Papadrakakis / Stefanou / Papadopoulos Computational Methods in Stochastic Dynamics

1. Auflage 2011
ISBN: 978-90-481-9987-7
Verlag: Springer Netherlands
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 22, 342 Seiten

Reihe: Computational Methods in Applied Sciences

At the dawn of the 21st century, computational stochastic dynamics is an emerging research frontier. This book focuses on advanced computational methods and software tools which can highly assist in tackling complex problems in stochastic dynamic/seismic analysis and design of structures. The book is primarily intended for researchers and post-graduate students in the fields of computational mechanics and stochastic structural dynamics. Nevertheless, practice engineers as well could benefit from it as most code provisions tend to incorporate probabilistic concepts in the analysis and design of structures. The book addresses mathematical and numerical issues in stochastic structural dynamics and connects them to real-world applications. It consists of 16 chapters dealing with recent advances in a wide range of related topics (dynamic response variability and reliability of stochastic systems, risk assessment, stochastic simulation of earthquake ground motions, efficient solvers for the analysis of stochastic systems, dynamic stability, stochastic modelling of heterogeneous materials). Numerical examples demonstrating the significance of the proposed methods are presented in each chapter.

Autoren/Hrsg.

Papadrakakis, Manolis

Stefanou, George

Papadopoulos, Vissarion

Inhaltsverzeichnis

1;Preface;6
2;Contents;12
3;Model Reduction and Uncertainties in Structural Dynamics;14
3.1;1 Introduction;15
3.2;2 Deterministic Procedures to Reduce Models;16
3.2.1;2.1 General Remarks;16
3.2.2;2.2 Guyan Reduction;16
3.2.3;2.3 Component Mode Synthesis;17
3.2.3.1;2.3.1 Fixed Interface Craig-Bampton Method;17
3.2.3.2;2.3.2 Component Modes for Unconstrained Components;19
3.2.4;2.4 Meta-Models;19
3.3;3 Stochastic Analysis;20
3.3.1;3.1 Uncertainty Modeling;20
3.3.2;3.2 Simulation Techniques;21
3.4;4 Model Reduction Taking into Account Uncertainties;23
3.4.1;4.1 Introduction;23
3.4.2;4.2 Karhunen-Loève Expansion of the Substructure Matrices;23
3.4.2.1;4.2.1 Introduction;23
3.4.2.2;4.2.2 Calibration Set for the Structural Matrices;24
3.4.2.3;4.2.3 Calibration Set for the Matrix of Eigenvectors and Constraint Modes;24
3.4.2.4;4.2.4 Karhunen-Loève Representation of the Matrices;25
3.4.2.5;4.2.5 Assembly of the Global Matrices;25
3.4.3;4.3 Random Combination of Substructures;26
3.4.3.1;4.3.1 Stochastic Reduced Basis Approximation (SRBA);26
3.4.4;4.4 Linear Interpolation of Substructure Matrices;27
3.4.4.1;4.4.1 Generation of the Sample Pool;27
3.4.4.2;4.4.2 Approximation of Structural Matrices and Eigenvectors;28
3.5;5 Numerical Example;29
3.5.1;5.1 Problem Statement;29
3.5.2;5.2 Results;30
3.5.2.1;5.2.1 Karhunen-Loève Expansion of the Substructure Matrices;30
3.5.2.2;5.2.2 Random Combination of Substructures;31
3.5.2.3;5.2.3 Stochastic Reduced Basis Approximation (SRBA);31
3.5.2.4;5.2.4 Linear Interpolation of Substructure Matrices;33
3.5.2.5;5.2.5 Computational Efforts;33
3.6;6 Conclusions;34
3.7;References;35
4;Resonant Damping of Flexible Structures Under Random Excitation;38
4.1;1 Introduction;39
4.2;2 Resonant Response Format;40
4.2.1;2.1 Structure and Resonator Representation;41
4.2.2;2.2 Feedback Filters;42
4.3;3 Root Locus Diagram;43
4.4;4 Optimal Parameters;45
4.5;5 Multi-Degree-of-Freedom Systems;48
4.5.1;5.1 Background Mode Correction in MDOF Systems;48
4.5.2;5.2 MDOF System with Acceleration Feedback;50
4.6;6 Damping of Wind Excited Building;53
4.6.1;6.1 Design of Resonant Control;53
4.6.2;6.2 Root Locus Analysis;55
4.6.2.1;6.2.1 Response Analysis;56
4.7;7 Conclusions;58
4.8;References;59
5;Importance Sampling of Nonlinear Structures Using Adapted Process;60
5.1;1 Introduction;61
5.2;2 Importance Sampling Method;63
5.2.1;2.1 Distribution Shifted to a Fixed Point;65
5.2.2;2.2 Distribution Shifted to an Adapted Process;65
5.3;3 Stochastic Control Approach;66
5.3.1;3.1 Bellman's Equation;68
5.3.2;3.2 Double Barrier Problem;71
5.4;4 Numerical Investigation;72
5.4.1;4.1 Control Law;72
5.4.2;4.2 Variance Reduction;74
5.5;5 Conclusions;77
5.6;References;77
6;Use of Time-Variant Spectral Characteristics of Nonstationary Random Processes in the First-Passage Problem for Earthquake Engineering Applications;79
6.1;1 Introduction;80
6.2;2 Central Frequency and Bandwidth Parameters of Non-Stationary Stochastic Processes;80
6.3;3 Spectral Characteristics of the Stochastic Response of SDOF/MDOF Linear Systems Subjected to Non-Stationary Excitations;83
6.3.1;3.1 Complex Modal Analysis;83
6.3.2;3.2 NGSCs of Response Processes of MDOF Linear Systems Using Complex Modal Analysis;84
6.3.3;3.3 Response Statistics of MDOF Linear Systems Subjected to Modulated Coloured Noise;86
6.4;4 Failure Probability Approximations for the First-Passage Reliability Problem;88
6.5;5 Benchmark Application;90
6.5.1;5.1 Benchmark Linear MDOF System: Three-Story Shear-Type Building;90
6.5.2;5.2 Earthquake Base Excitation;91
6.5.3;5.3 Spectral Characteristics of the Response Processes for the Benchmark Structure;93
6.5.4;5.4 Failure Probability Estimates for the Benchmark Structure;95
6.6;6 Conclusions;98
6.7;References;99
7;Stochastic Seismic Analysis of Large Linear Structural Systems Under Fully Non-stationary Spectrum Compatible Ground Motion;101
7.1;1 Introduction;101
7.2;2 Spectrum Compatible Ground Motion Models;102
7.2.1;2.1 Quasi-Stationary Model;103
7.2.2;2.2 Fully Non-stationary Model;105
7.3;3 Deterministic Seismic Analysis;108
7.4;4 Stochastic Seismic Analysis;109
7.5;5 Modal Correction Method for Stochastic Response;112
7.6;6 Numerical Results;114
7.7;7 Concluding Remarks;119
7.8;References;120
8;Soil Spatial Variability and Structural Reliability of Buried Networks Subjected to Earthquakes;122
8.1;1 Introduction;123
8.2;2 Pipe Modeling;123
8.2.1;2.1 Pasternak Model;123
8.2.2;2.2 Stiffness Matrix;124
8.2.3;2.3 Joint-Connection Stiffness;125
8.3;3 Modeling the Spatial Variability of Soils and Its Effects: The Correlation Length;126
8.4;4 Equations of Motion;128
8.5;5 Response Surface Method;129
8.5.1;5.1 Principle of the Method;130
8.5.2;5.2 General Algorithm of the Proposed Method ;131
8.6;6 Performance Functions and Limit States;132
8.7;7 Case Study;133
8.7.1;7.1 Characteristics of the Sewer's Section;133
8.7.2;7.2 Effect of the Correlation Length;133
8.7.3;7.3 Effect of the Joint-Pipe Stiffness Ratio;135
8.7.4;7.4 Effect of the Soil-Pipe Stiffness Ratio;135
8.8;8 Conclusions;137
8.9;References;138
9;An Efficient First-Order Scheme for Reliability Based Optimization of Stochastic Systems;139
9.1;1 Introduction;139
9.2;2 Reliability-Based Optimization Problem;141
9.2.1;2.1 Formulation;141
9.2.2;2.2 Application to Dynamical Systems;142
9.3;3 Proposed Approach;143
9.3.1;3.1 General Description;143
9.3.2;3.2 Descent and Feasible Directions;143
9.3.3;3.3 Direction Evaluation;144
9.3.3.1;3.3.1 Interior Design;144
9.3.3.2;3.3.2 Active Non-Linear Constraints;145
9.3.3.3;3.3.3 Active Linear Constraints;147
9.3.4;3.4 One Dimensional Search;147
9.3.5;3.5 Algorithm Description;148
9.3.6;3.6 Algorithm Properties;149
9.4;4 Implementation Issues;150
9.4.1;4.1 Reliability Assessment;150
9.4.2;4.2 Gradient Estimation;150
9.4.2.1;4.2.1 Performance Function Approximation;150
9.4.2.2;4.2.2 Sensitivity of Failure Probability Function;151
9.4.2.3;4.2.3 Directional Sensitivity of Failure Probability Function;152
9.4.3;4.3 Line Search;153
9.4.3.1;4.3.1 Deterministic Constraints;153
9.4.3.2;4.3.2 Reliability Constraints;154
9.4.4;4.4 Variability of Reliability Estimates;155
9.5;5 Application Problem;156
9.6;6 Conclusions;160
9.7;References;161
10;Application of a Mode-based Meta-Model for the Reliability Assessment of Structures Subjected to Stochastic Ground Acceleration;164
10.1;1 Introduction;164
10.2;2 Meta-Model for the Structural Modes;166
10.2.1;2.1 General Aspects;166
10.2.2;2.2 Meta-Model for the Modal Quantities;167
10.3;3 Numerical Example;168
10.3.1;3.1 Model Description;168
10.3.1.1;3.1.1 Uncertainty Modeling;169
10.3.2;3.2 Dynamic Analysis of the Nominal System;170
10.3.2.1;3.2.1 Modal Analysis;170
10.3.2.2;3.2.2 Modal Contribution;170
10.3.3;3.3 Set-up of the Meta-Model;171
10.3.4;3.4 Reliability Assessment Using a Mode-based Meta-Model;171
10.3.4.1;3.4.1 Case 1: Deterministic Loading;171
10.3.4.2;3.4.2 Case 2: Stochastic Ground Acceleration;174
10.4;4 Conclusions;178
10.5;References;178
11;Nonlinear Dynamic Response Variability and Reliabilityof Frames with Stochastic Non-Gaussian Parameters;180
11.1;1 Introduction;181
11.2;2 Force-Based Formulation of the Beam-Column Element;182
11.3;3 Stochastic Stiffness Matrix;184
11.4;4 Simulation of Uncertain Parameters Using Non-Gaussian Translation Fields;184
11.5;5 Numerical Examples;186
11.5.1;5.1 Response Variability and Reliability of the Frame;188
11.6;6 Conclusions;192
11.7;References;193
12;The Role of Uncertainties in Aeolian Risk Assessment;195
12.1;1 Introduction;195
12.2;2 Sources of Uncertainty in Wind Engineering;196
12.3;3 A Procedure for Aeolian Risk Assessment;200
12.4;4 The Relevance of Uncertainties with Reference to a Case Example;201
12.4.1;4.1 The Epistemic Uncertainty of the Aerodynamic Coefficients;202
12.4.2;4.2 The Model Uncertainty of Aeroelastic Forces;203
12.4.3;4.3 The Influence of Model Uncertainty on Risk Assessment;211
12.5;5 Concluding Remarks;215
12.6;References;215
13;The Method of Separation: A Novel Approach for Accurate Estimation of Evolutionary Power Spectra;217
13.1;1 Introduction;217
13.2;2 Relevant Elements of Stochastic Process Theory;218
13.3;3 Existing Methods for Evolutionary Spectrum Estimation;221
13.3.1;3.1 The Short-Time Fourier Transform;221
13.3.2;3.2 The Harmonic Wavelet Transform;223
13.3.3;3.3 The Wigner--Ville Transform;223
13.4;4 The Method of Separation: Evolutionary Spectrum Estimation of Separable Random Fields;225
13.4.1;4.1 Theory and Derivation;225
13.4.2;4.2 Performance Test with Kanai--Tajimi Benchmark Estimation;227
13.5;5 Stochastic Modeling of Imperfections in Structures: Estimation of Strongly Narrow-Band Power Spectra;229
13.5.1;5.1 Spectral Separability;230
13.5.2;5.2 Number of Samples and Spectral Smoothing;231
13.5.3;5.3 The Uncertainty Principle;232
13.5.4;5.4 Performance of Available Estimation Techniques;232
13.6;6 Summary and Conclusions;235
13.7;References;235
14;A Seismologically Consistent Husid Envelope Functionfor the Stochastic Simulation of Earthquake Ground-Motions;237
14.1;1 Introduction;238
14.2;2 Mathematical Formulation of the Envelope Function;241
14.2.1;2.1 Strong-Motion Dataset;242
14.2.2;2.2 Calibration of the Model;243
14.3;3 Prediction Models for the HEF Parameters;244
14.4;4 Sensitivity Analysis;245
14.5;5 Envelope Variance;247
14.6;6 Results;248
14.7;7 Prediction of Peak Ground Acceleration;250
14.8;8 Summary and Conclusions;252
14.9;References;253
15;Sparse Representations in Stochastic Mechanics;255
15.1;1 Introduction;256
15.2;2 Model Problem;256
15.2.1;2.1 Deterministic Problem;257
15.2.2;2.2 Stochastic Problem;257
15.3;3 Discretisation of Random Fields;258
15.3.1;3.1 Karhunen-Loève Expansion;259
15.3.2;3.2 Singular Value Decomposition;259
15.3.3;3.3 Low-Rank Approximation;260
15.4;4 Solution Methods and Tensorial Character of Solution;260
15.4.1;4.1 Direct Integration;261
15.4.2;4.2 Stochastic Collocation;262
15.4.3;4.3 Stochastic Galerkin;262
15.5;5 Structure of Discrete Equations;263
15.6;6 Sparse Approximations;263
15.6.1;6.1 Basic Iteration;263
15.6.2;6.2 A Sparse Low-Rank Start;264
15.6.3;6.3 SVD on the Fly to Keep It Lean;264
15.7;7 Implementation of Low-Rank Format Solvers;265
15.7.1;7.1 Truncation Operator and Strategies;265
15.7.2;7.2 Termination of the Iteration;267
15.7.3;7.3 Convergence and Stagnation;267
15.7.4;7.4 Numerical Examples;268
15.8;8 Conclusion;271
15.9;References;271
16;Auto-parametric Stability Loss and Post-critical Behaviourof a Three Degrees of Freedom System;274
16.1;1 Introduction;275
16.2;2 Semi-trivial Solution and Its Stability;277
16.3;3 Post-critical System Response: General Consideration;282
16.4;4 Large Excitation Amplitude;288
16.5;5 Medium and Small Excitation Amplitudes;290
16.6;6 Transition from Semi-trivial to Post-critical Response;292
16.7;7 Conclusions;294
16.8;References;295
17;Probability Based Size Effect Representation for Failurein Civil Engineering Structures Built of Heterogeneous Materials;297
17.1;1 Introduction and Motivation;297
17.2;2 Meso-Scale Model of Material Heterogeneities with Deterministic Material Parameters;300
17.2.1;2.1 Structured Mesh and Element Kinematics Enhancements;300
17.2.2;2.2 Operator Split Solution Procedure for Computing Interface Failure Modes;302
17.2.3;2.3 Comparison Between Structured and Unstructured Mesh Computations;303
17.3;3 Probability Aspects of Inelastic Localized Failure for Heterogeneous Materials;304
17.3.1;3.1 Geometry Description of Material Meso-Structure;305
17.3.1.1;3.1.1 Stochastic Integrals Computations;306
17.4;4 Probabilistic Characterization of Two-Phases Material at Macro-Scale;308
17.4.1;4.1 Determination of Statistical RVE Size;308
17.4.2;4.2 Simple Tension Test: Numerical Results and Discussion;309
17.5;5 Size Effect Representation;311
17.5.1;5.1 Random Fields for Material Properties and Their Karhunen-Loève Expansion;312
17.5.2;5.2 Size Effect and Correlation Length;314
17.6;6 Concluding Remarks;316
17.7;References;318
18;The Scatter of Eigenfrequencies in Beams Made of Metal Foam;320
18.1;1 Introduction;320
18.2;2 Determination of the Linear Elastic Properties;322
18.2.1;2.1 Generation of Structure;322
18.2.2;2.2 Modeling Different Foams;324
18.2.3;2.3 Including Random Inhomogeneities on the Mesoscale;328
18.2.4;2.4 Validation;330
18.3;3 Statistical Evaluation;331
18.3.1;3.1 Determination of the Distribution Function;331
18.3.2;3.2 Determination of Correlations;332
18.3.3;3.3 Power Spectral Density;334
18.3.4;3.4 Conclusions of Statistical Evaluation;336
18.4;4 Application;336
18.4.1;4.1 Generation of Realizations;337
18.4.1.1;4.1.1 Karhunen-Loève Expansion;337
18.4.1.2;4.1.2 Spectral Representation;340
18.4.2;4.2 Monte-Carlo Simulations;341
18.5;5 Conclusions;343
18.6;References;343
19;Index;345