Then^m = 2051/212.5 = 9.7 in (246 A mm). Since the area is symmetric with respect to
the y axis, this is also a centroidal axis. The intersection point G of the ;c and y axes is,
therefore, the centroid of the area.
- Compute the distance between the centroidal axis and the
reference axis
Compute k, the distance between the horizontal centroidal axis of each element and the x
axis. Only absolute values are required. Thus Ar 1 = 9.7 - 4.0 = 5.7 in (1448 mm); k 2 = 9.7 -
8.0 = 1.7 in (43.2 mm); k 3 = 13.4 - 9.7 = 3.7 in (94.0 mm).
- Compute the moment of inertia of the entire area—x axis
Let /o denote the moment of inertia of an element with respect to its horizontal centroidal
axis and A its area. Compute the moment of inertia Ix of the entire area with respect to the
jc axis by applying the transfer equation Ix 2/ 0 + ^AK^2. Thus
Element /o, in^4 (dm^4 ) Ak^2 , in^4 (dm^4 )
1 1 /36(16)(6)^3 = 96(0.40) 48(5.7)^2 =1560(6.49)
2 yi2(16)(4)^3 = 85(0.35) 64(1.7)^2 = 185 (0.77)
3 0.110(8)^4 = 451 (1.88) 100.5(3.7)^2 = 1376 (5.73)
Total 632(2.63) 3121(12.99)
Then, Ix = 632 + 3121 = 3753 in^4 (15.62 dm^4 ).
- Determine the moment of inertia of the entire area—y axis
For this computation, subdivide element 1 into two triangles having the y axis as a base.
Thus
Element / about y axis, in^4 (dm^4 )
1 2e/, 2 )(6)(8)^3 = 512 (2.13)
2 yi2(4)(16)^3 = 1365 (5.68)
3 /2(0.785XS)^4 = 1607 (6.89)
/,= 3484(14.5)
- Compute the polar moment of inertia of the area
Apply the equation for the polar moment of inertia J 0 with respect to G: JG = Ix + Iy =
3753 + 3484 = 7237 in^4 (30.12 dm^4 ).
- Determine the moment of inertia of the entire area—w axis
Apply the equation in step 5 to determine the moment of inertia Iw of the entire area with
respect to the horizontal axis w through A. Thus k= 15.0 - 9.7 = 5.3 in (134.6 mm); Iw =
1, + AJc^1 = 3753 + 212.5(5.3)^2 = 9722 in^4 (40.46 dm^4 ).
- Compute the polar moment of inertia
Compute the polar moment of inertia of the entire area with respect to A. Then JA=Iw + Iy
= 9722 ± 3484 = 13,206 in^4 (54.97 dm^4 ).
