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.
Table of Contents
Preface xxiii
About the Author xxv
1 Introduction to Finite ElementMethod andMatrix 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
2.7 Nodal Loads 40
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
3.5 Nodal Load 59
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
6.2 Antisymmetrical Load 110
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
8.9 Nodal Loads 153
8.10 Degradation of Isoparametric Elements 155
8.11 Numerical Integration 161
8.13 Stress Refinement and Stress Smoothing 168
8.14 Elemental Form and Layout 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
10.5.1 Plate onWinkler Foundation 248
10.5.2 Plate on Elastic Half Space 249
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 andTheir 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
13.9 Unsteady Seepage Flow 300
13.10 DynamicWater Pressure during Earthquake 301
13.12 Potential Flow Formulated by Stream Function 𝜓 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 Conductionof 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
16.7 Criterion of Loading and Unloading 382
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
23.6 Two-Dimensional High-Order Joint Element 593
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
29.7 Conjugate Gradient Method 788
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 byWeighted Residual Method 820
31.4 Twist of Elastic Column 824
31.5 Unsteady Temperature Field 828
31.6 Dynamic Response of Structure 832
Bibliography 834
Appendix A 835
Appendix B 839
Index 841
