## 3 Illustrative Example
### Parametric Analysis of Variance Function of Friction Angle  
In this section, a parametric analysis is carried out to explore the influence of variation in the parameters \(a\), \(b\), \(c\), and \(d\) of the variance function of the friction angle on slope stability. It can be seen from Equation (4) that parameter \(d\) is equal to the variance of \(\varphi_{p}\) at quite a large \(\sigma^{\prime}{}_{3}\), parameter \(a\) is the variance of \(\varphi_{p}\) at \(\sigma^{\prime}{}_{3}\) = 0 subtracting \(d\), and parameters \(b\) and \(c\) control the decreasing rate of the variance of \(\varphi_{p}\). The higher the parameters \(a\) and \(d\) are, the greater the variance of \(\varphi_{p}\) becomes, while the higher the parameters \(b\) and \(c\) are, the more rapidly the variance of \(\varphi_{p}\) decreases. In order to compare the effects of variation in the parameters on decreasing the slope stability, parameters \(a\) and \(d\) are increased by 20%, 40%, 60%, and 80%, and parameters \(b\) and \(c\) are decreased by 20%, 40%, 60% and  
Figure 8: The variation in 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 critical slip surface \⃝2 in Figure 7. (**a**) The minor principal stresses, \(\sigma^{\prime}{}_{3}\), mean values, \(M(\varphi_{p})\), and design values of \(\varphi_{p}\) along the slices. (**b**) The minor principal stresses, \(\sigma^{\prime}{}_{3}\), and variances, \(Var(\varphi_{p})\) along the slices. 80%. Figure 9 shows the influence of variation in parameters \(a\), \(b\), \(c\), and \(d\) on the shape of the variance function of the friction angle. It is observed that increasing the value of parameter \(a\) only significantly affects the variances, \(Var(\varphi_{p})\), at relatively low \(\sigma^{\prime}{}_{3}\) (less than 500 kPa), and increasing the value of parameter \(d\) slightly shifts the curve of the variances, \(Var(\varphi_{p})\), upward (because the initial value of parameter \(d\) is small). Decreasing the values of parameters \(b\) and \(c\) reduces the decreasing rate of the variances, \(Var(\varphi_{p})\), and the effect of decreasing the value of parameter \(c\) on the decreasing rate of the variances, \(Var(\varphi_{p})\), is more significant than that of parameter \(b\). Figure 9: _Cont._  
Based on the slip surface in Figure 4 and the critical slip surface 2 in Figure 7, the slope safety factors were calculated by varying the parameters \(a\), \(b\), \(c\), and \(d\) of the variance function of the friction angle and the results are plotted in Figures 10-13. Figure 10 shows the reduction in the safety factors with respect to the increase in parameters \(a\) and \(d\) and the decrease in parameters \(b\) and \(c\) for the slip surface in Figure 4. It is observed that the variation in parameters \(b\) and \(c\) affects the slope safety factors obviously more significantly than the variation in parameters \(a\) and \(d\). The reason is that the slip surface in Figure 4 is  
Figure 9: The influence of variation in parameters \(a\), \(b\), \(c\), and \(d\) on the shape of the variance function of the friction angle. (**a**) The values of \(b\), \(c\), and \(d\) are fixed at 0.1, 1.5, and 0.3, respectively. (**b**) The values of \(a\), \(c\), and \(d\) are fixed at 3.5, 1.5, and 0.3, respectively. (**c**) The values of \(a\), \(b\), and \(d\) are fixed at 3.5, 0.1, and 0.3, respectively. (**d**) The values of \(a\), \(b\), and \(c\) are fixed at 3.5, 0.1, and 1.5, respectively. deep beneath the embankment surface, thus the minor principal stresses, \(\sigma^{\prime}{}_{3}\), at the bases of the slices are relatively high (14 slices with \(\sigma^{\prime}{}_{3}\) greater than 500 kPa).