Using MATLAB and. Abaqus. Finite Element Analysis. Introduction to. Amar Khennane. Taylor & Francis Croup. Taylor & Francis Croup,


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1 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
2 List of Figures List of Tables Preface Author xiii xxv xxvii xxix Chapter 1 Introduction ' 1.1 Prologue Finite Element Analysis and the User Aim of the Book Book Organization 2 Chapter 2 Bar Element Introduction OneDimensional Truss Element Formulation of the Stiffness Matrix: The Direct Approach TwoDimensional Truss Element Global Stiffness Matrix Assembly Discretization Elements' Stiffness Matrices in Local Coordinates Elements' Stiffness Matrices in Global Coordinates Element 1 II Element Element Global Matrix Assembly Only Element 1 Is Present Only Element 2 Is Present Only Element 3 Is Present Global Force Vector Assembly Boundary Conditions General Case Solution of the System of Equations Support Reactions '7 2.7 Members' Forces Computer Code: truss.m Data Preparation Nodes Coordinates Element Connectivity Material and Geometrical Properties Boundary Conditions Loading Element Matrices Stiffness Matrix in Local Coordinates Transformation Matrix 22
3 v Contents Stiffness Matrix in Global Coordinates "Steering" Vector Assembly of the Global Stiffness Matrix Assembly of the Global Force Vector Solution of the Global System of Equations Nodal Displacements Element Forces Program Scripts Problems Problem Problem Analysis of a Simple Truss with Abaqus Overview of Abaqus Analysis of a Truss with Abaqus Interactive Edition Modeling Analysis Analysis of a Truss with Abaqus Keyword Edition 57 Chapter 3 Beam Element Introduction Stiffness Matrix Uniformly Distributed Loading Internal Hinge Computer Code: beam.m Data Preparation Nodes Coordinates Element Connectivity Material and Geometrical Properties Boundary Conditions Internal Hinges Loading Stiffness Matrix Assembly and Solution of the Global System of Equations Nodal Displacements Element Forces Problems Problem Problem Problem Analysis of a Simple Beam with Abaqus Interactive Edition Analysis of a Beam with Abaqus Keyword Edition 103 Chapter 4 Rigid Jointed Frames Introduction Stiffness Matrix of a BeamColumn Element Stiffness Matrix of a BeamColumn Element in the Presence of Hinged End 107
4 vii 4.4 Global and Local Coordinate Systems Global Stiffness Matrix Assembly and Solution for Unknown Displacements Computer Code: frame.m Data Preparation Nodes Coordinates Element Connectivity Material and Geometrical Properties Boundary Conditions Internal Hinges 1' Loading Ill Element Matrices Stiffness Matrix in Local Coordinates Transformation Matrix Stiffness Matrix in Global Coordinates "Steering" Vector Element Loads Assembly of the Global Stiffness Matrix Solution of the Global System of Equations Nodal Displacements Element Forces, Analysis of a Simple Frame with Abaqus Interactive Edition Key word Edition 132 Chapter 5 Stress and Strain Analysis Introduction Stress Tensor Definition Stress TensorStress Vector Relationships Transformation of the Stress Tensor Equilibrium Equations Principal Stresses von Mises Stress Normal and Tangential Components of the Stress Vector Mohr's Circles for Stress Engineering Representation of Stress Deformation and Strain Definition Lagrangian and Eulerian Descriptions Displacement Displacement and Deformation Gradients Green Lagrange Strain Matrix Small Deformation Theory Infinitesimal Strain Geometrical Interpretation of the Terms of the Strain Tensor Compatibility Conditions Principal Strains 152
5 viii Contents Transformation of the Strain Tensor Engineering Representation of Strain StressStrain Constitutive Relations Generalized Hooke's Law Material Symmetries ' Symmetry with respect to a Plane Symmetry with respect to Three Orthogonal Planes Symmetry of Rotation with to respect One Axis Isotropic Material Modulus of Elasticity Poisson's Ratio Shear Modulus Bulk Modulus Plane Stress and Plane Strain Solved Problems Problem Problem Problem Problem Problem Problem Problem Problem Chapter 6 Weighted Residual Methods Introduction General Formulation Galerkin Method Weak Form Integrating by Part over Two and Three Dimensions (Green Theorem) Rayleigh Ritz Method Definition Functional Associated with an Integral Form Rayleigh Ritz Method Example of a Natural Functional 185 Chapter 7 Finite Element Approximation Introduction General and Nodal Approximations Finite Element Approximation Basic Principles for the Construction of Trial Functions Compatibility Principle Completeness Principle TwoDimensional Finite Element Approximation Plane Linear Triangular Element for C Problems Shape Functions Reference Element Area Coordinates Linear Quadrilateral Element for C Problems 203
6 lx Geometrical Transformation Construction of a Trial Function over a Linear Quadrilateral Element Shape Functions of Some Classical Elements for Cu Problems OneDimensional Elements TwoNodded Linear Element ThreeNodded Quadratic Element TwoDimensional Elements FourNodded Bilinear Quadrilateral EightNodded Quadratic Quadrilateral ThreeNodded Linear Triangle SixNodded Quadratic Triangle ThreeDimensional Elements FourNodded Linear Tetrahedra TenNodded Quadratic Tetrahedra EightNodded Linear Brick Element TwentyNodded Quadratic Brick Element 210 Chapter 8 Numerical Integration Introduction Gauss Quadrature Integration over an Arbitrary Interval [a, b] Integration in Two and Three Dimensions Integration over a Reference Element Integration over a Triangular Element Simple Formulas Numerical Integration over a Triangular Element Solved Problems Problem Problem Problem Chapter 9 Plane Problems Introduction Finite Element Formulation for Plane Problems Spatial Discretization Constant Strain Triangle Displacement Field Strain Matrix Stiffness Matrix Element Force Vector Body Forces Traction Forces Concentrated Forces Computer Codes Using the Constant Strain Triangle Data Preparation Nodes Coordinates Element Connectivity Material Properties 243
7 Boundary Conditions Loading Main Program Element Stiffness Matrix Assembly of the Global Stiffness Matrix Solution of the Global System of Equations Nodal Displacements Element Stresses and Strains Results and Discussion Program with Automatic Mesh Generation Analysis with Abaqus Using the CST Interactive Edition Keyword Edition Linear Strain Triangle Displacement Field Strain Matrix Stiffness Matrix Computer Code: LST_PLANE_STRESS_MESH.m Numerical Integration of the Stiffness Matrix Computation of the Stresses and Strains Analysis with Abaqus Using the LST Interactive Edition Keyword Edition The Bilinear Quadrilateral Displacement Field Strain Matrix Stiffness Matrix Element Force Vector Computer Code: Q4_PLANE_STRESS.m Data Preparation Main Program Integration of the Stiffness Matrix Computation of the Stresses and Strains Program with Automatic Mesh Generation Analysis with Abaqus Using the Q4 Quadrilateral Interactive Edition Keyword Edition The 8Node Quadrilateral Formulation Equivalent Nodal Forces Program Q8_PLANE_STRESS.m Data Preparation Main Program Integration of the Stiffness Matrix Results with the Coarse Mesh Program with Automatic Mesh Generation Analysis with Abaqus Using the Q8 Quadrilateral Solved Problem with MATLAB 326
8 *' Strip Footing with the CST Element Strip Footing with the LST Element Bridge Pier with the Q8 Element 336 Chapter 10 Axisymmetric Problems Definition StrainDisplacement Relationship StressStrain Relations Finite Element Formulation Displacement Field Strain Matrix Stiffness Matrix Nodal Force Vectors Body Forces Surface Forces Vector Concentrated Loads Example Programming Computer Code: AXI_SYM_T6.m Numerical Integration of the Stiffness Matrix Results Computer Code: AXI_SYM_Q8.m Numerical Integration of the Stiffness Matrix Results Analysis with Abaqus Using the 8Node Quadrilateral 372 Chapter 11 Thin and Thick Plates Introduction Thin Plates Differential Equation of Plates Loaded in 379 Bending Governing Equation in terms of Displacement Variables Thick Plate Theory or Mindlin Plate Theory StressStrain Relationship Linear Elastic Finite Element Analysis of Plates Finite Element Formulation for Thin Plates Triangular Element Rectangular Element Finite Element Formulation for Thick Plates Boundary Conditions Simply Supported Edge Builtin or Clamped Edge Free Edge 11.6 Computer Program for Thick Plates Using the 8Node Quadrilateral
9 xij Contents Main Program: Thick_plate_Q8.m Data Preparation Stiffness Matrices Boundary Conditions Loading Numerical Integration of the Stiffness Matrix Results Determination of the Resulting Moments and Shear Forces Contour Plots Analysis with Abaqus Preliminary ThreeDimensional Shell Elements Axisymmetric Shell Elements Thick versus Thin Conventional Shell Simply Supported Plate ThreeDimensional 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
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