  In this paper we initiate a program of rigorous analytical investigation of
the paradoxical buckling behavior of circular cylindrical shells under axial
compression. This is done by the development and systematic application of
general theory of "near-flip" buckling of 3D slender bodies to cylindrical
shells. The theory predicts scaling instability of the buckling load due to
imperfections of load. It also suggests a more dramatic scaling instability
caused by shape imperfections. The experimentally determined scaling exponent
1.5 of the critical stress as a function of shell thickness appears in our
analysis as the scaling of the lower bound on safe loads given by the Korn
constant. While the results of this paper fall short of a definitive
explanation of the buckling behavior of cylindrical shells, we believe that our
approach is capable of providing reliable estimates of the buckling loads of
axially compressed cylindrical shells.
