Table 15.2 on p. 735 of "Roark's Formulas for Stress and Strain", Seventh Edition, by Warren C. Young and Richard G. Budynas, McGraw-Hill, 2002
The "classical" buckling formula is independent of the buckling mode shape. Therefore, for a finite element buckling model of the thin cylindrical shell (say r/t > 100) there will be many, many eigenvalues clustered near the "classical" value, provided the shell is thin enough, provided there is a dense enough finite element mesh and provided that the boundary conditions are properly formulated, that is, the cylindrical shell is free to undergo uniform Poisson radial expansion in the pre-buckling phase of the problem, and "classical" simple support is applied in the bifurcation buckling phase of the problem. These clustered eigenvalues will have mode shapes with various numbers of axial and circumferential waves.
NOTE: The buckling load of a uniformly axially compressed cylindrical shell is notoriously sensitive to initial geometric imperfections. Therefore, experimental buckling loads of actual cylindrical shells with their inevitable random imperfections will generally be much, much lower than the theoretical "classical" value givern here.
The next several slides show typical discrepancies between test and "classical" theory for the buckling of axially compressed cylindrical shells.
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