Higher Engineering Mathematics

388 INTEGRAL CALCULUS

Problem 18. Determine correct to 3 significant

figures, the second moment of area about axis

XXfor the composite area shown in Fig. 38.25.

X 4.0 cm X

1.0 cm 1.0 cm

8.0 cm

6.0 cm

T T

2.0 cm 2.0 cm

CT

Figure 38.25

For the semicircle,

IXX=

πr^4

8

=

π(4.0)^4

8

= 100 .5cm^4

For the rectangle,

IXX=

bl^3

3

=

(6.0)(8.0)^3

3

=1024 cm^4

For the triangle, about axisTTthrough centroidCT,

ITT=

bh^3

36

=

(10)(6.0)^3

36

=60 cm^4

By the parallel axis theorem, the second moment of

area of the triangle about axisXX

= 60 +

[ 1

2 (10)(6.0)

][

8. 0 +^13 (6.0)

] 2

=3060 cm^4.

Total second moment of area aboutXX

= 100. 5 + 1024 + 3060

= 4184. 5

=4180 cm^4 , correct to 3 significant figures

Problem 19. Determine the second moment of

area and the radius of gyration about axisXXfor

theI-section shown in Fig. 38.26.

Figure 38.26

TheI-section is divided into three rectangles,D,E

andFand their centroids denoted byCD,CEandCF

respectively.

For rectangle D:

The second moment of area aboutCD (an axis

throughCDparallel toXX)

=

bl^3

12

=

(8.0)(3.0)^3

12

=18 cm^4

Using the parallel axis theorem:

IXX= 18 +Ad^2

whereA=(8.0)(3.0)=24 cm^2 andd= 12 .5cm

HenceIXX= 18 +24(12.5)^2 =3768 cm^4.

For rectangle E:

The second moment of area about CE (an axis

throughCEparallel toXX)

=

bl^3

12

=

(3.0)(7.0)^3

12

= 85 .75 cm^4

Using the parallel axis theorem:

IXX= 85. 75 +(7.0)(3.0)(7.5)^2 =1267 cm^4.