[MUSIC] Okay, we've come to the last topic in the course for Mechanics of Materials part IV. And that's to go over again the theories of failure. And so the learning outcome is to review the maximum normal stress failure, which we've talked about in before. The maximum shear stress failure, which we've also talked about before. In fact, we talked about these in my mechanics of materials part three course. The maximum normal stress, we talked about all the way back to my mechanics of materials part one course. And then finally, we're going to explain something called the maximum distortion energy theory. And so the failure theories, we went all the way back to my mechanics of materials part one course. We did a simple tension test, and it was easy to perform and it provided very good results for a variety of materials. And so we put a tensile force, we looked at the stress strain curve. We could predict yield, we could predict fracture. We could find Young's modulus, a very nice test, very useful test. But what about when we have more complex loading conditions, when it's not just the simple axial load? We have biaxial loading or triaxial loading. Well in those cases, the cause of failure may be unknown. And there are several theories that can be used for predicting failure for these various types of loading, and I'm going to focus on just three. The first one we will review is the maximum normal stress theory. That says that when the failure stress, excuse me, when the normal stress in our actual engineering element is greater than what's defined as our failure stress, then we're going to experience failure. And it assumes that the material is subject to a combination of loads. And it can be a combination loads and it fails when the maximum normal stress at any point exceeds the axial failure stress as determined by that simple tension test. And we said this by experiences whose generally good for just brittle materials, it was not good for ductile materials like steel, aluminum, plastics etc. For those ductile materials, we also looked at the maximum shear stress theory but which is also called Tresca's yield criterion. And this said that failure occurs when the shear stress, the actual shear stress is greater than what we define as our failure shear stress. And this is good for ductile material. And that's because in yield ductile materials usually, the yield is caused by a slippage of crystal planes along the maximum shear stress surface. And so here's our simple tension test, here's a stress block with that tension on it. If we draw more circle, we see that tau failure is one-half of the normal failure, based on the simple tension test. And so failure will occur in this case when the maximum shear stress at any point, for a complex load, reaches the failure shear stress which is equal to one half the normal yield stress or failure stress, as determined by the simple tension test for the same material. So for the actual conditions, the more circle may look different but this is the condition for failure. Finally, another very common and useful theory of failure is the maximum distortion energy theorem or what's called von Mises yield criterion. Now I'm not going to go into the details of the development of the theory, you can do that on your own or take a more advanced class. But this is generally regarded now as the best yield criteria for ductile materials, better than the shear stress or Tresca criterion, in most cases. And this states that failure occurs when the strain energy of distortion, or the change of shape of the element reaches a critical value. Now when I talk about strain energy that's a concept that, again, I haven't covered in this course, you would have to cover it in a more advanced course. But alternatively you can think of this theory, and it gives the same results as yielding occurring when the shear stress and what we call octohedral planes, which is a critical value. Octohedral planes or any plane whose normal makes equal angles with the three principal axes. And so, if you're interested in this, I'd recommend that you look at more advanced discussion of the max distortion energy theorem or the von Mises yield criterion on your own. And if you do that, you'll find we're just dealing, in this course, with 2D plane stress. We're going to say that the outer plane stress is equal to 0. And so, by the maximum distortion energy theory, this is the equation we'll look at. The failure stress squared is equal to the two principle stresses squared minus the first principle stress times the second principle stress. Okay, and so now let's wrap up by applying these failure theories to the combined loading example that we did in the last module. This was the loading condition, this was the stress at point A. And then we looked at more circles to find an orientation and a prediction of the maximum principles, or the principal stresses and the maximum in-plane shear stress shown here. And so there are those values.. If we were using the various failure theories, let's see what we would say. Okay, for maximum normal stress theory, our maximum normal stress is 9.12 megapascals and tension. And that would need to be less than or equal to what we defined as the normal stress and failure. For the maximum shear stress theory, in this case we found the maximum shear stress for this complex loading was 4.79 megapascals. And that would need to be less than the failure shear stress as defined earlier, not earlier, but as defined for whatever material you're using. And then finally for the maximum energy distortion energy theory, I'm going to use that formula that I just showed on the last slide. I'll put the values in, and I find out that it ends up being, there should be an equal sigh here. And this is sigma failure, and it's equal to the square root of these values, and I've put those in. You find that the calculated stress for this theory must be less than, which is 9.36, must be less than or equal to whatever the failure our yield stress is, if yield is the definition of failure for the material that you're using. And so, that's a good overview of the theories of failure. [MUSIC]