Civil Engineering Reference
In-Depth Information
where
sc
is the number of sub-cells which equals to 3 for collocation point at a corner
node, and 4 for collocation point at a mid-node.
J
is the Jacobian of the transformation
from
[
,,
coordinates.
This transformation is given by
[
,
,
to
9
5
5
5
¦
¦
¦
[
N
(,, )
[K][
;
K
N
(,, )
[K]K 9
;
N
(,, )
[K]]
(15.39)
n
l n
()
n
l n
()
n
l n
()
n
1
n
1
n
1
Where
l(n)
is an array that indicates the local number of node
l
. For the sub-cell 2 in
Figure 15.8 for example
() (4,1,2,3,8)
ln
. More details can be found in [6].
The shape functions are defined as
1
1
N
(1
[
)(1
K]
) (1
)
;
N
(1
[
)(1
K]
)(1
)
1
2
8
8
1
1
N
(1
[
)(1
K]
)(1
)
;
N
(1
[
)(1
K]
)(1
)
(15.40)
3
4
8
8
1
(1
N
[
)
5
8
The Jacobian is defined as
www
www
www
www
www
www
[ K ]
[ [ [
[ K ]
K KK
[ K ]
]]]
J
(15.41)
where
5
w
[
w
N
¦
n
(
[K][
,
,
)
etc.
ln
()
(15.42)
w
[
w
[
n
1
The Jacobian tends to zero with
o(r
2
)
thereby cancelling out the singularity. Having
computed the residual
^
R
due to an initial stress state
^`
0
ı
we solve the problem for the
boundary unknowns
^
x
. The next step is to compute stress increments at the cell and
boundary nodes.
Search WWH ::
Custom Search