(function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); })(window,document,'script','dataLayer','GTM-5M6SH59');
9 minutes read
19 October 2017

FEA results verification with hand calculations: stability

9 minutes read

Last time I described 5 steps to FEA results verification. Among those steps were FEA results verification with hand calculations. I figured it makes sense to expand on this idea. Especially since Luke asked me about something on this topic via email.

Luke asks:

  • How is the eigenvalue calculated, can this be done longhand?
  • Can LBA analysis be calculated by a long hand method, I understand the software will show the failure area and the areas under most stress/strain/compression etc but can a value be derived without the software?

Let’s go a bit deeper into the topic of FEA results verification with hand calculations. To discuss this (and to answer Luke question), we need to consider what are we doing and why.

Can I solve my problem with hand calculations?

This is a great question! You need to ask it before you even start to model your problem. The reason for this is simple:

If you can solve a problem with hand calculations easily: don’t do FEA calculations!

Simply because FEA will take you longer than hand calculations in most cases. There are of course instances when such action makes sense. Maybe you want to check your hand calculations? Or you’re about to design hundreds of similar elements (in such case FEA + scripts can be a huge help!).

However, as a rule of thumb, it is better to solve problems with hand calculations when it is easy. It will simply take less time!

This seems obvious, but there is a second part to it, that is a bit harder to spot:

The fact that you do FEA calculations most likely mean that easy solution of the problem by hand does not exist!

This sounds a bit pessimistic, but no worries – this is not the end of the post!

Using FEA helps when there is no solution

Of course, not all problems can be solved with simple equations. In fact, most practical problems do not have a closed mathematical solution. This is why we use FEA after all right?

I would say there are 2 possibilities:

  • Problem you are trying to solve does not have a mathematical solution at all
  • The task has a mathematical solution but it is either inaccurate or takes a lot of effort to use (or both!).

In such situations, FEA shines! Simply because it allows you to calculate something that is otherwise impossible (or at least hard) to calculate. But this also means that you won’t be able to easily verify the outcomes you got. Well, at least not all the outcomes and not at one go!

Knowing all that let’s think about what we actually CAN do 🙂

Simplifying the problem to “estimate” the outcome

You already know that most likely you cannot perform an FEA results verification with hand calculations directly. But no one says you cannot estimate the answers. Most problems can be more or less solved without FEA. After all, there was engineering before the use of FEA was common!

It’s quite probable that you won’t be able to get a correct answer. But at least you will be able to more or less guess the range within which this answer should be contained. This way you will see if what you got from FEA make any sense at all! With some experience, you will actually know what outcomes to expect from FEA based on the simplified calculations.

This is especially true in stress design. But since Luke asks about stability I will go in that direction in this post. Just bear in mind that stability is much more complex. This means that most problems cannot be simplified in order to get an estimated answer! I will get back to stress analysis results verification on some other occasion!

Practical estimation: shell compression!

Since Luke asks about a specific problem let’s follow this path. Just be aware that the comments here fit different analyses as well.

In stability, there are several closed solutions like Euler’s column or Timoshenko shell. However, those closed solutions are few and far between. This makes FEA results verification a bit more tricky when it comes to LBA.

Let’s take a look at a problem that actually has a solution – and why it can serve us well!

CASE 1: Math solution is known

This is what I would call benchmarking. Imagine there is a problem you wish to analyze. If you know a closed math solution there is no reason to do FEA. Unless you want to check if you really get a proper answer using your favorite software/procedure/mesh size and type.

One of such cases can easily be a Timoshenko problem (uniformly compressed shell). Critical capacity (perfect elastic buckling load) for such shell is:

FEA results verification with hand calculations

It’s easy to calculate that for a steel (E=210 GPa) shell that is 1mm thick and has a 500mm radius you would get critical stress equal to 254MPa.

Simple right? This means that you don’t really need FEA to get this answer. But you can try to solve this problem with FEA! This way you will see if your method of solving such problems actually works!

You can model the shell and load it as if you would do for any problem similar to this. Simply use your favorite mesh settings etc. In the end, your model can look like this:

Compressed shell: FEA results verification with hand calculations

When calculated you would get:

Now you can compare the FEA outcome to the correct one and wonder if your model is satisfying. Maybe the mesh size is not ideal, or you should pick a different element type? Checking between a known solution and an FEA model allows you to search for answers to such questions.

Without a doubt, the answer I got is pretty close to the correct answer. This means that I may be confident that the method works. In the very worse case, I need to tweak things here and there.

This is why it makes sense to compare FEA outcomes to a known solution. You can verify if the selected method is actually correct in solving this and similar problems!

CASE 2: Math solution is unknown

This is, of course, a more difficult thing. We already know that Linear Buckling can correctly estimate the linear critical load of a shell. We can assume that if we model a similar problem we will get a proper answer:

Shell compression: FEA results verification with hand calculations

As you can see not a lot has changed in our problem. I simply changed support from uniform to 4 discrete supports. But this made the whole thing a lot more complicated.

There is no mathematical solution to this problem, and frankly, you won’t be able to do a lot here.

All I can do is hope that based on the previous example LBA algorithm gave me a correct answer. However, checking this particular case by hand is not really possible directly. There is simply no simplified way to calculate this value by hand!

All you could do here is to calculate the shell capacity. This has nothing to do with the LBA eigenvalue of this problem, however. But if we would calculate the shell’s capacity we could compare the outcome to a capacity given by our favorite design code. You can expect that the capacity obtained from FEA will be 30-50% higher than the one from the code (i.e. EN 1993-1-6). You can read more about it here.

Just note that the above check reference a highly nonlinear solution to the problem. Unfortunately, LBA simply cannot be verified this way. This is because there is no code procedure that would allow calculating the linear critical elastic load of such a shell.

When we talk about stress calculations things are simpler and a lot more can be verified. But this is a topic for another post 🙂

To answer the question

It is possible to calculate some eigenvalues by hand (just as I did in the first example). Some cases are already solved, but unfortunately, those are usually the simple ones.

In shells that would be a Timoshenko problem. In steel frames, Wood’s or Horne’s methods allow calculating the critical load for a frame or even a column in a more complicated frame system. Those methods usually give only estimates (more on some issues here), but still, they would be great for the verification of FEA outcomes.

This means that searching for simplified solutions to similar problems makes sense. Those will greatly help you to verify the outcomes of your LBA analysis. Just don’t spend too much time on the search. Not a lot of problems have solutions, so you may be searching for something that doesn’t yet exist!

However, more complicated stability problems are very difficult to verify. Also, there is no hand calculation method that solves any given problem. Usually, all you can do is benchmark your method with a problem with a known solution. You can also try analogies to “similar” cases that have known solutions.

Of course, when solving FEA you can do a lot of “along the way” checks like mesh convergence studies etc. But this has nothing to do with hand calculations!

To sum this up:

  • It is possible to verify some FEA calculations by hand
  • With stability (LBA) this is a bit more complicated as it is difficult to simplify the model
  • In order to verify the outcomes you would have to know the simplified mathematical solution. Sadly, not a lot of those are known
  • If mathematical solution is unknown, it is best to rely on benchmarking, convergence studies and experience when verifying stability analysis outcomes.

Want to learn more?

First of all, if you have a question related to FEA or structural steel feel free to leave it in the comments below, or send it via Linkedin. This way I will be able to answer your question, just as I did with the one I got from Luke!

Learning about results verification is a great step toward using FEA. If you like FEA, you can learn some useful things in my special free FEA course for my subscribers. You can get it below.

Author: Łukasz Skotny Ph.D.

I have over 10 years of practical FEA experience (I'm running my own Engineering Consultancy), and I've been an academic teacher for a decade. Here, I gladly share my engineering knowledge through courses, and on the blog!

Read more

10 Lessons I’ve Learned in 10 Years!

Get Essential FEA Course for Free!

Comments (9)

Giorgio - 2020-10-14 09:41:29

The number of cases that have an analytical close-form solution is extremely small, because, in real-world, structures are much more complex than beams, shells, etc..
Those cases where an exact solution is available, are also used to validate the code.

FEM models are by definition an approximation, so are their results, and most analysts (including myself) are usually just happy to get a value within 10 per cent from the hypothetical correct one.

One has to differentiate between the correct solution, and a plausible solution, which involves a lot of engineering judgement, which in most cases, unfortunately, comes from experience.
That is why I always try to have a physical sample of the part I am going to simulate, load it, deform it, and - if it is allowed- break it, and run a FEM on the CAD model of the same, to see how far reality and simulation s are set apart.
When this is not possible, my suggestion is to run a physical test on a scaled-down model of the real thing you need to evaluate.
Even a simplified model without many of the features that will make its manufacturing complex and costy will help you understand a lot about the behaviour of the structure.

Łukasz Skotny Ph.D. - 2020-10-16 10:11:36


I completely understand your approach, and of course, I do agree that close-formed solutions are few and far between.

It's awesome that you can test each specimen you investigate - we did several such tests so far (sometimes Customers did the tests beforehand, so we could calibrate based on those), but as you mentioned often it's engineering judgement that comes from experience!

Thank you for sharing your experience!

Alpha - 2018-08-31 09:25:02

Very practical approach for an an Engineer.


Łukasz Skotny Ph.D. - 2018-08-31 09:44:53


I'm really glad that you like it!

All the best

Larry - 2017-11-09 19:20:30

Here are a few handbook references for known structural stability solutions:
Bloom, 2001, Handbook of Thin Plate Buckling and Postbuckling, Chapman & Hall/CRC, NY, NY.
Samuelson and Eggwertz, 1992, Shell Stability Handbook, Taylor & Francis, Elsevier Science
Publishers, LTD, England.
Column Research Committee of Japan, 1971, Handbook of Structural Stability, Corona Publishing
Company, LTD, Tokyo, Japan. (This is a very extensive reference)
NACA, 1957, Handbook of Structural Stability, Part I Buckling of Flat Plates (NACA-TN-3781)
Part II Buckling of Composite Elements (NACA TN 3782)
Part III Buckling of Curved Plates and Shells (NACA TN 3783)
Part IV Failure of Plates and Composite Elements (NACA-TN-3784)
Part V Compressive Strength of Flat Stiffened Panels (NACA-TN-3785)
Part VI Strength of Stiffened Curved Plates and Shells (NACA TN 3786)

Ziggy - 2017-10-25 02:13:51

Hi Lukasz

You introduced αcr into calculations without explaining where is it coming from and why the value is 5.19


Łukasz Skotny Ph.D. - 2017-10-25 09:55:04

Hey Ziggy

αcr is a critical load multiplier - this is literally what you get from the LBA analysis. It's simply the outcome of the numerical procedure.

You can read more on LBA here: https://enterfea.com/linear-buckling-explained/
Load multipliers are explained in this post: https://enterfea.com/load-multiplier-basics/

Thank you for writing - if you have any more questions feel free to ask!

All the best

Mathias - 2017-10-20 15:42:44

Great stuff!

Łukasz Skotny Ph.D. - 2017-10-24 21:29:13

Hey Mathias!

Thank you - I'm glad that you like it :)

All the best


Sign up to newsletter

and get Free FEA Course!