First-order-second-moment

The precision uncertainty associated with field sampling is generally much larger than that associated with analytical technique, which is roughly 2% to the 67% confidence interval for the two compounds used as conservative and gas tracers. A technique to determine the precision uncertainty associated with field sampling and incorporated into the mean Kl estimate will therefore be propagated with the first-order, second moment analysis (Abernathy et al., 1985) ... 

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. 

Probabilistic response analysis consists of computing the probabilistic characterization of the response of a specific structure, given as input the probabilistic characterization of material, geometric and loading parameters. An approximate method of probabilistic response analysis is the mean-centred First-Order Second-Moment (FOSM) method, in which mean values (first-order statistical moments), variances and covariances (second-order statistical moments) of the response quantities of interest are estimated by using a mean-centred, first-order Taylor series expansion of the response quantities in terms of the random/uncertain model parameters. Thus, this method requires only the knowledge of the first- and second-order statistical moments of the random parameters. It is noteworthy that often statistical information about the random parameters is limited to first and second moments and therefore probabilistic response analysis methods more advanced than FOSM analysis cannot be fully exploited. 

The mean-centred First-Order Second-Moment (FOSM) method is presented as simplified FE probabilistic response analysis method. The FOSM method is applied to probabilistic nonlinear pushover analysis of a structural system. It is found that a DDM-based FOSM analysis can provide, at low computational cost, estimates of first- and second-order FE response statistics which are in good agreement with significantly more expensive Monte Carlo simulation estimates when the frame structure considered in this study experiences low-to-moderate material nonlinearities. 

DOLINSKI, K. First Order Second Moment Approximation In Reliability of Structural Systems Critical Review and Alternative Approach", Structural Safety K3) (1983) 211 -231. 

The present study shows that It is possible to evaluate the variability of statically determinate and statically indeterminate structures due to spatial variation of elastic properties without resort to finite element analysis. If a Green s function formulation is used, the mean square statistics of the indeterminate forces are obtained in a simple Integral form which is evaluated by numerical methods in negligible computer time. It was shown that the response variability problem becomes a problem Involving only few random variables, even if the material property is considered to constitute stochastic fields. The response variability was estimated using two methods, the First-Order Second Moment method, and the Monte Carlo simulation technique. 

T. Hisada, S. Nakagiri, and M. Mashimo, A Note on Stochastic Finite Element Method (Part 10) - On Dimensional Invariance of Advanced First-Order Second-Moment Reliability Index in Analysis of Continuum, Seisan-Kenkyu, vol. 37, no. 3, pp. 111-114, Institute of Industrial Science, University of Tokyo, Tokyo, Japan, 1985. 

The first-order second-moment (FOSM) method (e.g., Melchers 1999) can be used to compute the additional variance of collapse capacity due to uncertainty in the system parameters. The total variance of the collapse capacity, (T cc(TOT)> based on FOSM is (Ibarra and Krawinkler 2011) ... 

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. 

First-Order Second-Moment (FOSM) Method in Component Reliability... 

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. 

First, consider the generic performance function G(X), and let fx(x) denote the joint probability density function of X. Recall X = X / = 1 to n], and let fix. and crx, denote the mean and standard deviation of respectively. Further, the covariance of X, and Xj is denoted by Cov(X Xj). The first-order second-moment (shortly, referred to as FOSM) method approximates G to be a Gaussian distribution, using only the mean and covariance of X. 

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. 

The same numerical example can also be solved using the first-order second-moment method, by linearizing the limit state equation at the mean of the variables. The distribution information of the variables is not used, in this approach, = 38 x 54—1,500 = 552, and 4 = (3.8 X 54) + (2.7 X 38) + 15f which yields Gq = 241.37. Therefore, fi = 2.287, and hence, Pf= 0.011. Evidently, this is extremely erroneous because of the inaccuracy of the FOSM method. 

Next, it is necessary to use the design model to determine an estimate of system performance and the uncertainty in this estimate. As described in this section, this step requires a large amount of computing time by conventionally used Monte Carlo simulation. As an alternative, the first-order second moment (FOSM) method adopted within our framework reduces the computational burden. 

The first-order second moment method (FOSM) is the method adopted within the framework to propagate input parameter uncertainty through numerical models (26, 27). FOSM provides two moments, mean and variance of predicted variables. This method is based on Taylor series expansion, of which second-order and higher terms are truncated. The expected value of concentration, E[u] and its covariance, COV[u] are (25, 27),... 

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