[MUSIC] Welcome to Module 36 of Mechanics and Materials Part I. Today's learning outcome is to calculate the in-plane strains based on strain gage rosette measurements, and we talked about strain gage rosettes last module. So, here's a strain guage rosette generically. We are able to measure the normal strain in three different directions at three angles and we can come up with three equations then. The normal strains in the a, b and the c direction are measured and the three unknowns are on the right hand side are two normal strains epsilons of x and epsilons of y and the shear strain epsilons of xy. Once we know those, we can then solve for in-plane principal strains, the principal planes and the max in-plane shear strain. So here's an example. I have a strain rosette. It's going to be a 45-degree strain rosette placed on the surface of a critical point of an engineering part and we measured the following Ea, Ev, and Ec normal strains. And we want to determine the in-plane stresses in this module and then next module we'll go ahead and use more circle to determine the principal planes and maximum shear in-plane shear strength. Okay, so to start off with, for a 45 degree rosette, theta sub a is going to be 45, I am sorry theta sub a is 0 degrees, because it's aligned with the x-axis. Theta sub b is going to be 45 degrees and theta sub c is going to be 90 degrees. Similarly, if you had a 60 degree strain rosette, you'd have 0 degrees, 60 degrees and 120 degrees. Okay, so let's now go ahead and calculate, we know epsilon a, epsilon b and epsilon c. Let's go ahead and use the equation. So let's start with epsilon a. Epsilon a is measured to be 350 and its mu, millimeters per millimeters if its dimensionless. I'm going to leave the mu's off just as I calculate just a shorten my calculations but put it back on in my final answer. So epsilon sub a is 350 mu equals epsilon sub x which is unknown. Cosine squared epsilons of a we said was 0 degrees + epsilons of y sine squared 0 degrees + gamma xy sine 0 degrees cosine 0 degrees. And we know that cosine of 0 degrees is 1, so that's squared is 1. Sine squared is 0. Sine squared of 0 degrees is 0. And this should be 0 degrees, not theta. And sine of 0 is 0 degrees. Or sine of 0 degrees is 0. And so what we end up with is epsilon sub x. The in-plane normal strain in the x direction is 350 mu millimeters per millimeter, and so that's our one answer. Let's do the same thing now for epsilon sub c and so epsilon sub c is measured as 600 mu. That's equal to epsilon sub x times cosine squared 90 degrees now for epsilon sub c + epsilon sub y times sine squared 90 degrees + gamma xy sine of 90 degrees cosine of 90 degrees. And with that, we know that cosine squared or cosine of 90 degrees is 0. So cosine squared of 90 degrees is 0. Even though epsilon sub x has a value that term zeros out. Sine of 90 degrees is 1. So that squared is 1. And again, cosine of 90 degrees is 0. So this term zeros out. And we end with now epsilons of y equals 600 mu millimeters per millimeter. And that's our normal strain in the y direction in-plane. Finally, we'll use the epsilon b, the third equation, and so we have epsilon sub b is measured to be 400 mu, and 400 mu is equal to, I guess I was going to leave the mus off so let's erase that off and we'll put it in the answers. But we have 400 times epsilon x cosine squared 45 degrees and + epsilon y sine squared 45 degrees + gamma xy sine of 45 degrees cosine of 45 degrees. And we have epsilon sub x was 350. We've already measured that above or calculated that above. Sine squared of 45 degrees is 0.5. Epsilon sub y is 600. And sine squared of 45. You know what, this should be and I'll just correct this here. This should actually be a cosine, right? Cosine squared, 45 degrees for, we're using the b equation. But that still comes out to be 0.5, so there's no change there. Sine squared of 45 is also 0.5. And then sine of 45 times cosine of 45 is also 0.5. And now if you calculate the only unknown left in this equation is gamma xy and so gamma xy ends up equaling -150 mu and that's a radians because it's an angle. Dimension is radians and so that's our other answer. So we've gone ahead and calculated the end plane and this is not stresses, this should be strains. And that's it. We'll carry on with the second part, part B, next time.
