Eccentric loading on circular columns — rule for no-tension condition To ensure no tensile stress anywhere on the cross-section of a circular column under axial load with eccentricity, which empirical rule is typically applied to limit the eccentricity range?

Introduction / Context:
When a compressive load acts with eccentricity on a column section, bending stresses superimpose on direct compression. Designers often use simple “kern” rules to ensure the resultant remains within a central core so that tensile stresses do not develop (important for masonry and unreinforced concrete).

Given Data / Assumptions:

• Circular, homogeneous, isotropic cross-section of radius R.
• Axial compression with eccentricity in any direction.
• Linear elastic stress distribution σ = P/A ± M y / I.

Concept / Approach:

The kern (core) of a section is the locus of load resultants for which normal stress stays compressive over the entire section. For a circle, the kern is a concentric circle of radius R/4. Equivalently, the resultant load must lie within the “middle fourth” of the diameter—hence the middle fourth rule.

Step-by-Step Solution:

Verification / Alternative check:

By setting σ_min = P/A − M c / I ≥ 0 and using M = P e, you obtain e ≤ I/(A c) = R/4, confirming the rule analytically.

Why Other Options Are Wrong:

• Middle third rule applies to rectangles (kern limits at ±b/6, ±h/6).
• Middle half and middle eighth are not the correct kern sizes for circular sections.
• “None” is incorrect—there is a well-defined rule.

Common Pitfalls:

• Applying the rectangular middle-third rule to circular columns by mistake.
• Confusing diameter quarters with area quarters; the rule concerns diameter (R/4 from center).

Final Answer:

Middle fourth rule.