Lets take a look at this from a simplified design point of view. The barrel has two forces to resist: the radial gas pressure and the forward axial force caused by wad friction with the barrel walls. The most useful formula for resisting radial stress is the hoop stress formula - stress equals pressure times inside radius divided by wall thickness (there are good calculators available via google). One such calculator says for a chamber pressure of 12,500 psi, an inside diameter of 0.729", and wall thickness of 0.150" the stress would be 30,375 psi. I feel sure that tube makers would use steel with working strength well above that number. I have a sleever Woodward boxlock with 0.160" walls at the face and 0.115" at the seam 3.25" from the face. Safe? It carries 3T / sq inch proofs. Note that the original barrel stubs will, indeed, carry load. The sleeve steel will quickly expand to fill any void due to less than contact fit (think press fit). Then, the original stub will start to pick up load.
The axial (forward-backward) force will be caused by the wad to barrel wall friction. Since the front of the barrel is open, there will be no chamber-barrel pressure force in the forward direction. The wad drag is in the backward direction, so the barrel to stub joint must resist it. This resistance must come from the solder or weld plus (and this one is the big hitter) the steel to steel friction of the joint. If we have 10,000 psi of chamber pressure and the steel to steel friction coefficient is 0.2, we will have available 2,000 pounds of force per square inch of joint surface. Note also that this force will build up as the pressure builds in the chamber. That means that the resisting force will be available before the wad hits the barrel. I think the weld/solder is going to carry very little firing force.
DDA
