Hello. In this video about the beam grids, I will present you the principle of this kind of structures, and the type of load-bearing elements that it uses. We will have a look at the most used materials for the construction of beam grids and I will present you several possible geometries for this kind of structures. We have already seen, with truss grids, structures which are very similar to beam grids. There is a family of beams which span in one direction - here the orange beams - well, there are severals, and so on, until the end, and then in the other direction, we have another family of beams - which I draw here in green - which can take some loads. When we place a load at any point, it can go either in the direction of a green beam, either in the direction of an orange beam, either a part in one direction and a part in the other direction, in a general way. In the same way than for the truss grids, we are relatively free for the arrangement of the supports: here, we have a beam grid leaning on linear supports - they are not linear but point supports ; however, they are closely distributed so they can almost be considered as linear - or, on the opposite, we can have a beam grid leaning on point supports, with a small number of supports, which provides the structure with a larger flexibility given the reduced number of interferences with the load-bearing systems, but in principle, we still have the same thing, that is to say that the orange beams will carry in this direction and the green beams in the perpendicular direction. It should be noted that like for truss grids, if we do not have a lot of load-bearing elements, the load must be carried several times, that is to say that a load which goes from here will be able to go in both directions but once it will reach the end of this structure, it will have to be carried in both directions to reach the supports because here, we have not reached the supports yet, we have to go till there to reach the supports, and to do this, it is necessary that the loads are carried twice, so, the type of beam grids with point supports will have larger internal forces and will need more material than a beam grid on linear supports like this one on the left. The materials usually used for the construction of beam grids are on the one hand, reinforced concrete. Here, you can see a double gymnasium: there are indeed two gymnasiums side by side, with a beam grid roofing. It should be noted that here, we have a beam grid on linear supports since there are large concrete beams all around which are supported on these V-shaped supports, that is a quite audacious arrangement for this construction. Here, you have several examples of steel beam grids, first, on the top, the project of Mies van der Rohe for an office building for the rum producer Bacardi in Cuba, in 1957, - this project has not been built - but we can obviously notice that one of the biggest advantage of this beam grid is that the number of supports is very small: there are only eight supports on the rim for a large inner area, and on the left, a project which was built ten years later, the Art National Gallery in Berlin, which enabled Mies van der Rohe to make one of its largest steel beam grids. We can notice again that there are very few supports, which facilitates the use of such a large structure. Here you have a very original structure, so much that it has been used as logo for the course "The Art of Structures II"; that is a beam grid whose beams, in the horizontal direction in the picture - which we cannot see here - are steel beams, and then, in the other direction, there are UHPFRC beams, that is Ultra-High Performance... - I am going to note it for you to know it, even though that is not something I will ask you in the exam, that is just for you to know what it is. ...Fiber Reinforced Concrete. That is a very special concrete, extremely expensive, but which has the advantage to provide us with the opportunity to make very thin elements, I think these are at most 80mm thick, which would be impossible with normal concrete, just because of the requirements for the cover, but this Ultra-High Performance Fiber Reinforced Concrete is completely watertight, therefore it is not subjected to the same rules for cover of reinforcing bars. That is a very beautiful structure anyway, we all agree on that. The typology of beam grids is not limited to the ones we have seen, where we had two groups of beams with one going in a direction and the other one in another, well, that is what we have down here, for this industrial building in Lausanne, so here we have large beams in one direction, and then in the other direction, we have another set of large beams, with only two families of beams perpendicular to each other. Above, we have the art gallery of Yale University, where we can notice three beams systems: one which is more or less vertical in the picture, another which goes down leftwards, and a third one which goes down rightwards. So, the interest of such a system is that since we increase the number of beams, each of these beams carries a smaller load and then, we can make beams with smaller dimensions, but the formwork however, certainly see it, is especially complicated to make. On the right, we have a similar system made by Pier Luigi Nervi for the Naples train station. We have three families of beams: this family here, a family which goes down leftwards, and a family which goes down rightwards. And then finally, we have this example which comes from Madrid, for a metro station, in which we have a large number of beam families which interesct, so once again, beams in this direction, then in this direction, and in the opposite direction, but there is a fourth family of beams, here, which are the ones which are more or less horizontal in the drawing; so here, we have four families of beams which cross in this roofing. Finally, who said that the beam grids were limited to rectilinear systems? Actually, there is no reason, if you have a system which adapts well, that is an access ramp for a multistory parking garage, and here, we have beams which are circular in one direction and then which are rectilinear in another direction. A beam does not need to be a rectilinear element. Of course, that is simpler in a certain point of view, but if it is preferable to have another arrangement for the use we want to have, there is no fundamental problem to have a non-rectilinear beam grid. That is what we can see on the right, in another work of Pier Luigi Nervi, with several families of beams: we are going to say that we have beams which are directly connected to the supports, here, there and there, and then I am going to draw them with the same color in the other direction, all these beams get a little further from the supports but they directly reach them, following axes which directly link the posts straight ahead. Then, we have a second family of beams, here, which are also rectilinear beams, really rectilinear, but which are located at the interface between the panels about which we will talk a bit later. The rectilinear beams which do not extend till the supports, cross the beams which extend till the supports, then they create beautiful rectilinear lines in the whole structure; and then, finally, we have two additional families of beams: one family of curved beams which go in the direction of the support, and this corresponds well to what is happening inside a slab, we will see it later, so here, Nervi really used his knowledge in this field; and then, finally, we have a last family of beams, which are more or less circular around the column, and which perpendicularly cross - or almost perpendicularly - all the other beams of the structure. So, here, we can see several families of beams on which we place afterwards a very thin concrete flooring, which only has a small span, that is a system which is quite efficient. In this video, I have introduced to you beam grids: I first showed you that the load-bearing systems are similar to the ones seen for truss grids, I told you that these beam grids are generally made either out of reinforced concrete or steel, I did not tell you but I should have, we also have the possibility to have timber beam grids, even though this is not usual. Geometrically, beam grids can simply be orthogonal with two families of beams which perpendicularly intersect, or else we can have more than two families, three or four easily, which cross at regular angles, and which create an interesting space. Finally, we have seen that the beams themselves do not necessarily need to be rectilinear, they can be curved, in such a way that they correspond more to the geometry of the structure or that they more directly bring the internal forces to the columns.