## 3 Stability Analysis of Ridge Slope  
Numerical analysis was performed on the established ridge slope model to study the impact of ridge line inclination \(\alpha\), flank slope angle \(\beta\), and slope height \(H\) on the ridge slope destruction mode. The strength reduction method was used to calculate the displacement and plastic zone of the slope. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(H\) (m) & \(\beta\)(\({}^{\circ}\))\(\bullet\)(\({}^{\circ}\))  
Many test results showed that geotechnical materials followed unified intensity theory when the intermediate main stress coefficient (\(b\)) was 0.5 [37]. Based on the mechanical model and mathematical expression of the unified strength theory, a corresponding user interface subroutine UMAT (User-defined Material Mechanical Behavior) was written in Fortran language according to the basic elements of elastoplastic constitutive relation proposed in classical elastoplastic mechanics, which was connected to ABAQUS software. The main stress coefficient was 0.5; this was used to calculate the safety factor for the strength reduction method. The software environment used in this study was Windows 8, the hardware environment was Intel(R) Core (TM)2 Duo CPU, NVIDIA GeForce GT240/ATI HD4670, and the RAM was 8 GB. The version of ABAQUS was 6.13. In this study, the strength reduction method was used to study the stability of the ridge slope. In general, the strength envelope (\(l\)) of the ridge slope rock was in a state of separation from the stress circle of the current state, and the ridge slope was in a stable state with relatively small deformation. With the gradual increase in the reduction coefficient, when the strength envelope (\(l_{0}\)) was tangent to the Mohr circle of rocks on the ridge slope, the ridge slope was in the limit equilibrium state. With the further increase in the reduction factor, as shown in Figure 4, the strength envelope (\(l_{t}\)) intersected the stress circle of the current state. At this time, instability and failure of the ridge and slope had occurred, resulting in a large displacement. Through a strength reduction method, the ridge and slope became unstable. The ridge slope underwent a large displacement. ABAQUS software was used to simulate the stability of the ridge slope under different conditions. During the calculation process of ABAQUS software, the \"Nlgeom\" large deformation analysis was initiated in the analysis step. When the strength reduction was not performed or the strength reduction factor was small, the ridge slope was in a relatively stable state in which the displacement of the ridge slope was only a few millimeters. However, the stability of the ridge slope under different conditions cannot be obtained. Therefore, the displacements of ridge slope with a reduction factor of 30 was selected to study the stability of the ridge slope. [PERSON] et al. [38] also adopted ABAQUS software and obtained the displacement of the slope top up to 10.37-11.28 m using a strength reduction method to study the displacement field when the slope instability failure occurred. In unified strength theory, the failure mode of rock mass can be determined by the following equation:  
\[\left\{ \begin{array}{l}f=\frac{1}{1+b}\sigma_{1}+\frac{b}{1+b}\sigma_{2}-N_{ \varphi}\sigma_{3}+2c\sqrt{N_{\varphi}},\sigma_{2}\leq\frac{1-\sin\varphi}{2} \sigma_{3}+\frac{1+\sin\varphi}{2}\sigma_{1}\\ f^{\prime}=\sigma_{1}-\frac{b}{1+b}N_{\varphi}\sigma_{2}-\frac{1}{1+b}N_{ \varphi}\sigma_{3}+2c\sqrt{N_{\varphi}},\sigma_{2}\geq\frac{1-\sin\varphi}{2 }\sigma_{3}+\frac{1+\sin\varphi}{2}\sigma_{1}\\ f^{\prime}=\sigma_{3}-\sigma_{t}\end{array} \right. \tag{1}\]  
Figure 4: Principle of strength reduction method. where \(b\) is the failure criterion selection parameter, reflecting the influence of intermediate principal stress on material failure. \(N_{\varphi}=\frac{1+\sin\varphi}{1-\sin\varphi}\), \(\varphi\) is the friction angle of rock material, \(\sigma_{t}\) is the tensile strength of a rock material.