First-order second moment method

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. 

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. 

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),... 

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. 

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. 

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. 

Gangwall et al. [47] were the first to apply Fourier analysis for the evaluation of the transport parameters of the Kubin-Kucera model. Gunn et al. applied the frequency response [80] and the pulse response method [83] in order to determine the coefficients of axial dispersion and internal diffusion in packed beds from experiments performed at various Reynolds numbers. Bashi and Gunn [83] compared the methods based on the analytical properties of the Fourier and the Laplace transforms for the calculation of transport coefficients. MacDonnald et al. [84] discussed the applications of the method of moments to the analysis of the profiles of skewed chromatographic peaks. When more than two parameters have to be determined from one single run, the moment analysis method is less suitable, because only the first and second moments are reliable (see Figure 6.9). Therefore, only two parameters can be determined accurately. 

These methods are appealing since the fundamental equation of motion is for the phase-space distribution itself rather than for individual trajectories. The structure of the Fokker-Planck equation in effect carries out a number of averages that must otherwise be performed by generating suitable trajectory ensembles. A preliminary application of the Fokker-Planck method to gas-surface scattering has been made [3.37]. In this application it was assumed that the full phase-space distribution was Gaussian in character with time-dependent first and second moments. Consequently the Fokker-Planck equation produced a set of first-order differential equations for these moments [3.48]. Integration of these equations was essentially... 

However, since the power spectra of the excitation forces are available and the structure Is assumed to be linear, the power spectral method can be applied to determine first and second moment of the response. Hence, carring out a Fourier transform of these matrix equations, the mechanical transfer functions of the structure con be established, in order to reduce the computational effort only the mechanical transfer functions of the top floor deflections and the first floor load effects are determined explicitly. Considering these transfer functions the matrix of the power spectra of the response quantities can be determined as follows. 

As a consequence, field methods, which consist of computing the energy or dipole moment of the system for external electric field of different amplitudes and then evaluating their first, second derivatives with respect to the field amplitude numerically, cannot be applied. Similarly, procedures such as the coupled-perturbed Hartree-Fock (CPHF) or time-dependent Hartree-Fock (TDHF) approaches which determine the first-order response of the density matrix with respect to the perturbation cannot be applied due to the breakdown of periodicity. 

Applying a chromatographic method it is sometimes possible to separate copolymer molecules according to their size Z and composition [5]. The SCD found in such a way can be compared with that calculated within the framework of the chosen kinetic model. The first- and second-order statistical moments of SCD are of special importance. 

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