## 5 Exploring the Rupture Surface Forms of Soil Nail-Supported Structures
Four different internal friction angles (\(10^{\circ}\), \(20^{\circ}\), \(30^{\circ}\), and \(40^{\circ}\)) are analyzed for the rupture surface form of the soil nail-supported structure, as detailed in Table 2 and Figure 10.  
The following can be concluded from Table 2 and Figure 10:  
1. When \(\beta=80^{\circ}\), as the angle of internal friction (\(\varphi_{2}\)) continues to increase, the ratio of \(L/H\) gradually reduces, and the decrease is greater. The changes in the \(h/H\) ratio are reduced compared to in the previous case. 2. According to Table 2, under the premise of a certain slope angle of the pit or slope, different internal friction angles correspond to different rupture surface forms. In comparing Figure 2c,d, it can be seen that the form of slip fracture damage obtained through engineering experience is only the general shape of the rupture surface rather than all the forms of the slip fracture surface when \(L/H=0.3\) or \(0.4\). Therefore, when the soil nailing support structure is designed according to engineering experience, the length of soil nails may not meet the requirements [45]. This will lead to a significant reduction in the strength and safety of the soil nailing support structure, which can easily lead to engineering accidents. Therefore, in an actual project, when the slope angle has a certain value, it should be designed according to the corresponding rupture surface form of the internal friction angle of different soils, so as to ensure the safety and durability of the soil nailing support structure. (3) Through substituting \(\beta=70^{\circ}\) into Equation (22), the critical height at which damage to the soil nailing support structure occurs can be obtained as  
\[H=\frac{\overline{c}}{\gamma\tan(45^{\circ}-\frac{\varphi_{3}}{2})}\cdot f( \varphi_{3}) \tag{29}\] \begin{table}
\begin{tabular}{c c c c c} \hline \(\varphi_{2}\) & \(10^{\circ}\) & \(20^{\circ}\) & \(30^{\circ}\) & \(40^{\circ}\) \\ \hline \(f(\varphi_{2})\) & 5.87 & 5.7 & 5.75 & 6.12 \\ \(L/H\) & 0.38 & 0.28 & 0.21 & 0.14 \\ \(h/H\) & 0.34 & 0.35 & 0.35 & 0.33 \\ \hline \end{tabular}
\end{table} 
Table 2: Forms of rupture surfaces corresponding to different internal friction angles c (\(\beta=80^{\circ}\)). Figure 10: Geometric dimensions of failure surfaces corresponding to different internal friction angles \(\varphi_{2}\). (**a**) Graph of \(L/H\) and \(f(\varphi_{2})\) versus \(\varphi_{2}\); (**b**) graph of \(h/H\) and \(f(\varphi_{2})\) versus \(\varphi_{2}\). In combining this with the previously derived Equation (24), the values of _h/H_ and _L/H_ in this case can be obtained:  
\[\frac{h}{H}=\frac{2}{f(\varphi_{3})}\ ;\frac{L}{H}=[1-\frac{2}{f(\varphi_{3})}]. \sqrt{ka}-\cot 80^{\circ} \tag{30}\]  
From Equation (30), it can be seen that the rupture surface form of the slope is a function of the angle of internal friction \(\varphi_{3}\) when \(\beta\) is a constant. Four different internal friction angles (\(10^{\circ},20^{\circ},30^{\circ}\), and \(40^{\circ}\)) are analyzed for the rupture surface form of the soil nail-supported structure, as detailed in Table 3 and Figure 11.  
The following can be concluded from Table 3 and Figure 11:  
1. When the slope angle \(\beta\) = 70\({}^{\circ}\), both _L/H_ and _h/H_ start to decrease, while the angle of internal friction \(\varphi_{3}\) continues to increase;
2. A comparison of Table 3 with Figure 2c,d shows that the rupture surface form obtained from engineering experience differed significantly from the rupture surface form calculated in the present work. To better compare and analyze the variation rules of _L/H_ and _h/H_ in fracture surface forms under slope angles of different soil nailing structures and internal friction angles of different soils, the data in Tables 1-3 are sorted, and Figure 12 is obtained. Figure 12 shows the _L/H_ and _h/H_ curves of fracture surface forms under different working conditions.