Brian L. Vlcek

Georgia Southern University, Statesboro, Georgia

Erwin V. Zaretsky and Robert C. Hendricks

Glenn Research Center, Cleveland, Ohio

Test Population Selection From Weibull-Based, 

Monte Carlo Simulations of Fatigue Life

NASA/TM—2012-217669

August 2012
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Brian L. Vlcek

Georgia Southern University, Statesboro, Georgia

Erwin V. Zaretsky and Robert C. Hendricks

Glenn Research Center, Cleveland, Ohio

Test Population Selection From Weibull-Based, 

Monte Carlo Simulations of Fatigue Life

NASA/TM—2012-217669

August 2012

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

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Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2012-217669 1

Test Population Selection From Weibull-Based, Monte Carlo 

Simulations of Fatigue Life 

Brian L. Vlcek 

Georgia Southern University 

Statesboro, Georgia 30460 

Erwin V. Zaretsky* and Robert C. Hendricks 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

Fatigue life is probabilistic and not deterministic. Experimentally establishing the fatigue life of 

materials, components, and systems is both time consuming and costly. As a result, conclusions regarding 

fatigue life are often inferred from a statistically insufficient number of physical tests. A proposed 

methodology for comparing life results as a function of variability due to Weibull parameters, variability 

between successive trials, and variability due to size of the experimental population is presented. Using 

Monte Carlo simulation of randomly selected lives from a large Weibull distribution, the variation in the 

L10 fatigue life of aluminum alloy AL6061 rotating rod fatigue tests was determined as a function of 

population size. These results were compared to the L10 fatigue lives of small (10 each) populations from 

AL2024, AL7075 and AL6061. For aluminum alloy AL6061, a simple algebraic relationship was 

established for the upper and lower L10 fatigue life limits as a function of the number of specimens failed. 

For most engineering applications where less than 30 percent variability can be tolerated in the maximum 

and minimum values, at least 30 to 35 test samples are necessary. The variability of test results based on 

small sample sizes can be greater than actual differences, if any, that exists between materials and can 

result in erroneous conclusions. The fatigue life of AL2024 is statistically longer than AL6061 and 

AL7075. However, there is no statistical difference between the fatigue lives of AL6061 and AL7075 

even though AL7075 had a fatigue life 30 percent greater than AL6061. 

1.0 Introduction 

A common method of engineering design is to use deterministic equations for strength and life. That 

is, it is assumed that at loads and/or times below that calculated, no failure will occur. The calculated lives 

and load limits are usually coupled with a safety factor that is dictated by experience and design code 

requirements. Complicating the issue is that failure is extremely variable and dependent on materials, 

processing, and operating conditions. 

For a finite life where it is assumed that failure can occur, reliability is dictated by the design 

application. If a failure does occur a criteria is that it does not cause injury and secondary damage. A 

statistical distribution is assumed about the calculated value of life or strength. From this method an 

acceptable life and/or load is determined at a predetermined probability of survival. Usually for critical 

applications that affect human safety and third party liability, these assumptions and calculations are 

re-examined and modified based on field experience, condition monitoring and continued inspection of 

critical components. 

For most products that have an established design and application, product improvement is 

evolutionary. Incremental changes are made to improve product performance, life, reliability, and 

maintainability as well as to decrease cost of manufacture. As the criticality (usually with respect to 

safety) of the designed component increases, the required probability of survivability for a given 

 

*

Distinguished Research Associate. 
NASA/TM—2012-217669 2

application increases (Ref. 1). Usually, for critical applications with competing failure modes, fatigue 

becomes the failure mode that limits the components life. That is, if the component does not fail from 

other causes, fatigue is the limiting mode of component failure (assuming that the component has been 

operated at stresses above which fatigue can occur, that is, above the material’s fatigue limit). 

Fatigue life is probabilistic and not deterministic. For example, it is not uncommon to find the ratio of 

longest to shortest fatigue life of bearings to be 20 to 1 or greater (Ref. 2). A sound knowledge of material 

fatigue life is thus essential to a good design. Experimentally establishing the fatigue life of materials, 

components, and systems is both time consuming and costly. As a result, conclusions regarding fatigue 

life are often inferred from a dangerously insufficient number of physical tests. 

In 1939, Weibull (Ref. 3 to 5) developed a method and an equation for statistically evaluating the 

fracture strength of materials based upon small population sizes. The method can be and has been applied 

to analyze, determine and predict the cumulative statistical distribution or fatigue failure or any other 

phenomenon or physical characteristic that manifests a statistical distribution. 

Vlcek, Hendricks and Zaretsky (Ref. 6) virtually tested 31 400 rolling-element bearings that were 

randomly assembled by Monte Carlo (random) number generation. The method of Johnson (Ref. 7) was 

used to plot the virtual data on Weibull plots. The Monte Carlo results were compared with endurance 

data from 51 bearing sets comprising 5321 bearings. A simple algebraic relation was established for the 

upper and lower L10 life limits as a function of the number of bearings failed for any bearing geometry. 

There was a fifty percent (50 percent) probability that the resultant bearing life will be less that that 

calculated. The maximum and minimum variation between the bearing resultant life and the calculated 

life correlated with the 90-percent confidence limits for a Weibull slope of 1.5. 

From the above described Monte Carlo analysis, Vlcek, Hendricks and Zaretsky (Ref. 6) established 

rules to compare and distinguish tests of identical bearings either from two or more sources or made from 

different manufacturing sources and/or materials. These include variability due to Weibull parameters, 

variability between successive trials, and variability due to size of the experimental population with which 

the initial Weibull parameters were determined. 

Zaretsky, et al. (Ref. 8), extended this method to compare two computational models to determine the 

fatigue life and reliability of a commercial turboprop gearbox with each other and with field data from 

64 gearboxes. These models were (1) Monte Carlo simulation of randomly selected lives of individual 

bearings and gears comprising the system and (2) two-parameter Weibull distribution function for 

bearings and gears comprising the system using strict-series system reliability to combine the calculated 

individual component lives in the gearbox. A series of equations were empirically developed from the 

Monte Carlo simulation to determine the statistical variation in predicted life and Weibull slope as a 

function of the number of gearboxes failed. These results were successfully compared to field data from 

the 64 gearboxes. 

In view of the aforementioned, the following are objectives of the work reported herein: (a) to 

determine a methodology for comparing life results as a function of variability due to size of the 

experimental population; (b) to determine the variation in the fatigue lives of aluminum alloy AL6061 

rotating rod fatigue specimens and distribution parameters as a function of sample size using Monte Carlo 

simulation; and (c) compare the resultant fatigue life results from AL2024 and AL7075 to that of AL6061. 

Nomenclature 

L10 Life at 10% failure; 90% survival 

L Characteristic life, life at 63.2% failure; 36.8% survival 

L Life at indicated probability of survival 

j Sequential order number from 1 to n 

e Weibull slope 

n Subset Population Size 

S Survivabilty; percent samples to survive ; Failure =1– S 
NASA/TM—2012-217669 3

2.0 Apparatus, Specimens, and Procedure 

Fatigue lives of notched, rotating 12.7-mm- (1⁄2-in.-) diameter by 101-mm- (4-in.-) long aluminum 

AL6061 rods were determined using a commercial rotational fatigue tester. This test is commonly 

referred to as either the Krause, Budd, or Vishay fatigue test. Figure 1 is a schematic 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. 1 insert). 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- (1⁄2-in.-) diameter rotating collet. 

The test sample extended from the collet as a horizontal cantilever (Fig. 1). 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 extension. As a point on the surface of the test specimen rotates about the 

axis, it experiences a sinusoidal load that is totally reversing each rotation of the shaft. By 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 study, all tests were conducted at a bending stress of 

157.2 MPa (22.8 ksi). All tests were performed at room temperature and 7500 rpm, and each test was 

conducted to failure; there were no suspensions or censoring of the data. 

Figure 1.—Schematic of Vishay rotating fatigue tester 

and test specimen geometry insert. 
NASA/TM—2012-217669 4

3.0 Statistical Analysis 

3.1 Weibull Analysis 

In 1939, W. Weibull (Refs. 3 to 5) developed a method and equation for statistically evaluating the 

fracture strength of materials. He also applied the method and equation to fatigue data based upon small 

sample (population) sizes, where the two-parameter expression relating life, L, characteristic life, Lβ and 

probability of survival, S is 

1 ln ln ln where 0 ; 0 1

                  

L e LS

S L

 (1) 

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 (Fig. 2). 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 log of the time or cycles to failure (Fig. 3). The 

tangent of the line is designated the Weibull slope e, which is indicative of the shape of the cumulative 

distribution or the amount of scatter of the data. The method of using the Weibull distribution function for 

data analysis for determining component life and reliability was later developed and refined by 

L.G. Johnson (Ref. 7). 

Figure 2.—Generic Weibull plot where (Weibull)

slope of line is e; probability of survival, S, is 

36.8 percent at which L = L, or L/L = 1. 

Figure 3.—Weibull plot of fatigue life of aluminum alloy 

AL6061 rotating test rods. Number of failures, 357 

(failure points omitted from Weibull plot for clarity); 

bending stress range, 0 to 157.2 MPa (22.8 ksi); 

speed, 7500 rpm; temperature, room. 
NASA/TM—2012-217669 5

The Johnson-Weibull method has been in continual use by NASA for over 50 years to statistically 

analyze cumulative failure data such as occurs in bearings and gears for which an extensive data base now 

exists in the open literature (Ref. 9). The method allows for reasonable engineering estimates and 

comparisons of cumulative life distributions with small sample sizes or populations. The Weibull 

parameters were determined using a spreadsheet program that generates Weibull plots. 

The Weibull slope of the resultant Weibull plot approximates the statistical distribution of the data. 

As an example, a Weibull slope of 1 approximates an exponential distribution. A Weibull slope of 2 

approximates a Raleigh distribution. A Weibull slope of 3.57 approximates a Gaussian or normal 

distribution. The resulting values of life compare reasonably well with other statistical distributions such 

as log normal. However, the ease of use and consistency of results offers an advantage of the JohnsonWeibull method over these other distribution functions. 

3.2 Monte Carlo Analysis 

To determine the input parameters for the Monte Carlo simulation a total of 357 aluminum alloy 

AL6061 rotating test rods (Fig. 1) were tested to failure. A Weibull plot (Fig. 3) was computer generated 

for these data. The individual data points were omitted from Figure 3 for purposes of clarity. From these 

results, the Weibull parameters were as follows: Weibull slope e = 2.878, Lβ = 79 457 stress cycles, and 

L10 = 36 354 stress cycles. These values are summarized in Table 1 and will be referred to as the 

experimental “baseline” values for comparison purposes. 

For purposes of Monte Carlo simulation, it was assumed that there were 1000 virtual aluminum 

AL6061 rod specimens contained in a virtual bin. According to the method of Johnson (Ref. 7), each rod 

specimen in the bin is assigned an order number (1, 2, 3,...1,000) which is also the order in which they 

would fail. The life of each component at the probability of survival corresponding to the randomly 

selected rank can be determined using Equation (1), knowing the survivability (from the order and 

ranking), characteristic life, and Weibull slope from the experimental “baseline” values. 

Using Monte Carlo techniques (Ref. 10), various-sized groups or populations were randomly 

assembled from the virtual bin. This group or population represented failure distributions of varying sizes; 

sizes of 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 35, 40, 45, 50, 75, 100, 125, 150, 175, 

and 200 were arbitrarily selected. At population sizes greater than 30, the interval between population size 

was increased. Each random number generated within a population corresponded to an order number of a 

component in the virtual bin, with which an associated failure life was determined. From each population 

size n, a distribution of associated lives was thus determined. 

TABLE 1.—COMPARISON OF ROTATING ROD FATIGUE LIVES 

OF THREE ALUMINUM ALLOYS 

[Test conditions: Bending stress, 157.2 MPa (22.8 ksi); 

speed, 7500 rpm; temperature, room.] 

Aluminum alloy Experimental 

Weibull slope 

Characteristic 

life, 

Lβ (cycles) 

L10 Life 

(cycles) 

Comparison 

to baseline 

L10 life 

(Eq. (3)), 

percent a

AL6061 2.55 65,249 26,987 –25.8 

a

AL7075 2.89 76,226 35,029 –3.6 

a

AL2024 6.22 224,304 156,207 331 

b

Baseline 

AL6061 

2.878 79,457 36,354 ------- 

a

Ten (10) test specimens failed. 

b

From Weibull plot of Figure 3, 357 test specimens failed. 
NASA/TM—2012-217669 6

Next, this distribution of fatigue lives was sequentially ranked and then reordered according to the 

method of Johnson (Ref. 7). From the order number, a percent survivability was determined 

corresponding to each fatigue life using median ranks. 

A curve fit of a Weibull plot with the independent variable of the natural logarithm of the life (ln L) 

and a dependent variable of the natural logarithm of the natural logarithm of the inverse of the probability 

of survival (ln ln(1/S)) was created. A straight line was fitted to these n data points or number of 

specimens failed using the method of least squares. The Weibull slope and the L10 life were determined 

from the resultant line. 

For each value of n, the procedure was repeated 21 times to estimate variation between trials and to 

determine the maximum, minimum, and median values for each series of 21 trials. Ninety percent of the 

trials (19 trials) were bounded by the maximum and minimum L10 life values. This entire process in turn 

was repeated 10 times to establish statistical variation between repetitions. Thus, the lives and slopes 

reported herein are averaged values from these 10 repetitions. These results are shown in Figure 4. 

Best-fit curves obtained using the linear regression package of a commercial spreadsheet were fitted 

through the points for the minimum and maximum values shown in each of the plots of Figure 4. The 

maximum and minimum variation in the distribution of the fatigue data as measured by the variation in the 

Weibull slope (Fig. 4 (b)) decreased as the number of test samples in a population increased. 

The bounds on the maximum and minimum L10 life values (Fig. 4(a)) were derived for each set of the 

respective curves as follows: 

Maximum L10 Life = Baseline L10 Life (1 + 2n–0.53) (2a) 

 Minimum L10 Life = Baseline L10 Life (1 – 1.25n–0.4) 

 where n > 2 (2b) 

 Minimum L10 Life  0 where n  2 (2c) 

where n is the number of specimens failed. 

To facilitate comparison, the results of Figure 4(a) are presented using percent variation in L10 life. 

The percent variation in L10 life is then expressed as Equation (2): 

Percent variation in life 100

,baseline

,Monte Carlo ,baseline

x

L

L L

L

10

10 10

10

  (3) 

The normalizing value L10,baseline is 36,354 cycles. These results are plotted in Figure 5. 

Figure 4.—Maximum and minimum variation of L10 life and Weibull slope as a function of number of specimens 

failed for aluminum alloy AL6061 rotating test rods. (a) Fatigue life variation. (b) Weibull slope variation. 
NASA/TM—2012-217669 7

Figure 5.—Maximum and minimum variation of L10 life as 

a function of number of specimens failed normalized to 

L10 life equal 36354 cycles. (Data from Fig. 4(a)). 

3.3 Monte Carlo Verification 

In order to verify the Monte Carlo simulation, the experimental data for the aluminum alloy AL6061 

was ordered in the sequence in which they were tested, independent of the actual experimental lives. 

Using Weibull analysis, the L10 lives of 21 sequential sets of 3 each test specimens (n = 3) were 

determined for the first 63 specimens tested. Next, the L10 lives of 21 sequential sets of 5 each test 

specimens (n = 5) were determined for the next 105 specimens tested. Last, the L10 lives of 21 sequential 

sets of 9 each test specimens (n = 9) were determined for the next 189 specimens tested. The results were 

compared to the maximum and minimum values of Figure 4(a) and Equations (2a) and (2b). The L10 lives 

of each of the subgroups fell on or within the maximum and minimum values of Figure 4(a) as defined by 

Equation (2). There was very good agreement between the maximum and minimum values of the 

respective L10 lives. 

4.0 Results and Discussion 

Incremental changes are made to improve product performance, life, reliability, and maintainability as 

well as to decrease cost of manufacture. As the criticality (usually with respect to safety) of the designed 

component increases, the required probability of survival for a given application increases. Usually, for 

critical applications with competing failure modes, fatigue becomes the failure mode that limits the 

components life and reliability. Fatigue life is probabilistic and not deterministic. Experimentally 

establishing the fatigue life of materials, components, and systems is both time consuming and costly. As 

a result, conclusions regarding fatigue life are often inferred from a statistically insufficient number of 

physical tests. 

4.1 Comparing Life Results 

From the results of the Monte Carlo simulation of the L10 life of aluminum alloy AL6061 (Figs. 4 

and 5), there is an even (50 percent) probability that the resultant L10 test life for any randomly selected 

test group of the same material will be greater or less than its established baseline life. As the number of 

specimens in a test group increases, the magnitude of the variation in life decreases. From these 

observations rules can be established to compare and distinguish tests of identical or different aluminum 

alloys either from one another, two or more sources, or made from different processing methods. The 

following rules are suggested: 

(1) If the L10 life of the material is between the Maximum and Minimum L10 life variations, there can be 

no conclusion that there is a significant difference between the new material and the baseline material 

regardless of the ratio of the L10 lives. The new material is acceptable for its intended application (Fig. 6(a)). 
NASA/TM—2012-217669 8

Figure 6.—Rules for comparing material specimen fatigue life results to AL6061 (based on Fig. (5)). (a) New material 

is acceptable. (b) New material is not acceptable. (c) New material is superior to baseline material. (d) New 

material is acceptable. 

(2) If the L10 life of the new material is less than the minimum variation, the new material is not 

acceptable for its intended application (Fig. 6(b)). 

(3) If the L10 life of the new material is greater than the maximum variation, there exists a significant 

difference between the materials with the new material being superior for its intended application (Fig. 6(c)). 

(4) If the L10 life of the new material is greater than the baseline life (zero variation) but less than the 

maximum variation, there can be no conclusion that there is a significant difference between the new 

material and the baseline material regardless of the ratio of the L10 lives. The new material is acceptable 

for its intended application (Fig. 6(d)). 

4.2 Experimental Life Results 

With a reasonable understanding of how the Monte Carlo simulations responded to variations in 

inputs, it is possible to compare fatigue data for aluminums AL7075 and AL2024 with that of aluminum 

AL6061. Ten (10) each rotating notched rod fatigue tests were conducted for AL7075 and AL2024, 

respectively. Baseline verification tests were conducted with 10 additional specimens of aluminum 

AL6061. These tests were conducted at a bending stress of 157.2 MPa (22.8 ksi), room temperature and a 

speed of 7500 rpm. Each test was conducted to failure; there were no suspensions or censoring of the data. 

Using Equation (3) the resultant L10 lives of the 3 materials were compared to the L10 life from the baseline 

tests of the 357 test samples of AL6061 (L10 = 36 354 cycles). The results are summarized in Table 1.

Referring only to the resultant L10 lives based upon 10 test specimens for each aluminum alloy, the 

materials would be intuitively ranked in the order of their relative L10 lives. That is, AL2024 first, AL7075 

second, and AL6061 third. There is a strong suggestion that the AL7075 has a 30 percent higher life than 

AL6061. However, referring to the baseline tests of AL6061 and to the Monte Carlo results (Fig. 4), and 

using the criteria of Figure 5 to compare results, it can be reasonably concluded that AL2024 is 

statistically superior to the other 2 alloys. However, there is no statistical difference between AL6061 
NASA/TM—2012-217669 9

and AL7075. The variability of test results based on small sample sizes can be greater than actual 

differences, if any, that exists between materials and can result in erroneous conclusions. It is worth 

noting that the L10 life of even the baseline material (AL6061) differs between that determined for only 

10 trials (Table 1) and the baseline of 357 trials by 35 percent. 

4.3 Minimum Population Size and Probabilistic Variability 

Engineers and designers tend to consider fatigue lives as absolutes, when actually there is 

probabilistic variability associated with reported values. There is a significant range between the 

maximum and minimum 90-percent bounding fatigue lives for small population sizes of 2 to 10, where 

the percent variation exceeds 100-percent (Fig. 5). For population sizes ranging from 14 to 25, the range 

between the maximum and minimum L10 life continues to decrease, but nonetheless has differences of 

75 to 100 percent variation. It is not until 30 to 100 samples are tested that these differences in minimum 

life begin to level at 40 to 60 percent variation. 

The minimum life locus is most likely used in conservative designs and was thus focused on herein. 

Again, there is a significant variation between the simulated and calculated minimum lives for small 

population sizes of 2 to 6. This difference is nearly 100 percent. The difference decreases significantly 

with increasing sample size. Over the range of 10 to 28 specimens, the simulated minimum life differs 

from the known calculated life by about 50 percent. At approximately 30 specimens, the difference is less 

than 30-percent variation, and at 70 specimens the difference is less than a 20-percent variation. It should 

be noted that even for 200 test specimens, there is still a range observed between the upper and lower 

90-percent bounding curves. This range is within 15 percent. 

Vlcek, Hendricks, and Zaretsky (Ref. 6) performed a similar Monte Carlo simulation on a bearing 

system comprising three components. While the life of multiple components had to be taken into account, 

similar trends as those evident in Figures 4 and 5 were observed. Similar observations were also made by 

Zaretsky, Savage, Lewicki, and Vlcek (Ref. 8), where a Monte Carlo simulation was used to determine 

the 90-percent bounds on the L10 life of a 21 component gear box. 

The number of fatigue tests performed is dependent upon the probabilistic variability that can be 

tolerated. Figure 5 helps establish the range in variability associated with fatigue life and test population 

size. For most engineering applications where less than 30 percent variability can be tolerated in the 

maximum and minimum values, at least 30 to 35 test samples are necessary. Design decisions can be 

made based upon smaller population sizes, as long as the inherent increase in variability in the life value 

and the associated risk are recognized and accounted for. 

5.0 Summary of Results 

Fatigue life is probabilistic and not deterministic. Experimentally establishing the fatigue life of 

materials, components, and systems is both time consuming and costly. As a result, conclusions regarding 

fatigue life are often inferred from a statistically insufficient number of physical tests. A proposed 

methodology for comparing life results as a function of variability due to Weibull parameters, variability 

between successive trials, and variability due to size of the experimental population is presented. Using 

Monte Carlo simulation of randomly selected lives from a large Weibull distribution, the variation in the 

L10 fatigue life of aluminum alloy AL6061 rotating rod fatigue tests was determined as a function of 

population size. These results were compared to the L10 fatigue lives of small (10 each) populations from 

AL2024, AL7075 and AL6061. The following results were obtained: 

1. For aluminum alloy AL6061, a simple algebraic relationship was established for the upper and 

lower L10 fatigue life limits as a function of the number of specimens failed. 

2. For most engineering applications where less than 30 percent variability can be tolerated between 

the maximum and minimum L10 fatigue life values, at least 30 to 35 test samples are necessary. Design 

decisions can be made based upon smaller population sizes, as long as the inherent increase in variability 

in the life value and the associated risk are recognized and accounted for. 
NASA/TM—2012-217669 10

3. The variability of test results based on small sample sizes can be greater than actual differences, if 

any, that exists between materials and can result in erroneous conclusions. Based on a test population of 

10 specimens each, the fatigue life of AL2024 is statistically longer than AL6061 and AL7075. However, 

there is no statistical difference between the fatigue lives of AL6061 and AL7075 even though AL7075 

had a fatigue life 30 percent greater than AL6061. 

4. The maximum and minimum L10 life limits determined experimentally for Aluminum alloy 

AL6061 rotating rod fatigue specimens for test population sizes of 3, 5, and 9 correlated with those 

obtained from the Monte Carlo simulation. 

References 

1. Zaretsky, E.V., “Design for Life, Plan for Death,” Machine Design, 66(15), Aug. 8, 1994, pp. 55–59. 

2. Zaretsky, E.V., STLE Life Factors for Rolling Bearings, STLE SP34, STLE, Park Ridge, IL, 1992. 

3. Weibull, W. (1939), “A Statistical Theory of the Strength of Materials,” 

Ingenioersvestenskapsakademiens. Handl., 151. 

4. Weibull, W. (1939), “The Phenomenon of Rupture of Solids,” Ingeniorsvetenkapsakademiens. 

Handl., 153. 

5. Weibull, W. (1951), “A Statistical Distribution Function of Wide Applicability,” J. Appl. Mech. 

Trans. ASME, 18(3), pp. 293297. 

6. Vlcek, B.L., Hendricks, R.C., and Zaretsky, E.V., (2003), “Determination of Rolling-Element Fatigue 

Life from Computer Generated Bearing Tests,” STLE Tribology Trans., 46 (4), pp. 479–493. 

7. Johnson, L.G., The Statistical Treatment of Fatigue Experiments, Elsevier Publishing Co., 

Amsterdam, The Netherlands, 1964. 

8. Zaretsky, E.V., Savage, M., Lewicki, D., and Vlcek, B., “Determination of Turboprop Gearbox 

System Fatigue Life and Reliability,” STLE Tribology Trans., 50 (4) pp. 507–516, Oct. 2007. 

9. Zaretsky, E.V., Tribology for Aerospace Applications, STLE SP37, STLE, Park Ridge, IL, 1997. 

10. Mooney, C.Z., Monte Carlo Simulation, Sage Publications, Thousand Oaks, CA, 1997. 
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4. TITLE AND SUBTITLE

Test Population Selection From Weibull-Based, Monte Carlo Simulations of Fatigue Life 

5a. CONTRACT NUMBER

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6. AUTHOR(S)

Vlcek, Brian, L.; Zaretsky, Erwin, V.; Hendricks, Robert, C. 

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5f. WORK UNIT NUMBER

WBS 457280.02.07.03.04.02 

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National Aeronautics and Space Administration 

John H. Glenn Research Center at Lewis Field 

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Subject Category: 37 

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14. ABSTRACT

Fatigue life is probabilistic and not deterministic. Experimentally establishing the fatigue life of materials, components, and systems is both 

time consuming and costly. As a result, conclusions regarding fatigue life are often inferred from a statistically insufficient number of 

physical tests. A proposed methodology for comparing life results as a function of variability due to Weibull parameters, variability between 

successive trials, and variability due to size of the experimental population is presented. Using Monte Carlo simulation of randomly selected 

lives from a large Weibull distribution, the variation in the L10 fatigue life of aluminum alloy AL6061 rotating rod fatigue tests was 

determined as a function of population size. These results were compared to the L10 fatigue lives of small (10 each) populations from 

AL2024, AL7075 and AL6061. For aluminum alloy AL6061, a simple algebraic relationship was established for the upper and lower L10

fatigue life limits as a function of the number of specimens failed. For most engineering applications where less than 30 percent variability 

can be tolerated in the maximum and minimum values, at least 30 to 35 test samples are necessary. The variability of test results based on 

small sample sizes can be greater than actual differences, if any, that exists between materials and can result in erroneous conclusions. The 

fatigue life of AL2024 is statistically longer than AL6061 and AL7075. However, there is no statistical difference between the fatigue lives 

of AL6061 and AL7075 even though AL7075 had a fatigue life 30 percent greater than AL6061. 

15. SUBJECT TERMS

Fatigue failure; Weibull analysis; Monte Carlo simulation; Aluminum AL6061; Aluminum AL7075; Aluminum AL2024; 

Probabilistic failure; Variability 

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