# Seismic Stability of Dual Tunnels in Cohesive-Frictional Soil Subjected to Surcharge Loading
## 4 Results and Discussions  
The cases of double tunnels in \(c\)-\(\varphi\) soil without seismic action and single tunnels in \(c\)-\(\varphi\) soil with seismic action were compared with the previous literature [16; 28] and Optum G2 (a commercial FELA software) [42] to verify the reliability of the present model. It should be noted that the \(N_{cs}\) in Figure 3 represents the bearing capacity factor. A specific definition can be seen in the corresponding literature [16; 28]. It can be seen from Figure 3 that, for all cases, the magnitude and variation trends of UB and LB AFELA results are close to other results. Additionally, slight errors are caused by the different meshing strategy. The maximum error of LB and UB AFELA results occurs in Figure 3 with the \(\gamma D/c=0\) and \(S/D=8\). The UB and LB results are 7.735 and 7.22, respectively, and the greatest error is 6.89%, which is calculated by Equation (8):  
\[\text{Error}=\left|\frac{2(\text{UB}-\text{LB})}{\text{UB}+\text{LB}}\right| \cdot 100\% \tag{8}\]  
Figure 2: Detailed mesh arrangement in AFELA. The slight error means that this AFELA model is reliable. Additionally, to further reduce errors, this study uses the mean value of UB and LB AFELA results to assess the seismic bearing capacity. Figure 4 shows the effect of \(\gamma D/c\) on the seismic bearing capacity factor with different \(S/D\), \(H/D\) and \(\varphi\). As expected, the seismic bearing capacity decreases with the increase in \(\gamma D/c\). This is because the heavier soil mass would apply a greater load on the tunnels, and less cohesion means a decrease in soil strength. Furthermore, it can be seen that the curves of each case are nearly straight. This means that the \(\gamma D/c\) linearly affects the seismic bearing capacity. In view of this, \(\gamma D/c=0\) and \(\gamma D/c=1\) are adopted in subsequent investigations for simplicity. The seismic bearing capacity with any value of \(\gamma D/c\) can be predicted from two known results of \(\gamma D/c=0\) and \(\gamma D/c=1\). Figure 3: Comparison of \(N_{cs}\)-\(S/D\) curves with different \(\gamma D/c\) for double tunnels without seismic action. The interaction of tunnels significantly affects their stability. The estimation of critical interaction distance is the essential question in engineering practice. Figure 5 shows the effect of horizontal distance between tunnels with different influential factors. It can be determined that, for all cases, the seismic bearing capacity increases with the increase in \(S/D\) until the \(N_{cs}\) reaches an extremum value. This is because the further the distance between dual tunnels is, the weaker the interaction between the tunnels becomes. Additionally, the extremum value is the critical interaction distance mentioned above, beyond which the dual tunnels would not impact each other. It can be observed that the critical interaction distance of cases with a greater seismic coefficient \(k_{h}\) is greater. This demonstrates that the seismic action would intensify the interaction between dual tunnels. Additionally, the stronger the seismic action is, the greater the interaction becomes. Through comparisons between Figure 5a,b, it can be observed that the seismic bearing capacity of cases with a deeper tunnel depth is greater than those cases with a shallower tunnel depth. Furthermore, the critical interaction distance of \(H/D=1\) ranges from 3 to 4, whereas the critical interaction distance of \(H/D=3\) ranges from 7 to 14. Namely, the deeper the tunnels are located, the greater the critical interaction distance of the tunnels becomes.