Boundary Methods: Elements, Contours, and Nodes (Dekker Mechanical Engineering)

PREFACE

The finite element method is firmly established as a powerful and popular analysis tool. It is applied to many different problems of continua but is most widely used for structural mechanics. Accordingly, structural mechanics is emphasized in this book, with lesser excursions into other areas such as heat conduction. The finite element literature is very large. In a book this size it would scarcely be possible even to list all publications, let alone discuss all useful procedures. -This text is introductory and is oriented more toward the eventual practitioner than toward the theoretician. The book contains enough material for a twosemester course. We assume that the reader has the following background. Undergraduate courses in calculus, statics, dynamics, and mechanics of materials must be mastered. Matrix operations (summarized in Appendix A) must be understood. More advanced studies-theory of elasticity, energy methods, numerical analysis, and so on-are not essential. Occasionally these studies must be called upon, but only for their elementary concepts. the specific elements discussed are often quite good, but we do not claim that they are the best available. Rather. these elements illustrate useful concepts and procedures. Similarly, blocks of Fortran code in the book illustrate the steps of an element formulation, of an algorithm for equation solving, or of finite element bookkeeping, but they may not be the most efficient coding available. These blocks of code can form the basis of various semester projects if -so desired. However, the principal purpose of most of these blocks of code is to state precisely the content of certain procedures, and they thereby serve as aids to understanding. Software entitled FEMCOD is intended for use with the book. FEMCOD is a "framework" program for time-independent finite element analysis: it provides the machinery for input of data, assembly of elements, assignment of loads and boundary conditions, and solution of equations. The user may supply coding for a particular element and for postprocessing (such as stress calculation). To institutions that adopt this textbook, FEMCOD, with instructions for use and examples, is available on diskette from the publisher (John Wiley & Sons, Inc., 605 Third Avenue, New York, N.Y., 10158). Our presentation of structural dynamics is based partially on the finite element course notes of Ted Belytschko, We gratefully acknowledge his advice and assistance. The inspiration for the discussion of optimal lumping came originally from Isaac Fried. We are also grateful to T. J. R. Hughes, W. K. Liu, and V. Snyder for their insights. Not the least of our thanks is to Beth Brown, who typed and retyped with her usual intelligence and dependability, despite substantial other commitments, and without ever suggesting that the task might be tiresome.

Madison, Wisconsin October 1988

R.D.COOK D. S. MALKUS M. E. PLESHA

vii

CONCEPTS

AND· APPLICATIONS OF FINITE ELEMENT ANALYSIS

.,,

CONCEPT!';. AND' ,,~',

THIRD EDITION

APPLICATlt~Ns

OF

ANAt~YSIS

FINITE ELEMENT

ROBERT D. COOK , DAVID S. MALKUS ~ ~

and

;l,

..."

c

~~~~

_

Wi~::o,}"I','

;

-Mnrlison ,

hJS-&.s:IA~L E. PLESHA

., e!,:;~I~~,:~~'~~(~I~IFy0bf Wisconsin-Madison

1m WILEY

JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore

... ,.~!

OFlIHT

I

Copyright © 1974, 1981, 1989, by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Sections 107 and 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons. Library of Congress Cataloging in Publication Data:

Cook, Robert Davis. Concepts and applications of finite element analysis. Bibliography: p. Includes index. I. Structural analysis (Engineering) 2. Finite element method. 1. Malkus, David S. n. Plesha, Michael E. m. Title. TA646.C66 1989 624.1'71 88-27929

10 9 8 7 6 5 4 3 2 t

About the Authors Robert D. Cook received his Ph.D. degree from the University of Illinois in 1963. He then 'went to the 'University of Wisconsin-Madison, where he is Professor of Engineering Mechanics. His research interests include stress analysis and finite element methods. He is a member of the American Society of Mechanical Engineers. With Warren C. Young, he is coauthor of Advanced Mechanics of Materials (Macmillan, 1985). The first edition of Concepts and Applications of Finite Element Analysis was published in 1974 and the second in 1981, both with Dr. Cook as sole author. David S. Malkus received his Ph.D. from Boston University in 1976. He spent two years at the National Bureau of Standards and seven years in the Mathematics Department ofIlIinois Institute ofTechnology . He is now Professor of Engineering Mechanics and a professor in the Center for Mathematical Sciences at the University of Wisconsin-Madison. His research interests concern the application ofthe finite element method to problems of structural and continuum mechanics, in particular the flow of non-Newtonian fluids. He is a member of the Rheology Research Center (University of Wisconsin-Madison), the American Academy of Mechanics, the Society for Industrial and Applied Mathematics, and the Society of Rheology. Michael E. Plesha received his B.S. degree from the University of Illinois at Chicago, and his M.S. and Ph.D. degrees from Northwestern University, the Ph.D. degree in ]983. After a short stay at Michigan Technological University, he joined the Engineering Mechanics Department at the University of WisconsinMadison, where he is an associate professor. His research interests include constitutive modeling and finite element analysis of contact-friction problems, transient finite element analysis, and geomechanics.

y

PREFACE

The finite element method is firmly established as a powerful and popular analysis tool. It is applied to many different problems of continua but is most widely used for structural mechanics. Accordingly, structural mechanics is emphasized in this book, with lesser excursions into other areas such as heat conduction. The finite element literature is very large. In a book this size it would scarcely be possible even to list all publications, let alone discuss all useful procedures. -This text is introductory and is oriented more toward the eventual practitioner than toward the theoretician. The book contains enough material for a twosemester course. We assume that the reader has the following background. Undergraduate courses in calculus, statics, dynamics, and mechanics of materials must be mastered. Matrix operations (summarized in Appendix A) must be understood. More advanced studies-theory of elasticity, energy methods, numerical analysis, and so on-are not essential. Occasionally these studies must be called upon, but only for their elementary concepts. The specific elements discussed are often quite good, but we do not claim that they are the best available. Rather, these elements illustrate useful concepts and procedures. Similarly, blocks of Fortran code in the book illustrate the steps of an element formulation, of an algorithm for equation solving, or of finite element bookkeeping, but they may not be the most efficient coding available. These blocks of code can form the basis of various semester projects if .so desired. However, the principal purpose of most of these blocks of code is to state precisely the content of certain procedures, and they thereby serve as aids to understanding. Software entitled FEMCOD is intended for use with the book. FEMCOD is a "framework" program for time-independent finite element analysis: it provides the machinery for input of data, assembly of elements, assignment of loads and boundary conditions, and solution of equations. The user may supply coding for a particular element and for postprocessing (such as stress calculation). To institutions that adopt this textbook, FEMCOD, with instructions for use and examples, is available on diskette from the publisher (J ohn Wiley & Sons, Inc., 605 Third Avenue, New York, N.Y., 10158). Our presentation of structural dynamics is based partially on the finite element course notes of Ted Belytschko. We gratefully acknowledge his advice and assistance. The inspiration for the discussion of optimal lumping came originally from Isaac Fried. We are also grateful to T. J. R. Hughes, W. K. Liu, and V. Snyder for their insights. Not the least of our thanks is to Beth Brown, who typed and retyped with her usual intelligence and dependability, despite substantial other commitments, and without ever suggesting that the task might be tiresome.

Madison, Wisconsin October 1988

R.D.COOK D. S. MALKUS M. E. PLESHA

vii

CONTENTS

NOTATION Chapter} 1.1 1.2 1.3 1.4 l.S 1.6 1. 7 1.8

INTRODUCTION

The Finite Element Method 1 The Element Characteristic Matrix 7 Element Assembly and Solution for Unknowns 11 Summary of Finite Element History 14 Strain-Displacement Relations 15 Theory of Stress and Deformation 17 Stress-Strain-Temperature Relations 20 Warning: The Computed Answer May Be Wrong 24 Problems 25

Chapter Z THE STIFFNESS METHOD AND THE PLANE ___ ~_/-TRUSS 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11

2.12 2.13

31

Introduction 31 Structure Stiffness Equations 32 Properties of [K). Solution for Unknowns 34 Element Stiffness Equations 36 Assembly of Elements. Plane Truss Example 38 Assembly Regarded as Satisfying Equilibrium 40 Assembly as Dictated by Node Numbers 41 Node Numbering That Exploits Matrix Sparsity 44 Automatic Assignment of Node Numbers 47 Displacement Boundary Conditions 48 Gauss Elimination Solution of Equations 53 Stress Computation. Support Reactions 55 Summary of Procedure 57 Problems 59

,t~~p-t~-0J -~

3.1 3.2 3.3 3.4

1

STATIONARY PRINCIPLES, THE RAYLEIGH-RITZ METHOD, AND INTERPOLATION

69

Introduction 69 Principle of Stationary Potential Energy 70 Problems Having Many D.O.F. 73 Potential Energy of an Elastic Body 75

ix

X

3.5 3.6 3.7 3.8 3.9" 3.10 3.11 3.12 3.13

CONTENTS

The Rayleigh-Ritz Method 78 Comments on the Rayleigh-Ritz Method Based on Assumed Displacement Fields 81 Stationary Principles and Governing Equations 83 A Piecewise Polynomial Field 88 Finite Element Form of the Rayleigh-Ritz Method 90 Finite Element Formulations Derived from a Functional 93 Interpolation 95 Shape Functions for Co Elements 96 Shape Functions for C" Elements 99 Problems 101

Chapter 4 DISPLACEMENT·BASED ELEMENTS FOR STRUCTURAL MECHANICS 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

109

Formulas for Element Matrices [k] and {re} 109 Overview of Element Stiffness Matrices 113 Consistent Element Nodal Loads {re } 118 Equilibrium and Compatibility in the Solution 124 Convergence Requirements 126 The Patch Test 129 Stress Calculation 132 Other Formulation Methods 136 Problems 137

. Chapter 5 STRAIGHT-SIDED TRIANGLES AND TETRAHEDRA 5.1 5.2 5.3

5.4 5.5 5.6

Natural Coordinates (Linear) 147 Natural Coordinates (Area and Volume) 149 Interpolation Fields for Plane Triangles 153 The Linear Triangle 154 The Quadratic Triangle 157 The Quadratic Tetrahedron 159 Problems 159

Chapte~ 6 6.1 6.2 6.3 6.4 6.5 6.6

147

THE ISOPARAMETRIC FORMULATION

. Introduction 163 An Isoparametric Bar Element 164 Plane Bilinear Isoparametric Element 166 Summary of Gauss Quadrature 170 Computer Subroutines for the Bilinear Isoparametric Element Quadratic Plane Elements 176

163

173

xi

CONTENTS

6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14

Hexahedral (Solid) Isoparametric Elements 180 Triangular Isopararnetric Elements 182 Consistent Element Nodal Loads [r.} 185 The Validity of Isoparametric Elements 186 Appropriate Order of Quadrature 188 Element and Mesh Instabilities 190 Remarks on Stress Computation 194 Examples. Effect of Element Geometry 196' Problems 199

• Chapter 7.: COORDINATE TRANSFORMATION -.~~~---"

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

Introduction 209 Transformation of Vectors 209 Transformation of Stress, Strain, and Material Properties Transformation of Stiffness Matrices 213 Examples: Transformation of Stiffness Matrices 214 Inclined Support 216 Joining Dissimilar Elements to One Another 218 Rigid Links. Rigid Elements 220 Problems 222

~ Chapt:~~~ .~

'-

-

-.

8.'1

8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16

209

-~

.',-

TOPICS IN STRUCTURAL MECHANICS

D.O.F. Within Elements. Condensation 228 Condensation and Recovery Algorithms 231 Parasitic Shear. Incompatible Elements 232 Rotational D.O.F. in Plane Elements 236 Assumed-Stress Hybrid Formulation 239 A Plane Hybrid Triangle with Rotational D.O.F. User-Defined Elements. Elastic Kernel 244 Higher Derivatives as Nodal D.O.F. 246 Fracture Mechanics. Singularity Elements 247 Elastic Foundations 250 Media of Infinite Extent 252 Finite Elements and Finite Differences 256 Reanalysis Methods 256 Substructuring 257 Structural Symmetry 260 Cyclic Symmetry 262 Problems 263

242

211

228

xii

CONTENTS

. Chapter' 9 CONSTRAINTS Constraints. Transformations 272 Lagrange Multipliers 275 Penalty Functions 276 Naturally Arising Penalty Formulations. Numerical Integration and Constraints 278 Constraint Counting 283 Additional Techniques for Incompressible Media 285 Problems 288

9;1 9.2 9.3 9.4 9.5 9.6

\

272

Chapter 10/S0LIDS OF REVOLUTION ------_._~

.._-

10.1 10.2 10.3 10.4 10.5 10.6 10.7

Introduction 293 Elasticity Relations for Axial Symmetry 294 Finite Elements for Axial Symmetry 295 Fourier Series 298 Loads Without Axial Symmetry: Introduction 301 Loads Without Axial Symmetry: Element Matrices 304 Related Problems 307 Problems 308

GhapterAl "'---".~

11.1 11.2 11.3

11.4 11.5

293

-

..'---

BENDING OF FLAT PLATES

314

Plate-Bending Theory 314 Finite Elements for Plates 319 Mindlin Plate Elements 323 A Triangular Discrete Kirchhoff Element 328 Boundary Conditions and Test Cases 332

Problems

335

Chapter 12. SHELLS

12.r- ~Shell Geometry and Behavior. 12.2 12.3 12.4 12.5

340 Shell Elements

340

Circular Arches and Arch Elements 343 Flat Elements for Shells 351 Shells of Revolution 352 Isoparametric General Shell Elements 358

,~l~ms

362

Cha~' pte. r 13) FINITE ELEMENTS IN DYNAMICS AND ~~

13.1 13.2 13.3

VIBRATIONS

Introduction 367 Dynamic Equations. Mass and Damping Matrices Mass Matrices, Consistent and Diagonal 370

367

368

xiii

CONTENTS

13.4 13.5 13.6 13.7 13.8 13.9 13.10 13.11 13.12 13.13 13.14

Damping 376 Natural Frequencies and Mode Shapes 378 Time-History Analysis. Modal Methods 381 Mass Condensation. Guyan Reduction 387 Component Mode Synthesis 391 Time-History Analysis. Direct Integration Methods 395 Explicit Direct Integration Methods 397 Implicit Direct Integration Methods 405 Other Implicit and Explicit Methods. Mixed Methods 407 Stability Analysis. Accuracy of Direct Integration Methods 410 Concluding Remarks on Time-History Analysis 417 Problems 418

Chapter 14 STRESS STIFFENING AND BUCKLING 14.1 14.2 14.3 14:4 14.5 14.6 14.7 ~ ..

"'-

429

Introduction 429 Stress Stiffness Matrices for Beams and Bars 432 Stress Stiffness Matrix of a Plate Element 435 A General Formulation for [k".l 437 Bifurcation Buckling 441 Remarks on [K".l and Its Uses 444 Remarks on Buckling and Buckling Analysis 446 Problems 448 -\',

Chapter 15' WEIGHTED RESIDUAL METHODS /

15~r-'

15.2 15.3 15.4 15.5 15.6

Introduction 455 Some Weighted Residual Methods 455 Example Solutions 458 Galerkin Finite Element Method 461 Integration by Parts 466 Two-Dimensional Problems 468 Problems 470

Chapter 16 HEAT CONDUCTION AND SELECTED FLUID PROBLEMS 16.1 16.2 16.3 16.4 16.5 16.6 16.7

455

Introduction to Heat Conduction Problems 474 A One-Dimensional Example 475 Heat Conduction in a Plane 477 General Solids and Solids of Revolution 479 Finite Element Formulation 480 Thermal Transients 484 Related Problems. Fluid Flow 486

474

xiv

16.8 16.9

CONTENTS

Fluid Vibration and Waves, Pressure Formulation Fluid-Structure Interaction 491 Problems 495

CCh~pt;;1T ~--

17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8

488

AN INTRODUCTION TO SOME NONLINEAR PROBLEMS

Introduction 501 Some Solution Methods 502 One-Dimensional Elastic-Plastic Analysis 510 Small-Strain Plasticity Relations 515 Elastic-Plastic Analysis Procedures 519 Nonlinear Dynamic Problems 522 A Problem Having Geometric Nonlinearity 529 Other Nonlinear Problems 532 Problems 533

Chapter 18 NUMERICAL ERRORS AND CONVERGENCE 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 18.9

Introduction. Error Classification 542 Ill-Conditioning 543 The Condition Number 546 Diagonal Decay Error Tests 550 Residuals 552 Discretization Error: Analysis 553 Discretization Error: Estimation and Extrapolation Tests of Element Quality 563 Concluding Remarks 566 Problems 566

Modeling 573 Programming and Programs

542

558

Chapter 19 MODELING, PROGRAMS, AND PROGRAMMING 19.1 19.2

501

573

584

Appendix A MATRICES: SELECTED DEFINITIONS AND MANIPULATIONS

589

Appendix B SIMULTANEOUS ALGEBRAIC EQUATIONS

592

B.1 B.2

Introduction 592 Solution of Simultaneous Linear Algebraic Equations by Gauss Elimination 593

CONTENTS

Appendix C EIGENVALVES AND EIGENVECTORS C.I C.2 C.3 CA

C.S

xv 598

The Eigenproblern 598 The Standard Eigenproblem 598 The General Eigenproblem 599 Remarks on Special Forms 602 Solution Algorithms 603

REFERENCES

605

INDEX

623

"

NOTATION

What follows is a list of principal symbols. Less frequently used symbols, and symbols that have different meanings in different contexts, are defined where they are used .' Matrices and vectors are denoted by boldface type. MATHEMATICAL SYMBOLS Rectangular or square matrix. Column, row, and diagonal matrices. Matrix, transpose. Matrix inverse and inverse transpose; that is, ([ J-I)T "'" ([

II

{~~}

Y)~I.

Norm of a matrix or a vector. Time differentiation; for example, it = duldt, it = d 2u/dt2 • Partial differentiation if the following subscript(s) is literal; for example, w,x = ow/ax, W,xy = a2w/ax oy.

l

J , were h II . a scaI ' ar f unction

Represents -an -an ... -an

aal aa2 of al, a2, ... , Q".

oan

T

IS

LATIN SYMBOLS

A [AJ {a} B [B]

em

[CJ d.o.f. D

{D}, {d} E [E]

{F} G I

rIJ J [J] k [K], [k] [K u ] , [k u ] L,L T

f, m, n Il e q

[M], em]

Area or cross-sectional area. Relates {d} to {a}; [d] = [AHa}. Generalized coordinates. Bulk modulus, B = £/(3 - 6v). Spatial derivativets) of the field variable(s) are [B]{d}. Field continuity of degree In (Section 3.11). Damping matrix. Constraint matrix. Degree(s) of freedom. Displacement. Flexural rigidity of a plate or shell. Nodal d.o.f. of structure and element, respectively. Modulus of elasticity. Matrix of elastic stiffnesses (Section 1.7). Body forces per unit volume. Shear modulus. Moment of inertia of cross-sectional area. Unit matrix (also called identity matrix). Determinant of [J] (called the Jacobian). The Jacobian matrix. Spring stiffness. Thermal conductivity. Structure and element conventional stiffness matrices. Structure and element stress stiffness matrices. Length of element, length of structure. Direction cosines. 'Number of equations. Structure and element mass matrices.

xvii

xviii

NOTATION

[N], LNJ

o

[0],

{oJ

{P} q {R}

{r.]

s.

S, T

t

rn U,

u;

u,

V, W

{u} V,

v,

x,y, Z

Shape (or basis, or interpolation) functions. Order; for example, 0(11 2 ) = a term of order 112 . Null matrix, null vector. Externally applied concentrated loads on structure nodes. Distributed load (surface or line). Total load on structure nodes; {R} = {P} + 2: {r.}, Loads applied to nodes by element, for example, by temperature change or distributed load (Eq. 4.1-6). Surface, element surface. Temperature. Thickness. Time. Transformation matrix. Strain energy, strain energy per unit volume. Displacements, for example, in directions x, )', z. Vector of displacements; [u] = [u v w]T. Volume, element volume. Cartesian coordinates.

GREEK SYMBOLS a [r]

{e}, {eo} [K], {K} A II

~, 1], {

~l> ~2' ~3

TI P

{(T}, {(To}

4> {}

Coefficient of thermal expansion, penalty number. Jacobian inverse; [r] = [J] - '. Strains, initial strains. Matrix of thermal conductivities, vector of curvatures. Eigenvalue. Lagrange multiplier. Poisson's ratio of an isotropic material. Isoparametric coordinates. Area coordinates. A functional; for example, TIp potential energy. Mass density. Stresses, initial stresses. A dependent variable. Meridian angle of a shell. Surface tractions. Circular frequency in radians per second.

CHAPTER

1

INTRODUCTION

A brief overview of the finite element method and its concepts is presented.

Background information used for finite element applications in structural mechanics is discussed.

1.1 THE FINITE ELEMENT METHOD

The finite' element method is a numerical procedure for analyzing structures and continua. Usually the problem addressed is too complicated to be solved satisfactorily by classical analytical methods. The problem may concern stress analysis, heat conduction, or any of several other areas. The finite element procedure produces many simultaneous algebraic equations, which are generated and solved on a digital computer. Finite element calculations are performed on personal computers, mainframes, and all sizes in between. Results are rarely exact. However, errors are decreased by processing more equations, and results accurate enough for engineering purposes are obtainable at reasonable cost. The finite element method originated as a method of stress analysis. Today finite elements are also used to analyze problems of heat transfer, fluid flow, lubrication, electric and magnetic fields, and many oth~rs. Problems that previously were utterly intractable are now solved routinely. Finite element procedures are used in the design of buildings, electric motors, heat engines, ships, airframes, and spacecraft. Manufacturing companies and large design offices typically have one or more large finite element programs in-house. Smaller companies usually have access to a large program through a commercial computing center or use a smaller program on a personal computer. Figure 1. I-I shows a very simple problem that illustrates discretization, a basic finite element concept. Imagine that the displacement of the right end of the bar is required. The classical approach is to.writethe differential equation of the continu~slyji;~!~~1~i: solve-this equationfor axiaf'displacement-itis.a [unc-tion and finally substitute x = L T to find the required end displacement. The finite element approach to this problem does not begin with a differential equation. Instead, the bar is_discretized by modeling it as a series oi fini:« elements, each uniform but of -~~¥fferent cfoss-secii()na(a~i~AJFig~ 1. f-lb)~"in~eacK elellient, ijViirieain~;;fy~w{thi;"fU:~I:aQ[~J.9iU on the boundary of the region in such a way that interelement continuity of 4> tends to be maintained in the assemblage. . A finite element analysis typically involves the following steps. Again we will cite stress anaylsis and heat transfer as typical applications. Steps I, 4, and 5 require decisions by the analyst and provide input data for the computer program. Steps 2, 3, 6, and 7 are carried out automatically by the computer program.

Q ~

r I

t·)

J

vi.

Divide the structure or continuum into finite elements. Mesh generation programs, called preprocessors, help the user in doing thi~~~vork·.~~·-·-'--!:9rmulate the properties of each element. In stress analysis, this means det~rmillinKJl.odalL~ad§ associated with all elem~ntA$JQrmClt,ion statesthat m;;llo\ved. In heat t;;nSfer, it m~ans'dete-rmfning nodal heattl'ilX'eS"assO= dat~d~vith' all element temperature fields that are allowed.

v;. Assemble elements to obtain the finite element model of the structure. _.

-~

-'_J_"_c.

_.

•

• • • _ . • __ ,.,~_.

.~.

/4. Apply the known loads: nodal forces and/or moments in stress analysis, nodal heatfli.lxesTn·hea({iansfer~ .. .

v

./ ,j

5. In stress analysis, specify how the structure is supported. This step involves setting sever~1no~;l.~ispI~e:m~titsto!
"large;',?,fn'

11.;"/

.

Why Study the Theory of Finite Elements? Many satisfactory elements have already been formulated and reside in popular computer programs. The practitioner ~deSires to understand how various elements behave. Clearty~~giI]_~!::~~~~~Jnderstand analysis tools wilt be able to use them to better advantage and will be ~~~~~

_

~"",~~~"~~"_=~"~~,~~.:-r~~~_~C·~=-'-'-·~"'--""'''''''_'''~7-~,'''

~",.\)(t, :~%I~~I~. ti:,Jf~fs~~~~~\~~-'i~i;~d~o;r3-~~s~ae~~~~u~r~~~~iRfo~~~i:;~~,~f~~~~~ i11eOfy'to an adequate but not excessive degree. We recognize that for engineers the study of rinite elements is more than a theoretical study of mathematical foundations and formulation procedures for various types of finite elements. Complete computer codes need not be studied in detail, but concepts and assumptions' behind the coding should be mastered. Otherwise, the treatment of loads and boundary conditions may be confusing, the variety of program options and element types may be baffling, and error messages may provide no clue as to the source of difficulty or how to correct it.

Figure 1.1-6. A detailed model of half of an automobile frame, used to find deformations, stresses, natural frequencies, and mode shapes. (Courtesy of A. O. Smith Corp., Data Systems Division, Milwaukee, Wisconsin.)

The Element Characteristic Matrix

7

BEARING HOUSING MODEL·

DEFORNED NODEL

STRESS CONTOUR

Figure 1.1-7. Finite element mesh and computed deformations and stresses in a portion of a bearing housing. (Courtesy ofAlgor Interactive Systems 1I1C., Pittsburgh, Pennsylvania.i

1.2 THE ELEMENT CHARACTERISTIC !(~/J'Y' MATRIX ( .~._ ., ._. __

.~

.. __ ';o-. __

.'.

~

~-'

,-. c;

o~,

The element characteristic matrix h~LQ!ff~Lent.namesj!LdjJf~[~.IltPI()bl~Il}'!feas, In structural mechanics it iscalleda.~.tif1il.R~~m(11J·ix: it relates nodal displacements to nodal forces.' In"'1i~;t-~;;d~~ti~n it is called ;;?O~~dl;:::;Tl:jty'~;;;trrr:'-irrerates IlOdal temperatures to nodal fluxes. There are three important ways to derive an el~~ent characteristic matrix. St fo1p l,;;.y "."",' d - b4cJ,i

-:>

CO TheJ!Jl:e~.l_~1!.!!.!£c1.~s

b~~~!~d o!H?_b-y~lyal Tea~