Hello, in this video, we will see how to stabilize a bar in compression and then, I will introduce the concept of the buckling deflected shape. We are going to start with a bar in compression, we are going to stabilize it with guy-wires, in the way we learned -- we have seen it in "The Art of Structures I". And then, we are going to look at how these structures can remain in equilibrium when we deform them a bit. And then, I will introduce the principle of the critical buckling load. I will discuss the importance of the deflected shape and of the buckling length. In this video, you can see a vertical bar stabilized by two guy-wires on the left and on the right. These guy-wiresare chains, but, on the bottom, they have springs. I have first placed only one spring on each side, then, you can see that with only one spring, the structure deforms. However, here, I put two springs, and you can see that I can place my load and the structure holds. If I move it a bit leftwards, it holds and comes back to its initial position and if I move it a bit rightwards, likewise, it comes back to its initial position. Here, the structure is in a stable state. If I now add a second load, you can see the same behavior than the one we have seen before with a single spring, so the structures ends up giving way It gave way on the right while before it was on the left, that is not very important. Here, we have a structural scheme for this structure, so we are going to place a load of ten Newtons. We have recognized well in our... in our configuration this load of 10N. We will work on it in a force diagram later on, so I already introduce it. This is the initial position. What interests me is especially the deformed position. In the deformed position, what do we have? We have a bar which is compressed, because the force of 10N must be taken by an element. Then, at the same time, the left guy-wires is in tension. To simplify, but it does not change much, we are going to neglect the right wire to only consider the left one. Here, I look at a free-body, so I do not cut the right wire, because the right wire does not exist anymore, but I insert my force of 10N, Then in the direction parallel to the cable, I insert the internal force, the tension in the wire. We are going to get back to it later on. Then, finally, the polygon of forces is closed by the compressive internal force in the bar. Now, what is the value of this tension? What we can see is that, actually, the bar changed its length, here, initially, it was 17.5, and then, now, it is -- it lengthened, here, of a value delta L, it lengthened. Then, since we have a spring -- we are going to consider that the entire bar is a spring, which is not exactly the case in my experiment, but the additional internal force due to the lengthening of the shroud is equal to K, the spring constant in Newton per millimeter, times delta L. What happened during the first experiment I have just shown? Well, the value of K being too small, maybe we actually only reached a force which had this value here. And then, at this moment, equilibrium was not possible, since the compression needed to be larger that what could be offered by the system. Then, we have doubled: we have put two K by placing two springs in the lower part of the model, and then this equilibrium became possible. So, this is the equilibrium in a deflected position. The principle for stability: we always try to express the equilibrium of the structure in its deflected position. So, that is a good news, we have seen that our bar, which was unstable when we had not any guy-wires, can be stabilized. We call the critical buckling load, the load -- we are just going to draw it here on the right -- the critical load, that is the maximum load beyond which the structure is unstable. We have seen it in the experiment, when I tried to add a second load of 10N, the structure became unstable. So, somewhere between both, there was a load which was just acceptable, maybe 15N, I do not know, which was acceptable, and then beyond, the structure becomes unstable. So, we call this load, the critical buckling load, and obviously, it is important not to exceed it, because otherwise, we will have this instability phenomenon. Here, I have a second model which is similar, so, I have a bamboo, the one you have seen before with the cane, which is clamped at its base, and then, I place a load of 10N on top of it through a small steel element, and then, we can see that the structure is stable. If I proceed to the same experiment with a steel bar, it is a bit wider, but it could have the same size, we can see that the structure remains stable, and then, I add a whole series of weights, just to show that the structure is really much more stable. The type of material, and specifically the modulus of elasticity of the material, is very important for the stability phenomenon, that is why we can see more concrete columns, or even steel columns when the instability phenomenon is very significant. The buckling deflected shape is the shape that my column takes when I apply on it a compressive load. The structure is going to take this shape, it is thus going to move aside the pressure line -- the pressure line, in this case, that is very clear that it is going to extend vertically downwards. We move aside from it in this case. The buckling deflected shape is the shape taken by a compressed element under an applied load. You will maybe tell me: "If I have a column which is perfectly rectilinear, then I will not have this deflection." Maybe in theory, but it will be sufficient that, at the beginning, the structure is slightly imperfect for it to start moving aside at a given moment. We cannot assume to have a perfect column, because it will never be perfect, in reality, and, anyway, eventually, when we will be close to the critical load, then it will start deforming, but in practise, it starts much earlier. Another thing which is significant is the influence of the length of our column. I am just going to show it on the left, and then, for the others, you will do it mentally or you will add it in your notes, whatever. Here, I have a length h for my column, so I will have a quite low load. If I have a length h for my column, I will have a certain critical buckling load: Qcrit. If I take a column half as high -- so, here, I have h/2 -- I am going to have a quite increased critical load, I divide again the length of the column by two to reach h/4. I am going to increase again the critical load, and then, I divide again this length, then I will have a value which becomes very large, I cannot even represent it, but it is somewhere here on the top: h/8t. All this has a hyperbolic shape, because the critical load is proportional to one over the length of our column squared. We call this curve the Euler's curve, Euler was a Swiss mathematician who was interested in many problems including this stability problem. The critical load cannot exceed a certain value -- I am going to draw it, for example, around here. Here, this value corresponds to the strength of the cross-section, that is to say the area of the cross-section times the compressive strength of the material used. We can, in theory, go further with the Euler's curve, but we will be limited to a certain value of this critical load, but which would be very interesting to reach. What happens if we exceed the limit load? So, I place a second load, and my bamboo bendst erribly, but actually, it still does not break, but it is terribly deflected. We took off one load, it remained quite deformed, now, I put a third one, and then, here, the bamboo breaks, so indeed, the result, if we exceed the ultimate buckling load we are going to have a tremendous increase of deformations and then, as a result, a tremendous increase of internal forces, we will talk about internal forces in the next video, and then, in this case, the bamboo collapses without any hopes to save it. In this video, we have seen how the equilibrium of a compressed bar enables us to understand the stability phenomenon, and to understand how to really stabilize a bar. The importance comes from the stiffness of the springs, so stiffness is an important parameter for the stability phenomenon. Concerning buckling, we have introduced the concept of critical load, and then, that of deflected shape. We have not talked yet about its length, this is coming in the next video. Then, we have seen that, finally, if we exceed all the values, then the strength of the material is going to lead to the final collapse.