# Application of the Improved Entry and Exit Method in Slope Reliability Analysis
## 3 Limit Equilibrium Method
### Discussion on Reduced Searching Range
#### 4.3.2 Further Verification of the Reliability of the Reduced Searching Range  
The traditional entry and exit method with different trial slip surfaces is adopted for calculating the failure probabilities of Example 1 and Example 2. Table 7 shows the calculated results. As can be seen from Table 7, the failure probabilities increase with the number of increasing trial slip surfaces. However, the calculated failure probabilities obtained with 61 \(\times\) 61 \(\times\) 31 trial slip surfaces and 101 \(\times\) 101 \(\times\) 51 trial slip surfaces is anomalous. This is because the trial slip surfaces used to decide the critical slip surface are different. For example, the \(P_{f61\times 61\times 31}\) is larger than \(P_{f421\times 21\times 11}\) but smaller than \(P_{f41\times 41\times 21}\), because the set of the trial slip surfaces with 61 \(\times\) 61 \(\times\) 31 trial slip surfaces contains the set of the trial slip surfaces with 21 \(\times\) 21 \(\times\) 11 trial slip surfaces but intersects with the set of the trial slip surfaces with 41 \(\times\) 41 \(\times\) 21 trial slip surfaces. In addition, failure probabilities of Example 1 obtained by analyzing 81 \(\times\) 81 \(\times\) 41 trial slip surfaces is 0.0917, which is 16.9% larger than that obtained by analyzing 21 \(\times\) 21 \(\times\) 11 trial slip  
Figure 8: Difference values of potential critical slip surface and critical slip surface (Example 1 on the left, Example 2 on the right).  
surfaces. The failure probabilities of Example 2 obtained by analyzing \(81\times 81\times 41\) trial slip surfaces is 0.0201, which is 11.0% larger than that obtained by analyzing \(21\times 21\times 11\) trial slip surfaces. This is because the more trial sliding surfaces analyzed under the higher calculation accuracy, the smaller the safety factors will be. These results indicate that the calculation accuracy of the safety factor has an effect on the failure probability of slopes.  
The failure probabilities obtained by using the improved entry and exit method with a fixed reduced searching range (unilateral \(2D\) and unilateral \(4D\)) are compared with failure probabilities obtained by using the traditional entry and exit method, as shown in Figure 9. It can be seen that when reduced searching range is equal to unilateral \(2D\) and unilateral \(4D\), the failure probabilities are reasonably close to those obtained by using the traditional entry and exit method. The relative differences for unilateral \(2D\) between the results of Example 1 and Example 2 are about below 1.1% and 2.0%, respectively. The relative differences for unilateral \(4D\) between the results of Example 1 and Example 2 are about below 0.3% and 0.5%. These outcomes further show the rationality of the given reduced searching range.  
Regarding the research findings obtained in this discussion section, it is evident that the number of trial slip surfaces significantly affects the accuracy of computed failure probability results. The proposed improved entry and exit method can significantly reduce the computational cost while obtaining a more accurate result in calculating failure probability. This method is applicable for realizing rapid and practical reliability analysis assessments on a large number of slopes--both at the design stage and at the stage of planning the technology for the execution of geotechnical works. Note that by the means of the enhanced computational efficiency provided by the method, in the practice of slope reliability analysis, the reduced searching range for the second search calculation can be reasonably selected based on the requirements of calculation accuracy.