GATE 2023 ME – Question 59
A cylindrical bar has a length $L = 5$ m and cross section area $S = 10$ m$^2$. The bar is made of a linear elastic material with a density $\rho = 2700$ kg/m$^3$ and Young's modulus $E = 70$ GPa. The bar is suspended as shown in the figure and is in a state of uniaxial tension due to its self-weight.
The elastic strain energy stored in the bar equals _________ J. (Rounded off to two decimal places)
Take the acceleration due to gravity as $g = 9.8$ m/s$^2$.
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Show answer and explanation
Correct answer: 2.07 to 2.09
Explanation
At the distance $x$ from the top, the bar carries the weight of the part below it, so the stress is $\sigma(x) = \rho g(L - x)$. The strain energy is $U = \int_0^L\frac{\sigma^2}{2E}S\,dx = \frac{S\rho^2g^2}{2E}\int_0^L(L - x)^2dx = \frac{S\rho^2g^2L^3}{6E}$. With the numbers: $U = \frac{10 \times 2700^2 \times 9.8^2 \times 5^3}{6 \times 70 \times 10^9} = \frac{8.7516 \times 10^{11}}{4.2 \times 10^{11}} = 2.08$ J.