Civil Engineering Reference
In-Depth Information
3
A
2
A
1
P
A
3
2
Figure 6.10
Areas used to define area
coordinates for a triangular element.
1
L
1
0
3
3
1
2
L
1
P
L
1
1
P
2
2
1
1
(a)
(b)
Figure 6.11
(a) Area
A
1
associated with either
P
or
P
is constant.
(b) Lines of the constant area coordinate
L
1
.
where
A
is the total area of the triangle. Clearly, the area coordinates are not
independent, since
1
(6.41)
The dependency is to be expected, since Equation 6.40 expresses the location of
a point in two-dimensions using three coordinates.
The important properties of area coordinates for application to triangular
finite elements are now examined with reference to Figure 6.11. In Figure 6.11a,
a dashed line parallel to the side defined by nodes 2 and 3 is indicated. For any
two points
P
and
P
on this line, the areas of the triangles formed by nodes 2 and
3 and either
P
or
P
are identical. This is because the base and height of any tri-
angle so formed are constants. Further, as the dashed line is moved closer to node
1, area
A
1
increases linearly and has value
A
1
=
A
, when evaluated at node 1.
Therefore, area coordinate
L
1
is constant on any line parallel to the side of the tri-
angle opposite node 1 and varies linearly from a value of unity at node 1 to value
of zero along the side defined by nodes 2 and 3, as depicted in Figure 6.11b. Sim-
ilar arguments can be made for the behavior of
L
2
and
L
3
. These observations
L
1
+
L
2
+
L
3
=
