|
|
|
Six Probabilistic MethodsIn the most general case, the failure probability of a system may be expressed in the form
gj(x) is the j-th limit-state function expressed in terms of the vector of random variables, x. Ck denotes the k-th cut set representing a set of limit-states whose joint exceeding constitutes the failure of the system, where the union is overall all the cut sets. The special class of series systems is defined by Eq. (1) when Ck contains a single element for each cut set k. The special class of parallel systems is defined when there is only a single cut set, i.e., k=1.
UNIPASSTM provides the following six categories of probability methods to calculate the probability written in Eq. (1):
Generally speaking, FORM, SORM, RSM and MBM give approximate results while SM and ISM offer "exact" solutions. FORM and SORM make approximations on the limit-state functions around the MPPs and RSM and MBM make the approximation on the limit-state functions around the mean point. These six categories of probability methods differ in accuracy and efficiency. FORM provides first-order failure probability. SORM provides second-order failure probability. SM and ISM give the "exact" failure probability because true limit-state functions are used. The failure probability calculated by FORM and SORM are asymptotically correct with respect to the reliability index (the distance from the failure boundary to the origin in the standard normal space). Although RSM and MBM can give a rough failure probability, they are usually more effective in calculating the mean and standard deviation of the limit-state function.
There are several criteria that should be considered when choosing the proper method for an analysis: Efficiency: The number of limit-state function calculations used in the analysis usually measures the efficiency. This is because in most applications, each limit-state function calculation requires UNIPASS™ to interface with one or more time-consuming external software packages (e.g. a finite element software). Consequently, the calculation of limit-state functions expends the majority of CPU resources. If the closed limit-state function (i.e., without interfacing the external software) is used, very little CPU is needed in all the six preceding probability methods. Therefore, to improve the efficiency of an analysis, the number of the limit-state function calculations must be minimized. As a thumb of rule, the order of efficiency for these six methods is MBM, RSM, FORM, SORM, ISM, and SM. Accuracy: Depending on the purpose of the analysis, different methods provide different degrees of accuracy. In general, one should make the approximation in the area of interest in order to obtain accurate results. For example, if the mean and the standard deviation of limit-state function are of interest, the MVBM and RSM will give satisfactory results. This is because the mean and the standard deviation are the central measurements of the limit-state function in the random variable space, and the approximation to the limit-state function around the mean point made by MVBM and RSM leads to a better approximate limit-state function for computing the mean and standard deviation of the limit-state function. However, if the failure probability is of interest, FORM/SORM offer more accurate results because the failure domain is in the ‘tail’ of the random variable space and FORM/SORM approximate the limit-state function around the MPP, the point with the highest probability density in the failure domain. Invariance: The failure probability and the associated sensitivity measurements should be invariant to any arbitrary expression of the limit-state functions in terms of X. For example, the failure probability should not vary for the two equivalent limit-state functions, x1-x2 = 0 and log x1-log x2 = 0 . Generally, FORM, SORM, SM and ISM satisfy these criteria. Operability: The selected method should be able to accommodate any arbitrary number of random variables. In UNIPASS™, the RSM is limited to a maximum number of 16 random variables. The rest of the methods have no limitations on the number of random variables. Stability: If external software is called, it must provide enough precision for the calculation of derivatives by finite difference. Some external software may not produce enough significant digits in the outputs or may have some degree of noise in the results. These will strongly affect the accuracy in calculating the derivatives of the limit-state functions. In this case, FORM (except the non-gradient based MPPL method), SORM, and MBM may not suitable because they require gradient calculations of the limit-state functions. Feasibility: Not all methods can be used for all problem types and analysis types. While any of the methods can be used for probability analysis, only FORM, SM and ISM can be used for Inverse analysis in a component problem. To determine the applicability of a method for a particular problem and analysis type, refer to Table 1 and Table 2 below.
Depending on the analysis method used, the typical outputs are:
Table 1 Summary of UNIPASS™ Analyses and Problem Types l=Probability Analysis l=Inverse Analysis l=PDF/CDF Analysis
Table 2 UNIPASS™ Probabilistic Methods Selection Guide
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
Last Updated 02/08/10
Send mail to info@predictionprobe.com with
questions or comments about this web site.
|
