## 2 Establishment of Wedge Sliding Body Model
### Stability Coefficient of Tunnel Face  
Existing studies showed that increasing the sliding resistance and reducing the sliding stress are the main measures used to improve the stability of the tunnel surface. To improve the stability of the tunnel face, typical reinforcement measures, such as pipe roofing support, tunnel face bolting support, tunnel face grouting support and reserved core soil, are usually used to reinforce the tunnel excavation section in the rock mass engineering [25]. [PERSON] et al. [26] and [PERSON] et al. [27] conducted a quantitative analysis of pipe roofing support, bolting support and grouting support. The pressure reduction factor (\(a_{1}\)) of rock mass was used to quantitatively evaluate the vertical deformation pressure of disturbance section under pipe roofing support. The grouting support effect of tunnel face was evaluated using cohesive strength (\(c^{\prime}\)) to improve the cohesion of surrounding rock. When evaluating the support effect of bolting support, the minimum bearing capacity of each bolt in five failure modes was first calculated, and then the bearing capacity was accumulated to obtain the total supporting stress (\(P_{1}\)) provided by the bolting support. The calculation steps of the stability coefficient (\(K\)) of the tunnel face based on the reinforcement effects of pipe roofing support, tunnel face bolting support, tunnel face grouting support and reserved core soil are as follows. The vertical pressure (\(F^{\prime}\)) of wedge sliding body after pipe roofing support was applied can be expressed as:  
\[F^{\prime}\ =\ \alpha_{1}F, \tag{14}\]  
The cohesive strength (\(c^{\prime}\)) of surrounding rock after grouting reinforcement can be expressed as:  
\[c^{\prime}\ =\ \frac{c_{g}V_{g}+cV}{V}\ =\ \alpha_{2}c, \tag{15}\]  
\[\alpha_{2}\ =\ \begin{cases}1+\frac{c_{g}cI_{g}n}{cD\tan\beta}&\left(L_{g}<D \tan\beta\right)\\ 1+\frac{c_{g}c^{\prime}n}{c}&\left(L_{g}\geq D\tan\beta\right)\end{cases} , \tag{16}\]  
where \(c_{g}\) is the cohesion strength of mortar. \(V_{g}\) is the volume of mortar. \(V\) is the volume of wedge sliding body. \(\zeta\) is the filling rate of mortar. \(L_{g}\) is the grouting depth. \(n\) is the porosity of the surrounding rock. \(a_{2}\) is the strengthening coefficient of cohesion after grouting support. Then, by combining the cohesive strength (\(c^{\prime}\)) with Equations (11) and (12), the shear stress (\(T^{\prime}_{1}\)) on the sliding surface and shear stress (\(T^{\prime}_{2}\)) on the side of wedge sliding body after grouting reinforcement can be immediately obtained, respectively. A fiberglass anchor bolt is a typical bolt commonly used for face reinforcement. Its failure modes include the failure of bolt body by tension (\(P_{1i}\)), failure of non-anchored section by sliding of bolt body (\(P_{2i}\)), failure of non-anchored section by sliding between the mortar and hole wall (\(P_{3i}\)), failure of the anchored section by sliding between bolt body and hole wall (\(P_{4i}\)), and failure of the anchored section by sliding between mortar and hole wall (\(P_{5i}\)) [27]. The anchoring stress of single bolt is the minimum anchoring stress among the five failure modes considering construction safety. The calculation equations of anchoring stress can be expressed as [28; 29]:  
\[P_{1i}\ =\ \frac{\pi d^{2}f_{b}}{4}, \tag{17}\]  
\[P_{2i}\ =\ \pi d_{b}\text{I}_{i}f_{t}\left(2+\frac{11.7d_{b}}{\text{I}_{i}} \right), \tag{18}\]\[P_{3i} = \pi d_{b}\mathrm{I}_{\mathrm{II}}f_{t}\left(\frac{1+\lambda}{2}\sigma _{v}\tan\varphi+c\right), \tag{19}\] \[P_{4i} = \pi d_{b}\mathrm{I}_{\mathrm{II}}f_{t}\left(2+\frac{11.7d_{b}}{ \mathrm{I}_{\mathrm{II}i}}\right),\] (20) \[P_{5i} = \pi d_{h}\mathrm{I}_{\mathrm{II}}f_{t}\left(\frac{1+\lambda}{2} \sigma_{v}\tan\varphi+c\right), \tag{21}\]  
where \(f_{b}\) is the tensile strength of anchor bolt. \(d_{b}\) is the diameter of anchor bolt. \(f_{t}\) is the design value of tensile strength of mortar. \(\mathrm{I}_{\mathrm{II}}\) is the length of the \(i\)-th row of anchor bolt in non-anchored area.