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Chapter 2 Reliability and Confidence Levels of Fatigue Life 2.1 Introduction As is well known, fatigue lives of nominally identical specimens subjected to the same nominal cyclic stress display scatter as shown schematically in Fig. 2.1. This phenomenon reflects the stochastic nature of a fatigue damage process. Previous researches reveal that the uncertainty modeled by the stochastic variables can be divided in the following groups [1, 2]: (1) physical uncertainty, or inherent uncertainty, that is related to the natural randomness of a quantity, (2) measurement uncertainty, i.e., the uncertainty caused by imperfect measurements, (3) statistical uncertainty, which is due to limited sample sizes of observed quantities, (4) model uncertainty, one related to imperfect knowledge or uncertain idealizations of the mathematical models used or uncertainty related to the choice of probability distribution types for the stochastic variables. Based on this, many stochastic mathematical expressions for fatigue damage process have been developed. Due to the variations between individual specimens, fatigue data can be described by random variables to study the variability of fatigue damage and life and to analyze their average trends. With improvement in crack-size measurements, fatigue crack growth data can be depicted by random fields/stochastic processes in a random time–space and state-space to indicate local variations within a single specimen and to analyze the statistical nature of fatigue crack growth data. This has been done by a stationary lognormal process-based randomized approach of deterministic crack growth equation in power law and polynomial forms. In order to understand the stochastic nature of fatigue damage characterization and statistically meaningful data sets, it is desirable to have a technique that accounts for small sample numbers to determine structural fatigue life and performance, which is the focus of this chapter. J. J. Xiong and R. A. Shenoi, Fatigue and Fracture Reliability Engineering, Springer Series in Reliability Engineering, DOI: 10.1007/978-0-85729-218-6_2, Springer-Verlag London Limited 2011 27 28 2 Reliability and Confidence Levels of Fatigue Life Fig. 2.1 S–N curve 2.2 Basic Concepts in Fatigue Statistics It is well known that fatigue is a two-stage process, those of crack initiation and crack propagation; total fatigue life of a structural component is then equal to the sum of crack initiation and crack propagation lives. In general, fatigue life is governed by loading in three interval ranges as follows (1) Low life region, i.e., under large strain cycles, fatigue life of a specimen is less than about 104 cycles; (2) Medium life region, with fatigue life of a specimen amounts from 104 to 106 cycles; (3) Long life region, where under low stress cycles, fatigue life of specimen is greater than about 106 cycles. In general, fatigue lives in the long life region show a greater dispersion than those in the low life region. The factors influencing the dispersion of fatigue experiments, that are also termed as occasional factors, include: (1) measurement equipment uncertainty, (2) inhomogeneity of experimental material, i.e., the test specimens were cut along different orientations of the primary material, (3) inconsistency of specimen dimension and configuration, (4) inconsistency of specimen processing procedures, (5) variability of specimen during a heat-treating process, e.g., different positions of specimens in heat-treating furnace, (6) occasional changes in the experimental environment. Well planned experiments are necessary to determine how fatigue life data are influenced by these occasional factors. The test variables, e.g., fatigue life, fatigue load, fatigue limit and strength limit, etc., dependent on these kinds of occasional factors, are termed as the random variables. The population represents all subjects investigated, whereas the individual indicates a basic unit in a population. The behaviour of the population is dependent on the behaviour of the many individuals. Then it is essential to understand the behaviour of each individual to obtain the characteristics of the population. In this 2.2 Basic Concepts in Fatigue Statistics 29 there are two primary problems whereby: (1) a population is generally composed of a so large number on individuals, even infinite, that it is impossible to investigate all individuals. (2) for a few full-scale parts in industrial production, fatigue tests for determining fatigue lives of individuals are destructive since the tested parts cannot then be practical use, i.e. it is infeasible to perform destructive tests of whole parts. In general, some individuals are randomly sampled from the population to be tested for inferring the nature of population. These sampled individuals are called as the sample and the number of individuals in a sample is termed as the sample size. The components and parts for fatigue tests, or the small standard coupons for determining fatigue behaviour of material are generally known as the specimens. A determined value of fatigue life of a specimen refers to an individual and an experimental dataset of a set of specimens refers to a sample. For example, when sample size equals five, this means that the sample includes five observed values. The eigenvalues of the observed data representing statistical nature of a sample may be classified into two categories as: (1) the central position of data, e.g., mean and median, (2) the dispersion of data, including standard deviation, variance and coefficient of variation, etc. If a sample with a sample size of n is randomly sampled from a population to obtain n observed values of x1, x2, …, xn, then the mean of n observed values is the sample mean and is denoted as x ¼ n 1X xi n i¼1 ð2:1Þ The sample mean represents the central position of data. Besides the arithmetic mean x of sample, the geometric mean G of sample is also usually used in fatigue reliability analysis. In the case of n observed values of x1, x2, …, xn, then the geometric mean G of sample is G¼ n Y !1=n ð2:2Þ xi i¼1 where Q is the continued multiplication notation and n Q xi represents the contin- i¼1 ued multiplication of n observed values of x1, x2, …, xn. The logarithmic form of Eq. 2.2 becomes log G ¼ n 1X log xi n i¼1 ð2:3Þ From Eq. 2.3, it is clear that the logarithm of geometric mean G equals to the arithmetic mean of logarithm of each observed value. In general usage, the mean implies the arithmetic one. The median is also a characteristic value to depict the central position of data. Taking a set of data into sequential arrangement, then the mid value is called as the sample median of the dataset and denoted as Me. In the case of odd number of 30 2 Reliability and Confidence Levels of Fatigue Life observed data the sample median is the mid value, while in the case of even number of observed data the sample median is the mean of two mid values. As a way of measuring dispersion, the sample variance s2 is defined as n P s2 ¼ i¼1 ðxi xÞ2 ð2:4Þ n1 Or alternatively, n P s2 ¼ i¼1 x2i 1 n n P 2 xi i¼1 n1 ð2:5Þ where n is the number of observed values; (n - 1) is the freedom degree of Pn 2 P variance. ni¼1 x2i is the sum of squares of observed value and is the i¼1 xi squares of sum of observed value. The standard deviation is another characteristic value to describe the dispersion of observed data. The square root s of sample variance s2 is termed as the sample standard deviation, namely sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn xÞ2 i¼1 ðxi ð2:6Þ s¼ n1 or sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pn 2 1 Pn i¼1 xi n i¼1 xi s¼ n1 ð2:7Þ The formulations of variance and standard deviation can also be written as Pn 2 x nx2 2 s ¼ i¼1 i ð2:8Þ n1 rP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 2 x2 i¼1 xi n ð2:9Þ s¼ n1 Some characteristic features are as follows: (1) The standard deviation is an important index to indicate the dispersion of data, i.e., a greater standard deviation means a larger dispersion of observed data. (2) The standard deviation is positive with the same unit of observed value. (3) The standard deviation for a set of observed values, randomly sampled from the population, is termed as the sample standard deviation, which is different from the population standard deviation mentioned below. The standard deviation is calculated through the deviations of observed values from the mean. It depends only on the absolute deviation of each observed value 2.2 Basic Concepts in Fatigue Statistics 31 and is independent on the absolute value of each observed data. In order to consider the influence of the observed value on the standard deviation, dividing the standard deviation by the mean yields the characteristic value, namely, the coefficient of variation or coefficient of dispersion Cv as s Cv ¼ 100% x The coefficient of variation is an important index to indicate the relative dispersion of a dataset; it is a dimensionless unit and is generally used for comparing dispersions between two sets of observed values with possibly different features and units. As mentioned above, fatigue life, fatigue load, fatigue limit, strength limit, etc., are random variables, whose expected value n, is defined as Z 1 E ð nÞ ¼ xf ð xÞdx ð2:10Þ 1 E(n) represents the central position of the random variable distribution. If the variance of random variable n is denoted as Var(n), then this is Z 1 VarðnÞ ¼ ½x EðnÞ2 f ðxÞdx ð2:11Þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi The square root VarðnÞ is termed as the standard deviation of random variable n. The expression of variance Var(n) can be simplified into VarðnÞ ¼ E n2 2EðnÞEðnÞ þ ½EðnÞ2 Thus, VarðnÞ ¼ E n2 ½EðnÞ2 ð2:12Þ No matter which distributions two random variables n and g follow, and whether n and g are mutually independent or not, the mathematical expectation of n ? g is equivalent to the sum of the mathematical expectations of n and g. If E(n) and E(g) are known, then it can be shown that Eðn þ gÞ ¼ EðnÞ þ EðgÞ ð2:13Þ Similarly, it is possible to have the mathematical expression for the difference between two random variables n and g as Eðn gÞ ¼ EðnÞ EðgÞ ð2:14Þ Using Eq. 2.13 as an analogy, one can write the sum of the central positions of n random variables n1 ; n2 ; . . .; nn as Eðn1 þ n2 þ þ nn Þ ¼ Eðn1 Þ þ Eðn2 Þ þ þ Eðnn Þ ð2:15Þ 32 2 Reliability and Confidence Levels of Fatigue Life The variance of the sum of random variables n and g may be obtained as Varðn þ gÞ ¼ VarðnÞ þ VarðgÞ þ 2Covðn; gÞ ð2:16Þ In the case where two random variables n and g are mutually independent, it can be proved that the covariance of random variables n and g equals to zero, or Covðn; gÞ ¼ 0. Then Eq. 2.16 becomes Varðn þ gÞ ¼ VarðnÞ þ VarðgÞ ð2:17Þ Again, using the analogy of Eqs. 2.16 and 2.17, it is possible to have Varðn gÞ ¼ VarðnÞ þ VarðgÞ 2Covðn; gÞ ð2:18Þ Varðn gÞ ¼ VarðnÞ þ VarðgÞ ð2:19Þ From Eqs. 2.17 and 2.19, it can be concluded that no matter which distribution two random variables n and g follow, the variance of random variable n ? g is equivalent to that of n-g and equals to VarðnÞ þ VarðgÞ. However, in the case of two dependent random variables, it is necessary to know probability density function (PDF) p(n, g) of two-dimensional random variables (n, g) to obtain the covariance Cov(n, g) and to determine the variance of the sum (or difference) between two-dimensional random variables. Again, using the analogy of Eq. 2.17, it is possible to obtain the variance of the sum of n random variables n1, n2, …, nn, where if n random variables are mutually independent, then Varðn1 þ n2 þ þ nn Þ ¼ Varðn1 Þ þ Varðn2 Þ þ þ Varðnn Þ ð2:20Þ Since the sample mean n is a random variable function as n 1 1X n ¼ ð n1 þ n2 þ þ nn Þ ¼ n n n i¼1 i from Eq. 2.15, it can be deduced that 1 E n ¼ ½ E ð n1 Þ þ E ð n2 Þ þ þ E ð nn Þ n As all individuals (observed values) in a sample come from a same population, random variables n1, n2, …nn have the same PDF. Letting the mean of their same population be l, i.e., E ð n1 Þ ¼ E ð n2 Þ ¼ ¼ E ð nn Þ ¼ l then the expected value can be written as E n ¼l ð2:21Þ 2.2 Basic Concepts in Fatigue Statistics 33 Again from Eq. 2.20, it is possible to have the variance of the sample mean as VarðnÞ ¼ 1 ½Varðn1 Þ þ Varðn2 Þ þ þ Varðnn Þ n2 Letting the variance of identical populations of random variables n1 ; n2 ; . . .; nn be r2, we have, Varðn1 Þ ¼ Varðn2 Þ ¼ ¼ Varðnn Þ ¼ r2 Then the variance of the sample mean becomes VarðnÞ ¼ r2 n ð2:22Þ pffiffiffi and the standard deviation of sample mean is r= n. In case that the samples with a size of n are continuously random-sampled from a specific population to obtain their sample means, then it is almost certain that these sample means would follow a probability distribution with a population mean of E n and a population variance of VarðnÞ. Since the formulations of E n and VarðnÞ are deduced in the case of the unknown population distribution, no matter which probability distribution the population follows, as long as the pop ulation mean and variance of l and r2 are given, it is inevitable for E n and VarðnÞ to be l and r2 n respectively. It is worth noting that two concepts of mean and expected value should not be confused. The expected value in Eq. 2.10 is deduced from the mean, but the mean always is not the expected value. The mean generally includes more comprehensive implications. The expected value represents the mean of possible values of a random variable and is meaningful and significant only in the case of large sample size. When the PDF of a random variable is obtained from a large sample, the expected value determined by using Eq. 2.10 is the population mean and a constant, e.g., the population mean of normal distribution is a constant of value l. It is usual for the observed values of fatigue life N to be transferred into the logarithm form and then plotted into the histogram to clearly show the ordered change of data. Usually also, it is necessary to find a curve, i.e., experimental frequency curve to fit the histogram for statistical analysis. Although the investigated subjects of the histogram are varied, their experimental frequency curves display some common features as: (1) The ordinate of curve is always positive; (2) There is at least one peak on the centre portion of curve; (3) The two ends of the curve spread out along left and right directions until the ordinate of curve equals or is near to zero; (4) The area between the curve and the abscissa axis should be equal to 1. 34 2 Reliability and Confidence Levels of Fatigue Life Fig. 2.2 Experimental frequency curve With increasing observation frequency, the number of grouped data sets increases (shown in Fig. 2.2) and the shape of experimental frequency curve varies less and less until it reaches a stable state. In other words, in the case of n ? ?, the frequency may approximate the probability; the area between frequency curve and horizontal ordinate axis represents the probability; the ordinate of frequency curve depicts probability density and the frequency curve is then called as the probability density curve (PDC). Actually, it is impossible to conduct infinite observations. However, so long as one assumes that infinite individuals exist potentially, no matter whether they are observed one by one or not, the PDC exists objectively and consists of infinite individuals and represents the character of the infinite population. According to the characteristics of the various experimental frequency curves, several probability density functions can be proposed to depict the PDC; the mathematical representation for describing experimental frequency curve is termed as the theoretical frequency function, which is normally known as the PDC in statistics. The normal and Weibull PDFs are usually applied in fatigue reliability. The normal PDF is denoted as ðxlÞ2 1 f ð xÞ ¼ pffiffiffiffiffiffi e 2r2 r 2p ð2:23Þ or " # 1 ðx lÞ2 f ð xÞ ¼ pffiffiffiffiffiffi exp 2r2 r 2p ð2:23Þ where e = 2.718 is the base of a natural logarithm. l and r are constants. The function of f(x) is the normal PDF. As shown in Fig. 2.3, the normal PDC, i.e., the Gaussian curve, demonstrates the curve is bilaterally symmetric and is suitable for representing the observed value of logarithmic fatigue life. The Weibull PDF is also suitable for fatigue statistical analysis and is expressed as follows: h ib 0 b N N0 b1 NNN N f ðN Þ ¼ e a 0 ð2:24Þ Na N0 Na N0 2.2 Basic Concepts in Fatigue Statistics 35 Fig. 2.3 Normal probability density curve Fig. 2.4 Weibull probability density curve or ( ) b N N0 b1 N N0 b f ðN Þ ¼ exp Na N0 Na N0 Na N0 ð2:24Þ where N0, Na and b are three parameters. The Weibull PDC is shown in Fig. 2.4. Figure 2.4 shows that the curve is left–right asymmetric and it intersects the abscissa at N0. In certain cases, from the actual observed results, the Weibull PDC is seen to be representative of fatigue life N. In the inference of fatigue life, the statistics U, v2 and t, etc., are generally implemented for interval estimation. Assuming a random variable X follows the normal distribution and taking the following transformation: U¼ Xl r ð2:25Þ then function U is a random variable. Since X samples are in an interval of (-?, ?), with the sample span of U ¼ ðX lÞ=r also being from -? to ? too, and the PDF of U can then be written as 1 u2 uðuÞ ¼ pffiffiffiffiffiffie 2 ð1\u\1Þ 2p ð2:26Þ where U is the standard normal variable and u(u) is the standard normal PDF. By comparing Eq. 2.26 with Eq. 2.23, it is found that u(u) is the normal PDF with a population mean of 0 and a standard deviation of 1. Therefore, the standard normal or Gauss distribution is denoted as N(0;1). Equation 2.25 is termed the 36 2 Reliability and Confidence Levels of Fatigue Life Fig. 2.5 Standard normal probability density curve standardized substitution of normal variable and the standard normal PDC is shown in Fig. 2.5. The PDF of random variable v2 is expressed as 12m m x ð2:27Þ fm ð xÞ ¼ 2mx21 e2 ð0\x\1Þ C 2 where x is the sampled value of random variable v2. m is a parameter of the PDF of v2 and is termed as a degree of freedom. With increasing m, the PDC of v2 becomes of near symmetric form. The expected value and variance of random variable v2 can be derived as: a ð2:28Þ E v2 ¼ ¼ m b a Var v2 ¼ 2 ¼ 2m b ð2:29Þ The v2 distribution shows the following features as: (1) In case that U1, U2, …, Um are m mutually independent standard normal P variables, then vi¼1 Ui2 follows the v2 distribution with degrees of freedom m. (2) In case where v21 and v22 are mutually independent random variables of v2 with degrees of freedom m1 and m2 respectively, then v21 ? v22 is also a random variable following the v2 distribution with a degree of freedom m1 ? m2. Similarly, it can be deduced that the sum of finite mutually independent random variables of v2 is a random variable of v2, whose degree of freedom equals the sum of degrees of freedom of all random variables of v2. (3) In the case where s2x represents the variance of a random sample from a normal population N(l;r) with a sample size of n, then the random variable ðn 1Þs2x r2 follows the v2 distribution with a freedom degree of m = n - 1. v2 ¼ ðn 1Þs2x r2 ð2:30Þ 2.2 Basic Concepts in Fatigue Statistics 37 From the v2 distribution, it is possible to have the following random variable function: rffiffiffiffiffi v2 ð2:31Þ g¼ m qffiffiffiffi 2 As the sampled span of v2 is from 0 to ?, g ¼ vm samples from 0 to ? too. Thus the PDF of random variable function of g is obtained as m 2 2m 2 m1 1my2 ð2:32Þ gð yÞ ¼ m y e 2 ð0\y\1Þ C 2 If standard normal variables U and g are mutually independent, then the ratio between these two random variables is known as the t distribution with tx as the variable: tx ¼ U U ¼ qffiffiffiffi g v2 ð2:33Þ m The sampled span of random variable tx is from -? to ?. Assuming t0 be a sampled value of random variable tx, then the distribution function P(tx \ t0) becomes Z t0 Pðtx \t0 Þ ¼ hðtÞdt ð2:34Þ 1 with mþ1 2 C mþ1 t2 hðtÞ ¼ pffiffiffiffiffi 2 m 1 þ m pmC 2 ð2:35Þ where h(t) is the PDF of t. Because h(t) is an even function, the t PDC is similar to the standard normal PDC and is bilaterally symmetric relative to the ordinate axis. Further mathematical proof can be deployed to demonstrate that in the case of m ? ?, the t distribution is nearly the same as the standard normal distribution; in reality, in the case of m C 30, both distributions are very close. 2.3 Probability Distribution of Fatigue Life As mentioned above, the normal and Weibull PDFs are usually applied in reliability analysis of fatigue life. The normal PDC expressed by Eq. 2.23 is shown in Fig. 2.6. From Fig. 2.6, it is clear that the curve has a maximum of f(x), a symmetry axis at an abscissa value of l and two inflexions located on the curve at x = l ± r. Bilaterally symmetric sections of the curve spread out along an 38 2 Reliability and Confidence Levels of Fatigue Life asymptote to the abscissa. The shape of the curve depends on the population standard deviation r. The larger the r, flatter is the shape of curve. This implies greater dispersion. In turn, less the r, the sharper is the shape of curve and less is the dispersion. In the case that l and r are known, the normal PDC can be determined completely. The simple notation N(l;r) is implemented to expediently denote the normal distribution with the population mean and standard deviation of l and r separately. For a specific normal PDF, the distribution function F(xp) of the normal variable, i.e., the probability of the normal variable X being less than a value of xp can be obtained as: 1 F xp ¼ P X\xp ¼ pffiffiffiffiffiffi r 2p Zxp e ðxlÞ2 2r2 dx ð2:36Þ 1 where F xp ¼ P X\xp , in geometrical terms, implies the area between the curve from -? to xp and the abscissa axis, i.e., the dashed area as shown in Fig. 2.7. If logarithmic fatigue life follows the normal distribution, then F(xp) equals the rate of failure. In the case of a known normal PDF, Eq. 2.36 shows that the value of F(xp) is fully dependent on xp. With the ordinate and abscissa being F(xp) and xp, the distribution function curve can be drawn, see Fig. 2.8. It is seen that F(xp) increases with increasing of xp; this is owed to the increase in the dashed area between the curve to the left of xp and the abscissa (shown in Fig. 2.7). In the case of xp = l, the dashed area should be 0.5, or F(xp) = 0.5, whereas in the case of xp approaching -? or ?, the limits of F(xp) are 0 and 1 respectively. From the normal PDF, it is possible to have the cumulative frequency function, i.e., the probability of the normal variable X being greater than a value of xp: Z 1 ðxlÞ2 1 P X [ xp ¼ pffiffiffiffiffiffi e 2r2 dx ð2:37Þ r 2p xp Fig. 2.6 Normal probability density curve 2.3 Probability Distribution of Fatigue Life 39 Fig. 2.7 Normal probability density curve Fig. 2.8 Distribution function and cumulative frequency curves If x represents the logarithmic fatigue life, the cumulative frequency function, i.e., P(X [ xp) is equivalent to the reliability level p and is also a function of xp, which has the following relationship with P(X \ xp) as: P X [ xp þ P X\xp ¼ 1 ð2:38Þ The cumulative frequency curve is shown in Fig. 2.8 too. From Fig. 2.8, it is evident that P(X [ xp) decreases with increase in xp, while the reverse is true with P(X \ xp) increasing with decrease in xp. The cumulative frequency function P(X [ xp) of normal variable plays an important role in fatigue reliability analysis. The standardized substitution method of variable is employed to integrate Eq. 2.37; let u¼ du ¼ xl r dx ; dx ¼ rdu r ð2:39Þ 40 2 Reliability and Confidence Levels of Fatigue Life then Eq. 2.37 becomes P X [ xp 1 ¼ pffiffiffiffiffiffi r 2p Z1 e ðxlÞ2 2r2 xp 1 dx ¼ pffiffiffiffiffiffi 2p Z1 u2 e 2 du ð2:40Þ up From Eq. 2.39, the lower limit of the integral is up ¼ xp l r ð2:41Þ Equation 2.40 shows that the integrand function is transferred to be a standard normal PDF as: 1 u2 uðuÞ ¼ pffiffiffiffiffiffie 2 2p Hence, P(X [ xp) can be represented by the area between the normal (or the standard normal) PDC and the abscissa. up is called as the standard normal deviator pertaining to a reliability level of p. Obviously, the relationship between xp and p is established through up. In case of the reliability level p = 50%, then up = 0 and x50 ¼ l ð2:42Þ Evidently, the population mean of l equals to logarithmic fatigue life pertinent to a reliability level of 50%. Fatigue life N50 pertaining to a reliability level of 50% is termed as the median fatigue life, which is the antilogarithm of x50. N50 means that the lives of half the individuals among the population are greater than N50, whereas those of other half are less than N50. If X1 and X2 are statistically independent normal variables with the population means of l1 and l2 and population standard deviations of r1 and r2 respectively, then from Eqs. 2.13 and 2.17, it can be show that 1 ¼ X1 þ X2 inevitably becomes a normal variable too with an expected value of l1 ? l2, a variance of r21 ? r22 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi together with a standard deviation of r21 þ r22 . The PDF of 1 is ( ) 1 ½x ðl1 þ l2 Þ2 ð2:43Þ f ð xÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffi exp 2 2 r1 þ r22 r21 þ r22 2p Using the same method as in the above derivation, it can be deduced that the difference between two normal variables of X1 and X2, 1 ¼ X1 X2 ; is also the normal variable with a mathematical expectation of l1 - l2 and a variance of r21 ? r22. The above findings on the sum or difference between two normal variables can be extended to the case of n normal variables of X1, X2, …, Xn with the population means of l1, l2, …, ln and the population standard deviations of r1, r2, …, rn respectively. Letting 1 ¼ a 1 X1 þ a 2 X 2 þ þ a n Xn ð2:44Þ 2.3 Probability Distribution of Fatigue Life 41 then 1 is a normal variable with the following expected value: Eð1Þ ¼ a1 l1 þ a2 l2 þ þ an ln ð2:45Þ Varð1Þ ¼ a21 r21 þ a22 r22 þ þ a2n r2n ð2:46Þ and the variance: Therefore, it can be concluded that the homogeneous linear function 1 of statistically independent normal variables is also a normal variable with the expected value and variance determined by using Eqs. 2.45 and 2.46 respectively. If fatigue life is denoted as N, assuming a random variable X = log N, then the PDF of logarithmic fatigue life follows the normal distribution, or " # 1 ðx lÞ2 f ð xÞ ¼ pffiffiffiffiffiffi exp 2r2 r 2p From the above equation, it is possible to have the PDF of fatigue life as 1 e pðN Þ ¼ rN pffiffiffiffi 2p ln 10 ðlog NlÞ2 2r2 ð0\N\1Þ ð2:47Þ It is worth noticing that l and r in Eq. 2.47 are the population mean and standard deviation of logarithmic fatigue life respectively. The PDC of fatigue life N is shown in Fig. 2.9. From Eq. 2.47, one has the expected value, i.e., population mean lN of fatigue life N as 1 lN ¼ exp r2 ln2 10 þ l ln 10 2 or log lN ¼ ln lN 1 2 ¼ r ln 10 þ l ln 10 2 ð2:48Þ Equation 2.48 reveals the relationship between the population mean and variance of l and r2 of normal variable X and the population mean lN of random Fig. 2.9 Probability density curve of fatigue life 42 2 Reliability and Confidence Levels of Fatigue Life variable N. l is regarded as the population mean of logarithmic fatigue life (x = log N). If the sample size is denoted as n, then the estimator of l is 1 ^ ¼ ðlog N1 þ log N2 þ þ log Nn Þ l n ^ is equivalent to the estimator of logarithmic For the normal distribution, l fatigue life pertaining to a reliability level of 50%, alternatively ^ 50 ^ ¼ log N l ^ 50 is the estimator of fatigue life pertaining to a reliability level of 50% where N (i.e., median estimator of fatigue life) and equals the geometric mean of fatigue lives observed at values of N1, N2, …, Nn as 1 ^ 50 ¼ ðN1 N2 ; . . .; Nn Þn N However, lN represents the population mean of fatigue life N, whose estimator is the arithmetic mean of fatigue lives as 1 ^N ¼ ðN1 þ N2 þ þ Nn Þ l n ^ and l ^N , it is easy to find the By comparing the above two estimators of l differences between l and lN. Equation 2.48 shows that in the case of the logarithmic fatigue life following the normal distribution, the difference between log lN and l is 12 r2 ln 10; and the population variance of logarithmic fatigue life is r2. Furthermore, it can be proved that the reliability level p corresponding to logarithmic fatigue life of log Np is the reliability level pN pertaining to fatigue life Np. Moreover, from Eq. 2.47, it is possible to obtain the reliability level pN pertinent to a specific fatigue life Np as [3] 1 pN ¼ pffiffiffiffiffiffi r 2p ln 10 Z1 Þ2 1 ðlog Nl e 2r2 dN N ð2:49Þ Np As mentioned above, the normal distribution is suitable for the cases of medium and short life ranges, whereas the Weibull distribution fits better for fatigue life in long life range of greater than 106 cycles. The Weibull PDF has the advantage of a minimum safe life, i.e., the safe life corresponding to a reliability level of 100%, while from the normal distribution theorem, only in the case where the logarithmic safe life xp = log Np is near -?, or Np = 0, the reliability level is 100%. Evidently, this is in disagreement with the actual case. In order to overcome this drawback, it is necessary to add an undetermined parameter N0 to replace xp = log Np with xp = log (Np - N0); here N0 is the minimum safe life pertaining to a reliability level of 100%. The Weibull PDF can be allowed to depict the distribution law of fatigue life N as 2.3 Probability Distribution of Fatigue Life 43 ( ) b N N0 b1 N N0 b f ðN Þ ¼ exp ðN0 \N\1Þ Na N0 Na N0 Na N0 ð2:50Þ where N0 is the minimum life parameter, Na is the characteristic life parameter and b is the Weibull shape parameter (slope parameter). Due to the Weibull PDF being characterised by three parameters unlike the normal distribution having only two, i.e., l and r, the Weibull PDF may more perfectly fit the experiments than the normal distribution. In the case of b = 1, f(N) in Eq. 2.50 becomes a simple exponential PDF. In the case of b = 2, f(N) is the Rayleigh PDF and in the case of b = 3*4, f(N) is close to the normal PDF. The Weibull PDC is shown in Fig. 2.10. Figure 2.10 shows that the peak of curve always deviates to the left and the deviation varies with the change of b. For b [ 1, the curve intersects the abscissa at N = N0 and exists a high-positive minimum life of N0, the difference of (Na - N0) is greater, the curve becomes flatter and the dispersity is larger. The right end of curve spreads out to infinity along an asymptote to the abscissa. Furthermore, as shown below, it can be proved that like other PDFs, the Weibull PDF satisfies the condition of R1 f ð N ÞdN ¼ 1, i.e., the area between the curve and the horizontal ordinate axis N0 equals to 1. If b = 2, then the Weibull distribution becomes the Rayleigh PDF as f ðN Þ ¼ 2N NN 22 e a Na2 The random variable following the Weibull distribution, i.e., the Weibull variable, is denoted as Nn. From Eq. 2.50, one has the distribution function F(Np) of the Weibull variable, namely, the probability P(Nn \ Np) of Nn being less than a value of Np as Z Np f ðN ÞdN ð2:51Þ F Np ¼ P Nn \Np ¼ N0 Equation 2.51 represents the area between the curve from N0 to Np and the abscissa, as the dashed area shown in Fig. 2.11. Substituting Eq. 2.50 into Eq. 2.51 yields Fig. 2.10 Weibull probability density curve 44 2 Reliability and Confidence Levels of Fatigue Life Fig. 2.11 Weibull probability density curve F Np ¼ Z Np N0 ( ) b N N0 b1 N N0 b exp dN Na N0 Na N0 Na N0 ð2:52Þ Taking the following transformation N N0 b N N0 dN 1 1b 1 ¼ Z; ¼ Zb; ¼ Z b dZ Na N0 b Na N0 Na N0 then in the case of N = N0, Z = 0, and in the case of N = Np, Z ¼ h Np N0 Na N0 ib . Taking the integral transformation of Z with a lower limit of 0 and an upper h ib N N limit of Zp ¼ Npa N00 , then Eq. 2.52 becomes F Np ¼ Z 0 Zp b Na N0 1b Z b dZ ¼ Z b1 eZ Na N0 b Z 0 Zp Z eZ dZ ¼ eZ 0 p ¼ 1 eZp Substituting Zp into the above equation, one has the distribution function as ( ) Np N0 b F Np ¼ 1 exp ð2:53Þ Na N0 The distribution function curve is shown in Fig. 2.12 with an ordinate of P Nn \Np ¼ F Np and an abscissa of Np. It can be observed that P(Nn \ Np) increases with increasing of Np, this is because the area between the curve from N0 to Np and the abscissa increases with the shifting of Np to the right (shown in Fig. 2.11). Equation 2.53 reveals that in the case of Np ? ?, the limit of P(Nn \ Np) is 1 (shown in Fig. 2.12). Substituting P(Nn \ Np) = 1 and Np = ? into Eq. 2.51, it is possible to have Z 1 f ðN ÞdN ¼ 1 N0 The above equation demonstrates that the area between the curve and the horizontal ordinate axis equals unity. 2.3 Probability Distribution of Fatigue Life 45 Fig. 2.12 Distribution function and cumulative frequency curves The distribution function P(Nn \ Np) is equivalent to the rate of failure and cumulative frequency function P(Nn [ Np) equals the reliability level, or ( ) Np N0 b P Nn [ Np ¼ 1 P Nn \Np ¼ exp ð2:54Þ Na N0 The cumulative frequency function P(Nn [ Np) is denoted as the reliability level p as ( ) Np N0 b p ¼ exp ð2:55Þ Na N0 In the case of known parameters of N0, Na and b together with a specific reliability level p, from Eq. 2.55, one has a safe life of Np, i.e., fatigue life per tinent to a reliability level of p. The curve of p ¼ P Nn [ Np is shown in Fig. 2.12 too. From Fig. 2.12, it is clear that when Np = N0, p = 1, that is the minimum life N0 is the safe life pertaining to a reliability level of 100%. When Np = Na, from Eq. 2.55, it is possible to have ( ) Np N0 b 1 ¼ 36:8% p ¼ exp ¼ e1 ¼ 2:718 Na N0 This implies that the characteristic life parameter Na is fatigue life corresponding to a reliability level of 36.8% (shown in Fig. 2.12). Owing to the Weibull PDF requiring three parameters of N0, Na and b, but without both parameters of l and r2, so it is necessary to employ the three parameters of N0, Na and b to derive the parameters of l and r2. From Eq. 2.10, the definition of mathematical expectation of the Weibull variable Nn can be written as Z 1 EðNn Þ ¼ Nf ðN ÞdN N0 46 2 Reliability and Confidence Levels of Fatigue Life Fig. 2.13 Weibull probability density curve Substituting Eq. 2.50 into the above equation and taking the following transformation of variable as N N0 b N N0 dN 1 1b 1 ¼ Z; ¼ Zb; ¼ Z b dZ ð2:56Þ Na N0 b Na N0 Na N0 then N0 ðNa N0 Þ 1b 1 b1 Z b dZ b Zb þ Z b eZ b N N a 0 0 Z 1 1 Z ð1þbÞ1 eZ dZ þ N0 ¼ ðNa N0 Þ E ðN n Þ ¼ Z 1 0 From the definition of CðaÞ function, the integral item in the above equation becomes Z 1 1 1 Z ð1þbÞ1 eZ dZ ¼ C 1 þ b 0 Hence, it is possible to have the mathematical expectation of the Weibull population mean l as a function of the three parameters that define this distribution as 1 ð2:57Þ l ¼ EðNn Þ ¼ N0 þ ðNa N0 ÞC 1 þ b According to the geometric meaning of mathematical expectation, the population mean l is the centre position of form of the area between the PDC f(N) and the abscissa (shown in Fig. 2.13), while the population median N50 represents fatigue life Np corresponding to a reliability level of 50%. From Fig. 2.13, it can be observed that in the case of b = 1.74, the peak of curve deviates to the left and l [ N50, whereas for the normal population, because of the symmetry of the curve, both population mean and median are concurrent as demonstrated in Eq. 2.51: l = N50. Therefore, it is essential to implement the median fatigue life or strength, and not by the mean, to obtain fatigue behaviour of material. If the mean equals to the median, i.e., l = N50, then the Weibull PDF is close to the normal PDF and the shape parameter of the Weibull distribution b = 3.57. 2.3 Probability Distribution of Fatigue Life 47 From Eq. 2.12, it is possible to derive the variance Var(Nn) of the Weibull variable as 2 1 2 r ¼ VarðNn Þ ¼ ðNa N0 Þ C 1 þ C 1þ b b 2 2 ð2:58Þ Equation 2.58 is regarded as a measure of the population dispersion. Equation 2.58 shows that r2 increases with increasing (Na - N0) and decreases with increasing b. The Weibull distribution has a strong compatibility and flexibility to fit the experimental data, since the shape of PDC is capable of deviating to the left and right with the deviation being determined through a skew coefficient. 2.4 Point Estimation of Population Parameter Estimating the population parameters, e.g., l and r2 from a sample is termed as point estimation and a sample with sample size greater than 50 is regarded as a large sample. In the tests of fatigue life, only one value can be determined from one specimen. In many circumstances, due to time and resource constraints, it is infeasible to conduct extensive experimental investigations in order to generate large numbers of datasets required by classical statistical processes. In contrast, only small numbers of sample data (sample size n \ 50) can be provided. When the sample eigenvalues are taken as the estimators of population parameter, generally, it is necessary to satisfy the demands of consistency and unbiasedness. In the case of sample size n ? ?, the sample mean x becomes the expected value E(n) of random variable and the population distribution coincides with the distribution of random variable. This is because the sample mean x approximates uniformly to the population mean l. Similarly, in the case of sample size n ? ?, the sample variance s2 approximates uniformly to the random variable variance Var(n), i.e., population variance r2. Obviously, in ^ case where the sample mean x and variance s2 are regarded as the estimators l ^2 of population mean l and variance r2 respectively, then the estimators and r become closer to the truths of population parameter with increasing of sample size n. Unbiased estimator means that the expected value of estimator as a random variable determined from each sample with a sampled size of n should equal the estimated population parameter. For example, if the sample mean x is taken as the ^ of population mean l, then it is necessary for the expected unbiased estimator l ^ of popuvalue of sample mean to be equal to l. Thus, the unbiased estimator l lation mean should satisfy the following condition ^Þ ¼ l E ðl ð2:59Þ 48 2 Reliability and Confidence Levels of Fatigue Life As a random variable, the sample mean may be written as 1 n¼ n n X ni i¼1 From Eq. 2.21, the expected value of n just equals to l, or E n ¼l The above equation reveals that the sample mean x satisfies the unbiasedness condition as the estimator of population mean l, so x ¼ l ^ ð2:60Þ Letting the observed values of fatigue life to be N1, N2, …, Nn, then the estimator of normal population mean of logarithmic fatigue life is ^ ¼ x ¼ l n 1X log Ni n i¼1 For the normal distribution, it is possible to have l ¼ x50 ¼ log N50 ^ 50 of median fatigue life From the above two equations, one has the estimator N as ^ 50 ¼ log N n 1X log Ni n i¼1 or 1 ^ 50 ¼ ðN1 N2 . . .Nn Þn N In case that fatigue life follows the Weibull distribution, then the population mean l of the Weibull distribution can also be estimated by using the sample mean as N n X ¼1 ^¼N l Ni ð2:61Þ n i¼1 ^2 of population variance should satisfy the Similarly, the unbiased estimator r following condition as 2 ^ ¼ r2 E r ð2:62Þ The sample variance s2 satisfies the condition (2.62) as the unbiased estimator ^2 of population variance r2, thus r ^2 s2 ¼ r ð2:63Þ 2.4 Point Estimation of Population Parameter 49 Table 2.1 Correction coefficient ^k of standard deviation n 5 6 7 8 9 10 11 12 13 14 ^k n ^k 1.063 1.051 1.042 1.036 1.031 1.028 1.025 1.023 1.021 1.020 15 1.018 16 1.017 17 1.016 18 1.015 19 1.014 20 1.014 30 1.009 40 1.006 50 1.005 60 1.005 No matter which distribution the population follows, Eqs. 2.60 and 2.62 are suitable for the estimation of mean and variance parameters. Note that the estimators are not equivalent to the truths of population parameters of l and r2 absolutely. Only in the case of large enough sampling, is the estimator near the truth. The unbiased estimator of population variance is 2 Pn 2 1 Pn Pn xÞ2 i¼1 xi n i¼1 xi 2 2 i¼1 ðxi ^ ¼s ¼ ¼ r n1 n1 ^2 , i.e., The estimator of standard deviation is obtained by the square root of r ^ ¼ s. Strictly speaking, the sample standard deviation s is a biased estimator of r population standard deviation since the unbiased condition of E(sn) = r is not satisfied. In fatigue reliability design, the sample standard deviation s is usually corrected to find an unbiased estimator of population standard deviation to remove the bias by using v2 distribution. However, such unbiased estimator fits only for the normal population. The unbiased estimator of normal population standard deviation can be written as ^ ¼ ^ks r ð2:64Þ where ^k ¼ rffiffiffiffiffiffiffiffiffiffiffi n1 n 1C 2 2 C n2 ð2:65Þ ^k is the coefficient of correction of standard deviation. If the population follows the normal distribution, the unbiased estimator of population standard deviation can be obtained from Eq. 2.64. The coefficients of correction corresponding to different sample size are listed in Table 2.1. Table 2.1 shows that there is a small difference between the correction coefficient ^k and 1; as a result, no correction is ^ ¼ s is taken. This is especially conducted in general engineering application and r ^ so in the case of n [ 50; then k ! 1. Thereby, for large sample sizes, the sample standard deviation s always is the unbiased estimator of population standard deviation r. However, in fatigue reliability design of aeronautics and marine structural parts, it is desirable to correct the sample standard deviation s. Gener^ ¼ x of population mean l is suitable for no matter ally, the unbiased estimator l which distribution the population follows. Consequently, for the normal population, the population mean l is the population median and the sample mean is the 50 2 Reliability and Confidence Levels of Fatigue Life estimator of population median. In the case where the population follows the ^ ¼ x and r ^ ¼ ^ks into Eq. 2.41, it is possible to normal distribution, substituting l have the estimator of a percentile xp pertaining to a reliability level of p as ^xp ¼ l ^ þ up r ^ ¼ x þ up ^ks ð2:66Þ Thus the estimators of each fatigue life Ni pertinent to a reliability level could be determined from small number of samples. A sample with a sample size of n is random-sampled from a known population to obtain n observed values for arranging in sequential queue from smaller to greater as x1 \x2 \ \xi \ \xn where i is the arranged ordinal of the observed value from smaller to greater. If the PDF of the population is denoted as f(x), then the rate of failure F(xi) (distribution function) of ith observed value xi can be determined. No matter which distribution the random-sampled population follows, or which PDF f(x) is, the mathematical expectation of the rate of failure corresponding to xi is i/(n ? 1), which is termed as the mean rank and regarded as the estimator of population failure rate in engineering. Therefore, the estimator of population reliability level p pertaining to ith observed value xi becomes ^ p¼1 i nþ1 ð2:67Þ In case of only one specimen for fatigue test, i.e., n = 1, then from Eq. 2.67, the estimator of reliability level pertinent to fatigue life of specimen is only 50%, namely ^ p¼1 i 1 ¼1 ¼ 50% nþ1 1þ1 2.5 Interval Estimation of Population Mean and Standard Deviation In reality, no true population parameters, e.g., mean l and standard deviation r, etc., are known. And it is hard for the point estimator of population parameter determined from a sample with a finite sample size to amount to the theoretical truth obtained from infinite observed values. As a consequence, sometimes it is feasible to use an interval limit for estimating population parameter to indicate the error of estimation. At a specific probability, the location interval of population parameter can be estimated by using the sample eigenvalues and is called as the interval estimation of population parameter. Though the estimator x of population mean l satisfies the consistency and unbiasedness demands, it is possible for the 2.5 Interval Estimation of Population Mean and Standard Deviation 51 Fig. 2.14 Standard normal probability density curve sample mean x determined from a small sample with finite observed values to be close to, but impossible to equal to the population mean l. Therefore, it is imprecise for finite observed values to be applied to estimate the population mean, whereas it is feasible for the sample mean to be employed for estimating the location interval population mean pertaining to a specific probability, which is termed as the interval estimation of population mean. In the interval estimation of population mean l, assuming l be unknown, then the standard normal variable can be written as u¼ x l pffiffiffi r0 = n In general, it is possible to select a probability of c from the area between the standard normal PDC and the abscissa as the dashed area shown in Fig. 2.14. Thus the blank areas between two ends of curve and the abscissa are (1-c)/2 respectively; the corresponding uc can be determined from the numerical tabular representation of the normal distribution (listed in Table 2.2). So the standard normal variable locates in an interval of (-uc; uc) at a probability of c, that is uc \u\uc where is termed as the confidence level. We then have uc \ x l pffiffiffi\uc r0 = n or r0 r0 x uc pffiffiffi\l\x þ uc pffiffiffi n n ð2:68Þ Equation 2.68 is the interval estimation formula of the normal population mean l. Equation 2.68 demonstrates that the confidence level of the interval pffiffiffi pffiffiffi x uc r0 = n; x þ uc r0 = n including the population mean l equals to c, here the pffiffiffi pffiffiffi interval is called as the confidence interval, and x þ uc r0 = n and x uc r0 = n are termed as the confidence upper and lower limits respectively. The pre-condition 52 2 Reliability and Confidence Levels of Fatigue Life Table 2.2 Numerical tabular of and u u c u c u c 3.0 2.9 2.8 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 0.8664 0.8384 0.8064 0.7698 0.7286 0.6824 0.6318 0.5762 0.5160 0.4514 0.3830 0.3108 0.2358 0.1586 0.0796 4.753 4.265 3.719 3.090 2.576 2.326 1.960 1.645 1.282 1.036 0.842 0.674 0.524 0.385 0.253 0.126 0 0.999998 0.99998 0.9998 0.998 0.990 0.980 0.950 0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100 0.000 0.9974 0.9962 0.9948 0.9930 0.9906 0.9876 0.9836 0.9786 0.9722 0.9642 0.9544 0.9426 0.9282 0.9108 0.8904 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 to apply Eq. 2.68 is to know the population standard deviation, or to deal with large sample. However, sometimes it is hard to satisfy the above conditions. Actually, it is feasible to apply the t distribution theorem to treat a sample with a denotes sample size of greater than 5 from the practical experience. Assuming X the sample mean with a sample size of n random-sampled from the normal population N(l;r), then the standard normal variable becomes U¼ l X prffiffi ð2:69Þ n Substituting Eq. 2.69 into Eq. 2.33 yields pffiffiffi Xl n r pffiffiffiffiffi lÞ mn ðX pffiffiffiffiffi tx ¼ qffiffiffiffi ¼ v2 r v2 m s2x If denotes the sample variance randomly-sampled from the normal population N(l;r), then the freedom degree of v2 ¼ ðn 1Þs2x r2 is m = n - 1. Substituting v2 and m into the above equation gives the tx variable with a freedom degree of m = n - 1 as tx ¼ lpffiffiffi X n sx ð2:70Þ A confidence level of c is selected to determine two abscissa values of t and -tc (shown in Fig. 2.15), between which the area below the curve (i.e., the dashed area in Fig. 2.15) amounts to c. Thus, the tc is obtained as 2.5 Interval Estimation of Population Mean and Standard Deviation 53 Fig. 2.15 t probability density curve Z1 hðtÞdt ¼ 1c 2 ð2:71Þ tc Since the probability of the tx variable located in an interval of (-tc, tc) is equivalent to c (i.e., a confidence level of c), it is possible to have the following inequality tc \tx \tc ð2:72Þ Substituting Eq. 2.70 into Eq. 2.72 shows tc \ lpffiffiffi X n\tc sx where l is assumed to be an undetermined value. Assuming random variables X and sx be sampled as x and s respectively in a sampling, then the above in equation becomes x lpffiffiffi tc \ n\tc s ð2:73Þ Equation 2.73 can also be written as s s x tc pffiffiffi\l\x þ tc pffiffiffi n n ð2:74Þ Equation 2.74 is the interval estimation formula of the normal population mean l, demonstrating that the confidence level of the confidence interval sffiffi sffiffi p p x tc ; x þ tc including the population mean l equals c. n n From Fig. 2.15, it is observed that greater the confidence level c, greater is the value of tc ; and wider is the confidence interval. In fact, it is desirable for the confidence interval to be less and for the confidence level to be greater. However, the above theorem reveals that less the confidence interval, less the confidence 54 2 Reliability and Confidence Levels of Fatigue Life Fig. 2.16 v2 probability density curve level becomes. To address this contradiction, i.e., not only to decrease the confidence interval but also to keep the confidence level high, it is necessary to increase the sample size of n to reduce the value of tc psffiffin and then to lower the confidence interval of x tc psffiffin; x þ tc psffiffin : The transformation form of Eq. 2.74 becomes stc l x stc \ pffiffiffi pffiffiffi\ x x n x n ð2:75Þ where ðl xÞ=x is the relative error of sample mean x to population mean l. The relative error limit (absolute value) is denoted as d, or stc d ¼ pffiffiffi x n ð2:76Þ where d is a small quantity in a span from 1 to 10%. In the case where x, s and n satisfy Eq. 2.76, Eq. 2.75 demonstrates that the confidence level of the relative error of sample mean to population median being less than ±d is equal to c. Hence, by using Eq. 2.76, the least number of observed values (or effective specimens) is obtained. From Eq. 2.30, it is possible to conduct the interval estimation of population standard deviation. Similarly, a confidence level of c is selected to determine an interval of (v2c1 ; v2c2 ) (shown in Fig. 2.16), between which the area between the curve and the abscissa (i.e., the dashed area in Fig. 2.16) amounts to c. In the case that c and m are known, from Table 2.3 of v2 distribution. v2c1 and v2c2 are obtained to make the probability of the v2 variable locating in the interval of (v2c1 ;v2c2 ) to equal to c, alternatively, for a confidence level of c, one has v2c1 \v2 \v2c2 Substituting Eq. 2.30 into the above equation yields v2c1 \ ðn 1Þs2x \v2c2 r2 2.5 Interval Estimation of Population Mean and Standard Deviation 55 Table 2.3 Numerical tabular of vc2 c 0.0 0.80 t 0.90 0.95 0.98 0.99 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 7.815 9.488 11.070 12.592 14.067 15.507 16.919 18.307 19.675 21.026 22.362 23.685 24.996 26.296 27.587 28.869 30.144 31.410 32.671 33.924 35.172 36.415 37.652 9.348 11.143 12.823 14.449 16.013 17.535 19.023 20.483 21.920 23.337 24.736 26.119 27.488 28.845 30.191 31.526 32.852 34.170 35.479 36.781 38.076 39.364 40.646 11.345 13.277 15.068 16.812 18.475 20.090 21.666 23.209 24.725 26.217 27.688 29.141 30.578 32.000 33.409 34.805 36.191 37.566 38.932 40.289 41.638 42.80 44.314 12.838 14.860 16.750 18.548 20.278 21.955 23.589 25.188 26.757 28.300 29.819 31.319 32.801 34.267 35.718 37.156 38.582 39.997 41.401 42.796 44.181 45.558 46.928 2.366 3.357 4.351 5.348 6.346 7.344 8.343 9.342 10.341 11.340 12.340 13.339 14.339 15.338 16.338 17.338 18.338 19.337 20.337 21.337 22.337 23.337 24.337 6.251 7.779 9.236 10.645 12.017 13.362 14.684 15.987 17.275 18.549 19.812 21.064 22.307 23.542 24.769 25.989 27.204 28.412 29.615 30.813 32.007 33.196 34.382 Assuming r be an unknown value and random variable sx be sampled as s in a sampling, then the above inequality becomes v2c1 \ ðn 1Þs2 \v2c2 r2 Inverting the above inequality leads to 1 r2 1 \ \ 2 2 2 vc2 ðn 1Þs vc1 or sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffi n1 n1 s \r\s v2c2 v2c1 ð2:77Þ Equation 2.77 is the interval estimation formula of normal population standard deviation r, demonstrating that the confidence level of the confidence interval rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . . 2 s ðn 1Þ vc2 ; s ðn 1Þ v2c1 including the value of r is c. 56 2 Reliability and Confidence Levels of Fatigue Life 2.6 Interval Estimation of Population Percentile The normal population percentile xp is defined in probability form as Z 1 P X [ xp ¼ f ð xÞdx ¼ p xp where f(x) is the PDF of normal variable X. From Eq. (2.41), the value of xp = l ? upr corresponding to a reliability level of p is the population percentile, which represents the logarithmic safe life or safe fatigue strength. From Eq. 2.66, the percentile estimator is ^xp ¼ x þ up r ^ ^ ¼ ^ks. It is hard for the sample percentile (x þ up r ^) to just equal to the where r population true value (l ? upr); in other words, it is possible for the sample ^) to be greater or less than the population truth (l ? upr). Thus percentile (x þ up r ^ of normal there is an error between the predictions for sample percentile x þ up r random variable X, determined from Eq. 2.66, and the population true value l ? upr. Also the confidence level of statistical results estimated from small samples needs to be analysed. The confidence level is a statistical variable correlated with sample size. Usually, the confidence levels are fixed and the amount of data determines the width of the corresponding confidence intervals, i.e., the confidence interval decreases with increase in the amount of data (shown in Fig. 2.17). While the width of the corresponding confidence intervals is fixed and the amount of data determines the confidence levels, e.g., under a fixed width of corresponding confidence intervals, the fewer the samples, the less data items are available and thus the lower the confidence level in the statistical results. With larger sample numbers, there are more data items and thus the confidence level in ^ of normal random the statistical results is higher. The sample percentile x þ up r variable X pertinent to reliability level p, estimated using a large sample size, is near the population true value and the confidence levels are generally high. Thus it Fig. 2.17 Normal probability density curve 2.6 Interval Estimation of Population Percentile 57 is not necessary for the confidence level of large sample sets to be analyzed; only the reliability level need be estimated using Eq. 2.66. A log-normal distribution lends itself to a perfect theoretical solution to estimate the unbiased distribution parameter values, the confidence level and the confidence interval. It is thus an apt approach for fatigue life estimation and is widely applied in reliability-based design approaches using small sample numbers. Using the log-normal distribution and Eq. 2.66, the confidence level of safe fatigue life can be determined easily. In addition, it is also possible to predict the minimum number of specimens required for fatigue tests for a given confidence level. As is well known, the t-distribution is generally used for the calculation of confidence interval for the estimator of mean value and variance of a random variable. However, a theoretical solution of the confidence interval for the estimator of sample percentile of a random variable does not currently exist. Consequently, it is essential to establish the relationship between t-statistics and ^ of the normal random variable X and to derive a thesample percentile x þ up r ^ oretical solution to determine the confidence interval of sample percentile x þ up r of normal random variable X pertinent to reliability level p from small sample numbers. ^ can be written into the From Eqs. 2.65 and 2.66, the sample percentile x þ up r random variable function as þ up ^ksx f¼X and sx denote the random variables of sample mean and sample standard where X deviation respectively. In practical application, f is assumed to approximately follow the normal distribution. Therefore it is possible to calculate the expected value E(f) and variance Var(f). þ up ^ksx ¼ EðX Þ þ up E ^ksx E ð fÞ ¼ E X Þ ¼ l and r ^ ¼ ^ks is the From Eqs. 2.60 and 2.64, it can be show that EðX unbiased estimator of normal population standard deviation, i.e., E ^ksx ¼ r Thus, E(f) becomes E ð f Þ ¼ l þ up r ð2:78Þ The variance Var(f) can be written as þ up ^ksx VarðfÞ ¼ Var X or Þ þ u2p ^k2 Varðsx Þ VarðfÞ ¼ VarðX ð2:79Þ 58 2 Reliability and Confidence Levels of Fatigue Life From Eq. 2.22, one has 2 Þ ¼ r VarðX n ð2:80Þ Again, from Eq. 2.30, the v2 variable with degree of freedom m = n - 1 is ðn 1Þs2x ¼ v2 r2 ð2:81Þ r sx ¼ pffiffiffiffiffiffiffiffiffiffiffiv n1 ð2:82Þ r r2 Varðsx Þ ¼ Var pffiffiffiffiffiffiffiffiffiffiffiv ¼ Var ðvÞ n1 n1 ð2:83Þ Then, From Eqs. 2.12 and 2.83, it can be shown that Varðsx Þ ¼ o r2 n 2 E v ½EðvÞ2 n1 ð2:84Þ Based on Eq. 2.28, expected value of the v2 variable is Z 1 2 E v ¼ xfm ð xÞdx ¼ m 0 By means of the statistics, it is also possible to have pffiffiffiC mþ1 EðvÞ ¼ 2 2m C 2 Substituting the above two equations into Eq. 2.84 shows 8 " #2 9 = C n2 r2 < Varðsx Þ ¼ n 1 2 n1 ; n 1: C 2 Again, substituting Eqs. 2.80 and 2.85 into Eq. 2.79 yields 8 " #2 9 = C n2 r2 u2p ^k2 r2 < VarðfÞ ¼ þ n 1 2 n1 ; n n1: C 2 ð2:85Þ ð2:86Þ By using Eq. 2.65, the simplified form of above equation is deduced as 2 1 2 ^2 þ up k 1 VarðfÞ ¼ r ð2:87Þ n 2.6 Interval Estimation of Population Percentile 59 From Eqs. 2.78 and 2.87, the standard normal variable is derived as þ up ^ksx l þ up r X f E ð fÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ VarðfÞ r 1n þ u2p ^k2 1 ð2:88Þ In terms of Eq. 2.33, the tx variable is U tx ¼ qffiffiffiffi v2 m Substituting Eqs. 2.81 and 2.88 as well as m = n - 1 into the above equation allows þ up ^ksx l þ up r X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tx ¼ sx 1n þ u2p ^k2 1 Thus, in a sampling, the sampled value of tx variable is x þ up r ^ l þ up r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t¼ s 1n þ u2p ^k2 1 ð2:89Þ In the case of a specific confidence level of c and degree of freedom of m = n - 1, the tc value can be obtained in an interval of (-tc, tc) pertaining to a confidence level of c, namely, tc \t\tc or x þ up r ^ l þ up r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q tc \ \tc s 1n þ u2p ^k2 1 Transforming the above inequality yields qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 ^2 tc s 1n þ u2p ^k2 1 t s c x þ up r ^ l þ up r n þ up k 1 \ \ x þ up r x þ up r x þ up r ^ ^ ^ ð2:90Þ Equation 2.90 is the interval estimation formula of the normal population percentile. The minimum number of observed values (or specimens) can be determined ^) and (l ? upr) exceeds a limit of d by stipulating that no error between (x þ up r pertinent to a specific confidence level of c. The error limit is denoted as d, then qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tc s 1n þ u2p ^k2 1 d¼ ^ x þ up r 60 2 Reliability and Confidence Levels of Fatigue Life ^ ¼ ^ks, the function of error limit d with respect to the coefficient of Since r variation s=x is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tc xs 1n þ u2p ^k2 1 ð2:91Þ d¼ 1 þ up ^k xs For a reliability level of p = 50%, up = 0 and Eq. 2.91 degenerates into Eq. 2.76. The error limit d generally is selected from 1 to 10%. Thus it is possible to establish the minimum number of specimens to estimate the population percentile as s d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2:92Þ x t 1 þ u2 ^k2 1 0:05u ^k c n p p where s=x is calculated from n observed values. In the case where n satisfies Eq. 2.92 and there is no system error, the confidence level of the relative error of ^) to the population truth (l ? upr) being less than ±d, amounts to c. (x þ up r It is worth pointing out that the reliability level and confidence level are two different concepts, e.g., the confidence level of c = 95% implies that among 100 ^) pertinent to a reliability level of p, estimators of logarithmic safe life (x þ up r determined from 100 samples, the relative error of 95 estimated values to the truth (l ? upr) is less than ±5%. Obviously, the confidence level is proposed with regard to the sample, whereas the reliability level is defined regarding the individual. According to the t distribution theorem, one has the lower confidence limit of logarithmic fatigue life corresponding to a confidence level of c, with P t\tc ¼ c ð2:93Þ where tc is the c percentile of t distribution corresponding to a confidence level c. Substituting Eqs. 2.89 and 2.64 into Eq. 2.93 and taking transformation gives ( ) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 ^ ^ þ up k 1 \ l þ u p r P x þ up ks tc s ¼c ð2:94Þ n From Eq. 2.94, it is possible to have the one-sided lower confidence limit of logarithmic fatigue life pertaining to a reliability level of p and a confidence level of c as [4] rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ^ ^ ^xpc ¼ log Npc ¼ x þ up ks tc s þ u2p ð^k2 1Þ ð2:95Þ n Similarly, the one-sided upper confidence limit of logarithmic fatigue life pertinent to a reliability level of p and a confidence level of c is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ pc ¼ x þ up ^ks þ tc s 1 þ u2p ð^k2 1Þ ^xpc ¼ log N ð2:96Þ n References 61 References 1. Gao ZT (1981) Statistics applied in fatigue. National Defense Industry Press, Beijing 2. Gao ZT, Xiong JJ (2000) Fatigue Reliability. Beihang University Press, Beijing 3. Xiong JJ, Gao ZT (1997) The probability distribution of fatigue damage and the statistical moment of fatigue life. Sci China (Ser E) 40(3):279–284 4. Xiong J, Shenoi RA, Gao Z (2002) Small sample theory for reliability design. J Strain Anal Eng Des 37(1):87–92 http://www.springer.com/978-0-85729-217-9