[MUSIC] Hi. Welcome to module 12 of mechanics materials part three. Today, is the last kind of a theory before we start getting into problems next time. Today we're going to determine how to find and use another section property, which we call the section modulus s. And so recall that the maximum stress was given by this formula, where c was the distance farthest from the neutral axis so we could determine where the max stress was going to occur. We're going to define the section modulus s as the symbol s, which is equal to i over c. And so with this definition, we find that sigma max is equal to the moment over the section modulus. And for design now, we want the s that's available for our cross section to be greater than the maximum moment expected over the allowed or actual stress. And that's the maximum bending moment expected again. We often want to build in, we always usually want to build in a factor of safety and we've talked about factor of safety before. We want to factor safety that's greater than one, so that we avoid failure. And so the factor of safety is the failure stress over the actual stress, where sigma's the failure in this case, over sigma actual allowed, and so this is our actual allowed that we put in. And we find that s, the section modulus that we're going to be able to get from a cross section, is actually listed for typical I-beams and other cross sections in tables in the Manual of Steel Construction that's put out by the American Institute of Steel Construction. And most of these resources for finding section moduli for different cross-sections are free to the public, and you can find them online. But that's what we're going to use as we go forward and do actual design problems. And so, here's our elastic flexural formula. We put it in terms of our section modulus. We said that the area moment of inertia I, is equal to the integral over the r squared, or integral over the area of r squared. The A, we found out had a finite for composite shapes and standard shapes the last time. And it was a measure of a cross section resistance to bending about a certain axis. I want to look at one more analogy kind of putting things together for different courses and different concepts. Recall from my 2D and 3D dynamics courses, if you took them, that we defined the mass moment of inertia about an axis as shown here, and the maximal inertia about the z axis through a point p was how much mass was located how far from the axis rotation. And it was a resistance to angular acceleration. And I did an example with a skater, whereas the skater pulled in their arms, the mass moment of inertia became less and so they spin, they get less resistance to angular acceleration, so they spin became, accelerate even more. Same sort of a concept here for a area moment inertia. The more are we have further from the neutral actions provides greater resistance to bending in the case of beam bending. And so we want to get a lot of area far from the neutral axis and that's why we use cross section such as I beam shapes, etc. So maybe hopefully that makes a little bit more sense for you now, so we can say some on materials and get high efficiency for resistance to bending by putting more area further from the neutral axis. And again, the logical type of cross section for that is in I beam and that's why we see it often used in construction. And so, again, that's it for the theory. Next time we'll start up with some real world examples. >> [SOUND]