For a beam carrying a uniformly distributed load, the strain energy will be maximum when the beam is:
Try it first: pick an option, then read the explanation.
Asked in SSC JE 2012
Detailed Explanation
Strain energy in bending is U = ∫M²dx/(2EI), or for a given load, ½ × load × deflection. The more flexible the support arrangement, the larger the moments and deflections, and the more energy is stored.
For the same w and L: \[ U_{cant} = \frac{w^2L^5}{40EI},\quad U_{ss} = \frac{w^2L^5}{240EI} \]
Why “a cantilever” Is Correct
The cantilever is the least restrained arrangement: maximum moment wL²/2 and maximum deflection wL⁴/8EI. Its strain energy is six times that of a simply supported beam, and even more than the propped or fixed beams.
| Support (UDL w, span L) | Max |M| | Max deflection |
|---|---|---|
| Cantilever | wL²/2 | wL⁴/8EI |
| Simply supported | wL²/8 | 5wL⁴/384EI |
| Propped cantilever | wL²/8 | wL⁴/185EI |
| Fixed both ends | wL²/12 | wL⁴/384EI |
Why the Other Options Are Wrong
- B. simply supported: A simply supported beam has smaller moments (wL²/8) and stores less energy.
- C. a propped cantilever: The prop reduces moments and deflection further.
- D. fixed at both ends: A fixed beam is the stiffest and stores the least energy.
Key Point
More flexible support ⇒ larger M and δ ⇒ more strain energy. Cantilever is the maximum.
Key Concepts for Students
- U = ½ W δ for a single load.
- Fixed-ended beam: smallest deflection (wL⁴/384EI).
- Energy methods give deflection via Castigliano's theorem.
Practice more: this is question 41 in Slope and Deflection MCQs – Part 3. Try the whole set and check every answer instantly.