Weibull application

Gn L) is often difficult to determine for a given load distribution, but when is large, an approximation is given by the Maximum Extreme Value Type I distribution of the maximum extremes with a scale parameter, 0, and location parameter, v. When the initial loading stress distribution,/(L), is modelled by a Normal, Lognormal, 2-par-ameter Weibull or 3-parameter Weibull distribution, the extremal model parameters can be determined by the equations in Table 4.11. These equations include terms for the number of load applications, n. The extremal model for the loading stress can then be used in the SSI analysis to determine the reliability. 

The final reliability formulation for the interference of two 3-parameter Weibull distributions subjected to multiple load applications is given in equation 4.84 ... 

Weibull, W. 1951 A Statistical Distribution Function of Wide Applicability. Journal of Applied Mechanics, 73, 293-297. 

Statistical Methods for Nonelectronic Reliability, Reliability Specifications, Special Application Methods for Reliability Prediction Part Failure Characteristics, and Reliability Demonstration Tests. Data is located in section 5.0 on Part Failure Characteristics. This section describes the results of the statistical analyses of failure data from more than 250 distinct nonelectronic parts collected from recent commercial and military projects. This data was collected in-house (from operations and maintenance reports) and from industry wide sources. Tables, alphabetized by part class/ part type, are presented for easy reference to part failure rates assuminng that the part lives are exponentially distributed (as in previous editions of this notebook, the majority of data available included total operating time, and total number of failures only). For parts for which the actual life times for each part under test were included in the database, further tables are presented which describe the results of testing the fit of the exponential and Weibull distributions. 

Weibull distribution This distribution has been useful in a variety of reliability applications. The Weibull distribution is described by three parameters, and it can assume many shapes depending upon the values of the parameters. It can be used to model decreasing, increasing, and constant failure rates. 

Weibull, W. A Statistical Distribution Function of Wide Application. 7. Appl. Mech., Vol. 18, 1951, pp. 293. 

Equation (20.4.3) defines tlie pdf of the Weibull distribution. Tlie exponential distribution, whose pdf is given in Eq. (20.4.1), is a special case of the Weibull distribution witli p = 1. Tlie variety of assumptions about failure rate and tlie probability distribution of time to failure tliat can be accommodated by the Weibull distribution make it especially attractive in describing failure time distributions in industrial and process plant applications. 

The Weibull distribution will again find application in tlie case study reviewed in Section 21.6. 

In addition to tlie exponential and Weibull distributions, several otlier probability distributions figure prominently in reliability calculations and hazard risk analysis. Presented below are tlieir pdfs, principal characteristics, and an indication of their application. 

When n < 0.7, the ln[—ln(l — a)] against In t plots show curvature and linearity is improved if the latter parameter is replaced by t. This reduces the Weibull distribution to a log-normal distribution. Since both exponential and normal distributions are special cases of the more general gamma distribution, Kolar-Anic and Veljkovic [441] compared the applicability of the Weibull and the gamma distributions. The shape parameter of the latter (e) was shown to depend exclusively on the shape parameter of the former (n). 

These various distributions reflect the variations in reactivity of the reacting sites. Johnson and Kotz [444] discuss in detail the Weibull and other distributions which find application when conditions of strict randomness of the exponential distribution are not satisfied. From an empirical point of view, the power transformation is a practical and convenient method of introducing a degree of flexibility into a model. Gittus [445] has discussed some situations in which the Weibull distribution may be expected to find application, including nucleation and growth processes in alloy transformations. 

W. Weibull. A statistical distribution of wide applicability. J. Appl. Meehan. 1951, 38, 293-297. 

Several of the standard statistical distributions are described by Hahn Shapiro (Statistical Models in Engineering, 1967) with mention of their applicability. The most useful models are the Gamma (or Erlang) and the Gaussian and some of their minor modifications. As an illustration of something different the Weibull distribution is touched on in problem P5.02.18. These distributions usually are representable by only a few parameters that define the asymmetry, the peak and the shape in the vicinity of the peak. The moments are such parameters. 

Other degradation laws can be combined with a Weibull plot, particularly when a lower threshold of, for example, voltage is anticipated. An example of its application to electrical cable insulation lifetime is given in [9]. As well as service failures Weibull plots have proved useful in assessing the failure of fibres and fibre bundles. 

These examples illustrate the application of Weibull plots to service failure data in order to predict the probability of failure before the end of the service life, coupled with a power law to relate time to failure to the applied electric stress (voltage). 

In the work described earlier, the applicability of the Weibull model was further tested by assessing the precision of estimation [expressed by the CV defined as the standard error of estimates divided by the estimated value] and the relative accuracy of estimation of the model parameters (based on the difference of the estimates from the actual value, divided by the actual value). Regarding the precision of estimates, for data with SD = 2 the maximum CV value for Wo, b, and c was 13%, 52%, and 16%, respectively, whereas the corresponding numbers for data with SD = 4 were 33%, 151%, and 34%, respectively. As expected, the precision of the estimates decreases as the SD of the data increases, with the poorest precision for the b estimates and the best for the Wo estimates. Additionally, the maximum CV values were associated with low c values (c = 0.5). 

Weibull. W. (1951). A statistical distribution function of wide applicability. J. Appl. Mech. 18, 293-297. Wells, J.K. and Beaumont, P.W.R. (1985). Debonding and pull-out processes in fibrous composites. J. Mater. Sci. 20, 1275-1284. 

The Weibull growth model = a - b exp( - c x° ) is frequently used in biological and agricultural applications. According to the investigations in (ref. 22), the nonlinearity of this model is considerably reduced if fitted in one of the reparameterized forms... 

The investigation of failures of manufactured components and systems, especially in the electronics and aerospace industries, has generated a variety of statistical models on which data analysis may be based. Each model uses a specific distibution of failure probabilities, and it is important to select a model that matches the actual distribution inherent in the product concerned. In the case of dielectric breakdown, where a large number of quite different modes of failure are known to occur, sometimes even together, the application of a particular statistical failure model must be approached with great caution. Nevertheless, one treatment, based on a Weibull distribution of failure probability, has taken root, and is most generally used in practice. For a dielectric, the Weibull failure probability function has the form... 

We can regard a fiber as consisting of a chain of links. We assume that fiber failure occurs when the weakest link fails. This is called the weakest-link assumption. It turns out that such a weak-link material is well described by the statistical distribution known as the Weibull distribution (Weibull, 1939,1951). We first describe the general Weibull treatment for brittle materials and then describe its application for fibers. 

This distribution appears whenever g a) is given by a power law in (j, coming from the power law variation of the density of linear cracks g l) with their length 1. In the random percolation model considered here, this does not normally occur (except at the percolation threshold p = Pc)- However, for various correlated disorder models, applicable to realistic disorders in rocks, composite materials, etc., one can have such power law distribution for clusters, which may give rise to a Weibull distribution for their fracture strength. We will discuss such cases later, and concentrate on the random percolation model in this section. 

The Weibull distribution is a pure mathematical function, but in many applications two parameters are sufficient to describe the actual time dependency of a... 

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