 Okay, Fulton, Cantu, Barry, Thompson, Thompson, all right, I've found somebody not here. I've got to find somebody not here each day. Howdy. All right, we've been working on columns. We were working on single story structures originally. We knew we had multiple story structures available, but if you had single story structures, we had some pretty good pen conditions that worked nicely. They worked nicely theoretically in 305. They have been adjusted to work nicely practically in the tables, depending on how long they are effectively and whether they buckled about the week or the strong axis, we could design such a column and practice with several of them. In multi-story frames, rather than having a nice pen, which is never nice because you really can't make one and you probably don't even want one, and rather than a nice fix which you really can't make but you wish you could because it would make the column shorter effectively, these end restrictions, these end restraints are due to the girders that are running across the columns. And what they find is they are supported by the EI over L of the members that are entering the joints. Some of the people want the thing to buckle. They like to be pinned in. The column always wants to be pinned-pinned. You have to restrain it and tell it, no, you can't do that if you want it to be other than pinned-pinned. And then the girders also come in. Just a quick review of the table I just discussed. There are your different cases. This is for a frame that is braced. I know it's braced. I know it has bracing in it somewhere like that. These are braced because they're all braced by that one and the tops cannot translate horizontally because the top didn't go horizontally. Stayed directly above the bottom or very close. So this is a braced frame. They call it a sidesway inhibited. Here's another braced column because it is directly above this point. This is a sidesway not permitted inhibited. Here's one where sidesway is uninhibited. It's an unbraced column out of a frame that's unbraced. This one is right over that. That's braced, sidesway inhibited. Not braced, sidesway uninhibited. Unbraced, sidesway uninhibited. So you'll start seeing some graphs that say that the map is for sidesway inhibited frames or sidesway uninhibited. Uninhibited means it's unbraced and inhibited means it's braced. We'll get into that shortly. This is the 305 that I taught you. You remember we had a method of solving for moments in statically indeterminate structures of slope deflection. We did a lot of work to get to that point but basically we said that if you will go on the structure and stand at point A and look towards point B, the moment caused under your feet is equal to 2EI over L times the amount of rotation of the joint under your feet at A plus the rotation on the other end that you can see down there. You're looking at member AB and then if there was some delta between the two joints if one went up with respect to the other one. What I want to remind you is that the difficulty in rotating a joint which is what we call G is EI over L. In other words, to get a given theta your choice of what theta is. You tell me what theta is going to be and I'll multiply it times EI over L and perhaps some other constants. It's a function of EI over L. The stronger the material is you make it out of the harder it is to rotate. The bigger the member's moment of inertia is the harder it is to rotate. The longer the member is the easier it is to rotate. If I ever ask you how difficult is it to rotate a joint and you're going to tell me it's a function of EI over L. Now, let me show you why this is the G that we use. The G that we use is a large number if the columns have anything to say about it because they would like to buckle. The girders are supporting it and they don't like to rotate and the girders are coming in from the side. They're either small or medium or large. The larger they are, the tougher the joint is to cause it to rotate. And the ratio of those two, these people saying, why don't we roll this thing and this guy here says I don't want to roll. I want to just stay horizontal like I was. That ratio will tell you whether or not that particular joint has a tendency to not roll at all like a fixed end. Roll a little bit like a partially supported end. Roll just freely like a pinned end. Then we'll get into those graphs in a minute also. For example, here is a column AB out of a frame. This thing would like to buckle like a pinned pin and a pinned pin and a pinned pin. And if it were braced, then I can keep on just showing these things pretty much directly above each other. There would be a little motion, but not much. If it has no braces in there, then probably this one will do something and it'll move over some and it'll move over some and it'll move over some. You'll get a different shape of the column when it deflects. Now here are your columns. These are the bad guys. These are the guys that it could be aren't even connected very well to the girders. For example, if I have a column that looks like this and it's connected to a girder at the top with a little old weenie angle and bolted down, then I've got a beam, a girder coming in from this side and from this side and it's just sitting on what they call a seat angle. You've provided no resistance to rotation at all. That's a pure pinned pin. You do see how it's pinned because these support it. They don't let it buckle at that point. They'll let it move. You're not going to get any resistance to rotation out of these girders because you need to weld those rascals down. You need to really make it all one piece. If you want these girders, right now when this thing buckles like this the girders say I didn't even notice you. All you did is you just kind of closed up that gap a little bit. Don't even know there was a problem. If you'll put this piece up against there and weld it like crazy and you may have to put some plates in there to transmit that force on across then when this thing wants to rotate it'll give it a try and the minute it tries to roll this girder right here will say I don't think so. I don't want to roll and therefore you will support the column against rotation at these points. So first I need to know all the bad guys. I need you to total up all the people that say buckle, buckle, let's buckle. Those are the columns. Some of all of their stiffnesses at the end of the column under consideration. So in this column right here there's only one column coming into the joint. I would need this bad guy's EI over L. Resisting that tendency to roll are two good guys. They say look we don't think you ought to just roll. So I'll need to know his EI over L and his EI over L. Without a doubt in this class E will be the same for everybody and you probably would just see it drop out. The moment of inertia can very well change and the links can very well change. That could be 20 foot, could be 25 feet, could be 20 feet. So I have two good guys that go in the denominator. I don't know who decided to put the good guys in the denominator and the columns in the numerator. Had they done it upside down you'd just get an upside down number and we would be using that for our number. These are the people who resist rotation of the joint. They're all of the girders. Of all the stiffnesses of the girders coming into the column, coming into that joint for the column you're talking about. So for column AB for this joint you'll have two personal bad guys coming in and you'll have two personal good guys coming in. The bad guys go in the numerator the good guys go in the denominator and that ratio is called G for joint A. Then down on B you have two good guys you got two bad guys that would be G sub B. The people that hurt you are in the numerator they're the columns. People who help you are the beams depending on how many come in that's what you put in. This person has got two people trying to hurt you and make the joint roll and there's one guy resisting trying to not let it roll and making the effective length of these columns shorter saving you money. So here's an example. I'm looking at this column number two you'll notice that I needed a massive girder at this floor level don't know why maybe there were some exceptionally large loads even though they were large loads they could be transmitted through a relatively small column because whereas this one's bent wow tough on things this one's in pure axial wow very strong and therefore when I go for the E of the bad guy I of the bad guy L of the bad guy at the top then I will be putting in the denominator some really massive numbers because the I is much larger and I don't know about the links the links will influence that also low values are good because and I'll just show you the table we're going to use we'll get to it in a minute this is for the E we'll get to it in a minute this is for an uninhibited frame it's what they call an alignment chart sideways is uninhibited here are your G's and here are your fix I'll show you why those are where they are you'll notice this is K I would like K to be at least one for an unbraced frame an unbraced frame is very close to like our unbraced column remember what the K was for that to get the true effective length one no good guess two no but getting closer 2.1 right and it is two if you're still in 305 land but we're no longer in 305 land we're in real world 2.1 and so look what happens when G gets lower what do we say here low values are good if G for joint A is this number and G for joint B is this number there's your K right there if you say wait a minute I calculated that wrong GA is this and GB is that then you no longer get a 1.5 you've got to take a 2.9 that means 2.9 times the length of the column it's true length so obviously the more you can lower these G's really effectively without cheating and still be safe you come out way ahead here's one that I showed you before the G's coming in from the side this one would be zero and therefore you have all bad guys and no good guys so the G will be infinite and they got an infinite on there there's an infinite there's an infinite it's very possible I don't know you'd have to look a little further now I think that's it that's the only two beams we have coming in they're both pinned here's your G bottom the equation says tell me the EI over L for the bad guy coming into the joint tell me the EI over L for column number 7 coming into the joint here are the people that are supporting things but it's a really weenie beam even compared to the column so these numbers are going to be very low since these are high and these are low you're going to have some pretty large G's and those are bad for money Gamble Gamble how hard is it to roll a joint wait, wait a minute maybe I ought to rephrase that how far, how hard is it to rotate a joint okay, yeah didn't want anybody to get the wrong idea there more of them that's practically fixed fixed and by almost being fixed fixed you're going to get to use a 0.5 no, what? 0.6 or 0.65 I don't remember but you're right, not 0.5 not in the practical world and you're going to have to have some really stiff girders coming in to force that thing to buckle like that here's one where I haven't really showed you any sizes if they're weenie girders it's just like a pin pin column if you make these girders bigger they're harder to rotate so the column buckles like this where the effective length goes from 16 maybe down to 14 feet if you put some massive girders on there then this thing will come kind of like this and have the pins here and here so up to when you make this length what was it? 0.6 or 0.65? 0.65 of the original length that's about the best you could do same example K is 1 K is 0.8 K is 0. there it was right there 0.65 these are the bad guys they want to buckle they want to do this they want to be pin pin that's their goal in life L is the true length the true length of the girder and the true length of the column because if you did that you'd have a problem because you would be entering with the true length and finding out the effective length was 0.8 then you had to go back and say what if the effective length was 0.8 well then it would have dropped down to 0.7 something like that so they're using the true lengths in these things so I've got an example let me before I get into the example let's go back to these pages where the graphs show up where the nomograph shows up first this is out of the reference the specifications here's what he's asking you to calculate he doesn't really tell you but one's at the top and one's at the bottom you're going to get one for the top you're going to get one for the bottom or put them in bottom top it doesn't make any difference you get the same answer this is only if the frame is braced you see how the top A is right pretty much above point B so somebody puts some braces in there enough to brace the entire frame you'll notice this is in the commentary if you work out your EI over L sum for the columns over the sum of the EI over L for the girders you're going to get just a number in this case this particular person who drew that line has got tiny columns, little weenies and the girders are beasts so that's no rotation practically fixed and k for that is 0.5 or 0.65 practically well I don't know that because you depend on the other end too that's just g for the bottom if you have tiny columns you have bad guys over good guys they have no say so in the matter they're just too flimsy and therefore the girders control and you'd have up to 0 the other end of this column, column A on the other hand they have monstrous columns up there and they just make up your mind, beast columns, tiny columns beast columns tiny girders G-I-R-D-E-R-S that means this will be a big number and people will just say we can't resist so they will have low numbers that's just like it was pinned or close to it therefore large over nothing is like 50 or 60 or infinity the way you find the K for a fixed pinned column by girders is you put the GA on one joint, the GB draw a line low and behold 0.7 you will only use the K factor table in cases where you like have a single frame single story frame then those are very appropriate alright now I gotta remember that these charts are based on the assumptions of people made when they derived them one of them in this that everything in here is elastic that there are none of the columns have gone into the plastic range we'll talk more about that in a minute the second normal graph is for an unbraced frame notice that the upper joint is displaced with respect to the lower joint a reasonable amount an inch or so see no braces in there again elastic behavior of the columns only they are also called unbraced frames or side sway uninhibited is what they call it and then you go get the same thing you get the G at the top and the G at the bottom and you enter the numbers and for instance if you had a 1 for one end and if you had a 10 for the other end then you just come through here and you pick out the K value and of course those are done mathematically and somebody went in and said okay how strong is this column but it rotates so many degrees at the top and so many degrees at the bottom and how would that happen well there would be a ratio of the people coming in wanting to buckle and these people not wanting to buckle and they plot all the numbers and they give you a graph for it so this is for any frame where the side sway is allowed to occur you get the numerator of the G for the columns you get the denominator for the G for the girders and you enter that number then you do the same thing at the bottom and you get that number then having those two numbers you draw a straight line across it tells you the effective length of the column so let me see there's several practical things that he wants you to take into account we can't take them all into account there's 50 of them but the most important ones most of them if you don't take them into account you're on the safe side but so that's what he says in the text we're going to do because it's just impossible to do them all and still get into welds or anything else but he says I want you to note that for columns supported by but not rigidly connected with a G or a foundation theoretically it's a pin and G therefore is theoretically infinite reason being is the girders the columns coming in want to buckle they say we're rolling this joint down here and you say well let me see if I can resist no it's a pin that would be something over nothing that would be a G of infinity the truth is not that bad and they have found a long experience and a lot of testing and a lot of money unless you really put a pin down there you say look I drilled a hole in it I put a bearing and I greased the bearing monthly well that's a real pin you better use infinity for that design but if you just stick it on a column and bolt it down they found that you can use a 10 for practical designs so now let's see that is for sides way inhibited see this is the tail end of the previous page so in this one here instead of using zero like you and I did for this one you can use a G where is this I'm sorry instead of using an infinity like we did on this one you can drop it to a 10 it comes under the sections I don't think I have the previous page but it doesn't matter this is the one it works with the second thing he says is if it's rigidly attached to a properly designed footing then you may think that it is not able to rotate but he says in truth he says I'm willing to give you something you got coming to you I found by test he says but just because you rigidly attach it to a big old thick plate and bolt it to a properly designed footing it's still got some rolling to it I don't want you to go down to zero I want you to use a 1 he says if you show me some kind of calculation where it's just got to be zero then that's okay you can do it and then they got a bunch of other things I don't have the page on them but we omit everything the only other thing we do omit or we do include I'll show you in a minute I was mentioned earlier we said that the columns acted elastically interestingly a lot of the columns don't act elastically they are ready well towards plastic behavior before they even get ready to buckle so here's an example got a rigid frame shown in this figure he is written down what kind of columns and girders are coming into the joints he is studying this column right here here are two bad guys coming in at the top here are two good guys trying to support it they have different links they have different moments of inertia I went and got the moments of inertia for 12 by 96, 120 there is 833, 1070 1350 and 1080 and I told you what page they are on also have different links here is the G at the top the column is a 12 by 96 it has a 833 divided by L his L was 12 feet divided by 12 there is another column coming in under the bottom into that joint it's a W12 by 120 so you add 1070 and it is also 12 feet long you divide it by the people that are trying to stop this column from buckling the moment of inertia of the 24 by 55 1350 divided by its length and the 24 by 68 divided by its length it gives you a .94 this is a braced or an unbraced frame unbraced frame there you go so let's just see G sub A here is G sub A either end is fine we got a .94 is everybody happy here how did you know that because it says its side sway is inhibited this is braced the other guy said well I didn't notice the words but I noticed these two points are right above each other that's not an unbraced frame you're using the wrong nomograph not a problem I got another one G sub A is .94 .94 there's that probably there it is right there .94 on the other end there are two 12 by 120s coming in that's why you see these two numbers are the same there's I sub G here's our girders why is he only got one number written down here there's two of them coming in thoughts on the subject only got one that's right in other words what's coming here is a 24 by 55 what's coming in here is a 24 by 68 he says hey man I already worked the number out that's it right there .95 come over you can see the difference between the 94 and the 95 there got really good eyes and I say well I don't like this I don't think I'm getting enough accuracy you're getting plenty of accuracy there's plenty more inaccuracies that are a lot worse than these however they do give you the equation for these yes I didn't have that page they give you the equations there they are they give you on page 16.15 11 they give you the equations and I think I mentioned EES did I mention that it's available to you I did that's what I did when I said well I'm not getting enough accuracy you want to know what K is you simply type in EES this equation put your G sub A there put your G sub B there tell him up at the top and then give him this equation right here and he'll solve for K and he'll tell you the right answer the right answer is 1.1 1.2 1.3 the right answer is 1.3026411 perfect I love it the answer is 1.3 plenty close nobody's nothing's ever going to fail because you didn't get that thing picked off there perfectly believe me so there's an example what did he get he got a 1.3 I without a doubt in the world I took his number no no that's calculated that's calculated when I drew it on a graph it looked like it came out pretty close to 1.3 in a long frame like that you won't be able to just put one brace and have them all be held down you know some reasonable distance you're going to have to put another set of braces because after a while just slack and slop and things in the joint going to make these make these not really brace so they'll have you put more braces these are typical braces these are more efficient than these are you would think you would think yeah I would think however you'll get that when you run it in the computer program and the computer program says that lateral load is coming down here pushing horizontally on this column and then you will have to design this column not as a just a plain old column out in space but one that has a horizontal load on it and when you design it you'll find out that kind of hurt my column and the guy said well I don't know because it sure did help our windows and I say well I guess we got a question here and what you want to pay for that window he says well I don't know how big are these windows I said well they're right down here on the floor he says okay as an architect how much more money I say well about a quarter million he says that's not that bad on a $200 million building so do it it's his call all we do is we make sure it doesn't fall down oh they gotta be much much greater no question about it because they do have lateral loads on them you'll see when you right now you and I are designing columns without that we will get into designing now then it's going to be called a beam column because it has beam loads and column loads when we get designing into them it's more challenging and the beam has to be much bigger all right one condition usually not satisfied I'm on our Sugui page 150 is that if the slenderness ratio is less than the break point what is this break point for 50 KSI steel 113 that's right it's less than that you're down definitely in the inelastic range and some of your fibers are going inelastic and it turns out that if you don't take that into account you're missing a good bet and I'll show you why let me come back to the design or that equation in a minute here we go here are three columns first we have a small column with I guess we have the same column have 600 kips of load on it when you put 600 kips of load on it that's such a weenie column you don't really get much load on it as a matter of fact it's why you only get 600 kips of load on it at 600 kips of load you cause a uniform stress piece of view over area gross and this is the yield point when the column does tend to buckle it is connected nicely it's welded on there the column has all this capacity to tell the girder I think we ought to roll in other words he puts a moment on the girder that really is hard on the girder it really does want to rotate the girder and the girder strains not to let it roll there will be some rolling exists in this case I have a shorter column I don't know how much shorter but it buckled at 900 kips at 900 kips on this particular column incidentally this column when they tested it SWA credit was 600 and it buckled at 600 this one when they said it would go to 900 it really went to a thousand how come this is off well it's not off that pad but what happened was when you put the 900 on there your load over area has already used up a lot of the available stress in the fibers and instead of having a lot of stress left over to talk to the girder now then the stress you got available so you already used up half of the yield so you have this little area times its moment arm times its area and this one you have some moment to discuss with the girder the rolling situation but not nearly as much as this one man the girder knows this guy is here the girder says well I don't know let's negotiate when on the other hand you shorten it up so that you can supposedly put 1600 kips on it you put 1600 kips on it you test it it's way off you really can put 2200 kips on it I said why is that here's what happened when you put 1600 kips on it you darn near yield the thing just with the 1600 kips and then it started to buckle it started to roll once it started to roll it could talk to the girder with this little bit of added stress see how that stress times area times moment arm has a little talk little moment to it you know maybe 20 foot pounds on the bottom of the girder and the girder said he must be kidding I'm not rolling here we put 50 kip feet on the girder here we put 10 kip feet on the bottom of the girder here we put a kip foot on the bottom of the girder so what happens is this bad guy who wants to roll has far less say in the matter once these stresses get in the inelastic range and that's what they're trying to correct for that's why we got a thing called Towson B here's a more step by step thing of what happens is you put load on it you put load on it you get some stress and if it buckles at that stage here is your excess moment that you can try and roll the girder with whereas if you you already used up a lot of the stress because you're over in this region then all you got left is these little elastic things the truth is if you keep on rolling you know those fibers will go plastic till they all go plastic so there's a little more than just the linear part but still this has far less influence on the girders forcing the girder to roll at that joint than this one this one has far more effort far more effectiveness than this one does so we get what we call a stiffness reduction factor listed on page 4-31 they list it as Towson B now where they get this Towson B from is our page 151 in Segui he says do you agree that F criticals Pi squared E tangential divided by KL over R squared I say well I thought that was just E he says no no you agreed with me that if this thing didn't have just a beautifully defined yield point then when you got the loads up in here you had a tangential modulus elasticity you're supposed to be using I said well that's that's true but I thought these things were all elastic he says no no you just proved to me they're not all elastic we were just pretending the columns were elastic when we got those G values before well okay I guess you're right says you're real G that you should have written down the girders are elastic all the time in their effort not to roll but the columns on the other hand you really shouldn't have put E you should have put E tangential so it says the G you should have been written writing down and entering into the table it's not really an elastic G it's an inelastic G and the inelastic G has all of these numbers are unchanged I column may not be the same as I but you know the only change you made in that formula there was E over E you change it to E tangential over E so if you got a G elastic you really ought to be multiplying it by E tangential over E which means divide this stuff out that says F critical inelastic is to F critical elastic is a ratio of just with the tangential as opposed to the regular E the same idea the only change made was the E that's what we're going to call Towson B now past this point the derivation kind of goes downhill the reason being is about two pages of work was done by Glombos in 1998 if you go look it up then you can see he does show that F critical elastic can be represented by this elastic with this where lambda is that without seeing it there's not much reason to plow through all the rest of it let's just say basically a lot of people been through the work that Glombos did back then and it's correct I have no reason well I've been through it too there's no reason to suspect otherwise he's trying to get this ratio for tau into an equation form he says I'm going to give you a table sometimes people really want to use it in equation form and this is the only thing the equation is for is to get you an equation for tau what you do is it's four times the nominal load divided by the yield load one minus and in terms that everybody agrees on anytime you see an alpha it's a one the alpha is always there in the equations to accommodate the allowed stress people they are still not really doing it the way we're doing it they get little different answers but it's sometimes where you need some factor that basically really is entirely dependent on plastic action then they're going to have to be brought kicking and screaming into our world but if you see an alpha we're already in our world for LRFD it's a one for ASD they're basically adjusting their loads to bring them into the plastic range so that they can use the same equation we use our equation is just this one our piece of use the ultimate request that's what you're going to try and get your nominal load up to over a piece of Y fees involve the whole thing this is our Towson B equation then he's got an example where all he asked you is what's Towson B if you know the piece of you and the F sub Y and that stuff here's the table for the different steels unless you're going to use the equation for a 50 KSI steel all you have to do is you tell me the piece of you request and you tell me the gross area of the column that you're planning on using you tell me the stress caused let's say the stress is 30 and I'll tell you that that column has been hurt so bad that 4% of its strength 4% of its ability to talk to a girder has been lost that's good for you let's say that you ran the stress 40 KSI and a 50 KSI steel it's lost about 36% of its ability to force the girder to roll all you do is you take your G's and you multiply them times this number right here goes up to about 45 KSI out of 50 in which case your G's should be reduced to .36 of their values before or after you're finding K well no dude that's before you're finding K you got the G you pick up the nomograph and the guy says did you take into account that column is yielding I say oh no I haven't done that yet so what you do then is you apply this factor to the G's that'll give you a new K and that'll tell you how strong it is so you know piece of you that came from the load people you know the gross area because you have selected an initial thing and say I hope to double a W12 by 50 it's gonna work and now you're trying to find out if it works now the only time that you don't actually do that are the two practical cases that they have if you ever have a real fix column where they have insisted that you use G is equal to 1 instead of 0 or if you were permitted to bring this G for a pen column from infinite down to 10 those do not get multiplied by these tau factors they're the only ones that that's already included in here and he's got another example incidentally this is the frame got 12 by 14's across here 1, 2, 3, 4, 5 bays this is really what the structure looks like the solid one is the front face those are the ones in back what happened was the main loads were coming from this end the guy wanted the open windows he wanted windows everywhere so he can't do it if you want me to I can orient these columns such that the flanges on the left, the flanges on the right the webs like that when you push on it it will not fall down because these are welded at these joints however when you push on it this way that's going to bend it about the minor axis you just don't have much choice but to go ahead and put braces in there because bending this thing about its minor axis is going to provide very little of any strength and therefore the whole thing will fall down if you want to situate your column strong axis, weak axis your choice but strong axis you don't need the bracing you can live without them it would be cheaper if you put them but about the weak axis there's just not much else you can do but put braces and that's what's assumed alright, let's start there