## Buckling length in sway and non-sway structures

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 “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:

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 horizontal direction! This will result is a deformation like this:

Even though the type of the connection did not change (it is still rigid at one side and hinge 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 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 “fight” against this horizontal movement:

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 an elastic support at the top of the frame:

Of course, such a bracing is not the only way to get an 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:

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 structure without the bracings deforms 5 times more than the one with the bracings, this is a “non-sway” structure.

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 an 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:

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 influence 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 “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 a support in “Y” direction at the top. There is also an elastic support in the top right corner of the frame in “X” direction:

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. Critical load multiplier was 0.3.

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

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

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 system with bracings (B/A from one of the pictures above). In such 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 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 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 deformation ratio smaller than 5. The compressive force was then higher than 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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1. Tim Clark September 19, 2017 at 8:53 pm - Reply

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 September 20, 2017 at 7:11 am - Reply

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
Ł

2. Carmen September 21, 2017 at 6:32 pm - Reply

Nice and simple

• Łukasz Skotny September 21, 2017 at 10:05 pm - Reply

Hey, Carmen!

I’m glad that you like it 🙂

All the best
Ł

3. Igor May 10, 2018 at 9:36 am - Reply

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 May 10, 2018 at 11:53 am - Reply

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.

Have a great day!
Ł

4. sara May 29, 2018 at 8:41 am - Reply