 All right, we're rolling. OK, so what we're going to add to it today. Well, there's something we've kind of glossed over a little bit. It's this idea that when we stress an object, the stress is the same throughout. It's as if everything we looked at had a rigid cap on it. And then we stress it. And anywhere in between that we might happen to look, that stress we've looked at as if it's uniformly distributed. In fact, what we're actually saying is that that force is the same throughout. So we just calculated at any point an average stress as simply the force being exerted, distributed over the cross-sectional area. What really happens, or at least a more accurate picture, is that there isn't this virtual rigid cap so that when we do stress things with some low p, the object that will actually deform maybe something sort of like that as it absorbs this force being pushed into it. The sides will curve in a little bit. The inside will deform a lot. Of course, this is greatly exaggerated. It's easiest to see this stuff if you don't imagine that our material here is steel. But if you imagine it's rubber. And if you pushed on rubber with two point forces, it would do something like that. If we had some kind of grid on here that we could watch deform, it would be greatly deformed as the material is absorbing this force. So now if we look at the stress distribution, we see that as we go into the material, say that this is a width v, if we only go one quarter of that distance into the material. So if the material is 10 centimeters wide, we're now talking about only being two and a half centimeters into the material. What we'd originally seen in our sort of naivete through this class, this average stress, is really something much more like this, where before we had this average stress, but what we really see is some much, much higher peak stress because of this true point deformation of the material. If we go a little bit farther into the material, say one half v now, or remember v is the width of the material. So that was 10 centimeters. We're now five centimeters into the material with the same point load. We now see our average stress that we've been using, but just sort of assuming that we've had overall, is now not quite as severe, but still quite a bit more than the average stress. The peak stress has decreased from what it was earlier, but it's still quite a bit over the average stress that we've been calculating as we go along, the simple p over a, because of this point loading. And it's not really until we're approximately a depth v into the material that's equivalent to the width v. We need to be about one width worth into the material before we really start to see that our average stress calculation is not so bad. There's just not that much difference now between the peak stress and the average stress. And we find that this occurs not only in uniform cross-sectional area pieces, like I've got here where our point load. This also occurs any time there's a change in cross-sectional area, which are times like where we might have a hole in the material. For instance, there's a bolt fastened there, or it could be even something's attached there and needs to pull on that. We find that we get this growing stress distribution that evens out. But then once we reach this cross-sectional area, all that stress is now funneled through a very small area. And we need some time for that to even out again. There's a lot of stress. If you want to look at it, you can see that maybe there's lines of equal stress. And then they need to get funneled through these very narrow areas now causing, well, in fact, what we call it is a stress concentration. Because there is this necking, this narrowing of the cross-sectional area to absorb the force, we have the stress being concentrated. It also happens when there's a change in area, as in just simple narrowing of the piece. And in fact, you've seen very often the direct attempts to address this. If these corners are sharp, then the stress concentration at those corners is very great. So what helps a lot is to ease the transition into this narrower area. And you see this as fillets. This allows that stress concentration to occur much less abruptly. We don't have this piling up of these stress lines, if you will, at these corners. And in fact, if you want to test this yourself, it's very easy to do. Take a piece of paper, two pieces of paper, cut one like that, cut one like that, pull on the piece of paper, and it's going to tear very easily right there. You pull on it this other piece, and it's not going to tear very easily at that corner, because you reduce the stress concentration at this corner and allow the paper, more of the paper, to absorb the forces that are being put through it. So we need to know how to model these stress concentrations so that we can take them into account, the fact that the material is somewhat weakened by either these holes or these change in areas, these necking down areas, but not so much that we can't handle it. So we define the stress concentration factor as something like this. Given the letter capital K that stands for concentration, I guess, defined as the peak stress scene, remember that was the high point of that distribution, over the average stress that we've been calculating all along here. So very near where the force was applied, and then maybe it was a little early to erase that, very early close into where either the force is now applied or where we have these changes in area, we have the peak concentration, or the average, that we've been calculating all along. But now we know that there's some peak concentration that we need to address, and we can do it with this stress concentration factor. Perhaps it's easiest to show it in an example. We'll use this type of example of affiliated material. It's going to depend upon a couple factors, but it needs help. Jake will draw a little picture for him. He's doing pretty good at the technical free hand sketching. We don't need to use vanishing points, Jake, because it's just a small piece, we're looking at. For this type of change in area, it depends upon a couple of the parameters here. The width of the full piece before it next down, the width of the piece after the change in area, the radius of the fillet itself, and that can be either machined in like that, or it could be a weld. Either one will help distribute that stress, relieve some of the stress concentrations, and then the last of the factors on this is the thickness of the piece itself. We need that for the cross-sectional area that's absorbing this piece. So we can take into account how severe the change in area is, how broad the transition to that change in area is with the fillet. The fillets can be very small, very tight fillet, with only a little tiny bit of rounding, or they can be much more generous. The more generous rounding will absorb the stress, change is better, but it uses more material. If you're welding, it's tough to build up a lot of material with a weld sometimes, because you've got to bring in that extra material with the welding done itself. And it takes more time, which means you've got the material heated for longer, which can change its characteristics, all kinds of things go on with welding itself as a full other study. What's nice is all of this stuff has been done ahead of time, taking into account all these different changes in area, w down to h, what radius is put in there, what thickness is put in there. All of this has been done ahead of time. So all we have to do is read it off of the chart. This is a slightly different one than is in your book. Your book is figure 524. If you have your book, you can pop open to that. It's still pretty much the same thing. I just think this one's a little bit easier to read. Sorry, 424. That's on page 161 of the new edition. I'm not sure where it is in the other, the older edition. Here, go ahead and take a look at it. Pal up here with somebody you like. Go ahead. Come on. Frank, you've got to meet somebody by the end of the class anyway. Mine's starting out for spring break. Maybe there's somebody who's driving down at Daytona Beach, they'll take it. Look at that hat. Good man. Good man. Miss Kajiniali, right there. Oh, thank you. Notice that the labeling is a little bit different than our book. I got this from a different book. I think it's just a little bit clearer to read because there aren't quite as many lines on the graph. But the graphs are still read the same way, still basically the same thing. So we'll run through it real quick with a quick example. We'll call this fillet 10 millimeters. There's a height W back here of a major piece, 60. It nets down to 40 millimeters just to put some numbers on it and has a thickness of 10. And we want to do something like find the maximum force that we can apply to this axially. Look at the loading we have here. This is an axial loading. We have been looking at torsion for the last couple days. We're going to come back to that in a second and do the very same type of thing. We need to find the maximum allowable force for an allowable stress of, let's say, 165 mega-pascals. To do that, what we've done before is we took that allowable stress to be our average stress and just multiplied at times the area. What we have to do now, however, is take into account that that average stress is much smaller than is the peak stress by this factor of k. So I've just taken our definition of k and zipped it over there, replaced it with the average stress with that times the cross-sectional area. So that will allow us to find the peak stress, where this peak stress, now that's where we'll put in the material allowable stress. Before we put the average stress there, just assuming that the stress was the same everywhere. Also, we have two areas we're working with now. We have the cross-sectional area back here in the fat part and a cross-sectional area back here in the narrow part. We use the narrow area as our calculation. That makes all of this a little bit more conservative. k is always greater than 1, as you can see in the chart. It doesn't even go below 1. Would make sense that the peak stress was less than the average stress. So we reduce our allowable force by the stress concentration factor. We're going to pull off the chart in a second. We also use the smallest of the areas. So that gives us the smallest p allowable for all of the available calculations. So we just read it right off the chart, remembering that the chart uses a w here and an h here. It's just a choice of the author, what designation they use. But the deal is the same. Across the x-axis, we do the fillet radius over the width of the narrow part. So our chart says r over d. Your chart in the book will say r over h, just an arbitrary choice in letters. So we have what, 10 millimeters for the radius of the fillet, over 40 millimeters for the narrow width 0.25 on the chart, which is right here at the little endpoint there, almost at the end of the x-axis. And then we go up to the line that gives us the same major width to the minor width, which this chart shows big d over little d. Your book will show w over h, right? For those different lines coming across. And for our case, that's what? 60 millimeters over 40 millimeters, which is 2 and 1 half. Sorry, 1 and a half. Yes, Bob, I'm getting old. Thank you, it wasn't Taser, guess what I mean. So we go to the 0.25 ratio of the radius of the fillet to the minor width, up to the 1.5 line, which is that third line up from the bottom, and see where it crosses the chart. Looks like what? About 1.4, oh, sorry, wrong line. Fourth line up, just eyeball it, 1.63 or so. That look about right off that chart, 0.25 up to the 1.5 line. Where those two cross, we then go over and get our stress back. That actually looks a little more, maybe 1.66. Remember, we put a factor in safety in all of this anyway. So now we can finish and figure out what the allowable force will be to stay within the stress of the material, ultimate stress of the material, and using the stress concentration factor. So 165 times 10 to the, let's see, megapascals, 10 to the 6th for meter squared over the 1.66, and then times the cross-sectional area of the small section, which is 400 millimeters by 10 millimeters, so 400 times 10 to the, let's see, a millimeter is 10 to the minus 3, but it's squared, so it's 10 to the minus 6th millimeter squared. Just do that calculation, sorry, that's not a millimeter squared there, that's meters. Put in a 10 to the 6th. You missed that, Bob. You must be getting old. Before this, we would have simply calculated the stress over the area where we'd use our allowable limit, 165 times that area, or maybe even the bigger area, since we hadn't done this type of thing before, but typically we look in the most endangered part of the material, which would be the narrower one. What's that come out to be? 165 by 400, wake up, TJ? 40, it's less than this? Be greater than this, because we divide by the 166 unless this number is wrong. I don't think so. Tell me what here. Okay, so 66 kilometers, 50 days getting old. That's what we would have done before, putting in much more force than we really should have once we allow for the fact that there's stress concentrations specifically at these corners. So that kind of calculation we can no longer do, it's just not conservative enough, we need to be conservative to make sure that these materials don't fail. And you can see the type of things you'd wanna do to reduce the stress concentration factor, the smaller that is, the greater the load you can put in, you can do things like a bigger, radius, less severe necking down of the material, these kind of things you might think just through common sense anyway if you had any. But you're here to accumulate that. That's what comes with getting old, all right? There's other possibilities in there. We also have the same types of charts when there's a hole in the material and as you can imagine it has to do with the width across there, the width there, the diameter of the hole itself, and again, the thickness of the material. Those kind of charts are in there as well. There's also charts for tubular materials under axial stress, also charts for this type of thing. I don't think our book has them where there are bending moments on the piece itself. This is the type of thing if you had that, one of the simple types of beams we looked at where there's some kind of intermediate load cause it to bend like that, that's what a bending moment does and that's what we're gonna look at starting Friday, I believe. We also have stress concentration factors for torsion loads like the type we've been looking at for the last couple days. So we'll look at those now. Depends upon a couple things you can imagine, not too bad, don't you? Depends upon a couple things. The major diameter back here, the minor diameter that we come down to and the radius of the fillet and then of course the load applied to the piece in torsion as it is. So we have the same type of concentration factors that we had with the axial stress only now we'll be looking at the shear stress. Remember it was shear stress that was our concern in torsion loaded to the tune of Tc over dron and blank. J, not getting old and older, thank you. This is all we would have done before. We would have figured out the maximum stress at the outer radius. Remember where we determined that it was greater? But because of this change in area, we have stress concentrations there. So we need to calculate a new peak stress where we take the original calculation we would have made, which is to assume a uniform stress distribution. Oh, sorry, not normal stress, shear stress. And again, this is always greater than one and so we can now calculate an allowable stress, women. So we'll do a quick example with just this type of thing here. So this is called a step shaft. Let's say 900 RPM. That's typically what drive shafts do, they operate at some power level, transmitting power. We'll set an allowable stress, shear stress of 8 KSI. The fat end of the shaft is seven and a half inches in diameter, the parallel end of the shaft is 3.75. So we'll take it as a half increase in area and we'll put a radius on the fillet of 9.16 of an inch. Good old English units. Find the maximum power that shaft can transmit without violating the allowable stress of the shear stress of the material. So first thing you probably need to do is remind ourselves the relationship between power, RPM and the torque that can be transmitted. It's a combination of the speed with which it's running. Two pi has to do with, that actually comes from the conversion of radians because the units on this F for frequency is revolutions per second or what people have taken physics to should know as a Hertz. Right, that you don't look familiar to most people? Well, you know what a Hertz donut is, don't you? Pow, Hertz donut. There's lots of stuff that needs editing and the classes where you're here, Bob. Bob Phillips, student number 500, something, something, I'll look it up. Times the torque that's being applied. So we need to determine what's the maximum torque we can apply staying within this stress concentration factor in the allowable shear limits. Once we find what that is, then we can figure out what the maximum power is when this is running with that torque at 900 RPM. So most of it's plug and chug, however we need to know where to get that K and that comes from another picture in the book. This one happens to be on page figure 532 in the book, which is our section on the fork. I think it's section 58, if I remember. Yeah, section 58, wherever that is, so if you have the page for the new and the old. Slightly different picture than is in the book, just because I think this one's a lot easier to read when it's put up. The author of our book had, I think, more grid lines. They were a little bit darker and had more of these data lines on there. It just makes it harder to read, so. Either one is the same, they're the same chart. This one's just, I think, lightened up a little bit. So again, across the bottom, we need the radius of the fillet over the minor area. Sorry, the minor diameter. Yep, it is diameter, always check that. So we have 9-16ths of an inch, watch your units, because these ratios are unit lists, which they wouldn't be if we had feet and inches mixed up. So a lovely mix of fractions and decimals as we go in the English system. So whatever 9-16ths is divided by 3.75, that's what? 0.15. We're going to be midway between, of course, 0.1 and 0.2, so we're gonna be somewhere on this line wherever one of those three lines crosses. And those three lines are big diameter over the little diameter, 7.5 over 3.75, which is two. So we go 1.5 up to the two line, which is the top one, go over, looks like we're at 1.33 or something. Just eyeball it a little bit. If anything, be conservative with it. All right, now you can finish the calculation. So you do that, take over a little bit. However, I do want the power and horsepower since we're dealing in English units, that's the typical unit of power. A sort of a sub-unit might be pound inches per second. I need to lay down for a nap. Been working hard already this morning. Power, geritol drink is wearing off. I'm not springing. We're using the smallest c and smaller j. Yes, yeah, reduce that if you want. 1.5, c cubed, because remember, j and c are simply geometries of the cross-section. So if you put all those in there together, they do reduce to that, and you do that for the small shaft. Gerobor C is just a geometric factor. Some books even put it together to get another factor, but this is more stuff to remember. All right, so we've got K off the chart. That'll allow you to find the torque load that can be applied, and then torque running at a certain speed is power. Same chart as that in the book. More real. Use yourself, Brandon, ask tomorrow's calculator. TJ, don't give it to him until you know at least his name. Not that guy. Second, you got the same T here before you go into the power calculation. If you don't agree, no sense during the power calculation. Always check these things as you go along. And watch your units. The key we use is for the small shaft. Do you guys agree? See, once you know the horsepower, then you can go down and talk to our Earl at A's hardware to get at our such and such horsepower motor. What did you get, Joel? What was your horsepower? What did you get for a tele-force meter? 662? I had 62 points, something. I had 662. I may have written it down or I'm, you know, I'm feeble down separately. I had 10.3. That's what I got. 62.3. Kid mentions. For the torque radius. 62.3. Yeah, we're having a factor of 10 over here. I do that. What'd you do? I'm perfect. That's what I was doing. Honor the owner. I do most of the work. I do the people that I work with. Right, what did you get? The exact amount. I got 60,000. What'd you get? I'm pretty consistent. She's crazy. She's crazy about it. That's why you made this one. What are the units there? So I did make that one. No, that's a good one. That's a good one. Yeah, okay. So that's 63,000. Okay. Okay. All right, then something falls apart after that because I don't have that in support. That's a lot of horsepower. 82 is a lot of horsepower for a shaft and a bay. I'm going to say this to you, whatever you're going to say. Yeah. And it's just 80,000? Yes. Yeah. Can you change the amount of those? Yeah, yeah. Can you change the volume? Oh, yeah, yeah. Good. Okay. Did you check your... You had the same torque I had, P.J., but you didn't have the same power and horsepower I had. So watch your units. Basically. Does our book have conversion factors, the horsepower, the table of conversion factors on here? Out of work a horse can do in one day. Bless the English system. God is American anyway, so. You know what I got? Is that a killer horsepower or a vanilla horsepower? I got 890. Would you get 10,000? Yes. I don't know. It's about here. We're just trying to get you... What do you have, Colin? Let's check some numbers. Do you have... Do you have J over C? Did you do that separately? Something happened on this paper, but he doesn't know who did it. J over C, I guess. We've got J over C. J over C. 10.35 inches Q. All right? Yes. All right? J, then T is simply the... Write it down separately. T is the allowable stress over K times J over C. Watch your units. I think most of us now agree on something like 62.3. Kip inches? Yes. No? Do you have that, Bob? Yes, sir. You have that? Yes. And for the frequency, you have revolutions per second. Sixty? I had 15. Power, then, 2 pi times the frequency. 15 revolutions per second times the torque. We give us units of Kip inches per second, but we want it in foot pounds per second to use that conversion factor. So there's 1000 pounds per inches, inches, Kips, Kips. We have foot pounds per second, but then the foot pounds will take out with horsepower. And now those units all cancel. We get foot horsepower. I mean, what? 2 pi, 15, 6.5, 6.5, 7,000. We're going to go. I told the I-50 to it in your head. 890? So I went. That is not going to be edited. That stays as 890. Earl will stock an 890 horsepower motor. He won't stock a 10,000 horsepower motor and not taking special order. Earl's the man at Hayes Hardware, but I don't know. All right. Well, I think you purposely did it away in the last few minutes here, just so I couldn't get out of the last question. The only thing that would have done was, as you go through these type of calculations, if you were a designer here, you might then decide, well, we need to change the radius so you can redo the problem with a greater radius, a final design radius I was going to give you is 15, 16ths of an inch. That leads to a greater allowable power, about an 11% increase, 985 horsepower. So if you want to check that, you can. That's what I would have given you now if you hadn't squandered away all the time doing simple algebra. What? You're still not okay? I'm still okay. Colin? Get in here.