Energy_slides

M. Vable Intermediate Mechanics of Materials: Chapter 7
Energy Methods
• Minimum-energy principles are an alternative to statement of equilibrium
equations.
Displacements
1
Kinematics
External
Forces
and
Moments
Strains
4
Energy Methods
Equilibrium
Material Models
2
Internal
Forces
and
Moments
Static Equivalency
3
Stresses
The learning objectives in this chapter is:
• Understand the perspective and concepts in energy methods.
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
U o = Complimentary strain energy den
dU o = ε dσ
A
dσ
U o = Strain energy density
O
dε
Strain Energy: U = U o dV
[
V
ε
Strain Energy Density: U o
= ∫σdε
0
Units: N-m / m 3 , Joules / m 3 , in-lbs / in 3 , or ft-lb/ft. 3
Complimentary Strain Energy Density:
dU o = σ dε
∫
U o
=
σ
∫
0
εdσ
ε
7-2

M. Vable Intermediate Mechanics of Materials: Chapter 7
Uniaxial tension test:
Linear Strain Energy Density
U o
ε ε
Eε 2
= ∫σdε
= ∫( Eε) dε
= -------- =
2
0 0
1
--σε
2
1
U o
= --τγ
2
• Strain energy and strain energy density is a scaler quantity.
1
U o
= -- [ σ
2 xx
ε xx
+ σ yy
ε yy
+ σ zz
ε zz
+ τ xy
γ xy
+ τ yz
γ yz
+ τ zx
γ zx
]
1-D Structural Elements
y
A
z
x
dx
dV=Adx
Axial strain energy
• All stress components except σ xx are zero.
σ xx
du ( x)
= Eε xx
ε xx =
dx
1 2 1
U A ∫
--Eε
2 xx dV --E ⎛du⎞ 2 = = ∫ ∫ dA
dx
=
2 ⎝ dx
⎠
V
• U a is the strain energy per unit length.
L
A
1
U A
= ∫U a
dx
U a
--EA⎛du⎞ 2
=
2 ⎝ dx
⎠
L
∫
L
1 du
--
2⎝
⎛ dx⎠
⎞ 2 ∫ E d A
A
dx
1
U A = ∫U a dx
U a = ---------
N2
2EA
L
7-3

M. Vable Intermediate Mechanics of Materials: Chapter 7
Torsional strain energy
• All stress components except τ xθ in polar coordinate are zero
τ xθ
= Gγ xθ
γ xθ = ρ x
dφ ( x)
d
1 2 1
U T ∫
--Gγ
2 xθ
dV
2
--G ⎛ ρ dφ⎞ 2 = = ∫ ∫ dA
dx
=
⎝ d x ⎠
V
• U t is the strain energy per unit length.
L
A
1
U T
= ∫U t
dx
U t
--GJ⎛dφ⎞ 2
=
2 ⎝dx⎠
L
∫
L
1 dφ
--
2⎝
⎛ dx⎠
⎞ 2 ∫ Gρ 2 d A
A
dx
1
U T = ∫U t dx
U t = -- -------
T2
2GJ
Strain energy in symmetric bending about z-axis
There are two non-zero stress components, σ xx and τ xy.
σ xx
L
d v
= Eε xx ε xx = – y
dx 2
2
1 2 1 ⎛ d v ⎞ 2
U B = ∫
--Eε
2 xx dV = ∫ ∫
--E⎜y
2 ⎝ dx 2 ⎟ dA
dx
=
⎠
V
L
A
• where U b is the bending strain energy per unit length.
2
2
1 ⎛d v ⎞ 2
U B = ∫U b dx
U b = --EI
2 zz ⎜
⎝dx 2 ⎟
⎠
L
∫
L
2
1⎛d v ⎞ 2
-- ⎜
2⎝dx 2 ⎟ ∫ Ey 2 dA
⎠
A
dx
The strain energy due to shear in bending is:
As
τ max
2
1
M z
U B = ∫U b dx
U b = -- ----------
2
L
EI zz
« σ max
U s
« U B
1
U S =
∫
--τ
2 xy γ xy dV
=
V
1
-- τ 2
xy
∫
------ dV
2 E
V
7-4

M. Vable Intermediate Mechanics of Materials: Chapter 7
Table 7.1 Energy densities
Axial
Torsion of circular
shafts
Symmetric
bending of
beams
Strain energy density per
unit length
Complimentary strain
energy density per unit
length
1 du
U a = --EA⎛ ⎞ 2
1
2 ⎝ dx
⎠ U a = ---------
N2
2EA
1 dφ
U t
= --GJ⎛ ⎞ 2
1
2 ⎝dx⎠
U t = -- -------
T2
2GJ
2
1 ⎛d v ⎞ 2
1
U b
= --EI
2 zz ⎜
⎝dx 2 ⎟ U b -- M 2
z
= ----------
⎠
2EI zz
7-5

M. Vable Intermediate Mechanics of Materials: Chapter 7
Work
• If a force moves through a distance, then work has been done by the
force.
• Work done by a force is conservative if it is path independent.
• Non-linear systems and non-conservative systems are two independent
description of a system.
Loading Mode
P
dW
=
Fdu
δW
Work
=
Pδu L
p(x)
u(x)
u L
δW
=
L
∫
0
p( x)δux
( ) dx
T
δW
=
Tδφ L
t (x)
φ( x)
φ L
δW
=
L
∫
0
t( x)δφ( x) dx
δW
=
Pδv L
P
M
v L
θ
=
dv
dx
δW
=
Mδθ L
v(x)
δW
=
L
∫
0
p( x)δvx
( ) dx
p(x)
• Any variable that can be used for describing deformation is called the
generalized displacement.
• Any variable that can be used for describing the cause that produces
deformation is called the generalized force.
7-6

M. Vable Intermediate Mechanics of Materials: Chapter 7
Virtual Work
• Virtual work methods are applicable to linear and non-linear systems,
to conservative as well as non-conservative systems.
The principle of virtual work:
The total virtual work done on a body at equilibrium is zero.
δW = 0
• Symbol δ will be used to designate a virtual quantity
δW ext = δW int
Types of boundary conditions
y Displacement and rotation specified
at this end
T
x
P x
Internal forces and moment specified
at this end to meet equilibrium
T
N
V y
T
P x
z
M
ε
P y M
ε
P y
Kinematic variable
Statical variable
(Primary variable)
(Secondary variable)
or
u
N
φ
or
T
v
or
θ
Geometric boundary conditions (Kinematic boundary conditions)
(Essential boundary conditions):
Condition specified on kinematic (primary) variable at the boundary.
Statical boundary conditions
(Natural boundary conditions)
Condition specified on statical (secondary)variable at the boundary.
M z
V y
dv
=
or
M
dx
z
7-7

M. Vable Intermediate Mechanics of Materials: Chapter 7
Kinematically admissible functions
• Functions that are continuous and satisfies all the kinematic boundary
conditions are called kinematically admissible functions.
• actual displacement solution is always a kinematically admissible
function
• Kinematically admissible functions are not required to correspond to
solutions that satisfy equilibrium equations.
Statically admissible functions
• Functions that satisfy satisfies all the static boundary conditions, satisfy
equilibrium equations at all points, and are continuous at all points
except where a concentrated force or moment is applied are called
statically admissible functions.
• Actual internal forces and moments are always statically admissible.
• Statically admissible functions are not required to correspond to solutions
that satisfy compatibility equations.
7.3 Determine a class of kinematically admissible displacement
functions for the beam shown in Fig. P7.3.
A
x
L
Fig. P7.3
7.4 For the beam and loading shown in Fig. P7.3 determine a statically
admissible bending moment.
B
wL 2
w
L
C
7-8

M. Vable Intermediate Mechanics of Materials: Chapter 7
Virtual displacement method
• The virtual displacement is an infinitesimal imaginary kinematically
admissible displacement field imposed on a body.
actual displacement
x
kinematically admissible displacement
δv = virtual displacement
• Of all the virtual displacements the one that satisfies the virtual work
principle is the actual displacement field.
Virtual Force Method
• The virtual force is an infinitesimal imaginary statically admissible
force field imposed on a body.
• Of all the virtual force fields the one that satisfies the virtual work
principle is the actual force field.
7-9

M. Vable Intermediate Mechanics of Materials: Chapter 7
7.7 The roller at P shown in Fig. P7.7 slides in the slot due to the
force F = 20kN. Both bars have a cross-sectional area of A = 100 mm 2
and a modulus of elasticity E = 200 GPa. Bar AP and BP have lengths of
L AP = 200 mm and L BP = 250 mm respectively. Determine the axial stress
in the member AP by virtual displacement method.
B
A
110 o F
Fig. P7.7
7.8 A force F = 20kN is appled to pin shown in Fig. P7.8. Both
bars have a cross-sectional area of A = 100 mm 2 and a modulus of elasticity
E = 200 GPa. Bar AP and BP have lengths of L AP = 200 mm and
L BP = 250 mm respectively. Using virtual force method determine the
movement of pin in the direction of force F.
P
B
A
110 o P
F
40 o
Fig. P7.8
7-10