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

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...

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...

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