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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
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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