Topic of linear and nonlinear buckling is one of the most searched FEA topics. It is understandable, since lack of knowledge about the difference in both approaches can lead to great overestimation of model capacity and following real-life instability failure. In this post I will discuss the difference with a shell example.

#### Problem introduction

I think this debate is popular, because performing a nonlinear buckling analysis requires far more knowledge and sometimes even a better software than for the linear buckling. This leads to situation where sometimes linear buckling is the only solution that can be applied (i.e. when license costs are a limitation) and question arise whether accuracy of such algorithm is sufficient. This does not mean of course, that nonlinear buckling is simply better, since it requires far more computing time to perform the design. In general I see the difference between the two as follows:

**LBA: Linear Buckling – Positives:
**

- Short computing time
- Easy to define correctly
- No convergence problems
- Experience not required

**LBA: Linear Buckling – Negatives:**

- In many cases the outcome may be wrong

**GNA: Nonlinear Buckling – Positives:**

- Outcome is correct
- You can animate instability failure process
- Outcome is more robust
- Can interact with plasticity of the material (if needed)

**GNA: Nonlinear Buckling – Negatives:**

- Requires much more computing time
- Experience is required
- Convergence problems
- Far more difficult to set up

As you can see, I consider the nonlinear buckling analysis to be more difficult, and time consuming analysis that gives better results. If you are interested in Linear Buckling I wrote about it here. Basic concepts of geometric nonlinearity are described here.

#### Comparison example: Linear buckling

In previous part of the course I have performed an LBA (linear buckling) analysis and checked the optimal mesh size for the task. The outcome I have received is summarised at the picture below.

Please note the distinctive shell deformation after instability failure at load 36.55kN/m.

#### Comparison example: Nonlinear buckling

To perform a nonlinear buckling analysis I used an *arc-length* method (in this case modified Riks Algorithm). I used stability path method of determining model capacity, as well as geometry after failure.

It is possible to read from this *stability path* that critical load in case of nonlinear analysis is 30.6kN/m which is 16% smaller than the outcome from LBA. Also the deformed shape of the shell after collapse differs greatly from that obtained with LBA.

#### Summary

As shown on the above example difference between linear and nonlinear buckling is important. In simple problems difference in critical load between those two is in around 15% (as in presented example). From my experience I would say that the more complex problem, the higher difference in outcomes between those two analyzes is. I think that the highest difference I have obtained was around 100% – I will try to find such problem and post an example in the blog. It is also important to notice that LBA do not predict shape of the instability well (especially in shells). This is important not only as an information about collapse region – sometimes scaled LBA deformation is used as imperfections, and this may lead to falsified results.

I would say that the biggest drawback for linear buckling is inability to incorporate nonlinear material behaviour into this procedure. This is often a very important aspect of analysis and will be discussed later on.

Today there will be no clip showing how I did things in Femap: Next week I will discuss parameters of the nonlinear analysis I have used, and then show how to implement it. Make sure to subscribe so you don’t miss it 😉

Have a nice week!

RobertMay 15, 2016 at 11:47 amVery interesting article! I have been struggling for months to understand what you described about linear buckling. Thank you! Please continue to write good articles like that.

JohannesAugust 20, 2020 at 12:00 amVery interesting, thank you! Do you have a computational time comparison of the linear vs. nonlinear buckling?

Łukasz SkotnyAugust 20, 2020 at 1:36 pmHey Johannes!

Sadly, I don’t. But based on “experience” I would say that LBA runs for “a few minutes” (like 2-4min). The same model with nonlinear stuff would run for around 2 hours, depending on how you set up the solver etc.

Hope this helps 🙂

Ł