Brian L. Vlcek

Georgia Southern University, Statesboro, Georgia

Robert C. Hendricks and Erwin V. Zaretsky

Glenn Research Center, Cleveland, Ohio

Probabilistic Analysis for Comparing Fatigue Data 

Based on Johnson-Weibull Parameters

NASA/TP—2013-217633

April 2013
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Brian L. Vlcek

Georgia Southern University, Statesboro, Georgia

Robert C. Hendricks and Erwin V. Zaretsky

Glenn Research Center, Cleveland, Ohio

Probabilistic Analysis for Comparing Fatigue Data 

Based on Johnson-Weibull Parameters

NASA/TP—2013-217633

April 2013

National Aeronautics and

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Glenn Research Center

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NASA/TP—2013-217633 1

Probabilistic Analysis for Comparing Fatigue Data Based on 

Johnson-Weibull Parameters

Brian L. Vlcek

Georgia Southern University

Statesboro, Georgia 30460

Robert C. Hendricks and Erwin V. Zaretsky

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Summary

Leonard Johnson published a methodology for establishing 

the confidence that two populations of data are different. 

Johnson’s methodology is dependent on limited combinations 

of test parameters (Weibull slope, mean life ratio, and degrees 

of freedom) and a set of complex mathematical equations. In 

this report, a simplified algebraic equation for confidence 

numbers is derived based on the original work of Johnson. The 

confidence numbers calculated with this equation are compared to those obtained graphically by Johnson. Using the 

ratios of mean life, the resultant values of confidence numbers 

at the 99 percent level deviate less than 1 percent from those 

of Johnson. At a 90 percent confidence level, the calculated 

values differ between +2 and –4 percent. The simplified equation is used to rank the experimental lives of three aluminum 

alloys (AL 2024, AL 6061, and AL 7075), each tested at three 

stress levels in rotating beam fatigue, analyzed using the Johnson-Weibull method, and compared to the ASTM Standard 

(E739–91) method of comparison. The ASTM Standard did 

not statistically distinguish between AL 6061 and AL 7075. 

However, it is possible to rank the fatigue lives of different 

materials with a reasonable degree of statistical certainty 

based on combined confidence numbers using the JohnsonWeibull analysis. AL 2024 was found to have the longest 

fatigue life, followed by AL 7075, and then AL 6061. The 

ASTM Standard and the Johnson-Weibull analysis result in 

the same stress-life exponent p for each of the three aluminum 

alloys at the median, or L50, lives.

Introduction

If a machine or structural component is properly designed 

and manufactured, a phenomenon that may limit its usable life 

is fatigue. Fatigue failure is not deterministic, but probabilistic. Within a given population, the variability in fatigue lives 

can be one or two orders of magnitude. This variability is 

often the result of material manufacture and use.

Fatigue is a failure mode that occurs at a stress level below 

the ultimate tensile stress, and is a direct result of cyclic loading. Under fatigue loading, cracks can develop at surface 

defects, finish marks, and inclusions (Refs. 1 and 2) or in

high-stress regions of a structural body that are initially free 

from microscopic defects. Once cracking occurs, a dominant 

crack usually propagates progressively to fracture (Ref. 1).

For high-cycle components such as shafts and other highspeed rotating machinery components, the concept of a fatigue 

limit is employed in their design. That is, the machine element 

is designed where the stress to which the component is subjected is below that at which fatigue will occur. However, 

there are problems in defining a fatigue limit for some materials because factors such as heat treatment, surface condition, 

residual stresses, temperature, and environment affect fatigue 

life. As a further complication, aluminum and other nonferrous alloys do not exhibit a fatigue limit (Ref. 1). If an aluminum component is run long enough at any stress level it will 

eventually fail from fatigue.

In the real world there is always a probability that a part will 

fail. Instead of designing a component to never fail, machine 

components can be designed for a “safe” life. Hence, it becomes 

necessary to design for finite life with an acceptable risk or 

probability of failure. A knowledge and understanding of a 

material’s fatigue life, or probability of fatigue failure, becomes

essential to a good engineering design. 

As a result of the expense or time associated with fatigue 

testing, it has been practice to analyze fatigue data based on 

limited sample sizes—often, 10 or fewer fatigue tests are analyzed. It is essential to maximize the information gained from 

a limited amount of data to ensure that data distribution is factored into the analysis. Because of a limited understanding of 

probabilistic failure analysis, the first inclination of most practicing engineers is to determine a statistical mean or median 

life. The distribution of failures or scatter in the life is not necessarily taken into account. No consideration is given to the 

distribution in the fatigue lives. At best, error brackets based 

upon the standard deviation might be considered.

If the applied stresses corresponding to the fatigue failures 

are known, engineers often plot the data on a stress-life (S-N) 

curve and try to draw conclusions by comparing these curves.

This would be considered as the next level in complexity of a 

fatigue life analysis. In general, stress-life data are analyzed 

according to the ASTM E739–91, “Standard Practice for Statistical Analysis of Linear or Linearized Stress-Life (S-N) and 

Strain-Life (ε-N) Fatigue Data” (Ref. 3). Unfortunately, the 
NASA/TP—2013-217633 2

standard does not provide a method to rank the confidence that 

one S-N curve is superior to another. At best, the standard can 

be used to define 95-percent confidence bands. This pair of 

bands brackets the linearized curve-fit and establishes limits 

which 95 percent of the test data would fall. The only fact that 

can be inferred is whether or not competing 95-percent confidence bands bracket data sets with or without overlap. If there 

is no overlap, it is reasonable to infer that the data sets are 

different from each other at a 95-percent level of confidence.

The authors of the ASTM standard (Ref. 3) recognized the 

inherent limitations of the standard when stating that, “As 

alternate fatigue models and statistical analysis are continually 

being developed, later revisions of this practice may subsequently present analyses that permit more complete interpretation of S-N and ε-N data.” In fact, the ASTM standard refers 

to the Weibull distribution function as a possible alternative 

statistical analysis for such data. There are many examples in 

the literature of the use of the Weibull distribution function to 

determine the life and strength of materials, structural components, and systems/machines (Refs. 4 to 9). It is worth noting 

that Weibull analysis has also been used to evaluate preventative maintenance practices (Refs. 10 and 11). 

Leonard G. Johnson (Ref. 12), while with the GM Research 

Center in the 1950s and 1960s, is credited with the first practical engineering analysis technique for evaluating fatigue data 

based on the Weibull distribution function. Although Johnson 

developed a practical method for establishing confidence that 

two or more populations are different, it requires difficult interpolation between or extrapolation beyond his published 

plots. Additionally, the mathematical technique Johnson presents is somewhat difficult to follow for someone not familiar 

with statistics or gamma function solutions. Finally, Johnson’s 

work is documented in the 1964 text (Ref. 12) that is somewhat rare and difficult to find. As a result of these factors, his 

technique of ranking fatigue data populations based upon confidence numbers has fallen out of widespread use.

In view of the aforementioned, the objectives of the work 

reported herein are to (1) develop a simplified relationship to 

calculate confidence numbers independent of interpolation of 

Johnson’s graphical plots, (2) develop a simplified relation to 

calculate confidence numbers beyond those combinations of 

parameters used by Johnson (Ref. 12), and (3) apply these 

relations to rank the experimental lives of three aluminum 

alloys (AL 2024, AL 6061, and AL 7075), each tested at three 

stress levels in rotating beam fatigue.

Nomenclature

A0 parameter defined by Equation (3b)

B0 parameter defined by Equation (3c)

C confidence number

D parameter defined by Equation (5a)

DOF total degrees of freedom

L life, number of stress cycles or hr 

L10 10 percent life or life at which 90 percent of a 

population survives, number of stress cycles or hr 

L50 50 percent life or life at which 50 percent of a 

population survives, number of stress cycles of hr 

Lx life at which (1–S) samples have failed

Lβ characteristic life or life at which 63.2 percent 

of population fails, number of stress cycles or hr 

m Weibull slope 

MLR mean life ratio

MLR@99percent mean life ratio at a confidence number of 

99 percent

MLRexp experimental mean life ratio 

n number of data points in a data set 

N life, number of stress cycles

p stress-life exponent

S probability of survival, fraction or percent 

Y1 correction parameter

ε strain

σ bending stress, MPa (ksi)

Probabilistic Analysis 

Weibull Distribution Function

Weibull (Refs. 13 to 15) is credited with being the first to 

suggest a reasonable way to estimate fracture life with a statistical distribution function. The probability distribution function empirically identified by Weibull in 1939 is as follows: 

ln ; 0 ; 0 1 1 lnln < < ∞ < < 













 = β

L S

L

L

m

S x x (1)

This form is referred to as the two-parameter Weibull distribution function. Weibull empirically determined this function 

by fitting curves to fracture data. The variable S is the level of 

survivability (probability of survival) being considered. 

For example, if 15 percent of the samples have failed, then 

the survivability would be 0.85. Lx is the life in cycles or hours 

at which 1–S percent of samples have failed. In the case of S

equaling 90 percent, Lx is the life at which 10 percent of the 

samples have failed—this is typically referred to as the L10

life. Lβ is the characteristic life of the material, which is 

defined as the life at which 63.2 percent of the samples have 

failed. Finally, m is the Weibull parameter or slope, which is an 

indicator of the scatter or distribution in the data—the larger the 

number the smaller the amount of scatter. A Weibull slope of 

1.0 is indicative of an exponential distribution, 2.0 is Rayleigh 

distribution, and 3.57 is a normal, or Gaussian, distribution of 

the data.
NASA/TP—2013-217633 3

Figure 1 is a generic Weibull plot. When plotting the ln ln [1/S]

as the ordinate against the ln L as the abscissa, fatigue data are 

assumed to plot as a straight line. The ordinate ln ln [1/S] is 

graduated in statistical percent of components failed or 

removed for cause as a function of ln L, the natural logarithm 

of the time or cycles to failure. The tangent of the line is designated the Weibull parameter or slope m, which is indicative

of the shape of the cumulative distribution, or the amount of 

scatter of the data previously discussed. 

Johnson (Ref. 12) provided the first practical engineering 

method for data analysis and projections using the Weibull 

distribution function. Effectively, a distribution of a small 

number of fatigue lives is ranked, plotted, and fitted with a 

straight line. From this Weibull plot, the life at any probability 

of survival can be determined. 

The relative ranking of lives of competing materials can be 

ranked with a basic comparison of L10 lives determined with 

Weibull techniques (Ref. 16). Johnson (Ref. 12) carried this 

concept further, by establishing confidence numbers that established the probability of whether two data sets being compared 

were different. These confidence numbers were dependent upon 

knowing the Weibull slope, total degrees of freedom, and either 

a mean life or L10 life ratio. 

Confidence Numbers

Johnson (Ref. 12) developed a method based on Weibull 

parameters for establishing the probability, which he termed 

“confidence number,” that two values being compared were 

statistically different from each other. He defined the confidence number as being “the probability that the true mean life 

ratio of the populations is greater than unity.” In other words, 

the reasonable statistical certainty that population A is better 

than population B. The confidence number is calculated from 

a set of complex mathematical equations derived by Johnson 

and is a function of the degrees of freedom of the two data sets 

being compared, the ratio of the lives of the two data sets, and 

the Weibull slope m. Confidence numbers greater than 

90 percent are considered statistically significant. This means 

that if the tests were to be repeated 100 times, 90 out of 100 

times population A would be better than population B. 

Johnson (Ref. 12) published 10 graphs based on his equations for Weibull slopes of 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.5, 3.0, 

3.5, and 4.0 in order to graphically determine confidence 

numbers. Each graph incorporates total degrees of freedom of 

12, 24, 48, 96, 400, and 3000 and mean life ratios from 1.0 to 

at most 3.0. 

Figure 2, reproduced from Johnson (Ref. 12) for Weibull 

slope m = 1, is representative of the 10 confidence number 

graphs. To determine the confidence number, the appropriate 

graph is selected based upon the Weibull slope. At the appropriate mean life ratio, the intersection with the linear curve 

corresponding to the desired total degrees of freedom corresponds to the confidence number. 
NASA/TP—2013-217633 4

Total Degrees of Freedom

The number of degrees of freedom is considered to be the 

number of parameters that may be independently varied. That 

is, the degree of freedom is an expression of the number of 

options available. For a population with a fixed mean containing

ni failures, (ni – 1) members can be chosen arbitrarily (Ref. 12). 

Once (ni – 1) members have been selected, there is no choice of

the final selection since only one member remains. Thus for this 

case the number of degrees of freedom is said to (ni – 1). 

When trying to rank or compare two populations, Johnson 

(Ref. 12) identifies the total degrees of freedom DOF as 

DOF = (n1 −1)(n2 −1) (2)

where n1 and n2 are the total numbers of fatigue lives in the 

two data sets being compared. For example, if two data sets 

being compared have 10 data points each, the DOF will be 81.

Looking at each of Johnson’s graphs, the limitations of the 

select cases for which the Johnson graphs are applicable 

become apparent. For example, intersection with a linear curve 

with a total degree of freedom of 81 requires interpolation 

between 48 and 96. This interpolation for 81 degrees of freedom 

is shown in Figure 2. Weibull slopes that either fall between or 

above those for which graphs exist require interpolation between 

or extrapolation beyond the range covered by Johnson’s graphs. 

Alternately, use of Johnson’s mathematical solution for confidence numbers requires an understanding of Johnson’s mathematical methodology. Additionally, the mathematical technique 

Johnson presents is somewhat difficult to follow for someone not 

familiar with statistics or gamma function solutions.

Mean Life Ratio

In addition to being a function of DOF and Weibull slope, 

use of Johnson’s plots require knowledge of the mean life 

ratio (MLR) if the confidence number is to be determined. If 

the entire data set comprises fatigue failures (i.e., there are no 

suspended tests), then the mean life and average life are the 

same. Since Weibull distributions are generally skewed, the 

percentage failed or location of the mean typically differs from 

the 50 percent value (unless the Weibull slope is 3.57; i.e., a 

normal distribution), and decreases with increasing Weibull 

slope. For example, at a Weibull slope of 1.0 the mean life 

occurs at 63.2 percent probability of failure (36.8 percent 

probability of survival). For a Weibull slope of 2.0 the mean 

life is found at 54.4 percent probability of failure (45.6 percent 

probability of survival). Thus, care should be given since 

mean life and average life cannot be used interchangeably in 

most cases. The mean life must be determined from an appropriate Weibull analysis. When comparing two data sets, the 

MLR is the larger of the two mean lives, divided by the smaller. 

Determining Mean Life Ratios at a Confidence 

Number of 99 Percent

Because the limitations for the select cases for which the 

Johnson graphs are applicable have been established, it is 

desirable to have a simplified relationship for confidence 

number that is a function of the Weibull parameters and DOF. 

To do this, Johnson’s confidence number graphs were scaled 

and a single mathematical equation that reproduced Johnson’s 

linear plots was derived. 

First, Johnson’s 10 confidence number graphs (Ref. 12)

were enlarged. The MLR corresponding to a confidence number of 0.99 was read from each figure for each of the six DOF. 

By empirically optimizing linear parameters to minimize the 

percent difference between the calculated and graphically 

determined MLR at 99 percent (MLR@99percent), Equation (3a) 

was derived to provide a calculated value of MLR@99percent:

[ ln( ) ] 1 2 MLR@99percent = A0 DOF + B0 + (3a)

where

0.05584 0.0844

0 − − = m A (3b)

and

0.6729 1.2796

0 = + m B (3c)

where m is the Weibull slope, and Equations (3b) and (3c) are 

functions of m and used in Equation (3a). Equation (3a) fits 

Johnson’s graphs for 1 ≤ m ≤ 4 values with a deviation in MLR

of at most 6.5 percent with the potential for extrapolation. This 

calculated value of MLR@99percent will be used in the following 

section to determine the confidence number—the value of 

ultimate interest. While B0 is linear in 1/m, A0 deviates somewhat from linearity in the range 2 < m < 3 and is possibly better fitted as a cubic function, but then extrapolation would 

become highly questionable. 

The intercept values of the DOF locus at a confidence number of 99 percent were used to determine a calculated MLR to 

within 6.5 percent (Eq. (3a)). For those interpolating values 

within the range of Johnson’s figures and needing 

MLR@99percent to a deviation within 2.1 percent, Equation (3a) 

can be corrected as follows:

1

@99percent,fromEq.(3a)

@99percent, corrected 1 Y

MLR

MLR − = (4a)

where

( ) ( )

100

0.5588 ln 5.8506ln 12.923 2

1



  − +

=

DOF DOF

Y (4b)
NASA/TP—2013-217633 5

Equation (4b) is used to calculate a constant used in Equation (4a). Equation (4a) results in a smaller percent difference 

between the calculated and graphically determined MLR, but 

should not be used to extrapolate values beyond the combination of Weibull parameters used by Johnson. 

It is not known whether Johnson limited his design curves 

to the 10 slopes from 1.0 to 4.0 and 6 degrees of freedom from 

12 to 3000 for technical or practical reasons. Nevertheless, 

Johnson does not place any limits on Weibull slope or DOF

when using his mathematical methodology to determine a confidence number, and there are ample examples throughout his 

text where he interpolates values from his figures. 

Simplified Expression for Confidence Numbers 

at Any Reasonable DOF and Weibull Slope

Ultimately what is desired is a simplified equation that provides the confidence number Ci at any reasonable combination 

of Weibull slope, DOF, and MLR. Since all of the lines of a 

Johnson plot pass through a point at MLR = 1.0 and Ci = 0.5, 

the confidence number plots in Johnson (Ref. 12) can be recreated (Fig. 2) by drawing lines through this common point 

and points at the intersection of the 99 percent confidence 

number and MLR calculated using Equation (3a). These lines 

were fitted with an equation that minimized the deviation from 

the Johnson lines, and Equation (5a) is the resulting equation:

Ci = 1− 0.5exp[− D(MLRexp −1)] (5a)

where MLRexp equals the experimental MLR. The variable D is 

given by Equation (5b), where 

( )

( 1)

3.912

1

1 0.99

0.5 ln

@99percent

@99percent

− =

−





 





− =

MLR

MLR

D

(5b) (5b)

where MLR@99percent is found using Equation (3a) (or Eq. (4a) 

if trying to interpolate with greater accuracy). Equation (5b) is 

the calculation of a constant used in Equation (5a). 

Equation (5a) was used to calculate the confidence for a mean 

life ratio’s DOF and Weibull slopes corresponding to confidence numbers of 99 percent as read from Johnson’s 10 figures. 

The percent deviation of the calculated confidence number from 

that graphically determined is reported in Tables I through III.

The calculated values of confidence numbers in Table I differ from the graphically determined value of 0.99 by less than 

1.0 percent. At a 95 percent (0.95) confidence number (Table II), 

the calculated value differs between +1.5 and –2.0 percent 

from the graphically determined value. At a 90 percent (0.90) 

confidence number (Table III), the calculated value differs

TABLE I.—PERCENT DIFFERENCE BETWEEN THE 

CALCULATED CONFIDENCE NUMBER AND 

0.99 CONFIDENCE NUMBER (REF. 12) FOR 

10 WEIBULL SLOPES AND 6 DEGREES OF FREEDOM

Weibull

slope

Degrees of freedom

12 24 48 96 400 3000

4.0 0.22 0.42 0.54 0.81 0.82 0.06

3.5 0.03 0.25 0.46 0.59 0.52 –0.03

3.0 –0.21 0.02 0.14 0.39 0.48 0.05

2.5 –0.41 –0.22 –0.04 0.08 0.32 –0.03

2.0 –0.09 –0.10 –0.12 -0.06 -0.10 –0.64

1.8 0.04 –0.04 –0.14 0.03 –0.06 –0.56

1.6 –0.25 –0.14 0.03 0.02 –0.02 –0.65

1.4 –0.24 –0.09 -0.05 0.31 0.18 -0.36

1.2 –0.03 0.07 0.06 0.35 0.31 –0.06

1.0 –0.04 0.09 0.10 0.35 0.55 0.05

TABLE II.—PERCENT DIFFERENCE BETWEEN THE 

CALCULATED CONFIDENCE NUMBER AND 

0.95 CONFIDENCE NUMBER (REF. 12) FOR 

10 WEIBULL SLOPES AND 6 DEGREES OF FREEDOM

Weibull

slope

Degrees of freedom

12 24 48 96 400 3000

4.0 –0.24 0.13 0.54 0.71 0.49 –1.53

3.5 –0.80 –0.26 –0.09 0.38 0.36 –1.41

3.0 –1.40 –0.99 –0.71 –0.32 –0.32 –1.66

2.5 –1.89 –1.39 –0.98 –0.54 –0.28 –1.48

2.0 –0.55 –0.40 –0.50 –0.25 –0.39 –2.09

1.8 –0.01 –0.22 –0.55 0.09 –0.09 –2.04

1.6 –0.95 –0.49 –0.15 –0.17 –0.07 –2.37

1.4 –0.77 –0.36 –0.21 0.63 0.33 –1.10

1.2 –0.09 0.11 0.08 0.94 0.78 –0.31

1.0 –0.12 0.01 0.15 0.93 1.29 0.16

TABLE III.—PERCENT DIFFERENCE BETWEEN THE 

CALCULATED CONFIDENCE NUMBER AND 

0.90 CONFIDENCE NUMBER (REF. 12) FOR 

10 WEIBULL SLOPES AND 6 DEGREES OF FREEDOM

Weibull

slope

Degrees of freedom

12 24 48 96 400 3000

4.0 –1.31 –0.80 –0.47 –0.27 –0.69 –4.08

3.5 –1.95 –1.13 –0.89 –0.59 –0.44 –3.38

3.0 –2.37 –1.98 –1.87 –1.13 –1.56 –3.87

2.5 –3.55 –2.87 –2.28 –1.87 –1.65 –3.36

2.0 –1.01 –0.62 –0.81 –0.57 –0.37 –3.34

1.8 –0.04 –0.46 –0.82 –0.03 –0.23 –3.30

1.6 –1.53 –0.88 –0.36 –0.10 –0.25 –4.06

1.4 –1.25 –0.57 –0.42 0.70 0.09 –2.03

1.2 –0.19 0.17 0.24 1.25 1.32 –0.27

1.0 –0.19 0.08 0.16 1.30 1.66 –0.11

between +2.0 and –4.0 percent from the graphically determined value. Thus, confidence numbers can be determined 

within acceptable variances using Equation (5a) for any reasonable values of DOF and Weibull slope.
NASA/TP—2013-217633 6 

Combined or Composite Confidence Numbers

When more than two related data sets are available for 

comparison, a composite confidence number can be calculated 

that incorporates significantly more data in its calculation 

using the following equation from Johnson (Ref. 12):

∏( ) =

− −= −

n

i

i n C C

1

1 121 (6)

For example, if two S-N curves are being compared where 

the fatigue life (number of cycles) is known for each of the 

materials at seven stresses, Equation (5a) can be used to 

determine the confidence number that the two lives were different at each of the seven stresses. Rather than try to interpret 

the significance of each of the seven confidence number values separately, each of the individual confidence numbers (Ci) 

at each of the individual stress levels are incorporated into the 

combined confidence number using Equation (6) from 

Johnson (Ref. 12). For this example, n = 7 in Equation (6). 

Apparatus, Specimens, and 

Experimental Procedure 

Experimental fatigue lives of three rotating-beam aluminum 

alloys (AL 2024, AL 6061, and AL 7075), were determined 

experimentally and initially reported in Vlcek et al. (Ref. 17). 

The relative fatigue lives of the three aluminum alloys are 

ranked herein using the simplified expression (Eq. (5a)) to 

determine Johnson’s confidence numbers. 

Fatigue lives of notched, rotating 12.7-mm- (0.5-in.-) diameter by 101-mm- (4-in.-) long aluminum shafts (Fig. 3), were 

determined using a commercial rotational fatigue tester. This 

test is commonly referred to as the Krause, Budd, or Vishay 

fatigue test. Figure 4(a) is a sketch of the primary components 

of the rotational fatigue tester. 

The notches in the test sample were cut on a lathe using a tungsten carbide tool insert. No surface finish or polish was performed 

on the cut surface. The notches were located midway (50.8 mm, 

2 in.) along the length of the test sample (see Fig. 3). The reduced 

diameters of the shafts at the notches were 6.35 mm (0.25 in.), 

and the walls of the notch were at a 60° angle. 

For each test, one end of the test sample was mounted in a 

12.7-mm- (0.5-in.-) diameter rotating collet. The test sample 

extended from the collet like a horizontal cantilever 

(Fig. 4 (b)). The free end was then mounted in a collet that

was part of a symmetric, tapered shaft extension. The collet 

was rotated by a variable-speed electric motor. 

A bearing from which a constant load was suspended was 

placed on the free end of the tapered shaft. As a result, a constant, sinusoidal load that is totally reversing each rotation of 

the shaft was applied to the free end of the rotating shaft. By 
NASA/TP—2013-217633 7

sliding a weight along a balance arm on the apparatus, the 

bending stress at the minimum sample cross section can be 

varied from 0 to 900 MPa (130.3 ksi). For this axially symmetrically loaded specimen, a fatigue crack can initiate at any 

point around the periphery of the shaft.

All tests were performed at room temperature and 7500 rpm. 

Each material was tested at three different bending stresses:

157.2, 180, and 202 MPa (22.8, 26.1, and 29.3 ksi). Ten tests 

were conducted to failure at each stress level. 

Test Materials 

Three aluminum alloys, AL 2024, AL 6061, and AL 7075, 

were used in this study. Without alloying elements, aluminum is 

too soft or ductile for most engineering applications (Ref. 18).

Aluminum 2024 is used widely in the aircraft industry. At 3.8 

to 4.9 percent, copper is the principal alloying element in this 

material. The maximum yield stress of 324 MPa (47 ksi) and 

Brinell hardness of 120 were reported by the material supplier 

for AL 2024. It is worth noting that unprotected, this alloy does 

not have as good corrosion resistance as other aluminum alloys. 

Aluminum 6061 is among the series of alloys that are heat 

treatable because of the presence of magnesium silicide. The 

supplier provided that the yield stress of the AL 6061 alloy 

was 276 MPa (40 ksi) and the Brinell hardness was 95. While 

considered not as strong as other aluminum alloys, AL 6061 is

one of the most versatile and widely used of the aluminum 

alloys because of good formability and corrosion resistance. 

The major alloying element in AL 7075 is zinc at 5.1 to 

6.1 percent. Magnesium, chromium, and copper are three other 

alloying elements found in AL 7075. With a yield stress of 

503 MPa (73 ksi) and a Brinell hardness of 150, AL 7075 is

considered to be among the highest strength aluminum alloys. 

It is used widely in the fabrication of highly stressed components, and it is a material of choice for airframes.

Results and Discussions 

Rotating beam fatigue tests were carried out at three stress 

levels 157.2, 180, and 202 MPa (22.8, 26.1, and 29.3 ksi) on 

three aluminum shaft alloys—AL 2024, AL 6061, and 

AL 7075 (Ref. 17)—and are presented in Table IV. These test 

results are analyzed and discussed herein.

Test Results

Figure 5 shows stress-life (S-N) plots of the data of Table IV, 

plotted according to ASTM Standard E739–91 (Ref. 3). There is 

a lot of scatter in the data at each of the three stress levels. From 

curve fits of the life data shown in Figure 5, the median life L50

is determined at the various stress levels. These results are 

summarized in Table V. From these plots, the life L is inversely 

proportional to the stress σ raised to the stress-life exponent p, 

where 

p

L

σ

1 ~ (7)

The values for stress-life exponent p are shown in Figure 5(d) 

and summarized in Table VI. 

From Figure 5, AL 2024 again has the longest fatigue life of 

the three alloys followed by AL 7075 and AL 6061. However, 

computing the 95-percent confidence bands from the ASTM 

standard suggests that there is no statistical significance 

between the lives of AL 7075 and AL 6061. 

TABLE IV.—NUMBER OF STRESS CYCLES TO FAILURE AND STATISTICAL RESULTS OF 10 SAMPLES OF 

THREE ALUMINUM ALLOYS AT THREE BENDING STRESSES

Sample 

number

AL 2024 AL 6061 AL 7075

157.2 MPa

(22.8 ksi)

180.0 MPa

(26.1 ksi)

202.0 MPa

(29.3 ksi)

157.2 MPa

(22.8 ksi)

180.0 MPa

(26.1 ksi)

202.0 MPa

(29.3 ksi)

157.2 MPa

(22.8 ksi)

180.0 MPa

(26.1 ksi)

202.0 MPa

(29.3 ksi)

1 142 000 219 700 78 300 38 900 16 700 7 200 38 800 46 100 21 700

2 180 800 207 700 144 400 46 100 20 900 14 800 104 800 39 800 21 900

3 273 200 146 600 113 300 57 000 23 500 20 800 51 600 40 300 17 300

4 188 000 141 300 152 600 36 400 17 500 10 400 43 100 52 300 24 200

5 226 400 190 500 135 100 40 200 13 800 8 600 64 700 52 600 24 200

6 224 000 116 400 59 800 80 700 28 800 9 300 53 700 42 500 12 200

7 220 400 205 700 86 200 123 300 23 500 5 600 94 100 27 400 11 200

8 204 500 157 300 75 200 43 100 27 100 8 800 100 900 16 800 10 200

9 226 400 215 500 68 500 54 400 27 600 9 600 47 300 28 100 10 300

10 203 600 180 800 61 600 51 400 32 700 7 600 76 900 37 300 10 500

Average 208 930 178 150 97 500 57 150 23 210 10 270 67 590 38 320 16 370

Standard 

deviation 34 732 35 768 35 658 26 545 6 007 4 424 24 902 11 408 6 110

Maximum 273 200 219 700 152 600 123 300 32 700 20 800 104 800 52 600 24 200

Minimum 14 200 116 400 59 800 36 400 13 800 5 600 38 800 16 800 10 200
NASA/TP—2013-217633 8
NASA/TP—2013-217633 9
NASA/TP—2013-217633 10

TABLE V.—MEDIAN LIFE

[ASTM Standard E739–91.]

Stress level, 

MPa (ksi)

Calculated life (mean life)

AL 2024 AL 6061 AL 7075

157.2 (22.8) 223 951 54 147 67 848

180.0 (26.1) 146 016 21 567 31 719

202.0 (29.3) 101 268 9 811 16 550

TABLE VI.—STRESS-LIFE COMPONENTS DETERMINED 

FOR THREE ALUMINUM ALLOYS USING 

FOUR DIFFERENT S-N CURVES

S-N curve method Stress-life exponent, p

AL 2024 AL 6061 AL 7075

ASTM standard (Fig. 5) 3.2 6.8 5.6

Johnson-Weibull

 (Fig. 6(a)), L10

4.4 6.6 5.8

Johnson-Weibull

 (Fig. 6(b)), L50

3.1 6.8 5.6

TABLE VII.—WEIBULL PARAMETERS FROM FATIGUE 

LIVES OF THREE ALUMINUM ALLOYS AT 

THREE BENDING STRESSES 

[From Ref. 17.]

Alloy Weibull 

parameter

Bending stress

157.2 MPa

(22.8 ksi)

180.0 MPa

(26.1 ksi)

202.0 MPa

(29.3 ksi)

AL 2024

Slope 6.220 5.122 2.901

L10 156 207 124 678 50 701

L 50 211 468 180 100 97 063

Mean 208 503 177 876 98 208

Characteristic life 224 304 193 459 110 136

AL 6061

Slope 2.549 4.028 2.747

L10 26 987 14 635 5 118

L 50 56 510 23 362 10 162

Mean 57 922 23 201 10 333

Characteristic life 65 249 25 587 11 612

AL 7075

Slope 2.894 3.100 2.640

L10 35 029 20 898 7 980

L 50 67 160 38 373 16 290

Mean 67 965 38 624 16 631

Characteristic life 76 226 43 189 18 716

In aerospace applications, early lives at high reliability is of 

primary interest and not necessarily the average or median life, 

even for comparison purposes. Hence, it may be reasonably 

concluded that where there are differences in lives that are relatively small, the ASTM method does not provide a method to 

differentiate with any reasonable engineering or statistical certainty whether a material is superior to another based on a 

fatigue criteria. 

Weibull Analysis

Weibull analysis is ideally suited for determining the probable fatigue life of a larger population with a limited amount of 

data, and methodologies exist to rank the populations. Each 

data set in Table I was ranked and the Weibull parameters 

were determined according to Johnson (Ref. 12). These 

Weibull parameters are reported in Table VII. The L10 and L50

lives at each stress level for each material are plotted as function of stress in S-N plots of Figure 6. The stress-life exponents p are given in Figure 6 and summarize in Table VI. The 

ASTM Standard and the Johnson-Weibull analysis result in 

the same stress-life exponent p for each of the three aluminum 

alloys at the median or L50 lives.

The ranking of the fatigue lives of AL 6061 to AL 7075, AL 

6061 to AL 2024, and AL 7075 to AL 2024 was determined 

using Equation (5a). Then the combined confidence number, 

which is the probability that the 2 sets of 10 fatigue lives at 

each of the three stresses is different, is determined. 

First, the probability that the mean life of the AL 7075 is 

greater than that of the AL 6061 is determined by calculating 

the confidence number using Equation (5a). The necessary 

Weibull parameters (slope and mean life ratio, MLR) can be 

found in Table IV. Both data sets contain 10 fatigue failures;

thus the DOF is still 81. If the Weibull slopes for both sets of 

data were equal, only one confidence number would need to 

be determined. Since the AL 6061 and AL 7075 data sets have 

different Weibull slopes (Table VII) at a bending stress of 

157.2 MPa (22.8 ksi), the confidence numbers must be determined at slopes of both 2.894 and 2.549, and then the average 

confidence number is calculated. From Equations (3) and (5), 

and Table VII with MLRexp = 67965/57922 ~1.17, for a 

Weibull slope of 2.894 the confidence number is found to be 

0.851, and for a Weibull slope of 2.549 the confidence number 

is found to be 0.831. The confidence number (0.841) reported 

in Table VII is the average of these two values. 

If the method is repeated at the 180.0 and 202.0 MPa (26.1 

and 29.3 ksi) bending stresses, the confidence numbers are 

found to be 0.998 and 0.992, respectively. Thus, the life of the 

AL 7075 aluminum is greater than that of the AL 6061 at 

180.0 and 202.0 MPa (26.1 and 29.3 ksi) with greater than a 

99 percent confidence. However, the difference in the fatigue 

lives of the two materials at 157.2 MPa (22.8 ksi) are not statistically significant, with an 84 percent confidence.

Since multiple related data sets are available, a combined or 

composite confidence number can be determined. According 

to Johnson (Ref. 12), a “composite” or combined confidence 

number C can be determined that incorporates and compares 

all data sets (six in this case). 

Substituting in the values of Ci at the three bending stresses 

(0.831, 0.998, and 0.992) and solving for C using Equation (6) 

where in this case n = 3,
NASA/TP—2013-217633 11
NASA/TP—2013-217633 12

TABLE VIII.—CALCULATED CONFIDENCE NUMBERS FOR THREE ALUMINUM ALLOYS 

AT THREE BENDING STRESSES

Bending stress,

MPa (ksi)

Aluminum alloys 

compared

Mean 

life 

ratio

Weibull slope Degrees of 

freedom

Calculated 

confidence 

number

mAL7075 mAL6061 mAL2024

157.2 AL 6061 and AL 7075 1.17 2.894 2.549 ------ 81 0.841

180.0 AL 6061 and AL 7075 1.66 3.10 4.028 ------ 81 0.998

202.0 AL 6061 and AL 7075 1.61 2.64 2.747 ------ 81 0.992

157.2 AL 6061 and AL 2024 3.60 ------ 2.549 6.220 81 0.999

180.0 AL 6061 and AL 2024 7.67 ------ 4.028 5.122 81 0.999

202.0 AL 6061 and AL 2024 9.50 ------ 2.747 2.901 81 0.999

157.2 AL 7075 and AL 2024 3.07 2.894 ------ 6.220 81 0.999

180.0 AL 7075 and AL 2024 4.61 3.10 ------ 5.122 81 0.999

202.0 AL 7075 and AL 2024 5.91 2.64 ------ 2.901 81 0.999

( )( )( )( )

0.999989

1 2 1 0.831 1 0.998 1 0.992 3 1

=

= − − − − − C (8)

As a result of incorporating two-thirds more data, greater 

confidence is achieved in stating that one material is longer 

lived than another at these test conditions. In this case, the 

AL 7075 is longer lived than the AL 6061. 

The process is repeated to rank AL 2024 and AL 7075 and 

rank AL 2024 and AL 6061 by determining the probability 

and comparing the MLR of each. The individual confidence 

numbers for both of these comparisons at each of three bending stresses are reported in Table VIII. The combined confidence number in both of these cases is significantly greater 

than 0.99. The AL 7075 mean fatigue life is greater than that 

of AL 6061 with a probability more than 99 percent. The 

fatigue life of the AL 2024 is greater than both the AL 7075 

and AL 6061 with a probability or more than 99 percent. It is 

thus possible to rank the fatigue lives of different materials 

with a reasonable degree of statistical certainty.

Summary of Results

Probabilistic failure analysis is essential when analysis of 

S-N curves is inconclusive in determining the relative ranking 

of two or more materials. In 1964, Leonard Johnson published 

a methodology for establishing the confidence that two populations of data are different. Johnson’s methodology is dependent on a limited combination of test parameters (Weibull slope,

mean life ratio, and degrees of freedom) and a set of complex 

mathematical equations. To create a simplified method, 

Johnson’s original confidence number plots were fitted, and an 

equation was found for confidence number as a function of 

Weibull slope, degree of freedom, and mean life ratio. The 

confidence numbers calculated with this equation were compared to those obtained graphically from Johnson. The simplified equation was used to rank the experimental lives of three 

aluminum alloys (AL 2024, AL 6061, and AL 7075), each 

tested at three stress levels in rotating beam fatigue analyzed 

using the Johnson-Weibull method and compared to the 

ASTM Standard (E739–91) method of comparison. The following results were obtained: 

1. Using the ratios of mean life, the resultant values of confidence numbers at 99 percent deviated less than 1 percent from those of Johnson using the simplified algebraic 

equation. At a 90 percent (0.90) confidence number, the 

calculated values differed between +2 and –4 percent.

2. The ASTM Standard (E739–91) method did not statistically distinguish between AL 6061 and AL 7075. However, it is possible to rank the fatigue lives of different 

materials with a reasonable degree of statistical certainty 

based on combined confidence numbers using JohnsonWeibull analysis. AL 2024 was found to have the longest 

fatigue life, followed by AL 7075, and then AL 6061.

3. The ASTM Standard and the Johnson-Weibull analysis 

result in the same stress-life exponent p for each of the 

three aluminum alloys at the median or L50 lives.

Glenn Research Center

National Aeronautics and Space Administration

Cleveland, Ohio, April 29, 2013

References

1. Tipton, S.M.: Static and Fatigue Design. Mechanical Design 

Handbook, Harold A. Rothbart, ed., McGraw-Hill, New York, 

NY, sec. 7, 1996, pp. 7.3−7.73. 

2. Zaretsky, E.V.: STLE Life Factors for Rolling Bearings. STLE 

SP−34, 1992. 

3. Standard Practice for Statistical Analysis of Linear or Linearized 

Stress-Life (S-N) and Strain-Life (ε-N) Fatigue Data. Active 

Standard E739−91, vol. 03.01, ASTM International, 2004. 

4. Burrow, M.F. et al.: Analysis of Tensile Bond Strengths Using 

Weibull Statistics. Biomaterials, vol 25, no. 20, 2004, pp. 5031−5035. 

5. Ellis, F.V.; and Tordonato, S.: Failure Analysis and Life Assessment Studies for Boiler Tubes. ASME Pressure Vessels and Piping 

Division Publication, vol. 392, 1999, pp. 3−13. 

6. Tomimatsu, M.; Sakai, M.; and Kikuchi, M.: Fracture Toughness 

Evaluation Based on the Master Curve Procedure. ASME Pressure 

Vessels and Piping Division Publication, 1997, pp. 343−348. 
NASA/TP—2013-217633 13

7. Osborne, Nora; Graves, George G.; and Ferber, Matt K.: Dynamic 

Fatigue Testing of Candidate Ceramic Materials for Turbine 

Engines to Determine Slow-Crack-Growth Parameters. ASME 

Paper, 1995. 

8. Ostojic, P.; and Berndt, C.C.: Variability in Strength of Thermally 

Sprayed Coatings. Surf. Coat. Technol., vol. 34, no. 1, 1987,

pp. 43−50. 

9. Holland, F.A.; and Zaretsky, E.V.: Investigation of Weibull Statistics in Fracture-Analysis of Cast-Aluminum. J. Mech. Des., 

vol. 112, no. 2, 1990, pp. 246−254. 

10. Williams, Samuel R.; and Fec, Michael C.: Weibull Analysis of 

Reconditioned Railroad Roller Bearing Life Test Data. ASME 

Rail Transportation Division Publication, vol. 5, 1992, pp. 83−87. 

11. Summers-Smith, J.D.: Fault Diagnosis as an Aid to Process 

Machine Reliability. Quality and Reliability Engineering International, vol 5, no 3, 1989, pp. 203−205.

12. Johnson, L.G.: The Statistical Treatment of Fatigue Experiments. 

Elsevier Publishing Co., Amsterdam, The Netherlands, 1964. 

13. Weibull, W.: A Statistical Theory of the Strength of Materials. 

Ingenioersvetenskapsakad. Handl., no. 151, Stockholm, Sweden, 

1939. 

14. Weibull, W.: The Phenomenon of Rupture in Solids. Ingenioersvetenskapsakad. Handl., no. 153, Stockholm, Sweden, 1939. 

15. Weibull, W.: A Statistical Distribution Function of Wide Applicability. J. Appl. Mech. Trans. ASME, vol. 18, no. 3, 1951, pp. 293–297. 

16. Vlcek, B.L.; Hendricks, R.C.; and Zaretsky, E.V.: Comparative 

Fatigue Lives of Rubber and PVC Wiper Cylindrical Coatings. 

Trib. Trans., vol. 46, no. 1, 2003, pp. 101−110. 

17. Vlcek, B.L., et al.: Experimental Determination of Fatigue Lives of 

Rotating Aluminum Alloys Using Weibull Analysis. 125–

ISROMAC–11, Paper presented at the 11th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery Honolulu, HI, 2006. 

18. Aluminum Standards and Data. Sixth ed., Aluminum Association, 

Washington, DC, 1979.
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14. ABSTRACT

L. Johnson published a methodology for establishing the confidence that two populations of data are different. Johnson’s methodology is 

dependent on limited combinations of test parameters (Weibull slope, mean life ratio, and degrees of freedom) and a set of complex 

mathematical equations. In this report, a simplified algebraic equation for confidence numbers is derived based on the original work of L. 

Johnson. The confidence numbers calculated with this equation are compared to those obtained graphically by Johnson. Using the ratios of 

mean life, the resultant values of confidence numbers at the 99 percent level deviate less than 1 percent from those of Johnson. At a 90 

percent confidence level, the calculated values differ between +2 and -4 percent. The simplified equation is used to rank the experimental 

lives of three aluminum alloys (AL 2024, AL 6061, and AL 7075), each tested at three stress levels in rotating beam fatigue, analyzed using 

the Johnson-Weibull method, and compared to the ASTM Standard (E739-91) method of comparison. The ASTM Standard did not 

statistically distinguish between AL 6061 and AL 7075. However, it is possible to rank the fatigue lives of different materials with a 

reasonable degree of statistical certainty based on combined confidence numbers using the Johnson-Weibull analysis. AL 2024 was found to 

have the longest fatigue life, followed by AL 7075, and then AL 6061. The ASTM Standard and the Johnson-Weibull analysis result in the 

same stress-life exponent p for each of the three aluminum alloys at the median, or L50, lives.

15. SUBJECT TERMS

Probabilistic life prediction; Rolling-element bearings; Bearing life prediction; Weibull analysis 

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