## 4 Limiting Support Force of the Tunnel Face
### Three-Dimensional Numerical Model  
In the development of this model, it is assumed that the strata and ground are horizontal. Capitalizing on the tunnel's symmetry, only a half-section of the tunnel was modeled. The numerical model spans dimensions of 50 m in length, 21 m in width, and 56 m in height. By taking hydrological conditions into account, the water level was situated at a depth of 42 m from the model's base. Comprising approximately 74,100 zones and 79,894 grid points, the numerical model presumes that the strata conform to the Mohr-Coulomb strength criterion [29; 30]. The boundary conditions were delineated as follows: (1) The upper boundary of the numerical model is unrestricted, while the lower boundary is  
\begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Stratum** & **Thickness (m)** & **Density (kg/m3)** & \(E\) **(MPa)** & _c/kPa_ & \(\boldsymbol{\varphi}\) (\({}^{\circ}\))** & \(\boldsymbol{\
u}\) \\ \hline Yellow stratum \#1 & 4 & 1740 & 13.00 & 24.0 & 19.0 & 0.23 \\ Yellow stratum \#2 & 8 & 1940 & 14.00 & 24.4 & 20.0 & 0.30 \\ Paleosol stratum & 5 & 1880 & 11.20 & 27.6 & 20.1 & 0.32 \\ Silty clay stratum & 10 & 1890 & 15.60 & 33.7 & 23.2 & 0.28 \\ Fine sand stratum & 2 & 2020 & 25.24 & 0.0 & 30.5 & 0.25 \\ Medium sand stratum & 27 & 2060 & 31.00 & 0.0 & 31.1 & 0.25 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Stratum parameter.  
Figure 27: Three-dimensional formation model of shield excavation.  
fully constrained; (2) the model's lateral boundaries are fixed solely in the normal direction, with the remaining directions being unimpeded. Post-excavation, a shell element was applied to the tunnel's periphery to simulate the lining as a linear elastic shell element. Characterized by a Young modulus of 20 GPa, a Poisson ratio of 0.17, and a lining thickness of 0.3 m, the lining model is integral to the simulation's fidelity. The simulation employs a single-step excavation method, with a primary focus on assessing the stability of the tunnel face. Throughout the numerical simulation in this study, the tunnel was incrementally formed through the excavation of 25 m sections at a time. This incremental approach facilitates a granular evaluation of the tunnel face stability at each critical juncture of the excavation process, ensuring a comprehensive understanding of the structural integrity under dynamic conditions.  
The simulation here does not consider the influence of the excavation process on soil deformation, so the single-step excavation method was adopted in the numerical simulation. The \"shell\" unit was then installed on the soil around the excavated tunnel to simulate the lining. Then, a uniform support pressure (\(\sigma_{\text{N}}\)) was applied to the driving face to simulate shield thrust. The uniform support pressure is equal to the original horizontal earth pressure at the center of the tunnel face. In the numerical simulation, the support pressure gradually increases with the gradient \(\sigma_{\text{N}}/\text{N}\), and N is the simulation number. Each calculation can obtain a set of shield thrusts and tunnel face center horizontal position movements (S\({}_{\text{i}}\)). After N times of simulation, we could obtain N groups of shield machine thrust and the corresponding horizontal displacement (S\({}_{\text{i}}\)).  
According to the results of N times of simulation, the relationship between the thrust of the shield machine and the horizontal displacement (S\({}_{\text{i}}\)) of the corresponding tunnel face is obtained. With the decrease in thrust, \(\sigma_{\text{N}}\), the horizontal displacement, S\({}_{\text{i}}\), increases gradually. When the horizontal displacement suddenly increases under the thrust of a shield machine, the thrust of the shield machine at this time can be regarded as the maximum support pressure of the shield machine. Figure 28 shows the simulation flowchart.  
Figure 28: Flowchart of tunnel face stability simulation.