Nonlinear FEA Masterclass Sample Lesson

Welcome in the sample lesson of my Nonlinear FEA Masterclass!

I’m really happy that you are interested in my course!

As you can imagine, selecting a single sample lesson from 50+ in the course is not easy. In the end, I’ve decided to pict the one, that will nicely “fit” with the samle example I’ve selected.

This lesson is from the middle of the course, so it already assumes that students know a few things taught in previous lessons and examples, but I feel that you will be able to follow it even now!

And of course, I hope to see you in the course! Enjoy!

Getting capacity from nonlinear material

This will be FUN!

Right off the bat, I feel we need to make a very important distinction!

The ways the material can cause failures!

The material can fail on its own! This is the most “obvious” way of failure. Whatever material you choose, it will have a stress/strain curve, right? And as you go “along” that curve, at some point you reach the level of strains that is simply “too high” and the material will fail. That would be a classical “breaking”.

Material may start to act “weirdly”! This is my absolute favorite! Especially since it’s not that obvious to figure out! Apart from the fact that material can break at some strain level, it’s also true that the nonlinear behavior of the material may “soften” your model. After all, in many cases, the material becomes “less rigid” as the strain increase. And this can easily mean, that while the model will not “break” as such, the reduced rigidity on a substantial area will cause a failure!

I save the “weird behavior” of the material for the next lesson. Here, we will discuss the first possibility:

The obvious material failure

The coolest thing about the “obvious” material failure is the fact that it’s not so obvious!

I mean, we all understand that the material can take certain punishment, and then it will fail. And this “punishment” is usually measured as strain. Sure, many materials have allowable “stress” assigned to them, but this isn’t the best approach (surprisingly). Let’s consider our nice “normal steel” stress-strain curve:

Notice, that if we would define the “allowable stress” as f_y on the stress-strain chart above… this is “true” for both marked total strains e₁ and e₂. The problem is, that total strain e₁ is a completely different outcome than total strain e₂! But the stress under which they happen is the same!

Also, I must admit that using “total strains” isn’t super useful either! This is because some of the strain is elastic, and the rest of the strain is plastic. Usually, in the design rules and guidelines “plastic strain” is used as a measure. This means that the above chart (for plastic strain) would look like this:

Notice that now, the “lines” to e₁ and e₂ aren’t vertical – they are parallel to the elastic part of the stress-strain curve. This way, the elastic strain is removed, and this is actually how much strain would remain after unloading. I guess this is just one more reason to draw this chart “out of proportion” with the elastic range inclined way more than it should be… it makes things easier to read! I will assume for now, that in most cases we will simply talk about plastic strains.

But in the end, I want you to remember that measuring “hurt” by stress isn’t the best option. It may “fall short” in accuracy in many cases, mostly those that include yielding. Of course, I’m aware that this is what we are used to doing. Furthermore, it’s also the case, that many materials have a “constantly rising” stress-strain curve like this one:

For those materials, it doesn’t really matter if you will use stress or strain… simply because when you have the curve above, you can easily transform one into the other! But as a rule, think of strain as the more useful component – at least in metals.

The beginner, the expert, and the desperate

There is one little thing I managed to sneak on you – if you caught it, congratulations!

Let’s take a look at the chart I just used once more:

I guess it deserves a mention, that if we would allow stresses higher than yield, the problem I just described wouldn’t bother us at all! Of course, I don’t mean “stresses higher than yield” that you get from Linear FEA! I mean stresses that happen after total strain is higher than e2! The strengthening starts there, and again you could “transform” stress and strain easily. So while various strains would still happen for stress equal to fy… if we would be allowing higher stresses, that wouldn’t be much of a problem, right? After all, we can be sure that there IS a strengthening, and this is a well-researched phenomenon.

I well remember my professor who told us in one of the lectures that you can divide the stress-strain curve in steel into 3 zones. The elastic part is for beginners, the plastic part is for experts and the strengthening is for the desperate!

I well remember that lesson, and it seems that most design codes I’m familiar with agree on this. However, I feel that I should point out that the strengthening isn’t such a no-go zone as you may think. I would never use it in nonlinear FEA design, but at the same time, we use it all the time. For instance, all cold form cross-sections reached strengthening in the regions where they were bent into shape. And they still work. I mention this just to give you a perspective, that things aren’t always black and white.

But still, even knowing that I would never go into the strengthening of the “normal steel” in my FEA analysis… codes that limit the “allowable” plastic strain tend to agree with me on this one. It’s not easy to pull a single “allowable plastic strain” value. But if I would have to “guess” I would say that in general, the allowable plastic strain oscillates somewhere around 4-5% (for “normal structural steel” like S235). As I mentioned before, I prefer to use a 2-3% “limit” myself, and this is what we do in the analysis in my office.

And this brings us to a very interesting consideration:

The limitations you are willing to accept

You may be thinking that when I’m choosing 2-3% of plastic strains as a “limit”, I’m effectively “cheating” my Customers. After all, lower limits must lead to over-designed structures, right? Well… not so much really! As we already discovered in one of the examples, the actual increase in a plastic strain of around 1% (between 3% and 4%) usually happens when the load increases very slightly (perhaps around 1% as well, but this is of course case-dependent). So limiting the plastic strain to 2-3% means that the capacity of the models is only 1-2% smaller, but the reliability of the structure significantly increases – at least in many cases. This is a good thing of course, or at least this is how I see it. But I will also understand how someone may say “the codes allow for more so I’m going with more” – this is absolutely a valid idea!

Whether you agree with me or not, the truth is, that for each material you will need to know the “limit”. And this “limiting value” (be it strain, stress, or whatever else!) isn’t really connected with FEA that much! It’s just a limit that you are willing to (or allowed to!) use in the design.

So what strains should you allow (assuming that I managed to convince you to actually use strains as a measure!)? Of course, in “general” this is an impossible question. Heck, this even depends not only on the material but also on the application. After all, I just wrote to you, that I would easily allow for 2-3% of plastic strain in S235… But, for the same material in fatigue case – such plastic strains would be a suicide most likely! So, as you can easily see by this example even for the same material, clearly defining a “one and set” limit seems impossible.

Let’s wonder for a second, where could you search information specifically for your work about those:

• Industry standards/codes: I’m lucky enough, that I’m working in fields that are at least somewhat codified. So I can always refer to the code like EN 1993-1-6 to check, what the allowable strain is for the steel that I’m using. The same code (or group of codes) will provide me with safety factors I should use in my analysis and so on. This is definitely the easiest situation, where there is a set of well-accepted rules you can rely on!
• Laboratory Tests + Guessing: If you work in a different field, that simply isn’t codified at all things will be tricky! I guess that one of the reasonable ways would be to base your assumptions on laboratory tests of material specimens. Of course, it would be best if you could test the material yourself… but let’s be realistic here! Luckily scientific literature is quite full of various tests on materials, although access to those research papers might be quite expensive. Still cheaper than doing the tests yourself of course! The problem here is, that tests give you the stress-strain curve “as is”. This means that you still need to consider where you say that “it’s enough for me”, and you will still have to assume the safety factors for your analysis!
• Asking more experienced engineers: Guessing on your own is never a cool thing – I know. This is why asking older colleagues in your field for advice and “industry guidelines” seems like a decent idea. I must admit, that there are no guarantees that you will get any useful information – after all, we are talking about pretty advanced stuff here, and I would dare to say that most engineers never had to wonder about this (since they used linear FEA at best anyway!). But it’s also true that in various industries there are unwritten rules about what is allowed and what isn’t. I’ve learned that while I was training one engineering team, that told me that their company expects them to use a safety factor of 10 on everything… As you can imagine they had pretty responsible tasks to do!
• Old books: It’s funny, but at least where I’m at, people always recommend old books way more than new books. There are reasons for it I guess (since now, books are mostly written by scientists who want to advance their scientific career, so they are rarely practically useful). Whether you prefer old or new, searching for answers is never a bad idea. After all, if you manage to find some decent information in a book, it has more “weight” than your “engineering guess”. If you are very desperate, I can tell you that in “New Science of Strong Materials” (I have a reprint from 1991), J. E. Gordon mentions that “in practical engineering materials, strains nearly always lie in the range +/- 1.0%”, and a few pages afterward: “(…) the compressive strain in the hull plates [of a submarine] might be about 0.7%”. I know, that this is a super vague answer and prof. Gordon definitely didn’t assume I will quote his book in this way, at least it’s a guide of some sort… which may prove better than nothing at all! Just remember, that the same author in the same book writes: ” Sometimes nobody is quite as blind as the expert”, so always ponder if those limits make any sense in your case!
• Experience: And I know you won’t like this one! And I’m not even saying that being more experience makes you somehow magically “get the values from the thin air”. What I meant to say is, that when I began, my limit for plastic strains was “twice the elastic strain”… which is more or less 0.22% for S235! In time, I’ve learned and understood (and read!) more, and now I’m more comfortable with higher limits, different material models, and all the jazz! So if you are just starting out, don’t get too crazy! Give yourself the time to absorb the knowledge and expand what you already know. It took me a few years to “clear this” in my head.

I know, that the above may not be the answer that you were hoping for, but at least it’s true (if this is a consolation at all!). I would highly recommend that you read the standards of your industry (if you have any!), and ask around, to search for knowledge. After all, this is not an FEA “thing” but an engineering assumption… and we’ve been doing engineering for a LONG time! Someone has to know!

A small summary before we continue!

There are two different failure modes, a nonlinear material can cause. The first one is the more obvious one… and that is that the material itself will break! The second one (the one with the “weird behavior of the model”) will be discussed in the next lesson. But before we go there, let’s wrap this lesson up in a few points:

• Know the stress-strain curve: This may seem simple since this is obvious that you will need the data for the nonlinear material… to use it! However, it’s not as obvious as you will think. While we will discuss different material models, and how you can incorporate the stress-strain curve of your material into the FEA model… in the end, you still need to know the “numbers” for your particular materials. The more popular ones have the material models well described, but the more “exotic” ones may be tricky even at this stage!
• You need to measure material “hurt” somehow: Knowing the stress-strain curve is important, but you also need to know what the “limit” for your material is. And to define the limit you must know, whether you wish to limit stresses, strains, or even something completely different. As I wrote here, it seems that metal strains will do a better job, as you won’t get in trouble in a plastic plateau!
• It’s not easy to set borders: But at some point, you will have to. If you define the stress-strain curve for your material, you can easily run the nonlinear analysis. But after receiving the outcomes, you will have to make a call, if the material failed or not. And to do this call, you will have to know the “limit” of what you are willing to allow. I know perfectly well, that finding the information about your material in your application may be difficult. To help you out a bit, I listed several possibilities of where I would search for the answers.

Armed with the above, you could define the nonlinear material (although I admit, that we still haven’t covered different material models in FEA, so bear with me here!). After you will run the nonlinear material analysis, you will also have the tool (the “limit”!) to check if the capacity of the material is sufficient or not. And this basically means that you will be able to say if the material failed under the load you prescribed! This is a big step, but not the end of the road… after all, there is this “weird behavior” we should attend to now!