The difference between linear and nonlinear FEA

This is a 10min read. It will be more comfortable to read the full article on my blog :)

I remember my first presentation regarding my Ph.D. It was my first year on the doctoral program, and I was starting to be on a first-name basis with people that were my teachers just a semester ago. I went to the middle of the room and started talking about shell buckling... the problem was everything I did was calculated with LBA. I simply didn't know the difference between linear and nonlinear FEA! Don't worry... you don't have to follow my mistakes!

Linear Finite Element Method simplifies a lot of things. For instance, the material will never yield resulting in unrealistically high stresses in your model. Also, you may not predict buckling or membrane state (or do it very poorly) because nonlinear geometry is not taken into account. Nonlinear FEA, when defined correctly, takes care of all those problems for you.

If you want to avoid blunders I did as a Ph.D. researcher... definitely read on!

Linear or nonlinear geometry

This may not be the most "obvious" part of the nonlinearity but I start here... because this is how I have learned nonlinear FEA. You see, when you use linear analysis solver assumes that you will be within "small deformations". This actually means that 2 assumptions are made:

• Assumption 1: Deformation do not impact how the structure behaves (i.e. noting enters membrane state)
• Assumption 2: There is no stability failure

The two above are big ones... unless you are designing something solid in shape. I mean, if you really have to analyze a stocky solid it won't enter a membrane state, and it won't buckle either. In such a case nonlinear geometry won't do you any good...

... funny thing is, it won't do you any harm either! If you use a nonlinear geometry analysis with something that behaves linearly... you will get the same outcomes as from linear analysis. But of course, the setup and the analysis itself will take some additional time!

Things start to become more complicated when things deform... a LOT!

Imagine you have a string attached on both ends to the wall. If you put a load on it, the string will deflect like crazy! Simply put if you would treat it as a beam, such a beam is so "weak" due to bending that calculated deformations would be insane!

But it's actually not so simple to deflect so much. If the horizontal movement on both supports is blocked... string has to get longer to deform downward with this nice arc. Maybe even much longer!

Elongation is not "free" - you have to apply a normal force to elongate something. And with this in mind, the string "resists" getting longer and longer by developing the normal force inside it. This limits the deformation and allows the string to actually carry the load. You use this phenomenon every time you hang your laundry to dry!

While seemingly obvious, the linear analysis does not see this happen. Simply put, in linear analysis deformations of the system does not impact its response. In other words, the string gets longer to allow for very big deformations, but this does not generate tension in the string. And hence, you get really stupid outcomes!

Being stable

The second problem is with stability. As you most likely know you can pull a crate on a string, but you can't push the crate with the same string. What would happen is that the string would simply tangle up, and not carry any compression. In an extremely simplified form, this is bucking.

In essence, everything that is slender (thin and relatively long) have a "critical load". When you apply this load, instead of nicely carrying it with compression the element will simply "buckle". Usually, this will look like your element simply "bends into an arc". This actually is a form of failure... sadly linear analysis will gladly load your element way above its critical load and display nice results. Completely ignoring the fact that your element actually failed way before the applied load level!

Sometimes, buckling can catch even experienced professionals by surprise. Take a look at the shell below. Maximal stress is 140MPa (way below the yield of 355MPa). However what you don't see here is, that such a shell buckles under the compressive stress of around 80-100MPa (in this case) and that it will fail under load much smaller than the one obtained in the analysis... This means that we just failed to estimate the maximal capacity. Sadly, based on the outcomes of linear analysis alone, we can think that our shell can be loaded much more than in reality! I know, it's scary!

LBA for the rescue, nonlinear geometry, linear and nonlinear material... and far more :)

You can read the entire article on my blog!