More on the mis‐specification of the shape parameter with Weibull‐to‐exponential transformation

When lifetimes follow Weibull distribution with known shape parameter, a simple power transformation could be used to transform the data to the case of exponential distribution, which is much easier to analyze. Usually, the shape parameter cannot be known exactly and it is important to investigate the effect of mis‐specification of this parameter. In a recent article, it was suggested that the Weibull‐to‐exponential transformation approach should not be used as the confidence interval for the scale parameter has very poor statistical property. However, it would be of interest to study the use of Weibull‐to‐exponential transformation when the mean time to failure or reliability is to be estimated, which is a more common question. In this paper, the effect of mis‐specification of Weibull shape parameters on these quantities is investigated. For reliability‐related quantities such as mean time to failure, percentile lifetime and mission reliability, the Weibull‐to‐exponential transformation approach is generally acceptable. For the cases when the data are highly censored or when small tail probability is concerned, further studies are needed, but these are known to be difficult statistical problems for which there are no standard solutions. Copyright © 2000 John Wiley & Sons, Ltd.     

Lifetime statistics for single Kevlar 49 filaments in creep-rupture

Experimental data are presented for the lifetime of single Kevlar 49 filaments under moderate to high stress levels at standard ambient conditions (21°C, 65% r.h.). Filaments were drawn from two spools, A and B, taken from the same production lot. Previously we found that filaments from spool A were 7% lower in mean strength but much less variable in diameter than filaments from spool B; however, the respective variabilities in failure stress were equivalent. The lifetime data were interpreted in light of a previously developed kinetic model embodying Weibull failure statistics and power law dependence of lifetime on stress level. As predicted, lifetime data at each stress level generally followed a two-parameter Weibull distribution with a shape parameter value near 0.2. Based on absolute stress levels, the filaments drawn from spool B had a Weibull scale parameter for lifetime about ten times greater than those from spool A; however, when the stress-levels were normalized by the respective Weibull scale parameters for short-term strength, these differences disappeared. With respect to power law dependence of lifetime on stress level, three distinct time domains emerged, each marked by a different power law exponent. Similar behaviour was observed earlier for preproduction Kevlar 49/epoxy strands, and the values for the power law exponents for the filaments agree closely with those for the strands.     

Robust Regression using Probability Plots for Estimating the Weibull Shape Parameter

The Weibull shape parameter is important in reliability estimation as it characterizes the ageing property of the system. Hence, this parameter has to be estimated accurately. This paper presents a study of the efficiency of using robust regression methods over the ordinary least‐squares regression method based on a Weibull probability plot. The emphasis is on the estimation of the shape parameter of the two‐parameter Weibull distribution. Both the case of small data sets with outliers and the case of data sets with multiple‐censoring are considered. Maximum‐likelihood estimation is also compared with linear regression methods. Simulation results show that robust regression is an effective method in reducing bias and it performs well in most cases. Copyright © 2006 John Wiley & Sons, Ltd.     

Maximum-Likelihood Estimation of the Parameters of Gamma and Weibull Populations from Complete and from Censored Samples

Iterative procedures are given for joint maximum-likelihood estimation, from complete and censored samples, of the three parameters of Gamma and of Weibull populations. For each of these populations, the likelihood function is written down, and the three maximum-likelihood equations are obtained. In each case, simultaneous solution of these three equations would yield joint maximum-likelihood estimators for the three parameters. The iterative procedures proposed to solve the equations are applicable to the most general case, in which all three parameters are unknown, and also to special cases in which any one or any two of the parameters are known. Numerical examples are worked out in which the parameters are estimated from the first m failure times in simulated life tests of n items (m ≤ n), using data drawn from Gamma and Weibull populations, each with two different values of the shape parameter.     

Weibull-based methodology for condition assessment of cast iron water mains and its application

A qualitative and quantitative understanding of how cast iron water distribution pipes fail in service would facilitate a targeted approach to the management of rehabilitation in the water industry. This paper proposes a technique for assessing the condition of pipes, based on strength characteristics obtained from small samples; this offers an alternative way of estimating the likelihood of failure to current methodologies based on pit‐depth measurements. Examination of recovered pipe samples indicates that the strength of the cast iron pipe reduces over time as a result of corrosion, although other time‐dependent processes, such as fatigue, may also contribute to this degradation. Taken with previous work, this paper suggests that the variation in strength of small samples removed from cast iron water distribution pipes can be characterized using Weibull methods. It is argued that the Weibull modulus provides a useful indicator of the condition of the pipe. Using scaling arguments, inherent in the Weibull methodology, it is then possible to use data from small samples to predict the likely strength characteristics of water distribution pipes in the ground, which is reasoned to be a good measure of the potential performance of the pipe in service. The Weibull approach is applied to a number of different data sets obtained from testing samples extracted from a range of pipes, which have seen service at various locations in the Thames Water region. One of these data sets was from locations where failure had occurred in service. It is shown that the use of Weibull analysis can identify pipes in the network that have degraded the most significantly. A methodology is suggested whereby this information taken with other performance indicators can be used to identify the local regions where rehabilitation is required most urgently. Alternatively, it can be used to identify those regions of the network, which are in good condition and unlikely to need repair or replacement work.     

Estimation of Weibull Distribution Shape Parameter When no More than Two Failures Occur per Lot

An estimate, with limits of uncertainty, is given for the shape parameter of the Weibull distribution when failure experience results from lots having different quality levels. The data result from the following situation. Lots are removed from service at some target value, normally after experiencing no failures. Some lots experience one failure, and the remaining non-failed items remain in service until the target value is reached. Finally, there are a few lots in which two failures occur. In this instance, all remaining non-failed items are removed immediately after the second failure occurs. It is assumed that all lots have the same shape parameter, with lot differences, when they exist, appearing as differences in the scale parameter value.     

Reliability analysis for wind turbines with incomplete failure data collected from after the date of initial installation

Reliability has an impact on wind energy project costs and benefits. Both life test data and field failure data can be used for reliability analysis. In wind energy industry, wind farm operators have greater interest in recording wind turbine operating data. However, field failure data may be tainted or incomplete, and therefore it needs a more general mathematical model and algorithms to solve the model. The aim of this paper is to provide a solution to this problem. A three-parameter Weibull failure rate function is discussed for wind turbines and the parameters are estimated by maximum likelihood and least squares. Two populations of German and Danish wind turbines are analyzed. The traditional Weibull failure rate function is also employed for comparison. Analysis shows that the three-parameter Weibull function can obtain more accuracy on reliability growth of wind turbines. This work will be helpful in the understanding of the reliability growth of wind energy systems as wind energy technologies evolving. The proposed three-parameter Weibull function is also applicable to the life test of the components that have been used for a period of time, not only in wind energy but also in other industries.     

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.     

A study of time‐between‐events control chart for the monitoring of regularly maintained systems

Owing to usage, environment and aging, the condition of a system deteriorates over time. Regular maintenance is often conducted to restore its condition and to prevent failures from occurring. In this kind of a situation, the process is considered to be stable, thus statistical process control charts can be used to monitor the process. The monitoring can help in making a decision on whether further maintenance is worthwhile or whether the system has deteriorated to a state where regular maintenance is no longer effective. When modeling a deteriorating system, lifetime distributions with increasing failure rate are more appropriate. However, for a regularly maintained system, the failure time distribution can be approximated by the exponential distribution with an average failure rate that depends on the maintenance interval. In this paper, we adopt a modification for a time‐between‐events control chart, i.e. the exponential chart for monitoring the failure process of a maintained Weibull distributed system. We study the effect of changes on the scale parameter of the Weibull distribution while the shape parameter remains at the same level on the sensitivity of the exponential chart. This paper illustrates an approach of integrating maintenance decision with statistical process monitoring methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Fracture life prediction and sensitivity analysis for hollow extrusion dies

It has been reported in literature that extrusion dies most often fail by fatigue fracture. Experimental studies have shown that cracks pre‐exist in dies because of various factors including machining, heat treatment and surface hardening. High levels of repeated mechanical and thermal loads result in crack propagation leading to ultimate fracture failure. In an earlier work by the authors, a simplified approach of plate‐with‐edge‐crack was used to develop a fracture mechanics based fatigue life prediction model for tube dies. In the current work, extrusion die is modeled as a pressurized‐cylinder‐with‐internal‐crack, a more realistic approach for a hollow (tube) die. Stochastic nature of various fatigue‐related die parameters has been used to reflect their variability. Monte Carlo simulation has been performed to forecast fracture failure of extrusion dies under a given set of operating conditions and mechanical properties. Predicted mean‐time‐to‐failure is quite close to actual average extrusion die life data from the industry. Using tube die as a basis, fracture life of other hollow profiles can be estimated through their shape complexity values. Analysis has also been carried out to evaluate how sensitive fracture life of hollow extrusion dies is to material and process parameters. Major findings are that die life is highly sensitive to initial crack size, wall thickness, profile outer diameter and billet length; moderately sensitive to Paris constant and extrusion ratio; and only slightly sensitive to fracture toughness and ram speed. These results can contribute to a deeper understanding of the factors responsible for fracture failure of an extrusion die exposed to thermo‐mechanical fatigue environment.     

Using degradation data to assess reliability: a case study on train wheel degradation

Degradation experiments are usually used to assess the lifetime distribution of highly reliable products, which are not likely to fail under the traditional life tests or accelerated life tests. In such cases, if there exist product characteristics whose degradation over time can be related to reliability, then collecting ‘degradation data’ can provide information about product reliability. In general, the degradation data are modeled by a nonlinear regression model with random coefficients. If we can obtain the estimates of parameters under the model, then the failure‐time distribution can be estimated. In order to estimate those parameters, three basic methods are available, namely, the analytical, numerical and the approximate. They are chosen according to the complexity of the degradation path model used in the analysis. In this paper, the numerical and the approximate methods are compared in a simulation study, assuming a simple linear degradation path model. A comparison with traditional failure‐time analysis is also performed. The mean‐squared error of the estimated 100pth percentile of the lifetime distribution is evaluated for each one of the approaches. The approaches are applied to a real degradation data set. Copyright © 2008 John Wiley & Sons, Ltd.     

Cumulative Sum Control Charts for Monitoring Weibull‐distributed Time Between Events

The Weibull distribution can be used to effectively model many different failure mechanisms due to its inherent flexibility through the appropriate selection of a shape and a scale parameter. In this paper, we evaluate and compare the performance of three cumulative sum (CUSUM) control charts to monitor Weibull‐distributed time‐between‐event observations. The first two methods are the Weibull CUSUM chart and the exponential CUSUM (ECUSUM) chart. The latter is considered in literature to be robust to the assumption of the exponential distribution when observations have a Weibull distribution. For the third CUSUM chart included in this study, an adjustment in the design of the ECUSUM chart is used to account for the true underlying time‐between‐event distribution. This adjustment allows for the adjusted ECUSUM chart to be directly comparable to the Weibull CUSUM chart. By comparing the zero‐state average run length and average time to signal performance of the three charts, the ECUSUM chart is shown to be much less robust to departures from the exponential distribution than was previously claimed in the literature. We demonstrate the advantages of using one of the other two charts, which show surprisingly similar performance. Copyright © 2014 John Wiley & Sons, Ltd.     

单向纤维增强陶瓷基复合材料单轴拉伸行为

采用细观力学方法对单向纤维增强陶瓷基复合材料的单轴拉伸应力-应变行为进行了研究。采用Budiansky-Hutchinson-Evans（BHE）剪滞模型分析了复合材料出现损伤时的细观应力场,结合临界基体应变能准则、应变能释放率准则以及Curtin统计模型三种单一失效模型分别描述陶瓷基复合材料基体开裂、界面脱粘以及纤维失效三种损伤机制,确定了基体裂纹间隔、界面脱粘长度和纤维失效体积分数。将剪滞模型与3种单一失效模型相结合,对各个损伤阶段的应力-应变曲线进行模拟,建立了准确的复合材料强韧性预测模型,并讨论了界面参数和纤维韦布尔模量对复合材料损伤以及应力-应变曲线的影响。与室温下陶瓷基复合材料单轴拉伸试验数据进行了对比,各个损伤阶段的应力-应变、失效强度及应变与试验数据吻合较好。     

基于GM-噪声SVR方法的矿山设备可靠性参数估计

为了有效估计小子样条件下矿山设备的三参数威布尔分布可靠性模型参数,提出基于GM-噪声SVR的参数估计方法。该方法以灰色估计法(GM)为基础估计模型的位置参数,采用基于训练样本数量和噪声参数寻优的ε - 带支持向量回归机(ε-SVR)估计尺度参数和形状参数,并通过拟合的三参数威布尔分布函数分析预测和解决设备的可靠性问题。算例结果表明,GM-噪声SVR方法可以很好地用于矿山设备可靠性模型参数估计,估计某带式输送机三参数威布尔分布可靠性模型的位置参数、尺度参数和形状参数依次为3.1525、188.3763、1.0476,平均无故障时间为188 h,标准均方根误差NRMSE为0.0519。这表明该方法的可行性和有效性。     

Parametric Estimation

The exponential distribution is often used in reliability work to describe the distribution of time to “chance” failure and is characterized by a constant failure rate. In this paper the small sample powers are compared for four test statistics for the hypothesis of constant failure rate vs. the hypothesis of non-constant failure rate. The tests are compared for samples of size n = 10(5)50 using the Weibull distribution for the alternative distribution. The shape parameter of the Weibull is varied from 0.5 to 2.5. For the two test statistics which involve arbitrary grouping of the data the effect of group size and number was also examined.     

Robust Estimation for Weibull Distribution in Partially Accelerated Life Tests with Early Failures

Maximum likelihood estimation (MLE) is a frequently used method for estimating distribution parameters in constant stress partially accelerated life tests (CS‐PALTs). However, using the MLE to estimate the parameters for a Weibull distribution may be problematic in CS‐PALTs. First, the equation for the shape parameter estimator derived from the log‐likelihood function is difficult to solve for the occurrence of nonlinear equations. Second, the sample size is typically not large in life tests. The MLE, a typical large‐sample inference method, may be unsuitable. Test items unsuitable for stress conditions may become early failures, which have extremely short lifetimes. The early failures may cause parameter estimate bias. For addressing early failures in the Weibull distribution in CS‐PALTs, we propose an M‐estimation method based on a Weibull Probability Plot (WPP) framework, which leads a closed‐form expression for the shape parameter estimator. We conducted a simulation study to compare the M‐estimation method with the MLE method. The results show that, with early‐failure samples, the M‐estimation method performs better than the MLE does. Copyright © 2015 John Wiley & Sons, Ltd.     

A physical probabilistic model to predict failure rates in buried PVC pipelines

For older water pipeline materials such as cast iron and asbestos cement, future pipe failure rates can be extrapolated from large volumes of existing historical failure data held by water utilities. However, for newer pipeline materials such as polyvinyl chloride (PVC), only limited failure data exists and confident forecasts of future pipe failures cannot be made from historical data alone. To solve this problem, this paper presents a physical probabilistic model, which has been developed to estimate failure rates in buried PVC pipelines as they age. The model assumes that under in-service operating conditions, crack initiation can occur from inherent defects located in the pipe wall. Linear elastic fracture mechanics theory is used to predict the time to brittle fracture for pipes with internal defects subjected to combined internal pressure and soil deflection loading together with through-wall residual stress. To include uncertainty in the failure process, inherent defect size is treated as a stochastic variable, and modelled with an appropriate probability distribution. Microscopic examination of fracture surfaces from field failures in Australian PVC pipes suggests that the 2-parameter Weibull distribution can be applied. Monte Carlo simulation is then used to estimate lifetime probability distributions for pipes with internal defects, subjected to typical operating conditions. As with inherent defect size, the 2-parameter Weibull distribution is shown to be appropriate to model uncertainty in predicted pipe lifetime. The Weibull hazard function for pipe lifetime is then used to estimate the expected failure rate (per pipe length/per year) as a function of pipe age. To validate the model, predicted failure rates are compared to aggregated failure data from 17 UK water utilities obtained from the United Kingdom Water Industry Research (UKWIR) National Mains Failure Database. In the absence of actual operating pressure data in the UKWIR database, typical values from Australian water utilities were assumed to apply. While the physical probabilistic failure model shows good agreement with data recorded by UK water utilities, actual operating pressures from the UK is required to complete the model validation.     

Control Charts for Monitoring Field Failure Data

One responsibility of the reliability engineer is to monitor failure trends for fielded units to confirm that pre‐production life testing results remain valid. This research suggests an approach that is computationally simple and can be used with a small number of failures per observation period. The approach is based on converting failure time data from fielded units to normal distribution data, using simple logarithmic or power transformations. Appropriate normalizing transformations for the classic life distributions (exponential, lognormal, and Weibull) are identified from the literature. Samples of size 500 field failure times are generated for seven different lifetime distributions (normal, lognormal, exponential, and four Weibulls of various shapes). Various control charts are then tested under three sampling schemes (individual, fixed, and random) and three system reliability degradations (large step, small step, and linear decrease in mean time between failures (MTBF)). The results of these tests are converted to performance measures of time to first out‐of‐control signal and persistence of signal after out‐of‐control status begins. Three of the well‐known Western Electric sensitizing rules are used to recognize the assignable cause signals. Based on this testing, the ―X‐chart with fixed sample size is the best overall for field failure monitoring, although the individual chart was better for the transformed exponential and another highly‐skewed Weibull. As expected, the linear decrease in MTBF is the most difficult change for any of the charts to detect. Copyright © 2005 John Wiley & Sons, Ltd.     

Two‐Stage Reliability Sampling Plans with Bogey Tests

This article considers the design of two‐stage reliability test plans. In the first stage, a bogey test was performed, which will allow the user to demonstrate reliability at a high confidence level. If the lots pass the bogey test, the reliability sampling test is applied to the lots in the second stage. The purpose of the proposed sampling plan was to test the mean time to failure of the product as well as the minimum reliability at bogey. Under the assumption that the lifetime distribution follows Weibull distribution and the shape parameter is known, the two‐stage reliability sampling plans with bogey tests are developed and the tables for users are constructed. An illustrative example is given, and the effects of errors in estimates of a Weibull shape parameter are investigated. A comparison of the proposed two‐stage test with corresponding bogey and one‐stage tests was also performed. Copyright © 2012 John Wiley & Sons, Ltd.