This is Mechanics of Materials Part 4, and again we're moving well right along with the class. We've gone through beam curvature, we looked at singularity functions for doing beam deflections, we did beam deflections in the last prac using superposition techniques, and now we're going to look at statically indeterminate beams. So that's our learning outcomes, solve a statically indeterminate beam structure using superposition techniques. So here's our example, we want to solve for the force and the moment reactions in the beam. In this module, we'll use superposition techniques for finding the deflection. In the next module, we'll redo the problem using singularity functions to solve for the beam deflection. Because sometimes for the superposition techniques, you can't always find common beam tables for a particular loading situation, and you may want to just go with the singularity function techniques which will work for you. So how might we start this problem? What you should say is, well, let's at least start with the static equilibrium equations. So how do we do that? As always, always always always, I can't emphasize enough in all of my classes in engineering mechanics, let's draw that free body diagram. So draw the free body diagram on your own and come on back. So here's the free body diagram. What's next? So what you should say is, let's go ahead and solve for the reactions. Once again I'd like you to solve for the reactions on your own and come on back and see how you did. You should be old hat at this by now after going through my previous courses. So if we sum forces in the y direction, we can solve for an equation which relates Ay and By, and we'll call that equation one. Then if we sum moments about point O and set it equal to zero, we come up with the second equation, and it's in terms of the moment reaction at point A and By. So now I have two equations, that's the good news. The bad news is that we have three unknowns, Ay, By, and MR, the moment reaction at A. So that gives us a sad face. But we're going to need an additional equation, and so how might we go about this? If you recall my earlier classes, what we're going to need is, this is a statically indeterminate structure. So we're going to need an additional equation, we get that as being the deformation equation or what we call the compatibility equation. We did this in my Mechanics and Materials Part 1 course for an axial loading situation, we did it in my Part 2 course for a torsional loading situation, and now we're doing it for a beam bending situation. So here is my beam loading situation. I'm going to go ahead and write a deformation equation using superposition techniques. Based on this loading, I can superimpose several different loads. First of all, this is the loading that would be done by the distributed load of 73 kilonewtons per meter. Then in addition, we have a loading due to the 45 kilonewton point force here at 1.8 meters out, and we have a slope at the end, so this portion of the problem is similar to what we did in the last module with the Falling Water Beam example. Then finally, in addition to those two loads as we go out to the right, we have this load By that's going to have the forces backup for geometric compatibility. So that's going to give us our other equation, and we'll call that y4. So on the right-hand side, we go down because of the distributed load, we go down because of the point load of 45 kilonewtons, but we're going to have to be pushed back up by the By loads so that the total deflection at the right hand side is equal to zero. So yB again is equal to zero, it's equal to this y1, which is down negative, this y2 which is down negative, this y3 which is down negative, and then pushing back up the By is plus y4. If we substitute those values in, we're going to get this result. You can see now that the only unknown that we have is By. We can put in the numbers for everything else. We know L1, we know L2, L3, we're given E and were given I. We also know what theta is at the end, and we're going to assume that this is in the linear elastic range, you should double-check that to make sure that you're still in the linear elastic range looking at the stresses in the beam. If I put those values in, this is the equation I get, and I can solve for By, so that's one of my force reactions and it's equal to 94.2 kilonewtons up. Once I have that, it's quite straightforward to solve for the other two unknowns. I go back to my static equilibrium equations, I substitute back in, I can find the value for Ay, and I can find the value for the moment reaction at the left-hand side. That's the standard technique for solving statically indeterminate beam problems. You may come up with a lot of different loading conditions, different reactions that you want to solve for, but if you use this technique, you'll be able to work your way through and get the solution, and we'll see it next time.