Earth Pressure and Slip Surface of Narrow Backfills behind Retaining Walls: An Investigation on the Relative Density of Backfill

The development of slip surfaces within narrow backfills, crucial for active earth pressure calculation, remains a matter of debate. While the internal friction angle (φ) has been widely studied, the role of the dilation angle (ψ) has received limited attention. In this study, the relative density (Dr) of the backfill—a parameter directly linked to both φ and ψ—was selected as the primary variable. Small-scale physical model tests and complementary FLAC3D (version 3.0) numerical simulations were conducted. The results show that the wall displacement required to reach the fully active state increases with higher Dr, as does the extent of principal stress rotation within the narrow backfill. For semi-infinite backfill, the inclination (θ) of the slip surface decreases with a decreasing ψ. For narrow backfill, the inclination of the initial segment (θ1) of the slip surface matches that under semi-infinite conditions (θ1 = θ); the development of the subsequent segment (θ2) depends on ψ, with a critical value identified at approximately ψ = 0.6φ. When ψ exceeds this critical value, the slip surface propagates along the stable boundary (θ2 = 90°); conversely, when ψ is lower, the slip surface reflects at the boundary and continues within the backfill at an inclination similar to θ1 (θ2 = θ1). Finally, an empirical formula relating θ to φ and ψ is proposed and validated against experimental results. The proposed method provides improved predictions of active earth pressure for both semi-infinite and narrow backfill conditions compared with classical theories.