Civil Engineering Reference
In-Depth Information
1
4
(
1
N
2
=
−
ξ)(
1
+
η)
1
4
(
1
N
3
=
+
ξ)(
1
+
η)
(3.1)
1
4
(
1
N
4
=
+
ξ)(
1
−
η)
and these can be used to describe the variation of unknowns such as displacement or fluid
potential in an element as before.
The same shape functions can also often be used to specify the relation between the
global
(x, y)
and local
(ξ, η)
coordinate systems. If this is so the element is of a type
called “isoparametric” (Ergatoudis
et al
., 1968; Zienkiewicz
et al
., 1969), and the 4-node
quadrilateral is an example. The coordinate transformation is therefore,
x
=
N
1
x
1
+
N
2
x
2
+
N
3
x
3
+
N
4
x
4
=
{
x
}
y
=
N
1
y
1
+
N
2
y
2
+
N
3
y
3
+
N
4
y
4
[
N
]
(3.2)
=
[
N
]
{
y
}
where the [
N
] are given by (3.1) and
are the nodal coordinates.
In the previous chapter (e.g. equations 2.68 and 2.128), it was shown that element prop-
erties involve not only [
N
] but also their derivatives with respect to the global coordinates
(x, y)
which appear in matrices such as [
B
] and [
T
]. Further, products of these quantities
need to be integrated over the element area or volume.
Derivatives are easily converted from one coordinate system to the other by means of
the chain rule of partial differentiation, best expressed in matrix form for two dimensions by
{
x
}
and
{
y
}
∂
∂ξ
∂
∂η
∂y
∂ξ
∂x
∂ξ
∂
∂x
∂
∂y
∂
∂x
∂
∂y
=
=
[
J
]
(3.3)
∂x
∂η
∂y
∂η
or
∂
∂x
∂
∂y
∂
∂ξ
∂
∂η
[
J
]
−
1
=
(3.4)
where
[
J
]
is the Jacobian matrix. The determinant of this matrix, det
|
J
|
known as “The
Jacobian”, must also be evaluated because it is used in the transformed integrals as follows:
d
x
d
y
=
1
1
det
|
J
|
d
ξ
d
η
(3.5)
−
1
−
1
For three dimensions, the equivalent expressions are self-evident.
Degenerate quadrilaterals such as the one shown in Figure 3.3(a) are usually acceptable,
however reflex interior angles as shown in Figure 3.3(b) should be avoided as this will cause
the Jacobian to become indeterminate.
