The Finite Element Method for Mechonics of Solids with ANSYS Applicotions

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1 The Finite Element Method for Mechonics of Solids with ANSYS Applicotions ELLIS H. DILL 0~~F~~~~"P Boca Raton London New Vork CRC Press is an imprint 01 the Taylor & Francis Group, an Informa business

2 Preface Author xiii xv Chapter 1 Finite Element Concepts Introduction I 1.2 Direct Stiffness Method, Merging the Element Stiffness Matrices Augmenting the Element Stiffness Matrix Stiffness Matrix Is Banded The Energy Method Truss Example Axially Loaded Rod Example Augmented Matrices for the Rod Merge of Element Matrices for the Rod Force Method Other Structural Components SpaceTruss Beams and Frames General Beam Equations, Plates and Shells Two- or Three-Dimensional Solids Problems 26 References 28 Bibliography 28 Chapter 2 Linear Elasticity Basic Equations Geometry of Deformation Balance of Momentum Virtual Work Constitutive Relations Boundary Conditions and Initial Conditions Incompressible Materials Plane Strain Plane Stress Tensile Test Pure Shear Pure Bending Bending and Shearing 37 v

3 vi Properties of Solutions A Plane Stress Example with a Singularity in Stress Potential Energy Proof of Minimum Potential Energy Matrix Notation Axially Symmetrie Deformations Cylindrical Coordinates Axial Symmetry Plane Stress and Plane Strain Problems 50 References 51 Bibliography 52 Chapter 3 Finite Element Method for Linear Elasticity Finite Element Approximation Potential Energy Finite Element Equations Basic Equations in Matrix Notation Basic Equations Using Virtual Work Underestimate of Displacements Nondimensional Equations Uniaxial Stress General Equations for an Assembly ofelements Generalized Variational Principle Potential Energy Hybrid Displacement Functional Hybrid Stress and Complementary Energy Mixed Methods of Analysis Nearly Incompressible Materials Nearly Incompressible Plane Strain 78 Bibliography 79 Chapter 4 The Triangle and the Tetrahedron Linear Functions over a Triangular Region Triangular Element für Plane Stress and Plane Strain Plane Quadrilateral from Four Triangles Square Element Formed from Four Triangles Plane Stress Example: Short Beam Extrapolation of the Solution Linear Strain Triangles Four-Node Tetrahedron Ten-Node Tetrahedron Problems 99

4 vii Chapter 5 The Quadrilateral and the Hexahedron Four-Node Plane Reetang1e Stress Calculations Plane Stress Example: Pure Bending Plane Strain Example: Bending with Shear Plane Stress Example: Short Beam Improvements to Four-Node Quadrilateral Wilson-Taylor Quadrilateral Enhaneed Strain Formulation Approximate Volumetrie Strains Redueed Integration on the 1( Term Redueed Integration on the A Term Uniform Redueed Integration Example Using Improved Elements Numerieal Integration Coordinate Transformations Isoparametrie Quadrilateral Wilson-Taylor Element Three-Node Triangle as a Special Case of Reetangle Eight-Node Quadrilateral Nodal Loads Plane Stress Example: Pure Bending Plane Stress Example: Bending with Shear Plane Stress Example: Short Beam General Quadrilateral Element Eight-Node Block Twenty-Node Solid Singularity Element Mixed U-PElements Plane Strain Alternative Formulation for Plane Strain D Elements Problems 163 References 168 Bibliography 169 Chapter6 Errors and Convergence of Finite Element Solution General Remarks Element Shape Limits Aspect Ratio Parallel Deviation for a Quadrilateral Large Corner Angle Jacobian Ratio 175

5 viii 6.3 Patch Test Wilson-Taylor Quadrilateral 178 References 180 Chapter 7 Heat Conduction in Elastic Solids Differential Equations and Virtual Work Example Problem: One-Dimensional Transient Heat Flux Example: Hollow Cylinder Problems 188 Chapter 8 Finite Element Method for Plasticity Theory of Plasticity Tensile Test Plane Stress Summary of Plasticity Finite Element Formulation for Plasticity Fundamental Solution Iteration to Improve the Solution Example: Short Beam Problems 203 Bibliography 204 Chapter 9 Viscoelasticity Theory of Linear Viscoelasticity Recurrence Formula for History Viscoelastic Example Finite Element Formulation for Viscoelasticity Basic Step-by-Step Solution Method Step-by-Step Calculation with Load Correction Plane Strain Example Problems 219 Bibliography 220 Chapter 10 Dynamic Analyses Dynamical Equations LumpedMass Consistent Mass Natural Frequencies Lumped Mass Consistent Mass Mode Superposition Solution Example: Axially Loaded Rod 227

6 ix Exact Solution for Axially Loaded Rod Finite Element Model One-Element ModeL Two-Element Model Mode Superposition for Continuum Model of the Rod Example: Short Beam Dynamic Analysis with Damping Viscoelastic Damping Viscous Body Force Analysis of Damped Motion by Mode Superposition _ Numerical Solution of Differential Equations Constant Average Acceleration General Newmark Method General Methods Implicit Methods in General Explicit Methods in General Stability Analysis ofnewmark's Method Convergence, Stability, and Error Example: Numerical Integration for Axially Loaded Rod Example: Analysis of Short Beam Problems 251 Bibliography 253 Chapter 11 Linear Elastic Fracture Mechanics Fracture Criterion Analysis of Sheet Fracture Modes Mode Modc Mode III Determination of K by Finite Element Analysis Crack Opening Displacement Method J-Integral for Plane Regions Problems 267 References 268 Bibliography 268 Chapter 12 Plates and Shells Geometry of Deformation Equations of Equilibrium Constitutive Relations for an Elastic Material., 271

7 x 12.4 Virtual Work Finite Element Relations for Bending C1assical Plate Theory Plate Bending Example Problems 287 Referenees 288 Bibliography 289 Chapter 13 Large Deformations Theory of Large Deformations Virtual Work Elastic Malerials 293 '3.1.3 Mooney-Rivl in Model of an Ineompressible Material Generalized Mooney-Rivlin Model., Polynomial Formula Ogden's Function Blatz-Ko Model Logarithmie Strain Measure Yeoh Model Fitting Constitutive Relations 10 Experimental Data Volumetrie Data Tensile Test Biaxial Tesl Finite Elements for Large Displaeements Lagrangian Formulation Basie Step-by-Step Analysis Iteration Proeedure Updated Referenee Configuration Example I ~ Example Structure of Tangent Modulus Stability and Buekling Beam-Column Snap-Through Buekling Shallow Areh Problems 324 Referenees 326 Bibliography 326 Chapter 14 Conslraints and Contact Application of Constraints Lagrange Multipliers 327

8 xi Perturbed Lagrangian Method Penalty Functions Augmented Lagrangian Method Contact Problems Example: A Truss Contacts a Rigid Foundation Load F y > 0 Is Applied with F s = Loads Are Ramped Up Together: F, = 27a, Fy= 12.8a Lagrange Multiplier, No Friction Force I Stick Condition Slip Condition Lagrange Multiplier, with Friction Stick Condition Slip Condition Penalty Method Stick Condition Slip Condition Finite Element Analysis I Example: Contact of a Cylinder with a Rigid Plane Hertz Contact Problem Dynamic Impact Problems 347 References 348 Bibiiography 348 Chapter 15 ANSYS APDL Examples ANSYS Instructions ANSYS File Names Graphie Window Controls Graphics Window Logo Display ofmodel Display of Deformed and Undeformed Shape White on White Adjusting Graph Colors I.2.5 Printing from Windows Version of ANSYS I.2.6 Some Useful Notes ANSYS Elements SURFI53, SURFI Truss Example Beam Bending Beam with a Distributed Load One Triangle Plane Stress Example Using Triangles Cantilever Beam Modeled as Plane Stress 366

9 xii 15.9 Plane Stress: Pure Bending Plane Strain Bending Example Plane Stress Example: Short Beam Sheet with a Hole Solution Procedure Plasticity Example Viscoelasticity Creep Test Viscoelasticity Example Mode Shapes and Frequencies of a Rod Mode Shapes and Frequencies of a Short Beam Transient Analysis of Short Beam Stress Intensity Factor by Crack Opening Displacement Stress Intensity Factor by J-Integral Stretching of a Nonlinear Elastic Sheet Nonlinear Elasticity: Tensile Test Column Buckling Column Post-Buckling Snap-Through Plate Bending Example Clamped Plate Gravity Load on a Cylindrical Shell Plate Buckling Heated Reetangular Rod Heated Cylindrical Rod _ Heated Disk Truss Contacting a Rigid Foundation Compression of a Rubber Cylinder between Rigid Plates Hertz Contact Problem Elastic Rod Impacting a Rigid Wall Curve Fit for Nonlinear Elasticity Using Blatz-Ko Model Curve Fit for NonlinearElasticity Using Polynomial Model Bibliography 469 Chapter 16 ANSYS Workbench Two- and Three-Dimensional Geometry Stress Analysis Short Beam Example Short Beam Geometry Short Beam, Static Loading Short Beam, Transient Analysis FiJIeted Bar Example Sheet with a Hole 480 Bibliography 482 Index 483

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

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