[MUSIC] Welcome back to Module 23 Mechanics of Materials part I. Today's learning outcome is to represent the transformation of plane stress using Mohr's Circle. We said that Mohr's Circle was a graphical tool for the depiction of the transformation equations for plane stress. And we rearrange the transformation equation to look like this. We said it was in the form of a circle. The radius was the square root of the right hand side. This was the center. And we said that the angle of Mohr's Circle was two times the stress block angle. And that Mohr's Circle now is going to represent, at each point in our body the stresses, the normal stresses and the shear stresses on a particular plane for a single point. And so if I have this engineering body, whatever it maybe and I want to look at the stress at a particular point to see what it is. I can show the stress, shear stress and normal stress on any plane through a single point and all of those points are going to be represented for plane stress for two dimensional stress on Mohr's Circle. Here's the radius, here's the center. Our sign convention will be that normal stresses on Mohr's Circle will be the same as we've seen before. Tension will represent positive stress. Compression will be negative stress. As far as shear stress is concerned on Mohr's Circle, it's going to be a little bit different sign convention I'm going to use. And that is that the clockwise rotations, like the top and the bottom here, are going to be positive. Counterclockwise rotations are going to be negative, and I show those here. So here's a stress block that shows a known stress condition from our external loading on our engineering member. And I've shown one plane aa, I can show that plane aa at any angle. By my Mohr's circle sign convention on the horizontal face here we have sigma sub y intention, so that's positive. Sigma sub yx is clockwise, so that's also going to be positive by my Mohr's Circle sign convention. On the vertical face here, I've got sigma sub x shown as positive and tau sub xy causing a counterclockwise rotation, so that's negative. So now, let's start by plotting these two points on a graph of the normal stresses and shear stresses. So I have sigma sub y and positive tau sub yx. That's the horizontal face. And I'll label it as H with a circle around it. So this is sigma sub y, tao sub yx. We know that tao sub yx and tao sub xy are the same by equilibrium so I'll also plot the vertical face. I'll assume we'll work with actual numbers in the future for problems that we actually solve, but I'll assume for now that sigma sub x is greater than sigma sub y. And so come down here, and I've got my vertical face and it's sigma sub x minus tau sub xy. I can draw a line in between them and this will be my center. I can draw now a circle that's going to represent all the points for the normal and shear stress. And so our center is going to be at, Sigma sub average and 0. And sigmas of average, again, was sigmas of x plus sigmas of y over 2 and 0. We can then rotate to the principle planes where the shear stress equals 0. And so this will be 2 sigma sub p to the principle planes, and we find out now that this one is going to be sigma sub 1, our principle stress 1. And 180 degrees later, because we're working with 2 theta instead of theta, for theta on the stress blocks, it would be 90 degrees. But at 180 degrees, we have sigma sub 2, the second principal stress. We know that the radius, this value here is the same as tau max, the maximum shear stress. And so, that will occur right there and so, this is tau max and the angle to that would be 2 theta sub s. And we can go to any arbitrary plain. So let's go to an arbitrary plain AA. A counterclockwise rotation of theta sub A. So that's going to be 2 theta sub A on my Mohr circle. So if I go around 2 theta sub A, and again this is just arbitrary. I'm just giving this as an example. So this would be 2 theta sub A. And this would be now the theta stress, the normal stress, and the shear stress on a plain here AA. And so this would be, sigma sub A and tau sub A. And that's point A. And so that's a good representation, generic representation of Mohr's circle. Next module we'll work with some actual numbers and solve a problem. And we'll see you then. [MUSIC]