A First Course in the Finite Element Method, 5th ed.

A FIRST COURSE IN THE FINITE ELEMENT METHOD provides a simple, basic approach to the course material that can be understood by both undergraduate and graduate students without the usual prerequisites (i.e. structural analysis). The book is written primarily as a basic learning tool for the undergraduate student in civil and mechanical engineering whose main interest is in stress analysis and heat transfer. The text is geared toward those who want to apply the finite element method as a tool to solve practical physical problems. Cover Page 1 Copyright Page 5 Title Page 6 Contents 7 Preface 15 Notation 19 English Symbols 19 Greek Symbols 21 Other Symbols 22 Chapter 1: Introduction 23 Chapter Objectives 23 Prologue 23 1.1: Brief History 24 1.2: Introduction to Matrix Notation 26 1.3: Role of the Computer 28 1.4: General Steps of the Finite Element Method 29 1.5: Applications of the Finite Element Method 37 1.6: Advantages of the Finite Element Method 45 1.7: Computer Programs for the Finite Element Method 47 References 49 Problems 51 Chapter 2: Introduction to the Stiffness (Displacement) Method 53 Chapter Objectives 53 Introduction 53 2.1: Definition of the Stiffness Matrix 54 2.2: Derivation of the Stiffness Matrix for a Spring Element 54 2.3: Example of a Spring Assemblage 60 2.4: Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method) 62 2.5: Boundary Conditions 64 2.6: Potential Energy Approach to Derive Spring Element Equations 78 Summary Equations 87 References 88 Problems 88 Chapter 3: Development of Truss Equations 94 Chapter Objectives 94 Introduction 94 3.1: Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates 95 3.2: Selecting Approximation Functions for Displacements 101 3.3: Transformation of Vectors in Two Dimensions 104 3.4: Global Stiffness Matrix for Bar Arbitrarily Oriented in the Plane 107 3.5: Computation of Stress for a Bar in the x-y Plane 112 3.6: Solution of a Plane Truss 114 3.7: Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space 122 3.8: Use of Symmetry in Structures 131 3.9: Inclined, or Skewed, Supports 134 3.10: Potential Energy Approach to Derive Bar Element Equations 140 3.11: Comparison of Finite Element Solution to Exact Solution for Bar 151 3.12: Galerkin’s Residual Method and Its Use to Derive the One-Dimensional Bar Element Equations 155 3.13: Other Residual Methods and Their Application to a One-Dimensional Bar Problem 158 3.14: Flowchart for Solution of Three-Dimensional Truss Problems 163 3.15: Computer Program Assisted Step-by-Step Solution for Truss Problem 163 Summary Equations 166 References 167 Problems 168 Chapter 4: Development of Beam Equations 188 Chapter Objectives 188 Introduction 188 4.1: Beam Stiffness 189 4.2: Example of Assemblage of Beam Stiffness Matrices 199 4.3: Examples of Beam Analysis Using the Direct Stiffness Method 201 4.4: Distributed Loading 214 4.5: Comparison of the Finite Element Solution to the Exact Solution for a Beam 227 4.6: Beam Element with Nodal Hinge 233 4.7: Potential Energy Approach to Derive Beam Element Equations 240 4.8: Galerkin’s Method for Deriving Beam Element Equations 243 Summary Equations 245 References 246 Problems 247 Chapter 5: Frame and Grid Equations 257 Chapter Objectives 257 Introduction 257 5.1: Two-Dimensional Arbitrarily Oriented Beam Element 257 5.2: Rigid Plane Frame Examples 261 5.3: Inclined or Skewed Supports—Frame Element 280 5.4: Grid Equations 281 5.5: Beam Element Arbitrarily Oriented in Space 299 5.6: Concept of Substructure Analysis 312 Summary Equations 318 References 320 Problems 321 Chapter 6: Development of the Plane Stress and Plane Strain Stiffness Equations 350 Chapter Objectives 350 Introduction 350 6.1: Basic Concepts of Plane Stress and Plane Strain 351 6.2: Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations 356 6.3: Treatment of Body and Surface Forces 371 6.4: Explicit Expression for the Constant-Strain Triangle Stiffness Matrix 376 6.5: Finite Element Solution of a Plane Stress Problem 378 6.6: Rectangular Plane Element (Bilinear Rectangle, Q4) 389 Summary Equations 395 References 398 Problems 399 Chapter 7: Practical Considerations in Modeling; Interpreting Results; and Examples of Plane Stress–Strain Analysis 406 Chapter Objectives 406 Introduction 406 7.1: Finite Element Modeling 407 7.2: Equilibrium and Compatibility of Finite Element Results 420 7.3: Convergence of Solution 424 7.4: Interpretation of Stresses 427 7.5: Static Condensation 429 7.6: Flowchart for the Solution of Plane Stress–Strain Problems 433 7.7: Computer Program-Assisted Step-by-Step Solution, Other Models, and Results for Plane Stress–Strain Problems 433 References 439 Problems 442 Chapter 8: Development of the Linear-Strain Triangle Equations 459 Chapter Objectives 459 Introduction 459 8.1: Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations 459 8.2: Example LST Stiffness Determination 464 8.3: Comparison of Elements 467 Summary Equations 470 References 470 Problems 471 Chapter 9: Axisymmetric Elements 474 Chapter Objectives 474 Introduction 474 9.1: Derivation of the Stiffness Matrix 474 9.2: Solution of an Axisymmetric Pressure Vessel 485 9.3: Applications of Axisymmetric Elements 491 Summary Equations 496 References 498 Problems 498 Chapter 10: Isoparametric Formulation 508 Chapter Objectives 508 Introduction 508 10.1: Isoparametric Formulation of the Bar Element Stiffness Matrix 509 10.2: Isoparametric Formulation of the Plane Quadrilateral Element Stiffness Matrix 514 10.3: Newton-Cotes and Gaussian Quadrature 525 10.4: Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature 531 10.5: Higher-Order Shape Functions 537 Summary Equations 547 References 550 Problems 550 Chapter 11: Three-Dimensional Stress Analysis 556 Chapter Objectives 556 Introduction 556 11.1: Three-Dimensional Stress and Strain 557 11.2: Tetrahedral Element 559 11.3: Isoparametric Formulation 567 Summary Equations 575 References 578 Problems 578 Chapter 12: Plate Bending Element 594 Chapter Objectives 594 Introduction 594 12.1: Basic Concepts of Plate Bending 594 12.2: Derivation of a Plate Bending Element Stiffness Matrix and Equations 599 12.3: Some Plate Element Numerical Comparisons 604 12.4: Computer Solutions for Plate Bending Problems 606 Summary Equations 610 References 612 Problems 612 Chapter 13: Heat Transfer and Mass Transport 621 Chapter Objectives 621 Introduction 621 13.1: Derivation of the Basic Differential Equation 623 13.2: Heat Transfer with Convection 626 13.3: Typical Units; Thermal Conductivities, K; and Heat-Transfer Coefficients, h 627 13.4: One-Dimensional Finite Element Formulation Using a Variational Method 629 13.5: Two-Dimensional Finite Element Formulation 648 13.6: Line or Point Sources 657 13.7: Three-Dimensional Heat Transfer by the Finite Element Method 660 13.8: One-Dimensional Heat Transfer with Mass Transport 663 13.9: Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin’s Method 663 13.10: Flowchart and Examples of a Heat-Transfer Program 668 Summary Equations 672 References 675 Problems 676 Chapter 14: Fluid Flow in Porous Media and Through Hydraulic Networks; and Electrical Networks and Electrostatics 696 Chapter Objectives 696 Introduction 696 14.1: Derivation of the Basic Differential Equations 697 14.2: One-Dimensional Finite Element Formulation 702 14.3: Two-Dimensional Finite Element Formulation 714 14.4: Flowchart and Example of a Fluid-Flow Program 719 14.5: Electrical Networks 720 14.6: Electrostatics 724 Summary Equations 738 References 742 Problems 742 Chapter 15: Thermal Stress 750 Chapter Objectives 750 Introduction 750 15.1: Formulation of the Thermal Stress Problem and Examples 750 Reference 774 Summary Equations 775 Problems 777 Chapter 16: Structural Dynamics and Time-Dependent Heat Transfer 785 Chapter Objectives 785 Introduction 785 16.1: Dynamics of a Spring-Mass System 786 16.2: Direct Derivation of the Bar Element Equations 788 16.3: Numerical Integration in Time 792 16.4: Natural Frequencies of a One-Dimensional Bar 804 16.5: Time-Dependent One-Dimensional Bar Analysis 808 16.6: Beam Element Mass Matrices and Natural Frequencies 813 16.7: Truss, Plane Frame, Plane Stress, Plane Strain, Axisymmetric, and Solid Element Mass Matrices 820 16.8: Time-Dependent Heat Transfer 825 16.9: Computer Program Example Solutions for Structural Dynamics 832 Summary Equations 841 References 845 Problems 846 Appendix A: Matrix Algebra 851 Introduction 851 A.1: Definition of a Matrix 851 A.2: Matrix Operations 852 A.3: Cofactor or Adjoint Method to Determine the Inverse of a Matrix 859 A.4: Inverse of a Matrix by Row Reduction 861 A.5: Properties of Stiffness Matrices 863 References 864 Problems 864 Appendix B: Methods for Solution of Simultaneous Linear Equations 867 Introduction 867 B.1: General Form of the Equations 867 B.2: Uniqueness, Nonuniqueness, and Nonexistence of Solution 868 B.3: Methods for Solving Linear Algebraic Equations 869 B.4: Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methods 880 References 887 Problems 887 Appendix C: Equations from Elasticity Theory 889 Introduction 889 C.1: Differential Equations of Equilibrium 889 C.2: Strain/Displacement and Compatibility Equations 891 C.3: Stress-Strain Relationships 893 Reference 896 Appendix D: Equivalent Nodal Forces 897 Problems 897 Appendix E: Principle of Virtual Work 900 References 903 Appendix F: Properties of Structural Steel Shapes 904 Answers to Selected Problems 930 Chapter 2 930 Chapter 3 932 Chapter 4 937 Chapter 5 939 Chapter 6 943 Chapter 7 946 Chapter 8 947 Chapter 9 947 Chapter 10 949 Chapter 11 949 Chapter 12 951 Chapter 13 951 Chapter 14 952 Chapter 15 953 Chapter 16 954 Appendix A 957 Appendix B 958 Appendix D 958 Index 959 A 959 B 959 C 960 D 962 E 963 F 963 G 965 H 965 I 966 J 966 K 966 L 966 M 967 N 968 O 968 P 969 Q 970 R 970 S 970 T 974 U 975 V 975 W 976 X 976 Z 976 This third edition provides a simple, basic approach to the finite element method that can be understood by both undergraduate and graduate students. It does not have the usual prerequisites (such as structural analysis) required by most available texts in this area. The book is written primarily as a basic learning tool for the undergraduate student in civil and mechanical engineering whose main interest is in stress analysis and heat transfer. The text is geared toward those who want to apply the finite element method as a tool to solve practical physical problems.