ABSTRACT
It has been emphasized in this treatise that most problems in the mechanics of solids, when approached from an analytical viewpoint, can be represented locally by a system of differential equations or globally by an equivalent variational functional. Often the variational functional is formed first and used to establish the governing equations. In any event, once the mathematical (analytical) model has been established, solutions to particular problems are sought. Such solutions, which must satisfy the governing equations including the boundary conditions, can be derived from the functional or from the differential equations. As is immediately evident, contemporary mathematical methods can provide the exact solution to only the simplest forms of the governing equations. Hence, advantage is taken of the availability of the powerful digital computer by recasting problems in algebraic form. In so doing, the mathematical continuum model, which requires an infinite number of degrees of freedom (DOF) for its description, is replaced by a discrete model, which utilizes a finite number of DOF. This transformation usually involves some sort of approximation.
