Core (kernel) of a rectangular column: For a prismatic rectangular column of cross-sectional area A, what is the area of its core (also called kernel) within which the resultant compressive load must fall to avoid tension anywhere on the section?

Introduction / Context:
The core or kernel of a section is the locus of points where a resultant compressive load may be applied so that resultant stress over the entire section remains compressive (no tension). For a rectangular column, this is a smaller concentric rectangle controlling eccentric load placement in design of foundations and columns.

Given Data / Assumptions:

• Section is rectangular, prismatic, homogeneous, and elastic.
• Total section area is A = b * d.
• Only axial compression with possible eccentricity; no shear.

Concept / Approach:

For a rectangle of width b and depth d, the core dimensions are b/3 by d/3. Hence the core area equals (b/3) * (d/3) = (b d)/9 = A/9. This arises from the straight-line stress distribution condition ensuring that extreme fiber stress does not change sign when the load resultant lies within the central kernel.

Step-by-Step Solution:

Verification / Alternative check:

For a square (b = d), kernel is an inner square with side b/3, again giving area A/9, confirming the general rectangular result.

Why Other Options Are Wrong:

A/4 and A/6 are too large; A/16 is the circular section core area (r/4 radius), not applicable to rectangles; A/12 is incorrect for rectangles.

Common Pitfalls:

Confusing rectangular and circular cores; mixing up kernel dimensions (b/3, d/3) with mid-third rule for retaining walls.

Final Answer:

A/9