Teaching lecture PVW University of Notti

(1)

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

(2)

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

(3)

(4)

ds

₂

Background:

work by a force or by a couple

P

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:

(

)

θ

M

d

rd

F

ds

F

r

d

F

r

d

r

d

F

r

d

F

dW

=

⋅

=

+

⋅

+

⋅

−

=

2

1

r

Small displacement of a rigid body:

• translation to

A’B’

• rotation of

B’

about

A’

to

B”

(5)

Background:

external Vs internal work

(deformable bodies)

∫

=

y

e

Fdy

W

0

Fy

W

_e

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 truss

y

dy

y

Fdy

dW

_e

=

(

)

Fdy

dFdy

Fdy

dy

dF

F

dW

_e

=

+

⋅

=

+

≈

(6)

Internal Work

(linear systems) and Strain Energy (axial)

Hooke’s Law:

Stress:

Strain:

Final Deflection:

AE

L

N

Ny

U

W

i

2

1

2

=

ε

σ

=

E

A

N

=

σ

L

y

=

ε

AE

NL

y

=

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.

Background:

(deformable bodies)

y

σ

=

N

A

N

=

∫σ

dA

A

(7)

•

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

”.

i

e

W

=

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

Background:

(deformable bodies)

(8)

Background:

Def. of virtual displacement

Virtual displacement

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

Imagine the material to undergo a small displacement

δ

u

from the

current configuration

,

δ

u.

δ

u

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

.

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)

(9)

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

δ

y, then the force

F

does virtual work.

δ

W

=

F

δ

y

If at a particular location of a structure, we have a

real deflection

, y, and impose a

virtual force

δ

F

at

the same location and in the same direction of we

then have the virtual work

δ

W

=

δ

Fy

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

(10)

Generalizations (I) –

generalized internal work for a beam

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 /

U

N N

V

x z

y

x y

M M

T1

(11)

(12)

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:

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)

δ

W

(13)

Application 1

(14)

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

P

M

x

P

M

U

δ

δθ

δ

+

=

+

=

0

(

θδθ

)

δθ

3

sin

0

=

M

+

P

−

l

θ

sin

3

Pl

M

=

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

D

y

δ

θδθ

(15)

Application 2

(16)

Applications

– evaluation of a restrain force

(Principle of substitution of constrains)

Problem

For the following truss, calculate the vertical reaction in C

(17)

Solution

Firstly, set up the free-body-diagram of

the whole truss:

Next,

release the constraint corresponding to reaction

V

_C

and replace it by the unknown force

V

_C

and apply a

virtual displacement to the truss

δ

W

I

=0

δ

W

I

=

δ

W

E

-10

·

δ

y

/2+

V

C

·

δ

y

=0

V

C

=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:

(18)

Application 3

(19)

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/

∑

=

∆

⋅

udL

1

Virtual

Loads

Real

Displ.

(20)

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

δ

W

E

=

δ

W

I

(21)

Generalizations –

generalized internal work for a beam

U

N N

V

x z

y

x y

M M

T1

(22)

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

δ

W

E

=

δ

W

I

∆ ∙ 1

∙

௅

଴

∙

௅

଴

ଷ

3

E= elastic modulus

I= inertial moment of the beam

Adapted from:

(23)

Summary of the applicative concepts

• Virtual Work allows us to

calculate their deflections or the forces acting in structures

.

By making use of

virtual forces:

(24)

By making use of

virtual displacements :

• Virtual Work allows us to

calculate their deflections or the forces acting in structures

.

Summary of the applicative concepts

(25)

(26)

Some history

Bernoulli

(27)

Some history

(28)

Some history

•

Aristotele

speaking about

motion

•

Archimedes

speaking about

statics

3

rd

century

B.C.

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.

•

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 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

1564-1642

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

(29)

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

(30)

RESUME

o

Utility and

scientific relevance

of the PVW

o

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

history