Shear & Bending Moment Diagrams: 4 Shear And Bending Overhangs

So the final section in this chapter, the final chapter in this section rather on shear moment diagrams is about overhangs, which is cases 12 through 15. We've looked at simply supported cases, the first five, and we recognize that the bending moment diagram for any loading on a simple span is going to be all positive. And likewise, for cantilevers, regardless of what your load is, your moment diagram will be negative. And looking at overhangs, I'm not interested in the first two. It looks like the overhang is not loaded, so they're like simply supported. Instead, let me zoom in to condition 14. Let me calculate the reactions. I have kept the loads at 12 kips for problem 14. and I overhung 2 feet in addition to the 24 feet between the supports. So the reactions are 12 divided by 2, 6 kips, because it is symmetrical. So let's do a shear diagram very quickly. Concentrated loads take me down, so I go down minus 4 kips. There's no change. and then I hit a reaction and the reaction takes me up 6 kips. From minus 4 plus 6, I'm at plus 2 kips. And then there is no change in loading. It remains constant. And then I hit a minus 4 that takes me down to minus 2 kips. No change. Oops, that looks funny. No change. And then I hit a reaction of plus 6. It takes me from minus 2 to plus 4. So I'm at plus 4 now. And then, no change. And then finally, I go down 4 kips. So let's draw the moment diagram. and let's please recognize that there's going to be a maximum at this location, this location, and also this location. Negative to positive, passing through zero. Positive to negative, passing through zero. And then finally, negative to positive. So these are high points or low points, I don't know. So let me calculate the areas. The first area looks like it's, let me erase these, they're in my way now. So the first area looks like it's minus 4 kips times 2 feet. That's minus 8 kip foot. The next area has a height of 2 kips, a base of 12 feet. That's plus 24 kip foot. And this is minus 24 kip foot. And the last area is a height of 4 and a base of 2 feet, which is plus 8 kip foot. The plus 8 cancels the minus 8. The plus 24 cancels the minus 24. looks like everything is good. So let's start plotting the moment diagram and let's remember that when you have a constant or between concentrated loads your shear diagram is flat, it's like boxes, your moment diagram is sloping. If the area and the shear diagram is positive you slope upward. If the area in the shear diagram is negative you slope downward. So this first area I I encounter is negative, so I'm going to go down, down to minus 8. Minus 8 kip foot. And then there's a plus 24, and there's a point of zero shear, which means this is the high point. So minus 8 plus 24 is plus 16, so it looks like we're going up to plus 16. also in a straight line because the shear diagram is flat. The loading is symmetrical, must be the moment diagram is also symmetrical. So I'm going down and then I'm coming back to zero. And this is minus eight kip foot. And let's recap what happened. Wherever you cross zero, there's a high point, or in this case, a low point. I pass zero again, there's a high moment. and crossing zero one more time, there's a low moment. And very important to recognize that in an overhang, the overhang itself is going to have a negative moment. Between the supports, you're going to have a positive moment and then on the other overhang, you're going to have a negative moment. Wherever the moment diagram crosses zero and you're going from negative to positive is something called a point of inflection, or a point of contraflexure. And at this location, at POI, the moment is zero, at the point of inflection. So in an overhang, you have negative and positive areas. In cantilevers, the moment diagram is all negative. In simple spans, the moment diagram is all positive. Very good. So let's look at this last condition, and let's learn from the previous problem that this will be negative. This will be negative moment diagram. This will be positive. The maximum shear is at the support. The maximum bending moment is between the supports, wherever shear crosses zero. Excellent. So let's find the reactions. I kept the 0.5 kip per foot constant. I added 2 feet on the overhang and kept the 24 feet between the supports the same as previously. So now my total load is 14 kips, and my reactions are going to be 7 and 7. 7 kips, or half of 14 on each side, because it's symmetrical. So let's plot a shear diagram, recognizing that for a uniform load, we're going to slope downward. What is the value of shear over there? It's 2 feet. at a rate of 0.5 kip per foot, so that's minus 1 kip. And then I hit a 7 kip reaction, it takes me up to 6 kips. Here we are, at plus 6 kips. And what happens in the next 12 feet, I expect to be at 0 here, but in the next 12 feet, from here to here, I'm going to lose 12 feet at a rate of 0.5 kip per foot, I'm going to lose 6 kips. So I am at plus 6. I'm going to lose 6 due to the uniform load at a rate of 0.5 kip per foot for the next 12 feet. So my shear diagram is going to do that. It's going to end up at 0. Very good. Loading is symmetrical. You can mirror your shear diagram or else you continue dropping another 6 kips. And then you rise 7 kips. You're back at 1 kip. And then you lose over the next two feet, you lose 2 times 0.5, back to zero. So this is minus 6 kips here. So the maximum shear is either minus 6 kips, plus 6, minus 1, plus 1. It's 6 kips. Vmax equals 6 kips. And again, it's at the support. This area, the first triangular area, I'd like to draw the moment diagram. And before drawing the moment diagram, I will bring to your attention these three points. Because on the overhang, it's going to be negative. And between the support, it's going to be positive. And the shape of the curvature, let me just draw it first, has to be curved. And because it's a uniform load, and the curvature is always downward. That's what the moment diagram is going to look like for this one. Let's get the values. So the first area here is a height of 1 kip times a base of 2 feet divided by 2. That's 1 kip times 2 feet divided by 2 because it's a triangle. That's minus 1 kip foot. The next area is positive, and it's equal to a height of 6 kips. The base is 12 feet. 6 times 12 divided by 2 is plus 36 kip times foot. Minus 36 kip times foot and plus 1 kip foot. So it looks like my maximum negative moment is minus 1 kip foot. Add to that 36, you end up at plus 35 kip foot for a maximum positive moment. Then you go down to minus 1 kip foot and then back to 0. So point of inflection is the point where the moment diagram goes from positive to negative or negative to positive, passing through zero. And at this location, the moment is zero. And it's called the point of contraflexure or inflection. Very good. Now this diagram is extremely important for concrete because I need to know where I have a negative moment. Because wherever it is that I have a negative moment, I need to flip my rebar. Let me explain very quickly. I have a deflected shape, exaggerated, looks like that. So there is compression on the top, there is tension on the bottom. There is tension on the top, there is tension on the top. So I need to flip my rebar wherever there is tension, the rebar needs to follow. So wherever there is a negative moment from here to here, and from here to here the moment is negative, the rebar will be on the top. Wherever there is a positive moment from here to here, the rebar will be on the bottom. And at the point of inflection, the rebar is flipped. Very good. What else? So the cantilever has all negative moments. The simple span has all positive moment. The overhang has positive and negative. It's always negative on the overhang. It's always positive. Not always, but mostly between the supports. Because if I have a humongous load here, then the whole moment diagram might end up being negative. I don't know, but that's the exception. Good.