GEOMETRIC PROPERTIES OF AN AREA
Calculate the polar moment of inertia of the area in Fig. 13: (a) with respect to its cen-
troid, and (b) with respect to point A.
Calculation Procedure:
- Establish the area axes
Set up the horizontal and vertical coordinate axes u and y, respectively. - Divide the area into suitable elements
Using the American Institute of Steel Construction (AISC) Manual, obtain the properties
of elements 1, 2, and 3 (Fig. 13) after locating the horizontal centroidal axis of each ele-
ment. Thusy 1 =^2 / 3 (6) - 4 in (101.6 mm); y 2 = 2 in (50.8 mm); y 3 = 0.424(8) = 3.4 in (86.4
mm). - Locate the horizontal centroidal axis of the entire area
Let jc denote the horizontal centroidal axis of the entire area. Locate this axis by comput-
ing the statical moment of the area with respect to the u axis. Thus
Moment, in^3
Element Area, in^2 (cm^2 ) x Arm, in (cm) = (cm^3 )
1 0.5(6)(16) = 48 (309.7) 4 (10.2) = 192 (3,158.9)
2 4(16)= 64 (412.9) 8 (20.3) = 512 (8,381.9)
3 1.57(8)^2 = 100.5 (648.4) 13.4(34.9) = 1,347(22,045.6)
Total 212.5 (1,351.0) 2,051(33,586.4)FIGURE 13