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7.1.
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yxx
yxyxyxyxy
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e
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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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−=−
++=+
++=
++=
1
2
1
2
])1[(
1
])1[(
1
1
)21)(1(
2
1
)(2
1
)(
1
)(
)21)(1(
)(
2
1
)(
)(2
1
2
)(
2
1
)(
)(2
1
)(2
))((2
2)(
2)(
7.2.
+
−=+−
+
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+
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givesrelation ity compatibil intoresult thisusing and,2
0
0
equations, mequilibriu fromBut
1
2])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
2
2
2
2
2
2
2
2
2
2
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
==
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==
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y
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u
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7.4.
−=
=−=++−=
=+==
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=+==
−=−=++−=
=+==
++=+=
+=
Az
y
A
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c
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T
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x
A
xx
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ˆ
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where,axes) principal(for )(
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where,axes) principal(for )(
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:endsat momentsresultant zeroFor
ˆ
1
axes) principal(for
ˆ
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0)(0
:
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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
)(
000
2
1
0
000
2
1
0
:equationsity compatibil and equilbriumsatisfy must Stresses
22
2
2
ykxkxkxy
kxkxy
y
F
x
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y
ky
x
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y
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x
F
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y
x
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xy
x
+=+=+
+=
==+
+
−
−=+
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+
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=+
+
==
−
+
=+
+
7.6.
xyxyxyxy
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e
E
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e
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ee
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ee
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E
e
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+
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−
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+
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=
−
+
=−
+
−
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−=
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1
1
)(
1
)(
1
)(
1
2
)(
1
)(
1
)(
1
2
)(
1
)(
1
)(
1
)(
1
2
2
7.7.
+
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+
−=
+
+
+
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−=+
+
+
+
=+
+
+
=
−
+
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y
x
y
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x
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x
x
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e
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e
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e
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e
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x
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y
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v
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x
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x
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y
v
x
u
x
E
u
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v
y
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v
x
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x
v
y
uE
yy
v
x
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x
F
yx
x
v
y
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x
u
y
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y
v
x
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yx
y
x
y
x
xy
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xx
x
xy
x
y
x
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xyxyyx
xyy
x
xyxyxyyyxx
y
y
yy
yxy
x
x
xx
xy
x
xyyx
)1()(
givesrelation ity compatibil intoresult thisusing and,2
0
equations, mequilibriu fromBut
2)(
)1(
1
1
2)(
1
)(
1
2
1
,)(
1
,)(
1
:Equationity Compatibil Michell-Beltrami
0
)1(2
0
1
)1(2
0
1
)1(2
0
0
)1(2
0
)1(2
1
0
)1(2
1
0
:equations mequilibriu into Substitute
)1(2
,
1
,
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
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 , coordinate
plane-of-out on the depend willntsdisplaceme theimply that results These
functionsarbitrary are and where
2
1
2
1
),(
2
1
0
2
1
, offunction )(
1
that Note
functionarbitrary an is ),( where,),(
2
2
z
hg
x
f
x
e
x
f
x
e
z
u
x
w
z
u
yxgz
y
f
z
y
e
v
y
f
z
y
e
z
v
y
w
z
v
e
yxeee
yxfyxfzew
z
w
e
zz
xz
zz
yz
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
2
2
2
2
2
2
yxyxz
z
z
z
z
z
zz
zz
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yzxy
z
zzxxz
z
yz
z
y
ee
E
e
cbyaxe
ccbyyFbyFbxg
y
e
yFaxeayf
x
e
bxgayfxgyf
yx
e
xg
y
e
y
e
yf
x
e
x
e
yx
e
y
e
x
e
z
e
zyx
e
x
e
xz
e
z
e
x
e
y
e
zy
e
y
e
z
e
+
−
−=+
−=
++=
+==
==
+===
====
=
=
=
=
=
=
=
+
+
−
=
=
=
+
=
=
+
7.10.
case!either for changenot does modulusshear that theNotice
)1(2)1(2
1
)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(2
1
1
)
1
1(2
1
)1(2
)31)(1()21)(1(
)1(
1
)
1
21)(
1
1(
1
1
)21)(1(
:strain plane tostress Plane
2
2
2
2
2
2
2
2
=
+
=
++
+
+
+
=
+
+
+
+
=
+
=
−+
=
−+++
+
+
+
=
+
−
+
+
+
+
+
=
−+
=
=
+
=
+−
−
−
=
−
+
−
=
+
=
−+
=
−−+−
−
−
=
−
−
−
+
−
−
=
−+
=
EE
E
E
EE
E
E
EE
E
E
EE
E
E
7.11.
+
+−=+
+
−=+
+
−
−=+
=+
+
−
+
=+
+
+
+
−+
+
=+
+
++
F
F
F
F
y
F
x
F
F
y
v
x
u
x
E
u
F
y
v
x
u
x
EE
u
F
y
v
x
u
x
u
y
x
y
x
y
x
yx
x
x
x
)1()(
1
)(
1
1
)(: (7.1.7)Equation
(7.1.5)relation for Likewise
0)
)1(2
: toreduceswhich
0)
)1(2)1)(1(
(
7.10 Exercise from results theusing stress plane toConverting
0)(:(7.1.5)Equation (a)
22
2
2
2
2
2
1
7.12.
7.1. Exercisein given sexpression match withproperly Results
1
1
1
1
1
])1[(
1
1
1
)(
1
])1[(
1
1
1
)(
1
)31)(1(
,,
1
,
1
:Strain Plane Stress Plane (b)
7.6. Exercisein given sexpression match withproperly Results 1
22
)(
1
1
)(
)1)(1(
2)( :Likewise
)(
1
1
)(
)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
E
E
e
EEE
e
EEE
e
EE
E
E
ee
ee
E
e
E
ee
E
eee
ee
E
e
E
ee
E
eee
EE
E
+
=
−
−
+
→
+
=
−−
+
=
−
−
−
→−=
−−
+
=
−
−
−
→−=
−+
→→
−
→
−
→→
+
=→=
+
−
=
+
++
−+
→++=
+
−
=
+
++
−+
→++=
−+
→→
+
→
+
+
→→
7.13*.
bygiven are ntsdisplacemestrain plane and stress plane theof plots MATLAB, Using
identical. become ntsdisplaceme two the0, ratio sPoisson'When
1
2
)1(
/
][
2
)1(
1
2
1
/
][
2
:0) ( aixs- Along
12
)1(
,)1( :ResultsStrain Plane
1
,
1
:Strain Plane Stress Plane
,][
2
, :Results Stress Plane
2
2
2
.
22
2
.
2
2
.
22
.
222
2
2
2
222
→
−
−
=−
−
=
−
=−=
=
−+
−
−
=−−=
−
→
−
→→
−−+=−=
l
x
EIMl
v
lx
EI
M
v
l
x
EIMl
v
lx
EI
M
v
yx
lxy
EI
M
v
EI
Mxy
u
E
E
lxllxy
EI
M
v
EI
Mxy
u
StrainP
StrainP
StressP
StressP
-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
Dimensionless Displacement, v(x,0)/(M l 2/EI)
Exercise 7-13 Note |v-plane stess| > |v-plane strain|
plane stess
plane strain
n= 0.4
7.14*.
bygiven are ntsdisplacemestrain plane and stress plane theof plots MATLAB, Using
identical. become ntsdisplaceme two the0, ratio sPoisson'When
/
1
1
1
)1(
/
)(
,
/
1
)21()1(
/
)(
:lizingdimensiona-Non
1
11
1
21
)1(
)21(
)
1
1(
1
,
)1(
)21(
:Stress Plane Strain Plane
)21(
)1(
:ResultStrain Plane
111
.
111
.
2
1
2
1
2
2
2
1
→
+
+
−
+=
+−+=
+
+
−+
=
+
+
−
+
+ +
+
=
+
→
+
+
→→
+−
+
=
rrr
r
ETr
u
rrr
r
ETr
u
r
r
r
E
T
r
r
r
E
T
u
E
E
r
r
r
E
T
u
StressPrStrainPr
r
r
0 1 2 3 4 5 6 7 8 9 10
0
2
4
6
8
10
12
14
16
18
20
Dimensionless Distance, r/r1
Dimensionless Displacement, ur /(Tr1 /E)
Exercise 7-14 Note |ur-plane stess| > |ur-plane strain|
plane stess
plane strain
n= 0.4
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
0.5
1
1.5
2
2.5
Poisson Ratio,
Dimensionless Boundary Displacement, ur /(Tr1 /E)
Exercise 7-14 Note |ur-plane stess| > |ur-plane strain|
plane stess
plane strain
7.15.
V
y
V
x
V
V
x
V
y
y
F
x
F
y
V
x
V
x
y
y
x
y
V
F
x
V
F
yx
V
x
V
y
y
x
yx
yxxyyx
24
2
2
2
2
2
2
2
2
2
2
24
2
2
2
2
2
2
2
2
2
2
2
2
2
2
)1()1()(
)1()(:Stress Plane
1
21
1
1
1
,,,,
−−=
+
+=+
++
+
+−=+
−
−
−
−
−=
−=
−=+
=+
=
7.16.
−
+
=−++−=
+=−+=
=
+
+
+
+
−
−
+
+
+
+
+
+
−
−
+
=
++=
−
=
−
=
−=
=
=
=
+
+
+
+
−
−
+
=
+
+−
=
+
=
+
+
+
=+
=
=
−
−
+
=−
=
=
r
u
r
uu
r
eeee
u
rr
u
eeee
r
u
x
u
x
u
x
u
x
u
y
u
y
u
y
u
y
u
y
u
y
u
y
u
y
u
x
u
x
u
x
u
x
u
eeee
rryrrx
ryxryrx
x
u
x
u
x
u
x
u
y
u
y
u
y
u
y
u
uu
x
uu
yx
v
y
u
e
y
u
y
u
y
u
y
u
uu
yy
v
e
x
u
x
u
x
u
x
u
uu
xx
u
e
r
xyyxr
r
xyyx
r
r
r
r
r
r
r
r
r
xyyxr
r
r
r
r
rrxy
r
r
ry
r
r
rx
1
2
1
)sin(coscossincossin
1
cossin2cossin
, Likewise
cossin
coscossinsin
sinsincoscos
sincoscossinsin
cossinsincoscos
cossin2sincos
cos
sin,
sin
cos
cossin
cossin,
cos
sin,
sin
cos
coscossinsinsinsincoscos
2
1
)cossin()sincos(
2
1
2
1
coscossinsin)cossin(
sinsincoscos)sincos(
22
22
2
2
22
22
7.17.
02
2
11
2
1
2
1
,
1
,
:Relationsnt Displaceme-Strain
2
2
2
2
2
2
2
2
3
2
2
2
2
2
2
22
2
2
2
=
−
+
−
−
=
−
=
−+
=
−
+
+
−−=
+
+
+−=
−
+
=
−
+
=
+=
=
rrr
rrrrr
r
rr
r
r
r
r
rr
r
rr
r
r
e
r
e
r
r
e
r
e
r
r
r
e
r
e
r
u
r
r
u
r
u
ru
u
rr
e
r
e
r
r
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