Teaching lecture PVW University of Nottingham March 2014

Introducing the principle of virtual work

(PVW)Francesco PetriniSchool of Civil and Industrial Engineering, Sapienza University of Rome, Via Eudossiana 18 - 00184 Rome (ITALY), tel. [email protected]@stronger2012.com

The University of Nottingham, Department of Civil Engineering, Nottingham, 27 March 2014

PVW: relevance in the scientific field

University of Nottingham, 27 March 2014 Francesco Petrini

• 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.

Background


ds2

Background: work by a force or by a couple


P

PParticle

Particle

Work of a force (infinitesimal movement)

Work of a force (finite movement)B

A

TUDelft. Virtual Work. Aerospace Engineering lecture notes. available at: httpocw.tudelft.nlcoursesaerospace-engineeringstaticslectures7-virtual-work

Given a particle P

Adapted from:

( )θθ dMrdFdsFrdF

rdrdFrdFdW

===⋅=

+⋅+⋅−=

22

211r

r

rr

r

r

r

Small displacement of a rigid body:

• translation to A’B’

• rotation of B’ about A’ to B”

Given a rigid body

Background: external Vs internal work (deformable bodies)


∫= y

e FdyW0

FyWe 2

1=

Gradually Applied Force F Due to a Force small increment dF

External Work done by a Force

Caprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/

Given an axially loaded deformable body

This is exactly the area under the force-deformation diagram in the case of elastic behavior of the trussy dy

y

FdydWe =

( )Fdy

dFdyFdydy

dFFFdWe ≈+=⋅

++=

22

dF

dy

dF

Adapted from:


Internal Work (linear systems) and Strain Energy (axial)

Hooke’s Law:

Stress:

Strain:

Final Deflection:

AE

LNNyUW ii 22

1 2

===

εσ E=

AN=σ

Ly=ε AE

NLy =

Internal work Internal strain energy

N=F

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.



yAN=σ

N =∫σdA

A

Adapted from:


• 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”.

ie WW =

Thus:

� The external work done by external forces moving through external displacements is equal to the strain energy stored in the material


* We can consider a structure isolated once we have identified and accounted for all sources of restraint and loading

� On the basis of the following


Adapted from:

Background: Def. of virtual displacement


Virtual displacement

Virtual displacement are in general taken as infini tesimal (δ_). This is due to the factthat virtual displacements must be small enough such that the force directions are maintainedVirtual displacement need to be compatible with the existing restrains

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 isit related to the applied external forces – it does not actually occur physically.

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

Given a deformable body (the deformability is not necessary)

Adapted from:

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

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


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 /

Principle of Virtual Displacements: Virtual work is the work done by the actual forces acting on the body moving through a virtual displacement.

Principle of Virtual Forces: Virtual work is the work done by a virtual force acting on the body moving through the actual displacements

Generalizations (I) – generalized internal work for a beam



U

N

N

V

x z

y

x y

M M

T1

Adapted from:

PVW formulation and applications


The formulations PVW

University of Nottingham, 27 March 2014 Francesco PetriniCaprani C.(2010). Virtual Work. 3rd Year Structural Engineering lecture notes. available at: http://teaching.ust.hk/~civl113/download/

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:

A body is in equilibrium if, and only if, the virtual work of all forces acting on the body is zero

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) .

case for deformable bodies includes the case for rigid bodies in which the internal virtual work becomes zero

GENERAL FORMULATION

ALTERNATIVE FORMULATION FOR DEFORMABLE BODIES (Virtual displacement version)


δWI=δWE

Application 1


A set of rigid bodies

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

D

PM

xPM

UUU

δδθδδδ

+=

+==

0

0

( )θδθδθ sin30 lPM −+=

θsin3PlM =

virtual displacements

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/

Internal work is equal to zero

External work is equal to Internal work

Dy θδθδ

θθθθθ

θ

sin3

sin3cos

sincos3tan

cos3

lx

llxy

lx

D

DD

D

−=

===

=

Adapted from:

Application 2


Evaluation of a restrain force

Applications – evaluation of a restrain force(Principle of substitution of constrains)


Problem

For the following truss, calculate the vertical reaction in C

Adapted from:


Solution

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

δWI=0 δWI=δWE -10·δy/2+VC·δy=0 VC=5 kN

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.

virtual displacements

Adapted from:

Applications – evaluation of a restrain force(Principle of substitution of constrains)

Application 3


Unit load method

Unit load method


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/

∑=∆⋅ udL1

VirtualLoads

Real Displ.

Adapted from:

Applications


Mukherjee S., Prathap G. (2012). Lecture : Variational Principles in Computational Solid Mechanics. Available at: http://nal-ir.nal.res.in/5179/1/FEA_Lectures_2009_ICAST2.pdf

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:

Generalizations – generalized internal work for a beam



U

N

N

V

x z

y

x y

M M

T1

Adapted from:

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 ∆ ∙ 1 � � � � ∙ � � ��

�� ∙ � ��

�� 3

E= elastic modulusI= inertial moment of the beam

Adapted from:

Curvature Χ

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:


By making use of virtual displacements :

• Virtual Work allows us to solve determinate and indeterminate structures and to calculate their deflections or the forces acting in structures .

Summary of the applicative concepts

Adapted from:

Some history


Some history


Bernoulli


Some history



Adapted from:

Some history


• Aristotele speaking about motion

• Archimedes speaking about statics3rd

century B.C.

1717And 1724 A.C.

• The Swiss mathematicians Jean Bernoulli , was the firs that introduced the fundamental concept of infinitesimal magnitude for the virtual displacemen ts .

• In a successive scientific he unified the two approaches based either on velocities or on displacements

1763 And 1788 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

• Galielo Galilei re-elaborated the above mentioned applications and expressed the PVW in a more linear way, just referring to the gravitational loadso Still referring to the case of the levero Making reference on velocitiesso He started to refer to something of “virtual” velocities in its explanation

1564-1642 A.C.

• The extension of the PVW applications to other cases wi th 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

1596-1652 A.C.

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.archives-ouvertes.fr/docs/00/19/50/80/PDF/Virtualwork_leastaction.pdf)

RESUMEo Utility and scientific relevance of the PVWo Background

• work of a force or of a couple• internal Vs external work• connection between work and energy• virtual quantities and virtual work

o PVW formulation and application• Application for rigid bodies• Application for deformable bodies

o Some historyo Generalizations


Teaching lecture PVW University of Nottingham March 2014

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