## 1 Introduction
### An Algorithm for Determining the Values of the Friction Angle at the Bases of the Slices  
For coarse-grained soil, such as rockfills, the cohesion is ignored, and the values of the friction angle at the bases of the slices are determined by the stress-dependent mean value and variance of the friction angle. The spatial variability of the values of the friction angle is taken into account through the varying mean value and variance of the friction angle along the slip surface. According to [PERSON] and [PERSON] [16; 22], the mean value, \(M(\varphi_{p})\), and variance, \(Var(\varphi_{p})\), of the friction angle \(\varphi_{p}\) of rockfills are the functions of the minor principal stress, \(\sigma^{\prime}{}_{3}\):  
\[M\big{(}\varphi_{p}\big{)}=\varphi_{0}-\Delta\varphi\ln\left(\frac{\sigma^{ \prime}{}_{3}}{p_{a}}\right) \tag{3}\]  
\[Var\big{(}\varphi_{p}\big{)}=a\exp\left[-b\left(\frac{\sigma^{\prime}{}_{3}}{p _{a}}\right)^{c}\right]+d \tag{4}\]  
in which \(\varphi_{0}\), \(\Delta\varphi\), \(a\), \(b\), \(c\), and \(d\) are the parameters.  
In order to consider the adverse variation in \(\varphi_{p}\), the design values of \(\varphi_{p}\) at the bases of the slices are determined as follows:  
\[\varphi_{p}=M\big{(}\varphi_{p}\big{)}-k\sqrt{Var\big{(}\varphi_{p}\big{)}} \tag{5}\]  
in which \(\sqrt{Var\big{(}\varphi_{p}\big{)}}\) is the standard deviation of \(\varphi_{p}\), \(k\) is a coefficient, and in this study, \(k\) is taken as 3.  
In Equations (3) and (4), \(\sigma^{\prime}{}_{3}\) is calculated by (see Figure 3 for the derivation)  
\[\sigma^{\prime}{}_{3}=\sigma^{\prime}{}_{n}\Big{(}\sec^{2}\varphi_{f}-\tan \varphi_{f}\sec\varphi_{f}\Big{)} \tag{6}\]  
\[\tan\varphi_{f}=\frac{\tan\varphi_{p}}{F_{s}} \tag{7}\]  
in which \(\sigma^{\prime}{}_{n}\) is the stress normal to the failure arc, and \(\varphi_{f}\) is the friction angle corresponding to the sliding limit state of the slope. In terms of [PERSON]'s simplified method, \(\sigma^{\prime}{}_{n}\) is computed as  
\[\sigma^{\prime}{}_{n}=\frac{(W\pm V)}{b+b\tan\varphi_{f}\tan\alpha}-u \tag{8}\]  
Figure 2: The forces acting on the slice for [PERSON]’s simplified method and the dynamic distribution coefficient, \(\alpha\), where \(T_{i}\) is the resisting shear force, \(E_{i}\) and \(E_{i+1}\) are the normal interslice forces, \(H\) is the embankment height, \(h\) is the height of the centroid of the slice, and \(\alpha_{m}\) is the maximum value of \(\alpha\) located on the embankment crest.  
Iterations are required to obtain the design values of \(\varphi_{p}\) and the safety factor \(F_{s}\). The iterative process is given as follows:  
1. Assume an initial \(\varphi_{p}^{(0)}\) and \(F_{s}^{(0)}\); in this study, \(\varphi_{p}^{(0)}=\varphi_{0}\) and \(F_{s}^{(0)}=1\).
2. Determine \(\varphi_{f}\) by substituting \(\varphi_{p}^{(k)}\) and \(F_{s}^{(k)}\) (\(k=1,2,\cdots,n\)) in Equation (7), then \(\sigma^{\prime}_{3}\) is calculated using Equations (6) and (8).
3. Obtain \(\varphi_{p}^{(k+1)}\) from Equations (3)-(5).
4. Compute \(F_{s}^{(k+1)}\) using Equation (1).
5. Repeat steps (b) to (d) until \(\left|F_{s}^{(k+1)}-F_{s}^{(k)}\right|\leq\varepsilon\), where \(\varepsilon\) is the tolerance; in this study, \(\varepsilon=0.001\).