Problem 9.187

1
Problem 9.187
2
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.
9.12 in the book).
3
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 sections area.
4
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
Problem 9.187 Solution
Divide the area into sections.
C
y
C
x
B
B
x
C
x
C
6
Problem 9.187 Solution
y
C
Compute the moment of inertia of each section.
x
B
B
x
C
C
x
C
Moment of inertia with respect to the x axis
For section 2
(IBB)2 ( Ix)2 A d2
( Ix)2 (IBB)2 _ A d2 p a4 _ p a2 (
)2 p a4 _ a4
(Ix)2 pa4 a4
7
Problem 9.187 Solution
y
C
x
B
B
x
C
C
x
C
For section 1
(Ix)3 (Ix)2 p a4 a4
For section 3
Compute the moment of inertia of the whole area.
Moment of inertia of the whole area
(For a 20 mm)
Ix 1.268 x 106 mm4
8
y
Problem 9.187 Solution
C
x
B
B
x
C
C
x
C
Moment of inertia with respect to the y axis
(Iy)1 (2a) (2a)3 a4
For section 1
(Iy)2 p a4
For section 2
Moment of inertia of the whole area
Iy 339 x 103 mm4
(For a 20 mm)