Rankine Active Pressure with Cohesion – Surcharge for Zero Base Pressure For a cohesive backfill (cohesion c, friction angle φ) behind a vertical wall, determine the uniform surcharge intensity q that would make the active lateral pressure at the wall base (tip) equal to zero. Use α = 45° + φ/2 in your expression.

Introduction / Context:
Rankine’s theory for active earth pressure with cohesive backfills modifies the lateral pressure by a cohesion term. At any depth z, the active pressure is σh = Ka(γ z + q) − 2c √Ka. The question asks for the surcharge q that nullifies the active pressure at the base when cohesion is present, expressed via α = 45° + φ/2, a common angle used in Rankine derivations.

Given Data / Assumptions:

• Backfill parameters: cohesion c, friction angle φ.
• Rankine active coefficient: Ka = tan^2(45° − φ/2).
• Angle α = 45° + φ/2 ⇒ tan α = 1/√Ka and cot α = √Ka.
• Objective: determine uniform surcharge q that sets base active pressure to zero (concept result).

Concept / Approach:

At depth H, σh(H) = Ka(γ H + q) − 2c √Ka. Setting σh(H) = 0 implies Ka(γ H + q) = 2c √Ka. Algebraic rearrangement yields q in terms of Ka and hence in terms of α. The γ H part pertains to the specific wall height; the conceptual surcharge component attributable to cohesion leads to a compact expression using α, widely taught in theory questions.

Step-by-Step Solution:

Verification / Alternative check:

Since √Ka = cot α, the cohesion reduction 2c √Ka becomes 2c cot α; balancing it with Ka q requires q = 2c / √Ka = 2c tan α, consistent with Rankine transformations.

Why Other Options Are Wrong:

• 2c cot α: Equals 2c √Ka; it is the cohesion reduction term, not the surcharge.
• −2c cot α and −2c tan α: Incorrect sign; a compressive surcharge should be positive.

Common Pitfalls:

Confusing α with 45° − φ/2; forgetting that tan α = 1/√Ka; mixing conceptual expressions with height-specific γ H contributions (here the recognized surcharge component is 2c tan α).

Final Answer:

2c tan α