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Atena TheoryTRANSCRIPT

Cervenka Consulting Ltd. 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, August 24, 2009

iii

CONTENTS

1 CONTINUUM GOVERNING EQUATIONS 9

1.1 Introduction 9

1.2 General Problem Formulation 10

1.3 Stress Tensors 12

1.3.1 Cauchy Stress Tensor 12

1.3.2 2nd Piola-Kirchhoff Stress Tensor 12

1.4 Strain Tensors 14

1.4.1 Engineering Strain 14

1.4.2 Green-Lagrange Strain 14

1.5 Constitutive tensor 14

1.6 The Principle of Virtual Displacements 15

1.7 The Work Done by the External Forces 17

1.8 Problem Discretisation Using Finite Element Method 17

1.9 Stress and strain smoothing 19

1.9.1 Extrapolation of stress and strain to element nodes 19

1.10 Simple, complex supports and master-slave boundary conditions. 21

1.11 References 22

2 CONSTITUTIVE MODELS 23

2.1 Constitutive Model SBETA (CCSbetaMaterial) 23

2.1.1 Basic Assumptions 23

2.1.2 Stress-Strain Relations for Concrete 26

2.1.3 Localization Limiters 32

2.1.4 Fracture Process, Crack Width 33

2.1.5 Biaxial Stress Failure Criterion of Concrete 33

2.1.6 Two Models of Smeared Cracks 35

iv

2.1.7 Shear Stress and Stiffness in Cracked Concrete 37

2.1.8 Compressive Strength of Cracked Concrete 37

2.1.9 Tension Stiffening in Cracked Concrete 38

2.1.10 Summary of Stresses in SBETA Constitutive Model 38

2.1.11 Material Stiffness Matrices 39

2.1.12 Analysis of Stresses 41

2.1.13 Parameters of Constitutive Model 41

2.2 FracturePlastic Constitutive Model (CC3DCementitious, CC3DNonLinCementitious, CC3DNonLinCementitious2, CC3DNonLinCementitious2User, CC3DNonLinCementitious2Variable, CC3DNonLinCementitious2SHCC, CC3DNonLinCementitious3) 43

2.2.1 Introduction 43

2.2.2 Material Model Formulation 44

2.2.3 Rankine-Fracturing Model for Concrete Cracking 44

2.2.4 Plasticity Model for Concrete Crushing 46

2.2.5 Combination of Plasticity and Fracture model 50

2.2.6 Variants of the fracture plastic model 53

2.2.7 Tension stiffening 56

2.2.8 Crack spacing 56

2.2.9 Fatigue 57

2.2.10 Strain Hardening Cementitious Composite (SHCC, HPFRCC) material 60

2.2.11 Confinement-sensitive constitutive model 62

2.3 Von Mises Plasticity Model 66

2.4 Drucker-Prager Plasticity Model 69

2.5 User Material Model 70

2.6 Interface Material Model 70

2.7 Reinforcement Stress-Strain Laws 74

2.7.1 Introduction 74

v

2.7.2 Bilinear Law 74

2.7.3 Multi-line Law 75

2.7.4 No Compression Reinforcement 76

2.7.5 Cycling Reinforcement Model 76

2.8 Reinforcement bond models 77

2.8.1 CEB-FIP 1990 Model Code 78

2.8.2 Bond Model by Bigaj 79

2.9 Microplane material model (CCMicroplane4) 81

2.9.1 Equivalent localization element 81

2.10 References 85

3 FINITE ELEMENTS 89

3.1 Introduction 89

3.2 Truss 2D and 3D Element 91

3.3 Plane Quadrilateral Elements 95

3.4 Plane Triangular Elements 102

3.5 3D Solid Elements 103

3.6 Spring Element 114

3.7 Quadrilateral Element Q10 116

3.7.1 Element Stiffness Matrix 116

3.7.2 Evaluation of Stresses and Resisting Forces 119

3.8 External Cable 120

3.9 Reinforcement Bars with Prescribed Bond 121

3.10 Interface Element 123

3.11 Truss Axi-Symmetric Elements. 126

3.12 Ahmad Shell Element 128

3.12.1 Coordinate systems. 130

3.12.2 Geometry approximation 135

vi

3.12.3 Displacement field approximation. 136

3.12.4 Strain and stresses definition. 137

3.12.5 Serendipity, Lagrangian and Heterosis variant of degenerated shell element. 138

3.12.6 Smeared Reinforcement 144

3.12.7 Transformation of the original DOFs to displacements at the top and bottom of the element nodal coordinate system 144

3.12.8 Shell Ahmad elements implemented in ATENA 148

3.13 Curvilinear nonlinear 3D beam element. 149

3.13.1 Geometry and displacements and rotations fields. 149

3.13.2 Strain and stress definition 153

3.13.3 Matrices used in the beam element formulation 154

3.13.4 The element integration 160

3.14 Global and local coordinate systems for element load 162

3.15 References 165

4 SOLUTION OF NONLINEAR EQUATIONS 166

4.1 Linear Solvers 166

4.1.1 Direct Solver 167

4.1.2 Direct Sparse Solver 168

4.1.3 Iterative Solver 168

4.2 Full Newton-Raphson Method 172

4.3 Modified Newton-Raphson Method 173

4.4 Arc-Length Method 174

4.4.1 Normal Update Method 177

4.4.2 Consistently Linearized Method 178

4.4.3 Explicit Orthogonal Method 179

4.4.4 The Crisfield Method. 180

4.4.5 Arc Length Step 181

vii

4.5 Line Search Method 181

4.6 Parameter 182 4.7 References 183

5 CREEP AND SHRINKAGE ANALYSIS 184

5.1 Implementation of creep and shrinkage analysis in ATENA 184

5.1.1 Basic theoretical assumptions 184

5.2 Approximation of compliance functions ( , ')t t by Dirichlet series. 186 5.3 Step by step method 187

5.4 Integration and retardation times 188

5.5 Creep and shrinkage constitutive model 190

5.6 References 195

6 TRANSPORT ANALYSIS 197

6.1 Numerical solution of the transport problem spatial discretisation 201

6.2 Numerical solution of the transport problem temporal discretisation 208

6.2.1 -parameter Crank Nicholson scheme 209 6.2.2 Adams-Bashforth integration scheme 209

6.2.3 Reduction of oscillations and convergence improvement 210

6.3 Material constitutive model 211

6.4 Fire element boundary load 214

6.4.1 Hydrocarbon fires 214

6.4.2 Fire exposed boundary 215

6.4.3 Implementation of fire exposed boundary in ATENA 216

6.5 References 217

7 DYNAMIC ANALYSIS 219

7.1 Structural damping 222

8 EIGENVALUES AND EIGENVECTORS ANALYSIS 224

8.1 Inverse subspace iteration 224

viii

8.1.1 Rayleigh-Ritz method 225

8.1.2 Jacobi method 225

8.1.3 Inverse iteration method 227

8.1.4 Algorithm of Inverse subspace iteration 227

8.1.5 Sturm sequence property check 230

8.2 References 230

9 GENERAL FORM OF DIRICHLET BOUNDARY CONDITIONS 231

9.1 Theory behind the implementation 231

9.1.1 Single CBC 232

9.1.2 Multiple CBCs 234

9.2 Application of Complex Boundary Conditions 238

9.2.1 Refinement of a finite element mesh 238

9.2.2 Mesh generation using sub-regions 239

9.2.3 Discrete reinforcement embedded in solid elements 240

9.2.4 Curvilinear nonlinear beam and shell elements 241

9.3 References 242

INDEX 243

9

1 CONTINUUM GOVERNING EQUATIONS

1.1 Introduction This chapter presents the general governing continuum equations for non-linear analysis. In general, there exist many variants of non-linear analysis depending on how many non-linear effects are accounted for. Hence, this chapter first introduces some basic terms and entities commonly used for structural non-linear analyses, and then it concentrates on formulations that are implemented in ATENA.

It is important to realize that the whole structure does not have to be analyzed using a full non-linear formulation. However, a simplified (or even linear) formulation can be used in many cases. It is a matter of engineering knowledge and practice to assess, whether the inaccuracies due to a simplified formulation are acceptable, or not.

The simplest formulation, i.e. linear formulation, is characterized by the following assumptions:

The constitutive equation is linear, i.e. the generalized form of Hook's law is used.

The geometric equation is linear that is, the quadratic terms are neglected. It means that during analysis we neglect change of shape and position of the structure.

Both loading and boundary conditions are conservative, i.e. they are constant throughout the whole analysis irrespective of the structural deformation, time etc.

Generally linear constitutive equations can be employed for a material, which is far from its failure point, usually up to 50% of its maximum strength. Of course, this depends on the type of material, e.g. rubber needs to be considered as a non-linear material earlier. But for usual civil engineering materials the previous assumption is satisfactory.

Geometric equations can be considered linear, if deflections of a structure are much smaller than its dimensions. This must be satisfied not only for the whole structure but also for its parts. Then the geometric equations for the loaded structure can then be written using the original (unloaded) geometry.

It is also important to realize that a linear solution is permissible only in the case of small strains. This is closely related to material property because if strains are high, the stresses are usually, although not necessarily, high as well.

Despite the fact that for the vast majority of structures linear simplifications are quite acceptable, there are structures when it is necessary to take in account some non-linear behavior. The resulting governing equations are then much more complicated, and normally they do not have a closed form solution. Consequently some non-linear iterative solution s