Civil Engineering Reference
In-Depth Information
y
y
2
2
3
3
1
1
4
4
x
x
(a)
(b)
Figure 3.3
(a) Degenerate quadrilateral (b) Unacceptable quadrilateral
3.2.1 Numerical integration for quadrilaterals
Although some integrals of this type can be evaluated analytically, this has traditionally
been impractical for complicated functions, particularly in the general case when
(ξ, η)
become curvilinear (Ergatoudis
et al
., 1968). In most finite element programs (3.5) are
evaluated numerically, using Gauss-Legendre quadrature over quadrilateral regions (Irons,
1966a,b). The quadrature rules in two dimensions are all of the form
1
1
n
n
f(ξ,η)
det
|
J
|
d
ξ
d
η
≈
w
i
w
j
f(ξ
i
,η
j
)
−
1
−
1
i
=
1
j
=
1
nip
≈
W
i
f(ξ,η)
i
(3.6)
i
=
1
where
nip
=
n
2
(total number of integrating points),
w
i
and
w
j
(or
W
i
=
w
i
w
j
)are
weighting coefficients and
(ξ
i
,η
j
)
are sampling points within the element. These values
for
n
equal to 1, 2, and 3 are shown in Table 3.1, and complete tables are available in other
sources, for example Kopal (1961). The table assumes that the range of integration is
±
1,
hence the reason for normalising the local coordinate system in this way.
The approximate equality in (3.6) is exact for cubic functions when
n
=
2 and for
quintics when
n
=
3. Usually one attempts to perform integrations over finite elements as
accurately as possible, but in special circumstances (Zienkiewicz
et al
., 1971) “reduced”
integration, whereby integrals are deliberately evaluated approximately by decreasing
n
can
improve the quality of solutions.
3.2.2 Analytical integration for quadrilaterals
Computer Algebra Systems (CAS) such as “REDUCE” and “Maple” enable algebraic
expressions (e.g. the finite element shape functions) to be manipulated essentially “analyti-
cally”. Expressions can be differentiated, integrated, factorised and so on, leading to explicit
formulations of element matrices avoiding the need for conventional numerical integration.
Particularly for three-dimensional elements, this approach can lead to substantial savings
