Active earth pressure coefficient K_a: For a cohesionless backfill with friction angle φ (Rankine conditions), which expression correctly gives K_a?

Introduction / Context:
Rankine’s earth pressure theory provides closed-form coefficients for active and passive states in cohesionless soils. Recognizing the correct form of the active coefficient K_a is fundamental to retaining wall design.

Given Data / Assumptions:

• Backfill is cohesionless with internal friction angle φ.
• Wall yields sufficiently to mobilize the active state.
• Plane ground, no wall friction (Rankine assumptions).

Concept / Approach:

For Rankine conditions: K_a = tan^2(45° − φ/2). Equivalently, K_a = (1 − sin φ)/(1 + sin φ). Passive coefficient is K_p = tan^2(45° + φ/2) = (1 + sin φ)/(1 − sin φ). Selecting the correct sign and half-angle is crucial.

Step-by-Step Solution:

Verification / Alternative check:

Substituting φ = 0 gives K_a = 1 (horizontal at-rest for zero friction), confirming the correct trend for the active case under Rankine assumptions.

Why Other Options Are Wrong:

(b) is the passive expression; (c) is K_p; (d) is an equivalent correct form of K_a but the single best canonical form requested is (a); (e) is not a Rankine coefficient.

Common Pitfalls:

Mixing active and passive forms; forgetting the φ/2 half-angle.

Final Answer:

K_a = tan^2(45° − φ/2)