Civil Engineering Reference
In-Depth Information
and the pseudo-stress vector
V
is defined as
V
-
½
x
°
°
V
y
°
°
V
-
½
x
V
°
®
°
¾
z
V
for
3
D
and
V
V
for
2
D
®
¾
(9.43)
y
W
°
°
°
°
xy
W
°
°
¯
¿
xy
W
yz
°
°
W
¯
¿
xz
Matrices
S
and
R
are of dimension 3x2 for two-dimensional problems and of
dimension 6x3 for three-dimensional problems.
Matrix
S
is given by
1
S
S
S
ª
º
xxx
xxy
xxz
«
»
S
S
S
yyx
yyy
yyz
«
»
S
S
S
zzx
zzy
zzz
«
»
S
(9.44)
S
S
S
xyx
xyy
xyz
«
»
S
S
S
yzx
yzy
yzz
«
»
S
S
S
¬
¼
xzx
xzy
xzz
The coefficients of
S
are given by:
C
2
ª
º
S
C
(
G
r
G
r
G
r
)
(
n
1)
r r r
(9.45)
¬
¼
ijk
3
ki
,
j
kj
,
i
ij
,
k
,
i
,
j
,
k
n
r
Values x, y, z are substituted for i, j, k. Constants are defined in Table 9.1 for plane
stress/strain and 3-D problems and
Matrix
R
is given by
R
R
R
ª
º
xxx
xxy
xxz
«
»
R
R
R
«
»
yyx
yyy
yyz
«
R
R
R
»
zzx
zzy
zzz
R
(9.46)
«
»
R
R
R
xyx
xyy
xyz
«
»
R
R
R
yzx
yzy
yzz
«
»
R
R
R
¬
¼
xzx
xzy
xzz
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