## 3 Numerical Simulation and Analysis of Support Pressure on Tunnel Face  
### Simulation Model and Basic Assumptions  
The numerical simulations were carried out in close connection with the actual situation of the Y25-Y26 pipe-jacking project. The derivation of mathematical expressions and numerical simulations of tunnel-face support pressures explored in this study were all in the context of the same project. The length, width, and height of the numerical simulation model were taken as 100 m \(\times\) 80 m \(\times\) 60 m, as shown in Figure 5. The bottom surface of the model was constrained with vertical and horizontal displacements, while the top surface was free. The pipe jacking was constructed using three circular reinforced concrete pipes with an internal diameter of 3.0 m and an external diameter of 3.6 m. The spacing between neighboring pipes was 2.8 m, and the length of the pipe-jacking excavation was 80 m. The reinforced concrete pipe sheet was considered an elastic material with a thickness of 0.3 m. The pipe-jacking machine casing was made of steel, which was also considered to be an elastic material. The values of the calculated parameters were taken as shown in Table 1. The material properties of the strata and pipe joints were described using 3D cell entities. Partial components were meshed by means of geometric partitioning and then expanded with a 2D mesh. The pipe perimeter grid was 0.5 m. The model boundary calculation cell grid was set to 1.0 m in both the longitudinal and transverse directions. The pipe-jacking machine shield shell was modeled elastically, and its material properties were described using 2D plate cells, which were meshed by analyzing the outer surface of the iso-surrogate layer. The values of the stratigraphic parameters are taken as shown in Table 2, which comes from the survey report of the background project. In the geotechnical laboratory of the survey company, soil tests for the moisture content, liquid-plastic limit, straight shear, compression, and free swell and a uniaxial compression test of rock were completed. The coupling between the soil entities and the jacked tunnel entities was investigated using Boolean operations. The mesh size of the soil body was 5 m and the mesh size of the pipe jacking tunnel was 1 m. The numerical calculation process was divided into four stages, which were initial stress field equilibrium, pipe excavation, pipe jacking advancement, and pipe sheet application. At first, the initial boundary conditions and self-weight of the model were imposed. The initial displacement was zeroed out to start the analysis of the initial stress field. Secondly, a uniform digging force was applied to the tunnel face. The pipe jacking tunnel was passivated to simulate excavated soil. Then, the pipe jacking machine casing was activated and the friction force was activated at the same time. The casing of the pipe-jacking machine was blunted, the pipes were jacked, and the pipe sheets were applied. Finally, the soil of the next pipe section was excavated. The pipe-jacking machine casing was added and the excavation forces and friction forces exerted by the previous pipe section were blunted. For pipe-jacking projects crossing a river, the grouting pressure, resistance, and friction between the pipe and soil should be considered. In order to highlight the main issues and reduce the amount of tedious work, we chose to simplify the conditions. Pipe jacking forms a boundary inside the formation and changes the original stress field of the formation, which leads to a series of complex physical and mechanical effects in the formation around the construction area. This is closely related not only to the physical properties of the formation but also to the construction method, construction process, and related construction parameters. When using numerical software for three-dimensional calculations of multi-line parallel pipe jacking, it is difficult to take all the factors of pipe jacking into account to completely and realistically reflect the construction process.