# Dimensional Reduction-Based Moment Model for Probabilistic Slope Stability Analysis
## 5 Applications
### Example 1: Infinite Soil Slope
#### 5.1.2. Frictional Soil with Pore Pressure  
To further verify the proposed method, we assume \(tan\varphi^{\prime}\), \(tan\beta\), \(\gamma\), and \(u\) are the random variables. The other two parameters (\(c^{\prime}\) and \(H\)) are determinative. To simplify computation, let \(c^{\prime}=0\) and \(H=5\) m. The statistical parameters of random variables are listed in Table 6. In this case, the safety factor of slope can be estimated by the following equation.  
\[\text{FOS}=\frac{tan\varphi^{\prime}}{tan\beta}\Bigg{(}1-\frac{u\big{(}1+tan^ {2}\beta\big{)}}{\gamma H}\Bigg{)} \tag{19}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Random Variable** & **Distribution** & **Mean** & **Standard Deviation** \\ \hline \(tan\beta\) & Normal & 0.3250 & 0.0325 \\ \(tan\varphi\) & Normal & 0.5770 & 0.1732 \\ \(\gamma\) & Normal & 18.0000 & 0.5000 \\ \(u\) & Normal & 12.0000 & 1.2000 \\ \hline \hline \end{tabular}
\end{table}
Table 6: The statistical parameters of random variable for infinite slope with pore pressure.  
Figure 5: The accumulative density of safety factor.  
Figure 6: The relationship between probability failure and safety factor.  
The proposed method was adopted to estimate the reliability of the slope in this case. The input integration grid and cut component function value were listed in Table 7. The mean value and standard deviation of the safety factor are 1.5294 and 0.4903, respectively. The reliability index and failure probability of slope are 1.0796 and 0.1401 and are very close to the value 1.1014 and 0.1354 based on FORM. The maximum relative error of reliability index and failure probability is less than 2% and 3.5%, respectively. It shows MDRM can represent well the high-dimensional, nonlinear, complex, and implicit relationship between the safety factor and its influence factors. It is feasible to approximate the limit state function using the MDRM model. MCS, which may be implemented 10,000 times, was used to compare with the proposed method. The probability distributions are listed in Figure 7. It shows the probability distribution of the safety factor to be in good agreement with MCS's results. The proposed method obtained the exact result with 17 times computation which is far less than the 10,000 times of MCS. It shows the proposed method can improve efficiency dramatically. MDRM provides a good way to approximate the limit state function and reliability analysis of practical slope engineering.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Random Variables** & **Trial** & _tan\(\beta\)_ & _tan\(\varphi\)_ & _\(\gamma\)_ & \(u\) & **FOS** \\ \hline \multirow{8}{*}{_tan\(\beta\)_} & 1 & 0.2321 & 0.577 & 18 & 12 & 2.1362212 \\  & 2 & 0.2809 & 0.577 & 18 & 12 & 1.7583501 \\  & 3 & 0.3250 & 0.577 & 18 & 12 & 1.5136633 \\  & 4 & 0.3691 & 0.577 & 18 & 12 & 1.3265885 \\  & 5 & 0.4179 & 0.577 & 18 & 12 & 1.1646101 \\ \hline \multirow{8}{*}{_tan\(\varphi\)_} & 1 & 0.325 & 0.0822 & 18 & 12 & 0.2155666 \\  & 2 & 0.325 & 0.3422 & 18 & 12 & 0.8977175 \\  & 3 & 0.325 & 0.5770 & 18 & 12 & 1.5136633 \\  & 4 & 0.325 & 0.8118 & 18 & 12 & 2.1296092 \\  & 5 & 0.325 & 1.0718 & 18 & 12 & 2.81176 \\ \hline \multirow{8}{*}{_tan\(\varphi\)_} & 1 & 0.325 & 0.577 & 16.5715 & 12 & 1.4911026 \\  & 2 & 0.325 & 0.577 & 17.3222 & 12 & 1.5034222 \\ \cline{1-1}  & 3 & 0.325 & 0.577 & 18.0000 & 12 & 1.5136633 \\ \cline{1-1}  & 4 & 0.325 & 0.577 & 18.6778 & 12 & 1.5231612 \\ \cline{1-1}  & 5 & 0.325 & 0.577 & 19.4285 & 12 & 1.5329065 \\ \hline \multirow{8}{*}{_u_} & 1 & 0.325 & 0.577 & 18 & 8.5716 & 1.5884363 \\  & 2 & 0.325 & 0.577 & 18 & 10.3732 & 1.5491431 \\ \cline{1-1}  & 3 & 0.325 & 0.577 & 18 & 12.0000 & 1.5136633 \\ \cline{1-1}  & 4 & 0.325 & 0.577 & 18 & 13.6268 & 1.4781836 \\ \cline{1-1}  & 5 & 0.325 & 0.577 & 18 & 15.4284 & 1.4388903 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Input grid and corresponding safety factor.