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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 It is known that when the cylinder is thick enough, Euler's buckling formula is no longer valid.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!

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## Comments

## slenderness limit on 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.

## That makes sense

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?

## Engineering classification

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:

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

## Books and reference

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,

Hugh

## Introductory book on buckling

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!

Julian

## Julian J. Rimoli - absolutely correct answer

## where can I find that

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