Dr. Aslam Al-Omari
Statics
(805221)
Chapter 10: Moments of Inertia
Prepared by:
Dr. Aslam Al-Omari
Dr. Aslam Al-Omari
10.1) Definition of Moments of
Inertia for Areas
ƒ
The first moment of area:
ƒ
An integral of the second moment of an area is referred to as
the moment of inertia for the area
Ix =
ƒ
∫
y 2 dA
Iy =
Polar moment of Inertia:
∫
= ∫ (y
JO =
r 2 dA
2
+ x 2 )dA
= Ix + Iy
∫
∫ xdA
x 2 dA
∫
ydA
Dr. Aslam Al-Omari
10.2) Parallel-Axis Theorem
for an Area
I x = I x ' + Ad
2
y
I y = I y ' + Ad
2
J O = J C + Ad
x
2
Dr. Aslam Al-Omari
10.3) Radius of Gyration of
An Area
kx =
Ix
A
ky =
kO =
JO
A
Iy
A
Dr. Aslam Al-Omari
10.4) Moments of Inertia for an
Area by Integration
Example 10.1:
Determine the moment of
inertia for the rectangular
area with respect to:
a) The centroidal x′ axis
b) The axis xb passing through
the base
c) The pole or z′ axis
perpendicular to the x′ and
y′ plane and passing
through the centroid C
Dr. Aslam Al-Omari
10.4) Moments of Inertia for an
Area by Integration
a)
I x' =
∫
y' 2 dA
A
h
=
2
y' 2 (bdy ' )
∫
−h
2
h
= b
2
∫
−h
y' 2 dy '
2
1
=
bh 3
12
Dr. Aslam Al-Omari
10.4) Moments of Inertia for an
Area by Integration
b) I = I + Ad
x
x'
b
2
y
1
⎛h⎞
3
=
bh + bh ⎜ ⎟
12
⎝2⎠
1
= bh 3
3
2
Dr. Aslam Al-Omari
10.4) Moments of Inertia for an
Area by Integration
c)
I y'
1
=
hb 3
12
J C = I x' + I y'
1
1
3
=
bh +
hb 3
12
12
1
=
bh (h 2 + b 2 )
12
Dr. Aslam Al-Omari
10.5) Moments of Inertia for
Composite Areas
Example 10.5:
Compute the moment
of inertia of the
composite area shown
about the x-axis
Dr. Aslam Al-Omari
10.5) Moments of Inertia for
Composite Areas
Example 10.5:
=
Dr. Aslam Al-Omari
10.5) Moments of Inertia for
Composite Areas
Example 10.5:
For the circle (NEGATIVE):
I x = I x ' + Ad
=
1
π (25
4
2
y
)
4
[
+ π (25
) ](75 )
2
2
= 11 . 4 × 10
6
mm
4
For the Rectangle (POSITIVE):
I x = I x ' + Ad
=
1
(100
12
2
y
)(150 )
3
+
[(100 )(150 )](75 )
2
= 112 . 5 × 10
6
mm
For the Whole Area (SUMMATION):
I x = 112 . 5 × 10
6
− 11 . 4 × 10
6
= 101 . 1 × 10
6
mm
4
4
Dr. Aslam Al-Omari
10.5) Moments of Inertia for
Composite Areas
Example 10.6:
Determine the
moment of inertia
of the beam’s crosssectional area
shown about the xand y- centroidal
axes.
Dr. Aslam Al-Omari
10.5) Moments of Inertia for
Composite Areas
Example 10.6:
=
Dr. Aslam Al-Omari
10.5) Moments of Inertia for
Composite Areas
Examples:
Dr. Aslam Al-Omari
10.5) Moments of Inertia for
Composite Areas
Examples:
Dr. Aslam Al-Omari
10.5) Moments of Inertia for
Composite Areas
Examples:
Dr. Aslam Al-Omari
10.5) Moments of Inertia for
Composite Areas
Examples:
Dr. Aslam Al-Omari
10.5) Moments of Inertia for
Composite Areas
Examples: