Benchmark for nonlinear geometry!

During last week I've posted a tip in the FEA Guild about using Benchmarks. Some great pointers were in the comments, but one got my attention the most:

While there is no chance that I will build such database myself, I'm pretty sure we have a chance of making one together! I will try to add some benchmarks from time to time, for various problems. This way, when you will learn, you will have a chance to check if what you do "gives" correct answers.

If you have some examples solved in FEA, that would be good for this purpose let me know ([email protected], or in the comments below). Building such a database really would help us all!

I couldn't create a benchmark from the scratch in a day, but I dig out a benchmark I had in the archives and refresh it a bit!

Let's roll!

When it makes sense to do benchmarks:

• You try to solve a problem of a different kind for the first time or after a long break!
• When you bought a new software package or created one yourself.
• When you are just learning about something, and you are not sure if what you are doing is correct.
• Or maybe you have come up with a new approach to solving nonlinear problems and you would like to validate it.
• It's also possible that you just opened silo code EN 1993-4-1 and have no clue where to take the required comparison outcomes from.

In all of those cases making a benchmark is a good idea.

What is benchmarking?

If you have never met the term before, to do a benchmark means to solve a problem you already know the solution of (and that you are certain of it). This way, you can test your FEA setup, and hope that it will produce correct answers. This way, you may be sure, that the procedure you will use in other, similar problems works properly!

There are of course benchmarks for a lot of problems. Today I will discuss one for geometric nonlinearity in shells. This is most likely one of the most famous tests, widely used in a lot of scientific papers! It was originally published by prof. Rotter in 1989, but since then it was done and re-checked by a lot of people doing shell design.

Weld Type A

If you are interested original text it was published in Journal of Structural Engineering, Vol. 115, May 1989. It described a popular imperfection shape caused by welding depression dividing this shape into "Type A" and "Type B", hence the name of the benchmark (as well as one of the popular imperfection shapes used in the design of shells).

The problem itself was described very well and easily allowed for reproduction of this analysis, which is one of the reasons why it became so popular. The original text discusses outcomes from analyzes using those imperfections, but here we will focus on a specific case described in the paper (there were several there).

The task!

Create a shell model with the length to radius ratio L / r = 3. Radius to thickness ratio of the shell is r / t = 1000. Material is linearly elastic with E = 200 GPa and Poisson's ratio v = 0.3. Shell is simply supported at ends (S3), and even a mesh size is defined in the region of imperfection as 0.25√(rt).

This means that you can pick a lot for yourself. Just to give you an example, it may look like this:

At the top, you can support translations in horizontal directions, while at the bottom all 3 translations. If you want to use a cylindrical coordinate system, just remember to "point" Z-axis as I did here and all remains the same!

Naturally, load goes at the top edge (directly downwards). Value doesn't really matter (you will do a post-failure analysis anyway). Just make sure it is higher than capacity. If you don't know where to start, expected capacity is around 37kN/m. This means that the load of 50kN/m on the top circumference (directly downward to cause compression) should do the trick. You will see later where I got this from : )

Looks innocent right?

In all honesty, I hate symmetrical uniformy supported shells! They always give trouble with convergence. Analitically, discretely supported shell is almost impossible to solve... but in FEA it's actually the easier case!

This is of course not the end of "fun". After all... we are missing the imperfection!

Weld type A

As in all scientific research, there will be equations!. Imperfection amplitude is given by:

Where: delta_zero is the imperfection amplitude (it this benchmark equal to t, so in the "example" I've described above that would be 0.001m or 1mm). The "x" is a coordinate along shell length (x=0 in the middle of the shell where the imperfection is applied).

This means that the imperfection has a maximal amplitude in the middle of the shell, and it changes starting from the middle in both directions!

Additionally:

All of the equations above came straight from the paper. Originally an "x" coordinate is used for imperfection distribution. This is a bit unfortunate (as most likely you will have an x-axis in your model, and it won't "fit"). Just to avoid confusion, this is how imperfection should be located:

It's worth noticing, that the paper already suggests element size near the imperfection (this is the "important space"). So there is no need to do mesh convergence study. Just make elements near the imperfections QUAD4 squares with the edge length of 0.25√(rt). In our case that is 7.9mm. This means you will have around 800 elements along the circumference. That is quite a lot, don't be surprised : )

This is a "scientific paper", so everything is made dimensionless. But we already assumed a lot of dimensions (see the drawing above). This means, that I can actually calculate the equation, to show you how the imperfection will look like in our case:

Note:

The "x" coordinate in the vertical axis on the chart is the one shown in green on the drawing above. Maximal deformations are in the middle of the shell. Also, maximal amplitude is to the inside (this is why radious is smaller than 1 in that direction).

The challenge

I think it is self-explanatory at this stage. You should implement such a shell with such imperfections and load it as we discussed. The task is to calculate the critical capacity of the shell... so the load under which the shell will buckle.

Note, that you should use nonlinear geometry, but elastic material property (material is elastic)!

This is when the benchmark starts. If you want to test your "nonlinear geometry skills" give this task a swing. I admit it made me sweat when I attempted it for the first time! Good luck!

Hints, solution and additional ideas!

Visit my blog to learn more about this problem!

Want to learn more?

This is really awesome! I have a free nonlinear course just for you!