Euler's column theory — identify the foundational assumptions: Which of the following are assumed in Euler's elastic buckling theory for slender columns?
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AFailure occurs by elastic buckling alone
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BLength is very large compared to cross-sectional dimensions (slender)
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CMaterial obeys Hooke's law (linear elasticity)
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Dall of the above
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ELoad acts with large eccentricity by default
Answer
Correct Answer: all of the above
Explanation
Introduction:Euler's theory predicts the critical load for ideal slender columns. Knowing its assumptions clarifies when the formula Pcr = π^2EI/Le^2 applies.Given Data / Assumptions:
- Ideal geometry: straight column, prismatic section.
- Axial load through centroid with negligible initial curvature.
- Linear elasticity up to buckling.
Concept / Approach:The theory presumes elastic behavior and buckling as the governing limit state, valid for high slenderness ratios where inelastic yielding is not reached before instability.Step-by-Step Solution:
Recognize slenderness: length >> section sizeAdopt Hooke’s law: stress proportional to strainLimit state: failure by buckling, not crushing or yieldingVerification / Alternative check:Columns with low slenderness do not satisfy Euler's assumptions; inelastic or empirical formulas (e.g., Rankine-Gordon) are then used, confirming the boundary of applicability.Why Other Options Are Wrong:
- “Load acts with large eccentricity”: That violates the central-loading assumption in the classical derivation.
Common Pitfalls:Applying Euler's formula to stocky columns; ignoring end conditions when computing effective length Le; assuming plasticity while using elastic theory.Final Answer:
all of the above