# Dimensional Reduction-Based Moment Model for Probabilistic Slope Stability Analysis
## 5 Applications
### Example 1: Infinite Soil Slope
#### 5.1.1 Frictional/Cohesion Soil  
Now, we consider an effective stress analysis of the infinite slope with shear strength parameters (\(c^{\prime}\) and \(tang^{\prime}\)) and no pore pressure. In this case, the safety factor of the slope can be estimated by the following equation. \[\text{FOS}=\frac{c^{\prime}}{\gamma Hsin\beta\ cos\beta}+\frac{tang^{\prime}}{ tan\beta} \tag{18}\]  
Soil strength parameters (\(c^{\prime}\) and \(tang^{\prime}\)) were regarded as random variables with a normal distribution. The statistical parameters of random variables were listed in Table 2. The other parameters were determinative. The proposed method was adopted to estimate the reliability index of the slope. The input integration grid and cut component function value were listed in Table 3. To \(H\) = 5 m, \(\beta\) = 30, and \(\gamma\) = 17 kN/m\({}^{3}\), the mean value and standard deviation of safety factors are 1.2718 and 0.3114, respectively. MCS, which was implement 10,000 times, was used to verify the proposed method. The probability distributions are shown in Figure 3. It shows the probability distribution of safety factor is in excellent agreement with MCS's results. The proposed method obtained the exact result with nine times computation which is far less than the 10,000 times of MCS. It shows the proposed method can improve efficiency dramatically. Figure 2: The infinite slope. To further verify the proposed method, the reliability of slope was estimated in different depths of potential failure surface and slope inclination. Table 4 listed the reliability index and failure probability of slope in different \(H\). The values of the proposed method are in good agreement with the values based on FORM. The maximum relative error of the reliability index is less than 0.3%. Table 4 listed the results of reliability analysis in different slope inclination using the proposed method and FORM. The results are similar to the results in different \(H\). The maximum relative error of the reliability index is less than 0.18%. The above results show the proposed method has good accuracy and performance. \begin{table}
\begin{tabular}{c c c c} \hline
**Random Variable** & **Distribution** & **Mean** & **Standard Deviation** \\ \hline \(c^{\prime}\) & Normal & 10 & 3 \\ \(tan(\varphi^{\prime})\) & Normal & 0.5774 & 0.1732 \\ \hline \end{tabular}
\end{table} 
Table 2: The statistical parameters of random variables for infinite slope without pore pressure. Figure 3: Probability distribution of safety factor and its comparison with MCS for frictional soil. \begin{table}
\begin{tabular}{c c c c} \hline
**Random Variables** & **Trial** & \(\mathcal{C}^{\prime}\) & _tan(\(\varphi^{\prime}\))_ & **FOS** \\ \hline \multirow{6}{*}{\(c^{\prime}\)} & 1 & 1.42909 & 0.5774 & 0.860396 \\  & 2 & 5.93311 & 0.5774 & 0.973175 \\  & 3 & 10 & 0.5774 & 1.075007 \\  & 4 & 14.06689 & 0.5774 & 1.17684 \\  & 5 & 18.57091 & 0.5774 & 1.289619 \\ \hline \multirow{6}{*}{\(tan(\varphi^{\prime})\)} & 1 & 10 & 0.082573 & 0.368321 \\  & 2 & 10 & 0.342605 & 0.739685 \\ \cline{1-1}  & 3 & 10 & 0.5774 & 1.075007 \\ \cline{1-1}  & 4 & 10 & 0.812195 & 1.41033 \\ \cline{1-1}  & 5 & 10 & 1.072227 & 1.781694 \\ \hline \multicolumn{2}{c}{Fixed mean value} & 10 & 0.5774 & 1.075007 \\ \hline \end{tabular}
\end{table} 
Table 3: Input grid and corresponding safety factor. To evaluate the uncertainty of slope, the probability distribution was shown in Figure 4 in different H. Figure 4 shows the mean value of the safety factor will decrease with the increasing of \(H\). Table 4 also shows the mean value, standard deviation, and reliability index all decrease with the increasing of \(H\), and the failure probability increase with the increasing of \(H\). There shows the failure risk of slope will increase with the increasing of \(H\). This is consistent with the theory and practice of slope engineering. Table 5 also shows similar results. The goal of reliability analysis is for guiding the design and construction of slope.