Evaluation of fatigue life data by normalising procedures
The generally scarce and sparse number of fatigue results available from experimental programmes emphasizes the necessity of efficient procedures for evaluating the statistical parameters defining the S N field. This, in turn, has an influence on the cumulative damage calculation and prediction, since damage indices use the S N field as basic information. In this paper a consistent and compatible regression model for defining the S N field will be considered and normalising procedures are suggested for evaluating the model parameters. Different normalizing approaches are subsequently used for deriving damage indices for fatigue life prediction under variable loading and their comparative suitabilities are then discussed. 1.INTRODUCTION Structures and mechanical components are frequently submitted to loads of variable amplitudes and of a random nature. The corresponding fatigue life prediction has to be analysed by means of damage accumulation models which utilise as basic information the S N field of the material, determined from fatigue life tests conducted at various different constant stress ranges. Thus, the reliability of the life prediction under variable amplitude loading depends to a great extent on the quality of the estimation of the parameters related to the S N field. Accordingly, a statistical non linear regression analysis of the fatigue results is needed on account of the limited number of fatigue results spread over several stress ranges and of the considerable scatter of the results within each stress range. Since two random variables have to be considered the stress range s or the stress level s, depending on the material tested, and the number of cycles to failure N two different statistical distributions, F(N; s) representing the number of cycles to failure given the stress range s, or else E( s; N) representing the stress range given the number of cycles to failure, could be envisaged. Both distributions must fulfil physical and statistical conditions for the statistical model to be valid. In this paper, a consistent statistical model for analysing the S N field is presented as well as methods for estimating the model parameters, based on normalising test data. Additionally, damage indices identified as the normalised variables are defined and their interpretation discussed. 2.A MODEL FOR ANALYZING FATIGUE LIFE DATA In what follows the statistical model for analysing fatigue life data developed by Castillo et al. [1] will be considered. This model, based on the weakest link principle, arises from a functional equation after setting physical and statistical requirements (stability, compatibility and limit conditions) to the distributions F(N; s) and E(s; N) which prove to be three parameter Weibull distribution families for minima [1, 2 Thus, the cumulative distribution function (c.d.f. of the logarithm of the lifetime N, given the stress range si, is given by: s s A i i E D C log B N log exp 1 N log F (2) from which the percentile curves (see Figure 1) can be derived: E P 1 log D C log B N log A 1 s (1) where N is the lifetime measured in number of cycles to failure, si is the stress range and A, B, C, D and E are the model parameters to be determined, with the following meanings: A Weibull shape parameter; B Threshold value or limit lifetime; C Endurance limit; D Scale parameter; E Parameter determining the position of the zero percentile curve. As soon as the five parameters are determined, the analytical expression of the whole S N field is known, which allows for the probabilistic prediction of the fatigue failure under constant amplitude loading. As can be observed, the percentile curves are represented by equilateral hyperbolas. The Weibull parameters corresponding to the lifetime at a given stress range si can be related to the model parameters A, B, C, D and E through the expressions: C log ED B i i s s l (3) C log D i i s s d (4) A i s b (5) from which the mean and standard deviation values at each level can be derived: C log K B i 1 i s m (6) C log K i 2 i s s (7) where G E A 1 1 D K1 and G G A 1 1 A 2 1 D K 2 2 As a consequence, the mean and standard deviation curves are also represented by equilateral hyperbolas, though latter, logically, not identifiable with the percentile curves. 3.ESTIMATION OF MODEL PARAMETERS USING NORMALIZED VARIABLES To estimate the five parameters of the model, the analyst has pairs of values (N i,si) obtained in experimental tests carried out at several stress ranges. It is possible to estimate the five parameters simultaneously through maximization of the likelihood function, but this procedure generally encounters convergence and precision problems due to the presence of multiple relative maxima in the likelihood function. Alternatively, a more advantageous two step method for estimating the model parameters has been proposed by Castillo et al. [3,4,5] Equation (1) indicates that if the parameters B and C were known the random variable C log B N log s would follow a Weibull distribution with three parameters l' d' and b' depending only on A, D and E. This fact suggests estimating the five parameters in two steps, first B and C, then A, D and E. Once B and C have been estimated, the problem becomes a standard estimation of the three parameters of the Weibull distribution. However, while parameters A, D and E are constant over the whole S N field, the parameters li, di and bi are related to the stress range, si, so that their estimation is conditioned by the fatigue model proposed. Since all the data related to all participating stress levels affect the evaluation of the model parameters, a statistical normalisation seems to be a desirable procedure. This permits data pertaining to Weibull distributions with the same shape parameter b, but with different location and scale parameters l and d respectively (in the present case, the lifetime distributions for different stress ranges) to be pooled in a single distribution in order to evaluate subsequently its parameters with increased reliability This procedure is based on the fact that a Weibull distribution remains stable with respect to location and scale transformations. Thus, if the variable X follows a Weibull distribution for minima b d l W expressed as b d l W X the normalised variable Z, defined as b a X Z will also follow a Weibull distribution for minima (see Figure 2) b d l b b a W Z (8) In the fatigue analysis, log N is the original variable identifiable with X. Depending on the location and the scale transformation, i.e. on the "a" and "b" values, Z can result in different expressions which have to satisfy the fatigue model requirements given by Equations (6) and (7) A suitable choice for "a" and "b" will be crucial in the analysis of the reliability of the evaluation and in its interpretation. In this paper three different normalised Figure 1: S N field with percentiles curves in the fatigue model of Castillo et al. [1] logN C B log s