Finite Element Method: Linear Static and Dynamic Finite Element Analysis / Edition 1

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Overview

Directed towards students without in-depth mathematical training, this text is intended to assist engineering and physical science students in cultivating comprehensive skills in linear static and dynamic finite element methodology. Included are a comprehensive presentation and analysis of algorithms of time-dependent phenomena plus beam, plate, and shell theories derived directly from 3-dimensional elasticity theory. An ideal primer for more advanced works on this subject. Brief Glossary of Notations.
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Product Details

  • ISBN-13: 9780133170252
  • Publisher: Prentice Hall Professional Technical Reference
  • Publication date: 4/28/1987
  • Edition number: 1
  • Pages: 768

Table of Contents

Preface XV
A Brief Glossary of Notations XXII
Part 1 Linear Static Analysis
1 Fundamental Concepts; A Simple One-Dimensional Boundary-Value Problem 1
1.1 Introductory Remarks and Preliminaries 1
1.2 Strong, or Classical, Form of the Problem 2
1.3 Weak, or Variational, Form of the Problem 3
1.4 Eqivalence of Strong and Weak Forms; Natural Boundary Conditions 4
1.5 Galerkin's Approximation Method 7
1.6 Matrix Equations; Stiffness Matrix K 9
1.7 Examples: 1 and 2 Degrees of Freedom 13
1.8 Piecewise Linear Finite Element Space 20
1.9 Properties of K 22
1.10 Mathematical Analysis 24
1.11 Interlude: Gauss Elimination; Hand-calculation Version 31
1.12 The Element Point of View 37
1.13 Element Stiffness Matrix and Force Vector 40
1.14 Assembly of Global Stiffness Matrix and Force Vector; LM Array 42
1.15 Explicit Computation of Element Stiffness Matrix and Force Vector 44
1.16 Exercise: Bernoulli-Euler Beam Theory and Hermite Cubics 48
Appendix 1.I An Elementary Discussion of Continuity, Differentiability, and Smoothness 52
References 55
2 Formulation of Two- and Three-Dimensional Boundary-Value Problems 57
2.1 Introductory Remarks 57
2.2 Preliminaries 57
2.3 Classical Linear Heat Conduction: Strong and Weak Forms; Equivalence 60
2.4 Heat Conduction: Galerkin Formulation; Symmetry and Positive-definiteness of K 64
2.5 Heat Conduction: Element Stiffness Matrix and Force Vector 69
2.6 Heat Conduction: Data Processing Arrays ID, IEN, and LM 71
2.7 Classical Linear Elastostatics: Strong and Weak Forms; Equivalence 75
2.8 Elastostatics: Galerkin Formulation, Symmetry, and Positive-definiteness of K 84
2.9 Elastostatics: Element Stiffness Matrix and Force Vector 90
2.10 Elastostatics: Data Processing Arrays ID, IEN, and LM 92
2.11 Summary of Important Equations for Problems Considered in Chapters 1 and 2 98
2.12 Axisymmetric Formulations and Additional Exercises 101
References 107
3 Isoparametric Elements and Elementary Programming Concepts 109
3.1 Preliminary Concepts 109
3.2 Bilinear Quadrilateral Element 112
3.3 Isoparametric Elements 118
3.4 Linear Triangular Element; An Example of "Degeneration" 120
3.5 Trilinear Hexahedral Element 123
3.6 Higher-order Elements; Lagrange Polynomials 126
3.7 Elements with Variable Numbers of Nodes 132
3.8 Numerical Integration; Gaussian Quadrature 137
3.9 Derivatives of Shape Functions and Shape Function Subroutines 146
3.10 Element Stiffness Formulation 151
3.11 Additional Exercises 156
Appendix 3.I Triangular and Tetrahedral Elements 164
Appendix 3.II Methodology for Developing Special Shape Functions with Application to Singularities 175
References 182
4 Mixed and Penalty Methods, Reduced and Selective Integration, and Sundry Variational Crimes 185
4.1 "Best Approximation" and Error Estimates: Why the standard FEM usually works and why sometimes it does not 185
4.2 Incompressible Elasticity and Stokes Flow 192
4.2.1 Prelude to Mixed and Penalty Methods 194
4.3 A Mixed Formulation of Compressible Elasticity Capable of Representing the Incompressible Limit 197
4.3.1 Strong Form 198
4.3.2 Weak Form 198
4.3.3 Galerkin Formulation 200
4.3.4 Matrix Problem 200
4.3.5 Definition of Element Arrays 204
4.3.6 Illustration of a Fundamental Difficulty 207
4.3.7 Constraint Counts 209
4.3.8 Discontinuous Pressure Elements 210
4.3.9 Continuous Pressure Elements 215
4.4 Penalty Formulation: Reduced and Selective Integration Techniques; Equivalence with Mixed Methods 217
4.4.1 Pressure Smoothing 226
4.5 An Extension of Reduced and Selective Integration Techniques 232
4.5.1 Axisymmetry and Anisotropy: Prelude to Nonlinear Analysis 232
4.5.2 Strain Projection: The B-approach 232
4.6 The Patch Test; Rank Deficiency 237
4.7 Nonconforming Elements 242
4.8 Hourglass Stiffness 251
4.9 Additional Exercises and Projects 254
Appendix 4.I Mathematical Preliminaries 263
4.I.1 Basic Properties of Linear Spaces 263
4.I.2 Sobolev Norms 266
4.I.3 Approximation Properties of Finite Element Spaces in Sobolev Norms 268
4.I.4 Hypotheses on a(.,.) 273
Appendix 4.II Advanced Topics in the Theory of Mixed and Penalty Methods: Pressure Modes and Error Estimates 276
4.II.1 Pressure Modes, Spurious and Otherwise 276
4.II.2 Existence and Uniqueness of Solutions in the Presence of Modes 278
4.II.3 Two Sides of Pressure Modes 281
4.II.4 Pressure Modes in the Penalty Formulation 289
4.II.5 The Big Picture 292
4.II.6 Error Estimates and Pressure Smoothing 297
References 303
5 The C[superscript 0]-Approach to Plates and Beams 310
5.1 Introduction 310
5.2 Reissner-Mindlin Plate Theory 310
5.2.1 Main Assumptions 310
5.2.2 Constitutive Equation 313
5.2.3 Strain-displacement Equations 313
5.2.4 Summary of Plate Theory Notations 314
5.2.5 Variational Equation 314
5.2.6 Strong Form 317
5.2.7 Weak Form 317
5.2.8 Matrix Formulation 319
5.2.9 Finite Element Stiffness Matrix and Load Vector 320
5.3 Plate-bending Elements 322
5.3.1 Some Convergence Criteria 322
5.3.2 Shear Constraints and Locking 323
5.3.3 Boundary Conditions 324
5.3.4 Reduced and Selective Integration Lagrange Plate Elements 327
5.3.5 Equivalence with Mixed Methods 330
5.3.6 Rank Deficiency 332
5.3.7 The Heterosis Element 335
5.3.8 T1: A Correct-rank, Four-node Bilinear Element 342
5.3.9 The Linear Triangle 355
5.3.10 The Discrete Kirchhoff Approach 359
5.3.11 Discussion of Some Quadrilateral Bending Elements 362
5.4 Beams and Frames 363
5.4.1 Main Assumptions 363
5.4.2 Constitutive Equation 365
5.4.3 Strain-displacement Equations 366
5.4.4 Definitions of Quantities Appearing in the Theory 366
5.4.5 Variational Equation 368
5.4.6 Strong Form 371
5.4.7 Weak Form 372
5.4.8 Matrix Formulation of the Variational Equation 373
5.4.9 Finite Element Stiffness Matrix and Load Vector 374
5.4.10 Representation of Stiffness and Load in Global Coordinates 376
5.5 Reduced Integration Beam Elements 376
References 379
The C[superscript 0]-Approach to Curved Structural Elements 383
6.1 Introduction 383
6.2 Doubly Curved Shells in Three Dimensions 384
6.2.1 Geometry 384
6.2.2 Lamina Coordinate Systems 385
6.2.3 Fiber Coordinate Systems 387
6.2.4 Kinematics 388
6.2.5 Reduced Constitutive Equation 389
6.2.6 Strain-displacement Matrix 392
6.2.7 Stiffness Matrix 396
6.2.8 External Force Vector 396
6.2.9 Fiber Numerical Integration 398
6.2.10 Stress Resultants 399
6.2.11 Shell Elements 399
6.2.12 Some References to the Recent Literature 403
6.2.13 Simplifications: Shells as an Assembly of Flat Elements 404
6.3 Shells of Revolution; Rings and Tubes in Two Dimensions 405
6.3.1 Geometric and Kinematic Descriptions 405
6.3.2 Reduced Constitutive Equations 407
6.3.3 Strain-displacement Matrix 409
6.3.4 Stiffness Matrix 412
6.3.5 External Force Vector 412
6.3.6 Stress Resultants 413
6.3.7 Boundary Conditions 414
6.3.8 Shell Elements 414
References 415
Part 2 Linear Dynamic Analysis
7 Formulation of Parabolic, Hyperbolic, and Elliptic-Elgenvalue Problems 418
7.1 Parabolic Case: Heat Equation 418
7.2 Hyperbolic Case: Elastodynamics and Structural Dynamics 423
7.3 Eigenvalue Problems: Frequency Analysis and Buckling 429
7.3.1 Standard Error Estimates 433
7.3.2 Alternative Definitions of the Mass Matrix; Lumped and Higher-order Mass 436
7.3.3 Estimation of Eigenvalues 452
Appendix 7.I Error Estimates for Semidiscrete Galerkin Approximations 456
References 457
8 Algorithms for Parabolic Problems 459
8.1 One-step Algorithms for the Semidiscrete Heat Equation: Generalized Trapezoidal Method 459
8.2 Analysis of the Generalized Trapezoidal Method 462
8.2.1 Modal Reduction to SDOF Form 462
8.2.2 Stability 465
8.2.3 Convergence 468
8.2.4 An Alternative Approach to Stability: The Energy Method 471
8.2.5 Additional Exercises 473
8.3 Elementary Finite Difference Equations for the One-dimensional Heat Equation; the von Neumann Method of Stability Analysis 479
8.4 Element-by-element (EBE) Implicit Methods 483
8.5 Modal Analysis 487
References 488
9 Algorithms for Hyperbolic and Parabolic-Hyperbolic Problems 490
9.1 One-step Algorithms for the Semidiscrete Equation of Motion 490
9.1.1 The Newmark Method 490
9.1.2 Analysis 492
9.1.3 Measures of Accuracy: Numerical Dissipation and Dispersion 504
9.1.4 Matched Methods 505
9.1.5 Additional Exercises 512
9.2 Summary of Time-step Estimates for Some Simple Finite Elements 513
9.3 Linear Multistep (LMS) Methods 523
9.3.1 LMS Methods for First-order Equations 523
9.3.2 LMS Methods for Second-order Equations 526
9.3.3 Survey of Some Commonly Used Algorithms in Structural Dynamics 529
9.3.4 Some Recently Developed Algorithms for Structural Dynamics 550
9.4 Algorithms Based upon Operator Splitting and Mesh Partitions 552
9.4.1 Stability via the Energy Method 556
9.4.2 Predictor/Multicorrector Algorithms 562
9.5 Mass Matrices for Shell Elements 564
References 567
10 Solution Techniques for Eigenvalue Problems 570
10.1 The Generalized Eigenproblem 570
10.2 Static Condensation 573
10.3 Discrete Rayleigh-Ritz Reduction 574
10.4 Irons-Guyan Reduction 576
10.5 Subspace Iteration 576
10.5.1 Spectrum Slicing 578
10.5.2 Inverse Iteration 579
10.6 The Lanczos Algorithm for Solution of Large Generalized Eigenproblems 582
10.6.1 Introduction 582
10.6.2 Spectral Transformation 583
10.6.3 Conditions for Real Eigenvalues 584
10.6.4 The Rayleigh-Ritz Approximation 585
10.6.5 Derivation of the Lanczos Algorithm 586
10.6.6 Reduction to Tridiagonal Form 589
10.6.7 Convergence Criterion for Eigenvalues 592
10.6.8 Loss of Orthogonality 595
10.6.9 Restoring Orthogonality 598
References 601
11 Dlearn--A Linear Static and Dynamic Finite Element Analysis Program 603
11.1 Introduction 603
11.2 Description of Coding Techniques Used in DLEARN 604
11.2.1 Compacted Column Storage Scheme 605
11.2.2 Crout Elimination 608
11.2.3 Dynamic Storage Allocation 616
11.3 Program Structure 622
11.3.1 Global Control 623
11.3.2 Initialization Phase 623
11.3.3 Solution Phase 625
11.4 Adding an Element to DLEARN 631
11.5 DLEARN User's Manual 634
11.5.1 Remarks for the New User 634
11.5.2 Input Instructions 635
11.5.3 Examples 663
1. Planar Truss 663
2. Static Analysis of a Plane Strain Cantilever Beam 666
3. Dynamic Analysis of a Plane Strain Cantilever Beam 666
4. Implicit-explicit Dynamic Analysis of a Rod 668
11.5.4 Subroutine Index for Program Listing 670
References 675
Index 676
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