GATE 2024 CE (CE2) – Question 49
For the 6 m long horizontal cantilever beam PQR shown in the figure, Q is the mid-point. Segment PQ of the beam has flexural rigidity $EI = 2 \times 10^5$ kN.m$^2$ whereas the segment QR has infinite flexural rigidity. Segment QR is subjected to uniformly distributed, vertically downward load of 5 kN/m.
[Figure: a cantilever fixed at P, with 3 m from P to Q and 3 m from Q to R. The 5 kN/m load acts on QR.]
The magnitude of the vertical displacement (in mm) at point Q is ________ (rounded off to 3 decimal places).
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Correct answer: 1.171 to 1.191
Explanation
Since QR is infinitely rigid, it does not bend and its load acts on the flexible part PQ as a force and a moment at Q. The load is $5 \times 3 = 15$ kN and acts at 1.5 m from Q, which gives the moment $15 \times 1.5 = 22.5$ kN m. The tip deflection of PQ (length 3 m) is $\frac{PL^3}{3EI} + \frac{ML^2}{2EI} = \frac{15 \times 27}{3 \times 2 \times 10^5} + \frac{22.5 \times 9}{2 \times 2 \times 10^5} = 6.75 \times 10^{-4} + 5.0625 \times 10^{-4} = 1.18125 \times 10^{-3}$ m $= 1.181$ mm.