## 3 Calculation Formulas for Non-Limited Seismic Active Earth Pressure  
### Force Analysis and Non-Limited Active Earth Pressure Calculation  
The RB model exhibits multiple parallel linear slip zones within the entire sliding soil wedge, indicating that the soil undergoes layer-by-layer sliding from top to bottom until the lowermost slip zone is activated. The geometry of slip zones and sliding soil wedges in the RB model closely aligns with [PERSON]'s earth pressure theory, better satisfying its assumptions. Under RB mode, slip surfaces migrate downward as wall displacement increases. Lower parallel surfaces gradually reach limit states, becoming dominant slip zones. Specifically, multiple parallel \"potential slip surfaces\" exist within the backfill soil behind the wall, forming angles with the horizontal direction equal to the inclination angle defined in [PERSON]'s theory. These \"potential slip surfaces\" exhibit identical mobilized values of the soil's internal friction angle along their interfaces. As illustrated in Figure 8, one soil slice is selected for mechanical analysis. X-direction equilibrium equation:  
\[P_{\text{i}}+\cos\alpha\text{d}T_{\text{i}}=\text{d}N_{\text{i}}\sin\alpha+F_{ \text{ai}} \tag{9}\]  
Figure 8: Schematic diagram of force distribution on an oblique differential element in RB mode under seismic loading. Figure 7: Similarity transformation for the analysis of soil rupture angle behind seismic retaining walls. Y-direction equilibrium equation:  
\[Q_{\rm i}+\cos\alpha{\rm d}N_{\rm i}+\sin\alpha{\rm d}T_{\rm i}={\rm d}G \tag{10}\]  
In Equation (10), \(P_{\rm i}=\sigma_{\rm ni}{\rm d}z\), \(Q_{\rm i}=\tau_{\rm ni}{\rm d}z\), \(Q_{\rm i}=P_{\rm i}\tan\delta_{\rm m}\), \(T_{\rm i}=N_{\rm i}\tan\varphi_{\rm m}\), \({\rm d}G=\gamma z{\rm cotad}z\). \(P_{\rm i}\), \(Q_{\rm i}\) represent the normal and tangential forces acting on the right boundary of the inclined differential element. \(\sigma_{\rm ni}\), \(\tau_{\rm ni}\) denote the normal and shear stresses on the left boundary of the inclined differential element. \(N_{\rm i}\), \(T_{\rm i}\) correspond to the normal and tangential forces along the inclined edge of the element. \(F_{\rm ai}\) represents the horizontal inertial force of the soil slice. \(\delta_{\rm m}\) indicates the mobilized wall-soil friction angle under a non-limit active state. \(\varphi_{\rm m}\) signifies the mobilized internal friction angle of the soil under non-limit active state. The horizontal seismic force acting on the soil slice is  
\[F_{\rm ai}=\gamma\cot\alpha k_{\rm h}g\bigg{[}(f_{\rm a}-1)\bigg{(}z-\frac{z^ {2}}{2H}\bigg{)}+z\bigg{]}{\rm d}z=F(z){\rm d}z \tag{11}\]  
Combining Equations (9)-(11), we obtain  
\[\sigma_{\rm ni}=\frac{\tan(\alpha-\varphi_{\rm m})\ +k_{\rm h}g\big{[}(f_{\rm a }-1)\big{(}1-\frac{z}{2H}\big{)}+1\big{]}}{1+\tan\delta_{\rm m}\tan(\alpha- \varphi_{\rm m})}\gamma z\cot\alpha \tag{12}\]  
Let \(\sigma_{\rm ni}=k_{\rm m}\sigma_{z}=k_{\rm m}\gamma z\), then  
\[k_{\rm m}=\frac{\tan(\alpha-\varphi_{\rm m})\ +k_{\rm h}g\big{[}(f_{\rm a}-1) \big{(}1-\frac{z}{2H}\big{)}+1\big{]}}{1+\tan\delta_{\rm m}\tan(\alpha- \varphi_{\rm m})}\cot\alpha \tag{13}\]  
When \(k_{\rm h}=0\) in Equation (13), the result aligns with Reference [14]. By substituting \(\delta_{\rm m}\) (mobilized wall-soil friction angle) and \(\varphi_{\rm m}\) (mobilized soil internal friction angle) obtained from Equations (2) and (7) into Equation (13), the lateral pressure coefficient for the non-limit state \(k_{\rm m}\) is derived. Under non-limit conditions, the total horizontal earth pressure acting on the retaining wall is  
\[P_{\rm a}=\int_{0}^{H}\sigma_{\rm ni}{\rm d}z=\frac{1}{2}\gamma H^{2}K_{\rm m} \tag{14}\]  
where \(K_{\rm m}\) is the non-limit earth pressure coefficient.