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GATE 2022 ME (ME1) – Question 16

Mechanics of Materials · Euler theory of columns · 1 mark · Multiple choice

A uniform light slender beam AB of section modulus $EI$ is pinned by a frictionless joint A to the ground and supported by a light inextensible cable CB to hang a weight $W$ as shown. If the maximum value of $W$ to avoid buckling of the beam AB is obtained as $\beta\pi^2EI$, where $\pi$ is the ratio of circumference to diameter of a circle, then the value of $\beta$ is

a horizontal beam AB of length 2.5 m, pinned at the wall at A. A cable CB rises from B to a point C on the wall at 30° to the beam. The weight W hangs from B.
  1. 0.0924 m$^{-2}$
  2. 0.0713 m$^{-2}$
  3. 0.1261 m$^{-2}$
  4. 0.1417 m$^{-2}$

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Correct answer: (A) 0.0924 m$^{-2}$

Explanation

The vertical equilibrium of joint B gives $T\sin 30° = W$, so the cable tension is $T = 2W$. The horizontal component $T\cos 30° = 1.732W$ compresses the beam. The beam is pinned at both ends (the cable end acts as a pin), so the Euler load is $\frac{\pi^2EI}{L^2}$ with $L = 2.5$ m. Then $1.732W = \frac{\pi^2EI}{6.25}$ and $W = \frac{\pi^2EI}{10.825} = 0.0924\pi^2EI$.