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  • ervenka Consulting s.r.o. Na Hrebenkach 55 150 00 Prague Czech Republic Phone: +420 220 610 018 E-mail: cervenka@cervenka.cz Web: http://www.cervenka.cz

    ATENA Program Documentation Part 1 Theory Written by

    Vladimr ervenka, Libor Jendele,

    and Jan ervenka

    Prague, December 6, 2016

    mailto:cervenka@cervenka.czhttp://www.cervenka.cz/

  • Ackowledgements:

    The software was developed with partial support of Eurostars funding program.

    The following models were developed with the financial support of TA R: CC3DNonLinCementitious2HPRFC, CC3DNonLinCementitious2FRC, CC3DNonLinCementitious2SHCC, CCModelGeneral, CCTransportMaterial

    Trademarks:

    ATENA is registered trademark of Vladimir Cervenka.

    Other names may be trademarks of their respective owners.

    Copyright 2000-2016 ervenka Consulting s.r.o.

  • ATENA Theory i

    Contents 1 CONTINUUM GOVERNING EQUATIONS 1

    1.1 Introduction 1

    1.2 General Problem Formulation 2

    1.3 Stress Tensors 4

    1.3.1 Cauchy Stress Tensor 4

    1.3.2 2nd Piola-Kirchhoff Stress Tensor 4

    1.4 Strain Tensors 5

    1.4.1 Engineering Strain 5

    1.4.2 Green-Lagrange Strain 5

    1.5 Constitutive Tensor 6

    1.6 The Principle of Virtual Displacements 6

    1.7 The Work Done by the External Forces 8

    1.8 Problem Discretisation Using Finite Element Method 9

    1.9 Stress and Strain Smoothing 10

    1.9.1 Extrapolation of Stress and Strain to Element Nodes 11

    1.10 Simple, Complex Supports and Master-Slave Boundary Conditions. 12

    1.11 References 13

    2 CONSTITUTIVE MODELS 15

    2.1 Constitutive Model SBETA (CCSbetaMaterial) 15

    2.1.1 Basic Assumptions 15

    2.1.2 Stress-Strain Relations for Concrete 18

    2.1.3 Localization Limiters 24

    2.1.4 Fracture Process, Crack Width 25

    2.1.5 Biaxial Stress Failure Criterion of Concrete 25

    2.1.6 Two Models of Smeared Cracks 27

    2.1.7 Shear Stress and Stiffness in Cracked Concrete 29

    2.1.8 Compressive Strength of Cracked Concrete 29

    2.1.9 Tension Stiffening in Cracked Concrete 30

    2.1.10 Summary of Stresses in SBETA Constitutive Model 30

  • ii

    2.1.11 Material Stiffness Matrices 31

    2.1.12 Analysis of Stresses 32

    2.1.13 Parameters of Constitutive Model 33

    2.2 FracturePlastic Constitutive Model (CC3DCementitious, CC3DNonLinCementitious, CC3DNonLinCementitious2, CC3DNonLinCementitious2User, CC3DNonLinCementitious2Variable, CC3DNonLinCementitious2SHCC, CC3DNonLinCementitious3) 34

    2.2.1 Introduction 34

    2.2.2 Material Model Formulation 35

    2.2.3 Rankine-Fracturing Model for Concrete Cracking 35

    2.2.4 Plasticity Model for Concrete Crushing 38

    2.2.5 Combination of Plasticity and Fracture model 41

    2.2.6 Variants of the Fracture Plastic Model 43

    2.2.7 Tension Stiffening 47

    2.2.8 Crack Spacing 47

    2.2.9 Fixed or Rotated Cracks 48

    2.2.10 Fatigue 48

    2.2.11 Strain Hardening Cementitious Composite (SHCC, HPFRCC) material 52

    2.2.12 Confinement-Sensitive Constitutive Model 54

    2.3 Von Mises Plasticity Model 58

    2.4 Drucker-Prager Plasticity Model 61

    2.5 User Material Model 62

    2.6 Interface Material Model 62

    2.7 Reinforcement Stress-Strain Laws 66

    2.7.1 Introduction 66

    2.7.2 Bilinear Law 66

    2.7.3 Multi-line Law 67

    2.7.4 No Compression Reinforcement 68

    2.7.5 Cyclic Reinforcement Model 68

    2.8 Reinforcement Bond Models 69

    2.8.1 CEB-FIP 1990 Model Code 70

    2.8.2 Bond Model by Bigaj 71

    2.8.3 Memory Bond Material 73

  • ATENA Theory iii

    2.9 Microplane Material Model (CCMicroplane4) 73

    2.9.1 Equivalent Localization Element 74

    2.10 References 78

    3 FINITE ELEMENTS 83

    3.1 Introduction 83

    3.2 Truss 2D and 3D Element 85

    3.3 Plane Quadrilateral Elements 89

    3.4 Plane Triangular Elements 95

    3.5 3D Solid Elements 97

    3.6 Spring Element 108

    3.7 Quadrilateral Element Q10 110

    3.7.1 Element Stiffness Matrix 110

    3.7.2 Evaluation of Stresses and Resisting Forces 113

    3.8 External Cable 115

    3.9 Reinforcement Bars with Prescribed Bond 116

    3.10 Interface Element 121

    3.11 Truss Axi-Symmetric Elements. 124

    3.12 Ahmad Shell Element 126

    3.12.1 Coordinate Systems. 128

    3.12.2 Geometry Approximation 133

    3.12.3 Displacement Field Approximation. 134

    3.12.4 Strain and Stresses Definition. 135

    3.12.5 Serendipity, Lagrangian and Heterosis Variant of Degenerated Shell Element. 136

    3.12.6 Smeared Reinforcement 141

    3.12.7 Transformation of the Original DOFs to Displacements at the Top and Bottom of the Element Nodal Coordinate System 141

    3.12.8 Shell Ahmad Elements Implemented in ATENA 145

    3.13 Curvilinear Nonlinear 2D Isoparametric Layered Shell Quadrilateral Elements 146

    3.13.1 Geometry and displacements 147

    3.13.2 Connection of the shell2D to an ambient solid element 149

  • iv

    3.13.3 Green-Lagrange strains 152

    3.14 Curvilinear Nonlinear 2D Isoparametric Layered Shell Triangular Elements 157

    3.15 Curvilinear Nonlinear 3D Isoparametric Layered Shell Hexahedral Elements 158

    3.15.1 Geometry and displacements 160

    3.15.2 Green-Lagrange strains 161

    3.16 Curvilinear Nonlinear 3D Isoparametric Layered Shell Wedge Elements 167

    3.17 Curvilinear Nonlinear 3D Beam Element 168

    3.17.1 Geometry and Displacements and Rotations Fields 168

    3.17.2 Strain and Stress Definition 171

    3.17.3 Matrices Used in the Beam Element Formulation 171

    3.17.4 The Element Integration 178

    3.18 Curvilinear Nonlinear 3D Isoparametric Beam Element 180

    3.19 Curvilinear Nonlinear 1D element 181

    3.19.1 Connection of the beam1D to an ambient solid element 182

    3.20 Integrated forces and moments for shells 184

    3.21 Integrated forces and moments for beams 186

    3.22 Global and Local Coordinate Systems for Element Load 186

    3.23 References 189

    4 SOLUTION OF NONLINEAR EQUATIONS 191

    4.1 Linear Solvers 191

    4.1.1 Direct Solver 192

    4.1.2 Direct Sparse Solver 193

    4.1.3 Iterative Solver 193

    4.1.4 Parallel Direct Sparse Solver PARDISO 197

    4.2 Full Newton-Raphson Method 199

    4.3 Modified Newton-Raphson Method 201

    4.4 Arc-Length Method 202

    4.4.1 Normal Update Method 205

    4.4.2 Consistently Linearized Method 205

    4.4.3 Explicit Orthogonal Method 206

  • ATENA Theory v

    4.4.4 The Crisfield Method. 207

    4.4.5 Arc Length Step 208

    4.5 Line Search Method 208

    4.6 Parameter 209

    4.7 Band Width Optimization 211

    4.8 References 214

    5 CREEP AND SHRINKAGE ANALYSIS 217

    5.1 Implementation of Creep and Shrinkage Analysis in ATENA 217

    5.1.1 Basic Theoretical Assumptions 217

    5.2 Approximation of Compliance Functions ( , ')t t by Dirichlet Series. 219

    5.3 Step by Step Method 220

    5.4 Integration and Retardation Times 221

    5.5 Creep and Shrinkage Constitutive Model 223

    5.6 References 232

    6 DURABILITY ANALYSIS 235

    6.1 Carbonation 236

    6.1.1 Example of Carbonation 237

    6.2 Chlorides 237

    6.3 Diffusion coefficient for chlorides 239

    6.4 MODELS for PROPAGATION PHASE 241

    6.4.1 Carbonation during propagation phase 241

    6.4.2 Chloride ingress during propagation phase 242

    6.4.3 Cracking of concrete cover 242

    6.4.4 Spalling of concrete cover 242

    6.5 Alkali-Aggregate Reaction 243

    6.5.1 Introduction of alkali-aggregatea model for concrete 243

    6.5.2 Model for ASR kinetics 245

    6.5.3 Prediction of ASR swelling 248

  • vi

    6.5.4 Influence of moisture FM 249

    6.5.5 ASR for 3D conditiions 250

    6.5.6 2. Validation on free expansion 251

    6.5.7 3. Validation on uniaxially confined specimen 253

    6.5.8 4. Comments 254

    6.6 References 254

    7 TRANSPORT ANALYSIS 257

    7.1 Numerical Solution of the Transport Problem Spatial Discretisation 260

    7.2 Numerical Solution of the Transport Problem Temporal Discretisation 266

    7.2.1 -parameter Crank Nicholson Scheme 267 7.2.2 Adams-Bashforth Integration Scheme 267

    7.2.3 Reduction of Oscillations and Convergence Improvement 268

    7.3 Material Constitutive Model 268