## 3 Calculation Formulas for Non-Limited Seismic Active Earth Pressure
### Validation of the Calculation Method  
(1) When \(k_{\rm h}=0\), validation using experimental data from Reference [12]: Rigid wall with a vertical back face and height \(H=1.0\) m; the backfill is air-dried sandy soil with horizontal surface and no surcharge. The parameters used in the calculation are unit weight \(\gamma=15.34\) kN/m\({}^{3}\), internal friction angle \(\varphi=33.4^{\circ}\), initial soil friction angle \(\varphi_{0}=8^{\circ}\), soil-wall friction angle: \(\delta=25^{\circ}\), initial soil-wall friction angle: \(\delta_{0}=10^{\circ}\) and displacement \(S_{\delta}(z)=0.1\%\)\(z\). Using Equation (8), the calculated rupture angle \(\alpha\) is \(57.6^{\circ}\). The comparison between experimental and calculated values of the horizontal earth pressure intensity \(p_{\rm m}\) distribution along the wall height under different horizontal displacements of a rigid retaining wall in the RB displacement mode is shown in Figure 9a. From Figure 9b, it can be observed that the calculated and measured values of \(p_{\rm m}\) at various points along the wall height closely match under different horizontal displacements at the wall top. As the horizontal displacement at the wall top increases, the horizontal earth pressure continuously decreases. Due to the smaller displacement in the lower soil layer, the mobilized friction angles between the wall-soil interface and the soil itself exhibit relatively minor variations, resulting in a gentler reduction trend of \(p_{\rm m}\) in the lower section. Specifically, the soil pressure at the wall toe changes minimally as the rotation angle increases. In the RB displacement mode, the displacement at the wall base is approximately zero. The proposed method in this study yields an at-rest earth pressure at this location, which remains constant regardless of the wall top displacement. However, in reality, the wall base is constrained by the foundation, leading to complex soil stress conditions that require further investigation. When the horizontal displacement at the wall top under the RB displacement mode is small, slight soil arching effects are observed in the upper soil layer from the measured pressure distribution. Since these effects are not considered in the proposed calculation method, the calculated values for this region are slightly lower than the measured values. The relationships between the ratio \(h/H\) (the distance from the wall base to the horizontal earth pressure resultant force's acting point relative to the wall height), the horizontal earth pressure coefficient \(K_{\rm m}\), and the relative horizontal displacement \(S/H\) (or wall rotation angle) are illustrated in Figure 9b. Figure 9b shows that the calculated and measured values of \(h/H\) and \(K_{\rm m}\) agree well and follow consistent trends. The calculated relative position \(h/H\) of the resultant force aligns closely with experimental data. As the horizontal displacement at the wall top increases, \(h/H\) decreases gradually from an initial value of \(1/3\) (corresponding to the at-rest earth pressure state). This occurs because, under small displacements, the reduction rate of earth pressure in the upper wall section exceeds that in the lower section, shifting the resultant force downward. With further displacement, the upper soil reaches the active earth pressure state and ceases to decrease, while pressure near the wall base diminishes slowly. Only the mid-section continues to decrease, causing the resultant force to gradually rise. Ultimately, when the soil reaches the fully active limit state with linear pressure distribution, the resultant force returns to the \(1/3\) wall height position. The non-limit horizontal earth pressure coefficient \(K_{\rm m}\) decreases progressively with increasing wall top displacement, exhibiting rapid initial reduction followed by a slower  
Figure 9: Comparison between experimental and calculated (**a**) horizontal earth pressure distributions, (**b**) active earth pressure coefficients and resultant force positions in RB mode from Reference [12].