## 3 Calibration and Application
### Case 2: A Heterogeneous Slope
#### 3.2.1 Heterogeneous Slope Description
This is because the LEM calculates slope stability under random samples of shear strength parameter based on the pre-assumed critical sliding surface, and therefore its failure mode is not a genuine failure mode. Failure modes 1, 4 and 6 are obtained simultaneously by the FEM while the other three failure modes are not obtained. This is because the FEM needed to build a finite element model satisfying the static equilibrium condition, strain compatibility condition and constitutive relation of soil mass when analyzing slope stability. Therefore, on the basis of the known slope constitutive model, more accurate calculation results can be obtained than with the LEM or UBM. Unfortunately, the constitutive relation of soil mass is complex and there is no unified constitutive model to characterize it at present. In addition, the FEM contains more human factors when judging slope failure modes. 2. Six slope failure modes are obtained based on \(k_{F}(t_{m})\), \(k_{\gamma}(t_{m})\) and \(k_{m}(t_{m})\) by the UBM. There are slight differences in the failure probability of each failure mode, the failure risk of each failure mode and overall failure risk. Theoretically, there is only one slope limit state under the action of each random field, which corresponds to the only external overload coefficient, bulk overload coefficient and slope stability coefficient. However, due to the influence of calculation errors, rounding errors and calculation accuracy, the accumulation of subtle differences is caused. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **Failure Mode** & **Failure Area (m\({}^{2}\))** & **Failure Times** &
\begin{tabular}{c} **Failure** \\ **Probability (\%)** \\ \end{tabular} & **Failure Risk (m\({}^{2}\))** \\ \hline \multirow{6}{*}{\(R_{k}^{k_{f}(t_{m})-\mathrm{IFP}}\)} & Mode 1 & 373.92–514.15 & 12 & 0.60 & 5546.18 \\  & Mode 2 & 529.00–666.35 & 52 & 2.60 & 31,899.98 \\  & Mode 3 & 672.10–816.98 & 79 & 3.95 & 58,861.09 \\  & Mode 4 & 820.05–961.51 & 119 & 5.95 & 105,853.14 \\  & Mode 5 & 965.15–1110.83 & 52 & 2.60 & 53,284.80 \\  & Mode 6 & 1117.08–1260.54 & 11 & 0.55 & 12,957.99 \\ \hline \multirow{6}{*}{\(R_{k}^{k_{\gamma}(t_{m})-\mathrm{IFP}}\)} & Mode 1 & 361.25–501.78 & 6 & 0.30 & 2666.57 \\  & Mode 2 & 514.15–660.71 & 41 & 2.05 & 24,762.45 \\ \cline{1-1}  & Mode 3 & 663.39–807.83 & 81 & 4.05 & 60,028.04 \\ \cline{1-1}  & Mode 4 & 811.18–959.02 & 113 & 5.65 & 100,506.79 \\ \cline{1-1}  & Mode 5 & 966.10–1103.17 & 69 & 3.45 & 70,491.61 \\ \cline{1-1}  & Mode 6 & 1111.05–1260.54 & 15 & 0.75 & 17,560.83 \\ \hline \multirow{6}{*}{\(R_{k}^{k_{m}(t_{m})-\mathrm{IFP}}\)} & Mode 1 & 429.77–567.03 & 11 & 0.55 & 5789.23 \\ \cline{1-1}  & Mode 2 & 585.14–715.86 & 47 & 2.35 & 31,078.84 \\ \cline{1-1}  & Mode 3 & 719.18–862.89 & 88 & 4.40 & 70,239.95 \\ \cline{1-1}  & Mode 4 & 864.14–1008.95 & 105 & 5.25 & 98,484.41 \\ \cline{1-1}  & Mode 5 & 1008.92–1133.62 & 63 & 3.15 & 67,266.86 \\ \cline{1-1}  & Mode 6 & 1172.37–1296.33 & 11 & 0.55 & 13,509.97 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Statistics of heterogeneous slope failure modes. Figure 13: Histogram of s Heterogeneous type failure area. 3. Table 8 compares the calculation results of the UBM, LEM and FEM. It can be seen that the failure risk obtained by Equations (14) and (20) based on UBM are the same. However, it should be noted that Equation (14) is the slope failure risk calculation based on IFP. Therefore, it is necessary to first count all slope failure modes, including the failure risk coefficient and failure probability of each failure mode; Equation (20) is the slope failure risk calculation based on EFP by failure probability and failure risk coefficient of each discrete element, which can be easily obtained by the element failure function Equations (15)-(17) and the element failure probability Equation (18) proposed in this paper.