## Appendix A The Proposed Analysis  
For static conditions, the state of stress in Zone I is known completely, and it is admissible as long as the base AC of the prism is sufficiently rough [34]. The stress components in region ABO (Zone I) is given by Equation (A1). The steps for the derivation to these equations (for [PERSON]'s Zone I) can be found in Pipatpongsa and Vardhanabhuti [35]. \[\sigma_{x} =\gamma(z\cos\varphi^{\prime}-x\sin\varphi^{\prime})\cos\varphi^{ \prime}\] \[\sigma_{z} =\gamma(z-x\tan\varphi^{\prime})\Big{(}1+\sin^{2}\!\varphi^{ \prime}\Big{)}\] \[\tau_{xx} =\gamma(z\cos\varphi^{\prime}-x\sin\varphi^{\prime})\sin\varphi^{ \prime}\]  
Considering dynamic conditions, the above stresses should be recalculated as to include the influence of pseudo-dynamic forces. The equations of equilibrium in Zone I will then become:  
\[\left. \begin{array}{l}\frac{\partial\sigma_{x}}{\partial x}+\frac{\partial \tau_{xx}}{\partial z}=-k_{h}\gamma\\ \frac{\partial\tau_{xx}}{\partial x}+\frac{\partial\sigma_{z}}{\partial z}=(1 -k_{p})\gamma\end{array} \right\}.\] (A2)  
It is noted that the most unfavorable condition in the problem of earth pressures arises for positive seismic coefficients. The resistance term is not included in these equations, because Zone I is cohesionless, whilst, the friction resistance will be taken into account through the following Mohr's circles analysis. Zone I balances at its maximum friction angle. From Mohr-Coulomb criterion, it is:  
\[\tau_{xz}=\sigma_{x}\tan\varphi^{\prime},\] (A3)  
whilst,\(\sigma_{x}\) can be related to \(\sigma_{z}\) as follows:  
\[\frac{\sigma_{x}+\sigma_{z}}{2}-\sigma_{x}=\tau_{xz}\tan\varphi^{\prime}.\] (A4)  
Substituting Equation (A3) into Equation (A4) and solving as for \(\sigma_{z}\):  
\[\sigma_{z}=\sigma_{x}+2\tau_{xz}\tan\varphi^{\prime}=\Big{(}1+2\tan^{2}\varphi ^{\prime}\Big{)}\sigma_{x}.\] (A5)  
The stress relations of Equations (A3) and (A5) are then substituted into Equations (A2), giving  
\[\left. \begin{array}{l}\frac{\partial\sigma_{x}}{\partial x}+\tan\varphi\frac {\partial\sigma_{x}}{\partial z}=-k_{h}\gamma\\ \tan\varphi^{\prime}\frac{\partial\sigma_{x}}{\partial x}+\Big{(}1+2\tan^{2} \varphi^{\prime}\Big{)}\frac{\partial\sigma_{x}}{\partial z}=(1-k_{p})\gamma \end{array} \right\}.\] (A6)  
The first-order spatial derivatives of \(\sigma_{x}\) are, finally,  
\[\left. \begin{array}{l}\frac{\partial\sigma_{x}}{\partial x}=-\gamma Q_{ \rm tan}\\ \frac{\partial\sigma_{x}}{\partial z}=\gamma Q_{\rm cot}\end{array} \right\},\] (A7)  
where,  
\[Q_{\rm tan}=\left. \begin{array}{l}\frac{k_{h}\left(1+2\tan^{2}\varphi^{ \prime}\right)+(1-k_{e})\tan\varphi^{\prime}}{1+\tan^{2}\varphi^{\prime}}\\ Q_{\rm cot}=\frac{k_{h}\cot\varphi^{\prime}+(1-k_{e})\cot^{2}\varphi^{\prime} }{1+\cot^{2}\varphi^{\prime}}\end{array} \right\}.\] (A8)  
The differential of \(\sigma_{x}\) with respect to \(x\) and \(z\) is given as follows  
\[d\sigma_{x}=\frac{\partial\sigma_{x}}{\partial x}dx+\frac{\partial\sigma_{x}} {\partial z}dz.\] (A9)  
Integration of \(d\sigma_{x}\) can be carried out due to its simple form, with an integral constant \(C\) to be left after integration. \[\sigma_{x}=-\gamma Q_{\rm tan}\int dx+\gamma Q_{\rm cot}\int dz+C.\] (A10)The state of stress at the apex of the granular heap (\(x=0,z=0\)) is zero (\(\sigma_{x}=0\)), therefore, \(C=0\). The stress components \(\sigma_{z}\) and \(\tau_{xz}\) are obtained by substituting \(\sigma_{x}\) into Equations (A6) and (A5).