A damage-based evaluation of probability density distribution for rain-flow ranges from random processes

Abstract

A new method is proposed for the evaluation of a single Weibull distribution approximating the cycles turning out from the rain-flow counting of a random process. The founding hypothesis is a multi-modal distribution assumption, i.e. the probability density is a linear combination of several Weibull distributions; thus a simplified evaluation of a single distribution is proposed as an approximation of the most damaging part of the counted cycles instead of the most frequent range occurrences.

Introduction

In the reliability analysis of fatigue behaviour of real components under service loading, one of the fundamental issues is the evaluation of loading or stress spectra and the extrapolation of complete design spectra from short time measured or simulated histories.

In order to perform such estimation, by assuming that the investigated service loading condition is stationary and ergodic, there are mainly two possibilities:

• to regard the time histories as stochastic processes by determining their expected power spectral density or the rate of occurrence of peaks and troughs. Then, to evaluate the expected distribution of rain-flow cycles by means of one of the criteria proposed in literature: they are based on empirical relationships with the frequency domain characteristics or on the knowledge of the conditional probability density distribution (see [1], [2], [3], [4], [5] and reported references);

• to perform the deterministic counting of the time history and to evaluate the probability density of counted ranges (Fig. 1).

The first approach is more convenient from a methodological point of view, but a sound methodology applicable to several types of experimentally determined time histories (particularly when wide-band or non-gaussian processes are concerned) has not been developed yet [6], [7], [8].

On the other hand the deterministic counting procedure is more widely applicable since it can be performed on every kind of stochastic process. Thus it is necessary to use a probability density function in order to analyse obtained data and to evaluate the occurrence frequency of ranges in the analysed loading conditions. This choice and the methods used for its numerical evaluation are critical tasks since they strongly influence the estimation of probability of occurrence of high range cycles, which mainly affect component durability assessment.

In this framework, recent contributions [8], [9] have shown the possibility of using a particular multi-modal distribution to express probability density as a linear combination of different Weibull distributions. Basing themselves on this assumption, Nagode and Fajdiga [8] gave a numerical procedure for the accurate evaluation of unknown parameters of each single Weibull distribution (named “characteristic form”) of the multi-modal distribution. A problem in the practical application of that procedure is that, as already stated by the authors, usually at least three or four forms should be considered for a proper description of the range cycles distribution, because the first form of the multi-modal series describes the most frequent, usually low range, cycle occurrence. Moreover, there is not a unique solution, but it is possible to have disagreements in parameter estimations of higher order forms, by changing the values of control parameters in the numerical estimating procedure, so that it is possible to find different solutions achieving almost the same accuracy.

Since, for the reliability analysis, the range distribution is used for damage prediction, the frequent low cycle occurrence can be meaningless. Otherwise, by founding the parameter estimations on damage distribution, different approaches could be sketched in order to evaluate the loading spectra components. In particular, it is possible to start from the most damaging components (which are the most interesting from a design point of view) instead of starting from the most frequent (which can also be influenced by meaningless small amplitude structural vibrations or by unfiltered experimental noise on measured signals) in order to optimise the accuracy of the most interesting parameters.

This paper aims to investigate the relationships between the multi-modal distribution and the damage caused on a structural element. The task is to develop a simple procedure for a direct, and eventually approximated evaluation of the most interesting form of the multi-modal series, so that the complete distribution of counted cycle can be efficiently neared by a part of it having a simple Weibull distribution.