Horny Appetites

Gain a clear understanding of the basics of the finite element method (FEM) with this simple, direct, contemporary approach in Logan's A FIRST COURSE IN THE FINITE ELEMENT METHOD, Enhanced 6th Edition, SI Version. This unique presentation is written so you can easily comprehend content without the usual prerequisites, such as structural analysis. This book is ideal, whether you are a civil or mechanical engineering student primarily interested in stress analysis and heat transfer, or you need a foundation for applying FEM as a tool in solving practical physical problems. New and expanded real-world examples and problems demonstrate FEM applications in a variety of engineering and mathematical physics-related fields. Each chapter uses a consistent structure with step-by-step, worked-out examples, ideal for undergraduate or graduate-level study. A new WebAssign digital platform provides additional online resources to clarify concepts and assist you in completing assignments. Cover 1 Contents 9 Preface to the SI Edition 15 Preface 16 Digital Resources 18 Notation 22 Chapter 1: Introduction 25 Chapter Objectives 25 Prologue 25 1.1 Brief History 27 1.2 Introduction to Matrix Notation 28 1.3 Role of the Computer 30 1.4 General Steps of the Finite Element Method 31 1.5 Applications of the Finite Element Method 39 1.6 Advantages of the Finite Element Method 45 1.7 Computer Programs for the Finite Element Method 49 References 51 Problems 54 Chapter 2: Introduction to the Stiffness (Displacement) Method 55 Chapter Objectives 55 Introduction 55 2.1 Definition of the Stiffness Matrix 56 2.2 Derivation of the Stiffness Matrix for a Spring Element 56 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 79 Summary Equations 89 References 90 Problems 90 Chapter 3: Development of Truss Equations 96 Chapter Objectives 96 Introduction 96 3.1 Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates 97 3.2 Selecting a Displacement Function in Step 2 of the Derivation of Stiffness Matrix for the One-Dimensional Bar Element 102 3.3 Transformation of Vectors in Two Dimensions 106 3.4 Global Stiffness Matrix for Bar Arbitrarily Oriented in the Plane 108 3.5 Computation of Stress for a Bar in the x - y Plane 113 3.6 Solution of a Plane Truss 115 3.7 Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space 124 3.8 Use of Symmetry in Structures 133 3.9 Inclined, or Skewed, Supports 136 3.10 Potential Energy Approach to Derive Bar Element Equations 145 3.11 Comparison of Finite Element Solution to Exact Solution for Bar 156 3.12 Galerkin's Residual Method and Its Use to Derive the One-Dimensional Bar Element Equations 160 3.13 Other Residual Methods and Their Application to a One-Dimensional Bar Problem 163 3.14 Flowchart for Solution of Three-Dimensional Truss Problems 167 3.15 Computer Program Assisted Step-by-Step Solution for Truss Problem 168 Summary Equations 170 References 171 Problems 171 Chapter 4: Development of Beam Equations 193 Chapter Objectives 193 Introduction 193 4.1 Beam Stiffness 194 4.2 Example of Assemblage of Beam Stiffness Matrices 204 4.3 Examples of Beam Analysis Using the Direct Stiffness Method 206 4.4 Distributed Loading 219 4.5 Comparison of the Finite Element Solution to the Exact Solution for a Beam 232 4.6 Beam Element with Nodal Hinge 238 4.7 Potential Energy Approach to Derive Beam Element Equations 246 4.8 Galerkin's Method for Deriving Beam Element Equations 249 Summary Equations 251 References 252 Problems 253 Chapter 5: Frame and Grid Equations 263 Chapter Objectives 263 Introduction 263 5.1 Two-Dimensional Arbitrarily Oriented Beam Element 263 5.2 Rigid Plane Frame Examples 267 5.3 Inclined or Skewed Supports - Frame Element 285 5.4 Grid Equations 286 5.5 Beam Element Arbitrarily Oriented in Space 304 5.6 Concept of Substructure Analysis 319 Summary Equations 324 References 326 Problems 327 Chapter 6: Development of the Plane Stress and Plane Strain Stiffness Equations 361 Chapter Objectives 361 Introduction 361 6.1 Basic Concepts of Plane Stress and Plane Strain 362 6.2 Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations 366 6.3 Treatment of Body and Surface Forces 381 6.4 Explicit Expression for the Constant-Strain Triangle Stiffness Matrix 386 6.5 Finite Element Solution of a Plane Stress Problem 387 6.6 Rectangular Plane Element (Bilinear Rectangle, Q4) 398 Summary Equations 403 References 408 Problems 408 Chapter 7: Practical Considerations in Modeling; Interpreting Results; and Examples of Plane Stress/Strain Analysis 415 Chapter Objectives 415 Introduction 415 7.1 Finite Element Modeling 416 7.2 Equilibrium and Compatibility of Finite Element Results 429 7.3 Convergence of Solution and Mesh Refinement 432 7.4 Interpretation of Stresses 435 7.5 Flowchart for the Solution of Plane Stress/Strain Problems 437 7.6 Computer Program-Assisted Step-by-Step Solution, Other Models, and Results for Plane Stress/Strain Problems 438 References 444 Problems 445 Chapter 8: Development of the Linear-Strain Triangle Equations 461 Chapter Objectives 461 Introduction 461 8.1 Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations 461 8.2 Example LST Stiffness Determination 466 8.3 Comparison of Elements 468 Summary Equations 471 References 472 Problems 472 Chapter 9: Axisymmetric Elements 475 Chapter Objectives 475 Introduction 475 9.1 Derivation of the Stiffness Matrix 475 9.2 Solution of an Axisymmetric Pressure Vessel 486 9.3 Applications of Axisymmetric Elements 492 Summary Equations 497 References 499 Problems 499 Chapter 10: Isoparametric Formulation 510 Chapter Objectives 510 Introduction 510 10.1 Isoparametric Formulation of the Bar Element Stiffness Matrix 511 10.2 Isoparametric Formulation of the Plane Quadrilateral (Q4) Element Stiffness Matrix 516 10.3 Newton-Cotes and Gaussian Quadrature 527 10.4 Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature 533 10.5 Higher-Order Shape Functions (Including Q6, Q8, Q9, and Q12 Elements) 539 Summary Equations 550 References 554 Problems 554 Chapter 11: Three-Dimensional Stress Analysis 560 Chapter Objectives 560 Introduction 560 11.1 Three-Dimensional Stress and Strain 561 11.2 Tetrahedral Element 563 11.3 Isoparametric Formulation and Hexahedral Element 571 Summary Equations 579 References 582 Problems 582 Chapter 12: Plate Bending Element 596 Chapter Objectives 596 Introduction 596 12.1 Basic Concepts of Plate Bending 596 12.2 Derivation of a Plate Bending Element Stiffness Matrix and Equations 601 12.3 Some Plate Element Numerical Comparisons 606 12.4 Computer Solutions for Plate Bending Problems 608 Summary Equations 612 References 614 Problems 615 Chapter 13: Heat Transfer and Mass Transport 623 Chapter Objectives 623 Introduction 623 13.1 Derivation of the Basic Differential Equation 625 13.2 Heat Transfer with Convection 628 13.3 Typical Units; Thermal Conductivities, K; and Heat Transfer Coefficients, h 629 13.4 One-Dimensional Finite Element Formulation Using a Variational Method 631 13.5 Two-Dimensional Finite Element Formulation 650 13.6 Line or Point Sources 660 13.7 Three-Dimensional Heat Transfer by the Finite Element Method 663 13.8 One-Dimensional Heat Transfer with Mass Transport 665 13.9 Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin's Method 666 13.10 Flowchart and Examples of a Heat Transfer Program 670 Summary Equations 675 References 678 Problems 679 Chapter 14: Fluid Flow in Porous Media and through Hydraulic Networks; and Electrical Networks and Electrostatics 697 Chapter Objectives 697 Introduction 697 14.1 Derivation of the Basic Differential Equations 698 14.2 One-Dimensional Finite Element Formulation 702 14.3 Two-Dimensional Finite Element Formulation 715 14.4 Flowchart and Example of a Fluid-Flow Program 720 14.5 Electrical Networks 721 14.6 Electrostatics 725 Summary Equations 739 References 743 Problems 744 Chapter 15: Thermal Stress 751 Chapter Objectives 751 Introduction 751 15.1 Formulation of the Thermal Stress Problem and Examples 751 Summary Equations 776 Reference 777 Problems 778 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 Appendix B: Methods for Solution of Simultaneous Linear Equations 867 Appendix C: Equations from Elasticity Theory 889 Appendix D: Equivalent Nodal Forces 897 Appendix E: Principle of Virtual Work 900 Appendix F: Geometric Properties of Structural Steel Wide-Flange Sections (W Shapes) 904 Answers to Selected Problems 932 Index 962 WCN:,02-300 WCN: 02-300