Slenderness ratio of a long column (structural mechanics): Among the following definitions, identify the correct engineering definition of “slenderness ratio” used in Euler/Rankine buckling design. It should express how long (L) a member is relative to its stiffness against bending (radius of gyration, k), and should clearly indicate that the least radius of gyration (k_min) is used for safety in buckling calculations.

Introduction / Context:
Columns fail primarily by buckling rather than crushing when they are thin and long. To quantify how “long and thin” a column is, engineers use the slenderness ratio. This ratio is central to Euler and Rankine formulas and guides whether elastic buckling or inelastic behavior governs design.

Given Data / Assumptions:

• Column length considered between effective end conditions is L.
• Radius of gyration about an axis is k = sqrt(I/A), where I is the second moment of area and A is the cross-sectional area.
• Least radius of gyration k_min controls buckling since buckling occurs about the weakest (least stiff) axis.

Concept / Approach:

The slenderness ratio quantifies susceptibility to buckling and is defined as L/k. For safety we use the smallest k, hence L/k_min. Using k_min ensures we check buckling about the axis with minimum stiffness where buckling is most likely.

Step-by-Step Solution:

Verification / Alternative check:

When k decreases (section more “slender” about that axis), L/k increases, which correctly predicts a greater tendency to buckle. This aligns with Euler's critical load proportional to 1/L^2 and inversely to k^2 via I = A k^2.

Why Other Options Are Wrong:

• Area divided by k or least k is dimensionally inconsistent with a ratio used for buckling.
• k divided by area has no direct buckling meaning.
• Using “any axis” can miss the weakest axis; buckling checks must use k_min.

Common Pitfalls:

• Using overall length instead of effective length that accounts for end conditions.
• Checking only the strong axis and ignoring the weak axis (k_min).

Final Answer:

Length of column divided by least radius of gyration.