All buckled up!

I recently did a simulation product demo for a Norwegian customer and the question of buckling came up. I realised this was an excellent example of understanding the assumptions of the solutions you are producing plus the physical implications of your modelling methods.

We started off with a simple curved panel from a training example,

and used shell elements to perform a linear buckling study on a quarter of the model. You can see the edge restraint on the bottom right edge, and the other two symmetry restraints with a point load at one corner, equivalent to the centre of the full panel.

The output of interest from a buckling study is the buckling load factor, which was obtained by RMB on the Results folder, and was reported as 27.9. This means that the load must be increased by 27.9 times to make the panel buckle.

However this was a ‘linear’ buckling study, which assumes that the deformation occurring in the component before the onset of buckling must be small. If the panel shows large deflections before buckling occurs, then its stiffness may change. Our linear buckling study solves this in one step and has no opportunity to update the stiffness matrix, therefore giving us a potentially wrong answer.

To test this we ran a nonlinear buckling study on the same component, where the final load is applied in a series of incremental steps. This allows the stiffness matrix to be updated at each increment and so the stiffness of the panel is more realistic as it deforms.

In such a study we do not have a simple buckling load factor report, rather we can plot the displacement of the corner node where the load is applied in a Response Graph. The result looked like this:

where the curve needs to be read from right to the left since the displacement (on the x-axis) is in the negative Y direction, or downwards in our model.

What this shows is that the displacement increases as the load is applied up to a ‘snap-through’ point where it begins to change direction. This snap-through point provides the value of the incremental Load Factor at the onset of buckling and is equivalent to our buckling load factor. Here you can see that it is reading about 13.2, which is a lot less than the linear case of 27.9 above!

So you can see that with a nonlinear buckling study our model would buckle if we increased our load by 13.2 times, rather than 27.9 times; a much more sobering figure.

There is one final step that is worth considering too. Our model here is perfectly symmetrical and the solution is also symmetrical due to the way it has been constructed. However the real world is not symmetrical with manufacturing variations and real loads resulting in an asymmetrical setup.

To see what effect this has on our buckling load factors we re-ran the ‘full panel’ model with a slight offset of our load as an example of an inherent asymmetry. Here you can see a split line used to create a vertex on which the load is applied, slightly to one side of the middle of the panel.

The response graph of displacement at this point shows a different result again, however the buckling load factor is even lower, coming out around 11.7.

So you can see that buckling of ‘real-world’ structures should consider potential asymmetries that allow the structure to buckle sooner than in a symmetrical case.

All-in-all buckling is an interesting failure method and for an accurate solution you should consider both the real structure, your modelling methods, and the solution type to make sure it is as accurate as possible.