 Okay, we were talking about shearing capacity of bolted connections. Now then we're going to cover their tensile capacity. He shows you, for example, a hanger or a hanger bracket. Probably got a hole or several holes drilled in it so that you can connect to it. It's bolted to the bottom of a wide flange. Puts the bolts in pure tension. Some of the bolts, if you remember, we had a table which told us how much tension they insist we put in them. Maybe 18 kips, maybe you have 65 kips of load, maybe you have 4 bolts in all, maybe that adds up more or less. You know, as long as the load goes in the bolts and the bolts aren't overstressed, I'm not too worried about it. When you put this force in the bolt, when you crank down the nut and stretch the bolt, you put tension in the bolt, obviously. That also means that you put compression between these two surfaces. Now, one thing that's good about that is if you do decide to also add some shear stress on here, that load will be taken by the friction, unless you don't care, if you don't snug them or don't tighten them up tight enough, then they will slip and they'll bear against the bolts and that's okay too. We can design for any of those conditions. But how bolts work, if you will draw a free body just to the top half of this pair plates with the bolt connecting them. You'll notice that cutting through the bolts, you see a tension in the bolt. Your choice, 1 kip, 10 kip, 18 kips, 39 kips, whatever you decide to put in there. And that force has to equal to this force that's distributed around the hole. Now, I don't know how far this goes out and I doubt seriously if it looks like that. More than likely it's high around the edge of the hole and where the bolt is and then drops off somehow for some distance. Don't know for sure, it would depend on the thickness of the plates, the modulus of elasticity of the material, in this case a single number. And it would probably be circular around the hole. Drawing it kind of like this. If it looks kind of like that, then I say circular around the hole. Well, it probably wouldn't be circular around the hole because it looks like you ran out of steel before. And so, but somehow there's a force compressing these two plates together because you tensioned the bolt. Bottom half, same idea. Head, nut. When you did that, you got some deflection of the bolt. For example, if you had something that had, I don't know how many, 20 threads per inch. And you really put the muscle to it and rotated the nut one full rotation. Then you moved one twentieth of an inch. You put that much stretch in the bolt. Not quite true because you got a little deformation due to the shear inside the head. You got a little deformation inside of the threads where they're connected to the nut. You had a little compression of the plates, but pretty much you had about one twentieth of an inch of extension, elongation, delta of the bolt itself that's connecting these two plates. Now at that time, if I draw you a free body, I would show you that somebody else has decided that they are going to pull on the plate. And that's why you put the bolts in there and the hanger bracket, of course, with some force. When you do that, you're tending to open up these two plates. He doesn't show you a way to do it here. You're going to have to do it somehow. Somehow, very possibly, this goes on down the road. This was a T, maybe something like that. Here's the top plate. Looks like that top plate. You know, but somehow you're pulling these two plates apart around these bolts. And what's going to happen is when you do pull on this bolt, you may stretch the bolt a little more. Let's just say delta B. It was already stretched one twentieth of an inch because you cracked a full turn on the nut. So it guaranteed went one twentieth of an inch at twenty threads per inch. And I'm noticing that after you put this load on here, these two plates are still in contact. They haven't separated, which means you haven't put just a horrible load on there. You put enough load and you'll finally just separate the two plates. But under most loads, these two plates are still in contact. Therefore, you told me when you put a little load on here, maybe thirty kips, the bolt you feel may have stretched a little longer. And I say, could be. How much? I don't know. But if they, if the bolt did stretch, then the plate who used to be here has now moved to a new position down here because it was under compression and now you are decompressing it. Which means that the deflection of flange, I guess, this is the plate, is the same as the deflection in the bolt because they're still in contact. Now I've got some equations for delta B and I've got some equations for pieces of steel that have a thickness T or L and here those equations are right out of my 305 book. Did it today. Deflection of the plate is equal to P L over A E, where you subscript it. This is the deflection of the plate and the other thing we call it delta. Probably should have done it here. Deflection of the plate, length of the plate, area of the plate, E of the plate. This is the area of the item under compression. This is the property of the material being connected. This would be a thickness of the plate. That's how long that piece of steel was, how long that piece of T on the flange was and the change in load in the plate during you putting this external load on the plate. The corresponding deflection of the bolt was whatever delta load you put in the bolt, times the original length of the bolt, area of the bolt, E of the bolt. Now anytime you say I don't see that, just say I don't agree until I can talk with you about it some more but I'm telling you that when you pulled on the hanger, the hanger pulled the bolt and stretched it and by stretching it you allowed the two plates to decompress a little bit and those two deflections have to be the same. They won't be the same when the plate finally fully decompresses and has no pressure on it between the hanger, bracket and the flange of what it's connected to. But up until then, and I'm not going to be happy if you do that anyway, then I can tell you that P L over A E for the bolt is equal to P L over A E for the plate in generic terms. For your case, your mileage may vary. Then I saw for P, delta P of the bolt divided by delta P of the plate is equal to L of the plate over A plate E plate divided by L bolt area times area bolt E bolt. And I don't know why but it seemed like a good thing to go ahead and put a little more information here. This area of the bolt is the area of the bolt that's in tension which of course is just the entire area. And this area is the area of the plate that's in compression. That's that big round circle that got lopped off because it didn't go out far enough for a nice circle to develop. And I'm going to tell you that in general the length of the plate divided by the length of the bolt is probably relatively small. The length of the plate that we were looking at would be this long, that's the plate we're talking about. The length of the bolt is the length between the bottom and the top. The length of the matter is the bolt will be in tension up inside the head. There will be a little bit of stretch in there and there will be a little length stretched down inside the nut. So I'm not going to say what it is because I don't know, I'm just going to say it's small, probably at least be a half. Then there's the area in tension in the bolt and an area in the plate that was compressed after you tightened the bolt and before I brought the load to put on the hanger bracket. Here was the area of the bolt, here was the area of the plate. Plates weren't perfectly flat to begin with so the compression zone under there isn't perfectly round either. Probably pretty high compressive stresses in here, probably pretty low until they reach zero. But I'll tell you that this number is .485 square inches or pi D squared over four. The area here has got a radius, I don't know, stick your finger in there, let's tighten it down, you know, and see how far out before you say okay, it doesn't hurt when you tighten down a bolt, but that's probably got a radius of two inches, three inches. The bolts are three inches apart generally. So I would say that that number is very small. And the last number says E of the bolt over E of the plate, they're both the same so it's a one. So I'm telling you that as you change the force on the outside that the change in load in the bolt versus the change in load in the plate got to be a very small number. Now your text goes that far, so GUI goes that far, then he throws down about another page I think of numbers and calculations and ratios and this and that and the other. But all of us have to get down to the point where we say we don't know. Assuming this and assuming that and assuming this, then he can come up with what we think is probably right. Then he says, very verified by test. Are you going to verify by test that that ratio is a very small number? Well, I'm just going to do it now. I'll go ahead and I'm going to take sensors. I'm going to learn about how you find the load inside of a bolt with a strain gauge and I'll give it to somebody with some money and he's going to go take these things and he's going to do exactly what we just showed. We're going to take this thing. We're going to tighten up the bolt to 18 kips. While we're at it, we'll probably measure how accurate we were able to get the 18 kips so that somebody else can get a fee to go with that number. Once he's got the proper 18 kips measured in the bolt and the 18 kips measured in the bolt, 18 times 4, I know the compression between there, 18, 18, 36, 72. So there's 72 kips of load in here. I'm going to come out here and I'm going to pull on it with 65 kips and I'm going to see the change in load in the bolts. And what we find by experiment, the change in the bolt is only about 5 to 10 percent more because almost all of the difference, the equilibrium of the thing is because you decompress this compression area right here. As you put the load on the bolt, the bolt says, go ahead and do something. I said, I just put on 10 kips. He says, no, I don't think so. I didn't feel it, but you lost 10 kips right around in here. So you put 20 kips on there and he said, you feel that, don't you? He says, what? No, I don't feel anything. Since you start really pulling the load out of here, there isn't a lot left to take out of there. Well, some of the bolts are going to start, some of the load is going to stop coming out of the previously compressed pressure area and go into the bolt. But about 10 percent above installed is how much you're going to get. So when the plates finally separate under attention load, the tension now in the bolt is usually smaller than about 10 percent higher than installed. And they say, not enough to worry about. Not to mention the fact that that's when the plates separate and we're really not going to be happy if you do that. Now here's how they do it. Dr. Jones, he has imported strain gauges from Japan for years, sells them on the open market. One of the niffier ones they make, I don't know who else may make these. You can tell they're from Japan. Gauge series is recommendable if an ordinary strain gauge cannot be mounted on the bolt surface. What they do is they drill a little tiny hole inside the bolt. They get away from the threads and they get away from the head of the bolt. They stick some glue down there and they shove a strain gauge. I think I've discussed with you before what a strain gauge was because we were talking about checking something out. It's a very fine wire. Maybe generally they have 120 or 300 ohms. I'm not sure where those numbers came from. You can get them in other omeges. But a plus or minus five volts to it, when it's not strange, you get 300 ohms. When you put tension on the bolt, you stretch the bolt, you stretch the gauge. When you elongate the gauge, the wire is so fine, you get quite a difference in resistance and the change in resistance is directly be able to calibrate it to the tension stress or the tension force in the rod. Here they say you probably ought to get one of these if you can't put these on the bolt surface. Well, putting them on the bolt surface is a problem because if you put it on this side and you don't get uniform load under the head, well then you get more stretching here than you do there. So you put one on both sides and you figure out a way to take the average of these two or you put four of them. Usually cheaper just to buy one of these or either that or do it yourself. They'll sell you the little gauge. You can drill your own hole because a lot of times you're not testing a standard bolt that they make. Although you can send them to them and they'll make you one. Here's the good part about it. Well, number one, it's kind of nice that you get to design the bolt just based on your preload that you put in the bolt in the first place. So it makes the design very simple. Here is the load that I put on this connection. It rattled around. It went up and down and it reached maybe every now and then 22 kips, 21 kips, and then it's around in there. That was the biggest number I ever saw. This particular bolt had 28 kips of tension in it. And here is the load in the bolt because as you change the load, the only change in the load came between the two compressed surfaces, opening a little bit and closing a little bit. It didn't really open but it just decompressed. That means that the bolt doesn't feel any change in the load and so you don't have fatigue in the bolt. This really would be a problem otherwise. Or it could be, you can design against it. And there's this derivation. Delta is equal to PL over 8E, blah, blah, blah. Of course in the bolt about 10% larger under extreme conditions. Prying action is when you pull on this T, you tend to bend these surfaces. So to keep these in equilibrium, there's actually a moment in there. It's like everything else, there's no end to how many things we could cover. And in graduate school you'll still not cover 20% of everything out there. This is one of the things we don't cover is prying action. So if it says in the problem include prying action, ignore. It's able to be derived, able to be referred back to the specifications. Just like anything else, just like everything else we have done. Combine shear and tension. Now then rather than just having the bolt subject to shear stresses or just tension stresses, now they're going to be loaded about two axes. No, no, no, that's a beam, isn't it? They're going to be loaded in two modes, shear and tension. Generally speaking, having seen what these people always tend to do in the past, they usually talk about how much strength is available in one mode, bending about the YY axis. And did you take a percentage of that stress out of the system? That available? Yeah. Did you take it all? Yeah. Well, then you don't have any left for bending about the other axis. That it's okay. What is that, like an interaction between the two discussions, the two kinds of failure? Yeah, they do the same thing exactly in bolts. All they need to know is what interaction equation best fits the situation. And again, they're going to find that out by test. Here, for example, there's a 60 kip load, I don't know if it's factored or unfactored, I do know it's sitting at a 345 angle so that the tension request, perhaps yet to be factored, don't know, is four-fifths of the original load. And the shear is three-fifths of the original load. And if it goes through the centroid of the connection, I would be comfortable telling you that this load will be distributed equally among the four bolts because the tension load is applied right there. And if the load is right down that shearing surface, I would be comfortable telling you that the shear load will be carried equally by these four bolts because that's what we've been doing for the past week. We've been throwing those loads into the bolts equally. I usually, although I don't always get a response, but usually somebody will say, I don't think that's quite true. Say, because I remember you had some kind of a goofy diagram, had four numbers written there, four numbers there, four numbers there, four numbers there, and you had different loads in those bolts. Well, that's true. That was because someone was trying to see if you really knew what was going on, and they let one of these bearing capacities drop down below the bolt capacity. And that's OK. That's fair game in life, and it's fair game on an exam. But most of the time, in most designs, you don't do that. Most of the times you make the shear and the bolts control, and then you come back and you just check it for bearing stresses. And when all of a sudden, this little plug is going to pull out before that bolt shears, you usually change that bolt to move it over there so that bearing doesn't control. Makes the design easier. It makes more efficient use of everything there. So we would go into a problem from now on, probably saying something like bearing is not a factor. Or at the very least, bearing will be 2.4 diameter of the bolt, thickness of the plate, f sub u, namely crushing controls. And you move the plate. You move the bolts around so that plug shear doesn't control. I could live with that, too. But even in that case, if you ever found where this crushing strength controlled, you'd probably be better off putting some more bolts in there, rather than letting that plug shear control the design and losing the strength you have with the bolts. Either that or good. But smaller bolts are not going to carry it anyway. I mean, they're not going to be used anyway. Now they find, by a test, Chesson at all 50 years ago, proposed and has been found to be quite accurate, an elliptical interaction equation. Whereas always, something comes in at some number. This would be the permitted available tensile stress in the bolts. You have that information. You may have it. I may not. Here we go. Here's the nominal strength of fasteners and threaded barts. Here is the type of bolt. And there is the tensile strength. This nominal still needs a fee. You have those numbers. And so if you ever have a bolt subjected to shear and tension, and then all of a sudden you say, incidentally, there isn't any shear, well, then I don't want you coming any higher than the permitted tension. Even if you find some combination of tension and shear, here's your shear, here's your tension, I don't want you going any higher than that tension number. I don't care what the equation says. Lipses don't do that, so that's probably not a problem. So the ellipse comes in at the permitted available nominal tensile stress. And they can also be forced. It would be your choice. They're using stress symbols here. And you should not exceed the permitted available shear stress. This is permitted tension stress with no shear. Permitted available shear stress with no tension. And then what you do in between is your choice. Circle, parabola, ellipse. You can do your own testing. But an ellipse works perfectly from the data that they've received. Equation of that ellipse is, you tell me how much tension is requested, divided by how much is permitted, square it, plus. You tell me how much shear stress you're planning on putting my bolt, divided by how much you could put in the bolt in the absence of tension. This is tension in the absence of shear. Square them is equal to 1. Or smaller, your bolt's a good bolt. We'll all stand behind you. It's not your bolt that caused it to fall down. The 180-mile-an-hour wind. Now, what the specs do in AISC, rather than trying to get ellipse numbers and get this and square that and do so and so, don't ask me why. In other words, I would just assume have the ellipse, and they'll still let you use the ellipse if you wanted to. They say, well, look, let's just go ahead and take f sub t. Give them a little bit here. The test sold, that's not going to hurt. And then just take a straight line down to this point where you shoot this curve straight up, and we'll tell you the equation of that line. And you say, OK, we're going to give them a little. Here, we're going to take a little away. They don't like it. They're welcome to use the ellipse itself. This book uses, and almost everybody does, three straight lines. Just exactly like you use one straight line, two straight lines, and an Euler formula. I mean a Timoshenko formula for lateral torsional buckling. Now, all we've got to do is see what that line and those lines give you. Here's the ellipse. This is only in the commentary. It's not in the main specs. It's on that page. Discusses, if you like it, go for it. It tells you what the terms are. Small f is the requested tensile strength, or stress. And cap f is how much is available. That would be f nominal times phi. So if you see it like this, the truth is that's phi times the nominal. And then for shear, same type of terms. If you could do that, they would not mind that at all. I doubt seriously if you're, I don't think I've ever seen anybody do that. I mean, they use one or the other. You know, I see what you're talking about. You're saying right here, you know you might be able to use a little smaller bolt. And here, ours would be more restricted. So here, you could get more stress. But my guess is that's not what you'd be doing after you get out. We've done that before, because I remember this thing bringing it up out of the 305 book. We said is sigma y the limit. And everybody says, yeah, you think it is. The truth is that some materials fail by different failure criteria. And this is not a bad one for steel, whereas rather than just worrying about the stress you put in it, you find out how much distortion energy you put in it, which case you really can. If you only put sigma y in it, you only get this much till it fails. But if you put some sigma y and some sigma tau in it, then you get to put more stress in it. So it's not unusual that the curves are not just straight line. Basically, though, just like before, we've got places we can't go. You remember back on this LTB curve, when you were down in here, you had a C sub b. And you were told you got to be careful. If that C sub b tells you, take the number you got out of the table and multiply it times 1.1, that was OK. But if he told you you could multiply that number times 1.6, that was not OK. You couldn't go there, because it was above the plastic moment. Well, the same thing is going to happen to us here. We're going to have to have some rational way to get started and to continue. What we usually do is we just start with the shear. Take the shear force, find out how much shear stress you're going to put in it. If you, this would be your F sub v axis, if your shear stress is above how much is available, available shear with no tension, then you've got to go home. That's not fair. You can't do it. So first off, we just make sure the bolts will take the load without exceeding the allowed, the permitted, the available shear stress with no tension. And it's easy. We just keep adding bolts until that happens. So maybe that now drops our shear stress down to this number. Now then you say, OK, did you take care of the tension load? I say, no, I didn't know there was some. I say, yeah, I don't remember. It was at a 3, 4, 5 angle. It had those two components. There's a tension load on it also. And I say, well, hang on. I think you can have some tension load. I don't think you took all the strength out when you put the shear stress in the bolts. So I come over here, and I'm going to have to know the equation. Incidentally, there's the equation of that line. F intention requested over permitted plus F and shear stress and shear permitted over, requested over permitted, the ratios added together is 1.3, or less. I'll say, yeah, you've got some tension stress left. And you say, great. How much tension stress do I have left? I simply solve for it out of here. Find out how much tension stress you have left. Tell you how much. And you say, well, that's not enough. I say, well, that's not my problem. Go get some bigger bolts, or use more bolts, or say, oh, I got you. The problem is, in a connection where you just put not too much shear at all, when you start doing this way, you start out of this shear, and then you go up to that line, and you've violated everything on Earth right now. You cannot go above how much is permitted with no shear, available tension with no shear. That's all you've got, period, and tension. So that's what we have to be careful of. Here is the equation. Here are the terms. Little f is the tension stress that you're requesting. Little f shear is how much shear stress you have been requested. That's a P over A kind of a term, tension, and shear. Here is, these are specification numbers that come out of a table. Oh, yeah. I don't know where it is right now, but I guarantee it's on the next page. But it's that same table, and maybe it's the same table that I was just showing you a minute ago, but yeah, there it is. It's that table right there. And that's what he's, look at there, look at there. There's f, some nominal tension. And there's f nominal shear right there. And, well, it's very easy. When you pick a number of shear and ask how much tension can you have, he's going to say, whoa, you're going to have way out here. And you're going to check it against this number right here in the table. And you're going to say, OK, not acceptable. And incidentally, that number is not in the table. There is a problem here. In other words, the whole thing we've been done is based on these are the nominal available shear stresses. This number right here says f sub t. And it's really easy to see what's wrong there. This is f nominal tension. And you've got to take f nominal tension and multiply it times feed to turn it into how much is really available. This is not available. But I've got a picture. There you go right there, for example. That's how much, that's the nominal strength of a bolt and shear or tension or whatever this stands for. But you can't have it because some of your bolts are going to only have that much. So you're going to have to knock that down by feed. And that does cause some problems or some confusion. It's not a problem, really. All right, to avoid going above the line, obviously you stay down less than that. Now, you can solve for f tension. Now, here's what we're going to do. We're going to assume you're going to start with shear. You're going to make sure your bolts handle the shear. And then after you pick some number of bolts, you may throw a couple more in there to get the shear stress in the bolts down so you get some tension. And then you need to know how much tension stress is left to see if your tension load can live with that. So it says if equation 74 is solved for the, this isn't required, I don't think. I like the word better, the remaining available for you after shear stress has been, the guy says, no, you don't write books, man. You get too many words. After the shear stress has been accounted for. That's how much we're looking for. We want to know how much is left for us. Solving for this, you multiply through the equation by f sub t there and f sub t there. That'd be that times that. There it is. That's still there. That's times f sub t, 1.3 f sub t. And that's what he has right here. f sub t is equal to 1.3 f, big f sub t, minus big f sub t over big f sub v, big f t over big f v, little f sub v. So if you see me do anything illegal, holler. But if you see me not doing something illegal, you can't say, well, why are you doing this? I just had to be like your mom because that's it. Now then, he uses a symbol, whatever that thing is, so that he can use that kind of symbol in that omega kind of symbol so everybody is happy with it. Fui, the available strength is phi times the nominal strength. Well, that's always true. Here's the nominal strength, but you can't use it. You want to talk about available numbers. These are available numbers and you have to multiply it times phi. Then if you are given the nominal strength as we are in our tables, then the nominal strength would be the available strength divided by phi. Check it right here. Here's your available strength is this. If you want to know how much nominal strength I need out of the tables, you're going to have to divide it by phi so I can go get the table number, where so-and-so, so-and-so. So what he says is if you don't mind, I'm going to divide everything by phi. I'll divide that term by phi and divide that term by phi and divide that term by phi. I said, why? You probably are thinking instead, why? Here's why. First off, you agree it's legal. Number two, what this does by doing, dividing this by phi, it turns the F sub t into nominal available tensile stress left for you after you count for shear stress and nominal is what's given in the tables. Or what you're looking for. This number is not in the tables. And since it's not in the tables, I might have to see how you're going to get a number out of the tables so you can drop it in this equation. F sub t divided by phi, there's your F sub n, here's your available. This is the number in the table. If you'll divide the number by phi, then you'll have the table number that I can pull out of the table. This turns F sub t, which is what you have up here, into a nominal F sub nominal t. See the difference? Subtle F sub t, F nominal t. Different, how much? Phi. It is. And you say, so why don't they just give us a table with all those numbers times 0.75? Only thing I can think of, the guy said one more page and the people won't, you know, they can't carry it around. So they didn't put it in there. But what we're doing is right. And that's the way everybody designs these things. So this number will be in the table. It's on page 16.1-120. I got it on page 429 AD. That is the tension number listed in the table. This is the tension permitted listed in the table. This number right here is not in the table at all. We're gonna somehow find a way to get this guy with a fee in it also. But renumbering these two table numbers, see these don't have the nominal in them here. So, gently, I don't remember where we used that or needed it. Ah, here we go. This was available shear. See, it already had the fee in this table. Cause he says it's available. Means it's good to go. There's your available tensile strength in bolts, but it's not in terms of stress. It's not in terms of a continuous curve. Uh-huh. You start talking about that kind of stuff where you're talking about plugging it into some kind of a straight line equation or an ellipse. He drops down from available. Now he says these are nominal strengths. They have not been multiplied by fee. He may tell you what fee ought to be. Nope, he doesn't. He says, that's not my job. I'm telling you the nominal strength. Has the proper name, F nominal tension. Let me go back to where we kind of left off here. We left off with some kind of a weird equation that had F sub t over fees equal to 1.3 that over that, minus that over that, that and that. This number right here you'll notice is 1.3. See the difference, that says F sub t, but this is the nominal strength. Had to divide this by 0.75 to turn it into F sub nominal t. And F sub nominal t, when you look in the book, that's all you got. And I want these numbers. I want these numbers in the equation so I can use them. Same thing happened here. F sub t over fee, well of course it's the same term. So it's F sub nominal tension. I was left over with an F sub v that I still got a problem with. That's not listed in our table. And then this number here is again, the request. It changed from the book that I have all the notes in from F sub v to F sub r v, where little r stood for one bolt and then v meant shear. But that you just get right by taking load over area. Take the shear load divided by the area of the bolts. That's the number that you're looking for right here. Finally, I say I got trouble with this thing here, people. That's got some fees going on. He says, well, do you agree that if I gave you the nominal shear, I'd have to multiply it times fee in order to get this? I say, yeah. He says, well, then does it become pretty obvious what I want you to put here so that you can pull this out of the table number? I said, this is not in the table. He says, there are no numbers, but nominal numbers in the table. Okay, this number's in the table, so I'll replace this with that. I finally end up with this really, this number right here used to be listed as F sub tension. That's, he's gonna change it to a symbol F prime, nominal tension. And the reason being, it stands for something. It's how much tension is left for you after you take care of the shear problems. And here is that substituted fee in sub V. The final, I guess that is the final version. Here's the order of operations. Number one, go tell me how much shear stress you put in your bolts. If you put more than the number in the table, you can't do it. If you have not put that much and you multiply that times the number in the table divided by 0.75 times the shear number in the table, then you can subtract that from 1.3 ends of T then maybe that's what that stands for. No, it doesn't. Number in the table? Nah, it doesn't, it's nominal in tension. Then that much tension is left for you. You're on that straight line. If on the other hand, this number that we told you was left for you was bigger than how much you had ever then you got to back off because you're on that straight line up to the left where it can't go. That's the order of operations. It tells you what everything is. Note that this tension left for you cannot exceed obviously the net tension and of course F sub RV must not exceed the number in the table. V is 0.75, here's just a summary and he summarizes the equation. I find the equation a little easier to use if you actually take the NT out of both terms saying that the tension left for you is parentheses 1.3 minus how much shear stress you put in the boat with your load divided by a fee times the number in the table. Yeah, I know, I'm not ready. Now the commentary gives you the elliptical equation. It's the only place you find it if you want to use it evidently. Obviously so few people do use it that sit back in the commentary. In slip critical connections, they do not ask you to check interaction. It's a slip critical connection and you of course have to check it for bearing but if it's a slip critical connection then there will be no shear in the boats. Therefore they don't make you check interaction for those kind of connections. Again, the nominal stresses that we'll be using in those equations. Here's the equation that you and I just kind of halfway derived. The terms, what they mean. And then we're back to your book where I don't have any or very many notes. Already done it, already done it, already done it. Incidentally, where are those words? In slip critical connections, I told you you didn't have to investigate interaction. However, the tension that you put on there causes the clamping force to be reduced. Therefore it's gonna reduce the available slip capacity of your connection. So for example, if you have some 39 kip loads in your boats and you have a force that goes through the centroid of the connection you'd ought to be shown here not up in the air like that. These two pressure plates are pressured. 39 times looks to me like four. That's how much compression force is in there. That force times coefficient of friction times some other numbers gave you your slip resistance. If you come along and this number is 100 times the cosine of that angle, that's 80. You just took 80 kips of the compression out between those two plates. So you're gonna have to go back on your slip equation and admit that the pressure between the two plates is no longer full tension force supposed to be in the boats times four. You're gonna have to take off that 80 kips. And here's the factor by which you do it. Your slip strength has to be multiplied by k so that's C, one minus the tension applied to the connection by you incidentally, you did it, I watched you, divided by the old compression number that they thought was gonna be there. Namely, the tension you were gonna put in the boats minimum times 1.3, 1.13, because you're a nice guy and you always give us a little more times the number of boats. So for example, if this 39 times four was compressed in the plates before you put this load on it and you put 39 point times four tension, see how this will be yours, that's 39 times four. Here's your 39.4, that'll be one minus one, that's how much slip you got coming. There's no pressure between the two plates anymore. So the operation is first start checking shear and bearing usual strengths, see if the tension left for you is good enough to handle the tensile load if it's not more bolts or something. Slip critical connection, same ideas except you don't have to do the interaction, but you do have to worry about the slip critical load being reduced because you opened up the joint. Let's start on that example next time. It is ignored, it is ignored. It's not usually 10% when the joint finally really fully opens up, it's about 10%. We rarely get that high. Now, okay, I sure don't. Yes, sir. Um, uh-huh. What? I can't believe it, why didn't you tell us so we could all not come? I don't know, if I get them graded I will. But I'll put yours over there, just pick it up over there. If I've even handed them out Monday the way things are going, yes, sir. Okay, is it okay to miss my class? It's never okay to miss my class. Do I think you ought to come to my class rather than get long term employment? Sorry, but you know, I take what I can get out of you. But don't ask me if it's okay. Okay, will I count off? No.