## 4 Numerical Simulation
### Deformation of Tunnel Segments  
The structural safety of the tunnel is essential to prevent excessive disturbance to the mounted segments from the concentrated release of stress in the process of the shield tunneling under-passing of the culvert at a close distance. Moreover, the axial deformation of the segments requires significant control. In Figure 19, the AB and CD survey lines of the right tunnel along the tunnel axis were selected to analyze the displacement changes in the right tunnel segment before and after the excavation of the left tunnel. Specifically, the AB and CD survey lines were used to analyze the deformation in the \(Z\) and X directions, respectively. The displacement curves of the right segments before and after the excavation of the left tunnel in the \(Z\)-direction are shown in Figure 20. The maximum subsidence of the 19 th-21 st ring segment of the right tunnel was 25.1 mm before the excavation of the left tunnel in the unreinforced condition. Meanwhile, the maximum relative displacement of the adjacent segments was 17.4 mm. This indicates that a large subsidence of segments was caused by the concentrated release of stress when the shield tunnel was out of the culvert section, which might further lead to a large dislocation of the adjacent segments, threatening tunnel safety. In the unreinforced condition, additional longitudinal displacement of the 4 th-26 th ring segments of the right tunnel was caused by the excavation of the left tunnel. The maximum additional longitudinal displacement was \(-4.7\) mm. After reinforcement with the MJS method, an additional vertical displacement was found only at the 8 th-21 st ring segments of the right tunnel produced together with a significant drop in the displacement value. The maximum additional vertical displacement was only \(-0.5\) mm. Hence, the MJS method can markedly control the disturbance of the newly built tunnel to the proceeding tunnel. The vertical displacement of the segment corresponding to the right tunnel first increased and then decreased in the under-passing culvert process of the left tunnel. The maximum additional displacement could be observed at the 15 th ring segment at the center of the culvert. The displacement curve of the right segments before and after the excavation of the left tunnel in the X-direction is shown in Figure 21. In the absence of reinforcement, the maximum displacement in the X direction was \(-2.7\) mm before the excavation of the left tunnel; the maximum displacement in the X direction after the excavation of the left tunnel was \(-4.6\) mm; the maximum additional displacement was \(-2.1\) mm, which was located near the 13 th ring segment. In general, three disturbances might be generated during the construction process of the newly built tunnel to the preceding tunnel. More precisely, the first disturbance results from an increase in the soil pressure on the segment of the preceding tunnel owing to external loads, such as the excavation pressure and the jack force exerted by the shield machine on the heading face of the newly built tunnel. The second  
Figure 19: Drawing of prior-dug tunnel segment monitoring: (**a**) schematic of the right tunnel survey line; (**b**) position map of the survey line. disturbance stems from a decrease in the surrounding rock pressure of the preceding tunnel owing to soil mass unloading after the soil excavation of the newly built tunnel. The third disturbance emanates from the soil mass around the preceding tunnel rebounding locally after the shield machine leaves the heading face when the segment of the newly built tunnel is mounted. The above disturbances increase with increasing axial distance between the preceding tunnel and the newly built tunnel. This shows that the segment of the preceding tunnel is most affected by the second disturbance in the construction of the newly built tunnel, and the segment is displaced toward the newly built tunnel.