Civil Engineering Reference
In-Depth Information
15.3.5
Computation of stresses at cell nodes
For computation of the internal stress results the following equation is used
7,8
³
³
ı
P
S
P
,
Q
t
Q dS
R
P
,
Q
u
Q dS
a
a
a
S
S
(15.43)
³
ˆ
E
PQ
,
ı
QdV
F ı
P
a
0
0
a
V
which is derived in the same way as Eq. (13.59).
The coefficients of
E
are given by
C
ˆ
ijkl
ª
2
E
C
GG GG GG
C
G
r r
¬
3
ik
jl
jk
il
ij
kl
4
ij
,
k
,
l
n
1
r
Cr r
(15.44)
º
QG
G
r r
G
r r
G
r r
Cr r
G
Cr r r r
4
il
,
j
,
k
jk
,
i
,
l
ik
,
j
,
l
jl
,
i
,
k
4
kl
,
i
,
j
6 ,
i
,
j
,
k
,
l
¼
and those of
F
by
ª
º
FCC
GG GG
C
GG
(15.45)
ijkl
18
¬
12
jk
il
ik
jl
13
ij
kl
¼
where the constants are given in Table 15.3.
Constants for fundamental solution
E
Table 15.3
Plane strain
Plane stress
3-D
n
1
1
2
C
2
-1/4SQ
-(1+QS
-1/8SQ
C
3
1-2Q
(1-QQ
1-2Q
C
4
2
3
C
12
1
1
7-5Q
C
13
1-4Q
(1-3QQ
2-10 Q
C
18
-1/8(1-Q
-Q/8
-1/30(1-Q
The discretized form of (15.43) is
E
N
E
N
C
N
¦¦
¦¦
¦¦
ˆ
ee
ee
cc
ı
P
'
Ru
'
St
'
Eı F ı
P
(15.46)
a
n
n
n
n
n
0
n
0
a
en
11
en
11
cn
11
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