Engineering

Failure

Analysis

Contents lists available at SciVerse ScienceDirect

Engineering Failure Analysis

journal homepage: www.elsevier.com/locate/engfailanal

Probabilistic and testing analysis for the variability ■. oossMark

of load spectrum damage in a fleet ^

Xiaofan He a'*, Fucheng Suib, Bin Zhaia, Wenting Liua

a School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China b Shenyang Aircraft Design and Research Institute, Shenyang 110035, China

ARTICLE INFO

ABSTRACT

Article history:

Received 29 October 2012

Received in revised form 11 June 2013

Accepted 19 June 2013

Available online 26 June 2013

Keywords: Load spectrum Load damage

Engineering crack initiation life

Scatter

Variability

Scatter of fatigue life of a fleet is mainly caused by the variability in structures and load spectra. To ensure the safety in service, the probabilistic characterization of load spectrum variability should be researched in durability analysis and testing work. This paper investigates the variability of load damage rate of a fleet. Based on the flight historical parameters measured by individual aircraft tracking (IAT) from hundreds of aircrafts for a certain type of fighter in China, SWT formula and linear damage rule are used to evaluate the load damage, and then, one average and four other individual load spectra are selected corresponding to different damage severities. Fatigue tests are conducted with the Aluminum alloy 7B04-T74 specimens under five spectra and the Titanium alloy TA15M specimens under three of them. The engineering crack initiation lives are measured and the mean lives are estimated assuming the fatigue life following a log-normal distribution. An obvious difference of at least 2.4 times in the load damage rates is found in the fleet. The fatigue lives of a fleet of aircrafts are calculated by Neuber's approach, and the probabilities refer to damage severities of those 5 load spectra in a fleet are evaluated. The statistical analysis of the fatigue lives and the probabilities shows that a lognormal distribution can be used to describe the variability of load damage rate of a fleet. The variation of the load damage rate is in the same order of magnitude with that in structural properties.

© 2013 The Authors. Published by Elsevier Ltd. All rights reserved.

1. Introduction

To ensure the safe usage of aircraft structures, the reliability life should be evaluated analytically and tested in design stage according to the strength requirements for aircraft structure in specifications or standards [1-3]. It is well known that the fatigue life of a fleet is inherently varying due to various factors, which can be mainly divided into two categories: structural variability and load spectrum variability [4-6]. A need exists to account for the effects of both the variabilities on the fatigue life.

Structural variability refers to the statistical variability inherent in the fatigue performance of built-up structures which arises from the variability in material properties, manufacturing and assembly processes, etc, and it is usually quantified by the probability distribution of the fatigue life under a prescribed load spectrum. Since the reliability theory of fatigue was established many decades ago [7,8], several continuous probability functions and corresponding parameters have been used to delineate the variability of the fatigue properties of commonly used metallic structures [8-10].

q This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial-No Derivative Works License, which permits non-commercial use, distribution, and reproduction in any medium, provided the original author and source are credited. * Corresponding author. Tel.: +86(10) 82315738. E-mail address: buaa_he@hotmail.com (X. He).

1350-6307/$ - see front matter © 2013 The Authors. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfailanal.2013.06.016

Load spectrum variability of a fleet refers to the differences of load-time history among the aircrafts of the same type and usage. It may occur due to differences in the pilot's expertise, climate, runway, the aircraft's weight, etc. Comparing with structural variability, it is more difficult to determine the variability in load spectra because the conditions in service are complicated and the measurement of load parameters is time-consuming and costly [11]. In recent years, a large number of load-time historical parameters have been characterized with the wide utilization of individual aircraft tracking (1AT), and more attention is focused on the load spectrum variability. Generally, there are two ways to express it: (1) the variability of load-time historical parameters between individual aircrafts within a fleet [12-15]; (2) the variability of the damage rate related to load spectrum. The damage rate refers to the fatigue damage per flight hour which could be used to study the influence of load spectrum on the variability of fatigue life. Based on the load data from 1AT for key structures, the load spectrum damage rates of individual aircrafts in a fleet can be calculated and then a suitable random variable will be used to describe the variability of damage rate [15-20]. Since there is not an accepted distribution function for the load damage rate, it is necessary to investigate its variability by analysis and testing.

In order to develop a suitable load spectrum, lots of fatigue tests are conducted to draw a comparison and clarify the difference between various methods developed [21-23]. Fatigue tests are also used to assess the relative severity of a design spectrum against the actual service spectrum [24]. However, reports about systematic testing research on the variability of load damage rate for a fleet have not been found yet.

Based on the tracking load data from a certain type of fighter in China, in this study, individual load spectra for hundreds of aircrafts are developed, in which five load spectra corresponding to different damage severities are selected. The fatigue tests are conducted with the Aluminum alloy 7B04-T74 specimens under those five load spectra and the Titanium alloy TA15M specimens under three of them. The engineering crack initiation lives are then analyzed statistically to investigate the variability of load damage rate of a fleet.

2. Calculation and statistical analysis on the load damage rate

2.1. Collection of flight parameters

A recording system consists of sensors, data collection and processing device, signal tape recorder was used to measure flight parameters of a fighter in China. The signals from sensors are edited and then recorded by the magnetic tape recorder. After landing, the recorded information are imported into the computers on the ground at weekly or monthly intervals, and then, the recorded data are reverted to original historical parameters by using a special software for further processing and analysis.

The recording system performs a continuous scanning of several parameters related to load, mainly including flight time, airspeed, altitude, normal acceleration, lateral acceleration, fuel remained, attack angle, pitching angle, sideslip angle, grade angle, as well as the position of aileron, horizontal stabilizer, leading edge slot, rudder, etc. The sampling rate of above mentioned parameters is 1~8 times per second, specially, 8 for accelerations.

A very large set of recorded data have been accumulated for hundreds of aircrafts by using this system.

2.2. Data processing of flight parameters

The data processing of flight parameters for each fighter can be briefly described as the following steps:

(1) Data editing. The spurious data collected is detected and eliminated to ensure the reality and reliability of flight parameters of individual aircrafts;

(2) Data reduction. The middle points between the peak and trough accelerations are removed, and effective values at a series of peaks and troughs in a load parameter-time history are obtained.

(3) Data compression. After obtaining the data corresponding to above effective peak and trough points, in order to reduce the load cycles applied, some small acceleration cycles are filtered since their influence on the fatigue damage of airplane structures are insignificant.

(4) Effective acceleration analysis. Accounting for the weight of aircraft stores and the fuel consumed, the instantaneous effective weight corresponding to the acceleration nz(i) filtered in step (3) can be calculated and written down as G(i) which is used to correct the normal acceleration at the center of gravity. Supposing the standard weight of aircraft to be G(0), the effective normal acceleration accounting for the influence of aircraft weight will be evaluated as

nz,e(i)=nz(i) (1)

Therefore, the data for effective acceleration-time history can be determined.

Based on the data of flight parameter-time history measured by 1AT for hundreds of aircrafts in China, the effective acceleration spectra of all aircrafts are developed [25].

2.3. Load damage rates calculation of individual spectra

Based on the individual spectra developed in Section 2.1, the load spectrum damage rates can be simply calculated by using Smith-Watson-Toffer (SWT) formula and linear accumulation damage model. The SWT formula is applied to convert all load cycles to symmetric ones. As we know, the SWT formula is expressed as [26],

c c 1 R /Smax(Smax Smin)

S-1 = Smax y 2 =y-5--(2)

where Smax and Smin are the peak and trough of a stress cycle, respectively, R = Smin/Smax is the stress ratio and S_1 denotes the peak stress for converted symmetric cycle.

For simplification, it is assumed that the stress is linear with respect to effective normal acceleration at the center of gravity. Other loading conditions, such as wing root bending moments, are not considered. Therefore, the nominal stress level S will be

S = r(1g) nv (3)

where nze is the effective normal acceleration, and r(1g) is the stress level corresponding to unit normal acceleration. Thus, each asymmetric acceleration cycle could be converted to a symmetric acceleration cycle as follows

nz..e ,max(nz,e ,max — nz.e ,min) f A>, nz,e ,-1 = y-2--(4)

where nz e max and nz e min are the peak and trough of an acceleration cycle, respectively, and nz e -1 denotes the equivalent symmetric peak acceleration.

With a certain stress ratio, the S-N curve can be written in the form

SmN = C (5)

where N is the number of load cycles, C is a constant, and m is the curve slope with R = -1 in logarithm coordinates, usually taken as 3-5 . In this paper, m = 4 is chosen for general used metallic materials/structures [15]. The damage caused by i-th load cycle can be defined as

- N (6)

where Ni is the fatigue life corresponding to i-th load cycle.

Substituting Eq. (5) into Eq. (6) and considering Eq. (3), we have

di=c=k • nmA-i. i (7)

where nz e -1ii denotes the peak acceleration of the i-th symmetric acceleration cycle, k = rm(1g)/C is a constant.

The above-mentioned load spectra are transferred to discrete load cycles by rain flow counted method. According to Eq. (4), the symmetric cycles could be obtained, cycle by cycle. Assuming that a basic life period includes K cycles, the cumulative damage D per basic life period could then be calculated on the basis of linear cumulative damage model as follows

D = k nm_e.-1. i (8)

Since k is a constant, the summation in Eq. (8) can be considered as the equivalent load damage Deq given by

D eq = £ Ke--1. i (9)

Assuming that a basic life period corresponds to T0 flight hours, the equivalent load damage per flight hour which is referred to as load damage rate can be obtained as

D 0 ,eq = Deq/Ta =Z <e,-1, i/T0 (10)

Obviously, the load damage rate represents the damage severity of individual load spectra in a fleet.

2.4. Statistical analysis on the calculated load damage rate

Assuming the load damage rate follows a two-parameter lognormal distribution, LN{ßD, <j2d), we have the following linear relationship

lS10D0.eq.Pj = Id + upj rD (11)

where Pj is the probability, uPj is the standard normal variable corresponding to Pj, D0 eq Pj is the load damage rate corresponding to Pj, iD and rD denotes the expectation and standard deviation of lg10D0 eq, respectively.

According to Eq. (10), all the load damage rates D0eqPj(j = 1.2 . ■■■ ,n) with n being the number of aircrafts in a fleet are calculated and arranged in an ascending order. For each aircraft in the fleet, the corresponding Pj can be estimated as Pj = j and the values of uPj can be calculated. Then, the collected data for uPjandlg10D0 eq Pj are plotted in Fig. 1. From the figure, it is shown that the relationship between uPj and lg10D0 eqPj are quite linear. Obviously, the equivalent load damage rate calculated by SWT formula and linear damage model follows a two-parameter lognormal distribution.

3. Testing on the variability of load damage rate

3.1. Five individual load spectra

Among hundreds of load spectra available, 5 individual load spectra with different damage severities are selected to conduct fatigue testing, which are named as spectrum 1-5 according to their damage rates in an ascending order. Spectrum 2 denotes the spectrum with average damage rate in a fleet. In order to reduce the testing duration, only one spectrum with damage rate less than the average one, spectrum 1, is chosen. To show the relative severity of above 5 load spectra, relative damage rates are calculated as 0.898,1.0,1.1,1.77 and 1.892 by dividing all the five damage rates by the one of spectrum 2, respectively. The normal acceleration - cumulative exceedance curves corresponding to those 5 spectra for a life period of 1000 flight hours are shown in Fig. 2.

In this study, Spectrum 1 almost retains the actual load information. However, Spectra 2-5 are treated by removing the acceleration cycles with their peak value less than 2.0 g and extrapolating the overload cycle which occurs once in 1000 flight hours. Using extrapolation approach, it is needed to fit each peak acceleration - exceedance curve per 1000 flight hours by the following polynomial function.

lg10N = ao + alnzp + a2n\ P + a3n3z P + a4n\ P

where, N is the exceedance, nzp denotes the peak acceleration, and a(i = 0,1, 2, ■■■, 4) are the coefficients to be determined. A nonlinear curve fitting is used to solve the coefficients for each spectrum. When N =1, the peak overload, nz maxp, will satisfy the following equation.

0 = ao + ainz,max,P + a2n2>max P + ^Lax,? + a4n4,max,P

Above nonlinear equation can be solved by numerical approach. The overload cycle with a trough of 1.0 is inserted to a fight mission profile. A fraction of each spectrum obtained is presented in Fig. 3.

3.2. Specimens

Two types of specimens are designed and applied to conduct the fatigue testing.

Standard normal variable Up Fig. 1. Data of standard normal variable, uP , and lg10 load damage rate, lg10D0ei, [20].

Fig. 2. The acceleration- cumulative exceedance curves corresponding to five load spectra. In the figure 2, accleration is normalized by dividing the accelerations by the maximum acceleration factor corresponding to those 5 load spectra.

3.2.1. Aluminum alloy 7B04-T74 specimen

The geometry of the specimen with its functional section of 8 mm thickness and 42 mm width is shown in Fig. 4, in which there are three holes with a diameter of 8 mm. Surface roughness values of the holes is Ra1.6. The specimen is made of the forged Aluminum alloy 7B04-T74. Its properties are summarized in Tables 1 and 2.

3.2.2. Titanium alloy TA15M specimen

The geometry of the specimen is shown in Fig. 5. Its functional section has a thickness of 3.5 mm and a width of 42 mm. It is made of Titanium alloy TA15 M. The surface roughness of all the holes is also Ra1.6. TA15 M is an approximate a Titanium alloy, and its chemical compositions and mechanical properties are listed in Tables 3 and 4.

To reduce the influence on fatigue life due to the variability of material properties and manufacturing process, all specimens are cut from the same batch and manufactured strictly. The non-destructive detection is conducted carefully on all specimens before fatigue testing.

400 600 iV/cycle (a) fraction of spectrum 1

400 600

JV/cycle

(c) fraction of spectrum 3

400 600 N/cycle (b) fraction of spectrum 2

400 600 N/cycle (d) fraction of spectrum 4

400 600 JV/cycle (e) fraction of spectrum 5

Fig. 3. Fractions of five individual load spectra (vertical axis denotes the relative acceleration which is the ratio of acceleration to the maximum acceleration in 5 load spectra).

3.3. Testing results

The fatigue tests are completed using a closed loop servo-hydraulic controlled Material Testing System (MTS 880) with a load capacity of 100kN operating under load control model. The load control of the system is accomplished by using an external MTS TestStar lis controller. A variable amplitude loads in the form of sinusoidal wave are applied with a frequency of 8 Hz. The fatigue tests is conducted under the room temperature conditions (20-25 °C) with 30-40% relative humidity in air. The stress level per unit acceleration, r(1 g), is 41 MPa for 7B04-T74 specimens and 99.2 MPa for TA15 M specimens,

Fig. 4. Geometry of Aluminum alloy 7B04-T74 specimen (unit:mm).

Table 1

Chemical compositions and mass fraction of Aluminum alloy 7B04-T74.

Chemical composition Cu Mg Mn Cr Zn Fe Si Ti Ni Others Al

Mass fraction (%) 1.4-2.0 1.8-1.8 0.2-0.6 0.1-0.25 5.0-6.5 0.05- -0.25 60.1 60.05 60.1 60.1 Bal.

Table 2

Mechanical properties of Aluminum alloy 7B04-T74 (L-S).

Property Elastic module E (GPa) Yield stress rs (MPa) Ultimate strength rb (MPa) Possion ratio i Fracture toughness KiC/MPa x m1'2

Value 69 536 598 0.3 26.28

Fig. 5. Geometry of Titanium alloy TA15M specimen (unit:mm).

Table 3

Chemical compositions and mass fraction of Titanium alloy TA15M.

Chemical composition Al Zr Mo V impurity Ti

Mass fraction (%) 5.5-7.0 1.5-2.5 0.5-2.0 0.8-2.5 <0.7 Bal.

respectively. The features of fracture surfaces are examined after the testing, and the engineering crack initiation life is determined by quantitative fractography when the crack depth reaches to 0.8 mm which is usually considered as an initial crack length in damage tolerance analysis [3]. Assuming that the fatigue life under each load spectrum follows a lognormal

Table 4

Mechanical properties of Titanium alloy TA15M (L-S).

Property E (GPa) a s (MPa) ab (MPa) l Kic/MPa x m1/2

Value 110 900 980 0.33 88

distribution, then the estimated mean lives (geometric mean life) and the standard variation of lg10 fatigue lifer, can be estimated.

3.3.1. Testing results of 7B04-T74 specimens

The fatigue tests for 7B04-T74 specimen are conducted under all 5 load spectra. The effective numbers of specimens are 6, 6, 7, 7 and 5, respectively. The failure surfaces are shown in Fig. 6. it can be seen that the bands caused by fatigue loading occur on fracture surfaces clearly. The estimated mean lives and standard deviations of log10 life are listed in Table 5.

3.3.2. Testing results of Titanium alloy TA15M specimens

Fatigue tests are also conducted forTA15M specimens under spectrum 2, 3 and 5. For those spectra, 6, 7 and 7 specimens are used, respectively. Typical failure surfaces are shown in Fig. 7. it can be seen that there are clear fatigue bands on the fracture surface as well. Similarly, the mean engineering crack initiation lives and the standard deviations of log10 life are obtained and listed in Table 6.

(a) quarter-ellipse corner crack (b) semi-ellipse crack

Fig. 6. Typical failure surfaces of 7B04-T74 specimens.

Table 5

Testing results of Aluminum alloy 7B04-T74 specimens.

Spectrum Number of specimens Geometric mean life/flight hours Standard deviation of log10 life

1 6 11882 0.09

2 6 10668 0.08

3 7 9688 0.06

4 7 5206 0.07

5 5 4447 0.07

Fig. 7. Typical failure surfaces of TA15M specimens.

Table 6

Testing results of Titanium alloy TA15M specimens.

Spectrum Number of specimens Geometric mean life/flight hours Standard deviation of logi0 life

2 6 14421 0.04

3 7 11702 0.06

5 7 5847 0.05

Table 7

Fatigue life ratios and relative damage rates.

Spectrum Relative damage rates Aluminum alloy 7B04-T74 specimens Titanium alloy TA15M specimens

1 0.898 0.90

2 1.0 1.0 1.0

3 1.11 1.11 1.23

4 1.77 2.0

5 1.892 2.43 2.47

From Tables 5 and 6, it is shown that for each type of specimens the variations of engineering crack initiation lives under all spectra are close together. The difference of mean engineering crack initiation life between different spectra reflects the variability of load damage rate.

3.4. Comparison of calculated and experimental results

3.4.1. Fatigue life ratios and relative damage rates

To clarify the relative severity of those 5 spectra, fatigue life ratios are calculated by dividing the mean engineering crack initiation life under spectrum 2 by those under all spectra. For two types of specimens, the fatigue life ratios are listed in Table 7. In this table, the calculated relative damage rates in Section 3.1 are also included.

It is shown that there is an obvious difference of at least 2.4 times among the damage rates between individual aircrafts in the fleet.

3.4.2. Feasibility of SWT linear model

The relative damage rates are close to fatigue life ratios obtained from testing except for that under spectrum 5. As we known, the interaction and nonlinear effect of variable stress influence the fatigue damage, but those effects cannot be considered in the linear accumulative damage model [27]. The fatigue damage results from the SWT model would not be accurate enough, specially, in the cases where the local plastic effects induced by higher loads are ignored. However, although the SWT linear model for the fatigue life or damage assessment is irrelevant to the material properties and stress level, it could be used quite simply and conveniently.

3.4.3. Calculation of fatigue life by Neuber's method

3.4.3.1. For 7B04-T74 specimens. The yield strength of 7B04-T74 material is 536 MPa. The theoretical stress concentration factor of this specimen is 2.3. Hence, when the peak acceleration is greater than 536/41/2.3 = 5.68 g, the material behavior will be plastic, and the nonlinear effect must be taken into consideration. Therefore, the Neuber's method [28] which belongs to local stress-strain method can be applied to calculate the fatigue lives. Based on the material parameters selected [29], the fatigue notch coefficient, Kf, derived by the mean tested fatigue life under spectrum 2 is used to calculate the local stresses and strains of each load cycle, and the fatigue lives under other load spectra could be calculated. The ratios of the fatigue lives under above 5 spectra to that under Spectrum 2 are obtained as 0.80, 1.0, 1.10, 1.92 and 2.31, respectively. It is shown that the fatigue life ratios calculated by the Neuber's method agree well with the corresponding testing data, as shown in Table 7.

3.4.3.2. For TA15M specimens. The yield strength of TA15M material is 900 MPa. The plastic properties of the material will appear when the acceleration is greater than 900/99.2/2.3 = 3.94 g. With each acceleration spectrum, the nonlinear effects should be more severe for Titanium alloy TA15M than that for Aluminum alloy 7B04-T74. Similarly, the fatigue life ratios calculated by Neuber's method are equal to 1.0, 1.2 and 2.28, respectively, which are also acceptable compared with the tested fatigue life ratios (Table 7).

4. Hypothesis test on the distribution of load damage rate

4.1. Hypothesis test on the variability of fatigue life

4.1.1. Analysis on the calculated life

In order to evaluate the fatigue lives more accurately, the Neuber's method is applied to recalculate the fatigue lives under hundreds of load spectra mentioned in Section 2 for those two types of specimens. The analysis on the recalculated lives of a

fleet shows that the fatigue life under spectrum 2 is still the average one, and the values of rT are 0.18 for 7B04-T74 and 0.17 for TA15M specimens, respectively. The value of rT represents the variability of load spectra in this fleet. in addition, for general used metallic built-up structures, the standard deviation of lg10 fatigue life under a predicted load spectrum which represents the structural variability usually be in the region of 0.07-0.20 [2,30-32], it is obvious that the variation of load damage rate is close to that of structures.

4.1.2. Hypothesis test on the variability of testing life

If the fatigue life follows a lognormal distribution, the fatigue life, T, satisfies the following formula

lgioT = it + Up ot

. lSio T - It

where T is the fatigue life calculated by Neuber's method, iT is the expectation of lg10 T, and uP denotes the corresponding standard normal variable of lg10 T (see Eq. (11)).

To check up the probability characteristics of load damage, according to Eq. (14), for 7B04-T74 specimen, the values of uP calculated under those 5 load spectra are -0.25421, 0.0, 0.25179, 1.67239 and 2.14226 with the corresponding values of P being 39.9667%, 50%, 59.94%, 95.2776%, and 98.3914%, respectively. Similarly, for TA15M specimen, the values of uP under the 3 load spectra on test are 0.0, 0.53374, and 2.30625, with the corresponding values of P being 50%, 70.3239%, and 98.9452%, respectively.

in a sense, if the fatigue life of a fleet has a lognormal distribution, the tested mean lives and their corresponding probabilities P should have a linear relationship in lognormal coordinates. The data for 7B04-T74 specimens, uPj and log10 Ti(i = 1.2 . ••• .5), are plotted in Fig. 8. A correlation coefficient of linear regression is obtained and equal to 0.999. The data for TA15 M specimens, uPi and log10 Ti(i = 1.2.3), are plotted in Fig. 9. The correlation coefficient of linear regression is almost equal to 1.0.

14000 12000 10000

8000 6000 :

lgT=4.02878-0.18103uP

-0.5 0.0 0.5 1.0 1.5 2.0 2.5

Fig. 8. Fitting curve for the fatigue life distribution of 7B04-T74 specimens.

3 o -C

16000 14000 12000

^ 8000

lgr=4.159-0.17uj

0.0 0.5 1.0 1.5 2.0 2.5

Fig. 9. Fitting curve for the fatigue life distribution of TA15M specimens.

It can be shown that the distribution of fatigue life for the fleet agrees with lognormal distribution well. 4.2. Analysis of the variability of load damage rate

Load damage is inversely proportional to fatigue life, so, the load damage rate D0eq in Eq. (10) is also in inverse proportion to the fatigue life T because k is a constant. If the fatigue life follows a lognormal distribution, T ~ LN(iT, of), the Load damage rate would also follow a lognormal distribution, i.e. D0 eq ~ LN(id, o2D) with oD = oT.

5. Conclusions

(1) The load damage analysis and the fatigue testing for 5 individual load spectra show that an obvious difference of at least 2.4 times among the load damage rates occurs in the fleet investigated. The variability of load damage rates is significant and should be taken into account for fatigue life assessment and fleet management.

(2) The distribution of load damage rate can be delineated by a two-parameter lognormal distribution, and its variation is in the same order of magnitude with that of aircraft structures.

(3) It is simple and convenient to use the SWT formula and linear cumulative damage model in evaluating the relative load damage of a fleet. Though the load interaction and the nonlinear effect are not included in this model, the calculated relative damage severities could be acceptable. In order to accurately evaluate the relative damage, the local stress-strain method can be applied and the corresponding fatigue testing should be carried out.

Acknowledgement

The authors wish to acknowledge the support from the National Natural Science Foundation of China (Grant No. 11002009) and the support from the Fundamental Research Funds for the Central Universities.

References

[1] Anon. Joint service specification guide [JSSG-2006] Aircraft structure. 2nd ed. USA: Department of Defense; 2002 .

[2] Anon. Defense standard 00-970 part 1. Issue 2. Design and airworthiness requirements for service. Aircraft structures. UK Ministry of Defense; 1999.

[3] GJB 67A.6. Military airplane strength specification repeated loads, durability and damage tolerance. China; 2009 [in Chinese].

[4] Payne AO. The fatigue of aircraft structures. Engng Fract Mech 1976;8:157-203.

[5] Schijve J. Statistical distribution functions and fatigue of structures. IntJ Fatigue 2005;27:1031-9.

[6] Hoffman PC. Fleet management issues and technology needs. Int J Fatigue 2009;31:1631-7.

[7] Weibull W. Fatigue testing and analysis of results. Pergamon Press; 1961.

[8] Freudenthal AM, Gumbel EJ. On statistical interpretation of fatigue tests. Proc Roy Aeronaut Soc 1953;216:309-32.

[9] Cardrick AW, Mew AB. Scatter considerations in the interpretation of major fatigue tests. In: Proceedings of the ICAF symposium. Seattle, USA; 1999.

[10] Freudenthal AM. The scatter factor in the reliability assessment of aircraft structures. J Aircraft 1977;14:202-8.

[11] Abelkis PR, Potter JM. Service fatigue loads monitoring, simulation and analysis. In: A symposium sponsored by ASTM committee E-9 on fatigue. ASTMSTP-671, Atlanta, Ga; 1977.

[12] Socie DF, Pompetski MA. Modeling variability in service loading spectra. Editors: Jonhnson WS. Hillberry BM. Probabilistic Aspects of Life Prediction, ASTM STP 1450 2004; p. 46-57.

[13] Nagode M, Fadija M. On a new method for prediction of the scatter of loading spectra. Int J Fatigue 1998;20:271-7.

[14] Kaniss AM. Statistical review of counting accelerometer data for navy and a marine fleet aircraft from 1 Jan 1962 to 30 Jun 1977. RCS NADC 13920-2; 1977.

[15] De Jonge JB, Hol PA. Variation in load factor experience of Fokker 27 and F28 operational acceleration exceedance data. DOT/FAA/AR-96/114; 1996.

[16] De Jonge Jb. Monitoring load experience of individual aircraft. J Aircraft 1993;30:751-5.

[17] De Jonge JB. Load experience variability of fighter aircraft. In: Australian Aeronautical Conference (3rd: 1989: Melbourne, Vic.). Australian Aeronautical Conference, 1989: Research and Technology, the Next Decade, Melbourne, 9-11 October 1989; Preprints of Papers, The. Barton, ACT: Institution of Engineers, Australia, 1989: p. 102-8.

[18] Lincoln JW, Melliere RA. Economic life determination for a military aircraft. AIAA-98-25192; 1998.

[19] Meyer ES, Fields SS, Reid PA. Projecting aircraft fleet reliability. In: Aging Aircraft Conference, Orlando, Florida; 2001.

[20] Wang Z, Liu WT, Wang Lei. Study on the fatigue scatter factor for individual aircraft structure. J Mech Strength 2009;31:150-4 [in Chinese].

[21] Abelkis PR. Effect of transport aircraft wing loads spectrum variation on crack growth. In: Effect of load spectrum variables on fatigue crack initiation and propagation. ASTM STP 714; 1980. p. 143-69.

[22] Larson CE, White DJ, Gray TD. Evaluating spectrum effect in US air force attack/fighter/trainer individual aircraft tracking. In: Effect of load spectrum variables on fatigue crack initiation and propagation. ASTM STP 714; 1980. p. 218-27.

[23] Hill HD, Saff CT. Effect of fighter attack spectrum on crack, growth. AFFDL-TR-112; 1977.

[24] Barter S, Dixon B, Molent L. Assessing relative spectra severity using single fatigue test specimens. Eng Fail Anal 2009;16:863-73.

[25] Sui FC. Research on fatigue load spectra compilation methodologies for the evaluation of structural service life of new fighter. Doctor thesis in Beihang University; 2008 [in Chinese].

[26] Smith KN, Watson P, Topper TH. A stress-strain function for the fatigue of metals. J Mater 1970;5:767-78.

[27] Conle A, Topper T. Overstrain effects during variable amplitude service history testing. IntJ Fatigue 1980:130-6.

[28] Neuber H. Theory of stress concentration for shear-strained prismatic bodies with arbitrary nonlinear stress-strain law. J Appl Mech (Trans. ASME) 1961;28(12):544-9.

[29] Wu XR. Property handbook for the aeronautical material. China: Aviation Industry Press; 1998 [in Chinese].

[30] White P, Molent L, Barter S. Interpreting fatigue test results using a probabilistic fracture approach. IntJ Fatigue 2005;27:752-67.

[31] Hoffman ME, Hoffman PC. Corrosion and fatigue research - structural issues and relevance to naval aviation. IntJ Fatigue 2001;23(S1):1-10.

[32] Payne AO. Determination of the fatigue resistance of aircraft wings by full-scale testing. In: Proc symposium on full-scale fatigue testing of aircraft structures. Amsterdam; 1959. p. 76-132.