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

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