1) Strain energy is the energy stored in a material body due to elastic deformation caused by external loads. It is calculated using the area under the stress-strain curve. 2) Strain energy is calculated differently for different loading conditions like axial loading, torsion, and bending. For axial loading and torsion, strain energy is proportional to the load and length. For bending, strain energy is proportional to the bending moment squared and inversely proportional to the flexural rigidity. 3) The document presents an example calculation of strain energy in a beam under a combination of point loads using the bending moment formulae and integrating them along the length. The total strain energy is found to be 14.38 N

1. STRAIN
ENERGY
PRESENTED
BY
Name - Asif Rahaman
Roll No.- 34900721068
Dept. of Mechanical Engineering
Cooch Behar Govt. Engineering College

2. CONTENT
S
presentation title
• INTRODUCTION
• STRAIN ENERGY
• STRAIN ENERGY
CALCULATION
1) Under axial
loading
2) Under Torsion
3) Under bending

3. INTRODUCTION
Energy methods are widely used to
obtain solutions to elasticity problems
and determine deflection of structures.
3
• Like deflections of joint on a truss or
points on a beam or shaft.
In energy method, strain energy is
associated with it. Hence it is also
known as Strain energy method.

4. P
 Strain energy is the energy stored in the material due to
deformation under external load.
STRAIN ENERGY

5. P
F
P › F
STRAIN ENERGY

6. STRAIN ENERGY

7. • TORSION
• BENDING
7
STRAIN
ENERGY
CALCULATION
• AXIAL LOADING
DIFFERENT LOADING CONDITIONS

8. 8
STRAIN
ENERGY
CALCULATION
UNDER AXIAL LOADING
l
Let ,
P P
P = Force
applied
l = Length of the
bar
E = Young's
Modulus
A= Area of the
bar

9. 9
STRAIN
ENERGY
CALCULATION
UNDER AXIAL LOADING
•
Load
→
Deformation
→
P
∆l
Strain Energy [ U ] = Area under the curve OB
O
B
Strain Energy [ U ] = × ∆l × P

10. 1 0
STRAIN
ENERGY
CALCULATION
UNDER AXIAL LOADING
Strain Energy [ U ] = × ∆l × P
Since , ∆l =
Strain Energy [ U ] = × × P
Strain Energy [ U ] =
P = Force applied
l = Length of the
bar
E = Young's
Modulus
A = Area of the
bar
Where ,
J or N-m

11. 1 1
STRAIN
ENERGY
CALCULATION
UNDER TORSION
l
Let ,
T T
T = Applied
Torque
l = Length of the
bar
G = Modulus of
Rigidity
J = Polar moment of
inertia

12. 1 2
STRAIN
ENERGY
CALCULATION
UNDER TORSION
•
Load
→
Angle of
twist→
Τ
θ
Strain Energy [ U ] = Area under the curve OB
O
B
Strain Energy [ U ] = × θ × Τ

13. 1 3
STRAIN
ENERGY
CALCULATION
UNDER TORSION
Strain Energy [ U ] = × θ × Τ
Since , θ =
Strain Energy [ U ] = × × Τ
Strain Energy [ U ] = J or N-m
Τ = Applied Torque
l = Length of the
bar
G = Modulus of
Rigidity
J = Polar moment of
inertia
Where ,

14. 1 4
STRAIN
ENERGY
CALCULATION
UNDER BENDING
l
Let ,
M M
M = Applied
Moment
l = Length of the
bar
E = Young’s Modulus
I = Moment of inertia
θ = Radius of curvature

15. 1 5
STRAIN
ENERGY
CALCULATION
UNDER BENDING
l
Let ,
M M
M = Applied
Moment
l = Length of the
bar
E = Young’s Modulus
I = Moment of inertia
θ
θ = Radius of curvature

16. 1 6
STRAIN
ENERGY
CALCULATION
UNDER BENDING
•
Load
→
Angle of bending
→
Τ
θ
Strain Energy [ U ] = Area under the curve OB
O
B
Strain Energy [ U ] = × θ × M

17. 1 7
STRAIN
ENERGY
CALCULATION
UNDER BENDING
Strain Energy [ U ] = ×
𝑙
𝑅
× M
Since ,
𝑀
𝐼
=
𝐸
𝑅
Strain Energy [ U ] = × × M
Strain Energy [ U ] =
Since , 𝑙 = 𝑅𝜃
𝑀𝑙
𝐸𝐼
𝑀2
𝑙
2𝐸𝐼
This Equation is valid only for “ PURE BENDING
J or N-m

18. 1 8
STRAIN
ENERGY
CALCULATION
UNDER BENDING
dx
Suppose a bending stress W is acting on the bar.
Let , on the dx portion bending moment is M.
Strain Energy [ dU ] =
𝑀2
𝑑𝑥
2𝐸𝐼
l
W
Strain Energy [ U ] =
0
𝑙
𝑀2𝑑𝑥
2𝐸𝐼
For the whole bar

19. 1 9
EXERCISE
3KN
5KN
2 m 1 m
Find the strain energy (in N-m ) stored in the beam.
EI = 𝟏𝟎𝟒
.
x
O B
A
𝑀𝐴𝐵 = −3𝑥
Moment for the section AB will be __
Where x = 0 to 1
Similarly , Moment for the section OA will be __
𝑀𝑂𝐴= −3(𝑥 + 1) − 5𝑥
𝑀𝑂𝐴= −8𝑥 − 3
Where x = 0 to 2
x

20. 2 0
EXERCISE
3KN
5KN
2 m 1 m
Find the strain energy (in N-m ) stored in the beam.
EI = 𝟏𝟎𝟒
.
x
O B
A
x
Here the total strain energy U will be ___
𝑈 = 𝑈𝑂𝐴+ 𝑈𝐴𝐵
𝑈 =
𝑀𝑂𝐴
2
𝑑𝑥
2𝐸𝐼
+
𝑀𝐴𝐵
2
𝑑𝑥
2𝐸𝐼
𝑈 =
0
2
(−8𝑥 − 3)2 𝑑𝑥
2𝐸𝐼
+
0
1
(−3𝑥)2 𝑑𝑥
2𝐸𝐼

21. 2 1
EXERCISE
3KN
5KN
2 m 1 m
Find the strain energy (in N-m ) stored in the beam.
EI = 𝟏𝟎𝟒
.
x
O B
A
x
𝑈 =
1
2𝐸𝐼
[ 0
2
64𝑥2 + 9 + 48𝑥 𝑑𝑥 + 0
1
9𝑥2 dx ]
𝑈 =
1
2×104[
64×8
3
+ 9 × 2 + (48 × 2)]
𝑈 = 14.38 𝑁 − 𝑚