Retaining wall with uniform surcharge: When retained soil is subjected to a uniform superimposed (surcharged) load of intensity p over the backfill surface, what is the total horizontal active pressure on a vertical wall of height h? (Use Rankine's earth pressure theory with active pressure coefficient Ka.)

Introduction / Context:
In soil mechanics and retaining wall design, surcharge loads such as traffic, storage, or construction equipment add to the lateral earth pressure on the wall. Under Rankine’s active condition for level backfill, a uniform surcharge of intensity p produces an additional, height-uniform lateral pressure of magnitude Ka * p. Understanding how to compute the total horizontal force from this surcharge is essential for safe design.

Given Data / Assumptions:

• Vertical retaining wall of height h with level ground surface behind it.
• Uniform surcharge (superimposed load) of intensity p on the backfill.
• Active earth pressure condition per Rankine; coefficient Ka = (1 - sin φ) / (1 + sin φ).
• Backfill is homogeneous, dry, and cohesionless; wall friction neglected; no seismic effects.

Concept / Approach:
Under a uniform surcharge p, the lateral earth pressure intensity added by surcharge is constant with depth and equals Ka * p (same at all elevations). The total horizontal force due to this uniform pressure is the pressure intensity multiplied by the wall area per unit out-of-plane width, which is simply the height h. The resultant acts at mid-height (h/2 from the base).

Step-by-Step Solution:

Verification / Alternative check:
The shear resultant of a uniform pressure strip of height h is simply intensity times height. A quick units check: p has units of force/area; multiplying by h (length) gives force/length, which is correct for force per unit out-of-plane width. Ka is dimensionless, so units remain consistent.

Why Other Options Are Wrong:

• p * h ignores the reduction due to earth pressure coefficient Ka.
• Ka * p * h / 2 and p * h / 2 are triangular-resultant expressions; surcharge-induced pressure is uniform, not triangular.
• Ka * p * h^2 has incorrect dimensionality and arises from confusing pressure intensity with depth-varying self-weight pressure.

Common Pitfalls:
Confusing uniform surcharge pressure (constant with depth) with soil self-weight pressure (linear with depth); forgetting the coefficient Ka; placing the resultant at h/3 instead of the correct h/2 for a uniform load block.

Final Answer:
Ka * p * h