 It's not a bad turnout for Monday, President's Day, 4th of July, Yom Kippur, I understand where everybody is. Where the devil is everybody? Geez, man. All right, we have been working on net section fracture, gross section yee, oh no, that's not what we've been working on, is it? We've been working on beams, beams that would go into the plastic region perhaps. If you can keep the dang things stable while you try and put a lot of moment on them, if your beam does remain stable that would mean that it's not going to flop over to the side when you load it on the top. You may have to support the top compression flange to keep it from doing that. It also means that as you put a large moment on it the flanges don't personally buckle over about a six or eight inch length, that's called flange loco buckling. And you have to also make sure that the web does not buckle locally because in either of those last two cases the wide flange is no longer a wide flange, it's something else because the elements are not nice and square with the world anymore. But if you can prevent this thing from flopping over on the side called lateral torsional buckling and if neither your web nor your flanges buckle locally, if they're stable, then you can have as your nominal strength of the beam, you can have its full plastic moment M subplastic. We did a couple of examples where we calculated M plastic. He mentions as with compression members instability can be in an overall or a global sense. That's where the whole wide flange kicked out to the side from top to bottom or between nodes and then you had local buckling problems of the web and of the flanges or you had to see if they were going to give you a problem. Here is a beam that is going to be bent elastically. I'll let you go through the numbers. It's an 8 by 2 and an 8 by 2. This person must be an allowed stress design person if they're interested in this because they're going to scream foul the first time some fiber reaches yield. Here they have solved for the location of the elastic neutral axis. A1 Y1 plus blah, blah, blah turned out to be six and a half inches up. That'd be an inch and a half underneath the bottom of the flange. Somehow they know the stress on the top and stress on the bottom. Somebody's already calculated the moment of inertia about the neutral axis. Yeah, they got it written down here 290.67 down here. They did the work just in case you didn't know where that came from. Here's the moment of inertia about the elastic neutral axis. Remember that BH cubed over 12. However many you get, one for each rectangle plus AD squared, one for each rectangle. There are the calculations. Here you can still do that if not review this. The stress on the top fiber is we were told they intended to apply 2236 inch kips to this shape. So to get the stress on the top face they put MC. C is 8 and 2, that's 10. Subtract the distance to the elastic neutral axis of six and a half. That tells you how far it is from there to the top fiber. MC over I, get 26.92 KSI to go to the bottom fibers. Huh, look at that. That's what they were shooting for in the first place. They put that moment on there just so they got exactly F sub Y. That is probably when a loud stress person would scream foul. Fiber has reached yield. Bill and I would also be interested in that. We would call that the first yield moment. Sometimes we have to calculate it for work that we do in LRFD. Here is a person interested more in plastic design. They located their plastic neutral axis by drawing a line which divided the area above that line equal to the area below that line. Since they were both 8 by 2s they had an easy job of it. They had an 8 by 2 on top of an 8 by 1. They probably had to go ahead and guess the plastic neutral axis up in here somewhere and solve for its location such that the area cross hatched above that line was equal to the area below that line plus this area. That would be where the plastic neutral axis has to be. They went ahead and ran all their fibers up to F sub Y. They got Z for the shape. That's the plastic section modulus as opposed to S for an elastic section modulus. They got that simpler than we, they get theirs easier than we get ours. And Z is equal to 8 times 2 times the distance from the plastic neutral axis to the centroid of the area under discussion plus 2 times 8 times the distance from the plastic neutral axis to the centroid of the area under discussion. That's Z. 8 times 2 times 1 plus 8 times 2 times 4 is 80 cubic inches. Stress is equal to M over Z so M in this case we're running it up to plastic so M becomes M plastic is equal to Z times sigma but we're not using those sigmas anymore. Those are holdovers from our 205 class. We're using F sub yield. As the plastic moment is equal to Z times F sub Y. Z times F sub Y. 50 KSI steel yield, 4,000 inch kips and you'll notice that where as our friends only got 2236 and started screaming nomos, nomos, we can run it up to 4,000. We've got an increase of about 80% added strength that's always been there. It's never, it's not something we just found, it's always been there. We just hadn't used it until we got into this method of analysis. Here's another one I'll leave it for you to do. 50 KSI steel, a 10 by 2 and a 10 by 2, again the plastic neutral axis. Somebody didn't know any better. They just guessed it was here, why plastic? So they said 10 times 2 plus this area above the plastic neutral axis which is 2 times 8 minus Y plastic, that's the height. That shaded area plus that shaded area had to equal to this shaded area so they worked it out. Area above 10 times 2 plus 2 inches wide times 10 minus Y plastic, whatever Y plastic turned out had to be equal to 2 times the height Y plastic. They did all this work, they did all this work, guess what they found, 10 inches. They said well it was good practice because sometimes it's not always, you know, the area above is not equal to the area below. Regardless, you now know where the plastic neutral axis is, it's right there. Your Z is going to equal to 2 times 10 times 1 plus 2 times 10 times 5, 120. Your plastic moment is going to be 6,000 inch kips. That's another 1.8, I don't know what the other one was, it's about 80% more. Now wide flanges don't give you that much excess because a wide flange really does have some pretty high stresses in the top and bottom flanges and because they're thin, the stress is almost what we're going to put up there, but why? So about all we pick up really is just the web, the web of the beam for loud stress designed people. They'll have F sub Y, the beam is seen from the side, they'll have F sub Y, they'll almost have F sub Y here. We'll take F sub Y all the way down. So about all we're picking up is this piece right here and that's a relatively thin piece of metal, so using LRFD we pick up about 10% more. That is really always been there and is safe to use. This is back on the previous problem where we were trying to locate the centroid of this wide flange. I told you I thought that was a waste, I still do, but just to show you this was a WT5 by 30, it was cut from a wide flange, 10 by 60, and if you want to know where the centroid is, the centroid elastically is Y bar. Y bar is listed for that shape, Y bar, there's Y bar, just go down the table and pick it up. My way is just as good because I've got table numbers to do it with, he's probably right. Now those numbers that you and I just did are good numbers, those are in plastic, however if those flanges or webs were ever pretty thin then it's possible that the web themselves or the flange itself would locally buckle and those numbers can't, that M sub-plastic can't be reached. It's still there theoretically but if something buckles on the way you gotta quit putting load on it. By the same token if this was the wide flange you were studying here right here you put load on top of it, the entire top of this beam is going to seriously think it's a column. Those little fibers here say dang 10 KSI, well not in our case, 10, 20, 30, 40, 50 for an allowed stress guy, for you and me 50, 50, 50, 50, 50 KSI, he says man that's a column. He says I'd like to buckle up and down. Well the person down here is in tension, he won't allow that to happen, he won't allow it to buckle about this axis, however he has less to say if it buckles about this axis because that column will just kick out to the side. It'll be a while before he can say hey I've got a lot of tension, I don't like that, get back over here and it may only go that far but that's enough. That beam has laterally in torsion buckled, therefore the entire shape not just the flange and not just the web has become unstable and you got to quit, you can't put any more load on it. You say what happened to my mcp of 6,000 foot pounds, well it's there but unfortunately before you were able to get the beam fully plastic it popped over the side and failed. I don't mind if they do this as long as I know when they do that because I will back off from that number to make sure that the load never causes this to happen but I need a limit state for plastic moment where all the little fibers did go fully plastic. I need a limit state for when it's going to laterally torsionally buckle. I need a limit state when it's going to flange local buckle and I need a limit state for when it's going to web local buckle and I will find those four numbers and I will make sure nobody gets their load here to the lowest of that scent. Yes sir, but you ain't going to do that today. Once you know if we're going to study one where the beam column is a beam column where there are both compressive loads in the column and bending moments in the column, yes. Now one way you can stop this beam from laterally torsionally buckling is you can support the top flange, you don't care about the bottom flange, it's intention is to stay straight as it can be anyway, but you can prevent this thing from flopping over to the side by putting a concrete slab on top of it, welding little things that look like studs or nails about every two or three feet along the beam's length, pour the wet concrete on there, then let it set and then this thing won't move at all. It just can't, it will be completely supported. You can instead, now these things have to go probably about every six inches, four inches, every six inches to tie the slab to the steel, not just supported, but because when you load up a shape that's got concrete and you're working it in compression and you're using the beam for tension, there's a big shear stress in between them, a shear load and that's, these things have to be big enough and numerous enough to hold that. But if you ever do that, this thing is continuously supported. If you say, well I hadn't planned on doing that, up here it's not a concrete slab, it's another beam. Then over here there's some more beams. Well if you'll tie that beam off every one foot, then the thing will have to laterally flop over on a one foot unbraced length or maybe a five foot unbraced length or maybe a 17 foot unbraced length. There's some links that this thing is subject to, flopping over between supports, lateral supports and there's other times when they're close enough together that can't happen or it won't happen. You'll notice that this flange is shown moving to the right horizontally. This part of the flange is moving to the left horizontally and somebody's probably got a little wide flange or a little channel or some kind of a structural member stopping this from moving this way. Here the ends of the beams, they always have to be supported on the ends of the beam or the whole thing will roll over. It won't just ladle or retortional buckle, it'll flop over. You can do that also by putting a bracing section in there where if this wants to move to the right then it pulls on this, it pulls this tension thing to the right, that really can't happen. It won't allow that to happen. He's in such tension anyway, he stays straight. Even if he did he'd have to push this one over, they're boxed in nicely. If you put these every five feet or every 10 feet or every some number of feet, these things will not fail laterally, torsionally buckling. Also put flat diaphragm plates in there. Here's one where they did put some flat diaphragm plates in there but they still had a problem with the flange buckling moving up and down. They might have had some trouble on this compression side with the web popping out right and left, right and left, right and left. Local buckling on the web, those have to be checked also. Now here's the different kind of curves you get. You can read which one goes with which things. But basically if it fails before you ever even get to first yield, that's pretty much something that's got a lot of lateral torsional buckling in it and they just couldn't pick up much load. These later ones are better. The ones you really like are the ones that go way on up towards yield and have a long failure pattern that you have plenty of indication. Here's your deflection. Have a lot of deflection on them before failure really occurs. They'll give you more load carrying capability if you get the beam to behave in that fashion. Now when you and I did columns, we had a lambda where the radius of gyration just killed you. To the right of that point, Mr. Euler held all the cards and he told you right on the money what this number was according to tests. At least pretty close. There were imperfections in the column. They weren't all perfectly straight. The load wasn't perfectly down the center. There was a few things in there that we had to put a .877 on Mr. Euler's work to get a real honest value for how much stress you could put in the column. Here he just went nuts. It wasn't his fault. He just said it will not buckle till you reach this stress. But it didn't matter because some of the fibers yielded and dropped this down to the yield stress. We had two equations. We had one break point in a table. I don't remember where that table is. We'll probably run across it in a minute for comparison purposes. That's columns. You didn't know how good you had it. We used to have, I don't know, four or five things in here to try this out with. Then the engineer says, too much trouble. Not worth it. Give me two equations and let's see how much it costs. How much am I giving up just to have one break point and be on that side or that side and use the appropriate equation? Well, not a lot. He says, good. Go with it. Beams. Well, here we go. Beams, sadly, didn't work. They literally did have five ranges. They had four break points. Again, the engineer said, you can't live with that. Just too much work, too much problem. So cut it down to two like you did a column. And they cut it down to two. You lost so much strength that really was still in there. They said, OK. Make it a little harder and let's see how much we save. They decided they would have three ranges on the beams. There would be one range where the beam would be called compact. Then that would be controlled by one equation. We'll get into that later. Then there's a region where the beam is called, it's not compact. It's non-compact. And then there's an equation and a range beyond a certain break point where it is called slender. And so I'm sorry to tell you the bad news, but you got to take the bad with a good and they're going to pay you for it. You're going to have two break points in beams where you only had one break point to worry about in columns. There will be a lambda as a measure of that. Same thing you used up in here, lambda subar. This one will also have a lambda subar where the radius of gyration is just killing you and you're not getting a lot of load carrying capability out of it. But you get enough so that we do work down in that region. A lambda plastic below which all the fibers in the beam are in the plastic state. And then a region between the two here, they say, well, this is compact plastic. This is kind of plastic elastic. So we're going to call that the non-compact. There's a compact. There's a non-compact. And these suckers are slender just so we can name them. These failure things, they will be due to either the web's lambda for your beam you pick or the flange's lambda. Here, for instance, looks like someone said lambda mine. His was right there. His beam will take a full plastic moment. If your lambda for the web or the flange, doesn't matter which one. You've got to take the worst one. If any one of them falls in this region, your beam is non-compact. And if it's beyond lambda subar, it is slender. I'm dropping back to the old text because I got too much commitment to it here. But it's the same. The same number, same name, same word, same everything. AISC has compact, non-compact, or slender flanges and webs. They depend on how much they stick out versus how thick they are. For instance, on a wide flange, this is B sub F. This thing sticks out B sub F over 2. It doesn't want to buckle if it's thick. So the ratio of stick out divided by thickness is your lambda. The lambda that he'll tell you just use a B over T. The B is the stick out, B sub F over 2, and the T is the thickness of the flange. On the web, incidentally, that is only supported on one side. The web is supported on both sides. The web has an unsupported length of H. You can find that by taking D minus how far it is from there and going back to where this radius quits, K-design. You take off two of them and that will give you the H of the wide flange divided by the thickness of the web. Now this is more confusing. It's not hard. It's just confusing. And there's so much information about all you can do is just get in, start stroking, and when you swim to the other side, you're there. First off, we have a compact, we have non-compact, and we have slender regions. This is at lambda p, plastic, below which the fibers are all plastic, we have a region between lambda plastic and lambda r, radius of gyration, going to kill you. That's actually a straight line. This is actually a straight line. And then we have a curve. For beams that we are interested in, we're interested in a whole bunch of them. But for beams that we normally use, namely wide flanges and maybe channels, here is the set of numbers that you will find. First off, there's going to be about 20 cases in the table. So on page 16.1-17, I got a copy of it on 198H. You're going to find that they will discuss wide flanges, flange. You will be told to go get your b over t, for which you will use b sub f over 2, about about t sub f, and that's your number. If your number comes less than this breakpoint, this is your lambda sub p, then you're in the compact or the plastic region, you get full m sub p. Congratulations, full plastic moment. If, on the other hand, your stick out divided by how thick it is falls within these two breakpoints, then you have a non-compact shape, and you're going to have to do some adjusting to your plastic moment. It will be dropped to a smaller number. This one is lambda sub r. That's just like you had for a lambda sub r where the radius of gyration kills your column. If your lambda is out in this region, then you have a slender section, and you'll be given a different equation to use in that region. That's for a wide flange's flange. This is for a wide flange's web. This will be listed in case 10. This one will be listed as case 15 in the tables. Here are your breakpoints. And just laying it on a page like that makes a lot of sense if you've done it for years. But if it's not broken up into breakpoints, breakpoints, regions, regions, regions, it's a little hard to tell what's going on. So you'll be calculating your value for use. You'll see if it falls in which region. That'll tell you which equations to apply. Here it is a little bigger. The more I looked at it, and the more I wrote on it, the more I didn't like it. Here it is. This is for flanges of eye shape to be a wide flange channels. In case 10, it's on that page in your specs. Here's your thickness of the flange. There's your b sub f. You will be computing lambda is equal to b sub f over 2. That's the stick out. That's the resistance to being stuck out in the air. You'll enter your number somewhere along this axis. If it's below this breakpoint, then it is plastic, compact, inelastic, non-compact, or slender. This is for the webs of eye shapes. It's listed under case 15 in your tape. Compact, non-compact, slender. You see the breakpoints are different. Numbers are different because it's a different behavior. It's a flexure problem instead of just a pure old compression problem. Here is your h. Here is your k-design. That's in the table. The depth is in the table. h is d minus 2 of these k-design people. This is how you calculate yours. h over the thickness of the web. Here are these numbers. For a w21 by 93, there's its depth. So you can take off 2k designs and get little h. Little h is how high the web is unsupported, which you'll be dividing by the thickness of the web. There's the thickness of the web. Here it is calculated. The stick out divided by the thickness of the flange. Here's how far it sticks out. Sticks out 4.21 divided by 0.93. 4.21 divided by 2. 4.21 divided by 0.93. That's lambda for the flange. That's lambda for the web. Truth is, it's on the next page, so I don't know why I make you know how to calculate it. Other than I may just draw one and it won't be a 12 by 40 w. It'll be a Lowry 16 by 37.6. But there is your compact selection criteria. There's your lambda sub flange. And there's your lambda sub web for a w21 by 93. Obviously, most of the numbers we use are in the tables for all the convenience of the people who design these day after day and don't want to sit there with a calculator and divide them out. I'm showing you where Sugui got all this stuff. Is this in the commentary? No, it is not because you don't see the gray bar down one of the sides. The commentary is marked gray. This is in the main specs. On page 16.1-14, member properties, how you classify sections for local buckling. We already did this page. Four columns had non-slender and slender. Beam, we have compact, non-compact or slender. One of the restrictions is the flanges must be continuously connected to the web. It means you can't come underneath here and have two plates and just tack weld and tack weld and tack weld has to be welded all the way or it has to be a rolled shape. He tells you exactly when, which is what I told you, when it is compact, non-compact and slender. Unstiffened elements is like the flange because it isn't stiffened, it's just hanging out in the air. Obviously stiffened on one side, that's an unstiffened element. A stiffened element is like a web. It is connected to two pieces of metal tending to have it not buckle. Then all the rest of this, well, here is the column table that we had earlier. You had one lambda to calculate and you had one break point to compare. Here's how it looked, non-slender and slender at lambda sub when the radius of gyration is getting ready to kill you. And this was for wide flanges and this was for hollow rectangular HSS shapes and so on. Those were for these. These were the corresponding break points. You were told to just calculate B over T. The truth is what you used to calculate that was B sub F divided by 2 divided by T of the flange was your lambda. Here it is for beams. What you just saw was cases 1 through 9. Here's where we start on. You'll also notice this says for axial compression columns. Here you are for flexure. Remember subjected to bending. These are beams. Here's your case 10 for flanges of roll-dye shapes. Here is what you calculate. Here are your break points. Here is case 15 we talked about. There's your webs of a wide flange. Here is your calculation. H over T sub w. And here are your two break points. Then you have other things. You have angles. You have T's. There's a T. It's not funny. Here you have a T and here you have a T. They have a T. They have different break points. What he assumes is that you're using this as a beam, a standard beam, where the top is in compression and the bottom is in tension. Since the top is in compression, it's the flange that tends to buckle. These are the break points found by theory and test. If you're turning it upside down, he assumes that the top is in compression. Wow, look at that, man. That little thin thing sticking out there like that. And here is the tension side. You see the difference in the numbers. In other words, the break point here is pretty far to the left. The break point on this little thin guy in compression stuck out in the air like that is pretty high. Same way on the next number, although it doesn't make as much difference. Actually, well, yeah, if it's built up, you can build them up yourself. 10 and 11. Yeah, these are built-up shapes well. See, one of the main differences is on this one for the... This is for flanges. This is for webs. On this one, the H is the inner surface to the inner surface. And on this one, it's back here a little further. That's H is a K-diesel. D minus K-design times 2. I don't know about an FL. See me after class. That over there. All right, now, skip. Pending strings are compact shapes. You're kidding. How can you skip that? Oh, yeah, I know what's going on. That's the new text. Oh, yeah. There's the old text. Now, I need to discuss with you the fact that these people who write these books know 10 times what you and I know and been doing it 10 times longer. But there's a serious disconnect in what he means on this page. Section 5.5 is bending strength of all beams. That's what we're getting ready to discuss. We will compartmentalize. We will specialize a little further down the road. So this really ought to say bending strength of all beams. There's four ways a beam can fail. A beam can fail first by reaching in plastic and becoming fully plastic, or it can fail by lateral torsional buckling, either elastically or inelastically. We'll have those different equations available to us. Or the flange can buckle locally or the web can buckle locally. So there's four things that you have to check. If the maximum bending stress is less than f sub y, then the, I'm sorry, if the maximum bending stress is less than the proportional limit when buckling occurred, then since no fiber reached the yield state yet, that failure is going to be called elastic. Otherwise it'll be inelastic. He already said he's going to do them compact, non-compact or slender. We know how to do that because we know how to calculate lambda and compare the break points and so on. Same things I've already mentioned. Now, right here is where this needs to go. Of all the beams on the face of the earth, some of them are compact, meaning some of them will not have flanges or webs so thin that the flange or the web will buckle. Those are called compact shapes. So that title really ought to be right here and define when that state occurs. Compact shapes are when there is no local flange or web buckling because you are to the left of the appropriate break points. They are called compact shapes. We begin with compact shapes. And then we are taking a subset of everything we got. Right off the bat, if I tell you the thing is compact, you don't have to check web or flange local buckling. I'm guaranteed it won't work or somebody has. What makes sure that it is compact is that your measured stick out to thickness of the flange is less than the break point from case 10 and the numbers for your lambda for the web, that's lambda the flange, that's lambda of the web is less than the appropriate lambda for the web. Those numbers are listed in case 10 and 13, page 198 I in the notes that we just covered. Now, it turns out the web criteria is met by everything in the book. So truth is, if you just say that on an exam and don't actually check this break point, then and if you're using an f sub y smaller than 65 ksi, see there's your f sub y 65 ksi, everything in the book is smaller than that number and not all smaller than this one on the flange. Some of those do not meet this criteria. So he says in most cases all you got to do is check the flange. The non-compact shapes are also identified in the dimensions and property tables with a footnote. See if I've got a, here we go, here's one. See this guy here? He is not subject to flexural problems and he's not subject to compression problems. This guy's got a problem in flexure. This guy's gonna have some thin flanges. This guy's got some compression problems. Remember when we had a superscript with a C? Those had problems. We said we were going to avoid that one because we really didn't know how to handle them. Said we could use them if they were part of a table put together by people who knew how to handle them. That was this. So compression problems, compression problems, H. H says flanges stick as the dickens. Sometimes if the flange of something is really thick there are some special provisions that you have to handle. We won't get into those. These are great. These are good. That one's got a flexure problem. Footnote F. Superscript F, but it's footnoted down here. Shape exceeds compact limit for flexure when your F sub Y is 50 KSI. So you can tell if you've got problems. You don't have to actually calculate this number and calculate this number and see if it's less. You can go to the table for that beam and see if it's got problems. If it's got problems it's got problems. Now if the unbraced length is very short now we're into compact beams. This is how you make sure they are compact. This is the section on compact shapes. If it is compact and it is continuously supported horizontally on the top flange then I guarantee you the nominal strength will run right up to the plastic strength. You know that if you fall asleep with that lollipop sticking out of your mouth they're going to have to perform an appendectomy on you to get it out. Okay. Your mouth will still go in there but your eyes will close. I was a little afraid what would happen there if you fell on the desk. Alright. Here's your plastic moment. It's going to be with a resistance factor of 0.9 because you can't guarantee me you're really always going to get this for all the cases you've tested. And the plastic moment will be as we said before f sub y times z sub x. Z sub x is typically listed in the book so that you don't have to work them out unless you get one that's not in the book courtesy of some rotten prof in case you'd have to know how to calculate z sub x. They're also listed. I told them, I told them, they're just about to go. Here's page 3-19. I've got those in my notes, 1.99c and d. You already showed this one here. Here's your k-design. Here's your depth. Here are other values we will use. Oh, here is the plastic section modulus about the strong and about the weak axis. Here's r sub tuff stuff. R sub t0. There's a torsional j. It's a warping constant. We'll get into all of them. These are the two we've covered so far. Here's another section entirely. This is on page 1-25. This is on page 3-19. These are a listing of all the z's and the corresponding plastic moments about the strong axis including the .9. This is about the weak axis. This is something we'll use. Incidentally, these are your break points for your beams. If you put your brace points on this beam closer together than 14.9 feet, you've got to brace. Bracing the thing from against lateral buckling, then it will stay in the plastic region. If you go above this, it's going to be that elastic problem. And if in here it's going to be intermediate between the two. There are your two break point links. Here is the full chapters themselves. We'll get into all of these. There's your .9. Here is your yielding. There's your spectra. It tells you how much you get if it's fully plastic. This is how much you get if it's not fully plastic if you're between the two break points. And this is how much you get if your brace points are further than that break point. F critical is this. And not only that, you're stuck with the elastic section modulus of the beam. Not the plastic section modulus of the beam. More equations, more good stuff. I've put them all together just so we'll have them all in a bunch when we come back. Don't even have anything circled yet. Different shapes. Rectangular box shaped hollow structural sections. There's an example problem. And we'll do that next time. You had a question on something. Was it in this one? It mentions it down here, but I'm wondering what that is. Let's put it on here where everybody can see it. Which one again? F sub L, doubly and singly symmetric. These are built up so these are welded. F sub L. So it tells you what F sub L is. It's 0.75 F sub Y for major bending of a compact and an uncompact shape. Built up members where this restriction is in effect. And all you got to do is take F sub Y to 0.7. Now why is it F sub L? We really don't get into built up shapes at all. You know what that is. That's S sub X. No, you don't. That's the built up thing. See, they don't have to be symmetric. So you can have a section modulus about the X axis on the tension side and a section modulus about the X axis on the compression side. And it's got something to do with the fact that the neutral axis has moved down because this is smaller and therefore this has one kind of a tendency to do bad things to you and this one has a different one. I mean it's weird, you know. And graduate school will teach you all those little nuances. Yes. These L values? We're seeing that these. That's correct. In other words, we really haven't gotten into it yet but you see that 30 foot there? There is a length and then it's braced only at the ends. There is a length that if you stay below and it's called L sub plastic, you don't have to check buckling at all. There's a length beyond which it's going to have so much tendency to buckle that it's going to be way out on this side. They give you those Ls and we will have a developing equation for them in here. Yes, sir. The claim is that for some countries they pre-stretch it before they take it out to the field. Well, yeah, yeah. No. Then it will have. No. Strong tendency to. No, it will not. And the reason being all you got in there is F sub Y. So let's just say you wanted to use it as a column so you took it and you just pulled it like crazy so it had all kinds of or pushed on it like crazy so it had locked in tension stresses. Right. When you come and put the load on a column, you keep adding the load on the column and first thing you know the little fibers reach F sub Y in compression, that's all they were going to do anyway if you hadn't messed with them in the first place. So you can't pre-stress a piece of steel. You can pre-stress a piece of concrete so that when it gets bent, you know, the fibers feel compression due to the tensioning strands. Then they feel tension due to bending. Now you get the difference, but not in steel. We saw the equations about the stress and, you know, what the traps are. Correct. But who really looked at the strains associated with that? The strains? Well, the strain is equal to the yield strain. Now, if you want to know the strain, I can tell you immediately, you know, the strain, here's your curve, there's E, that's E, that's stress over strain, and so you are using the stress of 50, right? So E is equal to stress divided by strain and therefore if you want to know the strain is equal to sigma over E, if you want 50 KSI steel divided by 29,000 KSI, that is the strain at yield. And we don't discuss it much because, you know, it's what it is. It's always going to be that. Okay, so we take... Yeah. You know, because it's sort of easier to imagine or figure out, you know, strains, the changes in length, you know, how much is the steel kind of change in length? That's something... More than the internal stresses you can't see? Right. That's true, but this is the way our friends have been doing business and you and I are pretty much... Oh, yeah, we're spectacular. So with the buckling, like, how much does the buck allow? I guess it doesn't matter. It doesn't matter. The first time you see it buckle out, it's failed. So we're going to be way off of that number no matter what. Now, the first time it buckles out, you know, it may be restrained on the end, so it may buckle out just a little and then the people at the top say, no, I don't think so. So it may not actually fall down, but generally speaking, we're through with it. We don't have much left in the game. The worst case, of course, is that once it gets out of shape, once it gets a little over to the side, then there's a moment on the column. In other words, you see the load was straight down and this load was straight up. Now then you say it did this. Now then the load is coming. The load is eccentric. You know what happens when loads are eccentric? They put moments in. They put moments they want to bend in. But we take all that into account. Okay.