Columns & Diagonal Braces: Column & Brace Slenderness 2

So let's discuss a little bit further the slenderness ratio of a column or a diagonal brace, be it in wood, steel, or concrete. So this page is dedicated to that. So in a rectangular profile, solid profile, such as wood or concrete, the slenderness ratio is defined as the unbraced length of the diagonal brace or the column divided by the smallest dimension of its cross section. So in this case, the smallest dimension, if it were rectangular, would be this dimension. In a square, clearly it's either side. So let's apply this definition. The slenderness ratio is the unbraced length divided by the least dimension of cross-section. So looking at these three columns, A, B, C, it looks like column A has a slenderness ratio of L over D, unbraced length, divided by the smallest dimension. Which means this guy has a three-foot length divided by the smallest dimension. let's assume these are actual dimensions not nominal. So the dimension is four inches not three and a half for example. So three feet divided by four inches we need to make sure slenderness ratio is a dimensionless quantity it has no units. So I'm going to go ahead and multiply the three feet by 12 to make it into inches and therefore inches divided by inches is unitless, and that's what a ratio is. So it turns out this guy has a slenderness ratio of 9 versus the next one down has a slenderness ratio of unbraced length divided by the least dimension of section. And so that turns out to be its length is 10 feet, and its smallest dimension is 4 inches, and let's make that into inches. So it turns out this guy has a slenderness ratio of 30. And column C has a slenderness ratio of L-umbrace divided by the least dimension of section, which turns out to be the height or the umbrace length is 16 feet. Make that into inches. divided by the least dimension. This is a rectangular profile, and the 4 inches is less than the 6 inches. So this is the least dimension. The column could buckle about the 4 inch or the 6 inch, but it's more likely to buckle about the smaller dimension. That's why we're taking the 4, not the 6. Divided by the least dimension is 4 inches. So it turns out this is a slenderness ratio of 48. So we have three columns, and they have different heights. Their least dimension is the same in each case. It is four inches, actual dimension. So I'd like to take these three columns and put them on this graph that we have seen before in the slide presentation. this graph says if your column or diagonal brace has a slenderness ratio less than this much, whatever this number is, then you are dealing with pure crushing. And if it has a slenderness ratio more than whatever this number is, you are dealing with pure buckling. And ideally, you would like to be between pure buckling and pure crushing. So somewhere in the middle. So looking at wood specifically, the slenderness ratio is L over D. And ideally, it's somewhere between 11 and 50. Those are limits specified by the timber industry. Based on the species of wood, you might have a certain allowable compressive stress parallel to the grain. So based on the species, I have a certain number. So let's say just for simplicity, I looked it up, and for this species, I'm entitled to 1,000 pounds per square inch of compression parallel to the grain. That is a column or a diagonal brace or a top cord of a truss even. They are all subjected to compression parallel to the grain. So let's put our four columns or four pieces of timber on this diagram. The first one had a slenderness ratio of 9, somewhere here, a bit less than 11. Slenderness ratio equal 9. The second column or post or whatever we're calling these had a slenderness ratio of 30. So I don't know how much K is, but 30 looks like it could be somewhere here. Slenderness ratio equals 30. And then the last one had a slenderness ratio of 48. So here's the lesson from this chart. It says if you are short and fat, you can count on the full 1,000 PSI because there's no chance of buckling. But as you get taller and skinnier, then you have to reduce that 1,000. You're not allowed to use it because you're too tall and skinny. You're more likely to buckle than to crush. So at a slenderness ratio of 48, you're only allowed this much, whatever this value is. So that could be 100, it could be 200, but it's definitely not 1,000. And at 30, you're allowed this much, a little bit more. So the allowable compressive stress on an axially loaded member in compression depends on its slenderness. If it's short and fat, you can use the full value of compression. If it's tall and skinny, you have to reduce. And ideally, you would like to be somewhere between 11 and 50. Now, if you look at this curve, it's actually the Euler curve. It's made of two segments. Segment 1 does that kind of curve. Segment 2 does a different curve. Segment 2 does that. Both are parabolas, and both of them are facing upward down. They have different equations. So this one has a slight slope. This one has a steep slope. So the reduction in the second half after k, whatever k is, The reduction is more severe than B4K. It's a shallower slope. So anyway, that's the story of a compression member in timber. Now, looking at concrete, it could be cast in place or it could be precast. Anyway, I need to know the profile of the cross section. And there is a least dimension very similar to wood. and there is an unbraced length. Now, we'll get to this later. Okay, I'll get to it later. But in concrete, the limits for a short column is anything less than 32, and then for a tall column is 120. Okay, slenderness ratio for a rectangular shape is unbraced length divided by the least dimension, be it wood or concrete, But the limits of what is short and what is tall and skinny is different in wood than it is in concrete. Let's get to steel because that's where it makes a difference. And there's a little bit more complexity in steel. So in steel, the slenderness ratio is not defined as L over D, but rather L over R. So L over R, L is the embrace length, same as before. R is the least radius of gyration. Instead of the least dimension, it's the least radius of gyration. And if we look at this wide flange, if it's loaded excessively, it's going to buckle like that. There is no chance that it's going to buckle in the other dimension, right? Because the web is skinnier than the flange is. So there's two radii of gyration, one of them about the x-axis, the other about the y-axis. Everything is listed in the seal manual. But it's a similar concept to the least dimension. This time it's called the least radius of gyration, which is the distribution of the material with respect to the centroid. Is it more likely to buckle about the x-axis or the y-axis? It is extremely similar to L over D. Now, so the radius of gyration, by definition, is the square root of I, the moment of inertia, divided by the area. So again, everything's in the steel manual listed very nicely, but the point is, it is just a dimension that tells me how much mass is going about the x-axis versus how much mass is going about the y. axis. And then with HSS shapes, I mean, there's no mass in the middle, so it's basically the perimeter. And again, there's a moment of inertia about the x-axis and another one about the y-axis. And you take the smaller of the two, so it's the least radius of gyration. Now, what is this K? That's the complexity in steel. K describes the way the column or diagonal brace is attached. Are both ends pinned or are both ends rigid? Now, we didn't see that before because in wood, a column is always pinned at the bottom, pinned at the top, because you cannot make a rigid connection in wood. so period it's pin pin in cast in place concrete you have no choice because the bottom is rigid the top is rigid when we look at this uh picture the rebar is going into the foundation so this is a rigid connection and the rebar is sticking up there's going to be a beam or a slab or something coming up top, that one is also a rigid connection. So in cast and place concrete, I have no choice. It's rigid, rigid. Then in precast, no matter what you do, it's going to be pinned at the bottom and pinned at the top. So there is no choice. In steel, there is choice. In wood, it's pin, pin. In cast and place concrete, it's rigid, rigid. In precast concrete, it's pin, pin. They cannot be analyzed any other way. That's what they are. They're not rigid enough in case of wood and precast. And it's very rigid in the case of cast in place. But when it comes to steel, there's some options here. Because you recall maybe from a previous video, if the anchor rods are between the two flanges, here's one flange, here's another flange, Here's your non-shrink route. If the anchor rods are between the two flanges, then you have a pin. Looking at this one, the anchors are between the two flanges. This is a pin connection. We can also tell that it's pinned because that flange is sloping. Therefore, this dimension at the bottom is less than the dimension at the top. When you reduce the area, you're making a pin connection. here's a square tube and then they reduce the area to make it smaller because this too is a pin connection when we look at a diagonal brace they cut through the square tube and there's a gusset plate here and oops let me erase because we don't see it anymore and there is just one pin going through the gusset plate and the square tube. This is a pin connection. If, however, the anchor bolts are outside the two flanges, then you have a rigid connection. So there's these choices to make. When I think of this square tube, the anchor bolts are outside. When I think of this white flange, We're looking at the flange here. I see the anchor bolts outside the flanges, which means this one is a rigid connection. Had they been here, it would have been a pin. So all of these will be rigid because the holes are outside of the white flange or the square tube or whatever the column or diagonal brace is. So, in drawing form, we've seen this drawing before. If your anchor bolts are outside the flanges, then you have a rigid connection to the foundation. If they are in between, if they are in between, then pin. so in steel the tips of a column could be pinned or rigid and we need to make that distinction because that affects the length of the column, the unbraced length so let's take a look had these anchor bolts been somewhere here that would have been pinned but no, they are beyond the flanges Therefore, we're considering that to be a rigid connection. It prevents rotation. So there is the slenderness ratio we said is equal to, in steel, K L over R. Where R is the radius of gyration and there's two radii of gyration. It's the smaller of the two and L is the unbraced length. So K is a factor that describes the end conditions of a column. How is the column or diagonal brace attached? It could be pinned on both ends. That's condition D. Then your theoretical K value is 1. If a column is pinned on both ends, then K is 1. that's what happened with wood there was no KL over D because K was 1 in precast concrete K is 1 because it's pin-pin if your end conditions are rigid both ends rigid then your theoretical K value is 0.5 instead of 1 so let's have an example just understand that this table is important especially in steel because it describes the end conditions of a column. Let me have an example here. If the unbraced length of these three columns is 16 feet. And in the first case, you have both ends are rigid. In the second case, you have both ends are pinned. In the third case, you have a cantilever situation or a building coming out of its foundation or a single-story building where the bottom is anchored to the foundation, but the top is free. And that would be these conditions where K is equal to 2. So I have three columns. Each one is physically 16 feet long. But in case number A, in case A, K is 0.5, theoretical value. In case 2, it is 1.0. And in case 3, it is 2.0. Those are the values from those slenderness end conditions of a column. So KL in this case, KL is considered, is called the effective buckling length. So the physical length is 16 feet, but the effective buckling length is 8 feet. Versus the second condition with a K of 1.0 times the 16-foot length, KL, or effective buckling length, is 16 feet. And in the third condition, KL is 2 times 16 feet, or an effective length of 32 feet. So here's what this is saying. If the column is rigidly attached on both ends, then the deflected shape or the buckling shape, this is the same as a beam that is restrained on both ends. If it were, sorry, if it were pinned on both ends, then it would bow or buckle this way. And the effective buckling length is 16. it would buckle over the entire 16 feet. But if both ends were rigid, it would not do that. Instead, it's going to do something like that. And this is saying that the length that is subjected to buckling, this is the effective length, and it's equal to 0.5 times 16, or 8 feet. So buckles only over 8 feet because the top 4 feet and the bottom 4 feet belong to the support. These are part of the support, and therefore the length subject to buckling is 0.5 of 16 feet or 8 feet. But in case B, where both ends are pinned, it could rotate at both ends, and therefore the effective buckling length is 1k is equal to 1 times 16. the whole 16 feet would buckle. Versus condition C, where K is 2, theoretical value of K is 2, and the effective length is 32 feet, this one is as if it buckles over 32 feet. So it's very slender. And this is 32 feet is the effective buckling length. So we need to take into account the rigidity of the end conditions. Of course, there's other intermediate conditions that I didn't talk about that are less popular, which is rigid on the base and then pinned at the top, et cetera, et cetera. So back to here, if your slenderness ratio is 32 for the column that is rigid only at