Shear & Bending Moment Diagrams: 7 Shear And Bending Moment Q&A

So looking at some questions that have appeared on the ARE before, it's basically word questions that have appeared, and not so much so the math. So looking at two loadings, A and B, A is two simply supported beams, B is a single continuous beam. Now if these have the same span, so whatever this is, L, L, this is L, and L, but it's one beam. The other one is two beams. And if this is W, W, W, W, so for the same load, for the same span, which one of the two, a simply supported beam or a continuous beam, would have the greatest maximum shear, A or B, simply supported or continuous? So let's just do a quick tributary analysis and see what... Remember, please, that the maximum shear is equal to the reaction. So the question is basically which one has the greatest reaction. So this guy is going to carry that much. The other half goes to the right. Same here. Now for this one, we need to recognize that the guy in the middle is doing double duty. So essentially, this reaction is the greatest compared to the two end ones. And these are all equal. And condition B has a larger reaction. Therefore, it must be that it has the greatest shear at the middle support in loading B. Now, which one has the greatest maximum positive moment, simply supported or continuous? Here, I strongly encourage you to think of deflection whenever asked about moment. So if I look at question two, which one has the greatest maximum positive moment? My answer is the one that deflects more. So let's take a look at deflection instead of bending moment. So this one is going to do this. This one is going to do the same. This one is not going to do that. It's continuous, so it's going to do that. So, in fact, this one deflected a lot more than this one. Therefore, it must have had a greater maximum positive moment to have caused a greater deflection. So it looks like A. Then it says, question 3, which one has the greatest maximum negative moment? I should recognize that this one is all positive, compression on the top, tension on the bottom. This one has a negative moment over the support, a positive moment between the supports. So which one will have the greatest negative moment? It will be B. Excellent. Just for your information, I'd like to draw the shear and moment diagrams very quickly. This one is going to look like this, twice. So there's the shear diagram, twice, for simply supported. This reaction is smaller than this reaction. And this one we saw with continuous beams, it does something like that. And for the bending moment diagrams, the maximum is here, and it does that, and this one does that. And this one, there's a zero shear here in all of these cases. So this one, let's see how to do this. This one has a negative moment. And I'm trying to be accurate here. I didn't make it accurate. Let me erase and explain what I wanted to do. See this maximum, oops, see this maximum positive moment? Well, this one doesn't rise as much because the negative moment takes away from the positive moment. And quite frankly, that is equal to that. But now part of it is negative. So let me erase a little bit. So this one has a negative moment that took away from the positive. This one had a greater maximum positive moment than this one. This one is a smaller value. So it caused more deflection. Very good. Let's try another question also from the ARE. Given these four loading conditions, ABCD, which one has the greatest maximum shear, ABC or D? Let's please recall, there are no numbers here, so let's please recall just the concepts. The maximum shear is equal to the reaction. The maximum shear occurs at the support. Very good. So that, let me just throw down some numbers so that you can make sense of it. If this were 12 kips, and this were 12 kips, and this uniform load were 12 kips, and this one were 12 kips, then how much is the reaction in A? 6 and 6. How much is the reaction in C? 6 and 6. And in B, you know what? The 12 kip is going to two locations, 6 and 6. Yes, there's a moment and there's all this complication and a horizontal and all that stuff, but the 12 kip is going to two points. And the same with D. It's 6 and 6. So, all are equal in shear. And all are equal to 12 kips divided by 2. The maximum shear for cases A, B, C, D is 6 kips. Excellent. Now, which one has the greatest maximum positive moment, A, B, C, or D, if they have the same load and the same span? Which one has the greatest maximum positive moment? Very good. Let's please remember that whenever you're asked about positive moment, I need you to think of deflection. So let me pull out my brown pen, and let me say this one deflects so much. And this one is not going to deflect an equal amount. Instead, it's going to do that. Very good. Who deflects more, A or C? 12 kips in the middle or 12 kips uniform? 12 kip uniform would have done that. 12 kip concentrated is going to cause more deflection. So, so far, it looks like A has the greatest deflection, must be the greatest moment. I didn't do B, because B, if I compare it to A, here's A, well, B is going to have a fixed end, and so its deflection is going to be less than A. Very good. So the greatest deflection actually is due to a concentrated load. And if the supports in condition A allow rotation, then the deflection is greater than if they restrain rotation. So let me say the following. Deflection for A is greater than the deflection for C. The two simply support it. And then the two end restrained are going to deflect a lot less. but B is going to deflect more than A. Sorry, more than D. Therefore, I can go back and say the moment of A, positive moment, is greater than the positive moment of C, which is greater than the positive moment of B, which is greater than the positive moment of D. Does that make sense? Whenever asked about moment, please think deflection. We understand deflection very well. Moment, we're a little bit touchy on that, so let's just think deflection. Very good. So they all have the same shear, and then here's the order of positive bending moment. And the final question is, which one has the greatest maximum negative moment? We should right away eliminate A and C, because their negative moment does not exist. It's all positive, compression on the top, tension on the bottom. So the question is, which one has more negative moment? 12 kips in the middle, both ends restrained. Or 12 kips uniform, both ends restrained. I'm sure that the moment of B is going to be greater. Sorry, the negative moment of B is going to be greater than the negative moment of D. Very good. So just for your information, this one again is going to do a shear diagram that looks like that. Versus this one is going to do that. And this one is going to do very much the same thing as the one to its left, because we're plotting forces, that's all. So this one is going to do the same thing. So the shear diagrams are going to look the same. Sorry, I assume that's symmetrical. And the moment diagram here is going to look something like that. And for this one, it's going to look something similar, but you know what? I will... let me figure out how to explain this. If that's the positive maximum moment due to 12 kips in the middle, both ends are free to rotate, then this one does exactly the same magnitude, but it doesn't go as high, because part of it is negative due to this one. There is a moment reaction, and there's another moment reaction. So what I'm trying to say is this amount of moment is the same as this one, only some of it is negative, and a little bit less is positive, so this value is less than this one. Okay, and this one does a moment diagram that looks like that. And this one doesn't do that. It does something very similar. It just sinks down a little bit. Let me draw this as solid. And part of it is negative because of that and that. I hope this helps. Here's a totally stupid exercise that will help us understand the relationships between the shear and bending moment diagrams. There are absolutely no numbers. It's just a review. So, for the shear diagram, reactions take me up. Concentrated loads cause sudden drops in shear. Uniform loads cause a gradual sloping drop in shear. I have two overhangs. Therefore, the moment diagram in this area should be negative. In this area should be negative. Between the supports should be positive. Okay, so let's draw the shear diagram first. I start going from left to right. I have a uniform load, and these little arrows say go down in a slope. So we're going to go down. We don't have numbers. I don't have exact magnitude, so we go down a certain amount. Then the reaction takes me up. How much up? I have no idea. I have no numbers. How about that much? Very good. Then there is a uniform load, and the uniform load is the same uniform load as over here. It's the same over here. So this line should be parallel to the other green line. Then we hit a concentrated load, and it takes us down. How much down? I don't have numbers. Then there's no load, and then there's another concentrated load, and it takes me down a certain amount. And then there is a reaction that takes me up. And I need to go above the line because there is still one more concentrated load. And I know that I have to end up here. So I'm going to continue across. And then finally, the last concentrated load will take me down to zero. So that's the shear diagram. Let's draw the moment diagram. And let's first of all acknowledge points of zero shear. This is one. This is two. This is 3. With the shear changing signs from positive to negative or negative to positive. So these are points of maximum or minimum bending moment. There is a negative area here. There is a positive area here. Negative, negative, positive. We have to start here. We have to end here. In this zone, the uniform load is going to cause a curve. In this zone, it's going to cause a straight line. And in the next zone, it's going to cause a straight line. And in the final zone, it's also going to cause a straight line. Please remember, this is a stupid problem. You'll never have it on the ARE. It's just a recap of what we've learned so far. So curve here. And line, and line, and line. Now let me look at the areas. in this area, the shear diagram is down below. So, it's a negative area. So, the shear diagram is going to drop down. It's going to drop down further, then it's going to rise and end up here. Sorry, the moment diagram. I'm not sure what I said. So, the moment diagram will be dropping. Okay. So, let's go down a certain amount because I have a negative area in the shear diagram. I uniform load in the load diagram which means it's a curve. Next thing we're going to do, we're going to go up and it looks like a pretty large area so maybe up to here. So we're going to go up in a curve up to here. We've reached point of zero shear therefore this is the high moment and it's positive. This was an M max but it was negative. Very good. Then I have a negative area in shear and a box or a constant in shear, which means it's a slope in the moment diagram. How much? I don't have numbers again. And then there's another negative area and another straight line. But we have to go below because what is coming next is positive and we have to end up here. So I cannot go down that much and then up. That doesn't make sense. So let me go dip below. And then finally, there's a positive area, and it brings me back to zero. So this is another maximum moment, and it's negative. There is a point of inflection right here, and there is another point of inflection right here. For the overhang, for each overhang, there is a point of inflection. And the bending moment, sure enough, is negative for the first bay, for the first overhang, and it's also negative on the second overhang, and it's positive between the supports. And the maximum shear, I'm sorry I didn't mention, is here. I don't know if it's this one. I need numbers to figure it out. But these are the high points in the shear diagram. Very good.