# Working Face Stability of Box Shield Tunneling under Non-Uniform Support Pressure
## 2 Materials and Methods
### Modified Model  
Face stability in homogeneous ground can be assessed by considering the mechanism which consists of a wedge and a prism that extends from the tunnel crown to the surface. Figure 4 shows the profile of the proposed model. The width, height, and depth of the tunnel, respectively, are \(L\), \(D\), and \(C\). In the proposed model, the width of the partially supported area is \(B\). Due to symmetry, only block I, block II, and their interaction are necessary to analyze. There are interaction frictions, \(T_{12}\) and \(T_{21}\), and normal forces, \(N_{12}\) and \(N_{21}\), on the contact surface of block I and block II. When block II has a downward trend due to the reduction in the support force, it will produce downward friction \(T_{12}\) on block I which will cause the risk of instability. Therefore, the analysis of the limit equilibrium state of the two blocks and the interaction forces are used so that the minimum support force \(S_{2}\) to keep block I stable can be deduced when support force \(S_{1}\) is known. As shown in Figure 5, block I is acted upon by (a) the resultant vertical force of the prism on the top surface \(oa\); (b) the resultant normal forces and shear forces along the failure surfaces \(ode\), \(abe\), and \(bde\); (c) the support force \(S_{1}\) acting on surface \(oabd\); and (d) its weight. Figure 4: Modified model. Figure 5: The forces applied on block I. The resultant vertical force of the prism \(P_{1}\) on the top surface \(oae\) is given by:  
\[P_{1}=\gamma C\frac{(L-B)D\cot\theta}{4} \tag{1}\]  
where \(\theta\) denotes the wedge angle between slanted failure plane and the horizontal. The resultant normal force \(N_{1}\) and the shear force \(T_{1}\) along the failure surface \(bde\) are given by:  
\[T_{1}=\frac{c^{\prime}(L-B)D}{4\sin\theta}+N_{1}\tan\varphi^{\prime} \tag{2}\]  
The resultant normal force \(N_{3}\) and the shear force \(T_{3}\) along the failure surface \(abe\) are given by:  
\[N_{3}=\int_{0}^{D}K_{0}\sigma_{z}\frac{\cot\theta}{\sin\alpha}(D-z)\mathrm{d}z \tag{3}\]  
\[T_{3}=\frac{c^{\prime}D^{2}\cot\theta}{2\sin\alpha}+N_{3}\tan\varphi^{\prime} \tag{4}\]  
where \(\alpha\) denotes the bottom corner of isosceles trapezoid on the top of the wedge and can be solved as \(\alpha=\arctan\frac{2D\cot\theta}{L-B}\); \(K_{0}\) is a lateral stress coefficient and can be solved as \(K_{0}=1-\sin\varphi^{\prime}\); and the vertical stress \(\sigma_{z}\) along ab changes linearly with depth and can be solved as \(\sigma_{z}=\gamma(C+z)\). The support force \(S_{1}\) is defined as:  
\[S_{1}=s_{1}\frac{(L-B)D}{2} \tag{5}\]  
where \(s_{1}\) denotes the support pressure acting on surface \(oabd\). \[s_{1}=K_{\mathrm{a}}\gamma\bigg{(}C+\frac{D}{2}\bigg{)} \tag{6}\]  
where \(K_{\mathrm{a}}\) is the active earth pressure coefficient and can be solved as \(K_{\mathrm{a}}=\tan^{2}\left(\frac{\pi}{4}-\frac{\varphi^{\prime}}{2}\right)\). The weight of block I is:  
\[G_{1}=\gamma V_{1}=\frac{\gamma}{6}(L-B)D^{2}\cot\theta \tag{7}\]  
where \(V_{1}\) represents the volume of block I. The equilibrium conditions in the \(y\) and \(z\) directions lead to:  
\[\left\{ \begin{array}{l}S_{1}+T_{1}\cos\theta+T_{3}\sin\alpha\cos\theta=N_{3} \cos\alpha+N_{1}\sin\theta+T_{12}\cos\theta\\ P_{1}+G_{1}+T_{12}\sin\theta=N_{1}\cos\theta+T_{3}\sin\alpha\sin\theta+T_{1} \sin\theta\end{array} \right. \tag{8}\]  
Its solution reads as follows:  
\[T_{12}=-P_{1}\sin\theta-G_{1}\sin\theta-N_{3}\cos\alpha\cos\theta+T_{3}\sin \alpha+T_{1}+S_{1}\cos\theta \tag{9}\]  
\[N_{1}=(P_{1}+G_{1})\cos\theta+S_{1}\sin\theta-N_{3}\cos\alpha\sin\theta \tag{10}\]  
As is shown in Figure 6, block II is analyzed by the same method. Compared with block I, block II is acted upon by the support force \(S_{2}\) acting on surface \(oghd\). The vertical force at the wedge-prism interface \(oefg\) reads as follows:  
\[P_{2}=\sigma_{\rm v}BD\cot\theta \tag{11}\]  
where \(\sigma_{\rm v}\) denotes the vertical stress.