A First Course in the Finite Element Method, Sixth Edition, SI Version
By Daryl L. Logan
Contents:
Preface to the SI Edition ix
Preface x
Digital Resource xii
Acknowledgments xv
Notation xvi
1 Introduction 1
Chapter Objectives 1
Prologue 1
1.1 Brief History 3
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 21
1.7 Computer Programs for the Finite Element Method 25
References 27
Problems 30
2 Introduction to the Stiffness (Displacement) Method 31
Chapter Objectives 31
Introduction 31
2.1 Definition of the Stiffness Matrix 32
2.2 Derivation of the Stiffness Matrix
for a Spring Element 32
2.3 Example of a Spring Assemblage 36
2.4 Assembling the Total Stiffness Matrix by Superposition
(Direct Stiffness Method) 38
2.5 Boundary Conditions 40
2.6 Potential Energy Approach to Derive Spring Element Equations 55
Summary Equations 65
References 66
Problems 66
3 Development of Truss Equations 72
Chapter Objectives 72
Introduction 72
3.1 Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates 73
3.2 Selecting a Displacement Function in Step 2 of the Derivation
of Stiffness Matrix for the One-Dimensional Bar Element 78
3.3 Transformation of Vectors in Two Dimensions 82
3.4 Global Stiffness Matrix for Bar Arbitrarily Oriented in the Plane 84
3.5 Computation of Stress for a Bar in the x – y Plane 89
3.6 Solution of a Plane Truss 91
3.7 Transformation Matrix and Stiffness Matrix for a Bar
in Three-Dimensional Space 100
3.8 Use of Symmetry in Structures 109
3.9 Inclined, or Skewed, Supports 112
3.10 Potential Energy Approach to Derive Bar Element Equations 121
3.11 Comparison of Finite Element Solution to Exact Solution for Bar 132
3.12 Galerkin’s Residual Method and Its Use to Derive
the One-Dimensional Bar Element Equations 136
3.13 Other Residual Methods and Their Application to a One-Dimensional Bar Problem 139
3.14 Flowchart for Solution of Three-Dimensional Truss Problems 143
3.15 Computer Program Assisted Step-by-Step Solution for Truss Problem 144
Summary Equations 146
References 147
Problems 147
4 Development of Beam Equations 169
Chapter Objectives 169
Introduction 169
4.1 Beam Stiffness 170
4.2 Example of Assemblage of Beam Stiffness Matrices 180
4.3 Examples of Beam Analysis Using the Direct Stiffness Method 182
4.4 Distributed Loading 195
4.5 Comparison of the Finite Element Solution to the Exact Solution for a Beam 208
4.6 Beam Element with Nodal Hinge 214
4.7 Potential Energy Approach to Derive Beam Element Equations 222
4.8 Galerkin’s Method for Deriving Beam Element Equations 225
Summary Equations 227
References 228
Problems 229
5 Frame and Grid Equations 239
Chapter Objectives 239
Introduction 239
5.1 Two-Dimensional Arbitrarily Oriented Beam Element 239
5.2 Rigid Plane Frame Examples 243
5.3 Inclined or Skewed Supports—Frame Element 261
5.4 Grid Equations 262
5.5 Beam Element Arbitrarily Oriented in Space 280
5.6 Concept of Substructure Analysis 295
Summary Equations 300
References 302
Problems 303
6 Development of the Plane Stress and Plane Strain
Stiffness Equations 337
Chapter Objectives 337
Introduction 337
6.1 Basic Concepts of Plane Stress and Plane Strain 338
6.2 Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations 342
6.3 Treatment of Body and Surface Forces 357
6.4 Explicit Expression for the Constant-Strain Triangle Stiffness Matrix 362
6.5 Finite Element Solution of a Plane Stress Problem 363
6.6 Rectangular Plane Element (Bilinear Rectangle, Q4) 374
Summary Equations 379
References 384
Problems 384
7 Practical Considerations in Modeling; Interpreting Results; and Examples of Plane Stress/Strain Analysis 391
Chapter Objectives 391
Introduction 391
7.1 Finite Element Modeling 392
7.2 Equilibrium and Compatibility of Finite Element Results 405
7.3 Convergence of Solution and Mesh Refinement 408
7.4 Interpretation of Stresses 411
7.5 Flowchart for the Solution of Plane Stress/Strain Problems 413
7.6 Computer Program–Assisted Step-by-Step Solution, Other Models, and Results
for Plane Stress/Strain Problems 414
References 420
Problems 421
8 Development of the Linear-Strain Triangle Equations 437
Chapter Objectives 437
Introduction 437
8.1 Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations 437
8.2 Example LST Stiffness Determination 442
8.3 Comparison of Elements 444
Summary Equations 447
References 448
Problems 448
9 Axisymmetric Elements 451
Chapter Objectives 451
Introduction 451
9.1 Derivation of the Stiffness Matrix 451
9.2 Solution of an Axisymmetric Pressure Vessel 462
9.3 Applications of Axisymmetric Elements 468
Summary Equations 473
References 475
Problems 475
10 Isoparametric Formulation 486
Chapter Objectives 486
Introduction 486
10.1 Isoparametric Formulation of the Bar Element Stiffness Matrix 487
10.2 Isoparametric Formulation of the Plane Quadrilateral (Q4) Element Stiffness Matrix 492
10.3 Newton-Cotes and Gaussian Quadrature 503
10.4 Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature 509
10.5 Higher-Order Shape Functions (Including Q6, Q8, Q9, and Q12 Elements) 515
Summary Equations 526
References 530
Problems 530
11 Three-Dimensional Stress Analysis 536
Chapter Objectives 536
Introduction 536
11.1 Three-Dimensional Stress and Strain 537
11.2 Tetrahedral Element 539
11.3 Isoparametric Formulation and Hexahedral Element 547
Summary Equations 555
References 558
Problems 558
12 Plate Bending Element 572
Chapter Objectives 572
Introduction 572
12.1 Basic Concepts of Plate Bending 572
12.2 Derivation of a Plate Bending Element Stiffness Matrix and Equations 577
12.3 Some Plate Element Numerical Comparisons 582
12.4 Computer Solutions for Plate Bending Problems 584
Summary Equations 588
References 590
Problems 591
13 Heat Transfer and Mass Transport 599
Chapter Objectives 599
Introduction 599
13.1 Derivation of the Basic Differential Equation 601
13.2 Heat Transfer with Convection 604
13.3 Typical Units; Thermal Conductivities, K; and Heat Transfer Coefficients, h 605
13.4 One-Dimensional Finite Element Formulation Using a Variational Method 607
13.5 Two-Dimensional Finite Element Formulation 626
13.6 Line or Point Sources 636
13.7 Three-Dimensional Heat Transfer by the Finite Element Method 639
13.8 One-Dimensional Heat Transfer with Mass Transport 641
13.9 Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin’s Method 642
13.10 Flowchart and Examples of a Heat Transfer Program 646
Summary Equations 651
References 654
Problems 655
14 Fluid Flow in Porous Media and through
Hydraulic Networks; and Electrical Networks and Electrostatics 673
Chapter Objectives 673
Introduction 673
14.1 Derivation of the Basic Differential Equations 674
14.2 One-Dimensional Finite Element Formulation 678
14.3 Two-Dimensional Finite Element Formulation 691
14.4 Flowchart and Example of a Fluid-Flow Program 696
14.5 Electrical Networks 697
14.6 Electrostatics 701
Summary Equations 715
References 719
Problems 720
15 Thermal Stress 727
Chapter Objectives 727
Introduction 727
15.1 Formulation of the Thermal Stress Problem and Examples 727
Summary Equations 752
Reference 753
Problems 754
16 Structural Dynamics and Time-Dependent
Heat Transfer 761
Chapter Objectives 761
Introduction 761
16.1 Dynamics of a Spring-Mass System 762
16.2 Direct Derivation of the Bar Element Equations 764
16.3 Numerical Integration in Time 768
16.4 Natural Frequencies of a One-Dimensional Bar 780
16.5 Time-Dependent One-Dimensional Bar Analysis 784
16.6 Beam Element Mass Matrices and Natural Frequencies 789
16.7 Truss, Plane Frame, Plane Stress, Plane Strain, Axisymmetric, and Solid Element Mass Matrices 796
16.8 Time-Dependent Heat Transfer 801
16.9 Computer Program Example Solutions for Structural Dynamics 808
Summary Equations 817
References 821
Problems 822
Appendix A Matrix Algebra 827
Appendix B Methods for Solution of Simultaneous
Linear Equations 843
Appendix C Equations from Elasticity Theory 865
Appendix D Equivalent Nodal Forces 873
Appendix E Principle of Virtual Work 876
Appendix F Geometric Properties of Structural Steel Wide-Flange
Sections (W Shapes) 880
Answers to Selected Problems 908
Index 938