Moment of Inertia_Maj Sarwar

7/29/2019 Moment of Inertia_Maj Sarwar

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Since only mass has an INERTIA, The term MOMENT OFINERTIA OF AREA applied to an area is misnomer(inappropriate).

Last chapter (Chapter X- Centroid):The first moment of area, sometimes misnamed as thefirst moment of inertia:

First moment of area is commonly used in engineeringapplications to determine the centroid of an object orthe statical moment of area.

Chapter XI

Page: 206
http://en.wikipedia.org/wiki/Centroidhttp://en.wikipedia.org/wiki/Centroid


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From Centroid:

The moment of any area or First Moment of Inertia or First

Moment of Area is defined as the product of the area and

the perpendicular distance from the centroid of the area to

the moment axis. ; dAxAx dAxAy

Second moment of Inertia is defined as the product of thearea and square of the perpendicular distance from thecentroid of the area to the moment axis.

If Axis is in the plane of an area is called RectangularMoment of Inertia:

dAxIx2

dAyIy2


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Polar Moment of Inertia

If Axis is perpendicular to the area called

Polar Moment of Inertia:

dAydAxdArJ2

22

SHOW THE ANIMATED CLIP TO DEFINE

MOMENT OF INERTIA OF AREA


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Radius of Gyration

The Radius of Gyration k of an area is also amathematical conception defined by:

A rectangular and polar Radius of Gyration

Are respectably: I= k2A. andJ= k2A

kx =

Ix

A ky =

Iy

A kO =

JO

A


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yThe radius of gyration ofan areaAwith respect to

thex axis is defined asthe distance kx, where

Ix

= kx

A. With similar

definitions for the radii ofgyration ofA with respect

to they axis and withrespect to O, we have

x

kx

2

O

kx =IxA

ky =IyA

kO =JOA

A


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Why Second Moment of Inertia

also known as the Second Moment of the Areais a termused to describe the capacity of a cross-section to resistbending.

It is a mathematical property of a section concerned with a

surface area and how that area is distributed about thereference axis. The reference axis is usually a centroidal

axis.


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EXAMPLES


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Example 135 (Page: 208)

simple rectangular shape

I y dAz 2

bdydA

b

h/2

h/2

z

y

dy

12

883

3

3

33

2

2

3

2

2

2

bh

hhb

yb

bdyyI

h

h

h

hz

Centroidor Neutral axis


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APPLICATION


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I Is an Important value!

It is used to determine the state of stress in a section. It is used to calculate the resistance to bending.

It can be used to determine the amount of deflection in abeam.

b

h/2

h/2

z

y

y

b/2

b/2z

h

12

3bhI

z

12

3hbI

z>

Stronger section


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Some Practical Concept

Different types of Rectangular Section.Which one is more efficient to resistbending or deflection.

[Square/rectangular/hollow or box etc] Why we use corrugated tin instead of

plane sheet?

If a beam is rectangular not squared howto place it.

Why we use Truss in Mills or Factory?


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Example 136(Page: 208): Triangle

Determine the moment ofinertia of a triangle withrespect to its base.

A differential strip parallel to the xaxis is chosenfor dA.

dyldAdAydIx 2

For similar triangles,

dyh

yhbdA

h

yhbl

h

yh

b

l

Integrating dIx from y= 0 to y = h,

h

hh

x

yyh

h

b

dyyhyhbdy

hyhbydAyI

0

43

0

32

0

22

43

12

3bhIx


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Example 138: Circle

a) Determine the centroidal polarmoment of inertia of a circular

area by direct integration.

b) Using the result of part a,determine the moment ofinertia of a circular area withrespect to a diameter.

An annular differential area element is chosen,

rr

OO

O

duuduuudJJ

duudAdAudJ

0

3

0

2

2

22

2

4

2rJO

From symmetry, Ix= Iy,

xxyxO IrIIIJ 22

2 4

4

4rII xdiameter


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Next Class

Transfer Formula

Moment of Inertia_Maj Sarwar

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