Stationary Principle_AERSP497F

7/25/2019 Stationary Principle_AERSP497F

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AERSP 470

Energy MethodsEnergy Methods

The Stationary Principle ReviewThe Stationary Principle Review


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Energy Methods & The Stationary PrincipleEnergy Methods & The Stationary Principle

Energy Methods (Lagrangian Methods) vs. Newtonian Methods (based onor!e"Mo#ent E$%i&ibri%#)

Energy Methods' we deine Strain Energy and Eterna& *or+ (a&so ,ineti! Energy- ordyna#i! rob&e#s)

*hat is the dieren!e between rigid and e&asti! bodies/ No Strain in rigid body (idea&i1ation- no body is rigid)

Strain in e&asti! body

2s there strain energy asso!iate with 3rigid bodies/ 5 3e&asti! str%!t%res/

*hat is ,ineti! Energy/

6ow does a rigid body behave %nder the a&i!ation o &oads/ an it %ndergo trans&ation/ Rotation/ E&asti! deor#ation/

6ow does the behavior o an e&asti! body %nder the a&i!ation o &oads dier/


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Energy Methods & The Stationary PrincipleEnergy Methods & The Stationary Principle

*hen a or!e is a&ied to an e&asti! body- wor+ is done. 8hat wor+is stored as energy (Strain Energy)

onsider the o&&owing !ase'

*or+ done by or!e- - as % (instantaneo%s dis&a!e#ent) goes ro#

0$.


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Stationary PrincipleStationary Principle

Stationary Prin!i&e- or Prin!i&e o Mini#%# 8ota& Potentia& Energy

8he eterna& wor+ otentia& is deined as'

8he wor+ done by a syste# in eanding against or!es eerted ro# o%tside

9eine a s!a&ar %n!tion ($) 8ota& Potentia& Energy

or the sring rob&e#

The Stationary Principle states that among all

geometrically possible displacements, q, (q) is

a minimm !or the actal q"


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or the sring rob&e#- #ini#i1e '

8he or!e e$%i&ibri%# e$%ation obtained- ,$ : - as a res%&t o %sing

Energy Methods is the sa#e as what yo% wo%&d have obtained%sing Newtonian Methods. So the two #ethods are e$%iva&ent.

Now ea#ine a ;


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Newtonian Method =asi! or!e E$%i&ibri%#

>%n!tion ?'

>%n!tion ;'


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Lagrangian Method

: @ *


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@se Stationary Prin!i&e'

As with the sing&e


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Sa#&e Prob&e#' So&ving Sring

Syste# @sing Stationary Prin!i&e

8he tota& strain energy o the syste# o o%r srings is eressed in

ter#s o the noda& dis&a!e#ents and sring !onstants'

( ) ( ) ( ) 2343

2

232

2

121

2

1

2

12

2

1qqkqqkqqkUe +

+=

( ) ( ) ( )

[ ]44332211

2

343

2

232

2

121

2

1

2

12

2

1

qPqPqPqP

qqkqqkqqkWU

+++

+

+==

F4F2 00


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( )( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) 4343

4

343332223343232

3

232221112232121

2

1121

1

2212

2212

1

0

Pqqkdq

d

PUkUkkUkPqqkqqkdq

d

PUkUkkUkPqqkqqkdq

d

Pqqkdq

d

dq

d

i

=

++=+=

++=+=

=

=


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=

+

+

4

3

2

1

4

3

2

1

33

3322

2211

11

00

220

022

00

F

F

F

F

U

U

U

U

kk

kkkk

kkkk

kk

8a+e inverse o +B to so&ve or dis&a!e#ents


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#ontinm systems $ bars#ontinm systems $ bars

onsider a bar %nder an %ni


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8o deter#ine the strain energy- start by !onsidering a s#a&& seg#ento the bar o &ength dx

or!e E$%i&ibri%#'

%orce eqilibrim relation


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onsider an in!re#ent in eterna& wor+ by the a&ied or!e

asso!iated with a dis&a!e#ent in!re#ent-du

.

2n!re#ent in eterna& wor+ dW

Stress Strain Re&ation Strain 9is&a!e#ent Re&ation

Note that'


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8hereore- in!re#ent in eterna& wor+'

8h%s- in!re#ent in eterna& wor+ si#&y red%!es to'

P


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o#aring eressions and '- it !an be seen that'

2n!re#ent in eterna& wor+by a&ied or!e- dW

2n!re#ent in storedstrain energy dU

2n!re#ent in strain

energy er %nit

vo&%#e- dU*


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d@ and d@C are d%e to a s#a&& (in!re#enta&) strain dxx(or

dis&a!e#ent du)

: strain energy

er %nit vo&%#e


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8he strain energy stored in the entire bar'

Strain energy- @- or a %ni


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Eterna& *or+'

8ota& Potentia&'


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Sa#&e < Rod


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Sa#&e < Rod


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'eams nder 'ending oad


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'eams nder 'ending oad'eams nder 'ending oad

E%&er


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8h%s- we !an write the aia& and verti!a& dis&a!e#ents o generi!

oint P as'

@se these dis&a!e#ents to get strains'


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8hat &eaves %s with-

And the stress'

Now !onsider the Res%&tant aia& or!e on a !ross se!tion'


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And the Res%&tant =ending Mo#ent on a !ross


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ro# the above eressions- it is seen that Etension =ending

are de!o%&ed'

Re!a&& dis&a!e#ent o generi! oint- P'

=%t or %re bending rob&e# %oter# vanishes- so'


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Re!a&&- Strain Energy'

8his !o#es ro#'

or the bea# bending rob&e#'


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Eterna& *or+- W- or the bea# bending rob&e#'


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Sample Problem * 'eamSample Problem * 'eam

Si#&y s%orted bea# with stiness E2. 9eter#ine the de&e!tion o the

#id


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Sample ProblemSample Problem

8he strain energy- U- d%e to

bending o a bea# is given by

(Given in the rob&e#)

2

2

2

2

1

dz

vdEIM

dz

EI

MUL

=

=

*


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L

zvv B

sin=

2

2

2

21

dz

vdEIM

dzEI

MUL

=

=

3

24

0

2

4

42

4

sin2

L

EIvU

dzL

z

L

vEI

U

B

LB

=

=

L

z

L

v

dz

Lzvd

dz

vd BB

sin

sin

2

2

2

2

2

2

==


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8he otentia& energy is given by'

ro# the stationary rin!i&e o 8PE'

BB Wv

L

EIvVUTPE =+==

3

24

4

( )0

2 3

4

==+

WL

EIv

v

VU B

B

EI

WL

EI

WLv

EIWL

EIWLv

B

pBs

33

3

4

3

02083.048

02053.02

==

==

ro# =ea# =ending 8heory

Stationary Principle_AERSP497F

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