ENGINEERING MECHANICS (STATICS I)
2021
CHAPTER 6
| AREA MOMENTS OF INERTIA
PART 1: DEFINITION:
Moment of Inertia (I): is also known as the Second Moment of the Area is a term used
to describe the capacity of a cross-section to resist bending. The deflection of a beam under load
depends not only on the load, but also on the geometry of the beam's cross-section. This is why
beams with higher area moments of inertia, such as I-beams, are so often seen in building
construction as opposed to other beams with the same area.
b Rectangular and Polar Moments of Inertia
Consider the area A in the x-y plane in figure below.
 The first moment of area dA is given by:
 xdA or  ydA
A
A
 An integral of the second moment of an area is referred to
as the moment of inertia for the area.

Fig. 1
The moments of inertia of the element dA about x- and yaxes are:
=
…moment of Inertia about x-axis
=
…moment of Inertia about y-axis
 Then the Second moments or moments of inertia of the
area A with respect to the x and y axes,
b The expressions I x 
moments of inertia.
Lecture Note
2
2
 y IdA  I yy2 dA
 x dA

x
are
the 2 rectangular
I called

 x dA
y
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ENGINEERING MECHANICS (STATICS I)
2021
b Second moment of differential area dA about the pole O or z axis is called the polar
moment of inertia,
Jo 
. For the entire area:
r
2
dA
A
r 2  x2  y2
Jo 
 (x
2
 y 2 ) dA  I x  I y
A

Rectangular moments of Inertia
I x   y 2 dA

I y   x 2 dA
Polar moment of inertia
J 0   r 2 dA

Note: The second moment of area cannot be negative. It is always positive.
PART 2: RADIUS OF GRYATION
The entire area, in fig.1, may be concentrated at a single point (
moment of area for a given reference. The distances
,
,
) to give the same second
are called the radii of gyration.
Thus,
=√
=√
kO2  k x2  k y2
=√
Lecture Note
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ENGINEERING MECHANICS (STATICS I)
2021
PART 3: PARALLEL AXIS-THEOREM [TRANSFER AXIS]

It is often necessary to get moment of inertia of an area about axes parallel to Centroidal
axes. So this theorem provides relationship between Centroidal moments of inertia and
moments of inertia about parallel axes.

Consider the figure given. The x’-y’ axes pass through the
Centroid C. Our aim is to determine the moments of inertia
of the area about the parallel x-y axes.

By definition, the moment of inertia of the element dA
about x-axis is given by
=(

)
and for the entire area,
= ∫(
)
=∫
+
∫
+
Fig. 2
∫
The first integral is the moment of inertia about the Centroid.
I x   y2 dA

The second component is the first moment area about the Centroid, i.e. about x’ axis.
̅
=∫
on the x’ axis.
̅ = 0, since this is the y-coordinte of the centroid of the area located
∫

=0
The third term is simply
0
=∫
= ̅

+
Thus, the expression for
Lecture Note
∫
+
∫
and the similar expression for
become
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ENGINEERING MECHANICS (STATICS I)
2021
̅
̅
Parallel Axis Theorems
̅
=
Lecture Note
+
and
̅ =
̅ + ̅
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ENGINEERING MECHANICS (STATICS I)
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Examples
Lecture Note
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Lecture Note
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Exercises:
Lecture Note
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PART 4: MOMENT OF INERTIA OF COMPOSITE BODIES

The moment of inertia of a composite area A about a given axis is obtained by adding the
moments of inertia of the component areas A1, A2, A3…, with respect to the same axis.
The procedures are given below.
1. Divide the composite area into simpler parts.
2. Compute the moment of inertia of each simple body about its Centroidal axis
from table.
3. Transfer each Centroidal moment of inertia to a parallel reference axis using
parallel axis theorem.
4. The sum of the moments of inertia for each simple body about the parallel
reference axis is the moment of inertia of the composite body.
5. Any cutout area (hole) must be assigned a negative moment; all others are
considered positive.

Usually it is convenient to tabulate the results for each of the parts in terms of its area A,
its Centroidal moment of inertia ,̅ the distance d from its centroidal axis to the axis about
which the moment of inertia of the entire section is computed, and the product Ad2.
̅
̅
∑ ̅
∑ ̅
A1
A2
A3
∑
Lecture Note
∑
∑
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Examples
Lecture Note
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Lecture Note
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Radius of gyration:
=√
=√
Lecture Note
=√
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Lecture Note
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Exercises
Lecture Note
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Lecture Note
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