# Stability Analysis of the Landfill Slope with an Engineered Berm Under Composite Failure Mode
## 2 Computational Model
### Equations of Equilibrium When Over-Berm Failure Occurs
#### 2.2.2 Equations of Equilibrium About Log-Spiral Failure Body  
We perform limit equilibrium analysis on the log-spiral failure body under the over-berm failure mode and establish a moment equilibrium equation. The schematic diagrams  
Figure 3: Forces acting on the block wedge when over-berm failure occurs. of limit equilibrium analysis for log-spiral failure body are shown in Figure 4. Moreover, the mathematical equation of the log-spiral slip surface in polar coordinates and the definition of related parameters can be found in the paper of [PERSON] et al. [31]. The moment equilibrium equation about the log-spiral failure body for the poles (\(x_{c}\), \(y_{c}\)) can therefore be written as follows:  
\[M_{W_{\rm L}}=M_{E_{\rm HPL}}+M_{E_{\rm VPL}}+M_{C_{\rm L}} \tag{15}\]  
where \(M_{W_{\rm L}}\) is the moment stemmed from weight of the log-spiral failure body, \(W_{\rm L}\); \(M_{E_{\rm HPL}}\) is the moment generated by normal force from the passive wedge acting on the log-spiral failure body, \(E_{\rm HPL}\); \(M_{E_{\rm VPL}}\) is the moment produced by frictional force acting on the side of the log-spiral failure body next to the passive wedge, \(E_{\rm VPL}\); \(M_{C_{\rm L}}\) is the moment resulted from apparent cohesive force acting on the log-spiral slip surface, \(C_{\rm L}\). In Equation (15), \(M_{W_{\rm L}}\) represents the driving moment responsible for the occurrence of failure, while \(M_{E_{\rm HPL}}\), \(M_{E_{\rm VPL}}\), and \(M_{C_{\rm L}}\) denote the resisting moments counteracting the trend of instability. \(M_{W_{\rm L}}\) can be calculated by modifying the result of [PERSON] and [PERSON] [33] and combining the two different scenarios in Figure 4 to obtain the calculation formula of \(M_{W_{\rm L}}\), which can be written as Equations (18) and (20), respectively. \(M_{E_{\rm VPL}}\) and \(M_{E_{\rm HPL}}\) can be derived directly from the geometric relationships in Figure 4, namely Equations (16), (19), and (21). \(M_{C_{\rm L}}\) can be obtained by reorganizing the derivation result of [PERSON] et al. [34], as shown in Equation (17). Here are the specific forms of these equations:  
\[M_{E_{\rm VPL}}=E_{\rm VPL}\Big{(}Ae^{-\psi\beta_{1}}\sin\beta_{1}\Big{)} \tag{16}\]  
\[M_{C_{\rm L}}=\frac{c_{\rm SW}}{FS}\int_{\beta_{1}}^{\beta_{2}}\big{(}Ae^{- \psi\beta}\Big{)}^{2}d\beta \tag{17}\]  
There are two calculation formulas for \(M_{W_{\rm L}}\) and \(M_{E_{\rm HPL}}\), which are shown as follows:  
(1) when \(B<H\)cot\(\alpha\)  
\[ \begin{array}{l}M_{W_{\rm L}}=\gamma_{\rm SW}\int_{\beta_{1}}^{\beta_{2}} \big{(}Ae^{-\psi\beta}\cos\beta-Ae^{-\psi\beta_{2}}\cos\beta_{2}\big{)}\big{(} Ae^{-\psi\beta}\sin\beta\big{)}\big{(}Ae^{-\psi\beta}\big{)}(\cos\beta-\psi\sin \beta)d\beta\\ -0.5\gamma_{\rm SW}(H\cot\alpha-B)^{2}\tan\omega\big{[}Ae^{-\psi\beta_{1}}\sin \beta_{1}+(H\cot\alpha-B)/3\big{]}\end{array}  \tag{18}\]  
\[M_{E_{\rm HPL}}=E_{\rm HPL}\Big{[}Ae^{-\psi\beta_{1}}\cos\beta_{1}-(H-H\cot \alpha\tan\omega+B\tan\omega)/3\Big{]} \tag{19}\]  
(2) when \(B\geq H\)cot\(\alpha\)  
\[M_{W_{\rm L}}=\gamma_{\rm SW}\int_{\beta_{1}}^{\beta_{2}}\big{(}Ae^{-\psi \beta}\cos\beta-Ae^{-\psi\beta_{2}}\cos\beta_{2}\big{)}\Big{(}Ae^{-\psi\beta} \sin\beta\Big{)}\Big{(}Ae^{-\psi\beta}\Big{)}(\cos\beta-\psi\sin\beta)d\beta \tag{20}\]  
\[M_{E_{\rm HPL}}=E_{\rm HPL}\Big{[}Ae^{-\psi\beta_{1}}\cos\beta_{1}-H/3\Big{]} \tag{21}\]  
Figure 4: Forces acting on the log-spiral failure body when over-perm failure occurs. Substituting Equation (4) into Equation (16) and rearranging it, we can obtain  
\[M_{E_{\text{FPL}}}=C_{\text{PL}}\Big{(}Ae^{-\psi\hat{\beta}_{1}}\sin\beta_{1} \Big{)}/FS_{\text{V}}+E_{\text{HPL}}\tan\phi_{\text{SW}}\Big{(}Ae^{-\psi\hat{ \beta}_{1}}\sin\beta_{1}\Big{)}/FS_{\text{V}} \tag{22}\]  
When \(B<H\text{cot}\alpha\), substituting Equations (17)-(19) and (22) into Equation (15) and rearranging it, the calculation formula for \(E_{\text{HPL}}\) can be attained as follows:  
\[