# Dynamic Response of Railway Subgrade Under Train Load and Freeze-Thaw Action
## 3 Dynamic Response of Subgrade Under Different Train Speeds  
### Vertical Displacement  
Figure 5 depicts the vertical displacement-time history curves for points A, B, and C. The dynamic displacement-time curves of the subgrade exhibit a distinctive M-shaped pattern, with negative values signifying compression, which is in direct correlation with the passage of the moving train load. As energy is laterally transferred to points B and C, the vertical displacement of the subgrade progressively diminishes, resulting in a less pronounced M-shaped curve, which signifies the attenuation of vibration waves propagating laterally along the track subgrade structure due to the passage of trains. The initial and secondary extrema of the vertical displacement curve for point A, as depicted in Figure 5, represent the displacement induced by the front bogie of the train (as shown by the small red circle on the left), while the tertiary and quaternary extrema correspond to the displacement generated by the rear bogie of the train (as shown by the larger red circle on the right). Upon the passage of the front bogie over point A, the dynamic displacement is induced. Before this displacement has fully dissipated, the subsequent arrival of the rear bogie generates an additional dynamic displacement. The superposition of these two displacement events leads to the third and fourth displacement extrema exceeding the first and second extrema, thereby corroborating the findings reported by [PERSON] [20].  
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Figure 6 illustrates the variation in vertical displacement curves for points A, B, and C as a function of different train speeds. At a train speed of 180 km/h, the maximum vertical displacement recorded at point A is 0.272 mm. As the train speed is elevated to 220 km/h, the maximum vertical displacement at point A diminishes to 0.261 mm, the maximum vertical displacement at point B reduces from 0.245 mm to 0.232 mm, and the maximum vertical displacement at point C decreases from 0.207 mm to 0.196 mm. This result indicates that with the increase in train speed from 180 km/h to 220 km/h, the maximum vertical displacements at all monitored points exhibit a concomitant reduction.  
The vertical displacement of the subgrade in response to a moving constant load represents an instantaneous deformation, which, as a dynamic response, is inherently time dependent. As the train traverses the observation point, the dynamic load is transmitted to the railway track and subgrade structure. At a train speed of 180 km/h, the duration of applied pressure at the observation point is greater than at the speed of 220 km/h. With the increase in train speed, the duration of the applied train loads on the subgrade is diminished. Consequently, this results in smaller displacements at points A, B, and C. The vertical displacements at these points are further mitigated at higher train speeds, indicating that the instantaneous vertical displacement diminishes with the escalation of train speed.  
Figure 5: Vertical displacement–time history curves of points.  
Figure 6: Vertical displacement of points with train speed (A-B-C are observation points).