added this to the barrel dimension thread but thought others might be interested. We did this analysis on a barrel profile we are thinking of selling.

Very interesting. Strain gauge results? Boss caliber. Y-axis: yield strength over hoop stress ratio so from chamber 1.5 to transition 1.1 to 7 at muzzle. I did thick-walled pressure vessel calc's for 1 of my rifles with certain barrel contour. Chamber hoop stress at inner wall surface about 1.4x max chamber pressure sounds about right. Assumed AISI 4140 Rockwell C Hardness 35 barrel yield strength about 150 ksi.

150 ksi / (1.4 x 65,000 psi) = 1.6

Very interesting.
Is this graph the result of a calculation, or of measurements?

If calculated, how is the pressure profile determined?

Calculation. Krieger gave us their 4140 yield strength, used Quickload for the pressure vs barrel distance, thick wall equation for hoop stress. Wanted to show the critical area is the breech to linear taper transition and not the muzzle (maybe something else going on there). So I think when barrel makers have minimum muzzle wall thickness, it is that implication at the other end is the concern?
Below is stress simulation for 458 caliber at the breech end. We thought the quarter rib would help reduce the stress, did not know over what circumfrernce.

Interesting predicts hotter loading barrel damage yielding initiates at transition low 1.1 safety factor. Not to mention bolt lugs yielding possible risk from increased chamber pressure bolt thrust. Assuming bolt action barrel transition probably weak link if load pressure unwisely increased past specified maximum. Or brass strained to rupture which can also cause a wreck.

Interesting, but in all the actual testing, and local guys loading pistol power in their 243s, I have seen, the barrel never fails, nor does the receiver or bolt.
The brass cartridge case serves as a natural pressure relief valve, popping at well below the failure level of barrels, receivers, or bolts. Even those not made from 4140. .
That is how we get close to 100K psi on our M256 Tank Cannons; steel case with steel electric, screwed in primers.
I think our rifle systems are very strong, is the take away, for me at least.

Should be yield strength exceeded -> tensile/shear crack formation -> crack growth under load cycling -> catastrophic failure risk in time.

Thanks for the comments. What started us down this path is our integral barrel profile has a short transition contour (to help compensate with the qrib weight) compared to Douglas and others. The barrels will not “fail” per se but the maximum stress is along the circumference of the bore, read stretches, and when it exceeds the yield strength, does not return to normal. Cracking can occur and we did not want this to happen to expensive integral barrels

Very interesting.
Where does the Knox form end in relation to the case mouth? I had always felt that if the distance from the case mouth to the end of the Knox form is more than 1.5 x the wall thickness between the chamber and the Knox form OD, one should be OK, do you think it should be longer based on this?

Peter, we use Quickload as the calculator, the peak pressure from the bolt face:
505 Gibbs - 4.6"
458 Lott - 3.4" (non-bottleneck)?
416 Rem - 3.8"
375 H&H - 3.9"
338 WinMag- 4.2"
same powder, max. pressure, woodleigh softs.
I guess one could "rule of thumb" and say 4"
and use peak pressure and diameter of barrel at 4".
For us, we wanted a customer to be able to choose the next smaller muzzle diameter and still have some stress margin. For example, in the analysis above, the barrel was a #5 for the 458.

Barrel Number caliber (inch) (inch)
# 1 0.284 1.20 0.760 0.560
# 2 0.308 1.20 0.800 0.600
# 3 0.375 1.20 0.825 0.625
# 4 0.375 1.20 0.850 0.650
# 5 0.416 1.20 0.900 0.700
# 6 0.458 1.20 0.950 0.750
# 7 0.505 1.20 1.075 0.875

FYI, the distance from the bolt face to the end of the transition "the Knox" as you call it, for their sporter barrels:
PACNOR - 6"
Dakota - 3.6"
Mcgown - 5.25"
Douglas -4"
Krieger - 6"
Ours kinda follow the Dakota contour and is 3.25" hence our interest in the stresses

Enlightenment appreciated. Strain gauge readings to compare with calculations would be really cool.

Thanks Steve

However, I am getting mired in terminology here.
When you say "the end of the transition", do you mean the point at which the maximum-diameter cylindrical section over the chamber ends, or the point at which the straight taper heading toward the muzzle starts?

The point I am referring to is the end of the cylindrical section, in other words the point at which the barrel starts to become thinner.

Also while I realize that this is not the norm, I do feel that if one wants to minimize the profile for any given rifle, then the profile needs to be calculated with respect to where the chamber ends, or where the shoulder of the case is (depending on how much of a shoulder the case has). If the barrels are CNC milled (I'm guessing that's how you do it as I can't really see a good way of forming in integral rib using purely mechanical machines), then it should be very easy to accomplish?

Lastly, I think the actual muzzle diameter needs to have some allowance for mild bore obstructions (such as rain water or excessive oil). This would be difficult to calculate and I'm guessing beyond the abilities of Quickload or any such software, but would probably make little difference at the chamber end and be more and more relevant the closer to the muzzle you get?

Just for clarity, I am not a trained or qualified gunsmith or gunmaker. I design specialized industrial valves, and thus I have a lot of experience in designing pressure vessels. This is a much more regulated field than firearms design, and the 1.5x wall thickness is derived from the rules for pressure vessel design.

The point where the iinear taper starts. The peak pressure begins to decay at app. 4” faster than the barrel thinning from the linear taper, hence the start of the taper or the end of the radiused section is the max stress. Our section is 1” long, about 6 degrees whereas the other barrels almost 3” long and about 30 degrees. So looks like to us they wanted a bit more wall thickness past the 4”.
The 458 Lott pressure was set to max. Saami. For the #6 barrel, the stress margin is 1.25.
Saami proof loads for brass are 1.4x max.
CNC milling. Coincidentally, our first contour and quarter rib style we are calling the “African”.

A comment from an old stress analyst with significant thick-wall pressure vessel experience, including computer-based nonlinear finite element failure computations, verified by strain gauge instrumented burst testing.
Local ductile yielding in thick-wall open-ended cylinders, like rifle barrels, first occurs at the inside surface and is governed by the sum of hoop stress plus internal pressure, not just hoop stress, compared to yield strength.
On that basis, the results shown seem to indicate that local yielding would occur on the inside surface and extend at least a short distance through the barrel wall. Residual stresses would be left in the barrel, along with an extremely small permanent increase of exterior barrel radius.
Assuming sufficient ductility, probably only if the internal pressure were substantially higher, would ductile yielding progress all the way through the barrel wall and easily observable effects occur.
As already pointed out, crack propagation after repeated loadings followed by subsequent failure is another possibility to address.

Hoop stress expressed as -

Yep, thick wal and pressure is explicit in the result.Nice simple equation. The simulation was to see the effect of the loner outside radius due to 1/4 rib and the extent around the inside radius is stress was reduced. Every little bit helps!