The Finite Element Method in Engineering This page intentionally left blank The Finite Element Method in Engineering Fifth Edition Singiresu S. Rao Professor and Chairman Department of Mechanical and Aerospace Engineering University of Miami, Coral Gables, Florida, USA AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Butterworth-Heinemann is an imprint of Elsevier Butterworth–Heinemann is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK © 2011 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.
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Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. MATLAB® is a trademark of The MathWorks, Inc.
and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. Library of Congress Cataloging-in-Publication Data Application submitted.
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN: 978-1-85617-661-3 For information on all Butterworth–Heinemann publications, visit our web site at: www. Typeset by: diacriTech, Chennai, India Printed in the United States of America 10 11 12 13 14 10 9 8 7 6 5 4 3 2 1 This page intentionally left blank CONTENTS PREFACE. xiii PART 1 • Introduction CHAPTER 1 Overview of Finite Element Method .3 General Applicability of the Method .4 Engineering Applications of the Finite Element Method.5 General Description of the Finite Element Method .6 One-Dimensional Problems with Linear Interpolation Model.7 One-Dimensional Problems with Cubic Interpolation Model.8 Derivation of Finite Element Equations Using a Direct Approach .9 Commercial Finite Element Program Packages .10 Solutions Using Finite Element Software.
40 PART 2 • Basic Procedure CHAPTER 2 Discretization of the Domain .2 Basic Element Shapes .4 Node Numbering Scheme .5 Automatic Mesh Generation. 65 CHAPTER 3 Interpolation Models .2 Polynomial Form of Interpolation Functions.3 Simplex, Complex, and Multiplex Elements .4 Interpolation Polynomial in Terms of Nodal Degrees of Freedom .5 Selection of the Order of the Interpolation Polynomial .7 Linear Interpolation Polynomials in Terms of Global Coordinates .8 Interpolation Polynomials for Vector Quantities .9 Linear Interpolation Polynomials in Terms of Local Coordinates .10 Integration of Functions of Natural Coordinates. 109 CHAPTER 4 Higher Order and Isoparametric Elements .2 Higher Order One-Dimensional Elements .3 Higher Order Elements in Terms of Natural Coordinates .4 Higher Order Elements in Terms of Classical Interpolation Polynomials .5 One-Dimensional Elements Using Classical Interpolation Polynomials.6 Two-Dimensional (Rectangular) Elements Using Classical Interpolation Polynomials .8 Comparative Study of Elements. 148 CHAPTER 5 Derivation of Element Matrices and Vectors .3 Solution of Equilibrium Problems Using Variational (Rayleigh-Ritz) Method .4 Solution of Eigenvalue Problems Using Variational (Rayleigh-Ritz) Method .5 Solution of Propagation Problems Using Variational (Rayleigh-Ritz) Method .6 Equivalence of Finite Element and Variational (Rayleigh-Ritz) Methods.7 Derivation of Finite Element Equations Using Variational (Rayleigh-Ritz) Approach .8 Weighted Residual Approach .9 Solution of Eigenvalue Problems Using Weighted Residual Method.10 Solution of Propagation Problems Using Weighted Residual Method.11 Derivation of Finite Element Equations Using Weighted Residual viii (Galerkin) Approach .12 Derivation of Finite Element Equations Using Weighted Residual (Least Squares) Approach .13 Strong and Weak Form Formulations.
189 CHAPTER 6 Assembly of Element Matrices and Vectors and Derivation of System Equations.2 Assemblage of Element Equations.3 Incorporation of Boundary Conditions .5 Multipoint Constraints—Penalty Method .6 Symmetry Conditions—Penalty Method. 228 CHAPTER 7 Numerical Solution of Finite Element Equations .2 Solution of Equilibrium Problems .3 Solution of Eigenvalue Problems.4 Solution of Propagation Problems.5 Parallel Processing in Finite Element Analysis. 268 PART 3 • Application to Solid Mechanics Problems CHAPTER 8 Basic Equations and Solution Procedure .2 Basic Equations of Solid Mechanics.3 Formulations of Solid and Structural Mechanics .4 Formulation of Finite Element Equations (Static Analysis) .5 Nature of Finite Element Solutions. 303 CHAPTER 9 Analysis of Trusses, Beams, and Frames.2 Space Truss Element .4 Space Frame Element .5 Characteristics of Stiffness Matrices.
338 CHAPTER 10 Analysis of Plates .2 Triangular Membrane Element .3 Numerical Results with Membrane Element .4 Quadratic Triangle Element .5 Rectangular Plate Element (In-plane Forces) .6 Bending Behavior of Plates.7 Finite Element Analysis of Plates in Bending .8 Triangular Plate Bending Element .9 Numerical Results with Bending Elements .10 Analysis of Three-Dimensional Structures Using Plate Elements. 386 CHAPTER 11 Analysis of Three-Dimensional Problems.4 Analysis of Solids of Revolution. 413 CHAPTER 12 Dynamic Analysis.1 Dynamic Equations of Motion .2 Consistent and Lumped Mass Matrices .3 Consistent Mass Matrices in a Global Coordinate System .4 Free Vibration Analysis .5 Dynamic Response Using Finite Element Method .6 Nonconservative Stability and Flutter Problems. 461 PART 4 • Application to Heat Transfer Problems CHAPTER 13 Formulation and Solution Procedure .2 Basic Equations of Heat Transfer .3 Governing Equation for Three-Dimensional Bodies .4 Statement of the Problem .5 Derivation of Finite Element Equations.
480 CHAPTER 14 One-Dimensional Problems .2 Straight Uniform Fin Analysis .3 Convection Loss from End Surface of Fin .3 Tapered Fin Analysis .4 Analysis of Uniform Fins Using Quadratic Elements .5 Unsteady State Problems .6 Heat Transfer Problems with Radiation. 507 CHAPTER 15 Two-Dimensional Problems.3 Unsteady State Problems. 526 CHAPTER 16 Three-Dimensional Problems .3 Three-Dimensional Heat Transfer Problems.4 Unsteady State Problems. 541 PART 5 • Application to Fluid Mechanics Problems CHAPTER 17 Basic Equations of Fluid Mechanics .2 Basic Characteristics of Fluids .3 Methods of Describing the Motion of a Fluid .5 Equations of Motion or Momentum Equations .6 Energy, State, and Viscosity Equations .8 Inviscid Fluid Flow.
564 CHAPTER 18 Inviscid and Incompressible Flows .2 Potential Function Formulation .3 Finite Element Solution Using the Galerkin Approach .4 Stream Function Formulation. 584 CHAPTER 19 Viscous and Non-Newtonian Flows.2 Stream Function Formulation (Using Variational Approach) .3 Velocity–Pressure Formulation (Using Galerkin Approach) .4 Solution of Navier–Stokes Equations.5 Stream Function–Vorticity Formulation .6 Flow of Non-Newtonian Fluids. 607 PART 6 • Solution and Applications of Quasi-Harmonic Equations CHAPTER 20 Solution of Quasi-Harmonic Equations .2 Finite Element Equations for Steady-State Problems .3 Solution of Poisson’s Equation .4 Transient Field Problems. 622 CONTENTS PART 7 • ABAQUS and ANSYS Software and MATLAB®Programs for Finite Element Analysis CHAPTER 21 Finite Element Analysis Using ABAQUS.
632 CHAPTER 22 Finite Element Analysis Using ANSYS .2 GUI Layout in ANSYS .4 Finite Element Discretization .5 System of Units .6 Stages in Solution. 667 CHAPTER 23 MATLAB Programs for Finite Element Analysis .1 Solution of Linear System of Equations Using Choleski Method .2 Incorporation of Boundary Conditions .3 Analysis of Space Trusses.4 Analysis of Plates Subjected to In-plane Loads Using CST Elements .5 Analysis of Three-Dimensional Structures Using CST Elements .6 Temperature Distribution in One-Dimensional Fins .7 Temperature Distribution in One-Dimensional Fins Including Radiation Heat Transfer.8 Two-Dimensional Heat Transfer Analysis .9 Confined Fluid Flow around a Cylinder Using Potential xi Function Approach .10 Torsion Analysis of Shafts. 702 Appendix: Green-Gauss Theorem (Integration by Parts in Two and Three Dimensions). 707 This page intentionally left blank PREFACE The finite element method is a numerical method that can be used for the accurate solution of complex engineering problems.
Although the origins of the method can be traced to several centuries back, most of the computational details have been developed in mid-1950s, primarily in the context of the analysis of aircraft structures. Thereafter, within a decade, the potential of the method for the solution of different types of applied science and engineering problems was recognized. Over the years, the finite element technique has been so well established that today, it is considered to be one of the best methods for solving a wide variety of practical problems efficiently. In addition, the method has become one of the active research areas not only for engineers but also for applied mathematicians.
One of the main reasons for the popularity of the method in different fields of engineering is that once a general computer program is written, it can be used for the solution of a variety of problems simply by changing the input data. APPROACH OF THE BOOK The objective of writing this book is to introduce the various aspects of the finite element method as applied to the solution of engineering problems in a systematic and simple manner. It develops each of the techniques and ideas from basic principles. New concepts are illustrated with simple examples wherever possible.
An introduction to commercial software systems, ABAQUS and ANSYS, including some sample applications with images/ xiii output, is also presented in two separate chapters. In addition, several MATLAB programs are given along with examples to illustrate the use of the programs in a separate chapter. After studying the material presented in the book, the reader will not only be able to understand the current literature on the finite element method but also be in a position to solve finite element problems using commercial software such as ABAQUS and ANSYS, use the MATLAB programs given in the book to solve a variety of finite element problems from different areas, and, also, if needed, be able to develop short programs for the solution of engineering problems. NEW TO THIS EDITION In this edition some topics are modified and rewritten, and many new topics are added.
Most of the modifications and additions were suggested by the users of the book and by reviewers. Some important features of the current edition are the following: ● 135 illustrative examples are included compared to 37 in the previous edition. ● 680 problems are included compared to 350 in the previous edition. The solution of most of the problems is given in the Solutions Manual available to instructors who use the book as a textbook.
● 10 MATLAB programs, available at the web site of the book, are given for the solution of different types of finite element problems. ● Expanded coverage of finite element applications to different areas of engineering is given. ● Several new concepts and topics such as patch test, strong and weak formulations, penalty method, multipoint constraints, symmetry conditions, rigid elements and quadratic triangle element and rectangular element under inplane loads are presented with examples. PREFACE ORGANIZATION The book is divided into 23 chapters and an appendix.
Chapter 1 gives an introduction and overview of the finite element method. The basic approach and the generality of the method are illustrated through several simple examples. Chapters 2 through 7 describe the basic finite element procedure and the solution of the resulting equations.
