- Email: [email protected]
Safe-life analysis accounting for the loading spectra variability
Engineering Failure Analysis 17 (2010) 1213–1220
Contents lists available at ScienceDirect
Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Safe-life analysis accounting for the loading spectra variability Xiaofan He *, Bin Zhai, Yanmin Dong, Wenting Liu School of Aeronautical Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 10 November 2009 Received in revised form 9 February 2010 Accepted 10 February 2010 Available online 13 February 2010 Keywords: Safe-life Structure Load spectrum Scatter factor Reliability
a b s t r a c t The scatter of the variable load spectra is considered separately from the scatter of the aircraft structure to assess the safe-life of a fleet. Suppose the fatigue life under the specified load spectrum follows the log-normal distribution with a constant coefficient of variation, and the load spectra damage follows the log-normal distribution with a constant standard deviation, the theoretical distribution function of fatigue life of the fleet that accounts for the variability of the structure and the load spectrum is derived with the conditional probability model. Monte-Carlo simulation shows that the distribution of the fleet fatigue life can be described approximately by a log-normal distribution function, and the log expectation is the log mean life under variable load spectra and the variation equals the sum of the structural variation and the load spectra variation. The scatter factor method is then adopted to assess the safe-life. The scatter factor of the individual aircraft with definite load spectrum, as well as the scatter factor of the in-service fleet that takes the actual load scatter into consideration are listed. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction During its whole service life period, landings, takeoffs and maneuvers subject the structure of an aircraft to variable amplitude load, causing fatigue cracks to form in the structure. Fatigue is one of the primary mechanisms causing the deterioration of an aircraft structure, and fatigue life is thus a critical parameter related to the safety and economical operation. However, the fatigue life of aircraft structures is characterized by obvious randomness due to the influences of various random factors. In order to ensure the structural safety, the structural reliability analysis must be performed to determine the safe-life of the aircraft structure [1]. Regarding the life reliability research under the specified load spectrum, Weibull used Weibull distribution for the first time to describe the strength distribution of solid materials, applying the probability reliability theory to structural materials [2]; Freudenthal used the method of probability risk analysis to study the aircraft structural reliability and fatigue life variability, and presented the scatter factor to determine the safe-life of the structure [1,3]. A large number of theoretical analyses and experimental data show that the log-normal distribution LN(l, r2) can be used to describe the fatigue life distribution of military aircrafts [4]. The values of r for the commonly used metal materials and built-up structures are also given [4–6]. Meanwhile, the service reliability requirements for the aircraft structure are proposed in [7]. Therefore, the typical scatter factors used for fatigue analysis and full-scale fatigue test are gained [7–11]. This scatter factor method has been broadly and successfully applied in engineering, and has been written in civil specifications and military standards. With the further understanding of fatigue life variability, there has been growing recognition that the variable nature of fatigue life is not only related to the structural property, but also to the variability of the load spectra [12–14]. In [15], the scatter of load spectra is described. In [16], Barter et al. researched the differences of the damage caused by the predicted * Corresponding author. Tel.: +86 10 82315738 19. E-mail address: [email protected] (X. He). 1350-6307/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfailanal.2010.02.007
1214
X. He et al. / Engineering Failure Analysis 17 (2010) 1213–1220
service spectrum and the actual service spectrum. In [17], Hoffman shows the influence of the load scatter on the distribution of the fatigue life. In [18], the author listed DEF STAN 970 and JSSG-2006 that account for the scatter of the load. However, the effect of spectrum variation is not explicitly considered, the traditional scatter factor adopts the relatively large standard deviation value of the fatigue life, such as 0.176 or 0.2, which is considered to account for both the structural and the load variations to some extent, or the additional load scatter factor, such as 1.5 [7,17,19], to ensure the service safety of the fleet structures. However, the standard deviation of the log fatigue life under the specified load spectrum is usually 0.1–0.14 for the commonly used built-up metal structure, the authors also pointed out the large standard deviation of 0.2 had an unclear origin in [19], therefore the reliability is indefinite and so is the 1.5. With the current implementation of individual aircraft tracking (IAT) and individual aircraft monitoring (IAM), flight loads parameters are gathered by installing the flight data recorder(FDR) system. A large number of load-time history data of individual aircrafts of the same type and usage are recorded, and the load spectra variability is attracting more and more attention. On the basis of the available data of 200 F-15E, the Ref. [12] used the log-normal distribution to describe the scatter of the load damage. Therefore, it is possible to conduct the fatigue life reliability analysis that accounts for both the variability of structural property and the load spectra to assess the structural safe-life. According to the scatter factor method, the core of assessing the safe-life of the fleet is the distribution function with the parameters of the fatigue life. In [14], when the standard deviation of the fatigue life under the specified load spectrum is constant, the fatigue life distribution of a fleet is obtained by using conditional probability model. And the experiences have shown that the variation of the structural life is relative to the log mean life [20]. Taking this into consideration, this paper conducts an analysis on the life distribution accounting for the structural and load scatter, and gives convenience for assessing the safe-life of individual aircraft and in-service fleet based on the scatter factor method.
2. Assumptions In the following parts of the paper, we made some hypotheses spontaneously: a. The factors that will influence the fatigue life variation can be classified mainly into two types: inherent and external scatter characteristics. The inherent scatter characteristics refers to the structural variability that are caused by such factors as material, manufacturing, assembly, etc. The structural variability can be denoted as the life variability under the specified load spectrum. The external scatter characteristics refers to the variations in the operational conditions (load and environmental conditions), which can be described by the fatigue life variability under the variable load spectra. b. Both structural and load spectra variables could be described by continuous random variable. c. Load spectrum is determined by the structural property and operational methods. It is generally thought that with the specified structure, the operational method is the only factor affecting the life variability. And the structural and load spectra variable are independent of each other. d. Both of the two variables must be taken into account in determining the safe-life of fleet aircrafts.
3. Structural variability under the specified load spectra The variability of the structural fatigue life is generally described by using the normal distribution, log-normal distribution, Weibull distribution and Gumbel distribution. The two-parameter log-normal distribution is generally adopted to describe the fatigue life distribution of the fighter aircraft structure [4,21]. Suppose the fatigue life under the specified load spectrum is N, let X = lgN, then the probability density function (pdf) of X is as follows
( " 2 #) 1 1 x lS fS ðxÞ ¼ pffiffiffiffiffiffi exp 2 rS 2prS
ð1Þ
where lS is the log mean life under the specified load spectrum; rS is the standard deviation of X and reflects the structural variability; S represents the structure. Eq. (1) is a typical conditional probability distribution fS|L(x|lS), which refers to the log life distribution under the specified load spectrum, where L represents the load spectrum. When the fatigue life of the structure is affected by many random factors, and its failure follows the proportion or product model, log-normal distribution can be a very good description of the structural life distribution. The fatigue load that causes fatigue damage to engineering structures may vary in magnitude and direction, and the variation is often random for the aircraft (e.g., in the case of gust events) or semi-random (e.g., flight maneuver load). Two simple methods for modeling the variation of the log fatigue life for the aircraft structure under variable amplitude loading are as follows:
X. He et al. / Engineering Failure Analysis 17 (2010) 1213–1220
1215
a. rS = constant, that is, rS, the constant standard deviation, is unchanged regardless of the stress levels and the spectrum severity. This case is discussed in [14]. b. The constant coefficient of variation (cov) method assumes that the ratio of the standard deviation to the log mean life at any stress level is constant, i.e., k = rS/lS = constant [20]. This is a linear different variation phenomenon. The standard deviation is related to the log mean life of the structure. Therefore, the pdf of the log life is
( " 2 #) 1 1 x lS fS ðxÞ ¼ fSjL ðxjlS Þ ¼ pffiffiffiffiffiffi exp 2 k lS 2p k lS
ð2Þ
4. Load spectra variability 4.1. Reasons for load spectra variability Load spectra variability includes the following two aspects: first, the operational load spectra that the aircraft is subjected to almost always deviate from the assumed design usage spectra due to variations in the operational environments and situations. Second, even though the aircraft is put into service according to the design operational requirements, the operational environments of the fleet aircrafts will be different because of the differences in piloting technique, runway quality, aircraft weight, climate, etc. Therefore, the load spectra of individual aircrafts in a fleet are variable. In this paper, we focus on the second variability, referring to the usage variation under the specified usage requirement. 4.2. Load spectra variability With the implementation of IAT and IAM, considerable actual load-time historical parameters captured by instrumentations on an aircraft structure are collected. Calculation of the fatigue damage corresponding to per base life period as well as statistical analysis on the fleet aircraft damage shows that the damage follows a log-normal distribution and the pdf is
( " 2 #) 1 1 y l0 fL ðyÞ ¼ pffiffiffiffiffiffi exp 2 rL 2prL
ð3Þ
where y = lgD, refers to the log damage corresponding to a base life period; l0 is the log mean damage; rL is the standard deviation of the log damage. Using D50 ¼ 10l0 to represent the average damage, we can define the relative damage as DR = D/D50. It is obvious that DR follows a log-normal distribution and its standard deviation is equal to the standard deviation of lgD. The pdf of the relative damage (the severity ratio) of US Air Force (USAF) F-15E aircraft fleet is shown in Fig. 6 in [12]. Let T0 stand for a basic life period, the relationship of N and D is
N ¼ T 0 =D
ð4Þ
lg N ¼ lg T 0 lg D
Clearly, the life N follows the log-normal distribution with unchanged standard deviation, and l0 is still used to represent the log mean life. The load spectra variability can be described by the life under variable load spectra for the specified structures. Its pdf is as follows
( " 2 #) 1 1 lS l0 fL ðlS Þ ¼ pffiffiffiffiffiffi exp 2 rL 2prL
ð5Þ
where lS = lgN in Eqs. (1) and (2) is the log life under the specified load spectrum which is relative to the load spectra severity. l0 represents log mean life accounting for the variability of load spectra in a fleet or the log life under average spectrum. rL also represents the standard deviation of the specified structure under different spectra and relates to the operational environment. According to the literatures available, there are various values for rL. FAA cites a value of 0.12 for small airplane [13,22]; USAF proposed a value of 0.13 based on approximate 200 F-15E aircraft operational data [12]. If D e [0, 1], then lS 2[1, +1]. For Eq. (2), lS < 0 implies that rS < 0, which is unreasonable. Thus, the interceptive distribution of Eq. (5) must be taken as follows:
( " 2 #) 1 1 1 lS l0 pffiffiffiffiffiffi exp fL ðlS Þ ¼ 1 P 2prL 2 rL
lS P 0
ð6Þ
where according to Eq. (5), P ¼ PflS < 0g means the probability of lS < 0. Because the safe-life of modern military aircraft structures is 103 flight hours at least, it means that l0 > 3 and taking the commonly used values of rL, the probability of lS < 0 is <1010 as calculated by EXCEL, we can get directly that P = 0 in Eq. (6).
1216
X. He et al. / Engineering Failure Analysis 17 (2010) 1213–1220
5. Fleet aircraft life distribution accounting for both load spectra variability and structural variability 5.1. The conditional probability model If the random variables of the load spectra and the structure property are independent of each other, according to the conditional probability formula, the joint pdf, fS,L(x, lS), of load spectra and the random variable lgN can be obtained as follows:
fS;L x; lS ¼ fSjL ðxjlS Þ fL ðlS Þ
ð7Þ
By Eqs. (2), (6), and (7), we can get
( " 2 #) 1 1 x lS lS l0 2 ; fS;L ðx; lS Þ ¼ exp þ 2p k lS rL 2 k lS rL
lS P 0
ð8Þ
The pdf of the fleet aircraft life is established as follows:
f ðxÞ ¼
Z 0
1
fS;L ðx; lS ÞdlS ¼
Z 0
1
( " 2 #) 1 1 x lS lS l0 2 dlS ; exp þ 2p k lS rL 2 k lS rL
lS P 0
ð9Þ
It is obvious that the life distribution is related to k and l0. 5.2. Fleet aircraft life distribution 5.2.1. The hypothesis test on the life distribution For the convenient usage in engineering, it is necessary to simplify the Eq. (9), but it has no analytical solution. Taking the typical parameter values, we get the pdf mapped by Mathematica software (see Fig. 1). From Fig. 1, we can see that Eq. (9) is very close to a normal distribution function. We propose the following hypothesis: fleet aircraft life follows a log-normal distribution or the fleet aircraft log life follows a normal distribution. The Monte-Carlo numerical simulation technique is used to test this hypothesis, and the procedures are as follows: a. For the specified l0 and rL, the Monte-Carlo simulation samples are used to generate randomly M samples, that is, lS,i(i = 1, . . . , M) of lS by Eq. (6); b. For the specified lS,i, by using Monte-Carlo simulation, we get randomly K samples X, that is, xi,j (j = 1, . . . , K) by Eq. (2); c. Integrate (a) with (b) to get M K samples, that is, xi,j (i = 1, . . . , M; j = 1, . . . , K), and then use a chi-square test to check whether X is subject to a normal distribution. If take a = 0.05, e.g., as a significant degree level and M = K > 5000, the hypothesis test shows that X has a good correlation to the normal distribution by using numerical analysis software Mathematica. 5.2.2. Approximate theoretical distribution Suppose the fleet aircraft life follows the log-normal distribution LNðlS;L ; r2S;L Þ, the distribution parameters can be estimated approximately as follows:
Fig. 1. The probability density function corresponding to typical parameter values.
1217
X. He et al. / Engineering Failure Analysis 17 (2010) 1213–1220
(
lS;L ¼ EðXÞ r2S;L ¼ VarðXÞ
ð10Þ
where E is the expectation of random variable; Var refers to the variation of random variable. If k = 0.01, 0.02, rL = 0.06, 0.08, 0.11, 0.13, and l0 = 3, 4, 5, 6 the typical parameters estimated by Mathematica can be obtained and they are shown in Table 1. A comparison of the data in Table 1 can find that errors between l0 and lS;L , r2S;L and ðkl0 Þ2 þ r2L are small enough to be ignored. Thus, the fleet life distribution parameters accounting for both the load spectra and structural variability are as follows
(
lS;L ¼ l0 r2S;L ¼ r2L þ ðk l0 Þ2
ð11Þ
A comparison of the result with the condition of rS = constant as shown in [14] finds that the fleet aircraft life distribution functions have the same form. Theoretical pdf of the fleet aircraft life is as follows:
fN ðNÞ ¼
Z
1
2 lS 2 exp 12 lgkN þ lSrLl0 l S
2p ln 10 k lS rL N
0
dlS
ð12Þ
Probability density function of the approximate distribution is as follows:
ðlg Nl0 Þ2 exp 12 ðk l0 Þ2 þr2L fN ðNÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i 2p ðk l0 Þ2 þr2L ln 10 N
ð13Þ
Take the typical values of l0 , rL and k, the life distribution functions are shown in Fig. 2. respectively calculated by Eqs. (12) and (13). Table 1 Typical parameter values. k
rL
l0
lS,L
r2S;L
ðk l0 Þ2 þ r2L
0.01
0.06 0.08 0.11 0.13
3 4 5 6
3 + 6.8e12 4 5 + 1.0e12 6 + 1.0e12
0.0045004 0.0080006 0.0146012 0.0205017
0.0045 0.008 0.0146 0.0205
0.02
0.06 0.08 0.11 0.13
3 4 5 6
3 + 7e13 4 5 + 1.0e12 6 + 5.0e12
0.0072014 0.0128026 0.0177277 0.0072004
0.0072 0.0128 0.017725 0.0072
The comparison between the theoretical and approximate parameters.
Fig. 2. Comparision between the theoretical and approximate distribution functions.
1218
X. He et al. / Engineering Failure Analysis 17 (2010) 1213–1220
It is obvious that the theoretical and approximate distribution curves are very similar and even completely coincide with each other. Suppose l0 , rL and k are of the same value as in Fig. 3, the relative errors of the distribution variate for the specified probability between the theoretical and approximate distribution are shown in Fig. 3. The result shows that the relative errors are <5% when P e (0.00001, 0.99999), <2% when P e (0.001, 0.999), and the approximate distribution is thus acceptable. 6. The safe-life analysis In order to ensure the safe use of the aircraft structure, the safe-life NP corresponding to the reliability P must be determined, that is
PfN > NP g ¼ P
ð14Þ
According to [7,23–25], the failure probability of military aircraft structure in its total life should be controlled less than 103, that is, the acceptable level of reliability degree is P = 99.9%. 6.1. The scatter factor Scatter factor is widely used to determine the structural safe-life, and when the fatigue life follows the log-normal distribution, it can be defined as [1,8]
Lf ¼ N50 =N P
ð15Þ
where N50 is the mean life corresponding to 50% survival probability There are usually two kinds of expressions about the scatter factor, as follows:
Lf ¼ 10uP r0 Lf ¼ 10uP r0
ð16Þ pffiffiffiffiffiffi1 1þn
ð17Þ
where P is the survival probability, and usually P = 99.9%, uP is the normal distribution variate for the specified P, r0 is the standard deviation of the log fatigue life and n is the number of specimens used to test. 6.2. Test scatter factor The full-scale fatigue test is conducted under the spectrum that is considered to represent the average usage of the fleet, then the safe-life is determined by dividing the tested life by the test scatter factor. A large standard deviation based on the experience is taken, such as r0 = 0.15, 0.176, 0.20 and n = 1, 2, 3, P = 99.9%, the typical values of the test scatter factor calculated by Eq. (17) are shown in Table 2. Generally, the test scatter factor which accounts for the scatter of the structure and the load experientially is taken as Lf = 4–6. 6.3. Individual fatigue scatter factor When the load spectrum is specified, the scatter of the fatigue life of the individual aircraft is caused only by structural variability. As the standard deviation is 0.12, and the fatigue life is 104 flight hours, it follows that k = 0.12/4 = 0.03, take lS = 3, 4, 5. The individual fatigue scatter factors calculated by Eq. (17) are listed in Table 2.
Fig. 3. The relative errors of the distribution variate for the specified probability between the theoretical and the approximate distribution.
1219
X. He et al. / Engineering Failure Analysis 17 (2010) 1213–1220 Table 2 The fatigue scatter factors.
r0
rL
k
lS
rS,L
0.15 0.176 0.2
n
Notes
1
2
3
...
1
4.52 5.88 7.48
3.70 4.64 5.71
3.43 4.25 5.17
... ... ...
2.91 3.50 4.15
Fleet (test)
0 0 0
0.03
3 4 5
0.09 0.12 0.15
2.47 3.35 4.52
2.19 2.85 3.70
2.09 2.68 3.43
... ... ...
1.90 2.35 2.91
Individual aircraft
0.12
0.03
3 4 5
0.150 0.170 0.192
4.52 5.52 6.91
3.70 4.39 5.33
3.43 4.03 4.85
... ... ...
2.91 3.35 3.92
Fleet (in-service)
Three kinds of scatter factors. pffiffiffiffiffiffi1 pffiffiffiffiffiffi1 The scatter factors were calculated by Lf ¼ 10uP r0 1þn or Lf ¼ 10uP rS;L 1þn .
It can be seen clearly that the fatigue scatter factor of individual aircraft is far less than that of the fleet aircrafts because of taking no account of the scatter of the load spectrum, the typical value of individual scatter factor is 2.0 in JSSG-2006 for US Navy and USAF aircraft [24–26], 3.33 in DEF STAN 970 [7]. 6.4. The in-service scatter factor The in-service scatter factor should consider the variation of structure and the actual load spectra. Just as the r0, the rS,L, reflects the scatters of both the structure and the load in this situation and the scatter factor can be calculated by replacing the r0 with the rS,L. r0 is determined by historical service experience, and rS,L is related to the actual usage process. By taking the typical parameters, that is, k = 0.03, rL = 0.12, and lS = 3, 4, 5, the standard deviations of fleet fatigue life are calculated by Eq. (11), and the in-service scatter factor of the fleet are calculated by Eq. (17). The calculation results are included in Table 2. Compared with the experiential scatter factor, it is determined by the real situations. In consideration of the actual usage of the fleet aircrafts to assess the baseline service life of the aircraft, we need not only re-develop the baseline load spectrum because the actual spectrum differs from the predicted test spectrum [25], but also reevaluate the variation of the fleet fatigue life by Eq. (11) to adjust the safe-life according to the full-scale fatigue test result. 7. Conclusion On the basis of the discussion above, we can draw the following results: a. When the fatigue life under the specified load spectrum follows LNðlS ; ðk lS Þ2 Þ, and the life under variable load spectrum follows the LNðl0 ; r2L Þ, the life of a fleet can be subjected to a log-normal distribution LNðl0 ; ðk lS Þ2 þ r2L Þ, which means that log mean life accounting for both the load spectra and structural variability is equal to the log mean life under the variable load spectra, and the variation is equal to the sum of the variations of the fatigue life and the loading spectrum damage. Traditionally, it is considered that the fatigue life of the fleet follows the log-normal distribution with a large standard deviation, such as 0.176 or 0.2, which is usually adopted to represent the scatter of the fatigue life. However, the reason why the value of 0.176 or 0.2 was chose is not very clear [19]. In [5], Payne ever suggested to adopt the value of 0.2, but did not explain why the value of 0.2 was chose. On the basis that the load damage follows the log-normal distribution in the fleet, the present paper makes the conclusion that the fatigue life of the fleet accounting for the scatter of the structure property and the load spectrum can also be subjected to the log-normal distribution through deduction and numerical simulation. The distribution is in accordance with the conventional opinion and clarifies the relationship of the variations, providing a possible interpretation for the origin of the standard deviation; b. If the IAT is implemented, the individual fatigue scatter factor is far less than the scatter factor of fleet aircrafts because the spectrum of each aircraft is definite and the scatter factor of the fleet may be different from the test scatter factor. When performing the individual aircraft life management, the scatter factor should be re-estimated according to the actual scatter of the load spectrum to re-assess the safe-life; c. The safe-life assessed by full-scale fatigue test that is under the predicted spectrum and the experiential variation of the load is a baseline life. When the aircraft is put into service, the spectrum and the variation of the load spectrum should be different from the predicted one. If the aircraft structure is managed on the basis of the baseline safe-life, it may be unsafe or uneconomic. If possible, the IAT and IAM can be implemented to access the load scatter of the aircraft. When determining the baseline life of the aircraft, we should reconsider the fleet loading severity and the load spectra variability.
1220
X. He et al. / Engineering Failure Analysis 17 (2010) 1213–1220
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
Freudenthal AM. The scatter factor in the reliability assessment of aircraft structures. J Aircraft 1977(January):202–8. Weibull W. A statistical representation of fatigue failure in solids. Acta Polytech 1949;1(9). Freudenthal AM, Gumbel EJ. On statistical interpretation of fatigue tests. Proc Roy Aeronaut Soc 1953:216. Hoskin BC, Ford DG. Scatter factors in aircraft fatigue life estimation. ARL/SM 350, Melbourne: Aeronautical Research Laboratory; 1970. 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; June 1959. p. 76–132. Parish HE. Fatigue test results and analysis of 42 Piston Provost wings. Reports and Memoranda no. 3474; April, 1965. Anon. Defense standard 00-970 part I. Issue 3 design and airworthiness requirement for service. Aircraft structures. UK Ministry of Defense; 2003. Zhentong Gao. The fatigue scatter factor for aircraft structures[R]. Beijing University of Aeronautics and Astronautics; 1982. p. 3–9 [BH-B884] [in Chinese]. Fuze Zhang. The type and value of fatigue scatter factor. Acta Aeronaut Astronaut Sinica 1987;6(B):239–43 [in Chinese]. Naikang Shi. Normalize the fatigue life and standard variation. Acta Aeronaut Astronaut Sinica 1987;8(10):B467–474 [in Chinese]. Cardrick AW, Mew AB. Scatter considerations in the interpretation of major fatigue tests. In: Proceedings of the ICAF symposium, Seattle, USA; 1999. Lincoln JW. Economic life determination for a military aircraft. AIAA-98-25192; 1998. Michael Reyer W. Administration probability basis of fatigue evaluations in small airplanes. In: 9th joint FAA/DoD/NASA aging aircraft conference; March 8, 2006. Xiaofan He, Yanmin Dong, Wenting Liu. Reliability analysis on the fatigue life with the separated structural and load spetra variation. Acta Aeronaut Astronaut Sinica, in press [in Chinese]. Schijve J. Statistical distribution functions and fatigue of structures. Int J Fatigue 2005;27:1031–9. Barter S, Dixon B, Molent L. Assessing relative spectra severity using single fatigue test coupons. Eng Fail Anal 2009;16:863–73. Hoffman Paul C. Fleet management issues and technology needs. Int J Fatigue 2009;31:1631–7. Molent L, Barter SA. A comparison of crack growth behaviour in several full-scale airframe fatigue tests. Int J Fatigue 2007;29:1090–9. White Paul, Molent Loris, Barter Simon. Interpreting fatigue test results using a probabilistic fracture approach. Int J Fatigue 2005;27:752–67. White Paul. Review of methods and approaches for the structural risk assessment of aircraft. Australian; 2006 [DSTO–TR–1916]. Schijve J. Fatigue of structures and materials in the 20th century and the state of the art. Int J Fatigue 2003;25:679–702. OT/FAA/CT-91/20. General aviation aircraft-normal acceleration data analysis and collection project; February 1993. Payne AO. The fatigue of aircraft structures. Eng Fract Mech 1976;8:157–203. Anon. Joint service specification guide. Aircraft structures. USA: Department of Defense; 2002 [JSSG-2006]. MIL-STD-1530C. Department of Defense Standard Practice. Aircraft Structural Integrity Program (ASIP). USA: Department of Defense; November 2005. Hoffman Margery E, Hoffman Paul C. Corrosion and fatigue research-structural issues and relevance to naval aviation. Int J Fatigue 2001;23(S1):1–10.
Contents lists available at ScienceDirect
Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Safe-life analysis accounting for the loading spectra variability Xiaofan He *, Bin Zhai, Yanmin Dong, Wenting Liu School of Aeronautical Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 10 November 2009 Received in revised form 9 February 2010 Accepted 10 February 2010 Available online 13 February 2010 Keywords: Safe-life Structure Load spectrum Scatter factor Reliability
a b s t r a c t The scatter of the variable load spectra is considered separately from the scatter of the aircraft structure to assess the safe-life of a fleet. Suppose the fatigue life under the specified load spectrum follows the log-normal distribution with a constant coefficient of variation, and the load spectra damage follows the log-normal distribution with a constant standard deviation, the theoretical distribution function of fatigue life of the fleet that accounts for the variability of the structure and the load spectrum is derived with the conditional probability model. Monte-Carlo simulation shows that the distribution of the fleet fatigue life can be described approximately by a log-normal distribution function, and the log expectation is the log mean life under variable load spectra and the variation equals the sum of the structural variation and the load spectra variation. The scatter factor method is then adopted to assess the safe-life. The scatter factor of the individual aircraft with definite load spectrum, as well as the scatter factor of the in-service fleet that takes the actual load scatter into consideration are listed. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction During its whole service life period, landings, takeoffs and maneuvers subject the structure of an aircraft to variable amplitude load, causing fatigue cracks to form in the structure. Fatigue is one of the primary mechanisms causing the deterioration of an aircraft structure, and fatigue life is thus a critical parameter related to the safety and economical operation. However, the fatigue life of aircraft structures is characterized by obvious randomness due to the influences of various random factors. In order to ensure the structural safety, the structural reliability analysis must be performed to determine the safe-life of the aircraft structure [1]. Regarding the life reliability research under the specified load spectrum, Weibull used Weibull distribution for the first time to describe the strength distribution of solid materials, applying the probability reliability theory to structural materials [2]; Freudenthal used the method of probability risk analysis to study the aircraft structural reliability and fatigue life variability, and presented the scatter factor to determine the safe-life of the structure [1,3]. A large number of theoretical analyses and experimental data show that the log-normal distribution LN(l, r2) can be used to describe the fatigue life distribution of military aircrafts [4]. The values of r for the commonly used metal materials and built-up structures are also given [4–6]. Meanwhile, the service reliability requirements for the aircraft structure are proposed in [7]. Therefore, the typical scatter factors used for fatigue analysis and full-scale fatigue test are gained [7–11]. This scatter factor method has been broadly and successfully applied in engineering, and has been written in civil specifications and military standards. With the further understanding of fatigue life variability, there has been growing recognition that the variable nature of fatigue life is not only related to the structural property, but also to the variability of the load spectra [12–14]. In [15], the scatter of load spectra is described. In [16], Barter et al. researched the differences of the damage caused by the predicted * Corresponding author. Tel.: +86 10 82315738 19. E-mail address: [email protected] (X. He). 1350-6307/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfailanal.2010.02.007
1214
X. He et al. / Engineering Failure Analysis 17 (2010) 1213–1220
service spectrum and the actual service spectrum. In [17], Hoffman shows the influence of the load scatter on the distribution of the fatigue life. In [18], the author listed DEF STAN 970 and JSSG-2006 that account for the scatter of the load. However, the effect of spectrum variation is not explicitly considered, the traditional scatter factor adopts the relatively large standard deviation value of the fatigue life, such as 0.176 or 0.2, which is considered to account for both the structural and the load variations to some extent, or the additional load scatter factor, such as 1.5 [7,17,19], to ensure the service safety of the fleet structures. However, the standard deviation of the log fatigue life under the specified load spectrum is usually 0.1–0.14 for the commonly used built-up metal structure, the authors also pointed out the large standard deviation of 0.2 had an unclear origin in [19], therefore the reliability is indefinite and so is the 1.5. With the current implementation of individual aircraft tracking (IAT) and individual aircraft monitoring (IAM), flight loads parameters are gathered by installing the flight data recorder(FDR) system. A large number of load-time history data of individual aircrafts of the same type and usage are recorded, and the load spectra variability is attracting more and more attention. On the basis of the available data of 200 F-15E, the Ref. [12] used the log-normal distribution to describe the scatter of the load damage. Therefore, it is possible to conduct the fatigue life reliability analysis that accounts for both the variability of structural property and the load spectra to assess the structural safe-life. According to the scatter factor method, the core of assessing the safe-life of the fleet is the distribution function with the parameters of the fatigue life. In [14], when the standard deviation of the fatigue life under the specified load spectrum is constant, the fatigue life distribution of a fleet is obtained by using conditional probability model. And the experiences have shown that the variation of the structural life is relative to the log mean life [20]. Taking this into consideration, this paper conducts an analysis on the life distribution accounting for the structural and load scatter, and gives convenience for assessing the safe-life of individual aircraft and in-service fleet based on the scatter factor method.
2. Assumptions In the following parts of the paper, we made some hypotheses spontaneously: a. The factors that will influence the fatigue life variation can be classified mainly into two types: inherent and external scatter characteristics. The inherent scatter characteristics refers to the structural variability that are caused by such factors as material, manufacturing, assembly, etc. The structural variability can be denoted as the life variability under the specified load spectrum. The external scatter characteristics refers to the variations in the operational conditions (load and environmental conditions), which can be described by the fatigue life variability under the variable load spectra. b. Both structural and load spectra variables could be described by continuous random variable. c. Load spectrum is determined by the structural property and operational methods. It is generally thought that with the specified structure, the operational method is the only factor affecting the life variability. And the structural and load spectra variable are independent of each other. d. Both of the two variables must be taken into account in determining the safe-life of fleet aircrafts.
3. Structural variability under the specified load spectra The variability of the structural fatigue life is generally described by using the normal distribution, log-normal distribution, Weibull distribution and Gumbel distribution. The two-parameter log-normal distribution is generally adopted to describe the fatigue life distribution of the fighter aircraft structure [4,21]. Suppose the fatigue life under the specified load spectrum is N, let X = lgN, then the probability density function (pdf) of X is as follows
( " 2 #) 1 1 x lS fS ðxÞ ¼ pffiffiffiffiffiffi exp 2 rS 2prS
ð1Þ
where lS is the log mean life under the specified load spectrum; rS is the standard deviation of X and reflects the structural variability; S represents the structure. Eq. (1) is a typical conditional probability distribution fS|L(x|lS), which refers to the log life distribution under the specified load spectrum, where L represents the load spectrum. When the fatigue life of the structure is affected by many random factors, and its failure follows the proportion or product model, log-normal distribution can be a very good description of the structural life distribution. The fatigue load that causes fatigue damage to engineering structures may vary in magnitude and direction, and the variation is often random for the aircraft (e.g., in the case of gust events) or semi-random (e.g., flight maneuver load). Two simple methods for modeling the variation of the log fatigue life for the aircraft structure under variable amplitude loading are as follows:
X. He et al. / Engineering Failure Analysis 17 (2010) 1213–1220
1215
a. rS = constant, that is, rS, the constant standard deviation, is unchanged regardless of the stress levels and the spectrum severity. This case is discussed in [14]. b. The constant coefficient of variation (cov) method assumes that the ratio of the standard deviation to the log mean life at any stress level is constant, i.e., k = rS/lS = constant [20]. This is a linear different variation phenomenon. The standard deviation is related to the log mean life of the structure. Therefore, the pdf of the log life is
( " 2 #) 1 1 x lS fS ðxÞ ¼ fSjL ðxjlS Þ ¼ pffiffiffiffiffiffi exp 2 k lS 2p k lS
ð2Þ
4. Load spectra variability 4.1. Reasons for load spectra variability Load spectra variability includes the following two aspects: first, the operational load spectra that the aircraft is subjected to almost always deviate from the assumed design usage spectra due to variations in the operational environments and situations. Second, even though the aircraft is put into service according to the design operational requirements, the operational environments of the fleet aircrafts will be different because of the differences in piloting technique, runway quality, aircraft weight, climate, etc. Therefore, the load spectra of individual aircrafts in a fleet are variable. In this paper, we focus on the second variability, referring to the usage variation under the specified usage requirement. 4.2. Load spectra variability With the implementation of IAT and IAM, considerable actual load-time historical parameters captured by instrumentations on an aircraft structure are collected. Calculation of the fatigue damage corresponding to per base life period as well as statistical analysis on the fleet aircraft damage shows that the damage follows a log-normal distribution and the pdf is
( " 2 #) 1 1 y l0 fL ðyÞ ¼ pffiffiffiffiffiffi exp 2 rL 2prL
ð3Þ
where y = lgD, refers to the log damage corresponding to a base life period; l0 is the log mean damage; rL is the standard deviation of the log damage. Using D50 ¼ 10l0 to represent the average damage, we can define the relative damage as DR = D/D50. It is obvious that DR follows a log-normal distribution and its standard deviation is equal to the standard deviation of lgD. The pdf of the relative damage (the severity ratio) of US Air Force (USAF) F-15E aircraft fleet is shown in Fig. 6 in [12]. Let T0 stand for a basic life period, the relationship of N and D is
N ¼ T 0 =D
ð4Þ
lg N ¼ lg T 0 lg D
Clearly, the life N follows the log-normal distribution with unchanged standard deviation, and l0 is still used to represent the log mean life. The load spectra variability can be described by the life under variable load spectra for the specified structures. Its pdf is as follows
( " 2 #) 1 1 lS l0 fL ðlS Þ ¼ pffiffiffiffiffiffi exp 2 rL 2prL
ð5Þ
where lS = lgN in Eqs. (1) and (2) is the log life under the specified load spectrum which is relative to the load spectra severity. l0 represents log mean life accounting for the variability of load spectra in a fleet or the log life under average spectrum. rL also represents the standard deviation of the specified structure under different spectra and relates to the operational environment. According to the literatures available, there are various values for rL. FAA cites a value of 0.12 for small airplane [13,22]; USAF proposed a value of 0.13 based on approximate 200 F-15E aircraft operational data [12]. If D e [0, 1], then lS 2[1, +1]. For Eq. (2), lS < 0 implies that rS < 0, which is unreasonable. Thus, the interceptive distribution of Eq. (5) must be taken as follows:
( " 2 #) 1 1 1 lS l0 pffiffiffiffiffiffi exp fL ðlS Þ ¼ 1 P 2prL 2 rL
lS P 0
ð6Þ
where according to Eq. (5), P ¼ PflS < 0g means the probability of lS < 0. Because the safe-life of modern military aircraft structures is 103 flight hours at least, it means that l0 > 3 and taking the commonly used values of rL, the probability of lS < 0 is <1010 as calculated by EXCEL, we can get directly that P = 0 in Eq. (6).
1216
X. He et al. / Engineering Failure Analysis 17 (2010) 1213–1220
5. Fleet aircraft life distribution accounting for both load spectra variability and structural variability 5.1. The conditional probability model If the random variables of the load spectra and the structure property are independent of each other, according to the conditional probability formula, the joint pdf, fS,L(x, lS), of load spectra and the random variable lgN can be obtained as follows:
fS;L x; lS ¼ fSjL ðxjlS Þ fL ðlS Þ
ð7Þ
By Eqs. (2), (6), and (7), we can get
( " 2 #) 1 1 x lS lS l0 2 ; fS;L ðx; lS Þ ¼ exp þ 2p k lS rL 2 k lS rL
lS P 0
ð8Þ
The pdf of the fleet aircraft life is established as follows:
f ðxÞ ¼
Z 0
1
fS;L ðx; lS ÞdlS ¼
Z 0
1
( " 2 #) 1 1 x lS lS l0 2 dlS ; exp þ 2p k lS rL 2 k lS rL
lS P 0
ð9Þ
It is obvious that the life distribution is related to k and l0. 5.2. Fleet aircraft life distribution 5.2.1. The hypothesis test on the life distribution For the convenient usage in engineering, it is necessary to simplify the Eq. (9), but it has no analytical solution. Taking the typical parameter values, we get the pdf mapped by Mathematica software (see Fig. 1). From Fig. 1, we can see that Eq. (9) is very close to a normal distribution function. We propose the following hypothesis: fleet aircraft life follows a log-normal distribution or the fleet aircraft log life follows a normal distribution. The Monte-Carlo numerical simulation technique is used to test this hypothesis, and the procedures are as follows: a. For the specified l0 and rL, the Monte-Carlo simulation samples are used to generate randomly M samples, that is, lS,i(i = 1, . . . , M) of lS by Eq. (6); b. For the specified lS,i, by using Monte-Carlo simulation, we get randomly K samples X, that is, xi,j (j = 1, . . . , K) by Eq. (2); c. Integrate (a) with (b) to get M K samples, that is, xi,j (i = 1, . . . , M; j = 1, . . . , K), and then use a chi-square test to check whether X is subject to a normal distribution. If take a = 0.05, e.g., as a significant degree level and M = K > 5000, the hypothesis test shows that X has a good correlation to the normal distribution by using numerical analysis software Mathematica. 5.2.2. Approximate theoretical distribution Suppose the fleet aircraft life follows the log-normal distribution LNðlS;L ; r2S;L Þ, the distribution parameters can be estimated approximately as follows:
Fig. 1. The probability density function corresponding to typical parameter values.
1217
X. He et al. / Engineering Failure Analysis 17 (2010) 1213–1220
(
lS;L ¼ EðXÞ r2S;L ¼ VarðXÞ
ð10Þ
where E is the expectation of random variable; Var refers to the variation of random variable. If k = 0.01, 0.02, rL = 0.06, 0.08, 0.11, 0.13, and l0 = 3, 4, 5, 6 the typical parameters estimated by Mathematica can be obtained and they are shown in Table 1. A comparison of the data in Table 1 can find that errors between l0 and lS;L , r2S;L and ðkl0 Þ2 þ r2L are small enough to be ignored. Thus, the fleet life distribution parameters accounting for both the load spectra and structural variability are as follows
(
lS;L ¼ l0 r2S;L ¼ r2L þ ðk l0 Þ2
ð11Þ
A comparison of the result with the condition of rS = constant as shown in [14] finds that the fleet aircraft life distribution functions have the same form. Theoretical pdf of the fleet aircraft life is as follows:
fN ðNÞ ¼
Z
1
2 lS 2 exp 12 lgkN þ lSrLl0 l S
2p ln 10 k lS rL N
0
dlS
ð12Þ
Probability density function of the approximate distribution is as follows:
ðlg Nl0 Þ2 exp 12 ðk l0 Þ2 þr2L fN ðNÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i 2p ðk l0 Þ2 þr2L ln 10 N
ð13Þ
Take the typical values of l0 , rL and k, the life distribution functions are shown in Fig. 2. respectively calculated by Eqs. (12) and (13). Table 1 Typical parameter values. k
rL
l0
lS,L
r2S;L
ðk l0 Þ2 þ r2L
0.01
0.06 0.08 0.11 0.13
3 4 5 6
3 + 6.8e12 4 5 + 1.0e12 6 + 1.0e12
0.0045004 0.0080006 0.0146012 0.0205017
0.0045 0.008 0.0146 0.0205
0.02
0.06 0.08 0.11 0.13
3 4 5 6
3 + 7e13 4 5 + 1.0e12 6 + 5.0e12
0.0072014 0.0128026 0.0177277 0.0072004
0.0072 0.0128 0.017725 0.0072
The comparison between the theoretical and approximate parameters.
Fig. 2. Comparision between the theoretical and approximate distribution functions.
1218
X. He et al. / Engineering Failure Analysis 17 (2010) 1213–1220
It is obvious that the theoretical and approximate distribution curves are very similar and even completely coincide with each other. Suppose l0 , rL and k are of the same value as in Fig. 3, the relative errors of the distribution variate for the specified probability between the theoretical and approximate distribution are shown in Fig. 3. The result shows that the relative errors are <5% when P e (0.00001, 0.99999), <2% when P e (0.001, 0.999), and the approximate distribution is thus acceptable. 6. The safe-life analysis In order to ensure the safe use of the aircraft structure, the safe-life NP corresponding to the reliability P must be determined, that is
PfN > NP g ¼ P
ð14Þ
According to [7,23–25], the failure probability of military aircraft structure in its total life should be controlled less than 103, that is, the acceptable level of reliability degree is P = 99.9%. 6.1. The scatter factor Scatter factor is widely used to determine the structural safe-life, and when the fatigue life follows the log-normal distribution, it can be defined as [1,8]
Lf ¼ N50 =N P
ð15Þ
where N50 is the mean life corresponding to 50% survival probability There are usually two kinds of expressions about the scatter factor, as follows:
Lf ¼ 10uP r0 Lf ¼ 10uP r0
ð16Þ pffiffiffiffiffiffi1 1þn
ð17Þ
where P is the survival probability, and usually P = 99.9%, uP is the normal distribution variate for the specified P, r0 is the standard deviation of the log fatigue life and n is the number of specimens used to test. 6.2. Test scatter factor The full-scale fatigue test is conducted under the spectrum that is considered to represent the average usage of the fleet, then the safe-life is determined by dividing the tested life by the test scatter factor. A large standard deviation based on the experience is taken, such as r0 = 0.15, 0.176, 0.20 and n = 1, 2, 3, P = 99.9%, the typical values of the test scatter factor calculated by Eq. (17) are shown in Table 2. Generally, the test scatter factor which accounts for the scatter of the structure and the load experientially is taken as Lf = 4–6. 6.3. Individual fatigue scatter factor When the load spectrum is specified, the scatter of the fatigue life of the individual aircraft is caused only by structural variability. As the standard deviation is 0.12, and the fatigue life is 104 flight hours, it follows that k = 0.12/4 = 0.03, take lS = 3, 4, 5. The individual fatigue scatter factors calculated by Eq. (17) are listed in Table 2.
Fig. 3. The relative errors of the distribution variate for the specified probability between the theoretical and the approximate distribution.
1219
X. He et al. / Engineering Failure Analysis 17 (2010) 1213–1220 Table 2 The fatigue scatter factors.
r0
rL
k
lS
rS,L
0.15 0.176 0.2
n
Notes
1
2
3
...
1
4.52 5.88 7.48
3.70 4.64 5.71
3.43 4.25 5.17
... ... ...
2.91 3.50 4.15
Fleet (test)
0 0 0
0.03
3 4 5
0.09 0.12 0.15
2.47 3.35 4.52
2.19 2.85 3.70
2.09 2.68 3.43
... ... ...
1.90 2.35 2.91
Individual aircraft
0.12
0.03
3 4 5
0.150 0.170 0.192
4.52 5.52 6.91
3.70 4.39 5.33
3.43 4.03 4.85
... ... ...
2.91 3.35 3.92
Fleet (in-service)
Three kinds of scatter factors. pffiffiffiffiffiffi1 pffiffiffiffiffiffi1 The scatter factors were calculated by Lf ¼ 10uP r0 1þn or Lf ¼ 10uP rS;L 1þn .
It can be seen clearly that the fatigue scatter factor of individual aircraft is far less than that of the fleet aircrafts because of taking no account of the scatter of the load spectrum, the typical value of individual scatter factor is 2.0 in JSSG-2006 for US Navy and USAF aircraft [24–26], 3.33 in DEF STAN 970 [7]. 6.4. The in-service scatter factor The in-service scatter factor should consider the variation of structure and the actual load spectra. Just as the r0, the rS,L, reflects the scatters of both the structure and the load in this situation and the scatter factor can be calculated by replacing the r0 with the rS,L. r0 is determined by historical service experience, and rS,L is related to the actual usage process. By taking the typical parameters, that is, k = 0.03, rL = 0.12, and lS = 3, 4, 5, the standard deviations of fleet fatigue life are calculated by Eq. (11), and the in-service scatter factor of the fleet are calculated by Eq. (17). The calculation results are included in Table 2. Compared with the experiential scatter factor, it is determined by the real situations. In consideration of the actual usage of the fleet aircrafts to assess the baseline service life of the aircraft, we need not only re-develop the baseline load spectrum because the actual spectrum differs from the predicted test spectrum [25], but also reevaluate the variation of the fleet fatigue life by Eq. (11) to adjust the safe-life according to the full-scale fatigue test result. 7. Conclusion On the basis of the discussion above, we can draw the following results: a. When the fatigue life under the specified load spectrum follows LNðlS ; ðk lS Þ2 Þ, and the life under variable load spectrum follows the LNðl0 ; r2L Þ, the life of a fleet can be subjected to a log-normal distribution LNðl0 ; ðk lS Þ2 þ r2L Þ, which means that log mean life accounting for both the load spectra and structural variability is equal to the log mean life under the variable load spectra, and the variation is equal to the sum of the variations of the fatigue life and the loading spectrum damage. Traditionally, it is considered that the fatigue life of the fleet follows the log-normal distribution with a large standard deviation, such as 0.176 or 0.2, which is usually adopted to represent the scatter of the fatigue life. However, the reason why the value of 0.176 or 0.2 was chose is not very clear [19]. In [5], Payne ever suggested to adopt the value of 0.2, but did not explain why the value of 0.2 was chose. On the basis that the load damage follows the log-normal distribution in the fleet, the present paper makes the conclusion that the fatigue life of the fleet accounting for the scatter of the structure property and the load spectrum can also be subjected to the log-normal distribution through deduction and numerical simulation. The distribution is in accordance with the conventional opinion and clarifies the relationship of the variations, providing a possible interpretation for the origin of the standard deviation; b. If the IAT is implemented, the individual fatigue scatter factor is far less than the scatter factor of fleet aircrafts because the spectrum of each aircraft is definite and the scatter factor of the fleet may be different from the test scatter factor. When performing the individual aircraft life management, the scatter factor should be re-estimated according to the actual scatter of the load spectrum to re-assess the safe-life; c. The safe-life assessed by full-scale fatigue test that is under the predicted spectrum and the experiential variation of the load is a baseline life. When the aircraft is put into service, the spectrum and the variation of the load spectrum should be different from the predicted one. If the aircraft structure is managed on the basis of the baseline safe-life, it may be unsafe or uneconomic. If possible, the IAT and IAM can be implemented to access the load scatter of the aircraft. When determining the baseline life of the aircraft, we should reconsider the fleet loading severity and the load spectra variability.
1220
X. He et al. / Engineering Failure Analysis 17 (2010) 1213–1220
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
Freudenthal AM. The scatter factor in the reliability assessment of aircraft structures. J Aircraft 1977(January):202–8. Weibull W. A statistical representation of fatigue failure in solids. Acta Polytech 1949;1(9). Freudenthal AM, Gumbel EJ. On statistical interpretation of fatigue tests. Proc Roy Aeronaut Soc 1953:216. Hoskin BC, Ford DG. Scatter factors in aircraft fatigue life estimation. ARL/SM 350, Melbourne: Aeronautical Research Laboratory; 1970. 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; June 1959. p. 76–132. Parish HE. Fatigue test results and analysis of 42 Piston Provost wings. Reports and Memoranda no. 3474; April, 1965. Anon. Defense standard 00-970 part I. Issue 3 design and airworthiness requirement for service. Aircraft structures. UK Ministry of Defense; 2003. Zhentong Gao. The fatigue scatter factor for aircraft structures[R]. Beijing University of Aeronautics and Astronautics; 1982. p. 3–9 [BH-B884] [in Chinese]. Fuze Zhang. The type and value of fatigue scatter factor. Acta Aeronaut Astronaut Sinica 1987;6(B):239–43 [in Chinese]. Naikang Shi. Normalize the fatigue life and standard variation. Acta Aeronaut Astronaut Sinica 1987;8(10):B467–474 [in Chinese]. Cardrick AW, Mew AB. Scatter considerations in the interpretation of major fatigue tests. In: Proceedings of the ICAF symposium, Seattle, USA; 1999. Lincoln JW. Economic life determination for a military aircraft. AIAA-98-25192; 1998. Michael Reyer W. Administration probability basis of fatigue evaluations in small airplanes. In: 9th joint FAA/DoD/NASA aging aircraft conference; March 8, 2006. Xiaofan He, Yanmin Dong, Wenting Liu. Reliability analysis on the fatigue life with the separated structural and load spetra variation. Acta Aeronaut Astronaut Sinica, in press [in Chinese]. Schijve J. Statistical distribution functions and fatigue of structures. Int J Fatigue 2005;27:1031–9. Barter S, Dixon B, Molent L. Assessing relative spectra severity using single fatigue test coupons. Eng Fail Anal 2009;16:863–73. Hoffman Paul C. Fleet management issues and technology needs. Int J Fatigue 2009;31:1631–7. Molent L, Barter SA. A comparison of crack growth behaviour in several full-scale airframe fatigue tests. Int J Fatigue 2007;29:1090–9. White Paul, Molent Loris, Barter Simon. Interpreting fatigue test results using a probabilistic fracture approach. Int J Fatigue 2005;27:752–67. White Paul. Review of methods and approaches for the structural risk assessment of aircraft. Australian; 2006 [DSTO–TR–1916]. Schijve J. Fatigue of structures and materials in the 20th century and the state of the art. Int J Fatigue 2003;25:679–702. OT/FAA/CT-91/20. General aviation aircraft-normal acceleration data analysis and collection project; February 1993. Payne AO. The fatigue of aircraft structures. Eng Fract Mech 1976;8:157–203. Anon. Joint service specification guide. Aircraft structures. USA: Department of Defense; 2002 [JSSG-2006]. MIL-STD-1530C. Department of Defense Standard Practice. Aircraft Structural Integrity Program (ASIP). USA: Department of Defense; November 2005. Hoffman Margery E, Hoffman Paul C. Corrosion and fatigue research-structural issues and relevance to naval aviation. Int J Fatigue 2001;23(S1):1–10.