## 3 Pseudo-Dynamic Tests and Numerical Analysis of Model Structure
### Finite Element Modeling Program
#### 3.2.2 Model Elements and Constitutive Relationships of Materials  
A three-dimensional beam element B31 was employed to model the steel truss and diagonal braces. It has two nodes and each node has 6-degree-of-freedom. Shear deformation and finite axial strain are considered. For different types of cross sections, multiple integration points are used to export the stress and strain. In this paper, box-, I-, circular- and L-shaped cross sections were used, and the number of the corresponding integration points are 16, 13, 8 and 9 respectively. A quadrilateral shell element with reduced integration S4R was used to model the RC tubular columns. It is a general shell element with 4 nodes and each node has 6-degree-of-freedom. It could consider the transverse shear deformation. It follows the Mindlin shell theory with increase of the thickness, while follows the Kirchhoff shell theory with decrease of the thickness. It considers the finite membrane strain and arbitrarily large rotation, suitable for the large deformation analysis. The numerical integration method is adopted to calculate the stress and strain for each cross-sectional point along the thickness direction of the shell element. 5 integration points are assigned in the thickness direction so as to simulate the progressive failure process of the shell element [25; 26; 27; 28]. For steel components including the truss, A-shaped frames and diagonal braces, the stress-strain relationship of steel was simulated by the bilinear kinematic hardening model, which takes into account the Bauschinger effect under cyclic loading. The material parameters of the steel are given in Table 2.  
For concrete tubular columns, the concrete damaged plasticity (CDP) model provided in the material library of ABAQUS was adopted to simulate the mechanical behavior of concrete. The compressive strength, tensile strength and Young's modulus were introduced in Section 3.1. The uniaxial tensile and compressive damaged stress-strain relationships expressed by Equations (1)-(7) were used to determine the parameters of the damaged plasticity model of concrete according to the China design code of concrete structures [29]. \[\sigma=(1-d_{\mathrm{t}})E_{0}\varepsilon \tag{1}\]  
\[x=\varepsilon/\varepsilon_{\mathrm{t}0} \tag{2}\]  
\[d_{\mathrm{t}}= \begin{cases}1-\big{(}1.2-0.2x^{5}\big{)}&x\leq 1\\ 1-\frac{1}{\varepsilon_{\mathrm{t}}(x-1)^{1/2}+x}&x>1\end{cases}  \tag{3}\]  
\[\sigma=(1-d_{\mathrm{c}})E_{0}\varepsilon \tag{4}\]  
\[x=\varepsilon/\varepsilon_{\mathrm{c}0} \tag{5}\]  
\[d_{\mathrm{c}}=\left\{ \begin{array}{cc}1-\frac{n}{n-1+x^{\pi}}&x\leq 1\\ 1-\frac{\varepsilon_{\mathrm{c}}(x-1)^{2}+x}{x}&x>1\end{array} \right. \tag{6}\]  
\[n=\frac{E_{\mathrm{c}}\varepsilon_{\mathrm{c}0}}{E_{\mathrm{c}}\varepsilon_{ \mathrm{c}0}-f_{\mathrm{c}}} \tag{7}\]  
where \(d_{\mathrm{t}}\) and \(d_{\mathrm{c}}\) are respectively the uniaxial tensile and compressive damage evolution parameters for concrete; \(E_{0}\) is the initial elastic modulus of concrete; \(\varepsilon_{\mathrm{t}0}\) and \(\varepsilon_{\mathrm{c}0}\) are respectively the peak strains of the uniaxial tensile and compressive stress-strain curves for concrete; \(\alpha_{\mathrm{t}}\) and \(\alpha_{\mathrm{c}}\) are respectively the descending stage parameters of the uniaxial tensile and compressive stress-strain curves for concrete, equal to 1.95 and 1.94, respectively; \(f_{\mathrm{c}}\) is the uniaxial compressive strength of concrete. The energy equivalence hypothesis was adopted to establish the damage evolution rules, and it supposed that the elastic complementary energy of the damaged material has the same form with the undamaged material, except that the stress of the undamaged material should be replaced by the effective stress of the damaged material [30]. The elastic complementary energy corresponding to the undamaged material and the damaged material can be expressed by Equations (8) and (9), respectively.