Minimum barrel strength

What does a five inch long barrel, with 5mm wall thickness, need at minimum, when handling these calibers;

Black powder
1. .36, 12grain FFFG BP
2. .44, 20grain FFFG BP
3. .50, 25grain FFFG BP

Regarding;

Yield point ? N/mm2,

Tensile strength ? N/mm2,

Extension ? %

Black powder
1. .36, 12grain FFFG BP
2. .44, 20grain FFFG BP
3. .50, 25grain FFFG BP

That's really not enough info to calculate from; shot weight would also be necessary. In addition, I don't know of any reliable equations to calculate pressure from the data you supply. Small arms internal ballistics are not usually done by calculation but by experiment.

Using the usual hoop stress formulas and an assumed pressure of 20Ksi, you would get a stress of about 32Ksi in .36 cal, 35Ksi in .44 cal, and 38Ksi in .50 cal. Almost any steel would give you that.

But that is all paper doodlings. You really need to make a pressure testing device and determine the pressure first, then do some calculating that includes a safety factor, then design something and test it before you have anything approximating truth. And if it is a muzzle loader which would give the shooter the ability to make heavier loads than you design for, you probably need to use a larger than normal safety factor.

The British were able to exceed chamber pressur s of 100,000 psi in closed bomb tests using black powder. So, your pressure vessel (barrel) needs to be able to handle the normal expected pressure, and have a safety factor.

I would estimate at your quoted loads with a tight pure lead roundball,none of the requested pressures would exceed 5,000 psi.....All your loads would be light C&B revolver loads.......for instance in 44 cal/428 dia..........44-40 WCF,40 grains of 3f and a 200 gn bullet do not exceed 12,000 psi.chamber pressure.......which quickly falls to half once the bullet has gone 1" down the barrel................if you want to investigate "thinness"...look at the wall thickness of revolver chambers......very thin ,even in modern magnums,to keep weight down...........EDIT....all the info you need is in the Lyman Blackpowder Handbook.,with test pressures of hundreds of loads.

i might add,that you are asking for specific engineering advice,which could place a burden of liability on the giver......and furthermore be in breach of the many anti tr/sm laws that spring up like mushrooms worldwide................point in fact.....the cops here (Ipswich,Qld) have just busted a "tinkerer"....for making a poor(but apparently working) copy of an Ingram.........the guy has booked himself a stretch,no matter what....and the cops are claiming he used plans from the net..