# Limit Equilibrium Models for Tunnel Face Stability in Composite Soft-Hard Strata
## 3 Three-Dimensional Finite Element Modeling
### Numerical Model and Simulation Process  
In this study, numerical analyses were carried out employing the commercial software PLAXIS 3D. The utilization and validation of PLAXIS 3D by numerous researchers in investigating diverse challenges associated with tunnel faces underscore its reliability and credibility [10; 14; 38]. The model shown in Figure 4 was constructed to analyze the stability of the tunnel face in composite strata with varying proportions of soft and hard layers. The dimensions of the model, measured in length (X axis) by width (Y axis) by height (Z axis), were 50 m \(\times\) 60 m \(\times\) 70 m. The tunnel had a diameter, \(D\), of 15.2 m and was buried at a depth of \(C=15.2\) m. The soil was divided into two layers, with the upper layer consisting of soft soil and the lower layer composed of hard rock. In this study, the tunnel cover depth (\(C=15.2\) m) was kept constant while modifying the thickness of the upper soft soil layer to explore different SAs. The boundary conditions of the model were defined as follows: the top face was considered a free boundary, the four side faces were constrained in the normal direction, and the bottom face was set as a fixed boundary. Seepage effects were not considered, and the groundwater level was established at \(-70\) m. To accurately represent the tunnel face and the upper soft soil layer, mesh refinement was applied, resulting in a total of 32,898 mesh elements and 49,456 nodes.  
Shield tunneling construction is a dynamic process. This study focused on the influence of different SAs in composite strata on tunnel face stability. The tunneling process was not considered in this study. The specific simulation steps were as follows [10]:  
1. The numerical model was established, and the initial stresses were generated using the \(K_{0}\) process;
2. The excavation progressed in increments of a one-time advancing length, denoted as \(D\), while incorporating the cooling of the soil elements;
3. The support pressure on the tunnel face was set to be equal to the initial ground horizontal stress in the opposite direction;
4. The support pressure gradually decreased while the displacement of the soil ahead of the tunnel face increased. The simulation continued until the support pressure  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Soft Soil** & **Hard Rock** & **Parameter** & **Soft Soil** & **Hard Rock** \\ \hline Effective friction angle, \(\varphi^{\prime}\) (\({}^{\circ}\)) & 11 & 35 & \begin{tabular}{c} Reference shear modulus, \\ \(G_{0}\)\({}^{\mathrm{ref}}\) (MPa) \\ \end{tabular} & 20 & 240 \\ \hline Dilation angle, \(\Psi\) (\({}^{\circ}\)) & 0.1 & 0.1 & \begin{tabular}{c} Reference stiffness stress, \\ \(p^{\mathrm{ref}}\) (kPa) \\ \end{tabular} & 100 & 100 \\ \hline Lateral pressure coefficient, \(K_{0}\) & 0.81 & 0.43 & \begin{tabular}{c} Poisson’s ratio for \\ unloading/reloading, \(\gamma_{\mathrm{ur}}\) \\ \end{tabular} & 0.2 & 0.2 \\ \hline Effective cohesion, \(c^{\prime}\) (kPa) & 12 & 32 & \begin{tabular}{c} Unloading/reloading \\ stiffness, \(E_{\mathrm{ur}}\)\({}^{\mathrm{ref}}\) (MPa) \\ \end{tabular} & 7.2 & 150 \\ \hline Failure ratio, \(R_{\mathrm{f}}\) & 0.9 & 0.9 & \begin{tabular}{c} Power for stress-level \\ dependency of stiffness, \(m\) \\ \end{tabular} & 0.8 & 0.3 \\ \hline Secant stiffness, \(E_{\mathrm{go}}\)\({}^{\mathrm{ref}}\) (MPa) & 2.4 & 30 & \
\begin{tabular}{c} Shear strain corresponding to \\ initial shear modulus of \\ 70\%, \(\gamma_{0.7}\) \\ \end{tabular} & \(10^{-4}\) & \(10^{-4}\) \\ \hline
\begin{tabular}{c} Tangent stiffness, \\ \(E_{\mathrm{ood}}\)\({}^{\mathrm{ref}}\) (MPa) \\ \end{tabular} & 2.4 & 30 & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Soil parameter values for HSS.  
showed minimal changes and a substantial increase in horizontal displacement of the soil was observed. This indicated that the failure of the face had occurred, and the calculation was terminated.