## 3 Illustrative Example
### The Slope Stability Analysis of a Rockfill Embankment
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Not Considering the Variability in the Friction Angle**} & \multicolumn{4}{c}{**Considering the Stress-Dependent Spatial Variability**} \\  & \multicolumn{4}{c}{**of the Rockfills**} & \multicolumn{4}{c}{**in the Friction Angle of the Rockfills**} \\ \cline{2-9}
**Slice No.** & \(\sigma^{\prime}{}_{\textbf{3}}\) & _M(\(\varphi_{p}\))_ & _Var(\(\varphi_{p}\))_ & **Design Value of \(\varphi_{p}\)** & \(\sigma^{\prime}{}_{\textbf{3}}\) & _M(\(\varphi_{p}\))_ & _Var(\(\varphi_{p}\))_ & **Design Value of \(\varphi_{p}\)** \\  & **(kPa)** & (\({}^{\circ}\)) & (\({}^{\circ}\)) & (\({}^{\circ}\)) & (\({}^{\circ}\)) & (\({}^{\circ}\)) & (\({}^{\circ}\)) & (\({}^{\circ}\)) \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 1: A summary of the computed minor principal stresses, \(\sigma^{\prime}{}_{3}\), and the associated mean values, \(M(\varphi_{p})\), variances, _Var_(\(\varphi_{p}\)), and design values of \(\varphi_{p}\) at the bases of the slices for the static loading condition. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Not Considering the Variability in the Friction Angle**} & \multicolumn{4}{c}{**Considering the Stress-Dependent Spatial Variability**} \\  & \multicolumn{4}{c}{**of the Rockfills**} & \multicolumn{4}{c}{**in the Friction Angle of the Rockfills**} \\ \cline{2-9}
**Slice No. ** & \(\sigma^{\prime}{}_{\textbf{3}}\) & _M(\(\varphi_{p}\))_ & _Var(\(\varphi_{p}\))_ & **Design Value of \(\varphi_{p}\)** & \(\sigma^{\prime}{}_{\textbf{3}}\) & _M(\(\varphi_{p}\))_ & _Var(\(\varphi_{p}\))_ & **Design Value of \(\varphi_{p}\)** \\  & **(kPa)** & (\({}^{\circ}\)) & (\({}^{\circ}\)) & (\({}^{\circ}\)) & (\({}^{\circ}\)) & (\({}^{\circ}\)) & (\({}^{\circ}\)) & (\({}^{\circ}\)) \\ \hline
1 & 52.552 & 54.9 & 0 & 54.9 & 56.937 & 54.6 & 3.7 & 48.9 \\
2 & 179.975 & 50.1 & 0 & 50.1 & 189.809 & 49.9 & 3.0 & 44.7 \\
3 & 307.703 & 48.0 & 0 & 48.0 & 318.338 & 47.9 & 2.3 & 43.4 \\
4 & 430.573 & 46.7 & 0  
The formulation of the proposed approach indicates that design values of \(\varphi_{p}\) and minor principal stresses, \(\sigma^{\prime}{}_{3}\), are related to each other. Therefore, for the same loading condition and the same slice, due to different design values of \(\varphi_{p}\), the minor principal stresses are not the same in the cases of considering and not considering the variability in the friction angle of the rockfills. However, the minor principal stresses are insensitive to the design values of \(\varphi_{p}\). For most slices in the three loading conditions, the relative differences of \(\sigma^{\prime}{}_{3}\) are less than 2%, and the maximum relative difference of \(\sigma^{\prime}{}_{3}\) is within 8%. Figures 5 and 6 show the variations of the minor principal stresses, \(\sigma^{\prime}{}_{3}\), mean values, \(M(\varphi_{p})\), variances, \(Var(\varphi_{p})\), and design values of \(\varphi_{p}\) along the slices for the static loading condition and seismic loading condition I. It can be seen that both the mean values, \(M(\varphi_{p})\), and variances, \(Var(\varphi_{p})\), of \(\varphi_{p}\) at the bases of the slices decrease as the minor principal stress, \(\sigma^{\prime}{}_{3}\), increases. Furthermore, both \(M(\varphi_{p})\) and \(Var(\varphi_{p})\) dramatically drop in the range of \(\sigma^{\prime}{}_{3}\) from 0 to approximately 800 kPa, and there are much less variations in \(M(\varphi_{p})\) and \(Var(\varphi_{p})\) in the range of \(\sigma^{\prime}{}_{3}\) between 800 kPa and 1100 kPa. As shown in Figures 5 and 6, the stress-dependent spatial variability in \(\varphi_{p}\) makes the design values of \(\varphi_{p}\) more uniform than the mean values of \(\varphi_{p}\). In order to investigate the influence of the number of slices on the computational results, the slope safety factor for the static loading condition is computed with 10, 20, 30, 40, 50, and 100 slices and is equal to 2.616, 2.612, 2.611, 2.610, 2.610, and 2.610, respectively.