# On Two Approaches to Slope Stability Reliability Assessments Using the Random Finite Element Method
## 5 Slope Reliability Assessment--The Two Approaches
### Approach II
On the other hand,

Figure 9:0 Figure 9:1 Figure 9:2

where the derivation makes use of the generic form of Equation (12), i.e., for \(F_{R}\). That is, the reliability (i.e., \(1-p_{f}\)) can be assessed by letting \(F_{R}=F_{d}^{\text{base}}/F_{d}^{\text{target}}\) in Equation (9) and keeping \(\mu_{\ln F_{R}}\) and \(\sigma_{\ln F_{R}}\) as the values based on the base case. Therefore, the conclusions in this paper can be considered general for the two most commonly used distribution types, i.e., normal and lognormal distribution. ## 6 Conclusions

This paper focuses on the two approaches in RFEM assessments of slope reliability. A comprehensive analysis was conducted to explain the often confusing methods of analysis. Some researchers use one approach while others use the other and it is difficult to understand the second approach (e.g., how the scaling works) and even more so to link to the first approach in some cases. Based on an auto-search algorithm of the factor of safety in the strength reduction method, and a radiation scan method to locate the critical sliding surface and to quantify the potential sliding volume, the paper demonstrates that the two approaches are actually equivalent and bridges the gap in understanding the two approaches that are usually used by different researchers in isolation. The merits and disadvantages of each approach are highlighted. The unification where the two approaches meet is explained. That is, for a sufficiently small step size of the global factor of safety, the results of approach I are moving close to those of approach II, which are smooth for a sufficiently large number of realisations. However, when the step size becomes small, the advantage of Approach I being computationally efficient is compromised. Efficient implementations of both approaches are provided in the form of pseudocodes, as different combinations of parameters are usually required such as different coefficients of variation and different site spatial variability scenarios represented by different scales of fluctuation. Although the investigations and discussions are presented mainly in the case of a normal distribution of the realised factor of safety, the applicability of the underlying reasoning is also extended to a log-normal distribution of the realised factor of safety. Note that the reason of using a range of global factor of safety, \(F_{d}\) (i.e., corresponding to different mean soil properties), for the two approaches, is not because the mean soil property is a variable for a particular site, as it is not. Rather, it is meant for a range of characteristic values to be used for design, which could vary from designer to designer. That is, different designers may use different \(F_{d}\) to get their characteristic values. **Author Contributions:** Conceptualisation, Y.L. ; methodology, Y.L. ; software, Y.L. ; validation, C.Q., Y.L. and Z.F. ; formal analysis, C.Q. ; investigation, Y.L. ; resources, Z.F. ; data curation, Z.L. ; writing--original draft preparation, Y.L. ; writing--review and editing, Z.F. ; visualisation, Z.L. ; supervision, Y.L. ; project administration, Y.L. ; funding acquisition, Y.L. **Funding:** This research was funded by the National Natural Science Foundation of China, grant number 41807228; the Fundamental Research Funds for the Central Universities, grant number 2652017071; and the Open Foundation of Key Laboratory of Failure Mechanism and Safety Control Techniques of Earth-Rock Dam of the Ministry of Water Resources, grant number YK319004. The last author appreciates the supports of the National Key R&D Program of China (2017 YFC0405104) and the National Natural Science Foundation of China (51979173). **Conflicts of Interest:** The authors declare no conflicts of interest. ## Abbreviations

The following abbreviations are used in this manuscript: