Core (kernel) of a circular section – area ratio For a circular section of radius r in no-tension theory, what is the ratio of the total cross-sectional area to the area of its core (kernel)?

Introduction / Context:
The core (kernel) of a section in masonry or concrete design is the locus within which the resultant compressive load must act to avoid tensile stress anywhere on the section. For circular sections, the core is also circular but reduced in size.

Given Data / Assumptions:

• No-tension material model is assumed (e.g., unreinforced masonry under compression).
• Section is a full circle of radius r.
• Core radius for a circle is r/4 (a standard result from elementary strength of materials).

Concept / Approach:
The area ratio equals the area of the whole section divided by the area of the core. For a circle, areas scale with the square of the radius.

Step-by-Step Solution:
Total area A = πr^2Core radius rc = r/4, so core area Ac = π(r/4)^2 = πr^2/16Required ratio = A / Ac = (πr^2) / (π*r^2/16) = 16

Verification / Alternative check:
Choose r = 1. Then A = π, Ac = π/16. A/Ac = 16. The ratio is independent of r, as expected.

Why Other Options Are Wrong:
Values 4, 8, and 12 underestimate the reduction in core size; 20 overestimates it. Only 16 matches the known core radius of r/4 for a circular section.

Common Pitfalls:
Confusing core radius (r/4) with diameter or using r/3 (which applies to a rectangle’s core distance from the edge in a different context).

Final Answer:
16