كتاب The Finite Element Method
منتدى هندسة الإنتاج والتصميم الميكانيكى
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 كتاب The Finite Element Method

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تاريخ التسجيل : 01/07/2009
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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  

كتاب The Finite Element Method  T_f_w_11
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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