Hello. In this video, we will look at which shape takes a cable according to the type of load to which it is subjected. What we have seen until now, it the case of loads which are distributed along the axis of the structure, often horizontally. But it is not a necessity, as it was the example of the case of the Golden Gate bridge. There is also the possibility that the load is uniformly distributed along the length of the cable, and we will see that it leads to a slightly different shape which has a particular name. In this video, you can see a cable subjected to 13 loads uniformly distributed, and equal, with an equal spacing, which thus take the shape of a parabola. Then, we superimpose on this white cable, a red cable which is simply subjected to its self-weight. In the case of the self-weight, the self-weight acts regularly, not along the horizontal axis, but along the axis of the cable. The cable weighs, for example, one kilogram per meter, but per meter of length of the cable. If we take a close look on these two cables, we can indeed notice that the red cable is lower than the white cable, than the cable which has loads uniformly distributed along the horizontal axis. In the middle, they are approximately at the same level. This, it is OK. But here, on the left and on the right, it is always the red cable which is lower. Why ? Well, because if we look at a small segment of cable, I am going to draw it in red, which as a certain small weight which acts along its length, let's say one Newton per meter. Well, when this segment of cable is inclined, to cross the same horizontal distance, I would need a longer segment of cable. Thus, this cable here will be heavier for an equal horizontal distance, for example, the distance between two red weights. We can see this in this figure. On the left, we have a cable which is subjected to a uniform load, q, one kiloNewton per meter along the axis x. On the right, we have a load distributed along the length of the cable. If we represent it along x, we obtain, here, in the middle, the same load than on the left, q, one kiloNewton per meter. But as the inclination increases, we get a load which is larger than q, one kiloNewton per meter. And since this load is larger at the ends, well, it is logical that this cable should go down more vertically. On the left, we have a second order parabola which is well-known. On the right, we have a curve which we call the catenary. Not very surprising, since it is really the shape that a chain (catena in latin) takes under its self-weight. And it is a curve that some of you may have studied, which has a mathematical definition a little bit more complicated. Well, I do not want to go into further detail for this course. What is important is to be able to distinguish that according to the nature of the loads, from where they come, the shape of the structure, the proper shape that the structure should take, will be slightly different. If we compare these two shapes, we can notice that, for what I did on the blackboard, we have maximized, here, this distance between the black cable, so the second order parabola and the red cable, so the catenary. In reality, our structures will usually be much flatter, and the difference between a parabola and a catenary is quite small. So, I would like that you clearly keep in mind that these two shapes are not the same, and that they exist and you must be able to identify them. However, at the same time, if you are looking for the proper shape to give to a structure, it is not very important. If you you draw second order parabolas, and that afterwards we have to build catenaries, the geometric difference will be quite small. To conclude, two examples of structures. First, the exhibition hall of Portugal, for the Universal Exhibition of 1998 in Portugal. We have here, an extremely thin structure, with an approximately 130 millimeters thick blade of concrete, thus, something which is extremely slender, taut between two supports. What we can immediately notice, is that the supports themselves are significant. Well, it is not very surprising, since we know that if we have an extremely taut cable, like this, what does it mean ? It means that there will be large forces at the supports, particularly large horizontal components of the forces at the supports, and these horizontal components must be carried by structures. That is why these vertical structures have numerous walls, a lot of intermediate walls. But it is a very beautiful structure which covers a public space in a very enjoyable way. The second example it is this airport hall in the United States, with also the requires to have a large surface with the least amount of supports possible, which is made possible by a cable-type structure. Indeed, we have a structure with a quite low thickness. We do not see it well here, because there is an edge beam which has been introduced, but we have a structure with a quite low thickness which is taut between the supports that we have on the left and on the right. Note that the supports, the more or less vertical columns, are themselves quite significant to be able to carry, the horizontal component of the internal force which is very significant, since it is a structure which is also quite flat. These two structures take the shape, essentially, of catenaries. In this video, we have seen that the loads can be either distributed along the axis of the structure, either distributed along the length of the cable. It induces differences in the loads which act on the cable, and in the shape which it takes. If the load is uniformly distributed along the axis of the main structure, then the cable takes the shape of a of second order parabola, as we have seen in the previous video. If on the contrary, the loads are, in a predominant way, distributed along the length of the cable, then the cable takes the shape of a catenary. These two shapes are however quite similar and it is not very important, in any case at this stage of the design, to distinguish which one is which one. However, within the framework of this course, you will be asked questions about the type of shape that a cable takes. Thus, it is important to be able to make the distinction at the theoretical level.
