Rankine earth pressure relationship If the active earth pressure coefficient Ka is given as 1/3 for a cohesionless backfill, what is the corresponding passive earth pressure coefficient Kp for the same internal friction angle (Rankine theory)?

Introduction / Context:
In retaining wall and earth pressure problems for cohesionless soils, Rankine theory provides closed-form expressions for the active and passive earth pressure coefficients, Ka and Kp. These coefficients depend only on the soil's internal friction angle and are reciprocals under Rankine assumptions, which is frequently tested in exams and design interviews.

Given Data / Assumptions:

• Cohesionless (c = 0) backfill satisfying Rankine assumptions.
• Active pressure coefficient Ka = 1/3.
• Same soil and friction angle for passive condition.
• Level ground, no wall friction in Rankine idealization.

Concept / Approach:

For Rankine theory: Ka = (1 − sin φ) / (1 + sin φ) and Kp = (1 + sin φ) / (1 − sin φ). Hence Ka * Kp = 1. This reciprocal relationship immediately maps one coefficient to the other when φ is unchanged.

Step-by-Step Solution:

Verification / Alternative check:

If Ka = (1 − sin φ)/(1 + sin φ) = 1/3, then (1 + sin φ)/(1 − sin φ) = 3, which is exactly Kp by Rankine. Consistent.

Why Other Options Are Wrong:

1/3, 2/3, and 1 contradict the reciprocal rule; 3/2 is not the inverse of 1/3.

Common Pitfalls:

Mixing up Coulomb and Rankine assumptions; forgetting the Ka*Kp = 1 identity under Rankine; using wall friction or sloping backfill without updating formulas.

Final Answer:

3