7/29/2019 Moment of Inertia_Maj Sarwar
1/15
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/Centroid7/29/2019 Moment of Inertia_Maj Sarwar
2/15
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
7/29/2019 Moment of Inertia_Maj Sarwar
3/15
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
7/29/2019 Moment of Inertia_Maj Sarwar
4/15
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
7/29/2019 Moment of Inertia_Maj Sarwar
5/15
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
7/29/2019 Moment of Inertia_Maj Sarwar
6/15
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.
7/29/2019 Moment of Inertia_Maj Sarwar
7/15
EXAMPLES
7/29/2019 Moment of Inertia_Maj Sarwar
8/15
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
7/29/2019 Moment of Inertia_Maj Sarwar
9/15
APPLICATION
7/29/2019 Moment of Inertia_Maj Sarwar
10/15
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
7/29/2019 Moment of Inertia_Maj Sarwar
11/15
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?
7/29/2019 Moment of Inertia_Maj Sarwar
12/15
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
7/29/2019 Moment of Inertia_Maj Sarwar
13/15
7/29/2019 Moment of Inertia_Maj Sarwar
14/15
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
7/29/2019 Moment of Inertia_Maj Sarwar
15/15
Next Class
Transfer Formula