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GATE 2025 CE (CE2) – Question 38

Structural Analysis · Statically indeterminate structures: force and displacement methods · 2 marks · Multiple choice

The figure shows a propped cantilever with uniform flexural rigidity $EI$ (in N.m$^2$) and subjected to a moment $M$ (in N.m). Consider forces and displacements in the upward direction as positive.
Find the upward reaction at the propped support B (in N) when this support settles by $(-\Delta)$, given in metres.

a beam of length L fixed at A on the left and on a roller support B on the right, with a clockwise moment M applied at B.
  1. $\frac{3M}{2L} - \frac{6EI\Delta}{L^3}$
  2. $\frac{8M}{3L} - \frac{2EI\Delta}{L^3}$
  3. $\frac{3M}{2L} - \frac{3EI\Delta}{L^3}$
  4. $\frac{M}{L} - \frac{8EI\Delta}{L^3}$

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Correct answer: (C) $\frac{3M}{2L} - \frac{3EI\Delta}{L^3}$

Explanation

Remove the prop at B and use the cantilever fixed at A. The clockwise moment $M$ at the tip deflects it downwards by $\frac{ML^2}{2EI}$, and the upward prop reaction $R$ lifts it by $\frac{RL^3}{3EI}$. The support settles by $\Delta$ downwards, so the net downward deflection at B must be $\Delta$: $\frac{ML^2}{2EI} - \frac{RL^3}{3EI} = \Delta$. Then $R = \frac{3EI}{L^3}\left(\frac{ML^2}{2EI} - \Delta\right) = \frac{3M}{2L} - \frac{3EI\Delta}{L^3}$.