Hi. This is module 14 of Mechanics of Materials Part 4. We are now starting the portion of the course which is on column buckling. So our learning outcomes are to define column buckling and to derive what's called the differential equation for column buckling. We're going to look at a simple column, and it's going to be a long straight prismatic bar that is subjected to compressive axial loads as shown. If that beam remains straight, we'll analyze it using the techniques that I talked about and worked with in my first course, part one of mechanics materials, where we looked at bar loadings under axial loading. However, if the beam is long and slender, we can have buckling occur and if it starts to deform laterally and this deflection can become large and lead to catastrophic failure. So buckling is a large sudden deformation of a structure due to just a small increase of the existing load. I'd like to show you a demo at this time. This is a nice simple demo of column buckling and this is a typical yard stick or meter stick. If I put a compressive load on it, there is a region for small loads where if I release, push release, it returns back to its original condition. But at some point, we come to a point where with just a little bit more load, the yard stick or the meter stick begins to buckle. You can see that very easily on your own. You could actually do this on your own and get a feeling for this phenomenon. So we're looking at column buckling of a simple column, long straight prismatic bar, subjected to compressive axial loads. Here it isn't it's deflected shape. Buckling is when a stable equilibrium becomes unstable. So during the initial compression, if we do a slight perturbation laterally, the load could be removed and the column would return to its straight configuration. But there is a point at which it becomes unstable so that when buckling occurs, a critical value is reached which when the beam or the column is perturbed laterally, it will not return to its straight configuration. So for long slender columns, the critical buckling occurs at stress levels below the proportional limit of the material. So therefore, this type of buckling is in elastic phenomenon. We call this simple column buckling Euler buckling for long slender columns and it's named after Leonhard Euler, who was again a very famous Swiss mathematician. Here, I show being configuration pinned on both ends. If we put it under compressive loads, this shows the column and the shape is exaggerated when loaded. We want to find, what is the minimum axial compressive load that will cause buckling? So here again is the situation. I've added a set of coordinates at the center of the beam with X up and Y to the left, and I said that the beam above the origin here is l over two. We're going to say that the deflection at the end of the beam, at the top is going to be delta. Now let's look at a slice. Let's go ahead and cut this column. We'll cut it at a distance x. At x, we'll have some deflection y. So here is a free body diagram of the top portion above that cut. We have our P force down. We're going to have to have an equal P force up and we're going to have to have a moment reaction to keep this in static equilibrium. You should be able to go ahead and solve for the moment reaction using your static equilibrium techniques on your own and then come on back and see how you did. So first of all, I'm going to say, "Okay. Well, the distance from here to here, for the cut is l over two minus x and the distance from where this cut is on our beam out to where the P force is applied, is going to be delta minus y here. So that's a good sketch. I can now sum moments about this point down here. I have the differential equation for the elastic curve that we talked about before. If we sum moments, we have the moment that's balancing MR due to the force P is P times its moment arm or delta minus y. So we can rearrange that equation and now you can see that we have a differential equation for column buckling that is in terms of the coordinate y.