non-linear behaviour and load-carrying capacity of cfrp-strengthened lipped channel steel columns

Engineering Structures 30 (2008) 2613–2630www.elsevier.com/locate/engstruct

Non-linear behaviour and load-carrying capacity of CFRP-strengthenedlipped channel steel columns

Nuno Silvestrea, Ben Youngb, Dinar Camotima,∗

a Department of Civil Engineering, IST/ICIST, Technical University of Lisbon, Portugalb Department of Civil Engineering, The University of Hong Kong, Hong Kong

Received 27 August 2007; received in revised form 18 February 2008; accepted 19 February 2008Available online 11 April 2008

Abstract

This paper reports the results of an experimental and numerical investigation on the non-linear behaviour and load-carrying capacity of CFRP-strengthened cold-formed steel lipped channel columns, devoting special attention to their local-plate and/or distortional buckling behaviours. Apreliminary GBT-based study concerning the column elastic buckling behaviour is performed, focusing on determining the (i) sheet location (web,flanges and/or lips) and (ii) carbon fibre orientation (longitudinal, transverse or inclined) that optimize the strengthening procedure (enhancedbuckling behaviour vs. cost). Then, an experimental programme comprising a total of 19 short and long fixed-ended lipped channel columns isdescribed. The columns were strengthened with carbon fibre sheets (CFS) bonded at different outer surface locations (web, flanges or lips) andhaving the fibres oriented either longitudinally or transversally — since the aim of the study is to assess the influence of the CFS on the columnstructural response, bare steel specimens were also tested. The experimental results, which consist of non-linear equilibrium paths (applied load vs.axial shortening) and ultimate strength values (most of them associated with local-plate and/or distortional failure mechanisms), are subsequentlyused to calibrate and validate geometrically and physically non-linear numerical analyses based on shell finite element models and carried out inthe code ABAQUS. Finally, on the basis of both the experimental and numerical results obtained, some relevant conclusions are drawn concerningthe most effective CFS location and fibre orientation to strengthen lipped channel steel columns affected by local-plate and/or distortional buckling.c© 2008 Elsevier Ltd. All rights reserved.

Keywords: Cold-formed steel columns; Carbon fibre sheets; FRP-strengthening; Buckling behaviour; Post-buckling behaviour; Load-carrying capacity; Local-platefailure; Distortional failure; GBT analysis; Experimental tests; FEM simulation

1. Introduction

In the last decade, FRP (fibre-reinforced polymer) compositematerials have been increasingly employed in the constructionindustry, mainly in applications dealing with structuralstrengthening and repair. They are ideally suited for thispurpose, due to a combination of (i) very high stiffness-to-weight and strength-to-weight ratios and (ii) an excellentdurability in aggressive environments. Indeed, it has beenshown, both analytically and experimentally, that the additionof externally bonded FRP composites significantly improvesthe performance of a structural member, namely its stiffness,load-carrying capacity, durability and fatigue behaviour under

∗ Corresponding author. Fax: +351 21 8497650.E-mail address: [email protected] (D. Camotim).

0141-0296/$ - see front matter c© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2008.02.010

cyclic loadings [1]. However, due to limitations stemmingfrom several economical and design-related issues, thisstrengthening/repair technique has not yet been given thechance to be exploited to its full potential.

The applicability and cost-efficiency of the FRP-strengthening concept depends largely on the material be-haviour of the member to be strengthened. Up to now, theresearch activity in this area has been mainly focused onapplications involving members made of reinforced concrete(mostly) and soft metals (e.g. aluminium), basically because itis much more feasible (i.e. economical) to satisfy the follow-ing key requirement: the strengthening material must be stifferthan the base material of the strengthened member. Since cold-formed steel members are considerably stiffer than the mostcommonly used FRP composites, strengthening them requiresexpensive high-strength fibres and, thus, this procedure has

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2614 N. Silvestre et al. / Engineering Structures 30 (2008) 2613–2630

Notation

bw, b f , bl Width of the web, flanges and lipsE, G, ν Steel Young’s modulus, shear modulus and

Poisson’s ratioET , EL Longitudinal and transverse Young’s modulify Steel yield stressIF.MS, IF.TH Index failure associated with the Maximum

Stress and Tsai–Hill criteriaL Column lengthPb, Pcr Buckling and critical loadsPL , PD Local-plate and distortional critical loadsPcr.S Critical load of the strengthened columnPExp, PNum Experimental and numerical ultimate loadsPNIL Ultimate load of the bare steel columnr Radius of the cross-section cornert Thicknessu Axial shorteningx, s Longitudinal and transverse (mid-line) local

coordinatesα Fibre orientation with respect to column longitu-

dinal axis xε f Material elongation after fractureφk Amplitude function of deformation mode kσ0.2, σu Material static 0.2% proof stress and tensile

strengthσ adm

ll , σ admt t , τ adm

lt CFS admissible normal longitudinal,normal transversal and shear stresses

been generally deemed not advantageous. Despite this fact, ElDamatty et al. [2] have shown, both experimentally and nu-merically, that glass fibre (GFRP) sheets can be used to en-hance the load-carrying capacity of wide flange steel beams.Nevertheless, there is no doubt that carbon fibre sheets (CFS),clearly more effective (and expensive, of course) than theirglass fibre counterparts, are currently the ones most employedto strengthen/repair steel structures, thus improving their seis-mic response and fatigue resistance [3–5]. Indeed, given theirelevated stiffness and strength values, high-modulus carbon fi-bres are particularly well suited to be combined with steelplates [6]. As far as strength is concerned, the failure of theCFRP-strengthened cold-formed steel members may stem from(i) local, distortional or global buckling of the steel-CFRPmember, (ii) rupture or debonding of the CFS or (iii) a combi-nation of both. Thus, a rational (safe and economical) design ofCFRP-strengthened steel members must be preceded by stud-ies addressing all the above potential failure modes. Concerningthe failure due to debonding between CFRP and steel, severalinvestigations were carried out very recently, namely those byFawzia et al. [7], Schnerch et al. [8], Colombi et al. [9] and El-Emrani and Kliger [10]. Moreover, since the strength of a steel-CFRP member depends mostly on the bond between the steeland CFS, which almost always governs its failure, further at-tention must be devoted to accurately estimating the capacity ofthe adhesives [11]. Due to its high yield stress, steel is a difficultmaterial to strengthen, due to the fact that substantial steel-to-

CFS stress transfer can only occur after steel has started to yield(in general, the CFS remains elastic up to collapse). In the con-text of hot-rolled steel beams, Lam and Clark [12] showed, bothnumerically and experimentally, that CFRP-strengthening (i)significantly improves the load-carrying capacity but (ii) onlyleads to a fairly small elastic stiffness increase. On the otherhand, Haedir et al. [13] performed experimental tests on circularhollow section beams strengthened with CFRP sheets — theirobjective was to assess how the beam slenderness influences itsbending capacity. Very recently, Shaat and Fam [14–16] showedthat jacketing short and long square hollow columns with CFRPsheets significantly increases their stiffness and strength, due tothe fact that it raises their local and global buckling stresses.However, it should be pointed out that the vast majority of theabove numerical and experimental investigations involve mem-bers under bending [2,4,9,12,13,17,18] and only very few ofthem concern compressed members [14–16]. Moreover, none ofthese studies addresses the local, distortional or global bucklingbehaviour of thin-walled columns with open cross-sections.

The objective of this work is two-fold: (i) to report theresults of an experimental investigation aimed at assessinghow the CFRP-strengthening influences (enhances) the non-linear behaviour and load-carrying capacity of cold-formedsteel lipped channel columns and also (ii) to use theseresults to calibrate and validate numerical analyses based onshell finite element models. Initially, the paper presents anddiscusses a parametric numerical study concerning the bucklingbehaviour of lipped channel columns — these results areobtained by means of an orthotropic Generalized Beam Theory(GBT) formulation developed by Silvestre and Camotim[19,20] and special attention is paid to determining the (i)sheet location (combination of web, flanges and lips) and (ii)carbon fibre orientation (longitudinal, transverse or inclined)that optimize the strengthening procedure, in the sense that theyprovide the best compromise between structural enhancementand cost efficiency. Then, one presents the results of theexperimental programme carried out, which involved two(short and long) test series of fixed-ended lipped channelcolumns. The specimens were strengthened with carbon fibresheets bonded at different outer surface locations (web, flangesor lips) and having fibres oriented either longitudinally ortransversally. The experimental results, which consist of non-linear equilibrium paths (applied load vs. axial shortening) andultimate strength values (most of them associated with local-plate and/or distortional failure mechanisms), are subsequentlyemployed to calibrate and validate geometrically and physicallynon-linear numerical analyses based on shell finite elementmodels and carried out in the code ABAQUS [21], adoptingan elastic–plastic constitutive law to describe the steel materialbehaviour. Finally, on the basis of both the experimentaland numerical results obtained, some relevant conclusions aredrawn concerning the most effective CFRP sheet location andfibre orientation to strengthen lipped channel columns affectedby local-plate and/or distortional buckling.

N. Silvestre et al. / Engineering Structures 30 (2008) 2613–2630 2615

Fig. 1. Lipped channel (a) dimensions, (b) CFRP sheet properties and (c) GBTdiscretisation.

2. Preliminary GBT-based study

In order to acquire some preliminary information about thebenefits of the FRP strengthening of steel columns, as faras their local and global buckling behaviour is concerned, aGBT formulation developed to analyse orthotropic thin-walledmembers [19,20] is employed. One assesses the bucklingbehaviour of simply supported lipped channel cold-formed steelcolumns exhibiting the cross-section dimensions bw = 90 mm,b f = 60 mm, bl = 8 mm, t = 1.5 mm (see Fig. 1(a)) andthe material properties E = 200 GPa and ν = 0.3 — theyare strengthened by carbon fibre sheets (CFS) with a 0.165 mmthickness and characterized by the elastic constants E1 =

235 GPa, E2 = 10 GPa, G12 = 5 GPa and ν12 = 0.30 (axes1 and 2 are oriented along and normal to the fibre direction —see Fig. 1(b)). The GBT discretization of the lipped channelsection is shown in Fig. 1(c), involves 17 nodes (6 natural and11 intermediate) and leads to 17 deformation modes — Fig. 2depicts the in-plane shapes of the 13 most relevant modes.1

While the first four modes concern rigid-body motions (1 —extension; 2 and 3 — major/minor axis bending; 4 — torsion),all remaining ones involve cross-section in-plane deformation(5 and 6 — symmetrical and anti-symmetrical distortionalmodes; 7–13 — local-plate modes). By assuming a single-halfwave sinusoidal buckling mode shape (exact solution for simplysupported members — locally and globally pinned end sectionsthat can warp freely), one defines standard eigenvalue problemswhose solutions provide the buckling loads [19]. Finally, notethat all the buckling analyses are performed assuming (i) linearelastic member behaviour and (ii) perfect bonding (i.e. no slip)between the steel plates and CFRP sheets.2

We first determined the buckling behaviour of the bare cold-formed steel columns, which is depicted in Fig. 3(a)–(b): (i)curve providing the variation of the single half-wave bucklingload Pb with the length L (in logarithmic scale) and (ii) amodal participation diagram showing the contributions of eachindividual GBT deformation mode to the column bucklingmode. The observation of these results prompts the followingremarks:

(i) The curve exhibits two local minima, corresponding tolocal buckling (LB — Pb = 88.2 kN for L = 80 mm)

1 Note that the shapes of these GBT-based modes are not qualitatively alteredby the addition of the CFS.

2 Obviously, exact buckling analyses should include the non-linearconstitutive relations describing the behaviour of the adhesive (interface)between the CFRP sheets and the steel plates.

and distortional buckling (DB — Pb = 66.4 kN for L =

350 mm). As for the buckling mode, it is (i1) local (7+ abit of 9), for L < 120 mm, (i2) mixed local-distortional(5 + 7), for 120 mm < L < 250 mm, and (i3) distortional(5+ a bit of 7 and 3), for 250 mm < L < 800 mm.

(ii) For 800 mm < L < 2500 mm, Pb decreasescontinuously and buckling combines major axis bending(2), torsion (4) and anti-symmetrical distortion (6):flexural–torsional–distortional buckling (FTDB).3

(iii) For L > 2500 mm, Pb decreases continuously again and,depending on the length, the columns buckle in one oftwo global modes: either (iii1) a flexural–torsional mode(FTB — 2 + 4), for 2500 mm < L < 8000 mm, or (iii2) aminor axis flexural mode (FB — 3), for L > 8000 mm.

In order to assess how the addition of CFS influencesthe cold-formed steel column buckling behaviour, oneconsiders the 7 strengthening configurations shown in Fig. 4:WFL(CFS in web + flanges + lips), WF(web + flanges),WL(web + lips), FL(flanges + lips), W (web), F (flanges), L(lips). Fig. 5(a)–(c) provide the variation of the column criticalbuckling load Pcr.S

4 with L , for the (i) bare steel columns(Pcr.S ≡ Pcr − lower black curves) and (ii) strengthenedcolumns with (ii1) WF (red curves), W (blue curves) and F(green curves) configurations and (ii2) fibre orientations α = 0◦

(Fig. 5(a)), α = 45◦ (Fig. 5(b)) and α = 90◦ (Fig. 5(c)).After observing the results displayed in these figures, one mayconclude that:

(i) Local, distortional and flexural–torsional buckling occursfor L < 200 mm (1–2 half-waves), for 200 mm <

L < 1300 mm (1–3 half-waves) and for L > 1300 mm(single half-wave). It can be seen that the column FRPstrengthening always leads to a non negligible criticalbuckling load increase (regardless of the buckling modenature) but does not affect the length range correspondingto local, distortional and global buckling.

(ii) Local buckling mostly benefits from placing the fibers atα = 45◦ (Fig. 5(b)). For distortional buckling, it is moreadvantageous to have the fibers at α = 90◦ (Fig. 5(c)).

(iii) Concerning the location of the CFS in the cross-section,it is obvious that the (most expensive) WF configurationis the best one, as far as enhancing the column bucklingbehaviour is concerned. To reach some conclusionsconcerning the overall “quality” of each strengtheningconfiguration, one must look at column lengths that arerepresentative of each buckling mode type and compare(qualitatively) the material cost against the strengtheningbenefit.

The graphs presented in Fig. 6 provide Pcr.S/Pcr values (ratiobetween the strengthened and bare steel column critical loads),

3 Although FTDM may occur in isotropic columns (as in the present case),its relevance is much higher for orthotropic columns, particularly when the ratioET /EL is low (EL and ET are the longitudinal and transverse elastic moduli).

4 Since the critical local and distortional buckling modes may exhibit morethan one half-wave, the corresponding buckling loads are likely to vary a littlealong the column length.


Fig. 2. In-plane shapes of the 13 most relevant cross-section deformation modes.

Fig. 3. Single half-wave buckling behaviour of bare steel columns: Pb vs. Lcurve and (b) GBT modal participation diagram.

for (i) the columns lengths L = 80 mm (LB — Fig. 6(a)),L = 350 mm (DB — Fig. 6(b)) and L = 2000 mm (FTB —Fig. 6(c)) and (ii) each CFS configuration.5 Moreover, in orderto investigate the effect of the fibre orientation on the columncritical loads, one considers again α = 0◦ (longitudinal fibres),α = 90◦ (transverse fibres) and α = 45◦ (fibres equally inclinedwith respect to the two coordinate axes). The observation of thegraphs shown in Fig. 6(a)–(c) leads to the following comments:

(i) Obviously, the contribution of strengthening the lips to thePcr.S increase is always negligible (1%–2%), regardless ofthe buckling mode shape — this can be clearly confirmedby (i1) looking at the results concerning configuration Land (i2) comparing the Pcr.S/Pcr values related to theconfiguration pairs WFL-WF, WL-W and FL-F. Then, itis just logical to discard lip strengthening and to consideronly the strengthening configurations WF, W and F —note also that WF is the “sum” of configurations Wand F.

(ii) Fig. 6(a) clearly shows that the most effective fibreorientation for column local buckling is α = 45◦. Thisquite unexpected result stems from the fact that the weblongitudinal and transverse half-wavelengths are similar,

5 Note that, for these column lengths, all critical buckling modes exhibit asingle half-wave (i.e., one has Pcr ≡ Pb).

which makes it more important, as far as increasingthe Pcr.S value, to strengthen the steel fibres orienteddiagonally (within each half-wave). Moreover, the localcritical load increases by 21.3% (WF), 13.9% (W) or5.9% (F), which shows that web strengthening accountsfor most of the Pcr.S increase — recall that local bucklingis triggered by the web instability. Therefore, taking alsothe strengthening cost into consideration, it seems fair tosay that the “optimal” configuration is W — indeed, thecomparison with the WF configuration performance showsthat it uses about 43% of the CFS material to achieve aPcr.S increase of about 65%.

(iii) Concerning distortional buckling (Fig. 6(b)), the mosteffective fibre orientation is α = 90◦. Indeed, sincethis instability phenomenon is governed by the rigid-body rotation of the “flange-lip assemblies” about theweb-flange longitudinal edges, it becomes of paramountimportance to strengthen the web transverse fibres — theyprovide restraint against the above “flange-lip assembly”rotations. Moreover, the distortional critical load increasesby 18.2% (WF), 13.0% (W) and 3.6% (F), which showsthat, once again, W is the “optimal” configuration. Onthe other hand, if one follows the “intuitive approach”of merely strengthening the flanges, a 6.2% critical loadincrease can be achieved at best (for α = 45◦).

(iv) For the longer columns, which buckle in flexural–torsionalmodes (Fig. 6(c)), the most efficient fibre orientation is,obviously, α = 0◦ — longitudinal flexure and warpingtorsion play a major role. The critical load now increasesby 10.5% (WF), 7.9% (W) and 2.8% (F), which meansthat the “optimal” configuration is W (web-strengtheningonly) once more. Indeed, it leads to a linear stiffnessincrease (i.e., larger moment of inertia, warping constantand St. Venant’s constant) considerably higher than its F(flange-strengthening only) counterpart.

(v) The comparison between Fig. 6(a), (b) and (c) makes itpossible to conclude that, broadly speaking, the columnbuckling behaviour enhancement due to the addition ofCFRP sheets depends on the critical mode nature, sinceit is (v1) most beneficial for the shorter columns (localbuckling), (v2) moderately so for the intermediate ones(distortional buckling) and (v3) noticeably less relevant forthe longer columns (flexural–torsional buckling).

3. Non-linear behaviour and load-carrying capacity

As far as the stiffness is concerned, buckling analyses arestill a good predictor of the geometrically non-linear behaviourof structures. However, the existing materials never exhibit a


Fig. 4. Lipped channel column CFS strengthening configurations.

Fig. 5. Variation of Pcr.S with L , for each strengthening configuration and (a) α = 0◦, (b) α = 45◦ and (c) α = 90◦.6

purely elastic behaviour, which means that strength also playsa crucial role in the collapse of a given structure. The failureof a CFS-strengthened cold-formed steel member may stemfrom (i) local (local-plate and/or distortional) or global bucklingof the steel-CFS member, (ii) rupture or debonding of theCFS or (iii) a combination of both. Thus, an efficient (safeand economical) design of such members must be based onan in-depth knowledge concerning all these potential failuremodes. The objective of this section is two-fold: (i) to reportthe results of an experimental investigation aimed at assessinghow the CFS-strengthening influences (enhances) the non-linear behaviour and load-carrying capacity of cold-formedsteel lipped channel columns and also (ii) to use these results tocalibrate and validate numerical (shell finite element) analysesperformed in the code ABAQUS [21].

3.1. Experimental investigation

The test program comprised two test series, namely (i) oneinvolving 9 short columns made of G550 steel sheets and(ii) another consisting of 10 long columns made of G450steel sheets. Next, one describes the column test procedure,which was already successfully employed in the testing ofbare cold-formed steel lipped channel columns [22,23] — in

6 In order to be able to follow properly the references to the colors in thisfigure the reader must have access to the electronic version of the paper, whichcan be found in the journal web site. The same applies to Figs. 5, 6, 8, 13 and14.

particular, one addresses issues related to (i) the labelling andcharacterisation of the specimens, (ii) the determination ofthe steel and CFS material properties, (iii) the test rig andexperimental set-up, and (iv) the operations and measurementsinvolved in performing a test.

(I) Specimen labelling. The tests were conducted on cold-formed steel lipped channel columns strengthened withCFS, which are bonded to their outer surfaces in differentlocations (web, flanges and/or lips) and with the fibresoriented either longitudinally (α = 0◦) or transversally(α = 90◦)7 — for reference purposes, a few bare steelcolumns were also tested. Since the zinc coating used inthese cold-formed steel columns exhibits a very strongbond to the base metal, it was not removed prior to gluingthe CFRP sheets to the column specimens. The columnspecimen labelling provides information about the testseries and the location and orientation of the CFS: (i)the first letter indicates whether the specimen belongsto the short (S) or long (L) column test series, (ii) thefollowing letters indicate whether the specimen has nostrengthening (NIL), CFS located in the web (W), CFSlocated in the web and flanges (WF) or CFS located in theweb, flanges and lips (WFL), and (iii) the final numberindicates the CFS fibre orientation (0 for longitudinal

7 Unfortunately, it was not feasible to cut the sheets supplied diagonally tothe fibre direction, in order to obtain α = 45◦ CFS. Hence, only columnsstrengthened with α = 0◦ and α = 90◦ CFS were tested.


Fig. 6. Variation of Pcr.S/Pcr with the CFS strengthening configuration and fibre orientation for (a) local (L = 80 mm), (b) distortional (L = 350 mm) and(c) flexural–torsional (L = 2000 mm) buckling.

Table 1Measured dimensions of the short column specimens (in mm)

Specimen bw b f bl t r L

S-NIL 124.1 101.9 13.8 1.071 3.0 600S-F-0 124.0 102.0 13.7 1.075 3.0 600S-F-90 123.7 101.9 13.8 1.073 3.0 601S-W-0 124.1 101.9 13.7 1.074 3.0 600S-W-90 124.1 102.0 13.6 1.068 3.0 600S-WF-0 123.6 101.9 13.7 1.068 3.0 602S-WF-90 124.1 101.9 13.7 1.061 3.0 601S-WF-90-R 124.1 101.9 13.8 1.073 3.0 600S-WFL-0 124.0 101.9 13.7 1.070 3.0 599

Mean 124.0 101.9 13.7 1.070 3.0 –Sd. Dev. 0.002 0.000 0.005 0.004 0.000 –

and 90 for transversal). Finally, the letter R identifies arepeated test.

(II) Test specimens. The column specimens were brake-pressed from high strength zinc-coated grades G450 andG550 structural steel sheets conforming to the AustralianStandard AS 1397 [24]. They all have nominal web widthbw = 125 mm, flange width b f = 102 mm, lip widthbl = 14 mm (see Fig. 1(a)) and inside corner radiir = 3.0 mm. While the 9 short columns are made ofG550 steel sheets with nominal thickness t = 1.0 mmand length L = 600 mm, the 10 long ones are made ofG450 steel sheets with nominal thickness t = 1.5 mmand length L = 2200 mm — the dimensions effectivelymeasured for each column specimen are shown in Tables 1and 2. The tests were conducted on fixed-ended columnswith their end sections welded to 25 mm thick steel plates,which were subsequently bolted to the load bearing plates— this procedure ensures full contact between the columnspecimen ends and the bearing plates, i.e., full warpingrestraint.

(III) Steel properties. The material properties of the cold-formed steel column specimens were obtained by means

Table 2Measured dimensions of the long column specimens (in mm)

Specimen bw b f bl t r L

L-NIL 126.3 102.6 13.8 1.552 3.0 2199L-NIL-R 126.6 102.8 13.6 1.553 3.0 2200L-F-0 126.4 102.6 13.9 1.560 3.0 2199L-F-90 125.4 102.6 13.7 1.567 3.0 2199L-W-0 126.4 102.7 13.8 1.556 3.0 2202L-W-90 125.6 102.7 13.7 1.549 3.0 2200L-WF-0 125.8 102.7 13.8 1.556 3.0 2201L-WF-90 125.2 102.7 13.8 1.559 3.0 2199L-WFL-0 125.7 102.8 13.8 1.555 3.0 2200L-WFL-0-R 126.4 102.8 13.7 1.553 3.0 2200

Mean 126.0 102.7 13.8 1.556 3.0 –Sd. Dev. 0.004 0.001 0.006 0.003 0.000 –

of tensile coupon tests. The coupons were extracted fromthe centre of the web, in the longitudinal direction —their dimensions conformed to the Australian StandardAS 1391 [25] for the tensile testing of metals using12.5 mm wide coupons of gauge length 50 mm. A MTSdisplacement controlled testing machine using frictiongrips was used to conduct the coupon tests, also inaccordance with AS 1391 — a calibrated extensometer of50 mm gauge length was used to measure the longitudinalstrain of the coupon specimens. A data acquisition systemwas employed to record the load and strain readings atregular intervals during the tests — the static load wasobtained by pausing the applied strain for 1.5 min nearthe 0.2% proof stress and the ultimate tensile strength,a procedure that allowed the stress relaxation associatedwith plastic straining to take place. The measured valuesof Young’s modulus, static 0.2% proof stress, static tensilestrength and elongation after fracture are (i) E = 218 GPa,σ0.2 = 521 MPa, σu = 546 MPa, ε f = 11.4% (G450steel) and (ii) E = 207 GPa, σ0.2 = 610 MPa, σu =

626 MPa, ε f = 9.2% (G550 steel).


(IV) CFS properties. The strengthening carbon fibre sheets(CFS) had a thickness of 0.11 mm and their materialproperties are (i) σu = 4200 MPa (tensile strength),(ii) E = 235 GPa (tensile modulus) and ε f = 1.8%(elongation after fracture). The epoxy resin used toattach/glue the (single) CFS onto the zinc coated cold-formed steel column outer surfaces has tensile strengthand modulus values equal to σu = 30 MPa and E =

3.5 GPa — after having the carbon fibre ply attached, thecolumn specimens were completely cured for 7 days.

(V) Test rig and experimental procedure. Fig. 7 provides anoverall view of both the test rig and experimental set-up employed, namely (i) the DARTEC servo-controlledhydraulic compression test machine, (ii) the two steelplates welded to the specimen ends, (iii) the rigid flatbearing plate connected to the test machine upper support,bolted to the specimen top end plate and restrainedagainst minor/major axis flexural and twist rotations, and(iv) the lower special bearing, which is bolted to thespecimen bottom end plate and subsequently restrainedagainst flexural and twist rotations — once full contact isachieved, vertical and horizontal bolts lock the bearing inposition. These bearings materialise fixed-ended supportsand the upper one allows for longitudinal translations,making it possible to apply the compressive force. Threedisplacement transducers were used to measure the topend plate vertical motion, thus monitoring (i) the axialshortening of the column specimen and (ii) the effectivepreclusion of flexural rotations. The hydraulic actuatoris always driven at a constant speed of 0.2 mm/minby means of a displacement control scheme — thisloading arrangement makes it possible to test the columnspecimens well beyond the ultimate load level, i.e., todetermine the post-collapse descending branch of thecolumn equilibrium path. A data acquisition system wasadopted to record the applied load and displacementtransducer readings at regular intervals throughout theduration of the tests — the static loads were recorded bypausing the applied strain for 1.5 min and, to estimatethe column specimen ultimate strengths as accuratelyas possible, the recording frequency was increased afterdetecting a sizeable axial stiffness drop.

3.1.1. Test resultsFig. 8(a), (b) display equilibrium paths (P(u) curves) that

provide the variation of the column axial shortening withthe applied load for the short and long column specimens.Moreover, the experimental ultimate strength values PExp aregiven in Tables 3 (short columns) and 4 (long columns). Theobservation of the results presented in these figures and tablesprompts the following comments:

(i) All column failures were due to local-plate/distortionalbuckling mode interaction — this is illustrated inFig. 9(a)–(b), which concern a short (S-F-0) and a long(L-F-0) column specimens and provide clear evidencethat the collapse always takes place in a distortionalmechanism. Then, the excessive column wall deformations

Fig. 7. Overview of the test rig and experimental set-up.

Fig. 8. Experimental equilibrium paths (P(u) curves) obtained from the (a)short and (b) long column tests.


Fig. 9. Experimental and numerical failure modes of the (a) S-F-0 and (b) L-F-0 columns.


Table 3Buckling and experimental/numerical ultimate loads of the short columns

Specimen Experimental — P (kN) Numerical — P (kN) PNum/PExpPExp PExp/PNIL − 1 (%) PL PL/PL .NIL − 1 (%) PNum PNum/PNIL − 1 (%)

S-NIL 54.9 – 26.8 – 55.9 – 1.018S-F-0 56.0 2.0 28.5 6.3 57.4 2.7 1.025S-F-90 56.4 2.7 27.9 4.1 56.8 1.6 1.007S-W-0 55.9 1.8 27.6 3.0 57.3 2.5 1.025S-W-90 55.6 1.3 28.0 4.5 56.3 0.7 1.013S-WF-0 60.2 9.7 29.2 9.0 62.8 12.3 1.043S-WF-45 – – 30.9 15.3 60.8 8.8 –S-WF-90 63.2 15.1

29.3 9.3 61.4 9.80.972

S-WF-90-R 62.4 13.7 0.984S-WFL-0 61.4 11.8 29.5 10.1 63.4 13.4 1.026S-WFL-90 – – 29.4 9.7 62.3 11.5 –

Mean 1.012Sd. Dev. 0.022

Fig. 10. Debonding of the CFRP sheets: (a) initial (near the yielded line) and(b) total.

due to the buckling effects also causes CFS debonding,which invariably occurs after the curve peak (maximumload) is reached, i.e., in the post-ultimate range. Thedebonding failure mode was not quantitatively measured inthe experimental investigation. However, it was observedduring the tests that, as the applied load increased, theCFRP sheets started to detach from the zinc-coated steelsurface near the yield line zone (see Fig. 10(a)), anobservation confirmed by the occurrence of crackingsounds. In the post-peak stages, debonding failure wasattained and qualitatively assessed: the CFRP sheetsbecame totally separated from the steel yielded zones (seeFig. 10(b)).

(ii) There are some qualitative differences between theequilibrium paths describing the structural response of theshort and long columns, namely concerning the relationbetween the corresponding proportional limit and ultimateloads — the former provide the range of validity of thelinear relation u = PL/EA. Indeed, it is possible toconclude that, generally speaking:(ii.1) In the short columns, the proportional limit loads

are in the 50–55 kN range and fairly close to thecorresponding ultimate strength values. This meansthat there is no significant axial stiffness degradationprior to the vicinity of the failure load — inseveral cases, this stiffness degradation occurs quite

abruptly, as shown by the visible equilibrium path“kinks”. Moreover, there are also noticeable ductilitydifferences, in the sense that the ultimate load mayor may not be preceded by an ascending equilibriumpath branch with a rather small slope — it is worthnoting that the less ductile behaviour is exhibited byone of the columns with a higher ultimate load.

(ii.2) In the long columns, the proportional limit loadsfall in the 70–75 kN range8 and are quite lowerthan the corresponding ultimate strength values. Thisis because the axial stiffness degradation growsprogressively until the ultimate load is reached, asshown by the smoothly decreasing slope of theequilibrium paths. Once again, there are visibleductility differences and the columns with the higherultimate loads are the less ductile.

(ii.3) Finally, one should point out that, as expected, theaxial stiffness degradation takes place clearly earlierfor the bare steel column, both short and long.

(iii) The relationship between the CFS location and the ensuingultimate load increase, with respect to the bare steelcolumn (S-NIL), is quite different for the short and longstrengthened specimens. In the first case (short columns),the test results show that:(iii.1) Strengthening only the flanges or the web leads to

quite marginal ultimate load increases — indeed,the largest is equal to 2.7% and corresponds to theS-F-90 specimen.

(iii.2) When both the web and the flanges are strengthenedwith longitudinal fibres (S-WF-0), the ultimateload experiences a 9.7% increase. A furtherstrengthening of the lips (S-WFL-0) renders thisincrease a bit larger — 11.8%.

(iii.3) The most effective FRP-strengthening is associatedwith gluing the CFS transversely to both the web

8 Note that the long column specimen wall thickness is 50% higher than theshort column one. This explains the (apparently paradoxical) fact that the longcolumn proportional limit loads are higher than their short column counterparts.


and flanges (S-WF-90) — a 15.1% increase isachieved.9

On the other hand, the long column test resultsmake it possible to conclude that:

(iii.1) Strengthening only the flanges or the web now leadsto more substantial ultimate load increases — theyvary between 3.4% (L-W-90) and 12.1% (L-W-0).

(iii.2) When both the web and the flanges are strength-ened, one has ultimate load increases of either15.7% (L-WF-0) or 18.4% (L-WF-90).

(iii.3) The most beneficial FRP-strengthening correspondsto placing CFS longitudinally around the wholecolumn outer surface (L-WFL-0) — a 19.8%increase is attained.10

(iv) The remarks/conclusions presented in the previous itemare in qualitative accordance with those drawn before,on the basis of the GBT buckling analyses. Taking intoaccount that it was not feasible to test column specimensstrengthened with either (iv1) diagonal fibres (α = 45◦)or (iv2) transverse fibres (α = 90◦) bonded to thelips, the experimental ultimate loads show that the mostefficient strengthening, for both short and long columns,corresponds to gluing the CFS transversely in the web andflanges (S-WF-90 and L-WF-90). This is not surprising ifone realises that (iv1) all the columns exhibit distortionalfailure mechanisms and (iv2) the preliminary GBT-basedinvestigation led to the conclusion that the columndistortional buckling behaviour is mostly improved byplacing the CFS transversally (i.e., with α = 90◦ fibres).

(v) In order to assess the reliability of the experimental set-upand procedure, three tests were repeated — the differencesbetween the ultimate strengths obtained for each pairof (supposedly) identical column specimens are rathersmall, namely 1.3% (S-WF-90), 1.8% (L-NIL) and 1.7%(L-WFL-0).

3.2. Numerical investigation

In order to perform the post-buckling analyses of theCFRP-strengthened columns, one uses the finite element codeABAQUS [21] — recall that the buckling analyses are carriedout by means of a GBT-based approach. Hereafter, oneaddresses issues related to (i) the column discretisation, (ii) themodelling of the column end support conditions, applied loadsand material behaviour, (iii) the incorporation of the initialgeometrical imperfections and (iv) the numerical techniquesemployed to find the solution of either the buckling linear

9 Due to practical difficulties related to gluing the CFS transversely bonded tothe lips, no S-WFL-90 column specimen was tested. Nevertheless, it is logicalto expect that its ultimate strength would not be too far apart from that of theS-WF-90 specimen — no more than 2% (the numerical results differ by 1.4%).10 Once again, no column specimen with the CFS transversely bonded to the

web, flanges and lips (L-WFL-90) was tested. However, taking into accountthe ultimate load increase difference between the L-WF-90 and L-WF-0specimens, it seems fair to predict that the L-WFL-90 specimen would exhibitan ultimate load increase larger than 19.8% — indeed, the numerical analysisof this column showed a 20.2% increase.

eigenvalue problem or the system of algebraic non-linearequations providing the post-buckling equilibrium paths. Inparticular, the following issues deserve to be mentioned:

(I) FE discretisation. In order to analyse both the local andglobal behaviours of a given thin-walled member, onemust adopt a two-dimensional model to discretise its mid-surface, a task that can be adequately performed by meansof 4-node isoparametric shell element with full integration(S4 elements in the ABAQUS nomenclature). In the caseof the lipped channel columns dealt with in this work, itwas found that it suffices to discretise the cross-sectioninto 36 FEs (12 in the web, 10 per flange and 1 per lip) —this corresponds roughly to adopting 10 mm-wide FEs. Inthe short columns, a 8 mm × 10 mm (length–width) meshsize was used, leading to a total of 1462 elements, 1505nodes and 8622 degrees of freedom. In the long columns, a30 mm×10 mm mesh size was adopted, which led to 2250elements, 2625 nodes and 15 342 degrees of freedom. Atthis stage, it is worth mentioning that no attempt was madeto model the round corners uniting the member walls.

(II) End support conditions. In order to be able tomake meaningful comparisons between numerical andexperimental results, it is essential to ensure an adequatemodelling of the member end support conditions. In thisparticular case, this was achieved by attaching rigid platesto the column end sections, thus preventing all their localand global displacements and rotations: (i) rigid-bodymotions (with the obvious exception of the loaded endsection axial translation), (ii) warping and (iii) in-planedeformation — see Fig. 9(a2)–(b2). These rigid end plateswere modelled by means of 3-node R3D3 finite elements(again ABAQUS nomenclature).

(III) Loading. The compressive load was applied at the axiallyfree end section centroid and, in order to obtain the loadvs. axial shortening equilibrium path, the correspondingaxial displacement was assessed by using an ABAQUS

command termed “MONITOR”.(IV) Steel modelling. The column (steel) material behaviour

was always assumed to be homogeneous and isotropicand two constitutive relations were considered to modelit, namely (i) a linear elastic law (bifurcation analyses)and (ii) a linear-elastic/perfectly-plastic law with nostrain hardening (post-buckling analysis). The linearelastic behaviour is fully characterised by the nominalor measured values of Young’s modulus (E) andPoisson’s ratio (ν). As for the elastic–plastic one, it isdescribed by the well-known Prandtl-Reuss model (J2-flow theory), which combines Von Mises’s yield criterionwith the associated flow rule. Both these models areavailable in the ABAQUS material behaviour library andtheir implementation merely involves just providing themeasured or nominal steel elastic constants and yieldstresses — recall that, for the short and long columns, thelatter are (i) fy = 610 MPa and fy = 521 MPa (measured)and (ii) fy = 550 MPa and fy = 450 MPa (nominal).

(V) CFS modelling. The cross-section walls having CFSattached were modelled as double-ply plates with one ply


made of steel and the other consisting of the CFS. In orderto be able to assess a possible CFS failure, and since theABAQUS material behaviour library does not include any(a priori defined) constitutive relation for FRP compositelaminates that accounts for material degradation, oneresorted to the commands “FAIL STRESS”, “MSTRS”and “TSAIH” to implement the Maximum Stress and/orTsai–Hill failure criteria11 — these criteria involve failureindexes IF , which are given respectively by (e.g., [26])

IF = max

{∣∣∣∣∣ σll

σ admll

∣∣∣∣∣ ;∣∣∣∣ σt t

σ admt t

∣∣∣∣ ;∣∣∣∣∣ τlt

τ admlt

∣∣∣∣∣}

(1)

IF =

(σll

σ admll

)2

−σllσt t

(σ admll )2

+

(σt t

σ admt t

)2

+

(τlt

τ admlt

)2

(2)

where σ admll , σ adm

t t and τ admlt are the CFS admissible

(maximum) normal longitudinal, normal transversal andshear stresses, with values equal to 4200 MPa, 30 MPaand 15 MPa, respectively (the compressive and tensilevalues are deemed identical). It is worth noting that (i)while the Maximum Stress criterion assumes no interactionbetween the three failure modes, (ii) the Tsai–Hill one maybe viewed as an extension of Von Mises’s yield criterionto orthotropic materials. It is assumed that, as long asthe failure index remains below the unit value (i.e., ifIF < 1), no CFS collapse occurs — i.e., such a collapseonly happens for IF ≥ 1.

(VI) Initial imperfections. Initial geometrical imperfectionscan be incorporated in an ABAQUS member post-buckling analysis either (i) manually, by directly inputtingan arbitrary initial deformed configuration, or (ii)automatically, through the a priori definition of alinear combination of normalised buckling mode shapes,yielded by a preliminary buckling analysis based ona finite element mesh identical to the one adopted toperform the post-buckling analysis [27]. Since no columninitial geometric imperfections were measured, the onesconsidered in this work were included automaticallyand consist of different linear combinations of the mostrelevant (critical) local-plate and distortional bucklingmode shapes, incorporated into the column initialgeometry by means of a specific ABAQUS command.This approach involved performing several preliminaryelastic non-linear analyses, each of them associatedwith initial imperfections that (i) combine equally thenormalised local-plate and distortional buckling modeshapes and (ii) have different overall amplitudes. Thecriterion adopted to select the appropriate imperfectionto include in a particular test simulation (geometricallyand physically non-linear analysis) was the “closeness”

11 Although the Tsai–Wu failure criterion is also available in the ABAQUSmaterial behaviour library, its application to the problem under considerationyields results that practically coincide with those obtained through the Tsai–Hillcriterion — this explains why the Tsai–Wu criterion was not considered in thiswork.

between the initial portions of the experimental andnumerical equilibrium paths P(u) — it is worth notingthat the imperfection amplitudes yielded by this approachwere found to agree fairly well with the ones obtainedby adopting the methodology proposed by Schaferand Pekoz [28]. For long columns, one includes inthe analyses local-plate and distortional imperfectionamplitudes similar to d1 = 0.66t (type 1) and d2 =

1.55t (type 2), values corresponding to a cumulativedistribution function P(∆ < d) = 0.75. For shortcolumns, one considers mainly local-plate imperfectionamplitudes similar to d1 = 1.98t (type 1), value associatedwith P(∆ < d) = 0.96. Finally, one last word tomention that no residual stresses were incorporated inthe numerical analyses, since it is current practice toneglect their influence on the post-buckling behaviourand ultimate strength of cold-formed steel members —indeed, several researchers found that, unlike in hot-rolled or welded steel members, the residual stresseffects are only marginal in cold-formed steel members(e.g., [29–31]).

(VII) Solution techniques. Performing a buckling analysisrequires solving an eigenvalue problem, defined bythe column (discretised) elastic and geometric stiffnessmatrices. This task is performed by means of either (i)the “sub-space iteration method” (ABAQUS analyses) or(ii) Galerkin’s method, adopting trigonometric functionsof the type ak sin(nπx/L) sin(πx/L), where n is thehalf-wave number, to approximate the deformation modek displacement field (GBT-based analyses). As for thenon-linear (post-buckling) equilibrium paths, relatingthe applied compressive load with the column axialshortening, they are obtained by means of an incremental-iterative technique that employs Newton–Raphson’smethod. Since one wishes to determine the columnelastic–plastic post-failure behaviour, the analyses adoptRiks’s arc-length control strategy — this is doneautomatically in ABAQUS, on the basis of pre-defined“calibration” increment and tolerance parameters.

3.2.1. Presentation, validation and discussion of the resultsThe ABAQUS buckling analyses yielded the critical load

values given in Tables 3 and 4 (PL and PD — local-plate anddistortional buckling). From the observation of these values, aswell as from their comparison with the GBT-based ones, thefollowing conclusions may be drawn:

(i) In the short columns, the local-plate buckling load values(26.8 kN ≤ PL ≤ 29.5 kN) are much smaller than theirdistortional counterparts (PD ≈ 130 kN). The comparisonbetween the PL values of the CFRP-strengthened and baresteel columns shows that the maximum increase (10.1%)is achieved for the S-WFL-0 strengthening configuration.

(ii) In the long columns, the local-plate buckling loads(84.5 kN ≤ PL ≤ 89.6 kN) are very similar tothe distortional ones (84.4 kN ≤ PD ≤ 90.3 kN).The comparison between the PL and PD values of theCFRP-strengthened and bare steel columns shows that the


Table 4Buckling and experimental/numerical ultimate loads of the long columns

Specimen Experimental — P (kN) Numerical — P (kN) PNum/PExpPExp PExp/PNIL − 1 (%) PD PD/PD.N I L − 1 (%) PL PL/PL .NIL − 1 (%) PNum PNum/PNIL − 1 (%)

L-NIL 85.2 –84.4 – 84.5 – 89.0 –

1.045L-NIL-R 86.7 1.8 1.027L-F-0 90.1 5.8 84.7 0.4 87.8 3.9 91.8 3.2 1.019L-F-90 92.6 8.7 85.5 1.3 86.9 2.8 93.7 5.3 1.012L-W-0 95.5 12.1 86.5 2.5 85.6 1.3 94.3 6.0 0.987L-W-90 88.1 3.4 89.0 5.5 87.2 3.2 93.5 5.1 1.061L-WF-0 98.6 15.7 86.7 2.7 88.9 5.2 98.4 10.6 0.998L-WF-45 – – 87.9 4.1 93.0 10.1 99.1 11.3 –L-WF-90 100.9 18.4 90.3 7.0 89.6 6.0 99.8 12.1 0.989L-WFL-0 102.1 19.8

88.3 4.6 89.3 5.7 102.5 15.21.004

L-WFL-0-R 100.4 17.8 1.021L-WFL-90 – – 90.5 7.2 89.7 6.2 107.0 20.2 –

Mean 1.016Sd. Dev. 0.024

Fig. 11. Buckling mode shapes of the (a) long and (b) short columns.

maximum increases (6.0% and 7.0%) are achieved forthe L-WF-90 column — this agrees with the preliminaryparametric study, which showed that the distortionalbuckling load increase is higher for transverse CFSstrengthening.

(iii) Fig. 11(a)–(b) show the buckling mode shapes of theshort and long columns — unlike the buckling loadvalues, these shapes do not vary with the strengtheningconfiguration. One notices that the short column bucklingmode is basically local-plate (with 5 half-waves), whilethe long column one combines local-plate (17 half-waves)and distortional (3 half-waves) buckling modes — thisinevitably leads to a post-buckling behaviour affected bylocal-plate/distortional mode interaction, a phenomenonthat currently interests the cold-formed steel technical andscientific communities (e.g., [32–34]).

(iv) The buckling loads yielded by the GBT-based analysesagree quite well with the ABAQUS ones: PL = 27.2 kNand PD = 130.2 kN (S-NIL), PL = 30.1 kN andPD = 132.0 kN (S-WFL-0), PL = 85.6 kN andPD = 85.7 kN (L-NIL), PL = 91.1 kN and PD =

88.2 kN (L-WFL-0) — a maximum difference of 2% wasfound. Fig. 12(a)–(b) provide the variation of the GBTmode amplitude functions concerning the deformation

modes participating in the long and short column bucklingmode shapes, namely modes 5 (distortional), 7 and 9(both local-plate), also shown also in Fig. 2 — thenumbers of longitudinal half-waves exhibited by the bothbuckling modes further confirm the ABAQUS results.Moreover, the comparison between Figs. 12(a) and 11(a)illustrates the remarkable similarity of the combined local-plate/distortional buckling mode shapes yielded by GBTand ABAQUS analyses — in particular, it is worth notingthe decreasing amplitude of the local-plate half-waveswhen one moves from the column mid-span towards itsfixed end supports (remember that, for simply supportedmembers, all the local-plate half-waves share a commonamplitude).

For the particular case of column S-WFL-0, Fig. 13(a) showsa comparison between the experimental non-linear equilibriumpath P(u) (applied load vs. axial shortening) and severalnumerical ones, yielded by (i) a GBT-based analysis (elasticbehaviour) and (ii) three ABAQUS shell finite element analyses,(elastic–plastic behaviour) — they differ in the constitutiverelation adopted, which may be either (ii1) the true non-linear


Fig. 12. GBT mode amplitude functions φk (x) concerning the buckling mode shapes of the (a) long and (b) short columns.

stress–strain curve depicted in Fig. 13(b)12 (dotted curve) or(ii2) a linear-elastic/perfectly-plastic relationship characterisedby the nominal (thin solid curve) or measured (dashed curve)yield stress ( fy) values. This comparison prompts the followingremarks:

(i) The GBT-based non-linear equilibrium path is close to theABAQUS ones for P < 50 kN = 1.67 Pcr (Pcr = 30 kN).The column deformed configuration in the post-bucklingrange is governed by modes 1 (axial), mode 7 (local-plate) and, to a much lesser amount, also by modes 5(distortional), 9 and 13 (local-plate). For P > 50 kN,plasticity progressively affects the column post-bucklingbehaviour and the GBT and ABAQUS equilibrium pathsnaturally diverge and evolve differently.

(ii) Concerning the two ABAQUS non-linear equilibrium pathsbased on linear-elastic/perfectly-plastic relationships, onenotices that the path corresponding to the nominal fy iscloser to the experimental P(u) curve than the measuredfy one. With respect to the maximum load value (PExp =

61.4 kN), this difference is even more clear: the load valueobtained with the measured fy (PMeas = 68.0 kN) is 7%greater than the nominal fy one (PNom = 63.4 kN). Thisfact, illustrated here for the column S-WFL-0, also takesplace for all the other columns (both short and long).

(iii) As for the ABAQUS non-linear equilibrium path basedon the true non-linear stress–strain curve (dotted line),one readily observes that the ultimate load value obtained(PTrue = 68.4 kN) is quite close to the one determinedfrom the measured fy one. Furthermore, the post-ultimateload path lies a bit above the nominal and measuredfy ones — this most likely stems from the slight strainhardening exhibited by the true stress–strain curve.

(iv) In view of what was mentioned in the two previousitems, and in order to obtain a better correlation with

12 In order to incorporate this non-linear stress–strain curve into the ABAQUSanalysis, it was necessary to approximate it by means of several straightsegments, on the basis of a selected set of measured stress and strain values.Moreover, the true stress–strain curve was obtained from the engineering one(they are both shown in Fig. 13(b)), as described, for instance, in [32].

Fig. 13. (a) Comparison between several equilibrium paths P(u) of theS-WFL-0 column and (b) curve providing the (engineering) stress–strainrelationship measured during the short column test.


Fig. 14. Numerical P(u) curves determined for the (a) short and (b) longcolumns.

the experimental results, the numerical results shown inthe remainder of this paper are obtained with linear-elastic/perfectly-plastic stress–strain curves and nominalyield stress values.

Fig. 14(a)–(b) show the P(u) curves provided by the ABAQUS

non-linear (post-buckling) analyses of the short and longcolumns — the corresponding ultimate loads PNum are given inTables 3 and 4. Note that one also includes curves concerning S-WF-45 and L-WF-45 columns (i.e., “diagonally strengthened”short and long columns), which were not experimentallytested — this is because the GBT-based buckling investigationpresented earlier in the paper showed that the α = 45◦ CFSprovide the most substantial local buckling load increases.After observing the results presented in these figures andtables, as well as the experimental P(u) curves presented inFig. 8(a)–(b), the following comments seem appropriate13:

(i) Despite buckling in local-plate modes, the short columnsexhibit distortional failure mechanisms — e.g., Fig. 9(a2)

13 At this stage, one must mention that the numerical analysis do not modeladequately the post-failure column behaviour, which are inevitably bound to be(sooner or later) affected by the rupture and/or debonding of the CFS.

shows the collapse mode of column S-F-0, and oneclearly observes the formation of a yield-line mechanisminvolving the plastification of the column central zone.Moreover, note also the remarkable similarity betweenthe experimental (Fig. 9(a1)) and numerical (Fig. 9(a2))deformed configurations of column S-F-0 at the onset ofcollapse.

(ii) As mentioned before, the long columns exhibit almostpractically coincidental local-plate and distortional criticalbuckling modes. Then, it is not surprising to observethat the corresponding failure mechanisms are of thedistortional-type — they are very localised at the columnmid-span zone, as shown in Fig. 9(b2) for column L-F-0.Once more, one underlines the amazing similarity betweenthe experimental (Fig. 9(b1)) and numerical (Fig. 9(b2))deformed configurations of column L-F-0 at the brinkof collapse. Note also the (a bit hard to distinguish)local buckles appearing in the web of this column,both in the numerical and experimental configurations— they provide clear evidence of the occurrence oflocal-plate/distortional mode interaction at failure, evenif the collapse mechanism is predominantly distortional(e.g., [33–35]). Finally, Fig. 10(b) shows the FE yield-linemechanism in the column central zone, where the CFRPsheets began detaching from the steel.

(iii) Like the experimental curves (see Fig. 8(a)), the shortcolumn numerical equilibrium paths exhibit proportionallimit loads that are quite close to the correspondingultimate strength values, thus meaning that there is littleaxial strength degradation (due to local buckling effects)prior to failure — this stiffness degradation occurs quiteabruptly, which replicates the “kinks” observed in theexperimental equilibrium paths. Furthermore, note also theductility differences amongst the various curves and thefact that the columns with the two higher ultimate loadsexhibit virtually no ductility at all (this same feature waspartially observed in the tests).

(iv) Conversely, and also confirming the test results (see thecurves in Fig. 8(b)), the long column proportional limitloads (≈70 kN) are far from their ultimate counterparts,which means that there is progressive and fairly smoothaxial strength degradation before collapse. Note also that,with a single exception (the L-WFL-0 column, havingthe higher ultimate load), all equilibrium paths exhibitvery steep descending branches immediately after failure,followed by curves with fairly moderate slopes. Thisfeature does not appear in most short column equilibriumpaths, which have more or less well defined plateauspreceding descending branches with reasonably highslopes.

(v) In both the short and long columns, the ultimate loadincreases, due to CFRP-strengthening, detected by thenumerical analyses are quite similar to those found in thecourse of the experimental investigation — Tables 3 and4 make it possible to quantify this similarity. Concerningthe relationship between the buckling load and ultimatestrength increases due to the presence of the CFS, it isworth noting that:


Fig. 15. Comparison between the numerical and experimental P(u) curves for columns (a) S-W-90+ L-F-0 and (b) S-WF-90+ S-WF-90-R+ L-WF-0.

(v.1) In the S-WF-0 and S-WFL-0 columns, the ultimateload increase is about 3% points higher than thebuckling load one.

(v.2) In the S-WF-90 column, the ultimate load andbuckling load increases are practically the same.

(v.3) In the L-WF-0, L-WF-90 and L-WFL-90 columns,the ultimate load increase is more than 6% pointshigher than the buckling load one — in the lastcolumn, it reaches about 10%.

(vi) The numerical ultimate load values (PNum) compare fairlywell with the experimental ones (PExp). Indeed, themean and standard deviation values of the PNum/PExpratio read 1.012 and 0.022 (short columns), and 1.016and 0.024 (long columns) — moreover, the maximumdifferences are 4.3% (short columns) and 6.1% (longcolumns). Fig. 15(a)–(b) make it possible to compare theexperimental and numerical equilibrium paths concerninga representative sample of the columns analysed, namelythe S-W-90, L-F-0, S-WF-90, S-WF-90-R and L-WF-0columns. One observes that:(vi.1) Despite the fact that the long column cross-section

area is about 50% larger (thicker walls) than theshort column one, the latter columns exhibit ahigher axial stiffness in the initial loading stages— this somewhat unexpected difference is due toa combination of much smaller lengths and initialgeometrical imperfections.

(vi.2) While the numerical and experimental descendingbranches of the L-F-0, S-W-90 and S-WF-90columns correlate fairly well, the ones concerningthe L-WF-90 column are qualitatively different— this is most likely due to debonding betweenthe steel and CFS, a phenomenon not capturedby the numerical analysis and providing a logicalexplanation for the steeper initial descending branchof the experimental curve.

(vii) Concerning the columns with α = 45◦ fibres (S-WF-45 and L-WF-45), the results confirm that this CFSconfiguration provides the higher local buckling loadincreases. However, the non-linear analyses of thesecolumns show that their ultimate loads experience no suchincrease — indeed, this occurs for the short (S-WF-45) andlong (L-WF-45) columns and is due to the fact that bothfail in distortional mechanisms (recall that distortionalbuckling load is not affected by α = 45◦ CFS). Forinstance, the S-WF-45 column ultimate load is even lowerthan their S-WF-0 and S-WF-90 column counterparts.

Finally, one devotes some attention to the mechanical behaviourof the CFS. Since the ABAQUS analyses provide informationabout their failure indexes IF.MS and IF.TH , associated withthe Maximum Stress and Tsai–Hill failure criteria given inEqs. (1) and (2),14 its is possible to use them to assesshow close from rupture are CFS at the different applied loadlevels. Fig. 16 shows the variation of the maximum mid-spanIF.MS and IF.TH values with the axial shortening u, for thesample of representative columns S-F-90, L-F-0 and L-W-0 —for comparison purposes, one also presents the dotted curvesP(u)/Pmax, which obviously never exceed 1.0. The observationof Fig. 16 prompts the following remarks:

(i) Regardless of the failure criterion adopted, index IFalways increases with the axial shortening u. However, thisincrease is much more pronounced after the ultimate loadis reached — in fact, IF never exceeds 30% before thecolumn collapse, which means that the CFS remain fullyeffective at that stage.

(ii) In column S-F-90, one has IF.MS = IF.TH due to the (ii1)strong predominance and (ii2) low admissible value of thelongitudinal normal stress σxx (recall that α = 90◦) —

14 Obviously, one has IF.MS < IF.TH , since IF.MS does not account for thestress component interaction (as does IF.TH ).


Fig. 16. Variation of the Maximum Stress and Tsai–Hill failure indexes with u (axial shortening) for columns (a) S-F-90, (b) L-F-0 and (c) L-W-0.

Fig. 17. Distribution of the Tsai–Hill failure index along the mid-span cross-section mid-line of column S-WFL-0.

then, the ratio σxx/σadmxx is very high for both criteria and

the stress component coupling is marginal. Since σ admxx is

low, both IF.MS and IF.TH increase steeply immediatelyafter failure — e.g., one has IF.MS = IF.TH = 1 foru = 1.33 mm and P = 51.3 kN < Pmax = 56.8 kN(about 10% applied load drop). IF.MS = IF.TH > 1indicates rupture of the CFS in the longitudinal direction(i.e., transversally to the fibres), due to an excessive σxx

value.

(iii) In columns L-F-0 and L-W-0, the IF.MS and IF.TH

curves differ, due to stress component coupling — bothσxx and σss are relevant (recall that α = 0◦), whichmeans that the values of the transverse normal stress ratioσss/σ

admss are no longer negligible and “compete” with

their σxx/σadmxx counterparts. Then, the more conservative

Tsai–Hill criterion indicates that, for these columns, theCFS rupture (IF.TH > 1) only occurs for an axialshortening value much larger than the one of the S-F-90column.

Fig. 18. Distribution of the Tsai–Hill failure index along the mid-span cross-section mid-line of column L-WFL-0.

(iv) For illustrative purposes, Figs. 17 and 18 display thevariation of IF.TH along the mid-span cross-section mid-lines (coordinate s) of the S-WFL-0 and L-WFL-0columns, for different applied load levels — only halfcross-section is shown (due to symmetry) and the thickersolid curve is associated with the column ultimate load.One can conclude that:(iv.1) Since the S-WFL-0 column IF.TH values are

governed by local buckling, they grow with theapplied load mainly in the flange (mostly) and webareas close to the web-flange corners. This is dueto the predominance of the transverse normal stressratio σss/σ

admss , stemming mainly from the across-

section transverse wall bending.(iv.2) Since the L-WFL-0 column failure is governed by

local-plate/distortional mode interaction, its IF.THvalues increase significantly with the applied loadlevel, both in the web (near the web-flange corners)and in the lips (near their free ends). In the formercase, this stems mostly from the relevance of the


transverse normal stresses σss associated with webbending. In the latter case, the key role is playedby the significant longitudinal normal stresses σxxthat arise from the warping displacements due to thedistortional post-buckling behaviour [35].

(iv.3) Taking into account the content of the previous twoitems, it might be worth considering strengtheningonly the flange and web zones close to the web-flange corners (short columns) and those two zonesplus the lips (long columns) — these might proveto be optimal CFS locations, in the sense that theymaximise the column ultimate strength increase,while minimising the amount of CFS employed(minimum cost/benefit ratio).

(v) On the basis of the previous remarks and recalling thatthe CFS rupture was only experimentally detected alongthe equilibrium path descending branches, it seems fair toconclude that the two-layer (steel-CFS) numerical modeladopted in this work (v1) simulates the column preand post-buckling behaviour reasonably well, and (v2)

provides very satisfactory ultimate strength estimates.(vi) Finally, just two words to mention that the load-carrying

capacity of the columns stems mostly from the steelprofiles, since only one very thin CFS was bonded totheir outer surfaces. Although the use of more CFSlayers looks appealing at first sight (it would boost thecolumn strength), it might also entail problems related todebonding at the adhesive interfaces prior to collapse, dueto the possible presence of high shear and peeling stresses(e.g., [7–11]).

4. Conclusion

This paper presented experimental and numerical resultsconcerning an investigation aimed at assessing how theCFRP-strengthening improves the non-linear behaviour andload-carrying capacity of cold-formed steel lipped channelcolumns. Since cold-formed steel lipped channel columns areoften prone to local-plate and distortional buckling, specialattention was devoted to these instability phenomena. Initially, apreliminary GBT-based study, involving the buckling behaviourof several lipped channel columns with different strengtheningconfigurations and fibre orientations, was reported — its resultsmade it possible to draw the following main conclusions:

(i) The most effective fibre orientation depends on thenature of the column critical buckling mode (local-plate,distortional or global), i.e., on the member length.

(ii) While the local-plate buckling behaviour is mostlyenhanced by using CFS with fibres oriented diagonally(α = 45◦), employing CFS with transverse fibres (α = 90◦)was shown to be the best option to improve the columndistortional buckling behaviour.

Next, the results of an experimental program involvingfixed-ended short and long lipped channel columns werepresented — the columns were strengthened with carbonFRP sheets bonded at different outer surface locations

(web, flanges and/or lips) and having fibres oriented eitherlongitudinally or transversally. The experimental resultsconsisted of (i) non-linear equilibrium paths (applied loadvs. axial shortening), (ii) ultimate strength values and (iii)experimental evidence of the failure mechanisms (virtuallyall of them exhibited a predominantly distortional nature).The experimental results were subsequently used to calibrateand validate a geometrically and physically non-linear two-layer (steel-CFS) shell finite element model to perform thenumerical simulation of the column tests. The numericalanalyses were carried out in the code ABAQUS, adopting (i)a linear-elastic/perfectly-plastic isotropic constitutive law todescribe the steel material behaviour and (ii) a linear elasticorthotropic stress–strain relation to model the FRP compositelaminates, which accounted for material degradation (throughthe Maximum Stress and/or Tsai–Hill failure criteria). Finally,on the basis of both the experimental and the numerical resultsobtained, the following relevant conclusions could be drawn:

(i) The comparison between the ultimate loads of theotherwise identical columns with and without CFRP-strengthening showed that the presence of the single CFSmay increase the load-carrying capacity by up to (i) 15%,for the short columns, and (ii) 20%, for the long columns.Moreover, it is expected that the addition of more than oneCFS will lead to more substantial stiffness and strengthincreases (even if debonding at the adhesive interfaces maypose problems).

(ii) Unlike the short columns, the long ones were foundto exhibit a progressive and smooth axial strengthdegradation prior to collapse, which is immediatelyfollowed by a sudden strength drop — this behaviour,which can be partially attributed to the occurrence of local-plate/distortional mode interaction, was detected in boththe experimental and numerical studies.

(iii) For the long columns, it was observed, both experimentallyand numerically, that strengthening the whole columnouter surface with transverse CFS (L-WFL-90) leads tothe highest ultimate load increase, amounting to more than20%.

(iv) For the short columns, however, the experimental andnumerical results appear to be slightly contradictory: while(iv1) strengthening the column web and flanges withtransverse CFS (S-WF-90) appears to be the most efficientstrategy, according to the experimental study (Pu increaseof about 15%), (iv2) the numerical investigation indicatesthat the best option is employing longitudinal CFS (S-WF-0) (Pu increase of about 12.5%).15 Note, however, thatboth results make it clear that lip strengthening plays has aminor influence on the short column ultimate strength (lessthan 2%).

(v) The numerical approach adopted to simulate the non-linearbehaviour of the CFRP-strengthened steel columns (two-layer steel-CFS model) yielded fairly good results con-cerning the column pre-collapse post-buckling behaviour,

15 Of course, it is always possible to resort to the “easy explanation” ofattributing this qualitative difference to the initial imperfections, which werenot measured in the tested column specimens.


ultimate strength value and failure mechanism — the meanand standard deviation values of the PNum/PExp ratio ob-tained were 1.012 and 0.022 (short columns), and 1.016and 0.024 (long columns). Moreover, the satisfaction ofthe Maximum Stress and/or Tsai–Hill failure, providing ameasure for the structural integrity of the CFS, ensured thevalidity of the numerical results up to the column collapse.However, this numerical model cannot simulate the col-umn post-ultimate behaviour adequately, due to the factthat it does not account for the CFRP debonding — arigorous modelling of this phenomenon can be achievedthrough the use of “interface elements” between the steeland CFRP layers, which would lead to much more com-plex and time-consuming analyses.

(vi) Since it was detected that the maximum CFS failure indexvalues occur in the vicinity of the web-flange corners(short and long columns) and close to the lip free ends(long columns only), future work should be devotedto investigate whether an optimal strengthening shouldinvolve only addition of CFS at those specific locations,thus minimising the cost/benefit ratio.

Acknowledgments

The authors are indebted to BlueScope Lysaght (Singapore)Pte. Ltd. for supporting this work, by supplying the cold-formed steel column specimens used in the experimentalinvestigation. Moreover, the assistance of Mr. Wai-Ching Koand Mr. Wing-Him Lei during the test program is also gratefullyacknowledged.

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non-linear behaviour and load-carrying capacity of cfrp-strengthened lipped channel steel columns

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