978-0124081369 Chapter 2

subject Type Homework Help
subject Pages 9
subject Words 1794
subject Authors Martin H. Sadd Ph.D.

Unlock document.

This document is partially blurred.
Unlock all pages and 1 million more documents.
Get Access
page-pf1
2.1.
==
++
=+=
+===
==
=+=
===
+
==
+
+
=+=
+===
00)23(
2
1
00))(
2
1
)23(
2
1
)(
2
1
0
)(
2
1
2
02
)23(
2
1
)(
2
1
0
)(
2
1
)(,, (c)
0
2
1
2
12
1
0
2
12
1
2
1
0
)(
2
1
2
1
2
1
2
1
2
)(
2
1
,, (b)
0
0)(
2
1
)(
2
1
0
)(
2
1
0
0
)(
2
1
)(
2
1
)(,, (a)
2
23
223
,,
223
,,
2223
,,
,,
2
2
2
,,
2
,,
222
CxAyz
ByAz
CxAyzByAz
uu
Cz
Bxy
CxAyzByAz
uue
zxCwBxyvAyzu
CxzCyz
CxzBy
CyzBy
uu
Cxy
CxzBx
CyzByAx
uue
CxyzwBxyvAxu
CyBxzCx
CyBxzBzAx
CxBzAx
uu
CyBxz
CxBzAxAy
uue
yxCwBxzvAxyu
ijjiij
ijjiij
ijjiij
ijjiij
ijjiij
ijjiij
page-pf2
2.2.
0,)2(,)( 22
v
u
v
u
wyxkvyxku
==+=
page-pf3
2.3.
givesn integratio of constants theredefining and results individual theCombining
constant
)(00
2
1
)(00
2
1
and
constant00
2
1
),(0
)(
)(
plane-yin x,behavior ldimensiona- twoAssume
1
3
1
1
21
22
11
oz
zx
yz
xy
z
y
x
uyxCu
dh
ygh
x
h
z
u
x
w
e
xfh
y
h
y
w
z
v
e
daxgdayf
a
dx
dg
dy
df
dx
dg
dy
df
x
v
y
u
e
yxhw
z
w
e
xgyCvC
y
v
e
yfxCuC
x
u
e
+=
==
==
=
+
=
==
=
+
=
+=+=
====+=
+
=
==
=
+==
=
+==
=
page-pf4
2.4.
( )
oxy
ozx
oyz
zx
yz
xy
z
y
x
wyxyxABzw
vxzAzyv
uzyAzxu
dbycxyxAh
dbzaxg
xgbzgbxM
z
g
zgaxgazL
x
g
dczayf
yfzxNAxzfxNAx
z
f
zfayfazL
y
f
xNyxNbxMazL
y
yxN
xMzL
y
yxN
xMzLxMzL
yxN
z
f
Ax
x
h
z
f
Ax
x
h
z
u
x
w
e
xM
y
h
Ay
z
g
y
h
z
g
Ay
y
w
z
v
e
zL
x
g
y
f
x
g
y
f
x
v
y
u
e
yxhBzwBz
z
w
e
zxgAzyvAz
y
v
e
zyfAzxuAz
x
u
e
+++=
++=
++=
+++=
++=
+===
+===
+=
+==
+===
====
=
=
=
=
=
=
=
=
=
+
=
=
=
=
+=
+
=
=
=
=
+
=
+
=
+==
=
+==
=
+==
=
)(
2
1
givesn integratio of constants theredefining and results individual theCombining
.)(
2
:fashionsimilar In
. formeach satisfy To
)()(
and)()(
. formeach satisfy To
)()()(
and)()( So
)(),(,)(,)(0
),(
)()(
),(
)()(,)()( relations, threeprevious theFrom
),(0
2
1
)(0
2
1
)(00
2
1
),(
2
1
),(
),(
222
3
2
2
1
1
2
1
2
page-pf5
2.5.
yxww
xzvv
zyuu
dcxbyhdbzaxg
hgdczayf
yxfczfcyN
z
f
zxfayfazL
y
f
cyNbxMazLyNxMzL
yNxMxMzLyNzL
yN
z
f
x
h
z
f
x
h
z
u
x
w
e
xM
y
h
z
g
y
h
z
g
y
w
z
v
e
zL
x
g
y
f
x
g
y
f
x
v
y
u
e
yxhw
z
w
e
xzgv
y
v
e
zyfu
x
u
e
z
u
x
w
e
y
w
z
v
e
x
v
y
u
e
z
w
e
y
v
e
x
u
e
yxwwxzvvzyuu
xyo
zxo
yzo
zx
yz
xy
z
y
x
yyzx
xxyz
zzxy
zyx
xyozxoyzo
+=
+=
+=
++=++=
+=
+===
+===
====
=
=
=
=
=
=
=
=
+
=
+
=
=
=
=
+
=
+
=
=
=
=
+
=
+
=
==
=
==
=
==
=
=+=
+
=
=+=
+
=
=+=
+
=
=
==
==
=
+=+=+=
formmotion body rigid thegivesn integratio of constants
theredefining and results individual theCombining . and
givingfashion similar in following and for Forms . formeach satisfy To
),()( and),()(
)(,)(,)(0)()()(
)()(,)()(,)()( relations, threeprevious theFrom
)(00
2
1
)(00
2
1
)(00
2
1
),(0
),(0
),(0
0)(
2
1**
2
1
0)(
2
1**
2
1
0)(
2
1**
2
1
0
*
,0
*
,0
*
*,*,*
32
1
21
page-pf6
2.6.
+
+
+
+
+
+
=
==
=
=
2sin2cos
22
2cos2sin
2
2cos2sin
2
2sin2cos
22
cossin
sincos
cossin
sincos
ryation theo transformldimensiona- twoUse
cossin
sincos
xy
yxyx
xy
xy
xy
xy
xy
yxyx
yxy
xyx
r
rr
ij
e
eeee
e
ee
e
ee
e
eeee
ee
ee
eQQ
ee
ee
e
Q
2.7.
0017.0,002.0,0027.0 :strains for the Solving 2
3
4
1
4
3
004.0
150cos150sin2150sin150cos
90cos90sin290sin90cos
2
3
4
1
4
3
001.0
30cos30sin230sin30cos
22
22
22
===
+=
++=
++=
++=
++=
xyyx
xyyx
oo
xy
o
y
o
xc
y
oo
xy
o
y
o
xb
xyyx
oo
xy
o
y
o
xa
eee
eee
eeee
eeee
eee
eeee
page-pf7
2.8*.
be tofound are valuesmaximum The
2cos001.02sin003.0
2sin001.02cos003.0001.0
2sin001.02cos003.0001.0
xy
y
x
e
e
e
+=
=
++=
page-pf8
2.9*.
A
45o
x
y
B
t
n
are relations theseofplot MATLAB The
)(2
(2.3.3) Relations
sin
2
1
,cos,sin
2
1
:direction For
cos
2
1
,sin,cos
2
1
:direction For
shown. as plane shaded in the and directions Define
111111
2
1
2
1
2
1
222
111
+++++=
===
===
lnenmemlenemelee
nmlt
nmln
tn
zxyzxyzyxnn
page-pf9
2.10*.
MATLAB CODE
% Principal Value Problem
% Enter strain matrix
e=[2,-2,0;-2,-4,1;0,1,6]*0.001;
2.11.
o
oooof
o
f
o
V
V
VVVVVV
VdxdydzeeOdxdydzeeedxdydz
dzedyedxeV
dxdydzV
==+==
+=+++++=
+++==
==
)1(
)1()1(),(ofterms)(
)1()1()1( Element Deformed of Volume
Element d Undeformeof Volume
32
321
321
2.12.
ijkk
ijijij
ijkkij
ij
ee
eee
ee
e
ˆ
with associated changes c volumetriNo03/143/163/2
ˆ
10
3/1410
13/162
023/2
~
ˆ
:Strain Deviatoric
10
3/400
03/40
003/4
1
~
:Strain Spherical
10
610
142
022
3
3
3
=+=
==
==
=
page-pfa
2.13.
yx
e
x
e
y
e
yxx
v
yxy
u
yx
e
x
v
y
u
e
xy
v
x
e
y
v
e
yx
u
y
e
x
u
e
xyy
x
xy
xy
y
y
x
x
=
+
+
=
+
=
=
=
=
=
2
2
2
2
2
33
2
2
3
2
2
2
3
2
2
2
ationdifferenti oforder the
einterchangcan we,continuous and valued-single are ntsdisplaceme theIf
2
2
1
2.14.
( )
 
ipmqnjpqjimnmqjnimnpqjmjnimn
ikjkjkikijkkkkijij
kkjijllikijkijllklklkkllij
kmlpljkmlmkjipljkplpkjimlmkplpkmij
kmlp
lplmlj
kpkmkj
ipimij
kmlpjmpiklij
eee
lkeeee
lk
eeeeee
e
ee
====
−=
==+=
++=
+=
==
,,,
,,,,
,,,,,,
,
,,
with (2.6.1) as same theiswhich ,0)(
get weand drop, willfirst term thesame, thebe taken toare and indices If
)(
))()(
ee
ee
page-pfb
2.15.
( ) ( )
( )
cases 3 and2 for the Likewise
0)(
)(2:1
)(
)(
)(
2
2
2
23,1,233,122,31313,22
32,2,311,233,12122,33231,23221,33331,223,132,121,11,1
3,32,21,1,
2,1,233,122,3131,22,1331
1,3,122,311,2323,11,3223
3,2,311,233,1212,33,2112
12,1211,2222,11,3333
31,3133,1111,33,2222
23,2322,3333,22,1111
,
==
=+++=
+++++=++==
++=
+++==
+++==
+++==
+==
+==
+==
=
ii
eeee
eeeeeeei
eeeee
eeeee
eeeee
eeee
eeee
eeee
e
jj
iiijij
kmlpmpkl
kmlpmpkl
kmlpmpkl
kmlpmpkl
kmlpmpkl
kmlpmpkl
kmlpjmpiklij
2.16.
equationorder fourth third thegives)2.6.2(
equationorder fourth second thegives)2.6.2(
equationorder fourth first thegives)2.6.2(
,(2.6.2) equationsity compatibil last three theUsing
)()()(
)()()(
)()()(
)(22
)(22
)(22
equations,ity compatibil efirst thre with theStarting
6
2
5
2
4
2
4,5,6
3
22
4
3
22
4
3
22
4
2
4
22
4
22
4
2
2
2
2
2
2
4
22
4
22
42
2
2
2
2
2
4
22
4
22
4
2
2
2
2
2
yx
xz
zy
y
e
x
e
z
e
zyx
yx
e
abc
x
e
z
e
y
e
zyx
xz
e
cab
z
e
y
e
x
e
zyx
zy
e
bca
c
yxz
e
yz
e
yx
e
xz
e
z
e
x
e
b
xzy
e
xy
e
xz
e
zy
e
y
e
z
e
a
zyx
e
zx
e
zy
e
yx
e
x
e
y
e
zx
yzxy
z
yzxy
zx
y
xy
zx
yz
x
zxx
z
zxx
z
yz
z
yyz
z
y
xyy
x
xyy
x
+
+
=
+
+
+
=
+
+
+
=
+
=
+
=
+
=
+
=
+
=
+
=
+
page-pfc
2.17.
00)2.6.2(
3
2
46)22(2662(2.6.2)
(2.6.2) equationsity compatibil esatisfy thmust strainsgiven The
6,5,4,3,2
2
2
2
2
2
1
=
==+=+
=
+
BABAyxBAxAy
yx
e
x
e
y
exyy
x
2.18.
valued.-multi are
ntsdisplaceme h theeven thoug satisfied are equationsity compatibil all Thus
00
000)0(0
000)0(0
00
00
00
yield )2.6.2( equationsity compatibilsix The
4
)(tan
42
1
4
)(tan
42
1
0
2
1
,0,0,0
tan
2
,0
22
1
22
1
1
=
==
+
=
==
+
=
=
=
=
+
=
=
+
=
+
=
=
+
=
=
+
==
==
==
=
===
yx
e
y
e
y
xy
e
x
e
x
yx
yb
x
y
x
b
z
u
x
w
e
yx
xb
x
y
y
b
y
w
z
v
e
x
v
y
u
e
z
w
e
y
v
e
x
u
e
x
yb
wvu
yz
zx
zx
yz
zx
yz
xyzyx
page-pfd
2.19.
+
=
=
+
=
+
+
+
+
=
+
=
=
+
+
=
=
r
uu
rr
u
BACCABe
r
uu
r
rd
rddur
rd
rdurd
u
u
CA
CACA
AC
ACCA
e
r
u
dr
dr
r
u
dr
r
u
dr
AB
ABBA
e
r
r
r
r
r
r
r
1
2
1
)(
2
1
1
)(
)(
])()[(
geometry, figure theFrom
**
****
2/122
A(r,)
C
D
dr
rd
d
A'
B'
C'
D'
B
dr
r
u
A*
C*
+
d
u
u
rd
r
uu
rr)
1
(
page-pfe
2.20.
( )
0)sincos(
1
)sincos(
1
2
1
0cossincossin
1
,0 (c)
2
1
1
cos
2
,sin
1
, (a)
1
2
1
,
1
,
2
2
=
++=
=+==
=
==
+
=
+=
=
CrBA
r
CBA
r
e
BABA
r
ee
r
r
B
eB
r
A
r
e
r
A
e
r
u
r
u
u
r
e
u
u
r
e
r
u
e
r
r
rr
rr
r
rr
r
r

Trusted by Thousands of
Students

Here are what students say about us.

Copyright ©2022 All rights reserved. | CoursePaper is not sponsored or endorsed by any college or university.