Download STASA.applied Finite Element Analysis for Engineers.
Applied Finite Element Analysis for Engineers FRANK L. STASA Florida Institute of Technology
CBS Publishing -Japan Ltd. New York Chicago San Francisco Philadelphia Montreal Toronto London Sydney Tokyo Mexico City Rio de Janeiro Madrid
'J Cf 7l To Frank W. and Lena C. Stasa, Ann R. Holt, and to the memory of John A. Holt
HRW Series in Mechanical Engineering
L. S. Fletcher, Series Editor
APPLIED FINITE ELEMENT ANALYSIS FOR ENGINEERS ADVANCED DYNAMICS FOR ENGINEERS
Copyright © 1985 CBS Publishing All rights reserved Address correspondence to 383 Madison Avenue, New York. NY 10017
Library of Congress Cataloging in Publication Data Stasa, Frank L. . Applied finite eledJent analysis for engineers. (HRW series in mechanicalengineering) Includes bibliographies and index. I. Finite element method. I. Title. II. Series. TA347.F5Sn 1985 . 620'.001'515353 85-742 ISBN 0'<)3'-%2737.<) (US';~llege Edition) ISBN 0'<)3·910744·2 (CBS International Edition) This International Edition is not for sale in the United States of America. its dependencies or Canada. Printed in Japan. 1986 5678
CBS COLLEGE PUBLISHING The Dryden Press Saunders College Publishing
Preface Written for senior-year undergraduates and first-year graduate students with solid backgrounds in differential and integral calculus, this book is oriented toward engineers and applied mathematicians. A course in linear algebra is helpful but not essential. Courses in elasticity, heat transfer, and fluid mechanics should facilitate the student's understanding of the applications emphasized in this text. Overall, the author's approach represents a compromise between the purely mathematical and the purely applied developments. Stress or structural analysis is given about the same level of treatment as thermal and fluid flow analysis. However, this book is structured so that it may be used in courses in which the application area is strictly stress analysis or strictly thermal analysis. Moreover, it is also possible to use this book without covering any of the material related to variational calculus and variational formulations. This strategy is made possible by emphasizing the Galerkin weighted-residual method in thermal (and fluid flow) analysis and the principle of virtual displacements in stress analysis. Consequently, this book should be useful to instructors teaching a course on the finite element method to senior undergraduate students. If the nonvariational path is desired, the instructor should omit the following sections: the last part of Sec. 4-2, Sec. 4-3, parts of Sees, 4-4, and Sees. 4-8, 5-3, and 8-6. The first two chapters are introductory in nature. Chapter 1 contains a brief survey of what the finite element method is, as well as a brief history of the method. Chapter 2 contains a review of the necessary mathematical concepts of matrices, vectors, and determinants. The reason for including this review in the text proper was to establish a common ground from which the finite element method could be developed, regardless of the specific background of the engineering student. All aspects of the finite element are explored in Chapter 3 by way of one of the simplest of all engineering applications-the truss. The so-called direct approach is adopted for this purpose and each step in the finite element solution process is given in full detail. For this reason, all students must be exposed to (and indeed should master) Chapter 3, which is the only structural analysis chapter that must be covered by all users of this book. Both twoand three-dimensional trusses are covered, but the former is given broader, more comprehensive treatment. The student is also introduced to computer programming concepts and a two-dimensional truss program. Chapter 4 provides the g>neral framework. for the development of nearly all (nonstructural) finite element models. Here the student is introduced to the concepts of globally based approximations to the true solution of simple ordinary differential equations. Among the methods covered are the Ritz, Rayleigh-Ritz (variational), point collocation, subdomain collocation, least squares, and Galerkin methods. The last four of these methods comprise the class of approximate solution methods known as the weighted-residual methods. Both the Rayleigh-Ritz (variational) and the Galerkin (weighted-residual) methods are extended to piecewise continuous approximations and, hence, to the finite element method itself. The student is introduced, of course, to the concept of shape functions at this point. Chapter 4 concludes with an application in the thermal analysis area: a pin fin (a type of extended
surface). Instructors may wish to have their students review Chapter I before proceeding with the next chapter. Chapter 5 is devoted specifically to the development of finite element models in the stress (or structural) analysis area. A brief review of some of the more important concepts of elasticity is provided and may be skipped by those who have had a prior graduate course on this subject. Depending on the instructor's preference, either of two developments may be used: the principle of minimum potential energy (variational) or the principle of virtual displacement with a simple one-dimensional application: the uniaxial stress member. The results from Chapter 5 are used throughout Chapter 7, the primary stress application chapter. Chapter 6 is, in effect, a catchall chapter, which contains essential material that has not been covered adequately up to this point. Here the student is introduced more formally to the concept of parameter functions (such as displacement and temperature functions) and to the compatibility and completeness requirements that these functions should satisfy. Shape functions (CD-continuous only) are derived and presented for the following types of one-, two-, and three-dimensional elements: two-node lineal (I-D), three-node triangular (2-D), four-node rectangular (2-D), four-node tetrahedral (3-D), and eight-node brick (3-0). Local, normalized coordinates, such as length, area, and volume coordinates, as well as serendipity coordinates, are introduced. Axisymmetric elements are also presented. Three simple integration formulas are given in terms of length, area, and volume coordinates for integrations over lineal, triangular, and tetrahedral elements, respectively. Finally, an alternative to the matrix inversion technique is provided, namely, the active zone equation solver. This method is based on triangular decomposition, forward elimination, and backward substitution, and takes advantage of the banded, and often symmetric, nature of the assemblage stiffness matrix. This method requires that the assemblage stiffness matrix be stored as a column vector (instead of a square matrix). Chapter 7 is the main stress analysis application chapter. Among the topics covered are the following: two-dimensional stress analysis (plane stress and plane strain), axisymmetric stress analysis, three-dimensional stress analysis, and the analysis of beams. The notion of substructuring and condensation is also introduced and working equations are developed. The chapter concludes with a brief description of the development of a two-dimensional stress analysis program. The instructor is referred to the Instructor's Solutions manual for a listing (in FORTRAN) of one version of this program-called program STRESS (the user's manual to the program is also included in the Instructor's Solutions manual). Chapter 8 is the principal thermal and fluid flow analysis chapter, in which the following topics are covered: one-, two-, and three-dimensional thermal analysis, and axisymmetric thermal analysis. Material on variational formulations in two-dimensional problems is provided, but this also may be skipped by those preferring the nonvariational path. Other application areas in Chapter 8 include convective energy transport, two-dimensional potential flow, and two-dimensional incompressible fluid flow. The chapter concludes with a brief description of the development of a two-dimensional, steady-state thermal analysis program. Again the instructor is referred to the Instructor's Solutions manual for a listing (in FORTRAN) of one version of this program-called HEAT (the user's manual to the program is also included in the Instructor's Solutions manual). Chapter 9 introduces higher-order elements and numerical integration (quadrature). The one-, two-, and three-dimensional elements introduced in Chapter 6 are extended to quadratic and cubic order. Subparametric, isoparametric, and superparametric elements are also introduced. Two- and three-dimensional isoparametric formulations are developed for the quadrilateral and triangular elements as well as for the brick and tetrahedral elements. Special formulas are presented, which facilitate greatly the evaluation of the integrals that naturally arise.
In conclusion, Chapter lOis devoted to transient thermal analysis and dynamic structural analysis. The concept of partial discretization is presented and applied to stress analysis and thermal analysis. Lumped and consistent capacitance and mass matrices are discussed. Solution methods are developed based on the finite element method itself (i.e., in time) and on the finite difference method. The result is two- and three-point recurrence schemes for transient thermal analysis and dynamic structural analysis, respectively. The chapter concludes with a brief introduction to modal analysis. Appendix A contains the material property data to be used in the problems, unless otherwise noted in the problem statements. Appendix B contains a short user's manual to the (two-dimensional)truss program along with the program listing (in FORTRAN). Appendix C contains listings of subroutines ACTCOL and UACTCL (and function DOT), which have been used with the written permission of McGraw-Hill and which appear in Professor Zienkiewicz's third edition of The Finite Element Method. The length of the STRESS and HEAT programs and their respective user's manuals precluded their inclusion in this text. Instructors who would like to have copies of all of these programs on floppy disks are encouraged to write to the author. The disk formats available are IBM PC, Apple II series, and 8-inch IBM 3740 standard format for CP/M-based machines. *Other FORTRAN programs that can be obtained on these disk formats include: beam analysis, a TurboPascal** version of the two-dimensional truss program, fin analysis, transient one-dimensional thermal analysis, and transient two-dimensional thermal analysis. The following suggestions are made to instructors teaching on the quarter and semester system. For those teaching on the quarter system: Chapters I to 4 can be covered comfortably during the fall quarter. The winter quarter could be devoted to stress analysis with coverage of Chapters 5, 6, 7, 9, and 10 (omit Sees, 10-4, 10-7, and 10-8). The spring quarter could be devoted to thermal and fluid flow analysis with coverage of Chapters 6, 8, 9, and 10 (omit Sees. 10-3, 10-9, and 10-10). For students who take all three courses, some of the material is necessarily repeated. The author has found this repetition not to be a problem, because these students get a firmer grasp of the material the second time around. For those teaching on the semester system: Chapters I to 6 could be covered comfortably during the first semester, and Chapters 7 to 10 during the second semester. This pace allows sufficient class time for the discussion of computer programming techniques. A useful project for the first course is to have the students modify the two-dimensional truss program in Appendix B so that it could be used in three-dimensional truss applications. The Instructor's Solutions manual contains a listing (in FORTRAN) of a two-dimensional, static stress analysis program (for Chapter 8). The manual is available from the publisher upon proper written request. The instructors may find it useful to distribute copies of these programs to their classes. The author wishes to thank all of the students who have used the original notes and class-tested the text manuscript. Their comments and suggestions were taken seriously, and their words of encouragement will always be remembered. Deserving of special recognition is Jay A. Buckman, who did an outstanding job of proofreading the page proofs. A special debt of gratitude is owed to General Herbert McChrystal, who first suggested this project. Very special thanks are extended to U. Shripathi Kamath, who had the monumental task of providing the solutions manual to the text, and to Michael Weaver for converting the FORTRAN version of the truss program into Pascal. The author is also thankful to the editors
*CP/M is registered trademark of Digital Research, Inc. **TurboPascal is a trademark of Borland International.
John J. Beck, Lynn Contrucci, and Rachel Hockett for their help and cooperation in publishing the manuscript. Finally, and most important of all, the author is once again deeply indebted to his wife, Donna, who meticulously typed every page of the manuscript and without whose encouragement this book would probably not have been completed, and to Lisa Ann, for helping to put this project into proper perspective and for providing the perpetual light at the end of tunnel. Frank L. Stasa
Introduction What Is the Finite Element Method? Discretization Relationship to the Finite-Difference Method Advantages and Disadvantages of the Finite Element Method Brief History of the Finite Element Method Remarks References