Introduction to Finite Element Analysis Using MATLAB and Abaqus Amar Khennane Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an informa business
List of Figures List of Tables Preface Author xiii xxv xxvii xxix Chapter 1 Introduction ' 1.1 Prologue 1 1.2 Finite Element Analysis and the User 1 1.3 Aim of the Book 2 1.4 Book Organization 2 Chapter 2 Bar Element 5 2.1 Introduction 5 2.2 One-Dimensional Truss Element 5 2.2.1 Formulation of the Stiffness Matrix: The Direct Approach 5 2.2.2 Two-Dimensional Truss Element 7 2.3 Global Stiffness Matrix Assembly 9 2.3.1 Discretization 9 2.3.2 Elements' Stiffness Matrices in Local Coordinates 9 2.3.3 Elements' Stiffness Matrices in Global Coordinates 10 2.3.3.1 Element 1 II 2.3.3.2 Element 2 11 2.3.3.3 Element 3 12 2.3.4 Global Matrix Assembly 12 2.3.4.1 Only Element 1 Is Present 13 2.3.4.2 Only Element 2 Is Present 13 2.3.4.3 Only Element 3 Is Present 13 2.3.5 Global Force Vector Assembly 14 2.4 Boundary Conditions 15 2.4.1 General Case 15 2.5 Solution of the System of Equations 16 2.6 Support Reactions '7 2.7 Members' Forces 18 2.8 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 Contents 2.8.2.3 Stiffness Matrix in Global Coordinates 22 2.8.2.4 "Steering" Vector 22 2.8.3 Assembly of the Global Stiffness Matrix 23 2.8.4 Assembly of the Global Force Vector 23 2.8.5 Solution of the Global System of Equations 23 2.8.6 Nodal Displacements 23 2.8.7 Element Forces 23 2.8.8 Program Scripts 24 2.9 Problems 27 2.9.1 Problem 2.1 27 2.9.2 Problem 2.2 32 2.10 Analysis of a Simple Truss with Abaqus 35 2.10.1 Overview of Abaqus 35 2.10.2 Analysis of a Truss with Abaqus Interactive Edition 36 2.10.2.1 Modeling 36 2.10.2.2 Analysis 51 2.10.3 Analysis of a Truss with Abaqus Keyword Edition 57 Chapter 3 Beam Element 63 3.1 Introduction 63 3.2 Stiffness Matrix 63 3.3 Uniformly Distributed Loading 67 3.4 Internal Hinge 71 3.5 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 3.6 Problems 81 3.6.1 Problem 3.1 81 3.6.2 Problem 3.2 84 3.6.3 Problem 3.3 87 3.7 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 Chapter 4 Rigid Jointed Frames 107 4.1 Introduction 107 4.2 Stiffness Matrix of a Beam-Column Element 107 4.3 Stiffness Matrix of a Beam-Column Element in the Presence of Hinged End 107
vii 4.4 Global and Local Coordinate Systems 108 4.5 Global Stiffness Matrix Assembly and Solution for Unknown Displacements 109 4.6 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 1' 1 4.6.1.6 Loading Ill 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 4.7 Analysis of a Simple Frame with Abaqus 124 4.7.1 Interactive Edition 124 4.7.2 Key word Edition 132 Chapter 5 Stress and Strain Analysis 135 5.1 Introduction 135 5.2 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 5.3 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
viii Contents 5.3.8 Transformation of the Strain Tensor 153 5.3.9 Engineering Representation of Strain 153 5.4 Stress-Strain Constitutive Relations 154 5.4.1 Generalized Hooke's Law 154 5.4.2 Material Symmetries '55 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 to respect 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 5.5 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 Chapter 6 Weighted Residual Methods 175 6.1 Introduction 175 6.2 General Formulation 175 6.3 Galerkin Method 176 6.4 Weak Form 178 6.5 Integrating by Part over Two and Three Dimensions (Green Theorem) 179 6.6 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 Chapter 7 Finite Element Approximation 191 7.1 Introduction 191 7.2 General and Nodal Approximations 191 7.3 Finite Element Approximation 193 7.4 Basic Principles for the Construction of Trial Functions 195 7.4.1 Compatibility Principle 195 7.4.2 Completeness Principle 196 7.5 Two-Dimensional Finite Element Approximation 197 7.5.1 Plane Linear Triangular Element for C 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 C Problems 203
lx 7.5.2.1 Geometrical Transformation 203 7.5.2.2 Construction of a Trial Function over a Linear Quadrilateral Element 206 7.6 Shape Functions of Some Classical Elements for Cu 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 Chapter 8 Numerical Integration 211 8.1 Introduction 211 8.2 Gauss Quadrature 211 8.2.1 Integration over an Arbitrary Interval [a, b] 214 8.2.2 Integration in Two and Three Dimensions 215 8.3 Integration over a Reference Element 216 8.4 Integration over a Triangular Element 217 8.4.1 Simple Formulas 217 8.4.2 Numerical Integration over a Triangular Element 218 8.5 Solved Problems 219 8.5.1 Problem 8.1 219 8.5.2 Problem 8.2 221 8.5.3 Problem 8.3 226 Chapter 9 Plane Problems 231 9.1 Introduction 231 9.2 Finite Element Formulation for Plane Problems 231 9.3 Spatial Discretization 234 9.4 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
9.4.5.5 Boundary Conditions 243 9.4.5.6 Loading 243 9.4.5.7 Main Program 243 9.4.5.8 Element Stiffness Matrix 245 9.4.5.9 Assembly of the Global Stiffness Matrix 246 9.4.5.10 Solution of the Global System of Equations 246 9.4.5.11 Nodal Displacements 246 9.4.5.12 Element Stresses and Strains 246 9.4.5.13 Results and Discussion 247 9.4.5.14 Program with Automatic Mesh Generation 249 9.4.6 Analysis with Abaqus Using the CST 253 9.4.6.1 Interactive Edition 253 9.4.6.2 Keyword Edition 260 9.5 Linear Strain Triangle 263 9.5.1 Displacement Field 264 9.5.2 Strain Matrix 265 9.5.3 Stiffness Matrix 266 9.5.4 Computer Code: LST_PLANE_STRESS_MESH.m 266 9.5.4.1 Numerical Integration of the Stiffness Matrix 270 9.5.4.2 Computation of the Stresses and Strains 271 9.5.5 Analysis with Abaqus Using the LST 272 9.5.5.1 Interactive Edition 272 9.5.5.2 Keyword Edition 278 9.6 The Bilinear Quadrilateral 279 9.6.1 Displacement Field 280 9.6.2 Strain Matrix 281 9.6.3 Stiffness Matrix 282 9.6.4 Element Force Vector 282 9.6.5 Computer Code: Q4_PLANE_STRESS.m 284 9.6.5.1 Data Preparation 284 9.6.5.2 Main Program 287 9.6.5.3 Integration of the Stiffness Matrix 289 9.6.5.4 Computation of the Stresses and Strains 290 9.6.5.5 Program with Automatic Mesh Generation 291 9.6.6 Analysis with Abaqus Using the Q4 Quadrilateral 295 9.6.6.1 Interactive Edition 295 9.6.6.2 Keyword Edition 302 9.7 The 8-Node Quadrilateral 304 9.7.1 Formulation 304 9.7.2 Equivalent Nodal Forces 307 9.7.3 Program Q8_PLANE_STRESS.m 307 9.7.3.1 Data Preparation 307 9.7.3.2 Main Program 311 9.7.3.3 Integration of the Stiffness Matrix 314 9.7.3.4 Results with the Coarse Mesh 314 9.7.3.5 Program with Automatic Mesh Generation 315 9.7.4 Analysis with Abaqus Using the Q8 Quadrilateral 321 9.8 Solved Problem with MATLAB 326
*' 9.8.1 Strip Footing with the CST Element 326 9.8.2 Strip Footing with the LST Element 331 9.8.3 Bridge Pier with the Q8 Element 336 Chapter 10 Axisymmetric Problems 353 10.1 Definition 353 10.2 Strain-Displacement Relationship 353 10.3 Stress-Strain Relations 354 10.4 Finite Element Formulation 355 10.4.1 Displacement Field 355 10.4.2 Strain Matrix 355 10.4.3 Stiffness Matrix 356 10.4.4 Nodal Force Vectors 356 10.4.4.1 Body Forces 356 10.4.4.2 Surface Forces Vector 356 10.4.4.3 Concentrated Loads 357 10.4.4.4 Example 357 10.5 Programming 358 10.5.1 Computer Code: AXI_SYM_T6.m 359 10.5.1.1 Numerical Integration of the Stiffness Matrix 362 10.5.1.2 Results 363 10.5.2 Computer Code: AXI_SYM_Q8.m 365 10.5.2.1 Numerical Integration of the Stiffness Matrix 368 10.5.2.2 Results 370 10.6 Analysis with Abaqus Using the 8-Node Quadrilateral 372 Chapter 11 Thin and Thick Plates 379 11.1 Introduction 379 11.2 Thin Plates 379 11.2.1 Differential Equation of Plates Loaded in 379 Bending 11.2.2 Governing Equation in terms of Displacement Variables 382 11.3 Thick Plate Theory or Mindlin Plate Theory 383 11.3.1 Stress-Strain Relationship 384 11.4 Linear Elastic Finite Element Analysis of Plates 385 11.4.1 Finite Element Formulation for Thin Plates 385 11.4.1.1 Triangular Element 385 11.4.1.2 Rectangular Element 387 11.4.2 Finite Element Formulation for Thick Plates 388 11.5 Boundary Conditions 389 11.5.1 Simply Supported Edge 11.5.2 Built-in or Clamped Edge 390 11.5.3 Free Edge 11.6 Computer Program for Thick Plates Using the 8-Node Quadrilateral 389 390 390
xij Contents 11.6.1 Main Program: Thick_plate_Q8.m 390 11.6.2 Data Preparation 395 11.6.2.1 Stiffness Matrices 395 11.6.2.2 Boundary Conditions 395 11.6.2.3 Loading 396 11.6.2.4 Numerical Integration of the Stiffness Matrix 397 11.6.3 Results 398 11.6.3.1 Determination of the Resulting Moments and Shear Forces 398 11.6.3.2 Contour Plots 399 11.7 Analysis with Abaqus 400 11.7.1 Preliminary 400 11.7.1.1 Three-Dimensional Shell Elements 401 11.7.1.2 Axisymmetric Shell Elements 401 11.7.1.3 Thick versus Thin Conventional Shell 401 11.7.2 Simply Supported Plate 401 11.7.3 Three-Dimensional Shells 406 Appendix A: List of MATLAB Modules and Functions 419 Appendix B: Statically Equivalent Nodal Forces 445 Appendix C: Index Notation and Transformation Laws for Tensors 447 References and Bibliography 453 Index 455