The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 396

Recall from Chapter 5, Eq. (5.25), that the transformation matrix T jj has the property
T i jj T i jj = I , where I is an identity matrix.

Arrange the entries in the nodal displacement vector v i as

v i u y 3 ] T v i 1 v i 2 v i 3 ] T

= [ u x 1 u y 1 u x 2 u y 2 u x 3 = [

Then from Eq. (6.107)

= =

v i 1 v i 1

v i 2
v i 3

T i 11 00
0 i 22 0
00 i 33

v i v i 2 T i v i = (6.108)

v i 3

or v i T iT v i , since, Eq. (5.26), T i ) − 1 T iT . Similarly, p i T i p i and p i T iT p i . Sub- = ( = = =

stitution of Eq. (6.108) into the stiffness equation in the local system, k i v i p i , results in

=

k i

= k i T i v i

k i T i v i

v i = p i − p i . Premultiply by T iT to obtain T iT T iT p i k i v i p i

− = − with

k i T i

k i T iT and p i T iT p i , which have the same form as Eq. (5.28). = =

6.6 Numerical Integration

The development of element stiffness matrices involves the evaluation of definite integrals,
such as those that result from the principle of virtual work. For the elements treated in the
previous section it was possible to evaluate the integrals analytically. Often, exact integra-
tion expressions are not readily obtained and numerical integration becomes essential. The
numerical integration provides approximations to the integrals; however, it is contended
[Zienkiewicz, 1977] that the error resulting from numerical integration may compensate
for the modeling error due to the geometrical discretization of the structure. This error
compensation appears to result in an improved solution.
In general, the integral can be approximated by using a simple summation of terms
involving the integrand, evaluated at n specific points (integration points), and multiplied
by suitable weights. Namely, for the case of one, two, or three dimensions,

n

W ( n )
i

F (ξ ) d ξ ≈ F (ξ i ) (6.109)

L

i

n

W ( n )
i

F (ξ , η) d ξ d η ≈ F (ξ

i ,

η ) (6.110)

i

L

i

n

W ( n )
i

F (ξ , η , ζ) d ξ d η d ζ ≈ F (ξ i , η i , ζ i ) (6.111)

V

i

where W ( n )
i

is a weighting factor with superscript ( n ) indicating that n integration points

are employed, and F (ξ i ) ,F (ξ i , η i ) , and F (ξ i , η i , ζ i ) are the values of the function F at the

points ζ i .

Numerical integration techniques tend to differ in the method of establishing values of
W ( n )
i

ξ i , η i ,

i , as appropriate). Considerations, such as accuracy and computational
efficiency, are taken into account in selecting a method. Two types of numerical integration,
usually referred to as quadrature formulas , have been found to be particularly useful in finite
element calculations. They are considered in the following sections.

and ξ

i (and

η

i ,

ζ