# Thin-Walled CFST Columns for Enhancing Seismic Collapse Performance of High-Rise Steel Frames
## 2 Modeling and Analysis Methods  
### Simplified Fiber Element Models  
A fiber element program that has been developed is utilized to conduct the elasto-plastic analysis of structures under various loading procedures [16; 22]. The stiffness of cross sections is numerically integrated by dividing the section into a number of parallel layers referred to as stress fibers. Accurate nonlinear stress-strain models are applied to these fibers to define the mechanical behavior of each member. The law of fiber discretization for H-shaped steel and hollow section members is shown in Figure 1.  
On the longitudinal definition of elements, the plastic length at both ends of each element depends on the thickness and outer size of the steel tubes. In this study, the plastic length is determined as a constant value, but the stress-strain models are consistent with the assumption of the constant  
Figure 1: Fiber element discretization and plastic region.  
plastic length of the member depth D, as shown in Figure 1. In the frame analysis program, the geometric nonlinearity has been rigorously formulated by adopting the moving coordinate system for each element. Therefore, the program can trace the nonlinear behavior up to collapse phenomenon. For dynamic analysis, the time history response of high-rise building models is calculated by the average acceleration method (\(\beta=1/4\)), based on the [PERSON]'s method. The convergence is guaranteed based on Newton-Raphson method.  
The patterns of local buckling of HSS and CFST members are obviously different due to the confined effect by in-filled concrete. For unfilled rectangular or square CFSTs under axial compression, the inward buckling is effectively prevented. Thus, the local buckling pattern of steel tubes influences strength deterioration with internal restraint from the in-filled concrete.  
Figure 2 shows the stress-strain relation for steel material with and without deterioration effects. The post-buckling strength deterioration and the corresponding negative slope in compressive direction enlarges the ratio of residual to maximum strain. The stress-strain models for confined concrete in CFST columns are shown in Figure 3. The post-crushing strength degradation reduces the unloading stiffness if the ratio of residual to maximum strain is a constant value [23]. Note that the splitting crack of the confined concrete in CFST columns occurs when the associated tubular face is subjected to tensile force. In this loading condition, the CFST columns enable the support of plastic deformation without the local buckling in steel tubes. In addition, the tensile strength of the confined concrete is much less than the yielding strength of steel tubes. Thus, the splitting crack can be neglected in the stress-strain relation of the concrete model [24].  
In the numerical simulations of steel and CFST structures using the fiber element method, it is assumed that the cross section remains plane, and the strain distribution keeps linear along with the  
Figure 3: Stress–strain relation of confined concrete. (**a**) Model with post-crushing deterioration; (**b**) Model without post-crushing deterioration.  
Figure 2: Stress–strain relation of thin-walled steel sections. (**a**) Model with post-buckling deterioration; (**b**) Model without post-buckling deterioration.  
depth of the cross section. The stress should be located at the centroid of each parallel fiber element, and vertical discretization is neglected for torsion or bi-axial bending. Besides, the effects of creep and shrinkage of concrete were not in the scope of this paper. The confinement effect and slipping of concrete on the strength and deterioration are taken into account in the normalized stress-strain relationship of the confined concrete. Post local-buckling strength and stiffness deterioration of steel tubular walls is taken into account in the constitutive relationship of steel plate elements.