Civil Engineering Reference
In-Depth Information
The form of Equation 6.53 suggests that expression of the interpolation
functions in terms of the nodal coordinates is algebraically complex. Fortunately,
the complexity can be reduced by a more judicious choice of coordinates. For
the rectangular element, we introduce the normalized coordinates (also known as
natural coordinates
or
serendipity coordinates
)
r
and
s
as
x
−¯
x
y
−¯
y
r
=
s
=
(6.54)
a
b
where 2
a
and 2
b
are the width and height of the rectangle, respectively, and the
coordinates of the centroid are
x
1
+
x
2
y
1
+
y
4
x
¯
=
y
¯
=
(6.55)
2
2
as shown in Figure 6.14a. Therefore,
r
and
s
are such that the values range from
−
1
to
+
1
, and the nodal coordinates are as in Figure 6.14b.
Applying the conditions that must be satisfied by each interpolation function
at each node, we obtain (essentially by inspection)
1
4
(1
N
1
(
r
,
s
)
=
−
r
)(1
−
s
)
1
4
(1
N
2
(
r
,
s
)
=
+
r
)(1
−
s
)
(6.56a)
1
4
(1
N
3
(
r
,
s
)
=
+
r
)(1
+
s
)
1
4
(1
=
−
+
N
4
(
r
,
s
)
r
)(1
s
)
hence
(
x
,
y
)
=
(
r
,
s
)
=
N
1
(
r
,
s
)
1
+
N
2
(
r
,
s
)
2
+
N
3
(
r
,
s
)
3
+
N
4
(
r
,
s
)
4
(6.56b)
4
3
4 (
1, 1)
3 (1, 1)
s
s
r
r
y
y
1
2
1 (
1,
1)
2 (1,
1)
x
x
(a)
(b)
Figure 6.14
A four-node rectangular element showing
(a) the translation to natural coordinates, (b) the natural
coordinates of each node.
