# Progressive Formation of Retrogressive Landslide and the Lateral Length of Instability
## 3 Determination of Lateral Length of Traction Instability and Analysis of Influencing Factors  
### Determination of Lateral Length of Traction Instability  
During the construction of the slope, due to the effects of rainfall, the removal of lateral constraints, blasting [56], and other factors, the rock mass structure in the local area of the slope is destroyed, thereby reducing its mechanical index. The mechanical principle is complex, and it is very difficult to describe this process accurately and completely. Assuming that the bottom surface of the rock mass in layer \(i\) is a weak surface after the slope is excavated, the rock mass of the slope will slide along this bottom surface. For slopes whose excavation slope angle is greater than the dip angle of bedding, because of the removal of material by excavation, the failure mode of the slope may be a complete sliding failure along a certain level, or it may slide along certain levels. During the sliding process, some joints in the rock mass are pulled apart, showing gradual sliding failure from bottom to top. For the slope with the natural slope angle equal to the dip angle of bedding, the slope calculation model shown in Figure 12 is established. Let the dip angle of bedding be \(\alpha\) and the excavation slope angle be \(\beta\). The \(x\)-axis position of the coordinate system is taken as the bottom surface of the studied block and changes with the change of the studied block. The origin of the coordinate is located at the intersection of the bottom surface of the studied block and the excavation surface. Let the origin of the coordinates be \(A_{i}\), \(C\) is a point on the excavation slope, and the intersection of the vertical line and the \(x\)-axis is \(B_{i}\). \(E\) is the apex angle of the slope and let its \(x\) and \(y\) coordinate values be \(x_{0}\) and \(y_{0}\). \(F\) is any point on the bottom of the \(i\)-th layer. The distance from \(G\) to the origin of the coordinate system is \(L_{0}\). \(L_{0}\) is the width of the weakening area of the mechanical parameter, that is, the length of the influence area of the incised slope is \(L_{0}\). \(L_{0}\) is related to factors such as construction methods, rock mass unloading, and weathering, and can be determined according to engineering experience. From previous research, it can be seen that in the process of slope slip failure, the range of the part of the rock mass where the mechanical parameters of the rock mass decline are limited, generally near the empty surface of the slope. Second, in this area, the closer to the surface of the slope, the greater the magnitude of the decrease in the value of the mechanical parameters of the rock mass. The further away from the surface of the slope, the smaller the magnitude of the decrease in the value of the mechanical parameters of the rock mass, and the closer it is to the value in the natural state. It can be seen from Figure 12 that the mechanical parameters of point A are most affected, and the mechanical parameters of point G are not affected, which is natural. In order to calculate the lateral length of traction instability, three parameters need to be determined, namely friction coefficient, cohesion, and tensile strength. For point F, let \(f_{i}\) be the friction coefficient of the bottom surface of the \(i\)-th layer, \(c_{i}\) be the cohesion of the bottom surface of the \(i\)-th layer, and \(S_{ti}\) be the equivalent tensile strength of the rock mass above the bottom surface of the \(i\)-th layer. The three mechanical parameters of point G are \"\(S_{ri}\),\(f_{ni}\) and \(cri\)\", and point A are \"\(S_{pi}\),\(f_{pi}\) and \(c_{pi}\)\". Assuming that the influence of the incised slope on the mechanical parameters of rock mass decreases linearly, then the relationship between the distance \(x\) from point \(F\) to the origin of the coordinate system and the mechanical parameters of point \(F\) is:  
\[f_{i} = \begin{cases}k_{fi}x+f_{ri}&0\leq x\leq L_{0}\\ f_{pi}&x>L_{0}\\ \end{cases}  \tag{1}\] \[c_{i} =