 Let's talk a little bit about internal forces. And we'll describe what we mean by comparing internal forces to the external forces that we've already discussed. So let's start by considering a 2D system loaded and in static equilibrium. So it doesn't have to have any particular shape. But I'll go ahead and create this system. And we'll assume there's some loads on it. And even though it's not necessarily drawn that way, we will assume that the loads are in static equilibrium. So for each of the external forces, for each of the loads and or reactions that are on this system, there must be an equal and opposite force from inside the body opposing the external force. And somehow, there must be some sort of, since there is material in between, there must be some sort of connections between these forces that are opposing the external forces at the surface. So we're going to slice through the body. We're going to cut through the body and consider each portion of this body in a separate equilibrium. Just consider the equilibrium for each piece of the body. So here's the left-hand side. And here's the right-hand side. And notice they still have the external forces, the loads and reactions applied. But in order to keep equilibrium in each of our two pieces, we have to think about what's happening at this sliced area. We have to think about the possibility of some new reactions that will be present in order to maintain static equilibrium. So on one side, let's think of three possible reactions we could have. We could have a normal reaction, in other words, a reaction that is perpendicular to the slice. We can have a tangential reaction, a reaction that is parallel to the slice. And we might, in addition, have some sort of moment, an imbalanced moment that must be represented by holding or preventing that piece of the slice from rotating. Well, if those reactions exist on one side of the slice, they must be paired with equal and opposite reactions on the other side. Here we have a normal force that's equal and opposite. We have a tangential force that's equal and opposite. And then we can also have a moment. In this case, if the moment is rotating in a clockwise direction, then the moment in the other case must rotate in a counterclockwise direction. And notice these paired forces are considering acting at the same location, the same position on that slice. At this line here and or about that point in the center of the slice. They're located at the same location. So that's sort of a key element about what helps us differentiate between external forces and internal forces. When we think about the internal forces, we're going to think about them as being force pairs, acting together in equal and opposite directions. Let's look at each of these internal force pairs and think of them as acting. Instead of acting on both sides of a slice, we'll actually put a little thin slice of material in between the two surfaces so we can think about how we're interacting on this material that's inside. So first of all, let's look at each kind of force. For our normal direction, we actually have a name, a term for this kind of force. We have our normal force pushing in one direction and our normal force opposing it. And we can go ahead and think about that as acting on this little piece in the middle. But we have a term for that. We're going to call this an axial force when it's internal. You can think about this as being along the axis where an axis is considered to be some line that's perpendicular to a plane. So if we sliced along the plane, the axis is perpendicular to that plane and often centered in the middle of whatever body is being defined by the plane. If we think about our tangential forces, our tangential forces, notice we have the tangential force applied in one direction, must be applied in the equal and opposite direction on the other side of the slice. If we think about a little piece that's stuck inside, it must be equal and opposite. And notice it's sort of having this effect on it of pushing up on one side and down on another side. You can actually experience this a little bit. If you take your hands together, you take your two hands and you press them together kind as if you were, I guess, in the position of prayer, sometimes how it's described. But if you put your two hands together and hold them and then you move one hand up and the other hand down, that's essentially setting up this kind of force system. And this is a term that we call shear. And the way you can think about this is a little bit like a scissors. In other words, for scissors are shears. Like a pair of scissors, a pair of scissors basically shears uses the idea of shearing to cut through something by pushing up on one side and pushing down on the other side and slicing across there. The letter we often use for shear is the letter v. The letter we use for axial forces is often the letter p. You can think of p as perhaps being perpendicular. The v, I'm not certain where that actually comes from. But maybe the idea of sort of that slicing motion looks a little bit like the point of the v. That might help you to remember it. And then our third sort of reaction is that moment. And we distinguish between the moment that we see here, this pair of moments. If one is bending in one direction, the other is bending in the other direction, we distinguish between those by calling that bending moment. Noticing that this kind of internal reaction, if allowed to move, would create a bend in our system. And if we think about that little piece that's in the middle, we can see how equal and opposite forces applied just to that piece would tend to make it squish on the bottom and stretch on the top, which is our conditions of bending. One of the things about a system like this, about thinking about this equal and opposite, is it gets confusing very quickly. It can get confusing very quickly because we don't know which direction we're talking about, since we're talking about both directions at the same time. So we need some sign conventions, some methods of talking about which direction we are talking about for these kind of internal forces. So here are our standard sign conventions. For axial forces, the sign convention, the positive sign convention, is we usually assume that tension is positive and compression is negative. So if we draw that, and again, including that small piece in between, it's usually useful to include that piece because we can think about how the piece is being interacted with. If the two sides are pulling on each other, they are pulling on that piece in between, and that piece of being between wants to be stretched, we consider that tension, and we usually consider that to be positive with most standard conventions. Compression, on the other hand, we consider to be negative, and we can visualize that, again, with this small piece in between, as being interactions that are pushing against each other or squishing that little piece in between, compressing it, wanting to make it smaller or thinner. Our convention for shear looks something like this. If we apply our shear forces, we're going to apply down on the left side and up on the right side. That is our positive convention for shear. Let me draw the negative convention here on the other side. The negative convention would be up on the left and down on the right. Well, if we choose that particular convention, let's look at how that affects the little piece in the middle. That little piece in the middle, if it's having equal and opposite reactions, in the case of positive shear, it's going to be rotating a tendency to rotate the material clockwise. So the material rotates, or is being tried. The material is wanting to rotate clockwise. I'm going to put the term inside the material here, because you'll notice if you think about the space in between the material, that looks like it's rotating counterclockwise. And that's where it can get confusing, is you have to think about how the material itself is rotating versus how it's rotating in between the two. So if we consider clockwise inside, that is positive. And then on the other side, if you think about the material being rotated counterclockwise inside, that is the negative convention for shear. And then finally, if we think about bending moment, there are two possible directions we can consider our bending moment. We can consider a rotation. Let's see, here is a counterclockwise rotation on the left, and a clockwise rotation on the right. That's one way that we can do it. And the other convention is to go pair them up so they go in the other direction. Now notice again, it's easy to confuse these two things, because they're pushed together on the same point. They're really acting on the same point. So we can sort of think about how this is acting on, again, that little piece that's inside. If that piece is being compressed on top and stretched on the bottom by those moments, tension on the bottom, that is considered a positive bending moment, as opposed to the opposite situation, where we have tension on the top and compression on the bottom. Notice the positive convention essentially corresponds to a situation where we're loading by pushing down on something, a normal weight loading, where we're pushing down on something is going to tend to create tension on the bottom and compression on the top. And that's sort of our positive convention here. So key point, internal forces can and do vary inside of the material. And we'll observe that in the next videos where we discuss shear and moment diagrams.