Hello. We have seen in the previous video that a pretensioning cable is an efficient system to stabilize a cable. In this video, we are going to see some additional examples about shapes that the pretensioning cables can take, particularly we will finish with an expansion to the third dimension, which is very interesting, since it offers the possibility to cover spaces. In this previous video, we have seen that the combination of a load-bearing cable, we cannot really see well here, and of a pretensioning cable, enables to stabilize a structure, decreasing its deformations, which happens, for example, during the accumulation of snow or when the wind blows on the roof. Here, we have another example which is a small concrete pedestrian footbridge. The structure itself, it is a concrete strip, we call it a ribbon, which is very, very thin, and then its weight would not be sufficient to limit the vibrations created by persons crossing it. It is also very thin, which means that its stiffness is not enough to consider it as a stiffening beam, at least not completely. So, what we have, in addition, on this footbridge, is a stiffening cable, we see it better here, in the elevation. Thus the footbridge itself, the concrete part, in which there are cables, is in tension. It is the same solution than the solution used by Freyssinet for the Youth House in Firminy-Vert, except that in this case here, it has been designed for the people to walk on the cable, and, as in the solution of the cables beam, the stiffening is providing by a number of small diagonal cables, but all these cables being in tension, are almost invisible, if we look at the photo, it is difficult to distinguish them. Here, we can see in the lower part that the construction to which the cables are anchored is also monolithic, and is fixed to the rest of the abutment, but it could be independent. We can thus have a system of independent cables which also stabilizes a cable which needs it. This configuration, with multiple cables, which is characterized by a significant transparency, has been used for greenhouses, that is to say places which need to let enter a maximum of light. They are vertical elements which are essentially exposed to the wind pressure. The effect of the wind is a pressure which has the kiloNewton per square meter as unit of measure, we are not going to make any calculations here, but there are two types of configuration. When the wind blows towards the facade, we have a pressures which presses where the wind presses on the facade, and in this case, the structure which we have here horizontally works in the following way : the cable which is inside, I draw it in red, it is the load-bearing cable, and the cable which is outside, I draw it in orange, it is the pretensioning cable. Then, between these two elements, we have here compression. If, reversely, the wind blows in the other direction, we have an effect of suction, then the wind pulls on the facade, it can be a very important effect, especially for lightwight structures like this type of glass facades. Then, we have an inversion of the internal forces, the load-bearing cable becomes the one which is outside, and the cable inside works as pretensioning cable. The elements between these two cables are still in compression. If we consider the rest of the length of these elements, when the wind presses on the facade, these elements are in compression, these small elements here, as well as these parts here, while if the wind pulls on the facade, the elements are in tension. We can see that we have a part of the element which is in compression, and the other part which is in tension, it is absolutely possible. Note that for these loads to be vertically transfer into the ground, we have on the left and on the right, maybe I am going to draw them in pink to avoid the confusion, we have on the left and on the right, similar structures which also work with pretensioning cables and load-bearing cables. This structure is a much more complex structure in which we really use the third dimension. It is the Olympic skating rink in Munich, built in 1983. First, there is an element which we have not seen yet in this course, but as we are going to see it very soon, it is not a problem to introduce it. It is an arch, an element which works in compression, that is why I draw it in blue. Afterward, we have a series of cables. The cables which have curvature upwards are load-bearing cables. And, to prevent their movement, they are stabilized by cables which pull downwards, which are thus pretensioning cables. With all this, we have a network of cables which cross each other, but we do not have yet something which enables us to be sheltered to attend a match of hockey, and we can see in the picture on the right how it has been solved, we have put an fabric, above this network of cables which has hugged its shape and which offered watertightness when it was raining and where it was snowing, the spectators under were not wet, since the water was evacuated onto the edges. There are also some elements whose we did not have seen the function. All the load-bearing cables and the pretensioning cables, are anchored into the edges, into these cables that I draw here in green, which are edge cables. They are normal cables which are subjected to forces which comes from the internal forces in the load-bearing and pretensioning cables, and which are themselves supported by a system with elements here in tension, and elements in compression. We maybe should also add that there are also here edge cables, and small hangers which hang up the ends of these cables to the edge of the arch. With this, we have explained how this part here works. You can notice that there are lots of cables, and this, it is a problem with this kind of constructions, knowing that in addition, it will be necessary to carry out, to avoid that it moves, it will be necessary to carry out connections at the intersection of each of the cables. Thus, it becomes a relatively complex construction, and we have quite quickly searched how to simplify it, and this is the way we can simplify it : instead of using a real network of cables, we can directly use the fabric which we envisaged to stretch above as bearing element. Obviously, it will be necessary to reinforce it. By the way, we can notice that there are reinforcing bands in the fabric I have taken here as illustration, but let's look at the various bearing elements of this structure. We have a central pillar, which goes up until here and which hold this ring. We also have inclined lateral pillars, also one here, an so forth. We have cables which catch hold of these pillars. It looks a lot like the details which we had on the edge of the skating rink of Munich. We also have edge cables. And then, all the rest of the structure is constituted by this membrane, which is made in a fabric which can resist to tension in several directions, there is tension in the direction where we would normally have load-bearing cables, and then we also have tension in the direction where we would have pretensioning cables. Altogether this yields a quite light structure, very transparent since it has just very few bearing elements, and the construction cost is also radically lower since we did not have had to make all these cables and all these intersections of cables. We are going to summarize here the various pretensioning systems which we have seen. The first solution with an external cable and hangers in tension. The second solution that we have seen for the greenhouses of the "Parc Citroën", we have a second cable, which uses the same supports, and then connecting components which are in compression. The third solution, it is a combination of these two solutions, that is to say a cable which is sometimes below, so with hangers in tension, sometimes above, so with elements in compression to transfer the pretension. Fundamentally, it is a solution between the first one and the second one. We can also have a solution in which we call on external supports, to anchor the cables which are going to stabilize the structure, and finally, we can call on a three-dimensional system, as we have seen it. (here I make a sketch which looks a little bit like a camping tent, but it can be something more sophisticated as you have seen it), which enables the stiffening by means of networks of cables or of membranes. In this lecture, we have seen how to stabilize a cable by means of various pretensioning system, noticing that we could use multiple cables, and position them in various ways in relation to the load-bearing cables, it had as consequence that we would have either hangers, either elements in compression to introduce the internal forces. We have also seen that it is possible to make a three-dimensional stabilization using cables which are not on the same plane, but which are on perpendicular planes to the load-bearing cables, and that we can even use a membrane system, which is a well stabilized system when it is put in tension.
