Retaining wall earth pressure (active case): For a vertical retaining wall of height h (per unit length out of plane) retaining horizontal backfill with unit weight w, and active earth pressure coefficient Ka, what is the total thrust exerted by the retained soil on the back of the wall?

Introduction / Context:
This problem tests Rankine’s active earth pressure theory for a vertical retaining wall with level backfill. Knowing the total lateral thrust and its dependence on the active pressure coefficient Ka is fundamental for sizing retaining walls and checking sliding, overturning, and base pressures.

Given Data / Assumptions:

• Wall height = h (per unit run along the wall).
• Backfill is horizontal, dry, and homogeneous with unit weight w.
• Active earth pressure condition with coefficient Ka (soil yields away from the wall).
• No surcharge load and no water table.

Concept / Approach:
Under Rankine active conditions, the lateral pressure at a depth z is pa(z) = Ka * w * z (a linear distribution from zero at the top to Ka * w * h at the base). The total active thrust is the area of this triangular pressure diagram times unit length of wall.

Step-by-Step Solution:
At depth z: pa(z) = Ka * w * z.Resultant thrust Pa = area of triangle = (1/2) * base * height = (1/2) * (Ka * w * h) * h.Therefore, Pa = (1/2) * Ka * w * h^2.Point of action is at h/3 above the base (for a triangular distribution), though the question asks only for the total magnitude.

Verification / Alternative check:
Dimensional check: w has units of force/volume; multiplying by h^2 (area) and by Ka (dimensionless) gives force per unit run—consistent with thrust per unit length of wall.

Why Other Options Are Wrong:

• Ka * w * h: Misses the triangular integration (factor 1/2 and h^2).
• (1/3) * Ka * w * h^3: Wrong power of h; thrust scales with h^2, not h^3.
• (1/2) * w * h^2: Ignores soil friction angle effect embedded in Ka.
• w * h^2: Also ignores triangular distribution and Ka.

Common Pitfalls:
Forgetting the factor of 1/2 from the triangular diagram, neglecting Ka, or confusing active with at-rest (K0) or passive (Kp) conditions.

Final Answer:
(1/2) * Ka * w * h^2