Principles of Virtual Work: Integral Form of the Basic Equations - Mechanics of Structures: Variational and Computational Methods - page 75

2

Principles of Virtual Work:
Integral Form of the Basic Equations

The variational or energy methods of structural mechanics have been important tools for the
development of basic equations for more than a century. In particular, they have been useful,
and sometimes essential, for the derivation of governing differential equations of motion.
It has long been recognized that the fundamental relationships of the previous chapter, e.g.,
the equations of equilibrium or the strain-displacement equations, have equivalent energy
representations. Now, the variationally based integral forms of the basic equations are
emerging as important foundations for computational techniques of structural mechanics.
This chapter considers the classical variational principles of the theory of elasticity and then
treats the generalized variational principles which are particularly helpful in achieving a
better understanding of the interrelationships between the various methods of structural
mechanics. The study of variational methods should begin with a brief look at the basics
of the calculus of variations, a branch of mathematics dealing with the extremal values
of integrals. Numerous sources covering the calculus of variations are available; a short
summary is provided in Appendix I.

2.1 Fundamental Definitions of Work and Energy

Before proceeding to the variational principles of the mechanics of solids, it is essential to
understand the concepts of work and energy. It is useful to define work and energy in what
may appear to be somewhat abstract terms. The subsequent development of the variational
principles will clarify why such definitions are given.

2.1.1 Work and Energy

Work is defined as the product of a force and the displacement of its point of application
in the direction of the force. More specifically, define the differential work dW done, while
the force F moves through a differential displacement ds as the product of ds and F s , the
component of F in the direction of ds

dW = F s ds (2.1)

Energy is defined as a quantity representing the ability or capacity to perform work. We
say a structural system “possesses” energy, whereas the forces in the system may “perform”

77