Diaphragms: 3. Rigid Diaphragms 1

So we talked about flexible diaphragms, and now I'd like to get into rigid diaphragms. And here we had an example earlier of a flexible diaphragm made of regular plywood or OSB on top of joists. So if we think of cross-laminated timber, cross-laminated timber is in fact a two-way system. The two bytes are running in two perpendicular directions, which makes it a little bit more rigid. But between the pieces, it's still flexible when you think about it. Within the individual panel, the CLT panel, it is rigid. But then between panels, it's made of pieces. So what they do is they will add some acoustic insulation between floors. to deaden some of the vibration. And then they will pour a three-inch topping of concrete reinforced. Therefore, this just became more of a rigid diaphragm. The same with metal decking. We saw earlier, we have a metal deck with insulation and some roofing membrane. That's flexible. As soon as there is reinforced concrete in there, it became a rigid diaphragm. So floors in general are going to get concrete topping inside of the metal deck, which makes it a rigid diaphragm. But the roof itself is going to get, on top of, for example, open web joists, you're going to get a metal deck insulation, et cetera. So that's a flexible diaphragm on the roof versus rigid on the floor. And then precast concrete is flexible, but anything cast in place, post-tensioned, anything like that is going to be a rigid diaphragm. Very good. So moving on, let's see what my next slide is. Yes. So we talked about flexible diaphragms. We said they are made of pieces. Well, in the case of rigid diaphragms, it has to be monolithic. Examples were an un-topped metal deck versus a topped metal deck topped with concrete, clearly. Plywood OSB were flexible, but if you put a topping on top of CLT panels, then that is considered a rigid diaphragm. Precast concrete is flexible. anything cast in place is going to be rigid. So in a flexible diaphragm, the collectors are stronger than the diaphragm itself. The diaphragm itself bends versus in a rigid diaphragm, the diaphragm is stronger than the collectors. So the cords will bend and deflect in a flexible diaphragm. But then in a rigid diaphragm, the collectors will be the ones to bend and to deflect or drift. The corners of the diaphragm in plane will distort from 90 degrees versus in a rigid diaphragm, the corners of the collectors are going to be the ones that distort. So aspect ratio for a flexible diaphragm could be less than three times the width or equal to three times the width or greater than three times the width. But in a rigid diaphragm, you must have the aspect ratio be less than 3 to 1. If it's more than 3 to 1, that's it. It's flexible, as we said earlier. There is no torsion in a flexible diaphragm. the diaphragm is much weaker than the collector, so it cannot twist them in plain. But in a rigid diaphragm, the code requires that torsion be taken into consideration. Now, if the collectors are symmetrical, then there is theoretically no torsion, but the code requires a 5% accidental torsion, and we'll discuss that in the next slide. But if the collectors are not symmetrical, one side is stronger than the other, then there will be definitely torsion. One more item, important item. In a flexible diaphragm, the load distribution, as we saw with flexible diaphragms, is based on tributary load. So depending on the placement and spacing of collectors, the load is distributed according to that spacing. Well, in a rigid diaphragm, we need to take into account the rigidity of the collector. Is it a shear wall? Is it a moment frame, eccentrically braced frame, concentrically braced frame? What is its rigidity? Because the load is distributed based on the rigidity of the collector. Inflexible, the rigidity of the collector doesn't matter. It's mostly the spacing of the collector. But in a rigid diaphragm, we need to know the rigidity of each collector. Very good. So, oh, we have to look at animations. That is a reminder for me to look at animations. So let's look at some animations. I would like to start with no torsion. So I'll go here. And let's take a look at some collectors without torsion. So let's start this animation. So in a moment frame with a rigid diaphragm on top, it tends to distort the vertical collectors. So they're no longer 90 degrees. But if the two legs or the two collectors have equal rigidity, then each one is going to take half the load. And there's something called the center of rigidity. And the center of rigidity is the center of resistance, basically. Looking at the second one, we have two brace frames. The center of mass is usually the center of the diaphragm. But the center of rigidity is based on the rigidities of the collectors. So as long as the collectors are symmetrical, the diaphragm just pushes the collectors, and you have distortion of the plane of the collectors. But the diaphragm does not deform as much as the verticals. And if they are equal rigidities, each gets half of the load in shear. So looking at the next one, which is shear walls, so the load hits the diaphragm. The diaphragm is much stronger than the verticals, so they distort. And if there are equal rigidities, well, then each collector will get half the load. In all three cases, the upper left is moment frame, the one next to it is brace frame, and the bottom one is shear wall. In each case, if the legs are equal rigidities, then there is no torsion theoretically, but we have to account for accidental torsion, as we will see in the next slide. Okay, when we have two bays, upper left is moment frame, then brace frame, then shear wall, as long as the collectors are equal rigidities, then the diaphragm moves as a monolith, and it distorts the collectors underneath it. But if their rigidities are equal, then each one gets, in this case, a third of the load. In the next one, the brace frame, we see, sorry, upper right. Let's focus on the upper right one. I must have animated these not in the correct sequence. So looking at the brace frame, the diaphragm is a monolith and it pushes back and it distorts the angle of the brace frames. But whatever it is, if those rigidities are equal, then each one gets a third of the load. The same with the shear wall animation on the bottom. It has a fat wall in the middle and two thinner walls on either end. The rigidities are symmetrical. So there is no torsion, and theoretically no torsion, but we have to account for accidental torsion. In this case, with the middle wall thicker than the other two walls, its rigidity is more. Therefore, it will attract more load than the two end collectors. It'll attract load based on its rigidity. Very good. So if, however, we do have torsion, Then let's take a look at these animations. In the upper animation, it looks like this one. Let me start it. It looks like the right-hand side is twice braced versus the left-hand side is braced one time. So now all of a sudden the center of rigidity or the center of resistance is no longer in the middle and the right side is stronger than the left side. Therefore, the center of rigidity closer to the right. And in this case, we do have torsion. And in this example, it's clockwise because the right-hand side is stronger than the left-hand side. Therefore, the center of rigidity moved to the right. And we have a couple. The red arrow is equal to the green arrow, but they're not head-to-head. Therefore, there's some torsion. Looking at the second animation, It's made of two unequal length walls. Therefore, the center of rigidity is closer to the longer wall, and we will have torsion. So if we're attacked by 80 kips, then the longer wall is going to resist more load, and the center of rigidity and the center of mass do not coincide. Therefore, there's going to be this counterclockwise rotation. And the attachment of the diaphragm to the walls is the most critical thing, and it's based on the shorter wall. We need to make sure that the diaphragm stays on the wall. Looking at this next animation, one side is a shear wall, the other side is a brace frame. Clearly, there are different rigidities. So the more rigid wall, the center of rigidity will move closer to the more rigid wall, and there will be torsion in this case, and the load is distributed based on the rigidities of the members. And it looks like there's a counterclockwise rotation in this example. Now, looking at these animations for two bays and comparing the upper left, which we have seen already, the upper left has a more rigid wall in the middle, and once the load comes in, it is symmetrical. So the center wall, yes, attracts more shear, but the placement is symmetrical. Therefore, there is no twisting or torsion, theoretically. Versus moving that fat wall off to the side in animation two, we see that the center of rigidity is no longer in the middle. It's closer to the fat wall, and therefore there will be torsion. In this case, counterclockwise. It doesn't matter if it's clockwise or counter. It's just for visualization. Now, looking at the bottom animation, it has three equal rigidity walls. Therefore, each one will receive a third of the load, but the center of rigidity is not in the middle just because of the placement of the walls. So the center of mass is always in the middle, but the center of rigidity is closer to the pair. And therefore, we're going to have torsion in this case clockwise. Very good. So coming back to our animations, sorry, coming back to our slides, I'm trying to keep, I'm trying to keep these down to a limited amount of time. I'm sure I've exceeded it this time. Okay, so back to our previous example, there is a wind load of 60 kips, and the center of mass is basically, oops, the geometric center. Sorry about that. It looks like I went over, sorry. This is flexible diaphragms. So back here. So the center of mass is basically the intersection of the two diagonals. Once those two diagonals intersect, you've located the center of mass. And the center of rigidity, on the other hand, depends on the rigidities of the walls. So in this example, I have two equal rigidity collectors, each equal to one. Therefore, what we do with rigid diaphragms is we add the rigidities. So 1 plus 1 equal 2. Therefore, I'm going to receive one part of the 2. You're going to receive another one part of the 2. So if we're being attacked by 60 kips, then I'm going to resist 30. Oops, I'm going to resist 30 kips or half of 60. You're going to resist 30 kips, another half of 60. Now, the center of rigidity is on top of the center of mass, therefore there is no torsion, theoretically. But the code requires that we incorporate a 5% eccentricity, so that the code says move your center of rigidity 5% to each side, 5% of the dimension, 5% of 48 feet is 2.4 feet. So assume you have an eccentricity of 2.4 feet. Find the moment or the torsional moment and make sure you can handle it. It's a simple matter. The code says if you have a rigid diaphragm, you cannot design it for shear only. You cannot say 60 kips goes 30 and 30 only. you have to add some provision for twisting or torsion. And the code also says you need to also incorporate 5% in the other direction. So whatever this dimension is, let's say it's 24 feet, you need to account for eccentricity in the y direction as well. Very good. So let's figure out how to calculate the center of rigidity. I'd like to say that we add the rigidities. So I have two collectors, and the one on the left is a rigidity of 1. These numbers have to be given. They're based on the construction of the collector. So the total rigidities is 3. Therefore, the left collector will receive one part of 3. And we're trying to resist, for example, 60 Gibbs. So one third of 60 kips, and that turns out to be 20 kips versus two thirds. I am twice as rigid. The one on the right is twice as rigid times 60 kips or 40 kips. So at the end of the day, I'm resisting 40 kips. You're resisting 20 kips. But where is the green center? The green center is going to be located closer to the more rigid wall. So actually, it's somewhere here. Oops, it went to the background. Sorry about that. Undo. I will not do that. Instead, I'm going to put my X here. And I'm going to say that this must be, it's closer to the right-hand side. It must be in the proportion of rigidities. So if you have 48 feet, it's going to be one-third of 48 and two-thirds of 48 feet. So it turns out 16 and 32 feet is the location of the green center. That means I am 16 feet, you are 32 feet. That's where the center of rigidity is located, which allows me to calculate. Sorry, there's too much math here. I get it. But this dimension is 32 minus 24. So that's 8 feet. Now I can figure out my torsional moment. There is a green 60 here, 40 plus 20. And there is a red 60 here. And they're separated by 8 feet. So the torsional moment is force times distance. There's a couple of 60 kippers separated by 8 feet. So 60 kips times 8 feet. So there is a twist here of 480 kip foot of torsion. Let's look at the second example. The second example says rigidity of 3 on the left collector and rigidity of 1. It's a shorter wall like the animation we just saw. So the sum of rigidities is 3 plus 1 equal 4. well then the left will get three parts of the four the one on the right will get one part of the four based on their rigidities and we're being attacked by 60 kips so I will get three quarters of 60 while you get a quarter of 60 is 15 and three quarters oops I wanted those to be green is 45 kips. So I am resisting 15 kips. You are resisting 45 kips. And the center of rigidity moved to over here. Where is over here? It is again one quarter and three quarter, one quarter of 48 feet and three quarters of 48 feet. One quarter of 48 is 12, and this one is three quarters or 36 feet. So that's telling me that the center of rigidity is 12 feet from the left or 36 feet from the right. One more example very quickly. Rigidities of 1 and 5. Their sum is 1 plus 5 equals 6. And therefore, the left receives a rigidity of one part of 6. The one on the right receives five times as much. And we're being attacked by 60 kips. And the outcome is basically I will take 10 kips and you will take 50. So the right is getting 50 kips and the left is getting 10 kips of shear from the diaphragm to the top of the collector. But the green center is way over here this time, it is located at 1 6th and 5 6th of 48 feet. So how much is the math? 8 and 42 and 40, sorry. So this is 40 feet and this is 8 feet. And now I can calculate that moment arm so I can figure out how much torsion. And in this case, there's a lot of torsion. Very good. Let's take a break here and come back to this one.