1/8/2011

1

STATICS: CE201

Chapter 10

Moments of Inertia

Notes are based on Engineering Mechanics: Statics by R. C. Hibbeler, 12th Edition, Pearson

Dr M. Touahmia & Dr M. Boukendakdji

Civil Engineering Department, University of Hail

(Fall 2010)

10. Moments of Inertia________________________________________________________________________________________________________________________________________________

Chapter Objective:

1. Define the moments of inertia (MoI) for an area.

2. Determine the MoI for an area by integration.

Contents:

10.1 Definition of Moment of Inertia for Areas

10.2 Parallel-Axis Theorem for an Area

10.3 Radius of Gyration of an Area

10.4 Moments of Inertia for Composite Areas

Chapter 10: Moment of Inertia

1/8/2011

2

10.1 Definition of Moments of Inertia for Areas

Chapter10 - Moment of Inertia

Many structural members like

beams and columns have cross

sectional shapes like I, L, C, etc.

Some others are made of tubes

rather than solid squares or rounds.

Why do they usually not have solid

rectangular, square, or circular

cross sectional areas?

What primary property of these

members influences design

decisions? How can we calculate

this property?

10.1 Definition of Moments of Inertia for Areas

Consider three different possible cross sectional shapes and areas

for the beam RS. All have the same total area and, assuming they

are made of same material, they will have the same mass per unit

length.

For the given vertical loading F on the beam, which shape will

develop less internal stress and deflection? Why?

The answer depends on the MoI of the beam about the x-axis. It

turns out that Section (A) has the highest MoI because most of the

area is farthest from the x axis. Hence, it has the least stress and

deflection (σ = M.y/I); as I increases, σ or stress decreases.

Chapter10 - Moment of Inertia

1/8/2011

3

10.1 Definition of Moments of Inertia for Areas

Chapter10 - Moment of Inertia

The area moment of Inertia represents the second moment

of the area about an axis. It is frequently used in formulas

related to the strength and stability of structural members.

The moments of inertia of a differential

area dA about the x and y are:

The moment of inertia of dA about the

“pole” O or z axis is then:

For the entire area A, the moments of inertia are determined

by integration:

The polar moment of inertia is:

dAxIA

y 2 dAyI

Ax

2

dAydI x2 dAxdI y

2

dArdJO2

yxA

O IIdArJ 2

The step-by-step procedure for analysis is:

Chapter10 - Moment of Inertia

1. Choose the element dA: There are two choices: a vertical

strip or a horizontal strip. Some considerations about this

choice are:

2. Integrate to find the MoI.

a) The element parallel to the axis about

which the MoI is to be determined usually

results in an easier solution. For example,

we typically choose a horizontal strip for

determining Ix and a vertical strip for

determining Iy.

b) If y is easily expressed in terms of x

(e.g., y = x2 + 1), then choosing a vertical

strip with a differential element dx wide

may be advantageous.

1/8/2011

4

Area Moment of Inertia of Common Shapes:

Chapter10 - Moment of Inertia

A filled rectangular area with a base width of b and height h:

A filled rectangular area as above but with respect to an axis

collinear with the base:

A filled circular area of radius r:

10.2 Parallel-Axis Theorem for an Area

Chapter10 - Moment of Inertia

The parallel-axis theorem can be used to find the moment of

inertia of an area about any axis that is parallel to an axis passing

through the Centroid and about which the moment of inertia is

known ( ):

The moment of inertia for an area about an axis is equal to its

moment of inertia about a parallel axis passing through the area’s

Centroid plus the product of the area and the square of the

perpendicular distance between the axes.

2

yxx AdII

2

xyy AdII

2AdJJ CO

yI xI CJ

1/8/2011

5

10.3 Radius of Gyration of an area

Chapter10 - Moment of Inertia

For a given area A and its MoI, Ix , imagine that the entire

area is located at distance kx from the x axis:

This kx is called the radius of gyration of the area about the

x axis. Similarly:

AkI xx

2A

Ik x

x

A

kx

y

x

A

Ik

y

y

A

Jk O

O

Example 1

Chapter10 - Moment of Inertia

Determine the moment of inertia for the rectangle area

shown in the figure with respect to:

(a) The Centroid x’ axis.

(b) The axis xb passing through the

base of the rectangle.

(c) The pole or z’ axis perpendicular

to the x’- y’ plan and passing through the

Centroid C.

1/8/2011

6

Solution 1

Chapter10 - Moment of Inertia

(a) MoI about x’ axis: Integration from y’=- h/2 to y’= h/2

(b) MoI about xb:

(c) Polar MoI about point C:

3

2

2

2

2

2

22

12

1bhydybybdydAyI

h

h

h

hA

x

3

2

32

3

1

212

1bh

hbhbhAdII yxxb

3

12

1hbI y

22

12

1bhbhIIJ yxC

Example 2

Chapter10 - Moment of Inertia

Determine the moment of inertia for the shaded area shown

in the figure about the x axis.

1/8/2011

7

Solution 2

Chapter10 - Moment of Inertia

A differential element dA parallel to the x axis is chosen for

integration:

Integration with respect to y, from

y = 0 to y = 200 mm, yields:

46

mm200

0

42

m200

0

22

mm200

0

22

mm 10.107400

100y

400100y100

dyy

dyy

dyxydAyIA

x

dyxdA 100

10.4 Moments of Inertia for Composite Areas

A composite area is made by adding or subtracting a

series of “simple” shaped areas like rectangles,

triangles, and circles.

For example, the area on the figure can be made from a

rectangle minus a triangle and circle.

Chapter2 - Force Vectors

The MoI of these “simpler” shaped areas

about their Centroidal axes are found in most

engineering handbooks.

Using these data and the parallel-axis

theorem, the MoI for a composite area can

easily be calculated.

The MoI of the composite area is equal to the

algebraic sum of the moments of inertia of

each of its parts.

1/8/2011

8

Example 3

Chapter10 - Moment of Inertia

Determine the moment of inertia of the area shown in the

figure about the x axis.

Solution 3

Chapter10 - Moment of Inertia

Composite Parts:

1. Divide the given area into its simpler shaped parts.

2. Locate the Centroid of each part and indicate the perpendicular

distance from each Centroid to the desired reference axis.

=

1/8/2011

9

Solution 3

Chapter10 - Moment of Inertia

Parallel-Axis Theorem:

(a) Circle:

(b) Rectangle:

Algebraic Summation: The MoI for the entire area is the

algebraic summation of the individual MoI:

462222 mm 104.117525254

1 yxx AdII

46232 mm 105.1127515010015010012

1 yxx AdII

466 mm 101.101104.11105.112 xI