# A Partitioned Rigid-Element and Interface-Element Method for Rock-Slope-Stability Analysis
## 5 Conclusions  
In this paper, the PRE-IE method is proposed for analyzing the stability of rock slopes with joints, cracks, and weak interlayers. The structure is modeled as consisting of several rigid bodies and discontinuous structural planes, which are divided into partitioned rigid elements and interface elements to couple contact forces and displacements on the node pairs. The contact force of node pairs and the displacement of the rigid body centroid are taken as mixed variables, and the governing equations of PRE-IE are established using the Lagrange multiplier method based on the principle of minimum potential energy  
Figure 16: Failure model of the slope after the PRE-IE method is used combined with SRM (reduction coefficient is 1.464). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & **PRE-IE** & **Fellenius** & **Janbu** & **Bishop** & **PEF-IE** & **Improved LEM [38]** \\ \hline Safety factor & 1.464 & 1.15 & 1.25 & 1.30 & 1.42 & 1.48 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Values of safety factors from different methods. These equations are then solved using the nonlinear contact iterative method and the incremental method. * By incorporating the advantages of the PFE-IE method, the PRE-IE method performs nonlinear iteration only on potential discontinuous structural planes, significantly improving computational efficiency while minimally affecting calculation accuracy. * The criterion for slope failure, which considers the failure of all contact node pairs, provides an objective measure and eliminates the potential impact of personal bias on safety factor calculations. * Two numerical examples are utilized to verify the correctness and validity of the PRE-IE method. The results demonstrate that the calculated slope failure pattern is consistent with real situations. * Through a comparative analysis of the safety factor calculated using the PRE-IE method and those obtained from other different methods, it is shown that the PRE-IE method, in conjunction with the SRM, is well suited for analyzing rock slope stability problems. * While the PRE-IE method presented in this paper focuses on static stability problems, previous studies have applied the PFE-IE method to dynamic problems. In future studies, by modifying the equations of PRE-IE from static equilibrium to dynamic equilibrium and enhancing the nonlinear contact iterative method, the PRE-IE method can be extended to analyze the dynamic stability of structures such as slopes or dams. Furthermore, exploring the integration of the PRE-IE method with other innovative approaches for monitoring the stability of reservoir bank slopes and ensuring the structural safety of dams holds significant potential for future research. Conceptualization, T.S. and T.L. ; methodology, T.S. ; software, T.S. and H.Q. ; validation, T.S. and H.Q. ; formal analysis, T.S. ; investigation, T.S. and X.L. ; resources, T.L. and X.L. ; data curation, T.S. ; writing--original draft preparation, T.S. ; writing--review and editing, T.S. ; visualization, T.S. ; supervision, T.L. ; project administration, T.L. ; and funding acquisition, T.L. All authors have read and agreed to the published version of the manuscript. This research was financially supported by the National Key Research and Development Plan (SN: 2022 YFC3005403), the National Natural Science Foundation of China (SN: 52009035) and the Science and Technology Plan Project of Jiangxi Provincial Department of Water Resources (SN: 202224 ZDKT14). Not applicable. The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results. The following abbreviations are used in this manuscript: