# Dynamic Response Analysis of Parallel Twin Tunnels Under Different Train Loads
## 4 Results and Discussion
### Twin-Tunnel Structure Response
#### 4.1.1. Sleepers
Under the A\({}_{1}\) to A\({}_{8}\) conditions, the \(D_{\rm V}\) differences between monitoring points S\({}_{5}\) and S\({}_{1}\) are 0.20, 0.23, 0.26, 0.29, 0.32, 0.35, 0.28, and 0.41 mm, respectively. Analysis revealed that \(D_{\rm V}\) increases with distance from the vibration source, a trend more pronounced at higher loads. This phenomenon can be attributed to the stabilizing effect of the twin tunnel reinforcement on the central area, which results in points closer to the source exhibiting smaller vertical dynamic responses than those farther away. Figure 6 analysis shows that both the \(D_{\rm H}\) and \(D_{\rm V}\) of T2 sleepers exhibit similar trends across different conditions, with a significant drop under the A\({}_{7}\) condition followed by a sharp rise, a phenomenon that requires further investigation. Under equivalent loading conditions, the \(D_{\rm V}\) of T2 sleepers surpasses their \(D_{\rm H}\) by approximately threefold, indicating that \(D_{\rm V}\) analysis should be given greater attention due to its heightened criticality in structural assessment. Figure 7 presents the \(V_{\rm H}\) and \(V_{\rm V}\) of T2 sleepers under eight conditions, with velocity defined as the peak-to-peak differences between maximum and minimum values at monitoring points. Figure 7a, b show that the \(V_{\rm H}\) values across monitoring points exhibit high consistency under identical loading conditions, matching the observed \(D_{\rm H}\) patterns, which could be attributed to the structure's high stiffness. The minimum \(V_{\rm H}\) values under the A\({}_{1}\) to A\({}_{8}\) conditions are 13.54, 15.30, 17.09, 18.86, 20.69, 22.66, 19.10, and 26.21 mm/s, respectively. Notably, the velocity under the A\({}_{8}\) condition nearly doubles that under the A\({}_{1}\)  
Figure 6: The maximum and minimum displacement differences of T2 sleepers under various loading conditions: **(a)** The \(D_{\rm H}\) at different monitoring points; **(b)** The \(D_{\rm H}\) under different loads; **(c)** The \(D_{\rm V}\) at different monitoring points; **(d)** The \(D_{\rm V}\) under different loads. condition, indicating a significant amplification in the dynamic response of the adjacent tunnel structure by the overload. The \(V_{\rm V}\) of T2 sleepers under different conditions is illustrated in Figure 7c, d. Under the A\({}_{1}\) condition, the minimum and maximum \(V_{\rm V}\) values are 35.01 mm/s and 36.67 mm/s, increasing to 70.86 mm/s and 73.61 mm/s under the A\({}_{8}\) condition, respectively. Notably, all monitoring points consistently exceed 35 mm/s under all conditions. Under the A\({}_{1}\) to A\({}_{8}\) conditions, the differences in \(V_{\rm V}\) between S\({}_{5}\) and S\({}_{1}\) of T2 sleepers are 1.66, 1.82, 2.00, 2.17, 2.32, 2.44, 2.20, and 2.75 mm/s, respectively, demonstrating a positive correlation between the \(V_{\rm V}\) and the distance from the vibration source under identical loading conditions. Figure 7 shows that the \(V_{\rm V}\) values at all monitoring points surpass their horizontal counterparts by approximately threefold under identical conditions. Consequently, subsequent research should prioritize the analysis of \(V_{\rm V}\) due to its critical implications for structural assessment. Combined with Figures 6 and 7, the horizontal and vertical displacements and velocities of T2 sleepers exhibit a linear increase with the vibration load under the A\({}_{1}\) to A\({}_{6}\) conditions. However, under the A\({}_{7}\) condition, all parameters abruptly decrease before sharply rising again under the A\({}_{8}\) condition. This transition requires an in-depth investigation into the static and dynamic responses of T2 when the load exceeds that under the A\({}_{4}\) condition (92.0 kN), especially the mechanisms behind this shift. Furthermore, the \(D_{\rm V}\) is approximately three times the \(D_{\rm H}\), and the \(V_{\rm V}\) is twice the \(V_{\rm H}\).