Erwin V. Zaretsky, Jonathan S. Litt, and Robert C. Hendricks

Glenn Research Center, Cleveland, Ohio

Sherry M. Soditus

United Airlines Maintenance, San Francisco, California

Determination of Turbine Blade Life From 

Engine Field Data

NASA/TP—2013-217030

April 2013
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Erwin V. Zaretsky, Jonathan S. Litt, and Robert C. Hendricks

Glenn Research Center, Cleveland, Ohio

Sherry M. Soditus

United Airlines Maintenance, San Francisco, California

Determination of Turbine Blade Life From 

Engine Field Data

NASA/TP—2013-217030

April 2013

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
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at the NASA Glenn Research Center.

Level of Review: This material has been technically reviewed by an expert reviewer(s). 
NASA/TP—2013-217030 1

Determination of Turbine Blade Life From Engine Field Data

Erwin V. Zaretsky, Jonathan S. Litt, and Robert C. Hendricks

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Sherry M. Soditus

United Airlines Maintenance

San Francisco, California 94128

Summary

It is probable that no two engine companies determine the 

life of their engines or their components in the same way or 

apply the same experience and safety factors to their designs. 

Knowing the failure mode that is most likely to occur 

minimizes the amount of uncertainty and simplifies failure and 

life analysis. Available data regarding failure mode for aircraft 

engine blades, while favoring low-cycle, thermal-mechanical 

fatigue (TMF) as the controlling mode of failure, are not 

definitive. Sixteen high-pressure turbine (HPT) T–1 blade sets 

were removed from commercial aircraft engines that had been 

commercially flown by a single airline and inspected for 

damage. Each set contained 82 blades. The damage was 

cataloged into three categories related to their mode of failure: 

(1) TMF, (2) Oxidation/erosion (O/E), and (3) Other. From 

these field data, the turbine blade life was determined as well 

as the lives related to individual blade failure modes using 

Johnson-Weibull analysis. A simplified formula for calculating turbine blade life and reliability was formulated. The L10

blade life was calculated to be 2427 cycles (11 077 hr). The 

resulting blade life attributed to O/E equaled that attributed to 

TMF. The category that contributed most to blade failure was 

“Other.” If there were no blade failures attributed to O/E and 

TMF, the overall blade L10 life would increase approximately 11 

to 17 percent.

1.0 Introduction

The service life of an aircraft gas turbine engine is based on 

deterministic calculations of low-cycle fatigue (LCF) and 

previous field experience with similar engines. It is probable 

that no two engine companies determine the life of their engines 

in the same way or apply the same experience and safety factors 

to their designs (Ref. 1). Davis and Stearns (Ref. 2) and Halila, 

Lenahan, and Thomas (Ref. 3) discuss the mechanical and 

analytical methods and procedures for turbine engine and highpressure turbine (HPT) design. The designs of the engine 

components are based on life predictions by using material test 

curves that relate life in cycles or time (hr) as a function of 

stress. Six criteria for failure were presented: (1) Stress rupture; 

(2) Creep; (3) Yield; (4) LCF; (5) High-cycle fatigue (HCF); and

(6) Fracture mechanics. Not mentioned as probable failure modes

and/or cause for removal of rotating engine components in 

References 2 and 3 are oxidation, corrosion, and erosion (wear).

Turbine blade metal temperatures frequently reach 1040 to 

1090 °C (1900 to 2000 °F), only a few hundred degrees below 

the melting point of the alloys used. Only because of 

oxidation-protective coatings and internal forced cooling is it 

possible for metals to be used under such harsh conditions. All 

commercial aircraft gas turbine engines use some form of 

nickel- or cobalt-base superalloy that has been intentionally 

strengthened and alloyed to resist high stresses in a hightemperature oxidizing environment (Ref. 4).

Aircraft engine turbine blades are not life-limited parts; that 

is, they can be used until they are no longer repairable, unlike 

limited life parts that must be removed after a specified 

amount of time or cycles, even if they appear new. The blades 

undergo regular inspections that result in no action, repair, or 

removal for cause. In this paper, a blade is considered failed 

when it is no longer fit for service and must be either repaired 

or replaced. 

It is believed that the primary failure mechanism in turbine 

blades is thermal-mechanical fatigue (TMF). TMF cracks 

usually appear along the leading or trailing edge of the firststage HPT T–1 blade (Ref. 5). Also, because the turbine 

blades are exposed to highly corrosive and oxidizing 

combustion gases, the loss of metal by scaling, spalling, and 

corrosion can cause rapid failure. 

Turbine blade materials have creep-rupture resistance to 

minimize creep failure at high speed and temperature for 

extended periods. Initially, the time to removal of these blades 

is determined by a creep criterion that is deterministic or is not 

assumed to be probabilistic. Material test data are used to 

predict rupture life based on calculated stress and temperature. 

This criterion is dependent on time exposure at stress and 

temperature (Ref. 1).

Blade coating life is another time-limiting criterion for 

removal and repair. The blades usually are removed when the 

engine is removed from service for other reasons, and, as 

necessary, the remaining coating is removed by chemical 

stripping or machining and is replaced. The coating life usually 

does not dictate blade replacement, only repair (Ref. 1).

Besides the time-life limitation of creep, the limiting time 

for blade replacement is HCF life. As with LCF, HCF is 
NASA/TP—2013-217030 2

probabilistic. The blades are subject to vibratory stresses 

combined with mechanical stresses from centrifugal loads, gas 

aerodynamic loads, and thermal loads (Ref. 1).

The failure modes for each blade in a turbine blade set are 

competitive. Knowing the blade failure mode that is most 

likely to occur minimizes the amount of uncertainty and 

simplifies failure and blade life analysis. Available data 

regarding failure mode, although favoring low-cycle, TMF as 

the controlling mode of failure, are not definitive.

There are several other major contributors besides the 

competing failure modes that contribute to turbine blade set 

life uncertainty. First the data are quantal-response data. This 

means that the data are either censored on the left (sometime 

after failure occurs) or censored on the right (failure has not 

occurred by a defined time). This situation arises when each 

blade is inspected only once and is determined to have failed 

or not failed. For turbine blade data, this type of information 

can be useful for reliability studies if the failed blades can be 

clustered by age (time to failure) at inspection (and the range 

of ages is large relative to the part life) (Ref. 6).

In 1939, Weibull (Refs. 7 to 9) is credited with being the 

first to suggest a reasonable way to estimate fracture strength 

with a statistical distribution function. He also applied the 

method and equation to fatigue data based on small sample 

(population) sizes. Johnson (Ref. 10) while with the GM 

Research Center in the 1950s and 1960s is credited with 

coming up with a practical engineering analysis based on the 

Weibull distribution function (Refs. 7 to 9). Johnson, using the 

Weibull distribution function to evaluate fatigue data, provides 

a means to evaluate censored data and to extract from these 

data the lives of the individual components that affect the 

system life. 

In view of the aforementioned, it becomes the objectives of 

the work reported herein to (1) determine turbine blade life 

from turbine engine field data using Johnson-Weibull analysis, 

(2) determine the turbine blade life related to individual blade 

failure modes, and (3) provide a simplified formula for 

determining turbine blade life from field data for engine 

turbine blade sets.

Nomenclature

e Weibull slope

F probability of failure, fraction or percent

Fm mean probability of failure

k life at operational condition, number of stress 

cycles or hr

L life, number of stress cycles or hr

Lβ characteristic life or life at which 63.2 percent of 

population fails, 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

Lavg average life, total time divided by total number of 

components, number of stress cycles or hr

LM mean time to removal

Lm mean life of a population, number of stress cycles 

or hr

M total number of stress cycles at operating condition 

where M = pm

N total number of engine operational condition 

changes over flight profile

Neng total number of engines in overhaul in field data set

m number of stress cycles per interval

n number of blades in a set

nblade number of blade failures within a specified blade 

set of the population Neng

p number of intervals

S probability of survival, fraction or percent

t time to blade set removal, cycles or hr

V volume, m3 (in.

3

)

X fractional percent of components or blades failed 

from specific cause, or operating variable 

(Appendix A)

Xn fractional percent of time at operational condition

σ stress, Pa (psi) 

σu location parameter below which stress no failure 

will occur, Pa (psi)

σβ characteristic stress at which 63.2 percent of 

population fails, Pa (psi)

Subscripts

avg designation of average life

blade turbine blade of a blade set 

blade set turbine blade set

eng engine

fm cataloged failure mode

i i-th component out of n

k k-th engine operational condition within flight 

profile 
NASA/TP—2013-217030 3

m designation of mean life or probability of survival 

at mean life

n n-th component of a set of blades; number of 

blades in blade set 

mis mission or operational life

ref reference life or reference probability of survival

sys system probability of survival or system life

V volume

β designation of characteristic life

1,2 bodies 1, 2, etc.; failure mode 1, 2, etc.

2.0 High-Pressure Turbine T–1 Blade Sets

2.1 Engine Operation and Repair

When a new aircraft engine is introduced into an airline fleet, 

one of the first questions asked is what will be the average time 

(hr) between overhaul or refurbishment of the high-pressure 

turbine (HPT) T–1 blades. Typically, for a new engine program 

the airlines bring the engines in early for overhaul, for example, 

approximately at 10 000 hr. As the airlines gain experience and 

confidence with an engine type, the time to refurbishment is 

increased for first-run engines, for example, at 22 000 hr. After 

refurbishment, second-run engines probably get around 

15 000 hr on the wing. The hot section is typically overhauled 

when the engine is removed from service (Ref. 1).

The typical hour-to-cycle ratio depends on the airline 

operator. Short-haul airline operation typically runs between 1 

to less than 4 hr/cycle. Long-haul, coast-to-coast airline 

operation in the continental United States typically runs 

between 4 to 6 hr/cycle. For other airline operations, the average 

can be 6 to 13 hr/cycle. These numbers play an important part in 

the overhaul process. It is expected that for the shorter cycle 

engines there will be more deterioration on the hot-section parts 

on the engines that have a shorter time cycle, implying that the 

deterioration is cycle dependent rather than time dependent. 

When an aircraft engine is removed from service for cause 

and shipped to the refurbishment shop, the engine and the 

performance of its individual modules are evaluated and the 

root cause of removal determined. If the engine is removed for 

performance or hardware deterioration or major part failure, the 

engine will be, in most cases, completely broken down into 

modules: for example, compressor, turbine, auxiliary gearbox, 

and so forth. Each module will then be refurbished (Ref. 1).

2.2 HPT T–1 Blades

A photograph of the blade type studied in this report is 

shown in Figure 1(a). The blade is made from a single-crystal 

nickel-based alloy and plated using plasma vapor deposition 

(PVD) to provide oxidation and corrosion resistance. The 

blade material and coating chemistries are given in Table 1

(Ref. 11). The blade section is approximately 71 mm (2.8 in.) 

in height and has a cord length at the tip of approximately 

37 mm (1.46 in.). The height from the blade root to the blade 

tip is approximately 118 mm (4.65 in.). The blade weighs 

approximately 277 g (0.611 lb). There are 82 blades in a T–1 

blade set for the particular engine application studied. 

A total of 82 blades are inserted around a T–1 turbine disk. 

The resulting tip-to-tip diameter of the blades is approximately 

0.93 m (36.5 in.). The blades are spun at a speed of 

approximately 9126 rpm during cruise, or 84.5 percent of the 

maximum speed of 10 800 rpm. Loading on the blades is due 

to centrifugal load, thermal loads from heating of the blades, 

aerodynamic loads from impingement of the hot combustion 

gases against the blade, and vibratory loads due to blade 

rotation. A load and stress analysis of these blades was beyond 

the scope of this paper.

An engine is borescoped periodically to determine its 

health. It is not uncommon to find that the HPT blades 

deteriorate in service because of the extreme operating 

conditions they encounter. Even when an engine is operating 

properly, it can experience some form of hardware deterioration of the HPT T–1 blades. Such a failed blade is shown in 

Figure 1(b). At the time of removal this blade had run 

15 000 hr (2700 cycles). The condition is typical for this time 

period.

2.3 Blade Failure Criteria

For the purpose of this report, blade failure is defined as the 

blade being no longer fit for its intended purpose but still 

capable of functioning for a limited time until being removed 

from service. Depending on the condition of the deterioration, 

an engine may be allowed to remain in service on a decreasedcycle inspection interval until it is determined that the 

deterioration is beyond limits (or its exhaust gas temperature 

(EGT) margin is too small) and the engine must be removed 

from service (Ref. 1). Causes of blade failure and/or removal 

are as follows:

(1) Creep (stress rupture)

(2) Yield

(3) Thermal-mechanical fatigue (TMF)

(a) Low-cycle fatigue (LCF)

(b) High-cycle fatigue (HCF)

(4) Fracture mechanics (flaw initiated crack)

(5) Fretting (wear and fatigue)

(6) Oxidation

(7) Corrosion

(8) Erosion (wear)

(9) Foreign object damage (FOD)

(10) Wear (blade tip rub)
NASA/TP—2013-217030 4

Figure 1.—Comparison of unfailed and failed T–1 turbine blades used in study. (a) Example of unfailed T–1 

turbine blade. (b) Example of failed T–1 turbine blade.

TABLE 1.—T–1 TURBINE BLADE MATERIAL CHEMISTRY (REF. 11)

Chemical element,

wt%

Density,

kg/m3

(lb/in.3)

Ni Cr Ti Mo W Re Ta Al Co Hf Si Y 8.94×103

Bal. 5 0 2 6 3 8.7 5.6 10 0.1 --- --- (0.323)

Overlay (coating)

x1 x3 --- --- --- --- --- x4 x2 x5 x6 x7 ----------

x elements of proprietary composition, x1 > x2 ...etc.

For post-operation failure inspection of blade sets, the blade 

failures were cataloged under three categories:

(1) Thermal-mechanical fatigue (TMF)

(2) Oxidation/erosion (O/E)

(3) Other (creep, yield, fracture mechanics, fretting, 

corrosion, FOD, and wear)

The blades removed from service can generally be repaired or 

refurbished two or more times. The blades can be stripped of 

their coatings and recoated. There is a minimum wall thickness 

and aerodynamic shape that must be met before the blade can be 

recoated. They can have minor blend repairs and new abrasive 

tips installed, and the roots can be shot peened. Of those T–1 

blades that are scrapped, approximately 90 percent are due to 

under-platform stress corrosion. 
NASA/TP—2013-217030 5

3.0 Procedure

Sixteen high-pressure turbine (HPT) T–1 blade sets were 

removed from commercial aircraft engines that had been 

commercially flown by a single airline. These engines were 

brought to the maintenance shop for refurbishment or overhaul. 

The blades on each HPT T–1 blade set were removed and

inspected for damage. The damage was cataloged into three 

categories related to their mode of failure:

(1) Thermal-mechanical fatigue (TMF)

(2) Oxidation/erosion (O/E)

(3) Other

The technician had a preset order in which to look for 

failure modes. The blades were first inspected for TMF. If 

cracks were evident on the blade, and even if other failure 

modes were also evident, the cause for removal was cataloged 

as TMF. The blades not failed from TMF were inspected 

for O/E. As with those blades cataloged as being failed by 

TMF, those blades that exhibited O/E damage were so 

cataloged even where damage from other failure modes was

manifested on the blade. The blades not failed for TMF or

O/E were examined for damage for the other causes discussed 

previously. These other causes were not separately identified 

and were categorized and cataloged as “Other.” 

A list of the engine blade sets, their time at removal, their 

respective number of failures, and their failure modes are 

given in Table 2. Of a total of 1312 blades contained in the 

Neng = 16 blade sets, 111 were considered to have failed, or

approximately 8.5 percent of the population. Although each of 

these blade sets was to comprise all new blades when installed 

in the engine, three blade sets had a mix of new blades with 

previously run (older) blades. The failures that were reported 

for the mixed blade sets did not distinguish between the older 

and newer blades.

Ideally, the time to failure for each blade in a set should be 

known. More specifically, the time at which the first blade 

fails in a set should be known based on the assumption that at 

the time of the first failure, the entire set is no longer fit for its 

intended purpose. For these type data, these times are not 

available and will have to be estimated. However, once the 

time to first failure in a set is determined or estimated, the 

distributive lives of the blades can be determined as well as 

the resulting lives from each failure mode.

TABLE 2.—DATA SET FOR T–1 TURBINE BLADE SETS INCLUDING ESTIMATED TIME

TO FIRST BLADE FAILURE IN A SET AND CAUSES OF FAILURE

[Number of turbine blades in set, 82.]

Engine

number

Time of removal 

of blade set

Number 

of failures

Observed failure mode Estimated time 

to first blade 

failure,

cycles

hr cycles Oxidation/erosion

(O/E)

Thermal-mechanical

fatigue 

(TMF)

Other

Number of blades failed

1B 5 898 1327 1 --- --- 1 1327

2B 7 318 1404 5 3 2 --- 1017

3B 8 188 1675 2 --- --- 2 1443

4B a8 333 1747 3 --- 1 2 1391

5B 9 049 1827 4 3 --- 1 1379

6B 8 717 1843 41 --- 1 40 886

7B 9 600 1924 10 --- 10 --- 1228

8B 10 113 2043 4 1 1 2 1542

9B 7 770 2047 7 7 --- --- 1394

10B 10 675 2091 2 --- --- 2 1801

11B 7 690 2115 4 1 1 2 1596

12B 11 051 2175 2 --- 1 1 1873

13B 10 398 2184 12 4 --- 8 1348

14B 11 614 2292 5 1 1 3 1660

15B 10 238 2295 3 3 --- --- 1827

16B 14 083 2847 6 --- 4 2 1994

Total 111 23 22 66

Lavg 9 421 1990 1482 a

Estimated. 
NASA/TP—2013-217030 6

4.0 Statistical Analysis 

4.1 Weibull Analysis

In 1939, Weibull (Refs. 8 and 9) is credited with being the 

first to suggest a reasonable way to estimate fracture strength 

with a statistical distribution function. He also applied the 

method and equation to fatigue data based on small sample 

(population) sizes. The probability distribution function 

identified by Weibull is as follows:

ln where 0 ; 0 1 1 ln ln < < ∞ < < 













  =



 





β

L S

L

L

e

S (1)

This form of Equation (1) is referred to as the twoparameter Weibull distribution function. The derivation of this 

equation is given in Reference 12, and in Appendix A.

The variable S is the level of survivability being considered. 

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

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

which the fraction (1 – S) of samples have failed. In the case 

of S equaling 0.9 (90 percent), L is the life at which 10 percent 

of the samples have failed—typically referred to as the L10

life. Lβ is the characteristic life of the material, defined as the 

life at which 63.2 percent of the samples have failed. Finally, e 

is the Weibull parameter or slope, which is an indicator of the 

scatter or distribution in the data—the larger the slope, the 

smaller the amount of scatter. 

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 lnL, the natural log of the time or cycles to failure. 

The tangent of the line is the Weibull parameter or slope e,

which is indicative of the shape of the cumulative distribution 

or the amount of scatter of the data. A Weibull slope e of 1.0 

is indicative of an exponential distribution of the data, 2.0 is a 

Rayleigh distribution, and 3.57 is approximately that of a 

normal distribution of data. For convenience, the ordinate is 

graduated as the statistical percent of components failed.

There are many examples of the use of the Weibull distribution function to determine the life and strength of materials, 

structural components, and machines. The first use of the 

Weibull distribution function outside of Weibull’s original 

reported work (Refs. 8 and 9) was by Lundberg and Palmgren

(Ref. 13) for predicting the life of ball and roller bearings.

Burrow et al. (Ref. 14) used Weibull analysis to determine 

the reliability of tensile strength measurements on dental 

restorative materials. Ellis and Tordonato (Ref. 15) used 

Weibull analysis in their failure analysis and life assessment 

studies of boiler tubes. The fatigue life associated with 

corrosion fatigue cracking of welded tubing was predicted.

Figure 2.—Weibull plot where (Weibull) slope of 

tangent of line is e; probability of survival, Sβ, 

is 36.8 percent at which L = Lβ or L/Lβ = 1.

Tomimatsu, Kikuchi, and Sakai (Ref. 16) used Weibull 

analysis in their determination of the fracture toughness of two 

steels used in reactor pressure vessel fabrication. Weibull 

analysis and dynamic fatigue slow-crack-growth parameters

were used by Osborne, Graves, and Ferber (Ref. 17) to 

demonstrate a significant difference in the high-temperature 

behavior of two silicon nitrides (SN–88 and NT164). Ostojic and 

Berndt (Ref. 18) demonstrated that Weibull parameters such as 

slope and characteristic life were meaningful parameters when 

determining the variability of bond strengths of thermally 

sprayed coatings.

Holland and Zaretsky (Ref. 19) used Weibull statistics to 

determine the fracture strength of two different batches of cast 

A357–T6 aluminum. The mean fracture strengths for the two 

batches were found to differ by an insignificant 1.1 percent. 

However, using a Weibull analysis they determined at the 

99.9999 percent probability of survival (one failure in a million) 

that the actual fracture strengths differed by 14.3 percent.

Weibull analysis can also be used to evaluate preventive 

maintenance practices. Williams and Fec (Ref. 20) studying 

reconditioned railroad roller bearings determined with Weibull 

analysis that the current practice of inspecting bearings at 

200,000 miles was an acceptable practice. Summers-Smith

(Ref. 21) applied Weibull analysis to the service life obtained 

from maintenance records that identified the cause of failure of 

a hydrodynamically lubricated thrust bearing and a rollingelement bearing, and increased production reliability. Similarly, 

Vlcek et al. (Ref. 22) used Weibull analysis to rank the relative 

fatigue lives of PVC coatings used in a printing process. The 

fatigue life of one PVC coating over another was demonstrated 

using L10 lives, and the ratio of the L10 life of a developed PVC 

coating to the original was found to be 2.3. 
NASA/TP—2013-217030 7

The method of using the Weibull distribution function for 

data analysis for determining component life and reliability 

was developed and refined by Johnson (Ref. 10). The Johnson

(Ref. 10) method was used to analyze the data reported herein.

4.2 Strict-Series System Reliability Blade Life

System life prediction or the life of a single blade set can be 

determined using strict-series system reliability derived in 

Appendix B (Refs. 12 and 23). The reliability (or probability 

of survival), S, and the probability of failure, F, are related by 

F = (1 – S). For a given time or blade set life, the reliability of 

an individual blade set Ssys of independent blades making up 

the blade set is the product of the independent reliabilities of 

each individual blade in the blade set Si (i = 1, 2, ..., n), as 

shown in Equation (2):

n S = S × S ×⋅⋅⋅× S sys 1 2 (2)

where Ssys is the blade set reliability, and S1, S2...Sn is the 

reliability of each blade in the blade set. If all components

have the same reliability S1 = S2...= Sn (as is assumed here), 

then Equation (2) reduces to 

n

n S = S sys (3)

where n is the number of blades in the blade set. For our case, 

each engine blade set has a total of 82 blades. Thus, for one 

blade set, Equation (3) can be written as 

82

sys n S = S (4)

From Equation (2), the lives of each of the blades at a 

specified reliability can be combined to determine the 

calculated system Lsys life of the set using the two-parameter 

Weibull distribution function (Eq. (1)) for the blades 

comprising the system and strict-series system reliability

(Ref. 13) as follows:





















= + + n

e

n

e e e L L L L

1 1 1 1

1 2

1 2 sys

 (5)

where Lsys is the life of individual blade set and L1, L2...Ln are 

the lives of the individual blades. The derivation for 

Equation (5) is given in Appendix B (Ref. 12).

In this work, the 82 blades in a set are each assumed to have 

the same life, L, where L1 = L2 = ...= Ln and Weibull slope, e = 

e1 = e2 = ... = en. Accordingly, Equation (5) can be written for 

the 82 blades in a single blade set as 





















= n n e e L L

1 82

sys

(6)

The calculated system life is dependent on the resultant 

value of the system Weibull slope e.

4.3 Linear Damage Rule

The blade set life is calculated using Equation (6) for each 

operating condition of its engine operating profile. In 

Appendix C is a representation of a short-duration flight profile 

(Fig. C.1). In order to obtain the operational life of the blade set, 

the resulting system lives for each of the operating conditions 

(illustrated in Fig. C.1) are combined in Equation (7) using the 

linear damage (Palmgren-Langer-Miner) rule discussed in 

Appendix C (Refs. 24 to 27) where k Lsys is the life for 

condition k and Xk is the time fraction spent at condition k, 

(k = 1, 2, ..., N).

k N L

X

L

X

L

X

L

X

L

k N

sys sys sys

2

sys

1

mis

... 1

1 2

= + + + (7)

It is assumed that each of the blade sets in Table 2 have the 

same operational cycle. N = 16 changes in engine operational 

conditions over the flight profile for Figure C.1. (Operational 

cycle N is not to be confused with Neng , the number of engine 

overhaul blade repairs of Table 2, which also has 16 entries.) 

The most damaging condition is at takeoff with the cruise 

condition being the dominant time on the engine.

5.0 Results and Discussion

Sixteen high-pressure turbine (HPT) T–1 blade sets 

(Neng = 16) were removed from commercial aircraft engines that 

had been commercially flown by a single airline. These engines 

were brought to the maintenance shop for refurbishment or 

overhaul. The blades for these turbines were manufactured from 

a single-crystal nickel alloy whose chemical composition 

together with the chemical composition of the blade coating are 

given in Table 1. The blades on each HPT T–1 blade set were 

removed and inspected for damage. The damage was cataloged 

in three categories related to their mode of failure:

(1) Thermal-mechanical fatigue (TMF)

(2) Oxidation/erosion (O/E)

(3) Other (creep, yield, fracture mechanics, fretting, 

corrosion, FOD, and wear)

The blades were first inspected for TMF. If cracks were 

evident on the blade even if other failure modes were also 

evident, the cause for removal was cataloged as TMF. The 

remaining blades were inspected for O/E. As with those blades 

cataloged as being failed by TMF, those blades that exhibited 

O/E damage were so cataloged even where damage from other 

failure modes was manifested on the blade. The remaining 
NASA/TP—2013-217030 8

blades were examined for damage for the other causes. These 

other causes were not identified and categorized and cataloged 

as “Other.” The time of removal of the blade sets together 

with the cataloged failure mode of those blades in each set that 

failed is summarized in Table 2. 

5.1 Field Data Analysis

Weibull plots of these data for Neng = 16 based on the time 

of removal in flight hours and flight cycles are shown in 

Figure 3.

Theoretically, the time to failure of a turbine blade set is the 

time at which the first blade in the set fails regardless of the 

cause. This is analogous to a weak link in a chain. The chain is 

failed when the first link fails. The problem is that we do not 

know when the first blade failed nor do we know for sure the 

time when the most recent blade failure in a particular blade set 

has occurred. For purpose of the analysis, assume that the most 

recent blade that has failed in a set fails at the time the set is 

removed from the engine (Table 2, Time of removal, cycles). 

From this assumption, we need to determine the time at which 

the first blade has failed. We assumed the following scenario: 

The set starts out with all new (unused) blades.

The reliability, S(t), of the last blade that fails is 

S(t) = 1 – F(t) (8)

In this calculation, the reliability, S(t) was estimated from 

the median rank of the failures, according to Equation (9)

(Ref. 21), where i is the failure number and n is the number of 

individual blades in a set (in this case, n = 82):

0.4

0.3

+

− ≈ ni F (9)

Although the time of the last failure in a blade set cannot be 

known with any reasonable certainty, it can be assumed to have 

occurred at or shortly before the blade set is removed from 

service. The time of removal of a blade set is obtained from 

Table 2. Also, the number of failures in each engine blade set is 

given in Table 2. As an example, for engine number 2B, the time 

of removal L is 1404 cycles and there are nblade = 5 failures within 

the blade set. 

It is assumed for purposes of calculation that when the first 

blade failure in a blade set occurs, the blade set is no longer fit 

for its intended purpose even though it is still functioning. 

Accordingly, estimation of the time of the first failure in the 

blade set is a precondition for determining turbine blade life. In 

order to accomplish this task, it is first necessary to determine 

the probability of failure out of a large blade population that the 

most recent failure in the blade set represents. For engine 

number 2B, solving for F in Equation (9), where i = 5 and 

n = 82, 

F = (5 – 0.3)/(82 + 0.4) = 0.057 (10a)

From Equation (8),

S = 1 – 0.057 = 0.943 (10b)

Figure 3.—Weibull plots of turbine blade set removal time for high-pressure turbine T–1 blade sets from field 

data. Number of T–1 turbine blades to a set, 82. (a) Flight hours (coefficient of determination, r2 = 0.9583). 

(b) Flight cycles (r2 = 0.954).
NASA/TP—2013-217030 9

From the field data (Table 2, engine 2B), the time of 

removal of the blade set is 1404 cycles. From the Weibull plot 

of Figure 3(b), a Weibull slope e equal to 5.984 is obtained. 

These Weibull parameters are substituted in Equation (1) 

to solve for the characteristic life, Lβ, of the blades for 

engine 2B, which is Lβ = 2255 cycles.

Again referring to Equation (9), the value for F of the first 

failure in that blade set is determined. For i = 1, F = 0.0085. 

From Equation (8), S = 0.9915. Substituting the value for S 

together with the Weibull slope e = 5.984 and Lβ = 2255 in 

Equation (1), the estimated time to the first failure is 

determined to be 1017 cycles. These calculations were 

repeated for each engine blade set. The resulting values are 

summarized in the last column of Table 2. 

A Weibull plot of the estimated time to first failure of each 

of the Neng = 16 blade sets is shown in Figure 4 and is 

compared with the time of blade set removal from Figure 3(b). 

The Weibull parameters for Figure 4, “Time to first blade 

failure” are summarized in Table 3. The mean time of the 

turbine blade set, Lm blade set, of 1482 cycles (7016 hr), is 

based on the first turbine blade failure and is approximately 

26 percent less than the average blade set removal time. 

These lives are summarized in Table 3. There is an 

insignificant difference in the Weibull slopes between the two 

Weibull plots. For purposes of comparison, the slope of 5.984 

derived from the time of removal in cycles was used. 

5.2 Turbine Blade Life 

Knowing the life of the blade set based on the estimated 

time to the first failure on each blade set, it is possible to 

determine the distributive lives of the individual blades from 

Equation (6). It is assumed that the Weibull slope for each of

the individual blades is identical to the Weibull slope for the 

blade sets based on the time to first failure from Figure 4 and 

equals 5.235.

Based on the characteristic life, Lβ, for the blade set of 

1608 cycles (see Table 3 column, “Estimated time to first 

blade failure”), the calculated characteristic life, Lβ, for the 

individual blade is 3731 cycles (see Table 3 column, “All 

failure modes”).

Figure 4.—Estimated time to first blade failure in set 

compared to blade set removal time. Number of 

T–1 turbine blades to a set, 82. For plotted data, 

coefficient of determination, r2 = 0.9734.

TABLE 3.—SUMMATION OF LIVES OF T–1 TURBINE BLADE SETS AND INDIVIDUAL BLADES

BASED ON FAILURE MODE USING JOHNSON-WEIBULL ANALYSIS

Weibull

parameters

Blade set life

(from Table 2 data)

Individual blade life based on failure mode,

cycles

Time of removal Estimated 

time to first 

blade failure,

cycles

All failure 

modes

Oxidation/erosion

(O/E)

Thermal-mechanical 

fatigue

(TMF)

Other

hr cycles

L1 life 4 337 993 668 1550 2093 2113 1717

L5 life 5 873 1304 912 2116 2857 2884 2343

L10 life 6 714 1471 1046 2427 3278 3309 2688

L50 life 9 529 2015 1499 3478 4698 4742 3852

Mean life b9 406 c1987 d1482 d3434 d4638 d4582 d3803 a

Lβ 10 201 2142 1608 3731 5039 5086 4132

Weibull slope 5.379 5.984 5.235 5.235 5.235 5.235 5.235 a

Life at a 63.2 percent probability of failure, characteristic life. b

Life at a 47.6 percent probability of failure. c

Life at a 47.2 percent probability of failure. d

Life at a 47.7 percent probability of failure.
NASA/TP—2013-217030 10

Figure 5.—Calculated individual turbine T–1 blade life 

from estimated time to first blade failure in blade set. 

Number of T–1 turbine blades to a set, 82.

From Equation (1) all the other blade lives for each 

probability of failure (survival) can be calculated. These 

results are summarized in Table 3 and are represented by the 

Weibull plot in Figure 5 labeled “Individual blade life.” The 

mean individual blade life, Lm blade, is 3434 cycles (16 256 hr), 

or approximately 2.3 times the mean life of the blade set, 

Lm blade set, of 1482 cycles (7016 hr). The L10 individual blade 

life calculated from Johnson-Weibull analysis is 2427 cycles 

(11 077 hr) compared to 1046 cycles (4774 hr) for the blade set. 

The life of the blade set will always be less than the life of an 

individual blade at the same probability of survival (failure).

5.3 Failure Mode

The time of removal of the blade sets together with the 

cataloged failure mode of those blades in each set that failed is 

summarized in Table 2. As previously discussed, the blades 

were first inspected for TMF. If cracks were evident on the 

blade even if other failure modes were also evident, the cause 

for removal was cataloged as TMF. The remaining blades 

were inspected for O/E. As with those blades cataloged as 

being failed by TMF, those blades that exhibited O/E damage 

were cataloged even where damage from other failure modes 

was manifested on the blade. The remaining blades were 

examined for damage for the other causes. These other causes 

were not identified; the cause was categorized and cataloged 

as “Other.” The “Other” category can include creep (stress 

rupture), yield, fracture mechanics (flaw initiated crack), fretting 

(wear and fatigue), corrosion, foreign object damage (FOD), and 

wear (blade tip rub).

There were 111 cataloged blade failures out of a total of 1312 

blades (Table 2). The failed blades comprised 8.5 percent of the 

total number of blades in the 16 blade sets. TMF accounted for 

approximately 20 percent of the failures, or 1.7 percent of the 

blade population. Oxidation/erosion accounted for approximately 21 percent of the failures, or 1.75 percent of the blade 

population. The highest accounting for blade failures occurs 

under the “Other” category. This is approximately 59 percent of 

the failures, or 5 percent of the blade population.

With reference to the strict-series system reliability equation 

(Eq. (5)), the resulting blade lives associated with the various 

failure modes with respect to the actual blade life can be 

derived from the Lundberg-Palmgren model for system failure

(Ref. 13) and are expressed by Johnson (Ref. 10) as follows:

e

e

L

L X

fm

blade = (11)

where X is the fractional percent of components failed from a 

cataloged failure mode, Lblade is the individual blade life, and 

Lfm is the individual blade life resulting from a cataloged 

failure mode (fm). If each blade failure due to a cataloged 

failure mode is known as a percentage of the total number of 

failed blades, then the life of the blade related to that failure 

mode can be determined from Equation (11) and vice versa.

However, a condition precedent for using Equation (11) is that 

the individual Weibull slopes must be known or assumed with 

reasonable engineering and statistical certainty.

The results of this analysis are shown in Figure 6 and 

summarized in Table 3. The resulting blade life attributed to O/E

equaled that attributed to TMF. The category that contributed

Figure 6.—Turbine T–1 blade life based on failure mode.
NASA/TP—2013-217030 11

most to blade life was “Other.” That is, if for any reason there 

were no blade failures attributed to O/E and TMF, the overall 

blade L10 life would increase from 2427 cycles to 2688 cycles,

or approximately 11 percent. Because of statistical variance, this 

increase in life would probably never be noticed in an actual 

application.

Referring to engine number 6B in Table 2, there are 40 

failures attributed to “Other” and a single failure attributed to 

TMF. Assume for purposes of discussion that at a time of 

1843 cycles (from Table 2, engine 6B) a single blade failed 

from TMF and broke loose, causing secondary damage to 40 

other blades in the set. The estimated time to first blade failure 

for engine 6B would change from 886 to 1843 cycles. 

A Weibull analysis of the data was performed with the 

revised life value (1843 cycles) for engine blade set 6B. From 

recalculation of the data, the Weibull slope was increased 

from 5.235 to 6.237 and the blade L10 life was decreased from 

2427 cycles to 2339 cycles. These changes are considered 

insignificant. 

If the 40 blade failures cataloged under “Other” for engine 

blade set 6B in Table 2 are discarded, the number of failed 

blades categorized under “Other” for engine blade set 6B is 

reduced from 66 to 26. This will reduce the total number of 

failed blades in Table 2 from 111 to 71. The failed blade 

fractions for 71 failed blades for the three categories become 

0.324, 0.31, and 0.366, for O/E, TMF, and “Other,” 

respectively. 

The respective blade L10 life for each failure category in 

Table 3 was recalculated using Equation (11) based on a 

Weibull slope e = 6.237 and the revised blade L10 life of 

2339 cycles. For the blade life based on O/E, the L10 life 

decreased from 3278 to 2802 cycles. For TMF, the L10 life 

decreased from 3309 to 2748 cycles. However, for the failure 

modes cataloged under “Other,” the L10 life increased from 

2688 to 2822 cycles. In this scenario, if the failure modes 

related to O/E and TMF are eliminated, the blade L10 life would 

be increased from 2339 to 2748 cycles, or approximately 

17 percent. Again, as before, this increase in life would 

probably never be noticed in an actual application. 

5.4 Simplified Life Formula

As previously discussed, there are competing failure modes 

that affect turbine blade life. Because of this, there was no 

attempt to analytically perform a life analysis based on any 

single failure mode to compare with the results presented. We 

are unaware of any published analysis of the turbine blades 

discussed in this paper. However, it is possible based on the 

work presented herein to develop a simplified equation that 

will allow the user airline to estimate the life of their turbine 

blades for the purpose of maintenance and replacement.

Of the failure modes discussed, it is our opinion that only 

the failure mode associated with low-cycle fatigue (LCF) 

(i.e., TMF) can be measured in terms of cycles to failure with 

reasonable engineering certainty. High-cycle fatigue (HCF) is 

related to the frequency of cycling, which is variable based 

upon gas velocity and thermal fluctuation. Also, the rate of 

cycling cannot be assumed with any reasonable engineering 

certainty much less measured. A prudent approach to the 

problem of HCF as it relates to a turbine blade application 

would be to assume that it is time dependent for a given 

engine application and operating profile. All the other failure 

modes discussed are also assumed to be time dependent for a 

given engine application and operating profile.

From Johnson (Ref. 10) the mean time to failure or removal 

is a function of the Weibull slope e. From the Weibull analysis 

summarized in Table 3, for a Weibull slope e of 5.379, the mean 

time to blade set removal (Lm blade set) is 9406 hr (1987 cycles).

This occurs at a 47.6 percent probability of failure. The mean 

time per cycle is equal to 4.73 hr/cycle (9406 hr, or 1987 

cycles). Appendix D presents a derivation of a measure of 

consistency between engine life in cycles and in hours.

For purposes of comparison, as the dispersion or scatter in 

the data increases, the Weibull slope e decreases (Ref. 10). As 

an example, for a normal distribution where the mean time to 

failure occurs at a 50 percent probability of failure, the 

Weibull slope e equals 3.57. For a Rayleigh distribution where 

the Weibull slope e equals 2, the probability of failure is 

54.4 percent. For an exponential distribution, the probability 

of failure is 63.2 percent at a Weibull slope e equal to 1. From 

this trend, an empirical formula can be derived as follows:

Fm ≈ 0.621e

–0.172 (12a)

where Fm is the mean probability of failure as a fractional 

percent and e is the Weibull slope. From Equation (1),

( ) 















 =



 





− Lβ

L e F

M

m

ln

1

1 ln ln (12b)

or

( )

(1/ )

m

M 1

1 ln

e

F

L L 



 





− = β (12c)

where LM is the mean time to removal and Lβ is the 

characteristic life or the life at a 63.2 percent probability of 

failure.

From Table 2, the summation of the time of removal of the 

blade sets divided by the number of blade sets equals the 

numerical average of the time of blade set removal 

(Lavg-blade set) where Lavg-blade set = 9421 hr (1990 cycles). This 

numerical average of 9421 hr (1990 cycles) correlates to the 

mean value from the Weibull analysis of 9406 hr 

(1987 cycles) summarized in Table 3. Accordingly, the 

numerical average of the blade set removal time (Lavg-blade set)
NASA/TP—2013-217030 12

can be substituted for the mean time to blade set removal 

(Lm blade set ) from the Weibull analysis in further calculations. 

From Table 3 (column, “Estimated time to first failure”), the 

mean time to first blade failure in a set is 1482 cycles, or 

7016 hr (1482 cycles × 4.73 hr/cycle). The mean time, Lm blade

to first blade failure in a set in terms of the average blade set 

time to removal is

Lm blade = (7016 hr/9406 hr) Lavg-blade set = 0.742 Lavg-blade set (13)

An acceptable failure rate needs to be established for blade 

removal. As discussed, 111 blades (8.5 percent) failed of the 

1312 blades comprising the 16 blade sets. It is therefore 

assumed that a 10-percent failure rate (L10) would be 

acceptable as an upper failure limit. From Figure 2 and 

Equation (1),

e

L L

S S = −

−

2 1

2 1

ln ln

lnln (1/ ) lnln (1/ ) (14a)

This reduces to

[ln(1/S1)/ln(1/S2)] = [L1/ L2]

e (14b)

In Equation (14b), let

S1 = S90 = 0.90

S2 = Sm = (1 – 0.477) = 0.523

L1 = L10

and from Equation (13)

L2 = Lm = 0.742 Lavg-blade set

where 

Lavg-blade set = (Sum of time to removal of blade sets)/

(number of blade sets) (14c)

Substituting the above values into Equation (14b) and 

solving for the L10 blade set life for time to first blade failure 

in a set where Weibull slope e = 5.235 (from Fig. 4),

L10 blade set

= 0.742 Lavg-blade set [ln(1/S1)/ln(1/S2)]

1/e

= 0.742 Lavg-blade set [ln(1/0.90)/ln(1/0.523)]1/5.235 (15)

= 0.524 Lavg-blade set

Combining Equations (3) and (14), the following empirical 

equation for the L10 individual blade life can be written:

L10 blade = 0.524 Lavg-blade set (n)

1/e = 0.524 Lavg-blade set (n)0.191

(16a)

Equation (16a) can be further simplified where

L10 blade ≈ (Lavg-blade set /2) (n)

0.2 (16b)

Substituting Lavg-blade set = 1990 cycles and n = 82 into 

Equations (16a) and (16b), L10 blade = 2418 and 2401 cycles, 

respectively. This correlates to the individual blade L10 blade life 

from Table 3 of 2427 cycles. Assuming a Weibull slope of 

5.235, the value of the characteristic life Lβ for individual 

blades can be calculated from Equation (1). Knowing Lβ, the 

individual blade life at any reliability (probability of survival, 

S) can be calculated from Equation (1).

6.0 Summary of Results

Sixteen high-pressure turbine (HPT) T–1 blade sets were 

removed from commercial aircraft engines that had been 

commercially flown by a single airline. Each blade set 

contained 82 blades. These engines were brought to the 

maintenance shop for refurbishment or overhaul. The blades 

on each HPT T–1 blade set were removed and inspected for 

damage. The damage found was cataloged into three 

categories related to their mode of failure. These were 

(1) Thermal-mechanical fatigue (TMF), (2) Oxidation/erosion 

(O/E), and (3) Other. From these field data, the turbine blade 

life was determined as well as the lives related to individual 

blade failure modes using Johnson-Weibull analysis. From 

these data and analysis, a simplified formula for calculating 

turbine blade life and reliability was formulated. The 

following results were obtained:

1. The following empirical equation for the L10

individual blade life was formulated:

L10 blade ≈ (Lavg-blade set /2) (n)

0.2

where Lavg-blade set = (sum of time to removal of blade 

sets)/(number of blade sets) and n is the number of 

blades in a set.

2. The individual blade life, L10 blade, calculated from 

Johnson-Weibull analysis is 2427 cycles (11 077 hr) 

compared to L10 blade set life of 1046 cycles 

(4774 hr). The life of the blade set (blade set life is 

defined as the failure time of first blade in a blade 

set) will always be less than the life of an individual 

blade at any given probability of survival (failure).
NASA/TP—2013-217030 13

3. The resulting individual blade life attributed to O/E

equaled that attributed to TMF. The category that 

contributed most to individual blade failure was 

“Other,” which includes creep (stress rupture), yield, 

fracture mechanics (flaw initiated crack), fretting 

(wear and fatigue), corrosion, foreign object damage 

(FOD), and wear (blade tip rub).

4. If there were no blade failures attributed to O/E and 

TMF, the overall individual blade life, L10 blade, would

increase approximately 11 to 17 percent.

Glenn Research Center

National Aeronautics and Space Administration

Cleveland, Ohio, April 29, 2013

NASA/TP—2013-217030 15

Appendix A

Derivation of Weibull Distribution Function

According to Weibull (Refs. 7 to 9) and as presented in 

Reference 12 (see also Ref. 23), any distribution function can 

be written as

F(X ) = 1− exp{− [f(X )]} (A1)

where F(X) is the probability of an event (failure) occurring 

and f(X) is a function of an operating variable X. Conversely, 

from Equation (A1) the probability of an event not occurring 

(survival) can be written as

1− F(X ) = exp{−[f(X )]} (A2a)

or

1− F = exp{− [f(X )]} (A2b)

where F = F(X) and (1 – F) = S, the probability of survival.

If there are n independent components, each with a 

probability of the event (failure) not occurring (1 – F), the 

probability of the event not occurring in the combined total of 

all components can be expressed from Equation (A2b) as

( F) { [n (X )]} n 1− = exp − f (A3)

Equation (A3) gives the appropriate mathematical 

expression for the principle of the weakest link in a chain or, 

more generally, for the size effect on failures in solids. The 

application of Equation (A3) is illustrated by a chain 

consisting of several links. Testing finds the probability of 

failure F at any load X applied to a “single” link. To find the 

probability of failure Fn of a chain consisting of n links, one 

must assume that if one link has failed the whole chain fails. 

That is, if any single part of a component fails, the whole 

component has failed. Accordingly, the probability of 

nonfailure of the chain (1–Fn), is equal to the probability of 

the simultaneous nonfailure of all the links. Thus,

( )

n 1− Fn = 1− F (A4a)

or

n

n S = S (A4b)

Where the probabilities of failure (or survival) of each link 

are not necessarily equal (i.e., S1 ≠ S2 ≠ S3 ≠...), Equation (A4b)

can be expressed as

... 1 2 3 S = S ⋅ S ⋅ S ⋅ n (A4c)

This is the same as Equation (2) of the main text.

From Equation (A3) for a uniform distribution of stresses 

σ throughout a volume V,

FV = 1− exp{− [V f(σ)]} (A5a)

or

S = 1− FV = exp{− [V f(σ)]} (A5b)

Equation (A5b) can be expressed as follows:

( ) V

S

ln f ln 1 ln ln = σ + 



 



 (A6)

It follows that if ln ln (1/S) is plotted as the ordinate and 

ln f(σ) as the abscissa in a system of rectangular coordinates, a 

variation of volume V of the test specimen will imply only a 

parallel displacement but no deformation of the distribution 

function. Weibull assumed the form

( )

e

u

u

















σ − σ

σ − σ

σ =

β

f (A7)

Where e is the Weibull slope, σ is a stress at a given 

probability of failure, σu is a location parameter below which 

stress no failure will occur, and σβ is the characteristic stress at 

which 63.2 percent of the population will fail. Equation (A6) 

becomes

e ( ) e ( ) V

S u u ln ln ln 1 ln ln = σ − σ − σ − σ + 



 





β (A8)

If the location parameter σu is assumed to be zero, and V is 

normalized whereby lnV is zero, Equation (A8) can be written 

as

















σ

σ =



 





β

ln 1 ln ln e

S

where 0 < σ < ∞ and 0 < S < 1 (A9)

Equation (A9) is identical to Equation (1) of the main text.

The form of Equation (A9) where σu is assumed to be zero 

is referred to as “two-parameter Weibull.” 

Where σu is not assumed to be zero, the form of the 

equation is referred to as “three-parameter Weibull.”

NASA/TP—2013-217030 17

Appendix B

Derivation of Strict Series Reliability

As discussed and presented in References 12 and 23, 

Lundberg and Palmgren (Ref. 13) in 1947, using the Weibull 

equation (Appendix A) for rolling-element bearing life 

analysis, first derived the relationship between individual 

component lives and system life. The following derivation is 

based on but is not identical to the Lundberg-Palmgren 

analysis.

Referring to Figure 2, from Equation (A9) in Appendix A, 

the Weibull equation can be written as















 =

















Lβ

L e S

ln 1 ln ln

sys

(B1)

where L is the number of cycles to failure.

Figure B.1 is a sketch of multiple Weibull plots where each 

Weibull plot represents a cumulative distribution of a 

component in the system. The system Weibull plot represents 

the combined Weibull plots 1, 2, 3, and so forth. All plots are 

assumed to have the same Weibull slope e (Ref. 12). The slope 

e can be defined as follows:

ref

sys ref

ln ln

1 ln ln 1 ln ln

L L

S S e −





 



 −

















= (B2a)

or

e

L

L

S

S





 



 =





 





















ref

ref

sys

1 ln

1 ln

(B2b)

From Equations (B1) and (B2b),

e e

L

L

L

L

S S 















 =



 









 



 =

















sys ref ref β

1 ln 1 ln (B3)

and



































 = −

β

e

L

L S exp sys (B4)

where Ssys = S in Equation (B1). For a given time or life L, 

each component or stressed volume in a system will have a 

different reliability S. From Equation (A4c) for a series 

reliability system

... sys 1 2 3 S = S ⋅ S ⋅ S ⋅ (B5)

Figure B.1.—Sketch of multiple Weibull plots where each numbered 

plot represents the cumulative distribution of a component in 

system and system Weibull plot represents combined distribution 

of plots 1, 2, 3, etc. (all plots are assumed to have same Weibull 

slope e).
NASA/TP—2013-217030 18

Combining Equations (B4) and (B5) gives

exp exp exp ...

1 2

×



































 × −



































 = −





































−

β β β

e e e

L

L

L

L

L

L (B6a)





















+

















+

















+















 = −





































−

β β β β

exp exp ...

1 2 3

e e e e

L

L

L

L

L

L

L

L (B6b)

It is assumed that the Weibull slope e is the same for all 

components. From Equation (B6b)





















+

















+

















+















 = −

















− ... L

L

L

L

L

L

L

L e e e e

β β1 β2 β3

(B7a)

Factoring out L from Equation (B7a) gives

... 1 1 1 1

1 2 3

+

















+

















+

















=

















β β β β

e e e e

L L L L (B7b)

From Equation (B3) the characteristic lives Lβ1, Lβ2, Lβ3, etc., 

can be replaced with the respective lives L1, L2, L3, etc., at Sref

(or the lives of each component that have the same probability 

of survival Sref) as follows:

ref ref ref 1

ref 2 ref 3

1 1 11 ln ln

11 11 ln ln ...

           =        

      

+++              

e e

e e

SL SL

SL SL

(B8)

where, in general, from Equation (B3)

e e

L S L 



 









 



 =

















β ref ref

1 1 ln 1 (B9a)

and

,etc.

1 1 ln 1

1 ref 1

e e

L S L 



 









 



 =

















β

(B9b)

Factoring out ln(1/Sref) from Equation (B8) gives

e e e e

L L L L

1/

ref 1 2 3

... 1 1 1 1

















 +



 



  +



 



  +



 



  =



 



 (B10)

or rewriting Equation (B10) results in

∑=





 



  =



 



 n

i

e

i

e

L L ref 1

1 1 (B11)

Equations (B10) and (B11) are identical to Equation (5) of 

the text.

Equation (B10) can also be rewritten as follows:

















 +



 



  +



 



  +



 



 1 = ...

3

ref

2

ref

1

ref

e e e

L

L

L

L

L

L (B12)

From Equation (B12) and according to Johnson (Ref. 10) 

the fraction of failures due to each cataloged failure mode of a 

component is expressed as

(1) Percent fraction of failures resulting from cataloged 

failure mode 1, 

e

L

L X 



 



 =

1

ref

1 (B13a)

(2) Percent fraction of failures resulting from cataloged 

failure mode 2, 

e

L

L X 



 



 =

2

ref

2 (B13b)

(3) Percent fraction of failures resulting from cataloged 

failure mode 3, 

e

L

L X 



 



 =

3

ref

3 (B13c)

The form of Equation (B13) is the same as Equation (11) of 

the text. Substituting Equation (B13) into Equation (B12),

X1 + X2 + X3 + ... = 1 (B14)

From Equations (B13a) to (B13c), if the life of the 

component and the percent fraction of the total failures 

represented by each cataloged failure mode are known, the life 

of the component related to each cataloged failure mode can 

be calculated. Hence, by observation, it is possible to 

determine the failure modes of a component population and 

determine the component’s life related to each cataloged 

failure mode. (Refer to Eq. (11) of the text.) 
NASA/TP—2013-217030 19

Appendix C

Linear Damage Rule

Most machine components are operated under combinations 

of variable loading and speed. Figure C.1 shows an example 

of a typical flight profile for a commercial flight with the time 

of each segment given. 

Palmgren (Ref. 24) working with ball and roller bearings 

recognized that the variation in both load and speed must be 

accounted for in order to predict component life. Palmgren

(Ref. 24) reasoned: “In order to obtain a value for a 

calculation, the assumption might be conceivable that (for) a 

bearing which has a life of k million revolutions under 

constant load at a certain rpm (speed), a portion M/k of its 

durability will have been consumed. If the bearing is exposed 

to a certain load for a run of M1 million revolutions where it 

has a life of k1 million revolutions, and to a different load for a 

run of M2 million revolutions where it will reach a life of k2

million revolutions, and so on, we will obtain

1

3

3

2

2

1

1 + + + = k

M

k

M

k

M (C1)

In the event of a cyclic variable load we obtain a convenient 

formula by introducing the number of intervals p and 

designate m as the revolutions in millions that are covered 

within a single interval. In that case we have

( ) 1

3

3

2

2

1

1 =















+ + + k

m

k

m

k

m p (C2)

where k still designates the total life in millions of revolutions 

under the load and rpm (speed) in question (and M = pm).”

Equations (C1) and (C2) were independently proposed for 

conventional fatigue analysis by Langer (Ref. 25) in 1937 and 

Miner (Ref. 26) in 1945, 13 and 21 years after Palmgren

(Ref. 24), respectively. The equation has been subsequently 

referred to as the linear damage rule or the Palmgren-LangerMiner rule. For convenience, the equation can be written as 

follows:

k

k

L

X

L

X

L

X

L

X

L = + + +

3

3

2

2

1

1 1 (C3)

and

X1 + X2 + X3 +Xk = 1 (C4)

where L is the total life in stress cycles or race revolutions, 

L1...Lk is the life at a particular load and speed in stress cycles 

or race revolutions, and X1...Xk is the fraction of total running 

time at load and speed. From Equation (C1)

M1 = X1L, M2=X2L, M3=X3L, ... Mk= XkL (C5)

Because the flight profile is repeatable, for example, 

Figure C.1, it is reasonable to use the percent of time in each 

segment to determine engine component life using 

Equation (C3).

Equation (C3) is the basis for most variable-load fatigue 

analysis and is used extensively in bearing life prediction.

Figure C.1.—Example typical flight profile for a 

commercial gas turbine engine, as shown.

NASA/TP—2013-217030 21

Appendix D

Variation Between Engine Life in Cycles and in Hours

The relationship between engine hours and engine cycles 

varies depending on engine usage. Therefore, the distribution of 

failures as a function of hours and as a function of cycles, 

although closely related, will not necessarily have the same 

Weibull slope. As a result, the conversion of engine cycles to 

engine hours may differ with probability of survival.

When the Weibull slopes are equal, the ratio of engine hours 

to engine cycles is the same at all probabilities of survival. For 

Weibull analysis,

















=















Lβ

L

e

S

ln 1 ln ln (D1)

where S = 1 – F is the probability of component survival with

F as the probability of failure; e, the Weibull slope; Lβ, the 

characteristic life at S = 0.368 (or F = 0.632); and L,

component life.

For S hour = Scycle,

hour cycle

ln ln































 = 































β Lβ

L

e

L

L

e (D2a)

Therefore,

p

L

L

L

L





































= 















β β hour cycle

(D2b)

where p = ecycle/ehour. Solving for the ratio of life in hours to 

life in cycles in terms of characteristic life gives

( 1)

cycle cycle

hour

cycle

hour

−

β β

β





















































=















p

L

L

L

L

L

L

(D3)

However, where there is variation between the Weibull 

slopes, there is also a variation in the ratio of engine hours to 

cycles at a given probability of survival. This is illustrated in the 

data in Table 3. From Table 3, for L1, L5, L10, and L50 ( ) Lhour Lcycle = 4.37, 4.50, 4.56, and 4.73 hours/cycle, 

respectively. 

Note that as the Weibull slopes representing the data for 

engine cycles and hours approach each other; that is, as ecycle

⇒ ehour, p ⇒ 1. Thus from Equation (D3), ( ) Lhour Lcycle ⇒

( ) Lβ hour Lβ cycle ⇒ Constant (invariant).

The value of p = ecycle/ehour provides a metric for consistency 

between engine cycle and engine hour data.

NASA/TP—2013-217030 23

References

1. Zaretsky, Erwin V.; Hendricks, Robert C.; and Soditus, 

Sherry: Weibull-Based Design Methodology for 

Rotating Structures in Aircraft Engines. Int. J. Rotating 

Mach., vol. 9, 2003, pp. 313–325.

2. Davis, Donald Y.; and Stearns, E. Marshall: Energy 

Efficient Engine: Flight Propulsion System Final Design 

and Analysis; Topical Report. NASA CR–168219, 1985.

3. Halila, E.E.; Lenahan, D.T.; and Thomas, T.T.: Energy 

Efficient Engine High Pressure Turbine Test Hardware: 

Detailed Design Report. NASA CR–167955, 1982.

4. Manson, S.S.; and Halford, G.R.: Fatigue and Durability 

of Structural Materials. ASM International, Materials 

Park, OH, 2006, p. 401.

5. Sawyer, John W., ed.: Gas Turbine Engineering 

Handbook. Gas Turbine Publications, Inc., Stamford, 

CT, 1966, p. 235.

6. Nelson, Wayne: Applied Life Data Analysis. John Wiley 

& Sons, NY, 1982, pp. 405–432.

7. Weibull, Waloddi: A Statistical Theory of the Strength 

of Materials. Ingenioersvetenskapsakad. Handl., no. 151, 

1939.

8. Weibull, Waloddi: The Phenomenon of Rupture in Solids. 

Ingenioersvetenskapsakad. Handl., no. 153, 1939.

9. Weibull, Waloddi: A Statistical Distribution of Wide 

Applicability. J. Appl. Mech. Trans. ASME, vol. 18, 

no. 3, 1951, pp. 293–297.

10. Johnson, L.G.: The Statistical Treatment of Fatigue 

Experiments. Elsevier Publishing Co., Amsterdam, The 

Netherlands, 1964.

11. Cetal, A.D.; and Duhl, D.N.: Second-Generation NickelBase Single Crystal Superalloy. Superalloys 1988, The 

Metallurgical Society, Warrendale, PA, 1988, pp. 235–244. 

12. Melis, M.E.; Zaretsky, E.V.; and August, R.: 

Probabilistic Analysis of Aircraft Gas Turbine Disk Life 

and Reliability. J. Propul. Power, vol. 15, no. 5, 1999, 

pp. 658–666.

13. Lundberg, G.; and Palmgren, A.: Dynamic Capacity of 

Rolling Bearings. Acta Polytech., vol. 1, no. 3, 1947.

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

Using Weibull Statistics. Biomat., vol. 25, no. 20, 2004, 

pp. 5031–5035.

15. 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.

16. Tomimatsu, M.; Kikuchi, M.; and Sakai, M.: Fracture 

Toughness Evaluation Based on the Master Curve 

Procedure. ASME Pressure Vessels and Piping Division 

Publication, vol. 353, 1997, pp. 343–349.

17. Osborne, N.G.; Graves, G.G.; and Ferber, M.K.: 

Dynamic Fatigue Testing of Candidate Ceramic 

Materials for Turbine Engines to Determine Slow Crack 

Growth Parameters. J. Eng. Gas Turbines Power, ASME 

Trans., vol. 119, 1997, pp. 273–278.

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

Thermally Sprayed Coatings. Surf. Coat. Technol., 

vol. 34, 1988, pp. 43–50.

19. Holland, F.A.; and Zaretsky, E.V.: Investigation of 

Weibull Statistics in Fracture Analysis of Cast 

Aluminum. J. Appl. Mech. Trans. ASME, vol. 112, 

1990, pp. 246–254.

20. Williams, S.R.; and Fec, M.C.: Weibull Analysis of 

Reconditioned Railroad Roller Bearing Life Test Data. 

ASME Rail Transportation Division Publication, vol. 5, 

1992, pp. 83–87.

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

Process Machine Reliability. Qual. Reliab. Eng. Int., 

vol. 5, no. 3, 1989, pp. 203–205.

22. Vlcek, Brian L., et al.: Comparative Fatigue Lives of 

Rubber and PVC Wiper Cylindrical Coatings. Tribol. 

Trans., vol. 46, no. 1, 2003, pp. 101–110.

23. Poplawski, Joseph V.; Peters, Steven M.; and Zaretsky, 

Erwin V.: Effect of Roller Profile on Cylindrical Roller 

Bearing Life Prediction—Part I: Comparison of Bearing 

Life Theories. Tribol. Trans., vol. 44, no. 3, 2001, 

pp. 339–350.

24. Palmgren, A.: The Service Life of Ball Bearings. NASA 

Technical Translation TT–F–13460, 1971, (from 

Zectsckrift des Vereines Deutscher Ingenieure, vol. 68, 

no. 14, 1924, pp. 339–341).

25. Langer, B.F.: Fatigue Failure From Stress Cycles of 

Varying Amplitude. J. Appl. Mech. Trans. ASME, 

vol. 4, no. 4, 1937, pp. A160–A162.

26. Miner, Milton A.: Cumulative Damage in Fatigue. J. 

Appl. Mech. Trans. ASME, vol. 12, no. 3, 1945, 

pp. A159–A164.

27. Kapur, K.C.; and Lamberson, L.R.: Reliability in 

Engineering Design. John Wiley and Sons, Inc., New 

York, NY, 1977, pp. 8–53.
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4. TITLE AND SUBTITLE

Determination of Turbine Blade Life From Engine Field Data 

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Zaretsky, Erwin, V.; Litt, Jonathan, S.; Hendricks, Robert, C.; Soditus, Sherry, M. 

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13. SUPPLEMENTARY NOTES

Erwin V. Zaretsky is currently a Distinguished Research Associate. 

14. ABSTRACT

It is probable that no two engine companies determine the life of their engines or their components in the same way or apply the same 

experience and safety factors to their designs. Knowing the failure mode that is most likely to occur minimizes the amount of uncertainty 

and simplifies failure and life analysis. Available data regarding failure mode for aircraft engine blades, while favoring low-cycle, thermal 

mechanical fatigue as the controlling mode of failure, are not definitive. Sixteen high-pressure turbine (HPT) T-1 blade sets were removed 

from commercial aircraft engines that had been commercially flown by a single airline and inspected for damage. Each set contained 82 

blades. The damage was cataloged into three categories related to their mode of failure: (1) Thermal-mechanical fatigue, (2) 

Oxidation/Erosion, and (3) Other. From these field data, the turbine blade life was determined as well as the lives related to individual blade 

failure modes using Johnson-Weibull analysis. A simplified formula for calculating turbine blade life and reliability was formulated. The 

L10 blade life was calculated to be 2427 cycles (11 077 hr). The resulting blade life attributed to oxidation/erosion equaled that attributed to 

thermal-mechanical fatigue. The category that contributed most to blade failure was “Other.” If there were there no blade failures attributed 

to oxidation/erosion and thermal-mechanical fatigue, the overall blade L10 life would increase approximately 11 to 17 percent.

15. SUBJECT TERMS

Turbine blade; Failure; Field data; Gas turbine; Weibull 

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