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15 minutes read
5 February 2024

Practical use of Linear Buckling

15 minutes read

I will start by saying that buckling analysis can be very demanding. I guess this is why Linear Buckling is so popular!

Linear Buckling Analysis (LBA) is pretty simple. It allows you to calculate the critical load multiplier, which shows how many times your load would have to increase for the model to fail due to buckling.

This may seem like a perfect solution to a complex problem. Especially since Linear Buckling Analysis is straightforward to set up.

It’s as easy to run as a classical linear stress analysis (LA) – FEA will never get easier than this.

This article will be divided into two chapters:

  • Part 1: Problems with LBA – why it’s not as great as people think
  • Part 2: Practical use of LBA – what you can use this analysis for, and how to reasonably use it in design to assess the capacity of some of your models.

Problems with Linear Buckling

Every time I start talking about LBA, I feel I have to list issues connected with this analysis. Simply put, those are important to understand, and you can’t afford not to know this!

Without a doubt, a huge benefit of LBA is how simple it is to set up, and how fast it runs.

All you have to do is request the analysis and decide how many first “buckling shapes” you want to see. The rest is more or less automatic.

Furthermore, LBA runs only a bit longer than a typical linear analysis. It’s WAY faster than nonlinear FEA, so it’s a huge benefit.

In the end, you get outcomes that look like this:

I mean, what not to love, right?

It’s common knowledge, that the FEA outcomes always look smart, and I guess LBA is no exception. Showing this to someone indicates, that you did consider stability, and it seems that the problem is solved.

Sadly, this is not entirely the case!

You see, Linear Buckling is… well… linear!

This means, that while you analyze your buckling problem (which is a highly nonlinear thing), you will have to ignore some things.

Unfortunately, ignoring those nonlinearities will lead to overestimation of capacity of your model… so it’s NOT an error on the safe side!

Let’s briefly take a look at what LBA ignores:

Linear Geometry:

Ignoring nonlinearity here is a problem in most analyses, although it’s mostly visible in plate and shell structures. The buckling of those elements is highly nonlinear, and the failure mode (and capacity) predicted by LBA may be seriously overestimated!

Linear Material:

This is a tricky one. LBA simply analyzes an elastic buckling, which by definition doesn’t use nonlinear material. A good observation would be, that you also should do a linear stress analysis (LA) that checks “material capacity”.

The thing is, that as the model yields in reality, it loses rigidity, and this may lead to collapse failure. I admit that I would call this “plastic buckling”, although this is not a very precise term.

Neither LA nor LBA analysis could take that into account, and in many practically designed plate/shell structures, this is a serious issue (and a common failure mode).

Imperfections:

This is easier to explain, mostly because it’s likely that you never attempted to check what is the impact of imperfections on LBA outcomes.

In short… it’s very small, and sometimes they even increase the capacity instead of seriously decreasing it!

You could technically to an LBA analysis with imperfections: just add them to the model and run LBA! The problem is, that since LBA operates with linear geometry no second-order effects would take place… so adding imperfections is pointless.

But in reality (especially in shell structures!) imperfections can (and should!) reduce the capacity of the real structures by a LOT (like 2-3 times!). This is not something you can easily ignore… but LBA will not analyze this for you – it just can’t!

What this all means for practical use of LBA?

All of the above mean, that LBA will overestimate the capacity of your model.

The biggest issue is, that the degree of this overestimation varies, just as susectibility to various nonlinearities vary from model to model.

If your model is very slender (“thin and long”), it most likely won’t yield before it buckles (so ignoring material nonlinearity is not an issue)… but it may be way more sensitive to imperfections and nonlinear geometry effects.

However, more “bulky” models, while may not care all that much about imperfections (especially when there are eccentricities in the models already) may really struggle with lack of nonlinear material in collapse analysis (not to use the “plastic buckling” term here).

This all means, that you know that LBA overestimates the outcome… but you don’t know by how much!

Practical use of LBA in design

There are two things I want to discuss there:

  • How to use LBA to “check” your model
  • How to use LBA to assess capacity, and when this is even possible.

Let me start with the part that most likely is obvious, just to move it out of the way.

Using LBA to “check” your model:

This is a very well-used approach I think. Simply put, LBA computes quickly, and setup is minimal… plus it checks how your model “behaves”.

Sure, it may not predict the failure correctly, but you can run LBA to check if your model will compute.

The outcomes, as accurate as they are, will show you if the model is reasonably supported (doesn’t fly “to space”), if all things are properly connected (and nothing buckles insanely, because some nodes were not merged), etc.

Since it usually takes only a few minutes to calculate LBA, it’s simply a perfect tool to verify your model before you run more “serious” analyses.

The above shows you, what the LBA outcome of a poor model looks like.

I didn’t “connect” the mesh nodes between the two plates. This is a mistake that would not cause any warnings (although I admit there is a tool I could use to find this without running any analysis in Femap).

Still, the idea is, that you may have a lot of mistakes, that will not trigger warnings. Maybe as above, some elements did not “connect”, or you forgot to apply a portion of the load, or maybe one of several supports is not there.

In most of the above, LBA outcomes you will get will look “weird” – and this is how you can quickly find such mistakes!

It’s a useful thing, and we are running an LBA analysis of our models for this purpose almost every time we finish modeling something.

How to estimate the capacity based on LBA?

If you read my blog for a while I guess I can surprise you here… you actually could estimate the capacity with LBA in very specific conditions.

But of course, I have to start with a disclaimer!

There are no General Rules for LBA estimation of capacity! Be VERY careful!

From time to time, I’m doing a corporate training sessions. So many times before I’ve heard something along such lines:

“In different training we were told, that if LBA multiplier is above XYZ all is good”

Funny enough this “XYZ” value ranges anywhere from 1.1 to 20 (!!!) depending on the company.

This is NOT how this works! There is no simple limit that fits all models!

But you can try to develop value, for the typical models that you are solving!

Let’s see how!

You already know, that LBA misses a few important ingredients, that lead to an overestimated outcome.

The amount of overestimation depends on your model. Most things have a different sensitivity to nonlinear geometry, nonlinear material, or imperfections.

This is why you can have a “general rule” that would simply encompass all cases!

However, if you are usually solving similar models, you can check how sensitive those models are!

Sure, this will not be the “general LBA number” we all would love to have, but it will be your LBA number, you will be able to use in the design of your typical models!

But before I discuss this further… please be careful when you are sharing the value (share the method instead!). The value you will establish will be only for models that are similar to yours… it’s not a general guideline by any means!

How do you establish an acceptable LBA limit for your models?

This is pretty simple if you think about it. The whole thing takes only a few steps:

  • Firstly, you will need a way to establish the capacity of your model. Sadly, LBA will not help you here (at first). In the case of my office, we are using Nonlinear FEA to design things. It doesn’t matter how you do it, the important part is, that you are certain what the capacity of the model is.
  • Perform LBA analysis, to know the LBA outcome for your model. As I already wrote, the benefit is, that you can do this at the beginning and check if your model is ok at the same time. Also, I actually add LBA outcomes to my reports – many people expect to see them, but it’s also my way of showing why LBA alone is not a good idea!
  • Check what is the correlation between the “real capacity” and the LBA outcome for your model. You can note this somewhere, just to have the catalog in handy. I have to admit, I have never done such a catalog, but I feel that including a snapshot from LBA analysis and “real capacity” (with associated failure mode) would be a good idea – this way you will be able to assess if your “typical models” are similar enough to treat them as “similar” for this procedure.
  • Just perform this 10-20 times and draw conclusions. Just be careful of the scatter. My experience is, that this makes sense when the LBA and capacity relation is fairly consistent. So if you have a big discrepancy (whatever “big” may mean to you), maybe there are like two sub-groups of your typical models you should treat independently?

How does this work in practice?

I can show this in a simple example!

As you may know, my company does (among other things) a lot of pressure vessel design. While I’m not running statistics, I would say that we are in the “small hundreds” of designed vessels (very high tens at minimum).

This means, that I’ve seen a LOT of pressure vessels solved with Nonlinear FEA, as well as their LBA outcomes (that we include in our reports anyway, as I wrote).

I admit, that I never made any notes, nor analyzed this statistically. But I know by now, that if the vacuum load case of LBA shows a circumferential buckling of the vessel like the one below, and the multiplier is below 2.4-2.7… things will be bad.

I could go through the vessels that I’ve designed, and do some statistics on outcomes. I’m pretty certain there would be some rules to “discover there”.

You know, things like “if the diameter is bigger than the distance between heads, the LBA multiplier should be closer to 2.6”, or “if the thickness to diameter ratio is bigger than XYZ, we should expect a 2.4 LBA multiplier to work”, etc.

To be honest with you, I never felt the need to check those extra rules. Truth is, I will do nonlinear FEA anyway, so I don’t need to predict this exactly.

However, knowing this already saved me some work a few times. If I run an LBA check of my model and see an LBA multiplier of 2.0, I simply know there is very little chance this will actually work – so I will instantly reach out to the Customer to discuss potential solutions (without wasting time for nonlinear analysis).

And to me, that is the benefit.

But I know that some of the companies that I’ve trained are using this to much higher limits. If the time demands on your work are very high, then you may simply not have time to perform nonlinear analysis before you make the design call.

And then, having the relation between LBA and real capacity known for your typical models is a HUGE asset. You simply can predict the capacity of the system with much higher accuracy.

One last note…

This article was long in the making, but I must admit I was afraid to publish it. I know full well, that this approach can be greatly misused!

After all, you could imagine someone “figuring out” their LBA number for a given “typical problem”, and calling this typical problem “a benchmark”. Then such a person could try to convince you that this always works, and that “benchmark” proves this! And this is definitely not how this approach should be used!

I hope that this article puts enough light on the matter, to make you skeptical about such “magical general rules”, that I know do not exist.

You may also notice, that even the example I used is a very “safe” one. If you ever designed pressure vessels per ASME VIII div. 2 then you know, that ASME actually provides an LBA limit for circumferential stability, and it’s 2.5 according to that code.

Funny enough ASME VIII div. 2 is the only code I know that does this!

The funny thing is, that I already found quite a few vessels, that had an LBA higher than 2.5, but still would not pass the more accurate requirements of European codes when it comes to circumferential buckling.

This is why above, I gave a range of 2.4-2.7 as potentially problematic. If I had to make a recommendation (based on my experience) I would aim at an LBA of 2.7 – 2.8.

Still, I have to say, that the difference between 2.7 – 2.8 and 2.5 given in ASME is rather small, and this is definitely not an ASME critique (more like a curious note really). Plus ASME in the recent edition changed a few things about stability, so things are developing for sure.

Why I didn’t use a different example?

I hope that this part will give you something to think about, and maybe save you from some troubles.

Let’s face it, if you are in the pressure vessel design space, you may feel that I went the “easy way”.

After all, instead of circumferential stability, so nicely defined in ASME VIII div. 2 I could give you the LBA limit for so many other stability failures seen in pressure vessels.

It may seem, that I could easily use more complex example like:

  • Stability of the vessel domed ends, when the case is more complex:
  • The local buckling capacity failure, when the load is not uniform (which doesn’t happen all that often to be fair):
  • Or even uniform shells, but subjected to slightly more complex load patterns different than the “uniform circumferential compression and nothing else”:

The reason why I haven’t provided you with those is simple:

Those are too complex to use this way with Linear Buckling!

I mean, each of such cases is somewhat different, and will differently react to nonlinearities and imperfections. And this means that the relation between the real capacity and LBA outcome will greatly vary.

To the point, that while you could establish such a relation for a particular model, you couldn’t use it for a similar case.

Simply, other cases are not “sufficiently similar” to use this approach, unless you are making almost the same thing twice!

Summary

As you can see, LBA has some serious limitations.

In short, it will ignore several influences:

  • Nonlinear Geometry
  • Nonlinear Material
  • Imperfections

On the other hand, this doesn’t mean LBA is “useless”. You can use it in many ways, and considering the topic of this article, my two favorites are:

  • Checking if your model is ok. You will see if the load was applied correctly if the supports of the model are working as intended, and if all elements are connected as they should. I think you start to value this only after you wait for 20 hours for a nonlinear outcome just to learn, that something was wrong in the model!
  • You can use LBA to assess the capacity of your typical models. However, this is not as straightforward as you may think. First of all, you need to be designing “similar” models (that are similarly sensitive to things LBA ignores). Then you have to design quite a few of those typical models (LBA is NOT a design!), and compare the obtained outcomes with LBA multipliers for those structures. Finally, you may be able to establish a relation between the capacity of your typical models, and the outcome of LBA. This may aid you in quick estimates of model capacity, or even in design if you will be very time-pressed.

That would be it for this article, I hope you enjoyed it. And if you would like to learn more about FEA from me, definitely sign up for my free lecture using the box below!

See you around!

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!

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Comments (2)

Mohamed Salih - 2024-02-13 05:27:18

Thanks Lukasz, for a very clear insight into linear vs nonlinear buckling analysis and a well articulated summary.
Looking forward for more pearls of wisdom from you.
Best regards

Reply
Łukasz Skotny Ph.D. - 2024-02-13 08:24:36

Thanks Mohamed

I'm glad that you like this article :)

All the best!
Ł

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