## 2 Theoretical Analysis of Tunnel-Face Support Pressure
### Theoretical Calculation Principle of Tunnel-Face Support Pressure  
After simplifying and making assumptions about the background project, this study constructed a theoretical analysis model of different tunnel-face damage modes. For river-crossing pipe-jacking projects with significant features such as a small overburden thickness and a water level above the surface, the analytical model for calculating the support pressure at the tunnel face makes the following assumptions:  
1. The project site strata are homogeneous, continuous, and isotropic. Tunnel-face damage is caused by sliding within the stratum soil body. The soil on the sliding surface obeys the Mohr-Coulomb failure criterion. 2. The support pressure on the tunnel face is uniformly distributed. Its vertical variation and the effect of seepage from the ground at the tunnel face are not considered. 3. The circular pipe-jacking tunnel face is simplified to a retaining wall of equal area and no thickness. 4. When the pipe-jacking tunnel face is damaged, there is a wedge-shaped block in front. The effect of friction on the contact surface of the wedge block is neglected, the wedge block does not undergo internal deformation, and the static equilibrium condition is satisfied. The force on the wedge-shaped block in front of the pipe-jacking tunnel face is as shown in Figure 2a. The meaning of each parameter in the figure is as follows: P is the  
Figure 1: Destructive modes: (**a**) active damage in chimney-like shear zones; (**b**) overall downward sliding active destruction; (**c**) passive destruction. support pressure on the tunnel face; P\({}_{\rm w}\) is the hydrostatic pressure exerted on the tunnel face; P\({}_{\rm n}\) is the vertical force acting on the top of the wedge block; G is the self-weight of the wedge block; T and N are the frictional force (upward direction for active damage and downward direction for passive damage) and the normal stress on the diagonal section of the wedge block, respectively; and T\({}^{\prime}\) and N\({}^{\prime}\), respectively, represent the frictional resistance (upward direction for active failure, downward direction for passive failure) and normal force on the sliding surfaces on both sides of the wedge-shaped block. The force characteristics of the wedge-shaped block were analyzed by taking the example of passive damage occurring on the pipe-jacking tunnel face. Equations (1) and (2) can be obtained from the conditions of an equilibrium of forces in the horizontal and vertical directions, respectively. \[{\rm P}={\rm Tcos}\ \alpha+2{\rm T^{\prime}cot}\ \alpha+{\rm Nsin}\ \alpha+{\rm P}_{\rm W} \tag{1}\]  
\[{\rm P}_{\rm n}+{\rm G}={\rm Ncos}\ \alpha-{\rm Tsin}\ \alpha+2{\rm T^{\prime}sin}\ \alpha \tag{2}\]  
Equation (3) can be obtained from the relationship between the friction force T and the normal force N on the sliding surface of the block. \[{\rm T}=\frac{{\rm cB}^{2}}{{\rm sin}\ \alpha}+{\rm Ntan}\ \varphi \tag{3}\]  
Here, c is the soil cohesion, kPa; \(\alpha\) is the angle of inclination of the sliding block, \({}^{\circ}\), \(\alpha=45^{\circ}+\frac{\varphi}{2}\); \(\varphi\) is the angle of internal friction of the soil, \({}^{\circ}\); G is the self-weight of the wedge block, \({\rm G}=\frac{{\rm B}^{3}\gamma_{s}}{{\rm 2 tan}\ \alpha}\), kN; \(\gamma_{\rm s}\) is the volumetric weight of soil, kN/m\({}^{3}\); and B is the wide of wedge block, m.  
Equation (4) can be obtained by combining Equations (1)-(3) and simplifying.