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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Copyright 2009, Elsevier Inc. All rights reserved.
1-5.
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1-7.
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Copyright 2009, Elsevier Inc. All rights reserved.
1-12.
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Copyright 2009, Elsevier Inc. All rights reserved.
1-12.
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Copyright 2009, Elsevier Inc. All rights reserved.
1-12.
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Copyright 2009, Elsevier Inc. All rights reserved.
1-14.
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1-15.
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1-17.
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1-18.
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1-18. Continued
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2-2.
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2-3.
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2-4.
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Copyright 2009, Elsevier Inc. All rights reserved.
2-5.
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Copyright 2009, Elsevier Inc. All rights reserved.
2-6.
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2-8*.
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2-9*.
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Copyright 2009, Elsevier Inc. All rights reserved.
2-10*. MATLAB CODE % Principal Value Problem % Enter strain matrix e=[2,-2,0;-2,-4,1;0,1,6]*0.001; % Calculate principal values L and directions N [N,L]=eig(e); fprintf('Principal Values') disp(diag(L)') fprintf('Principal Directions') N1=N(:,1)' N2=N(:,2)' N3=N(:,3)' SCREEN OUTPUT >> Principal Values -0.0047 0.0026 0.0061 Principal Directions N1 = -0.2852 -0.9543 0.0893 N2 = 0.9570 -0.2784 0.0815 N3 = -0.0529 0.1087 0.9927 2-11.
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2-13.
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2-15.
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2-17.
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2-19.
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2-20.
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3-3.
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3-4.
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3-6.
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Copyright 2009, Elsevier Inc. All rights reserved.
www.mechanicspa.mihanblog.com
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3-9.
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Copyright 2009, Elsevier Inc. All rights reserved.
3-10*.
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Copyright 2009, Elsevier Inc. All rights reserved.
3-12.
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3-14.
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3-17.
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Copyright 2009, Elsevier Inc. All rights reserved.
3-19.
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3-22.
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Copyright 2009, Elsevier Inc. All rights reserved.
3-25.
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Copyright 2009, Elsevier Inc. All rights reserved.
4-1.
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21
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==+=+=
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==
Copyright 2009, Elsevier Inc. All rights reserved.
4-3.
ijijkkij
ijijkkjiijijkk
kljkiljlikklijklijklij
ee
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4-4.
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32 Since
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++++=
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4-5.
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3,-4 exercise From
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Copyright 2009, Elsevier Inc. All rights reserved.
4-7.
kkiieii
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thusand2 materials, isotropicFor 0)( then stress, for the directions and valuesprincipal theare and If
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4-8.
18.52MPa 264.16MPa 2)(
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Copyright 2009, Elsevier Inc. All rights reserved.
4-9.
plane in the beamr rectangula a of bending pure toscorrespond stress of state thisThus0 2
0 2)(
2)(
components stress plane-in following thegives law sHooke'
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4-11.
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211
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Copyright 2009, Elsevier Inc. All rights reserved.
4-12.
3-
3-
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3-
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10511000511000511
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Copyright 2009, Elsevier Inc. All rights reserved.
4-13.
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4-16.
[ ] [ ][ ] [ ][ ] [ ]
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4
4
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Copyright 2009, Elsevier Inc. All rights reserved.
4-17.
[ ][ ][ ] TT
Ee
TTTETE
e
TETE
e
e
yxzz
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zyxx
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4-18.
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Copyright 2009, Elsevier Inc. All rights reserved.
5-1.
p 40o
x
y
a
b
(a)
x
y S
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h
(b)
x
y T
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Copyright 2009, Elsevier Inc. All rights reserved.
5-1. Continued
5-2.
p
a b
r
(f)
r1
r2
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Copyright 2009, Elsevier Inc. All rights reserved.
5-3.
0),(,2),(
0),(,02),(
0),0(,02),0(
0)0,()0,((a)
=
+
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=
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5-4.
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),(),(,),(),(:Conditions Interface
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0),0(,0),0(
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0),0(,0),0(0)0,()0,(
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hxhxhxhxhxvhxvhxuhxu
yaya
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hhxShhx
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====
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x
y
S
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a
a
x
y
S
(a)
a
h1 x
y
h2
(1)
(2)
Copyright 2009, Elsevier Inc. All rights reserved.
5-4. Continued
),(),(,),(),(
),(),(,),(),(:Conditions Interface
0),(,),(:(2) Material
0at ntsdisplaceme & stresses Bounded:(1) Material(a)
2)2(
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====
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5-5.
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1)1(
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21)2(
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====
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hxhxhxhxhxvhxv
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a
h1 x
y
h2
(1)
(2)
Copyright 2009, Elsevier Inc. All rights reserved.
5-6.
TlhydyyTdyyTTldyyT
PlydyyTPldyyTdyyT
SwhxdxxTdxxTSwdxxT
h
h x
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5-7.
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FFF
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FuuuF
uuu
Copyright 2009, Elsevier Inc. All rights reserved.
5-9.
021
1
01)21(
2
0
)1(2
)1(2)21)(1(
0
0)(gives (5.4.3)Relation
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Copyright 2009, Elsevier Inc. All rights reserved.
5-10.
])([2
,,
gives )(0,0,at rotation andnt displaceme no of conditonsboundary theApplying
)(2
),(
0),(0)(),(),( shown that becan it relations above thefrom Also
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======
Copyright 2009, Elsevier Inc. All rights reserved.
5-11.
theory.SOMth exactly wimatch results elasticity the,2
,2
, constantsWith
)(2
,
theory materials ofstrength case, For this figure. in theshown as loading endunder andright theat supported problem bending beam cantilever arepresent toused be could field that thesuggests This
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l
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Copyright 2009, Elsevier Inc. All rights reserved.
5-12.
solution. elasticityproper a not are thusand ity,compatibilnot but mequilibriusatisfy stresses
check)not (does0)1(40)])(1([)]([)1(
0)()1(
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222
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x
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zyxx
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zyyxy
zxyxx
zxyzxy
zyx
Copyright 2009, Elsevier Inc. All rights reserved.
x
y / P
xy / P
y = 10a
x
y / P
xy / P
y = 100a
5-13*.
( )
belowshown are100 and10 of cases specific for the results MATALB following The identical. are (b) and (a) problems from stresses theloading, thefrom distances largeat Thus
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bx
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==
+
+++
+++++=
++
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+=+=+=
Copyright 2009, Elsevier Inc. All rights reserved.
6-1.
xxxx
x
x
x
x
eEeEdxdydz
dUU
dxdydzE
dxdydzE
d
dxdydzxuddudxdydzF
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22
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6-2.
( )221212212
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6-3.
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21
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ijijkkjjijijijkkijij
EE
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Copyright 2009, Elsevier Inc. All rights reserved.
6-4.
( ) ( )mnmnmnkkmnmnkkmnjj
ijjnimjnimijkkjnjmknkmjjmn
jnimmn
ij
ijmn
ij
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6-5.
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221
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Copyright 2009, Elsevier Inc. All rights reserved.
6-7.
( ) ( )221212212
122
1
21
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121)2(
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6-8.
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1
octoctoctd
zxyzxyxzzyyxoct
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+++++=
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6-9.
222
2222
1)2(21
)(2
)2(2
100000
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yxy
xyx
ij
EE
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+++=
++++=
=
6-10.
EIlM
EclMdyyl
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EcMUdVU
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cM
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c
c
l c
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=====
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Copyright 2009, Elsevier Inc. All rights reserved.
6-11.
)1(821
211
)(1 :EnergyStrain Total
)1(4)(1)22(
21
,,0
42422
4222
0 0
222
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2222
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+=+=
+=+=
++==
+=++=++=
======
LRELRE
RLE
rdrdrLE
dxdAyxE
UdVU
rEyxE
yxE
U
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R
L
AVT
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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(
Copyright 2009, Elsevier Inc. All rights reserved.
6-13.
EIlq
EIlqlwlxxl
EIxqw
EIlq
EIlqlwxlx
EIlqw
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l N
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24 issolution exact The
0104.096
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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
,
====
====
====
===
Copyright 2009, Elsevier Inc. All rights reserved.
7-1.
xyxyxyxyxy
xyy
yxx
yxyxyxyxy
yxyxyxyxx
yxyx
yxyx
yyxy
xyxx
Eee
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eeee
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+===
+=
+=
+++=++=
++++=+++=
=++=+++=
++=
1212
])1[(1
])1[(1
)(1)()21)(1()(21)(
)(212
)(1)()21)(1()(21)(
)(212
)(2))((2
2)(2)(
Copyright 2009, Elsevier Inc. All rights reserved.
7-2.
+
=+
+
=+
+
+
==+
+
+
==+
+
+
+=+
+=
+
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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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yF
xF
yF
xF
yxyx
xyyyF
Fyx
yxxxF
Fyx
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EyxExEy
yxe
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e
ye
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yv
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xxu
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yv
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xF
yx
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yu
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xF
yx
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xu
xu
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xu
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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(
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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
Copyright 2009, Elsevier Inc. All rights reserved.
7-3.
00(b)(a)
00(a)(b)
(b)
0)()(
(a)
0)()(
0)(,0)(:equationsNavier
4
4
22
22
22
22
22
==
==
+
=
=
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++
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=+
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vyx
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xF
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Fyv
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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
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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
Copyright 2009, Elsevier Inc. All rights reserved.
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
Copyright 2009, Elsevier Inc. All rights reserved.
7-7.
+
+=+
+
=+
+
+
==+
+
+
==+
+
+
+=++
+=
+
=+
+===
=+
+
+
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+
+
+
+
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+
+
+
+
=++
=+
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+
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+
=
+
=
yF
xF
yF
xF
yxyx
xyyyF
Fyx
yxxxFF
yx
yxyx
EyxExEyyxe
xe
ye
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Ee
Ee
Fyv
xu
yEv
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uyvE
xv
yxuE
Fxu
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yxv
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xF
yx
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xu
xEu
Fxyv
yuE
yxv
xuE
Fxv
yuE
yyv
xuE
xF
yx
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yuE
xu
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yv
xuE
yxyx
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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
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222
2
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2
2
22
Copyright 2009, Elsevier Inc. All rights reserved.
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
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yev
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yxfyxfzewzwe
zzxz
zzyz
yxz
zz
+
=
==
+
=
+
=
==
+
=
=+=
+==
Copyright 2009, Elsevier Inc. All rights reserved.
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
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2
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2
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22
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2
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2
yxyxz
z
z
zz
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zzxyzxyz
zzxxz
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eeE
e
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e
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zyxe
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ze
xe
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ye
ze
+=+=
++=+====
+===
======
=
=
==
=
+
+
=
=
=+
=
=
+
Copyright 2009, Elsevier Inc. All rights reserved.
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
Copyright 2009, Elsevier Inc. All rights reserved.
7-11.
+
=+
+
+=+
+
+=+
=+
+
++
=+
+
++=+
+
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xx
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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
Copyright 2009, Elsevier Inc. All rights reserved.
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
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2
2
xyxyxyxy
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yxxyxxyxx
EEEe
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eeEeEeeEeee
EEE
+=
+
+=
+=
=
+=
=
+
+==
+=++++++=
+=++++++=
++
++
Copyright 2009, Elsevier Inc. All rights reserved.
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
==
==
=
+==
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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
Copyright 2009, Elsevier Inc. All rights reserved.
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|
plane stessplane strain
n=0.4
Copyright 2009, Elsevier Inc. All rights reserved.
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|
plane stessplane strain
7-15.
VyV
xVV
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y
yF
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242
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2
)1()1()(
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121
11)(
11)(:Strain Plane
,,,,
=
+
+=+++
+
+=+
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+
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=+
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==+
=+=
Copyright 2009, Elsevier Inc. All rights reserved.
7-16.
+
=++=
+=+=
=
+
++
+
+
+
++
++
+
=++=
=
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121)sin(coscossincossin
1cossin2cossin
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Copyright 2009, Elsevier Inc. All rights reserved.
7-17.
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