Hello. In this video, we will see how we can proceed to stiffen arches. In the previous video, we have seen that arches, when they are subjected to variable loads, tend to take a shape which does not correspond to the one of the funicular polygon and thus, to become unstable. The purpose of this lecture is to see how we can stabilize arches, which is a process, as you will see, similar to what we have already seen to stabilize cables. The stiffening methods will be the same, with some methods which are not practicable for the arches, and a couple additional methods which are not used for cables, even if we could for some. Let's see first the effect of a load on an arch. We have a load g which is applied, it will also be present when we will have variable loads. And we want to look at the shape of the funicular polygon corresponding to the applied loads on this arch. Well, the shape of the arch, it was already drawn, I just draw it in blue because obviously the arch is in compression. It is an arch, here, with two concentrated loads, or below, an arch with a uniform load. It is exactly the symmetric of the figure which we have seen before with cables. When we add a variable load q, whether it be a concentrated load, or a distributed load on part of the span, there is a change of shape of the funicular polygon of the arch. The arch remains symmetric compared to what we had seen about the cable. Then, where the cable goes down, well, the funicular polygon goes up. Likewise for uniform loads. This is the shape of the funicular polygon of the loads, but when we apply a load on an arch, we have already seen it, the arch tends to go down under the load, as long as there is a distance which is creating, and which tends to increase between the position that the arch would like to take; I get a load then I want to go down, and the position that it should take to have the correct shape, according to the funicular polygon of the loads. Thus, we really have problem of stability. How can we solve that? Well, the first solution that we have seen, when we looked at cables, is that it was a good solution to add weight. Is it also true for the arches ? Well, no. If we remember of what we have seen, if we increase the loads on the arch, as soon as we add a load which is a little bit asymmetrical, the phenomenon of instability starts, the arch goes down of one millimeter, while the funicular polygon goes up of one millimeter, and it tends to increase. Therefore this solution is not possible to stabilize and arch. Now, you will see, we will see together that there are solutions where there are arches with lots of matter. That is true, but it is not mainly because of the weight that the arch becomes stable, but because of other properties which we will see shortly. The second solution which we had seen to stiffen a cable was to use a stiffening beam. This solution is also possible for an arch. The bridge over the Tschiel valley, which I have already showed you, has side parapets, that is to say the edges of the bridge which are used to prevent people or vehicles from falling down from the bridge. So, it is something useful, but there, they work as a stiffening beam to stabilize this arch which by itself is very thin. This is a solution which works to stabilize an arch. Let's look until the end at what happens if I apply an asymmetrical load on this arch with two loads. Well, we can see that the arch falls down. Then, it is really something serious. But let's look at what happens, just before the arch falls down. We measure the distance between the support and the upper node which is the farthest from this support. Here, this distance is, for me, roughly 240 millimeters. For you, you have another distance, but the reasoning stays valid. You can look at this distance. In the other direction, the distance is 290 millimeters. What does it mean ? It means that if I number the points A, B, C and D, well, the distance A-C decreases and the distance B-D increases. What can we do to prevent it from happening ? Well, in any case, we have good instruments to prevent the distance B-D from increasing. We can see in this video, I introduce an additional cable between B and D, and once this cable is in place, I can carry the additional load. The structure remains stable. Actually, I can even press quite hard, and the structure always remains stable. The stiffening by additional cables is thus a possible method of stiffening an arch. We have in this picture, the example of the Goum department store in Moscow, whose roof is constituted of a large number of metallic arches, which are just next to each other, very thin with a glass roof. So, we obviously notice that these arches, being as thin, should have a problem of stability. But actually, you can see, this structure is more than 100 years old; it still works very well. How does it work? We can look at that in the picture in the bottom right-hand corner. We have the arch corresponding to the structure, and in addition, we have a multitude of cables which link one of the support to the arch That is to say that if the arch wants to move away from this support, if the distance wants to increase, well, the cable is going to say "no, it is not possible". And thus, these cables stabilize our structure. We can find the same solution in a much more recent structure, in Switzerland. Here, once again additional cables with a link on the supports, and afterwards on various parts of the arch, like that, and indeed this system stiffens the arch which as we can see is very thin, by the way. It is a metallic tube, here, which is very thin and which then comes onto a post. Another solution is illustrated in this photograph of the bridge of Longeray, near Geneva. Here, the structure has a very significant thickness. It is a reinforced concrete bridge which dates from the end of World War II, then, relatively recent. And the concept, is that when the shape of the funicular polygon changes under the effect of variable loads, it moves, but it stays inside the matter. Thus, as long as there is a possibility for the funicular polygon to stay inside the matter, the structure will remain stable, this is stiffening by a significant thickness. The viaduct of Austerlitz in Paris shows us another stiffening solution which we have already seen for cables. That is to say that we can clearly notice that there are several arches, there is an arch which begins below, which passes above, and once again below, and finally which finishes above. A second arc above which it antagonistic to it, which is above when the other one is below, and vice versa. It absolutely looks like what we had for the bridge over the river Thames in London. Except that, obviously, it was a solution for cables. And as in London, we can also notice the presence of vertical and diagonal bars to stiffen the structure. It is thus, here, a stiffening by additional arches. I use the plural form, even if, here, there is only one additional arch, and on the other hand, we have a stiffening between both arches thanks to the diagonals, as we had it for Tower Bridge in London. Here, we have a very old structure which is almost 2000 years old, nowadays, and which still stands. So, we can ask ourselves the question. Why does this structure still stand ? Well, it is a question that we can ask ourselves in general. Why? Because the first thing that we notice, when we study the arches that the Romans have left us, is that they are very often what we call semicircular arches. These arches have two particularities, the first one is that they are circular, they have the shape of a circle. And the second particularity, is that they finish vertically on the supports, on the left and on the right. That is a good idea to finish vertically, as we will see it in the next video, but now, it is not a very good idea, a priori, to take a circular shape for an arch, since we have seen that, pretty often, the right shape would rather be a parabola, or possibly a catenary. So, why does this shape work ? Because, obviously, it works, if these structures are here since 2000 years. Well, let's look at what helps them. So, the arch itself, it is this zone which I indicate in yellow, but it is surrounded by all this system of masonry, on the left and on the right, which we call reinforcement masonry. The shape of this element, here, that is what we call a spandrel, and the stiffening is made by the reinforcement masonry. How does it work ? So, I am just going to talk about the case of the variable loads, the distributed loads. You will see, it can be solved in the same way than usually. So, if I have, here, a big variable load which acts, the funicular polygon, with all the loads, should go up here and go down here, but that is not the case. Now, what is the shape that the arch would tend to take ? The arch, when we add it a big force, it tends to want to go down, here, and to go up in the right part. But in the right part, we precisely have this reinforcement masonry which says : "no, no, I am not ready to let you go up". And thus, this masonry has as effect to add a pressure on the arch, and this pressure is going to stabilize the arch, making it come back to its place. If now, my load moves to come on the other side, then, that is the right part which now, tends to want to go down. But the left part would like to go up, but on the left, there is also this reinforcement masonry. Thus, the structure is stable. It also means that if you have such a bridge, well, there is no question of unwisely taking off this reinforcement masonry, because it could easily lead to its collapse. So, you have to proceed with caution during the restoration. We have, here, the example of a big Gothic cathedral which is surrounded by all a series of flying buttresses. We are going to see that these elements are very important for the stabilization of our cathedral. The vault, or the dome which we have, here, tends to exert an inclined outwards force. If there is nothing to resist to this force, in all likelihood, the structure is going to collapse. So, that is why the architects of the Middle-Age have invented this concept of flying buttresses, and why they have put them. The flying buttress is going to act by its weight; to simplify, I only make it act in the middle of the length of the flying buttress, but it obviously acts in a distributed way. What is the effect of this weight ? If we look at the funicular polygon, well, the force which comes from the dome pushes in this way, until it meets the yellow force, which is going to change its direction, and to make this force go down in an inclined way, and we can see that, the funicular polygon stays inside the matter and our structure remains stable. But you have already seen in the lower part, the line of the polygon of forces tends to want to get out of the matter. Then, how does it work ? Well, it works in two parts. There is first, here, the weight of this column which is not negligible. We can see that this column is quite thick, and moreover, here, above, we have an element which we often consider as decorative, but which actually has a structural function. It is called the pinnacle, and the function of the pinnacle is to procure an additional weight to the column. Both together have the effect of significantly changing the direction of the new funicular polygon, in such a way that it stays within the matter, and thus, a flying buttress is a stabilizing element for a vault, or a dome of cathedral. Approximately ten years ago, we could read in the local newspaper of Lausanne, "the cathedral stands by strings". What was happening ? If we look at the cathedral, we can see that in this zone, and it is the same thing on the other side, there is a certain number of flying buttresses which have the function of, precisely, stabilizing the main nave of the cathedral. However, these flying buttresses were made of a relatively friable rock which was degrading, and it was necessary to replace them. But, as I told you, it is not a good idea to remove flying buttresses. So, what did we do ? Well, we have replaced them by cables which I call internal cables. Why internal ? Because these cables link together two parts of the arch, but are not linked to the supports, unlike the first solution with cables which I showed you. Here we have a photo of the cathedral, when the cables were present in the cathedral. There were four cables according to the paper, but we can see three, here, and in the infographics of the newspaper, we can see, well, maybe it is not very accurate, we are not totally sure that these cables were located at these places, but they had the effect of exerting a inward force, in this way, on the cathedral, as a replacement for the effect of the flying buttresses which pressed in this way on the vault to stabilize it. That is clear that it is not exactly the same effect, but was sufficient during the maintenance works for the construction not to suffer, and for these flying buttresses to be replaced, and the cables have been removed since. We do not need to keep thinking in two dimensions. We can also look at what happens in the third dimension, and here, we have a very nice example, with the dome of the cathedral St. Peter of Rome, built in the 16th century. Soon after its construction, cracks were observed in this dome which were approximately horizontal. I am going to show you, later, how they were exactly. These cracks, if we look at a vertical section of the cathedral, we can see that actually, we have two domes, one internal dome, one external dome, that is the one which we can see, and between both, we have an empty space. This, it corresponds to the idea. It is necessary that the matter should be quite wide for the funicular polygon to be able to stay inside the matter. But it was not enough, and thus, it was observed, since there are stairs within the dome, that there were cracks in the internal dome. It lasted some time, and in the 18th century, the mathematician Poleni was given the task of studying this problem and of finding a solution. What he did was to look at how a sector of the cathedral works. A sector, it is a slice which has not a constant thickness. It is a little bit like a piece of cake, that is to say that in the middle it is very, very thin, and on the edges, it is very large. If we looked at that in a plan view, it would have a triangular shape. He represented the behavior of this arch, in a reverse way using a chain, on which he placed weights. You can see, here, circles of varying diameters which represent the various points. Thus, he has obtained, for this dome, this shape, which is the shape of the funicular polygon corresponding to the weight of the structure. Afterwards, he reversed this drawing, and on the basis of this reversal, what can we notice ? Well, we can notice that precisely, the funicular polygon corresponding to the loads gets out of the matter, which explains the presence of cracks. Now, the question is, how to repair this structure ? Naturally, we could have imagined to proceed as we proceeded for the cathedral of Lausanne, that is to say to pull cables through. But, I remind you, I have shown you before a photo, there is a splendid golden ceiling. It would not be an extremely suitable solution, we would really have lost a lot, from an artistic point of view. The solution which was found was to place cables which surround the dome, and which are put in tension. There are several levels of cables which encircle this dome, and thus, from outside, they have this effect of pushing inwards, and this has stabilized this dome. I do not want to tell you more about that, because you are going to have an exercise, this week about this dome. So, I let you exactly discover how it worked. To summarize the various stiffening methods which we have seen in this lecture, we have seen a stiffening by stiffening beam which acts in the same way than it acted for a cable. We have seen a stiffening by multiple cables which prevent the changing of shape of the arch, when the arch tends to want to lift up under the effect of unsymmetrical loads. We have seen a stiffening by a significant thickness, in which we enable the funicular polygon to move while remaining inside the matter. We have seen a stiffening by additional cables, as for the Tower Bridge in London. They are not additional cables, of course, they are additional arches, and, these arches themselves, were stiffened between themselves. We have seen a stiffening by reinforcement masonry. So, the reinforcement masonry is the masonry which is placed above an arch, and which once again, in a quite similar way than the cables of the second solution, prevents the arch from going up when it would like to do it. We have seen a stiffening by flying buttresses outside the structure. You will acknowledge that the flying buttresses take up quite a lot of space. That is probably one of the reasons why it worked well for a cathedral which was in a relatively unique place, and which had to remain a special place of the city. But, it is not something which we can often see. And finally, we have a stiffening by internal cables. An internal cable being simply a cable which is not linked to the supports, unlike the cables of the second solution. In this lecture, we have seen how arches need to be stiffened to behave properly. The purpose of the stiffening is to prevent the arch from deforming, and thus to ensure that the funicular polygon remain inside the matter. A solution which would only use the weight of the arch is not possible, since an instability would still occur with a small variable load. There are numerous solutions which call on cables, on an increase of the thickness, on several arches with a stiffening, on reinforcement masonry, on flying buttresses, or on internal cables.
