Locating the center of mass of a building A plan view of a typical floor of a multistory building is shown in Figure 3.3. Concrete shear walls resist lateral forces in both directions. Determine the location of the center of mass (CM) at the given floor level. The mass of the floor diaphragm is

Question

Locating the center of mass of a building
A plan view of a typical floor of a multistory building is shown in Figure 3.3. Concrete shear walls resist lateral forces in both directions. Determine the location of the center of mass (CM) at the given floor level. The mass of the floor diaphragm is uniformly distributed and the weights of the concrete shear walls are as noted in the figure and are tributary to the diaphragm—that is, half of the wall height below and half of the wall height above the diaphragm constitutes the weight of the wall for analysis. The noncantilevered portion of the floor diaphragm is 85 psf and the cantilevered portion of the diaphragm is 65 psf. The floors of the building are 6 in. thick concrete slab on metal deck and are idealized as a rigid diaphragm. The floor-to-floor height is 12 ft.

Accepted Answer

The center of mass coordinates can be determined by the following equations:
[latex]x_{cm}=\frac{\sum{w_{i,y}x_i}}{w_{i,y}}[/latex] (3.5)
[latex]y_{cm}=\frac{\sum{w_{i,x}y_i}}{w_{i,x}}[/latex] (3.6)
The weight and center of gravity for each building component with respect to the x- and y-directions are presented in Table 3.1.

Table 3.1 Tabulated weights and center of gravity for each building component
Building component	Weight [latex]W_i[/latex] (k)	[latex]X_i[/latex] (ft)	y (ft)	[latex]W_i X_i[/latex] (ft-k)	[latex]W_i Y_i[/latex] (ft-k)
Wall
A	35	0.5	50	17.5	1750
B	35	0.5	10	17.5	350
C	45	60	30	2700	1350
D	25	50	59.67′	1250	1491.75
E	25	30	[latex]\frac{4}{12}=[/latex] 0.333′	750	8.33
Diaphragm	[latex]\frac{60 imes 60 imes 85}{1000}=306^k[/latex]	30′	30′	9180	9180
Noncantilevered	[latex]\frac{20 imes 60 imes 85}{1000}=78^k[/latex]	70′	30′	5460	2340
Cantilevered	549			19375	16470

Then the coordinates for the center of mass can be readily determined as follows:
[latex]x_{cm}=\frac{\sum{w_{i,y}x_i}}{w_{i,y}}=\frac{19,375}{549}=35.3[/latex]

[latex]y_{cm}=\frac{\sum{w_{i,x}y_i}}{w_{i,x}}=\frac{16,470}{549}=30[/latex]
The location of the center of mass of the building is shown in Figure 3.4.

Question 3.1: Locating the center of mass of a building A plan view of a t...

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