## 3 Illustrative Example
### The Slope Stability Analysis of a Rockfill Embankment  
The rockfill embankment is assumed to be composed of a blasting-quarried granite rockfill material [16; 22] and lies on a rock foundation. It has a height of 100 m and an inclination ratio of 1:1.4. The cross section of the rockfill embankment is shown in Figure 4. An arbitrary slip arc is adopted to illustrate the slope stability analysis approach as shown in Figure 4, which has a radius of 221.02 m, and the \(x\) and \(y\) coordinates of the radius center are 84.41 m and 124.19 m, respectively. The slip block is divided into 20 slices with identical widths. The unit weight of the rockfills is equal to 20 kN/m\({}^{3}\). The rockfill material consists of angular or sub-angular particles with a \(D_{mxx}\) (maximum particle size) of 60 mm, a \(D_{10}\) (particle size at 10% finer) of 1.76 mm, a \(D_{50}\) (particle size at 50% finer) of 18.4 mm, and a \(D_{60}\) (particle size at 60% finer) of 26.7 mm. The mean values, \(M(\varphi_{p})\), and variances, \(Var(\varphi_{p})\), of the friction angle of the rockfills are as follows [22]:  
\[M\big{(}\varphi_{p}\big{)}=52.4-3.9\ln\left(\frac{\sigma^{\prime}_{3}}{p_{a}} \right) \tag{9}\]  
Figure 3: The derivation of the formula, Equation (6), for the minor principal stress, \(\sigma^{\prime}_{3}\)[41]. (10)  
Based on the slip arc, the slope safety factors for different loading conditions were computed using the proposed approach. Three loading conditions are considered herein: (1) the static loading condition, i.e., only takes into account the weights of the slices, (2) the seismic loading condition I, i.e., takes into account both the weights of the slices and the horizontal and upward vertical seismic inertia forces, and (3) the seismic loading condition II, i.e., takes into account the weights of the slices as well as the horizontal and downward vertical seismic inertia forces. The peak horizontal ground acceleration is \(a_{h}=0.4\) g, and the maximum dynamic distribution coefficient is \(\alpha_{m}=2.0\). The slope safety factors for the three loading conditions without considering the variability in the friction angle of the rockfills (i.e., the variance of the friction angle is equal to 0) are 2.874, 2.286, and 2.297, respectively, while if the stress-dependent spatial variability in the friction angle of rockfills is considered, the slope safety factors are reduced to 2.612, 2.060, and 2.080, respectively. 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 are listed in Tables 1-3. It is observed that the minor principal stresses, \(\sigma^{\prime}{}_{3}\), are basically proportional to the heights of the slices, and the maximum minor principal stress is located at the base of the intermediate slice. Because the horizontal seismic inertia forces towards the slope face are involved, the minor principal stresses for the seismic loading conditions are smaller than those for the static loading condition. Moreover, the minor principal stresses for seismic loading condition I (with upward vertical seismic inertia forces) are smaller than those for seismic loading condition II (with downward vertical seismic inertia forces). Because the mean values, \(M(\varphi_{p})\), and variances, \(Var(\varphi_{p})\), are dependent on the minor principal stresses, \(\sigma^{\prime}{}_{3}\), the values of \(M(\varphi_{p})\) and \(Var(\varphi_{p})\) at the bases of the slices differ from slice to slice and differ from loading condition to loading condition. Figure 4: A cross section of the rockfill embankment and the fixed slip surface for the computation of slope safety factors.