[MUSIC] Hi, and welcome back. In today’s module, we're going to cover Coulomb-Mohr theory. This is going to wrap up Unit 2, which is Static Failure Theory. So, in the next modules, we'll end up getting into Fatigue Failure Theory. So the learning objectives for today's module are to understand when to utilize Coulomb Mohr theory and how to apply this theory to an example problem. In this module, it's critical that you understand principal stresses, which you should have learned in your basic mechanics of materials or deformable bodies class. If you haven't seen principal stresses before, you can go back and review them here in Doctor Whiteman's modules in Mechanics of Materials I. So, principal stresses, that you should have learned of in your either Mechanics of Materials or Def. Bod.s class, is often the first failure theory learned. What you learn is that stress at a point is directional. So, if we have an object in a certain stress state and we look at a specific point, obviously, we're going to be able to easily calculate the stresses in the X and the Y directions, but the maximum stresses might not be acting in the X and Y direction. They might be acting at an angle and so, in order to determine the maximum stresses at a point in an object, you plug in to these principal stress, equations and those tell you what your maximum principal stresses are, and the angle at which they're acting at. And then you compare these principal stresses to a yield stress. So this is a very common theory that students learn. The problem with this theory is that it doesn't always safely predict failures in ductile materials, depending on the loading conditions. However, it's actually a great theory to utilize with brittle materials, especially when you use it in the Coulomb-Mohr theory. So let's get into the Coulomb-Mohr theory for brittle materials. Throughout the past couple modules, we've talked through a number of failure theories. So we started with failure theories for ductile materials, and particularly we focused here on von Mises Theory, which is also the distortion energy theory. Today what we're going to do is, we're talk about the Brittle Coulomb Mohr Theory, which is for brittle materials with a strain at fracture less than .05. Just a reminder that, if you also have a change in geometry when you're calculating the stresses for brittle material, it's always a good idea to incorporate a Kt value. So Brittle Coulomb Mohr Theory typically for brittle material the tensile strength is not equivalent to the compressive strength. There' also a Ductile Coulomb Mohr Theory and this is for ductile materials, whose compressive strength is different than its tensile strength. And if you learn Brittle Coulomb Mohr Theory, it's very easy to go in a textbook and learn Ductile Coloumb Mohr Theory, they're very close together. So, Brittle Coloumb Mohr Theory is a conservative theory for brittle materials. There's also a Modified Mohr theory, this is a little less conservative, and if you're interested in that you can check out any major machine design textbook. Okay, so, Brittle Coulomb Mohr Theory. Again you need static loading to utilize this theory. You need a brittle material, which we usually classify as a strain at fracture of less than 0.05 and you're going to utilize principal stresses. A reminder that you should always look at your operating conditions to determine if your material is actually behaving in a brittle or ductile manner. So the first step is to calculate your principal stresses with your principal stress equations or Mohr's Circle, depending on what you're more comfortable with. And the second is to order the principal stresses from largest to smallest. So you can see here that you have to order the principal stresses where sigma 1 is going to be your largest principal stress, and sigma 3 is going to be your smallest. And in some cases your sigma 1 will be 0 and in other cases your sigma 3 would be 0, it just depends on your loading conditions. So you're then always going to choose sigma a. Sigma a is going to be your largest principal stress. So sigma a is always going to be sigma 1, and we're going to assign sigma b to be your smallest principal stress, so sigma b is always going to be sigma 3. And then we have these failure conditions that we're going to apply based off of sigma a and sigma b. So case one, your sigma a and your sigma b are both greater than 0. So essentially, your principal stresses are all greater than 0. In that case, you're dealing with positive stresses, and so your failure condition is going to be if your largest positive stress, is greater than your ultimate strength in tension, you're going to fail, right? So that if your sigma a is greater than your ultimate strength in tension you're going to fail. And here's your factor safety equation where n is equal to your ultimate tensile strength divided by your sigma a. Case two, you have some principal stresses that are greater than 0 and some that are less than 0, so you're in both this compressive and tensile state. And so you're going to compare your largest positive stress to the ultimate tensile strength and your smallest stress. Which is going to be negative to the ultimate compressive strength and then, if this equation is greater than 1, then you've failed. And so, right here, is your factor of safety equation. And then for Case 3, what happens is that all of your principal stresses are negative. So all of your principal stresses are below 0, meaning that you're in a compressive state. And so now you're going to compare your smallest or your most negative stress. So this can get a little confusing, but it would be your greatest in magnitude or your most negative stress to the ultimate compressive strength. And your factor of safety will be your ultimate compressive strength divided by your most negative stress or the greatest magnitude. Okay, so, those are the conditions for the Brittle Coulomb Mohr Theory. Let's talk through a little bit of an example problem. So, in this example, typically in engineering, we have a lot of athletes and this is actually a very common failure mode in an athlete's bone. So if you have a femur bone which is that main bone in your thigh, in your leg it could have an outside diameter. Let's say, it's a smaller woman, so 24 millimeters. The tensile strength of this bone is right around 120 MPa and the ultimate compressive strength of the bone is 170 MPa. Note this is pretty common for brittle material that your compressive strength is higher than your tensile strength, so this is a common failure mode in both skiing and in figure skating. So if a figure skater is spinning in the air and then she goes to land a jump, she has to stop the rotation so her femur is going to feel a torque from trying to stop the rotation. And it's also going to feel a bending load from the weight of her coming down on the bone as she lands. The same thing happens in skiing, when people are turning, there's a torque due to turning. There's a bending load due to the weight of their body. And sometimes this can cause a pretty catastrophic failure in the bone. So we're going to go ahead and calculate that. So we're going to say this femur bone is subjected to a torque 80 N-m and bending moment of a 100 N-m. In this case, we're going to greatly simplify the properties of bone, we're going to say it's acting as a brittle material, it's a solid cylinder. It's isotropic and linear elastic. In reality bone can get a lot more complicated. And if you're really curious about seeing those types of models, you can take a graduate level biomechanics class and see all sorts of fun ways to model bone. But for this undergraduate level class, let's go ahead and use the Brittle Coulomb Mohr Theory and what we're going to do is we're going to find the factor of safety at A. And I've given you, here's point A, you can assume it's equidistant from the top and the bottom and it's acting. A is going to be on the YZ plane where Y is coming out of the page and it's acting on the very edge here. So what I'm going to let you do is go ahead and try to calculate this on your own and then we're going to go through the solution together in the next module. So that's all for today and I'll see you guys in the next module. [SOUND]