# Stability Analysis of Box Tunnel Working Face under Non-Uniform Support Pressure
## 2 Materials and Methods
### Theoretical Model Construction
As shown in Figure 4, considering the equilibrium of an infinitesimal slice with thickness \(dz\), forces acting on the slice include the self-weight \(dG\); the resultant \"supporting\" force \(V(z)\) exerted by the underlying ground _bcgf_; the resultant \"loading\" force \(V(z)+dV\) exerted by the overlying ground _adhe_; the resultant normal force \(dN\) and shear force \(dT\) at the inclined slip surface _cdhg_; and the resultant shear force _dTs_ at the two vertical slip surfaces _abcd_ and _efgh_. The equations of static equilibrium of the slice in the \(x\) and \(y\) directions can be respectively expressed as:  
\[dN=(dV+dG)\cos\theta \tag{2}\]  
\[2 dT_{s}+dT=(dV+dG)\sin\theta \tag{3}\]  
Self-weight \(dG\) is  
\[dG=\gamma BdA \tag{4}\]  
where \(\gamma\) is the weight of the soil, \(dA\) is the slice side area, and  
\[dA=z\cot\theta dz \tag{5}\]  
The maximum shear strength of the inclined sliding surface _cdhg_ is  
\[dT_{\max}=\frac{Bdz}{\sin\theta}c^{\prime}+dN_{\max}\tan\varphi^{\prime} \tag{6}\]  
From the idea of strength discounting (Equation (1)), the actual shear force can be found to be  
\[\tau=\frac{\tau_{\max}}{f_{s}} \tag{7}\]  
From Equation (7), the actual shear force on the inclined sliding surface _cdhg_ is  
Figure 2:0  
The maximum shear strength of the vertical sliding surface _abcd_ (_efgh_) on both sides is  
Figure 2:1  
Figure 2:2 Forces acting upon an infinitesimal slice. From Equation (7), the actual shear force on the vertical sliding surface _abcd_ (_efgh_) on both sides is  
Figure 2:3  
where \(K_{0}\) is the lateral pressure coefficient of the soil, and \(K_{0}\)\(=1-\sin\varphi^{\prime}\), \(\sigma_{z}\) is the vertical stress, taking into account Equation (5):  
Figure 2:4  
It is possible to obtain  
Figure 2:5  
By substituting Equations (2), (4), (5), (8), and (12) into Equation (3), the equilibrium equation parallel to the inclined plane _cdhg_ can be obtained:  
Figure 2:6  
of which  
Figure 2:7  
Figure 2:8  
Figure 2:9  
of which  
\[f_{s}=\frac{\tau_{\max}}{\tau} \tag{1}\]0  
Equation (12), which is a differential equation of vertical stress combined with boundary conditions, can be obtained:  
\[V(z)=\left\{ \begin{array}{l}\zeta e^{\frac{z}{\eta\xi(f_{s},\theta)\tan \theta}}\\ -\eta[\eta(2c^{\prime}-\mathfrak{f}B\xi(f_{s},\theta))z+\xi(f_{s},\theta)] \end{array} \right\} \tag{18}\]  
of which  
\[\zeta(f_{s},\theta)=\frac{\eta}{\cos\theta}\big{[}Bc^{\prime}+\eta\xi(f_{s}, \theta)\sin\theta\big{(}2c^{\prime}-\mathfrak{f}B\xi(f_{s},\theta)\big{)} \big{]} \tag{19}\]  
\[\eta=\frac{B}{2K_{0}\tan\varphi^{\prime}\tan\theta} \tag{20}\]  
Another boundary condition for the differential equation of vertical stress is \(V(D)=V_{\text{silo}}=\sigma_{v}BD\cot\theta\), of which  
\[V(D)=\left\{ \begin{array}{l}\zeta(f_{s},\theta)e^{\frac{D}{\eta\xi(f_{s}, \theta)\tan\theta}}\\ -\eta[\eta(2c^{\prime}-\gamma B\xi(f_{s},\theta))D+\zeta(f_{s},\theta)]\end{array} \right\} \tag{21}\]  
The load applied by the overburden is  
\[\sigma_{v}=\max\biggl{\{}0,\ \frac{R\gamma-c^{\prime}}{K_{0}\tan\varphi^{ \prime}}\Big{(}1-e^{-K_{0}\tan\varphi^{\prime}\frac{c}{K}}\Big{)}+\sigma_{s} e^{-K_{0}\tan\varphi^{\prime}\frac{c}{K}}\biggr{\}} \tag{22}\]where \(\sigma_{s}\) is the surface load, assumed to be 0 in this paper; and \(R\) equals the ratio of the volume of the soil column to its circumferential area, which is  
\[R=\frac{BD\cot\theta}{2(B+D\cot\theta)} \tag{23}\]  
From the boundary conditions and Equations (21) and (22), we obtain  
\[\sigma_{\triangleright}=\frac{\left\{ \begin{array}{l}\zeta(f_{s},\theta)e^{ \frac{D}{B\zeta(f_{s},\theta)\ln\theta}}\\ -\eta[\eta(2c^{\prime}-\gamma B\zeta(f_{s},\theta))D+\zeta(f_{s},\theta)] \end{array} \right\}}{BD\cot\theta} \tag{24}\]  
When the soil strength (\(c^{\prime}\),\(q^{\prime}\)) and tunnel dimensions (\(C\), \(L\), \(D\) and \(B\)) are given, \(\theta\) and \(f_{s}\) in Equation (24) are mutual functions and inverse functions, respectively.