# Solution for Active and Passive Earth Pressure on Rigid Retaining Walls with Narrow Backfill
## 2 Theoretical Analysis
### Passive Limit Earth Pressure
By setting the derivative equal to zero and rearranging, Equation (16) is derived. \[-A_{3}D_{3}\psi^{4}-2B_{3}D_{3}\psi^{3}+(A_{3}F_{3}-B_{3}E_{3}-3C_{3}D_{3}) \psi^{2}+(2A_{3}G_{3}-2C_{3}E_{3})\psi+(B_{3}G_{3}-C_{3}F_{3})=0 \tag{16}\]  
Equation (16) constitutes a unary quartic equation with four general solutions. Based on the range of soil parameters, these solutions typically include two imaginary roots and two real roots. Within the valid range of the rupture angle, the stationary point is presented in Equation (17). This stationary point is rigorously validated to be consistent with the rupture angle corresponding to the minimum value obtained by solving for \(E_{x2}(\theta)\) in MATLAB. \[\theta_{p}=\arctan\left(-\frac{b_{2}}{4a_{2}}-\frac{1}{2}\sqrt{\frac{{b_{2}}^ {2}}{4a_{2}^{2}}-\frac{2c_{2}}{3a_{2}}+\Delta}+\frac{1}{2}\sqrt{\frac{{b_{2}} ^{2}}{2a_{2}^{2}}-\frac{4c_{2}}{3a_{2}}-\Delta-\frac{-\frac{{b_{2}}^{3}}{{a_{2 }}^{3}}+\frac{4{b_{2}}c_{2}}{{a_{2}}^{2}}-\frac{8d_{2}}{{a_{2}}}}{4\sqrt{\frac{{ b_{2}}^{2}}{{a_{2}}^{2}}-\frac{2c_{2}}{3a_{2}}+\Delta}}\right) \tag{17}\]In which:  
\[ \begin{gathered} a_{2}=-A_{3}D_{3}\quad b_{2}=-2B_{3}D_{3}\quad c_{2}= A_{3}F_{3}-B_{3}E_{3}-3C_{3}D_{3}\\ d_{2}=2A_{3}G_{3}-2C_{3}E_{3}\quad e_{2}=B_{3}G_{3}-C_{3}F_{3}\quad \lambda_{2}={c_{2}}^{2}-3b_{2}d_{2}+12a_{2}e_{2}\\ \eta_{2}=2{c_{2}}^{3}-9b_{2}c_{2}d_{2}+27a_{2}d^{2}+27{b_{2}}^{2}e_{2}-72a_ {2}c_{2}e_{2}\\ \Delta_{2}=\frac{3\tilde{2}\lambda_{2}}{3a_{2}\sqrt[3]{\eta_{2}+ \sqrt{-4\lambda_{2}^{2}+\eta_{2}^{2}}}}+\frac{3\sqrt[3]{\eta_{2}+\sqrt{-4 \lambda_{2}^{2}+\eta_{2}^{2}}}}{3\sqrt[3]{2}a_{2}}\end{gathered}  \tag{18}\]  
By substituting \(\theta_{p}\) into Equation (13), the passive limit earth pressure resultant force on the retaining wall under finite soil conditions can be determined. As the backfill width behind the retaining wall (\(L\)) increases, the height of the inflection points on the slip surface (\(L\)tan\(\theta\)) increases synchronously until it aligns with the ground surface. At this point, the slip wedge transforms into a triangular wedge, with its width corresponding to the critical width. The specific expression is \(H=L\)tan\(\theta_{p}\), but this represents an implicit expression of the critical width. For ease of application, this study adopts the width of the triangular wedge in Coulomb theory as the critical width (\(L_{p}\)). \[ \begin{gathered} L_{p}=\frac{2 HA_{4}}{-B_{4}+\sqrt[3]{B_{4}}^{2}-4 A_{4}C_{4}}\\ A_{4}=4 cH(\cos\delta-\sin\delta\tan\varphi)+2\gamma H^{2}(\cos \delta\tan\varphi+\sin\delta)\\ B_{4}=4(\gamma H^{2}\tan\varphi+2 cH)(\cos\delta\tan\varphi+\sin \delta)\\ C_{4}=-2(\cos\delta-\sin\delta\tan\varphi)(\gamma H^{2}\tan\varphi+2 cH) \end{gathered}  \tag{19}\]  
In the case where the backfill behind the wall comprises cohesionless soil (\(c=0\)), the force on the slip wedge in the passive state is depicted in Figure 3. Similarly, this scenario can be considered a special case of the force on the clay slip wedge. It should be noted that when the backfill width behind the wall equals the critical width, the results obtained using the proposed method are consistent with the Coulomb result for cohesionless soil. However, for clay, the results obtained in this paper are slightly greater than the Coulomb results at the critical width.