 Hello, everyone. In this video, I want to do an example of taking a stress element that we've already solved and transferring that to a more circle and using that to better understand the system. So I'm going to go ahead and bring up my screen. And I've already got an example stress element shown here. And what I'm looking at is a basic stress element. On my X face, I have a 100 MPa scale normal stress my Y face. So Sigma Y, I have a 60 MPa scale normal stress. And then I have a shear stress of 45 MPa scales. And the orientation of my shear arrows here indicates it's a positive shear stress as we consider it for a sign convention on our stress element. So if I go ahead and draw a more circle axes, Sigma and tau axes, I can go ahead and actually plot the faces of the stress element on this this set of axes. Remember that that tension or positive shear stress for arrows pointing away from my stress element are represented positively. So that's to the right hand side of my tau axis. And then shear stress, the shear stress sign convention on a more circle is that if it causes clockwise rotation, it's positive. And counterclockwise rotation, it's negative. So if I start with my my Y face here, I have a positive 60. So I'll just kind of, you know, roughly mark out say 20, 40, 60, something like that. I'm just really roughly approximating. And my 45 is pointing to the right. So that means it's a positive stress. So I can go 60, 20, 45, and put a dot somewhere like that. Now this is that that first point 60, 45 on my axis. My X face has 100. So again, 20, 40, 60, 80, 100. But now my arrow is pointing up, which would be which would cause a counterclockwise rotation of my stress element. So I have a negative shear stress. So I can put that dot somewhere down here. And I'm going to label this point 100 minus 45. So now I can actually just draw my circle from this, right? I can now really try to figure it out. But someday I'll learn how to draw circles. My more circle would look something like this, it passes through those two points. But typically how I would want to figure this out or calculate things for the more circle would be to use those equations I had before. So I would find the center point, which is again, sigma average. And it's just the average of the two sigma stresses, sigma X and sigma Y. So 100 plus 60 over two equals 80 megapascals. Now, of course, we can see that, right? Because these two points are kind of, you know, opposite sides of this circle, and I can go somewhere right in between them, I would expect to get 80 for where that location is. I can also calculate the radius using the equation I had, which is square root of, in this case 100 minus 60 over two squared plus tau XY, which is 45 squared, all of that under a square root. So I can go ahead and calculate that out, plug it into my calculator or what have you. And I would get 49.24 megapascals. So what that tells me again is that this radius is 49.24. Now, I might also be interested in the orientation of my stress element. So one way that I can do that is I can say, well, let's dash in a line from the center out to my X face. Remember the 100 minus 45 is the X face. And I can see that my max stress occurs up here on the X axis. So I have an angle theta, which represents that. And typically, this is actually labeled theta sub p. Now I can work out the geometry for this, right? There's a lot of geometry going on here, but I can work out the geometry for this and say, well, tangent of two theta p. Now, you may ask, why did I do two theta p? Well, you have to remember that this rotation is actually two theta p. If I say theta p is the rotation of my stress element. So wherever that max stress element orientation is, it's two times it on the Mohr circle. So that's why I put that two there. And this is going to be equal to two tau x, y over sigma x minus sigma y. So really what I've done here is drawn a triangle that calculates or figures out the angle of all of this based on what my stresses are. Now, in reality, the triangle I'm using goes from one side up here, 6045 down to the 100 minus 45. That's why we have two tau x, y, because it's positive and negative. So two of them total on the one side. And then it's the difference between those two values is my, you know, X portion. So then I'm getting the angle based on this large triangle that I would inscribe in the circle. So taking that geometry, I say, okay, two times 45 over 100 minus 60. And you know, I don't know what that works out to be. I actually don't have that worked out in my head or on my notes. But I get two theta p equals 66.04 degrees, which therefore gives theta p equal to 33.02 or about basically 33 degrees. So what this means is that I have to rotate 66 degrees to get from this orientation to the max stress, which of course is out here on my Mohr circle. If I was rotating my stress element, it means that my stress element x axis where this 100 is pointed, would actually be oriented 33 degrees counterclockwise for that max stress element where the stress would be a maximum. And the shear stress is actually zero on that face then. So I would rotate that up by 33 degrees. Now, of course, I'm probably interested in, in these other values such as sigma max. And at this point, it's pretty easy to get that because if I know my center of my circle, which is 80, and I know my radius, which is 49.24, then of course I see that I can just add those two and get 129.24 megapascals. Now, I'm probably interested in my tau max, especially if I'm talking about a ductile material. And in this case, I can see my tau max is just my radius, right 49.24. So I'm just looking at my circle here, sigma min. In this case would be 80 minus 49.24. So that equals 30.76 megapascals. So great, I got all those values. Now, if I wanted to draw the stress element, what does that actually look like? Well, again, taking what I had previously oriented on an x and y axis, I'm rotating my stress element. Now, I'm not going to get the angle right here, but rotating my stress element such that this new axis, this would be x prime from what we talked about before, is rotated by 33.02 degrees. My normal stress on this is 129.24. My shear stress is zero. My normal stress on this other face is 30.76. We know that those correspond because they're 180 degrees apart on my more circle, which means they're 90 degrees apart on my stress element. So I can see what that looks like. And this would be that that rotated stress element, which corresponds to the max normal stress, 129.24 megapascals at 33 degrees of orientation. All right, I'll stop there. Thanks.