Civil Engineering Reference
In-Depth Information
can be used to write the following conditions satisfied by the area coordinates
when evaluated at the nodes:
Node 1:
L
1
=
1
L
2
=
L
3
=
0
Node 2:
L
2
=
1
L
1
=
L
3
=
0
(6.42)
Node 3:
L
3
=
1
L
1
=
L
2
=
0
The conditions expressed by Equation 6.42 are exactly the conditions that
must be satisfied by interpolation functions at the nodes of the triangular ele-
ment. So, we express the field variable as
(
x
,
y
)
=
L
1
1
+
L
2
2
+
L
3
3
(6.43)
in terms of area coordinates. Is this different from the field variable representa-
tion of Equation 6.36? If the area coordinates are expressed explicitly in terms
of the nodal coordinates, the two field variable representations are shown to be
identical. The true advantages of area coordinates are seen more readily in
developing interpolation functions for higher-order elements and performing
integration of various forms of the interpolation functions.
6.5.2 Six-Node Triangular Element
A six-node element is shown in Figure 6.12a. The additional nodes 4, 5, and 6 are
located at the midpoints of the sides of the element. As we have six nodes, a com-
plete polynomial representation of the field variable is
a
3
x
2
a
5
y
2
(
x
,
y
)
=
a
0
+
a
1
x
+
a
2
y
+
+
a
4
xy
+
(6.44)
L
2
0
L
1
0
3
L
3
1
3
1
2
1
2
L
2
L
1
1
2
5
L
3
5
6
6
L
2
1
L
1
1
L
3
0
2
2
4
4
1
1
(a)
(b)
Figure 6.12
Six-node triangular elements: (a) Node numbering convention.
(b) Lines of constant values of the area coordinates.
