Hello. In this video, we will see how to stiffen cables because, as you have already observed during the manipulations of the previous videos, when we place a load on a cable, it deforms a lot, and it is not very comfortable, and it can sometimes be problematic. We will see how the big deformations of the cables can be limited by using diverse types of stiffening, to improve the comfort of the users, it is not very enjoyable to cross a footbridge which moves a lot, but also the safety. For example, the deformations of a bridge while a high speed train passes are going to determine the maximum speed which is possible to avoid a risk of derailment. And for this, we will see various methods of stiffening which can often be combined together. In this video, we can see the effect that a load added on the right on the cable has on the entire cable. The right part of the cable, significantly goes down, while the left part goes up. There is no place along the cable which which does not move when we add a vertical load. We look at this example dealing with the case of two cables which have shapes relatively close to the cables of real structures: one with a polygonal shape, with only two permanent applied loads G, one with a parabolic shape with a load uniformly distributed along the axis x. And on the right, we will see how, when we add to these loads, a variable load, the shape of the cable is modified. On the left, the shape of the cable, we already know it. It is already drawn, by the way. It is rectilinear between the supports and the loads for both concentrated loads, and then it is parabolic, well, in that case, with very small changes of angles when we introduces loads, but it is pretty much a continuous shape. When we add an additional variable load Q, for example the passing of a vehicle, but it can also be persons who gather in a house, or even snow which piles up on a roof. We add a variable load Q, and we look at how these cables deform. Well, we have seen it in the video about the cable with concentrated loads. So, the cable goes down under the effect of the load Q, on the right, and the rest of the cable goes up. When we apply a load in addition to another distributed load, we have the same thing which happens. So, on the right part, the cable goes down, and on the left part, it tends to become at once flatter and a little bit higher. Everywhere, but particularly where we have applied the additional loads, we have a large displacement, also here, which is a problem since, one more time, it can scare someone who crosses a footbridge, that is the example which we can see, here. That is very funny to do that as sporting activity, but it is less enjoyable to do that if we are coming back to home with shopping or a school bag, it becomes dangerous. Likewise, if there are distributed loads which, would maybe be more characteristic of groups of persons, or of big trucks. In this video, we first see the cable under its self-weight, which has a certain shape, the shape of the catenary which we know well. If we add a weight to this cable, it takes a triangular shape with very significant displacements. In the second part of this video, we have the same shape, but for this shape, we have already added a series of rectangular red weights which symbolize the weight of a structure, since structures always have a self-weight. This structures, let's say that it is made of concrete, it is quite heavy. If now, we add to it one, then two loads, we can notice that yes, it always moves in the zone of the loads, but in a much less pronounced way. So, we have a way to decrease the deformations, if we start with a significant weight. The first solution of stiffening is thus a stiffening by a significant weight. If we first look at the case of the roof of the airport of Dulles, on the left, which we have already seen. Well, when we send a few people on the roof to do sealing works, for example, we do not wish that these people be in danger, respectively, that there are too much vibrations. So, this structure has to be stiff enough. Another case which often occurs, it is the presence of snow on the roof. And if there is wind, the snow will only pile up on one part of the roof. For example, on the right, as I have shown. The solution used for this roof to avoid the problems, is that it is quite heavy, in concrete. Thus, the weight of the persons is relatively small compared to the weight of the roof. Likewise, the snow load is relatively small. The structure on the right, built by Le Corbusier, in France, in Firminy-Vert, uses the same principle. We have a series of load-bearing cables; here, we can see them well; here, another pair. And then I just draw one of both following cables, and so forth. Note that this series of cable is not itself a roof. It is necessary to add something to get a roof. What Le Corbusier did, is that he has placed between these pairs of cables, precast concrete panels which we can clearly see, and these panels which touch each other, provide watertightness, and thus create the roof. Above, we have also added an external watertight cladding, but it is relatively lightweight. Then, we have the advantage that this solution has a certain weight. So if someone, once more time, has to go on the roof to do, for example, sealing works, well, there is no problem since the persons will be relatively light compared to the weight of these precast concrete panels. The picture on the top is a photo which was taken during the construction of the building. At the bottom, a more recent picture in which we can see that the structure has approximately the same appearance. We still distinguish the cables. In this video, I try to make roll one of my 10 Newton weights on a small board which is hung up from the cable. Note that under this board, there still are the red weights which symbolize the weight of the structure. A structure has always some self-weight, that is why I let these weights. But the novelty is the addition of the horizontal deck on which the weight of 10 Newtons rolls. And we can see that when I place a weight, or even two weights of 10 Newtons, well, the displacement where the load is, is quite small. If we turn this small timber board on its edge, it becomes even stiffer, and we can see that the deformation, whether it be with one or two loads is almost invisible. We have thus transformed our system into a very stiff system. It is, in this case here, a stiffening by a stiffening beam. We have already seen, in the bottom left-hand corner, the case of the Golden Gate bridge, on which we have a very large stiffening beam, and we have said that it is called a truss, with this system in zigzag which is very very common. And below, we have the Chain Bridge in Budapest, on which we also have the same system of stiffening which we can see enlarged in the bottom right-hand corner. Here, we can see this beam with the vertical and diagonal elements which constitute this stiffening beam. This system is very efficient, and has been quite used for the stiffening of the long span bridges. Another solution is to add cables, in orange here, in diagonal which we call stay cables; we can see that the stay cable on the right directly carries the load which is applied on it, whether we add one or two loads, with a quite small resulting deformation. We can see this system in the Brooklyn Bridge. So, on the Brooklyn Bridge, we can first notice that there is also a stiffening beam. I told you, the systems are often combined. And then, there is a series of stay cables which are cables which are hung from the top of the pillars, and which go down to various locations to be directly connected to the deck. It is here, a stiffening by a system of stay cables. If we look at the right part of the photo, we can recognize these stay cables which are hung on the top of the tower and which are linked to the deck. There are different families of stay cables. Here, I draw two of them which are hung from the central tower, but we have also a family of stay cables which are hung from the edge towers, on both sides We can see here, another family of cables which vertically goes down to also reach the level of the deck. These cable are classic hangers. As we can see them, here, in the view, on the left, they are classic hangers which link the deck to the load-bearing cable, since on this bridge, we also have, as bearing system, a parabolic cable similar to the one we know for the Golden Gate Bridge. Another solution to be able to carry variable loads, always with the presence of the self-weight of the structure, is to have multiple cables. We know that if we place the load more on the right part, well, a triangular shape which goes down, more on the right would be favorable. If we place the load more on the left, then a triangular shape with the lowest part on the left would be favorable. Then what we do, it is that we combine two or several cables to get a shape which is adapted. If the load is more on the left, it will be more carried, in our case, by the red cable. If it is more on the right, as in the case in which I added two loads, it will be more carried by the white cable. That is this system which we observe on the Tower Bridge in London, for the lateral spans of this bridge, where we have two cables, one which begins on the lower part and which goes up to the upper part, and the second one which is first above, then it is below. And these two cables collaborate together, the pink cable tending to carry more the loads when the vehicles are close to the support of the left abutment, while the blue cable will tend to carry the loads, when they will come closer to the intermediate tower. Let's also note the presence of vertical and diagonal elements between these two elements of cable, which are here to stiffen the system even further. It is here a stiffening by multiple cables. In the case of this bridge, we have two cables, but we could absolutely have a system with more cables. When we added weight to our structure, in the first solution, what did we do ? We placed internal forces in the cable of the top, the load-bearing cable, we increased the internal forces which acted in this cable, and the result of this increase of the internal forces was a decrease of the deformation. However, we had the drawback to have had to put lots of weight, which is at once expensive because we must pay this weight, and doubly expensive because we must have a cable which resists more. A solution would be to be able to add the internal forces without adding weight. Obviously, we should continue to pay for the increased internal forces. That is the solution which is presented here, with what we call a pretensioning cable, it is the lower red cable. This cable is oriented downwards. A priori, it would work well if the loads were upwards. Well, that is exactly the way they are. The orange hangers between both cables transmit a tensile internal force between both cables, the consequence is that both cables are in tension, even before the first applied load. That is why we call this a pretensioning system. And, we can see that when we add one or two loads to this system, it only deforms a little bit. Note that I did not add the self-weight on this model, but if I had add it, it would not have changed anything, it would have just deformed even less. It is thus a stiffening by pretensioning cable. Be careful to vocabulary: it is not a pretentious cable, it is a pre-tensioning cable. So, we observe, here, in this structure, the load-bearing cable, which I draw, here, in red. In the elevation, below, I also draw it in red. It reaches the pylon and it goes down into the ground, we can see it here, on the right with a double cable which is anchored in the ground. Then, there is a second cable which is this cable that I draw, here, in orange, which is in tension between both supports of the vertical cables which bring back the load-bearing cable to the ground. And between these two cables, we have hangers, which stabilize the system. And we finish with, obviously, the compression which is carried by two masts, on the left and on the right. We have then seen, here, the load-bearing cable, and below, the pretensioning cable. This second structure illustrates the same principle. That is what the engineer Jawerth has called a cables beam, but it is really a pretensioning system. What do we have in this system ? Well, that is very similar to what we had before. So, we have cables which are anchored in the ground, a load-bearing cable, a pretensioning cable. Then, instead of having hangers, we now have a system of diagonals which are all in tension, therefore they are all represented by cables, which makes for a very lightweight structure. Finally, we obviously still have masts to carry the vertical loads and bring them down to the ground. There are other types of pretensioning systems. There is a special video which follows this video which only deals with variations of the pretensioning systems. In this table, I am going to resume the six systems which we have seen to stiffen the cables. The first stiffening is made by means of a significant weight. What does it mean, significant ? Well, it means that it clearly has to be more significant that the vertical loads which apply on the structure. The second system, it was a stiffening beam. So, this beam is hung up from hangers which are hung up to the cables, and it offers an adequate stiffness to enable that, when a load is applied on the stiffening beam, it is going to be transmitted, not to only to one hanger, which would locally deforms the cable in a very significant way, but it going to be transmitted to several hangers, which is going to limit the deformations. It is a little bit similar to what we have seen before with the weight. The third solution, it is a solution with stay cables. Here again, this solution can be combined with a system of hangers. We have then, a similar system to the one used for the Brooklyn bridge, with hangers, and we add stay cables which directly link the top of the pylon to the places where the diverse hangers are connected. It is also possible to build this type of structure without having the load-bearing cable and the hangers, but simply the stay cables. We will see again this system a little bit later in this course. The fourth solution which we have seen, it is a solution with multiple cables, two or several cables. So, typically, a cable which is lower in the left part, and a cable which is lower in the right part. In practice, this solution is usually combined with the following solution. So, we have the same two cables, pink and blue, than before. But as we have seen it in the case of Tower Bridge in London, we add a stiffening system. Actually, it is a stiffening system of the cable, itself, which enables to the cable to resist when the loads are directly applied to it. This solution is actually a combination of the solution two and of the solution four. Finally, the last solution which we have seen, it is the solution with pretensioning cables. We hang a cable which has an opposite curvature to the load-bearing cable, with hangers which link both cables and which induce internal forces in the load-bearing cable, in such a way that this one, when we will add it a variable load, will just say : "yes, I have a variable load, but as I already have large internal forces, I am not going to deform a lot. I do not need to deform a lot to be able to carry them." During this lecture, we have seen a number of methods to stiffen cables. We have also seen, in the case of examples, that these methods are often combined, always with the presence of self-weight, because there is no structure which has no weight. Finally, in combination, stiffening systems by beam, by multiple cables, by stay cables, by pretensioning and by stiffening of the cable itself.
