## 2 Materials and Methods
### Quantitative Evaluation of Rockfall Hazards  
(1) Stability Calculations of Hazardous Rock Masses  
In the study area, the instability of hazardous rock masses primarily manifests in two failure modes: toppling and falling. Stability analysis was conducted under three distinct conditions: natural state, intense rainfall, and seismic loading. The computational models employed are as follows:  
1 Stability Analysis of Rockfall-Type Hazardous Rock Masses  
Based on the spatial geometry and structural plane configuration of hazardous rock masses in the study area, the WY1 rock mass is primarily characterized by a rockfall-type failure mode. (a) The parameter \(j\) is calculated using the following equation (Figure 4):\[F=\frac{(H-h)c}{G+Q_{y}} \tag{1}\]  
In the equation, the variables are defined as follows:  
\(F\) denotes the factor of safety. \(G\) represents the self-weight of the hazardous rock mass (kN). \(Q_{y}\) represents the horizontal or external additional force, expressed in kilonewtons (kN). \(c\) denotes the cohesion of the hazardous rock, expressed in kilopascals (kPa). \(H\) represents the vertical distance from the upper end of the rear-edge fissure to the lower end of the unpenetrated section (i.e., the height of the hazardous rock cantilever), expressed in meters (m). \(h\) denotes the depth of the rear-edge fissure, expressed in meters (m). (b) The stability of break-off and free-fall failure in rockfall-type hazardous rock masses is calculated using the following equation (Figure 5):  
\[F_{O}=\frac{\sigma_{t}(H-h)^{2}}{6[(G+Q_{V})\text{a}_{0}+Q_{h}\text{b}_{0}]+V[ 2h_{\text{r}}+3(H-h)]} \tag{2}\]  
In the equation, the variables are defined as follows:  
\(F_{O}\) denotes the flexural (tensile-break) factor of safety for the hazardous rock mass. \(Q_{V}\) represents the vertical surcharge acting on the rock mass (e.g., accumulated debris or vegetation), expressed in kilonewtons (kN). Figure 4: Stability Analysis of Undercutting-Induced Falling Failure in Rockfall-Prone Areas. Figure 5: Stability Analysis of Fracture-Induced Falling Failure in Rockfall-Prone Areas. \(Q_{h}\) represents the horizontal surcharge acting on the rock mass (e.g., wind or vibration forces), expressed in kilonewtons (kN). \(h_{v}\) is the vertical distance from the point of application of the sliding force to the fracture (break-off) point, expressed in meters (m). \(V\) denotes the sliding or shear force acting along the shear surface of the rock mass (e.g., seismic inertial force), expressed in kilonewtons (kN). \(\mathrm{a_{0}}\) and \(b_{0}\) denote the horizontal and vertical distances, respectively, between the rock-mass centroid and the midpoint of the rear-edge vertical plane, expressed in meters (m). \(\sigma_{t}\) represents the tensile strength of the rock mass, expressed in kilopascals (kPa). 2 Stability Analysis of Toppling-Mode Hazardous Rock Masses  
Based on the spatial geometry and structural plane configurations of hazardous rock masses in the study area, WY2, WY3, WY4, WY5, and WY6 are primarily characterized by toppling-type failure modes. (a) When the centroid of the hazardous rock mass is located on the inner side of the midpoint of its base, the basal break-off toppling stability is calculated using the following equation (Figure 6):  
\[F_{1}=\frac{\sigma_{t}b^{2}+6 Ge}{6\big{(}Q_{h}h_{0}-Q_{y}e\big{)}+2 Vh_{o}} \tag{3}\]  
(b) When the center of gravity of the hazardous rock mass extends beyond the midpoint of its base yet remains within the front edge, the toppling stability governed by rear-tension failure can be evaluated using the following equation:  
\[F_{2}=\frac{\sigma_{t}b^{2}}{6[Q_{h}h_{0}+(G+Q_{r})e]+2 Vh_{e}} \tag{4}\]  
In Equations (3) and (4), the variables are defined as follows:  
\(F_{1}\) and \(F_{2}\) represent the safety factors for break-off toppling stability under two conditions: (i) the center of gravity is located within the midpoint of the base (\(F_{1}\)), and (ii) it extends beyond the midpoint but remains within the front edge (\(F_{2}\)).