Nonlinear buckling is a really interesting analysis. It gives you enormous possibilities, at only a few minutes worth of work. At the start, it might be frustrating to set it up (there is a learning curve involved), but in the end, it is a default engineering tool for me right now! Possibilities are limitless!

Nonlinear buckling is a very powerful tool. It’s not as simple as linear buckling I wrote about lately. The nonlinear approach requires some additional settings in the solver. The results you get are well worth the hassle. To start off I will go with the simplest of cases: elastic buckling of a perfect model.

#### Nonlinear buckling (GNA: Geometrically Nonlinear Analysis)

I will start with the sweet stuff, so you will know what is here to gain. The outcomes from nonlinear buckling analysis are far more robust that those from linear buckling. What you get is:

Nonlinear buckling outcomes:

this is not the best part of course, but it is definitely fun! It also helps a lot as you can see how things are failing and in what sequence.Failure animation:is an incredible tool. It basically contains all the information about structure behavior of your model!Stability path:if you would multiply the loads you applied with this value, you would get a load causing failure. In case of GNA that would be failure due to buckling.Critical load multiplier:this is also very neat. You do not only get the outcomes until failure… you can actually calculate how the model will look like way after failure. A helpful thing if you wonder what will happen, is there any strengthening later on etc.Post-critical outcomes:while linear buckling only provides “shape” without real displacement values, nonlinear buckling does more! You can see the actual deformations of the model before and after failure!Actual deformations:

The typical animated outcome from GNA looks like this:

#### What nonlinear buckling actually do?

This is a **very** good question! In essence, GNA will divide the load you have applied in your model in a way that you requested in solver settings (below). There are several possibilities here. For now, let us assume that you have asked the solver to divide the load into 100 equal parts (often called *increments*).

During calculations, the solver will apply each part of the load one after another. In the first *increment*, the load will be 1% of the load you applied. In the second *increment*, the load is increased by **additional **1% for a total of 2% and so on). Such approach in linear static makes no sense (since all the outcomes are proportional). In nonlinear solution things are different. Deformation of the model from the previous* increment* influence the outcome for the next *increment*. This is why the division of the applied load into smaller pieces is important!

At the end of the analysis, you get something I call a “slideshow”. In our example that would be 100 different outcome sets, one for each *increment*. Usually in your software that will look just as if you had just made 100 linear analysis sets with gradually increasing load (but of course the outcome is different, especially at higher loads).

From this “slideshow” you can create the failure animation I mentioned above. Most post-processors simply have a button “animate” or “multi-step animate” for this purpose. You can also create stability path from this “slideshow” (more on this here). Sample outcomes may look like this:

#### How to set up nonlinear buckling analysis (GNA)

The setting of nonlinear buckling analysis is surprisingly “code independent”. Sure, depending on the software you are using, menus will look different, but the things you will be setting up are very similar.

**Required steps to set up nonlinear buckling:**

most per-processors assume this as default when you set up nonlinear analysis. But i.e. in Abaqus, you have to check the “Nlgeom” box.*Activate geometrical nonlinearity*:for now, we are keeping this simple. There are of course other possibilities I will describe in future posts.*Use linear material*:those algorithms are commonly referred to as arc-length methods. They allow you to trace a snap-threw problem. The standard solver will have huge convergence problem where buckling occurs. If this interests you read more about it here.*Pick Riks / Crisfield / Modified Riks algorithm*:You have to explain to the solver what amount of load should it take for the first increment. Each solver has its own idea how this should be done. In Abaqus, you simply give a multiplier of the applied load (like 0.01 which stands for “start with 1% of the load I applied). In NX Nastran you provide the number that the applied load should be divided by (so NINC = 100 is the Abaqus setting of 0.01). Additional increments are calculated automatically in arc-length methods.*Define initial increment*:it is so easy to forget that! I even once launched an analysis that computed for 10h, and forgot to mark what outcomes I want to get! Most software fills it for you, but usually, somewhere you have to mark that you want to have “intermediate” results. After 100 steps it would be nice to have 100 outcomes sets, not only the last one showing you the model in a deep post failure state!*Choose what outcomes you want*:you will notice that the nonlinear problems take significantly more time to compute. Sometimes this is 5 minutes, sometimes several hours. The quality of outcomes is easily worth the wait I think. Especially since you can run the large model in the evening to have outcomes in the morning 🙂*Calculate*:

This is the general outline of what you need to do. At the start setting this up might be frustrating (it was for me for sure!) but after you get some experience all the setting takes around 3 minutes!

#### Nonlinear buckling pros and cons

Nonlinear buckling positives:

The outcome you get is incredibly robust.Starting with the stability path and post-critical outcomes and finishing with actual model deformations and animations of failure.This analysis can take material nonlinearity into account.We haven’t discussed it here, but the fact that solver can use material nonlinearity is huge! In many cases taking both nonlinearities into account is a must!Can deal with contact.Another big one. Contact between objects can really impact stability outcomes. Being able to take that into account is great!

Of course, nothing is great without the drawbacks. Nonlinear buckling has several, mostly concerning the fact… that it is actually more difficult to use that linear buckling:

Nonlinear buckling negatives:

More difficult to set up :linear buckling is easier to do.Convergence problems :even if you set up everything all right, from time to time you will get convergence issues. Nothing too serious if you know how to restart the previous analysis!Require experience :there is a learning curve here. The more you use it, the faster you get the needed intuition. Not everything is obvious (like the amount of requiredincrementsetc.).Longer computing :the only drawback that is not directly connected with the skill level of the user. Calculation time is higher for sure. You can shorten it with proper memory management for solver and better gear. You can write scripts that will solve several problems one after the other during night… but in the end it will always compute longer than linear analysis.

#### My thoughts on nonlinear buckling

As you could feel while reading this post I really like nonlinear analysis. Most drawbacks go away with time and with the experience you gain. The only “real” drawback is the time required to do the calculations. It used to bother me much more, but I have learned to organize my work such, that the longest calculations are made when I sleep. Also, bare in mind that in a case of small analysis this “long computing time” can easily be 3 minutes. The average computing time for my problems is around 15min… You will always have time to make another tea!

I honestly believe that you should learn how to do nonlinear design even if you won’t use it often. In the case of most popular solvers the software, you use already have the needed tools. This can be a great stepping stone for your career (as it was for mine).

Knowing how to do such an analysis brings your skills (and value as a professional) to a completely new level! I really think it is well worth to learn it. I admit it will take time, and sometimes you won’t be able to convince your boss/customer that it makes sense. But when you understand this… it really changed the way I look at entire engineering!

Thank you for reading all of this. If you enjoyed it, consider sharing the news about my blog via your social networks!

Also, I have prepared a **free nonlinear FEA course** just for you! Get it below!

Nikos ZJanuary 10, 2017 at 10:58 amDear Łukasz, good morning!

Another great article in the series of the great articles you provide us with!

I would just like to comment on one of your phrases:

”But when you understand this… it really changed the way I look at entire engineering!”…

I deeply agree with what you say. Non-linear analysis and especially geometrical and boundary non-linearities make a student’s life difficult. It usually takes a lot of time in order to fully grasp the rationale behind and this does not often happen during the undergraduate years. I can remember myself back then at the time when all this was a pain in the neck. But after graduation, throughtout the upcoming years and after I had paid attention to this beautiful set of LOGIC (because it IS logical), I was able to see that the true value of engineering and the real truth behind the behaviour/response of the structures is in the field of non-linearities. This does NOT mean that linear analysis is of no use. On the contrary, without linear analysis and true grasp of it one does not go anywhere. And it is linear analysis that will guide you through this kingdom. It just happens that non-linear analysis requires more time to set inside an engineer and be able to grow. But when you understand it, you will only love it. And perhaps then it is a very good time for post-graduate study if you really love it, that will take your understanding to a higher level (for example, in my case this ended up to doing a deep dynamic non-linear Thesis in my post-graduate thesis).

Thanks again for your beautiful articles. I am always looking forward to your next inspirations.

Kind Regards,

Nikos

Łukasz SkotnyJanuary 10, 2017 at 12:11 pmHey Nikos,

Thank you for the kind words. It seems we share a similar story 🙂

Cheers

Łukasz

TienNovember 10, 2017 at 12:55 amhi Lukasz, I always love reading your blog, I suggest you can provide a course on nonlinear analysis, and provide some real case example (such as silo design). I’ll be one of the first to join such a course 😉