## 3 Analysis of Numerical Results
### Effect of Embankment Stiffness  
In this section, three different stiffnesses of embankment (\(E_{1}\), \(E_{2}\), and \(E_{3}\)) were chosen, to evaluate the effect of embankment stiffness on the dynamic response of railway structures, as shown in Figure 14. Figure 15 shows a comparison of \(a_{x}\) and \(d_{x}\) with the embankment stiffness in different paths. As shown in these figures, both \(a_{x}\) and \(d_{x}\) decrease with an increase of embankment stiffness, regardless of the monitoring paths. When comparing the changes of \(a_{x}\) and \(d_{x}\) under different paths, the values of \(a_{x}\) and \(d_{x}\) in Path 1 decrease significantly with an increase of embankment stiffness, followed by Path 2 and Path 3. This indicates that properly strengthening the embankment stiffness can reduce the influence of the train load on railway structure vibration.  
Figure 14: Comparisons of \(a_{x}\) and \(d_{x}\) cloud with different embankment stiffnesses: (**a**) \(a_{x}\), \(E_{1}\), \(v=83\) m/s; (**b**) \(d_{x}\), \(E_{1}\), \(v=83\) m/s; (**c**) \(a_{x}\), \(E_{2}\), \(v=83\) m/s; (**d**) \(d_{x}\), \(E_{2}\), \(v=83\) m/s; (**e**) \(a_{x}\), \(E_{3}\), \(v=83\) m/s; (**f**) \(d_{x}\), \(E_{3}\), \(v=83\) m/s.  
For a more straightforward presentation, Figure 16 shows changes in the normalized maximum \(a_{\rm z}\) and \(d_{\rm z}\) with the normalized embankment stiffness in different paths. Here, \(a_{\rm z}\) and \(d_{\rm z}\) at \(E_{2}\) in Path 1 are standard values, and the other values can be calculated using the ratio of the actual value to the standard value. It can be seen that the normalized maximum \(a_{\rm z}\) and \(d_{\rm z}\) decrease with an increase of the normalized embankment stiffness, irrespective of the monitoring paths. However, with an increase of embankment stiffness, the beneficial effect of strengthening the embankment stiffness on reducing \(a_{\rm z}\) and \(d_{\rm z}\) is weakened, especially in Path 3. This indicates that an increase in embankment stiffness in a specific range can reduce the dynamic response of a railway structure. However, the influence of embankment stiffness on the dynamic response is not as significant as the train speed, as shown in Figure 11.  
Figure 16: Relationship between normalized maximum \(a_{\rm z}\), \(d_{\rm z}\) and normalized embankment stiffness in different paths: (**a**) normalized maximum \(a_{\rm z}\); (**b**) normalized maximum \(d_{\rm z}\).  
Figure 15: Changes in \(a_{\rm z}\) and \(d_{\rm z}\) with the embankment stiffness in different paths: (**a**) \(a_{\rm z}\) in Path 1, (**b**) \(d_{\rm z}\) in Path 1; (**c**) \(a_{\rm z}\) in Path 2; (**d**) \(d_{\rm z}\) in Path 2; (**e**) \(a_{\rm z}\) in Path 3; (**f**) \(d_{\rm z}\) in Path 3.  
For the underlying ground, Figure 17 shows the changes in \(a_{x}\), \(d_{z}\), and \(\sigma_{x}\) with ground depth with different embankment stiffnesses in Path 4. As shown in Figure 17a, the \(a_{x}\) decreases with an increase of embankment stiffness, with a rapidly decreasing trend with the ground depth. For \(d_{z}\), as shown in Figure 17b, the \(d_{z}\) shows a decreasing trend at higher embankment stiffnesses, although the reduction trend of \(d_{z}\) with the ground depth is not as rapid as that of \(a_{x}\). More importantly, when considering the effect of embankment stiffness on \(\sigma_{x}\) in a railway ground (Figure 17c), similarly to the values of \(a_{x}\) and \(d_{z}\), the \(\sigma_{x}\) decreases with increasing embankment stiffness, irrespective of the ground depth. These factors indicate that increasing the embankment stiffness can slightly lessen the ground vibration, although the effect of embankment stiffness on the ground vibration is not significant.