflected in the L/D interactionmechanics of the two column types.

In order to provide a “pictorial feel” of the novel proposal, Fig. 12 joint-ly depicts (i) the local (fnl/fy) and distortional (fnd/fy) DSM design curves,defined by Eqs. (1a)–(1b) and (2a)–(2b), and also (ii) the NDL DSM de-sign curve (fndl/fy), defined by Eqs. (3a)–(3b) and assuming that fcrl=fcrd.Concerning the fndl* /fy values, provided by the novel proposal, they:

(i) Coincidewith the fnd/fy values for λdb1.5 (stocky column)— recallthat the existing DSM provisions were found to provide quite sat-isfactory ultimate strength predictions in this slenderness range.

(ii) Are located inside the “grey area” for λd≥1.5 (slender columns),comprised between (ii1) the existing DSMD curve (fnd/fy), appli-cable to the columns with Lcrd/Lcrl≤4, and (ii2) the DSM NDLcurve (fndl/fy), applicable to the columns with Lcrd/Lcrl≥8.

The application of the proposed DSM approach to the fixed col-umns analysed in this work leads to the fndl* values that are alsogiven in Tables B.1 and B.2 (Annex B). Table 3 provides the averageand standard deviations of these new sets of estimates — also includ-ed are the values corresponding to the other DSM estimates, alreadyshown in Table 2. Moreover, Fig. 13(a)–(b) shows the variations ofthe ratios (i) fndl* /fu and (ii) fu/fy and fndl* /fy with the distortional slen-derness λd for the 90 stocky and 93 slender columns — the currentDSM L and D design curves are also depicted in the latter. The obser-vation of these results makes it possible to draw the following conclu-sions, concerning the “quality” of the fndl* estimates:

(i) They are quite accurate and exhibit a low scatter — average/standard deviation of the fndl* /fu equal 0.94/0.06 (stocky col-umns) and 1.00/0.06 (slender columns).

(ii) The fndl* ultimate strength estimates are fairly accurate even forthe SL2 columns (Lcrd/Lcrl=5.7 and larger fndl* /fu values) —

out of the 18 estimates, 2 are safe and very accurate(0.98≤ fndl* /fu≤1.00), 11 are unsafe but accurate (1.00b fndl* /fu≤1.10) and the remaining 5 are not too much unsafe(1.10b fndl* /fu≤1.16).

f *ndl /fu

D bucklinggoverns

D or L/D Interactivebuckling governs

2.00

3.5 4.00.50

0.75

1.00

1.25

1.50

1.75

0.5 1.0 1.5 2.0 2.5 3.0

λd

fu /fy and f *ndl /fy

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 4.0

Exact results - fu /fy

DSM results - f *ndl /fy

fnl /fy (L curve)

fnd /fy (D curve)

D buckling governs

3.5

D or L/D Interactive buckling governs

λd

(a) (b)

Fig. 13. Variation of (a) fndl* /fy and (b) fu/fy and fndl* /fy with the distortional slenderness λd.

23N. Silvestre et al. / Journal of Constructional Steel Research 73 (2012) 12–30

(iii) Comparing the fndl* /fu values (Fig. 13(a)) with their fnd/fu andfndl/fu counterparts (Fig. 11(b)–(c)), it is clear that the pro-posed approach yields much more accurate ultimate strengthestimates (much closer to the horizontal line fndl* /fu=1), thatare never too unsafe (only 10 fndl* /fu values above 1.09).

(iv) Fig. 13(b) confirms that the proposed ultimate strength esti-mates (black dots) are on the DSM D design curve for (iv1)all stocky columns and (iv2) the slender columns with lowLcrd/Lcrl values. On the other hand, they lay well below thatcurve for the slender columns with moderate or high Lcrd/Lcrlvalues — those exhibiting relevant ultimate strength erosiondue to the L/D interaction.

(v) Fig. 13(b) also shows that the new proposal is able to capturethe “vertical dispersion” of the “exact” ultimate strengths(white dots) in the 1.50bλdb2.50 slenderness range.

Next, the ultimate strength estimated provided by the proposedDSM design approach are compared with the available experimentalresults concerning lipped channel columns affected by L/D interac-tion, which have been reported by Yang and Hancock [12] andKwon et al. [13,20]. Out of the 53 column tests carried out by theseauthors, only 8 correspond to “plain” lipped channel columns (allthe others column exhibited “v-shape” web and/or flange intermedi-ate stiffeners). The calculation of their local and distortional criticalbuckling stresses, by means of the code GBTUL [39,40] and consideringfixed end supports, revealed (quite surprisingly) that the above8 lipped channel column geometries correspond to fcrd/fcrl valuescomprised between 1.36 and 3.23 (average of with 2.00) — the localcritical buckling stresses are considerably lower than their distortion-al counterparts. This feature contrasts with the columns numericallyanalysed in this work, which have very similar local and distortionalcritical buckling stresses (0.90≤ fcrd/fcrl≤1.10).

Fig. 14(a)–(b) displays the variation of the experimental (fu/fy)and predicted (fndl* /fy) ultimate strength ratios with the distortional(Fig. 14(a)) and local (Fig. 14(b)) slenderness values — also depictedare the current DSM L and D design curves. The observation of thesefigures prompts the following remarks:

(i) Since the local critical stresses are much lower than the distor-tional ones, it would be logical to expect the DSM local buck-ling curve to match the experimental results (white circles).However, Fig. 14(b) shows that, with a single exception, allthe experimental results fall substantially below the DSM

local strength curve. Moreover, all the experimental resultsalso fall considerably below the DSM distortional strengthcurve. These two facts provide clear evidence of severe local/distortional interaction, also observed during the tests, whichleads to a substantial column ultimate strength erosion.

(ii) Again with the same single exception, the ultimate strengthpredictions yielded by the proposed DSM approach (black cir-cles) always overestimate the experimental values (whitedots). At this point, it should be recalled that the developmentof the proposed DSM design approach to account for L/D inter-action was based on numerical ultimate strengths concerningcolumns with nearly coincident local and distortional criticalbuckling stresses (0.90≤ fcrl/fcrd≤1.10), which is not the caseof the tested columns.

(iii) Due to the high fcrd/fcrl ratios, combined with very high yieldstresses (fy/fcrl values much larger than their fcrd/fcrl counter-parts), the tested columns (iii1) have “plenty of room” to evolveinto the elastic local post-buckling range and (iii2) reach thedistortional buckling stress level before plasticity becomes rel-evant. Therefore, the emergence of the geometrically non-linear distortional effects precludes the columns from reachingtheir local failure loads, leading to the ultimate strength erosionclearly observed in Fig. 14(b). Conversely, the presence of thegeometrically non-linear local effects precludes the columnsfrom reaching their distortional failure loads, as shown inFig. 14(a).

(iv) Strictly speaking, it may be argued that the tested columns donot exhibit local/distortional buckling mode interaction, if suchcoupling phenomenon is defined as associated with the close-ness of the local and distortional critical buckling stresses,which is clearly absent. Indeed, it may be more appropriateto talk about a secondary bifurcation phenomenon, which onlycomes into play because the column yield stress is “highenough” to allow its emergence and development. Neverthe-less, there is no argument about the existence of significant ul-timate strength erosion, with respect to the failure loadspredicted by the current DSM local strength curve.

(v) Due to relatively small column distortional post-bucklingstrength, when compared with its local counterpart, columnswith fcrd/fcrl ratios much lower than 1.0 are not expected to ex-hibit meaningful ultimate strength erosion, with respect to thedistortional failure loads predicted by the current DSM curve.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5

fu /fy and f *ndl /fy

DSM results - f *ndl /fy

fnd /fy (D curve)

λd

Kwon et al. [13,19]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5

fu /fy and f *ndl /fy

DSM results - f *ndl /fy

fnl /fy (L curve)

λl

Kwon et al. [13,19]

(a) (b)

Fig. 14. Variation of the experimental (fu/fy) and proposed (fndl* /fy) ultimate strength ratios with the (a) distortional and (a) local slenderness.

24 N. Silvestre et al. / Journal of Constructional Steel Research 73 (2012) 12–30

(vi) The above facts clearly show that further numerical and“controlled” experimental studies are needed in order (vi1)to acquire in-depth understanding on the post-buckling be-haviours of columns having local and distortional criticalbuckling stresses more than 10% apart and (vi2) to be ableto develop and calibrate a DSM design approach accountingfor L/D interaction effects, regardless of their “origin” (close-ness of fcrd and fcrl or fcrd/fcrl>1.0 and fy/fcrl> fcrd/fcrl). In thisrespect, it should be mentioned that the numerical and ex-perimental investigations recently carried out by Dinis etal. [41], Young et al. [42] and Loughlan et al. [43] providedvaluable contributions to understand the influence of thefcrd/fcrl value on the (elastic and elastic–plastic) post-buckling behaviour and strength of lipped channel columns.

Finally, it should be noted that the application of the proposedDSMapproach to the slender simply supported columns did not provide ac-curate ultimate strength estimates. The corresponding fndl* , presentedin Table A.2, is too unsafe— the average/standard deviation of the fndl* /fu values is equal to 1.29/0.12. In this respect, it is worth mentioningthat a recent numerical investigation carried out by Landesmann andCamotim [44] showed that the current DSM distortional strengthcurve significantly overestimates the slender column distortional fail-ure loads — this finding may have some bearing on the aboveoverestimation.

7. Conclusion

This paper dealt with the structural response and DSM design oflipped channel columns affected by strong local/distortional interac-tion. After providing the most important features of the buckling and(mostly) post-buckling behaviour of these columns, which includedidentifying the most detrimental initial imperfection shape, a total of93 simply supported columns and 183 fixed columns were selected,such that (i) their critical distortional and local buckling stresses werenot more than 10% apart and (ii) a wide distortional/local slendernessrange was covered in both cases. Then, the ultimate strengths of allthese columns were numerically (SFEA) determined, assuming (i)critical-mode (distortional) initial imperfections with small amplitudes(10% of the wall thickness t) and (ii) a linear-elastic/perfectly-plasticcarbon steel constitutive law— both the residual stresses and corner ef-fects were neglected in the analyses.

On the basis of the column “ultimate strength data” gathered dur-ing the above FEM-based parametric study, it was possible (i) to

assess the performance of the estimates provided by the currentDSM design curves, including the more recent NDL approach (com-bining the individual distortional and local DSM curves), and also(ii) to identify the need to include additional features in a novelDSM approach, specifically developed to take into account stronglocal/distortional interaction. The output of this assessment and de-velopment effort was the proposal of different DSM-based design ap-proaches for simply supported and fixed columns, both of whichreflect the findings of this work, namely that (i) the current DSM dis-tortional strength curve adequately predicts the stockier column ulti-mate strengths, (ii) the DSM NDL approach provides safe and accurateestimates for the slender simply supported columns and that (iii) it isnecessary to involve the Lcrd/Lcrl ratio to obtain satisfactory estimatesof the slender fixed column ultimate strengths. The proposed DSM de-sign approaches are defined by the following expressions:

(I) Simply supported columns

f nl−d ¼ f nd if λd≤1:4; ð5aÞ

f nl−d ¼ f ndl if λd > 1:4; ð5bÞ

(II) Fixed columns

f nl−d ¼ f nd if λd≤1:5; ð6aÞ

f nl−d ¼ f �ndl if λd > 1:5; ð6bÞ

with fndl* obtained from

f �nl ¼ f y ifLcrdLcrl

≤4; ð7aÞ

f �nl ¼ f y þ 1−0:25LcrdLcrl

� �f y−f nl� �

if 4bLcrdLcrl

b8; ð7bÞ

f �nl ¼ f nl ifLcrdLcrl

≥8: ð7cÞ

It is worth noting that the numerical ultimate strength data usedto validate/calibrate the above DSM design approaches were obtainedconsidering critical-mode initial geometrical imperfections exhibitingthe most detrimental shape (pure distortional) and fairly small am-plitudes (10% of the wall thickness), which were shown to lead toconsiderable strength erosion, due to the local/distortional

25N. Silvestre et al. / Journal of Constructional Steel Research 73 (2012) 12–30

interaction. However, the influence of the initial geometrical imper-fection shape and/or amplitude on the column ultimate strength ero-sion was not assessed in this work — nevertheless the initialgeometrical imperfections considered here were shown to agreewell with the DSM ultimate strength estimates concerning columndistortional failures. Moreover, further numerical and experimentalresults are required (i) to confirm the efficiency of the proposedDSM design approach for lipped channel columns affected by local/distortional interaction and also (ii) to assess whether it can be ex-tended to other cross-section shapes.

Table A.1Numerical (“exact”) and DSM ultimate strength values for the stocky simply supported col

FEA

ST1 bf L fy fcrl fcrd

bw=180, bs=20, t=3.4 mm 90 650 250 361 399350550

95 650 250 358 377350550

100 650 250 355 355350550

105 650 250 353 338350550

110 650 250 350 320350550

ST2 bw L fy fcrl fcrdbf=78, bs=30, t=2.8 mm 100 800 250 736 656

350550

105 800 250 680 641350550

110 800 250 630 625350550

115 800 250 581 611350550

120 800 250 538 596350550

ST3 bs L fy fcrl fcrdbw=100, bf=100, t=2.0 mm 22 950 250 317 285

350550

24 950 250 317 299350550

26 950 250 317 314350550

28 950 250 316 331350550

30 950 250 315 350350550

On the other hand, it was found that, strictly speaking, the ulti-mate strength erosion experimentally observed in the lipped channelcolumn tests reported by Kwon et al. [13,20] does not stem from thecloseness of the critical local and distortional buckling stresses, asthey are quite far apart. Instead, it seems to be due to a secondary bi-furcation phenomenon occurring when fcrlb fcrd (local buckling pre-cedes distortional buckling) and fcrdb fy (distortional bucklingprecedes yielding). The authors are currently investigating this mat-ter, both numerically and experimentally, and the correspondingfindings will be reported in the near future.

Annex A

umns.

DSM

fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu

231 0.79 239 1.03 221 0.96 215 0.93297 0.94 301 1.01 276 0.93 251 0.85379 1.17 406 1.07 360 0.95 302 0.80230 0.81 239 1.04 218 0.95 211 0.92287 0.96 300 1.05 270 0.94 245 0.85340 1.21 405 1.19 351 1.03 295 0.87222 0.84 238 1.07 213 0.96 206 0.93276 0.99 299 1.08 264 0.96 240 0.87322 1.24 404 1.25 342 1.06 287 0.89216 0.86 238 1.10 210 0.97 203 0.94266 1.02 298 1.12 259 0.97 235 0.88305 1.28 403 1.32 334 1.10 281 0.92211 0.88 237 1.12 206 0.98 199 0.94256 1.05 298 1.16 253 0.99 230 0.90290 1.31 402 1.39 326 1.12 274 0.94

fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu249 0.62 250 1.00 247 0.99 247 0.99345 0.73 350 1.01 324 0.94 324 0.94508 0.92 514 1.01 441 0.87 423 0.83248 0.62 250 1.01 246 0.99 246 0.99344 0.74 350 1.02 322 0.94 322 0.94503 0.93 501 1.00 438 0.87 412 0.82248 0.63 250 1.01 246 0.99 246 0.99344 0.75 350 1.02 320 0.93 320 0.93499 0.94 489 0.98 434 0.87 402 0.81248 0.64 250 1.01 245 0.99 245 0.99342 0.76 350 1.02 318 0.93 318 0.93493 0.95 476 0.97 430 0.87 392 0.80248 0.65 250 1.01 244 0.98 244 0.98341 0.77 342 1.00 316 0.93 311 0.91489 0.96 464 0.95 426 0.87 383 0.78

fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu226 0.94 230 1.02 197 0.87 187 0.83262 1.11 288 1.10 241 0.92 215 0.82275 1.39 388 1.41 308 1.12 255 0.93228 0.91 230 1.01 201 0.88 190 0.83270 1.08 288 1.07 246 0.91 219 0.81287 1.36 388 1.35 315 1.10 261 0.91224 0.89 230 1.03 204 0.91 193 0.86279 1.06 288 1.03 251 0.90 223 0.80318 1.32 388 1.22 323 1.02 267 0.84232 0.87 229 0.99 208 0.90 197 0.85288 1.03 288 1.00 257 0.89 228 0.79315 1.29 388 1.23 331 1.05 272 0.86234 0.85 229 0.98 212 0.91 200 0.85297 1.00 287 0.97 263 0.89 232 0.78340 1.25 387 1.14 339 1.00 279 0.82

Av. 1.08 0.96 0.88S. dv. 0.12 0.07 0.06

26 N. Silvestre et al. / Journal of Constructional Steel Research 73 (2012) 12–30

Annex B

Table A.2

Numerical (“exact”) and DSM ultimate strength values for the slender simply supported columns.

FSA FEA DSM

SL1 bf L fy Lcrl Lcrd fcrl fcrd fu fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fu

bw=100, bs=5, t=1.0 mm 45 260 250 80 250 103 113 112 157 1.40 131 1.17 102 0.91 131 1.17350 129 195 1.51 155 1.20 115 0.89 155 1.20550 152 260 1.71 192 1.26 134 0.88 192 1.26

47.5 260 250 80 260 102 108 104 156 1.50 128 1.23 100 0.96 128 1.23350 121 194 1.60 152 1.26 113 0.93 152 1.26550 145 259 1.79 188 1.30 131 0.90 188 1.30

50 270 250 80 270 102 102 99 156 1.58 125 1.26 98 0.99 125 1.26350 115 194 1.69 147 1.28 110 0.96 147 1.28550 136 259 1.90 182 1.34 127 0.83 182 1.34

52.5 280 250 80 280 101 96 94 156 1.66 121 1.29 95 1.01 121 1.29350 109 193 1.77 143 1.31 106 0.97 143 1.31550 128 258 2.02 176 1.38 123 0.96 176 1.38

55 300 250 80 290 101 91 91 156 1.71 118 1.30 92 1.01 118 1.30350 105 193 1.84 139 1.32 103 0.98 139 1.32550 122 258 2.11 171 1.40 120 0.98 171 1.40

SL2 bw L fy Lcrl Lcrd fcrl fcrd fu fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fubf=80, bs=10, t=1.3 mm 100 550 250 90 530 157 127 119 182 1.53 139 1.17 117 0.98 139 1.17

350 122 226 1.85 165 1.35 132 1.08 150 1.23550 126 303 2.40 205 1.63 153 1.21 183 1.45

110 550 250 95 535 135 121 114 172 1.51 136 1.19 111 0.97 136 1.19350 116 215 1.85 161 1.39 125 1.08 147 1.27550 121 287 2.37 199 1.64 145 1.20 179 1.48

115 550 250 95 540 124 118 112 167 1.49 134 1.20 108 0.96 134 1.20350 114 208 1.82 159 1.39 122 1.07 145 1.27550 122 278 2.28 197 1.61 141 1.16 176 1.44

120 550 250 100 540 115 115 109 163 1.50 132 1.21 105 0.97 132 1.21350 111 203 1.83 156 1.41 119 1.07 144 1.30550 124 271 2.19 194 1.56 138 1.11 176 1.42

125 550 250 100 545 107 113 106 159 1.50 131 1.24 103 0.97 131 1.24350 109 198 1.82 155 1.42 116 1.06 144 1.32550 123 263 2.14 192 1.56 135 1.10 177 1.44

130 550 250 105 550 100 110 105 155 1.48 129 1.23 101 0.96 121 1.15350 107 193 1.80 153 1.43 113 1.06 142 1.33550 123 257 2.09 189 1.54 131 1.07 174 1.41

SL3 bs L fy Lcrl Lcrd fcrl fcrd fu fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fubw=95, bf=80, t=0.95 mm 9 550 250 85 565 91 83 78 150 1.92 112 1.44 87 1.12 96 1.23

350 79 186 2.35 132 1.67 97 1.23 110 1.39550 82 248 3.02 163 1.99 112 1.37 131 1.60

9.5 600 250 85 590 91 87 82 150 1.83 115 1.40 89 1.09 96 1.17350 83 186 2.24 135 1.63 99 1.19 110 1.33550 87 248 2.85 167 1.92 115 1.32 130 1.49

10 600 250 85 610 91 91 86 150 1.74 118 1.37 91 1.06 97 1.13350 87 186 2.16 139 1.62 101 1.17 110 1.28550 90 248 2.76 171 1.90 117 1.30 130 1.44

10.5 650 250 85 630 92 96 90 151 1.68 121 1.34 93 1.03 97 1.08350 91 187 2.05 143 1.57 104 1.14 110 1.21550 94 249 2.65 176 1.87 121 1.29 130 1.38

11 650 250 85 655 92 100 94 151 1.61 123 1.31 95 1.01 97 1.03350 95 187 1.97 146 1.54 106 1.12 110 1.16550 97 249 2.57 180 1.86 123 1.27 129 1.33

Av. 1.93 1.44 1.06 1.29S. dv. 0.39 0.22 0.12 0.12

Table B.1Numerical (“exact”) and DSM ultimate strength values for the stocky fixed columns.

FSA FEA DSM

ST1 bf L fy Lcrl Lcrd fcrl fcrd fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fu

bw=180, bs=20, t=3.4 mm 90 2000 250 145 630 367 420 244 0.77 241 0.99 225 0.92 224 0.92 225 0.92350 314 0.91 302 0.96 282 0.90 261 0.83 282 0.90550 408 1.14 408 1.01 369 0.90 313 0.77 369 0.90

95 2000 250 145 650 365 382 240 0.81 240 1.00 218 0.91 218 0.91 218 0.91350 316 0.96 302 0.96 272 0.86 254 0.80 272 0.86550 403 1.20 407 1.01 353 0.88 303 0.75 353 0.88

100 2000 250 145 670 363 379 239 0.81 240 1.00 218 0.91 218 0.91 218 0.91350 310 0.96 301 0.97 271 0.87 253 0.82 271 0.87550 395 1.20 407 1.03 352 0.89 302 0.76 352 0.89

(continued)

FSA FEA DSM

ST1 bf L fy Lcrl Lcrd fcrl fcrd fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fu

105 2000 250 150 695 360 389 238 0.80 239 1.00 220 0.92 219 0.92 220 0.92350 294 0.95 300 1.02 274 0.93 254 0.86 274 0.93550 371 1.19 406 1.09 356 0.96 304 0.82 356 0.96

110 2000 250 150 720 358 372 235 0.82 239 1.02 217 0.92 216 0.92 217 0.92350 291 0.97 300 1.03 269 0.92 251 0.86 269 0.92550 363 1.22 405 1.12 349 0.96 299 0.82 349 0.96

ST2 bw L fy Lcrl Lcrd fcrl fcrd fu d fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fubf=78, bs=30, t=2.8 mm 100 2000 250 90 765 740 797 248 0.56 250 1.01 250 1.01 250 1.01 250 1.01

350 346 0.66 350 1.01 339 0.98 339 0.98 339 0.98550 530 0.83 515 0.97 472 0.89 452 0.85 472 0.89

105 2000 250 90 780 685 754 248 0.58 250 1.01 250 1.01 250 1.01 250 1.01350 346 0.68 350 1.01 335 0.97 335 0.97 335 0.97550 529 0.85 502 0.95 464 0.88 447 0.84 464 0.88

110 2000 250 95 785 633 728 248 0.59 250 1.01 249 1.00 249 1.00 249 1.00350 344 0.69 350 1.02 332 0.97 332 0.97 332 0.97550 507 0.87 489 0.96 458 0.90 432 0.85 458 0.90

115 2000 250 95 795 586 679 247 0.61 250 1.01 248 1.00 248 1.00 248 1.00350 340 0.72 350 1.03 327 0.96 327 0.96 327 0.96550 491 0.90 477 0.97 447 0.91 415 0.85 447 0.91

120 2000 250 100 800 543 693 247 0.60 250 1.01 248 1.00 248 1.00 248 1.00350 337 0.71 343 1.02 329 0.98 328 0.97 329 0.98550 479 0.89 466 0.97 450 0.94 407 0.85 450 0.94

ST3 bs L fy Lcrl Lcrd fcrl fcrd fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fubw=100, bf=100, t=2.0 mm 22 1600 250 100 850 319 338 235 0.86 230 0.98 210 0.89 204 0.87 210 0.89

350 290 1.02 289 1.00 259 0.89 236 0.81 259 0.89550 355 1.28 389 1.10 334 0.94 280 0.79 334 0.94

24 1600 250 100 910 319 338 232 0.86 230 0.99 210 0.91 204 0.88 210 0.91350 288 1.02 289 1.00 259 0.90 236 0.82 259 0.90550 365 1.28 389 1.07 334 0.92 280 0.77 334 0.92

26 1600 250 100 965 319 378 242 0.81 230 0.95 218 0.90 209 0.86 218 0.90350 320 0.96 288 0.90 271 0.85 243 0.76 271 0.85550 369 1.21 389 1.05 352 0.95 289 0.78 352 0.95

28 1600 250 100 1015 318 377 237 0.81 230 0.97 218 0.92 209 0.88 218 0.92350 304 0.96 288 0.95 270 0.89 242 0.80 270 0.89550 371 1.21 389 1.05 351 0.95 289 0.78 351 0.95

30 1600 250 100 1065 317 375 237 0.82 229 0.97 217 0.92 209 0.88 217 0.92350 303 0.97 288 0.95 270 0.89 242 0.80 270 0.89550 373 1.21 388 1.04 350 0.94 288 0.77 350 0.94

ST4 bs L fy Lcrl Lcrd fcrl fcrd fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fubw=100, bF=100, t=2.4 mm 25 2100 250 100 845 456 456 244 0.74 250 1.02 230 0.94 230 0.94 230 0.94

350 322 0.88 324 1.01 290 0.90 276 0.86 290 0.90550 417 1.10 439 1.05 382 0.92 334 0.80 382 0.92

22 2100 250 100 770 456 412 241 0.78 250 1.04 224 0.93 224 0.93 224 0.93350 316 0.92 324 1.03 280 0.89 266 0.84 280 0.89550 388 1.15 439 1.13 365 0.94 321 0.83 365 0.94

30 2100 250 100 960 453 501 247 0.71 250 1.01 235 0.95 235 0.95 235 0.95350 335 0.84 324 0.97 299 0.89 284 0.85 299 0.89550 466 1.05 438 0.94 397 0.85 346 0.74 397 0.85

ST5 bs L fy Lcrl Lcrd fcrl fcrd fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fubw=110, bf=100, t=1.9 mm 17.5 1760 250 105 755 262 262 211 0.98 216 1.02 191 0.91 174 0.82 191 0.91

350 242 1.16 270 1.12 232 0.96 200 0.83 232 0.96550 279 1.45 363 1.30 296 1.06 237 0.85 296 1.06

20 1760 250 100 825 263 284 224 0.94 216 0.96 197 0.88 180 0.80 197 0.88350 270 1.11 270 1.00 241 0.89 207 0.77 241 0.89550 300 1.39 364 1.21 308 1.03 246 0.82 308 1.03

16 1760 250 105 705 261 239 202 1.02 216 1.07 184 0.91 168 0.83 184 0.91350 232 1.21 270 1.16 223 0.96 192 0.83 223 0.96550 270 1.52 363 1.34 283 1.05 227 0.84 247 0.91

ST6 bs L fy Lcrl Lcrd fcrl fcrd fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fubw=110, bf=85, t=2.4 mm 15 940 250 95 540 457 458 238 0.74 250 1.05 230 0.97 230 0.97 230 0.97

350 302 0.87 325 1.08 290 0.96 276 0.91 290 0.96550 387 1.10 440 1.14 382 0.99 335 0.87 382 0.99

13 940 250 95 490 455 380 232 0.81 250 1.08 218 0.94 218 0.94 218 0.94350 287 0.96 324 1.13 271 0.94 258 0.90 271 0.94550 379 1.20 439 1.16 352 0.93 310 0.82 352 0.93

17 940 250 95 590 459 532 248 0.69 250 1.01 239 0.96 239 0.96 239 0.96350 342 0.81 325 0.95 305 0.89 290 0.85 305 0.89550 464 1.02 440 0.95 407 0.88 355 0.77 407 0.88

Table B.1 (continued)

(continued on next page)

27N. Silvestre et al. / Journal of Constructional Steel Research 73 (2012) 12–30

(continued)

FSA FEA DSM

ST1 bf L fy Lcrl Lcrd fcrl fcrd fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fu

bw=100, bs=10, t=2.4 mm 50 620 250 80 285 598 598 245 0.65 250 1.02 244 1.00 244 1.00 244 1.00350 336 0.76 350 1.04 316 0.94 316 0.94 316 0.94550 469 0.96 481 1.03 426 0.91 392 0.84 426 0.91

45 620 250 80 265 604 640 246 0.63 250 1.02 246 1.00 246 1.00 246 1.00350 338 0.74 350 1.04 322 0.95 322 0.95 322 0.95550 479 0.93 482 1.01 437 0.91 402 0.84 437 0.91

55 620 250 80 305 592 567 246 0.66 250 1.02 242 0.98 242 0.98 242 0.98350 334 0.79 350 1.05 311 0.93 311 0.93 311 0.93550 457 0.99 479 1.05 417 0.91 383 0.84 417 0.91

ST8 bw L fy Lcrl Lcrd fcrl fcrd fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fubf=80, bs=10, t=1.5 mm 115 1150 250 95 500 170 170 165 1.21 187 1.13 159 0.96 135 0.82 159 0.96

350 188 1.43 233 1.24 190 1.01 153 0.81 190 1.01550 192 1.80 312 1.63 239 1.24 179 0.93 222 1.16

120 1150 250 100 505 156 165 162 1.23 181 1.12 157 0.97 131 0.81 157 0.97350 182 1.45 226 1.24 188 1.03 149 0.82 188 1.03550 198 1.82 302 1.53 235 1.19 174 0.88 221 1.12

106 1150 250 90 495 193 177 171 1.19 195 1.13 162 0.95 140 0.82 162 0.95350 197 1.41 243 1.23 194 0.98 159 0.81 194 0.98550 204 1.76 326 1.60 243 1.19 187 0.92 224 1.10

Av. 1.06 0.95 0.87 0.94S. dv. 0.13 0.07 0.07 0.06

Table B.2“Exact” and DSM ultimate strength values for the slender fixed columns.

FSA FEA DSM

SL1 bf L fy Lcrl Lcrd fcrl fcrd fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fu

bw=100, bs=5, t=1.0 mm 45 1400 250 80 250 105 116 135 1.47 158 1.17 133 0.99 104 0.77 133 0.99350 156 1.74 196 1.26 157 1.01 117 0.75 157 1.01550 186 2.18 262 1.41 195 1.05 136 0.73 195 1.05

47.5 1400 250 80 260 105 111 132 1.50 158 1.20 130 0.98 102 0.77 130 0.98350 156 1.78 196 1.26 154 0.99 115 0.74 154 0.99550 187 2.23 261 1.40 190 1.02 133 0.71 190 1.02

50 1400 250 80 270 104 105 113 1.54 158 1.40 127 1.12 100 0.88 127 1.12350 150 1.83 196 1.31 150 1.00 112 0.75 150 1.00550 179 2.29 261 1.46 185 1.03 130 0.73 185 1.03

52.5 1400 250 80 285 103 100 116 1.58 157 1.35 124 0.97 97 0.84 124 1.07350 155 1.87 195 1.26 146 0.94 109 0.70 146 0.94550 187 2.35 260 1.39 180 0.96 126 0.67 180 0.96

55 1400 250 85 290 103 95 114 1.62 157 1.38 120 1.05 95 0.83 120 1.05350 153 1.92 195 1.27 142 0.93 106 0.69 142 0.93550 180 2.41 260 1.44 175 0.97 123 0.68 175 0.97

SL2 bw L fy Lcrl Lcrd fcrl fcrd fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fubf=80, bs=10, t=1.3 mm 100 2400 250 90 530 155 135 145 1.36 181 1.25 143 0.99 120 0.83 143 0.99

350 157 1.61 225 1.43 170 1.08 135 0.86 154 0.98550 176 2.02 301 1.71 211 1.20 157 0.89 188 1.07

110 2400 250 95 540 137 129 137 1.39 173 1.26 140 1.02 115 0.84 140 1.02350 149 1.65 215 1.44 166 1.11 129 0.87 152 1.02550 163 2.06 288 1.77 206 1.26 150 0.92 185 1.13

115 2400 250 95 540 126 126 133 1.41 169 1.27 138 1.04 112 0.84 138 1.04350 145 1.67 210 1.45 164 1.13 126 0.87 149 1.03550 165 2.09 280 1.70 204 1.24 147 0.89 182 1.10

120 2400 250 100 545 117 119 130 1.45 164 1.26 134 1.03 107 0.82 134 1.03350 143 1.71 204 1.43 159 1.11 121 0.85 147 1.03550 160 2.15 272 1.70 198 1.24 141 0.88 180 1.13

125 2400 250 105 550 109 120 125 1.44 160 1.28 135 1.08 106 0.85 135 1.08350 143 1.71 199 1.39 160 1.12 120 0.84 149 1.04550 163 2.14 265 1.63 199 1.22 139 0.85 183 1.12

130 2400 250 105 550 101 117 125 1.46 156 1.25 133 1.06 104 0.83 133 1.06350 137 1.73 194 1.42 158 1.15 117 0.85 147 1.07550 155 2.17 258 1.66 196 1.26 136 0.88 180 1.16

SL3 bs L fy Lcrl Lcrd fcrl fcrd fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fubw=95, bf=80, t=0.95 mm 9 2500 250 85 570 93 88 98 1.69 151 1.54 116 1.18 89 0.91 99 1.01

350 105 1.99 188 1.79 136 1.30 100 0.95 113 1.08550 128 2.50 251 1.96 168 1.31 116 0.91 135 1.05

9.5 2500 250 85 595 93 93 103 1.64 152 1.48 119 1.16 92 0.89 100 0.97350 107 1.94 188 1.76 140 1.31 103 0.96 114 1.07550 131 2.43 251 1.92 173 1.32 119 0.91 135 1.03

10 2500 250 85 610 94 94 105 1.63 152 1.45 120 1.14 93 0.89 99 0.94

Table B.1 (continued)

ST7

28 N. Silvestre et al. / Journal of Constructional Steel Research 73 (2012) 12–30

(continued)

FSA FEA DSM

SL1 bf L fy Lcrl Lcrd fcrl fcrd fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fu

350 107 1.93 188 1.76 141 1.32 104 0.97 112 1.05550 128 2.42 251 1.96 175 1.37 120 0.94 133 1.04

10.5 2500 250 85 635 94 104 107 1.55 152 1.42 126 1.18 97 0.91 101 0.94350 115 1.83 188 1.63 149 1.30 109 0.95 115 1.00550 136 2.30 251 1.85 184 1.35 126 0.93 136 1.00

11 2500 250 85 655 94 110 112 1.51 152 1.36 129 1.15 99 0.88 102 0.91350 126 1.78 188 1.49 153 1.21 112 0.89 115 0.91550 141 2.24 251 1.78 189 1.34 130 0.92 135 0.96

SL4 bw L fy Lcrl Lcrd fcrl fcrd fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fubF=80, bs=10, t=1.0 mm 95 2500 250 85 595 103 103 111 1.56 157 1.41 125 1.13 98 0.88 106 0.95

350 121 1.85 195 1.61 148 1.22 110 0.91 121 1.00550 136 2.31 259 1.90 183 1.35 127 0.93 145 1.07

88 2500 250 80 585 113 107 115 1.53 162 1.41 127 1.10 102 0.89 107 0.93350 124 1.81 202 1.63 150 1.21 114 0.92 121 0.98550 135 2.27 269 1.99 186 1.38 132 0.98 143 1.06

100 2500 250 85 595 95 101 109 1.57 152 1.39 124 1.14 96 0.88 104 0.95350 118 1.86 189 1.60 146 1.24 108 0.92 119 1.01550 135 2.33 252 1.87 181 1.34 125 0.93 142 1.05

SL5 bf L fy Lcrl Lcrd fcrl fcrd fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fubW=150, bs=10, t=1.2 mm 140 1430 250 150 830 54 54 80 2.15 125 1.56 90 1.13 64 0.80 80 1.00

350 95 2.54 154 1.62 105 1.11 71 0.75 93 0.98550 121 3.18 205 1.69 128 1.06 82 0.68 112 0.93

130 1430 250 140 790 58 60 83 2.04 128 1.54 95 1.14 68 0.82 85 1.02350 100 2.41 158 1.58 111 1.11 76 0.76 99 0.99550 126 3.02 210 1.67 136 1.08 87 0.69 119 0.94

150 1430 250 155 865 50 50 79 2.25 121 1.53 86 1.09 60 0.76 77 0.97350 94 2.66 150 1.60 100 1.06 67 0.71 89 0.95550 123 3.33 199 1.62 122 0.99 77 0.63 107 0.87

SL6 bs L fy Lcrl Lcrd fcrl fcrd fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fubw=200, bf=190, t=1.5 mm 12.5 1850 250 200 1105 46 46 74 2.33 117 1.58 82 1.11 57 0.77 73 0.99

350 89 2.76 145 1.63 96 1.08 64 0.72 85 0.96550 112 3.46 193 1.72 117 1.04 73 0.65 102 0.91

14 1850 250 195 1190 47 53 76 2.17 118 1.55 89 1.17 62 0.82 76 1.00350 90 2.57 146 1.62 104 1.16 69 0.77 88 0.98550 114 3.22 193 1.69 127 1.11 79 0.69 105 0.92

11 1850 250 200 1005 46 38 74 2.56 117 1.58 74 1.00 52 0.70 69 0.93350 88 3.03 145 1.65 86 0.98 58 0.66 80 0.91550 112 3.80 192 1.71 105 0.94 66 0.59 97 0.87

SL7 bs L fy Lcrl Lcrd fcrl fcrd fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fubw=235; bf=160, t=1.9 mm 12.5 1950 250 200 905 65 65 94 1.97 133 1.41 99 1.05 72 0.77 95 1.01

350 112 2.33 164 1.46 115 1.03 80 0.71 111 0.99550 144 2.92 218 1.51 141 0.98 92 0.64 136 0.94

14 1950 250 200 990 65 73 98 1.86 133 1.36 106 1.08 77 0.79 100 1.02350 118 2.20 165 1.40 124 1.05 86 0.73 117 0.99550 149 2.75 219 1.47 153 1.03 99 0.66 143 0.96

11 1950 250 200 835 64 56 92 2.12 133 1.45 91 0.99 67 0.73 90 0.98350 111 2.51 164 1.48 107 0.96 75 0.68 105 0.95550 141 3.14 218 1.55 130 0.92 86 0.61 129 0.91

SL8 bf L fy Lcrl Lcrd fcrl fcrd fu λd fnl fnl/fu fnd fnd/fu fndl fndl/fu fndl* fndl* /fubw=180, bS=10, t=1.9 mm 120 1030 250 155 620 111 111 124 1.50 161 1.30 130 1.05 103 0.83 130 1.05

350 148 1.78 200 1.35 154 1.04 116 0.78 153 1.03550 189 2.23 267 1.41 190 1.01 134 0.71 190 1.01

110 1030 250 150 590 114 121 133 1.44 162 1.22 135 1.02 107 0.80 135 1.02350 153 1.70 202 1.32 160 1.05 121 0.79 160 1.05550 200 2.14 269 1.35 199 1.00 141 0.71 199 1.00

130 1030 250 160 660 109 104 122 1.55 160 1.31 126 1.03 100 0.82 125 1.02350 144 1.83 199 1.38 149 1.03 112 0.78 148 1.03550 183 2.30 265 1.45 184 1.01 130 0.71 183 1.00

Av. 1.51 1.11 0.81 1.00S. dv. 0.19 0.12 0.10 0.06

Table B.2 (continued)

SL3

29N. Silvestre et al. / Journal of Constructional Steel Research 73 (2012) 12–30

30 N. Silvestre et al. / Journal of Constructional Steel Research 73 (2012) 12–30

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