Short hollow circular column — maximum eccentricity to avoid tension For a short column with hollow circular cross-section of external diameter D and internal diameter d, what is the greatest eccentricity e (measured from the centroid) that a compressive load W can have without causing tension anywhere on the cross-section?

Introduction / Context:
For concentrically compressed members with possible eccentricity, avoiding tensile stress anywhere on the section requires the load line to pass within the core (kern). The kern radius depends on the section geometry. For circular and hollow circular sections, closed-form expressions exist for the limiting eccentricity that ensures compressive stress across the whole section.

Given Data / Assumptions:

• Short column (no instability effects; pure elastic stress distribution).
• Hollow circular section with outer diameter D and inner diameter d.
• Linear stress distribution due to combined axial load and bending.

Concept / Approach:

The limiting eccentricity equals the kern radius r_k measured from the centroid. For a hollow circular section, r_k = (I)/(AR_max), where I is second moment of area about any axis through the centroid, A is area, and R_max is the outer radius (D/2). Evaluation gives r_k = (D^2 + d^2)/(8D). A load with e ≤ r_k keeps the resultant compressive over the entire section; e > r_k would induce tension at the far edge.

Step-by-Step Solution:

Verification / Alternative check:

For a solid circular section (d = 0), the formula reduces to r_k = D/8, matching the well-known solid circle core radius.

Why Other Options Are Wrong:

• (b) Uses D^2 − d^2, which corresponds to A terms, not the correct (R^2 + r^2) dependence.
• (c) Dividing by d instead of D gives a nonphysical increase as the hole shrinks.
• (d) The factor 16 is incorrect per derivation.
• (e) D/12 is not the kern radius for circular sections.

Common Pitfalls:

• Using Z = I/(c) or section modulus directly; kern uses I/(A*R), not I/c.

Final Answer:

e = (D^2 + d^2) / (8D).