Unlock document.
This document is partially blurred.
Unlock all pages and 1 million more documents.
Get Access
10.1.
22
4
224
2
2
2
2
2
2
16
4
2
1
,
2
1
lines previous Solving
)(
2
1
,)(
2
1
,
zz
zzzzzzzzzzyx
y
i
xzy
i
xz
zz
i
y
z
zy
z
zy
zzx
z
zx
z
zx
zzyzzxiyzziyxz
==
=
−
−
−
+
+
=
+
=
+
=
−
=
−
=
+
=
+
=
+
=
−=+=−=+=
10.2.
( ) ( ) ( ) ( )( )
( )
)((z)z)()((z)z(z)
2
1
)()()(,)()((z) Letting
)()()()()()()()(
2
1
)]()()()()()()()([
2
1
)]()()()([
2
1
)]()([
)]()([
2
1
)]([
)]()([
2
1
)]([
)]()()()([
function, reala bemust sinceNow
)()()()(
)()()()(
)()()()()()(
)()()()(
)(
0 (10.2.6) relation gIntegratin
___________
4
4
2
4
22
4
zRezzz
zfzhzzezg
zfzhzfzhzezgzzezgz
zfzfzhzhzezzzezzgzgz
zfzfzhzhzfzhRe
zezzzezzeRe
zzgzgzzgzRe
zfzhzzezgzRe
zfzhzzezgz
zfdzzdzzedzzcz
zdzezczzdzdzbzcz
z
zbzczbdzza
zz
za
zz
zz
+=
+++=
+=+=
+++++++=
+++++++=
+++=+
+=
+=
+++=
+++=
+++=
++=++=
+=+=
=
=
10.3.
gives and on intergrati of functionsarbitrary redefining and gRearrangin
)()()()()3()2(4again Integrate
)()()()3()2(4
relations twoprevious thefrom term thegEliminatin
)()(2 conjugatecomplex Next take
)()(2 result thisgIntegratin
0)(2
as written becan equation Navier complex theand , shown that becan it
1,-10 Exercise from relations and,,nt displacemecomplex theUsing
0)()(
formcomplex theconstruct to0)u()(u :EquationsNavier Use
______
______
______
__________
_____
_____
2
_____
22
2
+
+−+=+
+−
+=
+
=
+
++
=
+
++
=
+
++
+
=
+
−=+=
=
+
+++
+
++
=++
gf
zgzfzzfU
zfzf
z
U
z
U
zf
z
U
z
U
z
U
zf
z
U
z
U
z
U
z
U
z
U
zzz
U
z
U
z
U
y
v
x
u
ivuUivuU
y
v
x
u
y
vi
y
v
x
u
x
u
10.4.
)2(])()([)(2
)2(22 law, s Hooke'Using
)2(
2
1
])()([
])()([)()(
)()()(
2
1
)(2
)()()()(2
______________
______________
______________
____________
____________
xyyx
xyyxxyyx
xyxy
xyxy
ieeezzzivu
z
ieeei
izzz
zzzzzz
zzzz
y
i
x
ivu
z
zzzzivu
+−=
−
−=+
+−=+−
−−−=
−
−
−
−=
−
−=
−
−
+
=+
−
−=+
10.5.
( )
1
1
2
1
2
1
)](
2
1
)(
2
1
)([2
][2][22
][242
2
][24
2
2
)()(Re)()()()(
2
1
),(
2
2
2
2
2
22
2
2
2
2
2
2
22
2
2
2
2
2
22
2
2
2
2
___________
zzzzRe
(z) + (z)"z Re(z) + (z)Re
(z) + (z)"z
z
i
zz
i
zzzz
i
yx
zz
(z) + (z)
zz
z
zz
z
zzzz
x
z
zz
z
zzzz
y
zzzzzzzzzzz
xyxy
x
x
xyxy
xy
xy
yx
y
x
−
−
=
−
=
=
=+−
−
−=
−
+
−=
−=
+
=−
=
=+
+
+
=
+
+
=
=
+
−
−=
−
−
−=
=
+=
+++=
10.6.
−
+=−+=+−++=+
+−=+=
+−=++−=
−−+−=+−
+−=
−−=−
+=+
+
−
=
−
−
−
+
=
+
−
+
+
=
i
r
r
i
xyxyxyxy
xyxyrr
xyxyr
xyxyr
yxr
xy
xy
r
xy
yxyx
xy
yxyx
r
eivuiivuvuivuiuu
vuuvuu
eiii
iii
)()sin)(cos()cossin(sincos
cossin,sincos
BAppendix in lawsrmation nt transfodisplaceme theFrom
)2()2sin2)(cos2(
)2cos2(sin2)2sin2)(cos(2
2cos22sin)(2
2sin22cos)(
2cos2sin
2
2sin2cos
22
2sin2cos
22
3Chapter in lawsation transformstress theFrom
2
10.7.
( )
problem bending pure thesolve willfield stress , and of valuesindicated with the
16
3
3
16
),(
satisfied,000),(
00),(
00),(
00),(
be wouldconditionsboundary the
),,( beamr rectangula a of problem bending pure a gConsiderin
])3()1[()1(2
2
1
])3()1[()1(2
2
1
]2)[(]2)[(
))((2)2)((
2)()()()(2
:ntsDisplaceme
4,84 stresses normal for the Solving
4,8
)(8]22[2][22
)(8][4]22[2][2
,)(,)(
3
22
22
2222
2222
______
2
_____
2
____________
_____
22
RI
RR
c
cx
c
cx
Ixy
Ixy
Iy
RI
IR
IRRI
IRIR
IyRIx
IxyRxy
IRxyxy
IRyx
IR
AA
c
M
AMcAMydyyl
dyyl
Ayl
Acx
Acx
cyclxl
AyxxyAv
AyxxyAu
xyAAyxixyAAyx
AiAyxixyyxAiA
AizAizzAizzzzzivu
xAyAxA
yAyA
yiAyAAizAiz(z) + (z)"z i
xAyAzAiAizAizAiz(z) + (z)
iAAAAizzAizz
==−=
==
==
==
==
−−
−+++−
=
+−−++
−=
−−++−−
++++−−=
+−=−
−=+
−=−−=
==−
+=−=
=+−
+−=−=+=
=+
+=−==
−
−
10.8.
(8.4.3). relationsby given solution previous toidentical arewhich
1
)(
,
1
)(
gives tutingbacksubsti and andfor sexpression twoprevious theSolving
1
2)(
1
2)(
00
conditionsboundary with 8-8 Figurein shown problemcylinder icaxisymmetr heConsider t
1
2,
1
2
stresses normal for the Solving
1
,
2
)(
2
22][22
4][2
, where/)(,)(
2
1
2
2
2
2
21
2
1
22
1
2
2
12
2
2
2
1
2
1
2
2
2
2
21
2
1
22
1
2
2
12
2
2
2
1
2
2
2
22
1
2
1
11
22
22
2
2
22
22
rr
prpr
rrr
pprr
rr
prpr
rrr
pprr
BA
pB
r
Apr
pB
r
Apr
B
B
r
AB
r
A
B
r
B
r
iBB
r
e
er
B
(z)ee(z) + (z)"z i
A(z) + (z)
iBBBiAAAzBzAzz
r
RR
RRr
RRr
Ir
RRRRr
IrRr
IR
i
i
ii
rr
Rr
IRIR
−
−
+
−
−
−=
−
−
+
−
−
=
=+=
=+=
==
−=+=
−=−=−
+−=−
=
=+−
=
=+
+=+===
10.9.
1
2
:componentsnt displaceme individual for the Solving
])()()([)(2
:ntsDisplaceme
0,,
)( :ConditonsBoundary
0, stresses individual for the Solving
2
22][22
0][2
/)(/)(
0)()(0)(
2
2
2
2
2
2
2
2
2
22
22
2
−=−=−
−=+
==−=
−=−=
==−=
−=−=
=
=+−
=
=+
−=
=
=
=
=
−
−−
r
pa
r
A
re
A
ezzzzeiuu
r
pa
r
pa
paApa
r
A
r
A
e
er
A
(z)ee(z) + (z)"z i
(z) + (z)
zAzzAz
zzz
r
i
ii
r
rr
r
rr
i
i
ii
rr
r
10.10.
0,
2
0, :9-10 Exercise From
0,
2)( :ConditionBoundary
2)(2:9-10 Exercise From
2
2
=
−=−=
==−=
=
=
−==
−=−=+
rr
rr
r
r
rr
r
a
r
A
u
r
a
u
aAau
r
A
u
r
A
iuu
10.11.
2
)1(
2
)1(
)(log
)1()1(
)(log
)1(
circuitinterior clockwisefor ,])()()([)(
2r|z|when
20r0|z|when
log2
loglog)()()(2
)1(
let ,log
)1(
log
)1(
2
_____________
2
2
____________
=
+
+
−
+
−=
−
+
+
+
+
−
−+
+
=
+
+=+=+=
→→→
→→→
+−=
−+−=−
−=
+
=
+
+
iY + Xi
2
iY) (X
i
2
iY + X
i
ir
2
iY) (X
e
2
iY X
ir
2
iY + X
-i
zzzzidsiTTiYXF
U
U
eCrC
zCeCz CzzzzU
2
iY + X
Cz
2
iY) - (X
= (z) z ,
2
iY + X
- = (z)
C
i
C
C
n
y
n
xkkk
i
i
k
k
k
10.12.
B
A
B
A
B
A
B
A
B
A
B
A
B
A
n
x
n
y
zzzzzzRe
zzzzzz
y
y
x
x
ds
y
dy
x
dxdsyTxTM
)]()()([
)()()()(
2
1
2
1
)(
___________
−−=
++++
+
+
−=
+
−=−=
10.13.
2
2
2
2
2
2
_____________
3
2
3
2
23
2
3
2
3
3
22
2
3
3
2
2
2
,
log
,
log
form theof bemust inclusion theofmotion body -rigid theNow
2
log
)1(
)(2
2
1
)1(2
log
)1(
2)1(2
log
)1(2
1
)1(2
log
)1(2
)()()()(2
match Moments and ForceResultant
00)]()()([ COver Moment
)(
)1(
0
)1(
])()()([)( :Origin Enclosing Con ForceNet
at vanish stresses all that Note
)sincos(
)1(2
2)1(
,)sincos(
)1(2
)3(2
2
)sincos(
)1(2
2)1(
)sincos(
)1(
21
)1(
)(2
)1()1(
2
2
)1(2)1(2
1
)1(2
2
)]()([22
)1(
)sincos(21
)1(2
1
)1(2
2])()([2
2)1(2
log
)1(2
)(,log
)1(2
)(
M
a Y
v
a X
u
e
a
Mi
a
Y i + X
ivu
e
r
Mi
e
r
a
Y i X
r
Y i + X
z
Mi
z
a
Y i X
z
Y i + X
z
Y i X
zz
Y i + X
zzzzivu
MMzzzzzzRe
Y i + X
i
ii
zzzzidsiTT
Y X
r
a
r
Y X
r
r
a
r
M
X Y
r
a
r
Y+ X
r
a
r
r
Mi
e
r
Y i + X
a
e
r
Y i X
e
r
Y i + X
ee
r
Mi
e
r
a
Y i + X
e
r
Y i X
e
r
Y i + X
re
ez + z"z i
r
Y+ X
e
r
Y i X
e
r
Y i + X
z + z
z
Mi
+
z
a
Y i + X
+z
Y i - X
= zz
Y i + X
- = z
ooo
i
ar
ii
C
C
C
n
y
n
x
2 2
r
2
r
2
r
i
2
ii
iii
2
iii
i
rr
ii
r
2
2
=
−
=
−
=
+
+
−=+
+
−
+
−
+
+
−=
+
+
−
−
+
−
+
−
+
+
−=
−
−=+
−=+−=
−−=
−=
+
−+
+
−−=
+
+−=+
+
+
−−
=+
+
+−
=
−−
+
−−
=
+
−
=−
−
+
−
+
−
+
+
=
−
+
−
+
−
+
+
=
=+−
+
−=
+
−
+
+
−=
=+
+
+
+
=
−−
−−−−−
−
