# Limit Equilibrium Analysis of Landfill Instability Based on Actual Failure Surface
## 2 Theoretical Model and Analysis Method
### Analysis Method  
#### 2.4.1 Two-Wedge Model without Berm  
The traditional two-wedge method is shown in Figure 3a. The pore pressure acting on the failure surface acts on the sliding body as a resultant force, and the pore pressures acting on the interface between the wedge bodies are same in amplititude and opposite in direction.  
For passive wedges, considering the balance of vertical and horizontal forces, according to the limit equilibrium condition, calculations are as follows:  
Equilibrium of forces in Y direction \(\sum F_{Y}=0\) gives  
\[W_{P}+E_{VP}=U_{P}+N^{\prime}_{P}, \tag{1}\]  
\[E_{VP}=\frac{C_{sw}+E_{HP}\cdot tan\varphi_{sw}}{FS_{Y}}, \tag{2}\]Equilibrium of forces in \(X\) direction \(\sum F_{X}=0\) gives  
\[U_{HP}+E_{HP}=F_{P}, \tag{3}\]  
\[F_{P}=\frac{C_{P}+N_{P}^{\prime}\cdot tan\delta_{P}}{FS_{P}}, \tag{4}\]  
The horizontal component of the interface force can be expressed as  
\[E_{HP}=\frac{\left(W_{P}-U_{P}+\frac{C_{P}}{FS_{V}}\right)\cdot\frac{tan\delta_ {P}}{FS_{P}}+\frac{C_{P}}{FS_{P}}-U_{HP}}{1-\frac{tan\delta_{P}}{FS_{V}}\cdot \frac{tan\delta_{P}}{FS_{P}}}, \tag{5}\]  
For the active wedge, considering the balance of vertical and horizontal forces, according to the limit equilibrium condition, we have the following equations:  
Equilibrium of forces in \(Y\) direction \(\sum F_{Y}=0\) gives  
\[W_{A}=F_{A}\cdot sin\beta+\left(U_{A}+N_{A}^{\prime}\right)\cdot cos\beta+E_{VA}, \tag{6}\]  
\[F_{A}=\frac{C_{A}+N_{A}^{\prime}\cdot tan\delta_{A}}{FS_{A}}, \tag{7}\]  
\[E_{VA}=\frac{C_{sw}+E_{HA}\cdot tan\phi_{sw}}{FS_{V}}, \tag{8}\]  
Equilibrium of forces in \(X\) direction \(\sum F_{X}=0\) gives  
\[\left(U_{A}+N_{A}^{\prime}\right)\cdot sin\beta=F_{A}\cdot cos\beta+U_{HA}+E_{ HA}, \tag{9}\]  
substituting Equations (6)-(9) into Equation (5) gives  
\[E_{HA}=\frac{\left(W_{P}-\frac{C_{sw}}{FS_{V}}\right)\cdot\left(sin\beta-cos \beta\cdot\frac{tan\delta_{A}}{FS_{A}}\right)+U_{A}\cdot\frac{tan\delta_{A}}{ FS_{A}}-U_{HA}\cdot\left(cos\beta+sin\beta\cdot\frac{tan\delta_{A}}{FS_{A}} \right)-\frac{C_{A}}{FS_{A}}}{cos\beta+sin\beta\cdot\frac{tan\delta_{A}}{FS_{ A}}+sin\beta\cdot\frac{tan\delta_{P}}{FS_{V}}-cos\beta\cdot\frac{tan\delta_{A}}{FS_{A}} \cdot\frac{tan\delta_{P}}{FS_{V}}}, \tag{10}\]  
Figure 3: Two-wedge model. (**a**) Traditional two-wedge force analysis model. (**b**) Two-wedge force analysis model based on actual failure surface. \(W_{A}\), \(W_{P}\) are the bulk density of the active wedge and passive wedge. \(E_{HA}\), \(E_{VA}\), \(U_{HA}\), \(E_{HP}\), \(E_{VP}\), and \(U_{HP}\) are the horizontal and vertical components of the interface force between the active wedge and the passive wedge, and the water pressure, respectively. \(U_{A}\), \(U_{P}\), \(N_{A}\)’, \(N_{P}\)’, \(F_{A}\), and \(F_{P}\) are the resultant pore water pressure, effective positive pressure, and frictional resistance acting on the active wedge and passive wedge failure surfaces. \(C_{A}\), \(C_{P}\), \(C_{sw}\), \(\delta_{A}\), \(\delta_{P}\), and \(\phi_{sw}\) are the cohesion and friction angle of active wedge and passive wedge interfaces. \(FS_{A}\), \(FS_{P}\), and \(FS_{V}\) are the FS of active wedge sliding surface, passive wedge sliding surface, and active wedge and passive wedge interface. \(\beta\) is the angle between the failure surface and the horizontal plane.  
Since \(E_{HA}=E_{HP}\), the relational expressions about \(FS_{A}\), \(FS_{P}\), and \(FS_{V}\) can be obtained:  
\[E_{HA}=\frac{\left(W_{P}-\frac{C_{sw}}{FS_{V}}\right)\cdot\left(sin\beta-cos \beta\cdot\frac{tan\delta_{A}}{FS_{A}}\right)+U_{A}\cdot\frac{tan\delta_{A}}{ FS_{A}}-\frac{C_{A}}{FS_{A}}}{cos\beta+sin\beta\cdot\frac{tan\delta_{A}}{FS_{A}}+sin \beta\cdot\frac{tan\delta_{A}}{FS_{V}}-cos\beta\cdot\frac{tan\delta_{A}}{FS_{A}} \cdot\frac{tan\delta_{up}}{FS_{V}}}, \tag{11}\]  
where \(FS_{A}=FS_{P}\) (both on the failure surface), since the overall slip occurs first in the waste body, the internal stability is relatively high, and \(FS_{V}=2.0\) is used for calculation based on experience.