## 5 Parametric Analysis of Lateral Collision Effects
### Collision Stiffness  
Collision stiffness is an important parameter affecting the seismic performance of the anti-falling beam device. When it plays a role, the inertial force shared by the pier increases significantly. Now, the collision stiffness \(k_{\text{k}}\) is taken as 1 \(\times\) 10\({}^{4}\) kN/m, 5 \(\times\) 10\({}^{4}\) kN/m, 1 \(\times\) 10\({}^{5}\) kN/m, 3 \(\times\) 10\({}^{5}\) kN/m, 5 \(\times\) 10\({}^{5}\) kN/m, 7.5 \(\times\) 10\({}^{5}\) kN/m, and 1 \(\times\) 10\({}^{6}\) kN/m, respectively, and the relationship between collision stiffness and seismic response of cross-fault bridges is compared and discussed.  
Figure 18 is the relationship curve between the collision stiffness with the internal force of the pier bottom and the displacement of the bearing under the action of three seismic waves. It can be seen that the trend of curve change under different seismic waves is consistent. However, the influence of collision stiffness on the internal force of the pier bottom and the displacement of the bearing is inconsistent. With the increase in collision stiffness, the torque at the bottom of the pier increases first and then decreases, and the end remains level. The variation range of the torque at the bottom of the pier is very limited, and the maximum change is only 3.3%. It shows that the influence of collision stiffness on pier bottom torque is very small, and it can be ignored when selecting parameters. However, the impact stiffness has a significant effect on the bending moment of the pier bottom, and its overall trend is obviously increasing. Because in terms of the nature of the contact element, the collision stiffness directly determines the size of the collision force; with the increase in collision stiffness, the collision effect will undoubtedly be amplified, and the increased collision force will be transmitted to the pier through the cap beam, which will eventually lead to the increase in the bending moment response at the bottom of the pier. However, the bending moment at the bottom of the pier increases slowly in the later period. The displacement of the bearing is different from the internal force of the pier bottom. Due to the increase in the collision stiffness, the restriction ability of the shear key to the displacement of the main beam is also increased, so the displacement of the bearing decreases with the increase in the collision stiffness. However, according to the slope of the bearing displacement curve, its growth trend has gradually slowed down. It shows that with the increase in collision stiffness, the limit effect of the shear key drops sharply.  
Figure 17: Displacement response of the middle pier bearing of the fault-crossing bridge under different sliding displacements: (**a**) maximum displacement, (**b**) residual displacement.  
The above analyses show that when the collision stiffness is large, the degree of change in the seismic response of the structure is extremely small. In the smaller range of collision stiffness, the bending moment of the pier bottom and the relative displacement of the pier beam should be the key objects to be considered. Therefore, the selected collision stiffness is in the range of \(k_{\mathrm{k}}=5\times 10^{4}\)-\(1\times 10^{5}\) kN/m, and the shear key device has a better seismic effect.