First-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. 

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 ... 

The first order reliability method (FORM) is an approximate method for assessing the reliability of a structural system. Its basic assumption is to approximate the limit state function (g(Tj,g(x)) = 0) of the structural reliability problem by means of a hyperplane which is orthogonal to the design point vector note that this approximation is constructed in the standard normal space. Thus, the failure probability can be estimated using the Euclidean norm of X, i.e. 

The probabilities defined by Eqs. (33.26) and (33.27), can be calculated by integrating the joint probability function of the significant random variables using Monte Carlo or FORM numeric methods (first order reliability methods). A description of these methods can be found in Refs. 11 and 12. 

In the first-order reliability method, the limit-state surface in the standard normal space is replaced with the tangent plane at the point with minimum distance from the origin. The first-order estimate of the probability of failure, then, is... 

An important feature of the first-order reliability method is the facility to compute reliability sensitivity measures with respect to any set of desired parameters. The simplest such measure is the unit vector a = - VyG/l VyG I computed at the linearization point y, which represents the sensitivity of /3 with respect to variations in the linearization point [1,22], i.e.,... 

According to Eq. (12), as long as we calculate the value of Prob( nB), we can obtain the value of v t). Figure 3 illustrates the probability of ArtB in FORM approximation (First Order Reliability Method) in the space of the standard variables U = T(X) where T is an isoprobabilistic transformation. In this figure, t) and fi(t + At) are respective the reliability indices of the events A and B. n(t) and D(t -i- At) are respective the unit normal vectors of A and B. represents the probability density function of the binormal law. The probability AnB is represented by the integral of over the hatched volume. 

ABSTRACT Runway overrun is one of the main accident types in airline operations. Nevertheless, due to the high safety levels in the aviation industry, the probability of a runway overrun is small. This motivates the use of structural reliability concepts to estimate this probability. We apply the physically-based model for the landing process of Drees and Holzapfel (2012) in combination with a probabilistic model of the input parameters. Subset simulation is used to estimate the probability of runway overrun for different runway conditions. We also carry out a sensitivity analysis to estimate the influence of each input random variable on the probability of an overrun. Importance measures and parameter sensitivities are estimated based on the samples from subset simulation and concepts of the First-Order Reliability Method (FORM). 

Probabilistic structural reliability analysis is performed to ensure that a structure is able to withstand the required design loads. A realistic description of the environmental loads and structural response is a crucial prerequisite for structural reliability analysis of structures exposed to environmental forces. The concept of environmental contours is an efficient method of estimating extreme conditions as basis for design. See (Winterstein et d., 1993) and (Haver Winterstein 2009). It is widely used in marine structural design. See e.g., (Baarholm et al., 2010), (Fontaine et al., 2013), (Jonathan et al., 2011), (Moan 2009) and (Ditlevsen 2002). The traditional approach is to use the well-known Rosenblatt transformation introduced in (Rosenblatt 1952) to transform the environmental variables into independent standard normal variables and identify a sphere with desired radius in the transformed space. Environmental contours are then found by re-transforming the sphere back to the original space. This approach is closely related to the FORM-approximation (First Order Reliability Method), where the failure boundary in the transformed space is approximated by a hyperplane at the design point. 

For a general structural system and limit state, similar to the case of barrier crossing at a constant level of a single-degree-of-freedom oscillator, Gueri and Rackwitz (1986) included R as basic variables, together with X, and reduced the dynamic problem to a time-invariant one with n + m basic variables therefore, the first-order reliability method can be used, where m is the number of response variables. The required information includes the explicit form of the performance function, G the probability distributions not only of system parameters, X, but also of response variables, R and even the joint PDF,, R(x, r). The method is very complicated because of the number of response variables and the difficulty of determining the distribution of response variables of interest. For some limit states, it may not be possible to identify R and determine the performance function in the explicit form. 

First-order reliability method Monte Carlo sampling Probability of failure Reliability analysis Uncertainty... 

Note that, in general, G may be nonlinear unlike Eq. 1, and therefore, the concepts explained in section Calculation of Eailure Probability cannot be readily extended to this case. Eurther, underlying random variables, i.e., X, may not follow Gaussian distributions at all. In order to overcome these challenges, researchers have developed a suite of computational methods to efficiently evaluate the integral in Eq. 11 and compute the failure probability. While some of these methods are based on simulation, other methods are based on linearizing G and, hence, popularly known as first-order reliability methods. These computational methods are discussed in sections Simulation-Based Methods and Eirst-Order Reliabihty Methods, respectively. 

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. 

In addition to the first-order reliability methods and simulation-based techniques, there are other types of techniques and methods that have become popular for stractural reliability analysis, over the past two decades. The purpose of this section is to provide an overview of some of these approaches and list appropriate references that would aid in-depth understanding of these methods. 

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