Thank you Sherry. Good evening everyone. This is, as Sherry mentioned, this is the first course in pre-stress concrete design. It's really a basic course and the idea is to present basic concepts on pre-stressing. For those of you that may have been out of school for a while or that need a refresher, this is a perfect course for you. The other benefit of it is that it has been recently updated to the latest edition of the PCI Design Handbook, the 8th edition that just came out this year, or actually late last year, and it is also in compliance with ACI 318.14. It is a course that we have offered several times, that PCI has offered several times, and it really is based on the solution of a fairly simple design example. So let me just go through the first series of slides just to let you know what's covered in the course. These are just some statements that Sherry already alerted you to, so you probably already received a copy of this presentation as part of your registration to the course. So here's really the course organization and how we're going to be covering it throughout the period of six consecutive weeks on Thursdays. We will first go through three lectures that are intended to present to you the basic philosophy of design and some basic concepts of pre-stressing, introducing pre-stressing materials. Then that will be today. In lecture number two, we will start talking about pre-stressing examples and the principles of pre-stressing through some simple examples, and we'll talk a little bit about loading and how the codes are organized, both the ACI 318.14 and the PCI Design Handbook. And then we will really focus on starting in lecture three and going through lecture six. We'll really focus on the design example, and we will go into detail in how to design and detail a pre-stressed concrete beam, which as I mentioned, is fairly simple. The idea is to present the concepts that you can extend them to a more complex design example on your own, but the idea is to try to present as much depth and to keep the calculation simple. It'll be a simple beam that will be designed. So this first lecture, the idea is to talk about the basics of pre-stressing and introduce pre-stressing materials. Several of you may be working as producers or in a design firm. This first lecture is really intended as a very basic introduction to pre-stressed concrete design, and for that reason, you might find some of the graphics are quite simple, illustrative, but some of the concepts here are not necessarily that involved. So hopefully you'll bear with me and enjoy the presentation. As part of each of the sessions, we present you with a series of learning outcomes, and these are the learning outcomes that you might expect after completing this session. First is why pre-stressing is beneficial. Why is it beneficial compared with reinforced concrete design? Also introducing what is referred to throughout the course as the basic equation of pre-stressing. Talk a little bit about what the difference between pre-tension and post-tension concrete members. Then go through a virtual tour of a pre-cast, a pre-stressing plant. This course was offered at a pre-cast concrete producer facilities for several times, and what would come in here would be a tour of the facility. Since we are not doing it, since now we're doing it online, we have a series of pictures that actually illustrate the entire process from hitting the highlights from beginning to end of the construction of a pre-stressed concrete girder. Talk a little bit about lowering tensile stresses that occur after transfer of a pre-stressing force of members in a pre-casting facility. And then finally, the last bullet point here is to understand what are the basic properties, the fundamental properties of materials that are used in pre-stressed concrete design, and just point you to the relevant standards either from ASTM or the Ashtoiler deco. Let me just back up and start or just clarify for a second. If you have questions, I believe there's a box that you can type in your questions. There's a chat box, and if you type in your question, Sherry will look at that, and I will be checking periodically in my second screen to see if there's any questions. So feel free to type in questions as we go along, and I'll try to answer them as soon as they get to me. All right, so the first thing that I wanted to cover is what are the properties of the stress-strain properties of concrete. We need to talk a little bit about some of the properties of the materials that make up pre-stressed concrete, and the first thing is to talk about concrete. So this is a typical curve of a, the stress-strain curve of a concrete cylinder tested in compression. It involves a, when you test a concrete cylinder in compression, you usually break the cylinders just to determine strength. That strength is called here, denoted here as F prime C. It's normally referred to as the 28-day strength of concrete. So in testing facilities or in laboratory facilities that are just intended to test, you know, for quality control purposes, you don't necessarily construct the stress-strain curve as shown here in this graph. The important part of this graph is to define a couple of regions. One of them is how the behavior of concrete is in compression. So the upper curve here shown, and the other one is to show how it behaves in tension. So as you can see, concrete in compression is a highly non-linear material. It starts deviating from linearity at about 50% of F prime C. And again, F prime C is the 28-day strength. And then it drops gradually as you, after you reach the peak. This is the typical behavior of what's referred to as a quasi-brittle material. I'll come back to that in a second. The other thing that is shown in this figure is what we can define as the modulus of elasticity. So typically one can, for a linear elastic material, the modulus of elasticity is defined as the slope of the initial linear, initially linear curve of behavior in the material. However, for concrete, since it's a non-linear material, we have to choose a point through which to pass a line. So we start at the origin and typically that line passes through 50% of F prime C. And that's how we define modulus of concrete, modulus of elasticity of concrete. So you see that it really is not a linear material in compression, but we can, if it is, the stresses are low enough, we can treat it as if it were a linear material. Certainly if concrete is loaded in tension, uniaxially loaded in tension, it does behave like a linear material. However, one of the drawbacks of concrete is that it is very weak in tension. So this is graphically showing the full stress-strain behavior in compression and tension of a concrete cylinder. Also, I failed to mention that throughout the course, we have a series of polls. So I will periodically stop the presentation and ask Sherry to start a particular poll. So we're nearing one, but let me just show you one before we start the first poll. I'm going to drag onto my main screen a couple of pictures of a cylinder that's being tested in our laboratory. And some of you may already have conducted these types of tests. You know, the typical one is just to measure force, as I mentioned earlier, but the other component that's missing in this graph is that we also need to measure deformation or axial shortening of the concrete cylinder so that we can actually plot strengths. So how do we do that is shown here. So this is a concrete cylinder and we attach a compressometer through three rings that are connected to a couple of dial gauges. They may be digital gauges as well. So what we do in the laboratory is this machine is loading the cylinder. One can record how much the cylinder compresses and therefore, you know, load it up to about 50% of F prime C and then release it and then load it again just to get a couple of readings. And one can then determine modulus of elasticity that way. This is not commonly done in testing laboratories unless one wants to determine modulus. Typically you would only break the cylinder for strength. And if you do that, that's as many of you know, and I've tested cylinders, you know, that's one aspect of the cylinder after testing. You know, it's just after peak, after F prime C, after going past F prime C, this is the way the concrete cylinder would look. Okay. So for design purposes, we don't really need to use the entire curve. So as I mentioned earlier, the behavior of concrete in compression is denoted as a quasi brittle material. But for design, we are going to consider or assume that concrete behaves as a brittle material. So we won't really use this unloading curve, this gradual unloading curve. So, but at some point we will assume, let's say at some level, we will assume that concrete has crushed and therefore we can no longer sustain any more loading. And at that point we will call it a day. So for design, as mentioned here, concrete is considered a nonlinear brittle material. So there's the unloading branch of that curve will not be represented. So we will choose a strain at which we stop that unloading part of the stress-strain curve. So that brings us to the first poll of this evening, which I'm going to ask Sherry to see if she can start it. And the question for the first poll, will you read the question Sherry? Yes, it is. I will. What is the maximum compression strain that is assumed concrete can reach for design purposes? The poll should be available on your screen and you can answer it now. We will give you a few more seconds. A few of you need to vote. Still waiting for a few of you. We'll be closing the poll in a few seconds. I am closing the poll now. Poll is now closed. Sergio, 38% answered 0.002, 62% 0.003, and 0% 0.03. Great. All right. Thank you, Sherry. So as I hear those numbers, the majority voted that the maximum strain that is considered for design for failure of concrete and compression is 0.003, and that is correct. 0.002, as you see here in the graph, really corresponds to a strain that is close to the peak stress. So we do for design consider that concrete has unloaded slightly, but after this point, you know, this part of the stress-strain curve is eliminated. So we consider, as I mentioned earlier, concrete to be non-linear and then brittle. Essentially, we consider that concrete would drop load after this point, after a strain of 0.003. So that's an important consideration. Okay. So just let me keep going here. All right. So continuing with the basics, what happens then if a beam is fabricated with this brittle material, right? That doesn't resist tension very well. It only resists about 10% of the compressive strength in tension. So if we fabricate an unreinforced beam, and this is a typical modulus of rupture beam, short beam, and you can see here the supports, there's one support here, another support there. Let me just change to the pointer. Okay. So one support here, one support there, the loading points are here. So we have a region of constant moment between those two load points. And if you remember from strength of materials and noticing that concrete is weak in tension, once we get to the peak stress in the bottom fiber, which is where maximum tension stresses would occur, concrete would crack because of its low tensile strength. And that crack would propagate, move upward, and will split the beam in half. So this behavior is highly brittle. And furthermore, the stresses that are needed to break it are very low compared with the compression stress capacity of concrete. So that definitely is not a very good use of the material for structural purposes. So that's the basis for using reinforcement in reinforced concrete design. So this is going back again to the strength of materials approach. We have a beam that is rectangular in cross-section. We have strains. So strains are indicative of how the fibers are shortening or elongating if they're above or below the neutral axis. So for a beam as shown in the previous slide, the top half of the beam would be in compression. The lower half of the beam would be in tension. So we would have compressive strains on the top, tensile strains on the bottom. And since we're dealing with linear elastic properties of the material, notice that this is before cracking of concrete. So we're really near the origin of that stress-strain curve. So concrete can still carry tension, granted that this value of stress is very, very low. And it can also carry compression behaving as if it were a linear material. So this is a typical stress diagram based on a linear elastic behavior of a material. However, when we reach a stress here, the cracking stress of concrete, and that would be denoted by F prime T here, the beam then fails as shown in this picture. So in reinforced concrete design, as many of you probably remember, at the onset of cracking, if we have reinforcement, we place that reinforcement near the tension phase, of course, to carry those tension stresses that the concrete can no longer carry. So the concrete stresses are shown here as this lower triangle, but we now have rebar resisting the majority of those tension stresses. Before cracking, you know, in this condition, even if we had steel, pardon me, the steel would contribute very little because the amount of steel that's here is very small. And also the steel stress would be very small. It would be at about 5% of the yield stress. The steel area is about 1% of the area of the cross section of B times D. So before cracking, steel doesn't contribute much. And right after cracking, steel still doesn't contribute much, but it helps restrain the growth of that crack, that sudden opening of the crack and propagation towards the top. However, as we keep loading the beam, now the steel is there and carrying any tension that's generated below the neutral axis, we can then engage or produce stresses, compressive stresses in the concrete that take it into its non-linear range. And you might see that this curve here resembles a stress strain curve of a cylinder in compression. And in fact, it's quite close to it. So as there are compression strains above the neutral axis, now we can push the beam further into its non-linear range of behavior for concrete. And then the steel is going to take all the tension stresses that are generated below the neutral axis. In reality, one could think that, well, there are some tension stresses that the concrete still takes, but if I were to try to draw them to scale, they would be that small triangle that's shown there. Not very much. They wouldn't contribute much to the strength because this strain here corresponds to the cracking stress of concrete. So for all intents and purposes, the steel is carrying all the tension force. Okay. So this is a real beam tested in the laboratory. It's a reinforced concrete beam. So you can imagine there's longitudinal reinforcement. Here are two load points. The supports are outside of this picture. You cannot see them, but it's a beam under four-point bending, meaning two concentrated forces and two supports. And as you can see, this region here is a region of constant moment and zero shear. You can see nearly vertical cracks within that region. That really indicates that this region is only dominated by flexure, by bending. And you can see that as the cracks move up, they stay relatively tight and closed. And that's because of the reinforcement. The concrete tries to open up, but the reinforcement restrains it from opening and carries that tension force. As these cracks propagate, they will concentrate or generate a stress concentration region near the top. And that's what eventually will cause a failure of a beam in tension and then compression failure. So that failure would correspond to a strain of approximately 0.003 if we believe the assumption that we use for design. So in order to utilize the strength of reinforced concrete members, and going back here, utilize this compression strength, a beam will crack like that. So in order to utilize the reinforcement, there will be cracks that form in the beam. These cracks are more dramatic here in this picture than what they really are. They have been highlighted so that they can be seen in the picture, but they're still there. And some of the problems that they may cause is that first of all, people get uneasy, right? After seeing cracks there are unsightly. The presence of cracks causes a reduction in beam stiffness. So the beams will tend to deflect more after they're cracked. But more importantly, they allow chemicals to penetrate the beam. And some of those chemicals, you know, like chlorides and water and air, you know, would cause reinforcement to corrode if it's not epoxy coated. Water penetration into the cracks in cold environments can also cause freestyle damage. There's calcium hydroxide leachates that can also come up. You can see efflorescence or sulfates that penetrate the cracks. So one would like to just keep the cracks as closed as possible while still using the non-linear or the maximum strength of concrete. And or to completely eliminate the cracks altogether to make the concrete more durable. And so the potential solution would be to pre-stress the beam. The basic problem of reinforced concrete is that if it does, the beam does not crack, we don't use the materials very efficiently. But if the beam cracks, it causes all the problems that I just pointed out. So pre-stressing could be the way to avoid these undesirable effects of crack. So in the next few slides, we will start looking at fabrication and basic concepts on how a pre-stressed concrete beam is actually pre-stressed. So we, I'd like to just start, ask Sherry to start the second poll this night. So this poll has to do with the difference between pre-tensioning and post-tensioning. Okay, I will be launching that poll in one second. Okay, the poll is now available and it reads, are, sorry, let me see, I can't see the poll. Are you in general familiar with differences between pre-tensioning and post-tensioning construction? I must apologize, it's cut off on the screen, but you can still see the answers and please vote. Again, it is, are you in general familiar with differences between pre-tensioning and post-tensioning construction? Okay, we've got a few more people that we're waiting on. Okay, I am going to be closing the poll. 96% say yes, 4% say no. Okay, so for the benefit of those that said no, so I'm glad to see that the majority does. Did my screen disappear, Sherry? It's black on my side. Yes, it did. Okay, hold on, let me see if I, I just, it's doing something weird on you. I can see your mouth and I see it's cut off, it's very strange. Okay, let me see. There it is. Okay, okay, so I have a few sketches, just to go over the differences very quickly, since you, most of you know the differences. In, there's two ways to construct or to pre-stress a beam. So, all of these sketches are based on a beam. We'll first talk about pre-tensioning. In pre-tensioning, what you do is, there's a pre-stressing bed that has very stiff bulkheads at the end. This is just a couple of graphs. And what you do is, you tension strand in between those bulkheads, and essentially, strand behaves as a rubber band, right? When you first stretch it, it's a rubber band, essentially strand behaves as a rubber band, right? When you first stretch it, it wants to go back to its undeformed position, so that's why you need to hold it in place with these end chucks. By keeping the strand in place, after keeping the strand in place and with that tension, then forms are erected or molds are erected and concrete is cast within those molds, while the strand is still under tension. And then, after concrete has hardened and gained sufficient strength, crews come in and cut the strand, and thereby, through bond, they transfer the pre-stressing force because the strand wants to shorten back to its undeformed position, but the concrete that has been cast in that mold does not let it go back to its undeformed position. So, stresses are then transferred through bond, and this is just the pretensioning process in words, what I just described. And let me just show you a picture. I like to show this picture because it illustrates a seven-wire strand, and this is the surrounding concrete. So, you could imagine that if this element tries to shorten, it would have to twist around inside the concrete to be able to go back to its undeformed position. I have some props here as well. Many of you may have seen, if you're fabricators especially, you have seen seven-wire strand, right? That's the view of a seven-wire strand. So, this is what we saw cut concrete here in the laboratory with pre-stressing strand, and that's how it looks like inside of concrete. So, it's pretty hard for this strand to shorten back to its undeformed position once you release it from the bed. So, that's the premise behind pretensioning. In post-tensioning construction, instead of having a casting bed, what one does is you cast ducts, empty ducts, in a reinforced concrete element essentially, and that reinforced concrete element has other reinforcement such as reinforcement for shear and so on, and the ducts are to allow one to feed strands through those ducts and to then apply the pre-stressing force after the element has been cast. So, these strands are anchored in place using a chuck. I'll show you a chuck in a section, in a second, pardon me, and then there's a live end that is stretched using a high-capacity jack with an anchor. So, this strand is stretched and the anchor is placed here. Once there's enough tension in the strand, the jack is removed and the pre-stressing force is really applied through the end of the beam, at the end of the beam, using those chucks. This is the summary of what that process involves. Now, in post-tensioning, these are some of the characteristics. Notice that the strands or bars, they could be bars, they are tensioned against the concrete by jacking, by using a hydraulic jack. So, it requires some hardware such as plates and wedges at the end of the beams. There's also, since you're passing strand through ducts, if the duct doesn't have a horizontal or a straight configuration, say it's parabolic or it has some curves in it, as you stretch the strand, the strand will want to, or tension the strand, the strand will want to go, to become straight. So, it introduces friction against the ducts and that friction is one of the causes of loss of pre-stressing force that we will talk about later in this course. So, I have here a strand with a chuck at the end. So this is a chuck that's used for pre-tensioning. So going back here, this is this element here at the end of a casting bed, and I'll show you more pictures later. So what this has is inside the chuck, it has a wedge. There's a wedge that has that configuration, and the chuck itself is tapered. I don't know if you can see the taper through the opening. So this wedge fits into the chuck. The chuck is then fixed against the buttress. So the strands are fed through the chuck. As you can see, I can pull the strand to my right, but I can't pull it to the left again because the wedge is keeping it in place. So the crews can pull strand through it, through chucks, but the strand cannot go back until one torches the strand, and then it releases the force into, or you'd have to torch it here. It releases the force into the element. Okay, so within the post-tensioning way of constructing prestressed members, there are two categories. One of them is called bounded post-tensioning, in which there's grout injected into the voids between the duct and the strands. So after post-tensioning, you inject grout, and there's the other way of constructing post-tensioned members, and that is called unbounded post-tensioning construction, where the strands are left in the ducts without filling with grout. In terms of, so let me just, before we move on, let me just show you a couple of pictures here, examples of sections. This is a segmental section that's the reinforcement in a segment where one of the ducts is being placed. This is the end duct where the post-tensioned strands would be fed through. The same type of element with after concrete casting, so the reinforcement would be inside. And notice that this opening right here, here's another post-tensioning duct, and here's the grout injection port right here. So you may have ducts that can, where you can fit several strands at a time, and smaller ducts for a smaller number of strands. And finally, the end product, a cross-section like that, where all these, notice how big these boxes need to be, or these inserts need to be. And this is to allow the jacks to go in there and apply the post-tensioning force that's needed. Some of them, you have these pockets that are cast into the concrete, and that's what one would use to post-tension the strand in these segments. Okay, so in terms of calculations, are there differences for designing post-tensioning and pre-tensioning members? So this slide intends to just talk about any potential differences. When one checks the condition at transfer of pre-stressing force, there are different allowable stresses, depending on whether you're dealing with pre-tensioning or post-tension construction. There are limits on stresses in the concrete because of the anchorage devices that bear against the concrete. One needs to check that they don't actually crush the concrete in compression. If we are looking at service loads, so unfactored loads, and we'll talk about factored and unfactored loads in a future session, there's really no difference in the method. When we're talking about ultimate loads, to estimate the stress in the strand at ultimate, there are two different methods, depending on whether the element is pre-tensioned or post-tensioned, particularly for unbonded post-tensioned members. And when we calculate pre-stressing force losses, when we're looking at the long-term behavior of pre-stress elements, the force losses, the pre-stressing force losses are different for each method. So you have to account for certain conditions, such as the friction between the strands or the cables and the ducts, for the case of post-tensioning, and you wouldn't account for that, or you wouldn't have to consider that in pre-tensioning. Okay, so I sense that the majority of you are either producers or designers. I'm going to go through these series of pictures fairly quickly. This just takes you through the production of a typical pre-stress concrete element, and it's only for pre-tension construction. So at the facility, at the pre-casting facility, these are some of the bulkheads. This is an example of a bulkhead, and you typically have a strand pattern at a certain spacing. A very common spacing is a two-inch pattern. These holes are a template that are used to feed strands, and it looks here like this bulkhead or plate has been fabricated to construct beams that look like eye girders, so perhaps astro girders. And the reason I say that, you can see that there's a concentration of openings up here in the top flange, a concentration of strands in the bottom. And these are, the way that each of these strands are stressed, each of the strands is stressed individually using this jack, and we have the pump and the chuck assembly that I showed you earlier before. So the first thing is to stress the strand. Once the strand is stressed and tied against the bulkheads, the rest of the reinforcing cage is fabricated. This is the mild reinforcing cage. This consists of the stirrups and any top reinforcement that might be needed. Then forms are erected around the elements. So these are either tipped in or just built on the sides and then brought in. Concrete is batched typically at the facility. And then it's brought to the place where it's going to be cast using buggies or if you have an indoor facility, the concrete trucks might come in and just simply place or discharge the concrete directly into the forms. This concrete buggy is just discharging into the forms through the chute. As you can see, it's a concrete mix, could be SCC or to avoid vibration or normal concrete. Most of the time these days, people are using self-consolidated concrete. And then after casting, curing takes place. And since the beds need to be turned over, the pre-casting beds need to be turned over very quickly. It is fairly common to try to cure the concrete using steam curing. So that concrete gains strength rapidly in the first 12 to 16 hours after casting. Companion cylinders are cast as well. They're tested when people feel it's time to release the pre-stressing force. One wants to release the pre-stressing force after concrete has reached a certain strength. So typically, release strengths are in the range of 3000 PSI or greater. And once concrete has reached the design release strength, then strands can be then cut. And the pre-stressing force is now transferred into the element. This is the aspect of the strands after they have been cut. Notice how they flare out because they become uncoiled as the crews torch them. Here's one of the flame cutting tanks that would have been used to cut the strand at the ends. And then the element that has been fabricated is removed from the forms and lifted and taken to storage. So this picture shows a series of different pre-cast concrete elements. Some of them may be only pre-cast and some of them may be pre-cast pre-stressed. So column trees with corbels on their sides so that you can set perhaps these ledge beams or inverted T beams, double Ts. After, these are hollow core slabs. And after storing them on site, shipping takes place on these long flat beds with appropriate dunnage to avoid cracking of the elements. And that's essentially, that takes you through a full cycle. Once you complete the cycle, essentially once you remove the element from the molds, then you're ready to cast another element in those same molds, right? With hopefully in a 24 hour operation cycle to have it be efficient. So that's just a very brief tour, as I mentioned, and I'm sure that many of you were familiar with it already. So we're going to start looking more into the details of how pre-stressed concrete elements behave. And for that, we need to talk about some theory. And one of the theoretical aspects is the principle of elastic superposition. So that brings us to the third poll of this evening. And I wanted to ask Sherry if she could start it. And the question, I'm going to read it just in case it doesn't appear in her screen. The question is, do you remember what the principle of elastic superposition means? What it involves? Again, the poll is on the screen. Again, it is, do you remember what the principle of elastic superposition means? Yes or no? We have 79% of you who have voted. We have 79% of you who have voted. We need the, we're waiting on the rest of you. We will give you a few more minutes, a few more seconds. Okay, I'm getting ready to close the poll. Sergio, 65% said yes, 35% say no. And did you, okay. Yeah, something's going on with my computer. I need to refresh. Every time you close a poll, it just goes black. So I need to refresh the screen and I do it by just hitting some other program. Okay, all right. All right, so it's about two thirds yes and one third no. Let me just explain it this way. So the principle of elastic superposition is a tool that is very often used when we deal with materials that behave in their linear elastic range. So that could be when, for pre-stressed concrete in particular, it could be when the elements are within their service level. And as you remember, you might recall from the stress-strain curve of concrete, it's only at a very low level of stress that we can consider concrete to be linear. But we could probably take it up to 50% of its peak strength at prime C and not lose much accuracy. But the nice thing about the principle of elastic superposition is that you can compute effects such as stresses, for example, in a cross section separately by each load type. So for example, you can compute stresses caused by dead loads. And then you can compute stresses caused by live loads. And then you can add them together. So you don't need to take the combination of those stresses simultaneously to get to the solution you want, as long as, again, as long as the materials stay within their elastic range. So for a pre-stressed concrete design, we use this principle of elastic superposition very many, many times because we superimpose the effects of loads acting on a beam to loads generated by the pre-stressing force. So this illustrates that concept that I just mentioned. So imagine we have a beam that's simply supported, right? So we have two different types of loadings. We have the loading W, that's a uniformly distributed load on the top of the beam, and we have a load P. We don't know where that load comes from. In this particular case, let's say it's a pre-stressing force P. So that load compresses the beam, but at the same time, there's load acting, transverse load acting uniformly distributed on the top of the beam. So if we maintain the beam within its elastic range, we could use the principle of elastic superposition to compute stresses. So let me just show here using the pen. Imagine the cross section of that beam is rectangular. I apologize for my pause here. All right, so that's a rectangular beam. So these stresses right here are acting over the height of this beam, H, and so over the entire area. This beam is the cross section. So these stresses are acting. So we simply compute stresses. If the P force is acting concentrically, we calculate stresses as P over A, and that's this component of stress. And notice that all of them are compressive. So we are essentially compressing the beam uniformly throughout its height. Now, if we introduce loading W, we know that this type of loading will generate bending of the beam. So the top of the beam would be compressed. The bottom of the beam would be in tension. So in previous slides, we have seen the tension stresses come out from the side of the beam. So this is the same diagram, but this is also commonly used in strength of material. So how do we compute these stresses based on the loading? Well, we first compute a moment at some section. That moment is then used. And if we have the neutral surface here at mid height of this rectangular beam, the distance Y is measured from that neutral surface to any fiber. So that's what that Y means. And I is the moment of inertia. So we get the moment from the loading. Y is the distance from the neutral axis to any point. So any line here at that line, we would get that particular stress. So that's what that equation allows me to do. That's the strength of materials equation. So the principle of elastic superposition tells me that as long as the material stays in its elastic range, I can add this stress configuration to that stress configuration. So adding, including science algebraically, we get that configuration of stresses. So less compressive in the bottom because of these tension stresses in the bottom and more compression in the top because we're adding that compression force to that or the stress, pardon me, to that compression stress. Or we could have some tension near the bottom, depending on the magnitude of the moment, right? That induces these tension stresses. We could have very high tensions that overcome that compression. We're gonna have a little bit of tension near the bottom. So essentially what we're doing is we're adding in an equation format, we're adding these stresses, pardon me, these stresses to those stresses and we end up with the two terms in this equation. So that's one part of the pre-stressing equation. So we have P over A term and MY over I term. So now using, again, if we now use the principle of elastic superposition on a slightly more complex case, if now the pre-stressing force is eccentric with respect to the neutral axis shown here by a dashed line this loading configuration, now P is acting at a distance E from the neutral axis. W is again acting uniformly over the beam. We can replace this, the moment that P produces about the neutral axis, P times E by this moment here. So we can move P to the center like it was in the previous slide. And this is an equivalent moment whose magnitude is going to be P times E. So it's constant, right? We have constant eccentricities at the end of the beam. So again, by using the principle of elastic superposition we have P over A term, the stresses, compression from the eccentricity of the pre-stressing force. We have a negative moment. So tension on the top fibers. So that's what's shown here opposite to the compression force, compression stress, pardon me. And these are stresses generated by the loading and adding term by term using the principle of elastic superposition. We end up with diagrams, stress diagrams that might look like this depending on the relative magnitudes of those stresses, P over A, P E Y over I, M Y over I. So M Y over I represents the external loading and these first two terms represent the stresses induced by the pre-stressing force. So this brings us to the basic equation for pre-stressing which is just, I'm repeating again in this slide. And in order to calculate stress in the concrete at any level, going back one slide, at any level, right? It could be close to the neutral axis or farther from it. We need to look and calculate each of these terms. P over A is going to be constant but then we see that the level at which we calculate those stresses depends on these Y variable, right? That I illustrated was the distance or it got erased but it's the distance from the neutral axis or the centroid of the section to the fiber, the distance between the centroid of the section and the fiber at which I'm computing stresses. The notation in this equation is shown here. So I'll pause for a second so that you can read it. The key points is that the bending stress is at any point is given by this equation and all these variables up here are described here. E is the eccentricity of the pre-stressing force and notice that it's measured from the geometric centroid which is equal to the neutral axis for a linear beam. So that's one and the same. When you think of a centroid, it applies to a linear elastic beam. Otherwise it would have to be called the neutral axis but for our intents, it's just one of the same. And Y is also the distance from the centroid to the fiber where I want to compute stresses. A note on sign convention and this is a very important slide. We will try to use this convention where the, you'll notice if we go back a couple of slides that in some cases stresses subtract and in some cases they add it to the P over A term. So notice that the P over A term is always positive. It's compressive. In this case, tension stresses would subtract from the P over A term. And in this case, they would add to it. So the convention we're using is that the top sign in each of these terms corresponds to fibers above the neutral axis, above the centroid. The bottom sign would correspond to fibers below the centroid of the section. Compression is positive. So P is positive and tension stresses are negative. And notice here that Y is always measured positive either going up or down. If we use this convention, then Y is always measured positive and E is positive measured below the neutral axis or centroid and it's negative if it's above the neutral axis. And going back a couple of slides, this configuration would mean a positive E, right? If we follow then this sign convention. Okay, so just try to keep that handy or keep that straight in your mind. If you don't wanna see, if it's too complicated to remember, you can always think of compression negative, tension positive, and we'd have to just picture the diagrams in your mind. So this principle of elastic superposition can be extended into more complex cases, such as the case of a composite section. So a little bit of an explanation, composite section is one that is made, a section that is precast, pre-stressed at a plant. And this is a section that is non-composite on the left. And then typically a slab is cast in place and made composite with the rest of the section. So you'll notice that some of the stresses generated in this section occur before the slab is cast. And some of them occur after the slab is cast. So there's a change in geometric properties of the cross section. At first, the geometric properties correspond to these AASHTO type girder, AASHTO family of girders. And once the slab cures on top of it, that section becomes this more complex section for which we need to calculate area and moment of inertia, right? So we need to only account for superimposed loads or moments after the section is composite. So all of these terms, we can extend the basic equation of pre-stressing. We can separate it into non-composite behavior. All of these three terms, notice that pre-stressing force acting through its eccentricity act over the non-composite section. Part of the superimposed loads such as self-weight would act on the non-composite section. And then there are some moments that are generated on the composite section, and for which we would need to consider the moment of inertia of this section and no longer the AASHTO girder. So the principle of elastic superposition is very powerful because we can do calculations like this fairly simply by just accounting for differences in the geometry of the section. Okay, so going back to this figure, when we have the pre-stressing force applied eccentrically a distance E from the centroid of the section. So if we construct the moment diagram, I'm going to come back to this text in a second, but let's look at the moment diagrams that are generated first by the superimposed load W. And remember, these are simple supports. And then we'll look at the moment diagram generated by the moment that the pre-stressing force produces. So the moment produced by W has a parabolic shape. It has a maximum value of WL squared over eight at mid-span, and it's zero at the ends. And as a reminder, it is a positive moment, so it produces tension at the bottom fiber. Using the principle of elastic superposition, the pre-stressing force acting over this distance E produces a negative moment. So this is the line of zero moment, this horizontal line. The top line is zero. So we have a negative moment of magnitude equal to P times E coming from the pre-stressing force. So if we superimpose these two moments by using the principle of elastic superposition, we add fiber or section to section by section of these two moment diagrams, and we end up with an effective moment diagram that looks like this, where you have N moments that have a magnitude of P times E. And then from that new baseline, we have the parabola superimposed onto that baseline with a magnitude of WL squared over eight minus P times E at mid-span, right? Because we now have a new baseline that has shifted a negative amount PE. So notice now that we have negative moments at the ends that produce tension on the top of the beam. So that may cause a little bit of a problem. So notice that this is what's explained here in this text. If the strands are straight, so presumably the pre-stressing force is acting along that line because strands are straight. So we pre-stress them and we keep the strands straight. That's the line of application of the pre-stressing force. If the pre-stressing force is applied at a decentricity and it's constant, we will have these tension stresses, these N moments that might generate top tension stresses near the end of the beam. And in many cases, if we want to pre-stress with a relatively high force, since we don't have any self-weight of the beam helping us decrease those stresses, we may exceed the cracking strength of concrete. So top cracks may form. So these are areas of possible cracking. So what would correspond, what stresses would correspond at the end or what stress profile would correspond at the end of the beam? Well, a profile that looks like this, where you have top fiber stresses in this region, top of the beam and compression in the rest. The stresses at mid span look fine as we had illustrated before, right? Right here, we have a positive moment and therefore the beam is under compression throughout because the bottom stresses are controlled by the pre-stressing force. But then at the ends, at the two ends, we have similar situation of top fiber tension stresses. Now, the concrete will not crack unless the top tension stresses exceed this threshold of six root F prime CI. And this is an empirical value. It could be lower than that, or it could be higher than that. But just for ACI limits, the top fiber stresses to six root F prime CI. And notice that it's not called F prime C since the pre-stressing force is released a short time after the concrete element is cast, then the concrete strength might be lower than the 28 day strength. So this F prime CI corresponds to the concrete strength after release or at the time of release of the pre-stressing force. So unless we take some precautionary measures here at the end of the beam, we might see some top fiber cracking or top section cracking. So what are those precautionary methods? Well, the problem we're encountering is because we don't have the deadload that is helping me with generating these positive bending moments throughout the span right at the end, because deadload moments start at zero at the ends. So we could try to not transfer the total pre-stressing force right at the ends. And we could do that by a procedure called the bonding, so or blanketing. You can cover the strands with a plastic tube so that it doesn't bond to concrete. So essentially, if this is the end of the member, you start applying that pre-stressing force gradually from the ends towards the inside, gradually by the number of strands that are bonded actually to concrete. These blankets or pipes prevent the strand from bonding and transferring those stresses. So this method essentially decreases the magnitude of P at the end and gradually increases it towards the middle where you need it to be maximum, right? To avoid tension stresses near the bottom of the beam. So that's called the bonding strands. Now, there are certain limitations in some design guidelines. The number of strands that are de-bonded on each section is limited. So it's also a good idea to stagger the de-bonding. The fraction of the total number of strands that need to be de-bonded also sometimes is limited. There is contention about that there's no some there are some guidelines that tell you well you can only debond 25% of the total number of strands at a given section. Some others say well you can debond up to 50 and so on. So there's some discussion going on in that but just keep in mind that it's always good to have to distribute that debonding so that stress concentrations are not generated all of a sudden. Another way of lowering those end-top tension stresses is to instead of dealing with P the magnitude of the force you can deal with the eccentricity. So in other words instead of applying P at a constant eccentricity throughout the length of the beam you can change the profile of the pre-stressing strand by either draping the strand and it should be more the draping should look more like a parabolic drape or by harping the strand essentially depressing the strands at some points. And in essence what you do is you try to minimize the eccentricity at the end because it's really the P times E term that generates that end moment that negative end moment. So these deflected or these configurations are called draped or harped strands. Because of the way the geometric shape of these strands draped strands can only be used in post-tension construction because you can actually set the ducts in any shape you want within the concrete element. You can have them adopt a parabolic profile but that's not possible or shouldn't be very easily couldn't easily be done if you are using pre-tensioning. So the method that's most commonly done in pre-tensioning is using the harped configuration where you have to essentially provide a way to push down those strands at those bend points right so you have to have the some lock way to lock down the strands because as you tension them they will want to push back up and straighten up again. In this particular case for the post-tension member you have the concrete surrounding the strand so it doesn't go back to its straight condition. So those are the two methods either by by draping or harping or debonding. So draping and harping actually works by decreasing E as I mentioned earlier because what we are interested in reducing is the product of P times E. Debonding reduces P and changing the strand profile reduces E at the ends. So this is just a sample of one AASHTO girder for example at the mid span pattern where you might need the highest amount of pre-stressing force near the bottom to counteract service load stresses. You want most strands near the bottom and then at the ends you could harp those strands and have them move to the top and the harping points could be at a fraction of the span length L so this is 40% of L. There are different different places where you might harp them 40, 35, 30 percent of L. The key feature of course to keep in mind is that you can only harp the strands that lie within the web of a member right so if you have a section like this only the strands that will eventually can eventually be moved up to the top while still lying in the member are those that can be harped. So four of those strands here were harped towards the end and this notice the difference in eccentricity I don't know if you can see it in your screens but going from an eccentricity of 17.4 inches at mid span you go to an eccentricity of 9.7 inches at the ends so that decreases the P times E term. And these are some of the hold down an example hold down device that could be used you could see strands are coming here from the right and then are held down by that device that's tied against the casting bed the forms and then they come back up towards the end. There's a post here with the bulkhead that actually holds the strand in place and this hold down device is like a series of pulleys that let the strand slide through it. This is a six strand hold down device that would have to stay in place right we would have to leave in place after casting. Another view of the harp strand towards the end along with a mild reinforcement the stirrups and some reinforcement to keep the ends from cracking. Another view not very clear but the harp strands are coming up here and then there's the rest of the strands are in this bottom bulb of the this element. So this these are just conceptual pictures and I'm going to show you what happens when one harps or drapes strands or debonds them as well. So this is just an example it's a type 3 AASHTO girder I girder so it has this configuration of strands at every 2 inches the eccentricity is 17.4 inches. If one were to do the calculations that we will learn early or later the top of this beam would crack given this large prestressing force near the bottom at this large eccentricity. So to try to avoid it from cracking what one could try to choose strands to debond and for this particular example these were the strand patterns that were chosen. The circles would debond the strands for seven and a half feet measured from the end. The squares mean that you debond those strands for 15 feet measured from the end. So notice that we're staggering the debonding to control the amount of prestressing force that is transferred at those two sections at the end at seven and a half feet and at 15 feet. And then the other option to control those top fiber stresses would be to harp the strands or yeah harp the strands by moving four of the strands to the top or near the top of the beam and thereby reducing the eccentricity. So these plots let me just explain what this means this is a plot this is a 60 foot so all these sketches are for this beam that's being used over a 60 foot span. So what this plot represents is the plot of top tension stresses plotted as a function of distance from the supports of the span of the beam. So from 0 to 60 feet we're plotting top tension stresses as a positive quantity here on the vertical axis and also plotted here is what the allowable tensile stress in AASHTO or in ACI 318. So we'll come to that when we do the calculations but for now just believe me that close to 500 psi would be the allowable tensile stress. So if we do not harp nor debond strands through bonding the prestressing force doesn't transfer into the beam right at the end. All right it transfers over a short distance so that's why these stresses grow gradually up to a certain point. Let me pause for a section. I'm looking at a question that Sherry just emailed or sent me it says what exactly is the difference between pretensioning and prestressing? The difference is that are they the same thing? The difference is that prestressing is the general term used to encompass both pretension construction and post tension construction. Pretension and post tension construction are the ways in which the two different ways in which that prestressing force is applied. That's the difference. So prestressing is really the general term and pretension and post tension are terms that are used of how the prestressing force is applied to the member. All right so going back to to the current discussion here. So if we don't have harping nor debonding the stresses the top fiber stresses that would generate in the beam would exceed the allowable tensile stress. So we'd have to do something about that. By debonding some strands two strands up to seven and a half feet so the strands start transferring stress and then this notice that this curve is parabolic and this is when the self-weight of the beam is helping me. So up to this point the self-weight of the beam starts helping me and then as I bond new strands so the strands are debonded for seven and a half feet and then further in for 15 feet so we have only we have four strands that are debonded so we start decreasing the top fiber stresses because of self-weight. Then we let two of those four strands start to develop prestressing force and that's the plot that will result. Then self-weight oops self-weight of the beam would keep would help me lower those top fiber stresses further and then we bond the rest of the strands and eventually the the the plot of top fiber stresses joins the curve corresponding to no measure no preventive measure to control top fiber stresses. If instead of controlling the amount of prestressing force transferred at each section if we harp the strands to reduce e we may have near the ends of the beam we may have section top fiber stresses that are in compression and then eventually they go into tension towards the middle and start growing growing growing but then we get to the point where the self-weight of the beam will keep them under control. So in this case we do not encounter any top fiber stresses with the configuration that was chosen for this particular beam. Now different points harping points would result in these different configurations that's 0.45 L the green line 0.4 L the red and 0.35 L the blue. So the shorter the harp or the closer the harping point is to the end of the beam the less those top fiber stresses are compressive or the less the region the end regions of the beams are in compression at the top for the top fiber stress. So this is the effect this is what one would get after doing all the calculations throughout the length of the beam so you can see that there's benefits of using harping and debonding to control those top fiber stresses. Okay so the last few slides I have tonight are talking about the prestressed concrete materials that are used. So let's go briefly through these and these are very similar. The first thing very similar to reinforced concrete construction. So the first thing to clarify is that throughout this course we will use these two boxes when when we're referring to the PCI design handbook the 8th edition that was published last year as I mentioned we will use a box that has a solid line like this and just try to illustrate that with PCI 8th edition. So any provision that's written inside the box would refer to the PCI 8th edition in this particular case. If the box is dashed as shown here then the provision inside the box is referred or a section or whatever is included here refers to the ACI 318 14 building code. That's what we will use throughout this this course in the different sections. So what are the materials that we use for prestressed concrete? Of course it's concrete, prestressing steel, some mild reinforcement and structural steel for embedments. And what I want to just mention are specific properties or mechanical properties that are of interest for each of the materials and the relevant either ASTM or AASHTO standards. So in the PCI 8th edition handbook section 9.2 really doesn't explicitly say that you need special concrete for prestressed concrete. So the first bullet there says that no special concrete is required but it is often times you need to specify as a designer the strength at the time of release F'CI to be able to run calculations for top fiber stresses at the time of release. And then you also have to specify the design strength at F'CI. Usually the 28 days is what's normally specified but if you have concrete with pozzolanic materials then such as fly ash sometimes you specify 56 days strength because those concretes tend to gain strength at a lower rate that normal concrete. Type 3 cement is normally used because you want to gain strength rapidly and as was alluded to in the plant tour earlier steam or heat curing is many times used especially hot and cold weather pardon me so that you gain enough strength and you can release the prestressing force into an element rapidly. The elastic modulus as I mentioned is a secant modulus and what that means is that you draw a line starting at zero and crossing the stress-strain diagram at 50% of F'CI and if you use that definition the ACI code tells you that EC can be calculated using this equation where WC is the unit weight of concrete in pounds per cubic foot so 150 pounds per cubic foot for normal weight concrete and F'CI is the strength the 28 day strength of concrete in PSI units. So the factor 33 here already includes the unit so again pounds per cubic foot and PSI for normal weight concrete it results in 57,000 root F'CI. For the prestressing steel usually for the most widely used strand is the seven wire strand that I illustrated earlier today with the modules of elasticity of 28,500 KSI. There are two grades normally 250 and 270. 270 is now the most common and this is I'll refer to this grade in a second and the most common now is low relaxation and the PCI handbook actually removed the stress relief strand in its late or many of the design aids that were that showed both types of strands they only now show the low relaxation because stress relieved is no longer easily or easily found in the United States. The sizes vary of the strand and these are the specifications by the way 3 eighths, half an inch, 0.6 inch so I illustrated earlier a half inch I don't know if the picture is not great but on my right hand is 0.6 diameter it doesn't do it justice because of the perspective but I have a 0.6 and a 0.5 maybe it's better to show them here and that's a 3 eighths that's a very tiny wire as you can see so that's not wire it's a seven wire strand and the stress strain behavior of strand is shown here so it doesn't have a well-defined yield point so it starts going nonlinear at a fraction of what's referred to FPY the yield stress of strand the yield stress of strand is measured at 1% elongation because it doesn't have a well-defined yield point it looks here that it does but it really does that it has a it's it's better to define it because it has this this curve it's better to choose a strain at which we define yield so that's how it's defined stress relief would be the lower curve low relaxation would be the higher curve the sharper curve and the strand needs to satisfy a minimum elongation of three and a half percent and at that point that's where the the grade of strand is specified either 250 or 270 is specified at a three three and a half percent elongation so that's what what's shown here 250 or 270 for stress relieved or low relaxation the tensile strength at 1% strain through this specification a 416 it tells you that the minimum tensile strength FPY for let's go to the low relaxation it has to be 212 KSI or 243 KSI depending on whether the strand is 270 or 250 and the elongation minimum elongation at rupture is three and a half percent notice we'll use this area extensively in this course and that's why I listed here we'll use half inch diameter strand in our example and the air cross-sectional area of that strand is 0.153 square inches there's also mild reinforcing bars that are used in pre-stress concrete construction and those are used either for stirrups or top steel with you know grade 60 so ASTMA 615 is what governs that the specification that covers that steel usually we use grade 60 new billet steel and the AASHTO specification would be oops M31 so going back these are the material just highlights on the materials used for pre-stress concrete and with that with about eight minutes to go on the session tonight we conclude today's presentation that concludes that's the end of this lecture and I would like to open it up for questions if there's any more questions if you have any questions please type them into the chat box it's available now any questions okay I've got a couple questions coming in Sergio I will send you the first one now okay and I'll actually share it with the entire audience so everyone can see the question and you don't have to read it again there's the first one would I please explain the difference between strand types um I'm not sure I understand the question is it sizes or or different types of strands so so the one word we've been using here seven or I've been explaining is the seven wire strand which is six wires coiled around a center wire there's also the sizes are their nominal diameter so the half inch or 0.5 strand has a nominal diameter of half an inch I'm not understanding this question Sergio he Brian wrote curing temps recommendation and then he wrote steam curing temps exactly sure yeah I think the question is is actually asking what is the what are typical steam curing temperatures that are recommended so it really depends on on the application and I think what is typical at least what I have seen and I have to confess I have limited experience in this but I have seen steam curing temperatures in the order of 110 degrees Fahrenheit you know around 110 120 degrees Fahrenheit but there's no there's no I it has to do also with the the concrete mix design so I would I would encourage you to consult with a materials expert because it depends on the type of concrete you also don't want and the outside temperature you don't want to shock the concrete with very high temperatures and all of a sudden the more important thing in my mind is to not shock the concrete with changing the temperature from very high to very low say if you're casting something in freezing conditions and you're curing it and all of a sudden you remove the curing blankets that might produce a shock that would cause concrete cracking in the concrete then the next one is how to calculate the gross moment of inertia for composite sections and then they wrote for calculation of stresses okay so that's a that's a very let me see if I can explain it quickly I'll try to do a sketch here so if you have a composite section it is likely that you have a slab that has a different strength concrete strength that is from the the strength of the concrete strength of the of the beam so so f prime C might be different right let's say one and two so the top says two and therefore we will have two moduli of elasticity so you have two moduli of elasticity you want to need to depending on f prime C as we showed the module of elasticity equation so what you need to do this is we're dealing with linear elastic materials you have to transform typically what one does is transform the concrete in the slab to equivalent concrete of the pre stressing member so what that when that dot when we do that we typically decrease the width oops we decrease the width of the slab to reflect the fact that it's modulus of elasticity might be lower because of its lower concrete strength so if if the width of the slab is B the effective width would be I'm trying to go quickly here it would be B effective over n so that lower smaller slab width reflects the fact that the slab has a lower modulus of elasticity than the concrete member and after we transform the section into a now a homogeneous material now this section right here below it is a homogeneous material section because we have transformed the slab into concrete strength equal to the one of the pre stressing strength so now we can treat this section as if it were homogeneous and then calculate its centroid and calculate its moment of inertia about its centroid and that will be the centroidal properties of that section so hopefully you remember how to calculate area and moment of inertia of a section such like such like that otherwise I would I would suggest that you look at a I think in a strength of materials or a statics book they should cover cases like this of you know sections of moments of inertia of sections of dissimilar materials hopefully I answered that question okay we've got another one the diff what is the difference between stressful relieved and low relaxation strands well there are there are two possible answers that I can think of for that question the one the one that deals with the fabrication of the strand itself is in the way that the way that the strand is produced is results in in in whether it's stress relieved or low relaxation so in its production the stress-strain properties that result end up for an end from an engineering perspective what's important is to so the metallurgy may be the same are but but the way that the or similar but the way that they are fabricated is is especially heat treated if there's any heat treatment or cooling during the fabrication that's what results in stress relieved behavior or strand or low or low relaxation now from a stress-strain perspective the only difference is how sharp this curve is from the linear behavior into the nonlinear behavior that's one difference and the other difference between stress relieved and low relaxation is in calculation of stress losses low relaxation strands tend to have much like a rubber band so strands would lose pre stressing force over time if you hold a rubber rubber band tight between two fixed points eventually it'll lose its tension you you might have it see lacks with stranded it is not as pronounced but you it does lose some stress over time and the nice thing about low relaxation strand is that it loses less of that stress over time so it's that's why it's become more popular in fact stress relieved is not even used anymore so I think you should only you could safely assume that any strand you get from from a fabricator is for relaxation we have a follow-up question to the questions you've had before that and it's so what's the moment of value of fourth term in stress formula okay so the value here I'm assuming it has to do with this slide that's the fourth term so I composite would be the moment of inertia of the composite section calculated based on reducing the width of this slab to make it of the same material as the pre stress concrete member and M composite would be moments acting on the section after the slab has hardened so for example let's say this is a bridge so you cast the deck of a bridge on the bridge and the bridge really behaves as non-composite until the slab hardens so any other surfaces that are applied to the bridge such as a wearing surface or any other loads that are applied guard rails utilities that are applied to the new section that is formed with this concrete hardened would come into play in the composite moment calculation also live loads vehicles act on the composite section and not the non composite section so in this moment you have superimposed dead loads and live loads and you have the moment of inertia of the composite section and we have one last question if you have any other questions after this one please send them to my email I will pass them on to Sergio and then send the answers off to the class thank you Sergio last question besides area increase is there a benefit in using one and a half inch low relaxation special strand you get more area that's all so low relaxation you'll have less losses due to relaxation and the 0.5 special in comparison with a 0.5 normal so to speak just gives you more more cross-sectional area so more pre stressing force okay thank you so much Sergio and thank you everyone for attending tonight

pre-stressed concrete design

introduction

basic concepts

principles

pre-tensioning

post-tensioning

stress-strain properties

concrete

reinforcement

production process

fiber stresses

composite sections