GATE 2023 ME – Question 58

Mechanics of Materials · Bending and shear stresses, shear centre · 2 marks · Numerical answer

The area moment of inertia about the y-axis of a linearly tapered section shown in the figure is _____________ m$^4$.
(Answer in integer)

a linearly tapered plate of length 12 m along the x axis, starting at the y axis (x = 0) where its total height is 3 m (1.5 m on each side of the x axis) and growing to 6 m at x = 12 m (3 m on each side).

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Correct answer: 3024

Explanation

The height of the section at the distance $x$ from the y-axis varies linearly from 3 m to 6 m: $h(x) = 3 + \frac{3}{12}x = 3 + 0.25x$. The moment of inertia about the y-axis is $I_y = \int x^2\,dA = \int_0^{12}x^2(3 + 0.25x)\,dx = 3 \cdot \frac{12^3}{3} + 0.25 \cdot \frac{12^4}{4} = 1728 + 1296 = 3024$ m$^4$.