# A Partitioned Rigid-Element and Interface-Element Method for Rock-Slope-Stability Analysis
## 4 Numerical Example
### Stability Analysis of a Slope with Circular Slip Surface  
A circular slip surface is a common type of slip surface in slope stability analysis. The following numerical example is of a homogeneous slope with a circular slip surface. The schematic diagram of the circular slip surface is shown in Figure 14, and the slip surface equation is shown below:  
\[\frac{(x-40)^{2}}{1600}+\frac{y^{2}}{800}+\frac{(z-40)^{2}}{1600}=1 \tag{34}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Numerical Method** & **Single Iteration Time (s)** & **Total Time (s)** \\ \hline PRE-IE & \(1\) & \(5\) \\ PFE-IE & \(52/54/63/68/70\) & \(305\) \\ \hline \hline \end{tabular}
\end{table}
Table 7: Computation times of two numerical methods.  
Figure 14: Schematic diagram of the circular slip surface.  
Figure 13: Maximum and minimum y-direction displacements of the wedge on the XY cross section [27].  
The mesh model of the slope is shown in Figure 15. The model is divided into two rigid elements--the rotational sliding mass (gray) and the rock bed (blue)--and the interface elements (fuchsia) are on the circular slip surface between them.  
The material properties are listed in Table 8. The values of model parameters \(k_{1}\), \(n\), \(R_{f}\), and \(\varphi\) in Equation (31) are 4800, 0.56, 0.74, and 36\({}^{\circ}\), respectively. The tensile strength of the structural plane is 0.  
After using the PRE-IE method combined with SRM, the slope failure is shown to occur when the reduction coefficient is 1.464, and the failure model of slope is shown in Figure 16 (since the displacement is very small, it is magnified three times in order to make the failure model more obvious). It can be seen from the figure that the rotational sliding mass slides along the circular slip surface, which is consistent with the practical situation of slope failure.  
The safety factors calculated with different methods are shown in Table 9. The Felle-nius method does not consider the side force of the slice, so the safety factor calculated is significantly smaller than that of other methods. Both the Janbu method and the Bishop method take into account the side force between slices, and the Bishop method also considers the moment equilibrium, so the calculated safety factor is larger than that of the Sweden slice method. The above methods cannot satisfy horizontal force equilibrium, and the improved LEM proposed by [PERSON] [38] can satisfy all conditions of equilibrium and supposes all the slices are in limit equilibrium, so the maximum safety factor is calculated. The PFE-IE method regards the slope as an elastic body, while the PRE-IE method regards the slope as a rigid body. Neither methods needs to consider the distribution of the inter-slice forces, and each can satisfy all of the requirements for a statically admissible solution. Moreover, the kinematically admissible velocity discontinuities are permitted along the  
Figure 15: PRE-IE mesh model of the slope with a circular slip surface.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Material** & \(E/(\text{GPa})\) & \(v\) & \(c/(\text{kPa})\) & \(\varphi/(^{\circ})\) & \(\rho/(\text{kg/m}^{3})\) \\ \hline Rock mass & - & - & - & - & 2200 \\ Slip surface & 1 & 0.1 & 30 & 30 & 2000 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Material properties of the rock mass and slip surface.  
interface according to the nonlinear-contact iterative method. Therefore, the calculated safety factor is between the lower bound limit analysis and upper bound limit analysis, which is close to the practical condition. The above results prove that the PRE-IE method can effectively calculate the slope stability safety factor when the slip surface is circular.