The Finite Element Method in Engineering

Transcription

1 The Finite Element Method in Engineering Fifth Edition Singiresu S. Rao Professor and Chairman Department of Mechanical and Aerospace Engineering University of Miami, Coral Gables, Florida, USA Щ/ФМШ ELSEVIER AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO Butterworth-Heinemann is an imprint of Elsevier

2 PREFACE PART 1 CHAPTER 1 xiii Introduction Overview of Finite Element Method Historical Background General Applicability of the Method Engineering Applications of the Finite Element Method General Description of the Finite Element Method One-Dimensional Problems with Linear Interpolation Model One-Dimensional Problems with Cubic Interpolation Model Derivation of Finite Element Equations Using a Direct Approach Commercial Finite Element Program Packages Solutions Using Finite Element Software PART 2 CHAPTER 2 CHAPTER 3 CHAPTER Basic Concept 40 Basic Procedure Discretization of the Domain Introduction Basic Element Shapes Discretization Process Node Numbering Scheme Automatic Mesh Generation 65 Interpolation Models Introduction Polynomial Form of Interpolation Functions Simplex, Complex, and Multiplex Elements Interpolation Polynomial in Terms of Nodal Degrees of Freedom Selection of the Order of the Interpolation Polynomial Convergence Requirements Linear Interpolation Polynomials in Terms of Global Coordinates Interpolation Polynomials for Vector Quantities Linear Interpolation Polynomials in Terms of Local Coordinates Integration of Functions of Natural Coordinates Patch Test 109 Higher Order and Isoparametric Elements Introduction Higher Order One-Dimensional Elements Higher Order Elements in Terms of Natural Coordinates Higher Order Elements in Terms of Classical Interpolation Polynomials 130

3 .. :-'; ' ''; '. >-у/а,'?''..; 'УУ,'fyux'S'- : ". :' '. : '-'"ТЙ!.-. -' ^улукругууу'у;-,уу -у/у f 'У-у у У 'ХУК'УУуЯу CONTENTS 4.5 One-Dimensional Elements Using Classical Interpolation Polynomials Two-Dimensional (Rectangular) Elements Using Classical Interpolation Polynomials Continuity Conditions Comparative Study of Elements Isoparametric Elements Numerical Integration 148 CHAPTER 5 Derivation of Element Matrices and Vectors Introduction Variational Approach Solution of Equilibrium Problems Using Variational (Rayleigh-Ritz) Method Solution of Eigenvalue Problems Using Variational (Rayleigh-Ritz) Method Solution of Propagation Problems Using Variational (Rayleigh-Ritz) Method Equivalence of Finite Element and Variational (Rayleigh-Ritz) Methods Derivation of Finite Element Equations Using Variational (Rayleigh-Ritz) Approach Weighted Residual Approach Solution of Eigenvalue Problems Using Weighted Residual Method Solution of Propagation Problems Using Weighted Residual Method Derivation of Finite Element Equations Using Weighted Residual (Galerkin) Approach Derivation of Finite Element Equations Using Weighted Residual (Least Squares) Approach Strong and Weak Form Formulations 189 CHAPTER 6 Assembly of Element Matrices and Vectors and Derivation of System Equations Coordinate Transformation Assemblage of Element Equations Incorporation of Boundary Conditions Penalty Method Multipoint Constraints Penalty Method Symmetry Conditions Penalty Method Rigid Elements 228 CHAPTER 7 Numerical Solution of Finite Element Equations Introduction Solution of Equilibrium Problems Solution of Eigenvalue Problems Solution of Propagation Problems Parallel Processing in Finite Element Analysis 268 PART 3 Application to Solid Mechanics Problems CHAPTER 8 Basic Equations and Solution Procedure Introduction Basic Equations of Solid Mechanics 277

4 8.3 Formulations of Solid and Structural Mechanics Formulation of Finite Element Equations (Static Analysis) Nature of Finite Element Solutions 303 CHAPTER 9 Analysis of Trusses, Beams, and Frames Introduction Space Truss Element Beam Element Space Frame Element Characteristics of Stiffness Matrices 338 CHAPTER 10 Analysis of Plates Introduction Triangular Membrane Element Numerical Results with Membrane Element Quadratic Triangle Element Rectangular Plate Element (In-plane Forces) Bending Behavior of Plates Finite Element Analysis of Plates in Bending Triangular Plate Bending Element Numerical Results with Bending Elements Analysis of Three-Dimensional Structures Using Plate Elements CHAPTER 11 Analysis of Three-Dimensional Problems Introduction Tetrahedron Element Hexahedron Element Analysis of Solids of Revolution 413 CHAPTER 12 Dynamic Analysis Dynamic Equations of Motion Consistent and Lumped Mass Matrices Consistent Mass Matrices in a Global Coordinate System Free Vibration Analysis Dynamic Response Using Finite Element Method Nonconservative Stability and Flutter Problems Substructures Method 461 PART 4 Application to Heat Transfer Problems CHAPTER 13 Formulation and Solution Procedure Introduction Basic Equations of Heat Transfer Governing Equation for Three-Dimensional Bodies Statement of the Problem Derivation of Finite Element Equations 480 CHAPTER 14 One-Dimensional Problems Introduction Straight Uniform Fin Analysis Convection Loss from End Surface of Fin Tapered Fin Analysis Analysis of Uniform Fins Using Quadratic Elements 499

5 14.5 Unsteady State Problems Heat Transfer Problems with Radiation 507 CHAPTER 15 Two-Dimensional Problems Introduction Solution Unsteady State Problems 526 CHAPTER 16 Three-Dimensional Problems Introduction Axisymmetric Problems Three-Dimensional Heat Transfer Problems Unsteady State Problems 541 PART 5 Application to Fluid Mechanics Problems CHAPTER 17 Basic Equations of Fluid Mechanics Introduction Basic Characteristics of Fluids Methods of Describing the Motion of a Fluid Continuity Equation Equations of Motion or Momentum Equations Energy, State, and Viscosity Equations Solution Procedure Inviscid Fluid Flow Irrotational Flow Velocity Potential Stream Function Bernoulli Equation 564 CHAPTER 18 Inviscid and Incompressible Flows Introduction Potential Function Formulation Finite Element Solution Using the Galerkin Approach Stream Function Formulation 584 CHAPTER 19 Viscous and Non-Newtonian Flows Introduction Stream Function Formulation (Using Variational Approach) Velocity-Pressure Formulation (Using Galerkin Approach) Solution of Navier-Stokes Equations Stream Function-Vorticity Formulation Flow of Non-Newtonian Fluids Other Developments 607 PART 6 Solution and Applications of Quasi-Harmonic Equations CHAPTER 20 Solution of Quasi-Harmonic Equations Introduction Finite Element Equations for Steady-State Problems Solution of Poisson's Equation Transient Field Problems 622

6 PART 7 ABAQUS and ANSYS Software and MATLAB Programs for Finite Element Analysis CHAPTER 21 Finite Element Analysis Using ABAQUS Introduction Examples 632 CHAPTER 22 Finite Element Analysis Using ANSYS Introduction GUI Layout in ANSYS Terminology Finite Element Discretization System of Units Stages in Solution 667 CHAPTER 23 MATLAB Programs for Finite Element Analysis Solution of Linear System of Equations Using Choleski Method Incorporation of Boundary Conditions Analysis of Space Trusses Analysis of Plates Subjected to In-plane Loads Using CST Elements Analysis of Three-Dimensional Structures Using CST Elements Temperature Distribution in One-Dimensional Fins Temperature Distribution in One-Dimensional Fins Including Radiation Heat Transfer Two-Dimensional Heat Transfer Analysis Confined Fluid Flow around a Cylinder Using Potential Function Approach Torsion Analysis of Shafts 702 Appendix: Green-Gauss Theorem (Integration by Parts in Two and Three Dimensions) 705 Index 707