Hi folks! In case you haven’t noticed, the FEAsphere is buzzing. And the buzz in question?

Buckling analysis. Linear buckling in particular.

The buckling phenomenon was nicely explained by Cyprien from FEAforAll.

Cyprien makes the most awesome graphics too. Source above.

Simply put, yoy apply axial compression to a thin rod or beam, and of course it will bend. Everybody knows that.

The problem is, it shouldn’t! Or rather, a perfectly homogeneous material under perfectly axial load wouldn’t. However, things are rarely perfect. Maybe there are some minor deviations in the direction of the force, or maybe some slight variations in the material characteristics. In any case, it will buckle in real life.

We know it happens.

However, simulations are often done in those aforementioned perfect conditions. Hence, basic simulations might not predict buckling at all.

Think about it for a moment. According to the computer, everything’s fine. According to real life, things collapse. See why it can be dangerous?

Indeed, I’ve already written about the dangers of blindly trusting the software. The software does exactly what it’s been designed to do. Whether or not that corresponds to reality…that’s where real understanding comes in! That includes understanding the actual physical situation, and the simplifications made to model it with FEA.

Everything is better with randomness

The concept of buckling also ties nicely to my future topic: uncertainty quantification. Remember that buckling can happen due to minor imperfections. Almost by definition, we probably won’t know exactly where those imperfections are, or how large they are.

Well, uncertainty quantification consists of first determining some statistical characteristics of those imperfections. Once that is done, their effect on the outcome is analysed by stochastic simulations. For instance, the end goal might be computing the probability of failure due to buckling. But more of that in the future.

Modeling buckling

Linear buckling analysis is closely related to eigenvalue analysis, at least according to my limited understanding. Indeed, a generalized eigenvalue problem is solved for the stiffness matrix \mathbf{S}, and something called the stress stiffening matrix \mathbf{S}_\delta.

\mathbf{S} \mathbf{u} = \lambda \mathbf{S}_\delta \mathbf{u}

The eigenvectors \mathbf{u} of the problem then correspond to the buckling modes – how the system deforms. The eigenvalues \lambda can be used to determine critical loads, i.e. loads sufficient to buckle the system.

Cyprien gives a small example on his site, but for a more detailed overview I recommend you check out Łukasz’s explanation on EnterFEA. He goes through explaining the meaning of negative and positive eigenvalues, and the pros and cons of linear analysis.

Conclusion

This current buckling-buzz appeared seemingly overnight. One moment, I had barely heard of it. The next, that’s pretty much all anyone is talking about.

So, if structural mechanics is your thing, I highly recommend you take part! Comment on the articles I have linked. Or, contribute to the discussion in the various LinkedIn posts and discussion groups:

Cyprien’s post here.

Łukasz’s post here.

Łukasz’s group discussion here.

Case example here by another specialist.


-Antti


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Linear Buckling – Don’t Miss Out!

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