# Dynamic Response Analysis of Parallel Twin Tunnels Under Different Train Loads
## 4 Results and Discussion
### Twin-Tunnel Structure Response
#### 4.1.1. Sleepers
Differential displacements and velocities at the sleeper end increase with higher loads, suggesting excessive overloading may cause vertical misalignment and cracking in adjacent tunnel sleepers. Such significant differentials pose potential derailment risks for high-speed metro trains, threatening passenger safety and operational integrity. Figures 6 and 7 analyze the effects of train vibration loads on adjacent tunnel sleepers, and Figure 8 shows the differential between the maximum and minimum \(D_{\rm V}\) of primary tunnel (T1) sleepers under different conditions. Negative \(D_{\rm V}\) values represent the overall downward displacement of T1 sleepers, with zero being the maximum \(D_{\rm V}\). As demonstrated in Figure 8a, the \(D_{\rm V}\) of T1 sleepers generally increases with higher loads, except under the A\({}_{7}\) condition, where displacement decreases compared to under the A\({}_{6}\) and A\({}_{8}\) conditions. Under identical loading conditions, T1 sleepers show slightly higher \(D_{\rm V}\) values  
Figure 7: The maximum and minimum speed differences of T2 sleepers under different loads: **(a)** The \(V_{\rm H}\) at different points; **(b)** The \(V_{\rm V}\) under different loads; **(c)** The \(V_{\rm H}\) at different points; **(d)** The \(V_{\rm V}\) under different loads. at the right-side monitoring point (S\({}_{5}\)) than at the left side (S\({}_{1}\)), while T2 sleepers exhibit the opposite trend, where the left-side (S\({}_{1}\)) displacements exceed those at the right side (S\({}_{5}\)). This phenomenon is possibly due to the stabilizing effect of the tunnel linings in the central region between the twin tunnels, reducing displacements on the right side (S\({}_{5}\)) of T1 and left side (S\({}_{1}\)) of T2 under train loading. Figure 8b illustrates the \(D_{\text{V}}\) of T1 sleepers under varying loads, which generally increases with higher loads but decreases unexpectedly under the A\({}_{7}\) condition compared to that under the A\({}_{6}\) and A\({}_{8}\) conditions. Comparing Figures 6d and 8b, both tunnels exhibit similar \(D_{\text{V}}\) patterns under identical conditions. Under the A\({}_{7}\) condition, the \(D_{\text{V}}\) of T1 sleepers (\(-5.88\) mm) approximates that under the A\({}_{5}\) condition (\(-5.98\) mm), while the \(D_{\text{V}}\) of T2 sleepers (\(4.92\) mm) closely corresponds to that under the A\({}_{4}\) condition (\(4.97\) mm). Under the same load, the \(D_{\text{V}}\) of T1 sleepers at each monitoring point is slightly higher than that of T2. The \(V_{\text{V}}\) of T2 sleepers is more critical than its V\({}_{\text{H}}\) (Figure 7). Figure 9 shows the difference between the maximum and minimum \(V_{\text{V}}\) of T1 sleepers under various loading conditions, providing further insight into their dynamic responses. The \(V_{\text{V}}\) anomalously decreases under the A\({}_{7}\) condition compared to that under the A\({}_{6}\) and A\({}_{8}\) conditions. Under identical loading conditions, the right-side monitoring points consistently demonstrate higher \(V_{\text{V}}\) values than the left-side points, with more pronounced variation rates than in the \(D_{\text{V}}\). The comparison between Figures 7d and 9b reveals that both tunnels exhibit similar \(V_{\text{V}}\) patterns, with T1 sleepers consistently maintaining higher \(V_{\text{V}}\) values than T2 under various loading conditions. Figures 5-9 show that both T1 and T2 sleepers have similar patterns in vertical and horizontal displacement and velocity responses under all conditions, with T1 showing slightly more significant responses than T2. The difference in velocity responses between tunnels is greater than that in the displacement responses. Under the A\({}_{1}\) to A\({}_{6}\) conditions, all monitoring points display linear increases in vertical and horizontal displacements and velocities as loading increases.