Need help engineering a cannon barrel

I am building a scaled down replica of a civil war cannon and need help with the design, bore size, wall thickness, safety factor, etc. I am melting copper, tin, and zinc to make gunmetal (88% copper, 9% tin, 3% zinc) and casting the barrel. Doing it the old fashioned way! My size is limited by the capacity of my crucible which dictates about a 3.75" breech diameter tapering down to the muzzle. Minimum dimension is around 2.75" diameter, barrel length about 22". I was planning on drilling a 1" bore, which would leave me 1.375 wall thickness at the breech, and .875 minimum wall thickness. I've done a test pour and tensile strength tested the resulting metal which broke at 20,500 psi. I would use only black powder which has a max pressure of 27k psi.

I'm not an engineer, so I have very little knowledge on how to calculate the strength of this barrel with the above parameters. I found a hoop stress calculator which is typically used for static pressure in cylinders (https://www.easycalculation.com/physics/classical-physics/stress-circumferential-direction.php) and according to it the stress goes down below 20k psi just 3mm into the wall from the internal bore. Stress at the outside of the barrel is only 3800psi, well below my 20,500 breaking point. While this seems reasonable to a layman, I'd like to get input from somebody who really knows how to figure this out.

Let me also mention that I know this is not a "safe" cannon. 10% of historic cannons blew up injuring their operators. I never plan to fire this cannon without strict safety measures. Meaning, light the fuse and get behind a significant barricade that will protect me in case of failure. I may also use a remote triggering device. I never plan to fire this cannon with people standing in the open where someone could get hit with shrapnel in case of failure.

What Jim Said. A 1/4" liner of chrome moly steel with a properly press fitted and welded plug should be tremendously strong. I had a 3/8ths" liner turned for a 3" bored gun and a master metallurgist told me it was good for a cannon -by itself! Strength he estimated at over 90,000lbs. This is the way to go! Especially since you may (eventually) have family or the public in attendance. I once read where the U.S. Army conducted tests with black powder in the 1800's and were able to produce pressures approaching that figure. Normally, you would not begin to get anywhere near that pressure, but it's better to have lots of freeboard when it comes to artillery. Bronze is typically more forgiving, but can still rip open, and you don't want to be next to the rip! The liner will ease your mind, and not detract from your piece's beauty. Pour your barrel solid, then drill it. That way all the voids will be in the core. George.

What Jim and Treestalker said ! Make your liner the cannon. Tie it down to a wood pallet, sand bag it, and use a remote firing device.
This is to proof it. Where did you get the 10% figure ?

Thanks for your replies. Unfortunately, a liner defeats the purpose of my project. I know I can easily line it with 4140 and make it safe, but that's not building a historic bronze cannon. That's a modern steel cannon. Safer? Yes. I'm aware of the safety issues with the cannon, and as I said I will take precautions to mitigate any danger. 10% cannon failure came from a civil war guru that I know.

Jim said use yield strength and a safety factor of 4 times the wall thickness of what it can hold. I agree with using yield str, but I don't have a way to accurately measure that, only tensile. For bronze it looks like yield strength is typically about half of the tensile strength. So let's assume 10,000PSI yield strength and a 1" bore. What is the minimum wall thickness to hold 27k PSI chamber pressure? How do you calculate that? Then I can step up wall thickness by a factor of 4 to get a reasonable chance of success. Again, I will never shoot this cannon with people in line of sight from it. And once I'm done with it, I'll likely melt it down and make something else. I'm not going to leave it laying around for someone else to shoot not knowing the danger of it.

I guess I don't understand why you want to build it in the first place. If you are building and art piece, build it unfireable.. If you want to fire it , build it like someone who knows nothing may get their hands on it, because it may wind up being stolen, or left to your posterity after you pass away unexpectedly. Just sayin', Geo.

Safety first, all else second.

Talk to north South Skirmish Association and to people who make and sell bronze cannon about how they proof them. Proof your gun as if you intend to live fire it. Have you cast bronze before ?

Can anybody answer my question? Do you use a hoop stress calculation?

Hoop stress is part of it but perhaps a better idea is to use Lame's Equation for THICK wall tubing. (The thick wall tube formulas are for ductile materials such as brass.)

The formula given in the "Machinery's Handbook" for a closed ended cylinder for calculating wall thickness is

The square root of the [(allowed stress + internal pressure) divided by the (allowed stress - the internal pressure)] Subtract 1 from the answer. This new answer is to be multiplied by the radius of the bore to calculate the needed wall thickness.

Using your 20500 psi tensile and a 27000 psi pressure the answer of the first part of the calculation will be a negative and one cannot take the square root of a negative number (without using "i" which is another subject).

In other words, unless the tensile strength of the material is higher than the internal pressure, the formula doesn't work.

What Zonie said! I'm still trying to decipher what the scarecrow said to OZ about the triangle! ( the sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side) I know, the Pythagorean Theorem says the square of the hypotenuse is equal to the sum of the squares of the remaining (right angle) sides of a right triangle, the scarecrow got it wrong. You can trust Zonie though, he's dangerous, just hard for us backwoods boys to understand! Geo.

You really have not mentioned how you plan on casting your cannon. Casting parts as critical as a barrel are quite different than casting non critical parts like buttplates and trigger guards. If you lack the knowledge in this area you will likely end up with a casting full of defects like inclusions and porosity (think of the latter like a piece of foam). No matter how good the design advice you receive here, if your casting is not good you will be making a bomb.

The best advice so far was to sleeve your casting. It was good advice and will make your end project much safer if properly done.

A lot of good safe info here.
The 4140 liner would be my choice.