In structural engineering, the slenderness ratio of a compression member is defined as the ratio of its effective length to its least radius of gyration, used to assess buckling tendency.

Introduction / Context:
In steel and reinforced concrete design, compression members such as columns can fail by buckling well before the material reaches its compressive strength. The key non-dimensional parameter that signals buckling vulnerability is the slenderness ratio, which compares a member's effective length to its least radius of gyration.

Given Data / Assumptions:

• Member carries primarily axial compression.
• Effective length L_eff accounts for end restraints (fixed, pinned, free, etc.).
• r_min is the least radius of gyration among the principal axes.

Concept / Approach:
The radius of gyration r about an axis is r = sqrt(I/A). Slenderness ratio is defined as lambda = L_eff / r. Buckling is more likely as lambda increases; therefore the least radius of gyration r_min must be used because buckling occurs about the weakest (least stiff) axis.

Step-by-Step Solution:
Define r = sqrt(I/A)Identify r_min among available axesCompute L_eff based on boundary conditions (for example, pinned-pinned: L_eff = L; fixed-free: L_eff = 2L)Slenderness ratio lambda = L_eff / r_min

Verification / Alternative check:
If end conditions are changed, only L_eff changes; the expression remains L_eff / r_min, confirming the definition is independent of material strength and depends purely on geometry and restraint.

Why Other Options Are Wrong:

• r_min / L_eff reverses the definition and would decrease with increasing slenderness, which contradicts buckling behavior.
• A / I and I / A are unrelated ratios; they neither include length nor capture end restraint effects.

Common Pitfalls:

• Using actual length L instead of effective length L_eff.
• Using radius of gyration about the strong axis rather than the least (critical) axis.

Final Answer:
Effective length / least radius of gyration (L_eff / r_min)