Retaining walls — the lateral earth pressure at a given depth behind a vertical wall supporting level, cohesionless backfill is:

Introduction / Context:
Design of retaining walls requires understanding how lateral earth pressure varies with depth. For a granular backfill under Rankine or Coulomb active conditions, the pressure distribution is triangular, with intensity increasing linearly with depth, similar to hydrostatic pressure but scaled by an earth-pressure coefficient rather than by unity.

Given Data / Assumptions:

• Vertical wall, level backfill, cohesionless soil (c = 0).
• Active state is mobilized (wall yields sufficiently).
• Unit weight gamma and friction angle phi characterize the backfill.

Concept / Approach:

The lateral earth pressure intensity at depth z is sigma_h = K_a * gamma * z (plus K_a * q if a uniform surcharge q exists). Since sigma_h is directly proportional to z, the distribution is linear with depth. The total resultant thrust on the wall face is the triangular area under the pressure diagram, hence proportional to H^2, but the pressure at a given depth is proportional to depth, not its square. This distinction avoids common misinterpretation between intensity and resultant force.

Step-by-Step Solution:

Verification / Alternative check:

Integrate sigma_h from 0 to H: resultant P_a = 0.5 * K_a * gamma * H^2 acting at H/3 from the base, which confirms the linear distribution assumption.

Why Other Options Are Wrong:

(a) Confuses pressure with total mass. (c) refers to the resultant thrust scaling, not the pressure intensity. (d) ignores dependence on gamma and depth. (e) is incorrect because a simple proportionality exists.

Common Pitfalls:

Mixing up pressure intensity (linear in z) with total thrust (proportional to H^2); neglecting surcharge or cohesion when present.

Final Answer:

proportional to the depth of the soil