A First Course in the Finite Element Method (MindTap Course List)

Provide a simple, direct approach that highlights the basics with A FIRST COURSE IN THE FINITE ELEMENT METHOD, 6E. This unique book is written so both undergraduate and graduate students can easily comprehend the content without the usual prerequisites, such as structural analysis. The book is written primarily as a basic learning tool for the undergraduate students in civil and mechanical engineering who are primarily interested in stress analysis and heat transfer. The text offers ideal preparation for students who want to apply the finite element method as a tool to solve practical physical problems. Cover 1 Title 10 Statement 11 Copyright 12 Contents 13 Preface 21 Acknowledgments 24 Notation 25 Ch 1: Introduction 29 Ch 1: Chapter Objectives 29 Prologue 29 1.1: Brief History 31 1.2: Introduction to Matrix Notation 32 1.3: Role of the Computer 34 1.4: General Steps of the Finite Element Method 35 1.5: Applications of the Finite Element Method 43 1.6: Advantages of the Finite Element Method 49 1.7: Computer Programs for the Finite Element Method 53 Ch 1: Reference 55 Ch 1: Problems 58 Ch 2: Introduction to the Stiffness (Displacement) Method 59 Ch 2: Chapter Objectives 59 Ch 2: Introduction 59 2.1: Definition of the Stiffness Matrix 60 2.2: Derivation of the Stiffness Matrix for a Spring Element 60 2.3: Example of a Spring Assemblage 64 2.4: Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method) 66 2.5: Boundary Conditions 68 2.6: Potential Energy Approach to Derive Spring Element Equations 83 Ch 2: Summary Equations 93 Ch 2: Problems 94 Ch 3: Development of Truss Equations 100 Ch 3: Chapter Objectives 100 Ch 3: Introduction 100 3.1: Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates 101 3.2: Selecting a Displacement Function in Step 2 of the Derivation of Stiffness Matrix for the One-Dimensional Bar Element 106 3.3: Transformation of Vectors in Two Dimensions 110 3.4: Global Stiffness Matrix for Bar Arbitrarily Oriented in the Plane 112 3.5: Computation of Stress for a Bar in the x – y Plane 117 3.6: Solution of a Plane Truss 119 3.7: Transformation Matrix and Stiffness Matrix for a Barin Three-Dimensional Space 128 3.8: Use of Symmetry in Structures 137 3.9: Inclined, or Skewed, Supports 140 3.10: Potential Energy Approach to Derive Bar Element Equations 149 3.11: Comparison of Finite Element Solution to Exact Solution for Bar 160 3.12: Galerkin’s Residual Method and Its Use to Derive the One-Dimensional Bar Element Equations 164 3.13: Other Residual Methods and Their Application to a One-Dimensional Bar Problem 167 3.14: Flowchart for Solution of Three-Dimensional Truss Problems 171 3.15: Computer Program Assisted Step-by-Step Solution for Truss Problem 172 Ch 3: Sumary Equations 174 Ch 3: Problems 175 Ch 4: Development of Beam Equations 197 Ch 4: Chapter Objectives 197 Ch 4: Introduction 197 4.1: Beam Stiffness 198 4.2: Example of Assemblage of Beam Stiffness Matrices 208 4.3: Examples of Beam Analysis Using the Direct Stiffness Method 210 4.4: Distributed Loading 223 4.5: Comparison of the Finite Element Solution to the Exact Solution for a Beam 236 4.6: Beam Element with Nodal Hinge 242 4.7: Potential Energy Approach to Derive Beam Element Equations 250 4.8: Galerkin’s Method for Deriving Beam Element Equations 253 Ch 4: Sumary Equations 255 Ch 4: Problems 257 Ch 5: Frame and Grid Equations 267 Ch 5: Chapter Objectives 267 Ch 5: Introduction 267 Ch 5: Two-Dimensional Arbitrarily Oriented Beam Element 267 5.2: Rigid Plane Frame Examples 271 5.3: Inclined or Skewed Supports—Frame Element 289 5.4: Grid Equations 290 5.5: Beam Element Arbitrarily Oriented in Space 308 5.6: Concept of Substructure Analysis 323 Ch 5: Summary Equations 328 Ch 5: Problems 331 Ch 6: Development of the Plane Stress and Plane Strain Stiffness Equations 365 Ch 6: Chapter Objectives 365 Ch 6: Introduction 365 6.1: Basic Concepts of Plane Stress and Plane Strain 366 6.2: Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations 370 6.3: Treatment of Body and Surface Forces 385 6.4: Explicit Expression for the Constant-Strain Triangle Stiffness Matrix 390 6.5: Finite Element Solution of a Plane Stress Problem 391 6.6: Rectangular Plane Element (Bilinear Rectangle, Q4) 402 Ch 6: Sumary Equations 407 Ch 6: Problems 412 Ch 7: Practical Considerations in Modeling; Interpreting Results; and Examples of Plane Stress/Strain Analysis 419 Ch 7: Chapter Objectives 419 Ch 7: Introduction 419 7.1: Finite Element Modeling 420 7.2: Equilibrium and Compatibility of Finite Element Results 433 7.3: Convergence of Solution and Mesh Refinement 436 7.4: Interpretation of Stresses 439 7.5: Flowchart for the Solution of Plane Stress/Strain Problems 441 7.6: Computer Program–Assisted Step-by-Step Solution, Other Models, and Results for Plane Stress/Strain Problems 442 Ch 7: Problems 449 Ch 8: Development of the Linear-Strain Triangle Equations 465 Ch 8: Chapter Objectives 465 Ch 8: Introduction 465 8.1: Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations 465 8.2: Example LST Stiffness Determination 470 8.3: Comparison of Elements 472 Ch 8: Summary Equations 475 Ch 8: Problems 476 Ch 9: Axisymmetric Elements 479 Ch 9: Chapter Objectives 479 Ch 9: Introduction 479 9.1: Derivation of the Stiffness Matrix 479 9.2: Solution of an Axisymmetric Pressure Vessel 490 9.3: Applications of Axisymmetric Elements 496 Ch 9: Summary Equations 501 Ch 9: Problems 503 Ch 10: Isoparametric Formulation 514 Ch 10: Chapter Objectives 514 Ch 10: Introduction 514 10.1: Isoparametric Formulation of the Bar Element Stiffness Matrix 515 10.2: Isoparametric Formulation of the Plane Quadrilateral (Q4) Element Stiffness Matrix 520 10.3: Newton-Cotes and Gaussian Quadrature 531 10.4: Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature 537 10.5: Higher-Order Shape Functions (Including Q6, Q8, Q9, and Q12 Elements) 543 Ch 10: Summary Equations 554 Ch 10: Problems 558 Ch 11: Three-Dimensional Stress Analysis 564 Ch 11: Chapter Objectives 564 Ch 11: Introduction 564 11.1: Three-Dimensional Stress and Strain 565 11.2: Tetrahedral Element 567 11.3: Isoparametric Formulation and Hexahedral Element 575 Ch 11: Summary Equations 583 Ch 11: Problems 586 Ch 12: Plate Bending Element 600 Ch 12: Chapter Objectives 600 Ch 12: Introduction 600 12.1: Basic Concepts of Plate Bending 600 12.2: Derivation of a Plate Bending Element Stiffness Matrix and Equations 605 12.3: Some Plate Element Numerical Comparisons 610 12.4: Computer Solutions for Plate Bending Problems 612 Ch 12: Summary Equations 616 Ch 12: Problems 619 Ch 13: Heat Transfer and Mass Transport 627 Ch 13: Chapter Objectives 627 Ch 13: Introduction 627 13.1: Derivation of the Basic Differential Equation 629 13.2: Heat Transfer with Convection 632 13.3: Typical Units; Thermal Conductivities, K; and Heat Transfer Coefficients, h 633 13.4: One-Dimensional Finite Element Formulation Using a Variational Method 635 13.5: Two-Dimensional Finite Element Formulation 654 13.6: Line or Point Sources 664 13.7: Three-Dimensional Heat Transfer by the Finite Element Method 667 13.8: One-Dimensional Heat Transfer with Mass Transport 669 13.9: Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin’s Method 670 13.10: Flowchart and Examples of a Heat Transfer Program 674 Ch 10: Sumary Equations 679 Ch 10: Problems 683 Ch 14: Fluid Flow in Porous Media and through Hydraulic Networks; and Electrical Networks and Electrostatics 701 Ch 14: Chapter Objectives 701 Ch 14: Introduction 701 14.1: Derivation of the Basic Differential Equations 702 14.2: One-Dimensional Finite Element Formulation 706 14.3: Two-Dimensional Finite Element Formulation 719 14.4: Flowchart and Example of a Fluid-Flow Program 724 14.5: Electrical Networks 725 14.6: Electrostatics 729 Ch 14: Sumary Equations 743 Ch 14: Problems 748 Ch 15: Thermal Stress 755 Ch 15: Chapter Objectives 755 Ch 15: Introduction 755 15.1: Formulation of the Thermal Stress Problem and Examples 755 Ch 15: Summary Equations 780 Ch 15: Problems 782 Ch 16: Structural Dynamics and Time-Dependent Heat Transfer 789 Ch 16: Chapter Objectives 789 Ch 16: Introduction 789 16.1: Dynamics of a Spring-Mass System 790 16.2: Direct Derivation of the Bar Element Equations 792 16.3: Numerical Integration in Time 796 16.4: Natural Frequencies of a One-Dimensional Bar 808 16.5: Time-Dependent One-Dimensional Bar Analysis 812 16.6: Beam Element Mass Matrices and Natural Frequencies 817 16.7: Truss, Plane Frame, Plane Stress, Plane Strain, Axisymmetric, and Solid Element Mass Matrices 824 16.8: Time-Dependent Heat Transfer 829 16.9: Computer Program Example Solutions for Structural Dynamics 836 Ch 16: Sumary Equations 845 Ch 16: Problems 850 Appendix A: Matrix Algebra 855 Appendix B: Methods for Solution of Simultaneous Linear Equations 871 Appendix C: Equations from Elasticity Theory 893 Appendix D: Equivalent Nodal Forces 901 Appendix F: Properties of Structural Steel Shapes 908 Answers to Selected Problems 934 Index 966 WCN: 02-200-203 1. Introduction 2. Introduction to the Stiffness (Displacement) Method 3. Development of Truss Equations 4. Development of Beam Equations 5. Frame and Grid Equations 6. Development of the Plane Stress and Plane Strain Stiffness Equations 7. Practical Considerations in Modeling; Interpreting Results; and Examples of Plane Stress/Strain Analysis 8. Development of the Linear-Strain Triangle Equations 9. Axisymmetric Elements 10. Isoparametric Formulation 11. Three-Dimensional Stress Analysis 12. Plate Bending Element 13. Heat Transfer and Mass Transport 14. Fluid Flow in Porous Media and through Hydraulic Networks; and Electrical Networks and Electrostatics 15. Thermal Stress 16. Structural Dynamics and Time-Dependent Heat Transfer A. Matrix Algebra B. Methods for Solution of Simultaneous Linear Equations C. Equations from Elasticity Theory D. Equivalent Nodal Forces E. Principle of Virtual Work F. Properties of Structural Steel Shapes Discover a simple, direct approach that highlights the basics you need within A FIRST COURSE IN THE FINITE ELEMENT METHOD, 6E. This unique book is written so both undergraduate and graduate students can easily comprehend the content without the usual prerequisites, such as structural analysis. The book is written primarily as a basic learning tool for students, like you, in civil and mechanical engineering who are primarily interested in stress analysis and heat transfer. The text offers ideal preparation for utilizing the finite element method as a tool to solve practical physical problems