# Transient Response of Bridge Piers to Structure Separation under Near-Fault Vertical Earthquake
## 4 Numerical Simulation and Analysis
### The Influence of Near-Field Vertical Seismic Acceleration on the Structure
Figure 5: Longitudinal relative displacement of pier beam under different pier height: (**a**) Longitudinal relative displacement of pier beam under two conditions of separation or not when H = 9 m; (**b**) Longitudinal relative displacement of pier beam under two conditions of separation or not when H = 11 m.  
To calculate the bending failure of the pier, it is assumed that the pier is under eccentric compression. The collision diagram is shown in Figure 7. In order to consider the most unfavorable situation of the structure, it is assumed that the leading eccentricity and the maximum collision force occur at the same time. The shear stiffness of the bearing has no attenuation and remains unchanged within the calculation range. The bottom of the pier is composed of three bending moments: the bending moment \(M_{z}\) caused by forced resonance, the bending moment \(M_{\varphi}\) caused by the bearing shear force at the pier bottom, and the bending moment \(M_{\varphi}\) generated by eccentric collision. To simplify the calculation in this study, the influence of the plastic hinge produced at the pier bottom on the displacement of the pier top and the bending moment at the pier bottom is ignored. Ignoring the reset of the bearing during separation, the shear deformation is the longitudinal relative displacement of the pier and beam. The calculation formula of each bending moment is as follows:  
\[M_{c}=F_{max}\times\eta\times\Delta d;\ M_{\varphi}=K_{\varphi}\times\Delta d \times L;\ M_{z}=E_{r}I_{r}\partial^{2}U_{d}/\partial\xi^{2} \tag{39}\]  
where \(\ \eta\\) is the magnification factor of the eccentric collision. To better study the effect of separation on the structure, three types of piers are selected here: 9 m, 12 m, and 15 m. Figure 8 shows the changes in the bending moment at the bottom of the pier at three heights. Figure 8a shows that when the pier height H = 9 m, the high amplitude vertical acceleration reduces the allowable bending moment of the pier. Whether separated or not, the piers are in a safe zone. When the pier height H = 12 m, \(T_{s}\\) is close to the excitation period \(T\). The forced resonance of the bridge pier produces a  
Figure 6: Bending moment change of bridge pier under two conditions: (**left**) Vibration bending moment; (**right**) Eccentric compression moment. Figure 7: Force diagram at the bottom of the pier under eccentric compression. large bending moment, the high amplitude vertical impact force under the condition of separation reduces the allowable bending moment of the pier and causes damage to the pier. When the pier height H = 15 m, the moment of the pier is 1.66 MN and the structure is in the safe zone. When the system is separated, the bending moment increases to 2.94 MN, which increases by 77%, and the pier is damaged. Figure 8b shows the variation of various bending moments at the pier bottom under different conditions. It can be seen that the shear stiffness of rubber bearing is low, and the bending moment produced by bearing shear has little influence on pier failure in most cases. When H = 12 m, the pier failure is mainly due to forced resonance, and the bending moment caused by eccentric compression is significantly increased compared with H = 9 m. When H = 15 m, the seismic excitation period is far away from the longitudinal natural vibration period of the bridge, the seismic response of the bridge is low, and the structure is always in the safe range. When the structure is separated, \(M_{c}\) and \(M_{z}\) are immensely increased by the large expansion of longitudinal relative displacement. It can be seen that ignoring the bridge separation caused by high vertical acceleration not only underestimates the vertical contact force between the main girder and pier, but also may misestimate the seismic displacement response of the bridge.