Sadd Elasticity Solutions Manual

Solutions Manual

Elasticity: Theory, Applications and Numerics Second Edition

By

Martin H. Sadd

Professor Department of Mechanical Engineering & Applied Mechanics

University of Rhode Island Kingston, Rhode Island

Foreword

Exercises found at the end of each chapter are an important ingredient of the text as they provide homework for student engagement, problems for examinations, and can be used in class to illustrate other features of the subject matter. This solutions manual is intended to aid the instructors in their own particular use of the exercises. Review of the solutions should help determine which problems would best serve the goals of homework, exams or be used in class. The author is committed to continual improvement of engineering education and welcomes feedback from users of the text and solutions manual. Please feel free to send comments concerning suggested improvements or corrections to [email protected]. Such feedback will be shared with the text user community via the publishers web site. Martin H. Sadd January 2009

Copyright 2009, Elsevier Inc. All rights reserved.

1-1.

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1-15.

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4-7.

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4-8.

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103

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4-9.

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4-10.

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4-11.

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213

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4-12.

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4-13.

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4-14.

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4-15.

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4-16.

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4

4

3

43

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4-17.

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4-18.

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5-1.

p 40o

x

y

a

b

(a)

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y S

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(b)

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(d) (c)

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====

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5-1. Continued

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5-3.

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5-4.

),(),(,),(),(

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0),(,0),(

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0),(,),(:(2) Material

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:(1) Material(a)

1)2(

1)1(

1)2(

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====

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(b)

x

y

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a

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(a)

a

a

x

y

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(a)

a

h1 x

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(1)

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5-4. Continued

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====

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5-5.

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:(1) Material(a)

1)2(

1)1(

1)2(

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1)1(

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21)2(

21)2(

)1()1(

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====

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0),(),(,),(),(

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5-6.

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5-7.

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5-9.

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5-10.

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5-11.

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,

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5-12.

solution. elasticityproper a not are thusand ity,compatibilnot but mequilibriusatisfy stresses

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x

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y = 10a

x

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5-13*.

( )

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6-1.

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6-4.

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6-9.

222

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100000

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EEU

+++=

++++=

=

6-10.

EIlM

EclMdyyl

EcMdVy

EcMUdVU

yEcMy

cM

EEU

ycM

c

c

l c

cT

x

zxyzxyzyx

2

3

221

0

2

0 6

22

6

2

26

22

6

22

3

23)2(

89

8989

49

21

2

0,23 :field stressGiven

=====

===

======


6-11.

)1(821

211

)(1 :EnergyStrain Total

)1(4)(1)22(

21

,,0

42422

4222

0 0

222

0

2222

222222222222

+=+=

+=+=

++==

+=++=++=

======

LRELRE

RLE

rdrdrLE

dxdAyxE

UdVU

rEyxE

yxE

U

xy

R

L

AVT

yzxzxyzyx

6-12.

+==

==

+=+====

+=+

S V iiiiV ii

V iiV iiS ii

S V iiiiS iiS V iiiiS ii

iiiiiiii

S V iiiiS V iiii

dVxFdSxTK

dVeV

dVedVudSun

dVxFdSxTK

dSundVAxFdSAxTdSuKAn

TFuKAnTFAxu

dVuFdSuTdVuFdSuT

31

:Theorem Divergence theFrom313

,, as state second and , 3,0, :as statefirst Choose

:Theorem Reciprocal

,

)1()1()1(

)1()2()1()2()2()1()2()1(


6-13.

EIlq

EIlqlwlxxl

EIxqw

EIlq

EIlqlwxlx

EIlqw

EIlq

EIlqlw

lx

EIlqw

cEI

lqc

lcl

EIc

lqlcl

EIc

lcqlclcl

EI

dxlxc

lxcqdx

lxc

lxc

lEI

N

dxlxjcq

lxj

ljcEI

dxwqdxwdEI

lxjcw

lxjw

ooo

ooo

ooo

o

o

o

l

o

l

l N

jjo

N

jj

l

o

N

jjj

44233

442

4

5

4

5

4

25

4

1

24

4

2

14

4

1

122

214

4

210

2

2104

4

01

2

1

2

0

2

2

2

1

015.03845)2/( thusand,)2(

24 issolution exact The

0104.096

)2/(span -midat and,)(24

are 6.2 Example from Results

0131.04)2/(span -midat and,sin4

issolution eapproximat the thusand ,0,4 :tscoefficien for the Solving

0162

0

022

0

2821

2

2sinsin2sin4sin2

:Case 2

sinsin2

2

sinsin

==+=

===

==

=

==

==

==

+=

+

+==

=

=

==

==

=

6-14.

yIlqlyxlx

Iqy

dxwdE

yIlqly

lx

Ilqy

dxwdE

yIlqly

Ilqy

dxwdE

ydxwdE

dxwdEIM

IMy

oxx

ox

oxx

ox

oxx

ox

xx

2

max2

2

2

2

max3

2

2

2

2

max

2

2

2

2

2

2

2

125.0)2/()(,)(2

:SolutionExact

129.0)2/()(,sin4 :10-6 Exercise

0833.0)2/()(,12

:2-6 Example

,

====

====

====

===


7-1.

xyxyxyxyxy

xyy

yxx

yxyxyxyxy

yxyxyxyxx

yxyx

yxyx

yyxy

xyxx

Eee

Ee

Ee

EEe

EEe

eeee

eeeeee

+===

+=

+=

+++=++=

++++=+++=

=++=+++=

++=

1212

])1[(1

])1[(1

)(1)()21)(1()(21)(

)(212

)(1)()21)(1()(21)(

)(212

)(2))((2

2)(2)(


7-2.

+

=+

+

=+

+

+

==+

+

+

==+

+

+

+=+

+=

+

+

+

=

+

+=+=+=

=+

+

++

+

=+

+

++

+

=+

+

+

+

+

=+

+

=+

+

+

+

+

=++

+

=+

+=

++

=

yF

xF

yF

xF

yxyx

xyyyF

Fyx

yxxxF

Fyx

yxyx

EyxExEy

yxe

x

e

ye

Ee

Ee

Ee

Fyv

xu

yxv

yv

Fyv

xu

xxu

yu

Fyv

yv

xu

yxv

yu

xF

yx

Fxv

yu

yxu

yv

xu

xF

yx

xv

yu

yv

yv

xu

xu

yv

xu

yxyx

yxyxxy

xyyyy

yxy

xyxxx

xyx

yxxyyx

xyxyyx

xyyx

xyxyxyyyxx

y

x

yyyxy

xxxyx

xyyx

)()1(

givesrelation ity compatibil intoresult thisusing and,2

0

0

equations, mequilibriu fromBut

2)()1(

12])1[(1])1[(1

2

1,])1[(1,])1[(1:Equationity Compatibil Michell-Beltrami

0)()

0)()

02)()(0

0)(2)(0

)(,2)(,2)(

:Equations sNavier'

2

2

2

2

22

2

2

2

2

2

2

2

2

2

222

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2


7-3.

00(b)(a)

00(a)(b)

(b)

0)()(

(a)

0)()(

0)(,0)(:equationsNavier

4

4

22

22

22

22

22

==

==

+

=

=

+

+

++

+

+

+

++

=

=

+

+

++

+

+

++

=+

+

++=+

+

++

vyx

uyx

vy

ux

Fyv

xu

yv

yF

yv

xu

xu

x

vx

uy

Fyv

xu

yv

xF

yv

xu

xu

y

Fyv

xu

yvF

yv

xu

xu

yx

yx

yx

7-4.

=

==++==+==

=

==++==+==

==++=

=+==

++=+=+=

A zy

AyyAA z

A

czzA

Tzy

A zx

AxxAA z

A

czzA

Tzx

A zAA z

A

czzA

Tzz

czyxz

czz

Tz

xdAI

A

dAxIAIdACxBxyAxxdA

xdAxdAM

ydAI

B

dAyIBIdACyByAxyydA

ydAydAM

dAA

CACdACByAxdA

dAdAR

A

CByAx

2

2

)(

)(

2

2

)(

)(

)(

)(

)(

)()(

1

where,axes) principal(for )(

0)(0

1

where,axes) principal(for )(

0)(0

:endsat momentsresultant zeroFor 1axes) principal(for )(

0)(0

:section -cross with endsat forceresultant zeroFor

,)(

sum, assolution totalChoose


7-5.

( )( )

checks alsowhich ,)1( )()(

)(satisfy alsomust stress normal plane-of-out The

000)(1

1)(

000210

000210

:equationsity compatibil and equilbriumsatisfy must Stresses

22

2

2

ykxkxkxy

kxkxyyF

xF

kxy

kyx

Fyx

kyy

kxyx

Fyx

yx

yxz

yxyx

yyxy

xxyx

+=+=++=

==+

+

=+

==+

=++

==

+=+

+

7-6.

xyxyxyxy

xyyyxyxy

yxxyxyxx

yxyx

yxyx

xyy

yxx

eEE

e

eeEeeEeeE

eeEeeEeeE

eeE

eeE

Ee

Ee

+=+=

+=++=

+=+++=

+=+=+

==

11

)(1

)(1

)(1

2

)(1

)(1

)(1

2

)(1

)(1

)(1

)(1

2

2


7-7.

+

+=+

+

=+

+

+

==+

+

+

==+

+

+

+=++

+=

+

=+

+===

=+

+

+

=+

+

+

+

+

=+

+

+

+

+

=++

=+

+

+

=+

+

++

+

=+

+

+

+

+

=++

+

+=

+

=

+

=

yF

xF

yF

xF

yxyx

xyyyF

Fyx

yxxxFF

yx

yxyx

EyxExEyyxe

xe

ye

Ee

Ee

Ee

Fyv

xu

yEv

Fxy

uyvE

xv

yxuE

Fxu

yvE

yxv

yuE

xF

yx

Fyv

xu

xEu

Fxyv

yuE

yxv

xuE

Fxv

yuE

yyv

xuE

xF

yx

xv

yuE

xu

yvE

yv

xuE

yxyx

yxyxxy

xyyyy

yxy

xyxxx

xyx

yxxyyx

xyxyyxxyyx

xyxyxyyyxx

y

y

yyyxy

x

x

xxxyx

xyyx

)1()(

givesrelation ity compatibil intoresult thisusing and,2

0

0

equations, mequilibriu fromBut

2)()1(

1

12)(1)(12

1,)(1,)(1:Equationity Compatibil Michell-Beltrami

0)1(2

01)1(2

01)1(2

0

0)1(2

0)1(21

0)1(21

0

:equations mequilibriu into Substitute)1(2

,1

,1

2

2

2

2

22

2

2

2

2

2

2

2

2

2

222

2

2

2

2

22

2

2

2

2

2

2

2

2

22

22

2

2

2

2

22

2

2

2

2

22


7-8.

l.dimensiona- threebe willfield the thusand , coordinateplane-of-out on the depend willntsdisplaceme theimply that results These

functionsarbitrary are and where

),(210

21

),(210

21

, offunction )(1

that Note

functionarbitrary an is ),( where,),(

2

2

z

hg

yxhzxfz

xev

xfz

xe

zu

xw

zue

yxgzyfz

yev

yfz

ye

zv

yw

zve

yxeee

yxfyxfzewzwe

zzxz

zzyz

yxz

zz

+

=

==

+

=

+

=

==

+

=

=+=

+==


7-9.

problem. general afor satisfied benot willrelationsity compatibil thegintegratin fromresult theso and linear, benot willfieldstrain or stress plane-in thegeneralIn

)(1

)((7.2.2)Relation

constantarbitrary an is where,)()()(

)()(

)( and constant)(0)()(0

)(0

)(0

:gives results three thesengIntergrati

0

02

02

:aren formulatio stress plane in the includednot e which werrelationsity compatibil vanishing-non threeThe

3

2

2

2

2

2

22

2

22

2

2

2

2

2

22

2

2

2

2

yxyxz

z

z

zz

z

zz

zz

zzxyzxyz

zzxxz

zyzzy

eeE

e

cbyaxe

ccbyyFbyFbxgye

yFaxeayfxe

bxgayfxgyfyx

e

xgye

ye

yfxe

xe

yxe

ye

xe

ze

zyxe

xe

xze

ze

xe

ye

zye

ye

ze

+=+=

++=+====

+===

======

=

=

==

=

+

+

=

=

=+

=

=

+


7-10.

case!either for changenot does modulusshear that theNotice

)1(2)1(21

)1()21(

)1

1(2

)1()21(

)1(2

)1)(1()21)(1()1(

)1()21(

)1

21)(1

1(

1)1()21(

)21)(1(

:stress plane strain to Plane

)1(2)1(21

1)1

1(2

1)1(2

)31)(1()21)(1()1(

1)1

21)(1

1(

11)21)(1(

:strain plane tostress Plane

2

2

2

2

2

2

2

2

=+=+++

++=

++++

=+=

+=+++

+++=

++

++

++

=+=

=+=+

=

+=+=

+=+

=

+

=+

=

EEE

E

EEE

E

EEE

E

EEE

E


7-11.

+

=+

+

+=+

+

+=+

=+

+

++

=+

+

++=+

+

+

=+

+

+

+

+=+

+

+

=+

+

=+

=+

+

+

=+

+

++++

=+

+

++

yF

xF

yF

xF

yF

xF

Fyv

xu

xu

Fyv

xu

xEuF

yv

xu

x

E

u

Fyv

xu

xEu

yF

xF

yF

xF

yF

xF

Fyv

xu

xEu

Fyv

xu

xEEu

Fyv

xu

xu

yxyx

yxyx

yxyx

x

xx

x

yxyx

yxyx

yxyx

x

x

x

11)()

11()(

)1()( : (7.2.7)Equation

(7.2.5)relation for Likewise

0)(: toreduceswhich

0))1)(1(2

0))

11(2

1

1-7 Table from results theusingstrain plane toConverting

0))1(2

:(7.2.5)Equation (b)

)1()(

11

1)(

11)(: (7.1.7)Equation

(7.1.5)relation for Likewise

0))1(2

: toreduceswhich

0))1(2)1)(1(

(

10-7 Exercise from results theusing stress plane toConverting

0)(:(7.1.5)Equation (a)

22

2

2

2

22

2

21

22

2

2

2

2

21


7-12.

1.-7 Exercisein given sexpression match withproperly Results

1

1

111

])1[(11

1)(1

])1[(11

1)(1)31)(1(

,,1

,1

:Strain Plane Stress Plane (b)

6.-7 Exercisein given sexpression match withproperly Results1

22

)(11

)()1)(1(

2)( :Likewise

)(11

)()1)(1(

2)(

)1)(1(,,

1,

)1()21( :Stress Plane Strain Plane (a)

2

2

2

2

2

2

2

xyxyxyxy

xyxyxyy

yxyxyxx

xyxyxyxy

xyyyxyyxy

yxxyxxyxx

EEEe

EEEe

EEEe

EEE

Eee

eeEeEeeEeee

eeEeEeeEeee

EEE

+=

+

+=

+=

=

+=

=

+

+==

+=++++++=

+=++++++=

++

++


7-13*.

bygiven are ntsdisplacemestrain plane and stress plane theof plots MATLAB, Usingidentical. become ntsdisplaceme two the0, ratio sPoisson'When

12

)1(/

][2

)1(

121

/][

2

:0) ( aixs- Along12

)1(,)1( :ResultsStrain Plane

1,

1 :Strain Plane Stress Plane

,][2

, :Results Stress Plane

22

2.22

2

.

2

2.22

.

2222

2

2

222

==

==

=

+==

+==

lx

EIMlvlx

EIMv

lx

EIMlvlx

EIMv

yx

lxyEI

MvEI

Mxyu

EE

lxllxyEIMv

EIMxyu

StrainPStrainP

StressPStressP

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-0.5

-0.4

-0.3

-0.2

-0.1

0

Dimensionless Distance, x/l

Dim

ensi

onle

ss D

ispl

acem

ent,

v(x,

0)/(M

l 2/E

I)

Exercise 7-13 Note |v-plane stess| > |v-plane strain|

plane stessplane strain

n=0.4


7-14*.

bygiven are ntsdisplacemestrain plane and stress plane theof plots MATLAB, Usingidentical. become ntsdisplaceme two the0, ratio sPoisson'When

/1

11)1(

/)(

,/1)21()1(

/)(

:lizingdimensiona-Non

111

121

)1()21(

)1

1(

1,

)1()21( :Stress Plane Strain Plane

)21()1( :ResultStrain Plane

111

.

111

.

21

21

2

2

21

+++=

++=

+++=

+

+

+++

+=

++

+

++=

rrrr

ETru

rrrr

ETru

rrr

ET

rrr

E

Tu

EE

rrr

ETu

StressPrStrainPr

r

r

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

16

18

20

Dimensionless Distance, r/r1

Dim

ensi

onle

ss D

ispl

acem

ent,

u r /(

Tr1

/E)

Exercise 7-14 Note |ur-plane stess| > |ur-plane strain|


n=0.4


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.5

1

1.5

2

2.5

Poisson Ratio,

Dim

ensi

onle

ss B

ound

ary

Dis

plac

emen

t, u r

/(Tr

1 /E

)Exercise 7-14 Note |ur-plane stess| > |ur-plane strain|


7-15.

VyV

xVV

xV

y

yF

xF

VyV

xVV

xV

y

yF

xF

yVF

xVF

yxV

xV

y

yxyx

yxyx

yxxyyx

242

2

2

2

2

2

2

22

2

242

2

2

2

2

2

2

22

2

2

2

2

2

2

)1()1()(

)1()(:Stress Plane

121

11)(

11)(:Strain Plane

,,,,

=

+

+=+++

+

+=+

=

+

=+

++

+

=+

=

==+

=+=


7-16.

+

=++=

+=+=

=

+

++

+

+

+

++

++

+

=++=

=

=

==

==

++

++

+=

+

+=

+

=

+

++

=+=

=

+

==

=

ru

ruu

reeee

urr

ueeee

ru

xu

xu

xu

xu

yu

yu

yu

yu

yu

yu

yu

yu

xu

xu

xu

xu

eeeerryrrx

ryxryrx

xu

xu

xu

xu

yu

yu

yu

yu

uux

uuyx

vyue

yu

yu

yu

yuuu

yyve

xu

xu

xu

xuuu

xxue

rxyyxr

rxyyx

r

rr

rr

rr

rr

xyyxr

rr

rr

rrxy

rr

ry

rr

rx

121)sin(coscossincossin

1cossin2cossin

, Likewise

cossincoscossinsin

sinsincoscos

sincoscossinsin

cossinsincoscos

cossin2sincos

cossin,sincos

cossincossin,cossin,sincos

coscossinsinsinsincoscos21

)cossin()sincos(21

21

coscossinsin)cossin(

sinsincoscos)sincos(

22

22

2

2

22

22


7-17.

02

2

11

2

121,1,

:Relationsnt Displaceme-Strain

2

22

2

2

2

2

2

3

2

22

2

2

222

2

2

2

=

+

=

=

+

=

+

+=

+

+

+=

+=

+

=

+==

rrr

rrrrrr

rr

rr

rr

rr

rrr

rr

erer

rerer

r

rere

rur

ru

ruruu

rrerer

r

ru

ruruu

ru

ru

ruu

rr

r

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Sadd Elasticity Solutions Manual