## 3 Calibration and Application
### Case 2: A Heterogeneous Slope  
#### 3.2.1 Heterogeneous Slope Description  
Figure 8 shows a slope with a height of 24 m, a top width of 40 m and a slope ratio of 3:4. Many scholars have calculated and analyzed this slope [34; 35; 36]; however, there is no systematic analysis and research on the ultimate bearing capacity. For this reason, the ultimate bearing capacity and slope failure risk are studied based on the proposed method. The slope's undrained shear strength is set as a random quantity and assume to have lognormal distribution. The mean value of the undrained shear strength of the upper and lower layers of the slope is 70 kPa and 100 kPa, respectively, and the standard deviation is 21 kPa and 30 kPa, respectively. Using non-common-node finite elements discrete to the slope, 1508 finite elements, 4524 non-common nodes and 2409 discontinuities are obtained. The soil capacity \(\gamma\) is set to a constant value of 19.0 kN/m\({}^{3}\), and other calculation parameters are shown in Table 5. The quantity of random fields \(n_{m}=2000\), and Equation (7) was used to obtain the 2000 random fields. and slope stability coefficient. The main reason is that the FEM relies on iteration non-convergence, plastic zone penetration and node displacement mutation as judgment criteria when analyzing slope stability. Therefore, the results are not necessarily strictly accurate solutions and their values may be greater than the upper limit solutions obtained by the UBM. 2. Based on the \(k_{F}(t_{m})\), \(k_{\gamma}(t_{m})\) and \(k_{m}(t_{m})\), the slope failure probability can be calculated, which is close or equal. From the perspective of calculation efficiency, the above three methods in this paper can carry out large-scale parallel calculation. However, it required 2.16 h (roughly 86.4 h for single thread calculation), 2.19 h (roughly 87.6 h for single thread calculation) and 17.28 h (roughly 691.2 h for single thread calculation) to calculate the \(k_{F}(t_{m})\), \(k_{\gamma}(t_{m})\) and \(k_{m}(t_{m})\) of 1000 random field samples. It required 118.0, 122.3 and 472.6 h, respectively, to calculate the \(k_{F}(t_{m})\), \(k_{\gamma}(t_{m})\) and \(k_{m}(t_{m})\) by the FEM for the same sample. 3. Figures 10 and 11 reflect the process of gradually increasing the top load and bulk density to cause the slope to reach the limit state. It can be observed intuitively that with the gradual increase of top load \(P\) and bulk density \(\gamma\), the slope stability coefficient obtained via the UBM, LEM and FEM all show a decreasing trend, and the results obtained via the UBM and FEM are close to each other, which are all greater than the results calculated by the LEM. 4. Figure 11 gives the histogram of \(k_{F}(t_{m})\), \(k_{\gamma}(t_{m})\) and \(k_{m}(t_{m})\). It can be intuitively observed that \(k_{F}(t_{m})\), \(k_{\gamma}(t_{m})\) and \(k_{m}(t_{m})\) all present the characteristics of lognormal distribution, being high in the center and low on both sides. The LEM's default is that there is only one slope failure mode, as depicted by the blue line in Figure 12; three slope failure modes can be obtained simultaneously by the FEM, as depicted by the black line in Figure 12. The most significant advantage of the proposed method is that it can capture all the slope failure modes simultaneously. By calculating 1000 groups of slope random field samples, a total of three slope failure modes are obtained, as depicted by the pink area in Figure 12. The histogram of the slope failure area is shown in Figure 13. Table 7 lists the statistics of slope failure modes by IFP method. Table 4 lists the statistics of the slope failure risk. Figure 11: Histogram of \(k_{F}(t_{m})\), \(k_{\gamma}(t_{m})\) and \(k_{m}(t_{m})\) of heterogeneous slope. Figure 12: Heterogeneous Slope failure modes. The calculation results show that:  
1. Only failure mode 4 is obtained by the LEM while the other five failure modes are not obtained.