Say I have a spherical container with a certain gas inside, heated to a very high temperature, and thus a very high pressure. The gas will exert a force radially outwards onto the container, and at some critical point the container will no longer be able to contain the gas and deform or break.
At first, to solve for the maximum pressure a given container could hold, I thought that maybe I could use Young's Modulus, which related stress on an object to the strain felt by an object. The change in 'length' of the sphere I would get by considering changing the radius of the sphere a small amount $dr$ and seeing what happened to a small element (that was my plan anyway). Then I realized that since the force applied onto the object wasn't really in the direction of stretching, but rather perpendicular to it (as the metal should stretch in the direction of the blue arrows(I think)) I thought maybe a better substitute would be Shear Modulus. But that didn't really make sense in my head.
Finally I seemed to have a breakthrough, remember that Ultimate tensile strength exists! Ultimate tensile strength is given in units of pressure, so I thought that perhaps the answer to my question was simply the number listed on the Wikipedia article. Not only did that feel supremely unsatisfactory, it also seems wrong, because Ultimate tensile strength seems to imply that the force of stretching be in the same direction of stretching. No more good ideas have come to me in a while, so I thought I should ask, How do I calculate the maximum pressure a given container can contain?
Edit: I realize ultimate tensile strength can't be the answer (or the full answer anyway) because the thickness of the container must come into the equation at some point! (I think). So far none of my methods seem to have included this crucial aspect...