# Using modes from linear buckling analysis as imperfections

It first introduces us to linear buckling analysis, and then mentions that linear analysis can be unrealistic as it doesn't take imperfections into account. Then it is said:

One good strategy is to first perform a linearized buckling analysis and then use the computed mode shape as imperfection. The idea is that the structure will be most sensitive to this shape.

So if I understand correcty, the article suggests that to get a more realistic analysis we first compute the linear buckling modes, use this as initial imperfections and then do a non-linear analysis.

But I don't exactly understand why the structure is most sensitive to the buckling modes computed from linear analysis. What does this statement mean exactly?

• Good strategy or possible strategy? Commented Apr 13, 2020 at 12:07
• @Solar Mike Sorry? Commented Apr 13, 2020 at 12:32
• "One good strategy is to first perform a linearized buckling analysis" your English understanding or mine? Commented Apr 13, 2020 at 12:56
• @Solar Mike It is a direct quote from the website. What are you talking about? Commented Apr 13, 2020 at 13:24

I briefly glanced this article. Seems interesting but I haven't read it yet - hopefully I'll still be able to help.

First, lets say a few words regarding the non-linear buckling analysis. It is important to realize that contrary to linear analysis, it takes into account the following:

1. Geometrical non linearity - the model shape changes according to the applied loads (deformation) so the load distribution through the model is updated accordingly. Step by step
2. Material non linearity - As the stresses reached the yield point, the YOUNG modulus changes and so is the model stiffness matrix. Yielded zones will not carry excessive loads anymore and the load distribution will be updated
3. Imperfections - small deflections that make a difference between a "clean" FEA model to a real life case

In the sense of it, buckling modes are the shapes a body will tend to wear while being subjected to the a specific load pattern. There are shapes for bending loads, torsional loads and so on. For each mode there is critical load the model could not further withstand and that will eventually cause it to buckle.

While running a buckling analysis, we apply loads on a model and examine its behavior. We would be able to obtain the deflection it gets and to check whether buckling occurs under these loads. If we run a non-linear analysis, it is a good advice to add a small imperfection, in the shape of the relevant mode (i.e, a shape the will contribute to the deflection under the specific case). Do not forget that physically and mathematically, the model will first begin to deform when linearity governs (small deflections, constant stiffness) - so it will try to deform into one of the modes that can be acquired with linear buckling analysis.

Hope this helps...

It's worth remembering that buckling is caused by the bending moment generated by axial loads due to (possibly infinitesimal) imperfections.

Given that, it becomes simple to understand what the text is saying. The linear buckling analysis will show you the shape the column or structure will take upon buckling. The particular shape this gives you is, by definition, the shape which does the worst job resisting the bending moment generated in buckling; if another shape were worse, you'd get that shape instead.

So if you set your imperfections according to the linear buckling analysis, you'll basically be doing buckling's job for it.

• I still don't understand why the shape obtained from linear analysis does the worst job resisting the bending moment. Let's say we choose to introduce some kind of initial imperfection to our structure and then run a non-linear analysis. We could choose the shape of our initial structure according to the shape obtained from linear analysis or something arbitrary. Why is the shape from linear analysis worse in resisting the load than something else? Commented Apr 16, 2020 at 9:13
• @S.Rotos to answer a question with other questions: do you understand the derivation of Euler buckling (which is basically what linear analysis does)? If so, why do you think linear analysis gives the buckled shape it does? What is the meaning of that shape? I'm having a hard time answering your question here more directly because it seems quite intuitive to me that the shape found by linear analysis is the worst possible shape at resisting buckling.
– Wasabi
Commented Apr 16, 2020 at 13:39
• @S. Rotos - my answer was not clear enough? Those shape describe the tendencies of the body to deform. Commented Apr 17, 2020 at 6:49