chapter 4. ppt moment of inertia strength of material

noment of inertia

1. Chapter No. 04
Moment Of Inertia

2. C304.4
Determine Area Moment of Inertia of regular and composite
sections.

3. MOMENT OF INERTIA
Moment of Inertia:
 
The product of the elemental area and square of the
perpendicular distance between the centroid of area and the
axis of reference is the “Moment of Inertia” about the
reference axis.
 
I
xx
= ∫dA. y
2
I
yy
= ∫dA. x
2
x
y
x
y
dA

4. It is also called second moment of area because first
moment of elemental area is dA.y and dA.x; and if it is
again multiplied by the distance,we get second
moment of elemental area as (dA.y)y and (dA.x)x.

5. Polar moment of Inertia
(Perpendicular Axes theorem)
The moment of inertia of an area about an axis perpendicular
to the plane of the area is called “Polar Moment of Inertia”
and it is denoted by symbol I
zz
or J or I
p
. The moment of
inertia of an area in xy plane w.r.to z. axis is I
zz
= I
p
= J =
∫r
2
dA = ∫(x
2
+ y
2
) dA = ∫x
2
dA + ∫y
2
dA = I
xx
+I
yy
 
O
y
x
r
z
x
Y

6.  Hence polar M.I. for an area w.r.t. an axis perpendicular to
its plane of area is equal to the sum of the M.I. about any
two mutually perpendicular axes in its plane, passing
through the point of intersection of the polar axis and the
area.
PERPENDICULAR AXIS THEOREM

7. Parallel Axis Theorem
y
x
x
x
0
x
0*
G

8. Ixx = dA. y2
∫
_
= dA (d +y')
∫
2
_ _
= dA (d
∫
2
+ y'
2
+ 2dy')
_
= dA. d
∫
2
+ dAy
∫
΄
2
+ 2d.dAy'
∫
_
d
2
dA = A.(d)
∫
2

∫
dA. y'2 = Ix
0
x
0

_
2d dAy’ = 0
∫
( since Ist moment of area about centroidal axis = 0)
_
I
x x
= I
x
0
x
0
+Ad
2

9. Hence, moment of inertia of any area about an axis xx is
equal to the M.I. about parallel centroidal axis plus the
product of the total area and square of the distance
between the two axes.
Radius of Gyration
It is the perpendicular distance at which the whole area may be
assumed to be concentrated, yielding the same second
moment of the area above the axis under consideration.

10. I
yy
= A.r
yy
2
I
xx
= A.r
xx
2
r
yy
= √ I
yy
/A
And r
xx
= √ I
xx
/A

y
y
A
r
xx
A
x x
r
xx
and r
yy
are called the radii of gyration
r
yy

11. MOMENT OF INERTIA BY DIRECT INTEGRATION
d
dy
x
0
xx
d/2
x
0
y
b
M.I. about its horizontal centroidal axis :
G
.
RECTANGLE :
 

I
XoXo
=
-d/2
∫
+d/2
dAy
2
=
-d/2
∫
+d/2
(b.dy)y
2


= bd
3
/12
 About its base
I
XX
=I
XoXo
+A(d)
2
Where d = d/2, the
distance between axes xx
and x
ox
o

=bd
3
/12+(bd)(d/2)
2
=bd
3
/12+bd
3
/4=bd
3
/3

12. h
x
0
x
h/3
x
0
b
x
y
(h-y)
dy
(2) TRIANGLE :
 
 (a)
 M.I.
about its base :
 

I
xx
= dA.y

2
= (x.dy)y

2
From similar triangles
b/h = x/(h-y)


x = b . (h-y)/h

h

I
xx
= (b . (h-y)y

2
.dy)/h

0
= b[ h (y
3
/3) – y
4
/4 ]/h
= bh
3
/12

13. (b) Moment of inertia about its centroidal axis:
_
I
xx = I
x
0
x
0
+ Ad
2

_
I
x
0
x
0
= I
xx – Ad
2
= bh
3
/12 – bh/2 . (h/3)
2
= bh
3
/36

14. I
x
0
x
0
= dA . y

2

R 2

= (x.d .dr) r
  
2
Sin
2


0 0

R 2

= r
 
3
.dr Sin
2
d
 

0 0
R 2

= r

3
dr {(1-

Cos2 )/2} d
 

0
R
0
2


=[r
4
/4] [ /2 – Sin2 /4]
 
0 0
= R
4
/4[ - 0] = R
 
4
/4
I
XoXo = R

4
/4 =
D

4
/64
xx
x
0x
0
R

d

y=rSin

3. CIRCULAR AREA:
r

15. I
xx
= dA . y

2

R


= (r.d .dr) r
 
2
Sin
2


0
R
0


= r

3
.dr Sin

2
d
 

0 0
R

= r
 
3
dr (1- Cos2 )/2) d
 

0 0

=[R
4
/4] [ /2 – Sin2 /4]
 

0
= R
4
/4[ /2 - 0] = R
 
4
/8
4R/
3

y
0
y
0
xx
x
0
x
0
4. SEMI CIRCULAR AREA:
R

16. About horizontal centroidal axis:

I
xx = I
x
0
x
0
+ A(d)
2

I
x
0
x
0
= I
xx – A(d)
2
= R

4
/8 R

2
/2 .
(4R/3 )

2
I
x
0
x
0
= 0.11R
4

17. QUARTER CIRCLE:
I
xx = I
yy

R /2

I
xx = (r.d .dr).
  
r
2
Sin
2

0 0
R /2

= r

3
.dr Sin

2
d
 

0 0
R /2

= r3 dr (1- Cos2 )/2)
  
d


0 0
/2

=[R
4
/4] [ /2 – (Sin2 )/4]
 


0
= R
4
( /16 – 0) = R
 
4
/16
x x
x
0
x
0
y
yy
0
y
0
4R/3π
4R/3π

18. Moment of inertia about Centroidal axis,
_
I
x
0
x
0

= I
xx
- Ad
2
= R

4
/16 - R

2
. (0. 424R)
2
= 0.055R
4
The following table indicates the final values of M.I.
about X and Y axes for different geometrical figures.

19. Sl.No Figure I
x
0
-x
0
I
y
0
-y
0
I
xx
I
yy
1
bd
3
/12 - bd
3
/3 -
2
bh
3
/36 - bh
3
/12 -
3
R
4
/4 R
4
/4 - -
4
0.11R
4
R
4
/8 R
4
/8 -
5
0.055R
4
0.055R
4
R
4
/16R
4
/16
b
d
x
0
x
x
0
x
d/2
b
h
xx
x
0
x
0
h/3
x
0
x
0
y
0
y
0
O
R
y
0
y
0
xx
x
0
x
0
4R/3π
x
0
y
y
0
4R/3π
4R/3π
Y
Y X
o