## 4 Results and Discussion
### Dynamic Response of Train  
Due to the nonlinear behavior of the wheel-rail contact and the randomness of seismic motion, coupled with the superposition of low-frequency pulse components and high-frequency impulsive components in fault seismic motion, the seismic random analysis of the train-bridge coupled system is highly complex. To unveil the correlation between the dynamic indicators of the train and FCAs and PGRDs, the assumption is made that the earthquake occurs simultaneously while the train is on the bridge. The scenario before the complete formation of PGRD is defined as case 1, while the scenario after the complete formation of PGRD is defined as case 2. The Code for Design of High Speed Railways [44] and the High-Speed Test Train Power Car Strength Dynamic Performance Specifications [45] of China offer three indicators to assess responses of train wheel-rails: (1) the maximum lateral wheel-rail force Q, where Q \(\leq\) 10 + P\({}_{0}\)/3 and P\({}_{0}\) represents the static wheel load on the load-reduction side; (2) the derailment coefficient Q/P, where Q/P \(\leq\) 0.8 and P denotes the maximum vertical force of the wheel-rail; (3) the wheel-load-reduction rate \(\Delta\)P/P\({}_{\rm m}\), where \(\Delta\)P/P\({}_{\rm m}\)\(\leq\) 0.6, \(\Delta\)P refers to the reduction in wheel load on the load-reduction side, and P\({}_{\rm m}\) stands for the average static wheel load. In this study, these indicators were employed to evaluate the running safety of the high-speed train. When the FCA is 90\({}^{\circ}\), the fling-step effect is primarily characterized by lateral excitation. In Figure 12, it is clear that lateral excitation is fatal to train derailment, and all three indicators show a positive correlation with the PGRD. When the bridge crosses the fault at a small angle, the fling-step effect is primarily characterized by longitudinal excitation. However, longitudinal excitation has a minimal impact on train derailment. Nevertheless, there is no clear tendency between the derailment coefficient and PGRD, which suggests the intricate nature of the train-bridge interaction. Figure 11: Peak shear force in tower: (**a**) longitudinal direction at FCA = 15\({}^{\circ}\); (**b**) transverse direction at FCA = 15\({}^{\circ}\); (**c**) longitudinal direction at FCA = 90\({}^{\circ}\); (**d**) transverse direction at FCA = 90\({}^{\circ}\). Figure 12: Peak wheel–rail responses with different FCAs and PGRDs in case 1: (**a**) derailment coefficient; (**b**) wheel-load-reduction rate; (**c**) lateral wheel–rail force. During fault dislocation, the fault seismic motion is primarily dominated by low-frequency components, which control the internal forces and displacements of the bridge. Consequently, trains have the potential to pass safely during this phase. Figure 12 illustrates that when the bridge crosses the fault at a small angle, all three indicators (the dynamic indicators of the train) remain below the specified limit. This suggests that trains can safely traverse the bridge under such conditions. Similarly, when the bridge is oriented vertically across the fault, it is still safe for trains to pass as long as the PGRD does not exceed 0.4 m.  
Due to space limitations, only representative scenarios of the bridge crossing the fault at a small angle (FCA = 15\({}^{\circ}\)) and vertically crossing the fault (FCA = 90\({}^{\circ}\)) are selected for the comparison of case 1 and case 2, as shown in Figures 13 and 14. In Figures 13 and 14, it can be observed that the PGRD generation phase (case 1) accounts for only 40% to 80% of the influence on the wheel-rail dynamic indicators. After the formation of PGRD (case 2), the train is simultaneously affected by additional track irregularities and the high-frequency components of fault seismic motion. This results in a 31% to 48% increase in the derailment coefficient, a 45% to 142% increase in the wheel load reduction rate, and a 50% to 103% increase in the lateral wheel-rail force.