Computational Analysis of Randomness in Structural Mechanics: Structures and Infrastructures Book Series, Vol. 3, 1st Edition (Hardback) book cover

Computational Analysis of Randomness in Structural Mechanics

Structures and Infrastructures Book Series, Vol. 3, 1st Edition

By Christian Bucher

CRC Press

248 pages

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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.

Table of Contents

1 Introduction

  • 1.1 Outline
  • 1.2 Introductory examples
  • 1.2.1 Outline of analysis
  • 1.2.2 Static analysis
  • 1.2.3 Buckling analysis
  • 1.2.4 Dynamic analysis

2 Preliminaries in Probability Theory and Statistics

  • 2.1 Definitions
  • 2.2 Probabilistic models
  • 2.2.1 Random variables
  • 2.2.2 Some types of distributions
  • 2.2.3 Conditional distribution
  • 2.2.4 Functions of random variables
  • 2.2.5 Random vectors
  • 2.2.6 Joint probability density function models
  • 2.2.7 Marginal and conditional distribution
  • 2.3 Estimation
  • 2.3.1 Basic properties
  • 2.3.2 Confidence intervals
  • 2.3.3 Chi-square test
  • 2.3.4 Correlation statistics
  • 2.3.5 Bayesian updating
  • 2.3.6 Entropy concepts
  • 2.4 Simulation techniques
  • 2.4.1 General remarks
  • 2.4.2 Crude Monte Carlo simulation
  • 2.4.3 Latin Hypercube sampling
  • 2.4.4 Quasirandom sequences
  • 2.4.5 Transformation of random samples
  • 2.4.6 Simulation of correlated variables

3 Regression and Response Surfaces

  • 3.1 Regression
  • 3.2 Ranking of variables
  • 3.3 Response surface models
  • 3.3.1 Basic formulation
  • 3.3.2 Linear models and regression
  • 3.3.3 First- and second-order polynomials
  • 3.3.4 Weighted interpolation
  • 3.3.5 Moving Least Squares Regression
  • 3.3.6 Radial basis functions
  • 3.4 Design of experiments
  • 3.4.1 Transformations
  • 3.4.2 Saturated designs
  • 3.4.3 Redundant designs

4 Mechanical vibrations due to random excitations

  • 4.1 Basic definitions
  • 4.2 Markov processes
  • 4.2.1 Upcrossing rates
  • 4.3 Single-degree-of-freedom system response
  • 4.3.1 Mean and variance of response
  • 4.3.2 White noise approximation
  • 4.4 Multi-degree-of-freedom response
  • 4.4.1 Equations of motion
  • 4.4.2 Covariance analysis
  • 4.4.3 First passage probability
  • 4.5 Monte-Carlo simulation
  • 4.5.1 General remarks
  • 4.5.2 Central difference method
  • 4.5.3 Euler method
  • 4.5.4 Newmark method
  • 4.5.5 Digital simulation of white noise
  • 4.6 Fokker-Planck equation
  • 4.7 Statistical linearization
  • 4.7.1 General concept
  • 4.8 Dynamic stability analysis
  • 4.8.1 Basics
  • 4.8.2 Nonlinear stability analysis
  • 4.8.3 Linear stability analysis

5 Response analysis of spatially random structures

  • 5.1 Representation of Random Fields
  • 5.1.1 Basic Definitions
  • 5.1.2 Properties of the auto-covariance function
  • 5.1.3 Spectral Decomposition
  • 5.1.4 Conditional Random Fields
  • 5.1.5 Local Averages of Random Fields
  • 5.2 Geometrical Imperfections
  • 5.3 Stochastic Finite Element Formulation
  • 5.3.1 Elasticity (Plane Stress)
  • 5.3.2 Principle of Virtual Work
  • 5.4 Finite Element Method
  • 5.4.1 Element Formulation
  • 5.4.2 Structural response
  • 5.4.3 Stochastic Stiffness Matrix
  • 5.4.4 Integration Point Method
  • 5.4.5 Static Response - Perturbation Method
  • 5.4.6 Monte Carlo Simulation
  • 5.4.7 Natural Frequencies of a Structure with Randomly Distributed Elastic Modulus

6 Computation of failure probabilities

  • 6.1 Structural Reliability
  • 6.1.1 Definitions
  • 6.1.2 First Order - Second Moment Concept
  • 6.1.3 FORM - First Order Reliability Method
  • 6.2 Monte-Carlo-Simulation
  • 6.2.1 Definitions and Basics
  • 6.2.2 Importance Sampling (Weighted Simulation)
  • 6.2.3 Directional Sampling
  • 6.2.4 Asymptotic Sampling
  • 6.3 Application of RSM
  • 6.3.1 Basic concept
  • 6.3.2 Structural examples
  • 6.4 First Passage Failure
  • 6.4.1 Problem formulation
  • 6.4.2 Extension to Non-Linear Problems

About the Series

Structures and Infrastructures

Book Series Editor: Prof. Dan M. Frangopol, Lehigh University, PA, USA

Our knowledge to model, analyze, design, maintain, manage and predict the life-cycle performance of structures and infrastructures is continually growing. However, the complexity of these systems continues to increase and an integrated approach is necessary to understand the effect of technological, environmental, economical, social and political interactions on the life-cycle performance of engineering structures and infrastructures. In order to accomplish this, methods have to be developed to systematically analyze structure and infrastructure systems, and models have to be formulated for evaluating and comparing the risks and benefits associated with various alternatives. We must maximize the life-cycle benefits of these systems to serve the needs of our society by selecting the best balance of the safety, economy and sustainability requirements despite imperfect information and knowledge.

In recognition of the need for such methods and models, the aim of this book series is to present research, developments, and applications written by experts on the most advanced technologies for analyzing, predicting and optimizing the performance of structures and infrastructures such as buildings, bridges, dams, underground construction, offshore platforms, pipelines, naval vessels, ocean structures, nuclear power plants, and also airplanes, aerospace and automotive structures.

The scope of this book series covers the entire spectrum of structures and infrastructures. Thus it includes, but is not restricted to, mathematical modeling, computer and experimental methods, practical applications in the areas of assessment and evaluation, construction and design for durability, decision making, deterioration modeling and aging, failure analysis, field testing, structural health monitoring, financial planning, inspection and diagnostics, life-cycle analysis and prediction, loads, maintenance strategies, management systems, nondestructive testing, optimization of maintenance and management, specifications and codes, structural safety and reliability, system analysis, time-dependent performance, rehabilitation, repair, replacement, reliability and risk management, service life prediction, strengthening and whole life costing.

This book series is intended researchers, practitioners, and students world-wide with a background in civil, aerospace, mechanical, marine and automotive engineering, as well as people working in infrastructure maintenance, monitoring, management and cost analysis of structures and infrastructures. Some volumes are monographs defining the current state of the art and/or practice in the field, and some are textbooks to be used in undergraduate (mostly seniors), graduate and postgraduate courses. This book series is affiliated to Structure and Infrastructure Engineering (Taylor & Francis, ), an international peer-reviewed journal which is included in the Science Citation Index.
If you like to contribute to this series as an author or editor, please contact the Series Editor ( or the Publisher ( A book proposal form can be downloaded at

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