A cantilever of span l carries a concentrated load W at a distance a from the fixed end. The deflection under the point load is:
Try it first: pick an option, then read the explanation.
Asked in SSC JE 23/09/2019 (Evening)
Detailed Explanation
Between the fixed end and the load, the beam behaves exactly like a cantilever of length a with a tip load W. Beyond the load, the bending moment is zero, so that part simply stays straight and follows the slope at the load point.
Under the load: \[ y_B = \frac{Wa^3}{3EI} \] At the free end (for comparison): \[ y_C = \frac{Wa^3}{3EI} + \frac{Wa^2}{2EI}(l-a) \]
Why “Wa³/(3EI)” Is Correct
The deflection under the load is Wa³/(3EI). The free-end deflection is larger because the unloaded straight part tilts by θB = Wa²/2EI.
Why the Other Options Are Wrong
- A. W(l − a)³/(3EI): W(l − a)³/3EI uses the unloaded length.
- B. (l − a)Wa²/(3EI) + Wa³/(3EI): This uses Wa²/3EI instead of Wa²/2EI for the slope term, and it is in any case a free-end, not under-load, expression.
- D. Wl³/(3EI): Wl³/3EI applies only when the load is at the free end (a = l).
Key Point
Cantilever, load at a from fixed end: δ under load = Wa³/3EI; δ at tip = Wa³/3EI + Wa²(l − a)/2EI.
Key Concepts for Students
- Zero-moment segments stay straight.
- Tip deflection uses the slope at the load point times the remaining length.
- Load at mid-span: tip deflection = 5WL³/48EI.
Practice more: this is question 21 in Slope and Deflection MCQs – Part 2. Try the whole set and check every answer instantly.