research article fatigue damage assessment for concrete...

Research ArticleFatigue Damage Assessment for Concrete StructuresUsing a Frequency-Domain Method

Hongyan Ding123 Qi Zhu3 and Puyang Zhang123

1State Key Laboratory of Hydraulic Engineering Simulation and Safety Tianjin University Tianjin 300072 China2Key Laboratory of Coast Civil Structure Safety Tianjin University Ministry of Education Tianjin 300072 China3School of Civil Engineering Tianjin University Tianjin 300072 China

Correspondence should be addressed to Puyang Zhang zpy td163com

Received 22 October 2014 Revised 15 December 2014 Accepted 18 December 2014 Published 29 December 2014

Academic Editor Yang Tang

Copyright copy 2014 Hongyan Ding et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A fatigue damage assessment for concrete was carried out according to Eurocode 2 Three frequency-domain methods the levelcrossing counting (LCC) method the range counting (RC) method and a new proposed method were used for the damageassessmentThe applicability of these frequency-domainmethods was evaluated by comparison with the rainflow countingmethodin the time domain A preliminary numerical study was carried out to verify the applicability of the frequency-domain methodsfor stress processes with different bandwidths thus the applicability of the LCC method and the new method was preliminarilyconfirmed The fatigue strength of concrete had a minor effect on the fatigue damage assessment The applicability of the LCC andthe newmethods deteriorated for relatively low coefficients of variance of the stress process because the ultimate number of constantamplitude cycleswas sensitive to the range of the cyclesThe validity of the joint probability functions of the twomethodswas provenusing a numerical simulation The integration intervals of the two frequency-domain methods were varied to estimate the lowerand upper bounds on the fatigue damage which can serve as references to evaluate the accuracy of the time-domainmethod results

1 Introduction

Fatigue is the process of gradual damage to materials that aresubjected to continually changing stresses Concrete fatigue isprimarily a problem for offshore structures railway sleepersand bridges which are often exposed to alternating loadings[1] Unlike steel structures both the range and level of stressaffect fatigue damage in concrete structures [2] In this studythe European Standard Eurocode 2 [3] was applied to assessthe fatigue damage of concrete structures

A well-established procedure for fatigue damage assess-ment involves the time-domain analysis of the stress pro-cesses that a structure will be subjected to over its lifetime[4ndash6] Stress processes in structures that are induced bywind waves or road irregularities often occur on a fairlyirregular and random basis however only the ultimatenumber of stress cycles for a constant stress range and themean stress that can be sustained up to failure is knownThus a description of the cycles in the stress records in termsof parameters such as the number of counted cycles thedistribution of cycle amplitudes and the means is required

The counting method [7] was applied in this study Levelcrossing counting (LCC) peak counting (PC) simple rangecounting (RC) and rainflow counting (RFC) are the countingmethods most commonly used in engineering practice TheRFCmethod is widely accepted as themost efficient countingmethod available [8]

The fatigue damage of structures can also be assessedin the frequency domain The frequency-domain methodis faster and more cost efficient than the time-domainmethod [9ndash11] Variations in parameters and optimizationcan be rapidly executed for a properly functioning frequency-domain method The stress processes in the frequencydomain are represented using a spectral formulation suchas a power spectral density (PSD) function which providesinformation on the power distribution over a range offrequencies The first n (typically 4) moments of the PSDfunction are used to find the probability density function(PDF) for the peaks and valleys of the stress process (ie therange of the stress and mean stress) [12] The fatigue damagecan be calculated by integration using the PDF andPalmgren-Minerrsquos rule

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 407193 10 pageshttpdxdoiorg1011552014407193

2 Mathematical Problems in Engineering

In this study PDFs that are available in the literatureand a PDF developed by the authors were applied to cal-culate the fatigue damage of concrete structures The RFCmethod in the time domain is the standard method forfatigue assessmentThe results of different frequency-domainmethods were compared with the results of the RFC methodto select the most effective frequency-domain methods Apreliminary numerical study was carried out to verify theapplicability of the frequency-domain methods for stressprocesses with different bandwidths thus the applicabilityof the LCC method and the new method was preliminarilyconfirmed The integration intervals of the two frequency-domainmethods were varied to estimate the lower and upperbounds on the fatigue damage which can serve as referencesto evaluate the accuracy of the time-domain method resultsFew published reports on fatigue assessment for concrete inthe frequency domain are currently available thus the resultsof this study can serve as a useful reference in this field [13]

2 Basic Theory

21 Frequency-Domain Process A stress process ℎ(119905) isassumed to be a stationary Gaussian process The autocorre-lation function of ℎ(119905) is defined as follows [14]

119870 (120591) equiv ℎ (119905) ℎ (119905 + 120591) = lim119879rarrinfin

1

2119879int

119879

minus119879

ℎ (119905) ℎ (119905 + 120591) 119889120591 (1)

The Fourier transform of the autocorrelation function isexpressed as

119870 (120591) = int

infin

minusinfin

119878 (119891) 119890minus119894120596120591

119889119891 (2)

where the new function 119878(119891) is the PSD of ℎ(119905) 119878(119891) can alsobe written in terms of119870(120591)

119878 (119891) = int

infin

minusinfin

119870 (120591) 119890119894120596120591

119889120591 (3)

Equations (2) and (3) are known as the Wiener-Khintchine relations

We are typically not interested in distinguishing a PSDthat is associated with a negative frequency from one that isassociated with a positive frequency A single-sided PSD canbe defined as follows

119878(119891) = 119878 (119891) + 119878 (minus119891) (119891 ge 0) (4)

Variables that are encountered in engineering analysissuch as the stress are real The PSD is an even function forreal variables thus

119878(119891) = 119878 (119891) + 119878 (minus119891) = 2119878 (119891) (119891 ge 0) (5)

The 119899th moment of the PSD function can be obtainedfrom the characteristic of the PSD function The spectralmoments are defined as follows

119898119899= int

infin

0

119891119899119878(119891)119889119891 (6)

The first fourmoments (119898011989811198982 and119898

4) are themost

commonly used moments of the PSD function

The bandwidth of a signal is taken to be the frequencyinterval over which most of the signal power is concentratedThe bandwidth can be defined in terms of the spectralmoments [15]

1205721=

1198981

radic11989801198981

1205722=

1198982

radic11989801198984

0 le 1205721 1205722le 1

(7)

These two bandwidth parameters tend to unity for aldquonarrow-bandrdquo signal and zero for a ldquowide-bandrdquo signal

22 Fatigue Damage Assessment Method for Concrete Struc-tures from Eurocode 2 In Eurocode 2 Palmgren-Minerrsquos rule[16] is applied to calculate the total fatigue damage as givenin (8) A satisfactory fatigue resistance may be assumed forconcrete under compression if119863 le 1

119863 =

119898

sum

119894=1

119899119894

119873119894

(8)

where 119863 is the total fatigue damage 119898 is the number ofintervals with a constant amplitude 119899

119894is the actual number

of constant-amplitude cycles in the interval ldquo119894rdquo 119873119894is the

ultimate number of constant-amplitude cycles in the intervalldquo119894rdquo that can be carried out before failure

119873119894= 1014((1minus119864cdmax119894)radic1minus119877119894) 119877

119894=

119864cdmin119894

119864cdmax119894

119864cdmin119894 =120590cdmin119894

119891cdfat119864cdmax119894 =

120590cdmax119894

119891cdfat

(9)

where 119877119894is the stress ratio 119864cdmin119894 is the minimum stress

compression level 119864cdmax119894 is the maximum stress compres-sion level 120590cdmin119894 is the lowest stress in a cycle 120590cdmax119894 isthe highest stress in a cycle and 119891cdfat is the design fatiguestrength of the concrete 119873

119894is a function of both the peaks

and valleys of the stress process

23 Counting Methods LCC PC RC and RFC are thefour most commonly used counting methods in engineeringpractice

In the LCC method every crossing of the predeterminedstress levels is countedThen themost damaging level cycle iscounted for fatigue by constructing the largest possible cyclefollowed by the second largest cycle and so on until all of thelevel crossings have been considered

In the PCmethod the occurrence of a relative maximumor minimum stress value is identified Peaks (valleys) that areabove (below) the reference level are counted

In the RC method a range is defined as the differencebetween two successive reversals the range is positive whena valley is followed by a peak and negative when a peak isfollowed by a valley

The RFC method is the most frequently used cyclecounting method in engineering practice Starting from alocal loadmaximumMax

119896 twominima are identified before

and after Max119896 that is Min

119896minusand Min

119896+ The point with

Mathematical Problems in Engineering 3

the smallest deviation from Max119896is chosen as the rainflow

minimum Min119896RFC thus producing the kth rainflow cycle

(Min119896RFC Max

119896) The aforementioned procedure is repeated

for the entire stress process

3 Fatigue Damage Assessment for Concrete inthe Frequency Domain

31 Joint Probability Density Applied to Fatigue DamageAssessment As described in Section 22 119873

119894is a function of

both the peaks and valleys in the stress process thus thefatigue damage for concrete structures is also a function of thepeaks and valleys in the stress process The joint probabilitydensity of the counted cycles ℎ(119906 V) which is a function ofpeak ldquo119906rdquo (corresponding to120590cdmax119894 in (8)) and valley ldquoVrdquo (cor-responding to 120590cdmin119894 in (8)) is constructed by applying thespectralmoment parameters of the PSD function As the RFCmethod is the ldquostandardrdquo countingmethod we determine thejoint probability density of counted cycles using ℎRFC(119906 V)However no analytical solutions of ℎRFC(119906 V) are currentlyavailable and only approximate approaches can be used toobtain an accurate fatigue damage assessment

Tovo [17 18] calculated the joint probability density ofcounted cycles for a zero-mean Gaussian process from thedistribution of level crossing counted cycles as follows

ℎLCC (119906 V)

= [119901119901 (119906) minus 119901V (119906)] 120575 (119906 + V) + 119901V (119906) 120575 (119906 minus V) 119906 ge 0

119901119901 (119906) 120575 (119906 minus V) 119906 le 0

(10)

where 120575 is the Dirac delta function and 119901119901(119906) and 119901V(119906)

are the cumulative distributions of the peaks and valleysrespectively

119901119901 (119909) = Φ(

119909

120590119909radic1 minus 1205722

2

) minus 1205722119890minus1199092

21205902

119909Φ(1205722119909

120590119909radic1 minus 1205722

2

)

(11)

119901V (119909) = 119901119901 (minus119909) (12)

where 119909 is the stress 120590119909is the standard deviation Φ is the

cumulative Gaussian distribution and 1205722is the bandwidth

parameter as defined by (7)Tovo [17 18] also calculated a joint probability density for

the cycles in a Gaussian process where the peaks and valleysdistributions are given by (11) and (12) respectively from thedistribution of the counted cycles over the stress range

ℎRC (119906 V) =1

1205902119909120572222radic2120587

times 119890minus(1199062

+V2)41205902119909

(1minus1205722

2

)119890minus((119906minusV)241205902

119909

(1minus1205722

2

))((1minus1205722

2

)21205902

119909

)

times[[

[

119906 minus V

radic41205902119909(1 minus 1205722

2)

]]

]

(13)

Equation (13) can be rewritten as a function of the meanstress ldquo119898rdquo and stress range ldquo119904rdquo

ℎRC (119904 119898) =1

radic2120587120590119909(1 minus 1205722

2)

times 119890minus1198982

2120590119909(1minus1205722

2

) 119904

12059011990912057222

119890minus1199042

21205901199091205722

2

= 119901119898 (119898) 119901119904 (119904)

(14)

where 119904 = (119906 minus V)2 and119898 = (119906 + V)2Equation (14) shows that the joint probability density

is a product of 119901119898(119898) and 119901

119904(119904) thus it is reasonable to

assume that the mean stress ldquo119898rdquo and stress range ldquo119904rdquo are twoindependent variables Then other expressions for 119901

119904(119904) can

replace that in (14) to obtain a new joint probability densityThe Dirlik method [19] is considered the most accurateapproach for constructing an empirical expression for thePDF over the stress range ldquo119904rdquo [20] and is used to obtain a newproposed ℎNP(119904 119898)

ℎNP (119904 119898) =1

radic2120587120590119909(1 minus 1205722

2)

119890minus1198982

(2120590119909(1minus1205722

2

))119901Dirlik (119904) (15)

119901Dirlik (119904) =1

radic120590119909

[1198661

119876119890minus119885119876

+1198662119885

1198772119890minus1198852

21198772

+ 1198663119885119890minus1198852

2]

(16)

119885 =119904

radic120590119909

119909119898

=1198981

1198980

(1198982

1198984

)

12

(17)

where

1198661=

2 (119909119898

minus 1205722

2)

1 + 12057222

119909119898

=1 minus 1205722

2minus 1198661+ 1198662

1

1 minus 119877

119885 = 1 minus 1198661minus 1198662

119877 =1205722minus 119909119898

minus 1198662

1

1 minus 1205722minus 1198661+ 11986621

119876 =125 (120572

2minus 1198663minus 1198662119877)

1198661

(18)

32 Expressions for Fatigue Damage In this section (10)(14) and (15) are used to calculate the fatigue damage in thefrequency domain

The expected peak occurrence frequency 120592119901is defined as

[21]

120592119901= radic

1198984

1198982

(19)

The fatigue damage over a given time 119879 can be calculatedas follows

119864 (119863) = 119879120592119901∬

ℎ(119906 V)119873119894 (119906 V)

119889119906 119889V (20)

When ℎLCC(119906 V) (see (10)) is used the component relatedto 120575(119906 minus V) may be neglected in the calculation because this


function implies that 119906 = V that is there is no effect on thedamage When 119906 le 0 ℎLCC(119906 V) = 119901

119901(119906)120575(119906 minus V) thus

119864 (1198631015840) = 119879120592

119901∬

ℎ(119906 V)119873119894 (119906 V)

119889119906 119889V

= 119879120592119901∬

119901119901 (119906) 120575 (119906 minus V)119873119894 (119906 V)

119889119906 119889V

= 119879120592119901∬

0

minusinfin

119901119901 (119906) 120575 (119906 minus V)119873119894 (119906 V)

119889119906 119889V

= 119879120592119901int

0

minusinfin

(int

0

minusinfin

119901119901 (119906) 120575 (119906 minus V)119873119894 (119906 V)

119889119906)119889V

= 119879120592119901int

0

minusinfin

119901119901 (V)

119873119894 (V V)

119889V

(21)

From the definition of119873119894119873119894(V V) rarr infin thus 119864(1198631015840) rarr

0 and the component related to 120575(119906 minus V) clearly has no effecton the damage

Equations (10)ndash(15) and (21) are all based on a zero-mean process for a process with a non-zero-mean 119898

119909 these

equations can easily be modified using a variable shift In thisstudy we perform a fatigue damage assessment for concreteunder compression thus all of the stresses have the samesign (and are hereafter assumed to be positive)The followingequations are given for a non-zero-mean process

The fatigue damage can be calculated by applyingℎLCC(119906 V) as follows

119864 (119863LCC

) = 119879120592119901∬

ℎ(119906 V)119873119894 (119906 V)

119889119906 119889V

= 119879120592119901∬([119901

119901(119906 minus 119898

119909) minus 119901V (119906 minus 119898

119909)]

times 120575 ((119906 minus 119898119909) + (V minus 119898

119909))

times (119873119894 (119906 V))

minus1) 119889119906 119889V

= 119879120592119901int

infin

119898119909

(int

119906

minusinfin

(119901119901(119906 minus 119898

119909) minus 119901V (119906 minus 119898

119909)

times 120575 (119906 + V minus 2119898119909)

times (119873119894 (119906 V))

minus1) 119889V)119889119906

= 119879120592119901(int

2119898119909

119898119909

119901119901(119906 minus 119898

119909) minus 119901V (119906 minus 119898

119909)

119873119894(119906 2119898

119909minus 119906)

119889119906

+int

infin

2119898119909

119901119901(119906 minus 119898

119909) minus 119901V (119906 minus 119898

119909)

119873119894 (119906 0)

119889119906)

(22)

Because all of the stresses are positive the lower limit ofthe valley is zero When (2119898

119909minus 119906) le 0 the valley of a cycle is

set to zeroSimilarly the fatigue damage119864(119863RC

) can be calculated bythe RC method and 119864(119863

NP) can be calculated by the new

method developed by the authors that is by applying (14)(15) and (23)

119864 (119863) = 119879120592119901∬

ℎ(119904119898)

119873119894 (119904 119898)

119889119904 119889119898 (23)

As the stress record is assumed to be Gaussian almostall of the values will fall in the interval (119898

119909minus 5120590119909 119898119909

+

5120590119909) thus the following integration intervals can be used to

calculate 119864(119863RC

) and 119864(119863NP

) 119898 (119898119909minus 5120590119909 119898119909+ 5120590119909)

119904 (0 5120590119909) (The upper bound on the integration interval

for 119904 is estimated as 119904 = ((119898119909+ 5120590119909) minus (119898

119909minus 5120590119909))2 For

(119898119909minus 5120590119909) lt 0 the lower bound on the integration interval

is set to zero)

4 Numerical Simulation

Numerical simulations were performed to investigate theapplicability of the aforementioned frequency-domainmeth-odsThe fatigue damage was assessed using the time-domainmethod with RFC and Palmgren-Minerrsquos rule to obtain theldquostandardrdquo results The accuracy of the results obtained fromthe frequency-domainmethodswas evaluated by comparisonwith the accuracy of these ldquostandardrdquo results

The aforementioned joint PDFswere all constructed fromspectral moments and bandwidth parameters Preliminarynumerical simulations were carried out to investigate theeffect of the bandwidth parameters particularly 120572

2 on

the applicability of the frequency-domain methods SomePSDs with simple shapes (see Figure 1) were applied in thesimulationThe random stress processes that were used in theRFC methods in the time domain can be derived from thesePSDs (see Figure 2) [22 23]

The design fatigue strength of concrete119891cdfat was chosenas 191MPa using Eurocode 2 A given mean value 119898

119909of

50MPawas used for all of the spectra and random processesAll of the spectra had the same 119898

0of 10MPa2 which was

equal to the variances 1205902

119909of the stress processes The total

simulation time was 1000 sThe results of the preliminary numerical simulations are

shown in Figure 3 119863LCC 119863RC 119863NP and 119863RFC denote the

fatigue damage that was calculated using the LCC methodRC method the new method and RFCmethod respectivelyThe results of the three aforementioned frequency-domainmethods were normalized by the results obtained using theRFC method in the time domain (the order of magnitudeof the absolute fatigue damage was 10minus9) Figure 3 illustratesthat both the LCC method and the new method yieldedaccurate estimates of the fatigue damage compared withthe RFC method over the entire bandwidth of the stressprocess Both the LCC method and the new method yieldedconservative results where the values of 119863

LCC119863

RFC werebetween 20 and 120 and the values of 119863

NP119863

RFC werebetween 20 and 182 Both of these methods should be usedin engineering practice to obtain preliminary estimates ofthe fatigue damage in the initial design stage and parametricstudies The range method yielded poor estimates for mostbandwidths and more accurate results were obtained onlywhen using higher 120572

2values The orders of magnitude of


0 2 4 6 8 1000

02

04

06

08

10

Frequency (Hz)

PSD(

MPa

2H

z)

(a)

0 2 4 6 8 1000

02

04

06

08

10

Frequency (Hz)

PSD(

MPa

2H

z)

(b)

Figure 1 Examples of PSD curves

0 20 40 60 80 100

2

3

4

5

6

7

8

Stre

ss (M

Pa)

Time (s)

1205722 = 017

(a)

0 20 40 60 80 1001

2

3

4

5

6

7

8

9St

ress

(MPa

)

Time (s)

1205722 = 038

(b)

0 20 40 60 80 100

1

2

3

4

5

6

7

8

9

Time (s)

Stre

ss (M

Pa)

1205722 = 086

(c)

Figure 2 Examples of random processes with different bandwidth parameters


00 01 02 03 04 05 06 07 08 09 100

2

4

6

8

10

12

1205722

DLC

CD

RFC

(a) 119863LCC119863RFC

00 01 02 03 04 05 06 07 08 09 1002468

101214161820

1205722

DN

PD

RFC

(b) 119863NP119863RFC

00 01 02 03 04 05 06 07 08 09 10

minus10

minus8

minus6

minus4

minus2

0

2

1205722

log 1

0(D

RCD

RFC)

(c) 119863RC119863RFC

Figure 3 Results of the preliminary numerical simulations

00 02 04 06 08 100

2

4

6

8

10

12

14

16

DLC

CD

RFC

1205722

fcdfat = 191MPafcdfat = 256MPafcdfat = 313MPa

(a) 119863LCC119863RFC

00 02 04 06 08 100

2

4

6

8

10

12

14

16

18

20

DN

PD

RFC

1205722


(b) 119863NP119863RFC

Figure 4 Effect of fatigue strength on fatigue damage

119863RC

119863RFC were between 10minus2 and 10minus13 for values of 120572

2below

065The order ofmagnitude for119863RC119863

RFC approached unitywhen 120572

2approached unity Thus the applicability of the RC

method is limited to narrow-band stress processesWe further verified the applicability of the LCC method

and the new method by conducting parameter studies First

the effect of the fatigue strength on the fatigue assessmentwas investigated for a stress process with a mean of 119898

119909

and a variance of 1205902

119909 Fatigue strengths of 191 256 and

313MPa were considered Figure 4 illustrates that for thesethree different fatigue strengths the values of119863LCC

119863RFC and

119863NP

119863RFC were close to each other for all values of 120572

2 at


000 002 004 006 008 010 012Cv

DLC

CD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

100

101

102

103

104

(a) 119863LCC119863RFC

000 002 004 006 008 010 012Cv

DN

PD

RFC

101

102

103

104

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(b) 119863NP119863RFC

Figure 5 Effect of coefficient of variation on fatigue damage (in logarithmic scale)

some values of 1205722 the three values were nearly equal to each

other Thus the fatigue strength had only a minor effect onthe values of119863LCC

119863RFC and119863

NP119863

RFCThe effects of the coefficient of variation 119862V = 120590

119909119898119909of

the stress process on the applicability of these two methodswere investigated for the same 119868

119898= 119898119909119891cdfat ratio as

shown in Figure 5 Similar119863LCC119863

RFC and119863NP

119863RFCcurves

were obtained for the same 1205722value In some cases the

119863LCC

119863RFC and 119863

NP119863

RFC values increased dramaticallyas the coefficient of variation 119862V decreased Both the LCCmethod and the new method produced overly conservativeestimates of the fatigue damage

The applicability of the frequency methods deterioratedfor lower coefficients of variation because 119873

119894was defined

as an exponential function Figure 6 shows the cycle countversus the stress amplitude and the mean stress value ofa specified stress record that were obtained using the RFCmethod and the corresponding fatigue damage caused bypairs of the stress amplitude and mean stress Only a fewcycles had stress amplitudes near the maximum amplitude119903max and a mean stress value near the mean values of stressrange 119898

119909 whereas the corresponding fatigue damage was

fairly high with an order of magnitude that was nearlythe same as that of the fatigue damage of the entire stressprocess The value of 119873

119894was highly sensitive to the potential

maximum stress range of the cycle and the total fatiguedamage could be estimated by calculating the fatigue damage1198631 induced by 1 cycle with a pair of range and mean values

of 119904max and 119898119909 respectively (see (14)) As the stress record

was assumed to be Gaussian the limits of the stress interval(119898119909minus 119899120590119909 119898119909+ 119899120590119909) were typically used as estimates of the

upper and lower bounds respectively of the record thus it

was reasonable to assume that 119904max was equal to 119899120590119909 1198631 is

defined as follows

1198631=

1

119873119894(119904max 119898119909)

= 10minus14((1minus(119898

119909+119904max)119891cdfat)radic1minus(119898119909minus119904max)(119898119909+119904max))

= 10minus14((1minus(119898

119909+119899120590119909)119891cdfat)radic1minus(119898119909minus119899120590119909)(119898119909+119899120590119909))

= 10minus14((1minus(119898

119909+119899119862V119898119909)119891cdfat)radic1minus(119898119909minus119899119862V119898119909)(119898119909+119899119862V119898119909))

= 10minus14((1minus(1+119899119862V)119898119909119891cdfat)radic1minus(1minus119899119862V)(1+119899119862V))

= 10minus14((1minus(1+119899119862V)119868119898)radic2119899119862V(1+119899119862V))

(24)

The values of 1198631 were calculated for 119899 = 20 30 31 3540 and 50 for different 119862V values and a constant 119868

119898 The

ratios of 1198631(119898119909 119899120590119909) to 119863

1(119898119909 3120590119909) are shown in Figure 7

(on a logarithmic scale) These ratios increased or decreasedexponentially as119862V decreased for values of 119899 that were greateror less than 30The value of1198631(119898

119909 5120590119909)was approximately 9

orders ofmagnitude larger than1198631(119898119909 3120590119909) that is 1 cycle of

stress with a pair of the stress range and the stress mean valueof (119898

119909 5120590119909) produced the same damage as approximately

109 cycles of stress with a pair of the stress range and thestress mean value of (119898

119909 3120590119909) In the time domain the

number of cycles with high stress range could be countedas zero In the frequency domain the probability of theoccurrence of the high stress range was relatively negligiblebut was a nonzero value which had a tremendous effectwhen coupled to the damage that was induced by high stressrangeThus the fatigue damage that was calculated using the


145290 435 580 725

870135270 405 540 675

810

050

100150200250300350400

Mean (MPa)Amplitude (MPa)

Cou

ntin

g

(a) Cycle count versus stress amplitude and mean stress value

145 290 435 580 725 870135 270 405 540 675 810

01234567


Dam

age

times10minus9

(b) Corresponding fatigue damage

Figure 6 Cycle count versus stress amplitude and mean stress value and corresponding fatigue damage

00 01 02 03 04 05 06minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

n = 20

n = 31

n = 35

n = 40

n = 50

C

log 1

0(D

1(n120590xm

x)D

1(3120590xm

x))

Figure 7 Ratios of 1198631(119899120590119909 119898119909) to119863

1(3120590119909 119898119909) versus 119862V

frequency domain was larger than that calculated using thetime domain

We verified the applicability of these joint PDFs of theLCCmethod and the method for different 119862V values by usingan imaginary simple function 119873

119894= 119906 sdot V in the numerical

simulation Figure 8 shows the ratios of 119863LCC

119863RFC and

119863NP

119863RFC that were obtained Both of these two ratios

remained nearly constant for different 119862V values when thevalue of119873

119894was not highly sensitive to the potentialmaximum

stress ranges of the cycle Both ratios were approximately 10Thus both of these two joint PDFs described the distributionsof the ldquovalleysrdquo and ldquopeaksrdquo of the stress processes fairlyaccurately

Different integration intervals were used for the LCCmethod and the newmethod to calculate the fatigue damagewhich was normalized by the results of the RFC method(Figure 9) The effect of the integration intervals on thefatigue damage reflected the effect of the definition of 119873

119894on

the fatigue damage The fatigue damage was overestimated

using large intervals and underestimated using small inter-vals where the extent of overestimation or underestimationwas relatively higher for a lower 119862V As the joint PDFsdescribed the ldquovalleysrdquo and ldquopeaksrdquo of stress processes fairlyaccurately the upper and lower bounds on the fatigue damagecan be estimated by varying the integration intervals in thefrequency methods

The RFC method in the time domain is generally con-sidered the ldquostandardrdquo method for assessing fatigue damagehowever the stress process applied in the time-domainmethod is a fragment of the entire actual process Stressprocesses with different maximum ranges may producedifferent fatigue damage assessments Figure 10 shows afatigue damage assessment on a logarithmic scale using 10stress records which had the same coefficients of variationbecause theywere derived from the same PSDThemaximumfatigue damage was two orders of magnitude larger than theminimum fatigue damage Thus the RFC method in thetime domain does not serve as an appropriate ldquostandardrdquoand should be used with caution that is different stressprocesses should be used in engineering practice As notedabove varying the integration intervals in the two frequency-domainmethods can be used to estimate the lower and upperbounds of the fatigue damage thus the results derived fromthe frequency-domain methods could serve as a referenceto evaluate the accuracy of the results of the time-domainmethod

5 Conclusion

Three frequency-domain methods were used for a fatiguedamage assessment The underlying principle of thefrequency-domain method is the construction of a PDF forthe ldquopeaksrdquo and ldquovalleysrdquo of the stress process (ie the stressrange and mean stress) Two PDFs were taken from theliterature and one PDF was formulated by the authors Thisnovel PDF was constructed by replacing 119901

119904(119904) in (14) using

the Dirlik method (see (16)) and assuming that the stressrange ldquo119904rdquo and stress mean ldquo119898rdquo were independent variables


000 002 004 006 008 010 01200

05

10

15

20

25

30

C

DLC

CD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(a) 119863LCC119863RFC

000 002 004 006 008 010 01200

05

10

15

20

25

30

C

DN

PD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(b) 119863NP119863RFC

Figure 8 Applicability of two joint PDFs

000 002 004 006 008 010 012minus3

minus2

minus1

0

1

2

3

4

C

1205722 = 087 I1205722 = 087 II1205722 = 087 III

1205722 = 016 I1205722 = 016 II1205722 = 016 III

log 1

0(D

LCCD

RFC)

(a) Integration intervals for the LCC method I 119904(minus5120590119909 5120590119909) 119898(119898119909 minus5120590119909119898119909 + 5120590119909) II 119904(minus4120590119909 4120590119909) 119898(119898119909 minus 4120590119909119898119909 + 4120590119909) and III119904(minus3120590119909 3120590119909)119898(119898119909 minus 3120590119909119898119909 + 3120590119909)

000 002 004 006 008 010 012minus3

minus2

minus1

0

1

2

3

C

1205722 = 087 I1205722 = 087 II1205722 = 087 III

1205722 = 016 I1205722 = 016 II1205722 = 016 III

log 1

0(D

NPD

RFC)

(b) Integration intervals for the newmethod I 119906(119898119909 +infin) II 119906(119898119909119898119909 +4120590119909) and III 119906(119898119909119898119909 + 3120590119909)

Figure 9 Effects of integration intervals on damage assessment

0 1 2 3 4 5 6 7 8 9 10 11Number of samples

minus260

minus255

minus250

minus245

minus240

minus235

minus230

log 1

0(D

RFC)

Figure 10 Fatigue damage assessment based on 10 stress recordswith the same parameters (eg 120572

1 1205722119898119909 120590119909)

The applicability of these three methods was investigatedby comparing the results of the frequency-domain methodswith those from the RFC method in the time domain Apreliminary study demonstrated that both the LCC methodand the new method yielded accurate estimates over theentire range of bandwidths of the stress process whereas theRC method yielded poor estimates for low bandwidths

We carried out parametric studies to further verify theapplicability of the LCC method and the new method Thefatigue strength of concrete had a slight impact on the appli-cability of these two methods Decreasing the coefficient ofvariation deteriorated the applicability of these two methodsfor some cases This behavior was attributed to the definitionof 119873119894 The value of 119873


maximum stress ranges of the cycle particularly for relativelylow coefficients of variationThevalidity of the two joint PDFsfor the LCC method and the new method was evaluated bya numerical simulation The integration intervals of the two


frequency-domain methods can be varied to calculate thelower and upper bounds on the fatigue damage which canserve as references to evaluate the accuracy of the results ofthe time-domain method

In conclusion the RC method is only applicable to wide-band (120572

2gt 065) stress processes and the LCC method and

the new method are applicable for all bandwidths Both theLCCmethod and the newmethod should be used in practicaldesign formutual authenticationOnly pure compressionwasinvestigated in this study and119873

119894was defined for pure tension

and tension-compression [24 25] as an exponential functionas in pure compressionThus the conclusions of this study areonly applicable for the aforementioned cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge the support from theNational Natural Science Foundation of China (Grant nos51379142 and 51309179) the International SampT CooperationProgram of China (Grant no 2012DFA70490) and theTianjin Municipal Natural Science Foundation (Grant no13JCYBJC19100)

References

[1] F Goransson Fatigue assessment of concrete foundations forwind power plants [MS thesis] Chalmers University of Tech-nology Goteborg Sweden 2011

[2] F Olsson Fatigue assessment methods for reinforced concretebridges in Eurocode [MS thesis] Chalmers University of Tech-nology Goteborg Sweden 2010

[3] British Standards Institution EN 1992-22005 Eurocode 2Design of concrete structures Concrete bridges Design anddetailing rules 2005

[4] H Agerskov ldquoFatigue in steel structures under random load-ingrdquo Journal of Constructional Steel Research vol 53 no 3 pp283ndash305 2000

[5] Z Li J W Ringsberg and G Storhaug ldquoTime-domain fatigueassessment of ship side-shell structuresrdquo International Journalof Fatigue vol 55 pp 276ndash290 2013

[6] P R Thies L Johanning V Harnois H C M Smith and DN Parish ldquoMooring line fatigue damage evaluation for floatingmarine energy converters Fieldmeasurements and predictionrdquoRenewable Energy vol 63 pp 133ndash144 2014

[7] S Ariduru Fatigue life calculation by rainflow cycle countingmethod [MS thesis] Middle East Technical University AnkaraTurkey 2004

[8] MMatsuishi andT Endo Fatigue ofMetals Subjected toVaryingStress Japan Society of Mechanical Engineers Fukuoka Japan1968

[9] X Liu G Feng and H Ren ldquoStudy on the application of spec-tral fatigue analysisrdquo Journal of Marine Science and Applicationvol 5 no 2 pp 42ndash46 2006

[10] American Bureau of Shipping Spectral-Based Fatigue Analysisfor Floating Offshore Structures 2005

[11] G Petrucci M Di Paola and B Zuccarello ldquoOn the character-ization of dynamic properties of random processes by spectralparametersrdquo Journal of AppliedMechanics vol 67 no 3 pp 519ndash526 2000

[12] J V D Tempel Design of support structures for offshore windturbines [PhD thesis] Delft University of Technology DelftThe Netherlands 2006

[13] T Kukkanen and T P J Mikkola ldquoFatigue assessment byspectral approach for the ISSC comparative study of the hatchcover bearing padrdquo Marine Structures vol 17 no 1 pp 75ndash902004

[14] P Stoica andRMoses Spectral Analysis of Signals PrenticeHall2005

[15] M Mrsnik J Slavic and M Boltezar ldquoFrequency-domainmethods for a vibration-fatigue-life estimationmdashapplication toreal datardquo International Journal of Fatigue vol 47 pp 8ndash17 2013

[16] M Miner ldquoCumulative damage in fatiguerdquo Journal of AppliedMechanics vol 12 no 3 pp 159ndash164 1945

[17] R Tovo ldquoCycle distribution and fatigue damage under broad-band random loadingrdquo International Journal of Fatigue vol 24no 11 pp 1137ndash1147 2002

[18] D Benasciutti and R Tovo ldquoSpectral methods for lifetimeprediction under wide-band stationary random processesrdquoInternational Journal of Fatigue vol 27 no 8 pp 867ndash877 2005

[19] T DirlikApplications of Computers in Fatigue Analysis Univer-sity of Warwick Coventry UK 1985

[20] P Ragan and L Manuel ldquoComparing estimates of wind turbinefatigue loads using time-domain and spectral methodsrdquo WindEngineering vol 31 no 2 pp 83ndash99 2007

[21] J M Naser Analysis of vibration-induced fatigue cracking insteel bridges [MS thesis] Chalmers University of TechnologyGoteborg Sweden 2010

[22] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 no 4 pp 191ndash204 1991

[23] B Hu and W Schiehlen ldquoOn the simulation of stochasticprocesses by spectral representationrdquo Probabilistic EngineeringMechanics vol 12 no 2 pp 105ndash113 1997

[24] Det Norske Veritas DNV-OS-C502 Offshore concrete struc-tures 2012

[25] ldquoCEB-FIP Model Coderdquo 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


In this study PDFs that are available in the literatureand a PDF developed by the authors were applied to cal-culate the fatigue damage of concrete structures The RFCmethod in the time domain is the standard method forfatigue assessmentThe results of different frequency-domainmethods were compared with the results of the RFC methodto select the most effective frequency-domain methods Apreliminary numerical study was carried out to verify theapplicability of the frequency-domain methods for stressprocesses with different bandwidths thus the applicabilityof the LCC method and the new method was preliminarilyconfirmed The integration intervals of the two frequency-domainmethods were varied to estimate the lower and upperbounds on the fatigue damage which can serve as referencesto evaluate the accuracy of the time-domain method resultsFew published reports on fatigue assessment for concrete inthe frequency domain are currently available thus the resultsof this study can serve as a useful reference in this field [13]

2 Basic Theory

21 Frequency-Domain Process A stress process ℎ(119905) isassumed to be a stationary Gaussian process The autocorre-lation function of ℎ(119905) is defined as follows [14]

119870 (120591) equiv ℎ (119905) ℎ (119905 + 120591) = lim119879rarrinfin

1

2119879int

119879

minus119879

ℎ (119905) ℎ (119905 + 120591) 119889120591 (1)

The Fourier transform of the autocorrelation function isexpressed as

119870 (120591) = int

infin

minusinfin

119878 (119891) 119890minus119894120596120591

119889119891 (2)

where the new function 119878(119891) is the PSD of ℎ(119905) 119878(119891) can alsobe written in terms of119870(120591)

119878 (119891) = int

infin

minusinfin

119870 (120591) 119890119894120596120591

119889120591 (3)

Equations (2) and (3) are known as the Wiener-Khintchine relations

We are typically not interested in distinguishing a PSDthat is associated with a negative frequency from one that isassociated with a positive frequency A single-sided PSD canbe defined as follows

119878(119891) = 119878 (119891) + 119878 (minus119891) (119891 ge 0) (4)

Variables that are encountered in engineering analysissuch as the stress are real The PSD is an even function forreal variables thus

119878(119891) = 119878 (119891) + 119878 (minus119891) = 2119878 (119891) (119891 ge 0) (5)

The 119899th moment of the PSD function can be obtainedfrom the characteristic of the PSD function The spectralmoments are defined as follows

119898119899= int

infin

0

119891119899119878(119891)119889119891 (6)

The first fourmoments (119898011989811198982 and119898

4) are themost

commonly used moments of the PSD function

The bandwidth of a signal is taken to be the frequencyinterval over which most of the signal power is concentratedThe bandwidth can be defined in terms of the spectralmoments [15]

1205721=

1198981

radic11989801198981

1205722=

1198982

radic11989801198984

0 le 1205721 1205722le 1

(7)

These two bandwidth parameters tend to unity for aldquonarrow-bandrdquo signal and zero for a ldquowide-bandrdquo signal

22 Fatigue Damage Assessment Method for Concrete Struc-tures from Eurocode 2 In Eurocode 2 Palmgren-Minerrsquos rule[16] is applied to calculate the total fatigue damage as givenin (8) A satisfactory fatigue resistance may be assumed forconcrete under compression if119863 le 1

119863 =

119898

sum

119894=1

119899119894

119873119894

(8)

where 119863 is the total fatigue damage 119898 is the number ofintervals with a constant amplitude 119899

119894is the actual number

of constant-amplitude cycles in the interval ldquo119894rdquo 119873119894is the

ultimate number of constant-amplitude cycles in the intervalldquo119894rdquo that can be carried out before failure

119873119894= 1014((1minus119864cdmax119894)radic1minus119877119894) 119877

119894=

119864cdmin119894

119864cdmax119894

119864cdmin119894 =120590cdmin119894

119891cdfat119864cdmax119894 =

120590cdmax119894

119891cdfat

(9)

where 119877119894is the stress ratio 119864cdmin119894 is the minimum stress

compression level 119864cdmax119894 is the maximum stress compres-sion level 120590cdmin119894 is the lowest stress in a cycle 120590cdmax119894 isthe highest stress in a cycle and 119891cdfat is the design fatiguestrength of the concrete 119873

119894is a function of both the peaks

and valleys of the stress process

23 Counting Methods LCC PC RC and RFC are thefour most commonly used counting methods in engineeringpractice

In the LCC method every crossing of the predeterminedstress levels is countedThen themost damaging level cycle iscounted for fatigue by constructing the largest possible cyclefollowed by the second largest cycle and so on until all of thelevel crossings have been considered

In the PCmethod the occurrence of a relative maximumor minimum stress value is identified Peaks (valleys) that areabove (below) the reference level are counted

In the RC method a range is defined as the differencebetween two successive reversals the range is positive whena valley is followed by a peak and negative when a peak isfollowed by a valley

The RFC method is the most frequently used cyclecounting method in engineering practice Starting from alocal loadmaximumMax

119896 twominima are identified before

and after Max119896 that is Min

119896minusand Min

119896+ The point with




(Min119896RFC Max








ℎLCC (119906 V)


119901119901 (119906) 120575 (119906 minus V) 119906 le 0

(10)



119901119901 (119909) = Φ(

119909

120590119909radic1 minus 1205722

2

) minus 1205722119890minus1199092

21205902

119909Φ(1205722119909

120590119909radic1 minus 1205722

2

)

(11)

119901V (119909) = 119901119901 (minus119909) (12)





ℎRC (119906 V) =1

1205902119909120572222radic2120587


+V2)41205902119909

(1minus1205722

2

)119890minus((119906minusV)241205902

119909

(1minus1205722

2

))((1minus1205722

2

)21205902

119909

)

times[[

[

119906 minus V

radic41205902119909(1 minus 1205722

2)

]]

]

(13)


ℎRC (119904 119898) =1

radic2120587120590119909(1 minus 1205722

2)


2120590119909(1minus1205722

2

) 119904

12059011990912057222

119890minus1199042

21205901199091205722

2

= 119901119898 (119898) 119901119904 (119904)

(14)





119904(119904) can


ℎNP (119904 119898) =1

radic2120587120590119909(1 minus 1205722

2)

119890minus1198982

(2120590119909(1minus1205722

2

))119901Dirlik (119904) (15)

119901Dirlik (119904) =1

radic120590119909

[1198661

119876119890minus119885119876

+1198662119885

1198772119890minus1198852

21198772

+ 1198663119885119890minus1198852

2]

(16)

119885 =119904

radic120590119909

119909119898

=1198981

1198980

(1198982

1198984

)

12

(17)

where

1198661=

2 (119909119898

minus 1205722

2)

1 + 12057222

119909119898

=1 minus 1205722

2minus 1198661+ 1198662

1

1 minus 119877

119885 = 1 minus 1198661minus 1198662

119877 =1205722minus 119909119898

minus 1198662

1

1 minus 1205722minus 1198661+ 11986621

119876 =125 (120572

2minus 1198663minus 1198662119877)

1198661

(18)



[21]

120592119901= radic

1198984

1198982

(19)


119864 (119863) = 119879120592119901∬

ℎ(119906 V)119873119894 (119906 V)

119889119906 119889V (20)




119901(119906)120575(119906 minus V) thus

119864 (1198631015840) = 119879120592

119901∬

ℎ(119906 V)119873119894 (119906 V)

119889119906 119889V

= 119879120592119901∬

119901119901 (119906) 120575 (119906 minus V)119873119894 (119906 V)

119889119906 119889V

= 119879120592119901∬

0

minusinfin

119901119901 (119906) 120575 (119906 minus V)119873119894 (119906 V)

119889119906 119889V

= 119879120592119901int

0

minusinfin

(int

0

minusinfin

119901119901 (119906) 120575 (119906 minus V)119873119894 (119906 V)

119889119906)119889V

= 119879120592119901int

0

minusinfin

119901119901 (V)

119873119894 (V V)

119889V

(21)




119909 these



119864 (119863LCC

) = 119879120592119901∬

ℎ(119906 V)119873119894 (119906 V)

119889119906 119889V

= 119879120592119901∬([119901

119901(119906 minus 119898

119909) minus 119901V (119906 minus 119898

119909)]


119909))

times (119873119894 (119906 V))

minus1) 119889119906 119889V

= 119879120592119901int

infin

119898119909

(int

119906

minusinfin

(119901119901(119906 minus 119898

119909) minus 119901V (119906 minus 119898

119909)

times 120575 (119906 + V minus 2119898119909)

times (119873119894 (119906 V))

minus1) 119889V)119889119906

= 119879120592119901(int

2119898119909

119898119909

119901119901(119906 minus 119898

119909) minus 119901V (119906 minus 119898

119909)

119873119894(119906 2119898

119909minus 119906)

119889119906

+int

infin

2119898119909

119901119901(119906 minus 119898

119909) minus 119901V (119906 minus 119898

119909)

119873119894 (119906 0)

119889119906)

(22)







119864 (119863) = 119879120592119901∬

ℎ(119904119898)

119873119894 (119904 119898)

119889119904 119889119898 (23)


119909minus 5120590119909 119898119909

+



) and 119864(119863NP

) 119898 (119898119909minus 5120590119909 119898119909+ 5120590119909)



119909minus 5120590119909))2 For


is set to zero)




2 on



119909of








LCC119863


NP119863




0 2 4 6 8 1000

02

04

06

08

10

Frequency (Hz)

PSD(

MPa

2H

z)

(a)

0 2 4 6 8 1000

02

04

06

08

10

Frequency (Hz)

PSD(

MPa

2H

z)

(b)


0 20 40 60 80 100

2

3

4

5

6

7

8

Stre

ss (M

Pa)

Time (s)

1205722 = 017

(a)

0 20 40 60 80 1001

2

3

4

5

6

7

8

9St

ress

(MPa

)

Time (s)

1205722 = 038

(b)

0 20 40 60 80 100

1

2

3

4

5

6

7

8

9

Time (s)

Stre

ss (M

Pa)

1205722 = 086

(c)



00 01 02 03 04 05 06 07 08 09 100

2

4

6

8

10

12

1205722

DLC

CD

RFC

(a) 119863LCC119863RFC

00 01 02 03 04 05 06 07 08 09 1002468

101214161820

1205722

DN

PD

RFC

(b) 119863NP119863RFC

00 01 02 03 04 05 06 07 08 09 10

minus10

minus8

minus6

minus4

minus2

0

2

1205722

log 1

0(D

RCD

RFC)

(c) 119863RC119863RFC


00 02 04 06 08 100

2

4

6

8

10

12

14

16

DLC

CD

RFC

1205722


(a) 119863LCC119863RFC

00 02 04 06 08 100

2

4

6

8

10

12

14

16

18

20

DN

PD

RFC

1205722


(b) 119863NP119863RFC


119863RC


2below







119909




119863RFC and

119863NP


2 at


000 002 004 006 008 010 012Cv

DLC

CD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

100

101

102

103

104

(a) 119863LCC119863RFC

000 002 004 006 008 010 012Cv

DN

PD

RFC

101

102

103

104

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(b) 119863NP119863RFC




119863RFC and119863

NP119863


119909119898119909of


119898= 119898119909119891cdfat ratio as


RFC and119863NP

119863RFCcurves


119863LCC

119863RFC and 119863

NP119863



119894was defined










defined as follows

1198631=

1

119873119894(119904max 119898119909)

= 10minus14((1minus(119898


= 10minus14((1minus(119898


= 10minus14((1minus(119898




(24)


119898 The

ratios of 1198631(119898119909 119899120590119909) to 119863











145290 435 580 725

870135270 405 540 675

810

050

100150200250300350400


Cou

ntin

g


145 290 435 580 725 870135 270 405 540 675 810

01234567


Dam

age

times10minus9



00 01 02 03 04 05 06minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

n = 20

n = 31

n = 35

n = 40

n = 50

C

log 1

0(D

1(n120590xm

x)D

1(3120590xm

x))


1(3120590119909 119898119909) versus 119862V





119863RFC and

119863NP






119894on




5 Conclusion


119904(119904) in (14) using



000 002 004 006 008 010 01200

05

10

15

20

25

30

C

DLC

CD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(a) 119863LCC119863RFC

000 002 004 006 008 010 01200

05

10

15

20

25

30

C

DN

PD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(b) 119863NP119863RFC


000 002 004 006 008 010 012minus3

minus2

minus1

0

1

2

3

4

C

1205722 = 087 I1205722 = 087 II1205722 = 087 III

1205722 = 016 I1205722 = 016 II1205722 = 016 III

log 1

0(D

LCCD

RFC)


000 002 004 006 008 010 012minus3

minus2

minus1

0

1

2

3

C

1205722 = 087 I1205722 = 087 II1205722 = 087 III

1205722 = 016 I1205722 = 016 II1205722 = 016 III

log 1

0(D

NPD

RFC)




minus260

minus255

minus250

minus245

minus240

minus235

minus230

log 1

0(D

RFC)


1 1205722119898119909 120590119909)














Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(Min119896RFC Max








ℎLCC (119906 V)


119901119901 (119906) 120575 (119906 minus V) 119906 le 0

(10)



119901119901 (119909) = Φ(

119909

120590119909radic1 minus 1205722

2

) minus 1205722119890minus1199092

21205902

119909Φ(1205722119909

120590119909radic1 minus 1205722

2

)

(11)

119901V (119909) = 119901119901 (minus119909) (12)





ℎRC (119906 V) =1

1205902119909120572222radic2120587


+V2)41205902119909

(1minus1205722

2

)119890minus((119906minusV)241205902

119909

(1minus1205722

2

))((1minus1205722

2

)21205902

119909

)

times[[

[

119906 minus V

radic41205902119909(1 minus 1205722

2)

]]

]

(13)


ℎRC (119904 119898) =1

radic2120587120590119909(1 minus 1205722

2)


2120590119909(1minus1205722

2

) 119904

12059011990912057222

119890minus1199042

21205901199091205722

2

= 119901119898 (119898) 119901119904 (119904)

(14)





119904(119904) can


ℎNP (119904 119898) =1

radic2120587120590119909(1 minus 1205722

2)

119890minus1198982

(2120590119909(1minus1205722

2

))119901Dirlik (119904) (15)

119901Dirlik (119904) =1

radic120590119909

[1198661

119876119890minus119885119876

+1198662119885

1198772119890minus1198852

21198772

+ 1198663119885119890minus1198852

2]

(16)

119885 =119904

radic120590119909

119909119898

=1198981

1198980

(1198982

1198984

)

12

(17)

where

1198661=

2 (119909119898

minus 1205722

2)

1 + 12057222

119909119898

=1 minus 1205722

2minus 1198661+ 1198662

1

1 minus 119877

119885 = 1 minus 1198661minus 1198662

119877 =1205722minus 119909119898

minus 1198662

1

1 minus 1205722minus 1198661+ 11986621

119876 =125 (120572

2minus 1198663minus 1198662119877)

1198661

(18)



[21]

120592119901= radic

1198984

1198982

(19)


119864 (119863) = 119879120592119901∬

ℎ(119906 V)119873119894 (119906 V)

119889119906 119889V (20)




119901(119906)120575(119906 minus V) thus

119864 (1198631015840) = 119879120592

119901∬

ℎ(119906 V)119873119894 (119906 V)

119889119906 119889V

= 119879120592119901∬

119901119901 (119906) 120575 (119906 minus V)119873119894 (119906 V)

119889119906 119889V

= 119879120592119901∬

0

minusinfin

119901119901 (119906) 120575 (119906 minus V)119873119894 (119906 V)

119889119906 119889V

= 119879120592119901int

0

minusinfin

(int

0

minusinfin

119901119901 (119906) 120575 (119906 minus V)119873119894 (119906 V)

119889119906)119889V

= 119879120592119901int

0

minusinfin

119901119901 (V)

119873119894 (V V)

119889V

(21)




119909 these



119864 (119863LCC

) = 119879120592119901∬

ℎ(119906 V)119873119894 (119906 V)

119889119906 119889V

= 119879120592119901∬([119901

119901(119906 minus 119898

119909) minus 119901V (119906 minus 119898

119909)]


119909))

times (119873119894 (119906 V))

minus1) 119889119906 119889V

= 119879120592119901int

infin

119898119909

(int

119906

minusinfin

(119901119901(119906 minus 119898

119909) minus 119901V (119906 minus 119898

119909)

times 120575 (119906 + V minus 2119898119909)

times (119873119894 (119906 V))

minus1) 119889V)119889119906

= 119879120592119901(int

2119898119909

119898119909

119901119901(119906 minus 119898

119909) minus 119901V (119906 minus 119898

119909)

119873119894(119906 2119898

119909minus 119906)

119889119906

+int

infin

2119898119909

119901119901(119906 minus 119898

119909) minus 119901V (119906 minus 119898

119909)

119873119894 (119906 0)

119889119906)

(22)







119864 (119863) = 119879120592119901∬

ℎ(119904119898)

119873119894 (119904 119898)

119889119904 119889119898 (23)


119909minus 5120590119909 119898119909

+



) and 119864(119863NP

) 119898 (119898119909minus 5120590119909 119898119909+ 5120590119909)



119909minus 5120590119909))2 For


is set to zero)




2 on



119909of








LCC119863


NP119863




0 2 4 6 8 1000

02

04

06

08

10

Frequency (Hz)

PSD(

MPa

2H

z)

(a)

0 2 4 6 8 1000

02

04

06

08

10

Frequency (Hz)

PSD(

MPa

2H

z)

(b)


0 20 40 60 80 100

2

3

4

5

6

7

8

Stre

ss (M

Pa)

Time (s)

1205722 = 017

(a)

0 20 40 60 80 1001

2

3

4

5

6

7

8

9St

ress

(MPa

)

Time (s)

1205722 = 038

(b)

0 20 40 60 80 100

1

2

3

4

5

6

7

8

9

Time (s)

Stre

ss (M

Pa)

1205722 = 086

(c)



00 01 02 03 04 05 06 07 08 09 100

2

4

6

8

10

12

1205722

DLC

CD

RFC

(a) 119863LCC119863RFC

00 01 02 03 04 05 06 07 08 09 1002468

101214161820

1205722

DN

PD

RFC

(b) 119863NP119863RFC

00 01 02 03 04 05 06 07 08 09 10

minus10

minus8

minus6

minus4

minus2

0

2

1205722

log 1

0(D

RCD

RFC)

(c) 119863RC119863RFC


00 02 04 06 08 100

2

4

6

8

10

12

14

16

DLC

CD

RFC

1205722


(a) 119863LCC119863RFC

00 02 04 06 08 100

2

4

6

8

10

12

14

16

18

20

DN

PD

RFC

1205722


(b) 119863NP119863RFC


119863RC


2below







119909




119863RFC and

119863NP


2 at


000 002 004 006 008 010 012Cv

DLC

CD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

100

101

102

103

104

(a) 119863LCC119863RFC

000 002 004 006 008 010 012Cv

DN

PD

RFC

101

102

103

104

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(b) 119863NP119863RFC




119863RFC and119863

NP119863


119909119898119909of


119898= 119898119909119891cdfat ratio as


RFC and119863NP

119863RFCcurves


119863LCC

119863RFC and 119863

NP119863



119894was defined










defined as follows

1198631=

1

119873119894(119904max 119898119909)

= 10minus14((1minus(119898


= 10minus14((1minus(119898


= 10minus14((1minus(119898




(24)


119898 The

ratios of 1198631(119898119909 119899120590119909) to 119863











145290 435 580 725

870135270 405 540 675

810

050

100150200250300350400


Cou

ntin

g


145 290 435 580 725 870135 270 405 540 675 810

01234567


Dam

age

times10minus9



00 01 02 03 04 05 06minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

n = 20

n = 31

n = 35

n = 40

n = 50

C

log 1

0(D

1(n120590xm

x)D

1(3120590xm

x))


1(3120590119909 119898119909) versus 119862V





119863RFC and

119863NP






119894on




5 Conclusion


119904(119904) in (14) using



000 002 004 006 008 010 01200

05

10

15

20

25

30

C

DLC

CD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(a) 119863LCC119863RFC

000 002 004 006 008 010 01200

05

10

15

20

25

30

C

DN

PD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(b) 119863NP119863RFC


000 002 004 006 008 010 012minus3

minus2

minus1

0

1

2

3

4

C

1205722 = 087 I1205722 = 087 II1205722 = 087 III

1205722 = 016 I1205722 = 016 II1205722 = 016 III

log 1

0(D

LCCD

RFC)


000 002 004 006 008 010 012minus3

minus2

minus1

0

1

2

3

C

1205722 = 087 I1205722 = 087 II1205722 = 087 III

1205722 = 016 I1205722 = 016 II1205722 = 016 III

log 1

0(D

NPD

RFC)




minus260

minus255

minus250

minus245

minus240

minus235

minus230

log 1

0(D

RFC)


1 1205722119898119909 120590119909)














Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119901(119906)120575(119906 minus V) thus

119864 (1198631015840) = 119879120592

119901∬

ℎ(119906 V)119873119894 (119906 V)

119889119906 119889V

= 119879120592119901∬

119901119901 (119906) 120575 (119906 minus V)119873119894 (119906 V)

119889119906 119889V

= 119879120592119901∬

0

minusinfin

119901119901 (119906) 120575 (119906 minus V)119873119894 (119906 V)

119889119906 119889V

= 119879120592119901int

0

minusinfin

(int

0

minusinfin

119901119901 (119906) 120575 (119906 minus V)119873119894 (119906 V)

119889119906)119889V

= 119879120592119901int

0

minusinfin

119901119901 (V)

119873119894 (V V)

119889V

(21)




119909 these



119864 (119863LCC

) = 119879120592119901∬

ℎ(119906 V)119873119894 (119906 V)

119889119906 119889V

= 119879120592119901∬([119901

119901(119906 minus 119898

119909) minus 119901V (119906 minus 119898

119909)]


119909))

times (119873119894 (119906 V))

minus1) 119889119906 119889V

= 119879120592119901int

infin

119898119909

(int

119906

minusinfin

(119901119901(119906 minus 119898

119909) minus 119901V (119906 minus 119898

119909)

times 120575 (119906 + V minus 2119898119909)

times (119873119894 (119906 V))

minus1) 119889V)119889119906

= 119879120592119901(int

2119898119909

119898119909

119901119901(119906 minus 119898

119909) minus 119901V (119906 minus 119898

119909)

119873119894(119906 2119898

119909minus 119906)

119889119906

+int

infin

2119898119909

119901119901(119906 minus 119898

119909) minus 119901V (119906 minus 119898

119909)

119873119894 (119906 0)

119889119906)

(22)







119864 (119863) = 119879120592119901∬

ℎ(119904119898)

119873119894 (119904 119898)

119889119904 119889119898 (23)


119909minus 5120590119909 119898119909

+



) and 119864(119863NP

) 119898 (119898119909minus 5120590119909 119898119909+ 5120590119909)



119909minus 5120590119909))2 For


is set to zero)




2 on



119909of








LCC119863


NP119863




0 2 4 6 8 1000

02

04

06

08

10

Frequency (Hz)

PSD(

MPa

2H

z)

(a)

0 2 4 6 8 1000

02

04

06

08

10

Frequency (Hz)

PSD(

MPa

2H

z)

(b)


0 20 40 60 80 100

2

3

4

5

6

7

8

Stre

ss (M

Pa)

Time (s)

1205722 = 017

(a)

0 20 40 60 80 1001

2

3

4

5

6

7

8

9St

ress

(MPa

)

Time (s)

1205722 = 038

(b)

0 20 40 60 80 100

1

2

3

4

5

6

7

8

9

Time (s)

Stre

ss (M

Pa)

1205722 = 086

(c)



00 01 02 03 04 05 06 07 08 09 100

2

4

6

8

10

12

1205722

DLC

CD

RFC

(a) 119863LCC119863RFC

00 01 02 03 04 05 06 07 08 09 1002468

101214161820

1205722

DN

PD

RFC

(b) 119863NP119863RFC

00 01 02 03 04 05 06 07 08 09 10

minus10

minus8

minus6

minus4

minus2

0

2

1205722

log 1

0(D

RCD

RFC)

(c) 119863RC119863RFC


00 02 04 06 08 100

2

4

6

8

10

12

14

16

DLC

CD

RFC

1205722


(a) 119863LCC119863RFC

00 02 04 06 08 100

2

4

6

8

10

12

14

16

18

20

DN

PD

RFC

1205722


(b) 119863NP119863RFC


119863RC


2below







119909




119863RFC and

119863NP


2 at


000 002 004 006 008 010 012Cv

DLC

CD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

100

101

102

103

104

(a) 119863LCC119863RFC

000 002 004 006 008 010 012Cv

DN

PD

RFC

101

102

103

104

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(b) 119863NP119863RFC




119863RFC and119863

NP119863


119909119898119909of


119898= 119898119909119891cdfat ratio as


RFC and119863NP

119863RFCcurves


119863LCC

119863RFC and 119863

NP119863



119894was defined










defined as follows

1198631=

1

119873119894(119904max 119898119909)

= 10minus14((1minus(119898


= 10minus14((1minus(119898


= 10minus14((1minus(119898




(24)


119898 The

ratios of 1198631(119898119909 119899120590119909) to 119863











145290 435 580 725

870135270 405 540 675

810

050

100150200250300350400


Cou

ntin

g


145 290 435 580 725 870135 270 405 540 675 810

01234567


Dam

age

times10minus9



00 01 02 03 04 05 06minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

n = 20

n = 31

n = 35

n = 40

n = 50

C

log 1

0(D

1(n120590xm

x)D

1(3120590xm

x))


1(3120590119909 119898119909) versus 119862V





119863RFC and

119863NP






119894on




5 Conclusion


119904(119904) in (14) using



000 002 004 006 008 010 01200

05

10

15

20

25

30

C

DLC

CD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(a) 119863LCC119863RFC

000 002 004 006 008 010 01200

05

10

15

20

25

30

C

DN

PD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(b) 119863NP119863RFC


000 002 004 006 008 010 012minus3

minus2

minus1

0

1

2

3

4

C

1205722 = 087 I1205722 = 087 II1205722 = 087 III

1205722 = 016 I1205722 = 016 II1205722 = 016 III

log 1

0(D

LCCD

RFC)


000 002 004 006 008 010 012minus3

minus2

minus1

0

1

2

3

C

1205722 = 087 I1205722 = 087 II1205722 = 087 III

1205722 = 016 I1205722 = 016 II1205722 = 016 III

log 1

0(D

NPD

RFC)




minus260

minus255

minus250

minus245

minus240

minus235

minus230

log 1

0(D

RFC)


1 1205722119898119909 120590119909)














Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 1000

02

04

06

08

10

Frequency (Hz)

PSD(

MPa

2H

z)

(a)

0 2 4 6 8 1000

02

04

06

08

10

Frequency (Hz)

PSD(

MPa

2H

z)

(b)


0 20 40 60 80 100

2

3

4

5

6

7

8

Stre

ss (M

Pa)

Time (s)

1205722 = 017

(a)

0 20 40 60 80 1001

2

3

4

5

6

7

8

9St

ress

(MPa

)

Time (s)

1205722 = 038

(b)

0 20 40 60 80 100

1

2

3

4

5

6

7

8

9

Time (s)

Stre

ss (M

Pa)

1205722 = 086

(c)



00 01 02 03 04 05 06 07 08 09 100

2

4

6

8

10

12

1205722

DLC

CD

RFC

(a) 119863LCC119863RFC

00 01 02 03 04 05 06 07 08 09 1002468

101214161820

1205722

DN

PD

RFC

(b) 119863NP119863RFC

00 01 02 03 04 05 06 07 08 09 10

minus10

minus8

minus6

minus4

minus2

0

2

1205722

log 1

0(D

RCD

RFC)

(c) 119863RC119863RFC


00 02 04 06 08 100

2

4

6

8

10

12

14

16

DLC

CD

RFC

1205722


(a) 119863LCC119863RFC

00 02 04 06 08 100

2

4

6

8

10

12

14

16

18

20

DN

PD

RFC

1205722


(b) 119863NP119863RFC


119863RC


2below







119909




119863RFC and

119863NP


2 at


000 002 004 006 008 010 012Cv

DLC

CD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

100

101

102

103

104

(a) 119863LCC119863RFC

000 002 004 006 008 010 012Cv

DN

PD

RFC

101

102

103

104

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(b) 119863NP119863RFC




119863RFC and119863

NP119863


119909119898119909of


119898= 119898119909119891cdfat ratio as


RFC and119863NP

119863RFCcurves


119863LCC

119863RFC and 119863

NP119863



119894was defined










defined as follows

1198631=

1

119873119894(119904max 119898119909)

= 10minus14((1minus(119898


= 10minus14((1minus(119898


= 10minus14((1minus(119898




(24)


119898 The

ratios of 1198631(119898119909 119899120590119909) to 119863











145290 435 580 725

870135270 405 540 675

810

050

100150200250300350400


Cou

ntin

g


145 290 435 580 725 870135 270 405 540 675 810

01234567


Dam

age

times10minus9



00 01 02 03 04 05 06minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

n = 20

n = 31

n = 35

n = 40

n = 50

C

log 1

0(D

1(n120590xm

x)D

1(3120590xm

x))


1(3120590119909 119898119909) versus 119862V





119863RFC and

119863NP






119894on




5 Conclusion


119904(119904) in (14) using



000 002 004 006 008 010 01200

05

10

15

20

25

30

C

DLC

CD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(a) 119863LCC119863RFC

000 002 004 006 008 010 01200

05

10

15

20

25

30

C

DN

PD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(b) 119863NP119863RFC


000 002 004 006 008 010 012minus3

minus2

minus1

0

1

2

3

4

C

1205722 = 087 I1205722 = 087 II1205722 = 087 III

1205722 = 016 I1205722 = 016 II1205722 = 016 III

log 1

0(D

LCCD

RFC)


000 002 004 006 008 010 012minus3

minus2

minus1

0

1

2

3

C

1205722 = 087 I1205722 = 087 II1205722 = 087 III

1205722 = 016 I1205722 = 016 II1205722 = 016 III

log 1

0(D

NPD

RFC)




minus260

minus255

minus250

minus245

minus240

minus235

minus230

log 1

0(D

RFC)


1 1205722119898119909 120590119909)














Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00 01 02 03 04 05 06 07 08 09 100

2

4

6

8

10

12

1205722

DLC

CD

RFC

(a) 119863LCC119863RFC

00 01 02 03 04 05 06 07 08 09 1002468

101214161820

1205722

DN

PD

RFC

(b) 119863NP119863RFC

00 01 02 03 04 05 06 07 08 09 10

minus10

minus8

minus6

minus4

minus2

0

2

1205722

log 1

0(D

RCD

RFC)

(c) 119863RC119863RFC


00 02 04 06 08 100

2

4

6

8

10

12

14

16

DLC

CD

RFC

1205722


(a) 119863LCC119863RFC

00 02 04 06 08 100

2

4

6

8

10

12

14

16

18

20

DN

PD

RFC

1205722


(b) 119863NP119863RFC


119863RC


2below







119909




119863RFC and

119863NP


2 at


000 002 004 006 008 010 012Cv

DLC

CD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

100

101

102

103

104

(a) 119863LCC119863RFC

000 002 004 006 008 010 012Cv

DN

PD

RFC

101

102

103

104

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(b) 119863NP119863RFC




119863RFC and119863

NP119863


119909119898119909of


119898= 119898119909119891cdfat ratio as


RFC and119863NP

119863RFCcurves


119863LCC

119863RFC and 119863

NP119863



119894was defined










defined as follows

1198631=

1

119873119894(119904max 119898119909)

= 10minus14((1minus(119898


= 10minus14((1minus(119898


= 10minus14((1minus(119898




(24)


119898 The

ratios of 1198631(119898119909 119899120590119909) to 119863











145290 435 580 725

870135270 405 540 675

810

050

100150200250300350400


Cou

ntin

g


145 290 435 580 725 870135 270 405 540 675 810

01234567


Dam

age

times10minus9



00 01 02 03 04 05 06minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

n = 20

n = 31

n = 35

n = 40

n = 50

C

log 1

0(D

1(n120590xm

x)D

1(3120590xm

x))


1(3120590119909 119898119909) versus 119862V





119863RFC and

119863NP






119894on




5 Conclusion


119904(119904) in (14) using



000 002 004 006 008 010 01200

05

10

15

20

25

30

C

DLC

CD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(a) 119863LCC119863RFC

000 002 004 006 008 010 01200

05

10

15

20

25

30

C

DN

PD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(b) 119863NP119863RFC


000 002 004 006 008 010 012minus3

minus2

minus1

0

1

2

3

4

C

1205722 = 087 I1205722 = 087 II1205722 = 087 III

1205722 = 016 I1205722 = 016 II1205722 = 016 III

log 1

0(D

LCCD

RFC)


000 002 004 006 008 010 012minus3

minus2

minus1

0

1

2

3

C

1205722 = 087 I1205722 = 087 II1205722 = 087 III

1205722 = 016 I1205722 = 016 II1205722 = 016 III

log 1

0(D

NPD

RFC)




minus260

minus255

minus250

minus245

minus240

minus235

minus230

log 1

0(D

RFC)


1 1205722119898119909 120590119909)














Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










000 002 004 006 008 010 012Cv

DLC

CD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

100

101

102

103

104

(a) 119863LCC119863RFC

000 002 004 006 008 010 012Cv

DN

PD

RFC

101

102

103

104

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(b) 119863NP119863RFC




119863RFC and119863

NP119863


119909119898119909of


119898= 119898119909119891cdfat ratio as


RFC and119863NP

119863RFCcurves


119863LCC

119863RFC and 119863

NP119863



119894was defined










defined as follows

1198631=

1

119873119894(119904max 119898119909)

= 10minus14((1minus(119898


= 10minus14((1minus(119898


= 10minus14((1minus(119898




(24)


119898 The

ratios of 1198631(119898119909 119899120590119909) to 119863











145290 435 580 725

870135270 405 540 675

810

050

100150200250300350400


Cou

ntin

g


145 290 435 580 725 870135 270 405 540 675 810

01234567


Dam

age

times10minus9



00 01 02 03 04 05 06minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

n = 20

n = 31

n = 35

n = 40

n = 50

C

log 1

0(D

1(n120590xm

x)D

1(3120590xm

x))


1(3120590119909 119898119909) versus 119862V





119863RFC and

119863NP






119894on




5 Conclusion


119904(119904) in (14) using



000 002 004 006 008 010 01200

05

10

15

20

25

30

C

DLC

CD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(a) 119863LCC119863RFC

000 002 004 006 008 010 01200

05

10

15

20

25

30

C

DN

PD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(b) 119863NP119863RFC


000 002 004 006 008 010 012minus3

minus2

minus1

0

1

2

3

4

C

1205722 = 087 I1205722 = 087 II1205722 = 087 III

1205722 = 016 I1205722 = 016 II1205722 = 016 III

log 1

0(D

LCCD

RFC)


000 002 004 006 008 010 012minus3

minus2

minus1

0

1

2

3

C

1205722 = 087 I1205722 = 087 II1205722 = 087 III

1205722 = 016 I1205722 = 016 II1205722 = 016 III

log 1

0(D

NPD

RFC)




minus260

minus255

minus250

minus245

minus240

minus235

minus230

log 1

0(D

RFC)


1 1205722119898119909 120590119909)














Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










145290 435 580 725

870135270 405 540 675

810

050

100150200250300350400


Cou

ntin

g


145 290 435 580 725 870135 270 405 540 675 810

01234567


Dam

age

times10minus9



00 01 02 03 04 05 06minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

n = 20

n = 31

n = 35

n = 40

n = 50

C

log 1

0(D

1(n120590xm

x)D

1(3120590xm

x))


1(3120590119909 119898119909) versus 119862V





119863RFC and

119863NP






119894on




5 Conclusion


119904(119904) in (14) using



000 002 004 006 008 010 01200

05

10

15

20

25

30

C

DLC

CD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(a) 119863LCC119863RFC

000 002 004 006 008 010 01200

05

10

15

20

25

30

C

DN

PD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(b) 119863NP119863RFC


000 002 004 006 008 010 012minus3

minus2

minus1

0

1

2

3

4

C

1205722 = 087 I1205722 = 087 II1205722 = 087 III

1205722 = 016 I1205722 = 016 II1205722 = 016 III

log 1

0(D

LCCD

RFC)


000 002 004 006 008 010 012minus3

minus2

minus1

0

1

2

3

C

1205722 = 087 I1205722 = 087 II1205722 = 087 III

1205722 = 016 I1205722 = 016 II1205722 = 016 III

log 1

0(D

NPD

RFC)




minus260

minus255

minus250

minus245

minus240

minus235

minus230

log 1

0(D

RFC)


1 1205722119898119909 120590119909)














Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










000 002 004 006 008 010 01200

05

10

15

20

25

30

C

DLC

CD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(a) 119863LCC119863RFC

000 002 004 006 008 010 01200

05

10

15

20

25

30

C

DN

PD

RFC

1205722 = 016

1205722 = 036

1205722 = 054

1205722 = 062

1205722 = 077

1205722 = 087

(b) 119863NP119863RFC


000 002 004 006 008 010 012minus3

minus2

minus1

0

1

2

3

4

C

1205722 = 087 I1205722 = 087 II1205722 = 087 III

1205722 = 016 I1205722 = 016 II1205722 = 016 III

log 1

0(D

LCCD

RFC)


000 002 004 006 008 010 012minus3

minus2

minus1

0

1

2

3

C

1205722 = 087 I1205722 = 087 II1205722 = 087 III

1205722 = 016 I1205722 = 016 II1205722 = 016 III

log 1

0(D

NPD

RFC)




minus260

minus255

minus250

minus245

minus240

minus235

minus230

log 1

0(D

RFC)


1 1205722119898119909 120590119909)














Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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