To check COSMOS

NASA SP-8007 BUCKLING of THIN-WALLED CIRCULAR CYLINDER,
Sept. 1965, revised Aug. 1968, see sp8007.pdf

Use NASA external pressure analysis, section 4.2.3, starting page 8

Some definitions from NASA and parameter values for proposed STAR beam pipe

cylinder radius

cylinder length

skin thickness

Poisson's ratio

Young's modules

correlation factor to account for difference between classical theory and predicted instability loads

curvature parameter, isotropic cylinder

for now let

number of buckle waves in circumferential direction

number of buckle half waves in axial direction

This is the mode COSMOS shows for one end fully constrained and the other end free

buckle aspect ratio

wall flexural stiffness per unit width

buckling coefficient of cylinder subject to lateral pressure. NASA eq. 14

In the approach of NASA article they find b that gives a minimum

But here for this comparison we assumed a b (buckle aspect ratio) which is consistent with the the COSMOS analysis, see figure below, we get:

COSMOS critical buckling pressure see load factor below

very good agreement for this case where n = 1 and m = 0.5, but moving on to the next case.

Next case:
Both ends fixed then COSMOS give as shape consistent with

and

In this case the COSMOS result is:

now almost factor of 2 difference between NASA and COSMOS

Looking at NASA Fig. 4 and noting that:

Shows we are in the domain where the deformation of the tube is an oval and one should be use NASA eq. 18. In this domain b does not enter, is this where m=0 making b = infinity?

using NASA eq.18

This is to be compared with the COSMOS buckling pressure

The NASA article mentions after eq. 18 that restraining the edge against longitudinal movement and or rotation can increase the buckling pressure by as much as 50%. So this is probably the discrepancy.

COSMOS for both ends fixed:

COSMOS for slightly relaxed constraint, namely ends held round only:

very slight drop in buckling pressure

Asymmetry looks suspicious?

COSMOS can't be defined with a restraint with both ends free. It is necessary to have at least one end held round, so one can't compare COSMOS directly with the NASA eq. 18 result.

Final conclusion:
It appears that there is good agreement between the NASA publication and COSMOS for the simple cylinder when the two represent more or less the same boundary condition. Where a difference is found it would appear that the restraints are different.