Web under local loads – Hand calculations
It’s very hard to calculate the capacity of small sections of any structure under concentrated loads. Luckily EN 1993-1-5 gives us a decent solution for webs under concentrated loads!11 January 2021
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.
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!
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!
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?
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!
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:
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!
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