Chapter 5 presents methods for assessing structural reliability under incomplete probability information, i.e., when complete distributional information on the basic random variables is not available. First, second-moment methods are presented where the available information is limited to the means, variances, and covariances of the basic random variables. These include the mean-centered first-order second-moment (MCFOSM) method, the first-order second-moment (FOSM) method, and the generalized second-moment method. These methods lead to approximate computations of the reliability index as a measure of safety. Lack of invariance of the MCFOSM method relative to the formulation of the limit-state function is demonstrated. The FOSM method requires finding the “design point,” which is the point in a transformed standard outcome space that has minimum distance from the origin. An algorithm for finding this point is presented. Next, methods are presented that incorporate probabilistic information beyond the second moments, including knowledge of higher moments and marginal distributions. Last, a method is presented that employs the upper Chebyshev bound for any given state of probability information. The chapter ends with a discussion of the historical significance of the above methods as well as their shortcomings and argues that they should no longer be used in practice.

Chapter 7 describes the second-order reliability method (SORM), which employs a second-order approximation of the limit-state surface fitted at the design point in the standard normal space. Three distinct SORM approximations are presented. The classical SORM fits the second-order approximating surface to the principal curvatures of the limit-state surface at the design point. This approach requires computing the Hessian (second-derivative matrix) of the limit-state function at the design point and its eigenvalues as the principal curvatures. The second approach computes the principal curvatures iteratively in the process of finding the design point. This approach requires only first-order derivatives of the limit-state function but repeated solutions of the optimization problem for finding the design point. One advantage is that the principal curvatures are found in decreasing order of magnitude and, hence, the computations can be stopped when the curvature found is sufficiently small. The third approach fits the approximating second-order surface to fitting points in the neighborhood of the design point. This approach also avoids computing the Hessian. Furthermore, it corrects for situations where the curvature is zero but the surface is curved, e.g., when the design point is an inflection point of the surface. Results from the three methods are compared numerically.

Chapter 11 addresses time- and/or space-variant structural reliability problems. It begins with a description of problem types as encroaching or outcrossing, subject to the type of dependence on the time or space variable. A brief review of essentials from the random process theory is presented, including second-moment characterization of the process in terms of mean and auto-covariance functions and the power spectral density. Special attention is given to Gaussian and Poisson processes as building blocks for stochastic load modeling. Bounds to the failure probability are developed in terms of mean crossing rates or using a series system representation through parameter discretization. A Poisson-based approximation for rare failure events is also presented. Next, the Poisson process is used to build idealized stochastic load models that describe macro-level load changes or intermittent occurrences with random magnitudes and durations. The chapter concludes with the development of the load-coincidence method for combination of stochastic loads. The probability distribution of the maximum combined load effect is derived and used to estimate the failure probability.

Chapter 9 describes simulation or sampling methods for reliability assessment. The chapter begins by describing methods for generation of pseudorandom numbers for prescribed univariate or multivariate distributions. Next, the ordinary Monte Carlo simulation (MCS) method is described. It is shown that for small failure probabilities, which is the case in most structural reliability problems, the number of samples required by MCS for a given level of accuracy is inversely proportional to the failure probability. Thus, MCS is computationally demanding for structural reliability problems. Various methods to reduce the computational demand of MCS are introduced. These include the use of antithetic variates and importance sampling. For the latter, sampling around design points and sampling in half-space are presented, the latter for a special class of problems. Other efficient sampling methods described include directional sampling, orthogonal-plane sampling, and subset simulation. For each case, expressions are derived for a measure of accuracy of the estimated failure probability. Methods are also presented for computing parameter sensitivities by sampling. Finally, a method is presented for evaluating certain multifold integrals by sampling. This method is useful in Bayesian updating, as described in Chapter 10.

Chapter 2 provides a review of probability theory, focusing on the topics that are essential for the remainder of the book. Included are the elements of set theory, the axioms and basic rules of probability theory, the concept of a random variable, discrete and continuous random variables, univariate and multivariate probability distributions, reliability and hazard functions, expectation and statistical moments, distributions and moments of functions of random variables, and extreme-value distributions. Appendix A presents commonly used probability distribution models with their properties, for easy reference. Thorough mastery of the material in this chapter is essential for understanding the remainder of the book.

Chapter 3 describes models for multivariate distributions. Included are the multinormal distribution, the multi-lognormal distribution, multivariate distributions constructed as products of conditional distributions, and three families of multivariate distributions with prescribed marginals and covariances: the Nataf family, the Morgenstern family, and copula distributions. Several structural reliability methods require transformation of original random variables into statistically independent standard normal random variables. The conditions for such a transformation to exist and be reversible are described and formulations are presented for each of the described multivariate distribution models. Also developed are the Jacobians of each transform and its inverse, which are also used in reliability analysis. Analytical and numerical results to facilitate the use of Nataf and Morgenstern distributions are provided in this chapter.

Chapter 8 introduces the theory and computational methods for system reliability analysis. The system is defined as a collection of possibly interdependent components, such that the system state depends on the states of its constituent components. The system function, cut and link sets, and the special cases of series and parallel systems are defined. Methods for reliability assessment of systems with independent and dependent components are described, including methods for bounding the system failure probability by bi- or tri-component joint probabilities. Bounds on the system failure probability under incomplete component probability information are developed using linear programming. An efficient matrix-based method for computing the reliability of certain systems is described. The focus is then turned to structural systems, where the state of each component is defined in terms of a limit-state function. FORM approximations are developed for series and parallel structural systems, and the inclusion–exclusion rule or bounding formulas are used to obtain the FORM approximation for general structural systems. Other topics include an event-tree approach for modeling sequential failures, measures of component importance, and parameter sensitivities of the system failure probability.

Chapter 14 develops methods for reliability-based design optimization (RBDO). Three classes of RBDO problems are considered: minimizing the cost of design subject to reliability constraints, maximizing the reliability subject to a cost constraint, and minimizing the cost of design plus the expected cost of failure subject to reliability and other constraints. The solution of these problems requires the coupling of reliability methods with optimization algorithms. Among many solution methods available in the literature, the main focus in this chapter is on a decoupling approach using FORM, which under certain conditions has proven convergence properties. The approach requires the solution of a sequence of decoupled reliability and optimization problems that are shown to gradually approach a near-optimal solution. Both structural component and system problems are considered. An alternative approach employs sampling to compute the failure probability with the number of samples increasing as the optimal solution point is approached. Also described are approaches that make use of surrogate models constructed in the augmented space of random variables and design parameters. Finally, the concept of buffered failure probability is introduced as a measure closely related to the failure probability, which provides a convenient alternative in solving the optimization subproblem.

Chapter 6 describes the first-order reliability method (FORM), which employs full distributional information. The chapter begins with a presentation of the important properties of the outcome space of standard normal random variables, which are used in FORM and other reliability methods. The FORM is presented as an approximate method that employs linearization of the limit-state surface at the design point in the standard normal space. The solution requires transformation of the random variables to the standard normal space and solution of a constrained optimization problem to find the design point. The accuracy of the FORM approximation is discussed, and several measures of error are introduced. Measures of importance of the random variables in contributing to the variance of the linearized limit-state function and with respect to statistically equivalent variations in means and standard deviations are derived. Also derived are the sensitivities of the reliability index and the first-order failure probability approximation with respect to parameters in the limit-state function or in the probability distribution model. Other topics in this chapter include addressing problems with multiple design points, solution of an inverse reliability problem, and numerical approximation of the distribution of a function of random variables by FORM.

Chapter 10 describes Bayesian methods for parameter estimation and updating of structural reliability in the light of observations. The chapter begins with a description of the sources and types of uncertainties. Uncertainties are categorized as aleatory or epistemic; however, it is argued that this distinction is not fundamental and makes sense only within the universe of models used for a given project. The Bayesian updating formula is then developed as the product of a prior distribution and the likelihood function, yielding the posterior (updated) distribution of the unknown parameters. Selection of the prior and formulation of the likelihood are discussed in detail. Formulations are presented for parameters in probability distribution models, as well as in mathematical models of physical phenomena. Three formulations are presented for reliability analysis under parameter uncertainties: point estimate, predictive estimate, and confidence interval of the failure probability. The discussion then focuses on the updating of structural reliability in the light of observed events that are characterized by either inequality or equality expressions of one or more limit-state functions. Also presented is the updating of the distribution of random variables in the limit-state function(s) in the light of observed events, e.g., the failure or non-failure of a system.

Many problems in structural reliability require the use of a computational platform, such as a finite-element code, to evaluate the limit-state function. Chapter 12 describes the framework for such coupling between a finite-element code and FORM/SORM analysis. The chapter begins with a brief review of the finite-element formulation for inelastic problems. Because FORM requires the gradients of the limit-state function, it is necessary for the finite-element code to compute not only the response vector but also its gradient with respect to selected outcomes of the random variables. The use of finite-differences for this purpose is not practical because of accuracy issues and computational demand. The direct-differentiation method (DDM) presented in this chapter provides an accurate and efficient means for this purpose. It is shown that the DDM requires a linear solution at the convergence of each iterative step in the nonlinear finite-element analysis. Next, a method for discrete representation of random fields of material properties or loads in the context of finite-element analysis is presented. The chapter concludes with a review of alternative approaches for finite-element reliability analysis or uncertainty propagation, including the use of polynomial chaos and various response-surface methods with efficient selection of experimental design points.

Nonlinear stochastic dynamics is a broad topic well beyond the scope of this book. Chapter 13 describes a particular method of solution for a certain class of nonlinear stochastic dynamic problem by use of FORM. The approach belongs to the class of solution methods known as equivalent linearization. In this case, the linearization is carried out by replacing the nonlinear system with a linear one that has a tail probability equal to the FORM approximation of the tail probability of the nonlinear system – hence the name tail-equivalent linearization method. The equivalent linear system is obtained non-parametrically in terms of its unit impulse response function. For small failure probabilities, the accuracy of the method is shown to be far superior to those of other linearization methods. Furthermore, the method is able to capture the non-Gaussian distribution of the nonlinear response. This chapter develops this method for systems subjected to Gaussian and non-Gaussian excitations and nonlinear systems with differentiable loading paths. Approximations for level crossing rates and the first-passage probability are also developed. The method is extended to nonlinear structures subjected to multiple excitations, such as bi-component base motion, and to evolutionary input processes.