Hello. In this video, we are going to deal with truss towers and cantilevers, particularly with bracing systems. What is a bracing ? That is a structural system, so a structure, which is used to resist forces - initially wind bracings are used to resist the wind. - Generally, we can call bracings all the systems which enable to resist horizontal forces. It means, the wind on the building, such as this one here, we have wind which can act at every level. The wind is stronger on the top, and weaker on the bottom, but it acts in all the directions, and obviously, it can also act in the other direction. So the system must be able to resist forces which, indeed, act in all the directions. In this video, we will make some considerations about the shape and the type that bracings can take. We will see how they are integrated inside buildings, and we will calculate some internal forces for particular types. In this structure, the John Hancok Center in Chicago, - Chicago is known as the Windy City - it is very important to be able to resist wind forces, and what kind of structure can we recognize here ? Well, we recognize a truss. If I say that the loads act at the level of the nodes in this way, - well, I know, it is almost a constant depth truss - I will have tension in this element, I will have compression in these elements, in these diagonals, as well as in the chord here, and then, I will also have tension in these diagonals. Of course, the consideration would completely be reversed if the internal forces were acting in the other direction. So, it means that these elements must be able to resist tension and compression as well. We can see on this picture of the structure how these diagonals have been integrated in the global architecture of the structure. Obviously, some people have their window behind one of these diagonals, but they appreciate this building anyway, but to be able to do this, the facade of the building must be flat, more or less of constant depth, and then, without any recesses or particularities. This is a solution which is possible, but which is actually not very common, precisely because of these limitations. We have here a building which is more creative, we can see it here on the right, this building is the Library Tower in Los Angeles, where we have constant part, and elements which are gradually withdrawn. So, we have a structure, which is not constant, and furthermore, it is circular with a shape which is a bit softened. That is impossible to place a system of diagonals or of chords in these facades. So, how did we proceed ? We have introduced a core. That is this element here, which is located in the middle of the structure. Inside the core here, we typically place elevators, but also pipes and other types of ducts to bring or to evacuate fluids. We can also place some elevators inside. What type of structure can we see here ? We can see here vertical chords. That is quite easy to see them. And we can see a truss with K-shaped diagonals. We can see that these K-shaped diagonals are a bit particular, because that is not really a K, it is a kind of U or V which is a bit open. Why do we do that ? Because, in this way, we gave ourselves the freedom to create doors. - Here, that is a high building - I have said that we could put elevators in the cores. It is necessary to be able to enter in the elevators, so, creating this truss with quite particular K-shaped diagonals, we can enter in the core to reach the elevators, the stairs, and the other facilities which are inside the core. The whole being a large truss system which resists the horizontal loads. In this video, you can see the construction of a truss cantilever. It is fixed on a column on the left. Then, I place two loads, and then a third load to have 20 Newtons at the end of the cantilver and 10 Newtons on the intermediate node. If we want to solve this structure with the applet, it is absolutely possible to place a load of 10 Newtons here, and then, a load of 20 Newtons at the end of the cantilever, and to obtain the resolution. Here, we can see that the upper chord is rectilinear - we directly obtain the internal force here - and then the lower chord is curved. You remember the method used to directly obtain the internal forces: we could isolate this node to obtain these internal forces (on the basis of these internal forces here), but we are simply going to do it using a free-body. We are going to proceed to the complete analysis of this structure. I am first going to number the nodes. One, two, three, four and five. I am going to start by looking at the free-body around the node five. This free-body is subjected to a load of 20 Newtons: I haved already indicated it in the course's supports, that you also have. This is a load of 20 Newtons. Then, we are going to obtain the internal force in the bar three-five. This internal force goes leftwards, so that is a tensile internal force. Here we have a tensile internal force. Then, the internal force in four-five which goes up towards the beginning of the force of 20 Newtons. That is a compressive internal force. Here is the contribution of the node five to the global equilibrium. We take an interest in the node four, which is an unloaded node. This node is subjected to the internal force in four-five in the other direction, then to the internal force in three-four to go up, so that is a tensile internal force, three-four is in tension, then we come back to the beginning of four-five thanks to the internal force in one-four, which is a compressive internal force. Here is the contribution of the node four to the Cremona diagram. We now take an interest in the node three with the load of 10 Newtons. This node is subjected to the internal force in three-four in the other direction, to the internal force in three-five in the other direction, to the load of 10 Newtons, then to the internal force in two-three, which is a tensile internal force, and to the internal force in one-four, which is a compressive internal force. Here is the contribution of the node three to the Cremona diagram. So we can see that with the classic tools of the resolution of trusses, we obtain methods which are very similar to the ones we have already studied to obtain the internal forces. We can also notice that that if we look at the arch-cable, certain of the rules which we had already determined, that is to say... that the chord which is on the side of the cable is in tension, while the one which is on the side of the arch is in compression. We can see that this rule is still verified. We can also see that if the inclination of the diagonals looks like the one of the arch, then they are in compression, while if it is opposite, it is in tension. So, the things we have seen for the constant height trusses are valid for truss cantilevers. Here, we have an example of cantilever, of big cantilever, with an approximately 50 meters long span in Geneva for the Hall 6 of Palexpo. We can notice that we have diagonals which are sometimes simply N-shaped, and some other times X-shaped. Why this ? Because, because of this exhibition, this cantilever is outside, it is possible to have large wind loads which can lead to reversals of the internal forces, that is a bit similar to what happens for a high-rise building. Here is a particular example of cantilever designed by the architect Norman Foster. We can see a structural system in the central part of a building, here, with several stories, we have elements in tension. That is to say that all these floors are going to be carried by the central part by means of these elements in tension. These elements are going to go up into a system, which looks a bit like a crane, and then to go down in tension on this side here. Likewise, on the other side, we have elements in tension which go down, it occurs at every level where it is necessary. Of course, for it to be in equilibrium, we must have compression all along, in these elements. Then, these columns here are also in compression. So, why this system ? This system is quite complex, it is not often used, but it is interesting in this context, because the wish of the architect was to create large free areas. Here, we have this at several levels. These are zones where there are no supports or doors, but we can have internal yards or restaurants, with little gardens, so a big freedom. Instead of having a very high building - we are located at 30 or 40 stories over a garden or a patio - here, no, we are located at a few stories, it is just necessary to go down of a few stories to reach a more open area. In this lecture about truss towers and cantilevers, we have seen some examples of shapes and types of bracings: visible bracings on the facade, bracings located inside, integrated in what we call cores. We have seen how to determine the internal forces in this type of truss bracings. We have seen that the rules which we had already established, particularly about the sign of the internal forces for the trusses, still apply to the trusses in cantilever.