## 3 Calculation Formulas for Non-Limited Seismic Active Earth Pressure
### Validation of the Calculation Method
decline. When the rotation angle reaches a critical value and most of the soil behind the wall attains an active state, \(K_{\rm m}\) stabilizes. Overall, the calculated \(K_{\rm m}\) values are slightly higher than the measured values, which are conservative for engineering applications. (2) When \(k_{\rm h}=0\), validation using experimental data from Reference [44]: Rigid wall with a vertical back face and height \(H=0.55\) m. The parameters used in the calculation are \(\gamma=18.13\) kN/m\({}^{3}\), \(\varphi=47.8^{\circ}\), \(\varphi_{0}=15^{\circ}\), \(\delta=2\varphi/3=31.8^{\circ}\), \(\delta_{0}=23.9^{\circ}\), \(S_{\delta}(z)=0.6\%\)\(z\). Using Equation (8), the calculated rupture angle \(\alpha=66.4^{\circ}\). The horizontal earth pressure distribution under the RB mode calculated using the proposed method is compared with the model test measurements in Figure 10a. Figure 10b illustrates the variations in the relative position \(h/H\) of the horizontal earth pressure resultant force's acting point with the rotation angle, along with comparisons to experimental results. As shown in the figures, the calculated values closely align with the measured values. However, since the calculated earth pressure at the wall base is smaller than the experimental measurements, while exceeding the measured values within the range of 0.5 \(H\) to 0.85 \(H\) (where \(H\) is the wall height), the calculated relative position \(h/H\) of the resultant force is slightly higher than the measured value. Overall, the proposed non-limit earth pressure calculation method for the RB displacement mode demonstrates reasonable accuracy. (3) When \(k_{\rm h}>0\), the experimental data are adopted from Reference [45]. The parameters are \(H=1.02\) m, \(\gamma=16.1\) kN/m\({}^{3}\), \(\varphi=40.1^{\circ}\), \(\varphi_{0}=10^{\circ}\), \(\delta=2\varphi/3=26.7^{\circ}\), \(\delta_{0}=10^{\circ}\), \(S_{\delta}(z)=0.1\%z\), \(k_{\rm h}=0.215\), and \(f_{\rm a}=1.1\). Using Equation (8), the calculated rupture angle is \(\alpha=49.3^{\circ}\). Figure 11a,b compare the experimental and calculated distributions of the maximum horizontal dynamic earth pressure and minimum horizontal dynamic earth pressure, respectively, under the RB displacement mode at different wall displacements. As shown in Figure 11a,b, for the RB mode involving rotation about the wall base, the dynamic active earth pressure exhibits a nonlinear distribution with depth. Due to the small wall displacement, the earth pressure near the wall base remains relatively high, approaching the at-rest earth pressure, resulting in a lower acting point for the dynamic active earth pressure (below 1/3 of the wall height). In the maximum earth pressure measurements, higher pressures are observed near the wall base. For the minimum earth pressure measurements, when the horizontal displacement at the wall top exceeds a specific threshold, the earth pressure in the upper wall section is zero. This zero lateral stress indicates potential instantaneous separation between the wall and backfill, occurring when the lateral pressure induced by the negative inertial body forces of the backfill exceeds the lateral pressure from surcharge loads during near-active phases. Figure 10: Comparison between experimental and calculated (**a**) horizontal earth pressure distributions and (**b**) resultant force positions in RB mode from Reference [44]. To align with experimental conditions, the maximum earth pressure in the calculations is derived by orienting inertial forces outward from the retaining wall, while the minimum earth pressure corresponds to \(k_{\rm h}=0\). Figure 11 demonstrates good agreement between calculated and experimental curves. However, under dynamic loading, the measured maximum active stress at the mid-height of the wall is slightly lower than the values predicted by the Mononobe-Okabe method. Discrepancies between calculated and measured pressure curves arise because the proposed method does not account for soil arching effects.