[SOUND] Welcome to module 30 of Mechanics of Materials part 3. Today's learning outcome is to continue the design of an appropriate beam, using a real world engineering problem. And today we're going to look at what's called the maximum sheer stress failure theory. And so, here is our beam model and the worksheet. We said that the beam can experience elastic fluctual stress and transfer shear stress. In designing the beam, last time we talked about the maximum normal stress theory. Today, I want to talk about another failure theory, which is called the Maximum Shear Stress Theory. Or what's referred to sometimes as Tresca's Yield Criterion. And that was named after Henry Tresca, who was a French mechanical engineer. And so, for the Maximum Shear Stress Theory we now say that failure occurs when our shear stress experience is greater than our failure shear stress. And so recall, when we have design we've built in a factor of safety. The factor of safety is the failure stress over the actual stress. Now, instead of working with normal stresses, we're going to be working with shear stresses, and so we want to factor a safety that's greater than 1. That avoids failure. The Factor of Safety is something that we've talked about before. It's a design criteria for different types of engineering components or structures. And the designer defines what the failure is going to be or designs against what the failure mode is going to be. And so, the component, our structure, we want it to meet certain performance criteria, i.e., we don't want it to have some excessive deformation, or we don't want it to fracture, we don't want it to yield. That's the choice of the designer. And so, for the maximum shear stress theory, we will use shear stress, again, instead of normal stress. And so, for the normal stress theory we use the simple tension test, or what's called an STT, to look at the failure to design our stress strain diagram. And what we're going to assume now is that even in a complex loading condition, when we have non-uniform bending, and we see that we have flexure and shear, we're going to say that the material has the same capability as we found in the simple tension test. And so, here's our simple tension test. Here, was our failure. We pulled on it with a uniaxial load, but now we'll draw Mohr circle, and we can see from different planes at that stress block, the different orientations, we can also experience shear stress. And so, if we go out and draw our Mohr circle on the horizontal plane we have our failure normal stress. On the other plane we have 0 stress. And so, when we draw more circles, we see that there's an orientation for our stress block where we're going to have maximum shear stress, and that's what we're concerned about in the maximum shear stress theory, or the Trusca Theory. So Tau failure, let's say that our Sigma failure is the yield or the proportional limit. The failure is defined as the yielding or reaching the proportional limit of our material. And so, for Tau failure that's also going to be the yield or proportional limit. And so, that's going to be by our Mohr Circle. One-half of the sigma yield. And so, factor of safety is the failure stress over the actual stress. We're saying that our failure stress is defined as the proportional limit or yield stress, shear stress. That's going to go over our actual stress, and a factor of safety greater than 1 is going to go ahead and avoid failure. And so, as a recap, Tresca's Yield Criterion, Maximum Shear Stress Tau experienced is greater than Tau failure. This theory is particularly good for ductile materials. And this is like steel, aluminum, plastic, etc. And that's because yield in ductile materials is usually caused by a slippage of the crystal planes along the maximum shear stress surface. And so, now, we've talked about two different failure theories that we can use in design and we'll go back. And decide how are we going to proceed with our design of this engineering beam or engineering structure. See you next time. [SOUND]