Deflection comparison: two simply supported beams of equal length l and breadth b carry the same central point load W. Beam B has double the depth (2d) compared to beam A (depth d). State whether the deflection of beam B is “double” that of beam A.
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ACorrect
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BIncorrect
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CTrue only if E is doubled
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DTrue only for cantilevers
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EIncorrect unless load is halved
Answer
Correct Answer: Incorrect
Explanation
Introduction / Context:The midspan deflection of a simply supported beam under a central point load depends on span, stiffness, and load. Changing the cross-section (depth) changes the second moment of area I and therefore the deflection. This question checks understanding of how deflection scales with depth.
Given Data / Assumptions:
- Both beams: simply supported, same length l and breadth b.
- Same central point load W and same material (same E).
- Beam B has depth 2d; beam A has depth d.
Concept / Approach:The midspan deflection for a simply supported beam with a central point load is delta = W * l^3 / (48 * E * I). For a rectangle, I = b * d^3 / 12. Deflection is inversely proportional to I and thus to d^3 for constant b.
Step-by-Step Solution:
I_A = b * d^3 / 12.I_B = b * (2d)^3 / 12 = 8 * (b * d^3 / 12) = 8 * I_A.delta_B / delta_A = I_A / I_B = 1 / 8.Therefore, beam B deflects one-eighth of beam A, not double.Verification / Alternative check:Use section modulus or quick proportionality: deflection ∝ 1 / d^3 for rectangular sections; doubling depth reduces deflection by 2^3 = 8.
Why Other Options Are Wrong:“Correct”: opposite of the actual scaling.Conditions about E or cantilevers are irrelevant; the stated comparison is for like-for-like beams and loading.
Common Pitfalls:Thinking deflection scales with 1/d or 1/d^2 instead of 1/d^3; forgetting the cubic effect of depth on I.
Final Answer:
Incorrect