# Seismic Analysis of Steel Solid Web Girder-RC Tubular Column Hybrid Structure
## 3 Numerical Approaches and Validation  
### Simulation of RC Tubular Columns  
A general shell element with four nodes S4R was adopted to simulate the tubular column. It was a quadrilateral shell element with reduced integration. Transverse shear deformation was considered. For each cross-sectional point along the thickness direction of the shell element, the numerical integration method was adopted to calculate the stress and strain. To simulate the progressive failure process of the shell element, five integration points were assigned in the thickness direction [19; 20; 21; 22].  
The concrete damaged plasticity (CDP) model was adopted to simulate the mechanical behavior of concrete [23]. According to the China design code of concrete structures [24], the parameters of the CDP model can be determined using Equations (1)-(7).  
\[\sigma=(1-d_{\mathrm{t}})E_{0}\varepsilon \tag{1}\]  
\[x=\varepsilon/\varepsilon_{\mathrm{t}0} \tag{2}\]  
\[d_{\mathrm{t}}=\left\{\begin{array}{cc}1-\big{(}1.2-0.2x^{5}\big{)}&x\leq 1 \\ 1-\frac{1}{\alpha_{\mathrm{t}}(x-1)^{1.7}+x}&x>1\end{array}\right. \tag{3}\]  
\[\sigma=(1-d_{\mathrm{c}})E_{0}\varepsilon \tag{4}\]  
\[x=\varepsilon/\varepsilon_{\mathrm{c}0} \tag{5}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Component** & **Material** & \begin{tabular}{c} **Strength** \\ **Grade** \\ \end{tabular} & **Location** & **Cross Section (mm)** \\ \hline \multirow{4}{*}{Tubular column} & Concrete & C40 & \multicolumn{2}{c}{External diameter} & 4000 \\ \cline{3-5}  & & \multicolumn{2}{c}{Wall thickness} & 400 \\ \cline{2-5}  & Steel rebar & HRB400 & Longitudinal rebar & \(\Phi\)25 \\ \cline{2-5}  & & HPB300 & Circular stirrup & \(\Phi\)10 \\ \hline \multirow{4}{*}{Steel solid web girder platform} & \multirow{4}{*}{Steel} & \multirow{4}{*}{Q345} & Solid web girder & Main direction & H1300 \(\times\) 450 \(\times\) 16 \(\times\) 22 \\ \cline{3-5}  & & & \begin{tabular}{c} Solid web \\ girder \\ \end{tabular} & \begin{tabular}{c} Secondary \\ direction \\ \end{tabular} & H1300 \(\times\) 450 \(\times\) 16 \(\times\) 22 \\ \cline{3-5}  & & & \begin{tabular}{c} Brace of solid web girder \\ \end{tabular} & \begin{tabular}{c} TUB250 \(\times\) 250 \(\times\) 12 \\ \end{tabular} \\ \cline{3-5}  & Steel diagonal & & & \begin{tabular}{c} Diagonal brace \\ \end{tabular} & TUB450 \(\times\) 450 \(\times\) 20 \\ \cline{3-5}  & & & \
\begin{tabular}{c} Horizontal beam \\ \end{tabular} & HM390 \(\times\) 300 \\ \cline{3-5}  & frame & &
\begin{tabular}{c} Diagonal column \\ \end{tabular} & HW250 \(\times\) 250 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Strength grades of materials and cross sections of the main components.  
\[d_{\rm c}=\left\{\begin{array}{cc}1-\dfrac{n}{n-1+x^{n}}&x\leq 1\\ 1-\dfrac{1}{\alpha_{\rm c}{\rm(}x-1\big{)}^{2}+x}&x>1\end{array}\right. \tag{6}\]  
\[n=\dfrac{E_{\rm c}\varepsilon_{\rm c0}}{E_{\rm c}\varepsilon_{\rm c0}-f_{\rm c}} \tag{7}\]  
where \(d_{\rm t}\) and \(d_{\rm c}\) are respectively the uniaxial tensile and compressive damage evolution parameters for concrete; \(E_{0}\) is the initial elastic modulus of concrete; \(\varepsilon_{\rm t0}\) and \(\varepsilon_{\rm c0}\) are, respectively, the peak strains of the uniaxial tensile and compressive stress-strain curves for concrete; \(\alpha_{\rm t}\) and \(\alpha_{\rm c}\) 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;\(f_{\rm c}\) is the uniaxial compressive strength of concrete.  
Equations (8) and (9), respectively, give the expression of elastic complementary energy for the undamaged and damaged materials.  
\[W_{0}=\dfrac{\sigma^{2}}{2E_{0}} \tag{8}\]  
\[W_{D}=\dfrac{\overline{\sigma}^{2}}{2E_{0}}=\dfrac{\sigma^{2}}{2E_{0}{\rm(}1- D\big{)}^{2}}=\dfrac{\sigma^{2}}{2E_{D}} \tag{9}\]  
where \(\sigma\) and \(\