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First course in finite element method
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A First Course in the Finite Element Method,
Fifth Edition
Daryl L. Logan
Publisher, Global Engineering:
Christopher M. Shortt
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c 2012 Cengage Learning
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A FIRST COURSE
IN THE FINITE
ELEMENT METHODd
Fifth Edition
Daryl L. Logan
University of Wisconsin–Platteville
Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
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CONTENTSd
Preface xi
1 Introduction 1
Chapter Objectives 1
Prologue 1
1.1 Brief History 2
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 23
1.7 Computer Programs for the Finite Element Method 25
References 27
Problems 29
2 Introduction to the Stiffness (Displacement) Method 31
Chapter Objectives 31
Introduction 31
2.1 Definition of the Sti¤ness Matrix 32
2.2 Derivation of the Sti¤ness Matrix for a Spring Element 32
2.3 Example of a Spring Assemblage 38
2.4 Assembling the Total Sti¤ness Matrix by Superposition
(Direct Sti¤ness Method) 40
2.5 Boundary Conditions 42
2.6 Potential Energy Approach to Derive Spring Element Equations 56
Summary Equations 65
References 66
Problems 66
iii
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3 Development of Truss Equations 72
Chapter Objectives 72
Introduction 72
3.1 Derivation of the Sti¤ness Matrix for a Bar Element
in Local Coordinates 73
3.2 Selecting Approximation Functions for Displacements 79
3.3 Transformation of Vectors in Two Dimensions 82
3.4 Global Sti¤ness Matrix for Bar Arbitrarily Oriented in the Plane 85
3.5 Computation of Stress for a Bar in the x-y Plane 90
3.6 Solution of a Plane Truss 92
3.7 Transformation Matrix and Sti¤ness Matrix for a Bar
in Three-Dimensional Space 100
3.8 Use of Symmetry in Structure 109
3.9 Inclined, or Skewed, Supports 112
3.10 Potential Energy Approach to Derive Bar Element Equations 118
3.11 Comparison of Finite Element Solution to Exact Solution for Bar 129
3.12 Galerkin’s Residual Method and Its Use to Derive the One-Dimensional
Bar Element Equations 133
3.13 Other Residual Methods and Their Application to a One-Dimensional
Bar Problem 136
3.14 Flowchart for Solution of Three-Dimensional Truss Problems 141
3.15 Computer Program Assisted Step-by-Step Solution for Truss Problem 141
Summary Equations 144
References 145
Problems 146
4 Development of Beam Equations 166
Chapter Objectives 166
Introduction 166
4.1 Beam Sti¤ness 167
4.2 Example of Assemblage of Beam Sti¤ness Matrices 177
4.3 Examples of Beam Analysis Using the Direct Sti¤ness Method 179
4.4 Distributed Loading 192
4.5 Comparison of the Finite Element Solution to the Exact Solution
for a Beam 205
4.6 Beam Element with Nodal Hinge 211
iv d Contents
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4.7 Potential Energy Approach to Derive Beam Element
Equations 218
4.8 Galerkin’s Method for Deriving Beam
Element Equations 221
Summary Equations 223
References 224
Problems 225
5 Frame and Grid Equations 235
Chapter Objectives 235
Introduction 235
5.1 Two-Dimensional Arbitrarily Oriented Beam Element 235
5.2 Rigid Plane Frame Examples 239
5.3 Inclined or Skewed Supports—Frame Element 258
5.4 Grid Equations 259
5.5 Beam Element Arbitrarily Oriented in Space 277
5.6 Concept of Substructure Analysis 290
Summary Equations 296
References 298
Problems 299
6 Development of the Plane Stress
and Plane Strain Stiffness Equations 328
Chapter Objectives 328
Introduction 328
6.1 Basic Concepts of Plane Stress and Plane Strain 329
6.2 Derivation of the Constant-Strain Triangular Element
Sti¤ness Matrix and Equations 334
6.3 Treatment of Body and Surface Forces 349
6.4 Explicit Expression for the Constant-Strain
Triangle Sti¤ness Matrix 354
6.5 Finite Element Solution of a Plane Stress Problem 356
6.6 Rectangular Plane Element (Bilinear Rectangle, Q4) 367
Summary Equations 373
References 376
Problems 377
Contents d v
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7 Practical Considerations in Modeling;
Interpreting Results; and Examples
of Plane Stress–Strain Analysis 384
Chapter Objectives 384
Introduction 384
7.1 Finite Element Modeling 385
7.2 Equilibrium and Compatibility of Finite Element Results 398
7.3 Convergence of Solution 402
7.4 Interpretation of Stresses 405
7.5 Static Condensation 407
7.6 Flowchart for the Solution of Plane Stress–Strain Problems 411
7.7 Computer Program-Assisted Step-by-Step Solution, Other Models,
and Results for Plane Stress–Strain Problems 411
References 417
Problems 420
8 Development of the Linear-Strain Triangle Equations 437
Chapter Objectives 437
Introduction 437
8.1 Derivation of the Linear-Strain Triangular Element
Sti¤ness Matrix and Equations 437
8.2 Example LST Sti¤ness Determination 442
8.3 Comparison of Elements 445
Summary Equations 448
References 448
Problems 449
9 Axisymmetric Elements 452
Chapter Objectives 452
Introduction 452
9.1 Derivation of the Sti¤ness Matrix 452
9.2 Solution of an Axisymmetric Pressure Vessel 463
9.3 Applications of Axisymmetric Elements 469
Summary Equations 474
References 476
Problems 476
vi d Contents
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10 Isoparametric Formulation 486
Chapter Objectives 486
Introduction 486
10.1 Isoparametric Formulation of the Bar Element
Sti¤ness Matrix 487
10.2 Isoparametric Formulation of the Plane Quadrilateral Element
Sti¤ness Matrix 492
10.3 Newton-Cotes and Gaussian Quadrature 503
10.4 Evaluation of the Sti¤ness Matrix and Stress Matrix
by Gaussian Quadrature 509
10.5 Higher-Order Shape Functions 515
Summary Equations 525
References 528
Problems 528
11 Three-Dimensional Stress Analysis 534
Chapter Objectives 534
Introduction 534
11.1 Three-Dimensional Stress and Strain 535
11.2 Tetrahedral Element 537
11.3 Isoparametric Formulation 545
Summary Equations 553
References 556
Problems 556
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 Sti¤ness Matrixand 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 590
Contents d vii
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13 Heat Transfer and Mass Transport 599
Chapter Objectives 599
Introduction 599
13.1 Derivation of the Basic Di¤erential Equation 601
13.2 Heat Transfer with Convection 604
13.3 Typical Units; Thermal Conductivities, K; and Heat-Transfer
Coe‰cients, 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 635
13.7 Three-Dimensional Heat Transfer by the Finite
Element Method 638
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 641
13.10 Flowchart and Examples of a Heat-Transfer Program 646
References 653
Problems 654
14 Fluid Flow in Porous Media and Through
Hydraulic Networks; and Electrical Networks
and Electrostatics 674
Chapter Objectives 674
Introduction 674
14.1 Derivation of the Basic Di¤erential Equations 675
14.2 One-Dimensional Finite Element Formulation 680
14.3 Two-Dimensional Finite Element Formulation 692
14.4 Flowchart and Example of a Fluid-Flow Program 697
14.5 Electrical Networks 698
14.6 Electrostatics 702
Summary Equations 716
References 720
Problems 720
viii d Contents
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15 Thermal Stress 728
Chapter Objectives 728
Introduction 728
15.1 Formulation of the Thermal Stress Problem and Examples 728
Reference 752
Summary Equations 753
Problems 755
16 Structural Dynamics and Time-Dependent Heat Transfer 763
Chapter Objectives 763
Introduction 763
16.1 Dynamics of a Spring-Mass System 763
16.2 Direct Derivation of the Bar Element Equations 766
16.3 Numerical Integration in Time 770
16.4 Natural Frequencies of a One-Dimensional Bar 782
16.5 Time-Dependent One-Dimensional Bar Analysis 786
16.6 Beam Element Mass Matrices and Natural Frequencies 791
16.7 Truss, Plane Frame, Plane Stress, Plane Strain, Axisymmetric,
and Solid Element Mass Matrices 798
16.8 Time-Dependent Heat Transfer 803
16.9 Computer Program Example Solutions for Structural Dynamics 810
Summary Equations 819
References 823
Problems 824
Appendix A Matrix Algebra 829
Introduction 829
A.1 Definition of a Matrix 829
A.2 Matrix Operations 830
A.3 Cofactor or Adjoint Method to Determine the Inverse of a Matrix 837
A.4 Inverse of a Matrix by Row Reduction 839
A.5 Properties of Sti¤ness Matrices 841
References 842
Problems 842
Contents d ix
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Appendix B Methods for Solution of Simultaneous
Linear Equations 845
Introduction 845
B.1 General Form of the Equations 845
B.2 Uniqueness, Nonuniqueness, and Nonexistence of Solution 846
B.3 Methods for Solving Linear Algebraic Equations 847
B.4 Banded-Symmetric Matrices, Bandwidth, Skyline,
and Wavefront Methods 858
References 865
Problems 865
Appendix C Equations from Elasticity Theory 867
Introduction 867
C.1 Di¤erential Equations of Equilibrium 867
C.2 Strain/Displacement and Compatibility Equations 869
C.3 Stress–Strain Relationships 871
Reference 874
Appendix D Equivalent Nodal Forces 875
Problems 875
Appendix E Principle of Virtual Work 878
References 881
Appendix F Properties of Structural Steel Shapes 882
Answers to Selected Problems 908
Index 937
x d Contents
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PREFACEd
The purpose of this fifth edition is again to provide an introductory approach to the
finite element method that can be understood by both undergraduate and graduate
students without the usual prerequisites (such as structural analysis and upper level
calculus) required by many 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, although new material on
electrical networks and electrostatics has been included in this edition that should be of
interest to the electrical engineer as well. The concepts are presented in su‰ciently
simple form with numerous example problems logically placed throughout the book,
so that the book serves as a valuable learning aid for students with other backgrounds,
as well as for practicing engineers. The text is geared toward those who want to apply
the finite element method to solve practical physical problems.
General principles are presented for each topic, followed by traditional applications of these principles, which are in turn followed by computer applications where
relevant. This approach is taken to illustrate concepts used for computer analysis of
large-scale problems.
The book proceeds from basic to advanced topics and can be suitably used in a
two-course sequence. Topics include basic treatments of (1) simple springs and bars,
leading to two- and three-dimensional truss analysis; (2) beam bending, leading to plane
frame, grid and space frame analysis; (3) elementary plane stress/strain elements, leading
to more advanced plane stress/strain elements and applications to more complex plane
stress/strain analysis; (4) axisymmetric stress analysis; (5) isoparametric formulation of
the finite element method; (6) three-dimensional stress analysis; (7) plate bending analysis; (8) heat transfer and fluid mass transport; (9) basic fluid flow through porous media
and around solid bodies, hydraulic networks, electrical networks, and electrostatics
analysis; (10) thermal stress analysis; and (11) time-dependent stress and heat transfer.
Additional features include how to handle inclined or skewed supports, beam
element with a nodal hinge, the concept of substructure analysis, the patch test, and
practical considerations in modeling and interpreting results.
The direct approach, the principle of minimum potential energy, and Galerkin’s
residual method are introduced at various stages, as required, to develop the equations
needed for analysis.
Appendices provide material on the following topics: (A) basic matrix algebra
used throughout the text; (B) solution methods for simultaneous equations; (C) basic
xi
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