Hi and welcome back. In today's module, we're going to cover a quick review of bending stresses. The learning outcomes for today's module is to understand how to calculate a bending stress in an object. And if you are unfamiliar with bending stresses and did not cover that in the Mechanics of Materials class. This module is intended to be a quick review so I ask that you go ahead and dive into Dr. Whiteman's Mechanics of Materials Beam Bending modules which are linked here below. With that let's talk about the assumptions we're going to make for bending in a beam. We're going to assume that our beam is isotropic, that it's homogeneous, that it's operating at room temperature and that it conforms to Hooke's law where stress is equal to strain times modules of elasticity. A couple of other things we're going to assume that the beam is in pure bending which just makes life a little bit easier for us. The equation for bending stress is that sigma, bending stress is a normal stress is equal to negative M y over I. And down here, see we have a beam and we can see that we're going to assume that it's fixed along here at the wall and we see we have a load pushing it down. And so obviously that's going to cause bending stresses in the beam. So if we think through what these bending stresses are going to be like, the beam is going to get pushed down and it really wants to deform downward. And what happens is the top here gets pulled into tension and you can see these tensile of stresses running along here and the bottom gets pushed into compression. The bottom fibers on this beam are getting squished into compression so you see the compressional stresses occurring at the bottom of the beam. And then along the middle of the beam we have a neutral axis which is the X axis and a neutral plane. Which is the XZ plane, and you can see that plane here. And along that plane, there's no bending stress. Your bending stress is zero at the neutral axis and the neutral plane. Another thing to keep in mind, so we have M, which is the bending moment at whatever point you're at on the beam. Let's say I want to find the bending stress right here. And what I would need to do is I would have to calculate the moment at that point, using our bending moment diagram. And here a bit of a shortcut since the beam is not constrained anywhere before the force, you can just say the bending moment is the force times the length between the force and the point of interest. So y is going to be the distance between the neutral axis and the point of interest. And here the point of interest is at the surface of the beam. And then I is your mass, I'm sorry, it's your moment of inertia, your mass moment of inertia and here it's going to be with this cross section, it's going to be your moment of inertia there. One of the things over the years I've seen students get confused about is where is the neutral axis. And the neutral axis coincides with the centroidal axis of the cross section which sounds a little bit confusing. One of the ways I have students who are very 3D visual thinkers approach this is to look at where the load is. For example here the bar is getting pushed up, right? It's getting pushed up on the Y axis. And then, intuitively you know that the top of the bar then is going to get squished, right? It's going to be in compression and the bottom of the bar if we pretend this is in 3D. Down here, that's getting pulled into tension. With that, I'm going to let you think through. Where is the neutral axis on this problem? And here are a couple of options. We'll let you guys think or select where you think the neutral axis is and then we'll continue on. Okay, so here, for the top example that we just talked through, the neutral axis is going to happen right along the X plane, or I'm sorry, the X axis in the XZ plane. It's going to look like this, right? It's going to bisect the beam right along that XZ plane and if we look at where the neutral axis is in the lower example where now we're pushing the beam into the negative Z direction,right? The neutral axis we're going to have the front of the beam here is going to get stretched into tension. The filaments on the back of the beam are going to get squished into compression and the neutral axis always occurs between the compression and the tension. The neutral access will be right in between the two and you will be able to see how, again how you got these tensile stresses at the face of the beam closest to you and these compressive stresses further away from you. With that I'm going to leave you with an example problem to attempt on your own and then we'll work through it in the next lecture. And it's the same example problem we worked through for the axial and torsional stress cases. In this case, a load is also going to, one of these loads is also causing a bending stress. Again we're going to assume that rod AB is strong enough and that we don't need to perform any analysis on it. We're trying to find the bending stress. At point O and we have a load F acting in the negative X direction and a load P acting in the negative Y direction and so one of the things you immediately need to think about is what is the bending moment at point O and what force is causing that bending moment at point O. And so we're going to start out with this example with a hint that your bending moment at point O is going to be 250 newton meters. And so I'm going to leave you guys to work through this example. And get as far as you can and then next module we'll go through the entire example and we'll talk about why the bending moment is that value and how to calculate the stresses at point O. So I'll see you next time.