Hello, in this video, I will talk about vertical Vierendeel beams. In this configuration, we will see that on the one hand, we can compare this structure to superimposed frames or else, on the other hand, when we think about what we have seen with the Vierendeel beam in the previous video, to a Vierendeel cantilever. That will depend a bit on the loads, if they act vertically or horizontally. In this structure located in Paris which we are going to use as example, the headquarter of UNESCO, the structural concept has been developped by the engineer Nervi and we can see a peculiar construction. That is a tall building with columns whose width increases as we go towards the bottom of the building and at ground level, there are some very particular columns with a very inclined shape. Let's first look at the effect of loads on these frames starting by the top storey. For the sake of simplicity, I only place one load in the middle of the frame. We already know how to solve this, we have seen it in the lecture about frames. We know that the pressure line... That is a statically indeterminate structure, the pressure line is going to have this shape. And since it is necessary to bring back pressure inside the frame, we are going to apply compression in the diagonal which is going to enable us to straighten up the compression. We are going to have an arch structure, deviated by these internal forces in the diagonals and a cable structure which I am going to draw right now, which is added to this structure. There is nothing new compared to what we have seen about frames, that is just the top storey frame. Now, we are going to say that it is possible to place tensile reinforcements in the slab. I place it anywhere but that means that the thrust of the frame is carried by the slab. The consequence of this is that the support reactions under the slab are simply vertical. If we now look at the frame just below, it has the same load. We are going to assume that the live load is the same. And some loads which are roughly equal to half of the supports reactions of the previous storey, act on this frame So it has an influence on the shape of the pressure line since now we have these three loads so we are going to have something which is going to be like this. The left and right parts are influenced by the support reaction. And then the part which goes up towards the support line of the load. The result of this, qualitatively speaking, does not change a lot, but we can see that the eccentricity to be carried is smaller. We again have diagonals in compression and in a similar way, the internal forces in the cross-beam and the posts. Here, that is absolutely similar. I have been mistaken with this line before, we are going to correct it, the diagonal here is wrong. You probably had noticed it. Sorry for this mistake. And here, in this direction, there is obviously going to be tension. I again forgot, I did not put it here, we also have compression vertically to distribute the internal force between the arch system and the cable system. What we can see however is that, here, this part of the pressure line is more vertical and that will be even more the case as we go down the stories. Here on the right, I have just drawn the right part of the pressure line corresponding to the load on the first storey from the top, the second one from the top, the third one from the top, etc. We can see that because of the accumulation of forces which come from the upper stories, the pressure line becomes more and more vertical. So, I am going to write here: because of the weight of the upper stories, the internal forces in the posts, they are the columns of the frame, tend to become more vertical and do not get out anymore of the material. Here, I have only drawn the part in compression but obviously, we have seen the part in tension which we must also have. However, when we have something which is simply inclined inside 1the column and which does not get out of the material, we do not need to have tension at the same time. We are going to have a structure which is more economical. We can notice here that it is also made possible by the fact that the dimensions of the columns increase as we go towards the bottom of the building, which is logical since the internal forces increase as we go down the building. Here, in this video, I push horizontally on a vertical Vierendeel beam. We can imagine that it is a model of our building. We can see that I can push at the level of the nodes, whether it be at the upper level or at the lower level, they are still very deformable structures. Let's now look at the effect which we have here for the frame. If we look at the upper level, the effect of a horizontal force translates into tension and compression. So, we are going to have, we have already seen it before, we are going to have tension in the diagonal here and we are going to go till the other side diagonally and again we are going to have tension and then compression opposite to each other in the cross-sections. The compression must be diagonal. This is nothing new, we have already seen it when we studied frames. If we are now located several stories below, on the second floor in the present case, then we are going to have the sum or the resultant of both horizontal forces, since they are not equal, the resultant is a bit lower, in this case, than the middle of the segment between both forces, whatever, and the result is again going to be a system with tension and compression which are exactly identical to what we had before. I draw it very quickly. Tension in this diagonal, compression in the other one. Compression in the posts rightwards, going down rightwards. What is more interesting is to look at what happens if we consider the entire building. It is interesting to see that precisely, the dimensions which have been given for the columns on the ground floor have actually been chosen, in all likelihood, according to this configuration since at this moment, we can see that we have material, I just draw it in yellow the outside of these columns, which we cannot see very well, we can see that thanks to this, compression, particularly, can remain inside the column, which does not require any additional reinforcement, even though we are going to have to place reinforcement anyway for the case in which the wind would come from the other side. We can, like in the case of the Vierendeel beam, use the representation with teeth to understand the behavior of a Vierendeel beam. We take two parts which we separate. In these two parts, the teeth of the orange part tend to rotate upwards. The bottom parts do not move much. And then in the left part, teeth tend to go downwards, which is obviously not compatible. We are going to have to make it compatible, you can guess what I am going to do, I have already done it in the lecture about horizontal Vierendeel beams. To make these displacements compatible, we are going to have to push downwards on these orange cantilevers, here, and to push the pink cantilever upwards. As a consequence, we will have here compression and tension, here compression and tension, here compression and tension, and in the other direction, we are going to have here compression and tension, here compression and tension, here compression and tension. If I copy all this in the right drawing, this means that we are going to have here, in these posts, diagonals in compression in this direction, and in the other direction, they are going to be in tension. This, that was... I should have drawn it, for horizontal loads applied in this way on the posts. So, that is similar to what we have seen for a Vierendeel beam in the other direction. We could have made the complicated drawing like before but what I want is that you remember these principles and how we can understand the internal forces on the basis of this reflection with both half-structures which must be made compatible. To finish this, I show you an example of a building entirely built with Vierendeel beams. You can first see the quite significant amount of material. As it happens, this is a school, we do not need to make doors to get out everywhere, otherwise this would not be very convenient, because these elements here, in the lower part too, are load-bearing elements. Here, we can see at once a Vierendeel beam over several stories and a cantilever Vierendeel beam, it certainly works as a cantilever for the wind too, but it also works as a cantilever to let some space under these eaves for the schoolyard. In this video, we have seen that vertical Vierendeel beams can be considered either as superimposed frames, or as Vierendeel cantilevers, depending on the effect of horizontal and vertical loads. We have seen that it is also possible, there is a large freedom, to combine both types of Vierendeel beams, horizontal and vertical, there is no reason to deprive ourselves of this freedom.
