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| موضوع: كتاب The Finite Element Method السبت 24 أغسطس 2019, 11:39 am | |
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أخوانى فى الله أحضرت لكم كتاب The Finite Element Method Fundamentals and Applications in Civil, Hydraulic, Mechanical and Aeronautical Engineering Bofang Zhu Professor, China Institute of Water Resources and Hydropower Research and Academician, Chinese Academy of Engineering Beijing, China
و المحتوى كما يلي :
Contents Preface xxiii About the Author xxv 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 2.5.1 Isotropic Body: Plane Stress 32 2.5.2 Isotropic Body: Plane Strain 33 2.5.3 Anisotropic Body 34 2.6 Equivalent Nodal Force and Element Stiffness Matrix 35 2.7 Nodal Loads 40 2.7.1 Equivalent Nodal Loads of Distributed Boundary Forces 41 2.7.2 Nodal Loads of Uniform Volume Force 41 2.7.3 Nodal Loads Due to Potential of Volume Force 42 2.7.4 Nodal Loads Caused by Initial Strain 43 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 2.10.1 Stress Concentration near the Circular Hole 51 vi Contents 2.10.2 Stress Analysis of I Beam with a Hole in Web 51 2.10.3 Stress Analysis of the Concrete Gravity Dam 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 3.5.1 Distributed Volume Force 60 3.5.2 Distributed Surface Force 60 3.5.3 Initial Strain and Initial Stress 61 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 4.5.1 Multiple Substructures 89 4.5.2 Condensation of the Internal Degrees of Freedom of Substructures 90 4.5.3 Coordinate Transformation 90 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 5.3.1 6-Node Quadratic Triangular Element 100 5.3.2 10-Node 3-Order Triangular Element 101 5.3.3 3-Node 18 DOF Triangular Element 102 Bibliography 104 6 Axisymmetrical Problems in Theory of Elasticity 105 6.1 Stresses Due to Axisymmetrical Loads 105 6.1.1 Displacement Function 105 Contents vii 6.1.2 Element Strains 106 6.1.3 Element Stress 108 6.1.4 Element Stiffness Matrix 109 6.1.5 Nodal Loads 110 6.2 Antisymmetrical Load 110 Bibliography 114 7 Spatial Problems in Theory of Elasticity 115 7.1 Constant Strain Tetrahedral Elements 115 7.1.1 Displacement Function 115 7.1.2 Element Strain 117 7.1.3 Element Stress 118 7.1.4 Stiffness Matrix of the Element 119 7.1.5 Nodal Load 120 7.2 Volume Coordinates 121 7.3 High-Order Tetrahedral Elements 122 7.3.1 10-Node Linear Strain Tetrahedral Elements 122 7.3.2 20-Node Tetrahedral Element 123 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 8.5.1 Plane Coordinate Transformation 142 8.5.2 Spatial Coordinate Transformation 144 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 8.10.1 Degradation of 4-Node Plane Isoparametric Elements 155 8.10.2 Degradation of an 8-Node Space Isoparametric Element 158 8.10.3 Degradation of High-Order Elements 160 8.11 Numerical Integration 161 8.11.1 One-Dimensional Gauss Quadrature Formula 162 8.11.2 Two-Dimensional and Three-Dimensional Gauss Quadrature Formulas 163 8.12 Selection of the Numerical Integration Order 164 8.12.1 Conditions for Nonsingularity of the Global Stiffness Matrix [K] 164 8.12.2 Integral Order Ensuring the Calculation Precision 165 8.12.3 Reduced Integration and Selected Integration 167 8.13 Stress Refinement and Stress Smoothing 168 viii Contents 8.13.1 Stress Refinement 168 8.13.2 Stress Smoothing 169 8.14 Elemental Form and Layout 173 8.14.1 Effect of the Elemental Shape on Strain 173 8.14.2 Effect of Edge Node Spacing on Strain 175 8.14.3 Intensification of Computing Mesh of Isoparametric Elements 175 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 8.18.1 Direction Cosine 188 8.18.2 Scalar Product 188 8.18.3 Vector Product 188 8.18.4 Infinitesimal Area in Curvilinear Coordinate System 189 8.18.5 Infinitesimal Area of Spatial Curved Surface 190 8.18.6 Spatial Infinitesimal Volumes 191 8.19 Numerical Examples of Isoparametric Elements 191 8.20 Infinite Elements 192 8.20.1 Two-Dimensional Infinite Elements 192 8.20.2 Three-Dimensional Infinite Elements 196 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 9.3.1 Comparison of Different Computation Methods 210 9.3.2 The Effect of Foundation Deformation on the Displacement and Stress of Arch Dam 212 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 10.2.1 Displacement Function 229 10.2.2 Stiffness Matrix 231 10.2.3 Nodal Load 232 10.2.4 Example 233 10.2.4.1 Square Thin Plate Supported by Four Edges 233 10.2.4.2 Square Thin Plate Supported by Corner Points 233 10.3 Triangular Thin Plate Element 235 10.3.1 Displacement Function 235 Contents ix 10.3.2 Stiffness Matrix and Nodal Load 238 10.3.3 Smoothing Curvature 238 10.3.4 Example 239 10.3.4.1 The Square Plate Bearing Concentrated and Distributed Loads 239 10.3.4.2 The Distortion of the Square Plate 239 10.4 Plate Element with Curved Boundary and Deflection and Rotation Defined Respectively 241 10.4.1 Beam Element Considering the Shearing Deformation 241 10.4.2 Curved Plate Element with the Deflection and Rotation Interpolated Respectively 245 10.5 The Plate on Elastic Foundation 248 10.5.1 Plate on Winkler 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 13.1.1 Stress–Strain Relations for Solids 281 13.1.2 Stress–Rate and Strain Relations for Fluid 282 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 13.6.1 Generalized Darcy Law 285 13.6.2 Fundamental Equations 287 13.6.3 Discretization of the Problems 287 13.7 Free Surface Calculation for Seepage Analysis 290 13.7.1 Method of Mesh Revision 290 13.7.2 Method of Revision of the Conductivity Matrix 290 13.7.3 Residual Velocity Method 291 13.7.4 Initial Velocity Method 294 x Contents 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 Dynamic Water Pressure during Earthquake 301 13.11 Inviscid Fluid Flow Formulated by Potential Function Φ 303 13.11.1 Basic Equations 303 13.11.2 The Flow around Objects without Lift 306 13.11.3 The Flow around Objects with Lift 307 13.12 Potential Flow Formulated by Stream Function ???? 307 13.12.1 Basic Equations 307 13.12.2 The Flow around Objects without Lift 308 13.12.3 The Flow around Objects with Lift 310 13.13 Flow on the Free Surface 312 13.14 Viscous and Non-Newtonian Flow 316 13.14.1 Solution of the Stokes Equation 316 13.14.2 Solution of the Navier–Stokes Equations 317 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 14.2.1 Euler’s Equation 322 14.2.2 Variational Principle of Problem of Heat Conduction 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 14.6.1 Two-Dimensional Isoparametric Elements 329 14.6.2 Three-Dimensional Isoparametric Elements 331 14.7 Computing Examples of Unsteady Temperature Field 331 14.8 Temperature Field of Mass Concrete with Pipe Cooling 332 14.8.1 Concrete Cylinder Cooled by Water Pipe 332 14.8.2 Mass Concrete Cooled by Water Pipe 334 14.8.3 Mass Concrete Cooled by Water Pipe with Precise ????(????) 334 Bibliography 335 15 Methods for Nonlinear Finite Element Analysis 337 15.1 Incremental Method 338 15.1.1 Method of Starting Point Stiffness 338 15.1.2 Method of Midpoint Stiffness 339 15.2 Iterative Method 342 15.2.1 Direct Iterative Method 342 15.2.2 Newton Method 343 15.2.3 Modified Newton Method 344 15.2.4 Quasi-Newton Method 345 Contents xi 15.2.5 The Calculation of {Ψn} and Initial Stress Method and Initial Strain Method 347 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 16.3.1 Geometric Characteristics of Stress Space 357 16.3.1.1 The Hydrostatic Stress Axis 357 16.3.1.2 ???? Plane 358 16.3.1.3 Line L′ Parallel to the Line L 358 16.3.1.4 The Plane Parallel to ???? Plane 358 16.3.2 The Geometric Expression of Any Point 358 16.3.3 Principal Stresses 361 16.4 Decompose of Strain Tensor 362 16.5 Criterion of Yield 363 16.5.1 Tresca Yield Criterion 364 16.5.2 Mises Yield Criterion 365 16.5.3 Mohr–Coulomb Yield Criterion 367 16.5.4 Drucker–Prager Yield Criterion 368 16.5.5 Lade Yield Criterion 370 16.5.6 Bresler–Pister Yield Criterion 370 16.5.7 Ottosen Yield Criterion 371 16.5.8 Hsieh–Ting–Chen Four-Parameter Criterion 371 16.5.9 Mohr–Coulomb Criterion with the Maximum Tensile Stress 372 16.5.10 Willam–Warnke Criterion with Three and Five Parameters 373 16.5.10.1 Willam–Warnke Criterion with Three Parameters 374 16.5.10.2 Willam–Warnke Criterion with Five Parameters 376 16.5.11 Zhang–Lu Yield Criterion 378 16.6 Strain Hardening 379 16.6.1 Isotropic Strain Hardening Model 380 16.6.2 Flowing Strain Hardening Model 381 16.6.3 Mixed Strain Hardening Model 381 16.7 Criterion of Loading and Unloading 382 16.7.1 Loading and Unloading of Ideal Plastic Material 382 16.7.2 Loading and Unloading of Strain Hardened Materials 382 16.7.3 Strain Softening, Brittle Failure, and Residual Strength 383 16.8 The Finite Element Method in Elastic–Plastic Incremental Theory 384 16.8.1 The Elastoplastic Matrix of Incremental Theory 384 16.8.2 Symmetric Expression of Nonassociated Elastic–Plastic Stiffness Matrix 386 16.8.3 The Calculation of { ???????????? F } 387 16.8.4 Effective Stress, Effective Plastic Strain, and Calculation of ????F/???????? 389 16.8.4.1 The Effective Stress ???? i 389 xii Contents 16.8.4.2 The Effective Plastic Strain ???? i 389 16.8.4.3 The Calculation of ????F/???????? 390 16.8.5 Singular Points on the Yield Surface 391 16.8.6 Numerical Calculation Method 392 16.8.6.1 The Displacement Increment 392 16.8.6.2 Tentative Stress 392 16.8.6.3 The Scale Factor 393 16.8.6.4 The Plastic Stress Increment 394 16.8.6.5 Stress Back to the Yield Surface 395 16.8.6.6 Calculation Steps 396 16.8.7 Example 396 16.9 Finite Element Method in the Full Variable Theory of Plasticity 397 16.9.1 Basic Assumption of Full Variable Theory 397 16.9.2 The Stress–Strain Relationship of Yiliuxin 398 16.9.3 The Elastic–Plastic Matrix of Full Variable Theory 399 16.10 Practical Simplified Models for Nonlinear Problem of Material 399 16.10.1 Isotropic Model Containing One-Variable Modulus E(t) 400 16.10.2 Isotropic Model Containing Two-Variable Modulus K(t) and G(t) 400 16.10.3 Orthotropic Model and the Equivalent Uniaxial Strain 401 16.10.3.1 Orthotropic Constitutive Relations 401 16.10.3.2 Equivalent Uniaxial Strain 403 16.10.4 The Approximate Calculation of Strain Softening 404 Bibliography 404 17 Creep of Concrete and its Influence on Stresses and Deformations of Structures 407 17.1 Stress–Strain Relation of Concrete 407 17.1.1 Stress–Strain Relation of Concrete under Action of Stress in One Direction 408 17.1.2 Stress–Strain Relation Under Complex Stress Conditions 411 17.1.3 Modulus of Elasticity of Concrete E(????) 413 17.1.4 Unit Creep of Concrete 414 17.1.5 Formula for Preliminary Design 416 17.2 Influence of Creep on Stresses and Deformations of Linear Elastocreeping Body 416 17.3 Analysis of Elastocreeping Stresses of Concrete Structure 419 17.3.1 The Calculation of Strain Increment under Uniaxial Stress 420 17.3.1.1 The Elastic Strain Increment 420 17.3.1.2 The Increment of Creep Strain When C(t, ????) = ????(????)[1 − e−r(t − ????)] 420 17.3.1.3 The Increment of Creep Strain When C(t, ????) = ∑ ????j(????)[1 − e−rj(t − ????)] 422 17.3.2 The Calculation of Strain Increments under Complex Stress Conditions 423 17.3.3 Equilibrium Equations 423 17.4 Compound Layer Element for the Simulation Analysis of Concrete Dams 424 Bibliography 429 Contents xiii 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 18.1.1 The Stress–Strain Relation of Ideal Elastic Body (Hooke Body) 431 18.1.2 The Stress–Strain Relation of Ideal Plastic Body: The Dashpot 431 18.1.3 Maxwell Body 431 18.1.4 Kelvin Body 432 18.1.5 Standard three-Component Viscoelastic Body 433 18.1.6 Kelvin Chain 433 18.1.7 The Stress–Strain Relation When Stress Changes with Time 434 18.2 The Stress–Strain Relation under the Action of Complex Stresses 434 18.2.1 The Stress–Strain Relation When Poisson’s Ratio Is Constant 434 18.2.2 Different Law for Volume Deformation and Shear Deformation 435 18.3 Stress Analysis of Viscoelastic Body 436 18.3.1 Stress Analysis of Viscoelastic Body with Constant Poisson’s Ratio 437 18.3.2 Stress Analysis of Viscoelastic Body with Different Laws for Volume Deformation and Shear Deformation 437 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 18.5.1 Viscoelastic–Plastic Problems under Action of One-Dimensional Stress 441 18.5.2 Viscoelastic–Plastic Problems with Complex Stress States 444 18.5.3 Visco-Plastic Strain Increment 446 18.5.4 Stress Analysis of Viscoelastic–Plastic Bodies 446 18.5.5 The Choice of Time Interval Δt n 448 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 19.5.1 Rectangular Thin-Plate Element 462 19.5.2 Triangular Thin-Plate Elements 464 Bibliography 465 20 Problems in Analysis of Structures with Large Displacement 467 20.1 The Basic Method for Geometrical Nonlinear Problems 467 20.1.1 Basic Formulas 467 20.1.2 The Solution 469 20.1.3 The Elastic Stability Problem 470 20.2 The Plate Element of Large Deflection 471 xiv Contents 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 21.2.1 Displacement Method 484 21.2.2 Stress Method 486 21.3 J-Integral Method 486 21.4 Energy Method, Flexibility Method, and Bueckner Formula 490 21.4.1 Energy Release Rate G and the Related Formulas 490 21.4.2 Flexibility Method 491 21.4.3 Energy Method 492 21.4.4 Bueckner Formula 492 21.5 Stiffness Derivative Method 494 21.5.1 Plane Problem 494 21.5.2 Axial Symmetrical Problem 495 21.5.3 Space Problem 497 21.6 Singular Element of the Crack Tip 499 21.6.1 Triangular Singular Element 499 21.6.2 Circle Singular Element 500 21.6.3 Hybrid Singular Element 500 21.7 Singular Isoparametric Element (1/4 Length Midpoint Method) 502 21.7.1 Rectangular Singular Isoparametric Element 502 21.7.2 Triangular Degenerated Singular Isoparametric Element 503 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 22.2.1 Consistent Mass Matrix 517 22.2.2 Lumped Mass Matrix 517 22.2.3 Several Typical Element Mass Matrices 518 22.2.3.1 Beam Element 518 22.2.3.2 Plane Constant Strain Triangular Elements 518 22.2.3.3 Rectangular Plate Element 520 22.2.4 Comparison of Two Mass Matrices 520 22.3 Damping Matrix 522 22.3.1 Damping of Single Freedom System 522 22.3.2 Damping of System of Multidegree of Freedom 523 22.4 Natural Frequency and Vibration Mode of Structure 526 22.4.1 Natural Frequency and Vibration Mode 526 22.4.2 Orthogonality of Modes 529 Contents xv 22.4.3 Free Vibration Equation of Structure Represented by Flexibility Matrix 531 22.4.4 Effects of Zero Mass 532 22.4.5 Static Condensation 532 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 Iteration Method for Computing the Natural Frequency and Vibration Mode 538 22.7.1 Inverse Iteration Method: The Calculation of Lowest Frequency and Vibration Mode 539 22.7.2 Mode Clearance: Calculation of Other Frequencies and Modes 541 22.7.3 Shifting: To Improve the Convergence Speed 544 22.7.4 Positive Iterative Method: Calculation of the Maximum Frequency and Vibration Mode 545 22.8 Energy Method for Computing the Natural Frequencies of Structure 545 22.8.1 Rayleigh Energy Method 546 22.8.2 Ritz Energy Method 547 22.9 Subspace Iteration Method for Computing the Natural Frequencies and Vibration Modes of Structure 548 22.9.1 Subspace Iteration Method 549 22.9.2 Modified Subspace Iteration Method 553 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 22.13.1 Linear Acceleration Method 561 22.13.2 Wilson Method (???? Method) 563 22.13.3 Newmark Method 564 22.13.4 Calculation Stability, Precision, and the Selection of Time Step 566 22.13.4.1 Computational Stability 567 22.13.4.2 Calculation Accuracy 567 22.13.4.3 The Selection of the Time Step Δt 569 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 23.1.1 Rock Block 582 xvi Contents 23.1.2 Fault 582 23.1.3 Soft Layer 582 23.1.4 Joint 582 23.1.5 Crack 583 23.2 Equivalent Deformation Modulus 583 23.3 Two-Dimensional Linear Joint Element 584 23.3.1 Stiffness Matrix of Element 584 23.3.2 Nodal Force Due to Initial Stress 587 23.4 Stiffness Coefficients of Joint Element 587 23.5 Layer Element 591 23.6 Two-Dimensional High-Order Joint Element 593 23.6.1 6-Node Quadratic Joint Element 593 23.6.2 6-Node Curved Joint Element 595 23.7 Three-Dimensional Joint Element 597 23.7.1 6-Node Three-Dimensional Joint Element 597 23.7.2 Three-Dimensional Curved Joint Element 599 23.8 Infinite Joint Element 602 23.8.1 Infinite Joint Element in Plane Problem 603 23.8.2 Infinite Joint Element in Spatial Problem 604 23.9 Choice of Method for Stress Analysis in Rock 605 23.9.1 The Objective and Importance of Analysis 606 23.9.2 Character of Rock 606 23.9.3 Buried Depth of the Project 606 23.9.4 Original Data 606 23.10 Elastic Increment Method for Nonlinear Stress Analysis 606 23.10.1 Fracture and Slide of Rock 607 23.10.2 Fracture and Slide of Joint and Soft Layer 608 23.11 Initial Stress Method and No Tension Method 608 23.11.1 No Tension Method 609 23.11.2 Fracture and Slide of Stratified Rock 609 23.11.3 Fracture and Slide of Joint and Soft Layer 611 23.12 Elastic–Plastic Increment Method 612 23.12.1 Elastic–Plastic Computation for Integral Rock 612 23.12.2 Elastic–Plastic Computation for Rock with Weak Surface 613 23.12.3 Elastic–Plastic Calculation for Joint Element 614 23.13 Viscoelastic–Plastic Method 616 23.14 Computation of Anchor Bolt in Rock Foundation 618 23.15 Computing Examples in Rock Mechanics 621 23.15.1 Computing of Rock Slope 621 23.15.2 Antisliding Stability of Gravity Dam on Rock Foundation 623 Bibliography 626 24 Problems in Soil Mechanics 627 24.1 Nonlinear Elastic Model 627 24.2 Elastic–Plastic Model with Two Yield Surfaces 633 24.2.1 Yield Function and Elastic–Plastic Matrix 634 24.2.2 Plastic Coefficient 636 24.3 Interaction between Soil and Structure: Contact Element 637 Contents xvii 24.4 Consolidation of Soil 640 24.4.1 Terzaghi’s Consolidation Theory 640 24.4.2 Biot’s Consolidation Theory 643 24.4.3 Nonlinear Consolidation Problem 647 24.4.3.1 Nonlinear Elastic Consolidation Problem 648 24.4.3.2 Elastic–Plastic Consolidation Problem 648 24.4.3.3 Viscoelastic Consolidation Problem 648 24.4.3.4 Viscoelastic–Plastic Consolidation Problem 648 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 24.7.1 Computation Methods 652 24.7.1.1 Finite Element Strength Discount Method 653 24.7.1.2 Finite Element Increment Loading Method 654 24.7.2 Failure Criteria 654 24.7.3 Advantage of Finite Element Limit Analysis Method 655 24.7.4 Calculation Examples 655 Bibliography 657 25 Plain and Reinforced Concrete Structures 659 25.1 Constitutive Models of Concrete 660 25.1.1 Uniaxial Stress–Strain Relationship of Concrete 660 25.1.2 Constitutive Models of Concrete in Biaxial Stress State 662 25.1.3 Constitutive Models of Concrete in Triaxial Stress State 666 25.1.4 Equivalent Uniaxial Strain and Orthotropic Model for Concrete 668 25.2 Finite Element Models for Cracks in Concrete 672 25.2.1 Crack Inducement in Concrete 672 25.2.2 Discrete Crack Model 673 25.2.3 Smeared Crack Model 675 25.2.4 Thin-Layer Element for Crack 677 25.2.5 No-Tension Crack Model 679 25.2.6 Fracture Mechanics Model 680 25.2.6.1 The Sharp Crack Model 680 25.2.6.2 The Blunt Crack Band Model 681 25.2.6.3 Comparison of Concrete Crack Models 681 25.3 The Calculation of the Smeared Crack Model 682 25.3.1 Modes of the Concrete Failure and Constitutive Relations Before and After Failure 682 25.3.2 Concrete Crushing 684 25.3.3 The Split of Concrete Under the Plane Stress 684 25.3.4 Concrete Split Under the Plane Strain State 685 25.3.5 The Split of Concrete of the Spatial Problems 685 25.3.6 The Behavior of Concrete After Split 686 25.3.7 The Stress Adjustment and the Calculation Procedure when Concrete Splits 686 25.4 The Constitutive Relation and the Stress Calculation of the Steel 691 25.4.1 The Constitutive Relation of the Steel Bar 691 25.4.1.1 The Ideal Elastic–Plastic Model 691 xviii Contents 25.4.1.2 The Trilinear Model 691 25.4.1.3 The Complete Model 691 25.4.2 The Calculation of the Stress of the Steel Bar 692 25.5 The Finite Element Model of the Steel Bar 692 25.5.1 Line Element 692 25.5.2 Solid Element 693 25.5.3 Thin Membrane Element 693 25.6 The Connection of the Steel Bar and Concrete 693 25.6.1 Fixed Connection 693 25.6.2 The Linking Spring Element 693 25.6.3 The Contact Element 695 25.7 The Bond Stress between the Steel Bar and Concrete: The Stiffness Coefficient of the Linking Spring and the Contact Element 696 25.7.1 The Bond Stress between the Bar and the Concrete 696 25.7.2 The Stiffness Coefficient of the Linking Spring 697 25.7.3 The Stiffness Coefficient of the Contact Element 698 25.8 The Stiffness Matrix of the Reinforced Concrete Structure 698 25.9 The Calculation of Steel Bar in the Isoparametric Element 698 25.9.1 Plane Problem 699 25.9.2 The Axisymmetric Problems 701 25.9.2.1 The Calculation of the Bar in the Radial Plane 702 25.9.2.2 The Calculation of the Circumferential Bar 702 25.9.2.3 The Calculation of the Steel Plate Lining 703 25.9.2.4 The Circumferential Point Element 703 25.9.2.5 The Spatial Problems 703 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 26.2.1 The Optimization Method 713 26.2.2 The Approximate Reanalysis 713 26.2.3 Application of the Substructure Method 715 26.3 Elastic Displacement Back Analysis of Homogeneous Body and Proportional Deformation Heterogeneous Body 716 26.3.1 Inversion of Elastic Modulus 717 26.3.2 The Inversion of Initial Ground Stress in a Small Area 717 26.3.2.1 E Is Known: Inversion of q and p 718 26.3.2.2 q Is Known: Inversion of p and E 718 26.3.2.3 Inversion of E, p, and q at the Same Time 719 26.3.3 Back Analysis of Initial Ground Stress in a Wide Range 720 26.3.3.1 The First Method 720 26.3.3.2 The Second Method 721 26.4 Back Analysis of Material Parameters of Heterogeneous Elastic Body 722 26.4.1 The Difference State and Its Inverse Analysis 722 26.4.2 The Stiffness Matrix Decomposition Method 723 Contents xix 26.4.3 Optimization Method 725 26.4.4 Improvement of the Optimization Method 725 26.5 Back Analysis of Interaction of Elastic Structure with the Surrounding Medium 728 26.5.1 The Displacement Method 728 26.5.2 The Hybrid Method 731 26.6 Nonlinear Solid Back Analysis 733 26.6.1 The Solving Method 733 26.6.1.1 The Direct Search Method 733 26.6.1.2 The First-Order Taylor Expansion 734 26.6.1.3 The Second-Order Taylor Expansion 734 26.6.2 The Constitutive Model 734 26.6.2.1 The Nonlinear Elastic Model 735 26.6.2.2 Elastic–Plastic Model 735 26.6.2.3 Viscoelastic and Viscoelastic–Plastic Model 735 26.6.2.4 Partition Composite Model 736 Bibliography 737 27 Automatic Mesh Generation, Error Estimation, and Auto-adaptation Technique 739 27.1 Automatic Generation of Computing Grid 740 27.1.1 Isoparametric Transformation Method 740 27.1.2 Composite Function Method 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 28.2.1 Square Matrix 752 28.2.2 Row Matrix 752 28.2.3 Column Matrix 752 28.2.4 Scalar 752 28.2.5 Triangular Matrix 752 28.2.6 Diagonal Matrix 753 28.2.7 Unit Matrix 753 28.2.8 Zero Matrix 753 28.2.9 Transpose Matrix 754 28.2.10 Symmetric Matrix, Antisymmetric Matrix and Skew Symmetric Matrix 754 28.2.11 Band Matrix 755 28.3 Equality, Addition, and Subtraction of Matrices 755 28.3.1 The Equality of Matrices 755 28.3.2 The Addition and Subtraction of Matrices 756 28.4 Matrix Multiplied by a Number 756 xx Contents 28.5 Multiplication of Matrices 757 28.5.1 Compatible Matrix 757 28.5.2 Rules of Matrix Multiplication 757 28.5.3 Properties of Matrix Multiplication 758 28.5.4 The Positive Power of Square Matrix 760 28.6 Determinant 760 28.6.1 Definition of Determinant 760 28.6.2 Minors and Cofactors 761 28.6.3 Principal Minors 761 28.6.4 Expansion of the Determinant by One Row (Column) 762 28.6.5 Properties of Determinant 763 28.7 Inverse Matrix 763 28.7.1 The Definition of Inverse Matrix 763 28.7.2 The Adjoint Matrix 764 28.7.3 Inverse Matrix 764 28.7.4 The Inverse Matrix of the Diagonal Matrix 765 28.7.5 The Properties of the Inverse Matrix 766 28.8 Partitioned Matrix 766 28.8.1 Definition of the Partitioned Matrix 766 28.8.2 Addition and Subtraction of the Partitioned Matrix 766 28.8.3 The Multiplication of the Partitioned Matrix 767 28.8.4 The Inverse of the Partitioned Matrix 768 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 Iterative Method 778 29.3 Seidel Iterative Method 780 29.4 Over-Relaxation Iterative Method 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 30.1.1 The Extrema of One-Variable Functions 793 30.1.2 The Extrema of a Function with Several Variables 794 30.2 The Extrema of Functionals 795 30.3 Preliminary Theorems 796 30.4 Euler’s Equation of One-Dimensional Problems 797 Contents xxi 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 to Weighted Residual Method 813 31.2 Weight Function for Internal Residual Method 814 31.2.1 Collocation Method 814 31.2.2 Least Squares Method 816 31.2.3 Moment Method 817 31.2.4 Galerkin Method 818 31.3 Establish Fundamental Equations of Finite Element Method by Weighted 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
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