## 2 Methodologies
### Element Failure Probability of Slope  
Traditional slope reliability analysis generally adopts the integrated failure probability method [33] and uses the threshold value of slope stability coefficient for evaluation. It is  
Figure 1: Flow chart of a numerical solution using UBM for a slope. considered that when the slope stability coefficient \(k_{m}\geq 1.0\), then the slope remains stable; when the slope stability coefficient \(k_{m}<1.0\), then the slope becomes unstable. Therefore, the slope failure function based on the IFP of \(k_{m}\) is expressed as:  
\[I_{k_{m}}(t_{m})=\left\{ \begin{array}{c}0,\text{ if }k_{m}(t_{m})\geq 1.0\\ 1,\text{ if }k_{m}(t_{m})<1.0\end{array} \right. \tag{8}\]  
Further, according to the concept of slope in a limited state, it can be determined that if the slope has been destroyed without applying any external load or applying any opposite direction load, then the external overload coefficient and bulk density overload coefficient are both less than 0. Therefore, the slope-failure function based on the IFP of \(k_{F}\) can be expressed as:  
\[I_{k_{F}}(t_{m})=\left\{ \begin{array}{c}0,\text{ if }k_{F}(t_{m})\geq 0\\ 1,\text{ if }k_{F}(t_{m})<0\end{array} \right. \tag{9}\]  
The slope failure function based on IFP of \(k_{\gamma}\) can be expressed as:  
\[I_{k_{\gamma}}(t_{m})=\left\{ \begin{array}{c}0,\text{ if }k_{\gamma}(t_{m}) \geq 0\\ 1,\text{ if }k_{\gamma}(t_{m})<0\end{array} \right. \tag{10}\]  
The IFP of a slope is expressed as:  
\[P_{f}^{\text{M-IFP}}=\frac{\sum\limits_{t_{m}=1}^{n_{m}}I_{M}(t_{m})}{n_{m}} \times 100\% \tag{11}\]  
where \(P_{f}^{\text{M-IFP}}\) = the failure probability of the slope, of which \(M=k_{m},k_{F},k_{\gamma}\). Therefore, the equation calculation for slope failure risk (IPF-FR) based on the IFP [25] is:  
\[R^{\text{M-IFP}}=P_{f}^{\text{M-IFP}}C \tag{12}\]  
where \(R^{\text{M-IFP}}\) = the failure risk of the slope; \(P_{f}^{\text{M-IFP}}\) = the failure probability of the slope; \(C\) = the failure risk coefficient. Equation (10) is widely used in slope failure risk analysis, but the impact of multi-failure modes of slopes is not considered. In view of this, [PERSON] et al. [34] proposed a slope failure risk equation based on multi-failure modes, in which the failure risk of the \(k\)-th failure mode is:  
\[R_{k}^{\text{M-IFP}}=P_{f}^{\text{M-IFP}}C_{k} \tag{13}\]  
where \(R_{f}^{\text{M-IFP}}\) = the failure risk of the \(k\)-th failure mode; \(P_{f}^{\text{M-IFP}}\) = the failure probability of the \(k\)-th failure mode; \(C_{k}\) = the failure risk coefficient of the \(k\)-th failure mode, where \(k\) = 1,2,..., \(N_{r}\), \(N_{f}\) is the quantity of failure modes. The failure risk of a slope is:  
\[R^{\text{M-IFP}}=\sum\limits_{k=1}^{N_{f}}R_{k}^{\text{M-IFP}} \tag{14}\]  
where \(R^{\text{M-IFP}}\) = the slope failure risk by \(M\)-IFP. Compared with Equation (12), Equation (14) can calculate the slope risk assessment of single failure mode and multi-failure modes at the same time. However, both LEM and FEM must be used to conduct screening of complex failure modes in advance. It is worth mentioning that [PERSON] et al. [18] proposed the concept of element failure probability based on the slope stability coefficient (\(k_{m}\)-\(u_{c}\)-EFP), which uses the slope stability coefficient obtained by solving the mathematical programming model of the UBM and the failure condition of slope elements determined by a kinematically-admissible velocity field to obtain the failure condition of each area. The failure function is:  
\[I_{k_{m}}^{e}(t_{m})=\left\{ \begin{array}{rl}0\text{ if }k_{\text{m}}(t_{m}) \geq 1.0\\ 0\text{ if }u_{\text{c}}^{e}(t_{m})=0\text{ and }k_{\text{m}}(t_{m})<1.0\\ 1\text{ if }u_{\text{c}}^{e}(t_{m})~{}>~{}0\text{ and }k_{\text{m}}(t_{m})<1.0\end{array} \right. \tag{15}\]  
where \(I_{k_{m}}^{e}(t_{m})\) = the failure function of the element \(e\) in the \(t_{m}\)th random field; \(u_{\text{c}}^{e}(t_{m})\) = the velocity of the element in the \(t_{m}\)th random field.