Shear & Bending Moment Diagrams: 2 Bending Moment Simple Spans

Now that we've drawn the shear diagrams for these first five cases that are all simply supported, I would like to draw the bending moment diagram that corresponds to these five cases. I did say at the end of the previous video that the sum of areas in the shear diagram should equal to zero. The positive area should equal to the negative area. Sorry, that was wrong. The sum of positive areas should equal to the sum of negative areas. Very good. So the bending moment diagram is a plot of the sum of moments, which is force times distance along the span of a beam going from left to right. Another alternate definition that comes from calculus is a moment diagram is a plot of the sum of areas in the shear diagram. So it's the integration from calculus. So to draw the moment diagram, all we have to do is sum of the areas in the shear diagram, and then we're ready to plot a moment diagram. So one thing of note, though, is if your shear diagram is doing a constant, your moment diagram will do a positive or a negative straight line. So the moment diagram in this zone, since the shear diagram is constant, will be a sloping straight line. And when the shear diagram slopes for a uniform load, the moment diagram will be a parabola. I will explain all of this as I go, but I just wanted to give you a few hints before we get started. Also, wherever shear crosses zero going from positive to negative, that is where the maximum moment will occur. So, before I even plot the moment diagram, I know that the maximum will be at that location. For the next problem, there is zero shear areas go from positive to negative, well, that will be the high point of the moment diagram. There will be a high point in case 3 at that location. In case 4, that's zero shear and areas change from positive to negative, there will be a maximum moment. In case 5, that is a point of zero shear and there will be a maximum moment at that location. Please do not confuse the point of zero shear with the point of inflection. The point of inflection is on the moment diagram, where the moment is zero. That's a totally different concept. Okay, so let's get started. To plot the moment diagram, I'm very simply going to calculate the areas in the shear diagram, and then I'm going to sum them and draw a moment diagram. So this first area has a height. It's a rectangle. It has a height of 6 skips, and its length is 12 feet. So 6 kips times 12 feet is 72 plus 72 kip foot. Notice the units, the area in the shear diagram, the height of the rectangle is kips, the base is feet, the area is kip foot, that is a moment. So the second area is minus 6 kips times 12 feet or minus 72 kip foot. Looks like the work is correct because the sum of areas is 0. For problem number 2, the height is 8 kips of that first rectangle, 8 kips, height, and the base is 8 feet, so plus 64 kip foot. And the second area is minus 4 kips times 16 feet, equal minus 64 kip foot. That checks plus 64 minus 64. For problem number 3, I have an area here of plus 6 kips times 8 feet, or plus 48 kip foot. I have no area here. And then I have an area of minus 6 times 8 feet, or minus 48 kip foot. Checks, plus 48 minus 48. And for the case 4, let me erase a little bit of their stuff in my way. Okay, so this first area is 6 kips. How much is the base? 6 feet. So this is plus 36 kip foot. And this area here, this small area, is 2 kips. Times 6 feet is plus 12 kip foot. And then I have minus 12 kip foot. And I have minus 36 kip foot. Checks. Okay. The next one is a triangle, the next shear diagram, case number 5, is a triangle whose height is 6 kips and whose base is 12 feet. And that's the point of zero shear, so I'm going to calculate the area of the first triangle and then the second one. Area of a triangle is base times height over 2, so it's 6 kips times 12 feet divided by 2. So this is plus 36 kip times foot. and minus 36 kip times foot. So let's see what's going on with the moment diagrams. Please remember, I'm going to plot the sum of areas to the left of any location on the shear diagram. So at this location, I have 0 area to the left. By the way, for all these diagrams, we have to start at 0, we have to end at 0, just like we did with the shear diagram. We started at zero, we ended at zero. The same applies to the moment diagram. We have to start at these points and end at zero. Otherwise, something's wrong. So, looking at the first area, whenever you have a constant in your shear diagram, when do we have a constant in the shear diagram? Between concentrated loads. If we have a uniform load, the shear diagram is sloping downward. Right now, we don't have a uniform load. We have two boxes. Constant, then the moment diagram is going to be sloped. If your area is positive, we're going to slope upward. If your area is negative, we're going to slope downward. So, one foot after the support, I will have an area of one foot times a height of six skips. Two foot times six skips. Three foot times six skips. I'm summing areas as I go from left to right. 10 foot, 11 foot, 12 foot. At 12 foot from the left, I will reach the maximum moment. The maximum moment is this entire box to the left of zero shear. So the maximum moment here is 72 kip foot. And frankly, I went, to get up there, I went one foot, six skip foot is the area of that rectangle. 12, 18, 24. I'm adding six skips times one foot for every foot I travel. So I'm going 6, 12, 18, 24, 30, until I reach 72. That's a straight line. Whenever it's constant, your moment diagram will be sloping. If the area on the shear diagram is positive, you're sloping upward. Next, we lose minus 6 skips times 1 foot. We're going to go down to 66. Then 2 foot after mid-span, we're going to lose 2 feet times 6. We're going to lose 12 from 72. We're at 60. We're at 54. We're at 48. and we're dropping down in a straight line back to zero. That's the moment diagram for this loading and the shear diagram. Excellent. Maximum moment, 72 kip foot. Where is the maximum moment? At zero shear. How much is the maximum moment? It's the sum of areas to the left. That's how much it is, or to the right, frankly. It gives you the same answer. Okay. Let's go to the next one. I'm going up to 64 kip foot. If I have a constant in shear, I have a slope in the moment diagram. If the area is positive, I'm going to slope upward. All the way up to that first rectangle. 64 kip foot. Then I'm going to lose 4 kips times 1 foot. 4 kips times 2 foot. 4 kips times 3 foot. I'm accumulating negative area. And after I travel 16 feet, I will lose 64 kip foot. I'm at plus 64 and I lose 64. I'm back to zero. That's the moment diagram for this loading. Excellent. So whenever you have just concentrated loads, your shear diagram is constant, your moment diagram is sloping. There's no curves here. The curve comes with the uniform load. Excellent. Let's do the third one. I'm going to climb up to 48. Let me keep some kind of scale going. So if that's 64, that's 48. Okay. So I'm going to climb up to 48 kip foot. I'm going to climb 6, 12, 18, 24, 30, 36, 42, 48. Very good. That's a constant slope. of 6 kip foot every foot. Excellent. Then there's no new area. We're at zero. There's no rectangle here. So we continue constant. And then we lose 48 kip foot. And we lose it gradually. And we end up down here. That's the moment diagram for this Loti. Looking at the next one, where's zero shear? Here. How much is zero shear? If I don't want to mess with this too much, just add up the areas. It's 36 plus 12. That's how much the maximum shear is. So where's my 48? Here's my 48. The maximum shear looks like it's going to be 36 plus 12. That's 48 kip foot. How are we getting there? At this location, the moment is equal to the sum of areas to the left. So that's 36 kip foot, which is this much. And then we're going to add. There's still a plus. we're still in the positive zone, then we're going to add 12 to 36. So that's the shear, sorry, the moment diagram up to that point. By the way, if the loading is symmetrical, the shear diagram is symmetrical, the moment diagram is symmetrical, I can just mirror this guy and say that's the moment diagram. Very good. Maximum moment at zero shear. Maximum moment is equal to this much area, Or, it's the same, it's equal to this much area, but it's not negative, it's positive. Very good. Looking at the last one, I know that my maximum moment is located here. And I know that my maximum moment is 36 kip foot. It's the sum of areas to the left of zeros here. There's one triangle, it's plus 36 kip foot. If this is 48, then maybe this is 36, a little bit less. 36 kip foot is the point I'm going to rise up to. Excellent. So how do we get up there? The shear diagram is sloping downward. This area is positive. This area is negative. So in the moment diagram, when you have a slope, your moment diagram is always going to be a curve. It will never look like that. It will always curve downward. It will never hold water. Okay, so let's see, is it this curve or is it this curve? I said it'll never hold water, therefore it's not this one. Why is it not this one? Because if I look at the shear diagram, there's a huge amount of area in the beginning, and there's very little area towards the center. So it, let me repeat that very quickly, It rises quickly and then it tapers. So that's the shape of the moment diagram. It is symmetrical. There's very little area here, so I'm not going to do that. That's sudden. No, it's gradual. So it's gradual and then it gets steep. There's a lot of area towards the end. It's gradual and then it gets steep. And that's the moment diagram. Just the maximum moment is 36 kip foot for that last case. And let's summarize what we saw here. The moment diagram for a simply supported beam will always be positive. That's the first conclusion I want to make. Why? Because this beam is going to deflect like this, and it's going to do compression on the top, tension on the bottom. That's called a positive moment. This one is going to deflect like that. This one is going to deflect like that. So is this one. They all have compression on the top, tension on the bottom. Therefore, they are all going to have a positive moment diagram. Let me erase these because I need them for one more thing. Okay. Excellent. So that's one thing. The second thing is wherever you have zero shear, that's where your maximum moment is. Zero shear, maximum moment. Maximum moment, zero shear. Also, wherever you have high shear, you have low moment. Wherever you have high shear, you have low or zero moment. So those two are opposite. High shear, low moment. High moment, low shear. Excellent. Looking at the five cases, and let's remember, please, that we had a 12 kip load on a 24-foot span. It was just configured differently. Well, 12 kips in the middle was the ugliest at 72 kip foot. Sorry, as far as shear, they all had 6 kip reactions, unless it was not symmetrical. But as far as bending moment, case number 1 is 72 kip foot. Case number 5 is half as much. So a concentrated load causes a lot more damage than a uniform load. Also note, please, that for the 12 kip concentrated load in problem 1, nothing happened around the moment diagram. The 12 kip shows up in shear. It's a force. It's not a moment. It's force times distance. The distance to the left and to the right of that load is minuscule. So it's not going to make too much change in the moment diagram. It changes direction. It used to be positive going up. Now it's going down. But the value is right there at 72 kip foot. Excellent. So those are the moment diagrams. And it looks like this one has the second rank of moment at 64 kip foot. It's because it's 12 kips. It's going to make more moment than 6 kips. And 4 kips is going to be less than the 12 kip moment. And a uniform 12 is going to have the least moment. One quick note before I conclude this. There's one more diagram that I'm not going to draw that really tells the whole story. Whenever you're asked about bending moment, I personally prefer that you think of deflection. Bending moment causes deflection. There's one more diagram, which is the deflected shape divided by the stiffness, blah, blah, blah. We're not going to get into that. But if you can determine the deflected shape, then you know everything you need to know about the bending moment. I think that this one is going to deflect like so. Let me pick a color that works here. There are no good colors. Okay, so this one, the deflected shape is like that. If you were to carry a cable and push down on it or a necklace with some pendant, that's the concentrated load and the shape looks like that. If you move the pendant off to the side, that's your deflected shape. If you have two pendants, whatever, I'm getting ridiculous, but that's the deflected shape if it were on a chain or a cable or what have you with two equal concentrated loads, that's what it looks like.