[MUSIC] This is module 16 of mechanics of materials part three. We are moving along in the course, we finished the elastic bending portion of the course, and now we're going to start to look at inelastic bending. And so our learning outcome is to define inelastic bending for symmetric cross sections. And so here is our stress-strain diagram, which goes all the way back to part I of my mechanics material series of courses. And we can idealize this material as Idealized Elastoplastic Material. And so we have a linear region and then we go completely plastic. Now, this is a very good assumption for mild steel. I'll use this for some other materials, but it's particularly good for mild steel as the Normal Stress-Strain Diagram is shown here. And so, the yield stress and the proportional limit are assumed to be the same, so we go directly from the linearly elastic region to perfectly plastic. And so, for Symmetric Beams, we have symmetry about the x axis and about the y axis. And so here will be our fully elastic situations. So we have not yet reached an Inelastic condition, but as we continue to put a moment on the beam, we eventually get to a point where the outer fibers start to yield and get into that plastic range and that's shown here. So I have an elastic portion of the cross section and then I go plastic, and it's symmetric about the neutral axis, equal amounts, or equal distances above and below the neutral axis. And I can go to the point where the beam becomes fully plastic, so the top is fully in compression shown here, and the bottom is fully in attention. Now, I want to go just up to the point of where we go Fully Plastic. If I go Fully Plastic, then I'm not going to have to add any more load, and the beam will become a plastic hinge and fail. So when I say Fully Plastic, I mean just up to the point, just before it becomes a plastic hinge and goes past the point of plasticity. Okay, so for fully symmetric beams, the neutral axis here for all three cases remains at the centroid because of symmetry. And the other thing I want you to note, is that, for most materials in the elastic range, it's reasonable to assume that the tension and compression stress-strain curves are the same. So we're going to assume that Hooke's Law applies the same for both tension and compression above and below the neutral axis. And so the other assumptions that we made, that still hold true, are that plain sections remain plain. We have no twisting, no buckling and we have small deflections. And so we find, regardless of the material, that the strain is proportional to the curvature and varies linearly with the distance y from the neutral axis across the cross section. And we're going to find that that's very useful in solving actual problems as we go along. And so that's where I'm going to leave off this lecture and I'll see you at the next module. [MUSIC]