# Application of the Improved Entry and Exit Method in Slope Reliability Analysis
## 3 Limit Equilibrium Method
### Simplified Bishop Method and Failure Probability  
The simplified Bishop approach proposed by [34] is considered one of the best limit equilibrium approaches for calculating the safety factor of circular slip surfaces in slope stability analysis [35]. The factor of security \(F_{s}\) can be shown as follows:  
\[F_{s}=\frac{\sum\frac{1}{m_{ai}}[(W_{i}-\mu_{i}b)\tan\varphi^{\prime}+c^{\prime }b]}{\sum W_{i}\sin\alpha_{i}} \tag{6}\]  
where \(W_{i}\) is gravity force; \(c^{\prime}\) is the cohesion; \(\mu_{i}\) is the pore water pressure; \(\varphi^{\prime}\) means the internal friction angle; \(b\) represents the soil strip width; \(\alpha_{i}\) refers to the inclination of the bottom of the ith soil strip; \(m_{ai}=\cos\alpha_{i}+\tan\varphi\sin\alpha_{i}/F_{s}\).  
In this paper, the Karhunen-Loeve (KL) expansion method [33] is adopted to generate two-dimensional random fields. The improved entry and exit approach is employed to recognize the critical slip surface associated with the minimum safety factor. Monte Carlo simulations are utilized to derive the failure possibility of the slope, and the failure probability (\(N\!f/N\)), defined as the ratio of the number of failure samples (\(N\!f\)) to the sample size of Monte Carlo simulations (\(N\!\)). The process for computing the failure probability can be outlined in Figure 5 as follows:  
Figure 4: Search for the critical slip surface (Css) based on the reduced searching range.  
In Example 1, the number of random field samples adopted to perform Monte Carlo simulation is 50,000. In Table 2, the failure probability acquired by applying the traditional entry and exit approach is \(7.84\times 10^{-2}\), which is reasonably close to the calculation results of other scholars [18; 36; 37; 38]. The minimum relative difference between the calculated failure probability in this article and those from other scholars is about 0.76%.  
The failure probability of Example 1 is calculated by applying the traditional entry and exit approach and the improved entry and exit approach with the same search accuracy. Table 3 shows the calculated results. According to Table 3, 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 approach (\(81\times 81\times 41\) trial slip surfaces is analyzed) and the improved entry and exit method (the reduced searching range is equal to unilateral \(2D\)) is 1.1%. When the reduced searching range is equal to unilateral \(4D\), the failure probability obtained by using the improved entry and exit approach is slightly smaller (0.33%) than the target value. 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. The calculation efficiency of the improved entry and exit method is significantly better than traditional entry and exit method.