# Dynamic Response Analysis of Parallel Twin Tunnels Under Different Train Loads
## 4 Results and Discussion
### Twin-Tunnel Structure Response
#### 4.1.2 Liners  
Figure 10 presents the maximum resultant displacements (\(D_{\rm R}\)) and velocities (\(V_{\rm R}\)) at T2 liner monitoring points under various conditions. The calculation formulas for \(D_{\rm R}\) and \(V_{\rm R}\) are as follows:  
\[D_{\rm R}=\sqrt{D_{\rm H}^{2}+D_{\rm V}^{2}},\ V_{\rm R}=\sqrt{V_{\rm H}^{2}+V _{\rm V}^{2}} \tag{1}\]  
Figure 10: The \(D_{\rm R}\) and \(V_{\rm R}\) of T2 liners under different conditions: (**a**) The maximum \(D_{\rm R}\) at different angles; (**b**) The maximum \(D_{\rm R}\) under different loads; (**c**) The maximum \(V_{\rm R}\) at different angles; (**d**) The maximum \(V_{\rm R}\) under different loads. Figure 9: Comparison of the maximum and minimum values of the \(V_{\rm V}\) of T1 sleepers under different loads: (**a**) \(V_{\rm V}\) at different points; (**b**) \(V_{\rm V}\) under different loads. Figure 10a reveals that the maximum \(D_{\rm R}\) occurs at the 180\({}^{\circ}\) position adjacent to T1 and gradually decreases toward the 0\({}^{\circ}\) position opposite T1. This pattern indicates that \(D_{\rm R}\) negatively correlates with distance from the vibration source, with closer points showing stronger responses. The \(D_{\rm R}\) values at 180\({}^{\circ}\)are 2.82, 3.24, 3.65, 4.06, 4.48, 4.89, 4.39, and 5.71 mm under the eight conditions, generally increasing with higher loads. However, an anomalous reduction occurs under the A\({}_{7}\) condition, where the \(D_{\rm R}\) decreases to levels similar to under the A\({}_{5}\) condition. Figure 10b illustrates the \(D_{\rm R}\) at four key positions (0\({}^{\circ}\), 90\({}^{\circ}\), 180\({}^{\circ}\), and 270\({}^{\circ}\)) of T2 liners. Under identical conditions, the \(D_{\rm R}\) follows a consistent descending order: 180\({}^{\circ}>90^{\circ}>270^{\circ}>0^{\circ}\). This pattern indicates the influence of proximity to the vibration source and varying tunnel burial depth, which enhances stability and reduces the load impact at greater depths. Under the A\({}_{1}\) to A\({}_{8}\) conditions, the \(D_{\rm R}\) increases with higher train load (excluding the drop under the A\({}_{7}\) condition). Higher loads also exacerbate non-uniform displacement in T2 liners, potentially leading to microcracks and structural deterioration. Figure 10c presents the maximum \(V_{\rm R}\) at various angular positions of T2 liners under different conditions. The maximum \(V_{\rm R}\) consistently occurs at 210\({}^{\circ}\) (lower-left liner), while the minimum is at 15\({}^{\circ}\) (upper-right liner). Notably, these positions do not align with those of \(D_{\rm R}\), indicating that velocity measurements are more sensitive to burial depth variations and spatial deviations from displacement patterns. The comparison between Figure 10a, c reveals that the \(V_{\rm R}\) distribution across angular positions is more uniform than that of \(D_{\rm R}\). Despite similar maximum \(V_{\rm R}\) values across different angular positions, higher vibration load increases the difference between values. Figure 10d illustrates the \(V_{\rm R}\) variation at four critical positions (0\({}^{\circ}\), 90\({}^{\circ}\), 180\({}^{\circ}\), and 270\({}^{\circ}\)). Although \(V_{\rm R}\) follows the same order as \(D_{\rm R}\) (180\({}^{\circ}>90^{\circ}>270^{\circ}>0^{\circ}\)), the differences are smaller, as indicated by the nearly overlapping trend lines. Figures 6, 7 and 10 show similar patterns in displacement and velocity responses between T2 sleepers and liners under all conditions. Both structural components demonstrate consistent increases in displacement and velocity as the vibration load increases under the A\({}_{1}\) to A\({}_{6}\) conditions, followed by an abrupt reduction under A\({}_{7}\) and a rapid increase under the A\({}_{8}\) condition. Figure 11 shows the \(D_{\rm R}\) and \(V_{\rm R}\) of T1 liners under various conditions to investigate the impact of train vibration load on twin-tunnel liners.