Civil Engineering Reference
In-Depth Information
A parallel procedure for the interpolation functions associated with the other
three corner nodes leads to
1
4
(
r
N
2
(
r
,
s
)
=
+
1)(1
−
s
)(
s
−
r
+
1)
(6.59b)
1
4
(1
N
3
(
r
,
s
)
=
+
r
)(1
+
s
)(
r
+
s
−
1)
(6.59c)
1
4
(
r
N
4
(
r
,
s
)
=
−
1)(1
+
s
)(
r
−
s
+
1)
(6.59d)
The form of the interpolation functions associated with the midside nodes is
simpler to obtain than those for the corner nodes. For example,
N
5
has a value of
zero at nodes 2, 3, and 6 if it contains the term
r
−
1
and is also zero at nodes 1,
4, and 8 if the term
1
+
r
is included. Finally, if a zero value is at node 7,
(
r
,
s
)
=
(0, 1)
is obtained by inclusion of
s
−
1
.
The form for
N
5
is
1
2
(1
1
2
(1
r
2
)(1
N
5
=
−
r
)(1
+
r
)(1
−
s
)
=
−
−
s
)
(6.59e)
where the leading coefficient ensures a unity value at node 5. For the other mid-
side nodes,
1
2
(1
s
2
)
N
6
=
+
r
)(1
−
(6.59f )
1
2
(1
r
2
)(1
N
7
=
−
+
s
)
(6.59g)
1
2
(1
s
2
)
N
8
=
−
r
)(1
−
(6.59h)
are determined in the same manner.
Many other, successively higher-order, rectangular elements have been de-
veloped [1]. In general, these higher-order elements include internal nodes that,
in modeling, are troublesome, as they cannot be connected to nodes of other
elements. The internal nodes are eliminated mathematically. The elimination
process is such that the mechanical effects of the internal nodes are assigned
appropriately to the external nodes.
6.7 THREE-DIMENSIONAL ELEMENTS
As in the two-dimensional case, there are two main families of three-dimensional
elements. One is based on extension of triangular elements to tetrahedrons and
the other on extension of rectangular elements to rectangular parallelopipeds
