# Solution for Active and Passive Earth Pressure on Rigid Retaining Walls with Narrow Backfill
## 3 Experimental Verification  
### Model Test  
To investigate the influence of the backfill width on the active earth pressure, [PERSON] et al. [26] conducted a model test under rotation mode around the wall bottom (RB mode). The retaining wall had a vertical back face with a retained soil height of 0.5m. The backfill consists of cohesionless soil with an internal friction angle of 36.5\({}^{\circ}\) and a unit weight of 15 kN/m\({}^{3}\). Additionally, the wall-soil friction angle is 24.3\({}^{\circ}\) and Poisson's ratio is taken as 0.23. The backfill widths were set to 0.1 m, 0.15 m, 0.2 m, 0.25 m, and 0.35 m. In the Duncan model, the failure ratio (\(R_{f}\)) is set to 0.75, the modulus parameter (\(K\)) is set to 410, and the dimensionless index (\(n\)) is set to 0.58. The comparison between the proposed method and experimental results is depicted in Figure 4a. Figure 4: (a) Distribution of active earth pressure; (b) Comparison of earth pressure resultant force. In this mode, the wall displacement gradually decreases with increasing burial depth. The soil modulus is closely correlated with the confining pressure. Since the confining pressure at the wall top is zero, its modulus also becomes zero, resulting in a negligible influence of wall displacement on earth pressure in this region. As the burial depth increases, the soil modulus progressively increases, while the soil at the wall bottom remains undeformed. Consequently, the earth pressure at the bottom maintains a constant value. With the gradual increase of backfill width (\(L\)) behind the wall, the earth pressure also increases. When \(L>0.2\) m, the impact on the earth pressure decreases rapidly. At \(L=0.25\) m, the backfill width exerts almost no effect on the earth pressure, approaching the critical width (\(L_{a}=0.29\) m) calculated by Equation (10). When \(L=L_{a}\), the results from this study align with the Coulomb results. By integrating the model test results along the soil depth, the earth pressure resultant force behind the wall can be determined. This result is subsequently compared with those from Equations (3) and (8), as shown in Figure 4b. The Pearson correlation coefficient between the two is 0.993, and the trends are generally consistent; however, the results of this paper are slightly lower than the experimental results. The errors of the two gradually decrease as the backfill width decreases, ranging from a maximum of 18.45% to a minimum of 8.29%, indicating good numerical consistency. [PERSON] et al. [42] conducted model tests to investigate the influence of the backfill width on the passive earth pressure under translational mode. In this experiment, the retained soil height was 0.525 m with a vertical wall back face. The backfill consisted of cohesionless soil with an internal friction angle of 35\({}^{\circ}\), a unit weight of 14.6 kN/m\({}^{3}\), a wall-soil friction angle of 16.57\({}^{\circ}\), and Poisson's ratio was 0.24. The backfill widths were set to 0.56 m, 0.7 m, 1.05 m, 1.4 m, and 1.75 m. In the Duncan model, the failure ratio (\(R_{f}\)) was set to 0.78, the modulus parameter (\(K\)) was set to 390, and the dimensionless index (\(n\)) was set to 0.58. Comparative results between the two approaches under the limit state are shown in Figure 5a. In translational mode, the soil deformation is independent of the burial depth. The earth pressure at the top of the wall remains unchanged, as the confining pressure is zero, resulting in a tangent modulus of zero. As the burial depth increases, the confining pressure and the tangential modulus of the soil increase progressively, leading to a non-linear rise in earth pressure. In the actual project, the stiffness of the retaining wall is considerably greater than that of the fill behind it. When the retaining wall displaces, the earth pressure behind it generates a stress concentration, leading to the development of the soil arching effect [43].