## 5 Results and Discussion
### Effects of Train Speed on the Response of the Bridge  
This subsection investigates the effects of the train speed on the bridge's response. Firstly, the bending moment time histories at both the left and right bridge piers were analyzed. Figure 9 presents a comprehensive comparative analysis of these structural responses under two distinct train speeds--300 km/h and 30 km/h--and considers the flexibility of the soil. This figure employs a dual time-scale representation on the horizontal axis to facilitate direct comparison between the two velocity scenarios. The bottom scale spans 0-8 s, corresponding to the high-speed case (300 km/h), while the top scale extends from 0-80 s, representing the low-speed passage (30 km/h). This scaling approach allows for a direct visual comparison of the structural behavior under these dramatically different loading conditions. The structural response is depicted through color-coded lines: black lines represent the bending moment at the deck position directly above the left pier, while red lines indicate the response above the right pier. Furthermore, the line style differentiates between speed cases--solid lines represent the high-speed scenario (300 km/h) while dashed lines correspond to the low-speed case (30 km/h). Several significant observations are made from this analysis and presented in Table 2.  
This analysis effectively demonstrates the significant role of train speed in bridge response characteristics, emphasizing the importance of dynamic analysis in the design and assessment of high-speed rail bridges. Additionally, the time history of the vertical displacement at the midspan of the bridge's central span was analyzed and is presented in Figure 10. Similar to the previous examination of bending moments, the analysis considered the bridge's response under two distinct train speeds--30 km/h and 300 km/h. Figure 9: Bending moment time histories for train speeds of 30 km/h and 300 km/h. The key observations from the displacement time history analysis are as follows: At the lower train speed of 30 km/h, the maximum vertical displacement was measured as 14.56 mm. In contrast, the higher train speed of 300 km/h resulted in a larger maximum displacement of 16.80 mm, representing an approximately 15% increase. Additionally, the displacement response exhibited distinct dynamic characteristics between the two train speed cases. The low-speed scenario (30 km/h) showed a more gradual and damped behavior, with a smooth transition and minimal oscillations. Conversely, the high-speed case shows a more gradual and damped behavior, with a smooth transition and minimal oscillations. The high-speed case demonstrates more complex dynamic interactions, evidenced by the higher-frequency content in the response. The response amplitudes vary between the left and right piers due to the asymmetric loading effects of the train's structure. \begin{table}
\begin{tabular}{p{142.3 pt} p{142.3 pt} p{142.3 pt}} \hline \hline  & \(\bullet\) & The train’s speed demonstrates a notable influence on the maximum bending moment \\ Speed Impact on Maximum Bending Moment & \(\bullet\) & At the right pier, the maximum value increases substantially from 21,462 kNm at 30 km/h to 26,004 kNm at 300 km/h, representing an approximately 21\% increase in peak response. \\ \hline  & \(\bullet\) & Low-speed case (30 km/h): The response exhibits clear damping characteristics with minimal oscillatory behavior. The bending moment follows a more quasi-static pattern, showing smooth transitions without significant positive-negative fluctuations. \\ Dynamic Response Characteristics & \(\bullet\) & High-speed case (300 km/h): Both piers display pronounced oscillatory behavior, characterized by distinct alternating positive and negative moment values. This dynamic amplification is indicative of significant inertial effects and structural vibration. \\ \hline  & \(\bullet\) & The high-speed case demonstrates more complex dynamic interactions, evidenced by the higher-frequency content in the response.