Braced Frames: 3b Bf Discussion

So let's examine what happens in a brace frame when you load it with gravity loads or with lateral loads. Pursuant to the moment frame, where we just discussed where the 10 kip load went, in a brace frame, it is important to recognize that any gravity loads are handled by the post and beam, not by the diagonal. So for 10 kip load gravity wise the diagonal is doing nothing and the 10 kip load is going five and five and just like the post and beam we saw earlier nothing different whatsoever. So the vertical reactions are five kips and five kips half of 10 on each side. There is no lateral activity, so the horizontals are zero, the diagonal is zero. Same with this example where you have a fixed base. The ten kip is still going five and five on each side, and there is no moment activity, there is no horizontal activity, so all these reactions are zero. I'll repeat again. A brace frame is pinned usually between the horizontal and verticals, Therefore, it makes no sense to put a fixed connection, sorry, a fixed end support. Okay, so what happens with the lateral load? That's where the diagonal really kicks into action. And we have two kinds of diagonals. We have a tension only diagonal and a tension and or compression. diagonal. Okay, so a tension only diagonal is basically a cable, a tie rod, or a galvanized strut, and a tension or compression member is heftier. It has more cross-sectional area, it has more section modulus, more moment of inertia, and typically it's a pair of angles, a pair of channels. Those are not totally symmetrical. More in the symmetrical range is a wide flange, a rectangular tube, better still, a square tube, or ideally a pipe. Now those get to be more expensive and harder to detail, but structurally, the more symmetrical the cross-section, the better it serves under axial loading, which is what diagonal bracing does. Moment frame does bending. Diagonal bracing does axial. Therefore, it's much more efficient and it's smaller. It's smaller in cross-section. So I'd like to take a quick break here and take a look at these pictures. And to compare the moment frame with diagonal bracing, I'll bring to your attention this image here that has a high-rise building and we recognize immediately the pin connections, sorry, the rigid connections with this, we recognize the moment connection with these web stiffeners. And please note the following. There is a wind load coming in from here, or a seismic load, a lateral load, and there is no bracing, there is no shear wall, so it must be moment resisting or a moment frame, which is good up to a maximum of 20 floors. Very good. So if you look at the depth of the member here, you see how hefty it is compared to the next bay, two bays over is a gravity bay, and the beam is much smaller, and here it's much deeper because it has to deal with wind load and it doesn't have any bracing or shear wall. The same with the column. it is very large compared to the toothpick carrying the gravity load. Gravity on a post and beam is no big deal. Wind load cannot be sustained, so we go to a moment frame. But that's why the moment frame is so much more expensive, because the members have to be hefty to resist all the bending, both the columns and the beams, versus bracing. Bracing is a lot cheaper because you don't need as much cross-sectional area as we see here in the example of the New York Times by Renzo Piano, where we have this fork bracing where you have two diagonals and two diagonals stacked in between them, changing directions at the intersection. Or the John Hancock Tower. I mean, these braces are relatively small compared to if you didn't have those braces. all the members would have to be a lot bigger, and therefore much more expensive. Steel, of course, is sold by weight. Excellent. So that's the basic difference between bracing and moment frame. And here are some examples of tension-only bracing, a cable, or a galvanized strap here. But these are tension-only. So if you put them in compression, they don't work, as we see here in this example, it bulge. So if you go to something a little bit heftier, then you end up with tension and or compression, as we see in this chevron bracing or the inverted chevron bracing. Bracing could be eccentric or could be concentric. If we look at these diagonals, they're going to the same point on the beam. Versus if we look at these diagonals, there is a gap between them called a link beam. And that link is not braced. You see what's happening here is there's a triangle and there's another triangle. And those two triangles are very strong and stable. but the link beam in between is not braced to give the frame a little bit more ductility because when a member bends, it provides ductility versus when it is totally braced in a symmetrical manner, then you have two triangles like this without a link beam. This is much more rigid and less ductile. So here's a little detail where we see that the brace comes in at a certain distance, and that's my link beam. And it's there in order to bend and provide ductility. Very good. So let's go back and try to understand where that 12 kip load goes. And just a reminder, please, with the moment frame, we had two legs, and if they were equal stiffness, each leg took half of 12 kips. If they were unequal stiffness, then the 12 kip is distributed based on rigidity of each member. So if this were a rigidity of 4 and this were a rigidity of 2, that's a total of 6. Then the column on the left would take in proportion 2 out of the total of 6, and this one would take 4 out of the total of 6 of the 12 kips. So the left column would probably carry 4 kips, and the one on the right would carry roughly 8 kips. Very good. So now let's go back to diagonal bracing. I have 12 kips, and I have a pin joint. The pin joint says I don't do lateral, so it passes the load off, and the load comes to here. Now this diagonal is in tension. And it takes the 12 kips and pulls it back to point A. So the reaction at A is 12 kips. And the reaction at B is 0. There is no diagonal at B. So it does not see any of the 12 kip load. None whatsoever. So essentially what happened is the 12 kip went on this triangle. It just ended up at point A. In this second example, the 12 kip is pushing. and this member is in compression and cannot be attention-only bracing. It has to have more substance. Let me fix that and put two lines to say, okay, this guy has some area, a section modulus, etc., and can handle compression. Now, the 12-kip load went down the slide and ended up at B. So at B, we have to fight. The attack is left to right. Well, the reaction is right to left. And this has to be 12 kips. So, and sorry, at A, there is no horizontal reaction. It's zero. Now, are the vertical reactions zero or not? We saw earlier with the post and beam that if you have a vertical load, the horizontals are zero. The same with, in this case, for the brace frame, there is a lateral load. Once there is a lateral load, there is an overturning moment. That means that the vertical reactions are not zero. The pivot for this example is B, and therefore there is a tie-down at A, and it has to have balance on the other side, equal and opposite. So these two actually make a couple. The attack is clockwise. Well, then the response from these up and down reactions at A and B, respectively, is counterclockwise. so they create a couple. They are equal and opposite, separated by a certain distance, creating a couple, which gives me counterclockwise rotation and keeps the frame from rotating. Okay, let's make sure we're clear on this one, please. Here's my 12 kip load, A, B, C, D. Now, the diagonal is going to be in tension. And if the load were on this side at D, it would still be in tension, and the reaction at A is 12 kips to the left in either of these cases. The reaction at B is 0. And in case, I didn't number them, but 1, 2, 3, 4, for example, In case number one, the overturning is about point B, so the tie down is at A and the pushback is at B. In case two, the pivot is still at point B and the tie down is at A and the pushback is at B. As long as that force is in the same direction, the pivot is B in these examples. The same with condition 3 and condition 4. The pivot of rotation is still B because the frame is going to do that. So it's going to rotate clockwise about B. So the tie-down needs to be at A. So if the pivot is at B, then the tie-down is at A in all of these examples. And in condition 3, this has to be 12 kips, because A does not have a diagonal, so it's 0. And in condition 4, also, the diagonal takes the load and buries it at B. So this one, oops, is going in that direction, and this one is 0, and this is 12 kips. In condition 3, the diagonal itself is being pushed by the 12-gip load, so this one is in compression. Likewise, in condition 4, if you don't like seeing the 12 here, you can move it anywhere on its axis and put it over here, and then we see that actually this is compression. In the previous example, condition 2, that was tension. Looking at case 5, 6, 7, and 8, I just flipped the arrows, but just to make sure that we're clear on everything, the diagonal is going from D to A. Must be the reaction at A. Oops, wrong arrowhead. Is 12. And in case number 6, the load of 12 kips is pushing on that diagonal, sending it down to A. This is 12 kips. And in condition 7, it's pulling on that diagonal. And so the reaction has to be at point D where the diagonal terminates in the foundation. And the other reaction is 0. So, in condition 8, it looks like the 12 kip load travels horizontally, and then the reaction at B picks it up, and this is 12 kips, this is 0. That's for shear. That's the distribution. The 12 kip is doing shear. Now, let me make sure I label all the diagonals. This one is in compression. Why? Because I'd like to think of that 12 kip as this condition 5. I'd like to think of the 12 kips at deep pushing. And the same with condition 6, it's compression. In condition 7, it's being pulled. And also in condition 8, I can move this force to here and pretend that this is its location, and therefore the diagonal goes into tension. Excellent. Now let's talk about rotation, overturning moment, and where the pivot is. As long as the arrowhead is left or right, the pivot is B. Example 2, same thing, pivot is B. And in example 3, the pivot is B. In example 4, the pivot is B. Now I reverse the arrowhead, it looks like the pivot, this thing wants to do that. So the pivot now is at A. Likewise with condition 6 and condition 7 and condition 8. All of these are rotating counterclockwise due to the 12-kip load about point A. Excellent. So with this load, there is an overturning moment that is clockwise. So these two have to make counterclockwise, which they did, to resist that overturning moment of clockwise about point B. Same thing with this example. We have a counterclockwise couple that is created between the vertical reaction at A and at B to counter the overturning moment, which is clockwise. As long as the overturning moment is clockwise, these first four examples, it's clockwise, then the tie-down is on the left, the pushback is on the right. they make a counterclockwise moment to resist the clockwise overturning moment. Looking at condition 5678, it looks like the overturning moment is about point A, and it is counterclockwise this time. And we need the vertical reactions to respond to this overturning moment. So I think this one is going to be a tie-down because this frame is going to lift. So we need somebody to tie it down. That's the downward reaction at B. And it needs an opposite one on the other side at A. So these two now are going to make clockwise because the attack is counterclockwise. Likewise, in example 6, it looks like there needs to be a tie down at B and a push back at A. And in condition 7, same thing. As long as that arrowhead is in the same direction, we're going to get the tie down at B and the pushback at A. Very good. So in summary, a lateral load makes overturning. And that overturning causes the vertical reactions not to be zero, but to be equal and opposite, making a couple to take care of that overturning moment. Excellent. So that's it for bracing. So to be clear on everything, let's make sure we understand these different variations. They're all the same in my book, but just to make sure that there's no tricks on the ARE. I would like to take each one of these and address the diagonal, make sure it's in tension or compression. And then we'll look at where the 12 kip lateral load goes, the shear. And also look at the overturning moment about which pivot and where the tie down is, where the pushback is. Please, let's just make sure we understand that a brace frame has pin joints. So without the diagonal, it's a post and beam. You're adding a diagonal to a post and beam with pin joints. That's the brace frame. Okay, so in this first example, the 12-kip comes to joint C. Joint C is a pin. It transfers the load. And it's as if the load of 12 kips is at a D. And it's the same now as example 2 next to it. This reaction, oops, let's not do that. Let's just stick with tension compression. So this one is in tension. And in example 2, it's being pulled also. So it's in tension. Versus example 3, the diagonal is being pushed. And therefore, it is in compression. and I would like to put another line here to say, okay, it needs more beef. The same with example four. I can move this force and put it here, and now all of a sudden I see that it is in compression. That's transmissibility. You can move a force anywhere on its line of action. If you don't change the arrowhead, the effect is the same. So this one is also a compression member. And going down to example five, I'm not clear what this 12-kip is doing. I think it's better if I see it over here. Now, all of a sudden, I see it pushing on the diagonal AD. That one is compression. The same with example 6. It's in compression. It's being pushed. And in example 7, it's being pulled, so this one is tension. And in example 8, I don't see this. Let me move it to here. I see it pulling on the diagonal. That's tension. Okay, so I did tension or compression. Let me talk about shear. Shear, ARE likes to call it, NCARB likes to call this horizontal reaction base shear. So, calculate the base shear in example 1. Calculate the base shear at location at support B. B has no diagonal. That's it. The base shear is 0. At A, it has to be the full 12 kips. The attack is left to right. Well, then the reaction is right to left. So reaction at A is 12 kips, B is 0. Looking at example 2, B is still 0. There's no diagonal here. It's part of a post and beam. It just does up-down. So this one is 0. This one is 12 kips going this direction. Looking at example 3, the load goes from C down to B, pushes on the support at B. It has to push back. So this one is 12, this one is 0, there's no diagonal at A. Same with example 4, this is 12, this is 0. And then example 5, let's see, the load is going this way, the response has to be that way, that's the reaction, it's direction. Now where's the diagonal dying? It's dying at A, must be, this is the 12 kip, B does not have a diagonal, it's 0. The load pulled on CD, CD compressed DA, and the load of 12 kip ended up at A. Looking at 6, same thing. This one is pushing back with 12 kips because the load went down this way. And the reaction base shear at B is 0. Looking at example 7, the diagonal is being pulled. Well, at B we have to respond with an opposite. 12 kip because the diagonal sends the load from C to B. B has to tie back or push back. Okay, so this one is 0 at A. There's no diagonal. There's no diagonal here. This must be 0. This one is 12 kips. Very good. So I did the base shear. Now I'd like to do the overturning and identify the pivot. Let me switch to purple. So with the load going in this direction, the pivot is B, because this frame wants to do that. So the pivot is B in the first example. As long as that force is going left to right, the pivot is B. And the overturning moment is clockwise in each one of these examples. And that overturning moment, we'll get to it later, but it's equal to 12 kips times that height. How high up is the 12 kip load? That's what's causing the overturning moment. Looking at examples 5 through 8, it looks like the load is going this way. Well, this frame wants to do that, and so the pivot this time is A. This is the pivot, this is the pivot, as long as that load is in that direction, the pivot is at A. If the pivot is at A or B, the tie down we need to figure out. So we've done the overturning moment and the pivot. It looks like this needs to be the tie-down because the frame is going up. It wants to rotate like so. And so this point went up. We need to bring it down. The tie-down is at A. And equal and opposite is a pushback at B. These two verticals have to be equal. They were created to resist the overturning moment. The overturning moment in example 1 is clockwise. Well, then these two come together and make counterclockwise, and they make a couple. And the moment of a couple is equal to one of the two forces, remember they are equal, times the distance between them. That's the moment of the couple that is supposed to resist the 12th kip at the height given. Again, the attack is clockwise. Well, then the response is counterclockwise, and these two are equal at A and B. And example 3, same thing. There's a tie down at A, a push back at B. Same with example 4, tie down and push back. Now looking at example 5, the overturning moment is this way, OTM at A versus in the previous examples, the OTM was at B. Okay, so with an overturning moment at A, this frame is going to do that, and we need to tie down at B. So this is a tie down equal to a certain amount of force. This is equal and opposite. Now the attack was counterclockwise. The response between these two together is clockwise. Looking at example 6, same thing. There's an overturning moment at the pivot A, and there's a tie-down at B and a pushback at A. More of the same, overturning moment this way, and the tie-down at B, the pushback at A. And finally, number eight, same thing, overturning moment is counterclockwise. Well, the response needs to be clockwise. So this one is down, this one is up. And these two are equal and opposite, and they make a couple separated by that much distance. The wider the base, the more the couple, the more moment you can handle. The narrower the base, the harder it is. Okay, so conclusion to this page is the base shear is where the diagonal terminates. That's the most important thing. In the first, forget it.