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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 publisher’s web site. Martin H. Sadd January 2009

Copyright © 2009, Elsevier Inc. All rights reserved.

mailto:[email protected]�

1-1.

(scalar)5401

(matrix)402000201

(scalar)7400000201

(vector)243

(matrix)350

10180461

110240111

110240111

(scalar)251104160111

(scalar)6141 (a)

332211

332313

322212

312111

333323321331322322221221311321121111

332211

333332323131232322222121131312121111

332211

=++=++=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

=++++++++=

++++++++=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=++=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

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⎣

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⎥⎥⎥

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⎡=

=++++++++=

++++++++==++=++=

bbbbbbbbbbbbbbbbbbbbbbbbbb

bb

bbabbabbabbabbabbabbabbabbabba

babababa

aa

aaaaaaaaaaaaaaaaaaaaaaaa

ii

ji

jiij

iiijij

jkij

ijij

ii

(scalar)6114

(matrix)112112224(scalar)17240120044

(vector)634

(matrix)8160480261

240120021

240120021

(scalar)304160140041

(scalar)5221 (b)

332211

332313

322212

312111

333323321331322322221221311321121111

332211

333332323131232322222121131312121111

332211

=++=++=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

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⎡=

=++++++++=

++++++++=

⎥⎥⎥

⎦

⎤

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⎡=++=

⎥⎥⎥

⎦

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=++++++++=

++++++++==++=++=


bb


babababa

aa


ii

ji

jiij

iiijij

jkij

ijij

ii


(scalar)2011

(matrix)000011011

(scalar)3000001011

(vector)112

(matrix)1841931722

410201111

410201111

(scalar)251610401111

(scalar)5401 (c)

332211

332313

322212

312111

333323321331322322221221311321121111

332211

333332323131232322222121131312121111

332211

=++=++=

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++++++++=

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⎦

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⎡

⎥⎥⎥

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⎡=

=++++++++=

++++++++==++=++=


bb


babababa

aa


ii

ji

jiij

iiijij

jkij

ijij

ii

1-2.

conditions eappropriat esatisfy th andclearly011101110

21

231381112

21

)(21)(

21)a(

][)( ijij

jiijjiijij

aa

aaaaa

⎥⎥⎥

⎦

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⎣

⎡

−−−+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

−++=


020

21

450542022

21

)(21)(

21)b(

][)( ijij

jiijjiijij

aa

aaaaa

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−++=



21

831302122

21

)(21)(

21)c(

][)( ijij

jiijjiijij

aa

aaaaa

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−−+

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⎢⎢⎢

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⎡=

−++=

1-3.

0011100100

831302122

41 :2(c)-1 Exercise From

0030302

020

450542022

41 :2(b)-1 Exercise From

0011101110

231381112

41 :2(a)-1 Exercise From

002

][)(

][)(

][)(

=⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

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=⎟⎟⎟⎟

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=⎟⎟⎟⎟

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⎝

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⎦

⎤

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⎣

⎡

−−−

⎥⎥⎥

⎦

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⎢⎢⎢

⎣

⎡=

=⇒=⇒−=−=

T

ijij

T

ijij

T

ijij

ijijijijijijjijiijij

traa

traa

traa

bababababa

1-4.

ij

jkij

iiiijij

aaaaaaaaaa

aaaaaaaaaa

aaaa

aaaaaaaaa

aaaa

=⎥⎥⎥

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⎦

⎤

⎢⎢⎢

⎣

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⋅⋅⋅⋅⋅⋅

δ+δ+δδ+δ+δδ+δ+δ=δ

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

δ+δ+δδ+δ+δδ+δ+δ

=δ+δ+δ=δ

333231

232221

131211

331323121311321322121211311321121111

3

2

1

333232131

323222121

313212111

332211


1-5.

333231

232221

131211

312232211331233321123223332211

332112322311312213322113312312332211

332112213322311132312213321

322113312312312231332211123321

)()()(

)det(

aaaaaaaaa

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

aaaaaaaaa

aaaaaaaaaaaaa kjiijkij

=

−+−−−=−−−++=

ε+ε+ε+

ε+ε+ε=ε=

1-6.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=

⎥⎥⎥

⎦

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⎢⎢⎢

⎣

⎡

−⎥⎥⎥

⎦

⎤

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⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−==′

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

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⎥⎥⎥

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−==′

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−−−

=⎥⎥⎥

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⎢⎢⎢

⎣

⎡

−⎥⎥⎥

⎦

⎤

⎢⎢⎢

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⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−==′

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

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⎣

⎡

−==′

⎥⎥⎥

⎦

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⎢⎢⎢

⎣

⎡

−−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−⎥⎥⎥

⎦

⎤

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⎣

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⎥⎥⎥

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−==′

⎥⎥⎥

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⎢⎢⎢

⎣

⎡=

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⎣

⎡

⎥⎥⎥

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⎤

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⎣

⎡

−==′

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=⇒

5.05.12/25.25.32/2

021

2/22/202/22/20

001

410201111

2/22/202/22/20

001

2/22/2

1

011

2/22/202/22/20

001 :1(c)-1 Exercise From

5.05.105.15.40221

2/22/202/22/20

001

240120021

2/22/202/22/20

001

02

2

112

2/22/202/22/20

001 :1(b)-1 Exercise From

120140

021

2/22/202/22/20

001

110240111

2/22/202/22/20

001

22

1

201

2/22/202/22/20

001 :1(a)-1 Exercise From

2/22/202/22/20

001 axis-about xrotation 45 1

o

T

pqjqipij

jiji

T

pqjqipij

jiji

T

pqjqipij

jiji

ij

aQQa

bQb

aQQa

bQb

aQQa

bQb

Q


1-7.

⎥⎦

⎤⎢⎣

⎡

θ+θθ+−θθ−θθ−−θθ−θθ−−θθ+θθ++θ

=

⎥⎦

⎤⎢⎣

⎡θθ−θθ

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡θθ−θθ

==′

⎥⎦

⎤⎢⎣

⎡θ+θ−θ+θ

=⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡θθ−θθ

==′

⎥⎦

⎤⎢⎣

⎡θθ−θθ

=⎥⎦

⎤⎢⎣

⎡

θθ+θ−θ

=⎥⎦

⎤⎢⎣

⎡′′′′

=

2222112

211

2122211

221

2212211

212

2222112

211

2221

1211

21

21

2

1

2212

2111

coscossin)(sinsincossin)(cossincossin)(cossincossin)(cos

cossinsincos

cossinsincos

cossinsincos

cossinsincos

cossinsincos

cos)90cos()90cos(cos

),cos(),cos(),cos(),cos(

aaaaaaaaaaaaaaaa

aaaa

aQQa

bbbb

bb

bQb

xxxxxxxx

Q

T

pqjqipij

jiji

o

o

ij

1-8.

ijjpippqjqipij aQaQaQQa δ==δ=δ′' 1-9.

jkiljlikklijlmknjnimkmjnimlpkpjmim

npmqnqmppqmnlqkpjnimjkiljlikklij

QQQQQQQQQQQQ

QQQQ

δγδ+δβδ+δαδ=γ+β+α=

δγδ+δβδ+δαδ=δ′δ′γ+δ′δ′β+δ′δ′α

ln

)('''

1-10.

klijlikjljkiijkl

jkiljlikklijjkiljlikklijijkl

C

C

=δδ+δδβ+δαδ=

δδ+δδβ+δαδ=δγδ+δβδ+δαδ=

)(

)(

1-11.

321

3

2

1

3132213

1

3

2

2

1

321

3

2

1

000000

00

00

00

000000

If

λλλ=λ

λλ

=

λλ+λλ+λλ=λ

λ+

λλ

+λ

λ=

λ+λ+λ==

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

λλ

λ=

a

a

iia

III

II

aI

a


1-12.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −==′

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

±=⇒===⇒

=++=−

=+−⇒=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−

=λ

±=⇒±==⇒

=++=

=+−⇒=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

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⎡−

−

=λ

−±=±=−=⇒=++

==+

⇒=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=λ=λ=λ−=λ⇒

=−λ+λλ⇒=−λ+λλ⇒=λ+λ−λ−∴

=−=−=⇒⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−=

100000002

2/200011011

100011011

2/200011011

21

and2/200

011011

2/2by given ismatrix rotation The

)1,0,0(1,01

0202

0000021012

:Case1

)011)(2/2(2/2

10

00

100011011

:Case0

)0,1,1)(2/2(,2/21

00

0300011011

:Case21,0,2 Roots

0)1)(2(0)2( 02 isEqn sticCharacteri

0,2,1100011011

(a)

)3()3(321

2)3(3

2)3(2

2)3(1

)3(2

)3(1

)3(2

)3(1

3

2

1

3

)2(21

2)2(3

2)2(2

2)2(1

)2(3

)2(2

)2(1

3

2

1

2

)1()1(2

)1(1

2)1(3

2)1(2

2)1(1

)1(3

)1(2

)1(1

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1

321

223

T

pqjpipij

ij

aaaij

aQQa

Q

nnnnnn

nnnn

nnn

,,nn

nnnn

nn

nnn

nnnnn

nnn

nnn

IIIIIIa

n

n

n


1-12.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−

⎥⎥⎥

⎦

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⎢⎢⎢

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⎡ −==′

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

±=⇒===⇒

=++=−

=+−⇒=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−

=λ

±=⇒±==⇒

=++=

=+−⇒=

⎥⎥⎥

⎦

⎤

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⎣

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⎥⎥⎥

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−

−=λ

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==+

⇒=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=λ=λ−=λ−=λ⇒

=+λ+λλ⇒=+λ+λλ⇒=λ−λ−λ−∴

==−=⇒⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−=

000010003

2/200011011

000021012

2/200011011

21

and2/200

011011


)1,0,0(1,01

0202

0000021012

:Case0

)011)(2/2(2/2

10

00

100011011

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00

0300011011

:Case30,1,3 Roots

0)1)(3(0)34( 034 isEqn sticCharacteri

0,3,4000021012

(b)

)3()3(321

2)3(3

2)3(2

2)3(1

)3(2

)3(1

)3(2

)3(1

3

2

1

3

)2(21

2)2(3

2)2(2

2)2(1

)2(3

)2(2

)2(1

3

2

1

2

)1()1(2

)1(1

2)1(3

2)1(2

2)1(1

)1(3

)1(2

)1(1

)1(3

)1(2

)1(1

1

321

223

T

pqjpipij

ij

aaaij

aQQa

Q

nnnnnn

nnnn

nnn

,,nn

nnnn

nn

nnn

nnnnn

nnn

nnn

IIIIIIa

n

n

n


1-12.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −==′

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

±=±==

±=⇒−==⇒=++

=+−⇒=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−

=λ=λ

−±=±=−=⇒=++

==+

⇒=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=λ=λ=λ−=λ⇒

=+λλ=λ−λ−∴

==−=⇒⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−=

000000002

2/200011011

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1-14.

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=∇⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=∇

=++=∇

−−−=∂∂∂∂∂∂=×∇

++=++=⋅∇++=

+=∇⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=∇

=++=∇

+−=∂∂∂∂∂∂=×∇

++=++=⋅∇

++=

++=∇⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=∇

=++=∇

+−=∂∂∂∂∂∂=×∇

++=++=⋅∇++=

uu

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1-15.

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knkmki

jnjmji

inimii

jkimnijkiimn

jkijki

aa

aaaaa

aaaa

aa

ε−=∴

−=+−=−−=

δδ−δδ−=δδδδδδδδδ

−=εε−=ε

ε−=

)(21)(

21

)(21

21

21

21

1-16.

)()()(2)()(

2)()()(

)()( (a)

,,,

22

,,,,,,,,,,,,,,2

,,,

uuu ⋅∇φ+⋅φ∇=φ+φ=φ=φ⋅∇ψ⋅φ+ψ∇φ+ψφ∇=

ψφ+φψ+ψφ=ψφ+ψφ+ψφ+φψ=ψφ+φψ=φψ=φψ∇

ψ∇φ+φψ∇=ψφ+φψ=φψ=φψ∇

kkkkkk

kkkkkkkkkkkkkkkkkkk

kkk

uuu∇∇

φ∇=φ=φ=φ⋅

=φε=φε=φ×

×⋅−×⋅=ε+ε=+ε=ε=×⋅

×φ+×φ=φε+φε=φ+φε=φε=φ×

2,,,

,,,

,,,,,

,,,,

)(

inryantisymmetandsymmetryofbecause0)(

)()()()()

)()))( (b)

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vuuvvuvuvu

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∇∇

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∇∇(∇

∇∇((∇

vuuvvu

uuu

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u

u

∇⋅−⋅∇=

−=δδ−δδ=εε=εε=××∇−⋅=

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=ε=ε=×⋅

)(21

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1-17.

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⎟⎠⎞

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+⎟⎠⎞

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−∂∂

+⎟⎠⎞

⎜⎝⎛

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−θ∂

∂=×

∂∂

+θ∂

∂+⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

=∇

∂∂

+θ∂

∂+

∂∂

=⋅

∂∂

+θ∂∂

+∂∂

=

∂∂

+θ∂∂

+∂∂

=

=∂∂

=θ∂

∂=

∂∂

=∂∂

=∂∂

−=θ∂

∂=

θ∂∂

=θ+θ−=θ+θ====⇒+θ+=

=ξθ=ξ=ξ

θθ

θ

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∇

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ξ∂∂

=

=⇒=ξ∂∂

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=

=⇒=φ+θφ+θφ=ξ∂∂

ξ∂∂

=

ξξ=ξξξ=ξξξ=

θ=ξφ=ξ=ξ

sinsin)(

)(

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:factorsScalecos,sinsin,cossin

,,:scoordinateSpherical

322

332

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1

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321

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hxxh

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kk

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φθθθ

φφφ

φ

φ

φ−=θ∂

∂=

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=∂∂

φ=θ∂

∂−=

φ∂∂

=∂∂

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∂=

φ∂∂

=∂∂

θ+θ−=

φ−φθ+φθ=φ+φθ+φθ=

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R

R

r

2

2

2222

2

222

22

22

222

2

sin1)(sin

sin11

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1(sin1)(sin

sin1sin

sin1

)(sin1)(sin

sin1)(1

)(sin1)sin(

sin1)sin(

sin1

sin1ˆ1ˆˆ

sin1ˆ1ˆˆ

(1.9.16) - (1.9.12) Using

θ∂∂

φ+

φ∂∂

φφ∂∂

φ+⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

=

θ∂∂

φθ∂∂

φ+

φ∂∂

φφ∂∂

φ+⎟

⎠⎞

⎜⎝⎛

∂∂

φ∂∂

φ=∇

θ∂∂

φ+φ

φ∂∂

φ+

∂∂

=

θ∂∂

φ+φ

φ∂∂

φ+φ

∂∂

φ=⋅

θ∂∂

φ+

φ∂∂

+∂∂

=

θ∂∂

φ+

φ∂∂

+∂∂

=

⇒

θφ

θφ

φ

θφ

fR

fRR

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fR

fRR

fRRR

f

uR

uR

uRRR

RuR

uRR

uRRR

fR

fRR

ff

RRR

R

R

R

R

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∇

∇

∇


1-18. Continued

θ

φR

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e

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ˆ)(1sin1ˆ)(sin

sin1

ˆ)()[(1

ˆ)]sin()([sin1ˆ)]()sin([

sin1

2

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

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⎛φ∂

∂−

∂∂

+

⎥⎦

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⎡∂∂

−θ∂

∂φ

+⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛θ∂

∂−φ

φ∂∂

φ=

⎟⎟⎠

⎞⎜⎜⎝

⎛φ∂∂

−∂∂

+

⎟⎟⎠

⎞⎜⎜⎝

⎛φ

∂∂

−θ∂∂

φ+⎟⎟

⎠

⎞⎜⎜⎝

⎛θ∂∂

−φφ∂∂

φ=×

φ

θφ

θ

θφ

φθφθ

R

R

R

RR

uRuRR

RuRR

uR

uu

R

uRuRR

uRR

uR

RuuRR

∇


2-1.

−−

−−

−−

=−=ω

⋅⋅⋅

++

=+=

+===

−

−−

=−=ω

⋅⋅

⋅=+=

===

+−

−−−

−−

=−=ω

⋅⋅+⋅

+

=+=

+===

00)23(21

00))(21

)23(21)(

210

)(21

202

)23(21)(

210

)(21

)(,, (c)

021

21

210

21

21

210

)(21

2121

212

)(21

,, (b)

0

0)(21

)(210

)(21

00

)(21

)(21

)(,, (a)

2

23

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,,

223

,,

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,,

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2

2

2

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2

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