Euler's buckling formula

Euler's buckling formula is based on Euler-Bernoulli beam theory, which does not account for the effect of transverse shear deformation.  This effect is significant for non-slender beams, or in this case, non-slender columns.  So, to find the buckling load for non-slender columns (if it exists), you will need a beam theory for non-slender beams (e.g. Timoshenko).  It would seem to me that the same slenderness limit on Euler-Bernoulli beam theory would apply to Euler's buckling formula.

Dear Hugh Wang,

For a column of length L and cross section radius of gyration rho, it is possible to distinguish 3 failure regimes under compression:

1. Long column range: for large values of L/rho (usually above 60) the failure is elastic and Euler’s equation is valid.
2. Short column range: for intermediate values of L/rho (usually between 20 and 60) the failure is inelastic and Euler’s equation is no longer valid. Several semi-empirical methods have been developed to describe failure in this region from the engineering point of view, e.g. Johnson’s equation. The book “Analysis and Design of Flight Vehicle Structures” by E. F. Bruhn has a good compendium of methods and design tables.
3. Block compression range: for small values of L/rho (usually under 20) the failure is completely plastic and can be assumed to be, in absence of cross sectional instabilities, equal to the compressive yield strength of the material.
   

For unstable cross-sections, e.g. thin C beams, local buckling may appear, thus reducing the strength of the column.

Well, I hope this helps.

Regards,

Julian