Diaphragms: 2. Flexible Diaphragms

So summarizing what we saw in the introductory video on diaphragms, some terminology, collectors are basically your vertical lateral resisting system. So if you have partition walls, of course, those don't count as collectors, but a brace frame, a moment frame, a shear wall, those will act as collectors. They'll take the lateral load from the diaphragm and transfer it down to the foundation. We said in a flexible diaphragm, there is a deflection in the plane of the diaphragm, and therefore, wherever the load hits, there is a compression cord, and the backside is extending, and therefore, it is called a tension cord. the drag strut takes the load from the diaphragm and spreads it to vertical members. Now most importantly is that we go over the definition of flexible versus rigid diaphragm. So if the diaphragm itself experiences a change in the angles and a deflection then if that deflection in the diaphragm is more than twice the drift in the collectors then the diaphragm is weaker than the verticals and that's considered the flexible diaphragm versus if there is very little deflection in the diaphragm and there is more deflection or drift in the vertical lateral resisting system or the collectors, then that is considered a rigid diaphragm if the deflection of the vertical system is more than twice that of the horizontal system. Very good. So saying this again is just so important. The diaphragm deflection is more than twice the drift of the collectors. That's flexible. Versus if the collector or the vertical lateral resisting system deflects more than the diaphragm, then that is considered a rigid diaphragm. Here are some examples we saw earlier. OSP on top of joists or an un-topped metal deck with insulation. Those are flexible. Versus as soon as you put concrete in that metal deck for a floor framing, for example, A roof framing doesn't get concrete typically. So if there is cast-in-place concrete, then it's considered a rigid diaphragm. So those are some more pictures. We saw them earlier. Anything precast concrete is a bunch of pieces, and therefore it's considered a flexible diaphragm. And in steel, an un-topped deck is considered flexible. And pretty much anything in wood framing is considered a flexible diaphragm. So recapping, just in summary, it looks like if it's made of pieces, then it's a flexible diaphragm. And as soon as there is cast in place concrete or something that makes it monolithic, then it's a rigid diaphragm. Also, a flexible diaphragm, the horizontal component, is weaker than the vertical components. Therefore, it cannot twist them. It's too weak to twist the verticals. That's called torsion. A rigid diaphragm, on the other hand, is stronger than the legs it sits on. Therefore, it can incorporate torsion. And that's something I will get into when I get to rigid diaphragms. Very good. We need to talk about something called the aspect ratio of the diaphragm. Aspect ratio. So if the dimensions of the diaphragm are such that one leg is more than three times larger than the other leg, then that's it. it's a flexible diaphragm, period, even if it has cast-in-place concrete. It is too long and skinny to behave as a rigid diaphragm. So if the aspect ratio is greater, if the length is greater than the width by three times, then it is considered flexible immediately. To be rigid, then it must be less than 3W, three times the width. Then it is considered rigid, but it has to have cast in place concrete. So please, something important here. If it's less than, if the length is less than three times the width and it's cast in place concrete, then it will be rigid. If it's less than three times the width and it does not have cast in place concrete, then it's flexible. If it's more than three times the width, then it is a flexible diaphragm, whether or not it has cast in place concrete. Very good. The next thing to talk about, so let me put some dimensions on this diaphragm here. If this is 16 feet, for example, then 3 times 16 is 48 feet. Anything less than 48 feet could be a rigid diaphragm if it has cast-in-place concrete. But if it's more than 48 feet compared to the 16, then it doesn't matter if there's cast-in-place concrete or not. That one is considered flexible. Very good. So the next thing to talk about is with flexible diaphragms, the distribution of load from the horizontal to the collectors. Diaphragm has a G in there. Okay. The flexible diaphragm works by tributary load distribution. So, what's going to happen with this flexible diaphragm is if it has two collectors, then each is going to get 50%, each is going to get one half. So, assuming that this, for example, is 1.25 kip per foot, please, if you're math phobic, don't worry. We're going to start easy on the math. Whenever it gets nasty, I'll give you a warning. So, this is little w. And big W will be 1.25 kip per foot times 48 feet or 60 kips. So if this is 60 kips, because every linear foot of this diaphragm is getting 1.25 kips, 48 feet later that adds up to 60 kips. Well, flexible diaphragms will spread the load based on tributary load. So this half goes to the left collector. This half goes to the right collector. We take the 60 kips and we end up with 30 kips or half of 60 on each collector. Very good. So let's look a little bit more in detail. Please don't panic. This is a simple page. The next page is a little bit uglier. But right now, if we have 60 kips, the point of this sheet is to illustrate that no matter the rigidity of the collectors, no matter what they're made of, the load distribution depends on tributary analysis, not on the rigidity. So if you have 60 kips, each collector is going to get 30. Whether it's braced frame or not braced, if it's twice braced on this side, once braced on this side, it doesn't matter. You're getting 30 kips each because tributary load distribution controls in a flexible diaphragm. So if you have a shear wall made of masonry versus a brace, the rigidities are totally different. But with a flexible diaphragm, it does not matter. You take half. I take the other half. With rigid diaphragms, we will do the same problems, but then it's going to depend on the rigidity. So in this case, all of these will have 30 kips. As long as there's two collectors, it's 60 divided by 2, whether it's a short wall, a long wall, a thick wall versus a thin wall. It doesn't matter. Each of you gets half of 60. Very good. So flexible diaphragms distribute the load based on tributary analysis. Very good. Let's go to two bays. Now when you put two bays together, you will notice this one will get X. This one will get X and X, or 2X. And this one will get X based, again, on tributary load distribution. So if we have 1.25 kip per foot. Oops, I should be in red. 1.25 kip per foot. And if this is 48 feet, like the previous example, this is 48 feet and this is 16 feet. Well, then this time we're going to have 120 kips. Or 60 and 60. 60 on this side and 60 on this side. So tributary load distribution says I will take 30. Oops. I will take. What am I doing wrong? Sorry. I am taking 30 kips. What is going on here? Oh, sorry. Okay, I get it. So I'll take 30 kips. I'll take 30 kips. I'll take 30 kips. And you'll take 30 kips. So the first 60 got divided into two 30s. The second 60 got divided into two 30s. So bottom line is I got 30 kips on this collector. I got two 30s on the middle one or 60 kips. And the end one got 30 kips as well. So now if we move the intermediate collector and we put it at 24 feet, this used to be 96 feet, 48 plus 48. Now, if we put this collector at 24 feet and the next collector at 72 feet, then we're going to do tributary load, basically. We're going to split the 24 feet, and we're going to split the 72 feet, and the guy in the middle always carries more. If there is a thick wall in the middle, it doesn't matter. You're getting 30, you're getting 30 and 30, and you're getting 30. But if we move the wall, these two cases are based on tributary load. So in the second case, if I put this at 16 feet versus 80 feet, well then, each collector is going to get a load based on tributary analysis. And I would rather do this frankly in plan. So I have this drawing here, which is the same as the previous one. It's just if this diaphragm was 96 feet and its center is at 4848. And I said the first. So the collectors in this example, in the first example, the collectors are at 24 feet, 24 feet, and 72 feet. This is 96 feet, just like the initial dimension we had. So now we're going to take half of 24 and call it 12 feet. And we're going to take half of 72 and we're going to call it 36 feet. And the guy in the middle is doing 12 plus 36, or 48 feet for the guy in the middle. So, collector 1, collector 2, and collector 3. Collector 1 gets 12 feet of the 96, and Collector 2 gets 12 plus 36, or 48 feet of the total 96. And Collector 3 is going to get half of 72, or 36 feet, divided by 96 feet. Please, the math is not difficult yet. We're just doing tributary load. And we're trying to resist 120 kips. Is that right? That's what the attack was, 60 plus 60, or 120 kips. Capital W from the previous page. So this one will be times, we have to provide, the three collectors have to provide a resistance of 120 kips. So it is distributed in the proportion of tributary load. I'm sorry, I didn't do the math ahead of time, so I'll have to do it live. Sorry for taking the time. 1 eighth of 120 is 15. Let me erase a little bit. This is getting messy. Okay. 12 over 96 times 120 is 15 kips. as I said earlier, 15 kips. 36 over 96 turns out to be 45 kips. And the guy in the middle is going to do 16 kips. So that's the share of each one of these collectors. Let me just check my math, 36 divided by 96 times, that's 3 eighths times 120 should be good. So 45 kips. So the total, of course, is 120 kips. That resists the red 120 kips. So now I would like to do the second example, which says, okay, now if your collectors, if the middle collector is moved, And we place it at 16 feet from the right, which gives me 80. From the left, for a total of 96. I'm just moving that middle collector. Initially, we started on the previous page with that collector in the middle. But now we're moving it just to see the impact and how much load each collector gets. One, two, and three. Collector 1 is at 80 feet, therefore it's going to get 40 feet. And Collector 2 is 16 feet from the right from Collector 3, therefore it's going to get half of 16 feet from the right. and the guy in the middle is going to get 40 plus 8 or 48 feet. Please notice that the 48 feet tributary width on collector 2 is not changing. No matter where you put it, it's going to get half the load. What is changing is collectors 1 and 3, they get different amount based on the spacing, but collector 2 is always carrying half the load. So, collector 1 gets 40 feet of the total 96. Collector 2 gets 48 feet of the 96. And collector 3 gets a total of 8 out of 96 times 120 kips. So this turns out to be 1, 12, 10 kips. This turns out to be 50 kips. And this guy carries half the load. So please, again, the middle collector is doing the same amount of work regardless of where we place the middle collector. So in summary, this is important to understand that the load distribution from horizontal to vertical depends on tributary spacing of the collectors. Okay, let's see what the next page is because... Oops. I wanted to do chord force. Okay, that's good enough. So calculating the chord force in the flexible diaphragm is the segment that has a little bit of math. Not too much, so don't panic. This is not high math yet. So the 60 kip load on this flexible diaphragm induces a compression cord and a tension cord, and they form a couple. And the couple is separated between the compression zone and the tension zone, is separated by 16 feet between that red arrow, the compression cord, and the tension cord. So they make a couple. And if we think of the diaphragm, which is that whole horizontal piece that is 16 feet by 48 feet, really it looks like a regular beam that is spanning 48 feet between two collectors, Maybe I should color these green to symbolize the two walls. And it is subjected to a uniform load of 1.25 kip per foot or 60 kips total. And that gave me my reactions of 30 and 30. Now, in addition, this was separated by 48 feet. So, in addition to the shear of 30 kips to each collector, the diaphragm is bending and deflecting. And therefore, there is a bending moment on this diaphragm. And this bending moment is equal to, come on, change, okay. The bending moment of a uniform load is WL over 8. W is 60 kips. The span of the diaphragm between the collectors is 48 feet, and that's the thing that is subjected to bending and deflection, the whole diaphragm. And divided by 8, this is the formula for a uniform load to get the maximum moment it's subjected to. It's a standard load. You need to know this one. So that turns out to be 360 kip foot of bending moment on the diaphragm that is to be resisted by a couple that is formed between the compression cord and the tension cord. So actually, the compression should equal to the tension. Otherwise, the diaphragm fails. And in fact, the amount of tension in the tension cord times 16 feet should equal to the amount of compression in the compression side of the diaphragm times 16 feet. And we need to resist 360 kip times foot. That's the bending moment on the diaphragm that is to be resisted by the couple between compression and tension. separated by 16 feet, which tells me that the amount of tension should equal to the amount of compression, and that should equal to 360 kip times foot, divided by 16 feet separating the two components of this internal couple in the diaphragm. So it turns out we need to design this diaphragm so that its compression cord and its tension cord can handle 22.5 or 360 divided by 16. So it should be able to handle 22.5 kips. Okay, so that's the cord force for a flexible diaphragm. a little bit of math, but I think the important thing is you will probably definitely, if you're taking the ARE, you must know the uniform load moment when a beam is subjected to a uniform load on a span. Very good. Продолжение следует...