## Appendix A The Proposed Analysis
Thus, the three components become:  
\[\sigma_{x} =(-Q_{\text{tan}}x+Q_{\text{cot}}z)\gamma \tag{11}\] \[\sigma_{z} =\left(1+2\tan^{2}\varphi^{\prime}\right)(-Q_{\text{tan}}x+Q_{ \text{cot}}z)\gamma\] \[\tau_{xz} =\tan\varphi^{\prime}(-Q_{\text{tan}}x+Q_{\text{cot}}z)\gamma\]  
The equation of borderline OB is  
\[z=\frac{x_{OB}}{\tan\left(45^{o}-\frac{\varphi^{\prime}}{2}\right)}. \tag{12}\]  
Substituting \(z\) from above, the three stress components given in Equation (A11) on the borderline OB will become  
\[\sigma_{x,OB} =\left(-Q_{\text{tan}}+\frac{Q_{\text{cot}}}{\tan\left(45^{o}- \frac{\varphi^{\prime}}{2}\right)}\right)x_{OB}\gamma \tag{13}\] \[\sigma_{z,OB} =\left(1+2\tan^{2}\varphi^{\prime}\right)\left(-Q_{\text{tan}}+ \frac{Q_{\text{cot}}}{\tan\left(45^{o}-\frac{\varphi^{\prime}}{2}\right)} \right)x_{OB}\gamma\] \[\tau_{xz,OB} =\tan\varphi^{\prime}\left(-Q_{\text{tan}}+\frac{Q_{\text{cot}} }{\tan\left(45^{o}-\frac{\varphi^{\prime}}{2}\right)}\right)x_{OB}\gamma\]  
Stresses in Zone II cannot yet be characterized because the direction of principle stresses and slide planes are both unknown. It is known, however, that the shear stress on the line OC is zero and on OB \(\tau_{xz}=\tau_{xz,OB}\). The variation of shear stress between these two values is assumed to follow the power function of \(x\) given by Equation (11) (used along with the \(f(m)\) function of Equation (12)). The rationality of Equation (11) and the role of the function \(f(m)\) have already been discussed in Sections 7 and 10.1. Substituting \(\tau_{xz,OB}\) and \(x_{OB}=z\tan(45^{o}-\varphi^{\prime}/2)\) to Equation (11),  
\[\tau_{xz}=f(m)\gamma z\tan\varphi^{\prime}\left(Q_{\text{cot}}-Q_{\text{tan}} \tan\left(45^{o}-\frac{\varphi^{\prime}}{2}\right)\right)\left(\frac{x}{z\tan \left(45^{o}-\frac{\varphi^{\prime}}{2}\right)}\right)^{m}, \tag{14}\]  
with partial derivatives  
\[ \begin{split}\frac{\partial\tau_{xz}}{\partial x}&=f( m)m\gamma\frac{\tan\varphi^{\prime}}{\tan\left(45^{o}-\frac{\varphi^{\prime}}{2} \right)}\left(\frac{x}{z\tan\left(45^{o}-\frac{\varphi^{\prime}}{2}\right)} \right)^{m-1}.\\ &\quad\cdot\left(Q_{\text{cot}}-Q_{\text{tan}}\tan\left(45^{o}- \frac{\varphi^{\prime}}{2}\right)\right),\end{split}  \tag{15}\]  
and  
\[ \begin{split}\frac{\partial\tau_{xz}}{\partial z}&=f( m)\gamma\tan\varphi^{\prime}(1-m)\left(\frac{x}{z\tan\left(45^{o}-\frac{ \varphi^{\prime}}{2}\right)}\right)^{m}.\\ &\quad\cdot\left(Q_{\text{cot}}-Q_{\text{tan}}\tan\left(45^{o}- \frac{\varphi^{\prime}}{2}\right)\right).\end{split}  \tag{16}\]  
Working first for \(\sigma_{z}\), from Equation (A2)  
\[\frac{\partial\sigma_{z}}{\partial z}=(1-k_{v})\gamma-\frac{\partial\tau_{xz}}{ \partial x}, \tag{17}\]from where, integrating as for \(z\)  
\[\sigma_{z}=\gamma z\bigg{(}(1-k_{\rm p})-f(m)Q_{1}\bigg{(}\frac{x}{z}\bigg{)}^{m- 1}\bigg{)}+f(x), \tag{111}\]  
where,  
\[Q_{1}=\frac{m}{2-m}\frac{\tan\varphi^{\prime}}{tan^{m}\big{(}45^{ \circ}-\frac{\varphi^{\prime}}{2}\big{)}}\bigg{(}Q_{\rm cot}-Q_{\rm tan}\tan \bigg{(}45^{\circ}-\frac{\varphi^{\prime}}{2}\bigg{)}\bigg{)}. \tag{112}\]  
On the borderline OB, it is \(x=x_{OB}\), \(z=x_{OB}\tan(45^{\circ}+\varphi^{\prime}/2)\) and \(\sigma_{z}=\sigma_{z,OB}\), thus,  
\[f(x) =\sigma_{z,OB}-\gamma z\bigg{(}(1-k_{\rm p})-f(m)Q_{1}\bigg{(} \frac{x}{z}\bigg{)}^{m-1}\bigg{)}= \tag{113}\] \[=x_{OB}\gamma\left( \begin{array}{c}\Big{(}1+2\tan^{2}\varphi^{ \prime}\Big{)}\bigg{(}-Q_{\rm tan}+\frac{Q_{\rm out}}{\tan\big{(}45^{\circ}- \frac{\varphi^{\prime}}{2}\big{)}}\bigg{)}-\\ \tan\bigg{(}45^{\circ}+\frac{\varphi^{\prime}}{2}\bigg{)}\bigg{(}(1-k_{\rm p}) -f(m)Q_{1}\bigg{(}\frac{1}{\tan\big{(}45^{\circ}+\frac{\varphi^{\prime}}{2} \big{)}}\bigg{)}^{m-1}\bigg{)}\end{array} \right).\]  
For \(x_{OB}=0\), \(\sigma_{z,OB}\) is also zero, thus \(f(x)=0\). On the vertical line OC \(x=0\), thus, Equation (111) simplifies to  
\[\sigma_{z,\rho}=\gamma z\left(