Total active thrust for dry level backfill (no surcharge) For a retaining wall with dry, horizontal backfill and no surcharge, the resultant active earth pressure acts at H/3 above the base. The magnitude of the total active thrust Pa is directly proportional to which power of wall height H?

Introduction / Context:
Estimating the total active earth thrust on a retaining wall is fundamental in geotechnical design. For a dry, horizontal backfill with no surcharge, the lateral earth pressure increases linearly with depth, producing a triangular pressure diagram. Converting this pressure distribution to a resultant force clarifies how thrust scales with wall height.

Given Data / Assumptions:

• Dry, cohesionless backfill; no surcharge.
• Rankine or Coulomb (no cohesion) with level backfill.
• Lateral pressure at depth z: σ_h = Ka * γ * z.
• Resultant acts at H/3 above the base.

Concept / Approach:

The pressure varies linearly from zero at the top to KaγH at the base, forming a triangle. The total thrust is the area of the triangle. Because the base ordinate is proportional to H, the area—and thus the thrust—is proportional to H^2.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional consistency: γ has units of force/volume, multiplied by H^2 gives force per unit out-of-plane width, as expected.

Why Other Options Are Wrong:

H or √H underestimates growth; H^3 overestimates; log H is not physically relevant for triangular pressure without special conditions.

Common Pitfalls:

Confusing pressure (∝ H) with thrust (∝ H^2); forgetting the 1/3 location of the resultant is about line of action, not magnitude scaling.

Final Answer:

H^2