Hypothetical Hollow Steel Sphere: collapse from outside pressure

[thedan16]

Hello physics forums,

Say you had a hollow steel sphere of thickness 1 mm and diameter of 1 meter (from outside to outside)?

Inside the sphere is gas at 1 atm pressure. Outside is 1 atm of pressure. How much gas would I have to remove from the inside until the sphere collapsed from outside atmospheric pressure?

I'd have to use bulk modulus correct?

 

[AlephZero]

This http://www.engr.uconn.edu/~cassenti/AnsysTutorial/Modules_APDL/Module%205%20Buckling%20Sphere.pdf

gives a formula $$P = \frac{2Et^2}{r^2\sqrt{3(1-\mu^2)}}$$

But since it doesn't give any information about to how it was derived and what assumptions were made, I have no reason to believe it's correct. This is one of those problems which is very easy to describe, but very hard to solve.

The only thing one can say confidently is that it will collapse by buckling, not by compressive failure of the material.

 

[thedan16]

Yes, I'm curious to how this formula was derived! What about a cylinder or a cube?

 

[256bits]

That is the the Zoelly-Van Der Neut formula for buckling of spherical shells and goes as far back as 1915. It is the theoretical limit to the buckling failure of a sphere, whereas from experimental data most actual spheres will fail with loads 1/3 to 1/4 of that value, due to say manufacturing and assembly defects giving a sphere different from that of a perfectly smooth one.

If your library has the book by Timoshenko, Theory of Elastic Stability, you will find a derivation using linear methods.
Nonlinear methods of solution for this problem are difficult to solve.

Research pays off and here are some pdf's of interest:
http://traktoria.org/files/pressure_hull/spherical/buckling_of_spherical_shells.pdf
http://www.dtic.mil/dtic/tr/fulltext/u2/610809.pdf

The first link gives a different formula that is said to agree more with experimental results,
Pcr = 0.37E/ m^2

where m is the radius/thickness ratio.

The first formula in the same format becomes Pcr = 1.21 E / m^2, using a Poission ration of 0.3.

I suppose for your sphere, you will have to make some design choices on the differences fom an ideal shere