# Solution for Active and Passive Earth Pressure on Rigid Retaining Walls with Narrow Backfill
## 2 Theoretical Analysis
### Passive Limit Earth Pressure  
There are slight differences in the forces acting on the slip wedge between the passive and active states. In the passive state, where the wedge undergoes rightward movement, the rock (or structure) maintains intimate contact and interacts with the slip wedge. Consequently, the earth pressures at the same depth above the slip surface inflection point are postulated to be equal on both sides of the wedge. For computational simplification, the earth pressure is considered to follow a triangular distribution. Furthermore, the cohesion (\(c\)) is uniformly distributed on the slip surface. The corresponding mechanical model of the slip wedge under these conditions is shown in Figure 2.  
In the schematic, \(E_{x2}\) represents the resultant force on the retaining wall. Both \(E_{x2}\) and the counterforce \(S\) form an angle \(\delta\) (wall-soil friction angle) with the horizontal plane. The soil beneath the slip surface also exerts a counterforce (\(R\)) on the slip wedge, where the angle between R and the normal to the slip surface equals the internal friction angle (\(\varphi\)). The resultant force of cohesion (C) along the slip surface forms an angle \(\theta\) (rupture angle) with the horizontal plane. In this case, the rupture angle \(\theta\) is constrained within the range (\(10^{\circ}\), \(40^{\circ}\)). The expressions for the self-weight (\(W\)), the resultant force of cohesion (\(C\)), and the counterforce (\(S\)) acting on the slip wedge are given in Equation (11). Figure 2: Force analysis of the clay slip wedge in the passive state. \[W=\gamma(HL-\frac{1}{2}L^{2}\tan\theta)\quad C=cL/\cos\theta\quad S=E_{x2}\frac{ \left(H-L\tan\theta\right)^{2}}{H^{2}} \tag{11}\]  
According to the static equilibrium conditions in both horizontal and vertical directions for the slip wedge under the passive state, Equation (12) is derived. \[ \begin{array}{c}R\sin(\theta+\varphi)+S\cos\delta+C\cos\theta=E_{x2}\cos \delta\\ R\cos(\theta+\varphi)=S\sin\delta+C\sin\theta+W+E_{x2}\sin\delta\end{array}  \tag{12}\]  
The expression relating \(E_{x2}\) with respect to the rupture angle \(\theta\) can be derived from Equations (11) and (12), as illustrated in Equation (13). \[E_{x2}=\frac{H^{2}cL\tan\theta\tan(\theta+\varphi)+H^{2}W\tan(\theta+\varphi) +H^{2}S\cos\delta+H^{2}cL}{H^{2}\cos\delta-\left(H-L\tan\theta\right)^{2}\cos \delta\tan(\theta+\varphi)-H^{2}\sin\delta\tan(\theta+\varphi)} \tag{13}\]  
Equation (13) explicitly demonstrates that \(E_{x2}\) is a unary function of the rupture angle \(\theta\). By introducing the substitution \(\psi=\tan\theta\), it simplifies this to Equation (14). \[ \begin{array}{c}E_{x2}=\frac{A_{3}\psi^{2}+B_{3}\psi+C_{3}}{D_{3}\psi^{3}+E_ {3}\psi^{2}+F_{3}\psi+G_{3}}\\ A_{3}=H^{2}cL-\frac{1}{2}\gamma H^{2}L^{2}B_{3}=\gamma H^{3}L-\frac{1}{2}\gamma H ^{2}L^{2}\tan\varphi\\ C_{3}=\gamma H^{3}L\tan\varphi+H^{2}cLD_{3}=L^{2}(\cos\delta\tan\varphi-\sin \delta)\\ E_{3}=2 HL(\sin\delta-\cos\delta\tan\varphi)-L^{2}(\sin\delta\tan\varphi+\cos \delta)\\ F_{3}=2 HL(\cos\delta+\sin\delta\tan\varphi)-2H^{2}\sin\delta G_{3}=-2H^{2}\sin \delta\tan\varphi\end{array}  \tag{14}\]  
For a given soil, parameters \(A_{5}\) through \(G_{5}\) are constant in the function. The passive limit earth pressure resultant force (\(E_{p}\)) corresponds to the minimum value of \(E_{x2}(\theta)\) within the defined parameter range. The minimum value should also be the stationary point of \(E_{x2}(\theta)\). Thus, taking the derivative of Equation (14) yields Equation (15). \[\frac{dE_{x2}}{d\psi}=\frac{(2A_{3}\psi+B_{3})(D_{3}\psi^{3}+E_{3}\psi^{2}+F_{3 }\psi+G_{3})-(A_{3}\psi^{2}+B_{3}\psi+C_{3})(3D_{3}\psi^{2}+2E_{3}\psi+F_{3}) }{\left[D_{3}\psi^{3}+E_{3}\psi^{2}+F_{3}\psi+G_{3}\right]^{2}} \tag{15}\]  
The passive limit earth pressure behind the retaining wall definitely exists and its corresponding rupture angle is the stationary point of \(E_{x2}(\theta)\) within the range of \(\theta\). Furthermore, the denominator in Equation (15) is consistently greater than or equal to 0.