Weibull definition

Theodore et al. employed Monte Carlo metliods in conjunction with the binomial and Weibull distributions to estimate out-of-compliance probabilities for electrostatic precipitators on tlie basis of observed bus section failures. The following definitions apply (see Fig. 21.6.1). 

Model-dependent comparison of two time profiles is best achieved in terms of the semi-invariants discussed earlier in the section on Characterization of Semi-invariants ( Moments ). This treatment is in accordance with the Level B definition of IVIVC, as proposed in several official guidelines. It makes full use of the underlying model that the data are presented by a distribution function, but no specific function is required. Although derived function parameters (e.g., Weibull, polyexponential, etc.) may be used, the computation may also be performed numerically on the observations as such. 

A statistical definition of brittleness can be formulated in terms of the Weibull distribution of fracture probability for a material (Derby et al., 1992). The Weibull modulus m (see Eq. 2) can range from zero (totally random fracture behaviour, where the failure probability is the same at all stresses, equivalent to an ideally brittle material) to infinity (representing a precisely unique, reproducible fracture stress, equivalent to an ideally non-brittle material). 

The Master Curve methodology uses a mathematical model to describe the probability of cleavage fracture initiation in a material containing a distribution of postulated fracture initiators (flaws). The model includes the temperature dependence of Kj, which was estimated empirically from a data set including various ferritic structural steels. The scatter definition based on the Weibull distribution, the size adjustment and the definition of the temperature dependence are the basic elements of the Master Curve methodology as described in ASTM E 1921. 

As with most estimators, these Weibull estimators have the property of yielding "biased" or offset estimates. The definition of bias is most easily illustrated using Figure 1 (generated during an earlier study on analysis of fracture origin positions) which... 

Goda (Fig. 38.4) exhibits the difference between the distribution functions of the partial duration and annual maximum series data in terms of return period for a given wave height. The 50-year wave height is set at 8.0 m. Three distributions FT-II and two Weibulls are shown with their converted aimual maximum distributions. Because of the definition of annual maxima, its retmn period must be greater than 1.0 year. Except for the zone of short return period, the distribution for the annual maximum series exhibits little difference from that for the partial duration series. 

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