## 2 Seismic Active Earth Pressure Analysis
### Analysis of Active Earth Pressure Forces on a Retaining Wall Under Seismic Loading  
Real seismic ground motions exhibit significant three-dimensional spatial characteristics, where \(P\)-waves, \(S\)-waves, and surface waves form complex wave field interferences during their propagation. Due to this complexity, the sinusoidal wave model, as the most fundamental mathematical representation of ground motion, maintains an irreplaceable position in earthquake engineering. This simple periodic function is widely applied in various fields, including seismic building design, earthquake early warning system development, and ground motion parameter studies. Within the theoretical framework of the Fourier transform method, any complex ground motion time history can be decomposed into the superposition of sinusoidal waves with different frequencies, amplitudes, and phases. This mathematical property establishes sinusoidal waves as the fundamental unit for the frequency-domain analysis of ground motions. As shown in Figure 2, the height of a rigid retaining wall is \(H\), while the retaining wall slope angle is \(\theta\). The backfill cohesion is \(c\) and the internal friction angle is \(\varphi\). The unit weight of the backfill is \(\gamma\). The friction angle between the backfill and the retaining wall is \(\delta\). The angle between the failure surface of the backfill and the horizontal bottom plane is \(\alpha\). The backfill slope angle is \(i\). Under sinusoidal shaking conditions, the amplitude of the horizontal and vertical seismic acceleration at the bottom is \(k_{\mathrm{h}}g\) and \(k_{\mathrm{v}}g\), respectively. The following basic assumptions are proposed: (1) the sliding surface is planar, and the failure wedge passes through the heel of the retaining wall; (2) the seismic acceleration varies with the time and depth; (3) the backfill soil is homogeneous and isotropic, with a constant shear modulus along the wall height; (4) the retaining wall is assumed rigid, with its deformation negligible.; (5) the seismic acceleration amplification (attenuation) coefficient of the backfill soil follows a linear distribution along the elevation; (6) the seismic vibration of the retaining wall-soil system is characterized as sinusoidal steady-state vibration. The detailed derivation process of Equations (1) and (2) is provided in Appendix A, and the definitions of all symbols are listed in Appendix B.  
The lateral active force acting on the retaining wall \(P_{\mathrm{ae}}(t)\) can be defined as follows:  
\[P_{\mathrm{ae}}(t)=\frac{1}{2}\gamma H^{2}K_{\mathrm{ae}}(t)-K_{\mathrm{ac}} cH+\frac{2 Kc^{2}}{\gamma} \tag{1}\]  
The active earth pressure coefficient \(K_{\mathrm{a}}(t)\) can be defined as follows:  
\[K_{\mathrm{a}}(t)=\frac{2P_{\mathrm{ae}}(t)}{\gamma H^{2}} \tag{2}\]  
From Equations (A1)-(A18) in Appendix A, it can be observed that the active earth pressure coefficient \(K_{\mathrm{a}}(t)\) is a function of dimensionless expressions like \(\alpha\), \(t/T\), \(H/\lambda\), \(H/\eta\), and \(c/\gamma H\).  
At a certain time \(t\) in the shaking period, there is a corresponding sliding surface with an inclination angle \(\alpha\) and an active earth pressure coefficient \(K_{\mathrm{a}}(t)\). The peak active earth pressure coefficient coincides with the critical stress state of backfill soil under seismic loading, with the corresponding angle \(\alpha\) defining the inclination of the critical failure plane.  
In special cases, when \(c\
eq 0\), \(\varphi\
eq 0\), \(\theta=0\), \(\delta=0\), \(k_{\mathrm{h}}=0\), \(k_{\mathrm{v}}=0\), \(\alpha=\pi/4+\varphi/2\), \(K_{\mathrm{ae}}(t)\) reduces to \(\tan^{2}\left(\pi/4-\varphi/2\right)\), \(K_{\mathrm{ac}}\)  
It is interesting to find that Equation (3) is the same as the Rankine equation for cohesive backfills in the active earth pressure state. Furthermore, Equation (3) can be reduced to the common Rankine equation for cohesionless backfills when \(c=0\).