Hi, this is module 15 of Applications in Engineering Mechanics. Today we're going to continue with what we started last time, and actually sketch a shear force diagram for a multiforce member. So here's the generic beam situation that I've, I, I laid out last time. We looked at the free body diagram of a differential element, and we came up with the following relationships. We found that the negative value of the load at any point equals the, the slope or the rate of the change of the shear diagram. And we found that the change in shear between two points equals the negative the area under the load curve. And so what I'd like to do now is to apply those that theory, if you will, to this beam, as, as loaded, as, as shown. And come up with the shear force diagram. So we know on this, on this beam, all of these applied external forces and moments. The only thing we don't know at this point is the reactions at the roller constraint and the pin constraint. So I'd like you to start off by determining the external reactions at point C here at the roller, and point F at the pin on your own. And by now you ought to be real experts of this, given your background from the first course introduction engineering mechanics, and what we've done so far in this course. So you should be able to knock this out quite, quite handily when you have that done come on back. Okay here is the result you should have come up with and, and the equations of equilibrium are used to get them. I found Fs of Y a 11,000 pounds out here on the right. And Cs of Y is equal to 21,000 and since there is no forces in the x direction, there was no X reaction force at point F. Okay, now knowing those there's, there's my beam with all the loads and, and this moment being applied. We're going to work on the bending moment diagram in the next modules. But for this case now, we'll use these relationships for the shear diagram. And so for the shear diagram, as we come along and we're looking at the material to the right, we have zero up until we get to the end of the beam. So we start off at zero, and at the end of the beam, we see that we have this 10,000 pound shear force. And it's acting counterclockwise on the right side of my material, just like this situation over here. So that's going to be negative. So the shear force immediately drops down, to minus 10,000. And now we don't have any change in the shear between points A and C. There's no other shear forces being applied. There is a moment, but that's not going to affect the shear force. And so, we go out to point C here and the shear force diagram stays at minus 10,000 pounds. At that point, we have our external reaction from the roller that's up 21,000 pounds. And again we're, we're, we keep looking to the right on the material. So that's up, it's going to be a positive shear force, and so we're going to change from minus 10,000 up to A positive 11,000. And then we're going to go from point C now to point D. During that interval, we know that the change in shear is going to be equal to the area under the load curve. And I found the area under the load curve, it's just a triangular area. And the total load under the area under the load curve is going to be 6000 pounds. So, we know that at point D the shear force is going to be 6,000 pounds less than 11,000, or that will be 5,000 pounds. Now the shape of that curve in between those two points, we know at any point along that distance minus the value of the, the load is equal to the chan, or the slope of the shear diagram. And so the slope of the shear diagram that, the, the, the load here is, is zero and it goes up to minus 6,000 so it gets greater and greater and greater in a negative sense. And so here we start off with a slope of zero, and it becomes more and more negative until we get down to 4,000. And So we've integrated a ramp function here. So this becomes a parabola function. I'm going to label that a P for parabola. By the way, down here in going from point A to point C. That was a straight line, so I'm just going to label that with S. Now again between D and E we have no shear forces and so, no shear force changes, and so we're going to go from D to E as a straight line. And then going from E to F, the area under the load curve is 16,000. So that means the change in shear will be from 5,000 down to minus 11,000. We're going to be down here at minus 11,000. And between those two points, the negative of the value of load stays constant at 4,000 so the, the slope of the shear diagram is going to stay constant. And it's going to go down with a straight line here. And, so that's a straight, curve. And then finally we're going to go up at the end 11,000 that's positive, and it closes off at zero. And you should close off at zero on both ends of your shear diagram. The only other thing you might want to do is you might want to find this distance where, the, shear force ended up switching through and was zero. And so let's use these similar triangles to find that distance. We'll use this triangle, and this triangle. And so we've got this side, this length is four. And we're going to compare it to this length here, which we'll call x. So x is to four as this drop was 16,000 actually, we'll just call it 16. We'll keep the thousands off, that'll be easy, just to 5,000. I mean you could do 16 to 16,000 to 5,000, or five to 16, either way is, is fine. If you solve for x, then you find this distance x is 1.25 feet. The total distance is four. So that means this distance is 2.75 feet. So now we know the value of the shear. The internal value of the shear force anywhere along the beam, and we would use that in designing the beam. You can see on your own, where would be the cases where you'd have the maximum shear that you'd have to design for? Take a minute and think about that. Come on back. ' Kay, those, those locations, the, the worst case is here [COUGH] at point C, where I have a shear force of 11,000. Or here at point F, where I have a shear force of minus 11,000. Here's another large shear force. you can see we actually have no shear, as I said, at this point right here. But we've got a very good sense of how this shear varies over the entire length of the beam. And so this is just a clean copy of what I just did so that you can see it. And that's it for shear force diagrams.
