In this video, we are going to talk about tridimensional trusses. We will see that these structures are very interesting, but that they bring some difficulties because of their nodes. I will show you some quite exciting examples of structures which use tridimensional trusses. The structure here on the right is familiar to the students of EPFL, since this is the typical structure of the elements of EPFL built during the first stage of construction. The principle is quite simple. If we have here a column which gives us a support reaction, this column will induce a compressive force on the four bars which it supports. So that would be here. And here. So these four bars are in compression. To have an horizontal equilibrium, it is then necessary to have tension at the upper level. This is the basic principle of the support elements. So here, we have tension. The rest of the structure essentially behaves as a truss. So the next diagonal is going to be in tension, the one after in compression, and so forth, till the middle of the span. The upper part, the upper chord, is going to be in compression. And the lower chord in tension. This is a bit more complicated than what you have seen in two dimensions, but the principles are the same. And we could, we are not going to do it, but we could develop a graphical statics similar to the one we have seen, but in three dimensions. Here on the left, you can see two variants of the same principle: on the bottom, the system such as it is used at EPFL, with always an outflank beyond the columns, then, above, a system which never outflanks so which exactly covers the surface delimited by the four columns. This system can also be applied for bridges with the same principles: a tensioned lower chord, compressed upper chords, and diagonals which are alternately tensioned and compressed. I am maybe going to copy it here, on the bridge. So the lower chord is in tension. Here, these diagonals are in tension. So the upper chord is in compression. And then these diagonals here too. This is a very pretty system which is used for this bridge in Lully, in the canton of Fribourg, with a concrete deck for the vehicles, absolutely classic. Here, a structure which is much less classic, a geodesic dome. This is the invention of Mr Buckminster Fuller, whose we can see the same, on the bottom. He propagated this type of very interesting structures. This cover of the building of the United States for the Universal Exhibition in Montreal in 1967 is actually not a cover. It is only an open truss. The real building is behind the trees, inside the dome. A particularity of these geodesic domes is that they are mainly constituted of hexagons. There are a lot of hexagons, such as the ones you can see on the surface of a soccer balloon. In some places, it is necessary to introduce not hexagons, but pentagons. I have not noticed others in this part here, but there are some on this entire dome. I have not any other photos to show it, but this dome is obviously made of tridimensional elements. It is not only curved, but it has a certain thickness, made with tridimensional elements. A problem with this type of structure: the nodes turn out to be very complex. And the lengths of the bars are highly variable. This is why this Mero system, that is a brand among others, has been developped with standardized spherical links. They have a certain number of threaded holes. They enable, look at the scheme in the bottom left-hand corner, to add a certain number of bars, not using all the holes, but without being obliged to make nodes to measure. On the right, we can see on the bottom a detail of one of these Mero type spheres, with metalic elements for the connexion, and timber elements. This is used for a small cover of a pedestrian pathway at ETH, in Zurich. The fact that these structures are composed of trusses, that is to say that they are not very heavy, and that on the other hand they can cover surfaces, makes that they are often used for large dimensions structures. On the left, the cover of a stadium in Saoudia Arabia, created with tridimensional trusses, covered by membranes which ensure a protection against the sun, this is the only function of this roof. But it must also obviously resist wind. On the right, the Globe Arena in Stockholm, an entertainement structure, in which entertainement activities take place, created on the basis of the same principle, with one difference: till now, all the solutions were based on hexagons, and actually triangles, here, the solution is based on rectangles. So it is possible to do it. It is a bit more complicated, since actually they are not exactly rectangles. They rather are trapeziums which become smaller gradually when we move towards the top of the structure. The dimensions of the facade elements had to be adapted, we can see it well on the pictures on the top which shows the completed structure. Here, again a project a bit more radical, the Eden Project in Cornwall, in Great Britain. These are huge greenhouses which are sheltered under geodesic domes. Since they are geodesic domes, we can recognize, in some places, that there are pentagons. Here, there is a pentagon for example. And here there is another one. While most of the covers are constituted, in a classic way, of hexagons. On the left, a connexion sphere system. This is not the same system. We can see that the holes have been drilled where bars arrive. The cover is constituted of a double-membrane made by light plastic, filled with air under pressure. There is always an air pressure which is brought, in such a way that it is used as insulation. Here, a view of the inside where we can notice the presence of a tridimensional truss. And we can also notice the presence of a palm tree, which means that this greenhouse satisfactorily works, since it is located in Great Britain, a place where the palm trees do not naturally grow. Finally, to talk about spatial trusses, let's take an interest into the space station. If we zoom in on the space station, we can quickly figure out that there are tridimensional truss elements. Here, this is a module which is not liveable, since it is exposed, but all the rest of the station, such as these parts here, where the astronauts live, are built on the same principle. Why? Because it is an external structure which offers strength, it is just necessary to close it for it to offer in addition, a liveable zone for the astronauts. Why use trusses for the space station? Because they are the most efficient types of structures, and which thus require the least weight possible to be put into orbit. In this lecture, we have seen real tridimensional structures which use the three dimensions of space to create trusses. We have seen that they are interesting structures but which require particular nodes, as well as construction details, and variable bar lengths. However, they offer a large freedom for the design of structures, particularly for the large dimensions structures.
