## 2 Numerical Finite Element Model
### Bearing Supports and Stoppers  
The bearing support layout including fixed and roller bearings is shown in Figure 4. The movements of the fixed bearing supports (F) are restrained in the longitudinal and transverse radial directions. Force-displacement relationship for fixed bearing supports is shown in Figure 5a. Roller bearing supports (R) can freely move in longitudinal direction, but they are restrained in transverse radial direction. Coulomb friction force is considered in numerical analysis for roller bearings, which are represented by using a trilinear element (Figure 5b). The structural properties of the steel bearing supports are shown in Table 3. The dispersed steel stoppers (D1-D4) are installed at a spatial multipoint near each roller bearing to limit excessive displacement of rollers. In this analysis, the bounding phenomenon at steel stopper is simulated by a modified Hertz-damp model, as shown in Figure 6a. The model can consider the energy dissipation in the bounding process, which can accurately simulate the bounding process. A nonlinear damper is activated during the approach period of the bounding to simulate the process of energy dissipation, which takes place mainly during that period in Figure 6b. During the restitution period, no energy dissipation occurs. The bounding force, \(F\), for this type of bounding element is expressed as  
\[F=\left\{\begin{array}{ccc}0&\text{for }\delta<0\ (\text{no contact}),\\ K_{\delta}\delta^{3/2}+C\delta&\text{for }\delta\geq 0,\overset{\bullet}{ \delta}>0\ (\text{contact}-\text{approach period}),\\ K_{\delta}\delta^{3/2}&\text{for }\delta\geq 0,\overset{\bullet}{ \delta}\leq 0\ (\text{contact}-\text{restitution period}),\\ \delta=x_{1}-x_{2}-d_{0}\ C=2\overline{\xi}\sqrt{K_{\text{s}}\sqrt{\frac{g _{1}m_{2}}{m_{1}+m_{2}}}},\end{array}\right. \tag{1}\]  
where C is the bounding element's damping, \(K_{\text{s}}\) is the bounding stiffness parameter, and the value is 780 MN/m; \(x_{1}\), \(x_{2}\) denote displacements of the structural members; \(d_{0}\) presents the initial distance between separation gap and the value is 4 cm, 6 cm, and 8 cm, respectively; \(m_{1}\) (1.86 \(\times\) 10\({}^{5}\) kg), \(m_{2}\) (7.38 \(\times\) 10\({}^{4}\) kg) are the mass of superstructure and pier, respectively; \(\overline{\xi}\) is the damping ratio, and the value is 0.02.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Type of** \\ **Damper** \\ \end{tabular} } & \multirow{2}{*}{**Attenuation Coefficient**} & \multirow{2}{*}{**Power Exponent**} & \multirow{2}{*}{**Stiffness**} & \multirow{2}{*}{**Stigper = 4 cm**} & \multirow{2}{*}{**Stigper = 6 cm**} & \multirow{2}{*}{**Stigper = 8 cm**} \\ \cline{4-4}  & & & & & **Resistance Force** & & **Resistance Force** \\ \cline{4-4}  & & & & & **F\({}_{1}\) (MN)** & & **Force F\({}_{1}\) (MN)** \\ \hline D0 & — & — & — & — & — & — \\ D100 & 117 & 0.22 & 90 & 0.105 & 0.108 & 0.111 \\ D300 & 350 & 0.22 & 130 & 0.313 & 0.324 & 0.333 \\ D500 & 583 & 0.22 & 200 & 0.521 & 0.539 & 0.555 \\ D1000 & 1165 & 0.22 & 400 & 1.041 & 1.077 & 1.109 \\ D2000 & 2330 & 0.22 & 800 & 2.082 & 2.154 & 2.218 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Structural properties of viscous dampers.  
Figure 4: Elevation view of finite element model of curved viaduct.