 So, I may do that and then you can just download, you know, if you come in 20 minutes late, you can just download the first third, but I'm not sure. So, you know, this is the first time we've done this. So, we'll see, but they're there for you. They'll download faster on campus, I think, than at home. You know, those connections are generally better. All right, so let's see. No, no open questions we need to deal with first. All right, we've only got two things so far. Well, not in this order, reverse order, but the only two things we've got stress-intensive. We're working with some very, very simple, very straightforward ideas. We've looked at the possibility that an object of some length, say L, can be loaded in some way, whether it's compression or in tension, doesn't really matter. We'll work with tension today just because of what's gonna follow, but it doesn't matter, it can be done either way. So, we can load some objects and we've looked at the fact that depending upon what the cross-section is that's supporting this load, we get some idea of what the stress is then in the material. And that's just simply the ratio of the load that's being supported by that area. We also looked at the deformative response of this material to, for example, here a tension that there's the possibility that the material will elongate by some distance del, greatly exaggerated here, of course. Well, I guess it could be, you know, it's a pretty elastic material, pretty stretchy material like rubber or happens to be very thin or both, it could actually deform by that kind of margin. But then we looked at the strain in the material as the actual deformation. Itself, notice that that's independent of that cross-sectional area. It's also independent, at least that value itself is independent of the load that's being opposed on it. So, we're gonna get to the point today where we're gonna put these two together. And if I had $50,000, we'd actually do this, but I can't just buy $50,000 for a one-day demonstration. So, we'll just toss through the demonstration. It's a very straightforward one, so it's a very easy one to imagine, but it is kind of fun to do it. Well, I'll tell you why it's fun to do it when we get there. So, imagine we take some kind of engineering material and we'll look at a sample piece of this and the sample piece will look something like this. Top of the bottom, it's got some threaded portion and then it narrows down to a very standard test length in here where we have our constant cross-section. So, that will be our cross-sectional area A and we'll have in there a test-length L. Between this L or during this test-length L, it is a constant cross-section and the reason it nets down like that to this narrower region is so that that indeed is the test region. Everywhere else, there's a bigger cross-sectional area. So, we know that if the cross-sectional area gets bigger, the stress gets smaller. So, in this way, with this being the minimum cross-sectional region, we have the maximum stress in that region so that focuses the test on just that piece. Everything outside of that is going to be at a lower stress and so is not going to be the area of concern that way. Now we have a focus on this narrow cross-sectional region area and we have a test length there. So, we've essentially set, fixed these two values with that test piece. Then the threaded parts, we clamp into some machine and now we can use that machine to either start pulling the piece apart if we want to test it in tension or we can start compressing it if we want to test it in compression. And this is a very standard set size piece of material and can be ordered in all kinds of different engineering materials, different types of steel alloys, different types of aluminum, all kinds of different things. Anything we want to test, any material we want to test. And in fact, a manufacturer will do this for us if we're going to order some A36 structural steel I-beams then we trust that the manufacturer has taken some of that A36 steel and tested it for us and will publish for us the results we're about to look at now. So we're going to take this test piece, we can run it in compression order tension, we'll run it in tension just cause I've already pictured that. So the machine will put this piece under increasing tension. As it does so, it can monitor two things for us of course. As the force increases, we already know what the cross-sectional area is because that's standard for a test piece so it's something you can just, you can buy these test pieces, read off the cross-sectional area and enter it into the machine. So the machine will apply a force P, will have entered into it the value of the cross-sectional area from the test piece. So the machine will actually measure then for us the stress in the piece. Pretty convenient. Machine will put out a big graph for us. Also, the machine will already know what the length is cause we can type that in as part of the initiation of the machine, you know, you start with the machine. First thing it does is ask, I guess, for the area of the test piece and the length of the test section. It's usually very standard, maybe an inch or so. It depends on the size of the machine. And as the machine applies this load, it notices that it's two clamped jaws there will start getting farther apart. Well, that's what Dell is then. So the machine will be able to measure Dell, already knows what Dell is, it will calculate then as well as it does this test. The strain in the material, just very easy for a machine to know what the P is, doesn't know what the stress is, to know what the test length is, doesn't know what the strain is automatically. And so it'll put out for us a graph of stress on the x-axis, strain on the y, sorry, strain on the x-axis, stress on the y-axis. In a way, if you look at it, this is the load being put on the material, the stress is the load being put on the material, the strain is the material response to that load. Notice here in here is our load, in here is our response to that load, the actual deformation there. And so it's very easy then to put in those input values, turn on the machine and let it start applying some load, very little load at first, and then it gets greater and greater and greater, and we see what the material response to that is. This is where all these values in the back of the book are, we'll look at them in some detail in a second. So to start with, our material will take a lot of load, and not very much response yet. So we're just putting in some load, it's pretty easy to increase that load, and the material really doesn't respond yet very much. So we keep it going. And in fact, for quite some time, we find that this is a linear response. As we increase the load as determined in the stress for the machine, and it's output, there's just not that much response yet. That's no different than you grabbing a piece of the metal and trying to put some effort into it, and it's not gonna do very much. And in fact, as we're gonna find, I have to exaggerate this part of the graph just to get it on here. It's really extremely steep and almost right up along the y-axis. I have to exaggerate it here. We're gonna have to change scales as we go here because what happens later would compress this so much that we can't even tell the difference between it and the y-axis. Continues on for a good linear portion here where the stress is directly proportional to the strain as this load is being increased. Not very much strain response yet. The material's just not increasing that much. And we could run this test right here if I have to have it, about $50,000. However, right about now, most of you are feeling, what a great chance to take a coffee break. So we decide, let's turn off the machine and we'll come back in a little bit while we're all gonna go out for coffee. Who came in? Frank came in last so he's buying. So we turn off the machine. We don't instantly release the load. We let the load back off a little bit. As it does so, we notice it goes right back down this same line to a point where there's no load and no strain. No stress, no strain. We get right back down to the origin. That's kind of, that's very useful in that the material will go right up a linear that's always the best kind of relationship to come across in science and physics and engineering because it's so easy to understand. We go right up that, if we release the load, we come right back down and back down the very same line. We come back from our coffee break. Thanks Frank, appreciate that. Come back from our coffee break. Turn the machine back on, let the load start going back up. Start increasing this peak again. Start increasing this stress again. Notice what the response is. We go right back up the very same line. That's useful. It's extremely then predictable what the material's gonna do. Whatever this material is in here, maybe it's A36 structural steel, whatever it is, it's very predictable what it's going to do. That's great to us as engineers. That means that we can choose a particular material, put it under load, we know what it's going to do. We know what's gonna happen when that load is taken off. We know what's gonna happen when that load's put back on. We know what the material's going to do. That makes the engineering a lot easier to do. Within limits, within limits. We get up to some point finally where we start to lose the linear portion. We start to get the graph to curl over a little bit. Notice that that's a place of where the load's not increasing very much. So the stress isn't increasing very much. But the strain is increasing. Now the material is starting to give to that load. It's actually starting to stretch a little bit. Where we lose this linearity, we call that the yield stress. So we'll put a little sigma sub y here. That's where we start to get yield. That's a very important point to us as engineers. If we want to keep the predictable response here, we need to stay under that stress. If we want the material to yield predictably, lose any strain that got in it as we take off the load and go back up and down this line several times, then we need to stay below that yield stress. And if you look in your books, that's one of the items that essentially the middle column calls it the yield strength, sigma sub y gives it both tension, compression, and for shear. Because we can redo the test as a shear test. So you've got some pretty good values. Notice for all the steels, they're the same in compression as they are in tension. That's pretty good if we have a structural piece that's going to be stretched and then compressed, and then stretched and then compressed. It responds in the same way to either side. So that can be kind of useful. All right, since this is linear and predictable, we call this region the elastic region. The material is behaving exactly as a linear spring behaves. As we keep going a little bit, we start to get a curl over at least for certain types of materials. These are, we get a different curve from here on for different types of materials, for ductile materials. Those that have this kind of response at room temperatures, which are most of our structural steels are ductile materials. So let me let it go a little bit farther as we start to see the type of things it does. Bound in here, it tends to really increase the mechanical stretching of the material. So much so, it'll stretch so much so that it actually will reduce the load in here. It stretches so much that the machine now feels that there's quite a bit of stretching that's gone on, the jaws here are separated quite a bit. To the point it actually reduces some of the load here. Continue with the test. We see that it starts to come back up here. Any time in here, from this yield point beyond, if we turn off the machine or completely take off the load, we never get back to down here. We've permanently, permanently now stretch the material. We'll never get it back. You've all seen that happen with springs. You stretch them too far, they'll never go back to their original length and you ruin them. That's why we don't let students play with springs and physics last very much, because you ruin them. Get them out there, all the guys gotta take the springs and they start doing this Charles Atlas thing and seeing who can bust the springs and who can't. And then if we take it a bit farther, we actually can get up above the previous stress limits to some point then where we peak out a little bit and come down and finally the material actually ruptures. That's the exciting part of the experiment because it happens with an incredibly loud bang. Suddenly the material has separated here, physically separated and the jaws now are at full load, but there's nothing to hold them together anymore and they slam apart and makes a really loud bang. If you're standing next to the machine when it does that and the lab assistant didn't warn you it was gonna happen, you're gonna get ringing in your ears for the rest of your life. Thank you very much, lab assistant at Portland State University. If I knew your name, I'd sue you. Jake, yeah, hanged up? Like when the stress is increasing, like that second curve. In here? Yeah, is that because the area is getting smaller? It's part of it. That's what's really happening out in this region. We call it necking. It's where the material is actually physically thinning now and so the area is starting to drop as the load goes up. Obviously this point is important. This is the ultimate stress. It's the most it can withstand. That kind of point, generally we don't wanna do any engineering near that for our structures. We wanna stay way back here where everything's elastic and recoverable and predictable but there are times when you want a piece that you actually want it to fail because then it protects other more expensive components in the equipment so you do need to sometimes engineer out at this region. So that a small, cheap, easily replaced part will fail, protecting then all the more expensive or harder to get to equipment. It's the type of thing if you do any snow blowing, which most of us do, you've got those big augers in the front attached to an axle and then there's shear pins holding in that auger. If a rock gets caught in those augers, you want the shear pins to fail rather than have those augers jam hard to a stop, slam the engine to a stop and then the engine's rolling. Much nicer to lose a buck and a quarter on a shear pin than $500 on the engine of the snow blower. So it's that kind of idea that sometimes we do wanna engineer, actually engineer in that region. So sort of as a review in this region, we're actually saying what we call yield. This region is called strain hardening. Anytime if you release the load there, turn off the machine or unload the structure or whatever it is, you'll come back down to some level where now there's some built in strain to the material. Sometimes that's useful. We're not gonna look at that very much. We're gonna look very much. Spend most of our concern and time in this elastic region. Now to just give you some of the idea of the numbers of, by the way, ductile is a material that will rupture at room temperature and follows this kind of curve. So that's most of the structural steels that we'll be interested in as engineers. You may end up in some part of the industry where you've gotta worry a lot more about some of the other things going on. For the most part, we'll worry about just this elastic region. So let me just put some numbers on here to show you what we're talking about. These are numbers for a low carbon steel. A steel without very much carbon in the substructure. If you take a course called Material Science, you'll look at what makes an alloy, how they do it, and what's the response of the materialist to different alloying percentages and the like. So just to get some values in here for a common steel, this strain limit on the yield is something like 0012, and the yield strength is up around maybe 40. These values, by the way, KSI, kips per square inch, and then we don't necessarily need units on the strain. This value here where we kinda hit a low point is .02. So you can see the scale difficulty of trying to get all this on the same drawing. This is very, very small compared to this. And so it's difficult to get the whole thing on a graph that's useful, and we'll talk about that in a second. This value out here being something like .2, and then rupture it around .25. So that's a pretty good amount of stretching there. That's a quarter inch over one inch of material. And ultimate, maybe up here around 60. So those are some of the numbers we're getting. Our book, and because of that, we will not even deal with the rupture strength. It's only got listings for the yield strength, the end of the linear region, and this ultimate strength if you wanna get out there to the absolute maximum. Correct. A quick look at some of the other kind of things that can go on. Look at this stress-strain curve. Other materials that aren't ductile, for example, aluminum, will have a curve that looks a lot more like this. Again, a nice linear region, and then it tends to respond like that, and then just reach a rupture point. It doesn't have this sort of dip into the curve that most of the steels have. And again, the real graph look more like that, where that linear region, it's indistinguishable from the y-axis. If we did it the same scale all the way across. If we go up to some point past this linear region, say up to here, and then release the load, we tend to come back down parallel to that original linear region, and it's sort of like the material now starts over from there, and you can go up or down now that line. So in a way, you've increased the yield, the yield limit, you've increased the point where the linearity stops, because we're a little bit higher up here than we were down here, but you've permanently built in some strain offset, which might be exactly the kind of thing you want to do. It's not a typical to put in a permanent strain offset of about 0.2% for aluminum to get a little bit more yield out of it than before. Other materials of interest, ceramics and brittle materials, glass, brick, stone, they have not very much of a linear region, and kind of just go up and then rupture. Extremely useful in compression, very hard to engineer in tension, so a lot of point in that is ignored with those things. All right, so there's sort of a cartoon picture of the response of some of the materials to what we got going on here. Any question for, we started looking at some more detail, especially in this engineering region of interest down here where things are so predictable and repeatable. All right, let's leave that graph up there because we're gonna work something with that one. Oh, wait, what we do? Let's take a quick peek of what the book does. How many of you have a book with you? Okay, so it is worth it if I put it up there on the projector. Just to see what they do with these graphs. There's different ways to handle this problem. The fact that this region is incredibly small, and if we did the whole thing to the same scale, we wouldn't even see that line. So the book does a couple different things with it to make it so it's usable. Here's one of, notice, here's the basic curve that I just sketched out first. More to the true scale of it. So what they've done is taken this linear region of interest and redrawn it to a different scale. And that's then what's shown in the light blue numbers. Notice that you can follow that with the light blue numbers all the way out to here, then that point stops and restarts on a different scale there. So that's one way to handle this problem, to graphically handle the problem that is very, very difficult to get all of this on the same scale and make it usable and visible where we can see some actual numbers. So it's not really two different materials, two different tests, it's just this picture stretches out and then reverts back to a grosser scale, starts again here and then finishes the curve. So that's the way, that's the main way our book handles this. So that was a good example as any of them in here. So yeah, you can see other problems where they do the very same thing. So I guess that's one of the few instances where it's okay to use color and graphics. My students have had me before, remember I very seldom recommend that you ever use color on a graph, but that might be one of the places where it's worth it. The trouble is, of course, if that gets photocopied, then that the clarity of that is lost and so you might wanna do other things. One thing that could be done is to make two graphs out of this where there's one graph for the elastic region, just like they did there with the lower portion of the diagram, continue the graph with a different scale that pushes this back into there, but then lets the rest of it go out. That's sort of like what they did there, only taking this graph out from underneath and sliding it over here. That way you wouldn't have to use color on these and you could use a much different scale. It can still be obvious that the two went together. Maybe even continue a dotted line over here to show that it continues. So there's different ways to handle it, different books through different things. But again, the bulk of our concern is going to be with this linear region so that we can stay below the point where the structural materials start to yield. It could be that once they start to yield, remember this is actually a physical lengthening of the material. It could be that things don't fit quite right anymore. A machine might not run properly or doors might stop closing if we're talking about structural members that undergo this type of stretching. All kinds of dastardly things can happen. So we're going to focus then on this linear region. It wouldn't be too much of a surprise that when we have a linear region like this, especially one that so politely goes through the origin, which of course this does, if there's no load, there'll be no responsible material. It's just sitting there, you just got it out of the box. No cars going across the bridge, whatever the situation might be. It shouldn't be a great surprise to you then that the slope of that line is very important. For different materials, there's a different slope there. This slope is known as Young's modulus. If it's Young's modulus, we give it the symbol E. Actually that comes from the fact it's also known as the elastic modulus because this is the elastic region in which we're concerned. Remember, the material will return directly back down that line if the load is relieved. So it's a perfectly elastic response. In fact, we'll see in a second that you've actually been here before in certain ways. So that's the stress at any point in the test divided by the strain at any point in the test. It's stress, strain, little bit of simple mathematics on this. Canceling like terms, top and bottom, we're left with S over A to simplify the ratio. But I'm bummed, a bunch of old men. When I heard that 30 years ago, and this was also a Portland State University, I thought that was fun. In fact, I had to wait until I came here and taught this class the first time. I could finally use what might possibly be the greatest engineering joke ever. Well actually, at the time I thought it was, I've seen some of my own that are even better. We haven't gotten to them yet. I don't remember the old man, that's what I got in here on YouTube. So let's double check what we got here. That's the stress over the strain Now this is, remember, a material characteristic. This is something the manufacturer should publish. It's the type of thing of course that we can look up in the back of the book. Once it's been done for a particular material, it's considered no different than density or color or even cost. It's a material characteristic that the manufacturers have let you know about. Now let's just take this just for a second to show you that this is actually rather familiar to you. So I'll just do a little bit of a quick math to rearrange things a little bit. Let's see, is that the, yeah, that's right. That's the piece I'm looking for. All right, so if I rearrange that just a little bit, I'm going to put P over here and everything else over there, which will be E times A over L. Times del, that algebra is okay, I think. Here's why that's important. Notice that everything right here, what can you tell me about that quantity, E A over L? Everything in there is a constant. Remember this is a material characteristic. You tell me you have A36 steel, I'll look it up. I know exactly what that is. The area is already determined by the piece, in this case it was a test piece, but it could be a structural piece that we have L that, remember, is the original length of the material, then this is the variable response to some load. So I can rewrite this as F equals K. That look familiar? It sure should. This is, of course, hooks law for linear springs, just like you saw in physics class. Just to further emphasize a couple things, one is that especially this young's modulus is a constant as far as we're concerned. It's a design point that we can use in calculations and that the material response that we're talking about in here is indeed elastic because that's exactly what you looked at when you looked at hooks law with first springs, which you probably didn't. We didn't happen to do it in physics one. I don't think you do it in physics two. Did you? Actually did hooks law on some springs. You might, because then you looked at harmonic motion, you need to know what K is for harmonic motion. So it might have been worth actually figuring out what K is, but it's such a simple experiment. You just hang increasing masses and measure the length of the spring as you go. So we've got this very predictable, very usable point in terms of the material response itself, the material characteristics. So that again is a piece we're gonna need in the books. So if you don't bring your book, at least photocopy this full inside both of them, because one's the English table, one's the SI table, so that you have this with you at all times. And then notice one of the things in there is let's see, second column, modulus of elasticity E, elastic modulus as I happen to write it. All right, let's put that to some use if we can. All right, so here's a simple design of some kind with which we might concern ourselves. There's a three-foot cable there holding up one end, simply pinned beam, feet along here. So that'll be two feet, missile beam, and then another three feet. We hang a weight of some kind. In fact, you're gonna have to determine what that is. That's part of what we're gonna do here is determine what that is. Once that weight is put on there, the response of the system is a deflection caused by the stretching of that support cable there, so that the system drops down to there. Just for help, we'll label a couple of these points for reference, a strain response in the support cable DE due to the load put on there at point C. For now, however, we will not later have this luxury. For now, we're gonna assume that that beam does not bend in any way. Not too long from now, we will look at the response of other structural materials. Right now, we're only looking at tension and compression. The bending is due to a transverse load, so we'll be looking at that a little bit. All right, and this cable here is four feet long. All right, you need to find a couple things. One, we're gonna need to find the weight. Also want to find the strain in both cables. We have one support cable at DE and one cable at EC holding the load itself, which we don't know yet. Both cables are A36 steel, the things you need. This downward displacement here is 0.025 inches. And the cables have a cross-sectional area. Remember, that's important to us now, because that's what is determined and used in the stress cross-sectional area of 0.002 square inches. All right, so we can get a start on this one and we'll finish it up on tomorrow morning. What's the easiest thing to find, the thing that we find almost right off the start? Gotta find three things. The strain in the two cables, the one cable we have here, the one cable we have here, and then we also need to find out what is the load that will cost this system to respond in this way. DJ, you said the strain in this cable is the easiest thing you've got, everything you need there to find that right away. And cable BC? We've got a weight hanging from it. I bet there's a strain in it. So there is a belt. I don't know what. Now remember, when it's a concern how much this load moves, don't forget there's strain in the cable and a deflection there too, if that was a concern. We have to look at both of those things. Double check, I gave you all the pieces. Be careful, make sure that not only do you use the right set of units, whether it's SI or English, but that you also read the table properly. Most of these elastic modulus numbers are fairly large. Once you find the strain and you know Young's modulus, you can find the stress in that cable ED. Once you find the stress in ED, you can find the load in ED. Once you find the load in ED, you can then find what this load is using our static equilibrium. So that's most of your game plan there with that one.