## 2 Engineering Overview and Finite Element Model Establishment
### Bridge Finite Element Modeling
#### 2.2.2 Simulation of Lateral Collision  
Existing studies have shown that [34; 35] the contact element method is an effective way to simulate the collision effect between structural components. The commonly used mechanical models for collision between structures are the linear spring model, Kelvin-Voigt model, Hertz model, and Hertz-Damp model. Among them, the linear spring model and the Hertz model cannot consider the energy consumption during the collision process when simulating the collision reaction of the structure; although the Hertz-Damp model [36] can consider the energy dissipation problem of collision, the contact element used is a highly nonlinear element, which limits its application range. The Kelvin-Voigt model [37; 38] can not only consider the collision energy consumption but also the element form is concise, and the simulation effect can also meet the analysis requirements. Therefore, it is the most widely used in the study of seismic collision problems of bridge structures.  
Therefore, based on the Kelvin-Voigt model, this paper considers the initial gap between the shear key and the beam. The lateral collision is simplified as the model shown in Figure 7, which consists of a spring and a damping unit in parallel and then connected in series with the initial gap. The relationship between the collision force and the displacement is shown in Figure 8.  
In Figures 7 and 8, \(m_{1}\) and \(m_{2}\) are the masses of the main girder and pier, respectively. \(k_{\rm k}\) is the collision stiff-ness; \(g_{\rm P}\) is the initial gap of the shear key; \(C_{\rm k}\) is the damping coefficient; \(u_{1}\) and \(u_{2}\) are the displacement of the main beam and the shear key, respectively; \(F_{\rm max}\) is the maximum contact force; \(u_{\rm max}\) is the maximum displacement.  
Figure 8: Relationship between collision force and displacement.  
Figure 7: Equivalent model of lateral collision.  
In the process of transverse collision, the collision force \(F_{\rm c}\) between the shear key and the main beam can be expressed as:  
\[F_{\rm c}=\left\{\begin{array}{ll}k_{\rm k}\big{(}u_{1}-u_{2}-g_{\rm P}\big{)}+ C_{\rm k}\big{(}\dot{u}_{1}-\dot{u}_{2}\big{)},&u_{1}-u_{2}-g_{\rm P}\geq 0\\ 0,&u_{1}-u_{2}-g_{\rm P}<0\end{array}\right. \tag{3}\]  
where \(\dot{u}_{1}\) and \(\dot{u}_{2}\) is the speed of the main beam and the shear key; \(C_{\rm k}\) is the damping coefficient. According to the energy conservation relationship in the collision process, the relationship between the damping coefficient \(C_{\rm k}\) and the recovery coefficient \(e\) can be established as follows:  
\[C_{\rm k} =2\zeta\sqrt{k_{\rm k}\frac{m_{1}m_{2}}{m_{1}+m_{2}}} \tag{4}\] \[\zeta =-\frac{\ln e}{\sqrt{\pi^{2}+(\ln e)^{2}}} \tag{5}\]  
where \(\zeta\) represents the collision damping coefficient, which is related to the collision recovery coefficient \(e\); when \(e=1\), it means that the collision is completely elastic, and when \(e=0\), it means that the collision is completely plastic. In this paper, the energy loss in the process of concrete collision is considered, and the collision recovery coefficient \(e\) is 0.65.