Statistical uncertainty in the fatigue threshold stairca test method
Kim R.W.Wallin ⇑
VTT Materials for Power Engineering,P.O.Box 1000,FI-02044VTT,Espoo,Finland
a r t i c l e i n f o Article history:出行必备
Received 16April 2010
Received in revid form 13September 2010
Accepted 14September 2010
Available online 22September 2010Keywords:Stairca test Fatigue
Threshold stress
Maximum likelihood Binomial probability
a b s t r a c t
A popular method of estimating a materials fatigue threshold is the so called stairca test,where a rel-atively small number of test specimens are ud to estimate the materials fatigue strength.Usually the test results are analyd using the maximum likelihood method (MML),either directly or by using the approximation by Dixon and Mood.There has been veral studies looking at the bias and confidence of both the mean estimate as well as the standard deviation,but a comprehensive study of the reliability of the estimate has been missing.Here,the accuracy of the MML estimate is studied in detail.It is shown that the MML method is not suitable to estimate the scatter of the fatigue strength from a stairca test.An optional analysis method allowing for a better estimate of confidence bounds,bad on binomial probability is prented.Even this new analysis m
ethod suffers from similar problems as the MML esti-mate.The conclusion is that the stairca test cannot be ud to estimate the scatter in fatigue strength.
Ó2010Elvier Ltd.All rights rerved.
1.Introduction
The stairca test is considered an effective method for the esti-mation of the fatigue strength [1].In the test,shown in Fig.1,first a starting stress level (s 0)is lected.If the specimen fails before the predefined fatigue life N c ,the specimen is designated as failed and a cond specimen is tested at a stress decremented by a constant amount (d ).If,on the other hand the specimen does not fail,it is designated as run-out and the next stress is incremented by the amount (d ).This is repeated until the data is considered sufficient.Usual stairca tests consist 30specimens.The starting stress level should preferable be taken as clo to the mean threshold stress as possible.Since this value is not known before-hand,the result is that the steps will oscillate in an interval of i Ád Áð s th Æ0...1=2Ád Þ.The limiting situations are when one of the steps coincides with the mean threshold stress (side)and when the mean threshold stress is in the middle of two steps (mid).The two situations are shown in Fig.2.All other possible loca-tions fall between the two limiting situations.
Several methods of analysing stairca test data have been pro-pod [3].The prently most commonly ud method is the one developed by Dixon and Mood [4],which is an approximation of a maximum likelihood (MML)analysis with the assumption that the threshold stress follows a normal distribution.The MML esti-mate has been explored in numerous studies,[2,5–7]and a common census is that the test provides a good estimate of the mean fatigue strength,but that there is a considerable uncertainty in the estimate of the scatter.Despite a considerable amount of studies,the uncertainty in the scatter estimate has not previously been quantified to enable a true asssment of the applicability of the stairca test method.
The main problem with the stairca test method is highlighted in Fig.3.Here,fatigue strengths of 30virtual specimens have been generated form a normal distribution with mean 100and standard deviation 10.The true mean and standard deviations for the data t are 97.2and 8.2,respectively.This virtual data t has been ex-pod to six different virtual stairca tests,with three different step sizes and the two limiting step locations.As en from Fig.3,the resulting description of the threshold distribution is strongly dependent on the step size and relative location.It is thus expectable that the parameters also affect the uncertainty of the MML estimates.
2.The maximum likelihood method
Normally experimental data is analyzed by the method of least squares.This is a special ca of a maximum likelihood analysis where the error distribution follows the normal distribution.The basic maximum likelihood method does not make u of a cumulative probability distribution.It us the probability density function directly.This way,no information regarding individual probabilities is needed.The maximum likelihood method exam-ines the likelihood that a certain probability density function describes a data t correctly.This is achieved by calculating a combined likelihood by multiplying all different discrete he maximum likelihood estimate is defined as shown
玉米简笔画0142-1123/$-e front matter Ó2010Elvier Ltd.All rights rerved.doi:10.1016/j.ijfatigue.2010.09.013
Tel.:+358505114126.
E-mail address:Kim.Wallin@vtt.fi
L¼
Y n
i¼1
fðx iÞð1Þ
The constants in the distribution function are lected so as to provide a maximum value for L.For example,let us assume that the distribution function in question is the normal distribution in the form of
fðxÞ¼
1
rÁ
ffiffiffiffiffiffiffi
2p
pÁexpÀ
1
2
Á
xÀ x
r
2
()
ð2Þ
when Eq.(2)is ud in Eq.(1),it is easily en that Eq.(1)becomes maximum,when the sum of squares forð½xÀ x =rÞ2becomes when Eq.(3)obtains a minimum value.
S¼
X n
i¼1
x iÀ x
r
2
ð3Þ
This leads automatically to the least squares procedure.For the general ca however,it is advisable to apply the maximum likeli-hood argument in the form of Eq.(1).The estimation is generally simplified by,instead of maximising the likelihood(L)directly, maximising the logarithm of the likelihood in the form of
ln L¼
X n
i¼1
ln f fðx iÞgð4Þ
The standard maximum likelihood method does not include so called censoring,but censoring is easily implemented by making u of the survival distribution function,S(x).Censoring means that any data in the data t not corresponding to failure,is not a part of the probability density function,but the survival distribution func-tion.This corresponds to the run-outs in the stairca test.This leads to a conditional probability including both the probability density function as well as the survival function.This is equivalent to censoring when the censored value is lower than the true value. The likelihood gets in this ca the form of
L¼
Y n
i¼1
fðx iÞd iÁSðx iÞ1Àd ið5Þ
The parameter d i=1for uncensored value and d i=0for censored value.Eq.(5)is maximized similarly to Eq.(1).
Nomenclature
祁字怎么读BP binomial probability
d staircas
e test step size
f(x i)probability density function
i order number
L likelihood
MML maximum likelihood
n number of significant tests
n f number of failed tests
n ro number of run-outs
N c fatigue life
p discrete probability
p5%binomial probability estimate corresponding to5%con-fidence level
p50%median binomial probability
p95%binomial probability estimate corresponding to95% confidence level
P conf confidence level of probability p
P(x)cumulative failure probability for parameter x
P{X=r}probability of r events
r number of tests related to specific failure s stress sd true standard deviation of threshold stress
sd95%upper bound threshold stress standard deviation esti-mate for95%confidence level
sd MML MML estimate of the threshold stress standard devia-tion
s mean threshold stress
s0starting stress level
S sum of squares
S(x)survival function for parameter x
x mean value for parameter x
b bias of MML standard deviation estimate
d censoring parameter
r standard deviation
r^s standard deviation in the estimate of the mean thresh-old stress
r sd standard deviation in the estimate of the threshold stress standard deviation
R n R n ro+R n f
R n f sum of failures corresponding to stress s or lower
R n ro sum of run-outs corresponding to stress s or higher
K.R.W.Wallin/International Journal of Fatigue33(2011)354–362355
A general likelihood expression for random censoring when the censored value is either lower or higher than the true value has the form of
L¼
Y n
i¼1
fðx iÞi iÁSðx iÞj iÁPðx iÞk ið6Þ
The power i i=1(j i=0,k i=0)if the value is not censored,j i=1(i i=0, k i=0)if the censored value is lower than the true value and k i=1 (i i=0,j i=0)if the censored value is higher than the true value.
Eq.(6)is the general maximum likelihood expression and it can be ud with any kind of data t.The accuracy of the estimate is related to the number of data belonging to the probability density function,f(x),the data belonging to the cumulative distributions only have a small effect on the accuracy.The stairca test is a spe-cial ca,where it is only recorded whether a proof criterion is fu
l-filled or not.In this ca all tests correspond to either of the cumulative distributions P(x)or S(x).Since the confidence of the estimate is mainly dependent on the number of data belonging to the probability density distribution,the maximum likelihood estimate in the ca of the stairca test is rather vague.It should be emphasized that the u of the maximum likelihood method al-ways requires an assumption of the underlying distribution.
a normal distribution is assumed and in reality the data follows a Weibull distribution,the estimate will be affected by an additional error.
356K.R.W.Wallin/International Journal of Fatigue33(2011)354–362
3.MML uncertainty
The uncertainty in the MML estimate was for simplicity studied for a best ca he true threshold data follows a nor-mal distribution and the stairca test data is analyd with the normal distribution.In each ca,5000data ts were generated through Monte Carlo simulation.It was found,as also en in pre-vious [6],that the estimate of the mean threshold stress is innsitive to step size and location.The MML method provides an un-biad estimate of the mean threshold stress.The uncertainty in the estimate(in terms of the standard deviation) is shown in Fig.4.Also this estimate is comparatively innsitive to step size and location.For large step sizes(d/sd>1)the location starts to affect the standard deviation,but the effect is compara-tively small.The standard deviation in the estimate of the mean is simply a function of number of specimens and the true fatigue threshold scatter.It can be expresd in the form of Eq.(7).It is worth noting that the uncertainty in the estimate of the mean is nearly independent of step size.The main parameter is the true scatter of the material.This emphasizes the importance of the accuracy of the estimate of the scatter.
r^s%1:4Ásd
ffiffiffi
n
pð7Þ
The mean estimate of the scatter is,unlike the estimate of the mean clearly biad,as en from Fig.5.The bias is both a function of number of data as well as the relative step size.Again,for large step sizes(d/sd>1)also the location starts to affect the bias,but the effect is comparatively small.If the effect of step size and loca-tion is neglected,the MML estimate bias can be approximated in the simple form of Eq.(8).This is very clo to the bias estimated for the Dixon and Mood analysis,propod by Svensson et al.[8] and reported in[5].In[5],an improved bias estimate is given, which also accounts for the relative step size,but that bias esti-mate requires the u of a table.The bias of the mean estimate of the scatter is less important than the accuracy of the estimate of the scatter.Fig.6shows the standard deviation of the estimate of the true standard deviation.It can be approximated in the form of Eq.(9),where b refers to
MML¼b%nÀ3:5ð8Þ
r sd
%1:5þ0:5Ásd
Á
b
ffiffiffiffiffiffiffiffiffiffiffiffi
nÀ5
pð9Þ
The uncertainty in the standard deviation estimate is thus a
function of the true standard deviation,the step size and the num-
ber of data.The problematic factor in the estimate is the strong ef-
fect of the true standard deviation which is unknown.Usually,the
goal is to u a step size which is clo to the true scatter.Even
though Eq.(9)does not directly allow a check of the actual relative
step size,it is logical to assume that the step usually is not less than
half of the true scatter.This enables the construction of a conrva-
tive estimate of the true scatter from Eq.(9).Assuming that sd/
d63,a95%upper bound estimate for the scatter can be expresd
in the form of Eq.(10),where sd MML is the MML standard deviation
estimate and n is the number of significant tests in the stairca
test.
sd95%6MMLÁ1þ
4:9
ffiffiffiffiffiffiffiffiffiffiffiffi
nÀ5
p
ð10Þ
Eq.(10)is applicable as long as d/sd95%is larger than0.5.
In ca that the validity criterion for Eq.(10)is not fulfilled,or if
a more accurate estimate is desired,another slightly more compli-
cated expression in the form of Eq.(11)can be ud.
sd95%6
ð
ffiffiffiffiffiffiffiffiffiffiffiffi
nÀ5
p
安全事故观后感þ2:46ÞÁsd MML
ffiffiffiffiffiffiffiffiffiffiffiffi
nÀ5
p
À1:64Ásd MML
d
ð11Þ
Eq.(11)is derived from Eq.(9),by using a safety factor of2be-
tween the true standard deviation and the estimate.Eq.(11)is
王勃代表作
K.R.W.Wallin/International Journal of Fatigue33(2011)354–362357
shown graphically in Fig.7,from where it is clear that a too small step size produce a strong underestimation of the true threshold stress scatter.
dota2国服
As en from the above,the MML method combined with the stairca test provides a very poor estimate of the fatigue thresh-olds true scatter.Even a best ca estimate,where the true fatigue threshold distribution is known,a conrvative estimate of the scatter requires a safety factor of the o
rder of2(or more).An op-tional analysis method that makes no assumptions regarding the underlying distribution is bad on binomial probability.This is de-scribed next.
4.Binomial probability
The binomial distribution shown in Eq.(12)is often ud in proof type testing,where a certain fraction of results fail a certain value.It gives the probability that there are exactly r events in a t of size n,when the discrete probability of the event is equal to p.
P f X¼r g¼
n
r
Áp rÁð1ÀpÞnÀrð12Þ
The problem with Eq.(12)is that the probability of the event(p) is assumed to be known.In a situation where n tests have been made and r events have been found,the question is reverd to as what the discrete probability(p)may be with some confidence (P conf).A cumulative probability expression
for the confidence can be written in the form of Eq.(13)[9].P conf,reprents the desired confidence level.For example P conf=0.5reprents the median binomial probability estimate.Generally p is solved numerically for a desired P conf.The median binomial probability estimate can however be expresd in the approximate form of Eq.(14)[9].
P conf f p6x g¼R x
p¼0
p rÁð1ÀpÞnÀrÁdp
R
p¼0
p rÁð1ÀpÞnÀrÁdp
ð13Þ
p 50%%
rþ0:684
nþ1:368
ð14Þ
Eq.(14)acts basically as a bias adjustment to the experimental probability,being equal to r/n.For large data ts,the bias adjust-ment becomes zero and p%r/n.Small data ts,however,needs to be adjusted by Eq.(14)and the uncertainty in the probability can be estimated by Eq.(13).It should be emphasized that the bino-mial probability estimate is not the same as the commonly ud rank probability estimate.
As an example,the virtual stairca tests shown in Fig.3have been assd by the binomial probability method.Tables1–6con-tain the outcome of the data ts in Fig.3.In the tables,the col-umns n ro and n f are the individual numbers of run-outs and failures corresponding to stress level s.The are the values ud in a normal MML analysis.The column R n ro contains all run-outs corresponding to s or higher and R n f all failures corresponding to s or lower.The column R n is the sum of R n ro and R n f.The columns R n f and R n are the basis for the binomial probability estimates.
The binomial probability estimates are obtained from Eq.(13) by tting P conf equal to0.05,0.5and0.95and solving numerically
Table1
Binomial probability analysis of data t in Fig.3a.
s n ro n f R n ro R n f R n p5%p50%p95%
8090150150.4 4.317.1
10069691539.159.277.3
120060151582.995.799.6
Table2
Binomial probability analysis of data t in Fig.3b.
s n ro n f R n ro R n f R n p5%p50%p95%
7030150150.4 4.317.1等比数列性质公式总结
地震中的父与子9011312315922.541.7
1101111141573.689.797.7
130010151582.995.799.6
Table3
Binomial probability analysis of data t in Fig.3c.
s n ro n f R n ro R n f R n p5%p50%p95%
8010150150.4 4.317.1
909114115 2.316.434.4
100595101545.265.382.2
110050151582.995.799.6
Table4
Binomial probability analysis of data t in Fig.3d.
s n ro n f R n ro R n f R n p5%p50%p95%
853*******.4416.2
95103133168.521.239.6
10538311145676.190.3
115030141481.995.499.6
Table5
Binomial probability analysis of data t in Fig.3e.
s n ro n f R n ro R n f R n p5%p50%p95%
9420140140.4 4.618.1
964212214 5.717.436.3
9855871527.946.966.7
100363131660.478.891.5
102030161683.89699.6
Table6
Binomial probability analysis of data t in Fig.3f.
s n ro n f R n ro R n f R n p5%p50%p95%
9540150150.4 4.317.1
97741141513.228.648.4
99474111551.671.486.8
101040151582.995.799.6 358K.R.W.Wallin/International Journal of Fatigue33(2011)354–362
