Shallow foundations on cohesionless soil — compare ultimate bearing capacities Two footings, one circular and one square, rest on the surface of a purely cohesionless soil. The diameter of the circular footing equals the side length of the square footing. What is the ratio of their ultimate bearing capacities (circular to square)?

Introduction / Context:
Ultimate bearing capacity for shallow foundations in cohesionless (c = 0) soil at ground surface depends strongly on footing shape. Terzaghi's general bearing capacity expression uses shape factors that differ for circular and square footings. This question checks whether you recall the relative magnitudes of the shape coefficients for the N_gamma term, which governs bearing for c = 0 at D_f = 0.

Given Data / Assumptions:

• Soil is purely cohesionless; c = 0 and φ > 0.
• Footings are at the ground surface; surcharge term q = 0.
• Diameter of circular footing = side of square footing = B.
• Ultimate capacity governed by the N_gamma term.

Concept / Approach:
For D_f = 0 and c = 0, qult ≈ 0.5 * γ * B * N_gamma * s_gamma, where s_gamma is the shape factor. Commonly used values: s_gamma ≈ 0.4 for a square footing and ≈ 0.3 for a circular footing (strip footing baseline is 0.5). Therefore the ratio (circular / square) reduces to 0.3 / 0.4.

Step-by-Step Solution:
1) Write qult (square) = 0.4 * γ * B * N_gamma.2) Write qult (circular) = 0.3 * γ * B * N_gamma.3) Take ratio = (0.3)/(0.4) = 0.75.4) Express as a fraction → 3/4.

Verification / Alternative check:
As shape becomes more compact (circular), the effective contribution of the N_gamma term decreases compared to square of the same breadth; many design handbooks list exactly these multipliers.

Why Other Options Are Wrong:
4/3 and 1.3 incorrectly imply the circular footing has a larger capacity than the square for the same breadth. 1.0 ignores shape effects.

Common Pitfalls:
Confusing area-based effects with shape factors; applying strip footing coefficients to square or circular footings.

Final Answer:
3/4