Finite element modelling of concrete-filled steel stub columns under axial compression

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Finite element modelling of concrete-filled steel stub columns under axial
compression
Article  in  Journal of Constructional Steel Research · October 2013
DOI: 10.1016/j.jcsr.2013.07.001
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1 
 
Finite element modelling of concrete-filled steel stub columns 
under axial compression 
Zhong Tao a,*, Zhi-Bin Wang b,c, Qing Yu c 
a Institute for Infrastructure Engineering, University of Western Sydney, Penrith, NSW 2751, Australia 
b College of Civil Engineering, Fuzhou University, Fuzhou, Fujian Province, 350108, China 
c Department of Construction Management, Tsinghua University, Beijing, 100084, China 
 
ABSTRACT 
Due to the passive confinement provided by the steel jacket for the concrete core, the behaviour of the 
concrete in a concrete-filled steel tubular (CFST) column is always very challenging to be accurately 
modelled. Although considerable efforts have been made in the past to develop finite element (FE) 
models for CFST columns, these models may not be suitable to be used in some cases, especially when 
considering the fast development and utilisation of high-strength concrete and/or thin-walled steel tubes 
in recent times. A wide range of experimental data is collected in this paper and used to develop refined 
FE models to simulate CFST stub columns under axial compression. The simulation is based on the 
concrete damaged plasticity material model, where a new strain hardening/softening function is 
developed for confined concrete and new models are introduced for a few material parameters used in 
the concrete model. The prediction accuracy from the current model is compared with that of an existing 
FE model, which has been well established and widely used by many researchers. The comparison 
indicates that the new model is more versatile and accurate to be used in modelling CFST stub columns, 
even when high-strength concrete and/or thin-walled tubes are used. 
Keywords: Concrete-filled steel tubes; Stub columns; Axial compression; Finite element analysis; 
Stressstrain model; Passive confinement. 
* Corresponding author at: Institute for Infrastructure Engineering, University of Western Sydney, Penrith, NSW 
2751, Australia. Tel.: +61 2 4736 0064; fax: +61 2 4736 0054. E-mail address: z.tao@uws.edu.au (Z. Tao).
 
 2
Nomenclature 
Ac Cross-sectional area of concrete 
As Cross-sectional area of the steel tube 
B Overall width of square or rectangular section 
D Overall diameter of circular section or overall depth of rectangular section 
Ep Strain hardening modulus of steel 
Es Modulus of elasticity of steel 
fb0 Compressive strength of concrete under biaxial loading 
fc Cylinder compressive strength of concrete 
fcu Cube compressive strength of concrete 
fl Confining pressure 
fy Yield strength of steel 
fu Ultimate strength of steel 
GF Fracture energy of concrete 
Kc 
Ratio of the second stress invariant on the tensile meridian to that on the 
compressive meridian 
L Specimen length 
N Axial load 
Nu Ultimate strength of a column 
Nuc FE predicted ultimate strength 
Nue Experimental ultimate strength 
t Wall thickness of the steel tube 
 Axial shortening of a column 
 Strain 
c0 Peak strain of unconfined concrete 
cc Peak strain of confined concrete 
 Stress 
ξc Confinement factor 
 Dilation angle of concrete 
 
 
 
 3
1. Introduction 
Due to the excellent composite action between the steel tube and concrete, concrete-filled steel 
tubular (CFST) columns are becoming increasingly popular and used in various structures 
throughout the world. Extensive experimental and analytical studies have been conducted to 
understand the behaviour of the composite columns mainly from the 1960s (Shanmugam and 
Lakshmi, 2001 [1]). From these investigations, different design codes have been formulated to 
reflect the design philosophies and practices in the respective countries, such as Australia, China, 
Japan, USA and European countries (Tao et al., 2008 [2]). 
 
In recent few decades, finite element (FE) technique is becoming increasingly popular for 
modellingCFST columns thanks to the existence of many commercially available software, such as 
ABAQUS and ANSYS. FE analysis allows the direct modelling of the composite action between 
the steel and concrete components, and different factors, such as local and global imperfections, 
residual stresses and boundary conditions, can be considered more precisely. The prediction 
accuracy of a FE model, however, is greatly affected by the input parameters, especially by the 
selection of a suitable concrete model. 
 
Nowadays, high-performance construction materials, such as high-performance concrete and steel, 
are being increasingly used in engineering structures as a result of the continued advancement of 
materials technology. These high-performance materials often exhibit high strength as well. As far 
as CFST columns in real structures are concerned, the highest cylinder compressive strength and 
yield strength reported so far are 130 MPa and 690 MPa for the concrete and steel, respectively 
(Godfrey Jr., 1987 [3]; Uy, 2011 [4]). FE analysis is now used routinely for design and research 
problems. To embrace the development of materials, new FE models may need to be developed to 
 
 4
improve the prediction accuracy. To serve this purpose, sufficient test data need to be collected and 
used to verify the prediction accuracy. 
 
In the past, numerous tests have been conducted on CFST stub columns. A database was used by 
Tao et al. (2008) [2] to check the applicability of different codes in calculating the strength of CFST 
columns. In that database, 484 test results for circular stub columns and 445 test results for 
rectangular stub columns (square sections mainly) were included. Only ultimate strength, however, 
was reported in the literature for the majority of these test results. It is worth noting that different 
authors might give different definitions of the ultimate strength. This may affect the magnitudes of 
the ultimate strength, especially for those columns without softening post-yield response (Tao et al., 
2011 [5]. Meanwhile, the capacity of predicting full-range loaddeformation curves is also very 
important in evaluating FE models. In this paper, only test results of axial load (N)axial strain () 
or axial shortening () curves reported are collected and used to make consistent comparisons. 
 
Among many FE models, the FE model developed by Han et al. [6] has been widely used to 
simulate CFST columns, where the concrete damaged plasticity model in ABAQUS was adopted. 
Default values, however, were used in [6] for many material parameters, such as the dilation angle 
(), the ratio of the compressive strength under biaxial loading to uniaxial compressive strength 
(fb0/fc'), and the ratio of the second stress invariant on the tensile meridian to that on the compressive 
meridian (Kc). This paper aims to develop a refined FE model in simulating CFST columns, in 
which new models, if required, will be introduced to determine these concrete material parameters. 
Meanwhile, a new strain hardening/softening rule will be developed for concrete. The wide range of 
test data collected will be used to calibrate the new FE model. 
 
 
 
 5
2. FE modelling 
2.1. General 
The general-purpose finite element program ABAQUS version 6.12 (ABAQUS, 2012 [7]) was used 
in the present study to build a FE model for CFST stub columns. Four-node shell elements with 
reduced integration (S4R) and 8-node brick elements with three translation degrees of freedom at 
each node (C3D8R) were used to model the steel tube and concrete core, respectively. 
 
Mesh convergence studies were conducted to determine optimal FE mesh that provides relatively 
accurate solution with low computational time. It was found that the aspect ratio of elements has 
neglectable influence on the N curves if this ratio is smaller than 3. Therefore, element size in the 
axial direction was selected as 2.5 times that in the lateral direction. Based on the mesh convergence 
studies, element size across the cross-section was chosen as D/15 for a circular column or B/15 for a 
rectangular column, where D and B are the overall diameter of the circular tube and the overall 
width of the rectangular tube, respectively. For a typical specimen, the mesh contains a total of over 
4000 elements. 
 
Surface-to-surface contact is usually used for the interaction simulation of the steel tube and 
concrete. A contact surface pair comprised of the inner surface of the steel tube and the outer 
surface of concrete core can be defined. “Hard contact” in the normal direction can be specified for 
the interface, which allows the separation of the interface in tension and no penetration of that in 
compression. The tangent contact can be simulated by the Coulomb friction model. For CFST stub 
columns, there is little or no slip between the steel tube and concrete since they are loaded 
simultaneously. For this reason, the column’s behaviour is not sensitive to the selection of friction 
coefficient between steel and concrete (Lam et al., 2012 [8]). Friction coefficients of 0.25, 0.3 and 
 
 6
0.6 were used by Schneider (1998) [9], Lam et al., 2012 [8] and Han et al. (2007) [6], respectively. 
 
In the current FE model, the surface-based interaction continued to be used to model the 
concrete-steel tube interface. A coefficient of friction between the steel tube and concrete was taken 
as 0.6, which agrees with most test results (Rabbat and Russel, 1985 [10]). 
 
The Poisson’s ratios for steel and concrete were taken as 0.3 and 0.2, respectively. These values 
have been used widely in FE numerical simulation. 
 
It has been well documented that initial local imperfections and residual stresses have apparent 
influence on the behaviour of hollow tubes. For CFST stub columns, however, the effects of local 
imperfections and residual stresses are minimised by concrete filling, and were therefore ignored in 
the current FE simulation. This was confirmed by the research conducted by Tao et al. (2011) [5], 
which explained that the out-of-plane deformation of the steel tube caused by the concrete 
expansion plays a similar role as the initial imperfections. 
 
2.2. Boundary conditions 
A stub column was normally placed into a testing machine and the load was applied on the 
specimen directly. To minimise the influence of end conditions on CFST stub columns, most 
researchers used end plates and/or stiffeners welded to both ends of a tube. Some tests, however, 
were conducted on specimens without end plates or stiffeners (Yamamoto et al., 2000 [11]; 
Giakoumelis and Lam, 2004 [12]). If no stiffening method is used, local buckling of the steel tube is 
more likely to be initiated at the ends, which may affect the overall performance of the specimen. 
 
For stub columns with welded end plates and/or stiffeners, there is no need to include the end plates 
or stiffeners in the model. Instead, the top and bottom surfaces of the steel tube and concrete can be 
 
 7
fixed against all degrees of freedom except for the displacement at the loaded end (clamped end 
condition). The result obtained is the same as that of the model with end plates and/or stiffeners. For 
unstiffened stub columns, the steel and concrete are usually in contact with the stiff platens of the 
testing machine directly. Considering the end friction provided by the testing machine, all three 
translational degrees of freedom can be restrained for the ends of the column except the vertical 
displacement at the loaded end. The rotational degrees of freedom for both ends of the steel tube, 
however, are not restrained, which is referred to as pinned end condition. In most cases, the 
boundary conditions have very minor influence on N or N curves. But some influence can be 
observed for square columns, especially when compact tubes are used. Fig. 1 compares the obtained 
N curves whenpinned and clamped ends are used for the square specimen SSH-1-2 reported in 
[13]. The thickness of the end plates for this specimen is 20 mm, and other parameters are given in 
the figure, where B is the overall width of the square tube, t is the wall thickness, L is the specimen 
length, fy and fc are the yield strength of steel and cylinder compressive strength of concrete, 
respectively. Computed local buckling occurs at the mid-height when a clamped support condition 
is used. But local buckling occurs only near the ends when a pinned support condition is used. In 
the latter case, the predicted ultimate strength decreases by 6.4% and the corresponding peak strain 
increases significantly, as shown in Fig. 1. The comparison highlights that a stiffening method may 
be essential in some cases to eliminate/minimise the influence of end conditions. 
 
In conducting tests, different L/D ratios of specimens were often used by different researchers. For 
columns with a larger L/D ratio, the global imperfections may somewhat affect the column’s 
behaviour. The circular specimen 3HN reported in [14] is used as an example to investigate the 
influence of the L/D ratio. To consider the global imperfections, an eigenvalue buckling analysis 
was carried out first. The distribution of the overall imperfections was defined to be in the form of 
 
 8
the lowest buckling mode. The imperfection amplitude was taken as L/1000. N curves of this 
specimen with various L/D ratios are shown in Fig. 2. It is found that the ultimate strength decreases 
significantly when the L/D ratio increases from 1 to 1.5. This is owing to the fast diminishing end 
effects. When L/D ratios lie in the range of 2-5, very close N curves are obtained. Once the L/D 
ratio exceeds 5, obvious lateral deflection will develop and the ultimate strength begins to decrease 
again. Since L/D ratios reported in the past for almost all CFST stub columns are in the range from 
2 to 5, the global imperfections can be ignored in a simulation. 
 
2.3. Material modelling of steel 
Different stress ()strain () models have been used for the steel material by different researchers, 
including elastic-perfectly plastic model (Schneider, 1998 [9]; Hu et al., 2003 [15]), and 
elastic-plastic model with linear hardening (Guo et al., 2007 [16]; Zha and Xiong, 2007 [17]) or 
multi-linear hardening (Han et al., 2007 [6]). At strains of general structural interest (normally less 
than 5%), steel exhibits no significant strain hardening. Very close axial load (N)axial strain () 
curves are obtained using different stressstrain models for steel. Fig. 3 is depicted to illustrate this, 
where two duplicate circular specimens, i.e. H-58-1 and H-58-2, reported by Sakino and Hayashi 
(1991) [18] were used as examples. The FE model developed in this paper is used for the simulation 
in the following unless otherwise specified. For the elastic-plastic model with linear hardening, the 
strain hardening modulus (Ep) was taken as 0.005Es (Guo et al., 2007 [16]), where Es is the steel 
modulus of elasticity. Clearly, the selection of a  relationship of steel has negligible influence 
on the ultimate strength and only affects the load-deformation curve slightly in a later stage. 
 
From the comparison shown in Fig. 3, it can be concluded that different  models of steel have 
little influence on loaddeformation curves of CFST columns at strains of general structural interest. 
Sometimes, however, engineers may be interested in structural behaviour subject to severe 
 
 9
deformation. Statistical analysis indicates that high strength steel exhibits less strain hardening than 
normal strength steel (Tao et al., 2013 [19]). A  model was proposed by Tao et al. (2013) [19] 
for structural steel with a validity range of fy from 200 MPa to 800 MPa. This model was used to 
simulate the steel material in circular CFST columns, which is expressed as follows 























uu
up
pu
u
yuu
pyy
ys
)(
0







f
fff
f
E
p (1) 
in which fu is the ultimate strength; y is the yield strain, y=fy/Es; p is the strain at the onset of 
strain hardening; u is the ultimate strain corresponding to the ultimate strength; p is the 
strain-hardening exponent, which can be determined by 











yu
pu
p ff
Ep

 (2) 
in which Ep is the initial modulus of elasticity at the onset of strain-hardening, and can be taken as 
0.02Es. p and u are determined using Eqs. (3) and (4), respectively. 






MPa800MPa300)]300(018.015[
MPa30015
yyy
yy
p ff
f


 (3) 






MPa800MPa300)]300(15.0100[
MPa300100
yyy
yy
u ff
f


 (4) 
 
A schematic diagram of this model is shown in Fig. 4, where only three parameters, i.e. yield 
strength (fy), ultimate strength (fu) and modulus of elasticity (Es), are required to determine the 
full-range stressstrain curve. If the value of Es was not reported for a test specimen, it was taken as 
200,000 MPa in the modelling. Likewise, the following equation proposed by Tao et al. (2013) [19] 
was used to determine fu from fy if fu was not available. 
 
 
 10






 

MPa800MPa400)]400(1075.32.1[
MPa400MPa200)]200(1026.1[
yyy
4
yyy
3
u fff
fff
f (5) 
 
Not like circular CFST columns, a rectangular CFST seldom demonstrates strain-hardening 
behaviour. This is owing to the easier local buckling of the rectangular steel tube and less effective 
confinement provided to the concrete. For this reason, an elastic-perfectly plastic model of steel 
gives a better prediction of the descending branch of the N curve than other models incorporating 
strain hardening. Therefore, the elastic-perfectly plastic model is used to simulate the steel material 
in rectangular CFST columns. 
 
2.4. Material modelling of concrete 
For a CFST column under axial compression, the concrete core expands laterally and is confined by 
the steel tube. This confinement is passive in nature, and can increase the strength and ductility of 
concrete. This mechanism is well understood and is often referred to as “composite action” between 
the steel tube and concrete (Han et al., 2007 [6]). It is believed that the confined concrete is in a 
triaxial stress state and the steel is in a biaxial state after interaction between the two components 
occurs. FE analysis is a powerful method allowing the composite action to be considered provided a 
rational and accurate concrete model is available to describe the behaviour of concrete under 
passive confinement. 
 
2.4.1. Concrete damaged plasticity model 
The concrete damaged plasticity model available in ABAQUS was used. Since this paper only deals 
with columns under monotonic loading, damage variables were not defined. Therefore, concrete 
nonlinearity was modelled as plasticity only. In this model, key material parameters to be 
determined include the ratio of the second stress invariant on the tensile meridian to that on the 
compressive meridian (Kc), dilation angle (), and strain hardening/softening rule. Other parameters 
 
 11
include the modulus of elasticity (Ec), flow potential eccentricity (e), ratio of the compressive 
strength under biaxial loading to uniaxial compressive strength (fb0/fc'), viscosity parameter and 
tensile behaviour of concrete. For the FE model presented by Han et al. (2007) [6], constant values 
of 30º, 0.1, 1.16 and 2/3 were used for , e, Kc and fb0/fc', respectively. Considering the complex 
nature of passively confined concrete, constant values may not be suitable to be used in some cases. 
New models, if required, are introduced for these material parameters in the following. 
 
The empirical equation (6) recommended in ACI 318 (2011) [20] was adoptedto calculate Ec, 
where fc is in MPa. Default values of 0.1 and 0 were used for the flow potential eccentricity and 
viscosity parameter, respectively. These two parameters have no significant influence on the 
prediction accuracy. 
'4700 cc fE  (6) 
 
Test results of equibiaxial concrete strength (fb0) are still very scarce. Based on test data collected 
from 14 references, Papanikolaou and Kappos (2007) [21] proposed the following equation to 
predict the ratio of fb0/fc: 
075.0
ccb0 )'(5.1'/
 fff (7) 
The above equation was used in this paper to determine the ratio of fb0/fc. When the concrete 
strength fc is 30 MPa, fb0/fc is 1.162 according to Eq. (7). But when fc increases to 100 MPa, fb0/fc 
drops to 1.062. In general, the calculated ultimate strength Nu will increase slightly if a higher ratio 
of fb0/fc is used. For example, fb0/fc was taken as 1.166 for the circular specimen 3HN shown in Fig. 
2. If fb0/fc is changed to 1.062 for this specimen, the decrease in Nu is only 2.7%, which is not 
significant. 
 
Although CFST stub columns are not sensitive to tensile behaviour of concrete when subjected to 
 
 12
axial compression, tension stiffening need to be defined in ABAQUS. In the current model, the 
uniaxial tensile response was assumed to be linear until the tensile strength of concrete was reached, 
which was taken as 0.1fc. Beyond this failure stress, the tensile softening response was 
characterised by means of fracture energy (GF). The following equation was used to define GF for 
tensile concrete (FIP, 1993 [22]; Bažant and Becq-Giraudon, 2002 [23]): 
7.0
c
max
2
maxF 10
'
)265.00469.0( 





f
ddG N/m (8) 
where fc is in MPa, dmax is the maximum coarse aggregate size (in mm). If dmax had not been 
reported in a reference, it was taken as 20 mm. 
 
2.4.2. Determining Kc 
The ratio of the second stress invariant on the tensile meridian to that on the compressive meridian 
(Kc) is one of parameters for determining the yield surface of concrete plasticity model. Test results 
indicate that the range of Kc varies from 0.5 to 1 (Seow and Swaddiwudhipong, 2005 [24]). The 
default value of Kc used in ABAQUS is 2/3, which has been adopted by many researchers. A 
sensitivity analysis was carried out to investigate the influence of Kc on N curves of two 
specimens tested by Tomii et al. (1977) [14]. The results of sensitivity analysis are shown in Fig. 5. 
It can be found that Kc has no influence on the initial stage of the N curve. After the yielding of 
the column, the influence of Kc becomes significant. The ultimate strength increases with 
decreasing Kc. This highlights that the value of Kc should be determined very carefully. 
 
Assuming the ratio of fb0/fc to be 1.16, it was derived by Yu et al. (2010a) [25] that Kc could be 
taken as 0.725 using a model proposed by Teng et al. 2007 [26] to calculate the strength of confined 
concrete. Since fb0/fc is expressed as a function of fc in this paper as shown in Eq. (7), Kc will also 
be related to fc. According to Yu et al. (2010a) [25], the following equation can be deduced to 
 
 13
calculate Kc: 
b0c
b0
c 5'3
5.5
ff
f
K

 (9) 
By introducing Eq. (7) into Eq. (9), Kc can be determined by 
075.0
c
c )'(25
5.5
f
K

 (10) 
From this equation, it can be found that Kc decreases slightly from 0.725 to 0.703 when fc increases 
from 30 MPa to 100 MPa. 
 
2.4.3. Determining  
Dilation angle () is one of parameters required for ABAQUS to define the plastic flow potential. 
The allowed value of  ranges from 0 to 56 in ABAQUS. Different constant values of  have 
been used by different researchers in the past. Most researchers adopted a value of 20 or 30 for 
confined concrete. 
 
The influence of  on N curves is shown in Fig. 6 based on sensitivity analysis. Once again, the 
two specimens tested by Tomii et al. (1977) [14] were used as examples. Four different  values 
were selected, i.e., 0.01, 20, 30 and 40. Since  cannot be taken as 0 in ABAQUS, a small value 
of 0.01 was used to represent this level. The dilation rate of concrete decreases with decreasing , 
which affects the interaction between the steel tube and concrete. As  increases, stronger 
interaction will be developed and the concrete can be confined by a higher confining stress at a later 
stage. As a result of this, the computed ultimate strength increases with an increase in . For the 
square column, however,  has little influence on the N curve once  is greater than 20. When  
was taken as 0.01 for this specimen, a convergence problem occurred in calculating the strain 
softening branch. It is also found that the initial stage of the N curve is not affected by the 
selection of . 
 
 14
 
As pointed out by Yu et al. (2010b) [27],  is affected by the confining stress and plastic 
deformation of concrete. For concrete confined by a steel tube, the variation in confining stress is 
less significant after the yielding of the steel tube. Since  does not affect the initial stage of the 
N curve, it is valid to assume a constant  for a column in calculating the full-range N curve. 
Since the dilation of concrete decreases with increasing confinement, it is assumed that  is a 
function of the so-called “confinement factor” c. Numerical tests were conducted to identify 
suitable values of  for specimens with different c. The following equation is then proposed based 
on regression analysis to determine  for circular columns. 







 5.0for 672.6
5.0for )1(3.56
c
64.4
4.7
cc
c 


e
 (11) 
where the confinement factor c is expressed as 
'cc
ys
c fA
fA

 (12) 
in which As and Ac are the cross-sectional areas of the steel tube and concrete, respectively. 
 
As far as rectangular columns are concerned, it is found that a constant value of 40 can be used for 
. As mentioned earlier, N curves of rectangular columns are not very sensitive to  when is 
greater than 20. A value of 40 gives the best prediction of the ultimate strength for rectangular 
columns. 
 
2.4.4. Strain hardening/softening rule 
For unconfined concrete under compression, the strain hardening/softening rule can be determined 
by the uniaxial  curve of unconfined concrete. Almost identical slopes of strain softening 
branches, however, will be obtained if this  curve continues to be used in ABAQUS for concrete 
confined by different constant active pressures (Yu et al., 2010a [25]). This is not consistent with 
 
 15
experimental observation that the descending branch becomes less steep when the confining 
pressure increases. To cope with this limitation of ABAQUS software, Yu et al. (2010a) [25] pointed 
out that the hardening/softening function should be related not only to the plastic strain, but also to 
the confining pressure. A user-defined subroutine USDFLD was used by Yu et al. (2010a) [25] to 
define the dependence of strain hardening/softening on the confining pressure for fibre 
reinforcement polymer (FRP)-confined concrete. To do this, a model is required to describe the 
axial strainlateral strain relationship of concrete under various confining pressures. 
 
For CFST columns under axial compression, it is believed that there is no or negligible interaction 
between the steel tube and concrete in the initial loading stage. A small gap may appear since the 
initial lateral expansion of the concrete is smaller than that of the steel tube due to the difference in 
Poisson’s ratio between the steel and concrete. As axial strain increases, the lateral expansion of the 
concrete gradually becomes greater than the expansion of the steel until the two components are incontact again. After that, contact pressure and interaction develop between the steel tube and 
concrete. This mechanism highlights the complexity of the interaction in CFST columns. It is very 
difficult to measure the lateral expansion and confining pressure of concrete in a steel tube during 
the loading process. For this reason, no accurate model is available till now to describe the axial 
strainlateral strain relationship of concrete in CFST columns. 
 
Based on numerical tests, researchers proposed different compressive  models to be used for FE 
modelling of concrete confined by steel tubes. These models can be used to determine the strain 
hardening/softening function directly. Therefore, no user-defined subroutine is required. This 
reduces the modelling complexity, increases the computation efficiency and avoids the potential 
convergence problems. 
 
 16
 
Different from that proposed by Han et al. (2007) [6], a new three-stage model is proposed in this 
paper to represent the strain hardening/softening rule of concrete confined by steel tubes, as shown 
in Fig. 7. In the initial stage (from Point O to Point A), there is no or very little interaction between 
the steel tube and concrete. Therefore, the ascending branch of the stressstrain relationship of 
unconfined concrete is appropriate to be used to represent the curve OA until the peak strength fc is 
reached. After that, a plateau (from Point B to Point C) is included to represent the increased peak 
strain of concrete from confinement. During this stage, any strength increase of concrete from 
confinement will be captured in the simulation through the interaction between the steel tube and 
concrete. Beyond Point C, a softening portion with increased ductility resulting from confinement is 
defined. 
 
A model proposed by Samani and Attard (2012) [28] is used to describe the ascending curve OA: 
c02
2
c
0
)1()2(1'
 



XBXA
XBXA
f
 (13) 
where c0X ; 'c
0cc
f
E
A

 ; 1
55.0
)1( 2



A
B . The strain at peak stress under uniaxial compression 
c0 is calculated according to the relationship in Eq. (14). This equation was proposed by De Nicolo 
et al. (1994) [29] based on regression analysis of uniaxial compression tests results from 17 
references, in which fc' ranged from 10 MPa to 100 MPa. 
7
cc0 10)33.4'626.0(00076.0
 f (14) 
where fc' is expressed in MPa. 
 
The strain at Point B (cc) for the concrete model is determined by the following equation proposed 
by Samani and Attard (2012) [28]: 
 
 17
 
'002.03124.0
c
B
c
c0
cc
c
'
'00367.09224.2,
f
k
f
f
fke










 (15) 
where fB is the confining stress provided to the concrete at Point B. 
 
The circular specimen 3HN in Fig. 5a is further used as an example to illustrate the development of 
confining stress on concrete (fl), which is shown in Fig. 8. It should be pointed out that fl is the 
average value at the interface between the steel tube and concrete. In the elastic stage, there is no 
confining stress as explained before. Just before and after the yielding of the steel, the confining 
stress increases very fast. Once the ultimate strength is reached, fl keeps stable or increases very 
slowly depending on the value of c. This example highlights the fact that fl changes with axial 
strain (). To determine the confining stress fB corresponding to Point B in Fig. 7, it is assumed that 
at this point the ultimate strength is reached, as shown in Fig. 8. 
 
Numerical tests were conducted to calculate fB for circular columns with different parameters. It is 
found that fB increases with increasing fy or decreasing D/t ratio. Although fc' has no significant 
influence on the development of fl in a later stage, there is a delay in the development of interaction 
and fl for columns with higher strength concrete. Therefore, fB also decreases with increasing fc'. On 
the basis of regression analysis, Eq. (16a) is proposed to determine fB for circular columns. For 
rectangular CFST columns, the concrete core is subjected to uneven confinement, and fl at the 
corners will be higher than those at other parts. It is found that a reduction factor of 0.25 can be 
applied to Eq. (16a) for rectangular CFST columns and reasonable N curves can be obtained. 
Therefore, fB determining from Eq. (16b) can be viewed as an equivalent confining stress. 
8.4
c
10
02.0
y
B )'(6.11
)027.01(
fe
ef
f
t
D


 

 (circular CFST) (16a) 
 
 
 18
8.4
c
10
02.0
y
B )'(6.11
)027.01(25.0
22
fe
ef
f
t
DB


 

 (rectangular CFST) (16b) 
 
For the descending branch of the concrete model (BC) shown in Fig. 7, an exponential function 
proposed by Binici (2005) [30] was used, which is defined by: 
cc
cc
rcr exp)'( 
















  fff (17) 
in which fr is the residual stress as shown in Fig. 7;  and  are parameters determining the shape of 
the softening branch. The expression for fr is proposed as: 


 


CFST)r rectangula('1.0
 CFST)(circular '25.0')1(7.0
c
cc
38.1
r
c
f
ffe
f

 (18) 
The parameter  is determined as: 






 
CFST)r rectangula(0075.0005.0
 CFST)(circular 
1
036.0
04.0
c
49.308.6 c

 e (19) 
Meanwhile,  can be taken as 1.2 and 0.92 for circular and rectangular columns, respectively. It 
should be noted that fr,  and  cannot be derived from tests directly. Hence, different trial values 
were used until best-fit values were obtained to ensure predicted N curves match with measured 
curves. It was found that fr and  can be expressed as functions of c. Eqs. (18) and (19) were then 
developed on the basis of regression analysis. 
 
3. Verification of the current FE model 
Through an extensive literature search, N or N curves of 142 circular, 154 square and 44 
rectangular specimens were collected and used to verify the proposed FE model. The parameters of 
these specimens are summarised in Tables 1 and 2 for circular and rectangular specimens, 
respectively, where D is the overall diameter of a circular section or overall depth of a rectangular 
 
 19
section, B is the overall width of a square or rectangular section. It should be noted that parameters 
of square specimens (D/B=1) were also summarised in Table 2. For simplicity, square columns are 
considered as a special case of rectangular columns in the following, unless otherwise specified. 
The 340 tests in total given in Tables 1 and 2 are from 30 references and were reported by Gardner 
and Jacobson (1967) [31], Gardner (1968) [32], Tomii et al. (1977) [14], Sakino and Hayashi 
(1991) [18], O'Shea and Bridge (1998) [33], Schneider (1998) [9], Tan et al. (1999) [34], 
Yamamoto et al. (2000) [11], Huang et al. (2002) [35], Han and Yao (2004) [36], Giakoumelis and 
Lam (2004) [12], Sakino et al. (2004) [37], Han et al. (2005) [38], Gupta et al. (2007) [39], Liew 
and Xiong (2010a) [40], Liew and Xiong (2010b) [41], Lu and Kennedy (1994) [42], Uy (2000) 
[43], Varma (2000) [44], Han et al. (2001) [45], Han (2002) [46], Liu et al. (2003) [47], Lam and 
Williams (2004) [48], Liu (2005) [49], Liu and Gho (2005) [50], Tao et al. (2005, 2008, 2009) 
[51-53], Liew et al. (2011) [13] and Chen et al. (2012) [54]. It is worth noting that the majority of 
the above references have received extensive citations. 
 
For the circular columns shown in Table 1, the ranges of different parameters are: fy=186-853 MPa; 
fc=18-185 MPa; D=60-450 mm, L/D=1.8-4.3, and D/t=17-221. The parameter ranges for the 
rectangular columns in Table 2 are: fy=194-835 MPa; fc=13-164 MPa; D=60-500 mm, L/D=2.8-4.8, 
D/t=11-150; and D/B=1-2. As can be seen, the parameter ranges of the collected data are very broad 
and cover the current practical ranges. Itshould be noted that values of fc for some specimens given 
in Tables 1 and 2 were not available. Instead, a compressive strength (fcu) of 150 mm cubes was 
reported. To make consistent comparisons, Eq. (20) proposed by L’Hermite (Mirza and Lacroix, 
2004 [55]) was used to convert fcu to equivalent cylinder strength. The obtained results are quite 
close to those determined by using a conversion table presented by Yu et al., (2008) [56]. 
 
 
 20
cu
cu
10c 6.19
log2.076.0' f
f
f 










 (20) 
where fc' and fcu are both in MPa. 
 
As observed from tests, peak loads for some stocky CFST specimens are often associated with very 
high plastic strains. In some cases, no descending branches of N or N curves can be observed. 
This is particularly true for circular columns. For comparison purposes, the ultimate strength (Nu) is 
defined as the first peak load in this paper. But if the strain corresponding to the first peak load is 
great than 0.01 or there is no descending branch, Nu is then taken as the load at a strain of 0.01. It 
should be noted that  was taken as /L if N curves were reported instead of N curves in some 
references. 
 
The test data summarised in Tables 1 and 2 are used to verify the prediction accuracy of the current 
FE model. The predicted ultimate strengths (Nuc) from the current FE model and Han et al.’s FE 
model are compared with the measured ultimate strengths (Nue) of all collected data in Figs. 9 and 
10, respectively. Table 3 shows both the mean value (Mean) and standard deviation (STD) of the 
ratio of Nue/Nuc for specimens with different cross-sections. 
 
For the current FE model, the mean values of Nue/Nuc are 1.024, 1.009 and 1.037 for circular, square 
and rectangular columns, respectively; whilst the standard deviations of Nue/Nuc are 0.065, 0.078 
and 0.079, respectively. As can be seen, slightly conservative predictions are obtained from the 
current FE model but with reasonable accuracy. For square columns, predictions of ultimate 
strengths from the new model are very close to those given by Han et al.’s model. But for circular 
and rectangular columns, the current model is superior to Han et al.’s model in predicting ultimate 
strength. In particular, Han et al.’s model underestimates the ultimate strengths of circular 
specimens with small c values, whilst overestimate those of circular specimens with large c values. 
 
 21
This is owing to the fact that concrete in a circular tube is more effectively confined, but constant 
material parameters were used by Han et al. (2007) [6] except the strain hardening/softening 
function. The current model gives good predictions for circular columns with either large or small 
c values, which can be seen from Fig. 9a. 
 
The influence of fy, fc and D/t ratio on prediction accuracy of the current FE model is further 
checked. In general, no change in prediction accuracy is found as fy, fc or the D/t ratio varies. The 
only exception is found regarding the influence of fc for square columns, as shown in Fig. 11. A 
variation in prediction accuracy is found when the concrete strength is greater than 75 MPa. For 
example, the current FE model overestimates the ultimate strength by 7% on average for the 4 
square specimens with an fc of 110 MPa reported by Varma (2000) [44]. On the other hand, Liew et 
al. (2011) [13] reported test results of 9 square columns, where the concrete strength fc varied from 
148 MPa to 164 MPa. On average, the current model underestimates the ultimate strength by 15.5% 
for these specimens. Despite this, the shape of the descending branch of N curves for these 
specimens is still predicted very well. A possible explanation of the variation in prediction accuracy 
is that high-strength concrete is very sensitive to curing conditions. It is very difficult to ensure the 
curing condition of concrete cylinders is the same as that of concrete inside a tube. To reduce the 
variation in strength, however, high-strength concrete cylinders may also be cured in a sealed 
condition and kept in a same environment as the specimens. 
 
In terms of the prediction accuracy of N curves, both the current FE model and Han et al.’s model 
give reasonable predictions for normal columns. This is illustrated in Fig. 12, where two specimens 
tested by Tomii et al. (1977) [14] are used as examples. This explains why Han et al.’s model has 
been used widely. 
 
 22
 
Fig. 13 shows the predicted N curves of three specimens with stocky sections, including a 
circular specimen CC8-A-8 (D/t =16.7), a square specimen sczs2-1-4 (B/t=20.5) and a rectangular 
specimen No. 4 (D/t=28) reported by Sakino et al. (2004) [37], Han et al. (2001) [45] and Lu and 
Kennedy (1994) [42], respectively. Han et al.’s FE model gives unsafe predictions of post-yielding 
strength for stocky circular columns as shown in Fig. 13a. The predictions from the current FE 
model agree well with the test results of all the three specimens. 
 
The predictions of Han et al.’s and the current FE models for specimens with thin-walled tubes and 
high-strength concrete are shown in Figs. 14 and 15, respectively. The test curves of thin-walled 
specimens S16CS80A and UNC-H shown in Fig. 14 were presented by O'Shea and Bridge (1998) 
[33] and Tao et al. (2009) [53], respectively; whilst those of specimens with high-strength concrete 
in Fig. 15 were reported by Liew and Xiong (2010a) [40], Liew et al. (2011) [13] and Liu and Gho 
(2005) [50], respectively. For circular thin-walled columns and circular CFST columns with high 
strength concrete, both models give reasonable predictions for the ultimate strength, but the current 
FE model is more satisfactory in predicting the strain softening branch than Han et al.’s FE model. 
For the N curves of square or rectangular specimens shown in Figs. 14 and 15, both models give 
reasonable predictions. 
 
 
In general, the strain-hardening or softening branch is more accurately predicted using the current 
FE model. Meanwhile, the current model gives reasonably accurate predictions for columns in a 
wider parameter range. Apart from simulating normal CFST columns, the new FE model is also 
suitable to be used for CFST columns with high-strength concrete and/or thin-walled steel tubes. 
Compared with Han et al.’s model, the improvement of the current FE model in predicting the 
 
 23
full-range N or N curves is more significant than the improvement in predicting the ultimate 
strength. This is especially true for circular specimens. 
 
4. Conclusions 
In this paper, a wide range of experimental data was collected and used to develop a new FE model 
in simulating CFST stub columns under axial compression. This model was compared with an 
existing FE model developed by Han et al. (2007) [6] in terms of prediction accuracy. The 
following conclusions can be drawn based on the results of this study: 
(1) Han et al.’s FE model gives reasonable predictions for normal CFST columns, although it 
underestimates the ultimate strengths of circular columns with small confinement factors c, and 
overestimates those of circular specimens with large c values. It was also found that the 
prediction of post-peak curves of circular columns with high-strength concrete or thin-walled 
tubes is less satisfactory using this model. 
(2) A new FE model was developed in this paper for CFST stub columns, where the dilation angle 
used in the concrete damaged plasticity model was calibrated against test data. Meanwhile, a 
new concrete strain hardening/softening function was developed to be used in modelling CFST 
columns. 
(3) The predictions from the new FE model were compared with the test data collected, which 
indicates that the new model is more versatile and accurate in modelling CFST stub columns. 
Apart from modelling normal CFST columns, the new FEmodel is also suitable to be used for 
CFST columns with high-strength concrete and/or thin-walled tubes. 
 
It should be noted that passively confined concrete is always very challenging to be accurately 
modelled. The strain hardening/softening function for confined concrete should be related not only 
 
 24
to the plastic strain, but also to the confining pressure. ABAQUS and most other software have a 
limitation in reflecting this. Further research is required to measure the lateral expansion of concrete 
inside a steel tube during the loading process. A model may be put forward accordingly to describe 
the axial strainlateral strain relationship of the core concrete in CFST columns. 
A user-defined material subroutine can then be developed for ABAQUS to define a more rational 
strain hardening/softening function. In this way, the confining pressure between the steel tube and 
concrete can be determined from the interaction between the two components, and the strain 
hardening/softening rule will not directly related to the geometric parameters of the cross-section. 
 
Acknowledgements 
This work is supported by the Australian Research Council (ARC) under its Future Fellowships 
scheme (Project No: FT0991433). The financial support is highly appreciated. 
 
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40413. 
[32] Gardner NJ. Use of spiral welded steel tubes in pipe columns. ACI J 1968; 65(11): 93742. 
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strength concrete. Aust Civil Eng Trans 1998; 40: 1527. 
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behavior of stiffened concrete-filled steel columns. J Struct Eng, ASCE 2002; 128(9): 
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135777. 
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steel-tube short columns. J Struct Eng, ASCE 2004; 130(2): 1808. 
[38] Han LH, Yao GH, Zhao XL. Tests and calculations for hollow structural steel (HSS) stub 
columns filled with self-consolidating concrete (SCC). J Constr Steel Res 2005; 61(9): 
124169. 
[39] Gupta PK, Sarda SM, Kumar MS. Experimental and computational study of concrete filled 
steel tubular columns under axial loads. J Constr Steel Res 2007; 63(2): 182–93. 
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Proceedings of the 4th International Conference on Steel & Composite Structures, Sydney, 
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[41] Liew JYR,Xiong DX. Experimental investigation on tubular columns infilled with ultra-high 
strength concrete. In: Proceedings of the 13th International Symposium on Tubular Structures, 
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29 
 
Captions for Tables 
 
Table 1 Summary of test data of circular CFST columns. 
Table 2 Summary of test data of rectangular CFST columns. 
Table 3 Comparison results of FE predictions with measured ultimate strength. 
 
Captions for Figures 
 
Fig. 1. Influence of boundary conditions on predicted N curves. 
Fig. 2. Influence of L/D ratio on N curves. 
Fig. 3. Effect of different stressstrain models of steel on N curves. 
Fig. 4.  model proposed by Tao et al. (2013) [19] for structural steel. 
Fig. 5. Influence of Kc on N curves. 
Fig. 6. Influence of  on N curves. 
Fig. 7. Stressstrain model proposed for confined concrete. 
Fig. 8. Development of confining stress (fl) as a function of axial strain (). 
Fig. 9. Comparison between measured ultimate strength and predicted strength using current FE model. 
Fig. 10. Comparison between measured ultimate strength and predicted strength using Han et al.’s model. 
Fig. 11. Effect of concrete strength on the prediction accuracy of current model for square columns. 
Fig. 12. Comparison between predicted and measured N curves for normal specimens. 
Fig. 13. Comparison between predicted and measured N curves for specimens with compact sections. 
Fig. 14. Comparison between predicted and measured N curves for concrete-filled thin-walled columns. 
Fig. 15. Comparison between predicted and measured N curves for CFST columns with high-strength 
concrete. 
 
 
 
 
30 
 
Tables 
Table 1 Summary of test data of circular CFST columns. 
 
Number of 
specimens 
D 
(mm) 
t 
(mm) 
D/t 
 
L/D 
 
fy 
(MPa) 
fc 
(MPa)
Source 
8 76-153 1.7-4.1 30-48 2.0 363-605 21-34 Gardner and Jacobson, 1967 [31] 
1 169 2.6 65 1.8 317 37 Gardner, 1968 [32] 
27 150 2.0-4.3 35-75 3.0 280-336 18-29 Tomii et al., 1977 [14] 
12 174-179 3.0-9.0 20-58 2.0 248-283 22-46 Sakino and Hayashi, 1991 [18] 
10 165-190 0.9-2.8 59-221 3.5 186-363 41-80 O'Shea and Bridge, 1998 [33] 
2 141 3.0-6.5 22-47 4.3 285-313 24-28 Schneider, 1998 [9] 
14 108-133 1.0-7.0 18-125 3.5 232-429 92-106 Tan et al., 1999 [34] 
6 102-319 3.2-10.3 31-32 3.0 334-452 23-52 Yamamoto et al., 2000 [11] 
3 200-300 2.0-5.0 40-150 2.8-4.2 266-342 27-31 Huang et al., 2002 [35] 
4 100-200 3.0 33-67 3.0 304 50 Han and Yao, 2004 [36] 
7 114-115 3.8-5.0 23-30 2.6 343-365 26-95 Giakoumelis and Lam, 2004 [12] 
6 108-450 3.0-6.5 17-152 3.0 279-853 41-85 Sakino et al., 2004 [37] 
26 60-250 1.9-2 30-134 3.0 282-404 75-80 Han et al., 2005 [38] 
8 89-113 2.7-2.9 33-39 3.0-3.8 360 28-33 Gupta et al., 2007 [39] 
4 219 6.3 35 2.7 300 149-175 Liew and Xiong, 2010a [40] 
4 219 4.9-9.7 23-45 2.7 377-381 54-185 Liew and Xiong, 2010b [41] 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 31
Table 2 Summary of test data of rectangular CFST columns. 
 
Number of 
specimens 
B 
 (mm) 
D 
 (mm) 
D/B 
 
t 
(mm) 
D/t 
 
L/D 
 
fy 
(MPa) 
fc 
(MPa) 
Source 
27 150 150 1.0 2.0-4.3 35-75 3.0 294-341 18-36 Tomii et al., 1977 [14] 
4 152 152-254 1.0-1.7 4.8-9.5 16-40  377-432 45-49 Lu and Kennedy, 1994 [42] 
11 77-127 127-153 1.5-2.0 3.0-7.5 17-51 4.0-4.8 312-430 24-30 Schneider, 1998 [9] 
3 156-306 156-306 1.0 3.0 52-102 2.9 300 38-50 Uy, 2000 [43] 
4 303-305 303-305 1.0 5.8-8.9 34-52 3.9-4.0 259-660 110 Varma, 2000 [44] 
6 100-301 100-301 1.0 2.3-6.0 44-50 3.0 300-395 27-48 Yamamoto et al., 2000 [11] 
20 120-200 120-200 1.0 3.8-5.9 21-37 3.0 321-330 13-46 Han et al., 2001 [45] 
24 70-135 90-160 1.1-1.75 2.9-7.6 21-52 3.0 194-228 50.3 Han, 2002 [46] 
3 200-300 200-300 1.0 2.0-5.0 40-150 2.8-4.2 266-342 27-31 Huang et al., 2002 [35] 
8 80-181 101-183 1.0-2.0 4.2 24-44 3.0 550 61-72 Liu et al., 2003 [47] 
11 100-101 100-101 1.0 4.1-9.6 11-24 3.0 289-400 25-89 Lam and Williams, 2004 [48] 
2 200 200 1.0 3.0 67 3.0 304 49.5 Han and Yao, 2004 [36] 
6 119-323 119-323 1.0 4.4-6.5 18-74 3.0 262-835 41-80 Sakino et al., 2004 [37] 
24 60-250 60-250 1.0 1.9-2.0 30-134 3.0 282-404 42-71 Han et al., 2005 [38] 
6 80-120 106-180 1.0-2.0 4.0 27-45 3.0 495 60-89 Liu, 2005 [49] 
10 100-120 120-180 1.0-1.8 5.8 21-31 3.0 300 83-106 Liu and Gho, 2005 [50] 
2 128-249 129-250 1.0 2.5 52-100 3.0 234 50-55 Tao et al., 2005 [51] 
6 190-250 190-250 1.0 2.5 76-100 3.0 270-342 49-57 Tao et al., 2008 [52] 
2 250 250 1.0 2.5 100 3.0 338 20-42 Tao et al., 2009 [53] 
15 150 150 1.0 8-12.5 12-19 3.0 446-779 148-164 Liew et al., 2011 [13] 
4 410-500 410-500 1.0 10-16 26-50 3.0 358-389 42.5 Chen et al., 2012 [54] 
 
 
Table 3 Comparison results of FE predictions with measured ultimate strength. 
Section type Number of specimens Han et al.’s model Current FE model 
Mean STD Mean STD 
Circular 142 0.989 0.098 1.024 0.065 
Square 154 1.020 0.075 1.009 0.078 
Rectangular 44 1.065 0.086 1.037 0.079 
 
 32
Figures 
0
2000
4000
6000
8000
0 3 6 9 12 15 18
A
xi
al
 lo
ad
 N
(k
N
)
Test [13]
Clamped
Pinned
 
Fig. 1. Influence of boundary conditions on predicted N curves. 
 
0
300
600
900
1200
1500
0 10000 20000 30000
A
xi
al
 lo
ad
 N
 (
kN
)
 
Fig. 2. Influence of L/D ratio on N curves. 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 3. Effect of different stressstrain models of steel on N curves. 
 
 
 
 
 
 
Square specimen SSH-1-2 
Axial strain  ()
L/D=1 L/D=1.5 
L/D=2 L/D=3 
L/D=6 
B=150 mm, t=8 mm, L=450 mm,
fy=779 MPa, fc'=157 MPa 
D=150 mm, t=3.2mm, 
fy=287.4 MPa, fc'=28.7 MPa 
Axial shortening  (mm) 
Axial strain  ()
0
400
800
1200
1600
2000
0 10000 20000 30000
A
xi
al
 lo
ad
 N
(k
N
)
T est(H-58-1) [18]
Test(H-58-2) [18]
Elastic-perfectly plastic
Linear hardening
Multi-linear hardening
D=174 mm, t=3 mm, L=360 mm,
fy=265.8 MPa, fc'=45.7 MPa 
(Ep=0.005Es) 
 
 33
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 4.  model proposed by Tao et al. (2013) [19] for structural steel. 
 
 
0
300
600
900
1200
1500
0 10000 20000 30000 40000
A
xi
al
 lo
ad
 N
 (k
N
)
0
300
600
900
1200
1500
0 10000 20000 30000 40000
A
xi
al
 lo
ad
 N
 (
kN
)
 
 (a) Circular specimen 3HN [14] (b) Square specimen 3MN [14] 
Fig. 5. Influence of Kc on N curves. 
 
 
0
300
600
900
1200
1500
0 10000 20000 30000 40000
A
x
ia
l 
lo
ad
 N
 (
k
N
)
0
300
600
900
1200
0 10000 20000 30000 40000
A
xi
al
 lo
ad
 N
 (
k
N
)
 
 (a) Circular specimen 3HN [14] (b) Square specimen 3MN [14] 
Fig. 6. Influence of  on N curves. 
 
 
 
arctan(Es) 
arctan(Ep) 
 y p u 
 Strain  
fu 
fy 
St
re
ss
 
 
Axial strain  () Axial strain  () 
=0.01 
=20 
=30 
=40 
=0.01 
=20 
=30 
=40 
Axial strain  () 
Kc=0.6 
Kc=2/3 
Kc=0.725 
Kc=0.8 
B=150 mm, t=3.2 mm, L=450 mm, 
fy=300.1 MPa, fc'=27.8 MPa 
Axial strain  () 
Kc=0.6 
Kc=2/3 
Kc=0.725 
Kc=0.8 
D=150 mm, t=3.2 mm, 
L=450 mm, fy=287.4 MPa, 
fc'=28.7 MPa 
D=150 mm, t=3.2 mm, L=450 mm, 
fy=287.4 MPa, fc'=28.7 MPa 
B=150 mm, t=3.2 mm, L=450 mm,
fy=300.1 MPa, fc'=27.8 MPa 
 
 34
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 7. Stressstrain model proposed for confined concrete. 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 8. Development of confining stress (fl) as a function of axial strain (). 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (a) Circular section (b) Square and rectangular sections 
Fig. 9. Comparison between measured ultimate strength 
and predicted strength using current FE model. 
 
A 
St
re
ss
 
 
Strain  
fc 
cc 
B 
c0 
f r 
Confined concrete 
Unconfined concrete 
O 
C
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2 2.5 3 3.5
Confinement factor  c
N
ue
/N
uc
-10%
10%
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2 2.5 3 3.5
Confinement factor  c
N
ue
/N
uc
Square section
Rectangular section
-10%
10% 
0
2
4
6
8
0 10000 20000 30000 40000
Circular specimen 3HN [14] 
Axial strain  () 
C
on
fi
ni
ng
 s
tr
es
s 
f l 
(M
P
a)
 
Reaching ultimate strength 
 Yielding of steel 
 
B
 
 35
 
 
 
 
 
 
 
 
 
 
 
 
 
 (a) Circular section (b) Square and rectangular sections 
Fig. 10. Comparison between measured ultimate strength 
and predicted strength using Han et al.’s model. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 11. Effect of concrete strength on the prediction accuracy of current model for square columns. 
 
 
0
300
600
900
1200
1500
0 10000 20000 30000 40000
A
xi
al
 lo
ad
 N
 (k
N
)
Test [14]
Han et al.
Current model
0
300
600
900
1200
1500
0 10000 20000 30000 40000
A
xi
al
 lo
ad
 N
 (
kN
)
Tes t [14]
Han et al.
Current model
 
 (a) Circular specimen 3HN (b) Square specimen 3MN 
Fig. 12. Comparison between predicted and measured N curves for normal specimens. 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 40 80 120 160 200
N
ue
/N
uc
-10%
10%
Concrete strength fc (MPa) 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2 2.5 3 3.5
Confinement factor  c
N
ue
/N
uc
-10%
10%
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2 2.5 3 3.5
Confinement factor  c
N
ue
/N
uc
Square section
Rectangular section
-10%
10% 
D=150 mm, t=3.2 mm, L=450 mm,
fy=287.4 MPa, fc'=28.7 MPa 
B=150 mm, t=3.2 mm, L=450 mm, 
fy=300.1 MPa, fc'=27.8 MPa 
Axial strain  () Axial strain  () 
 
 36
 
 
 
0
1000
2000
3000
4000
5000
0 10000 20000 30000 40000
A
xi
al
 lo
ad
 N
(k
N
)
Tes t [37]
Han et al.
Current model
0
400
800
1200
1600
2000
2400
0 10000 20000 30000 40000
A
xi
al
 lo
ad
 N
(k
N
)
Tes t [45]
Han et al.
Current model
 
 (a) Circular specimen CC8-A-8 (b) Square specimen sczs2-1-4 
0
1750
3500
5250
7000
0 5000 10000 15000 20000
A
xi
al
 lo
ad
 N
(k
N
)
Test [42]
Han et al.
Current model
 
(c) Rectangular specimen No. 4 
Fig. 13. Comparison between predicted and measured N curves 
 for specimens with compact sections. 
 
 
 
0
700
1400
2100
2800
3500
0 5000 10000 15000 20000
A
xi
al
 lo
ad
 N
(k
N
)
Test [33]
Han et al .
Current model
0
1000
2000
3000
4000
0 5000 10000 15000 20000
A
xi
al
 lo
ad
 N
 (
kN
)
Test [53]
Han et al.
Current model
 
 (a) Circular specimen S16CS80A (b) Square specimen UNC-H 
Fig. 14. Comparison between predicted and measured N curves 
for concrete-filled thin-walled columns. 
D=108 mm, t=6.47 mm, D/t =16.7, 
L=324 mm, fy=853 MPa, fc'=77 MPa 
B=120 mm, t=5.86 mm, B/t =20.5, 
L=360 mm, fy=321 MPa, fc'=43.6 MPa 
B=152 mm, D=253 mm, t=9 mm, D/t =28, 
L=759 mm, fy=394 MPa, fc'=49 MPa 
D=190 mm, t=1.52 mm, D/t =125, 
L=664 mm, fy=306 MPa, fc'=80.2 MPa 
B=250 mm, t=2.5 mm, B/t =100, 
L=750 mm, fy=338 MPa, fc'=41.8 MPa 
Axial strain  () Axial strain  () 
Axial strain  ()
Axial strain  () Axial strain  () 
 
 37
 
0
2000
4000
6000
8000
0 6 12 18 24
A
xi
al
 lo
ad
 N
(k
N
)
Test [40]
Han et al .
Current model
0
2000
4000
6000
8000
10000
0 3 6 9 12 15 18
A
xi
al
 lo
ad
 N
(k
N
)
Tes t [13]
Han et al.
Current model
 
 (a) Circular specimen CS-1 (b) Square specimen SSH-1-2 
 
0
600
1200
1800
2400
0 3 6 9 12
A
xi
al
 l
oa
d 
N
(k
N
)
Test [50]
Han et al.
Current model
 
(c) Rectangular specimen A5-1 
Fig. 15. Comparison between predicted and measured N curves 
for CFST columns with high-strength concrete. 
 
 
 
 
D=219 mm, t=6.3 mm, L=600 mm,
fy=300 MPa, fc'=163 MPa 
B=150 mm, t=8 mm, L=450 mm,
fy=779 MPa, fc'=157 MPa 
B=100, D=130 mm, t=5.8 mm, 
L=390 mm, fy=300 MPa, fc'=106 MPa 
Axial shortening  (mm) Axial shortening  (mm) 
Axial shortening  (mm) 
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