A STATISTICAL INVESTIGATION OF THE FATIGUE LIVES OF Q235 STEEL-WELDED JOINTS

An investigation into the fitting of six assumed distributions (three-parameter Weibull, two-parameter Weibull, extreme minimum value, extreme maximum value, normal and lognormal distributions) of 23 groups of fatigue life data for Q235 steel-welded joints is performed in terms of linear regression analyses. The results reveal that the fatigue life distribution shapes mostly tend to be positively skewed. Therefore, the extreme minimum value and normal distributions are not the most appropriate distributions to assume for a fatigue life evaluation. The three-parameter Weibull distribution may give misleading results in fatigue reliability analyses because the shape parameter is often lesss than 1. This means that the hazard rate decreases with fatigue cycling. This is contrary to the general understanding of the behaviour of welded joints. Reliability analyses may also be affected by slightly non-conservative evaluations in tail regions of the three-parameter Weibull distribution. The two-parameter Weibull distribution does not give as good a fit as either the extreme maximum value distribution or the lognormal distribution. On the other hand, the extreme maximum value and lognormal distributions can be safely assumed in reliability analyses due to the good total fit effects and the conservative evaluations in tail regions. In addition, the extreme maximum value distribution is in good agreement with the general physical understanding of the structural behaviour of welded joints.     

On the relationship between statistical distributions of defect size and fatigue life in 7050-T7451 thick plate and A356-T6 castings

The relationship between the distributions for the size of fatigue-initiating defects and fatigue life of 7050-T7451 thick plate and A356-T6 alloy castings reported previously in the literature were analyzed. Results showed that (i) the size of fatigue-initiating defects in all four datasets follow the Gumbel distribution, (ii) the fatigue life model based on the Paris-Erdoğan law for crack propagation provides respectable fits to fatigue life data, and (iii) the statistical distribution for fatigue life based on the Gumbel distribution of defect size and the fatigue life model provides excellent fits to all datasets, (iv) this statistical distribution for fatigue life performs better than the lognormal and Weibull distributions commonly used to model variability in fatigue life of aluminum alloys.     

两种国产涡轮盘材料疲劳寿命概率分布规律初探

基于两种国产发动机轮盘材料GH4133和1Cr11Ni2W2MoV低循环疲劳实验数据,选择三种常用的分布函数(对数正态分布、双参数Weibull分布和三参数Weibull分布),用概率图法和柯尔莫哥洛夫-斯米尔诺夫检验方法进行拟合优度检验,以考察分布函数的总体拟合效果.通过经验失效率的变化趋势来评估分布函数是否符合疲劳失效机制.结果表明:形状参数大于1的三参数Weibull分布函数是建立国产轮盘材料GH4133和1Cr11Ni2W2MoV疲劳寿命分布的理想数学模型.对数正态分布函数在本工作所研究的轮盘材料低循环疲劳寿命范围内是合理的.双参数Weibull分布函数用作国产轮盘材料疲劳寿命分布模型时必须进行假设检验.     

Probabilistic cumulative damage model to estimate fatigue life

This paper introduces a numerical model to estimate fatigue life under step‐stress conditions, using the Weibull and lognormal distributions. The maximum likelihood method was used to estimate the free parameters of the distributions. The model was fitted to an experimental data on fatigue life in the specimens of steel SAE 8620, by using evolutionary computation to optimize the likelihood function. Results are reported on the values of the parameters and their confidence interval. Also, a validation of the model is discussed using analysis of residuals.     

基于统计与粗糙集理论的飞机垂尾抖振载荷分布假设选择与评价方法

研究飞机垂尾抖振问题的主要任务之一是编制抖振疲劳载荷谱、估算飞机结构的抖振疲劳寿命并校核飞机结构的强度等,一般通过统计方法来建立其抖振疲劳载荷时程的峰(谷)值分布模型,而分布模型的优劣对确定各飞行状态下的极限工况及抖振响应的循环次数与幅值分布等信息影响显著。通过分析五种常用于描述抖振疲劳载荷峰(谷)值的概率分布假设模型：正态分布、对数正态分布、威布尔分布、瑞利分布和极值分布,给出了基于参数估计的概率分布规律,并提出了一种采用各概率分布假设所对应模型的“拟合优劣指标”作为评价和选择的依据。同时,结合各分布模型的特性对飞机抖振载荷时程处理要求的匹配程度,运用粗糙集理论确定了系统评价指标的最小分辨距离与最大分辨率,来消除由于误差引入导致的评价指标数值差异而造成的误判。算例分析表明,该方法可合理且高效地实现对飞机抖振载荷概率分布假设的正确评价与选择     

STATISTICAL INVESTIGATION OF THE BEHAVIOUR OF MICROCRACKS IN CARBON STEELS

Abstract— In order to clarify the statistical behaviour of microcracks, rotating bending fatigue tests of plain specimens were carried out using two kinds of carbon steel with different microstructures, normalized 0.21% C steel and heat-treated 0.45% C steel. The emphasis is to investigate separately the statistical properties of the initiation and propagation of microcracks through successive observations of the specimen surface. Results show that the microcrack initiation life distributions of normalized and heat-treated steels are expressed by a two-parameter Weibull distribution and a Weibull distribution of the mixed type, respectively. On the other hand, the microcrack propagation life distributions of these steels are represented by three-and two-parameter Weibull distributions, respectively. The three-parameter Weibull distribution is well fitted to the crack length distribution for both the steels.     

Probabilistic analysis of the uncertainty in the fatigue capacity of welded joints

This paper presents the results of a study of the uncertainty in the fatigue capacity (constant amplitude fatigue life) of welded steel joints, due to uncertainties related to geometrical and material parameters.An efficient method of probabilistic fracture mechanics analysis is described and applied. A linearelastic fracture mechanics model and the Paris-Erdogan law of crack propagation were adopted. Stressintensity factors were evaluated by employing an influence function method, which is very cost-effective. The main parameters were treated as stochastic variables. Data for weld and crack geometry of the non-load carrying fillet weld cruciform joint selected as the example joint in the study, were recorded from specimens. Other data were compiled from the literature. The uncertainties associated with the basic variables were transformed into a measure of uncertainty of the fatigue capacity by employing the Monte Carlo simulation technique. The relative contributions to the uncertainty in the fatigue capacity from the various factors were also compared.The $\text{S-N}$ data established analytically compared fairly well with test data obtained with 42 specimens. The probabilistic fracture mechanics analysis provided a sufficient sample of data to allow a test of analytical probability distributions to the fatigue life. The fit of two- and three-parameter lognormal and Weibull distributions was examined. Only the three-parameter lognormal pdf passed the chi-square test on the 5% confidence level.     

LZ50钢断裂韧度的合理统计模型

试验研究了LZ50车轴钢的断裂韧度的合理统计模型。结果表明,LZ50钢是一种偏脆性的材料,应当严格控制铁道车辆车轴的制造表面质量。同时证实了现有正态分布模型不能合理描述试验数据。为此,通过比较三参数Weibull、两参数Weibull、正态、对数正态、极小值和极大值6种常用统计分布对试验数据的拟合优度、数理一致性和尾部安全性,说明了极小值分布是良好统计模型。建立了断裂韧度的极小值概率测定方法,测定了典型存活概率和置信度下的材料断裂韧度值。     

SCATTER CHARACTERISTICS OF FATIGUE LIFE AND THE BEHAVIOUR OF SMALL CRACKS

Usually, there is large scatter in fatigue data and this should be evaluated quantitatively when fatigue data are applied to the design of machines and structures. Consequently it is important to clarify the physical basis of scatter in fatigue life. In this present study, rotary bending fatigue tests were performed on an annealed 0.21% carbon steel. At least sixteen smooth specimens were fatigued at each of three stress ranges and successive observations of the surface were studied for all the specimens using the plastic replica method. By examining the initiation and propagation behaviour of cracks the physical basis of scatter in fatigue life is analysed and discussed. To estimate the scatter characteristics quantitatively, the distributions of crack initiation life, propagation life, fatigue life and crack length were individually studied by assuming a Weibull distribution for each set of data.     

FATIGUE LIFE DISTRIBUTION AND GROWTH OF CORROSION PITS IN A MEDIUM CARBON STEEL IN 3%NaCl SOLUTION

Abstract— Statistical fatigue tests have been conducted on a structural medium carbon steel, S45C, in room air and in 3%NaCl solution, using five cantilever-type rotary bending fatigue testing machines which were specially manufactured for the purpose of the present study. Fatigue life distribution was examined at three and five stress levels in air and in 3%NaCl solution, respectively, and twenty specimens were allocated to each stress level. In room air, it was found that fatigue life distributions followed the three-parameter Weibull distribution, which were closely related to fracture morphology. In 3%NaCl solution, they also followed the Weibull distribution, but the scatter in fatigue life was smaller in comparison to that in air. It is suggested that the decrease in the scatter of fatigue life may be attributed to a smaller fraction of crack initiation life in 3%NaCl solution. The growth of corrosion pits was investigated using a laser microscope. The distribution of corrosion pit depths followed the log-normal distribution, and the corrosion pit depths increased with increasing time or the number of cycles. It was found that the growth of corrosion pits was accelerated by stress cycling and the depths increased with increasing stress level. Based on these results, a growth law of corrosion pits, including the effect of stress cycling, is proposed.     

Fatigue reliability under multiple-amplitude loads

A method to determine the fatigue of structures subjected to multiple-amplitude loads is presented. Unlike the more common cumulative damage methods, which are usually based on fatigue life data, the proposed method is based on tensile strength data. Assuming the Weibull distribution for the initial tensile strength and the fatigue life, the probability distributions for the residual tensile strength in both the crack initiation and the crack propagation stages of fatigue are determined. The method is illustrated for two-amplitude loads by means of experimental results obtained by testing specimens of a structural steel and is shown to be more accurate than the Palmgren-Miner cumulative damage method.     

Probabilistic anisomorphic constant fatigue life diagram approach for prediction of P–S–N curves for woven carbon/epoxy laminates at any stress ratio

A general engineering methodology to construct a family of anisomorphic constant fatigue life (CFL) diagrams with probability of failure as the parameter that allows efficiently predicting P–S–N curves at any stress ratios is developed and validated for a plain weave fabric carbon/epoxy laminate. Constant amplitude fatigue tests are first performed to obtain statistical samples of fatigue life at different stress levels and stress ratios, respectively. Static tensile and compressive strength data are also collected. The Kolmogorov–Smirnov and Anderson–Darling goodness-of-fit tests suggest that both two-parameter lognormal and Weibull distributions are acceptable as the distributions for the static strength and fatigue life data, respectively, at the significance level of 5%. Then, we attempt to develop a methodology for efficient construction of the anisomorphic CFL diagrams for different constant values of probability of failure. It requires the P–S–N curves for any percentile points of the distribution for the critical stress ratio. To come up with this requirement, a probabilistic scaling law is formulated. It takes account of the probability-of-failure dependence of the critical stress ratio and the stress-ratio dependence of the P–S–N curve for the critical stress ratio. Finally, the anisomorphic CFL diagrams for different constant values of probability of failure are predicted using the proposed methodology, and they are shown to be in good agreement with the experimental results. It is also demonstrated that the P–S–N curves can efficiently and accurately be predicted for the woven CFRP laminate at any stress ratios using the proposed probabilistic anisomorphic CFL diagram approach.     

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.     

Statistical study of strength and fatigue life of composite materials

Fatigue life scattering of glass fibre-reinforced epoxy composites has been studied at 300 K and 77 K. The static strength and the fatigue life distributions were studied using normal, log-normal and two-parameter Weibull distribution functions; the value of mean fatigue life was analysed using mean fatigue life, mean log fatigue life and expected value of the Weibull distribution function. The effect on fatigue life of two stress levels (low-high and high-low) was investigated using the distribution functions; a modification of Miner's rule is made to predict the resulting scattering.     

On fatigue life variability in bone cement

Fatigue life of Simplex P bone cement was tested at three different stress amplitudes by using specimens produced by two different mixers. Fatigue life data showed high variability in all instances. Statistical analysis showed that fatigue life was not affected by the type of mixer. Analysis of fracture surfaces showed that fatigue life variability could be attributed to the presence of defects, such as bubbles and mixing defects. Both Weibull and Tiryakio?lu distributions provided excellent fits to the fatigue life data. Moreover, the Gumbel parameters for fatigue initiator size data estimated in the Tiryakio?lu distribution agreed closely with fractographic measurements.     

A new type of extreme value distributions

The three classical pairs of extreme value distributions correspond to random variables with ranges of values unbounded from either one or both sides. Some applications of statistics of extremes, however, deal with variables, which are bounded on both sides (local values of specific fracture energy in crack diffusion theory is one such example). In this paper, we derive a fourth pair of extreme value distributions, which are supported on a finite segment (one for maxima and one for minima). First, we propose a derivation of the three known maximal value distributions, which lends itself to a generalization (everything is done for maxima, since transition to minima is standard). The derivation is then extended to a slightly more general setting, and the fourth distribution is obtained. It is explained that certain fact concerning groups of transformations of the real line prevents any further generalization, i.e. the extended list of extreme value distributions is complete. The three classical maximal value distributions can be obtained as limits of the new one. A possible criterion of when one may expect the new distribution to be more adequate than the Weibull distribution is offered. An illustrative numerical example is considered, in which the scatter of sample minima is modeled by both Weibull and the new distribution. Another example shows that when the modeling of data requires very high values of the shape parameter of the Weibull distribution, the new distribution may be expected to have much smaller “shape parameter” values. The modeling of experimentally observed scatter of crack arrest length, using the Weibull distribution, is compared to that using the new distribution.     

Statistical aspects of fatigue life and endurance limit

Based on the assumption that the fatigue life and endurance limit is mainly controlled by the materials particle distribution, different possible statistical distributions to describe the fatigue have been assessed. Some distributions lead to high statistical uncertainty or physically non‐realistic size effects. It is shown that, if the maximum particle size distribution is described by a Frechet distribution, both the fatigue life, as well as the endurance limit are described by a Weibull distribution.     

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.     

On estimating Weibull modulus by moments and maximum likelihood methods

The effect of true Weibull modulus and sample size on Weibull modulus estimated by moments and maximum likelihood methods was investigated. Results indicated that the value of true Weibull modulus had no effect on estimated modulus for the maximum likelihood method, and a strong effect for the moments method, especially when sample size was less than 30. In addition, the distribution of Weibull modulus estimated by both methods was investigated using the modified Anderson–Darling statistics for goodness of fit. It was found that the distribution was not normal, lognormal, 3-parameter Weibull, or 3-parameter log-Weibull for the maximum likelihood method, as reported in previous studies. For the moments method however, the distribution of normalized Weibull moduli was found to be lognormal for sample sizes of 40 and above. The other three distributions showed a significant level of lack-of-fit at all sample sizes. An erratum to this article can be found at