Good evening, everyone, again. This is, as Sherry mentioned, this is the second of six sessions on the basic pre-stress concrete design course. And just to begin, let me just, these are statements that already Sherry gave you, so I'm going to skip through those. And immediately, just to remind you of the course organization, this is our second session. And in this session, we will spend most of the time talking about two simple examples to start to illustrate concepts in pre-stress concrete design. And then we will give a very brief overview of how the code, the ACI code and the PCI design handbook are organized. So essentially, in this session, we'll really just illustrate some very fundamental concepts that really refer to topics in strength of materials. But it's worth reviewing them before we start with a more realistic example. As you can see, our next session, next week, we will then finish talking about specific code requirements. And then we will start, we will introduce the design example. And from then on out, we will illustrate all calculations with reference to the design example. So that's what we have in front of us. So today, as I mentioned, we will spend most of our time talking about these two simple design examples. After completing the session, these are the learning outcomes that we are hoping that you will be able to achieve. The idea is to be able to demonstrate through these two simple examples, what the advantages of pre-stressing concrete are, and then also become familiarized in case you are not yet about how the ACI code, the ACI 318 code is organized after 2014. I will also show you the contents of the newly released PCI Design Handbook, 8th edition, that is going to be used throughout the rest of this course. All right, so on with the examples. The first thing we're going to try to do is, as I said, these are two very simple examples that are mostly academic, not that we are going to use them or that you normally use them in a pre-stressed concrete environment. I think these are good examples that illustrate the general concepts. So the first thing is we're going to use a concrete block that is 10 inches by 10 inches, and that has six pre-stressing strands that are located as shown, symmetrically as shown. Now these pre-stressing strands have a nominal diameter of half an inch, and the first step we will take is that we will assume that these strands are really non-pre-stressed. So in other words, what they are doing is they're just inside of this concrete block, and they're just sitting there as if they were reinforcing bars, but they have this strand configuration, right? The coiled wires are on the center wire, there are seven wire strands. The material properties of this small element, it is constructed with concrete with a compressive strength of 5,000 PSI. We are assuming that the tensile strength of this concrete is 500 PSI, so 10% of the compressive strength, and the modulus of elasticity is 4 million PSI. So we will often refer to that modulus in KSI units to drop three zeros, and that the strands that are shown there, the half-inch strands, are 270 KSI low relaxation strength. So these are the strands that are used for this example. And the first idea is to find the cracking load. So if we apply a load axially to this element, we want to find how much load it takes for us to crack the concrete. Now keep in mind that because of the strand, the block will not fail. Well, it will form a crack, right, somewhere along the length of this element, but it won't necessarily fail. But we want to find what it takes to crack this element. And initially, we will assume that the prestressing strand is not prestressed. So that's what this slide indicates. We're saying that the steel strands are not prestressed, so they're just sitting there as part of the reinforcement inside the element. And we first need to calculate some cross-sectional properties to be able to answer our question. So this is, in all these slides, just to keep you, to guide you where we are, the title reflects what we're going to be covering because the block itself, we're going to examine it in different conditions. So the gross cross-sectional area of this block, remember, is 10 inches by 10 inches. So that's 100 square inches. The area of the prestressing strand, it's a half inch strand. So we have each strand has 0.153 square inches times six. It's 0.918 square inches of steel. And the concrete area is only the area that's not occupied by the strand. So we have the gross area and we subtract from it the prestressing strand area. And that gives us a net concrete area of 99.1 square inches. So those are the relevant prestressed concrete properties. Now what we're going to do is we're going to use an approach that's commonly used in strength of materials. When you're combining two materials that have different modulus of elasticity. So in this case, recall that the concrete has a modulus of 4 million PSI, as shown in this denominator here. I'll come to that equation in a second. But the prestressing strand does not have that same modulus. So we're going to use an approach that is often referred to as deformation compatibility. And if I go back a couple of slides, we're going to see how this element stretches, elongates under the application of this axial force. And we're going to assume that the strands that are inside the cross section elongate as much as the concrete. So they're perfectly bonded. There's no slip between the concrete, the strands and the concrete surrounding. So having said that, expressed in terms of strain. We are saying that the strain in the concrete, strain as a measure of elongation, is equal to the strain in the prestressing strand. Now if you remember from last session, let me just sketch it again. When we're talking about small strains, and we're talking about this concrete block being tensioned, we are saying, and I'm sketching something quick here, that that's the stress strain behavior of concrete. So first the first quadrant right here is in compression. What really would like to point out is this region down here where that refers to the tension side of concrete. So the vertical axis in this curve corresponds to the stress. The horizontal axis corresponds to the strain. So in the region that is mostly linear, we can write an equation for stress following Hooke's law as shown. That's the relationship or the equation that describes stress and strain, and they're connected by modulus of elasticity of the concrete. And since we're tension in the concrete, it behaves linearly. So it's just the E modulus is the slope of this diagram. So that's what we're going to be doing, right? So recall, if we now express that equation I was just showing you, stress equal E times epsilon, solving for epsilon, the strain in the concrete, gives me the strain at which that block would crack. So recall that the tensile strength of concrete that we're using is 500 PSI, and being careful with using the same units in numerator and denominator, we divide the stress, the cracking stress of the concrete by its modulus, and we get the cracking strain. Now how do we get to the load? The load, as I said, we're using this approach of deformation compatibility. So that means that the strain in the concrete that we just calculated has to be equal to the strain in the steel that's inside of this element. So we're going to calculate the forces in each of the elements of this block individually. So we'll talk about the force in the prestressing strand, or the steel, and we're going to add to that the force that is exerted in the concrete, and adding those two components will give me the total cracking force at a strain of 0.000125. So we use the concept of stress, average stress, in that force equals to stress times area from strength of materials, so that's the concept of average stress. Note that in all this discussion we're using the lowercase letters F to represent stress. So we have the force in the prestressing strand, or in the steel in this case, is the stress in the prestressing strand times its area, plus the force in the concrete is the stress in the concrete times its area. Now the stress in the prestressing strand, again from linear behavior, I often refer to this as Hooke's law, right, to resemble, or to, it's also used in springs. So stress equals strain times modulus, the strain from that deformation compatibility is the same as that that causes cracking in the concrete, so 0.000125, and we multiply that times the modulus of elasticity of the prestressing strand in KSI, 28,500. This is from last week, and we get a stress in the prestressing strand of 3.56 KSI. We multiply that times the cross-sectional area of the strand. We add to that the stress that causes cracking in the concrete, right, because this stress is what's acting in the concrete at cracking, times the net area of concrete, and we get a force roughly equal to 53 kips, and that's the force, that's the answer, right, the force that would cause cracking of this non-prestressed block. So it's fairly simple, the main concept we illustrated here is that we satisfied equality of strains, assuming that there's perfect bond between strand and concrete. So we're going to build up on that slightly. Now reconsidering the same problem, but now we're going to add prestress, right, this same block with prestress, and prestress is going to be acting at first axially, and prestress acts to compress the block initially. So we're going to add a compressive strain initially, and then we're going to try to calculate how much force it takes for that prestress block to crack it again, right, so we prestress it first, and then we come back and try to crack it. The amount of prestressing force that we're going to apply is equivalent to adding a stress of a fraction of FPU, remember that FPU is the grade of the strand that we're using, so we're using 270 ksi strand, specifications that we will use later on and we will illustrate limits the amount of prestressing stress in strand to a fraction of FPU, in this case we're assuming 75% of FPU. So again, we're using grade 270 ksi strand, we limit the initial stress in the strand to 75% of that, and we multiply that times the area of the strand, the six half inch strands, and that gives us a value that's referred to as the initial prestressing force, P sub I, that's why we have the I. So we are compressing that block with 186 kips of force prior to applying any external force to try to crack it again. So this prestressing force, as I tried to illustrate with my hands, will shorten under the application of that force. Now this shortening, right, that's illustrated here as a strain, is going, assuming perfect bond, the steel will shorten the same amount as the concrete does. So upon application of the prestressing force, the block will shorten, so in turn we will lose some of that prestressing force. The prestressing stress that we mentioned earlier, the 75% of FPU, is like the rubber band that we stretched and we were holding and then we release it and then the block shortens a little bit, so we, some of that stress that we initially built into the strand is lost because we're applying it to a non-rigid body, which is this block that we're prestressing. So what happens is that we have a net reduction or axial reduction in length of that element that we will call epsilon CP and that equals to the change in strain in the prestressing strand. Epsilon CP is the shortening of the concrete due to prestressing, that's what the two subscripts here mean. So the strain in the concrete due to prestressing is equal to the shortening that the strand goes through after axial shortening of the block. So this phenomenon, and that's why this is capitalized, is called elastic shortening. Since the concrete is behaving in its elastic range, it's going to shorten elastically and will in turn create a loss of prestressing force that we'll discuss later. So we want to find what the stress in the concrete is due to the prestressing force P sub I. So the change in strain in the strand needs to be found in order for us to find the stress in the concrete. And this change in strain is equal to epsilon CP, the strain in the concrete due to prestressing. So these numbers, which are equal, these strains, have to be found initially. So let's take a step back and look at our problem. We have a block, prestressing force, and no other external force acting. So we have a system that is in internal equilibrium. And before the session started, I made a sketch of that system. So we cut the block in half and we expose the interior of that system. And it's this block. And what you can see here, let me just move it closer to the screen, if we cut it in half, we have the force in the concrete, P sub C, has to be equal to all the force in the prestressing strand in order for horizontal force equilibrium to be satisfied. So we have summation of horizontal forces equals zero. We say then that PC minus PP is equal to zero. So the force in the concrete PC is equal to the force in the prestressing strand with opposite sides, right? We have, in some case, we have compression, and in some other case, we have tension. So that's that first equation you see here. This comes from force equilibrium. Okay. So how do we express each of these terms? First, the prestressing force in the strand is going to be equal to the initial prestressing force, P sub I, right, whatever, what we stretched to the initial stretch, 75% of FPU times the area of the strand minus the change in the prestressing force, which is caused by that shortening of the concrete after the force of the strand is released. So if we still consider elastic behavior, we have that PI, we know what that is. We calculated it earlier, and here it is again, 186 kips, 75% of FPU times AP. Now delta P sub P is the stress in the strand, E times the change in strain, so E times epsilon times the area. So these two numbers here give me a stress, and the middle one, we multiply times the area to give me a force. So that's the net force in the prestressing strand after the block shortens. So that's P sub P. P sub C is now looking at the concrete, the modulus of concrete times the strain of the concrete due to prestressing force times the area of concrete, A sub C. So we have a stress here times area gives me a force. And recall that PP from this equation, PP equals PC. The definitions for each of these forces are here to remind you. So what we're going to do is we're going to set PP equal to PC, and then we're going to solve, since we know that PP is a function of PI, EP, AP, and delta epsilon P, we're going to equate delta epsilon P to epsilon CP. These are the two equal quantities that come from the change in strain in the two materials. That occurs in the next slide. So if we do that, and solving for one of those strains, we end up with this equation. I will, rather than discussing it here, I'm going to spend some time later on discussing how this equation is derived, but for now, you just need to know, let me just go back, that we equate these two forces, and we solve them for delta epsilon P, which is equal to epsilon CP, and that's why we solve for that, for those two quantities that are equal, and we get this expression. The initial pre-stressing force, P sub I, we know its value, 186 kips, and we have in the denominator quantities that are known, E of the concrete, area of the concrete, E of the pre-stressing strength, area of the pre-stressing strength. So plugging in, we could have done explicitly, we could have just plugged in 186 kips here, going back, that's what this calculation is, and in the denominator, watch for units, of course, 4,000 KSI for E of the concrete, the net area of concrete, 99.1, E of the pre-stressing strength, 28,500 KSI, and area of the pre-stressing strength. During the calculation, we end up with a strain, a shortening strain, of 0.00044. So that's the strain that is induced in the block after the pre-stressing force is applied. Remember, I don't know if you remember the definition of strain in terms of deformation, but one definition, simple definition of strain, is the change in length of the element divided by its original length. So delta L over L, if some of you may remember that. So that's what we're talking about here. It's a quantity that's dimensionless because it refers to a change in length divided by a length. Okay, so again, using linear behavior, that's the strain. What is the stress then in the concrete? Well, we have the modulus of elasticity of concrete here, again, through linear behavior. We have that strain we just calculated. We conduct the operation, and we end up with 1.76 KSI in compression. So remember that we have a concrete that can resist a 5 KSI compressive strain, the stress, pardon me. So that's the stress in the concrete after the application of the pre-stressing force. The approach we used was, again, a deformation-based approach, so a strain approach. Now, we talked about, going back here, we said that there is going to be a loss, right? A loss of pre-stressing force. How do we calculate that? Well, we calculate that from this calculation. We said, okay, P sub P was the pre-stressing force is now the, or the leftover pre-stressing force is the initial pre-stressing force minus the force after the block deforms. So we, again, use the same type of calculation, EFP, EP is 28.5, AP, we calculated that before, and the change in strain is the same as the strain in the concrete, 0.00044. So we have a total of pre-stress force after shortening of the block of 175 kips, roughly. The stress in the strand is 174.5 divided by the total area. We have 190 KSI in tension. So notice that the initial pre-stressing force of 186 kips, that's what the strand is feeling, and then it decreases, the force in it is reduced, so it still has a tension force of 174.5. So the initial stress of the strand was 202.5 KSI, that's 75, 0.75 of FPU. So there has been some stress loss, and we'll talk about stress losses later in this course, but we're introducing the concept of a stress loss, and keep in mind that one of the sources of losses is due to elastic shortening. That's what we're doing right now. How do we estimate that, how much we've lost? It's usually expressed in percent. So we have the initial stress of 75% of FPU. We subtract the stress in the strand after losses. So we have, so what's left, right, this is what the stress in the strand that's left, we're rounding up in this case, we had 190, so we subtract that, and we divide by the original stress in the strand, and we get 6.2% loss from elastic shortening. Okay, so we are now in a condition that this block that was originally this long, the shaded area, now we need to crack it, we'd like to investigate how much force it would take for it to crack it. So now it's in a pre-compressed state, and what we're going to do is first we're going to take it to its undeformed condition prior to application of the prestressing force, that would set it to, let's say, to zero, let's say, zero deformation, and then we're going to take it into longer deformation, axial deformation, until we crack the concrete. So the way we're approaching this is, again, by deformation compatibility. We first remove the axial strain that we've generated in it by prestressing the block, so we have to remove this strain of .00044, and then we have to apply an initial cracking strain of .000125. So with this strain, remember, that was the cracking strain of the unstressed block. So we know that in order to crack a block that has zero strain in it, we needed to apply a strain of .000125, so that's what we're doing. So the total strain, then, that we need to apply is the summation of these two strains, so we have a total strain to induce cracking, in this case, of .000565. So to calculate the cracking force, then, we need to estimate how much force the concrete is carrying and how much force the prestressing strand will carry when we strain it, the total strain of .000565. So that's the total force that we need to overcome to crack the concrete block. So using the same approach as before, we now factor out the common strain that we need to apply, the total strain, and notice that the multiplication of this term right here times each of the moduli of concrete and prestressing strand gives me a stress first in the concrete and then in the prestressing strand. That stress multiplied times the corresponding areas gives me the force term, each of these force terms that I'm looking for. So we have the common strain, the modulus of concrete, the area of the concrete, that gives me a force in the concrete, then the strain times modulus of prestressing strand times area of prestressing strand, that gives me the force in the prestressing strand. So it takes 239 kips now to crack that block. If we compare the cracking load, this cracking load we just determined, this is the force that it would take to crack. Now compare that with the force that we would come up with if we added the, instead of using deformation compatibility, using forces. If we add the amount of prestressing force, P sub I, initial prestressing force prior to losses, and we add to that the force that we calculated first of the concrete block without prestressing, which are these two terms, right, 3.56 and .918, we would end up with approximately the same value. In this slide, these are identical values. There are some round off errors. So either using a deformation approach or a force approach would give you the same answer. The big difference here is that we would use directly the initial prestressing force to come up with the same force. And the other key point here is that if we compare these 235 kips with the 53 kips it took to crack the unstressed block, we see that we now, by prestressing it, we have increased the cracking strength of this block nearly five times. So this first illustrates, you know, the first concept of prestressing. By prestressing the concrete, now we, it would take five more times force before we crack it. And remember that we want to avoid cracking of these elements to avoid corrosion of the steel and so on. So that's one big advantage of prestressed concrete. Now like I said, this is an academic exercise. Okay, going back. It seems like we're going back. But I want to, before we do that, so let's go back and we're going to solve this problem a slightly different way. We're going to, oops, sorry. We're going to go back again and try to come up again with this number in a slightly different way. And it looks like we're taking a step backwards, but we're going to illustrate some, a very important concept in the next slide. So before I do that, let me just ask Sherry to start the first poll of the night, because I want to get a, have an idea of whether, you know, the concept that's coming up. Okay. I will be launching the poll in a second, but I will read it very quickly. Do you remember the transformed section technique to calculate stresses in members made of two different materials? The poll is now available on your screen. Please answer now. I will be closing in a few seconds. If you haven't voted, please do so now. I am closing the poll. Sergio, 60% responded yes, 40% responded no. All right. Thank you, Sherry. So, so it's about half and half, right? And those that perhaps responded no, may not have even seen it before. So those, the majority does remember. So let me just go through it at least in some, some level of detail so that everyone understands what we're doing. So these equations are exactly the same as before, right? We start with horizontal force equilibrium, recognizing the, that the force in the concrete needs to be equal to the force in the prestressing steel after transfer of the prestressing force. So those two forces need to add up to zero. And we also have discussed that the prestressing force P sub P is equal to the initial prestressing force minus the change in that force because of the elastic shortening of the element. And we set that equal to the force in the concrete and that we calculate that based on the strain induced in the block. And the second term in this first equation is also calculated based on the strain induced in the block. Delta epsilon P equals epsilon CP. So we use these two equations, PC and PP to plug into this first equation. And as you can see, these terms come directly from this equation where we have a negative sign here, minus P sub I, and then that negative sign becomes a positive sign, EPA P delta EP equals. So the next slide, the first equation is just transcribed from the previous slide. So we have it in front of us and we remember, we recall that those two strains here and there are exactly the same because the strand is not slipping relative to the concrete. So we can factor that number out, that value out, epsilon sub CP say, let's call it epsilon sub CP. And then it multiplies EA of the concrete plus EP, EA, pardon me, of the prestressing strand and we move the P sub I term to the right-hand side. Then we solve for epsilon CP and we end up with this equation that if you go back a few slides, that's the equation we started with that I told you we were going to take, take it step-by-step and derive it. So we have a force divided by terms that look like axial stiffnesses, E times A terms. So let's, I have a slightly different approach here. So let's say that, let me just write it here and I'm going to, as I'm writing it here, I'm going to say what I'm doing and then I'll show it to you on the screen. We have epsilon sub CP equals P sub I divided by EC AC plus EP AP. So what I'm doing here is I'm going to factor EC in the denominator. And then I'm going to multiply that times AC plus EP. And then I divide EC by EC the second term so that I don't modify the equation. So what I'm doing here is as follows. So I'm ending up with that second way of expressing the denominator. So P sub I divided by EC times AC plus EP over EC times AP. And that we can then write as PI divided by EC times that quantity that we call the transformed area. So the transformed area, notice that we are transforming the prestressed steel area into an equivalent concrete area by multiplying it times this quantity EP over EC. So we call that A transformed. So that's what it is, P sub I divided by EC times A transformed. So that concept is used, or that idea of transforming an area of another material into the material of interest, in this case concrete, is called the transformed area concept. And the ratio of EP over EC is called the modular ratio and often denoted as N, N as in men's, all right? Okay, so a slightly different approach here is shown in the bottom here, the bottom equations. The denominator is obtained the way I showed you in my sheets here. And if we multiply this equation here above by the modulus of concrete, you can see that each of these terms is simply multiplied times the modulus of concrete, right? And we can see that this right-hand side is identical to the denominator. That's why we call the left-hand side EC times A transformed. I find it clearer to just show it the way I derived it in the piece of paper. So you have it both ways. So now epsilon sub CP is PI over EA, a much simpler equation, but we need to remember what A transformed corresponds to. So notice that up to here now, remember that epsilon sub CP is the strain in the concrete due to the prestressing force only, right? So it is PI, the prestressing force acting over the transformed area, acting on the transformed area. So we have a P over A as a stress that's acting in the transformed area of concrete. Okay. So moving on. So if we now use the same concept, so for PI, if we then say, well, what happens if we apply a force P? Does the strain in the concrete change? Well, it does, but it's going to be calculated easily using that concept of transformed area. If we now apply an external force P, we could call that force the cracking force, so to speak, right? Then we could compute the strain, the applied strain in pretty much the same way that we did for the prestressing force strain. If it is a tension force then to cracking, then it would be of opposite side. So that's of opposite side to the prestressing force strain. So remember that since we're dealing with linear behavior, we can use the principle of elastic superposition. So what that means is that we can add strains. So notice that in the two equations, this one and the one before, like the only thing that the denominator is the same. The only thing that changes is PI or P. And if this is an applied tension force, PI is a compression force, we can then say, well, the total strain is going to come up. It's going to end up being the applied force P minus the initial force, prestressing force over EA transformed. So all of these are properties of concrete, or if we want to express this equation in terms of stress, we move the Young's modulus to the left-hand side, and we have stress equals P minus PI over A transformed. So these are the strain and stress in the concrete. If we now look at the strain and stress in the prestressing strand, we can then say, we can then again, okay, well, in the prestressing strand, remember that the strain in the concrete is equal to the strain in the strand. So that's that first term that comes out from this equation. And then we add to that the strain in the prestressing strand that was locked in, right? The strain that we induced by prestressing. So that's the only difference here. We again use the principle of superposition. This is the applied strain, right, due to the change in the shortening of the block. That's that strain induced in a prestressing strand. But this strain is the one that is induced when we prestretch the strand before we apply the prestressing force to the concrete. And again, if we want to then write this equation in terms of stress, we multiply everything times the modulus of elasticity of the prestressing strand, epsilon P. So this epsilon P times EP gives me EP over EC times FC. Notice that this term right here, and I'm going to circle it, let's see, I'm going to circle it, this term is the stress. So that's FC. We multiply times EP because we're multiplying the whole equation times EP, and then we divide by EC because that's in the denominator. We add then EP times epsilon PI, that's FPI, the initial prestressing stress. Notice again that this quantity appears, EP over EC, the modular ratio. So again, highlighting it with a pen, this number right here, we're going to refer to as N. And it appears again. So it not only appears in the transformed area concept, but it also appears in the transformed area, it appears in the stress values. So N times FC, N times the stress in the concrete, plus FPI gives me the total stress in the strand. Okay? That's an alternate to calculating stresses using deformation compatibility. Okay, let's see how we can use these equations. So first we calculate N, EP over EC, because that's important, right? So we calculate 28,500 KSI for modulus of strand divided by 4,000, modulus of concrete gives me 7. And we usually round it up, round it down, or round it to the nearest integer, right? If it's 7.8, I would use 8. If it's 7.2, I would use 7. Then we also calculate the transformed area, the area of the concrete plus NAP. And remember, going back two slides, the transformed area is the area of the concrete times N, times NP, pull out, pardon me, plus N times AP. So that's what's shown here. So area of the concrete, as we had before, N, 7. Notice that we're rounding, so we are losing some accuracy, that's okay. A sub P is the area of the strand, gives me a transformed area of 105.5 square inches. So notice that this is greater than the 10 by 10 square inches that this block represents. And it is so because we're transforming steel that has a higher modulus into equivalent concrete, right? By a factor of seven. What that says is that for each square inch of steel, the equivalent concrete would be seven square inches. That's what that number is telling us. Okay, so using the transformed area, we can then find, and using P sub I, we can find the stress in the concrete as P over A transformed, PI over A transformed. We have 1.76 KSI in compression. And the prestressing strand stress is FPI minus NFC, the initial prestressing stress. And because of the signs, remember that the initial prestressing stress was, the stress in the strand was a tensile stress, but then as we, the stress in the concrete is compressive. So that's why we have different signs here, positive for tension in this particular case. So we have 202.5 KSI minus seven times 1.76 that we just calculated, 190 KSI in tension. So in a single slide, we're calculating stresses in the concrete and in the pre-stressing strand by using transformed area concept of what took us several slides to develop by using deformation compatibility. So we can then use these two stresses and multiply the stresses times the corresponding areas to get the forces. So if we have this compression stress of 100, pardon me, 1.76 KSI in compression, that's due to the pre-stressing force. If we add to that, the amount that the concrete can take before it cracks, and that's shown here, 1.76 plus 0.5 KSI, that's the tensile strength of concrete. And we add them algebraically, we have a total tension force or a total stress, pardon me, that would be needed to crack concrete. So using that total stress and plugging it into this equation, P over A transformed, we can find the force that it would take, that this block could take, which is now pre-stressed to crack it. And notice that we plug in the transformed area. So we have 239 kips as we found before. In two slides, we have found the total force. So the power of transforming the section is that we transformed its properties into equivalent single material properties, in this case, concrete, and then we come up with the answer much faster. And the other thing is that we don't have to, so losses now are embedded into this result. So remember that this number came out of considering losses, so in other words, considering the shortening of the concrete before we have to stretch it back again. This number already reflects that. So by transforming this section, we're already accounting for that deformation. Okay, so that finishes the first example that is the simplest one. So the second one, since the calculations would become much more complex if we were to use deformation compatibility, we're going to use that concept of transformed properties in order to deal with this block. So, but the first thing is we're going to approach it in the same way that we did before. But this block, you'll notice it's a 10 by 10 inch block where the strands, the six strands are concentrated now near the bottom, near the tension face. So the centroid of the pre-stressing strand is located two inches from the tension face. We're going to subject this block to bending. So in other words, we're going to apply a bending moment that has that sign, tension on the bottom. We're going to apply a moment that inducing tension on the bottom of the block. And we want to first calculate how much moment or what the cracking moment is for this block. And we will first assume what we did before that these six strands are not tension. So they're sitting there non-pre-stressed. They're just in the concrete section. So we're going to use a result from strength of materials again. We're going to use the bending formula. So this formula, which I'm skipping this formula above it, and I will come back to that in a second. This formula comes out from the bending formula, which some of you may be more familiar with it if we express it in terms of the stress F equals MC over I. You may have seen that before. So if we solve for moment, the cracking moment, that's what we're doing here. So this comes from strength of materials. So the cracking moment is calculated as, and I apologize, I just realized there's a small error here. It should be F prime T, not F prime sub C. That's the tensile strength of the concrete, F prime T times its moment of inertia of this block divided by C, which is the distance from the centroid of this block to the extreme fiber in tension. So it is 10 divided by two is five inches. So going back, remember that the moment of inertia of this rectangular block, the square block is B times H cube divided by 12. That's 833 inches to the fourth power. So we end up with a cracking moment of this mildly reinforced block with six strands is 83.3 kip each. Now there's no strand that we have considered here. The area is so small compared with the rest of the concrete that we're neglecting it for this calculation, right? If we were doing it very strictly from an academic point of view, we would have to consider it, but we're not doing it. We're just neglecting it. So the number to keep in mind is 83.3. Okay. So now notice that the strand is eccentric. So the pre-stressing force is going to be applied at the centroid of the strand. And roughly, you know, that would take place right here at that level. That's where the centroid is. So the pre-stressing force is now, I'm going to dash where the centroid of that strand is, of that block is. So there is some eccentricity of the pre-stressing force. It's applied at the centroid of the pre-stressing strand. And there's some distance between the centroid, which is two inches from the bottom and the centroid of the shape, which is five inches from the bottom. Okay. So there's two deformation components. There's going to be some shortening due to axial force, where we're going to separate the deformations, shortening due to the axial force P, that I, by changing slide, my force disappeared. And there's also going to be rotation at the end due to the component P times E. This shortening, we've already calculated as the pre-stressing force, initial pre-stressing force, divided by E of the concrete and the air transformed area. So we already derived this expression. So now we need to concentrate on deriving an expression that lets me estimate how much shortening there is caused by rotation of a section when we induce that moment that is equal to PI times E. That's going to, we are going to do that by using this formula. We're going to say, well, what is the moment that the pre-stressing strand induces in the section? Well, we're going to do the same approach. We're going to say, well, the initial moment due to PI times E minus however moment we lose because of that rotation. Okay, so that's what we're going to do. And so delta M sub P is the loss of moment induced by that rotation. So we're going to express that in terms of the force. So first let's estimate the stress at the centroid, the stress in the concrete induced by bending. So we're using the bending formula and this is calculated at the centroid, at the level corresponding to the centroid of the strand. So we have the moment in the pre-stressing strand times the eccentricity. So instead of having C here, we have E, that's the distance from the centroid of the section to the centroid of the pre-stressing strand and divided by the moment of inertia of the cross section, which if we rewrite, we write in terms of M sub P. We simply move this to the left-hand side, E and IC, and we get M sub P and that is equal to MI, the initial moment due to the pre-stressing force minus the change in moment in the pre-stressing force due to that moment, right, that the pre-stressing force generates. So we have a stress again at that level, E sub P times delta epsilon P induced by bending, that's the stress, times the area of the pre-stressing strand that gives me a force, a force times an eccentricity gives me a moment and that's that moment that I'm looking for right here. Delta M P is this term right here. And the initial moment is that term right there. Okay, so definitions are here. Anything else just to highlight the eccentricity of the pre-stressing steel is measured from the centroid of the section to the centroid of the pre-stressing steel. And the rest of the definitions are shown. Okay, so let's look at FCB. FCB is calculated the stress in the concrete at the level of the pre-stressing strand induced by bending is the modulus of concrete times the strain in the concrete at that level. And recall that we are assuming perfect bond, so the strain in the concrete has to be equal to the change in strain in the pre-stressing strand. So we equate these two numbers from last slide. Epsilon sub CB is that number. And we have P I times, pardon me, that's the moment, right? That's the equation. Let me just go back. Oops. That's this term is M P minus, M P is M I minus delta M P. So that's that right hand side. We just simply transfer it here. So we have in front of us and we start manipulating this equation. We first express this stress FCB as E times epsilon CB. That's the stress in the concrete at the level of the pre-stressing strand equals the moment induced, the initial moment induced by the pre-stressing strand, P times E, P I times E minus the change in moment because of the rotation of the ends. And then we factor, we move the eccentricity to the right-hand side. That's where these squares appear. We factor out epsilon CB, knowing that epsilon CB and delta epsilon PB are the same. So we factor that out and then we solve for epsilon CB and we end up with this equation. Using the same idea as we have before, we can express the denominator in this particular case as the product of E sub C times I transform. So we're using the transformed concept, transformed area concept by factoring out E sub C and finding I of the concrete plus E P divided by EC times AP. Times E squared, that becomes the transformed moment of inertia. So that's how we, one derives it, the transformed moment of inertia in the denominator. Notice that we're calculating the strain at that level at the center of the strand. So we have the numerator PI times E squared and the denominator, the derivation for E I transformed is shown here. So let me just go through a couple of terms. So what's EC here? EC, pardon me, IC, not EC. IC is I gross, the gross cross section, the gross moment of inertia plus the area of that block times the distance between the centroid of that block minus the distance to the centroid of the transformed section squared. This is, you may remember the parallel axis theorem to calculate moments of inertia. That's what this term represents right here. Now this other term, we haven't talked about this negative sign here, but the fact that instead of using, so N we remember that we recall that it's EP over EC, but we subtract the number one because we have to recognize that the strand is taking space or displacing concrete from that space. So that's why we subtract minus one here. This essentially reflects that there's no concrete in this region. So we have to subtract it from its contribution to moment of inertia. So that's shown here in this slide. Those of you that have dealt with reinforced concrete design or service level calculations, that's what you would do if the concrete is not cracked. So long story short, the strain at the pre-stressing strand centroid can be calculated using the transformed moment of inertia using this equation. So this is the strain induced by bending. And here are some definitions. CTR is the centroid of the transformed section and CG is the centroid of the growth section, okay? Now this will facilitate our calculations because remember we can use the principle of superposition. So we add the total strains. The total strains in the concrete are now going to be PI over EA transformed plus PE squared over EI transformed. So we have factored the one over EC. So we add the shortening induced from direct axial force and we add to that the shortening induced by rotation. That's what that equation is doing, adding those two components of deformation, okay? And that in particular, since we're using E here, this is the strain at the level corresponding to the centroid of the strand. But if we want to calculate the strain at any other level, we would replace from this term, we would replace one of the E's that are here shown as squared by the distance from the centroid to that fiber. And we call that Y. So we can, at any strain, at any point from the centroid of the section, we can calculate strains by multiplying it times Y. You might remember that this P times E is the moment induced by the eccentricity of the prestressing force. So that is a moment. And the other E in the equation above it is just to get the stress at that level or the strain in this case, since we're dividing by E sub C. So if we then move this modulus to the left-hand side, we can express this equation as a stress equation, P over A transformed plus P E Y over I transformed. So this has the form of the general prestressings equation that we talked about last time. If we then induce an applied moment, we add a third term and we can say, well, we can then express the total stress by using the principle of superposition by adding or subtracting the moment, depending on what fiber we are interested in. In this case, positive means compression. And the top sign corresponds to top fibers or fibers above the centroid. The negative sign corresponds to fibers below the centroid. So we have these three terms. And so this illustrates the concept of transformed area on a section that is bending. And I believe that all of these notes here, they're just repeated here, but in our first class, we discussed these over, or the sign convention, I should say. Okay. So now to the calculation. So all of these is background for, to be able to allow us to do the calculation. Now we're going to look at the block with prestress. And remember going back many, many slides now, that's the original problem, right? We are, we want to know by using this deformation idea, we want to know how much moment it takes now to crack this block if it is prestressed. So now we're going from non-prestressed to prestressed. So in this case, we again use the tensile strength of the concrete 500 PSI. We calculate the modular ratio EP over EC at seven. We recall that. And we define the distance from the centroid of the section to the tension fiber as 4.84. And therefore since the centroid of the steel is at two inches from the bottom, the eccentricity from this centroid is 2.84. Okay. The transformed area we calculated before, and the moment, the transformed moment of inertia is 881 inches to the fourth power. Next slide will illustrate these two quantities, YB and 881. So how do we calculate YB? Well, we, it's a centroid calculation. So this 100 is the gross section of concrete times it's distance from, so these are all taken the moments of these areas are taken about the base of the block. So a hundred square inches times five inches, that's to the centroid of the square. Plus we add now the strength and notice that we transform the strength by using the modular ratio. Seven minus one, that's N times the area of the strand. This converts the area of strength to equivalent concrete. And this area is located two inches from the base. That's the moment of this area about the base. And we divide by the total transformed area. That's a centroid calculation that ends up being 4.84 inches from the bottom, measured from the bottom. That's the centroid of the transformed area. So notice that it's not five inches, but the presence of strand pulls the centroid downward. The centricity, as I mentioned before, is 4.84 minus two inches, 2.84. So again, the centricity is measured from the centroid. Transformed area, we have calculated that before. So this is repeated here for your convenience. And then transform moment of inertia. We have the gross moment of inertia of concrete, 833 inches to the fourth power. That's BH cubed over 12. We have the transformed area of this transformed area of total area, I should say, times the distance between the centroid of the original area minus the centroid of the transformed area squared. So again, parallel axis theorem. And then modular ratio minus one to accounting for concrete being displaced. The area of strands and 2.84 squared, that is the distance between the strand and the centroid or the eccentricity. Again, parallel axis theorem calculations. Area times distance squared gives me moment of inertia. So the total is 881. That's what's shown in the previous slide. Okay, so going back to our equation. So if we recall the sign convention, we have P over A, P E Y over I, and I'm dropping the subscripts, you can read them. The negative sign corresponds to tension. These two are positive for compression. This corresponds to tension because I'm interested in a fiber that is below the neutral axis or the centroid of the section. And what I'm interested in finding is how much moment it takes to crack the section. And the section will crack at a stress of 500 PSI, five KSI. So we replace this M by M cracking, and we will replace this F prime sub T by the 500 PSI stress in tension. So it's going to have a negative sign here in a second. So as you can see, the equation here is identical. The only thing we've done is we have replaced M by M cracking, and then we solve for M cracking by algebra, right? So we multiply first times I transformed and divide by Y and move all the terms associated that are not part of this term to the left-hand side. So we end up in summary, we end up with this equation. P, I transformed, A transformed, Y, et cetera. I'll let you look at that, and then you can derive it if you would like. But then repeat it here for our convenience. The cracking moment is now going to be PI, I transformed. So I transformed is 881, PI is 186 kips. PI is 186 kips. That's the force, the pre-stress, initial pre-stressing force before losses. Since we're using transformed area, we can account to losses afterwards. I mean, we don't need to account for elastic shortening losses, pardon me. The distance from the centroid to the tension phase is 4.84 inches. A transformed is 105.5, I believe, so we round it up. PI times E, you know what those are. F prime T, I mentioned that the negative sign would appear. There it is. And it's a double negative, right? Negative because of our solving for that stress, for that, for M cracking, and we have a negative in the parentheses. So make sure that we use both negatives, so that term will become positive. I transformed, 881, and 4.84 corresponds again to the distance between the center of the section and the extreme fiber in tension. So we end up with a cracking moment of 938 kippage. And again, another academic exercise, but it illustrates that when we apply the pre-stressing force eccentrically, we went from 83.3 to 938 kippage. 11 times the cracking moment for the unstressed concrete section. So the calculations were facilitated by using transformed area concept. Otherwise we would have to have taken explicitly all the deformation components and made sure we didn't make any mistakes. So as you see that with these two examples, we have illustrated that transforming the section is powerful and we can use that, and we can then come up with the basic equation for pre-stressing. Okay, so those are the two examples. If there's any questions, we have the second part of this class. We are going to talk a little bit about code requirements, but mostly we're going to talk about the code organization. So I want to ask Sherry to launch the second poll of the night and then also open it up for questions. So why don't we answer the second poll? Okay, the second poll for tonight. I will read it and then launch it. Please indicate the latest version of the ACI 318 code that you have used. And the poll was launched now. And please answer on your screen. Okay. Waiting for a few more people. Okay, I'm going to close the poll now. Okay, 5% answered ACI 318-05 or before, 15% ACI 318-08, 5% ACI 318-11, 65% ACI 318-14, and 10% have never used ACI 318. Okay, thank you, Sherry. So before going into the code requirements, so let me just make a comment. Those of you that have never used ACI 318, if you have used the PCI handbook, you have indirectly used ACI 318 because some of the provisions in the PCI handbook come directly from ACI 318. Now it's good to see that most of you are already familiar with 318-14. The basis of this course, as I will mention in a few seconds, are the ACI 318-14 code and the PCI design handbook, eighth edition. So it looks like you're going to get a lot of out of this course by the design example, the majority of you. And for some of you, it might be a little bit of repetition. Before going into the code organization and highlights, I'd like to just pause, and since we just finished the simple examples, open it up for questions rather than waiting till the end. Just a few questions if you have any right now, otherwise we can move on and then open it up again at the end of the session. Sergi, I don't have any questions right now. Okay, so let's keep moving then. And then again, so we'll pause again at the end for any questions you might have. So let's talk about code requirements. So the idea is to be able to understand how ACI 318-14 is, it's what its philosophy is and how it applies to pre-stress concrete. And also between this session and next, we want to be able to illustrate the difference between service loads and factored loads. We won't specifically address those right now. We will address those next week and also talk about 318-14 design philosophy. Those are the learning outcomes from the code section of this course. Keep in mind that pre-stress concrete design, if you're talking about what governs design of pre-stress concrete, ACI 318 is actually the governing body. So anything, any code refers to ACI 318 in some version or another. Most, so lately, if your state has already adopted the IBC 2015, it refers to 318-14 as the concrete for, as the code for pre-stress reinforced concrete. So ACI 318 is the code that is used for design of not only pre-stress concrete, but also reinforced concrete, plain concrete. So it's an all-encompassing structural concrete code and it represents a unified approach or presents a unified approach for design of structural concrete. And it turns out that pre-stress and reinforced concrete share many of the same concepts. So a little bit of background now. In the PCI 8th edition, the PCI design handbook that we are using for this course, there are references to ACI 318-14 throughout. If you have a copy of the 8th edition, and if you don't, I encourage you to get one, section 5.1 has an introduction of the PCI design handbook as it references ACI 318-14 and also IBC 2015. These are the two governing codes that are referenced in the PCI 8th edition design handbook. Now, there were some exclusions that apply only to pre-stress concrete that came out from the PCI standard design practice. And those used to be in the 7th edition. There used to be in chapter 14, section 4.1, 14.1, pardon me. There was a section, the PCI standard design practice was there. So if you have a copy of the 7th edition, you can look at those exclusions in that section. The 8th edition does not have that section or does not have the PCI standard design practice anymore, but it will be published in a future edition of the PCI journal. So stay tuned for the standard design practice, the PCI standard design practice that actually refers to exclusions that one can use for design of pre-stress concrete that are not included in the ACI 318-14. Now, another note that I would like to make and then point you to is that also in the 7th edition, there used to be a 22-page appendix that were impacts of the ACI 318-08 had on pre-stress concrete. So it had a 22-page long appendix illustrating point by point what that code, how that code impacted pre-stress concrete. This 8th edition design handbook does not have the appendix, but the provisions in the ACI 318-14 remain largely unchanged from, oh wait, there are some changes, so don't get me wrong, there are changes, but the 22-page long appendix largely still applies. If you have a copy of the 7th edition, you should look into that just to see how this code in 08 affected the design of pre-stress concrete. Okay, so let me just give you a comparison now to more recent codes. In ACI 318-2011, the organization of it had two chapters that pertained to precast and pre-stress concrete. So you would go into chapter 16 or 18 and would find provisions for either precast concrete or pre-stress concrete, but you also had to navigate other sections of the code, such as chapter 11, for example, for design of shear and pre-stress concrete. So there were specifications and there were requirements spread throughout the code. Now in 2014, 65% of you already know this, the code was completely reorganized. And the idea was to make the code more user-friendly. So it was reorganized into sections that were called member chapters. So say sections related to beam design, column design, et cetera, and other chapters that are referred to as toolbox chapters, toolbox chapters, where general provisions are listed. So as you can see, the code is now divided into structural components. So you open the chapter on beams and there are all the requirements for design of a beam. And if they're not specifically there, they will point you directly to other sections in the code that you have to satisfy. So the idea is that by reading a specific chapter, you would not forget about any provision if you're designing, say, a beam or a column. That's why it was reorganized. Now, going back here, in chapter 18 of ACI 318.11, there used to be a series of exclusions for prestressed concrete that did not apply for prestressed concrete that were in the rest of the code. Now, those exclusions, as I said, as it's listed here, used to be in chapter 18, section 1.3. Now, those exclusions are spread throughout the ACI 14 code. That's why this is called ACI 318.11 old exclusions. So many of them, I track them down in the new code and I give you the section. So for example, let me just, I'm going to read this one, for example. The position of floor joist construction joints does not apply to prestressed systems. And that is in this particular section of the code. Two-way and one-way joist systems do not apply to prestressed construction. That is now in 8.8 and 9.8. And they are in two different chapters because chapter 8 refers to two-way systems and chapter 9 refers to one-way systems, slabs and beams, respectively, and so on. I won't read all of them. I'll let you browse through them, but I've listed several. Chapter 11, chapter 11 is design of walls. So the design of walls now includes non-prestressed and prestressed walls. So everything, since that chapter is specific for walls, everything is now in that chapter. An important exclusion, section 6.3.2.3 in the new code, ACI code, allows you using the definition of the T-beam in the ACI code for prestressed members. What is how the effective flange of a T-beam is calculated. The code allows you to use that for prestressed members as well. But then in the commentary, it says that for a prestressed member, the effective flange width for a prestressed T-beam is left to the experience and judgment of the licensed design professional. That recognizes that past experience in prestressed concrete has indicated that the effective flange width may be different from what is calculated based in using these two sections in the ACI code. So it's left to the design engineer or to the experience of the design professional. And these are some of the T-beam requirements for non-prestressed members. Notice that referring back to the seventh edition in particular here of the PCI design handbook, the PCI standard design practice indicated the same thing, right? That for thin flange members, so one of the issues with design experience, right? Thin flange members, a double T section, uses the entire flange as effective. So I should point you to an example in the PCI eighth edition design handbook, example 5214, that actually illustrates how one deals with the entire width of a double T because it's a thin flange member, it uses the whole width of the double T as the effective flange width and not using these provisions from the ACI code for the overhanging flange. Just a reminder, I'm sure many of you know this, right? Precast and prestressed concrete, what is the difference? Well, the terminology is that precast is concrete that is cast in a factory. It could be prestressed, it could be non-prestressed. And if it is prestressed, it could be pre-tension or post-tension. So that's precast. Precast is the universe of fabricated concrete in a factory, right? Not cast in situ. And prestressed concrete is the class of concrete that has stress in it. So the ones that we've been discussing today. And finally, just to give you a roadmap, right? Of the eighth edition chapters, this is the design handbook chapters. You'll see that you have chapters on analysis, design, design of connections, et cetera. And what's new in the PCI eighth edition design handbook, there's three appendices that are not covered in this course but it's important to point them out because they come out of a significant investment by PCI on research. Blast design of precast prestressed concrete component is in appendix A. Structural integrity requirements and design for disproportionate collapse is in appendix B. And another resource that is now in the eighth edition, this is brand new, is design of concrete diaphragms. And so perhaps we'll be developing courses on these topics, advanced topics later. So just stay tuned as I've said before. Okay, so let me just briefly go through the code organization and we won't go into too much detail here. Notice, as I mentioned before, that the first six chapters in the ACI code are general chapters. That's what's included in the first six chapters of the ACI 314 code. We will start particularly, we'll start with the chapter five loads in our design example. We'll also refer to structural analysis. So we'll also use this, one of these general chapters five and six. Then chapter seven through 13 are referred to various structural elements, one-way slabs, beams, two-way slabs, et cetera. Chapter 14 talks about plain concrete and so on. 18 is reserved for earthquake resistant design. Durability and design of concrete as a material is chapter 19. And then the toolbox chapter, these are, pardon me, chapter 21, we will also refer to this chapter because that's where it includes the strength reduction factors that are used for design. And 22 through 25 are the toolbox chapters that are used by each of these general chapters or design of structural elements chapters. And you can see the rest, you can read the rest. So with that and this very brief introduction of how the ACI 318-14 is organized, I'd like to conclude the session this week. And again, open it up again for questions. So questions on the first part or the second part? Just wanted to let you know the chat box is open. Please type any questions in there. We'll give you all a few minutes. Nothing so far, Sergio, but I'll give it a few more minutes. Okay. Every time I say nothing, it's always a question. Yeah. Oh, got a question. What is the reasoning of leaving chapter 14 in PCI 7th edition out of 8th edition? Not the entire chapter was left out, only the section that referred to the PCI standard design practice. To be honest, I think what I have heard, and I don't know this for a fact, is that the standard design practice, the PCI standard design practice still had to go through a review through the technical advisory committee review. And there was a need to actually publish the PCI design handbook because it had been slated for publication back in 2017, right? So I think it was decided to leave that for now so to not delay the publication of the handbook anymore and to publish it in a future edition of the PCI journal. That was, I think that's what the reason for that. So only the section that referred to the PCI standard design practice was left out. Nothing. Okay. Well, we have light questions tonight, Sergio. Well, maybe people need time to think about things and they'll probably email you, Sherry. Yes, if you think of any questions after tonight, just email me to snotnetpci.org and I'll pass those on to Sergio and share with everyone. Thank you for attending tonight. Please have a very safe holiday weekend and we will see you next Thursday night at six o'clock p.m. central time. Thank you and good night. Good night, everyone.

pre-stressed concrete design

organization of the course

design examples

concrete block properties

material properties

calculate strains and stresses

deformation compatibility

transformed area

stress loss

advantages of pre-stressed concrete

ACI 318 code

PCI design handbook chapters