The Variational and Energy Methods, and Some General Principles of Structural Analysis

Abstract

In the previous analyses, we have continually relied upon the differential equations with appropriate boundary conditions as the basic mathematical equipment to obtain solutions to various stress-strain problems. In many cases, however, these equations turn out to be so complicated that it becomes very difficult not only to solve them analytically but even to obtain satisfactory numerical solutions. One of the possible alternative ways to get around these difficulties is to proceed from the energy considerations, as we did, for instance, in Section 12.5, when a law of conservation of energy was used to evaluate the shear deformations in beam bending. Much more general and powerful methods for solving structural problems exist, however, than simply equating the external work to the strain energy of the system. These are variational methods, and it is these methods that will be addressed in this chapter. The variational methods do not use differential equations and have as their basis some of the most fundamental principles of mechanics. Although, in the majority of cases, they provide only approximate solutions, in effect, any desirable accuracy can be achieved.