## 5 Impact Coefficient Analysis of the Bridge
### Stress Impact Coefficient  
Currently, the primary method for solving bridge stresses is the moving force method [32], which simplifies the train load as a series of vertical moving concentrated forces, with magnitudes and arrangements matching the actual train configuration. In this study, dynamic stress calculations were performed using a coupled simulation approach involving ABAQUS and SIMPACK. In SIMPACK, the bridge and track are \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Velocity/(km/h)**} & **Single-Track or** & **Quarter Point of** & **Side Span** & **Quarter Point of** & **Main Span** \\  & **Double-Track** & **the Side Span** & **Mid-Span** & **the Main Span** & **Mid-Span** \\ \hline \multirow{2}{*}{140} & Single & 1.025 & 1.030 & 1.029 & 1.035 \\  & Double & 1.028 & 1.035 & 1.028 & 1.036 \\ \hline \multirow{2}{*}{160} & Single & 1.029 & 1.033 & 1.032 & 1.038 \\  & Double & 1.029 & 1.036 & 1.035 & 1.039 \\ \hline \multirow{2}{*}{180} & Single & 1.032 & 1.037 & 1.036 & 1.042 \\  & Double & 1.033 & 1.037 & 1.040 & 1.043 \\ \hline \multirow{2}{*}{200} & Single & 1.035 & 1.043 & 1.040 & 1.047 \\  & Double & 1.039 & 1.045 & 1.044 & 1.052 \\ \hline \multirow{2}{*}{220} & Single & 1.039 & 1.048 & 1.045 & 1.053 \\  & Double & 1.040 & 1.047 & 1.049 & 1.056 \\ \hline \multirow{2}{*}{240} & Single & 1.044 & 1.052 & 1.050 & 1.059 \\  & Double & 1.045 & 1.050 & 1.056 & 1.063 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Maximum value of bridge displacement. connected via force elements, and the force variations over time for each force element are computed as the train passes. These time-varying concentrated forces are then applied to the corresponding locations on the bridge in ABAQUS. To verify the accuracies of the different methods and the reliability of the simulation model, a comparison was made between the moving force method, the coupled simulation, and field measurement results, as shown in Figure 9. The moving force method was calculated using the ABAQUS subroutine vdload, with an integration time step of 0.001 s. The train load was simplified into 32 moving concentrated forces, with an electric multiple unit (EMU) axle load of 132 kN and a trailer axle load of 11.8 kN. Due to space constraints, Figure 10 only presents the comparison results for the tie rod C12C13 and the suspension rod C13E13. Figure 9: Time history of dynamic stress for different components (180 km/h): **(a)** the lower chord member E4E5; **(b)** the web member E5A4; **(c)** the tie rod C12C13; **(d)** the suspension rod C13E13. Figure 10: Dynamic stress impact factors of structural members at different speeds: (**a**) the lower chord member; (**b**) the upper chord member; (**c**) the straight web member; (**d**) the diagonal web member; (**e**) the tie rod; (**f**) the suspension rod. From this figure, it can be seen that among the four types of members, the suspension rod experienced the highest stress, followed by the lower chord and tie rod, and the web member exhibited the lowest stress. The results from the coupled simulation were closer to the field measurement results and show a more consistent variation pattern compared with the moving force method, further validating the reliability of the vehicle-bridge coupling analysis approach used in this study. Although the moving force method partially accounted for the dynamic effects of the train, it did not consider the interaction between the vehicle and the bridge, nor did it include the effects of the track irregularities. As a result, the discrepancies between the moving force method and the measured results were relatively large. The impact coefficients of the bridge members under different train speeds are shown in Figure 10. From this figure, it can be observed that as the speed increased, the stress impact coefficients of different types of members exhibited an upward trend.