 Let's talk a little bit today about stress states. So we've already talked about one dimension. One dimension limits the force combinations that we can possibly have. For example, if I apply a force pair, and our stress is in terms of a force pair in one dimension, I pretty much have tension or compression. And again, what we call axial stress. We add some possibilities when we move into two dimensions, where we can kind of take two things that would normally be collinear and make them not collinear and add the possibility of shear. For example, if we push up on one end of the bar and down on the other end of the bar, we end up with shear stress, which, as you can see, is sort of like axial stress, but shifted so it's no longer collinear. One's moved over to one side or another. We also added, in two dimensions, this idea of bending moment, where we can apply moment in two different locations. Or moment can also be considered as being the application of a series of one or more of these non-collinear forces to create internal stresses inside the different beam. Well, we can also think about, and we'll need to think about, shear states in three dimensions. States in three dimensions get much more complex, or at least they add some additional complexity here. If I can think about something and I take the moments, as we had over here, but instead of applying them in the same plane, we shift them so they're applied in a different plane. For example, we introduce one moment around this end of the beam, but we introduce a different moment around that end of the beam. Instead of introducing it in the same plane, this leads to a condition known as torsion. Here, for example, this would be bending. We've already seen this in bending. But if I take and introduce a twist in this direction, you'll get something, a stress state known as torsion. And you'll can see here that we actually end up with shear. If you remember our shear motion is this kind of sliding side to side. We end up with shear on the outside of the pool noodle here. We're going to have shear on the outside of the material. But that same amount of shear goes through the entire material, except it has sort of a neutral status to shear through the middle of the pool noodle. So things get a little bit more complex when we move in to three dimensional space. However, if we think like calculus about an infinitely small piece of material, we can simplify enormously by considering the direction. For example, in 2D, when we think about shear, here, for example, is a teeny tiny piece that we've zoomed in on so that we're very close. And it's getting to be infinitely small, which is one of the key ideas in calculus. And we apply shear to it. I'm going to take that teeny tiny piece and use a physical example here. Here's the teeny tiny piece. And if I push up, well, it's naturally doing itself. If I push up on one side and push down on the other side, you'll notice it tends to deform in a certain way. Well, that shear can be applied. The same action can be applied by instead of thinking about this as being two forces, one pushing up and one pushing down, instead I can think about it as being one force pushing out and one force pulling out in this direction or both of them pulling away. Tension along this diagonal. Or I can consider it as being compression along this diagonal and get the same type of response. In other words, we could simplify this idea of shear simply by rotating ourselves on this infinitely small piece. And instead of thinking about it being shear, we think about it as being a combination of some tension and some compression. So being able to rotate ourselves as we think about each of these infinitesimal pieces is really useful, because now we only have to think about tension and compression. And similarly, as we looked at with bending moment, when you think about bending moment, you can simplify it by not thinking about it as moment, but instead about thinking about it as a series of different small forces applied equal and opposite to each other. And if we get down to be infinitely small, instead of it thinking about it as being a variation of force, we can just think about it as being just different stresses at each level. Notice this also applies to three dimensions. If I now think about material instead of being a little square, but being a little teeny, tiny, three-dimensional cube, I might have some combination of forces acting on the faces and edges and side of that cube trying to deform it. But as it turns out, no matter what combination of forces I use, I can pick an appropriate orientation. And if I shrink small enough, I can actually think about this. Well, let me reorient it here. I can pick an appropriate orientation and reduce all of my forces to some combination of axial forces. We'll call this our stress in an xx direction, but we'll call it x prime, x prime. This will be our stress in the y direction, and this will be our stress in the z direction. But we'll use prime to represent directions that have been rotated so that we've reduced and gotten rid of all shear and all moments and can simply think about it as being three stresses. These stresses have a term. They're called principal stresses. So at any point in the material, there is a set of three principal stresses that are either in tension or compression that represent the stress state of the material. These principal stresses are all perpendicular to each other, and if any one of these stresses becomes too big in any one direction, the material will fail. And by failing, it means that the chemical bonds that hold the material together will tear apart or otherwise change or shift to permanently change the material. Notice this thinking is kind of like we think about vector components, that if you have a particular vector and you want to describe it in terms of components, well, if you describe it in components in one direction, there might be two components. If you choose a different basis to describe it in, one of the components becomes much shorter and one becomes much longer. And if you pick the appropriate basis, you end up simply describing it as one component in one dimension and a zero component in the other direction. Same idea with principal stresses. If we rotate ourselves in three dimensional space the appropriate way, we can think of all the stress as being tension or compression. Let's take a look at an example. Here I have a crayon. This also can work pretty well with chalk. Here I have a crayon. This can also work pretty well with chalk. If I take the crayon and subject it to a bending moment, so I'm going to push down on the ends of the crayon and up in the middle, it will eventually fail. It'll break apart. And when I do so, you'll notice that it pulled apart and it pulled apart at not exactly a 90 degree angle, but pretty close. And then once the 90 degree angle had started breaking, then actually it had bent a little bit and then it pulled apart at a 90 degree angle to what was bent. So there was initial sort of 90 degree angle here near the top and then that bent a little bit more. If instead I take a crayon and twist it, here I have the crayon and I'm going to apply torsion to the crayon by twisting it and spinning it. If I do that, when I break the crayon, if I look carefully, you'll notice I do not yet a nice 90 degree break. I get a bit of a spin to the crayon. You see there's sort of a point that's formed there? That angle that's there is actually a 45 degree angle because what's happening if you remember when I twisted this pool noodle, I created shear. Well, that shear was the same as tension along this line and compression along the other line. Well, the crayon in tension breaks along that 45 degree angle. So when I spin the crayon, it creates this new point. That's at roughly a 45 degree angle. Not perfectly because the material might not necessarily fail in all the same locations, but effectively you get a 45 degree angle. This demonstration also works particularly well with chalk. If you have a piece of sidewalk chalk and you attempt to twist it, you'll similarly get that point. So knowing this relationship between axial force, we can study our material response by considering, we can study our material response by considering axial loads to determine yield stress. In other words, we can take a long bar of material, kind of like our crayon, pull on both ends of the material until eventually it fails. It breaks apart either by snapping apart or by stretching and then snapping apart. And that will determine for us some sort of yield stress. And then once we've determined the yield stresses for particular materials, for our structure materials, then we build our designs to avoid concentrations of that stress. For example, if I had some sort of structure that was bolted in, there might be places where we determine that the stress is particularly high. And we try to make sure that the maximum stresses at those stress concentration locations is less than the yield stress. We may not necessarily know the directions of that stress, but as long as we're certain what the values are for the tension, for the tensile yield stress, and we can make estimates as to the maximum possible stress in whatever we design, then we can be reasonably assured that the design will not fail.