Buckling analysis terminology: The “equivalent length” of a column is defined as the length of a hinged–hinged column of the same material and section that has the same Euler crippling load as the given column with its end conditions. Is this definition correct?
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ATrue
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BFalse
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COnly valid for very short columns
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DOnly valid for very long columns
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EValid only when load is eccentric
Answer
Correct Answer: True
Explanation
Introduction / Context:In column buckling, end restraints change the effective buckling length and thus the critical load. Using an equivalent length allows all end conditions to be treated with a single Euler formula by adjusting the effective column length.
Given Data / Assumptions:
- Elastic buckling per Euler theory.
- Same cross-section and material for real and equivalent columns.
- End conditions captured via an effective length factor K.
Concept / Approach:Euler's critical load: P_cr = pi^2 * E * I / (L_e)^2, where L_e = K * l. The equivalent (effective) length L_e is defined so a hypothetical hinged–hinged column of length L_e buckles at the same load as the actual column with given end fixity.
Step-by-Step Solution:
Identify actual end condition ⇒ determine K (e.g., 0.5 for fixed–fixed, 1.0 for pinned–pinned, 2.0 for fixed–free).Compute L_e = K * l.Use Euler formula with L_e to get the same P_cr as the real column.Verification / Alternative check:Mode shapes for different end conditions show different effective half-wavelengths; the K factors are derived from boundary-condition solutions of the buckling differential equation.
Why Other Options Are Wrong:Restriction to short or long columns is incorrect; the definition is general within Euler's elastic range (slender columns).
Common Pitfalls:Confusing actual length with effective length; misapplying K, leading to large errors in P_cr.
Final Answer:
True