[MUSIC] Welcome to Module 19 of Mechanics and Materials part three. Today's learning outcome is to continue on with that inelastic beam bending problem that we started with last time. So last time we solved Part A. Now we're going to solve Part B. Where we have a string gauge located on the top of the beam that measures the strain in the X direction and it finds it to be 3,250 mu, millimeters per millimeter. And we want to find the applied moment that caused this situation where this strain was measured at that amount. And so, remember that strain is proportional and independent of material to the curvature and varies linearly with a distance y from the neutral axis across the cross section. And so, here is my strain that I measure at the top that is 0.00325, that's given. And I want to find out where I go from plastic to elastic because when it's 0.002, my beam has gone beyond that, it's gone into the plastic region. So I want to find out where this is 0.002. Well we know that the total distance to top from the neutral axis is 150 plus 12.5 or 162.5 millimeters and we want to find this distance and I'll call it X. And so we can use a similar triangles, geometry here, I've got x is to 162.5 as 0.002 is to 0.00325. If I solve for X I find that it's equal to 100 Millimeters. And so the distance up to where my beam starts to go into the plastic region is 100mm. And that occurs symmetrically at the bottom as well, in tension. At the top it's going to be in compression, at the bottom it's going to be in tension of the loading situation. And so, here is my strains, and we know that above the .002 range where we go into the plastic region, is what's shown, and then below that, we're in the elastic region. And so here's my elastic region where my stress stays in the elastic zone, and then my plastic region. And the other thing we need to notice is, in the plastic region, that there is a difference in area above where the flange and the web connect, and below. And so, I've got three different sections here that I'm going to calculate the total force that's acting on the cross section. And so the total force at the top, is going to be the plastic stress times the area. And the area of the flange up here is 200 times 12.5 where there's your total newtons. Then I have a section from where the flange and the web meet down to a point, 100 millimeters above the neutral axis, so that's going to be 410 times the width, which is 12.5, in this case, down 50 because that leaves 100 millimeters up from the neutral axis, and so, there is my value for that force. And then finally I find the value, the force in the elastic region, which is going to be a triangle one half the base which is 410 Newton millimeters times the area of cross section which is 12.5 times 100 or again 256, so that's the stress times the area. One half of that for the triangle and that gives me the force there. So now let's go ahead and find the applied moment that causes this situation. So, the moment is equal to, lets find the moment for the top section here. The force is 1,025,000 Newtons and that's times its moment arm which is now, going up from the neutral axis, 150 millimeters plus one half of 12.5. So that's going to be 150 plus 12.5 over 2. I'd like you to do the same thing now for the other two forces. And if you do that, you're going to get 256,250 times it's moment arm which is, in this case, we said that it goes into the plastic region at a 100 millimeters and so, then we're going to go half way up into the plastic region below the flange so it's going to be 50 divided by 2 or 25. Okay, and then finally we have to find the moment for this elastic portion. So the force is 256,250. And it's times the moment arm for a triangle like this. The moment arm is two thirds of 100. And that takes care of that top portion. But what about the bottom portion? And so what you need to do is we also need to take into effect the bottom portion so, we're going to multiply this whole thing by 2. And I end up with 418,542,000 Newton meters. That's for a partially plastic situation. If we compare that, just to the fully elastic applied moment, we found that to be 379,000 Newton meters. And so, you can see that we've gotten a little bit more, we have a little bit more applied moment for the situation where it goes into the partially plastic region. And so we'll come back next time and solve part C. [SOUND] [MUSIC]