Moment Frames: 2b Mf Discussion

So examining the moment frame and hopefully you've seen in the toys video, I would like to do two things in this video. One is to explain the deformed shape of the moment frame that we saw with the toys and two is the direction of the reactions to figure out the direction of the reactions. Now, first of all, with the post and beam, we had a pin between the horizontal and the vertical. Here, we have a rigid connection between horizontal and vertical. That's the black blob versus the white blob. Excellent. So, under a 10 kip, for example, uniform load on the horizontal, gravity downward, We saw with the toys that this guy is going to come out a little bit, but stay square for the angle. And rotation is possible down here. That angle changed from 90 degrees. And then the horizontal joins these two. Very good. So that's the deformed shape under a gravity load. And for the fixed end, it looks exactly like that, with a slight difference in that this bottom part of the column is to remain square. And now it has been claimed by the foundation. And the remaining portion of this will start to behave like image 2A. So it's going to go up a little bit and bend. It's going to bend from a certain point up. It's going to start bending. The angle with the horizontal shall remain constant. And then the beam comes in and does that. And we have two points of inflection. Whether it's pinned at the base or rigid at the base, the horizontal beam has two points of inflection because of the two black blobs surrounding the beam. Excellent. So looking at compression and tension, just like we did with the toys, let's start with the tension. It looks like this part, the outside of the column and the top part of the beam up to the point of inflection, looks like it's in tension. The bottom of the beam is in tension. And compression begins here. There's compression. The top of the beam is in compression. and the inside of the columns is in compression. Looking at the fixed end situation, very similar. Up to here, there is compression. The top of the beam is also in compression, and the rest is in tension. The bottom of the beam is in tension, and the outside of the column is in tension. And that's what the deformed shape looks like, which portions are in compression, which portions are in tension. Very good. So, basically, what I'm saying in this video is the column here has that length versus in 2a, it has a full length. The whole length is subjected to bending moment and rotation versus in 2b, this portion of the column, the lower portion of the column belongs to the foundation, and the column now is much shorter, and so it's much stiffer than 2A. Excellent. Looking at a lateral load of 12 kips, for example, what happens is the column is allowed to rotate at the base because it's not fixed, and that angle shall remain 90 degrees between horizontal and vertical, and then the horizontal segment goes down first and then goes back up. There is a point of inflection here. There is nothing on the column. It's bowing in one direction. Okay. Looking at the same condition, but with the base fixed this time in 2D, it cannot do what it did earlier. It can't do that because the base is not going to allow rotation. So what it ends up doing is it stays put for a while and then it starts to bend. And that angle shall remain 90 degrees. And the horizontal goes down and then up again. There is a point of inflection on the horizontal and also on the verticals. There's a point of inflection. So let me pull out my color pens. Here's for tension. And it looks like this segment is all in tension. And once I reach a point of inflection, I flip. Now the top of the beam and the outside of the column is in tension. And as far as compression goes, this side of the column and up to the point of inflection is in compression. And the inside of the column and the lower part of the beam is in compression. So, looking at the one with the fixed end, it's pretty much the same, only some of the compression, the point of inflection on the vertical is going to disrupt the compression and tension on the vertical. Excuse me. So, this side is in compression, and the rest is in tension. So this portion is in tension, this portion is in tension, and the remaining part of the column is also in tension. So we did the deformed shape. I saw who is in tension, who is in compression. Now it's time to look at the reactions. First of all, it's important to recognize that with a moment frame, whether it's pinned at the base or the base is fixed, there's too many unknown reactions. In 2a, I have two horizontal green reactions and two vertical red reactions. That's a total of four. I only have three equations. So it's indeterminate to the first degree. There's one extra unknown compared to equations, four versus three. In the second example, 2b, there are six unknown reactions, and all I have is three equations. That's too many unknowns. This is where you need a computer to do indeterminate analysis, or else an engineer, one and the same, they'll figure it out, not on the ARE. Excellent. So let's look at the reactions. Where did the 10 kip vertical load go? So it went into two vertical reactions. Each carries half of 10 kips. And we saw with the toys that this thing wants to do spreading. So there's some thrust kicking out and we need to bring that point back. So we're going to need a reaction that brings this point back to here. So I'm going to need a horizontal reaction to bring back that leg and keep it in place. Likewise, I need another one here. So that's the direction of my horizontal reactions. They're inward. They are to contain the thrust and keep the frame from spreading. When we look at the second example, pretty much the same thing. The 10 kip is going 5 and 5. 5 kips on each side of the verticals. That's a safe assumption. Sum of vertical forces equals zero, the two legs are equal, each is fighting as much as the other. Likewise, the thrust has to be contained, just like in 2a. In 2b, we're keeping the legs from spreading. Now, if we look at the deformed shape from the previous example that I have already erased, but this one did something like this. it did under a gravity load it did that and that so now the moment reaction this one Has to push that leg and bring it back to vertical Because the support is fixed and the other moment reaction needs to bring back that leg to vertical. So my reactions are, this thing would have done that, so I have to fight you back, so that's a counterclockwise reaction. And this other one has to bring back that leg to vertical, so it must be clockwise. So that's the moment reactions and their directions. Very important just to recognize the directions. The magnitude is beyond the scope of ARE. it's more on the PE than the ARE. Excellent. Now looking at the lateral load of 12 kips in 2C and 2D, I need to recognize that in 2C there's two legs, leg A and leg B, and they are equal stiffness, and they will share in resisting the 12 kip load, so each one will take 6 kips. Now, if these stiffnesses were different, then the reaction is distributed proportionately. But right now, they're equal. Each takes half. Now, a lateral load makes overturning, overturning moment. We saw that with the post and beam. So, the vertical reactions need to respond because this frame is going to do that. with fixed connections between horizontal and vertical, that's not the same as with a post and beam. Now what happens is the left end at A lifts, and the frame kicks down at B. So A, we need somebody to bring this point down, must be this reaction is a tie down. And now we have a vertical. We didn't have verticals in the picture before. There was only a 12-kip lateral load, but it necessitated a tie-down at A, which means that the reaction at B has to be equal and opposite to that tie-down. So that's the direction of the reactions in 2C. Looking at 2D, it's exactly the same. This one has to be half of 12 and half of 12 for two equal stiffness legs. There is an overturning moment, which means this one has to be a tie-down, and this one has to be equal but opposite in the opposite direction. Now the attack here, the overturning moment, looks like it is clockwise. Well, then the moment reactions need to react and need to be opposite, so they are both going to be counterclockwise. price.