Kern (Core) of a Circular Section — No-Tension Condition For a short column with a circular cross-section, to avoid tensile stress anywhere at the base, the line of action of the resultant load must lie within a concentric “kern” circle whose diameter is what fraction of the main circle's diameter?

Introduction:
The “kern” (or core) of a section is the locus of points through which a compressive resultant can act without causing tension anywhere on the base. For masonry and short columns, keeping the load within the kern ensures uniform compression. This question focuses on the kern size for a circular section.

Given Data / Assumptions:

• Short column (no significant slenderness effects).
• Linear elastic stress distribution under eccentric compression.
• Circular cross-section of radius r (main diameter = 2r).

Concept / Approach:
For a circle, the kern is a concentric circle of radius r/4. If the load resultant stays within this radius from the centroid, the entire base remains in compression. Therefore, the kern diameter equals 2 * (r/4) = r/2.

Step-by-Step Solution:
Let main radius = r ⇒ main diameter = 2r.Kern radius for a circle = r/4.Kern diameter = 2 * (r/4) = r/2.Fraction of main diameter = (r/2) / (2r) = 1/4.Hence, kern diameter = one-fourth of the main diameter.

Verification / Alternative check:
Classical results for kerns: rectangle → middle third; circle → concentric circle radius r/4; triangle → centroidal regions defined by medians. The circular case consistently yields r/4 radius.

Why Other Options Are Wrong:

• one-half: This is the kern diameter in absolute units (r/2) but expressed as a fraction of the main diameter (2r) it becomes 1/4, not 1/2.
• one-third, one-eighth, one-fifth: Do not match the r/4 kern radius result.

Common Pitfalls:
Confusing kern diameter with kern radius or mixing absolute dimensions with fractional comparisons.

Final Answer:
one-fourth