Moments of Inertia

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โมเมนต์�ที่��สองของพื้��นที่�� • Stress within the beam varies linearly with the

distance from an axis passing through the centroid C of the beam’s cross-sectional area

σ = kz• For magnitude of the force acting

on the area element dA

dF = σ dA = kz dA

Definition of Moments of Inertia for Areas• Since this force is located a distance z from the y

axis, the moment of dF about the y axis

dM = dF = kz2 dA• Resulting moment of the entire stress distribution

= applied moment M

• Integral represent the moment of inertia of area about the y axis

dAzM 2

Moment of Inertia • Consider area A lying in the x-y plane• Be definition, moments of inertia of the

differential plane area dA about the x and y axes

• For entire area, moments of inertia are given by

Ay

Ax

yx

dAxI

dAyI

dAxdIdAydI

2

2

22

Moment of Inertia • Formulate the second moment of dA about the

pole O or z axis• This is known as the polar axis

where r is perpendicular from the pole (z axis) to the element dA

• Polar moment of inertia for entire area,

yxAO

O

IIdArJ

dArdJ

2

2

Parallel Axis Theorem for an Area• For moment of inertia of an area known about an

axis passing through its centroid, determine the moment of inertia of area about a corresponding parallel axis using the parallel axis theorem

• Consider moment of inertia

of the shaded area• A differential element dA is

located at an arbitrary distance

y’ from the centroidal x’ axis

• The fixed distance between the parallel x and x’ axes is defined as dy

• For moment of inertia of dA about x axis

• For entire area

• First integral represent the moment of inertia of the area about the centroidal axis

AyAyA

A yx

yx

dAddAyddAy

dAdyI

dAdydI

22

2

2

'2'

'

'

• Second integral = 0 since x’ passes through the area’s centroid C

• Third integral represents the total area A

• Similarly

• For polar moment of inertia about an axis perpendicular to the x-y plane and passing through pole O (z axis)

2

2

2

0;0'

AdJJ

AdII

AdII

ydAydAy

CO

xyy

yxx

Moments of Inertia for an Area by Integration

Example 10.1Determine 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 rectangular, and (c) the pole or z’ axis perpendicular to the x’-y’ plane and passing through the centroid C.

Solution

Part (a)• Differential element chosen, distance y’ from x’

axis• Since dA = b dy’

3

2/

2/

22

12

1

''

bh

dyydAyIh

hAx

Solution

Part (b)• Moment of inertia about an axis passing through

the base of the rectangle obtained by applying parallel axis theorem

32

3

2

3

1

212

1bh

hbhbh

AdII xxb

Solution

Part (c)• For polar moment of inertia about point C

)(12

1

12

1

22

'

3'

bhbh

IIJ

hbI

yxC

y

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Example 10.5

Compute the moment of

inertia of the composite

area about the x axis.

Solution

Composite Parts• Composite area obtained

by subtracting the circle form the rectangle

• Centroid of each area is located in the figure

Solution

Parallel Axis Theorem• Circle

• Rectangle

4623

2'

46224

2'

105.1127515010015010012

1

104.117525254

1

mm

AdII

mm

AdII

yxx

yxx

Solution

Summation• For moment of inertia for the composite area,

46

66

10101

105.112104.11

mm

I x

Example 10.6

Determine the moments

of inertia of the beam’s

cross-sectional area

about the x and y

centroidal axes.

Solution

Composite Parts

• Considered as 3 composite areas A, B, and D

• Centroid of each area is located in the figure

Solution

Parallel Axis Theorem• Rectangle A

4923

2'

4923

2'

1090.125030010010030012

1

10425.120030010030010012

1

mm

AdII

mm

AdII

xyy

yxx

Solution

Parallel Axis Theorem• Rectangle B

493

2'

493

2'

1080.160010012

1

1005.010060012

1

mm

AdII

mm

AdII

xyy

yxx

Solution

Parallel Axis Theorem• Rectangle D

4923

2'

4923

2'

1090.125030010010030012

1

10425.120030010030010012

1

mm

AdII

mm

AdII

xyy

yxx

Solution

Summation• For moment of inertia for the entire cross-

sectional area,

49

999

49

999

1060.5

1090.11080.11090.1

1090.2

10425.11005.010425.1

mm

I

mm

I

y

x