Fundamentals of finite element analysis : linear finite element analysis /: linear finite element analysis. (2017)

• Record Type: Book
• Further Information: Note: Ioannis Koutromanos.
• Authors: Koutromanos, Ioannis, 1982-
• Contents: Preface xiv About the Companion Website xviii 1 Introduction 1 1.1 Physical Processes and Mathematical Models 1 1.2 Approximation, Error, and Convergence 3 1.3 Finite Element Method for Differential Equations 5 1.4 Brief History of the Finite Element Method 6 1.5 Finite Element Software 8 1.6 Significance of Finite Element Analysis for Engineering 8 1.7 Typical Process for Obtaining a Finite Element Solution for a Physical Problem 12 1.8 A Note on Linearity and the Principle of Superposition 14 References 16 2 Strong and Weak Form for One-Dimensional Problems 17 2.1 Strong Form for One-Dimensional Elasticity Problems 17 2.2 General Expressions for Essential and Natural B.C. in One-Dimensional Elasticity Problems 23 2.3 Weak Form for One-Dimensional Elasticity Problems 24 2.4 Equivalence of Weak Form and Strong Form 28 2.5 Strong Form for One-Dimensional Heat Conduction 32 2.6 Weak Form for One-Dimensional Heat Conduction 37 Problems 44 References 46 3 Finite Element Formulation for One-Dimensional Problems 47 3.1 Introduction—Piecewise Approximation 47 3.2 Shape (Interpolation) Functions 51 3.3 Discrete Equations for Piecewise Finite Element Approximation 59 3.4 Finite Element Equations for Heat Conduction 66 3.5 Accounting for Nodes with Prescribed Solution Value (“Fixed” Nodes) 67 3.6 Examples on One-Dimensional Finite Element Analysis 68 3.7 Numerical Integration—Gauss Quadrature 91 3.8 Convergence of One-Dimensional Finite Element Method 100 3.9 Effect of Concentrated Forces in One-Dimensional Finite Element Analysis 106 Problems 108 References 111 4 Multidimensional Problems: Mathematical Preliminaries 112 4.1 Introduction 112 4.2 Basic Definitions 113 4.3 Green’s Theorem—Divergence Theorem and Green’s Formula 118 4.4 Procedure for Multidimensional Problems 121 Problems 122 References 122 5 Two-Dimensional Heat Conduction and Other Scalar Field Problems 123 5.1 Strong Form for Two-Dimensional Heat Conduction 123 5.2 Weak Form for Two-Dimensional Heat Conduction 129 5.3 Equivalence of Strong Form and Weak Form 131 5.4 Other Scalar Field Problems 133 Problems 139 6 Finite Element Formulation for Two-Dimensional Scalar Field Problems 141 6.1 Finite Element Discretization and Piecewise Approximation 141 6.2 Three-Node Triangular Finite Element 148 6.3 Four-Node Rectangular Element 153 6.4 Isoparametric Finite Elements and the Four-Node Quadrilateral (4Q) Element 158 6.5 Numerical Integration for Isoparametric Quadrilateral Elements 165 6.6 Higher-Order Isoparametric Quadrilateral Elements 176 6.7 Isoparametric Triangular Elements 178 6.8 Continuity and Completeness of Isoparametric Elements 181 6.9 Concluding Remarks: Finite Element Analysis for Other Scalar Field Problems 183 Problems 183 References 188 7 Multidimensional Elasticity 189 7.1 Introduction 189 7.2 Definition of Strain Tensor 189 7.3 Definition of Stress Tensor 191 7.4 Representing Stress and Strain as Column Vectors—The Voigt Notation 193 7.5 Constitutive Law (Stress-Strain Relation) for Multidimensional Linear Elasticity 194 7.6 Coordinate Transformation Rules for Stress, Strain, and Material Stiffness Matrix 199 7.7 Stress, Strain, and Constitutive Models for Two-Dimensional (Planar) Elasticity 202 7.8 Strong Form for Two-Dimensional Elasticity 208 7.9 Weak Form for Two-Dimensional Elasticity 212 7.10 Equivalence between the Strong Form and the Weak Form 215 7.11 Strong Form for Three-Dimensional Elasticity 218 7.12 Using Polar (Cylindrical) Coordinates 220 References 225 8 Finite Element Formulation for Two-Dimensional Elasticity 226 8.1 Piecewise Finite Element Approximation—Assembly Equations 226 8.2 Accounting for Restrained (Fixed) Displacements 231 8.3 Postprocessing 232 8.4 Continuity—Completeness Requirements 232 8.5 Finite Elements for Two-Dimensional Elasticity 232 Problems 251 9 Finite Element Formulation for Three-Dimensional Elasticity 257 9.1 Weak Form for Three-Dimensional Elasticity 257 9.2 Piecewise Finite Element Approximation—Assembly Equations 258 9.3 Isoparametric Finite Elements for Three-Dimensional Elasticity 264 Problems 287 Reference 288 10 Topics in Applied Finite Element Analysis 289 10.1 Concentrated Loads in Multidimensional Analysis 289 10.2 Effect of Autogenous (Self-Induced) Strains—The Special Case of Thermal Strains 291 10.3 The Patch Test for Verification of Finite Element Analysis Software 294 10.4 Subparametric and Superparametric Elements 295 10.5 Field-Dependent Natural Boundary Conditions: Emission Conditions and Compliant Supports 296 10.6 Treatment of Nodal Constraints 302 10.7 Treatment of Compliant (Spring) Connections Between Nodal Points 309 10.8 Symmetry in Analysis 311 10.9 Axisymmetric Problems and Finite Element Analysis 316 10.10 A Brief Discussion on Efficient Mesh Refinement 319 Problems 321 References 323 11 Convergence of Multidimensional Finite Element Analysis, Locking Phenomena in Multidimensional Solids and Reduced Integration 324 11.1 Convergence of Multidimensional Finite Elements 324 11.2 Effect of Element Shape in Multidimensional Analysis 327 11.3 Incompatible Modes for Quadrilateral Finite Elements 328 11.4 Volumetric Locking in Continuum Elements 333 11.5 Uniform Reduced Integration and Spurious Zero-Energy (Hourglass) Modes 337 11.6 Resolving the Problem of Hourglass Modes: Hourglass Stiffness 339 11.7 Selective-Reduced Integration 346 11.8 The B-bar Method for Resolving Locking 348 Problems 351 References 352 12 Multifield (Mixed) Finite Elements 353 12.1 Multifield Weak Forms for Elasticity 354 12.2 Mixed (Multifield) Finite Element Formulations 359 12.3 Two-Field (Stress-Displacement) Formulations and the Pian-Sumihara Quadrilateral Element 367 12.4 Displacement-Pressure (u-p) Formulations and Finite Element Approximations 370 12.5 Stability of Mixed u-p Formulations—the inf-sup Condition 374 12.6 Assumed (Enhanced)-Strain Methods and the B-bar Method as a Special Case 377 12.7 A Concluding Remark for Multifield Elements 381 References 381 13 Finite Element Analysis of Beams 383 13.1 Basic Definitions for Beams 383 13.2 Differential Equations and Boundary Conditions for 2D Beams 385 13.3 Euler-Bernoulli Beam Theory 388 13.4 Strong Form for Two-Dimensional Euler-Bernoulli Beams 392 13.5 Weak Form for Two-Dimensional Euler-Bernoulli Beams 394 13.6 Finite Element Formulation: Two-Node Euler-Bernoulli Beam Element 397 13.7 Coordinate Transformation Rules for Two-Dimensional Beam Elements 404 13.8 Timoshenko Beam Theory 408 13.9 Strong Form for Two-Dimensional Timoshenko Beam Theory 411 13.10 Weak Form for Two-Dimensional Timoshenko Beam Theory 411 13.11 Two-Node Timoshenko Beam Finite Element 415 13.12 Continuum-Based Beam Elements 418 13.13 Extension of Continuum-Based Beam Elements to General Curved Beams 424 13.14 Shear Locking and Selective-Reduced Integration for Thin Timoshenko Beam Elements 440 Problems 443 References 446 14 Finite Element Analysis of Shells 447 14.1 Introduction 447 14.2 Stress Resultants for Shells 451 14.3 Differential Equations of Equilibrium and Boundary Conditions for Flat Shells
• Edition: 1st
• Publisher Details: Hoboken, New Jersey : John Wiley & Sons, Inc
• Publication Date: 2017
• Extent: 1 online resource
• Subjects: 518.25; Finite element method
• Languages: English
• ISBNs: 9781119260127; 9781119260141
• Related ISBNs: 9781119260080
• Notes: Note: Description based on CIP data; resource not viewed.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.