Engineering Mechanics

(^116) „„„„„ A Textbook of Engineering Mechanics
Moment of inertia of the section about vertical axis passing through the centroid of the section
We know that moment of inertia of the rectangular section about the vertical axis passing
through its centre of gravity,
33
64
1
150 (120)
21.6 10 mm
G 12 12
db
I
×
== =× ...(i)
and area of one semicircular section with 50 mm radius,
22
(50) 3927 mm 2
22
r
a
ππ
== =
We also know that moment of inertia of a semicircular section about a vertical axis passing
through its centre of gravity,
IG2 = 0.11 r^4 = 0.11 × (50)^4 = 687.5 × 10^3 mm^4
and distance between centre of gravity of the semicircular section and its base
4450
21.2 mm
33
r ×
== =
ππ
∴ Distance between centre of gravity of the semicircular section and centre of gravity of
the whole section,
h 2 = 60 – 21.2 = 38.8 mm
and moment of inertia of one semicircular section about centre of gravity of the whole section,
(^) =+ =IahG 22223 (687.5 10 )× + ×[3927 (38.8) ]^264 =×6.6 10 mm
∴ Moment of inertia of both the semicircular sections about centre of gravity of the whole
section,
= 2 × (6.6 × 10^6 ) = 13.2 × 10^6 mm^4 ...(ii)
and moment of inertia of the whole section about a vertical axis passing through the centroid of the
section,
= (21.6 × 10^6 ) – (13.2 × 10^6 ) = 8.4 × 10^6 mm^4 Ans.
Example 7.14. Find the moment of inertia of a hollow section shown in Fig. 7.18. about an
axis passing through its centre of gravity or parallel X-X axis.
Solution. As the section is symmentrical about Y-Y axis,
therefore centre of a gravity of the section will lie on this axis.
Let y be the distance between centre of gravity of the section
from the bottom face.
(i) Rectangle
a 1 = 300 × 200 = 60 000 mm^2
and 1
300
150 mm
2
y ==
(ii) Circular hole
22
2 (150) 17 670 mm
4
a
π
=× =
and y 2 = 300 – 100 = 200 mm
We know that distance between the centre of gravity of the section and its bottom face,
11 2 2
12

• (60000 150) – (17670 200)
  
  
  • 60000 – 17670
  
  
  
  

ay a y

y

aa

××

== = 129.1 mm

∴ Moment of inertia of rectangular section about an axis through its centre of gravity and parallel

to X-X axis,

3

64

1

200 (300)

450 10 mm

12

IG

×

==×

Fig. 7.18.