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.

Flooding of large lowland rivers is dependent upon seasonal variability in Earth’s general circulation, in addition to large-scale atmospheric teleconnections. Large lowland rivers are unique in that local-scale hydrologic and geomorphic controls also influence floodplain inundation, creating challenges to government management organizations. River bank erosion is a key geomorphic process linked to seasonal discharge pulses, as well as channel bank sedimentology. Flood sedimentary processes are influenced by older floodplain geomorphology atop negative relief floodplains that provides a topographic framework for sediment dispersal. River deltas form downstream of fluvial hinge-lines, with the apex at the main channel avulsion node. At the terminus of drainage basins, the geometry of deltaic sedimentary deposits is influenced by the dominance of either fluvial, tidal, or wave processes. Fluvial dominated deltas are characterized within the ‘delta cycle,’ a conceptual model that provides important insights to the development of sediment management strategies to address coastal land loss caused by sea level rise, subsidence, and reduced sediment flux because of upstream dams.

Millions of dams fragment and degrade Earth’s riparian landscapes. This chapter examines linkages between dams, rivers, and the environment and is subdivided into three sections, including riparian impacts of dams, dam removal, and reservoir sediment management strategies. The latter is crucial to sustain downstream fluvial environments and water resource infrastructure. Trapping of fluvial sediments in upstream reservoirs results in downstream degradation of riparian environments by channel bed incision and terrestrialization of aquatic habitat. In North America and western Europe dam removal for environmental restoration is occurring at a brisk pace, but in Southeast Asia, South America, southeast Europe, and parts of Africa dam construction for hydropower is rapidly occurring. Dam removal is an emerging science with great potential to improve downstream riparian ecosystems. It is essential that government agencies develop strategies to sustainably manage reservoirs so that fluvial sediments can be reactivated and transported to downstream riparian and deltaic environments to offset subsidence and sea level rise.

Many of Earth’s large lowland rivers are heavily impacted by land change and hydraulic engineering to support a range of societal demands. Dams and river engineering for flood control have disconnected rivers from floodplains, reduced coastal sediment flux, and driven land subsidence of deltaic wetlands because of reduced sediment loads. Structural modification of lowland rivers alters hydrologic and sedimentary processes, resulting in unintended geomorphic and environmental adjustments that require decades to unfold. Such problems not only degrade associated riparian ecosystems but also increase human vulnerability to flooding. Integrated approaches to lowland river management are needed to reduce flood risk, conserve and restore riparian environments, while ensuring that lowland rivers continue to meet societal demands.

The final chapter on human impacts to lowland rivers logically ends at the basin terminus, at the coast. Large flood basins and deltas are the most challenging environments to manage because of being impacted by ground subsidence and coastal storm surge events, particularly large populations in delta cities. Artificial floodwater diversion mirrors natural flood pulses, although it also disturbs aquatic ecosystems that requires further management. Challenges with floodwater diversion for environmental management of the lower Sacramento basin links to historic hydraulic gold mining. A major geomorphic phenomenon that influences flood basin environments is channel avulsion, which can be set up by embanked floodplain sedimentation and subsidence. Ground subsidence problems are particularly acute as regards flood control infrastructure, increasing the risk of dike failure during coastal storm surge events. Urban flood basin subsidence and flood control challenges is examined in a review of the epic 2011 Chao Phraya River flooding of Bangkok and the 2005 coastal storm surge flooding of New Orleans by Hurricanes Katrina-Rita.