Column behaviour terminology The ratio of the effective length of a column to the minimum radius of gyration of its cross-section is known as:

Introduction / Context:
Columns fail predominantly by buckling rather than crushing when they are slender. Quantifying slenderness is therefore central to column design in steel, reinforced concrete, timber, and aluminum structures. The slenderness ratio provides a non-dimensional measure linking member length and section stiffness distribution.

Given Data / Assumptions:

• Effective length kL reflects end restraints (pinned, fixed, etc.).
• Minimum radius of gyration r_min = sqrt(I_min/A) where I_min is the least second moment of area.
• Elastic buckling governed by Euler theory for ideal columns; design codes modify with imperfections and inelasticity.

Concept / Approach:

The slenderness ratio is defined as λ = (effective length)/(minimum radius of gyration) = kL / r_min. It captures both the global instability tendency (longer members buckle more easily) and cross-sectional stiffness (larger r resists curvature). Design curves and interaction checks are parameterized by λ.

Step-by-Step Solution:

Verification / Alternative check:

Euler critical stress σ_cr = π^2 E / (λ^2) (when expressed with λ = kL/r); this shows buckling sensitivity increases rapidly with λ, confirming why the term is central in column stability.

Why Other Options Are Wrong:

• Buckling factor and crippling factor are not standard definitions for this ratio.
• “Stability index” is used in frame stability (second-order analysis), not for the simple ratio asked.
• “None of these” is incorrect since a precise, named term exists.

Common Pitfalls:

• Using r about the wrong axis; always take the minimum radius of gyration.
• Ignoring end conditions; k often differs from 1.0.

Final Answer:

slenderness ratio.