## 2 Theoretical Analysis of Tunnel-Face Support Pressure
### Pressure Calculation and Analysis of Optimal Support Pressure Value  
The method utilized in this study is the limit equilibrium method, which requires high conditions for the prototype. Therefore, the conditions of the model are simplified to be relatively simple. Mud-water-balanced pipe jacking uses a balancing medium, such as mud and water, to obtain a certain pressure in the working silo and thereby balance out the  
Figure 3: The trapezoidal wedge model in front of the pipe-jacking tunnel face.  
pressure of the underground water and soil layer in front of the tunnel face. The sum of soil and water pressures in the soil above the front of the pipe-jacking machine was assumed to be P\({}_{1}\). The corresponding tunnel-face support pressure was taken as P\({}_{2}\), and the sum of soil and water pressures in the bottom soil at the front of the pipe-jacking machine was assumed to be P\({}_{3}\). The corresponding tunnel-face support pressure was P\({}_{4}\), as illustrated in Figure 4. Theoretically, P\({}_{1}\) = P\({}_{2}\) and P\({}_{3}\) = P\({}_{4}\) when the mud-water-balanced jacking tunnel face is in equilibrium.  
In order to simplify the analysis and to highlight the main contradictions of the problem (in the actual construction process), the following assumptions were made: (1) the tunnel-face support pressure and resistance are trapezoidal in distribution, and the point of action is located at the center of gravity of the trapezoidal area; (2) the stratum in front of the pipe-jacking tunnel face is a semi-infinite body of homogeneous isotropy; and (3) the influence of the top pipe digging speed and friction between the cutter and the tunnel face are not considered.  
Based on the above assumptions, the tunnel-face support pressure calculation model is constructed, as shown in Figure 2b. The static soil pressure at the point of action of the ground pressure at the tunnel face is used as a datum. Soil and water sub-calculations are adopted, and Equations (8) and (9) are obtained to calculate the optimal value of the tunnel-face support pressure.  
\[\text{P}_{01}=[\text{K}_{0}\gamma_{s}(\text{H}_{\text{a}}+\text{H}_{\text{w}}) +\gamma_{\text{w}}\text{H}_{\text{w}}]\pi\text{D}^{2}/4 \tag{8}\]  
\[\text{P}_{02}=[\text{K}_{0}\gamma_{s}(\text{H}_{\text{b}}+\text{H}_{\text{c}}) +\gamma_{\text{w}}\text{H}_{\text{r}}]\pi\text{D}^{2}/4 \tag{9}\]  
Here, P\({}_{01}\) is the optimal support pressure for the tunnel face in the area without river water influence, kPa; P\({}_{02}\) is the optimal support pressure for the tunnel face in the area with river water influence, kPa; K\({}_{0}\) is the static earth pressure coefficient; \(\gamma_{\text{w}}\) is the volumetric weight of water, kN/m\({}^{3}\); H\({}_{\text{a}}\) is the distance from the underground water level to the ground surface, m; H\({}_{\text{b}}\) is the pipe-jacking burial depth, m; H\({}_{\text{c}}\) is the distance from the point of ground-pressure action on the pipe-jacking tunnel face to the top axis of the digging machine (which took the value of 2/3D), m; H\({}_{\text{r}}\) is the river depth, m; and H\({}_{\text{w}}\) is the distance from the point of soil pressure action to the underground water level, m.  
Figure 4: Mechanical equilibrium principle and calculation model of mud–water-balanced pipe-jacking tunnel face.