The ratio of the free-end deflections of a cantilever due to an isolated load placed at 1/3 and at 2/3 of the span (measured from the fixed end) is:
Try it first: pick an option, then read the explanation.
Asked in SSC JE 23/01/2018 (Evening)
Detailed Explanation
For a load W at distance a from the fixed end, the tip deflection = deflection under the load + slope at the load × remaining length.
\[ \delta_{tip} = \frac{Wa^3}{3EI} + \frac{Wa^2}{2EI}(L-a) = \frac{Wa^2}{6EI}(3L - a) \]
a = L/3: \( \dfrac{W L^2/9}{6EI}\cdot\dfrac{8L}{3} = \dfrac{4WL^3}{81EI} \) a = 2L/3: \( \dfrac{4WL^2/9}{6EI}\cdot\dfrac{7L}{3} = \dfrac{14WL^3}{81EI} \)
\[ \text{Ratio} = \frac{4}{14} = \frac{2}{7} \]
Why “2/7” Is Correct
The ratio is 2/7. Moving the load from 1/3 to 2/3 of the span increases the tip deflection 3.5 times.
Why the Other Options Are Wrong
- A. 1/7: 1/7 drops the factor 2 (uses a³ ratios incorrectly).
- C. 3/7: 3/7 has no basis.
- D. 2/5: 2/5 ignores the slope contribution of the unloaded part.
Key Point
Cantilever tip deflection for a load at a: Wa²(3L − a)/(6EI).
Key Concepts for Students
- Maxwell's reciprocal theorem: deflection at C due to load at B equals deflection at B due to load at C.
- Ratio of deflections under the loads (a³ ratio) would be 1/8.
- Load at the tip (a = L) gives WL³/3EI.
Practice more: this is question 28 in Slope and Deflection MCQs – Part 2. Try the whole set and check every answer instantly.