## 5 Exploring the Rupture Surface Forms of Soil Nail-Supported Structures  
The calculations in the previous subsection resulted in two more succinct scaling relationships, which are detailed in Equation (24), so that the form of the rupture surface when damage occurs to the soil nailing support structure at different slope angles can be analyzed. (1) Through substituting \(\beta=90^{\circ}\) into Equation (22), the critical height at which damage to the soil nailing support structure occurs can be obtained as  
\[H=\frac{\overline{c}}{\gamma\tan(45^{\circ}-\frac{\varphi_{1}}{2})}.f(\varphi _{1}) \tag{25}\]  
where \(f(\varphi_{1})=3(1-\sin\varphi)+\sqrt{9\sin^{2}\varphi-2\sin\varphi+5}\). In combining this with Equation (24), which had been derived earlier, we obtain  
\[\frac{h}{H}=\frac{2}{f(\varphi_{1})};\;\frac{L}{H}=[1-\frac{2}{f(\varphi_{1}) }].\sqrt{ka} \tag{26}\]  
From Equation (26), it can be seen that the rupture surface form of the slope is a function of the angle of internal friction (\(\varphi_{1}\)) when \(\beta\) has a constant value. Four different internal friction angles (\(10^{\circ}\), \(20^{\circ}\), \(30^{\circ}\), and \(40^{\circ}\)) were used to analyze the rupture surface form of the soil nail-supported structure, as detailed in Table 1 and Figure 9.  
The following can be concluded from Table 1 and Figure 9:  
1. As the angle of internal friction \(\varphi_{1}\) continues to increase, the rupture surface extends to the top of the pit or slope, thereby reducing the horizontal distance from the edge of the soil nailing support structure. Moreover, the tension cracks in the upper part of the soil nailing support structure gradually increase. 2. When \(\varphi_{1}=30^{\circ}\), \(L/H=0.29\) and \(h/H=0.51\). Under these conditions, the value of \(L\) is very close to that of \(L\) in the rupture surface form of the \"Methodology of Cheng Liangkui\" (mentioned in Figure 2c). However, \(h/H=0.75\) (given in Figure 2c), which is much different from that calculated using the method of the present work. 3. When \(\varphi_{1}=20^{\circ}\), Table 1 shows that \(L/H=0.41\). Under these conditions, the value of \(L\) is very close to the value of \(L\) in the form of rupture surface of the \"Methodology of Beijing University of Technology\" (mentioned in Figure 2d). This indicates that with the present method, when \(\theta=45^{\circ}+20^{\circ}/2=55^{\circ}\), the distance from the top edge of the pit or slope to the rupture surface extending to the horizontal ground is close to \(0.4H\), and regardless of the internal friction angle, the value of \(L\) is \(0.4H\). 2. Through substituting \(\beta=80^{\circ}\) into Equation (22), the critical height at which damage to the soil nailing support structure occurs can be obtained as \[H=\frac{\overline{c}}{\gamma\tan(45^{\circ}-\frac{\varphi_{2}}{2})}.f(\varphi _{2})\] (27) In combining this with the previously derived Equation (24), the values of \(h/H\) and \(L/H\) in this case can be obtained: \[\frac{h}{H}=\frac{2}{f(\varphi_{2})};\frac{L}{H}=[1-\frac{2}{f(\varphi_{2})} ].\sqrt{ka}-\cot 80^{\circ}\] (28) \begin{table}
\begin{tabular}{c c c c c} \hline \(\varphi_{1}\) & \(10^{\circ}\) & \(20^{\circ}\) & \(30^{\circ}\) & \(40^{\circ}\) \\ \hline \(f(\varphi_{1})\) & 4.70 & 4.29 & 4.02 & 3.80 \\ \(L/H\) & 0.48 & 0.41 & 0.29 & 0.21 \\ \(h/H\) & 0.43 & 0.47 & 0.51 & 0.53 \\ \hline \end{tabular}
\end{table} 
Table 1: Forms of rupture surfaces corresponding to different internal friction angles c (\(\beta=90^{\circ}\)). Figure 9: Geometric dimensions of failure surfaces corresponding to different internal friction angles \(\varphi_{1}\). (**a**) Graph of \(L/H\) and \(f(\varphi_{1})\) versus \(\varphi_{1}\); (**b**) graph of \(h/H\) and \(f(\varphi_{1})\) versus \(\varphi_{1}\). From Equation (28), it can be seen that the rupture surface form of the slope is a function of the angle of internal friction \(\varphi_{2}\) when \(\beta\) is a constant value.