Nonlinear behaviour of built-up cold-formed steel section battened columns

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Nonlinear behaviour of built-up cold-formed steel section battened columns Mohamed Dabaon, Ehab Ellobody , Khaled Ramzy Department of Structural Engineering, Faculty of Engineering, Tanta University, Tanta, Egypt abstract article info Article history: Received 26 October 2014 Accepted 12 March 2015 Available online 28 March 2015 Keywords: Built-up columns Cold-formed steel FE modelling Nonlinear behaviour Steel structures design This paper discusses nonlinear behaviour and design of built-up cold-formed steel section battened columns. The built-up columns were pin-ended and consisted of two cold-formed steel channels placed back-to-back and were connected using batten plates. Nonlinear 3-D nite element models were developed to simulate the structural performance of the axially loaded columns. The nonlinear material properties of at and corner portions of the channels, initial geometric imperfections, actual geometries and boundary conditions were carefully considered in the models. The nite element models were veried against tests, recently conducted and reported by the au- thors, on the same form of construction. The column strengths, failure modes, deformed shapes at failure, load- lateral displacement and load-axial strain relationships were predicted from the nite element analyses and compared well against the test results. In addition, the validated nite element models were used to perform an extensive parametric study investigating different parameters affecting the behaviour of the columns com- prising different slenderness, column lengths, cross-section geometries, steel strengths, spacing between chan- nels and different batten plates spacing. Furthermore, the column strengths predicted in the parametric study were compared with design strengths calculated using the North American Specication, Australian/New Zealand Standard and European Code for cold-formed steel columns. © 2015 Elsevier Ltd. All rights reserved. 1. Introduction Cold-formed steel structures are commonly used in a wide range of structures mainly owing to its superior strength-to-self-weight ratio, ease of construction, and economic design. Major research develop- ments in cold-formed steel structures were summarised by Yu [1] and Hancock [2]. However, in a recent survey by the authors [35], it is found that numerous extensive experimental investigations were re- ported in the literature on cold-formed steel single columns having symmetrical or asymmetrical cross-sections. On the other hand, there is a lack in the tests carried out on built-up cold-formed steel section columns, with examples of main previous investigations on built-up columns [617] that focused on columns with hot-rolled steel sections. Therefore, the authors conducted a series of new tests [35] on built-up cold-formed steel section battened columns to augment experimental data on this form of construction. However, to date, there is no detailed nite element models reported in the literature on built-up cold-formed steel section battened columns leading to the current investigation. Finite element analysis of built-up cold-formed steel section battened columns can compensate the lack of test data on this form of construc- tion as well provide better understanding for the complex buckling behaviour of the built-up columns. This study uses a consistent and ef- cient nonlinear 3-D nite element modelling approach for analyzing the buckling behaviour of cold-formed steel columns [1820], stainless steel columns [2123] and composite columns [2426]. A review of the main ndings of the nite element modelling approach was previously pre- sented in [27,28]. The nonlinear material properties of at and corner portions of cross-sections, initial local and overall geometric imperfec- tions and residual stresses were carefully considered in the modelling approach. In all the studies [1828], the numerical results were com- pared against test results and good agreement has been achieved. The failure modes, deformed shapes at failure, load-displacement relation- ships and column strengths were predicted from the nite element anal- yses. In addition, the numerical results were compared with design strengths calculated using current codes of practice. However, the nite element approach was not extended to study the behaviour of built-up cold-formed steel section battened columns, which is credited to this study for the rst time. Behaviour of cold-formed steel sections is different from that of hot- rolled steel sections. Steel produced by hot rolling has usually a sharp yielding plateau and a steel yield stress dened by the level at which the stress-strain curve becomes horizontal. On the other hand cold- formed steels have reduced or cold-worked gradual yielding plateau. In addition cold-formed steel products are quite thin, which makes it very sensitive to initial local and overall geometric imperfections as well as residual stresses resulting from manufacturing and handling Journal of Constructional Steel Research 110 (2015) 1628 Corresponding author. Tel./fax: +20 40 3315860. E-mail address: [email protected] (E. Ellobody). http://dx.doi.org/10.1016/j.jcsr.2015.03.007 0143-974X/© 2015 Elsevier Ltd. All rights reserved. Contents lists available at ScienceDirect Journal of Constructional Steel Research

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Nonlinear behaviour of built-up cold-formed steel sectionbattened columns

Transcript of Nonlinear behaviour of built-up cold-formed steel section battened columns

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    r bndes. Nadepeennsd-aest results. In addition, the validated nite element models were used to perform

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    Journal of Constructional Steel Research 110 (2015) 1628

    Contents lists available at ScienceDirect

    Journal of Constructiosteel section battened columns leading to the current investigation.Finite element analysis of built-up cold-formed steel section battened

    columns can compensate the lack of test data on this form of construc-

    Behaviour of cold-formed steel sections is different from that of hot-rolled steel sections. Steel produced by hot rolling has usually a sharpyielding plateau and a steel yield stress dened by the level at whichTherefore, the authors conducted a series of new tests [35] on built-upcold-formed steel section battened columns to augment experimentaldata on this form of construction. However, to date, there is no detailednite elementmodels reported in the literature on built-up cold-formed

    strengths calculated using current codes of practielement approach was not extended to study thecold-formed steel section battened columns, whstudy for the rst time.symmetrical or asymmetrical cross-sections. On the other hand, thereis a lack in the tests carried out on built-up cold-formed steel sectioncolumns, with examples of main previous investigations on built-upcolumns [617] that focused on columns with hot-rolled steel sections.

    pared against test results and good agreement has been achieved. Thefailure modes, deformed shapes at failure, load-displacement relation-ships and column strengthswere predicted from the nite element anal-yses. In addition, the numerical results were compared with designtion as well provide better understanding fo

    Corresponding author. Tel./fax: +20 40 3315860.E-mail address: [email protected] (E.

    http://dx.doi.org/10.1016/j.jcsr.2015.03.0070143-974X/ 2015 Elsevier Ltd. All rights reserved.the authors [35], it isl investigations were re-l single columns having

    portions of cross-sections, initial local and overall geometric imperfec-tions and residual stresses were carefully considered in the modellingapproach. In all the studies [1828], the numerical results were com-found that numerous extensive experimentaported in the literature on cold-formed stee1. Introduction

    Cold-formed steel structures are costructures mainly owing to its superioease of construction, and economic dments in cold-formed steel structuresHancock [2]. However, in a recent suprising different slenderness, column lengths, cross-section geometries, steel strengths, spacing between chan-nels and different batten plates spacing. Furthermore, the column strengths predicted in the parametric studywere compared with design strengths calculated using the North American Specication, Australian/NewZealand Standard and European Code for cold-formed steel columns.

    2015 Elsevier Ltd. All rights reserved.

    y used in a wide range ofgth-to-self-weight ratio,ajor research develop-mmarised by Yu [1] and

    behaviour of the built-up columns. This study uses a consistent and ef-cient nonlinear 3-D nite element modelling approach for analyzing thebuckling behaviour of cold-formed steel columns [1820], stainless steelcolumns [2123] and composite columns [2426]. A review of the mainndings of the nite element modelling approach was previously pre-sented in [27,28]. The nonlinear material properties of at and cornerSteel structures design an extensive parametric study investigating different parameters affecting the behaviour of the columns com-FE modellingNonlinear behaviour

    lateral displacement and loacompared well against the tNonlinear behaviour of built-up cold-formbattened columns

    Mohamed Dabaon, Ehab Ellobody , Khaled RamzyDepartment of Structural Engineering, Faculty of Engineering, Tanta University, Tanta, Egypt

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

    Article history:Received 26 October 2014Accepted 12 March 2015Available online 28 March 2015

    Keywords:Built-up columnsCold-formed steel

    This paper discusses nonlineabuilt-up columnswere pin-econnected using batten plateperformance of the axially lochannels, initial geometric imin themodels. The nite elemthors, on the same form of cor the complex buckling

    Ellobody).steel section

    ehaviour and design of built-up cold-formed steel section battened columns. Thed and consisted of two cold-formed steel channels placed back-to-back andwereonlinear 3-D nite element models were developed to simulate the structurald columns. The nonlinear material properties of at and corner portions of therfections, actual geometries and boundary conditions were carefully consideredt models were veried against tests, recently conducted and reported by the au-truction. The column strengths, failure modes, deformed shapes at failure, load-xial strain relationships were predicted from the nite element analyses and

    nal Steel Researchthe stress-strain curve becomes horizontal. On the other hand cold-formed steels have reduced or cold-worked gradual yielding plateau.In addition cold-formed steel products are quite thin, which makes itvery sensitive to initial local and overall geometric imperfections aswell as residual stresses resulting from manufacturing and handling

  • Loading plate

    End batten plate901506

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    Batten platesy

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    D = 100 mm

    y1

    y1y1

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    ri = 4bb1

    bb

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    Intermediatebatten plate

    901006

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    L = 2210 mm

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    b = 30process. Current design codes of practice usually propose empirical for-mulas to account for initial geometric imperfections. Behaviour and de-sign of built-up cold-formed steel section battened columns is morecomplicated, compared with that of single cold-formed steel columns.This is attributed to presence of local and overall buckling, connectionsamong different components, interfaces among these components anddistribution of loads bending moments in the column components.Therefore assessment of current design codes predictions against niteelement results for built-up cold-formed steel section battened columnsis addressed and highlighted in this study.

    The main objective of this study is to develop nonlinear 3-D niteelement models, using ABAQUS software [29], simulating the struc-tural performance of axially loaded built-up cold-formed steel sec-tion battened columns. The nonlinear material properties of atand corner portions of the channels, initial geometric imperfections,actual geometries and boundary conditions were accounted for inthe models. The nite element models were veried against test re-sults of the same form of construction. The column strengths, failure

    x

    1505

    (a) Elevation

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    Fig. 1. Details and denition of symbols for the built-up cold-formed stee

    Table 1Material properties of cold-formed steels used in this study.

    Test material Position Measured

    Eo(GPa)

    0.2(MPa)

    u(MPa)

    u(%)

    Normal strengthsteel [3-5]

    Flat portion 210 310 436 24Corner 205 357 502 7

    High strengthsteel [18]

    Flat portion 208 500 530 11Corner 203 575 610 3modes, deformed shapes at failure, load-lateral displacement andload-axial strain relationships were predicted from the nite ele-ment analyses and compared against the test results. The veried -nite element models were used to perform parametric studieshighlighting the effects on the column behaviour and strengthowing to the change column slenderness, column lengths, cross-section geometries, steel strengths, spacing between channels anddifferent batten plates spacing. The nite element strengths were

    l section battened column test specimens investigated in this study.

    Fig. 2. General layout and test setup for the built-up cold-formed steel section battenedcolumn test specimens investigated in this study.

  • compared with design strengths calculated using the NorthAmerican Specication [30], Australian/New Zealand Standard [31]

    thickness and ri is the internal radius. The batten plates were welded to

    rigid plates were attached to both ends of the specimens to ensure uni-form load distribution over the two channels as shown in Fig. 1.

    Fig. 3. Atypical nite element mesh for a 3.52m built-up cold-formed steel section battened column.

    18 M. Dabaon et al. / Journal of Constructional Steel Research 110 (2015) 1628the channels using Shielded Metal Arc Welding process. Two loadingand European Code [32] for cold-formed steel columns.

    2. Summary of experimental investigation

    The nite element models developed in this study were veriedagainst the tests, recently conducted and reported by the authors in[35], on built-up cold-formed steel section battened columns shownin Fig. 1. The tests provided the column strengths, failure modes, load-axial shortening, load-lateral displacement and load-axial strain relation-ships. The built-up columns were compressed between pin-ended sup-ports and had a nominal column length (L) of 2210 mm. The built-upcolumns consisted of two cold-formed steel channels placed back-to-back at varying spacing. The two channels were connected using battenplates, with varying longitudinal spacing. The cold-formed steel channelsections were brake-pressed from at strips having a plate thickness of2mm and made of carbon steel. Each channel had dimensions(D b t ri), shown in Fig. 1, that are equal to 100 30 2 4mm, where D is the overall depth, b is the channel width, t is the plateFig. 4. Boundary condition and load appThe material properties of steel used in the test specimens were de-termined by tensile coupon tests. Longitudinal coupons were takenfrom the nished specimens belonging to the same batch of specimensas the column tests. The stress-strain relationship obtained from tensilecoupon tests reected the average behavior of thematerial through thethickness. The tensile coupon test specimens were taken from the cen-ter of the longitudinal direction of the at portion of the cross-sectiondepth of the nished untested specimens. The coupon specimen dimen-sions conformed to the Australian Standard AS 1391 [33] for the tensiletesting of metals using 12.5 mm wide coupons of gauge length 50 mmand a thickness of 2mm. Themeasuredmaterial properties are the stat-ic 0.2% (0.2) proof stress, the static tensile strength (u), as well as theinitial Youngsmodulus (Eo), and the elongation after fracture (u) basedon a gauge length of 50 mm. The measured yield and ultimate stresseswere 310 and 436 MPa, respectively, while, the initial Youngs moduluswas 210 GPa. The measured elongation after fracture (u) based on agauge length of 50 mmwas 24%. The material properties in the cornerof the channel section were extrapolated from the material propertiesof 1.5 mm thickness cold-formed steel plain angle column detailed inEllobody and Young [18], according to the measured 0.2% tensile proofstress (0.2) in the at portions of the channel section. The measuredlication on a 2.2m built-up column.

  • yield and ultimate stresses were 310 and 436 MPa, respectively, while,the initial Youngs modulus was 210 GPa. The aforementioned valueswere the averagemeasured values for the ve tensile coupon test spec-imens conducted and detailed in Ref. [5]. The measured and predictedmaterial properties of the at and corner coupons are summarized inTable 1. A hydraulic testing machine was used to apply compressiveaxial force to the test specimens. To prevent the exural bucklingabout x-axis, a lateral support was constructed and installed on themain testing machine, as shown in Fig. 2. The support prevented theoutside movement along the y-axis at the mid-height of the column.Prior to testing a theodolite was used tomeasure initial overall geomet-ric imperfections and to judge the direction of lateral displacementmeasurement. The axial compressive force was applied to the specimenand increased with displacement control. The displacement controlallowed the test to continue in the post-buckling range. Further detailsregarding the tests are found in Refs. [35].

    3. Finite element model

    3.1. General

    The nite element program ABAQUS [29] was used to simulate thebehaviour of the built-up cold-formed steel section battened columns.Themodel used themeasured geometry and nonlinearmaterial proper-ties of the cross-section. Finite element analysis for buckling requirestwo types of analyses. The buckling modes of the columns are estimat-ed, rst, through the Eigenvalue analysis. This is a linear elastic analysisperformed using the (*BUCKLE) procedure available in the ABAQUSlibrary with the load applied within the step. The Eigenvalue analysiswas performed for a number of buckling modes and the adequatebuckling mode predicted from Eigenvalue analysis was used. A load-

    displacement nonlinear analysis is, then, carried out. In this analysis,the initial imperfections, residual stresses and material nonlinearityare included. From this analysis, the ultimate loads, failuremodes, later-al displacements, axial strains and axial shortenings are determined.

    3.2. Finite element mesh and type

    The cold-formed steel channels, batten plates and the end loadingplates were dened individually. S4R shell element, available inABAQUS [29] element library, was used to model the slender cold-formed steel channels (chords) and batten plates. The S4R elementhas six degrees of freedom per node and provides an accurate solution

    19M. Dabaon et al. / Journal of Constructional Steel Research 110 (2015) 1628Fig. 5. Initial geometric imperfection modes (Eigenmode 1) for a 3.52m built-up column.

    (a) Overall buckling and (b) Local buckling.for most applications. The upper and lower end plates were modelledusing a ne mesh of three dimensional four-node bilinear rigid quadri-lateral shell elements (R3D4) available in ABAQUS [29] element library.Convergence studies were performed to determine the correct meshthat provides adequate accuracy and minimum computational time inmodeling the cold-formed steel built-up section battened columns. Itis found a nemesh size of 5 5mm(lengthwidth) provided reason-able accuracy in modelling the built-up section columns, with a nermesh was used at the corners as shown in Fig. 3. A reference pointwas dened at a distance perpendicular to the plane of each end plateand facing the center of the plate was selected to act as a referencepoint (RP) as shown in Fig. 4. The reference points dene the effectivelength of each specimen between the two hinged supports. The bound-ary conditions were assigned to the RPs, while the load was assigned tothe RP of the upper end plate. The upper loading plate is constrainedwith the cold-formed steel channels by Tie constraint, to ensure thatthe displacements and rotations of the connected elements were keptthe same in the whole loading process. Also, tie-constraint can be usedin ABAQUS [29] to ensure that three-dimensional shell meshes arecoupled automatically to three-dimensional shell meshes.

    3.3. Material modelling

    The measured stress-strain curves of at and corner portions of thechannels [35] were incorporated in the nite element models. Theelastic-plastic material model provided by ABAQUS [29] allows for anonlinear stress-strain curve to be used. The rst part of the nonlinearcurve represents the elastic part up to the proportional limit stresswith measured Youngs modulus (Eo), and Poissons ratio in the elasticpart was set to 0.3. Since the analysis of post-buckling involves large in-elastic strains, the nominal (engineering) static stress-strain curve wasconverted to a true stress and logarithmic plastic strain curve. The truestress true and plastic true strain truepl , as required by ABAQUS [29],were calculated using Eqs. (1) and (2):

    true 1 1

    pltrue ln 1 trueE

    2

    Table 2Comparison of test and nite element results for the built-up cold-formed steel sectionbattened columns investigated in this study.

    Specimen Test results FE results Test/FE

    PTest.(kN)

    Failuremode

    PFE(kN)

    Failuremode

    PTest./PFE

    B2B25-300 109.9 F 107.6 F 1.02B2B50-300 119.1 F + L 121.7 F + L 0.98B2B75-300 125.3 L 126.9 L 0.99B2B50-150 133.1 F 132.65 F 1.00B2B50-400 112.3 L 111.5 L 1.01Mean --- --- --- --- 1.00COV --- --- --- --- 0.015

  • where and are themeasured nominal (engineering) stress and strainvalues. The nonlinear plastic hardeningmodel assumes an initial isotro-pic yield surface described by the Von Misses criterion and expands, orhardens, isotropically.

    3.4. Boundary conditions and load application

    The built-up cold-formed steel section battened columns investi-gated in this study were pin-ended columns. In order to simulate theupper and lower pin-end supports, the displacements and rotations(boundary conditions) were assigned to the upper and lower Refer-ence Points (RPs), respectively, as shown in Fig. 4. The lateral supportswere simulated by assigning boundary conditions to prevent thelateral movement of points at the half length of column. The load wasapplied to the RP of the upper end plate. The load was applied in incre-ments using the modied RIKS method available in the ABAQUS [29]library.

    3.5. Modelling of initial local and overall geometric imperfections

    Local andoverall buckling behaviour of built-up cold-formed steel sec-tion battened columns depends on many factors comprising channeldepth-thickness ratio (D/t), channel width-thickness ratio (b/t), localbuckling length around minor individual channel axis (Lz), local slender-ness around axis y1 (Lz/ry1) where ry1 is the radius of gyration about y1axis, overall slenderness around x and y axis (rx and ry) and batten plates

    geometry and its connections with the channels. Both initial local andoverall geometric imperfections are found in columns as a result offabrication and transportation processes. Hence superposition of localbucklingmodes as well as overall bucklingmodeswithmeasuredmagni-tudes is recommended for accurate nite element analysis. The bucklingmodes can be obtained by carrying Eigenvalue analysis of the columnwith very small and very large thickness to ensure local and overall buck-ling occurs, respectively, for a built-up column. The shape of a local buck-ling mode as well as overall buckling mode is the lowest buckling mode(Eigenmode 1) in the Eigen Value buckling linear analysis provided byABAQUS [29]. This technique is used to model the initial local and overallgeometric imperfections of the built-up columns investigated in thisstudy since it has been adopted successfully in previous studies detailedin [1820]. The technique also accounts for the actual number of battenplates, relative rigidity of batten plates tomainmember, combined bend-ing and shear rigidity of batten plates and the actual battened column ge-ometries, which affect the elastic buckling of the columns as detailed byChen and Li [34]. For the tested specimens, the average measured overallimperfections were found to be 1/1100 of the specimen length [35]. Thelocal imperfectionswere taken as 0.5% of the channel thickness as recom-mended in [18]. Since all buckling modes predicted by ABAQUS [29]Eigenvalue analysis are generalized to 1.0, the buckling modes arefactored by the measured magnitudes of the initial local and overall geo-metric imperfections. Fig. 5 shows the unfactored local and overall imper-fection buckling modes for a 3.52m built-up cold-formed steel sectionbattened column used in the tests [35].

    Nu

    20 M. Dabaon et al. / Journal of Constructional Steel Research 110 (2015) 1628(a) Experimental (b)

    Fig. 6. Deformed shape of built-up columerical (c) Close-up

    mn B2B25-300 failing by F mode.

  • 3.6. Modelling of residual stresses

    Residual stresses can be incorporated into the nite element modelas initial state using the ABAQUS (*INITIAL CONDITIONS, TYPE =STRESS) option. However, previous studies detailed in [1820] haveshown that it has a negligible effect on the column strength, stiffnessof the column, load-axial shortening behaviour and failure modes.Therefore, residual stresses were not included in the model to avoidthe complexity of the analysis.

    4. Verication of nite element model

    The nite element models developed in this study for the built-upcold-formed steel section battened columns were veried against thetests conducted and detailed by the authors in [35]. The columnstrengths, failure modes, deformed shapes at failure, load-lateral dis-placement and load-axial strain relationships predicted numericallywere compared against that measured experimentally as summarisedin Table 2 and presented in Figs. 6-13. The column strengths obtainedin the tests (PTest) and nite element analysis (PFE) were compared, asshown in Table 2 as well as presented in Fig. 13, and good agreementwas achieved. The mean value of the (PTest./PFE) ratio is 1.00 with thecorresponding coefcient of variation (COV) of 0.015, as shown inTable 2 and presented in Fig. 13.

    Twomodes of failurewere predicted numerically and conrmed ex-perimentally for the built-up cold-formed steel section battened col-umns investigated in this study as summarised in Table 2. The failuremodes were exural buckling (F) and local buckling (L), which was

    observed in the channel anges between batten plates. Looking atTable 2, which compared the experimental and numerical failuremodes, it can be seen that specimen B2B25-300 and B2B50-150 failedmainly by exural buckling failure mode F, specimens B2B50-400 andB2B75-300 failed mainly by local buckling failure mode L and specimenB2B50-300 failed by a combined F + L failure mode. The experimentaland numerical failure modes showed logical and expected behaviourwith slender built-up column B2B25-300 experienced elastic exuralbuckling. As the spacing between channels increased from 25 to50mm in specimen B2B50-300, the column slenderness decreased andcollapsemechanismwasdriven by combinedexural and local bucklingfailure modes. By increasing the spacing between channels again from50 to 75mm in specimen B2B50-300, failure was dominated by localbuckling. Similar behaviour was observed for specimens B2B50-150and B2B50-400, with local buckling lengths (Lz) 150 and 400mm, re-spectively. As the local buckling length increased, failure mode wasmainly driven by the slenderness of a single channel rather than theoverall slenderness and L failure mode occurred. Looking at Fig. 6,which shows the deformed shapes at failure obtained from the testsand nite element analysis for the built-up column B2B25-300, it canbe seen that the column failedmainly owing to elastic exural buckling.It can also be seen that there is a good agreement between both exper-imental and numerical deformed shapes. The deformed shapes showedthat the column buckled around y-y axis, which is coincident with elas-tic buckling mode shown in Fig. 5. The experimental and numerical de-formed failure modes were also compared for built-up column B2B50-300 as shown in Fig. 7. The column failed by a combined F + L mode.Once again, it can be seen that there is a good agreement between

    (

    21M. Dabaon et al. / Journal of Constructional Steel Research 110 (2015) 1628(a) Experimental

    Fig. 7. Deformed shape of built-up columb) Numerical (c) Close-up

    n B2B50-300 failing by F + L mode.

  • 22 M. Dabaon et al. / Journal of Constructional Steel Research 110 (2015) 1628both experimental and numerical deformed shapes. The deformedshapes showed that the column buckled around y-y axis and also expe-rienced ange local buckling near the upper end batten plates. Finally,Fig. 8 showed a comparison between the experimental and numericaldeformed failure modes for built-up column B2B75-300. The columnfailed mainly by an L mode. It can be seen that there the experimentaland numerical deformed shapeswere in good agreement. Thedeformedshapes showed that the column anges buckled locally near the columnquarter points.

    Figs. 911 plotted examples of the load-lateral displacement rela-tionships obtained experimentally and numerically at the mid-lengthpoint for columns (B2B25-300 and B2B50-300) and at the quarter-length point for column (B2B50-300), respectively. Generally, it can beseen from Figs. 911 that there is a good agreement between the ex-perimental and numerical curves before and after buckling occurrence.

    (a) Experimental (Fig. 8. Deformed shape of built-up colu

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    Fig. 9. Load-lateral displacement at mid-length point for column B2B25-300.The ultimate load and corresponding lateral displacements obtainedfrom nite element analysis for columns B2B25-300, B2B50-300 andB2B50-300 were (107.6 kN and 6.1 mm), (121.7 kN and 3.3 mm) and(121.7 kN and 2.1 mm), respectively, compared with (109.9 kN and12.7mm), (119.1 kN and 4.1 mm) and (119.1 kN and 2.5 mm), respec-tively, measured in the tests.

    Finally, the load-axial strain relationships obtained experimentallyand numerically for built-up column B2B25-300 were also compared,as an example as shown in Fig. 12. The relationships were plotted atthemid-length point at the side in compression of the column. General-ly, it can be seen from Fig. 12 that there is a good agreement betweenthe experimental and numerical curves, which were approximately lin-ear up to 82% of the ultimate load. The ultimate load and correspondingaxial strain (Micro strain) obtained from nite element analysis for

    b) Numerical (c) Close-upmn B2B75-300 failing by L mode.

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    Fig. 10. Load-lateral displacement at mid-length point for column B2B50-300.

  • summarising the nite element analysis results for the built-up coldformed steel section (C2) battened columns, it can be seen that built-up

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    Fig. 11. Load-lateral displacement at quarter point for column B2B50-300.

    110Test/FE = 0.98

    23M. Dabaon et al. / Journal of Constructional Steel Research 110 (2015) 1628columnB2B25-300were (107.6 kNand 982.62), respectively, comparedwith (109.9 kN and 1297.7) measured in the tests.

    5. Parametric study

    The veried nite element models were used to perform an exten-sive parametric study highlighting the effects of different parameterson the behaviour and strength of built-up cold-formed steel section bat-tened columns. A total of 60 built-up cold-formed steel section battenedcolumns were analysed in the parametric study. The built-up columnswere divided to eight series C1B25, C1B50, C1B75, C1B100, C2B50,C2B100, C2B150 and C2B200. The parameters investigated comprisedifferent (B1/D) ratios, different buckling length (Lz) and differentcolumn lengths. Column C1B50L3000-180 denotes a built-up cold-formed steel section battened column having two channels of (C1) sec-tion, placed back-to-back at a spacing of 50mm (B50), column length(L3000) and local buckling length Lz of 180mm, which is the numberafter the dash. The back-to-back channel spacing-to-overall channeldepth (B1/D) ratios varied from 0.251.0. Two cold-formed plain chan-nels (chords) (C1 and C2) were considered in this study. Channel C1had dimensions (D b t ri), see in Fig. 1, that are equal to100 30 2 1 mm, where D is the overall depth, b isthe channel width, t is the plate thickness and ri is the internal radius.Channel C2 had the dimensions (D b t ri) that are equal to200 80 2.5 6mm. The intermediate batten plates had dimensions(bb ab tb), see Fig. 1 and Tables 3-4, while the end batten plates haddimensions (bb1 ab tb). The cold formed steel considered in the para-metric study varied from normal strength steel tested by the authors in[35] to high strength steel used by Ellobody and Young [18], respective-ly for Channels C1 and C2. The measured and predicted material proper-ties of at and corner portions of the channels were summarised inTable 1. The built-up column lengths varied from 600-9760mm, whilethe local buckling lengths (Lz), see Fig. 1, varied from 1501216mm.0

    20

    40

    60

    80

    100

    120

    140

    0 200 400 600 800 1000 1200 1400

    TestFEM

    Load

    (kN)

    Axial strain (Micro strain)

    Fig. 12. Load-axial strain at mid-length point for the built-up column B2B25-300.The maximum initial overall geometric imperfection magnitude wastaken as 1/1100 of column length for the columns having lengths lessthan 3000mm, 1/1500 of column length for the columns having lengthsfrom 3000-5000mm and 1/2000 of column length for the columns hav-ing lengths more than 5000mm. The local imperfections were taken as0.5% of the channel thickness as recommended in [18].

    The column strengths (PFE) and failure modes obtained from the -nite element analyses for the built-up cold-formed steel section bat-tened columns investigated in the parametric study are summarisedin Tables 5 and 6. Looking at Tables 5 and 6 it can be seen that the col-umn local and overall slenderness have clearly identied the failuremodes of the built-up columns. Two slenderness valuesweremonitoredin this study to judge the built-up column buckling behaviour as well asthe failure mode as summarised in Tables 5 and 6. The rst slendernessis the nondimensional critical slenderness (c) calculated using theNorth American Specication [30] and Australian/New Zealand Standard[31]. While, the second slenderness is the nondimensional critical slen-derness () calculated according to the European Code [32]. Looking atTable 5 summarising the nite element analysis results for the built-upcold formed steel section (C1) battened columns, it can be seen thatbuilt-up columns having c 0.77 failed mainly by L failure mode,built-up columns having c 1.15 failed mainly by F failure mode andthe remaining built-up columns failed by a combined F + L failuremode except for column C1B50L3000-600 having Lz = 600mm. Built-up column C1B50L3000-600 had c of 1.41 and failed by F + L since theincreased local buckling length resulted in local ange buckling in parallelwith overall exural buckling. Similarly, it can be seen that built-up col-umns having 0.6 failed mainly by L failure mode, built-up columnshaving c 0.99 failed mainly by F failure mode and the remainingbuilt-up columns failed by a combined F + L failure mode except forcolumn C1B50L3000-600. On the other hand, looking at Table 6

    100100 110 120 130 140

    Test/FE = 1.0Test/FE = 1.02

    PFE (kN)

    Fig. 13. Verication of the nite element model.120

    130

    140

    Test resultsP Tes

    t (kN)columns having c 0.88 failed mainly by L failure mode, built-up col-umns having c 1.65 failedmainly by F failuremode and the remainingbuilt-up columns failed by a combined F+L failuremode, once again, ex-cept for some columns with higher Lz. Similarly, it can be seen that built-up columns having 0.55 failedmainly by L failure mode, built-up col-umns having c 1.25 failedmainly by F failuremode and the remainingbuilt-up columns failed by a combined F + L failure mode.

    6. Design rules

    6.1. General

    The built-up column strengths predicted from the parametric study(PFE) were compared with the unfactored design strengths calculatedusing the North American Specication (PNAS) [30], Australian/New

  • Table 3Details and dimensions of built-up cold-formed steel section battened column having a channel section C1.

    Series Specimen Channel cross-section L B1 B1/D Lz Batten plates

    D b t ri ab bb bb1 tb

    (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm)

    C1B25 C1B25L600-300 100 30 2 1 600 25 0.25 300 65 100 150 6C1B25L1000-300 100 30 2 1 1000 25 0.25 300 65 100 150 6C1B25L1400-300 100 30 2 1 1400 25 0.25 300 65 100 150 6C1B25L2200-300 100 30 2 1 2200 25 0.25 300 65 100 150 6C1B25L3000-300 100 30 2 1 3000 25 0.25 300 65 100 150 6C1B25L4200-300 100 30 2 1 4200 25 0.25 300 65 100 150 6C1B25L5000-300 100 30 2 1 5000 25 0.25 300 65 100 150 6

    C1B50 C1B50L600-300 100 30 2 1 600 50 0.5 300 90 100 150 6C1B50L1000-300 100 30 2 1 1000 50 0.5 300 90 100 150 6C1B50L1400-300 100 30 2 1 1400 50 0.5 300 90 100 150 6C1B50L2200-150 100 30 2 1 2200 50 0.5 150 90 100 150 6C1B50L2200-300 100 30 2 1 2200 50 0.5 300 90 100 150 6C1B50L2200-400 100 30 2 1 2200 50 0.5 400 90 100 150 6C1B50L3000-180 100 30 2 1 3000 50 0.5 180 90 100 150 6C1B50L3000-300 100 30 2 1 3000 50 0.5 300 90 100 150 6C1B50L3000-600 100 30 2 1 3000 50 0.5 600 90 100 150 6C1B50L4200-300 100 30 2 1 4200 50 0.5 300 90 100 150 6C1B50L5000-300 100 30 2 1 5000 50 0.5 300 90 100 150 6

    C1B75 C1B75L600-300 100 30 2 1 600 75 0.75 300 115 100 150 6C1B75L1000-300 100 30 2 1 1000 75 0.75 300 115 100 150 6C1B75L1400-300 100 30 2 1 1400 75 0.75 300 115 100 150 6C1B75L2200-300 100 30 2 1 2200 75 0.75 300 115 100 150 6C1B75L3000-300 100 30 2 1 3000 75 0.75 300 115 100 150 6C1B75L4200-300 100 30 2 1 4200 75 0.75 300 115 100 150 6C1B75L5000-300 100 30 2 1 5000 75 0.75 300 115 100 150 6

    C1B100 C1B100L600-300 100 30 2 1 600 100 1 300 140 100 150 6C1B100L1000-300 100 30 2 1 1000 100 1 300 140 100 150 6C1B100L1400-300 100 30 2 1 1400 100 1 300 140 100 150 6C1B100L2200-300 100 30 2 1 2200 100 1 300 140 100 150 6C1B100L3000-300 100 30 2 1 3000 100 1 300 140 100 150 6C1B100L4200-300 100 30 2 1 4200 100 1 300 140 100 150 6C1B100L5000-300 100 30 2 1 5000 100 1 300 140 100 150 6

    Table 4Details and dimensions of built-up cold-formed steel section battened column having a channel section C2.

    Series Specimen Channel cross-section L B1 B1/D Lz Batten plates

    D b t ri ab bb bb1 tb

    (mm) (mm) (mm) B1/D (mm) (mm) (mm) (mm) (mm)

    C2B50 C2B25L3520-384 200 80 2.5 6 3520 50 0.25 384 170 240 320 6C2B25L3520-800 200 80 2.5 6 3520 50 0.25 800 170 240 320 6C2B25L5600-800 200 80 2.5 6 5600 50 0.25 800 170 240 320 6C2B25L7680-320 200 80 2.5 6 7680 50 0.25 320 170 240 320 6C2B25L7680-800 200 80 2.5 6 7680 50 0.25 800 170 240 320 6C2B25L7680-1216 200 80 2.5 6 7680 50 0.25 1216 170 240 320 6C2B25L9760-800 200 80 2.5 6 9760 50 0.25 800 170 240 320 6

    C2B100 C2B50L3520-384 200 80 2.5 6 3520 100 0.5 384 220 240 320 6C2B50L3520-800 200 80 2.5 6 3520 100 0.5 800 220 240 320 6C2B50L5600-800 200 80 2.5 6 5600 100 0.5 800 220 240 320 6C2B50L7680-320 200 80 2.5 6 7680 100 0.5 320 220 240 320 6C2B50L7680-800 200 80 2.5 6 7680 100 0.5 800 220 240 320 6C2B50L7680-1216 200 80 2.5 6 7680 100 0.5 1216 220 240 320 6C2B50L9760-800 200 80 2.5 6 9760 100 0.5 800 220 240 320 6

    C2B150 C2B75L3520-384 200 80 2.5 6 3520 150 0.75 384 270 240 320 6C2B75L3520-800 200 80 2.5 6 3520 150 0.75 800 270 240 320 6C2B75L5600-800 200 80 2.5 6 5600 150 0.75 800 270 240 320 6C2B75L7680-320 200 80 2.5 6 7680 150 0.75 320 270 240 320 6C2B75L7680-800 200 80 2.5 6 7680 150 0.75 800 270 240 320 6C2B75L7680-1216 200 80 2.5 6 7680 150 0.75 1216 270 240 320 6C2B75L9760-800 200 80 2.5 6 9760 150 0.75 800 270 240 320 6

    C2B200 C2B100L3520-384 200 80 2.5 6 3520 200 1 384 320 240 320 6C2B100L3520-800 200 80 2.5 6 3520 200 1 800 320 240 320 6C2B100L5600-800 200 80 2.5 6 5600 200 1 800 320 240 320 6C2B100L7680-320 200 80 2.5 6 7680 200 1 320 320 240 320 6C2B100L7680-800 200 80 2.5 6 7680 200 1 800 320 240 320 6C2B100L7680-1216 200 80 2.5 6 7680 200 1 1216 320 240 320 6C2B100L9760-800 200 80 2.5 6 9760 200 1 800 320 240 320 6

    24 M. Dabaon et al. / Journal of Constructional Steel Research 110 (2015) 1628

  • el se

    PNAS

    (kN

    15814713188.451.226.919.116115715013713112410910483.763.145.416316015614613110281.4163162159153

    25M. Dabaon et al. / Journal of Constructional Steel Research 110 (2015) 1628Table 5Comparison of nite element results and design strengths for the built-up cold-formed ste

    Series Specimen FE Results

    PEF. Failure mode

    (kN)

    C1B25 C1B25L600-300 157.2 LC1B25L1000-300 156.5 LC1B25L1400-300 130.2 F + LC1B25L2200-300 90.8 FC1B25L3000-300 57.7 FC1B25L4200-300 31.4 FC1B25L5000-300 22.7 F

    C1B50 C1B50L600-300 155.9 LC1B50L1000-300 156.9 LC1B50L1400-300 154.5 LC1B50L2200-150 137.2 F + LC1B50L2200-300 136.7 F + LC1B50L2200-400 131.1 F + LC1B50L3000-180 120.5 FC1B50L3000-300 112.4 FC1B50L3000-600 86.3 F + LC1B50L4200-300 69.4 FC1B50L5000-300 51.5 F

    C1B75 C1B75L600-300 153.2 LC1B75L1000-300 158.3 LC1B75L1400-300 158.7 LC1B75L2200-300 156.2 LC1B75L3000-300 144.3 F + LC1B75L4200-300 110.0 FC1B75L5000-300 86.9 F

    C1B100 C1B100L600-300 151.1 LC1B100L1000-300 151.8 LC1B100L1400-300 151.7 LC1B100L2200-300 150.8 LZealand Standard (PAS/NZS) [31] and European Code (PEC3) [32] for cold-formed steel columns. Tables 5 and 6 compare the nite elementstrengths and design strengths of the built-up columns cold-formedsteel section battened columns investigated in the parametric study.In addition, Figs. 14-21 plotted the nite element strengths and designstrengths of the built-up columns.

    6.2. Design rules specied in the NAS Specication and AS/NZS Standard

    The unfactored design strength of concentrically loaded compres-sion members calculated using the NAS Specication and AS/NZS Stan-dard (PNAS&AS/NZS) is calculated as follows:

    PNAS&AS=NZS Ae Fn 3

    Where, Ae is the effective area and Fn is the critical buckling stress.The critical buckling stress (Fn) can be calculated as follows:

    For c 1.5

    Fn 0:6582c

    Fy 4

    For c N 1.5

    Fn 0:8772c

    Fy 5

    C1B100L3000-300 150.2 L 143C1B100L4200-300 128.9 F + L 125C1B100L5000-300 111.0 F 109

    Mean --- --- --- ---COV --- --- --- ---ction battened columns having a channel section C1.

    Design strengths Design / FE Results

    &AS/NZS c PEC3 PNAS&AS=NZSP FE

    PEC3P FE

    ) (kN)

    .0 0.55 152.8 0.33 1.01 0.97

    .0 0.72 142.0 0.54 0.94 0.91

    .4 0.92 122.9 0.76 1.01 0.941.36 78.9 1.20 0.97 0.871.81 49.0 1.63 0.89 0.852.50 27.1 2.28 0.85 0.862.96 19.7 2.72 0.84 0.86

    .8 0.48 152.8 0.21 1.04 0.98

    .2 0.56 152.8 0.35 1.00 0.97

    .4 0.67 146.6 0.49 0.97 0.95

    .2 0.85 122.6 0.76 1.00 0.89

    .1 0.93 122.6 0.76 0.96 0.90

    .9 1.00 122.6 0.76 0.95 0.94

    .5 1.15 93.7 1.04 0.91 0.78

    .5 1.20 93.7 1.04 0.93 0.831.41 93.7 1.04 0.97 1.091.63 58.8 1.46 0.91 0.851.92 44.0 1.74 0.88 0.86

    .1 0.45 152.8 0.15 1.06 1.00

    .6 0.50 152.8 0.25 1.01 0.97

    .9 0.57 152.8 0.35 0.99 0.96

    .1 0.73 141.0 0.56 0.94 0.90

    .5 0.92 123.1 0.76 0.91 0.85

    .4 1.22 91.6 1.06 0.93 0.831.43 72.9 1.27 0.94 0.84

    .6 0.44 152.8 0.12 1.08 1.01

    .1 0.47 152.8 0.20 1.07 1.01

    .8 0.52 152.8 0.28 1.05 1.01

    .0 0.63 149.7 0.44 1.01 0.99Where c = nondimensional critical slenderness calculated asfollows:

    c Fy=Fe

    q6

    where, Fy is yield stresswhich is equal to the 0.2%proof stress (0.2) and Feis least of the elastic exural, torsional, and exural torsional bucklingstress determined in accordance with Sections C4.1.1C4.1.2 of the NASSpecication and Sections 3.4.13.4.4 of the AS/NZS Standard. All the cal-culations were based on the modied slenderness ratio, specied for thedesign of built-up compression members, which is calculated as follows:

    KLr

    m

    KLr

    2o Lz

    ry1

    !2vuut 7where, (KL/r)m is the modied slenderness ratio of built-up column,(KL/r)o is the overall slenderness ratio of entire section about built-upmember axis, Lz is the local buckling length about the minor axis of anindividual built-up section component that is taken as the intermediatefastener or spot weld spacing, see Fig. 1, and ry1 is the minimum radiusof gyration of full unreduced cross-sectional area of an individual built-up section component.

    6.3. Design rules specied in the EC3 Code

    The unfactored design strength of concentrically loaded compres-sion members calculated using the EC3 Code (PEC3) is also dependenton the effective section area. Table 5.2 of the EC3 (BS EN1993-1-1[35]) was used to classify the cross-sections, which were of Class 4.

    .6 0.77 137.9 0.60 0.96 0.92

    .3 0.99 115.6 0.83 0.97 0.90

    .7 1.15 98.9 0.99 0.99 0.89--- --- --- 0.97 0.92--- --- --- 0.063 0.076

  • Table 6Comparison of nite element results and design strengths for the built-up cold-formed steel section battened columns having a channel section C2.

    Series Specimen FE Results Design strengths Design / FE Results

    PEF. Failure mode PNAS&AS/NZS c PEC3 PNAS&AS=NZSP FE

    PEC3P FE

    (kN) (kN) (kN)

    C2B50 C2B25L3520-384 272.9 F + L 333.6 1.11 359.0 0.83 1.22 1.32C2B25L3520-800 273.3 F + L 313.7 1.20 359.0 0.83 1.15 1.31C2B25L5600-800 200.6 F 183.6 1.80 212.2 1.32 0.92 1.06C2B25L7680-320 151.9 F 118.7 2.38 126.9 1.81 0.78 0.83C2B25L7680-800 134.5 F 115.2 2.42 126.9 1.81 0.86 0.94C2B25L7680-1216 128.3 F 110.2 2.49 126.9 1.81 0.86 0.99C2B25L9760-800 92.2 F 78.1 3.06 82.7 2.30 0.85 0.90

    C2B100 C2B50L3520-384 387.2 L 405.8 0.79 432.9 0.57 1.05 1.12C2B50L3520-800 354.2 F + L 381.5 0.90 432.9 0.57 1.08 1.22C2B50L5600-800 256.4 F + L 291.0 1.29 333.1 0.91 1.13 1.30C2B50L7680-320 243.6 F 210.5 1.65 230.3 1.25 0.86 0.95C2B50L7680-800 202.9 F 198.8 1.71 230.3 1.25 0.98 1.13

    183140438411352301282254210454427386358335301277------

    26 M. Dabaon et al. / Journal of Constructional Steel Research 110 (2015) 1628C2B50L7680-1216 193.2 FC2B50L9760-800 150.5 F

    C2B150 C2B75L3520-384 323.2 LC2B75L3520-800 392.3 LC2B75L5600-800 320.9 F + LC2B75L7680-320 308.8 F + LC2B75L7680-800 254.9 F + LC2B75L7680-1216 230.3 F + LC2B75L9760-800 209.0 F

    C2B200 C2B100L3520-384 301.4 LC2B100L3520-800 408.4 LC2B100L5600-800 371.4 LC2B100L7680-320 376.4 F + LC2B100L7680-800 304.4 F + LC2B100L7680-1216 268.2 F + LC2B100L9760-800 253.4 F + L

    Mean --- --- ---COV --- --- ---According to EC3 (BS EN1993-1-3 [32]), the unfactored design strength(PEC3) is calculated as follows:

    PEC3 Ae Fy forclass4crosssections 8

    where, Ae = effective area, Fy = yield stress which is equal to the 0.2%proof stress (0.2) and =the reduction factor for the relevant bucklingmode.

    1

    22

    q but1:0 9

    0:5 1 0:2 2h i 10

    0

    50

    100

    150

    200

    0 50 100 150 200

    F.E. Results(NAS&AS/NZS) / FE = 1.0

    P NAS

    &AS

    /NZS

    (kN

    )

    PFE (kN)

    Fig. 14. Comparison of the nite element strengths and design strengths for the built-upcold-formed steel battened columns having a channel section C1..1 1.80 230.3 1.25 0.95 1.19

    .9 2.14 158.9 1.59 0.94 1.06

    .1 0.61 464.7 0.43 1.36 1.44

    .9 0.76 464.7 0.43 1.05 1.18

    .9 1.03 402.6 0.69 1.10 1.25

    .5 1.25 322.8 0.94 0.98 1.05

    .0 1.33 322.8 0.94 1.11 1.27

    .0 1.45 322.8 0.94 1.10 1.40

    .7 1.65 244.2 1.20 1.01 1.17

    .4 0.51 482.0 0.35 1.51 1.60

    .1 0.68 474.4 0.35 1.05 1.16

    .9 0.88 438.5 0.55 1.04 1.18

    .3 1.01 383.0 0.75 0.95 1.02

    .1 1.11 383.0 0.75 1.10 1.26

    .8 1.25 383.0 0.75 1.13 1.43

    .5 1.35 317.7 0.96 1.10 1.25--- --- --- 1.04 1.18--- --- --- 0.149 0.151 Ae FyNcr

    s Lcr

    i

    Ae=Ag

    q1

    11

    Where, Lcr is the buckling length in the plane considered, i is the ra-dius of gyration about the relevant axis, determined using the proper-ties of the gross cross-section, is the imperfection factor and Ncr isthe elastic critical force for the relevant buckling mode based on thegross sectional properties.

    1 EFy

    s12

    0

    50

    100

    150

    200

    0 50 100 150 200

    F.E. Results(EC3) / FE = 1.0

    P EC3

    (kN

    )

    PFE (kN)

    Fig. 15. Comparison of the nite element strengths and design strengths for the built-upcold-formed steel section battened columns having a channel section C1.

  • 0100

    200

    300

    400

    500

    0 100 200 300 400 500

    F.E. Results(NAS&AS/NZS) / FE = 1.0

    P NAS

    &AS

    /NZS

    (kN

    )

    PFE (kN)

    Fig. 16. Comparison of the nite element strengths and design strengths for the built-upcold-formed steel section battened columns having a channel section C2.

    020406080

    100120140160180

    0 0.4 0.8 1.2 1.6 2 2.4 2.8

    Finite elementNAS&AS/NZS

    P NAS

    &AS

    /NZS

    (kN

    )

    Nondimensional slenderness c

    Fig. 18. Comparison of the nite element strengths and design strengths for the built-upcold-formed steel battened columns having a channel section C1.

    27M. Dabaon et al. / Journal of Constructional Steel Research 110 (2015) 16286.4. Comparisons with design rules

    Looking at Tables 56 that summarised the nite element built-upcolumn strengths (PFE) and design strengths (PNAS) and (PAS/NZS) aswell as looking at Figs. 1417 that plotted the built-up columnstrengths, generally, it can be seen that the specications were conser-vative for the built-up cold-formed steel section C1 battened columns,except for some columns failing by L failure mode. The mean value of(PNAS&AS/NZS/PFE) and (PEC3/PFE) ratios are 0.97 and 0.92, respectively,with corresponding coefcients of variation (COV) of 0.063 and 0.076,respectively. On the other hand, the specications were generallyunconservative for the built-up cold-formed steel section C2 battenedcolumns. The mean value of (PNAS&AS/NZS/PFE) and (PEC3/PFE) ratios are1.04 and 1.18, respectively, with corresponding coefcients of variation(COV) of 0.149 and 0.151, respectively.

    Figs. 1821 plotted the nite element strengths and designstrengths against the critical nondimensional slenderness (c) calculat-ed according to the NAS Specication [30] and AS/NZS Standard [31 aswell as against the critical nondimensional slenderness () calculatedaccording to the EC3 Code [32]. Generally, it can be seen that the speci-cationswere unconservative for the built-up cold-formed steel sectionbattened columns failing mainly by local buckling, while the specica-tions were conservative for the built-up columns failing mainly by elas-tic exural buckling.

    7. Conclusions

    Nonlinear 3-Dnite elementmodels highlighting the buckling behav-iour and strength of built-up cold-formed steel section battened columnshave been developed and reported in this paper. The nite elementmodels carefully accounted for the nonlinear material properties ofat and corner portions of cold-formed cross sections, initial local andoverall geometric imperfections, actual geometries and actual boundary0

    100

    200

    300

    400

    500

    0 100 200 300 400 500

    F.E. Results(EC3) / FE = 1.0

    P EC3

    (kN

    )

    PFE (kN)

    Fig. 17. Comparison of the nite element strengths and design strengths for the built-upcold-formed steel section battened columns having a channel section C2.conditions. The column strengths, failuremodes, deformed shapes at fail-ure, load-lateral displacement and load-axial strain relationships werepredicted numerically and compared against that measured experimen-tally by the authors. The comparison of test and nite element resultshave shown that good agreement existed and themodels accurately rep-resented the complex buckling behaviour of the built-up columns. Theveried nite element models were used to perform an extensive para-metric study investigating the effects on the built-up column strengthand behaviour owing to the change in column cross-section geometries,column lengths, column local and overall slenderness and cold-formedsteel strengths. The column strengths predicted from the nite elementanalyses were compared with the design strengths calculated using theNorth American Specication, Australian/New Zealand Standard andEuropean Code for cold-formed steel columns. Generally, it has beenshown that the specications were unconservative for the built-upcold-formed steel section battened columns failing mainly by local buck-ling, while the specications were conservative for the built-up columnsfailing mainly by elastic exural buckling.

    NomenclatureAe Effective cross-section areaab Width of batten platesB Overall cross-section widthB1 Channel back-to-back spacingb Width of an individual channelbb Intermediate batten plate widthbb1 End batten plate widthCOV Coefcient of variationD Overall cross-section depthEo Initial Youngs modulus of cold-formed steelF Elastic exural buckling failure modeFe Least of the elastic exural, torsional, and exural torsional

    buckling stress020406080

    100120140160180

    0 0.4 0.8 1.2 1.6 2 2.4 2.8

    Finite elementEC3

    P EC3

    (kN

    )

    Nondimensional slenderness ( )

    Fig. 19. Comparison of the nite element strengths and design strengths for the built-upcold-formed steel battened columns having a channel section C1.

  • 0100

    200

    300

    400

    500

    600

    0 0.4 0.8 1.2 1.6 2 2.4 2.8

    Finite elementNAS&AS/NZS

    P EC3

    (kN

    )

    Nondimensional slenderness c

    Fig. 20. Comparison of the nite element strengths and design strengths for the built-upcold-formed steel battened columns having a channel section C2.

    28 M. Dabaon et al. / Journal of Constructional Steel Research 110 (2015) 1628Fn Critical buckling stressFy Yield stressFEM Finite element modelK Buckling factorL Local failure modeL Column lengthLz Local buckling lengthNcr Critical buckling (Euler) loadPFE Column strength predicted numericallyPAS/NZS Design strength calculated using the Australian and New

    Zealand StandardPEC3 Design strength calculated using the European CodePNAS Design strength calculated using the North American

    SpecicationPTest Test strengthr Radius of gyrationri Internal radiusry1 Minimum radius of gyration around the individual channel

    minor axis y1t Thickness of channelstb Thickness of batten platesy Yield stressu Ultimate stress0.2 0.2% proof stress1 Buckling factorc Critical slenderness Overall nondimensional slenderness1 Lateral displacement at quarter-length2 Lateral displacement at mid-lengthu Strain at fracture Buckling factor Imperfection factor0

    100

    200

    300

    400

    500

    600

    0 0.4 0.8 1.2 1.6 2 2.4 2.8

    Finite elementEC3

    P EC3

    (kN

    )

    Nondimensional slenderness ( )

    Fig. 21. Comparison of the nite element strengths and design strengths for the built-upcold-formed steel battened columns having a channel section C2.Acknowledgement

    The authors would like to acknowledge the support and funding bythe Department of Structural Engineering, Faculty of Engineering, TantaUniversity. The authors are grateful to the Concrete and Heavy StructuresLaboratory staff, Faculty of Engineering, Tanta University, for their assis-tance and technical support.

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    Nonlinear behaviour of built-up cold-formed steel section battened columns1. Introduction2. Summary of experimental investigation3. Finite element model3.1. General3.2. Finite element mesh and type3.3. Material modelling3.4. Boundary conditions and load application3.5. Modelling of initial local and overall geometric imperfections3.6. Modelling of residual stresses

    4. Verification of finite element model5. Parametric study6. Design rules6.1. General6.2. Design rules specified in the NAS Specification and AS/NZS Standard6.3. Design rules specified in the EC3 Code6.4. Comparisons with design rules

    7. ConclusionsNomenclatureAcknowledgementReferences