## 3 Theoretical Analysis of Tunnel Portal Stability  
The soil body of the tunnel portal is subjected to the upper soil body load and ground load \(P\). When the sealing door is removed, the tunnel portal soil body equilibrium is broken and a sliding surface may be formed. Thus, damage occurs, as shown by the dashed line in Figure 4. This sliding surface is assumed to be a circular arc, with the top of the tunnel portal as the center and the diameter \(D_{1}\) of the tunnel portal as the radius. The sliding moment is greater than the resistance moment on the circular arc surface. To enhance the stability of the soil body in the tunnel portal area, it is necessary to reinforce the soil body and enhance the resistance moment on the sliding surface. The sliding moment of the soil body consists of three parts, which are the sliding moment caused by the ground load \(P\), the sliding moment caused by the self-weight of the upper soil body, and the sliding moment caused by the self-weight of the sliding soil body. The resistance moment of the soil body is composed of the resistance moment of the soil body before reinforcement and the increasing resistance moment after reinforcement of the soil body. The increased resistance moment after reinforcement of the soil body is especially critical for tunnel portal stability. These expressions can be computed as followed. \[\text{Sliding moment }M,\qquad M=M_{1}+M_{2}+M_{3} \tag{1}\]  
\(M_{1}\)(kN\(\cdot\)m) is the sliding moment due to ground load \(P\);  
\(M_{2}\)(kN\(\cdot\)m) is the sliding moment caused by the self-weight of the upper soil;  
\(M_{3}\)(kN\(\cdot\)m) is the sliding moment caused by the sliding soil body. \[\text{Sliding resistance moment }M_{d},\qquad M_{d}=M_{r}+\Delta M_{r} \tag{2}\]  
\(M_{r}\)(kN\(\cdot\)m) is the resistance moment before soil reinforcement;  
\(\Delta M_{r}\)(kN\(\cdot\)m) is the increased resistance moment after soil body reinforcement. Increased resistance moment after soil body reinforcement \(\Delta M_{r}\):  
\[\Delta M_{r}=\Delta c\cdot\theta\cdot D_{1}^{2}=\Delta c\cdot\arcsin\frac{L} {D_{1}}\cdot D_{1}^{2} \tag{3}\]  
\(\Delta c\)(kPa) is the increased value of cohesion after soil body reinforcement;  
\(\theta\) is the center angle of the circle corresponding to the reinforcement body, \(\theta\leq\pi/2\);  
\(D_{1}\)(m) is the diameter of the tunnel portal;  
\(L\)(m) is the longitudinal length of the rectangular freezing soil mass. Figure 3: The schematic diagram of the Wuhu River-Crossing Tunnel end reinforcement. The equilibrium condition for the stable soil mass:  
\[KM=M_{d} \tag{4}\]  
\(K\) is the slip stability coefficient. The relationship between slip stability coefficient \(K\) and \(\theta\) is as follows. \[K=\frac{M_{r}}{M}+\frac{\Delta c\cdot D_{1}^{2}}{M}\cdot\theta \tag{5}\]  
When the rectangular freezing soil mass longitudinal length \(L\) is smaller than the diameter of tunnel portal \(D_{1}\), i.e., \(\theta\leq\pi/2\), the center angle of circle \(\theta\) increases with the increase in longitudinal length \(L\). According to Equation (5), the stability coefficient of the tunnel portal also increases with increase in the center angle of the circle \(\theta\). When the longitudinal length \(L\) is greater than the diameter of tunnel portal \(D_{1}\), the center angle of circle \(\theta\) equal \(\pi/2\) and the slip stability coefficient will be constant. From the Equations (3) and (5), it is known that with the reinforcement width increasing, the resistance moment of the soil body will not increase, and the stability coefficient of the tunnel portal will remain unchanged. Compared with the freezing soil mass length \(L\), the influences of freezing soil mass width on the resistant moment are very small. The influences of freezing soil mass width were ignored, and the width was kept the same in the paper. From the above analysis, it is clear that the magnitude of the sliding resistance moment is determined by the central angle \(\theta\).