Hello, in this lecture, we are going to talk about superimposed beams. Actually, we have already partially dealt with this topic during the last lecture, and I have told you that when we placed two beams on top of each other, actually, it had not the same effect at all than when we had a beam with a double depth. We are going to revisit this subject, to understand a bit what happens, especially sliding between two superimposed beams, which is really the cause of the problem I told you about, then we will look at solutions to prevent this sliding, and thus to improve the performance of superimposed beams. So there is first a beam on the bottom with a depth h, which is subjected to two loads Q which are at the first and of third quarter points of the beam. Then, we have two beams of depth h, they are the two superimposed beams of the last time, I have just spreaded them a bit, to be able to see a bit better what happens, then we finally have a beam which has a depth h. On the right, we can see the result of a calculation made with a calculation program for engineer, regardless how we have actually calculated it, but we can see the results. What is interesting is to notice the amplitude difference. I can measure here, the maximum deflection of the lower beam, with a depth h, for me it is roughly equal to 20 millimeters, while the maximum deflection of the upper beam is significantly smaller since it is approximately equal to 3 millimeters. Yes, we have seen that the stiffness, we have seen in the previous lecture, that if we have a stiffness which is equal to one here, the stiffness will be equal to 8 here, so it means that the deformation will be 8 times smaller. In a non-surprising way, here, we have a vertical deflection which will be equal to one half, that is to say, we approximately had 20 mm, we roughly have 10 milimeters measured here. Obviously, these deformations are not the ones we will have in reality, they have been exaggerated by the calculation software, otherwise we would not see anything. It is interesting to notice what happens in the zone where we have the two beams, because if we take a point which was maybe initially just in the lower part of the upper beam, under the beam, then another point which was at the same place, but in the upper part of the lower beam, so what we can see is that these points tend to move away from one another. This point here moves in this direction, while this one moves in this direction. That is very clear I think, we can see at the ends of the beams, where there is really a relative displacement between the top of the lower beam, and the bottom of the upper beam. That is what we call sliding. Here, I have replaced these things together, you remember what we have seen the last time, when we have a monolithic beam, we have compressive stresses in the upper part, tensile stresses in the lower part, while when we have two superimposed beams, in the upper part of each of the beams, there will be compression, and then in the lower part, there will be tension. And then, along this interface here, we will have a sliding, this part slides in this way, this part slides in the other way, it is also true here. And then in the other direction, the same thing. That is what we call sliding. And twe can first notice on the basis of this observation, it is clear, that reinforcements will be difficult. Imagine that you have, for example, in a building, a beam which would be, for example, the upper beam, which tends to deform too much, and perhaps cannot carry enough loads, and at this moment, we consider how to reinforce it by placing a second beam underneath. At this moment, they are not going to be linked, they are going to slide on top of one another, and thus, instead of having eight times less deformations, there will be much more. You can notice that here the ratio 8 is not represented because we wanted to have something from which we could see the rotation of both beams. We can try to fight against sliding, that is what we do here, in the example of a composite steel-concrete beam. What have we done? On the metallic beam, you can see on the left, we add these elements which are welded here on the bottom, or fixed in a very strong way, but generally, that is welded, these elements, we call them dowels. These dowels have the function of limiting sliding which can occur between the steel part, we are going to say the first beam, and the concrete part, the second beam, which we just cast on it in a second phase. This is an efficient solution. We will soon see an example about how to pre-dimension such a composite beam. Another solution, which is also an interesting, I show it here, in this video, where we can clearly see that the beam which I hold in front of me, is constituted of a certain number of elements which are superimposed on top of one another. So, in principle, it should not work well, but afterwards, these timber strips are glued together, in a system which works at high pressure and at a quite high temperature. As a result, sliding at the interface between these elements is not possible anymore. Here, we can see these various elements in more details, so we can see here a timber element, a strip which is separated by these two yellow lines, and here another strip, and so forth. The interest of these strips is that we can make them from relatively young trees, here, we can see the annual growth rings. Here, that is a tree which is maybe a couple decades old, but which is not particularly old, and which will enable, being combined with other trees, here, I am going to draw the rings for the lower strip. So using relatively young trees, and glueing them together, we can obtain beams with sizable dimensions, like the one I have in front of me. This solution thus enables us to create beams with quite large dimensions, here, I indicate the dimensions, and also with quite free dimensions, you can see here, a beam which has been created has a variable depth, it is deeper in the part which is far from us, and shallower in the part which is close to us. This has been made with a material which is just initially a juxtaposition of strips on top of each other. If we look at all the beams here, it is maybe a bit more than one meter deep, if we had to extract this beam from a tree, this tree should be hundreds of years old. There are not a lot of these trees in nature. Here, composing this cross-section with strips, we have the chance to dispose of almost infinite dimensions of timber, and then it has enabled a very rational use of young timber, and the availability of timber beams with very large dimensions. In this lecture, we have revisited the principle of superimposition of beams, explaining why sliding is what prevents two superimposed beams to be really efficient. We have looked at two methods to prevent sliding, a method with dowels, this is the solution which is used for the composite steel-concrete structures, or else, the other method, that is a gluing system, in the case of glued-laminated timber which enables us to obtain beams with quite large cross-sections, made from relatively young trees.
