## 4 Conclusions  
It is important to study the vibration of a track formation caused by a moving train load, for the dynamic stability and vibration control of a railway structure. In this study, numerical simulation based on TDM was adopted to evaluate the dynamic response of the railway formation of a high-speed railway. The main conclusions include the following aspects:  
1. The dynamic acceleration and displacement in the vertical direction in railway structures are mainly generated under a moving train load, which should be considered in the railway structural design. Moreover, with increasing train speeds, the soil vibration along the direction of train movement appears to experience a significant fluctuation effect, which is manifested in the peak value of the vertical displacement of the track lagging behind the position of the wheel load, while the vertical displacement of the track at the rear of the wheel load is not equal to that at the front of the wheel load. The maximum vertical displacement occurs near the middle of the front and rear wheels of the two carriages, which indicates that the numerical model can consider the superimposition effect of the dynamic load from different wheels of the train. 2. The vertical displacement and vertical stress in a high-speed railway structure increase as the train speed increases. However, with the increased depth of ground, the attenuation trends of the vertical displacement and vertical stress under a high-speed train load condition become faster. Thus, the influenced area becomes smaller. 3. Since the embankment stiffness of a ballastless track structure is much greater than the subsoil stiffness, the increase of embankment stiffness has little effect on the vertical stress transferred from the track to the ground soil, which indicates that for a ballastless track structure, the enhancement of the stiffness of the sub-structure cannot effectively reduce the propagation of dynamic stress with soil depth. In code design, it is essential to increase the stiffness of the upper and lower soil layers in the same proportion. In particular, when the railway foundation contains a soft soil with low stiffness, the improvement of the stiffness of the soft soil should be emphasized. 4. When the train speed is low, the vertical stress in the ground soil obtained from TDM is less than that calculated using SSM. When the train speed is high, the vertical stress of the soil calculated by TDM within 5 m below the ground surface is greater than that calculated by SSM, although the former is smaller than the latter when the ground depth exceeds 5 m. This indicates that this design code may underestimate dynamic stresses on the railway formation in high-speed situations (i.e., \(v=112\) m/s). However,  
Figure 18: Relationship between vertical stress and ground depth with different train speeds and embankment stiffnesses: (**a**) train speed; (**b**) embankment stiffness. for lower velocities and greater soil depths, the vertical stress obtained by SSM is significantly greater than that obtained by TDM, and the proposed model may be limited in application under these conditions. It should be emphasized that the method presented in this paper can effectively calculate the influence of train speed on the dynamic response of a railway structure. Compared with the SSM (vehicle load is equivalent to soil column) for this specification, the results of the TDM were more consistent with the stress transfer process of the soil in the process of train movement, and more accurate results can be obtained. Especially for high-speed trains, the dynamic stress of the soil shows a fluctuation effect, and the transient dynamic method is more practical. However, when simulating a train load, the calculation method in this paper does not consider the influence of track irregularities and vibration frequency on the vibration characteristics of the railway structure, which needs to be further considered in future research. Conceptualization, J.Y. ; methodology, J.Y., Y.Z. and P.M.; software, Y.Z.