Finite Element Method in Engineering: 3rd Edition (3rd Edition)

Author: Chandrasekharappa
Publisher:
Publish Date: 2005-04-01
Features: This book is divided into 12 chapters, with its fundamental objective being to provide a clear theoretical foundation, modeling methods, and specific computer implementation programs for the finite element method. Many chapters in the book expand on practical examples and exercises, with the added theoretical content and computer programs covering acoustics, axisymmetric quadrilateral elements, conjugate gradient algorithms, and eigenvalue problems. The book also includes 3 additional programs, all developed on the Windows platform and sharing the same programming structure. The accompanying CD-ROM provides the source code for all the computer programs. The book is divided into 12 chapters:
Chapter 1 briefly introduces the historical background and basic concepts of the finite element method, reviewing equilibrium equations, stress-strain relationships, strain-displacement relationships, and the principle of potential energy, while introducing the concept of the Galerkin method.
Chapter 2 covers the properties of matrices and determinants, introducing Gauss elimination, discussing the solution of symmetric banded matrix equations and the handling of banded matrix "skyline" methods, and also touching on Cholesky decomposition and the conjugate gradient method.
Chapter 3 introduces the fundamental concepts and expressions of the finite element method through the analysis of one-dimensional problems, covering the main steps of finite element analysis: shape function representation, derivation of the element stiffness matrix, formation of the global stiffness matrix, handling of boundary conditions, solution of equations, and stress calculation. It also provides expressions based on the potential energy method and the Galerkin method, as well as considerations for temperature effects.
Chapter 4 presents the finite element formulation for plane and three-dimensional truss problems, providing the assembly of the global stiffness matrix in both banded matrix and "skyline" matrix forms, along with computer programs for solving based on these forms.
Chapter 5 introduces the constant strain triangular (CST) element for solving two-dimensional plane stress and plane strain problems, detailing the modeling process and boundary condition handling, and also providing corresponding methods for orthotropic materials.
Chapter 6 covers the modeling process for axisymmetric bodies under axisymmetric external loads, presenting the corresponding triangular element expressions and offering solutions to several practical problems.
Chapter 7 introduces the basic concepts of isoparametric quadrilateral elements and higher-order elements, as well as the numerical method for area integration using the Gauss method, providing expressions for axisymmetric quadrilateral elements and solutions based on the conjugate gradient method.
Chapter 8 discusses beam elements and the application of Hermite shape functions, covering two-dimensional and three-dimensional frame structures.
Chapter 9 focuses on three-dimensional stress analysis, including tetrahedral and hexahedral elements, and introduces the wavefront method for solution and implementation.
Chapter 10 provides a detailed treatment of scalar field problems. In other chapters, the Galerkin method and energy principles are used as the fundamental basis for deriving the finite element method.
Chapter 11 covers dynamic problems, presenting expressions for the element mass matrix and discussing the solution of eigenvalues (natural frequencies) and eigenvectors (modal shapes) for general eigenvalue problems, including methods such as inverse iteration, Jacobi method, tridiagonalization, and explicit shifting.
Chapter 12 introduces the concepts of pre-processing and post-processing, detailing the principles and implementation methods for automatic mesh generation in two-dimensional problems, providing least-squares methods for calculating node stresses from element values for triangular and quadrilateral elements, and also covering contour line techniques in post-processing.
For undergraduate students, some of the more advanced content in the book can be omitted, or the material can be adopted as needed based on a new, comprehensive content framework. It is recommended and encouraged to begin using the programs in Chapter 12 after completing Chapter 5, as this will help readers efficiently prepare data for various finite element analyses.