Web under local loads – Nonlinear FEA
The more complex a problem is, and the higher the accuracy needed, the more it makes sense to employ Nonlinear FEA. Will it make sense to use it in solving local web loads? Let’s find out!8 February 2021
Linear FEA calculations are the most common type of static analysis done with finite elements. Last week we have discussed when it is safe to ignore material nonlinearity to use this simple yet robust tool. Today I will describe when we can safely ignore geometric nonlinearity.
There are several things you should consider when you want to use linear static design. We won’t discuss dynamics here, but “linear” means that you are omitting some key things in linear FEA.
Depending on whom you ask there are several “aspects” that can be treated as linear in the analysis:
As I mentioned before, last week we have discusses when it is safe to ignore material nonlinearity. Today I will shortly describe when you can ignore geometric nonlinearity. Since follower forces are closely connected with geometric nonlinearity they come “free” with this topic!
This is a very tricky question. If you know how geometrical nonlinearity works you understand that it is more subtle than material nonlinearity. The easiest way to check would be to perform a geometrically nonlinear analysis and simply compare the outcomes. Of course, that would be a very ineffective check. There are easier ways to do it, even if they require certain experience.
The easiest would be to perform a linear buckling analysis. If you want to learn more about it, you can try my free online course:
Usually, when something is susceptible to nonlinear geometry (large deformations) it is connected with small bending rigidity. The analyzed element cannot withstand the load in a “rigid enough” way and change the “form” in which it carries the load. Such effects will result in a low eigenvalue from Linear Buckling Analysis (LBA). If you have certain experience and you know what you are looking for, analyzing LBA outcomes seems like a great start.
To be more specific you should be cautious when the eigenvalue of your model is smaller than 10. This is a “popular” value. It is even listed as one of the checks in Eurocode 3 for steel structures. This code also mentions that if you are using the plastic design you should aim for eigenvalues higher than 15.
I’m always reluctant to give such information without further comment. You see – this is not a “perfect solution”. I will show it in a simple example. Imagine you are designing a steel truss with reasonable beams. All checks according to EC (including the buckling of elements) are ok, and the capacity ratio is maximally 0.96:
For short members in compression limit capacity is around 2-2.5 times lower than the ideal Euler force (due to imperfections etc.). This means that if you have a simple truss you will get a lot of eigenvalues around 2.0 – 2.5. To be exact first… 96 forms have eigenvalues between 2.01 – 5.4 that look more or less like this:
This doesn’t mean however that your truss is susceptible to large deformation… even though this seems so based on the criteria I just described. It is, of course, easy to prove. All I have to do is to make a linear calculation. Note that the maximal deformation is 64.4mm (maximal capacity ratio as mentioned above is 0.96):
Then I will do a geometrically nonlinear analysis. Note that maximal deformation is 67.8mm:
The maximal capacity ratio is in this case 100%.
In summary, I got almost 100 eigenvalues below 5.4 which are far below recommended 10 (lowest being 2.01). This would suggest that nonlinear geometry should be significant in the model. This is however not the case since with nonlinear geometry both deformations and stresses increased around 5%. Not an astonishing change isn’t it?
This is the problem with geometric nonlinearity – it requires experience to spot when it can be ignored. This is why I usually use geometric nonlinearity calculations in FEA… just in case. I still get surprised after 10 years of experience in the FEA field! When using LBA for testing it is always good to check if your FEA solver includes warping-torsion commonly referred to as a “7th degree of freedom”. If you don’t have such possibilities, your LBA won’t be able to find lateral torsion buckling in beam models. It will work just fine in plate/shell/solid models.
Another good idea is to fulfill code criteria for maximal deformations. Sure there are no criteria for many elements. Luckily you can find similarities of what you are calculating to other defined cases. This way when you know you are within “allowable” deformations for your type of structure this means that you are safe from large deformations problems (unless the code you are using defines it differently of course).
Usually, those criteria provide allowable deformation. Usually, this is the length of the element divided by a certain value (like L/250). Such approach is very common in civil engineering. According to Eurocodes such checks are called SLS (serviceability limit state). When you do those checks just remember that for cantilevers the allowable deformation is twice as high!
When I’m in doubt about what limit to use I follow a simple guideline:
The above is just a rule of thumb obviously but should give you the general “range” of allowed deformation.
In a Linkedin discussion, a friend of mine Leonardo Rosa pointed out that there is a popular “5% strain” rule. According to it, if in the linear analysis you get more than 5% strain you should perform a geometrically nonlinear analysis.
Definitely an interesting concept – it never hurts to make this additional check!
To check if you can ignore geometric nonlinearity:
Definitely check out my FREE FEA course. You can get it by subscribing below.
If you have a spare 15 seconds write a comment with your thoughts on the matter or any questions you might have. I have a good history of replying to each and every comment 🙂
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