a critical comparison of two models for assessment of fatigue data

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International Journal of Fatigue 30 (2008) 45–57

JournalofFatigue

A critical comparison of two models for assessment of fatigue data

Enrique Castillo a,*, Antonio Ramos b, Roland Koller c, Manuel Lopez-Aenlle b,Alfonso Fernandez-Canteli b

a Department of Applied Mathematics and Computational Sciences, University of Cantabria and University of Castilla-La Mancha, Spainb Department of Construction and Manufacturing Engineering, University of Oviedo, Spain

c Empa Dubendorf, Switzerland

Received 7 March 2006; received in revised form 6 December 2006; accepted 26 February 2007Available online 2 March 2007

Abstract

A comparison and a critical discussion of the results obtained for the analysis of fatigue data using the up-and-down method and theWohler field based method proposed by Castillo et al. are done and some practical conclusions are given. In addition, the paper intro-duces an alternative Markov model for analyzing the up-and-down method which improves that proposed by Chao and Fuh [Chao MT,Fuh CD. Bootstrap methods for the up and down test on pyrotechnics sensitivity analysis. Stat Sinica 2001; 1–21.] in that it keeps one ofthe characteristic features of the up-and-down method consisting of the fact that some stress levels are not possible for odd steps andsome are not possible for even steps. In addition, the method is valid for any parent distribution including the most common normaland Weibull models. In particular the case of a Weibull parent distribution is used for fatigue data and two estimation methods are pre-sented: maximum likelihood and least squares. In particular, it is shown that the up-and-down method neglects important informationcontained in the data making it very expensive and inefficient when compared with other alternative models. The methods are illustratedusing data from Dixon and Mood and an experimental program carried out at Empa.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Fatigue data evaluation; S–N field modeling; Weibull distribution; Normalization; Size effect; Censored data; Up-and-down method; Boot-strap method; Markov chains; Experimental results

1. Introduction

Optimization of planning, testing and evaluation of fati-gue data are one recursive subject appearing in the existingliterature on which laboratories and research groups dealingwith fatigue design are interested on (see Cullimore [14],Edwards and Picard [17], Fernandez-Canteli et al. [19],Huck et al. [21], Tide and Van Horn [22] and Warner andHulsbos [23]). No specific model seems to be generallyaccepted by the community, although the up-and-downmethod, despite of its limitations, finds good acceptanceboth by the academic milieu as well as by research groupsconcerned with industrial applications [2–4]. This can bedue to the simple test strategy and assessment procedure

0142-1123/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijfatigue.2007.02.014

* Corresponding author. Tel.: +34 942201722; fax: +34 942201703.E-mail address: [email protected] (E. Castillo).

required in the practical application of the up-and-downmethod, but also because the limited information providedby it, the fatigue limit, appears to some people as sufficientinformation for practical design. However, since thismethod does not fully utilize the information contained inthe data, its legitimation seems to rest more on the fact thatthe up-and-down method is described in an ASTM standard[1], than in its adequacy to fatigue analysis. In addition, nostatistical background justifies the assumption of a normalor log-normal distribution for the stress range for givennumber of cycles to failure. Further, data results corre-sponding to the high cycle fatigue region, i.e., the oneinvolved in the up-and-down method, are to be completedby additional results in the medium cycle fatigue if the wholeS–N field is required for the current fatigue life design.

In this paper, a comparison, from a practical andtheoretical perspective, is performed between the fatigue

mailto:[email protected]

log

log N

B

C

P=0.95

P=0.5

P=0.05

P=0

Fig. 1. S–N field with curves representing the same probability of failure.

46 E. Castillo et al. / International Journal of Fatigue 30 (2008) 45–57

analysis resulting by application of the model proposed byCastillo et al. and the up-and-down method to the evalua-tion of practical fatigue results on a certain material con-ducted at the Empa (Swiss Federal Laboratories forMaterials Testing and Research). The applicability and reli-ability for these two procedures is analyzed and the advan-tages and disadvantages of both methods are discussed.

In Sections 2 and 3, the models considered here areintroduced, in Section 4, the experimental program carriedout and the statistical evaluation according to both modelsare presented. Section 5 presents a comparative analysis ofthe results and finally in Section 6, the conclusions from theinvestigation are presented.

2. The model of Castillo et al.

2.1. Modeling the S–N field

The S–N field deals with two related variables: The fati-gue life, N, and the stress range, Dr. The problem consistsof developing a non-linear regression model to describe theS–N field and to estimate the model parameters. In thispaper, we follow the model of Castillo et al. [3,6,7,11],which is a Weibull model for the analysis of fatigue resultsable to consider specimens with different lengths. The selec-tion of this model is not arbitrary, but based on the follow-ing assumptions which lead to a functional equation (seeCastillo et al. [5]):

(1) Weakest link principle. If a longitudinal element isdivided into n sub-elements, its fatigue life must bethe fatigue life of the weakest element.

(2) Independence. The fatigue strengths of two non-over-lapping subelements are independent randomvariables.

(3) Stability. The cumulative distribution function (cdf)model must be valid for all lengths, but with differentparameters.

(4) Limit value. The cdf should encompass extremelengths, i.e., the case of a length going to infinity.Thus, the cdf must belong to a family of asymptoticfunctions.

(5) Limited range of the random variables involved. Thevariables Dr and N have a finite lower end, whichmust coincide with the theoretical lower end of theselected cdf.

(6) Compatibility. In the S–N field, the cumulative distri-bution function, G(N*;Dr*), of the lifetime givenstress range should be compatible with the cumula-tive distribution function of the stress range givenlifetime, F(Dr*;N*). Though in standard tests Dr* isfixed and the associated random lifetime N* is deter-mined, here Dr* is interpreted as the random stressthat needs to be applied to produce failure at N*.

Only the Weibull distribution and the following S–N fieldsatisfies the above six conditions:

ðlog N �BÞðlogDr�CÞ ¼ kþ d �L0

Lilogð1� PÞ

� �1=b

; ð1Þwhere N is the fatigue life measured in cycles, Dr is thestress range, P = F(logN;Drk) is the probability of failure,L0 is the reference length, Li is the specimen length and b,B, C, d and k are the parameters to be estimated with thefollowing meaning:
b s hape parameter of the Weibull distribution, B t hreshold value for N or limit number of cycles, C t hreshold value for Dr or endurance limit, d s cale parameter, k p arameter fixing the position of the zero probabil-
ity curve.

As it can be observed in Fig. 1, the isoprobability curves, i.e.,the curves joining points with the same probability of failure,are represented by equilateral hyperbolas. The analyticalexpression of the S–N field allows the probabilistic predic-tion of fatigue failure under constant amplitude loading.The cdf of the fatigue life N at the stress range Drk becomes:

F ðlog N ; DrkÞ ¼

1� exp � Li

L0

ðlog N � BÞðlog Drk � CÞ � kd

� �b" #

: ð2Þ

2.2. Parameter estimation

The model allows, in a first step, the estimation of B andC which define the hyperbolas asymptotes in the model.Next, in a second step, the other parameters b, d and kcan be estimated. Substituting the mean value of theWeibull distribution

l ¼ kþ d � C 1þ 1

b

� �; ð3Þ

into expression (1) the following expression is obtained:

ðlk � BÞðlog Drk � CÞ ¼ kþ d � C 1þ 1

b

� �� L0

Li

� �1=b

¼ Ki;

ð4Þ

E. Castillo et al. / International Journal of Fatigue 30 (2008) 45–57 47

where Ki is a length dependent constant, and lk is the meanvalue of logN at stress range Drk. This means that forevery length the probability curve associated with the meanvalue is also represented by a hyperbola. Expression (4)suggests to estimate first B and C by minimizing:

QðB;C;K1;...;KtÞ¼Xt

i¼1

Xn

k¼1

Xm

j¼1

logNikj�B� Ki

logDrk�C

� �2

;

ð5Þ

with respect to B, C and K1, K2, . . .,Kt, where t is the num-ber of different lengths considered here, n is the number ofstress ranges and m the number of tests conducted at eachstress range. Once the values of the parameters B and C areknown, Eq. (2) shows that the random variable

V ¼ ðlog N � BÞðlog Dr� CÞ; ð6Þhas a three parameter Weibull distribution.

This allows us to pool all the fatigue data results for dif-ferent stress ranges (see Fig. 2), into a unique populationand helps to overcome the limitation of the low numberof results at each different test level (see Castillo et al.[10]), so that better estimates are thus achieved.

In addition, the b, d and k Weibull parameters can beestimated by standard methods. The model is also applica-ble to specimens of different lengths (Castillo et al. [12]).

Provided that the variable logN for the length Li followsa Weibull distribution W(kk,dik,b) at the stress range ‘‘k’’,the Weibull cdf for the new variable V is given by:

F ðV ; k; di; bÞ ¼ 1� exp � V � kdi

� �b" #

: ð7Þ

Once k, d1,d2, . . .,dt and b are known, the model parame-ters b, d and k can be calculated. It becomes apparent thatthe parameters d and k depend of the reference length. Theestimation of k, d1,d2, . . .,dt and b can be done using themaximum likelihood method (see Castillo et al. [8–10]and Lopez-Aenlle [20]). Since the probability density func-tion (pdf) of a Weibull distribution is given by:

Fig. 2. S–N field showing schematically the pdfs of logN for differentstress ranges and their conversion to the normalized distribution.

f ðV ;k;d;bÞ ¼ exp � V � kdi

� �b" #

� b � 1

di� V � k

di

� �b�1

: ð8Þ

The logarithm of the pdf becomes:

log½f ðV ; k; di; bÞ� ¼ �V � k

di

� �b

þ log b

þ ðb� 1Þ logV � k

di

� �� log di; ð9Þ

and the value of the parameters k, d1,d2, . . .,dt and b can beobtained by maximizing with respect to k, d1,d2, . . .,dt andb the expression:

L ¼ �Xn

i¼1

V i � kdi

� �b

þ ðb� 1ÞXn

i¼1

logV i � k

di

� �

þ n log b�Xn

i¼1

log di: ð10Þ

The application of this procedure does not require the inde-pendence assumption. Nevertheless, once the parametersare estimated, the independence assumption has to bechecked so that if this condition is not fulfilled the analysismust be repeated by making the adequate corrections totake into account the existing dependence. Other methodsfor the estimation of the model parameters have beendeveloped by Castillo et al. [8,9].

2.3. The case of censored data (runouts)

Experimental programs in fatigue usually involve thepresence of censored data, i.e., tests interrupted beforefailure of the specimen occurs, due to accidental causesor because the limit number of cycles has been reached.This type of data are called censored data or data withrunouts. In such cases it is possible to resort to specificstatistical techniques, as for instance the E–M algorithm,based on an iterative process. This technique consists of:

(1) Estimate the model parameters considering only theresults associated with failures.

0.8

1

1.2

1.4

1.6

1.8

2

0 10 20 30 40 50 60

heig

ht

Test number

Runout

Failure

1

2

3

4

5

10

7

8

9

11

12 14 16 18 20 30 32 34

35

38 40 44 46

47

50 52 54

21 23 25 27 29 31 33 37 39 41 43 45 49 51 53

56

55 57

58 60

59

22 24 28 36 42 48

6

13 15 17 19

26

Fig. 3. Illustration of the up-and-down method using the Dixon andMood data and showing the five stress levels.


(2) Assign to the censored results their expected failurevalues, based on the estimated model parameters.

(3) Re-estimate the model parameters considering thedata associated with real failures plus the expectedones associated with the runouts.

(4) Repeat Steps 2 and 3 until convergence of theprocess.

To assign to the censored results their expected values, thefollowing method must be applied.

Denoting the limit number of cycles as N0, the normal-ized variable at this point becomes:

V 0 ¼ ðlog N 0 � BÞðlog Dr� CÞ; ð11Þand since the Weibull distribution conditioned by V P V0 is:

F ðV jv P V 0Þ ¼ 1� exp � V 0 � kdi

� �b

� V � kdi

� �b" #

;

V jv P V 0; ð12Þ

and the expected value of the rth order statistic of a sampleof size q from an uniform distribution U(0,1) is r/(q + 1),the censored result V0 can be replaced by the V solutions,obtained from:

1� exp � V 0 � kdi

� �b

� V � kdi

� �b" #

¼ rqi þ 1

;

r ¼ 1; 2; . . . qi; ð13Þwhere qi is the number of runouts coinciding at the samestress range, Drk, and the same limit number of cycles, N

for r = 1,2, . . .,qi. Thus:

V ¼ kþ diV 0 � k

di

� �b

� log 1� rqi þ 1

� �" #1=b

;

r ¼ 1; 2; . . . ; qi: ð14Þ

3. The normal and the Weibull up-and-down methods

3.1. Dealing with fatigue limit or strength

In daily practice some continuous variables appear suchthat they cannot be measured directly. This is for examplethe case of the endurance limit, i.e., the stress level belowwhich the failure does not occur, or the fatigue limit, i.e.,the fatigue strength at a given number of cycles (normally

..

.

p2;tþ1

p1;tþ1

p0;tþ1

p�1;tþ1

p�2;tþ1

..

.

0BBBBBBBBBBB@

1CCCCCCCCCCCA¼

� � � 1� a2 0 0 0 0

� � � 0 1� a1 0 0 0

� � � a2 0 1� a0 0 0

� � � 0 a1 0 1� a�1 0

� � � 0 0 a0 0 1� a�� 0 0 0 a�1 0

� � � 0 0 0 0 a�2

0BBBBBBBBB@

2 · 106 or 107 cycles). Unfortunately, once an specimensubject to a fatigue experiment at the stress level Dr hasfailed it cannot be tested again at a lower stress level tosee if failure occurs at that level. Thus, the population var-iable is characterized by a continuous variable – the fatiguelimit – which cannot be measured in practice. All we can dois to select some stress level and determine whether the fati-gue limit is below or above such a level. This type of situ-ation arises in many fields of actual research.

There are several common procedures to deal with theproblem of estimation, as for example the up-and-downmethod (see ASTM [24], Bruce [26] and Dixon and Mood[30], Choi [29], Dixon [31–33] and Lipnick et al. [34]).

In this paper, we deal with some modifications of the up-and-down method. The technique consists of choosing sev-eral stress levels:

. . . ;Dr2;Dr1;Dr0;Dr�1;Dr�2; . . . ;

and start the test at level Dr0. If the specimen fails, wemove downward to Dr�1, and upwards to Dr1, otherwise.Then, the process is repeated a number of times t. The re-sults of the experiment consists of the stress levels Dri andthe binary values ci where 1 means survival and 0, failure.

In the standard version the steps are of the same magni-tude, though alternative options are possible.

The result of such an experiment can be represented asin Fig. 3, where we have used data reported by Dixonand Mood [30] and the failures (asterisks) are the datapoints followed by data points in a lower level, and the sur-vivals (squares) are the data points followed by anotherdata points in an upper level.

3.2. The up-and-down method

The up-and-down method can be modelled by means ofa Markov chain (see Chao and Fuh [28]). The states are thestress levels, and the transition probabilities are the proba-bilities of moving between the adjacent stress levels.

Let pit be the probability of the test number t to be con-ducted at stress level i, i.e., we assume that t = 1,2, . . .,n isthe test number and i = . . .,�2,�1,0,1,2, . . ., refers to theselected initial stress levels for the test with i = 0 being the ini-tial stress level, and the level i has associated stress Dri. Let ai

be the probability of a test run at level i to lead to a failure.Then, we have the following transition equation:

� � ��

2 � � ��

1CCCCCCCCCA

..

.

p2;t

p1;t

p0;t

p�1;t

p�2;t

..

.

0BBBBBBBBBBB@

1CCCCCCCCCCCA; ð15Þ


which in compact form can be written asptþ1 ¼ Tpt; ð16Þ

and pt is a row matrix that gives the probability mass func-tion associated with the t step of the test, and T is the tran-sition probability matrix.

Then, we have

pt ¼ Ttp0; ð17Þwhere p0 is the initial probability, i.e., the probabilitiesassociated with the first test (if one decides in a determinis-tic manner to start at level t = 0, it would be a matrix withall zeroes but a one in the t = 0 position).

Since the ai probabilities of very high and very low levelstends to one and zero, respectively, Eq. (15) for a givenvalue of i = k can be replaced by equation

pk;tþ1

..

.

p2;tþ1

p1;tþ1

p0;tþ1

p�1;tþ1

p�2;tþ1

..

.

p�k;tþ1

0BBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCA

¼

0 � � � 0 0 0 0 0 � � � 0

1 � � � 0 0 0 0 0 � � � 0

� � � � � � � � � � � � � � � � � � � � � � � � � � �0 � � � 1� a2 0 0 0 0 � � � 0

0 � � � 0 1� a1 0 0 0 � � � 0

0 � � � a2 0 1� a0 0 0 � � � 0

0 � � � 0 a1 0 1� a�1 0 � � � 0

0 � � � 0 0 a0 0 1� a�2 � � � 0

0 � � � 0 0 0 a�1 0 � � � 0

0 � � � 0 0 0 0 a�2 � � � 0

� � � � � � � � � � � � � � � � � � � � � � � � � � �0 � � � 0 0 0 0 0 � � � 1

0 � � � 0 0 0 0 0 � � � 0

0BBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCA

pk;t

..

.

p2;t

p1;t

p0;t

p�1;t

p�2;t

..

.

p�k;t

0BBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCA

; ð18Þ

i.e., the infinite matrix T can be replaced by a finite matrix.Note that we have included two reflectant barriers at twolevels (one high and one low).

We have preferred this approximation to the one pro-posed by Chao and Fuh [28]:

pk;tþ1

..

.

p2;tþ1

p1;tþ1

p0;tþ1

p�1;tþ1

p�2;tþ1

..

.

p�k;tþ1

0BBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCA

¼

e � � � 0 0 0 0

1� e � � � 0 0 0 0

� � � � � � � � � � � � � � � � � �0 � � � 1� a2 0 0 0

0 � � � 0 1� a1 0 0

0 � � � a2 0 1� a0 0

0 � � � 0 a1 0 1� a�1

0 � � � 0 0 a0 0

0 � � � 0 0 0 a�1

0 � � � 0 0 0 0

� � � � � � � � � � � � � � � � � �0 � � � 0 0 0 0

0 � � � 0 0 0 0

0BBBBBBBBBBBBBBBBBBBBBB@

because ours keeps one of the characteristic features of theup-and-down method consisting of the fact that some stresslevels are not possible for odd steps and some are not pos-sible for even steps. For example, if we consider the twoapproximations below

b¼

0 0:05 0 0 0

1 0 0:5 0 0

0 0:95 0 0:95 0

0 0 0:5 0 1

0 0 0 0:05 0

0BBBBBBB@

1CCCCCCCA

; b� ¼

0:001 0:05 0 0 0

0:999 0 0:5 0 0

0 0:95 0 0:95 0

0 0 0:5 0 0:999

0 0 0 0:05 0:001

0BBBBBBB@

1CCCCCCCA;

ð20Þ

where the asterisk refers to the matrix used by Chao andFuh, the corresponding probabilities of the test to be con-

ducted at the different stress levels when starting from thecentral level after four and five steps are

0 � � � 0

0 � � � 0

� � � � � � � � �0 � � � 0

0 � � � 0

0 � � � 0

0 � � � 0

1� a�2 � � � 0

0 � � � 0

a�2 � � � 0

� � � � � � � � �0 � � � 1� e0 � � � e

1CCCCCCCCCCCCCCCCCCCCCCA

pk;t

..

.

p2;t

p1;t

p0;t

p�1;t

p�2;t

..

.

p�k;t

0BBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCA

; ð19Þ


p4 ¼

0:025

0

0:95

0

0:025

0BBBBBB@

1CCCCCCA

; p5 ¼

0

0:5

0

0:5

0

0BBBBBB@

1CCCCCCA

; p�4 ¼

0:0249988

0:000024975

0:949953

0:000024975

0:0249988

0BBBBBB@

1CCCCCCA

;

p�5 ¼

0:00002625

0:49995

0:00004745

0:49995

0:00002625

0BBBBBB@

1CCCCCCA;

for the proposed and the Chao and Fuh approximations,respectively. Note that even though both transition matri-ces are very close, the second destroys the characteristicfeature of impossibility of some stress levels in odd andeven steps, that holds in the up-and-down method.

The expected values of the number of tests to be con-ducted at all levels can be calculated as follows. Let Xt bea row matrix containing as elements, xit, the number oftests conducted at stress level Dri after t steps of the up-

T ¼

0 1� a1 0 0 0

1 0 1� a0 0 0

0 a1 0 1� a�1 0

0 0 a0 0 1

0 0 0 a�1 0

0BBBBBB@

1CCCCCCA;

ðCTÞ�1 ¼

1 a1a�1

ð�1þa1Þð�1þa�1Þ1 a1a0

�1þa0þa�1�a0a�1

�1 0 1a1a0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia�1�a0ð�1þa1þa�1Þpð�1þa0Þð�1þa�1Þ

1 a�1

�1þa�11 a0ð�1þa1þa�1Þ

�1þa�1

�1 0 1 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia�1 � a0ð�1þ a1 þ a�1

p1 1 1 1

0BBBBBBBB@

Q ¼

�1 0 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia0 � a1a0 þ a�1 � a0a�1

p0

0 0 0 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia0 � a1a0 þp

0BBBBBB@

U ¼

1�ð�1Þ1þt

20 0 0 0

0 1 0 0 0

0 0 1þ t 0 0

0 0 0 � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia0�a1a0þa�1�a0a�1pð Þ1þt�1� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia0�a1a0þa�1�a0a�1p �1

0

0 0 0 0 ða0�a1a0þa�1�affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia0�a1a0þa�1p

0BBBBBBBB@

and-down method. Then, the expected value E(Xt) of Xt,including the initial step, is

EðXtÞ ¼ Iþ Tþ T2 þ . . .þ Tt� �

p0: ð21Þ

Since the matrix T can be written as

T ¼ CQC�1; ð22Þwhere Q is a diagonal matrix containing its eigenvalues andC is a matrix containing its eigenvectors as columns, Eq.(21) can be written as

EðXtÞ ¼ CðIþQþQ2 þ � � � þQtÞC�1p0

¼ CUC�1p0; ð23Þ

where U is a diagonal matrix whose elements are

uii ¼ktþ1

i �1

ki�1if ki 6¼ 1

t þ 1 if ki ¼ 1

(ð24Þ

where ki are the eigenvalues of T.

Example 1 (Five levels). For the particular case of fivelevels with extreme (1 and 0), ai probabilities in the exteriorlevels, we have

a1a0

�1þa0þa�1�a0a�1

a1a0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia�1�a0ð�1þa1þa�1Þp�1þa0þa�1�a0a�1

a0ð�1þa1þa�1Þ�1þa�1ffiffiffi

Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia�1 � a0ð�1þ a1 þ a�1Þ

p1

1CCCCCCCCA;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia�1 � a0a�1

1CCCCCCA;

0a�1Þ1þt

2 �1ffiffiffiffiffiffiffiffiffiffiffi�a0a�1�1

1CCCCCCCCA:


Note that this is practically the case of Dixon and Mood[30], where a2 = 0.999767 � 1 and a�2 = 0.00620967 � 0.

3.3. Parameter estimation

The up-and-down method in the standard version isanalyzed assuming an underlying normal distribution. Thiscannot be sustained in the fatigue-life case in which thenumber of cycles to failure for a given stress range followsa Weibull distribution, thus the possibility of consideringany other distribution must be considered. In this sectionwe give some method for estimating the parameters inde-pendently of the distribution assumed.

3.3.1. Maximum likelihood

A very natural method of estimation is the maximumlikelihood method in which the likelihood of the observedpath in the up-and-down experiments is maximized withrespect to the parameters to be estimated (see Vagero[35]). The corresponding likelihood is

L ¼Yt

i¼1

arii ð1� aiÞsi ; ð25Þ

where ri and si are the numbers of tests conducted at levelDri with broken and surviving specimens, respectively.Since ai is a function of the parameter vector h, the estima-tion problem is equivalent to the optimization problem (seeBazaraa et al. [25] and Vanderplaats [36]).

Maximizeh

log L ¼Xt

i¼1

½ri log F DrðDri; hÞ þ si logð1

� F DrðDri; hÞÞ�; ð26Þ

where FDr(Dri;h) is the cdf of Dr at the point Dri.

3.3.2. Least squares

An alternative to the maximum likelihood method con-sists of minimizing the sum of squares of the differencesbetween the expected and the observed levels (see (23)and(24)), that is:

Table 1Estimates and their properties for the Dixon and Mood data using the maxim

Estimation meth

Maximum likelih

Objective function logL = 22.994l 1.31r 0.160

Covariance matrix0:00143 �0�0:00074 0:

�l-Confidence interval (1.247,1.385)r-Confidence interval (0.044,0.214)Bias½l� 0.0048MSE½l� 0.00145Bias½r� �0.0133MSE½r� 0.00239

Minimizeh

Xn

i¼1

½EðX iðhÞÞ � ri � si�2

¼Xn

i¼1

½fCðhÞUðhÞC�1ðhÞp0gi � ri � si�2: ð27Þ

3.4. Application to the Dixon and Mood data

To illustrate the method, we apply it to the Dixon andMood data in Fig. 3, assuming a normal distributionN(l,r) for the fatigue limit. Then, the optimization prob-lem (26) becomes

Maximizeh

log L ¼Xt

i¼1

ri log UDri � l

r

� ��

þsi log 1� UDri � l

r

� �� ; ð28Þ

where U(Æ) is the cdf of the standard N(0,1) random vari-able, and the ri and si, from Fig. 3, are the elements ofthe following vectors:

r ¼ f1; 10; 19; 2; 0g; s ¼ f0; 0; 8; 18; 2g:

The results have been obtained using two computer pro-grams in Mathematica and GAMS, for a double check,using the very simple programs included in Appendix A,and the results are given in Table 1. The estimates are prac-tically the same as those given by Dixon and Mood [30]:l ¼ 1:32; r ¼ 0:17.

The covariance matrix of the estimates and the confi-dence intervals for l and r were obtained by the bootstrapmethod with 10,000 simulations, and are shown in thetable.

From the simulations we have also obtained the valuesof Bias½l�; Bias½r�; MSE½l� and MSE½r�, where MSErefers to the mean squared error, given in the table.

For the least squares method, the optimization problem(27) becomes

um likelihood and the least squares method

od

ood Least squares

Q = 0.3571.3190.187

:0007400222

�0:00162 �0:00001�0:00001 0:00232

� �(1.238,1.393)(0.123,0.299)�0.00160.001620.0100.00242


Minimizeh

Xn

i¼1

½fCðl; rÞUðl; rÞC�1ðl; rÞp0gi � ri � si�2:

ð29ÞThe results are also shown in Table 1, which are very sim-ilar to the ML solution.

4. Experimental programs

Two different experimental fatigue programs carried outby Empa are presented and evaluated with two differenttypes of steels.

4.1. Experimental program using the 42CrMo4 material

With the aim of comparing the fatigue assessment pro-vided by the model proposed by Castillo et al. and theup-and-down method, a specific experimental fatigue pro-gram was launched at the Empa (Swiss Federal Laborato-ries for Testing and Research) in Dubendorf (Zurich).According to this two different strategies were applied:the first one aiming to define the whole S–N field whereasthe second one restricted to the study of the fatigue

850

875

900

925

950

975

1000

1025

1e+05 1e+06 1e+07

Max

. str

ess

valu

e [M

Pa]

Number of cycles

1

2

3, 16

4, 15

7, 5

68

9

10, 12, 14

1113

Failure

Runout

885

908

931..6

955..8

0 1 2 3 4 5 6 7 8 9 10 11 12

Max

. str

ess

valu

e [M

Pa]

Test number

6

7

8

9

10

11

12

13

14

15

16

FailureRunout 10 cycles7

10 cycles7a

b

Fig. 4. Results for the 42CrMo4 material. (a) Wohler field and (b)up-and-down sequence.

strength distribution for a limit number of 2 millions cycles.At least 20 specimens were tested in both methodologies.

A low-alloy steel 42CrMo4 (material number codedDIN-1.7225) with a nominal value of the ultimate strength,Rm = 1097 MPa and a yield strength, Rp0,2 = 1059 MPa,was used. This kind of material is used in structural con-struction because of its satisfactory behavior in mechanicalservice and fatigue life.

All the tests were conducted under constant amplitudeloading with a stress ratio R = 0.1. High frequency vibro-phore machines running at a maximum frequency of80 Hz were used for testing. This means that only 34.7 h,respectively, 6.9 h, are needed to complete a test when alimit number of cycles of 10 millions, respectively, 2millions.

The temperature in the specimen was continouslychecked to keep it low in order not to influence the failuremechanism during testing.

In planning the S–N field program, a complementarystrategy to the existing results was applied. Due to the pri-mary aim of the program, to compare the results of thefatigue strength for both methods at a limit number ofcycles of 2 millions, the results are predominantly clumpedin the lower region of the S–N field. To provide prior infor-mation to the up-and-down method, five tests (launchingstage) were carried out in the first stage of the experimentalprogram. This allows us an initial estimation of the Wohlerfield that supplies the stress range step between tests andthe starting stress level. A minimum of 10 tests were per-formed following the up-and-down method to guaranteereliable parameter estimation.

The fatigue test results obtained from the Wohler fieldlaunching stage and from the up-and-down sequence areshown together in Fig. 4, the tests being labelled with thetest number to facilitate identification. The techniquesdescribed in Section 2 were applied to the parameter esti-mation of the Castillo et al. model. The GAMS tool wasused to solve the optimization problem. In the solution

830

850

870

890

910

930

950

970

990

1010

1030

1050

1070

1e+05 1e+06 1e+07

Number of cycles

95% CI of P=0.5

Failure

Runout

Expected failure

P=0

P=0.05

P=0.5

P=0.95

Max

. str

ess

valu

e [M

Pa]

Fig. 5. The Wohler field estimated from Castillo’s model for the 42CrMo4material.

Table 2Sequence analysis of fatigue test and estimates of the fatigue limit at eachstage, including the five launching data points, for the 42CrMo4 material

No. Fatigue limit (MPa) at 2 · 106 cycles

Castillo’s model Up-and-down

P = 0.05 P = 0.5 P = 0.95 P = 0.05 P = 0.5 P = 0.95

1 – – – – – –2 – – – – – –3 846 858 866 – – –4 893 907 952 – – –5 893 907 950 – – –6 892 919 941 – 932 –7 900 917 969 801 943 11108 894 913 940 905 940 9759 891 908 938 901 932 963

10 889 906 936 889 922 95611 885 900 948 894 920 94612 885 900 948 890 915 93913 878 898 938 893 914 93414 879 899 936 892 911 93015 881 903 937 894 910 92716 881 900 938 896 912 928


of the up-and-down method, the GAMS tool, as describedin Section 3, and the R language were utilized.

4.1.1. Using the model of Castillo et al.

From the program the following estimates of the Cas-tillo et al. model were obtained:

B ¼ 3:57; C ¼ 20:40; b ¼ 1:61; d ¼ 0:38; k ¼ 2:07:

The variability of these parameter estimates was analyzedusing the bootstrap technique (see Efron and Tibshirami[18] and Chernik [13]). Two thousand bootstrap simula-tions using Java and GAMS were performed. Thus, twothousands values of the parameters were obtained, fromwhich the 95% confidence interval (CI) for the quantilecurve P = 0.5 was obtained, and is shown in Fig. 5 (shad-owed region), which also displays the Wohler field as wellas the fatigue test results, including the expected lives ofthe runouts using expression (14).

4.1.2. Using the normal and Weibull up-and-down methodsThe estimated mean and standard deviation of the up-

and-down method assuming an underlying normal distri-bution were:

l ¼ 20:63; r ¼ 0:025;

and applying the bootstrap method (Mathematica andGAMS) (see Davison and Hinkley [15] and Dixon andMassey [16]) with 2000 simulations the resulting 0.95 con-fidence intervals for l and r were, respectively:

ð20:6070; 20:6499Þ and ð0:0025; 0:047Þ;while the three parameters of the corresponding Weibulldistribution applied to the up-and-down method were:

b ¼ 1:60 d ¼ 0:0435; k ¼ 20:59:

Also the bootstrap method was used to find out the 0.95confidence intervals for b, d and k, respectively:

0

0.2

0.4

0.6

0.8

1

860 870 880 890 900 910 920 930 940 950

P

Castillo’s modelUp-and-down (Normal)Up-and-down (Weibull)

Max. stress value [MPa]

Fig. 6. Cdf’s adjusting the fatigue data with these models for the42CrMo4 material.

ð1:3607; 21:9161Þ; ð0:035; 0:0927Þ and ð20:55; 20:5911Þ:

Fig. 6 shows a comparison of the cdf’s obtained for theCastillo’s model (Weibull distribution) and the up-and-down method using two different underlying distributions(Normal and three-parameter Weibull).

Small discrepancies can be observed among the cdf fromthe Castillo’s and the other models. This can be explainedbecause the Castillo model uses the test of the launchingstage to calculate the model parameters whereas the up-and-down uses them only to guess the step with and thestarting point.

The two Weibull distributions obtained, one with theup-and-down and the other with the Castillo’s model havethe same lower end stress value. The most significant differ-ence can be observed below a probability of P = 0.05among the normal distribution and the two Weibull distri-butions due to the fact that for a normal distribution itslower end value is at infinity. After all, the three cdf’sobtained were very similar (the difference in P = 0.5 is lessthan 10 MPa).

Table 2 illustrates a comparison between the values ofthe fatigue limit calculated for 0.05%, 0.5% and 0.95%probability of failure at 2 million of cycles using the modelof Castillo et al. and the up-and-down method. The resultsfrom Castillo’s model after only 6 results have a relativeerror with respect to its final value less than 2%. On theother hand, if the up-and-down method is used to have arelative error below 2% at least 10 tests must be performed.Note that before the up-and-down method starts, the fati-gue limit from the Castillo’s model has a relative error lessthan 4%.

It can be observed that the number of tests needed bythe up-and-down method to have a relative error below0.5% is 10, while the Castillo’s model needs two tests less.

250

275

300

325

350

375

400

425

450

475

1e+05 1e+06 1e+07

Number of cycles

1

2

3

47

6, 109

5, 17

13, 18 11, 12, 14, 1915, 16

8

Failure

Runout

266

270

274

278

282

286

290

294

1 2 3 4 5 6 7 8 9 10 11 12 13

Test number

6

9

10

8

12

13

14

15

16

17

5

18

19

Failure

Runout

10 cycles7

10 cycles7

14

Max

. str

ess

valu

e [M

Pa]

Max

. str

ess

valu

e [M

Pa]

Fig. 7. Results for the Steel-Alloy M1 material. (a) Wohler field and (b)up-and-down sequence.

230

240

250260270280290300310320330340350360370380390400410420430440450460470

1e+04 1e+05 1e+06 1e+07 1e+08

Number of cycles

95% CI of P=0.5Failure

Runout

Expected failure

P=0

P=0.05

P=0.95

P=0.5

Max

. str

ess

valu

e [M

Pa]

Fig. 8. Wohler field estimated using the Castillo model. Steel-Alloy M1material.


4.2. Experimental program using the Steel-Alloy M1

material

The results of an experimental fatigue program, carriedout at Empa (Swiss Federal Laboratory for Testing andResearch) in Dubendorf (Switzerland), on Steel-Alloy,called from here M1, were evaluated.1 The aim of thisexperimental program was to find out the Wohler fieldand the fatigue limit with a fixed number of tests, 6/8 testsfor the Wohler field and 12/14 tests for the up-and-downmethod.

The test facilities, the optimization tools and tech-niques were the same as those used with the experimentalprogram explained in Section 4.1. The nominal value ofthe ultimate strength was Rm = 1135 MPa and the yieldstrength was Rq0,2 = 948 MPa. All tests were conductedunder constant amplitude loading with a stress ratio ofR = �1, and notched specimens. In a first stage 7 testswere run to have an estimation of the Wohler field dueto the same reason as that explained for the other mate-rial. Analyzing these results, the step width and the initialstress level were chosen and the up-and-down sequencewas completed after 12 tests. In this case two of the testsfrom the first stage were reused in the up-and-down anal-ysis to enlarge the final number of tests incorporated tothe up-and-down method.

The test strategy is primarily subject to the determina-tion of the fatigue limit using the up-and-down method.This explains why most of the fatigue results are concen-trated in the lower region of the S–N field as it occurredwith the other material.

The fatigue test results obtained from the Wohler fieldtest planning and from the up-and-down sequence areshown in Fig. 7, the tests being labelled with the test num-ber to facilitate identification.

4.2.1. Using the model of Castillo et al.

The parameters of the Castillo’s model estimates were:

B ¼ 8:09; C ¼ 19:21; b ¼ 13:80; d ¼ 1:77; k ¼ 0:1:

The variability of the parameter estimates was checkedusing the bootstrap technique as was made with the othermaterial. Two thousand bootstrap simulations were per-formed. Thus, two thousand values of the parameters wereobtained and from them the corresponding 95% confidenceregion of the quantile curve P = 0.5 were obtained, and isshown in Fig. 8 (shadowed region), which also displaysthe Wohler field with the fatigue test results, includingthe expected lives of the runouts using expression (14).

1 Due to contract conditions between the company of the secondmaterial and Empa, we are not allowed to mention the real materialdesignation.

4.2.2. Using the normal and Weibull up-and-down methods

The calculated mean and standard deviation of the up-and-down method using and underlying normal distribu-tion were:

l ¼ 19:450; r ¼ 0:0170;

Table 4Relative error of each method in %

No. Castillo’s model Up-and-down method

2 25.53 –3 2.19 –4 2.28 –5 2.28 –6 0.49 –7 1.8 2.038 3.8 2.379 2.53 2.03

10 1.77 1.3411 1.25 0.9912 0.9 0.9913 0.62 0.7614 0.13 0.2415 0 0.0516 0 0.1317 0 0.0518 0 0.11

0

0.2

0.4

0.6

0.8

1

260 265 270 275 280 285 290

P

Castillo’s modelUp-and-down (Normal)Up-and-down (Weibull)

Max. stress value [MPa]

Fig. 9. Cdf’s adjusting the fatigue data with these models. Steel-Alloy M1material.


and applying the bootstrap method (Matlab and GAMS)(see Castillo et al. [27]) with 2000 simulations the following0.95 confidence intervals for l and r, respectively:

ð19:4369; 19:4644Þ and ð0:0021; 0:0306Þ;meanwhile the three parameters of the corresponding Wei-bull distribution applied to the up-and-down method were:

b ¼ 52:3; d ¼ 19:457; k ¼ 0:

Also the bootstrap method was used to find out the 0.95confidence intervals for b, d and k, respectively:

ð1:0; 60:2Þ; ð0:001; 19:5Þ and ð0; 19:4Þ:Fig. 9 shows the cdf’s obtained the Castillo’s model and

the up-and-down method using two different underlyingdistributions (Normal and three-parameter Weibull). In

Table 3Sequence analysis of fatigue tests and estimates of the fatigue limit at eachstage, including the seven launching data points, for the Steel-Alloy M1material

No. Fatigue limit (MPa) at 10 · 106 cycles

Castillo’s model Up-and-down

P = 0.05 P = 0.5 P = 0.95 P = 0.05 P = 0.5 P = 0.95

1 – – – – – –2 – – – – – –3 206 206 206 – – –4 281 283 285 – – –5 269 271 277 – – –6 269 271 277 – – –7 270 276 279 – – –8 264 272 278 267 273 2809 255 266 276 267 272 278

10 260 270 277 268 273 27911 261 272 279 268 275 28212 264 274 280 269 276 28413 265 275 281 269 276 28314 266 275 282 269 277 28415 266 277 284 270 278 28716 267 277 285 270 279 28917 267 277 284 270 279 28918 267 277 283 270 279 28819 267 277 284 271 279 288

this case, all the distributions have the same shape with adifference of only 3 MPa between the cdf’s from the up-and-down method – both underlying distributions – andthe cdf from the Castillo’s model.

Table 3 shows a comparison between the values of thefatigue limit calculated for 0.05%, 0.5% and 0.95% proba-bility of failure at 10 million cycles using the model of Cas-tillo et al. and the up-and-down method.

In Table 4 the evolution of the relative error of eachmethod can be observed. The error after nine tests remainsbelow 2%, but with the difference that from the third testthe Castillo’s model has an error below 4% and the up-and-down method needs at least seven tests.

5. A comparative analysis

(1) In the up-and-down method, another method isneeded to set up the initial variables of the method:stress level and step width.

(2) The scatter of the regression model depends on thenumber of tests and the material’s properties. In theup-and-down the scatter not only depends on thesefactors but strongly on the initial variables selected:start stress level and step width.

(3) Both methods give good results for the mean value ofthe fatigue limit (see Tables 2 and 3).

(4) The number of tests needed to obtain the same rela-tive error by the Castillo’s model are approx. 30% lessthan those required by the up-and-down method (seeTable 4).

(5) The up-and-down gives only the fatigue limit at acertain number of cycles and this result cannot beextrapolated to a different number of cycles. Ifthe fatigue limit for another number of cycles isneeded another fatigue test program must be car-ried out.


(6) The regression model gives the fatigue limit at anynumber of cycles below the maximum number ofcycles tested (see the Wohler field from the Castillo’smodel in Figs. 5 and 8).

6. Conclusions

The main conclusions of this work are the following:

(1) The proposed regression model appears as a goodfatigue model to analyze fatigue data and superiorto the up-and-down method.

(2) The regression model can be used to set up the initialparameters of the up-and-down method to improvethe accuracy of the initial variables.

(3) Using the regression model the fatigue limits areobtained with a relative error less than 3% after only6/7 tests.

Appendix A (Computer programs) In this appendix, the comput

calculations are included. Mathematica Program:

Needs["Statistics‘ContinuousDistributions’]

X={2,1.7,1.4,1.1,0.8}n={1,10,19,2,0}m={0,0,8,18,2}F[mu_,sigma_]=-Sum[n[[i]]* Log[CDF[NormalDis

X[[i]]]]+m[[i]]*Log[1-CDF[NormalDistributX[[i]]]],{i,1,Length[X]}];

FindMinimum[F[mu,sigma],{mu,1.3},{sigma,0.2}]

GAMS Program:

$title UpandDown

file out/UpandDown.out/;

put out;

SET I number of levels/1*5/;

PARAMETERS

X(I)/1 2, 2 1.7,3 1.4,4 1.1,5 0.8/

N(I)/1 1,2 10,3 19,4 2,5 0/

M(I)/1 0,2 0,3 8,4 18,5 2/;

VARIABLES z,mu;

POSITIVE VARIABLES sigma;

EQUATIONS zdef;

zdef.z=e=Sum(I,N(I)*log(errorf((X(I)-mu)/sigmMODEL UD/zdef/;

mu.l=1.3; sigma.l=0.1;

SOLVE UD USING nlp MAXIMIZING z;

put ‘‘z=",z.l:12:8,’’ modelstat=",UD.modelstat

put ‘‘mu=",mu.l:12:8,’’ sigma=",sigma.l:12:8/;

(4) To obtain the fatigue limit with a probability of 50%both methods the up-and-down and the regressionmodel give good results.

(5) A new underlying distribution as the three parametricWeibull distribution is used successfully within theup-and-down method.

Acknowledgements

The authors are indebted to the Spanish Ministry of Sci-ence and Technology (Project BIA2005-07802-C02-01), tothe Spanish Ministry of Education, Culture and Sportsfor partial support of this work. The authors are also grate-ful to the Editor and the referees, who made interestingsuggestions allowing the improvement of the paper.

er programs in Mathematica and GAMS used for the

tribution[mu, sigma],

ion[mu,sigma],

a))+M(I)*log(1.0-errorf((X(I)-mu)/sigma)));

," solvestat=",UD.solvestat/;


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a critical comparison of two models for assessment of fatigue data

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