# A Partitioned Rigid-Element and Interface-Element Method for Rock-Slope-Stability Analysis
## 4 Numerical Example
### Stability Analysis of the Symmetrical Wedge  
The stability analysis of a rock slope containing a symmetrical wedge is a classic three-dimensional limit equilibrium problem, which is often used to verify the correctness and rationality of a new numerical method. [PERSON] [37] and [PERSON] [27] have solved this problem using different methods. The overall model is a homogeneous rock slope composed of bedrock and a symmetric wedge, as shown in Figure 10. The structural model of the symmetric wedge is shown in Figure 11. The maximum height of the wedge is 64.89m, and its attitude is listed in Table 4. The left and right sides of the wedge are weak structural planes, on which the interface elements are built. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Surface** & **L-WSP** & **R-WSP** & **Top Surface** & **Front Surface** \\ \hline Dip angle/(°) & 45 & 45 & 10 & 60 \\ Dip direction/(°) & 115 & 245 & 180 & 180 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The attitude of the symmetric wedge. Figure 9: Safety factor search method. Both the bedrock and the wedge are considered rigid elements, so only their density needs to be considered. The material properties are listed in Table 5. The values of model parameters \(k_{1}\), \(n\), \(R_{f}\), and \(\varphi\) in Equation (31) are 4800, 0.56, 0.74, and 36\({}^{\circ}\). The tensile strength of the structural plane is 0. The displacement distribution of the symmetric wedge on the YZ cross-section is shown in Figure 12 when the slope failure occurs with the PRE-IE method. Since the wedge is regarded as a rigid body, the displacement can be decomposed into translational displacement along the intersection of the left and right structural planes and rotational displacement around the centroid of the wedge. The following observations can be made based on the figure. The wedge mainly slides along the intersection of the left and right structural planes. At the same time, due to gravity, the lower part of the wedge has a larger vertical displacement compared to the upper part, which causes the wedge to have a small rotation around the centroid. This displacement model is consistent with the practical situation of slope failure. Figure 11: Structural model of the symmetric wedge [27]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Material** & \(E\)/(**GPa**) & \(v\) & \(c\)/(**kPa**) & \(\varphi\)/(\({}^{\circ}\)) & \(\rho\)/(**kg/m\({}^{3}\)**) \\ \hline Rock mass & - & - & - & - & 2600 \\ Structural plane & 1 & 0.3 & 50 & 20 & 2000 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Material properties of the rock mass and structural plane. Figure 10: Rock slope model containing a symmetrical wedge [27]. The values of the safety factor are shown in Table 6. The analytical solution of the slope safety factor given by [PERSON] is 1.556. [PERSON] treats the slope as an elastic body and uses the PFE-IE method with SRM to obtain the safety factor of 1.554. In this paper, the safety factor 1.554 is calculated via the PRE-IE method with SRM, which uses the slope failure criterion proposed above. Figure 13 shows the maximum and minimum y-direction displacements of the wedge on the XY cross-section: (a) is the result calculated using the PRE-IE method proposed in this paper, and (b) is the result calculated using the PFE-IE method. Compared with the PFE-IE method, the PRE-IE method treats the wedge as a rigid body, and its elastic deformation was not taken into account; the maximum displacement increased by 3.98%, the minimum displacement increased by 5.45%, and the displacement distribution was the same. It can be seen from the above results that although there are some differences in the displacement calculated using the two methods, the safety factor is the same and very close to the analytical solution.