GATE 2024 ME – Question 53

Mechanics of Materials · Shear force and bending moment diagrams · 2 marks · Numerical answer

A horizontal beam of length 1200 mm is pinned at the left end and is resting on a roller at the other end as shown in the figure. A linearly varying distributed load is applied on the beam. The magnitude of maximum bending moment acting on the beam is _______ N.m. (*round off to 1 decimal place*)

A simply supported beam 1200 mm long with a pin at the left end and a roller at the right end. A downward load rises linearly from zero at the left end to 100 N/m at the right end.

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Correct answer: 4.58 to 4.62

Explanation

The total load is $W = \frac{1}{2} \times 100 \times 1.2 = 60$ N, acting at $\frac{2}{3}L$ from the left. The reactions are $R_A = \frac{W}{3} = 20$ N and $R_B = \frac{2W}{3} = 40$ N. The load intensity at distance $x$ from the left is $\frac{100x}{1.2}$, so the shear force is $V = 20 - \frac{100x^2}{2 \times 1.2}$, which is zero at $x = \frac{L}{\sqrt{3}} = 0.693$ m. The moment there is $M = 20x - \frac{100x^3}{6 \times 1.2} = \frac{WL}{9\sqrt{3}} = \frac{60 \times 1.2}{15.59} = 4.6$ N·m.