Civil Engineering Reference
In-Depth Information
where
3
3
w
[
w
N
w
[
w
N
¦
¦
n
n
(
[
,
K
)
[
,
(
[
,
K
)
K
n
n
w
[
w
[
w
K
w
K
n
1
n
1
(6.53)
3
3
w
K
w
N
w
K
w
N
¦
¦
n
n
(
[
,
K
)
K
,
(
[
,
K
)
K
n
n
w
[
w
[
w
K
w
K
n
1
n
1
Table 6.3
Local node number
l(n)
of subelement nodes,
P
i
at mid-side nodes
P
i
at
node
Subelement 1
Subelement 2
Subelement 3
n=
n=
n=
n=
n=
n=
n=
n=
n=
5
4
1
5
2
3
5
3
4
5
6
1
2
6
3
4
6
4
1
6
7
4
1
7
2
3
7
1
2
7
8
1
2
8
3
4
8
2
3
8
The
Jacobian
is given by
w
[
w
K
w
K
w
[
(6.54)
J
det
J
w
[
w
K
w
[
w
K
[
the
Jacobian
is zero. Without modification, the
integration scheme is applicable to elasticity problems. In equations (6.46) we simply
replace the scalars
U
and
T
with matrices
U
and
T
.
The reader may verify that for
1
6.3.8
Subdivision of region of integration
As for the plane problems we need to implement a subdivision scheme for the
integration. In the simplest implementation we subdivide elements into sub regions as
shown in Figure 6.16. The number of sub regions
N
[
in [ and
N
K
in K direction is
determined by
N
INT RL
[/
RL
/
] ;
N
INT RL
[/
RL
/
]
(6.55)
[
[
K
K
min
min
Equation 6.47 is replaced by
NN
Ml K j
()
( )
[
K
¦¦¦¦
e
ni
'|
U
N
[K
,
U
P Q
,
[K
,
J
J W W
n
m
k
i
m
k
m
k
(6.56)
l j mk
NN
Ml K j
11 1 1
()
( )
[
K
¦¦¦¦
e
ni
'|
U
N
[K
,
T
P Q
,
[K
,
J
J W W
n
m
k
i
m
k
m
k
l
11 1 1
j
mk
Search WWH ::
Custom Search