Applicability of Rankine’s column formula Rankine’s empirical formula is valid for:

Introduction / Context:Rankine’s formula blends crushing failure (dominant in short columns) and elastic buckling (dominant in long columns) to estimate critical load. It is widely used because it provides a smooth transition across slenderness ratios where neither pure Euler nor pure crushing alone is adequate.

Given Data / Assumptions:

• Empirical constants derived from tests.
• Applies to prismatic columns with typical end conditions (via effective length).
• Material behaves elastically up to failure modes considered.

Concept / Approach:Rankine’s formula: P_cr = (sigma_c * A) / [1 + a * (L_e / r)^2], combining material crushing strength sigma_c and slenderness (L_e / r). For very small slenderness, denominator ≈ 1 (crushing). For large slenderness, it trends toward Euler behavior.

Step-by-Step Solution:Identify effective length L_e and radius of gyration r.Compute slenderness lambda = L_e / r.Use Rankine formula across lambda values to obtain a single P_cr expression applicable to both regimes.

Verification / Alternative check:Limiting cases confirm: lambda → 0 ⇒ crushing; lambda → large ⇒ Euler-like response through 1/(lambda^2) behavior.

Why Other Options Are Wrong:Restricting to only short or only long columns ignores the blended nature. End condition exclusivity is handled via L_e, not by the formula’s scope. “Weak columns only” is not a standard category.

Common Pitfalls:Using Euler for stocky columns; ignoring effective length factors; misusing crushing stress without stability check.

Final Answer:both short and long columns