[MUSIC] >> Hi. This is module 35 of Mechanics and Materials part one. We're moving right along in the course. We just completed plane strain, Mohr's circle for plane strain, and now we're moving on to strain transformation and measuring strains. And so today's learning outcome is to calculate the in-plane strains based on experimental analysis techniques to measure that strain. I've talked about this earlier. Strains are often easier to measure than stress. So we're going to use what we call experimental analysis techniques to measure those strains. Strains are often measured, usually measured, on a free, unstressed surface of a member. And so, if you'll recall, a free, unstressed surface of a member is the condition called Plane Stress. And so here's the Plane Stress review from earlier in the course. With plane stress we have no out of plane stresses is in the z direction. And even though there is no stress in the z direction, there is, however, going to be a strain in the z direction. And it is also a principal strain. There are no shear strains on the z face since it is a free surface. And so here you see the shear strains on the z surface, and the normal strain. Now the shear strains are equal to 0. However, the normal strain does have a value. However, for small deformations you can show that the in-plane normal strains, the 2D normal strains, are not effected by the out-of-plane displacement, or the strain in the z direction. And the plane strain equations, and Mohr's circle for plane strain, are valid for the case of plane stress when the measurements are made on a free surface. And so, the very important result is that the transformation equations of plane strain apply to strains in plane stress. And also, the transformation equations of plane stress can be used for stresses in plane strain. So this is very convenient when we're analyzing problems using 2D modeling. And so, to measure strain we use electrical resistance strain gauges. And I'm going to show you one of these in a second. With a strain gage, the changes in the length of the strain gage are measured and attached to the specimen. And they are going to cause a change in electrical resistance that is measured. We calibrate these electrical resistance measurements so that it correlates with some known strain. And then once it's calibrated, any reading can be converted to strain on any object. And so, here again, we calibrate the electrical resistance measurements to measure strain. Strain gages can measure normal strain. However, we also want to get in plane shear strain. And so, shear strain can be calculated by measuring normal strains in three different directions and then using the strain transformation equations to calculate the shear strain. And so, what we're going to use are called strain gage rosettes, or what are used are called strain gage rosettes. Standard gages are 45 degrees and 60 degrees, and the positive gage orientation angle is always measured counterclockwise from the x axis. So, here is a couple of examples of strain gage rosettes. This one is 45 degrees on the left. So you start with one of the strain gages oriented along the x axis, and then you have two more 45 and 90 degrees counterclockwise from the x axis. Similarly, here's a standard 60 degree strain gage rosette. And now I want you to look at this beverage can where we've actually got a strain gage rosette, a 45 degree strain gage rosette, attached to the beverage can to measure strains. And you'll see that there's electrical wires hooked up to measure the electrical resistance changes. And so, a good practical example of how we measure strain. And so, here's our strain gage. Again, depending on which strain gage rosette orientation we use, we'll have an angle theta sub a, an angle theta sub b, and an angle theta sub c. You can recall now the normal strain transformation equations that we came up with in the previous modules. And so, what we can do is write three equations for the three strain measurements that we are able to get from our specimen in the A direction, the B direction, and the C direction. And so those are all measured. We then have three equations and three unknowns. The unknowns are the normal strain in the x and y direction, and the shear strain. And so we have three equations, three unknowns. We can solve for those in plane strains. And so, here again, the measured strains, three equations, three unknowns. And the nice part then is we can solve then for the in plane principal strains, principal planes, and the max in plane shear strain. And we'll do that, an actual example, in the next module. [MUSIC]