GATE 2021 AE – Question 32

Structures · Strength of materials: Strain energy, Castigliano's principles, three-dimensional Hooke's law, plane stress and strain, Euler buckling · 1 mark · Numerical answer

In a thin-walled closed cylindrical pressure vessel, hoop strain is three times axial strain. Poisson’s ratio is

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Correct answer: 0.19 to 0.21

Explanation

For a thin-walled closed cylinder with internal pressure $p$, radius $r$ and thickness $t$, the stresses are
$$\sigma_h=\frac{pr}{t},\qquad\sigma_a=\frac{pr}{2t}\;\Rightarrow\;\sigma_h=2\sigma_a .$$
The radial stress is negligible.

**Strains** from Hooke's law (plane stress):
$$\epsilon_h=\frac{\sigma_h-\nu\sigma_a}{E}=\frac{\sigma_a(2-\nu)}{E},\qquad\epsilon_a=\frac{\sigma_a-\nu\sigma_h}{E}=\frac{\sigma_a(1-2\nu)}{E}.$$

**Given** $\epsilon_h=3\epsilon_a$:
$$2-\nu=3(1-2\nu)=3-6\nu\;\Rightarrow\;5\nu=1\;\Rightarrow\;\nu=\mathbf{0.2}.$$