Civil Engineering Reference
In-Depth Information
Table 3.1 Coordinates and weights in Gauss-
Legendre quadrilateral integration formulae
(ξ
i
,η
j
)
w
i
,w
j
W
i
n
nip
1
1
(
0
,
0
)
(
2
,
2
)
4
3
3
,
±
2
4
±
(
1
,
1
)
1
3 4@)
5
9
,
9
5
,
±
25
81
±
5
,
0
9
,
9
40
81
(2@)
±
0
,
±
5
9
,
9
40
81
(2@)
9
,
9
64
81
(1@)
(
0
,
0
)
in integration times. A further point is that for some elements (e.g. a 14-node hexahedron
described later in this Chapter) the shape functions are so complex algebraically that it is
doubtful if they could be isolated at all without the help of computer algebra.
For finite elements in the context of plane elasticity, the element stiffness matrix has
been shown in Chapter 2 (e.g. 2.69) to be given by integrals of the form
[
B
]
T
[
D
][
B
] d
x
d
y
[
k
m
]
=
(3.7)
where [
B
] and [
D
] represent the strain-displacement and stress-strain matrices respectively.
In the case of quadrilateral elements, if the element is rectangular with its sides parallel
to the
x
-and
y
-axes, the term under the integral consists of simple polynomial terms which
can be easily integrated in closed form by separation of the variables, resulting in compact
terms like (2.63). In general however, quadrilateral elements will lead to very complicated
expressions under the integral sign which can only be tackled numerically.
Noting that “2-point” Gaussian quadrature, that is
nip = 4
, leads in most cases to
accurate estimates of the stiffness matrix of a 4-node general quadrilateral, a compromise
approach is to evaluate the contribution to the stiffness matrix coming from each of the
four “Gauss-points” algebraically and add them together, thus:
4
|
J
|
i
(
[
B
]
T
[
D
][
B
]
)
i
[
k
m
]
≈
W
i
det
(3.8)
i
=
1
where det
is the Jacobian described previously.
This at first leads to rather long expressions, but a considerable amount of cancelling
|
J
|
and simplification is possible (e.g. the 1
/
√
3 term that appears in the sampling points of the
integration formula disappears in the simplification process). The algebraic expressions can
