## 3 Analysis of Numerical Results  
### Dynamic Response of the Railway Structure  
For the analysis model shown in Figure 2, three monitoring points (Point A, Point B, and Point C) and four monitoring paths (Path 1, Path 2, Path 3, and Path 4) were selected, to study the dynamic response of the railway structure under a train load. Here, Point A is in the middle of two rails on the slab surface, with the \(x\)-axis coordinate of 200 m; Point B is directly below Point A, at the interface between the foundation and subgrade; Point C is at the embankment toe, and Points A, B, and C are on the same _zoy_ plane. In this case, the train speed (\(v\)) of 83 m/s and the time (\(t\)) of 0.996 s were selected as the analysis conditions. The reason for choosing these positions as monitoring points was to observe the vibration behavior of the soil during the train running, ignoring the effect of boundary conditions in the \(x\) direction. Moreover, the vibration at the center of the slab surface, the foundation surface, and the embankment toe can effectively reflect the maximum vibration value and the attenuation law of the vibration in the depth and width directions of the entire railway structure under a train load. Figure 6 presents the variations of acceleration (\(a\)) and displacement (\(d\)) with time at different monitoring points, respectively. As shown in this figure, both a and d in the z direction are larger than in the x and y directions. This indicates that the vibration of the railway track under a train load is mainly generated in the vertical direction. Therefore, the dynamic response in the vertical direction will be emphasized in the subsequent analysis. As for the a on track surface, there are four distinct acceleration peaks (Figure 6a), due to four pairs of concentrated loads being applied to the rail. Moreover, at point A, the amplitude of the positive a is much larger than that of the negative acceleration. However, as the vibration is transmitted into the ground, the two acceleration peaks at the middle position are gradually superposed, and the positive and negative amplitudes tend to become equal, as shown in Figure 6c,e. On the other hand, a peak displacement is also generated, due to the superposition of the displacement near to the position of the middle two pairs of loads, as shown in Figure 6b,d,f. In general, without considering the effect of train speed, the vertical displacement caused by symmetric concentrated loads is symmetrically distributed along the longitudinal direction of the track. However, an asymmetry of the vertical displacement in railway structures is caused by vibrations of the track structure under a moving load. Moreover, due to the effect of structural damping, the peak vertical displacement lags at the load position, and the vertical displacement caused by the wheel load at the train's rear is not equal to the vertical displacement caused by the wheel load at the front of the train. Figure 5: Vertical stress in a cross-section of the railway ground. Considering that the dynamic response of a railway structure is mainly generated in the vertical direction, Figure 7 shows the vertical acceleration (\(a_{\rm z}\)) and vertical displacement (\(d_{\rm z}\)) at Points A, B, and C. By comparing the \(a_{\rm z}\) and \(d_{\rm z}\) at these points, it can be seen that the maximum values of \(a_{\rm z}\) and \(d_{\rm z}\) at Point A are the largest, followed by Point B and Point C. This means that the vertical acceleration and displacement along the track surface and into railway ground are gradually attenuated. In addition, both \(a_{\rm z}\) and \(d_{\rm z}\) at Point B are larger than those at Point C, indicating that the soil below the center of the rail is mainly affected by the train loads. To further analyze the trend of \(a_{\rm z}\) and \(d_{\rm z}\) in the railway structure, Figure 8 shows the normalized maximum \(a_{\rm z}\) and \(d_{\rm z}\) at different monitoring points.