The Ritz Method of Approximation

Abstract

The principle of virtual work is a beautiful alternative to the Newton-Cauchy view of mechanics. This beauty notwithstanding, the principle of virtual work, in its basic form, is not very useful. The sunple truth is that it is impossibly difficult to implement the part of the principle that says, “for all ū ∈ ℱ(ℬ).” Furthermore, the displacement u(x) that solves the problem may not be one of the named and tabulated functions of classical mathematics (e.g., polynomial, trigonometric, and exponential). For a continuous system, the “for-all” statement implies proving that the functional is zero for an infinite number of virtual displacement functions. This aspect of the continuous system stands in stark contrast to a discrete system of AT degrees of freedom where the for-all statement means to prove it for N linearly independent vectors, a decidedly finite operation. It is the nonfinite aspect of the principle of virtual work that causes problems for practical computations.