Hello, in this video, we will deal with truss grids, which are the first extension in three dimensions of truss structures. Of course, real structures are always tridimensional. When we create a truss, that is for example to use it as the roof of a factory, as the cover of a garage, or maybe as the load-bearing system of a bridge crossing a river. In every instance, the structure that we are going to use is tridimensional. The question is: "how can we move from 2D to 3D for trusses?". And in this video, we will see that we can do it by juxtaposition of trusses or by crossing of trusses. Here is the plan of the entire course "The Art of Structures". Of course for you, in "The Art of Structures II", we started with the subject of trusses, but after having entirely visited the subject of trusses, we want to go here, below, the transition to the third dimension, where we will look at various solutions, before moving to the next topic of the course. The first solution to create a tridimensional surface, to cover a surface by trusses, is simply to juxtapose these trusses. So here I have a first truss. Then I create a second one next to it, which is exactly the same, then a third one, and so forth. Obviously, that is not enough to cover the space. Afterwards, it is necessary to introduce a secondary load-bearing system which is going to be used to support the roof elements. If we look at the example here on the right,this is the roof of the Lausanne's station, we have roof-shaped trusses, with a slightly variable depth in the upper part, but we can clearly identify trusses, and we have a series of these trusses which are relatively spaced from one another. And between these trusses, we can recognize other trusses, - here we can very clearly see them, here too, also here - and then between these trusses, to have a complete load-bearing system, we are going to add secondary load-bearing elements, -we can see them well in this zone where there is the roof- secondary load-bearing elements and then, finally, tertiary load-bearing elements, here, until we manage to carry the glass panels, and here the roof. So we first have a juxtaposition system, then we add secondary and tertiary elements, etc., until we can finally install the roof. So, this is not really a tridimensional system, I would call this system a "2D+ system". That is clearly something which fulfils a 3D function but where the structure itself is not tridimensional, this is a juxtaposition, a superimposition of structures which are in one direction, then in the other, and so forth, a certain number of trusses and then other types of structures. Here we have a solution which is closer to 3D, where we have two families of beams, the first one which I draw in orange, they are beams which are supported by two supports, simple beams, so; and then, in the other direction, in the same plane, and with the same depth, the same importance, the same capacity to carry loads, we have beams in the perpendicular direction. These green beams, unlike the pink beams of the previous drawing, they really have the same dimension, a comparable load-bearing capacity. Let's now look at what happens when I apply a load Q here, in the middle of this system, the load can decide to go in the direction of the green beam, or in the direction of the orange beam. If the spans and the dimensions of the cross-sections are exactly the same in both directions, then we can say that the internal forces are divided by two. So I put, "under conditions", since, I just said it, that supposes that the spans are the same, that there are supports everywhere, etc. If we go on with this reasoning, when the load which left here in the direction of the green beam, reaches this point, there is again an intersection where it can decide to keep going in the direction of the green beam, or to go in the direction of the orange beam. Likewise, if the load keeps going in the direction of the green beam, it will reach the support; if, however, it goes in the direction of the orange beam, it will reach again another intersection, where it will be able to decide wheter to go in the direction of the green beam, or to keep going in the direction of the orange beam. And so forth. These movements are similar to the ones of a tower in a chess game, which moves in this way. On the right, we have a similar system, but instead of having placed supports at the end of all the beams, there are only supports at the end of beams which are on the edge. This is really advantageous for the use, since there are much less obstacles for people who would like to take a walk under this structure, or for the vehicles which would pass under, or for the machines which we would like to install. However, there is indeed a higher complexity. So here I have two families of beams, the orange beams and the green beams, these orange and green edge beams are supported, the other beams are indirectly supported, leaning on, respectively, for the orange beams, the green beams, or conversely for the green beams, on the orange beams. Let's now look at what happens for my load Q, that I place again in the middle of the structure. Well, from this point here, it will indeed be able to decide whether to move onto the green beam or onto the orange beam, at the intersections, it will also be able to decide to keep going; however, when it will arrive here, on the edge, there is no support, so the load will have to be carried over by the orange beam. If the load passed by here, so likewise, when it will arrive here, it will have to be carried over by the orange beam. So for this type of trusses, each load is carried twice. That is logical, before we had beams which were spanning in one direction and which could find a support, now we go in one direction, we do not find any supports, we must go in the other direction; or if we use the other route, that is the same thing. So the internal forces will rather be larger. Like before, the displacement is the one of a tower in a chess game. We still did not reach a real tridimensional system, I call this kind of solution "2.5D" solutions. There is admittedly a behavior in which the orange beams cannot not work without the green beams, especially because at the end of these orange beams, there is a green beam to carry them and vice versa, but however loads always follow a preferred route, either in the direction of the orange beams, either in the direction of the green beams. Here we have an example of this type of construction. That is the USM factory in Münsinger, in Switzerland, where we have here the orange edge beams, which are supported by posts; the green edge beams, which are also supported by posts; and then the orange intermediate beams, there are two, we cannot see them well, the second one especially; and then the green intermediate beams, and there are also two. Otherwise, we can notice that this truss is a truss with N-shaped diagonals in tension, which is why this structure is very transparent. Here we have a structure which is much more complex, that is the Palexpo exhibition hall, in Geneva. It is in particular used every year for the Geneva car show. The interest of this hall, is that it only has a very few posts, here you have a post and the others are almost out of sight. And you can see that it creates a very long distance, which enables at once to place the cars quite freely, and to avoid the visitors having to skirt all the time walls, columns, partitions. Here we have a picture of the construction of this hall, and we can clearly recognize trusses which go in on direction and others, in the other direction. On this cross-section, we can only see the orange truss, the other one is perpendicular to it, we cannot see it. This cross-section has been made in the same state than above. Afterwards, the structure has been raised up and set down onto these pillars to offer a large free area for the exhibitions. If we look on the right, that is not easy to recognize these structures, because the roof is very dark. We can however recognize a truss which goes in this direction, and then we can guess other trusses which go in this direction. But that is very difficult to clearly identify them. The interest of this type of structure, is that it provides us with a wide range of possibilities for the use, there are also sport events which are organized inside these halls which offer a very large flexibility. In this course about grids of trusses, we have seen how, thanks to a juxtaposition or a crossing of trusses, it is possible to move from two to three dimensions, or at least to create structures which cover tridimensional surfaces. In the next video, we will see real solutions of tridimensional trusses.