How to do FEA by hand – Part 2: 2D PLATES!
Find out how to do FEA by hand for 2D elements. Learn to construct a global stiffness matrix and calculate displacements and stresses.17 October 2022
I used to think about the role of math and software in FEA as about two different phenomenons. But recently I realized that those are just two sides of the same coin.
You may think you need to understand the math behind FEA to be good at using it in design. Software packages will try to convince you that you don’t need to know a thing since they will do everything for you. Of course, neither is true.
In this article, I will tackle both approaches. Also, I will show you what I think is the most important part of using FEA in practical design. I think I know what that might be… I’ve been doing FEA design for years now!
It seems that we started living in a world where everything is supposed to be “easy”.
I’m not sure if I like this, to be honest. Without a doubt, I’m not in search of unneeded challenges. But the fact that some things are difficult to obtain… makes them “worth it” at the same time. Well, at least when they are useful!
So let me start with this: FEA is difficult!
It’s difficult to gain skills that allow you to use FEA in design on a professional level. You will most likely spend months (or even years) gaining this skill. There is no “magic pill” to cheat that.
Sure, you can “speed this up” with courses, having a mentor, or a good environment at work that encourages learning. But, in the end, you will put a LOT of effort into this before it will work for you…
… and however bad this may seem, I’m happy that this is the case! The fact that FEA is difficult, means that FEA skills are also highly valued!
There are many reasons why FEA is difficult. I will try to make a short list, just so you know what’s in my head:
Of course, there are other reasons as well, but I think that the above are the biggest obstacles. At least those would be the biggest for me, and the engineers/companies I observed and trained.
I know that some (maybe even many…) software vendors will try to argue that I’m wrong here. That FEA is simple and their soft has this wonderful “wizard” to set up this or that analysis, and you don’t need to worry about all of the “difficult things”. Let me say at the very start that I strongly disagree with such an approach.
I don’t know if it is popular anymore, but some time ago FEA democratization was a rather big “movement”. The idea was (originally) to empower more and more engineers to use FEA consciously. I’m a huge fan of such actions and I support them by spreading practical FEA knowledge on my blog! I mention this, so there is no doubt that I care for engineers and their knowledge! It would be great if everyone would simply know more and “got better” at their craft.
However, (there is always a however, right?)… what turned out later, is a different thing altogether. The whole movement moved from “empowering engineers to understand more” to “dumbing things down so people can use the tools”. And this is where I’m super critical about the thing!
FEA is difficult, and the fact that it got integrated into a friendly CAD environment doesn’t change that. You still need to understand what you are doing! Sadly a lot of engineers see it as “since I can use the software, I know FEA”. Sadly, this is not the case, and I will develop this idea later.
I will close this introduction with a weird statement. I’m very happy that FEA is difficult! After all, this is what makes FEA worthwhile to pursue!
It cost me a LOT of effort to learn it. If FEA would be simple, everyone would do it on an expert level. It’s likely that my FEA design company would be in trouble if this would happen.
Plus, the fact that FEA is difficult, gives me the chance to train people in it as well. I have to admit I really enjoy this! If it would be simple, there wouldn’t be much interest I guess, so my inner-teacher would be constantly sad.
I also think, that gaining a difficult-to-obtain skill makes us feel proud, and rightly so! It’s simply good for you to develop and grow. Investing time in something that is difficult and sought after is a great idea in my opinion! Not only do you get to develop your career, but you also feel good about yourself in the process!
In short: FEA is difficult…
… but it’s a good thing!
I feel that I should be fair to many software vendors here (as I started quite negatively). FEA is difficult, sure! But this doesn’t mean that all software vendors are cheating you!
I admit, that there are a lot of things that you need to understand! Simply put software won’t “be an engineer” instead of you! If someone tries to convince you otherwise (“don’t you worry, just click here”), they are cheating you…
But there are also a lot of things you need to manually do in FEA. Things like meshing, sometimes loading, etc. Some of those things tend to be pretty time-consuming (and frustrating!). Software publishers work really hard to make those things easier to do. And when they say that “our soft makes this-or-that easier” – this is what they mostly mean.
So do not believe that after buying great software you will automatically become an expert. After all, a good sportscar doesn’t make you a good driver! But when you are a good driver, you really need a good car. Either to show off your skills or at least to feel better while driving it. So having good software is great, and there is definitely a reason for getting better software later on, etc.
On the same note, when you are just starting, you definitely don’t need it. In fact, it may even be intimidating and confusing. I feel that it’s better to start with simpler FEA programs. They may not have a lot of possibilities when it comes to super complex analysis… But when you don’t know how to use those possibilities anyway, you are not really missing anything!
I started learning FEA in Abaqus (one of the best solvers out there in many fields). It was a nightmare (everything seemed so complex and overloaded with options!). Then, we bought RFEM for my office, and things became so much easier! Sure, I couldn’t do various analyses in RFEM because it doesn’t have the capabilities. But I didn’t have the skills needed to do those analyses anyway, so it was not a problem!
In time, I developed my skills. I started noticing that the soft that I’m using misses options I would like to use. This is a good signal that it’s time to think about a better soft. When you already know which features are missing and you know why you need them. This took me several years (and a Ph.D.!) to get to that stage!
The cool thing is, that I bought Femap/Nastran with the money I’ve earned doing FEA in RFEM! So such an approach makes business sense as well since the simpler FEA packages tend to be way cheaper too!
I admit that now, with open-source FEA packages, you can start quite easily! You don’t even have to buy an FEA solver to start. And you can learn/upgrade as you go! The “natural” barriers of “buying expensive software” are melting in a sense. And this for sure is a good thing! Perhaps that is also a part of the democratization of the FEA movement. If it is – it’s a good one!
So far Nastran is sufficient for me as far as FEA goes. But to be honest, I hope that one day I will find things that I need that aren’t there… This will mean, that I will get new software. I really hope this will happen one day. It will show me that I’m still developing in some sense (or at least doing new things!).
And finally, there is one last thing I have to mention here. Being super good at using FEA software doesn’t make you an FEA expert! Yea… it’s great that you know all the “tricks” of your FEA package – this is a really useful skill. Just don’t think that this means that you know how to solve complex problems. It just helps you to do stuff quicker, not understand things more!
To sum this up:
FEA software plays a role of course, but it’s not critical I think!
The software won’t think for you nor make any decisions! It’s your role to understand what you are doing sufficiently to use to software properly. This means that it’s YOU who have to learn!
As a famous XXI philosopher said: “It doesn’t matter what’s under a hood, the only thing that matters is who’s behind the wheel!”
But also, you don’t have to start with the state-of-art FEA packages. Use something simple to use (nowadays, we even have easy to access open-source FEA).
When you will get to the point where you understand what features are missing in your soft and how you wish to use those features… it’s time to search for a better package!
And finally – being super good with FEA software doesn’t make you an FEA expert!
It would be awesome to know how FEA math works…
And I’m honest about it. If you would offer me a “magic trick” that just “gives this” to me, I would be willing to pay good money for that.
I think that as far as engineering is concerned, it is always better to know something than not to know something! FEA math is not an exception at all. I can only guess that it must feel awesome to know so much about this subject (obviously I don’t!). The problem is, that there is no “magic trick”, and you have to work really hard to “get it”! So, if this requires a lot of effort – is it really worth it?
There is one super cool thing in engineering (in general, and especially in FEA)! Whatever you become an expert in, will be useful! And with that in mind, being an expert in FEA math is a great career choice! I think that software companies are constantly searching for specialists that can do the “math magic”. Folks who can make the software faster, better, or more able in this way or another.
I can only guess how much the top folks in this space earn – think about the leverage! You do a single program better… and then you sell it: day, after day, after day! I’m pretty sure that if you would go to Ansys, Abaqus, or Nastran and told “I can make this algorithm of yours twice better because I’m so good at FEA math”… if that really is the case – you won’t be poor till the rest of your life!
This all means, that FEA math is a great field to work in! With good money in sight and some cool opportunities!
However, (you already know that there always is a “however”!)… will it help you in using FEA for actual design?
This is a tricky question. I think that people who say “absolutely yes” associate mathematics with understanding how FEA works. But to me, this is not an obvious relation.
Because for me, FEA math is something way more specific than that. This is why I will answer the question this way: “being able to write the equations FEA solver does, and then solve them yourself is useless in design“. And this is why I think this way:
Please note, that if you do not agree, I’m more than happy to discuss this with you. After all, if there are things that you think this skill gives – I would love to learn about them too! Just please, be aware of one thing…
First, let me tell you a short story:
In 7th grade, I went to a country-wide math competition. Yup, there was a point in my life I was good at it… I just grew out of it I guess 😛
It was a cool gig, because if you were in 7th or 8th grade and you “won” (well it was more like an exam, we didn’t compete against each other) then you didn’t have to take a high school exam! You could just pick a school you like, and you would be accepted. It was a pretty big deal, since I’m dyslexic, and my score on the “polish exam” would be pitiful, and I wanted to go to a good school!
I remember this event way better than the 8th grade one, where I succeed really. Mostly because I learned something in the 7th-grade edition, that really influenced my life (on a math competition… I know!).
There was one super complex problem to solve,= there. It required you to calculate what is the value of “a+b” in a set of 2 complex equations. I failed this one, although I was good at algebra. Mostly because I spend an hour trying to calculate what is the value of “a” or value of “b”. My teacher, after the competition, told me the thing that stays with me to this day:
To know the sum of two numbers… you don’t have to know those numbers!
All you need to know is their sum!
The idea was so obvious, and so painful at the same time. I was able to solve this task in a few minutes knowing this “secret”. All I had to do is to treat “a+b” as a variable, and compute its value! And this has A LOT to do with my attitude toward understanding FEA!
If you follow me on Linkedin you know, that I take every occasion, to discuss “FEA math” with whoever happens to say that you need to know the math to be able to do FEA design.
And I usually ask the same question: “why?”. Well, the more elaborate version of this question is “please, give me a specific example where knowing math really helped you in FEA design?”. Usually, I get answers that go along those lines (if you have more, please! leave them in the comments below – this is such an important topic!):
The problem with those is, that each of the first 3 while correct is not related to FEA math! It’s related to understanding how FEA works… and the fourth one is plain wrong!
So let’s start from the end. We live in a super complex world, and it’s ludicrous to assume that you can know everything about something! After all, I’m typing this on a computer that I do my FEA on. I couldn’t fix anything or build my own computer if my life depended on it!
Heck, some folks in “laptop service” don’t know how to fix this stuff. They just replace broken components with new ones! And they check what is broken by replacing stuff one by one to see what will “fix it”! So understanding the ins and outs of the tools we use is a luxury we don’t have anymore.
If you don’t agree with this, let me know in the comments, of course. But also wonder if you understand how international coal and oil transports are being made! Because this is also a part of the “computer usage” you need for FEA. Or even check if you know how the calculator you use works, while you still do your math on it!
But back to the first three from the list above. You already know that to know the sum of “a+b” you don’t need to know “a” and “b”… you just need to know the sum! And this is where the disagreement lies I think!
Many people feel that understanding math is needed to understand how FEA works… and I just don’t agree!
I’m comfortably fine to say, that I know how iterations work in nonlinear FEA. Also, I understand what arc-length methods do. I know how to calculate a model with nonlinear material and a lot of other things. And yet I have no idea how the equations governing those things look like (not to mention how to solve them!). Just as I have no idea how the graphic card in my computer works (and I still use it to display FEA outcomes).
I don’t have to know how the solver does matrix operations to understand that when I will use bad mesh, the outcomes will be bad! But… I absolutely need to know that when I will use bad mesh, the outcomes will be bad!
The thing is, that you can understand that after seeing a few examples of mesh convergence! It’s way quicker and more effective to learn about meshing this way. You could dig into the math for months to draw similar conclusions, but it would take way more effort. Ęven more! This assumes that you would actually manage to punch through all that math to draw those conclusions!
The same can be said for other things, like load incrementation, yielding, etc. You don’t need to know what happens under the “hood”… this happens completely independently of you!
All you need to know is which parameters control this process, what they mean, how they behave, and how to adjust them depending on the situation.
I admit, that learning all of the math (and then assimilating it into “general conclusions”) can lead to learning that, sure! But this is not the only way, and I think it’s way more time-consuming than alternatives! After all, I know those things, and I use all of that effectively in design… and yet I have no idea how “math under the hood” works!
But this is not the only problem I have with math in FEA… there is more!
A quick note:
You don’t really need to understand the mathematics, but you definitely need to know “how FEA works”, what is important, and how to decide which parameters to set, etc.
This is a fantastic problem, and I think it’s rarely discussed (I will talk about why later). You see… understanding FEA math doesn’t make you an expert FEA designer either!
Even assuming that knowing FEA math is indeed useful, it would be a very small help in a big problem. The problem of “how to ask a proper question that FEA will give me the answer that I need!”.
Even if you perfectly understand how the computer is calculating the answer you will receive… you still don’t know which question you SHOULD ASK!
Quite a few people I discussed this over the years said something along the lines of “math is what makes you an FEA design expert (as opposed to “just” an FEA user)”. And… I completely do not agree! And I have a very simple question here, that shows it. Which equation the solver uses, tell you how to support or load your model?
In essence, there are only two things you needed to know in FEA:
Sadly, the discussions I’m having about this usually revolve around mathematics “as the only way to enlightenment”. You know with statements like: “without understanding equations, you should not even try to use FEA”. And this is a “trap”. There is a LOT outside of FEA math, that you need to know in order to do an FEA design. And this is engineering knowledge!
“FEA math” doesn’t make you an expert… after all, knowing how computer calculates the answer doesn’t tell you a thing about what you should model and how!
So the “trap” is quite an elaborate thing. You’re saying that without understanding the math you can’t use FEA. And this gives students (and professors!) a feeling, that when they know the math… they can easily use FEA!
Sadly this is not the case…
I already mentioned it here briefly, but let’s expand on the idea.
Let’s use a car analogy when it comes to FEA. I would start with a few definitions to make it quick:
If you look at this from such an easy-to-understand perspective, some things become instantly clear!
If you are like me, you will easily understand what I mean. I’m a driver, but I have only a very rough idea of how the engine in my car works. Heck, when it comes to more technical details, I’m totally clueless!
And you know what – that’s totally fine!
But the opposite is true as well! The fact that someone knows how to build/repair a car engine, doesn’t make him a good driver! Hell, it doesn’t make him a driver at all!
This is a completely different skill set. And while one person can have both (driving skills, and engine design) it’s totally not required. You don’t need to know the traffic rules to design engines. Just as you don’t know how to build an engine to drive a car.
So why there is so much “hype” about FEA math?
Mostly, because there is an overlap. If you think about it, a driver should know when to switch gears while driving. That would be the equivalent of knowing how to set up a solver or read and understand outcomes.
One could argue, that in order to understand the “gravity” of the decision on which gear to drive in any given situation… you MUST know how the engine works!
And truth be told, if you would learn all there is about the clutch, gearbox, and engine you could really derive from that when to switch gears.
So at the Uni you will hear:
In order to use FEA, you have to know FEA math. After all you can’t drive if you don’t know how to shift gears!
But driving is a practical skill, right?
When you went to driving school, they briefly explained how the engine works. But there were not many details, because they simply told you when to switch gears, right? And then, you had to drive around with the teacher, that taught you the skills in the practical setting.
And you know what… this works! I would bet that most drivers have no idea how in detail all of the stuff under the hood works. But, they perfectly know when to switch gears!
The driving schools perfectly understand that you don’t need to know “a” and “b”. All you need is to know the “a+b” value. And since driving teachers are usually good drivers, they are happy to tell you what the “a+b” is! Without all that “dancing around” with checking what is the value of “a” and then “b”.
The same of course can be done with FEA. But it’s way harder to teach! Just as in driving it requires practical lessons and experienced teachers who did FEA for a long time. So instead, Universities often prefer to teach you FEA math. Simply because it’s easier for them (and oftentimes, they just don’t have the resources to do otherwise!).
This is mostly why we are taught FEA in this theoretical way. To be honest, I don’t think it will change quickly.
I mean, C’mon! It would be severely disappointing if you wouldn’t think about the automatic gearboxes when reading the previous part! After all, engineers usually search for problems and blind spots!
And this is when the thing makes a full circle. As we are back to the software pushed by some (definitely not all!) vendors. They may say “don’t you worry about all this driving – our car has this awesome automatic gearbox! It will do this for you!”
And you instantly think – well automatic gearbox may be a cool thing… but they don’t make me a driver, right?
Because shifting gears is not the only thing you need to know! Even the ability to shift them super fast, or automatically doesn’t make you a good driver (or driver at all!).
You also need to know the traffic rules! And no car (so far at least!) can learn those in your place!
In our car analogy, knowing the traffic rules would be the analogy of being a good engineer, and having good design skills. In the end, this is what makes you a good driver!
You most likely started learning to drive on some beat-up old car. Then, as you got better, you changed it for something better (maybe a few times already). And we all understand why!
It would be a bad idea to put a young inexperienced driver in a Ferrari. It’s so much easier to have an accident in such a case. I know for sure, that I will buy my kids some old beat-up cars first. Simply so they won’t get the feeling that they are the “race drivers” straight after the exams!
And this is the problem I have with some of the marketing for FEA packages. They are offered as a “wonder car” that drives itself… and this is just not the case! In every situation, you need to have good engineering skills (be a good driver), and there is no running away from this.
As I already quoted once:
It doesn’t matter what’s under a hood, the only thing that matters is who’s behind the wheel!
And this is what I meant when I wrote in the beginning that this is a coin with two sides. Many folks at Universities would tell you that you need to know all the FEA math. That without it you can’t even think about using FEA. Some vendors, on another hand, would tell you that you don’t need to know anything. They would suggest that the software “does this for you”.
The reality is, that they are wrong all the same. You need to know a LOT to use FEA... it’s just not the math! What you need are conclusions that can be derived from knowing math. You know, things like how to properly mesh a model, how to interpret outcomes, etc. And you also need to have a lot of design engineering skills. You simply need to know what and why you want to calculate!
And here we are at the end! I’m sitting on a train going to my Customer to help them with the ASME audit. I felt I really want to share this with you (while killing time on 7h train ride!).
It’s a very long article, but I hope I managed to clearly convey what I feel and believe. Also, if you feel the same, I wanted to help you a bit. I hope that I gave you some arguments and examples (and car analogies!) you could use as arguments. Of course, only if you will ever end up discussing this matter with someone.
And if you scrolled the whole thing just to read the conclusions, those are:
Phew… this is a long one! I’m really curious to learn what you think about this all – please, let me know in the comments below.
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