## 4 Calculation Model of \"Wedge-Prism\"  
### Considering Anisotropy of Soil Strength  
Using the modified \"wedge-prism\" limit equilibrium model in Section 2 of this paper, and considering the loosening earth pressure of the soil column above the wedge, the calculation model of the ultimate supporting force is established as shown in Figure 1.  
The ultimate supporting force of the excavation face can be obtained by taking wedge for force analysis, as shown in Figure 6.  
The forces acting on the wedge include:  
(1) Resultant force \(V\) of vertical overburden pressure acting on surface cdef. \(V=\sigma_{v}BL\), where, \(\sigma_{v}\) is the earth pressure at the top of the tunnel, \(L\) is the length of the top of the sliding block, and \(B\) is the \begin{table}
\begin{tabular}{c c c c c c} \hline
**Researcher** & **Research Method** & **C/D** & **H** & \(L\) & **H/L** \\ \hline \multirow{3}{*}{[PERSON] (1994)} & \multirow{3}{*}{Centrifuge Model Test} & 0.5 & 0.5D & 0.46\(D\) & 1.09 \\  & & 1 & 0.75D & 0.5D & 1.5 \\  & & 2 & 0.85\(D\) & 0.5D & 1.7 \\ \hline [PERSON] (2004) & Centrifuge Model Test & 2 & 0.59\(D\) & 0.27\(D\) & 2.19 \\ \hline [PERSON] (2006) & 1 g Model Test & 2 & 1.19\(D\) & 0.6\(D\) & 1.98 \\ \hline [PERSON] (2013) & 1 g Model Test & 2 & 1.5\(D\) & 0.75\(D\) & 2 \\ \hline [PERSON] (2014) & Centrifuge Model Test & 0.5 & 0.5D & 0.3D & 1.67 \\  & & 1 & 0.5D & 0.3D & 2 \\  & & 2 & 0.5D & 0.3D & 2 \\ \hline [PERSON] (2016) & Centrifuge Model Test & 0.5 & 0.5D & 0.4D & 1.25 \\  & & 1 & 0.65D & 0.4D & 1.63 \\ \hline [PERSON] (2019) & Centrifuge Model Test & 2 & 1.3\(D\) & 0.6\(D\) & 2.17 \\ \hline \end{tabular}
\end{table} 
Table 1: Statistics on the height of unstable “prism” with different tunnel depth ratios. Figure 6: Analysis of force balance of wedge. equivalent shield diameter [33]. Assuming that the area of the square and the circle are equal, we can get \(B=\sqrt{\pi}D/2\), where \(D\) is the diameter of the shield. (2) The weight of the block \(G\). \(G=\pi\gamma D^{3}/8\cot\beta\), where \(\gamma\) is the bulk density of the soil and \(\beta\) is the \"wedge angle\". (3) \(p\) is the horizontal force acting on the excavation face abcd. \(\sigma_{s}\) is the ultimate supporting force acting on the excavation face abcd, \(P=\frac{\pi D^{2}}{4}\sigma_{s}\). (4) The frictional resistance \(T_{1}\) and normal force \(N_{1}\) on the sliding surface of the wedge. \(T_{1}=c_{\alpha=\zeta-\frac{\pi}{4}}B^{2}/sin\ \beta+N_{1}tan\varphi\), where \(c\) is the cohesive force of soil. (5) The frictional resistance \(T_{2}\) and normal force \(N_{2}\) on the side of the wedge. \(T_{2}=B^{2}\Big{(}c_{\alpha=-\frac{\pi}{4}}+K_{0}\sigma_{z}^{\prime}\ tan\ \varphi\Big{)}/(2 tan\ \beta)\), where \(K_{0}\) is the lateral earth pressure coefficient of the prism [33]. \(K_{0}=1-sin\ \varphi\), where \(\varphi\) is the angle of internal friction in soil. \(\sigma_{z}^{\prime}=(2\sigma_{v}+B\gamma)/3\) is the vertical average stress of the wedge [6]. The magnitude of the ultimate supporting force \(\sigma_{s}\) is determined by analyzing the force of the sliding block. It can be obtained from the horizontal force balance of the block:  
\[P+T_{1}cos\beta+2T_{2}cos\beta=N_{1}sin\beta \tag{3}\]  
From the vertical force balance can be obtained:  
\[V+G=T_{1}sin\beta+2T_{2}sin\beta+N_{1}cos\beta \tag{4}\]  
The calculation equation of the minimum horizontal force acting on the excavation face is obtained as follows:  
\[P=\varepsilon(V+G)-\Bigg{(}\frac{c_{\alpha=\zeta-\frac{\pi}{4}}B^{2}}{sin\beta }+2T_{2}\Bigg{)}(\varepsilon sin\beta+cos\beta) \tag{5}\]  
Then the effective limit supporting force acting on the excavation face is as follows:  
\[\sigma_{s}=\frac{4}{\pi D^{2}}\varepsilon(V+G)-\Bigg{(}4\frac{c_{\alpha= \zeta-\frac{\pi}{4}}B^{2}}{\pi sin\beta D^{2}}+\frac{8T_{2}}{\pi D^{2}}\Bigg{)} \left(\varepsilon sin\beta+cos\beta\right) \tag{6}\]  
where \(\varepsilon=tan(\beta-\varphi)=tan(45^{\circ}-\varphi/2)\). The variable \(\beta\) is the angle between the slip surface of the wedge and the horizontal direction.