# Dynamic Response Analysis of Parallel Twin Tunnels Under Different Train Loads
## 4 Results and Discussion
### Twin-Tunnel Structure Response
#### 4.1.1. Sleepers  
Figure 5 illustrates the time-history curves of vertical displacement (\(D_{\rm V}\)) at the midpoint (S\({}_{3}\)) of T2 sleepers under three conditions: A\({}_{1}\) (\(F_{1}=62.4\) kN), A\({}_{4}\) (\(F_{4}=92.0\) kN), and A\({}_{7}\) (\(F_{7}=121.6\) kN). The label A\({}_{1}\)-S\({}_{3}\) represents the \(D_{\rm V}\) of the midpoint (S\({}_{3}\)) of the sleeper under the A\({}_{1}\) condition. The \(D_{\rm V}\) values are observed to fluctuate in response to varying train loads. During 0-2.7 s, the \(D_{\rm V}\) amplitude first decreases and then increases, stabilizing after 2.7 s, indicating model equilibrium. As the load increases, the positive \(D_{\rm V}\) value first rises and subsequently decreases, while the absolute magnitude of the negative \(D_{\rm V}\) value increases until stabilization, suggesting that overloading causes settlement in adjacent tunnel sleepers. Notably, the \(D_{\rm V}\) of A\({}_{1}\)-S\({}_{3}\) is the smallest, while those of A\({}_{4}\)-S\({}_{3}\) and A\({}_{7}\)-S\({}_{3}\) are nearly identical. The maximum \(D_{\rm V}\) values recorded under each condition are \(-2.64\), \(-3.80\), and \(-4.07\) mm, with those of A\({}_{4}\)-S\({}_{3}\) and A\({}_{7}\)-S\({}_{3}\) increasing by 43.94% and 54.17% compared to that of A\({}_{1}\)-S\({}_{3}\). This suggests that the \(D_{\rm V}\) increases as the load rises under the A\({}_{1}\) to A\({}_{4}\) conditions, but the pattern under the A\({}_{4}\) to A\({}_{7}\) conditions remains unclear and requires further investigation. To further investigate the displacement pattern of the sleepers in T2, the \(D_{\rm H}\) and \(D_{\rm V}\) of T2 sleepers are compared under different loads (A\({}_{1}\)-A\({}_{8}\)), as shown in Figure 6. The displacements are peak-to-peak values, representing the difference between the maximum and minimum displacements at the monitoring point. Five monitoring points on T2 sleepers are monitored: left endpoint (S\({}_{1}\)), left quarter point (S\({}_{2}\)), midpoint (S\({}_{3}\)), right quarter point (S\({}_{4}\)), and right endpoint (S\({}_{5}\)). S\({}_{2}\) and S\({}_{4}\) serve as the train action points during T2 operation. The \(D_{\rm H}\) of each monitoring point of T2 sleepers is shown in Figure 6a. Under the A\({}_{7}\) condition, the \(D_{\rm H}\) values closely resemble those observed under the A\({}_{4}\) condition, but they exhibit a significant increase as the load transitions under the A\({}_{4}\) to A\({}_{7}\) conditions. For a more comprehensive comparison, the \(D_{\rm H}\) values of T2 sleepers under different loads are compiled in Figure 6b. The \(D_{\rm H}\) curves of each monitoring point completely overlap, indicating that the influence of train load on the \(D_{\rm H}\) of T2 sleepers is consistent, which can  
Figure 5: T2 sleeper midpoint \(D_{\rm V}\) time–history curve. be attributed to the high stiffness of concrete sleepers. The average \(D_{\rm H}\) values under the A\({}_{1}\) to A8 conditions are 1.07, 1.23, 1.38, 1.54, 1.69, 1.85, 1.55, and 2.15 mm, with differences of 0.16, 0.15, 0.16, 0.15, 0.16, \(-\)0.3, and 0.60 mm between adjacent loads, respectively. These results indicate that the \(D_{\rm H}\) increases linearly under the A\({}_{1}\) to A\({}_{6}\) conditions but decreases to the level under the A\({}_{4}\) condition under the A\({}_{7}\) condition, then sharply rises under the A\({}_{8}\) condition, reaching 1.38 times the value observed under the A\({}_{7}\) condition. This special displacement phenomenon has not been previously reported. The \(D_{\rm V}\) curves of T2 sleeper monitoring points under varying loads are presented in Figure 6c. The maximum and minimum \(D_{\rm V}\) are 3.54 mm and 3.34 mm under the A\({}_{1}\) conditions, increasing to 7.21 mm and 6.80 mm under the A\({}_{8}\) condition. The \(D_{\rm V}\) under the A\({}_{7}\) condition closely matches that under the A\({}_{4}\) condition. A visual comparison of \(D_{\rm V}\) across monitoring points is graphically represented in Figure 6d.