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 

The three-parameter Weibull cumulative distribution fimction, F t), that predicts the cumulative probability of failure up to a specific time, t, is mathematically expressed by Equation 6.11. The probability density fimction,/(t), which is a derivative of the cumulative distribution fimction, is expressed by Equation 6.12  

A sample Weibull plotting. (Reprinted from ReUaSoft. With permission.) 

In a reliability analysis the most relevant parameters are reflected by the reliability and hazard functions. The reliability function can be mathematically expressed by Equation 6.14  

The hazard fxmction reflects the instantaneous failure rate at a specific time t. For a two-parameter Weibull, it is mathematically expressed as Equation 6.15  

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Weibull analysis Weibull distribution Weibull function Weibullmodulus... 

Today there are many tools available to aid in problem solving or f ure analysis. These include the Weibull Analysis, Failure Mode i Effect Analysis, and Fault Tree Analysis, to name a few. One of the m widely accepted is the Weibull analysis. This method can provide accurate engineering analysis based on extraordinary small samples [1]. 

Abemethy, R. B, Weibull Analysis for Improved Reliability and Maintain ability for the Chemical Industries, 3rd International Conference and Exhi bition on Improving Reliability in Petroleum Refineries and Chemical Plants, Organized by Gulf Publishing Co. and Hydrocarbon Processing, Houston, TX, November 1994. 

Considering a uniformly stressed volume, or, in the context of Weibull analysis, a single link, the invariant I2 corresponds to the magnitude of the stress component in the direction of di7 as shown in Fig. 11.1.12 corresponds to the shear stress on the face normal to dt. The invariants /3 and /4 are the... 

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). 

The ratios of the MLE/BMC were not great, ranging from a mean of 1.39 for a probit analysis of the 10% response to 3.02 for a Weibull analysis of the 10% response. It is important to note that when using the probit analysis, the LOAEL/MLE ratios were equal to or greater than 1 in 99%, 94%, and 71% of the cases for the 1%, 5%, and 10% responses, respectively. The MLE would probably be protective at the 1% response level but not for the 5% and 10% response levels. Similar numbers of 99%, 97%, and 76% were observed for the Weibull analysis. 

While the size effect is very important in craze breakdown and fracture and necessitates the use of the full Weibull analysis in making comparisons between samples of different size, the median plastic strain for fibril breakdown — e is a useful relative indicator of the breakdown resistance of crazes in different polymer... 

The graphical results of various Ss on the Weibull analysis can be seen in Fig. 6.17. If all curves are taken together, the result is the bathtub curve. 

Weibull analysis case study - anisotropic conductive film in flip-chip bonded devices... 

Various assumptions are used in applying the Weibull analysis to strength prediction. Name one. 

This has been demonstrated for a commercial glass ceramic [38], and for alumina [39]. It has been suggested that the microstructure and the defect populations themselves may vary throughout the volume of a component, causing variations in strength unpredictable by Weibull analysis [40,41]. An attempt to verify the minimal specimen size required for strength determination was carried out lately [42]. 

Both functions describe the experimentally determined part of the distribution very well, but they show major differences at low failure probabilities, where no experimental data exist [see Eq. (10)]. This range of low probabilities, however, is exactly the range where distribution curves are applied to predict the reliability of components. For the above example, the R = 99.9999% design stress determined by Weibull analysis is 348 MPa, yet when determined on the basis of the Gaussian... 

The tensile strengths of electroplated Ni continuous carbon fibers have been determined and examined by Weibull analysis [46]. 

Soni PR, Rajan TV, Ramakrishnan P, Continuous nickel electroplating of carbon fibers and Weibull analysis of their tensUe strengths. Metals Mater Proc, 8(2), 187-191, 1996. 

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

Methodologies Used Including Weibull Analysis for Understanding and Explaining the Problem... 

There are a number of studies on the application of Weibull analysis to synthetic fibers, while very limited studies on natural fibers. As this is not under the scope of this book, the former will not be covered. In the following section, some of the studies carried out in respect of natural fibers are discussed. 

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... 

Kulkami AG, Satyanarayana KG, Rohatgi PK (1983) Weibull analysis of strength of coir fibers. Eibre Sci Technol 19 59-76... 

Wu HF, Netravali AN (1992) Weibull analysis of strength-length relationships in single Nicalon SiC fibres. J Mater Sci 27 3318... 

Zhang Y, Wang X, Pan N, Postle R (2002) Weibull analysis of the tensile behavior of fibers with geometrical irregularities. J Mater Sci 37(7) 1401-1406... 

Costa LL, LoiolaRL, Monteiro SN (2010) Tensile strength of bamboo fibers Weibull analysis to characterize the diameter dependence. In Proceedings of biomaterials symposium. First TMS-ABM international materials congress, Rio de Janeiro, Brazil, 26-30 July 2010, pp 1-10 (to be published in CD)... 

Ferreira AS, Monteiro SN, Lopes FPD et al. (2009) Curaua fiber mechanical properties evaluation by the Weibull analysis (in Portuguese). In Proceedings of 64 rd international... 

Nascimento DCO, MottaLC, Monteiro SN (2010) Weibull analysis of tensile tested piassava fibers with different diameters. In Proceedings of characterization of minerals, metals and materials - TMS 2010, Seattle, USA, 21-25 Feb 2010, pp 1-8... 

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