Thank you, Sherry, for the introduction and welcome back everyone to the third session. Today we're going to talk, we have a pretty tight schedule and we have, so I'm going to try to go through the first part of this presentation fairly quickly because most of it or there are portions of it that pertain to general structural engineering and then since we're going to start introducing the design example today, I'm going to try to spend more time on the design example than the code part of the session. So this is what we have in stock for today. We're going to keep talking about code requirements and then we're going to start introducing the design example and we're going to begin to do some calculations on the design example. I'm going to start off, in order for me to have a feel of how quickly I can go through the code requirements, I want to just start the first poll of the night. So this is a very, very rapid poll and it starts right away. The idea is to get an understanding of what your background is. So if I'm going to turn it back to Sherry to start the, you know, to launch the first poll. Thank you, Sergio. The first poll, I will read it and then launch it. Which one of the following categories best describes your current position? Precast pre-stress concrete producer, structural engineer, private practice, state DOT engineer, academic, or other? The poll is now in process and please answer on your screen. Great. I am going to close the poll now. Sergio, 32% precast pre-stress concrete producer, 42% structural engineer or private practice, 0% state DOT, 16% academic, and 11% other. Okay. Thank you, Sherry. So the idea after this session will be to familiarize yourself in case you are not familiar already on the HCI 318.14 organization. We talked about this last time, but we'll be specifically talking about certain requirements that apply to precast pre-stress concrete beams. And then we're going to move on, like I said, to the design example where we will introduce the example and we will start determining the number of strands needed for a pre-stress concrete member, pre-stress pretension, and we will be able to start talking about calculating pre-stress force losses and adjusting strengths. So first we have to go through several definitions, even though these are not per se HCI definitions, they are used throughout the structural engineering. There are some definitions. Service loads are loads that are expected to be acting on a structure on a day-to-day basis. A factored load is a service load that is multiplied by a load factor. And these load factors are given in HCI 318.14 section 531. So they are applied directly to a service load or a service load effect. A load effect is defined in general terms as the effect of the forces or loads acting on a member. So for example, the effect could be moments as a result of the applied loads, shears, torsions, or stresses. So anything, any effect, it could also be a deflection, for example. Any effect that is generated by an applied load is called a load effect. A service load effect is that which is generated with loads acting at their service level. And a factored load effect is an effect that is generated by the loads acting at factored level or loads acting with their load factors. And then the other idea or concept for structural engineering is that what we need to do is we need to combine all possible load categories or load types that are acting on a structure and we need to combine them in a manner in which they reflect the likelihood of simultaneous application of those loads on a structure and the likelihood that those loads are going to be acting at their maximum levels. So that's what the load combinations really involved in HCI 318.14. That's the section 531 where the factored load combinations are included in HCI 318.14. And service load combinations are really, they don't exist in HCI 318. But one could simply add several load types without load factors and define that as a service load combination. These other two definitions, these are not necessarily used directly in HCI. They're most common. One uses the AASHTO LRFD, the concept of a limit state. So a limit state is in any condition that when achieved, the structure ceases to fulfill its intended purpose. So when we reach, say, for example, a service limit state, that means that we have reached a condition at the service level, service load level, for which the structure is no longer satisfying or no longer meets the design requirements. And a strength limit state is that condition but under factored loads. HCI 318 requires members to resist all applicable loads. So that means dead loads, live loads, wind loads, earthquake loads, et cetera. And it also is required that, or HCI 318 states that a structure must be checked for any service limit states or strength limit states that are applicable to the structure. In HCI 318.14, the philosophy is that the strength limit state is checked first and service limit states, such as deflections, are verified after the structure is initially designed. However, the typical design for prestressed concrete construction is to first design a structure or a component under the service limit state and then check the strength limit state to see if it is satisfied. So HCI 318 tells the designer that all limit states have to be satisfied and met by any structure being designed. As I mentioned, HCI 318 does not really provide specific load combinations. But section 4.12.21 states that for prestressed concrete members, all of them, all of these members shall be, their design shall be based on strength and on behavior at service conditions at all critical stages during the life of the structure. So this means during fabrication, this is important for prestressed concrete because they are, prestressed pretension are fabricated in a facility, in a plant. So during fabrication, handling, shipping, erection, etc. So the engineer who, the design engineer is responsible for checking all these different stages and making sure that the structure or the component being designed satisfied the limit state, the limit states that pertain or that apply to a particular component. If one wants to try to find out what those service load combinations are, one could resort to ASCE 710. It has requirements or recommendations on how to define service load combinations. As I mentioned earlier, one could use dead live wind and combine them directly without any load factors, especially when one is dealing with extraordinary events such as wind or earthquake. There used to be a time when one was able to reduce the effect of those events to consider them being applied at the service level. So ASCE 7 has more guidance on that. I'm going to go through these slides very quickly. These are some guidelines on how an engineer can decide what types of loads are applicable simultaneously. These are some of the questions that the engineer might want to answer before deciding whether two loads need to be applied simultaneously. Are loads variable? Are they constant? Do some loads, when they are applied, mitigate the effect of others? If so, should one then apply that load that mitigates the effect of others at a reduced level to ensure that we don't over mitigate other loads? So these are some recommendations. These are the primary load combinations that are listed in HCI 318.14. Each of the coefficients in each of the equations corresponds to the load factor corresponding to each of the load types. For example, in equation 531A, the ultimate load effect is calculated as 1.4 times the dead load effect. That equation could be listed, since we're talking about effects, it could be moments, ultimate moments equals 1.4, the dead load moment, or it could be the load itself. So the factored load, W sub U, could be equal to 1.4 times W sub dead. So that is the equation that is used when only dead loads are acting on a structure. The second equation, 531D, is the most common equation for dead and live loads, for gravity loading. And then 531D is most commonly used when wind is the principal action, principal load acting on the structure, where the load factor for live load could be different from 1.0. There are several notes in HCI 318 that tells you when each of these combinations, when these load factors are different from the general way that they're listed in these series of equations. Notice also that the PCI 8th edition, 426, has a copy or also includes the same load combinations as the HCI code does. The definitions of the load effect or loads are given in this slide. So some of them are obvious, D is dead load, E is seismic or earthquake load, etc. So the interesting part here is that D could be used to characterize either an effect, a load effect, or a load itself. This I think is obvious, the load combinations can be used either applied directly to loads or to load effects. So in general, what happens is when we use a program, for example, for analysis, one specifies the loads at their service load level, and then one creates the combinations by factoring those loads by the appropriate load factors. And then the programs calculate the effect. So factored moment, factored shear, factored axial force, etc., based on those defined load combinations. So this slide lists some of the exclusions on load factors and some of the definitions or how the effects of E, for example, earthquake or wind are calculated, and these are based on ASCE 710. E, for example, is the load effect of an earthquake, and it is based on a return period of 2,475 years. This means this is a return period that is considered a factored level earthquake. So E already is at the factored level, same as wind is at the factored level. That's why in these two, when you use equations for earthquake and wind, and going back a couple of slides here, the equation for earthquake and wind, their load factors are one, because the way we define the loads now in ASCE 7 are already at the factored level. If you use an earlier version of ASCE 7, wind was at the service level, and therefore the load combination used to be multiplied, the wind load combination used to be multiplied by a load factor of 1.6. So this is the general design procedure in the ACI code. This one determines all load types in the structure. You calculate the load effects. You multiply the load effects by a specified load factor, depending on the load combinations. You can combine the loads as required to get maximum effects. Then on the resistance side, you calculate the resistance of a member. Then you find a reduction factor of that resistance to apply to the resistance of the member, and then you apply the general design equation, which tells you that the ultimate or the factored load effect cannot exceed the factored resistance. And some of the – these are the most common fee factors. These are, in fact, all of the fee factors that are included in ACI 3.18.14 and also the PCI 8th edition. A fee factor of 0.9 for tension-controlled members. A fee factor of 0.75 for compression-controlled members with spirals, 0.65 for compression-controlled members that do not contain spirals, 0.75 for shear and torsion, and 0.75 for strong time-offs. So members that are classified as transition members will use a fee factor between 0.9 and 0.75 or 0.65, depending on whether they're spirally reinforced or not. And so I'm not sure if all of you or many of you understand what a tension-controlled or a compression-controlled member is. So I wanted to ask Sherry if she could start – ask that through a poll. It's just a yes-no poll in this case. Again, I will launch the poll and read it for you now. Again, as Sergio said, please indicate if you know what a tension-controlled or compression-controlled section is. The poll is now available on your screen. Please respond now. We have 100 percent voted, and I will close the poll now. Ninety-one percent said yes, nine percent said no. Great. So the majority of you do know what that means, and I'll just – then I'm going to just review very quickly and make reference to prestressed concrete members, how this applies to prestressed concrete members. A tension-controlled member, you might recall from reinforced concrete design, is a member in which the net tensile strain – so this strain here is called the net tensile strain – exceeds 0.005. A compression-controlled member is one in which the net tensile strain is less than the yield strain, which is calculated as FY over the modulus of elasticity of the steel. Typically the yield strain is 0.002 for grade 60 reinforcement, and that's also what's used for prestressed concrete reinforcement, so prestressing strength. And a transition member is one in which the net tensile strain lies between that limit of the yield strain and 0.005. So that's how one defines the net tensile strain. So at the onset or the condition of flexural failure of a member – this shows a T-section where the flexural failure is indicated by crushing of the concrete at a strain of 0.003 in compression – the strain at the farthest layer from the compression phase, at the time when the compression phase is crushing, that strain is referred to as the net tensile strain. And that's how one determines whether it's a tension-controlled, compression-controlled, or transition section. And a plot of the feed factor that was listed in the previous slide tells me that a tension-controlled member has a feed factor of 0.9 – that would be this line right here – a compression-controlled member. It depends on whether the member is tied or spirally reinforced, so 0.65 or 0.75. And then the feed factor would vary linearly between those two limits, and that would be referred to as a transition section. Now we need to look back and think about the case of prestressed concrete. At a strain of 0.002, it makes sense to call it a compression-controlled member, because that's the yield strain of both mild reinforcement and one could also use that for prestressed concrete elements. The strain of 0.005 makes sense for tension-controlled members that are mild, that have grade 60 reinforcement. But remember, you might not remember, I'll show you a slide in a second, that the yield of prestressing steel is approximately 0.01, 1%. So does that mean that the prestressed steel does not yield for a tension-controlled member? So to answer that question, let's run a quick exercise. Let's think of a beam that has eccentric prestressing like we did last time, and imagine that the only thing that's acting on this beam is a prestressing force. There's no gravity yet. So the condition, the strain condition for that beam would be at the top we have a top fiber strain, a tension strain, right, because the beam is cambering upward, so there's strain at, there are tensile strains at the top. And near the bottom, the concrete is feeling compression, but the prestressing strand is at, has some tension strain, epsilon P1, and epsilon P1 is the initial strain applied during pretensioning, so when we pretension the strand, that's epsilon I, minus the change in strain due to elastic shortening of that member. We talked about these elastic shortening effects on the simple examples we were talking about last time. And look at the dimension D sub T. D sub T is the distance from the compression phase to where the prestressing strand, extreme prestressing strand is located, so that's called D sub T where epsilon P is measured for this condition. So now we're going to start applying load, and when we apply load, a small amount of load is applied, we go from this gray strain line, we start applying load so the beam starts decom, so detensioning from the top and losing a little bit of compression on the bottom, some compressions on the right side in this figure, and this bar represents the strain, represents the strain in the prestressing steel. So as we apply loading, the beam goes from this strain condition to, pardon me, to this solid strain condition, so all of it is in compression with some zero compression at the top and a lot of compression at the bottom, and the strain in the strand increases. If we keep continuing the load, applying load, then the strain, you'll notice how it changed from this condition now to that condition, now all the beam is in compression from top to bottom, the strand strain increased, the top through bottom of the beam is in uniform compression, the strain in the strand increased a little bit more to epsilon P4, now the top of the beam starts increasing the strain in compression and the bottom is beginning to see some tension, and notice that right at the level where the prestressing strand is located, the beam has now undergone from being in compression to zero. So this condition is called the decompression state. So it means that the beam that used to be in compression here, because of the applied loading, now has zero compression at that level, and the strand is still increasing to accumulate strain, and of course the strains are also growing here in the top part of the beam. At this stage, one wants to evaluate, or we could evaluate, what the value is for epsilon sub P6 is at this condition. So at this condition, epsilon sub P6 would be almost identical to epsilon I if there were no losses of prestressing force, but there are some losses. So typically at this condition, at the decompression condition, epsilon P6 is roughly about 0.005 to 0.006 for a typical jacking stress of 0.75 FPU. So if we continue loading then, we now increase from epsilon P6 to epsilon P6 plus epsilon T, the net tensile strain at failure of the beam. The strain at the top of the beam is at its crushing strain, so now we have the total strain in the strand being equal to epsilon P6 plus epsilon T. So if we add these two components together, that results, that ends up being about 0.01 to 0.011. So that means that the prestressing strand is clearly past yield, and not only that, epsilon T is going to be above, typically would be above 0.005, which is the definition of a tension-controlled section. So in this slide, it is important to note that the only part that we compare to the 0.005 limit will be the strains generated after the condition of decompression of the beam. And this is the slide that shows stress versus strain plots for prestressing strand. These are the ranges of where, for prestressing strand, what would correspond roughly to a compression-controlled, a tension-controlled, or a transition region. Notice however that in this plot, we're talking about total strains in the strand. So that means the initial strain plus the strain applied after decompression. So that's why in the previous slide, we said that typically for a tension-controlled section, the strains, the total strain will exceed 0.01, 0.011 roughly. So also note here that the limit for compression-controlled is just above the linear range of behavior, the proportional limit, and that the tension-controlled limit is in the hardening region of this stress-strain curve. So all of this, so this approach for design was first proposed by MAST in 1992. This is a landmark paper that proposed this first approach because it unified the design of both reinforced concrete and prestressed concrete flexural and compression members. So if you're the one that, like me, that likes to collect landmark papers, this is a good one to have in your library, in your virtual library. Okay, so changing topics here, still about code requirements. Another important concept in prestressed concrete design is going to be the definition of what development length means. And the development length for pretension elements that are bonded to the concrete is the distance that is required of the strand to be embedded in concrete for it to develop its full stress, the full magnitude of stress. So notice that stresses are transferred between the concrete and the strand through bond. And I think I showed you, you may remember I showed you a picture of a strand embedded in concrete with the little, you know, with the wire that twists around the center core. And it's through those logs, so to speak, that the concrete and steel bond together. So in this equation, this entire equation is the one that's given in the ACI code in this reference section, and as well in the PCI design handbook, where the development length is given by two terms, FSC over three times DB, DB is the diameter of the strand, FPS and FSC are also, and then there's, pardon me, there's a second term here that is also multiplied by DB. In this equation, these stress, the stresses are all in KSI units. If you are used to using a previous edition of the PCI design handbook, you might recall that there used to be a 3,000 here instead of a three, and this second term used to be divided by 1,000. So note the units that are new for the PCI 8th edition, FSC and FPS are in KSI, and these are how they are defined. FSC is the effective prestressing stress. This is the prestressing stress after losses. So we pretension the strand, then apply the, or release the prestressing force onto the element, calculate losses, and then after loss calculations, that's what's called the effective prestressing stress, and FPS, which is different from the effective prestressing stress, is the stress generated in the strand at nominal moments. So that is beyond FSC. So when we pretension a beam, a beam is at FSC after we calculate losses, but then we can bring the strand or the beam to failure, and the strand stress will increase up to a level called FPS. Of this equation for L sub D, the first term is often referred to as the transfer length. So the transfer length plus the second term will give me the development length. So the development length is the length required to develop the strand to FPS at failure, whereas the transfer length is the length that the strand needs to transfer the effective prestressing force into the beam. So we will often see L sub TR for transfer length as FSC over 3 times dB, dB being the nominal diameter of the prestressing strength. But note that the ACI code does not really define transfer length. There is some commentary in ACI 318 that it does discuss that this term corresponds to the transfer length. And a plot of how the steel stress is developed with distance from the end of a pretension member is shown here. So as you can see at the end of a member, there's no stress in the steel. And as we get into the concrete over a distance called L sub TR, right, L sub TR would be FSC over 3 times dB. That's the transfer length. The strand is able to develop a stress of FSC, the effective prestressing stress. And then if we load the beam, then the strand should be able to develop to FPS, but it needs a little bit more length to do so. And the total length from beginning to where it reaches FPS, the stress at flexural strength, that is called L sub D. So FPS can be estimated. We'll learn how to do that in our design example in a few sessions. But there's also a useful design aid in the PCI design handbook, design aid 15-4 that lists various transfer lengths and development lengths for various values of FSC and FPS. Now there's also equations given in the PCI design handbook. This is taken directly from the handbook, figure 5-7, where if you want to estimate well at what stress a strand is, if it's not fully developed. Let's say if we are at a section of a beam that is not at the full development length. So one has to follow the approach of using these two lines to find the stress in the strand for partial development. And that equation or this dot right here is calculated using this equation that you can derive yourselves from just the use of these lines, the equations of these two lines. Or if FPD were to lie, let's say the section were to lie below the transfer length LTR, then it's just a linear relationship with distance, right? So LX would be somewhere here up to a point, and it's a fraction of LTR times FSC. So this is useful when you want to determine at what stress a strand is when it hasn't reached the transfer length or the development length. Okay, so all of that is presented because when we have in the region where the strand is not fully developed, so in this region right here, the strength reduction factor, we cannot use a tension-controlled strength reduction factor because there's a possibility of strand pullout. So the strength reduction factor used if we are within the region, within the transfer length region, end of the member, the fee factor should be 0.75. Let me just jump to C here. Past the development length, the fee factor is 0.90, and this is an ultimate. And between the transfer length and the development length, one can increase linearly from 0.75 to 0.90, the fee factor for flexural strength. So this is a particular exception for the tension-controlled and compression-controlled definitions for flexural strength that apply to sections of a member that lie within the region that is at the end of the members where the strand is not fully developed. So that shows you a variation of fee factor as a function of development length. Okay, so as I said, I went very quickly through the first part so that we can spend the rest of tonight talking about the design example and being more positive in the design example. So let me first introduce what we're going to do. This is our design example that we will use for the rest of this course. It's a rectangular building that shows a plan view of a rectangular building. Four spans, four 20-foot spans in one direction, one 32-foot long span in the other, in the vertical direction. And as you can see, when we cut a section AA, we have hollow core planks in the longitudinal direction resting on rectangular beams that are going to be pre-stressed. It lists here that there are 12 RB28s. I'm going to explain what those are in case you don't know. And then if we take another section, section B, those 12 RB28s are supported on a 30-foot center to center bearing span for a total length of those rectangular beams of 32 feet. So it's a very simple example, but it's nice that it will let us develop concepts fairly easily while being rigorous with all our calculations. So our task throughout the course will be to design the pre-tensioned interior 12 RB28 beams, which are typical, right? So we're going to first load those beams, say a typical interior beam, and then start designing them. Okay. So let's first define the problem, keep defining the problem. As I said, four bays, 20 feet by 32 feet. The hollow core planks are six inches deep by four feet wide. They are supported on 12 RB28 beams. The beams are 32 feet long and supported on 30 feet on center. The live load for our case is going to be a hundred PSF so that we know what the load factor will be for this case. It will be a factor of 1.0 in the gravity load combination of dead plus live. The material properties, we're going to use a 4,000 PSI concrete strength at the time of release and a 28-day strength of 5,000 PSI. The pre-stressing strand that we'll use is a half-inch grade 270 low relaxation strand, and we'll use grade 60 mild reinforcement in case we need it, say for stirrups. And when we look back here, say for example, in when we select the material properties, we have to consult with the producer fabricator. Those of you that are producers know this perfectly well, what your facility can handle. Producers need to be in touch with their producer to make sure that you can handle that the producer can actually come up with a 4,000 PSI strength at the time of release, that they have half-inch strand in stock. More commonly these days, I think producers are stocking 0.6-inch strand, but it's often good to check with a producer to see if they do have 0.6-inch if you're going to specify that. The newer or the larger strand these days is 0.7, so again, check with the producer to see what they have in terms of material availability. And there's good information about material properties in Chapter 9 of the PCI design handbook. So these are some general notes about material selection. So we have come to the last poll of this evening, and just to have an idea of what you have used in the past in terms of the design handbook. Thank you, Sergio. This is the last poll. What edition of the PCI design handbook do you commonly use? The poll is now launched and is on your screen. Give people a few more seconds. I am closing the poll now. Sergio, 48%, 8th edition, 29%, I mean, 29%, 6th or 7th edition, 0%, the 5th edition or earlier, 24% not familiar with the PCI design handbook. Great. So many are already using the 8th edition, which is very good. All right. So I'm glad to hear that all of what we, so there are some charts that we're going to be using throughout the design example, and we'll use the charts that appear in the 8th edition of the PCI design handbook. So the approach for design, of course, as we always do in structural engineering, when you first have to load the element that you are designing. So the first thing will be to calculate the slab weight. We will use a tributary live load that's acting on that center, the typical interior beam. Then we will use the beam section and find the self-weight of that beam section. So we're starting the problem by saying it's a 12RB28. So we have predefined this, but that doesn't necessarily need to be defined a priori. Let me just show you how we can do it. Let's imagine that we haven't defined that yet. So the first thing is to look at the plank, right? So we are going to use a 6-inch hollow core, 6-inch thick hollow core plank, 4 foot wide. And we're going to use, we're going to assume that it doesn't have topping. So the dead weight, the dead load is 49 PSI, PSF, pardon me, for that type of hollow core. And this table is taken directly from page 326 of the 8th edition design handbook. We know that the tributary width of that interior, typical interior beam is 20 feet. So the slab self-weight is simply 20 feet. To convert to a line load, we apply that width. We multiply that width times 49 PSF. That gives me 980 pounds per foot. The live load we said was 100 PSF, again, times the tributary width. That's 2000 pounds per foot. The total superimposed load that's dead from slab self-weight and live load from the applied live load on the floor is 2980. We are just adding them together without any load factors. This means that this is a service load combination. Okay, so one possible way to do a preliminary design of the beam that we're going to use is going to the beam load tables that appear in the PCI 8th edition. So in this particular case, we're using the one that appears in 339, where I'm going to zoom into several regions here in a second, so you don't need to try to squint and try to read what's in it. So what this lower part will let us define is how much, what size of beam we'll need for the given span, depending on the superimposed load that we just determined. So this is referred to as the superimposed load table. Okay, so if we zoom into the upper portion of these charts, we see that note number two says that the loads included in the load table are assumed based on 50% superimposed dead load and 50% live load. Of course, live load is always superimposed, so it doesn't say superimposed. So the section is given here, F prime C, the 28-day strength of 5,000 PSI, and the grade of the strand we're using is 270 for half-inch strands, so this is the appropriate chart to use. But this is a very important statement, so we need to see how our case applies, how well our case complies with this statement. So the superimposed dead loads means dead loads applied other than self-weight of the beam. So in our case, we have defined that the dead load is 980, the superimposed dead load is 980 pounds per foot, and the live load is 2,000 pounds per foot. So it's clear that it's not 50-50, right? They're not equal. So one way of trying to come up with an equivalent superimposed service load that we can enter in the tables for preliminary design is by doing the following. Since what we want to do is we want to design our beam eventually for strength at ultimate, we can use the gravity load combination, 1.2 dead plus 1.6 live, 1.2 times 980 plus 1.6 times 2,000, that would be the ultimate or the factored load for gravity that we would need to design the beam at strength, for ultimate strength. So that will give me a 4,376 factored load. If we equate that now to the same load combination, but we express the load combination as 50% dead load and 50% live load, with the appropriate load factors for each, you'll note that we can solve for the superimposed load that would correspond to the same factored load that we would get from our design. So the superimposed load of 3,126 pounds per foot is what we should be looking for in that superimposed load table. So 50% of that is 1,563 and 50% live load is 1,563. That would give me exactly the same factored load at ultimate as our actual condition. So when we do that, we enter the chart at using the span, 30 foot span. We go down this road, this column, pardon me. And in each column, in each cell, there are three entries. The first entry corresponds to the superimposed load that we're targeting or that we should be targeting. The second row corresponds to camber, estimated camber. And the third row corresponds to the estimated long time camber. So since we're shooting for, we're trying to find a superimposed load of 3,126 pounds per foot for a 30 foot span. This entry here satisfies that because this beam, whatever this beam is in this row, can carry 3,320, which exceeds our 3,126 superimposed load that we need. And that corresponds to a 12RB28. Notice that this row also gives me the number of strands needed, 12 strands, and the distance to the center of the strand, four inches. So 12 half inch strand. This would be a good starting point. So we now, what we use is the dimensions of these beam, the 12RB28 to be able to calculate self-weight, 350 pounds per foot. So now we have beam self-weight and we finally have all the loads that we need acting on the beam. So we have slab, beam self-weight and live load listed here. And now we can start by checking some conditions based on all those loads. So what we have to check in accordance with ACI, we need to check every condition that this beam will encounter during its service life. So we start by checking release. That's one condition. We also have to check service load stresses, which means the stresses that typically generate in that beam throughout its service life. And we will also have to check the ultimate load condition, the ultimate limit state. So these would be at ultimate. We won't follow these checks in this order necessarily. We will see, we will check them in the way that is most efficient for design. And that is by looking at what conditions present or represent a multiple or are present in the beam multiple times throughout its life. And that is service, the service load condition might be present in the beam several times throughout its life. The ultimate would only occur once, right? If we each ultimate, we reach ultimate at one point in time, then the beam would fail. But so these are two critical conditions that we must check. The other one that we have to check is release, but that's not a very permanent condition. It's just an instantaneous condition. It shouldn't be thought of as a controlling condition. So we can start by checking service load levels and then check release and adjust if necessary. So that's the approach that we are going to follow. First design for service, check the release condition and adjust the number of strengths if necessary. And then we're going to start checking ultimate load conditions for once we have determined the number of strengths. Okay, this is what I just said in a more organized way. We will assume that the service load condition controls our design. And of this service load condition, we will assume that tensile stresses will control. There are some assumed tensile stresses that are specified in the code that we cannot exceed. We will then calculate the number of strands that are needed to satisfy this tensile stress control limit. We will then calculate the pre-stressing loss that occurs at the time of release and then over time. And then we will check the service load condition for all tensile and compressive stresses that are indicated in the code. And we must adjust the strands as necessary. Then we will check the release condition. This means at the time the pre-stressing force is released into the beam. And again, we will adjust any parameters that are needed. And then we will check ultimate strength, flexural strength, and then we will check shear strength. And then we'll finish our design. This is the whole, what we have left in this course, we have all of these steps to do. Okay, first things first. So in order for us to take the first step, assuming that tensile stress is controlled, we need to assume, we need to decide what we want our section to behave as. Whether we want it to behave as an uncracked section, and that is classified as a class U section in the ACI code. Whether we want it to behave as a transition section or as a cracked section. And this classification refers to a beam depending on the level of tension stresses that are generated in it. So if the tension, the maximum tension stress at service is below 7.5 root F prime C, the section is classified as uncracked. So this 7.5 root F prime C is assumed to be the cracking strength of concrete under bending. So that's why any tension stresses that lie below this limit in absolute value is referred to as uncracked. In transition, when the tension stresses exceed 7.5, but lie below 12, that's referred to as a transition or a beam in transition. And a cracked beam is one that exceeds a tension stress of 12 root F prime C. So there's no requirement of how big this F sub T is. So remember when we started talking about the benefits of pre-stressing, we said that one of the benefits is to try, that under service conditions, the sections remain uncracked. So that's usually what one does. We try to maintain a beam in its uncracked condition so that there's no water ingress and no corrosion to the reinforcement. So that's what we'll do here. We'll assume that our section, the beam we're trying to design is uncracked. And this is a table taken directly from the PCI 8th edition handbook, which is also given in ACI 318. And it tells you for the different classes of pre-stressed concrete elements, a class U on cracked, transition or cracked, what type of section properties one should use. So when calculating the cross-sectional area, moment of inertia, et cetera, what assumptions to use, it tells you, well, use the growth section if your section is uncracked or visit section 24.522 in ACI 318 for guidance. And what the tensile stress at service loads, what the limit is for those tensile stresses according to 24.521 in ACI 318. And this is what we had just said in the previous slide that for the section to be uncracked, it has to have a tensile stress of 7.5 root F prime C or less. Okay, in terms of compressive stresses, the extreme compressive stress in the top of the beam under service conditions or in the compression side of the beam under service conditions shall not exceed 0.45 F prime C when we're talking about the pre-stressing force plus sustained loads or 0.6 F prime C when we're talking about the pre-stressing force plus the total load. So that means that plus line. So these are the compression stress limits for the service load check for classes U and T. And there's no limit for class C in terms of compressive stress limits. Okay, so let's, as I mentioned earlier, let's design the beam for class U uncracked and that tension, the service load tension will control. So we will try to limit the tension stresses in the bottom fiber to 7.5 root F prime C. So we start calculating section properties. We could also use the tables in the PCI design handbook, but in this case, it's much easier to calculate section properties. The area of that beam 12 by 28 is 336 square inches. And the moment of inertia goes moment of inertia is BH cubed over 12, 12 times 28 inches cubed over 12. That's 21,952 inches to the fourth power. And we will assume that the initial pool of the strand is 75% of FPU. This is the normal pool, which is 202.5 KSI. This should be verified with a producer. Sometimes they like to pull the strand to 70% of FPU. In terms of initial tension, steel tension, there's also another check one must make. We must ensure that this 0.75 FPU does not exceed 94.94 FPY or 0.80 FPU. So this is automatically satisfied, right? We just pulled to 75% of FPU, but this 94 FPY is 228.4 KSI, which is less than 202 KSI that we had specified here. Okay, I think this design date is not necessary in any, I mean, if you remember statics and how to construct your shear and bending moment diagrams, but these design dates are similar to the AISC design dates where you have bending moments and shear force diagrams for different beams. In our case, we have a simply supported beam with uniform loading, and it gives me the moment equation as a function of X for this beam, X being the station measured from the left support into the span. So we can use an equation like this to determine moments at any section X, WX over two times L minus X. And as you can see from these design dates, you can also get maximum deflection at mid span, a shear force diagram as a function of X as well. So they're useful tables to have. Okay, so let's look at the moments that are induced by each of the load components. Our self weight, and all of these moments are calculated at mid span, 39.4 kip feet, slab weight generates 110 kip feet, live load generates 225 kip feet for a total service load moment of 375 kip feet roughly. Notice that the span we're going to be using for the service load checks is 30 feet rather than the total beam length of 32 feet. And this is because you might recall that the effective span length measured from center of support to center of support is 30 feet and not 32. So these moments are calculated based on a 30 foot span. So we can calculate what this moment acting on the beam would generate in terms of stress by using the bending formula. Remember that we're assuming the beam to stay uncracked. So it should be linear. So we use the MC over I formula from strength of materials, 374.6 kip feet is the moment, C is the distance from the centroid of the section to the tension fiber. So that's a 28 inch beam. Half of that is 14 inches. And this is a conversion factor from inches to feet. And the moment of inertia of 21,952 inches to the fourth power. So the bottom stress would be 2.87 KSI in tension. So if these beam were non-pre-stressed, it would be clear that the beam would crack, right? It's a very high tension stress, almost three KSI. All right, so we need pre-stress. And so what we just calculated is the superimposed load tension stress right here. To that, we're going to add the effect of pre-stress, P over A and N, so PEY over I, in order to reduce these tension stresses and to keep it under the limit we want. So you'll notice that the parameters we're looking for are the force P and its eccentricity with respect to the centroid of the beam. So we need to assume one of the two parameters. And the first, the easiest one to assume will be the eccentricity. So we start with the eccentricity of the strand and going back again, notice that it's measured from the centroid of the beam to the centroid of the pre-stressing force. So we have 14 inches here from the centroid of the beam to the extreme tension fiber. So we need to estimate that value where the centroid of the strand will be. So let's assume that the centroid of the strand will be three inches from the bottom of the beam upward. So from 14 inches, we subtract three inches. So the eccentricities, assumed eccentricity will be 11 inches. This will be affected, of course, by the number of strands that we end up designing. So we'll have to check that in a second or later. So FCBL, the stress induced by the superimposed loads in the bottom fiber is this number here. We just calculated and it's negative because it's a tensile stress. That's 2.87 KSI. We have P over A, P over the area of the beam, PEY over I, P times the assumed eccentricity of 11 inches. Y is the distance to the tension fiber. And the moment of inertia of the growth section is 21,900 roughly. And this has to exceed a tension stress of 7.5 root F prime C of 5,000 PSI and divided by 1,000 to convert from PSI to KSI. So why is the greater sign used here? Well, because positive values here mean compression. So if we want to stay below cracking, this means that we have to stay to the right-hand side, let's say on a numerical, on a number scale, right? Where negatives are on the left and positives are on the right. We have to stay to the right. So we have to stay greater than the 7.5 root F prime C, which corresponds to the cracking stress of concrete, okay? So if we solve this equation, you can solve that. This is just algebra. We get that we need a prestressing force that exceeds 234 kips so that the beam doesn't crack, okay? So how do we determine the number of strands for that? Well, we have to, since we're checking a service condition, we have to assume some losses have occurred over time. And for now, we haven't calculated losses yet. We'll do that in a few slides. So the first thing is we had mentioned that the initial pool was gonna be 75% of FPU. So that's 75% of 270. And we're going to assume that the losses are 20% for now. That's a fairly common value to use, a 20% loss. So instead of using the initial pool to calculate the number of strands, we're going to reduce that by 20%. So that's what this one minus 0.2 represents. And we're using half-inch strand. And each half-inch strand has an area of 0.153 square inches. So this corresponds to the initial pool. 80% of that, that includes losses. And 0.153, that corresponds to what each strand will contribute. So that's 24.8 kips per strand after losses. So the number of strands that we need is the minimum force, pre-stressing force, 234 kips, divided by the contribution from each strand, 24.8 kips per strand. So we need 9.44 strands to satisfy our design limit or tension limit. So we round up, right? Since P has to be at least 234, we have to round up to 10. So that's what our initial design will look like. We have a 12 by 28. And notice, we are distributing the 10 strands in two layers, five strands per layer. And the spacing of strands or layers of strands is at two inches. And also laterally, the spacing of strands would be at two inches. And notice that our assumed eccentricity of three inches was a very good assumption because now the centroid of this strand pattern is exactly at the assumed location of the eccentricity we had chosen. So we do not need to adjust for that difference in eccentricity. If at this point, our strand pattern were different from what we had, gave me a different strand centroid than what we had assumed, we'd have to adjust E for the actual E, and then go back and revise it, see if we needed more strands to reflect the actual strand pattern. But we don't need to do that for this case. Notice also that the table that we used to choose the beam preliminarily, at least, said that we needed 12 strands. And according to our calculations, we only need 10. So that's why those tables are only used for preliminary design. Okay, so we now have a number of strands, and now we're in a condition that we can calculate pre-stress losses. We don't need to assume them anymore, we can calculate. So the initial pre-stressing force will be lost, and the losses will take place not only due to elastic shortening, we explained elastic shortening last time, but there's also going to be other sources of losses, such as creep, shrinkage, and other sources of losses for non-pretension, so post-tension members. As I mentioned, elastic shortening, we already discussed last time. So recall that we assumed 20% initial pre-stressing force loss. These are all the sources of losses that are mentioned in the PCI design handbook. Elastic shortening for pre-tensioning, creep, shrinkage, relaxation, that's relaxation loss of a strand, so loss of stress over time of the tension steel. And these two other sources of losses are friction losses are only included in post-tension members, and anchored setting losses are also only included in post-tension members. So the only four sources of losses that we will include in our design example are these first four. So ACI 318.14 does not have a way of calculating losses. It doesn't tell you how to calculate losses. It just tells you that you must calculate them. And thankfully, or fortunately, the PCI design handbook does have a method by which one can calculate losses, and one can determine each of these sources of losses independently. So ACI provides you freedom on what method to use to calculate losses. There are other methods such as the AASHTO method that you would also satisfy ACI if you use that method to calculate losses. So any acceptable analytical method is allowed by ACI. So the total loss, which is gonna be given as a stress, a loss of stress, is going to be the stress loss, loss by elastic shortening, the stress loss by creep, by shrinkage, and by relaxation. So we'll go term by term here. The elastic shortening loss is going to be calculated using this formula. And I'm going to explain each of the formulas when if they have a mechanics type of background. So you may remember last time that we defined the modular ratio as the ratio of the modulus of the pre-stressing strand to modulus of concrete. So that's what this ratio represents here. So this is the modular ratio. So this could be thought of as N. FCIR is the stress at the level of the pre-stressing, at the center of the pre-stressing strand when the pre-stressing force is released. So that is the stress in the concrete member at the center of the pre-stressing strand. If we multiply that stress times the modular ratio N, that will give me a stress in the strand. So I'm converting stress in concrete to stress in strand by using the modular ratio here. And K sub ES we'll define in a second in the next slide. So how do we calculate FCIR? We derived this expression here last time, right? P sub I over A gross plus PIE squared over I gross. That's the axial load shortening of the element. This is the, I should say stress. This is the stress induced at the level of the pre-stressing strand from bending. So that's why we have E squared. We have P times E, times E again, because E gets me to the center of the member. And then we have a negative value here from any superimposed loads acting at the time when we're calculating these elastic shortening losses. What is that time? The time of release. So that's why we have the modulus of elasticity of concrete at the time of release here. And that's why we have here the initial pre-stressing force. And so this KCIR, there's a factor here that did not appear last time. This factor, we'll notice here that last time we referred to, we used the transformed section properties in these calculations. So this K factor, rather than transforming the section, this KCIR factor adjusts for the fact that we're using gross properties instead of transformed section properties. So KES, going back, this number is taken as one. So it essentially disappears for pre-tensioned members. FCIR, we talked about that, what that is. It's calculated at the centroid of the pre-stressing strand. Moduli of elasticity of pre-stressing strand and concrete. This constant, this value accounts for slip of strand, relaxation, and the difference between gross and transformed section properties. And it's typically used or taken as 0.9 for pre-tensioned members. These two quantities are clear, I guess. And N sub G, going back one more here, this is the applied moment on the beam at the time of pre-stress force release. That's the moment induced by self-weight of the beam. That's the only load that's acting on the beam when the pre-stressing force is released into it. Okay, so let's calculate some of these numbers for the elastic shortening loss. Now we have 10 strands, the area per strand, 0.153, and the initial pull. So we have an initial force of 310 kips. The moment induced by self-weight, WL squared over eight. And this is at center of the beam, right? So since we're checking the section at mid span. 0.35, that's self-weight of the beam, 32 feet squared divided by eight. We do not use 30 feet here because the beam has just been poured and we have just transferred the force in the pre-stressing force into it. It's still not sitting on its bearings. It's sitting on the casting bed. And as we transfer the force, it is assumed that it can rotate about its end. So the clear span is 32 feet. That's why we use 32 feet here. So the moment induced by self-weight is 44.8 kip feet or 538 kip each. FCIR, we plug in all the values we know. 310 over 336, et cetera. 0.94 KCIR. And I think the rest is self-explanatory, right? The eccentricity of 11 inches. So we get a stress at the center of pre-stressing strand of 2.10 KSI. So this is a stress that occurs at release. And this is the node I meant, this is what I just noted here about the span of 32 feet. So having KFCIR, we can calculate the modular ratio and of the E of strand, E of concrete. Remember that we're using the concrete strength at the time of release, 4,000 PSI. So that's why we calculate the modulus for 4,000 PSI. So we calculate an elastic shortening loss of 16.6 KSI. So that's the first source of loss. The creep loss is calculated in a similar manner, except that instead of using FCIR, we use the difference between FCIR, which is the stress in the strand at the time of release, minus the stress at the center of the strand for sustained loads. FCDS is a stressing concrete at the pre-stressing steel centroid due to superimposed pre-stressed or permanent loads, I should say. Creep is a phenomenon that occurs under sustained loads. So that's why we need to use the difference between the stresses that are acting at the beam at the time of release, minus the stresses that will be there under sustained loads. That's why this term is a subtraction. So FCDS is calculated as MSD, which is the moment due to all superimposed permanent loads. So without including member self-weight, because that has already been used at the time of release, that are applied after pre-stressing, okay? That's the only new definition here. And there's a new factor here, KCR. So there's a few notes about the sign convention here. FCIR is assumed to be compressive. FCDS is tensile. Tensile because under sustained loads at the level of the pre-stressing strand, the beam will start sagging, right? Or increasing in stress at that level. So that's why it's assumed net tensile. So this negative sign already assumes that this term will be net tensile. So FCIR minus FCDS is compressive. So typically, if we get a positive sign here from this subtraction, we are going to have a net compressive stress. If that does not happen, so read along here, we would have to change the signs, right? But the equation already has the negative sign. We do not need to input, in this particular case, we do not need to input FCDS as a negative value. Okay? All right. So what are the sustained loads? Well, the dead loads are the sustained loads. And those are the self-weight of the hollow core plants. Recall that they weighed 980 pounds per foot. And now the span that we're using to calculate the moment, since we're looking in long-term here, we're talking about the beam already mounted on its supports. So it's already on a 30 foot span. So the moment is 110 kip feet, and that's a kip inch units. FCDS now, the sustained load component, is calculated using the Fletcher formula again, using this moment, 1323, the eccentricity and the moment of inertia. That gives me 0.66 KSI. Notice that this is net tensile, right? So it's assumed to be tensile, but we do not do the double sign here, negative, negative, because the negative already accounted for the fact that in the majority of the cases, this will correspond to a tension. And KCR is listed, is given as two for pretension members. And again, the modular ratio here. Now, what we do for the modulus of concrete, we use the long-term modulus of concrete, which is the 28 day strength concrete modulus. So using 5,000 PSI here, that gives me 4 million PSI. And when we perform the calculation, the creep loss is 20.5 KSI. And as we move on, when we talk about shrinkage and relaxation losses, these two equations are more empirically based than the first two. So the shrinkage losses, you'll see a coefficient here. There's another shrinkage coefficient, which is taken as one. EPS is the modulus of the prestressing strand. And these are factors that include what is important for shrinkage loss. So shrinkage occurs through loss of moisture, right? So this term here calculates, estimates the volume to surface load, to surface ratio of the beam. And this number here will correspond to relative humidity, the ambient relative humidity, which is highly variable, right? So we have to make some educated guesses for those, for RH, and we can estimate V over S. So let's assume that relative humidity is 70%. V over S can be estimated as the area of the cross section divided by its perimeter. That's, we are neglecting the length of the member. So that's a fair approximation. So we neglect the length in both numerator and denominator. So we have the area divided by perimeter. That gives me a V over S of 4.2. This coefficient, which is empirical, right? It's applied there. The modulus of strand, one minus 0.064.2 times 100 minus 70% relative humidity. Notice that the shrinkage loss is much less than the elastic or creep loss, 5.2 KSI. And finally, the last source of loss, relaxation, is computed using this factor, K, for relaxation. And this J factor that multiplies all the previously determined sources of losses or losses summed together, where we can pull out values for K, R, E, J, and C from tables in the PCI design. So for example, just very quickly, for low relaxation strand, we have, for our case, 270. K, R, E is 5,000 PSI, and J is 0.04. And for our case of low relaxation strand, equal to 75% of FPU, the C factor shown here is one. There are other values, and I'm pointing you to the corresponding table in the eighth edition PCI design notebook. So again, K, R, E is 5 KSI. This table lists it in PSI units. J is 0.04, C is 1.0. So we plug in the values, recalling elastic shortening loss of 16, creep of 20 KSI, shrinkage of 5.24. Doing the calculation, we have a relaxation loss of 3.3, and adding all the losses together gives me a total loss of 45.6 KSI. Compared, so that's the total loss in terms of stress, divided by the initial pool will give me a 22.5% real loss after calculation. Remember that we initially assumed 20%, but the real loss now is 22.5, but we rounded the needed number of strands from 9.24 to 10. We still need to check whether that works for the real loss. We're losing a little bit more stress, pre-stressing stress than we originally assumed. So we have the initial pool minus the total loss of 45.6. We have the effective stress in the strand is 156.9 KSI. So we have 10 strands. The area per strand is 0.153 times 156.9, which is now using the actual stress in the strand after these 22.5% losses. That gives me 240 kips. And remember that we needed at least 234. So since we rounded up to 10, we're still okay, even though our losses are a little bit higher than what we originally assumed. If we didn't satisfy it at this point, then we'd have to increase the number of strands, recompute losses, and check again to make sure that we satisfy the effective pre-stressing force that we need based on our initial calculation. So as you can tell, I was flying throughout this night, but we ended with about five minutes to go. So this is the end of today's lecture, and I'd be happy to answer any questions you might have. If you have any questions, please, oh, we have a question, and I will send it like I started last week to the entire group so everyone can see the question. Hold on just a second. Sergio, that's the question. So I'm going to reread it again. Suppose we have a beam reinforced with strands and rebar. What is the minimum horizontal spacing between the strands and rebar? ACI 318.14 has explicit numbers for each type, strand, wire, rebar, but not for a combination. I would say that the spacing between strand and rebar has to satisfy the minimum spacing of bars of bars that the ACI code provides for, for say the maximum spacing that governs whether it is for the rebar that's being used or the strand. So typically for rebar, one uses at least one bar diameter spacing between mild reinforcing bars. So if that's what governs over two inches, that's what should be used, right? So that we want to leave that space because those spacings are specifically to allow placement of concrete. So that's a minimum spacing requirement. So you have several requirements, right? You have a bar diameter, one and a half times the maximum aggregate size. And I believe, I forget what the third requirement is, but the key here is that the spacing is, the bar spacing is given so that concrete can be placed. So we want to leave enough spacing to be able to consolidate concrete around the bars and the strand. A second question just came to you. Why is a transformed section not used in calculations of elastic shortening? It isn't because it's easier to just use the growth section properties. And since we're calculating losses here, when we calculated the elastic shortening losses, this is when it was important to use transformed section properties so that we didn't have to account for, or to use the deformation compatibility approach to elastic shortening losses. This factor already includes a modification from gross to transform section property. So it's easier to just use gross section properties and then use this factor for losses. And also we don't use it because remember that at the very beginning, or not very beginning, but initially when we looked at defining our section, for uncracked sections, we are permitted to use gross section properties in our stress calculations. So the ACI code allows us to use gross section properties without necessarily transforming the section. Well, thank you very much. I haven't received any other questions. We are now at close to our ending time. We've got one minute. So I would just like to end tonight's session. If you have any.

pre-stressed concrete beams

load combinations

limit states

tension-controlled members

compression-controlled members

development length

pre-tensioned interior RB28 beams

beam load tables

superimposed dead loads

live loads

equivalent superimposed service load

uncracked beam