

An introductory textbook covering the fundamentals of linear finite element analysis (FEA) This book constitutes the first volume in a twovolume set that introduces readers to the theoretical foundations and the implementation of the finite element method (FEM). The first volume focuses on the use of the method for linear problems. A general procedure is presented for the finite element analysis (FEA) of a physical problem, where the goal is to specify the values of a field function. First, the strong form of the problem (governing differential equations and boundary conditions) is formulated. Subsequently, a weak form of the governing equations is established. Finally, a finite element approximation is introduced, transforming the weak form into a system of equations where the only unknowns are nodal values of the field function. The procedure is applied to onedimensional elasticity and heat conduction, multidimensional steadystate scalar field problems (heat conduction, chemical diffusion, flow in porous media), multidimensional elasticity and structural mechanics (beams/shells), as well as timedependent (dynamic) scalar field problems, elastodynamics and structural dynamics. Important concepts for finite element computations, such as isoparametric elements for multidimensional analysis and Gaussian quadrature for numerical evaluation of integrals, are presented and explained. Practical aspects of FEA and advanced topics, such as reduced integration procedures, mixed finite elements and verification and validation of the FEM are also discussed. Provides detailed derivations of finite element equations for a variety of problems. Incorporates quantitative examples on onedimensional and multidimensional FEA. Provides an overview of multidimensional linear elasticity (definition of stress and strain tensors, coordinate transformation rules, stressstrain relation and material symmetry) before presenting the pertinent FEA procedures. Discusses practical and advanced aspects of FEA, such as treatment of constraints, locking, reduced integration, hourglass control, and multifield (mixed) formulations. Includes chapters on transient (stepbystep) solution schemes for timedependent scalar field problems and elastodynamics/structural dynamics. Contains a chapter dedicated to verification and validation for the FEM and another chapter dedicated to solution of linear systems of equations and to introductory notions of parallel computing. Includes appendices with a review of matrix algebra and overview of matrix analysis of discrete systems. Accompanied by a website hosting an opensource finite element program for linear elasticity and heat conduction, together with a user tutorial. Fundamentals of Finite Element Analysis: Linear Finite Element Analysis is an ideal text for undergraduate and graduate students in civil, aerospace and mechanical engineering, finite element software vendors, as well as practicing engineers and anybody with an interest in linear finite element analysis. 


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 OneDimensional Problems 17 2.1 Strong Form for OneDimensional Elasticity Problems 17 2.2 General Expressions for Essential and Natural B.C. in OneDimensional Elasticity Problems 23 2.3 Weak Form for OneDimensional Elasticity Problems 24 2.4 Equivalence of Weak Form and Strong Form 28 2.5 Strong Form for OneDimensional Heat Conduction 32 2.6 Weak Form for OneDimensional Heat Conduction 37 Problems 44 References 46 3 Finite Element Formulation for OneDimensional 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 OneDimensional Finite Element Analysis 68 3.7 Numerical Integration Gauss Quadrature 91 3.8 Convergence of OneDimensional Finite Element Method 100 3.9 Effect of Concentrated Forces in OneDimensional 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 TwoDimensional Heat Conduction and Other Scalar Field Problems 123 5.1 Strong Form for TwoDimensional Heat Conduction 123 5.2 Weak Form for TwoDimensional 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 TwoDimensional Scalar Field Problems 141 6.1 Finite Element Discretization and Piecewise Approximation 141 6.2 ThreeNode Triangular Finite Element 148 6.3 FourNode Rectangular Element 153 6.4 Isoparametric Finite Elements and the FourNode Quadrilateral (4Q) Element 158 6.5 Numerical Integration for Isoparametric Quadrilateral Elements 165 6.6 HigherOrder 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 (StressStrain 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 TwoDimensional (Planar) Elasticity 202 7.8 Strong Form for TwoDimensional Elasticity 208 7.9 Weak Form for TwoDimensional Elasticity 212 7.10 Equivalence between the Strong Form and the Weak Form 215 7.11 Strong Form for ThreeDimensional Elasticity 218 7.12 Using Polar (Cylindrical) Coordinates 220 References 225 8 Finite Element Formulation for TwoDimensional 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 TwoDimensional Elasticity 232 Problems 251 9 Finite Element Formulation for ThreeDimensional Elasticity 257 9.1 Weak Form for ThreeDimensional Elasticity 257 9.2 Piecewise Finite Element Approximation Assembly Equations 258 9.3 Isoparametric Finite Elements for ThreeDimensional 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 (SelfInduced) 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 FieldDependent 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 ZeroEnergy (Hourglass) Modes 337 11.6 Resolving the Problem of Hourglass Modes: Hourglass Stiffness 339 11.7 SelectiveReduced Integration 346 11.8 The Bbar 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 TwoField (StressDisplacement) Formulations and the PianSumihara Quadrilateral Element 367 12.4 DisplacementPressure (up) Formulations and Finite Element Approximations 370 12.5 Stability of Mixed up Formulations the infsup Condition 374 12.6 Assumed (Enhanced)Strain Methods and the Bbar 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 EulerBernoulli Beam Theory 388 13.4 Strong Form for TwoDimensional EulerBernoulli Beams 392 13.5 Weak Form for TwoDimensional EulerBernoulli Beams 394 13.6 Finite Element Formulation: TwoNode EulerBernoulli Beam Element 397 13.7 Coordinate Transformation Rules for TwoDimensional Beam Elements 404 13.8 Timoshenko Beam Theory 408 13.9 Strong Form for TwoDimensional Timoshenko Beam Theory 411 13.10 Weak Form for TwoDimensional Timoshenko Beam Theory 411 13.11 TwoNode Timoshenko Beam Finite Element 415 13.12 ContinuumBased Beam Elements 418 13.13 Extension of ContinuumBased Beam Elements to General Curved Beams 424 13.14 Shear Locking and SelectiveReduced 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 452 14.4 Constitutive Law for Linear Elasticity in Terms of Stress Resultants and Generalized Strains 456 14.5 Weak Form of Shell Equations 464 14.6 Finite Element Formulation for Shell Structures 472 14.7 FourNode Planar (Flat) Shell Finite Element 480 14.8 Coordinate Transformations for Shell Elements 485 14.9 A Clever Way to Approximately Satisfy C1 Continuity Requirements for Thin Shells The Discrete Kirchhoff Formulation 500 14.10 ContinuumBased Formulation for Nonplanar (Curved) Shells 510 Problems 521 References 522 15 Finite Elements for Elastodynamics, Structural Dynamics, and TimeDependent Scalar Field Problems 523 15.1 Introduction 523 15.2 Strong Form for OneDimensional Elastodynamics 525 15.3 Strong Form in the Presence of Material Damping 527 15.4 Weak Form for OneDimensional Elastodynamics 529 15.5 Finite Element Approximation and SemiDiscrete Equations of Motion 530 15.6 ThreeDimensional Elastodynamics 536 15.7 SemiDiscrete Equations of Motion for ThreeDimensional Elastodynamics 539 15.8 Structural Dynamics Problems 539 15.9 Diagonal (Lumped) Mass Matrices and Mass Lumping Techniques 546 15.10 Strong and Weak Form for TimeDependent Scalar Field (Parabolic) Problems 549 15.11 SemiDiscrete Finite Element Equations for Scalar Field (Parabolic) Problems 555 15.12 Solid and Structural Dynamics as a Parabolic Problem: The StateSpace Formulation 557 Problems 558 References 559 16 Analysis of TimeDependent Scalar Field (Parabolic) Problems 560 16.1 Introduction 560 16.2 SingleStep Algorithms 562 16.3 Linear Multistep Algorithms 568 16.4 PredictorCorrector Algorithms RungeKutta (RK) Methods 569 16.5 Convergence of a TimeStepping Algorithm 572 16.6 Modal Analysis and Its Use for Determining the Stability for Systems with Many Degrees of Freedom 583 Problems 587 References 587 17 Solution Procedures for Elastodynamics and Structural Dynamics 588 17.1 Introduction 588 17.2 Modal Analysis: What Will NOT Be Presented in Detail 589 17.3 StepbyStep Algorithms for Direct Integration of Equations of Motion 594 17.4 Application of StepByStep Algorithms for Discrete Systems with More than One Degrees of Freedom 608 17.4 Problems 613 References 613 18 Verification and Validation for the Finite Element Method 615 18.1 Introduction 615 18.2 Code Verification 615 18.3 Solution Verification 622 18.4 Numerical Uncertainty 627 18.5 Sources and Types of Uncertainty 629 18.6 Validation Experiments 630 18.7 Validation Metrics 631 18.8 Extrapolation of Model Prediction Uncertainty 633 18.9 Predictive Capability 634 References 634 19 Numerical Solution of Linear Systems of Equations 637 19.1 Introduction 637 19.2 Direct Methods 638 19.3 Iterative Methods 640 19.4 Parallel Computing and the Finite Element Method 644 19.5 Parallel Conjugate Gradient Method 649 References 653 Appendix A: Concise Review of Vector and Matrix Algebra 654 A.1 Preliminary Definitions 654 A.2 Matrix Mathematical Operations 656 A.3 Eigenvalues and Eigenvectors of a Matrix 660 A.4 Rank of a Matrix 662 Appendix B: Review of Matrix Analysis for Discrete Systems 664 B.1 Truss Elements 664 B.2 OneDimensional Truss Analysis 666 B.3 Solving the Global Stiffness Equations of a Discrete System and Postprocessing 671 B.4 The ID Array Concept (for Equation Assembly) 673 B.5 Fully Automated Assembly: The Connectivity (LM) Array Concept 680 B.6 Advanced Interlude Programming of Assembly When the Restrained Degrees of Freedom Have Nonzero Values 682 B.7 Advanced Interlude 2: Algorithms for Postprocessing 683 B.8 TwoDimensional Truss Analysis Coordinate Transformation Equations 684 B.9 Extension to ThreeDimensional Truss Analysis 693 Problem 694 Appendix C: Minimum Potential Energy for Elasticity Variational Principles 695 Appendix D: Calculation of Displacement and Force Transformations for RigidBody Connections 700 Index 706 


IOANNIS KOUTROMANOS, PHD, is an Assistant Professor in the Department of Civil and Environmental Engineering at the Virginia Polytechnic Institute and State University. His research primarily focuses on the analytical simulation of structural components and systems under extreme events, with an emphasis on reinforced concrete, masonry and steel structures under earthquake loading. He has authored and coauthored research papers and reports on finite element analysis (element formulations, constitutive models, verification and validation of modeling schemes). He is a voting member of the joint ACI/ASCE Committee 447, Finite Element Analysis of Reinforced Concrete Structures. Read more 


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