MOMENTS OF INERTIA
MRS SITI KAMARIAH BINTI MD SA’AT
SCHOOL OF BIOPROCESS ENGINEERING
UNIVERSITI MALAYSIA PERLIS
At the end of this topics, student should able to:
1.
2.
Develop method for determining the
moment of inertia for an area
Determine the mass moment of inertia.
 Or
called second moment of area, I
 Measures the efficiency of that shape its
resistance to bending
 Moment of inertia about the x-x axis and
y-y axis.
Ix 

y 2 dA
A
Iy 

x 2 dA
A
Jo 

r 2 dA  I x  I y
A
2
r2  x  y2
Unit :
m4, mm4 or cm4
3
I xx
y
h
x
bh

12
x
3
b
y
I yy
hb

12
b
v
h
h
x
u
u
v
x
n
n
b
Rectangle at one edge
 Iuu=
bh3/3
 Ivv= hb3/3
Triangle
 Ixx=
bh3/36
 Inn= hb3/6
y
B
y
h x
x H
x
x
b
y
y
 Ixx= Iyy
=
πd4/64
 Ixx
= (BH3-bh3)/12
 Iyy = (HB3-hb3)/12
 Used
to find the moment of inertia of an
area about centroidal axis.

I x  ( y ' d y ) 2 dA
A

2
 y' dA  2d  y'd  dA
2
y
A
y
A
I x  I x '  Ad y
A
2
I y  I y '  Ad x 2
J o  J c  Ad 2
 Calculate
the moment of inertia at z-z axis.
x
x
d
z
z
 b=150mm;h=100mm; d=50mm
bh3 150 (100 )3
I xx 

 12 .5x10 6 mm 4
12
12
I zz  I xx  Ad 2  12 .5x10 6  (150 x100 )( 50 ) 2  50 x10 6 mm 4
 Calculate
axis
the moment of inertia about x-x
x
x
12 mm
400 mm
24 mm
200 mm
d= 212 mm
 Ixx
of web = (12 x 4003)/12= 64 x 106 mm4
 Ixx
of flange = (200 x 243)/12= 0.23 x 106 mm4
 Ixx
from principle axes xx = 0.23 x106 + Ad2
Ad2 = 200 x 24 x 2122 = 215.7 x 106 mm4
 Ixx
from x-x axis = 216 x 106 mm4
 Total Ixx
= (64 + 2 x 216) x106 =496 x 106 mm4
 Unit
of length
 Used
in design of
columns in structure.
kx 
Ix
A
ky 
Iy
ko 
Jo
A
A
 Many
cross-sectional areas consist of a
series of connected simpler shapes, such
as rectangles, triangles, and semicircles.
 In order to properly determine the moment
of inertia of such an area about a specified
axis, it is first necessary to divide the area
into its composite parts and indicate the
perpendicular distance from the axis to
the parallel centroidal axis for each part.
 Use the moment of inertia of an area or
parallel axis theorem.
1.
2.
3.
4.
Subdivide the cross-section into threepart A,B,D
Determine moment of inertia of each
part, for rectangular, I=bh3/12.
Use the parallel axis theorem formula
for each part.
Summation for entire cross-section.
 Try
• Fundamental problems
• Problems: 10-49 till 10-56.
 To
use this method, first determine the
product of inertia for the area as well as
its moments of inertia for given x, y axes.

I xy  xy dA
A
 Units: m4, mm4.
 Product
of inertia may either +ve, -ve or
zero depending on the location and
orientation of the coordinate axes.
 If the axis symmetry for an area, product
of inertia will be zero.
 Passing
through the
centroid of the area.

I xy  ( x ' d x )( y ' d y )dA
A




A
A
A
A
 x ' y ' dA  d x y ' dA  d y x ' dA  d x d y dA
I xy  I x ' y '  Ad x d y
 Determine
y centroid
the product of inertia about the x and
1.
2.
Subdivide the cross-section into three-part
A,B,D
Determine product moment of inertia
 Total
up the product moment of inertia
 Try
• Example 10.7
• Problems: 10-71,10-75-78,10-82
A
measure of the body’s resistance
to angular acceleration.
 Used in dynamics part, to study
rotational motion.
 Mass moment of inertia of the
body:
2
I  r dm

m
• Where r= perpendicular distance from
the axis to the arbitrary element dm.
 The
axis that is generally chosen for
analysis, passes through the body’s mass
center G
 If the body consists of material having a
variable density ρ = ρ(x, y, z), the
element mass dm of the body may be
expressed as dm = ρ dV
 Using volume element for integration,
I   r 2 dV
V
 When
ρ being a constant,
I    r 2 dV
V
Shell Element
 For a shell element having height z,
radius y and thickness dy, volume dV =
(2πy)(z)dy
Disk Element
 For disk element having radius y,
thickness dz, volume dV = (πy2) dz
 For
moment of inertia about the z axis,
I = IG + md2
 For
moment of inertia expressed using k,
radius of gyration,
I  mk
2
I
or k 
m
Try:
• Problems: 10-89 -95,96,97,99,100
1.
2.
3.
4.
The definition of the Moment of Inertia
for an area involves an integral of the …
SI units for the Moment of Inertia for an
area.
The parallel-axis theorem for an area is
applied to ….
The formula definition of the mass
moment of inertia about an axis is …
5.
Calculate the moment of inertia of the
rectangle about the x-axis
3cm
2cm
2cm
x
GOOD LUCK !!!