Bayesian Methods For P-S-N Curves Estimating And Statistical Modeling Of Fatigue Loads

There are uncertainties in all aspects of fatigue analysis due to the lack of necessary information,and these uncertain factors lead to the uncertainties of structural fatigue design performance.These uncertainties of structural fatigue performance are difficult described accurately by theoretical analysis so that the prediction accuracy of structural fatigue life is always not satisfactory.The uncertainties of structural fatigue life prediction are mainly sourced from the uncertainties of complex and variable external loads,the dispersion of material fatigue properties and the uncertainties of structural geometry caused by manufacturing processes.In order to establish the probabilistic model of structural fatigue performance and fatigue load effect,and introduce the Bayesian estimation method to consider the uncertainties of model parameters,the main research contents are as follows:(1)The small sample characteristics of the fatigue test data lead to a large uncertainty in the fatigue design curves estimation.The normal linear model or the Weibull generalized linear model uses the experimental data of all the stress levels to estimate the distribution characteristics of the fatigue life.The structural prior of hierarchical Bayesian model not only obtains the estimation error of hyperparameters,but also avoids the selection of hyperparameters,and can also incorporate structural prior and subjective prior simultaneously.This research focuses on the application of the stratified Bayesian model in estimating the fatigue life curves for minimizing the number of specimens required.A hierarchical Bayesian model with accumulated prior information is proposed to estimate the fatigue design curves,and then the P-S-N curves are generated from the fatigue life predictive distributions,involving both the randomness of parameters and the scatter of observations,and calculated by an identical hierarchical structure.In addition,the convergence of the Markov chain Monte Carlo method,the hierarchical Bayesian model test method,the processing of missing data,the selection of non-information priors and information priors,and how to consider the non-information priors and information priors simultaneously are discussed in detail.The maximum likelihood estimation results are used to compare to verify the method validity.The results show that the hierarchical Bayesian normal linear model obtains safer fatigue design curves for considering the model parameter error.A generalized linear model and a hierarchical Bayesian model are introduced to estimate the P-S-N curves in this research,and the fatigue probability design curves are generated by the survivor function or the resulting predictive distributions.The scatter of fatigue life is often described by a Weibull distribution,and the linear regression model cannot be added directly into the distribution parameters for the Weibull distribution does not have a mean parameter.Therefore,the fatigue failure analysis is treated as a survival analysis in each stress levels such that the fatigue lives can naturally be seen as survival time.A Weibull GLM with a hierarchical Bayesian framework has been established for the P-S-N curves estimation,and then a generalized linear model is employed to estimate the P-S-N curves which are plotted by the survivor function with given survivor probabilities.(2)Currently,random fatigue load analysis often uses stochastic processes or cumulative continuous probability density to establish its prediction model.The stochastic process method can accurately describe the statistical properties of the fatigue load over time,but for these long-term non-stationary loads that are difficult to establish in physical models or the loads series with uncertain state space,in most instances,unfortunately,are difficult to apply.Therefore,this research focuses on establishing a continuous probability distribution of accumulative fatigue load history to study its statistical properties.The effect of high-level stress on fatigue life increases exponentially,while the high-level stress amplitude in fatigue load appears less frequently,and its probability density is at the tail of the load probability distribution.Therefore,when modeling the fatigue load using continuous probability density function,the fitting accuracy of the tail region puts forward higher requirements.The Bayesian finite mixture model is chosen as the continuous probability distribution for studying the cumulative effect of loads in this research,and the distribution function estimation error is obtained by the posterior distribution of the model parameters.In addition,the fatigue random load is affected by many environmental excitations,and the mixture number of the mixture model is difficult to determine at the very beginning.Therefore,the Dirichlet process mixture model with infinite or uncertain mixture number is introduced to automatically obtain the number of the mixture.However,when the number of the mixture is large,it is not easy to converge because of the label switching problems.In order to solve the estimation error of the high-level stress amplitude with lowfrequency in the fatigue stress spectrum,the equivalent stress amplitude spectrum method is proposed in this research.The equivalent stress amplitude spectrum converts all stress amplitudes in the traditional stress spectrum into the equivalent cycle count of the reference stress amplitude by using the fatigue design curve and the mean stress empirical curves.The equivalent stress amplitude spectrum method can reduce the estimation error of the amplitude of the low-frequency and high-stress levels,and can directly describe the distribution characteristics of the intensity of different stress amplitudes on the fatigue life in the traditional stress spectrum.It can also be used to calculate the fatigue life directly.In summary,the main idea of this research is to introduce the Bayesian models to take the uncertainty of parameters in statistical models of some structural or component fatigue design problems into consideration,and try to learn as much or control these uncertain information in the fatigue design process,in order to meet safer structural fatigue design or life prediction.