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22 August 2017

# Buckling length basics

I’m preparing materials for another training about stability and I figured I have never written anything about frame stability on the blog! It’s high time, so I decided to make a small roundup of methods for calculating buckling length in a frame system.

## Buckling length – do I need it?

Let’s start with the obvious: you need the buckling length for the stability design of the steel elements. This statement was accurate for many years, now it should more or less sound like:

You can use buckling length of elements to design them for stability.

Why the change? Eurocode actually introduced several different methods of design, and some of them don’t include a definition of buckling length. You either calculate the critical elastic load multiplier and plastic capacity. Thanks to those you can calculate slenderness without establishing the buckling length. Of course, you can also use sets of imperfections and a nonlinear approach to design, but this is sometimes very time-consuming.

In this post, I will assume that you want to calculate something in a more “classical” way. In such a way establishing buckling lengths is a way to go! Note that most software solutions still use a “classical” approach to design – this does not mean you need to do hand calculations!

## Buckling length coefficient

I think that you remember from the first course in static design that there are some “typical” buckling lengths. The value that everybody memorized and now gladly use.

To be honest I’m not very fond of those, mostly because using them may lead to problems as I will show you in this post. But I want to show them here, simply so you have the reference.

This all boils down to a “buckling coefficient” that you should multiply with the column length. This coefficient depends on the supports/boundary conditions at the end of the column. Basically, this looks like this:

Unfortunately, what is easy to forget is, that this works in non-sway systems. This means that if the top end can move in a horizontal direction then the outcomes aren’t perfect anymore!

## What buckling length really is?

This is a very good question. Somehow in all textbooks buckling length is given as an element length multiplied with buckling length coefficient. This is, of course, correct, but if you think about it, every distance is a certain length and a correct coefficient… isn’t it?

This brings us closer to the truth:

Buckling length is a length of a half-sine wave your element deforms into when buckling.

With this in mind, let’s look once more at the “classical” values:

This seems like a trivial thing doesn’t it? I mean technically this is the same outcome. However, there is a difference in perspective here. Imagine a system, where a very high, slender beam stands on the column, and in the perpendicular direction only small beams are present. It would look like this:

If we would think about coefficients for buckling length, we would most likely assume that the column length L should be to the “big” beam axis and that the coefficient is 1.0 (since this is a hinged connection). But in situations like that, it doesn’t really happen. If you would consider for a second how the column will deform, it would be easy to realize, that it has 2 different buckling lengths, depending on the buckling plane.

In the plane of the “big beam,” the column will form an arc starting under the bottom flange, while in a perpendicular direction the arc will be longer reaching the “smaller” beams. Finally, it would look like this:

I’m afraid that such situations are difficult to describe in a “coefficient” way. Sure, if you know the buckling length you can easily derive the coefficient from it… but how to learn what the coefficient is without doing so?

## Why should you care?

I’m pretty convinced, that wherever you look, buckling length is always explained with the coefficients. This is how people teach engineering I think. Sure there must be exceptions, but in general, this is how it looks like.

When you see it only this way, it is easy to make fatal mistakes. The best example of such a mistake is a humble steel frame:

The first thing students think is “rigid-hinge = 0.7”. Of course, this is completely not the case! In fact, it is far worse than that! The reality is that such a frame deforms in this way:

If deformation of the column looks similar to the cantilever this is not an accident 🙂

The above example is very simple, and if we would play with the way where hinges and rigid connections are in this frame this might get an awful lot more complicated. But I will get back to it in another post!

## But I don’t know how my structure will deform!

I agree that knowing how things will deform while buckling is a nice trick to pull off. But this is not so easy, especially in the more complex system (that has more than 3 beams in them…). This is why Linear Bifurcation Analysis (LBA) is getting more and more popular in structural steel design. This is an FEA algorithm that simply shows you how the structure will look like after failure due to buckling.

Not only this but it also helps you with getting critical force from it. This means that you don’t have to measure structure shape with a ruler on your screen… but I admit I did that on a few occasions 😀

If you are interested in learning FEA, definitely take a look at my free FEA course below this post:

## Last few words

Stability is a wonderful thing! At first, it is a bit intimidating I agree, but it gets easier as you go. If you are interested in subjects like this one, let me know. I will try to include more of them in the future.

Also, since I’m launching a Q&A column on the blog, be sure to send me any questions you may have at enter[email protected]. You can also leave them in the comments 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!

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### Join the discussion

Sunny - 2023-05-07 14:30:33

Hi. Your content was enlightening. Thanks. Id like to ask how one can determine the appropriate length of a hollow pipe for conducting buckling analysis as it relates to pipe thickness.

Łukasz Skotny Ph.D. - 2023-05-08 09:51:52

This may be very problematic, depending on the case. This is why my answer will be vague.

1. If your pipe is "relatively thick" to be in cross-section class 1-3, then at least things are decent. You can simply treat it as a normal beam in buckling. In those cases, it would not differ from I section all that much (apart from having different cross-section characteristics of course).

2. If you have a "relatively thin" tube as a cross-section in class 4, then you are in the world of hurt... you just entered a "shell bucking" region, and there isn't much I can tell you in a single comment outside of this is *very* complicated, and that you will most likely need a nonlinear FEA to solve this decently (or a lot of hand calculations if you will find the good formulas - just be careful with those, it's actually super hard to describe shell buckling with equations, and a LOT of limitations for each equation exist).

All the best!
Ł

Dele - 2021-05-17 12:36:24

Quite enlightening, in the most simplistic manner.
God bless you richly.

Łukasz Skotny Ph.D. - 2021-05-30 16:23:19

I'm glad that you like it Dele :)

Michal - 2021-03-24 08:52:17

Hello Lukasz,

Your article was very well written and in combination with your article regarding sway and non sway structures gives valuable information. Forgive my ignorance on the subject (im not a civil engineer) but i only ask because my program calculates very high buckling factors (as much as 10) on its default settings (sway enabled). What conditions cause a member to have such high buckling factors? Am i wrong to assume that buckling ratios cant be much greater than 2? Thanks for any feedback, Sincerly

Michal

Łukasz Skotny Ph.D. - 2021-03-26 20:54:40

Hey Mate!

Well... you can have "very high" buckling coefficients... and my "VERY" I mean 1000, not 7. While they don't happen in the structures I design, they definitely can happen. Think about a brick standing tall in the hydraulic press. This is a compressed element, that has a length - technically it can "buckle".. but the buckling load would be insanely high (much higher of course than the crushing load!). This is why we are not used to bricks buckling under compression of course.

But... you could analyze such a brick and to an LBA analysis of the problem. And if you apply the crushing load, I think that you could easily get a ridiculously high critical load multiplier.

And something you could think about at the end - it's not the buckling ratio (critical load multiplier) that "counts". Imagine a beam that buckles under 200kN of compressive load... and you load it with 100kN of compression. The critical load multiplier is 2 because 2x100 = 200kN, which is the buckling load. Easy right? But if you would load the same beam with 10kN of compression, the critical load multiplier would be 20 since 20x10 = 200kN which is the critical load. And if you would apply only 1kN of compression, the multiplier would be 200!

So multiplier alone rarely is meaningful. While in a "simple truss" the bracings that are "well designed" usually have the LBA outcome under the decisive load of around 2-3, this is not a "general rule" - if a shell I would be designing would display such a critical multiplier I would be very worried!

Hope this helps!
Ł

Milan Stojanovic - 2020-07-06 12:37:32

Hello Łukasz,

I read this post and the other one about "sway" and "non way"... I am working on precast concrete industrial hall. Columns are fixed in footings (both directions) and all connections between beams and columns are hinged. I have steel bracings in longitudinal direction (in exterior longitudinal walls of hall). When I calculate effective length factors for some columns in direction about weaker axis, with and without bracings (and with variation of bracing stiffness), I almost always get number that is larger than 2. Can you elaborate on those cases when we have effective length factors >2. Is it indicator that my columns are too slender?

Łukasz Skotny Ph.D. - 2020-07-06 19:38:47

Hey Milan!

Thank you for writing :) To fully answer your question I would need to know more about the loads in your load case that you use in calculations. After all, you most likely have different force values in different columns (and you use it to calculate the critical force right?). So when you do LBA, a "single" LBA case reflects buckling of a given column (the one that deforms) and the load from that column would be considered to calculate critical force. I hope you are following such approach :)

There are some things in the loading, that can screw up the buckling length a bit, but with hinges everywhere and only a single rigid connection at a foundation that would be a "classical" cantilever (disregarding the bracings). So I would assume you should obtain a "nice 2.0" as an outcome.

Perhaps if I would know more about your structure I would be able to help you more, but for now this is what I can write about it :)

All the best!
Ł

gabriel - 2020-06-18 23:09:25

Your blog is awesome. Keep up with the great work.
One question that may not be 100% related to the subject.
In your example, you show a beam with a pinned connection on top of a column. Is it possible to restrain rotation and make a fixed connection on beams running on top of the columns?
In other words, when a beam run onto the side of a column, it's clear that if we fix the web and flanges of the beam, it will be a fixed connection, however when the beam runs on top of the column I'm not sure how to make a fixed connection.
Thanks

Łukasz Skotny Ph.D. - 2020-06-19 06:14:09

Thank you for the kind words Gabriel!

and this: https://enterfea.com/how-to-calculate-rigidity-of-semi-rigid-connection-of-steel-members/

And construction wise, if you would stiffen the web of the beam with vertical stiffeners, with relatively thick end plates (and proportionally thick bottom flange of the beam) you can make this into a rigid connection. I've marked only 2 axial bolts in my example, so this is why I treated it as a hinge (while in fact it can carry a slight moment as well, due to the compression of the flange + tension in bolts pair of forces. But the rigidity of such connection is low anyway :)

Hope this helps!
Ł

Kannan - 2019-07-01 09:03:08

Is the Effective Buckling Length is applicable for liquid equipment only or for gas carrying equipment also?

Łukasz Skotny Ph.D. - 2019-07-02 10:39:30

I'm not sure what do you mean. Buckling length is a parameter for the compressed elements, I'm not sure where "liquid equipment" or "gas equipment" comes in play in your question...

Mangesh - 2017-08-29 17:15:30

Very informative thanks

Łukasz Skotny Ph.D. - 2017-08-29 20:24:50

Hey, Mangesh

I'm happy that you liked the post!
It is nice to hear that you enjoyed it!

All the best
Ł

Shirish Phatak - 2017-08-28 02:41:43

Please write more on such topics of Engineering and common mistakes occurring

Łukasz Skotny Ph.D. - 2017-08-28 03:22:14

Hi,

I will do my best to include few more topics like this one :)

All the best
Ł

Shirish Phatak - 2017-08-28 02:38:50

Very nice article and very informative in simple words

Łukasz Skotny Ph.D. - 2017-08-28 03:22:11

Hey Shirish!

I,m very happy that you like it :)

All the best
Ł

Krzysiek - 2017-08-26 17:51:21

Thank you for this text. It is an excellent example of beam buckling and the factor of length.

Łukasz Skotny Ph.D. - 2017-08-27 08:41:19

Hey, Krzysiek!

Thank you for kind words - I'm glad that you like the post :)

Cheers
Ł

Francisco Castillo M - 2017-08-24 13:34:02

That's interesting how easy explanation was. Thanks for sharing.

Łukasz Skotny Ph.D. - 2017-08-24 17:35:40

Hey, Francisco!

Thank you for kind words, I'm glad that you liked the post!

All the best
Ł