For a cantilever retaining wall with a granular backfill, how does the pressure distribution of active earth pressure differ from the at-rest pressure, and what causes this difference?

For a cantilever retaining wall with a granular backfill, the pressure distribution of active earth pressure differs significantly from the at-rest pressure primarily due to the degree of lateral movement allowed for the wall, which in turn affects the mobilization of shear strength within the soil. A cantilever retaining wall is a structure that holds back soil, and granular backfill refers to soil like sand or gravel, which has internal friction but negligible cohesion.

At-rest earth pressure (often denoted by the coefficient K₀) is the lateral pressure exerted by the granular backfill when there is absolutely no lateral movement or strain of the retaining wall relative to the soil. In this condition, the soil mass is constrained and cannot expand or contract horizontally. The horizontal stress develops purely as a response to the vertical overburden stress (the weight of the soil above a certain depth) without any lateral yielding. For a granular soil, the coefficient of at-rest earth pressure, K₀, is typically in the range of 0.4 to 0.6, and for normally consolidated soils, it can be approximated as 1 - sin(φ'), where φ' is the effective angle of internal friction of the soil. The pressure distribution for at-rest conditions is generally triangular, increasing linearly with depth, with the maximum pressure occurring at the base of the wall. This means the horizontal pressure at any depth is K₀ multiplied by the vertical effective stress at that depth.

Active earth pressure (often denoted by the coefficient Kₐ) is the minimum lateral pressure exerted by the granular backfill when the retaining wall moves *awayfrom the backfill just enough to allow the soil mass to reach a state of plastic equilibrium or incipient failure. This required outward movement is relatively small, typically on the order of 0.1% to 0.4% of the wall's height for granular soils. When the wall moves outwards, it permits the granular soil particles to rearrange and expand slightly in the lateral direction. This lateral expansion mobilizes the shear strength of the soil along a potential failure plane, meaning the soil's internal friction is fully engaged to resist the tendency of the soil mass to push the wall. The mobilization of this shear strength reduces the normal horizontal stress acting on the wall. For a vertical wall and horizontal backfill, the coefficient of active earth pressure, Kₐ, is (1 - sin(φ')) / (1 + sin(φ')), which is typically in the range of 0.2 to 0.35 for granular soils. Like at-rest pressure, the active pressure distribution is also generally triangular, increasing linearly with depth, with the maximum pressure at the base of the wall.

The primary cause of the difference between active and at-rest pressure is the lateral strain or movement of the wall. At-rest pressure represents a state of zero lateral strain, where the soil's shear strength is not fully mobilized in the horizontal direction, leading to higher lateral stresses. Active pressure, however, requires a small outward lateral movement of the wall. This movement allows the granular soil to expand laterally, fully mobilizing its internal shear strength along a potential failure surface. The mobilized shear strength then contributes to supporting the soil mass, effectively reducing the net horizontal force exerted on the wall. Consequently, the magnitude of active earth pressure is always significantly *lessthan the at-rest pressure for the same depth, even though both typically exhibit a triangular pressure distribution.