Table Of Content

CO· ROM Included INTRODUCTION TO FINITE ELEMENTS ENGINEERING IN THIRD EDITION Tirupathi R. Chandrupatla Ashok D. Belegundu Programs Included in the CD-ROM cn. . cn. . CH·1 CH·l CH·J CH·5 GAUSS FEMlD TRUSS2D CST AXISYM caSOL TRUSSKY SKYLINE CH·7 CH·8 CH·9 CH·IO CH·ll CH·U QUAD BEAM TETRA3D HEATlD INVITR MESHGEN QUADCG FRAME2D HEXAFRQN HEAT2D JACOBI PL0T20, BESTFIT AXIQUAD FRAME3D TORSION BEAMKM BESTFITQ CSTKM CONTOURA GENEIGEN CONTOURB CD-ROM Contents Direetory Deseription IQBASIC Programs in QBASIC \FORTRAN Programs in Fortran Language Ie Programs in C IVB Programs in Visual Basic IEXC'ELVB Programs in Excel Visual Basic IMATLAB Programs in MATLAB IEXAMPLES Example Data Files (.inp extension) List of Key Symbols Used in the Text Symbol Description u(.t,y,:) "" [u(x,y, z), v(x, y, l,), displacements along coordinate directions at point (x, y, zj w(x,y,Z)]T r, [[,,f,,a components of body force per unit volume, at point (x, y. z) T"= [T"T,,1~P components of traction force per unit area, at point (x, y,z) on the surface i == [f.,ey,i"}""Y,,,Yq]T strain eomponents;t: are normal strains and}' are engineering shear strains u =- {u"u"U"1',,,1',. . T.t,]T stress components; u are normal stresses and T are engineering shear stresses n Potential energy = U + W P, where U == strain energy, W P = work potential • vector of displacements of the nodes (degrees of freedom or DOF) of an element, dimension (NDN·NEN,l)---see next Table for explanation of NDNand NEN Q vector of displacements of ALL the nodes of an element, dimension (NN·NDN, I)-see next Table for explanation of NN and NDN k element stiffne~ matrix: strain energy in element, U, == !qTkq K glohal stiffness matrix for entire structure: n = I Q1 KQ _ Q IF r body force in element e distributed to the nodes of the element T' traction fOfce in element e distributed to the nodes of the element ""(x,y, ~) • virtual displacement variable: counterpart of the real displacement u(x, y, z) vector of vinual displacements of the nodes in an element; cOWlterpart of q N, D,and B shape functions in t1)( coordinates, material matrix, strain-displacement malril. respectively. u = Nq, € = Bq and (J' = DBq structure ofInpnt Files' TITLE (*) PROBLEM DESCRIPTION (*) NN NE NM NOIM NEN NON (*) 4 2 2 2 3 2 1 Line of data, 6 entries per 1ine NO Nl NMPC (*) 5 2 0 --- 1 Line of data, 3 entries Node. Coordinate'! Coordinate#NDIH (*) ~, O~2 4 g; } ---NN Lines of data, (NlJIM+l)entries Eleml Nodell NodelNEN Mat' Element Characterois.)ti cstt (*) of 1 4 1 2 1 0.5 ---HE Lines data. 2 3 4 2 2 O.S O. (NEN+2+#ofChar.)entries OO~~F' speCggified}DiSPlacement (*) ---NO Llnes of data, 2 entries 8 0 DOF' Load (*) 4 -7500 ) ---NL Lines of data, 2 entries 3 3000 MAn Mat.erial Propertiestt (*) 1 30e6 0.25 12e-6 } --!.NM Lines of data, (1+ , of prop.)entries 2 20e6 0.3 O. 81 i 82 j B3 (Multipoint constraint: Bl*Qi+B2*Qj=B3) (OJ } ---NMPC Lines of data, 5 entries tHEATlD and HEATID Programs need extra boundary data about flux and convection. (See Chapter 10.) (~) = DUMMY LINE - necessary No/eo' No Blank Lines must be present in the input file t'See below for desCription of element characteristics and material properties Main Program Variables NN = Number of Nodes; NE = Number of Elements; NM = Number of Different Materials NDIM = Number ofCoordinales per Node (e.g..NDlM = Uor2·D.or = 3for3.D): NEN = Number of Nodes per Element (e.g., NEN '" 3 for 3-noded trianguJar element, or = 4 for a 4-noded quadrilateral) NDN '" Number of Degrees of Freedom per Node (e.g., NDN '" 2 for a CiT element, or '" 6 for 3-D beam element) ND = Number of Degrees of Freedom along which Displacement is Specified'" No. of Boundary Conditions NL = Number of Applied Component Loads (along Degrees of Freedom) NMPC = Number of Multipoint Constraints; NO '" Total Number of Degrees of Freedom = NN ~ NDN .......... Mlterial PToperties Element Cbancterisdcs E FEMlD, TRUSS, TRUSSKY Area, Temperature Rise E,II,a CST,QUAD Thickness, Temperature Rise AXISYM Temperature Rise E. ". Ct E FRAME2D Area, Inertia, Distributed liJad E FRAME3D Area, 3·ioertias, 2-Distr. Loads E,II,a TETRA, HEXAFNT Temperature Rise Thermal Conductivity, k HEATID Element Heat Source E,p BEAMKM Inertia,Area CSTKM Thickness E,I',a,p 1 I Introduction to Finite Elements in Engineering Introduction to Finite Elements in Engineering T H I R D EDITION TIRUPATHI R. CHANDRUPATLA Rowan University Glassboro, New Jersey ASHOK D. BELEGUNDU The Pennsylvania State University University Park, Pennsylvania Prentice Hall, Upper Saddle River, New Jersey 07458 Chandl'llpalla,TIl'lIpathi R., Introouction to finite elements in engineering ITIl'lIpathi R. ClIandrupatla,Ashok D. Belegundu.--3rd ed. p.em. Includes bibliographical references and index. ISBN 0-13-061591-9 I. finite element method.2. Engineering mathematics. I. Belugundu,Ashok D., Il.TItle TA347.F5 003 2001 620'.001 '51535--d0:21 Vice President and Editorial Director,ECS: Marcial. Horton Acquisitions Editor; Laura Fischer Editorial Assistant: Erin Katchmar Vice President and Director of Production and Manufacturing, ESM: David W Riccardi E~ecutive Managing Editor: Vince O'Brien Managing Editor, David A. George Production SeMiicesiComposition: Prepart,lnc. Director of Creative Services: Pa ... 18elfanti Creative Director: Carole Anson Art Director: kylU! Cante Art Editorc Greg O. .. lIes Cover Designer: Bmu &nselaar Manufacturing Manager: Trudy PisciOlli Manufacturing Buyer: LyndD Castillo Marketing Manager; Holly Stark © 2002 by Prentice·Hall, Inc. Upper Saddle River, New Jersey 07458 All rights reseMied. No part of this book may be reproduced in any form or by any means, without permission in writing from the publisher. The author and publisher of this book have used their besl efforts in preparing this hook. These efforts include the development, re search,and testing of the theories and programs to determine their effectiveness. The author and publisher make no warranty of any kind. expressed or implied. with regard 10 thele programs Or Ihe dOf,:Umentation conlained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with,or arising out of, the furnishing, perfonnance,or use of Ihese programs. "Visual Basic~ and "'ExcelH are registered trademarks of the Microsoft Corporation, Redmond, WA. "MATLAB'" is a registered trademark of The MalhWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760.-21)98. Printed in the United State. of America 109876 ISBN D-13-0b1S~1-~ Pearson Education Ltd .. Lond"" Pearson Education Australia P1}". Ltd .Sydn<,y Pearson EdUcation Singapore, Pte. Ltd. Pearson Education North Asia Ltd. . Hong Kong Pearson Education Canada. Inc .• Toronto Pearson Educadon de Me~ico, S.A.de C.V. Pear'lOf) Educalion--Japan. 7okyo Pearson 'oducation Malaysia, Pte. Ltd. Pearson EdUcation. Upper Soddle Ri~er, New Jersey To our parents Contents PREFACE xv 1 FUNDAMENTAL CONCEPTS 1 1.1 Introduction 1 1.2 Historical Background 1 1.3 Outline of Presentation 2 1.4 Stresses and Equilibrium 2 1.5 Boundary Conditions 4 1.6 Strain-Displacement Relations 4 1.7 Stress-Strain Relations 6 Special Cases, 7 1.8 Temperature Effects 8 1.9 Potential Energy and Equilibrium; The Rayleigh-Ritz Method 9 Potential Energy TI, 9 Rayleigh-Ritz Method, 11 1.10 Galerkin's Method 13 1.11 Saint Venant's Principle 16 1.12 Von Mises Stress 17 1.13 Computer Programs 17 1.14 Conclusion 18 Historical References 18 Problems 18 2 MATRIX ALGEBRA AND GAUSSIAN EUMINATION 22 2.1 Matrix Algebra 22 Rowand Column Vectors, 23 Addition and Subtraction, 23 Multiplication by a Scalar. 23 vii • viii Contents Matrix Multiplication, 23 Transposition, 24 Differentiation and Integration, 24 Square Matrix, 25 Diagonal Matrix, 25 Identity Matrix, 25 Symmetric Matrix, 25 Upper Triangular Matrix, 26 Determinant of a Matrix, 26 Matrix Inversion, 26 Eigenvalues and Eigenvectors, 27 Positive Definite Matrix, 28 Cholesky Decomposition, 29 2.2 Gaussian Elimination 29 General Algorithm for Gaussian Elimination, 30 Symmetric Matrix. 33 Symmetric Banded Matrices, 33 Solution with Multiple Right Sides, 35 Gaussian Elimination with Column Reduction, 36 Skyline Solution, 38 - Frontal Solution, 39 2.3 Conjugate Gradient Method for Equation Solving 39 Conjugate Gradient Algorithm, 40 Problems 41 Program Listings, 43 3 ONE-DIMENSIONAL PROBLEMS 45 3.1 Introduction 45 3.2 Fmite Element Modeling 46 Element Division, 46 Numbering Scheme, 47 3.3 Coordinates and Shape Functions 48 3.4 The Potential.Energy Approach 52 Element Stiffness Matrix, 53 Force Terms, 54 3.5 The Galerkin Approach 56 Element Stiffness, 56 Force Terms, 57 3.6 Assembly of the Global Stiffness Matrix and Load Vector 58 3.7 Properties of K 61 3.8 The Finite Element Equations; Treatment of Boundary Conditions 62 Types of Boundary Conditions, 62 Elimitwtion Approach, 63 Penalty Approach, 69 Multipoint Constraints, 74 .... ......_ _____________ ~l ~,,~. Contents Ix 3.9 Quadratic Shape Functions 78 3.10 Temperature Effects 84 Input Data File, 88 Problems 88 Program Listing, 98 4 TRUSSES 103 4.1 Introduction 103 4.2 Plane 1fusses 104 Local and Global Coordinate Systems, 104 Formulas for Calculating € and m, 105 Element Stiffness Matrix, 106 Stress Calculations, 107 Temperature Effects, 111 4.3 Three-Dimensional'Ihlsses 114 4.4 Assembly of Global Stiffness Matrix for the Banded and Skyline Solutions 116 Assembly for Banded Solution, 116 Input Data File, 119 Problems 120 Program Listing, 128 S TWO·DIMENSIONAL PROBLEMS USING CONSTANT STRAIN TRIANGLES 130 5.1 Introduction 130 5.2 Finite Element Modeling 131 5.3 Constant-Strain Triangle (CST) 133 lsoparametric Representation, 135 Potential· Energy Approach, 139 Element Stiffness, 140 Force Terms, 141 Galerkin Approach, 146 Stress Calculations, 148 Temperature Effects,150 5.4 Problem Modeling and Boundary Conditions 152 Some General Comments on Dividing into Elements, 154 5.5 Orthotropic Materials 154 Temperature Effects, 157 Input Data File, 160 Problems 162 Program Listing, 174