Page 513 - Aircraft Stuctures for Engineering Student

Matrix methods of

structura I analysis

Actual aircraft structures consist of numerous components generally arranged in an
irregular manner. These components are usually continuous and therefore, theoreti-
cally, possess an infinite number of degrees of freedom and redundancies. Analysis is
then only possible if the actual structure is replaced by an idealized approximation or
model. This procedure has been discussed to some extent in Chapter 9 where we noted
that the greater the simplification introduced by the idealization the less complex but
more inaccurate became the analysis. In aircraft design, where structural weight is of
paramount importance, an accurate knowledge of component loads and stresses is
essential so that at some stage in the design these must be calculated as accurately
as possible. This accuracy may only be achieved by considering an idealized structure
which closely represents the actual structure. Standard methods of structural analysis,
some of which we have discussed in the preceding chapters, are inadequate for coping
with the necessary degree of complexity in such idealized structures. It was this
situation which led, in the late 1940s and early 1950s, to the development of matrix
methods of analysis and at the same time to the emergence of high-speed, electronic,
digital computers. Conveniently, matrix methods are ideally suited for expressing
structural theory and for expressing the theory in a form suitable for numerical
solution by computer.
A structural problem may be formulated in either of two different ways. One
approach proceeds with the displacements of the structure as the unknowns, the
internal forces then follow from the determination of these displacements, while in
the alternative approach forces are treated as being initially unknown. In the language
of matrix methods these two approaches are known as the stijiness (or displacement)
method and the flexibility (or force) method respectively. The most widely used of
these two methods is the stiffness method and for this reason, we shall concentrate
on this particular approach. Argyris and Kelsey ', however, showed that complete
duality exists between the two methods in that the form of the governing equations
is the same whether they are expressed in terms of displacements or forces.
Generally, as we have previously noted, actual structures must be idealized to
some extent before they become amenable to analysis. Examples of some simple
idealizations and their effect on structural analysis have been presented in Chapter
9 for aircraft structures. Outside the realms of aeronautical engineering the represen-
tation of a truss girder by a pin-jointed framework is a well known example of the