Determination of sample size for estimation of fatigue life by using Weibull or log-normal distribution

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Abstract

A method of sample size determination to estimate the fatigue life at desired probability of survival, confidence level, and maximum acceptable error is presented. The method is used for the cases where the data follows log normal or Weibull distribution. The error factors for determination of associated error from sample data are also given at 50, 90 and 95 percent probabilities, 90, 95, 97.5, 99 and 99.5 percent confidence levels for the sample size of 3 to 25 both for Weibull and log normal distribution. The practical application of the error factors are illustrated with examples. The accuracy of the method is studied by comparing them to some existing standard methods available in the literature.

Introduction

Fatigue life of any specimen, component or structure is the number of stress or strain cycles to cause failure in a anticipated stress amplitude. It is a function of many variables, including environmental and metallurgical conditions of the material, and exhibits scattered results, even where the specimens are from the same lot and the applied stress or strain cycles are equal. Due to this scattered nature it is difficult to predict the fatigue life or to establish the prediction interval from a limited number of sample data.

Many researchers have developed graphical and analytical methods to evaluate the fatigue life or strength and S-N curve from a limited amount of data [1], [2], [3], [4], [5]. Most of the analytical methods are based on normal or log normal distribution. Some of these methods can not be used for the prediction of life or strength at higher levels of probability. These days there is increasing demand from the design engineers for data concerning statistical fatigue life of various material to assess the reliability of a machine or components. As the fatigue testing is time consuming and costly, setting of the minimum sample size required to extract the statistical information is of great importance. This problem has been considered by many authors and in a variety of applications. Some of the works are those by Parida et. al. [2], Gao [4], Wilks [6], Lawless [7] etc.

In this paper an attempt has been made to derive a relationship between sample size, probability, confidence level and distribution parameters. Both log normal and Weibull distribution functions have been considered. Extensive tables for error factors for different probability and confidence level have also been presented in this paper. These tables can be directly used to determine the maximum error involved in estimating the fatigue life from the experimental data. Step wise procedure to determine the number of specimens required at a pre determined stress amplitude to estimate the fatigue life within a acceptable error at a given probability and confidence level is given. Suitability of the paper is also studied. Method of use of the presented tables is illustrated with examples.

Section snippets

Error factors for sample size determination by Weibull distribution

If fatigue life Nf follows Weibull distribution with cumulative distribution functionFNf(Nf)=1−expNfθβwhere θ is characteristic fatigue life, and β is the Weibull slope of the two parameter Weibull distribution model, X=ln(Nf) will then follow Extreme Value distribution with CDF asFx[ln(Nf)]=1−expx−ξδwhere δ=(1)/(β) and ξ=ln(θ)

The safe fatigue life at α percent of probability and γ percent of confidence level can be determined from the relation asμx=x̄+(d+λ)δ̄where x̄=ξ̄+λ·δ̄ and λ=ln[−ln(1−α)]

Error factors and determination of sample size by log normal distribution

The fatigue life Nf follows normal distribution, then y=log10(Nf) will follow log normal distribution with PDF asf(y)=1σ·exp−(y−μ)22σ2where σ is the population standard deviation of log fatigue life. The safe life can be estimated from the sample data using the relationμy=log(Nf)+u.ψ.swhere log(Nf) is mean value and ψ is a correction factor for sample standard deviation, s and given by [2]ψ=(n−1)2·Γ(n−1)/2Γn/2It can also been shown that [log(Nf)]/[χ2/(n−1)] and [(n−1)s2]/[σ2] have t and χ2

Procedure for determination of sample size

Step wise procedure for determination of sample size for the estimation of fatigue life for a given probability, confidence and acceptable error is given below.

  • 1.

    Test THREE samples at a pre-determined stress amplitude, S and determine the distributional parameters, x̄, s, θ̄, β̄, ξ̄, and δ̄

  • 2.

    Find φw or φN as per the distribution assumed to model the data using , .

  • 3.

    Select the probability, confidence level, and minimum acceptable error, R0 at which fatigue life prediction is to be made.

  • 4.

    Using Table 1,

Applying the proposed method to fatigue data

Fatigue life of 0.20 carbon steel at 240 MPa stress level are presented in Table 6. Twenty five test results are obtained by the rotating bending fatigue test at room temperature.

Use of the presented tables for the determination of the sample size required to estimate the fatigue life at 240 MPa at 90 percent probability and 90 percent confidence level with acceptable error of 5 percent is illustrated below.

With sample size n=8, the estimated distributional parameters are,

x̄=4.503, s=0.3899, β̄

Conclusions

A simple method to determine the sample size required to estimate the fatigue life for a given probability and confidence level is presented. The presented tables of error factors can be directly used for the determination of sample size at any probability and confidence level. Examples presented here illustrates the procedure. The methods are more generally applicable than the example presented.

References (10)

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There are more references available in the full text version of this article.

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