## 4 Active Earth Pressure Analysis under Seismic Action
### Seismic Active Lateral Pressure Coefficient for Cohesive Soil  
Soil arching effects exist objectively [36; 37], especially in limited soil and deep foundation pits. In the system under consideration in the present study, due to the lateral movement of the retaining structure and the influence of the wall-soil friction angle, the stress redistribution phenomenon will be more obvious (Figure 3a). In addition, this research is aimed at cohesive soil. Based on the research of [PERSON] and [PERSON] [15], and combined with the Mohr circle for initial liquefaction given by [PERSON] et al. [38], the stress Mohr circle when the soil behind the retaining structure reaches the active limit state under the earthquake is shown in Figure 3b. In Figure 3, \(\psi_{1}\) and \(\psi_{2}\) are the angles between the minor principal stress surface and the horizontal plane at the retaining structure and the slip surface, respectively; \(R\) is the radius of the minor principal stress arch; \(z\) is the distance from any point inside the soil to the surface; \(dz\) is the thickness of the thin-layer element; \(q\) is the uniform surcharge on the soil surface; \(dN\) is the vertical force at any point of the minor principal stress trajectory at depth \(z\); \(\tau_{w}\) is the shear stress on the retaining structure; \(\delta\) is the wall-soil friction angle;  
Figure 3: Arching effects in limited backfill: (**a**) trajectory of minor principal stress; (**b**) Mohr circle for stresses at the retaining structure. is the vertical seismic coefficient; \(K\) is the lateral earth pressure coefficient; and the other parameters are as mentioned above. From Figure 3b, the following relationship is given:  
\[\sigma^{\prime}=\sigma+\sigma_{0}=\sigma+c_{t}\cot\varphi_{t}, \tag{11}\]  
where \(\sigma\) and \(\sigma^{\prime}\) are the stress values in the new and old coordinate systems, respectively. When the soil reaches the active limit equilibrium state, combined with the stress Mohr circle, the principal stress deflection angle in the soil at the retaining structure can be obtained [15]:  
\[\psi_{1}=\tan^{-1}\left[\frac{1-K_{a}+\sqrt{\left(1-K_{a}\right)^{2}-4K_{a} \tan^{2}\delta}}{2K_{a}\tan\delta}\right], \tag{12}\]  
where \(K_{a}\) is the Rankine active earth pressure coefficient, and \(K_{a}=\tan^{2}(\pi/4-\varphi_{t}/2)\). The angle between the action surface of the major principal stress and the failure surface is \(\beta_{f}=\pi/4+\varphi_{t}/2\). Therefore, the angle between the minor principal stress surface and the horizontal plane at the slip surface can be given by:  
\[\psi_{2}=\frac{\pi}{2}-\beta_{f}+\alpha=\frac{3\pi}{4}+\frac{\varphi_{t}}{2}-\theta. \tag{13}\]  
As shown in Figure 3a, the average vertical stress acting on the thin-layer element can be obtained:  
\[\overline{\sigma^{\prime}}_{v}=\frac{\int_{\phi_{1}}^{\phi_{2}}\mathrm{d}N}{ R(\cos\psi_{1}-\cos\psi_{2})}=\sigma^{\prime}_{1}\bigg{[}1-\frac{1-K_{a}}{3}( \cos\psi_{1}+\cos\psi_{2}+\cos\psi_{1}\cos\psi_{2})\bigg{]}. \tag{14}\]  
Additionally, from Figure 3b, and considering the uniform surcharge on the soil surface, the following is obtained:  
\[ \begin{cases}\sigma^{\prime}_{x}=\sigma^{\prime}_{1}\cos^{2}\psi_{1}+\sigma^{ \prime}_{3}\sin^{2}\psi_{1}\\ \sigma_{1}=(1-k_{v})(\gamma z+q)\end{cases} , \tag{15}\]  
where \(\gamma\) is the unit weight of the retained soil.