# Seismic Stability of Dual Tunnels in Cohesive-Frictional Soil Subjected to Surcharge Loading
## 5 Failure Mechanisms  
The scope of the collapse and the failure patterns of the double tunnels are the key issues in engineering practice. Figures 9-12 present the distribution of shear dissipation for a better understanding of how the failure patterns evolved with different conditions. Figure 9 shows the failure mechanisms of \(k_{h}=\)0.1, \(H/D=3\), \(\gamma D/c=1\) and \(\varphi=10^{\circ}\) with different \(S/D\). Three typical failure patterns of the double tunnels are revealed: (1) cross-shaped failure, which is depicted in Figure 9a,b; (2) half-cross-shaped failure, which is depicted in Figure 9c; and (3) single side-wall failure, which is depicted in Figure 9d. It can be seen from Figure 9a,b that, when the tunnels locate close to each other, the majority of failure curves are at the pillar between the tunnels, and then the collapse extends from the tunnels to the ground surface. The cross-shaped failure zone, which comprises of two wedges indicates that the interaction of tunnels is dominant, and this strong interaction can significantly reduce seismic bearing capacity. With the increase in \(S/D\) (depicted in Figure 9c), one failure curve of crossed failure zone rotates clockwise, and there is onlyone failure curve between the double tunnels. This half-cross-shaped failure means that the interaction between the double tunnels becomes weaker with the increase in \(S/D\). Additionally, when \(S/D=9\), only two failure curves extend to the surface ground from the side-wall of a single tunnel. Namely, double tunnels have no influence on each other. Furthermore, it can be observed that the scope of influence on the ground surface broadens with a greater \(S/D\) until the failure pattern becomes a single side-wall failure. Figure 10 shows the effect of the tunnel depth on the failure mechanisms. Clearly, the failure patterns shift to the cross-shaped failure from a single side-wall failure with the increase in tunnel depth. In view of this, it can be speculated that the deeper the tunnel depth is, the stronger the interaction between the double tunnels is. Additionally, the scope of influence clearly becomes wider with the deeper depth of the tunnels. Figure 9: \(Cont.\)  
Figure 10: Upper bound failure mechanisms with \(S/D=3\), \(\gamma D/c=1\), \(k_{h}=0.1\) and \(\varphi=10^{\circ}\) for (**a**) \(H/D=1\); (**b**) \(H/D=2\); and (**c**) \(H/D=3\). Figure 11: Upper bound failure mechanisms with \(S/D=7\), \(H/D=3\), \(\gamma D/c=1\), and \(\varphi=10^{\circ}\) for (**a**) \(k_{y}=0\); (**b**) \(k_{y}=0.1\); (**c**) \(k_{y}=0.2\); and (**d**) \(k_{y}=0.3\). Figure 12: Upper bound failure mechanisms with \(S/D=7\), \(H/D=3\), \(\gamma D/c=1\) and \(k_{h}=0.1\) for (a) \(\varphi=5^{\circ}\); (b) \(\varphi=10^{\circ}\); (c) \(\varphi=15^{\circ}\); and (d) \(\varphi=20^{\circ}\). Figure 11 shows the failure patterns of \(S/D=7\), \(H/D=3\), \(\gamma D/c=1\) and \(\varphi=10^{\circ}\) with different seismic factor \(k_{h}\). Figure 11a shows a single side-wall failure. It can be seen from Figure 11b-d that the failure patterns change to the half-cross-shaped failure. Additionally, the connection curve between the double tunnels becomes more distinct with a greater seismic factor. This phenomenon indicates that the interaction of the tunnels would be stronger with the increase in \(k_{h}\). Based on this observation, it can be predicted that, for cases with multi tunnels/voids, the system is vulnerable to seismic actions, and its seismic bearing capacity would be further undermined by the side effect of additional interactions between vulnerable joints (e.g., underlying tunnels/caves). Furthermore, through the comparisons of Figure 11a-d, it can be observed that the failure curves tend to rotate clockwise with the increase in \(k_{h}\). This rotation phenomenon would broaden the failure curves on the surface ground, illustrating that stronger earthquakes can broaden the sphere of collapse. Figure 12 presents the effect of the internal friction angle on the failure mechanism.