 And the questions, remember what we were doing yesterday? I actually did a report, I posted it but I forgot to go check it later on. You know what, we're sort of like the end value is the elasticity of the elasticity. How do you know which material to put on the top versus the bottom versus the bottom of the progression? It doesn't matter which you put on the top or the bottom, as long as you apply, as you apply that in the right direction. The deal was, and it's true, no matter what the shape, the deal was we have two different materials and our regular method for solving for the stresses does not apply for two different materials. We set all that up assuming that the cross sections were prismatic and isotropic. In other words, all one material. I don't remember if I actually stated that, it's just all we had. I did state that about the prismatic nature of the beams, the symmetry about the y-axis. But I don't think I stated specifically that it's all one material because we didn't have anything else but all one material. So now that we're talking about the possibility of a second material, that means in one of them's a tougher material, if you will, one of them's a softer material. And it doesn't matter really which is which, as long as you understand that if we're going to take, let's say this is the tougher one, the one with the higher modulus, the stronger modulus, this one with the weaker modulus, for example, steel and wood. They look very, very different in terms of their elastic moduli. You can transform that in either way, and it doesn't matter what n is. It doesn't matter whether n is greater than one or less than one, as long as you do it properly in the right way. And you might do better with a picture and then you just know what to do with n. You can transform it either by taking out this stronger material and you're going to have to put in a lot more of the weaker material so that they can do the same thing. So you would then leave the lower material the same because we're not replacing that. We're going to take out that with a much weaker material so we're going to need a whole bunch more of it. So clearly you need to increase the dimension there be by some factor. And the only factor we have is n, so here n must be greater than one or it's not going to work in any way. You could also transform the beam by taking out the weaker material, putting in the much stronger, but since it's much stronger we don't need very much of it. So in that case we'd leave the original, stronger material the same and replace it with very little of the weaker material. Sorry, very little of the stronger material to replace that because we don't need as much of it. And so here now if this same dimension there is b, this is now got to be either b over n if n is greater than one or if you had n flip then that would be n beat as well. So it doesn't matter as long as what you do with it results in the same replacement. And we're only changing the width, are we not changing the length? No. And that's the nature of our prismatic beams, the symmetry about the y-axis but not the symmetry about the other axes. It might be symmetric about the other axes, but there's no need for it to be here. Alright, so we're going to take up with that, continue that in a very specific application, one you've all heard of and that's the use of reinforced concrete. Now the basic setup here is given any kind of prismatic beam and the easiest one to draw of course is just the rectangular one like that so we'll use that. So there's a rectangular beam drawn with the techniques we get from technical freehand sketching so it's got that beautiful perspective to it. So it's, you know, I know you guys won't go to a movie now but it's not a 3D movie so I'm making classes 3D so that it leaps out of the page anyway. But let's view it in terms of our standard type of bending where we have bending in the z direction only. Remember that's a right hand rule application. And we'll look at it in terms of our typical type of bending expected for a load bearing beam which is where reinforced concrete is used. So that's the type of bending that would be what we call positive bending whether it's actually by applied moments there or whether there's load across this beam that causes it to bend like that, it doesn't matter it's the same thing either way for our purposes because our concern here is that once we look at the stresses on this now kind of blowing it up in much larger cross-section or much larger side view we know that somewhere there's a neutral axis that depends upon the cross-sectional shape there in this case it happens to be right down the center so that's convenient but it doesn't need to be and across the top we have tension and across the top we have compression across the bottom we have tension so that's nothing new there that's just a review from when we started looking at this design of prismatic beams the trouble comes with concrete well the reason concrete is very useful to use as a building material is you can make it on site you can transport it in those big cement trucks and then when you get to the site pour it into the girder shape you don't have to carry it in the girder shapes down the highways trying to make those sharp bends through New York City and the like with these huge long beams on the trucks that's not true with steel it has to be done it has to be fabricated elsewhere then transported to the site as is also concrete's not terribly expensive for the most part it's made out of crushed sand and stone and limestone and other things like that so it's also steel that stuff we dig out of the earth as well there's a whole other very energy intensive step in steel fabrication that's not there with concrete it's also true that concrete can be poured into shapes that might be aesthetically pleasing as well if we have the bottom of these beams exposed if they were here we can make interesting shapes out of them that might be more pleasing so maybe we make the beam like that just so as we see the room as users there's a softer edge to it all kinds of options like that are possible with concrete that aren't as easily done with steel so that's why concrete can be a very attractive beam material however, as I've mentioned before concrete's terrible intention it's very good in compression so it's excellent up here it's terrible down here where there's tension so what we do with these type of beams is the top being in compression is very good so we leave that as concrete and then as we pour these beams of concrete we insert at the bottom where all the tension is some steel reinforcing bars what you know as rebar and you've seen those those are those big long bars actually if you ever see a building that's been torn down or if you look at any of those pictures from Tokyo this last week of the buildings that collapsed you'll see these spaghetti-like strands all over the place that's rebar that has blown out of the concrete girders that they were in then got all tangled as everything collapsed so we let the rebar take all of the tension that's down here because steel is very good in tension there's still tension in this concrete but most of the tension is absorbed by the rebar so we need to then model that in terms of our composite beam technique that we started with yesterday because once again to figure out what the stresses are we need an isotropic cross-section a single material cross-section and we don't have that anymore now with the rebar in there so we're going to do the same type of thing we did yesterday by transforming that cross-section into a slightly different shape we'll leave the... I don't want to come down to this neutral axis because it might not be that same place as we'll see if you remember when we transformed the cross-sections yesterday we had a new neutral axis because we had a new cross-sectional shape so we'll leave concrete up here because that's so good in compression and that's where the compression is and down here we'll replace this rebar that's so strong in tension with a whole bunch of concrete like we were doing yesterday we took out the tough material put in a whole bunch of the weaker material to compensate for it and remember this is just our modeling that we're doing this we don't actually do this so we're going to put in a whole bunch of concrete right there where the rebar was and in fact the area will be if we have that much area of the steel rebar and that's just the cross-sectional area the little rebars if you cut through them and expose that area that's the amount of area of the steel that's holding that steel holding that tension and then we'll put in an equivalent area of concrete but end times more of it because we need so much more of the concrete to take the same tension as just this little bit of rebar was taking and this is our artificial beam then for our design purposes just for our analysis purposes the rest of the concrete all that concrete we completely take out of the model it's so weak in tension that we don't even count it as contributing anything to the strength of the beam even this little bit up here where it's just a tiny bit in tension we take it completely out to and just say the whole thing is the whole thing is as if it wasn't even there because concrete is so bad in tension we just take it down to the neutral axis notice that the neutral axis of the rectangular section is right here but the neutral axis doesn't account in any way for the fact that we have different materials once we put in different materials for all kinds of purposes shifts the neutral axis so we have a new neutral axis a new place of zero the transition from compression above to tension below because we have dissimilar materials so the neutral axis geometrically was here but remember we only figure that location of the neutral axis based on areas not on what the materials themselves are so when we do put in this different material it does shift the neutral axis so I guess our tension model would be something like this then our stress model would be something like this this is for isotropic beam we put in the rebar that shifts the stress profile shifts the neutral axis to a place where we don't know where it is now we have to figure out what that is but we completely replace the steel with a whole bunch of concrete and then completely eliminate the rest of the concrete just assuming that it'll be in tension and it'll fail and we won't even count it it's a factor of safety measure if you will who needs that hand up? do we do what the NAS is the width and then the height of it is the same as the diameter? the width we don't care about we're just taking into account the total area the area of the rebar is replaced by n times that area of concrete at the same level we're not concerned with what that thickness is that's I guess more detail than we want to go into as we do it but we do need to locate this new neutral axis so I'll call that distance x and the easiest way to do this now is to simply sum up y bar a for anything above the neutral axis and it must equal the y bar a for anything below the neutral axis that's actually what we were doing with our other calculations when we're using that table it just we didn't do it like that but the geometry, the algebra is all exactly the same so we'll walk through it with an example then so here's a concrete beam 5.5 inches, 4 inches down we put rebar that is excuse me that is 6 inches on center 4 inches down and that's not where I'm putting my dimension figures the rebar is 4 inches down from the top so there's another inch and a half of concrete below that 6 inch on center with this rebar and the rebar is 5 eighths in diameter picture you're okay, everybody see what we've got there 5.5 inches tall 4 inches down from the top we placed some rebar that's 6 inches on center so our cross section we can actually save ourselves a little bit of trouble here we don't need to do the entire width of the beam since it just repeats as it goes along for example an entire floor you don't have to cast it as just a single beam and then lay floor on top of that you can do rebar reinforced concrete entire floors so what we'll do is take just a section midway between two rebar and then just do that much of the model so all we really need to look at since it repeats, actually let me draw more to scale we'll take a piece that's 12 inches across the top because that's the distance between adjacent rebar to the other but still includes at least two pieces of the rebar within so now we have 3 inches in from the side we have rebar 6 inches to the next rebar and then another 3 inches there that way we don't have to do the entire width of the thing we can use it it's sort of like a unit cell then this analysis doesn't need to be changed for however why we want to do it because it's all just the same thing repeating itself over and over so we'll just do this one little section and it'll be sufficient to represent as much width of this as we want to do that fair enough picture then so we've got 4 inches down here and then there's another inch and a half below that analyze a subset of the entire beam because it just keeps repeating itself so we'll be fine with that so we apply our magic arrow of transformation where we take out this rebar and replace it with a whole bunch more concrete so we'll leave that as concrete because that's the top where the compression is and we'll replace the rebar with a whole bunch more concrete of area n times the area of the steel that was there let's see put in some numbers for n here what do I have the way we're doing it will keep n as greater than 1 so it gives us a whole bunch more of the concrete so I'll put the steel on top 29 times 10 to the sixth for the steel 0.6 times 10 to the sixth for the concrete so you can see the concrete is an awful lot lower in the modulus of elasticity what's that 8.06 so we've taken out this little tiny bit of area of the rebar and put in 8 times more of that same area of the concrete to be able to absorb that much so let's see I have that area the area of the that's two bars in my analysis cell times the area of a 5.8 inch diameter so that total area of rebar you know that's just a pi r squared but then times two because I have two of them so the equivalent area of the concrete I don't know if I even have that number on hand here in the little picture no not quite but it's 8.06 times that number that's the area of this replacement concrete okay so far the next thing we need to do of course is find out just how thick that piece is we need to locate that neutral axis because we're assuming that takes all of the compression above and then all of the tension below is in the neutral axis alright so we'll use this this tends to be a little bit simpler I think for these kind of shapes so let's see why bar above we know where the centroid of that is so that's pretty easy to do so that's a distance x over 2 down we'll take the top as the reference y bar above x over 2 that's where the centroid of the upper piece is by above I mean above the neutral axis which we're trying to locate with that value x and it's area it was 12 inches so it's 12 by 6 sorry x 12 by x so that's y bar a above and the summation means just in case we had other shapes here we might have had a T shape here or something in fact we'll do that kind of example in a second we just add them up take the total area total of y bar a product above and that's got to equal the same y bar a below now let's see a is easy that's NAS yes we already got that number where it's located though it's located a distance let's see this is 4 inches down we're 4 inches down to that rebar and so the distance is 4 inches minus x below the x axis oh I'm sorry this we're not really doing this with reference we're referring to things from the neutral axis is that 4 inches so it's 4 inches from the top is where the rebar is you're right to the middle of the rebar so this distance then what do you mean to the middle of the beam because that 4 and the other drawing is from the top from the top down to the rebar that's what I drew here from the top down to the rebar it's in the middle of the rebar however don't worry again about the thickness of that piece we already know what its area is we don't care what its width is or its thickness we care what its area is and that's what we got and so we're essentially assuming it's very very thin so this the distance below the neutral axis is then 4 minus x and so y bar a below the neutral axis y bar is 4 minus x and a is n as alright so that's with reference to this neutral axis we're trying to locate not from the top like I think I said like I know I said alright that's an equation with a single unknown x we know what n is that's the ratio of the moduli we know what a s is that's the original area of the rebar and so we can then determine that x is less than so x equals 1.45 inches that's the distance down from the that's the thickness of this upper part of concrete that's taking all of the compression and that's our new neutral axis and then now we can finish the rest of the calculation to find these stresses the next part we need if we're looking for the maximum stress of the sea the next part we need is that moment of inertia moment of area second moment of area of the transformed cross section that's this piece here of the cross section we'll assume m is given in fact I'll just give it to you in a second when we need it it's just one of those plug-in numbers that we've got alright this is a little bit trickier than well it's not trickier it's actually easier except that the little bit of step we might miss is is an easy one to forget remember we have to do this using the parallel axis theorem for each of the parts of our cross section in this case it's just the concrete above the neutral axis and the replaced steel below the neutral axis I see remember is there a centroidal moment of inertia so that's if you remember for rectangular pieces this is one-twelfth b h cubed where in this case h for the upper part is x so it's one-twelfth b which is our twelve inch width times x cubed and that we have we just found the x plus the parallel axis theorem part which is b times x and then d is the location of the centroid from the neutral axis which is one-half x in this case because x is the width of that upper part and then that's squared because that's a d squared that's for just this upper part b is the width twelve x is the thickness remember we just located the neutral axis by saying however much above and however much below the sums of those area moments so that's just the piece above we need to do the same thing with the part below for its centroidal moment of inertia is and I'm going to put that just as I see for right now I'm not going to do one-twelfth b h cubed because we don't know how wide this is we don't know how thick it is I told you to ignore that we do know its area it's n as and then d squared is the four minus x that we had before and then that's squared so that's a d squared right there four minus x is the d squared there's a d squared right there so that's the contribution below the neutral axis which remember is located distance x from the top above neutral axis below neutral axis alright here comes the thing that actually makes things much easier well not much it's just some easier but it's also easy to forget this moment of inertia of the concrete replacement of the steel we're going to take as so thin we don't even care what its thinness is we're going to take it as so thin that its moment of inertia is zero and we won't even count it we won't even bother calculating it it just makes things a little bit easier since we don't know what its thickness is anyway we don't know what its width is we could I guess pick values but that's as artificial as any we're doing anyway so all these numbers we have b is twelve x is the one point four five n is eight point oh six area is the point six one four we've got all those numbers so we can just plug everything in and calculate those numbers should come out to forty four point four inches to the fourth remember that's the moment of inertia of the transformed cross section and so then now we can the maximum stresses each of the pieces the top is in compression the bottom is in tension so we're doing that in the concrete remember that's what we left there so we just do this straight MCI where c is actually x the distance to the top of the concrete from the neutral axis we're looking at just the concrete in compression above and I'll give you a moment now to just hang all this on forty inches and so we've got all these numbers now forty is the moment c for the upper part of the concrete here in compression for the distance the farthest distance of the material from the neutral axis to remember the farther away we are from the neutral axis the distance grows so that c is the same as x in this case 1.45 so that's part of the ease of calculating x from the top is we're going to need that number several times and then I is what we just calculated the 44.4 inches to the fourth so we'll get kips per square inch the compressive stress of the concrete per square inch 1.31 in compression now for the steel in tension below the we're going to assume the steel takes all of that tension remember that was our transformation we assume that none of the concrete below can absorb any of the tension even though it can that's going to be n times the very same numbers except that we have a different c this is the concrete c, this is the steel c because they're different distances from the neutral axis the steel can hold a lot more stress so you know we have to multiply by n and we already have all the units worked out from the same number above this steel distance from the neutral axis is the 4 minus x which is what 2 point and so that comes out to be 18.5 much much more than the sorry the concrete could stand earlier sneeze can't get more than one bless you a day because the steel can hold a lot more stress than the concrete can so you know it must be bigger that's why we put the steel down there because it can hold so much more stress if you divide by 8.06 or have the n upside down then this is going to be a very very small number and your calculation is going to be that there's no stress anticipated down here you might as well put in Kleenex which is not well known as a structural material you know from the direction these things should be this has got to be a lot bigger because that's the steel the steel can hold a lot more stress so now that the stress profile looks something like a maximum of 1.3 sorry the tension in the concrete 18.5 that's the tension is our profile look like that now remember we're assuming that this concrete that's down here takes actually since we're assuming that this concrete takes none of it our stress profile is actually like this sorry the let me try to make it a little bit more accurate here's the bottom of the beam here's the steel it's the steel that can take this 18.5 the steel itself is really only that thick that's the part that's in tension because remember we're assuming that the rest of that concrete can't hold any of the tension we've taken it completely out of the calculation that's our true stress profile for our design purposes because we're assuming that the rest of this concrete can't absorb any of that tension anyway certainly not this much so we just take it out of the calculation entirely man a lot went by there yeah no no because the width it just repeats itself you know there's more load as it gets wider but we have more material holding the load by the same amount so this is the cross section the depth we assume to be the same all the way along in real design you're going to have to take into account that things change at the sides because there's different supports here loads expected there this is adequate for the center of the beam we don't need to change any of the numbers for the increasing width because it just it just repeats itself over and over okay see we got ah we got 10 minutes for you to do your own problem back on the boat concrete TV which is here on those width of the flanges 18 inches top to bottom 6 inch thickness down there which is up from the bottom we have 2 3 bar this is a typical floor beam that bottom part is going to be in tension top part is going to be in compression so the concrete is going to be in compression so maximum allowable stress is 3 ksi first that will be the top part in compression and for the rebar assume a tension maximum allowable stress of 40 ksi nowhere near what the concrete can handle alone so that's the purpose of the rebar so the rebar diameter 1 inch so find the maximum allowable moment then so the type of thing where you've got these beams available they're right out of the catalog easily manufactured so you want to find out what the maximum moment oh sorry oh no we've got this 22 inches across the top but that's the 8 plus the 8 plus the 6 sorry no yeah same no I've got slightly different numbers for some reason 29 times 10 to the 6 psi over 3.8 times 10 to the 6 psi so for whatever reason slightly different ones there'll be a factor of safety in all of these calculations anyway I can speed you up with some of the numbers the area of the rebar those 2 1 inch diameter rebar we'll have an area of 1.571 times AS which will be the area of the concrete you replace 11 point we're going to leave the concrete alone up top above the neutral axis and we don't know where that is remember because that's the neutral axis of the transform section figure it's going to be somewhere in there and then we take out all the bottom concrete the rebar and replace it with a big long fat strip of concrete to absorb the stress so we need to figure out where that neutral axis is and remember that's by balancing Y bar A above which is now a composite shape but it's 2 simple rectangles and that's got to be equal to Y bar A below which will allow you to place the neutral axis in fact of all the calculations that's probably the toughest one so let's do that for now I happen to do it like this we already know that thickness here is 4 inches so what we really need is fat distance there as the only unknown so if you do it that same way then we can prepare numbers easily of course makes that distance 16 minus X because that's where the rebar is 16 inches down from the bottom of the flange it's 2 inches of concrete below the rebar we're throwing out of the calculation in fact if you want to see this business in action it's very easy in New York State because all of our bridges are crumbling if you drive under the bridges you'll see a lot of this concrete cracked and breaking away and even rust colored because now the rebar is getting exposed to water and salt and so underneath all the bridges you'll see the big chunks of this concrete is popping off and just the rebar is left so if you might want to become religious as you go over in New York State Bridges you can get for X that seems to be the the trickiest part of the calculation I is a little bit tricky you remember I is for the transformed cross-section and that D is measured from this this neutral axis there's a transform cross-section there's some fast on this stuff you must remember that X as I calculated is from the bottom of that T so if you do that same thing we can compare numbers get confused I'm finding a Y bar without doing a little chart thing if that's easier for you to do it that way do it that way so Y bar at the top will be 4 plus X for the top Y bar A will break it into two pieces that piece and that piece and so above we know that Y bar for piece 1 is going to be X plus half of that which is 2 2 plus X and then the whole area of that piece 1 is 22 by 4 Y bar is X over 2 we don't know that's what we're looking for and its area is 6 by X that's piece 2 do you think that makes sense so it's once you write out it's not really worth the table but you can certainly do it that way if you'd like and then for the bottom piece its Y bar is 16 minus X and its area is NAS and then that'll give you an equation for X because if you don't draw it like that what else would you draw just like the straight theme across you have to keep some of that C in there well I guess it's possible the neutral axis could be actually up here but if you do that I guess the X is now negative you sort of visually are balancing these things if you cut these out of cardboard then they balance at some place about where X is and remember this is under drawn by quite a bit that's actually almost 12 square inches that Y bar piece 1 is 2 plus X that's a piece 1 that's Y bar it's the distance of the centroid of section 1 from the neutral axis which is X plus 2 because that's 4 so it's halfway across that piece so it's 2 plus X down to the neutral axis so we don't have to reference this yeah we don't have to reference this from the top or the bottom neutral axis yeah I think I misplayed that on the very first part when I was introducing this and then piece 2 is the same alright we're at the end of time I gotta run to the other class so let me give you these numbers as I have you can double check them X drawn from the bottom of the flange there is 0.1573 inches and then I remember the centroidal moment of inertia of this strip you take as 0 so you'll have pieces 1, 2 and then that bottom piece to contribute to the moment of inertia 35, 35, 7 so see if you get those same numbers inches to the fourth and then you have to calculate maximum moment for each of these because one of them is going to fail before the other and then you have to take whichever one is lowest so the maximum moment for the concrete is 2551 based on the concrete limit 5 inches then you have to redo that calculation for the steel remember n will be in there and there will be a different c distance and then it relaxes that that you get 1170 so that will be the controlling one because it's smaller if you put a moment of 2551 the concrete will be okay but the steel will fail means this class will be 4 hours on 3