 Hello everyone. In this video, I want to talk about the Mohr's Circle. So as we previously mentioned, when we place our stresses on a stress element, we kind of pick the orientation of that stress element randomly. And the element can actually be oriented at any possible arrangement as I rotate it around. And a tool that we have available to us to help us understand that is the Mohr's Circle. So if I go ahead and bring up my screen, when we plot a Mohr's Circle, typically we're plotting on a normal stress shear stress axis. So I'll have normal stress on the x and shear stress on the y. And I'm probably going to label these axes with our notation from before, which is x prime y prime for shear and x prime or y prime for x axis. So a Mohr's Circle is, well, exactly as it sounds, it's a circle. So if I go ahead and pick a center point and then draw a circle around it, pretend it's a circle, and I say that this center point has a location c for the center and the circle has a radius r, then I can use that to define my circle. My circle is always centered vertically on this x axis. So that center point really defines the position. And what I'm really doing here, then, is representing all possible combinations of stress, of normal stress and shear stress, as it relates to the orientation of my stress element. So as I rotate my stress element, I'm traveling around the circle. And basically 180 degrees rotation of my stress element brings me back to the original position. So it's 360 degrees rotation on my Mohr's Circle. And so I can define then a few points on this. So I can say that my minimum stress, normal stress appears at the left-hand side of my circle. My center point is going to represent the average. And then my maximum stress is going to be out here at the right-hand side of the circle. And I can define those things. So what's useful, then, is I can use this notation to define what these points are. So the center of my circle, which I'll call c, is equivalent to the average normal stress. And the average normal stress is just going to be the average of my two normal stresses from my stress element, sigma x and sigma y. And it actually really doesn't matter what orientation these are at, because any position on this circle is going to have a sigma x and a sigma y that allow me to find what that average is. Because again, every position is on that circle. So I can write that. That's easy enough. And now I can define my radius. And my radius, I can define using geometry. And basically what I'm doing is I'm finding every, for any possible point on the circle, that point represents a combination of sigma x, sigma y, and my shear stress, tau x, y. And if I work out the geometry of this, I could find the radius, which is like the hypotenuse of a right triangle in this case. And I can go ahead and write that, where it's going to be equal to sigma x minus sigma y over two squared plus tau x, y squared. So really this is just coming from a from a Pythagorean theorem, you know, c squared equals a squared plus b squared type equation, but substituting in all the different values from from the geometry of my more circle. So this gives me a basic foundation for for how I prescribe a more circle. Now I want to talk just quickly about a general procedure then that we can use to use a more circle to actually to actually solve for problems. So if I go ahead and give myself some white space here, we have a few kind of key points to keep in mind, or key procedure maybe. One, we want to find our primary stresses, or our primary loads, I should say. So that's axial shear bending moment and torsion. And that's at the the cross section of interest that we're looking at. From that, we can find our corresponding stresses. So those primary stresses that relate to those, so sigma a, tau v, sigma b, tau t, and those again come from those from those primary loads. Next we want to come up with our stress element. So figure out how those primary stresses get represented on our stress element, and then use that stress element to draw our more circle. So that more circle gives us then some max stress that we're probably interested in in the normal direction. It gives us a max shear stress. And interestingly enough, which I'll show in an example, we can get the orientation of the stress element that corresponds to those maximum values, which can be useful. And then our last step, of course, would be to check this state of stress against our failure criteria, which I'll be talking about in a later video. So a few notes or things to keep in mind, and just kind of remember, every orientation of that stress element has a point on this circle, and can be represented as a point on that circle. And I'll show that in an example. I also want to talk for just a second about sign convention, and what the sign convention is for for a more circle and how it relates to a stress element. So if I talk about my stress element versus my more circle, and I have tau, and I, or excuse me, sigma and tau, the stress element is shown as positive when it is in tension. So when we have our square stress element, if we have our stress arrows positively pointing out, that is a positive direction, it's in tension. And on the more circle, it's the exact same thing. Tension is positive. So I had it plotted on an axis and the positive position on that axis is tension. If it was negative, I would be on the opposite side, the left hand side of the of the vertical axis. For tau, on my stress element, we generally consider stresses, a shear stress that has arrows that look like this, to be positive. So if it's being sheared up into the right, that would be a positive direction. If those arrows were pointing oppositely down into the left, then that would be a negative shear stress. Now on a more circle, we actually separate these two arrows. So in this case, something that would tend to cause clockwise rotation is considered positive, and something that would cause counterclockwise rotation is considered negative. So if I'm looking at the two sides of my stress element, this one has a positive shear, the one on top, and the one on the right has a negative shear. So if I have positive stress element, shear stress, as shown in my first column, then that gets separated into one positive and one negative component when I talk about my more circle. I also mentioned that we can get the orientation of our stress element from this, and the key distinction or key thing to keep in mind is that as my stress element rotates, so as I rotate this stress element, eventually this side rotates all the way around to become the right hand side, rotates all the way around to become the left hand side, and because it's a square, that's symmetric. Now it's not 90 degrees symmetric, meaning as soon as this right hand side is on the top, that isn't necessarily the same thing, but because of equal and opposite reaction forces and reaction stresses, as it rotates all the way around, it's symmetric, so 180 degrees of symmetry. Now to get to the same position on a more circle, we have to rotate all the way around 360 degrees. So a theta rotation for a stress element becomes a two theta rotation on more circle. So it's just something to keep in mind that the angle of rotation gets doubled on more circle compared to what it is when we rotate the stress element. All right, the last thing I wanted to talk about in this video is three-dimensional stress elements and how that relates to our more circle. So so far, I've really only been showing a planar case where we have a square and we can rotate that around an axis to represent our stress element. Of course, we live though in a 3D world, so we actually have three-dimensional stress elements would be a closer representation of reality. So if I go ahead and draw a 3D stress element and I know you're thinking, yeah, let's see how this turns out, so am I. Then I can represent the stresses on that as I would before. So sigma y, sigma x, and sigma z, assuming my three-sided or 3D coordinate system looks like this. Then I also have three-dimensional shear stresses. Now, this gets a little messy because I have on each face, I have two possible shear stresses in two different Cartesian directions. And the naming convention for these, so if I pick this one out and name it, would be, for example, tau yx for this case. And this first letter denotes the face, and the second letter denotes the direction. So I have tau on the y face pointing in the x direction. What about this one do you think? Pause the video and give a guess. Just kidding. This one is on the x face, one of these times I'm going to say this wrong, pointing in the z direction. So it's tau xz. So as you can see then this stress element has six shear stresses and three normal stresses that we would need to consider. And this is because of course this stress element is positioned in a rigid body, that's a three-dimensional object. Now when we want to then transfer this over to a Mohr circle, we have say for example a normal, I'm going to draw this a little bigger, a normal Mohr circle might look like this. I, this is just a planar case, right? I'm going to get a different color here. I can inscribe my three-dimensional components within this. And the reason we need a three-dimensional Mohr circle is because this 3D stress element can actually rotate in three directions, right? I can rotate it around in the screen, I can rotate it this way, and I can rotate it this way. So that's three dimensions of rotation. Now typically when I draw my 3D Mohr circle really all I need to do is start at the max out here and prescribe a circle there and prescribe one here. Now this is actually we're making some assumptions when we do this and it's a special case when this happens. And the reason we've prescribed the circles this way is because we want them both to pass through zero. And the reason we have them pass through zero is because typically when we're looking at state of stress, we're looking at the stress on the surface of a rigid body. And if we're looking at a stress element on the surface of a rigid body, because it's on the surface there can't be any stress in normal stress in the third dimension, say the z-axis dimension. Because if there was stress, because it's open to free space, it would just deform. It would move and relieve that stress. So because it's not contained, that stress has to go to zero and therefore always be zero. Now in the example that I've drawn here, this means that nothing really changes, right? My black circle was my original one, and the two circles I've prescribed in red inside of it don't change the max normal stress or the max shear stress that nothing's happened. But if I, for example, had an original Mohr's circle that looked like this, and that was my original Mohr's circle before I considered three dimensions, once I come in here I need to prescribe my additional two circles for the other two directions of rotation. So I need to make connections at the shared point here over to zero, and the other shared point on the axis all the way over to zero. And this is important, right? Because if I look at my new three-dimensional Mohr's circle, what I can see is that my max normal stress didn't change, but my max shear stress went way up. Now this is, you know, probably an exaggerated example, but my max shear stress went way up in this three-dimensional rotation. And remember that ductile materials typically fail by shear stress. So this is problematic, right? If I didn't consider this three-dimensional case, it means that I would be under predicting the actual failure. I would be not accurately estimating what that failure would be. I'd be actually pretty far off. So in some cases, we need to consider this three-dimensional stress situation because it may alter when we expect our part to fail. All right, so I'm going to stop there, and my next video will show some examples of how we use Mohr's circle or solve for Mohr's circle. Thanks.