Civil Engineering Reference
In-Depth Information
6.6 RECTANGULAR ELEMENTS
Rectangular elements are convenient for use in modeling regular geometries, can
be used in conjunction with triangular elements, and form the basis for develop-
ment of general quadrilateral elements. The simplest of the rectangular family of
elements is the four-node rectangle shown in Figure 6.13, where it is assumed
that the sides of the rectangular are parallel to the global Cartesian axes. By con-
vention, we number the nodes sequentially in a counterclockwise direction, as
shown. As there are four nodes and 4 degrees of freedom, a four-term polynomial
expression for the field variable is appropriate. Since there is no complete four-
term polynomial in two dimensions, the incomplete, symmetric expression
(
x
,
y
)
=
a
0
+
a
1
x
+
a
2
y
+
a
3
xy
(6.50)
is used to ensure geometric isotropy. Applying the four nodal conditions and
writing in matrix form gives
1
2
3
4
1
x
1
y
1
x
1
y
1
a
0
a
1
a
2
a
3
1
x
2
y
2
x
2
y
2
=
(6.51)
1
x
3
y
3
x
3
y
3
1
x
4
y
4
x
4
y
4
which formally gives the polynomial coefficients as
−
1
a
0
a
1
a
2
a
3
1
x
1
y
1
x
1
y
1
1
2
3
4
1
x
2
y
2
x
2
y
2
=
(6.52)
1
x
3
y
3
x
3
y
3
1
x
4
y
4
x
4
y
4
In terms of the nodal values, the field variable is then described by
−
1
1
x
1
y
1
x
1
y
1
1
2
3
4
1
x
2
y
2
x
2
y
2
(
x
,
y
)
=
[1
xyx
]
{
a
} =
[1
xyx
]
1
x
3
y
3
x
3
y
3
1
x
4
y
4
x
4
y
4
(6.53)
from which the interpolation functions can be deduced.
4 (
x
4
,
y
4
)
3 (
x
3
,
y
3
)
y
x
1 (
x
1
,
y
1
)
2 (
x
2
,
y
2
)
Figure 6.13
A four-node
rectangular element defined in
global coordinates.
