## 4 Results and Analyses
### Internal Force of the Lining of the Preceding Tunnel
#### 4.4.2 Bending Moment of the Lining of the Preceding Tunnel  
Figure 11 shows the distribution of the bending moment of the measured lining of the preceding tunnel, where the bending moment refers to the moment induced by all axial forces acting on the cross section of the tunnel lining. In the figure, the positive value of the bending moment corresponds to the case that the outer side of the lining is in tension, or the inner side is in compression.  
Figure 11a shows the bending moment of the preceding tunnel lining upon the completion of the following tunnel when the soil layer is reinforced with the strategies of the RR. With the increase in \(t\), the bending moments at the tunnel vault and tunnel intrados gradually decrease, while the bending moments at the right and left haunch gradually increase. These phenomena are closely related to the deformation presented in Figure 8a. When \(t=0.00\)\(D\), the shape of the preceding tunnel lining resembles a standing \"duck egg\", and the maximum bending moment of the lining is basically located on the arch crown and arch invert. With the increase in \(t\), the shape of the lining shown in Figure 8a gradually changes to the oblique \"duck egg\"; the maximum bending moment is basically located on the left and right haunch. When \(t=0.00\)\(D\), the bending moment at 90\({}^{\circ}\) is 74.4 kN\(\cdot\)m/m. When \(t=0.30\)\(D\), the bending moment at 90\({}^{\circ}\) is 49 kN\(\cdot\)m/m, and reinforcement of the soil layer around the preceding tunnel results in a reduction of about 33.8%.  
The bending moment distributions of the preceding tunnel lining upon the completion of the following tunnel when the soil layer is reinforced with the strategies of the LR are  
Figure 11: Bending moment of the lining of preceding tunnel: (**a**) RR; (**b**) LR; (**c**) TR; (**d**) CR.  
shown in Figure 11b. With the increase in \(t\), the variation trend of the bending moment is similar to the cases of the RR. From Figure 8b, it can be found that, as \(t\) increases, the convergence decreases at a faster rate in the LR than in the RR, and the same is the case for the variation rate of the bending moment. The position of the maximum bending moment in the cases of the LR is gradually shifted from \(90^{\circ}\) or \(270^{\circ}\) to \(165^{\circ}\) or \(345^{\circ}\) of the lining. When \(t=0.30\)\(D\), the maximum bending moment at \(90^{\circ}\) is \(42.1\) kN\(\cdot\)m/m, which decreases by about \(43.4\%\) in comparison with the case of \(t=0.00\)\(D\).  
Figure 11c shows the bending moments for the cases of the TR. It can be clearly seen that the characteristic of the bending moments is the same as that of the LR. When \(t=0.30\)\(D\), the maximum bending moment at \(90^{\circ}\) is \(41.3\) kN\(\cdot\)m/m. Compared with the case of \(t=0.00\)\(D\), the maximum bending moment decreases by about \(44.5\%\).  
Figure 11d presents the bending moments for the cases of the CR. Similar to the axial force response presented in Figure 10d, the reinforcement strategy of the CR seems to have little effect on reducing the bending moment of the lining of the preceding tunnel. When \(t=0.30\)\(D\), the bending moment at \(90^{\circ}\) is \(71.8\) kN\(\cdot\)m/m. Compared with \(t=0.00\)\(D\), it decreases by only \(3.0\%\).  
Figure 12 summarizes the variation of the maximum axial force and bending moment at \(90^{\circ}\) of the preceding tunnel lining with \(t\). It can clearly be seen that, with the increase in \(t\), the decreasing rate in the cases of the LR is similar to that in the cases of the TR, while the decreasing rate in the cases of the CR is the lowest. During the excavation of the following tunnel, the transfer of construction disturbances can be effectively hindered only by the reinforcement of the following tunnel, and the effect of the reinforcement of only the preceding tunnel is less significant. Reinforcement of the soil layer between the two tunnels seems to have a very weak effect in terms of controlling the disturbance.