## 4 Summary and Conclusions  
This paper proposes an approach for the slope stability analysis of rockfill embankments considering stress-dependent spatial variability in the friction angle. The approach is formulated in the framework of [PERSON]'s simplified method, but the values of the friction angle at the bases of the slices are determined by the stress-dependent mean values, \(M(\varphi_{p})\), and variances, \(Var(\varphi_{p})\), of the friction angle, \(\varphi_{p}\), which are the functions of the minor principal stresses, \(\sigma^{\prime}\), \(3\), at the bases of the slices. The spatial variability in the values of \(\varphi_{p}\) is taken into account through the varying variance of \(\varphi_{p}\) along the slip surface. The slope stability analysis of a homogeneous rockfill embankment indicates that the slope safety factors decrease when considering the adverse variation in \(\varphi_{p}\), and 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 for the same slip surface. The extent of the variation in \(Var(\varphi_{p})\) along the slip surface diminishes with the decrease in the depth of the slip surface, since the variation in the minor principal stresses, \(\sigma^{\prime}\), \(3\), along the slices becomes smaller. Regarding the illustrative example, the shape of the variance function of \(\varphi_{p}\) is significantly influenced by the variation in parameters \(a\), \(b\), and \(c\) (\(a\), \(b\), \(c\) and \(d\) are the parameters of the variance function in Equation (4)). The parametric analysis reveals that the variation in parameters \(b\) and \(c\) affects the slope safety factors obviously more significantly than the variation in parameter \(a\) for the deep slip surface, since the minor principal stresses, \(\sigma^{\prime}\)\(3\), at the bases of the slices are relatively high, and the influences of the variation in parameters \(b\) and \(c\) on \(Var(\varphi_{p})\) are quite more significant than those of parameter \(a\) at a relatively high \(\sigma^{\prime}\)\(3\), while the slope safety factor is only sensitive to the variation in parameter \(a\) for the shallow slip surface, since the minor principal stresses, \(\sigma^{\prime}\)\(3\), at the bases of the slices are relatively low, and only the influence of the variation in parameter \(a\) on \(Var(\varphi_{p})\) is significant at relatively low \(\sigma^{\prime}\)\(3\). The illustrative example demonstrates that the stress-dependent spatial variability of friction angle is obvious in the rockfill embankment, and its effects on slope stability should be considered. However, the proposed approach deals with the stress-dependent spatial variability of the friction angle in a simple way. The combination of large-scale tests or determination methods of material parameters with the proposed approach is expected to  
Figure 13: Variation in the variances, \(Var(\varphi_{p})\), along the slices for the critical slip surface ��⃝2 in Figure 7.  
be investigated in the future [43; 44]. In addition, there are still shortcomings in this study for embankments with complex topographical and geological conditions or reinforcement, and more advanced numerical calculation methods may offer better performance [45]. Conceptualization, C.R. and Z.W. ; Methodology, C.R., Z.Z., X.L., B.G. and Z.W. ; Software, Z.Z. and Y.J. ; Validation, B.G. ; Formal analysis, C.R., B.G. and Y.J. ; Investigation, Z.Z., X.L. and Y.J. ; Resources, Z.Z. and Z.W. ; Data curation, X.L. ; Writing--original draft, C.R. ; Writing--review & editing, Z.W. ; Visualization, Y.J.. All authors have read and agreed to the published version of the manuscript. This research is supported by the National Natural Science Foundation of China (No. 52309162) and the Fundamental Research Funds for the Central Universities (YJ202287). Not applicable. The data presented in this study are available on request from the corresponding author.