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Referring to von Guiricke's early experiment to hold vacuum using Magdeburg hemispheres, what is the best known shape to hold say 10 atmosphere vacuum (external pressure) using any material or material combination without buckling its walls? I.e., using the least amount of material so that the vacuumed vessel is as lightweight as possible.

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Material choice relaxed in question as better materials are now available compared to Magdeburg test time.

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  • $\begingroup$ Pressure vessels have no instability mode of collapse like vacuum vessels. $\endgroup$ – Narasimham Jul 20 '15 at 18:14
  • $\begingroup$ Material choice usually depends heavily on size of finished vessel. You shouldn't make a vessel less than 5mm thick for anything larger than a few cm across to deal with things like handling and shipping - do you have a size that you are looking for specifically? $\endgroup$ – Mark Jul 20 '15 at 18:20
  • $\begingroup$ Say 2 meter dia /length for example, At least theoretically there is Euler Buckling formula for long slender columns, but none for negative pressure so far? $\endgroup$ – Narasimham Jul 20 '15 at 18:27
  • $\begingroup$ Shells are notoriously complicated when it comes to buckling - they usually will fail as low as 30% below what theory predicts they will fail. This is largely attributed to imperfections and a slew of research is being done to explain it. For engineering purposes, NASA SP-8007 covers most scenarios with a nice empirical "knockdown" factor - a fudge factor, to make safe design decisions. It's not neat, but it does the job, and is still to this day considered state-of-the-art. $\endgroup$ – Mark Jul 20 '15 at 18:43
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When I design pressure vessels to hold vacuum, here are the preferences I usually give to customers:

  1. Spherical is always the best in terms of minimizing the thickness, but not the most economical in therms of amount it can hold or for being friendly to the manufacturer
  2. Ellipsoidal (2:1) heads on a shell whose length is equal to the diameter is second best. It comes out a bit oblate compared to the sphere (1.5:1), but per given packing area holds plenty of volume. The ellipsoidal heads work out to equal the thickness requirements of the shell, but is not structurally as efficient as the sphere.
  3. Avoid Flat areas - especially flat bottoms. If you need it to be flat on the bottom, weld a skirt to the bottom.
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