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508 lines
11 KiB
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# A First Course in the Finite Element Method
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Fourth Edition
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# Daryl L. Logan
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University of Wisconsin–Platteville
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THOMSON
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# THOMSON
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# A First Course in the Finite Element Method, Fourth Edition by Daryl L. Logan
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Associate Vice-President and Editorial Director:
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Evelyn Veitch
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Publisher:
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Chris Carson
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Kamilah Reid Burrell/ Hilda Gowans
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Interior Design:
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RPK Editorial Services
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Cover Design:
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Andrew Adams
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Compositor:
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International Typesetting and Composition
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Printer:
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R. R. Donnelley
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Cover Images:
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Courtesy of ALGOR, Inc.
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COPYRIGHT # 2007 by Nelson, a division of Thomson Canada Limited.
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Printed and bound in the United States 1 2 3 4 07 06
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For more information contact Nelson, 1120 Birchmount Road, Toronto, Ontario, Canada, M1K 5G4. Or you can visit our Internet site at http://www.nelson.com
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Library of Congress Control Number: 2006904397
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ISBN: 0-534-55298-6
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ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transcribed, or used in any form or by any means—graphic, electronic, or mechanical, including photocopying, recording, taping, Web distribution, or information storage and retrieval systems— without the written permission of the publisher.
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For permission to use material from this text or product, submit a request online at www.thomsonrights.com
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Every effort has been made to trace ownership of all copyright material and to secure permission from copyright holders. In the event of any question arising as to the use of any material, we will be pleased to make the necessary corrections in future printings.
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North America
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Nelson
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1120 Birchmount Road
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Toronto, Ontario M1K 5G4
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Canada
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Asia
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Thomson Learning
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5 Shenton Way #01-01
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UIC Building
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Singapore 068808
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Thomson Learning
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102 Dodds Street
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Southbank, Victoria
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Thomson Learning
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Thomson Learning
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Seneca, 53
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Mexico
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Spain
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Paraninfo
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Calle/Magallanes, 25
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28015 Madrid, Spain
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# Contents
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# 1 Introduction 1
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Prologue 1
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1.1 Brief History 2
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1.2 Introduction to Matrix Notation 4
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1.3 Role of the Computer 6
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1.4 General Steps of the Finite Element Method 7
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1.5 Applications of the Finite Element Method 15
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1.6 Advantages of the Finite Element Method 19
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1.7 Computer Programs for the Finite Element Method 23
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References 24
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Problems 27
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# 2 Introduction to the Stifness (Displacement) Method 28
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Introduction 28
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2.1 Definition of the Sti¤ness Matrix 28
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2.2 Derivation of the Sti¤ness Matrix for a Spring Element 29
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2.3 Example of a Spring Assemblage 34
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2.4 Assembling the Total Sti¤ness Matrix by Superposition (Direct Sti¤ness Method) 3 7
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2.5 Boundary Conditions 39
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2.6 Potential Energy Approach to Derive Spring Element Equations 5 2
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References 60
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Problems 61
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# 3 Development of Truss Equations
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Introduction 6 5
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3.1 Derivation of the Sti¤ness Matrix for a Bar Element in Local Coordinates 66
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3.2 Selecting Approximation Functions for Displacements 72
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3.3 Transformation of Vectors in Two Dimensions 75
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3.4 Global Sti¤ness Matrix 78
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3.5 Computation of Stress for a Bar in the x-y Plane 82
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3.6 Solution of a Plane Truss 84
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3.7 Transformation Matrix and Sti¤ness Matrix for a Bar in Three-Dimensional Space 92
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3.8 Use of Symmetry in Structure 100
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3.9 Inclined, or Skewed, Supports 103
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3.10 Potential Energy Approach to Derive Bar Element Equations 109
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3.11 Comparison of Finite Element Solution to Exact Solution for Bar 120
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3.12 Galerkin’s Residual Method and Its Use to Derive the One-Dimensional Bar Element Equations 124
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3.13 Other Residual Methods and Their Application to a One-Dimensional Bar Problem 127
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References 132
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Problems 132
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# 4 Development of Beam Equations
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Introduction 151
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4.1 Beam Sti¤ness 152
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4.2 Example of Assemblage of Beam Sti¤ness Matrices 161
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4.3 Examples of Beam Analysis Using the Direct Sti¤ness Method 163
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4.4 Distributed Loading 175
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4.5 Comparison of the Finite Element Solution to the Exact Solution for a Beam 188
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4.6 Beam Element with Nodal Hinge 194
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4.7 Potential Energy Approach to Derive Beam Element Equations 199
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4.8 Galerkin’s Method for Deriving Beam Element Equations 201
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References 203
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Problems 204
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# 5 Frame and Grid Equations 214
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Introduction 214
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5.1 Two-Dimensional Arbitrarily Oriented Beam Element 214
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5.2 Rigid Plane Frame Examples 218
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5.3 Inclined or Skewed Supports—Frame Element 237
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5.4 Grid Equations 238
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5.5 Beam Element Arbitrarily Oriented in Space 255
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5.6 Concept of Substructure Analysis 269
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References 275
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Problems 275
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# 6 Development of the Plane Stress and Plane Strain Stiffness Equations 304
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Introduction 304
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6.1 Basic Concepts of Plane Stress and Plane Strain 305
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6.2 Derivation of the Constant-Strain Triangular Element Sti¤ness Matrix and Equations 310
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6.3 Treatment of Body and Surface Forces 324
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6.4 Explicit Expression for the Constant-Strain Triangle Sti¤ness Matrix 329
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6.5 Finite Element Solution of a Plane Stress Problem 331
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References 342
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Problems 343
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# 7 Practical Considerations in Modeling; Interpreting Results; and Examples of Plane Stress/Strain Analysis 350
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Introduction 350
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7.1 Finite Element Modeling 350
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7.2 Equilibrium and Compatibility of Finite Element Results 363
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7.3 Convergence of Solution 367
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7.4 Interpretation of Stresses 368
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7.5 Static Condensation 369
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7.6 Flowchart for the Solution of Plane Stress/Strain Problems 374
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7.7 Computer Program Assisted Step-by-Step Solution, Other Models, and Results for Plane Stress/Strain Problems 374
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References 381
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Problems 382
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# 8 Development of the Linear-Strain Triangle Equations 398
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Introduction 398
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8.1 Derivation of the Linear-Strain Triangular Element Sti¤ness Matrix and Equations 398
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8.2 Example LST Sti¤ness Determination 403
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8.3 Comparison of Elements 406
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References 409
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Problems 409
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# 9 Axisymmetric Elements 412
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Introduction 412
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9.1 Derivation of the Sti¤ness Matrix 412
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9.2 Solution of an Axisymmetric Pressure Vessel 422
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9.3 Applications of Axisymmetric Elements 428
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References 433
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Problems 434
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# 10 Isoparametric Formulation 443
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Introduction 443
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10.1 Isoparametric Formulation of the Bar Element Sti¤ness Matrix 444
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10.2 Rectangular Plane Stress Element 449
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10.3 Isoparametric Formulation of the Plane Element Sti¤ness Matrix 452
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10.4 Gaussian and Newton-Cotes Quadrature (Numerical Integration) 463
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10.5 Evaluation of the Sti¤ness Matrix and Stress Matrix by Gaussian Quadrature 469
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10.6 Higher-Order Shape Functions 475
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References 484
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Problems 484
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# 11 Three-Dimensional Stress Analysis 490
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Introduction 490
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11.1 Three-Dimensional Stress and Strain 490
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11.2 Tetrahedral Element 493
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11.3 Isoparametric Formulation 501
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References 508
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Problems 509
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# 12 Plate Bending Element 514
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Introduction 514
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12.1 Basic Concepts of Plate Bending 514
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12.2 Derivation of a Plate Bending Element Sti¤ness Matrix and Equations 519
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12.3 Some Plate Element Numerical Comparisons 523
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12.4 Computer Solution for a Plate Bending Problem 524
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References 528
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Problems 529
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# 13 Heat Transfer and Mass Transport 534
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Introduction 534
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13.1 Derivation of the Basic Di¤erential Equation 535
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13.2 Heat Transfer with Convection 538
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13.3 Typical Units; Thermal Conductivities, K; and Heat-Transfer Coe‰cients, h 539
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13.4 One-Dimensional Finite Element Formulation Using a Variational Method 540
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13.5 Two-Dimensional Finite Element Formulation 555
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13.6 Line or Point Sources 564
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13.7 Three-Dimensional Heat Transfer Finite Element Formulation 566
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13.8 One-Dimensional Heat Transfer with Mass Transport 569
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13.9 Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin’s Method 569
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13.10 Flowchart and Examples of a Heat-Transfer Program 574
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References 577
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Problems 577
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# 14Fluid Flow 593
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Introduction 593
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14.1 Derivation of the Basic Di¤erential Equations 594
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14.2 One-Dimensional Finite Element Formulation 598
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14.3 Two-Dimensional Finite Element Formulation 606
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14.4 Flowchart and Example of a Fluid-Flow Program 611
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References 612
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Problems 613
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# 15 Thermal Stress 617
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Introduction 617
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15.1 Formulation of the Thermal Stress Problem and Examples 617
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Reference 640
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Problems 641
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# 16 Structural Dynamics and Time-Dependent Heat Transfer647
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Introduction 647
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16.1 Dynamics of a Spring-Mass System 647
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16.2 Direct Derivation of the Bar Element Equations 649
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16.3 Numerical Integration in Time 653
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16.4 Natural Frequencies of a One-Dimensional Bar 665
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16.5 Time-Dependent One-Dimensional Bar Analysis 669
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16.6 Beam Element Mass Matrices and Natural Frequencies 674
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16.7 Truss, Plane Frame, Plane Stress/Strain, Axisymmetric, and Solid Element Mass Matrices 681
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16.8 Time-Dependent Heat Transfer 686
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16.9 Computer Program Example Solutions for Structural Dynamics 693
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References 702
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Problems 702
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# Appendix A Matrix Algebra
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708
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Introduction 708
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A.1 Definition of a Matrix 708
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A.2 Matrix Operations 709
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A.3 Cofactor or Adjoint Method to Determine the Inverse of a Matrix 716
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A.4 Inverse of a Matrix by Row Reduction 718
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References 720
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Problems 720
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# Appendix B Methods for Solution of Simultaneous Linear Equations
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722
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Introduction 722
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B.1 General Form of the Equations 722
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B.2 Uniqueness, Nonuniqueness, and Nonexistence of Solution 723
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B.3 Methods for Solving Linear Algebraic Equations 724
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B.4 Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methods 735
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References 741
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Problems 742
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# Appendix C Equations from Elasticity Theory
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744
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Introduction 744
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C.1 Di¤erential Equations of Equilibrium 744
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C.2 Strain/Displacement and Compatibility Equations 746
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C.3 Stress/Strain Relationships 748
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Reference 751
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Appendix D Equivalent Nodal Forces 752
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Problems 752
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Appendix E Principle of Virtual Work 755
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References 758
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Appendix F Properties of Structural Steel and Aluminum Shapes 759
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Answers to Selected Problems 773
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Index 799
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