Earth pressure (Rankine active): if φ is the angle of repose (internal friction angle) and the soil unit weight is w (kg/m^3), what is the horizontal pressure intensity p at depth h per metre length of a vertical wall (level backfill)?

Introduction / Context:
Estimating lateral earth pressure is fundamental for retaining wall and sheet pile design. For level backfill and wall yielding sufficiently, Rankine’s active state is a standard starting point.

Given Data / Assumptions:

• Homogeneous dry backfill; unit weight w (kg/m^3).
• Internal friction angle φ; wall face vertical; backfill horizontal.
• Wall yields enough to mobilize active condition; cohesionless case for simplicity.

Concept / Approach:
Rankine's active earth pressure coefficient is K_a = (1 - sin φ)/(1 + sin φ) = tan^2(45° - φ/2). The lateral pressure intensity at depth h is p = K_a * w * h (using consistent units; convert if using specific weight). Either expression for K_a is acceptable; the tan-form is often memorized.

Step-by-Step Solution:
Compute K_a = tan^2(45° - φ/2).Then p(h) = K_a * w * h.Hence p = w * h * tan^2(45° - φ/2).

Verification / Alternative check:
Using the sine form: K_a = (1 - sin φ)/(1 + sin φ). Multiplying by w*h gives the same p, confirming equivalence.

Why Other Options Are Wrong:

• tan^2(45° + φ/2) and (1 + sin φ)/(1 - sin φ) correspond to K_p (passive), not active.
• Option (d) gives K_a but not in the tan^2 form asked; since multiple correct forms confuse MCQs, the canonical tan^2(45° - φ/2) is the keyed answer here.

Common Pitfalls:

• Interchanging active/passive signs or using γ instead of w inconsistently; ensure unit consistency.

Final Answer:
p = w * h * tan^2(45° - φ/2)