Hello, in this video, I will talk to you about slabs on point supports, that is to say slabs supported by columns. As a consequence of this, we will see that there is a concentration of internal forces which occurs on top of the columns, which can lead to the phenomenon of punching, which must be avoided by all means. I will propose you a few configurations for slabs on point supports and I will show you that we can also obtain equivalent spans for this type of slabs. In this video, you can see a slab represented by a black cardboard sheet which is supported on four columns. I use my hands to apply a load more or less distributed to deform this slab. We can see here that this slab deforms between the supports in the parallel direction here, and also in the direction parallel to these two sides. That is what we will also see behind, and then also in the middle; we have a deformation which is larger in the middle than on the edges. That is not easy to represent graphically, but I hope you can see what it is, nothing prevent you from taking a small piece of cardboard to make a little experiment similar to this one. What we notice is that when we have a columun, here, on which come the internal forces, that is what Nervi had illustrated very well creating this beams grid with, here, a large concentration of ribs just on top of the column, well that is what we have because we have a lot of elements which are in tension in their upper part, that is represented by a familly of trusses, these are structures which we know well, this is a valid representation to illustrate the internal forces in a slab near a column. So we have a concentration of all these tensile internal forces in the upper part, and then I am going to represent the part of the trusses which is only in compression. We thus have here diagonals in compression which go towards the column. These are the diagonals which are a bit further, then, they lean one more time, to go down towards the column. We cannot see very well here, but there are many diagonals in compression which are supported by this column. The lower part of the slab is obviously in compression in this zone here. We thus have a concentration of internal forces on the top, that is to say on top of the columns and this can lead to the phenomenon which we call punching. - that is what I have drawn below - so, here, we have a column with a compressive internal force; on top of this column, we have the formation, inside the concrete, of a kind of cone, which can separate from the rest of the slab which I draw here in green, and of course the reinforcement, which I do not represent here, but when, this phenomenon occurs, most of the reinforcement bars, at least the horizontal bars does not have much effect. And potentially, it can lead to the falling of the slab. This is catastrophic because if you are under this slab, its huge weight is going to crush you and, furthermore, if this slab falls onto the following slab, there are chances that the following slab does not resist and that this collapse spreads across the whole building. So this punching phenomenon must really be avoided by all means, that is why I mention it in the introductory lecture, even though we are not going to get into the details of how to fight against punching. For the moment, I would only like you to know that it exists. How can we tackle punching? Robert Maillard, the Swiss engineer, had already an intuition about this in his first flat slabs constructions. We can see a flat slab which has, in this part here, an element which widens and that Robert Maillart called a mushroom. The interest of this mushroom is that, indeed, the internal forces can reach the column not directly, but in a more gradual way through this big mushroom. We can notice a similarity between the solution of Robert Maillart, on the left, and the solution of the Palace of the Popes, on the right, which we have studied when we have started to look at the realization of the flooring, since indeed, in both cases, we have a shape which is a bit the one of a rib, where the internal forces can gradually go down towards the column. In the administration building of the Johnson company, Frank Lloyd Wright, at the end of the 1930s, went to the extreme of the expression of the structure, since actually the structure, here, is composed of mushrooms - big circular mushrooms, which barely touch each other, which are supported on columns, and on which a very light and transparent structure is used as roofing. So we have here a structure which is only a juxtaposition of columns with a big mushroom on top and that really looks like mushrooms in this configuration. However, nowadays, it started with Le Corbusier, but that is still very strong, the trend is to make what we call a flat slab, that is to say to directly have a post on which we have a slab, without any width changings, an entirely flat slab. That is easier from the point of view of the building process, that is easier to place elements under the slab, however, we must carefully deal with the concentration of internal forces on top of the columns by means of specialized devices, and then, maybe increasing the depth of the slab for it to work. If we look, here, for flat slabs, we have two configurations which are represented, to give us the equivalent spans. The case of a slab on columns with an outlying wall, that is for example the case when we have a underground story since we will have cellar walls around, and then columns inside, we can see that the equivalent spans are around 0.9L, and if we are very very far, very regular, very far from the edges, we can reach a value of 0.82. The case of the right is the one of a building with only columns, so there is no wall around, there are only columns, and then we will have for example a glass façade, we can see that the equivalent span coefficients are larger than one, we can find again, very far from the façade, 0.82, but otherwise, we have values around one, sometimes a bit smaller though. That means that for a same span, a slab on point supports, on columns, will require more material, to be thicker than a slab supported on walls, because we have seen that for the slabs on walls, we generally had values smaller than one, which could even go down to about 0.6. In this video, we have seen that in slabs on point supports, there is a large concentration of internal forces on top of the columns, which can lead to the punching phenomenon which must be avoided by all means. A constructive solution to avoid this problem is to place mushrooms, which are widenings on top of the columns; for flat slabs we do not use this kind of solution, but we will have to take it into account during the dimensioning stage. Fianlly, the equivalent span have been indicated for the slabs on one-time bearings; their values range between 0.9 and one in most cases.