Columns under eccentric load: core for circular sections Limit of eccentricity e for no tensile stress in a circular column of diameter d equals:

Introduction / Context:
The core (or kern) of a section is the locus of load application points that produce compressive stress everywhere with no tension. For masonry or concrete columns where tension is undesirable, ensuring the load resultant lies within the core prevents cracking.

Given Data / Assumptions:

• Circular cross-section, diameter d.
• Linear elastic stress distribution (Bernoulli hypothesis).
• Axial load with eccentricity e about one principal axis.

Concept / Approach:
For a section with axial load P and moment M = P * e, extreme fiber stress is sigma = (P / A) ± (M * y / I). For no tension, the minimum stress must be ≥ 0. The core radius for a circle is r_core = d / 8, meaning any e ≤ d / 8 keeps the entire section in compression.

Step-by-Step Solution:
For a circle: A = pi * d^2 / 4; I = pi * d^4 / 64; c = d / 2.Set sigma_min = P / A - M * c / I ≥ 0.Substitute M = P * e and rearrange to e ≤ (I / (A * c)).Compute I / (A * c) = (pi d^4 / 64) / [(pi d^2 / 4) * (d / 2)] = d / 8.Therefore, limit of eccentricity for no tension is e ≤ d / 8.

Verification / Alternative check:
For a rectangle b × h, core limits are ±b/6 and ±h/6; the circular case giving d/8 is consistent with standard kern values.

Why Other Options Are Wrong:
d/4 is too large and would cause tensile stress; d/12 and d/16 are conservative but not the theoretical limit; d/6 pertains to rectangles, not circles.

Common Pitfalls:
Using rectangular core limits for all shapes; forgetting to use I/(A*c) for the specific section geometry.

Final Answer:
d / 8