Rankine Passive Earth Pressure – Functional Dependence on φ For a horizontal backfill (no wall friction), the passive earth pressure coefficient Kp under Rankine theory varies with the angle of internal friction φ as:

Introduction / Context:
Earth pressure coefficients link in-situ stress states to backfill friction angle under limiting equilibrium. For Rankine conditions (no wall friction, planar ground), passive pressure corresponds to the largest lateral resistance mobilized by compressing the soil against the wall.

Given Data / Assumptions:

• Planar ground surface, no surcharge.
• Wall friction neglected (Rankine assumption).
• Soil is homogeneous and isotropic with internal friction angle φ.

Concept / Approach:

Rankine’s coefficients are K_a = tan^2(45° − φ/2) for active, and K_p = tan^2(45° + φ/2) for passive. These follow from Mohr–Coulomb stress circles at limiting states relative to the principal stress directions.

Step-by-Step Solution:

Verification / Alternative check:

For φ = 0°, Kp = tan^2(45°) = 1, which is consistent; for φ = 30°, Kp = tan^2(60°) = 3, a standard textbook value.

Why Other Options Are Wrong:

(a) and (c) correspond to K_a (active), not K_p. (b) uses φ instead of φ/2. (e) is not the Rankine expression.

Common Pitfalls:

Mixing active and passive forms; forgetting the half-angle; misreading units (degrees vs radians when coding calculators).

Final Answer:

Kp ∝ tan^2(45° + φ/2)