## 3 Theoretical Analysis of Tunnel Portal Stability
In addition, the angle affects the stability coefficient of the tunnel portal. Figures 5 and 6 show the theoretical calculation model diagram and the trapezoidal freezing soil mass local enlargement diagram, respectively. The effects of upper side \(L_{1}\) and lower side \(L_{2}\) of trapezoidal freezing soil mass on the central angle \(\theta\) are analyzed as follows. The trapezoidal freezing soil mass, which is shown in Figure 6a, is separated from Figure 5. The dimensions in Figure 6a have the following relationship. \[\frac{d}{2d+D_{1}}=\frac{a_{1}}{L_{2}-L_{1}} \tag{6}\]  
\[a=L_{1}+a_{1} \tag{7}\]  
\[\text{Sliding moment }M,\qquad M=M_{1}+M_{2}+M_{3} \tag{1}\]0  
\[\text{Sliding moment }M,\qquad M=M_{1}+M_{2}+M_{3} \tag{1}\]1 The stability calculation model of the tunnel portal with rectangular freezing soil mass. The part in the red box of Figure 6a was isolated, as shown in Figure 6b. The tunnel portal with trapezoidal freezing soil mass was analyzed separately. The dimensions in Figure 6b have the following relationship. \[\text{Sliding moment }M,\qquad M=M_{1}+M_{2}+M_{3} \tag{1}\]2  
\[\text{Sliding moment }M,\qquad M=M_{1}+M_{2}+M_{3} \tag{1}\]3  
\[\text{Sliding moment }M,\qquad M=M_{1}+M_{2}+M_{3} \tag{1}\]4  
Substitute Equations (13) and (14) into Equation (15), Equation (16) is obtained. \[\text{Sliding moment }M,\qquad M=M_{1}+M_{2}+M_{3} \tag{1}\]5  
Substitute Equations (10), (11), and (12) into Equation (16), the center angle of the circle can be computed by Equation (17). \[\text{Sliding moment }M,\qquad M=M_{1}+M_{2}+M_{3} \tag{1}\]6  
\(L_{1}\)(m) is the upper side length of the trapezoidal freezing soil mass;  
\(L_{2}\)(m) is the lower side length of the trapezoidal freezing soil mass;  
\(d\)(m) is the length from upper side of the trapezoidal freezing soil mass to the top of the tunnel;  
\(D_{1}\)(m) is the diameter of the tunnel portal, \(D_{1}\) = 14.5 m.  
Firstly, the lower side's length \(L_{2}\) of the trapezoidal freezing soil mass is 4 m, while the upper side's length \(L_{1}\) ranges from 2 m to 4 m. The upper side length of the trapezoidal soil mass is 2 m, while the lower side's length \(L_{2}\) ranges from 2 m to 4 m. Figure 7 shows the value of the central angle \(\theta\) for different \(L_{1}\), and \(L_{2}\). The relationship between \(\theta\) and \(L_{1}\) can be obtained by linear fitting with \(\theta=0.012105L_{1}+0.23107\) when the correlation coefficient is above 0.99. The relationship between \(\theta\) and \(L_{2}\) can be obtained by linear fitting with \(\theta=0.05845L_{2}+0.02143\), and the correlation coefficient is also above 0.99. From the above fitting formulas, it can be seen that the central angle \(\theta\) increases with the increase in \(L_{1}\) and \(L_{2}\), respectively. The increasing ratio of \(\theta\) with the \(L_{2}\) increasing is 4.83 times greater than that with the \(L_{1}\) increasing. It means that the central angle \(\theta\) is more sensitive to \(L_{2}\) than to \(L_{1}\), and that \(L_{2}\) has greater influences on the central angle \(\theta\). From  
\[\text{Sliding moment }M,\qquad M=M_{1}+M_{2}+M_{3} \tag{1}\]7 The value of the central angle \(\theta\) for different \(L_{1}\), \(L_{2}\). Equation (5), it is known that the tunnel portal stability coefficient is proportional to the central angle \(\theta\). Therefore, increasing \(L_{2}\) is more effective than increasing \(L_{1}\) for tunnel portal stability. To improve the reinforcement effect, the upper side length of the freezing soil mass can be reduced and the lower side length can be increased. Thus, the trapezoidal freezing soil mass can be used to reinforce the tunnel portal, and its reinforcement effects are superior to the rectangular freezing soil mass. From the above analysis, it is known that the freezing reinforcement scheme of the Wuhu River-Crossing Tunnel can be optimized. The optimized scheme is as follows. The upper side length of the freezing soil mass is reduced (from 3 m to 2 m), while the lower side length freezing soil mass is increased (from 3 m to 4 m).