Locating the center of rigidity For the building discussed in Example 3.1, determine the location of the center of rigidity (CR) and the value of the torsional rigidity, J, for the lateral force resisting system configuration as shown.

Question

Locating the center of rigidity
For the building discussed in Example 3.1, determine the location of the center of rigidity (CR) and the value of the torsional rigidity, J, for the lateral force resisting system configuration as shown.

Accepted Answer

The walls of the multistory building in Example 3.1 have a floor slab at the bottom and at the top of the wall at each level, which prevents rotation of the top of the wall and creates a fixed wall condition. Using deflection Equation 3.4 for a fixed wall, as previously discussed, the rigidity of each of the walls is determined as given in Table 3.2.

Table 3.2 Tabulated wall deflections and rigidities
Wall	Thickness b (in.)	Height h (ft)	Length L (ft)	Deflection δ	Rigidity
A	12	12	20	0.168	5.95
B	12	12	20	0.168	5.95
C	18	12	20	0.112	8.93
D	10	12	20	0.2016	4.96
E	10	12	20	0.2016	4.96

[latex]\delta_{fixed}=\delta_m+\delta_v=\frac{1}{b}\left(\frac{h}{L}\right)^3+\frac{3h}{bL}[/latex] (3.4)
The center of rigidity coordinates can be determined by the following equations:
[latex]x_{cr}=\frac{\sum{R_{i,y}x_i}}{R_{i,y}}[/latex] (3.7)
[latex]y_{cr}=\frac{\sum{R_{i,x}y_i}}{R_{i,x}}[/latex] (3.8)
The wall rigidities for each wall with respect to their direction of resistance are given in Table 3.3 and Figure 3.5.

Table 3.3 Tabulated wall rigidities
Wall	[latex]R_{iy}[/latex]	[latex]R_{ix}[/latex]	[latex]X_i[/latex] (ft)	[latex]Y_i[/latex] (ft)	[latex]R_{iy}X_i[/latex]	[latex]R_{ix}y_i[/latex]
A	5.95	0	0.5′		2.98
B	5.95	0	0.5′		2.98
C	8.93	0	60′		535.8
D	0	4.96		0.33′		1.64
E	0	4.96		59.67′		259.96
	20.83	9.92			541.76	297.60

Then the coordinates for the center of rigidity is determined as follows (Figure 3.6):
[latex]x_{cr}=\frac{\sum{R_{i,y}x_i}}{R_{i,y}}=\frac{541.76}{20.83}=26[/latex] ft
[latex]y_{cr}=\frac{\sum{R_{i,x}y_i}}{R_{i,x}}=\frac{297.60}{9.92}=30[/latex] ft

Question 3.2: Locating the center of rigidity For the building discussed i...

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