# Solution for Active and Passive Earth Pressure on Rigid Retaining Walls with Narrow Backfill
## 1 Introduction
[23] modeled the soil slip surface as a bilinear plane and analyzed the active earth pressures of narrow backfills through force balance analysis. In addition, the retaining wall foundation affects the earth pressure under finite backfill conditions. [PERSON] et al. [24] observed that the slip surface develops simultaneously toward the rock surface and the ground ahead of the wall, leading to the proposal of a slip-line computational model. To distinguish finite soils from semi-infinite soils, [PERSON] et al. [10] proposed a critical ratio of 0.5 between backfill width and wall height under an active state, while [PERSON] et al. [25] suggested a threshold ratio of three for passive conditions, beyond which the effect of backfill width on earth pressure becomes negligible. However, the classification based on experimental results or empirical data lacks sufficient accuracy, and the influence of various parameters on the critical width is not considered. The deformation mode of flexible retaining walls differs significantly from that of rigid retaining walls. [PERSON] et al. [26] developed a computational approach for active earth pressure under drum deformation by refining the horizontal layer analysis method, but this methodology was restricted to cohesionless soils. [PERSON] et al. [27] quantified the combined influence of horizontal shear stresses and the soil arching on passive earth pressure using the arched differential element method. For the rotation mode around the wall bottom,Wang et al. [28] investigated the effect of the backfill width on the active earth pressure using the inclined thin-layer element method. [PERSON] et al. [29] investigated active earth pressure distributions in unsaturated soil walls using the thin-layer element method. [PERSON] and [PERSON] [30; 31] examined the coupled effects of intermediate principal stress, soil arching, and unsaturated seepage on the active earth pressure under finite backfill width conditions. For retaining walls that rotate about their base, [PERSON] et al. [32] proposed a prediction model for active earth pressure based on the arch differential element method, incorporating backfill width effects. In the case of finite backfill width, the slip surface behind the wall has a certain curvature. [PERSON] et al. [33] characterized it as a cycloid through the wall heel. The stress deflection caused by wall-soil friction is considered by the arc-shaped minor principal stress trajectory [7], and the active earth pressure solution for cohesionless soil is obtained by the horizontal differential element method. [PERSON] et al. [34] assumed that the slip surface follows a log-spiral surface and proposed a semi-analytical method; nevertheless, its computational complexity limits practical engineering applications. The limit equilibrium method is widely employed in various earth pressure calculation scenarios [35; 36], but it neglects the effect of soil dilation [37] on the results. [PERSON] et al. [38] extended this method to cohesionless soils with finite width. The previous analysis reveals that existing studies are less relevant for clay and there is a lack of research on passive earth pressure in finite soils. Furthermore, the debate regarding the accurate definition of semi-infinite and finite soils persists and warrants further in-depth discussion. Building upon prior research, this study hypothesizes the development of a trapezoidal slip wedge during soil failure under finite soil conditions. Cohesion is uniformly distributed along the slip surface. In the passive state, the earth pressure on both sides above the slip surface inflection point is equal. The ratio of earth pressures on the left and right lateral boundaries in the active state is called \(a_{s}\). By utilizing the static equilibrium condition of the slip wedge, a generalized formulation for the limit earth pressure in finite backfill was derived. As the backfill width gradually increases, the trapezoidal wedge ultimately transitions into a triangular wedge.