## 2 Engineering Background
### Establishment of Numerical Simulation Model  
Based on site engineering survey and full consideration of tunnel excavation process stability, the calculation model was reasonably simplified, and the following basic assumptions were made in the finite element numerical simulation calculation: (1) it was assumed that the rock and soil mass (using the Mohr-Coulomb constitutive model) is homogeneous and isotropic in the same stratum; (2) the groundwater is in a lower position, without considering the seepage effect and the soil deformation's time effect, that is, without considering the secondary consolidation and creep effect of the stratum, and with assuming that the shield advance is a change in spatial displacement; (3) it was assumed that the shield advancing process is continuous, and the advancing length of each shield ring is 1.5 m; (4) in the calculation process, some parameters were adjusted appropriately to make the calculation more reasonable. The three-dimensional numerical simulation modeling of the shield tunnel is carried out using ABAQUS nonlinear finite element analysis software. In this study, the stratum soil, shield, and lining segments were simulated using three-dimensional solid 8-node hexahedral linear reduced integration elements (C3D8R), making up a total number of 184,380 numerical model elements. The numerical model was defined such that the x-axis is along the radial direction of the tunnel, the y-axis is along the tunnel's axial direction, and the z-axis is along the depth direction of the stratum. Among them, the maximum principal stress \(\sigma_{1}\) of the tunnel with advancing TBM face is along the tunnel axis (y-axis), and the minimum principal stress \(\sigma_{3}\) is along the horizontal direction (x-axis) perpendicular to the tunnel axis, which is orthogonal to the principal stress \(\sigma_{1}\) in azimuth, as shown in Figure 3. Since the length of each ring lining segment of the shield tunnel project is 1.5 m, the actual project was advanced at a speed of 4 rings (6 m/d). In order to simulate the actual excavation process, the y-axis simulation range of the numerical model was set to be 60 m. The excavation area unit was divided into 20 sections during numerical modeling, with each section being 3 m long. Considering the influence of the boundary effect in the numerical modeling, the x-axis simulation range of the numerical model was set to 60 m, and the simulation range on both sides of the tunnel was set about 5 times the diameter of the tunnel. It can be considered that the boundary conditions on both sides of the model do not affect the simulation results [49; 50; 51; 52]. The upper surface of the model was taken as the ground surface, the buried depth of the shield was set to 20 m, the length of the shield was defined as 6 m, the width of the lining segment was set to 1.5 m, the outer diameter was defined as 6.0 m, the inner diameter defined as 5.0 m, and the thickness was taken as 0.5 m. The z-axis simulation range of the numerical model was set to 50 m, and the influence of the bottom boundary was ignored. In the overall numerical model, the model dimensions are 60 m in length, 60 m in width, and 50 m in height, all of which have considered the boundary effects on the surrounding soil during tunnel excavation. Therefore, the upper surface of the model was set as a free boundary, horizontal constraints were applied to the front, rear, left and right sides. and vertical and horizontal constraints were applied to the lower surface of the model. According to the actual engineering geological survey data on-site, the strata of the numerical simulation section were set from top to bottom as plain fill, muddy gravel sand, plastic gravel clay, hard plastic gravel clay, and completely weathered biotite granite, and the vertical distribution of each stratum was uneven. Each stratum soil's volume equivalent substitution method was appropriately treated in the model's longitudinal range.