 Hi everyone. I want to talk a bit about rivets, welding, and bonding. These are joining methods, joining techniques that we won't spend a ton of time on, but they are worth mentioning a little bit. So rivets first, I'm not going to have any visuals for this, but rivets generally are useful because they're cheap. They ensure some level of, I don't want to say permanence, but difficulty to a disassemble, which sometimes is a design feature. You might not want someone in the field or a consumer taking something apart. So you might intentionally make it difficult to do so. They are also useful in that you can join two things that you only have access to one side. So if you have two overlapping materials and you want to join them, but you can't get access to the inside, only the outside rivets could do that for you. And they're generally relatively strong for what they do. We can usually analyze them as cylinders or hollow cylinders, depending on what the geometry actually is. But other than that, they're not terribly complicated. They may introduce some weakening into the materials because you have to pierce the thing that you're joining with a hole, but many, many fastening methods would, of course. Next, I want to talk about welding. So welding you're probably fairly familiar with. It can be as strong or stronger than the materials being welded. So that often isn't too much of a concern. You're making an assumption there that it's a good quality weld and appropriate materials were selected and such. One thing I will kind of mention that might not always be clear is when we're looking at welds, so say I've got a plate welded to another plate and a perpendicularly oriented, and I might be a little exaggerated, but I have a weld in here like this, a nice convex weld. Usually what I want to do when I analyze this this type of joint is I have a dashed line, which connects the two points here. And then I take, all right, we're just going to leave that line on there, I take and measure the distance here. And I call this the throat width or eventually the throat area. And it's going to be equal to this distance times the length of my weld. So the length in this case would be into and out of the screen. And the throat area that I want to that I want to know something about is this distance T, which is often going to be equal to 0.707h, where h is the side length our weld. And that side length is usually dictated by the thickness of the plates that we're welding together. So a good weld, you know, by whatever standard would have a certain side length. And that would be designed appropriately or we could design it actually, depending on our strength requirements. And then from that we can get this, this width here T, which is typically, which would be 0.707h. And that's just based on geometry, assuming both sides are the same h here and h at the bottom. You know, we can use geometry to get that. And then we call this our throat area. And this throat area then is what we use for our calculations of stress. So just kind of getting some terminology out of the way there, something to keep in mind. Now, depending on what kind of loading we're talking about, our welds may have multiple kinds of stress that they are subject to. So they can have bending and torsion. So we might apply our standard normal stress and torsional shear stress equations when we're analyzing our welds. You know, through superposition, we can have both happening at the same time. And we can, you know, put those things together when we actually do analysis. Now, one thing to think about is when we have multiple welds being put together, you know, we kind of have to consider those all as a group. But for ease of actually doing the calculations, we're probably going to be doing those separately first. And I'll give kind of an example of that. So suppose I had a weld, and I draw my best weld here, which isn't very good. I have a weld of an angular piece. So it has two sides to it, a top and a side. And those are welded to get those two welds go together, then to attach a plate to another plate or something like that. Now this as one whole object can be kind of difficult to do a calculation for with my stresses. So instead, I might, you know, kind of break these apart and treat them as two straight lines in order to analyze those stresses individually. And if I go ahead and look at an example of that. So suppose I've got my straight line weld as it's shown here. So these kind of gray circles represent the weld. And then to do my analysis, I need to be able to understand the geometry of this weld. And part of it is figuring out the moment of inertia because I need that for my stress calculations, both for my bending stress and for my torsional stress. So there's a couple things that are being talked about here. So this G prime, as it's labeled right here, is the center of gravity of this portion of the weld. So again, we're talking about a kind of a two sided well as I just drew. This G prime is the center of gravity of the one side. And of course, if this is just a straight line, basically we place that center of gravity right in the middle. That's relatively simple enough. Now, if we remember that I actually would have, you know, a second segment of weld across the top here that I'm not showing, that means that my center of gravity for those two as a whole is going to be up here. And when I do stress analysis, I need to be considering the center of gravity of the whole thing in order to do that analysis. And typically how I do that is I use something called the parallel axis theorem. So you may, you know, recall parallel axis theorem from previous coursework. But the general idea is, if we know the equation for the moment of inertia about a relatively simple piece of geometry, like in this case, we have a straight line of a certain, you know, length and a certain width, then we can translate that from its own center of gravity to some offset center of gravity using the parallel axis theorem. So for a rectangular shape like this, which has a width t and a length l, I can write my equations such that I'm calculating the moment of inertia and the standard or the generalized moment of inertia equation looks something like this. So I'm taking the moment of inertia about the x-axis and I'm measuring the distance between the x-axis and all points y, multiplying those times the area and summing those up. So for a straight line like this, I end up with something like this, 0 to l over 2. So because of symmetry, I'm just doing half and then I'm multiplying it by 2. So I'm going from 0 where x prime is 0 all the way up to l over 2 and plugging in y squared. So y, you know, as indicated on the drawing and my area then is just t dy. So I'm taking that differential piece of area and for each little differential piece of area, it's going to have a width t, a thickness differential y and if I evaluate this, I get l to the third t over 12. What about Iy prime? Well, we can calculate that but generally I'm actually going to say in this case, ly prime is equal to 0 and that's because t is relatively small. So because t is small, my contributions as I move away from the y-axis are very small. So I'm going to come up with a, I'd come up with a very small equation here and it's effectively negligible. Now, this is great. This is moments of inertia around that center of gravity, g prime. Now, if I want the moment of inertia around xy, which are centered at g, the center of gravity of the whole group, then I use the parallel axis theorem and that means I'm going to take x prime plus ab squared. So this is the contribution from the parallel axis theorem that I'm making use of and if I plug in what I know, I just calculated l to the third t by 12. My parallel axis theorem term is just the area l t b squared. So if I know where this new center of gravity g is, which is located by b, I can add that in. And the same thing in the y direction where I've taken into account the distance between those two axes in the x direction, which is a now. And I already said that this was zero and I have l t small a squared. Great. So I've figured out my equations for my moments of inertia about those two axes, about my center of gravity for the entire weld components put together. We might also need that polar moment of inertia when we're doing torsion. It's actually equal to the summation of these two. So I can take again that l third t over 12 and then these two parallel axis terms. I'm going to separate out the l t a squared plus b squared. And this should actually, you know, kind of make sense this portion right here, a squared plus b squared. Because really what it's giving you, if I kind of scroll up just a little bit again, what it's giving you is this distance here, right? So it's a Pythagorean theorem thing. a squared plus b squared square root would be this distance. So it's factoring in kind of that distance because we're rotating around that in a rotational pattern rather than bending. So great. We got these moments of inertia and then we would go ahead and use those in our standard stress equations that we already talked about. Now there is fatigue that we might need to consider in in welds. And when we have to consider fatigue, typically what we do for welds is we have a fatigue factor that we multiply our stress by in order to factor in fatigue. And really what this is doing again, like in regular fatigue that we've already talked about, it's factoring in material imperfections and things like that. So it's experimentally determined. We have values ranging from 1.2 to 2.7 depending on the type of weld. And it builds in those experimental failures as a multiplier on the stress. So it gives us an indication of how fatigue might play a role. The last thing I'm going to talk about that I won't spend much time on is adhesives and bonding. This is glues, tapes, those sorts of things. Lots of benefits to using adhesives. And if we're talking about industrial adhesives, they're going to provide strong bonding. They don't require any modification to our part. We don't have to drill holes usually and things like that. We might have to do some surface preparation, but that's usually a little bit less intensive. Though they are sometimes sensitive to that surface preparation, so we have to be careful with what we're doing. They do come with downsides as well, so sometimes there's environmental concerns. A lot of adhesives give off gas, different things that are dangerous to humans. The production methods might not be the cleanest depending on what it is. So there's some trade-offs there, of course, as there is with anything. All right, so I'll go ahead and stop there.