Direct Methods for Limit and Shakedown Analysis of Structures (eBook)

Advanced Computational Algorithms and Material Modelling

Paolo Fuschi, Aurora Angela Pisano, Dieter Weichert (Herausgeber)

eBook Download: PDF

2015 | 2015
XII, 313 Seiten
Springer International Publishing (Verlag)
978-3-319-12928-0 (ISBN)

Infos

Articles in this book examine various materials and how to determine directly the limit state of a structure, in the sense of limit analysis and shakedown analysis. Apart from classical applications in mechanical and civil engineering contexts, the book reports on the emerging field of material design beyond the elastic limit, which has further industrial design and technological applications. Readers will discover that 'Direct Methods' and the techniques presented here can in fact be used to numerically estimate the strength of structured materials such as composites or nano-materials, which represent fruitful fields of future applications.

Leading researchers outline the latest computational tools and optimization techniques and explore the possibility of obtaining information on the limit state of a structure whose post-elastic loading path and constitutive behavior are not well defined or well known. Readers will discover how Direct Methods allow rapid and direct access to requested information in mathematically constructive manners without cumbersome step-by-step computation.

Both researchers already interested or involved in the field and practical engineers who want to have a panorama of modern methods for structural safety assessment will find this book valuable. It provides the reader with the latest developments and a significant amount of references on the topic.

Inhalt

Foreword 5
Preface 8
Contents 9
A Stress-Based Variational Model for Ductile Porous Materials and Its Extension Accounting for Lode Angle Effects 11
1 Introduction 12
2 Formulation of the Statical Limit Analysis for Porous Materials 13
2.1 Associated Plasticity and Superpotentials 14
2.2 Stress-Based Variational Approach: Application of the Statical Limit Analysis to Porous Materials 15
3 Stress-Based Variational Approach of Ductile Porous Materials in Axisymmetric Loading Case [4] 18
3.1 Proposed Axisymmetric Trial Stress Field 19
3.2 Macroscopic Yield Criterion Under Axisymmetric Loading 20
3.3 Flow Rule and Porosity Evolution Under Axisymmetric Loading 22
4 Extension to a Load Depended Stress Variational Model 23
4.1 Proposed Non-axisymmetric Trial Stress Field 23
4.2 Macroscopic Yield Criterion Under Non-axisymmetric Loading 24
4.3 Macroscopic Flow Rule Under Non-axisymmetric Loading 27
4.4 Illustration of the Macroscopic Yield Criterion and Void Growth Under Non-axisymmetric Loading 28
5 Conclusion 30
6 Appendix: Illustration and Validation for the Macroscopic Model Under Axisymmetric Loading 30
References 34
Limit Analysis and Macroscopic Strength of Porous Materials with Coulomb Matrix 36
1 Introduction 36
2 The Hollow Spheroid Model 38
3 LA Methods and Coulomb Criterion 40
3.1 The Static Method 40
3.2 The Mixed Kinematic Method 41
3.3 The Coulomb Material 41
4 Numerical Implementation of the Axisymmetric Static Method 42
4.1 The Stress Field 42
4.2 The SA Conditions 43
4.3 The PA Conditions 43
4.4 The Post-analysis Process 44
5 Numerical Implementation of the Axisymmetric Mixed Method 44
5.1 The Virtual Velocity Field 45
5.2 Formulation of the Mixed Kinematic Method 45
6 Application to Sphere and Spheroid Models 48
6.1 Comparison with Exact Results for Spherical Voids 48
6.2 Numerical Results for Axisymmetric Loadings in the Spherical Void Case 49
6.3 Porous Coulomb Material with Oblate Voids 50
7 Conclusion 54
References 54
A Direct Method Formulation for Topology Plastic Design of Continua 56
1 Introduction 56
2 The Static Method of Limit Analysis 58
3 Finite Element Formulation of the Static Problem 59
4 Topology Optimization of Continua 60
5 Topology Optimization Problem Formulation 61
5.1 The Continuous Design Problem Formulation 62
5.2 The Discrete Problem Formulation 64
6 Numerical Examples 64
6.1 The Long Cantilever Beam 65
6.2 Medium Cantilever Beam 68
6.3 The MBB Beam 70
7 Conclusion 71
References 71
The Influence of Limited Kinematical Hardening on Shakedown of Materials and Structures 73
1 Introduction 73
2 Lower Bound Shakedown Analysis Accounting for Limited Kinematical Hardening 75
2.1 Melan's Statical Shakedown Theorem for Elastic-Perfectly Plastic Materials 75
2.2 Two Surface Model for Limited Linear Kinematical Hardening by Weichert and Groß-Weege (WGW) 76
2.3 Extension of WGW-Model for Limited General Nonlinear Kinematical Hardening by Pham 77
2.4 Overlay-Model for Limited Nonlinear Kinematical Hardening by Stein et al. 78
2.5 Description of the Loading Domain 78
2.6 Discretization 79
2.7 Resulting Nonlinear Optimization Problem 80
3 Effect of the Underlying Kinematical Hardening Law 81
4 Sample Under Constant Tension and Alternating Torsion 82
5 Conclusions 86
References 87
Theoretical Basis and a Finite Element Formula for the Direct Calculation of Steady Plastic States 89
1 Introduction 89
2 Governing Equations for a Plastic Body Under Cyclic Loading 92
3 Material Behaviour Description 94
4 Cyclic Limit State Theorems 96
4.1 Individual Theorems for the Elastic Shakedown Boundaries 97
4.2 Individual Theorems for Inelastic Shakedown Boundaries 99
5 Formula for a Steady Cyclic State Under Prescribed Loading 101
6 Direct Finite Element Computations of a Steady Cycle 102
6.1 Temporal and Spatial Discretization 102
6.2 Simple Computational Approach 105
6.3 Test Problem 106
7 Conclusions 109
References 110
On the Statistical Determination of Yield Strength, Ultimate Strength, and Endurance Limit of a Particle Reinforced Metal Matrix Composite (PRMMC) 112
1 Introduction 112
2 Limit and Shakedown Analysis of Random PRMMC Material 114
2.1 Micromechanical Homogenization of Elasto-Plastic Materials 114
2.2 Static Theorem and Its Numerical Reformulation 115
2.3 Solving the Quadratically Constrained Programming (QCP) by Primal-Dual Interior Point Method 120
3 Numerical Results 121
3.1 Finite Element Models for Statistical Analysis 121
3.2 Comparative Study Between Plane Stress, Plane Strain and 2.5D Models 123
3.3 Statistical Analysis and Study on the Size Effect 125
4 Conclusions 127
References 127
A New Starting Point Strategy for Shakedown Analysis 130
1 Introduction 131
2 Shakedown Analysis Under von Mises Yield Criterion 132
3 Starting Point Strategy 134
3.1 The Three Problem Types 135
3.2 The New Starting Point Strategy 136
3.3 Initial Values for the Lagrange Multipliers 138
4 Numerical Examples 139
4.1 Square Plate with Central Hole 139
5 Conclusion 145
References 146
Yield Design of Periodically Heterogeneous Plates 149
1 Introduction 149
2 Homogenization Theory in Yield Design for Periodic Plates 150
2.1 Initial Heterogeneous Yield Design Problem 150
2.2 Homogeneous Yield Design Plate Problem 151
2.3 Definition of the Macroscopic Strength Criterion Form the Solution of a Yield Design Auxiliary Problem 152
3 Numerical Resolution of the Auxiliary Problem 153
3.1 Finite Element Discretization of the Unit Cell 153
3.2 Fulfillment of the Strength Criterion Using Conic Constraints 154
3.3 Formulation of the Optimization Problem 156
4 Illustrative Applications 156
4.1 Validating Example 156
4.2 Fire Resistance of Reinforced Concrete Slabs 159
5 Conclusions 163
References 164
RSDM-S: A Method for the Evaluation of the Shakedown Load of Elastoplastic Structures 165
1 Introduction 165
2 Cyclic Elastoplastic States 167
3 The Residual Stress Decomposition Method for Shakedown Analysis (RSDM-S) 169
3.1 Initial Load Factor 171
3.2 Development of the Procedure 171
3.3 Numerical Strategy 174
4 Application Examples 174
4.1 Plate with a Central Hole 175
4.2 Grooved Rectangular Plate Under Varying Tension and Bending 177
5 Concluding Remarks 179
References 180
An Efficient Algorithm for Shakedown Analysis Based on Equality Constrained Sequential Quadratic Programming 182
1 Introduction 182
2 A Mathematical Programming Formulation for the Shakedown Analysis 184
2.1 The FEM Discrete Equation for the Static Shakedown Theorem 184
3 Shakedown Analysis Using Dual Decomposition Methods 187
3.1 The Proximal Point Method and the Pseudo-elastoplastic Step 187
3.2 The Dual Decomposition Solution of the Pseudo Elastoplastic Step 189
3.3 The Optimization Point of View and the Motivation for a New Strategy 191
4 A New Solution Scheme for the Pseudo Elastoplastic Step 191
4.1 The Linearized Equations for the Elastoplastic Step and the Sequential Quadratic Programming (SQP) Formulation 191
4.2 The EC-SQP Formulation 192
4.3 A Final Remark 196
5 Numerical Results 196
5.1 Description of the Test Problems 197
6 Conclusions 201
References 201
Limit Analysis on RC-Structures by a Multi-yield-criteria Numerical Approach 203
1 Introduction 203
2 Limit Analysis: Basic Concepts and Numerical Issues 205
2.1 Upper Bound Evaluation via LMM 206
2.2 Lower Bound Evaluation via ECM 208
3 Yield-Criteria for RC Structures 210
4 Three-Yield-Criteria LMM 212
5 Three-Yield-Criteria ECM 215
6 Applications 217
7 Concluding Remarks 221
References 222
Shakedown Analysis Within the Framework of Strain Gradient Plasticity 224
1 Introduction 224
2 The Constitutive Model 226
2.1 Thermodynamics Basis 227
2.2 A Choice for the Strengthening Potential 229
2.3 Plasticity Evolution Laws 231
3 The Shakedown Problem 232
4 Melan-Type Static Shakedown Theorem 234
5 Koiter-Type Kinematic Shakedown Theorem 238
6 Lower Bound and Upper Bound Theorems 241
6.1 Extended Lower Bound Theorem 241
6.2 Extended Upper Bound Theorem 242
6.3 The Shakedown Limit Load Problem 242
7 Application 245
7.1 Small ? Values: Alternating Plasticity 246
7.2 Higher ? Values: Ratchetting 249
7.3 Results and Comments 251
8 Conclusions 252
References 252
Shakedown Analysis of 3D Frames with an Effective Treatment of the Load Combinations 256
1 Introduction 256
2 The 3D Beam Model 259
2.1 Beam Kinematics and Statics 259
2.2 The Finite Element for the Beam 260
3 Shakedown Analysis Based on the Proximal Point Method and Dual Decomposition 262
3.1 The Envelope of Elastic Stresses 262
3.2 The Plastic Admissibility Conditions for Shakedown 262
3.3 The Pseudo-elastoplastic Step for Shakedown Analysis 263
3.4 The Dual Decomposition Solution of the Pseudo Elastoplastic Step 265
4 The Elastic Domain of the Beam Section 266
4.1 Evaluation of the Support Functions of the Beam Elastic Domain 266
4.2 The Approximation of mathbbE Using a Minkowski Sum of Ellipsoids 268
4.3 Simplified Evaluation of the Elastic Envelope 270
4.4 The Closest Point Projection Problem in Terms of Minkowski Sums of Ellipsoids 272
5 Numerical Results 274
5.1 Yield Function and Elastic Envelope Vertexes for an L-Shaped Section 274
5.2 Shakedown Analysis of a Continuous Beam 276
5.3 Shakedown Analysis of a 3D Frame 277
6 Conclusions 278
References 279
Uncertainty Multimode Failure and Shakedown Analysis of Shells 281
1 Introduction 281
2 Deterministic Shakedown Analysis of Shells 282
2.1 Governing Equations 282
2.2 Upper Bound Shakedown Analysis 284
3 Probabilistic Formulation 285
3.1 Definition of the Limit State Function 286
3.2 First- and Second-Order Reliability Methods 286
4 Multimode Failure 288
4.1 Bounds for the System Probability of Failure 289
4.2 First-Order System Reliability Analysis 290
4.3 Calculation of the Multiple Design Points 292
5 Numerical Applications 293
5.1 Plastic Collapse of a Pipe-Junction Subjected to Internal Pressure and Bending Moment 293
5.2 Shakedown of a Pipe-Junction Subjected to Internal Pressure and Bending Moment 297
6 Conclusions 299
References 299
Three-Dimensional Shakedown Solutions for Cross-Anisotropic Cohesive-Frictional Materials Under Moving Loads 301
1 Introduction 301
2 Problem Definition 303
3 Melan's Lower-Bound Theorem 305
4 Shakedown Analysis 305
5 Numerical Technique and Results 307
5.1 Homogenous Materials 307
5.2 Layered Materials 310
6 Conclusions 314
References 314

Produkt-Details

• Erscheint lt. Verlag: 6.1.2015
• Reihe/Serie: Solid Mechanics and Its Applications
• Reihe/Serie: Solid Mechanics and Its Applications
• Zusatzinfo: XII, 313 p. 102 illus., 66 illus. in color.
• Verlagsort: Cham
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Maschinenbau
• Schlagworte: Direct Methods • limit states • Numerical Algorithms • Optimization Methods • Plastic Design
• ISBN-10: 3-319-12928-7 / 3319129287
• ISBN-13: 978-3-319-12928-0 / 9783319129280