Variational Methods

Abstract

Energy or variational methods have an important place in Solid Mechanics both as an alternative to the more direct method of solving the governing partial differential equations and as a means of developing convergent approximations to analytically intractable problems. They are particularly useful in situations where only a restricted set of results is required — for example, if we wish to determine the resultant force on a cross-section or the displacement of a particular point, but are not interested in the full stress and displacement fields. Indeed, such results can often be obtained in closed form for problems in which a solution for the complete fields would be intractable.

From an engineering perspective, it is natural to think of these methods as a consequence of the principle of conservation of energy or the first law of thermodynamics. However, conservation of energy is in some sense guaranteed by the use of Hooke’s law and the equilibrium equations. Once these physical premises are accepted, the energy theorems we shall present here are purely mathematical consequences. Indeed, the finite element method, which is one of the more important developments of this kind, can be developed simply by applying arguments from approximation theory (such as a least-squares fit) to the governing equations introduced in previous chapters. For this reason, these techniques are now more often referred to as Variational Methods, meaning that instead of seeking to solve the governing partial differential equations directly, we seek to define a scalar function of the physical parameters which is stationary (generally maximum or minimum) in respect to infinitesimal variations about the solution.