Modified Weibull Distribution Function Research Articles

A new flexible Weibull distribution

Many of studies have suggested the modifications on Weibull distribution to model the non-monotone hazards. In this paper, we combine two cumulative hazard functions and propose a new modifiedWeibull distribution function. The newly suggested distribution will be named as a new flexible Weibull distribution. Corresponding hazard function of the proposed distribution shows flexible (monotone or non-monotone) shapes. We study the characteristics of the proposed distribution that includes ageing behavior, moment, and order statistic. We also discuss an estimation method for its parameters. The performance of the proposed distribution is compared with existing modifiedWeibull distributions using various types of hazard functions. We also use real data example to illustrate the efficiency of the proposed distribution.

Probabilistic modeling of the corrosion of steel structures in marine water-development works

Considering that corrosion takes place as a random process over time, a a probabilistic approach was utilized in this paper. The corrosion of metallic sheet piling employed in the fascia wall of a bulwerk is considered as an example. A stochastic model is constructed on the base of a modified Weibull distribution function with consideration of parameters of the corrosion process as a function of time. One of the factors defining the corrosion rate of the sheet piling is the degree of access of a section of the wall to the zone of variable water level, or the underwater zone. The type of corrosion-continuous or local-is another factor. The accuracy of corrosion prediction in the underwater zone is higher than that in the zone of variable water level.

Characterization of plastic shrinkage cracking in fiber reinforced concrete using image analysis and a modified Weibull function

Numerous strategies have been advocated to reduce the potential for plastic shrinkage cracking in concrete through mixture proportioning, curing methods, or the use of fiber reinforcement. The effectiveness of each approach must be adequately quantified to determine whether the additional initial cost of each strategy is justified. The majority of current research to characterize plastic shrinkage cracking in concrete relies on manual crack observation and measurement that is typically only performed at select locations. These manual measurements of crack width may provide only limited information and may be subject to operator bias. This paper describes a systematic methodology that uses a semi-automated image analysis approach to accurately quantify the salient features of the plastic shrinkage crack patterns. A restrained slab specimen with a stress riser was used to amplify the potential for plastic shrinkage cracking. The specimens described in this paper were exposed to an accelerated drying environment for the first 6 hours after casting. At an age of 24 hours, the crack was assessed using a series of thirteen gray level images that were acquired along the crack path. Crack contours were extracted from these images using gray level intensity thresholding to create binary images. Subtracting the binary image from a uniformly spaced grid enabled the crack width to be determined at numerous locations. This image analysis technique permits the crack widths to be assessed rapidly at numerous locations without operator bias. The measured crack widths were statistically analyzed using a modified Weibull distribution function. This enabled the crack width distribution to be fully described using only two or three parameters. It is anticipated that this approach can be used to better quantify the effects of short randomly distributed fiber reinforcement on plastic shrinkage crack formation.

A model describing the time course and variability in toxicity of hydrophobic chemicals to aquatic organisms

The growth of Chlorella marine, Nannochloris oculate, Pyramimonaos sp., Platymonas subcordiformis and Phaeodactylum tricornutum exposed to chlorobenzene, 1,2-dichlorobenzene, 1,2,3,4-tetrachlorobenzene and pentachlorobenzene was tested. The Boltzman equation was used to describe organism growth. The time course for uptake of hydrophobic organic chemicals (HOCs) by aquatic organisms was expressed by incorporating growth and, if desired, the effect of metabolism into the HOC bioconcentration process. The probability of any given concentration of HOCs in the organisms causing a specified toxic endpoint was expressed with a modified Weibull distribution function. The combined bioconcentration and probability equations were tested with data for time course of incubation of algae exposed to chlorinated benzenes (CBs). A set of parameters, including the uptake rate constant k 1, the elimination rate constant k 2 and thereafter the bioconcentration factor on a dry weight basis, BCF D, the critical HOC concentration in the organism resulting in a specified toxic endpoint, C A * , and the spread factor, S, could be obtained by fitting only experimental data for percent growth inhibition(%)-time-CB exposure concentration. The average coefficients of variation within CBs were 15.2% for BCF D, 21.0% for k 1, 18.3% for k 2, 8.1% for C A * and 9.7% for S. The variability in toxicity (such as EC10, EC50, EC90) derived from the model equations agreed well with those experimentally observed.

Analysis of Size Dependence of Ceramic Fiber and Whisker Strength

The strengths of ceramic fibers and whiskers have been observed to increase with decreasing fiber diameter and length. Typically, both surface flaws and volume flaws exist in ceramic fibers and whiskers, which makes it impossible to characterize the strength dependence of both the diameter and the length with a single‐modal Weibull distribution function. Our data also show that the single‐modal Weibull distribution is inadequate to characterize the strength of fibers with varying diameters even in the case of a constant fiber length. In addition, experimental data also show that, for sapphire whiskers whose surface flaws were removed by chemical polishing, the whisker strength has a much stronger size dependence on diameter than predicted by the single‐modal Weibull function, which indicates that factors other than those characterized by the Weibull function also play a role in the strength of sapphire whiskers. In this paper, the factors affecting the strengths of ceramic fibers and whiskers are analyzed in terms of Weibull statistics, fracture mechanics, and flaw size density variation with varying fiber diameters. A three‐parameter modified Weibull distribution, which combines the above strength‐affecting factors, is proposed to characterize both the diameter and the length dependence for ceramic fibers and whiskers with or without surface flaws. Characterization of the strength data of sapphire whiskers and Nicalon SiC fibers with varying diameters shows the validity of the modified Weibull distribution function.

Modelling of underwater light in freshwater lakes using survival and failure time analysis

1. This study presents a new approach to modelling subsurface irradiance using concepts from survival and failure time analysis. The model applies a modified Weibull distribution function to predict downwelling irradiance. Data sets from forty‐seven Norwegian sites show extremely high coefficients of determination, up to 99.99%, when analysed by the Weibull model.2. The uncritical use of a single kd value to model underwater light conditions is likely to result in poor estimates of received irradiance. This error may amount to several hundred per cent. The practice of force‐fitting linear least‐squares regressions to log‐transformed irradiance data inevitably leads to highly biased estimates of the true fraction of incident irradiance entering the water.3. Wave effects causing fluctuations of subsurface irradiance are modelled with synthetic data and compared with field observations. Fluctuations of surface elevation by waves produce skewed frequency distributions of the underwater light field. The result of these effects, which are to reduce the accuracy of estimated model parameters, can be largely eliminated by carefully designing field procedures used for the acquisition of subsurface light data.

An equation describing the time course and variability in uptake and toxicity of narcotic chemicals to fish

AbstractAn equation is proposed to express the time course of uptake of organic chemicals by fish. The chemicals' octanol/water partition coefficients are used to describe equilibrium partitioning, whereas the clearance rate constant, estimated as a function of water‐ and organic‐phase series resistances to transport, is used to describe the time course of uptake. The probability of the concentration of chemical in the fish causing a defined toxic end point is expressed by using a modified Weibull distribution function that contains an adjustable parameter describing variability in organism response. The effect of metabolism can be included if desired. The combined uptake and probability equations are tested by using mortality data for fathead minnows exposed to 18 narcotic chemicals. A single set of parameters is obtained to fit the entire data set. The equation highlights the importance of considering the kinetics of toxicant accumulation when interpreting toxicity results, especially for hydrophobic chemicals, and reinforces the fact that a common organism residue level (in either molar volume or molarity) is associated with 50% mortality in acute bioassays with narcotic organic chemicals. Assorted features of the equations and their applicability to toxicants with specific modes of action are discussed.

AN EQUATION DESCRIBING THE TIME COURSE AND VARIABILITY IN UPTAKE AND TOXICITY OF NARCOTIC CHEMICALS TO FISH

An equation is proposed to express the time course of uptake of organic chemicals by fish. The chemicals' octanol/water partition coefficients are used to describe equilibrium partitioning, whereas the clearance rate constant, estimated as a function of water- and organic-phase series resistances to transport, is used to describe the time course of uptake. The probability of the concentration of chemical in the fish causing a defined toxic end point is expressed by using a modified Weibull distribution function that contains an adjustable parameter describing variability in organism response. The effect of metabolism can be included if desired. The combined uptake and probability equations are tested by using mortality data for fathead minnows exposed to 18 narcotic chemicals. A single set of parameters is obtained to fit the entire data set. The equation highlights the importance of considering the kinetics of toxicant accumulation when interpreting toxicity results, especially for hydrophobic chemicals, and reinforces the fact that a common organism residue level (in either molar volume or molarity) is associated with 50% mortality in acute bioassays with narcotic organic chemicals. Assorted features of the equations and their applicability to toxicants with specific modes of action are discussed.

Analysis of static and dynamic fatigue of polycrystalline alumina

The fatigue failure of polycrystalline alumina in a moist air environment at 30° C has been analysed in terms of a modified Weibull distribution function using fracture mechanics theory. The good correlation obtained between the fatigue test data and fracture mechanics theory indicates that fatigue is controlled by the slow crack growth of pre-existing flaws. The distribution of these pre-existing flaws can be represented by the modified Weibull distribution which provides an upper and a lower limit strength and thus is more realistic for the physical phenomena it represents. Comparison of proof-test predictions with experiment indicate that the proof test can be effective in eliminating weak samples from the population.

A new modified Weibull distribution function for the evaluation of the strength of silicon carbide and alumina fibres

The strength distributions of silicon carbide and alumina fibres have been evaluated using a modified Weibull distribution function. The function provides an upper and lower strength limit and is characterized by two shape and location parameters. The sum of squares was used as a measure of fit between the distribution function and the data. The result showed good agreement between the two. In addition, the strength distribution and the average value at a different gauge length were extrapolated from the parameters estimated at the original gauge length. In this case also, the proposed function accurately predicted the data points.

Evaluation of Concurrent Flaw Populations in Silicon Carbide in Terms of a Modified Weibull Distribution Function

Concurrent surface‐ and edge‐flaw populations are often observed in strength testing of ceramics The overall strength distribution in such ceramic samples has been analyzed in terms of a modified Weibull distribution function. The junction provides an upper and lower strength limit and is characterized by two shape and location parameters. To evaluate the parameters of the distribution function, the type of flaw that causes failure need not be identified. The applicability of the function has been evaluated in terms of experimental results on SiC specimens. The analysis also shows that the relative values of shape parameters determine which flaw type dominates.

Evaluation of single-fibre strength distribution from fibre bundle strength

A new modified Weibull distribution function has been suggested for analysing the strength of fibres used as reinforcements for advanced composites. The function provides an upper and a lower strength limit and is characterized by two shape and two location parameters. A method for determining the parameters of this distribution from the analysis of the tensile curves of fibre bundles has also been developed. Application of the method to the experimental results on Thornel-300 carbon fibres shows that the shape parameters become modified in the case of bundles.

A Calibration Technique for Size Exclusion Chromatography

Abstract A modified Weibull distribution function is shown to be useful in calibrating the molecular size separation capabilities of both rigid and swellable gel packing materials. Two parameters are used in this function which are related to a packing material's micropore volume distribution. The calibration curves of a set of different packing materials connected in series were predicted from the Weibull calibration functions obtained for each individual packing material.

Versatile Kinetic Approach to Analysis of Dissolution Data

A new kinetically based dissolution equation is presented that considers dissolution of polydisperse systems and disintegrating solid dosage forms. The equation is applicable under sink as well as nonsink conditions and enables the specific dissolution rate parameter, the dispersion parameter, the disintegration lag time, and a newly introduced parameter, the dissolution availability, to be evaluated simultaneously and directly from percent of label claim dissolved versus time data. The equation showed excellent fit to dissolution data for aminophylline tablets. The kinetic significance of the estimated parameters of the equation is discussed. The method of analysis is compared to an approach employing an empirical equation based on a modified Weibull distribution function.