STATISTICAL TREATMENT OF FATIGUE TEST DATA

by

D. T. Raske

DISCLAIMER

-- UIJU- wcc crewred ai an account of ivtyk soonsoreo" by an agency of ine United States . _. _Nwither tne United State-. Government nor any agency thereof, nor any Of their employees, make*•vaujniy. e«c'Mi or implied, or auumej any legal lability or rexoonvbility tor m» dccui(!rjmj;r(j[ene». of usefulness of ^ny rn'Qrrnatfon. app^'atus, product. 0r procesi 0i5c/osertreo'ejenis irtai i d u * vioulo not mt'inae oriraroi^ oiwiea rrghri Reference tte^e-r

'"lerrixi"eceisarily

Swtp-

r'oieO. ,e t>y i

.rVWCrifci'K manufaCTu^er of otherwiseimply its endQisement, recommendaiion. or Favoring Dv i

tncv ihereol. The viewt and OD'n^ons of authors expressed *»trem do "reflect those o( the Umred Siile* Gn»«-««-nt or any agency

Prepared for

USDOE/UKAEA Exchange Meeting

on

Mechanical Properties for FBR Structural Materials

Warringtonr England

September 22-24, 1980

&

UltC-Ut-USOOEARGONNE NATIONAL LABORATORY, ARGONNE, ILLINOIS

Operated under Contract W-31-109-Eng-33 for the

U. S. DEPARTMENT OF ENERGY

Statistical Treatment of Fatigue Test Data

D. T. Raske

Materials Science DivisionARGONNE NATIONAL LABORATORY

I. INTRODUCTION

In general, the stress- or strain-life fatigue curves and correlations in

the Codes, handbooks, and standards used for the design of structural com-

ponents for power-generating plants are developed by engineers rather than

statisticians. As a consequence, rigorous statistical procedures for treating

fatigue data are not always followed. This results in fatigue curves which,

while still useful, may lack the necessary properties to make predictions within

and/or beyona the range of observations. Thus, the purpose of this report is to

discuss several guidelines and procedures for the treatment of data used to

develop fatigue curves for design use.

II. SCOPE

This report contains a discussion of several pertinent and vexing aspects

of fatigue data analysis. These topics include the choice and transformation

of the dependent variable, the treatment of data from suspended tests (run-

outs), dealing with outlying observations, and strain-life relations.

III. DEPENDENT/INDEPENDENT VARIABLES

In a least squares regression analysis of experimental data, the dependent variable

is always the uncontrolled variable and as a consequence contains random measure-

ment errors. Then, the calculated least squares estimators will be unbiased

and consistent, and the random measurement errors will be accounted for by the

1 2error-terms* of the regression model. ' Applied to fatigue data, cycles-to-

failure (N-) are therefore the dependent variables with the experimental scatter

being the measurement errors. This is important because if a test

variable with measurement errors** is used as the independent variable, this

variable is then correlated with the error-term and consequently the calculated

least squares estimators are biased (toward zero) and inconsistent. Thus the

regression model will systematically overestimate and/or underestimate the

dependent; variable (an example illustrating this point is given in Appendix A).

Therefore if anything other than cycles-to-failure is used as the dependent

variable in the analysis of fatigue data for stress- or strain-life curves,

no statistical significance can nor should be given to the least squares

2estimators, confidence limits, or coefficient of determination (R ).

IV. TRANSFORMATIONS ON N

A. Normal Error Model

In order to make significance tests and calculate confidence intervals

for least squares regression models, the collection of error-terms are assumed to be

1 2independent and normally distributed with zero mean and constant variance. * This means

that the distribution of error-terms will be approximately constant over all

observations, the least squares estimators will have minimum variance, and the

results of normal-distribution theory can be used to analyze the data. When

the distribution of error-terms is not constant and not normal an inappropriate

model is indicated. In these cases, a new model is needed, or the dependent

*The error-term is the difference between the observed and predicted dependentvariable.

**Although there are always small measurement errors associated with the controlledfatigue test parameters, these are generally not significant when compared to thevariability of fatigue life. For example, the strain error in a test of this typeis about an order of magnitude less than the fatigue life error.

variable may be transformed or weighed. Several standard transformations are:

Y T • ft, Y T = log Y, and Y T = 1/Y (see Refs. 1 and 2). In terms of fatigue data

where Y is usually defined as log N , these transformations effectively compress

the high-cycle regime as shovn in Fig. 1. One of the most common non-constant

error term distributions associated with elevated-temperature structural

materials fatigue data appears to be an increasing trapezoidal form. This

T T

indicates that the correct transformation is Y = log Y which results in N = log

(log N-) in the regression model.

B. Variability of Fatigue Data

1. Background

The above discussion cr error-term properties implies that the dependent

variable itself be normally distributed with constant variance. It is common

knowledge however that the variability or scatter of fatigue life Nf is not2

constant, but increases as the mean life increases. Thus, the variance, a , of fatigue

life and subsequently the distribution of error terms from a regression analysis

on these data are not constant. Therefore, around 1950 a considerable effort

was made to determine a transformation on Nf which would result in values that

were normally distributed with a constant variance. As a result, a number of

theoretical arguments, as well as experimental evidence, were put forth3—8

favoring the logarithmic transformation. An example of this is given in9

Fig. 2, in which histograms of fatigue test data on an aluminum alloy are

plotted on both a linear and a logarithmic scale. As shown, the linear distribution

is obviously not normal and is skewed to the left, as is typical of data of

this type. On the logarithmic scale, however, a chi-square test indicates

that the distribution is indeed normal. Thus, the logarithmic transformation

on fatigue life was not seriously questioned until recently when the available

low-cycle and creep-fatigue data on Type 304 stainless steel were analyzed.

In this study, the log Nf transformation resulted in a trapezoidal-type error-

term distribution which indicated that a transformation on log N- was required.

-1/2Consequently, a transformation of the form (log NL) was employed. Since

this is an extremely strong transformation which greatly compresses the high-

cycle regime relative to the shorter lives, the question of fatigue data

variability was reexamined with the aim of analytically determining the optimum

transformation.

2. Results

A literature search for fatigue-test data with replicates at given load or

strain levels resulted in 317 data sets on 24 materials employing 6 different

test methods. Of this total, 26 data sets were rejected as outliers when the

mean fatigue lives calculated on N~ were correlated with the mean levels calculated

on log N-. The remaining 291 data sets representing 4618 separate tests that

constituted the final data base are summarized in Table 1. The smallest number

of tests per set was 3 and the largest was 235.

The data base was analyzed to obtain the functional relation

s2 = f(Nf) , (1)

2where s is the observed variance of Nf. Once this relation is known, the

T 30appropriate transformation, N. , can be estimated from the relation

dN,

(2)T f dNf

N fT * -===f J *T(O

2 2 T

For example, if s « N, the resulting transformation would be Nf = log Nf.

As previously stated, series of transformations based on the log-function are

given in Fig. 1. These scales depict the increasingly stronger degree to which

the high-cycle regime is compressed relative to the low-cycle regime. Thus, it

should be possible to transform the scale for Nf such that the variability is

approximately constant for all lives using one of the forms shown in Fig. 1.

Using all of the data in Table 1, the best log-function type transformation

was log(log N-). Individually, the strain-controlled data were best fit to the

transformation (log Nf) , which is exactly halfway between (log N,.) and log(log N,)

in Fig. 1. Similarly, the data from load-controlled tests followed the log(log Nf)

-1/2transformation while the deflection-controlled test data followed (log N-)

A statistical comparison of the error-terms for s, the standard deviations,

indicated that the mean residual values for the strain- and load-controlled

data were not significantly different from each other, but were significantly

different from those for the deflection-controlled data. Thus for the data in

Table 1, the data from deflection-controlled tests have somewhat less scatter

than data from either strain- or load-controlled tests. Nevertheless, the

actual differences are small and for the purpose of this study the transformation

log(log Nf) is considered adequate.

In terms of the "best" description of Eq. (1) [or of the square root of

Eq. (1)] without the log-function requirement, the relation

a = 0.08 N f1 > 1 0 6 (3)

2provides the highest regression-analysis coefficient of determination (R ). Then,

by Eq. (2), the estimated transformation becomes

„ T „ -0.106 „.Nf = Nf . (4)

Compared to the log-function transformations in Fig. 1, Eq. (4) would generally

1/2lie between the (log Nf) and log (log N^) lives. Although this transformation

2did result in a somewhat better all-around fit (higher R , etc.) when applied to

fatigue data, the form Is not as convenient to use as the log(log Nf)

form. A comparison of the relation given by Eq. (3) and the actual daca is

ehown in Pig. 3. The consistency of the data is very good considering the

diverse data base and the fact that the fatigue lives spanned nine orders of

tnagnitude.

V. DATA FROM SUSPENDED TESTS (RUN-OUTS)

In the past, engineers dealing with fatigue data which contains run-outs

had two options to accommodate these data without resorting to involved

statistical procedures. The run-out data could be ignored and excluded from

the analysis, or they could be treated as failures. Either option usually

results in models which underestimate the true mean fatigue life, particularly

in the critical high-cycle regime. Thus it is desirable to treat these data

in some other manner, such as the recently developed iterative least squares

31(ILS) approach. This method initially treats the run-ouc data as failures

and successively reestimates the fatigue life from the relation

„ i+1 „ i . s1 f(z) ,_ .% = Nf + 1 - F(z) ( 5 a )

where

NfR° - N,1

2 - t (5b)8

with "f GO the ordinate and F(z) the area of a unit normal distribution at z.

RO ' iIn these relations N, is the fatigue life for the run-out, N- the estimated

fatigue life, and s the standard deviation for the i-th estimate of Nf.*

*Note that the transformed value of Nf should be used in these relations.

Thus, the initial estimate, Nf , would be determined from the model using Nf

as part of the data base. The same procedure applies for data containing more

than one run-out. In practice, Nf > Nf and z is an increasingly larger

negative number so f (z) and F(z) -*• 0 which means that Eq. (5a) will converge.

The rate of convergence depends upon the data base, and experience indicates

that the number of iterations is usually between four and fifty. An example

of this procedure applied to data for ERNiCr-3 weld metal is shown in Fig. A.

In this case the ILS method resulted in a strain-life curve which is significantly

more optimistic in the high-cycle regime when compared to the curve for the

data without the run-out. Convergence of the least squares estimators to three

decimal places was obtained in eleven iterations.

VI. OUTLYING OBSERVATIONS

Occasionally, one or several observations will result in error terms

which appear to lie far beyond the pattern exhibited by the remaining values.

In such cases one is tempted to reject these observations, but cannot do so

without direct physical evidence that the result was truely inappropriate or

incorrect or without statistical guidance that the result is probably inappropriate

or incorrect. A simple and effective method is to compare the ratio of the

suspected error value to the total error standard deviation with a critical

value. Tabulations of these critical values are given in the Refs. 33 ane 34

for sample sizes up to ̂ 150. For larger sample sizes (<1000), these values tend

to converge to ̂ 3.5 at the a = 0.05 level. In any event, the judgement of

suspect data must be carefully considered.

VII. STRAIN-LIFE RELATIONS

A number of strain-life models which have been used for fatigue data are

described in Appendix B. The list may not be complete, but is intended to

illustrate the advantages and disadvantages for a reasonable variety of models.

In many cases, the relation is nonlinear and special algorithms are required

for a least squares analysis.

VIII. SUMMARY

This report discussed several aspects of fatigue data analysis in order to

provide a basis for the development of statistically sound design curves-

Included is a discussion on the choice of the dependent variable, the assumptions

associated with least squares regression models, the variability of fatigue

data, the treatment of data from suspended tests and outlying observations,

and various strain-life relations.

REFERENCES

1. J. Neter and W. Wasserman, Applied Linear Statistical Models, RichardD. Irwin, Inc., 1974.

2. C. Daniel and F. S. Wood, Fitting Equations to Data, Wiley-Interscience,1971.

3. A. M. Freudenthal, "Planning and Interpretation of Fatigue Tests,"pp. 3-13 in Symposiim on Statistical Aspects of Fatigue, SpecialTechnical Publication No. I2l3 American Society for Testing andMaterials, 1951.

4. G. M. Sinclair and T. Jo Dolan, "Effect of Stress Amplitude onStatistical Variability in Fatigue Life of 75S-T6 Aluminum Alloy,"Trans. ASMS, 75:867-872 (1953).

5. A. M. Freudenthal and ii. J. Gurabel, "On the Statistical Interpreta-tion of Fatigue Tests," Proc. R. Soc. London Ser. A, 216:309-332(1953).

6. R. Roeloffs and F. Garofalo, "A Review of Methods Employed in theStatistical Analysis of Fatigue Data," Am. Soc. Test. Mater. Proc.56:1081-1090 (1956).

7. M. N. Torrey and G. R. Gohn, "A Study of Statistical Treatments ofFatigue Data," Am. Soc. Test. Matev. Proc. 56:1091-1123 (1956).

8. American Society for Testing and Materials, A Guide for FatigueTesting and the Statistical Analysis of Fatigue Data, ASTM STP91-A (1963).

9. T. J. Dolan and H. F. Brown, "Effect of Prior Repeated Stressingon the Fatigue Life of 75S-T Aluminum," Am. Soc. Test. Matev. Pvoc.52:733-742 (1952). . .

10. D. R. Diercks and D. T. Raske, Elevated-temperature, Strain-con-trolled Fatigue Data on Type 304 Stainless Steel -- A Compilation,Multiple Linear Regression Model, and Statistical Analysis,Argonne National Laboratory, ANL-76-95, 1976.

!!• D. R. Diercks, A Compilation of United States and British Elevated-Temperature, Strain-Controlled Fatigue Data on Type 316 StainlessSteel, Argonne National Laboratory, ANL/MSD-78-4, March 1978.

12. M. I. deVries, B. van der Schaff, and J. D. Elen, Results of 300 LowCycle Fatigue Tests at 723 K and 823 K on Irradiated and UnirradiatedStainless Steel DINl.4948 Plate and Welded Joints, NetherlandsEnergy Research Foundation, ECN-67, July 1979.

13. D. T. Raske, unpublished work. ' "

14. Personal communication, G. E. Korth, EG&G Idaho, Inc., to D. T.Raske, Argonne National Laboratory, June 1979.

15. D. T. Rsske, Section and Notch Size Effects in Fatigue, Universityof Illinois, T&AM Report No. 360, August 1972.

16. T. Endo and JoDean Morrow, "Cyclic Stress-Strain and Fatigue Be-havior of Representative Aircraft Metals," J. Mater. A(1):159-175(March 1969).

17. Personal communication, R. W. Landgraf, Ford Motor Co. s to D. T.Raske, Argonne National Laboratory, September 1976.

18. Personal communication, C. E. Jasfce, Battelle Columbus Laboratories, toD. T. Raske, Ar,̂ .me National Laboratory, June 1979.

19. W. Illg, Fatigue Tests on Notched and Unnotched Sheet Specimens of2024-TS and 7075-T6 Aluminum Alloy and of SAE 4130 Steel withSpecial Consideration to the Life -Range from 2 to 10,000 Cycles3National Advisory Committee for Aeronautics, Technical Note 3866,December' 1956.

20. P. J. Haagensen, Statistical Aspects of Coexisting Fatigue FailureMechanisms in OFHC Copper, University of Toronto Institute forAerospace Studies, UTIAS Technical Note No. 112, June 1967.

21. G. K. Korbacher, "On the Modality of Fatigue-endurance. Distribu-tions," Exp. Mech. 11(12): 540-547 (December 1971).

22. p. Soo and J. G. Y. Chow, Correlation of High- and Low-Cycle FatigueData for Ineoloy-80011, Brookhaven National Laboratory, BNL-NUREG-50574, NRC-8, October 1976.

23. E. Epremian rnd R. F. Mehl, Investigation cf Statistical Nature ofFatigue Properties, National Advisory Committee for Aeronautics,Technical Note 2719, June 1952.

24. G, E. Dieter and R. F. Mehl, Investigation of the Statistical Natureof the Fatigue of Metals, National Advisory Committee for Aero-nautics, Technical Note 3019, September 1953.

25. H. T. Corten and G. M. Sinclair, "A Wire Fatigue Machine for In-vestigation of the Influence of Complex Stress Histories," Am. Soo.Test. Mater. Proa. 56:1124-1137 (1956). • •

26. H. F. Hardrath and E. C. Utley, Jr., An Experimental Investigationof the Behavior of 24S-T4 Aluminum Alloy Subjected to RepeatedStresses of Constant and Varying Amplitudes, National AdvisoryCommittee for Aeronautics, Technical Note 2798, October 1952.

27. W. S. Hyler, R. A. Lewis, and H. J. Grover, Experimental Investiga-tion of Notch-Size Effects on Rotating-Beam Fatigue Behavior of7SS-T6 Aluminum Alloy, National Advisory Committee for Aeronautics,Technical Note 3291, November 1954.

28. J. A. Bennett and J. G. Weinberg, "Fatigue Notch Sensitivity of SomeAluminum Alloys," J. Res. Nat, Bur. Stand. 52(5) :235-245 (May 1954).

29. J. Schijve and F. A. Jacobs, Fatigue Crack Propagation in Unnotchedand Notched Aluminum Alloy Specimens, National Aero- and Astro-nautical Research Institute, Amsterdam, NLR-TR M.2128, May 1964.

30. M. S. Bartlett, "The Use of Transformations," Biometri.cs 3(1):39-52(Harch 1947).

31. J. Schmee and G. J. Hahn, "A Simple Method for Regression Analysis withCensored Data," Technometrics 21(4):417-432 (1979).

32. ASTM Standard E178-75, "Standard Recommended Practice for Dealing withOutlying Observations," 1976 Annual Book of ASTM Standards, Park 41,pp. 183-211.

33. Statistical Methods in Research and Production, 0. L. Davies and P. L.Goldsmith eds., Oliver and Boyd, Tweeddale Court, Edinburgh, 1972, p. 49.

34. D. T. Raske and Jo Dean Morrow, "Mechanics of Materials in Low Cycle FatigueTesting," Manual on Low Cycle Fatigue Testing, ASTM STP 465, AmericanSociety for Testing and Materials, 1969, pp. 1-25.

Table 1. Data Vied for the Analyai* of Scatter In Fatigue Data

TeatControl Material*

Kuabcr ofData Seta

Hunter

of Texts Kef.

Strain

Loud Ax

Rot

Kcv B

Rev T

304 t 316 SS"Alloy 71SC

SAT. 43M>HR1X, SAH 950X 4 980X2024 t 7075 Al

316 SSb

Alloy 718C

SAE 4340 I 41302024 I 7075 AlOHIC Co .Incoloy 60011

Alloy 718SAE 4340SAK 1045 I 1050ASTM A2HSHlld Slrc-1fiit. SteelStcrl wireAl alloysonic cu

SAF. 4340Araeo Fc2024 Al

Araco Fc vlrcAl vlreCu vlrpAx (R'0) 2024 AlAx + Cot II Alloy 71(1Ax (US) Incoloy D001lb

2911112918

31

12IB

72

12

1161•S

381

««»

ID1010511

1763959

263»4

10S

5993

8816

940

17372

235151140393

20

1*878

ISO

200.

aoo200

1934

10-1314

15-1617

15.16

1814

15,1915,1920,21

22

143

S.2333

2425

4,26-28

233

24

333

291422

Ml. Rev B Bronze

Totals 291 4618

*Ax » axial loading, hot & • rooting henrffnr.. Kcv R - reversed bcndlnc, Rev Trrvorr.cil torsion. Ax (U-0) - i<-ro-to-n.ix. axial loaillnp,. Ax + Kot B " axialplus rotating bending, nnd Ax (US) " ultrasonic axial loading*

EK'valcd-tcnpcraturc tests.

- »tid elcvatc-d-lcepcrature

Ocdcctlon-conLrollcd tests.

10N}= (log Nf)

I0 ! 0 3 I 0 4 IO5IO6 10 •

N { = log (log N

j= (log N

N}=logNf

\10 I0 2 IO3 I0 4 IO5 I06 I07 IO1

CYCLES TO FAILURESf

,8

Fig. 1. Relative Scales for Various Transformations.

20r-

5

ao.to

UiCO

10

log N{ * 7.212 ( • )S - 0.413

,RV\\\

Fig. 2. Distribution of Fatigue Life for 57 Type 75 S-TAluminum Specimens.

I I I I IAUSTEN ITiC STAINLESS STEELSALUMINUM ALLOYSSAE IOXX STEELSSAE 4XXX STEELSWIRE a EUTECTOID STEELSA5TM A285 STEELMILD STEELARMCO IRONHRLC.SAE 95OX a 98OX STEELSCOPPERBRONZE

= O.O8N f

(R2 = 0.973)

TEST TYPESOLIDOPENHALF-SOLID

STRAIN-CONTROLLOAD-CONTROLDEFLECTION-CONTROL

10 10 I03

CYCLES TO FAIUJRE.Nf

Fig. 3. Variability of Fatigue-test Data.

IO

V)

3I

O.I

i Cr • 3

F/iia.1<?s//n<tare o-f //re

•f&r run- oar~

'ou \ Scurve

i I,3 /£k4-

FA T/CC/e

Fig. 4. Comparison of Strain-life Curves for DataContaining a Run-out.

APPENDIX A

Regression Models Using Different Dependent Variables—An Example

A simple, linear, strain-life model was devised to demonstrate the

results obtained when using either the fatigue life or strain range &s the

dependent variable. The basic assumptions were:

1. The fatigue lives are log-normally distributed with constant

variance.

2. The controlled strains have insignificant measurement errors .

Based on the above, hypothetical data was provided for 3 strain ranges

2(2.0, 0.774, and 0.3%) and 6 corresponding fatigue lives (log 10 + 0.524,

log 10 + 0.524, and log 10 + 0.524). These data were fit to the relation,

Y = B + B,Xo 1

where Y and X assumed values of log Nf and log Ae and vice versa.

The results, given in the attached table and figure illustrate the

principals previously discussed. Namely, the least squares estimators for

the model e = f(N) are biased toward zero and both underpredict and overpredict

the actual data. In addition, while the distribution of error terms for the

model N = f(e) is uniform, the model e = f(N) produces a distribution which is

skewed. This implies that the error terms are correlated which indicates

that the model needs an additional term (see Ref. 1). Moreover, if the true

curve were not known beforehand, a comparison of these results would seem to

indicate that since the model E = f(N) produced the lowest apparent MSE, it

provides the best description of the data. This conclusion is not true however,

as evidenced by the statistics comparing the least squares fit to the true

curve. In this case, there is no doubt that N = f(e) Is the correct model.

Results of Example

ModelLeast Squares Fit to 6 Data Points Actual Fit to True Curve

MSESlope,B,

Intercept,

oon Log Nf on Nf

MSE

N = f (E) 0.907 0.642 0.412 -0.206 0.713

E = f(N) 0.907 0.126 0.016 -0.187 0.636

1.0 1.0 0

0.989 0.633 0.021

= coefficient of determination, s = error standard deviation, MSE = error mean square.

N = log Nf, e = log Ac, f ( ) = B Q + B ^

Coefficients when written as e = f(N).

).

APPENDIX B: Strain-life Relations

BASQUIN-COFFIN-MANSON RELATION (B-C-M)

Form

Ae ^(2Nf)c + -^ (2Nf)

b

Advantages

Al. This is a realistic model since the parameters are roughly analogous to

the tensile strength and ductility parameters (see Ref. 34).

A2. Extrapolation into the long-life regime will result in conservative

estimates.

A3. Used extensively - fits data on many structural materials very well.

A4. When used with log (Ae ) and log (Ae ) as the independent variables,

simple linear regression analysis is adequate. For more complex models,

multiple linear regression analysis is easily accomplished with e' c, etc.

as functions of temp., hold-time, etc.

A5. Can be used with indicator variables (see Ref. 1) to account for material

differences such as those due to heat-treatment, etc.

Disadvantages

M.. This relation intrinsically assumes that the distribution of fatigue

lives is log-normal with constant variance. For some materials, this

can lead to regression models which have non-constant error-term

distributions.

D2. The finite terminal slope, b, may be large enough to seriously under-

predict subsequent long-life data-particularly if the initial fit based

on a predominately high strain fatigue data base for a material which

is cyclically unstable (see D3).

D3. When the material does not exhibit the same cyclic hardening or softening

characteristics at all strain ranges, the relations Ae = e'(2N,.) and

°f b P

Ae = -=- (2NC) may not be linear and therefore the B-C-M form is invalide i. i

as stated.

D4. When D3 is true the relation must be used with total strain as the

independent variable and therefore a direct algebraic solution for Nf

is not • oc sible.

D5. When total strain is used as the independent variable, the relation must

be numerically inverted, and a nonlinear regression analysis algorithm

is required.

D6. The treatment of run-outs when the elastic and plastic strains are used

is unclear since their value at N_/2 is unknown at the outset and moreover,

since they will change in value after each iteration, the relation z + e = eP e

may not make sense. Run-outs can be accommodated when the total strain

nonlinear form is used; however, this may be extremely cumbersome.

MODIFIED LANGER RELATION (M-L)

FormBi

Ae = BQNf !

Advantages

Al. This, like the B-C-M relation, is realistic since the parameters can

be physically interpreted in relation to the strain-life curve.

A2. Since this relation predicts an endurance strain (B»), it is representative

of many structural materials.

A3. A solution for the dependent variable, log N , is straightforward and

successive iterations are easily accomplished when treating data which

contains run-outs.

A4. This relation uses the total strain range and therefore, the problem of

determining the appropriate elastic and plastic components is avoided.

Disadvantages

. Dl. The existence of an endurance strain can result in non-conservative

fatigue life predictions when high-cycle data is not available.

D2. This relation is nonlinear, and requires specialized techniques or

computer packages to determine the minimum variance least squares

estimators.

D3. Like the B-C-M relation, this model intrinsically assumes the distribution

of fatigue lives are log-normal with constant variance.

D4. This model is best suited to a single set of independent variables;

hence, the analysis of a data base which has different temperatures,

etc. may be computationally difficult or the results imprecise.

D5. The relation has only three parameters, and may not be sufficiently

flexible for the representation of some data.

LOG-LOG FORM (L-L)

FORM

Ae = BQ(log Nf)

IO

Advantages

Al. This relation uses the transformation log(log N,) which is effective

in stabilizing the variance on data which have a trapezoidal error

distribution (see Ref. 1) when fit with a log Nf transformation.

A2. In addition to the above, all the advantages of the Modified Langer

Relation apply.

Disadvantages

Dl. Since this transformation compresses the high-cycle regime in relation

to the low-cycle regime, the fit to low-cycle data may not be as accurate

as with the usual log N_ transformation.

D2. All of the disadvantages (except D3) of the Modified Langer Relation

apply.

r..'

POLYNOMIAL RELATIONS (POLY)

FORM

N = B + B.S + B_S2 orO 1 2

where,

N = log Nf & S = log Ae; or

N = log (log N f), or = (log Nf

etc.

v-1/2

Advantages

Al. These relations are simple multiple linear regression models.

A2. They are ideal for the analysis and description of multiple data sets

where the constants become functions of temperature, etc. Indicator

variables are also easily used vith these relations.

A3. The cubic form can provide a very accurate fit to the data.

A4. Any necessary transformation on strain or fatigue life is easily

accommodated.

Disadvantages

Dl. Extrapolation beyond the data base is, at best, risky.

D2. The use of search methods such as Stepwise to determine the best model

can result in polynomial relations that are statistically incorrect

(see Ref. 1).

D3. Total dependence on polynomial regressions, can easily lead to large

unwieldly models which probably cause more problems than they solve.

HYPERBOLIC RELATION (HYP)

«*»

FORM

N2 + B-jNS + B2S2 + B»N + B^S + B = 0

where

N = log N. & S = log Ae, or

N = log (log N ) etc.

Advantages

Al. The five parameters in this relation enable it to accurately fit all

types of strain-controlled fatigue data, including data on materials

which exhibit an endurance strain limit.

A2. The fO¥m of this relation suggests no specific distribution of fatigue

life or strain, and therefore any appropriaLe transformation on these

Variables may be used.

^Disadvantages

Dl, The relation is non-linear, and makes no physical sense in relation to

the actual strain-life curve.

D2. Some of the parameters may be highly correlated; hence, the confidence

intervals on their true values will be larger than desired for purposes

of extrapolation.

D3. Like the modified Langer relation, this model is probably best suited

to the analysis of a single strain-life curve.