# Numerical Simulation of Tunnel Face Support Pressure in Layered Soft Ground
## 3 Numerical Simulation Results and Discussion
### Equivalent Friction Angle and Equivalent Cohesive Force of Composite Formation  
[PERSON] et al. [24] studied the influence of different soil strength parameters, the covering diameter ratio, and tunnel diameter on tunnel excavation under a uniform soil  
Figure 9: The influence of the change in friction angle on the displacement in group e.  
layer. On the basis of 140 numerical analyses, a new design formula was derived to calculate the collapse pressure of the working face required for tunneling above the water table. \[P=\gamma D\left(\frac{1}{8\sin\varphi}-0.12\right)-\mathrm{c}\left(\frac{1.1}{ \sin\varphi}-0.5\right) \tag{1}\]  
The limitations of this equation are as follows:  
1. When \(20^{\circ}\leq\varphi\leq 40^{\circ}\) and \(\mathrm{c}\geq 0\);
2. When \(D\leq 10m\) and \(C/D\geq 0.5\). This formula is only for use in a uniform formation. Although [PERSON] briefly discussed the situation in a composite formation, they did not give a specific formula. Based on the study of different composite strata, this section establishes a formula for correcting the support pressure of the excavation face according to the composite strata. First, assuming that there is an equivalent friction angle \(\varphi^{*}\) of composite formation with constant cohesion, \(\varphi^{*}\) can be equivalent to the combined effect of the friction angle of the upper soil and the lower soil, so that it can be equivalent to a uniform formation. Based on the simulated composite formation data and support force, the equivalent friction angle of the composite formation under different conditions is obtained through the reverse calculation of [PERSON]'s formula. When the upper soil changes, the five types of conditions are combined with the changes in the friction force of the upper soil, and the results are shown in Figure 10. It can be seen that when the cohesion and the friction force of the underlying soil are constant, the equivalent friction angle of different layering conditions changes linearly. Similarly, as can be seen from Figure 10, when the underlying soil mass changes, the equivalent friction angle of different layering conditions also presents a linear change. Therefore, the formula of the equivalent friction angle is given as follows:  
\[\varphi^{*}=N_{0}\varphi_{0}+N_{1}\varphi_{1}+N_{10} \tag{2}\]  
where \(N_{0}\), \(N_{1}\), and  
Assuming that the distance from the ground boundary line to the bottom of the tunnel is \(X\) and the tunnel diameter is \(D\), the relationship between \(N_{0}\), \(N_{1}\), and \(X/D\) is as shown in Figure 11. It can be seen that they have linear relations. According to the fitting results, we can derive the calculation formulas of \(N_{0}\) and \(N_{1}\):  
\[N_{0}=0.8\frac{X}{D} \tag{3}\]  
\[N_{1}=-0.73\frac{X}{D}+0.85 \tag{4}\]  
The value of \(N_{10}\) is related to the intercept in the figure above. After plugging in the values of \(N_{0}\) and \(N_{1}\), respectively, the value of \(N_{10}\) can be obtained:  
\[N_{10}=-1.45\frac{X}{D}+2.8 \tag{5}\]  
Then, consider the change in the cohesion force. According to the above, the cohesion force has a similar change law to the friction force. Similarly, the formula can be obtained by operating in accordance with the flow of friction force change for the case of cohesion change:  
\[c^{*}=M_{0}c_{0}+M_{1}c_{1}+M_{10} \tag{6}\]  
where \(M_{0}\), \(M_{1}\), and \(M_{10}\) are the equivalent coefficient of the cohesive force of the lower soil mass, the equivalent coefficient of the cohesive force of the upper soil mass, and the layered coefficient of the cohesive force of the soil mass, respectively.