Shear & Bending Moment Diagrams: 3 Shear And Bending Cantilevers

Okay, so let's take a look at the next five cases, which are cantilever cases. And before doing that, let's just recap what we did with the simply supported beams. I'm interested in one fact, and that fact is that all the moment diagrams for a simply supported beam are positive, and that the maximum moment occurred where shear crossed zero line going from positive to negative or negative to positive. And I've highlighted in the shear diagram how much that maximum moment is equal to. It's equal to the sum of areas up to zero shear, either from the left or from the right. And then the dashed line is a deflected shape. Now, before starting the cantilever cases, I would like to say that this one will be negative. This one will be negative. All these moment diagrams need to be negative because they're a cantilever. And the cantilever does that. The top goes into tension. Bottom goes into compression, no matter how you load it. So, I expect the moment diagrams all to be negative. Very good. So, as we did before, I have kept the load at 12 kips, and I have kept the span at 24 feet, just for comparison with the simply supported, if I need that. Excellent. So let's do the shear diagrams first, and to do that, I need to calculate the reactions. The sum of vertical forces equals zero. There's not two supports to split the load. It's all going on the left. The horizontal reaction is zero because the loads are vertical, so I'm not even gonna Draw it from now on it's zero. There is a moment reaction. I'm not interested in it yet. It's purple It goes with the moment diagram, so for the shear diagram. I really don't care how much that is Excellent, so for the next problem. I still have 12 kips It's not in the tip of the over of the cantilever D-versal is 12 kips. Here I have 6 plus 6, the vertical reaction is 12. Here I mirrored it, I don't care, this reaction is 12 kips. And for a uniform load, total 12 kips, the reaction is 12 kips. So let's draw a shear diagram, just like we did earlier with the simply supported beams. The shear diagram is a plot of the sum of forces, nothing to do with moments. Those show up in the moment diagram. So let's plot forces for the shear diagram here. Starting from the very left, I go up 12 kips, plus 12, and then nothing happens, absolutely nothing. So there's no loads, there's nothing. So I continue at 12 kips, and then suddenly I lose 12 kips and this area is positive. Very good. So next diagram, I go up 12 kips and then I travel over, there's no new loads and then I lose 12 kips. So this was the reaction here, it went up 12 kips and then the load brought it down 12 kips and that's it, there's no loading here till the end of the cantilever. Very good. Looking at the next one, I go up 12 kips. Oops, color. I go up 12 kips. And then there's no new loads until I hit this one. And it drops me down. I'm at plus 12. It drops me down 6, so I'm at plus 6 now. And then nothing happens. And then I lose another 6 kips. Excellent. Back to 0. just like we did with simply supported beams. They start at 0, they end at 0. For the next one, I'm going left to right. The first thing I encounter is a downward 12, minus 12 kips. And then nothing happens. Oops. And then the reaction takes me back to 0. I go up 12 kips. And this area is negative for what it's worth. Okay. And then for the last one, I go up 12 kips. And then there's a uniform load, and its rate is 0.5 kip per foot. So every linear foot, I'm going to lose 0.5 kips. So 12, 11.5, 11, 10.5, 10, etc. I'm losing 0.5 kip per foot. 24 feet later, I will lose 24 times 0.5, or I will lose 12 kips. I'm starting with 12 kips red, I'm losing 12 kips green, I end up at 0. So this one is doing that, and ends up at 0. Excellent. So I need to compare the shear diagram for a cantilever versus the shear diagram for a simple span. With the simple span, you have positive and negative values of shear. For a cantilever, you have only positive or negative, not both. In both the simple span and the cantilever, the maximum shear is at the support. In both the simple span and the cantilever, the maximum shear is equal to the reaction. In the simple span, it was 6 kips as long as it was symmetrical. In the cantilever, it's 12 kips. That's the total load. It's all going to the left or to the right to one support. Very good. So that's the comparison between these two. Let's do the moment diagram now. And to do the moment diagram, I'm going to need this moment reaction. Let's please remember that a fixed end support is equal to 1, 2, 3 reactions. The horizontal is 0 for all these problems because I'm giving you a vertical load. We've calculated the vertical reaction. Now we have to buckle down and calculate this purple moment reaction. So I'm looking at this beam and I'm seeing 12 kips, and the 12 kip wants to do clockwise of 12 times 24, because it's force times distance, 12 kips times 24 is equal to 288 kip foot. So I need to be 288 kip foot of counter-moment, otherwise this beam rotates and that angle changes from 90 degrees, which is not what a fixed end does. It remains 90 degrees. Excellent. So I have a moment reaction of 288 kip foot. For the next problem, I see 12 kips at 8 feet. 12 kips at 8 feet from the support, and so that's 96 kip foot of clockwise. So I need to be 96 kip foot of counterclockwise. For the next one, I see two loads of 6 kips. One is at 24 feet, and the other is making clockwise also at 6 times 12. So this is 72, and this is 144. Add them up, that's 216. So we're being attacked by 216 kip foot of clockwise. Well, I need to be 216 kip foot of counterclockwise. Very good. Next one. The attack is counterclockwise about the pivot, or the support, and it's equal to 12 kips times 24 feet, which is equal to 288 kip foot. And the attack is counterclockwise. This one must be clockwise, 288 kip foot. Okay. For the last one, the resultant is 12 kips. It's located in the middle of the 24, which means it's at 12 feet. And so, the moment created here is 12 kips at 12 feet, or 144 kip foot. So, this one has to be 144 kip foot of counter moment. Very good. So, let's plot the moment diagrams. And please keep in mind that this clockwise is positive, and counterclockwise is negative. That's the convention. So now we're going to plot any concentrated moments. That's what this is, a concentrated moment. It's in purple. It goes on the moment diagram. It has nothing to do with the shear diagram. So we're going to plot concentrated moments plus areas in the shear diagram. Earlier, we did not have any of that business. That's why the moment diagram started at zero. Now it's going to start at zero, but it's going to have a sudden burst clockwise or counterclockwise. It will go up or it will go down based on the moment reaction. Very good. So I'm seeing counterclockwise here. Counterclockwise is negative. This is saying go down 288. Minus 288 kip foot. That's the sudden burst of moment, the moment reaction. The one next to it does minus 96. The one next to it does minus 216. Minus being counterclockwise. Kip foot, not kips. This one is on the right-hand side, so it's going to do clockwise. And it's going to be on this side, and clockwise is positive, and it's worth 288 kip foot. The last one, condition 11, has counterclockwise, which is negative. And it's 144. So if this is 288, this is 144. And it's going down because it's counterclockwise. Very good. So this is minus 144. So let me start adding the areas in the shear diagram. So I'm starting with minus 288 kip foot. And then I'm going to add the area in the shear diagram. How much is that area? I don't know. It's 12 kips tall, and it's 24 feet. So it's 12 kips. Let me erase this thing. It's 12 kips times 24 feet, or plus, because it's above zero, plus 288 kip foot. Very good. I'm reassured right now that the area in the shear diagram, plus 288, plus the moment reaction of minus 288 is zero. Well, that's reassuring. Okay. When you have a constant, a flat in the shear diagram, when you have a constant in the shear diagram, your moment diagram is sloping. If the area is positive, it slopes upward. If the area is negative, it slopes downward. Very good. Let me keep these clean. So now I have a positive, meaning I'm going to rise. I'm at minus 288. I'm going to add plus 288. I'm going to end up at zero. When the shear diagram is constant, the moment diagram is sloping. That's the moment diagram for this one. Looking at the next one, I'm at minus 96 kip foot because the moment reaction was counterclockwise. And then I have a rectangle. in the shear diagram means the moment diagram is going to do that. And how much is that area? I need to make sure it's equal to plus 96 to cancel the minus 96. Well, this is 12 kips tall and 8 feet wide. 12 kips times 8 feet equals plus 96 kip foot. Very good. That cancels the minus 96 and I end up at 0. There's no new area. Moment diagram remains flat. Looking at the next one, I have two areas this time. Very good. Let me erase a little bit. And let me calculate these areas. The first one is 12 kips tall by 12 feet wide. 12 kips times 12 feet equal plus 144 kip times foot. The next area is 6 kips tall, and it's 12 feet long, and it's plus 72 kip foot. So what happens here? If you have a constant in the shear diagram, your moment diagram is sloping. First area is 144, so it slopes steeply, and then 72. So how much is the value of moment at this location? It's minus 216 plus 144, which is minus 72 kip foot, and then I add plus 72 kip foot, the remaining area, and that takes me to zero. Great. Next one. I have minus 12 kips at the tip. That does nothing in the moment diagram. But as you start adding area in the shear diagram, adding negative area, then you're going to go down. And how much are you going down? The height of this rectangle is 12 kips negative, and its base is 24 feet. and that's minus 288 kip foot. So, I'm going down. The shear diagram is constant and negative. Therefore, the moment diagram is slope and going downward, it's going like that. Down to minus 288, and then the reaction is clockwise, the moment reaction, so it takes me up plus 288 back to zero. Looking at the last one, this is confusing me a little bit, I needed it for reactions. Now I'm going to take it out because I have my reactions. I just want to look at the shear diagram, and I know that my moment reaction is counterclockwise. I went down 144 kip foot. Now I'm going to start adding area in the shear diagram. How much is the area? The height is 12 kips. The base is 24 feet. And it's divided by 2 because it's half of a rectangle. That's why it's a triangle, it's half of a rectangle. So divide by 2, so that makes plus 144 kip times foot. And I have a momentary action of minus 144 kip foot. That cancels with the area of shear. Everything looks good. I'm going to end up here. Now, for a uniform load, we saw with simply supported beams, the moment diagram is curved. And I said in that video that you will never see that on a moment diagram. It's always curving downward. So here, I cannot go that way. That is wrong. The curvature has to be downwards. So let me fix this. There's a lot more area here than there is here. So it's going to start steep and then taper off. So, again, what was predicted at the beginning of the video turned out true. The moment diagram for all these shapes is negative. Which one is the ugliest of the ugly? The concentrated load at the tip of the cantilever. 12 kips at a distance of 24 makes a huge moment compared to 12 kips spread uniformly across the 24 feet. That one did half as much, 288 versus 144. Very good. If you've ever seen a bracket for a shelf, it either looks like this or like this. That's the shape of a cantilever. There's not much activity here at the tip. There's hardly any load. And then it accumulates towards a support. And as I said in the beginning of these lectures, A maximum shear needs square inches of area. Maximum moment needs maximum section modulus, which is the depth. So the cantilever is shaped very much like that. It's going to be deeper at the support. It's going to be very thin at the tip of the cantilever. And it increases gradually, either in a straight line or in a curve based on the amount of moment. Very good. I think that concludes this segment.