[MUSIC] Hi this is Module 22 of Mechanics of Materials part III on Beam Bending. And today's learning outcome is to continue that problem, the inelastic beam bending problem for unsymmetrical beams. And so part b is to determine the maximum moment the beam can support just before it reaches the fully plastic condition. So just before it becomes a perfectly plastic hinge. And so, as a reminder or a recall, we looked at beams that are symmetrical but we're looking at beams that are symmetrical about the y-axis so if I put a mirror here on the y-axis, it's the same on either side. But it's unsymmetrical about the x-axis, and we said for a fully elastic condition we can find the neutral axis, we know how to do that, in fact we did that last time. When we go partially plastic, the neutral axis starts to migrate up, because the extreme fiber starts to go plastic and so neutral axis goes up until the entire region becomes plastic and at that point, the area above the neutral axis and below the neutral axis are equal. And the force are equal, that means that the force are equal because I have the stress times the area and compression and then in tension that force has to be equal and so that means therefore that we have to have equal areas. Okay so, again I mentioned before that for the fully plastic or the inelastic region that the tension and compression can be different, stress strain diagrams. But these differences can be reasonably neglected for most real problems. And so here is our situation, for the fully plastic, again, the area above the neutral axis is equal to the area below, and so where would the neutral axis occur? And it should occur right here at the junction between the phalange and the web, and so here is our stress with compression on the top and tension on the bottom fully plastic. And so I can find the force at the top which is the stress times that area which is 150 times 50. And I can find the stress at the bottom which is again 250 times 150 times 50. And you'll notice as I mentioned before that these forces are equal. And now what I'd like you to do is find the fully plastic moment. We want to determine that moment, maximum moment that beam can support just before it reaches a fully plastic condition. So do that on your own, come on back, and when you do that you should start off. Let's start with the top here, I've got the moment is the force and I call counter-clockwise Positive. And so I've got the force which is 1,875,000 Newtons times its moment R which is just going to be one-half of 50. And then as you continue on, you've got the bottom portion times its moment arm, which is 150 divided by 2. Solving that gives you the maximum moment for the fully plastic condition. And that takes care of the problem. So, we'll come on back and continue on next module. [SOUND]