Basic Equations and Solution Procedure

The finite element method is used in the field of solid and structural mechanics. Various types of problems solved by the finite element method in this field include the elastic, elastoplastic, and viscoelastic analysis of trusses, frames, plates, shells, and solid bodies. This chapter presents general equations of solid and structural mechanics and their derivation and solution. The primary aim of any stress analysis or solid mechanics problem is to find the distribution of displacements and stresses under the stated loading and boundary conditions. If an analytical solution of the problem is to be found, one has to satisfy the basic equations of solid mechanics. The number of unknown quantities is equal to the number of equations available. The finite element equations can also be derived by using either a differential equation formulation method (for example, Galerkin approach) or variational formulation method (for example, Rayleigh-Ritz approach). In the case of solid and structural mechanics problems, each of the differential equation and variational formulation methods is classified into three categories, which are displacement, force, and displacement-force method. The chapter describes the displacement method (or equivalently the principle of minimum potential energy) for deriving the finite element equations. Variational formulation methods are based on the principle of minimum potential energy, principle of minimum complementary energy, and the principle of stationary Reissner Energy.