Earth pressure at rest K₀: For a cohesionless soil (φ = angle of internal friction), identify the correct proportional expression for the at-rest earth pressure coefficient. If none of the listed tan-based forms is valid, choose ‘‘none of these’’.

Introduction / Context:
In geotechnical engineering, three standard lateral earth pressure states are active (K_a), passive (K_p), and at-rest (K_0). While K_a and K_p follow Rankine or Coulomb relations involving tan^2(45° ± φ/2), the at-rest state K_0 describes the lateral stress that exists without wall yield or strain in the soil mass. This question probes whether one of the offered “tan” expressions matches K_0.

Given Data / Assumptions:

• Granular soil with angle of internal friction φ.
• At-rest condition, i.e., no lateral strain (wall not moving).
• We compare standard closed-form coefficients.

Concept / Approach:

For normally consolidated soils, the widely used empirical expression is K_0 ≈ 1 − sin φ (Jaky’s relation). By contrast, Rankine’s active and passive coefficients are K_a = tan^2(45° − φ/2) and K_p = tan^2(45° + φ/2). Note the crucial “φ/2” inside the trigonometric function and the squared tangent form. None of the provided options includes the “/2”, and none equals 1 − sin φ, so no option represents K_0 correctly.

Step-by-Step Solution:

Verification / Alternative check:

If, instead, the question had asked for active or passive pressure, K_a and K_p would involve tan^2(45° ∓ φ/2). The absence of “/2” confirms the mismatch with Rankine relations, and K_0 is not a tan expression at all.

Why Other Options Are Wrong:

(a), (b), (c), (d) are not recognized forms for K_0; (c) and (d) resemble Rankine forms but lack the φ/2 term and do not apply to at-rest conditions.

Common Pitfalls:

Confusing at-rest with active/passive states; forgetting the empirical nature of K_0 and the φ/2 term in Rankine relations.

Final Answer:

none of these