Slenderness ratio basics: The slenderness ratio λ of a column is defined as L_e / r. Under what condition would the slenderness ratio be zero?
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AWhen the effective length is zero (L_e = 0)
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BWhen the effective length equals the radius of gyration (L_e = r)
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CWhen the member is continuously supported on all sides throughout its length
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DWhen measured only between points of zero bending moment
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EWhen radius of gyration is zero
Answer
Correct Answer: When the effective length is zero (L_e = 0)
Explanation
Introduction / Context:Slenderness ratio λ = L_e / r is a key geometric parameter in column stability. It influences the buckling capacity and the applicability of Euler or inelastic buckling curves.
Given Data / Assumptions:
- L_e is the effective length accounting for end restraints.
- r is the radius of gyration about the buckling axis.
Concept / Approach:By definition, λ approaches zero only if the numerator L_e → 0, since r is positive for any real cross-section. Physically, L_e = 0 is a limiting conceptual case (e.g., an element of zero effective length), not realized in practice but useful to understand the mathematics.
Step-by-Step Solution:
Recall λ = L_e / r.Set λ = 0 ⇒ L_e must be 0 (r ≠ 0 for a real section).Hence the condition is L_e = 0.Verification / Alternative check:Other conditions (e.g., L_e = r) give λ = 1, not zero; making supports “continuous” reduces L_e but not to zero in real structures.
Why Other Options Are Wrong:L_e = r yields λ = 1; “continuously supported” reduces but does not nullify L_e; “between points of zero moment” defines buckling curve but not λ = 0; r = 0 is non-physical for a finite cross-section.
Common Pitfalls:Confusing “very small slenderness” with “zero”; neglecting direction-specific r and end-restraint factors in effective length calculation.
Final Answer:When the effective length is zero (L_e = 0)