A simply supported beam of span L and flexural rigidity EI carries a unit point load at its centre. The strain energy in the beam due to bending is:
Try it first: pick an option, then read the explanation.
Correct answer: C. L³/(96EI)
Asked in SSC JE 02/03/2017 (Morning)
Detailed Explanation
For a single gradually applied load, the strain energy equals the work done: half the load times the deflection under it.
\[ U = \frac12 P\Delta = \frac12 P\cdot\frac{PL^3}{48EI} = \frac{P^2L^3}{96EI} \xrightarrow{P=1} \frac{L^3}{96EI} \]
Check by integration: \( U = 2\int_0^{L/2}\frac{(Px/2)^2}{2EI}dx = \frac{P^2L^3}{96EI} \).
Why “L³/(96EI)” Is Correct
With P = 1 the strain energy is L³/(96EI).
Why the Other Options Are Wrong
- A. L³/(48EI): L³/48EI is the deflection under a unit load, not the energy (missing ½).
- B. L³/(192EI): L³/192EI is half the correct value.
- D. L³/(16EI): L³/16EI has no basis.
Key Point
Single load: U = ½PΔ. SS central load: U = P²L³/96EI.
Key Concepts for Students
- Castigliano: ∂U/∂P = PL³/48EI = deflection under P.
- Energy is proportional to the square of the load.
- Cantilever tip load: U = P²L³/6EI.
Practice more: this is question 42 in Slope and Deflection MCQs – Part 3. Try the whole set and check every answer instantly.