Probabilistic Tolerance Method for Omitting Small Fatigue Loads

，，2*，，

1.State Key Laboratory of Mechanics and Control of Mechanical Structures，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China；2.Key Laboratory of Fundamental Science for National Defense-Advanced Design Technology of Flight Vehicle，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China

（Received 10 June 2020；revised 20 July 2020；accepted 1 September 2020）

Abstract:The simplification of fatigue load spectrum，which can effectively reduce experimental cost，is of great importance for structural fatigue tests.By introducing random variables，the probabilistic tolerance method of removing small amplitude cycles proposed in this paper takes into account the randomness of both load and fatigue limit.The probability of the damage occurrence caused by the removed small loads is calculated to ensure that it cannot exceed the given probabilistic tolerance.Accordingly，the omission level is obtained and the truncated spectrum is formed.The unnotched aluminum sheet specimens are used to perform the fatigue test on the original fatigue spectrum and truncated fatigue spectrum of a transporter.The test results show that there is no statistical difference between the test life of the truncated spectrum and that of the original spectrum，which demonstrates the validity of the small-load-omitting method that considers randomness.

Key words：load spectrum truncation；P-S-N curve；exceedance frequency curve of load；probabilistic tolerance value

0 Introduction

The load-time histories of aircraft structures often contain a large percentage of small amplitude cycles［1-5］.These numerous but nearly“non-damaging”small loads take up a large amount of test time.Removing these small cycles is of great significance for saving time and cost of fatigue test.

The process of fatigue failure contains crack initiation and crack propagation，and the load of the crack initiation stage accounts for a large proportion.A lot of research on the omission of small fatigue loads at this stage has been carried out.Heuler and Seeger［6］regarded 50% of the endurance limit（107 cycles）as the omission level.Jonge et al.［7］conducted fatigue tests on 2024-T3 notched sheet specimens under the transport wing standard（TWIST）and Mini-TWIST load sequences，finding out that the omission of the lowest gust load cycles（81% of the fatigue limit）resulted in a crack initiation life increase by a factor of 2.4.Yan et al.［8］carried out the fatigue test of 45 steel notched specimens under three kinds of block spectrums and came to the conclusion that small loads below fatigue limit had no effect on fatigue life and could be removed directly.Bao et al.［9］studied the low-load omission level of crack propagation of aluminum alloys 2324-T39 and 7050-T7451 under transport spectrum and indicated that an omission level of 11.72% or 13.98% was reasonable.Through the fatigue tests of LY12CZ and 30CrMnSiNi2A specimens under fighter spectra，Zhang et al.［10］demonstrated that the omission of low amplitude cycles with an overload ratio（σmax，max/σmax）r≥2.5 had no distinct influence on crack initiation life.Schubbe studied the crack behavior of Al-Li alloy that an omission level of 30%will increase fatigue life by 15%［11］.

In the above-mentioned omission methods which give a direct cut to the small loads，the dispersive characteristic of fatigue loads is out of consideration.In this paper，a cluster of load-frequency curve which reflects the load spectrum of a fleet with the same transporter composition is studied.Fig.1［12］demonstrates the dispersion of a fleet spectrum.

Fig.1 Fleet load spectrum of Fokker 27[12]

Due to dispersion，the corresponding fatigue load under a certain load frequency is a random variable.Some of these loads are comparatively large，which could cause fatigue damage，and some are small，causing no damage.In this paper，random variables are used to describe the fatigue load and fatigue limit.The probability of damage occurrence caused by the corresponding load at a fixed frequency is calculated to guarantee that it cannot exceed the given probabilistic tolerance value.Accordingly，the omission level is obtained and the truncated spectrum is formed.This method takes into account both the external dispersion of fatigue loads and the inherent dispersion of material fatigue limit.Under the condition of a given probabilistic tolerance value，the original fatigue load spectrum was truncated.Fatigue tests were performed and the rationality of this method was verified.

1 Probability Tolerance Method for Small Load Omission

The load-frequency curve is recorded asS-Ecurve，whereSis the stress（maximum stress or stress amplitude），Eis the exceedance frequency corresponding to stressS，and the fatigue curve of the material isS-Ncurve，whereNis the fatigue life.The fatigue limit of the material is assumed to beSe.Due to the randomness of load and material properties，the stressSand fatigue limitSeare both random variables，as shown in Fig.2.

Fig.2 Randomness of variables S and Se

According to the fatigue theory，when the random variableSis greater thanSe，fatigue damage occurs，and when the random variableSis less thanSe，no fatigue damage occurs.The occurrence of damage under a certain stressSis assumed to be eventA.When the load group（S，Se）is located in the shadow region in Fig.3，eventAoccurs，that is，A=｛（S，Se）|S＞Se｝.ThefS（S|E）andgSe（Se|NL）in Fig.3 are the probability density function of loadScorresponding to a fixed exceedance frequencyEand the probability density function corresponding to fatigue limitSe，respectively.In general，both are considered to obey normal distribution law.From the area of the shadow region in Fig.3，it can be seen that the probability of damage occurrence gradually increases with the external loadS.When loadSis small enough，the probability of damage approaches zero.When loadSis developed to a certain value（load corresponding to surpassing frequencyE3），the probability of damage occurrence can be infinitely close to 1.

Fig.3 Schematic diagram of the occurrence of event A

Let the occurring probability of eventAbeP（A），then the calculating formula forP（A）is

whereP（A）is the function of loadS，and the probability of damage occurrence can be different with different load values.The integral regionΩin Eq.（1）is shown in Fig.4.

Fig.4 Integral region

When the omission level of load isSomit，the equivalent probabilityδis defined to describe the average probability of damage produced by loads withinSomitrange.

The effect of omitted small loads on fatigue life is considered acceptable if the equivalent probability is limited below the probability tolerance valueδCR.The omission criterion for small fatigue loads is

2 Examples and Test Verification

2.1 Measurement of material fatigue properties

Fatigue tests were conducted using unnotched sheet specimens made of LC4CS（7A04T6）aluminum alloy which is widely used in aircraft structures.The configuration of the specimen is shown in Fig.5.

Fig.5 Configuration of unnotched sheet specimen

It is assumed that theS-Ncurve with different reliability can be expressed by a three-parameter power function formula

After logarithmic operations on both sides

After the test，the double weighted least square method［13］was used for fitting theS-Ncurves under different reliabilities.The curve parameters are shown in Table 1.The curves with different reliabilities are obtained and shown in Fig.6.

It can be concluded from the above that the mean value of fatigue limit is 40.6 MPa，and the standard deviation is 0.7 MPa.

Table 1 P-S-N curve parameters with different reliabilities

Fig.6 P-S-N curve of LC4CS unnotched specimen

2.2 Truncation of load spectrum and test verification

The original load spectrum is derived from the actual center-of-gravity overload spectrum of a transport aircraft.Each spectrum block represents 2 000 flights and the number of load cycles is 431 681.The stress at the flight state of 1-gis 210 MPa.Since the nominal stress and the center-of-gravity overload have a linear relationship，the center-ofgravity overload spectrum is converted into stress spectrum，and the exceedance curve of the original spectrum is

In addition，the ground-air-ground cycles are also included.The valley stress isSg=0 MPa，and the peak stress，which occurs once in each flight，is determined as 296.7 MPa.The ground-air-ground cycle remains unchanged during the truncation of load spectrum.It’s worth noting that the exceedance curve mentioned above only considers the average of a fleet usage，and the spectrum of fleet cannot be acquired，so the dispersion of load spectrum is not considered in the verification.

Different omission levelsSomitare taken to calculate the corresponding equivalent probabilities，and the calculation results are shown in Fig.7.The equivalent probability is a monotonically increasing function of the omission levelSomit，and the desired omission level can be obtained by finding theSomitvalue corresponding to the probability toleranceδCRon their relationship curve.The probability tolerance，which is selected according to specific requirements in engineering application，is the maximum equivalent probability that the omitted small loads can cause damage.

Fig.7 Change of tolerance with omission level

The small-load-omission criterion was verified by experiment.Based on the TWIST method［14-15］，the truncated spectrum with omission levels 22，31，41，52 and 64 MPa were compiled to be flight-byflight testing spectrum，which were recorded asL1，L2，L3，L4 andL5，respectively.The detailed information of the five truncated spectrum and the original spectrumL0 are given in Table 2.The five truncated spectrum were tested in fatigue experiment and the results are shown in Table 3.It can be seen that the fatigue life did not change much after the omission level was reduced to 41 MPa，thus the test life of theL1 spectrum can be used as the fatigue life of the original spectrum.

T-test method was used to test the mean value of crack formation life under different load spectra，judging whether the logarithmic life expectancy of spectraL2—L5 had a large difference with that of spectrumL1.The significance levelα=0.05 was taken，and the calculated and critical values of the statistic parameterTare shown in Table 4.It can be seen that the value ofTsatisfiesTα/2＜T＜T1-α/2for spectraL2—L3，and the mean logarithmic lives of spectrumL2 and spectrumL3 exhibit no significant difference with that of spectrumL1.

Table 2 Spectra L0—L5

Table 3 Fatigue test results of spectra L1—L4

Table 4 Test results of mean logarithmic fatigue life under different load spectra

TheF-test method was used to test the variance of the crack formation life under different load spectra，judging whether the log-normalized standard deviation of spectraL2—L5 was significantly different from that of spectrumL1.The calculated and critical values of the statistic parameterFare listed in Table 5.It can be concluded that the value ofFsatisfiesFα/2＜F＜F1-α/2for all load spectraL2—L5，so the logarithmic life standard deviation of the truncated spectra has no significant difference from that of spectrumL1.

Table 5 Test results of logarithmic standard deviation of fatigue life under different load spectra

According to the results ofT-test andF-test，it can be concluded that the fatigue life distribution of spectrumL2 and spectrumL3 is the same as that of spectrumL1，which means that the omission level set to 41 MPa is reasonable，and the corresponding probability toleranceδCRis found to be 0.10 in Fig.7.

3 Conclusions

（1）A probabilistic tolerance omission method for fatigue loads is proposed.When the equivalent probability is equal to the probabilistic tolerance，the omission level can be obtained，and the truncated spectrum is further formed.

（2）The proposed omission method takes into account the effect of load dispersion and fatigue limit dispersion.If the structuralP-S-Ncurve，the original load spectrum，and the probabilistic tolerance value are given，the omission level as well as the corresponding truncated spectrum can be achieved.

（3）The test results demonstrate the validity of the proposed small-load-omission method in engineering applications，which is useful for saving both fatigue test time and cost.

AcknowledgementsThis work was supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions（PAPD）and National Natural Science Foundations of China（Nos.52075244，52002181）.

AuthorsDr.LIN Hanyu received the B.S.degree in aerospace engineering from Nanjing University of Aeronautics and Astronautics，Nanjing，China，in 2016，where she is now pursuing the Ph.D.degree.Her current research interests include the truncation and omission of fatigue load spectra，structure fatigue and reliability analysis.

Prof.YAO Weixing received the B.S.，M.S.and Ph.D.degrees in Northwestern Polytechnical University and then became a teacher in Aerospace Engineering，Nanjing University of Aeronautics and Astronautics.His main research interests are structure fatigue，composite material structure design，structural optimum design and thermal protection system.

Author contributionsDr.LIN Hanyu designed the study，complied the models，conducted the analysis，interpreted the results and wrote the manuscript.Dr.XU Lipu contributed to data and conducted the test.Prof.YAO Weixing contributed to the design and discussion of the study.Prof.HUANG Jie contributed to the discussion and background of the study.All authors commented on the manuscript draft and approved the submission.

Competing interestsThe authors declare no competing interests.