Page 41 - Handbook of Civil Engineering Calculations, Second Edition

1.24 STRUCTURAL STEEL ENGINEERING AND DESIGN

Then y m 2051/212.5 9.7 in. (246.4 mm). Since the area is symmetric with respect
to the y axis, this is also a centroidal axis. The intersection point G of the x and y axes is,
therefore, the centroid of the area.
4. 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 k 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).
5. Compute the moment of inertia of the entire area—x axis
Let I 0 denote the moment of inertia of an element with respect to its horizontal centroidal
axis and A its area. Compute the moment of inertia I x of the entire area with respect to the
2
x axis by applying the transfer equation I x I 0 AK . Thus

4
2
4
4
Element I 0 , in (dm ) Ak , in (dm )
4
1 1 /36(16)(6) 96 (0.40) 48(5.7) 1560 (6.49)
3
2
2
2 1 /12(16)(4) 85 (0.35) 64(1.7) 185 (0.77)
3
3 0.110(8) 451 (1.88) 100.5(3.7) 1376 (5.73)
2
4
_________
__________
Total 632 (2.63) 3121 (12.99)
4
4
Then, I x 632 3121 3753 in (15.62 dm ).
6. 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
4
4
Element I about y axis, in (dm )
1 2( 1 / 12)(6)(8) 0512 (2.13)
3
3
2 1 / 12(4)(16) 1365 (5.68)
3 1 / 2(0.785)(8) 1607 (6.89)
4
__________
I y 3484 (14.5)
7. Compute the polar moment of inertia of the area
Apply the equation for the polar moment of inertia J G with respect to G: J G I x I y
4
4
3753 3484 7237 in (30.12 dm ).
8. Determine the moment of inertia of the entire area—w axis
Apply the equation in step 5 to determine the moment of inertia I w 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); I w
4
2
2
4
I x Ak 3753 212.5(5.3) 9722 in (40.46 dm ).
9. Compute the polar moment of inertia
Compute the polar moment of inertia of the entire area with respect to A. Then J A I w
4
4
I y 9722 ± 3484 13,206 in (54.97 dm ).