Journal of Constructional Steel Research 67 (2011) 1567–1577

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Journal of Constructional Steel Research

Elasto-plastic analysis of circular concrete-filled steel tube stub columns

Fa-xing Ding a, Zhi-wu Yu a, Yu Bai b, Yong-zhi Gong a,⁎a School of Civil Engineering and Architecture, Central South University, Changsha, Hunan Province, 410075, P. R. Chinab Department of Civil Engineering, Monash University, Clayton, VIC 3168, Australia

⁎ Corresponding author. Tel.: +86 731 82656540; faxE-mail address: gyzh@yahoo.cn (Y. Gong).

0143-974X/$ – see front matter. Crown Copyright © 20doi:10.1016/j.jcsr.2011.04.001

a b s t r a c t

a r t i c l e i n f o

Article history:Received 14 December 2010Accepted 4 April 2011Available online 5 May 2011

Keywords:Concrete-filled steel tubeAxial loading behaviorElasto-plastic analysisStress–strain relationshipUltimate capacityConfinement effect

This paper presents a full-range elasto-plastic analysis using continuummechanics on circular concrete-filledsteel tube (CFT) stub columns under concentric loading condition, covering concrete strengths from 30 to120 MPa and diameter-to-wall thickness ratio (D/t) greater than 20. Firstly, a constitutive model wasemployed for laterally-confined concrete under axial compression. A continuum mechanics model was thenestablished and the corresponding elasto-plastic analysis was performed through a FORTRAN program. Thismodel is able to present full-range stress-strain developments in axial, radial and perimeter directions andfurther clarify the load sharing pattern between the steel tube and the concrete core. Based on the proposedmodel, parametric analysis was conducted to investigate the effects of material strengths and sectional steelratio on the triaxial stress–strain developments and the load sharing pattern. In addition, the model wassimplified to predict the ultimate capacity and the load-axial strain relationship of CFT composite sections andthe results are in good agreement with experiments. Further comparisons were made of the approachdeveloped by Han et al. 2004 and the existing international standards.

Crown Copyright © 2011 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Concrete-filled steel tube (CFT) columnsprovide excellent structuralbenefits for seismic resistance such as high ductility and large energyabsorption capacity. In addition, construction time can be considerablyreduced due to the elimination of permanent formwork. As a result,numerous studies on CFT columns with normal and high-strengthconcrete have been conducted in recent years [1–7]. The enhancementof CFT columns in structural properties is due to the composite actionbetween individual constituent elements. The steel tube acts as bothlongitudinal and transverse reinforcement and also provides a confiningpressure to the concrete,which introduces the concrete core to a triaxialstress state. On the other hand, the steel tube is stiffened by the concretecore. The resulting ultimate strength of CFT columns is determined bythe constituent material properties such as the compressive strength ofconcrete, the yield strength of steel, and the steel ratio.

Initially, focuswasmainly given to the ultimate capacity estimationof CFT stub columns under concentric compression [8–10]. Althoughlater on, analytic studies have included the confinement effects on theconcrete core and the stress transfer between the steel tube and theconcrete core [3–5], the load sharing pattern has not been clarifiedbetween the steel tube and the concrete core for a full range of loading.It appears that this load sharing pattern can be understood through anelasto-plastic analysis based on continuummechanics, and the triaxial

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stress–strain developments (axial, radial and perimeter directions)can be further predicted in a full elasto-plastic range. A recent modeldeveloped by Bai et al. [11] theoretically formulated the confinementeffects provided by the steel tube and the improved load-carryingcapacity of concrete core through a continuum mechanics method,only ultimate state was considered however.

The aimof this study is therefore to employ continuummechanics toachieve a full-range elasto-plastic analysis on circular CFT stub columnsubjected to axial compressive loading. For this purpose, a propermaterial constitutive model for concrete was adopted first. Then acontinuum mechanics model was established for the entirely-loadedCFT section. The modeling results were further verified againstexperimental data. Finally the influences of the compressive strengthof concrete, the yield strength of steel, and the steel ratio on the stress-strain development and load sharing pattern between steel tube andconcrete core were discussed. In addition, based on the elasto-plasticanalysis, a simplified model was developed for calculating the ultimatecapacity and the axial stress–strain relationship of circular CFT stubcolumns and comparedwith theapproachproposedbyHanet al. (2004)[12], and the existing standards such as AISC-360 (2005) [13] andEurocode 4 (2004) [14].

2. Material properties and constitutive models

The mechanical model for the circular stub column is establishedbased on the following assumptions: 1) there is no slip between thesteel tube and concrete core, and 2) the concrete core and steel tubeare in full contact with each other and there is no local buckling of the

ghts reserved.

Nomenclature

Ac Cross-sectional area of concrete coreAs Cross-sectional area of steel tubeAsc Cross-sectional area of CFT stub columnD Outer diameter of circular steel tubeEc Elastic modulus of concrete estimatedE0 Secant modulus of concrete at the peak stress under

uniaxial compressionEf Secant modulus of concrete at the peak stress in the

major directionEs Steel modulus of elasticityEsc Composite modulus of elasticity for CFT stub columnfc Uniaxial compressive strength of concretefcc Axial compressive strength of concrete subjected to

lateral confining pressurefcu Cubic compressive strength of concretefcyl Compressive cylinder strength of concretefs Yield strength of steelfsc Composite stress of CFT stub columnfsc,p Nominal axial limit stress of elasticity of CFT stub

columnfsc,u Ultimate strength of CFT stub columnfsc,y Residual capacity of CFT stub columnfu Tensile strength of steelJ2 The second stress invariantsL Length of CFT stub columnN Axial loadNc External compressive load on concrete coreNs External compressive load on steel tubeNu Axial ultimate capacityNy Residual capacityuL,c Axial displacement of concrete coreuL,s Axial displacement of steel tubeur,c Radial displacement of concrete coreur,s Radial displacement of steel tubet Wall thickness of steel tubeya Stress ratio in elastic limit of concreteσL,c Axial stress of concrete subjected to lateral confining

pressureσL,s Axial stress of steel tubeσr,c Confining pressure around the concreteσr,s Radial stress of steel tubeσθ,c Perimeter stress of concrete coreσθ,s Perimeter stress of steel tubeσi Equivalent stressεc Strain at peak uniaxial compression of concreteεcc Strain at peak axial compression of concrete subjected

to lateral confining pressureεi Equivalent strainεL,c Axial strain of concrete subjected to lateral confining

pressureεL,s Axial strain of steel tubeεL Average axial strain of CFT stub column at mid-heightεL,p Nominal limit axial strain of elasticity of CFT stub

columnεsc,0 Strain at peak axial compression of CFT stub columnεθ,c Perimeter strain of concrete coreεθ,s Perimeter strain of steel tubeεr,c Radial strain of concrete coreεr,s Radial strain of steel tubeεy Yield strain of steelεθ,s Perimeter expansion strainv0 Initial Poisson's ratio of concrete

vc Poisson's ratio of concretevf Peak Poisson's ratio of concretevs Poisson's ratio of steelvsc Strain ratio of steel tubeρ Cross-sectional steel ratio of CFT columnΦ Confinement index, = fsAs/(fcAc)

1568 F. Ding et al. / Journal of Constructional Steel Research 67 (2011) 1567–1577

stub column taking place. Constitutive models of concrete and steeltube were proposed and discussed as follows.

2.1. Concrete

The relationships between fc and fcu and between εc and fcu wereestimated by the following equations in Yu and Ding (2003) [15]:

fc = 0:4f 7=6cu ð1Þ

εc = 383f 7=18cu × 10−6 ð2Þ

It should be noted that different function forms of such relation-ships may exist, results from the adopted ones (Eqs. 1 and 2) are inconsistence with others.

When concrete was subjected to a lateral confining pressure, theaxial compressive strength fcc and the corresponding strain εcc becamemuch higher than those of the unconfined concrete. Based on theexperimental data of the laterally-confined normal strength concrete(NC) and high-strength concrete (HSC) cylinders under axialcompressive loading [16–21], the relationship between fcc and fcwas estimated as follows:

fcc = fc + 3:4σ r;c: ð3Þ

Eq. (3) is applicable for concrete strength ranging from 30 tol20 MPa. The comparison between the predicted and reported data isshown in Fig. 1, a reasonable agreement can be found.

For concrete subjected to triaxial loading condition, Ottosen(1979) [22] proposed the following equation to estimate the secantmodulus at the peak stress in the axial direction (Ef):

Ef =E0

1 + 4 A1−1ð Þx ð4Þ

where, x =ffiffiffiffiJ2

p=fc

� �f−1=

ffiffiffi3

p, E0= fc/εc, A1 is the variable in the

ascending branch of uniaxial stress–strain relationship. For the

0.0 0.4 0.8 1.2 1.6 2.00

2

f cc/

f c

4

6

8

cc 1 r,c

c c

f

f f

Candappa et al. [18] 9 points

Sfer et al. [20] 12 points

Ansari et al. [16] 17 points

Xie et al. [21] 33 Points

Imran et al. [19] 42 points

Attard et al. [17] 38 points

3.4

r,c/fcσ

Fig. 1. Axial compressive strength of confined concrete.

0.00 0.01 0.02 0.03 0.04 0.05 0.060

20

40

60

80

100

120

140

Richart et al. [23]Predicted curves

fc=25.2MPa

7.52MPa

13.9MPa

28.2MPa

Lateral pressure 3.79MPa

εL,c

σ L

,c(M

Pa)

Fig. 2. Axial compressive stress–axial strain relationship for confined concrete.

1569F. Ding et al. / Journal of Constructional Steel Research 67 (2011) 1567–1577

laterally-confined concrete under axial compressive loading, Eq. (4)was simplified as follows:

Ef =E0

1 + 5:54 A1−1ð Þσ r;c=fc:ð5Þ

However Eq. (5) could not satisfy the experimental data on thetriaxial compressive loading condition. When the confining pressureis small, Eq. (5) overestimates the value of Ef in comparison withexperimental data. From the test results by Richart et al. (1929) [23],Eq. (5) was modified as follows:

Ef =E0

1 + 4:8 A1−1ð Þ σ r;c=fc� �0:5 ð6Þ

where, A1 was estimated according to Yu and Ding (2003) [15]:

A1 = 9:1f−4=9cu : ð7Þ

In this way the relationship between εcc and εc can be estimated by

εcc = εc 1 + 3:4σ r;c

fc

� �1 + 4:8 A1−1ð Þ σ r;c

fc

� �0:5� �: ð8Þ

When the laterally-confined concrete is subjected to an axialcompression, the axial stress–strain relationship could be describedby the following expressions:

y =

A2x + B2−1ð Þx21 + A2−2ð Þx + B2x

2 x≤1

xα2 x−1ð Þ2 + x

x N 1

8>>><>>>:

ð9Þ

where y=σL,c/fcc, x=εL,c/εcc, and

A2 = A1 1 + 4:8 A1−1ð Þ σr;c

fc

� �0:5� �: ð10Þ

A2 is determined by Eq. (10) for the ascending branch of the axialstress–strain relationship of laterally-confined concrete. B2 is aparameter to control the decrease of elastic modulus at the ascendingbranch of the axial stress-strain relationship. According to thedefinition of elastic modulus, when σL,c is within the ascendingbranch and less than 0.4 fc, the curve can be considered as a straightline. Therefore, when x=0.4 fc/(fccA3) and y=0.4fc/fcc, the expressionfor B2 can be obtained as

B2 =y + A2−2ð Þxy− A2−xð Þx

1−yð Þx2 ð11Þ

and

α2 = 0:15 ð12Þ

for the descending branch of the axial stress–strain relationship.Comparison between the experimental results of axial stress–

strain relationship for laterally-confined concrete under axial com-pression and the description using Eq. (9) is shown in Fig. 2. As can beseen, the model gives a reasonable fitting of the experimental curves.A typical axial stress–strain relationship is given in Fig. 3 (a).

The secant value of Poisson's ratio of concrete (vc) under uniaxialcompression varies from 0.15 to 0.22 when concrete is in elastic stage,and a representative value of 0.2 is adopted. Poisson's ratio of concreteincreases gradually as the axial strain increases, and vc could begreater than 0.5 when the stress approaches the uniaxial compressive

strength. In this study, Poisson's ratio of concrete vc was assumed asfollows:

vc =

v0 x≤1; y≤ya

vf− vf−v0� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1− y−ya1−ya

� �2s

x≤1; yaby≤1

vf x N 1; yb1

8>>>>><>>>>>:

ð13Þ

where, vf=1−0.0025(fcu−20) and ya=0.3+0.002(fcu−20) basedon [22]. It can be seen that the expression of vf also reflects theinfluence of concrete strength. One difference from [22] is that thevalue of vf in this study could be continuously great than 0.5 becauseof the confinement effect on concrete core offered by steel tube.

2.2. Steel tube

Typical stress–strain relationship for steel consists of four stages asshown in Fig. 3(b). Detailed expressions were given as:

σ i =

Esεi εi≤εyfs εybεi≤εstfs + ζEs εi−εstð Þ εstbεi≤εufu εi N εu

8>><>>: ð14Þ

where Es=206,000 MPa, fu=1.5fs, Est=ζEs, εu=εst+0.5fs/(ζEs), andζ and εst are two coefficients. In this paper, it was considered thatεst=12εy and εu=120εy, and the resulting value of ζ is 1/216.

Poisson's ratio of steel denoted as vs was assumed as follows:

vs =0:285 εi≤0:8εy1:075 σ i=fs � 0:8ð Þ + 0:285 0:8 εybεi≤εy0:5 εi N εy

:

8<: ð15Þ

3. Mechanics model

3.1. Formulation in elastic stage

When a CFT stub column is subjected to the axial compressiveloading condition and the load is applied evenly across the cross-sectional steel and concrete, themechanicsmodel of concentric cylinderof steel tube and concrete core can be established. As shown in Fig. 4, incase of small deformation, the axial strains were identical for concretecore and steel tube (i.e. εL,c=εL,s=εL), and the axial symmetric plane

(a) concrete (b) steel

Fig. 3. Material stress–stain relationship.

1570 F. Ding et al. / Journal of Constructional Steel Research 67 (2011) 1567–1577

strain simplification becomes applicable. According to Airy stressfunction – Γ=C1lnr+C2r

2lnr+C3r2+C4 (where C1, C2, C3, and C4 are

coefficients) – the following equations could be obtained:

(1) For concrete core section [0b r≤(D/2-t)]:Stress

σr ;c = σθ ;c = 2C3 ð16Þ

σL ;c = EcεL + 4νcC3 ð17Þ

Fig. 4. Mechanics model of CFT stub column.

Strain

εr ;c = εθ ;c = 2C3 1−νc−2ν2c

� �= Ec−νcεL ð18Þ

Displacement

ur ;c = r 2C3 1−νc−2ν2c

� �= Ec−νcεL

h ið19Þ

uL ;c = LεL ð20Þ

(2) For steel tube section [(D/2−t)≤ r≤D/2]Stress

σr ;s = C10= r2 + 2C3

0 ð21Þ

σθ;s = −C10= r2 + 2C3

0 ð22Þ

σL;s = EsεL + 4νsC30 ð23Þ

Strain

εr;s =1 + vs

Es

C01

r2+

2C03 1−vs−2v2s� �

Es−vsεL

εθ;s = −1 + vsEs

C01

r2+

2C03 1−vs−2v2s� �

Es−vsεL

8>>>>><>>>>>:

ð24Þ

Displacement

ur;s = r2C0

3 1−vs−2v2s� �

Es−vsεL

24

35−1 + vs

Es

C01

r

uL;s = LεL

8>>><>>>:

ð25Þ

In Eqs. (16)–(25), Ec is estimated by the equation Ec=9500fcu1/3

according to Yu and Ding (2003) [15] and C1′ and C3′ are coefficients.When full contact between concrete core and steel tube is

assumed, according to Saint-Venant's principle, the stress conditionfor the stub column at mid-height can be written as

σL ;cAc + σL ;sAs = N ð26Þ

1571F. Ding et al. / Journal of Constructional Steel Research 67 (2011) 1567–1577

Let fsc=N/Asc, where Asc=Ac+As, the following equation can beobtained:

fsc = 1−ρð ÞσL;c + ρσL ;s ð27Þ

where ρ=As/Asc≈4 t/D.Based on continuum mechanics, the following boundary condi-

tions can be considered for axially-loaded cylinders of steel tube andconcrete core in the mid-height region:

(1) Stress boundary condition

σr j r =D=2 = 0 ð28Þ

(2) Stress and displacement compatible condition

σ r j −r= D=2−tð Þ = σ r j þ

r= D=2−tð Þ ð29Þ

ur j −r= D=2−tð Þ = ur j þ

r= D=2−tð Þ ð30Þ

According to Eqs. (16)–(25) and Eqs. (28)–(30), the expressionsof C1′, C3′ and C3 can be obtained as:

C01 =

D2

4vc−vsð Þ 1−ρð ÞQ⋅EsεL

C03 = −1

2vc−vsð Þ 1−ρð ÞQ⋅EsεL

C3 =12

vc−vsð ÞρQ⋅EsεL

8>>>>>>><>>>>>>>:

ð31Þ

where Q=[(1−vc−2vc2)nρ+(2−2vs2−ρ+ρvs+2ρvs2)]−1 and n=Es/Ec.

Substituting Eq. (31) for Eqs. (16) to (18) and Eqs. (21) to (24), thefollowing equations can be obtained:

σ r;c = σθ;c = ρ vc−vsð ÞQ⋅EsεLσL;c = Ec + 2ρvc vc−vsð ÞQEs½ �εL

ð32Þ

εr;c = εθ;c = ρ 1−vc−2v2c� �

vc−vsð ÞQn−vch i

⋅εL ð33Þ

σ r;s = D=2rð Þ2−1h i

vc−vsð Þ 1−ρð ÞQ⋅EsεLσθ;s = − D=2rð Þ2 + 1

h ivc−vsð Þ 1−ρð ÞQ⋅EsεL

σL;s = 1−2vs vc−vsð Þ 1−ρð ÞQ½ �⋅EsεL

8>><>>: ð34Þ

εr;s = D=2rð Þ2 1 + vsð Þ− 1−vs−2v2s� �h i

⋅ 1−ρð Þ vc−vsð ÞQεL−vsεL

εθ;s = − D=2rð Þ2 1 + vsð Þ + 1−vs−2v2s� �h i

⋅ 1−ρð Þ vc−vsð ÞQεL−vsεL

8<:

ð35Þ

Substituting the expressions of σL,c in Eq. (32) and σL,s in Eq. (34)for Eq. (27), the composite stress–strain relationship of the stubcolumn in elastic stage can be written as:

fsc = EscεL ð36Þ

Esc = 1−ρð ÞEc + ρEs + 2 vc−vsð Þ2 1−ρð ÞρQEs: ð37Þ

3.2. Formulation in inelastic stage

For the steel tube subjected to both lateral pressure and axialcompression, its internal edge (where r=D/2−t) would yield first,and the plastic region would expand rapidly to the whole section ofsteel tube considering that t≪D . When analyzing the stress of thesteel tube in plastic stage, the radial stress of steel tube was ignored.According to Eq. (34), the stresses of steel tube after yield can bewritten as:

σ θ;s = −2 vc−vsð Þ 1−ρð ÞQt⋅EtsεLσ L;s = 1−2vs vc−vsð Þ 1−ρð ÞQt½ �⋅EtsεL

8<: ð38Þ

where

Qt = 1−vc−2v2c� �

ntρ + 2−2v2s−ρ + ρvs + 2ρv2s� �h i−1

;

nt = Ets=Etc; E

tc = Et−2ρvc vc−vsð ÞQtE

ts:

Est is secant modulus of steel calculated by Eq. (14) and Et is secant

modulus of concrete calculated by Eq. (9).The strains at mid-height of steel tube after yield were:

εr;s = 1 + vsð Þ= 1−ρ=4ð Þ− 1−vs−2v2s� �h i

⋅ 1−ρð Þ vc−vsð ÞQtεL−vsεL

εθ;s = − 1 + vsð Þ= 1−ρ=4ð Þ + 1−vs−2v2s� �h i

⋅ 1−ρð Þ vc−vsð ÞQtεL−vsεL:

8<: ð39Þ

The strains at external surface of steel tube can be calculated byEq. (34) just by replacing the symbol Q with Qt.

For the concrete core, the non-linear behavior appears as the loadincreases. The equations for the stresses and strains of concrete corein inelastic stage were in the same form as those in elastic stage,while the symbols Ec, Es and Q should be replaced by Ec

t, Est and Qt

respectively. As a result the composite stress–strain relationship ofthe CFT stub column in inelastic stage can be written as:

fsc = EtscεL ð40Þ

where Esct =(1−ρ)Ect +ρEst+2(vc−vs)2(1−ρ)ρQ tEs

t.Therefore, the composite stress-axial strain relationship for the

stub column in the full elasto-plastic range can be written as:

fsc =EscεL εL≤εL;p

EtscεL εL N εL;p:

(ð41Þ

3.3. Comparison with experimental data

Based on the above formulations, an elasto-plastic analysis wasdeveloped using a Fortran 90 computer program. The modelingresults are compared with experimental results in Fig. 5 for the load-dependent strain developments, where a satisfactory agreement canbe found. A divergence between the predicted and experimentalresponses in the post-elastic region was noticed, which may becontributed by the inaccurate representation of the material proper-ties in the constitutive models. However the resulting difference inload estimation is less than 9% and has no significant consequence onthe overall performance of the proposed model.

Due to the planar stress state simplification, the externalcompressive load (Ns) versus axial strain (εL) relationship for thesteel tube was calculated by Wang et al. (2006) [6] based on the testresults, therefore the load on concrete core (Nc) can be separated fromthe total load (N). These results from [6] are shown in Fig. 6 for acomparison with the modeling results. Analytical results in Fig. 6

0

100

200

300

400

Wang et al. [6]

Predicted curves

θ ,s

L,cσσ

L,sσ

M-C-1-120h

(MPa

)

Lε

σ

0.000 0.003 0.006 0.009 0.012

Lε

0.000 0.003 0.006 0.009 0.0120

300

600

900

1200

1500

1800

Wang et al. [6]

Predicted curves

fs=325.3MPa , f

cu=57.77MPa

D×t×L=140.3×3.62×418mm

s

c

N

N

N

N(k

N)

(a) stresses-axial strain relationship (b) distribution of axial forces

Fig. 6. Comparisons between predicted curves and experimental ones.

-0.010 -0.005 0.000 0.005 0.010 0.0150.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

D×t×L=219×4.78×650mmfs=350MPa , f

cu=50.5MPa

SZ5S4A1a

Yu et al. [7]

Predicted curves

εθ,s εL

-0.015 -0.010 -0.005 0.000 0.005 0.010 0.0150.0

0.3

0.6

0.9

1.2

1.5

1.8

Wang et al. [6]

predicted curves

εθ,s

M-C-1-120h

D×t×L=140.3×3.62×418mmfs=325.3MPa

fcu

=57.8MPa

N (

MN

)

εL

N (

MN

)

Fig. 5. Comparisons between predicted curves and tested ones on axial load–strains relationship.

1572 F. Ding et al. / Journal of Constructional Steel Research 67 (2011) 1567–1577

further demonstrated that for the concrete core confined by steeltube, its axial stress and load-bearing capacity increase considerablyand the ductility improves as well, with the decrease of the axial stressof steel tube; therefore load sharing mechanism is achieved throughthe axial stress transferring from steel tube to concrete core.

3.4. Parametric analysis and discussion

Through the numerical analysis, the typical predicted compositestress-axial strain relationship for the CFT stub column are summa-rized in Fig. 7 considering different steel ratios, concrete strength andsteel strength. The influences of those parameters on stress–axialstrain relationships for the steel tube and concrete core aresummarized in Figs. 8 to 10. As seen in Figs. 7 to 10, the stresses insteel tube and concrete core were significantly influenced during the

0

20

40

60

80

100t=4mm, =1.441

t=3mm, =1.046

t=2mm, =0.676

Lε

t=1mm, =0.327

D=100mm fs=235MPa f

cu=40MPa

f sc (

MPa

)

f sc (

MPa

)

f (

MPa

)

0

40

80

120

160

200

t=5mm, =1.143

t=4mm, =0.884t=3mm, =0.642t=2mm, =0.415

t=1mm, =0.201

D=100mm fs=420MPa f

cu=100MPa

0.00 0.01 0.02 0.03 0.04

Lε0.00 0.01 0.02 0.03 0.04

(a) Steel ratio (for NC) (b) Steel ratio (for HSC) (cΦ

Φ

ΦΦ

ΦΦΦΦΦ

Fig. 7. Typical predicted composite stress–axia

loading process. For the concrete core confined by steel tube, the axialstress of concrete core is improved greatly and the ductility isenhanced. At the same time, the axial stress of steel tube is reducedbecause of lateral stress. When other parameters were fixed, theconfinement effect of the column is strengthened with the steel ratioand the strength of steel, and the ultimate strength and ductility of thecolumn improve as well; while with the increase of concrete strength,confinement effect and the ductility of the column decrease howeverthe ultimate strength of the column increases.

4. Simplified model

Fig. 11 shows a typical composite stress (fsc) and axial strain (εL)relationship for CFT stub columns under axial compression predictedthrough the numerical elasto-plastic analysis above. Generally, the fsc-εL

sc f sc (

MPa

)

0

20

40

60

80

fcu

=60MPa =0.421fcu

=50MPa =0.521fcu

=40MPa =0.676fcu

=30MPa =0.945D=100mm t=2mm f

s=235MPa

Lε0.00 0.01 0.02 0.03 0.04

Lε0.00 0.01 0.02 0.03 0.040

20

40

60

80

100

120

fs=600MPa =1.075

fs=420MPa =0.752

fs=390MPa =0.699

fs=345MPa =0.618

fs=235MPa =0.421

D=100mm t=2mm fcu

=60MPa

) Strength of concrete (d) Strength of steel

ΦΦΦΦ

ΦΦΦΦΦ

l strain relationship for CFT stub columns.

0

20

40

60

80

100

t=4mm, =1.441

t=3mm, =1.046

t=2mm, =0.676t=1mm, =0.327

D=100mm fs=235MPa f

cu=40MPa

(MPa

)L

,cσ

(MPa

)L

,sσ

(MPa

)σ

(MPa

)r,

cσ

0

6

12

18

24

t=4mm, =1.441

t=3mm, =1.046

t=2mm, =0.676

t=1mm, =0.327

D=100mm fs=235MPa f

cu=40MPa

0

50

100

150

200

250

t=4mm, =1.441t=3mm, =1.046

t=2mm, =0.676t=1mm, =0.327

D=100mm fs=235MPa fcu=40MPa 0

50

100

150

200

250

,s θ

t=4mm, =1.441t=3mm, =1.046

t=2mm, =0.676

t=1mm, =0.327

D=100mm fs=235MPa fcu=40MPa

(a) Axial stress of concrete (b) Radial stress of concrete (c) Axial stress of steel (d) Perimeter stress of steel

Lε0.00 0.01 0.02 0.03 0.04

Lε0.00 0.01 0.02 0.03 0.04

Lε0.00 0.01 0.02 0.03

Lε0.00 0.01 0.02 0.03

ΦΦ

ΦΦ

Φ

Φ

ΦΦ Φ

Φ

ΦΦ

ΦΦ

ΦΦ

Fig. 8. Influence of steel ratio on stresses of the stub columns.

0

10

20

30

40

50

60

70

fcu

=60MPa =0.421fcu

=50MPa =0.521fcu

=40MPa =0.676fcu

=30MPa =0.945D=100mm t=2mm fs=235MPa

0

3

6

9

12D=100mm t=2mm f

s=235MPa

fcu

=30MPa =0.945fcu

=40MPa =0.676fcu

=50MPa =0.521fcu

=60MPa =0.4210

50

100

150

200

250

D=100mm t=2mm fs=235MPa

fcu

=60MPa =0.421fcu

=50MPa =0.521fcu

=40MPa =0.676fcu

=30MPa =0.945

0

50

100

150

200

250D=100mm t=2mm f

s=235MPa

fcu

=30MPa =0.945fcu

=40MPa =0.676fcu

=50MPa =0.521fcu

=60MPa =0.421

(a) Axial stress of concrete (b) Radial stress of concrete (c) Axial stress of steel (d) Perimeter stress of steel

(MPa

)L

,cσ

Lε0.00 0.01 0.02 0.03 0.04

Lε0.00 0.01 0.02 0.03 0.04

(MPa

)L

,sσ

(MPa

)σ

(MPa

)r,

cσ

,s θ

Lε0.00 0.01 0.02 0.03

Lε0.00 0.01 0.02 0.03

ΦΦΦΦ

ΦΦΦΦ

ΦΦΦΦ

ΦΦΦΦ

Fig. 9. Influence of strengths of concrete on stresses of the stub columns.

0

20

40

60

80

100

120

fs=600MPa =1.075

fs=420MPa =0.752

fs=390MPa =0.699

fs=345MPa =0.618

fs=235MPa =0.421

D=100mm t=2mm fcu

=60MPa0

4

8

12

16

20

24fs=600MPa =1.075

fs=420MPa =0.752

fs=390MPa =0.699

fs=345MPa =0.618

fs=235MPa =0.421

D=100mm t=2mm fcu

=60MPa0

100

200

300

400

500

600

D=100mm t=2mm fcu

=60MPa

fs=600MPa =1.075

fs=420MPa =0.752

fs=390MPa =0.699

fs=345MPa =0.618

fs=235MPa =0.421

0

100

200

300

400

500

600fs=600MPa =1.075

fs=420MPa =0.752

fs=390MPa =0.699

fs=345MPa =0.618

fs=235MPa =0.421

D=100mm t=2mm fcu

=60MPa

(a) Axial stress of concrete (b) Radial stress of concrete (c) Axial stress of steel (d) Perimeter stress of steel

Lε0.00 0.01 0.02 0.03 0.04

Lε0.00 0.01 0.02 0.03 0.04

Lε0.00 0.01 0.02 0.03

Lε0.00 0.01 0.02 0.03

(MPa

)L

,cσ

(MPa

)L

,sσ

(MPa

)σ

(MPa

)r,

cσ

,s θ

ΦΦΦΦΦ

ΦΦΦΦΦ

ΦΦΦΦΦ

ΦΦΦΦΦ

Fig. 10. Influence of strengths of steel on stresses of the stub columns.

1573F. Ding et al. / Journal of Constructional Steel Research 67 (2011) 1567–1577

relationship of the stub column appears to display strain-hardening,elastic-perfectly plastic, and strain-softening behaviors, which aredominated by the parameters such as strength of concrete, strength of

Fig. 11. Typical composite stress (fsc) versus axial strain (εL) relationship.

steel, and steel ratio. The effects of these parameters on the behavior ofCFT columns can be simplified for the ultimate capacity Nu (or ultimatestrength fsc,u), composite modulus of elasticity (Esc), nominal axial limitstress of elasticity (fsc,p) the corresponding axial strain (εL,p), strain atpeak axial compression (εsc,0), and residual capacity Ny (or residualstrength fsc,y), as discussed below.

4.1. Ultimate capacity

The maximum load is defined as the ultimate capacity (Nu) orultimate strength (fsc,u) of the column, i.e. the point when dfsc/dεL=0in the fsc-εL relationship. Analytical results based on the elasto-plasticmodel indicated that the ultimate capacity of CFT stub column couldbe achieved when the concrete core reaches its ultimate strength ataxial direction. Meanwhile, the steel tube is within its elastic-perfectlyplastic or strain-hardening stage.

If the steel tube is at its perfectly plastic state and subjected to theVon-Mises yield condition, the strength criterion for concrete core oflinear expression is given as:

fcc = fc + kσ r;c ð42Þ

Table 1Influence of k on Nu,2/Nu,3.

k 3 3.2 3.4 3.6 3.8 4.0 4.1 4.3

Nu,2/Nu,3 1.018 1.010 1.005 1.002 1.000 1.000 1.000 1.000

1574 F. Ding et al. / Journal of Constructional Steel Research 67 (2011) 1567–1577

where k is a constant. According to Eqs. (32) and (38), the relationshipbetween σr,c and σθ,s in inelastic stage can be expressed as:

σ r;c =ρ

2 1−ρð Þσθ;s: ð43Þ

According to the Von-Mises yield condition for steel tube, thefollowing equation was obtained:

σ2L;s+ σ L;sσ θ;s + σ2

θ;s = f 2s : ð44Þ

Substituting Eqs. (42) and (43) with Eq. (44), σL,s was expressed as:

σL;s

fs=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− 3

Φ2

σr;c

fc

� �2s

− 1Φσr;c

fcð45Þ

where Ф is the confinement index and

Φ =fsAs

fcAc=

ρ1−ρ

fsfc:

ð46Þ

The ultimate capacity of the stub column in the mid-height regioncan be written as

Nu = fccAc + σ L ;sAs ð47Þ

Substituting Eqs. (45) and (42) with Eq. (47), the resulting ultimatecapacity of the stub column Nu,1 can be expressed as:

Nu;1 = Acfc 1 +

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΦ2−3

σ r;c

fc

� �2s

+ k−1ð Þσ r;c

fc

24

35 ð48Þ

and it can be found that

dNu;1

dσr;c= 0⇒

σr;c

fc=

Φ k−1ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 + 3 k−1ð Þ2

q:

ð49Þ

0 1 2 3 40

2

4

6

8

10

Eq. (5

Luksha and Nesterovich [25] 8 pointsGiakoumelis and Lam [10] 8 pointsO'Shea and Bridge [26] 15 pointsPrion and Boehme [27] 4 pointsSakina and Hayashi [28] 9 pointSchneider [4] 3 pointsYu et al. [7] 6 points

Cai [8] 25 pointsCai and Gu [9] 4 pointsHan and Yao [12] 4 poiHan et al. [24] 26 pointHuang et al. [2] 3 point

N u/f c

A c

Φ

(a) <5

Fig. 12. Comparisons between predicted results a

Substituting Eq. (49) for Eq. (48), the maximum ultimate capacity ofthe stub column Nu,2 can be expressed as:

Nu;2 = Acfc 1 + Φ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 + k−1ð Þ2

3

s24

35: ð50Þ

For the CFT stub column, when the load was applied to theconcrete section only, the corresponding ultimate capacity of the stubcolumn Nu,3 can be expressed as:

Nu;3 = fcAcð1 + kΦ=2Þ: ð51Þ

According to the triaxial tests of laterally-confined concrete underaxial compression, the value of k was usually from 3 to 4.3. Afterexamination of different values of k, Nu,2 is found to be almostidentical to Nu,3 (see Table 1). If k=3.4, numerical results through theelasto-plastic analysis indicated that Nu,1 was usually less than Nu,3.That is, the ultimate capacity of the CFT stub column could be usually10% lower than its maximum value Nu,2 or the value Nu,3 of the CFTstub column.

For a lower strength of concrete core, it can be identified that thevalue of Nu,1/Nu,3 is almost equal to 1.With the strength of concrete andsteel ratio increase, thevalue ofNu,1/Nu,3 decreases andwouldbe close to0.9. However, such a decrease may not always be observed in theexperimental investigation, because of the strain-hardening of steel inplastic stage and the resulting enhancement of the confinement effect.From this point of view, both the simplified limit equilibrium analysisand the full elasto-plastic analysis method are able to predict theultimate capacity of CFT stub column reasonably. Thus the equation forthe ultimate capacity (Nu) or ultimate strength (fsc,u) of CFT stub columncan be expressed as:

Nu = fcAcð1 + 1:7ΦÞfsc;u = 1−ρð Þ 1 + 1:7Φð Þfc:

(ð52Þ

Fig. 12 shows the comparison between the predicted ultimatecapacities using the simplified equation (Eq. (52)) and the measuredultimate capacities of 115 CFT stub columns obtained from 12references. For all the measured ultimate capacities, the necessaryparameters of the specimens are diameter-to-tube wall thicknessratio D/t ranging from 20 to 220, the length-to -diameter ratio L/Dranging from 2 to 4.5, concrete cubic strength fcu between 30 and120 MPa, and steel yield strength fs between 180 and 540 MPa. Thereferred specimens were tested in a consistent condition with the one

5

2)

ntsss

N u/f c

A c

0.0 0.3 0.6 0.9 1.2 1.50.0

0.6

1.2

1.8

2.4

3.0

3.6

Eq. (52)

Φ

(b) <1.5

nd experimental measurements for Nu/(fcAc).

1575F. Ding et al. / Journal of Constructional Steel Research 67 (2011) 1567–1577

considered in the modeling (load was applied to both steel andconcrete section) and a good agreement was found, as shown inFig. 12.

4.2. Composite modulus of elasticity

For the compositemodulus of elasticity given in Eq. (37), numericalresults indicated that the value of 2(vc−vs)2(1−ρ)ρQEs was verysmall and therefore can be neglected (i.e. 0b2(vc−vs)2(1−ρ)ρQEsbb(1−ρ)Ec+ρEs), as a result Eq. (37) can be simplified as:

Esc = 1−ρð ÞEc + ρEs: ð53Þ

The composite stiffness defined as EscAsc can be written as:

EscAsc = AcEc + AsEs: ð54Þ

As shown in Table 3, the composite modulus of elasticity predictedusing Eqs. (37) and (53) are compared with the measured resultsfrom three CFT stub columns presented in Yu et al. (2007). In Table 2,Esco and Esc

c are measured and predicted composite modulus ofelasticity, respectively. The results in Table 2 indicated that thepredicted results between Eqs. (37) and (52) are similar to each other,while overestimations are noticed in comparison to experimentalresults (about 4% to 13% respectively).

4.3. Strain at peak axial compression

The strain corresponding to the ultimate capacity of the CFT stubcolumn is named as strain at peak axial compression (εsc,0). When theCFT stub column reaches its ultimate capacity, concrete coreapproaches its ultimate strength, and εsc,0 becomes εcc, thus one has

εsc;0 = εcc = εc 1 + 3:4σr;c

fc

� �1 + 4:8 A1−1ð Þ σr;c

fc

� �0:5� �: ð55Þ

Substituting Eqs. (42) and (45)with Eq. (54), εsc,0 can be expressed as:

εsc;0 = 1 + 1:7ξΦð Þ 1 + 3:4ffiffiffiffiffiffiffiξΦ

qA1−1ð Þ

� �εc ð56Þ

where ξ=σθ,s/fs. ξ is mainly a function of strength of concrete core, aregression calculation based on the elasto-plastic model indicatedthat ξ=0.9-0.005fcu.

4.4. Nominal limit capacity of elasticity and the corresponding strain

The nominal limit capacity of elasticity and the correspondingstrain in the fsc-εL relationship of the stub columns are denoted as Np

(fsc,p) and εL,p respectively, and we have

Np = EscεL;pAsc ð57Þ

fsc;p = EscεL;p ð58Þ

εL;p = θfs = Es ð59Þ

Table 2Comparison of composite modulus of elasticity between test and predicted results.

Researchers Specimen Esco (104MPa) Esc

c (104MPa) by Eq. (37)

Yu et al. (2007) SZ5S4A1a 4.52 4.98SZ5S4A1b 4.75 4.96SZ5S3A1 4.16 4.80

where the parameter θ can be obtained by using a regression calculationbased on the elasto-plastic model as shown in the next equation:

θ = 0:48 + 3:91 × ρ1:62 +0:52−3:91 × ρ1:62

5:5 × 10−3 fs=fcð Þ2:6 + 1:ð60Þ

4.5. Residual capacity

Similarly, the residual capacity (Ny) or residual capacity (fsc,y) inthe fsc-εL relationship of the stub columns can be obtained by aregression calculation based on the elasto-plastic model:

Ny = 0:3 + 2:2Φð ÞfcAc ð61Þ

fsc ;y = 1−ρð Þ 0:3 + 2:2Φð Þfc ð62Þ

or

Ny =0:3þ2:2Φ1þ1:7Φ

Nu ð63Þ

fsc;y =0:3þ2:2Φ1þ1:7Φ

fsc;u: ð64Þ

4.6. Axial load–axial strain relationship

As seen in Fig. 11, the axial load–strain relationship of the CFT stubcolumn can be categorized into several stages:

1) when εL=0, fsc=0;2) ∀ 0≤εLbεL,p, dfsc/dεL=Esc;3) ∀εL,p≤εLbεsc,0, d2fsc/dεL2b0, and the slope of the curve decreases

monotonously and no inflection point appears;4) when εL=εsc,0, fsc= fsc,u, dfsc/dεL=0; e) at εi≥εL, d2fsc/dεL2=0, an

inflection point results;5) at εL→∞, fsc→ fsc,y, dfsc/dεL→0.

Therefore, the following non-dimensional mathematical functionsfor the axial load–strain relationship of centrally-loaded CFT stubcolumns can be proposed:

y =

A3x x≤xBb3a3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia23− 1−xð Þ2

q−b3 + 1 xB≤x≤1

x3 α3 x−1ð Þ2 + β3 x−1ð Þ3 + x3h i−1

x N 1

8>>>>><>>>>>:

ð65Þ

where y= fsc/fsc,u(y=N/Nu), x=εL/εsc,0, xB=εL,p/εsc,0, variable A3 isthe ratio of the composite modulus of elasticity Esc to the secantmodulus at the ultimate strength (Esc,p= fsc,u/εsc,0), i.e.

A3 = A1

1−ρð Þ + nρ½ � 1 + 1:7ξΦð Þ 1 + 3:4ffiffiffiffiffiffiffiξΦ

pA1−1ð Þ

h i1−ρð Þ 1 + 1:7Φð Þ : ð66Þ

Escc (104MPa) by Eq. (53) Esc

o /Escc by Eq. (37) Esco /Escc by Eq. (53)

4.97 0.908 0.9094.95 0.958 0.9604.78 0.867 0.870

1576 F. Ding et al. / Journal of Constructional Steel Research 67 (2011) 1567–1577

a3 and b3 are the parameters given by

a3 = A3−1ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1−xBA3 A3xB + A3−2ð Þ

sð67Þ

and a3 is valid only when A3xB+A3−2N0.

b3 =A3xB + A3−A2

3xB−1A3xB + A3−2

ð68Þ

The parameter β3 was obtained according to Stage 5) as shown inthe next equation:

β3 =0:7−0:5Φ0:3 + 2:2Φ

ð69Þ

and α3 can be obtained through a regression analysis:

α3 = 0:5=Φ4 Φ≤10:5 Φ N 1:

ð70Þ

Comparisons of N-εL relationship among the tests, the elasto-plastic model and the simplifiedmodel (Eq. (65)) are shown in Fig. 13.A good agreement is achieved again.

5. Comparison with standard provisions and othersimplified methods

The approaches introduced in the international standards, such asACI (2005) [29], AIJ (2008) [30], AISC-360 (2005) [13], and Eurocode 4(2004) [14], were applied to predict the ultimate capacity and theresults were compared with the experimental data in a number ofliterature, such as Han and Yao (2004) [12], Han et al. (2005) [24], andGiakoumelis and Lan (2004) [10]. A general conclusion has beenmadeby Lu and Zhao [31] that ACI (2005) [29], AIJ (2008) [30], and AISC-360 (2005) [13] are relatively conservative for predicting the ultimatecapacities of specimens, while Eurocode 4 (2004) [14] normally givesreasonable estimates. For a demonstration, the approaches inEurocode 4 (2004) [14] and AISC-360 (2005) [13], along with thesimplified model by Han et al. (2005) [24] and the proposed methodsin this paper were applied to predict the ultimate capacity of CFTcolumns, and were compared with experimental results presented inthe literature. For comparison with standard approaches, no material

0.0

0.4

0.8

1.2

1.6

2.0

2.4

CU-40

Huang et al. [2] Theoretical model Similified model

D×t×L=200×5×600mmfs=265.8MPa , f

cu=33.94MPa

N (

MN

)N

(M

N)

N (

MN

)N

(M

N)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Huang et al. [2] Theoretical model Similified model

CU-150

D×t×L=300×2×900mmfs=341.7MPa , f

cu=34.04MPa

0.0

0.4

0.8

1.2

1.6

2.0

2.4

Han and Yao [12] Theoretical model Similified model

scsc2-1,scsc2-2D×t×L=200×3×600mmfs=303.5MPa , f

cu=58.5MPa

0.00

0.15

0.30

0.45

0.60

0.75

0.90

Han and Yao [12] Theoretical model Similified model

D×t×L=100×3×300mmfs=303.5MPa , f

cu=58.5MPa

ca1,ca2

Lε0.00 0.01 0.02 0.03 0.04

Lε0.00 0.01 0.02 0.03 0.04

Lε0.00 0.01 0.02 0.03 0.04

Lε0.00 0.01 0.02 0.03 0.04

Fig. 13. Comparisons among the theoretical model, simplified model and

partial safety factors should be considered, and the relationshipbetween cylinder compressive strength of concrete (fcyl) and cubiccompressive strength fcu were according to CEB-FIP-1990 [32].

The ultimate capacities predicted using these methods arecompared in Fig. 12 and Table 3 with the test results of 115 CFTstub columns from Cai (1988) [8], Cai and Gu (1996) [9], Giakoumelisand Lam (2004) [10], Han and Yao (2004) [12], Han et al. (2005) [24],Huang et al. (2002) [2], Luksha and Nesterovich (1991) [25], O'Sheaand Bridge (2000) [26], Prion and Boehme (1994) [27], Sakina andHayashi (1991) [28], Schneider (1998) [4], and Yu et al. (2007) [7].Those experimental results were chosen based on the same loadingcondition and boundary condition, comparable ranges of columnslenderness, fcu and D/t.

The ultimate capacities from literature are further compared withthe predicted results from Eurocode 4 (2004) [14], AISC-360 (2005)[13], simplified model by Han et al. (2005) [24], and the proposedmethods in this paper as shown in Table 3. In Table 3, Nu

o and Nuc are

experimental and predicted ultimate capacity, respectively. ForEq. (51) and Eurocode 4 (2004) [14], the predicted values are ingood agreement with the test results (the resulting ratio ranging from1.008 to 1.065 with the standard deviations from 0.073 to 0.119). ForAISC-360 (2005) and the simplified model by Han et al. (2005) [24],the predicted values are relatively conservative (the resulting ratioranging from 1.361 to 1.361 with the standard deviations from 0.173to 0.115).

6. Conclusions

This paper presents a full elasto-plastic model and a simplifiedmodel for CFT stub columns with concrete strengths ranging from 30to 120 MPa and diameter-to-wall thickness ratio (D/t) greater than 20when under concentrically loaded conditions. After the validation by alarge number of experimental results, the following conclusions maybe drawn:

(1) Based on experimental results, a constitutivemodel for laterally-confined concrete under axial compression was proposed. Thepredicted curves are in reasonable agreement with the experi-mental results and could be used in the numerical analysis toinvestigate the behavior of CFT stub columns under concentricloading condition.

(2) Based on continuum mechanics, a mechanical model wasestablished for the concentric cylinders of circular steel tube with

N (

MN

)N

(M

N)

N (

MN

)N

(M

N)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Yu et al. [7] Theoretical model Similified model

D×t×L=219×4.72×650mmfs=350MPa , f

cu=50.5MPa

SZ5S4A1b

0.0

0.4

0.8

1.2

1.6

2.0

2.4

Yu et al. [7] Theoretical model Similified model

D×t×L=165×2.73×510mmfs=350MPa , f

cu=77.2MPa

SZ3S6A1

0.0

0.4

0.8

1.2

1.6

2.0

2.4

Sakina and Hayashi [28] Theoretical model Similified model

H-32-1,H-32-2

D×t×L=179×5.5×360mmfs=249MPa , f

cu=51.7MPa

0.0

0.4

0.8

1.2

1.6

2.0

Sakina and Hayashi [28] Theoretical model Similified model

H-58-1,H-58-2

D×t×L=174×3×360mmfs=266MPa , f

cu=53.7MPa

Lε0.00 0.01 0.02 0.03 0.04

Lε0.00 0.01 0.02 0.03 0.04

Lε0.00 0.01 0.02 0.03 0.04

Lε0.00 0.01 0.02 0.03 0.04

experimental measurements on axial load–axial strain relationship.

Table 3Comparison between predicted ultimate capacities and test results.

No. D(mm) D/t fs(MPa) fcu(MPa) Nuo/Nu

c Numberof tests

Reference

EC4(2004)

AISC(2005)

Han et al.(2004)

Eq. (52)

Mean S. D. Mean S. D. Mean S. D. Mean S. D.

1 96-273 34-102 240-419 32-47 1.218 0.144 1.577 0.200 1.287 0.163 1.053 0.105 25 Cai (1988) [8]2 166-219 28-37 280-377 75-85.5 1.071 0.058 1.370 0.082 1.156 0.048 1.009 0.031 4 Cai and Gu (1996) [9]3 114.3-115 23-31 343-365 31-105 1.024 0.092 1.386 0.173 1.132 0.128 0.955 0.073 8 Giakoumelis and Lam (2004) [10]4 100-200 33-67 303.5 38.5 1.016 0.062 1.308 0.087 1.134 0.068 0.984 0.062 4 Han and Yao (2004) [12]5 60-250 32-125 282-404 85.2-90 1.018 0.040 1.261 0.066 1.121 0.040 1.001 0.034 26 Han et al. (2005) [24]6 200-300 40-150 265-341 34-39 0.991 0.045 1.295 0.077 1.118 0.052 0.945 0.023 3 Huang et al. (2002) [2]7 159-1020 31-92 291-382 36-54 1.012 0.030 1.312 0.060 1.150 0.029 0.982 0.016 8 Luksha and Nesterovich (1991) [25]8 165-190 58-221 185-363 49-116 1.022 0.047 1.235 0.063 1.118 0.061 1.032 0.069 15 O'Shea and Bridge (2000) [26]9 152 92 270-328 81-93 1.014 0.061 1.221 0.062 1.106 0.058 1.010 0.043 4 Prion and Boehme (1994) [27]10 174-179 20-58 249-283 29.8-53.7 0.959 0.041 1.324 0.044 1.114 0.079 0.960 0.065 9 Sakina and Hayashi (1991) [28]11 140-141.4 21-47 285-537 30-35 1.169 0.160 1.478 0.200 1.177 0.110 0.961 0.083 3 Schneider (1998) [4]12 165-219 45-61 350 42-77 1.071 0.037 1.390 0.081 1.216 0.053 1.030 0.023 6 Yu et al. (2007) [7]Total range 60-1020 20-221 185-537 30-116 1.065 0.119 1.361 0.173 1.167 0.115 1.008 0.073 115

Note: “S. D.” for standard deviation.

1577F. Ding et al. / Journal of Constructional Steel Research 67 (2011) 1567–1577

concrete core loaded in the entire section, and an elasto-plasticanalysis was developed using a Fortran program. The modelreasonably predicted the mechanical responses of CFT stubcolumns.

(3) Analytical results indicated that the axial stress of concrete coreincreases and the ductility is enhanced due to the concrete corebeing confined by steel tube; at the same time, the axial stress ofsteel tube decreases because of the confinement effect and thestresses transferred from steel tube to concrete core. Theconfinement effect, ultimate capacity and ductility of the columnimprovewith the increase of steel ratio and the strength of steel;while with the increase of the strength of concrete, theconfinement effect and the ductility of the column decrease,the ultimate capacity of the column increases however.

(4) Based on the elasto-plastic analysis, a simplified model wasproposed using limit equilibrium method and regressioncalculations. The simplified method provides a good predictionfor the load-bearing capacity of centrally-loaded CFT columns.

Acknowledgment

This research work was financially supported by the NationalNatural Science Foundation of China, Grant No. 50808180, and the Ph.D. Programs Foundation of Ministry of Education of China, Grant No.200805331064.

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