Hello, in this video, we will see how the horizontal loads are carried by frames. Like in the previous video, we will see what is the effect of hinges and what is the shape of the pressure line when we have loads which act horizontally on a frame, and we will thus see how the internal forces distribute inside frames. We will also see a little bit what happens in the case of a multiple frame. We again have the 4 same frames than in the previous video. A frame with three hinges, a frame with two hinges, a frame with one hinge and a frame without any hinges. Here 3 hinges, 2 hinges, 1 hinge and no hinges. We are going to have in all cases, a load which acts horizontally from the left, a load which I am going to call H, because it is horizontal. That is for example the wind. That could also be the effect of an earthquake which mainly translates into horizontal loads. To understand well how the internal forces work, we are going to first look at the case of a frame with three hinges. In this case, we really have a cantilever with an upper support and then a lower support here, if we rotate everything of 90 degrees. It means that in this part here, we are going to have compression. And in this part here, we are going to have tension. That is quite different from what we had previously. Although in the right part, we are going to have compression in the diagonal to make the pressure line go up then we are going to have compression inside the post and in the lower part, in the cantilever. In the upper part, that is going to be exactly as we have seen it in the case of a vertical load. In the left part, we are going to have a cable here. If we want to make a cable go up, we are going to pull on it in a diagonal direction, which is going to make it go back inside the cross-section. We are going to have tension on the intrados and then on the other side, we are logically going to have compression. What happens in the case of a frame with two hinges? We have not any hinges at this level here but remember: I said that it must also pass by the line of action of the force. Actually, the pressure line is identical to what we had before. That means that the result will also be the same. That is the same than the left one so I am not going to draw it but we could determine it if we wanted, exactly in the same way. If we have an arch with one hinge, we are going to have compression in this way but we do not know where it goes. It is going to go onto the external part here and then likewise, tension like this on this side. On the right part, that is similar to what we had already seen for vertical loads: compression in the diagonal, compression which changes side in the post and then which remains under the hinge in the central part. And then for the upper part, for tension, that is similar, simply in a symmetrical way. What about the left part? Here we have a cable which we will need to pull with tension to bring it back inside the cross-section. And then we will have this configuration, this tensile internal force in the diagonal which must be offset by compression in the external and upper part of the cross-beam, in the corner. If we look at the right part, it is again going to be a pressure line which is determined by the height of the force on the left and then which is going to get out of the cross-section at the level of the supports. That will again be exactly the same thing than what we have seen in the left part. We can see quite fundamentally different configurations, although this one reminds us what I have shown you in the previous lecture when we had an eccentric load on the frame with tension in the lower part on the left and compression in the lower part on the right. Let's look again at this curved frame under the effect of a lateral wind this time. We are going to place horizontal wind, constant because the structure is not very high -lowercase w- and we are going to look at the internal forces in this structure. We use this applet. I am not going to encourage you to try to do it because that is quite difficult to obtain this shape, because the applet is pushed here to the limit of its capacity. And then we can see that we have a very large eccentricity in the part which receives the wind, with tension on the intrados, and obviously the corresponding compression on the extrados an then on the other side we also have a very significant eccentricity because this is a straight line so we do not have a curved shape like before, so the effect of this wind load on the left leads to very large compressions on the intrados. I get back to this picture. We can remember what we have just seen previously, that is to say compression here in a diagonal direction and then tension with a parabolic shape like this. What does it mean? It means that in this part of the cross-section, we are going to have tension inside. For this tension to come inside, we will need to have a bit of transversal tension, and then we will obviously have compression on the other side of the cross-section. And then that will be almost symmetrical on the other side, with compression on the intrados, transversal compression and tension like this. We thus can see that in such a frame, the load case with lateral wind is a very important load case because it leads to very large internal forces eccentricities both on the upwind side and on the downwind side. Finally, we want to look at an example of multiple frame. In the present case, that is the Parthenon in Athens. They are elements which are made out of marble, so by stone. That is an element which does not have a very large tensile strength. How does it work? Usually, it works in this way: we have columns with vertical internal forces, we are going to have a rather significant arch effect because these cross-beams have a large depth compared to their span and we are going to have a little bit of tension in the lower part because marble resists a bit in tension anyway. That is the usual case. So that was for vertical loads downwards. If we now have a load case, it is drawn but I am going to draw it bigger, with horizontal wind, what is the effect going to be? Well, the internal forces are going to be inclined in the columns. This inclination enables to support the horizontal component of the wind force. So there is a horizontal component, we can see here that each support reaction is inclined. Concerning the cross-beam, the functioning mode remains identical with an arch and then, a contribution of the tensile strength of the material which enables to carry at the same time the vertical loads. In all likelihood, in the course of History, what happens is the creation here of a crack in one of these stone cross-beams. That is not really possible to imagine that there are not any cracks in these structure - there are some ; if you had visited these structures, you probably observed it yourself. So, what happens in this case? Well, when there is this crack, we still can have an arch effect, compression being able to pass through a crack, but obviously, we cannot have a tensile internal force from the left to the right since tension could not cross the crack. So this arch itself now creates a thrust leftwards and a thrust rightwards, such as any arches, which was usually supported by tension in this case. This time, it cannot be the case, but this thrust can be taken by inclining the support reaction in the columns. And then, since these frames touch each other, making the other columns also have an inclined support reaction, an inclined force, the other cross-beams keep working normally, that is to say that they have a tensile internal force in their lower part, and thus equilibrium is possible. We can see that there are really a lot of columns here. We could even imagine that several, not all, but only several, possibly 2 or 3, of these cross-beams are cracked and that in spite of that equilibrium is still possible . By the way, it is necessary to understand this in this way because this building is located in Greece which is a country with high seismic activity and it stands since more than 2 000 years so that is not one or two cracks that one can see there which condemned it to collapse. In this video, we have looked at the effect of the horizontal loads on frames. Depending on the number of hinges, there are variations of the pressure line and the distributions of internal forces inside the posts and the cross-beams are quite different. Finally, we have looked at the example of the Parthenon: how multiple frames can resist gravity loads, horizontal loads and even cracking of one of the cross-beams without any possibilities to take any tensile internal forces.
