Chapter 4 presents the basic formulation of the structural reliability problem. It starts with the so-called R-S problem with R denoting a capacity (resistance, supply, strength, etc.) value and S denoting a measure of the corresponding demand (load, stress, etc.), both modeled as random variables. Solutions in integral form are presented for the failure probability by conditioning on R or S, or using formulations in terms of the safety margin or safety factor that lead to the introduction of the concept of reliability index. Exact solutions are presented for specific distributions of R or S. This allows examination of the so-called tail-sensitivity problem, i.e., the sensitivity of the failure probability to the selected probability distributions. It is shown that small failure probabilities are sensitive to the shape of the selected distributions in the tail. The formulation of the structural reliability problem is then generalized and presented in terms of a limit-state function of basic random variables. Using this formulation, the probability of failure is expressed as a multifold integral over the outcome space of the basic random variables. Descriptions of several example applications of the generalized structural reliability formulation conclude the chapter.

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 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.