Eccentric loading — kern (core) of a rectangular column section For a rectangular column with cross-section b × h (about the centroidal axes), the core (kern) region within which a compressive load must act to avoid tensile stress anywhere on the section has what shape and size?

Introduction / Context:
The kern (or core) of a section is the locus of load application points that keep the stress over the entire section compressive (no tension) when the member is axially compressed with eccentricity. For masonry and unreinforced concrete columns, staying within the kern avoids cracking from tension, making core geometry crucial in design checks and detailing guidelines.

Given Data / Assumptions:

• Cross-section is rectangular with breadth b and depth h.
• Material behaves linearly elastic; stress varies linearly across the section.
• Load is compressive with possible biaxial eccentricities.

Concept / Approach:
The kern is defined by the condition that the resultant stress does not change sign anywhere on the section. For a rectangular section, the no-tension condition leads to eccentricity limits of ±b/6 about the x-axis and ±h/6 about the y-axis. Combining these for biaxial loading yields a rhombus in the cross-sectional plane whose diagonals are b/3 and h/3, centered at the centroid. Any load resultant within this rhombus produces compressive stress over the entire section.

Step-by-Step Solution:

Verification / Alternative check:
Using the linear stress distribution formula, sigma = P/A ± M_x/Z_x ± M_y/Z_y, set the edge stress to zero to recover the e ≤ b/6 and h/6 bounds, confirming the rhombus kern.

Why Other Options Are Wrong:

• Rectangle b/2 × h/2 or square b/2: overly conservative and not the theoretical kern.
• Rhombus with side length h/2: dimensionally and conceptually incorrect.
• None: incorrect because the rhombus with diagonals b/3 and h/3 is standard.

Common Pitfalls:
Confusing the kern with the centroidal core used in reinforced sections; mixing up diagonals versus side lengths; forgetting that the kern shrinks as section shape changes (e.g., circular sections have a circular kern of radius r/4).

Final Answer:
Rhombus whose diagonals are b/3 and h/3