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9 minutes read
31 August 2017

Buckling length in sway and non-sway structures

9 minutes read

One of the few difficult things about buckling lengths is, that you need to know if your structure is a “sway” structure or a “non-sway” structure. Commonly the simple check is made with displacements, but unfortunately, it may lead to serious mistakes!

Why should you care about sway and non-sway system

Firstly, I want to show you why the “type” of a system is crucial in buckling design. In the previous post about buckling length, I have shown a chart that is very popular. It is often used in textbooks when buckling length is described:

This set only works in non-sway structures, and is it so easy to forget that! To make it simple if you have a frame like this, everything is fine:

Buckling length in a non-sway frame

Unfortunately, when you remove the horizontal support from the top, problems start to happen. The column is still rigidly connected at the bottom, with a hinge at the top. This suggests that the buckling length coefficient should be 0.7 as in the picture above. But the top connection is made to… nothing really. The entire top (with the beam) will move in a horizontal direction! This will result in a deformation like this:

Even though the type of the connection did not change (it is still rigid at one side and hinged at the other) the buckling length did! And the change is huge from 0.7H to 2H.

This is the reason why you want to be certain what type of structure you are dealing with!

Bracings to the rescue!

The “sway” and “non-sway” problem is one of the reasons why we like to use bracings in our structures. A simple “X” bracing will prohibit the movement of the top part, meaning that we will have a “non-sway” system. But is it always so?

Let’s think a second about what bracings really do. In a “typical X” bracing one part of the bracing elongates under tension (while the other part buckles under compression). This elongation of the bracing part means, that a normal force appears in the rod. The more we try to “sway” our structure, the longer the part in tension must become. This also means that the normal force will be higher and higher. Hence, our bracing rod “fights” against this horizontal movement:

Bracing in a frame forcing a smaller buckling length

However, to block the movement a certain force in the rod must be generated. This means that the bracing is effective but after a while. Furthermore, the smaller the cross-section of the bracing, the more horizontal movement will be possible.

At this stage, I have to ask: where is the “allowable” limit of this translation before the frame is a “sway” frame anyway? Whenever I approach a problem it is hard to solve I try to imagine 2 extremes.

Whenever I approach a problem it is hard to solve I try to imagine 2 extremes. In this case, the diameter of the rod can be 100mm (one extreme). In such a case there will be no movement at all and a structure is clearly a “non-sway” structure. On another extreme, the rod can be 0.1mm in diameter. Before it will elongate enough to carry any substantial force the structure will collapse. In such a case it is clearly a “sway” structure.

We can think about it in a simplified way: a bracing is nothing more than elastic support at the top of the frame:

Of course, such bracing is not the only way to get elastic support at the top. You can for instance also think about a short hall building with very rigid end walls and horizontal longitudinal trusses along the roof. Then those trusses support the horizontal movement of each truss in an elastic manner. This is shown in a simplified way below:

End walls influence on the sway frame

It’s time to ask: How to check if my structure is a “sway” structure. Unfortunately, this is not an easy question.

Classical way of checking if you have a “sway” structure

In a lot of books, a “classical” method of checking if you have a “sway” structure is described. According to this method, all you have to do is to check the deformations of the structure with and without the bracings. Then you compare those deformations. If a structure without the bracings deforms 5 times more than the one with the bracings, this is a “non-sway” structure.

Deformations of a sway and non-sway frame

The reasoning behind this method is simple. If the deformations with the bracings are much smaller, then the bracings are “effective”. If so, we can assume that the top won’t move, and the structure won’t “sway”.

But as always with such methods, a reasonable question should be asked… why 5 times? Why not 10 or 2 times? And in the end, is the 5 times a correct value?

Unfortunately, it seems that it isn’t, and I will show it to you with our frame as an example.

To sway, or not to sway – that is the question!

Let’s take a look at what we know so far. You already know that we can treat bracings as elastic support with stiffness K. Also if there are no bracings (K=0) the buckling coefficient in our case is 2.0. On another side, if the bracings are indestructible (K = infinity) the buckling coefficient is 0.7. What we don’t know is, what happens between those two states. It is easy to show this on a simple drawing:

Sway and non sway frames and buckling length

Luckily for us, LBA can provide us with all the answers that we need. I will simply make a model of such frame, and I will check the critical load multiplier for different stiffness of this elastic support. This way we can draw conclusions about how bracing stiffness influences the buckling length of the columns!

To do the check I have created a model that would resemble a wall bracing in a hall building. The columns are from I sections (but with a “strong axis” in the other direction. The horizontal element is a round pipe (acting as a part of a bracing system). I ignored implementing roof rafters etc. I simply assumed that they are correct, and I have made support in the “Y” direction at the top. There is also elastic support in the top right corner of the frame in the “X” direction:

Buckling length in a semi sway frame (LBA)

It’s high time to do the analysis!

Sway me more…

First, two states are “easy”. Firstly I did an analysis without this elastic support (K = 0 kN/m). Horizontal deformation from a test load I applied was 38mm. The critical load multiplier was 0.3.

The second “easy” case was with the infinitely rigid “X” support. In such a case, horizontal deflection is of course 0mm. The critical load multiplier is 1.68.

Of course, I also did some intermediate results. You can see some of them below:

Buckling lengths in sway and non sway frames

With several outcomes, I did some charts, showing how things change. As a reference point, I used the ratio of horizontal displacement between unbraced system and the system with bracings (B/A from one of the pictures above). On such a scale 5 means that it is a non-sway system according to the classical approach.

As you can see critical force changes “fluidly” between a sway system (that would be 1 on the horizontal axis) and the non-sway system (theoretically a 5, basically on the right side of the chart). Changes in critical buckling force of course mean, that the buckling length coefficient is changing as well, and this change can be seen below:

However both of the values may seem a bit “abstract”, so I made a simple example, and below you can see how the capacity ratio changes. This is for the purely compressed column in my frame, and in different systems, this will be different of course. But at least it is a good reference point:

There are no outcomes for elements with a deformation ratio smaller than 5. The compressive force was then higher than the critical buckling load and it is non-designable. Also, note that the vertical axis starts from 0.8, not from 0.

Conclusions and thoughts

This is getting a bit long, but let’s try to summarize everything you have read a bit:

Thoughts on “sway” and “non-sway” systems:

  • It is very important to remember that the popular buckling length coefficients from textbooks are for non-sway structures only!
  • If the structure is in a sway mode, things are much worse when it comes to stability.
  • Sometimes it is hard to estimate if your structure is in a sway mode or not.
  • If you use bracings that seem “normal” most likely you are fine. Usually, deformation ratio for those is around 10. You can see on the charts above that for my frame that is easily in the “comfort zone”.
  • However, sometimes you will use very slender bracings, or count on other elements (like trusses in a perpendicular direction). In such cases answer to a sway and non-sway question isn’t simple. It is best to model the structure and do the LBA analysis to check what is going on. If in doubt aim for high deformation ratio. I would say that 10 seems reasonable. Just be aware that “classical” value is 5.
  • If you were using the ratio of 5 (the classical one) to differentiate sway and non-sway systems this may or may not be problematic. In my example, the difference in capacity ratio was 35% between a perfect “non-sway” system and system with deformation ratio of 5. The difference of 35% is big, but not catastrophic most likely. You will have to judge yourself if this is significant in your projects. Just be aware that it can be far better or worse! Capacity due to buckling depend on a lot of factors and it is impossible to make general conclusions.
  • Remember that there are always 2 sides of the coin. If you were calculating non-sway systems your outcome may be off in the “dangerous” direction. However, if you assumed a sway system, most likely you will get an answer that is a bit conservative. Using LBA allows checking how things are in your case. Even if LBA is not always the perfect choice for buckling length estimation, it should get the job done. You may simply spend some time to watch all those eigenforms before you find what you are searching for!
  • The charts above are for my example. They are not general in nature by far! I just wanted to show you how this works, and how things can be analyzed!

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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 (20)

Hamza B. - 2020-07-01 14:30:36

Hi Lukas,

I think the effort that was put into this article in order to clarify the never ending questions about sway vs non-sway or braced vs non-braced is very much appreciated.
However I think that this article adds more to the confusion (props to my colleague who detected it), let me explain it based on how it is explained in the background document of ECCS 119 (chapter 4.3.2):

Sway/Non-sway/Braced/Unbraced are four independent classifications of frames. This article makes it seem that as soon as a frame is braced it is inherently considered as non-sway which is not true acc. to EC3. A frame can be perfectly braced (reducing the horizontal deformations by 80%) and still be considered as a sway structure (Alpha_cr Criteria for braced => 80% reduction in lateral flexibility thanks to the bracing system
--> Criteria for sway => Alpha_cr<10 (elastic), 15 (plastic)

Consider this as feedback and not as an effort to bash your article down, there should be more likeminded engineers like you who dare to clarify problems we face, keep doing you :)


Łukasz Skotny Ph.D. - 2020-07-04 20:03:20

Hey Hamza!

I will start from the end: how could I not treat your honest comment as "bashing" - I really appreciate the time and thought you put into writing this all, and I'm really happy that in our small tribe are people like you willing to spend the time and effort to make things better! This is highly appreciated!

So now to the matter at hand!

First of all, with regret I never read the ECCS 119 document - I guess I have some reading to catch up to.

I must confess I never did my Ph.D. in the sway frames (shell buckling for life!) but I also was never a fan of the "sway" criterion from the EC. I mean - think about it this way - alpha_cr is a multiplier of the applied load that causes buckling. Imagine the frame that is a "classical sway system" - there are some columns and you have a total of 60kN that yields an alpha_cr of 3.333. Then, you reduce that load to "only 10kN". This is the same load distribution, the same frame, the same everything... suddenly the load multiplier is 20 (3.333*6) simply because the load you multi[ply is lower... so what? Does this mean that the buckling length of the columns should change? Well... for me, it shouldn't.

I understand what you mean, but at the same time I'm not a big fan of all "structural divisions" - as you saw in the post :) But if I would have to put a "call" on sway/non-sway dilemma, I would definitely go with the "non-sway" when the buckling length is "classically good" (so both top and bottom node can be treated as supports) then "sway" as the "classically bad" case where the top node is not horizontal support at all, with a huge "in-between no man's land!"

I'm super curious what you think about it :)

All the best!

Timur - 2020-05-04 01:56:30

Hello Lucasz. Thank you for your posts. I am sure they helps all engineers all over the worlds. For example i am from Kazakhstan and your posts give me not only ready answers, but they open new directions for development ,but the most important thing- show what I had not thought about before. and it can be important thing in our business.
Lucasz, in our country sway and non-sway strusctures are called free and non free structures. In our building norms and rules (analog EN1993,but this is not him) there is no exact criterion by which it is possible to determine which structures sway or non sway.There are schemes but there are not numerical values. It is generally accepted that systems with vertical bracing are non-sway.
From this year in our country canceled former building norms and rules and introduce EN1993. And now many engineers interpret a rule αcr >10 as criterion for determining which will be frame - sway or not sway. An example of this was in comment above. It seems so consider not only in Kazakhstan. But i am personally think it is not right.The rule αcr> 10 is only needed to determine whether second-order effects should be taken into account.Personally i am think the easiest and correctly way is to look from the elastic buckling mode of a structure and if there is support from the top move in horizontal direction, то frame sway.
Correct me, if i mistake.

Łukasz Skotny Ph.D. - 2020-05-04 14:17:28

Hey Timur!

First of all, thank you for your kind words! I'm really glad that you find my blog useful!

As to your question, yea... I wouldn't base the "sway/non-sway" decision on the critical load multiplier.

After all, you can have a simple cantilever that has a critical multiplier higher than 10... and still, it's buckling length is 2H... since it's a freaking cantilever!

First of all, I really don't like the division for sway and non-sway structures. As with everything in life, there are no clear borders (man I need to make training on structural steel one day...). There are "a bit" sway structures, and those "quite" sway as well! This all depends on the bracing system and the geometrical arrangement of elements.

While if you look and stability forms from LBA one could raise an argument that if first models FOR A GIVEN ELEMENT are local than global it's a non-sway structure, but I find that awkward.

I would much rather do an LBA, and based on the outcome calculate the critical load in the beams/columns I'm not sure about. Then I simply know what the critical force is, and I can use it in design. This must be the easiest way to move around the subject... and during the procedure, I don't have to worry about the sway/non-sway dilemma... it sorts itself within the LBA analysis :)

Hope this helps!

CubeIn BD - 2019-07-25 12:00:07

Its a really good website. It will be very helpful for peoples.

Łukasz Skotny Ph.D. - 2019-07-25 17:04:06

Thanks :)

Moustafa - 2019-05-03 17:19:42

Hi mr lukasaz
Very nice article indeed .
I would be grateful if you can elaborate more on how to get the effective length after carrying out buckiling analysis

Łukasz Skotny Ph.D. - 2019-05-03 20:14:17

Hey Moustafa!

Thank you for your kind comment.

The short answer is, buckling analysis produces critical load multiplier. This multiplied with the force in the element gives you the critical force. Using this you can back-calculate the effective length.

The problem is, this works in the simple cases, but in the complex ones... well things get a bit more difficult with torsional buckling, changes in cross-section, different load patterns and all that. But to be honest the entire thing is like a 2-day training so it's impossible to discuss this "briefly". But I hope that the short version will be at least a bit useful for you!

All the best

Dan - 2019-04-10 16:57:20

Hello Lukasz,

We share the same Eurocodes with some differencies in national annexies. Really interesting read, pretty much fan of your blog. I have a few things to point out while trying really keep it simple as possible:

1) Never heard of rule of 5time larger deformation and I researched many sources around in this topic. Not saying that it doesnt make sense! if the deformation is small enough - theres no sway right :).

2) On the other hand I heard and read (from reputable sources - books from SPRINGER publisher or ECCS or some papers) many times about if αcr < 10 you can neglect second order effects and therefore structure is not sway. Its basically mentioned in EC3. Can you comment on that?

3) Can you please comment on frame with concrete columns? How to approach such problem for lets say frame with really tall (25m) concrete columns? I am aware of kmod < 10% to neglect 2nd order effects.What about sway? How would you approach it?

Thank you and have a nice day!


Łukasz Skotny Ph.D. - 2019-04-11 05:42:04

Hey Dan!

1. Funny, maybe that is a Polish thing? prof. Rykaluk book on stability (sadly in Polish) mentioned this, and I was taught that at Uni even before that book was published by someone else, so it's an "old thing" here :)

2. I never did a lot of studies on this topic, but I'm reluctant to interpret it this way. My thinking is: if a load multiplier is > 10 you still can use it to calculate critical force and extract buckling length from that. I doubt it would be much shorter than in the case of the swaying structure. Not to mention you can have 2 structures one 9.999 and the other 10.001... are they really that different? I think this means that if you have multiplier>10 you can ignore II order analysis (but not the fact that this is a sway structure). In such a case there would be merit there... but it's so easy to use II order analysis now (it's just a button) that checking if you can ignore II order take longer than using it!

3. I don't really design concrete stuff, and I never had to think about such problems. But stability is stability. I don't see how you can "ignore" the sway reality if you have a sway frame - I never saw compelling arguments, but also you are a first person to ever raise doubts about that. I would say that if you have a sway frame it's a sway frame... regardless if the columns are from wood, concrete or steel. The material is not at the "core" of what "sway" means in my opinion, but be aware that there might be some specific concrete rules in some Eurocodes that simplify stuff in some cases, and I'm just not aware of them... If that is the case, let me know for sure :)

I'm really glad that you like my blog!

All the best, and see you around!

sara - 2018-05-29 08:41:51

thanks and appreciate your efforts

Łukasz Skotny Ph.D. - 2018-05-29 10:00:48

Hey Sara!

I'm glad that you like it!

All the best

Igor - 2018-05-10 09:36:58

Thank you very much, Lukasz. it is a very interesting and useful article. I am from Russia and we usually use the same approach, but i have never faced case about 5 times in Russian books. may be it is my mistakes, but could you please share any European books about this topic? thank you in advance

Łukasz Skotny Ph.D. - 2018-05-10 11:53:03

Hey Igor!

I'm really happy that you like it! It's great to hear!
Sadly, the only book that comes to my mind would be prof. Rykaluk "Stateczność Konstrukcji Stalowych". I wrote "sadly" as it is in Polish :/

I've never read Wood's works in original - I have no idea if he himself has mentioned it.

I'm afraid I won't help you much more there.

Have a great day!

Carmen - 2017-09-21 18:32:20

Nice and simple

Łukasz Skotny Ph.D. - 2017-09-21 22:05:04

Hey, Carmen!

I'm glad that you like it :)

All the best

Tim Clark - 2017-09-19 20:53:10

Dear Lukasz, I see that you have a clear point to make but the explanation is extremely long and the point about 5x and 35% is not clear. The charts are also unclear where the variables of capacity and deformation have no units and the deformation attempts to represent braced and unbraced conditions. I am also in the business of helping people learn about design issues of all kinds, including the visuals and keep imagining how it is for someone to understand my message without knowing what I know..... not easy, so you are both generous and courageous to offer this presentation so openly. I look forward to seeing your additional materials - you are welcome to check me out on LinkedIn, best wishes, Tim Clark M.ASCE, SEI, RIBA, FRAS.

Łukasz Skotny Ph.D. - 2017-09-20 07:11:28

Hey, Tim!

Thank you so much for the honest feedback. I do my best to explain stuff, I'm sorry to hear that it is not clear for you. I will do my best to clarify:

1. In many books, it is written that if the deformation of a structure without its bracings is 5 times higher than the deformation of the same structure but with its bracings it is a "non-sway" structure. This is a classical approach.

2. There can be no units on the charts. Capacity is a "clear value", sometimes given as a percentage (as 1 = 100%). Horizontal axis do not have deformation but a ratio of deformation (deformations of the structure without bracing divided by deformation of the structure with bracings) - as such it has no units as well. This means that 5 on the horizontal axis means "classical sway structure".

3. That 35 % is the difference between capacity in a structure that was "classically" adopted as a "non-sway" structure, and the same structure but calculated accurately. On the last chart you can see that at deformation ratio 20 (let's call this the "ultimate non-sway structure") the design ration is 1.0, but if the deformation ration is 5 (so "classically" this should be calculated as a "non-sway" structure) capacity ratio is 135%... so a 35% difference.

I write rather long posts, I agree. This is simply as I try to be very clear and precise (but also in some magical way when I write something it usually ends up as 1100 words no matter the circumstances...). I hope you will find a blog that has a style that better suits you - I haven't seen one, but if I stumble upon it I will definitely let you know!

Thank you for kind words - sharing knowledge is fun for me. I admit that at first, it was a bit stressful, but now I simply enjoy it ;)

All the best

Dan - 2020-05-31 18:16:17

How does the presence of equally spaced purlin on the compression flange along the lenght affect this deformation shown on the portal. Will this bring the ratio away from unity?
Likewise can blockwall along the building lenght be relied on to provide lateral restraints for buckling of the column.

Łukasz Skotny Ph.D. - 2020-05-31 19:20:07

Hey Dan!

Well... purlins alone won't help the roof rafter in the "in-plane" deformations. With the wall, it would be perpendicular to the shown movement, so not very effective. But of course, that would depend on the wall construction etc.

All the best!


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