## 2 Establishment of Wedge Sliding Body Model  
### Limit Equilibrium Model of Wedge Sliding Body  
The soil in front of a tunnel face has a sliding tendency when a shallow tunnel is excavated, causing instability in the tunnel face [18] and forming a wedge sliding body. In tunnel engineering, the cross-sectional profile of the tunnel is generally composed of multiple central circles, and the cross-sectional shape is also different due to the differences in tunnel structure design. For the convenience of calculation, we make the tunnel cross-section shape equivalent to a rectangle (AabB). As shown in Figure 1a, the triangular prism (ABC-abc) is the wedge sliding body in front of the tunnel face, and the quadrangular prism (DCcd-EFe) is the upper soil body, and \(H\) represents the buried depth of tunnel. AaBb is the equivalent excavation surface of the wedge sliding body, \(D_{0}\) represents the tunnel height, and \(l_{0}\) represents the equivalent width. The excavation area is the section area of tunnel in the case of a full-section excavation, and the excavation area is the upper step area in the case of step method. EeF is the ground surface of quadrangular prismatic soil. BbDd is the unsupported section and its length \(L_{e}\) (DB, as shown in Figure 1b) is related to the actual excavation progress. AaCc is the slip surface of wedge sliding body, and the relationship between sliding angle (\(\beta\)) and internal friction angle (\(\varphi\)) satisfies:  
\[\beta = \frac{\pi}{4}-\frac{\varphi}{2}, \tag{1}\]  
To establish the limit equilibrium model of wedge sliding body, the following assumptions need to be made. The soil body is a homogeneous and isotropic rigid-plastic material and the weighted average method can be used for the multi-layer soil body. 1 The soil material obeys the Mohr-Coulomb failure criterion (\(\tau=c+\sigma\tan\varphi\), \(c\) is the cohesion, \(\sigma\) is the normal stress), and the influence of groundwater is not considered. 2 The damage range of excavation section is composed of wedge sliding body (ABC-abc) and prismatic body (DCcd-EFe). 3 The tunnel face is equivalent to a rectangle (ABba) with the same area and height for a calculation based on the principle of area equivalence. The vertical stress on the side of the wedge body linearly increases with the depth, and the stress on the top surface and inclined sliding surface is uniformly distributed. Figure 2 shows the schematic diagram of wedge sliding body stress. \(P\) is the active stress that ensures tunnel face stability and the active force applied by the tunnel face support measures, and the direction is the vertical tunnel face and points to the inside of wedge sliding body. \(F\) is the compressive stress of the soil body acting on wedge sliding body. \(G\) is the self-gravity of the wedge sliding body (\(\gamma\) indicates density), facing a vertically downward direction, and its size depends on the volume of the wedge sliding body and the density of the surrounding rock. \(N_{1}\) and \(T_{1}\) are the compression stress and shear stress on sliding surface (A(a)C(c)), respectively. \(N_{2}\) and \(T_{2}\) are the compression stress and shear stress on the wedge sliding body (ABC). The lateral abc has the same stress as the lateral ABC. Combined with the limit equilibrium model of the wedge sliding body, the following are the solutions for the self-gravity (\(G\)), compressive stress (\(F\)), shear stress (\(T_{1}\)) and shear stress (\(T_{2}\)), which are used to obtain the active stress (\(P\)). Figure 1: Schematic diagram of limit equilibrium model: (**a**) 3D computational model, (**b**) Plan sketch. When the wedge sliding body is in the limit equilibrium state, the stress balance equation can be expressed as:  
\[