Statics, fourteenth edition by r c hibbeler section 10 1

DEFINITION OF MOMENTS OF INERTIA FOR AREAS, RADIUS OF GYRATION OF AN
AREA

Today’s Objectives:
Students will be able to:

In-Class Activities:

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

•

Check Homework, if any

b) Determine the MoI for an area by integration.

•

Reading Quiz

•

Applications

•

MoI: Concept and Definition

•

MoI by Integration

•

Concept Quiz

•

Group Problem Solving

•

Attention Quiz

Statics, Fourteenth Edition
R.C. Hibbeler

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READING QUIZ

1. The definition of the Moment of Inertia for an area involves an integral of the form
A) ∫ x dA.

2
B) ∫ x dA.

2
C) ∫ x dm.

D) ∫ m dA.

2. Select the correct SI units for the Moment of Inertia for an area.
3
A) m
4
B) m
2
C) kg·m
3
D) kg·m

Statics, Fourteenth Edition
R.C. Hibbeler

Copyright ©2016 by Pearson Education, Inc.
All rights reserved.


APPLICATIONS

Many structural members like beams and columns have cross sectional shapes like an I, H, C, etc..

Why do they usually not have solid rectangular, square, or circular cross sectional areas?

What primary property of these members influences design decisions?

Statics, Fourteenth Edition
R.C. Hibbeler

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APPLICATIONS (continued)

Many structural members are made of tubes rather than solid
squares or rounds.
Why?

This section of the book covers some parameters of the cross
sectional area that influence the designer’s selection.
Do you know how to determine the value of these parameters for a
given cross-sectional area?

Statics, Fourteenth Edition
R.C. Hibbeler

Copyright ©2016 by Pearson Education, Inc.
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DEFINITION OF MOMENTS OF INERTIA FOR AREAS
(Section 10.1)

Consider a plate submerged in a liquid. The pressure of a liquid at a
distance y below the surface is given by p = γ y, where γ is the
specific weight of the liquid.

The force on the area dA at that point is dF = p dA.
The moment about the x-axis due to this force is y (dF).

2
The total moment is ∫A y dF = ∫A γ y dA =

2
γ ∫A( y dA).
This sort of integral term also appears in solid mechanics when determining stresses and deflection.
This integral term is referred to as the moment of inertia of the area of the plate about an axis.

Statics, Fourteenth Edition
R.C. Hibbeler

Copyright ©2016 by Pearson Education, Inc.
All rights reserved.


DEFINITION OF MOMENTS OF INERTIA FOR AREAS

10cm

3cm
10cm
10cm

1cm

P

3cm

x

(A)

(B)

(C)

R

S

1cm

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 P on the beam shown on the right, 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.

Statics, Fourteenth Edition
R.C. Hibbeler

Copyright ©2016 by Pearson Education, Inc.
All rights reserved.

DEFINITION OF MOMENTS OF INERTIA FOR AREAS

For the differential area dA, shown in the figure:
d Ix =

2
y dA ,

d Iy

2
x dA , and,

=

d JO =

2
r dA , where JO is the polar moment of inertia about the pole

O or z axis.

The moments of inertia for the entire area are obtained by integration.
Ix =

2
∫A y dA ;

JO =

∫A r

2

dA =

Iy =
∫A ( x

2
∫A x dA
2

2
+ y ) dA =

Ix +

Iy

4
The MoI is also referred to as the second moment of an area and has units of length to the fourth power (m or
4
in ).

Statics, Fourteenth Edition
R.C. Hibbeler

Copyright ©2016 by Pearson Education, Inc.

All rights reserved.


MoI FOR AN AREA BY INTEGRATION

For simplicity, the area element used has a differential size in only one
direction
(dx or dy). This results in a single integration and is usually simpler than
doing a double integration with two differentials, i.e., dx·dy.

The step-by-step procedure is:
1. Choose the element dA: There are two choices: a vertical strip or a horizontal strip. Some considerations about this
choice are:
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 I x and a vertical strip for determining Iy.

Statics, Fourteenth Edition
R.C. Hibbeler

Copyright ©2016 by Pearson Education, Inc.
All rights reserved.


MoI FOR AN AREA BY INTEGRATION (continued)

2
b) If y is easily expressed in terms of x (e.g., y = x + 1), then choosing a
vertical strip with a differential element dx wide may be advantageous.

2.

Integrate to find the MoI. For example, given the element shown in the figure above:
Iy =

2
2
∫ x dA = ∫ x y dx

Ix =

∫ d Ix =

and

3
∫ (1 / 3) y dx (using the parallel axis theorem as per Example 10.2 of the textbook).

Since the differential element is dx, y needs to be expressed in terms of x and the integral limit must also be in terms of x.
As you can see, choosing the element and integrating can be challenging. It may require a trial and error approach, plus
experience.

Statics, Fourteenth Edition
R.C. Hibbeler

Copyright ©2016 by Pearson Education, Inc.
All rights reserved.


EXAMPLE

Given: The shaded area shown in the figure.
Find: The MoI of the area about the
Plan:

•

x- and y-axes.

Follow the steps given earlier.

(x,y)

Solution:
Ix

=

2
∫ y dA

dA =

(1 – x) dy =

Ix

0∫ y

=

2

1

(1 – y

= [ (1/3) y

3

(1 – y

3/2

) dy

) dy

– (2/9) y

Statics, Fourteenth Edition
R.C. Hibbeler

3/2

9/2

]0 =

4

0.111 m

1

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EXAMPLE (continued)

Iy

• (x,y)

=

2
∫ x dA =

∫x

=

2 2/3
∫ x (x ) dx

=

8/3
∫

x
dx
0

2

y dx

1

= [ (3/11) x
=

11/3

0.273 m

]0

1

4

In the above example, Ix can be also determined using a vertical strip.
Then Ix =

3
2
4
∫ (1/3) y dx = 0∫ (1/3) x dx = 1/9 = 0.111

m
1

Statics, Fourteenth Edition
R.C. Hibbeler

Copyright ©2016 by Pearson Education, Inc.
All rights reserved.


CONCEPT QUIZ
1. A pipe is subjected to a bending moment as shown. Which property
M

of the pipe will result in lower stress (assuming a constant cross-

M

y

sectional area)?

x
A) Smaller Ix
C) Larger Ix

B) Smaller Iy

Pipe section

D) Larger Iy

2. In the figure to the right, what is the differential moment of inertia of the

y

3
y=x

element with respect to the y-axis (dIy)?
2
A) x y dx
C) y

2

x dy

B) (1/12) x

3

x,y

dy

D) (1/3) y dy
x

Statics, Fourteenth Edition

R.C. Hibbeler

Copyright ©2016 by Pearson Education, Inc.
All rights reserved.


GROUP PROBLEM SOLVING

Given: The shaded area shown.
Find:

Ix and Iy of the area.
(x,y)

Plan:

Follow the procedure described earlier.
•

y
dx

Statics, Fourteenth Edition
R.C. Hibbeler

Copyright ©2016 by Pearson Education, Inc.
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GROUP PROBLEM SOLVING (continued)

Solution:
3
The moment of inertia of the rectangular differential element about the x-axis is dIx = (1/3) y dx (see Case 2 in Example
10.2 in the textbook).
 

(x,y)

 
•

y
dx

 

Statics, Fourteenth Edition
R.C. Hibbeler

Copyright ©2016 by Pearson Education, Inc.
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GROUP PROBLEM SOLVING (continued)

The moment of inertia about the y-axis
 
(x,y)

 

where
•

y
dx

 

 

Statics, Fourteenth Edition
R.C. Hibbeler

Copyright ©2016 by Pearson Education, Inc.
All rights reserved.


ATTENTION QUIZ

1. When determining the MoI of the element in the figure, dI y equals
A) x

2

dy

B) x

3

C) (1/3) y dx D) x

2

2.5

dx

(x,y)

2
y =x

dx

2. Similarly, dIx equals
A) (1/3) x

1.5

C) (1 /12) x

3

2

dx

B) y

dy

D) (1/3) x

Statics, Fourteenth Edition
R.C. Hibbeler

dA
3

dx

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End of the Lecture
Let Learning Continue

Statics, Fourteenth Edition
R.C. Hibbeler

Copyright ©2016 by Pearson Education, Inc.
All rights reserved.