# 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
\begin{cases}k_{ci}x+c_{ri}&0\leq x\leq L_{0}\\ c_{pi}&x>L_{0}\\ \end{cases} \] \[S_{ti} = \begin{cases}k_{si}x+S_{ri}&0\leq x\leq L_{0}\\ S_{pi}&x>L_{0}\\ \end{cases} \]  
In the formula, \(k_{fi}\), \(k_{ci}\), and \(k_{si}\) are the corresponding coefficients defined as:  
\[ \begin{cases}k_{fi}=\frac{f_{pi}-f_{ri}}{L_{0}}\\ k_{ci}=\frac{c_{pi}-c_{pi}}{L_{0}}\\ k_{si}=\frac{5_{pi}-5_{ri}}{L_{0}}\\ \end{cases}  \tag{2}\]  
Figure 12: Slope calculation model. If the first sliding part is the front triangle, the following relationships can be derived for the triangle \(A_{i}B_{i}C\) from simple goniometry:  
\[ \begin{array}{l}|A_{i}B_{i}|=x\\ |B_{i}C|=x\tan(\beta-\alpha)\end{array} \]  
Let \(W_{AiBiC}\) be the weight of triangle \(A_{i}B_{i}C\),  
\[W_{A_{i}B_{i}C}=\frac{1}{2}\overline{\gamma}\left|A_{i}B_{i}\right|\cdot \left|B_{i}C\right| \tag{3}\]  
where \(\gamma\) is the average bulk density of overlying strata on layer \(I\) in the formula. On the pull-crack surface \(B_{i}C\), let \(F_{si}\) be the tensile force of the tensile strength of the rock mass to the triangle \(A_{i}B_{i}C\). \[F_{si}=\left|B_{i}C\right|S_{ti} \tag{4}\]  
On the sliding surface \(A_{i}B_{i}\), \(F_{qi}\) is the tensile force of rock mass on the triangle \(A_{i}B_{i}C\). \(F_{qi}\) is mainly composed of a friction coefficient and cohesion on the sliding surface. \[F_{qi}=W_{A_{i}B_{i}C}\cos\alpha\cdot\overline{f}_{i}+x\cdot\overline{c}_{i} \tag{5}\]  
where \(\overline{f}_{i}\), \(\overline{c}_{i}\) are the average friction coefficient and cohesion on the sliding surface. Let \(F_{Di}\) be the sliding force of the triangle \(A_{i}B_{i}C\):  
\[F_{Di}=W_{A_{i}B_{i}C}\cdot\sin\alpha \tag{6}\]  
Add Equations (4) and (5) to obtain the total anti-sliding force of block \(A_{i}B_{i}C\), denoted as \(F_{Ri}\):  
\[F_{Ri}=F_{si}+F_{qi}=\left|B_{i}C\right|S_{ti}+W_{A_{i}B_{i}C}\cos\alpha\cdot \overline{f}_{i}+x\cdot\overline{c}_{i} \tag{7}\]  
If \(F_{Di}>F_{Ri}\), the landslide has occurred. If \(F_{Di}<F_{Ri}\), the landslide will not occur. If:  
\[F_{Ri}-F_{Di}=0 \tag{8}\]  
The slope is in the limit equilibrium state, about to slide but not yet. The unknown \(x\) in the equation at this time is the lateral length \(L\) of the traction instability, denoted as \(L_{Hi}\). Substituting Equations (1)-(7) into Equation (8) gives a univariate quadratic equation, which can be solved:  
\[L_{Hi}= \begin{cases}\frac{-A_{12}+\sqrt{A_{12}^{2}-4A_{11}A_{13}}}{2A_{11}}& (L_{Hi}\leq L_{0})\\ \frac{-A_{22}+\sqrt{A_{22}^{2}-4A_{21}A_{23}}}{2A_{21}}&(L_{Hi}>L_{0})\end{cases}  \tag{9}\]  
In the formula:  
\[ \begin{cases}A_{11}=\frac{1}{4}\overline{\gamma}\tan(\beta-\alpha)\cos\alpha \cdot k_{fi}\\ A_{12}=\frac{1}{2}\tan(\beta-\alpha)\cdot(2k_{si}+\overline{\gamma}\cos\alpha \cdot f_{ri}-\overline{\gamma}\sin\alpha)+\frac{1}{2}k_{ci}\\ A_{13}=\tan(\beta-\alpha)\cdot S_{ri}+c_{ri}\\ A_{21}=\frac{1}{2}\overline{\gamma}\tan(\beta-\alpha)\cdot(\cos\alpha\cdot f _{pi}-\sin\alpha)\\ A_{22}=\frac{1}{4}\tan(\beta-\alpha)[4S_{pi}+\overline{\gamma}\cos\alpha(f_{ri} -f_{pi})L_{0}]+c_{pi}\\ A_{23}=\frac{1}{2}(c_{ri}-c_{pi})L_{0}\end{cases} \]If the length of the first sliding part is long, the sliding block becomes a trapezoidal shape, and the weight of the sliding block cannot be calculated according to the triangular formula. The calculation formula for \(L_{Hi}\) becomes:  
\[L_{Hi}= \begin{cases}\frac{-C_{12}+\sqrt{C_{12}^{2}-4C_{11}C_{13}}}{2\lambda_{11} }&(L_{Hi}\leq L_{0})\\ \frac{-C_{22}+\sqrt{C_{22}^{2}-4C_{21}C_{23}}}{2\lambda_{21}}&(L_{Hi}>L_{0}) \end{cases}  \tag{10}\]  
In the formula:  
\[