[MUSIC] >> Hi, and welcome back to mechanics of materials part one. In the next few modules we're going to be going through a bit of mathematics to develop some very important relationships. The beauty of these recordings is that even though I'll go rather quickly, you can go back and stop and start them to make sure that you understand every step in the process. So today's learning outcome is to find the maximum and minimum in-plane principal stresses. And so, here's where we left off last time. We have transformation equation for stress where we know stress on a, in certain directions, and we can find the normal and the shear stresses on any other plane using this transformation equations. And we found an angle to what we call, where we find the maximum or minimum normal stresses to occur with this tangent relationship where theta sub P is the angle to which is defying the principle plains. And notice that on the block itself, we're working with theta sub P. In the formula, we're working with 2 theta sub P. And so there's that relationship. And so let's look at this graphically. Since it's a tangent, opposite would be tau xy over adjacent, which would be sigma sub x minus sigma sub y over two. Knowing those two perpendicular sides by the Pythagorean Theorem, we can find the hypotenuse. And so let's now consider both tau xy and sigma x, and sigma y, sigma x minus sigma y over two being the same sign. So lets first consider them to both be positive as I've shown here. And if they're going to both be positive, we see that 2 theta sub P has to be between 0 and 90 degrees. They can also have the same sign if they're both negative, on the top and the bottom. And then we're going to be between 180 degrees and 270 degrees because we have a negative in the tau xy, or the shear stress direction, and a negative in the sigma x minus sigma y over 2 direction. And so, there are also two values of theta sub P on the stress block itself, and they're going to be one half the magnitude of the 2 theta sub P. So, the theta sub P's, the angles to the principal planes, will be 0 to 45 degrees, and half of this, which is 90 degrees or greater. And the rotation is counterclockwise. Okay. Now let's look at the case where these two values, tau xy and sigma sub x minus sigma sub y over 2 have, end up being a negative sign. You could have a negative tau xy in the numerator, and a positive sigma sub x minus sigma sub y over 2 value in the denominator. Or you could have a positive tao sub xy value in the numerator and a negative sigma sub x minus sigma sub y over 2 value in the denominator. And if that occurs, tangent of 2 theta sub pi is going to be negative. And so, therefore, we're going to have the, one would be negative, the other one would be positive. And so, we would be somewhere between 0 and minus 90 degrees, or somewhere between minus 180 and minus 270 degrees. And so, this time the rotation would be clockwise. Okay. So here is our relationship for tangent 2 theta sub P. And it can be plus or minus this value. Here it is shown graphically. And so by trigonometry, we can find the expression for cosine, which is the adjacent side over the hypotenuse, or the sine of 2 theta sub P, which is the opposite side over the hypotenuse. And now we can take our strain transformation, excuse me, stress transformation equation for sigma sub n, the normal stress, and we can substitute in cosine 2 theta sub P. So this should be theta sub P now. And sine 2 theta sub P, which will end up being theta sub P now. So these, now we're talking about specific angles, so these will be theta sub P and theta sub P. And this is the result I get by substituting those in. Okay. Plus or minus. We can, this value and this value is the same, so we can factor that out. And we have these two terms times this entire value here. We see that we have sigma sub x minus sigma sub y over 2 squared, and then the square root of that. So that leaves a square root in the numerator. Okay? And so here is our expression boiled down. And that now gives us the normal stresses on these principal planes. We have two of them, and I've labeled them principal stress number one and principal stress number two. Here it is again. So, at some angle, theta, we rotate our block. And at that angle theta we end up with what's called principal stresses. And I've shown them. They can be both positive. They can be both negative. One could be positive, one could be negative. But you'll note in this algebraic, in this development, I've considered max and mins to be algebraic quantities. And so, as I said, they could both be positive. I might have plus 1,500 and plus 500. Or I might have plus 800 and minus 200. Or I might have minus 400 and minus 1700. But in our calculations for our engineering problems, when we use the term maximum we're going to refer to the stresses with the largest absolute value, or the largest magnitude. And so, for example, I have sigma sub 1, maybe a 700 megapascals. Sigma sub 2 is minus 1200 megapascals. This one is in tension. This one is in compression. But if I'm talking about the maximum normal stress, I'm going to refer to the one that is the maximum absolute value when we do engineering problems. Okay. Here's our result again. We'll see from this that we can end up with what we call a stress invariant. If I add these two sigma sub 1s and sigma sub 2s together, I get sigma sub 1 plus sigma sub 2 on the left hand side. On the right hand side, this plus and minus will cancel out this part. And so, I'll have sigma sub x plus sigma sub y over two plus sigma sub x plus sigma sub y over 2, or sigma sub x plus sigma sub y. That's a very important result. It's the stress invariance. So what it's saying is, on any two orthogonal planes the sum of the normal stresses are going to be constant. And so, no matter how we turn the block, the sum of the normal stresses on two orthogonal planes is going to be invariant, or constant. And so that's where we will leave off this time. Some important relationships. And we'll continue on next time. [MUSIC]