Introduction

The introductory section starts with a motivating example demonstrating various random effects within the context of a simple structural analysis model. Subsequently, fundamental concepts from continuum mechanics are briefly reviewed and put into the perspective of modern numerical tools such as the finite element method. A chapter on probability theory, specifically on probabilistic models for structural

analysis, follows. This chapter 2 deals with the models for single random variables and random vectors. That includes joint probability density models with prescribed correlation. A discussion of elementary statistical methods – in particular estimation procedures – complements the treatment. Dependencies of computed response statistics on the input random variables can

be represented in terms of regression models. These models can then be utilized to reduce the number of variables involved and, moreover, to replace the – possibly very complicated – input-output-relations in terms of simple mathematical functions. Chapter 3 is devoted to the application of regression and response surface methods in the context of stochastic structural analysis. In Chapter 4, dynamic effects are treated in conjunction with excitation of structures

by random processes. After a section on the description of random processes in the time and frequency domains, emphasis is put on the quantitative analysis of the random structural response. This includes first and second moment analysis in the time and frequency domains. Chapter 5 on the analysis of spatially random structures starts with a discussion

of random field models. In view of the numerical tools to be used, emphasis is put on efficient discrete representation and dimensional reduction. The implementation within the stochastic finite element method is then discussed. The final chapter 6 is devoted to estimation of small probabilities which are typically

found in structural reliability problems. This includes static and dynamic problems as models. In dynamics, the

first passage probabilities over response thresholds plays an important role. Priority is given to Monte-Carlo based methods such as importance sampling. Analytical approximations are discussed nonetheless. Throughout the book, the presented concepts are illustrated by means of numerical

examples. The solution procedure is given in detail, and is based on two freely available software packages. One is a symbolic maths package called maxima (Maxima 2008) which in this book is mostly employed for integrations and linear algebra operations. And the other software tool is a numerical package called octave (Eaton 2008) which is suitable for a large range of analyses including random number generation and statistics. Both packages have commercial equivalents which, of course, may be applied in a similar fashion. Readerswhowant to expand their view on the topic of stochastic analysis are encour-

aged to refer to the rich literature available. Here only a few selected monographs are mentioned. An excellent reference on probability theory is Papoulis (1984). Response surface models are treated in Myers and Montgomery (2002). For the modeling and numerical analysis of random fields as well as stochastic finite elements it is referred to VanMarcke (1983) and Ghanem and Spanos (1991). Random vibrations are treated extensively in Lin (1976), Lin and Cai (1995), and Roberts and Spanos (2003). Many topics of structural reliability are covered inMadsen, Krenk, and Lind (1986) as well as Ditlevsen and Madsen (2007).