Strength of materials – Buckling: The slenderness ratio of a column is defined as the ratio of the effective length of the column to its least radius of gyration (l_e / k_min). Identify the correct definition.

Introduction / Context:
The slenderness ratio is a fundamental measure in column buckling used to judge whether a column will fail by crushing (short column) or by elastic buckling (long/medium column). A correct understanding of this ratio helps engineers select suitable design formulas such as Euler's or Rankine–Gordon.

Given Data / Assumptions:

• The column has an effective length l_e determined by end conditions.
• The least radius of gyration is k_min = sqrt(I_min / A), where I_min is the minimum second moment of area and A is the cross-sectional area.
• Material is homogeneous and straight; initial imperfections are neglected for the definition.

Concept / Approach:
The slenderness ratio, commonly denoted by λ, compares geometric slenderness to sectional stiffness: λ = l_e / k_min. A larger λ indicates a more slender member and a higher propensity for elastic buckling at lower loads.

Step-by-Step Solution:

Verification / Alternative check:
Dimensional analysis: l_e has units of length and k_min has units of length, so λ is dimensionless, validating the ratio.

Why Other Options Are Wrong:

• Area to k_min: incorrect dimensional form and not used in buckling criteria.
• k_min to area: also dimensionally inconsistent with a pure ratio used in codes.
• k_min to length: reciprocal of the correct ratio and not standard.

Common Pitfalls:
Using the radius of gyration about the wrong axis; always use the least (weak) axis value to capture the critical buckling mode.

Final Answer:
Length of column to least radius of gyration