Higher Engineering Mathematics

SOME APPLICATIONS OF INTEGRATION 389

H

For rectangle F:

IXX=

bl^3

3

=

(15.0)(4.0)^3

3

=320 cm^4

Total second moment of area for theI-section
about axisXX,

IXX= 3768 + 1267 + 320 =5355 cm^4

Total area ofI-section

=(8.0)(3.0)+(3.0)(7.0)+(15.0)(4.0)

=105 cm^2.

Radius of gyration,

kXX=

√

IXX

area

=

√(

5355

105

)

=7.14 cm

Now try the following exercise.

Exercise 153 Further problems on second

moment of areas of regular sections

1. Determine the second moment of area and
   radius of gyration for the rectangle shown in
   Fig. 38.27 about (a) axisAA(b) axisBBand
   (c) axisCC.
   ⎡

⎣

(a) 72 cm^4 ,1.73 cm

(b) 128 cm^4 ,2.31 cm

(c) 512 cm^4 ,4.62 cm

⎤

⎦

Figure 38.27

2. Determine the second moment of area and
   radius of gyration for the triangle shown in
   Fig. 38.28 about (a) axisDD(b) axisEE
   and (c) an axis through the centroid of the
   triangle parallel to axisDD.
   ⎡

⎣

(a) 729 cm^4 ,3.67 cm

(b) 2187 cm^4 ,6.36 cm

(c) 243 cm^4 ,2.12 cm

⎤

⎦

Figure 38.28

3. For the circle shown in Fig. 38.29, find the
   second moment of area and radius of gyra-
   tion about (a) axisFF and (b) axisHH.

[

(a) 201 cm^4 ,2.0cm

(b) 1005 cm^4 ,4.47 cm

]

Figure 38.29

4. For the semicircle shown in Fig. 38.30, find
   the second moment of area and radius of
   gyration about axisJJ.

[3927 mm^4 , 5.0 mm]

Figure 38.30

5. For each of the areas shown in Fig. 38.31
   determine the second moment of area and
   radius of gyration about axisLL, by using
   the parallel axis theorem.
   ⎡

⎣

(a) 335 cm^4 ,4.73 cm

(b) 22030 cm^4 ,14.3cm

(c) 628 cm^4 ,7.07 cm

⎤

⎦