Braced Frames: 3c Bf Math

Let's look at braced frames, and just like we did with the moment frame, I would like to discuss these examples. And the four questions apply. Calculate the base shear at A and B. Identify the pivot. calculate the overturning moment, and calculate the tie-down force at the pivot. Okay, not at the pivot, but opposite, wherever it is, identify it is the point. Excellent. So starting with the first one and recognizing that unlike a moment frame, this time I don't have a rigid connection, I have a pin connection between the horizontal and the vertical. So the 3 kip load pushes on this member and sends it into compression. So this member is in compression. Now the 3 kip load is over here, and it pulls on the member AD, sending it into tension. This one is in tension. And where's the load? It went from C to D back to A. So now the reaction at A is 3 kips. And there is no diagonal at B. So that's part of the post and beam. It doesn't do lateral. There is no diagonal. So the whole 3 kips went at A, none at B. Now the pivot is down here at B, because the frame wants to do that. Push down on B, lift at A. So what's going to happen here is there will be an overturning moment of 3 times 10 clockwise or 30 kip foot. And there is a tie down at A. There is a push back at B. Together they make a couple. So, summing moments about B, the reaction at B passes through B, it makes no moment. There is an overturning moment due to the 3 kip at 10 feet, which is 30 kip foot. And it has to be resisted by A, making that kind of moment equal to 30 kip foot. So, if this is 15 feet, A times 15 should give me 30 kip foot of counterclockwise, which means A is 30 kip foot divided by 15 foot, which is 2 kips. So A is a 2 kip tie down, and B is a 2 kip push back. And the two together, A and B, worth 2 kips each, will make a couple of 2 kips times 15 feet, 30 kip foot, counterclockwise, to battle the external force times distance moment of 3 kip at a height of 10 feet, which is a 30 kip foot clockwise. Very good. Let's go down to the next one. Let me erase a little bit so that we can work on this one. Delete. Very good. So looking at the next one, 3 kip on a cable. Cable does not do compression. This is a post and beam that has pretty much racked. You have no ability to do compression on a cable. So let's promote this to become a strut. A strut is a member that can handle compression or a diagonal brace that has a little bit more area than a cable or a tie rod. The more symmetrical that strut is, the better it is for taking the lateral load or the axial load because the load is going to travel along the axis of the member. Unlike with the beam, it's perpendicular to the axis. Now, it's along the axis. So, now what happens? Okay. I wanted to talk about this on that previous example. I didn't. But, this reaction turned out to be downward, which means this is a tension member. And this reaction turned out to be upward, which means it's pushing at B, and this member is in compression. And the diagonal was in tension. Okay, so back here, let's see what happens. Where did the 3-kip load go? It went wherever the diagonal sent it. It was being, the diagonal was being pushed at C. Then the load is transferred through the diagonal BC and ends up at B. and this is 3 kips and there is no diagonal at A, it does zero. There is an overturning moment of 3 times 10 or 30 kip foot clockwise and the pivot is at B which means this is a tie down, this is a pushback. Now the load is still 3 kips at 10 feet, the overturning moment is still 30 kip foot from the previous example, I know that this is two kips and two kips. A pair of two kippers separated by 15 feet gives me 30 kip foot. So this must be two kips and two kips. And those are my reactions due to the lateral load of three kips. I have a base shear of three kips. I have a tie down of two kips at point A, and a pushback at point B worth two kips. Let's go to two stories. No, let's color this one first. Is the member AC, the column AC, being pulled or pushed? Again, I see a tie down here. Must be this member is in tension. Here we go. This member is in tension. The strut itself is being pushed and so it's in compression. And frankly, I took care of the load. I don't need these. They're doing nothing. In this loading, they are doing nothing. But let's keep in mind that lateral loads reverse. So if it comes this way, then the whole situation changes. But right now, there is no work for these two members to do. If you look at truss analysis, that's what this is. Diagonal bracing is the same as a truss, but in the vertical direction instead of in the horizontal. Very good. Let's go down to this example where we have a two-story brace frame and three kips at C and six kips at E. So let's do the base shear first. We have an attack of 3 plus 6 equals 9 kips, and the attack is this way. And the load is going... Let me trace the load. It's going this way. It wants to jump off the truss. The diagonal brings it back. Now the 6 kip is here. It picks up the 3 kip and travels down CD and wants to jump off the truss, the diagonal brings it back. So this is 9 kips. This guy doesn't have a diagonal. It's 0. Very good. So that's the path of the load. The path of the load is as follows. I just erased it. It's like this. It comes this way, back, back, and back to A. Very good. So, let's see. Where are we doing? Okay. So, we took care of the base shear. Now for rotation, the pivot again. There is an overturning moment, and the pivot is point B. So, how much is the overturning moment? It's essentially due to 3 kips and 6 kips. The two lateral loads are making overturning about the pivot. The 3 kip is at 10 feet, and the 6 kip is at 20 feet. The moment is force times distance, 3 times 10, 6 times 20, is equal to 30 plus 120 kip foot. And so, we have an overturning moment of 150 kip foot of clockwise. So now, these two guys are no longer 0 because there's an overturning moment. And I know that this is the tie down on the left at A. This is the pushback at B. The moment of a couple, sorry, A and B are equal. That's the whole assumption because the sum of vertical forces needs to be equal to 0. So the distance between them is 15 feet. So A times 15 should equal to B times 15, that's the moment of the couple, should also fight the 150 kip foot. Which gives me A is equal to B is 150 kip foot divided by 15 feet. So A is equal to B, each one of them is equal to 10 kips. So I have a 10 kip tie down, I have a 10 kip push back, sum of vertical equals 0. And these two make a moment, moment of a couple. And the moment here is one of the tens times the 15 feet in between, or 150 kip foot, of counterclockwise. We were attacked with clockwise. The response is a pair of 10 kippers separated by 15 feet, making a couple of 150 kip foot counterclockwise. Very good. Let's color this, and let's please remember red is compression, and blue is tension. So, this member went into compression, the top one, EF, compression. Now, the 6 skip is at F, and the diagonal kicks in, it's in tension, and brings the 6 skips to C. Now, the 6 and the 3 together push on CD, sending it into compression. Now, the 9 kip, 6 plus 3, is at D. The diagonal picks it up and buries it at A. What happened, what colors are the remaining two sets of columns? It looks like if I think about it for a minute, this frame wants to do that, exaggerated. That's what it wants to do. So it looks like the left columns are in tension. The right columns are in compression. So let me color them. The right columns are being pushed down into the ground at B. the columns EC and AC are being pulled down. They're stretching. So that's the final coloring of this diagonal brace. Let's move on to the next one. Now I'm flipping one of the diagonals, and I flipped the 3-kip and 6-kip just to give you a hard time. It still looks like the lateral load is 3 plus 6, or 9, going this time right to left. And B has no diagonal, it's 0. And it looks like at A, we need to fight that 9 kips with 9 kips in the opposite direction. So that's the base shear at A. This time, the pivot, this frame wants to rotate like that. So it's going to rotate about pivot A and lift at point B. At, yeah, at point B. So this one is a tie down at B, and this one is a pushback. At the pivot, you have a pushback. Very good. So let's do the overturning moment. I don't think it's any different than the previous one, because the 6 kip and the 3 kip are in the same direction. Now they're still in the same direction, so they're going to make an overturning moment of 150 kip foot, or 3 times 10 plus 6 times 20. Very good. But the direction this time is counterclockwise. So OTM at 150 kip foot, and counterclockwise. Very good. And these guys need to get together and make clockwise, which is what they're doing. And again, the distance, or the couple arm as it's called, is 15 feet. So each one of us, like the previous problem, each one of us needs to be 10 kips at 15 feet to give me 150 kip foot of clockwise stabilizing moment. So let's recap these two problems, please, and understand that if the load is the same, the overturning moment is the same, if I flip the arrowhead, then the rotation is different and the pivot is different and the tie down and push back reverse. but their magnitude is still the same because the 15 feet between them is the same, and the requirement in moment, 150 kip foot, is the same. Well, then those two reactions are 10 kips. Their arrowheads are different than the previous problem because I reversed the arrowheads of the 3 kip and 6 kip lateral modes. Excellent. Let's graduate to this problem, which is three stories, 3, 6, 9 kips. Did I color this one? I didn't color this one. Let me color this one. So, red is compression, blue is tension. This member is being pushed. So, this is compression. And the 6 kip is now at E, and the diagonal ED picks it up and brings it back to point D. That one is in tension. And now, the 6 kip is joined with the 3 kip, and we have 9 kip, And there is a direct path to the foundation. I will take it. This one is a compression member. And it's carrying 9 kips. The diagonal, ED, is carrying 6 kips. And the diagonal, AD, is carrying 6 plus 3. Very good. So that's the colors. Let's color the columns. This time, the frame is doing that. exaggerated. It's doing that. So the right hand side is stretching and the left hand side is being compressed. So this one is stretching and the left hand side is shrinking or compressing. Very good. What about CD? It looks like the load did not go by CD. It looks like the load did this. It went from here to here to here. And that's it. It's buried. So we didn't use that member CD. So this one is doing nothing. And you can take it out. And if you take it out, you still have a triangle. That's why that member is a redundant member, as it's called. Please, to understand all this stuff, you must look at truss analysis. Anyway, just for the sake of clarity, these two members turned out to be tension, the other two members compression. Very good. So now we can graduate to this problem. Before I start, I would like to trace the load path and say that's what's happening. Okay? So we should get a feel for this by now. Excellent. So let's do the base shear first. And let's look at the total load is 3 plus 6 plus 9, or 18 kips. So we have a lateral load total of 18 kips. A has no diagonal, so it doesn't partake in that load. B has to be 18 kips. The attack is right to left. Well, then the reaction is left to right. So the base shear at B is 18 kips, at A is 0. Excellent. Overturning moment. There is an overturning moment. These three arrows are in the same direction, and they're doing counterclockwise. And the amount of moment is 3 kips times the distance to the pivot, 6 kips times the distance to the pivot, and 9 kips times the distance to the pivot. So we're looking at 10 feet, 20 feet, and 30 feet. Very good. 3 times 10, 6 times 20, and 9 times 30. Like I said, I simplified the math so that we can focus on the concept. So it looks like the total is 300, 420 kip foot of overturning moment. 420 kip foot of OTM. Excellent. So the pivot this time is at A, which means the pushback is at A, the tie down is at B. Excellent. I just need to say this. I hope you will not get confused. But if this 6 skip were the other way, then we just subtract its moment. Because we have 2 to the left, 1 to the right, 2 making counterclockwise, 1 making clockwise. We look at the net. Very good. So, back to our problem. We have 420 kip foot of overturning moment at A, the tie down at B, and the push back at A. Well, what's the distance between them? The distance between them is 15. So, A times 15 feet should equal to B times 15 feet should equal to 420 kip foot. That's what we want from the couple. Do the math, you will end up with A equal to B is equal to 28 kips. So we need a pushback of 28 kips. We need a tie-down of 28 kips. Excellent. That's it. I solved all these questions. And now I'd like to color it. The diagonals are all in tension. So this one, this one, and this one are in tension. And diagonal GF is in charge of 9 kips. Diagonal ED is in charge of 9 plus 6. And diagonal BC is in charge of 9 plus 6 plus 3. That's the way it is. It's accumulating. This is a cantilever truss, and it's accumulating the load until it's buried at B, the lateral load. Very good. So, we found the reactions, and the horizontal reaction, or the base shear, at B turned out to be 18 kips. Let's color this truss. So, this one is in compression. It's being pushed by 9 kips. The diagonal has two components. The horizontal component, oops, that's in the wrong direction. The horizontal component has to be 9 kips. And the vertical component is based on the slope, 10 to 15, or two-thirds of that. 10 over 15 times 9 kips equals 6 kips. I don't want to solve this, but I'm just warning you that the horizontal component of that tension diagonal is 9 kips. Therefore, the force in the diagonal is more than 9. It's 9 square plus 6 square square root. Forget that math. Sorry, I was coloring. Now all of a sudden I'm doing math. Please don't get confused. This one is in compression. Now member GH is in charge of 9 kips. It's in 9 kips compression. Member EF is in compression and 9 plus 6. And then member CD is also in compression, and it's carrying 9 plus 6 plus 3. And that's the load path. With the overturning moment, it looks like these guys are in compression. It looks like these guys are in tension. They're stretching. So let's color the remaining members. So this one here is in tension. And the other one is in compression, the other set of columns. Very good. So that concludes this one. Looking at the next question, it says, oops, it says, here's a gravity load. There's no lateral load on this problem. It's just two kips going down that column axially and six kips going down the right column axially. So here the reactions are very simply six and two. And most important, you're not doing anything. You don't do gravity. This is just a column and another column. Very good. As soon as you put a lateral load, the diagonals kick into action. But these reactions right now are zero. There is no lateral load. The point of this exercise is when you get something like this, which is the last problem for the brace frames, when you get something like this, it would really help to look at the vertical loads, get the reaction. look at the lateral loads separately, get the reactions due to those, and then add everything up. Very good. So what I'm saying is the reactions from this problem plus this problem should equal to the next one. Let's see how this works. Excellent. So this reaction due to the gravity loads only is 2, this one is 6. And due to the lateral loads, we did this here, and we found the reactions to be 28, a pair of 28s. The one on the right is down, the one on the left is up, and a base shear of 18 kips at B. So this one is 18 kips at B, going this way. And this one is 0. Now, this one was 28 kips. I don't know what happened here. Maybe I'm in white. No, I'm in green. Okay, so this one is 28 kips. And there was a tie down here of 28 kips. So at the end of the day, it looks like I need 28 plus 2 at A. I need a reaction of 30 kips. And I need a reaction here of 22 kips tied down. So, that's it. As far as color, it's all the same. I need to, the diagonals are blue, the horizontals are red, and then the columns, I just need to make sure that the right column, the series of columns on the right are tied down, the ones on the left are a pushback.