A simply supported uniform rectangular bar of breadth b, depth d and length L carries a load W at mid-span. The same bar extends by e under the same load applied as axial tension. The ratio of the maximum deflection to the elongation is:
Try it first: pick an option, then read the explanation.
Correct answer: C. (L/(2d))²
Asked in SSC JE 22/01/2018 (Evening)
Detailed Explanation
Write both displacements in terms of W, L, E, b and d, then divide. The load, material and breadth cancel.
\[ e = \frac{WL}{AE} = \frac{WL}{bdE} \qquad \delta = \frac{WL^3}{48E\,(bd^3/12)} = \frac{WL^3}{4Ebd^3} \]
\[ \frac{\delta}{e} = \frac{WL^3/(4Ebd^3)}{WL/(bdE)} = \frac{L^2}{4d^2} = \left(\frac{L}{2d}\right)^2 \]
Why “(L/(2d))²” Is Correct
The ratio is (L/2d)². For a bar 20 times longer than its depth this is 100 — bending is far more flexible than axial stretching.
Why the Other Options Are Wrong
- A. L/d: L/d is missing the square and the factor 2.
- B. L/(2d): L/2d is missing the square.
- D. (L/(3d))²: (L/3d)² comes from a circular section (δ/e = L²/3d², see a similar question), not a rectangle.
Key Point
Rectangular bar: central-load deflection / axial elongation = (L/2d)².
Key Concepts for Students
- For a circular bar of diameter d the ratio is L²/(3d²).
- I = bd³/12 = A·d²/12 for a rectangle.
- Bending stiffness depends on d³, axial stiffness on d.
Practice more: this is question 20 in Slope and Deflection MCQs – Part 2. Try the whole set and check every answer instantly.