 Oh The faithful few all right in your LRFD manual You find that you have a bunch of sections on how things work So that's kind of nice when you're on an exam and you don't remember how they work It's always a girl. I don't know why but it's always a girl less Well, maybe not maybe it's just a happy guy All right And so I recommend to you section 7 dash 6 you see it's not In the 16 point thing so it's not a spec, but it is a how things work We've had a centrally loaded bolt groups It gives you how to work both kinds Whether or not it's elastic or whether it's plastic eccentricity first in the plane of the feying surface that's where you have the Flange of the column you're going to bolt it to here You have the connection Here you put some bolts in it and you put some load on it and it rolls Actually first you put the load Actually, you put the load first over here Down all the bolts And then you put the moment on the load But you'll notice that the bolts are being sheared in the feying surface nothing's going on in there But shear and shear So those you can just add together vectorially and solve for the forces in the bolts The elastic method assumes that the bolt group And the whole plate to which is attached by the bolt group are going to Uh Roll about the instantaneous. I'm sorry the elastic centroid and we found the elastic centroid For beams and we found the elastic centroid for a group of bolts So you'll know where that moment is being applied about Now second Thing still with the rotation being In the feying surface or about sliding across that plane Rather than Rolling about the centroid of the bolt group Elastically Right there What we consider is that when you take the load and put the force down through the centroid It actually moves the plate down a little bit and deforms the bolts Then when you put the moment on it, it does a second number to it. It rolls the plate About some point And you find out that that points out in here somewhere Which you put the load on there the whole plate at certain instances That plate rotates about a different point than the centroid Now that's the same thing as you did when you had like a t-beam I told you that we were going to say that failure occurred When one of the fibers reached first yield You said okay, that means every all the fibers are elastic. I said yes You said all right first then let me find the elastic centroid of that shape Let's just say that was 12 inches Perhaps this was two inches Perhaps this was 12 inches And this was two inches Then you multiplied 12 times two times 13 Plus 12 times two times six Some of those numbers up and divided by 12 times two plus 12 times two and you found the centroid of the neutral axis Don't know why but I think there's an example of that somewhere. I think it turned out to be Nine and a half inches wherever it is. That's where it is And as long as you keep all the fibers elastic Then indeed it will bend about this neutral axis But if you keep on adding moment to it you say well now wait wait wait wait This guy right here's got 36 ksi. You can't put anymore. I said, oh, yeah, we're not going to play the elastic game We're going plastic design And you say well, okay, but that's going to shift this elastic neutral axis I say, okay, it could be probably is because a lot of these fibers are now plastic I said, where is it going to shift it? You say well now then rather than solving for the elastic centroid We saw for the plastic neutral axis and the plastic neutral axis will be here Where this area above the plastic neutral axis equals this area below the plastic neutral axis So what we didn't mention was that while all this is going on when you had everything elastic the neutral axis was here When you put a little more moment and a few of the fibers went plastic the neutral axis shifted to here Then to here then to here then to here and finally when all the fibers went plastic it went to the pna As opposed to the ena elastic neutral axis So it's been moving around You say well, you know, I don't really care. We don't have any design things to do in here. We either design elastic or plastic Well exactly the same thing happens with this bolt group If you keep all the bolts in the elastic region And if you will say you must stop putting load on it when the first bolt reaches What we say it can reach elastically Then the neutral axis or the centroid of the group will stay here But if you say go ahead pour it on let's see what happens And I say well, I'll tell you what's going to happen that bolt over there instead of going on up like this It's going to go on up like this and up and up and up and up And you say, okay, that's just going to mean that this point right here is going to shift Going to shift going to shift going to shift going to shift going to shift Who knows going to move around But the right answer for where this now is called the instantaneous center of rotation See the instantaneous center of rotation elastic was here Then when the first bolt started going plastic Then it shifted and it shifted and it shifted When the first bolt reaches what we're going to flat say hey, that really is it that's the end of things That's going to be at 0.34 inches of deformation And that deformation doesn't just occur in the bolt. It also occurs in the plate. It also occurs. I don't care what causes deformation Once the sum of all of those deformations reaches 0.34 they say quit and so you must quit And they saw for the equation of this curve. They got the experimental data And they find that's at 72.6 Now if you just keep on blowing it away And go ahead and get maybe an inch worth of deformation or a half inch of deformation It'll peak out at 74 and just kind of stay there We just we won't let you go there, but that would be called r sub u And you'd have to stay back to whatever they say Now that's called the instantaneous center of rotation method It takes into account the true nature of the way the bolts Deformed they really don't come up to the yield stress in the metal and then they just pop over and then they come on up and Do something like this. They really have that tendency The equation that they have found fits that curve nicely. We mentioned it before Take the ultimate load. You can put in the bolt. Well, there's a bunch of bolts. You got a325s You got threads included threads not included or excluded One-inch bolts five-inch bolts nine-inch diameter bolts. They got all kinds of things So they did their tests on a325 bolts three-quarter inch diameter with threads in included in the shear plane And that's the one for 74 Now if you want to know what this number is this number right here is for your bolt Well, then you just take 74 and adjust it for the new bolt You got a one-inch bolt. You don't have a three-quarter inch bolt Multiply this load times one squared divided by three-quarter squared That'll proportion it up to a one-inch bolt threads in the included in the plane of shear You don't want to know what it is when the threads all of a sudden got Excluded from the plane of shear you just go find the stress permitted in a included case Versus the stress in an excluded case multiply it times that ratio And that's the number that you'll be using in your equation for that kind of a bolt And it works quite well I guess the main thing is it works Because if it didn't work they would be giving you other factors to take care of Going from a three-quarter inch bolt to a one and a half inch bolt Here is that curve plotted out. I was just wondering what it really looked like And all I did is I put different deformations in the computer let him figure out what this thing said For our ultimate I used the number 74 For delta. I used a whole bunch of numbers across the deltas. He calculated the resistance In the bolt now this is uh, going to be nominal You're still going to need a fee on this thing And as you see it does really at 0.34 inches it comes right up around 72.6 kips And then it finally peaks out right around 74 kips Although that may have a while to go before it would reach there You'd have to literally put infinity before you probably will get that 74. Yes, sir The deformation in the shear plane that would include that when you look at the plate You see probably the worst The worst hole the worst bolt on there Maybe he's going to be this one down here And you see the plate is really messed up Oh, yeah, but it's not just that because when you look at the bolt the dang bolt Everything you know the bolt looks like that. So here you got point one and here you got point two three and someplace else you got something else and Distortion of the plate itself you name it they counted it they just they didn't even look they said did it deform In other words as this thing right here rolled enough such that the deformation there is this number They said quit now why they did it pretty obvious Here's someone doing an elastic design And i'm going to put just a pure moment on it just to demonstrate it You know you could do the moment with a with a force you'd have to take the Shear forces vertical and then also add in the shear forces horizontal And put in the shear forces and the shear forces and get that would be your answer But just with a pure moment elastic Causes this kind of stresses our forces inside of these bolts Somebody's going to scream uncle first and they're going to tell you stop stop stop stop and you say why You say because i've reached table two point j point something's number for a three quarter inch of threads included bolt And you say i okay i'll quit Here's how much moment you would have Elastically you'd have the area of the bolt multiplied by i just put s in here. So that's s s s s spacing You'd have the area of the bolt times It would be times a moment arm of 2.5 times whatever your spacing is Times the yield stress of the metal Then you'd have a pair and there'll be two of them. Here's two of them down here one on the top one on the bottom Then you're going to have this pair of bolts And they're one and a half s apart. So there's your moment arm The force is straight line. So whatever this f sub y is in this one This one's going to be one two three four five is going to be one two is going to be three fifths as far out So it's going to have three fifths as much stress in it Then you'll have one here that's with fifth as far out and it's got a moment arm of 0.5 s And two of everything in sight you crunch out those numbers you get seven times Whatever the area of your bolt is times whatever the spacing is times Whatever f sub y is for your bolt. That's elastic. That's first yield Then plastic you would say Crank that moment on there and just don't listen to anybody just let them scream And when this one here finally reaches f sub y Then that's enough screaming and we'll stop at that point Now this doesn't tell the whole story obviously because if up here you have more than point Three four inches you'd have to stop earlier than that In other words, you might get this much in the bolt and you might get that much in the bolt And then this bolt may not be able to give you that much in that bolt Because you somebody else that stopped due to deformation But just the theory behind it is Running these things up now to f sub y And i'm not sure where these numbers came from we were doing something I have no idea what Uh your moment plastic would be again the area of the bolt The moment arm would be the same The outside bolts would still have The f sub y Then the next interior bolts still have an only three-fifths of f sub y they're going to have f sub y also And then the ones that are real close if you can get them up to that before somebody down here says Look you got so much deformation. You're not meeting specs Then this one will reach f sub y It'll have a moment arm of this it'll reach f sub y times two instead of having seven we got nine And that's always why we do this is we always pick up more Honest ability to handle load that is handled in a safe fashion We're using um plastic design rather than we do elastic design Now i'm back to the Not the specs, but where he's telling you what's going on On the previous page, he gave you the equation He said cost a million bucks to get this equation. Please use it nicely Then he says where r is the nominal shear strength Don't know why that doesn't have an n on it because it is nominal This is the ultimate shear strength of the bolt still nominal Uh the deformation delta is whatever the deformation you have on your bolt Not to exceed point three four inches on any bolt He is he tells you if you don't know e And he says that this information was obtained from a three-quarter inch diameter a 325 bolt in He doesn't actually say they were Included, I don't know why but if you go check the 74 you find out that's what they that's what they were using It also shows you where your elastic center of gravity has changed to some instantaneous center of rotation When you put the ultimate load on it He says if you just put the load straight down Then this is more or less what the bolts look like if this thing is going to rotate about this centroid Then this person has a vector location Of right there and the force in that bolt is assumed to be and probably is perpendicular to that Ray And he says you're going to get the right answer He says now I won't guarantee you this is where the instantaneous center of rotation is He says I don't know if it's here or here or here or here or here He says I think it occurs such I think it occurs on this line through the cg Because if I draw you a c Instantaneous center up here, then these things don't balance out I say well, what do you mean balance out? He says you see this leftward component Says that just perfectly balances that rightward component. This left balances that right This left is different than that one, but it balances this right this left balances that right I said why do they have to balance he says well your original force had no open no sideways force So when you get the final answer for this thing here All of these little left components of all those forces has to balance out either zero Or has to balance out this thing times the cosine of theta I say what that makes sense. Otherwise the plate would not sit still says see They how about the vertical forces? He says well the vertical of this plus the vertical component of that plus plus plus plus plus minus this Has to equal zero Okay, that makes sense. He says not only that You see this moment right here. You see this p sub u go away. You bother me You see that p sub u If you multiply p sub u times the eccentricity from the center of gravity A lot of times we measure from there Plus how far out the instantaneous center of rotation ended up being that clockwise moment has to equal Him times where does he live plus her times? Where does she live plus it times? Where does the cow live the dog the kangaroo all of Have to balance out to this clockwise moment Is this otherwise one of your equations of statics is not working? It's not right What is the lo? Yeah, that's that's actually this one Well, now see when you put the load on there, it only had one gram of force in it And it was placed about the center of gravity So that number is fixed It's kind of like here's your column Kind of like here's your column. Here's the plate that you're going to bolt onto the side Here's your column. Here's the flange Here you have a beam that looks like this And you have decided to pretend that the load inside of the between the beam and the plate is in the middle Okay, that is The distance to the centroid of the bolt group. That's this e eccentricity That's on the plans Now then it doesn't roll about that point unless it's an elastic design it rolled out here So he took e and he added l sub c A distance from the cg to wherever this ended up being No No If they did it would be wonderful because then we'd have all the problems solved Let me show you why it's not When you just put almost no load on there, but you put a little It really does go about this point right here When you put a little more it still goes about this point when you put a little more It goes about this point because everybody's elastic Now then when you put some more load it goes about this point. Ah, what happened? Let me repeat that. Ah, what happened? Somebody started what? Somebody yielded that's exactly one of the bolts yielded And so as we put the added load on this bolt here says go away I'm not taking anymore. I quit. That's it. I'm through He's reached 70. I don't know what he's reached some but he's yielding And so somebody else had to pick it up that moved it to here and then it moved to here and then it moved to here Do I see I just keep on going and that finally ended up here At that time this guy who started yielding first says Ah, that's it. I said you said that was it a while ago. I said, no, no, no, I just started yielding now. I'm at 72.4 magic number Quit That's when we quit now then so see You can't say you think that's equal to that because you didn't complain here or here or here or here or here or here The moment arm when I'm interested in it is E plus l sub c And I don't know what l sub c is But I tell you what I can do. I can put it here Where this is six inches and this is four inches. I can just put it there And then I'll run it through the computer and he'll say that balances that that balances that that balances that that's all you're doing good That and that and that and that and that and that and that that those all balance He says, oh, you're some of your moments equations off like crazy Say, well, what do you want to do? He says, I don't care if you do anything I'm just telling you some of the moments not equal zero I put the instantaneous center of rotation here He says, okay, the vertical's still balanced the horizontal's still balanced But the moment was not as bad as it was but it's a little off I said, okay, put it here. He says, well now it's off the other way Uh, dang, okay I put it here First thing you know, I finally get it. So it's perfect. It's not too hot. It's not too cold. It's Just right And all of my three equations of statics balance All right Again, he shows you what I showed you he shows you the curve I have no idea why it comes in at 10 kips You know because the equation doesn't come in at 10 kips Unless they put 10 kips of load on it You know before uh, just to firm everything up in the test But anyway the curve that we're interested won't be down in here. Anyway, it's going to be up in here around 74 A 72.6 There's the equation. This is who they were Derived from There's numbers in the book are just flat wrong Could have been the elastic. I kind of doubt it's a nice even number like that I don't believe it. I think they were just lazy This is right above it. What does it say? Uh, I got two of those For four supplies illustrated as soon as it's rolling in direct share so-and-so So it's okay to both groups. Oh, yeah, that's interesting Each boat's similar to those shared direct share and share the eccentric moment proportional to its distance from the center of gravity Yeah, but I don't want to see a 10 in there And not only that the numbers are wrong. It's just they're just flat wrong Uh, because I know dang good and well this number comes out 74 and he's got it shown between 30 and 40 I'm sorry to repeat that When he goes plastic Oh, I'm sorry. No, no, no, he's talking about the elastic method This is not an elastic method This is just discuss the elastic method There you go Now then he's got a figure he wants to put in here. There he puts it there But that's a good point. That's very confusing because it's not the elastic method This is the uh instantaneous center of rotation method All right, so He shows you an example piece of view. Here's the cg. This might be elastic method Here's the point 34 and there's our equation Then he discusses the shear per bolt For how much You can put on the bolts or how much would be on the bolts you take the ultimate force divided by end This is what we did before And then this is this is the elastic method now And how much moment we get this eye polar moment of inertia C is the radial distance from a center of gravity the center of the bolt group my my my no doubt about it We're talking elastic method Eye polar remember when we get out got eye polar. We got some of the x squared plus y squared take a square root Determine the resultant and we've done all this we've done all this loud stress stuff He's telling us if you don't have your notes in there and you need to know how that's how Now then let's say we have an eccentricity normal to the plane of the fading surface This one has a Uh eccentricity parallel to the faying surface Here's one where we have An eccentricity normal to the faying surface our faying surface is where the thing tends to slide in shear This is has a moment perpendicular to that plane Now then instead of causing Put this right on the centroid causes shear This over a number of bolts load over bolts Plus moment shear and then adding the shears together vectorially This one's going to have put the force right on this plane And it will have the same as before Shearing force in the bolt total ultimate request divided by number of bolts But then when you put the moment on there you're going to cause tension in these top bolts And you're going to cause well, it would be compression in the bottom bolts and believe it or not in some of them It is actually compression Because if you go out to the parking lot you will see a concrete pedestal And you'll see a bolt coming up out of here And you'll see the nut run down the bolt threads right about there And one right about there and then you'll see the big old thing that Post is welded to Like that and you'll see another nut on the top Then you'll see the post or the light pole is Welded onto this and I got a hole in there where you can get in there and tinker with that stuff sir Yeah, this is con this is concrete The bolt the the bolt is down in there. That's right. And this does not rest on the concrete Why do you think they would do such a thing when the world don't just bolt it to the concrete? Yes, sir There's a gap. Have you seen that gap? Well, just metal to metal where the nut touches the plate No, actually they do it so you can level the dang thing because when they put it on Don't you see how bad they look how badly they built that thing? That's about a five degree slope on a concrete and the guy looked he says Did you notice your poles look like they're all drunk? They're all at different angles the guy said No, I don't see that So they can level the thing up So if you put a moment on this one you actually will put this in compression And you will put that in tension now It's gonna have a hard time leaving that on this case because you probably got a Million kips of tension between these two things here anyway due to all the tension in the bolts So when you do this the tension in these You know won't even start to occur until you open up that face At which time you get the full tensile capacity of the bolt And then down in here you're just going to have to make some kind of assumption And they have different ways of doing that our book does it one way The code gives you other ways that are acceptable Here's one of them where they tell you about how far this ought to be the depth over six They think that ought to be about right And they tell you to go ahead and put the plate in compression like that Count all the tension forces of the bolts above that point. It's a good method nothing wrong with it. It's not what we do He says and what we're going to do is we're just going to go ahead and pretend the centroid is at the group center And we're going to put any of the bolts above that at their full tensile capacity And we're going to pretend the bolts down at the bottom are in compression I don't care who's really down there. I don't care if the whole thing is in compression We're just going to assume there's bolts down there too So you're going to take this bolt times a half a space plus a space plus a space. That's his moment arm The force in that pair of bolts times three halves of s And that times a half of s where this is spacing spacing spacing spacing We'll analyze them like that. We'll be a little on the we'll be more than a little on the conservative side But you know, that's they're just bolts. They're not that expensive And if you say look i'm trying to save just two bolts Here you go. This is your guy right there All right, so here's segui's take on this whole thing Um the same equation 74 for the ultimate you can't ever get to 74 because that by then the deformation is just too too much. You can't live with it So the biggest number you're ever going to get out of that's going to be 72.6 kibbs Out of the 74 that could be there with deltas equal to infinity uh, the lambda is a regression coefficient and the Poisson's ratio is that what that is? I don't remember is a regression coefficient Those have been determined experimentally And they work nicely for our bolts of any size that you normally use With or without threads in the shear plane I already mentioned this already mentioned. There's all your data I already mentioned. There's the maximum You put 0.75. I don't know what you get to tell you the truth You will assume That once one of your bolt screams and says that's it and you say what do you mean? That's it I got a 72.6 inside of me Then you will assume that this bolt right here has a deformation. Let me check right quick Let me talk to this guy. What is your deformation? Zero point what thank you 0.35 inches I remember was it three five three four three three four three four There are little errors like that never killed anybody Then this guy's deformation right here is his distance Excuse me divided by that distance. That's his deformation. That's not his force Because his force looks to me like he's about half as far out. So let's just send that bolt up About half as far If the bolt screams 72.6 stop, please Half a point three four is point one seven Point one seven is about here Wow That is by no means half the force That's near all the force about 65 Kips a force in a three-quarter inch bolt And that's where we pick up a lot of the strength in these connections I already showed you that because I couldn't resist There was your elastic. There was your plastic between the two. It was moving on you as soon as the first fiber yielded Same way here. This is your elastic little more load. That's your elastic Then first thing you know once one of the bolts starts going non-linear the Plastic Centroid moves where I don't know If it's a pure down force, then it will just move left Because otherwise all the little horizontal components won't balance How I find that point I go just stick it anywhere and I punch a button and the computer says So I move it to here and he says I said, okay, so I move it over to here and he says And I move it to there and he says And that's where it is right there Here's an example I guess it Is not this case is this got four bolts vertical Here's a case where I've got six bolts The centroid is at the middle of the group of bolts This bolt is five inches up and this bolt is five inches down And these bolts are four inches apart. So that's two inches and that's two inches I'm assuming that the center Instantaneous center rotation is there and so it is there For all intents and purposes during this study It is here now whether everything will balance out is Got zero probability of happening. That's not my problem I just want to know What the answers are for if it were there Well first thing I'm going to go find the worst bolt Well in this method, it's real easy to find the first the worst bolt You don't have to wonder maybe some forces go this way some go that way The worst bolt is the one that's the furthest out Because if you go to one that's not as far out Then it's going to have that distance divided by that distance It's going to have less deformation And if it's got less deformation, then it's going to have less load and it's got load It's not the critical bolt. So the one that's the furthest Position vector or ray from the assumed instates center rotation. He's it Tell you what about him. He's got that much deformation Tell you what else about him. He got 72.6 You say well now these bolts are two inch diameter Oh, okay Well, the deformation is still 0.3 quarters or still point is still 0.34 But the load now will be jacked up by The area of a two square inch bolt versus the area of a three quarter inch bolt Other than that that'll be the same now then Since I know this is 72.6 and since I know that's 0.34 and since I know the distance to this bolt is Some jerk just decided it was four have no idea what hubris So it is four up and they assumed it from here It they assumed it was six to the left of this bolt Looks like they're using this as a reference bolt a reference point bottom left bolt Six to the left and nine up Six to the left and nine up that was an assumption Therefore this bolt is nine down And six plus four across that's nine by ten That's right. Look at here nine by ten And if it's nine by ten then it's hypotenuse is 13.45 So this dimension is 13.45 inches This bolt on the other hand is down four and over ten So it's got a slope of four by ten Four squared plus ten squared took a square roots 10.77 it's 10.77 inches out From the instantaneous center of rotation Since it's not as far out 13.45 versus 10.77 it doesn't have as much deformation But the deformation is in proportion to how far out it is Therefore the delta of bolt two the delta of bolt one was 0.34 The delta of bolt two is 0.34 times R2 over r1 You shouldn't have any trouble remembering which way to do this It's got less deformation because it's got less dimension out It's bound to be calculated somewhere around here. Here we go Delta two is equal to 0.34 times the 10.77 over 13. It's deformed 0.27 Find out how much force it's got in it. I come over here at 0.27 And it's got 71 kips of force in it. See what it really is See what it really is Since it has a deformation of 0.27 We plug it into our equation 74 kips is our ultimate nominal 1 minus 10 to a regression coefficient of minus 10 There's your 0.27 raised to a regression coefficient of 0.55 That's a pretty good number. Look at that how close I got 71.22 kips So now then I do that for all the bolts And I go back and I say You have a rightward component and you have a incidentally his Leftward component Is 71.22 kips there he is Times what is his leftward component times what? Here's all the numbers you should need right there I want to multiply 71.22 times something in there Oh my god multiply by Okay, it looks like that doesn't it because that's the x dimension However, gotcha And you know first responders are always the first to die So, you know, you just got to you know, if you want to get on with the glass You just got to be a first responder and you got to have everybody laugh at you like they knew You're looking for a guy that's perpendicular to a 4 by 10 vertical horizontal And if you kind of look at as you can see it doesn't have much x component So you're going to multiply it times 4 over 10.77 to get his horizontal component And then you're going to come down to this one and you're going to multiply 72.6 times I want the horizontal so I get the one that doesn't look like I want nine Out of 13.45. That's his horizontal Then you get his horizontal her horizontal its horizontal that was horizontal and guess what you do with all the horizontals You add them all up And I don't think the real load was even given in this one So I don't know what it was But if the horizontal load component you put on there doesn't equal all those horizontal forces Then you got the wrong spot The computer will tell you instantly you'll say click he has a no You move click no Click well no not quite and then first thing you know you'll put it somewhere and he'll say Well within three decimals I say say yes, that's close. That's fine. Let it go Same way with the vertical components same way with the moments. Yes, sir Of course Of course Excel has a very nice routine. It has a what if analysis It'll let you say What if what if I want this to be something? What would this over here have to be? But you have three equations that you need to work with Some of the moments some of the forces horizontal some forces vertical So you're going to have to go to what he has called a solver And a lot of times that's not installed on your computer But you go down there's instructions on the class. We're not going to do it. We just don't have time But you can stick that solver in there and then you can say Um, I don't know x and I don't know y for this And you say well, what do you let me tinker? What do you want to happen? I say well when you get all the x people I want them to add up to the loads x And the same way with y In the same way with a moment and he says hit a button and he does he goes puts it right on the mark All right, that's how it works They do have a table and we'll show you how to do that next time And you don't have to hand in any more homework until you feel like it and Whenever you get it done get it done and give it to me Did you already hand something in? Now yet, well, you know, it's real simple you may have just learned how to use it for all I know But I ain't no doubt about it. I've got two choices. I can really show you how things work And then Yeah All right, if you would strip off enough for you and him and hand them over Who's going to be around for the next class? Can take this down to the office for me Nobody you will shall I know? Well, I will but I wasn't planning on it