# Analysis of the Failure Area of the Slope Using the Slip Line Method
## 2 Slip Line Method
### Recurrence Formula of the Slip Line Method  
To solve various geotechnical problems by applying the numerical analysis method of the slip line method, three boundary problems (Cauchy, Riemann, mixed) and the basic calculation method are applied. The basic calculation method determines the intersection of the \(\alpha\) line passing through point \(A\) and the \(\beta\) line passing through point \(B\) using the given average stress \(p\) and \(\theta\) values at points A and B. Through this, the average stress \(p\) and \(\theta\) values at point \(P\) can be obtained.  
A limitation of the basic calculation method is that one cannot integrate along a characteristic line unless the positions of the \(\alpha\) line through point A and the \(\beta\) line through point B are known. Therefore, it is possible to obtain a solution by combining the stress equation and the slip line equation to obtain the position of the P point (\(X_{p}\), \(Y_{p}\)) and \(\theta_{p}\), \(P_{p}\) as shown in Figure 4.  
The recurrence formula can be derived from the stress equation (Equations (1) and (2)) derived from Figure 1 and the slip line equation (Equations (3) and (4)) derived from Figure 3. This is shown in Equations (5)-(8).  
\[-sin2\mu(P_{P}-P_{A})+(R_{P}+R_{A})(\theta_{P}-\theta_{A})=-\gamma sin(2\mu)(x _{P}-x_{A})-\gamma cos(2\mu)(y_{P}-y_{A}) \tag{1}\]  
\[sin2\mu(P_{P}-P_{B})+(R_{P}+R_{B})(\theta_{P}-\theta_{B})=\gamma sin(2\mu)(x _{P}-x_{B})-\gamma cos(2\mu)(y_{P}-y_{B}) \tag{2}\]  
Figure 4: Basic calculation method [27].  
Figure 3: Principal stress trajectory and slip lines for coulomb material [27].  
\[y_{P}-y_{A}=tan\left(\frac{\theta_{P}+\theta_{A}}{2}-\mu\right)(x_{P}-x_{A})=( \frac{dy}{dx})_{1}(x_{P}-x_{A}) \tag{3}\]  
\[y_{P}-y_{B}=tan\left(\frac{\theta_{P}+\theta_{B}}{2}+\mu\right)(x_{P}-x_{B})=( \frac{dy}{dx})_{2}(x_{P}-x_{B}) \tag{4}\]  
\[\theta_{p}=\frac{\gamma sin2\mu(x_{A}-x_{B})-\gamma cos2\mu(2y_{P}-y_{A}-y_{B}) +sin2\mu(P_{B}-P_{A})+(R_{P}+R_{A})\theta_{A}+(R_{P}+R_{B})\theta_{B}}{(2R_{P}+ R_{A}+R_{B})} \tag{5}\]  
\[P_{P}=\frac{\gamma sin2\mu(x_{P}-x_{B})-\gamma cos2\mu(y_{P}-y_{B})+P_{B}sin2 \mu-(\theta_{P}+\theta_{B})sin\varnothing-2c(\theta_{P}+\theta_{B})cos\varnothing}{ sin2\mu+sin\varnothing(\theta_{P}-\theta_{B})} \tag{6}\]  
\[y_{P}=y_{B}+\left(\frac{dy}{dx}\right)_{2}(x_{P}-x_{B}) \tag{7}\]  
\[x_{P}\frac{(y_{B}-y_{A})+(\frac{dy}{dx})_{1}\;x_{A}-(\frac{dy}{dx})_{2}\;x_{B }}{\left(\frac{dy}{dx}\right)_{1}-\left(\frac{dy}{dx}\right)_{2}}\;\;\;(\text{ When two points are known}) \tag{8}\]  
\[x_{P}=x_{A}+\frac{y_{P}-y_{A}}{tan(\theta_{P}+\theta_{A}/2-\mu)}\;\;\text{or }\;x_{P}=x_{B}+\frac{y_{P}-y_{B}}{tan(\theta_{P}+\theta_{B}/2-\mu)}\;\;(\text{ When one point is known})\]  
In Equations (1), (2), (5), and (6), \(\gamma\) means unit weight.  
In this paper, the slip line method was applied to the slope. Therefore, if the \(\theta\) and \(p\) values of two different pairs of slip lines are known, the plastic area can be analyzed by applying the Riemann problem, which can find a solution within the area of influence of the slip line. Figure 5a shows a schematic diagram of the Riemann boundary problem to be solved using the slip line method, where curves OA and OB are two different slip lines. The \(x\), \(y\), \(p\), \(\theta\) values of points \(P_{12}\) and \(P_{21}\) in Figure 5b can be calculated using the basic calculation method. The approximation of each point in the area of influence can be calculated in the same way (\(P_{13}\), \(P_{31}\), etc.).