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2
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7.3.
00(b)(a)
00(a)(b)
(b)
0)()(
(a)
0)()(
0)(,0)(:equationsNavier
4
4
22
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1
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7.7.
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givesrelation ity compatibil intoresult thisusing and,2
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2)(
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1
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1
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,
1
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1
2
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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
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1
, offunction )(
1
that Note
functionarbitrary an is ),( where,),(
2
2
z
hg
x
f
x
e
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f
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+
−
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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
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+
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====
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7.10.
case!either for changenot does modulusshear that theNotice
)1(2)1(2
1
)1(
)21(
)
1
1(2
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)21(
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)1(
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)21(
)
1
21)(
1
1(
1
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: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
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:strain plane tostress Plane
2
2
2
2
2
2
2
2
=
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+
+
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+
+
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E
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7.11.
+
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+
−
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+
−
+
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+
+
+
−+
+
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F
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F
y
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x
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F
y
v
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u
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x
u
x
u
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y
x
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x
x
)1()(
1
)(
1
1
)(: (7.1.7)Equation
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0)
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: toreduceswhich
0)
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(
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)(
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,,
1
,
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:Stress Plane Strain Plane (a)
2
2
2
2
2
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2
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+
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+
→
+
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+
=
−
−
−
→−=
−−
+
=
−
−
−
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−+
→→
−
→
−
→→
+
=→=
+
−
=
+
++
−+
→++=
+
−
=
+
++
−+
→++=
−+
→→
+
→
+
+
→→
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
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M
v
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u
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E
lxllxy
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M
v
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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
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)
1
1(
1
,
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)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
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V
x
V
y
y
x
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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.
−
+
=−++−=
+=−+=
=
+
+
+
+
−
−
+
+
+
+
+
+
−
−
+
=
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−
=
−
=
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=
=
=
+
+
+
+
−
−
+
=
+
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+
=
+
+
+
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−
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=
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r
u
r
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u
x
u
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x
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u
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u
y
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y
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x
u
x
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rryrrx
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x
u
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x
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x
u
y
u
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v
y
u
e
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v
e
x
u
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u
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u
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r
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r
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r
r
r
r
r
r
r
r
r
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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
r
u
r
u
r
u
u
r
u
r
u
r
u
u
r
r
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r
u
r
u
r
ue
r
r
u
r
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r
e
u
u
r
e
r
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e
7.18
( )
==
−
−
+
=
+=
=
r
u
dr
d
rere
dr
d
e
r
u
r
u
r
u
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