## 2 Establishment of Wedge Sliding Body Model
### Stability Coefficient of Tunnel Face
\(\mathrm{I}_{\mathrm{II}i}\) is the length of the \(i\)-th row of anchor bolt in anchored area. \(d_{h}\) is the drilling diameter. \(\lambda\) is the lateral pressure coefficient of soil body. The total supporting stress (\(P_{1}\)) of tunnel face can be expressed as:  
\[ \begin{cases}P_{i}&=~{}m_{i}\times\min(P_{1i},P_{2i},P_{3i},P_{4i},P_{5i})\\ P_{1}&=~{}\sum\limits_{i=1}^{i}P_{i}\end{cases} , \tag{22}\]  
where \(m_{i}\) is the number of transverse anchor bolts in row \(i\). \(n\) is the number of rows of anchor bolt. Reserved core soil is a common measure to ensure the stability of tunnel face, which is realized by dividing the tunnel face into several parts and then optimizing the excavation sequence of the tunnel face. Figure 4a shows the schematic diagram of reserved core soil. The tunnel face is divided into three areas: the annular excavation area, reserved core soil, and excavation area of the lower section. As shown in the figure, the annular excavation area is first excavated to form an arc advance guide pit at the tunnel outline during tunnel excavation. The reserved core soil in the center of tunnel face is retained to support against tunnel face, and then increase the tunnel face stability. The quantitative supporting method of reserved core soil is introduced based on the existing research. A simplified schematic diagram of reserved core soil size is shown in Figure 4b, where \(b_{1}\) is the bottom width, \(\mathrm{b}_{2}\) is the top width and \(\mathrm{h}\) is the height of reserved core soil. It is assumed that the mechanical action of the wedge sliding body on the reserved core soil causes the reserved core soil to be in a critical sliding state. That is, the supporting stress reaches the maximum value, and the maximum supporting stress can be determined by the passive soil pressure. The supporting stress (\(P_{2}\)) provided by the reserved core soil is expressed as:  
\[ \begin{cases}P_{2}&=~{}\frac{1}{2}(b_{1}+b_{2})\left(\frac{1}{2}K_{p}\gamma h ^{2}+2\sqrt{K_{p}}ch\right)\\ K_{p}&=~{}\tan^{2}(45^{\circ}+\frac{1}{2}\varphi)\end{cases} , \tag{23}\]  
where \(K_{p}\) is the Rankine passive soil pressure coefficient. Figure 4: Schematic diagram of reserved core soil support: (**a**) Schematic diagram of reserved core soil, (**b**) Schematic diagram of reserved core soil size. It is assumed that the toe angle of the reserved core soil is the sliding angle (\(\beta\)) of the surrounding rock. The top width (\(b_{2}\)) of reserved core soil can be expressed as:  
\[b_{2}\ =\ b_{1}-2h\tan\beta, \tag{24}\]  
The supporting stress (\(P_{2}\)) can be obtained by the simultaneous Equations (23) and (24):  
\[P_{2}\ =\ -\frac{1}{2}K_{p}\gamma h^{3}\tan\beta+\left(\frac{1}{2}K_{p}\gamma b_{ 1}-2\tan\beta\sqrt{K_{p}c}\right)h^{2}+\sqrt{K_{p}}cb_{1}h, \tag{25}\]  
From the stable conditions of wedge sliding body, it can be seen that when the sliding stress generated by self-gravity and compressive stress are greater than anti-sliding stress generated by surrounding rock, then the failure of sliding instability will occur. In order to achieve a quantitative description of the stability of the wedge sliding body, the ratio of anti-sliding force to the sliding force of the wedge sliding body is defined as the stability coefficient (\(K\)). The ratio of anti-sliding force to sliding force without any supports is defined as the initial stability coefficient of tunnel face (\(K_{0}\)). Based on the limit equilibrium model of wedge sliding body in Figure 1, initial stability coefficient (\(K_{0}\)) of tunnel face can be expressed as:  
\[K_{0}\ =\ \frac{T_{1}+2T_{2}}{(F+G)\cos\beta}, \tag{26}\]  
If the reinforcement effects of pipe roofing support, tunnel face bolting support, tunnel face grouting support and reserved core soil are considered, the stability coefficient (\(K\)) of the tunnel face after reinforcement can be expressed as:  
\[K\ =\ \frac{T_{1}^{\prime}+2T_{2}^{\prime}+(P_{1}+P_{2})\sin\beta}{(\alpha_{1} F+G)\cos\beta}, \tag{27}\]