## 4 Active Earth Pressure Analysis under Seismic Action
### Establishment of Seismic Active Earth Pressure Formula  
The thin-layer element method is used in this part. The selected element is shown in Figure 4. When the element thickness \(\mathrm{d}z\) is sufficiently small, \(\overline{bc}\) can be approximated as a straight line.  
In Figure 4, \(\overline{\sigma}_{v}\) and \(\overline{\sigma}_{v}+\mathrm{d}\overline{\sigma}_{v}\) are the average vertical stress acting on the top and bottom of the thin-layer element, respectively; \(k_{h}\) and \(k_{v}\) are the horizontal and vertical seismic coefficients, respectively; \(\mathrm{d}G\) is the self-weight of the thin-layer element; \(\sigma_{x}\) is the horizontal reaction force of the retaining structure; \(\sigma_{r}\) is the reaction force of the lower soil body to the upper soil; and \(\tau_{s}\) is the shear stress at the slip surface.  
According to Figures 2 and 4, the geometric relationship of the element is:  
\[\begin{cases}\overline{ab}=r_{0}\{\exp[(\theta-\theta_{0})\tan\varphi_{t}]\cos \theta-\exp[(\theta_{a}-\theta_{0})\tan\varphi_{t}]\cos\theta_{a}\}\\ \overline{cd}=\overline{ab}-\cot\alpha\mathrm{d}z;\ \overline{ad}=\mathrm{d}z;\ \overline{bc}=\csc\alpha\mathrm{d}z\end{cases}. \tag{20}\]  
The self-weight of the element is (higher-order infinitesimals omitted):  
\[\mathrm{d}G=\gamma S_{abcd}=\gamma\overline{ab}\mathrm{d}z. \tag{21}\]  
The shear stresses on both sides of the element are taken as:  
\[\begin{cases}\tau_{w}=c_{a}+\sigma_{x}\tan\delta=(c_{t}\cot\varphi_{t}+\sigma _{x})\tan\delta\\ \tau_{s}=c_{t}+\sigma_{x}\tan\varphi_{t}\end{cases}, \tag{22}\]  
where \(c_{a}\) is the adhesion between the retaining structure and soil.  
The stress equilibrium equation in the horizontal and vertical directions of the thin-layer element are:  
\[\begin{cases}\sigma_{x}\overline{ad}+\tau_{s}\cos\alpha\overline{bc}=k_{h} \mathrm{d}G+\sigma_{r}\overline{bc}\sin\alpha\\ \overline{\sigma}_{v}\overline{ab}+(1-k_{v})\mathrm{d}G=(\tau_{s}\sin\alpha+ \sigma_{r}\cos\alpha)\overline{bc}+(\overline{\sigma}_{v}+\mathrm{d}\overline {\sigma}_{v})\overline{cd}+\tau_{w}\overline{ad}\end{cases}. \tag{23}\]  
By combining Equations (20)-(23) and with higher-order infinitesimals omitted, the following differential equation is obtained:  
\[\frac{\mathrm{d}\overline{\sigma}_{v}}{\mathrm{d}z}+A_{1}\overline{\sigma}_{v }=A_{2}, \tag{24}\]  
where  
\[\begin{cases}A_{1}=\frac{K^{\prime}(\tan\delta+\tan\theta)-\tan(\theta- \varphi_{t})}{\mathrm{d}\overline{b}}\\ A_{2}=(1+k_{h}\tan\theta-k_{v})\gamma-\frac{c_{t}\{1+K^{\prime}\cot\varphi_{t}( \tan\delta+\tan\theta)+\tan\theta[\tan(\theta-\varphi_{t})-\cot\varphi_{t}] \}}{\mathrm{d}\overline{b}}\end{cases}. \tag{25}\]  
Equation (24) is solved with the following boundary condition, \(\overline{\sigma}_{v}|_{z=z_{c}+\tau_{0}\sin\theta_{0}}=(1-k_{v})(\gamma z_{c }+q)\), yielding:  
\[\overline{\sigma}_{v}=B\exp[A_{1}(z_{1}-z)]+A_{2}/A_{1}, \tag{26}\]  
Figure 4: Horizontal thin-layer element model.  
where  
\[\begin{cases}B=(1-k_{v})(\gamma z_{c}+q)-A_{2}/A_{1}\\ z_{1}=z_{c}+r_{0}\sin\theta_{0}\end{cases}. \tag{27}\]  
In summary, the distribution formula of the seismic active earth pressure can be expressed as:  
\[\sigma_{x}=K\{B\exp[A_{1}(z_{1}-z)]+A_{2}/A_{1}\}. \tag{28}\]