Hello, in this video, I will talk to you about predimensioning of slabs. For this, I will introduce the concept of equivalent span then we will see what we precisely want when we deal with predimensioning of slabs. On the basis of the support conditions of slabs, we will look at how they behave. Then, we will get to the concept of equivalent span which will enable us to proceed to the predimensioning of slabs on a systematic basis. This predimensioning will simply consist in obtaining the minimal depth to give to each element of the slab. The predimensioning of the amount of tensile reinforcement which should be placed inside the slab will not be dealt with. That is a bit different from what we have done before with beams, since in the case of beams, we have also determined the amount of reinforcement. We are going to look at some characteristic configurations of slabs. Firs of all, this slab on the left, with a support on the left and a support on the right so a simple support on each side represented with one line. The distance between both these supports is equal to L and then, on the top and on the bottom, no supports, so a free edge. The internal forces inside a slab beam which we extract of this slab are represented here. We then have an arch in compression, together with a cable in tension, and the span to be taken into account here, is directly the distance between the supports L. If we look at the right side, we have a slab which is clamped on both sides so with a double line on the left and on the right. However, on the top and on the bottom, it is still free so we draw a free edge. Again, isolating a small piece of beam from this slab and using the representation which I have introduced within the framework of the video about continuous beams, we get a configuration which, in theory, would remain within the depth of the slab. Well, that is not exactly the case in my drawing, but we obtain a configuration of the slab in which we can recognize cantilever elements on the left and on the right. And then in the central part, we have an element which strongly looks like what we have seen before, that is to say a classic arch-cable. And here, between these two elements, we have the equivalent span which we note Leq. So I also write Leq on the top. In this case, it is approximately equal to 0.67 times the span. If we know how to dimension this slab elements with this span, we will also know how to predimension this slab element with a smaller equivalent span. In this picture, we have two other configurations, the case of a slab which is clamped on the right and simply supported on the left, with free edges on the top and on the bottom. Here, we have the arch which goes from the left support but which stops before the right support. We will have an equivalent span smaller than L but larger than what we had before, approximately, Leq = 0.8 * L. That is roughly the same thing when we have a cantilever with obviously three free edges and then a clamped support on the left. The equivalent span in the case of a cantilever must be taken at 1.69*L. This picture summarizes for the case of a square slab, so its dimensions are L times L and the equivalent span is equal to a k factor which is given in the table which we are going to look at, times the span L. So one more time, to be able to read well this figure, it is necessary to remember that we have here a free edge, here a supported edge and here a clamped edge which does not allow rotation. We can notice that in this table, there is a family of configurations for which the k factor is larger than one. There are a few cases. It should be noticed that this case is absolutely unstable, since we only have a support on the left and not enough supports so the structure here is going to fall. But we have included it in the table to be complete. Here, we have a certain number of cases, especially the cantilever which we have seen before or else cases which are similar to the cantilever, where the equivalent span to take into account is larger than the distance between the supports. And then, on the opposite, here on the bottom, we have a configuration where the slab is clamped on all its sides and where the equivalent span to take into account is only 0.56 times the span of the slab. We have in this picture equivalent span factors for a rectangular slab. It still has a small side which is equal to L, and the long side is equal to 1.5*L. If your slab is between both, you can interpolate or use the most cautious value to proceed to dimensioning. The equivalent span is always defined as k times L where L is the smaller of two spans. Here, you can see values which sometimes are larger than two in a certain number of cases, and then which are a little bit larger than before. which is logical since the influence of the longer span can be felt in this configuration. Once we have obtained the equivalent span, it is necessary to proceed to the dimensioning and I am going to do a reminder. This topic has been seen in the course "The Art of Structures I", but there are mainly two ways to proceed to the dimensioning which must be done simultaneously, or let's say, one after each other. That is, on the one hand, the Serviceability Limit State, in which we care about the behavior of the structure which must be satisfactory. For example, we avoid accumulations of water like on the structure on the left. And then, on the right, the Ultimate Limit State, that is to say the collape. It is obviously unacceptable that a structure collapses onto its occupants. These two criteria, the Serviceability Limit State (SLS) and the Ultimate Limit State (ULS), have been considered by Professor Muttoni in his book to create this figure which gives, as a function of the equivalent span of a slab, the allowable slenderness ratio which it can have. This, for two types of loadings. A light type of loading (2kN/m²), it would correspond to an accomodation building. A heavier type (5kN/m²), it would maybe correspond to a school or to a shopping center. If you have special structures such as libraries for example, or something else, this figure should not be applied because that would not be sufficient or it would be necessary to decrease the slenderness ratio in a corresponding way. And then here, we have the vertical deflection which we are going to accept. In most cases, we can accept a vertical deflection equal to 1/300th of the span. But if we have fragile elements, like glass elements, etc. which are connected to the structure, we will want to limit the vertical deflection to 1/500th of the span. On the right, I have drawn simple beams which have, respectively, a slenderness ratio of 1/30, 1/20 and 1/15 to give you an idea about the slenderness ratio which we can expect. If we take a slab with an equivalent span of 6 meters and the conditions which are the least limitative, that is to say 2 kN/m² of load and a vertical deflection of 1/300, we can read here that the ratio is equal to 19.5. What does it mean? It means that in this case, the minimal depth of the slab is equal to 6 mètres... ... so 6000 mm / 19.5 that is to say around 310 mm. We will see other examples of application in one of the videos which follow. As I said in the introduction, in a concrete slab, there is of course concrete but there are also reinforcing bars. We can see here the lower reinforcement and the upper reinforcement. What we do during the predimensioning here is only determining the total depth of the slab, the minimal depth that we will have to give to it. It will then be necessary, for a dimensioning, to proceed to the calculation of the amount of reinforcement required, but this is out of the scope of this course. In this video, we looked at predimensioning of building slabs. We have seen that, on the basis of the types of support, we can determine an equivalent span for slabs and that on the basis of this equivalent span, inserting the types of loads and the deflection requirements, we can obtain the minimal depth to give to slabs. I have also insisted on the fact that it is only the depth. Later, to build a concrete slab, it will also be necessary to determine the amount of reinforcement but this course does not deal with this.
