First-order second moment method FOSM

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),... 

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. 

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. 

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) ... 

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. 

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. 

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