Rankine–Gordon (Rankine) column formula — applicability across slenderness ranges The Rankine–Gordon formula accounts for both direct compressive (crushing) stress and elastic buckling. For which range of column lengths is this semi-empirical formula considered applicable in practical design?

Introduction / Context:The Rankine–Gordon (often called Rankine) column formula blends crushing strength (short-column behavior) with Euler buckling (long-column behavior). It is widely used in civil and mechanical engineering to estimate the safe load for columns with a broad range of slenderness ratios.

Given Data / Assumptions:

• Columns may be short, intermediate, or long based on slenderness ratio.
• Material is homogeneous and isotropic; end conditions are known.
• Rankine formula combines two failure modes: crushing and buckling.

Concept / Approach:The Rankine formula is typically written in the form:P_rankine = P_crushing / (1 + P_crushing / P_euler)or equivalently using stresses:sigma_allow = sigma_c / (1 + (sigma_c / sigma_euler))It transitions smoothly: for very short columns, P_euler is very large so the denominator tends to 1 and crushing dominates; for very slender columns, P_euler becomes small and governs.

Step-by-Step Solution:

Verification / Alternative check:Design codes and textbooks show Rankine matching test data reasonably across ranges, especially useful where Euler alone is unconservative for intermediate columns.

Why Other Options Are Wrong:

• Only very long/long/short/intermediate: each restricts the applicability incorrectly.

Common Pitfalls:Using Euler for intermediate columns without checking crushing; ignoring end conditions when computing Euler load.

Final Answer:All of the above (works across short to long ranges)