 Any questions, remember what we were putting together? Now these two things that what we've done in the week before with the normal stresses on beams, doesn't matter what the loading is, we almost always put the loading down, but it doesn't need to be, it's whatever the loading is. We just tend to do it this way because one that's more natural way for support beams to be, that's certainly what almost all the beams in this building are doing, they're loaded from above, and so they will bend in the way we've been looking at. But we've been looking the two weeks before this, we looked at the normal stresses in the x direction, using that. And yesterday we added to the fact that since we've got shear stresses across the beam that that even causes shear stresses down the length of the beam, we started looking at that yesterday. And that was a direct factor of that shear, that's what that b is. But then very much like the normal stresses, everything else is really a consequence of the geometry. That's what certainly the y is, it's the distance up or down the beam from the neutral axis. Q is, well we'll review that in just a second. And then both of them have to do with the moment of inertia of the cross section, whatever its shape, that's what the i is. It's the same i for both of these. And then remember what T was? The cross sectional width at the point of intersection. And I want to just review that for a little bit. We're looking at various cross sections here now that we're looking at these shear stresses because what it gives us is an idea of what kind of shear stresses there are that are gonna try to take this beam apart, mostly at these places of intersection. If we were building a simple plate i-beam like this, you would stack up these three plates and you weld it at those junctions right there in the hope that that would hold that beam together as it bends because if those were not, if we weren't careful enough, then the beam would actually split apart at those different places. I guess it wouldn't look like that, it would get longer at those places, it would get shorter. But the point being that if we don't carefully weld it along there, then we lose all of the structural integrity of having those three things together. All we're getting then is some additional bending support just because I goes up here, but we're not getting any additional support from avoiding the shear stresses or reducing the shear stresses by attaching these different plates together so that we can withstand that shear. So we look at what are the shear stresses at these interfaces. There's a little bit of welding buildup, typically that's not counted in the calculation itself, it's a little difficult to know just what that is. Plus what this queue of concern is is the area above the area from the neutral axis beyond the neutral axis above that interface. So for these two interfaces, the queue is calculated on that area beyond those interfaces, beyond where we're calculating those shear stresses. There's lots of different types of beams. It's not uncommon to take an I-beam, which actually looks something like that in profile. They're a little bit rounded because they're either molded or extruded. It's not uncommon to take something like that and then add another beam to it for extra strength and then maybe even rivet through there. Add strength to it, maybe do that on the bottom, maybe not, it depends upon where that beam is, just what is needed, what you need it to do. But then again, the interface here, where we do this calculation of the shear stresses is right there and we take the area beyond that, which in this case is that entire channel cross section, even including those little lips down there, but it's everything beyond the interface above the neutral axis, where we do this calculation of this queue. Remember, that's the first moment of area of the area beyond the interface. So it's pretty obvious with these ones, it may not be as obvious with this one that you just also include these little lips over. That's usually pretty easy if these are stock and standard channel beams that you've riveted on there, because the manufacturer will give you the moment of inertia of those areas and the queue and T, the thickness, which in this case is the interface, thickness of the interface itself. What's maybe not so obvious is another type of built up beam and we'll do this one out of wood. Maybe make a beam that's perhaps a box beam. Could be just a beam like that, but usually it's finished off where we have wood as all these pieces, maybe plywood on these outer pieces. It might be somewhat decorative, might not, is that a hand up Jake? By symmetry we were in where the neutral axis is. So now the fasteners we put in there, maybe screws or nails and or glue, the interfaces of concern are going to be then these ones here and we want to figure out what the shear stresses is along those planes, make sure that we put in enough nails and or big enough nails or screws to withstand those shear stresses. Then the queue calculation is based upon these areas here. The areas enclosed by our, so that one's not quite as obvious what areas used for queue as the other two examples are. So we'll do a couple problems work through those, one that kind of get us warmed up and going, another one to actually look at that type of thing that we have there. So we'll take a sort of a semi-I-beam shape, at least non-symmetric top-to-bottom we'll use. It's very simply loaded. We've gone through lots and lots of that kind of stuff figuring out what the shear in the moment are and the reactions all that, we still need that kind of stuff. So we'll put two 1.5 kilonewton loads there. So it's maybe something like maybe a grand piano or something you're trying to support. So it's on its two legs at that point. Symmetrically loaded just to keep things simple so we can focus on the new stuff, 24 meters from each end and two meters between the two. And we wanna find out what the average shear stress is. So that's the type of thing we're looking at before. The cross-section we'll use for this is a little bit different kind of like an I-beam. Just the lower flange is a little bit narrower. For whatever reason. With all dimensions here in millimeters. So 100 there, 20 there, 80. Height on that web, 60 and 20 down here. Remember the purpose or the benefit of these I-beam type things is this greater area that's farther away from the neutral axis increases the moment of inertia. That increase, if we can increase the moment of inertia since it's on the bottom that decreases the stresses that we've been calculating both normal end and shear stresses that we've been calculating the last couple weeks here. So I want to find shear stress at two places. Both here, we'll call that a reference and here we'll call that a B. Wanna find the shear stress in both of those places. Just to speed things up a little bit I'll give you the location of the neutral axis from the bottom it's 68.3 millimeters. Remember though you'll be expected to find those type of things but we've done that for several weeks so I don't want to labor that point now when there's some other things we can look at. And the moment of inertia is 8.63 times 10 to the minus six, not four, minus six meters to the fourth. So I'd like you to find the average shear stress. That's the same thing as this x, y we've seen. Find it at some place along here or click only on the other side. You should almost instantly know what the shear is across there. Remember we need to know what the shear load is. In the beam itself, these internal shear loads that's what causes the shear stresses and that's what we're trying to protect against. Yeah, both loads are. So you should almost instantly know what the reactions are. Should be able to picture the shear diagram in your head. It's maximum shear on these two outer parts. In fact, if somebody wants to volunteer to find the average shear stress in this middle part, it's, ah, you win. That's why you're my favorite student. Oh, see Bobby left so I could say that. Oh, it's on tape, darn it. Hi mom. By the way, this quantity vq over i itself is what's known as the shear flow. It has characteristics of it very much like fluid flow through a channel. I myself don't see that comparison. I've looked at it through all the books. Have a background in fluid mechanics. I don't see how that looks like a flow but these mechanics guys who do this stuff have meant they do. So just putting into practice some of the things we saw yesterday for two different spots in the beam you can come up with is what the shear is and I hope you're all ready for that. And then you need q for the two different spots. You'll need a qa and a qv. That's all the difference between these two calculations. The surface between the two or of the cross section of Bob that was looking for q. T is this little distance here where the shear stress is happening. That is this thickness here which is the same as the other ones is 20. That's what we use for the t. So this is mostly just a practice in calculating q. I'm gonna make sure you know what you're doing with each of those. Maybe check that with neighbors and everything else is the same for both calculations be i and t. Already done Frank? It's just coffee from the students early for that. This professor just walked by some coffee. You could knock her down and take it. What other? Well, think about it. You should, you're getting enough experience now that you can almost instantly come up with the shear diagram of this. So does that answer your question then? I don't think so. What fun do we do here? Well, let's, let's, Is it various? Well, yeah, sort of the shear diagram. What are the reactions? Since it's symmetrically loaded, you know that instantly, you know that there's gonna be a reaction up of 1.5 kilonewtons. You know from our experience that that gives us the first point on the shear diagram. And the slope of the shear diagram is the load curve, which is zero. So the slope is zero, your shear diagram. Now your question is answered, isn't it? Yeah, definitely. Sorry. Oh no, that's okay. Will Bobby can tolerate you in class? I can. In class, there's always a student somewhere holding back, right Bob? If it worked, there'd be another employee holding you back. You get married. There's another adult in the household who's holding you back. One kid, at least one of them will hold you back. It's just life, and you gotta deal with it. Deal with anybody? Jake, what's going on? Bang QV the same? No, remember Q is Y bar A. Hey, and it's different for both of those. Each of those products, parts of the product is different for each of the pieces. You know we had to do what? How did you pursue it? You know there's a, well, you'd have to get up early, but there, we have a coffee pot over there. Just not, nobody's done anything when it's over there. If you guys wanna make a pot off, you're welcome to. You pay for it, you clean it. Don't expect us to be upset. Sorry? Just add the Q's after we put it to the. Add them to each other? Yeah, we do the whole thing. No. Each one of these that individually goes into its own calculation for the shear stress there. V and I don't change, but Q and T change for, oh no, T doesn't change for these, not this particular beam. Same thickness in both places. So all you need to really calculate, all you need to check with each other is the Q, hopefully the rest of it's kind of trivial. Each of them have their own distance from the videos themselves, it's all out of focus. Are you feeling okay to sleepy? When'd you go to bed? Ever? Last night? Last night. Late? Well, late for an adult like me and very late for a young punk like you. Two? Two AM? What were you doing until two AM? Anything that's been going to bed? What? Homework for me? All right, there's some kind of thing on Angel when you have to log in every half hour so I know how late I'm keeping you up. And I can get up in the morning and check the logs. Got their life screwed up. What's up? My only suckers. So check the Qs, how we doing on those? That's the only thing you need to check. I think everything else is pretty much done. How come you got them? You agreed? Perfectly? Jake, what'd you have for QA? Let me check, let me check. Yeah, no, yeah, 83. And then for QB, what's somebody have? 10 to the minus sixth, stick in and calculate. And for the lower part, all now that we're warmed up in penetrable fog. So here's the beam du jour. Obviously made out of wood. Why else would I bring out the crown chalk? Twice at once, that's what we always say. Millimeters. Inside one is 250 by 30. The shear expected at this point and the spacing between the nails. We're going to have to go back to yesterday and bring that back, that's 100 millimeters. Each nail diameter is two millimeters calculated yesterday. What to do on this one, because I'm not going to give you I, this is both of which you need, that's the delta X, H. Delta X is given, that's the spacing between the nails. And so we only need this delta H. This one, we don't have to need that little T, which is okay, remember, that's the distance across the thickness of the beam. And this is a different shape being put together in different ways. If you're being crossing that little mistake earlier, 25 millimeters from the top edge. What about, what is it? You do a Q without that. As long as you have Y bar, okay. You can do a Q, but you come in a little bit late when you went out first. You have I now, all I need is Q. And then shade it more. Shading portions to the Q. There it is. There's one right there. It's 379. And Q now. The area. So Q would be the next thing to check. We were something just to keep it separate from the first Y bar we calculated. Once you know what the neutral axis is, then everything's measured from there. You found where the neutral axis is. You don't use any measurements from the bottom. And every measurement then is from the neutral axis. Once you know where the neutral axis is, that's your reference point. When you choose to do a reference from the top or the bottom, that's perfectly fine, your choice. Once you know where the neutral axis is, then everything's measured from there. Do you think that's like the d value? And that's how it's measured? Yeah, kind of. It's the distance to the centroid of your Q area. Or no, that's not what we called it. Remember that big script A. The centroid of that area is from the distance from the neutral axis, which is like d. But you don't do the parallel axis there on this. But it is the same thing you would have used. Same number. Got it? Final numbers? You're forgetting. The thing you're forgetting is the total shear. But there's nails on each side. So you have to divide that in half because each side takes one half of that. And at the end of time, we're gonna check, let me put up some of the intermediate numbers and make sure you got them. Over two sides, 97.2 megapascades, quite get to that.