Changing beam proportions — effect on central deflection under a midspan point load A simply supported beam with a central point load has a rectangular cross-section of width b and depth d (depth is vertical). If the width and depth are interchanged (new width = d, new depth = b), the central deflection will change in the ratio:
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Ab/d
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Bd/b
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C(d/b)^2
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D(b/d)^2
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E(b/d)^3
Answer
Correct Answer: (d/b)^2
Explanation
Introduction / Context:The central deflection of a simply supported beam with a central load depends inversely on the flexural rigidity E I. For rectangular sections, the second moment of area I depends strongly on the depth (about the bending axis) as I ∝ b d^3. Interchanging width and depth changes I markedly, thus changing deflection significantly.
Given Data / Assumptions:
- Simply supported span, single central point load P.
- Initial section: width b (horizontal), depth d (vertical).
- After swap: width = d, depth = b.
- Same material (E), same load P, same span L.
Concept / Approach:
For a central point load, δ = P L^3 / (48 E I). With a rectangular section bending about the strong axis, I = b d^3 / 12. After swapping, I′ = d b^3 / 12. Since deflection is inversely proportional to I, the ratio δ′/δ equals I/I′.
Step-by-Step Solution:
I = b d^3 / 12.I′ = d b^3 / 12.δ ∝ 1/I and δ′ ∝ 1/I′ ⇒ δ′/δ = I / I′ = (b d^3) / (d b^3) = (d^2)/(b^2) = (d/b)^2.Verification / Alternative check:
If d > b (a common case), (d/b)^2 > 1, so swapping makes the beam much more flexible (deflection increases)—consistent with intuition because the “depth” has been reduced from d to b.
Why Other Options Are Wrong:
- b/d or d/b underestimate the cubic sensitivity of I to depth.
- (b/d)^2 inverses the correct ratio.
- (b/d)^3 has no basis for deflection ratio in this swap scenario.
Common Pitfalls:
- Forgetting that I for a rectangle varies with the cube of the depth about the bending axis.
Final Answer:
(d/b)^2.