Modified Numerical Modeling of Axially Loaded Concrete-Filled Steel Circular-Tube Columns

Predicting the behavior of concrete in a ConcreteFilled Steel Tubular (CFST) column is challenging due to the sensitivity to input parameters such as the size of the crosssection, the material modeling, and the boundary conditions. The present paper proposes a new modified finite element model to predict the behavior and strength of a CFST subjected to axial compression. The development is based on the concrete damaged plasticity model, with its stress-strain relationship revised from the available model. The predicted accuracy of the modified model is verified via a wide range of experimental tests. The proposed model has more accuracy than the available models in predicting the ultimate compression strength. The results show good agreement with the test data, allowing its use in modeling CFST columns. Keywords-CFST columns; axial compression; finite element modeling; stress-strain relationship; steel; concrete

INTRODUCTION Concrete-Filled Steel Tubular (CFST) columns are widely used in modern constructions. Many studies have been conducted in order to understand the behavior of these composite columns, either experimental [1,2] or analytical [3,4]. Based on the outcomes from these investigations, different guidelines and codes have been published for the design procedure [5,6]. CFST columns combined with steel-framed structures [7,8] form very powerful structures. Authors in [9] used fiber reinforced polymer tubes filled with recycled materials and concrete for pile foundations. Using Finite Element (FE) models has become a popular technique to model CFST via software such as ANSYS and ABAQUS. This approach can model the interaction between concrete core and steel tube accurately. In this study, the FE software ABAQUS is used to FE model CFST. Recently, authors in [10] studied the influence of geometric and material properties on the behavior of the Concrete-Filled Double Skin Steel Tube (CFDST) member with M16 studs under bending. The material model is an input parameter that is mainly affected by the behavior of CFST. Several concrete and steel stress-strain ߪ − ߝ models have been developed [11][12][13][14][15][16]. Almost all models are developed from experimental data based on regression analysis. Authors in [17,18] proposed FE modeling for analyzing box and circular CFST columns. Authors in [16] developed a simplified model to calculate the capacity of the cross-section and the axial force via the load-deformation ܰ − ߝ curve based on 50 test data of Self-Consolidating Concrete (SCC) filled Hollow Structural Steel (HSS) stub columns. Similarly, authors in [14,15] proposed a simple and effective model for CFST subjected to extreme loadings. In [11], the steel ߪ − ߝ relationship was revised based on the available models by [19]. In the present study, the concrete ߪ − ߝ curve developed in [12] is revised via the modification of the plateau branch to represent the actual behavior of the material. This paper aims to develop an FE model of circular CFST columns under axial compression loading. A modified ߪ − ߝ model is proposed for the concrete material. The purpose of this study is to improve the accuracy of the FE model using ABAQUS. The accuracy and effectiveness of the new model are verified against the test data from [3,20]. In the future, focus may be given to the behavior of CFST with steel fiber reinforcement [21], concrete using fly ash [22], or concrete with cement paste including silica oxide nanoparticles.

A. Steel Material Model
In this study, the material model proposed in [19] is adopted for the characteristics of the stress-strain curves, as illustrated in Figure 1. The relationship of ߪ − ߝ can be expressed as follows: where ‫ܧ‬ ௦ is Young's modulus of steel, ݂ ௬ is the yield strength corresponding to the yield strain ߝ ௬ ൌ ா ೞ , ݂ ௨ is the ultimate strength corresponding to the ultimate strain ߝ ௨ , ‫‬ is the hardening exponent that can be calculated as: where ‫ܧ‬ is the initial elastic modulus at the onset of strain hardening, and can be taken as 0.02E s . Yield strain ߝ and ultimate strain ߝ ௨ can be expressed based on the range of yield strength ݂ ௬ which is 200MPa-800MPa:

B. Concrete Material Model
Characteristics of stress-strain curves for concrete have been studied in [1,24]. Most researchers focused on the development of a ߪ − ߝ model based on the experimental results. For example, in [25], ߪ − ߝ cuves for the fiber model of concrete have been developed to predict the monotonic and cyclic force-deformation behavior of CFST columns. Similar studies have been performed in [2,26]. In these works, empirical models were proposed for the strain-softening response for CFST. In this paper, the damaged plasticity model of concrete material is utilized. This model is available in ABAQUS and can be easily used by defining some key material parameters. These factors are calibrated and verified with the extensive test results of [27][28][29] and they are briefly introduced as follows: Young's modulus of concrete is predicted based on the concrete strength ሺ݂ ᇱ ሻ [30]: According to [31], the ratio of compressive strength under biaxial loading to the uniaxial compressive strength ሺ݂ /݂ ᇱ ሻ is obtained based on the available data in the literature as follows: The fracture energy ሺ‫ܩ‬ ி ሻ that is defined as the tensile softening response beyond the failure stress, depends on the maximum coarse aggregate size ሺ݀ ௫ ) and concrete strength ሺ݂ ᇱ ሻ. The following equation can be deduced to determine ‫ܩ‬ ி [32,33]: Authors in [29] stated that the compressive meridian ሺ‫ܭ‬ ሻ that is used to determine the yield surface of the material model, varied in the range from 0.5 to 1, and it is expressed as a function of equibiaxial concrete strength ሺ݂ ሻ and concrete strength ሺ݂ ᇱ ሻ [34]: The dilation angle ሺψሻ factor is used to define the plastic flow potential. This parameter is affected by the confining stress and plastic deformation of the material [27]. The factor ψ is determined as the function of the confinement factor ሺߦ ሻ based on regression analysis [1]: in which the confinement factor is calculated as: where ‫ܣ‬ ௦ and ‫ܣ‬ are the cross-sections of the steel tube and concrete respectively.
For the strain hardening-softening rule, authors in [12] proposed a ߪ − ߝ curve for confined concrete through the tests. This model is revised in the present paper. The details of the proposed model are described below.

C. The Proposed Stress-Strain Relationship for Concrete
The ߪ − ߝ model developed in [12] is modified to describe the effective ߪ − ߝ relationship of confined concrete by the steel tube, as illustrated in Figure 2. The model includes three main stages, corresponding to ascending branch (OA), plateau branch (AB), and descending branch (BC). The initial stage OA is described as follows: The ultimate strain ε cc of the confined concrete is expressed using the model of [12]. In this model, ε cc depends on the compressive strength ݂ ᇱ , and confining stress at the point B, ݂ : It is worth mentioning that the confining stress only happens before and after the yielding location of the material. In this study, the confining stress of the circular column is determined based on the regression analysis as follows: The constant value of confining stress ݂ remains from ߝ to ߝ ᇱ corresponding to the plateau branch (AB) that represents the increment of the peak strain of the concrete due to confinement. For the last stage of the concrete model (BC), the model of [36] is used, which is expressed as follows: where ݂ is the residual stress [1] and ߙ and ߚ are factors determining the shape of the softening stage. The parameter ߚ can be taken as 1.2 [1], and ߙ is calculated as follows:

A. Finite Element Modeling
The current paper used the available model developed in [37]. The diameter and length of the CFST are 300mm and 900mm respectively. The thickness of the steel tube is 3.2mm. Concrete with 27.2MPa compressive strength and 0.351 of ultimate strain is assigned for the concrete core, while steel with yield strength of 232MPa is assigned for the steel tube. The finite element model of Figure 3 is developed in ABAQUS. The steel tube and concrete are simulated by a 4node bilinear element with reduced integration with hourglass control (CAX4R). The mesh size of each element ranges from D/20 to D/10 which can provide accurate simulation. For the interaction between the steel tube and the concrete core, the surface-to-surface contact with Hard contact condition is used to model linear behavior, and the Column friction model is used to simulate the nonlinear behavior with a coefficient of 0.6 [3]. For boundary condition, the bottom surfaces are assumed to be fully fixed against all degrees of freedom. The incremental load is applied on the top of the column using the displacement control method to solve.

B. Effect of the Mesh Size on Finite Element Analysis
The accuracy and required computing time of the FE model are investigated in this section. Different meshing approaches, including the axisymmetric model and 3D model, are generated. Figure 4 illustrates the stress distribution from each model under the axial force. It can be seen that there is no obvious effect on the predicted Von Mises stress for the CFST column. For better consideration, the axial load-strain ሺܰ − ߝሻ relationship is obtained, as shown in Figure 5. The predicted responses from the two meshing approaches are almost the same as those from the experiment-tested CU-070 [20]. An important finding is that the computational time in the axisymmetric model is only 22sec which is less by a factor of 20 when compared to the 3D model. It can be concluded that the FE axisymmetric model can give accurate and efficient computation.  Influence of the meshing and element types.

A. Ultimate Strength
The data of 663 circular CFST columns from the previous studies are selected for validation. The ultimate strength ܰ ௨ is defined as the peak load of the ܰ − ߝ curve. In this paper, the ultimate strength from the proposed model ܰ ௨ is compared with those obtained from the available models in [3,20]. ܰ ௨ is normalized with the measured ultimate strength ܰ ௨ , which is denoted as ܰ ௨ /ܰ . The comparison is summarized in Table I. It can be seen that the result obtained from the FE model is more conservative than those produced from the available models. Additionally, the smallest value of standard deviation for the proposed model is found to be 0.055. The results indicate that the predicted outcomes from the current FE model are more reasonable than the previous models.

B. ܰ − ߝ Curves of the CFST Column
In terms of the prediction of the ܰ − ߝ curves, the proposed model is compared to the available models from [3,20]. The comparison is performed for different specimens obtained from [2,16,20]. Figure 6 illustrates the predicted ܰ − ߝ relationship of normal CFST, with its modeling parameters taken from [20]. The observed result from the FE model agrees with the experiment very well. N-ε curves of CFST column for normal specimens. fc' = 77.00 MPa; fy = 853.00 MPa; ξ ξ ξ ξc = 3.22 Figure 7 shows the prediction of the ܰ − ߝ curve of the specimen reported in [2]. In this test, the ratio ‫ݐ/ܦ‬ varies from 21.7 to 150. As can be seen, the unsafety in the prediction is obtained for the model in [3] in the post-yielding strength branch. On the contract, the modified FE model gives a better prediction with the test. Furthermore, the predicted ܰ − ߝ relationship was also performed for specimens with thin-walled steel tube and high-strength, as shown in Figure 8. The test curve is shown in [16]. The results indicate that the prediction from the FE modeling matches very well with those obtained from Han et al.'s model [16] and the test. In contrast, Hu et al.'s model [20] is an unsafe prediction in the post-yielding strength stage. The above discussion leads to the conclusion that the developed FE model can be used in simulating the CFST column under axial compression for accuracy, effectiveness, and safety predictions. V. CONCLUSION The current paper developed a FE modeling for axially loaded circular concrete-filled steel tube columns. The proposed model is compared with the available models in terms of accurate prediction. A new FE model is used to analyze the CFST subjected to axial compression. The model revised the ߪ − ߝ curve in [12]. The proposed model is more accurate than the models in [3,20]. in predicting the ultimate strength. The predicted ܰ − ߝ curve from the new model matches well with the test data, for both normal materials and thin-walled steel tubes, which indicates the versatility of the developed modeling in calculating the load-deformation curves. Regarding future work, the behavior of CFST columns with steel fiber reinforcement [21], concrete using fly ash [22], concrete with cement paste with including silica oxide nanoparticles [20], etc. can be studied by performing FEM simulation.