Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Principles of virtual work
A. Hlod
CASA
Center for Analysis, Scientific Computing and Applications
Department of Mathematics and Computer Science
21-June-2006
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Conclusions
Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Outline
1
Introduction
2
Work and energy
Work and potential energy of the internal forces
Work and potential energy of the applied loading
Virtual work
3
Principle of virtual work
4
Principle of complementarity virtual work
5
Conclusions
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Conclusions
Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Conclusions
Formulation of principle of virtual work
Sum of works of the internal and external forces done by virtual
displacements is zero
−δWi − δWe = 0.
Virtual displacements are infinitesimal change in the position
coordinates of a system such that the constraints remain
satisfied.
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Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Conclusions
Example standard way
To find a relation between W and
P we need to solve the system of 8
equations

P = F11




F

 22 = F21


F32 = F31



F42 = F41
 F11 = F41




F42 = F22




F = F31

 21
F31 + F41 + F32 + F42 = W
and get W = 4P.
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Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Conclusions
Example principle of virtual work
From the principle of virtual
work,
−δWi −δWe = W δz−PδV = 0.
Geometrically admissible
virtual displacements δV and
δz are related according to
δV = 4δz. Thus,
W = 4P.
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Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Work & energy. Elastic extension of a bar
Z
N̄ = −N = −
σx dA = −σx A
A
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Conclusions
Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Work & energy. Elastic extension of a bar
Internal work done in a differential element dx is
−σx dx Adx = −σx dx dV
with the strain dx = d(du/dx) and the volume element
dV = Adx.
The total work of the bar is
Z L Z x
Wi = −
σx dx Adx.
0
0
Using the Hooke’s law σx = Ex we obtain
Z
Wi = −
L
Z
A
0

0
x
1
Ex dx dx = −
2
Z
0
L
AE2x dx.
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Conclusions
Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Conclusions
Work of the internal forces in 3D
Consider a parallelepiped (dxdydz). Then the work during the
strain increment dx , dy ,...,dγyz would be
(σx dx + σy dy + ... + τyz dγyz )dxdydz.
The stress tensor σij and the strain tensor ij are

σx

τxy
σij =
τxz
τxy
σy
τyz




x
γxy /2 γxz /2
τxz
y
γyz /2  .
τyz  , ij =  γxy /2
γxz /2 γyz /2
z
σz
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Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Conclusions
Work of the internal forces in 3D
Then the work for the whole body of volume V becomes
Z Z
Wi
= −
V
Z
(σx dx + σy dy + ... + τyz dγyz )dxdydz =
Z Z σij dij dV = −
σ T ddV ,
(0,0,...,0)
Z ij
= −
V
(x ,y ,...,γyz )
0
V
0
where σ = [σx , σy , σz , τxy , τxz , τyz ]T and
= [x , y , z , γxy , γxz , γyz ]T .
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Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Using the Hooke’s law σij = Eijkl kl we arrive at
Z
Z
1
1
Wi = −
Eijkl ij kl dV = −
T EdV =
2 V
2 V
Z
Z
1
1
σij ij dV = −
σ T dV .
= −
2 V
2 V
Next we introduce the strain energy density as
U0 () =
1 T
E = G 211 + 222 + 233 +
2
ν
1 2
2
2
(11 + 22 + 33 )2 + (γ12
+ γ13
+ γ23
)
1 − 2ν
2
which is a positive definite function. The total strain energy is
Z
Ui = −Wi =
U0 ()dV .
V
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Conclusions
Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Conclusions
Complementarity strain energy density
Complementarity strain energy density is
U0∗ = σij ij − U0 ().
In case of Hooke’s law
1
1 1
U0∗ (σ) = σ T − σ T = σ T E−1 σ =
(σ11 + σ22 + σ33 )2 +
2
2
2E
2
2
2
(σ12
+ σ13
+ σ23
− σ11 σ22 − σ22 σ33 − σ22 σ33 )
With a linear material law
U0 =

1
1 T
E = σ T E−1 σ = U0∗
2
2
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Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
The complementarity strain energy is
Z
Z
1
∗
∗
Ui =
U0 (σ)dV =
σ T E−1 σdV .
2
V
V
The complementarity work of the internal forces is given by
Z Z σij
Z Z σ
Wi∗ = −
ij dσij dV = −
dσdV .
V

V
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Conclusions
Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Conclusions
Extension of a bar
The external work performed until the final configuration uF is
reached
Z u
F
We =
N̄du.
0
If the displacement is proportional to the load u = k N̄ we have
Z
uF
We =
Z
N̄du =
0

0
uF
1 uF2
1
u
du =
= N¯F uF .
k
2 k
2
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Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
External work
The external work done by the prescribed forces p̄i , p̄Vi is
Z
ui
Z
We =
ui
Z Z
p̄i dui dS +
Sp
0
V
0
p̄Vi dui dV .
For linear material
1
We =
2

Z
1
p̄i ui dS +
2
Sp
Z
V
p̄Vi ui dV .
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Conclusions
Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Complementarity external work
The complementarity external work is
Z Z pi
We∗ =
ūi dpi dS
Su
0
For linear material
We∗

1
=
2
Z
pi ūi dS.
Su
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Conclusions
Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Energy for loading
Energy of external forces
Z Z ui
Z Z
Ue = −
p̄i dui dS −
Sp
0
V
ui
0
p̄Vi dui dV .
Complementarity energy of external forces
Z Z ui
∗
Ue = −
pi ūi dS.
Su

0
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Conclusions
Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Internal&external virtual work
The internal virtual work is given by
Z
σij δij dV ,
δWi = −
V
and the external virtual work is
Z
Z
δWe =
p̄i δui dS +
p̄Vi δui dV ,
Sp
V
where δui are the virtual displacements.
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Conclusions
Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Conclusions
Internal&external complementarity virtual work
The internal complementarity virtual work is given by
Z
δWi∗ = −
ij δσij dV ,
V
and the external virtual work is
Z
∗
δWe =
ūi δpi dS.
Su
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Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Conclusions
Principle of virtual work (1)
The principle of virtual work can be derived form the equations
of equilibrium and vice versa. The condition of equilibrium for a
solid body under the prescribed body forces is
σij,j + p̄Vi = 0 in V .
The force boundary conditions are given on the part of the
surface Sp
pi = p̄i on Sp
where pi = σij aj . On the other part of the boundary Su
(S = Su ∪ Sp ) displacements are prescribed.
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(1)
(2)
Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Conclusions
Principle of virtual work (2)
Next we multiply (1) and (2) by the virtual displacement δui and
integrate the first relation over V and the second one over Sp .
Z
Z
− (σij,j + p̄Vi )δui dV +
(pi − p̄i )δui dS = 0.
(3)
V
Sp
The virtual displacements are kinematically admissible if the
displacements ûi = ui + δui satisfy the geometric boundary
conditions (ui = ûi on Su ),
δui = 0 on Su .
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Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Conclusions
Principle of virtual work (3)
We use the Gauss integral theorem to receive
Z
Z
Z
σij,j δui dV =
pi δui dS −
σij δui,j dV .
V
S
From the definition of Sp we have
Z
Z
Z
pi δui dS =
pi δui dS −
Sp
S
pi δui dS.
V
Sp
pi δui dS = 0, (6)
Su
where δij = 1/2δ(ui,j + uj,i ).

(5)
Su
We substitute (4) and (5) into (3)
Z
Z
Z
Z
σij δij dV − p̄Vi δui dV −
p̄i δui dS −
V
(4)
V
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Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Conclusions
Principle of virtual work (4)
Because the variations are kinematically admissible δui = 0 on
Su which gives us
Z
Z
Z
σij δij dV −
p̄Vi δui dV −
p̄i δui dS = 0
(7)
V
V
Sp
or in terms of internal and external virtual work
−δWi − δWe = 0.
(8)
(7) or (8) are called the principle of virtual work. This principle
is also known as the principle of virtual displacements.
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Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Conclusions
Principle of complementarity virtual work (1)
We start from the local kinematic conditions
ij =
1
(ui,j + uj,i ) in V
2
(9)
and
ui = ūi on Su .
(10)
Multiply (9) by a virtual stress field δσij and integrate over the
volume. Multiply (10) by the virtual force δpi and integrate over
the surface Su . The sum of two gives
Z
Z
(ij − ui,j )δσij dV −
(ūi − ui )δpi dS = 0.
(11)
V

Su
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Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Conclusions
Principle of complementarity virtual work (2)
Here we introduced the concept of statically admissible
stresses σˆij = σij + δσij which satisfy the equilibrium equations
in V and static boundary conditions on Sp . Then, it follows that
δσij,j = 0 in V ,
(12)
δpi = 0 on Sp .
(13)
Using the Gauss integral theorem, (12), (13) and δpj = ai δσij
we get
Z
Z
ui δpi dS =
Su

ui,j δσij dV .
V
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Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Conclusions
Principle of complementarity virtual work (3)
By combining (11) and (8) we receive
Z
Z
ij δσij dV −
ūi δpi dS = 0,
V
Su
or in terms of external and internal complementary virtual work
−δWi∗ − δWe∗ = 0.
(7) or (8) are called the principle of complementary virtual work.
This is also known as the principle of virtual stresses and as
the principle of virtual forces.
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Introduction
Work and energy
Principle of virtual work
Principle of complementarity virtual work
Conclusions
Work and potential energy of the internal forces
Work and potential energy of the applied loading
Principle of virtual work
Principle of complementary virtual work
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Conclusions