Hello, in this video, we will deal with support conditions and I will introduce the concept of buckling length which is essential for the theory of stability. We will see two fundamental cases. Actually, we have already seen them, but we will see them again. We will introduce, within the framework of these fundamental cases, the concept of buckling length, then we will see the effect of supports on the buckling length, as well as the effect of intermediate supports. In the previous two videos, you have seen two cases of stability. On the left, you have a cane, which bends taking the shape of half a wave. And then, on the right, you have a vertical column, clamped on its base, which bends taking the shape of a quarter wave. Why do I talk about waves? Because the shape a column takes when it reaches its buckling critical load is the shape of the sine function. So, here that is half a sine, if you want, between 0 and 180°, if you know a bit about trigonometry, but it is not necessary to know trigonometry for the following part of this course. Here, I then have a column, with a height h on the left. We are rather going to say a height 2h. And then, the right column has a height h. Under the effect of a load applied on the top, as we have seen it in the second video, it is going to deflect in this way. We call the buckling length, the length of the half wave. So, for the left column, the buckling length which I identify by L_fl, is equal to 2h. What is about the right column? Here, we can see that we have a quarter wave. Actually, the complete wave, or rather the half wave, which would correspond to the other example, would have this shape, and then we would have a buckling length such as for the left case, which is equal to 2 times the height of the column. So what we can see here is that the buckling length of a column depends on its support conditions. The left columns has a hinged support on the bottom and a hinged support on the top, whereas the right column is clamped on its base but it does not have any supports on the top so it can move transversally; consequently, as we can see, its buckling length is larger, it is actually twice its height. We are going to see a whole series of cases, all these columns have the same height this time but they have varying support conditions. Here, I have a series of columns with the same height h and then, the leftmost one, that is the case we have previously seen, so that is the case of a column clamped on its base. And then below, we have the other half wave with a buckling length of 2h. If the column, in addition to be clamped on its base, is held on its top, what happens? On the top, it will not be able to move transversally, but it will be able to rotate. However, it will not be able to rotate at the base, it is thus going to deform according to this shape. We try to identify the wave. Maybe I am going to take another color to facilitate the identification. Here, you can see in orange, we find the half wave. And then, for this column, with these particular support conditions, we will have a buckling length smaller than h. Typically, the buckling length is 0.7h. If I now keep the support on the top, the same than here but now I enable rotation at the base, we are going to have a deflected shape which we have already seen. We thus have a wave, I mean, a half wave which is like this. And then that is the fundamental case of the cane we saw before, the buckling length is equal to the height of the column. If we have a column clamped at its base like both these columns here, but which, in addition, cannot rotate on the top, thus the column can go down but cannot rotate, this means that there will no rotation here, the deflected shape will be like this and there will not be any rotation at the base either. I take the orange color again to identify that we have the half wave in the middle part and thus the buckling length of this column is equal to half its height. Finally, we have here a case in which we do not enable rotation either; but we enable displacement. What is going to happen? This column cannot rotate on the top, it cannot rotate on the bottom, but it can transversally move on the top. How are we going to find the wave? We will have here a half wave, such as the one we had in this case, and then we would have the other half wave below. As a result, the buckling length for these support conditions is equal to the height of the column. But that is quite different from this case because here, that was also the height of the column, but the buckling deflected shape, the shape taken by the column is fundamentally different from this case here. Why is it important? Because the critical load, as I said before, depends on the height or more exactly, the critical load is inversely proportional to the square of the buckling length. That is very important. That means that if we divide the buckling length by two, then the critical load is multiplied by 4. You remember, at the beginning, I proposed you two configurations. The configuration with the Romans. This Roman temple essentially has this configuration. The columns are not really held by what is on top of the structure and that means that we can have a displacement with a buckling length of about 2h while the solution of Norman Foster rather locates here, with a buckling length which is the half. We thus have a factor 4 on the buckling length, so the solution of Romans and the solution of Norman Foster.. that means that Norman Foster, for the same column, would have a critical load 16 times larger. You remember, his column does not have the same size because he does not need such a large critical load. It is much smaller. And moreover, that is amplified by the fact that in the case of Norman Foster, steel was used, a material with a very very large strength while Romans used stone, which has a smaller strength. That really explains the possibility, if we choose well the support conditions, to decrease the dimensions of a column for a given length. There is another case which is interesting, which is the one of an intermediate support on a column. Here I have the basic column, the one we have already seen, that is the same cane than before. It deflects in this way. But if now I hold it in an intermediate way; it can for example be a facade which leans on the slabs of a building; then, the facade element is going to deform in this way under the effect of buckling. And then, if we had at first, a buckling length which was equal to the height of the structure; I am going to add the total dimension, the total length of my column which is h. Here, I have a buckling length which is equal to h. Already in the second case, we can see that we have two half waves. Actually, we have a complete sine in the second configuration. Here I have a buckling length which is equal to h/2. That means that the critical load, if we keep everything else identical, the critical load is already 4 times larger. If now I manage to have two intermediate supports, then the deflected shape is going to have this appearance, with a buckling length which is equal to the length between the supports, the distance between the supports, and in this case, the bucklin length is equal to h/3, which means that the critical load will be 3 squared so 9 times larger. If we manage to hold in an intermediate way a compressed element, it will significantly increase its buckling resistance and it will therefore decrease the risk of instability. In this video, we have seen two fundamental cases of stability, a column which is hinged on the top and on the bottom, and a column which is only clamped at its base. We have introduced the concept of buckling length and we have seen the effect that the supports have on this buckling length and, as a consequence, on the critical buckling load. We have also seen that intermediate supports along a compressed element contribute in a significant way to increase this critical load by reducing the buckling length.