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Introduction to finite element analysis using MATLAB and abaqus
Mathematics Khennane Introduction to Introduction to Finite Element Analysis Using MATLAB and Abaqus ® There are some books that target the theory of the finite element, while others focus on the programming side of things Introduction to Finite Element Analysis Using MATLAB® and Abaqus accomplishes both This book teaches the first principles of the finite element method It presents the theory of the finite element method while maintaining a balance between its mathematical formulation, programming implementation, and application using commercial software The computer implementation is carried out using MATLAB, while the practical applications are carried out in both MATLAB and Abaqus MATLAB is a high-level language specially designed for dealing with matrices, making it particularly suited for programming the finite element method, while Abaqus is a suite of commercial finite element software Introduction to Finite Element Analysis Using MATLAB® and Abaqus introduces and explains theory in each chapter, and provides corresponding examples It offers introductory notes and provides matrix structural analysis for trusses, beams, and frames The book examines the theories of stress and strain and the relationships between them The author then covers weighted residual methods and finite element approximation and numerical integration He presents the finite element formulation for plane stress/strain problems, introduces axisymmetric problems, and highlights the theory of plates The text supplies step-by-step procedures for solving problems with Abaqus interactive and keyword editions The described procedures are implemented as MATLAB codes, and Abaqus files can be found on the CRC Press website K16894 an informa business w w w c rc p r e s s c o m 6000 Broken Sound Parkway, NW Suite 300, Boca Raton, FL 33487 711 Third Avenue New York, NY 10017 Park Square, Milton Park Abingdon, Oxon OX14 4RN, UK w w w c rc p r e s s c o m Introduction to Finite Element Analysis Using MATLAB® and Abaqus “A very good introduction to the finite element method with a balanced treatment of theory and implementation.” — F Albermani, Reader in Structural Engineering, The University of Queensland, Australia Finite Element Analysis Using MATLAB and Abaqus ® Amar Khennane Introduction to Finite Element Analysis Using MATLAB and Abaqus đ â 2013 by Taylor & Francis Group, LLC © 2013 by Taylor & Francis Group, LLC Introduction to Finite Element Analysis Using MATLAB and Abaqus ® Amar Khennane Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business â 2013 by Taylor & Francis Group, LLC MATLABđ is a trademark of The MathWorks, Inc and is used with permission The MathWorks does not warrant the accuracy of the text or exercises in this book This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2013 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Version Date: 20130220 International Standard Book Number-13: 978-1-4665-8021-3 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com © 2013 by Taylor & Francis Group, LLC Contents List of Figures xiii List of Tables xxv Preface xxvii Author xxix Chapter Introduction 1.1 1.2 1.3 1.4 Chapter Prologue Finite Element Analysis and the User Aim of the Book Book Organization Bar Element 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Introduction One-Dimensional Truss Element 2.2.1 Formulation of the Stiffness Matrix: The Direct Approach 2.2.2 Two-Dimensional Truss Element Global Stiffness Matrix Assembly 2.3.1 Discretization 2.3.2 Elements’ Stiffness Matrices in Local Coordinates 2.3.3 Elements’ Stiffness Matrices in Global Coordinates 10 2.3.3.1 Element 11 2.3.3.2 Element 11 2.3.3.3 Element 12 2.3.4 Global Matrix Assembly 12 2.3.4.1 Only Element Is Present 13 2.3.4.2 Only Element Is Present 13 2.3.4.3 Only Element Is Present 13 2.3.5 Global Force Vector Assembly 14 Boundary Conditions 15 2.4.1 General Case 15 Solution of the System of Equations 16 Support Reactions 17 Members’ Forces 18 Computer Code: truss.m 19 2.8.1 Data Preparation 20 2.8.1.1 Nodes Coordinates 20 2.8.1.2 Element Connectivity 20 2.8.1.3 Material and Geometrical Properties 20 2.8.1.4 Boundary Conditions 20 2.8.1.5 Loading 21 2.8.2 Element Matrices 21 2.8.2.1 Stiffness Matrix in Local Coordinates 21 2.8.2.2 Transformation Matrix 22 v © 2013 by Taylor & Francis Group, LLC vi Contents 2.8.2.3 Stiffness Matrix in Global Coordinates 2.8.2.4 “Steering” Vector 2.8.3 Assembly of the Global Stiffness Matrix 2.8.4 Assembly of the Global Force Vector 2.8.5 Solution of the Global System of Equations 2.8.6 Nodal Displacements 2.8.7 Element Forces 2.8.8 Program Scripts 2.9 Problems 2.9.1 Problem 2.1 2.9.2 Problem 2.2 2.10 Analysis of a Simple Truss with Abaqus 2.10.1 Overview of Abaqus 2.10.2 Analysis of a Truss with Abaqus Interactive Edition 2.10.2.1 Modeling 2.10.2.2 Analysis 2.10.3 Analysis of a Truss with Abaqus Keyword Edition Chapter Beam Element 63 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Chapter 22 22 23 23 23 23 23 24 27 27 32 35 35 36 36 51 57 Introduction 63 Stiffness Matrix 63 Uniformly Distributed Loading 67 Internal Hinge 71 Computer Code: beam.m 73 3.5.1 Data Preparation 73 3.5.1.1 Nodes Coordinates 73 3.5.1.2 Element Connectivity 74 3.5.1.3 Material and Geometrical Properties 74 3.5.1.4 Boundary Conditions 74 3.5.1.5 Internal Hinges 74 3.5.1.6 Loading 75 3.5.1.7 Stiffness Matrix 76 3.5.2 Assembly and Solution of the Global System of Equations 76 3.5.3 Nodal Displacements 76 3.5.4 Element Forces 77 Problems 81 3.6.1 Problem 3.1 81 3.6.2 Problem 3.2 84 3.6.3 Problem 3.3 87 Analysis of a Simple Beam with Abaqus 90 3.7.1 Interactive Edition 90 3.7.2 Analysis of a Beam with Abaqus Keyword Edition 103 Rigid Jointed Frames 107 4.1 4.2 4.3 Introduction 107 Stiffness Matrix of a Beam–Column Element 107 Stiffness Matrix of a Beam–Column Element in the Presence of Hinged End 107 © 2013 by Taylor & Francis Group, LLC vii Contents 4.4 4.5 4.6 4.7 Chapter Global and Local Coordinate Systems 108 Global Stiffness Matrix Assembly and Solution for Unknown Displacements 109 Computer Code: frame.m 109 4.6.1 Data Preparation 109 4.6.1.1 Nodes Coordinates 110 4.6.1.2 Element Connectivity 110 4.6.1.3 Material and Geometrical Properties 110 4.6.1.4 Boundary Conditions 110 4.6.1.5 Internal Hinges 111 4.6.1.6 Loading 111 4.6.2 Element Matrices 112 4.6.2.1 Stiffness Matrix in Local Coordinates 112 4.6.2.2 Transformation Matrix 113 4.6.2.3 Stiffness Matrix in Global Coordinates 113 4.6.2.4 “Steering” Vector 113 4.6.2.5 Element Loads 113 4.6.3 Assembly of the Global Stiffness Matrix 113 4.6.4 Solution of the Global System of Equations 114 4.6.5 Nodal Displacements 114 4.6.6 Element Forces 114 Analysis of a Simple Frame with Abaqus 124 4.7.1 Interactive Edition 124 4.7.2 Keyword Edition 132 Stress and Strain Analysis 135 5.1 5.2 5.3 Introduction 135 Stress Tensor 135 5.2.1 Definition 135 5.2.2 Stress Tensor–Stress Vector Relationships 137 5.2.3 Transformation of the Stress Tensor 139 5.2.4 Equilibrium Equations 139 5.2.5 Principal Stresses 140 5.2.6 von Mises Stress 141 5.2.7 Normal and Tangential Components of the Stress Vector 141 5.2.8 Mohr’s Circles for Stress 143 5.2.9 Engineering Representation of Stress 144 Deformation and Strain 144 5.3.1 Definition 144 5.3.2 Lagrangian and Eulerian Descriptions 145 5.3.3 Displacement 146 5.3.4 Displacement and Deformation Gradients 147 5.3.5 Green Lagrange Strain Matrix 148 5.3.6 Small Deformation Theory 149 5.3.6.1 Infinitesimal Strain 149 5.3.6.2 Geometrical Interpretation of the Terms of the Strain Tensor 150 5.3.6.3 Compatibility Conditions 152 5.3.7 Principal Strains 152 © 2013 by Taylor & Francis Group, LLC viii Contents 5.4 5.5 Chapter Weighted Residual Methods 175 6.1 6.2 6.3 6.4 6.5 6.6 Chapter 5.3.8 Transformation of the Strain Tensor 153 5.3.9 Engineering Representation of Strain 153 Stress–Strain Constitutive Relations 154 5.4.1 Generalized Hooke’s Law 154 5.4.2 Material Symmetries 155 5.4.2.1 Symmetry with respect to a Plane 155 5.4.2.2 Symmetry with respect to Three Orthogonal Planes 157 5.4.2.3 Symmetry of Rotation with respect to One Axis 157 5.4.3 Isotropic Material 158 5.4.3.1 Modulus of Elasticity 160 5.4.3.2 Poisson’s Ratio 160 5.4.3.3 Shear Modulus 160 5.4.3.4 Bulk Modulus 160 5.4.4 Plane Stress and Plane Strain 162 Solved Problems 163 5.5.1 Problem 5.1 163 5.5.2 Problem 5.2 164 5.5.3 Problem 5.3 167 5.5.4 Problem 5.4 168 5.5.5 Problem 5.5 170 5.5.6 Problem 5.6 171 5.5.7 Problem 5.7 172 5.5.8 Problem 5.8 174 Introduction 175 General Formulation 175 Galerkin Method 176 Weak Form 178 Integrating by Part over Two and Three Dimensions (Green Theorem) 179 Rayleigh Ritz Method 183 6.6.1 Definition 183 6.6.2 Functional Associated with an Integral Form 183 6.6.3 Rayleigh Ritz Method 183 6.6.4 Example of a Natural Functional 185 Finite Element Approximation 191 7.1 7.2 7.3 7.4 7.5 Introduction 191 General and Nodal Approximations 191 Finite Element Approximation 193 Basic Principles for the Construction of Trial Functions 195 7.4.1 Compatibility Principle 195 7.4.2 Completeness Principle 196 Two-Dimensional Finite Element Approximation 197 7.5.1 Plane Linear Triangular Element for C0 Problems 197 7.5.1.1 Shape Functions 197 7.5.1.2 Reference Element 199 7.5.1.3 Area Coordinates 202 7.5.2 Linear Quadrilateral Element for C0 Problems 203 © 2013 by Taylor & Francis Group, LLC ix Contents 7.5.2.1 7.5.2.2 7.6 Chapter Numerical Integration 211 8.1 8.2 8.3 8.4 8.5 Chapter Geometrical Transformation 203 Construction of a Trial Function over a Linear Quadrilateral Element 206 Shape Functions of Some Classical Elements for C0 Problems 207 7.6.1 One-Dimensional Elements 207 7.6.1.1 Two-Nodded Linear Element 207 7.6.1.2 Three-Nodded Quadratic Element 207 7.6.2 Two-Dimensional Elements 207 7.6.2.1 Four-Nodded Bilinear Quadrilateral 207 7.6.2.2 Eight-Nodded Quadratic Quadrilateral 208 7.6.2.3 Three-Nodded Linear Triangle 208 7.6.2.4 Six-Nodded Quadratic Triangle 208 7.6.3 Three-Dimensional Elements 208 7.6.3.1 Four-Nodded Linear Tetrahedra 208 7.6.3.2 Ten-Nodded Quadratic Tetrahedra 209 7.6.3.3 Eight-Nodded Linear Brick Element 209 7.6.3.4 Twenty-Nodded Quadratic Brick Element 210 Introduction 211 Gauss Quadrature 211 8.2.1 Integration over an Arbitrary Interval [a, b] 214 8.2.2 Integration in Two and Three Dimensions 215 Integration over a Reference Element 216 Integration over a Triangular Element 217 8.4.1 Simple Formulas 217 8.4.2 Numerical Integration over a Triangular Element 218 Solved Problems 219 8.5.1 Problem 8.1 219 8.5.2 Problem 8.2 221 8.5.3 Problem 8.3 226 Plane Problems 231 9.1 9.2 9.3 9.4 Introduction 231 Finite Element Formulation for Plane Problems 231 Spatial Discretization 234 Constant Strain Triangle 235 9.4.1 Displacement Field 236 9.4.2 Strain Matrix 237 9.4.3 Stiffness Matrix 237 9.4.4 Element Force Vector 237 9.4.4.1 Body Forces 238 9.4.4.2 Traction Forces 238 9.4.4.3 Concentrated Forces 239 9.4.5 Computer Codes Using the Constant Strain Triangle 240 9.4.5.1 Data Preparation 241 9.4.5.2 Nodes Coordinates 243 9.4.5.3 Element Connectivity 243 9.4.5.4 Material Properties 243 © 2013 by Taylor & Francis Group, LLC Appendix A: List of MATLAB Modules and Functions A.47 T6_mesh.m % This module generates a mesh of the linear strain triangular element % global nnd nel geom connec XIG YIG global Length Width NXE NYE X_origin Y_origin dhx dhy % % nnd = 0; k = 0; for i = 1:NXE for j=1:NYE k = k + 1; n1 = (2*j-1) + (2*i-2)*(2*NYE+1) ; n2 = (2*j-1) + (2*i-1)*(2*NYE+1); n3 = (2*j-1) + (2*i)*(2*NYE+1); n4 = n1 + 1; n5 = n2 + 1; n6 = n3 + ; n7 = n1 + 2; n8 = n2 + 2; n9 = n3 + 2; % geom(n1,:) = [(i-1)*dhx - X_origin (j-1)*dhy - Y_origin]; geom(n2,:) = [((2*i-1)/2)*dhx - X_origin (j-1)*dhy - Y_origin ]; geom(n3,:) = [i*dhx - X_origin (j-1)*dhy - Y_origin ]; geom(n4,:) = [(i-1)*dhx - X_origin ((2*j-1)/2)*dhy - Y_origin ]; geom(n5,:) = [((2*i-1)/2)*dhx - X_origin ((2*j-1)/2)*dhy - Y_origin ]; geom(n6,:) = [i*dhx - X_origin ((2*j-1)/2)*dhy - Y_origin ]; geom(n7,:) = [(i-1)*dhx - X_origin j*dhy - Y_origin]; geom(n8,:) = [((2*i-1)/2)*dhx - X_origin j*dhy - Y_origin]; geom(n9,:) = [i*dhx - X_origin j*dhy - Y_origin]; % nel = 2*k; m = nel -1; connec(m,:) = [n1 n2 n3 n5 n7 n4]; connec(nel,:) = [n3 n6 n9 n8 n7 n5]; max_n = max([n1 n2 n3 n4 n5 n6 n7 n8 n9]); if(nnd y1) theta=2*atan(1); else theta=-2*atan(1); end else theta=atan((y2-y1)/(x2-x1)); end % % Construct the transformation matrix % C = [cos(theta) -sin(theta) 0 ; sin(theta) cos(theta) 0 ; 0 cos(theta) -sin(theta) ; 0 sin(theta) cos(theta) ]; % %%%%%%%%%%%%%%%% end function truss_C %%%%%%%%%%%%% A.49 truss_g.m function[g] = truss_g(i) % % This function forms the steering vector for element i % global connec nf % % retrieve the nodes of element i % node_1=connec(i,1); node_2=connec(i,2); % % Form the steering vector from element’s degrees % of freedom % g=[nf(node_1,1); nf(node_1,2); nf(node_2,1);nf(node_2,2)]; % %%%%%%%%%%%%%%%%%% end function truss_g %%%%%%%%%%%%%%%% A.50 truss_kl.m function[kl] = truss_kl(i) % % This function forms the element stiffness matrix % in local coordinates % global geom connec prop % % retrieve the nodes of element i % node_1=connec(i,1); node_2=connec(i,2); % % % Retrieve the x and y coordinates of nodes and % x1=geom(node_1,1); y1=geom(node_1,2); x2=geom(node_2,1); y2=geom(node_2,2); % % Evaluate length of element i % © 2013 by Taylor & Francis Group, LLC Appendix A: List of MATLAB Modules and Functions L = sqrt((x2-x1)^2 + (y2-y1)^2); % % Retrieve section properties of element i % E= prop(i,1); A=prop(i,2); % % Calculate element stiffness matrix in its % local coordinates % kl=[E*A/L -E*A/L ; 0 0 ; -E*A/L E*A/L ; 0 0 ]; % %%%%%%%%%%%%%%%%%% End function truss_kl%%%%%%%%%%%% © 2013 by Taylor & Francis Group, LLC 443 © 2013 by Taylor & Francis Group, LLC Appendix B: Statically Equivalent Nodal Forces Statically equivalent nodal loads –P –P 2 PL –PL 8 Actual load P L –qL –qL2 12 q q –qL qL2 12 –3qL 20 –qL2 30 q –7qL 20 qL2 20 5qL 3qL qL2 FIGURE B.1 Common beam loadings 445 © 2013 by Taylor & Francis Group, LLC © 2013 by Taylor & Francis Group, LLC Appendix C: Index Notation and Transformation Laws for Tensors C.1 INDEX NOTATION FOR VECTORS AND TENSORS C.1.1 VECTOR AND TENSOR COMPONENTS Operations on Cartesian components of vectors and tensors can be expressed very efficiently and clearly using index notation The index notation refers to vectors or tensors by their general term, with the indices ranging over the dimensions of the vector or the tensor Let u be a vector and a a second-order tensor defined in a Cartesian basis Using matrix notation, they can be represented by their Cartesian components as ⎧ ⎫ ⎨u1 ⎬ u = u2 ⎩ ⎭ u3 ⎛ a11 a = ⎝a21 a31 a12 a22 a32 ⎞ a13 a23 ⎠ a33 (C.1) Using index notation, the vector u and the tensor a can be expressed in a compact manner as u = ui a = aij (C.2) C.1.2 EINSTEIN SUMMATION CONVENTIONS Under the rules of index notation, if an index is repeated in a product of vectors or tensors, summation is implied over the range of the repeated index For example, for a range from to 3, the following expressions can be developed as bi = a1 b1 + a2 b2 + a3 b3 (C.3) ⎧ ⎫ ⎨a11 x1 + a12 x2 + a13 x3 ⎬ ci = aik xk = a21 x1 + a22 x2 + a23 x3 ⎩ ⎭ a31 x1 + a32 x2 + a33 x3 (C.4) λ = aij bij = a1 b1 + a1 b2 + a1 b3 + a2 b1 + a2 b2 + a2 b3 + a3 b1 + a3 b2 + a3 b3 (C.5) cij = aik bkj = ai1 b1j + ai2 b2j + ai3 b3j ≡ [C] = [A] × [B] (C.6) aij = bji ≡ [A] = [B]T (C.7) Expression (C.6) is equivalent to the product of two matrices 447 © 2013 by Taylor & Francis Group, LLC 448 Appendix C: Index Notation and Transformation Laws for Tensors C.1.3 THE KRONECKER DELTA AND THE PERMUTATION SYMBOL In the index notation, two special quantities, the Kronecker delta and the permutation factor, must be defined for use in the various operations involving vectors and tensors The Kronecker δij is defined as δij = i=j i=j (C.8) Thus δ11 = δ22 = δ33 = (C.9) δ12 = δ21 = δ13 = δ31 = δ23 = δ32 = (C.10) and In matrix notation, the equivalent of the Kronecker delta is the identity matrix The Kronecker delta can be used as a substitution operator, since δij bj = δi1 b1 + δi2 b2 + δi3 b3 = bi (C.11) δij aik = δ1j a1k + δ2j a2k + δ3j a3k = ajk (C.12) The permutation factor eijk is defined as ⎧ ⎫ ⎨ i, j, k = 1, 2, 2, 3, 3, 1, ⎬ eijk = −1 i, j, k = 3, 2, 2, 1, 1, 3, ⎩ ⎭ otherwise (C.13) We can observe that eijk = ejki = ekij = −eikj = −ekji = ejik (C.14) ekki = (C.15) eijk eimn = δjm δkn − δjn δmk (C.16) Using these definitions, the cross-product of two vectors can be written as u × v = eijk uj vk (C.17) C.1.4 RULES OF INDEX NOTATION There three important rules in index notation, which are as follows: An index may occur either once or twice in a given term When an index occurs unrepeated in a term, that index is understood to take all the values of its range Unrepeated indices are known as free indices Free indices appearing on each term must agree • When an index appears twice in a term, that index is understood to take all the values of its range and the resulting terms summed Repeated indices are often referred to as dummy indices • © 2013 by Taylor & Francis Group, LLC 449 Appendix C: Index Notation and Transformation Laws for Tensors • Free and dummy indices may be changed without altering the meaning of the equation The number and the location of the free indices reveal the exact tensorial rank of the quantity expressed The following expressions are valid Aik xk , = Aki Bkj xj + Cik uk Aij Bik Cnk , but xi = Aij , xj = aik uk , xj = Aki Bkj uj are meaningless C.2 COORDINATE TRANSFORMATIONS C.2.1 TRANSFORMATION LAWS FOR VECTORS Given two arbitrary coordinate systems e1 , e2 , e3 and e1 , e2 , e3 in a three-dimensional Euclidean space Any change of coordinate system is characterized by a Jacobian matrix [J], which helps express the vectors of the new base in terms of the ones in the old base: ⎡ ∂e ⎢ ∂e1 ⎢ ⎢ ∂e ∂ei =⎢ [J] = ⎢ ∂e ∂ej ⎢ ⎣ ∂e ∂e1 ∂e1 ∂e2 ∂e2 ∂e2 ∂e3 ∂e2 ∂e1 ⎤ ∂e3 ⎥ ⎥ ∂e2 ⎥ ⎥ ∂e3 ⎥ ⎥ ∂e ⎦ (C.18) ∂e3 If the Jacobian does not vanish, expression (C.18) possesses a unique inverse The coordinates systems represented by e1 , e2 , e3 and e1 , e2 , e3 are completely general and may be any curvilinear or Cartesian systems In the case of Cartesian systems as shown in Figure C.1, the Jacobian takes the form of a constant tensor lij or, because of the identity between second-order e3 e΄ e΄ e2 e1 e΄ FIGURE C.1 Transformation of coordinates © 2013 by Taylor & Francis Group, LLC 450 Appendix C: Index Notation and Transformation Laws for Tensors tensors and square matrices, a constant matrix [Q], which is called the transition matrix from the old basis to the new basis In index notation, the transformation takes the form ei = lij ej with (C.19) ⎛ lij = cos ei , ej l11 = ⎝l21 l31 l12 l22 l32 ⎞ l13 l23 ⎠ l33 (C.20) In matrix notation, it takes the form: {e } = [Q]{e} with (C.21) ⎡ [Q] = cos ei , ej Q11 = ⎣Q21 Q31 Q12 Q22 Q32 ⎤ Q13 Q23 ⎦ Q33 (C.22) Example: Anticlockwise Rotation around the Axis In the case of an anticlockwise rotation as shown in Figure C.2, the relation between the bases is written as e1 = cos(ψ)e1 + sin(ψ)e2 + × e3 (C.23) e2 = − sin(ψ)e1 + cos(ψ)e2 + × e3 (C.24) e3 = × e1 + × e2 + × e3 (C.25) The matrix [Q] takes the form ⎡ cos(ψ) [Q] = ⎣− sin(ψ) e3 sin(ψ) cos(ψ) ⎤ 0⎦ (C.26) e΄ e΄ e2 ψ e1 FIGURE C.2 Rotation around the third axis © 2013 by Taylor & Francis Group, LLC e΄ Appendix C: Index Notation and Transformation Laws for Tensors 451 The matrix [Q] is an orthonormal matrix and has the following properties: [Q]T = [Q]−1 (C.27) In index notation, the relationship (C.27) is given as lik ljk = δij (C.28) Note that in index notation, and by analogy to matrix notation, you cannot write (C.28) as lji lij = δij This is completely erroneous in index notation since the repeated indices in the first term imply summation; therefore, the first term is a scalar and the second a tensor Given an arbitrary vector v represented in the base (e1 , e2 , e3 ) as v = v1 e1 + v2 e2 + v3 e3 = vj ej (C.29) The same vector can also be represented in the base (e1 , e2 , e3 ) as v = v1 e1 + v2 e2 + v3 e3 = vi ei (C.30) Using Equation (C.19), Equation (C.30) is rewritten as: v = vi lij ej (C.31) Comparing Equations (C.29) and (C.31) reveal that the vector components in the primed and unprimed basis are related by vj = vi lij = lij vi (C.32) {v} = [Q]T {v } (C.33) vi = vi lij (C.34) {v } = [Q]{v} (C.35) in matrix notation The inverse transformation is defined as or in matrix notation as C.2.2 TRANSFORMATION LAWS FOR TENSORS Given two arbitrary vectors u and v represented in the base (e1 , e2 , e3 ) respectively as u = u1 e1 + u2 e2 + u3 e3 (C.36) v = v1 e1 + v2 e2 + v3 e3 (C.37) Now suppose the existence of a linear application between the two vectors defined by u = f (u) and expressed in index notation as ui = aij vj © 2013 by Taylor & Francis Group, LLC (C.38) 452 Appendix C: Index Notation and Transformation Laws for Tensors or in matrix notation as {u} = [a]{v} (C.39) In another base, say (e1 , e2 , e3 ), the vectors u and v are expressed as u = u1 e1 + u2 e2 + u3 e3 (C.40) v = v1 e1 + v2 e2 + v3 e3 (C.41) and the relationship u = f (u) is expressed in index notation as ui = aij vj (C.42) {u } = [a ]{v } (C.43) and in matrix notation as The problem is to find a relationship between the tensors a and a Using (C.32), Equation (C.38) is rewritten as ui = aij lmj vm = lmj aij vm (C.44) Substituting in (C.44) for ui using (C.32) leads to lki uk = aij lmj vm = lmj aij vm (C.45) Multiplying both sides of the equations by lni , and noting that lni lki = δnk , Equation C.46 becomes δnk uk = lki lmj aij vm (C.46) uk = lki lmj aij vm (C.47) akm = lki lmj aij (C.48) That is, Comparing (C.42) and (C.47), it follows Using matrix notation, and after substituting Equation (C.33), Equation (C.39) becomes {u} = [a][Q]T {v } (C.49) Replacing the vector {u} by [Q]T{u }, Equation (C.49) becomes [Q]T{u } = [a][Q]T {v } (C.50) and premultiplying both sides of the equation by the matrix [Q], and noting [Q][Q]T = [I], yields the result {u } = [Q][a][Q]T {v } (C.51) Comparing Equations (C.43) and (C.51) yields the result [a ] = [Q][a][Q]T (C.52) [a] = [Q]T [a ][Q] (C.53) The inverse relation is expressed as © 2013 by Taylor & Francis Group, LLC References and Bibliography In the course of writing this present work, many books on matrix structural analysis and the theories of elasticity and finite element methods have been consulted Some of these books have been explicitly cited while others not An exhaustive list of all the books consulted is given below REFERENCES Dhatt G and Touzot G Une Présentation de la Méthode des Éléments Finis, Deuxième édition Maloine S.A Editeurs, Paris, France, 1984 Hammer P.C., Marlowe O.J., and Stroud A.H Mathematical tables and other aids to computation, American Mathematical Society, 10(55), 130–136, 1956 Smith I.M and Griffiths D.V Programming the Finite Element Method, 2nd edn Wiley, Chichester, U.K., 1988 Timoshenko S and Goodier J Theory of Elasticity, 3rd edn McGraw-Hill, New York, 1970 Timoshenko S and Woinowsky-Krieger S Theory of Plates and Shells McGraw-Hill, New York, 1959 Zienkiewicz O.C The Finite Element Method, 3rd edn McGraw-Hill, York, London, 1977 BIBLIOGRAPHY Chandrupatla T.R and Belegundu A.D Introduction to Finite Elements in Engineering, 3rd edn Prentice-Hall, Upper Saddle River, NJ, 2002 Cook R.D Finite Element Modeling for Stress Analysis Wiley, New York, 1995 Kwon Y.W and Bang H The Finite Element Method Using Matlab, 2nd edn CRC Press, London, U.K., 2000 Logan D.L A First Course in the Finite Element Method Using Algor, 2nd edn Brooks/Cole Thompson Learning, Pacific Groove, CA, 2001 Mase G.E Schaum’s Outline Series: Theory and Problems of Continuum Mechanics McGraw-Hill, New York, 1970 McGuire M., Gallagher G.H., and Ziemian R.D Matrix Structural Analysis, 2nd edn Wiley, New York, 2000 Meek J.L Computer Methods in Structural Analysis E & FN SPON, London, U.K., 1991 Reddy J.N An Introduction to the Finite Element Method, 3rd edn McGraw-Hill, New York, 2006 Saada A.S Elasticity: Theory and Applications, 2nd edn Krieger Publishing, Melbourne, FL, 1993 453 © 2013 by Taylor & Francis Group, LLC © 2013 by Taylor & Francis Group, LLC Mathematics Khennane Introduction to Introduction to Finite Element Analysis Using MATLAB and Abaqus ® There are some books that target the theory of the finite element, while others focus on the programming side of things Introduction to Finite Element Analysis Using MATLAB® and Abaqus accomplishes both This book teaches the first principles of the finite element method It presents the theory of the finite element method while maintaining a balance between its mathematical formulation, programming implementation, and application using commercial software The computer implementation is carried out using MATLAB, while the practical applications are carried out in both MATLAB and Abaqus MATLAB is a high-level language specially designed for dealing with matrices, making it particularly suited for programming the finite element method, while Abaqus is a suite of commercial finite element software Introduction to Finite Element Analysis Using MATLAB® and Abaqus introduces and explains theory in each chapter, and provides corresponding examples It offers introductory notes and provides matrix structural analysis for trusses, beams, and frames The book examines the theories of stress and strain and the relationships between them The author then covers weighted residual methods and finite element approximation and numerical integration He presents the finite element formulation for plane stress/strain problems, introduces axisymmetric problems, and highlights the theory of plates The text supplies step-by-step procedures for solving problems with Abaqus interactive and keyword editions The described procedures are implemented as MATLAB codes, and Abaqus files can be found on the CRC Press website K16894 an informa business w w w c rc p r e s s c o m 6000 Broken Sound Parkway, NW Suite 300, Boca Raton, FL 33487 711 Third Avenue New York, NY 10017 Park Square, Milton Park Abingdon, Oxon OX14 4RN, UK w w w c rc p r e s s c o m Introduction to Finite Element Analysis Using MATLAB® and Abaqus “A very good introduction to the finite element method with a balanced treatment of theory and implementation.” — F Albermani, Reader in Structural Engineering, The University of Queensland, Australia Finite Element Analysis Using MATLAB and Abaqus ® Amar Khennane .. .Introduction to Finite Element Analysis Using MATLAB and Abaqus ® © 2013 by Taylor & Francis Group, LLC © 2013 by Taylor & Francis Group, LLC Introduction to Finite Element Analysis Using MATLAB. .. & Francis Group, LLC Introduction to Finite Element Analysis Using MATLAB and Abaqus 1.3 AIM OF THE BOOK The many textbooks that describe the theory of the finite element and/ or its engineering... displacements correspond to known forces and known displacements correspond to unknown forces © 2013 by Taylor & Francis Group, LLC 16 Introduction to Finite Element Analysis Using MATLAB and Abaqus Finally,