[SOUND] Welcome to module 16 of Mechanics of Materials Part I. We're moving right along in the course. I've check of the topics that we have covered. And today we're going to look at stresses on inclined planes. And we actually started to discuss the stresses on inclined planes in module four but we're going to look at it in more general this time for the case of plane stress. And we're also going to define the sign convention for stress on inclined planes in general. And so if you remember back to module five, well, I said that for more complicated structures, we may have a stress distribution that may not be uniform. But we can shrink down to an infinitesimal small cube and the stress distribution will approach uniformity. And so here's a graphic of that cube. And I've shown the stresses, both the normal and shear stresses acting on this state of stress at a point in their positive directions. And then we, in module six, went ahead and we said, okay, let's let the out of plane stresses in the z direction equal 0. And we'll come up with the case of plane stress. And so, again, back at module four, this was what we did on the normal and shear stresses on an inclined plane for uniaxial loading for an angle theta. We found this was the normal stress and this was an expression for the shear stress. We found out that the max normal stress occurs at an angle of 0 or 180 degrees. The max shear stress occurs at an angle of 45 or 135 degrees. We also noted that as theta became greater than 90 degrees, the shear stress and the shear force vector change directions. And one final note was important that the magnitude of the normal stress, the maximum normal stress, was equal to two times the magnitude of the maximum sheer stress. And so, now let's extend that for the more general case of plane stress where we now have, on our cube in two dimensions, both sheer stresses and normal stresses. And we're going to want to find the sheer stress and normal stress on an arbitrary plane at an angle theta. And in general, if you recall back to module 15 of my course, Applications in Engineering Mechanics, we were able to draw what we called shear force and bending moment diagrams. And you may want to go back and review those modules in Applications for Engineering Mechanics, but I have a structure, a beam, with different types of vertical loads. I can also even have an axial load if I'd load like. But I can cut the beam at any point and find the internal forces acting on an infinitesimally small part. And so, for instance, on this beam here, I could go ahead and have a certain type of loading, I could make a cut. And I would be able, in general, to find these stresses, the normal stresses, and the shear stresses from the external loading. But I would also like to find what the shear stresses or the normal stresses are on the arbitrary plane. And the reason for that is, that we may want to know what plane does the max stress, normal stress, or shear stress occurs, because we want to make sure we avoid failure. And I have a part here, this is a piston with a connecting rod. This is actually from a mini Baja vehicle that one of our teams here at Georgia Tech built. But you can see that there's a failure. And whenever we have a situation with loading, we want to know what the shear stress is and normal stresses are on various plains, to make sure that we avoid failures. And so here, again, I take a cut on an inclined plain. I'm going to go ahead and define coordinates for the inclined plain in the normal and the tangential direction. And we'll go ahead and I show the stresses here. The sign convention will be the same as what we've used before. Normal stress will be positive in tension or negative in compression. Shear stress is positive if it is in the positive direction of the coordinate axis. So, for instance, the positive normal face and the positive tangential direction is the definitions of our positive shear stress. And the angle that we'll measure on this inclined plane will be counterclockwise, taken as positive from the reference x-axis. And so, that's where we'll leave off. And we'll pick up with solving for the stresses on inclined planes next module. [SOUND]