Statistical Weibull Analysis

The BMCs were generally lower than the NOAELs when analyzed with either statistical estimate. The mean NOAEL/BMC ratios for the 1%, 5%, and 10% response were 1.60, 1.16, and 0.99, respectively, when using a probit analysis, and 3.59, 1.59, and 1.17, respectively, when using the Weibull analysis. It is interesting to note that comparable means from a Weibull analysis of developmental toxicity data were considerably greater, the developmental toxicity means of the NOAEL/BMC ratios were 29, 5.9, and 2.9 (Allen et al. 1994). 

Keywords Fractography Morphology and properties Plant fibers Statistical evaluation Weibull analysis... 

First, simple statistical analysis (arithmetic average and standard deviation) was applied to three fibers, namely curaua, ramie, and sisal fibers. In other studies, Weibull analysis program was used to analyze the obtained fracture strength values of different diameters. In both the cases, the results indicated that two types of variation of strength of fibers could be possible. Thinner fibers showed increased... 

Future work must address two areas to provide the foundation for statistically based analyses of high-cycle CF (as well as environmental LCF and FCP). For simple laboratory conditions, the Weibull analysis of mechaniccil HCF failure probability [82] must be extended to include CF. Second, variable load, temperature, and environment chemistry histories are likely to be complex in applications and significantly affect CF Hfe. Such history effects have not been studied. The scaling of Basquin relationship data to predict the Ufe of a structure is qualitative and uncertain. Either the local strain approach to CF crack formation/eeurly growth life or the fracture mechanics analysis of CF propagation provide a better foundation for life prediction and failure analysis. 

Outline Life of product population is predicted with the help of Weibull analysis in which a statistical distribution is attempted to fit into life data from a representative sample of units. Then same data set can be used for estimation of important life parameters/cbaracteristics such as reliability or probability of failure at a specific time, the mean life, and the failure rate. For Weibull data analysis, the following information are required ... 

Statistical data analysis of operation time till failure shows that operation time till failure T as random variable follows Weibull distribution (according to performed goodness of fit tests). The parameters k and p are assumed as independent random variables with prior probability density functions p x)—gamma pdf with mean value equals to prior (DPSIA) estimate of k and variance—10% of estimate value, / 2(j)— inverse gamma (as conjugate prior (Bernardo et al, 2003 Berthold et al, 2003)) pdf with mean value equals to prior (DPSIA) estimate of p and variance—10% of estimate value. Failure data tj, j =1,2,. .., 28. Thus, likelihood function is... 

The most important statistical subjects relevant to reverse engineering are statistical average and statistical reliability. Most statistical averages of material properties such as tensile strength or hardness can be calculated based on their respective normal distributions. However, the Weibull analysis is the most suitable statistical theory for reliability analyses such as fatigue lifing calculation and part life prediction. This chapter will introduce the basic concepts of statistics based on normal distribution, such as probability, confidence level, and interval. It will also discuss the Weibull analysis and reliability prediction. 

The reliability theory is heavily dependent on statistics and probability, but was developed apart from mainstream statistics and probability to help insurance companies in the nineteenth century. It also heavily relies on the theory of interference. Part fimctional reliability will be discussed below in terms of safety margins and Weibull analysis. 

The Weibull distribution was first formulated in detail by Walloddi Weibull in 1951, and thus it bears his name. It more accurately describes the distribution of life data, such as fatigue endurance, compared to other statistical distributions, such as the normal distribution which fits better for hardness and tensile strength. Weibull analysis is particularly effective in life prediction. It can provide reasonably accurate failure analyses and failure predictions with few data points, and therefore facilitates cost-effective and efficient component testing. Weibull analysis is widely used in many machine design... 

In many applications, the Weibull analysis is applied to predict the part reliability or unreliability based on limited data with the help of modern computer technology. The limited data will inevitably introduce some statistical uncertainty to the results. Figure 6.11 shows the unreliability as a function of time in a Weibull plot. The six data points reasonably fit on the straight line and verify that the data are a Weibull distribution. The hourglass curves plotted on each side of the Weibull line represent the bounds of 90% confidence intervals for fhis analysis. The width of the intervals depends on the sample size it narrows when more samples are analyzed. 

The analysis of covariance between a continuous variable (P is the curve shape parameter from the Weibull function) and a discrete variable (process) was determined using the general linear model (GLM) procedure from the Statistical Analysis System (SAS). The technique of the heterogeneity of slopes showed that there was no significant difference (Tables 5 and 6). 

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... 

A statistical analysis of the liber tensile strength values determined on a series of fiber samples can be easily made by using the two-parameter Weibull distribution described above. Using the form of the Weibull expression given in Eq. (10.2), we can write the probability of failure F(o) of the liber at a stress a, as... 

As we will see in the appropriate sections of the next two chapters, the precise ranges of the validity of the Weibull or Gumbel distributions for the breakdown strength of disordered solids are not well established yet. However, analysis of the results of detailed experimental and numerical studies of breakdown in disordered solids suggests that the fluctuations of the extreme statistics dominate for the entire range of disorder, even very close to the percolation point. 

All these quantities are readily determined. However, the determination of lc and Of (/c) necessitates a statistical analysis using the Weibull model. Gf (/c) cannot be measured directly, since /c is usually less than 0.5 mm. Therefore, it is determined from the tensile strength Gf (/) at higher gauge lengths using Equation 20... 

In this paper the Weibull theory is applied to very small specimens. The analysis follows the ideas presented in [13]. The relationships between flaw population, size of the fracture initiating flaw and strength are discussed. It is shown that a limit for the applicability of the classical fracture statistics (i.e. Weibull statistics based on the weakest link hypothesis) exists for very small specimens (components). 

Lamon J. Ceramics reliability statistical analysis of multiaxial failure using the Weibull approach and the multiaxial elemental strength model. Journal of the TVmeiican Ceramic Society 1990 73(8) 2204-2212. 

CARES (Ceranfics Analysis and Reliability Evaluation of Structures) is a public-domain program from the National Aeronautic and Space Agency (NASA) that incorporates Weibull statistics. The program was formally known by the less friendly acronym SCARE (Structural Ceranfics Analysis and Reliability Evaluation). 

A systematic analysis has been made for the statistical approach to describe secondary drop size distributions. Two groups were identified. An empirical one based on the Weibull distribution where the scale and shape parameters can change according to the degree of control desired over the size and frequency range. The second group is semiempirical and is associated with a log-normal distribution function. The statistical meaning of the log-normal expresses the multiplicative nature of the secondary atomization process. 

Modeling. In order to carry out the analysis of the nature of the operational phenomena in facilities and equipment, it is very useful to use statistics as a support for the quantification of the parameters. The phenomena s historical behavior is characterized based on operation and failure periods that have occurred since the commissioning time. The conditions that characterize the equipment operational time data are so numerous that it is not possible to say when exactly the next failure will occur. However, it is possible to express which will be the probability that the equipment is in operation or out of service at any given time. These times are associated with a cumulative distribution function of the random variable, which is defined as the addition of the probabilities of possible values of the variable that are lower or equal to a preset value. The mentioned random variable is constituted by the operating times and downtime of equipment or system in a given period. For its parameterization Weibull distribution is very appropriate as it is very effective and relatively simple to use in the reliability evaluation of a system by quantifying the probability of failure in the performance of the system s duties from the failure probabilities of its components based on the operation times. There are three different parameters ... 

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