Strength of Material, How to set CO PO PRO

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1. AE/ME/PG/PT/FG: 22306: Strength
of Materials: Moment of Inertia:
CO1_Compute MI of symmetric
and asymmetric sections

Anil Deshmukh_LAM_GP Nashik
MSBTE LEAD

2. Written by
Strength of Materials
MSBTE LEAD

3. Moment of Inertia

4. UO1_Calculate MI of the given
standard shape

5. August 31, 2024Maharashtra State Board of Technical EducationPage 5
What we will learn today
1.Concept of M. I.
Key takeaways
Formulae of M. I.

6. August 31, 2024Maharashtra State Board of Technical EducationPage 6
Concept Map
Calculate MI of
the given
standard shape
Explain with sketches
effect of change in
MI in case of given
beam and column
Calculate polar MI
and Radius of
gyration for the
given section
Calculate MI of the
given simple
composite shape
Compute MI of
symmetric and
asymmetric
sections

7. August 31, 2024Maharashtra State Board of Technical EducationPage 7
Moment of Inertia
It is a geometrical quantity. It
is It is a geometrical quantity.
It is required for analysis and
design of the components /
parts of machine subjected to
moments such as bending
moment, twisting moment
and buckling.
subjected to moments such as
bending moment, twisting
moment and buckling.

8. August 31, 2024Maharashtra State Board of Technical EducationPage 8
Moment of Inertia of an elemental area about
the reference axis is the product of the
elemental area and square of perpendicular
distance of reference axis from elemental area.
Moment of Inertia is also known as second
moment of area.
i
x
= da * y
2
i
Y
= da * x
2


Ix = Σ da * y
2
Iy = Σ da * x
2

I is used for moment of inertia.
S. I. Unit of moment of Inertia is m
4
. Generally
we use mm for dimensions, if so then unit of MI
is mm
4
x
y
da
A
X
Y
Moment of Inertia

9. August 31, 2024Maharashtra State Board of Technical EducationPage 9
Ix = Σ da * y
2
Ix = ∫ da * y
2
Ix =
-d/2
∫
d/2
b*dy * y
2
Ix =
-d/2
∫
d/2
b* y
2
* dy
Ix = [b* y
3
/3]
d/2
–d/2

Ix = b/3*[ d
3
/8 + d
3
/8]
Ix = bd
3
/12
M. I. of Rectangle
Similarly I
Y
= db
3
/12
b
y
dy
d
XX C
Y
Y

10. August 31, 2024Maharashtra State Board of Technical EducationPage 10
•Similarly M. I. of circular section about Z
axis can be obtained from basic principle
I
Z
= ∫ da * r
2
I
Z
=
0
∫
R
2πr*dr * r
2
I
Z =
0
∫
R
2π r
3
dr
I
Z = [2π r
4
/4]
R
0

I
Z = π/2 *[ R
4
- 0] = π R
4
/2
Iz = π * d
4
/32
Y
Y
XX
Z
Z
d
R
r
dr

11. August 31, 2024Maharashtra State Board of Technical EducationPage 11
M. I. of some symmetrical sections
B
b
dD
C XX
Y
Y
I
x
= (BD
3
-bd
3
)/12
I
y
= (DB
3
-db
3
)/12
I
y
=Σ (db
3
)/12 of all
three rectangles
Y
Section not symmetrical
about Y axis
B
D
d
b
XX
B
D
d
b/2 b/2
XX

12. August 31, 2024Maharashtra State Board of Technical EducationPage 12
Example
400
mm
230 mm
X X
Y
Y
Ix = bd
3
/12
Ix = 230*400
3
/12=1.23x10
9
mm
4

I
Y
= 400*230
3
/12=0.405x10
9
mm
4

I
Y
= db
3
/12

13. August 31, 2024Maharashtra State Board of Technical EducationPage 13
►This presentation include formulae for some symmetrical sections.
►Practice sheet is attached with. Solve the problems.
►Next presentation includes two theorems of M. I. and formulae for M. I.
of remaining standard/basic sections.
►Thanks and be prepared for the quiz.