## 3 Analysis of Numerical Results
### Effect of Train Speed  
To evaluate the effect of train speed on the dynamic responses of the railway structure, three different train speeds were analyzed in this study; that is \(v=56\) m/s (200 km/h), \(v=83\) m/s (300 km/h), and \(v=112\) m/s (400 km/h). Figure 10 represents the cloud images for the \(a_{x}\) and \(d_{z}\) at different train speeds. As shown here, the maximum vertical displacement occurs near the middle of the front and rear wheels of the two carriages, which indicates that this numerical model can consider the superimposition effect of the  
Figure 8: Comparisons of normalized maximum \(a_{x}\) and \(d_{x}\) at different points. Figure 9: Variations of \(a_{x}\) and \(d_{z}\) with ground depth. dynamic load from different wheels of the train. In addition, Figure 11 shows the variations of \(a_{x}\) and \(d_{x}\) with train speeds in different paths. From these figures, it can be seen that both \(a_{x}\) and \(d_{x}\) increase remarkably with the increase of the train speed, regardless of the monitoring paths. In addition, due to the effects of velocity and damping, the asymmetry of \(a_{x}\) and \(d_{x}\) is more obvious, which is reflected by the wavelength and amplitude of the displacement in front of the load action point being less than those behind the load. These indicate that the train speed impacts the dynamic response of the whole railway structure. To further analyze the effect of train speed on \(a_{\mathrm{z}}\) and \(d_{\mathrm{z}}\), Figure 12 presents the variations of the normalized maximum \(a_{\mathrm{z}}\) and \(d_{\mathrm{z}}\) with train speed in different paths. Here, the \(a_{\mathrm{z}}\) and \(d_{\mathrm{z}}\) at \(v=83\) m/s in Path 1 are standard values, and other values can be calculated from the ratio of the actual value to the standard value. As illustrated in Figure 12a, the normalized maximum \(a_{\mathrm{z}}\) increases remarkably with increasing train speed, although it reduces rapidly with an increase of depth (Path 2 and Path 3). In addition, the normalized maximum \(d_{\mathrm{z}}\) increases linearly with the increase of train speed. However, its attenuation trend with depth is not remarkable compared to the normalized maximum \(a_{\mathrm{z}}\). This indicates that an increment of the train speed significantly enhances the vibration response of a railway structure within the shear wave velocity of soft soil. The influence of train speed should be emphasized in the railway structure design of high-speed railways. Figure 13 illustrates the variations of \(a_{\mathrm{z}}\), \(d_{\mathrm{z}}\), and \(\sigma_{\mathrm{z}}\) with ground depth in Path 4. For the selected period data shown in Figure 13a, the \(a_{\mathrm{z}}\) increases with the train speed and decreases rapidly with increased ground depth (Path 4). It is noted that both positive and negative \(a_{\mathrm{z}}\) occurred at the train speed of 112 m/s. This indicates an increase of train speed boosts the vibration of the ground soil. In addition, in Figure 13b, \(d_{\mathrm{z}}\) increases gradually in the wake of an increased train speed, while the decreasing trend of \(d_{\mathrm{z}}\) along with depth at a high train speed is faster than that at a low train speed. More importantly, when comparing the \(\sigma_{\mathrm{z}}\) in Path 4, as illustrated in Figure 13c, the \(\sigma_{\mathrm{z}}\) near the ground surface is larger at higher train speeds. However, with an increase of depth, \(\sigma_{\mathrm{z}}\) has a more rapid attenuation trend, which causes the value of \(\sigma_{\mathrm{z}}\) at a high train speed to be less than that at a low train speed, after a certain depth. These points indicate that the vertical displacement and stress in a specific range of subgrade increase as the train speed increases. However, with an increase of depth, the attenuation trends of vertical displacement and vertical stress under the high-speed conditions become faster, and the affected area gradually decreases.