The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 332

(1943) and of Prager 2 and Synge 3 (1947). However, computers were not available then to
implement the necessary extensive numerical operations.
According to Clough (1990), who was a summer employee of the Structural Dynamics
Unit of Boeing Airplane Company in Seattle, WA, “by 1952 aircraft structural analysis had
advanced to the point where a complex structure idealized as an assemblage of simple
truss, beam, or shear panel elements could be analyzed by either the force or displacement
method formulated as a series of matrix operations and using an automatic digital com-
puter to carry out the calculations.”
The fundamental breakthrough in the U.S. development occurred in the 1952-1953 win-
ter when M.J. Turner, the head of Boeing's Structural Dynamics Unit, conceived of a novel
model of panels for a wing. As described by Clough, “the essential idea in the proposed
Turner procedure was that the deformations of any plane stress element be approximated
by assuming a combination of simple strain fields acting within the element. The idea
is applicable to both rectangular and triangular elements, but the use of triangular ele-
ments was given greater emphasis because an assemblage of triangular elements could
serve to approximate plates of any shape. In modeling a triangular plate, the deformations
were approximated by three constant strain fields: uniform normal strains in the x and y
directions combined with a uniform x - y shear strain. Based on these strain patterns, the
force-displacement relationships for the corner nodal points could be calculated using
Castigliano's theorem, or the equivalent principle of virtual displacements.” It was not
until September 1956 that this 1953 work appeared in print (Turner, et al, 1956). The name
finite element method was first used in Clough (1960).
During this development period, there were significant advances in the matrix analysis
of structures. In particular, Argyris (1954 and 1955) published a series of articles organizing
the matrix formulation of structural mechanics. The dual nature of the transformations of
the force and displacement methods was identified, as were the relationships between the
displacement method and the principle of virtual work and between the force method and
the principle of complementary virtual work.
The finite element method serves as the analysis component of the design process (Fig. 6.1).
This, of course, is the same roll played by the structural analyses of Chapter 5. As indicated
in Fig. 6.1, the design of a system may involve repeated analyses and the introduction of
design criteria. An important initial step is the idealization of the structure, resulting in a
model that can be analyzed.
The structure is idealized into a model composed of a number of elements of finite size
(Fig. 6.2). The model provides the name of the analysis technique. The connections between
these so-called “finite elements” are point locations, i.e., nodes. The locations of the nodes
are identified in the global coordinate system X, Y, Z. The distribution of displacements
and forces are represented by values of these variables at the nodes, i.e., the finite element
method involves a discretization process. The particular names given to the finite element
method depend on the variables selected as unknowns at the nodes. As in Chapter 5, if the
2 William Prager (1908-1980) was born in Karlsruhe, Germany, and received his doctorate in engineering sciences
in Darmstadt in 1926. Between 1929 and 1932 he worked with Prandtl in G ottingen. He moved to the Technical
Institute of Karlsruhe, only to be fired by Hitler. The next eight years were spent in Turkey. In 1941 he accepted
a position at Brown University in the United States. The center of gravity of applied mechanics seemed to move
to Brown University. It is said that Prager followed Einstein's philosophy of scientific work being “as simple
as possible-but no simpler.” He is credited with being influential in the introduction of more mathematics and
research into US engineering schools.
3 John Lighton Synge, born in 1897, was from an Anglo-Irish family that can be dated to the fifteenth century. The
name Synge has been traced to Henry VIII saying to a choirboy “Synge, Millington, synge.” He served on faculties
of universities in Dublin, Toronto, and the United States. He was a visiting professor at Brown University in 1941.
He is most famous for his geometrical insight into the theory of relativity.