Three-parameter Weibull Distribution Function Research Articles

Experimental Study on Flexural Fatigue Resistance of Recycled Fine Aggregate Concrete Incorporating Calcium Sulfate Whiskers

In order to study the flexural fatigue resistance of calcium sulfate whisker-modified recycled fine aggregate concrete (RFAC), flexural fatigue cyclic loading tests at different stress levels (0.6, 0.7, and 0.9) considering a calcium sulfate whisker (CSW) admixture as the main influencing factor were designed. Furthermore, the fatigue life was analyzed, and fatigue equations were established using the three-parameter Weibull distribution function theory. In addition, the micro-morphology of CSW-modified recycled fine aggregate concrete was observed and analyzed through Scanning Electron Microscopy (SEM), and the strengthening and toughening mechanisms of CSW on recycled fine aggregate concrete were further explored. The test results demonstrate that the inclusion of recycled fine aggregate reduces the fatigue life of concrete, while the incorporation of CSW can effectively improve the fatigue life of the recycled fine aggregate concrete, where 1% of CSW modification can extend the fatigue life of recycled fine aggregate concrete by 56.5%. Furthermore, the fatigue life of concrete under cyclic loading decreases rapidly as the maximum stress level increases. Fatigue life equations were established with double logarithmic curves, and P-S-N curves considering different survival probabilities (p = 0.5, 0.95) were derived. Microscopic analyses demonstrate that the CSW has a “bridging” effect at micro-seams in the concrete matrix, delaying the generation and enlargement of micro-cracks in the concrete matrix, thus resulting in improved mechanical properties and flexural fatigue resistance of the recycled fine aggregate concrete.

Appropriateness of Reduced Modified Three-Parameter Weibull Distribution Function for Predicting Gold Production in Ghana

Appropriateness of Reduced Modified Three-Parameter Weibull Distribution Function for Predicting Gold Production in Ghana

Three-parameter Weibull distribution with upper limit applicable in reliability studies and materials testing

The Weibull plot with concave shape of an experimental curve mostly indicates the necessity to use the three-parameter Weibull distribution with nonzero location parameter, which represents the lower limit of studied data set. The paper proposes a three-parameter Weibull distribution function with the localization parameter representing the upper limit of the studied data set. This function is suitable for the Weibull plot with convex shape of experimental curve. Several examples of applying the newly proposed function are presented: for lifetime or time-to-breakdown in reliability studies and for strength properties in materials testing.

Flexural fatigue behaviors of high-content hybrid fiber-polymer concrete

The gaining effect of fibers with different scales on the crack resistance of concrete is different. Microscale fibers play an essential role in preventing cracking, while macro scale fibers mainly produce crack prevention effects during crack propagation. This paper aims to investigate the flexural static and fatigue performances of high-content hybrid fiber-polymer concrete (HCHFPC). The results show that the microstructure of HCHFPC is more compact than plain concrete (P.C.), the micro-cracks and holes are significantly reduced, and the flexural strength is about three times that of P.C. Through the test of the three-parameter Weibull distribution function, it was found that there is a good correlation between the fatigue life of each specimen. Based on this, a fatigue life prediction model with different failure probability (PSN) is established. Finally, by comparing the flexural fatigue performance of mainstream high-performance concrete, it is confirmed that HCHFPC has high toughness and flexural fatigue durability.

High-throughput experiments for rare-event rupture of materials

High-throughput experiments for rare-event rupture of materials

Ultra-high three-point bending fatigue fracture characteristics of CFRP modified by MWCNTs and fatigue life data analysis

An uninterrupted constant amplitude ultrasonic three-point bending fatigue test was carried out on plain weave composites with different Multi-wall carbon nanotubes (MWCNTs) content. S-N curves under different reliability levels, Weibull distribution curves with different stress levels, and fatigue life survival probability distribution were plotted. MWCNTs were a kind of nano-particles that significantly improved the anti-fatigue performance of CFRP. It increased the interface strength of carbon fiber and epoxy resin and ultimately prolonged the ultra-high cycle fatigue life of CFRP. In addition, the three-parameter Weibull distribution function was more in line with the distribution characteristics of CFRP fatigue life data.

Three-parameter (3P) weibull distribution for characterization of strength of ceramics showing R-Curve behavior

Strength distribution of advanced ceramics is commonly characterized by two-Parameter (2P) Weibull distribution. However, deviation of strength distribution from 2P-Weibull distribution may occur in ceramics due to various mechanisms. R-curve behavior is one of these mechanisms where increase of fracture resistance with the extension of crack occurs. In such cases, 2P-Weibull distribution may not be the best fitting distribution function based on the goodness-of-fitness tests. This article examines the effectiveness of three-parameter (3P) Weibull distribution function for fitting the strength variation due to R-curve effect by using experimental and virtual strength data. The effect of Weibull parameters, degree of increase in crack resistance and number of samples on effectiveness of fitting via 3P-Weibull distribution is investigated. It is reported that 3P-Weibull distribution function fits the strength distribution better than 2P-Weibull distribution function for materials showing R-curve behavior when the crack resistance curve is steep and Weibull modulus is high. Furthermore, it is shown that at least 100 samples should be used for a reliable estimate when the material exhibits R-curve behavior.

Defect-based probabilistic fatigue life estimation model for an additively manufactured aluminum alloy

A probabilistic model to estimate the fatigue life of an additively manufactured material with polished surface was developed based on the statistical size distribution of grains and various manufacturing defects such as pores and unfused powder particles embedded in the microstructure. The probabilistic model includes the prospect of each microstructural feature to exist on the specimen surface and its potential impact on fatigue crack initiation through defect-grain interaction. The model was applied to as-built and peak-hardened Al-1.5Cu-0.8Sc-0.4Zr (wt.%) alloy developed by laser powder bed fusion. Bending fatigue experiments performed on both as-built and peak-hardened samples confirmed the trends predicted by the probabilistic model. The cumulative probability distribution plotted against fatigue life fitted well to a three-parameter Weibull distribution function. The indicator for the scatter in fatigue life from the Weibull fit suggested that peak hardening of the material narrowed fatigue life bounds. The probabilistic model was further used as a predictive tool to estimate fatigue life for a preferred microstructure with a lower pore number density and pore size.

A standardized measurement method and data analysis for the delamination strengths of YBCO coated conductors

Of crucial importance for the application of high temperature superconducting coated conductors (CCs) is the delamination strength due to thermal stress and Lorentz force during their operation. This paper reports the mechanical delamination strengths of YBCO CCs at room temperature and 77 K, as well as the electro-mechanical delamination strength form a modified anvil tensile method. Compared to previous measurement setups, our anvil device only has one degree of freedom of transverse tension. Moreover, we use a vertical soldering technique to ensure the data reliability. Since we eliminate the influence of the measurement method on the experimental results, a naturally discrete property of the delamination strength is obtained. To describe these experimental data, a three-parameter Weibull distribution function has been suggested and a new criterion for determining delamination strength is proposed from the reliability function, which can be conveniently referred to for engineering design.

Long-term effects of salvage logging on stand composition in seminatural spruce forests

The tree diameter distribution was used to characterize the impact of forest disturbances and the ecosystem dynamics during the process of regeneration in the absence of any management 44 years after the event. In addition, the impact of salvage logging after windthrow remains unknown, especially on long-term responses. In this study, comparisons were made between the “unsalvaged” and “salvaged” post-storm management scenarios. The study was conducted in Slitere Natural Park in the Northwest part of Latvia, where large-scale windthrow occurred on November 2, 1969. The three-parameter Weibull function was used to describe the diameter distribution of Norway spruce (Picea abies (L.) Karst.) stands formed after large-scale wind storm in two site types (on fertile mineral soils: with normal moisture regime (fresh)—Oxalidosa—and with periodic excess water (wet)—Myrtilloso-sphagnosa) and where salvage logging was or was not carried out (management scenarios). The naturally regenerated stands remained Norway spruce dominated and were independent of the post-storm management scenarios; still, they were significantly affected by a forest type. The three-parameter Weibull distribution function suited well to actual diameter distributions of the sample plots. Diameter distributions significantly differed between the forest types: In Myrtilloso-sphagnosa (wet) forest stands, it was a negative exponential and in Oxalidosa (fresh) stands unimodal and positively skewed. We did not detect distribution differences caused by salvage after a large-scale disturbance, indicating that other factors (like ecological importance of deadwood, a risk of insect outbreaks to neighboring areas, etc.) need to be considered, when deciding on the post-storm management.

Response of wind speed probability distribution parameters to urban expansion

Recent studies have indicated that wind speed probability distribution can be described closely using three-param- eters Weibull distribution, which has come to be widely used in the description of wind speed probability distribution over the past several years. The fitting effect of probability density curve would be better than the universal three-parameter Weibull distribution in calm wind by changing the universal three-parameter Weibull distribution function definition domain condition [ r , ∞) to a limited three-parameter Weibull distribution function definition domain [0, ∞), which would be more applicable to actual wind speed distribution. Timely observation wind speed of Beijing, Nanjing, Guangzhou, and other 5 large urban stations and their comparative county stations, and corresponding NCEP/NCAR Reanalysis-1 wind speed data collected from 1985–2014 were divided into sets, each of which had an average of six periods. The three parameters ( k , c , r ) of Weibull distribution of wind speed probability distribution are fitted using numerical iteration to analyze the changes in wind speed probability distribution parameters with respect to differences in wind speed section, and synthetic variations of those parameters can reflect changes in different wind speed sections, such as calm wind, light wind, and strong wind probability under urbanization, which can provide thorough understanding of the relationship to city size indexes and wind speed probability distribution parameters. Data analysis and mathematical derivation were here used to draw the following conclusions: (1) the relationship of urban-rural wind speed difference D and rural wind speed V x can be expressed as D = b 1 V x + i . Under these conditions, city wind speed parameters k , c , and r and the corresponding rural wind speed parameters c x , k x , and r x have the following relationship: c =( b 1 + 1) c x , r =( b 1+1) r x + i , k = k x . (2) As shown, parameter c is affected by unitary surface friction function, parameter r by the combination of various kinds of factors, and the change in parameter k is only an expression of climatic random fluctuation produced by constructing relative variables of c / c x = b 1 + 1, r / r x = b 1+1+ ir x , k / k x =1. (3) It has a correlation with the degree of variation in wind speed probability distribution parameters and the speed of city expansion. Urbanization resulted in significant variations in location parameter r , scale parameter c comes second and shape parameter k did not show any significant changes. (4) Parameter c decreased with city build-up area enlarges. Seven stations’ regression equations pass the significant tests and the variation rates are between −0.0203–−0.0016 km−2, which means that the expansion of city scale will narrow the distribution range of the wind speed probability density curve. Parameter r presents a rising trend (the value changes from negative to positive) along with the extension of city build-up area. Six stations’ regression equations showed significant results, and the rates of variation were between 0.0014 and 0.0171 km−2, which indicated that the likelihood of calm wind decreased as the urban area expanded. Analysis also showed a close relationship between parameter r and latitude. (5) Using reanalysis data to fit the comparison parameters c x and r x produced the relative variable c / c x , and r / r x , analyses showed results similar to the consequences of c and r . Shape parameter k showed a non-significant relationship with the city size index in comparison.

High-Cycle Microscopic Severe Corrosion Fatigue Behavior and Life Prediction of 25CrMo Steel Used in Railway Axles

The effects of environmental media on the corrosion fatigue fracture behavior of 25CrMo steel were investigated. The media include air, and 3.5 wt % and 5.0 wt % NaCl solutions. Experimental results indicate that the media induces the initiation of corrosion fatigue cracks at multiple sites. The multi-cracking sites cause changes in the crack growth directions, the crack growth rate during the coupling action of the media, and the stress amplitude. The coupling effects are important for engineering applications and research. The probability and predictions of the corrosion fatigue characteristic life can be estimated using the three-parameter Weibull distribution function.

Using the three‐parameter Weibull distribution in assessment of threshold strength of pressboard impregnated by different liquid dielectrics

This study presents the results of the measurements of electrical strength of pressboard specimens impregnated with three different liquid dielectrics: mineral oil, natural ester and synthetic ester. In the experiment carried out in accordance with IEC 60243 standard, two types of pressboard specimens of thicknesses 0.5 and 1 mm, respectively, were considered. The data obtained during the measurements in the form of breakdown voltages were used for calculation of the electrical strength. Both standard-based approach where median value was taken into account in calculations and statistical approach using three-parameter Weibull distribution function where location parameter V 0 was considered were applied. The results obtained on the basis of both approaches showed that both the specimens impregnated with mineral oil and ester liquids gave the satisfactory results within the electrical strength determined, which in all cases was higher than minimum required for the impregnated pressboards. The differences indicating slightly better impregnation properties of mineral oil, when the same impregnation conditions are used, are noticeable especially in the case of location parameter of Weibull distribution function. However, these differences are small and in relation to the requirements for the impregnated pressboards may be treated as negligible.

Probabilistic failure analysis for glass components under general loading conditions

In this work, an experimental programme consisting in four-point bending and coaxial double ring tests is performed on glass specimens with different dimensions. The results are analyzed using a novel probabilistic model aiming at proving the suitability of the three-parameter Weibull distribution function to predict failure of structural glass elements under general loading cases. The tests evaluation consists in the calculation of the stress distribution for each test modality using a finite element commercial code, in the election of a reference parameter and a failure criterion and in the estimation of the shape, location and scale Weibull parameters defining the failure cumulative distribution function for the reference parameter and the effective area. The satisfactory results obtained confirm the applicability of the procedure.

Análisis probabilístico de elementos de vidrio recocido mediante una distribución triparamétrica Weibull

Glass and ceramics present a brittle behaviour so a large scatter in the test results is obtained. This dispersion is mainly due to the inevitable presence of micro-cracks on its surface, edge defects or internal defects, which must be taken into account using an appropriate failure criteria non-deterministic but probabilistic. Among the existing probability distributions, the two or three parameter Weibull distribution is generally used in adjusting material resistance results, although the method of use thereof is not always correct.Firstly, in this work, the results of a large experimental programme using annealed glass specimens of different dimensions based on four-point bending and coaxial double ring tests was performed. Then, the finite element models made for each type of test, the adjustment of the parameters of the three-parameter Weibull distribution function (cdf) (λ: location, β: shape, δ: scale) for a certain failure criterion and the calculation of the effective areas from the cumulative distribution function are presented.Summarizing, this work aims to generalize the use of the three-parameter Weibull function in structural glass elements with stress distributions not analytically described, allowing to apply the probabilistic model proposed in general loading distributions.

Etching Process Effects on Surface Structure, Fracture Strength, and Reliability of Single-Crystal Silicon Theta-Like Specimens

The etching processes used to produce microelectromechanical systems (MEMS) leave residual surface features that typically limit device strength and, consequently, device lifetime and reliability. In order to optimize MEMS device reliability, it is therefore necessary to determine the effects that these etching processes have on MEMS component strength. The microscale theta specimen, which is shaped like the Greek letter Θ, acts as a tensile test specimen when loaded in compression by generating a uniform tensile stress in the central web region of the specimen. Three sets of single-crystal silicon theta specimens are fabricated using two deep reactive ion etching recipes and a temperature-controlled cryogenic plasma etching recipe, each set resulting in a different specimen surface structure. The resulting strength distributions are analyzed in two ways. First, the strength data are fit to a three-parameter Weibull distribution function to determine the lower bound, or threshold strength, of each distribution. Second, the strength data are used in conjunction with various loading schemes to assess their effect on the lifetime spectrum of the device. In both approaches, the theta specimen is used to great effect to gain quantitative insight into the role of etching-induced surface features on the manufacturing yield and operational reliability of MEMS components.

A New Life Theory for Rolling Bearings—By Linkage between Rolling Contact Fatigue and Structural Fatigue

Based on a linkage between rolling contact fatigue and structural fatigue, a new life theory for rolling bearings has been proposed using the three-parameter Weibull distribution function. A minimum life is introduced as the third parameter, which may be considered independent from the statistical strength of material. The following are the new findings: (1) Reliability functions and life equations were introduced for the rolling bearing races and the life test specimens made of the same material (JIS SUJ2/AISI 52100), and the linkage between them has successfully been established by using linkage factors. (2) The 90% load rating and rating life formulas for the races as well as for the whole bearing were developed. (3) Newly introduced 100% load rating and rating life formulas were postulated by introducing the minimum lives for the life tests and theoretical research. (4) For the ratios of each load rating between the test results on the grease-lubricated deep-groove ball bearings and the new life theory, the linkage factors were determined as β = 2.4 and β′ = 2.16 for 90 and 100% load ratings, respectively. (5) A reduction factor λ for 100% load rating was determined as λ = β′/β = 0.9.

Weibull Distribution Function Application to Static Strength and Fatigue Life of Materials

A rigorous study on the Weibull distribution function proposed by Weibull in 1939 and the dynamic capacity and the life formula for rolling bearings postulated by Lundberg-Palmgren in 1947 is performed especially with reference to the stressed volume, which relates to the number of individual probability in the product law of probabilities used in reliability function and probabilistic stress-life (P-S-N) relationship for structural materials as well as probabilistic force-life (P-F-L) relationship for rolling bearings. By means of the three-parameter Weibull distribution function with a minimum strength/life as the third parameter independent of statistical strength, a new statistical theory for treating the static strength/fatigue life of materials, a linkage between structural fatigue and rolling contact and P-F-L curve of ball bearings, is discussed and examined. As a result, the stressed volume proposed by Weibull for structural materials in the equal stress field yields a stressed volume, stressed area, stressed length, and/or stressed point corresponding to a critical stress of failure depending on the acting stresses, because they are expressed by the number of product law of probabilities. It has been established that a 50% stress rating for static strength and a 90% stress rating for fatigue life of materials as well as a 90% stress rating for the race of ball bearings could be specified. The linkage example can be shown by using the alternating torsion P-S-N test for JIS SUJ2/AISI 52100 steel and the race of ball bearings. Furthermore, it is also found that the stress-life/load-life exponent that depends on the strength of the materials is independent of the Weibull slope as well as the stressed volume, which were introduced in Lundberg-Palmgren's theory.

An iterative method to obtain the specimen-independent three-parameter Weibull distribution of strength from bending tests

Brittle materials, such as glass and ceramics, usually present a large strength scatter. Among other probability distributions, the Weibull distribution is widely used to characterize their resistance. Often the two-parameter model is employed, omitting the consideration of a threshold stress, leading to a simplified estimation method. For the sake of generality the present work uses fracture data from bending tests to obtain a three-parameter Weibull distribution function valid for a uni-axially and uniformly tensioned material element. The variable stress state prevailing in the flexural specimen and the size effect are simultaneously accounted for by means of an iterative fitting procedure. The method is extended to account for bimodal flaw distributions, discriminating between edge and surface failure results based on experimental observation. Finally, the model is applied to simulated data sets, obtaining satisfactory results.

On the Weibull cost estimation of building frames designed by simulated annealing

This paper proposes a general methodology to determine the number of numerical tests required to provide a solution for a heuristic optimization problem with a user-defined accuracy as compared to a global optimal solution. The methodology is based on the extreme value theory and is explained through a problem of cost minimization for reinforced concrete building frames. Specifically, 1000 numerical experiments were performed for the cost minimization of a two-bay and four-floor frame using the Simulated Annealing (SA) algorithm. Analysis of the results indicates that (a) a three-parameter Weibull distribution function fits the results well, (b) an objective and general procedure can be established to determine the number of experiments necessary to solve an optimization problem with a heuristic which generates independent random solutions, and (c) a small number of experiments is enough to obtain good results for the structural engineer.