Hello, in this video, we are going to deal with hinges, particularly with hinges in arches. We will see what are the hinges, how we can make them, how we build them for the various building materials. We will see what influence the presence of hinges has on the behavior of arches. Finally, we will conclude by a study of the mode of failure of arches. Before continuing to talk about arches with hinges, let's note that, whether it be during the time of the Romans, or like here, during the Middle-Age, arches were historically built without any hinges. On the Ponte Vecchio, we have an arch, quite slender by the way, which goes directly from one pier to the other one. It does not have the possibility to turn and it works very well. A hinge is a mechanical device which enables a relative rotation between two elements. The supports which we have already used for the cables and for the arches enable a rotation of the element which is supported, which can go up or go down a little bit. Obviously, cables are hinged but this is not really important since we do not need to use hinges. A cable is flexible enough for this to automatically happen. That is not the case for arches. We can see in this picture on the left the bottom of a big arch. We will see a complete photograph of this arch later. We have the part which comes from the bridge and we can see the support device, a metallic system, which then is supported by a large system, probably in concrete with masonry. Between these two elements, we can see a big element which I indicate in red which corresponds to the element of the hinge. On the right, we have another example of a very similar bridge, by the way designed by the same design office, it is the Eiffel office in Paris. We have the part which comes from the bridge and the part which is linked to the foundation and between both, we have this hinge. Note the scale which is given by the person who stands, looking at the bridge. It is thus an extremely significant element and we can imagine that the forces which pass through this hinge are also very large. By the way, that is why we can see elements here, in diagonal, certainly under compression which bring the internal force which is in the arch to the hinge. What is sure is that all around the hinge, there is only air. Thus, a hinge is an obligatory point of passage for internal forces. For us, that simplifies in a way the reflection and it was probably one of the reasons why hinges were introduced in the first place during the industrial revolution. Another reason for the introduction of hinges is that they enable more easily to link between elements, for example, which would have been made separately, transported on the construction site, and assembled on site. Here, we have an example of hinges for a concrete structure. We have the hinge at the key. What I draw in green is the edge of the concrete which goes down, and here which goes up, which has a very little empty space between both, and here we have the concrete hinge. We can see that there is a reinforcement which goes through these hinges. Obviously, there is also concrete. The key is the highest part of an arch while the springings are its support. Here we have a hinge at the springing, once again, I quickly indicate the surface of the concrete, and we have a hinge just in the middle. Concrete is different from steel because we can very easily give it any shape. In the concrete, which is poured in the fresh state, it is necessary to place elements which are going to give it its shape. What was used here is cork. Cork is the material which is used to make stoppers. It is a very compressible material because the rotation will tend to close this joint and in the lower part which tends more to open, at the key, we used wood. And at the springing, we use exactly the opposite placement, with cork below and wood above. This is a construction detail, how to build a concrete hinge. Note that we have modern materials which would facilitate this work. We would probably use polystyrene to make this kind of detail nowadays. When the structure is in timber, it is quite difficult to create a timber hinge. Generally, we will have a timber part on the left, another timber part on the right. Here, this is a hinge at the key once again, and we are going to use a metallic plate, which we are going to nail to each part, the left part and the right part. And we will place the hinge afterwards, we can see it here, in a sectional view, this is a cylinder, a bolt indeed. We will place this hinge and the rotation will occur thanks to that. That is a quite similar detail to what we have in steel construction since actually they are steel pieces which are assembled together. Note that precisely, as I said, this enables to transport the left half-arch and the right half-arch separately to the construction site, which can make the execution simpler. Thus, sometimes, we will use a hinge simply to be able to assemble the structure. We have the same detail, when we arrive on the support with two steel parts, an upper part and a lower part, and a hinge, once again a big bolt, with a very similar detail to what we had for steel construction. This video shows the behavior of an arch with three hinges but subjected to its self-weight which is not small, and then I exert on it an additional concentrated load and we can see that the arch deforms. If we take a closer look, we can see that the internal forces must inevitably pass through these three points since just next to these elements, there is only air thus the internal forces must pass here and somewhere here. The funicular polygon must necessarily pass by these three elements. When we do a graphical resolution, we need to find a funicular polygon corresponding to the loads and passing by these three elements. It is not very difficult. Sometimes, it is necessary to do an iteration to obtain this result, but is is absolutely feasible. Here, I show you an applet which solves this arch with three hinges. So, to model the hinge at the base, I use supports and I introduce the weight of each of the arch stones, that is how these elements of vault are called, which weigh three Newtons. And I introduce this weight on the center of gravity of each arch stone. Now, it is necessary that the funicular polygon should pass by this point. So, if I activate the solving as usually, I can find a solution which passes by this point. But actually, the polygon must always pass by this point, even if, for example, I add a new force. So here, I should do an iterative work to each time find the good position. It is not necessary, the applet has a solution to this. Using this button, here it is written : "initial tangent or point of passage of the polygon", you have maybe already activated it sometimes, seeing a red ball which passes and here you can see that the funicular polygon always passes, even if I go to the other side in tension, by the place I prescribe. Note that there are places, for example here on the right, which are impossible and the applet sometimes gets lost. If you get lost with the applet, simply come back to the mode "initial tangent" and you will manage to get it back. Here, I place the hinge at the proper position and now, we can see that if I have a variable load, well, this variable load makes change the shape of the funicular polygon but always make it pass by the three prescribed points, both supports and the hinge in the middle of the span. Note that in this case here, we can see that the funicular polygon prettily remains inside the matter. But if we had a different load, maybe it would be different. We have here examples of arches with three hinges. We have already seen these structures. So, first, we have for the bridge over the Salginatobel, hinges at the springing and at mid-span. The viaducts of Austerlitz has a hinge at mid-span and actually, hinges which are not at the springing, maybe for reasons related to the construction process. But in any case, it is also an arch with three hinges. There is no obligation about the actual positioning of hinges. Let's now look at what happens for an arch with two hinges. We can see that its behavior is very similar, actually, to what we had seen before. Except that there are only two points which are very clear, both supports by which the funicular polygon must always pass. Here, I have the applet for the second case, the arch with two hinges. I activate the funicular polygon and now I do not have any constraints. So, I know that the funicular polygon has to pass either by the lowermost part, either by the uppermost part, either by any intermediate position. Actually, we do not know. And it can vary according to the loads. If I put, for example, a big load of maybe five Newtons on the left, then maybe the good solution will be that the funicular polygon is quite high on the left and relatively low on the right. While if I place it on the other side, well, the polygon will rathr be, low on the right and higher on the left. But each time with an infinite number of solutions. Note that if we are close to the supports, the shape of the funicular polygon just changes a little according to the rotation because actually, precisely, it does not have a huge amount of place to go. Thus, we could remove a part of the matter on the left and on the right. And that is what we are going to see here, in the examples of two arches with two hinges built by the Eiffel design office at the end of the 19th century. The first one is the one for which we had taken a closer look at the supports, the bridge over the Douro in Porto. And indeed, we have a shape which enables the funicular polygon to be in all likelihood, just in the middle when there is no live loads and when there are very high loads on the left, that is a railway bridge therefore we can have a locomotive which passes here, then the arch will tend to go up here and to go down on the right. And conversely on the other side. That is thus the same principle between this lower arch and this upper arch, there is an infinity of positions which can be taken by the arch depending on the applied loads. Let's conclude this lecture by a failure study of an arch. Here it was the arch without hinges which I pushed until it falls down. So, there is no surprise. But let's look at what happens just before the arch collapses. In the middle, on the top, it is the arch without hinges. But what can we see ? We can see that the internal forces only pass here, since there is air between both, only passes here, only passes here, only passes here. What we can see is that just before the failure, we have four hinges. Note that we do not have the failure yet when we have these four hinges. Here I was quite far from the failure when I did the test with the arc with three hinges but we also had four hinges. And in the arch with two hinges, we also have a point of passage here, a point of passage here. What does it means ? It means that the funicular polygon must pass in this way. It must inevitably, well, I do not know exactly its shape, it is going to depend on the weight of the structure, but we are anyway quite sure of where it passes since it must respect these four points of passage. We can see that in the three cases, finally, the shape of the arch is quite similar with cracks which open either on the internal side, either on the external side. Here, we can also see that it opens on the external side here, internal here, external here, internal here. So, if we go in a Romanesque cathedral with a vault shape and if we see longitudinal cracks parallel to the axis, well, we will be able to see that maybe, we are looking at the beginning of a failure phenomenon. It does not mean yet that it is falling down. It is really necessary that the four hinges should be activated and that the load should be maintained for the collapse to occur. A last thing that I wanted to tell you is, what happens during the failure ? Precisely, there are large internal forces which must pass just by very limited places. What is possible is that the stone strength, if it is a stone arch, it is often the case, the stone strength is reached and that there is a crushing in this zone. And at this moment, we would have a crushing, and then the collapse further to the spreading of this failure. In this video about hinges, we have seen what is a hinge, an obligatory point of passage for the internal forces, we have seen how to build them in the various building materials, we have seen what happens in arches with two and three hinges. I would like to take this opportunity to mention that we can absolutely make an arch with only one hinge but obviously we cannot build an arch with four hinges because it would not be stable. Finally, we have seen that precisely, failure occurs in an arch when four hinges develop and at this moment, it leads to the collapsing of the arch.
