Civil Engineering Reference
In-Depth Information
Table 9.2
Element Stress Components (psi) for Four Elements Sharing a Common Node
in Example 9.4
x
y
xy
e
Element 1
2553.5
209.71
179.87
2475.8
Element 2
1922.7
351.69
43.55
1774.8
Element 12
1827.5
264.42
154.44
1731.5
Element 99
2189.0
249.14
480.57
2236.4
for the common node. The last row of the table lists the average values of the
three stress components at the common node. Clearly, the nodal stresses are not
continuous from element to element at the common node. As previously dis-
cussed, the magnitudes of the discontinuities should decrease as the element
mesh is refined.
In contrast, the
element
stress components for the same four elements are
shown in Table 9.2. The values listed in the table are computed at the element
centroid and include the equivalent (Von Mises) stress as defined previously.
While not included in the table, the principal stress components are also avail-
able from the solution. In general, the element stresses should be used in results
evaluation, especially in terms of application of failure theories.
9.8 PRACTICAL CONSIDERATIONS
Probably the most critical step in application of the finite element method is the
choice of element type for a given problem. The solid elements discussed in this
chapter are among the simplest elements available for use in stress analysis.
Many more element types are available to the finite element analyst. (One com-
mercial software system has no fewer than 141 element types.) The differences
in elements for stress analysis fall into three categories: (1) number of nodes,
hence, polynomial order of interpolation functions; (2) type of material behavior
(elastic, plastic, thermal stress, for example); and (3) loading and geometry of the
structure to be modeled (plane stress, plane strain, axisymmetric, general three
dimensional, bending, torsion).
As an example, consider Figure 9.12, which shows a flat plate supported at
the corners and loaded by a pressure distribution
p
(
x
,
y
) acting in the negative
z
direction. The primary mode of deformation of the plate is bending in the
z
direc-
tion. To adequately describe the behavior, a finite element used to model the plate
must be such that continuity of slope in both
xz
and
yz
planes is ensured. There-
fore, a three-dimensional solid element as described in Section 9.8 would not be
appropriate as only the displacement components are included as nodal variables.
Instead, an element that includes partial derivatives representing the slopes must
be included as nodal variables. Plate elements have been developed on the basis
of the theory of thin plates (usually only covered in graduate programs) in which
z
p
(
x
,
y
)
x
y
Figure 9.12
Example of a thin
plate subjected to
bending.
