This book presents a novel approach to the classical scientific discipline of Structural Engineering, which is inspired by numerous current applications from domains of Civil, Mechanical or Aerospace Engineering. The main goal of this book is to help with making the best choice between accuracy and efficiency, when it comes to building the most suitable structural models by practising engineers using modern computational tools available in commercial software products (SAP, FEAP, ANSYS …) for which we have carried out many developments that have been become the main reference in the field.

封面 1

目录 9

1 Introduction 14

1.1 Motivation and Objectives 14

1.2 Main Topics Outline 18

1.3 Further Studies Recommendations 24

1.4 Summary of Main Notations 25

2 Truss Model: General Theorems and Methods of Force, Displacement and Finite Elements 28

2.1 Truss Model-Strong Form and Weak Form 28

      2.1.1 Strong or Differential Form: Analytic Solution 29

      2.1.2 Weak or Integral Form 33

2.2 General Theorems of Structural Mechanics on Truss Model 35

      2.2.1 Principle of Virtual Work 35

      2.2.2 Principle of Complementary Virtual Work 38

      2.2.3 Principle of Minimum of Total Potential Energy 40

      2.2.4 Applied General Theorems 44

2.3 Castigliano's Theorems, Force and Displacement Methods 47

      2.3.1 Castigliano's Theorems-Stiffness and Flexibility 47

      2.3.2 Force and Displacement Methods 50

2.4 Finite Element Method Implementation for Truss Model 56

      2.4.1 Local or Elementary Description 56

      2.4.2 Consistence of Finite Element Approximation 61

      2.4.3 Equivalent Nodal External Load Vector 62

      2.4.4 Higher Order Finite Elements 63

      2.4.5 Role of Numerical Integration 65

      2.4.6 Finite Element Assembly Procedure 69

3 Beam Models: Refinement and Reduction 72

3.1 Reduced Models of Solid Mechanics: Planar Beams of Euler, Timoshenko and Reissner 72

      3.1.1 Euler-Bernoulli Planar Beam Model 72

      3.1.2 Solid Mechanics Versus Beam Model Accuracy for Planar Cantilever Beam 82

      3.1.3 Timoshenko Planar Beam Model 85

      3.1.4 Brief on Reissner Planar Beam Model 89

3.2 Beam Model Refinement and Reduction 94

      3.2.1 Method of Direct Stiffness Assembly for 3D Beam Elements 96

      3.2.2 Beam Model Refinement: Flexibility Approach for Reduced Model in Deformation Space 100

      3.2.3 Beam Model Reduction: Joint Releases and Length Invariance 104

3.3 Curved Shallow Beam and Non-locking FE Interpolations 116

      3.3.1 Two-Dimensional Curved Shallow Beam: Linear Kinematics 116

      3.3.2 Non-locking Finite Element Interpolation for Shallow Beam 119

      3.3.3 Illustrative Numerical Examples and Closing Remarks 127

4 Plate Models: Validation and Verification 132

4.1 Finite Elements for Analysis of Thick and Thin Plates 132

      4.1.1 Motivation: Timoshenko Beam Element Linked Interpolations 133

      4.1.2 Reissner-Mindlin Plate Model and FE Discretization 135

      4.1.3 Illustrative Numerical Examples and Closing Remarks 145

4.2 Discrete Kirchhoff Plate Element Extension with Incompatible Modes 150

      4.2.1 Reissner-Mindlin Plate Model and Enhanced FE Interpolations 151

      4.2.2 Illustrative Numerical Examples and Closing Remarks 154

4.3 Validation or Model Adaptivity for Thick or Thin Plates Based on Equilibrated Boundary Stress Resultants 157

      4.3.1 Thick and Thin Plate Finite Element Models 159

      4.3.2 Model Adaptivity for Plates 167

      4.3.3 Illustrative Numerical Examples and Closing Remarks 176

4.4 Verification or Discrete Approximation Adaptivity for Discrete Kirchhoff Plate Finite Element 186

      4.4.1 Kirchhoff Plate Bending Model 192

      4.4.2 Kirchhoff Plate Finite Elements 196

      4.4.3 Error Estimates for Kirchhoff Plate Elements Based Upon Equilibrated Boundary Stress Resultants 203

      4.4.4 Implementation of Equilibrated Element Boundary Tractions Method For DKT Plate Element 207

      4.4.5 Examples on Error Indicators Comparison and Closing Remarks 212

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures 224

5.1 Solids with Drilling Rotations: Variational Formulation 224

      5.1.1 Strong Form of the Boundary Value Problem 226

      5.1.2 Variational Formulation, Stability Analysis and Regularization 228

      5.1.3 Alternative Variational Formulations, Extension to Nonlinear Kinematics and Closing Remarks 236

5.2 Membranes with Drilling Rotations: Discrete Approximation 243

      5.2.1 Discrete Approximations with Quadratic and Cubic Displacement Fields 244

      5.2.2 Illustrative Numerical Examples and Closing Remarks 255

5.3 Shells with Drilling Rotations: Linearized Kinematics 260

      5.3.1 Geometrically Linear Shallow Shell Theory 261

      5.3.2 Incompatible Modes Based Finite Element Approximation 268

      5.3.3 Illustrative Numerical Examples and Closing Remarks 274

6 Large Displacements and Instability: Buckling Versus Nonlinear Instability 286

6.1 Large Displacements and Deformations in 1D Truss with Instabilities 286

      6.1.1 Large Strain Measures for 1D Truss 286

      6.1.2 Strong and Weak Forms for 1D Truss in Large Displacements 290

      6.1.3 Linear Elastic Behavior for 1D Truss in Large Displacements 292

      6.1.4 Finite Element Method for 1D Truss in Large Displacements 295

      6.1.5 Buckling, Nonlinear Instability and Detection Criteria 301

6.2 Geometrically Nonlinear Curved Beam and Nonlinear Instability 311

      6.2.1 Curved Reissner's Beam: Nonlinear Kinematics 312

      6.2.2 Finite Element Implementation for Curved Reissner's Beam and Comment on Objectivity 317

      6.2.3 Control of Nonlinear Instability 328

6.3 Buckling of (Heterogeneous) Euler's Beam 334

      6.3.1 Euler Instability Problem: Two Alternative Formulations 336

      6.3.2 Analytic Solution Based on Strong Form 340

      6.3.3 Numerical Solution Based on Reduced Models with Finite Element Method 342

6.4 Buckling Analysis of Complex Structures with Refined Models of Plates and Shells 348

      6.4.1 Buckling Problems for Plates and Shells 349

      6.4.2 Finite Element Shell Approximation Including Drilling Rotations 353

      6.4.3 Illustrative Numerical Examples and Closing Remarks 358

6.5 Buckling Problems for Coupled Thermomechanical Extreme Conditions 365

      6.5.1 Linear Thermoelasticity 1D 367

      6.5.2 Linearized Instability for Thermoelastic Coupling 369

      6.5.3 General Linear Eigenvalue Problem Solution Procedure 373

      6.5.4 Thermomechanical Coupling Model: Illustrative and Validation Examples 376

      6.5.5 Brief on Instability for Thermomechanical Coupling in 1D Finite Elasticity 381

7 Inelasticity: Ultimate Load and Localized Failure 386

7.1 Stress Resultants Finite Element Model for Reinforced-Concrete Plates 386

      7.1.1 Plate Element for Reinforced-Concrete Slabs 387

      7.1.2 Stress-Resultants Constitutive Model for Reinforced-Concrete Plates 391

      7.1.3 Illustrative Numerical Examples and Closing Remarks 399

7.2 Stress Resultants Plasticity for Metallic Plates 406

      7.2.1 Variational Formulation and Discrete Approximation for Metallic Plates 407

      7.2.2 Stress Resultants Plasticity Formulation for Metallic Plates 413

      7.2.3 Illustrative Numerical Examples and Closing Remarks 419

7.3 Plasticity Criterion with Thermomechanical Coupling in Folded Plates and Non-smooth Shells 425

      7.3.1 Theoretical Formulation of Shell Model for Folded Plates and Non-smooth Shells 427

      7.3.2 Finite Element Implementation with Shell Element 430

      7.3.3 Stress Resultants Constitutive Model of Saint-Venant Plasticity 433

      7.3.4 Thermomechanical Coupling 436

      7.3.5 Operator Split Solution Procedure with Variable Time Steps 439

      7.3.6 Illustrative Numerical Examples and Closing Remarks 442

7.4 Stress Resultants Plasticity and Localized Failure of Reissner's Beam 450

      7.4.1 Reissner's Beam with Localized Elastoplastic Behavior 451

      7.4.2 Stress Resultant Plasticity Discrete Approximations and Computations 458

      7.4.3 Illustrative Numerical Examples and Closing Remarks 466

8 Brief on Mulitscale, Dynamics and Probability 470

8.1 Mulitscale Approach to Quasi-brittle Fracture in Dynamics 470

      8.1.1 Geometrically Exact Shear Deformable Beam as Cohesive Link 472

      8.1.2 Micro and Macro Constitutive Models for Dynamic Fracture 475

      8.1.3 Dynamics of Lattice Network and Time-Stepping Schemes 480

      8.1.4 Illustrative Numerical Examples and Closing Remarks 483

8.2 Stochastic Upscaling, Size Effect and Damping Replacement 487

      8.2.1 Stochastic Upscaling in Localized Failure 488

      8.2.2 Probability-Based Size Effect in Ductile Failure 500

      8.2.3 Damping Model Replacement of Rayleigh Damping 506

8.3 Reduced Stochastic Models for Euler Beam Dynamic Instability 509

      8.3.1 Duffing Oscillator: Reduced Model for Euler Instability 510

      8.3.2 Instability Studies in Dynamics Framework 513

      8.3.3 Stochastic Solution to Euler Instability Problem 520

References 530

封底 546

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