[SOUND] Welcome to module 16 of Mechanics of Materials, Part I. We're moving right along in the course. I've checked off the topics that we have covered and today, we're going to look at stresses on inclined planes. And we actually started to discuss stresses on inclined planes in module four, but we're gonna look at it in more general this time for the case of plane stress. And, we're also gonna define the sign convention for stresses on inclined planes in general. And so, if you remember back to module five, 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 infinitesimally small cube and the stress distribution will approach uniformity. 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 zero 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 the 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 changed directions. And one final note was important that the magnitude of the normal stress, the maximum normal stress, was equal to 2 times the magnitude of the maximum shear 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 shear stresses and normal stresses, and we're going to want to find the shear 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 and Engineering Mechanics, we were able to draw what we called shear force and bending moment diagrams. And you may wanna 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 like, but I can cut the beam at any point and find the internal forces acting on an infinitesimally small part. 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 an arbitrary plane. And the reason for that is that we may wanna know where the max, what plane does the max stress, normal stress or shear stress occur, because we wanna 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 Baha vehicle that one of our teams here at Georgia Tech built. But you can see that there is a failure. And whenever we have a situation with loading, we wanna know what the shear stresses and normal stresses are on various planes to make sure that we avoid failures. And so here, again, I take a cut on an inclined plane. I'm going to go ahead and define coordinates for the inclined plane 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 have 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, this is 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 counter-clockwise 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 incline planes next module. [SOUND]
