# Calculation of Nonlimit Active Earth Pressure against Rigid Retaining Wall Rotating about Base
## 4 Derivation of the Analytical Solution for Active Earth Pressure
### Definition of Small Principal Stress Traces Parameters  
#### 4.2.1 Definition of Deflection Angle of Principal Stress  
The soil wedge ABC behind the wall has three boundaries. AB represents the wall back, BC represents the sliding surface of soil mass, and AC represents the top surface of fill. It is assumed that the principal stress trace intersects with the wall back and sliding surface at points D and E, respectively. As illustrated in Figure 2a, assuming that the included angle between the direction of small principal stress and the vertical direction is \(\theta_{a}\) at the wall-soil contact, Equations (6) and (7) can be obtained from the static equilibrium condition. \[\sigma_{1m} =\sigma_{1}^{a}\cos^{2}\theta_{a}+\sigma_{3}^{a}\sin^{2}\theta_{a} \tag{6}\] \[\tau_{1m} =(\sigma_{1}^{a}-\sigma_{3}^{a})\sin\theta_{a}\cos\theta_{a} \tag{7}\] \begin{table}
\begin{tabular}{c c c} \hline \multicolumn{2}{c}{**Soil Types and Properties of Materials**} & \multicolumn{1}{c}{\(\mathbf{K_{0}}\)} \\ \hline \multicolumn{2}{c}{Gravelly soil} & \multicolumn{1}{c}{0.17} \\ \hline \multirow{4}{*}{Sand soil} & e = 0.5 & 0.23 \\  & e = 0.6 & 0.34 \\  & e = 0.7 & 0.52 \\  & e = 0.8 & 0.6 \\ \hline \multirow{2}{*}{Silty soil and powdered clay} & w = 15–20\% & 0.43–0.54 \\  & w = 25–30\% & 0.60–0.75 \\ \hline \multirow{2}{*}{Clay} & Hard clay & 0.11–0.25 \\  & Compact clay & 0.33–0.45 \\  & Plastic clay & 0.61–0.82 \\ \hline \multirow{2}{*}{Peat soil} & High organic matter content & 0.24–0.37 \\  & Low organic matter content & 0.40–0.65 \\ \hline \multicolumn{2}{c}{Sandy silty soil} & \multicolumn{1}{c}{0.33} \\ \hline \end{tabular}
\end{table} 
Table 1: Static earth pressure coefficient \(\mathbf{K}_{0}\). where \(\sigma_{1m}\) = normal stress at the wall-soil contact surface; \(\tau_{1m}\) = shear stress in the soil at the wall-soil contact; \(\sigma_{1}^{a}\) = maximum principal stress at the wall-soil contact surface; and \(\sigma_{3}^{a}\) = minimum principal stress at the wall-soil contact surface. In addition, \(\theta_{a}\) is the deflection angle of small principal stress. The stresses at the wall-soil contact are subject to the Mohr-Coulomb criteria, which can be expressed as  
\[N=\frac{\sigma_{1}^{a}}{\sigma_{3}^{a}}=\tan^{2}(45^{\circ}+\frac{\varphi}{2}) \tag{8}\]  
Figure 2: (**a**) Definition of small principal stress traces parameters. (**b**) Definition of arc trace radius. \[\tau_{1m}=\sigma_{1m}\tan\delta_{m} \tag{9}\]  
where \(N\) is the ratio of maximum principal stress to minimum principal stress, \(\varphi\) is the soil friction angle, and \(\delta_{m}\) is the developed value of interface friction angle. By solving Equations (6)-(9), Equation (10) can be obtained as follows  
\[\theta_{a}=\arctan\left[\frac{\left(N-1\right)+\sqrt{\left(N-1\right)^{2}-4N \tan^{2}\delta_{m}}}{2\tan\delta_{m}}\right] \tag{10}\]  
where \(\theta_{a}\) is the included angle between the direction of small principal stress and the vertical direction at the wall-soil contact surface, and \(\delta_{m}\) is the mobilized value of interface friction angle. It is assumed that \(\theta_{b}\) is the included angle between the direction of small principal stress and the vertical direction at the potential slip plane. Equations (11) and (12) can be established:  
\[\sigma_{2m}=\sigma_{1}^{b}\sin^{2}(\theta_{b}-\alpha_{s})+\sigma_{3}^{b}\cos ^{2}(\theta_{b}-\alpha_{s}) \tag{11}\]  
\[\tau_{2m}=(\sigma_{1}^{b}-\sigma_{3}^{b})\sin(\theta_{b}-\alpha_{s})\cos( \theta_{b}-\alpha_{s}) \tag{12}\]  
where \(\sigma_{2m}\) = normal stress at the potential slip plane; \(\tau_{2m}\) = shear stress at the potential slip plane; \(\sigma_{1}^{b}\) = maximum principal stress at the potential slip plane; \(\sigma_{3}^{b}\) = minimum principal stress at the potential slip plane.