Hello, in this video I am going to talk to you about Gerber trusses which are directly issued from what we have seen in the previous lecture about the superposition of internal forces when we add a cantilever to a truss. You indeed remember that adding a cantilever to a truss, almost all internal forces decreased in the beam, in the initial truss and thus this layout interests us since if we can decrease the internal forces, we can also decrase the size of the elements to set up. We will also see that this Gerber truss layout is favorable for the deformations and then I will show you that it corresponds quite well to, at least, some methods of construction. Let's imagine that we are in the 19th century and that we want to build a bridge to cross a large river. Since we are in the 19th century, I am going to make... I am going to make a masonry abutment on each side, on each embankment and I have the idea to have a solution with 3 very simple trusses, which are drawn here. So I am going to have 2 piers in the river, still in masonry given the time it was built, but that is not a problem. Then a competitor which has studied well the lecture about trusses with cantilever arrives and says : "No, but I think we can do better." So, on both sides, I agree, I am going to make an end support in masonry. I am also going to place two piers in the river, but you should notice that I stagger them a little bit compared to the piers of my competitor and I bet you that my solution is more efficient. Here, we have the river which we must cross. You can notice that both these solutions are absolutely acceptable to cross the river. There is even, in the second solution, a part which is more open in the middle, which can be favorable for navigation, but we are going to consider that for the use we want to make, they are approximately equal. If I imagine that I have a load which is distributed over the whole length of my structure, uniformly, to simplify the reasoning, so I know that I am going to have, in the first solution, compressive internal forces in the end diagonals, and tensile internal force in the intermediate part. In the second configuration, this part here is a normal truss, which we know well, with compression on the top and tension on the bottom. We are going to add a cantilever which works in an inverted manner. On the top, we are going to place a truss which is absolutely normal. If we compare both families of trusses, we can notice that the trusses here have smaller spans than the trusses which were proposed here. Here is my solution with 2 trusses with cantilever and 1 simple truss in the middle. In this solution, as I said before, the trusses here have smaller spans. We can see that it is also the case for the truss in the middle and in addition, we have cantilevers, on the left for the one, on the right for the other which are going to reduce even more these internal forces. That is quite clear that this solution, the second solution, with 2 trusses with cantilever and 1 simple truss will be more economical, we will be able to use less material while solving the problem in a very similar manner. Of course, if you want to make a train pass on this construction, that is not going to be possible, it will be necessary to build it in a slightly more practical way, but it does not pose unsolvable problems. We are going to build these two trusses with cantilever. Here, that is going to be the part with the cantilever, we are going to have tension here on the bottom and instead of placing the intermediate truss on the top, we are going to place an inverted truss, so we are going to have tension here, here and here, compression on the top but that is going to give us the same solution. Then, maybe you will tell me: "Here, the train needs to jump." Obviously not! We are going to add a bar, actually two bars, which simply aim to let the train run, but there is no problem, these bars are carried by some parts of the truss. Please note that we have here two hinges in this configuration, two hinges which are important elements of the structure. Let's now quickly determine if the structure is statically determinate or not. Here, we are going to have two support reactions. Here one, here one and here one. So we are going to have 2, 3, 4, 5 supports reactions. We are going to add, as usual, the number of bars, so 1, 2, 3, 4, 5, 6, 7, 8, 9... I do not count this bar here because it is not part of the structure... 10, 11, 12, 13, 14, 15, 16... still not this one, 17, 18, 19, 20, 21, 22, 23, 24, 25 bars. And now, we are going to count the number of nodes. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 13, 14 and 15. 2 times 15 equal to 3, 2 times 15 nodes. We can see that 5 + 25 is equal to 30, therefore this structure is statically determinate. That is important if we talk about the 19th century, since actually at that time, we preferred statically determinate structures because they were more or less the only ones which we could calculate in a satisfying manner. For us, of course, that is interesting but at that time, that was even more important. So, here is the practical realization of our Gerber truss. We can see that we can obtain a structure which is similar to the one which was initially planned, structure which will be more efficient. Here, that is called the Gerber truss, because it has been invented by a German engineer of that name. Here, we can clearly recognize on this bridge over the Main in Germany, cantilever elements, here in the upper part of the cantilever, and in this lower part as well, we have a slightly lenticular beam with compression here. And if we only have loads on the cantilever, we are going to have internal forces which will have this shape. Of course, there will not only be the case of the loads on the cantilever, so we will not always have tension over the whole length but in this configuration, if I had only placed loads on the central part, we would obtain internal forces like this. Here, again, we can recognize the presence of hinges and when we have a hinge, you can notice that the cable and the arch cross each other. That is necessary if there is a hinge because the hinge is the only point through which the internal forces can pass. We have already seen this when we have looked at this kind of construction elements in the first course: "The Art of Structures I". We first want to look at the case... That is something similar to what we have looked at on the small drawing, but now you can have a bigger view to have a better understanding of what happens and maybe also to gain approach tools which will enable us to more easily understand what happens in this type of structure and in similar structures thereafter. Here, I have 4 simple beams. And if I have a simple beam, I know that each time, for the indicated loads, I have an arch-cable. I first draw all the arches. I have 4 arches which follow each other and I obviously have a cable underneath. The internal forces which act in the chords depend on the rise, the depth between the arch and the cable. Here we can see that this rise is constant over all the spans. We can see that this structure has been calculated by means of the applet i-Cremona. So we can see, we have calculated the whole structure in one go, we have introduced 1, 2, 3, 4, 5 forces for each beam and we have introduced the effect of the intermediate supports, each time by means of a load. The result is indeed this shape where we always have the arch above and the cable below, with some variations, the largest internal forces each time acting in the middle of the arch. Underneath, we have a different configuration where we have here a beam with a cantilever, a beam with a double cantilever, a cantilever on the left, a cantilever on the right, and here, we cannot see the following part, but at least, a beam with a cantilever on the left. Between these cantilevers, we have each time placed a truss which is hung up. We have each time a hinge when we pass from a yellow truss to a blue truss, or from a blue truss to a yellow truss. Let's see how to represent this using the applet. Here, each time we have a hinge, - there is a hinge here, another here - we made sure that the arch and the cable cross. The arch and the cable also cross on some other points but they at least cross each time there is a hinge. We have been able to solve this case using the applet in a very similar way, you can simply notice that now, the support reactions which we had to insert are variable for us to respect this condition. We have again a series of arches. We have here a cable, or rather a series of cables, they are lined up, we can say only one cable, which represents the part in tension. Like before, we can see that there is a rise, a depth between the arch and the cable. We can see that these rises, f2, f3, f4, are clearly much smaller than f because they are divided into a part where the arch is above the cable and a part where the arch is below the cable. The internal forces are always proportional to this distance and we can see that they are divided: there will be certain internal forces on this support, some other internal forces in the span and so on. The sum of the rises always remains equal: if we consider this distance here, the rises on the left and on the right, plus the rise in the middle, that will always be equal to f. However, the internal forces are significantly reduced. So, that is advantageous to have a construction with Gerber beams. Here, I show again the crossing points of the arch and the cable. The arch and the cable cross on each hinge. FInally, here I have the results of an electronic calculation. We could have calculated it by hand, but here that is an electronic calculation of these two configurations. Here, we have a simple truss, or a series of simple trusses and below, we have a serie of Gerber trusses. Of course, there are many Gerber configurations, they are the ones we have seen before, with each time a cantilever here on the right, here on the left and on the right and so forth. And here, what is drawn is the deformations, the shape which the structure takes under the effect of these loads which are indicated here in red. This deformation is very strongly exaggerated. That is the advantage of these calculation programs, we can show these deformations in a very exaggerated way, they are much smaller in reality, but what is interesting is to see that the vertical deflection here is quite larger than that of the Gerber truss. So we also really have an advantage. The Gerber truss has smaller vertical deflections. So, the solution with trusses with cantilever not only reduces the internal forces in most of the bars, but also the deformations, the vertical deflections of the structure. Here, we have the example of the Forth bridge, which you have seen on the first slide of this lecture. What is interesting is this illustration which had been made by the designers of the load-bearing system. We can clearly notice that here, there is some tension in their arms. Thanks to the human being, we have small deviations of the internal forces which are quite nice. Obviously, there is compression in these sticks as well as in the seat of the chair under. We will obviously have tension here until these elements having a significant weight, which are used as anchors to avoid that the arm here can lift up. We are obviously going to have, somewhere through the spinal column, a compression which is going to go down and which is going to end up being transmitted to the chair legs, and then, in the intermediate part with this person which is suspended, we will have a hangers system. And then here, really, if we had to look at it in detail, we have a small beam, a small arch-cable for it to work. If we have a look on the top, that is exactly like this that this structure has been designed with two large cantilevers. They were built in a symmetrical way to facilitate their setting up, and the upper part is in tension here too. We will have compression here, vertically in these piers and the intermediate part is a system which works a bit like I have drawn it before for the intermediate person, with compression on the top and tension on the bottom. Here, we can see an elevation of this structure, I want to tell you how this structure was built. I am going to take off the central part here. How did we make this? Well, we had built this part with tension here on the top. I am not going to draw it over the whole length of the bridge. Here, we had some tension, we had some compression on the bottom and some compression here, as well as here, and since this bridge crosses a large river or an inlet, we have taken the liberty of coming with the last structural element, which has this shape, approximately like this, which we have been able to make float on a barge, and then we have simply rised this element up thanks to cables up to its final level, which is a big simplification because this element here is not small, it is about fifty meters long, we have been able to build it on the embankment in very good conditions, while the work conditions in the other parts of the structures were slightly tougher. What is also interesting is to look at the internal forces which we have in the diagonals. If we look at here, these diagonals here, which are inclined in this direction, are in compression. That is the same here. And if we look at the structure, here is a real photo of the structure, we can see that these diagonals here, which have this inclination, are indeed composed of large tubes, such as these ones... We can see that there is at once a lot of material and a large stiffness to manage to have this, while the diagonals in the other direction are in tension, here, and we can see that these diagonals are indeed constituted of very light and very transparent trusses. Last comment: you can see that this bridge is used to carry a railway track approximately sixty meters above the ground and we can see this railway track here. That gives us a very good idea about the scale of the structure. Here, we have a train which passes on this bridge. You can see the hugeness of the structure just to make a train pass. That is a very large structure, a very beautiful record of this time of the large truss structures. In this lecture about the Gerber trusses, we have seen the advantages of a layout with Gerber joints, hinges, we can see that this has positive consequences on the internal forces in most of the elements, as well as on the deformations of the entire structure. I have also shown you that it can be quite interesting for certain construction methods enabling to assemble certain elements in a separated way, which are added to the structure afterwards.
