Civil Engineering Reference
In-Depth Information
y
y
3
2
(0,b)
2
3
1
4
1
4
x
(a,0)
x
(a)
(b)
Figure 3.1
(a) Plane rectangular element and (b) Plane general quadrilateral element
3
2
x, h
P(
)
1
4
Figure 3.2 Local coordinate system for quadrilateral elements
algebraic expressions will result, which are best generated by computer algebra packages
(Griffiths, 1994b, 2004).
Traditionally, the approach has been to work in a local coordinate system as shown in
Figure 3.2, originally proposed by Taig (1961), and to evaluate resulting integrals numer-
ically. The general point
P(ξ,η)
within the quadrilateral is located at the intersection of
two lines which cut opposite sides of the quadrilateral in equal proportions. For reasons
associated with subsequent numerical integrations it proves to be convenient to “normalise”
the coordinates so that side 12 has
ξ
=−
1, side 34 has
ξ
=
1, side 41 has
η
=−
1, and
side 23 has
η
=
1. In this system, the intersection of the bisectors of opposite sides of the
quadrilateral is the point
(
0
,
0
)
, while the corners 1, 2, 3, and 4 are
(
−
1
,
−
1
)
,
(
−
1
,
1
)
,
(
1
,
1
)
,and
(
1
,
−
1
)
respectively.
When this choice is adopted, the shape functions for a 4-noded quadrilateral with corner
nodes take the simple form
1
4
(
1
N
1
=
−
ξ)(
1
−
η)
