Columns & Diagonal Braces: Column & Brace Slenderness 1

In this video, I'd like to talk about compression members, basically columns, diagonal braces, or anything that is in compression. And if it's slender enough, it could buckle. That's the theme of this presentation. So I'm not going to do the math. I just need to have this equation in my face so I remember what to talk about. It's Euler's Buckley equation. It says the critical buckling load, or the load that would make a column buckle, is equal to pi squared over L squared multiplied by the stiffness. So E is the modulus of elasticity of the material. I is the moment of inertia of the profile or the cross-section. And EI is known as stiffness. E is the stiffness of the material. I is the stiffness of the shape. And L is the unbraced length of the membrane compression. So when we look at this chart specifically, it says there is something called a slenderness ratio. And the slenderness ratio dictates how much allowable compressive stress or load you can put on a member. So it's basically the slenderness ratio is a proportion of how tall and skinny a member is or how short and fat. So if it's short and fat, it tends to crush. There's no room for buckling. But if it's tall and skinny, then it's more likely to buckle than it is to crush. So Euler's equation in graph form looks something like this. Regardless of material, if a column is short and fat, it crushes. If a column is tall and skinny, then it's going to buckle. And ideally, a column wants to be in this range, where there's a mix of crushing and buckling. So that's Euler's equation in graph form. And this is the critical load. Beyond it, there's probably more buckling than crushing. Okay, so let's go to the next one. Sorry about that. Very good. So regardless of material, this equation holds, and this graph of how much load you can put on a column in proportion to its slenderness ratio, the shape of the curve is the same. It's just what determines slenderness ratio of a column that crushes versus the slenderness ratio of a column that is pure buckling. Depends on the material. So in wood, for example, the slenderness ratio is L over D. We'll talk about that shortly in another video. And in steel, it's KL over R. But here are the limits. in a steel column, wood column, or a concrete column below this slenderness ratio is considered pure crushing. And above this slenderness ratio, it's pure buckling. And ideally, we want to be somewhere in between these limits. So looking at this diagram here, I want to change my pen a little bit to something thinner. There we go. And I don't know if the color is good. I'll try yellow. So, the unbraced length of a column is basically the height between horizontal braces, and the slenderness ratio of a rectangular profile or a square profile is the unbraced length divided by the smaller dimension of the cross-section. So, when we look at this profile here, whatever that is, it's a wall, then it has this dimension is D2, but that's the thickness, but this dimension is D1. And so the slenderness ratio is the unbraced length divided by the smaller of the two, D1 or D2. It looks like clearly this dimension, D1, sorry, D2, I apologize, this is D2. So it would be, the slenderness ratio would be the umbraced length divided by the smaller of the two, in this case d2. So that's the slenderness ratio of a solid rectangular cross-section. So why is this happening? Okay, there we go. So this wall is going to buckle like that on the smaller dimension. It is not likely to buckle on the longer dimension. It's highly unlikely. So slenderness ratio is the unbraced length divided by the smaller of the two dimensions of the cross section. If we look at this column here, it has a height or an unbraced length, and it has its round, let's say, in profile. Therefore, its diameter is its least dimension. So L over D. In this case, this is a short column, and this is a very tall and skinny. Okay. I'm going to show a bunch of examples so you'll see a little bit better what is happening here. Okay, so let's look at the atrium we have at our School of Architecture at UNC Charlotte. When I think of these columns, they are 22 feet tall, these columns here. And it's not going to buckle as shown in this red dotted line. It will not buckle over the full 22 feet because there is something tying it at mid-height. So instead, this column is not 22 feet tall. It's 9 foot 2. This is 9 foot 2. There's another 9 foot 2. And this segment of the column has no chance of buckling. It's braced. So this is, in fact, a 9'2'' column, a 9'2'' and not a 22' column. So it's more likely to buckle, as shown in this diagram, with the dashed red lines. Here are some examples of slender columns in the beginning here. This is at the Kennedy Center in Washington, D.C. You will notice that the Greeks gave us something called enthesis, which is the column gets fatter in the middle to avoid buckling. Well, even at the Kennedy Center, they've added some profile to the column in the middle to reduce the likelihood of buckling. Versus these columns here, they look awfully skinny. I doubt that they're carrying any load. They're just ornament. We see them in this picture. This is a Philip Johnson building in, I think, Virginia. Anyway, the column is ornamental. It cannot be 10 floors tall and carry anything significant. Like the atrium I just showed, this column here is not the full height, but rather it's braced in the middle, so it would be one, two lengths of column. Here's this yardstick that is being pushed down in compression, and clearly it buckles on the short dimension, regardless of what the other dimension is. It's going to buckle on the thinner side. This column looks awfully skinny for its height, the unbraced length, versus the diameter or the least dimension of section. This column here looks like it's pretty chunky and it's crushing. Okay, so these are some examples. Let's look at this picture. It says as the load increases, the size of the column is increasing. So this is one floor, the column is this dimension. Three floors, the column is thicker. Five floors, the column is even thicker. And by the time you get to the bottom, it is increasing in thickness because it's carrying more and more floors. The unbraced length in each case is just basically the height, the unbraced length of the column, divided by its least dimension. If it's round, then it's the diameter. Okay, how tall is a column in a wood frame or a mass timber construction? What is the unbraced length? Please, this discussion is for columns or diagonal braces. It doesn't matter. As long as it's in compression, it could buckle. And then it has that critical load of buckling from the Euler equation. So a column in wood frame is one floor tall. Whatever that floor is, it's one floor tall. The diagonal brace is just from here to here. It is not continuous in this case. They didn't do that as the diagonal, but rather just one floor. Very good. So, looking at this image here from Brock Commons dorm at the University of British Columbia in Vancouver. Again, you can see the column is one floor tall. Versus precast. How tall is a column in precast concrete? And what is its unbraced length? in precast concrete it's going to be cast in a casting yard and then shipped to the construction site so it makes sense to make the column as tall as possible as long as it can fit on a truck for shipping so they might do a column which is two floors three floors tall whatever fits on the truck the bed could be 60 feet 70 feet somewhere in that neighborhood then they'll make it the longest possible. In this case, this column here is two floors tall. What is its unbraced length? There is a horizontal member coming in here. There's another horizontal member coming in here. Unbraced length, that much or that much, whichever is greater. Okay, so in precast concrete, it's whatever fits on the truck is the length, but depending on where horizontal bracing comes in, that determines the unbraced length. So this wall, I don't know how tall it is, five, six floors, but its unbraced length is wherever this double T comes in and defines the unbraced length that is subjected to buckling. In cast-in-place concrete, it's one floor. The column is one floor tall. If there's a floor taller than the other floors, then that's the critical unbraced length. and the cross-sectional dimension. If it's a rectangle, the smaller of the two. If it's a square, either one. If it's round, it's a diameter. So in steel construction, how tall is a column and what is its unbraced length? Again, a steel column is to be shipped from the fabricator all the way to the construction site. So they will try to make the column as tall as possible to fit on a truck. So it could be two and a half floors or three and a half floors. And I'm saying half because you don't want to splice a steel column where the floor is and the other beams are coming in. It's too congested. So they'll do like two and a half floors and then they'll start splicing at the half floor and then continue with the column just to avoid all kinds of congestion in a connection. Okay, let's look at this animation here of splicing a column. In this case, there's a tab on the web. They come in and they weld, and now the two columns have become one. And that is usually done at the height of half a floor. Same here. This is images for that animation. It could be a plate on the web, or it could be a plate on the flanges, pairs of plates on the flanges, to splice it to make it into one column. Looking at this crazy column, if I zoom in here, you can see that the flange thickness is 5 inches thick. Crazy. And the depth of this column is 23 inches. and the width of the flange is 18 plus inches. It's a monster column and there's the cell phone for a scale figure right next to the thickness of the flanges. Here, they had to splice this column and this is insane because they had to do a full penetration weld on the site and they had to do continuous welding for I think it was 30 hours. Just insane. Very expensive. Remember, it's a five inch thick flange and it has to be done twice. It was done with two welders. It ended up looking like this as if nothing. But the site welding is extremely expensive compared to shop welding. So it's always preferred to do welding in the shop and bolting. in the field. Okay. I like this picture a lot because it says if you have web stiffeners, then you are making a moment connection. This is a moment frame and it's taking wind load or earthquake load. There is no bracing and there is no shear wall. So clearly it must have a lateral strategy. It's not a shear wall. There is no bracing. So all these connections are moment connections. And notice, please, that this bay can handle lateral forces. This bay can handle lateral forces. But this one is gravity. So we have two bays that are handling lateral loads. and you can tell the column is so much fatter than the gravity column. The beam is so much deeper than the gravity beam because the gravity beam is taking loads from one floor, live load and dead load. But for the two end bays, we're handling wind load, and wind load is cumulative, and it's taking a lot more than gravity per floor. It's adding all the floors, so the members do get bigger. And if you were to put a diagonal, then the members become smaller, but you lose your view out and things like that. So the option to go with moment frame is better in certain situations, but it's always more expensive. So one has to see which battles to fight. Okay, how tall is a column? From one floor to one floor, a brace length is one floor. So diagonal bracing is basically a column at an angle, and diagonal bracing has to be provided in two perpendicular directions. You can't brace in one direction only, because then it's like dominoes in the other direction. Okay, how long is this diagonal? Whatever that length is, divided by the smallest dimension of section. There is something called tension-only bracing, which is these cables, ties, or straps. They cannot handle any form of compression. So in a metal building, it's very cheap. So it's pretty cheap to have diagonal bracing that is cables. And since it's tension only, you have to provide it in pairs. If the lateral load comes this way, this guy goes into compression and freaks out, but the other guy goes into tension and takes the load. But if the load comes from this direction, then this guy is doing the work in tension, and the other guy is not. And let me get an eraser very quickly. Typically, your lateral bracing system needs to be provided in two perpendicular directions. So, and probably every fourth bay, so this is a braced bay, this is a gravity bay, I am lateral. and then gravity, and then another gravity, and then you can see the next bay is braced in the ceiling all the way down to the other side. So every fourth bay, typically, minimum, should have some kind of lateral force resistance. So if my load is coming this way, a wind load, then it comes to here and it pushes on the red member, but it also pulls on the blue member. But if the load is coming right to left, then this guy is in compression. This guy is in tension. Okay. We have Chevron bracing. Same concept. The bracing is to handle lateral loads. It's not there for gravity. The post and beam. The post and beam is perfectly capable for gravity. But if you have a wind load coming this way, I will go into compression. You will go into tension. But if the wind load is coming from left to right, then I go into compression, you go into tension. Chevron bracing versus inverted chevron bracing. Again, these are all for wind loads. And you will notice that this bracing is in this direction, but this one in the background is in a perpendicular direction. And we need bracing in two perpendicular directions. Very ugly, very cheap, but always pin connected. Diagonal bracing is pin connected, not rigidly connected. We have some K bracing or knee bracing in the other direction because it seems there's a storefront or they needed visibility on this side versus on this side, maybe they can hide that diagonal with some kind of cladding. So heavier loads are going to require chunkier bracing. And this is called concentric bracing because that diagonal points to the intersection of beam column. And so does that diagonal point to the same intersection. So this is called concentric bracing versus if it didn't go to that intersection point, it would have been called eccentric bracing. So again, I need to know the length unbraced and the least dimension of section in order to determine the slenderness ratio. And in order to fall on this chart that says, here's my slenderness ratio, here's the critical buckling load, and you either have more crushing or you have more buckling. Very good. So sometimes diagonals don't intersect very nicely. So that's a pretty nasty situation, nasty and cheap, versus the New York Times building in New York, where Renzo Piano provided this very beautiful, elegant bracing across the whole height of the building. And you will notice that this pair of diagonals is they're on top of each other versus this other pair, they're side by side, and so nobody really occupies the center. There is no intersection there, and they pass each other very elegantly. Okay, so here's that building, the elevation, and it had these ceramic rods across the full height of the building,