A Load History Generation Approach for Full-scale Accelerated Fatigue Tests

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Engineering Fracture Mechanics 75 (2008) 3226–3243

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A load history generation approach for full-scaleaccelerated fatigue tests

J.J. Xiong a, R.A. Shenoi b,*

a Aircraft Department, Beihang University, Beijing 100083, People’s Republic of Chinab School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ, UK

Received 20 August 2007; received in revised form 23 October 2007; accepted 3 December 2007Available online 23 December 2007

Abstract

This paper seeks to establish a load history generation approach for full-scale accelerated fatigue tests. Primary focus isplaced on the load cycle identification such as to minimize experimental time while having no significant effects on the newgenerated load history. The load cycles extracted from an original load history are identified into three kinds of cyclesnamely main, secondary and carrier cycles. Then the principles are presented to generate the load spectrum for acceleratedtests, or a large percentage of small amplitude carrier cycles are deleted, a certain number of secondary cycles are merged,and the main cycle and the sequence between main and secondary cycles are maintained. The core of the generationapproach is that explicit criteria for load cycle identification are established and equivalent damage calculation formulaeare presented. These quantify the damage for accelerated fatigue tests. Three validation examples of its application for thegeneration approach of accelerated load histories are given in the paper. Good agreement of experimental lives between theoriginal and generated load histories is obtained. Finally, the generation approach of accelerated load histories is appliedto the full-scale accelerated fatigue test of helicopter tail, demonstrating the practical and effective use of the proposedapproach.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Load cycle; Load history; Equivalent damage; Rainflow count; Original load history; Generated load history; Main cycle;Secondary cycle; Carrier cycle

1. Introduction

In practice, the Palmgren–Miner linear accumulation damage rule [1,2], the S–N curve representing thematerial performance determined from constant-amplitude tests, and load cycles defined using the rainflowalgorithm [3] are often used for fatigue life prediction of a structure subjected to a random load. Althoughin most cases this is the best available method, the accuracy of the approach is often quite low. Consequently,the method is generally used in the design stage, when the accuracy of fatigue life predictions is less important

0013-7944/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engfracmech.2007.12.004

* Corresponding author. Tel.: +44 23 8059 2316; fax: +44 23 8059 3299.E-mail address: [email protected] (R.A. Shenoi).

mailto:[email protected]

Nomenclature

a0 initial crack lengthacr critical crack length related to the stress levelC undetermined constant of fatigue crack growth rate da/dN–DK curveD(Sa,Sm) damage resulted from a stress cycle of (Sa,Sm)K stress intensity factorKc fracture toughnessKm mean value of stress intensity factorm1 exponent of fatigue da/dN � DK formulam2 exponent of fatigue da/dN � DK formulaN fatigue lifeN(Sa,Sm) fatigue life with regard to fatigue stress level (Sa,Sm)Q transformation variableR stress ratioSa stress amplitudeSm mean stressSmax stress peakSmin stress valleya material constantb(a) correction coefficient of stress intensity factor for the plate with finite widthr stressrB tensile ultimate strengthrs yield stressm Poisson ratioDK stress intensity factor rangeDKth fatigue crack growth threshold value

J.J. Xiong, R.A. Shenoi / Engineering Fracture Mechanics 75 (2008) 3226–3243 3227

and when the experiments can be relatively cheap. For final evaluation of fatigue life, fatigue tests on compo-nents or whole constructions under variable amplitude loading are performed extensively [4]. All of this testinginvolves generating field or service loading using lengthy complex variable amplitude histories. These historiesmay be generated by means of one of the following two methodologies.

(1) Using standardised spectra such as the FALSTAFF [5,6] or TWIST [7,8], WASH I [9], load–time his-tories in the aircraft industry (in the design phase) are generated on the basis of a mission analysis[10–13]. However, the histories relative to the missions are obtained by instrumenting componentsand vehicles, which are then operated under varying service conditions. As is well known [14–18], theactual load–time histories often contain a large percentage of small amplitude cycles where the fatiguedamage associated with these small amplitude cycles can be small. As a result, in many cases, smallamplitude cycles are deleted from these histories in order to produce representative and meaningfulyet economical testing. Schijve et al. [14] investigated the removal of small load ranges from the manoeu-vre-dominated FALSTAFF load sequence. The deleting methodologies are either stress based [15,16] orstrain based [17,18]. These techniques attempt to remove cycles with negligible changes to the fatiguedamage or try to quantify a percentage change in the damage due to their deleting. However, deletingof variable amplitude histories can drastically affect both crack initiation life and crack growth lifeand this is highly dependent upon material, history, load level and sequence [18,19]. When generatinga load history it is essential that besides parameters such as amplitude, mean and the number of loadcycles, the sequence of the load cycles is taken into account. This sequence can often substantially influ-ence a fatigue crack’s growth rate.

3228 J.J. Xiong, R.A. Shenoi / Engineering Fracture Mechanics 75 (2008) 3226–3243

(2) A new load history is generated on the basis of a distribution of load cycles extracted from the originalload history. The methods range from classical autoregressive methods [20,21] which are used to gener-ate uncompressed load histories, to Markov methods [22,23] which are used to generate compressed loadhistories. A common weakness of these methods is that the generation of new load histories is not basedon the rainflow counting method, the method most often used for extracting load cycles when an esti-mation of the correlation between dynamic loads and the structure’s fatigue life is performed. An advan-tage of the rainflow method is not only that the load cycles extracted from a complex, random sequencewith this method correspond to the closed hysteresis loops in the diagram, but it also that it enables thesequence of load cycles to be taken into consideration when calculating the fatigue damage. Rychlik andco-workers [23,24], for example, suggest that a rainflow matrix, which represents the distribution of loadcycles, should first be transformed into a corresponding Markov matrix; the latter is then used for thegeneration of the new load history. With this approach the new load history is composed directly fromload cycles. However, these methods have two weaknesses. Firstly, the generated load history can becomposed only of those load cycles that were extracted from the original load history and secondly,the information about the sequence of load cycles is lost.

The problem is that none of the above-mentioned methods alone fulfils all the requirements that should be(at least formally) considered when generating new random load histories. These requirements are as follows:

� load cycles, extracted from original history, should correspond to the rainflow algorithm, because rainflowcycle counting is a standard procedure for determining damage events in variable amplitude loadings;� the information about the sequence of load cycles should be maintained;� the damage resulting from the new generating load history should be the same as that from the original load

history;� the new generated load histories should be of shorter length than the original load history to decrease the

test time, through deleting the small load cycles and merging the smaller ones.

Hence, it is desirable to develop a new method for the generation of load histories, which will pay regard toall of these requirements. This work seeks to establish a load history generation approach for full-scale accel-erated fatigue tests through deleting small amplitude carrier cycles and merging a certain number of secondarycycles. Primary focus is placed on load cycle identification and equivalent damage calculation.

2. Load history generation principle

An original load history (shown in Fig. 1) always consists of the representative load cycles (shown in Fig. 2aand b), which are termed as the compressed load histories [24,25]. Using the rainflow counting method, as sug-gested by Amzallag et al. [24], the load cycles can be extracted from compressed load histories. The rainflowcounting method can be chosen to extract the remaining load cycles from a residuum that remains after count-ing the load history [25]. By means of the rainflow counting method a load cycle is extracted from the

Fig. 1. An original load history.

Fig. 2. Representative load cycle: (a) hanging load cycle; (b) standing load cycle.


compressed load history if four consecutive points A, B, C and D in the compressed load histories fulfil thefollowing relations (see also Fig. 2a and b):

jxC � xBj 6 jxB � xAj ð1ÞjxC � xBj 6 jxD � xCj ð2Þ

The amplitude and the mean of the load cycle that is extracted with the rainflow method then are

Sa ¼jxC � xBj

2ð3Þ

Sm ¼xC þ xB

2ð4Þ

with the reversal point C being a half-point of the load cycle. In this way the load cycle is completely definedwith a pair of reversal points (B, C) from the compressed load histories. The relative position of these tworeversal points defines an orientation of the load cycle. If the reversal point B is higher than the reversal pointC, then the reversal point C corresponds to a valley and the load cycle is included in the examined history as ahanging load cycle in the growing branch of a larger load cycle. If the reversal point B is lower than the rever-sal point C, then the reversal point C corresponds to a peak and the load cycle is included in the examinedhistory as a standing load cycle in the falling branch of a larger load cycle shown in Fig. 2b. It is obvious thatall of the load cycles can be extracted from an original load history using the rainflow count method men-tioned above. These extracted load cycles can be classified into three kinds of load cycles, namely main, sec-ondary and carrier cycles.

Main cycles are the few larger load cycles, which probably cause significant damage to a structure; thus themain cycle should be maintained during the new load history generation. An original history often contains alarge percentage of small amplitude cycles and the fatigue damage associated with these small amplitude cyclescan be small. These small amplitude and high frequency load cycles are termed as carrier cycles. As a result, in

Fig. 3. Deleted carrier cycle of hanging load cycle from original load history.


many cases, carrier cycles are deleted from the original history in order to produce a representative and mean-ingful yet economical testing load history (shown in Fig. 3). From Fig. 3, it is shown that the principle is pro-posed to delete a carrier cycle as follows:

(a) Based on the rainflow counting, two adjacent and sequential secondary cycles (A,D) and (B, C) can beextracted from an original load history A–B–C–D.

(b) According to the criterion of deleting small loads, the carrier load cycle (B, C) can be deleted; the remain-der load cycle is (A, D). This remainder cycle (A,D) is used to replace the two original secondary cycles(A,D) and (B,C).

(c) The new load history A–D is used to replace the original load history A–B–C–D.

As mentioned above, it is possible to define a secondary cycle as one larger than a carrier cycle but smallerthan a main cycle. Because of the great numbers of secondary cycles in any original load history, adjacent andsequential secondary cycles should be merged into a new secondary cycle (shown in Fig. 4). From Fig. 4, onecan deduce a procedure to merge the adjacent and sequential secondary cycles as below.

(a) Based on the rainflow counting procedure, two adjacent and sequential secondary cycles (A, D) and(B, C) can be extracted from an original load history A–B–C–D.

(b) According to the equivalent damage formulations given below, in case of the minimum stress of the newsecondary cycle being equal to or less than one of the two adjacent and sequential secondary cycles, anew secondary cycle (A,D0) can be determined from above two secondary cycles (A,D) and (B, C). Thisnew secondary cycle (A,D0) is used to replace the two original secondary cycles (A,D) and (B,C); themerging of two adjacent and sequential secondary cycles (A,D) and (B, C) is thus accomplished.

(c) The new load history A–D0 is used to replace the original load history A–B–C–D.

It is worth noting that if the new merged secondary cycle becomes a main cycle then it is necessary to stopthis merging. Thus information about the sequence of load cycles is maintained in order to maintain a similarinteraction effect. Further, the new generated load history is shorter in length than the original load history.The generation process of a new load history can be represented schematically by a block diagram as shown inFig. 5.

Fatigue cracks generally appear at the surface of the material or at large inclusions, resulting from highstresses, surface roughness, fretting, corrosion, etc. Fatigue crack growth on a macroscopic level usuallyoccurs perpendicular to the main principal stress and is dependent on the material, the material thicknessand the orientation of the crack relative to principal material directions. Furthermore, the crack growthdepends on the cyclic stress amplitude, the mean stress and the environment. The crack growth rate, denotedda/dN, has become an important ‘‘material property” to characterize fatigue crack propagation for constantamplitude loading. Generally, the crack growth rate is presented as a function of the stress intensity factorrange DK for different stress ratios R, material thicknesses, and different environments. Various deterministicfatigue crack growth rate functions have been proposed in the literature. The functions can be represented by ageneral form [26,27]:

Fig. 4. A new secondary cycle merged from two adjacent and sequential secondary cycles.

Fig. 5. A block diagram of the load history generation for accelerated tests.


daðtÞdt¼ F ðDK;Kmax;R; S; aÞ ð5Þ

where da/dt is crack growth per flight time or cycle and F(DK,Kmax,R,S,a) is a non-negative function. Somecrack growth rate functions, such as Paris–Erdogan model [28], Trantina–Johnson model [29], Walker model

Fig. 6. da/dN–DK curve.


[30], Forman model [31], and generalized Forman model [32], are commonly used. However, the simplest crackgrowth model that describes the crack growth rate under complex spectrum loading is a power law relation [28]:

dadN¼ CðDKÞn ð6Þ

where C and n are constants. The Paris model is still one of the most used expressions for the crack growthrate, due to its simplicity.

Normally, three regions of crack growth rate are identified as shown in Fig. 6. Region 1 is usually referredto as the near threshold region due to the threshold stress intensity range, DKth, below which fatigue crackgrowth will not occur. This is believed to be true for many materials but for some material-environment com-binations the slope of the da/dN versus DK relationship has been found to be finite even for growth rates aslow as 10�7 mm/cycle. Therefore, the stress cycle pertaining to lower stress intensity range than threshold DKth

(shown in Fig. 6) can be regarded as a carrier cycle and should be deleted from the original load historyaccording to above-mentioned deleting procedure of carrier cycle. Region 2 is usually referred to as the stableor linear crack growth rate region, since the Paris relation usually fits the data in this region very well. From anengineering viewpoint, the stable or linear crack growth rate region generally includes a range between10�6 mm/cycle and 10�4 mm/cycle (shown in Fig. 6). Thus, the stress cycle pertinent to the stable or linearcrack growth rate region is deemed as a secondary cycle and the adjacent and sequential secondary cyclesshould be merged into a new secondary cycle to shorten the test time by using above-mentioned merging pro-cedure of secondary cycle. Finally, region 3 is often referred to as the unstable crack growth region, since crackgrowth rate increase very rapidly as the maximum stress intensity factor approaches the fracture toughness KC

(shown in Fig. 6). From a fatigue life point of view, the stress cycle pertinent to the unstable crack growthregion is defined as the main cycle and needs to be maintained. By using the quantifying criterion mentionedabove, it is possible to identify all load cycles extracted from an original load history into three kind of stresscycle: main, secondary and carrier cycles.

3. Quantification criteria to identify load cycle

As crack growth rates were developed for a wider range of rates, it was recognized that the linear relation(in a log–log scale) represented by Eq. (6) could not describe the crack growth rate for all possible stress


intensity ranges. The Walker equation, a slight modification to the Paris formulation and one that accountsfor stress ratio effects, for fatigue crack growth is presented [30]

da=dN ¼ CðDKÞm1ð1� RÞm2 ð7Þ
According to elasto-plastic fracture mechanics [33], the stress intensity factor K can be written as
K ¼ X ðrÞ � Y ðaÞ ð8Þ
where X(r) is a function with respect to stress r, and Y(a) is a function with respect to crack length a. Ingeneral
Y ðaÞ ¼ffiffiffiffiffiffipap

� bðaÞ ð9Þ
in which b(a) is the correction coefficient of stress intensity factor for a plate with finite width. By allowing forthe correction for a plastic zone at crack tip, it is possible to obtain
X ðrÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� paðr=rsÞ2

q ð10Þ

For a plane stress state, a = 1/(2p) and for a plane strain state, a = (1 � 2m)2/(2p).Because a fatigue stress cycle is defined by two components namely amplitude Sa and mean Sm, the stress

intensity factor range DK then becomes

DK ¼ Kmax � Kmin ¼ ½X ðSmaxÞ � X ðSminÞ� � Y ðaÞ ð11Þ

Since Smax = Sm + Sa and Smin = Sm � Sa, from Eqs. (10) and (11), one has

DK ¼ Sm þ Saffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� pa½ðSm þ SaÞ=rs�2

q � Sm � Saffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� pa½ðSm � SaÞ=rs�2

q8><>:

9>=>;Y ðaÞ ð12Þ

Substituting Eq. (12) into Eq. (7) yields

dadN¼ C

Sm þ Saffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� pa½ðSm þ SaÞ=rs�2

q � Sm � Saffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� pa½ðSm � SaÞ=rs�2

q8><>:

9>=>;

m1

½Y ðaÞ�m12Sa

Sm þ Sa

� �m2

ð13Þ

Letting (da/dN)f = 10�6 mm/cycle and (da/dN)T = 10�4 mm/cycle, and substituting (da/dN)f and (da/dN)T

into Eq. (13), it is possible to obtain (a) a filter limit equation to determine a carrier cycle and (b) a thresholdequation to identify a main cycle. Then, the criteria for load cycle identification for accelerated test can beestablished as

Sm þ Saffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

s � pa Sm þ Sað Þ2q � Sm � Saffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2s � paðSm � SaÞ2

q8><>:

9>=>;

m1

Sa

Sa þ Sm

� �m2

6

da=dNð Þf2m2 C rs � Y a0ð Þ½ �m1

ð14Þ

Sm þ Saffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

s � pa Sm þ Sað Þ2q � Sm � Saffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2s � pa Sm � Sað Þ2

q8><>:

9>=>;

m1

Sa

Sa þ Sm

� �m2

Pda=dNð ÞT

2m2 C½rs � Y ða0Þ�m1ð15Þ

where a0 is the initial crack size. From an engineering viewpoint, a0 can be generally chosen as a visible anddetectable macro-crack size of about 1.00 mm, a value chosen on the basis of practical considerations.

If the load cycle extracted by Eqs. (1)–(4) satisfies the inequality (14), then this load cycle can be identifiedas a carrier cycle. If the load cycle extracted by Eqs. (1)–(4) satisfies the inequality (15), then this load cycle isas a main cycle. If a load cycle does not satisfy either of inequalities (14) and (15), then it can be regarded as asecondary cycle.


For a stress cycle of a secondary cycle in the elastic range, it is not necessary to correct for the plastic zoneat crack tip. Thus, Eqs. (12) and (13) become respectively

DK ¼ 2SaY ðaÞ ð16ÞdadN¼ 2ðm1þm2ÞC � Sðm1þm2Þ

a ðSa þ SmÞ�m2 ½Y ðaÞ�m1 ð17Þ

0 5 10 15 200

20

40

60

80

100

120

140

160

180

200

220

Nom

inal

Stre

ss (M

Pa)

Time (s)

0 1 2 3 4 50

20

40

60

80

100

120

140

160

180

200

220

Nom

inal

Stre

ss (M

Pa)

Time (s)

a

b

c

17090

303

40

1.5

Fig. 7. (a) The specimen (size unit: mm). (b) Original load history. (c) Generation load history for accelerated tests.


and the inequalities (14) and (15) become

TableExperi

Specim

12345678

MeanTotal tMean

S m1þm2ð Þa Sa þ Smð Þ�m2

6

da=dNð Þf2 m1þm2ð ÞC Y a0ð Þ½ �m1

ð18Þ

S m1þm2ð Þa Sa þ Smð Þ�m2 P

da=dNð ÞT2 m1þm2ð ÞC Y a0ð Þ½ �m1

ð19Þ

4. Equivalent damage formulations

Separating the variables of Eq. (17) and integrating, the crack growth life under the stress cycle (Sa,Sm) ofsecondary cycle in the elastic range can be expressed as

N ¼ Sa þ Smð Þm2

2ðm1þm2ÞCSðm1þm2Þa

Z acr

a0

½Y ðaÞ��m1 da ð20Þ

Substituting Eq. (9) into Eq. (20) gives

N ¼ Q � Sa þ Smð Þm2

Sðm1þm2Þa

ð21Þ

with

Q ¼R acr

a0a�m1

2 b að Þ½ ��m1 da

2 m1þm2ð Þpm12 C

ð22Þ

where acr is the critical crack length. The latter can be determined using the following equation as

acr � bðacrÞ ¼Kc

ðSm þ SaÞffiffiffipp ð23Þ

It is seen that Eq. (21) describes the Sa � Sm � N fracture surface in three-dimensional coordinate system. TheSa � Sm � N surface is also termed as the generalized S � N surface.

According to the Palmgren–Miner rule [1,2], the damage resulting from a stress cycle of (Sa,Sm) is

DðSa; SmÞ ¼1

NðSa; SmÞð24Þ

where N(Sa,Sm) is determined from Eq. (21). From Eqs. (21) and (24), the equivalent damage of a new mergedstress cycle comprising two adjacent and sequential secondary cycles of ðSa1; Sm1Þ and ðSa2; Sm2Þ shown in Fig. 4can be determined as

1mental lives of LY12 aluminum alloy specimens

en number Original load history Generated load history

(cycles) (blocks) (cycles) (blocks)

249,603 1653 58,940 1592227,708 1508 52,133 1409256,851 1701 57,830 1562226,500 1500 71,891 1943206,568 1368 59,681 1613

56,425 152556,017 151359,239 1501

life 1546 1594.75est time 21 h 37 min 8 h 45 mintest time 4 h 19 min 1 h 5 min


ðSaÞðm1þm2Þeq

½ðSaÞeq þ ðSmÞeq�m2¼ Sðm1þm2Þ

a1

ðSa1 þ Sm1Þm2þ Sðm1þm2Þ

a2

ðSa2 þ Sm2Þm2ð25Þ

0 20 40 60 80

-100

0

100

200

300

400

500

Nom

inal

Stre

ss (M

Pa)

Time (s)

0 5 10 15 20

-100

0

100

200

300

400

500

Nom

inal

Stre

ss (M

Pa)

Time (s)

a

b

c



Letting (Smin)eq = Smin2 = Sm2 � Sa2, H ¼ Sðm1þm2Þa1

ðSa1þSm1Þm2 þSðm1þm2Þa2

ðSa2þSm2Þm2 , from Eq. (25), it is possible to obtain

TableExperi

Specim

123456789

MeanTotal tMean

ðSaÞeq ¼ ½H � ðSa2 þ Sm2Þm2 �1

m1þm2 ð26Þ

and

ðSmÞeq ¼ Sm2 � Sa2 þ ½H � ðSa2 þ Sm2Þm2 �1

m1þm2 ð27Þ

5. Experimental verification

In order to verify the criteria for load cycle identification and the generation principles for accelerated loadhistories, three kinds of specimens made of LY12 aluminum alloy, 40CrNiMoA and 30CrMnSiNi2A alloyedsteels were used in fatigue comparative tests between the original and accelerated load histories.

Test 1 concerns LY12 aluminum alloy specimens of a shape and size shown in Fig. 7a. The test is carriedout on an MTS-880-50KN fatigue testing machine at loading frequency of 15 Hz under room temperature andatmospheric conditions to verify the generation approach of load history for accelerated test. The original loadhistory is shown in Fig. 7b. From the literature [34], the da/dN–DK curve of LY12 aluminum alloy is obtainedas

dadN¼ 1:19� 10�2 DKð Þ3:83ð1� RÞ�1:43 ð28Þ

with DK th ¼ 2:8 MPaffiffiffiffimp

at the stress ratio of R = 0.05.By means of Eqs. (1)–(4), (18), (19) and (28), all load cycles can be extracted from the original load history

given in Fig. 7b and identified into the carrier, secondary and main cycles. According to the procedure shownin Fig. 5, the carrier cycles are deleted, the secondary cycles are merged based on Eqs. (26)–(28), and then anew load history for accelerated test is generated as shown in Fig. 7c. From Fig. 7b and c, it can be ascertainedthat there are 151 load cycles in a block of the original load history, and 37 cycles in a block of new generationload histories. In Fig. 7b and c, all carrier cycles are deleted, the secondary cycles are merged largely, and themain cycle and the sequence between main and secondary cycles are maintained.

Five and eight specimens are used for fatigue testing using the original and accelerated load histories,respectively, and experimental results are shown in Table 1. From Table 1, it is found that the total test timefor five specimens under the original load history is 1297 min (or 21 h and 37 min) and the mean test time forevery specimen is 259 min (or 4 h and 19 min). The total test time for eight specimens under the acceleratedload history is 525 min (namely 8 h and 45 min) and the mean test time for every specimen is 65.6 min (or 1 hand 5 min). The relative deviation in the mean life of the accelerated test from the original load history test isabout j1594:75�1546j

1594:75� 100% ¼ 3:15%, and the average saved test time for every specimen is 193.4 min (3 h and

2mental lives of 40CrNiMoA alloyed steel specimens



1,091,937 1793 213,173 1533834,330 1370 203,770 1465984,758 1617 168,739 1213841,635 1382 216,563 1558878,765 1442 169,031 1216760,587 1247 206,972 1489

194,739 1401220,988 1589177,781 1279

life 1475.17 1415.89est time 149 h 46.7 min 49 h 13 mintest time 24 h 57.8 min 5 h 28 min


13 min). This implies adequate close agreement between the shortened or accelerated test programme and theoriginal or extended test programme for engineering application.

Test 2 concerns 40CrNiMoA alloyed steel specimens of a shape and size shown in Fig. 8. All specimenshave an initial prefabricated crack of 0.5 mm through linear cutting and polishing. The tests are carriedout on an MTS-880-500KN fatigue testing machine at a loading frequency of 10 Hz under room temperatureand atmospheric pressure conditions. As in the previous test, the original load history is shown in Fig. 8b.



From the literature [34], the da/dN–DK curve of 40CrNiMoA alloyed steel at the stress ratio of R = 0.1 isobtained as

TableExperi

Specim

12345

MeanMean

dadN¼ 1:56� 10�4ðDKÞ2:95 ð29Þ


.Again, by means of Eqs. (1)–(4), (18), (19) and (29), all load cycles can be extracted from the original load

history given in Fig. 8b and identified into the carrier, secondary and main cycles. A similar process is followedfor the load history for the accelerated test programme shown in Fig. 5 resulting in the extraction of the car-rier, secondary and main cycles in Fig. 8c. From Fig. 8b and c, it can be deduced that there are 609 load cyclesin a block of the original load history, and 139 cycles in a block of the accelerated test load history.

Six and nine specimens are used for fatigue testing using the original and accelerated load histories, respec-tively, and experimental results are shown in Table 2. From Table 2, it is found that total test time for sixspecimens under the original load history is 8986.69 min (or 149 h and 46.7 min) and the mean test timefor every specimen is 1497.78 min (or 24 h and 57.8 min). The total test time for nine specimens under theaccelerated load history is 2952.9 min (or 49 h and 13 min) and the mean test time for every specimen is328.1 min (or 5 h and 28 min). The relative deviation in the mean life of the accelerated crack propagation testfrom the original load history test is about 1475:17�1415:89j j

1475:17� 100% ¼ 4:02%, and the average saved test time for

every specimen is 1169.68 min (19 h and 30 min). Again, it is evident that the new approach using acceleratedtest data gives results that agree very well with the actual test programme for engineering application.

Test 3 concerns 30CrMnSiNi2A alloyed steel specimens of a shape and size shown in Fig. 9. The tests arecarried out on an MTS-880-500KN fatigue testing machine at a loading frequency of 10 Hz under room tem-perature and atmospheric pressure conditions. As in the previous test, the original load history is shown inFig. 9b. From the literature [34], the da/dN–DK curve of 30CrMnSiNi2A alloyed steel is obtained as

dadN¼ 3:011� 10�8ðDKÞ2:45ð1� RÞ�0:98 ð30Þ


at the stress ratio of R = 0.1.By means of Eqs. (1)–(4), (18), (19) and (30), all load cycles can be extracted from the original load history

given in Fig. 9b and identified into the carrier, secondary and main cycles. A similar process is followed for theload history for the accelerated test programme shown in Fig. 5 resulting in the extraction of the carrier, sec-ondary and main cycles in Fig. 9c. From Fig. 9b and c, it can be shown that there are 6906 load cycles in ablock of the original load history, and 1663 cycles in a block of the accelerated test load history.

Three and five specimens are used for fatigue testing using the original and accelerated load histories,respectively, and experimental results are shown in Table 3. From Table 3, it is calculated that the total testtime for the three specimens under the actual load spectra is 1820 min (or 30 h and 20 min) and the mean testtime for every specimen is 607 min (or 10 h and 6 min). The total test time for the five specimens under theaccelerated load spectra is 728 min (or 12 h and 8 min) and the mean test time for every specimen is121 min (or 2 h and 1 min). From Table 3, it is evident that the relative deviations in the mean value of fatiguelife under the accelerated spectra compared to that under the actual spectrum test is 53:25�52:31j j

53:25�

3mental lives of 30CrMnSiNi2A alloyed steel specimens



367,340 53 87,307 50297,193 43 65,194 37427,556 61 115,307 66

106,088 6062,739 35

life 52.3 53.25test time 10 h 6 min 2 h 1 min

Fig. 10. (a) Full-scale fatigue test of helicopter tail. (b) Fatigue load application points for full-scale test of helicopter tail. (c) Fatigue loadspectrum in transverse direction. (d) Fatigue load spectrum in vertical direction. (e) Fatigue crack occurrence site. (f) Fatigue crack growthcurve.


-200

0

200

400

600

800

1000

1200

20 30 40 50 60 70 80 a mm

T h

e

f

Fig. 10 (continued)


100% ¼ 1:82%. The saved test time for every specimen is about 451 min (7 h and 31 min). Again, it is evidentthat the new approach using accelerated test data gives results that agree very well with the actual test pro-gramme for engineering application.

6. Application in full-scale fatigue test of helicopter tail

A full-scale fatigue test was carried out on a helicopter tail (shown in Fig. 10a) at loading frequency of15 Hz under room temperature and atmospheric conditions. Fatigue load spectra in transverse and verticaldirections were applied at tail rotor as shown in Fig. 10b to model the actual flight loads of tail rotor. Usingthe finite element method (FEM), the critical sections and hazardous locations can be searched and foundfrom the actual measured load spectra. Subsequently, by means of Eqs. (1)–(4), (18) and (19) and fracture per-formances of material of hazardous location, all load cycles can be extracted from the actual measured loadhistory and identified into the carrier, secondary and main cycles. According to the procedure shown in Fig. 5,the carrier cycles were deleted, the secondary cycles were merged based on Eqs. (26) and (27) and fracture per-formances of material of hazardous location, and then the new load histories for accelerated test in transverseand vertical directions were generated as shown in Fig. 10c and d. During fatigue test, all critical sections andhazardous locations responses were monitored and observed at various test times to find fatigue crack


occurrence. After 10,512 blocks of load history (or 22,506 flight hours), an S-shape crack with a length of32 mm appears at the skin surface of bottom helicopter tail near the right region of tail-lamp shade (shownin Fig. 10e). Fatigue crack growth curve with experimental time were observed and recorded as shown inFig. 10f. The full-scale fatigue test has spend an experimental time of 3658.9 h (or 11,253 blocks of load historyor 22,506 flight hours) until the crack reaches a length of 84.5 mm.

7. Conclusions

The focus of this paper has been to present new generation approach of load history for full-scale acceler-ated tests. Explicit criteria for load cycle identification and equivalent damage calculation are presented. Theapplicability of the new approach has been shown for three validation examples. Good agreement of exper-imental lives is achieved between the original and generated load histories using the newly developedapproaches. Finally, the generation approach of load history is applied to full-scale accelerated fatigue testof helicopter tail, demonstrating the practical and effective use of the proposed approach.

Acknowledgement

This project was supported by the National Natural Science Foundation and the Aeronautics ScienceFoundation in China, and the Engineering and Physical Sciences Research Council in UK.

References

[1] Miner MA. Cumulative damage in fatigue. J Appl Mech 1945;12:159–64.[2] Palmgren A. Die Lebensdauer von Kugellagern. Z Vereins Deut Ingenieure 1924;68:339–41.[3] Matsuishi M, Endo T. Fatigue of metals subjected to varying stress. In: Proceedings of the Kyushu branch of Japan society of

mechanics engineering, Fukuoka, Japan, 1968. p. 37–40 [in Japanese].[4] Stephens RI, Dindingert PM, Gunger JE. Fatigue damage deleting for accelerated durability testing using strain range and SWT

parameter criteria. Int J Fatigue 1997;19:599–606.[5] Aicher W, Branger J, Van Dijk GM, Ertelt J, Huck M, De Jonge JB. Description of a fighter aircraft loading standard for fatigue

evaluation FALSTAFF. Common Report of F + W Emmen, LBF, NLR, IABG, 1976.[6] Mitchenko EI, Prakash RV, Sunder R. Fatigue crack growth under an equivalent FALSTAFF history. Fatigue Fract Engng Mater

Struct 1995;18(5):583–95.[7] Schutz D, Lowak H, De Jonge JB, Schijve J. A standardised load sequence for flight simulation tests on transport aircraft wing

structures. LBF-report, B-106, NLR-Report TR 73, 1973.[8] Sieg J, Schijve J, Padmadinata UH. Fractographic observations and predictions on fatigue crack growth in an aluminium alloy under

MiniTWIST flight-simulation loading. Int J Fatigue 1991;13(2):139–47.[9] Schutz W, Klatschke H, Huck M, Sonsino CM. Standardized load sequence for offshore structures-WASH I. Fatigue Fract Engng

Mater Struct 1990;13(1):15–29.[10] Fowler KR, Watanabe RT. Development of jet transport airframe fatigue test spectra. In: Potter JM, Watanabe RT, editors.

Development of fatigue loading spectra. ASTM STP 1006. Philadelphia: American Society for Testing and Materials; 1989. p. 36–64.[11] Wanhill RJH, Hart WGJ, Schra L. Flight simulation and constant amplitude fatigue crack growth in aluminum–lithium sheet and

plate. In: Proceedings of the 16th international committee on aeronautical fatigue symposium, Tokyo, Japan, May 1991.[12] Schutz W. Standardized stress–time histories—an overview. In: Potter JM, Watanabe RT, editors. Development of fatigue loading

spectra. ASTM-STP 1006. Philadelphia (PA): American Society for Testing and Materials; 1989. p. 3–16.[13] Heuler P, Klatschke H. Generation and use of standardised load spectra and load–time histories. Int J Fatigue 2005;27(8):974–90.[14] Schijve J, Vlutters AM, Ichsan A, Kluit JCP. Crack growth in aluminium alloy sheet material under flight-simulation loading. Int J

Fatigue 1985;7(3):127–36.[15] Yan JH, Zheng XL, Zhao K. Experimental investigation on the small-load-omitting criterion. Int J Fatigue 2001;23(5):403–15.[16] Schon J. Spectrum fatigue loading of composite bolted joints—small cycle elimination. Int J Fatigue 2006;28(1):73–8.[17] Dowling NE. Estimation and correlation of fatigue lives for random loading. Int J Fatigue 1988;10:179–85.[18] Socie DF, Artwohl PJ. Effect of history editing on fatigue crack initiation and propagation in a notched member. In: Bryan DF,

Potter IM, editors. Effect of load history variables on fatigue crack initiation and propagation. ASTM STP 714. Philadelphia: Amer-ican Society for Testing and Materials; 1980. p. 3–23.

[19] Pompetzki TH, Topper TH, DuQuesnay DL. The effect of compressive underloads and tensile overloads on fatigue damageaccumulation in SAE 1045 steel. Int J Fatigue 1990;12:207–13.

[20] Klemenc J, Fajdiga M. An improvement to the methods for estimating the statistical dependencies of the parameters of random loadstates. Int J Fatigue 2004;26(2):141–54.


[21] Xiong JJ, Shenoi RA. An integrated and practical reliability-based data treatment system for actual load history. Fatigue FractEngng Mater Struct 2005;28(10):875–89.

[22] Anthes RJ. Modified rainflow counting keeping the load sequence. Int J Fatigue 1997;19:529–35.[23] Rychlik I. Simulation of load sequences from rainflow matrices: Markov method. Int J Fatigue 1996;18(7):429–38.[24] Amzallag C, Gerey JP, Robert JL, Bahuaud J. Standardization of the rainflow counting method for fatigue analysis. Int J Fatigue

1994;16:287–93.[25] Klemenc J, Fajdiga M. A neural network approach to the simulation of load histories by considering the influence of a sequence of

rainflow load cycles. Int J Fatigue 2002;24(11):1109–25.[26] Miller MS, Gallagher GP. An analysis of several fatigue crack growth rate descriptions. Measurement and data analysis, ASTM STP

738, 1981. p. 205–51.[27] Hoeppner DW, Krupp WE. Prediction of component life by application of fatigue crack growth knowledge. Engng Fract Mech

1974;6:47–70.[28] Paris PC, Erdogan F. A critical analysis of crack propagation laws. J Basic Engng, Trans ASME (Ser D) 1963;85:528–34.[29] Trantina GG, Johnson CA. Probabilistic defect size analysis using fatigue and cyclic crack growth rate data. Probabilistic fracture

mechanics and fatigue methods, ASTM STP 798, 1983. p. 67–78 .[30] Walker EK. The effect of stress ratio during crack propagation and fatigue for 2024-T3 and 7075-T6 aluminum. Effects of

environment and complex load history on fatigue life, ASTM STP 462, 1970. p. 1–14.[31] Forman RG et al. Numerical analysis of crack propagation in cyclic loaded structures. J Basic Engng, Trans ASME (Ser D)

1967;89:459–65.[32] Newman Jr JC. A crack opening equation for fatigue crack growth. Int J Fract 1984;24:131–5.[33] Hwang KC, Yu SW. Elastic–plastic fracture mechanics. Beijing: Hsinghua University; 1985.[34] Gao ZT, Jiang XT, Xiong JJ, et al. Experimental design and data treatment for fatigue performance. Beijing: Beihang University

Press; 1999.

A Load History Generation Approach for Full-scale Accelerated Fatigue Tests

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