## 2 Methods
### Scheme of the Calculation Model  
In this chapter, four scenarios were proposed to be calculated to investigate the effects of layers, dip angle, cohesion, and internal friction angle of multi-layered muddied interlayer on the stability of bedding cataclastic rock slope containing multi-muddy interlayer.  
Scheme 1: In order to study the impact of the number of exposed muddy interlayers, denoted as \(n\), on the stability of bedding cataclastic rock slopes, the interlayers are arranged at equal distances starting from the slope foot, with a dip angle \(\theta\) of \(22^{\circ}\), and \(n\) is respectively 2, 4, 6, 8, and 10. Therefore, there are five scenarios in this scheme.  
Scheme 2: In order to analyze the influence of the dip angle \(\theta\) of multi-muddy interlayer on the stability of bedding cataclastic rock slope, the interlayer is arranged from the toe of the slope, and the interlayer spacing \(d\) is taken as 2 m. Main calculation scheme: slope angle \(\beta\) = 45\({}^{\circ}\), \(\theta\) takes 0\({}^{\circ}\), 10\({}^{\circ}\), 20\({}^{\circ}\), 25\({}^{\circ}\), 30\({}^{\circ}\), 35\({}^{\circ}\), 40\({}^{\circ}\), 50\({}^{\circ}\), 60\({}^{\circ}\), 70\({}^{\circ}\), 75\({}^{\circ}\), 80\({}^{\circ}\), 85\({}^{\circ}\), and 90\({}^{\circ}\) in order, giving 15 arithmetic examples. Auxiliary calculation scheme: slope angle \(\beta\) = 30\({}^{\circ}\), \(\theta\) in order to take 0\({}^{\circ}\), 10\({}^{\circ}\), 15\({}^{\circ}\), 20\({}^{\circ}\), 25\({}^{\circ}\), 30\({}^{\circ}\), 40\({}^{\circ}\), 45\({}^{\circ}\), 50\({}^{\circ}\), 60\({}^{\circ}\), 65\({}^{\circ}\), 70\({}^{\circ}\), 75\({}^{\circ}\), 80\({}^{\circ}\), and 90\({}^{\circ}\), giving 15 arithmetic examples. Slope angle \(\beta\) = 60\({}^{\circ}\), \(\theta\) in order to take 0\({}^{\circ}\), 10\({}^{\circ}\), 20\({}^{\circ}\), 30\({}^{\circ}\), 40\({}^{\circ}\), 45\({}^{\circ}\), 50\({}^{\circ}\), 60\({}^{\circ}\), 70\({}^{\circ}\), 75\({}^{\circ}\), 80\({}^{\circ}\), 85\({}^{\circ}\), and 90\({}^{\circ}\), giving 13 arithmetic examples. The material parameters of the above three different inclination angle \(\theta\) calculation schemes were shown in Table 1. In order to calculate all the cases properly, the shear strength parameters of both the fractured rock body and the muddied interlayer were first increased by 30% (Table 2), and then the slope angle \(\beta\) = 60\({}^{\circ}\) and \(\theta\) was taken as 0\({}^{\circ}\), 10\({}^{\circ}\), 20\({}^{\circ}\), 30\({}^{\circ}\), 35\({}^{\circ}\), 40\({}^{\circ}\), 45\({}^{\circ}\), 50\({}^{\circ}\), 60\({}^{\circ}\), 70\({}^{\circ}\), 75\({}^{\circ}\), 80\({}^{\circScheme 3: An analysis of the effect of cohesion \(c\) of multi-muddy interlayer on the stability of bedding cataclastic rock slope. The interlayer was arranged from the toe of the slope, the inclination angle of the interlayer \(\theta\) was taken as 22\({}^{\circ}\), and the spacing \(d\) was taken as 2 m. The material parameters are shown in Table 1, where the internal friction angle of the muddied interlayer was taken as 16\({}^{\circ}\) and \(c\) was taken as 0.015 MPa, 0.02 MPa, 0.025 MPa, 0.03 MPa, 0.035 MPa, 0.04 MPa, 0.045 MPa, and 0.05 MPa, respectively, i.e., there were eight arithmetic cases in this scheme.  
Scheme 4: The influence of internal friction angle \(\varphi\) of multi-muddy interlayer on the stability of bedding cataclastic rock slope. The material parameters are shown in Table 1, where the cohesion of the muddied interlayer was taken as 0.03 MPa and \(\varphi\) was taken as 8\({}^{\circ}\), 10\({}^{\circ}\), 12\({}^{\circ}\), 14\({}^{\circ}\), 16\({}^{\circ}\), 18\({}^{\circ}\), 20\({}^{\circ}\), 22\({}^{\circ}\), 24\({}^{\circ}\), 26\({}^{\circ}\), 28\({}^{\circ}\), and 30\({}^{\circ}\), respectively, i.e., there were 12 calculations in this scheme.