## 3 Data and Methods
### Numerical Simulation Software and Steps
#### 3.2.1 The Basic Principle of Limit Equilibrium Method  
The traditional Limit Equilibrium Method (LEM) is often a simplified approach based on the assumption that the safety factor (K) of the slope can be determined. When the shear strength parameters of the slope are reduced by a factor of FFF (the slope safety factor), a critical slip surface, representing the most dangerous potential sliding surface, is assumed to exist within the slope under limiting equilibrium conditions. LEM typically assumes that failure occurs along a predefined sliding surface within the rock or soil mass. Based on the static equilibrium conditions of the sliding mass and the Mohr-Coulomb failure criterion, the method evaluates the likelihood of sliding along this surface, quantified by the safety factor or the probability of failure. Multiple potential sliding surfaces are systematically analyzed in the same manner to calculate the corresponding safety factors or failure probabilities. The sliding surface with the smallest safety factor or highest failure probability is identified as the most likely failure surface. Commonly used LEM approaches include the Swedish slice method, [PERSON]'s method, [PERSON]'s method, and the Morgenstern-Price (M-P) method. In this stability analysis, the M-P method was employed for calculations. The calculation diagram is shown in Figure 2. \begin{table}
\begin{tabular}{c c} \hline
**Geotechnical Parameters** & **Numerical Value** \\ \hline Heavy \(\gamma\) (kN/m\({}^{3}\)) & 16.7 \\ Cohesive force C (kPa) & 8 \\ Internal friction angle \(\Phi\) (\({}^{\circ}\)) & 9 \\ relative density & 2.7 \\ Plastic limit (\(\omega\) p/\(\%\)) & 27.3 \\ Liquid limit (\(\omega\) l/\(\%\)) & 56.1 \\ Plasticity index (Ip) & 28.8 \\ Original soil dry density (\(\rho\) d/g/cm\({}^{3}\)) & 1.67 \\ Undisturbed soil moisture content (\(\omega\)/\(\%\)) & 54.2 \\ \hline \end{tabular}
\end{table} 
Table 1: Basic physical and mechanical parameters of surface fill in the dump. Figure 2: M–P method calculation diagram. [PERSON] and [PERSON] proposed a general approach, assuming that:  
\[X_{i}/E_{i}=\tan\beta=f_{0}\left(x\right)+\lambda f_{1}\left(x\right) \tag{1}\]  
Here, \(\lambda\) is to be determined, and \(f_{1}(x)\) is an artificially assumed function (\(f_{1}(x)=kx+m\), where \(k\) and \(m\) are constants). 1. The deadweight of the soil strip \(\Delta W_{i}\), its magnitude, position and direction of action point are known. 2. The normal stress at the bottom of the strip is \(N_{i}\), and the tangential stress is \(T_{i}\). Assume that \(N_{i}\) and \(T_{i}\) act at the midpoint of the sliding surface, and their magnitudes are unknown. 3. The normal forces on both sides of the soil strip are \(E_{i}\) and \(E_{i}+\Delta E_{i}\), and the vertical shear forces are \(X_{i}\) and \(X_{i}+\Delta X_{i}\). 4. \(\Delta Q_{i}\) is the horizontal inertial force (i.e., horizontal seismic force), and the distance between its action point and the soil strip is \(h_{ei}\). 5. The vertical load \(q\Delta x\) on the slope surface. 6. The angle between the normal line (i.e., the radius of the arc) of the sliding surface of soil strip \(i\) and the vertical line is \(\alpha_{i}\); the cohesion of the soil on the sliding surface is \(c\), and the internal friction angle is \(\varphi\). Horizontal force balance:  
\[T_{i}\cos\alpha_{i}-N_{i}\sin\alpha_{i}=\Delta Q_{i}-\Delta E_{i} \tag{2}\]  
Vertical force balance:  
\[T_{i}\sin\alpha_{i}+N_{i}\cos\alpha_{i}=\Delta W_{i}+\Delta V_{i}-\Delta X_{i} \tag{3}\]  
Satisfy the limit equilibrium condition:  
\[T_{i}=c_{e}\Delta xsec\alpha_{i}+N_{i}\tan\varphi_{e} \tag{4}\]  
Eliminating \(T_{i}\) and \(N_{i}\), we can obtain:  
\[\tan\left(\varphi_{e}-\alpha_{i}\right)=c_{e}\;xsec^{2}\;\alpha_{i}+\left( \Delta W_{i}+\Delta V_{i}\right)(tan\varphi_{e}-\tan\alpha_{i})-\Delta Q_{i} \left(1+\tan\varphi_{e}\tan\alpha_{i}\right) \tag{5}\]  
When \(\Delta x\to 0\):  
\[