Jerry,
The hoop stress in a thin walled pressure vessel (i.e. the ratio of the inside diameter to the wall thickness is at least 10) is as follows"
S (stress) = P (pressure) x r (inside radius) / t (wall thickness)
In your example, if you wanted to calculate a safe wall thickness, you would rearrange the equation to:
t (min wall thickness) = P (pressure) x r (inside radius) / S (allowable stress)
For things in my work (that don't have to fly), I usually use the yield strength of the material divided by two (i.e. safety factor of 2) as the allowable working stress. Using 40,000 psi for the annealed yield strength of 1117, I would use an allowable stress of 20,000 psi. Assuming a shotgun max pressure of 12,000 psi, and using the ID radius of .540 / 2 = .270, the min wall thickness I would use would be:
(12,000 x .270) / 20,000 = 0.162 inches. This would be at the breech only as the pressure drops off rapidly in the barrel. If the breech thickness were doubled to .324, the barrel would have a safety factor of 4 instead of 2. At a breech wall thickness of 0.243, the safety factor would be 3, and so on. Now all of this assumes homogeneous properties of the steel and no flaws in the steel at the breech area (a safe bet with modern steels and never a safe bet with forge welded tubes - hence the practice of proof testing.)
To answer your question more directly, if the wall thickness were 0.162 and the ultimate strength of the material was 62,000 psi (here we would use the ultimate strength of the material with no safety factor) and the bore was .540, the wall should fail at:
t x Ultimate tensile / ID radius
[(0.162) x 62,000] / 0.270 = 37,200 psi
If the wall was 0.200 inches, it would fail at 45,926 psi ..........and so on.
Also, as the material starts to fail, it obviously yields first. As it yields, it work hardens and gets stronger (same effect you get by drilling too slowly through a material that work hardens badly). So the material strength would probably be higher than the 62,000 psi and the actual failure would occur at an even higher pressure.
Just to confuse things a little, all of the above is for "thin" walled pressure vessels. For a gun barrel, especially in the breech area, the ratio of the ID to the wall is more like 3, which puts it into the category of a thick walled pressure vessel. The calculations for that are more complicated, but I have found that the thin wall analysis is adequate for these purposes.
Hope this helps.
Dave C