Download Reliability and Confidence Levels of Fatigue Life

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
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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
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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
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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.
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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
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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
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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
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