Civil Engineering Reference
In-Depth Information
1
2
A
[
a
1
=
1
(
y
2
−
y
3
)
+
2
(
y
3
−
y
1
)
+
3
(
y
1
−
y
2
)]
(6.35)
1
2
A
[
a
2
=
1
(
x
3
−
x
2
)
+
2
(
x
1
−
x
3
)
+
3
(
x
2
−
x
1
)]
Substituting the values into Equation 6.32 and collecting coefficients of the nodal
variables, we obtain
[(
x
2
y
3
−
+
(
y
2
−
+
(
x
3
−
1
x
3
y
2
)
y
3
)
x
x
2
)
y
]
1
2
A
+
[(
x
3
y
1
−
+
(
y
3
−
+
(
x
1
−
2
x
1
y
3
)
y
1
)
x
x
3
)
y
]
(
x
,
y
)
=
(6.36)
+
[(
x
1
y
2
−
x
2
y
1
)
+
(
y
1
−
y
2
)
x
+
(
x
2
−
x
1
)
y
]
3
Given the form of Equation 6.36, the interpolation functions are observed to be
1
2
A
[(
x
2
y
3
−
N
1
(
x
,
y
)
=
x
3
y
2
)
+
(
y
2
−
y
3
)
x
+
(
x
3
−
x
2
)
y
]
1
2
A
[(
x
3
y
1
−
N
2
(
x
,
y
)
=
x
1
y
3
)
+
(
y
3
−
y
1
)
x
+
(
x
1
−
x
3
)
y
]
(6.37)
1
2
A
[(
x
1
y
2
−
N
3
(
x
,
y
)
=
x
2
y
1
)
+
(
y
1
−
y
2
)
x
+
(
x
2
−
x
1
)
y
]
where
A
is the area of the triangular element. Given the coordinates of the three
vertices of a triangle, it can be shown that the area is given by
1
x
1
y
1
1
2
A
=
(6.38)
1
x
2
y
2
1
x
3
y
3
Note that the algebraically complex form of the interpolation functions
arises primarily from the choice of the element coordinate system of Figure 6.8.
As the linear representation of the field variable exhibits geometric isotropy,
location and orientation of the element coordinate axes can be chosen arbitrarily
without affecting the interpolation results. If, for example, the element coordi-
nate system shown in Figure 6.9 is utilized, considerable algebraic simplification
results. In the coordinate system shown, we have
x
1
=
y
1
=
y
2
=
0, 2
A
=
x
2
y
3
,
3 (
x
3
,
y
3
)
y
Figure 6.9
Three-node
triangle having an element
coordinate system attached
to the element.
x
2
(
x
2
, 0)
1 (0, 0)
