# The Finite Element Method: Fundamentals and Applications in Civil, Hydraulic, Mechanical and Aeronautical Engineering

ISBN: 978-1-119-10732-3

Apr 2018

872 pages

Select type: O-Book

## Description

A comprehensive review of the Finite Element Method (FEM), this book provides the fundamentals together with a wide range of applications in civil, mechanical and aeronautical engineering. It addresses both the theoretical and numerical implementation aspects of the FEM, providing examples in several important topics such as solid mechanics, fluid mechanics and heat transfer, appealing to a wide range of engineering disciplines.  Written by a renowned author and academician with the Chinese Academy of Engineering, The Finite Element Method would appeal to researchers looking to understand how the fundamentals of the FEM can be applied in other disciplines. Researchers and graduate students studying hydraulic, mechanical and civil engineering will find it a practical reference text.

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Preface xxiii

1 Introduction to Finite Element Method and Matrix Analysis of Truss 1

1.1 Introduction to Finite Element Method 1

1.2 Truss Analysis Overview 5

1.3 Stiffness Matrix of Horizontal Bar Element 8

1.4 Stiffness Matrix of Inclined Bar Element 10

1.5 Coordinate Transformation 11

1.6 Nodal Equilibrium Equation and Global Stiffness Matrix 14

1.7 Treatment of Boundary Conditions 15

Bibliography 23

2 Plane Problems in Theory of Elasticity 25

2.1 Discretization of Continuous Medium 25

2.2 Displacement Function 28

2.3 Element Strain 30

2.4 Initial Strain 31

2.5 Element Stress 32

2.6 Equivalent Nodal Force and Element Stiffness Matrix 35

2.8 Nodal Equilibrium Equation and Global Stiffness Matrix 43

2.9 Establish the Global Stiffness Matrix by the Coding Method 48

2.10 Calculation Example 51

Bibliography 51

3 Element Analysis 53

3.1 Principle of Virtual Displacement 53

3.2 Element Displacement 56

3.3 Element Strain and Stress 57

3.4 Nodal Force and Element Stiffness Matrix 57

3.6 Application Examples of the Principle of Virtual Displacements: Beam Element 61

3.7 Strain Energy and Complementary Strain Energy 64

3.8 Principle of Minimum Potential Energy 65

3.9 Minimum Complementary Energy Principle 69

3.10 Hybrid Element 70

3.11 Hybrid Element Example: Plane Rectangular Element 73

3.12 Mixed Energy Principle 75

3.13 Composite Element 77

Bibliography 79

4 Global Analysis 81

4.1 Nodal Equilibrium Equation 81

4.2 Application of the Principle of Minimum Potential Energy 82

4.3 The Low Limit Property of the Solution of Minimum Potential Energy 84

4.4 The Convergence of Solutions 85

4.5 Analysis of the Substructure 88

Bibliography 91

5 High-Order Element of Plane Problem 93

5.1 Rectangular Elements 93

5.2 Area Coordinates 97

5.3 High-Order Triangular Element 100

Bibliography 104

6 Axisymmetrical Problems in Theory of Elasticity 105

6.1 Stresses Due to Axisymmetrical Loads 105

Bibliography 114

7 Spatial Problems in Theory of Elasticity 115

7.1 Constant Strain Tetrahedral Elements 115

7.2 Volume Coordinates 121

7.3 High-Order Tetrahedral Elements 122

Bibliography 124

8 Shape Function, Coordinate Transformation, Isoparametric Element, and Infinite Element 125

8.1 Definition of Shape Functions 125

8.2 One-Dimensional Shape Functions 126

8.3 Two-Dimensional Shape Function 127

8.4 Three-Dimensional Shape Function 130

8.5 Coordinate Transformation 136

8.6 Displacement Function 145

8.7 Element Strain 147

8.8 Stiffness Matrix 151

8.10 Degradation of Isoparametric Elements 155

8.11 Numerical Integration 161

8.12 Selection of the Numerical Integration Order 164

8.13 Stress Refinement and Stress Smoothing 168

8.14.1 Effect of the Elemental Shape on Strain 173

8.15 Inconsistent Elements 176

8.16 Patch Test 179

8.17 Triangular, Tetrahedral, and Prismatic Curved-Side Elements 183

8.18 Vector Computation in Isoparametric Elements 187

8.19 Numerical Examples of Isoparametric Elements 191

8.20 Infinite Elements 192

Bibliography 199

9 Comparison and Application Instances of Various Planar and Spatial Elements 201

9.1 Comparison and Selection of Various Planar Elements 201

9.2 Comparison and Selection of Various Spatial Elements 205

9.3 Analysis of Stresses in Arch Dam 209

9.4 Analysis of Stress in Buttress Dam 215

9.5 Analysis of Spatial Effect of Gravity Dam 217

9.6 Analysis of Spatial Effect of Earth Dam 217

9.7 Analysis of Stress on Tunnel Lining 220

Bibliography 221

10 Elastic Thin Plate 223

10.1 Bending of Elastic Thin Plate 223

10.2 Rectangular Thin Plate Element 228

10.3 TriangularThin Plate Element 235

10.4 Plate Element with Curved Boundary and Deflection and Rotation Defined Respectively 241

10.5 The Plate on Elastic Foundation 248

Bibliography 252

11 Elastic Thin Shell 255

11.1 Element Stiffness Matrix in Local Coordinate System 255

11.2 Coordinate Transformation: Global Stiffness Matrix 259

11.3 Direction Cosine of Local Coordinate 261

11.4 Curved-Surface Shell Element 264

11.5 Shell Supported or Reinforced by Curved Beam 268

11.6 Example 271

Bibliography 271

12 Axisymmetric Shell 273

12.1 Linear Element 273

12.2 Curved Element 277

Bibliography 280

13 Problems in Fluid Mechanics 281

13.1 Relation between Stress and Strain for Newtonian Fluids 281

13.2 Equation of Motion 283

13.3 Continuity Equation 284

13.4 Energy Equation 284

13.5 State and Viscosity Equations 284

13.6 Fundamental Equations for Steady Seepage Flow and Their Discretization 285

13.7 Free Surface Calculation for Seepage Analysis 290

13.8 Substitution of the Curtain of Drainage Holes by the Seeping Layer for Seepage Analysis 296

13.10 DynamicWater Pressure during Earthquake 301

13.11 Inviscid Fluid Flow Formulated by Potential Function Φ 303

13.12 Potential Flow Formulated by Stream Function �� 307

13.13 Flow on the Free Surface 312

13.14 Viscous and Non-Newtonian Flow 316

Bibliography 318

14 Problems in Conduction of Heat in Solids 321

14.1 Differential Equation: Initial and Boundary Conditions for Conduction of Heat in Solids 321

14.2 Variational Principle for Conduction of Heat in Solids 322

14.3 Discretization of Continuous Body 323

14.4 Fundamental Equations for Solving Unsteady Temperature Field by FEM 324

14.5 Two-Dimensional Unsteady Temperature Field, Triangular Elements 327

14.6 Isoparametric Elements 329

14.7 Computing Examples of Unsteady Temperature Field 331

14.8 Temperature Field of Mass Concrete with Pipe Cooling 332

Bibliography 335

15 Methods for Nonlinear Finite Element Analysis 337

15.1 IncrementalMethod 338

15.2 Iterative Method 342

15.3 Mixed Method 349

15.4 Application of Substructure Method in Nonlinear Analysis 349

Bibliography 351

16 Problems in Theory of Plasticity 353

16.1 One-Dimensional Stress–Strain Relation 353

16.2 Decompose of Stress Tensor and Stress Invariant 355

16.3 Haigh–Westergaard Stress Space 357

16.4 Decompose of Strain Tensor 362

16.5 Criterion of Yield 363

16.6 Strain Hardening 379

16.8 The Finite Element Method in Elastic–Plastic Incremental Theory 384

16.9 Finite Element Method in the Full VariableTheory of Plasticity 397

16.10 Practical Simplified Models for Nonlinear Problem of Material 399

Bibliography 404

17 Creep of Concrete and its Influence on Stresses and Deformations of Structures 407

17.1 Stress–Strain Relation of Concrete 407

17.2 Influence of Creep on Stresses and Deformations of Linear Elastocreeping Body 416

17.3 Analysis of Elastocreeping Stresses of Concrete Structure 419

17.4 Compound Layer Element for the Simulation Analysis of Concrete Dams 424

Bibliography 429

18 Stress Analysis for Viscoelastic and Visco-Plastic Bodies 431

18.1 The Stress–Strain Relation of Viscoelastic Body under the Action of Unidirectional Stress 431

18.2 The Stress–Strain Relation under the Action of Complex Stresses 434

18.3 Stress Analysis of Viscoelastic Body 436

18.4 Effective Modulus Method and Equivalent Temperature Method for Simple Harmonic Temperature Creep Stress Analysis of Concrete at Late Ages and Viscoelastic Body 439

18.5 Stress Analysis for Visco-Plastic Bodies 441

18.6 Combined Viscoelastic–Plastic Models 449

Bibliography 451

19 Elastic Stability Problem 453

19.1 Geometrical Stiffness Matrix of the Beam Element 453

19.2 Geometrical Stiffness Matrix of Plate Elements 457

19.3 Global Analysis 459

19.4 Cases of Beam System 461

19.5 Computing Examples of Elastic Stability of Thin Plate System 462

Bibliography 465

20 Problems in Analysis of Structures with Large Displacement 467

20.1 The Basic Method for Geometrical Nonlinear Problems 467

20.2 The Plate Element of Large Deflection 471

20.3 Three-Dimensional Solid Element of Large Displacement 476

20.4 Double Nonlinearity: Elastoplastic Large Displacement Problem 478

Bibliography 478

21 Problems in Fracture Mechanics 481

21.1 Introduction 481

21.2 Direct Method 484

21.3 J-Integral Method 486

21.4 Energy Method, FlexibilityMethod, and Bueckner Formula 490

21.5 Stiffness DerivativeMethod 494

21.6 Singular Element of the Crack Tip 499

21.7 Singular Isoparametric Element (1/4 Length Midpoint Method) 502

21.8 Blunt Crack Zone Model 506

21.9 Elastic–Plastic Fracture 509

21.10 Extended Finite Element Method for Fracture Analysis 512

Bibliography 514

22 Problems in Structural Dynamics 515

22.1 Equations of Motion 515

22.2 Mass Matrix 516

22.3 Damping Matrix 522

22.4 Natural Frequency and Vibration Mode of Structure 526

22.5 Mode Superposition Method for Analyzing the Structure of Forced Vibration 535

22.6 Dynamic Response of Structure under the Action of Earthquake Solving by Vibration Mode Superposition Method 536

22.7 Vector IterationMethod for Computing the Natural Frequency and Vibration Mode 538

22.8 Energy Method for Computing the Natural Frequencies of Structure 545

22.9 Subspace Iteration Method for Computing the Natural Frequencies and Vibration Modes of Structure 548

22.10 Ritz Vector Superposition Method for Solving Forced Vibration of Structure 554

22.11 Modified Ritz Vector Superposition Method 556

22.12 Dynamic Substructure Method 557

22.13 Direct Integration Method for Solving the Equation of Motion 560

22.14 Coupled Vibration of Solid and Fluid 570

22.15 Seismic Stress of Gravity Dam 571

22.16 Seismic Stress of Buttress Dam 574

22.17 Vibration of Arch Dam 575

22.18 Seismic Stress of Earth Dam 575

22.19 Seismic Stresses of Cylindrical Shell 577

22.20 Nonlinear Dynamic Responses of Underground Structures 578

Bibliography 580

23 Problems in Rock Mechanics 581

23.1 Structure of Rock 581

23.2 Equivalent Deformation Modulus 583

23.3 Two-Dimensional Linear Joint Element 584

23.4 Stiffness Coefficients of Joint Element 587

23.5 Layer Element 591

23.7 Three-Dimensional Joint Element 597

23.8 Infinite Joint Element 602

23.9 Choice of Method for Stress Analysis in Rock 605

23.10 Elastic Increment Method for Nonlinear Stress Analysis 606

23.11 Initial Stress Method and No Tension Method 608

23.12 Elastic–Plastic Increment Method 612

23.13 Viscoelastic–Plastic Method 616

23.14 Computation of Anchor Bolt in Rock Foundation 618

23.15 Computing Examples in Rock Mechanics 621

Bibliography 626

24 Problems in Soil Mechanics 627

24.1 Nonlinear Elastic Model 627

24.2 Elastic–Plastic Model with Two Yield Surfaces 633

24.3 Interaction between Soil and Structure: Contact Element 637

24.4 Consolidation of Soil 640

24.5 Stress, Deformation, and Stability of Earth Dam 648

24.6 Computation of Rockfill Dam with Concrete Face Slab 649

24.7 Limit Analysis in Rock and Soil Mechanics 652

Bibliography 657

25 Plain and Reinforced Concrete Structures 659

25.1 Constitutive Models of Concrete 660

25.2 Finite Element Models for Cracks in Concrete 672

25.3 The Calculation of the Smeared Crack Model 682

25.4 The Constitutive Relation and the Stress Calculation of the Steel 691

25.5 The Finite Element Model of the Steel Bar 692

25.6 The Connection of the Steel Bar and Concrete 693

25.7 The Bond Stress between the Steel Bar and Concrete:The Stiffness Coefficient of the Linking Spring and the Contact Element 696

25.8 The Stiffness Matrix of the Reinforced Concrete Structure 698

25.9 The Calculation of Steel Bar in the Isoparametric Element 698

25.10 The Layered Element of the Reinforced Concrete Plates and Shells 706

Bibliography 709

26 Back Analysis of Engineering 711

26.1 General Principles of Back Analysis 711

26.2 Back Analysis of the Seepage Field 712

26.3 Elastic Displacement Back Analysis of Homogeneous Body and Proportional Deformation Heterogeneous Body 716

26.4 Back Analysis of Material Parameters of Heterogeneous Elastic Body 722

26.5 Back Analysis of Interaction of Elastic Structure with the Surrounding Medium 728

26.6 Nonlinear Solid Back Analysis 733

Bibliography 737

27 Automatic Mesh Generation, Error Estimation, and Auto-adaptation Technique 739

27.1 Automatic Generation of Computing Grid 740

27.2 Error Estimation 742

27.3 Auto-adaptation Technique: h Method 745

27.4 Auto-adaptation Technique: p Method 746

Bibliography 748

28 Matrix 751

28.1 Definition of Matrix 751

28.2 Principal Types of Matrix 752

28.3 Equality, Addition, and Subtraction of Matrices 755

28.4 Matrix Multiplied by a Number 756

28.5 Multiplication of Matrices 757

28.6 Determinant 760

28.7 Inverse Matrix 763

28.8 Partitioned Matrix 766

28.9 Orthogonal Matrix 770

28.10 Positive Definite Matrix 771

28.11 Derivative of Matrix 772

28.12 Integration of Matrix 774

Bibliography 775

29 Linear Algebraic Equation Set 777

29.1 Linear Algebraic Equation Set 777

29.2 Simple IterativeMethod 778

29.3 Seidel Iterative Method 780

29.4 Over-Relaxation IterativeMethod 781

29.5 Block Over-Relaxation Iterative Method 781

29.6 Direct Solution Method 783

29.8 Comparison of Several Kinds of Commonly Used Method 790

29.9 Homogeneous Linear Equations 791

Bibliography 792

30 Variational Method 793

30.1 The Extrema of Functions 793

30.2 The Extrema of Functionals 795

30.3 Preliminary Theorems 796

30.4 Euler’s Equation of One-Dimensional Problems 797

30.5 Euler’s Equation for Plane Problems 800

30.6 Euler’s Equations of Spatial Problems 803

30.7 Ritz Method for Solving Variational Problems 806

30.8 Finite Element Method for Solving the Variational Problems 809

Bibliography 811

31 Weighted Residual Method 813

31.1 Introduction toWeighted Residual Method 813

31.2 Weight Function for Internal Residual Method 814

31.3 Establish Fundamental Equations of Finite Element Method by Weighted Residual Method 820

31.4 Twist of Elastic Column 824