# Working Face Stability of Box Shield Tunneling under Non-Uniform Support Pressure
## 2 Materials and Methods
### Modified Model
Considering the contribution of arching, \(\sigma_{\rm v}\) can be calculated as:  
\[\sigma_{\rm v}=\max\biggl{\{}0,\,\frac{R\gamma-c^{\prime}}{K_{0}\tan\varphi^{ \prime}}\Bigl{(}1-e^{-K_{0}\tan\varphi^{\prime}\frac{C}{R}}\Bigr{)}+\sigma_{\rm s }\,e^{-K_{0}\tan\varphi^{\prime}\frac{C}{R}}\biggr{\}} \tag{12}\]  
where \(\sigma_{\rm s}\) denotes the surface load and is set equal to zero; \(R\) equals the ratio of the volume of the prism to its side area: \(R=\frac{BD\cot\theta}{2(B+D\cot\theta)}\). The resultant normal force \(N_{2}\) and the shear force \(T_{2}\) along the failure surface \(defh\) are given by:  
\[T_{2}=\frac{c^{\prime}BD}{\sin\theta}+N_{2}\tan\varphi^{\prime} \tag{13}\]  
The support force \(S_{2}\) is defined as:  
\[S_{2}=s_{2}BD \tag{14}\]  
where \(s_{2}\) denotes the support pressure acting on surface \(oghd\). The self-weight of block II is:  
\[G_{2}=\gamma V_{2}=\frac{\gamma}{2}BD^{2}\cot\theta \tag{15}\]  
where \(V_{2}\) represents the volume of block II. The resultant normal force and the shear force of the two lateral slip surfaces, \(oed\) and \(gfh\), are equal:  
\[\left\{ \begin{array}{l}T_{23}=T_{21}\\ N_{23}=N_{21}\end{array} \right. \tag{16}\]  
Taking into account Equation (14) and the equilibrium conditions in the \(y\) and \(z\) directions leads to:  
\[\left\{ \begin{array}{l}N_{2}\sin\theta=(T_{2}+2T_{21})\cos\theta+S_{2}\\ P_{2}+G_{2}=(T_{2}+2T_{21})\sin\theta+N_{2}\cos\theta\end{array} \right. \tag{17}\]  
Its solution reads as follows:  
\[T_{21}=\frac{(P_{2}+G_{2})\sin\theta-T_{2}-S_{2}\cos\theta}{2} \tag{18}\]  
Figure 6: Forces applied on block II. \[N_{2}=(P_{1}+G_{1})\cos\theta+S_{2}\sin\theta \tag{19}\]  
Due to interaction frictions \(T_{12}=T_{21}\), combining Equation (9) and Equation (18) makes it possible to eliminate \(T_{12}\) and \(T_{21}\); the required support force \(S_{2}\) can now be found from:  
\[(2S_{1}+S_{2})\cos\theta=[2(P_{1}+G_{1})+P_{2}+G_{2}]\sin\theta+2N_{3}\cos \alpha\cos\theta-2T_{1}-T_{2}-2T_{3}\sin\alpha \tag{20}\]  
Due to Equations (1)-(7) and (10)-(15), we obtain from Equation (20):  
\[\left. \begin{array}{ll}s_{2}&=\gamma D\left\{\begin{array}{ll} \left[\frac{C(L-B)}{2 BD}+\frac{2L+B}{6B}+\frac{R\left(1-e^{-K_{0}\tan\varphi} \frac{C}{R}\right)}{DK_{0}\tan\varphi^{\prime}}\right]\frac{1-\cot\theta\tan \varphi^{\prime}}{1+\tan\theta\tan\varphi^{\prime}}\\ &\quad+K_{0}\frac{3C+D}{3B}\Big{(}\cot\theta\cot\alpha-\frac{\tan\varphi^{ \prime}}{\sin\theta(1+\tan\theta\tan\varphi^{\prime})}-K_{0}\Big{(}\frac{C}{ D}+\frac{1}{2}\Big{)}\Big{(}\frac{L}{B}-1\Big{)}\end{array}\right\}\\ &-c^{\prime}\left[\frac{1-e^{-K_{0}\tan\varphi^{\prime}}\frac{C}{R}}{K_{0} \tan\varphi^{\prime}}\frac{1-\cot\theta\tan\varphi^{\prime}}{1+\tan\theta \tan\varphi^{\prime}}+\frac{2D\cos\theta+L+B}{B\sin 2\theta(1+\tan\theta\tan \varphi^{\prime})}\right]\end{array} \right\} \tag{21}\]  
This simplifies to:  
\[s_{2}=\gamma DN_{\gamma}-c^{\prime}N_{c} \tag{22}\]  
where  
\[ \begin{array}{ll}N_{\gamma}&=\frac{\varepsilon^{-}}{\varepsilon^{+}}\Big{[} \frac{C(L-B)}{2 BD}+\frac{L}{3B}+\frac{R\kappa}{D}+\frac{1}{6}\Big{]}\\ &\quad+K_{0}\frac{3C+D}{3B}\Big{(}\cot\theta\cot\alpha-\frac{\tan\varphi^{ \prime}}{\tau\sin\theta}\Big{)}-K_{0}\Big{(}\frac{C}{D}+\frac{1}{2}\Big{)} \Big{(}\frac{L}{B}-1\Big{)}\\ &\quad\quad\quad\quad\quad N_{c}=\frac{\varepsilon^{-}}{\varepsilon^{+}} \kappa+\frac{2D\cos\theta+L+B}{B\varepsilon^{+}\sin 2\theta}\end{array}  \tag{24}\]  
Here, the shorthand notation is introduced:  
\[\kappa=\frac{1-e^{-K_{0}\tan\varphi^{\prime}}\frac{C}{R}}{K_{0}\tan\varphi^{ \prime}} \tag{25}\]  
\[\varepsilon^{-}=1-\cot\theta\tan\varphi^{\prime} \tag{26}\]  
\[\varepsilon^{+}=1+\tan\theta\tan\varphi^{\prime} \tag{27}\]  
As is shown in Equation (22), the formula consists of two elements. The first element is related to the weight of the soil. \(N_{\gamma}\) is a coefficient related to \(\gamma\). The second element is related to the cohesion of the soil. \(N_{c}\) is a coefficient related to \(c^{\prime}\).