# A First Course in the Finite Element Method Fourth Edition # Daryl L. Logan University of Wisconsin–Platteville THOMSON  # THOMSON  # A First Course in the Finite Element Method, Fourth Edition by Daryl L. Logan Associate Vice-President and Editorial Director: Evelyn Veitch Publisher: Chris Carson Developmental Editors: Kamilah Reid Burrell/ Hilda Gowans Permissions Coordinator: Vicki Gould Production Services: RPK Editorial Services Copy Editor: Harlan James Proofreader: Erin Wagner Indexer: RPK Editorial Services Production Manager: Renate McCloy Creative Director: Angela Cluer Interior Design: RPK Editorial Services Cover Design: Andrew Adams Compositor: International Typesetting and Composition Printer: R. R. Donnelley Cover Images: Courtesy of ALGOR, Inc. COPYRIGHT # 2007 by Nelson, a division of Thomson Canada Limited. Printed and bound in the United States 1 2 3 4 07 06 For more information contact Nelson, 1120 Birchmount Road, Toronto, Ontario, Canada, M1K 5G4. 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Mexico Spain Paraninfo Calle/Magallanes, 25 28015 Madrid, Spain # Contents # 1 Introduction 1 Prologue 1 1.1 Brief History 2 1.2 Introduction to Matrix Notation 4 1.3 Role of the Computer 6 1.4 General Steps of the Finite Element Method 7 1.5 Applications of the Finite Element Method 15 1.6 Advantages of the Finite Element Method 19 1.7 Computer Programs for the Finite Element Method 23 References 24 Problems 27 # 2 Introduction to the Stifness (Displacement) Method 28 Introduction 28 2.1 Definition of the Sti¤ness Matrix 28 2.2 Derivation of the Sti¤ness Matrix for a Spring Element 29 2.3 Example of a Spring Assemblage 34 2.4 Assembling the Total Sti¤ness Matrix by Superposition (Direct Sti¤ness Method) 3 7 2.5 Boundary Conditions 39 2.6 Potential Energy Approach to Derive Spring Element Equations 5 2 References 60 Problems 61 # 3 Development of Truss Equations Introduction 6 5 3.1 Derivation of the Sti¤ness Matrix for a Bar Element in Local Coordinates 66 3.2 Selecting Approximation Functions for Displacements 72 3.3 Transformation of Vectors in Two Dimensions 75 3.4 Global Sti¤ness Matrix 78 3.5 Computation of Stress for a Bar in the x-y Plane 82 3.6 Solution of a Plane Truss 84 3.7 Transformation Matrix and Sti¤ness Matrix for a Bar in Three-Dimensional Space 92 3.8 Use of Symmetry in Structure 100 3.9 Inclined, or Skewed, Supports 103 3.10 Potential Energy Approach to Derive Bar Element Equations 109 3.11 Comparison of Finite Element Solution to Exact Solution for Bar 120 3.12 Galerkin’s Residual Method and Its Use to Derive the One-Dimensional Bar Element Equations 124 3.13 Other Residual Methods and Their Application to a One-Dimensional Bar Problem 127 References 132 Problems 132 # 4 Development of Beam Equations Introduction 151 4.1 Beam Sti¤ness 152 4.2 Example of Assemblage of Beam Sti¤ness Matrices 161 4.3 Examples of Beam Analysis Using the Direct Sti¤ness Method 163 4.4 Distributed Loading 175 4.5 Comparison of the Finite Element Solution to the Exact Solution for a Beam 188 4.6 Beam Element with Nodal Hinge 194 4.7 Potential Energy Approach to Derive Beam Element Equations 199 4.8 Galerkin’s Method for Deriving Beam Element Equations 201 References 203 Problems 204 # 5 Frame and Grid Equations 214 Introduction 214 5.1 Two-Dimensional Arbitrarily Oriented Beam Element 214 5.2 Rigid Plane Frame Examples 218 5.3 Inclined or Skewed Supports—Frame Element 237 5.4 Grid Equations 238 5.5 Beam Element Arbitrarily Oriented in Space 255 5.6 Concept of Substructure Analysis 269 References 275 Problems 275 # 6 Development of the Plane Stress and Plane Strain Stiffness Equations 304 Introduction 304 6.1 Basic Concepts of Plane Stress and Plane Strain 305 6.2 Derivation of the Constant-Strain Triangular Element Sti¤ness Matrix and Equations 310 6.3 Treatment of Body and Surface Forces 324 6.4 Explicit Expression for the Constant-Strain Triangle Sti¤ness Matrix 329 6.5 Finite Element Solution of a Plane Stress Problem 331 References 342 Problems 343 # 7 Practical Considerations in Modeling; Interpreting Results; and Examples of Plane Stress/Strain Analysis 350 Introduction 350 7.1 Finite Element Modeling 350 7.2 Equilibrium and Compatibility of Finite Element Results 363 7.3 Convergence of Solution 367 7.4 Interpretation of Stresses 368 7.5 Static Condensation 369 7.6 Flowchart for the Solution of Plane Stress/Strain Problems 374 7.7 Computer Program Assisted Step-by-Step Solution, Other Models, and Results for Plane Stress/Strain Problems 374 References 381 Problems 382 # 8 Development of the Linear-Strain Triangle Equations 398 Introduction 398 8.1 Derivation of the Linear-Strain Triangular Element Sti¤ness Matrix and Equations 398 8.2 Example LST Sti¤ness Determination 403 8.3 Comparison of Elements 406 References 409 Problems 409 # 9 Axisymmetric Elements 412 Introduction 412 9.1 Derivation of the Sti¤ness Matrix 412 9.2 Solution of an Axisymmetric Pressure Vessel 422 9.3 Applications of Axisymmetric Elements 428 References 433 Problems 434 # 10 Isoparametric Formulation 443 Introduction 443 10.1 Isoparametric Formulation of the Bar Element Sti¤ness Matrix 444 10.2 Rectangular Plane Stress Element 449 10.3 Isoparametric Formulation of the Plane Element Sti¤ness Matrix 452 10.4 Gaussian and Newton-Cotes Quadrature (Numerical Integration) 463 10.5 Evaluation of the Sti¤ness Matrix and Stress Matrix by Gaussian Quadrature 469 10.6 Higher-Order Shape Functions 475 References 484 Problems 484 # 11 Three-Dimensional Stress Analysis 490 Introduction 490 11.1 Three-Dimensional Stress and Strain 490 11.2 Tetrahedral Element 493 11.3 Isoparametric Formulation 501 References 508 Problems 509 # 12 Plate Bending Element 514 Introduction 514 12.1 Basic Concepts of Plate Bending 514 12.2 Derivation of a Plate Bending Element Sti¤ness Matrix and Equations 519 12.3 Some Plate Element Numerical Comparisons 523 12.4 Computer Solution for a Plate Bending Problem 524 References 528 Problems 529 # 13 Heat Transfer and Mass Transport 534 Introduction 534 13.1 Derivation of the Basic Di¤erential Equation 535 13.2 Heat Transfer with Convection 538 13.3 Typical Units; Thermal Conductivities, K; and Heat-Transfer Coe‰cients, h 539 13.4 One-Dimensional Finite Element Formulation Using a Variational Method 540 13.5 Two-Dimensional Finite Element Formulation 555 13.6 Line or Point Sources 564 13.7 Three-Dimensional Heat Transfer Finite Element Formulation 566 13.8 One-Dimensional Heat Transfer with Mass Transport 569 13.9 Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin’s Method 569 13.10 Flowchart and Examples of a Heat-Transfer Program 574 References 577 Problems 577 # 14Fluid Flow 593 Introduction 593 14.1 Derivation of the Basic Di¤erential Equations 594 14.2 One-Dimensional Finite Element Formulation 598 14.3 Two-Dimensional Finite Element Formulation 606 14.4 Flowchart and Example of a Fluid-Flow Program 611 References 612 Problems 613 # 15 Thermal Stress 617 Introduction 617 15.1 Formulation of the Thermal Stress Problem and Examples 617 Reference 640 Problems 641 # 16 Structural Dynamics and Time-Dependent Heat Transfer647 Introduction 647 16.1 Dynamics of a Spring-Mass System 647 16.2 Direct Derivation of the Bar Element Equations 649 16.3 Numerical Integration in Time 653 16.4 Natural Frequencies of a One-Dimensional Bar 665 16.5 Time-Dependent One-Dimensional Bar Analysis 669 16.6 Beam Element Mass Matrices and Natural Frequencies 674 16.7 Truss, Plane Frame, Plane Stress/Strain, Axisymmetric, and Solid Element Mass Matrices 681 16.8 Time-Dependent Heat Transfer 686 16.9 Computer Program Example Solutions for Structural Dynamics 693 References 702 Problems 702 # Appendix A Matrix Algebra 708 Introduction 708 A.1 Definition of a Matrix 708 A.2 Matrix Operations 709 A.3 Cofactor or Adjoint Method to Determine the Inverse of a Matrix 716 A.4 Inverse of a Matrix by Row Reduction 718 References 720 Problems 720 # Appendix B Methods for Solution of Simultaneous Linear Equations 722 Introduction 722 B.1 General Form of the Equations 722 B.2 Uniqueness, Nonuniqueness, and Nonexistence of Solution 723 B.3 Methods for Solving Linear Algebraic Equations 724 B.4 Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methods 735 References 741 Problems 742 # Appendix C Equations from Elasticity Theory 744 Introduction 744 C.1 Di¤erential Equations of Equilibrium 744 C.2 Strain/Displacement and Compatibility Equations 746 C.3 Stress/Strain Relationships 748 Reference 751 Appendix D Equivalent Nodal Forces 752 Problems 752 Appendix E Principle of Virtual Work 755 References 758 Appendix F Properties of Structural Steel and Aluminum Shapes 759 Answers to Selected Problems 773 Index 799