# Working Face Stability of Box Shield Tunneling under Non-Uniform Support Pressure
## 3 Results  
### Comparisons of the Support Pressure  
In this section, the finite element limit analysis code OptumG3 is used for numerical validation. OptumG3 is favorable for 3D analysis and has the advantage of automatically searching for optimal faults through mathematical optimization [43]. The failure mechanisms of tunnels with different strength parameters and geometric parameters have been investigated. The results of the proposed model--the value of the collapse pressure and the failure geometry--are compared to those of a numerical simulation. Due to symmetry, half of the tunnel is selected for computation. Figure 7 illustrates the numerical model constructed to validate the analytical solution. The dimensions of the model are 30 m \(\times\) 24 m \(\times\) 40 m. The tunnel is excavated to half of its length (20 m). The number of grids is 3000. Standard boundary conditions are used. The support pressure \(s_{1}\) acting on block I is calculated by Equation (6). Accordingly, the support pressure \(s_{2}\) acting on block II is calculated. The cover depth of the tunnel is 6.0 m, i.e., \(D=6.0\) m. In addition, the surface surcharge is zero. The partially supported ratio \(B/L\) varies from 0.4 to 0.7. For shallowly buried tunnels, the depth ratio \(C/D\) varies from 1 to 2. For box tunnels, the width-height ratio \(L/D\) varies from 1.5 to 2.0. The soil is set as an elastic-plastic material following the Mohr-Coulomb failure criterion and associated flow rule. Typical non-cohesive soil is considered. The mechanical parameters of the soil are listed in Table 1.  
Figure 8 indicates the variation in the normalized support pressure with increasing \(\varphi^{\prime}\). Both the proposed model and the numerical model predict that \(s_{2}/\gamma D\) decreases nonlinearly as \(\varphi^{\prime}\) increases. At constant \(\varphi^{\prime}\), \(s_{2}/\gamma D\) associated with a higher \(B/L\) has a larger result. This is because the increase in \(B/L\) makes the fully supported area on the working face smaller and the partially supported area larger, thus leading to an increase in the risk of instability. At constant \(\varphi^{\prime}\) and \(B/L\), the proposed model predicts a higher \(s_{2}/\gamma D\) than the numerical model does. The results of the proposed model are more secure for the engineering design. Figure 8a,b shows that \(s_{2}/\gamma D\) with different depth ratios decreases in a similar pattern with an increasing \(\varphi^{\prime}\). A higher \(C/D\) makes a higher \(s_{2}/\gamma D\), which can be observed by comparison. Table 2 shows the variation in the ratio of the two support pressures \(s_{2}/s_{1}\) with an increasing \(\varphi^{\prime}\). It can be seen that \(s_{2}/s_{1}\) decreases in a similar pattern as \(\varphi^{\prime}\) changes with \(s_{2}/\gamma D\) in Figure 8. However, the changes in \(s_{2}/\gamma D\) in Figure 8 are more pronounced than the corresponding changes in \(s_{2}/s_{1}\) in Table 2. It is known that \(s_{1}\) is related to \(\varphi^{\prime}\) by Equation (6). \(s_{1}\) decreases with the increase in \(\varphi^{\prime}\), while \(\gamma D\) is constant. Under the same conditions, \(s_{2}/s_{1}\) is larger than \(s_{2}/\gamma D\), especially when \(\varphi^{\prime}\) is greater. Table 2 shows that \(s_{2}/s_{1}\) with different depth ratios decreases in a similar pattern with an increasing \(\varphi^{\prime}\). A higher \(C/D\) makes a lower \(s_{2}/s_{1}\), as can be seen by comparison. This is because \(s_{1}\) is related to \(C\) by Equation (6). A higher \(C/D\) makes a higher \(s_{1}\). The results of the proposed model are greater than the results of the numerical model. The reliability of the proposed model can be verified. \begin{table}
\begin{tabular}{c c c c c} \hline \(\gamma\)**/kN/m3** & **E/MPa** & \(\
u\) & \(c^{\prime}\)**/kPa** & \(\varphi^{\prime}\)**/\({}^{\circ}\)** \\ \hline
19.8 & 30 & 0.25 & 0 & 15–45 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters for numerical modeling. Figure 7: Geometry of the numerical model built in OptumG3.