Rankine's empirical design: according to Rankine's formula, what is the minimum depth D of foundation embedment in terms of soil unit weight γ, allowable bearing pressure q (at foundation level), and soil friction angle φ?

Introduction / Context:
Rankine's depth of foundation formula provides a minimum embedment to prevent shear failure and ensure the overburden confining pressure mobilizes adequate passive resistance. It relates the foundation depth to soil strength parameters in a simple closed form widely used for preliminary sizing.

Given Data / Assumptions:

• Soil unit weight = γ (kN/m^3).
• Allowable bearing pressure at foundation level = q (kN/m^2).
• Effective stress friction angle = φ (degrees).
• Groundwater effects and cohesion are neglected for simplicity.

Concept / Approach:
Rankine's active earth pressure coefficient K_a = ((1 − sin φ) / (1 + sin φ))^2. The classic embedment relation may be written as D = (q / (γ K_a)), which rearranges to D = (q / γ) * ((1 − sin φ) / (1 + sin φ))^2. This ensures that vertical overburden at the base provides the confining stress consistent with allowable pressure.

Step-by-Step Solution:
Compute K_a = ((1 − sin φ) / (1 + sin φ))^2.Use D = (q / (γ K_a)).Substitute K_a: D = (q / γ) * ((1 − sin φ) / (1 + sin φ))^2.Thus, option (a) matches Rankine's expression.

Verification / Alternative check:
For φ = 0°, K_a = 1, giving D = q / γ, which is dimensionally consistent and physically reasonable for purely frictionless soil in this simplified context.

Why Other Options Are Wrong:

• (b) Linear (not squared) ratio; does not reflect Rankine's K_a.
• (c) Inverts the ratio, yielding unrealistic shallow/negative depths.
• (d) and (e) use tan(45° ± φ/2) forms without the square; not Rankine's embedment formula.

Common Pitfalls:
Confusing K_a (active) with K_p (passive); omitting the square on ((1 − sin φ)/(1 + sin φ)).

Final Answer:
D = (q / γ) * ((1 - sin φ) / (1 + sin φ))^2