## 5 Conclusions
\begin{array}{l}Q_{\rm h}(t)=\frac{\gamma k_{\rm h}\lambda IK(f_{\rm a}-1)}{4 \pi^{2}}\Big{[}(2\pi H-\frac{\lambda^{2}}{\pi H})\cos 2\pi(\frac{t}{T}-\frac{H}{ \lambda})+2\lambda\sin 2\pi(\frac{t}{T}-\frac{H}{\lambda})+\frac{\lambda^{2}}{ \pi H}\cos 2\pi t\Big{]}+\\ \frac{\gamma k_{\rm h}\lambda I}{4\pi^{2}}\Big{[}2\pi H\cos 2\pi(\frac{t}{T}- \frac{H}{\lambda})+\lambda\sin 2\pi(\frac{t}{T}-\frac{H}{\lambda})-\lambda\sin \frac{2\pi t}{T}\Big{]}\end{array} \] (A10)  
where \(\lambda\) is the wavelength of the shear wave, \(T\) is the period of shaking, and \(\omega\) is the angular frequency. Similarly, the vertical inertial force generated by the vertical ground motion in the failure wedge can be expressed as follows:  
\[ \begin{array}{l}Q_{\rm v}(t)=\frac{\gamma k_{\rm x}\eta IK(f_{\rm a}-1)}{4 \pi^{2}}\Big{[}(2\pi H-\frac{\eta^{2}}{\pi H})\cos 2\pi(\frac{t}{T}-\frac{H}{ \eta})+2\eta\sin 2\pi(\frac{t}{T}-\frac{H}{\eta})+\frac{\eta^{2}}{\pi H}\cos \frac{2\pi t}{T}\Big{]}+\\ \frac{\gamma k_{\rm x}\eta I}{4\pi^{2}}\Big{[}2\pi H\cos 2\pi(\frac{t}{T}-\frac{H}{ \eta})+\eta\sin 2\pi(\frac{t}{T}-\frac{H}{\eta})-\eta\sin\frac{2\pi t}{T}\Big{]} \end{array} \] (A11)  
where \(\eta\) is the wavelength of the primary wave. The lateral active force acting on the retaining wall, \(P_{\rm ae}(t)\), is obtained by considering the equilibrium of the wedge,  
\[P_{\rm ae}(t)=\frac{W\sin(\alpha-\varphi)+Q_{\rm h}(t)\cos(\alpha-\varphi)-Q_{ \rm v}(t)\sin(\alpha-\varphi)-c_{\rm w}L_{\rm a1}\sin(\alpha-\varphi-\theta)- cL_{\rm a2}\cos\varphi}{\cos(\delta+\theta+\varphi-\alpha)}\] (A12)  
where \(c_{\rm w}\) is the soil-wall adhesion. Based on experience, it is usually assumed that \(c_{\rm w}/c=\tan\delta/\tan\varphi\). \(L_{\rm a1}\) is the effective adhesive length between the backfill and wall, and \(L_{\rm a2}\) is the effective length of sliding surface in the backfill. As shown in Figure 2,  
\[L_{\rm a1}=\frac{H-z_{\rm c}}{\cos\theta}\] (A13a) \[L_{\rm a2}=\frac{\cos(\theta-i)(H-z_{\rm c}/2)}{\sin(\alpha-i)\cos\theta}\] (A13b)  
where \(z_{\rm c}\) is the depth of the tension crack. Based on [PERSON]'s analysis [36], in static conditions, \(z_{\rm c}\) can be computed as follows:  
\[z_{\rm c}=\frac{2c}{\gamma}\tan(\frac{\pi}{4}+\frac{\varphi}{2})\] (A14)  
Substituting Equations (A10), (A11), and (A13) into Equation (A12) yields  
\[P_{\rm ae}(t)=\frac{1}{2}\gamma H^{2}K_{\rm ae}(t)-K_{\rm ac}cH+\frac{2 Kc^{2} }{\gamma}\] (A15)where  
\[K_{\rm ae}(t)=\frac{2W\sin(\alpha-\varphi)}{\gamma H^{2}\cos(\delta+\theta+ \varphi-\alpha)}+\frac{2Q_{\rm h}(t)\cos(\alpha-\varphi)}{\gamma H^{2}\cos( \delta+\theta+\varphi-\alpha)}-\frac{2Q_{\rm v}(t)\sin(\alpha-\varphi)}{\gamma H ^{2}\cos(\delta+\theta+\varphi-\alpha)} \tag{16}\]  
\[K_{\rm ac}=\frac{\sin(\alpha-\varphi-\theta)\tan\delta}{\cos(\delta+\theta+ \varphi-\alpha)\cos\theta\tan\varphi}+\frac{\cos(\theta-i)\cos\varphi}{\cos( \delta+\theta+\varphi-\alpha)\sin(\alpha-i)\cos\theta} \tag{17}\]  
\[K=(\frac{\sin(\alpha-\varphi-\theta)\tan\delta}{\cos(\delta+\theta+\varphi- \alpha)\cos\theta\tan\varphi}+\frac{\cos(\theta-i)\cos\varphi}{2\cos(\delta+ \theta+\varphi-\alpha)\sin(\alpha-i)\cos\theta})\tan(\frac{\pi}{4}+\frac{ \varphi}{2}) \tag{18}\]  
The distribution of active earth pressure on the retaining wall can be expressed as follows:  
\[