11 energy methods

7/17/2019 11 Energy Methods

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MECHANICS OF

MATERIALS

Third Edition

Ferdinand P. Beer

E. Russell Johnston, Jr.

John T. e!ol"

Le#ture Notes$

J. !alt Oler

Te%as Te#h &ni'ersit(

CHAPTER

© 2002 The McGraw-Hill Companies, Inc. All rights

11Energy Methods



© 2002 The McGraw-Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALST h i r d Beer ) Johnston ) e!ol"

Energy Methods

Strain Energy

Strain Energy DensityElastic Strain Energy for Normal Stresses

Strain Energy For Shearing Stresses

Sample Problem 11.2

Strain Energy for a General State of Stress

Impact LoaingE!ample 11."#

E!ample 11."$

Design for Impact Loas

%or& an Energy 'ner a Single Loa

Deflection 'ner a Single Loa

Sample Problem 11.(

%or& an Energy 'ner Se)eral Loas*astigliano+s ,heorem

Deflections by *astigliano+s ,heorem

Sample Problem 11.




MECHANICS OF MATERIALST h i r d Beer ) Johnston ) e!ol"

/ 0 niform ro is sbecte to a slo3ly increasing

loa/ ,he elementary work one by the loa P as the ro

elongates by a small dx is

3hich is e4al to the area of 3ith dx ner the loa-

eformation iagram.

work elementarydx P dU ==

/ ,he total work one by the loa for a eformation x15

3hich reslts in an increase of strain energy in the ro.

energy strainwork total dx P U

x

=== ∫ 1

"

11212

121

"

1

x P kxdxkxU

x

=== ∫

/ In the case of a linear elastic eformation5

Strain Energy

T




MECHANICS OF MATERIALSTh i r d Beer ) Johnston ) e!ol"

Strain Energy Density/ ,o eliminate the effects of si6e5 e)alate the strain-

energy per nit )olme5

densityenergy straind u

L

dx

A

P

V

U

x

x

==

=

∫

∫

1

1

"

"

ε

ε σ

/ 0s the material is nloae5 the stress retrns to 6ero bt there is a permanent eformation. 7nly the strain

energy represente by the trianglar area is reco)ere.

/ 8emainer of the energy spent in eforming the material

is issipate as heat.

/ ,he total strain energy ensity reslting from the

eformation is e4al to the area ner the cr)e to ε 1.

MECHANICS OF MATERIA ST





Strain-Energy Density

/ ,he strain energy ensity reslting from

setting ε 1 = ε R is the modulus of toughness.

/ ,he energy per nit )olme re4ire to case

the material to rptre is relate to its

ctility as 3ell as its ltimate strength.

/ If the stress remains 3ithin the proportional

limit5

E

E d E u x

22

21

21

"

1

1σ ε

ε ε

ε

=== ∫

/ ,he strain energy ensity reslting from

setting σ 1 = σ Y is the modulus of resilience.

resilienceof modulus E

u Y Y ==

2

2σ

MECHANICS OF MATERIALST





Elastic Strain Energy for Normal Stresses

/ In an element 3ith a nonniform stress istribtion5

energystraintotallim" ===∆∆= ∫ →∆

dV uU dV dU

V U u

V

/ For )ales of u 9 uY 5 i.e.5 belo3 the proportional

limit5

energy strainelasticdV E U x 2

2

∫ == σ

/ 'ner a!ial loaing5 dx AdV A P x ==σ

∫ =

L

dx

AE

P U

"

2

2

AE

L P U

2

2

=

/ For a ro of niform cross-section5



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Elastic Strain Energy for Normal Stresses

I

y M x =σ

/ For a beam sbecte to a bening loa5

∫ ∫ == dV EI

y M dV E

U x2

222

22σ

/ Setting dV dA dx!

dx EI

M

dxdA y EI

M dxdA

EI

y M U

L

L

A

L

A

∫

∫ ∫ ∫ ∫ =

==

"

2

"

2

2

2

"2

22

2

22

/ For an en-loae cantile)er beam5

EI

L P dx

EI

x P U

Px M

L

#2

2

"

22

==

−=

∫






/ For a material sbecte to plane shearing

stresses5

∫ = xy

xy xy d u

γ

γ τ

"

/ For )ales of τ xy 3ithin the proportional limit5

""u

xy xy xy xy

2

2

212

21

τ γ τ γ ===

/ ,he total strain energy is fon from

∫ ∫

=

=

dV "

dV uU

xy

2

2τ






#

$ xy

ρ τ =

∫ ∫ == dV "#

$ dV "

U xy

2

222

22 ρ τ

/ For a shaft sbecte to a torsional loa5

/ Setting dV dA dx!

∫

∫ ∫ ∫ ∫

=

==

L

L

A

L

A

dx"#

$

dxdA"#

$

dxdA"#

$

U

"

2

"

2

2

2

"2

22

2

22 ρ

ρ

/ In the case of a niform shaft5

"#

L$ U

2

2

=





Sample Prolem !!"#

a< ,a&ing into accont only the normal

stresses e to bening5 etermine the

strain energy of the beam for the

loaing sho3n. b< E)alate the strain energy &no3ing

that the beam is a %1"!(5 P = ("

&ips5 L = 12 ft5 a = ft5 % = ; ft5 an E

= 2;!1"# psi.

S7L',I7N>

/ Determine the reactions at A an & from a free-boy iagram of the

complete beam.

/ Integrate o)er the )olme of the

beam to fin the strain energy.

/ 0pply the particlar gi)en

conitions to e)alate the strain

energy.

/ De)elop a iagram of the bening

moment istribtion.





Sample Prolem !!"#

S7L',I7N>

/ Determine the reactions at A an & from a free-boy iagram of the

complete beam.

L

Pa R

L

P% R & A ==

/ De)elop a iagram of the bening

moment istribtion.

' L

Pa M x

L

P% M == 21





Sample Prolem !!"#

'

L

Pa M

x L

P% M

=

=

2

1

?D5 portion7)er the

0D5 portion7)er the

( in2(:&si1"2;

in.1":in.#a

in.1((&ips(

=×=

==

==

I E

%

L P

/ Integrate o)er the )olme of the beam to fin

the strain energy.

( )%a EIL

%a P %aa%

L

P

EI

dx x L

Pa

EI dx x

L

P%

EI

d' EI

M dx

EI

M U

%a

%a

+=

+=

+

=

+=

∫ ∫

∫ ∫

2

22222

2

2

"

2

"

2

"

22

"

21

#2

1

2

1

2

1

22

EIL

%a P U

#

222

=

( ) ( ) ( )

( )( )( )in1((in2(:&si1"2;#

in1":in#&ips("(

222

×=U

&ipsin:;. ⋅=U





Strain Energy for a $eneral State of Stress

/ Pre)iosly fon strain energy e to nia!ial stress an plane

shearing stress. For a general state of stress5( ) (x (x y( y( xy xy ( ( y y x xu γ τ γ τ γ τ ε σ ε σ ε σ +++++=

21

/ %ith respect to the principal a!es for an elastic5 isotropic boy5

( )[ ]

( )

( ) ( ) ( )[ ] istortiontoe12

1

change)olmetoe#

21

22

1

222

2

222

=−+−+−=

=++−

=

+=

++−++=

acc%%ad

c%a'

d '

acc%%ac%a

"u

E

'u

uu

E u

σ σ σ σ σ σ

σ σ σ

σ σ σ σ σ σ ν σ σ σ

/ ?asis for the maximum distortion energy failre criteria5

( ) specimentesttensileafor#

2

"uu Y

Y d d σ

=<




MECHANICS OF MATERIALSThi r d Beer ) Johnston ) e!ol"

%mpact &oading

/ *onsier a ro 3hich is hit at its

en 3ith a boy of mass m mo)ing

3ith a )elocity '".

/ 8o eforms ner impact. Stresses

reach a ma!imm )ale σ m an then

isappear.

/ ,o etermine the ma!imm stress σ m

- 0ssme that the &inetic energy istransferre entirely to the

strctre52"2

1 m'U m =

- 0ssme that the stress-strainiagram obtaine from a static test

is also )ali ner impact loaing.

∫ = dV E

U mm2

2σ

/ @a!imm )ale of the strain energy5

/ For the case of a niform ro5

V

E m'

V

E U mm

2"2

==σ

MECHANICS OF MATERIALST h




E'ample !!"()

?oy of mass m 3ith )elocity '" hits

the en of the nonniform ro &)*.

Ano3ing that the iameter of the

portion &) is t3ice the iameter of

portion )*5 etermine the ma!imm

)ale of the normal stress in the ro.

S7L',I7N>

/ De to the change in iameter5 thenormal stress istribtion is

nonniform./ Fin the static loa P m 3hich proces

the same strain energy as the impact.

/ E)alate the ma!imm stress

reslting from the static loa P m





E'ample !!"()

S7L',I7N>

/ De to the change in iameter5

the normal stress istribtion is

nonniform.

E

V dV

E

m'U

mm

m

22

22

2"2

1

σ σ ≠=

=

∫

/ Fin the static loa P m 3hich proces

the same strain energy as the impact.( ) ( )

L

AE U P

AE

L P

AE

L P

AE

L P U

mm

mmmm

1#

1#

(

22 222

=

=+=

/ E)alate the ma!imm stress reslting

from the static loa P m

AL

E m'

AL E U

A

P

m

mm

2"

:

1#

=

=

=σ




MECHANICS OF MATERIALShi r d Beer ) Johnston ) e!ol"

E'ample !!"(*

0 bloc& of 3eight % is roppe from a

height h onto the free en of the

cantile)er beam. Determine the

ma!imm )ale of the stresses in the beam.

S7L',I7N>

/ ,he normal stress )aries linearly alongthe length of the beam as across a

trans)erse section.


the same strain energy as the impact./ E)alate the ma!imm stress






E'ample !!"(*

S7L',I7N>

/ ,he normal stress )aries linearly

along the length of the beam as

across a trans)erse section.

E

V dV

E

+hU

mm

m

22

22 σ σ ≠=

=

∫


the same strain energy as the impact.

For an en-loae cantile)er beam5

2

#

#

L EI U P

EI

L P U

mm

mm

=

=

/ E)alate the ma!imm stress


( ) ( )22

##

c I L

+hE

c I L

E U

I

Lc P

I

c M

m

mmm

==

==σ





Design for %mpact &oads/ For the case of a niform ro5

V

E U mm

2

=σ

( )

( ) ( ) ( )

V

E U

V Lccc Lc I L

c I L E U

mm

mm

2(

BB

#

(12

(12(

(12

2

=

===

=

σ

π π

σ

/ For the case of the cantile)er beam

@a!imm stress rece by>/ niformity of stress

/ lo3 mols of elasticity 3ith

high yiel strength

/ high )olme

/ For the case of the nonniform ro5

( ) ( )

V

E U

AL L A L AV

AL

E U

mm

mm

:

2B2B2B(

1#

=

=+=

=

σ

σ





+or, and Energy nder a Single &oad

/ Pre)iosly5 3e fon the strain

energy by integrating the energy

ensity o)er the )olme.

For a niform ro5

( )

AE

L P dx A

E

A P

dV E

dV uU

L

22

2

21

"

21

2

==

==

∫

∫ ∫ σ

/ Strain energy may also be fon from

the 3or& of the single loa P 15

∫ =1

"

x

dx P U

/ For an elastic eformation5

11212

121

""

11

x P xk dxkxdx P U x x

==== ∫ ∫

/ Ano3ing the relationship bet3een

force an isplacement5

AE

L P

AE

L P P U

AE

L P x

2

211

121

11

=

=

=





+or, and Energy nder a Single &oad

/ Strain energy may be fon from the 3or& of other types

of single concentrate loas.

EI

L P

EI

L P P

y P dy P U

y

#

21

1

121

112

1

"

1

=

=

== ∫

/ ,rans)erse loa

EI

L M

EI

L M M

M d M U

2

211

121

112

1

"

1

=

=

== ∫ θ θ

θ

/ ?ening cople

#"

L$

#"

L$ $

$ d $ U

2

211

121

112

1

"

1

=

=

== ∫ φ φ

φ

/ ,orsional cople





Deflection nder a Single &oad

/ If the strain energy of a strctre e to a

single concentrate loa is &no3n5 then the

e4ality bet3een the 3or& of the loa an

energy may be se to fin the eflection.

l Ll L &* &) :."#." ==

From statics5

P , P , &* &) :."#." −=+=

From the gi)en geometry5

/ Strain energy of the strctre5

( ) ( )[ ] AE

l P

AE

l P

AE

L ,

AE

L ,

U &* &* &) &)

22

22

#(."2

:."#."

22

=+

=

+=

/ E4ating 3or& an strain energy5

AE

Pl y

y P AE

L P U

&

&

$2:."

#(."21

2

=

==





Sample Prolem !!".

@embers of the trss sho3n consist of

sections of alminm pipe 3ith the

cross-sectional areas inicate. 'sing

E = $ GPa5 etermine the )erticaleflection of the point E case by the

loa P.

S7L',I7N>

/ Fin the reactions at 0 an ? from afree-boy iagram of the entire trss.

/ 0pply the metho of oints to

etermine the a!ial force in each

member.

/ E)alate the strain energy of the

trss e to the loa P .

/ E4ate the strain energy to the 3or&

of P an sol)e for the isplacement.




MECHANICS OF MATERIALSi r d Beer ) Johnston ) e!ol"

Sample Prolem !!".S7L',I7N>

/Fin the reactions at 0 an ? from a free- boy iagram of the entire trss.

:21:21 P & P A P A y x ==−=

/ 0pply the metho of oints to etermine

the a!ial force in each member.

P ,

P ,

)E

*E

:1

:1$

+=

−=

"

:1

=

+=

)*

A)

,

P ,

P ,

P ,

)E

*E

:21

(

−=

= "= A& ,




MECHANICS OF MATERIALSir d Beer ) Johnston ) e!ol"

Sample Prolem !!".

/ E)alate the strain energy of the

trss e to the loa P .

( )2

22

2;$""2

1

2

1

2

P E

A

L ,

E E A

L , U

i

ii

i

ii

=

== ∑∑

/ E4ate the strain energy to the 3or& by P

an sol)e for the isplacement.

( )( );

2

21

1"$

1"("1"$.2;

22;$""22

×

××=

==

=

E

E

E

y

E P

P P U y

U Py

↓= mm2$.1# E y

MECHANICS OF MATERIALST h i





+or, and Energy nder Se/eral &oads

/ Deflections of an elastic beam sbecte to t3o

concentrate loas5

22212122212

21211112111

P P x x x

P P x x x

α α

α α

+=+=+=+=

/ 8e)ersing the application se4ence yiels

21111221

22222

1 2 P P P P U α α α ++=

/ Strain energy e!pressions mst be e4i)alent.

It follo3s that α 12=α 21 C Maxwell-s reci.rocal

theorem<.

22222112

21112

1 2 P P P P U α α α ++=

/ *ompte the strain energy in the beam by

e)alating the 3or& one by slo3ly applying

P 1 follo3e by P 25






Castigliano0s Theorem

22222112

21112

1 2 P P P P U α α α ++=

/ Strain energy for any elastic strctre

sbecte to t3o concentrate loas5

/ Differentiating 3ith respect to the loas5

22221122

12121111

x P P P

U

x P P P

U

=+=∂∂

=+=∂∂

α α

α α

/ )astigliano-s theorem> For an elastic strctresbecte to n loas5 the eflection x / of the

point of application of P / can be e!presse as

an /

/ /

/ /

/$

U

M

U

P

U x

∂∂

=∂∂

=∂∂

= φ θ


B J h t ! l"




MECHANICS OF MATERIALSr d Beer ) Johnston ) e!ol"

Deflections y Castigliano0s Theorem

/ 0pplication of *astigliano+s theorem is

simplifie if the ifferentiation 3ith respect to

the loa P / is performe before the integration

or smmation to obtain the strain energy U .

/ In the case of a beam5

∫ ∫ ∂

∂=

∂

∂==

L

/ / /

L

dx P

M

EI

M

P

U xdx

EI

M U

""

2

2

/ For a trss5

/

in

i i

ii

/ /

n

i i

ii

P

,

E A

L ,

P

U x

E A

L , U

∂

∂=

∂

∂==

∑∑ == 11

2

2


B J h t ! l"





Sample Prolem !!"1

@embers of the trss sho3n

consist of sections of alminm

pipe 3ith the cross-sectional areas

inicate. 'sing E = $ GPa5

etermine the )ertical eflection of

the oint ) case by the loa P .

/ 0pply the metho of oints to eterminethe a!ial force in each member e to 0.

/ *ombine 3ith the reslts of Sample

Problem 11.( to e)alate the eri)ati)e

3ith respect to 0 of the strain energy of

the trss e to the loas P an 0.

/ Setting 0 = "5 e)alate the eri)ati)e

3hich is e4i)alent to the esire

isplacement at ) .

S7L',I7N>

/ For application of *astigliano+s theorem5introce a mmy )ertical loa 0 at ) .

Fin the reactions at A an & e to the

mmy loa from a free-boy iagram of

the entire trss.


B J h t ! l"





Sample Prolem !!"1

S7L',I7N>

/ Fin the reactions at A an & e to a mmy loa 0 at ) from a free-boy iagram of the entire trss.

0 &0 A0 A y x (

( ==−=

/ 0pply the metho of oints to etermine the a!ial

force in each member e to 0.

0 , ,

0 , ,

, ,

&* A&

)* A)

*E )E

("

"

"

−==

−==

==

MECHANICS OF MATERIALST h i r

Beer Johnston e!ol"




Sample Prolem !!"1

/ *ombine 3ith the reslts of Sample Problem 11.( to e)alate the eri)ati)e3ith respect to 0 of the strain energy of the trss e to the loas P an 0.

( )0 P E 0

,

E A

L , y i

i

ii) (2#("#

1 +=∂∂

= ∑

/ Setting 0 = "5 e)alate the eri)ati)e 3hich is e4i)alent to the esire

isplacement at ) .

( )Pa1"$

1"("("#;

×

×=

1 y) ↓= mm#.2) y

11 energy methods

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