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Behavior of biaxially-loaded rectangular concrete-filled steel tubularslender beam-columns with preload effects

Vipulkumar Ishvarbhai Patel a, Qing Quan Liang a,n, Muhammad N.S. Hadi b

a College of Engineering and Science, Victoria University, P.O. Box 14428, Melbourne, VIC 8001, Australiab School of Civil, Mining and Environmental Engineering, University of Wollongong, Wollongong, NSW 2522, Australia

a r t i c l e i n f o

Article history:Received 23 June 2013Received in revised form15 November 2013Accepted 15 February 2014Available online 12 March 2014

Keywords:Concrete-filled steel tubesBiaxial bendingLocal bucklingNonlinear analysisSlender beam-columns

a b s t r a c t

In composite construction, rectangular hollow steel tubular slender beam-columns are subjected topreloads arising from construction loads and permanent loads of the upper floors before infilling of thewet concrete. The behavior of biaxially loaded thin-walled rectangular concrete-filled steel tubular (CFST)slender beam-columns with preloads on the steel tubes has not been studied experimentally andnumerically. In this paper, a fiber element model developed for CFST slender beam-columns with preloadeffects is briefly described and verified by existing experimental results of uniaxially loaded CFSTcolumns with preload effects. The fiber element model is used to investigate the behavior of biaxiallyloaded rectangular CFST slender beam-columns accounting for the effects of preloads and local buckling.Parameters examined include local buckling, preload ratio, loading angle, depth-to-thickness ratio,column slenderness, loading eccentricity and steel yield strength. The results obtained indicate that thepreloads on the steel tubes significantly reduce the stiffness and strength of CFST slender beam-columnswith a maximum strength reduction of more than 15.8%. Based on the parametric studies, a design modelis proposed for axially loaded rectangular CFST columns with preload effects. The fiber element anddesign models proposed allow for the structural designer to efficiently analyze and design CFST slenderbeam-columns subjected to preloads from the upper floors of a high-rise composite building duringconstruction.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Rectangular concrete-filled steel tubular (CFST) slender beam-columns have been widely used in the construction of modernhigh-rise composite buildings because of their high strength andstiffness performance. During the construction of a high-risecomposite building, hollow steel tubes are subjected to preloadsarising from construction loads and permanent loads of the upperfloors before infilling of the wet concrete. The stresses anddeformations in the hollow steel tubes induced by the preloadsmight significantly reduce the ultimate strengths of CFST slenderbeam-columns. Experimental and theoretical studies on biaxiallyloaded thin-walled rectangular CFST slender beam-columns withpreload effects have not been reported in the literature. Therefore,an effective fiber element model for the nonlinear analysis anddesign of biaxially loaded thin-walled CFST slender beam-columnsincorporating preload effects is much needed.

Extensive experimental studies on the behavior of CFST col-umns without preload effects have been conducted [1–11]. Testresults demonstrated that the confinement effect provided by therectangular steel tube did not increase the compressive strength ofthe concrete core but considerably improved its ductility [1]. Inaddition, local buckling of the steel tubes was found to remarkablyreduce the ultimate strength and stiffness of thin-walled CFSTcolumns [12–17]. Moreover, Liang et al. [16,17] utilized the finiteelement method to investigate the local and post-local bucklingbehavior of steel plates in double skin composite panels and thin-walled CFST columns under axial load and biaxial bending. Theyproposed a set of formulas for determining the initial localbuckling stresses and post-local buckling strengths of steel platesin CFST columns under stress gradients. These formulas can beincorporated in numerical models to account for local bucklingeffects in the nonlinear analysis of thin-walled CFST columnsunder biaxial bending. Mursi and Uy [18] conducted tests to studythe local and global buckling behavior of square CFST short andslender columns under biaxial bending without preload effects.These specimens were made of high strength steel tubes withyield strength of 701 MPa and low strength concrete. Uniqueorthogonal knife edge end supports were designed to simulate

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Thin-Walled Structures

http://dx.doi.org/10.1016/j.tws.2014.02.0130263-8231 & 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ61 3 9919 4134.E-mail address: Qing.Liang@vu.edu.au (Q.Q. Liang).

Thin-Walled Structures 79 (2014) 166–177

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the free rotations in the two orthogonal planes of bending. Thedepth-to-thickness ratio of the steel tube was varied from 24 ofcompact sections to 54 of relatively slender sections. This was animportant study as it provided test results on biaxially loadedhigh-strength CFST slender columns made of slender steel sec-tions, which can be used to validate numerical models.

There have been very limited experimental investigations onCFST columns with preload effects. The effects of preload on theultimate axial strengths of eccentrically loaded circular CFSTslender beam-columns were studied experimentally by Zha [19]and Zhang et al. [20]. Han and Yao [21] described experimentalinvestigations on eccentrically loaded normal strength rectangularCFST columns including preload effects. The constant preload wasapplied to the steel tube by pre-stressing bars before filling thewet concrete. Test parameters included the preload ratio, columnslenderness and loading eccentricity. Test results indicated thatthe preload on the steel tube caused initial deflections whichreduced the ultimate axial strengths of CFST beam-columns. Liewand Xiong [22] undertook tests on axially loaded circular CFSTshort and slender columns with preloads on the steel tubes. Theirstudy showed that the preload on the steel tube might reduce theultimate axial strength of the circular CFST slender beam-columnby 15%, provided that it was greater than 60% of the ultimate axialstrength of the hollow steel tube. The strength and behavior ofshort circular CFST columns were not affected by preloads.

Analytical and numerical models have been developed forsimulating the nonlinear inelastic behavior of CFST columns with-out preload effects [23–25]. A semi-analytical model was pre-sented by Lakshmi and Shanmugam [26] that simulates thebehavior of CFST slender columns under axial load and biaxialbending. A generalized displacement control method wasemployed in the model to solve nonlinear equilibrium equations.Mursi and Uy [27] developed a numerical model for the nonlinearanalysis of biaxially loaded CFST short and slender columnsincorporating local buckling effects. Their work was a goodattempt to address the local and global interaction bucklingproblem of CFST slender columns under biaxial loads. In theirmodel, the elastic local buckling of steel plates under stressgradients was considered, but the post-local buckling strength ofplates was determined by the effective width formulas for steelplates under uniform compression. In addition, the progressivepost-local buckling of steel tubes has not been modeled byredistributing the stresses on the steel tube walls. Moreover, theunconfined concrete model adopted in their study did not accountfor the confinement effect on the concrete ductility while theconfined concrete model used was shown to give a significantincrease in the strengths. Furthermore, the numerical solutionmethod has not been described in their paper. Liang [28,29]developed a performance-based analysis (PBA) technique forpredicting the ultimate strength and ductility of biaxially loadedthin-walled CFST short beam-columns without preload effects. ThePBA technique incorporated effective width formulas proposed byLiang et al. [17] to account for the effects of progressive localbuckling. Moreover, numerical models that do not consider pre-load effects were developed by Patel et al. [30,31] and Liang et al.[32] that simulate the load–deflection responses and load–moment interaction diagrams of high strength thin-wall rectan-gular CFST slender beam-columns under combined axial load andbending.

Xiong and Zha [33] utilized the finite element analysis ABAQUSprogram to study the behavior of eccentrically loaded circular CFSTslender beam-columns considering preload effects. It wasobserved that the ultimate axial strengths of CFST columnsdecreased with increasing the preloads on the steel tubes. Theyproposed a simplified formula for calculating the ultimate strengthof CFST beam-columns subjected to axial or eccentric loading.

However, their model and formula are applicable only to circularCFST slender beam-columns. Liew and Xiong [22] proposed adesign method for determining the strength of axially loadedcircular CFST slender columns incorporating preload effects. How-ever, their model yields very conservative prediction of axiallyloaded rectangular CFST columns with preload ratio greater than0.4. A numerical model was proposed by Patel et al. [34] forpredicting the load–deflection behavior of eccentrically loadedcircular CFST slender beam-columns accounting for the effects ofpreloads, initial geometric imperfections, concrete confinementand second order.

The above literature review clearly shows that there have beenvery limited experimental and numerical studies on the behaviorof uniaxially loaded rectangular CFST slender beam-columns withpreloads effects. No investigations have been reported on biaxiallyloaded rectangular CFST slender beam-columns considering theeffects of preload and local buckling. In this paper, a fiber elementmodel developed for the nonlinear analysis of biaxially loadedrectangular CFST slender beam-columns incorporating the effectsof preload and local buckling is briefly described. The fiberelement model is verified by comparisons with existing test resultsand used to investigate the behavior of biaxially loaded CFSTslender beam-columns with various important parameters. Adesign model for axially loaded rectangular CFST columns con-sidering preload effects is proposed and compared with the designmethod given by Liew and Xiong [22].

2. Fiber element model

2.1. Fiber element method

The inelastic behavior of the cross-section of a CFST beam-column is modeled by the fiber element method [28]. The columncross-section is divided into fine fiber elements as shown in Fig. 1.It is assumed that the plane section remains plane after deforma-tion. This results in a linear strain distribution throughout thedepth of the section as depicted in Fig. 1. Each fiber element can beassigned either steel or concrete material properties. Fiber stressesare calculated from fiber strains using material uniaxial stress–strain relationships. Axial forces and bending moments carried bythe cross-section are determined as stress resultants in thecomposite section.

Fig. 1. Fiber strain distribution in CFST beam-column section under axial load andbiaxial bending.

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2.2. Stress–strain relationships for concrete

The general stress–strain relationships for concrete in rectan-gular CFST columns suggested by Liang [28] as shown in Fig. 2 areadopted in the fiber element model. The confinement effect whichincreases the ductility of concrete in rectangular CFST columns isincluded in the stress–strain curve for concrete. The concretestress from O to A on the stress–strain curve is calculated basedon the equations given by Mander et al. [35] as:

sc ¼f 0ccλðεc=ε0ccÞ

λ�1þðεc=ε0ccÞλð1Þ

λ¼ EcEc�ðf 0cc=ε0ccÞ

ð2Þ

Ec ¼ 3320ffiffiffiffiffiffiffiffiffiγcf

0c

qþ6900 ðMPaÞ ð3Þ

where sc is the compressive concrete stress, f 0cc is the effectivecompressive strength of concrete, f 0c is the compressive strength ofconcrete cylinder, εc is the compressive concrete strain, ε0cc is thestrain at f 0cc . The Young's modulus of concrete Ec is given by ACI318-11 [36]. The effective compressive strength f 0cc is taken as γcf

0c ,

where γc is the strength reduction factor proposed by Liang [28] toaccount for the column section size effects and is expressed by

γc ¼ 1:85D�0:135c ð0:85rγcr1:0Þ ð4Þ

where Dc is taken as the larger of ðB�2tÞ and ðD�2tÞ for arectangular cross-section and t is the thickness of the steeltube wall.

The strain ε0cc corresponding to f 0cc is taken as 0.002 when theeffective compressive strength of concrete is under 28 MPa while

it is taken as 0.003 for concrete strength over 82 MPa. The strainε0cc is determined by the linear interpolation for the effectivecompressive strength of concrete between 28 and 82 MPa.

The concrete stress from A to D on the stress–strain curve iscalculated using on the following equations given by Liang [28]:

sc ¼f 0cc for ε0ccoεcr0:005βcf

0ccþ100ð0:015�εcÞðf 0cc�βcf

0ccÞ for 0:005oεcr0:015

βcf0cc for εc40:015

8><>:

ð5Þwhere βc was proposed by Liang [28] based on experimentalresults presented by Tomii and Sakino [37] to account for theconfinement effect on the post-peak behavior and is given by

βc ¼1:0 for Bs

t r24

1:5� 148

Bst for 24oBs

t r48

0:5 for Bst 448

8>><>>: ð6Þ

where Bs is taken as the larger of B and D for a rectangular cross-section.

The stress–strain curve for concrete in tension is shown inFig. 2. In the theoretical model, the tensile strength of concrete istaken as 0:6

ffiffiffiffiffiffif 0cc

qwhile the ultimate tensile strain is taken as 10

times of the strain at concrete cracking.

2.3. Stress–strain relationships for structural steels

Fig. 3 shows the stress–strain curves for structural steels. Thesteel generally follows the same stress–strain relationship underthe tension and compression. For mild structural steels, anidealized tri-linear stress–strain curve is assumed as shown inFig. 3. A linear-rounded-linear stress–strain curve is used for cold-formed steels but the rounded part is replaced by a straight line forhigh strength steels. The rounded part for cold-formed steels asshown in Fig. 3 can be modeled by the formula given by Liang [28].For high strength and cold-formed steels, the hardening strain εstis taken as 0.005 while it is considered to be 10εsy for mildstructural steels in the fiber element model. The ultimate strain εsuof steels is assumed to be 0.2.

2.4. Local and post-local buckling of steel plates

Thin steel plates in contact with concrete are constrained tobuckle locally outward when subjected to compressive stressgradients as depicted in Fig. 4. Local buckling significantly reducesthe strength and stiffness of CFST beam-columns with large depth-to-thickness ratios. Liang et al. [17] proposed formulas thatconsider the effects of geometric imperfections and residualstresses for estimating the initial local buckling stresses andpost-local buckling strengths of thin steel plates under stressgradients. Their formulas have been implemented in the numer-ical model to account for the effects of progressive local and post-local buckling. A complete treatment of the progressive local andpost-local buckling of CFST columns was given by Liang [28].

2.5. Equilibrium equations

The load control analysis procedure proposed by Patel et al.[34] is used to determine the mid-height deflection umo of thehollow steel tube subjected to preload. In the analysis, the internalmoment Mmi at the mid-height of the hollow steel tube isdetermined from the load–moment–curvature relationship. Thecurvature ϕmo at the mid-height of the steel tube is adjusted untilthe moment equilibrium is obtained at its mid-height. The deflec-tion umo at the mid-height of the hollow steel tube can be

Fig. 2. Stress–strain curve for concrete in rectangular CFST columns.

stusty0

s

fsu

fsy

0.9fsy

Fig. 3. Stress–strain curves for structural steels.

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calculated from the curvature ϕmo. It is assumed that the hollowsteel tube does not buckle under the given preload. The preloadratio is defined as

βa ¼Ppre

Pusð7Þ

where Ppre is the preload applied to the hollow steel tube and Pus isthe ultimate axial strength of the hollow steel tube including localbuckling effects.

The curvature at the mid-height of the CFST beam-column canbe obtained as

ϕm ¼ πL

� �2um ð8Þ

where L is the effective length of the CFST beam-column and um isthe mid-height deflection of the CFST beam-column.

The mid-height deflection umo of the steel tube caused bypreloads is treated as the initial geometric imperfection in theCFST beam-column. The external moment at the mid-height of theCFST beam-column with preload effects can be determined as

Mme ¼ PðeþuoþumoþumÞ ð9Þ

where P is the applied load, e is the eccentricity of the appliedload, uo is the initial geometric imperfection at the mid-height ofthe hollow steel tube and umo is the mid-height deflection of thehollow steel tube under the preload.

In the nonlinear analysis of CFST slender beam-column, themid-height deflection um of the CFST slender beam-column underaxial load and biaxial bending is gradually increased until theultimate axial strength is obtained or deflection limit is reached.The curvature ϕm at the mid-height of the CFST beam-column canbe calculated from the deflection um. For this curvature, theapplied axial load P is determined by solving for the internal forcethat satisfies the equilibrium condition in which the externalmoment Mme is equal to the internal moment Mmi at the mid-height of the CFST beam-column. The Müller's method [38] is usedto adjust the depth and orientation of the neutral axis in order tomaintain the moment equilibrium at the mid-height of the CFSTslender beam-column.

The equilibrium equations for beam-columns under biaxialloads are given as

Mme�Mmi ¼ 0 ð10Þ

tan α�My

Mx¼ 0 ð11Þ

where Mmi is the resultant internal moment which is calculated as

Mmi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2

x �M2y

q, and α is the angle of the applied load with

respect to y-axis, and Mx and My are the moments about the x andy axes respectively, as depicted in Fig. 1.

2.6. Computational algorithms

The computational procedure for determining the load–deflec-tion responses of CFST slender beam-columns incorporating theeffects of preloads and local buckling is given as follows:

(1) Input data.(2) Discretize the composite section into fine fiber elements.(3) Calculate the mid-height deflection umo of the hollow steel

tube using the load control procedure.(4) Set uo ¼ uoþumo.(5) Initialize the mid-height deflection of the CFST beam-col-

umn: um ¼Δum.(6) Calculate the curvature ϕm at the mid-height of the CFST

beam-column.(7) Adjust the neutral axis depth dn using the Müller's method.(8) Compute stress resultants P and Mmi considering local

buckling.(9) Calculate the residual moment ram ¼Mme�Mmi.

(10) Repeat Steps (7)–(9) until jramjoεk.(11) Compute bending moments Mx and My.(12) Adjust the orientation θ

� �of the neutral axis using the

Müller's method.(13) Calculate the residual moment rbm ¼ tan α�ðMy=MxÞ.(14) Repeat Steps (7)–(13) until jrbmjoεk.(15) Increase the deflection at mid-height of the CFST beam-

column by um ¼ umþΔum.(16) Repeat Steps (6)–(15) until the ultimate axial load Pn is

obtained or the deflection limit is reached.(17) Plot the load–deflection curve.

In the above computational procedure, the convergence toler-ance εk is set to 10�4. The numerical solution scheme based on theMüller's method [38] has been developed to iterate the neutralaxis depth ðdnÞ and orientation ðθÞ in the cross-sectional analysis.The Müller's method requires three initial values of the dn and θ tostart the iterative process. In the present model, three initial valuesof the neutral axis depth dn;1, dn;3 and dn;2 are taken as D=4, D andðdn;1þdn;3Þ=2 respectively while the initial values for the orienta-tion of the neutral axis θ1, θ3 and θ2 are assigned to α=4, α andðθ1þθ3Þ=2 respectively. The force and moment residuals rm;1, rm;2

and rm;3 corresponding to these three initial values are calculated.The depth and orientation of the neutral axis are iterativelyadjusted. Full details on the computational algorithms can befound in the papers by Patel et al. [30] and Liang et al. [32].

2.7. Development of strength envelopes

The load–deflection analysis procedure can be used to generatestrength envelopes by varying the loading eccentricity. The ulti-mate axial strength Pn is calculated by specifying the eccentricity eof the applied load in the load–deflection analysis. The ultimatebending strength Mn for a given loading eccentricity e andultimate axial load Pn is calculated as Mn ¼ Pn � e. A pair of theultimate bending strength Mn and the ultimate axial load Pn isused to plot the strength envelope. Note that the pure axial loadPoa is calculated by specifying the eccentricity of the applied load

Fig. 4. Effective and ineffective areas of steel tubular cross-section under axial loadand biaxial bending.

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to zero in the load–deflection analysis procedure. However, theload–deflection procedure is not efficient for determining the purebending strength. Therefore, the pure bending strength of aslender beam-column is obtained by specifying the axial load tozero in the load–moment interaction analysis procedure proposedby Liang et al. [32].

3. Comparisons of computer solutions with experimentalresults

3.1. Ultimate axial strengths

The ultimate axial strengths of CFST slender beam-columns underaxial load and biaxial bending without preload effects were deter-mined using the multiscale model proposed by Liang et al. [32] andcompared with experimental results presented by Mursi and Uy [18]in Table 1. The depth-to-thickness ratio of these specimens variedfrom 24 to 54. For b/t ratios greater than 30, the effects of localbuckling were considered in the numerical analyses. It can be seenfrom Table 1 that the numerical predictions are in good agreementwith experimental results. The mean ultimate axial strength ðPu:numÞpredicted by the multiscale model is 95% of the correspondingexperimental value ðPu:expÞ. The standard deviation of Pu:num=Pu:exp is0.09 while its coefficient of variation is 0.1.

Experimental results of uniaxially or concentrically loaded squareCFST slender columns with or without preload effects given by Hanand Yao [21] were used to verify the accuracy of the fiber elementmodel developed. In the present study, an initial geometric imperfec-tion of L=1000 at the mid-height of the beam-columns was taken into

account in the analyses of Specimens L-2 to LP-6. It was assumed thatthe eccentricity of the preload on the hollow steel tube was the sameas that of the applied compressive load on the CFST beam-column. Thecompressive strength of concrete cylinder was taken as 0.85 times theconcrete cube strength. As shown in Table 2, good agreement betweencomputational results and test data is obtained. The mean value of thepredicted to the experimental ultimate axial strength ðPu;num=Pu;expÞ is0.93 with a standard deviation of 0.04 and a coefficient of variation of0.04. The fiber element model yields conservative predictions of theultimate axial strengths of rectangular CFST slender beam-columnsconsidering preload effects.

3.2. Load–deflection curves

Fig. 5 shows a comparison of predicted and experimental axialload–deflection curves for the Specimen SL-C260 tested by Mursiand Uy [18]. The figure demonstrates that the numerical modelpredicts well the initial stiffness and ultimate strength of theSpecimen. However, the experimental curve departs from thecomputation one after the load is higher than about 1500 kN.The discrepancy between predicted and experimental results islikely attributed to the uncertainty of the actual concrete com-pressive strength and stiffness as the average concrete compres-sive strength was used in the numerical analysis.

The load–deflection responses of specimens tested by Han and Yao[21] were simulated by the fiber element model. Fig. 6 shows acomparison of predicted and experimental axial load–deflectioncurves for Specimen LP-1. The figure demonstrates that the fiberelement model predict well the initial stiffness of the specimen.However, after the loading level of about 170 kN, the experimental

Table 1Ultimate axial strengths of biaxially loaded CFST slender beam-columns without preload effects.

Specimens B� D� t (mm) D=t L (mm) ex (mm) ey (mm) e (mm) α (1) f 0c (MPa) uo f sy (MPa) f su (MPa) Es (GPa) Pu:exp (kN) Pu:num (kN) Pu: num=pu: exp Ref.

SL-C110 120�120�5 24 2800 10 10 14.14 45 17.62 0 701 800 220.225 765 847.72 1.11 [18]SL-C160 170�170�5 34 2800 15 15 21.21 45 22.88 0 701 800 220.225 1858 1676.54 0.90SL-C210 220�220�5 44 2800 20 20 28.28 45 22.88 0 701 800 220.225 2784 2509.61 0.90SL40-C210 220�220�5 44 2800 40 40 56.57 45 22.88 0 701 800 220.225 2188 2022.49 0.92SL-C260 270�270�5 54 2800 25 25 35.35 45 22.88 0 701 800 220.225 3634 3271.39 0.90Mean 0.95Standard deviation (SD) 0.09Coefficient of variation (COV) 0.10

Table 2Ultimate axial strengths of square CFST beam-columns with preload effects.

Specimens B�D� t (mm) D/t L (mm) uo (mm) e (mm) f 0c (MPa) f sy (MPa) f su (MPa) Es (GPa) Ppreload (kN) Pu:exp (kN) Pu:num (kN) Pu: num=pu: exp Ref.

S-1 120�120�2.65 45.2 360 0 0 17.09 340 439.6 207 0 640 579.07 0.90 [21]SP-1 120�120�2.65 45.2 360 0 0 17.09 340 439.6 207 211 664 580.31 0.87S-2 120�120�2.65 45.2 360 0 14 17.09 340 439.6 207 0 533 478.67 0.90SP-2 120�120�2.65 45.2 360 0 14 17.09 340 439.6 207 211 538 480.85 0.89S-3 120�120�2.65 45.2 360 0 0 30.6 340 439.6 207 0 816 754.39 0.92SP-3 120�120�2.65 45.2 360 0 0 30.6 340 439.6 207 211 812 753.41 0.93S-4 120�120�2.65 45.2 360 0 14 30.6 340 439.6 207 0 600 610.29 1.02SP-4 120�120�2.65 45.2 360 0 14 30.6 340 439.6 207 211 622 613.28 0.99SP-5 120�120�2.65 45.2 360 0 14 30.6 340 439.6 207 127 650 613.54 0.94SP-6 120�120�2.65 45.2 360 0 31 30.6 340 439.6 207 198 500 481.38 0.96L-2 120�120�2.65 45.2 1400 L/1000 0 30.6 340 439.6 207 0 769 716.93 0.93LP-2 120�120�2.65 45.2 1400 L/1000 0 30.6 340 439.6 207 194 730 715.31 0.98L-1 120�120�2.65 45.2 1400 L/1000 14 30.6 340 439.6 207 0 590 536.83 0.91LP-1 120�120�2.65 45.2 1400 L/1000 14 30.6 340 439.6 207 194 560 527.24 0.94LP-3 120�120�2.65 45.2 1400 L/1000 14 30.6 340 439.6 207 272 552 522.52 0.95LP-4 120�120�2.65 45.2 1400 L/1000 31 30.6 340 439.6 207 194 452 403.89 0.89L-5 120�120�2.65 45.2 1400 L/1000 31 25.5 340 439.6 207 0 412 385.47 0.94LP-6 120�120�2.65 45.2 1400 L/1000 31 25.5 340 439.6 207 117 397 378.81 0.95Mean 0.93Standard deviation (SD) 0.04Coefficient of variation (COV) 0.04

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curve slightly departs from the computational one. In addition, thefiber element model also predicts well the stiffness of the testedspecimen in the post-peak range. The discrepancy between predictedand experimental results is likely attributed to the uncertainty of theactual concrete compressive strength and stiffness as the averageconcrete compressive strength was used in the analysis. The predictedand experimental axial load–deflection curves for Specimen LP-6 aredepicted in Fig. 7. As indicated in this figure, the results obtained bythe fiber element model are generally in good agreement with testdata. In the post-peak range, the fiber element model also yields goodpredictions of the stiffness of the tested specimen. However, it is notedthat the initial stiffness of the specimen predicted by the numericalmodel is slightly higher than that of the experimental one. This islikely caused by the uncertainty of the actual concrete stiffness andstrength of the specimen.

4. Fundamental behavior with preload effects

4.1. General

The fiber element model developed was used to investigate theeffects of local buckling, preload ratio, depth-to-thickness ratio ðD=tÞ,column slenderness ratio ðL=rÞ, loading eccentricity ratio ðe=DÞ, steel

yield strength ðf syÞ and loading angle on the behavior of biaxiallyloaded normal and high strength thin-walled CFST slender beam-columns. The column slenderness ratio is the ratio of the columneffective length to the radius of gyration of its cross-section. It wasassumed that the eccentricity of the preload on the steel tube was thesame as that of the applied axial load on the CFST beam-column. Inaddition to this, local buckling of the steel tubes under preloads isprevented in the study as it would not be expected in the realconstruction of CFST columns. The initial geometric imperfection at themid-height of the hollow steel tube was considered to be L=1000. TheYoung's modulus of the steel tubes was 200 GPa.

4.2. Effects of local buckling

The fiber element model developed was utilized to study theeffects of local buckling on the behavior of normal strength rectangularCFST slender beam-columns. The beam-columnwith a cross-section of600�600 mm and under the eccentric load applied at an angle of 451with respect to the y-axis was analyzed. The D=t ratio of the sectionwas 90 while the L=r ratio was 40. The e=D ratio of 0.2 was specified inthe analysis. Normal strength concrete with compressive strength of50 MPawas filled into the steel tube. The yield and tensile strengths ofthe steel tube were 300 MPa and 430MPa respectively. The preload

0

100

200

300

400

500

600

0 5 10 15 20 25 30 35

Axi

al lo

ad (k

N)

Mid-height deflection um (mm)

Experiment (LP-1)

Fiber element analysis

Fig. 6. Comparison of computational and experimental axial load–deflectioncurves for Specimen LP-1.

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25 30 35

Axi

al lo

ad (k

N)

Mid-height deflection um (mm)

Experiment (LP-6)

Fiber element analysis

Fig. 7. Comparison of predicted and experimental axial load–deflection curves forSpecimen LP-6.

0

2000

4000

6000

8000

10000

12000

0 30 60 90 120 150

Axi

al lo

ad (k

N)

Mid-height deflection um (mm)

Local buckling ignored

Local buckling considered

Fig. 8. Effects of local buckling on the axial load–deflection curves for CFST slenderbeam-column ðβa ¼ 0:3Þ.

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20

Axi

al lo

ad (k

N)

Mid-height deflection um (mm)

Experiment

Numerical analysis

Fig. 5. Comparison of predicted and experimental axial load–deflection curves forSpecimen SL-C260.

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ratio of 0.3 was considered. The CFST slender beam-column wasanalyzed by considering and ignoring local buckling effectsrespectively.

Fig. 8 depicts the predicted load–deflection curves for the CFSTslender beam-column. The figure indicates that local buckling remark-ably reduces the stiffness and ultimate axial strength of the CFSTslender beam-column with the same preload ratio. The ultimate axialload of the slender beam-column is overestimated by 8.7% if localbuckling was not considered. Fig. 9 illustrates the effects of localbuckling on the strength envelope of the CFST slender beam-column.In each interaction curve shown in Fig. 9, the ultimate axial strengthwas normalized to the ultimate axial load ðPoaÞ of the axially loadedslender column including local buckling effects, while the ultimatemoment was normalized to the pure bending moment ðMoÞ of theslender beam-column with local buckling effects. It can be observedthat for the same preload ratio, the ultimate bending strength isreduced by local buckling when the axial load is greater than 30% ofthe ultimate axial strength Poa.

4.3. Effects of preloads

The effects of the preload ratio on the behavior of a normalstrength steel tube filled with high strength concrete were

investigated herein. A CFST beam-column with a cross-section of700�700 mm was considered. The D=t ratio of the column cross-section was 30. The angle of the applied load was 301 for biaxialbending. The steel yield strength of the welded mild steel tube was300 MPa and its tensile strength was 430 MPa. The compressivestrength of infill concrete was 60 MPa. The L=r ratio of 40 and thee=D ratio of 0.2 were specified in the analysis. The preload ratios of0.0, 0.4 and 0.8 were considered in the investigations.

Load–deflection curves for the CFST slender beam-column withvarious preload ratios are presented in Fig. 10. The figure showsthat the ultimate axial strength and flexural stiffness of the CFSTslender beam-column decrease with increasing the preload ratio.In addition, it is seen that the higher the preload ratio, the largerthe mid-height deflection in the CFST beam-column at theultimate load. Increasing the preload ratio from 0.0 to 0.4 reducesthe ultimate axial strength of the CFST beam-column by 6%.However, this strength reduction could be up to 14.1% when thepreload ratio is increased from 0.0 to 0.8. The strength envelopesof the CFST slender beam-column with preload ratios of 0.0 and0.4 are given in Fig. 11, where Pop is the ultimate axial load of theaxially loaded slender column including preload effects. It can beseen from Fig. 11 that considering preload effects narrows thestrength envelope of the slender beam-column. It should be notedthat pure bending strength of the slender beam-column is notaffect by the preload on the hollow steel tube.

4.4. Effects of preloads and depth-to-thickness ratio

The fundamental behavior of a rectangular CFST slender beam-column is influenced by the D=t ratio. The CFST slender beam-columns with a square cross-section of 800�800 mm wereconsidered in the analysis. The D=t ratio of the steel section wasvaried from 20 to 50 by changing the thickness of the section. TheL=r ratio was 80 while the e=D ratio was taken as 0.2 in theanalysis. The axial load was applied at an angle of 601 for biaxialbending. The yield and tensile strengths of the steel tubes were690 MPa and 790 MPa respectively. The concrete compressivestrength of 40 MPa was considered. The preload ratios of 0.0,0.3 and 0.6 were used in the analysis.

The effects of D=t ratio on the ultimate axial strengths of CFSTslender beam-columns with various preload ratios are demon-strated in Fig. 12. It can be seen from the figure that the ultimateaxial strengths of rectangular CFST slender beam-columnsdecrease considerably with increasing the D=t ratio under the

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

Axi

al lo

ad P

n/Poa

Moment Mn/Mo

Local buckling ignored

Local buckling considered

Fig. 9. Effects of local buckling on the strength envelopes for CFST slender beam-columnðβa ¼ 0:3Þ.

a=0.8

a=0

a=0.4

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

0 100 200 300 400 500

Axi

al lo

ad (k

N)

Mid-height deflection um (mm)

Fig. 10. Effects of preloads on the axial load–deflection curves for CFST beam-columns.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

Axi

al lo

ad P

n/Pop

Moment Mn/Mo

Preload effect ignored

Preload effect considered

Fig. 11. Effects of preloads on the strength envelopes for CFST slender beam-column.

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same preload ratio. This may be attributed to the fact that thecross-sectional area of steel and the post-local buckling strength ofthe steel tube decrease with an increase in the D=t ratio. Fig. 13shows the reduction in the ultimate axial strength caused by

preloads. The corresponding strength reduction with the D=t ratioof 20 and the preload ratio of 0.3 is 5.3% while for a CFST slenderbeam-column with the D=t ratio of 50, the preload with a ratio of0.6 causes a 12% reduction in its ultimate axial strength.

4.5. Effects of preloads on column strength curves

The effects of preloads on the column strength curves for highstrength steel tubes filled with high strength concrete wereexamined. The width and depth of the rectangular steel tube were600 mm and 800 mm respectively. The L=r ratio varied from 0 to100 while the dimension of the section was not changed. The e=Dratio of 0.2 was specified in the analysis of the CFST beam-columnswith various L=r ratios. The angle of the applied axial load was 301while the D=t ratio of 50 was specified in the analysis. The CFSTbeam-columns were made of high strength steel tubes with theyield and tensile strengths of 690 and 790 MPa respectively filledwith high strength concrete of 70 MPa. The preload with a ratio of0.0, 0.3 and 0.6 was applied to the steel tube respectively.

The normalized column strength curves for biaxially loadedCFST beam-columns with various preload ratios are presented inFig. 14, in which Pn is the ultimate axial load of the biaxially loadedCFST beam-column and Po is the ultimate axial strength of thecolumn section under axial compression. It appears from Fig. 14that the ultimate axial strength of CFST slender beam-columnsunder the same preload ratio is strongly affected by the columnsslenderness. Increasing the column slenderness ratio significantlyreduce the ultimate axial load of CFST beam-columns regardless ofthe preload ratios. The strength reduction-column slendernesscurves with preload ratios of 0.3 and 0.6 are illustrated in Fig. 15.The figure shows that the strength reduction increases withincreasing the column slenderness ratio as well as the preloadratio. As presented in Fig. 15, the preload having a ratio of0.6 reduces the ultimate axial strength of the CFST column withan L=r ratio of 100 by 15.8%. However, for the short beam-columnwith an L=r ratio of 22, the preload with a ratio of 0.3 reduces theultimate axial strength of the CFST column by only 0.7%. Thismeans that the preload effect can be ignored in the design of CFSTshort beam-columns with an L=r ratio less than 22.

4.6. Effects of preloads and loading eccentricity ratio

The behavior of normal strength slender steel tubes filled withnormal strength concrete and with various e/D ratios were studied.A CFST slender beam-column with a cross-section of 700�800mm

a=0

a=0.3

a=0.6

0

5000

10000

15000

20000

25000

30000

35000

40000

0 10 20 30 40 50 60

Axi

al lo

ad (k

N)

Depth-to-thickness ratio D/t

Fig. 12. Effects of preloads and diameter-to-thickness ratio on the ultimate axialstrengths.

a=0.3

a=0.6

0

2

4

6

8

10

12

14

0 10 20 30 40 50 60

Stre

ngth

redu

ctio

n (%

)

Depth-to-thickness ratio D/t

Fig. 13. Strength reduction caused by preloads in CFST beam-column with variousdepth-to-thickness ratios.

a=0

a=0.3

a=0.6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100 120

Ulti

mat

e ax

ial l

oad

P n/P

o

Slenderness ratio L/r

Fig. 14. Effects of preloads on the column strength curves of CFST beam-columns.

a=0.3

a=0.6

0

2

4

6

8

10

12

14

16

18

20

0 20 40 60 80 100 120

Stre

ngth

redu

ctio

n (%

)

Slenderness ratio L/r

Fig. 15. Strength reduction caused by preloads in CFST beam-columns with variousslenderness ratios.

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and loading eccentricities ranged from 0 to 2 were analyzed using thecomputer program developed. The angle of the axial load was fixed at451 with respect to the y-axis. The depth-to-thickness ratio of thesection was 50 while the L=r ratio of the beam-column was 80. Theyield and tensile strengths of the steel tubes were 300 MPa and430MPa respectively. The concrete compressive strength was 40MPa.The preload ratios were taken as 0.0, 0.3 and 0.6 in the analyses.

Fig. 16 shows the ultimate axial load ratio as a function of thee=D ratio. The ultimate axial strengths of CFST slender beam-columns are shown to decrease with an increase in the loadingeccentricity ratio regardless of preload ratios. The strength reduc-tion–loading eccentricity curves for CFST slender beam-columnswith preload ratios of 0.3 and 0.6 are given in Fig. 17. The figureillustrates that the strength reduction increases with increasingthe loading eccentricity ratio from 0 to 0.4. After reaching themaximum value, the reduction in the ultimate axial load decreaseswith an increase in the loading eccentricity ratio. It is interestingto note that the preload on the hollow steel tube causes amaximum reduction in the ultimate axial strength of a CFSTbeam-column when the e=D ratio is equal to 0.4. For the axiallyloaded CFST slender column with preload ratios of 0.3 and 0.6, thestrength reduction caused by the preload is 1.2% and 2.7%

respectively. However, the preload having a ratio of 0.6 leads toa reduction of 8.2% in the ultimate axial load of the CFST slenderbeam-column with an e=D ratio of 0.4.

4.7. Effects of preloads and steel yield strengths

In this section, the effects of steel yield strengths on thestrength and behavior of square steel tubes with various yieldstrength filled with high strength concrete of 80 MPa werestudied. A column cross-section of 650�650 mm was consideredin the parametric study. The depth-to-thickness ratio of 65, thecolumn slenderness ratio of 80 and the loading eccentricity ratio of0.2 were used to examine the effects of preloads and steel yieldstrengths. The steel yield strength was varied from 250 MPa to690 MPa. The applied axial load angle was fixed at 601 withrespect to the y-axis. The preload ratios were taken as 0.0,0.3 and 0.6 in the analysis.

Fig. 18 gives the normalized ultimate axial strengths of CFSTslender beam-columns with various preload ratios and steel yieldstrengths. It can be seen that regardless of the preload ratio, theultimate axial strength ratio of CFST slender beam-columnsincreases as the steel yield strength increases from 250 MPa to450 MPa. However, the strength ratio tends to decrease when the

a=0

a=0.3

a=0.6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

Ulti

mat

e ax

ial l

oad

P n/P

o

Eccentricity ratio e/D

Fig. 16. Effects of preloads and loading eccentricity ratio on the ultimate axialstrength.

a=0.3

a=0.6

0

1

2

3

4

5

6

7

8

9

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

Stre

ngth

redu

ctio

n (%

)

Eccentricity ratio e/D

Fig. 17. Strength reduction caused by preloads in CFST beam-columns with variousloading eccentricity ratios.

a=0a=0.3

a=0.6

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 100 200 300 400 500 600 700 800 900

Ulti

mat

e ax

ial l

oad

P n/P

o

Steel yield strength fsy (MPa)

Fig. 18. Effects of preloads and steel yield strengths on the ultimate axial strength.

a=0.3

a=0.6

0

2

4

6

8

10

12

14

0 100 200 300 400 500 600 700 800

Stre

ngth

redu

ctio

n (%

)

Steel yield strength fsy (MPa)

Fig. 19. Strength reduction caused by preloads in CFST beam-columns with varioussteel yield strengths.

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steel yield strength increases from 450 MPa to 690 MPa. This isattributed to the fact that the ultimate axial strengths of both theCFST column section and the CFST column are increased byincreasing the steel yield strength, but CFST column section hasa greater increment in its ultimate axial strength than the CFSTcolumn. The percentage reduction in the ultimate axial strengthcaused by preloads is shown in Fig. 19. It can be observed from thisfigure that for a CFST slender beam-column with the steel yieldstrength of 250 MPa, the preload with a ratio of 0.3 causes a 3.6%reduction in its ultimate axial strength. However, when increasingthe steel yield strength from 250 MPa to 690 MPa and the preloadratio from 0.3 to 0.6, the strength reduction could be as muchas 13.2%.

4.8. Effects of preload and loading angle

The fiber element analyses were undertaken on square CFSTslender beam-columns with a cross-section of 500�500 mm. Thebasic parameters examined in the study included the depth-to-thickness ratio of 60, loading eccentricity ratio of 0.2, slendernessratio of 60 and concrete compressive strength of 80 MPa. The yieldand tensile strengths of the steel were 690 MPa and 790 MPawhile the preload ratio was 0.3.

Fig. 20 shows the effects of the loading angle on the ultimateaxial strength of CFST slender beam-columns. The ultimate axialstrength increases as the loading orientation increases from 01 to451. However, the ultimate axial strength decreases with anincrease in the loading angle when the loading angle is greaterthan 451. This confirms the experimentally observed behavior ofCFST slender beam-columns under biaxial loads reported byBridge [39].

5. Design models

5.1. Liew and Xiong's design model

Based on Eurocode 4 [40], a design method for determining theultimate axial strength of axially loaded CFST columns with pre-load effects was proposed by Liew and Xiong [22] as

Npre;Ek ¼ χpre Npl;Rk ð12Þ

where Npl;Rk is the cross-section plastic resistance of the CFSTcolumn, given by

Npl;Rk ¼ Asf syþAcf0c ð13Þ

where As is the cross-sectional area of the steel tube and Ac is thecross-sectional area of the concrete core.

The column buckling strength reduction factor χpre is calculatedas follows:

χpre ¼1

ϕpreþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiϕ2

pre�λ2

q ð14Þ

ϕpre ¼1þ0:21 ðξpreλ�0:2Þþλ

2

2ð15Þ

in which λ is the relative slenderness of a CFST column, calcu-lated as

λ ¼ffiffiffiffiffiffiffiffiffiffiffiffiNpl:Rk

Ncr

sð16Þ

where Ncr is the Euler buckling load of the pin-ended CFST columnand expressed by

Ncr ¼π2ðEIÞef f

L2ð17Þ

where ðEIÞef f is the effective flexural stiffness of a CFST section,given as

ðEIÞef f ¼ EsIsþ0:6EcmIc ð18Þ

where Es and Ecm is the Young's modulus of the steel tube andconcrete respectively, and Is and Ic are the second moments of areaof the hollow steel tubular section and the concrete core respec-tively. The Young's modulus of concrete Ecm may be calculated as

Ecm ¼ 22000f 0cþ810

� �1=3

ð19Þ

In Eq. (15), ξpre is the preload influence factor, given as

ξpre ¼1�ðPpre=NEkÞ

1�βað20Þ

5.2. Proposed design model

The ultimate axial strength of a CFST column under axialcompression is influenced by the preload ratio ðβaÞ, relativeslenderness ðλÞ, and geometric imperfections. The geometricimperfection of L/1000 at the mid-height of the CFST column hasbeen incorporated in the numerical analysis. Based on the resultsof the fiber element analyses and Eurocode 4 [40], a design modelfor determining the ultimate axial strength of a CFST slendercolumn with preload effects is proposed by using the nonlinearregression analysis as follows:

Pup ¼ χprgPuo ð21Þ

where Puo is the ultimate axial strength of the column sectionunder axial compression, which is expressed by

Puo ¼ f syAsþ0:85f 0cAc ð22Þ

The column strength reduction factor χprg accounts for the effectsof preload ratio, relative slenderness and geometric imperfectionson the ultimate strength of axially loaded CFST slender columnsand is proposed as follows:

χprg ¼1

ϕprgþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiϕ2

prg�λ2

q ð23Þ

0.422

0.423

0.424

0.425

0.426

0.427

0.428

0 15 30 45 60 75 90

Ulti

mat

e axi

al lo

ad (P

n/Po)

Loading angle a (o)

Fig. 20. Effects of preloads and loading angle on the ultimate axial strength.

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ϕprg ¼1þ1:1αprg½ð1þβaÞλ�0:05�þ½ð1þζÞλ�2

2ð24Þ

αprg ¼1:2�βa

11:5ð1:2�βaÞ�4:3ð1:2�βaÞ2�1:5þ60ζð25Þ

ζ ¼0 for βar0:4βa �0:4

4 for 0:4oβar0:8

(ð26Þ

where the relative slenderness λ is calculated using Puo in Eq. (16).

5.3. Comparisons of design models

The column strength curves for axially loaded CFST columnswith preload ratios of 0.4 and 0.8 predicted by the fiber elementmodel and the proposed design model are given in Figs. 21 and 22.It can be observed that the column strength curves predicted bythe proposed design model agree very well with numerical resultsfor preload ratios ranging from 0.0 to 0.8. Fig. 23 presents acomparison of the column strength curves with a preload ratio of0.4 computed by Liew and Xiong's design method [22] and theproposed design model. Close agreement between these twodesign methods for the preload ratio of 0.4 is achieved, butLiew and Xiong's design method generally gives conservative

predictions of the ultimate axial strengths of CFST columns.Further comparisons of column strength curves for a preload ratioof 0.8 estimated by both design methods are provided in Fig. 24.It appears that Liew and Xiong's design method is very conserva-tive for estimating the ultimate axial strengths of axially loadedCFST columns with a preload ratio of 0.8 when compared with theproposed design model and the fiber element model. For the CFSTcolumn with a preload ratio of 0.8, the maximum strengthreduction computed by Liew and Xiong's design method is 41.5%.

6. Conclusions

This paper has presented the nonlinear analysis, behavior anddesign of biaxially loaded rectangular CFST slender beam-columnswith preload effects. A fiber element model has been developedfor simulating the load–deflection responses and strength envel-opes of biaxially loaded CFST slender beam-columns incorporatingthe important effects of preloads on the steel tubes and ofprogressive local buckling of the steel tube walls under stressgradients. Comparisons of computer solutions with existingexperimental results were made. The fiber element model was

0

0.2

0.4

0.6

0.8

1

1.2

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3

prg

Fiber element model

Proposed design model

Relative slenderness

Fig. 21. Comparison of proposed design model with fiber analysis for preload ratioof 0.4.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3

prg

Fiber element model

Proposed design model

Relative slenderness

Fig. 22. Comparison of proposed design model with fiber analysis for preload ratioof 0.8.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3

prg

Liew and Xiong's model [22]

Proposed design model

Relative slenderness

Fig. 23. Comparison of proposed design model with Liew and Xiong's model forpreload ratio of 0.4.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3

prg

Liew and Xiong's model [22]

Proposed design model

Relative slenderness

Fig. 24. Comparison of proposed design model with Liew and Xiong's model forpreload ratio of 0.8.

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utilized to study the fundamental behavior of biaxially loadedthin-walled CFST slender beam-columns by varying various impor-tant parameters. Based on the parametric studies, a design modelfor determining the ultimate axial strengths of axially loaded CFSTcolumns including preload effects has been proposed and com-pared with the design method given by Liew and Xiong [22].

The comparative studies demonstrate that the fiber elementmodel developed predicts well the experimentally observed beha-vior of CFST columns with preload effects. The parametric studiesindicate that preloads on the steel tubes have significant effects onthe strength and stiffness of CFST slender beam-columns withlarge preload ratios. However, the effects of preloads on CFST shortcolumns or CFST slender beam-columns with small preload ratioscan be ignored in the design. The preload has the most pro-nounced effect on CFST slender beam-columns with an e=D ratio of0.4. The proposed design model is shown to provide good predic-tions of the ultimate axial strengths of axially loaded CFST slendercolumns incorporating preload effects. It is found that the designmethod given by Liew and Xiong [22] is very conservative forestimating the ultimate axial strengths of axially loaded rectan-gular CFST slender columns with high preloads on the steel tubes.

References

[1] Furlong RW. Strength of steel-encased concrete beam columns. J Struct DivASCE 1967;93(5):113–24.

[2] Knowles RB, Park R. Strength of concrete-filled steel tubular columns. J StructDiv ASCE 1969;95(12):2565–87.

[3] Shakir-Khalil H, Zeghiche J. Experimental behaviour of concrete-filled rolledrectangular hollow-section columns. Struct Eng 1989;67(19):346–53.

[4] Shakir-Khalil H, Mouli M. Further tests on concrete-filled rectangular hollow-section columns. Struct Eng 1990;68(20):405–13.

[5] Schneider SP. Axially loaded concrete-filled steel tubes. J Struct Eng ASCE1998;124(10):1125–38.

[6] Varma AH, Ricles JM, Sause R, Lu LW. Seismic behavior and modeling of high-strength composite concrete-filled steel tube (CFT) beam-columns. J ConstrSteel Res 2002;58(5–8):725–58.

[7] Sakino K, Nakahara H, Morino S, Nishiyama I. Behavior of centrally loadedconcrete-filled steel-tube short columns. J Struct Eng ASCE 2004;130(2):180–8.

[8] Fujimoto T, Mukai A, Nishiyama I, Sakino K. Behavior of eccentrically loadedconcrete-filled steel tubular columns. J Struct Eng ASCE 2004;130(2):203–12.

[9] Ellobody E, Young B, Lam D. Behaviour of normal and high strength concrete-filled compact steel tube circular stub columns. J Constr Steel Res 2006;62(7):706–15.

[10] Liu D. Behaviour of eccentrically loaded high-strength rectangular concrete-filled steel tubular columns. J Constr Steel Res 2006;62(8):839–46.

[11] Lue DM, Liu JL, Yen T. Experimental study on rectangular CFT columns withhigh-strength concrete. J Constr Steel Res 2007;63(1):37–44.

[12] Ge HB, Usami T. Strength of concrete-filled thin-walled steel box columns:experiment. J Struct Eng ASCE 1992;118(11):3036–54.

[13] Bridge RQ, O'Shea MD. Behaviour of thin-walled steel box sections with orwithout internal restraint. J Constr Steel Res 1998;47(1–2):73–91.

[14] Uy B. Strength of concrete filled steel box columns incorporating localbuckling. J Struct Eng ASCE 2000;126(3):341–52.

[15] Han LH. Tests on stub columns of concrete-filled RHS sections. J Constr SteelRes 2002;58(3):353–72.

[16] Liang QQ, Uy B, Wright HD, Bradford MA. Local buckling of steel plates indouble skin composite panels under biaxial compression and shear. J StructEng ASCE 2004;130(3):443–51.

[17] Liang QQ, Uy B, Liew JYR. Local buckling of steel plates in concrete-filled thin-walled steel tubular beam-columns. J Constr Steel Res 2007;63(3):396–405.

[18] Mursi M, Uy B. Behaviour and design of fabricated high strength steel columnssubjected to biaxial bending, Part I: experiments. Advanced Steel Construction2006;2(4):286–315.

[19] Zha XX. The theoretical and experimental study on the behaviour of concretefilled steel tubular members subjected to compression-bending-torsion andinitial stresses on the steel tubes [Ph.D. Dissertation]. Harbin University of CivilEngineering and Architecture: Harbin, China; 1996 [in Chinese].

[20] Zhang XQ, Zhong ST, Yan SZ, Lin W, Cao HL. Experimental study about theeffect of initial stress on bearing capacity of concrete-filled steel tubularmembers under eccentric compression. J Harbin Univ Civil Eng Archit 1997;30(1):50–6 [in Chinese].

[21] Han LH, Yao GH. Behaviour of concrete-filled hollow structural steel (HSS)columns with pre-load on the steel tubes. J Constr Steel Res 2003;59(12):1455–75.

[22] Liew JYR, Xiong DX. Effect of preload on the axial capacity of concrete-filledcomposite columns. J Constr Steel Res 2009;65(3):709–22.

[23] Hu HT, Huang CS, Wu MH, Wu YM. Nonlinear analysis of axially loadedconcrete-filled tube columns with confinement effect. J Struct Eng ASCE2003;129(10):1322–9.

[24] Choi KK, Xiao Y. Analytical studies of concrete-filled circular steel tubes underaxial compression. J Struct Eng ASCE 2010;136(5):565–73.

[25] Portolés JM, Romero ML, Filippou FC, Bonet JL. Simulation and designrecommendations of eccentrically loaded slender concrete-filled tubularcolumns. Eng Struct 2011;33(5):1576–93.

[26] Lakshmi B, Shanmugam NE. Nonlinear analysis of in-filled steel-concretecomposite columns. J Struct Eng ASCE 2002;128(7):922–33.

[27] Mursi M, Uy B. Behaviour and design of fabricated high strength steel columnssubjected to biaxial bending, Part II: analysis and design codes. Adv SteelConstr 2006;2(4):314–54.

[28] Liang QQ. Performance-based analysis of concrete-filled steel tubular beam-columns, Part I: theory and algorithms. J Constr Steel Res 2009;65(2):363–72.

[29] Liang QQ. Performance-based analysis of concrete-filled steel tubular beam-columns, Part II: verification and applications. J Constr Steel Res 2009;65(2):351–62.

[30] Patel VI, Liang QQ, Hadi MNS. High strength thin-walled rectangular concrete-filled steel tubular slender beam-columns, Part I: modeling. J Constr Steel Res2012;70:377–84.

[31] Patel VI, Liang QQ, Hadi MNS. High strength thin-walled rectangular concrete-filled steel tubular slender beam-columns, Part II: behavior. J Constr Steel Res2012;70:368–76.

[32] Liang QQ, Patel VI, Hadi MNS. Biaxially loaded high-strength concrete-filledsteel tubular slender beam-columns, Part I: multiscale simulation. J ConstrSteel Res 2012;75:64–71.

[33] Xiong DX, Zha XX. A numerical investigation on the behaviour of concrete-filled steel tubular columns under initial stresses. J Constr Steel Res 2007;63(5):599–611.

[34] Patel VI, Liang QQ, Hadi MNS. Numerical analysis of circular concrete-filledsteel tubular slender beam-columns with preload effects. Int J Struct Stab Dny2013;13(3):1250065 (23 pages).

[35] Mander JB, Priestley MJN, Park R. Theoretical stress–strain model for confinedconcrete. J Struct Eng ASCE 1988;114(8):1804–26.

[36] ACI-318. Building code requirements for structural concrete and commentary.Detroit (MI): ACI; 2011.

[37] Tomii M, Sakino K. Elastic-plastic behavior of concrete filled square steeltubular beam-columns. Trans Archit Inst Jpn 1979;280:111–20.

[38] Müller DE. A method for solving algebraic equations using an automaticcomputer. Math Tables Other Aids Comput 1956;10(56):208–15.

[39] Bridge RQ. Concrete filled steel tubular columns. School of Civil Engineering,Research Report No. R283. Sydney, Australia: The University of Sydney; 1976.

[40] EN 1994-1-1. Eurocode 4: Design of composite steel and concrete structures.Part 1-1, General rules and rules for buildings, CEN; 2004.

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