Chapter 8 - Basic Equations and Solution Procedure

Abstract

The basic equations of solid and structural mechanics, the methods of formulating solid and structural mechanics problems, the formulation of finite element equations for static problems, and the nature of the resulting finite equations are discussed. The equilibrium equations of an infinitesimal element, Hooke's law, and strain-displacement relations of a solid body are derived. The number of available equations and the number of unknowns are given for three-, two-, and one-dimensional problems. Additional equations that need to be satisfied in order to find the correct or exact solution of the problem are also presented; these include the equilibrium equations of the overall body, boundary conditions of the body, and compatibility equations. The methods of formulating solid and structural mechanics problems using variational approaches, including the principle of minimum potential energy, principle of minimum complementary energy and principle of stationary Reissner energy for static problems, and Hamilton's principle for dynamic problems, are indicated. The general formulation of finite element equations for static analysis of a solid mechanics problem, in matrix form, is outlined based on the principle of minimum potential energy.