Hello. In this lecture, I am going to introduce to you a new type of structures, called arch-cables, which will be the last type of structures which we will see within the framework of this course. We will see that arch-cables are a type of structures in which the arches' thrust, which we know well, is taken in a different manner, which is why we will need to introduce a mobile support. We will see how to solve this kind of structures and also how to model them using the applet i-Cremona. What is a mobile support? Well, in my model, it is a support which looks a little bit like a roller skate. So, here, we have casters, which are going to be able to roll on the horizontal surface on which I am going to place my cart. I call that a cart. What can we notice? A displacement is possible in the horizontal direction. This means that there cannot be a force at the support in this direction. Thus, the only force at the support, here, will be a vertical support force. We can notice that, as before, this kind of support also allows a rotation. Let's also note that, even if we will not necessarily see often within the framework of this course, we can very well have a mobile support which rolls on a surface which is not horizontal, for example a vertical surface, but also, possibly, an inclined surface. We are going to see, obviously, in the rest of this course, what the chain is used for. Here, I have introduced the mobile support on the right in my arch. When I have introduced two loads on my arch, obviously, the arch wants to push, and as it cannot push since the mobile support moves, the whole structure collapses. The solution, as we have already seen it in the picture, is to link the mobile support to the fixed support by means of a chain. In this case, we can see that we can normally load the arch. Let's look at the internal forces which act in our arch. So, we have loaded 10 Newtons on our arch. Here, I already solve the case of the arch and we will see afterwards the one of the arch-cable to be able to compare. So, for the arch, I have introduced 2 forces of 10 Newtons, And the equilibrium, I do it very quickly, because you have already done it several times, and we have done it together. So the internal force in the right leg and in the left leg, as well as in the horizontal part of the arch. Thus, this arch is only under compression. Then, if we take an interest in the forces at the support, using the force in the right leg and in the left leg, in the other direction, we obtain the usual forces at the support, that is to say an horizontal thrust and a vertical force, which is equal to 10 Newtons of course, here. And then, we have a force, V and a force H at each support. How is it with the arch-cable? We are also going to load it with twice 10 Newtons. Do not forget that there is a cable because, otherwise, with the mobile support... You can see the symbol for the mobile support on the right, we will come back to it in a little bit, it is a triangle which is on 1 or 2 casters, here, 2 casters, it symbolizes the fact that it can move transversely, it is quite similar to my little cart which you have seen before. The Cremona diagram, with twice 10 Newtons, is very similar to the one of the arch because, if we make the equilibrium of the points of application of the loads, we will obtain the same thing. So, we are going to first take an interest in the free-body which is just next to the support on the right. What do we have in this free-body ? Let's solve this free-body. We obviously have an inclined leg of the arch, we have the horizontal cable, and what we know is that, we can only have a vertical force at the support, which comes from the ground. Well, let's come back to the Cremona diagram, it is this force, here, in the leg along which we are going to travel in the other direction. Then, we are going to add the tensile internal force in the cable, and a vertical force at the support. So, here, this is correct, we only have a vertical force on this support. Let's now look at the free-body close to the left support, which I draw again. What we have, acting on this free-body, is then an inclined leg, which comes from the arch, the cable in tension and, certain ground reactions which we will see a little bit later. What we know is that here, we have compression which acts and here, we have tension, the force T which we have just determined before. So, we have, acting on this free-body, the force T, well, I must draw it again in the Cremona diagram, this time, it is not possible to find a combination. To this force, we add afterwards the diagonal internal force in the leg and finally we obtain that the vertical force at the support is the only component which we have here, thus we only have a vertical component. It is interesting to see that, just by introducing a single mobile support, actually, we get the effect that on both supports, there are only vertical reactions. Now, be careful, we got this effect because there only are vertical loads acting on the system. If we had an inclined load which acted on the system, of course the support on the right could only support a vertical component, however, the support on the left could support an inclined component. Let's see what happens if I now add an asymmetrical load on my arch. Well, as on a normal arch, we can see that the arch tends to go down under the additional load and that actually it is necessary to give it the opposite shape, that is to say to make the arch go up here, where we apply the load for it to be stable. What this means, is that arch-cables behave, for the arch part, exactly as arches. So, the stability problems are still here, and we still need to stiffen them but we have seen before that we know how to do it. So, let's look at how to support the thrust and let's see how it can be advantageous. We have, here, a cathedral and if I indicate, here, the internal space of the cathedral, we can see that it is actually quite limited. There is a main nave and 2 secondary naves. Everything we have around is essentially used for supporting the thrust. On the right, I have drawn the same cathedral, with still the same volume which is useful for people who are inside, and we have taken off everything which is used for supporting the thrust. And, we can notice that it correspond to a lot of matter. Of course, it makes the beauty and the majesty of the cathedral, my purpose is not to take off the flying buttresses, where they are, but if we want to make modern and economical structures, it is interesting to know that it is thrust. If I simply introduce a cable, here, the forces at the supports, under vertical loads, will only be vertical and can thus directly go down in the columns which we have on the left and on the right. And all the flying buttresses and the other systems of buttress are not necessary anymore. We do not need either to introduce a mobile support, there are no casters because in this case here, these columns, which are very long, very high, they are flexible and thus they enable a slight transversal movement. By the way, I think that you have a feeling, if we take off all the matter of the flying buttresses, we feel a little bit that the arch will want to open. Yes, it will want to open but when it will start to open, the cable will be put in tension and it will say : "I do not go further, there will only have vertical internal forces from now on". Let's now look at how to model an arch-cable with the applet i-Cremona. I start by modelling an arch. Well, you can see that it is really very similar. So, I introduce a support on the left, a support on the right. I activate the funicular polygon and I give it the adequate shape. There we go, it is probably the shape of the funicular polygon of this vault. Now, if I want to introduce a mobile support, I erase the second support. If we introduce a mobile support, it will always be the second support. I take it off, and I press the button "control" on the keyboard or I click on the button "control" of the applet. And I introduce, well, I am going to introduce it on the right, in this way you can see well its symbol, a mobile support, which does not have the small casters but we clearly see that the support is separated from the ground, it can slide a little bit as on an air cushion, it is another symbol. We can now place this support just at the place where it was before. And, we can notice that, indeed, the thrust is supported as for the other elements. If I place my cursor in the middle of the cable, I obtain the internal force which acts in this cable. Another example of arch-cable structure, by necessity, it is a an arch, it has been used for the shuttering of a big bridge, a quite imposing structure. And because this bridge crosses a river, the builders of that time have decided to create this arch and to transport it making it float on 2 barges which we have on the left and on the right. So, here, we have a barge, and, here, we have another barge. Obviously, in this case, the only load which acts on this arch is its self-weight but it is enough to create a thrust. But, in this case, it is very, very clear, the only things that the barges can offer, they are ships, are vertical thrust, it is the Archimedes's force, which is necessarily upwards. So, it is absolutely impossible to support the thrust. The solution which has been adopted in this case, and we can see it in the water, with small marks, small waves, is to introduce a series of cables, which link these 2 barges, in such a way that the thrust is compensated and that there is only a vertical thrust, since obviously, it was impossible to take an horizontal thrust with water. Here is a picture of the finished bridge, you can see the very ingenious aspect of the solution which has been chosen, the barge not only enables to move the formwork but it also enables, once we have finished the arches, lowering the formwork, to move it, to bring it at the place where we want to build the second one, and so forth, and thus to have a system which has certainly been very economical for the construction of this bridge. Until now, we do not have, actually, seen yet how to really make mobile supports. You have, on the left, the symbol for the mobile support. So, we can enable, here, an horizontal movement, and we will only have a vertical force, perpendicularly to the surface of the support. In practice, we sometimes use supports shaped like a roller but it is important to say that if the displacement which we must have, must be significant, it is not very easy and on the other hand, the surface on which the rollers roll can easily becomes a little bit dirty, if there are small stones which fall, if there is a little bit of dirt, and thus these supports get fulled up and do not necessarily work very well in the long run. So, we have tried to find new solutions. On the left, I show you a relatively old solution, that is what we call a pendulum support. By all appearance, we can see something that we know well, something which is circular. But actually, you can see that there is a small tooth, on the top and on the bottom, that is a shape which is even more particular, but there is a small tooth which enables to control the way this support moves. So, how does it work in reality ? We have fixed to the ground a plate with a slit, likewise, there is the same plate with a slit which is fixed to the bridge. And, our system, is placed between both, so, that is a circle with a tooth which enters into this slit. When a displacement occurrs, both slabs remained where they were before, but the system has moved diagonally the slits are still aligned, in such a way that the system does not get skewed. Thus, we have a displacement. This displacement is caused by the fact that the cable must lengthen to enable to support the internal forces, on the one hand, and on the other hand, by the thermal effects, if the bridge lengthens or shortens, further to temperature differences. Then mobile supports are common solutions for bridges and various types of structures. A more modern system, that is what we call a sliding support, a system which is quite common nowadays. So, for this support, we have the same system, we have a piece which is fixed in a permanent way to the foundation, another one which is fixed in a permanent way to the bridge. It is generally steel. Inside, we have, each time, a block of rubbery products, it is generally a synthetic product, like neoprene, a rubber which is very hard and which is very durable. Then, between both, we have two layers of a material which we have already seen, which enables well the sliding, which is Teflon. Well, both, neoprene and Teflon, are protected brands, there are non-protected versions of these materials, but they really enable the support to slide, without having any friction or almost without having any friction. In this lecture, we have seen a new type of structure, the arch-cable, which is the combination of an arch and a cable, in which the cable supports the thrust. The use of an arch-cable requires generally the use of a mobile support, which is a support which enables a movement alongside the line between the supports of the arch. The consequence of the use of an arch-cable is that the forces at the supports are vertical, in the presence of vertical loads. We have also seen how to use the applet i-Cremona to model an arch-cable.
