# Stability of Reinforced Retaining Wall under Seismic Loads
## 3 Seismic Loads  
Under seismic loads, the shear wave velocity \(V_{s}\) and compression wave velocity \(V_{p}\) are characterized as:  
\[V_{s}=\sqrt{\frac{G}{\rho}} \tag{2}\]  
\[V_{p}=\sqrt{\frac{G(2-2v)}{\rho(1-2v)}} \tag{3}\]  
where \(G\), \(\rho\), and \(v\) are shear modulus, density, and Poisson's ratio of soils, respectively. In this study, we assumed \(v\) is 0.3 for soils, therefore:  
\[\frac{V_{p}}{V_{s}}=1.87 \tag{4}\]  
The period for the horizontal seismic load \(T\) is:  
\[T=\frac{2\pi}{\omega}=\frac{4\mathrm{H}}{V_{S}} \tag{5}\]  
Figure 1: Illustration of failure mechanism of reinforced wall by the horizontal slice method (HSM) (modified after [PERSON] et al. 2018).  
Figure 2: Forces acting on the \(i\)th slice (modified after [PERSON] et al. 2018).  
\(\alpha_{h}\) and \(\alpha_{v}\) are horizontal and vertical seismic acceleration, respectively. We assumed that the direction of propagation of seismic waves is vertical. Then the acceleration of z deep location from the surface of the retaining wall at moment \(t\) is:  
\[\alpha_{h}(z,t)=\alpha_{h}\times\sin\left[\omega(t-\frac{\mathrm{H}-z}{V_{S}})\right] \tag{6}\]  
\[\alpha_{v}(z,t)=\alpha_{v}\times\sin\left[\omega(t-\frac{\mathrm{H}-z}{V_{p}})\right] \tag{7}\]  
The body mass of the element in z with \(m\) depth and \(dz\) height \(m_{i}(z)\) is:  
\[m_{i}(z)=\frac{\gamma}{g}\times\frac{\mathrm{H}-z}{\tan\alpha_{i}}\times dz, \tag{8}\]  
while the self-weight of the \(i\)th slice is:  
\[Wi=\gamma\times(\frac{l_{i}+l_{i+1}}{2})\times dz \tag{9}\]  
where \(\gamma\) is the unit weight of backfill soil, and \(l_{i}\) and \(l_{i+1}\) are the length of the top and bottom side of the \(i\)th slice, respectively.  
Horizontal inertia force on the \(i\)th element is:  
\[q_{hi}=m_{i}(z)\times\alpha_{h}(z,t) \tag{10}\]  
Then the total horizontal seismic load on the slice \(Q_{h}\) is:  
\[\begin{array}{c}Q_{h}=\int_{0}^{\mathrm{H}}q_{hi}dz=\int_{0}^{\mathrm{H}}m_ {i}(z)\times\alpha_{h}(z,t)dz=\int_{0}^{\mathrm{H}}\frac{\gamma}{g}\times \frac{\mathrm{H}-z}{\tan\alpha_{i}}\times\alpha_{h}\times\sin\left[\omega(t- \frac{\mathrm{H}-z}{V_{S}})\right]dz\\ =\frac{\lambda\chi\gamma\times Q_{h}}{4\pi^{2}\tan\alpha_{i}}[2\pi\mathrm{H} \cos\omega\xi+\lambda(\sin\omega\xi-\sin\omega t)]\end{array} \tag{11}\]  
The vertical inertia force on the \(i\)th element is:  
\[q_{vi}=m_{i}(z)\times\alpha_{v}(z,t) \tag{12}\]  
Then the total horizontal seismic load on the retaining wall \(Q_{v}\) is  
\[\begin{array}{c}Q_{v}=\int_{0}^{\mathrm{H}}q_{vi}dz=\int_{0}^{\mathrm{H}}m_ {i}(z)\times\alpha_{v}(z,t)dz=\int_{0}^{\mathrm{H}}\frac{\gamma}{g}\times \frac{\mathrm{H}-z}{\tan\alpha_{i}}\times\alpha_{v}\times\sin\left[\omega(t- \frac{\mathrm{H}-z}{V_{p}})\right]dz\\ =\frac{\lambda\chi\gamma\times Q_{v}}{4\pi^{2}\tan\alpha_{i}}[2\pi\mathrm{H} \cos\omega\psi+\lambda(\sin\omega\psi-\sin\omega t)]\end{array} \tag{13}\]  
where \(\lambda\) is the vertical wavelength of the shear wave (\(\lambda=\mathrm{T}\times V_{S}\)), and \(\eta\) denotes the vertical wavelength of the compression wave (\(\xi=t-\frac{\mathrm{H}}{V_{S}}\); \(\psi=\frac{\mathrm{H}}{V_{p}}\)). By considering \(q_{hi}\) and \(q_{vi}\) on the critical direction of the element, the limits of \(Q_{h}\) and \(Q_{v}\) according to [PERSON] and [PERSON] [14] are, respectively:  
\[\lim_{v\gg\to\infty}(Q_{h})_{\mathrm{max}}=\frac{\gamma\times\mathrm{H}^{2} \times\alpha_{h}}{2g\times\tan\alpha}=\frac{\alpha_{h}}{g}\times W=k_{h}\times W \tag{14}\]  
\[\lim_{v\gg\to\infty}(Q_{v})_{\mathrm{max}}=\frac{\gamma\times\mathrm{H}^{2} \times\alpha_{v}}{2g\times\tan\alpha}=\frac{\alpha_{v}}{g}\times W=k_{v}\times W \tag{15}\]  
where \(k_{h}=\alpha_{h}/g\) and \(k_{v}=\alpha_{v}/g\overline{X}\). When\[\sum\limits_{i=1}^{n}T_{i}+\sum\limits_{i=1}^{n}N_{t,i}\times\cos\alpha_{i}- \sum\limits_{i=1}^{n}N_{N,i}\times\sin\alpha_{i}-\sum\limits_{i=1}^{n}Q_{h,i}=0 \tag{17}\]