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:

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:

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.