An alternative rule for placement of empirical points on Weibull probability paper

This paper proposes an alternative rule for placing empirical points on Weibull probability paper. Following the standard or old rule, one takes the mean or median value of the order statistics first, then performs the Weibull transformation. The new rule is to make the Weibull transformation first, then take the expected value of the transformed order statistics. Although the new rule for placing points requires much more complicated calculations, a simple Monte Carlo experiment shows that the new rule estimator of the shape parameter is unbiased, whereas the old rule estimator turns out to be heavily biased. Copyright © 1999 John Wiley & Sons, Ltd.     

A modified Weibull extension with bathtub-shaped failure rate function

Models with bathtub-shaped failure rate function are useful in reliability analysis, and particularly in reliability related decision making and cost analysis. The traditional Weibull distribution is, however, unable to model the complete lifetime of systems with a bathtub-shaped failure rate function. In this paper, a new model, which is useful for modeling this type of failure rate function, is presented. The model can also be seen as a generalization of the Weibull distribution. Parameter estimation methods are studied for this new distribution. Examples and results of comparison are shown to illustrate the applicability of this new model.     

A study of Weibull shape parameter: Properties and significance

The two-parameter Weibull distribution has been widely used for modelling the lifetime of products and components. In this paper we study the effect of the shape parameter on the failure rate and three variables of importance in the context of maintenance and reliability improvement. These variables are (i) time to failure, (ii) age at replacement based on risk and (iii) residual life. We propose a classification scheme for the distribution based on the shape parameter and discuss the application of the results.     

Weibull prediction of a future number of failures

This paper presents simple new prediction limits for the number of failures that will be observed in a future inspection of a sample of units. The past data consist of the cumulative number of failures in a previous inspection of the same sample of units. Life of such units is modelled with a Weibull distribution with a given shape parameter value. Copyright © 2000 John Wiley & Sons, Ltd.     

n-fold Weibull multiplicative model

In this paper we study the n-fold multiplicative model involving Weibull distributions and examine some properties of the model. These include the shapes for the density and failure rate functions and the WPP plot. These allow one to decide if a given data set can be adequately modelled by the model. We also discuss the estimation of model parameters based on the WPP plot.     

Sample size dependence of flaw distributions for the prediction of brittle solids strength using additive Weibull bimodal distributions

A method to evaluate the mean strength of brittle solids at short gauge length from experiments performed at higher gauge length is proposed for bimodal fracture behavior. The method based on additive Weibull bimodal distributions takes the evolution of the relative proportion of flaws along the gauge length into account via a gauge length dependent mixing parameter. A linear dependence of this parameter vs. the gauge length is proposed. The approach is assessed using experimental results on carbon and E-glass fibers. The new method provides values for the average tensile strength of the fibers at 100 μm up to 27% higher than those calculated using the classical approach. The underestimation of the classical approach can be attributed to the weight of the severe category of flaws at short length that is considered to be the same as that determined at the experimental gauge length, which can be several orders higher. A simplified approach taking into account solely the more severe category of flaw is shown to be applicable for the prediction of the strength at short gauge length independently of the nature of the fiber.     

Experimental evaluation of the strength distribution of fibers under high strain rates by bimodal Weibull distribution

This paper proposes a bimodal Weibull distribution model for strain- rate- and temperature-dependent fiber strength. The relationships of the mechanical quantities between fiber and fiber bundles at different strain rates and temperatures under tensile impact are established. A method for determining mechanical parameters of fibers by tensile impact tests of fiber bundles is established. Experiments on E-glass bundles have been performed at six strain rates (90, 300, 800, 1100, 1300 and 1700 s⁻¹) at three different temperatures (−70, 14, 80°C). According to the statistical analysis and models, the mechanical parameters for the fiber and their relationships with strain rate and temperature are obtained from the tensile impact experimental results. The emulated stress/strain curves from the model are in good agreement with the test data. The theoretical model and test results show that the shape parameters, β^d₁ and β^d₂, are not only strain rate independent but also temperature independent. The scale parameters σ^d₀₁ and σ^d₀₂, which change with strain rate and temperature, are not constant.     

On the validity of the Weibull failure model for brittle particles

The Weibull theory of material strength and fracture assumes that the Weibull modulus m is a material parameter, which does not depend on shape and size of the loaded object. Based on large data sets from single-particle fracture experiments with brittle materials (glass, clinker cement, limestone), the authors show that the Weibull modulus of nearly spherical particles seems to decrease with increasing particle diameter. A possible explanation is that the inner structure of the particles depends on their size so that small particles are much stronger than large ones. Received: 9 December 1999     

Loading rate effects on parameters of the Weibull stress model for ferritic steels

This study investigates the effects of loading rate on parameters of the Weibull stress model for prediction of cleavage fracture in a low strength, strongly rate-sensitive A515-70 pressure vessel steel. Based on measured, dynamic fracture toughness data from deep- and shallow-cracked SE(B) specimens, the calibrated Weibull modulus (m) at shows little difference from the value calibrated previously using static toughness data. This newly obtained result supports the hypothesis in an earlier study [Gao X, Dodds RH, Tregoning RL, Joyce JA. Weibull stress model for cleavage fracture under high-rate loading. Fatigue Fract Engng Mater Struct 2001;24:551-64] that the Weibull modulus likely remains rate independent for this material over the range of low-to-moderate loading rates. Additional experimental and computational results for higher rates show that a constant m-value remains applicable up to the maximum loading rate imposed in the testing program . Rate dependencies of the scale parameter (σ_u) and the threshold parameter (σ_w-min) are computed using the calibrated m, and the results indicate that σ_u decreases and σ_w-min increases with higher loading rates. The predicted cumulative probability for cleavage fracture exhibits a strong sensitivity to small changes in σ_u. Consequently, σ_u must be calibrated using dynamic fracture toughness data at each loading rate of interest in an application or selected to make the Weibull stress model predict a dynamic master curve of macroscopic toughness for the material.     

Bayesian estimation of mixed Weibull distributions

Estimation of mixed Weibull distribution by maximum likelihood estimation and other methods is frequently difficult due to unstable estimates arising from limited data. Bayesian techniques can stabilize these estimates through the priors, but there is no closed-form conjugate family for the Weibull distribution. This paper reduces the number of numeric integrations required for using Bayesian estimation on mixed Weibull situations from five to two, thus making it a more feasible approach to the typical user. It also examines the robustness of the Bayesian estimates under a variety of different prior distributions. It is found that Bayesian estimation can improve accuracy over the MLE for situations with low mixture ratios so long as the prior on the weak subpopulation's characteristic life has an expected value less than or equal to the true characteristic life.     

A study of two estimation approaches for parameters of Weibull distribution based on WPP

Least-squares estimation (LSE) based on Weibull probability plot (WPP) is the most basic method for estimating the Weibull parameters. The common procedure of this method is using the least-squares regression of Y on X, i.e. minimizing the sum of squares of the vertical residuals, to fit a straight line to the data points on WPP and then calculate the LS estimators. This method is known to be biased. In the existing literature the least-squares regression of X on Y, i.e. minimizing the sum of squares of the horizontal residuals, has been used by the Weibull researchers. This motivated us to carry out this comparison between the estimators of the two LS regression methods using intensive Monte Carlo simulations. Both complete and censored data are examined. Surprisingly, the result shows that LS Y on X performs better for small, complete samples, while the LS X on Y performs better in other cases in view of bias of the estimators. The two methods are also compared in terms of other model statistics. In general, when the shape parameter is less than one, LS Y on X provides a better model; otherwise, LS X on Y tends to be better.     

Comparison of Weibull small samples using Monte Carlo simulations

The evaluation of the functional reliability of different designs is a common task and times to failure can be compared using the likelihood ratio test. In the microelectronics industry, as in many others, the high cost of testing places severe restrictions on the sample size. Moreover, the products in these tests are often new and do not have previous reliability histories. These factors make the selection of the Type I and Type II errors in comparison tests very difficult. This paper presents the Monte Carlo simulation results of Type II errors for the likelihood ratio test of comparison as a function of the Type I error and the (small) sample size. Our conclusions are summarized as follows: (1) the common microelectronics industry standard sample size of 32 is often insufficient to reach satisfactory conclusions; (2) small sample tests should only be used for prescreening for significant differences; and (3) when only small samples are available, the Type I and the Type II errors must be selected carefully to prevent misleading conclusions. Copyright © 2006 John Wiley & Sons, Ltd.     

An investigation of the loading rate dependence of the Weibull stress parameters

This paper examines the dependence of the Weibull stress parameters on loading rate for a 22NiMoCr37 pressure vessel steel. Extensive fracture tests, including both quasi-static and dynamic tests, are conducted using deep- and shallow-cracked SE(B) specimens. The fracture specimens are carefully prepared to ensure the crack fronts are placed at the location where the material is homogeneous. Three dynamic loading rates (in terms of the stress intensity factor rate, in the low-to-moderate range are considered. The load-line velocities for the dynamic tests are chosen so that the resulted values for the deep- and shallow-cracked specimens are the same. Independent calibrations performed at each loading rate (quasi-static and the three dynamic loading rates) using deep- and shallow-cracked fracture toughness data show that the Weibull modulus, m, is invariant of loading rate. The calibrated m-value is 7.1 for this material. Rate dependencies of the scale parameter (σ_u) and the threshold parameter (σ_w-min) are computed using the calibrated m and the results indicate that σ_u decreases and σ_w-min increases with higher loading rates. The demonstrated loading rate invariant of m, when combined with the master curve for dynamic loading, can provide a practical approach which simplifies the process to estimate σ_u as a function of loading rate.     

Conditional Weibull Control Charts Using Multiple Linear Regression

Because in Weibull analysis, the key variable to be monitored is the lower reliability index (R(t)), and because the R(t) index is completely determined by both the lower scale parameter (η) and the lower shape parameter (β), then based on the direct relationships between η and β with the log‐mean (μ_x) and the log‐standard deviation (σ_x) of the analyzed lifetime data, a pair of control charts to monitor a Weibull process is proposed. Moreover, because the fact that in Weibull analysis, right censored data is common, and because it gives uncertainty to the estimated Weibull parameters, then in the proposed charts, μ_x and σ_x are estimated of the conditional expected times of the related Weibull family. After that both, μ_x and σ_x are used to monitor the Weibull process. In particular, μ_x was set as the lower control limit to monitor η, and σ_x was set as the upper control limit to monitor β. Numerical applications are used to show how the charts work. Copyright © 2016 John Wiley & Sons, Ltd.     

Fault diagnosis as an aid to process machine reliability

Identification of the reasons for failures of process plant machines is the first step in obtaining increased reliability. Weibull analysis of the service life obtained from maintenance records requires a minimum of only five incidents and provides an insight into the failure mechanism. Two examples are given where Weibull analysis of maintenance records helped identify the cause of failure and was instrumental in achieving enhanced reliability of process plant machines.     

A new approach to MLE of Weibull distribution with interval data

We study the two-parameter maximum likelihood estimation (MLE) problem for the Weibull distribution with consideration of interval data. Without interval data, the problem can be solved easily by regular MLE methods because the restricted MLE of the scale parameter β for a given shape parameter α has an analytical form, thus α can be efficiently solved from its profile score function by traditional numerical methods. In the presence of interval data, however, the analytical form for the restricted MLE of β does not exist and directly applying regular MLE methods could be less efficient and effective. To improve efficiency and effectiveness in handling interval data in the MLE problem, a new approach is developed in this paper. The new approach combines the Weibull-to-exponential transformation technique and the equivalent failure and lifetime technique. The concept of equivalence is developed to estimate exponential failure rates from uncertain data including interval data. Since the definition of equivalent failures and lifetimes follows EM algorithms, convergence of failure rate estimation by applying equivalent failures and lifetimes is mathematically proved. The new approach is demonstrated and validated through two published examples, and its performance in different conditions is studied by Monte Carlo simulations. It indicates that the profile score function for α has only one maximum in most cases. Such good characteristic enables efficient search for the optimal value of α.     

On the dependence of the Weibull exponent on geometry and loading conditions and its implications on the fracture toughness probability curve using a local approach criterion

In a previous paper the authors assessed the probability of failure of a three point bend specimen, SE(B), using a local approach criterion. In that paper the Weibull exponent, m, was derived from tests performed on round notch bars in traction, RNB(T), following the procedure suggested by Mudry. In the present study, it is addressed the issue of the dependence of the Weibull exponent m on geometry and loading conditions. It is shown that the amplitude and shape of the notch tip stress field and, in particular, the triaxiality characterising the stress state determines the value of the exponent m. Tests performed on RNB(T) specimens of carbon steel 22NiMoCr37, type A 508 Cl 3, at temperatures ranging from –18 °C to –196 °C actually indicate that m varies from 6 to 40, depending on the notch depth and root radius while for specimens carrying sharp cracks its value drops down to 4. This last result seems to be consistent with the Wallin hypothesis of a theoretical value equal to 4 for fracture mechanics specimens with high constraint, such as C(T) or SE(B), with positive values of the Q-stress or T-stress and triaxiality factor, TF, approaching 2.5. Temperature, in as long as it does not modify the stress state from plane strain to plane stress and the TF, has no effect on the value of m which is independent of the material as well.     

Control charts for monitoring the mean and percentiles of Weibull processes with variance components

In this article, we study Shewhart and exponentially weighted moving average control charts for monitoring the mean or, equivalently, the percentiles of a Weibull process when additional sources of variation, also known as variance components, are present. We adopt a frailty model to describe the monitored process. We derive analytical properties for this model and use them to develop control charts. We consider charts for the sample mean and exponentially weighted moving averages. We compare their average run length performances to their traditional counterparts when they do not account for variance components.     

Analysis of field failure data when modeled Weibull slope increases with time

Weibull time‐to‐fail distributions cannot be correctly estimated from field data when manufacturing populations from different vintages have different failure modes. To investigate the pitfalls of ongoing Weibull parameter estimation, two cases, based upon real events, were analyzed. First, a time‐to‐fail distribution was generated assuming the same Weibull shape parameter representing an increasing failure rate for each monthly batch or vintage of production. The shape parameter was estimated from simulated field data at regular periods as the population accumulated service time. Estimates of the shape parameter were not constant, but gradually decreased (as had occurred in a real system) with added service time. In the second case, field reliability performance was modeled to match the actual historical data for one product from a disk drive manufacturer. The actual data was proprietary and was not directly available for analysis. A production schedule was modeled with a mix of two failure characteristics. The population reaching the field in the first 12 months had a low, constant failure rate. For the second and third years of production, higher volumes were introduced that had the higher, increasing failure rates of the first case. Assessment of the mixed population at each month of calendar time resulted in an increasing Weibull shape parameter estimate at each assessment. When the two populations were separated and estimated properly, a better fit with more accurate estimates of Weibull shape parameters resulted. Copyright © 2010 John Wiley & Sons, Ltd.     

Bias correction for the least squares estimator of Weibull shape parameter with complete and censored data

Estimation of the Weibull shape parameter is important in reliability engineering. However, commonly used methods such as the maximum likelihood estimation (MLE) and the least squares estimation (LSE) are known to be biased. Bias correction methods for MLE have been studied in the literature. This paper investigates the methods for bias correction when model parameters are estimated with LSE based on probability plot. Weibull probability plot is very simple and commonly used by practitioners and hence such a study is useful. The bias of the LS shape parameter estimator for multiple censored data is also examined. It is found that the bias can be modeled as the function of the sample size and the censoring level, and is mainly dependent on the latter. A simple bias function is introduced and bias correcting formulas are proposed for both complete and censored data. Simulation results are also presented. The bias correction methods proposed are very easy to use and they can typically reduce the bias of the LSE of the shape parameter to less than half percent.