# Stability Analysis of the Landfill Slope with an Engineered Berm Under Composite Failure Mode
## 2 Computational Model  
### Basic Assumptions  
When analyzing static stability of the landfill slope with an engineered perm against the composite failure mode, the failure body in the limit equilibrium state can be divided into three parts: the block wedge (region \(L_{0}L1_{1}\), as in Figure 1; or region \(L_{0}L1_{1}L2_{2}\), as in Figure 2), the passive wedge (region \(OKLL_{0}\), as in Figures 1a and 2a; or region \(OKJLL_{0}\), as in Figures 1b and 2b), and the log-spiral failure body (region \(OGJK\), as in Figures 1a and 2a; or region \(OGK\), as in Figures 1b and 2b). The limit equilibrium analysis of the block wedge, log-spiral failure body, and passive wedge is carried out sequentially, and equilibrium equations are established separately to obtain the calculation formulas for normal force from passive wedge acting on block wedge (i.e., \(E_{\text{HPB}}\)) and normal force from passive wedge acting on log-spiral failure body (i.e., \(E_{\text{HPL}}\)), as well as the expression for the relationship between normal forces from block wedge and log-spiral failure body acting on passive wedge (i.e., \(E_{\text{HBP}}\) and \(E_{\text{HLP}}\)). Based on the relationship between \(E_{\text{HPB}}=E_{\text{HBP}}\) and \(E_{\text{HPL}}=E_{\text{HLP}}\), the implicit function equation for evaluating the safety factor of the landfill with an engineered perm against the composite failure mode is constructed. Considering the geometric parameters of the landfill slope studied, the limit equilibrium analysis for sliding bodies can be divided into two situations, namely (**a**): \(B<H\text{cot}\alpha\) (as in Figures 1a and 2a) and (**b**): \(B\geq H\text{cot}\alpha\) (as in Figures 1b and 2b). In Figures 1 and 2, \(O(0,\,0)\) is the origin of Cartesian coordinates; \((x_{\text{c}},\,y_{\text{c}})\) is the pole of polar coordinates in Cartesian coordinates; \(W_{\text{B}}\), \(W_{\text{L}}\) and \(W_{\text{P}}\) are the weights of block wedge, log-spiral failure body, and passive wedge, respectively; \(N_{\text{P}}\) and \(F_{\text{P}}\) are the normal and frictional forces,  
Figure 1: Forces acting on three wedges in the landfill when over-perm failure occurs. Figure 2: Forces acting on three wedges in the landfill when under-perm failure occurs. respectively, acting on the bottom of passive wedge; \(N_{\rm B}\) and \(F_{\rm B}\) are the normal and frictional forces, respectively, acting on the bottom of block wedge; \(E_{\rm HLP}\) and \(E_{\rm VLP}\) are the normal and frictional forces, respectively, acting on the passive wedge from log-spiral failure body; \(E_{\rm HPL}\) and \(E_{\rm VPL}\) are the normal and frictional forces, respectively, acting on the log-spiral failure body from passive wedge; \(E_{\rm HPB}\) and \(E_{\rm VPB}\) are the normal and frictional forces, respectively, acting on the block wedge from passive wedge; \(E_{\rm HBP}\) and \(E_{\rm VBP}\) are the normal and frictional forces, respectively, acting on the block wedge from passive wedge; \(E_{\rm HBP}\) and \(E_{\rm VBP}\) are the normal and frictional forces, respectively, acting on the passive wedge from block wedge; \(H\) is the height of back slope; \(H_{\rm B}\) is the height of engineered berm; \(B\) is the top width of waste mass; \(D_{\rm B}\) is the top width of the engineered berm; \(\omega\) is the angle of the front slope measured from the horizontal; \(\alpha\) is the angle of the back slope measured from the horizontal; \(\theta\) is the angle of the bottom of the passive wedge measured from the horizontal; \(\xi\) is the angle of the back slope of the engineered berm measured from the horizontal; \(\eta\) is the angle of the front slope of the engineered berm measured from the horizontal; \(\beta_{1}\) and \(\beta_{2}\) are the angles of point \(O\) and point \(G\) in the polar coordinates, respectively. To successfully build up an analytical model for analyzing static stability of the landfill with an engineered berm against composite failure mode, it is necessary to make some reasonable assumptions.