Hello, in this video, I am going to talk to you about slabs on linear supports. First of all, I will talk about the bending behavior of slabs, an extension of the bending behavior of beams, I will insist a lot on the deflected shape, the shape that the slab takes when subjected to loads, I will talk to you about the types of supports, since there are additional ones for slabs, and finally, I will talk about continuous slabs, in relationship with their deflected shape. In this video, you can see a slab symbolized by a black cardboard which is supported on two gray walls on the left and on the right and on which I push with my right hand. We can see that the slab deflects in the middle taking this shape. What we cannot see well, but I ask you to believe me, you can also try at home, is that this slab also deflects in the parts which are in the back, which means that the slab also deflects longitudinally. Actually, if we compare a flooring to a slab, the difference is obvious. On the left, I have a flooring for which the boards are independant from each other. If I apply a load on this board, then only the board on which the load acts deflects. The other boards do not move, they do not feel anything. As a consequence, the board on which the load acts deflects significantly more, since the others do not participate. On the opposite, with a slab, if we apply a load, all boards - they do not exist anymore, I have just drawn these small lines to represent zones - all the zones of the slab, as we have already observed on the model before, all zones of the slab deflect as well in the longitudinal direction as in the transversal direction. That is the essential difference between a flooring and a slab. Obviously, all materials cannot be used to make slabs, for example, timber boards do not easily let themselves linked to each other laterally, that is not completely impossible but that is difficult, so typically, with timber, we will make floorings; with a material such as concrete, we will make slabs. In this picture, you can see a unidirectional slab such as the one we have seen before in the model. This means that we have a slab which is supported on a wall on the right, for us that is simply a fixed support for example, which acts over the entire length of the wall and on the left we would have a mobile support. I am not going to draw it behind, but that is also a mobile support which extends over the whole length. The other two edges are not supported. We have here a free edge, this edge is free too. Here we have a supported edge. How can we represent this slab on a plan view from the top? The supported edges are represented here by continuous lines, there is one on the left and one on the right, and the free edges are indicated by dashed lines. There is one on the top and one on the bottom. Here, we have a free edge, and there too, which we can find here. How is this structure going to carry loads? It cannot carry loads in the direction where there are no supports. The loads are thus going to be transmitted in the direction of the supports to the left and to the right. That is why we call it a unidirectional slab. If we put a load in the middle of this slab, the entire slab will collaborate. However, when we place a uniformly distributed load as I did here, we get a unidirectional behavior. All these small boards, which are not disjointed, will exactly have the same deflection. In this video, you can see the example of a slab which is not supported on two facing walls anymore, but on four walls located all around it. I am going to accentuate the deflected shape because we cannot see it well on the black cardboard of the slab. We have a deflected shape in this direction, but also in the other direction, like this. This time, this bidirectional slab is loaded by a uniformly distributed load over its entire surface. We can see well the deflected shape of the slab: it deflects in this way in the middle. That is what we have also seen in the previous video. And the intermediate lines will always deflect a little bit less... in this direction or in the other direction. Now, we have a slab which is supported on all its edges. We have here a support system like before. We will see that for slabs, the difference between fixed supports and mobile supports is not really significant. One of the reasons is that the walls on which the slab is supported are flexible. So we do not necessarily make the distinction between fixed and mobile supports. This said, in some cases, it is necessary to have constructive details which ensure that the slabs can indeed move on top of the walls with small joints. That is something I will not talk about more within the framework of this course. Here, we have a supported edge, and also here. If we look on the right, of course all edges of this slab are supported. Here is how this slab carries loads: the loads always try to go towards the supports in the most direct way, so from the middle, we are located at the same distance from all supports, so the slab is going to carry in both directions. However, near the edges, if we have a load on the corner, the closest edges are in the diagonal direction, so there is also a load-carrying mode in the direction of the diagonal, and as a consequence of the fact that there are several load-bearing modes, slabs supported on four edges need less material to resist the loads we apply to them. We will get back to it in one of the next videos. Here we have the example of the Savoye villa, designed by the architect Le Corbusier, near Paris. Inside this villa, as it can be seen here, we have several linear supports, small beams, which support a slab; 1, 2, 3, 4 spans, so this is a continuous slab. In this case, we can see that the spacing between the beams is constant. What does it mean? If we look, under a uniformly distributed load, at the form of the deflected shape, the maximum deflection is larger in the edge spans on the left and on the right, because these spans do not have the correct length. We have seen it for the continuous beams, we would use 0.8 times the span in the middle. That would be more favorable. That is not what we have here, we have the same span everywhere, that can be understandable fron an architectural point of view. We thus have a deflected shape where the largest vertical deflections are located in the edge spans. That is very important to have the deflected shape in mind, the shape that the slab takes under the loads when we study the rest of these videos. We are going to look at how slabs are supported. We will see that depending on the detail we use, such as for example concrete or masonry walls, or else beams, the behavior is different and that is necessary to take it into account. We start here with the probably most usual solution, that is to say masonry walls on which the slab rests. If we have a slab edge, that is to say an end wall of a building, and the slab stops, then we are going to have a deflected shape which will essentially look like this. This corresponds to a supported edge from a structural point of view. If, however, the slab continues on the left and on the right, as we have seen it in the villa of Le Corbusier, then the deflected shape will be like this and from a structural point of view, we will have a clamped system leftwards and rightwards. Pay attention please! For graphical reasons, I have drawn a clamping on the left, but actually, the clamping is located on the axis of the wall. Both are on top of each other, the clampings leftwards and rightwards. If I have a small cantilever, I will have the same behavior than if I have a simple support. Here, I have a supported edge. Generally, during a predimensioning, we can neglect a small cantilever. In slabs, we can also have a free edge. The slab is going to be able to go up and down freely. I do not have any symbols, I just have the slab which keeps going. The edge of the slab is free here. Finally, if I have a long cantilever, that means L2 larger than approximately 1/3 of L, here, L2 is the length of the cantilever, and L, the length up to the next wall. In this case, the deflected shape is going to be close to what we have in a clamped case, and the behavior as well. We are going to have a system where the left part of the slab is clamped and the cantilever is also clamped. It is obviously necessary to be careful to cantilevers which are too long. I would say that it would not be reasonable to exceed half of L for a cantilever because otherwise the behavior of the structure is going to be quite different. It is possible, but that is a particular detail which is not dealt with within the framework of this course. We are now going to look at the case of slabs supported on reinforced concrete walls which are sufficiently thick, without defining what is "sufficiently thick". but we cannot build extremely thin walls and expect to get this behavior. If we make very thin walls, then we can consider them as masonry walls. If the wall is sufficiently thick, we are going to get a behavior similar to the one of a clamping. The slab thus behaves as if it had a clamped edge. If the slab is continuous, it does not change anything compared to the masonry wall in terms of deflected shape. We again have a system which is clamped on the left and on the right. And if we have a cantilever, any kind of cantilever, we are going to have, thanks to the wall, the same behavior than on the left with the edge of the slab. We are going to have very small deflections in the cantilever if it is short, more if it is long. In any case, we will have a behavior with a double clamping leftwards and rightwards. Finally, it is quite frequent that, like the Savoye villa which I have shown you before, slabs are not supported by walls, but rather by beams, generally made of reinforced concrete, but they could be made of steel, that would not change a lot. In this case, we are going to have a behavior where the beam, except if it is extremely stiff, will tend to rotate with the slab, which means, from a structural point of view, that we have a simply supported edge; if obviously, we have a continuous slab on the beam, thus the behavior is going to be like in all the other cases, that is to say with a clamping leftwards and rightwards. If we have a small cantilever, that is going to be the same thing than what we have seen before, that is to say that the beam is going to rotate, which corresponds to a simply supported system. If however, we have a long cantilever, then we can also end up in the configuration which we have seen before where there is no rotation at the level of the beam and then the system is clamped leftwards and rightwards. In this video, I introduced to you the bending behavior of slabs, the deflected shape they take under a point load and under a uniformly distributed load. We have seen that the behavior of slabs can be uni- or bidirectional depending on the way they are supported. We have talked about continuous slabs, insisting on the importance of the deflected shape. Then, we have seen various types of supports, that is to say concrete and masonry walls, or else beams, which provide supports for the slabs.