https://youtu.be/ai4nS1t7LeY?si=AtG-J9R1uI6kPONo
Hey gang, welcome back. We're talking today about Column Buckling. Want to introduce that to you? What does that mean? What's involved here? How do I find out how much load a column can handle before it fails?
There are some things we need to know about Column. Let's start off with number 1. That is the detail. This is the difference between long column and short column.
Number 1. All the problems that you're going to see in your book are going to be long column. Because long columns buckle in a different way than short columns do. This is a yardstick. This is what I consider a long column. It's long, it's skinny, and when I match on it look how it buckles.
Now imagine my hand up here is a pin connection. It allows this thing to rotate freely on my hand. When I match on it I get this buckling. Can you see it? That's a long column buckle.
Now what is a long column?
A long column has a slenderness ratio. The slenderness ratio is the smallest dimension which would be the thickness of my yardstick here. Not the width but the thickness of it. Divide it into the length. How many times does this thickness go into the length? If it goes, now the jury is out on this.
What's the slenderness ratio by the way? What is that number that the length gets divided by the smallest diameter?
Some say 12, some say 30. I think most people would agree it's 30 or above to be, and I've seen some that say 120. 120 times into that thickness. I think most of us say 30 times is the number for the slenderness ratio. That is a long column.
Like I said, all the problems that you're going to see in the book, all the problems that you're going to work out on your homework are going to be long column. If there's long column there must be a short column.
What's a short column? It might be something like this cook can. This cook can is a short column because think about its dimension. How many times does the width go into the height?
Well certainly not 30. It's maybe like two. Short fat column here. How does this buckle? Let's just test it out. I'm going to put on the floor. We'll put man size weight on it. Let me go really slow. See if I can actually stand on it.
Empty, you think I can?
Oh! I don't think I don't think I can.
Now look how short columns fail. They don't bow. They don't buckle. They crush. They collapse upon themselves. That's a short column failure. That's totally different than a long column failure. Now we know that.
Like I said, we're going to be dealing with long columns. Now the next thing I need to know about long columns and the next thing that makes a difference in buckling is how these ends are connected. If I'm pin connected, you can do this. If you have one of those little metal rulers, little 12 inch ruler, keep your hands open and buckle it and see how easy it is. We call that pin pin. Each one of those ends is pinned. Now these are called the end fixity conditions. There's a new vocabulary word for you. The fixity conditions. How the ends are fixed. If it's pin pin, we have this k value, which is the end fixity condition, of one. Now this is also called the ideal condition. So pin pin is ideal. And ideal means it's equal to one. And I'll show you what happens when we have that little later on. Go to number two. Fixed fixed. Fixed is a fixed support.
Now I want you to grab your little ruler and make a fist on both ends. Now it's fixed. I'm not going to allow the ends to rotate. It'll still buckle, but mash it together now. Don't let the ends rotate. I can't do it. It takes a lot more force. Because now, when this thing buckles, before when it buckled, it buckled like this. But now with the ends fixed, it might buckle something like this. It's still buckle, but the ends won't. This takes, and you can do it with your hand.
Now I've got to dent my hand. Thank you very much. It takes a considerable amount of force to buckle it with this end condition versus that one. So the K-value goes to 0.5 on that one. Now pin fix means that one end is free rotate. The other end is fixed. So it's kind of in the middle of those two. It's kind of a hybrid connection. The K-value for that is 0.7. And then there's fixed free, which is like a telephone pole. Now it has a wire hook to the top of it, so it can bow, it can buckle. Also, and the K-value for fixed free is, it's the easiest to buckle of all of those kinds. K-values 2 on that.
Now, we're in the hacker. We're going to use this K-value. When we're talking about buckling, one of the main things that we want to know is this. What is that critical force? That when I compress that system is going to make that column buckle, it's going to make it fail. And what we do on that is we use this. We use Euler's buckling equation. It's not Euler. Don't say that wrong. I want you to look really good in front of our bias, say, Euler. Because he's like, you know, he would be mad if you mispronounce his name. This is the buckling equation. P-critical equals pi squared E. That's a modularity, right? That tells us about the bendiness of the material. Sometimes I, I as moment, every moment of inertia, that tells us about the geometric properties of the cross section. Is it a diving border? Is it an I beam, right? And then down here on the bottom.
Now, this is where I see most of the mistakes on this mate is right there with that K-value. That K is inside the parentheses. Now, a lot of these times you'll see this equation and you'll see it written in the ideal form, which means K is 1. And so you won't see that K there. And you'll just see L squared. And so then when you do put a K in there, I've seen a lot of people that say, oh, the K must go outside. No, it's quantity squared. The K is inside the parentheses, okay? So don't mess that up. Depending on what your in conditions are, that multiplies times the free length. This L right here is the unsupported free length. This beam didn't have anything in the middle of it. If it would have been pin connected in the middle, then the free length would have been from one end to the middle where it's supported, right? So this L is not just the overall length of the beam. It's the free length of the beam. The unsupported length of the beam. That's imported, okay?
Now, so the two things that I see messed up, number one, getting the wrong infixity conditions. And we're going to do an example problem.
Next video, watch that. The infixity conditions change depending on the direction you look at the problem. Check that out, okay? That's the number one mistake I see. And number two is this I. I can be confusing, okay? Because let's look, for instance, at this problem here, okay?
Find the critical buckling force for the 10-foot long column. Assume pin-pin ideal connections, right? So I've just got what we would call an i-beam.
Okay, question of the day. Why do we call it an i-beam? I know what you're saying right now. Because it looks like an i. Okay, smarty. Okay, smarty. How about this? Here's same beam.
Okay, here's same beam. How come we don't call it an h-beam? How can we call it an i-beam? I'm going to tell you why, okay? Now, when we're talking about bending or buckling, okay? We have this guy here, a neutral axis, okay? And if you remember, I, remember we discussed I when we first came up with it. The farther away that I can get the centroid of half the part, let's say half the part would be like this guy right here, the blue here, okay? This part right here, okay? The farther I can get the centroid of that, here's the cent, here's the same thing over here, okay? In the i-configuration, is that much, right? The farther I can get the centroid of this shape away from the neutral axis, the less likely it is to bend, right? The stronger it's going to be in bending. When this configuration, the centroid of this may be like right there, right? So, this distance right here is not very far, if I wanted to call that d. It's not very far away. Whereas over here, I've got all of this area which brings that centroid maybe to like there, right? Same shape, but look how big d is now compared to that. So, if you have a column, it is always going to fail in what we call the weak axis. This is the strong axis. This is the weak axis of bending, okay? So, it's always going to fail around the weak axis. And that's why we call this an i-beam, and not an h-beam because it's strongest in the i-direction. If you go on a job side and you see somebody putting a beam up and they're putting it, they're welding it horizontally, let's say. And they've got it in the h-direction, you need to go up and tap them on the shoulder and say, hey, you're installing that wrong. It's not strong that way, okay? It's not. If you ever mess with an i-beam, it'll sag, it'll bend. If you pick it up like this versus picking it up like that, okay? Now, let's just demonstrate that real quick, okay? Let's just do this little trick. Alright, let's try this. This also is going to be i-ax, right? But it's completely different. What used to be the base here is the height over here, and what used to be the height here is now the base over there. Let's see if we can do this.
Now, I think we're going to have to use the parallel axis theorem, okay? Which is going to be 1, 12. Now, on this one, right, what I'm going to do on this one, I think, is I think I'll do the whole thing and then subtract out these two windows here, how about that, okay? So, 1, 12, the base, which is 12, right? Tums the height, which the whole thing is 12, cubed, minus the windows. Oh, let's see. How big is that? Oh, this is 1 inch right here, okay? So, this is 10 inches by 11 inches, right? So, 110 inches? So, what are you doing? This is a-d squared. Remember parallel axis theorem? A-d squared? Right? No! No, no, no, no. No parallel axis theorem this time. Because, where is the centroid of these pieces? The windows, the centroid is on, so d is zero, isn't it? So, we don't need the parallel axis theorem. We got smudges. Smudges! Okay, so minus the windows. One, 12, the base of the windows is 11. The height of the windows is 10. Okay, cubed. So here we go. That is equal to, what is that?
Hey, check this out. We got 12 on the bottom or 12 on the top. So that's 12 cubed. 1728 minus, oh, what is that? 10 cubed equals times 11. divided by 12 equals 916.67. And so 1728 minus 916.67 is 811.33. And that's inches to the fourth. Let's do this one over here.
Okay, this one probably, I'm going to have to use a parallel axis there, right? Because I've got to subtract out these windows here now, right? So let's just do, oh my goodness.
Okay, I'm going to do the same thing, whole square minus the other part.
Okay, so the whole square is 1, 12. The base is 12. The height is 12 cubed minus. And here comes the next part, right? Is these guys? There's two of them.
Okay, and I'm going to do this, John Perinthese. And here we go. One of those guys is this. The 1, 12. The base is 10, right? The height is, let's see, this height here is six. That's one, so that's a half. So the height is five and a half, 5.5, BH cubed minus A, that area at 5.5 times 10, that's 55 times D. And remember, D's from the neutral axis to the center of that thing, right? So if that's 5.5, let's see, what is half of 5.5? divided by 2, 2.75. So it's 2.75, but that's here. And then I've got to add a half to that, right? So plus 0.5. So that centrally lives at 3.25 squared. Okay, now how much is that? Okay, this is 1728 again, that's what that is. And then what is all of that business? 1728 minus two times, okay, here we go, 5.5 cubed equals times 10 equals divided by 12 equals 138.64 minus 3.25 squared times 55 equals 580. Okay, and that's 580.9. Now wait a minute, 138, that's not minus is it. It's plus 80 squared, not my, oh, that was almost bad, wasn't it? So plus 138.64 equals times 2 equals 1439, this part right here is 1439.15. Subtract it off of that. So 1728 minus answer is 288.8.
Okay, so the moment of inertia, look, it's a lot smaller than that is, isn't it?
Okay, so it's going to fail in that direction, not in this direction. This is the strong axis, weak axis. Now check this out, pi, never going to change.
E, the material, I don't care which configuration, that doesn't change. I, that does change, right? And then K, of course the inconditions can change. The free length here is 10 feet. So I think we have everything we need to calculate the critical force. Let's do it.
P, critical for this particular column is equal to pi squared times E, which is given as 29 times 10 to the 3 K, which is K, per inch squared, right? Unit, so we got to get it all over units to work out. Now we're going to use this I times 288.8 inches to the fourth divided by, okay? Now this is pin, pin. So our K value here is going to be one. And then of course our length is 10 feet, what are we going to do with that? Multiplot by 12, 120 inches, okay? And then we're going to what? We got to square that, don't we? Okay? All right. And guess what? The inches squared, that one goes to the bottom. I got another inches squared going to cancel out with that inches to the fourth, going to leave us with P in Kips, isn't it? Let's see what we get, gang. Here we go. 3.14159 squared times 2912 3000, times, times 288.8 equals divided by 120 squared equals bam. 5,740 Kips, okay? Now remember, I Kip is a thousand pounds, so we're talking about a lot of weight, aren't we? But that's a 12-inch beam, or I beam with one-inch flanges. That dude is going to carry some weight now, okay? So there you go. That's how you use Euler's buckling equation. Like I said, the most, most missed thing is miscalculation. Which high do I choose? Okay? Which high do I choose? What depends on which way it's buckling? Which way you're going to calculate? This one we're going to choose the week because that's how it's going to fail first. And then which k-dol I choose? What are my end conditions? So what's that next video? There's one that kind of, the end conditions change depending on which way you're looking at it, so that can be tricky. All right. I hope this is a good introduction to column buckling. I hope this helps, and I'll see you on the next video.