# Application of the Improved Entry and Exit Method in Slope Reliability Analysis
## 3 Limit Equilibrium Method
### Illustrative Example 2: \(c\)-\(\varphi\) Slope  
Example 2 is a \(c\)-\(\varphi\) slope used by several scholars to validate the application of their method [18; 37; 38; 39]. The same example is reanalyzed herein to verify the feasibility of the approach. The soil parameters of Example 2 are shown in Table 4, and Figure 7 shows the geometry of Example 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **Failure Probability \(P_{f}\)** & **Source** \\ \hline MCS (100,000) & 7.60 \(\times\) 10\({}^{-2}\) & [18] \\ MRSM (500,000) & 7.90 \(\times\) 10\({}^{-2}\) & [36] \\ MCS (10,000) & 7.73 \(\times\) 10\({}^{-2}\) & [37] \\ RSSs+MCS (10,000) & 7.50 \(\times\) 10\({}^{-2}\) & [38] \\ MCS (50,000) & 7.84 \(\times\) 10\({}^{-2}\) & This article (\(21\times 21\times 11\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Failure probabilities of Example 1. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Calculation** & \multicolumn{5}{c}{**Reduced Searching Range (Unilateral)**} & \multicolumn{3}{c}{**Traditional Method**} \\ \cline{2-7}
**Results** & **1D** & **2D** & **3D** & **4D** & **5D** & **6D** & **(Target Value)** \\ \hline Average \(F_{s}\) & 1.2512 & 1.2495 & 1.2491 & 1.2488 & 1.2487 & 1.2486 & 1.2482 \\ \(P_{f}\) & 0.0894 & 0.0907 & 0.0912 & 0.0914 & 0.0914 & 0.0915 & 0.0917 \\ Number of slip surfaces (simulation once) & 5580 & 9764 & 20,476 & 40,788 & 73,772 & 122,500 & 269,001 \\ Calculation time (\(h\)) & 0.54 & 0.97 & 2.08 & 4.09 & 7.23 & 11.67 & 25.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Effects of reduced searching range on the calculated failure probabilities. The number of random field samples adopted to perform the Monte Carlo simulation in Example 2 is 50,000. In Table 5, the failure probability acquired by applying the traditional entry and exit approach is \(1.81\times 10^{-2}\), which is reasonably close to the calculation results of other scholars [18; 37; 38; 39]. The minimum relative difference between the calculated results in this article and those from other scholars is about 3.2%. The failure probability of Example 2 is calculated by applying the traditional entry and exit approach and the improved entry and exit approach with the same search accuracy. The computation outcomes are shown in Table 6. According to Table 6, failure probability increases with the increasing reduced searching range. When the reduced searching range is equal to unilateral \(2D\), the calculated failure probability by applying the improved entry and exit method is close to the target value. The relative difference between the failure probabilities obtained by applying the traditional entry and exit method (\(81\times 81\times 41\) trial slip surfaces is analyzed) and the improved entry and exit method (the reduced searching range is equal to unilateral \(2D\)) is 2.0%. When the reduced searching range is equal to unilateral \(4D\), the failure probability acquired by applying the improved entry and exit method is almost equal to that obtained by applying the traditional entry and exit approach. Furthermore, when applying the improved entry and exit approach (the reduced searching range is equal to unilateral \(2D\)) to decide the critical slip surfaces, the number of slip surfaces (9764 trial slip surfaces) is about 3.6% of the traditional entry and exit method (269,001 trial slip surfaces) in each simulation. When using the improved entry and exit method (the reduced searching range is equal to unilateral \(4D\)) to decide the critical slip surfaces, the number of slip surfaces (40,788 trial slip surfaces) is about 15% of the traditional entry and exit method (269,001 trial slip surfaces) in each simulation.