mechanics of materials, energy methods

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M. Vable

Intermediate Mechanics of Materials: Chapter 7

Energy Methods
• Minimum-energy principles are an alternative to statement of equilibrium equations. Displacements

Ki

External
Forces
and
Moments

ati
cs

els

Strains

hod

Mod

ium

M et

eria
l

2

Mat

rgy

ilibr
E qu

En e

4

1
ne
m

s

Internal
Forces
and
Moments

Static Equivalency
Stresses
3

The learning objectives in this chapter are:
• Understand the perspective and concepts in energy methods. • Learn the use of dummy unit load method and Castigliano’s theorem for calculating displacements in statically determinate and indeterminate structures.

7-1

M. Vable

Intermediate Mechanics of Materials: Chapter 7

Strain Energy
• The energy stored in a body due to deformation is called the strain energy.
• The strain energy per unit volume is called the strain energy density and is the area underneath the stress-strain curve up to the point of deformation.
σ

Uo = Complimentary strain energy den
A

dUo = ε dσ


Uo = Strain energy density

O


Strain Energy:

dUo = σ dε
U =

∫ Uo

ε

dV [

V
ε

Strain Energy Density:

Uo =

∫ σ dε
0

Units:

N-m / m3, Joules / m3, in-lbs / in3, or ft-lb/ft.3
σ

Complimentary Strain Energy Density: U o =

∫ ε dσ
0

• The strain energy density at the yield point is called Modulus of Resilience. 7-2

M. Vable

Intermediate Mechanics of Materials: Chapter 7

• The strain energy density at rupture is called Modulus of Toughness. σ
Yield
Point
Modulus of
Resilience
ε
σ

Modulus of
Toughness
Rupture
Stress

ε

σ

Stronger Material
Tougher material

ε

7-3

M. Vable

Intermediate Mechanics of Materials: Chapter 7

Linear Strain Energy Density
ε

ε

2


1
Uniaxial tension test: U o = ∫ σ dε = ∫ ( Eε ) dε = -------- = -- σε 2
2
0

0

1
U o = -- τγ
2
• Strain energy and strain energy density is a scaler quantity. 1
U o = -- [ σ xx ε xx + σ yy ε yy + σ zz ε zz + τ xy γ xy + τ yz γ yz + τ zx γ zx ] 2
1-D Structural Elements
A
y
x
z
dV=Adx

dx

Axial strain energy
• All stress components except σxx are zero.
σ xx = Eε xx
UA =

1 2
∫ -- Eεxx dV =
2

V

UA =

ε xx =

du
(x)
dx

1 ⎛ du⎞ 2
∫ -- E ⎝ d x⎠ dA dx =
2



L A


L

1 ⎛ du⎞ 2
-E dA dx
2 ⎝ d x⎠ ∫
A

1
du 2
U a = -- EA ⎛ ⎞
2 ⎝ d x⎠

∫ Ua dx
L

• Ua is the strain energy per unit length.
UA =

∫ Ua
L

2

dx

1N
U a = -- ------ 2 EA

7-4

M. Vable

Intermediate Mechanics of Materials: Chapter 7

Torsional strain energy
• All stress components except τxθ in polar coordinate are zero τ xθ = Gγ xθ
UT =

1 2
∫ -- Gγxθ dV =
2

∫ Ut

1 ⎛ dφ⎞ 2
2
-Gρ dA dx
2 ⎝ d x⎠ ∫



L A

UT =

L

A

1 ⎛ dφ⎞ 2
U t = -- GJ
2 ⎝ d x⎠

dx

L




(x)
dx

1 ⎛ dφ⎞ 2
∫ -- G ⎝ ρ d x⎠ dA dx =
2



V

γ xθ = ρ

Ut is the strain energy per unit length.
2

1T
U t = -- ------ 2 GJ

∫ Ut dx

UT =

L

Strain energy in symmetric bending about z-axis
There are two non-zero stress components, σxx and τxy.
2

σ xx = Eε xx
1 2
U B = ∫ -- Eε xx dV =
2
V

ε xx = – y
2



dv
dx

2

1 ⎛ d v⎞
∫ -- E ⎜ y d x2 ⎟ dA dx =
2 ⎝


L A

2
2


L

1⎛d v ⎞
-- ⎜ 2 ⎟
2⎝dx ⎠
2

UB =

∫ Ub
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