Moment Frames: 2c Mf Math

So let's look at a moment frame instead, and let's look only at a lateral load. And let's remember the difference between a brace frame and a moment frame is the way the horizontal is attached to the verticals. So looking at this moment frame that is pinned at the base versus the next one that is fixed at the base, I recognize that this one has four unknown reactions, two horizontals in green and two red vertical reactions. And that makes this problem indeterminate. There's four unknowns and there's three equations of equilibrium, which means we have one extra unknown. So if I can make one assumption plus three equations, then I will be able to solve this problem. So my assumption will be the following. If you have three kips and your legs are equal stiffness, then the left horizontal and the right horizontal should be equal. Makes sense. It's an assumption, but it makes sense. So half of 3 is going here, and the other half of 3 is going here. So that does it for the horizontal reactions, or the base shear, as NCARB calls it, due to the 3-kip load. Now let me play a little bit devil's advocate, and let me say that if this leg is twice as stiff, twice as rigid, So it has a rigidity of 2, and this one has a rigidity of 1. I add the rigidities, 2 plus 1 equals 3, and so you will take 2 thirds the load, you will take 1 third the load. 2 thirds of 3 kips, and 1 third of 3 kips. So we end up with 2 kips on this side, and 1 kip on this side. That's the assumption. If they're equal rigidities, they partake equally in sharing the load and resisting the load. Each gets 1.5. But if the left column is twice as rigid as the right column, it ends up taking twice as much 2 kips versus 1 kip. Very good. So we're done with that. I want to go back to a regular equal stiffness. And let's assume that horizontal reactions are equal. Each is 1.5. Very good. Now, I still have an overturning moment to deal with. And the overturning moment causes those vertical reactions not to be zero. And as we said in a previous video, the frame is going to do that. It's going to push at point B, and it's going to lift at point A. So there is a tie down at A, and that tie down is that much. And there is a pushback at B, which is the pivot. And the overturning moment is equal to 3 kips at a distance from the pivot of 10 feet. So there is an overturning moment of 30 kip times foot. And that overturning moment is clockwise. And the role of these two reactions at A and B is to be equal and opposite and to make a couple that resist the 30 kip foot. So if I sum moments about point B, which is the pivot, there is a 30 kip foot of clockwise, which means that this guy has to be 30 kip foot to fight back. Now, how much is that tie-down reaction if this dimension is 15 feet? I have a force A at 15 feet to be equal to 30 kip foot in the opposite direction, which gives me that A is 2 kips. So A is 2 kips downward, a tie down of 2 kips, which means sum of vertical forces equals 0. Well, then the other one is also 2 kips. Now, you have a pair of 2 kippers. Here's what they're doing. I'm running out of room. There's two kips, sorry, undo. So there is a two kip upward, and there's a two kip downward. And they are separated by 15 feet. And they will generate a counterclockwise moment of one of the forces times the distance in between. So they come up with 30 kip foot of counterclockwise, which resists the 30 kip foot force times distance, the external moment of 3 kips at a height of 10 feet. Very good. So let me clean up a little bit, and let me say... Oops, yeah, take that out, and take all of this out. Delete. Of course, the whole thing went... Okay, can't do it. So let me erase then, I apologize. So, at the end of the day, it looks like my reactions are as follows. This one turned out to be 1.5, and this one turned out to be 1.5 or half of 3. The tie down here was 2 kips, and the pushback here was 2 kips. And that's it for this problem. It was indeterminate to the first degree, but I was able to resolve quite a bit by assuming that the two legs were equal and they shared equally in resisting the 3-kip load. Very good. Let's go to the next one, which is the same 3-kips, 15-foot and 10-foot and a moment frame, but this time the base is fixed. With a fixed base, you have three reactions at A and three reactions at B. and that is highly indeterminate. So in this situation, I have six unknown reactions, two horizontal, two vertical, and two moment reactions, and I only have three equations. So this is where you need higher math, and it will never be on the ARE. But we should be able to guess the directions of the reactions. So let me go into that a little bit. it's indeterminate. But it's still safe to assume that the 3 kip is going to two legs. Each is going to take 1.5. I'm pretty sure about that. Now, the overturning moment is still 3 times 10, and it's going that way. 3 kips at a height of 10 feet equal 30 kip foot. Very good. The pivot is still at B. If the pivot is at B and the overturning moment is clockwise, well then these guys need to be counterclockwise to fight back. So that's the arrowhead on the moment reactions. And the vertical reactions are going to do pretty much what they did in the earlier picture, down on the left, up on the right. It's just I don't know how much the value is because this moment plus this moment plus the couple should equal to 30 kib foot. How much is each? I have no idea. It's indeterminate. I need a computer or an engineer. Okay, leave this alone. They'll do higher math, the portal method or moment distribution or whatever. That's not on the ARE. All that is on the PE. But please keep in mind the arrowheads. Now, we can go to multi-story and the same logic would apply. 1 plus 2 plus 3 is 6 kips going to the right. Well, then I need to be half of 6 kips, and I need to be the other half. If the legs are equal, if they're not equal, you do that rigidity thing. You add the rigidities and you proportion each leg accordingly. Also, this 1 plus 2 plus 3 is making a moment. If these heights are 10 feet each, then there is a pivot at B, 1 kip times 10 feet. Let me start writing. 1 kip times 10 feet, about point B, plus 2 kips is at a height of 20 feet. That's the distance from here to here is 20 feet, from here to here is 10, and this one is 30. And 3 kips times 30 feet, which gives me 10 plus 40 plus 90. 10 and 40, yes, 140. 140 kip foot. You divide by, so the overturning moment is 140 kip foot, we cannot divide by 15, as I was about to say, because there's two moment reactions. And they take part of the 140, and this is indeterminate. But please recognize the three kips and the three kips, half of six. Now, when it goes to two bays, it becomes much harder. But the essence of this is you look at how many lateral bays you have. This is a lateral bay. This is a lateral bay. And you divide your 1 plus 2 plus 3 equals 6 kips. You divide it by the number of bays. So each bay here is going to take half of 6 kips. So if you look at the first 3 kips, it goes down at A. And the other half of 3 kips goes down at B. In other words, 1 1⁄2, 1 1⁄2. When we look at the second bay, it has two legs, B and C. This one takes 1 1⁄2, and this one takes 1 1⁄2, or half of 3, which is half of the original 6. So, bottom line here is this is X, this is 2X, this is X. The column line in the middle is shared by the two bays, so it's going to do double duty, X to X, X. and the rest of the reactions are the attack is clockwise, the overturning moment is clockwise. Well, then all the moment reactions are counterclockwise. And the pivot is at C, which means I need to push back and I need to tie down. And the guy in the middle cannot help with rotation, but it'll do 3 kips or 2x of shear. When you go to three bays, well, I don't care, it's not much different. It will be x, 2x, 2x, and x. So you take your six kips and you divide by how much is that? By 6x, and you find that this guy resists 1 over 6. And this one will resist two of them, 1 and 1, or two kips. This one will resist two kips, and this one will resist one kip. And the reactions are pretty similar. This one, sorry, the attack is clockwise. The overturning moment, the pivot, is D. And therefore, all the moment reactions are counterclockwise because the attack is clockwise. And where's the center of these bays? Somewhere here.