Cover image for A first course in the finite element method using Algor
Title:
A first course in the finite element method using Algor
Personal Author:
Edition:
2nd ed
Publication Information:
Pacific Grove, CA : Brooks/Cole, 2001
ISBN:
9780534380687

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30000004605808 TA347.F5 L634 2001 Open Access Book Book
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Summary

Summary

Daryl Logan's clear and easy to understand text provides a thorough treatment of the finite element method and how to apply it to solve practical physical problems in engineering. Concepts are presented simply, making it understandable for students of all levels of experience. The first edition of this book enjoyed considerable success and this new edition includes a chapter on plates and plate bending, along with additional homework exercise. All examples in this edition have been updated to Algor� Release 12.


Table of Contents

1 Introductionp. 1
Prologuep. 1
1.1 Brief Historyp. 2
1.2 Introduction to Matrix Notationp. 3
1.3 Role of the Computerp. 6
1.4 General Steps of the Finite Element Methodp. 6
1.5 Applications of the Finite Element Methodp. 13
1.6 Advantages of the Finite Element Methodp. 18
1.7 Computer Programs for the Finite Element Methodp. 19
Referencesp. 22
Problemsp. 25
2 Introduction to the Stiffness (Displacement) Methodp. 26
Introductionp. 26
2.1 Definition of the Stiffness Matrixp. 26
2.2 Derivation of the Stiffness Matrix for a Spring Elementp. 27
2.3 Example of a Spring Assemblagep. 32
2.4 Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method)p. 35
2.5 Boundary Conditionsp. 37
2.6 Potential Energy Approach to Derive Spring Element Equationsp. 50
Referencesp. 58
Problemsp. 59
3 Development of Truss Equationsp. 63
Introductionp. 63
3.1 Derivation of the Stiffness Matrix for a Bar Element in Local Coordinatesp. 63
3.2 Selecting Approximation Functions for Displacementsp. 69
3.3 Transformation of Vectors in Two Dimensionsp. 71
3.4 Global Stiffness Matrixp. 74
3.5 Computation of Stress for a Bar in the x-y Planep. 78
3.6 Solution of a Plane Trussp. 80
3.7 Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Spacep. 87
3.8 Use of Symmetry in Structurep. 92
3.9 Inclined, or Skewed, Supportsp. 95
3.10 Potential Energy Approach to Derive Bar Element Equationsp. 101
3.11 Comparison of Finite Element Solution to Exact Solution for Barp. 112
3.12 Galerkin's Residual Method and Its Application to a One-Dimensional Barp. 116
Referencesp. 119
Problemsp. 120
4 Algor Program for Truss Analysisp. 137
Introductionp. 137
4.1 Overview of the Algor System and Flowcharts for the Solution of a Truss Problem Using Algorp. 137
4.2 Algor Example Solutions for Truss Analysisp. 145
Referencesp. 177
Problemsp. 177
5 Development of Beam Equationsp. 186
Introductionp. 186
5.1 Beam Stiffnessp. 187
5.2 Example of Assemblage of Beam Stiffness Matricesp. 192
5.3 Examples of Beam Analysis Using the Direct Stiffness Methodp. 194
5.4 Distributed Loadingp. 203
5.5 Comparison of the Finite Element Solution to the Exact Solution for a Beamp. 214
5.6 Beam Element with Nodal Hingep. 220
5.7 Potential Energy Approach to Derive Beam Element Equationsp. 225
5.8 Galerkin's Method for Deriving Beam Element Equationsp. 228
5.9 Algor Example Solutions for Beam Analysisp. 230
Referencesp. 268
Problemsp. 269
6 Frame and Grid Equationsp. 276
Introductionp. 276
6.1 Two-Dimensional Arbitrarily Oriented Beam Elementp. 276
6.2 Rigid Plane Frame Examplesp. 280
6.3 Inclined or Skewed Supports--Frame Elementp. 299
6.4 Grid Equationsp. 300
6.5 Beam Element Arbitrarily Oriented in Spacep. 317
6.6 Concept of Substructure Analysisp. 322
6.7 Algor Example Solutions for Plane Frame, Grid, and Space Frame Analysisp. 328
Referencesp. 362
Problemsp. 363
7 Development of the Plane Stress and Plane Strain Stiffness Equationsp. 386
Introductionp. 386
7.1 Basic Concepts of Plane Stress and Plane Strainp. 387
7.2 Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equationsp. 392
7.3 Treatment of Body and Surface Forcesp. 406
7.4 Explicit Expression for the Constant-Strain Triangle Stiffness Matrixp. 411
7.5 Finite Element Solution of a Plane Stress Problemp. 413
Referencesp. 423
Problemsp. 423
8 Practical Considerations in Modeling; Interpreting Results; and Use of the Algor Program for Plane Stress/Strain Analysisp. 429
Introductionp. 429
8.1 Finite Element Modelingp. 430
8.2 Equilibrium and Compatibility of Finite Element Resultsp. 440
8.3 Convergence of Solutionp. 442
8.4 Interpretation of Stressesp. 443
8.5 Static Condensationp. 445
8.6 Flowchart for the Solution of Plane Stress/Strain Problems and Typical Steps Using Algorp. 449
8.7 Algor Example Solutions for Plane Stress/Strain Analysisp. 453
Referencesp. 484
Problemsp. 485
9 Development of the Linear-Strain Triangle Equationsp. 497
Introductionp. 497
9.1 Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equationsp. 497
9.2 Example LST Stiffness Determinationp. 502
9.3 Comparison of Elementsp. 505
Referencesp. 507
Problemsp. 508
10 Axisymmetric Elementsp. 511
Introductionp. 511
10.1 Derivation of the Stiffness Matrixp. 511
10.2 Solution of an Axisymmetric Pressure Vesselp. 521
10.3 Applications of Axisymmetric Elementsp. 528
10.4 Algor Example Solutions for Axisymmetric Problemsp. 531
Referencesp. 554
Problemsp. 555
11 Isoparametric Formulationp. 560
Introductionp. 560
11.1 Isoparametric Formulation of the Bar Element Stiffness Matrixp. 560
11.2 Rectangular Plane Stress Elementp. 566
11.3 Isoparametric Formulation of the Plane Element Stiffness Matrixp. 569
11.4 Gaussian Quadrature (Numerical Integration)p. 578
11.5 Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadraturep. 581
11.6 Higher-Order Shape Functionsp. 587
Referencesp. 591
Problemsp. 591
12 Three-Dimensional Stress Analysisp. 595
Introductionp. 595
12.1 Three-Dimensional Stress and Strainp. 595
12.2 Tetrahedral Elementp. 597
12.3 Isoparametric Formulationp. 604
12.4 Algor Example Solutions of Three-Dimensional Stress Analysisp. 609
Referencesp. 629
Problemsp. 630
13 Heat Transfer and Mass Transportp. 634
Introductionp. 634
13.1 Derivation of the Basic Differential Equationp. 635
13.2 Heat Transfer with Convectionp. 638
13.3 Typical Units; Thermal Conductivities, K; and Heat-Transfer Coefficients, hp. 639
13.4 One-Dimensional Finite Element Formulation Using a Variational Methodp. 640
13.5 Two-Dimensional Finite Element Formulationp. 654
13.6 Line or Point Sourcesp. 663
13.7 One-Dimensional Heat Transfer with Mass Transportp. 666
13.8 Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin's Methodp. 667
13.9 Flowchart of a Heat-Transfer Programp. 671
13.10 Algor Example Solutions for Heat-Transfer Problemsp. 672
Referencesp. 692
Problemsp. 692
14 Fluid Flowp. 701
Introductionp. 701
14.1 Derivation of the Basic Differential Equationsp. 701
14.2 One-Dimensional Finite Element Formulationp. 706
14.3 Two-Dimensional Finite Element Formulationp. 714
14.4 Flowchart of a Fluid-Flow Programp. 718
14.5 Algor Example Solutions for Two-Dimensional Steady-State Fluid Flowp. 719
Referencesp. 727
Problemsp. 728
15 Thermal Stressp. 732
Introductionp. 732
15.1 Formulation of the Thermal Stress Problem and Examplesp. 732
15.2 Algor Example Solutions for Thermal Stress Problemsp. 753
Referencep. 772
Problemsp. 773
16 Structural Dynamics and Time-Dependent Heat Transferp. 778
Introductionp. 778
16.1 Dynamics of a Spring-Mass Systemp. 778
16.2 Direct Derivation of the Bar Element Equationsp. 780
16.3 Numerical Integration in Timep. 784
16.4 Natural Frequencies of a One-Dimensional Barp. 796
16.5 Time-Dependent One-Dimensional Bar Analysisp. 800
16.6 Beam Element Mass Matrices and Natural Frequenciesp. 805
16.7 Truss, Plane Frame, Plane Stress/Strain, Axisymmetric, and Solid Element Mass Matricesp. 810
16.8 Time-Dependent Heat Transferp. 814
16.9 Algor Example Solutions for Structural Dynamics and Transient Heat Transferp. 821
Referencesp. 856
Problemsp. 857
17 Plate Bending Elementp. 862
Introductionp. 862
17.1 Basic Concepts of Plate Bendingp. 862
17.2 Derivation of a Plate Bending Element Stiffness Matrix and Equationsp. 866
17.3 Some Plate Element Numerical Comparisonsp. 871
17.4 Algor Example Solution for Plate Bending Problemsp. 872
Referencesp. 878
Problemsp. 879
Appendix A Matrix Algebrap. 883
Introductionp. 883
A.1 Definition of a Matrixp. 883
A.2 Matrix Operationsp. 884
A.3 Cofactor or Adjoint Method to Determine the Inverse of a Matrixp. 891
A.4 Inverse of a Matrix by Row Reductionp. 893
Referencesp. 895
Problemsp. 895
Appendix B Methods for Solution of Simultaneous Linear Equationsp. 897
Introductionp. 897
B.1 General Form of the Equationsp. 897
B.2 Uniqueness, Nonuniqueness, and Nonexistence of Solutionp. 898
B.3 Methods for Solving Linear Algebraic Equationsp. 899
B.4 Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methodsp. 910
Referencesp. 916
Problemsp. 917
Appendix C Equations from Elasticity Theoryp. 919
Introductionp. 919
C.1 Differential Equations of Equilibriump. 919
C.2 Strain/Displacement and Compatibility Equationsp. 921
C.3 Stress/Strain Relationshipsp. 923
Referencep. 926
Appendix D Equivalent Nodal Forcesp. 927
Problemsp. 927
Appendix E Principle of Virtual Workp. 930
Referencesp. 933
Appendix F Basics of Algorp. 934
Introductionp. 934
F.1 Hardware Requirements for Windows Installationp. 934
F.2 Conventionsp. 934
F.3 Getting Around the Menu Systemp. 935
F.4 Function Keysp. 935
F.5 Algor Processor Namesp. 937
F.6 File Extensions Generated by the Algor Systemp. 938
F.7 Checking Model for Defects by Using Superviewp. 938
Answers to Selected Problemsp. 943
Indexp. 965