## 5 Conclusions
; writing--review and editing, Z.S. and H.L. ; supervision, Z.S. ; project administration, W.W.; funding acquisition, W.W. All authors have read and agreed to the published version of the manuscript. **Funding:** This work was Supported by Open Research Fund of Anhui Provincial Key Laboratory of Building Earthquake Disaster Mitigation and Green Operations, Anhui Institute of Building Research & Design, Grant NO. 2022-JKYL-002; The Natural Science Foundation of Hubei Province, China, Grant No. 2022 CFB417. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The raw data supporting the conclusions of this article will be made available by the authors on request. **Conflicts of Interest:** The authors declare no conflicts of interest. Appendix A Derivation of the Formulas for the Active Earth Pressure Coefficient and the Distribution of the Active Earth Pressure  
As shown in Figure 2, it is assumed that the peak seismic acceleration of the failure wedge is linearly amplified along the height, and the amplification factors of the horizontal and vertical directions at point C are both \(f_{\rm a}\). \[a_{\rm E}=\frac{(f_{\rm a}k_{\rm h,g}-k_{\rm h,g})(H-z)}{H}+k_{\rm h,g}=[(f_{ \rm a}-1)(1-\frac{z}{H})+1]k_{\rm h,g} \tag{1}\]  
By the sine theorem,  
\[\frac{\rm AB}{\sin(\alpha-i)}=\frac{\rm BC}{\sin(i+90^{\circ}-\theta)} \tag{2}\]  
Find the vertical projection of BC as follows:  
\[L=\frac{\cos(\theta-i)\sin\alpha}{\cos\theta\sin(\alpha-i)}H \tag{3}\]  
From the geometric relationship, the horizontal seismic acceleration at point D can be expressed as follows:  
\[a_{\rm D}=\frac{(f_{\rm a}k_{\rm h,g}-k_{\rm h,g})(H-z)}{L}+k_{\rm h,g}=[( \frac{\cos\theta\sin(\alpha-i)}{\cos(\theta-i)\sin\alpha})(f_{\rm a}-1)(1- \frac{z}{H})+1]k_{\rm h,g} \tag{4}\]  
The acceleration in the shade slice in Figure 1 is linearly distributed from D to E, and the mass distribution of the slice is uniform. The representative peak acceleration can be expressed as the average acceleration of D and E. When the sine wave is input from the base, the horizontal acceleration at the \(z\)-depth of the slope can be expressed as follows:  
\[a_{\rm h}(z,t)=[(\frac{\cos\theta\sin(\alpha-i)+\cos(\theta-i)\sin\alpha}{2 \cos(\theta-i)\sin\alpha})(f_{\rm a}-1)(1-\frac{z}{H})+1]k_{\rm h,g}\sin\omega \bigg{(}t-\frac{H-z}{V_{\rm g}}\bigg{)} \tag{5}\]  
Similarly, the vertical acceleration at the \(z\)-depth of the slope can be expressed as follows:  
\[a_{\rm v}(z,t)=[(\frac{\cos\theta\sin(\alpha-i)+\cos(\theta-i)\sin\alpha}{2 \cos(\theta-i)\sin\alpha})(f_{\rm a}-1)(1-\frac{z}{H})+1]k_{\rm v,g}\sin\omega \bigg{(}t-\frac{H-z}{V_{\rm p}}\bigg{)} \tag{6}\]  
The self-weight of the slice is as follows:  
\[m(z)=\frac{\gamma(H-z)(1+\tan\alpha\tan\theta)\cos\alpha\cos(\theta-i)}{g\sin (\alpha-i)\cos\theta}{\rm d}z \tag{7}\]The self-weight of the failure wedge obtained by integrating the Equation (A7) is as follows:  
\[W=\frac{\gamma H^{2}(1+\tan\alpha\tan\theta)\cos\alpha\cos(\theta-i)}{2\sin( \alpha-i)\cos\theta}\] (A8)  
The horizontal inertial force generated by horizontal shaking in the shade slice can be derived as follows:  
\[ \begin{array}{l}Q_{\rm h}(t)=\int\limits_{0}^{\rm H}m(z)a_{\rm h}(z,t)=\\ \int\limits_{0}^{\rm H}\frac{\gamma(H-z)(1+\tan\alpha\tan\theta)\cos\alpha\cos (\theta-i)}{g\sin(\alpha-i)\cos\theta}[(\frac{\cos\theta\sin(\alpha-i)+\cos (\theta-i)\sin\alpha}{2\cos(\theta-i)\sin\alpha})\\ (f_{\rm a}-1)(1-\frac{z}{H})+1]k_{\rm h}g\sin\omega(t-\frac{H-z}{V_{\rm s}}) \rm d\it{z}\end{array} \] (A9)  
Make \(J=\frac{(1+\tan\alpha\tan\theta)\cos\alpha\cos(\theta-i)}{\sin(\alpha-i)\cos\theta}\) and \(K=\frac{\cos\theta\sin(\alpha-i)+\cos(\theta-i)\sin\alpha}{2\cos(\theta-i) \sin\alpha}\), \(\lambda=TV_{\rm s}\), \(\omega=2\pi/T\); Equation (A9) can be simplified as follows:  
\[