mechanics of materials, energy methods

Page 1 of 29

mechanics of materials, energy methods

By | Feb. 2014
Page 1 of 29
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...