GATE 2023 AE – Question 53
Consider a thin-walled cylindrical pressure vessel made of an alloy with yield strength of 300 MPa. The vessel has end caps to contain the pressure. The ratio of radius of the vessel to its wall thickness is 100. As per the von Mises yield criterion, the internal pressure that would cause the failure of the vessel is _____ MPa. (round off to two decimal places)
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Correct answer: 3.44 to 3.48
Explanation
For a thin-walled cylindrical pressure vessel with closed ends, internal pressure $p$, radius $r$ and wall thickness $t$:
**Stresses.**
- Hoop stress: $\sigma_\theta=\dfrac{pr}{t}=100\,p$ (since $r/t=100$).
- Longitudinal stress: $\sigma_L=\dfrac{pr}{2t}=50\,p$.
- Radial stress: negligible (about 0).
**Von Mises criterion** for a biaxial state:
$$\sigma_{eq}=\sqrt{\sigma_\theta^2-\sigma_\theta\sigma_L+\sigma_L^2}=p\sqrt{10000-5000+2500}=p\sqrt{7500}=86.6\,p .$$
**Yield:** $\sigma_{eq}=\sigma_y=300$ MPa:
$$p=\frac{300}{86.6}=\mathbf{3.46\ MPa}.$$