Shear Walls: 4 Sw Math

So in the case of a concrete shear wall, as this example will illustrate, there is a deadload that we need to take into account, and this deadload here is the result of the weight of the wall. Unlike a moment frame and a brace frame, those deadloads are relatively negligible. But in the case of concrete, the dead load is pretty appreciable, and we need for the 12-kip lateral load in this example to overturn the wall, it must overcome the dead load plus any imposed loads. So where did the 11 1⁄4 come from? This is a 10-foot wall whose base is 15 feet and whose thickness is half a foot. and the density of concrete is 150 pounds, or its weight per cubic foot, 150 pounds per cubic foot, that gives you the 11 and a quarter. Now where is it located? It's located at the centroid of that wall, which is halfway on the 15-foot dimension. 7 and a half, 7 and a half. Very good. So let's address the base shear, then we'll address the overturning moment as we did with the moment frame and the brace frame. The base shear is coming from the 12 kip load, and the 12 kip load is spread over the length. So here's the lateral resistance to the 12 kip load, and this lateral resistance is basically 12 kips spread over that length. That's the contact with the foundation, which turns out to be 0.8 kip per foot, or 800 pounds per every linear foot. So here's my resistance, 800 pound per foot, times 15 feet. That takes care of the 12,000 pounds, or 12 kips. Excellent. Let's talk about overturning. For the overturning moment, it is caused by the lateral load. So this lateral load wants to rotate the wall about pivot A. It wants to make it do that. And let's see how much the overturning moment is. It looks like 12 kips times 10 feet. at pivot A equals 12 kips times 10 feet, or 120 kip foot, because that's the height, that's the location of the 12 kip load above the pivot. That's the dimension, not the 15. The 15 is the shear, it's spread over 15 feet. The overturning moment is 12 times 10, or 120 kip foot, And it looks like the overturning moment is in the counterclockwise direction. Counterclockwise. So there is a stabilizing moment. This one is counterclockwise. So anybody clockwise shall be in the stabilizing moment. And looking at this, the 2-kip is on top of A. It's not going to do anything. It cannot participate in moment. So we have, in the stabilizing moment, we have the 8, sorry, let me write that out, because the 2 kip is nothing. It has no moment arm, so it's doing no rotation. Versus the 8 kips, the 8 kips is at a distance of 15 feet from the pivot. And there is the dead load at 7 1⁄2 feet. So add to that 11 1⁄4 kips times 7 1⁄2 feet. So this is 120. And the math on this one gives me 84.375. So add these up and you get 204.375. Now, I have an overturning moment of 120 kip foot, and it is counterclockwise, and I have a stabilizing moment in the clockwise direction worth 204.375. So, I have a factor of safety of how much? The code says the stabilizing moment is equal to a factor of safety times the overturning moment. In this example, the factor of safety is equal to the stabilizing moment divided by the overturning moment, or 204.375 divided by 120. So, I apologize, I haven't done the math on that. Let me do it very quickly. I have 204.375 divided by 120. So 1.7. The code says we need to have at least a factor of safety of 1.67 for buildings and 1.5 for retaining walls. Well, if this wall is a building, this is for retaining walls. and the other one is for structures. It looks like I'm barely above the minimum. Very good. Now what happens if that 12th kip is reversed? The arrowhead is reversed because wind or earthquake can come from any horizontal direction. So let's take a look at what changes now. The overturning moment is still the same. overturning moment is 12 kips at a height of 10 feet or 120 kip foot. The arrow reversed, so this one is going to be clockwise. And in the counterclockwise direction, sorry, where's the pivot? B, because it's going to do that. So the overturning moment at B is 120 kip foot. The 8 kip load this time is not going to make any moment because it is right on top. Let me, sorry, let me write stabilizing moment. The stabilizing moment due to the 8 kip is nothing. It has no perpendicular distance. The 2 kip is at a distance from the pivot of 15 feet And the 11 and a quarter is at the centroid of the wall or at 7 and a half feet. So 11 and a quarter times 7 and a half feet. So this is 30 kip foot and this was 84.375. For a total of 114.375 kip foot. No good. This is a stabilizing moment of 114 and an overturning moment of 120. The wall overturns. No good. So what do we need to do? We need to increase the thickness, increase the width of 15 feet, or lower the height of 10 feet. None of these are acceptable to an architect, so maybe we'll add some concrete in the foundation and just make the 11 and a quarter heavier. How much heavier does it need to be? We need to be stabilizing moment equal factor of safety minimum 1.67 times the overturning moment. So we're going to need a stabilizing moment of at least 1.67 times the overturning moment. which is 1.67 times 120, which is 200. That's what we had earlier, 200.4 kip foot. So I need to make sure I increase one of these dimensions or enough weight in the foundation to get a stabilizing moment of at least 200.4. I'm at 114.375. No good.