# Analysis of Slope Stability in Ion-Adsorption Rare Earth Mine Under In Situ Leaching Condition
## 2 Methodology
### Stability Analysis Method  
The research methods of slope stability can be divided into five categories: limit equilibrium analysis method, numerical analysis method of stress and strain, reliability probability analysis method, graphical method, and modern analysis methods based on nonlinear science, such as neural network method, fuzzy mathematics method, etc. The limit equilibrium method is widely used because of its simple calculation and sufficient results to meet the of engineering needs. The analysis methods of the limit equilibrium method mainly include the Fellenius method, the simplified [PERSON] method, the Janbu Simplified method, the Spencer method, the Morgenstern-Price method, etc., which are mainly divided into many methods according to the shape of the slope slip surface and the assumption of the force between slices. Because the safety factor value of slope arc failure is calculated by the simplified Bishop method, the Morgenstern-Price method and Spencer method can reflect the actual state of slope stability and have the advantages of fast calculation speed and reliable results, which are adopted by the majority of scientific and technological personnel, the above three analysis methods are the main methods of this stability research and evaluation. The Bishop method is a method to modify the Swedish method, considering the effect of inter-strip force [32]. In this study, the safety factors estimated for all potential failures are determined using [PERSON]'s simplified method, which is the most common method recommended in the design practice. In the method, the sliding surface is assumed as a quarter circle, and the foot of a certain slope is taken as the origin. The safety factors evaluated by [PERSON]'s simplified method are written in Equations (2) and (3), respectively. \[F_{s}=\sum_{i=1}^{n}\frac{1}{m_{ai}}[W_{i}tan_{qi}+c_{i}l_{i}cos\alpha_{i}]/ \sum_{i=1}^{n}w_{i}sin\alpha_{i} \tag{2}\]  
\[m_{ai}=cos\alpha_{i}+tan\alpha_{i}sin\alpha_{i}/FS \tag{3}\]  
where \(w_{i}\) is the gravity of the soil slice \(i\), \(\varphi_{i}\) is the internal friction angle of soil slice \(i\) on the sliding surface; \(c_{i}\) is the cohesion of slice \(i\) in the sliding surface; \(l_{i}\) is the circular length of slice \(i\); \(\alpha_{i}\) is the included angle between the tangent of the point in slice \(i\), circular, and the horizontal line. Figure 3: Profile of the studied in situ leaching slope. The Morgenstern-Price method is a limit equilibrium method for slope stability analysis [33]. The Morgenstern-Price method considers that there are normal forces and tangential forces between two adjacent soil strips. It is assumed that there is a certain functional relationship between the two forces in the horizontal direction, which can be expressed as Equation (4). \[X=\lambda f(x)E \tag{4}\]  
where \(X\) is the tangential force between the soil strips; \(E\) is the normal force between soil strips; \(\lambda\) is an undetermined constant. When utilizing the Morgenstern Price method, the calculation result is the same as the simplified Bishop method. The Spencer method is a rigorous limit equilibrium analysis method proposed by [PERSON] in 1967 [34]. It adopts the form of total stress, satisfies the balance conditions of force and moment, and is applicable to any sliding surface. It assumes that the ratio of horizontal force to vertical force on both sides of the soil strip is constant, that is, the resultant force between each soil strip is parallel to each other. Then, the stability safety factor of the soil slope is solved, which is expressed as Equation (5). \[X_{i}/E_{i}=X_{i+1}/E_{i+1}=tan\theta \tag{5}\]  
where \(X_{i}\), \(X_{1+i}\) are the inter-strip forces in the vertical direction; \(E_{i}\), \(E_{1+i}\) are the normal inter-strip forces; \(\theta\) is the Angle of the force between the strips. Geo-Studio, renowned as a premier geotechnical analysis software worldwide, features the SEEP/W and SLOPE/W modules. The SEEP/W is used to analyze groundwater seepage and pore water pressure dissipation.