# Solution for Active and Passive Earth Pressure on Rigid Retaining Walls with Narrow Backfill
## 1 Introduction  
The traditional Rankine theory is predicated on the assumption that all points of the soil behind the wall are in the limit equilibrium state. Coulomb theory posits that a triangular slip wedge forms when the soil behind the wall fails. With the advancement of modern infrastructure, retaining structures are increasingly situated near pre-existing structures or rock mass [1, 2]. Under finite backfill conditions, the restricted backfill width prevents full development of the slip surface, resulting in a significant divergence in earth pressure from semi-infinite soil scenarios, as evidenced by prior studies [3, 4]. Therefore, it is imperative to develop earth pressure calculation methods that incorporate the impact of backfill width. To investigate the earth pressure behind the retaining structures under finite backfill width conditions, [PERSON] and [PERSON] [5] and [PERSON] and [PERSON] [6] conducted model tests. Their results were significantly different from the theoretical predictions of earthpressure in semi-infinite soil. Moreover, the arching effect phenomenon [7; 8] also affects the distribution of earth pressure. [PERSON] and [PERSON] [9] additionally examined the impact of the retaining wall displacement pattern on earth pressure under finite backfill width conditions. As demonstrated by [PERSON] et al. [10] through slip surface analysis in active states, the complete development of Coulomb's theoretical triangular slip wedge is impeded by geometric constraints imposed by finite backfill dimensions. Similarly, field measurements by [PERSON] et al. [11] demonstrated that the active earth pressure under limited backfill widths is less than that predicted by Rankine theory, resulting in inherent conservatism in conventional design approaches. The advancement of computational mechanics has created novel research avenues for investigating earth pressure mechanisms. For finite soils, [PERSON] and [PERSON] [12] demonstrated that the active earth pressure behind the wall is significantly lower than that in the Coulomb solution. [PERSON] and [PERSON] [13] conducted systematic DEM investigations elucidating the coupled influence of backfill width and soil strength parameters on the active earth pressure. Through statistical characterization of DEM simulations, [PERSON] et al. [14] classified the failure mechanisms into three categories according to the backfill width behind the wall. In cases where the backfill is cohesionless soil, [PERSON] et al. [15; 16] examined the variation pattern of the passive earth pressure in translational and rotational modes around the wall bottom through FELA analysis. Notably, their simulations revealed the progressive development of quadrilateral slip wedges. [PERSON] et al. [17] investigated the active earth pressure of clay in the rotational mode around the wall top. Based on adaptive finite element analysis, [PERSON] et al. [18] investigated the failure mechanism of an inverted T-shaped retaining wall in the active state. [PERSON] et al. [19] found that the slip surface of the backfill originated from the wall heel, and its development was influenced by the backfill width behind the wall. Moreover, the slip surface curvature was found to correlate strongly with soil-structure interface friction and soil strength. When the rock slope is adjacent to the retaining wall and its inclined surface intersects with the wall heel, [PERSON] and [PERSON] [20] treated the rock slope surface as the slip surface and applied the wedge element method to determine the active earth pressure distribution. For translational retaining walls with a finite backfill width, [PERSON] et al. [21] developed a theoretical framework to calculate the active earth pressure of cohesionless soil using the limit equilibrium method. Both the geometric configuration and interfacial friction properties of the finite soil significantly affect the active earth pressure behind the wall, as demonstrated by [PERSON] et al. [22]. [PERSON] et al.