Teaching lecture PVW University of Notti
Introducing the
principle of virtual work
(PVW)
Francesco Petrini
School of Civil and Industrial Engineering, Sapienza University of Rome,
Via Eudossiana 18 - 00184 Rome (ITALY), tel. +39-06-44585072
francesco.petrini@uniroma1.it
Francesco.petrini@stronger2012.com
The University of Nottingham, Department of Civil Engineering, Nottingham, 27 March 2014
PVW: relevance in the scientific field
• Virtual Work allows us to solve determinate and indeterminate structures and to
calculate their deflections. That is, it can achieve everything that all the other
methods together can achieve.
• Virtual Work provides a basis upon which vectorial mechanics (i.e. Newton’s laws) can
be linked to the energy methods (i.e. Lagrangian methods) which are the basis for
finite element analysis and advanced mechanics of materials.
• Virtual Work is a fundamental theory in the mechanics of bodies. So fundamental in
fact, that Newton’s 3 equations of equilibrium can be derived from it.
• A rigorous and exhaustive demonstration of the PVW has not been provided at today
Mechanics Magazine published in London in 1824.
University of Nottingham, 27 March 2014
Francesco Petrini
Background
University of Nottingham, 27 March 2014
Francesco Petrini
Background: work by a force or by a couple
Given a particle P
Work of a force (infinitesimal movement)
P
Particle
Work of a force (finite movement)
B
P
A
Particle
Given a rigid body
ds2
Small displacement of a rigid body:
• translation to A’B’
• rotation of B’ about A’ to B”
r r r r
r
dW = − F ⋅ dr1 + F ⋅ (dr1 + dr2 )
r r
= F ⋅ dr2 = F ds2 = F rdθ = M dθ
Adapted from:
TUDelft. Virtual Work. Aerospace Engineering lecture notes. available at: httpocw.tudelft.nlcoursesaerospace-engineeringstaticslectures7-virtual-work
University of Nottingham, 27 March 2014
Francesco Petrini
Background: external Vs internal work (deformable bodies)
External Work done by a Force
Given an axially loaded deformable body
dF
dy
Gradually Applied Force F
Due to a Force small increment dF
dWe = Fdy
y
dW e =
We = ∫ Fdy
0
y
We =
1
Fy
2
This is exactly the area
under the force-deformation
diagram in the case of
elastic behavior of the truss
F + (F + dF )
dFdy
⋅ dy = Fdy +
≈ Fdy
2
2
y
dy
dF
Adapted from:
Caprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Background: external Vs internal work (deformable bodies)
Internal Work (linear systems) and Strain Energy (axial)
A
Stress:
Hooke’s Law:
Strain:
σ =N A
Final Deflection:
σ = Eε
ε=y L
NL
y=
AE
N =∫σdA
y
1
N 2L
W i = U i = Ny =
2
2 AE
Internal work
σ =N A
N=F
Internal strain energy
Dotted area underneath the load-deflection curve. It represents the work
done during the elongation of the element. This work (or energy, as they
are the same thing) is stored in the spring and is called strain energy
and denoted U.
Adapted from:
Caprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Background: external Vs internal work (deformable bodies)
On the basis of the following
•
The external work is an manifestation of external energy (added or removed to the structural
system)
•
As previously stated, the internal work is equivalent to the variation of the internal strain energy
(for elastic systems without dissipations)
•
Law of Conservation of Energy: “Consider a structural system that is isolated* such it neither
gives nor receives energy; the total energy of this system remains constant”.
Thus:
The external work done by external forces moving through external displacements is equal to the
strain energy stored in the material
We = Wi
* We can consider a structure isolated once we have identified and accounted for all sources of restraint and loading
Adapted from:
Caprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Background: Def. of virtual displacement
Given a deformable body (the deformability is not necessary)
Imagine the material to undergo a small displacement δu from the current configuration, δu.
δu is a virtual displacement, meaning that it is an imaginary displacement, and in no way is
it related to the applied external forces – it does not actually occur physically.
Virtual displacement are in general taken as infinitesimal (δ_). This is due to the fact
that virtual displacements must be small enough such that the force directions are
maintained
Virtual displacement need to be compatible with the existing restrains
Virtual displacement
Adapted from:
Kelly, P. Solid mechanics part III. available at:
http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_3_Stress_Mass_Momentum/Stress_Balance_Principles_09_Virtual
_Work.pdf
University of Nottingham, 27 March 2014
Francesco Petrini
Background: Def. of virtual work
Given any real force, F, acting on a body to which
we apply a virtual displacement. If the virtual
displacement at the location of and in the direction
of F is δy, then the force F does virtual work.
δW=Fδy
Principle of Virtual Displacements:
Virtual work is the work done by the actual
forces acting on the body moving through a
virtual displacement.
If at a particular location of a structure, we have a
real deflection, y, and impose a virtual force δF at
the same location and in the same direction of we
then have the virtual work
δW=δFy
Principle of Virtual Forces:
Virtual work is the work done by a virtual force
acting on the body moving through the actual
displacements
Zhen Y.(2012). Lecture : Energy Methods (II) — Principle of Virtual Work and Unit Load Method available at:
http://am.hit.edu.cn/courses/mechmat2012/Courseware_files/27_uni_presentation.pdf /
Caprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Generalizations (I) – generalized internal work for a beam
N
U
N
y
V
z
M
x
x
M
y
T1
Adapted from:
Zhen Y.(2012). Lecture : Energy Methods (II) — Principle of Virtual Work and Unit Load Method available at:
http://am.hit.edu.cn/courses/mechmat2012/Courseware_files/27_uni_presentation.pdf /
University of Nottingham, 27 March 2014
Francesco Petrini
PVW formulation and
applications
University of Nottingham, 27 March 2014
Francesco Petrini
The formulations PVW
Based upon the Principle of Minimum Total Potential Energy, we can see that any small
variation about equilibrium must do no work. Thus, the Principle of Virtual Work states that:
GENERAL FORMULATION
A body is in equilibrium if, and only if, the virtual work of all
forces acting on the body is zero
ALTERNATIVE FORMULATION FOR DEFORMABLE BODIES (Virtual displacement version)
External virtual work is equal to internal virtual work made by
equilibrated forces and stresses though unrelated virtual
displacements and strains (compatible with the restrains) .
δWI=δWE
case for deformable bodies includes the case for rigid bodies in which the internal virtual
work becomes zero
Zhen Y.(2012). Lecture : Energy Methods (II) — Principle of Virtual Work and Unit Load Method available at:
http://am.hit.edu.cn/courses/mechmat2012/Courseware_files/27_uni_presentation.pdf /
Caprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Application 1
A set of rigid bodies
University of Nottingham, 27 March 2014
Francesco Petrini
Applications – set of rigid bodies
Determine the magnitude of the couple M required to
maintain the equilibrium of the mechanism.
SOLUTION:
• Apply the principle of virtual work
δU = 0 = δU M + δU P
0 = Mδθ + PδxD
Internal work is equal to zero
External work is equal to Internal work
virtual displacements
xD = 3l cos θ
y D = xD tan θ = 3l cos θ
yD
sin θ
= 3l sin θ
cos θ
δxD = −3l sin θδθ
0 = Mδθ + P(− 3l sin θδθ )
M = 3Pl sin θ
Adapted from:
E. Russel Jhonstone Jr.(2010). Method of Virtual Work. Vector mechanics for Engineers: Statics. McGraw-Hill, Ninth ed.. Lecture Notes by J. Walt Oler available
at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Application 2
Evaluation of a restrain force
University of Nottingham, 27 March 2014
Francesco Petrini
Applications – evaluation of a restrain force
(Principle of substitution of constrains)
Problem
For the following truss, calculate the vertical reaction in C
Adapted from:
Caprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Applications – evaluation of a restrain force
Solution
(Principle of substitution of constrains)
Firstly, set up the free-body-diagram of
the whole truss:
Next, release the constraint corresponding to reaction
VC and replace it by the unknown force VC and apply a
virtual displacement to the truss
virtual
displacements
no internal virtual work is done since the members do not undergo virtual deformation. The truss
rotates as a rigid body about the support A.
δWI=0
Adapted from:
δWI=δWE
-10·δy/2+VC·δy=0
VC=5 kN
Caprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Application 3
Unit load method
University of Nottingham, 27 March 2014
Francesco Petrini
Unit load method
1 ⋅ ∆ = ∑udL
Virtual
Loads
Real
Displ.
Adapted from:
E. Russel Jhonstone Jr.(2010). Method of Virtual Work. Vector mechanics for Engineers: Statics. McGraw-Hill, Ninth ed.. Lecture Notes by J. Walt Oler available
at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Applications
Problem
For the following beam, calculate the vertical tip deflection Δ
BENDING
MOMENTS
We take the real set
of displacements
We take the virtual
set of forces
δWE = δWI
Adapted from:
Mukherjee S., Prathap G. (2012). Lecture : Variational Principles in Computational Solid Mechanics. Available at: http://nalir.nal.res.in/5179/1/FEA_Lectures_2009_ICAST2.pdf
University of Nottingham, 27 March 2014
Francesco Petrini
Generalizations – generalized internal work for a beam
N
U
N
y
V
z
M
x
x
M
y
T1
Adapted from:
Zhen Y.(2012). Lecture : Energy Methods (II) — Principle of Virtual Work and Unit Load Method available at:
http://am.hit.edu.cn/courses/mechmat2012/Courseware_files/27_uni_presentation.pdf /
University of Nottingham, 27 March 2014
Francesco Petrini
Applications
Problem
For the following beam, calculate the vertical tip deflection Δ
BENDING
MOMENTS
We take the real set
of displacements
We take the virtual
set of forces
Curvature Χ
δWE = δWI
∆∙1
E= elastic modulus
I= inertial moment of the beam
ଷ
∙
∙
3
Adapted from:
Mukherjee S., Prathap G. (2012). Lecture : Variational Principles in Computational Solid Mechanics. Available at: http://nalir.nal.res.in/5179/1/FEA_Lectures_2009_ICAST2.pdf
University of Nottingham, 27 March 2014
Francesco Petrini
Summary of the applicative concepts
• Virtual Work allows us to solve determinate and indeterminate structures and to
calculate their deflections or the forces acting in structures.
By making use of virtual forces:
Adapted from:
Caprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Summary of the applicative concepts
• Virtual Work allows us to solve determinate and indeterminate structures and to
calculate their deflections or the forces acting in structures.
By making use of virtual displacements :
Adapted from:
Caprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Some history
University of Nottingham, 27 March 2014
Francesco Petrini
Some history
Bernoulli
Mukherjee S., Prathap G. (2012). Lecture : Variational Principles in Computational Solid Mechanics. Available at: http://nalir.nal.res.in/5179/1/FEA_Lectures_2009_ICAST2.pdf
University of Nottingham, 27 March 2014
Francesco Petrini
Some history
Adapted from:
Mukherjee S., Prathap G. (2012). Lecture : Variational Principles in Computational Solid Mechanics. Available at: http://nalir.nal.res.in/5179/1/FEA_Lectures_2009_ICAST2.pdf
University of Nottingham, 27 March 2014
Francesco Petrini
Some history
•
3rd
century •
B.C.
Aristotele speaking about motion
Archimedes speaking about statics
•
Galielo Galilei re-elaborated the above mentioned applications and expressed the PVW in
a more linear way, just referring to the gravitational loads
o Still referring to the case of the lever
o Making reference on velocitiess
o He started to refer to something of “virtual” velocities in its explanation
15961652
A.C.
•
The extension of the PVW applications to other cases with respect to the lever is due
to the French scientist Cartesio (Renè Des Cartes), which applied the principle to the
inclined plane. He preferred to refer to displacements instead of velocities
1717
And
1724
A.C.
•
The Swiss mathematicians Jean Bernoulli, was the firs that introduced the fundamental
concept of infinitesimal magnitude for the virtual displacements.
In a successive scientific he unified the two approaches based either on velocities or on
displacements
1763
And
1788
A.C.
•
15641642
A.C.
•
Luigi Giuseppe Lagrange, was highly devoted to the clarification of the concepts of Virtual
entities and Work, partially introduced by the previous scientists. He tried to demonstrate
the PVW with a partial success
University of Nottingham, 27 March 2014
Francesco Petrini
Generalizations
PVW can be applied or extended to a large number of problems:
• In non-linear problems
• In dynamic problems
• In presence of thermal loads
• In presence of magnetic fields
• In presence of residual stresses
• In presence of stochastic phenomena (http://hal.archivesouvertes.fr/docs/00/19/50/80/PDF/Virtualwork_leastaction.pdf)
University of Nottingham, 27 March 2014
Francesco Petrini
RESUME
o
o
o
o
o
Utility and scientific relevance of the PVW
Background
• work of a force or of a couple
• internal Vs external work
• connection between work and energy
• virtual quantities and virtual work
PVW formulation and application
• Application for rigid bodies
• Application for deformable bodies
Some history
Generalizations
University of Nottingham, 27 March 2014
Francesco Petrini
principle of virtual work
(PVW)
Francesco Petrini
School of Civil and Industrial Engineering, Sapienza University of Rome,
Via Eudossiana 18 - 00184 Rome (ITALY), tel. +39-06-44585072
francesco.petrini@uniroma1.it
Francesco.petrini@stronger2012.com
The University of Nottingham, Department of Civil Engineering, Nottingham, 27 March 2014
PVW: relevance in the scientific field
• Virtual Work allows us to solve determinate and indeterminate structures and to
calculate their deflections. That is, it can achieve everything that all the other
methods together can achieve.
• Virtual Work provides a basis upon which vectorial mechanics (i.e. Newton’s laws) can
be linked to the energy methods (i.e. Lagrangian methods) which are the basis for
finite element analysis and advanced mechanics of materials.
• Virtual Work is a fundamental theory in the mechanics of bodies. So fundamental in
fact, that Newton’s 3 equations of equilibrium can be derived from it.
• A rigorous and exhaustive demonstration of the PVW has not been provided at today
Mechanics Magazine published in London in 1824.
University of Nottingham, 27 March 2014
Francesco Petrini
Background
University of Nottingham, 27 March 2014
Francesco Petrini
Background: work by a force or by a couple
Given a particle P
Work of a force (infinitesimal movement)
P
Particle
Work of a force (finite movement)
B
P
A
Particle
Given a rigid body
ds2
Small displacement of a rigid body:
• translation to A’B’
• rotation of B’ about A’ to B”
r r r r
r
dW = − F ⋅ dr1 + F ⋅ (dr1 + dr2 )
r r
= F ⋅ dr2 = F ds2 = F rdθ = M dθ
Adapted from:
TUDelft. Virtual Work. Aerospace Engineering lecture notes. available at: httpocw.tudelft.nlcoursesaerospace-engineeringstaticslectures7-virtual-work
University of Nottingham, 27 March 2014
Francesco Petrini
Background: external Vs internal work (deformable bodies)
External Work done by a Force
Given an axially loaded deformable body
dF
dy
Gradually Applied Force F
Due to a Force small increment dF
dWe = Fdy
y
dW e =
We = ∫ Fdy
0
y
We =
1
Fy
2
This is exactly the area
under the force-deformation
diagram in the case of
elastic behavior of the truss
F + (F + dF )
dFdy
⋅ dy = Fdy +
≈ Fdy
2
2
y
dy
dF
Adapted from:
Caprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Background: external Vs internal work (deformable bodies)
Internal Work (linear systems) and Strain Energy (axial)
A
Stress:
Hooke’s Law:
Strain:
σ =N A
Final Deflection:
σ = Eε
ε=y L
NL
y=
AE
N =∫σdA
y
1
N 2L
W i = U i = Ny =
2
2 AE
Internal work
σ =N A
N=F
Internal strain energy
Dotted area underneath the load-deflection curve. It represents the work
done during the elongation of the element. This work (or energy, as they
are the same thing) is stored in the spring and is called strain energy
and denoted U.
Adapted from:
Caprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Background: external Vs internal work (deformable bodies)
On the basis of the following
•
The external work is an manifestation of external energy (added or removed to the structural
system)
•
As previously stated, the internal work is equivalent to the variation of the internal strain energy
(for elastic systems without dissipations)
•
Law of Conservation of Energy: “Consider a structural system that is isolated* such it neither
gives nor receives energy; the total energy of this system remains constant”.
Thus:
The external work done by external forces moving through external displacements is equal to the
strain energy stored in the material
We = Wi
* We can consider a structure isolated once we have identified and accounted for all sources of restraint and loading
Adapted from:
Caprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Background: Def. of virtual displacement
Given a deformable body (the deformability is not necessary)
Imagine the material to undergo a small displacement δu from the current configuration, δu.
δu is a virtual displacement, meaning that it is an imaginary displacement, and in no way is
it related to the applied external forces – it does not actually occur physically.
Virtual displacement are in general taken as infinitesimal (δ_). This is due to the fact
that virtual displacements must be small enough such that the force directions are
maintained
Virtual displacement need to be compatible with the existing restrains
Virtual displacement
Adapted from:
Kelly, P. Solid mechanics part III. available at:
http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_3_Stress_Mass_Momentum/Stress_Balance_Principles_09_Virtual
_Work.pdf
University of Nottingham, 27 March 2014
Francesco Petrini
Background: Def. of virtual work
Given any real force, F, acting on a body to which
we apply a virtual displacement. If the virtual
displacement at the location of and in the direction
of F is δy, then the force F does virtual work.
δW=Fδy
Principle of Virtual Displacements:
Virtual work is the work done by the actual
forces acting on the body moving through a
virtual displacement.
If at a particular location of a structure, we have a
real deflection, y, and impose a virtual force δF at
the same location and in the same direction of we
then have the virtual work
δW=δFy
Principle of Virtual Forces:
Virtual work is the work done by a virtual force
acting on the body moving through the actual
displacements
Zhen Y.(2012). Lecture : Energy Methods (II) — Principle of Virtual Work and Unit Load Method available at:
http://am.hit.edu.cn/courses/mechmat2012/Courseware_files/27_uni_presentation.pdf /
Caprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Generalizations (I) – generalized internal work for a beam
N
U
N
y
V
z
M
x
x
M
y
T1
Adapted from:
Zhen Y.(2012). Lecture : Energy Methods (II) — Principle of Virtual Work and Unit Load Method available at:
http://am.hit.edu.cn/courses/mechmat2012/Courseware_files/27_uni_presentation.pdf /
University of Nottingham, 27 March 2014
Francesco Petrini
PVW formulation and
applications
University of Nottingham, 27 March 2014
Francesco Petrini
The formulations PVW
Based upon the Principle of Minimum Total Potential Energy, we can see that any small
variation about equilibrium must do no work. Thus, the Principle of Virtual Work states that:
GENERAL FORMULATION
A body is in equilibrium if, and only if, the virtual work of all
forces acting on the body is zero
ALTERNATIVE FORMULATION FOR DEFORMABLE BODIES (Virtual displacement version)
External virtual work is equal to internal virtual work made by
equilibrated forces and stresses though unrelated virtual
displacements and strains (compatible with the restrains) .
δWI=δWE
case for deformable bodies includes the case for rigid bodies in which the internal virtual
work becomes zero
Zhen Y.(2012). Lecture : Energy Methods (II) — Principle of Virtual Work and Unit Load Method available at:
http://am.hit.edu.cn/courses/mechmat2012/Courseware_files/27_uni_presentation.pdf /
Caprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Application 1
A set of rigid bodies
University of Nottingham, 27 March 2014
Francesco Petrini
Applications – set of rigid bodies
Determine the magnitude of the couple M required to
maintain the equilibrium of the mechanism.
SOLUTION:
• Apply the principle of virtual work
δU = 0 = δU M + δU P
0 = Mδθ + PδxD
Internal work is equal to zero
External work is equal to Internal work
virtual displacements
xD = 3l cos θ
y D = xD tan θ = 3l cos θ
yD
sin θ
= 3l sin θ
cos θ
δxD = −3l sin θδθ
0 = Mδθ + P(− 3l sin θδθ )
M = 3Pl sin θ
Adapted from:
E. Russel Jhonstone Jr.(2010). Method of Virtual Work. Vector mechanics for Engineers: Statics. McGraw-Hill, Ninth ed.. Lecture Notes by J. Walt Oler available
at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Application 2
Evaluation of a restrain force
University of Nottingham, 27 March 2014
Francesco Petrini
Applications – evaluation of a restrain force
(Principle of substitution of constrains)
Problem
For the following truss, calculate the vertical reaction in C
Adapted from:
Caprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Applications – evaluation of a restrain force
Solution
(Principle of substitution of constrains)
Firstly, set up the free-body-diagram of
the whole truss:
Next, release the constraint corresponding to reaction
VC and replace it by the unknown force VC and apply a
virtual displacement to the truss
virtual
displacements
no internal virtual work is done since the members do not undergo virtual deformation. The truss
rotates as a rigid body about the support A.
δWI=0
Adapted from:
δWI=δWE
-10·δy/2+VC·δy=0
VC=5 kN
Caprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Application 3
Unit load method
University of Nottingham, 27 March 2014
Francesco Petrini
Unit load method
1 ⋅ ∆ = ∑udL
Virtual
Loads
Real
Displ.
Adapted from:
E. Russel Jhonstone Jr.(2010). Method of Virtual Work. Vector mechanics for Engineers: Statics. McGraw-Hill, Ninth ed.. Lecture Notes by J. Walt Oler available
at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Applications
Problem
For the following beam, calculate the vertical tip deflection Δ
BENDING
MOMENTS
We take the real set
of displacements
We take the virtual
set of forces
δWE = δWI
Adapted from:
Mukherjee S., Prathap G. (2012). Lecture : Variational Principles in Computational Solid Mechanics. Available at: http://nalir.nal.res.in/5179/1/FEA_Lectures_2009_ICAST2.pdf
University of Nottingham, 27 March 2014
Francesco Petrini
Generalizations – generalized internal work for a beam
N
U
N
y
V
z
M
x
x
M
y
T1
Adapted from:
Zhen Y.(2012). Lecture : Energy Methods (II) — Principle of Virtual Work and Unit Load Method available at:
http://am.hit.edu.cn/courses/mechmat2012/Courseware_files/27_uni_presentation.pdf /
University of Nottingham, 27 March 2014
Francesco Petrini
Applications
Problem
For the following beam, calculate the vertical tip deflection Δ
BENDING
MOMENTS
We take the real set
of displacements
We take the virtual
set of forces
Curvature Χ
δWE = δWI
∆∙1
E= elastic modulus
I= inertial moment of the beam
ଷ
∙
∙
3
Adapted from:
Mukherjee S., Prathap G. (2012). Lecture : Variational Principles in Computational Solid Mechanics. Available at: http://nalir.nal.res.in/5179/1/FEA_Lectures_2009_ICAST2.pdf
University of Nottingham, 27 March 2014
Francesco Petrini
Summary of the applicative concepts
• Virtual Work allows us to solve determinate and indeterminate structures and to
calculate their deflections or the forces acting in structures.
By making use of virtual forces:
Adapted from:
Caprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Summary of the applicative concepts
• Virtual Work allows us to solve determinate and indeterminate structures and to
calculate their deflections or the forces acting in structures.
By making use of virtual displacements :
Adapted from:
Caprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/
University of Nottingham, 27 March 2014
Francesco Petrini
Some history
University of Nottingham, 27 March 2014
Francesco Petrini
Some history
Bernoulli
Mukherjee S., Prathap G. (2012). Lecture : Variational Principles in Computational Solid Mechanics. Available at: http://nalir.nal.res.in/5179/1/FEA_Lectures_2009_ICAST2.pdf
University of Nottingham, 27 March 2014
Francesco Petrini
Some history
Adapted from:
Mukherjee S., Prathap G. (2012). Lecture : Variational Principles in Computational Solid Mechanics. Available at: http://nalir.nal.res.in/5179/1/FEA_Lectures_2009_ICAST2.pdf
University of Nottingham, 27 March 2014
Francesco Petrini
Some history
•
3rd
century •
B.C.
Aristotele speaking about motion
Archimedes speaking about statics
•
Galielo Galilei re-elaborated the above mentioned applications and expressed the PVW in
a more linear way, just referring to the gravitational loads
o Still referring to the case of the lever
o Making reference on velocitiess
o He started to refer to something of “virtual” velocities in its explanation
15961652
A.C.
•
The extension of the PVW applications to other cases with respect to the lever is due
to the French scientist Cartesio (Renè Des Cartes), which applied the principle to the
inclined plane. He preferred to refer to displacements instead of velocities
1717
And
1724
A.C.
•
The Swiss mathematicians Jean Bernoulli, was the firs that introduced the fundamental
concept of infinitesimal magnitude for the virtual displacements.
In a successive scientific he unified the two approaches based either on velocities or on
displacements
1763
And
1788
A.C.
•
15641642
A.C.
•
Luigi Giuseppe Lagrange, was highly devoted to the clarification of the concepts of Virtual
entities and Work, partially introduced by the previous scientists. He tried to demonstrate
the PVW with a partial success
University of Nottingham, 27 March 2014
Francesco Petrini
Generalizations
PVW can be applied or extended to a large number of problems:
• In non-linear problems
• In dynamic problems
• In presence of thermal loads
• In presence of magnetic fields
• In presence of residual stresses
• In presence of stochastic phenomena (http://hal.archivesouvertes.fr/docs/00/19/50/80/PDF/Virtualwork_leastaction.pdf)
University of Nottingham, 27 March 2014
Francesco Petrini
RESUME
o
o
o
o
o
Utility and scientific relevance of the PVW
Background
• work of a force or of a couple
• internal Vs external work
• connection between work and energy
• virtual quantities and virtual work
PVW formulation and application
• Application for rigid bodies
• Application for deformable bodies
Some history
Generalizations
University of Nottingham, 27 March 2014
Francesco Petrini