 Okay. Problem one, getting ready for the later problems. I ask you just to get the easy part. Just to find the plastic section modulus Z for the shape shown. Z is nothing more. You have something that looks like this and this cuts the two pieces into equal parts. Z is nothing more than a1 times that distance plus area 2 times that distance plus area 3 times that distance. So if you can calculate the area of a rectangle and if you can calculate how far it is from its center you should get full credit on this problem. So first thing I did is I found, I don't remember, I had some channels in there and I didn't know what I was going to do to tell you the truth. So I got elastic moments of inertia, elastic moments of inertia. I got a plastic moment of inertia of the channel. I got where the channel was, the area of a channel, then I said okay maybe I'll see what I really need here. I'm running out of paper. Z sub xx, number one, about the xx axis. The xx axis due to symmetry obviously splits the part into two equal pieces. So that is not only the elastic neutral axis but the plastic neutral axis. Then it was my job to go find, for example, the section modulus of a channel about its strong axis. I said I think that's already in the book and I looked up here and I said sure enough there's z sub xx is for the channel. So I just wrote that down. There's only one of them so it's going to have to be multiplied by, because there's two of them, two later on. Then I have a plate and the plate is up here. The size of the channel is a 12 by 30 They're not all exactly 12 inches or whatever the number is but this one happens to be, most of them are. So that was 12 inches. The plate was sitting on top of a 12 inch item. It was a 1 by 10 plate, which meant it was 10 square inches of steel. So there's the area two. I wrote down 10 square inches of steel. I had to go six inches to get to the bottom and a half inch to get to the center of the plate. Half a 12 plus a half of an inch. Added all up multiplied out. I got z sub xx 197.6. I'm now two minutes deep in the quiz. Now I need, now I know it takes you longer than that because it takes a while. What does this person asking for? It does. And I'm ahead of you on that game because I think about it and then when I write it down I know what I'm asking. Next I'm going to find the plastic moment, the section, not the plastic moment. I didn't mind if you got that for me, but I didn't really grade it at this stage. You needed it on the next page so shouldn't have points taken off. And that's one thing that you may find. That's the main place you ought to be looking for points. If you got the wrong steel and I put a big mark on it and wrote dumb person and took off three here, you know I grade all the problem one and then I grade all the problem two. If I did the same thing to you on the second sheet you already paid for your sins here. So you need to watch if you've got the wrong steel on the next one. You didn't even need the steel on this one that you didn't get counted off twice for getting the wrong steel. So then I draw the plastic neutral axis where it divides the two pieces evenly and I see there's a plate that has a plastic section about the yy axis. I didn't remember that equation so turn it the other way the way I had it. So I just took the area of the top half of a plate times the distance from its centroid that's his personal z. There's one, two, three, there's four of those. So I had to multiply one times five times two and a half times four. Then to get the z of the channel there's the area we're discussing. It is not at five. It's actually out here at five minus b sub f in the manual plus x bar the distance from the back to the centroid of the shape. Now yeah that requires a little special knowledge something that not everybody can do but you sure should be able to. This thing centroid lies at this point. Now that cross-sectional area times the distance from its centroid to the neutral axis plastic neutral axis is its z. You say well where's the z about its own axis? Well now you're getting into elastic type of work where you add ad squared depending on where something is and moment of inertia of its own thing about its own axis. That doesn't work in plastic. Plastic is just where what's the area and where is it off of the axis. Those two numbers multiplied is the final answer for its z contribution. So you had four of these plates and two of these channels. I had a five by one times two and a half plate. There's four of them in the section. Then I had a 8.81 square inch channel. One of them was five minus the b sub f plus it would came a little too far back. You had to go back out x bar and then there were times two of them. Gave you 94. Told you of course which the weak axis was. Off by a factor of two or three. So the y y axis is the weak axis. Here is where I got the dimensions from. Here's the area of a c12 by 30. There's its 12 depth. Here's that b sub f that we came back 3.17 from the tip of the channel. Here are elastic moments of inertia if you ever happen to need them. Here is that x bar when we came to the edge of the plate. That was five. We subtracted the 3.17. We came back too far. Had to go back out x bar .674 inches to get that distance. Some of you, it's okay with me. You calculated the s, the section modulus is z for the channel two rather than using the one in the book. And that's okay. I'll take it either way. It takes a little more time. I had a 3.17 flange. It was that thick. It had a 12 inch minus .501 plate in it. Times the distance which is half of that distance up. If you run the numbers out, you get 33.68 and they list 33.8. Same numbers as far as engineering is concerned. Both are fine. There's is more accurate because this really kind of looks like this. But not a big deal. And I broke that out of this problem hoping that telling in or giving you a chance because it would be too late if you couldn't do the plastic moment. I said if you just can't get the plastic moment and you have no idea what I'm even talking about, you know, mine doesn't say that. Wasn't gonna, wasn't gonna give you that. But yours says the welded section shown plastic section much about x and y. You needed it because we're getting ready to use bending about both axis and the interaction equation. There's what the channels. There's the plates not to scale. If you just can't do it, you can put z is 1000 for x and 200 for y. I mean they weren't right. And I want you know, you know you're gonna lose some points if you can't do that. But if you say I can't work to because I can't get one, that's a fair argument. So and many of you did this, a couple of you got it right and you used mine anyway, said why take a chance? He's obviously if I get it wrong, I may get something else wrong. If I use his numbers bound to be acceptable from that point forward, not a bad, not a bad point of view. Problem two, these are both a 36 steel because those of you that use 50 ksi steel, they can be bought in 50 ksi steel, but that's not what I wanted. And it said on the quiz. I don't know where it says on the quiz. But anyway, supposed to use the right kind of steel, no matter. Taking that off me thought by now I didn't have to put that on there. Got a uniform vertical service load of 10 kip per foot and kip per foot. That's not right. It's another change that I made on yours. It says I didn't want you to have to factor it. So just one less thing you have to do. It's got a uniform factored already gravity load of 10 kip per foot. In other words, that's 1.2 dead plus 1.6 live. Okay, somebody did that work before you got here 10 kip per foot. It also includes the way to the beam. Have you compute the way to the beam? Didn't do it. It's a factored concentrated horizontal load applied at the center of the beam. Call it Q or whatever you like. Then I said, because it's a closed shape with thick elements, you know, I really didn't have to tell you why, but that is why it's got tremendous torsional resistance because it's closed like a tube and such thick things 1-H thick stuff like that is not going to have any buckling. But regardless, and the loads go through the shear center. You can assume it's compact. It's not subject to lateral torsional buckling, flanged local buckling, or web local buckling. Somebody designed the bearing pads so you don't have to check web shear. There won't be any buckling down here around the ends. All of that taken care of by someone else. How do I know you're learning something in here? You did a marvelous job of checking for shear and designing a base plate even though it wasn't a column. Concrete I'll write it on his exam. All of a sudden I see 1.7 FC prime. Isn't any concrete for miles around here? Either problem. So here we go. First, we know z sub x from problem one, either mine or yours. You're going to have to give me the right f sub y for a 36 steel. You're going to have to take nine tenths so that you can tell me the capacity of the beam to handle a moment about the x-axis. I was wondering, I was no somewhere in here, you gave me those amplification factors. Man, I had amplification factors coming out my ears and I was wondering, did I tell you not to use amplification factors? You know, I didn't think about it. There's no axial load. So there's nothing to amplify. So from problem one, the capacity of the beam about the x-axis, nine tenths, proper f sub y, right off the previous page, 200 or 1,000, take your choice, got this many inch kips. Any of you mixed inch kips and foot kips and quite a few of your calculations, you got to be careful with that. Then about the y-axis, same equation. And you have 30, 49. Now the weak axis bending has a little side requirement on it. It's on page 16.1-55 that not only can you not overload the beam, you can't over deflect the beam. And it is 1.6 times a yield times the elastic section modulus. Elastic section modulus is I over C. So I am going to have to go get the elastic moment of inertia about the y-axis before it's over with. Let me see. Oh, here it is. It's down on this page. So it's okay. It's coming up. So here it is. Here's the elastic section modulus about the weak axis. I have moment of inertia elastic for the channels because they're here about their own personal centroids. I have some elastic moment of inertia for the channels due to their transfer moment of inertia. Looking back at a little bigger picture, here's the y-axis. I have a moment of inertia of this channel about its own personal centroid. And I have an AD squared, not an AD as in plastic, an AD squared for that channel moment of inertia. And the moment of inertia of the plates are just base times height cubed over 12. And then of course those numbers will be times 2. Here's the moment of inertia of the channel just because it's there. 5.14 out of the table. Transfer moment of inertia, the AD squared is the area of the channel. This is the number we found was the distance from the centroid of the channel to the neutral axis. So I just pulled it again from problem 1, AD squared. And the moment of inertia of a single plate BH cubed over 12, those of you that got these numbers reversed, I think you got 0.833 or something like that. A plate standing up about its tall axis like that's not going to have little numbers like 0.8 base height cubed over 12. Total of everything in sight and admitting there's another plate on the other side and another channel on the other side, 287.4 inches to the 4th. So I can calculate the elastic section modulus I over C. That's 287.4. When you look at the plate or at the section bent about the weak axis, the C distance, how far can you go out without running out of steel? 5 inches. 5 inches. There's your elastic section modulus. You already thought you could have about the weak axis 3049 to prevent deflection problems 0.9 times 1.6 times 36 times 57.48. Turns out it did control slightly. You couldn't tell the difference in the real world, but I'd still want you to show me you calculated the number. So that's our capacity about the y-axis. Have the capacity about the strong 6400. Have the capacity about the weak 97, 29, 79 inch kips. Now we need to know our requests. Who wants to suck strength out of this beam? It should be so simple to tell me the requests. So uniform load, the equation for, you didn't have to draw a shear in a moment diagram. You can. W squared over 8. It's on page 3-213. Gives you moment, gives you a max moments in beams loaded in all kinds of ways. The load was W, 10. The length was 16 feet divided by 8, 3840, or 320 kip feet. You've got to make sure these units are matching up with whatever you're working on, whatever you're doing. And the request for extraction of bending strength, P L over 4 for a concentrated load in the middle of a beam. Of course, we don't know P, so we've just got to let it go along for the ride. It would be 48 P. This would be an inch kips divided by 12. This would be in foot kips, just as long as you're all of the numbers in your interaction equation are consistent. Doesn't matter. Those of you that said P sub U over P, P sub axial, P sub nominal, is greater than 0.2, you're on the wrong road. There is none. Those of you who wrote that with P equals to zero, and therefore this is the interaction equation, that's good. You got an eight nights in your final work for me. You got something wrong there. Here's your request in kip feet or inch kips. Here are the supplies that we've had available. And P is still running around loosening our equations, setting it less than or equal to one, setting it equal to one, 24.8 kips. Finally, so you don't know anything about plastic stuff. Well, you should know the elastic part, and there is some part of our work that we do elastically, kind of. We have columns. Columns have strengths. They have break points, and they have equations to be used if you're to the left or to the right of the break points. The break point is 4.71 square root of E over F sub y. So you have a break point of 134 on this kL over R axis. All you got to know is R. And as you notice on mine, I don't have that. I thought maybe they won't remember. Hint, R is the square root of I over A. Those of you who thought, why do you tell me such stupid things? Why don't you tell me how to get Z? So square root of I over A, you already got I. If you didn't run that equation before, then you'll have to run the moment of inertia calculations now. We got 287.4 from problem 2. Cross section area is 2 channels plus 2 plates. R is 2.76. So the kL over R is pen-pinned. Those of you who pulled numbers out of the tables in the back, that's fine, but it's still pen-pinned. You say, well, I was using practical instead of theoretical. It's still 1.0 pen-pinned. Length was 16 feet times 12 inches in a foot. Some of you didn't put the 12 inches in a foot. I don't mind that. But you're now in feet. You must change this to feet. And anything else down in here to find the breakpoint, you're going to have to find it in feet. Your units have to be consistent whatever you decide on. So 69.5 was your kL over R. The breakpoint for this steel was 134. So you were down here about halfway. So you're in that equation that has the thing to the power, as opposed to the .877 ff Euler. This one. You knew pi squared. You knew E. You knew 69.5 for the kL over R. It was 52.56. You knew the critical buckling stress was this. And you run all that out. You get 27 for the critical. The nominal then is equal to the critical times the area. It's 10.50. And then you got to put a .9 on it. It gives you a load carrying capacity of the column 945 kips. A couple of you said, if you don't mind, I'd like to not do all that work. How did you not do all that work? Well, the person who was smart enough not to do all that work is not going to brag about it or they didn't come. There's a table. It tells you the answer. It says, well, it doesn't have a breakpoint. It's in there. I don't know where it is. But once the kL over R gets to a certain number, he stops using one equation. He starts using the other equation. As soon as you found the kL over R, 69.5, if you want to know the final critical stress, including the fee, there it is. For 36 ksi between 69 and 70, we had 69.5. I can even do the interpolation. 25.1. Therefore, all I do is I take 25.1 times the area 944 kips. That table's always been there. Okay. Now, the grades, just because the grades were curved, I mean the grades are still, or in any way cast in stone, still open to any problems. I know there's no doubt that I've bound to gotten some of you because I caught myself doing it. Every time I saw you use the wrong steel, I said minus three, minus three every time. Then I'd go back when I'd be adding up the points. I said, didn't I just take off points for that? Yeah. And there was something else. I forget what else. Okay. Every now and then I would catch where you got the wrong plastic moment and you paid for your sins. And then later on, when you put those plastic moments in your calculations, I said, that's wrong, but you know, you already paid for them. So if you see that you have points taken off because you got the wrong plastic moment in problem one, and then they bit you again in problem two, you wrote some refund. Are you going to go to heaven? I hope. I hope not right away, but I'm not sure where that came from, but oh yeah. Well, by that, I mean, you know, you already lost points on the problem once for that mistake. You shouldn't lose points for the same mistake a second time. Yeah. All right. Nobody made a hundred. I couldn't believe that. The person who made 95, I'm really embarrassed for you. I really am. You know who you are. Dumb mistakes. Questions. How about how something works? All right. I will see you, Monday. That's okay. Well, not quite a few late. It's quite a few not going to come today. Oh, is that what it is? All right. Last name. Oh, wait, that's not your name. No, that's not your paper. But I got to give, do something to you for coming in late. And I can't get away with that twice. Park. Way below what you capable of. Name. Okay. Let me, let me get rid of the quizzes. Yes, sir. Name 42. I can answer that. Okay. Now we're back to this class. Yes, sir. That's, that's not expected. That's statistic from what happened in the past. You will notice that people who have that grade, if the last pulled out of B, they just decided to start. There weren't a high number of them. No. But what can I say about you personally? Nothing. All I can tell you is to help you make your decision in the past, what you got on A plus what you got on B average together. This was the mean of those people. Some did better, some did worse. I don't know. Go look at the, read the syllabus. Do you, have you not read the table where it says what chance you may make a certain grade in this class? I'm sorry. Have you, yes or no? What? Have you, yes or no? You have read that little, that little table? Yeah. Whatever that says, that was the statistical mean of what people about two or three hundred of them have done in this class in the past. Okay. Your mileage may vary. Yes. If you email me. Yes, I'm the one who sent you the information. You didn't give me all the information. Not yet. As soon as you do, I'll, I'll add you an override. Okay. I was a couple CRNs just in case they filled up or. Yeah, that'd be a good idea. Okay. Well, that's what, well, I may not have said it on that, but on the force request thing, it always says, give me the CRNs you want in order. In order. And you're right. And then I go in and I take your first one and it's full. Then I take the second one and it doesn't meet your schedule. Well, then I quit. Okay. But, you know, and I'll try them in the order that you give me. Now, you got to make sure that you can meet the Friday lab. That, that has to be done. Yes. And you are graduating. Yes. Okay. It's supposed to be. You will, you will. They'll find a way, one way or another. Okay. Okay. Sure thing. Yes, sir. Okay. You didn't care how it was worked, huh? Last name. Reeves, Reeves, Reeves, Reeves, Reeves. Wait, that's not yours. Okay. Reeves, Reeves. That isn't you. What happened? You're going to give me, you're going to give me some more studying on the final, aren't you? Oh, of course. Okay. All right.