Hello. In this video, we want to look at structural solutions that can be used to solve the problem of instability, that is to say how to design structures which will be sufficiently stable. We will see that there is a great similarity between what happens with instability and what happens with bending deformations, we will look at the internal forces in these structures, and we will déduce the types of cross-sections which are favorable. We will also see that the shape given to the element can contribute to increase its stability. Here, we have two columns: one we have already seen within the framework of this lesson, which is subjected to a vertical load and which deforms under the effect of instability; and then another one which is subjected to a horizontal load. This is a cantilever in bending, a type of structure which we have also seen within the framework of this course. Let's look at what happens at the bottom of these columns: on the side of the center of curvature, or inside the curve, we have compression while outside, we have tension. That is not exactly identical, but it looks alike a lot. And what does it mean? For us, we can directly learn lessons from it: the cross-sections which resist well to bending are also favorable from a stability point of view. We had seen that what is important is the distance between the material and the axes of the cross-section. In the intermediate part here, I have drawn a certain number of cross-sections which all use the same amount of material. So, if these are steel shapes, they all have the same weight per meter. What about their axes? If we look here, we have the axes - I am going to draw the axes for all these cross-sections - the axis of the center of gravity. We can see that both these cross-sections here, - the compact cross-sections - are not favorable since material is too close to the axes. However here, these tubular cross-sections are favorable since the material is on average very far from the axes. There are small differences between the solutions, but that is clear that the material is much further from the axis than for these two solutions. And then here, the solution with a double T beam, is more or less favorable since actually that is favorable around this axis and then less favorable around the other axis. This type of structure can still be used if in one direction, the one for which it is less favorable, there are more intermediate supports than in the other direction. In this case, we make an efficient use of this kind of cross-section. However, it is better to use a tube. Let's look at this again, purely from the point of view of bending. We have this fundamental solution with an element in compression and an element in tension. Fundamentally, we have already seen the solution, it is an arch-cable. We have also already seen this within the framework of this lesson, that is a guy-wire. You remember that the first structure which I stabilized, I stabilized it with external wires or cables. The use of external cables is thus a good solution, a good structural variant because we have something which resists well horizontal loads and which also deals well with the instability issues. We want to find other variants for this structure. That will be in the form of a truss, for example a truss like this one. This kind of truss is a favorable type of structure to stabilize a structural element if we have loads which act downwards. And then finally, all sorts of beams are also favorable for this. Here, I am going to draw an example of a beam with diagonals in compression. And then I have tension of course on this side here and then in the posts. This is also a favorable solution. You should note that in the three cases, we need space. The horizontal distance between the support which is in tension and the support which is in compression is very large. Here, we thus have structures which resist well horizontal forces and which are favorable for stability too. Let's see with this example , which accompanied us during most of the semester, what we can understand in these large trusses First of all, what we can see here is that both these elements are in tension. - I think you know it now - and they are very thin elements. Likewise, these elements of the lower chord, there are also two, they are very thin, because they are in tension so there are no instability problems. That is also true for these diagonals and then obviously for the rest of the lower chord. Conversely, if we look at the upper element here, there are two tubes actually, we can see one behind the other. And this is really thick. Why? Because we have taken two tubes. Here in the red part, we managed to take elements with material very close to the axis because they are in tension, there are no instability problems. on the contrary, for the upper chord and then for the diagonals in compression, we must use thick elements to insure stability. So, these elements are thick but they are tubes. So actually, they are not obviously much heavier than the red elements, however, they are a little bit more complicated to make. On the external facade of this building, we can recognize here elements which are extremely thick, with large dimensions and, at the same time, which work together with very thin elements. Here we have a type of structure, which, working with the remaining part of the facade truss, - there are these inversed V-shaped elements which are crucial - can support horizontal loads induced by the wind or by an earthquake and which would come from the left. We have the same structure upside down for the case of an earthquake or of wind which would come from the right. To conlude, we are going to look at the structure of the Milenium Dome in London, which was built for the 2012 Olympic Games. We can see that all this roofing is suspended from cables which are fastened to top of the masts. So the load which act on these masts is relatively large. There loads which come from the left, from the right, with many different angles. But we essentially have a compressive internal force in this element. This element looks like my cane and it is going to bend in this way when the critical buckling load will be reached. What we can see is that the curvature will be maximal in the central part. So what was done for this solution, was to give it a maximal depth - the depth is the distance between the left part and the right part - where the internal forces are the largest. You maybe remember other structures which we have seen which did not have a constant prismatic shape over their entire length, well here, we have spread these elements since it is where bending will tend to be larger. We thus can find a solution to the stability issue by choosing, that is important, a good cross-section, tubular cross-sections for example, cross-sections with material away from the axes; and then, making the longitudinal shape of the element vary for it to deflect as little as possible. In this video, we have seen the internal forces and the deformations which are linked to stability. We have seen that it is relatively similar to what we have already seen for bending. This leads to the fact that the cross-section with shapes which are favorable to resist bending are also favorable to resist instability and, just like in the case of bending, we can adapt the shape of the element to give it a larger effective depth where the internal forces are larger.