First Course in the Finite Element Method

ISBN-10: 0534929648

ISBN-13: 9780534929640

Edition: 2nd 1992

Authors: Daryl L. Logan

List price: $163.95
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This third edition provides a simple, basic approach to the finite element method that can be understood by both undergraduate and graduate students. It does not have the usual prerequisites (such as structural analysis) required by most available texts in this area. The book is written primarily as a basic learning tool for the undergraduate student in civil and mechanical engineering whose main interest is in stress analysis and heat transfer. The text is geared toward those who want to apply the finite element method as a tool to solve practical physical problems.
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Book details

List price: $163.95
Edition: 2nd
Copyright year: 1992
Publisher: Brooks/Cole
Publication date: 3/27/1992
Binding: Hardcover
Pages: 640
Size: 7.09" wide x 9.84" long
Weight: 2.750
Language: English

Introductionp. 1
Prologuep. 1
Brief Historyp. 2
Introduction to Matrix Notationp. 3
Role of the Computerp. 6
General Steps of the Finite Element Methodp. 6
Applications of the Finite Element Methodp. 13
Advantages of the Finite Element Methodp. 18
Computer Programs for the Finite Element Methodp. 19
Referencesp. 22
Problemsp. 25
Introduction to the Stiffness (Displacement) Methodp. 26
Introductionp. 26
Definition of the Stiffness Matrixp. 26
Derivation of the Stiffness Matrix for a Spring Elementp. 27
Example of a Spring Assemblagep. 32
Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method)p. 35
Boundary Conditionsp. 37
Potential Energy Approach to Derive Spring Element Equationsp. 50
Referencesp. 58
Problemsp. 59
Development of Truss Equationsp. 63
Introductionp. 63
Derivation of the Stiffness Matrix for a Bar Element in Local Coordinatesp. 63
Selecting Approximation Functions for Displacementsp. 69
Transformation of Vectors in Two Dimensionsp. 71
Global Stiffness Matrixp. 74
Computation of Stress for a Bar in the x-y Planep. 78
Solution of a Plane Trussp. 80
Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Spacep. 87
Use of Symmetry in Structurep. 92
Inclined, or Skewed, Supportsp. 95
Potential Energy Approach to Derive Bar Element Equationsp. 101
Comparison of Finite Element Solution to Exact Solution for Barp. 112
Galerkin's Residual Method and Its Application to a One-Dimensional Barp. 116
Referencesp. 119
Problemsp. 120
Development of Beam Equationsp. 137
Introductionp. 137
Beam Stiffnessp. 138
Example of Assemblage of Beam Stiffness Matricesp. 143
Examples of Beam Analysis Using the Direct Stiffness Methodp. 145
Distributed Loadingp. 154
Comparison of the Finite Element Solution to the Exact Solution for a Beamp. 165
Beam Element with Nodal Hingep. 171
Potential Energy Approach to Derive Beam Element Equationsp. 176
Galerkin's Method for Deriving Beam Element Equationsp. 179
Referencesp. 181
Problemsp. 181
Frame and Grid Equationsp. 188
Introductionp. 188
Two-Dimensional Arbitrarily Oriented Beam Elementp. 188
Rigid Plane Frame Examplesp. 192
Inclined or Skewed Supports--Frame Elementp. 211
Grid Equationsp. 212
Beam Element Arbitrarily Oriented in Spacep. 229
Concept of Substructure Analysisp. 234
Referencesp. 240
Problemsp. 240
Development of the Plane Stress and Plane Strain Stiffness Equationsp. 264
Introductionp. 264
Basic Concepts of Plane Stress and Plane Strainp. 265
Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equationsp. 270
Treatment of Body and Surface Forcesp. 284
Explicit Expression for the Constant-Strain Triangle Stiffness Matrixp. 289
Finite Element Solution of a Plane Stress Problemp. 291
Referencesp. 301
Problemsp. 301
Practical Considerations in Modeling; Interpreting Results; and Examples of Plane Stress/Strain Analysisp. 307
Introductionp. 307
Finite Element Modelingp. 308
Equilibrium and Compatibility of Finite Element Resultsp. 318
Convergence of Solutionp. 320
Interpretation of Stressesp. 321
Static Condensationp. 323
Flowchart for the Solution of Plane Stress/Strain Problemsp. 327
Computer Program Results for Some Plane Stress/Strain Problemsp. 328
Referencesp. 331
Problemsp. 332
Development of the Linear-Strain Triangle Equationsp. 344
Introductionp. 344
Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equationsp. 344
Example LST Stiffness Determinationp. 349
Comparison of Elementsp. 352
Referencesp. 354
Problemsp. 355
Axisymmetric Elementsp. 358
Introductionp. 358
Derivation of the Stiffness Matrixp. 358
Solution of an Axisymmetric Pressure Vesselp. 368
Applications of Axisymmetric Elementsp. 376
Referencesp. 380
Problemsp. 381
Isoparametric Formulationp. 386
Introductionp. 386
Isoparametric Formulation of the Bar Element Stiffness Matrixp. 386
Rectangular Plane Stress Elementp. 392
Isoparametric Formulation of the Plane Element Stiffness Matrixp. 395
Gaussian Quadrature (Numerical Integration)p. 404
Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadraturep. 407
Higher-Order Shape Functionsp. 413
Referencesp. 417
Problemsp. 417
Three-Dimensional Stress Analysisp. 421
Introductionp. 421
Three-Dimensional Stress and Strainp. 421
Tetrahedral Elementp. 423
Isoparametric Formulationp. 430
Referencesp. 436
Problemsp. 436
Plate Bending Elementp. 441
Introductionp. 441
Basic Concepts of Plate Bendingp. 441
Derivation of a Plate Bending Element Stiffness Matrix and Equationsp. 445
Some Plate Element Numerical Comparisonsp. 450
Computer Solution for a Plate Bending Problemp. 452
Referencesp. 454
Problemsp. 455
Heat Transfer and Mass Transportp. 458
Introductionp. 458
Derivation of the Basic Differential Equationp. 459
Heat Transfer with Convectionp. 462
Typical Units; Thermal Conductivities, K; and Heat-Transfer Coefficients, hp. 463
One-Dimensional Finite Element Formulation Using a Variational Methodp. 464
Two-Dimensional Finite Element Formulationp. 478
Line or Point Sourcesp. 487
One-Dimensional Heat Transfer with Mass Transportp. 490
Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin's Methodp. 491
Flowchart and Examples of a Heat-Transfer Programp. 495
Referencesp. 499
Problemsp. 499
Fluid Flowp. 508
Introductionp. 508
Derivation of the Basic Differential Equationsp. 508
One-Dimensional Finite Element Formulationp. 513
Two-Dimensional Finite Element Formulationp. 521
Flowchart and Example of a Fluid-Flow Programp. 526
Referencesp. 527
Problemsp. 528
Thermal Stressp. 532
Introductionp. 532
Formulation of the Thermal Stress Problem and Examplesp. 532
Referencep. 553
Problemsp. 554
Structural Dynamics and Time-Dependent Heat Transferp. 559
Introductionp. 559
Dynamics of a Spring-Mass Systemp. 559
Direct Derivation of the Bar Element Equationsp. 561
Numerical Integration in Timep. 565
Natural Frequencies of a One-Dimensional Barp. 577
Time-Dependent One-Dimensional Bar Analysisp. 581
Beam Element Mass Matrices and Natural Frequenciesp. 586
Truss, Plane Frame, Plane Stress/Strain, Axisymmetric, and Solid Element Mass Matricesp. 591
Time-Dependent Heat Transferp. 595
Computer Program Example Solutions for Structural Dynamicsp. 602
Referencesp. 609
Problemsp. 610
Matrix Algebrap. 616
Introductionp. 616
Definition of a Matrixp. 616
Matrix Operationsp. 617
Cofactor or Adjoint Method to Determine the Inverse of a Matrixp. 624
Inverse of a Matrix by Row Reductionp. 626
Referencesp. 628
Problemsp. 628
Methods for Solution of Simultaneous Linear Equationsp. 630
Introductionp. 630
General Form of the Equationsp. 630
Uniqueness, Nonuniqueness, and Nonexistence of Solutionp. 631
Methods for Solving Linear Algebraic Equationsp. 632
Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methodsp. 643
Referencesp. 649
Problemsp. 650
Equations from Elasticity Theoryp. 652
Introductionp. 652
Differential Equations of Equilibriump. 652
Strain/Displacement and Compatibility Equationsp. 654
Stress/Strain Relationshipsp. 656
Referencep. 659
Equivalent Nodal Forcesp. 660
Problemsp. 660
Principle of Virtual Workp. 663
Referencesp. 666
Answers to Selected Problemsp. 667
Indexp. 689
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