So, welcome back everyone, this, as Sherry mentioned, this is our fifth session of this basic pre-stress concrete design course, and we will continue working on our design example. We have, let me just flip through these couple of introductory slides, so it's part number three of our design example, our fifth session, and today we are going to start talking about another one of those limit states that we discussed early in the course. We have so far dealt with service limit states, we have checked stresses in, at different locations along the beam, due to pre-stressing and, you know, for both a rectangular beam with straight strands and a rectangular beam with harp strands. Today we will focus all of our session in computing ultimate moment, and we will see how that is done for both cases. We will, as we did last time, we will present two different conditions, one with straight strands and one with harp strands. So at the end of the session, the idea is that you are going to be able to compute the nominal moment and check for flexural strength at critical locations along the beam, and you will also, if you don't already understand the concept of tension control, compression control and transition sections, and also, as I mentioned earlier, the effect of harping strands on moment capacity. So this is where we are now. We have checked all these different service load conditions, including the release condition, which turns out to be critical for top tension stresses, and we have designs that we now need to move on and check the ultimate limit states for this beam. The first one to check will be flexural limit state, and then that's the last one for which we would need to adjust strands if we need to. And the last session next week will focus on shear design and a few other items such as camber and deflections. So having said this, I'd like to ask Sherry to launch the first poll of the night, just to get a feel for where we stand regarding flexural strength. Okay, Sergio. Our first poll, are you familiar with assumptions and calculations to determine flexural strength of reinforced concrete beams? I am launching the poll now, and please answer on your screen. I'm waiting on a few more people to vote. We'll leave it open for a few more seconds. If you haven't voted, please vote now. Okay, Sergio, I'm going to close the poll. 43 percent, yes, very comfortable, 43 percent, yes, but a bit rusty, have not reviewed them in a while, 14 percent, no, I've always used allowable stress method for reinforced concrete design. Okay, thank you, Sherry. So there's at least a little bit more than half, about 60 percent of people that have either not used this method of flexural strength, and a few, about 40 percent that have not seen it in a while. So let me just go, I will spend the time, my apologies to those that are quite comfortable, it will be a very quick review, but the idea is to get everyone up to speed. So first of all, we need to talk about reinforced concrete sections. In pre-stressed concrete design, we will use exactly the same definition of failure as is done in reinforced concrete design at ultimate conditions. So if we have a rectangular beam of width B and depth, effective depth B with an area of steel near the tension phase, it's a reinforced concrete section. Flexural failure for this reinforced concrete section is defined by using the strain profile. So flexural failure is the condition that this cross-section exhibits when the concrete reaches a strain in the top fiber and compression equal to 0.003. At that condition, at that strain, there's going to be a neutral axis depth that we'll define as C here from the top fiber and compression to the point where strains are 0, and at this ultimate condition, this reinforcing steel will be at a strain that we refer to as epsilon steel. This strain will receive a very specific name in a few slides, and I'll mention that in the future. So the basic definition of flexural failure is given as a strain reached in the top fiber and compression of the concrete of 0.003, and at that strain, let me see if I can figure out how to use the pen here. Let's see, I used it before. I can't find it. Let me just, yeah, let me just move on. At that point, when we reach flexural failure here, the concrete will be past, oh, I see it now. I apologize. Concrete will be past its linear range. So if I project the horizontal line representing the neutral axis, concrete stresses in compression will now resemble the parabola that we find when we test a concrete cylinder in compression. Instead of dealing with these complex stresses or nonlinear stresses in the cross section, we replace those with a block, a rectangular block that has a constant stress of 0.85 f prime C over a distance A, which is a fraction of the total distance C from the top fiber in compression to the neutral axis. So this block, equivalent stress block, is used to simplify the complex stress distribution in compression of the concrete at flexural failure of the section. So this distance A is equal to beta 1 times C, where beta 1 is just what's referred to as an equivalent stress block parameter that is a function of the concrete strength that is used. So for more details on beta 1, I should refer you to the ACI 318 code. The reason we do this replacement is that with this block, we can easily find the compression force in this rectangular section as being equal to 0.85 f prime C times B, the width of the beam, and times A, the depth of the equivalent stress block. So that's C. And then the tension force has to be in equilibrium with this compression force, and the tension force is the area of steel times the stress in the steel that's going to depend on this strain in the steel at failure of the section. So the distance between the compression and tension forces then is simply found by subtracting from D, the effective depth, mid-height of A, so A over 2. And this is because of the uniform stress distribution that we're using as a simplification for this compression stress block. So this is really the basis for flexural strength of reinforced concrete sections. So the key here is that we need F sub S, the stress in the steel. Okay, so if we have now a pre-stress section, the only thing we need to change, and this is again a section that is rectangular, the thing we need to change is we need to change from A sub S to APS, APS being the area of pre-stressing steel. So if we look at that horizontal force equilibrium equation I was alluding to in a previous slide, we have a compression equal tension, and instead of having AS, FS here on the right-hand side on the tension side, we have APS, the area of pre-stressing steel, and FPS. And on the left-hand side, we have the compression force, which is identical to the one for reinforced concrete design. The key here is that FPS is going to be different from FS here in reinforced concrete. In reinforced concrete, the material properties for mild reinforcement are such that we can simply use FY, and we'll see that later. Now this equation also considers the presence of compression steel, which we're assuming here to be equal to zero, there's no compression steel here on the top, and it also assumes that there might be some mild reinforcement near the bottom of the beam, which we're also, as we see from our cross-section, we're also taking as equal to zero. So if we then solve from this equilibrium equation, we solve for the depth A of the equivalent stress block, we can find that depth, and then we can compute the nominal moment of a pre-stressing, which is the ultimate moment without a strength reduction factor yet, which is equal to the area of steel, pre-stressing steel, times the stress of that pre-stressing steel at failure of the section, at flexural failure, times the lever arm between compression and tension, which is equal to that lever arm that I illustrated in the previous slide. Because tension and compression forces are equal, we can also express that equation as a function of the compression force instead of the tension force. So this figure on the right is taken directly from the PCI design handbook, and the only thing that you can see here is that I've crossed out the force associated with compression steel, and the force associated with mild reinforcement. Okay, so moving on. So the difference between reinforced concrete beams, and is, or the properties of reinforced concrete beams is that the steel that we use is mild steel, and we can assume that it's elastoperfectly plastic, and in reinforced concrete design, it is very common to have sections that are called, used to be called under-reinforced, and now are called tension controlled. Now what this means is that we're allowing the steel to yield before reaching that strain of 0.003 in the extreme fiber and compression. So if we allow the steel to yield, then we can just model that steel that is elastoperfectly plastic, so the condition at failure would mean that the stress that I've been indicating for the reinforced concrete section, that stress will be equal to FY, because if I draw a plot of steel stress versus steel strain, if we assume it to be perfectly, elastoperfectly plastic, this plateau would be equal to FY, and if we allow the steel to yield at flexural failure, we'll be at some point along this plateau, but it doesn't really matter as long as we are past yield. This would be the yield point right there. In prestressing steel, we don't have that yield plateau, and so that's what's shown here, and furthermore, HCI 318 allows the section to be designed as if it were compression controlled or in transition. Remember that we talked about tension compression controlled and transition sections in a previous presentation, but I think that the most important part is that the stress strain behavior of prestressing steel is quite different from mild reinforcement, and you can see plots like this in the PCI design handbook. First of all, there's no well-defined yield point. It's a curved, it's a rounded corner here between the elastic portion and the post-yield portion of the strand, and furthermore, there's no plateau, so after yielding, and we can define yielding at 1% strain, that is FPY, the stress in the strand continues to grow, so we need to find a method by which we can estimate FP, the stress in the strand at ultimate when the concrete is reaching its crushing strain in compression, and again, going back to the first paragraph here, the ACI code also allows prestressed concrete sections to be over-reinforced, which is now the terms that are used are compression controlled or in transition. So in order to determine that stress, which is important for our flexural strength calculations, there are two possible methods that one could use. We can use an approximate method that's described in ACI 318.14, section 23231, or you can use a strain compatibility approach that is included in the PCI design handbook in this particular section. So this strain compatibility approach, you actually try to find the actual strain in this steel caused by loading at ultimate, and in order to find that, you have to enter again the stress-strain curve for prestressing strand and find at what point the strand would be at the failure condition and use that as FPS or the stress in the strand at failure of the section. So this is an important point. This stress is past the linear region of the stress-strain curve, and it is in the hardening region. But for now, what we will use, I will just make a, you know, in the future, in a future slide, we will just show very quickly what this strain compatibility method involves, so we will use, for this example, we will use the approximate method in ACI 318. The PCI design handbook also has a flowchart. I don't intend for you to be able to read any of this flowchart, but in this, my intent here is to point you to a figure that actually takes you through the entire process on how to calculate flexural strength of a given section, and you can follow this flowchart to get to the finish line. There's even, you know, very involved equations here at the bottom that you can then eliminate terms that do not apply to your particular problem. So that's all I intend to show you in this slide, just that there is a flowchart. So again, the approximate method in the ACI 318 code involves using an approximate equation for, to estimate FPS, the stress in the pre-stressing strand at ultimate flexural strength of the section given by this formula, and this formula has been shown to be reasonably accurate by experimental testing. So it is commonly used if the effective pre-stressing stress exceeds 50% of FPU. So recall that FPU is the ultimate stress in the strand, so that would be in our case 270 KSI strand, and FSE is the effective pre-stressing stress after losses of, you know, the strand. So this effective pre-stressing stress is calculated after losses once the pre-stressing force is transferred into the concrete section. I will define all these parameters in the next slide, but one thing that's worth noting is that if you're using a previous version of ACI 318 prior to 14, this equation, and I don't remember exactly in what ACI version, whether it was 2011 or even before that, but instead of using the reinforcement ratio in this equation, and I'll define that in a second, the equation involved using what's termed the mechanical reinforcing ratio, omega, which is related to this rho ratio, reinforcement ratio, by, through multiplying rho by FY and dividing by F prime C, and similarly for the compression steel, rho prime would be, pardon me, omega prime would be rho prime times FY over F prime C. These quantities are called the mechanical reinforcement ratios. So if there's only pre-stressed tensile reinforcement and there's no mild reinforcement, this entire term here, the second term in the equation disappears. This equation, this term is only applicable if you have mild reinforcement in the section. The terms without prime correspond to tension steel. The terms with prime correspond to compression steel. So as you can see, the equation here at the bottom is when the section only has pre-stressing steel in it, and that would be our case. So these are the definitions. An important one is FPS and FPU, we already know what those are. FPY is the yield stress of the pre-stressing strand, the concrete, design concrete, compressive strength, and this, since we're dealing with ultimate conditions, this is the 28-day strength. Gamma sub P is a pre-stressing constant that depends on the type of steel used to pre-stress. And that is for bars, it's this value for stress relief strand, 0.85, and for low relaxation strand, it's 0.9, that's what we're using. And remember, I mentioned the stress block constant for flexural strength, beta 1. It's 0.85 for F prime C of 4,000 PSI or less, 0.65 for F prime C of 8,000 or greater. And we interpolate in between these two values. So for ours, our case, beta 1 is going to be 0.80, because we're dealing with 5,000 PSI concrete. Going back here, this is where gamma sub P is, that's our 0.9. And these are the other definitions of reinforcement ratio of the pre-stressing strand, of mild reinforcement, of compression steel, and the mechanical reinforcements for tension steel and compression steel of mild reinforcement, if they exist. Notice that in defining the reinforcement ratios, rho, one uses the actual area of either pre-stressing strand or mild reinforcement, and divides it by the width of the compression zone and the effective depth to that steel. So in this case, DP and D. And when we define that for compression steel, we don't use D prime. We still use the effective depth to the tension steel. Okay. So sometimes the quantity in the brackets of the FPS equation could be expressed instead of using rho and rho prime, you could use the definitions shown in the previous slide and just use AS over B times D, or AS prime over B times D. And you'll see that instead of having D, we are now left with DP, because the Ds in each of the definitions for rho and rho prime would cancel out with this other D outside of the parentheses. So these equations are exactly the same. These are just the quantity within the brackets for that equation. If we want to consider, if there is compression steel in our section and we want to use this value, we need to ensure that the whole quantity in the brackets exceeds .17. Otherwise, compression steel has to be, is neglected, shall be neglected. And one thing to keep in mind is that if we do neglect compression steel, there are two things that I like to mention always, is that first of all, neglecting compression steel is conservative when we calculate flexural strength. And furthermore, the difference between considering it and not considering it acting really gives me very little additional flexural strength. So it's not necessary to include it. Most of the time, in fact, it's neglected. So that's what I mentioned in the previous slide. In our case, the top steel is there primarily to control cracks. It would contribute to as compression steel at flexural strength, but we don't need to consider it since it is conservative to neglect it. All right. So after this preamble and discussion, let's start doing some numbers here. So recall that in order to use FPS, we need to satisfy that the effective prestressing stress is greater than 50% of FPU. And this is the effective prestressing stress from our previous session. That's what we are left with, 158 after losses, which is greater than 50% of FPU of 135. So the approximate method in ACI 318 can be used. This is to remind you of what our cross-section looks like. For the straight strand case, we have two layers of five strands each. Their centroid is three inches from the bottom. The cross-section is 28 inches deep. So we have 14 inches from the centroid of the section to the tension fiber. And then we subtract three inches. So that gives me an eccentricity of 11 inches to the centroid of the strand. This is all numbers that we had used before. The total area of strand is 10 strands times the area of a half-inch strand. We have 1.53 square inches. D sub P is measured to the centroid of the strand, but it is measured from the compression phase now. It's the effective depth, so that's that distance. So it's 11 inches plus 14, that's 25. And the reinforcement ratio for a prestressing strand is calculated using this ratio, area of prestressing steel divided by the effective concrete section. That gives me a reinforcement ratio of 0.0051. So that's from the previous slide, F prime CA 5000, beta 1 we already discussed. Gamma sub P is 0.28. I mentioned it was 0.9, and let me just go back, I made a mistake there. It's 0.28. What I should have highlighted is this number and not the 0.9 here. This is just the upper limit. So it's 0.28. I apologize for that. So coming back again to the slide, 0.28. So we plug in all these quantities into the approximate equation, and that gives me that at flexural failure of the section, the stress in the strand will be at 244 KSI. This is again shown to be satisfactory from an experimental perspective, so it's been proven by experiments, and that's the stress of the strand at flexural failure. Once we have that, we can compute the depth of the equivalent stress block from horizontal force equilibrium using that value of 244. That gives us an equivalent block depth of 7.32 inches by substituting all the appropriate quantities into this relationship. The nominal flexural strength is given by AP times FPS times D minus A divided by 2, and that gives me 664 kipfeet of nominal flexural strength, and the fee factor, the strength reduction factor for flexural strength limit state is 0.9 for tension-controlled sections. We assume this for now, and we'll check it later by remembering how this varies depending on whether the section is tension-controlled or compression-controlled during transition. So for now we have, if it is tension-controlled, we have a 597 design strength that we now need to compare against our acting moment. So how do we find the moment demand, the factored moment demand? We look at ACI, and this is a factored load combination where these are load factors being applied to dead load moments and live load service moments. And what we do is since our critical section for the simply supported beam is at mid-span where the moment is maximum, we use the self-weight of the beam. So this is WL squared divided by 8, self-weight of the beam, superimposed dead loads, and span of the beam to centerline of supports divided by 8. So this quantity is squared, of course. Those are the service load moments, dead load moments. These are the service load, live load moments at mid-span, and when we factor them using the appropriate load combination for gravity loads, we get 540 kipfeet. And notice that this is m sub u, the ultimate moment of 540 is less than our design strength of 597. So this beam is okay at mid-span. So one has to check also other sections along the span, but in our particular case, we're dealing with a very simple beam where the strands are constant throughout, so the flexural strength is going to stay constant except at the ends where we're developing the strand. We'll come back to that concept later. So the moment demand increases from zero at the end parabolically to a maximum at mid-span. So if we have constant or straight strands with constant design strength, this is my design strength, the beam should be okay everywhere, and as long as we are past the development length. So we will check that region, the development length region at the ends of the beam. We will check that in a few slides. All right, so what is this phi factor that we assumed as 0.9? So this goes back a few sessions where we had to find what a tension-controlled, a transition section or a compression-controlled section were defined. And let's start first with reminding ourselves of what a tension-controlled section is. It is a section where a particular strain that we refer to as the net tensile strain, this is a strain caused by loading only, exceeds 0.005 at flexural failure of the section. If that number exceeds 0.005 at flexural failure of the section, then the section is called tension-controlled and the phi factor is 0.9. If we jump to the bottom row here, if this net tensile strain now at flexural failure of the section is less than 0.002, the section is called a compression-controlled and the phi factor for it will be 0.65. And then we interpolate it between these two values based on net tensile strain. So the question here is, well, what is epsilon t for prestressing strand in our design example? Epsilon t is easy to determine for reinforced concrete because it's only the strain induced by loading, but we have strands in our section that are prestressed. So there is some strain in them at the beginning. So two things to remember is that we have now called epsilon t, the key here is t, and that is associated with a depth to the layer of steel that's farthest from the compression face. So that's why it's called t, it's to the extreme tension layer, right? And initially what we have is a strain in the strand that is of this value, right? This is the strain that is locked in the strand because remember that we tension the strand, then we release it into the cross-section and the cross-section is now subjected to a compression strain that is equal to epsilon p0, which is the strain in the concrete at the level of the steel due to prestressing only, prior to any loading acting on it, okay? So there's this strain in the strand at the extreme layer of strand dt. So then when we start loading, the concrete section, which is in compression near the bottom and is in tension near the top, is going to start losing some of that compression near the bottom as we load it. So the next slide illustrates that. We go through phases, and there are more steps in a previous presentation, where we now start applying load to the beam prior to failing it in flexure, and we go through the decompression state where, notice now that we are referring to the outermost layer of strand, and we have now built into the strand more strain. We had its initial strain that was locked in in the beam as it applied the prestressing force into the beam, and we have now decompressed the bottom part of the beam. So we have added this epsilon p0 strain to the strand. So we're proceeding here from, if we think of the stress-strain curve of strand like this, we're proceeding from the effective strain, or the strain associated with the effective stress, FSE. We're marching along now as we load it. We've added an epsilon p0. This distance here is equal to epsilon p0, but we haven't failed the section. We still need to get to a point associated with flexural failure, right? So we need to determine how much this strain is before we fail the section in flexure. Okay, so the total strain in the strand at this point is the sum of these two quantities. So abiding by our definition of flexural failure, that's the condition where the top fiber of the concrete reaches a strain of 0.003, and at that condition, we have a neutral axis depth, C, that is quite deep in this graph. And then we have added an additional strain that is referred to as the net tensile strain. So this is the quantity that we need to compare with those limits to define whether our section is tension-controlled, compression-controlled, or in transition. But remember that the total strain in the strand now is from here to there. It's a much larger strain than just epsilon sub t, all right? So we need to check whether this epsilon sub t is greater than 0.005 so that we can use the proper value of phi of the tension-controlled section. So let's do that. Let's look at our quantities. Dt is 26 inches. Remember that the effective depth was 25, but we add one inch to place ourselves in the bottom layer of strand. We have already computed where the equivalent stress block lies at flexural failure. That's from a previous slide of 7.32 inches. We already know that beta 1 is 0.8, so we can find the neutral axis depth by dividing or by using the definition of A, which is equal to beta 1 times C. We divide A by beta 1, and we get C of 9.15. And by using similar triangles on the strain profile, we can now find epsilon t. And what similar triangles am I alluding to here is that this triangle here, which has a base of 0.003 and a height of C, is similar to this other triangle, which has a base of epsilon t and has a depth, pardon me, not that outer triangle, but this smaller triangle here has a depth of Dt minus C. So that equation gives me this result. So I can now find epsilon t equals 0.0055, and it is greater than 0.005, so we have just verified that our section is tension controlled, and we can use a feed factor of 0.9. Okay, so that's fine. We have checked that the section is fine throughout the length because we have straight strands and we have verified that a feed factor of 0.9 is okay. There is another check that used to be called the ductility provision. It's no longer called that, but we need to check that at flexural strength, the flexural strength of my section exceeds 1.2 times its cracking strength. The reason there's this limit in ACI 318 section 9621 is to ensure that in the case that the beam were to crack under perhaps overloads beyond service, let's say our beam is very stocky and we need very little strand to hold the service loads and ultimate loads, the service loads I should say. If we have an overload and the beam does not have enough flexural strength to beyond cracking, we run the risk of failing the section, fracturing the strength. So the idea that the flexural strength should be at least 1.2 times the cracking strength is to prevent a section with very light reinforcement to fail suddenly after it cracks. So we need to determine what the cracking moment is to compare it with our design flexural strength. And so we go back since this cracking moment is a service check, we go back to the general pre-stressing equation, P over A plus PEY over I and minus the additional moment that we need to crack the section. And we equate that to the modules of rupture of the concrete we're using. Remember that positive signs here mean compression and negative means tension. And we're looking at the bottom fiber, which would be the fiber that would tend to crack if we overload the beam. So notice that we're back to service load checks for now. Okay. So the effective for pre-stressing force, we had found earlier that it was 241. So we plug that in here in PE. We also recall that our strength, concrete strength is 5,000 PSI. So we also plug that in and the rest are just properties of the section that we had determined. 11 is the eccentricity of our strength. That's the moment of inertia. 336 is the cross-sectional area. And the distance to the extreme fiber intention is 14 inches. The quantity we're trying to determine is M cracking, the cracking moment. And that is done by solving this equation to find M cracking. And that would be 384 kip feet when I multiply that quantity times 1.2, I find that to be 461. And my design strength is 597, which exceeds 1.2 times the cracking strength. So the section passes what this used to be called a ductility check. It's now really a minimum reinforcement check. Okay. So this check could be waived if you have a beam that has shear and flexural strength that is at least twice the demand. So in other words, let's say we have a beam for which the maximum moment is 100 kip feet and the design strength is 220 kip feet. Since it is at least twice 100, we don't require to check this 1.2 times M cracking condition. It also does not apply to beams with unbonded tendons. The next series of slides just illustrates what happens when the beam is made composite with some decking that is cast on site. So this applies mostly in bridge construction, but it could also apply to building construction where you have a topping slab on the beam. So in this case, the calculations we did earlier, this applied for a beam that's non-composite. That means that the beam will act alone in service, you know, the way it was fabricated in the plant. In the case of a composite section, we have to consider the effect of this cast-in-place concrete. And this equation is given in PCI 5215, in the PCI design handbook section 5215, where the cracking moment is calculated using this equation, where these are the section moduli defined as the section modulus in general is defined as I, the moment of inertia, divided by the distance to the extreme tension fiber. So this would be for the section modulus for composite section, and this would be the section modulus for the non-composite section. And these quantities here are again for composite and non-composite, and these are the moments acting on the non-composite section. So you can apply this formula and find the cracking moment by plugging in the values in this particular equation. I'm not going to go into discussing how that formula is derived. I think you can do it yourselves if you follow these couple of slides. It's by using the general prestressing equation, and instead of of lumping the cracking moment into a single term here, you define it as the cracking moment minus the moment that's acting on the beam prior to the condition of it being made composite. So the moments that are already in the beam are first added, I should say here, to the general prestressed concrete equation, and then subtracted from the additional moment we need to place on the beam for it to crack. So this whole term corresponds to the composite section, and the rest is for the non-composite section. And if you follow the derivation in the following slide, this is the same equation. We just express, instead of writing it in terms of c and i, c is the distance to the extreme tension fiber. I illustrated it as y, but c is to the extreme tension fiber. If we then use that, then we can replace i over c by the section moduli, s, in each of the individual terms, and the rest is algebra. You just group terms and keep working on it, solve for m cracking, and end up with the equation that was shown earlier. That's the equation that was in the first slide. So it's just a good mental exercise to understand where all the terms come from. All right, so we have been using, so notice that in this equation, the PCI design handbook assumes that the bottom fiber is in tension. All right, so we have been using strains or strain diagrams that are linear across the depth of the section. So I haven't really stopped and asked whether everyone understands why we are able to do that. This figure shows you three strain profiles at different loading conditions. So I wanted to ask, as the second poll of tonight, I wanted to ask if everyone understood why we are able to use a linear strain profile along the height of our section. Okay, I will launch the poll now. Again, do you know why we can draw a straight line through the depth of a beam to represent strain? Please answer the poll on your screen. It is available now. I'll give you a few more seconds. We're still waiting for several people to respond. I'm going to close the poll now. 72% of you said that you are not sure why we are able to use a linear strain profile. 28% yes, very comfortable. 28% no. Okay, thank you, Sherry. For those that answered no, let me try to go over it again, I guess. The main reason for this is you may have heard, if you took strength and materials a long time ago, that we normally, for flexural design, we normally used, of sections under service conditions, we used the assumption that a plain section that was plain before it started bending remains plain after bending. So when we apply a moment, a section that was straight prior to application of that moment simply rotates with respect to that vertical line. And as a result, the top fibers of this bending element are going to shorten, the bottom fibers are going to lengthen, but the important part is that the amount they shorten or they lengthen is proportional to distance, to their distance from the neutral axis. So that proportionality allows us to say, well, a plain section that was straight before bending remains straight after bending, and the strains, or the amount that they either elongate or compress in this particular case, is going to be a linear function, so just proportional to distance. That's the way we can use that. This often also is referred to as a plain section assumption. Okay, so having said that or tried to clarify that, let's look now at the strain compatibility approach very briefly, what it involves. And these figures show, this series of figures show the three conditions we discussed earlier. The condition of pre-stress transfer into the section, the decompression condition of the section, and the condition at ultimate. And notice that the strain in the strand at the centroid of the strand now, not at the extreme tension layer, notice that it's at DP, where we consider the centroid to be located. The total strain is adding the strains for each of these particular conditions. So the total strain in the strand is EP, or epsilon P, plus epsilon P0, plus what we refer to here as epsilon star P, to distinguish it from epsilon T. It's not epsilon T because we're not talking about the extreme layer of steel, but we're talking about the strain in the strand at the centroid of its location. So the strain compatibility approach uses this strain to enter the stress-strain relation of steel. So depending on where we are, let's say we're at failure of the section, we would enter, let's say we were at 0.02 total strain, we would enter a graph like this, and wherever we cut the corresponding graph, in our case, low relaxation, we would try to read the stress in the steel at that condition. So that is what is considered the strain compatibility approach in PCI. And there's a, I should refer you to this example where you can see how it's applied, the procedure, and to this particular design aid that's in the back of your handbooks. Okay, so moving on to checking the regions, the development length regions at the end of the beam. So we have talked about this equation several times now, where the development length of the strand is that length required to develop the stress in the strand at ultimate flexural failure. So the first term corresponds to the transfer length, the second term adds to the transfer length, the length required to reach the development length of the strand. So we need to calculate how much that is, so that we know at what section, our section, at what point along the beam, our section can develop its full flexural strength. So this equation right here is essentially plotted in figure 527 of a PCI design handbook. There are equations that represent this initial linear portion and this post transfer length portion. So if you're at a location along the beam for which you have not reached L sub D, the stress in the strand cannot be, you cannot use FPS as your stress in the strand. You have to use this FPD, so it's called the, I guess, the prestressing stress for partial development. And these equations here allow you to do that for any distance x from the start of the strand. So in our case, the start of the strand would be at the end of the beam. Okay, so I won't delve much into these equations, but these equations will, would let you calculate what the stress is if you haven't reached the full development length. So let's calculate for our beam what the full development length is. So we have the effective prestressing stress, we used this number before, and we have FPS using the approximate method of 244 KSI. And recall that we're using half inch strength. So the full development length is 69.4 inches. And FPS of 244 KSI is developed at 69.2 inches from the end of the beam. Since our beam is 32 feet long end to end, that is 192 inches. So of course, this distance is greater than 69. So at mid span, we, our strand is fully developed. So we don't need to worry about not reaching the full capacity that we require to use the full design flexural strength. Okay, this is really in words what I just said. I'll let you read through that at your convenience. I don't think there's anything, anything particularly important here that we haven't discussed yet. Recall that the critical section here is at mid span, that's important. Our maximum moment sections at mid span. And that's why we use 192 inches to compare that with our development length of 69.4 inches. There are these few sections in the, in ACI 318 that are important for development length. These do not apply to our particular example, but I wanted to show you those in case you encounter them in a future design. Let me see if there's anything that I'd like to comment on. So there's an exception here where there's a, one can invest. So the idea here is that one needs to investigate every section along the length of the beam, including the portions where we're still developing the strength. So you are allowed to use a distance less than LD, as long as the stress that you require does not exceed the stress given by the bilinear curve shown here, right? So again, this does not apply to our beam, none of these conditions. So I won't talk too much about that. One last check is that the fee factor varies also when you're in the transfer length. So you cannot use the tension controlled fee factor within the transfer length. We have to use a much smaller fee factor of 0.75. Beyond the development length, we use 0.9 and we interpolate between 0.75 and 0.9 if we're between those two cross sections to compute the design flexural strength. In our case, since we're using straight strand, we can still check these two reductions. And since it's a simply supported beam, within the transfer length, we're at a very low moment region. So even if we use a 0.75 fee factor for our nominal strength, the beam will satisfy flexural strength. So by checking all the sections, this is a plot of ultimate moment. Notice that it increases parabolically. And this is a plot of how it develops its design strength within the transfer length. Beyond that, into the development length and the full flexural design resistance. So as long as this curve is outside of our moment demand, our section is okay. So this is a full check of the section at different regions with including the end regions of the beam. So we have now satisfied the flexural strength check of our beam with straight strands. All right, so let me just move on here because we are now going to move on to the beam with harped strand. And then we'll have some time at the end of this session for questions. And or if you want to type a question right now, feel free. I keep checking my screen to see if there's any. Okay, so we're now going to see how these equations apply to the condition of our beam with harped strands. This is our design at mid-span. We have replaced the two layers of five strands each. We have moved six strands into upper layers. And that's our configuration at mid-span. We have an eccentricity of 10.2 inches. So that's what it is. And recall that at the end, these three strands here, these upper three strands were moved towards the top two inches from the top. So the first thing we need to do is we need to check whether we can use the approximate method to calculate FPS that is supported by ACI. So for the harped condition, the effective prestressing stress was 161 KSI, which is still greater than 0.5 times 270 of 135. So we can still use the approximate procedure. These calculations are the same as before. Notice now that the reinforcement ratio is calculated now. There's a problem here with this quantity here, and I apologize. We should have used here, please make the correction. We should have used 24.2 to calculate rho sub P of 0.0053. And I think it's, then it would be a little bit lower, but still, you know, close to this number. Let me do, yeah. So it's A sub P is the total area of prestressing strand. 12 inches is the width of the beam, the compression zone, and D sub P is the effective depth. Notice that the effective depth changed because we changed the pattern. So now we have an eccentricity of 10.2 plus 14. So that gives me 24.2, which is what we should plug in here. Okay. So from the previous slide, we have this number, assuming it's correct. I'd have to check it, but I think it is. All these parameters are the same as before. So we have an FPS for the approximate formula of 243 KSI. Notice that this value is not significantly different from the 244 that we computed for the straight strand case. And this is because there's just a minor modification here on the strand pattern. Okay. So now knowing that value of 243, we can again find the depth to the equivalent stress block and the moment capacity given by AP FPS, the effective depth of 24.2 minus A over 2, that gives me 637. Again, we assume a phi factor of 0.9, that gives me 573 for the design strength. And we again have the same ultimate moment. This doesn't change from before. It's 540 for the ultimate moment, which is less than my design moment of 573. So the section is fine at mid span. Now we have a little bit of a challenge here because we do have that pattern at mid span, but remember that the strand was harped, and my pulse is not really good here, but was harped towards the top, leaving two inches from the harped strand from the top. So how do we deal with that for the ultimate moment check? The ACI codes states that FPS should be found using strain compatibility if part of the pre-stress reinforcement is in the compression zone. So the compression zone is going to be defined as the zone above the neutral axis depth, which can be found from A. We know 7.29 is the equivalent stress block. We can find the neutral axis depth by dividing this number by beta 1. But instead of doing that in this example, instead of just trying to find how much of that harped strand is in the compression zone, let's take a simpler approach and say, okay, let me just check what the moment, let's assume that the harped strand, we neglect the harped strand in these regions. Let's assume that we only have it in the straight portion. And then let's calculate what the moment is at that section and that section. So let's calculate then at that section without the harped strand, what the strand pattern would be looking like, and then what the moment demand is at that section and in that section. And if this section can take the ultimate moment, then any contribution from the harped strand as it starts moving up will simply add to what I need, right? But I don't need, as soon as it moves above it, above the neutral axis depth, then I should neglect it, right? So, okay, so we have, if we neglect its presence altogether, we have only seven strands on the tension side neglecting the presence of harped strand. And that's the area of each half-inch strand. So we have a modified area of prestressing strand. We have also a modified area of a modified D. So that's 25.1 for this pattern here without any harped strand. And we have a rho sub p of 0.0036 now using this area and this Dp. And we have an FPS for this fictitious pattern of 252 KSI, a little bit higher than before. And the nominal flexural strength, we've calculated it as before with the new values we just computed. So we have a design flexural strength of 454 assuming the harped strand does not exist beyond this point towards the end of the beam, okay? So let's now calculate what the flexural demand is at that section. So we have a strength of 454. So from zero, say, to 12 feet and from 18 feet to the end of the beam, we have 454. And from 12 feet to 18 feet, we have 573. So that's 573, that's 454 plotted on top of the parabolic distribution or the parabolic, yeah, distribution of moment demand along the span of the beam. And we see that if we neglect the harped strand, we are violating the code in those regions. But we do have harped strand. So let's try to estimate how much it would contribute to the flexural strength so that we can avoid using the strength compatibility approach in our design calculations. Okay, so that's what this slide indicates, that we would be violating the flexural strength at 9 feet and 21 feet. But we are ignoring the harped strand presence. So let's try to include the harped strand at 9 feet now. So at 9 feet, notice that what I'm doing here is, let's see where we start violating code. We start violating code at a section corresponding to 9 feet and 21 feet from the end of the beam, okay, or from the start of the beam, I should say. So let's see how much the harped strand would contribute at that section. And if it contributes enough to bring this curve up beyond what my demand is, then the section will be okay at other points behind that. So that's what we're going to do. We're trying to estimate now at 9 feet what the contribution of the harped strand is. And fortunately, the harped strand at 9 feet is still located on the tension side of the beam, which is considered at mid-height here, okay, so just for simplicity. So we need to find first where it is located. How do we find that it's 6.75 above the second layer of strand? Well, you might recall that last time we estimated this angle as 7.3 degrees. So what we do is we start counting from this point onward how much the strand has gone down. It starts 2 inches from the top. And remember that the distance between the centerline of the support and the end of the beam is 1 foot. So that's why we use 10 feet. We use the sine of 7.3 degrees to see how much that strand has dropped. And this is simply a conversion from feet to inches. So from this point, 10 feet into the beam, the strand has dropped, the harped strand has dropped 15.25 inches. So from 28 inches, we subtract those 15.25 inches, and we also subtract 2 inches from the top, and we subtract these two 2-inch regions, or pardon me, these two 2-inch spacings of bottom strand to give me the distance between the top strand here and the location of the strand that I'm considering. So that's what gives us 6.75 inches. The eccentricity of this strand pattern is calculated as shown here. So that gives me 8.28 inches from mid-height of the section. And d sub p is again measured from the top to the centroid of the strand, which is to this newly eccentricity. So that's 14 inches plus 8.78. That gives me 22.78. All right. So having dealt with the geometry, we now have 10 strands again. That's why we use the total area. We have a depth to effective depth of a prestressing strand of 22.78. And we calculate again the parameters, rho sub p and fps for this section now. We've done this calculation before several times, so I don't think we need to repeat the details of it. So we now have fps of 241 ksi for this section with this strand pattern. We calculate the equivalent stress block depth, 7.23 again, for now this pattern. And we use ap fps, the effective depth for this strand pattern, minus a over 2. That gives me 583 kip feet. 0.9 times that gives me 525. Recall that at that section, if we go back, we originally had estimated 454 without considering the harp strand. Now by considering the harp strand at that section, we're at 525. So if we update our plot at that section, we're at 525. At mid span section, we're at 573. And that now overlaps the moment demand. And so our section should be okay. So one correction I should have made here that I didn't make is that the capacity at the end of the section is not built up suddenly from one foot into the beam into this full flexural strength. Remember that we had a slide that indicated how that was stepped up through the transfer length and beyond the development length. So that steps up from here to about five feet into the beam, something like that. That's what this plot should look like. All right. And then we have the stepped function for the rest of the beam. Okay. So we have again ended quite early. We've gone quite quickly through these slides. So I will entertain any questions you might have on either the straight strand part of this presentation or the harp strand part of this presentation. We have about 20 minutes left in tonight's lecture. As always, if you have any questions, please put them in the chat box and I will pass them on to Professor Brynja and the entire audience so that you can read your fellow attendees' questions. The chat box is open now. And we'll give everyone a minute to get their questions in. Sergio, I haven't gotten any questions yet. All right. That must have been very clear tonight. Did everyone understand what I said about this stepped function not to go to the full capacity at a section at the center line of the support? If you understood and don't want to type into the chat box, you can just raise your hand. Use the raise your hand function. Oh, I see several raising of your hands coming up. Okay. That's great. This is something I need to update on this slide. I will do that as well. For those of you, if you don't see the raised hand, it will be in your GoToTraining pane. And if you can't see the raised hand, just tell me, yes, in the chat box. We're getting a lot of raised hands. Okay. Sergio. So, everyone must have understood tonight's session. Great. If for any reason, we are finished a little early tonight, which is fine. It happens. But if for any reason you do think of a question later tonight or tomorrow or before next week's session, please email them to me. I will then pass them on to Sergio, and we will either get the answers to you or he will answer them. So, if you have any questions, feel free to or he will answer them before we start next week's session. Again, thank you so much for participating in the fifth session. And we look forward to seeing you next week.

pre-stressed concrete design

ultimate moment calculation

rectangular beam

straight strands

harp strands

flexural strength

tension control

compression control

transition sections

stress-strain behavior

ACI 318 code