In this paper, a numerical model for the analysis of bond-slip occurring in concrete filled steel tube (CFT) columns is introduced. Unlike the classical bond-link or bond-zone element using double nodes, the introduced model considers the bond-slip effect without taking double nodes by incorporation of the equivalent steel stiffness. Upon constructing the equations system on the basis of the force equilibrium and the compatibility condition at each nodal point, the deformations in a steel tube are determined, followed by evaluation of the bond-slip. Moreover, as a part of solving the equations system to evaluate the slip behavior, the mechanical properties for steel and bond-slip have been changed and updated through an iteration procedure. Finally, the validity of the introduced numerical model is verified by comparing the experimental data with the analytical results for CFT columns subjected to axial force and bending moment.

Concrete-filled steel tube (CFT) columns, which consist of a circular or rectangular steel tube filled with concrete, have been increasingly applied to column design and popularly adopted in buildings and bridges due to their excellent resisting characteristics. Since the steel tube provides triaxial confinement in much the same way as a stirrup in a reinforced concrete (RC) column, while carrying a portion of the applied axial force and bending moment, advantages in structural behavior such as increased strength, stiffness and ductility can be expected relative to ordinary RC columns (Shams and Ala Saadeghvaziri, 1997; Shanmugam and Lakshmi, 2001; Susantha et al., 2001; Bradford et al., 2002; Portolés et al., 2011). An additional advantage can also be obtained with respect to the construction sequence because the steel tube serves as formwork during the pouring of in-filled concrete (Schneider, 1998; Goto et al., 2010).

To examine the overall performance of CFT columns, various experimental
studies have been performed (Choi et al., 2017; Chen et al., 2017; McCann et
al., 2015) after Schneider (1998), and Huang et al. (2002) tested CFT
columns having circular and square cross sections to investigate the effect
of the steel tube wall thickness and the diameter-to-thickness (

In parallel with experimental studies, reliable numerical approaches have also been conducted (Goto et al., 2010; Hu et al., 2005; Moon et al., 2012) with the introduction of analytical models. The development of an analytical model for the response of CFT columns, however, requires an exact simulation of the contact behavior between the concrete core and the steel tube, because the accuracy of the numerical results is dominantly affected by both the bond-slip behavior and the confinement effect induced by the contact behavior between the concrete core and the steel tube. Many researchers, including Hibbitt and Sorensen, Inc. (2000), have simulated the contact behavior along the interface with the use of a proper friction coefficient because of its simplicity in application. However, this approach requires an indirect implementation of the confinement effect on the in-filled concrete because direct delivery of the stress between the in-filled concrete and the steel tube cannot be defined (Goto et al., 2010; Moon et al., 2012).

To remove this limitation and to directly consider the confinement effect with the bond-slip behavior, the bond-link element, which connects a node of a concrete finite element with a node of an adjacent steel element, was introduced. This element (Kwon et al., 2005; Kwak, 1994; Yin and Lu, 2010) has been commonly used in the finite element analysis of CFT columns. A few numerical analyses considering the bond-slip effect have been conducted by using the bond-link element (Bashir et al., 2010; Kwak and Kim, 2010; Li et al., 2012). However, owing to the complications in the numerical modeling with a considerable increase in the number of degrees of freedom caused by the use of double nodes, most finite element studies of CFT columns do not take into account the bond-slip developed at the interface between the concrete core and the steel tube (Johansson and Gylltoft, 2002; Kwon et al., 2005). Notably, the use of a fine mesh in modeling CFT columns increases the difficulty in the numerical modeling.

To address this issue, this paper introduces an improved numerical approach that can consider the bond-slip effect without the use of double nodes at the interface between the concrete core and the steel tube by incorporating the equivalent steel stiffness. The basic concept of this numerical design was introduced by Kwak and Kim (2001) and was derived from the bond-slip relation of a RC beam. Kwak and Kim applied the adjust stiffness matrix found through iteration that simulates a similar strain to the actual bond-slip behavior. However, since the Kwak and Kim model was developed for beam elements and the loading condition was limited to uniaxial load, an improved bond-slip model should be suggested to simulate CFT columns. In the bond-slip model proposed in this paper, after calculating the nodal displacements, the deformation of steel at each node can be found through the back-substitution technique from the first to the final steel element using a governing equation constructed on the basis of the force equilibrium at each node of the steel tube and the compatibility condition between the concrete core and the steel tube. The developed algorithm can also be implemented in many commercialized programs including ABAQUS (2013) and ADINA (2015) as a user defined material model, and the validity of the proposed numerical modes is established through correlation studies between the numerical results and the experimental data for CFT columns subjected to various monotonic loadings.

Stress-strain relation of concrete.

Since concrete is mainly used for compression members, the stress-strain relation of concrete in compression should be considered significantly, and the monotonic envelope curve of stress-stain for concrete introduced by Kent and Park (1971) and later modified by Scott et al. (1982) is adopted in this paper, because of its simplicity and computational efficiency. This model describes the monotonic concrete stress-strain relation in compression as a second-degree parabola accompanying the linear descending branch after reaching the compressive strength, as shown in Fig. 1.

On the other hand, the tensile behavior of concrete is assumed to be linear
elastic until reaching the tensile strength. After the tensile strength, the
tensile stress decreases linearly with an increase of the principal tensile
strain up to reach

In addition, as a part of considering the confinement effect, the use of
augmented compressive strength and corresponding strain through the
introduction of material coefficients has been considered in previous
studies while defining the stress-strain relation of concrete (Goto et al.,
2010; Hu et al., 2005; Hibbitt and Sorensen, Inc., 2000; Gupta et al., 2014), where the
material coefficients are usually defined as the ratio of the diameter of
in-filled concrete

Biaxial concrete strength envelope (Lee and Fenves, 1998).

Moreover, the multi-axial behavior of concrete must be defined to simulate
the three-dimensional behavior of concrete induced from the confinement by
the steel tube. Under combinations of multi-axial loading, concrete exhibits
strength and stress-strain behavior that are different from those under
uniaxial loading conditions. The strength envelope proposed by Lee and
Fenves (1998) and defined in ABAQUS (2013) is adopted in this paper. This
envelope, which is defined for the compression-compression region, is
slightly different from Kupfer's failure envelope (Kupfer and Gerstle, 1973)
obtained through a panel test. As shown in Fig. 2,

Stress state in a CFT column.

Free body diagram of a CFT column.

When a compressive force is applied to a circular CFT column, both the in-filled concrete and the steel tube will be subjected to not only normal stress but also stress in the radial direction, which develops a confinement effect. The interaction behavior between the in-filled concrete and the steel tube can then be derived on the basis of a perfect bond, where it is assumed that all the displacements of the in-filled concrete and the steel tube developed at the interface are identical.

Figure 3 shows the stress state of an in-filled concrete core and a steel
tube in the equilibrium condition at an arbitrary location at a distance

To evaluate the radial stress in the steel tube (

Differently from the perfect bond case, however, to consider the bond-slip effect it is necessary to define the different displacement fields between the in-filled concrete and the steel tube, and to this end the bond-link element defined by a spring element connecting a concrete node and an adjacent steel node is usually adopted. Since the link element has no physical dimension, the two connected nodes originally occupy the same location in the finite element (FE) mesh of the structure before deformation.

As shown in Fig. 4, an axial force applied to a CFT column is carried partly
by the in-filled concrete (

An idealized steel strut with bond-slip.

The use of the bond-link element in the FE analysis of RC structures, however, mandates the use of a double node to represent the relative slip between the in-filled concrete and the steel tube. In a complex structure, this requirement leads to not only a considerable increase in the number of degrees of freedom but also greater complexity of the mesh definition. To address these limitations in using the bond-link element, an improved numerical model is proposed in this paper.

To determine the equivalent modulus of elasticity

By considering the steel degrees of freedom in Eq. (2), the following
relation between concrete displacements and corresponding forces results in

Once the displacement increments at each node

Modeling of steel tube for matrix method.

Flow chart of bond-slip algorithm.

Gages on the Steel Tube.

Stress Distribution at the Steel Tube (

If the steel tube is assumed to be subdivided into

Modeling of a circular CFT column.

Dimensions and Material Properties of Specimens (Moon et al., 2012; Elremaily and Azizinamini, 2002).

Relation between load and displacement.

ABAQUS 6.13 (2013) is used in the numerical analyses, and 8-node 3-D solid
elements (named C3D8R element in ABAQUS) are adopted in the numerical
modeling of both the in-filled concrete matrix and the steel tube,
respectively. Moreover, to reserve consistency in the numerical modeling of
all of the specimens considered in this paper, the mesh size of each finite
element is based on an equal length of 20

In order to investigate the validity of the proposed bond-slip model, the
experimental results of a CFT column tested by Kwon et al. (2005) have been
used. This column is subjected to uniform compressive pressure on the
in-filled concrete only, and Fig. 8 shows the layout for the gages and the
geometric configuration for the specimen. The compressive strength of
concrete is 57.1

Figure 9 shows the distribution of steel stresses along the specimen at

To validate the effectiveness of the proposed numerical model, four
experimental results from several circular CFT columns tested by Schneider
(1998) and Huang et al. (2002) are investigated first, and these
experimental data have been widely used by many researchers (Moon et al.,
2012) in correlation studies between numerical results and experimental
data. The specimens are subjected to axial force only and the configuration
and the corresponding dimensions of the specimens can be found in Fig. 10
and Table 1, respectively. Numerical analyses are conducted with
displacement control on the top of the columns to achieve similar loading
conditions to those of the experiment. Since the resisting capacity of the
CFT column is dominantly affected by the ratio of

Lateral force-displacement relation under axial force and bending moment.

Figure 11 compares the analytical results with the measured load-displacement
response of the specimens. As shown in this figure, the perfect bond
assumption overestimates the energy absorption capacity, evaluated by the
area under the load-displacement curve, as well as the ultimate load, and
the overestimation appears to be enlarged with a decrease in the ratio of

The use of spring elements still gives reliable predictions for the
structural response up to reaching the ultimate load. However, the maximum
load is slightly overestimated at the post-yielding stage, and this
over-estimation is consistently maintained with an increase of the bending
deformation. This result appears to be caused by the use of the constant
slip modulus

Configuration of test specimen (unit: mm).

The proposed numerical model is also applied to two CFT columns subjected to
axial force and bending moment. These specimens were tested by Elremaily and
Azizinamini (2002) with the purpose of quantifying the contribution of the
steel tube to an increase of the energy dissipation capacity and ductility
in CFT columns. The configuration of the test specimens can be idealized as
a cantilevered member, as presented in Fig. 10, and the corresponding
dimensions and material properties of the specimens can be found in Table 1.
The applied constant axial forces are

The analytical responses for the horizontal drift

Test set-up details with actuator and cyclic load.

A correlation study for a CFT column subjected to double curvature bending
is conducted for a specimen that was designed and experimentally evaluated
by the authors. The dimensions and details of the test specimen are shown in
Fig. 13. The strengthening by the steel tube was limited to the length of

Details of 3-D model for test specimen.

Failure mode and load-displacement relationship of test specimen.

Development of slip at the interface between in-filled concrete and steel tube.

The loads were applied to the specimen by actuators placed at the top face
and at the mid-span of the specimen, as shown in Fig. 14b (Moon and Lee,
2014). The load at the mid-span is then transformed to the lateral load at
the top face of the specimen through the parallelogrammatic apparatus, which
was designed to prevent the rotational deformation accompanied by the
horizontal drift at the top face of the specimen (see Fig. 14c). The
applied axial load maintained a constant value of

As shown in Fig. 15, full modeling of the experimental specimen is
considered and fixed end boundary conditions are introduced at both end
faces. Upon e introduction of the self-weight and a constant axial force of

This paper introduces a slip model to take into account the bond-slip behavior in CFT columns. Differently from the classical spring element (bond-link element), which has many restrictions in numerical modeling because of the use of double nodes to represent the relative slip between the in-filled concrete and the steel tube, the introduced model, which does not use double nodes, can significantly reduce the number of nodes required to account for the bond-slip effect and will remove the difficulty arising in constructing a FE mesh in three-dimensional FE modeling. In advance, the introduced model can be implemented into many commercialized programs including ABAQUS (2013).

Through a comparison of numerical results and experimental data to verify the efficiency and applicability of the introduced slip model, the following conclusions have been drawn: (1) the bond-stress distribution shows a similar trend with the experimental results throughout the entire length of test specimens and is as reliable as the classical spring element; (2) when axial and lateral loads are applied separately, the load-displacement relationship of the test specimens shows good agreement with experiments; and (3) the load-displacement relationship of CFT columns subjected to a combined load of axial and bending is described well even with consideration of the secondary effect, and the failure mode and slip behavior simulated with the proposed bond-slip model are consistent with the experimental results. Accordingly, the introduced slip model, even with small error, can effectively be used to adequately predict the slip behavior as well as the ultimate load of CFT columns subjected to axial force and bending moment.

Data can be made available upon reasonable request. Please contact Hyo-Gyoung Kwak (kwakhg@kaist.ac.kr).

HGK suggested overall concept of this paper and details of the numerical bond-slip model. Also, he did the final examine of the whole paper. JYH applied the bond-slip model to CFT FEM model and verified with experiments and he set-up and performed the experiment to verify the suggested model.

The authors declare that they have no conflict of interest.

This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIP)(No. 2017R1A5A1014883) and the Korea Agency for Infrastructure Technology Advancement (KAIA) grant funded by the Ministry of Land, Infrastructure and Transport (Grant 13IFIP-C113546-01). Edited by: Amin Barari Reviewed by: two anonymous referees