Civil Engineering Reference
In-Depth Information
Chapter
15
/
Virtual Work and
Energy Methods
The majority of the structural problems we have encountered so far have involved
structures in which the support reactions and the internal force systems are statically
determinate. These include beams, trusses, cables and three-pinned arches and, in the
case of beams, we have calculated displacements. Some statically indeterminate struc-
tures have also been investigated. These include the composite structural members in
Section 7.10 and the circular section beams subjected to torsion and supported at each
end in Section 11.1. These relatively simple problems were solved using a combination
of statical equilibrium and compatibility of displacements. Further, in Section 13.6, a
statically indeterminate propped cantilever was analysed using the principle of super-
position (Section 3.7) while the support reactions for some cases of fixed beams were
determined by combining the conditions of statical equilibrium with the moment-area
method (Section 13.3). These methods are perfectly adequate for the comparatively
simple problems to which they have been applied. However, other more powerful
methods of analysis are required for more complex structures which may possess a
high degree of statical indeterminacy. These methods will, in addition, be capable of
providing rapid solutions for some statically determinate problems, particularly those
involving the calculation of displacements.
Themethods fall into two categories and are based on two important concepts; the first,
the principle of virtual work
, is the most fundamental and powerful tool available for
the analysis of statically indeterminate structures and has the advantage of being able
to deal with conditions other than those in the elastic range, while the second, based
on
strain energy
, can provide approximate solutions of complex problems for which
exact solutions may not exist. The two methods are, in fact, equivalent in some cases
since, although the governing equations differ, the equations themselves are identical.
In modern structural analysis, computer-based techniques are widely used; these
include the flexibility and stiffness methods. However, the formulation of, say, stiff-
ness matrices for the elements of a complex structure is based on one of the above
approaches, so that a knowledge and understanding of their application is advanta-
geous. We shall examine the flexibility and stiffness methods in Chapter 16 and their
role in computer-based analysis.
Other specialist approaches have been developed for particular problems. Examples of
these are the slope-deflection method for beams and the moment-distribution method
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