A First Course in the Finite Element Method, SI Edition, 6th Edition

Discover a simple, direct approach that highlights the basics you need within A FIRST COURSE IN THE FINITE ELEMENT METHOD, 6E. This unique book is written so both undergraduate and graduate students can easily comprehend the content without the usual prerequisites, such as structural analysis. The book is written primarily as a basic learning tool for students, like you, in civil and mechanical engineering who are primarily interested in stress analysis and heat transfer. The text offers ideal preparation for utilizing the finite element method as a tool to solve practical physical problems.

Table of contents :
Cover......Page 1
Contents......Page 8
Preface to the SI Edition......Page 15
Preface......Page 16
Notation......Page 19
Prologue......Page 22
1.1 Brief History......Page 24
1.2 Introduction to Matrix Notation......Page 25
1.3 Role of the Computer......Page 27
1.4 General Steps of the Finite Element Method......Page 28
1.5 Applications of the Finite Element Method......Page 36
1.6 Advantages of the Finite Element Method......Page 42
1.7 Computer Programs for the Finite Element Method......Page 46
References......Page 48
Problems......Page 51
Introduction......Page 52
2.2 Derivation of the Stiffness Matrix for a Spring Element......Page 53
2.3 Example of a Spring Assemblage......Page 57
2.4 Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method)......Page 59
2.5 Boundary Conditions......Page 61
2.6 Potential Energy Approach to Derive Spring Element Equations......Page 76
Summary Equations......Page 86
Problems......Page 87
Introduction......Page 93
3.1 Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates......Page 94
3.2 Selecting a Displacement Function in Step 2 of the Derivation of Stiffness Matrix for the One-Dimensional Bar Element......Page 99
3.3 Transformation of Vectors in Two Dimensions......Page 103
3.4 Global Stiffness Matrix for Bar Arbitrarily Oriented in the Plane......Page 105
3.5 Computation of Stress for a Bar in the......Page 110
3.6 Solution of a Plane Truss......Page 112
3.7 Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space......Page 121
3.8 Use of Symmetry in Structures......Page 130
3.9 Inclined, or Skewed, Supports......Page 133
3.10 Potential Energy Approach to Derive Bar Element Equations......Page 142
3.11 Comparison of Finite Element Solution to Exact Solution for Bar......Page 153
3.12 Galerkin’s Residual Method and Its Use to Derive the One-Dimensional Bar Element Equations......Page 157
3.13 Other Residual Methods and Their Application to a One-Dimensional Bar Problem......Page 160
3.14 Flowchart for Solution of Three-Dimensional Truss Problems......Page 164
3.15 Computer Program Assisted Step-by-Step Solution for Truss Problem......Page 165
Summary Equations......Page 167
Problems......Page 168
Introduction......Page 190
4.1 Beam Stiffness......Page 191
4.2 Example of Assemblage of Beam Stiffness Matrices......Page 201
4.3 Examples of Beam Analysis Using the Direct Stiffness Method......Page 203
4.4 Distributed Loading......Page 216
4.5 Comparison of the Finite Element Solution to the Exact Solution for a Beam......Page 229
4.6 Beam Element with Nodal Hinge......Page 235
4.7 Potential Energy Approach to Derive Beam Element Equations......Page 243
4.8 Galerkin’s Method for Deriving Beam Element Equations......Page 246
Summary Equations......Page 248
References......Page 249
Problems......Page 250
5.1 Two-Dimensional Arbitrarily Oriented Beam Element......Page 260
5.2 Rigid Plane Frame Examples......Page 264
5.3 Inclined or Skewed Supports—Frame Element......Page 282
5.4 Grid Equations......Page 283
5.5 Beam Element Arbitrarily Oriented in Space......Page 301
5.6 Concept of Substructure Analysis......Page 316
Summary Equations......Page 321
References......Page 323
Problems......Page 324
Introduction......Page 358
6.1 Basic Concepts of Plane Stress and Plane Strain......Page 359
6.2 Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations......Page 363
6.3 Treatment of Body and Surface Forces......Page 378
6.4 Explicit Expression for the Constant-Strain Triangle Stiffness Matrix......Page 383
6.5 Finite Element Solution of a Plane Stress Problem......Page 384
6.6 Rectangular Plane Element (Bilinear Rectangle, Q4)......Page 395
Summary Equations......Page 400
Problems......Page 405
Introduction......Page 412
7.1 Finite Element Modeling......Page 413
7.2 Equilibrium and Compatibility of Finite Element Results......Page 426
7.3 Convergence of Solution and Mesh Refinement......Page 429
7.4 Interpretation of Stresses......Page 432
7.5 Flowchart for the Solution of Plane Stress/Strain Problems......Page 434
7.6 Computer Program–Assisted Step-by-Step Solution, Other Models, and Results for Plane Stress/Strain Problems......Page 435
References......Page 441
Problems......Page 442
8.1 Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations......Page 458
8.2 Example LST Stiffness Determination......Page 463
8.3 Comparison of Elements......Page 465
Summary Equations......Page 468
Problems......Page 469
9.1 Derivation of the Stiffness Matrix......Page 472
9.2 Solution of an Axisymmetric Pressure Vessel......Page 483
9.3 Applications of Axisymmetric Elements......Page 489
Summary Equations......Page 494
Problems......Page 496
Introduction......Page 507
10.1 Isoparametric Formulation of the Bar Element Stiffness Matrix......Page 508
10.2 Isoparametric Formulation of the Plane Quadrilateral (Q4) Element Stiffness Matrix......Page 513
10.3 Newton-Cotes and Gaussian Quadrature......Page 524
10.4 Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature......Page 530
10.5 Higher-Order Shape Functions (Including Q6, Q8, Q9, and Q12 Elements)......Page 536
Summary Equations......Page 547
Problems......Page 551
Introduction......Page 557
11.1 Three-Dimensional Stress and Strain......Page 558
11.2 Tetrahedral Element......Page 560
11.3 Isoparametric Formulation and Hexahedral Element......Page 568
Summary Equations......Page 576
Problems......Page 579
12.1 Basic Concepts of Plate Bending......Page 593
12.2 Derivation of a Plate Bending Element Stiffness Matrix and Equations......Page 598
12.3 Some Plate Element Numerical Comparisons......Page 603
12.4 Computer Solutions for Plate Bending Problems......Page 605
Summary Equations......Page 609
References......Page 611
Problems......Page 612
Introduction......Page 620
13.1 Derivation of the Basic Differential Equation......Page 622
13.2 Heat Transfer with Convection......Page 625
13.3 Typical Units; Thermal Conductivities, K; and Heat Transfer Coefficients, h......Page 626
13.4 One-Dimensional Finite Element Formulation Using a Variational Method......Page 628
13.5 Two-Dimensional Finite Element Formulation......Page 647
13.6 Line or Point Sources......Page 657
13.7 Three-Dimensional Heat Transfer by the Finite Element Method......Page 660
13.8 One-Dimensional Heat Transfer with Mass Transport......Page 662
13.9 Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin’s Method......Page 663
13.10 Flowchart and Examples of a Heat Transfer Program......Page 667
Summary Equations......Page 672
References......Page 675
Problems......Page 676
Introduction......Page 694
14.1 Derivation of the Basic Differential Equations......Page 695
14.2 One-Dimensional Finite Element Formulation......Page 699
14.3 Two-Dimensional Finite Element Formulation......Page 712
14.4 Flowchart and Example of a Fluid-Flow Program......Page 717
14.5 Electrical Networks......Page 718
14.6 Electrostatics......Page 722
Summary Equations......Page 736
References......Page 740
Problems......Page 741
15.1 Formulation of the Thermal Stress Problem and Examples......Page 748
Summary Equations......Page 773
Reference......Page 774
Problems......Page 775
Introduction......Page 782
16.1 Dynamics of a Spring-Mass System......Page 783
16.2 Direct Derivation of the Bar Element Equations......Page 785
16.3 Numerical Integration in Time......Page 789
16.4 Natural Frequencies of a One-Dimensional Bar......Page 801
16.5 Time-Dependent One-Dimensional Bar Analysis......Page 805
16.6 Beam Element Mass Matrices and Natural Frequencies......Page 810
16.7 Truss, Plane Frame, Plane Stress, Plane Strain, Axisymmetric, and Solid Element Mass Matrices......Page 817
16.8 Time-Dependent Heat Transfer......Page 822
16.9 Computer Program Example Solutions for Structural Dynamics......Page 829
Summary Equations......Page 838
References......Page 842
Problems......Page 843
A.1 Definition of a Matrix......Page 848
A.2 Matrix Operations......Page 849
A.3 Cofactor or Adjoint Method to Determine the Inverse of a Matrix......Page 856
A.4 Inverse of a Matrix by Row Reduction......Page 858
A.5 Properties of Stiffness Matrices......Page 860
Problems......Page 861
B.1 General Form of the Equations......Page 864
B.2 Uniqueness, Nonuniqueness, and Nonexistence of Solution......Page 865
B.3 Methods for Solving Linear Algebraic Equations......Page 866
B.4 Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methods......Page 877
Problems......Page 884
C.1 Differential Equations of Equilibrium......Page 886
C.2 Strain/Displacement and Compatibility Equations......Page 888
C.3 Stress/Strain Relationships......Page 890
Reference......Page 893
Problems......Page 894
Appendix E: Principle of Virtual Work......Page 897
References......Page 900
Appendix F: Geometric Properties of Structural Steel Wide-Flange Sections (W Shapes)......Page 901
Answers to Selected Problems......Page 929
Index......Page 959

Details

Title	A First Course in the Finite Element Method, SI Edition, 6th Edition
Author	Daryl L. Logan (Author)
Language	English
ISBN	ISBN-13: 978-1305637344 ISBN-10: 1305637348
Size	45 MB
Download Method	Direct Download
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