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

dσ

Uo = Strain energy density

O

dε

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

Eε

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

•

dφ

(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