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E-raamat: Computational Analysis of Randomness in Structural Mechanics: Structures and Infrastructures Book Series, Vol. 3

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(Center of Mechanics and Structural Dynamics, Vienna University of Technology, Vienna, Austria)
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Computational Analysis of Randomness in Structural Mechanics: Structures and Infrastructures Book Series, Vol. 3Suurem pilt
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Proper treatment of structural behavior under severe loading - such as the performance of a high-rise building during an earthquake - relies heavily on the use of probability-based analysis and decision-making tools. Proper application of these tools is significantly enhanced by a thorough understanding of the underlying theoretical and computational concepts as provided by this book.

Detailing the computational aspects of stochastic analysis within the field of structural mechanics, this book first presents a few motivating examples that demonstrate the various random effects within the context of simple structural analysis models. It moreover briefly reviews the fundamental concepts from continuum mechanics and puts them in the perspective of modern numerical tools, such as the finite element method. More advanced topics are developed step by step while gradually increasing the complexity of the structural and probabilistic analyses.

This volume is intended for structural analysts and advanced students who wish to explore the benefits of stochastic analysis. It will provide researchers and decision makers working on structural and infrastructural systems with the necessary probabilistic information needed for strategic developments in construction, inspection and maintenance.
Editorial IX
About the Book Series Editor XI
Foreword XIII
Preface XV
About the Author XVII
1 Introduction 1
1.1 Outline
1
1.2 Introductory examples
2
1.2.1 Outline of analysis
2
1.2.2 Static analysis
2
1.2.3 Buckling analysis
5
1.2.4 Dynamic analysis
7
1.2.5 Structural analysis
9
2 Preliminaries in probability theory and statistics 13
2.1 Definitions
13
2.2 Probabilistic models
16
2.2.1 Random variables
16
2.2.2 Some types of distributions
19
2.2.3 Conditional distribution
25
2.2.4 Functions of random variables
26
2.2.5 Random vectors
28
2.2.6 Joint probability density function models
30
2.2.7 Marginal and conditional distribution
35
2.3 Estimation
36
2.3.1 Basic properties
36
2.3.2 Confidence intervals
40
2.3.3 Chi-square test
41
2.3.4 Correlation statistics
44
2.3.5 Bayesian updating
45
2.3.6 Entropy concepts
48
2.4 Simulation techniques
50
2.4.1 General remarks
50
2.4.2 Crude Monte Carlo simulation
50
2.4.3 Latin Hypercube sampling
51
2.4.4 Quasirandom sequences
54
2.4.5 Transformation of random samples
56
2.4.6 Simulation of correlated variables
56
3 Regression and response surfaces 59
3.1 Regression
59
3.2 Ranking of variables
62
3.3 Response surface models
67
3.3.1 Basic formulation
67
3.3.2 Linear models and regression
68
3.3.3 First- and second-order polynomials
69
3.3.4 Weighted interpolation
70
3.3.5 Moving least squares regression
71
3.3.6 Radial basis functions
72
3.4 Design of experiments
76
3.4.1 Transformations
76
3.4.2 Saturated designs
77
3.4.3 Redundant designs
78
4 Mechanical vibrations due to random excitations 81
4.1 Basic definitions
81
4.2 Markov processes
83
4.2.1 Uperossing rates
85
4.3 Single-degree-of-freedom system response
87
4.3.1 Mean and variance of response
87
4.3.2 White noise approximation
92
4.4 Multi-degree-of-freedom response
95
4.4.1 Equations of motion
95
4.4.2 Covariance analysis
96
4.4.3 First passage probability
104
4.5 Monte-Carlo simulation
107
4.5.1 General remarks
107
4.5.2 Central difference method
107
4.5.3 Euler method
109
4.5.4 Newmark method
110
4.5.5 Digital simulation of white noise
112
4.6 Fokker-Planck equation
118
4.7 Statistical linearization
120
4.7.1 General concept
120
4.8 Dynamic stability analysis
125
4.8.1 Basics
125
4.8.2 Nonlinear stability analysis
127
4.8.3 Linear stability analysis
129
5 Response analysis of spatially random structures 137
5.1 Representation of random fields
137
5.1.1 Basic definitions
137
5.1.2 Properties of the auto-covariance function
139
5.1.3 Spectral decomposition
141
5.1.4 Conditional random fields
142
5.1.5 Local averages of random fields
146
5.2 Geometrical imperfections
148
5.3 Stochastic finite element formulation
150
5.3.1 Elasticity (Plane stress)
150
5.3.2 Principle of virtual work
152
5.3.3 Finite element formulation
153
5.3.4 Structural response
156
5.3.5 Stochastic stiffness matrix
157
5.3.6 Integration point method
162
5.3.7 Static response – perturbation method
163
5.3.8 Monte Carlo simulation
166
5.3.9 Natural frequencies of a structure with randomly distributed elastic modulus
168
6 Computation of failure probabilities 171
6.1 Structural reliability
171
6.1.1 Definitions
171
6.1.2 First order – second moment concept
172
6.1.3 FORM – first order reliability method
174
6.2 Monte Carlo simulation
179
6.2.1 Definitions and basics
179
6.2.2 Importance sampling (weighted simulation)
179
6.2.3 Directional sampling
187
6.2.4 Asymptotic sampling
190
6.3 Application of response surface techniques to structural reliability
195
6.3.1 Basic concept
195
6.3.2 Structural examples
200
6.4 First passage failure
210
6.4.1 Problem formulation
210
6.4.2 Extension to non-linear problems
213
Concluding remarks 219
Notations 221
Bibliography 223
Subject Index 229
Structures and Infrastructures Series 231
Center of Mechanics and Structural Dynamics, Vienna University of Technology, Vienna, Austria Center for Advanced Technology for Large Structural Systems, Lehigh University, Bethlehem, PA, USA
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