The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 371

TABLE 6.5
Elements and Some Structures that can be Formed Using the Elements
Structure

Straight Curved Twisted Rectangular Thick-Walled

Element Type Beam Beam Beam Plate Cylinder

Beam X X X

Plate (Membrane)

X ∗

X X

(In-plane loading)
Plate (Out-of-plane loading) X X X

Solid X X X X X

∗ Use plane strain option.

of elements and some structures that can be formed with these elements. For example, a
solid element can be used in a straight beam, curved beam, or twisted beam, rectangular
plate, etc. The loadings in the test problem should be simple yet capable of generating all
possible deformations. In MacNeal and Harder (1985), a set of standard benchmark tests
for some widely used elements are given. With these tests, the advantages and weaknesses
of the elements can be detected. Results of actual tests tend to show considerable variations
in accuracy for what may appear to be reasonable elements. In finite element development,
these are the kinds of benchmark test problems that are helpful in locating and correcting
the element errors and weaknesses.
The following two examples from this reference show the test design for quadrilateral
and solid elements.

EXAMPLE 6.4 Accuracy Test for a Four-Node Quadrilateral Element
Consider a four-node quadrilateral element shown in Fig. 6.24a. The following cases should
be tested.
Element shape:

1. Regular (the element shape has not been intentionally distorted)—square or rectan-
gular
2. Irregular—distorted

Structures which employ this element are

1. Straight beam
2. Curved beam
3. Thick-walled cylinder

Loading:

1. For cantilevered straight and curved beams—tip loading including extensional and
shearing forces
2. For thick-walled cylinder—pressure at the inner radius

The test results should be compared to the theoretical solutions of the test structures
with the load cases. The theoretical solutions can be obtained from Pilkey (1994), as well
as from other structural mechanics topics. Typical problems occur in finding, for example,
that the element geometry significantly influences the accuracy for a straight beam under
bending load. Also, errors are often large for a thick-walled cylinder when Poisson's ratio
approaches 1/2.