# Seismic Response and Damage Analysis of an Isolated Bridge Subjected to Near-Fault Ground Motion
## 2 Finite Element Model of Bridge
### Bearing Analysis Model  
The LRB isolation element model, developed by [PERSON] et al. [28], was used in this analysis. The horizontal bi-directional coupling effect can be considered using the Bouc-Wen model improved by [PERSON] [29]. The restoring force is calculated as follows:  
\[\left[\begin{array}{c}F_{1}\\ F_{2}\end{array}\right]=\alpha k_{\text{b}}\left[\begin{array}{c}U_{1}\\ U_{2}\end{array}\right]+(1-\alpha)F_{\gamma}\left[\begin{array}{c}Z_{1}\\ Z_{2}\end{array}\right]+c_{\text{b}}\left[\begin{array}{c}\dot{U}_{1}\\ \dot{U}_{2}\end{array}\right]. \tag{4}\]  
where \(F_{1}\) and \(F_{2}\) are the restoring forces of lead rubber bearings in the X-direction and the Z-direction, respectively; \(U_{1}\) and \(U_{2}\) represent the relative displacement of the lead rubber bearing in the X-direction and the Z-direction, respectively; \(\alpha\) is the ratio of post-yield stiffness to pre-yield stiffness; \(k_{\text{b}}\) is the initial stiffness (pre-yield stiffness); \(c_{\text{b}}\) is viscous damping of lead rubber bearings; \(Z_{1}\) and \(Z_{2}\) are hysteretic displacements in the X-direction and the Z-direction, respectively, satisfying the following relationship:  
\[U_{\gamma}\left[\begin{array}{c}\dot{Z}_{1}\\ \dot{Z}_{2}\end{array}\right]=[G]\left[\begin{array}{c}\dot{U}_{1}\\ \dot{U}_{2}\end{array}\right], \tag{5a}\] \[G=\left[\begin{array}{cc}A-\beta\text{sgn}\Big{(}\dot{U}_{1}Z_{1}+\dot{U}_{2}Z_ {2}\Big{)}Z_{1}{}^{2}-\gamma Z_{1}{}^{2}&-\beta\text{sgn}\Big{(}\dot{U}_{1}Z _{1}+\dot{U}_{2}Z_{2}\Big{)}Z_{1}Z_{2}-\gamma Z_{1}Z_{2}\\ -\beta\text{sgn}\Big{(}\dot{U}_{1}Z_{1}+\dot{U}_{2}Z_{2}\Big{)}Z_{1}Z_{2}- \gamma Z_{1}Z_{2}&A-\beta\text{sgn}\Big{(}\dot{U}_{1}Z_{1}+\dot{U}_{2}Z_{2} \Big{)}Z_{2}{}^{2}-\gamma Z_{2}{}^{2}\end{array}\right]. \tag{5b}\]  
where: \(U_{\gamma}\) is the yield displacement of lead rubber bearing; \(A\), \(\gamma\), and \(\beta\) are parameters that control the shape and size of the restoring force-displacement hysteresis loop of the lead rubber bearing, generally taking 1, 0.5 and 0.5, respectively; sgn is a symbolic function.  
The restoring force-displacement relationship of the isolation bearing model is shown in Figure 8.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(E_{s}\) (GPa) & \(f_{y}\) (MPa) & \(b\) & \(\varepsilon_{\text{th}}\) & \(\varepsilon_{\text{cr}}\) & \(D_{\text{cr}}\) & \(D_{0}\) & \(\alpha\) \\ \hline
210 & 303 & 0.01 & 0.259 & 1.4 & 0.065 & 0.0 & 0.2175 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of the steel material.  
Figure 7: The stress–strain relationship of the steel material.  
The force of the LRB isolation element model in the vertical direction (\(\gamma\)-direction) is calculated as follows:  
\[F_{\mathrm{v}}=K_{\mathrm{v}}\cdot U_{\mathrm{v}}. \tag{6}\]  
where: \(F_{\mathrm{v}}\) is the vertical force of the lead rubber bearing; \(K_{\mathrm{v}}\) is the vertical stiffness of the lead rubber bearing; \(U_{\mathrm{v}}\) is the vertical displacement of the lead rubber bearing.  
The overlap area method [30] is used to determine the bearing capacity (critical load) of the bearing under a given lateral displacement:  
\[P^{\prime}_{cr}=\frac{A_{r}}{A_{b}}P_{cr}. \tag{7}\]  
where: \(P^{\prime}_{cr}\) is the bearing capacity (critical load) of lead rubber bearing considering the influence of lateral displacement; \(A_{r}\) is the area of the overlapping part of the upper and lower sections of the bearing; \(A_{b}\) is the cross-sectional area of lead rubber bearing; \(P_{cr}\) is the bearing capacity (critical load) of lead rubber bearing without lateral displacement.