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10. Moments of Inertia

HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien

Engineering Mechanics – Statics 10.01 Moments of Inertia

Chapter Objectives

• To develop a method for determining the moment of inertia for

an area

• To introduce the product of inertia and show how to determine

the maximum and minimum moments of inertia for an area

• To discuss the mass moment of inertia

HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien

Engineering Mechanics – Statics 10.02 Moments of Inertia

§1.Definition of Moments of Inertia for Areas

- The moment of inertia of a differential

area 𝑑𝐴 about an axis is given by

𝑑𝐼𝑥 = 𝑦2𝑑𝐴

- For the entire area, the moment of

inertia is obtained by integration

𝐼𝑥 =

𝐴

𝑦2𝑑𝐴

- Likewise, the moment of inertia about the 𝑦-axis is

𝑑𝐼𝑦 = 𝑥2𝑑𝐴

𝐼𝑦 =

𝐴

𝑥2𝑑𝐴

HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien

Engineering Mechanics – Statics 10.03 Moments of Inertia

§1.Definition of Moments of Inertia for Areas

- We may also take the moment of

inertia about an axis perpendicular to

the 𝑥 − 𝑦 plane through the point 𝑂

𝑑𝐽𝑂 = 𝑟2𝑑𝐴

𝑟: the perpendicular distance from 𝑑𝐴to the to the axis

- Integrating over the entire area we

obtain the polar moment of inertia as

𝐽𝑂 =

𝐴

𝑟2𝑑𝐴 = 𝐼𝑥 + 𝐼𝑦

- The moment of inertia is always a positive value with unit

𝑚4,𝑚𝑚4, …

HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien

Engineering Mechanics – Statics 10.04 Moments of Inertia

§1.Definition of Moments of Inertia for Areas

Tables and Manuals

- In practice we generally don’t need to do the integration

- Most Engineering manuals present Moments of Inertia of

common shapes that have been compiled in some chart or

table

HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien

Engineering Mechanics – Statics 10.05 Moments of Inertia

§2.Parallel-Axis Theorem for an Area

- The Parallel Axis Theorem enables us to find the moment of

inertia about any axis parallel to an axis about which the

moment of inertia is known

For the 𝑥 axis

𝐼𝑥 = 𝐼𝑥′ + 𝐴𝑑𝑦2

For the 𝑦 axis

𝐼𝑦 = 𝐼𝑦′ + 𝐴𝑑𝑥2

For the polar moment of inertia

𝐽𝑂 = 𝐽𝐶 + 𝐴𝑑2

HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien

Engineering Mechanics – Statics 10.06 Moments of Inertia

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§3.Radius of Gyration of an Area

- If the moments and areas are known, the radius of gyration

about an axis is given by

𝑘𝑥 =𝐼𝑥𝐴

For the 𝑦 axis

𝑘𝑦 =𝐼𝑦

𝐴

For the polar moment of inertia

𝑘𝑂 =𝐽𝑂𝐴

HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien

Engineering Mechanics – Statics 10.07 Moments of Inertia

§3.Radius of Gyration of an Area

- 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 of the rectangle, and (c)

the pole or 𝑧′ axis perpendicular to the 𝑥′ − 𝑦′ plane and

passing through the centroid 𝐶

Solution

𝐼𝑥′ = 𝐴

𝑦′2𝑑𝐴= −ℎ2

ℎ2𝑦′2𝑏𝑑𝑦′ =𝑏

−ℎ2

ℎ2𝑦′2𝑑𝑦′ =

1

12𝑏ℎ3

𝐼𝑥𝑏 = 𝐼𝑥′ + 𝐴𝑑𝑦2 =

1

12𝑏ℎ3 + 𝑏ℎ

ℎ

2

2

=1

3𝑏ℎ3

𝐼𝑦′ =1

12𝑏ℎ3, 𝐽𝐶 = 𝐼𝑥′ + 𝐼𝑦′ =

1

12𝑏ℎ(ℎ2 + 𝑏2)

HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien

Engineering Mechanics – Statics 10.08 Moments of Inertia

§3.Radius of Gyration of an Area

- Example 10.2 Determine the moment of inertia for the shaded

area about the 𝑥 axis

Solution

𝐼𝑥 = 𝐴

𝑦2 𝑑𝐴

= 0

200

𝑦2 100 − 𝑥 𝑑𝑦

= 0

200

𝑦2 100 −𝑦2

400𝑑𝑦

= 0

200

100𝑦2 −𝑦4

400𝑑𝑦

= 107 × 106𝑚𝑚4

HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien

Engineering Mechanics – Statics 10.09 Moments of Inertia

§3.Radius of Gyration of an Area

- Example 10.3 Determine the moment of inertia with respect to

the 𝑥 axis for the circular area

Solution

𝐼𝑥 = 𝐴

𝑦2 𝑑𝐴

= 𝐴

𝑦22𝑥𝑑𝑦

= −𝑎

𝑎

𝑦2 2 𝑎2 − 𝑦2 𝑑𝑦

=𝜋𝑎4

4

HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien

Engineering Mechanics – Statics 10.10 Moments of Inertia

§4.Moments of Inertia for Composite Areas

- A composite area is a series of connected simpler shapes

- The moment of inertia of a composite area is the algebraic

sum of the moments of inertia of the constituent parts

HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien

Engineering Mechanics – Statics 10.11 Moments of Inertia

§4.Moments of Inertia for Composite Areas

- Example 10.4 Determine the moment of inertia of the area

about the 𝑥 axis

Solution

The area can be obtained by subtracting the

circle from the rectangle. The centroid of

each area is located in the figure

• Circle

𝐼𝑥 = 𝐼𝑥′ + 𝐴𝑑𝑦2

=1

4𝜋(25)4+𝜋(25)2(75)2= 11.4 × 106𝑚𝑚4

• Rectangle

𝐼𝑥 = 𝐼𝑥′ + 𝐴𝑑𝑦2

= (100)(150)3+(100)(150)(75)2=112.5×106𝑚𝑚4

⟹ 𝐼𝑥 = −11.4 × 106 + 112.5 × 106 = 101 × 106𝑚𝑚4

HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien

Engineering Mechanics – Statics 10.12 Moments of Inertia