Second-order reliability method

These are generally called Second Order Reliability Methods, where the use of independent, near-Normal variables in reliability prediction generally come under the title First Order Reliability Methods (Kjerengtroen and Comer, 1996). For economy and speed in the calculation, however, the use of First Order Reliability Methods still dominates presently. 

Over the last two decades, there has been increasing interest in probabilistic, or stochastic, robust control theory. Monte Carlo simulation methods have been used to synthesize and analyze controllers for uncertain systems [170,255], First- and second-order reliability methods were incorporated to compute the probable performance of linear-quadratic-regulator... 

Equation 1 expresses the fact that the failure domain D is measured by means of probability measure. It is not easy to calculate Pf using Equation 1, therefore many techniques are developed in the literature. The well known approaches are the FORM/SORM (respectively, First Order Reliability Methods and Second Order Reliability Methods) that consists in using a transformation to change variables into an appropriate space where vector U = T X) is a Gaussian vector with uncorrelated components. In this space, the design point, , is determined. Around this point, Taylor expansion of the limit state function is performed at first order or second order respectively for FORM or SORM method (Madsen et al). In the case of FORM, the structure reliability index is calculated as ... 

Approximate Methods First and Second Order Reliability Methods (FORM and SORM). 

Examples of probabilistic response analysis using the mean-centred First-Order Second-Moment (FOSM) approximation, time-invariant (First- and Second-Order Reliability Methods, FORM and SORM) and time-variant (mean outcrossing rate computation) reliability analyses are provided to illustrate the methodology presented and its current capabilities and limitations. 

The main effort in the first or second-order reliability method is in finding the minimum-distance point(s), denoted y (x in X space, s in S space). This is formulated as a constrained optimization problem ... 

Der Kiureghian, A. 2005. First- and Second-Order Reliability Methods. In Nikolaidis, E., Ghiocel, D M. Singhal, S. (eds.) Engineering Design Reliability Handbook. CRC PressINC. 

Finally the expected value and variation of the performance must be evaluated. Some, as described below have developed the first- and second-order reliability methods for this task. The reliability index approach followed herein provides similar information but appears to be much more efficient. 

Implementation of the fixst-order reliability method in a finite element code requires two main algorithmic features. The first is an ability to compute the load effects for the sequence of points selected in the standard normal space for solution of the optimization problem, q. 9. Each point requires a single conventional run of the finite element code with the load space variables determined from the inverse of the probability transformation, Eq. 5. (As stated earlier, this transformation is usually in a triangular form, which facilitates its inversion.) The second is a capability to efficiently compute the Jacobian of the mechanical transformation, Since the size of X is usually large, a straightforward finite-difference approach is obviously impractical, as it requires repeated runs of the code for each element of the Jacobian. 

In (Hasofer Lind 1974), Hasofer and Lind introduced the reliability index technique for calculating approximations of the desired integral with reduced computation costs. The reh-abUity index has been extensively used in the first and second order rehabiUty methods (FORM (Hohenbichler Rackwitz 1983) and SORM (Fiessler, Neumann, Rackwitz 1979)). 

The evaluation of the integral in Eq. 1 can be computationally difficult some examples are as follows fx is often not well-defined because of the incompleteness of the statistical information available G(X) may have a nonlinear form the computation of the multifold integral can be very difficult if the number of tmcertain parameters is high. Various methods have been proposed for solving the integral form in Eq. 1. These approaches range from the classical moment methods for structural reliability (e.g., first-order second-moment reliability method) to the simulation-based approaches (i.e., Monte Carlo family of methods), and also the PEER approach, which is quite different compared to the other two techniques. In this entry, alternative methods for estimating the probability of failure are described. 

This section discusses a class of methods known as the first-order reliability methods to compute the probability of failure of structural systems. These methods are based on the first-order Taylor s series expansion of the performance function G(X). The first-method, known as the first-order second-moment (FOSM) method, focuses on approximating the mean and standard deviation of G and uses this information to compute Pf. Then, the FOSM method is extended to the advanced FOSM method in two steps first, the methodology is developed for the case where all the variables in X are Gaussian (normal) and, second, the methodology is extended to the general case of non-normal variables. 

The section discussed the use of first-order reliability methods in order to estimate the reliability of structures. First, the first-order second-moment (FOSM) method was presented and then extended to the advanced FOSM method. The concept of most probable point (MPP) was introduced. It was derived that the distance from the origin to the MPP, in standard normal space, is equal to the safety index or reliability index, denoted by ft. Information regarding the gradient at the MPP can be used to identify the sources of uncertainty that are significant contributors to the failure of the structure. 

An important goal of this study is to characterize the noncentrosymmetric arrangement of the azobenzene branches in dendrimers and the effect of these dendrimers on macroscopic second-order susceptibihty. The NLO activity of the dendrimers can be characterized by their molecular hyperpolarizabihty. The hyper-Rayleigh scattering (HRS) method is a reliable way of measuring the mol-... 

This type of electrode is a particularly powerful analytical tool since by performing steady-state measurements alone, it can measure faster rate constants than any other method. For a second-order reaction, the RRDE can reliably and reproducibly determine rate constants as fast as 10 mol dm ) s, while the maximum first-order rate constant measurable with the RRDE is about 10 s . A further advantage of the RRDE is the way that steady-state currents are measured (see below), whereas other methods of determining such high values ofk require the measurement of transients. 

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