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I understand how a circle (or sphere if you want 3D) is the best shape for holding a vacuum inside a container, but what if you wanted to have a large positive pressure inside of the container instead of outside? Would a circle/sphere still be the best shape as far as material required to hold X amount of pressure?

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Despite being difficult to manufacture, a sphere is the best shape for a pressure vessel, but due to the manufacturing difficulties are more costly to make.

From the American Society of Mechanical Engineers website:

Theoretically, a spherical pressure vessel has approximately twice the strength of a cylindrical pressure vessel with the same wall thickness

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Other sources agree with this, adding:

Spherical pressure vessels is preferred for storage of high pressure fluids. A sphere is a very strong structure. The even distribution of stresses on the sphere's surfaces, both internally and externally, generally means that there are no weak points.

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Due to the fact that the pressure differential is the same everywhere around the surface of a sphere, it must be the strongest, therefore hold the greatest pressure, and withhold the greatest pressure, too. A circle won't hold pressure, as it is only 2 dimensional, and everything would escape!

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It depends what you mean by 'best'

For a homogeneous material like steel a sphere will (in theory) be the lightest structure which will contain a given volume at a given pressure. However spheres do have certain disadvantages.

-They don't stack well, so if you are distributing industrial gases, for example, they are difficult to store and transport and take up a lot of space compared to their useful volume. - They are difficult to manufacture, especially in composites. - Spheres are often not a convenient shape to handle, especially in applications like breathing apparatus and fire extinguishers where this is very important.

In practice the most useful shape for a pressure vessel is often a cylinder with hemispherical or ellipsoidal ends, often with one end concave to allow it to stand on its end. This is not quite as efficient in term of the minimum thickness of material required for the walls but this is often outweighed by the practical considerations already noted.

Most general engineering textbooks will have a section on algebraic formulae for calculating the stresses in various different types of pressure vessel.

It's also worth noting that composite pressure vessels are increasingly common. These are often made by winding a resin impregnated yarn around a former and as well as cylinders toroidal vessels can be made in this way, which come close to being as weight efficient as spheres but are generally more convenient. With composite materials it is also possible to orientate the yarn so as to have different strength in different directions, enhancing material efficiency, which is easier to do with a cylinder or torus than a sphere.

Scale matters as well so for very large scale bulk storage (eg LPG container ships) a lot of the problems with constructing spheres may become less relevant.

In terms of internal vs external pressure the most important difference is that external pressure will impose compressive forces on the vessel so some geometries may be subject to buckling in addition to simple stresses but for common types of pressure vessel this won't make a huge difference in general design.

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It's all about the fact that spherical shell would not experience bending if the load is evenly distributed over the surface (which is the case when pressure is the only load). The very idea of using shells is to create them of such shape that stresses due to bending would be low, and the load bearing capacity of the shell would be provided by the tensile forces (or by compression) arising in the shell (tensile/compression forces lead to much lower stresses than bending moments do). An interesting example devoted to load bearing capacity of shells is presented on the following webpage (it is devoted to so-called "shallow" shells, which have low curvature, i.e. they closely resemble plates, but can withstand much larger loads, exactly because compression in the material plays much bigger role than bending in this case):

http://members.ozemail.com.au/~comecau/quad_shell_shallow_shell.htm

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