The chances and issues with 7 DoF (7th Degree of Freedom)

I assume that at this point, most structural engineers have heard about the mythic 7th Degree of Freedom (7 DoF). And for a good reason!

7th Degree of Freedom in your beam models allows you to analyze Lateral Torsional Buckling! It will also help you with torsional stability, and some other useful things. But this doesn’t mean it’s perfect, and today we will take a closer look at this!

I’m really allergic to “brand new things” that “simplify and automate” things for you. Not because they are bad as such, but because Companies usually promote those as the “final thing” that will solve all the problems for you.

Without a doubt, 7 DoF is pictured in this way, as a tool that finally will solve all the stability issues for you. You will never have to think about this again, and we will all live happy ever-after.

Of course, nothing is so simple – so let’s dive in.

The issue 7 DoF solves for you

Let’s start with the positives (heck, I can be an optimist for once!).

The main use of 7 DoF in static analysis is, that it can “see” Lateral Torsional Buckling of beams. This means, that your Linear Buckling Analysis can finally give you a critical load multiplier for bending!

This in turn should (at least in theory) greatly simplify calculation of the Critical Bending Moment M_cr. A huge victory, right?

It seems so big that we easily forget about other benefits. Models with 7 DoF nicely handle torsion. They can also predict torsional and lateral-torsional buckling. Sure, in “normal” structures those things don’t happen as often as Lateral Torsional Buckling, but they do. And they were also irritating to deal with.

In short, 7 DoF offers more than just an comfortable way of dealing with M_cr in your calculations. But on the other side, if it would somehow only addressed the M_cr issue, it would be plenty already!

This is why I will focus on M_cr here – to address the issues you will surely encounter when using 7 DoF. Issues that may greatly impact your design accuracy, while hidden in the “shiny (not so)new method”.

To explain what I mean, we need to take a look first at why we hate calculating M_cr in the first place!

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Why M_cr is so pesky to calculate?

I don’t want to turn this into a book on steel design, so I assume that you know why you have to calculate M_cr in the first place!

If you don’t, then please check out my Eurocode Steel Design: Part 1: Members course. We go quite in depth about this there.

Just so we can continue, let me only tell you this

If you design a beam in bending it is extremely likely it will be prone to Lateral Torsional Buckling.

This phenomenon can easily reduce the beam capacity due to bending by 50% or more. This is why you can’t simply ignore it.

Currently this design is based on the slenderness of your beam. The most commonly used method of calculating slenderness is based on M_cr value!

There are many methods of calculating M_cr. Some are more complex or time consuming than others. But all of them have something in common:

You need to know the boundary conditions of the beam in question!

This seems primitively simple at first. After all, you know the supports of your beam and/or connections to other elements. So you instantly think about the BC that would impact the bending moment diagram on your beam.

And this indeed is simple, I will give you that.

However, to calculate M_cr you need to know more than that!

Some of the things are intuitive and easy to come by. Like the place where you will apply load no your beam:

This is why your program ask you if the load is applied to the “top flange” or in the center – just in case you were wondering.

But very quickly we get to the murky areas! I admit that bending out of plane is still somewhat intuitive, even if we tend to go either “free” or “rigid” here:

The tricky part comes with the warping! This sneaks outside of the typical way we think about our beams, and it’s not super intuitive to “feel” what is the correct answer here:

This gets even worse, if you realize how HUGE impact those last two have! It stops being a question of, whether this is “simply” pinner or “simply” rigid. The rigidity of those “supports” play a huge role and may sway outcomes crazily!

This is exactly what we hate about M_cr

Our software in the module for beam dimensioning asks as for parameters (usually denote as k and k_w). And we don’t fully understand how to assign them.

This gets further complicated with the “torsional buckling length” we may have to provide.

The parameters we provide simply describe the boundary conditions I just discussed for the solver. The thing is, we would have to first know how those look like, to assign them reasonably.

Sure, you can be conservative here. But this will GREATLY underpredict capacity, leading to a huge overdesign!

I feel that nowadays we can’t really allow huge overdesign in our structures. This meant that most engineers struggled to provide those complex torsional and out of plane parameters. They do that in hope that somehow, they will land “close to the truth”.

7 DoF riding in on a white horse!

Seeing the above issue, it’s no surprise at all, that 7 DoF was so enthusiastically welcomed. After all, the promise is huge: You don’t have to deal with those freaking parameters any more!

But is this promise real?

Sadly, it’s not so simple. Without a doubt 7 DoF can help you with some things, but it’s just a method of solving beams in a bit more advance way. It saves you some time (i.e. in following equations etc.) but in the end, you still need to define the boundary conditions!

And this touches on the thing I really dislike about 7 DoF. Not the boundary conditions of course! To me it’s obvious, that it’s the engineers responsibility.

The beef I have with 7 DoF comes from how it is advertised. You can hear a lot of ads, telling you that 7 DoF will “fix this” for you! This is absolutely NOT TRUE!

The problem of deciding what Boundary Conditions your beams have remains just the same. It’s just “hidden” in software setup somewhere. And this means that you can miss it!

Don’t get me wrong 7 DoF is great, and can speed things up. But do not think for a second that it “solves” stability for you! It absolutely doesn’t!

It simply allows you to do some things more automatically. But of course, as with most of such things, proper model setup is key!

Let’s take a look at what you can (and should!) set up in your software. But what I feel is even more important, what traps wait for you in the 7 DoF kingdom!

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Complexities you have to understand!

I will start by saying that 7 DoF solvers aren’t magical. They are not “smart” and they can’t “design” stuff. Regardless how they are advertised!

But they do one thing decently (I don’t feel “good” fits here, as you will learn below!). They can see warping of beam elements. This means, that LBA will really see torsional effects, be it in compression or bending.

This is a lot in some sense, as it is really an elegant thing. But also, it’s “just that” and nothing more!

Let’s see what complexities and issues are in store for us here!

“Small” 7 DoF

This is something that can trip you, so I think we have to address this first. I will also refer to this later so pay attention here.

7 DoF solvers are tricky to set up, but not impossible to do. They simply have an inherited issue that is not very elegant. We will discuss this issue later, but for now let’s just say that it’s better to do a “single beam” with 7DoF than a whole structure!

And this is why, there are programs that can nicely calculate a single span beam using the 7 DoF. The most famous of those must be the free software LTBeamN delivered by CTICM.

This is a great tool, that allows you to precisely define boundary conditions of a single span beam, and calculate it using 7 DoF to receive critical bending moment.

What “small” 7 DoF does for you?

I feel that this is the most critical question we may ask here.

• Modeling a single beam with 7 DoF is more or less a glorified way of solving a Critical Bending Moment Equation!

• You need to input directly all Boundary Conditions! Admittedly, it is a bit easier, as you can simply provide “support rigidity” as oppose to “equivalent length coefficient” needed for the equation. So it may be seen a bit easier, or at least more intuitive. But however you wish to call it, you have to directly provide rigidity of each support, including the 7 DoF (warping) rigidity of support!

• When you will apply all the BC, it calculates M_cr for you! This looks great on the surface, but Equation for critical moment will do exactly the same thing! After all, you need to know all the inputs!

• 7 DoF has one huge advantage! Equation for M_cr address only the simple cases. Usually for “free” out-of-plane and warping supports at both ends. For more complex cases, you won’t be able to use the equation, as you won’t be able to obtain needed C₁ and C₂ parameters (they are undefined for more complex cases). Solving the beam with 7 DoF does not have this limitation – so complex cases can be calculated accurately!

And… that is about it!

If you have a simple case, “small” 7 DoF doesn’t do anything for you. If you have a complex case, it helps (since equation won’t have necessary parameters defined).

But regardless of the case, you still need to define everything! Just like you would do when you would be using an equation. So it’s not “simpler” in any way!

The issue with “small” 7 DoF

If world would include only single beams, I would not write this section. But it doesn’t… so here we are!

To show you what I mean, lets use an example that is a bit more familiar. Instead of using 7 DoF imagine we want to calculate the normal bending moment in a beam.

We have a nice program, that can calculate this for us, for a single beam. It works like this:

Of course, this is NOT the only option – it would be just a “default setting”. As we were defining our beam, program assumed (in our name) that it’s pinned-pinned. This of course gave the answer above.

But as a reasonable user, we can define the support rigidity. Depending on the choice we would get:

The first two are obvious. Support is either pinned or rigid. The third one seems a bit more “advanced”, as we defined the actual rigidity of the support. This is also why it’s not obvious where the maximal moment is. Depending on the support rigidity it can be in the span, or at the support.

So far so good right?

Here is the kicker! In a structure we tend to associate rigidity of beam support with rigidity of the connection – and this is WRONG!

Imagine a structure, that is a bit more complex. Our single beam software can’t really solve it – there are more beams after all. But I feel you know the answer.

As you can see, the connection is “infinitely rigid” in the top corners. This could suggest (wrongly) that if we would like to use our “single beam software” we should take the second one (rigid supports).

But of course this is not the case! While the connection is rigid, the columns we connected the beam to will rotate under bending… making the support semi-rigid.

To use our “single-beam software” we would have to somehow calculate that rigidity, and provide it to the software as an input. Not a simple thing… unless you model the whole structure!

Of course it’s the same (or even worse!) with 7 DoF.

This is the biggest issue with “small” 7 DoF implementation:

Sure it does everything… but on every step you need to calculate impossibly complicated rigidities to make it work!

And those rigidities do not only include connection rigidity – rigidity of the structure the beam is attached to will play a role as well.

Often rigidity of the structure plays a much bigger role than the connection rigidity itself!

The promise of “large” 7 DoF

Of course, this could potentially be solved. After all, all you need is to implement the 7 DoF to a solver that can solve 3D structures.

Then it will take into account the rigidity of the structure, so you only have to worry (or simplify) the connection rigidity.

This was of course done, but it doesn’t really work as you think it does!

Sure, you’re solver with 7 DoF most likely can solve big 3D structures. I mean, this is how it looks like:

It looks almost as if it works, right? Sadly, it doesn’t!

I don’t want to get into the details here (this would demand at least another blog post!), but it’s not that simple!

You see… you can’t transfer warping (the 7 DoF) to a perpendicular element! There is no proper Degree of Freedom to do that!

Think about it. In the previous example we had a bending in the horizontal beam… that was transfer on the column via rigid connection. And as a result in the column a bending moment appeared! This bending moment in the column can be on column strong or weak axis (depending on how the column is positioned).

And if we would transfer warping… what would be applied to the column? Well… nothing! We would need additional Degrees of Freedom to map this, and we don’t have those!

Consequence of this is severe!

Every time you use a “large” 7 DoF, you need to support each end of each member… to tell the solver what is the “rigidity” of warping constraint.

But this is NOT only the connection rigidity – it also has to take into account the rigidity of the entire structure… as the warping will simply “disappear” from your model!

This doesn’t mean of course, that “large” 7DoF has no uses. It do! In a continous straight beam it can see the “warping continuity” over the support for instance. So it’s a very neat tool to solve beams that have more than a single span.

But as soon as you connect to elements that are at an angle to your beam… you’re in trouble. This is a serious limitation.

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… but even if “large” 7 DoF would work… it still wouldn’t!

This is not where the story ends!

Imagine that you have a solver that has those additional DoFs. Warping is indeed transferred to perpendicular elements! You can calculate Lateral Torsional Buckling directly in your model!

The thing is, that this is still done in Linear Buckling Analysis. If you have a model with 100 beams (it’s a small one!), you would need to calculate HUNDREDS of forms to hope that you have a form for each element.

And then you will reach a problem… how to assign which form is for which element!

I’m not saying this is impossible to do, but it would be very difficult. This means that if someone will tell you that “their software” does stability design automatically for you – ask them how they map the eigenforms on the elements to decide which form is representing failure of which element.

And you know what I suspect? That they will tell you this is NOT how they are doing it!

Instead, they have an automated tool, that analyze each single beam separately. In short, even through you’ve calculated your whole model, those forms will not be used!

Instead the program will assign artificial Boundary Conditions to each beam (doing exactly what “small” 7 DoF is doing with all its faults!), and calculate each beam this way.

This is a nice solution, as you don’t have to calculate hundreds if not thousands of LBA forms. But this means that someone has to define the boundary conditions for each beam separately to be used in this process! How do you think, who would that be?

So yes, the process is automated, but you still have to go through each beam one by one, and mark how this “small” single beam 7 DoF model has to be supported for this automation to work! There is only a small difference between that, and simply doing this with buckling lengths!

And you will end up with the same issue as always:

How to assess how rigid the supports are, especially due to warping. And how to include the rigidity of the rest of the structure into that “support rigidity”.

Is this all a lie?

I hope that I shoed you how complicated things are. But there is a huge last thing I want to say:

7 DoF is NOT a lie! This is not a TRICK! It can tremendously help you better design your structures and will be applicable to MOST structures.

As soon as you start to have elements that have more than a single span, some uncertain bracings or simply are subject to torsional stability (or you’re not sure if they are) – 7DoF will save you big time.

I actually have an online course that teaches you how to do that! I discuss there in detail how to approach this, how to set up your calculations, what to pay attention to and how to benefit from this great tool. You can check out the course here: Eurocode Steel Design – Part 1: Members.

Regardless if you join the course to learn the details or not, please remember:

Solvers with 7 DoF are not “magical”.

They will not solve the stability problems for you automatically.

Sure, they will help you a lot in cases you could not solve otherwise
(there are many of those, that were just guessed prior to 7 DoF!)

But in the end, you still have to understand everything (and even more than before!) to use this tool effectively.

The same issues you always had in buckling design will be there still!
You simply have a tool now that can help you solve those better… at a cost of some extra complexity!

I really hope you enjoyed this! See you around!