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INTRODUCTION TO THE
FINITE ELEMENT METHOD
A Numerical Method for Engineering Analysis
DESAIbull ABEL
+ +
INTRODUCTION TO THE FINITE ELEMENT METHOD
INTRODUCTION TO THE
FINITE ELEMENT METHOD
A NUMERICAL METHOD FOR ENGINEERING ANALYSIS
CHANDRAKANT S DESAI Department of Civil Engineering
Virginia Polytechnic Institute and State University Blacksburg Virginia
JOHN F ABEL
School of Civil and Environmental Engineering Cornell University Ithaca New York
CBS CBS PUBLISHERS amp DISTRIBUTORS PVTLTD
New Delhi bull Bengaluru bull Chennai bull Kochi bull Mumbai bull Puna
ISBN 81-239-0895-4
First Indian Edition middot 1987 Reprint 2000 2002 2004 2005
Original English Language Edition Published by Litton Educational Publishing Inc now owned by Wadsworth Publishing Company a division of Wadsworth Inc IO Davis Drive Belmont California 94002 USA
Copyright 1972 by Litton Educational Publishing Inc now owned by Wadsworth Publishing Company a division of Wadsworth Inc to Davis Drive Belmont California 94002 USA
All rights reserved No part of this book 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
Sales Area India only
Published by Satish Kumar Jain and produced by VK Jain for CBS Publishers amp Distributors Pvt Ltd CBS Plaza 4819XI Prahlad Street 24 Ansari Road Daryaganj New Delhi - 110002 India bull Website wwwebspdcom e-mail delhiebspdcom cbspubsairtelmailinPh 23289259 23266861 23266867 bull Fax Ol 1-23243014
Branches
bull Rengaluru Seema House 2975 17th Cross KR RoadRansankari 2nd Stage Bengaluru - 560070bull Ph +91-80-2677167879 bull Fax +91-80-26771680bull E-mail cbsbnggmaiLcom bangaJorecbspdcomPune Bhuruk Prestige Sr No 52122+ l + 32Narhc Havcli (Near Katraj-Dehu Road By-pass) Pune 411041bull Ph +91-20-6470405859 020-32392277 bull E-mail punecbspdcom
bull Koclli 361 i 4 Kalluvilakam Lissie Hospital RoadKechi - 682018 Kerala bull Ph +9 l-484-4059061-65bull Fax +91-484-4059065 bull E-mail cochincbspdcom
bull Chennai 20 West Park Road Shenoy Nagar Chennai - 600030Ph +91-44-26260666 26208620 bull Fax +91-44-42032115bull E-mail chennaicbspdcom
bull Mumbai 83-C Ist Floor Dr E Moses Road Worli Murnbai-400 018Maharashtra Ph +91-9833017933 022-2490234024902341bull E-mail mumbai(cbspdcom
Printed at
Nazia Printers Delhi
PR FAC
High-speed electronic digital computers have enabled engineers to employ various numerical discretization techniques for approximate solutions complex problems The finitemiddot element method is one such technique It was originally developed as a tool for structural analysis but the theory and forshymulation have been progressively refined and generalized that the method has been applied successfully to such other fields heat seepage hydroshydynamics and rock mechanics As a result of this broad applicability and the systematic generality of the associated computer codes the method has gained wide acceptance designers and research engineers It now taught to both students and practicing engineers at many universities and will soon be a regular part of the curriculum at almost all colleges of engineering
In order to apply the finite element method to complex modern problem engineers must be familiar with the methods fundamental theory assumpshytions limitations However beginner may encounter difficulties in acshyquiring this familiarity becamiddot ISC there is a scarcity of unified introductory treatments that are sufficiently elementary This text therefore is designed to prepare undergraduates practicing engineers with bachelors both to solve their specific problems and to read further in the current finite element literature It is assumed that the reader has some background in the diverse mathematical associated with finite element method matrix algebra mechanics variational methods and computer skills
The book is divided into three parts introduction and background material a description of method and applications relatively simple computer code included Appendix I At the end of chapter bibliographic in-formation is provided in two sections References and Further Reading A brief descriptive commentary is included with most of the references cited Abbreviations in the references are summarized in Appendix
Part A is incorported both to guide the reader to useful texts on the back- ground material and to summarize without detailed derivations the methods middot and equations are employed Parts B C B is detailed description of the theory of the finite element method To provide a physical insight the formulation of the theory as well as the examples in the first three chapters this section are based largely the method of analysis which is common in structural and soils engineering Howevermiddot the last chapter in this section is devoted to the generalization of the theory
ACKNOWiEDGMENTS
Thls book is a synthesis of the work of many investigitors Among the indishyviduals whose contributions have greatly influenced the authors are Professors J H Argyris E B Becker R W Clough RH u-llagher H C Martin J T Oden T H H Pian E P Popov L C Reese B Fraeijs de Veubeke E L Wilson C Zienkiewicz and Dr C A Felippa
Many persons at the US Army Engineer Waterways Experiment Station have provided inspiration to the authors We wish to express gratitude to J P Sale S J Johnson W C Sherman Dr G H Keulegan Col LA Brown F R Brown and W R Martin for encouragement and sustained interest during the preparation of this book
Thanks are due to Dr G S Orenstein for carefully reading the manuscript and for offering valuable suggestions We also thank Professors E P Popov D W Murray and R S Sandhu and J Crawford D C Banks C J Huval JB Palmerton and JE Ahlberg for helpful comments
Finally we are deeply grateful to our wives P Desai and V L Abel for reading the entire manuscript for making a number of suggestions toward improving the presentation of the material and for assisting in the prepashyration of the book
Authorship of any sort is a fantastic indulgence of the ego lt is well no doubt to reflect on
how much one owes to others
J Galbraith The Affluent Society HoughtonMifflin-Co Boston Mass
vii
LIST OF SYMBOLS
The following list defines the principal symbols used in this book and gives the sections where they are explained further Other symbols are defined in context Rectangular matrices are indicated by brackets [ ] and column vectors by braces (A thorough description of the matrix notation employed is given in Section 2-1) Overdots indicate differentiation with respect to time and primes usually denote differentiation with respect to the space variable For forcing parameters used in variational principles an overbar indicates a prescribed or known value (Section 4-3) whereas within the finite element equations this notation indicates a condensed form (Sections 2-3 and 5-9)
(A]
B [B] [Ba] C
[C]
[C] [cP] [CP] D
[D]
e
ep E E E I F
Parameters for natural coordinate systems (5-4) Cross-sectional area (1-3) general functional which is the integral of another functional (4-1) area of a two-dimenshysional element (5-4) General rectangular matrix (2-1) coefficient matrix in eigenvalue problem of standard form (2-3) transformation relating generalized and nodal displacements (5-3) Semiband width of coefficient matrix (2-2 6-1) Transformation relating strains and displacements (5-S) Cohesive strength (3-S) Viscous damping matrix (2-4 11-3) constitutive (stressshystrain) matrix (3-3) Constitutive (stress-strain) matrices (3-5 7-5) The maximum value for an assemblage of the largest difference between the node numbers of an element (6-1) Constitutive (strain-stress) matrix (3-3) general diagonal matrix (2-2 11-3 l l-4) Superscript indicating elastic (3-5) subscript or supershyscript indicating element (6-3) Superscript indicating elastic-plastic (3-5) Youngs modulus of elasticity (l-3 3-3) Secant and tangent moduli (3-5) Yield surface (3-5) number of degrees of freedom at a node (6-1) General functional (4-1)
ix
LIST OF SYMBOLS [F] g g G h H l1 l2 l3 [I] li l2 J3 [J] [k ] [k ]
L
L [L1 ] [m] [ma] middot [M]N[N]O [OJP pqp Q (Qo
rrRs
Flexibility matrix (2-3 9-2) Acceleration due to gravity subscript indicating global coordinate system (6-5) Vector of gradients of field va1iable ( 1 2- 1 ) Shear modulus (3-3) Thickness of two-dimensional dement (5- 1) Hydraulic head (8-2) hydrostatic stress (9-3) Strain invariants (3-2) Identity matrix Stress invariants (3- 1 ) Jacobian matrix (5-4) Element elastic stiffness matrix ( 1 -3 5-6 5-7) general property matrix (8- 1) Element geometric stiffness matrix (7-6) Bulle modulus (3-3) Assemblage elastic stiffness matrix ( 1 -3 6-5) Length of one-dimensional element (l-3 5- 1) subscript ind icating local coordinate system (6-5) Lagrangian functional (4-2) natural coordinate for oneshydimensional element (5-4) geaeral differential operator (8-4) Natural coordinates of triangle or tetrahedron (5-4) Unit lower triangular matrix (2-2) Element mass matrix ( 1 1 -2) Assemblage mass matrix (2-4 1 1 -2) Number of equations for assemblage (2-2) Coefficient matrix for interpolation field variable (disshyplacement) model (5-3 5-4) Null vector and matrix Arbitrary load ( l -2) superscript indicating plastic (3-5) Vector of nodal field variables (displacements) for element ( 1 -3 5-3) Vector of nodal actions (forces) for element ( 1 -3 5-6 5-7) Vector of nodal actions (forces) due to initial effects (initial strains) (7- 1 ) Radial coordinate (3-4) natural coordinate for hexa-hedral element (5-4) Vector of nodal field variables (djsplacements) for assembshylage ( 1 -3 6-5) Vector of nodal actions (forces) for assemblage ( 1 -3 6-5) Natural coordinate for quadrilateral or hexahedral elements (5-4)
[S]
t
[t] T
T T T
T
[T]
u v w
W W
Wp
Wpc
x y z X Y Z
X [X] ex
11
11
y l
A
amp
e 8
AI
lAJmicroV
UST OF SYMBOLS xi
Transformation for skewed coordinates at ilh node (6-6) Surface areas over which tractions and displacements are prescribed respectively (4-2) Overall transformation for skewed coordinates (6-6) general symmetric matrix (2-2) Time natural coordinate for quadrilateral or hexahedrat element (5-4) Transformation from local to global coordinates (6-5) Superscript indicating transpose (2- 1 ) kinetic energy functional (4-2) temperature (7- 1) Surface traction components (4-2) Surface traction vector (4-2) Overall transformation from local to global coordinates (6-5 7-6) Displacements in cartesian coordinates (3-2) general field variables (8- 1 ) Strain energy and complementary strain energy (4-2) General volume (4-2) volume of a tetrahedral element (5-4) Work and complementary work (4-2) Potential and complementary potential of loads (4-2) Cartesian Cltr inates Components of body force intensity (4-2) Body force intensify vector (4-2) Modal matrix (2-3 1 1 -4) Coefficient of thermal expansion (7- 1 ) Generalized coordinates (5-3) Vector of element generalized coordinater (5-3) Shear strain (3-2) Variational operator (4-1 ) Prefix indicating a finite increment (7-2) Normal strain (3-2) Vector of strains (3-2 5-5) Circumferential coordinate (3-4) slope middot of a beam (5-3) angle of twist per unit length in torsion () 2-4) Eigenvalue (2-3 7-6) Lame constant (3-3) Spectral matrix (2-3 1 1-4) Lam constant (3-3) viscosity ( 14-4) Poisson ratio (3-3) Tangent Poisson ratio (3-5) Total potential and complementary potential functionals (4-2)
Il11 p (1
a t
tp p [p] lt (I)
Reissners functional (4-2) Mass density (4-2 1 1 -2) Normal stress (3- 1) Stress vector (3- 1 5-5 9-2) Shear stress (3-1) Angle of internal friction (3-5) potential function (14-3) Coefficient matrix for generalized coordinate model (5-3) General field variable (8-3) Vector of nodal field variables ( 12-1) Frequency (2-3 1 1 -4)
CONTENTS
PREFACE
ACKNOWLEDG MENTS
LIST O F SYM BOLS
PART A I NTRODUCTION AN D BACKGROUN DMATER IAL
1 INTRODUCTION
V
vii
ix
1
3 1 - 1 Background and appl ications 31 -2 General description of the method 51 -3 Summary of the ana lysis procedure 1 01 -4 Fu ndamentals for the understanding of the method 1 5
References 1 6Further read ing 1 7
2 MATRIX TECH NIQU ES 2-1 Matrix notation 2-2 Solution of large systems of a lgebra ic equations2-3 Eigenvalue problems 2-4 otution of propagation problems
References Further read ing
3 BASIC EQUATIONS FROM SOLI DMECHANICS
3-1 Stress 3-2 Stra in a 1 1d kinematics 3 -3 Linear constitutive equations 3-4 Two-dimensional special izations of elasticity3-5 Nonlinear materia l behavior 3-6 Material characterization
References Further reading
1 8 1 81 922242628
29 2930323541 495061xiii
Introduction To The Finite ElementMethod
Publisher CBS Publications ISBN 9788123908953 Author Chandrakant Desai
Type the URL httpwwwkopykitabcomproduct10373
Get this eBook
INTRODUCTION TO THE FINITE ELEMENT METHOD
INTRODUCTION TO THE
FINITE ELEMENT METHOD
A NUMERICAL METHOD FOR ENGINEERING ANALYSIS
CHANDRAKANT S DESAI Department of Civil Engineering
Virginia Polytechnic Institute and State University Blacksburg Virginia
JOHN F ABEL
School of Civil and Environmental Engineering Cornell University Ithaca New York
CBS CBS PUBLISHERS amp DISTRIBUTORS PVTLTD
New Delhi bull Bengaluru bull Chennai bull Kochi bull Mumbai bull Puna
ISBN 81-239-0895-4
First Indian Edition middot 1987 Reprint 2000 2002 2004 2005
Original English Language Edition Published by Litton Educational Publishing Inc now owned by Wadsworth Publishing Company a division of Wadsworth Inc IO Davis Drive Belmont California 94002 USA
Copyright 1972 by Litton Educational Publishing Inc now owned by Wadsworth Publishing Company a division of Wadsworth Inc to Davis Drive Belmont California 94002 USA
All rights reserved No part of this book 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
Sales Area India only
Published by Satish Kumar Jain and produced by VK Jain for CBS Publishers amp Distributors Pvt Ltd CBS Plaza 4819XI Prahlad Street 24 Ansari Road Daryaganj New Delhi - 110002 India bull Website wwwebspdcom e-mail delhiebspdcom cbspubsairtelmailinPh 23289259 23266861 23266867 bull Fax Ol 1-23243014
Branches
bull Rengaluru Seema House 2975 17th Cross KR RoadRansankari 2nd Stage Bengaluru - 560070bull Ph +91-80-2677167879 bull Fax +91-80-26771680bull E-mail cbsbnggmaiLcom bangaJorecbspdcomPune Bhuruk Prestige Sr No 52122+ l + 32Narhc Havcli (Near Katraj-Dehu Road By-pass) Pune 411041bull Ph +91-20-6470405859 020-32392277 bull E-mail punecbspdcom
bull Koclli 361 i 4 Kalluvilakam Lissie Hospital RoadKechi - 682018 Kerala bull Ph +9 l-484-4059061-65bull Fax +91-484-4059065 bull E-mail cochincbspdcom
bull Chennai 20 West Park Road Shenoy Nagar Chennai - 600030Ph +91-44-26260666 26208620 bull Fax +91-44-42032115bull E-mail chennaicbspdcom
bull Mumbai 83-C Ist Floor Dr E Moses Road Worli Murnbai-400 018Maharashtra Ph +91-9833017933 022-2490234024902341bull E-mail mumbai(cbspdcom
Printed at
Nazia Printers Delhi
PR FAC
High-speed electronic digital computers have enabled engineers to employ various numerical discretization techniques for approximate solutions complex problems The finitemiddot element method is one such technique It was originally developed as a tool for structural analysis but the theory and forshymulation have been progressively refined and generalized that the method has been applied successfully to such other fields heat seepage hydroshydynamics and rock mechanics As a result of this broad applicability and the systematic generality of the associated computer codes the method has gained wide acceptance designers and research engineers It now taught to both students and practicing engineers at many universities and will soon be a regular part of the curriculum at almost all colleges of engineering
In order to apply the finite element method to complex modern problem engineers must be familiar with the methods fundamental theory assumpshytions limitations However beginner may encounter difficulties in acshyquiring this familiarity becamiddot ISC there is a scarcity of unified introductory treatments that are sufficiently elementary This text therefore is designed to prepare undergraduates practicing engineers with bachelors both to solve their specific problems and to read further in the current finite element literature It is assumed that the reader has some background in the diverse mathematical associated with finite element method matrix algebra mechanics variational methods and computer skills
The book is divided into three parts introduction and background material a description of method and applications relatively simple computer code included Appendix I At the end of chapter bibliographic in-formation is provided in two sections References and Further Reading A brief descriptive commentary is included with most of the references cited Abbreviations in the references are summarized in Appendix
Part A is incorported both to guide the reader to useful texts on the back- ground material and to summarize without detailed derivations the methods middot and equations are employed Parts B C B is detailed description of the theory of the finite element method To provide a physical insight the formulation of the theory as well as the examples in the first three chapters this section are based largely the method of analysis which is common in structural and soils engineering Howevermiddot the last chapter in this section is devoted to the generalization of the theory
ACKNOWiEDGMENTS
Thls book is a synthesis of the work of many investigitors Among the indishyviduals whose contributions have greatly influenced the authors are Professors J H Argyris E B Becker R W Clough RH u-llagher H C Martin J T Oden T H H Pian E P Popov L C Reese B Fraeijs de Veubeke E L Wilson C Zienkiewicz and Dr C A Felippa
Many persons at the US Army Engineer Waterways Experiment Station have provided inspiration to the authors We wish to express gratitude to J P Sale S J Johnson W C Sherman Dr G H Keulegan Col LA Brown F R Brown and W R Martin for encouragement and sustained interest during the preparation of this book
Thanks are due to Dr G S Orenstein for carefully reading the manuscript and for offering valuable suggestions We also thank Professors E P Popov D W Murray and R S Sandhu and J Crawford D C Banks C J Huval JB Palmerton and JE Ahlberg for helpful comments
Finally we are deeply grateful to our wives P Desai and V L Abel for reading the entire manuscript for making a number of suggestions toward improving the presentation of the material and for assisting in the prepashyration of the book
Authorship of any sort is a fantastic indulgence of the ego lt is well no doubt to reflect on
how much one owes to others
J Galbraith The Affluent Society HoughtonMifflin-Co Boston Mass
vii
LIST OF SYMBOLS
The following list defines the principal symbols used in this book and gives the sections where they are explained further Other symbols are defined in context Rectangular matrices are indicated by brackets [ ] and column vectors by braces (A thorough description of the matrix notation employed is given in Section 2-1) Overdots indicate differentiation with respect to time and primes usually denote differentiation with respect to the space variable For forcing parameters used in variational principles an overbar indicates a prescribed or known value (Section 4-3) whereas within the finite element equations this notation indicates a condensed form (Sections 2-3 and 5-9)
(A]
B [B] [Ba] C
[C]
[C] [cP] [CP] D
[D]
e
ep E E E I F
Parameters for natural coordinate systems (5-4) Cross-sectional area (1-3) general functional which is the integral of another functional (4-1) area of a two-dimenshysional element (5-4) General rectangular matrix (2-1) coefficient matrix in eigenvalue problem of standard form (2-3) transformation relating generalized and nodal displacements (5-3) Semiband width of coefficient matrix (2-2 6-1) Transformation relating strains and displacements (5-S) Cohesive strength (3-S) Viscous damping matrix (2-4 11-3) constitutive (stressshystrain) matrix (3-3) Constitutive (stress-strain) matrices (3-5 7-5) The maximum value for an assemblage of the largest difference between the node numbers of an element (6-1) Constitutive (strain-stress) matrix (3-3) general diagonal matrix (2-2 11-3 l l-4) Superscript indicating elastic (3-5) subscript or supershyscript indicating element (6-3) Superscript indicating elastic-plastic (3-5) Youngs modulus of elasticity (l-3 3-3) Secant and tangent moduli (3-5) Yield surface (3-5) number of degrees of freedom at a node (6-1) General functional (4-1)
ix
LIST OF SYMBOLS [F] g g G h H l1 l2 l3 [I] li l2 J3 [J] [k ] [k ]
L
L [L1 ] [m] [ma] middot [M]N[N]O [OJP pqp Q (Qo
rrRs
Flexibility matrix (2-3 9-2) Acceleration due to gravity subscript indicating global coordinate system (6-5) Vector of gradients of field va1iable ( 1 2- 1 ) Shear modulus (3-3) Thickness of two-dimensional dement (5- 1) Hydraulic head (8-2) hydrostatic stress (9-3) Strain invariants (3-2) Identity matrix Stress invariants (3- 1 ) Jacobian matrix (5-4) Element elastic stiffness matrix ( 1 -3 5-6 5-7) general property matrix (8- 1) Element geometric stiffness matrix (7-6) Bulle modulus (3-3) Assemblage elastic stiffness matrix ( 1 -3 6-5) Length of one-dimensional element (l-3 5- 1) subscript ind icating local coordinate system (6-5) Lagrangian functional (4-2) natural coordinate for oneshydimensional element (5-4) geaeral differential operator (8-4) Natural coordinates of triangle or tetrahedron (5-4) Unit lower triangular matrix (2-2) Element mass matrix ( 1 1 -2) Assemblage mass matrix (2-4 1 1 -2) Number of equations for assemblage (2-2) Coefficient matrix for interpolation field variable (disshyplacement) model (5-3 5-4) Null vector and matrix Arbitrary load ( l -2) superscript indicating plastic (3-5) Vector of nodal field variables (displacements) for element ( 1 -3 5-3) Vector of nodal actions (forces) for element ( 1 -3 5-6 5-7) Vector of nodal actions (forces) due to initial effects (initial strains) (7- 1 ) Radial coordinate (3-4) natural coordinate for hexa-hedral element (5-4) Vector of nodal field variables (djsplacements) for assembshylage ( 1 -3 6-5) Vector of nodal actions (forces) for assemblage ( 1 -3 6-5) Natural coordinate for quadrilateral or hexahedral elements (5-4)
[S]
t
[t] T
T T T
T
[T]
u v w
W W
Wp
Wpc
x y z X Y Z
X [X] ex
11
11
y l
A
amp
e 8
AI
lAJmicroV
UST OF SYMBOLS xi
Transformation for skewed coordinates at ilh node (6-6) Surface areas over which tractions and displacements are prescribed respectively (4-2) Overall transformation for skewed coordinates (6-6) general symmetric matrix (2-2) Time natural coordinate for quadrilateral or hexahedrat element (5-4) Transformation from local to global coordinates (6-5) Superscript indicating transpose (2- 1 ) kinetic energy functional (4-2) temperature (7- 1) Surface traction components (4-2) Surface traction vector (4-2) Overall transformation from local to global coordinates (6-5 7-6) Displacements in cartesian coordinates (3-2) general field variables (8- 1 ) Strain energy and complementary strain energy (4-2) General volume (4-2) volume of a tetrahedral element (5-4) Work and complementary work (4-2) Potential and complementary potential of loads (4-2) Cartesian Cltr inates Components of body force intensity (4-2) Body force intensify vector (4-2) Modal matrix (2-3 1 1 -4) Coefficient of thermal expansion (7- 1 ) Generalized coordinates (5-3) Vector of element generalized coordinater (5-3) Shear strain (3-2) Variational operator (4-1 ) Prefix indicating a finite increment (7-2) Normal strain (3-2) Vector of strains (3-2 5-5) Circumferential coordinate (3-4) slope middot of a beam (5-3) angle of twist per unit length in torsion () 2-4) Eigenvalue (2-3 7-6) Lame constant (3-3) Spectral matrix (2-3 1 1-4) Lam constant (3-3) viscosity ( 14-4) Poisson ratio (3-3) Tangent Poisson ratio (3-5) Total potential and complementary potential functionals (4-2)
Il11 p (1
a t
tp p [p] lt (I)
Reissners functional (4-2) Mass density (4-2 1 1 -2) Normal stress (3- 1) Stress vector (3- 1 5-5 9-2) Shear stress (3-1) Angle of internal friction (3-5) potential function (14-3) Coefficient matrix for generalized coordinate model (5-3) General field variable (8-3) Vector of nodal field variables ( 12-1) Frequency (2-3 1 1 -4)
CONTENTS
PREFACE
ACKNOWLEDG MENTS
LIST O F SYM BOLS
PART A I NTRODUCTION AN D BACKGROUN DMATER IAL
1 INTRODUCTION
V
vii
ix
1
3 1 - 1 Background and appl ications 31 -2 General description of the method 51 -3 Summary of the ana lysis procedure 1 01 -4 Fu ndamentals for the understanding of the method 1 5
References 1 6Further read ing 1 7
2 MATRIX TECH NIQU ES 2-1 Matrix notation 2-2 Solution of large systems of a lgebra ic equations2-3 Eigenvalue problems 2-4 otution of propagation problems
References Further read ing
3 BASIC EQUATIONS FROM SOLI DMECHANICS
3-1 Stress 3-2 Stra in a 1 1d kinematics 3 -3 Linear constitutive equations 3-4 Two-dimensional special izations of elasticity3-5 Nonlinear materia l behavior 3-6 Material characterization
References Further reading
1 8 1 81 922242628
29 2930323541 495061xiii
Introduction To The Finite ElementMethod
Publisher CBS Publications ISBN 9788123908953 Author Chandrakant Desai
Type the URL httpwwwkopykitabcomproduct10373
Get this eBook
INTRODUCTION TO THE
FINITE ELEMENT METHOD
A NUMERICAL METHOD FOR ENGINEERING ANALYSIS
CHANDRAKANT S DESAI Department of Civil Engineering
Virginia Polytechnic Institute and State University Blacksburg Virginia
JOHN F ABEL
School of Civil and Environmental Engineering Cornell University Ithaca New York
CBS CBS PUBLISHERS amp DISTRIBUTORS PVTLTD
New Delhi bull Bengaluru bull Chennai bull Kochi bull Mumbai bull Puna
ISBN 81-239-0895-4
First Indian Edition middot 1987 Reprint 2000 2002 2004 2005
Original English Language Edition Published by Litton Educational Publishing Inc now owned by Wadsworth Publishing Company a division of Wadsworth Inc IO Davis Drive Belmont California 94002 USA
Copyright 1972 by Litton Educational Publishing Inc now owned by Wadsworth Publishing Company a division of Wadsworth Inc to Davis Drive Belmont California 94002 USA
All rights reserved No part of this book 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
Sales Area India only
Published by Satish Kumar Jain and produced by VK Jain for CBS Publishers amp Distributors Pvt Ltd CBS Plaza 4819XI Prahlad Street 24 Ansari Road Daryaganj New Delhi - 110002 India bull Website wwwebspdcom e-mail delhiebspdcom cbspubsairtelmailinPh 23289259 23266861 23266867 bull Fax Ol 1-23243014
Branches
bull Rengaluru Seema House 2975 17th Cross KR RoadRansankari 2nd Stage Bengaluru - 560070bull Ph +91-80-2677167879 bull Fax +91-80-26771680bull E-mail cbsbnggmaiLcom bangaJorecbspdcomPune Bhuruk Prestige Sr No 52122+ l + 32Narhc Havcli (Near Katraj-Dehu Road By-pass) Pune 411041bull Ph +91-20-6470405859 020-32392277 bull E-mail punecbspdcom
bull Koclli 361 i 4 Kalluvilakam Lissie Hospital RoadKechi - 682018 Kerala bull Ph +9 l-484-4059061-65bull Fax +91-484-4059065 bull E-mail cochincbspdcom
bull Chennai 20 West Park Road Shenoy Nagar Chennai - 600030Ph +91-44-26260666 26208620 bull Fax +91-44-42032115bull E-mail chennaicbspdcom
bull Mumbai 83-C Ist Floor Dr E Moses Road Worli Murnbai-400 018Maharashtra Ph +91-9833017933 022-2490234024902341bull E-mail mumbai(cbspdcom
Printed at
Nazia Printers Delhi
PR FAC
High-speed electronic digital computers have enabled engineers to employ various numerical discretization techniques for approximate solutions complex problems The finitemiddot element method is one such technique It was originally developed as a tool for structural analysis but the theory and forshymulation have been progressively refined and generalized that the method has been applied successfully to such other fields heat seepage hydroshydynamics and rock mechanics As a result of this broad applicability and the systematic generality of the associated computer codes the method has gained wide acceptance designers and research engineers It now taught to both students and practicing engineers at many universities and will soon be a regular part of the curriculum at almost all colleges of engineering
In order to apply the finite element method to complex modern problem engineers must be familiar with the methods fundamental theory assumpshytions limitations However beginner may encounter difficulties in acshyquiring this familiarity becamiddot ISC there is a scarcity of unified introductory treatments that are sufficiently elementary This text therefore is designed to prepare undergraduates practicing engineers with bachelors both to solve their specific problems and to read further in the current finite element literature It is assumed that the reader has some background in the diverse mathematical associated with finite element method matrix algebra mechanics variational methods and computer skills
The book is divided into three parts introduction and background material a description of method and applications relatively simple computer code included Appendix I At the end of chapter bibliographic in-formation is provided in two sections References and Further Reading A brief descriptive commentary is included with most of the references cited Abbreviations in the references are summarized in Appendix
Part A is incorported both to guide the reader to useful texts on the back- ground material and to summarize without detailed derivations the methods middot and equations are employed Parts B C B is detailed description of the theory of the finite element method To provide a physical insight the formulation of the theory as well as the examples in the first three chapters this section are based largely the method of analysis which is common in structural and soils engineering Howevermiddot the last chapter in this section is devoted to the generalization of the theory
ACKNOWiEDGMENTS
Thls book is a synthesis of the work of many investigitors Among the indishyviduals whose contributions have greatly influenced the authors are Professors J H Argyris E B Becker R W Clough RH u-llagher H C Martin J T Oden T H H Pian E P Popov L C Reese B Fraeijs de Veubeke E L Wilson C Zienkiewicz and Dr C A Felippa
Many persons at the US Army Engineer Waterways Experiment Station have provided inspiration to the authors We wish to express gratitude to J P Sale S J Johnson W C Sherman Dr G H Keulegan Col LA Brown F R Brown and W R Martin for encouragement and sustained interest during the preparation of this book
Thanks are due to Dr G S Orenstein for carefully reading the manuscript and for offering valuable suggestions We also thank Professors E P Popov D W Murray and R S Sandhu and J Crawford D C Banks C J Huval JB Palmerton and JE Ahlberg for helpful comments
Finally we are deeply grateful to our wives P Desai and V L Abel for reading the entire manuscript for making a number of suggestions toward improving the presentation of the material and for assisting in the prepashyration of the book
Authorship of any sort is a fantastic indulgence of the ego lt is well no doubt to reflect on
how much one owes to others
J Galbraith The Affluent Society HoughtonMifflin-Co Boston Mass
vii
LIST OF SYMBOLS
The following list defines the principal symbols used in this book and gives the sections where they are explained further Other symbols are defined in context Rectangular matrices are indicated by brackets [ ] and column vectors by braces (A thorough description of the matrix notation employed is given in Section 2-1) Overdots indicate differentiation with respect to time and primes usually denote differentiation with respect to the space variable For forcing parameters used in variational principles an overbar indicates a prescribed or known value (Section 4-3) whereas within the finite element equations this notation indicates a condensed form (Sections 2-3 and 5-9)
(A]
B [B] [Ba] C
[C]
[C] [cP] [CP] D
[D]
e
ep E E E I F
Parameters for natural coordinate systems (5-4) Cross-sectional area (1-3) general functional which is the integral of another functional (4-1) area of a two-dimenshysional element (5-4) General rectangular matrix (2-1) coefficient matrix in eigenvalue problem of standard form (2-3) transformation relating generalized and nodal displacements (5-3) Semiband width of coefficient matrix (2-2 6-1) Transformation relating strains and displacements (5-S) Cohesive strength (3-S) Viscous damping matrix (2-4 11-3) constitutive (stressshystrain) matrix (3-3) Constitutive (stress-strain) matrices (3-5 7-5) The maximum value for an assemblage of the largest difference between the node numbers of an element (6-1) Constitutive (strain-stress) matrix (3-3) general diagonal matrix (2-2 11-3 l l-4) Superscript indicating elastic (3-5) subscript or supershyscript indicating element (6-3) Superscript indicating elastic-plastic (3-5) Youngs modulus of elasticity (l-3 3-3) Secant and tangent moduli (3-5) Yield surface (3-5) number of degrees of freedom at a node (6-1) General functional (4-1)
ix
LIST OF SYMBOLS [F] g g G h H l1 l2 l3 [I] li l2 J3 [J] [k ] [k ]
L
L [L1 ] [m] [ma] middot [M]N[N]O [OJP pqp Q (Qo
rrRs
Flexibility matrix (2-3 9-2) Acceleration due to gravity subscript indicating global coordinate system (6-5) Vector of gradients of field va1iable ( 1 2- 1 ) Shear modulus (3-3) Thickness of two-dimensional dement (5- 1) Hydraulic head (8-2) hydrostatic stress (9-3) Strain invariants (3-2) Identity matrix Stress invariants (3- 1 ) Jacobian matrix (5-4) Element elastic stiffness matrix ( 1 -3 5-6 5-7) general property matrix (8- 1) Element geometric stiffness matrix (7-6) Bulle modulus (3-3) Assemblage elastic stiffness matrix ( 1 -3 6-5) Length of one-dimensional element (l-3 5- 1) subscript ind icating local coordinate system (6-5) Lagrangian functional (4-2) natural coordinate for oneshydimensional element (5-4) geaeral differential operator (8-4) Natural coordinates of triangle or tetrahedron (5-4) Unit lower triangular matrix (2-2) Element mass matrix ( 1 1 -2) Assemblage mass matrix (2-4 1 1 -2) Number of equations for assemblage (2-2) Coefficient matrix for interpolation field variable (disshyplacement) model (5-3 5-4) Null vector and matrix Arbitrary load ( l -2) superscript indicating plastic (3-5) Vector of nodal field variables (displacements) for element ( 1 -3 5-3) Vector of nodal actions (forces) for element ( 1 -3 5-6 5-7) Vector of nodal actions (forces) due to initial effects (initial strains) (7- 1 ) Radial coordinate (3-4) natural coordinate for hexa-hedral element (5-4) Vector of nodal field variables (djsplacements) for assembshylage ( 1 -3 6-5) Vector of nodal actions (forces) for assemblage ( 1 -3 6-5) Natural coordinate for quadrilateral or hexahedral elements (5-4)
[S]
t
[t] T
T T T
T
[T]
u v w
W W
Wp
Wpc
x y z X Y Z
X [X] ex
11
11
y l
A
amp
e 8
AI
lAJmicroV
UST OF SYMBOLS xi
Transformation for skewed coordinates at ilh node (6-6) Surface areas over which tractions and displacements are prescribed respectively (4-2) Overall transformation for skewed coordinates (6-6) general symmetric matrix (2-2) Time natural coordinate for quadrilateral or hexahedrat element (5-4) Transformation from local to global coordinates (6-5) Superscript indicating transpose (2- 1 ) kinetic energy functional (4-2) temperature (7- 1) Surface traction components (4-2) Surface traction vector (4-2) Overall transformation from local to global coordinates (6-5 7-6) Displacements in cartesian coordinates (3-2) general field variables (8- 1 ) Strain energy and complementary strain energy (4-2) General volume (4-2) volume of a tetrahedral element (5-4) Work and complementary work (4-2) Potential and complementary potential of loads (4-2) Cartesian Cltr inates Components of body force intensity (4-2) Body force intensify vector (4-2) Modal matrix (2-3 1 1 -4) Coefficient of thermal expansion (7- 1 ) Generalized coordinates (5-3) Vector of element generalized coordinater (5-3) Shear strain (3-2) Variational operator (4-1 ) Prefix indicating a finite increment (7-2) Normal strain (3-2) Vector of strains (3-2 5-5) Circumferential coordinate (3-4) slope middot of a beam (5-3) angle of twist per unit length in torsion () 2-4) Eigenvalue (2-3 7-6) Lame constant (3-3) Spectral matrix (2-3 1 1-4) Lam constant (3-3) viscosity ( 14-4) Poisson ratio (3-3) Tangent Poisson ratio (3-5) Total potential and complementary potential functionals (4-2)
Il11 p (1
a t
tp p [p] lt (I)
Reissners functional (4-2) Mass density (4-2 1 1 -2) Normal stress (3- 1) Stress vector (3- 1 5-5 9-2) Shear stress (3-1) Angle of internal friction (3-5) potential function (14-3) Coefficient matrix for generalized coordinate model (5-3) General field variable (8-3) Vector of nodal field variables ( 12-1) Frequency (2-3 1 1 -4)
CONTENTS
PREFACE
ACKNOWLEDG MENTS
LIST O F SYM BOLS
PART A I NTRODUCTION AN D BACKGROUN DMATER IAL
1 INTRODUCTION
V
vii
ix
1
3 1 - 1 Background and appl ications 31 -2 General description of the method 51 -3 Summary of the ana lysis procedure 1 01 -4 Fu ndamentals for the understanding of the method 1 5
References 1 6Further read ing 1 7
2 MATRIX TECH NIQU ES 2-1 Matrix notation 2-2 Solution of large systems of a lgebra ic equations2-3 Eigenvalue problems 2-4 otution of propagation problems
References Further read ing
3 BASIC EQUATIONS FROM SOLI DMECHANICS
3-1 Stress 3-2 Stra in a 1 1d kinematics 3 -3 Linear constitutive equations 3-4 Two-dimensional special izations of elasticity3-5 Nonlinear materia l behavior 3-6 Material characterization
References Further reading
1 8 1 81 922242628
29 2930323541 495061xiii
Introduction To The Finite ElementMethod
Publisher CBS Publications ISBN 9788123908953 Author Chandrakant Desai
Type the URL httpwwwkopykitabcomproduct10373
Get this eBook
ISBN 81-239-0895-4
First Indian Edition middot 1987 Reprint 2000 2002 2004 2005
Original English Language Edition Published by Litton Educational Publishing Inc now owned by Wadsworth Publishing Company a division of Wadsworth Inc IO Davis Drive Belmont California 94002 USA
Copyright 1972 by Litton Educational Publishing Inc now owned by Wadsworth Publishing Company a division of Wadsworth Inc to Davis Drive Belmont California 94002 USA
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PR FAC
High-speed electronic digital computers have enabled engineers to employ various numerical discretization techniques for approximate solutions complex problems The finitemiddot element method is one such technique It was originally developed as a tool for structural analysis but the theory and forshymulation have been progressively refined and generalized that the method has been applied successfully to such other fields heat seepage hydroshydynamics and rock mechanics As a result of this broad applicability and the systematic generality of the associated computer codes the method has gained wide acceptance designers and research engineers It now taught to both students and practicing engineers at many universities and will soon be a regular part of the curriculum at almost all colleges of engineering
In order to apply the finite element method to complex modern problem engineers must be familiar with the methods fundamental theory assumpshytions limitations However beginner may encounter difficulties in acshyquiring this familiarity becamiddot ISC there is a scarcity of unified introductory treatments that are sufficiently elementary This text therefore is designed to prepare undergraduates practicing engineers with bachelors both to solve their specific problems and to read further in the current finite element literature It is assumed that the reader has some background in the diverse mathematical associated with finite element method matrix algebra mechanics variational methods and computer skills
The book is divided into three parts introduction and background material a description of method and applications relatively simple computer code included Appendix I At the end of chapter bibliographic in-formation is provided in two sections References and Further Reading A brief descriptive commentary is included with most of the references cited Abbreviations in the references are summarized in Appendix
Part A is incorported both to guide the reader to useful texts on the back- ground material and to summarize without detailed derivations the methods middot and equations are employed Parts B C B is detailed description of the theory of the finite element method To provide a physical insight the formulation of the theory as well as the examples in the first three chapters this section are based largely the method of analysis which is common in structural and soils engineering Howevermiddot the last chapter in this section is devoted to the generalization of the theory
ACKNOWiEDGMENTS
Thls book is a synthesis of the work of many investigitors Among the indishyviduals whose contributions have greatly influenced the authors are Professors J H Argyris E B Becker R W Clough RH u-llagher H C Martin J T Oden T H H Pian E P Popov L C Reese B Fraeijs de Veubeke E L Wilson C Zienkiewicz and Dr C A Felippa
Many persons at the US Army Engineer Waterways Experiment Station have provided inspiration to the authors We wish to express gratitude to J P Sale S J Johnson W C Sherman Dr G H Keulegan Col LA Brown F R Brown and W R Martin for encouragement and sustained interest during the preparation of this book
Thanks are due to Dr G S Orenstein for carefully reading the manuscript and for offering valuable suggestions We also thank Professors E P Popov D W Murray and R S Sandhu and J Crawford D C Banks C J Huval JB Palmerton and JE Ahlberg for helpful comments
Finally we are deeply grateful to our wives P Desai and V L Abel for reading the entire manuscript for making a number of suggestions toward improving the presentation of the material and for assisting in the prepashyration of the book
Authorship of any sort is a fantastic indulgence of the ego lt is well no doubt to reflect on
how much one owes to others
J Galbraith The Affluent Society HoughtonMifflin-Co Boston Mass
vii
LIST OF SYMBOLS
The following list defines the principal symbols used in this book and gives the sections where they are explained further Other symbols are defined in context Rectangular matrices are indicated by brackets [ ] and column vectors by braces (A thorough description of the matrix notation employed is given in Section 2-1) Overdots indicate differentiation with respect to time and primes usually denote differentiation with respect to the space variable For forcing parameters used in variational principles an overbar indicates a prescribed or known value (Section 4-3) whereas within the finite element equations this notation indicates a condensed form (Sections 2-3 and 5-9)
(A]
B [B] [Ba] C
[C]
[C] [cP] [CP] D
[D]
e
ep E E E I F
Parameters for natural coordinate systems (5-4) Cross-sectional area (1-3) general functional which is the integral of another functional (4-1) area of a two-dimenshysional element (5-4) General rectangular matrix (2-1) coefficient matrix in eigenvalue problem of standard form (2-3) transformation relating generalized and nodal displacements (5-3) Semiband width of coefficient matrix (2-2 6-1) Transformation relating strains and displacements (5-S) Cohesive strength (3-S) Viscous damping matrix (2-4 11-3) constitutive (stressshystrain) matrix (3-3) Constitutive (stress-strain) matrices (3-5 7-5) The maximum value for an assemblage of the largest difference between the node numbers of an element (6-1) Constitutive (strain-stress) matrix (3-3) general diagonal matrix (2-2 11-3 l l-4) Superscript indicating elastic (3-5) subscript or supershyscript indicating element (6-3) Superscript indicating elastic-plastic (3-5) Youngs modulus of elasticity (l-3 3-3) Secant and tangent moduli (3-5) Yield surface (3-5) number of degrees of freedom at a node (6-1) General functional (4-1)
ix
LIST OF SYMBOLS [F] g g G h H l1 l2 l3 [I] li l2 J3 [J] [k ] [k ]
L
L [L1 ] [m] [ma] middot [M]N[N]O [OJP pqp Q (Qo
rrRs
Flexibility matrix (2-3 9-2) Acceleration due to gravity subscript indicating global coordinate system (6-5) Vector of gradients of field va1iable ( 1 2- 1 ) Shear modulus (3-3) Thickness of two-dimensional dement (5- 1) Hydraulic head (8-2) hydrostatic stress (9-3) Strain invariants (3-2) Identity matrix Stress invariants (3- 1 ) Jacobian matrix (5-4) Element elastic stiffness matrix ( 1 -3 5-6 5-7) general property matrix (8- 1) Element geometric stiffness matrix (7-6) Bulle modulus (3-3) Assemblage elastic stiffness matrix ( 1 -3 6-5) Length of one-dimensional element (l-3 5- 1) subscript ind icating local coordinate system (6-5) Lagrangian functional (4-2) natural coordinate for oneshydimensional element (5-4) geaeral differential operator (8-4) Natural coordinates of triangle or tetrahedron (5-4) Unit lower triangular matrix (2-2) Element mass matrix ( 1 1 -2) Assemblage mass matrix (2-4 1 1 -2) Number of equations for assemblage (2-2) Coefficient matrix for interpolation field variable (disshyplacement) model (5-3 5-4) Null vector and matrix Arbitrary load ( l -2) superscript indicating plastic (3-5) Vector of nodal field variables (displacements) for element ( 1 -3 5-3) Vector of nodal actions (forces) for element ( 1 -3 5-6 5-7) Vector of nodal actions (forces) due to initial effects (initial strains) (7- 1 ) Radial coordinate (3-4) natural coordinate for hexa-hedral element (5-4) Vector of nodal field variables (djsplacements) for assembshylage ( 1 -3 6-5) Vector of nodal actions (forces) for assemblage ( 1 -3 6-5) Natural coordinate for quadrilateral or hexahedral elements (5-4)
[S]
t
[t] T
T T T
T
[T]
u v w
W W
Wp
Wpc
x y z X Y Z
X [X] ex
11
11
y l
A
amp
e 8
AI
lAJmicroV
UST OF SYMBOLS xi
Transformation for skewed coordinates at ilh node (6-6) Surface areas over which tractions and displacements are prescribed respectively (4-2) Overall transformation for skewed coordinates (6-6) general symmetric matrix (2-2) Time natural coordinate for quadrilateral or hexahedrat element (5-4) Transformation from local to global coordinates (6-5) Superscript indicating transpose (2- 1 ) kinetic energy functional (4-2) temperature (7- 1) Surface traction components (4-2) Surface traction vector (4-2) Overall transformation from local to global coordinates (6-5 7-6) Displacements in cartesian coordinates (3-2) general field variables (8- 1 ) Strain energy and complementary strain energy (4-2) General volume (4-2) volume of a tetrahedral element (5-4) Work and complementary work (4-2) Potential and complementary potential of loads (4-2) Cartesian Cltr inates Components of body force intensity (4-2) Body force intensify vector (4-2) Modal matrix (2-3 1 1 -4) Coefficient of thermal expansion (7- 1 ) Generalized coordinates (5-3) Vector of element generalized coordinater (5-3) Shear strain (3-2) Variational operator (4-1 ) Prefix indicating a finite increment (7-2) Normal strain (3-2) Vector of strains (3-2 5-5) Circumferential coordinate (3-4) slope middot of a beam (5-3) angle of twist per unit length in torsion () 2-4) Eigenvalue (2-3 7-6) Lame constant (3-3) Spectral matrix (2-3 1 1-4) Lam constant (3-3) viscosity ( 14-4) Poisson ratio (3-3) Tangent Poisson ratio (3-5) Total potential and complementary potential functionals (4-2)
Il11 p (1
a t
tp p [p] lt (I)
Reissners functional (4-2) Mass density (4-2 1 1 -2) Normal stress (3- 1) Stress vector (3- 1 5-5 9-2) Shear stress (3-1) Angle of internal friction (3-5) potential function (14-3) Coefficient matrix for generalized coordinate model (5-3) General field variable (8-3) Vector of nodal field variables ( 12-1) Frequency (2-3 1 1 -4)
CONTENTS
PREFACE
ACKNOWLEDG MENTS
LIST O F SYM BOLS
PART A I NTRODUCTION AN D BACKGROUN DMATER IAL
1 INTRODUCTION
V
vii
ix
1
3 1 - 1 Background and appl ications 31 -2 General description of the method 51 -3 Summary of the ana lysis procedure 1 01 -4 Fu ndamentals for the understanding of the method 1 5
References 1 6Further read ing 1 7
2 MATRIX TECH NIQU ES 2-1 Matrix notation 2-2 Solution of large systems of a lgebra ic equations2-3 Eigenvalue problems 2-4 otution of propagation problems
References Further read ing
3 BASIC EQUATIONS FROM SOLI DMECHANICS
3-1 Stress 3-2 Stra in a 1 1d kinematics 3 -3 Linear constitutive equations 3-4 Two-dimensional special izations of elasticity3-5 Nonlinear materia l behavior 3-6 Material characterization
References Further reading
1 8 1 81 922242628
29 2930323541 495061xiii
Introduction To The Finite ElementMethod
Publisher CBS Publications ISBN 9788123908953 Author Chandrakant Desai
Type the URL httpwwwkopykitabcomproduct10373
Get this eBook
PR FAC
High-speed electronic digital computers have enabled engineers to employ various numerical discretization techniques for approximate solutions complex problems The finitemiddot element method is one such technique It was originally developed as a tool for structural analysis but the theory and forshymulation have been progressively refined and generalized that the method has been applied successfully to such other fields heat seepage hydroshydynamics and rock mechanics As a result of this broad applicability and the systematic generality of the associated computer codes the method has gained wide acceptance designers and research engineers It now taught to both students and practicing engineers at many universities and will soon be a regular part of the curriculum at almost all colleges of engineering
In order to apply the finite element method to complex modern problem engineers must be familiar with the methods fundamental theory assumpshytions limitations However beginner may encounter difficulties in acshyquiring this familiarity becamiddot ISC there is a scarcity of unified introductory treatments that are sufficiently elementary This text therefore is designed to prepare undergraduates practicing engineers with bachelors both to solve their specific problems and to read further in the current finite element literature It is assumed that the reader has some background in the diverse mathematical associated with finite element method matrix algebra mechanics variational methods and computer skills
The book is divided into three parts introduction and background material a description of method and applications relatively simple computer code included Appendix I At the end of chapter bibliographic in-formation is provided in two sections References and Further Reading A brief descriptive commentary is included with most of the references cited Abbreviations in the references are summarized in Appendix
Part A is incorported both to guide the reader to useful texts on the back- ground material and to summarize without detailed derivations the methods middot and equations are employed Parts B C B is detailed description of the theory of the finite element method To provide a physical insight the formulation of the theory as well as the examples in the first three chapters this section are based largely the method of analysis which is common in structural and soils engineering Howevermiddot the last chapter in this section is devoted to the generalization of the theory
ACKNOWiEDGMENTS
Thls book is a synthesis of the work of many investigitors Among the indishyviduals whose contributions have greatly influenced the authors are Professors J H Argyris E B Becker R W Clough RH u-llagher H C Martin J T Oden T H H Pian E P Popov L C Reese B Fraeijs de Veubeke E L Wilson C Zienkiewicz and Dr C A Felippa
Many persons at the US Army Engineer Waterways Experiment Station have provided inspiration to the authors We wish to express gratitude to J P Sale S J Johnson W C Sherman Dr G H Keulegan Col LA Brown F R Brown and W R Martin for encouragement and sustained interest during the preparation of this book
Thanks are due to Dr G S Orenstein for carefully reading the manuscript and for offering valuable suggestions We also thank Professors E P Popov D W Murray and R S Sandhu and J Crawford D C Banks C J Huval JB Palmerton and JE Ahlberg for helpful comments
Finally we are deeply grateful to our wives P Desai and V L Abel for reading the entire manuscript for making a number of suggestions toward improving the presentation of the material and for assisting in the prepashyration of the book
Authorship of any sort is a fantastic indulgence of the ego lt is well no doubt to reflect on
how much one owes to others
J Galbraith The Affluent Society HoughtonMifflin-Co Boston Mass
vii
LIST OF SYMBOLS
The following list defines the principal symbols used in this book and gives the sections where they are explained further Other symbols are defined in context Rectangular matrices are indicated by brackets [ ] and column vectors by braces (A thorough description of the matrix notation employed is given in Section 2-1) Overdots indicate differentiation with respect to time and primes usually denote differentiation with respect to the space variable For forcing parameters used in variational principles an overbar indicates a prescribed or known value (Section 4-3) whereas within the finite element equations this notation indicates a condensed form (Sections 2-3 and 5-9)
(A]
B [B] [Ba] C
[C]
[C] [cP] [CP] D
[D]
e
ep E E E I F
Parameters for natural coordinate systems (5-4) Cross-sectional area (1-3) general functional which is the integral of another functional (4-1) area of a two-dimenshysional element (5-4) General rectangular matrix (2-1) coefficient matrix in eigenvalue problem of standard form (2-3) transformation relating generalized and nodal displacements (5-3) Semiband width of coefficient matrix (2-2 6-1) Transformation relating strains and displacements (5-S) Cohesive strength (3-S) Viscous damping matrix (2-4 11-3) constitutive (stressshystrain) matrix (3-3) Constitutive (stress-strain) matrices (3-5 7-5) The maximum value for an assemblage of the largest difference between the node numbers of an element (6-1) Constitutive (strain-stress) matrix (3-3) general diagonal matrix (2-2 11-3 l l-4) Superscript indicating elastic (3-5) subscript or supershyscript indicating element (6-3) Superscript indicating elastic-plastic (3-5) Youngs modulus of elasticity (l-3 3-3) Secant and tangent moduli (3-5) Yield surface (3-5) number of degrees of freedom at a node (6-1) General functional (4-1)
ix
LIST OF SYMBOLS [F] g g G h H l1 l2 l3 [I] li l2 J3 [J] [k ] [k ]
L
L [L1 ] [m] [ma] middot [M]N[N]O [OJP pqp Q (Qo
rrRs
Flexibility matrix (2-3 9-2) Acceleration due to gravity subscript indicating global coordinate system (6-5) Vector of gradients of field va1iable ( 1 2- 1 ) Shear modulus (3-3) Thickness of two-dimensional dement (5- 1) Hydraulic head (8-2) hydrostatic stress (9-3) Strain invariants (3-2) Identity matrix Stress invariants (3- 1 ) Jacobian matrix (5-4) Element elastic stiffness matrix ( 1 -3 5-6 5-7) general property matrix (8- 1) Element geometric stiffness matrix (7-6) Bulle modulus (3-3) Assemblage elastic stiffness matrix ( 1 -3 6-5) Length of one-dimensional element (l-3 5- 1) subscript ind icating local coordinate system (6-5) Lagrangian functional (4-2) natural coordinate for oneshydimensional element (5-4) geaeral differential operator (8-4) Natural coordinates of triangle or tetrahedron (5-4) Unit lower triangular matrix (2-2) Element mass matrix ( 1 1 -2) Assemblage mass matrix (2-4 1 1 -2) Number of equations for assemblage (2-2) Coefficient matrix for interpolation field variable (disshyplacement) model (5-3 5-4) Null vector and matrix Arbitrary load ( l -2) superscript indicating plastic (3-5) Vector of nodal field variables (displacements) for element ( 1 -3 5-3) Vector of nodal actions (forces) for element ( 1 -3 5-6 5-7) Vector of nodal actions (forces) due to initial effects (initial strains) (7- 1 ) Radial coordinate (3-4) natural coordinate for hexa-hedral element (5-4) Vector of nodal field variables (djsplacements) for assembshylage ( 1 -3 6-5) Vector of nodal actions (forces) for assemblage ( 1 -3 6-5) Natural coordinate for quadrilateral or hexahedral elements (5-4)
[S]
t
[t] T
T T T
T
[T]
u v w
W W
Wp
Wpc
x y z X Y Z
X [X] ex
11
11
y l
A
amp
e 8
AI
lAJmicroV
UST OF SYMBOLS xi
Transformation for skewed coordinates at ilh node (6-6) Surface areas over which tractions and displacements are prescribed respectively (4-2) Overall transformation for skewed coordinates (6-6) general symmetric matrix (2-2) Time natural coordinate for quadrilateral or hexahedrat element (5-4) Transformation from local to global coordinates (6-5) Superscript indicating transpose (2- 1 ) kinetic energy functional (4-2) temperature (7- 1) Surface traction components (4-2) Surface traction vector (4-2) Overall transformation from local to global coordinates (6-5 7-6) Displacements in cartesian coordinates (3-2) general field variables (8- 1 ) Strain energy and complementary strain energy (4-2) General volume (4-2) volume of a tetrahedral element (5-4) Work and complementary work (4-2) Potential and complementary potential of loads (4-2) Cartesian Cltr inates Components of body force intensity (4-2) Body force intensify vector (4-2) Modal matrix (2-3 1 1 -4) Coefficient of thermal expansion (7- 1 ) Generalized coordinates (5-3) Vector of element generalized coordinater (5-3) Shear strain (3-2) Variational operator (4-1 ) Prefix indicating a finite increment (7-2) Normal strain (3-2) Vector of strains (3-2 5-5) Circumferential coordinate (3-4) slope middot of a beam (5-3) angle of twist per unit length in torsion () 2-4) Eigenvalue (2-3 7-6) Lame constant (3-3) Spectral matrix (2-3 1 1-4) Lam constant (3-3) viscosity ( 14-4) Poisson ratio (3-3) Tangent Poisson ratio (3-5) Total potential and complementary potential functionals (4-2)
Il11 p (1
a t
tp p [p] lt (I)
Reissners functional (4-2) Mass density (4-2 1 1 -2) Normal stress (3- 1) Stress vector (3- 1 5-5 9-2) Shear stress (3-1) Angle of internal friction (3-5) potential function (14-3) Coefficient matrix for generalized coordinate model (5-3) General field variable (8-3) Vector of nodal field variables ( 12-1) Frequency (2-3 1 1 -4)
CONTENTS
PREFACE
ACKNOWLEDG MENTS
LIST O F SYM BOLS
PART A I NTRODUCTION AN D BACKGROUN DMATER IAL
1 INTRODUCTION
V
vii
ix
1
3 1 - 1 Background and appl ications 31 -2 General description of the method 51 -3 Summary of the ana lysis procedure 1 01 -4 Fu ndamentals for the understanding of the method 1 5
References 1 6Further read ing 1 7
2 MATRIX TECH NIQU ES 2-1 Matrix notation 2-2 Solution of large systems of a lgebra ic equations2-3 Eigenvalue problems 2-4 otution of propagation problems
References Further read ing
3 BASIC EQUATIONS FROM SOLI DMECHANICS
3-1 Stress 3-2 Stra in a 1 1d kinematics 3 -3 Linear constitutive equations 3-4 Two-dimensional special izations of elasticity3-5 Nonlinear materia l behavior 3-6 Material characterization
References Further reading
1 8 1 81 922242628
29 2930323541 495061xiii
Introduction To The Finite ElementMethod
Publisher CBS Publications ISBN 9788123908953 Author Chandrakant Desai
Type the URL httpwwwkopykitabcomproduct10373
Get this eBook
ACKNOWiEDGMENTS
Thls book is a synthesis of the work of many investigitors Among the indishyviduals whose contributions have greatly influenced the authors are Professors J H Argyris E B Becker R W Clough RH u-llagher H C Martin J T Oden T H H Pian E P Popov L C Reese B Fraeijs de Veubeke E L Wilson C Zienkiewicz and Dr C A Felippa
Many persons at the US Army Engineer Waterways Experiment Station have provided inspiration to the authors We wish to express gratitude to J P Sale S J Johnson W C Sherman Dr G H Keulegan Col LA Brown F R Brown and W R Martin for encouragement and sustained interest during the preparation of this book
Thanks are due to Dr G S Orenstein for carefully reading the manuscript and for offering valuable suggestions We also thank Professors E P Popov D W Murray and R S Sandhu and J Crawford D C Banks C J Huval JB Palmerton and JE Ahlberg for helpful comments
Finally we are deeply grateful to our wives P Desai and V L Abel for reading the entire manuscript for making a number of suggestions toward improving the presentation of the material and for assisting in the prepashyration of the book
Authorship of any sort is a fantastic indulgence of the ego lt is well no doubt to reflect on
how much one owes to others
J Galbraith The Affluent Society HoughtonMifflin-Co Boston Mass
vii
LIST OF SYMBOLS
The following list defines the principal symbols used in this book and gives the sections where they are explained further Other symbols are defined in context Rectangular matrices are indicated by brackets [ ] and column vectors by braces (A thorough description of the matrix notation employed is given in Section 2-1) Overdots indicate differentiation with respect to time and primes usually denote differentiation with respect to the space variable For forcing parameters used in variational principles an overbar indicates a prescribed or known value (Section 4-3) whereas within the finite element equations this notation indicates a condensed form (Sections 2-3 and 5-9)
(A]
B [B] [Ba] C
[C]
[C] [cP] [CP] D
[D]
e
ep E E E I F
Parameters for natural coordinate systems (5-4) Cross-sectional area (1-3) general functional which is the integral of another functional (4-1) area of a two-dimenshysional element (5-4) General rectangular matrix (2-1) coefficient matrix in eigenvalue problem of standard form (2-3) transformation relating generalized and nodal displacements (5-3) Semiband width of coefficient matrix (2-2 6-1) Transformation relating strains and displacements (5-S) Cohesive strength (3-S) Viscous damping matrix (2-4 11-3) constitutive (stressshystrain) matrix (3-3) Constitutive (stress-strain) matrices (3-5 7-5) The maximum value for an assemblage of the largest difference between the node numbers of an element (6-1) Constitutive (strain-stress) matrix (3-3) general diagonal matrix (2-2 11-3 l l-4) Superscript indicating elastic (3-5) subscript or supershyscript indicating element (6-3) Superscript indicating elastic-plastic (3-5) Youngs modulus of elasticity (l-3 3-3) Secant and tangent moduli (3-5) Yield surface (3-5) number of degrees of freedom at a node (6-1) General functional (4-1)
ix
LIST OF SYMBOLS [F] g g G h H l1 l2 l3 [I] li l2 J3 [J] [k ] [k ]
L
L [L1 ] [m] [ma] middot [M]N[N]O [OJP pqp Q (Qo
rrRs
Flexibility matrix (2-3 9-2) Acceleration due to gravity subscript indicating global coordinate system (6-5) Vector of gradients of field va1iable ( 1 2- 1 ) Shear modulus (3-3) Thickness of two-dimensional dement (5- 1) Hydraulic head (8-2) hydrostatic stress (9-3) Strain invariants (3-2) Identity matrix Stress invariants (3- 1 ) Jacobian matrix (5-4) Element elastic stiffness matrix ( 1 -3 5-6 5-7) general property matrix (8- 1) Element geometric stiffness matrix (7-6) Bulle modulus (3-3) Assemblage elastic stiffness matrix ( 1 -3 6-5) Length of one-dimensional element (l-3 5- 1) subscript ind icating local coordinate system (6-5) Lagrangian functional (4-2) natural coordinate for oneshydimensional element (5-4) geaeral differential operator (8-4) Natural coordinates of triangle or tetrahedron (5-4) Unit lower triangular matrix (2-2) Element mass matrix ( 1 1 -2) Assemblage mass matrix (2-4 1 1 -2) Number of equations for assemblage (2-2) Coefficient matrix for interpolation field variable (disshyplacement) model (5-3 5-4) Null vector and matrix Arbitrary load ( l -2) superscript indicating plastic (3-5) Vector of nodal field variables (displacements) for element ( 1 -3 5-3) Vector of nodal actions (forces) for element ( 1 -3 5-6 5-7) Vector of nodal actions (forces) due to initial effects (initial strains) (7- 1 ) Radial coordinate (3-4) natural coordinate for hexa-hedral element (5-4) Vector of nodal field variables (djsplacements) for assembshylage ( 1 -3 6-5) Vector of nodal actions (forces) for assemblage ( 1 -3 6-5) Natural coordinate for quadrilateral or hexahedral elements (5-4)
[S]
t
[t] T
T T T
T
[T]
u v w
W W
Wp
Wpc
x y z X Y Z
X [X] ex
11
11
y l
A
amp
e 8
AI
lAJmicroV
UST OF SYMBOLS xi
Transformation for skewed coordinates at ilh node (6-6) Surface areas over which tractions and displacements are prescribed respectively (4-2) Overall transformation for skewed coordinates (6-6) general symmetric matrix (2-2) Time natural coordinate for quadrilateral or hexahedrat element (5-4) Transformation from local to global coordinates (6-5) Superscript indicating transpose (2- 1 ) kinetic energy functional (4-2) temperature (7- 1) Surface traction components (4-2) Surface traction vector (4-2) Overall transformation from local to global coordinates (6-5 7-6) Displacements in cartesian coordinates (3-2) general field variables (8- 1 ) Strain energy and complementary strain energy (4-2) General volume (4-2) volume of a tetrahedral element (5-4) Work and complementary work (4-2) Potential and complementary potential of loads (4-2) Cartesian Cltr inates Components of body force intensity (4-2) Body force intensify vector (4-2) Modal matrix (2-3 1 1 -4) Coefficient of thermal expansion (7- 1 ) Generalized coordinates (5-3) Vector of element generalized coordinater (5-3) Shear strain (3-2) Variational operator (4-1 ) Prefix indicating a finite increment (7-2) Normal strain (3-2) Vector of strains (3-2 5-5) Circumferential coordinate (3-4) slope middot of a beam (5-3) angle of twist per unit length in torsion () 2-4) Eigenvalue (2-3 7-6) Lame constant (3-3) Spectral matrix (2-3 1 1-4) Lam constant (3-3) viscosity ( 14-4) Poisson ratio (3-3) Tangent Poisson ratio (3-5) Total potential and complementary potential functionals (4-2)
Il11 p (1
a t
tp p [p] lt (I)
Reissners functional (4-2) Mass density (4-2 1 1 -2) Normal stress (3- 1) Stress vector (3- 1 5-5 9-2) Shear stress (3-1) Angle of internal friction (3-5) potential function (14-3) Coefficient matrix for generalized coordinate model (5-3) General field variable (8-3) Vector of nodal field variables ( 12-1) Frequency (2-3 1 1 -4)
CONTENTS
PREFACE
ACKNOWLEDG MENTS
LIST O F SYM BOLS
PART A I NTRODUCTION AN D BACKGROUN DMATER IAL
1 INTRODUCTION
V
vii
ix
1
3 1 - 1 Background and appl ications 31 -2 General description of the method 51 -3 Summary of the ana lysis procedure 1 01 -4 Fu ndamentals for the understanding of the method 1 5
References 1 6Further read ing 1 7
2 MATRIX TECH NIQU ES 2-1 Matrix notation 2-2 Solution of large systems of a lgebra ic equations2-3 Eigenvalue problems 2-4 otution of propagation problems
References Further read ing
3 BASIC EQUATIONS FROM SOLI DMECHANICS
3-1 Stress 3-2 Stra in a 1 1d kinematics 3 -3 Linear constitutive equations 3-4 Two-dimensional special izations of elasticity3-5 Nonlinear materia l behavior 3-6 Material characterization
References Further reading
1 8 1 81 922242628
29 2930323541 495061xiii
Introduction To The Finite ElementMethod
Publisher CBS Publications ISBN 9788123908953 Author Chandrakant Desai
Type the URL httpwwwkopykitabcomproduct10373
Get this eBook
LIST OF SYMBOLS
The following list defines the principal symbols used in this book and gives the sections where they are explained further Other symbols are defined in context Rectangular matrices are indicated by brackets [ ] and column vectors by braces (A thorough description of the matrix notation employed is given in Section 2-1) Overdots indicate differentiation with respect to time and primes usually denote differentiation with respect to the space variable For forcing parameters used in variational principles an overbar indicates a prescribed or known value (Section 4-3) whereas within the finite element equations this notation indicates a condensed form (Sections 2-3 and 5-9)
(A]
B [B] [Ba] C
[C]
[C] [cP] [CP] D
[D]
e
ep E E E I F
Parameters for natural coordinate systems (5-4) Cross-sectional area (1-3) general functional which is the integral of another functional (4-1) area of a two-dimenshysional element (5-4) General rectangular matrix (2-1) coefficient matrix in eigenvalue problem of standard form (2-3) transformation relating generalized and nodal displacements (5-3) Semiband width of coefficient matrix (2-2 6-1) Transformation relating strains and displacements (5-S) Cohesive strength (3-S) Viscous damping matrix (2-4 11-3) constitutive (stressshystrain) matrix (3-3) Constitutive (stress-strain) matrices (3-5 7-5) The maximum value for an assemblage of the largest difference between the node numbers of an element (6-1) Constitutive (strain-stress) matrix (3-3) general diagonal matrix (2-2 11-3 l l-4) Superscript indicating elastic (3-5) subscript or supershyscript indicating element (6-3) Superscript indicating elastic-plastic (3-5) Youngs modulus of elasticity (l-3 3-3) Secant and tangent moduli (3-5) Yield surface (3-5) number of degrees of freedom at a node (6-1) General functional (4-1)
ix
LIST OF SYMBOLS [F] g g G h H l1 l2 l3 [I] li l2 J3 [J] [k ] [k ]
L
L [L1 ] [m] [ma] middot [M]N[N]O [OJP pqp Q (Qo
rrRs
Flexibility matrix (2-3 9-2) Acceleration due to gravity subscript indicating global coordinate system (6-5) Vector of gradients of field va1iable ( 1 2- 1 ) Shear modulus (3-3) Thickness of two-dimensional dement (5- 1) Hydraulic head (8-2) hydrostatic stress (9-3) Strain invariants (3-2) Identity matrix Stress invariants (3- 1 ) Jacobian matrix (5-4) Element elastic stiffness matrix ( 1 -3 5-6 5-7) general property matrix (8- 1) Element geometric stiffness matrix (7-6) Bulle modulus (3-3) Assemblage elastic stiffness matrix ( 1 -3 6-5) Length of one-dimensional element (l-3 5- 1) subscript ind icating local coordinate system (6-5) Lagrangian functional (4-2) natural coordinate for oneshydimensional element (5-4) geaeral differential operator (8-4) Natural coordinates of triangle or tetrahedron (5-4) Unit lower triangular matrix (2-2) Element mass matrix ( 1 1 -2) Assemblage mass matrix (2-4 1 1 -2) Number of equations for assemblage (2-2) Coefficient matrix for interpolation field variable (disshyplacement) model (5-3 5-4) Null vector and matrix Arbitrary load ( l -2) superscript indicating plastic (3-5) Vector of nodal field variables (displacements) for element ( 1 -3 5-3) Vector of nodal actions (forces) for element ( 1 -3 5-6 5-7) Vector of nodal actions (forces) due to initial effects (initial strains) (7- 1 ) Radial coordinate (3-4) natural coordinate for hexa-hedral element (5-4) Vector of nodal field variables (djsplacements) for assembshylage ( 1 -3 6-5) Vector of nodal actions (forces) for assemblage ( 1 -3 6-5) Natural coordinate for quadrilateral or hexahedral elements (5-4)
[S]
t
[t] T
T T T
T
[T]
u v w
W W
Wp
Wpc
x y z X Y Z
X [X] ex
11
11
y l
A
amp
e 8
AI
lAJmicroV
UST OF SYMBOLS xi
Transformation for skewed coordinates at ilh node (6-6) Surface areas over which tractions and displacements are prescribed respectively (4-2) Overall transformation for skewed coordinates (6-6) general symmetric matrix (2-2) Time natural coordinate for quadrilateral or hexahedrat element (5-4) Transformation from local to global coordinates (6-5) Superscript indicating transpose (2- 1 ) kinetic energy functional (4-2) temperature (7- 1) Surface traction components (4-2) Surface traction vector (4-2) Overall transformation from local to global coordinates (6-5 7-6) Displacements in cartesian coordinates (3-2) general field variables (8- 1 ) Strain energy and complementary strain energy (4-2) General volume (4-2) volume of a tetrahedral element (5-4) Work and complementary work (4-2) Potential and complementary potential of loads (4-2) Cartesian Cltr inates Components of body force intensity (4-2) Body force intensify vector (4-2) Modal matrix (2-3 1 1 -4) Coefficient of thermal expansion (7- 1 ) Generalized coordinates (5-3) Vector of element generalized coordinater (5-3) Shear strain (3-2) Variational operator (4-1 ) Prefix indicating a finite increment (7-2) Normal strain (3-2) Vector of strains (3-2 5-5) Circumferential coordinate (3-4) slope middot of a beam (5-3) angle of twist per unit length in torsion () 2-4) Eigenvalue (2-3 7-6) Lame constant (3-3) Spectral matrix (2-3 1 1-4) Lam constant (3-3) viscosity ( 14-4) Poisson ratio (3-3) Tangent Poisson ratio (3-5) Total potential and complementary potential functionals (4-2)
Il11 p (1
a t
tp p [p] lt (I)
Reissners functional (4-2) Mass density (4-2 1 1 -2) Normal stress (3- 1) Stress vector (3- 1 5-5 9-2) Shear stress (3-1) Angle of internal friction (3-5) potential function (14-3) Coefficient matrix for generalized coordinate model (5-3) General field variable (8-3) Vector of nodal field variables ( 12-1) Frequency (2-3 1 1 -4)
CONTENTS
PREFACE
ACKNOWLEDG MENTS
LIST O F SYM BOLS
PART A I NTRODUCTION AN D BACKGROUN DMATER IAL
1 INTRODUCTION
V
vii
ix
1
3 1 - 1 Background and appl ications 31 -2 General description of the method 51 -3 Summary of the ana lysis procedure 1 01 -4 Fu ndamentals for the understanding of the method 1 5
References 1 6Further read ing 1 7
2 MATRIX TECH NIQU ES 2-1 Matrix notation 2-2 Solution of large systems of a lgebra ic equations2-3 Eigenvalue problems 2-4 otution of propagation problems
References Further read ing
3 BASIC EQUATIONS FROM SOLI DMECHANICS
3-1 Stress 3-2 Stra in a 1 1d kinematics 3 -3 Linear constitutive equations 3-4 Two-dimensional special izations of elasticity3-5 Nonlinear materia l behavior 3-6 Material characterization
References Further reading
1 8 1 81 922242628
29 2930323541 495061xiii
Introduction To The Finite ElementMethod
Publisher CBS Publications ISBN 9788123908953 Author Chandrakant Desai
Type the URL httpwwwkopykitabcomproduct10373
Get this eBook
LIST OF SYMBOLS [F] g g G h H l1 l2 l3 [I] li l2 J3 [J] [k ] [k ]
L
L [L1 ] [m] [ma] middot [M]N[N]O [OJP pqp Q (Qo
rrRs
Flexibility matrix (2-3 9-2) Acceleration due to gravity subscript indicating global coordinate system (6-5) Vector of gradients of field va1iable ( 1 2- 1 ) Shear modulus (3-3) Thickness of two-dimensional dement (5- 1) Hydraulic head (8-2) hydrostatic stress (9-3) Strain invariants (3-2) Identity matrix Stress invariants (3- 1 ) Jacobian matrix (5-4) Element elastic stiffness matrix ( 1 -3 5-6 5-7) general property matrix (8- 1) Element geometric stiffness matrix (7-6) Bulle modulus (3-3) Assemblage elastic stiffness matrix ( 1 -3 6-5) Length of one-dimensional element (l-3 5- 1) subscript ind icating local coordinate system (6-5) Lagrangian functional (4-2) natural coordinate for oneshydimensional element (5-4) geaeral differential operator (8-4) Natural coordinates of triangle or tetrahedron (5-4) Unit lower triangular matrix (2-2) Element mass matrix ( 1 1 -2) Assemblage mass matrix (2-4 1 1 -2) Number of equations for assemblage (2-2) Coefficient matrix for interpolation field variable (disshyplacement) model (5-3 5-4) Null vector and matrix Arbitrary load ( l -2) superscript indicating plastic (3-5) Vector of nodal field variables (displacements) for element ( 1 -3 5-3) Vector of nodal actions (forces) for element ( 1 -3 5-6 5-7) Vector of nodal actions (forces) due to initial effects (initial strains) (7- 1 ) Radial coordinate (3-4) natural coordinate for hexa-hedral element (5-4) Vector of nodal field variables (djsplacements) for assembshylage ( 1 -3 6-5) Vector of nodal actions (forces) for assemblage ( 1 -3 6-5) Natural coordinate for quadrilateral or hexahedral elements (5-4)
[S]
t
[t] T
T T T
T
[T]
u v w
W W
Wp
Wpc
x y z X Y Z
X [X] ex
11
11
y l
A
amp
e 8
AI
lAJmicroV
UST OF SYMBOLS xi
Transformation for skewed coordinates at ilh node (6-6) Surface areas over which tractions and displacements are prescribed respectively (4-2) Overall transformation for skewed coordinates (6-6) general symmetric matrix (2-2) Time natural coordinate for quadrilateral or hexahedrat element (5-4) Transformation from local to global coordinates (6-5) Superscript indicating transpose (2- 1 ) kinetic energy functional (4-2) temperature (7- 1) Surface traction components (4-2) Surface traction vector (4-2) Overall transformation from local to global coordinates (6-5 7-6) Displacements in cartesian coordinates (3-2) general field variables (8- 1 ) Strain energy and complementary strain energy (4-2) General volume (4-2) volume of a tetrahedral element (5-4) Work and complementary work (4-2) Potential and complementary potential of loads (4-2) Cartesian Cltr inates Components of body force intensity (4-2) Body force intensify vector (4-2) Modal matrix (2-3 1 1 -4) Coefficient of thermal expansion (7- 1 ) Generalized coordinates (5-3) Vector of element generalized coordinater (5-3) Shear strain (3-2) Variational operator (4-1 ) Prefix indicating a finite increment (7-2) Normal strain (3-2) Vector of strains (3-2 5-5) Circumferential coordinate (3-4) slope middot of a beam (5-3) angle of twist per unit length in torsion () 2-4) Eigenvalue (2-3 7-6) Lame constant (3-3) Spectral matrix (2-3 1 1-4) Lam constant (3-3) viscosity ( 14-4) Poisson ratio (3-3) Tangent Poisson ratio (3-5) Total potential and complementary potential functionals (4-2)
Il11 p (1
a t
tp p [p] lt (I)
Reissners functional (4-2) Mass density (4-2 1 1 -2) Normal stress (3- 1) Stress vector (3- 1 5-5 9-2) Shear stress (3-1) Angle of internal friction (3-5) potential function (14-3) Coefficient matrix for generalized coordinate model (5-3) General field variable (8-3) Vector of nodal field variables ( 12-1) Frequency (2-3 1 1 -4)
CONTENTS
PREFACE
ACKNOWLEDG MENTS
LIST O F SYM BOLS
PART A I NTRODUCTION AN D BACKGROUN DMATER IAL
1 INTRODUCTION
V
vii
ix
1
3 1 - 1 Background and appl ications 31 -2 General description of the method 51 -3 Summary of the ana lysis procedure 1 01 -4 Fu ndamentals for the understanding of the method 1 5
References 1 6Further read ing 1 7
2 MATRIX TECH NIQU ES 2-1 Matrix notation 2-2 Solution of large systems of a lgebra ic equations2-3 Eigenvalue problems 2-4 otution of propagation problems
References Further read ing
3 BASIC EQUATIONS FROM SOLI DMECHANICS
3-1 Stress 3-2 Stra in a 1 1d kinematics 3 -3 Linear constitutive equations 3-4 Two-dimensional special izations of elasticity3-5 Nonlinear materia l behavior 3-6 Material characterization
References Further reading
1 8 1 81 922242628
29 2930323541 495061xiii
Introduction To The Finite ElementMethod
Publisher CBS Publications ISBN 9788123908953 Author Chandrakant Desai
Type the URL httpwwwkopykitabcomproduct10373
Get this eBook
[S]
t
[t] T
T T T
T
[T]
u v w
W W
Wp
Wpc
x y z X Y Z
X [X] ex
11
11
y l
A
amp
e 8
AI
lAJmicroV
UST OF SYMBOLS xi
Transformation for skewed coordinates at ilh node (6-6) Surface areas over which tractions and displacements are prescribed respectively (4-2) Overall transformation for skewed coordinates (6-6) general symmetric matrix (2-2) Time natural coordinate for quadrilateral or hexahedrat element (5-4) Transformation from local to global coordinates (6-5) Superscript indicating transpose (2- 1 ) kinetic energy functional (4-2) temperature (7- 1) Surface traction components (4-2) Surface traction vector (4-2) Overall transformation from local to global coordinates (6-5 7-6) Displacements in cartesian coordinates (3-2) general field variables (8- 1 ) Strain energy and complementary strain energy (4-2) General volume (4-2) volume of a tetrahedral element (5-4) Work and complementary work (4-2) Potential and complementary potential of loads (4-2) Cartesian Cltr inates Components of body force intensity (4-2) Body force intensify vector (4-2) Modal matrix (2-3 1 1 -4) Coefficient of thermal expansion (7- 1 ) Generalized coordinates (5-3) Vector of element generalized coordinater (5-3) Shear strain (3-2) Variational operator (4-1 ) Prefix indicating a finite increment (7-2) Normal strain (3-2) Vector of strains (3-2 5-5) Circumferential coordinate (3-4) slope middot of a beam (5-3) angle of twist per unit length in torsion () 2-4) Eigenvalue (2-3 7-6) Lame constant (3-3) Spectral matrix (2-3 1 1-4) Lam constant (3-3) viscosity ( 14-4) Poisson ratio (3-3) Tangent Poisson ratio (3-5) Total potential and complementary potential functionals (4-2)
Il11 p (1
a t
tp p [p] lt (I)
Reissners functional (4-2) Mass density (4-2 1 1 -2) Normal stress (3- 1) Stress vector (3- 1 5-5 9-2) Shear stress (3-1) Angle of internal friction (3-5) potential function (14-3) Coefficient matrix for generalized coordinate model (5-3) General field variable (8-3) Vector of nodal field variables ( 12-1) Frequency (2-3 1 1 -4)
CONTENTS
PREFACE
ACKNOWLEDG MENTS
LIST O F SYM BOLS
PART A I NTRODUCTION AN D BACKGROUN DMATER IAL
1 INTRODUCTION
V
vii
ix
1
3 1 - 1 Background and appl ications 31 -2 General description of the method 51 -3 Summary of the ana lysis procedure 1 01 -4 Fu ndamentals for the understanding of the method 1 5
References 1 6Further read ing 1 7
2 MATRIX TECH NIQU ES 2-1 Matrix notation 2-2 Solution of large systems of a lgebra ic equations2-3 Eigenvalue problems 2-4 otution of propagation problems
References Further read ing
3 BASIC EQUATIONS FROM SOLI DMECHANICS
3-1 Stress 3-2 Stra in a 1 1d kinematics 3 -3 Linear constitutive equations 3-4 Two-dimensional special izations of elasticity3-5 Nonlinear materia l behavior 3-6 Material characterization
References Further reading
1 8 1 81 922242628
29 2930323541 495061xiii
Introduction To The Finite ElementMethod
Publisher CBS Publications ISBN 9788123908953 Author Chandrakant Desai
Type the URL httpwwwkopykitabcomproduct10373
Get this eBook
Il11 p (1
a t
tp p [p] lt (I)
Reissners functional (4-2) Mass density (4-2 1 1 -2) Normal stress (3- 1) Stress vector (3- 1 5-5 9-2) Shear stress (3-1) Angle of internal friction (3-5) potential function (14-3) Coefficient matrix for generalized coordinate model (5-3) General field variable (8-3) Vector of nodal field variables ( 12-1) Frequency (2-3 1 1 -4)
CONTENTS
PREFACE
ACKNOWLEDG MENTS
LIST O F SYM BOLS
PART A I NTRODUCTION AN D BACKGROUN DMATER IAL
1 INTRODUCTION
V
vii
ix
1
3 1 - 1 Background and appl ications 31 -2 General description of the method 51 -3 Summary of the ana lysis procedure 1 01 -4 Fu ndamentals for the understanding of the method 1 5
References 1 6Further read ing 1 7
2 MATRIX TECH NIQU ES 2-1 Matrix notation 2-2 Solution of large systems of a lgebra ic equations2-3 Eigenvalue problems 2-4 otution of propagation problems
References Further read ing
3 BASIC EQUATIONS FROM SOLI DMECHANICS
3-1 Stress 3-2 Stra in a 1 1d kinematics 3 -3 Linear constitutive equations 3-4 Two-dimensional special izations of elasticity3-5 Nonlinear materia l behavior 3-6 Material characterization
References Further reading
1 8 1 81 922242628
29 2930323541 495061xiii
Introduction To The Finite ElementMethod
Publisher CBS Publications ISBN 9788123908953 Author Chandrakant Desai
Type the URL httpwwwkopykitabcomproduct10373
Get this eBook
CONTENTS
PREFACE
ACKNOWLEDG MENTS
LIST O F SYM BOLS
PART A I NTRODUCTION AN D BACKGROUN DMATER IAL
1 INTRODUCTION
V
vii
ix
1
3 1 - 1 Background and appl ications 31 -2 General description of the method 51 -3 Summary of the ana lysis procedure 1 01 -4 Fu ndamentals for the understanding of the method 1 5
References 1 6Further read ing 1 7
2 MATRIX TECH NIQU ES 2-1 Matrix notation 2-2 Solution of large systems of a lgebra ic equations2-3 Eigenvalue problems 2-4 otution of propagation problems
References Further read ing
3 BASIC EQUATIONS FROM SOLI DMECHANICS
3-1 Stress 3-2 Stra in a 1 1d kinematics 3 -3 Linear constitutive equations 3-4 Two-dimensional special izations of elasticity3-5 Nonlinear materia l behavior 3-6 Material characterization
References Further reading
1 8 1 81 922242628
29 2930323541 495061xiii
Introduction To The Finite ElementMethod
Publisher CBS Publications ISBN 9788123908953 Author Chandrakant Desai
Type the URL httpwwwkopykitabcomproduct10373
Get this eBook
Introduction To The Finite ElementMethod
Publisher CBS Publications ISBN 9788123908953 Author Chandrakant Desai
Type the URL httpwwwkopykitabcomproduct10373
Get this eBook
