Civil Engineering Reference
In-Depth Information
collectively known as the
flexibility method,
in which the unknowns are the
forces and the knowns are displacements. The finite element method, in its most
often-used form, corresponds to the
displacement method,
in which the un-
knowns are system displacements in response to applied force systems. In this
text, we adhere exclusively to the displacement method. As will be seen as we
proceed, the term
displacement
is quite general in the finite element method and
can represent physical displacement, temperature, or fluid velocity, for example.
The term
finite element
was first used by Clough [6] in 1960 in the context of
plane stress analysis and has been in common usage since that time.
During the decades of the 1960s and 1970s, the finite element method was
extended to applications in plate bending, shell bending, pressure vessels, and
general three-dimensional problems in elastic structural analysis [7-11] as well
as to fluid flow and heat transfer [12, 13]. Further extension of the method to
large deflections and dynamic analysis also occurred during this time period
[14 , 15]. An excellent history of the finite element method and detailed bibliog-
raphy is given by Noor [16].
The finite element method is computationally intensive, owing to the required
operations on very large matrices. In the early years, applications were performed
using mainframe computers, which, at the time, were considered to be very pow-
erful, high-speed tools for use in engineering analysis. During the 1960s, the finite
element software code NASTRAN [17] was developed in conjunction with the
space exploration program of the United States. NASTRAN was the first major
finite element software code. It was, and still is, capable of hundreds of thousands
of degrees of freedom (nodal field variable computations). In the years since the
development of NASTRAN, many commercial software packages have been in-
troduced for finite element analysis. Among these are ANSYS [18], ALGOR [19],
and COSMOS/M [20]. In today's computational environment, most of these
packages can be used on desktop computers and engineering workstations to
obtain solutions to large problems in static and dynamic structural analysis, heat
transfer, fluid flow, electromagnetics, and seismic response. In this text, we do not
utilize or champion a particular code. Rather, we develop the fundamentals for
understanding of finite element analysis to enable the reader to use such software
packages with an educated understanding.
1.5 EXAMPLES OF FINITE ELEMENT
ANALYSIS
We now present, briefly, a few examples of the types of problems that can be
analyzed via the finite element method. Figure 1.7 depicts a rectangular region
with a central hole. The area has been “meshed” with a finite element grid of two-
dimensional elements assumed to have a constant thickness in the
z
direction.
Note that the mesh of elements is irregular: The element shapes (triangles and
quadrilaterals) and sizes vary. In particular, note that around the geometric dis-
continuity of the hole, the elements are of smaller size. This represents not only
