tailieunhanh - Ebook The finite element method in engineering (4th edition): Part 2

(BQ) Part 2 book "The finite element method in engineering" has contents: Basic equations and solution procedure; basic equations and solution procedure, analysis of plates, analysis of three dimensional problems, dynamic analysis, formulation and solution procedure,.and other contents. | APPLICATION TO SOLID MECHANICS PROBLEMS This Page Intentionally Left Blank 8 BASIC EQUATIONS AND SOLUTION PROCEDURE INTRODUCTION As stated in Chapter 1, the finite element method has been most extensively used in the field of solid and structural mechanics. The 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. Both static and dynamic analysis have been conducted using the finite element method. We consider the finite element elastic analysis of one-, two-, and three-dimensional problems as well as axisymmetric problems in this book. In this chapter, the general equations of solid and structural mechanics are presented. The displacement method (or equivalently the principle of minimum potential energy) is used in deriving the finite element equations. The application of these equations to several specific cases is considered in subsequent chapters. BASIC EQUATIONS OF SOLID MECHANICS Introduction 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 following basic equations of solid mechanics: Number of equations Type of equations Equilibrium equations Stress–strain relations Strain–displacement relations Total number of equations In 3-dimensional problems In 2-dimensional problems In 1-dimensional problems 3 6 6 2 3 3 1 1 1 15 8 3 279 280 BASIC EQUATIONS AND SOLUTION PROCEDURE The unknown quantities, whose number is equal to the number of equations available, in various problems are given below: In 3-dimensional problems Unknowns Displacements Stresses u, v, w σxx , σyy , σzz , σxy , σyz , σzx εxx , εyy , εzz , εxy εyz , εzx Strains Total number of unknowns 15 In 2-dimensional problems In .