What is the deflection at the free end of a cantilever of length L carrying a point load P at its free end? (E = Young's modulus, I = moment of inertia)
Try it first: pick an option, then read the explanation.
Correct answer: D. PL³/(3EI)
Asked in SSC JE 23/03/2021 (Evening); also 22/01/2018 (Evening) and SSC JE 2011
Detailed Explanation
This is the most used deflection result in strength of materials. It comes from integrating EI y'' = −P(L − x) twice with zero slope and deflection at the fixed end.
\[ EIy = -P\left(\frac{Lx^2}{2} - \frac{x^3}{6}\right) \Rightarrow y(L) = -\frac{PL^3}{3EI} \]
Why “PL³/(3EI)” Is Correct
The free-end deflection is PL³/(3EI), and the free-end slope is PL²/(2EI).
Why the Other Options Are Wrong
- A. PL²/(3EI): PL²/3EI has the dimensions of a slope.
- B. PL³/(6EI): PL³/6EI is half the correct value.
- C. 4PL³/(3EI): 4PL³/3EI is the deflection of a cantilever of length 2L divided by 2 — not this case.
Key Point
Cantilever, tip load: δ = PL³/3EI, θ = PL²/2EI.
Key Concepts for Students
- Deflection is proportional to L³ — doubling the length makes it 8 times larger.
- Stiffness of a cantilever at the tip = 3EI/L³.
- A simply supported beam with central load deflects 1/16 as much (WL³/48EI).
Practice more: this is question 12 in Slope and Deflection MCQs – Part 1. Try the whole set and check every answer instantly.