Discussion
The internal pressure \(p_{Y}\) at yield, based on the von Mises yield criterion, if the vessel is floating in space, is approximately
Answer: D
The internal pressure at yield is approximately 20 MPa.
In a thin spherical shell the membrane stresses in every tangential direction are equal, and the radial stress is negligible:
\[ \begin{aligned} \sigma_{1} &= \sigma_{2} = \frac{pr}{2t} \\ \sigma_{3} &\approx 0 \end{aligned} \]
For this biaxial state the von Mises stress is \(\sqrt{\sigma_{1}^{2} - \sigma_{1}\sigma_{2} + \sigma_{2}^{2}} = \sigma_{1}\), so yield occurs when \(\sigma_{1} = \sigma_{Y}\):
\[ \begin{aligned} p_{Y} &= \frac{2t\sigma_{Y}}{r} = \frac{2 \times 0.010 \times 500}{0.5} \\ &= 20\ \text{MPa} \end{aligned} \]
The larger options equate the pressure itself with or close to the material's yield stress; the thin wall (r/t = 50) means the stress in the wall is 25 times the pressure, so the vessel yields at a much lower pressure.
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