Hello, in this video, we are going to look at the influence of the cross-section of a beam on its strength. We will see how to arrange the material, how to optimize the shape of the cross-section and we add some general comments about the optimization of the shape of a beam over its entire length. We are going to look again at an example which we have already seen a short time ago. We have a beam here, which has a width of 2 h and a depth of h, and here is another beam which has a width of h and a depth of 2 h. What we know, if that is the cross-section of a simple beam with vertical downwards applied loads, is that we are going to have compression in the upper part of the beam and then tension in the lower part. Compression here, tension... and the effective depth is equal to h / 2 on the left, and to h on the right. So, clearly, with the same amount of material, we have a more efficient structure. Actually, what we have seen, is that moving from one solution to the other, we have multiplied the strength by 2 and the stiffness by 8. So that is something which is very significant and we can thus say that the cross-section which has the largest depth is largely more efficient. We have here a square cross-section with a width of h and a depth of h. We can simply think that putting it on its side, we are going to have a larger depth, which will be equal to h times the square root of 2, and then a larger width which will also be equal to h times the square root of 2. Is this cross-section really more efficient? As for the distribution of stresses, we know that we have compression in the upper part on the left and that is going to be the same thing on the right. And then tension in the lower part, as well as on the right. Let's try to calculate z. On the left, we know that z here, the effective depth, is equal to h / 2. Here, we want to find the center of gravity of the triangle. We have a triangle which has a depth h times the square root of 2 over 2, so its center of gravity is located at h times the square root of 2 over 2 over 3. So in total, the distance between both centers of gravity, of the lower and the upper part, is equal to h times the square root of 2 divided by 3, that is roughly equal to 0.47 h. So, we can see that the strength is almost identical actually. We had 0.5, 1 / 2 ; now we have 0.47. The strength is almost identical. Actually, if we look at the stiffness... We are not going to do it, that is a calculation which goes beyond the content of this course. But the stiffness is smaller. So, the principle of maximizing the depth is not the right principle. The principle that we will have to follow is the following one: we have to take the material away from the neutral axis. I had not drawn the neutral axis, I am going to draw it now. We have here the neutral axis of the cross-section. And clearly, if we had used this criterion, we would have seen that in the right part, there is a lot of material which is quite close to the neutral axis since we have this triangle. There is only a small part of the material which is really located at a distance of h times the square root of 2 over 2, while here, the material is more compact. But we will see better methods thereafter. If I have a certain amount of material, on the one hand, so I have a depth which is equal to h, and then for example, a depth which is equal to 2 h. So, I have a certain amount of material, what we have done till now is arranging the material to give the cross-section a rectangular shape with, obviously, tension on the top and compression on the bottom. It is still necessary to respect the depth of 2 h, but however, what is important, is to take the material away from the neutral axis. I am maybe going to rewrite this principle. Take the material away from the neutral axis. So, I can think about a solution to do so. I am going to up to the top but I am going to spread the material laterally. At the end, what I have just drawn here, that is approximately the same surface than here, and likewise, for the tension, I am going to spread the material in the lower part. If we look at the effective depth z of the cross-section, it is not exactly equal to 2 h, but almost. So z tends towards 2 h, it will never be able to reach it because a certain thickness of material is required anyway while here, unfortunately, or let's say, that was normal, we only had z equal to h. So we can see that arranging the material according to this principle, - as far as possible from the neutral axis - we can obtain a structure which will be significantly more efficient. That is what is used, for example, for the steel sections here, which we call shapes... with large flanges and we also call them double T shapes. Some of you would maybe like to call them H shapes, I will come back to why this is not a good idea later. Let's look here, if we have a cross-section with a depth h, let's look at the distribution of stresses. We are going to have compression in the upper part till here and then tension in the lower part. If we think about where is located the center of gravity of the material, we probably have something like this, so we have a quite favorable z which is maybe equal to, I don't know, 0.7-0.8 times h. If however we have one of these cross-sections and that we arrange the material in the form of H... That is why I do not like the shape of an H, because if we arrange it in the form of H, so we will have compression here and then a little bit of compression in the part which is very close to the neutral axis, and likewise, a little bit of tension very close to the neutral axis, and then most of the tension here, vertically, and here, what we can see is that the effective depth... the center of gravity is roughly here so the effective depth is approximately h / 2, even a bit less since the material is a bit close to the axis in the horizontal part of the H. That is why we talk about double T shapes. There is a T here, and then an inverted T below. That is much more correct. A H shape is actually a double-T shape which is not positioned well. Sometimes, we have to position them in this way, but the right way to position them, that is in the double T position. If we now look at the effective depth of various cross-sections, so we have here a solid circular cross-section with compression on the top and tension on the bottom and then the center of gravity of half a circle, we can find them in a textbook, that is not very important, we can see that the effective depth is clearly smaller than h over 2. If we do not arrange the material in a solid circular cross-section, but in a tubular circular cross-section, actually the center of gravity is going to be similar, so z... here, I should have indicated z, so z here is going to be quite similar however the material is efficiently arranged because a big part of the material is far from the neutral axis while here, not at all. We have a cross-section below, the tubular cross-section which is a quite efficient cross-section. We do not have to only use circular tubes, we can also have square or rectangular tubes and these tubes are quite efficient since we can see here: most of the material tends to be really far from the axis. There is a vertical part but there are also horizontal parts which are very far from the axis, and thus, the effective depth is quite significant, that is about the same than what we have here. Of course, we can also have double T or tubular cross-sections which have a larger depth, as you can see on the right. Here, that is even more favorable, because here, there are the vertical elements but the horizontal elements which we call the flanges for the shape with wide flanges, these elements are very far from the neutral axis. They are very efficient cross-sections. This, that is for the steel construction. We are going to say that z is probably from there to there and then just slightly smaller for tubular shapes. They are a bit lower because there are two vertical elements which we call webs and then the center of gravity is a bit more on the inside, next to the axis. This way to proceed can be used for other cross-sections than the cross-sections used in steel construction. Here, we have the construction of a box girder bridge. You can see on the right that it built as a cantilever. If this is a cantilever, then we have compression in the lower part of the bridge and tension in the upper part. That is what we recognize here in the cross-section on the left. We have a very thick part in compression, which is about one meter thick, here, which is able to support very large compressive internal forces. There will also be a part of the compression which will be carried by these vertical elements. And then in the upper part where we will have tension, we can actually see many holes into which we are going to insert prestressing cables, so they are elements which have a very high strength to carry tension. There are also some reinforcements, there is also a little bit on the bottom because we always put some reinforcements in reinforced concrete elements. And what we can see, that is the same principle than for a tubular cross-section where we have a compressed element, vertical elements because we have to separate them and then a horizontal element which is again strongly in tension. Finally, we want to look a bit at the statical efficiency, that is to say: what can we gain varying the cross-section along the beam? What we have seen so far, is the solution which I am going to call solution 1. We have seen rectangular cross-sections. That is what we have here, in this graph, and then here, in this other graph. Let's look at what this graph indicates us as a function of the slenderness ratio. You remember, the slenderness ratio, that is the ratio between the span and the depth of the cross-section. The more the cross-section slender is, the more it is going to need material, that is very clear. That is what we can see here: the volume, which is normalized, but the volume of material which increases very quickly, quicker than all other systems. Conversely, if we look at the stiffness, w divided by L, we can see that a beam with a constant cross-section will deform less. So here, we have the curve 1 and also the curve 1 here. On this curve 2, we have the idea to make the depth of the beam vary, that is to say that in the middle, where we have the largest internal forces, we place a maximal depth, and in the direction of the supports, we are going to make it decrease. We can find again this curve 2 here and here. And we can see that, we need reasonably less material to make a beam which has the same performance than a constant depth beam. Here, we have this curve here for this configuration number 2. Configuration number 3 for a wide flanges shape and a constant cross-section. We can see that this solution is quite efficient, requiring even less material than what we have seen before with a deformation which will however be larger, that is logical since we really have very little material in such a cross-section. Solution number 4 is a solution which is similar to number 2 where we have a maximal amount of material at mid-span, and we remove material without decreasing the depth, but decreasing the width of the cross-section. So here, we have solution number 4 and we can see that it is quite efficient and it requires even less material and it rather deforms a bit less than the solution in which we made the depth of the cross-section vary. The last solution is solution number 5, which is a combination of solution number 4 and solution number 3, so we have a beam which has a width which varies along the cross-section, but this beam is initially composed of large flanges profiles, but we have cut a part of the flanges so we have less material. We can find again this solution here and here. Thus we clearly use less material than previously and the deformations remain quite reasonable, but they are almost as large as those of the beam with variable depth anyway. What can we learn from this? Well, what is interesting is to compare the results which we just got with the good results which we have obtained before. Here, we have a solution with a truss, in yellow. What we can see is that a beam with a constant cross-section requires much more material than a truss. Likewise, it is going to deform considerably less. If now we look at a beam with an optimized shape, like solution 5, so an optimized beam with large flanges. We can see that we reach an efficiency which is comparable to that of a truss with variable depth. So it is possible to get extremely efficient structures in the form of beams, but they cannot have a constant cross-section, their cross-section must vary along the beam according to the internal forces. Let's write down these conclusions. Beams with a constant depth require more material than trusses. So, if we want to have a structure which is light, which uses little material, beams with a constant cross-section are something which is less favorable. However, we have also seen that it is easier to make a beam with a constant cross-section, for example if we want to make it out of concrete, because the formwork will be relatively easy to make. Steel beams with a double T shape are comparable to trusses with constant chords regarding their efficiency, they are relatively efficient. And finally, with regard to variations, it is favorable to make the cross-section vary, especially to make the width of the cross-section vary. We have seen than the depth has a less significant effect. In this lecture, we have seen that the way to arrange the material in a beam has an influence. Changing the orientation of the cross-section, when it was a square cross-section, was not very efficient, but however, to put a rectangular cross-section with its largest side in the direction of the depth, that is something which is very efficient. We have also seen that the shape of the cross-section can be optimized, using the general principle of taking the material away from the neutral axis. Finally, we have done some comments about the efficiency of the load-bearing systems and we have compared them to that of trusses. For beams with a rectangular cross-section, they are clearly less efficient than trusses, beams with a double-T cross-section are relatively efficient, and the best is to have the cross-section vary along the length of the beam in addition to that, according to the intensity of the internal forces.
