[SOUND] Hi this is Module 22 of Mechanics of Materials part one. Today's learning outcome is to show that the transformation equations for plane stress can be expressed in the form of an equation for a circle. And we call this Mohr's Circle. So as a recap, we found the principle stresses where the shear stress equalled zero. We found the angle to the principle stresses which were called the principle planes. We found the maximum in-plane shear stress. And the angle to that maximum plane shear stress, and we found out that the planes in which those occur are 45 degrees apart. And we had a couple other important relationships. The stress invariant, that says that the sum of the normal stresses on two orthogonal planes are same, or constant. And we found that the maximum shear stress was equal to the principal stress one minus principal stress two, divided by two. So, let's take and rearrange this first equation. I've just carried this term to the other side and this is the result I get. Here's my second equation rewritten. And now I can square these equations and add them together. And again, I'm going to go through some math here. I'll go rather quickly, but you can come back on your own and make sure you understand each step as I go through it. But when I square these and add them, this is what I get. And I see that I have sigma sub x minus sigma sub y squared over 2 sigma sub x sigma sub y over 2 squared, times cosine squared theta, and the same value, sigma sub x minus sigma sub y, over 2 squared, times sine sub squared 2 theta. So I could factor that out, that term, and I get that times sin squared theta plus cosine squared theta, which we know equals one. And so this term, these two terms become this, I can do the same over here factor out taos of xy squared, times sine squared of two theta, plus cosine squared of two theta, again, sine squared plus cosine squared is one, and so I end up with tau sub xy squared. And so this is the result I come up with. So here's our result again. I'm going to rewrite it slightly. Instead of saying tau sub nt squared, I'm going to just put in a tau sub nt minus 0 squared. And the reason I do that is I want to recall the equation of a circle. What you should remember from your days of geometry. Here's the equation, here is what the circle looks like. And so, I can see now that these equations are the same form, okay? So, I have sigma sub n is the same as x. A is the same as this value. Tau sub n is the same as this. B is zero and the right-hand side is the radius squared. So I have a center of my circle, a radius, and so I can express this equation in the form of the equation for a circle. So we call this Mohr's circle. It was an observation of Otto Mohr, a German engineer. It's a graphical tool for the depiction of the transformation equations for plane stress. Here's our equation for a circle and what it looks like. We see that the stress transformation takes that same form. So the radius would be the square root of this right hand side for our circle. The center would be this value, which would be sigma sub x plus sigma sub y, over two, and zero for b, and we call this sigma sub x plus sigma sub y over two, sigma of average. And so where the average normal stress is shown here, and so the stress transformation equation, remember, is based on an angle of two theta. The stress blocks angle is theta, so therefore the angle on Mohr's circle is two times the stress block angle. And we'll come back and use these equations to actually solve problems, and graphically depict the transformation equations for plain stress in future modules. [SOUND]