 So let's talk a little bit about shear stress and strain and how they differ from axial stress and strain. Although internal shear forces are generated by different condition than axial forces, for example, axial forces are generated from normal application of forces, in other words, perpendicular application of forces, whereas shear is generated from tangential application of forces, in other words, forces that are parallel to the surface that they're being applied to. And although they're generated by different conditions, shear stress and shear strain follow very similar rules to axial stress and axial strain. In fact, there are parallels in almost every aspect of these, at least when we consider the standard simplifications. So for example, if we apply a normal load to a bar, we can calculate something we've defined as being axial normal stress. There's our stress as being the force divided by a cross-sectional area, or this area we've identified is the area across the section of the bar. Well, similarly, if we instead apply tangential forces and create shear, and again, we've usually labeled shear as being v, we can use that same cross-sectional area and define something called the shear stress, usually denoted with tau, and that's the total force due to shear divided by the cross-sectional area. Notice, however, the cross-sectional area in this case is parallel to the application as opposed to being perpendicular to the application, as it is in the case of the normal or axial stress. Similarly, we defined axial strain by looking at the total length between the application of two forces and how much change there was, delta l. In this case, I'm going to call it delta x because we're going to distinguish it between changes in the y direction. But we defined our strain epsilon as being this change in length, which again I'm using as an x in this case, divided by the overall total length. Well, if we take the same type of bar, a long, thin bar, and we consider the application of a shear stress, we're going to get some sort of deformation. But the deformation isn't going to result in the lengthening of the bar. The deformation is going to result in sort of shifting the bar sort of in a parallelogram fashion. And this is greatly exaggerated, but we'll assume that it shifts it in some dimension. We'll say it's delta y here, this dimension down below. And again, we consider the original length of the beam. And we define a shear strain with the Greek letter gamma as the ratio between this change in the y dimension and the perpendicular dimension parallel to the application of the force divided by that original length. Notice this is essentially an angle for very small values of delta y. This is essentially equivalent to an angle. So we have seen that we have a stress and a strain. And we have equivalents of shear stress and shear strain. These are similarly related for relatively small loads. We know that there is a linear relationship if we plot a stress-strain curve for an axial load. We'll notice that there is a linear relationship such that the stress is equal to some modulus that we call the modulus of elasticity or Young's modulus times the strain. Well, we can consider a similar set of circumstances for shear. If we plot shear strain versus shear stress, we generally will also see the same type of linear behavior. Although the slope of that relationship will be a different value that will relate the shear stress to the shear strain. Typically, that value is represented with the letter g. And it's called the shear modulus. In addition, as we will also observe in axial loading, there is some point at which the material yields, sigma yield. We will also generally observe some point in shear where the material yields. And we can identify that as the yield stress for shear. Notice there's parallels between both of these situations. Here we have tau and gamma for shear. And above we have, and to the left, we have sigma and epsilon for axial loading. We're not going to look as carefully in this course at all of the different applications of shear. However, you should be able to recognize some conditions where shear does arise by looking for loadings that might be perpendicular to cross sections. Here's a couple of shear examples or conditions where we might observe shear. For example, if I attempt to sketch the links of a chain. And there is a cross section of the chain that I've tried to sort of sketch in there. And you might have some force pulling on the chain. We'll say there's an overall force p, which is distributed in two parts on either side of the link, p over 2, and must be countered by two parts on either side of the link pulling down. Well, in a case like that, you might be able to see where there's shear being exerted on each link because it's being applied perpendicular to the cross sectional area of the chain. And in that particular case, we would see how our shear stress would be equal to, well, in this case, it is p because it's axial there. But down here, we can roughly approximate that and say that the shear stress at that point is equal to p over 2. So if we put the v over a, it ends up being p divided by 2a for a problem like this. And most shear examples, we're going to have to be able to think in three-dimensional space to sort of see where the application of the area is perpendicular to the application of the force. Another example where it's common to see shear in action is the use of nails. If I take a large board and I want to nail it to another board, basically those boards are being kept together. If one is being loaded and tried to pull away from the other one, then the nail that is placed in there to hold the boards together will be experiencing some form of shear across the nail. And the area that we would be concerned about that's holding it together and experiencing that shear would be the cross-sectional area of the nail. Another interesting example, we might consider something like the head of a nail or a tack. Let's consider something that looks a bit like a nail or a tack, but let's assume that it's sort of square in shape and that it has something sort of extending from the bottom. I'm going to sort of draw where this would be extending through. So in other words, I have some sort of plate that might be something like the head of a nail. And inside that plate, there is some sort of rod or bar extending down from the bottom of the plate. Let me sort of finish sketching that in three dimensions, extending down through the bottom of the plate. And if I actually place some sort of force on the bottom of this piece that's being extended, notice that if we're trying to figure out, here's the force p, if we're trying to figure out the axial stress on this particular piece, we would see that that axial stress would be applied to this area that's the cross-section of that extended piece to this small area here. And if we decide to give this dimensions width times length for this little piece that's extending below, then that area of the length times width and the axial stress experience would be p divided by a or p divided by length times width. But notice if we sort of think about this as shear, we notice that p must be pulling down here. There must be a total amount of distributed force, but totaled up to be an equal and opposite force in the other direction, which we'll label with p. And that force is being applied on these little small faces, the place where that extended piece is being attached to the larger plate on top. And each of those faces has their own dimension. If this is my width and this is my length, we also in this case would care about the thickness of the plate, which represents the sort of height of that little piece. And in this case, the shear would be applied to a face that was parallel to the force. Here's the force p being applied. So the shear is going to act on this face here and on this face here and additionally on the two that are on the left side and in the back. So our total shear force, in a case like this, would be equal to the applied force. I'm sorry, our shear stress would be equal to the applied force divided by the combined areas. Two sides that have a thickness of t and a width of w plus two sides that have a thickness of t and a length of l. Notice, depending on the dimensions, the length, the width, or the thickness, it is likely that this value here, the shear stress, is going to be smaller. That will obviously have a large dependency upon the thickness to which it's attached. So again, we're not going to be putting a particular amount of detail into the analysis of shear in this course, but you should be able to recognize conditions under which shear arises.