 So today, let's talk about a structural phenomenon known as buckling. So yes, I've drawn a picture of a seat belt buckle here, but it's actually a very different type of phenomenon. First of all, I want to consider what happens. Consider the idea of if you take two forces and you apply them along a single line. We know there's two ways to do that. We can do that in tension, or we can do that in compression. Now it turns out that tension and compression, while they act very similarly on this point, it turns out the tension and compression, while they act very similarly on this point from the perspective of statics, actually have a very different sort of overall effect once you start putting material in between them. First of all, as we've learned from moment, if you have one of those off by just a little bit, if, for example, I make them so they're no longer collinear, then there's going to be a tendency for whatever material you have to rotate, because you've now created a moment. See how there's a very small moment arm here? Well, that same thing will occur in tension, that if I displace one of my forces by just a little bit, my material will rotate and have a tendency to rotate based on that very small moment arm. But there's a big difference between how these two things behave. In the case of the compression, you'll notice that the rotation that's being created here is in a counterclockwise direction in this picture, which is going to tend to make the piece that we had rotate even more and create a larger moment arm, which creates even more moment and more tendency to spin. Whereas in the case of the tension, the rotation created here is going to tend to pull the piece that we had back into line again and reduce the amount of moment and actually return it back into a collinear situation. So those two situations are very different. In that particular case, what we're thinking about is when they are in line, even if we create an equilibrium for the two forces being in line, one of the types of equilibrium is what we call a stable equilibrium, whereas the other one is an unstable equilibrium. Because even though it is an equilibrium, the slightest nudge, the slightest displacement, will tend to create conditions that make it harder and harder to maintain that equilibrium. There are a couple of parallels to this if you think about a physics situation. For example, if I take a bowl and I put a marble in it, if I nudge the marble, it will tend to roll up the side of the bowl, but then gravity will pull the marble back down toward the center of the bowl. On the other hand, if I balance a marble on top of a bowl and I'm very careful, I can balance it and have it in equilibrium, but the slightest nudge will tend to plummeting off the side of the bowl and it will not be returned to its equilibrium state. So the difference between compression and tension are very similar. In compression, you have unstable equilibrium. If you get nudged, it's going to get worse, whereas in tension, you have a stable equilibrium. If you get nudged, it's probably going to return to equilibrium. This is an important concept in the idea of buckling. So for buckling, first of all, we identify a certain set of conditions. We're going to call a beam element under a compressive load. So it's being compressed on the top of the bottom, some compressive load, where its length is significantly larger than either its width or its depth. Usually, we say in order of magnitude larger. So the length is greater than 10 times the width or depth of the beam. And we're going to call this a column. And you've seen many structural columns holding up a number of buildings. And in a situation like this, your axial displacement, how much you actually squish the beam is going to be very small. But there is some possibility that you might get a little bit, if the column's long enough, of lateral deflection, some sort of motion where the beam begins to bend just a little bit. Why does it begin to bend? Well, perhaps it's due to some imperfections, or things in the beam aren't exactly perfectly straight. And once you start getting that little bit of bending, you're creating conditions whereby it's a little less supportive, and then you get more bending and more bending, and eventually the column fails in something we call buckling. Usually, it's a pretty sudden and dramatic failure. Let me sort of demonstrate here with a straw. If I take my straw and I push on both ends of the straw, I can continue to push, but you see the slight bend? Well, if I'm pushing at that time, once that slight bend happens, we have a sudden kink and dramatic failure of the straw. That's the condition called buckling. Notice that doesn't occur if I pull on the straw. As much as I pull on the straw, even when I was bent entirely, if I pull on the straw, I'm actually pulling it back into its unbent state and making it straight again. So buckling does not occur in tension. It's also not entirely limited to columns. If I take something like a soda can, we've all seen or crushed a soda can, and I stand on top of it if I push it, eventually I can crush the soda can. I'm not able to do that with my hands, but if I stand on it, a smallest little indentation here where it starts bending and then the soda can crushes, this is again buckling. Notice I had to initiate that little indentation, but if I have a strong enough, if I just step on it with a strong enough force, I can still induce the condition of buckling. Buckling occurs at something called a critical load. So I'm going to identify this P with PCR, PCR, P critical. And it isn't a perfect science as to when it's going to occur. However, observations have shown and testing has shown that there are a few things that this critical load depend upon. And this critical load tends to be the boundary between when it's stable and when it's unstable. Once you're above the critical load, it's very likely that the smallest amount or the smallest amount of deflection laterally will result in your critical buckling failure. What are the factors? Well, the first most important factor is the length of the column. The longer the column, the more likely it is to experience buckling. Second factor is the stiffness of the column. If you remember when we've done axial loading, the stiffness of the column can be represented by the Young's modulus or the modulus of elasticity. That same modulus where our axial load is equal to the stiffness times the axial strain. That same value E is an important factor in buckling. The type of material affects the load, but generally that's reflected in this stiffness E. Also, the shape of the material in the cross-section. What this shape looks like? Is it round? Is it shaped like a square? Is it hollow? What does the cross-sectional shape look like? And finally, how it's held at the ends. In many of our analyses, we've considered beams that have a pin that are allowed to rotate at the ends, or we can consider them as being fixed where they're not allowed to rotate at the ends. And the choice that you make in how they're supported will also affect the buckling load, the critical buckling load. I'm going to focus a little bit on this idea of the shape of the cross-section. If you remember, in bending, we had a geometric measurement called the second moment of area, or the area moment of inertia. And we noticed that material that was furthest away from the center of the material, or furthest away from the neutral axis, had the greatest effect on resisting bending in the material. So if I'm attempting to bend the material, the stuff near the top or near the bottom is going to oppose the bending the most. Well, buckling is very similar to bending. That once you start with that little bit of bending, it looks very similar to a set of applications as if you would apply the forces, something like this. In fact, it essentially becomes a type of bending problem. So our second moment of area, or our area moments of inertia, become important in the calculations of buckling. Here are the relationships. We have three possible end cases, three ways that we can actually leave the ends of the column. They can be pinned, allowed to rotate freely. They can be fixed, not allowed to rotate freely and held in place. Or they could be free, in which case they are neither pinned or fixed, but allowed to move in space, and there's no reaction forces. Let's look at some combinations of these cases. Possible combinations. There is the pinned-pinned combination. Let me actually draw a pin on each end. This is a double-pinned combination, two pins. We have the combination where both ends are fixed, two fixed. We have the combination where one end is pinned and the other end is fixed. And then finally, the combination where one end is free and the other end is fixed. So this is pinned and fixed, and this one is free and fixed. Hopefully, you can see and understand why the combination of pin and free is not included here. In a case like that, the slightest rotation would cause the entire thing to just immediately fall over. So for each of these possible combinations, if we apply our axial compressive load, we will reach a critical buckling load. We'll reach a critical buckling load according to the following buckling load. We will reach a critical buckling load at each of the following values. In the first case, pi squared EI over L squared. That's the relationship. You'll notice it takes the Young's modulus into account. It takes the moment of inertia into account, and it takes the length into account. In this case, we have something called the effective length, which is length L here. In the case of two fixed supports, it's almost the same equation, but it has a factor of 4 applied, 4 pi squared EI over L squared. Essentially, what that's saying is we'll often consider two fixed supports as having a L over 2 effective length. The full length is still L, but notice if we replace the L in the first case with L over 2, that creates the factor 4. So often, what the terminology used is that the two fixed supports has an effective length of L over 2, or half the total length. In the case of the pin in the fixed, 2.045 pi squared EI over L squared. This is a critical value that's been observed. And the effective length in this case is 0.699 L. And in the final case of the free end, you end up with pi squared EI over 4 L squared, where the effective length is actually 2 times L. The key idea here for calculating buckling is if you would like to know what critical load above which you're likely to experience buckling, you need to know these pieces of information, the Young's modulus, the moment of inertia, the length, and then the fixed conditions, which affect the coefficient that's out in front. Notice you might have a different moment of inertia, Ixx or Iyy. If your beam is not symmetric, if it's not round or square, for example, if it's something like an Ibeam, you might have a different moment of inertia. Well, what you usually want to use is the smaller of those two values to determine what the smaller critical load is, because buckling is liable to occur in that direction. As a small demonstration of buckling in action, I've taken a piece of modeling clay here, rolled it out to make sort of a dowel shape. And now I'm going to compress it on both the top and the bottom. And notice if I'm very careful and compress it very lightly, it's able to resist. But if I start compressing it a little bit harder, eventually you start seeing that bend and you get failure in buckling.