Hello. In this video, I am going to talk to you about Vierendeel beams. Vierendeel is the name of a Belgian engineer, and you can see here one of his bridges, built in Belgium at the beginning of the twentieth century. And since he is Belgian, I am tempted to quote a saying by another Belgian man, the painter Magritte, to say : this is not a truss! That is true that it looks like a truss, that is like a truss, in which we would have forgotten, or in which we would not have put any diagonals. In this lecture, we will see the configuration which is necessary for a Vierendeel beam to work, we will see the differences between a truss and a Vierendeel beam, and then I will also talk to you about the Vierendeel beams with several stories. In this video, you can see that I have a Vierendeel beam, that is to say a truss where we have taken off the diagonals, that I load with our usual load, and there, the truss deforms a lot. So for a Vierendeel beam to work, I am going to have to clamp all nodes; usually, the nodes, in the trusses, are hinges, the bars can rotate one in relation to each other; here, this is not the case, I tighten a lot all my hinges: obviously, this construction was not planned, originally, to work as a Vierendeel beam, but you can see that now, I managed to put my load of 10N and to have a valid structure in the form of Vierendeel beam. Let's look at a detail of the bridge of Mister Vierendeel here which I have shown you at the beginning of this lecture, and clearly, we can see that there are no hinges. On the opposite, near where there would be hinges -- we are going to say near the nodes between the bars -- some additional material has been placed. We are going to say : here, this is clearly not a hinge. That is more a clamping of the chords in the posts. I remind you that the posts are the vertical elements. As a consequence, and we can see here the number of... -- at that time, these were rivets; that is a solution which is more expensive, and which is more complicated to build than a truss. but it must be said that there are advantages: if we look at this example of a building which was built relatively recently, quite close to EPFL, in Saint-Sulpice, we can see here a structure which, at first sight, is quite surprising, but if we look at where are these load-bearings elements, it is interesting to see that there are here load-bearing elements -- there is a second building behind, which is not connected. But if we look at the first building, which is in the foreground, which hosts apartments -- well, we have, on the right, in the back, two columns, on the left, only one column, and then all this space, here, is a free space. So it has been partly used to put small bushes, and partly to put parking places; but that is clear that we could not have parked cars if we had had a lot of columns; we could have maybe put columns, but that is another story. If you look at pictures of the construction of this building, there, again, we have this huge free space -- that is about thirty meters long, that is really the span of a bridge. And we can clearly see, here, that there is a lot of material which has been added, we can see steel elements with quite significant dimensions. That is not a transparent structure at all, unlike some of the trusses we have seen before. On the opposite, that is quite a heavy structure. But with some advantages, especially, this free space. If we compare a Vierendeel beam to a truss, what we can see is that in the truss, we can have internal forces in the diagonals. And we are going to see in this lecture, that it has a decisive effect, that is a bit like what we have seen before with frames. Here, as usual, the internal forces must remain inside the material, so we will have additional internal forces, like in frames. But this is not only in the field of internal forces that the difference between Vierendeel beams and trusses is significant, that is also the case for deflections: here, we have the deflections of a Vierendeel beam and of a truss; we are going to add "strongly exaggerated", because you should not think that it really deforms as much as here. I used a calculation software to calculate this; and we can see here that the deflection w for the Vierendeel beam, is tremendously larger than the deflection w for the truss. Now, I would like to clarify that it depends on certain parameters which have been chosen for the calculations, but in all cases, the deflection of a Vierendeel beam will be passably larger than that of a truss. It will never be smaller. As a consequence, if we want the same deflection, we will need more material with a Vierendeel beam. That is also what we have seen in the bridge of Mr Vierendeel, in the first picture: there is passably more material in this structure. Maybe the deflections of trusses are too small; we could accept slightly larger deflections, but the deflections of Vierendeel beams are, very generally, very large. Let's see how we can understand the internal forces in a Vierendeel beam; so, I should first add loads; we are going to have, the same loads on all these beams. I only draw them for the beam on the top. So loads which act vertically, at the level of the intermediate posts -- I do not put any on the first posts, simply to simplify the reasoning, but there could absolutely be some. How is this beam going to work? Well, I am going to imagine an arch-cable. So on top, I have an arch. And then I am going to give it a lenticular shape, because this structure is symmetrical, so there is no reason for the internal forces to be, a priori, rectilinear in the lower part, but we are going to get back to this point later. Vertically, between both, well, we should have compressive internal forces to transmit a part of the load -- half -- of the arch, towards the frame. We can see that, in the central part, all this can work very well because the internal forces remain inside the material. However, in the part, shall we say, of both openings at the ends, on the left and on the right, well the arch and the cable pass through the air, and then that is not possible, it will be necessary to correct this. How are we going to be able to correct this? We are going to do it as we did before, with frames; that is to say that we are going to insert compressions in the diagonals, as a consequence, the internal forces in the arches will be deviated, in such a way that they remain inside the material, and then we are going to do the same to deviate the internal forces in the cables, in such a way that they also remain inside the material. In the central part, we will not need this anymore, we are only going to use the arch-cable. I am now going to insert the missing internal forces, to reach the complete equilibrium of this structure. Here, I must have tension on the top. So that is something which is quite complex, I mean the functioning of a Vierendeel beam. I will not ask you -- there will not be a lot of questions about Vierendeel beams in the exercises, that is not a fundamental topic of the course. I will ask you some questions sometimes, but they will be theoritical questions, quite general questions. So, I finish this construction here, for it to be complete. So we can see that the left and right parts of this structure essentially behave like a frame; and then in the central part, on the opposite, we have an arch-cable and obviously, between the arch and the cable, we have elements which are in compression; in the lower part, I will now show you to you another way to understand the behavior of a Vierendeel beam. We have cut this beam in two, with a lower part and an upper part; obviously, we are going to bring them back together thereafter, for them to work together. Half of the loads act on the top half-beam, half of the loads act on the bottom half-beam. We are not going to get into all the details, but we are going to look at how this beam deforms. Under the effect of these loads, the beam on the bottom is going to deform in this way, it is going to bend, and, because of the rotation, we can see that these teeth, which represent the fact that the Vierendeel beam has been cut in two, these teeth tend to rotate, and they rotate inwards, at least the teeth which are at the ends, in the middle, we can see that they do not rotate much. Then again, when we reach the other side, we have larger rotations. Likewise, the top half-beam -- we are going to assume that it has its own supports -- it also deforms, but there, you can see that the teeth rotate outwards. Well, in the central part, it does not rotate much either, like in the lower part; but in the external parts, it rotates more. Actually, in reality, of course, both structures must be compatible. What does it mean? Well, for them to be compatible, it is necessary that the deformations, here, be the same. So, to have this, we are going to push on the pink teeth, at least, let's say, the first three ones, in this way, to make them get inside; and then we are going to push on the orange teeth in the other direction, to make them get out a bit, so that they can meet. What is the result of this? Well, if we look at this small element, we can see that it is a cantilever. What are the internal forces in such a cantilever? Well, in this direction, we are going to have compression, and in this direction, tension. Likewise for the following cantilever: compression, tension, the following cantilever, compression, tension. If we look at the lower part, we have a cantilever oriented in this way, so we have compression here, tension there, compression here, tension there. And so on: compression, tension. If we look at the other end of the beam, well, we are going to push in this way, here, and to push in the other way in the orange part; and likewise, we are going to find compression and tension. That is the same for the lower part, compression, tension, tension, compression, tension. If I draw outside what we have just observed, well, what we can observe is that in this part, we have diagonals which are inclined in this way, in compression, in the other way, they are in tension. And then on the other side, diagonals in this way are in compression, and in this way, they are in tension. But if we look at the illustration on the top, we can see that we have exactly observed the same thing: diagonals in compression in this direction, in tension in the other direction, and likewise here, diagonals in compression and diagonals in tension. So both these approaches confirm what we have seen about the behavior of Vierendeel beams. In the central part, there is not a big need to offset, so there are only small internal forces to insert; so we have an arch-cable with compressed elements, a very quiet part, and then, in the left and right parts, the posts are subjected to both tension and compression. If we have this in the posts, we will also have it, obviously, in the chords of this Vierendeel beam. I am not going to ask you to get into the details of the compression of Vierendeel beams within the framework of this course. I am giving you an introduction, I will ask you some questions in the exercises, which will be questions of general understanding about how Vierendeel beams work, but I ask you to know a little bit about what is happening inside these beams. Vierendeel beams, as I told you before, deform a lot and cannot carry much load, because large internal forces are induced. A solution to make them efficient is to create Vierendeel beams over several stories. Of course, it would also be necessary to limit the loads which act on them, this can be achieved by having Vierendeel beams, for example, for the facade system, and then another type of load-bearing element, for example walls, inside the structure. I am not going to get into the details of this structure, but that is clear that it is more efficient, it is certainly going to deform less than a one-storey Vierendeel beam. Here, I have a short video which show the behavior of a structure with Vierendeel beams over several stories; so, to be honest, I had to tighten up less my nodes for it to work because there are many more nodes so it works a bit better, but you can see that if again I push on it with one finger, I still can deform this structure, it remains very deformable. In this first lecture about Vierendeel beams, we have seen that if a Vierendeel beams looks like a truss without any diagonals, that is far to be a truss. There will be much more significant deformations in this type of structure, and the internal forces are very different also. Instead of having, like in a truss, only tension and compression in the structural elements, we have at once compression and tension in many of the structural elements. One of the possible solutions to increase the efficiency of Vierendeel beams is to place them over several stories.
