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Euler's buckling formula

Hi, everyone,

I am new here. Now I am doing a research which is related to the Euler's buckling formula.  It is known that when the cylinder is thick enough, Euler's buckling formula is no longer valid. I want to know in which region of slenderness(aspect ratio) of a cylinder or column that the Euler's buckling formula is valid. Is there any analytical study or numerical results about it? Is there any suggestion or paper I can find?

 Thank you very much! 


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.

If course, it is of note that the less slender a member becomes, the less susceptible it is to buckling. Is there an obvious case where one would expect buckling to be the failure mode in a member for which Bernoulli-Euler beam theory is unreasonable?

Julian J. Rimoli's picture

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range: for intermediate values of L/rho (usually between 20 and 60) the failure is inelastic and Euler’s equation is no longer valid.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.



Dear Julian,

First, thank you so much for your answers.

Second, is there any book or paper regarding to your answer? Since I am writing a paper and I need accurate and precise reference. Thank you.

Best regards,


Julian J. Rimoli's picture

Dear Hugh,

Any undergraduate book on aerospace structures should cover the basis of the topic. The book I mentioned before (Analysis and Design of Flight Vehicle Structures by E.F. Bruhn) is useful if you are interested in a "manual like" book for design purposes. Nevertheless, it is probably not the best book for learning the subject from scratch.

The book Buckling of Bars, Plates, and Shells by Robert M Jones is fully available online through Google Books and is probably better as an introductory book. That could be a good starting point for you.

Good luck!



Buckling effect

Dear Julian J. Rimoli, 

Thank you for you answer.

But I want to know the scale and marks of the axes. It is very important.

And where does this plot appear? Can I find it?

Thank you very much.

Best regards,

Hugh Wang

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