Über die Autorin bzw. den Autor

Jie Li is a Professor of Civil Engineering at Tongji University, specializing in the area of earthquake engineering and stochastic mechanics. He has worked on uncertainty quantification, response analysis, and reliability evaluation of structural systems involving randomness -- integrating both for system parameters and excitations -- for more than 15 years. He has authored six monographs and published over 200 papers in peer reviewed journals. Li holds executive positions in China's major architectural, vibration engineering, and disaster prevention societies and laboratories. He is the Editor-in-Chief of the Journal of Tongji University (Natural Science Series) and is on the editorial board of over 10 international and Chinese journals, including the International Journal of Nonlinear Mechanics and Earthquake Engineering and Engineering Vibrations. He has received a variety of national and provincial-level awards for Advancement in Science and Technology. Li holds a Ph.D. in Civil Engineering from Tongji University.

Jianbing Chen is an Associate Professor of Civil Engineering at Tongji University and serves at the State Key Laboratory in Disaster Reduction in Civil Engineering. He specializes in earthquake engineering and stochastic mechanics. Awards include the MOE's National Science Award, National Excellent Doctoral Thesis, Shanghai City's Excellent Young Teacher Award, and acceptance into the MOE's Excellent Scholars Program. He holds a B.S. from Northeastern University and a Ph.D. from Tongji University, both in Civil Engineering.

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In Stochastic Dynamics of Structures, Li and Chen present a unified view of the theory and techniques for stochastic dynamics analysis, prediction of reliability, and system control of structures within the innovative theoretical framework of physical stochastic systems. The authors outline the fundamental concepts of random variables, stochastic process and random field, and orthogonal expansion of random functions. Readers will gain insight into core concepts such as stochastic process models for typical dynamic excitations of structures, stochastic finite element, and random vibration analysis. Li and Chen also cover advanced topics, including the theory of and elaborate numerical methods for probability density evolution analysis of stochastic dynamical systems, reliability-based design, and performance control of structures.

Stochastic Dynamics of Structures presents techniques for researchers and graduate students in a wide variety of engineering fields: civil engineering, mechanical engineering, aerospace and aeronautics, marine and offshore engineering, ship engineering, and applied mechanics. Practicing engineers will benefit from the concise review of random vibration theory and the new methods introduced in the later chapters.

"The book is a valuable contribution to the continuing development of the field of stochastic structural dynamics, including the recent discoveries and developments by the authors of the probability density evolution method (PDEM) and its applications to the assessment of the dynamic reliability and control of complex structures through the equivalent extreme-value distribution."
―A. H-S. Ang, NAE, Hon. Mem. ASCE, Research Professor, University of California, Irvine, USA

"The authors have made a concerted effort to present a responsible and even holistic account of modern stochastic dynamics. Beyond the traditional concepts, they also discuss theoretical tools of recent currency such as the Karhunen-Loève expansion, evolutionary power spectra, etc. The theoretical developments are properly supplemented by examples from earthquake, wind, and ocean engineering. The book is integrated by also comprising several useful appendices, and an exhaustive list of references; it will be an indispensable tool for students, researchers, and practitioners endeavoring in its thematic field."
―Pol Spanos, NAE, Ryon Chair in Engineering, Rice University, Houston, USA

Source code for readers and lecture supplements for instructors available at [www.wiley.com/go/stochdyn]

Aus dem Klappentext

In  Stochastic Dynamics of Structures, Li and Chen present a unified view of the theory and techniques for stochastic dynamics analysis, prediction of reliability, and system control of structures within the innovative theoretical framework of physical stochastic systems. The authors outline the fundamental concepts of random variables, stochastic process and random field, and orthogonal expansion of random functions. Readers will gain insight into core concepts such as stochastic process models for typical dynamic excitations of structures, stochastic finite element, and random vibration analysis. Li and Chen also cover advanced topics, including the theory of and elaborate numerical methods for probability density evolution analysis of stochastic dynamical systems, reliability-based design, and performance control of structures.

Stochastic Dynamics of Structures presents techniques for researchers and graduate students in a wide variety of engineering fields: civil engineering, mechanical engineering, aerospace and aeronautics, marine and offshore engineering, ship engineering, and applied mechanics. Practicing engineers will benefit from the concise review of random vibration theory and the new methods introduced in the later chapters.

"The book is a valuable contribution to the continuing development of the field of stochastic structural dynamics, including the recent discoveries and developments by the authors of the probability density evolution method (PDEM) and its applications to the assessment of the dynamic reliability and control of complex structures through the equivalent extreme-value distribution."
—A. H-S. Ang, NAE, Hon. Mem. ASCE, Research Professor, University of California, Irvine, USA

"The authors have made a concerted effort to present a responsible and even holistic account of modern stochastic dynamics. Beyond the traditional concepts, they also discuss theoretical tools of recent currency such as the Karhunen-Loève expansion, evolutionary power spectra, etc. The theoretical developments are properly supplemented by examples from earthquake, wind, and ocean engineering. The book is integrated by also comprising several useful appendices, and an exhaustive list of references; it will be an indispensable tool for students, researchers, and practitioners endeavoring in its thematic field."
—Pol Spanos, NAE, Ryon Chair in Engineering, Rice University, Houston, USA

Source code for readers and lecture supplements for instructors available at [www.wiley.com/go/stochdyn]

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Stochastic Dynamics of Structures

By Jie Li Jianbing Chen

John Wiley & Sons

Copyright © 2009 John Wiley & Sons, Ltd
All right reserved.
ISBN: 978-0-470-82424-5

Chapter One

Introduction

1.1 Motivations and Historical Clues

Structural dynamics deals with the problems of response analysis, reliability evaluation and system control of any given type of structure subjected to dynamic actions. Structures (such as buildings, bridges, aircraft, ships and so on) refer to those bodies or systems composed of various materials in a certain way that are capable of bearing loads and actions. On the other hand, when we say an action applied on structures is dynamic, this not only indicates that the action is time varying, but also that the induced inertial effects cannot be ignored. For example, earthquakes, wind, sea waves, jet noise and turbulence in the boundary layer and the like are typical dynamic actions. The task of dynamic response analysis of structures is to capture the internal forces, deformations or other state quantities of structures when they are subjected to dynamic actions. At the same time, we may need to study whether the structural response meets some specified limit in a sense, which is generally referred to as reliability evaluation. Furthermore, to make a structure subjected to dynamic actions response in a desired way to an extent is what to be done in system control.

Most dynamic actions exhibit appreciable randomness. Actually, investigators frequently find that the results observed under almost identical conditions have obvious deviation, but simultaneously exhibit some statistical rules. In essence, the randomness results from the uncontrollability of causation of the realized phenomenon. For example, consider wind turbulence in the atmospheric boundary layer. It is well known that the observed wind speeds recorded at the same position but during different time intervals are quite different (Figure 1.1). However, if the statistics of a large number of samples are examined, then we find that the probabilistic characteristics of the wind speed are relatively stable (Figure 1.2). In fact, the randomness involved stems from a complicated physical mechanism in the wind flows, say the mechanism of turbulence. The underlying reason is the uncontrollable nature of the motion of air molecules.

In addition, the randomness involved in the physical parameters of structures is also one of the sources that induce randomness in the dynamic responses of structures. For instance, in the dynamic response analysis of building structures, the soilstructure interaction is one of the basic problems where the properties of soil must be considered in the establishment of a reasonable structural analysis model. Evidently, it is impossible to measure the physical properties of soil completely at all points in the groundwork. Thus, a reasonable modeling approach is to regard the physical properties of soil, such as the shear wave speed and the damping ratio, as random variables or random fields. This will lead to the structural analysis involving random parameters, usually known as stochastic structural analysis.

Stochastic dynamic response analysis, reliability evaluation and system control compose the basic research scope of the stochastic dynamics of structures.

Although the studies on stochastic dynamical systems can be dated back to the investigations on statistical mechanics by Gibbs and Boltzmann (Gibbs, 1902; Cercignani, 1998), it is generally considered more reasonable to regard the studies on Brownian motion by Einstein (1905) as the origin of the discipline.

In 1905, Einstein studied the problem of the irregular motion of particles suspended in fluids, which was first observed by the Scottish botanist Robert Brown in 1827 (Figure 1.3). Einstein believed that Brownian motion of the particles was induced by the highly frequent random impacts of the fluid molecules. Based on this physical interpretation, Einstein made the following assumptions:

(a) the motion of different Brownian particles is mutually independent;

(b) the motion of Brownian particles is isotropic and no external actions except the collision of fluid molecules are applied;

(c) the collision of fluid molecules is instantaneous, such that the time of collision can be ignored (rigid collision).

Based on the above assumptions, the probability density of the particle group at two different instants of time can be derived by examining the phenomenological evolution process of the particle group; that is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

where f(x, t + τ) is the probability density of the position of the particles at time t + τ, f (x + r, t) is the probability density by transition of the particles with distance r during the time interval τ, and Φ(r) is the probability density of displacement of the particles.

Using the rigid collision assumption, expanding the functions by using the Taylor series and retaining the first-order term with respect to f (x, t + τ) and the second-order term with respect to f (x + r, t) will yield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

Clearly, Equation (1.2) is a diffusion equation, where D is the diffusion coefficient.

In 1914 and 1917, Fokker and Planck respectively introduced the drift term for a similar physical problem, leading to the so-called Fokker–Planck equation (Fokker, 1914; Planck, 1917; Gardiner, 1983), of which the rigorous mathematical basis was later established by Kolmogorov (1931).

We note that, initially, the studies on Brownian motion were based on physical concepts; however, a statistical phenomenological interpretation was subsequently introduced in the deductions. In this book, we call this historical clue the Einstein–Fokker–Planck tradition or phenomenological tradition. In this tradition, a large number of studies on the probability density evolution of stochastic dynamical systems have been done (Kozin, 1961; Lin, 1967; Roberts and Spanos, 1990; Zhu, 1992, 2003; Lin and Cai, 1995). However, for the multidegree-of-freedom (MDOF) systems or multidimensional problems, advancement is still quite limited (Schuëller, 1997, 2001).

Soon after Einstein's work, Langevin (1908) came up with a completely different research approach. In his investigation, the physical interpretation of Brownian motion is the same as that of Einstein, but Langevin contributed to two basic aspects. He:

(a) introduced the assumption of random forces;

(b) employed Newton's equation of motion to govern the motion of the Brownian particles.

Based on this, he established the stochastic dynamics equation, which was later called the Langevin equation:

m[??] = γ[??] + ε(t) (1.4)

where m is the mass of the Brownian particles, [??] and [??] are the acceleration and velocity of motion respectively, γ is the viscous damping coefficient and ξ(t) is the force induced by the collision of the fluid molecules, which is randomly fluctuating.

Using the ensemble average, Langevin obtained a diffusion coefficient identical to that given by Einstein.

In contrast to the diffusion equation derived by Einstein, the Langevin equation is more direct and more physically intuitive. However, the physical features of the random forces are not completely clear in Langevin's work.

In 1923, Wiener proposed a stochastic process model for...