Once again welcome back to the drawing board with me Steven Lloyd, you may remember a little while ago we talked about the design of waling beam with combined effects, we had a look at von mises stresses, we had a think about plastic and elastic properties of the beam.
Now what determines whether you consider a beam to be plastic or elastic, well that’s it's classification and today we will be looking at the classification of steel sections. This only applies to structural steel; stainless steel and aluminium have different material properties so the criteria involved is slightly different there. So this applies to structural steel, grades s235, s355 and so on, the kind that are covered in Eurocode part 3.
First thing we do is take a comparison of the length of out outstands or the depth of internal sections and compare that to its thickness. In the case of a circular hollow section we are comparing the depth to the thickness of the circle. Once we have got that ratio we compare it and make sure it is less than a certain number of these epsilons. Now epsilons 4s235 is 1 and the value of epsilon reduces the higher the yield strength of the material is. The reason for that becomes clear when you start talking about plastic hinges forming. For now we will just take that as granted. If you have a very low ratio of depth over thickness or outstand over thickness then you're a plastic or a class 1 section. If you have a slightly less thickness or slightly larger diameter for instance you'll gradually step up this ladder from plastic to compact, semi compact to slender in British Standard Terms. In Eurocode speak it is Class 1, Class 2, Class 3 and Class 4 - not quite so exciting terminology there.
So let's talk about plastic hinges. A class 1 section, class 2 sections, class 3 and class 4 and the stress distributions at failure are slightly different each time. Now a Class 1 section, or plastic section, you can start bending that beam and you can bend it and it will form a hinge and you can keep bending it, and keep bending it, and keep bending it and that hinge will maintain a certain level of strength even as you continue to cause more rotation. If you can get to the end of the world with this rotation that's a plastic section, it will continue to remain plastic all the way around. Just short of that you've got a Class 2, you'd also use plastic properties with this. Now once you get to that point of yield and your stress distribution looks like that it will form a plastic hinge and it will allow itself to rotate but you've got a limited amount of rotation before that area buckles. This is why we are comparing thicknesses to lengths, it’s all about the local buckling because what happens of a Class 3 section is that it can get to its elastic limit a perhaps it can get a little bit further.
If your section is just on the boundary between Class 2 and Class 3 you might be able to get quite a bit of plastic distribution somewhere between here and here. If you're right on the border between Class 3 and Class 4 you'll just barely be able to get to your full elastic distribution before you end up buckling something. So if you have a very thin, very long outstand flange, by the time you've got that stress at the top and bottom edge your flanges are wanting to buckle. If you've got a very tall, very slender web, by the time you have started bending it your web wants to collapse. Class 4 section, it tends to sit in a little world of its own. A Class 4 section is the underdog, it can't even make its full elastic distribution of stress before parts of it start buckling. It's my favourite type of section because you have to apply special rules to it extra special rules are always fun when you're doing design. I want to look at our Class 4 section which is our 1220 by 16mm SuperTube. On a steel prop you would normally get a lot of compression, you've got the earth of either side and it wants to compress that tube. The self weight of it also wants it to sag so you've got a little bit of bending. Bit of bending, bit of compression, you get stress distribution looking like this.
Now over here I only talking about bending and the criteria are slightly different if you're talking about compression as they are to bending which makes sense if you're thinking about local buckling. So let's have a look at this area here. Under compression, anything that is not close to a junction or a joint in the ‘I’ section for instance, will potentially buckle. If you're right in this corner here you're in an area of strength. Anything that has two intersecting pieces of steel is unlikely to buckle.
As soon as you get out into this no mans land, on the very long slender web, that's the part that's going to buckle, that is the part that is not going to offer you the strength that you hope it will. Or you use this little channel section here as another example; at the corners you've probably got some strength and you won't be buckling, at these areas here where there is not so much stability, that's the area you're likely to buckle. When you're bending it's a different area because if you've got tension on one side and compression on the other you're pulling on a section you can't buckle something in tension so there is no point ignoring that area there in tension zone, that's not going to buckle as it's not in compression. That area that isn't going to buckle will probably help support the area next door to it that will. So you can understand why there is slightly different criteria on bending, slightly different criteria on compression, you wouldn't have any criteria if it was a tension member because you don't get tension buckling. Well you do but not in this case.
So what do we do with the Class 4 section? This is what a started talking about. You're using effective area so you don't consider the whole stress distribution over the whole area, what you say to yourself is "Well which are the parts that are likely to buckle?" If you look at, for instance, Eurocode 3 part 1 part 5, it gives you these instructions on plated structures and that is what determines these areas and how much of the steel you consider to be a loss cause. In the case of our Class 4 circular hollow sections, the very large tubes, we consider the area most in compression, this bit at the top - that the bit that we think will buckle first so if it is going to buckle, let’s just write it off and pretend it isn't there. So what you end up is designing around a section that although it's a circular you actually treat it as this horseshoe shape. Now I've got a little note down here telling you to go have a look at Groundforce's technical note 7, it is a great little document explaining our ideas behind the Class 4 circular hollow section. It comes from a little bit of research from somebody else about elliptical hollow sections and it explains why we treat our sections the way that we do. An affected area, generally speaking, is an area times by a reduction factor and this explains what reduction factor we use and where we get that from. Same with your effective section modulus for bending. It's the section modulus, usually an elastic section modulus multiplied by a reduction factor. I have not really explained in detail but I really like you to have a look at this technical note 7 as that does go into a bit more detail. Hopefully you'll find something interesting and you'll learn a little bit about why we classify sections and why we use elastic or plastic section properties.