# Solution for Active and Passive Earth Pressure on Rigid Retaining Walls with Narrow Backfill
## 2 Theoretical Analysis
### Active Limit Earth Pressure
\begin{array}{c}A_{1}=-(\frac{1}{2}\gamma H^{2}L^{2}+cH^{2}L)\quad B_{1}=\gamma H ^{3}L+\frac{1}{2}\gamma H^{2}L^{2}\tan\varphi\\ C_{1}=-(\gamma H^{3}L\tan\varphi+cH^{2}L)\quad D_{1}=a_{s}(L^{2}\sin\delta-L^{2} \cos\delta\tan\varphi)\\ E_{1}=a_{s}(2 HL\cos\delta\tan\varphi-2 HL\sin\delta-L^{2}\cos\delta-L^{2}\sin \delta\tan\varphi)\\ F_{1}=(1-a_{s})H^{2}\cos\delta\tan\varphi+2a_{s}HL\cos\delta+(1+a_{s})H^{2} \sin\delta+2a_{s}HL\sin\delta\tan\varphi\\ G_{1}=(1-a_{s})\cos\delta H^{2}-(1+a_{s})H^{2}\sin\delta\tan\varphi\end{array}  \tag{5}\]  
In the scenario where the soil behind the wall is in the active limit state, the resultant active earth pressure force (\(E_{a}\)) on the retaining wall is the maximum of the function \(E_{x1}(\theta)\). By analyzing numerous plots of the function \(E_{x1}(\theta)\) generated using Mathematica, it was observed that the point of maximum point within the range of the rupture angle is also the stationary point of the function. Therefore, to find the stationary point, the derivative of Equation (4) is taken, resulting in Equation (6). \[\frac{dE_{x1}}{d\psi}=\frac{(2A_{1}\psi+B_{1})(D_{1}\psi^{3}+E_{1}\psi^{2}+F_{1 }\psi+G_{1})-(A_{1}\psi^{2}+B_{1}\psi+C_{1})(3D_{1}\psi^{2}+2E_{1}\psi+F_{1})} {\left(D_{1}\psi^{3}+E_{1}\psi^{2}+F_{1}\psi+G_{1}\right)^{2}} \tag{6}\]  
In Equations (4) and (6), parameters \(\delta\), \(\varphi\), and \(x(\tan\theta)\) each possess distinct significance. Within the range of rupture angle values, it can be demonstrated that the denominator of the derivative of \(E_{1}(\theta)\) remains positive. To identify the stationary point, the derivative is set to zero as expressed in Equation (7). \[-A_{1}D_{1}\psi^{4}-2B_{1}D_{1}\psi^{3}+(A_{1}F_{1}-B_{1}E_{1}-3C_{1}D_{1})\psi ^{2}+2(A_{1}G_{1}-C_{1}E_{1})\psi+B_{1}G_{1}-C_{1}F_{1}=0 \tag{7}\]  
The above equation constitutes a quartic equation with four general solutions. However, given the explicit physical significance of the rupture angle \(\theta\) which must satisfy the condition \(\theta>0\), the specific analytical expression for the rupture angle is ultimately determined as depicted in Equation (8). \[\theta_{a}=\arctan\left(-\frac{b_{1}}{4a_{1}}+\frac{1}{2}\sqrt{\frac{{b_{1}}^ {2}}{4{a_{1}}^{2}}-\frac{2c_{1}}{3a_{1}}}+\Delta_{1}-\frac{1}{2}\sqrt{\frac{{b _{1}}^{2}}{2{a_{1}}^{2}}-\frac{4c_{1}}{3a_{1}}-\Delta_{1}+\frac{-\frac{{b_{1}} ^{3}}{{a_{1}}^{2}}+\frac{4b_{1}c_{1}}{{a_{1}}^{2}}-\frac{8d_{1}}{{a_{1}}}}{4 \sqrt{\frac{{b_{1}}^{2}}{{a_{1}}^{2}}-\frac{2c_{1}}{3a_{1}}+\Delta_{1}}}}\right) \tag{8}\]  
The coefficients are presented in Equation (9). \[ \begin{array}{c}a_{1}=-A_{1}D_{1}b_{1}=-2B_{1}D_{1}c_{1}=A_{1}F_{1}-B_{1}E_ {1}-3C_{1}D_{1}\\ d_{1}=2A_{1}G_{1}-2C_{1}E_{1}e_{1}=B_{1}G_{1}-C_{1}F_{1}\lambda_{1}=c_{1}{}^{ 2}-3b_{1}d_{1}+12a_{1}e_{1}\\ \eta_{1}=2{c_{1}}^{3}-9b_{1}c_{1}d_{1}+27a_{1}{d_{1}}^{2}+27b_{1}{e_{1}}-72a_ {1}c_{1}e_{1}\\ \Delta_{1}=\frac{3\gamma 2\lambda_{1}}{3a_{1}\sqrt[3]{\eta_{1}}+\sqrt{-4\lambda_{1} }^{3}+\eta_{1}{}^{2}}+\frac{\sqrt[3]{\eta_{2}}+\sqrt{-4\lambda_{1} }^{3}+\eta_{1}{}^{2}}{3\sqrt[3]{2}a_{1}}\end{array}  \tag{9}\]  
Based on the variation range of the soil parameters behind the wall, it was determined that Equation (7) typically has two imaginary and two real solutions. Among these solutions, Equation (6) exhibits opposite signs on either side of the real stagnation point, with the stationary point value exceeding zero. This stationary point coincides with the rupture angle corresponding to the maximum value of the function \(E_{x1}(\theta)\) computed using MATLAB, thereby confirming it as the maximum value point. By substituting \(\theta_{a}\) into Equation (3), the active limit earth pressure resultant force (\(E_{a}\)) acting on the retaining wall can be obtained under the condition of finite backfill width. \[