# 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  
To use the methods in Sections 3 and 4 to calculate the response of the bridge under earthquake action, it is necessary to select an appropriate number of modes and time-step increments. Due to the effect of structural damping, the high-order modal response has little effect on the structure. The number of modes selected in this paper is \(n=5\). The shortest time for the bending wave and axial wave penetration structure of the main beam is \(\Delta t=L/c_{b}=10.8\times 10^{-3}\) s. The shortest time for the pier's bending wave and axial wave to penetrate the system is \(\Delta t=L/c_{b}=4.38\times 10^{-3}\) s. Therefore, the chosen time step must be less than \(4.38\times 10^{-3}\) s, and the selected time step in this article is \(18\times 10^{-3}\) s.  
Figure 3 shows the calculation flow chart of the vertical displacement of the bridge in the case of possible vertical separation. The calculation time was 2 s. By recording the time of each separation and recontact in the vertical calculation, and substituting it into the longitudinal seismic calculation, the longitudinal displacement response of the bridge considering the separation condition was obtained. When the seismic excitation period T = 0.2 s, two types of V/H are selected. One is 0.67 specified in the specification, and the other is given by Equation (1), where the value is 1.0. Figure 4 shows the seismic response of the bridge under two V/H values. When \(\lambda=0.67\), it can be seen that the main girder and the pier are always in contact, and the maximum vertical contact force is 20.4 MN. When \(\lambda=1\), the main girder and the bridge pier will be divided, and the main girder and the bridge pier will be separated six times in 2 s. The vertical contact force increased to 26.6 MN, an increase of 30.4%. Compared with the vertical contact force, the separation has a more significant impact on the longitudinal displacement response of the structure. When the height of the pier is H = 15 m (\(T_{s}>T_{v}\), \(T_{s}\) and \(T_{v}\) are the horizontal and vertical natural vibration periods of the bridge, respectively), the maximum longitudinal relative displacement of the pier and beam is 14.9 mm when the separation is ignored. When the structure is separated, the maximum longitudinal relative displacement increases to 29.9 mm, nearly two times. Figure 5 shows the effect of separation on the displacement response of the structure when the pier height H = 9 m (\(T_{s}<T_{v}\)) and 12 m (\(T_{s}\approx T_{v}\)). When H = 9 m, the separation caused the maximum longitudinal displacement of the pier beam to increase from 6.06 mm to 7.09 mm, increasing by 17%. When H = 12 m, the separation caused the maximum longitudinal displacement of the pier beam to increase from 17.83 mm to 21.1 mm, increasing 21.9%. The effect of separation on the displacement response of the structure is significantly lower than when the pier height is 15 m. It can be seen that the partition has a more significant impact on the longitudinal displacement of the bridge when \(T_{s}>T_{v}\). Figure 3: Flow chart showing bridge vertical displacement calculation procedure. Figure 6 shows the change of bending moment of bridge pier in two cases. It can be seen that when the pier is higher, the separation has a greater impact on the bending moment. Figure 4: Seismic response of the bridge under different \(\lambda\): (**a**) The vertical displacement and contact force response of the bridge when \(\lambda\) = 0.67; (**b**) The vertical displacement and contact force response of the bridge when \(\lambda\) = 1; (**c**) Longitudinal relative displacement of pier beam when \(\lambda\) = 0.67; (**d**) Longitudinal relative displacement of pier beam when \(\lambda\) = 1.