10. Strain Energy Methods and
Failure Theories with Applications
10.1 Introduction
- Part A discusses the concept of strain energy and its application to stress and deformation determinations in members subjected to impact loading
10.2 Strain Energy
- Strain energy by axial loading
0
2
0 0 0
2
2
Work done by external force:
Work stored in the bar:
2 Strain energy intensity:
2 Modulus of resilience:
2 Modulus of toughness:
k
k
y R
T
W Pd
W U ALd AL d AL d AL
E E
u E
u E
u d
δ
ε ε σ
δ
σ ε σ ε σ σ σ
σ σ
σ ε
=
= = = = =
=
=
=
∫
∫ ∫ ∫
0 2 0
2 2
For shear loading:
2
Strain energy: ,
2 2
R
V V
u d
G
U dV U dV
E G
ε
γτ γ τ
σ τ
= =
= =
∫
∫
∫ ∫
10.3 Elastic strain energy for various loads
1) Strain energy for axial loading
( )
22 2 2
0 0
2 2 2 2
L L
V
P A P P L
U dV Adx dx
E E AE AE
σ
= = = =
∫ ∫ ∫
2) Strain energy for torsional loading
( )
2( )
2 2
2 0 2
2 2
0
2 2 2
2 2
L
V V A
L
T J T
U dV dV dA dx
G G GJ
T T L
GJ dx GJ
τ ρ ρ
= = =
= =
∫ ∫ ∫ ∫
∫
10.3 Elastic strain energy for various loads
3) Strain energy for bending loads
( )2
( )
2 2
2 0 2
2 0
2 2 2
2
r L r
V V A
L r
M y I M
U dV dV y dA dx
E E EI
M dx EI
σ −
= = =
=
∫ ∫ ∫ ∫
∫
4) Strain energy for transverse shear
( )2
2 2 2
2 2
2 2 0 2
r L r
V V A
V Q It V Q
U dV dV dA dx
G G GI t
τ
= = =
∫ ∫ ∫ ∫
10.3 Elastic strain energy for various loads
•
Example Problem 10-1
The modulus of elasticity E for both bars is 200 GPa. If the axial load P is 50 kN,
- Strain energy for the assembly
10.3 Elastic strain energy for various loads
•
Example Problem 10-2
Segment AB and BC are of the same material (G = 11,000 ksi). If T1 = 50 kip⋅in.
and T2 = 70 kip ⋅in.
- Strain energy for the assembly
10.3 Elastic strain energy for various loads
•
Example Problem 10-3
The cantilever beam has a constant cross section and is subjected to a concentrated force P at its free end.
- Strain energy in the beam due to bending
- Strain energy in the beam due to transverse shear - Compare results of above
10.4 Impact loading
- Dynamic force (loading): the force necessary to produce a change in
acceleration of a body. Ex) pressure on the airplane wings, collision of two automobiles, and a man jumping on a diving board
- Impact load: a suddenly applied load.
10.4 Impact loading
1) Strain Energy Method
( )
22
2
2
2
2 2 0
2
2 ,
1 1 2
,
1 1 2
st
st
st st st
st
st
AL L
W h E E
AL WL WhE
W W WhE L
A A AL E
WL WL WhL WL
AE AE AE AE
h
L
L WL
E AE E
h
σ σ
δ δ
σ σ
σ δ σ
δ δ
δ δ δ
σ
δ σ δ
σ σ δ
+ = ← =
− − =
= + + ← =
= + + =
→ = + +
= = =
→ = + +
2
2 2
0 2
2
2 2
2 2
2 2
st st
st
st st st st
When h
When h
h h
E E WL E
h Wh
L L AE AL
E mV mEV
AL AL
δ δ σ σ
δ
δ δ δ δ δ
σ δ
σ
=
=
=
= + + ≈
= ≈ =
= =
10.4 Impact loading
2) Work-kinetic Energy Method
( )
1 2 2 1
2 2
1 0
2
2 2 0
1 1 2
1 1 2
same as the result of the previous case by the strain-energy method
st
st
st
st
st st
P W
U T T
W h P
P WP W h P W h
h P W
δ δ
δ δ
δ δ
δ δ δ δ δ
→
=
= −
+ − =
− − =
= + +
= + + =
→
10.4 Impact loading
•
Example Problem 10-4
30-mm-diameter aluminum alloy (E = 70 GPa) rod 1 m long is fitted with a flange at the bottom.
- The max. mass that the collar may have if the yield strength (σy = 270 MPa) of the rod must not be exceeded.
10.4 Impact loading
•
Example Problem 10-8
The beam is of 75 mm wide and 25 mm deep. Each of the supporting coil spring has a modulus of 9 kN/m.
- The max. stress in the beam when the block of weight W = 45 N is dropped from a height h = 15 mm. The modulus of elasticity of the beam is 200 Gpa.
