 Okay. Estus. Estus. How might a beam fail? One of many. How might a beam fail? What kind of buckling? It's true. Well, okay. What kind of local? Two flavors. Pick one. What piece? These questions. Well, the web will be fine. The web could locally buckle. Good. Boucher. Boucher. Anything even close? Pretty close. I forgot what I was going to ask. You're off the hook. By the time I get to Vickers, I'll remember what I was going to say. Vickers. Vickers not here. Wiley. Sir? Wiley? Yes, Mr. Wiley. I remember now. I think they said the web could buckle. Does the web ever buckle for the stuff we do with our steels, with our shapes? That does not. They are all too thick and too short. If you make one up yourself and weld it together, then you have to check it, because it may very well, the web could buckle. The flange could buckle. Suduki? Not here. Rotts? Not here. Stevens. Good. Thank you for coming. That's a nice break. Stevens, since the web can buckle, and I guess that means the flange could locally buckle, both local bucklings, what else could go wrong with this beam? It could fail. How? And don't pick one of the two that we just picked. Give you a hint. Do that again. It could fall over. That's right. It could laterally, torsionally buckle. If you'll go, this is your assignment now. I want you to go get a yard stick or a meter stick. Once you stand it up like this and support it on both ends, I want you to push it on the middle until it breaks. You'll find that it won't break. It'll just keep flopping over to the side. That's lateral torsional buckling. That's because the top half of the beam is in horrible compression. So the whole top of the beam says, I'm a little column. I'm a little column. I'm a little column. And it wants to pop out. It doesn't care up or down or left or right, but it can't go up and down because the bottom is in tension. It holds it, doesn't let it go up and down. But it can pop over to the side even though the tension side doesn't like that. And so it laterally twists torsion buckling. Cook, Mr. Cook, what else? How else can a beam fail? They already got all the exotic kinds. There's just one basic kind. Okay, this is a rupture failure. It's not a rupture though, actually. It's a yielding failure. And that would be by plastic moment. All the little fibers, all the top half of the beam is in compression to F sub y. All the bottom fibers below the half of area mark are in tension and it just gives up. It has nothing left to give. It does. It could go in the ultimate range, but we don't go there. All right. Here's one way you can stop your problem. I see you have a problem. Yeah, lateral torsional buckling. Here's how you can stop it. Just weld these little studs. They do it with a little gun. Pop, pop, pop, pop. And then pour concrete on the top of it. And then the top flange, even though it is in hard high compression, it can't pop over to the side because it would have to take the concrete slab with it and all the other beams and the walls and so it just can't buckle laterally. You don't actually have to put these every six. We got to put them about every six inches because their real purpose is to tie the concrete to the steel. The concrete takes the compression, the steel takes the tension. But if you were just going to use them for lateral torsional buckling for this beam right here, you could probably put them every maybe four or five feet or maybe every seven or eight feet. It depends on the beam how badly it wants to buckle laterally and the more it wants to, the closer these little studs would have to be placed together. We don't usually use those. If we have those, we use them to make this thing not subject to lateral torsional buckling, but more than often you put braces coming in from the side like I showed you earlier and we just brace the top flange so that it doesn't buckle. And they don't have to have one every inch. It depends on the beam, but you can make them maybe six feet, seven feet apart, and then you need another one seven feet down the road. This particular beam is a 16 by 31 of 50 KSI steel. It's got a concrete floor that provides continuous lateral support like this of the compression flange. It's got a service load. Service loads are the unfactored loads. So if they tell you you've got a service load, then you're going to have to apply a factor. For a dead load, the factor is 1.4. For a dead load in combination with a live load, the service dead load would be factored by 1.2. 450 pounds per foot is superimposed on the beam, doesn't include the weight of the beam, so we're going to have to put another 31 pounds a foot of dead load on this number. Service unfactored load is 550 for the service live load. Does this beam have enough strength to get the job done? Your choices will always be, and you can say no at any time, you can prove it. Will it get a plastic moment? Well, the answer is always, that's a possibility. The only real question is, does one of these come up first? Would it laterally, torsionally buckle? This is a global failure. Somebody wrote no here, I don't know why, but they must have already done some work and they put a no on there. Could the web buckle locally? They probably wrote down no right from the start because they know this is a standard roll shape and it's less than, I believe, the number could run all the way up to 55 or 60, still would not in our book, none of them, the web would buckle. And could it locally buckle on the flange? Then you work out each of these problems and make sure that it doesn't fail for the load you're going to put on it at all. First off, here's someone tinkering with the flange. Their lambda for the flange is to be measured by B over T, generically B over T. B over T is the stick out of the flange, that's this total length from here to, total length from here to here divided by two because that's the stick out. And even with this concrete here, this thing could buckle, could buckle down, buckle down and buckle down and will do so if it's too thin or if it sticks out too far. Our stick out is half of the total width of the flange divided by the thickness of the flange. You can check these numbers on page one dash 22 or on page 23. 6.28 is your lambda. Here's your lambda marked on the graph right here. Your 6.28 down the road. Your lambda flange, oh that's just him calculating the same thing I did. Your lambda for the flange has two break points. One is a plastic break point below which you are a compact section, above which you will not get as much strength, you'll have to do some non-compact work, above which lambda when the radius of gyration kills you, you're in the slender ratio, you'll have a third equation controlling the situation. So to figure them all out, here's lambda plastic break point for the flange. You would go to one of these tables, you would look for flanges, this is a lecture about the weak axis, you'd look for flanges, you'd look for flanges, cement built up shapes, you'd look for okay, there it is right there. Flanges of rolled I shaped sections, channels and T's. Your plastic break point is 0.38, square root of E over FY. Your lambda sub R break point where the radius of gyration is going to force you into a different equation. I guess that's kind of like the radius of gyration of just the piece that sticks out, 1 over square root of E over FY. Here is our break point for the flange lambda plastic. Here is our, where's our break point for the web. I don't even see it. Well, I know what's happened here. This number right here, the 0.38 turned out to be 9.15, since this was 9.15, we know we have a compact section as far as the flange is concerned, so they didn't bother calculating this number. Although it's given right here for lambda sub R, would be this number right here. You would need it if your lambda, your lambda was out in here somewhere, because you wouldn't know if you were to the right of this point until you calculate it. For the web, this is a waste, we understand, we're doing it just because we enjoy it. The web has a lambda plastic out of the table. The web, the web has a lambda plastic of 3.76. 3.76 E over FY, it's 90.55 for the web. Our personal H over T sub W, if you remember, H is equal to D minus 2K designs, but you say, yeah, but I'm never going to bother with that, because it's listed in the book for both the proper H and the thickness of the web. For our shape, you go and pull this number right off of page 1-23. So here's the page that you can check out of our notes. If somebody wants to see something, I'll go back to that page of our notes. This was case 10. Here's the page number out of your user's manual, your specs. This one was given by case 15. There's his page, and I got one of those on page 198i. This one came from 1-23. This one came from 1-23. Let me see, 1-23. A 200C, that would probably be worth taking a quick look at. The C says here that you don't have to do any of this stuff because it is compact because there's no footnotes saying you've got flexor problems. On page 200C, it's on page 1-23 in the book. But if you're going to do that, you've got to tell me that you've checked it. You can't just look over and say, ah, good, great, this one's compact because I don't think you didn't check it for compactness, and I'll take off, well, I won't either. I'll take off 10 and I'll say no check for being compact. You'll come up and you'll be crying, and you'll say, oh, I did check it for a kind of, I saw it, I looked, I checked very carefully. Oh, you really did, huh? Yeah, okay. All right, well, I'll split it with you. Well, that was all I wanted anyway, you know. So if you're going to find some good piece of information you've got to put down. Shape is compact and you've got to tell me why you think it's compact. One way you can check the break points. One way you can see it doesn't have a footnote. Here's that 16 by 21. You see it's got no little foot, oops, that's the wrong page. Those footnotes appear back here. Oh, it does have a footnote. Uh-oh. Comments? Oh, so I'm sorry? It's the wrong kind, right. It's not flexure flange, that's right. It is, it's got problems if you're using it as a column. Then the little elements are pretty thin and stick out a little too far. Yes. But for us, it is a compact bending section. Then, since we now know it's compact, we can say lateral torsional buckling, no. That, I'm not sure now. Is that what we say from what we've learned so far? Maybe we can't say anything. Can we say anything there? Three things. Plastic moment, lateral torsional buckling, web-local buckling, flange-local buckling. Can we write down from the study we've done so far? Last two. That's right. It's not going to locally buckle in the web, not going to locally buckle in the flange. But is it going to globally buckle over the whole length of the beam? Well, let me give you a hint, continuously support it. So is it going to lateral torsional buckle? It can't because you supported the flange. And therefore, that's where that no comes from right there. Now then, it's going to fail someday if you put enough load on it because it's just going to make the whole thing go into the plastic range and you'll get m sub p. Is that the design strength? No, it's not. What do you do? What is this called? The nominal strength. That's correct. It's m sub n. M sub n could be a whole bunch of different things depending on what you find is the worst case. In our case, this is it. So this is the nominal strength which happens to be the plastic strength. But all nominal strengths, you can't guarantee me you're going to get them. There'll be one out of a hundred that will not do the job. And so you're going to have to multiply this times what? Okay. And what's the name of that point nine? Resistance factor. The more you become familiar with these terms, believe me, the better the easiest going to be for you because it comes up constantly in the book. And all of a sudden he says times of resistance factor and you say what, what, what, what was that all about? All right. So now then we're going to do our plastic moment. It is compact. Shape can be identified as compact. There's no footnote shown. Otherwise, it's compact, lateral supported. Nominal flexural strength is. Plastic moment is the nominal moment. It's equal to the yield stress times the plastic section modulus. What is the symbol for the elastic section modulus? S that is correct. S is what you used in 305 elastic section modulus. You'll use it a lot of times in here too. So you need to be familiar with those terms. 50 KSI. You got this 54 probably out of a Z table. I already got it off of the off of the dimensions table. Let's see where I've got it here. First page it shows up is probably where I got it. First table, dimensions table. I don't see any Z's. Dimensions table. Here we go. There's your elastic. There's your plastic. 54. That's what he used right there. Also some Z tables they don't do anything but Z. They're very handy sometimes also, but I don't see one right off hand. So we multiply the F sub Y times Z. We get 2700 inch kips, 225 foot kips. Compute the maximum bending moment to see if we're going to exceed. This is no good just yet. This still needs a buddy, a fee. The load is 450 pounds per foot plus the weight of the beam, 41 pounds a foot. Maximum moment on a simply supported beam, WS squared over 8. Therefore the dead moment is WS squared over 8 for dead and WS squared over 8 for live. Then you will multiply 1.4 times this and see how big it comes out. Then you multiply 1.2 times this plus 1. Okay, there's no doubt who's going to win this game. 1.2 times this plus 1.6 times that. You're going to take the larger of the two. Unless there's five or six other possibilities including snow and stuff, in which case you'll check them all out and take the largest. Those are our break points for local buckling of flange and web. There are dimension tables. We've got a lot of good stuff on them. There are your lambda break points right here or your break points. They obviously change as the section changes. I hate this. I don't know. These guys have written more books than I have. He says the dead load is less than eight times the live load, so load computation two controls. If you can remember that, fine. I don't know why. Just multiply 1.4 dead and see what you get. Multiply 1.2 dead plus 1.6 live and see what you get and pick the bigger. It so happens that if the dead load is less than eight times the live load, then the live load contribution equation will control. We get 164 kip feet of request for a moment. That would be M sub u. It says, alternately, if you would rather, you could come in and put the dead and the live loads together and then put them together. Once you get the load that's the maximum, then you put it in your WL squared over rate equation and obviously get the same answer. The design moment is 225 from the previous page times .9. It gives you 203. Your ultimate request goes 164, so your design moment is greater than your request. Good to go. Beam works. It may not be the lightest, but we still be happy to take some dollars per hour to check some more beams and make sure we get the lightest one. It goes fast. These are the pages where I have committed to a lot of writing and everything. We're getting back into Segui's pages here. We just did that. The moment of compact shapes is also a function of the unbraced length. L sub b. You are L sub b. That's you. In other words, you go and you look at a beam and you say, I like that beam. I think I'll try that beam. That beam has an L sub b when you build it. There are some break point L's that tell you what the beam is after you build it. Whether it is going to be a plastic beam, whether it's going to be a last-o plastic beam, whether it's going to be a slender beam, elastic. Again, you're going to have break points and you're going to have your numbers. You are L sub b. You find it's the distance between the points of lateral support. You choose them. You pick them. That's that number. Now, in this book, he'll try and remember to put little X's wherever you're going to brace the beam. This particular beam is a simply supported beam, braced only at the ends. All beams are by specification to be braced at the ends, or otherwise the whole thing will just fall over when you put load on it. That's the minimum you can put and that's what they did. I just put some pixel numbers out of the air here. If there's 30 feet between those braces, that's your L sub b. You decided what to do that. I asked you, why did you do that? You said, well, architect wanted it 30 feet long between the column lines. Then I understand why else it would be as 30 feet. If the thing has a real problem and you're having to use a really big beam to get that done, you might come in and brace it someplace. In the middle would be best because then they'd both be 15 feet long. But you say, well, I can't get the brace there because there's some pipes in the way. But you can brace it here to help. Okay, brace it here. Then this is L sub b in your study, not this one because it's shorter. Here's what the curve is going to look like once we find out how strong these beams are when you bend them. First you already know this. They're going to come in here at M plastic. And of course, once you start really seeing these, they're going to go ahead and have the fee attached. So it will be M design, how much moment is permitted. Then at some point you will find that up to this point, all the fibers were working for you at the plastic limit, at the yield limit. Sadly, a little further down the road, the thing started having this tendency to flop over on the side. And the strength of the beam dropped. It really got to L sub, when the radius of gyration really takes over. It's kind of nice. It all got the same idea behind them, all these curves we got. And Mr. Timoshenko has a really nice equation, which was first proposed for how strong beams are when they flop over on the side, when they laterally, torsionally buckle for wide flanges and channels and all kinds of good things. Now when he says this is a compact shape, he means that the curve generally looks like this, but compact shape means there is no local flange buckling and no local web buckling. Because of these numbers will be reduced if the flange or the web buckles before you can get this full strength out of the beam itself. So up to here, these things are laterally torsionally buckling. Below this point, there is no lateral torsional buckling. There's no instability of any kind. And with these words here, we're assuming that you don't have any of that kind of instability anywhere on this particular beam. Uh huh, now it gets into it. Like I say, it's not hard, but there's a lot of stuff going on here. Curves will look like this. They'll start off with a plastic moment on the beams. They all have some plastic moment, even if it's a little short piece, and then they start to fail laterally. That's listed as MP. It's your nominal moment strength in the vertical direction. At this particular equation is AISC F2.1-1. It's on page 16.1-47. And I've got it on page 199H if we want to see one of them. I want to see that equation. And all it says is F sub yz sub x. Then what we found is there's a lot of data in here. We couldn't really write an equation that was any better than just a straight line going through the data up to this point. So that's exactly what they did. They went and they saw for elsewhere. They found out how strong it is as they plotted it. Then they made a few beams that were a little shorter, a little shorter, a little shorter, a little shorter. And here, when they made the beam that long, it came out of M sub plastic strong. And they said, you know, straight line is as good as anything. I mean, this isn't true. This isn't true straight line. And then he says the straight line doesn't really work. It gets on Mr. Tumashenko's equation that he gave us. It's long. It's nasty. It's ugly, but it does a beautiful job. And we're going to work mostly with tables anyway so we won't have too much calculation on them to do. This equation is on page 16.147. It's AISC F2.1, F2.2. This is AISC F2.3. And then along with this, you'll need some more equations in F2.4 to go with it. One of them just says stress times the section modulus. And then you have to go get the stress out of another equation 2-4. In this region, your beams are plastic. There is no global lateral torsional buckling, no instability. In this region, your fibers are inelastic. I mean, some of the fibers are going to be yielded. And if you get the load back off of it, you're going to have a bent beam. If you go past L sub R and it starts to buckle and you can run up there and get the load off of it, it'll pop back perfectly straight because all of the fibers in it were elastic. This is your bracing length. Who? H.U.? Who here today? Johnson? Johnson? Sorry. I had something all planned out here for H.U.B. And when H.U.B. wasn't here? Yes, sir. Yes. No, no, it's just going to buckle. It'll buckle laterally. The only thing is once it buckles, if you, maybe the load falls off because it's kind of buckled. Well, when it pops back and it will, then it'll pop back bent. Because you have, you have plasticized some of the fibers. This one right here, if the load rolls off because it torsionally buckled over the side so bad, and they see you going down screaming and they say, oh, wow, man, that's a shame. But when you're back off the beam, it'll pop back perfectly straight because all the fibers were elastic. The first region, every fiber is plastic. It's a mess. But it doesn't buckle. That's correct. No, they apply if it's a compact shape, but you have more grief. I don't want to hit you with all the grief at one time. You have more corrections to make. Now, I think that's what I was really getting ready to ask you. I was going to say, I'm going to put the pencil somewhere and you're going to tell me if the fibers will pop back after you take the load off of it. And I was going to point here and you were going to say no. Or here, you're going to say, yeah, it'll just pop back like a rubber band. All right, now here this is spelled out. It's got a great book, all the words you need to get the job done are there, but you know, there's, there's no divisions of it. It doesn't say what he's talking about. So that's where I come in. This is the elastic region. This is this region down in here. Pick one. I don't care which one. I've got to talk about them first, something first. We're going to talk about things that your LB is to the right of L sub R. It's an elastic region. L sub B is greater than L sub R. Equation for the elastic, the lateral torsion buckling can be found in theory of elastic stability by Mr. Timoshenko back in 61. I think it was proposed long before then, but they published it here. And most people glommed on to it at that time. The equation says you need a critical buckling stress times the elastic section modulus about the x-axis. And incidentally, I don't care what this equation or any modification of it says, if this thing ever comes out greater than the plastic moment, they're lying. There's no moment on a beam bigger than the plastic moment. If somebody says, well, you didn't do this as bad as Timoshenko said you could, and so I'm going to correct it, and it comes out bigger than this number, it's not right. You can go up to this number, but no higher. Where F critical is the elastic buckling stress and is given by the accuracy of pi over L sub B, S sub X square root of E, I sub Y, GJ plus pi E over L sub B, that's you, that's you. Hear your name? Squared times the moment of inertia about the elastic moment of inertia about the y-axis times our warping constant. Oh yeah, square root. You know, I don't know how they ever figured that out, but they did. And that is the critical buckling stress at which this thing will when multiplied by that give you a nominal moment at which it will buckle. It's just exactly like Euler's equation, except it's got a lot more stuff going on. Now this equation, when they derived it and tested it, they derived it assuming that the moment throughout the beam would be constant from one end to the other. That happens, for instance, if you have a simply supported beam and you put a moment, concentrate a moment on each end. It also occurs in a simply supported beam if it's got a load P and a load P and they're evenly spaced. And you have a brace and a brace and a brace and a brace. Because this is the worst moment, you'll be studying this region between these two brace points and it's a constant moment. Timoshenko's thing is perfect for that case. What Timoshenko's equation is not good for, because he didn't test it and he didn't have a proposed solution and somebody else came along, is if you have a brace, have a brace, no brace, no brace. The reason Timoshenko's equation doesn't work well now is because, yes, there's a top piece of a beam here that just screaming, I'm a column, I'm a column, I'm a column, I want to buckle. But all these people, they say, what's his problem? What's his problem? I don't feel like I want to buckle. So you want to buckle? No, I don't want to buckle. You should get some relief from Timoshenko's equation, unless all the fibers are screaming for columns. We're f sub y. Here's all the things. Here's the page numbers. The constants were defined when we discussed torsional lateral buckling, which we didn't. Sometimes during the semester we have time to check it at the end, but it doesn't matter G, you know what G is. J is a polar moment of inertia and C sub W is a warping constant. That's all they told us before anyway. It's valid as long as the bending moment is uniform. Non-uniform moments are accounted for by a correction factor, a Christmas present, in bending because you weren't as bad to the beam as you could have been. And then you get an f critical times the section margins about the s sub x axis. But please, no matter what somebody tells you, do not think you could go above in plastic. You cannot do it. Where the f critical is defined not by this equation, but by this equation. Now this equation, excuse me, this equation and this equation are the same. This is the one that was published. We found a few fewer terms, a little easier way to get things done, so the AISC people wrote it in a different version, but it gives you identical numbers. C sub B has been added. Timoshenko didn't know about such a thing because he figured that was it. That's let's go. Well, we're going to correct this to take into account the fact if you do not hurt the beam as badly as Timoshenko had planned. J's polar moment of inertia. C is on the next page. It's defined. Here's a beam that should not have Timoshenko's full wrath. Not as bad as it could be. C sub B takes into account this non-uniform bending moment. Our subtest stuff is this thing here. Page 199 C gives you these numbers for the beams, and now we're through with the elastic region. Yeah, I mean, I didn't even start it. And I mean, when you're just reading it, all of a sudden it says, if the moment's on, so blah, blah, blah, blah, and all of a sudden you're lost, what's going on? Big ol' red line. New stuff. If the moment when lateral torsion buckling curse is greater than the moment corresponding to first yield, then we find out where not in the elastic behavior region, because you went past first yield, you're in the inelastic region. And we got a new theory. Let's Timoshenko's again. First off, there is a thing called M sub R. It could have been called M sub Y for M sub yield, because it really is M sub yield. The only thing is that's M sub yield. See how stress time section modulus, yield stress, that would be M yield. And because when they roll these beams, this thing's hot, hot, hot, hot, hot, hot, hot, hot, hot, hot. Then these flanges coo pretty quickly, because there's three sides exposed to the air, and the web coos pretty quickly. And when this coos, it solidifies. And this stuff here is still not solidified. So it's still trying to get shorter. And these people says, who's over there pulling on us trying to get shorter? And these guys say, well, the guy in the middle, he's still hot. He's trying to shrink and we don't want to shrink. You get residual stresses in the beam, so that when you bend it, it's at the worst place too on the outside fibers, when you bend the beam, you really don't get F sub Y S sub X at yield. What you really get is you get about seven tenths of that. They find that by test. So we can't hardly call this M yield, because that's M yield. So they call it M sub R. It corresponds, it corresponds to the point where L sub R occurred, where the radius of gyration was causing problems and causing this curve not to be correct anymore. And the numbers are, first off, this is L sub R. Again, by theory and test, the length of the beam past which you are in the Timoshenko range of doing business 1.95 blotty blotty blot square, blotty blot square root of square root of square root squared. That's how long the beam is. Probably being inches, because you're going to multiply this in inches, pounds per square inch, inches cubed inches. So you'll have to take that divided by 12 to get numbers that are normally listed for you in the book. And then here's your equation. All you got to know is the plastic moment. That makes a lot of sense, because that's where we're starting at plastic moment. Then you should drop straight line. You should drop rise over run. The rise is M sub P minus M sub R. You say it's a drop divided by the run is L sub R minus L sub P. It's just the slope of that line. And so here it is M sub P start drop starting value is M sub P minus M sub R. That's the height. And here's how far you're going to go across. You're going to go L sub B minus L P. So you're only going to go maybe to here. Here is your L B L B minus L P. You're only moving that far down that slope. And therefore I want you to drop that slope, but I want you to drop only that that much drop. Here's where you see it right here. Here's your rise. Here's your run. And this is how far down that slope you're going to go. Nothing but the equation of a straight line. However, you see this bozo right here. What is that? It's a correction factor. It can correct you right out of business. I mean, you put the right loads on there and this correction factor can be like a six. So he'll take a little moment and tell you you can multiply it by six and it'll be twice the plastic moment. These people got to end up less than M plastic. Period. Note M sub R. Okay, some people want to just go find this out of the table and it's in the table. But this equation, you'll notice doesn't have the fees in it yet. The fees come later on. So if you look this up in the book, you may miss the fact that on table three dash 19, you see M sub R, but it's M sub R times point nine. This doesn't have the point nine applied yet. Be careful. You don't pick this number up and put it here. And L sub P is wow, look at that. That one got easy. That's nice. 1.76 razor gyration about the y axis, weak axis, square root of E over F sub Y. That's where L sub P is located. You need L sub P. So you know this, you need L sub R. So you got this. So you know which equation to use. Then finally, he didn't even have it. But the last thing on this road, you know, we were, we were working with this, then we work with this. Now we need this. So if it's a summary, here's a summary. Im nominal, it's equal to plastic, it's equal to this. There's the page you'll find these equations on. There's where I've got those equations. And the equation listed is F two dash one. And it's good for this region where your bracing is closer together than LP. I'll leave it for you to read the summary. Summary is nothing but what we already did. All right, he wants to know the flexural strength of 14 by 68, 8242 steel. What is F sub Y for 8242 steel? One of those steel where it depends on how thick the flange is. Some metals, when you roll them down, they get stronger and stronger and stronger because you exclude more of the impurities and you roll the grains along the direction of the load. Some don't aren't sensitive in that respect. 8242 steel, you're gonna have to go tell me how thick the flanges are before I can tell you the stress, the yield stress on that one. He has three cases. Continuous lateral support, unbraced length of 20, unbraced length of 30. C sub B is a one. I say, where'd you get that from? He says, look, just let me assume it's one now. We'll get into cases where it's not one. All right. First, determine the yield stress of this. We refer to table 2.4. That's not table 2.4. That's not table 2.4. There's table 2.4. There's a 242 steel. It's got three choices, 42, 46 and 50, Jkl. J says that the flange is greater than two inches. K is if it's between one and a half and two. I is if it's less than or equal to one and a half. So let's go look at a W14 by 68, W14 by 68, W14 by 68. Here we go. W14 by 68 has a web less than one and a half, has a 10 inch flange. What we're looking for? No, what are we looking for? The thickness, that's what we rolled down to that thickness. That's right. Less than an inch and a half. Therefore, that flange will be quite strong. You have a high F sub y. The F sub y for that one would be 50 Ksi steel. Here he's going to check his ratios to see if it's compact. You'll find it is compact. He'll also see it has no little mark on the beam. It's compact for all shapes, for webs. It's for this one, it's compact, continuous, supported. All we're going to do is use this equation. We're going to have the full 50 times the full Z sub x listed in the table. It gives us this much strength nominally. Then we will take our nominal strength and get it into a design strength 431. Did he tell us the load? No, he's just asking us for the flexural strength. That's all he wants. Now, let me show you. This is Timoshenko. This is how he tested. Moment signal on each end gives you no shear diagram because there's no reactions. Every little fiber is up here with a moment on it and the entire top of the entire beam is just screaming and trying to buckle like crazy. That's because your stresses are 20, 20, 20, 20, 20, 20, 20. C Cb is a one you get no present. You deserve what you get. You deserve Timoshenko's full equation. Now then if you put M on this end and M over 2 on this end, the M caused 20 ksi, so M over 2 would cause 10. So these fibers are screaming that they're dying and want to pop over to the side to relieve the pressure. But these people down here, they don't know what the problem is. They don't think there's a problem. And now then the stresses are 20, 18, 15, 12, 10. It's not nearly as severe. You have a present coming. Your C Cb should be greater than one and you should be able to put more load on the beam than Timoshenko suggested because of that. And we'll get into that next time. I'll show you the C Cb equation. I can find it before it's time to go. Yeah, C Cb, Cb, Cb, Cb, you can prepare C Cb. Here it is. C Cb 12.5 m max divided by 2.5 m max plus three times the moment at the quarter point plus four times the moment in the middle plus three times the moment at the three quarter point. All right. He's in a hurry. We need to go. Yes, sir. Sure. Sure. Sure. Timoshenko's equation. Yeah, it's in here. It's this one right there. Yeah. No, no, no, no. That's the straight line. Okay, then it must be. Well, now you're not going to see Timoshenko's. Remember, I told you that they found one of rearranging the terms and things they liked better. This is it right here. It's identical. You run Timoshenko's app with the numbers in. You run this and that gets the same number. Cool. It looks the same. No. Okay. So when do you choose whether to use f2.4 or f2.3 in this section? Well, it depends on L sub b. If your bracing length is in there, you use... Right. That's when you use this one. If it's in there, you use f2. And if it's in here, you use f3. No, no, f3 recalls on f4. See the f-critical? That's where you get it from. So that's a pair. They come as a pair. Yes, sir. I got to be any later once you just hand it in Friday. Well, it's not okay. I'm just saying it won't be any less late. I know how you feel. I lost my watch the other day, man. I thought I was going to die. Who's going to tell me when to go home? I don't know. Yes, sir. Equations are only valid for... That is, that's correct. Now, it's still valid with a correction. But without a correction, it's no good at all. Okay. So we have a distributed load where we have a varying moment. Then we're going to use Timoshenko with a correction. Yeah. Because you didn't bend all the stresses as badly as you could have. And Timoshenko's equation assumed that you did.