A First Course in the Finite Element Method

Title Page......Page 1
Copyright......Page 2
Contents......Page 3
Prologue......Page 19
1.1 Brief History......Page 20
1.2 Introduction to Matrix Notation......Page 22
1.3 Role of the Computer......Page 24
1.4 General Steps of the Finite Element Method......Page 25
1.5 Applications of the Finite Element Method......Page 33
1.6 Advantages of the Finite Element Method......Page 37
1.7 Computer Programs for the Finite Element Method......Page 41
References......Page 42
Problems......Page 45
2.1 Definition of the Stiffness Matrix......Page 46
2.2 Derivation of the Stiffness Matrix for a Spring Element......Page 47
2.3 Example of a Spring Assemblage......Page 52
2.4 Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method)......Page 55
2.5 Boundary Conditions......Page 57
2.6 Potential Energy Approach to Derive Spring Element Equations......Page 70
References......Page 78
Problems......Page 79
Introduction......Page 83
3.1 Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates......Page 84
3.2 Selecting Approximation Functions for Displacements......Page 90
3.3 Transformation of Vectors in Two Dimensions......Page 93
3.4 Global Stiffness Matrix......Page 96
3.5 Computation of Stress for a Bar in the x-y Plane......Page 100
3.6 Solution of a Plane Truss......Page 102
3.7 Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space......Page 110
3.8 Use of Symmetry in Structure......Page 118
3.9 Inclined, or Skewed, Supports......Page 121
3.10 Potential Energy Approach to Derive Bar Element Equations......Page 127
3.11 Comparison of Finite Element Solution to Exact Solution for Bar......Page 138
3.12 Galerkin’s Residual Method and Its Use to Derive the One-Dimensional Bar Element Equations......Page 142
3.13 Other Residual Methods and Their Application to a One-Dimensional Bar Problem......Page 145
Problems......Page 150
Introduction......Page 169
4.1 Beam Stiffness......Page 170
4.2 Example of Assemblage of Beam Stiffness Matrices......Page 179
4.3 Examples of Beam Analysis Using the Direct Stiffness Method......Page 181
4.4 Distributed Loading......Page 193
4.5 Comparison of the Finite Element Solution to the Exact Solution for a Beam......Page 206
4.6 Beam Element with Nodal Hinge......Page 212
4.7 Potential Energy Approach to Derive Beam Element Equations......Page 217
4.8 Galerkin’s Method for Deriving Beam Element Equations......Page 219
References......Page 221
Problems......Page 222
5.1 Two-Dimensional Arbitrarily Oriented Beam Element......Page 232
5.2 Rigid Plane Frame Examples......Page 236
5.3 Inclined or Skewed Supports—Frame Eleme......Page 255
5.4 Grid Equations......Page 256
5.5 Beam Element Arbitrarily Oriented in Space......Page 273
5.6 Concept of Substructure Analysis......Page 287
Problems......Page 293
Introduction......Page 322
6.1 Basic Concepts of Plane Stress and Plane Strain......Page 323
6.2 Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations......Page 328
6.3 Treatment of Body and Surface Forces......Page 342
6.4 Explicit Expression for the Constant-Strain Triangle Stiffness Matrix......Page 347
6.5 Finite Element Solution of a Plane Stress Problem......Page 349
References......Page 360
Problems......Page 361
7.1 Finite Element Modeling......Page 368
7.2 Equilibrium and Compatibility of Finite Element Results......Page 381
7.3 Convergence of Solution......Page 385
7.4 Interpretation of Stresses......Page 386
7.5 Static Condensation......Page 387
7.7 Computer Program Assisted Step-by-Step Solution, Other Models and Results for Plane Stress/Strain Problems......Page 392
References......Page 399
Problems......Page 400
8.1 Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations......Page 416
8.2 Example LST Stiffness Determination......Page 421
8.3 Comparison of Elements......Page 424
Problems......Page 427
9.1 Derivation of the Stiffness Matrix......Page 430
9.2 Solution of an Axisymmetric Pressure Vessel......Page 440
9.3 Applications of Axisymmetric Elements......Page 446
References......Page 451
Problems......Page 452
Introduction......Page 461
10.1 Isoparametric Formulation of the Bar Element Stiffness Matrix......Page 462
10.2 Rectangular Plane Stress Element......Page 467
10.3 Isoparametric Formulation of the Plane Element Stiffness Matrix......Page 470
10.4 Gaussian and Newton-Cotes Quadrature (Numerical Integration)......Page 481
10.5 Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature......Page 487
10.6 Higher-Order Shape Functions......Page 493
Problems......Page 502
11.1 Three-Dimensional Stress and Strain......Page 508
11.2 Tetrahedral Element......Page 511
11.3 Isoparametric Formulation......Page 519
References......Page 526
Problems......Page 527
12.1 Basic Concepts of Plate Bending......Page 532
12.2 Derivation of a Plate Bending Element Stiffness Matrix and Equations......Page 537
12.3 Some Plate Element Numerical Comparisons......Page 541
12.4 Computer Solution for a Plate Bending Problem......Page 542
References......Page 546
Problems......Page 547
Introduction......Page 552
13.1 Derivation of the Basic Differential Equation......Page 553
13.2 Heat Transfer with Convection......Page 556
13.3 Typical Units; Thermal Conductivities, K; and Heat-Transfer Coefficients, h......Page 557
13.4 One-Dimensional Finite Element Formulation Using a Variational Method......Page 558
13.5 Two-Dimensional Finite Element Formulation......Page 573
13.6 Line or Point Sources......Page 582
13.7 Three-Dimensional Heat Transfer Finite Element Formulation......Page 584
13.9 Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin’s Method......Page 587
13.10 Flowchart and Examples of a Heat-Transfer Program......Page 592
Problems......Page 595
Introduction......Page 611
14.1 Derivation of the Basic Differential Equations......Page 612
14.2 One-Dimensional Finite Element Formulation......Page 616
14.3 Two-Dimensional Finite Element Formulation......Page 624
14.4 Flowchart and Example of a Fluid-Flow Program......Page 629
References......Page 630
Problems......Page 631
15.1 Formulation of the Thermal Stress Problem and Examples......Page 635
Reference......Page 658
Problems......Page 659
16.1 Dynamics of a Spring-Mass System......Page 665
16.2 Direct Derivation of the Bar Element Equations......Page 667
16.3 Numerical Integration in Time......Page 671
16.4 Natural Frequencies of a One-Dimensional Bar......Page 683
16.5 Time-Dependent One-Dimensional Bar Analysis......Page 687
16.6 Beam Element Mass Matrices and Natural Frequencies......Page 692
16.7 Truss, Plane Frame, Plane Stress/Strain, Axisymmetric, and Solid Element Mass Matrices......Page 699
16.8 Time-Dependent Heat Transfer......Page 704
16.9 Computer Program Example Solutions for Structural Dynamics......Page 711
Problems......Page 720
A.1 Definition of a Matrix......Page 726
A.2 Matrix Operations......Page 727
A.3 Cofactor or Adjoint Method to Determine the Inverse of a Matrix......Page 734
A.4 Inverse of a Matrix by Row Reduction......Page 736
Problems......Page 738
B.1 General Form of the Equations......Page 740
B.2 Uniqueness, Nonuniqueness, and Nonexistence of Solution......Page 741
B.3 Methods for Solving Linear Algebraic Equations......Page 742
B.4 Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methods......Page 753
References......Page 759
Problems......Page 760
C.1 Differential Equations of Equilibrium......Page 762
C.2 Strain/Displacement and Compatibility Equations......Page 764
C.3 Stress/Strain Relationships......Page 766
Reference......Page 769
Problems......Page 770
Appendix E: Principle of Virtual Work......Page 773
References......Page 776
Appendix F: Properties of Structural Steel and Aluminum Shapes......Page 777
Answers to Selected Problems......Page 791
Index......Page 817