# Transient Response of Bridge Piers to Structure Separation under Near-Fault Vertical Earthquake
## 4 Numerical Simulation and Analysis
### Seismic Response of Bridge Pier under Different Excitation Periods  
The seismic excitation period affects the bridge displacement response. The research range selected in Figure 9 is \(T\) = 0.05 s\(\sim\)0.5 s. The change of seismic acceleration under a long period, the characteristic site period \(T_{g}\) = 0.3 s, is selected in this study. It can be seen that only when the excitation period is close to the vertical first-order natural vibration period of the bridge that the structure will separate, and the higher-order modes have little effect on the structural separation. The full longitudinal relative displacement of the pier beam will peak at \(T=T_{s}\), but this crest may not be the maximum value. When the pier height H = 15 m, the ultimate value appears at T = 0.25 s, and the excitation period at this time is close to \(T_{v}\) and far away from \(T_{s}\). It can be seen that the most unfavorable conditions caused by the forced vibration of the bridge pier are not entirely in the interval \(T=T_{s}\). When the pier is higher (\(T_{v}>T_{s}\)), the structural separation produced in the interval \(T=T_{v}\) will increase the seismic response of the bridge and cause the pier to be bent damaged. Figure 8: Various bending moments and allowable bending moments at the pier bottom: (**a**) Changes of various bending moments at the bottom of the pier at three different heights; (**b**) Longitudinal relative displacement of pier beam under different pier height. The bending failure of the pier is affected by the relative displacement of the pier and beam. The change of axial force not only affects the allowable bending moment, but also changes the bending moment generated by eccentric compression. Figure 10 shows the vertical contact force between the main girder and the pier under different excitation cycles. It can be seen that the peak value of the vertical contact force is concentrated in the \(T\approx T_{s}\) interval. Compared with the specification \(\lambda=0.67\), calculated by Equation (1), it has a more considerable peak value in a short period and a smaller peak value in an extended period. The structure separation of the bridge will occur under both conditions, but the excitation period of the structural separation calculated by Equation (1) is shorter. The maximum vertical contact force of the two is the same, in the range of 2.5 \(F_{c}\sim\)3\(F_{c}\). Figure 11 shows the variation of the bending moment of the pier under different seismic excitation periods. Under different excitation periods, when H = 9 m, the bending moment produced by eccentric compression is shallow, which can be ignored. When H = 12 m, \(M_{c}\) has a peak at \(T\) close to \(T_{s}\), which is due to the more enormous displacement response produced by resonance. When H = 15 m, \(M_{c}\) has two extremes, one is when \(T\) is close to \(T_{s}\), this is because the resonance produces a larger displacement response. One is  
Figure 10: Various bending moments and allowable bending moments at the pier bottom: (**left**) Changes of various bending moments at the bottom of the pier at three extra heights; (**right**) Longitudinal relative displacement of pier beam under extra pier height. Figure 9: Longitudinal relative displacement of pier beam under different excitation periods. in the separation interval, which is due to the expansion of the relative displacement due to structural separation. It can be seen from Figure 11b,c that the variation trend of \(M_{z}\) and \(M_{\gamma}\) in different seismic periods is the same as that of \(M_{c}\). When the height of the pier is low, the bending moment of the pier under the earthquake action is small. When the pier height is higher, the bending moment at the bottom of the pier is greater. In addition, when considering the structure separation, there will be two wave peaks, which need to consider the pier damage in two cases.