Civil Engineering Reference
In-Depth Information
or
L
1
L
2
r
1
+
L
1
L
2
r
2
+
L
1
L
2
L
3
r
3
d
A
I
=
16
A
L
1
L
2
r
1
+
L
1
L
2
r
2
+
L
1
L
2
L
3
r
3
d
A
−
8
A
Application of the integration formula, Equation 6.49, to each of the six integrals repre-
sented here (left as an exercise), we find
=
A
I
315
(6
r
2
−
4
r
1
−
2
r
3
)
6.10 NUMERICAL INTEGRATION:
GAUSSIAN QUADRATURE
Previous chapters show that integration of various functions of the field variable
are required for formulation of finite element characteristic matrices. Chapter 5
reveals that the Galerkin method requires integration over the element domain
(and, as seen, physical volume), once for each interpolation function (trial solu-
tion). In fact, an integration is required to obtain the value of every component of
the stiffness matrix of a finite element. In addition, integrations are required to
obtain nodal equivalents of nonnodal loadings.
In this chapter, we focus primarily on polynomial representations of the dis-
cretized representations of the field variable. In subsequent formulation of ele-
ment characteristic matrices, we are faced with integrations of polynomial forms.
A simple polynomial is relatively easy to integrate in closed form. In many cases,
however, the integrands are rational functions, that is, ratios of polynomials; and
these are quite tedious to integrate directly. In either case, in the finite element
context, where large numbers of elements, hence huge numbers of integrations,
are required, analytical methods are not efficient. Finite element software pack-
ages do not incorporate explicit integration of the element formulation equations.
Instead, they use numerical techniques, the most popular of which is
Gaussian
(or
Gauss-Legendre
)
quadrature
[10].
The concept of Gaussian quadrature is first illustrated in one dimension in
the context of an integral of the form
x
2
=
(6.97)
I
h
(
x
)d
x
x
1
Via the change of variable
r
b
, Equation 6.97 can be converted to
=
ax
+
1
=
(6.98)
I
f
(
r
)d
r
−
1
