For a rectangular column section, within what central zone (kern) must the line of action of the compressive load fall to avoid any tensile stress on the cross-section?

Introduction / Context:
The kern (core) of a section is the region in which a compressive load must act so the resultant stress remains compressive across the entire section. This concept is central to masonry, concrete, and column design to prevent cracking due to tension.

Given Data / Assumptions:

• Rectangular cross-section.
• Axial compressive load possibly with small eccentricity.
• Linear elastic stress distribution (Navier’s formula) applies.

Concept / Approach:
For a rectangle, the kern along each principal direction extends to one-sixth of the corresponding dimension from the centroid. If the load resultant passes within this middle-third zone (i.e., within ± L/6), the extreme fiber stress does not become tensile.

Step-by-Step Solution:
Rectangular section side = b (consider one axis).Kern limit from centroid along that axis = b/6.Thus the total permissible central zone along base direction = one third of base, i.e., load must lie within ± b/6 from centroid → colloquially stated as 'one sixth of the base on either side of centroid'.

Verification / Alternative check:
Using stress formula: σ = P/A ± M*y/I At kern limit, compression at the opposite extreme reaches zero; solving yields y = b/6.

Why Other Options Are Wrong:

• One-half, one-fourth, one-fifth: do not satisfy the zero-tension condition from elastic theory for rectangles.
• One-third of base: describes full middle-third width, but the phrasing needed is the limit from centroid; the correct option states 'one sixth on either side'.

Common Pitfalls:

• Confusing the total middle-third width (b/3) with the kern limit from the centroid (± b/6).

Final Answer:
one sixth of the base on either side of centroid