To be honest, I don't think you would ever use a tube where the wall thickness was thin enough where a compression failure would occur.... There are formulae that allow you to calculate the failure point of a "pillar" (an object in compression), which usually occurs in one of two modes.... buckling or material yield.... Without referring to Google, IIRS the buckling load is primarily a function of geometry (a factor called the moment of inertia, which governs the stiffness in bend) and the Young's Modulas (stiffness) of the material.... The yield point in lbs. of compression is the cross-sectional area of the part in sq.in. times the Yield strength in psi....

My barrel tensioning systems are adjustable by compressing Belleville disc springs (there to minimize temperature change effects) up to about 1300 lbs.... It seems to take about 800 lbs. of tension in the barrel (compression in the shroud) to get the group size to settle down to the point where additional tension doesn't shrink it a lot.... Tom at AAA found about the same thing.... I think engineering the system for 1000 lbs. of load, with a 2:1 safety factor in yield would be plenty.... If you were using 6061-T6 aluminum, for example, with a 1" OD and a 0.035" wall....

Wall area = (1.000^2 - 0.965^2) x PI / 4 = 0.054 sq.in.... Yield point of 6061-T6 = 40,000 psi.... 0.054 x 40,000 = 2160 lbs....

If the tube is larger in OD, or has a greater wall thickness, or a stronger material.... the force before the end of the tube started to yield would increase.... I can't be bothered looking up and calculating the "pillar" strength, but the last time I did it, it was much greater than the yield strength....

There is another consideration for the shroud material in a tensioning system.... The thermal expansion of using a material different than the barrel should be considered.... You can look up the thermal expansion coefficients (PPM/deg.) and calculate the length change for the barrel and shroud for, say, a temperature change of 50*F (from 50* to 100*).... or some other value.... The calculation takes the form of the constant times the temperature change times the length.... It is not uncommon to get a 0.010" difference in the expansion of the barrel and shroud.... Without Bellevilles (or some other springy system), even a small change in relative length can drastically change the tension force in the barrel.... If the shroud expands less, the tension reduces.... if it expands more than the barrel, it increases.... If, for example, at 50*F you had 1000 lbs. of tension in the barrel, which stretched it 0.010".... and the shroud expanded 0.010" more than the barrel.... you would no longer have any tension.... If the Bellevilles were compressed 0.050" to achieve that 1000 lbs. tension.... then with the same 0.010" of expansion of the shroud (relative to the barrel), you would still have 800 lbs. of tension remaining.... The more Bellevilles you run (in series) the longer they need to be compressed to reach a given tension.... so the less any thermal expansion difference would cause in the force.... I use 5 Bellevilles as a good compromise.... but if you use identical materials for the barrel and shroud you could use 3 (or even 1).... If on the other hand, the thermal expansion coefficients were vastly different, you might be better off with 7 or 9 Bellevilles, to minimize the tension change....

Here are some approximate thermal expansion coefficients (PPM/deg.C).... Multiply by the length and temperature change to get the length change....

Aluminum ~ 23

Stainless Steel ~ 17

CrMoly ~ 12

Steel ~ 11

Titanium ~ 9

Carbon Fiber ~ 0

Note that metals can be engineered for lower (or higher) thermal expansion, as can CF composites.... Let's say you had a CrMoly barrel and a shroud, both 24" long.... For a 30 deg.C change in temperature (54*F)....

CrMoly (barrel).... 12/1000000 x 24 x 30 = 0.0086"

Titanium.... 9/1000000 x 24 x 30 = 0.0065"

Aluminum.... 23/1000000 x 24 x 30 = 0.0166"

Carbon Fibre.... 0/1000000 x 24 x 30 = 0.0000"

Titanium would be a good choice, because over a 30*C temperature change, the barrel stretch would only decrease by 0.002".... whereas with Aluminum it would increase by 0.008".... and with CF, the barrel tensioning would drop by about 0.009".... You can see why I like to use Bellevilles to reduce the force change over those length changes....

The other thing to consider is properly supporting the shroud at the receiver, so that the OD of the shroud helps stiffen the barrel at that point.... Ignoring that will give you a stiff "club" of a barrel, hinged at the front of the receiver.... and still prone to the fundamental mode of vibration (muzzle waving back and forth).... even though the tension damped out the harmonics (by raising their frequency above the critical range).... If possible, a close fit of the shroud over a stub on the receiver of at least 2 diameters is desired.... although over 1 diameter will probably work.... I would not use less support than that.... A properly supported shroud, with the barrel in tension inside it.... effectively increases the barrel diameter, and hence its stiffness....

Bob