1. Problem 9.187 y Determine the moments of inertia of the shaded areax y
Determine the moments of
inertia of the shaded area
shown with respect to the x
and y axes when a = 20 mm.

2. a x y
Solving Problems on Your Own
Determine the moments of
inertia of the shaded area
shown with respect to the x
and y axes when a = 20 mm.
Problem C
1. Compute the moments of inertia of a composite area with
respect to a given axis.
1a. Divide the area into sections. The sections should have a
shape for which the centroid and moments of inertia can be easily
determined (e.g. from Fig in the book).

3. a x y
Solving Problems on Your Own
Determine the moments of
inertia of the shaded area
shown with respect to the x
and y axes when a = 20 mm.
Problem C
1b. Compute the moment of inertia of each section. The moment
of inertia of a section with respect to the given axis is determined
by using the parallel-axis theorem:
I = I + A d2
Where I is the moment of inertia of the section about its own
centroidal axis, I is the moment of inertia of the section about the
given axis, d is the distance between the two axes, and A is the
section’s area.

4. a x y
Solving Problems on Your Own
Determine the moments of
inertia of the shaded area
shown with respect to the x
and y axes when a = 20 mm.
Problem C
1c. Compute the moment of inertia of the whole area. The
moment of inertia of the whole area is determined by adding the
moments of inertia of all the sections.

5. Divide the area into sections.x y
Problem Solution
Divide the area into sections. C y
4 a
3 p 2 C’ x’ B B x C 1
4 a
3 p
x’’
C’’ 3

6. ( Ix’)2 = (IBB)2 _ A d2 = p a4 _ p a2 ( )2 = p a4 _ a4y y
Problem Solution
4 a
3 p 2 C’ x’
Compute the
moment of
inertia of each
section. B B C x C 1
4 a
3 p
x’’
C’’ 3
Moment of inertia with respect to the x axis
(IBB)2 = p a4 1 8
For section 2:
(IBB)2 = ( Ix’)2 + A d2 1 8 1 2
4 a
3 p 1 8 8
9 p
( Ix’)2 = (IBB)2 _ A d2 = p a4 _ p a2 ( )2 = p a4 _ a4
(Ix)2 = ( Ix’)2 + A d2 = p a4 _ a p a2 ( a )2
4 a
3 p 1 8 8
9 p 1 2 5 8 4 3
(Ix)2 = pa a4

7. Ix = (Ix)1 + (Ix)2 + (Ix)3 = a4 + p a4 + a4 + p a4 + a4 4 3y y
Problem Solution
4 a
3 p 2 C’ x’ B B C x C 1
4 a
3 p
x’’
C’’ 3
(Ix)1 = (2a) (2a)3 = a4 1 12 4 3
For section 1: 5 8 4 3
For section 3:
(Ix)3 = (Ix)2 = p a a4
Compute the moment of
inertia of the whole area.
Moment of inertia of the whole area:
Ix = (Ix)1 + (Ix)2 + (Ix)3 = a p a a p a a4 4 3 5 8
(For a = 20 mm)
Ix = 4 a pa4 = x 106 mm4 5 4
Ix = x 106 mm4

8. Iy = (Iy)1 + (Iy)2 + (Iy)3 = a4 + p a4 + p a4 = a4 + p a4 4 3 1 8 x y y
Problem Solution
4 a
3 p 2 C’ x’ B B C x C 1
4 a
3 p
x’’ C’ 3
Moment of inertia with respect to the y axis 1 12 4 3
For section 1:
(Iy)1 = (2a) (2a)3 = a4 1 8
For section 3:
(Iy)3 = p a4 1 8
For section 2:
(Iy)2 = p a4
Moment of inertia of the whole area:
Iy = (Iy)1 + (Iy)2 + (Iy)3 = a p a p a4 = a p a4 4 3 1 8
Iy = 339 x 103 mm4
(For a = 20 mm)