mechanics of solid

1. STRAIN
ENERGY
PRESENTED
BY
ASIF
RAHAMAN

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

6. P
F
P › F
STRAIN ENERGY

7. STRAIN ENERGY

8. • TORSION
• BENDING
8
STRAIN
ENERGY
CALCULATION
• AXIAL LOADING
DIFFERENT LOADING CONDITIONS

9. 9
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

10. 1 0
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

11. 1 1
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

12. 1 2
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

13. 1 3
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

14. 1 4
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 ,

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 = Radius of curvature

16. 1 6
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

17. 1 7
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

18. 1 8
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

19. 1 9
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

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
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

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
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

22. 2 2
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

23. THANK YOU