# Application of the Improved Entry and Exit Method in Slope Reliability Analysis
## 3 Limit Equilibrium Method
### Discussion on Reduced Searching Range  
This section provides the number of potential critical sliding surfaces (with the global minimum safety factor) that were omitted due to the reduced searching range during the second search. Its primary purpose is to discuss the fluctuation degree of the difference value in positions between potential critical sliding surfaces and critical sliding surfaces under two different search accuracies. This also fundamentally validates the result in the example analysis that a larger reduced searching range leads to a more accurate estimation of failure probability.  
3.1 The Difference of the Location between Critical Slip Surface and Potential Critical Slip Surface  
The traditional entry and exit method is used to analyze Example 1 and Example 2. The slip surface with the lowest safety factor calculated by analyzing \(81\times 81\times 41=269\),001 trial slip surfaces is the critical slip surface. The slip surface with the lowest safety factor calculated by analyzing \(21\times 21\times 11=4851\) trial slip surfaces is illustrated by using the term \"potential critical slip surface\". The number of random field samples adopted to perform the Monte Carlo simulation in the two examples is 1000. The potential critical slip surface key point (key entry point, key exit point, and key tangent line) positions are subtracted from the critical slip surface key points (key entry point, key exit point and key tangent line) positions, and the difference value is obtained, as shown in Figure 8.  
In the calculation results of Example 1 in Figure 8, there are 123 exit points, 135 entry points, and 36 tangent lines whose difference values exceeded the length of \(2D\). There are 39 exit points, 42 entry points, and 22 tangent lines whose difference values exceeded the length of \(4D\). In the calculation results of Example 2 in Figure 8, there are 163 exit points, 1 entry point, and 3 tangent lines whose difference values exceeded the length of \(2D\). There are only 32 exit points whose difference values exceeded the length of \(4D\). The results indicate that most difference values do not exceed the range of \(2D\), and fewer exceed \(4D\). This means that when using the improved entry and exit method, compared with the reduced searching range based on unilateral \(2D\), the critical slip surface obtained for unilateral \(4D\) is closer to the accurate results of the global range search situation. However, the search results for the critical slip surface on the basis of unilateral \(2D\) are also good.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Calculation**} & \multicolumn{4}{c}{**Reduced Searching Range (Unilateral)**} & \multicolumn{2}{c}{**Traditional Method**} \\ \cline{2-7}  & \multicolumn{1}{c}{\(\mathbf{1}\)**\(\mathbf{D}\)**} & \multicolumn{1}{c}{\(\mathbf{2}\)**\(\mathbf{D}\)**} & \multicolumn{1}{c}{\(\mathbf{3}\)**\(\mathbf{D}\)**} & \multicolumn{1}{c}{\(\mathbf{4}\)**\(\mathbf{D}\)**} & \multicolumn{1}{c}{\(\mathbf{5}\)**\(\mathbf{D}\)**} & \multicolumn{1}{c}{\(\mathbf{6}\)**} & \multicolumn{1}{c}{**(Target Value)**} \\ \hline Average \(F_{s}\) & 1.1934 & 1.1929 & 1.1926 & 1.1925 & 1.1925 & 1.1925 & 1.1924 \\ \(P_{f}\) & 0.0195 & 0.0197 & 0.0200 & 0.0201 & 0.0201 & 0.0201 & 0.0201 \\ Number of slip surfaces (simula- (simula- tion once) & 5580 & 9764 & 20,476 & 40,788 & 73,772 & 122,500 & 269,001 \\ Calculation time (\(h\)) & 0.52 & 0.96 & 2.09 & 4.16 & 7.36 & 12.09 & 24.31 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Effects of reduced searching range on the calculated failure probabilities.