Strength, interactive failure and design of web-stiffened lipped channelcolumns exhibiting distortional buckling

Pedro B. Dinis a, Ben Young b, Dinar Camotim a,n

a Department of Civil Engineering, ICIST, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugalb Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China

a r t i c l e i n f o

Available online 15 October 2013

Keywords:Cold-formed steel web-stiffenedlipped channel columnsDistortional bucklingLocal–distortional interactionExperimental resultsNumerical simulationsDSM (Direct Strength Method)ultimate strength prediction

a b s t r a c t

The paper reports experimental and numerical results concerning the post-buckling behaviour, strengthand failure of fixed-ended cold-formed steel web-stiffened lipped channel columns that buckle indistortional modes. The experimental results, obtained from the tests carried out at The University ofHong Kong, (i) include initial imperfection measurements, equilibrium paths and failure loads andmodes, and (ii) provide evidence of the occurrence of flange-triggered local-distortional interaction. Afterpresenting and discussing a comparison between some test results and the values yielded by thecorresponding ABAQUS shell finite element numerical simulations, the experimental failure loads obtainedin this work, together with additional data reported in the literature, are used to assess whether theavailable Direct Strength Method (DSM) design approaches are capable of predicting them efficiently(safely and accurately). It is found that this is not the case, mainly because of the fact that the mechanicsof the flange-triggered and web-triggered local-distortional interactions are quite different. Althoughfairly good failure load predictions are provided by a new DSM design approach proposed in this work,further research is required on the mechanics of local–distortional interaction in web-stiffened lippedchannel columns.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Most thin-walled cold-formed steel columns are known tobe highly susceptible to local, distortional or global (flexural orflexural–torsional) buckling (see Fig. 1(b)–(e)) � depending onthe geometry (cross-section dimensions and length) and supportconditions, any of these instability phenomena may be the criticalone. Moreover, commonly used column geometries often lead tosimilar local, distortional and/or global buckling stresses, whichmeans that their post-buckling behaviour, ultimate strength andfailure mechanism are affected by interaction effects involving twoor more of the above three buckling mode types.

It is well known for a long time that thin-walled membersexhibit stable local and global post-buckling behaviours with highand low/marginal post-critical strength reserves, respectively.On the other hand, relatively recent studies (e.g., [1,2]) showedthat the distortional post-buckling behaviour fits in between theprevious two (intermediate post-critical strength reserve) and,moreover, exhibits a non-negligible asymmetry with respect to thesense of the flange-lip motions (outwards or inwards). Concerningmode interaction phenomena affecting the column post-buckling

behaviour, those involving local and global buckling are, by far, thebetter understood, as attested by their inclusion in virtually all textbooks and current design specifications for hot-rolled and cold-formed steel structures – these mode coupling effects are takeninto account either through the classical “plate effective width”concept or by means of the much more recent Direct StrengthMethod (DSM – e.g., [3]). Nevertheless, a considerable amount ofresearch work has been devoted to investigate local–distortional(L–D) interaction in cold-formed steel thin-walled columns – thiswork involves mostly lipped channel columns and comprisesexperimental investigations, numerical simulations and designproposals (e.g., [4–13]).

It was found that the local–distortional interaction effectsare relevant when the ratio between the column distortionaland local buckling loads is either (i) in the close vicinity of 1.0(at least comprised between 0.9 and 1.1), which corresponds tothe occurrence of a “true L–D interaction” (the coupling effectsgradually evolve as loading progresses and take place regardlessof the yield stress value), or (ii) above 1.0 (possibly by a largemargin), provided that the squash load exceeds the distortionalbuckling load by a “large enough” amount to allow for thedevelopment of significant L–D interaction effects prior to failure,which corresponds to “L–D interaction due to a secondary(distortional) bifurcation”. While the first finding stemmedfrom research work involving plain (no intermediate stiffeners)

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Thin-Walled Structures

0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.tws.2013.09.016

n Corresponding author. Tel.: þ351 21 8418403; fax: þ351 21 8497650.E-mail address: dcamotim@civil.ist.utl.pt (D. Camotim).

Thin-Walled Structures 81 (2014) 195–209

lipped channel, hat-section, zed-section and rack-section columns[10,14], the second one was based on investigations concerningonly plain lipped channel and rack-section columns [6,12,15].Concerning columns for which the critical distortional bucklingload precedes its local counterpart, the information is muchmore scarce – however, very recent numerical investigationsinvolving exclusively plain lipped channel columns [16] seem toindicate that the influence of the L–D interaction effects is lessrelevant than in the two situations addressed before, particularlywhen the difference between the two buckling loads is sizeable(and regardless of the yield stress value). At this stage, it is worthnoting that, in all the columns mentioned above, the L–D interac-tion effects are triggered by the web, which exhibits considerabledeformation due to both the local and distortional critical bucklingmodes – this is not the case for the web-stiffened lipped channelcolumns dealt with in this work, as the L–D interaction effects aretriggered by the flanges (the local critical buckling mode causesvirtually no deformation in the web – see Fig. 1(b)).

Although the amount of research work devoted to web-stiffenedlipped channel columns is not comparable to that concerning plainlipped channel columns, there exist available experimental, numer-ical and design results [6,7,17,18], namely concerning columns buck-ling in distortional modes and/or experiencing L–D interaction effectscausing a perceptible ultimate strength erosion. The fact that severalof the tests performed by Yap and Hancock [7,18] (i) involvedcolumns that buckle in distortional critical modes (the distortionalcritical buckling load is visibly lower than its local counterpart) and(ii) provided clear experimental evidence of the occurrence of flange-triggered L–D interaction raised some suspicion on the ability ofthe current DSM distortional design curve to estimate adequately thecorresponding failure loads – moreover, it provided the motivationfor this research effort, which deals with the fixed-ended web-stiffened lipped channel columns buckling in distortional modes.

Therefore, the objective of this work is to report experimental,numerical and design results concerning the post-buckling andultimate strength behaviour of cold-formed steel fixed-endedweb-stiffened lipped channel columns selected to buckle indistortional modes, in the sense that the critical distortionalbuckling stress is below (but not excessively so) its local andglobal (flexural–torsional) counterparts. The experimental results,obtained from the tests carried out at The University of Hong

Kong, (i) include initial imperfection measurements, equilibriumpaths, failure loads and collapse modes, and (ii) provide experi-mental evidence of L–D interaction – before presenting anddiscussing the test results, the experimental set-up and procedureare briefly described. Next, the results concerning some ofthe tested specimens are compared with the values yielded bythe corresponding numerical simulations, performed by means ofthe ABAQUS [19] shell finite element analyses (SFEA) – an in-depthnumerical investigation addressing the column imperfection-sensitivity is also presented and discussed. Then, the obtainedexperimental failure loads, together with additional ultimatestrength data reported in the literature [6,7,17,18], are used toassess and discuss the ability of the current DSM distortionaldesign curve, as well as other DSM design approaches available inthe literature, to predict efficiently (safely and accurately) theultimate strength of web-stiffened lipped channel columns that(i) buckle in distortional critical modes and (ii) are susceptible toexhibit L–D interactive failures. It is found that this is not the caseand it is shown that the inability is due to the occurrence ofultimate strength erosion stemming from flange-triggered L–Dinteraction, which is mechanically quite different from its web-triggered counterpart. It is worth noting that the vast majority ofthe existing numerical/experimental investigations and designconsiderations/proposals concerning L–D interactive failuresinvolve exclusively columns with either (i) plain webs and flangesor (ii) stiffened webs and flanges, all of which exhibit web-triggered L–D interaction (e.g., [4,5,8–16]) – the exceptions arethe investigation reported by Kwon et al. [6] and Yap and Hancock[7]. Although a new DSM design approach proposed in this work,which combines two existing interactive strength curves, providesfairly good predictions of the experimental failure load available(including those reported herein), it is also concluded that furtherresearch is required in order to acquire in-depth knowledge on themechanics of flange-triggered local–distortional interaction inweb-stiffened lipped channel columns.

2. Buckling behaviour – column geometry selection

The column geometries (cross-section dimensions and length)were selected to ensure critical distortional (D) buckling loads that

d1 bl

d2bw

bf

Fig. 1. Web-stiffened lipped channel (a) geometry and cross-section deformed shapes associated with column, (b) local (flange-triggered), (c) distortional, (d) flexural–torsional and (e) flexural buckling.

10 100 1000L(cm)0

Pcr (kN)150

100

50 LD-E=215 cm Pcr=86.6 kN

LL-D=75.6 cm Pcr=126.7 kN

Local-Distortional

55 15 1.2

ν =0.3

85 14 7

(mm )

E=210 GPa

Distortional

Global

Fig. 2. (a) Column critical buckling curve Pcr vs. L, (b1) LL–D column local–distortional interactive buckling mode and (b2) LD–E column distortional and global buckling modes.

P.B. Dinis et al. / Thin-Walled Structures 81 (2014) 195–209196

are not too far below their local (L – flange-triggered) and global(G – flexural–torsional) counterparts. The selection was achievedthrough trial-and-error buckling analysis sequences, carried out inthe code GBTUL [20]. Fig. 2(a)–(b) illustrates this procedure for theweb-stiffened lipped channel with the cross-section dimensionsand elastic constants indicated: they show (i) the Pcr (critical load)vs. L (length in logarithmic scale) curve and (ii) the criticalbuckling modes of the columns with lengths LL–D (Fig. 2(b1)) andLD–E (Fig. 2(b2)), which corresponds, respectively, to (i) practicallyidentical local and distortional buckling loads PbL¼PbD¼126.7 kN(i.e., fbL¼ fbD¼457.3 MPa), associated with 14 and 1 half-wavebuckling modes, and (ii) very close distortional and global bucklingloads PbD¼86.6 kN (fbD¼312.6 MPa) and PbE¼88.1 kN (fbE¼317.9 MPa), associated with 3 and 1 half-wave buckling modes.In order to lower PbD, thus ensuring a distortional critical bucklingbehaviour, it suffices to (i) select a length such that LL–D oLoLD–Eand/or (ii) increase the web width bw. For instance, a columnexhibiting bw¼100 mm and L¼200 cm has buckling loadsPbL¼130.9 kN, PbD¼85.3 kN and PbE¼141.3 kN.

3. Experimental investigation

This section reports the experimental investigation carriedout at The University of Hong Kong, which involved a total often fixed-ended cold-formed steel web-stiffened lipped channelcolumns with eight different geometries – to check the repeat-ability, two pairs of almost identical specimens were tested. Afterbriefly describing the test specimens, set-up and procedure, whichincludes providing the specimen geometries, material properties(stress–strain curves) and displacement measurements, someexperimental results are presented – they consist of (i) initialdisplacements, (ii) non-linear equilibrium paths, including therespective collapse loads, and (iii) failure modes.

3.1. Test specimens, set-up and procedure

The cold-formed steel web-stiffened lipped channel columnspecimens had nominal lengths ranging from 2100 to 3000 mmand were brake-pressed from high strength zinc-coated gradesG500 and G550 structural steel sheets with nominal thickness andyield stress of (i) t¼1.2 mm and fy¼500 MPa (G500) and (ii)t¼1.0 mm and fy¼550 MPa (G550). The base metal thicknesstn was measured after removing the zinc coating by acid-etching– the coating thickness measured, on each section side, 25.5 μm(t¼1.0 mm) and 21.5 μm (t¼1.2 mm). The specimens had nominalweb widths bw ranging from 97 to 118 mm, flange widths bfranging from 52 to 67 mm, lip width bl of 17 mm, stiffener depthd1 and width d2 of 10 and 20 mm, and inside corner radiiof 2.0 mm – dimensions shown in Fig. 1(a). All the measured

specimen cross-section dimensions and lengths are given inTable 1: five specimens with nominal t¼1.0 mm, labelled IS-1–IS-4, and five specimens with nominal t¼1.2 mm, labelled IS-5–IS-8 � note that, except for the web width (19% higher) and stiffenerdimensions, specimen IS-5 has cross-section dimensions similar tothose shown in Fig. 2(a). Moreover, (i) all but one of the specimensbuckle in 3 half-wave distortional modes (IS-8 buckles in a 4 half-wave distortional mode) and (ii) the PcrL/PcrD and PcrE/PcrD ratiosare in the 1.21–1.47 and 1.27–1.41 ranges, respectively (globalstands for flexural–torsion buckling) � note that the distortionaland global critical buckling loads are not too far apart (for speci-mens IS-5 and IS-8, PcrE is even lower than PcrL).

Tensile coupon tests were conducted to determine the cold-formed steel material properties. Longitudinal coupons wereextracted from the central web and flange regions of specimensIS-1 and IS-5 – since the specimens were fabricated from only twobatches of steel, the material properties are assumed to be thesame for the specimens sharing a given nominal plate thickness.For the IS-1 specimens, the measured Young's modulus wasE0¼216 GPa and the static 0.2% proof stress varied betweens0.2¼606–625 MPa (G550 steel), whereas for the IS-5 specimensthe material properties obtained were E0¼214 GPa and s0.2¼591–599 MPa (G500 steel) – in both cases, the lower values correspondto web coupons. Fig. 3(a) shows the stress–strain curves obtainedfrom the tensile coupon tests concerning the IS-1 (G550 steel) andIS-5 (G500 Steel) specimens, whereas Fig. 3(b) presents the firstpart provided by the IS-5 specimen coupon test.

A servo-controlled hydraulic testing machine was used to applythe compressive axial force to the column specimens, whosetop end plate were bolted to the rigid flat bearing plate, fullyrestrained against warping and minor-axis, major-axis and twistrotations – hence, the column top end was indeed completelyfixed. Concerning the column bottom end plate, it rested initiallyon a special bearing that was free to rotate in any direction, thusenabling an adequate positioning of the specimen, in order topreclude (or, at least, minimise) the presence of eccentricities. Theram of the actuator was first moved slowly until the columnbottom end plate was in full contact with the special bearing, for asmall load of approximately 1 kN – this procedure eliminated anypossible gaps between the column bottom end plate and thetest machine special bearing. After full contact was achieved,the bottom end plate was bolted to the special bearing, whichwas subsequently fully restrained against all rotations by means ofvertical and horizontal bolts – hence, the column bottom endbecomes completely fixed prior to testing (Fig. 4(a)-(b) providesdetailed views of the specimen bottom and top end supports).

Displacement control was used to drive the hydraulic actuatorat a constant speed of 0.2 mm/min in all cases, thus allowingthe tests to be carried out into the post-ultimate range, and3 displacement transducers were used to measure the column

Table 1Column specimen (i) geometries, (ii) squash and critical (local, distortional, global) buckling loads, (iii) initial geometrical imperfection amplitudes, (iv) test failure loads and(v) observed failure mode natures.

Specimen bw bf bl d1 d2 L tn Py PcrL PcrD PcrE PcrL

PcrD

PcrE

PcrD

Py

PcrL

Δ0 δ0/L PExp Obs. failuremode(mm) (mm) (mm) (mm) (mm) (mm) (mm) (kN) (kN) (kN) (kN) (mm) (kN)

IS-1 96.5 52.2 16.6 10.8 19.9 2097 1.005 134.6 92.1 70.1 97.8 1.31 1.39 1.46 1.9 �1/11008 63.26 LþDIS-2-1 106.1 57.2 16.6 10.5 20.0 2499 1.007 145.5 84.2 63.9 87.2 1.32 1.37 1.73 1.3 1/1468 59.80 LþDIS-2-2 107.3 57.2 16.5 10.0 19.8 2500 0.989 143.1 80.0 60.9 86.0 1.31 1.41 1.79 �1.8 1/3281 58.99 LþDIS-3 105.4 62.2 16.8 9.5 19.4 2702 1.001 149.3 73.6 60.7 77.6 1.21 1.28 2.03 0.2 �1/11820 57.51 LþDIS-4 115.0 62.4 16.9 9.6 19.7 3001 1.002 155.1 75.2 58.7 74.8 1.28 1.27 2.06 0.6 �1/11815 55.93 LþD

IS-5 101.4 57.2 16.6 9.7 20.0 2147 1.196 153.6 134.8 91.9 126.7 1.47 1.38 1.14 0.8 �1/2285 83.16 LþDIS-6-1 105.3 62.0 16.6 9.9 19.6 2402 1.205 163.1 125.1 87.9 116.2 1.42 1.32 1.30 1.3 1/9457 83.37 LþDIS-6-2 105.9 62.2 16.6 10.0 19.9 2402 1.198 162.8 122.8 86.6 116.3 1.42 1.39 1.33 1.0 �1/4728 80.86 LþDIS-7 111.7 67.4 16.7 10.1 19.9 2701 1.193 172.0 110.4 80.9 106.9 1.37 1.32 1.56 �3.8 1/226 75.51 LþDIS-8 117.6 67.3 16.5 10.3 19.4 2820 1.225 180.1 121.2 83.1 111.2 1.46 1.34 1.49 1.2 �1/17081 77.88 LþD

P.B. Dinis et al. / Thin-Walled Structures 81 (2014) 195–209 197

axial shortening. Furthermore, 7 transducers were used to mea-sure the deformation of the column mid-height cross-section,as schematically indicated in Fig. 4(c): (i) three located in theweb, one 10 mm away from the web-stiffener edge and the othertwo 10 mm away from the web-flange corners (due to spacelimitations, the former was effectively placed at a cross-section60 mm above the mid-height one), and (ii) two in each flange,10 mm away from the web-flange and flange-lip corners.The three web transducers make it possible to assess the columnlocal (web transverse bending) and global (minor-axis bending

and torsion) deformations. On the other hand, the flange transdu-cers provide information about the deformations caused by major-axis bending and cross-section distortion, respectively. A dataacquisition system was used to monitor and record the appliedload and the displacement transducer readings at regular timeintervals during the performance of the tests.

In order to assess the column specimen initial geometricalimperfections, two displacements were measured and recordedat the column mid-height cross-section prior to testing, namelyΔ0 and δ0 (see Fig. 4(d)) � all measured Δ0 and δ0 displacements

600

400

200

000

800

600

400

200

0

� (MPa)

� (MPa)G550

G500

2 4 6 8 � (%) � (%)0.2 0.4 0.6

Fig. 3. Test specimen coupon stress–strain curve: (a) complete curve (G550 and G500 steel coupons) and (b) initial part (G500 steel coupon).

Special Bearing

Top End Plate

Transducer

δ0

0

0

Fig. 4. (a) Fixed-ended column test set-up, (b) detailed views of the specimen bottom and top end supports (the former includes the special bearing), (c) displacementtransducer arrangement (transducer locations at the mid-height cross-section) and (d) test specimen mid-height initial displacement measurements.

P.B. Dinis et al. / Thin-Walled Structures 81 (2014) 195–209198

are given in Table 1, with the latter normalised w.r.t. the columnlength L. The Δ0 values provide information about the initialdistortional deformation and correspond to half the differenceamid the distances, measured parallel to the web, betweenthe flange-lip and web-flange corners (a positive Δ0 indicatesoutward flange-lip motions) – the maximum Δ0 measured was3.8 mm (specimen IS-7). On the other hand, the δ0 values maystem from various combinations of initial (i) minor-axis flexureand (ii) torsional rotation (recall the shear centre location) – notethat, due to its intermediate stiffener, the web barely exhibits anytransverse bending due to local or distortional deformations.In order to assess δ0 as accurately as possible, it was measuredin relation to a straight line uniting the corresponding points ofcross-sections located near the column end supports, used asreference – a theodolite was employed to relate the positions ofthe three points and the maximum δ0/L measured was, by far,1/226 (specimen IS-7). It is worth noting that positive δ0/L valuesare associated with minor-axis flexural curvatures towards thelips. Finally, it should be mentioned that no initial displacementprofiles were recorded, thus precluding any knowledge about theinitial imperfection shape.

3.2. Test results

The column experimental failure loads (PExp) are given in Table 1,which also provides the nature of the associated failure modes –

note that two tests were repeated (specimens IS-2-2 and IS-6-2 areidentical to the IS-2-1 and IS-6-1 ones). Fig. 5(a)–(d) shows thedeformed configurations at the onset of collapse for specimens IS-2-2, IS-4, IS-7 and IS-8 – Fig. 6(a) shows a detail of the specimen IS-4top flange deformed configuration. On the other hand, Fig. 6(b) shows the final (after unloading) deformed configurations ofspecimens IS-1, IS-2-1, IS-2-2, IS-3 and IS-4. Lastly, Fig. 7(a),(b) depicts the equilibrium paths recorded during the IS-4 andIS-8 specimen tests: (i) load vs. axial shortening (Fig. 7(a)), and(ii) the load vs. displacements measured by transducers 1 and 7

(Fig. 7(b) � see Fig. 4(c)). The observation of the experimentalresults presented prompts the following remarks:

(i) Regardless of the initial Δ0 value “sign” (inward or outwardmid-height flange-lip initial imperfection) and critical(distortional) buckling mode half-wave number (specimenIS-8 buckles in a 4 half-wave mode), all specimens failed in3 half-wave distortional modes with mid-height inwardflange-lip motions (see Fig. 5(a)–(d)). Moreover, localdeformations were also observed in the flanges of allspecimens near the collapse (see Fig. 6(a)), thus providingevidence of the occurrence of flange-triggered L–D inter-action1. However, these local deformations are not verypronounced and, most of all, are mainly elastic � theycease to be visible after the load is removed, as illustratedin Fig. 6(b).

(ii) The flange-triggered L–D interaction is not due to the nearcoincidence of the column local and distortional criticalbuckling loads – Table 1 shows that the PcrL/PcrD values ofall tests are in the 1.21–1.47 range. Instead, the L–Dinteraction stems from a secondary (local) bifurcation anddevelops because Py is “larger enough” than PcrL (the Py/PcrLvalues are in the 1.14–2.06 range) – the effects onlyemerge and grow as the loading nears the PcrL applied loadlevel [16].

(iii) Although the distortional and global critical buckling loadsare not too far apart (the PcrE/PcrD values are in the 1.27–1.41range), no specimen exhibited any trace of flexural–torsionaldeformations up until the onset of collapse. However, minor-axis flexure was observed in all tests at the advanced post-buckling stages � most likely, it stems from cross-section“effective centroid shifts” (along the major axis and towardsthe web) due to the post-buckling stress redistributionassociated with distortional (and also local) deformations(note the minor-axis flexural buckling load is considerablyhigher – 2.53–3.18 times higher than PcrD).

(iv) The failure loads of the two pairs of almost identical testswere fairly close: differences of 1.4% (specimen IS-2) and 3.0%(specimen IS-6), which demonstrate the test repeatability.

4. Numerical simulations

This section deals with the numerical simulation of some of theobtained experimental results by means of ABAQUS SFEA that

Fig. 5. Observed failure mode of specimens (a) IS-2-2, (b) IS-4, (c) IS-7 and (d) IS-8.

Fig. 6. (a) Evidence of flange local deformations near the collapse of specimen IS-4and (b) deformed configurations of specimens IS-1, IS-2-1, IS-2-2, IS-3 and IS-4after the removal of the applied load.

1 As discussed ahead in the paper, minor-axis flexural displacements were alsoobserved during the tests.

P.B. Dinis et al. / Thin-Walled Structures 81 (2014) 195–209 199

(i) adopt column discretisations into fine 4-node isoparametricelement meshes (length-to-width ratio close to 1), and (ii) modelthe column supports by attaching rigid plates to their end sectioncentroids. The numerical results presented, discussed and com-pared with the corresponding test results concern the non-linearbehaviour, ultimate strength and collapse of specimens IS-4 andIS-5. It is important to begin by stating the assumptions adopted toperform the numerical simulations reported herein.

(i) The steel elastic–plastic material behaviour is isotropic anddescribed by a multi-linear model that (i1) assumes E¼214 GPa,ν¼0.3 and fy¼591 or 606 MPa (web-flange minimum couponvalues), and (i2) approximates the experimental stress–straincurve, prior to the yield plateau, by linear segments connectingthe points concerning four stresses (f¼0.75 fy, 0.90 fy, 0.98 fy andfy) – Fig. 8 shows a comparison between the experimentallymeasured stress–strain curve and the multi-linear approxima-tion adopted in the numerical simulations.

(ii) The column end sections are fully fixed: with the soleexception of the axial translation of the loaded end section,which is free, all displacements and rotations are completelyprevented.

(iii) Both the residual stresses (not measured in the tested speci-mens) and corner effects are neglected � in cold-formed steelmembers, such effects have been shown to have a smallimpact on the column failure load [21].

(iv) The initial geometrical imperfections consist of combinations ofdistortional, local and global (flexural–torsional or minor-axisflexural) buckling mode shapes with prescribed amplitudes.Since (iv1) no initial displacement profiles were recorded priorto the tests, and (iv2) Δ0 and δ0 were measured only at two andone column mid-height cross-section points (see Fig. 4(d)), afairly extensive numerical study, aimed at assessing theimperfection-sensitivity of the column post-buckling behaviourand failure load/mode, is carried out in this work.

4.1. Imperfection-sensitivity investigation

The imperfection-sensitivity investigation consists of compar-ing the post-buckling behaviours, failure loads and collapsemechanisms of columns with the geometry of specimen IS-5 andcontaining 19 different initial geometrical imperfections, whichexhibits the characteristics provided in Table 2 and correspond to:

(i) One, two and three half-wave distortional buckling modeshapes, denoted D1, D2 and D3 (the last one is the columncritical buckling mode), with amplitude Δ0¼0.80 mm (experi-mentally measured value – recall that a positive Δ0 indicatesoutward flange-lip motions, as shown in Fig. 4(d)). In order toaccount for the distortional post-buckling asymmetry, the D1and D3 shapes are also considered with negative Δ0 values(i.e., mid-span inward flange-lip motions).

(ii) A 38 half-wave local buckling mode shape, denoted here asL, with amplitude equal to Δ0.

(iii) Combinations of the D1, D2, D3 and L shapes with flexural–torsional (FT) or minor-axis flexural (F) buckling mode shapes,with amplitudes equal to the experimentally measured δ0value (�0.94 mm � recall that a negative δ0 value corre-sponds to minor-axis flexural curvatures towards the web).

Fig. 9(a) shows the experimental and numerical equilibriumpaths P vs. d1, where d1 is the displacement measuredby transducer 1 (see Fig. 4(c) – inward displacements are positive).The numerical equilibrium paths concern columns with the19 initial imperfections defined in Table 2, which are labelledN1–N19. Moreover, Table 2 also displays the (i) ultimate strengthratios PNum/PExp (recall that PExp¼83.16 kN) and (ii) failure modenatures of all the columns analysed – note that D3i and D3oidentify three half-wave distortional failure modes with mid-height inward and outward flange-lip motions, respectively.Finally, Fig. 9(b1)–(b6) shows the experimentally observed failure

0

10

20

30

40

50

60

0,00

Displacement (mm)

Load

(kN

)

0

10

20

30

40

50

60

-5

Displacement (mm)

Load

(kN

)

T1

T7

d7

d1

0

10

20

30

40

50

60

70

80

90

0,00 6,00

Displacement (mm)

Load

(kN

)

0

10

20

30

40

50

60

70

80

90

-5

Displacement (mm)

Load

(kN

)T1

T7

d7

d1

0,50 1,00 1,50 2,00 2,50 3,00 3,50 4,00 4,50 5,00 0 5 1510 20 25 30

1,00 2,00 3,00 4,00 5,00 0 5 10 15 20 25 30

Fig. 7. Experimental equilibrium paths concerning specimens (1) IS-4 and (2) IS-8: (a) load vs. axial shortening (top transducer measurement average) and (b) load vs.transducers 1 and 7 displacements.

P.B. Dinis et al. / Thin-Walled Structures 81 (2014) 195–209200

mode (Fig. 9(b1)) and those obtained from the numerical simula-tions of columns N1, N2, N10, N15 and N18 (Fig. 9(b2)–(b6)).The observation of this set of column non-linear equilibriumpaths, numerical-to-test ultimate strength ratios and failuremodes lead to the following conclusions:

(i) The “sign” of the initial imperfection distortional component (i.e., whether the mid-height cross-section flange-lip assemblieshave initially moved inwards or outwards) has a very clearinfluence on the column ultimate strength value and collapsemechanism. Indeed, this asymmetry is quite pronounced andthe columns with outward initial mid-height flange-lip motionshave considerably more strength – for instance, the failure loadof the N1 column (outward D1 imperfection) is 15% higher thanits N4 column (inward D1 imperfection) counterpart � theirPNum/PExp values are 1.21 and 1.06, respectively. This strengthdifference, which was experimentally observed by Yap andHancock [7,18], is due to the much higher compressive stressesdeveloping in the web-flange junctions of the column withinward initial mid-height flange-lip motions. It is worth pointingout that, in plain lipped channel columns, the ultimate strengthbehaviour is reversed: the most detrimental distortional initialimperfection shape (in the sense that it leads to the lowestultimate strength) involves inward initial mid-height flange-lip

motions (e.g., [10]) – however, note that the distortional post-buckling strength asymmetry is considerable less pronounced inthe plain lipped channel columns than in their web-stiffenedcounterparts.

(ii) Regardless of the initial imperfection, all the columns analysednumerically fail in three half-wave distortional modes involvingoutward (D3o) or inward (D3i) mid-height flange-lip motions –they may also include flexural–torsional, minor-axis flexural orlocal deformations.

(iii) Only columns N1–N3 and N7–N9, which contain D1 and D3initial imperfections involving outward initial mid-heightflange-lip motions, collapse in D3o modes (modes involvingoutward mid-height flange-lip motions), which are charac-terised by the fact that the plastic strains concentrate aroundthe two “most inwardly deformed” cross-sections, as illustratedin Fig. 9(b2). In columns N2þN3 and N7þN8, the addition ofglobal (flexural–torsional or minor-axis flexural) initial imper-fections to the “pure” distortional ones naturally leads to afailure load drop. Such drop is more relevant when FT initialimperfections are added (columns N2 and N8) and, moreover,the collapse mode also involves FT deformations (it switchesfrom D3o to D3oþFT � see Fig. 9(b3)).

(iv) All the remaining 13 columns collapse in D3i modes (inwardmid-height flange-lip motions), which may have asymmetricconfigurations, as was also observed experimentally � comparethe deformed configurations shown in Fig. 5(a)–(d) and Fig. 9(b1) with those in Fig. 9(b4)–(b6), concerning the collapse ofcolumns N10, N15 and N18 (the last two have initial imperfec-tions with anti-symmetric components, namely 2 half-wavedistortional and 38 half-wave local components � obviously,the anti-symmetry level of the former is much higher).

(v) The columns whose failure modes involve inward mid-heightflange-lip motions (D3i) have ultimate strengths that are quiteclose to the experimental value (the PNum/PExp values arecomprised between 0.97 and 1.06). Obviously, the columns with“pure” distortional or local initial imperfections have higherfailure loads. The lowest amongst them concerns column N16,which has pure local initial imperfections (which are not akin tothe critical buckling mode) – the columns with pure distortionalinitial imperfections have PNum/PExp values equal to 1.03 (D3 –

column N10) and 1.06 (D1 and D2 – columns N4 and N13).(vi) The largest failure load drops (out of the 13 columns collap-

sing in D3i modes) occur when the “pure” initial imperfec-tions are combined minor-axis flexural ones (drops of about3%), thus confirming the relevant contribution of minor-axisflexural deformations to the collapse of the columns ana-lysed. It is worth noting that the initial imperfection combin-ing local and minor-axis flexural modes with amplitudesequal to Δ0 and δ0, respectively (column N18), leads to an“excessively flexible” equilibrium path, thus suggesting thatthe amplitude of the initial imperfection local componentshould be considerably lower than Δ0. Indeed, reducing thelocal component amplitude to 0.1 Δ0 brings the numericalequilibrium path much closer than its experimental counter-part and, moreover, PNum/PExp¼1.00 is obtained � thisequilibrium path corresponds to column N19.

(vii) Although this is not clearly shown in Fig. 9(b6), all thecolumns having local initial imperfections (columns N16–N19) collapse in modes that combine distortional and local(in the flange) deformations (LþD3i modes). This assertioncan be confirmed by looking at Fig. 10(a)–(d), which make itpossible to compare the (amplified) top flange deformedconfigurations of columns N10, N13, N16 and N19 at theonset of collapse. While the first two, which concern columnswithout an initial imperfection local component, exhibit notrace of local deformations at failure, the opposite occurs in

600

400

200

σ (MPa)

0 ε (%)0 0.2 0.4 0.6

Exp Num

fy0.98 fy0.90 fy0.75 fy

Fig. 8. Comparison between the experimentally measured stress–strain curve andthe multi-linear approximation adopted in the numerical simulations performed inthis work.

Table 2IS-5 column 19 initial imperfections, ultimate strength ratios PNum/PExp and failuremode natures.

Column Initial imperfection

Mode PNum

PExp

Failure mode

D1 D2 D3 L FT F

N1 Δ0 – – – – – 1.21 D3oN2 Δ0 – – – δ0 – 1.10 D3oþFTN3 Δ0 – – – – δ0 1.18 D3oN4 �Δ0 – – – – – 1.06 D3iN5 �Δ0 – – – δ0 – 1.06 D3iN6 �Δ0 – – – – δ0 1.03 D3iN7 – – Δ0 – – – 1.19 D3oN8 – – Δ0 – δ0 – 1.07 D3oþFTN9 – – Δ0 – – δ0 1.17 D3oN10 – – �Δ0 – – – 1.03 D3iN11 – – �Δ0 – δ0 – 1.03 D3iN12 – – �Δ0 – – δ0 1.01 D3iN13 – Δ0 – – – – 1.06 D3iN14 – Δ0 – – δ0 – 1.06 D3iN15 – Δ0 – – – δ0 1.04 D3iN16 – – – Δ0 – – 1.02 LþD3iN17 – – – Δ0 δ0 – 1.02 LþD3iN18 – – – Δ0 – δ0 0.97 LþD3iN19 – – – 0.1Δ0 – δ0 1.00 LþD3i

P.B. Dinis et al. / Thin-Walled Structures 81 (2014) 195–209 201

columns N16 and N19 – note, however, that the local defor-mations visible in these columns are basically elastic (theplastic deformations are essentially distortional), which con-firms the deformed configurations observed in the test speci-mens when the applied load was removed after the collapse,both in this work (see Fig. 6(b)) and in the experimentalstudy reported by Yap and Hancock [7].

4.2. Additional numerical simulations

The conclusions drawn from the above IS-5 column imperfection-sensitivity investigation, namely those concerning the relevance of

the local and minor-axis flexural components of the initial geome-trical imperfections, were found to be valid also for the remining IScolumns � in order to illustrate this statement, the numerical resultsconcerning specimen IS-4 are presented, discussed and comparedwith the corresponding test results. Fig. 11(a)-(b) shows the numer-ical and experimental equilibrium paths (i) P vs. d1 and P vs. d7 (d1and d7 are the measurements of transducers 1 and 7 � in both cases,the inward displacements are positive), and (ii) P vs. ε (ε is the axialshortening). These equilibrium paths concern columns containinginitial imperfections that combine (i) D1 and F mode shapes (N1column), with amplitudes equal to the experimentally measured Δ0(�0.60 mm) and δ0 (�0.25 mm) values, respectively, and (ii) L and Fmode shapes (N2 column), with amplitudes equal to Δ0 and δ0,

d 1 d1 d1d1

d1d1

d1(mm)

d1(mm)

d1(mm)

d1(mm)

N1-N6 Col. IS-5 Specimen

N2N4 N5 N6

N4N590

70

50

P (kN)

N4N5N6

N2

N1N3

N7-N12 Col.

N2

15 0 3015

N12N6N8

90

70

50

P (kN)

N8

N8N7N9

15 0 3015

N18

Exp

Num

90

70

50

P (kN)

N15N14N13

N13N14N15

N13-N15 Col. N16-N19 Col.

N10N11N12

d1d1

90

70

50

P (kN)

N16N19N18

N17 N19N17N16N18

Exp

Num

15 0 3015 15 0 3015

Fig. 9. IS-5 specimen/columns: (a) P vs. d1 paths, (b1) experimentally observed failure mode and numerical failure modes of columns (b2) N1, (b3) N2, (b4) N10, (b5) N15 and(b6) N18 (distinct initial imperfections).

P.B. Dinis et al. / Thin-Walled Structures 81 (2014) 195–209202

respectively – the table in Fig. 11(b) gives the correspondingnumerical failure loads PNum. As for Fig. 11(c), it displays the collapsemode observed experimentally for the test specimen IS-4 andthe corresponding numerical simulation concerning the N2 column– the latter includes the plastic strain distribution. The comparativeanalysis of these numerical and experimental results leads to thefollowing comments:

(i) There is a very good correlation between the numerical(N2 column) and experimental failure modes depicted inFig. 11(c): both exhibit dominant 3 half-wave distortionaldeformations with mid-height inward flange-lip motions(identical positive d1 and d7 values). However, note that (i1)the presence of local deformations at the collapse of bothspecimens is not perceptible and (i2) neither of the two failuremodes is symmetric, which is probably due to the asymmetryof the initial geometrical imperfections – indeed, the mostinward flange-lip motions/deformations do not occur atthe mid-height cross-section and, moreover, occur in differentcross-section for the top and bottom flange-lip assemblies.

(ii) The correlation between the numerical and experimentalfailure loads is also very good: the differences are of 2.1%and 1.9% for the N1 and N2 columns, respectively.

(iii) However, the very good correlation found for the failure loadand collapse mode does not extend to the equilibrium paths:although they follow the same general trends, (iii1) thenumerical P vs. ε curve is stiffer then the experimental one,and (iii2) the numerical P vs. d1 and P vs. d7 curves exhibitmuch more ductility prior to failure (displacements atcollapse about five times larger than those measured in thetest). Only through a more accurate modelling of the realinitial geometrical imperfections (not measured in the testspecimen) would it be possible to bring the numerical andexperimental equilibrium paths closer together.

(iv) As already happened for the IS-5 specimen, only the numer-ical simulation of the N2 column, which contains local initialimperfections, provides a collapse mode exhibiting distor-tional and local (flange) deformations (LþD3i mode) � thisfact can be confirmed by comparing the top flange deformedconfigurations of the columns N1 and N2, displayed in Fig. 12(a)-(b).

Since the numerical simulations carried out showed that thecolumns analysed in this work, all which buckle in distortionalcritical mode, may exhibit collapse modes that combine distor-tional (dominant) and local (flange) deformations, provided that

Fig.11. Numerical and experimental results concerning the IS-4 specimen: (a) P vs. d1, P vs. d7 and (b) P vs. ε equilibrium paths, and (c) observed and determined (N2 column)collapse modes.

Fig. 10. Top flange deformed shapes at collapse of the IS-5 columns (a) N10, (b) N13, (c) N16, and (d) N19.

P.B. Dinis et al. / Thin-Walled Structures 81 (2014) 195–209 203

the initial geometrical imperfection has a local component, it maybe argued that they are affected by some kind of L–D interaction.However, it should be noted that such type of L–D interaction hastwo distinct features, namely the facts that (i) the column distor-tional critical buckling load clearly (although not excessively)precedes its local counterpart and (ii) the interaction effects aretriggered by the flanges (not the web). To the authors' bestknowledge, in-depth information on the mechanics associatedwith such a coupling phenomenon is not yet available and furthernumerical and/or experimental investigations are definitelyrequired to acquire it.2 Finally, it is still worth noting that it wasalso found that the presence of minor-axis flexural initial imper-fections has a visible influence on the column post-bucklingbehaviour and ultimate strength � this relevance may havesomething to do with the fact that most of the stress redistribu-tion, occurring at the advanced post-buckling stages, takes placein the flanges and, therefore, entails “effective centroid shifts”(e.g., [22]) that are also responsible for minor-axis flexuraldeformations.

5. DSM design considerations

The current DSM strength curves for cold formed steel columndesign are defined by “Winter-type” expressions that (i) werecalibrated against fairly large numbers of experimental and/ornumerical failure loads and (ii) predict efficiently (safely andaccurately) the ultimate strengths of columns failing in local,distortional and global (flexural or flexural–torsional) modes, onthe sole basis of elastic buckling and yield stress values [3] � theDSM expression providing the column nominal strengths againstlocal (fNL), distortional (fND) and global (fNE) failures are given by

f NL ¼f y if λLr0:776

f yf crLf y

� �0:41�0:15 f crL

f y

� �0:4� �

if λL40:776

8><>:

where λL ¼ffiffiffiffiffiffiffif yf crL

s; ð1Þ

f ND ¼f y if λDr0:561

f yf crDf y

� �0:61�0:25 f crD

f y

� �0:6� �

if λD40:561

8><>:

where λD ¼ffiffiffiffiffiffiffiffif yf crD

s; ð2Þ

f NE ¼ð0:658λ2E Þf y if λEr1:5

0:877λ2E

� �f y if λE41:5

8><>: where λE ¼

ffiffiffiffiffiffiffiffif yf crE

s: ð3Þ

Moreover, in order to capture the local-global interactive fail-ures, the current DSM replaces fy by fNE in the fNL expressions, thusproviding fNLE estimates. On the other hand, several DSM-basedapproaches to estimate the ultimate strength of columns experi-encing mode interaction phenomena involving distortional buck-ling (mostly local–distortional interaction) have been proposed inthe literature – those mentioned next are relevant for the workpresented in this paper (also included is a new proposal that isdeveloped in the same spirit as one of its predecessors).

(i) In order to capture L–D interaction effects, Schafer [23]originally proposed to replace fy either (i1) by fNL in the fNDequations, providing the NDL approach (fNDL), or (i2) by fND inthe fNL equations, yielding the NLD approach (fNLD). The latterwas subsequently adopted to estimate the ultimate strengthof lipped channel columns with “v-shape” web and/or flangeintermediate stiffeners undergoing L–D interaction [4,7] –

reasonably accurate predictions were obtained, but the col-umns considered exhibited PcrL/PcrD values both above andbelow 1.0 (i.e., columns buckling in distortional or local criticalmodes).

(ii) A few years later, Kwon et al. [6] proposed a slight modifica-tion of the above NLD approach and employed it to estimatethe failure loads of plain, web-stiffened and web/flange-stiffened lipped channel columns undergoing L–D interaction– due to the modification, the vast majority of the failure loadpredictions became safe/conservative.

(iii) In 2011, Yap and Hancock [7] considered additional DSMapproaches to predict the failure loads of web-stiffened andweb/flange-stiffened lipped channel columns experiencingL–D interaction, which were compared with test resultsobtained at the University of Sydney – note that such testresults concerned columns with PcrL/PcrD values above andbelow 1.0. The first approach consisted of replacing fy by fNEin the fND equations, thus providing fNDE estimates (NDEapproach). The second approach involved the proposition ofa novel local strength/design curve, which replaces Eq. (1) andis given by the expression

f NLn ¼f y if λLr0:673

f yf crLf y

� �0:51�0:22 f crL

f y

� �0:5� �

if λL40:673

8><>:

where λL ¼ffiffiffiffiffiffiffif yf crL

s: ð4Þ

Then, fy is replaced by fNDE (prediction of the first approach) inEq. (4), thus providing an NLDE approach (fNLnDE). With thefirst approach, the fNDE estimates were found to be slightlymore accurate (less unsafe) than the fND ones. The secondapproach, on the other hand, naturally provided very con-servative/safe predictions, even if highly scattered (0.20standard deviation). In both the approaches, practically all

Fig. 12. Top flange deformed configurations of the IS-4 columns (a) N1 and (b) N2.

2 Up to now, virtually all the investigations, either numerical or experimental,concerning columns undergoing L–D interaction involved columns geometries(shapes, cross-section dimensions and lengths) associated with (i) local criticalbuckling loads that either precede (possibly by a fairly large amount), are almostidentical or just exceed their distortional counterparts, and (ii) interaction effectstriggered by the web � this is invariably the case if the web is not stiffened (e.g., forthe plain lipped channel, zed-section, hat-section and rack-section columnsinvestigated in [10,12,14,15]).

P.B. Dinis et al. / Thin-Walled Structures 81 (2014) 195–209204

the lower test-to-predicted failure load ratios corresponded tocolumns buckling in distortional critical modes (i.e., with PcrL/PcrD41.0).

(iv) Recently, Silvestre et al. [10] proposed an alternative and moreefficient DSM design approach for fixed-ended lipped channelcolumns with local and distortional critical buckling stressesnot more than 10% apart (0.9rPcrL/PcrDr1.1), which was latershown to be valid also to hat-section, zed-section and rack-section columns under the same circumstances [14]. However,it should be noted that this approach was never comparedagainst failure load data concerning columns with PcrL/PcrDvalues more than 10% apart from 1.0.

(v) Even more recently, Young et al. [12], on the basis of experi-mental data obtained at the University of Hong Kong, foundout that the ultimate strength of fixed-ended lipped channelcolumns can be fairly well predicted by the lowest of fNLE andfNLD (termed fNL* – NLn approach) – moreover, this approachhas just been also successfully applied to rack-section col-umns with PcrL/PcrD values moderately larger than 1.1 [15].However, it should be noted out that these ultimate strengthestimates were never compared against failure load dataconcerning columns buckling in distortional critical modes(i.e., with PcrL/PcrD41.0).

(vi) Adapting the rational behind the approach described in theprevious item, a new one is proposed here for fixed-endedcolumns buckling in distortional critical modes: to predicttheir ultimate strengths by means of the lowest of fNDE (NDEapproach proposed by Yap and Hancock [7]) and fNDL (NDLapproach proposed by Schafer [23]) – the correspondingultimate strength estimates are termed fNDn (NDn approach).

Next, the ability of the various DSM-based design approachesmentioned above to predict adequately the ultimate strength ofweb-stiffened lipped channel columns that (i) buckle in distor-tional critical modes and (ii) are susceptible to exhibit L–Dinteractive failures is assessed and discussed. The first steptowards achieving this goal consists of putting together a columnultimate strength data bank comprising (i) the experimentalvalues obtained from the investigation reported in this work (IScolumns), and (ii) additional test results available in the literature.The experimental ultimate strengths are compared with theirpredictions provided by the current DSM distortional design curve(fND) and the proposed approaches fNLD, fNDL, fNL*, fND* and fNLnDE.

5.1. Additional test results available in the literature

The number of experimental results involving web-stiffenedlipped channel columns that clearly buckle in distortional modes(i.e., with PcrL/PcrD41.0) is scarce and correspond to tests carriedout at the University of Sydney or the University of Yeungnam,namely (i) 4 (out of 6) specimens tested by Kwon and Hancock[17], with lengths varying between 800 and 1000 mm andt¼1.1 mm, (ii) 2 (out of 12) specimens tested by Kwon et al. [6],with L¼1200 mm and t¼0.6 or 0.8 mm, and (ii) 8 (out of 11)specimens tested by Yap and Hancock [7,18], with lengths varyingbetween 1000 and 2000 mm and thickness t¼1.065 mm3. Table 3provides, for all the above specimens, the (i) cross-section dimen-sions and length, (ii) squash and critical (local, distortional, global)buckling loads, (iii) experimental failure loads and (iv) observedfailure mode natures. It is worth noting that except for (i) threetests reported in [17], which failed in pure distortional modes, and(i) one test reported in [6], which failed in a local-distortional-global mixed mode, all the tested specimens included in Table 3(i) failed in modes combining distortional and local deformations,and (ii) exhibit PcrL/PcrD, PcrE/PcrD and Py/PcrL ratios in the 1.05–1.60,1.27–42.59 and 1.35–4.21 ranges, respectively. Moreover, note thatthe PcrE/PcrD values of the 4 tests reported in [17] are much higherthan the remaining ones, including those exhibited by the IScolumns dealt with in this work (see Table 1)4 – while these fourvalues are comprised between 19.49 and 42.59, the IS columnvalues are all grouped in the 1.27–1.41 range and the valuesconcerning the other column tests available vary from 1.71 to10.80.

5.2. Assessment of the DSM ultimate strength estimates

Table 4 shows, for each test specimen, (i) the distortionalslenderness λD, comprised between 1.41 and 2.52, (ii) the DSMultimate strength estimates (fND, fNLD, fNDL, fNLE, fNDE, fNL*, fND* and

Table 3Web-stiffened lipped channel column tests available: specimen (i) geometries, (ii) squash and critical (local, distortional, global) buckling loads, and (iii) experimental failureloads and mode natures.

bw bf bl d1 d2 t L Py PcrL PcrD PcrE PcrL

PcrD

PcrE

PcrD

Py

PcrL

PExp Obs. failure mode(mm) (mm) (mm) (mm) (mm) (mm) (mm) (kN) (kN) (kN) (kN) (kN)

Kwon and Hancock [17] 120.5 90.2 7.0 10.0 20.0 1.100 800 203.8 48.1 32.2 1370.6 1.49 42.59 4.24 66.1 D120.5 89.8 7.2 10.3 19.5 1.100 1000 203.9 48.1 30.0 881.7 1.60 29.41 4.24 64.5 D120.0 90.0 8.0 10.2 20.0 1.095 1000 203.6 48.4 34.8 887.3 1.39 25.49 4.21 64.0 D116.0 90.0 10.0 10.3 20.0 1.105 1000 205.7 50.6 44.7 870.2 1.13 19.49 4.06 68.1 LþD

Kwon et al. [6] 80.0 40.0 10.0 7.0 14.0 0.600 1200 69.9 23.0 20.6 84.8 1.12 4.12 3.00 22.2 LþD50.0 50.0 10.0 6.0 12.0 0.800 1200 86.7 34.8 33.3 56.9 1.05 1.71 2.49 33.8 LþDþE

Yap and Hancock [7,18] 120.0 65.0 12.0 10.0 20.0 1.065 1000 177.4 72.5 63.5 685.7 1.14 10.80 2.45 58.4 LþD1000 177.0 71.3 63.4 684.5 1.12 10.80 2.48 58.6 LþD1000 176.6 71.1 63.2 682.9 1.12 10.80 2.48 59.2 LþD1300 177.2 71.3 54.1 406.0 1.32 7.50 2.48 53.6 LþD1300 177.1 71.3 54.1 405.9 1.32 7.50 2.48 57.4 LþD1300 177.0 71.2 54.1 405.5 1.32 7.50 2.48 58.4 LþD2000 177.4 71.4 46.9 172.5 1.52 3.68 2.48 49.0 LþD2000 176.9 71.2 46.7 172.0 1.52 3.68 2.48 46.4 LþD

3 None of the remaining web-stiffened columns tested by these researchersbuckled in distortional critical modes.

4 It should be pointed out that this experimental programme was carried out inthe context of a study on the post-buckling and ultimate strength behaviour oflipped channel columns. Although L–D interaction was not the primary purpose ofthis investigation, it is mentioned that 3 (out of 6) tested specimens failed in modesexhibiting both distortional and local deformations � one of such specimens isincluded in Table 3.

P.B. Dinis et al. / Thin-Walled Structures 81 (2014) 195–209 205

Table 4Experimental ultimate strengths and their various DSM predictions (stresses in MPa).

Specimen Test ND NLD NDL NLE NDE

fExp fExp/fy λD fND fExp/fND fNLD fExp/fNLD fNDL fExp/fNDL fNLE fExp/fNLE fNDE fExp/fNDE

IS-1 246 0.41 1.45 325 0.76 290 0.85 274 0.90 288 0.86 230 1.07IS-2-1 215 0.36 1.58 299 0.72 259 0.83 245 0.88 249 0.87 198 1.09IS-2-2 216 0.36 1.61 294 0.73 254 0.85 240 0.90 246 0.88 195 1.10IS-3 202 0.33 1.65 287 0.70 240 0.84 230 0.88 218 0.93 179 1.13IS-4 189 0.31 1.71 277 0.68 233 0.81 221 0.85 207 0.91 167 1.13IS-5 261 0.44 1.41 328 0.80 306 0.85 285 0.92 304 0.86 236 1.11IS-6-1 247 0.42 1.48 312 0.79 283 0.87 265 0.93 273 0.90 213 1.16IS-6-2 240 0.40 1.49 310 0.77 281 0.85 263 0.91 272 0.88 212 1.13IS-7 212 0.36 1.59 291 0.73 256 0.83 241 0.88 241 0.88 190 1.12IS-8 209 0.35 1.60 289 0.72 258 0.81 241 0.87 244 0.86 188 1.11

Kwon and Hancock [17] 190 0.32 2.52 (177) (1.07) (139) (1.37) (130) (1.46) (289) 0.66 (172) (1.10)185 0.32 2.61 (171) (1.08) (135) (1.37) (125) (1.48) (283) 0.65 (163) (1.13)184 0.31 2.42 (185) (0.99) (143) (1.29) (135) (1.36) (284) 0.65 (177) (1.04)194 0.33 2.15 211 0.92 158 1.23 154 1.26 287 0.68 201 0.96

Kwon et al. [6] 199 0.32 1.83 267 0.75 209 0.95 203 0.98 293 0.68 226 0.88247 0.39 1.61 306 0.81 245 1.01 240 1.03 259 0.95 219 1.13

Yap and Hancock [7,18] 207 0.33 1.67 294 0.70 239 0.87 231 0.90 367 0.56 278 0.74208 0.33 1.67 294 0.71 237 0.88 230 0.90 365 0.57 278 0.75211 0.34 1.67 294 0.72 237 0.89 230 0.92 365 0.58 278 0.76190 0.30 1.81 271 0.70 225 0.84 214 0.89 348 0.55 248 0.77204 0.32 1.81 271 0.75 225 0.91 214 0.95 348 0.59 248 0.82207 0.33 1.81 271 0.77 225 0.92 214 0.97 348 0.60 248 0.84174 0.28 1.95 251 0.69 214 0.81 199 0.87 296 0.59 203 0.85165 0.26 1.95 251 0.66 214 0.77 199 0.83 296 0.56 203 0.81

Mean0.78 0.94 0.99(0.74) (0.88) (0.92)

Sd. Dv.0.12 0.18 0.19(0.06) (0.10) (0.09)

Max.1.08 1.37 1.26(0.92) (1.23) (1.26)

Min.0.66 0.77 0.83(0.66) (0.77) (0.83)

Specimen Test NL* ND* NLnDE

fExp fExp/fy fNL* Failure fExp/fNL* fND* Failure fExp/fND* fNLnDE fExp/fNLnDE

IS-1 246 0.41 288 LþFT 0.86 230 DþFT 1.07 211 1.16IS-2-1 215 0.36 249 LþFT 0.87 198 DþFT 1.09 181 1.19IS-2-2 216 0.36 246 LþFT 0.88 195 DþFT 1.10 178 1.21IS-3 202 0.33 218 LþFT 0.93 179 DþFT 1.13 161 1.26IS-4 189 0.31 207 LþFT 0.91 167 DþFT 1.13 152 1.24IS-5 261 0.44 304 LþFT 0.86 236 DþFT 1.11 225 1.16IS-6-1 247 0.42 273 LþFT 0.90 213 DþFT 1.16 202 1.22IS-6-2 240 0.40 272 LþFT 0.88 212 DþFT 1.13 200 1.20IS-7 212 0.36 241 LþFT 0.88 190 DþFT 1.12 176 1.20IS-8 209 0.35 244 LþFT 0.86 188 DþFT 1.11 177 1.18

Kwon and Hancock [17] 190 0.32 (139) (LþD) (1.37) (130) (LþD) (1.46) (124) (1.53)185 0.32 (135) (LþD) (1.37) (125) (LþD) (1.48) (120) (1.54)184 0.31 (143) (LþD) (1.29) (135) (LþD) (1.36) (126) (1.45)194 0.33 158 LþD 1.23 154 LþD 1.26 138 1.40

P.B.Dinis

etal./

Thin-Walled

Structures81

(2014)195

–209206

fNLnDE) and (iii) the corresponding experimental-to-predicted ulti-mate strength ratios – the average, standard deviation, maximumand minimum values are also given. Since three tests reported byKwon and Hancock [17] exhibited pure distortional failure modes,it was also decided to calculate the averages and standard devia-tions without considering those three tests5 � they are givenbetween parentheses in Table 4 and Fig. 14(a)–(f).

Fig. 13 compares the experimental fExp/fy values with thecurrent distortional DSM design curve (fND/fy vs. λD), while Fig. 14(a)–(f) plots the test-to-predicted ultimate strength ratios fExp/fND,fExp/fNLD, fExp/fNDL, fExp/fNL*, fExp/fND* and fExp/fNLnDE against λD � thesefigures also show the corresponding averages and standarddeviations. The comparative analysis of these experimental resultsleads to the following comments:

(i) Generally speaking, the fExp/fy value distributions concerningthe IS columns dealt with in this work and the test datacollected from the literature is fairly similar � in Fig. 13, thetwo sets of symbols are “perfectly” aligned with each other,with the exception of the four tests reported by Kwon andHancock [17] (recall that pure distortional failure modeswere observed in three of them). The ultimate strengthscorresponding to three out of these four tests are on or abovethe distortional design curve, which is in agreement with theobserved pure distortional failures – the ultimate strengthconcerning the fourth test, observed to fail in a modecombining distortional and local deformations, is a bit belowthe curve, thus evidencing a certain (small) amount of L–Dinteraction.

(ii) Because they account for possible interaction phenomenainvolving distortional buckling, all the remaining ultimatestrength estimates are lower than fND. This automaticallyimplies that including the four tests reported in [17] has thenet effect of increasing the various test-to-predicted ultimatestrength ratio averages and standard deviations.

(iii) The fND values consistently overestimate the experimentalultimate strengths by a very large margin – indeed, the fExp/fND average and standard deviation are equal to 0.78 and0.12. The strong overestimation is quite surprising, since itdid not occur in recent numerical investigations on plainlipped channel columns exhibiting clear distortional criticalbuckling (i.e., PcrL/PcrD41.1) [16,24] � the fND values pro-vided accurate and mostly safe predictions of numericalultimate strengths (as expected, since the current DSMdistortional design curve was developed on the basis of alarge plain lipped channel ultimate strength data). It is worthnoting, however, that no local–distortional interactioneffects were detected in the vast majority of such columns

Kwon

etal.[6]

199

0.32

209

LþD

0.95

203

LþD

0.98

171

1.16

247

0.39

245

LþD

1.01

219

DþFT

1.13

180

1.37

Yapan

dHan

cock

[7,18]

207

0.33

239

LþD

0.87

231

LþD

0.90

211

0.98

208

0.33

237

LþD

0.88

230

LþD

0.90

210

0.99

211

0.34

237

LþD

0.89

230

LþD

0.92

210

1.01

190

0.30

225

LþD

0.84

214

LþD

0.89

195

0.98

204

0.32

225

LþD

0.91

214

LþD

0.95

195

1.05

207

0.33

225

LþD

0.92

214

LþD

0.97

195

1.07

174

0.28

214

LþD

0.81

199

LþD

0.87

171

1.01

165

0.26

214

LþD

0.77

199

LþD

0.83

171

0.96

Mea

n0.95

1.08

1.19

(0.90)

(1.03)

(1.14)

Sd.D

v.0.17

0.18

0.17

(0.09)

(0.12)

(0.13)

Max

.1.37

1.48

1.54

(1.23)

(1.26)

(1.40)

Min.

0.77

0.83

0.96

(0.77)

(0.83)

(0.96)

fExp/fy

0.00.20.40.60.81.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0λD

D

- IS Columns - Kwon & Hancock [17]- Kwon et al. [6]- Yap & Hancock [7,18]

Fig. 13. DSM distortional design curve and variation of fExp/fy with the distortionalslenderness λD.

5 It is worth noting that the ultimate strengths of these three specimens arevery well predicted by the current DSM distortional design curve � see Table 4.

P.B. Dinis et al. / Thin-Walled Structures 81 (2014) 195–209 207

(and they were found to be negligible in the remainingones).

(iv) Both the fNLD and fNDL values provide less pronouncedultimate strength overestimations than the fND ones. Amongthem, the latter are clearly the less inaccurate, as attested bythe fExp/fNLD and fExp/fNDL averages and standard deviations:0.94 and 0.18, and 0.99 and 0.19, respectively.

(v) Concerning the fNL* values, they coincide with (v1) the fNLEones, for all the IS columns, and (v2) the fNLD ones, for all theremaining columns – this means that it is predicted that allthe IS columns fail in local–global (flexural–torsional) inter-active modes, while all the observed collapses involveddistortional and local deformations (see Table 1). The qualityof the fNL* estimates falls in between those exhibited by thefNLD and fNDL ones, as attested by the fExp/fNLD, fExp/fNDL andfExp/fNL* averages and standard deviations: 0.94 and 0.18, 0.99and 0.19, and 0.95 and 0.17, respectively.

(vi) As for the fND* values, they coincide with (vi1) the fNDE ones,for all the IS columns and one column reported by Kwonet al. [6], and (vi2) the fNLD ones, for all the remainingcolumns – this means that it is predicted that all the IScolumns fail in distortional–global (flexural–torsional) inter-active modes, while all the observed collapses involveddistortional and local deformations (see Table 1). However,the quality of the fND* estimates is clearly higher than thatexhibited by the fNDL ones, even if the scatter is a bit higher,as attested by the fExp/fNDL and fExp/fND* averages andstandard deviations: 0.99 and 0.19, and 1.08 and 0.18,respectively.

(vii) The NLnDE approach proposed by Yap and Hancock [7],which takes into account triple interaction effects, naturallyprovides the lowest ultimate strength estimates and, there-fore, several of them are very much on the conservative side– on the other hand, there are only very slightly unsafefailure load predictions. The fExp/fNLnDE average and standarddeviation are 1.19 and 0.17.

(viii) Out of the various DSM approaches considered, it seemsfair to argue that the ND* is the most efficient (safe andeconomical) one. Indeed, it has an average closely above 1.0

(1.08), a reasonable scatter (0.18 standard deviation) andmaximum/minimum values equal to 1.48/0.83 – if the threecolumn failing in pure distortional modes, reported in [17],are excluded, the above indicators become 1.03, 0.12 and1.26/0.83. Nevertheless, it should be pointed out that the fND*overestimate the ultimate strengths of all the columns testedby Yap and Hancock [7], while underestimating all thecolumns dealt with in this work – this is most likely due tothe difference between the PcrE/PcrD values exhibited by thetwo sets of columns (the latter are much lower, whichexplains the prediction of failure modes combining distor-tional and flexural–torsional deformations � see Table 4).

6. Conclusion

An experimental (mostly) and numerical investigation on thepost-buckling behaviour, ultimate strength and DSM design offixed-ended cold-formed steel web-stiffened lipped channel col-umns buckling in distortional modes was reported. The experi-mental results, obtained at The University of Hong Kong, providedclear evidence of the occurrence of local–distortional interactivefailures: all specimens failed in 3 half-wave distortional modeswith mid-height inward flange-lip motions and exhibiting localdeformations in the flanges. After performing and discussing anin-depth numerical (ABAQUS SFE) investigation on the imperfection-sensitivity of web-stiffened lipped channel columns, numericalsimulations concerning two tests were presented and shown tocompare fairly well with the corresponding experimental results.Then, the experimental failure loads obtained in this work,together with additional ultimate strength data reported inliterature and concerning exclusively web-stiffened lipped channelcolumns that buckle in distortional critical modes, were used toassess the merits of their predictions provided by (i) the currentDSM distortional design curve, (ii) four available DSM designapproaches specifically developed to capture the ultimate strengtherosion due to L–D interaction and (iii) one such approach newlyproposed in this work and combining two existing strengthcurves.

Fig. 14. Variation with λD of the test-to-predicted ultimate strength ratios (a) fExp/fND, (b) fExp/fNLD, (c) fExp/fNDL, (d) fExp/fNL*, (e) fExp/fND*, and (f) fExp/fNLnDE.

P.B. Dinis et al. / Thin-Walled Structures 81 (2014) 195–209208

Among the various conclusions drawn from the investiga-tion just reported, the following ones deserve to be speciallymentioned:

(i) The experimental results, obtained from tests carried out atThe University of Hong Kong, provide novel experimentalevidence of the occurrence of L–D interaction and its impacton the column failure load erosion – this evidence wasperfectly in line with the test results reported earlier by Kwonet al. [6] and Yap and Hancock [7].

(ii) The flange-triggered L–D interaction occurring in web-stiffened lipped channel columns is mechanically differentfrom its web-triggered counterpart, which takes place in plainlipped channel columns. The imperfection-sensitivity studycarried out in this work made it possible to unveil therelevance of the local initial imperfections on the columnultimate strength and failure mode nature – it may be said tobe the main cause of the occurrence of L–D interaction. An in-depth investigation on the mechanics of the web-stiffenedlipped channel column post-buckling is required, in order toshed further light on the above behavioural feature.

(iii) The failure load predictions provided by (iii1) the current DSMdistortional design curve and (iii2) the NLD, NDL and NL* DSMapproaches were fairly similar and mostly unsafe – however,the fNDL values provide the least pronounced ultimatestrength overestimations.

(iv) The most efficient (safe and accurate) failure load estimateswere obtained by means of the ND* DSM approach proposed inthis work, which consists of selecting the lowest of the fNDE andfNDL ultimate strength estimates (proposed by Yap and Hancock[7] and Schafer [23], respectively). However, it should also bepointed out that the NLnDE DSM approach, proposed by Yapand Hancock [7] and accounting for triple interaction effects,provides almost always safe ultimate strength estimates – butseveral of them are overly conservative.

(v) In spite of the fairly good performance of the proposed NDn

DSM approach, in predicting the column experimental failureload data considered in this work, the authors are aware thatfurther investigations are required in order to confirm and/orextend the findings reported. In particular, parametric studiesaimed at (v1) acquiring a better understanding of themechanics of the flange triggered L–D interaction and (v2)gathering numerical ultimate strength data concerning web-stiffened lipped channel columns with a wider variety ofgeometries and slenderness values (but all buckling in dis-tortional critical modes) are required � such parametricstudies are planned for the near future.

Acknowledgements

The authors are indebted to the Emeritus Professor GregoryJ. Hancock for his invaluable comments and suggestions on severalaspects of the work reported above – they have certainly con-tributed to improve considerably the quality of this paper. More-over, the authors gratefully acknowledge the BlueScope LysaghtSingapore, for supplying the cold-formed steel test specimensused in the experimental investigation, and Mr Wai-Kin Lam,

for his assistance during the performance of the experimentalprogramme.

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