Research Article Survival Analysis of Fatigue and Rutting ...downloads.hindawi.com/journals/je/2016/8359103.pdf · Research Article Survival Analysis of Fatigue and Rutting Failures

Research ArticleSurvival Analysis of Fatigue and Rutting Failures inAsphalt Pavements

Pabitra Rajbongshi and Sonika Thongram

Civil Engineering Department National Institute of Technology Silchar Silchar Assam 788010 India

Correspondence should be addressed to Pabitra Rajbongshi prajbongshiyahoocom

Received 3 February 2016 Accepted 14 April 2016

Academic Editor Ilker Bekir Topcu

Copyright copy 2016 P Rajbongshi and S Thongram This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Fatigue and rutting are two primary failure mechanisms in asphalt pavementsThe evaluations of fatigue and rutting performancesare significantly uncertain due to large uncertainties involved with the traffic and pavement life parameters Thereforedeterministically it is inadequate to predict when an in-service pavement would fail Thus the deterministic failure time which isknown as design life (119910119903) of pavement becomes random in nature Reliability analysis of such time (119905) dependent random variableis the survival analysis of the structure This paper presents the survival analysis of fatigue and rutting failures in asphalt pavementstructures It is observed that the survival of pavements with time can be obtained using the bathtub concept that contains aconstant failure rate period and an increasing failure rate periodThe survival function (119878(119905)) probability density function (pdf) andprobability distribution function (PDF) of failure time parameter are derived using bathtub analysis It is seen that the distributionof failure time follows three parametric Weibull distributions This paper also works out to find the most reliable life (119884119903

119877) of

pavement sections corresponding to any reliability level of survivability

1 Introduction

Fatigue and rutting are considered as primary modes offailure in asphalt pavements Inmechanistic-empirical (M-E)design of pavements a design solution is obtained so that theestimated pavement life (fatigue and rutting lives) is not lessthan the total predicted traffic repetitions during its designperiod Both pavement life (119873) and traffic (119879) parametersshow significant uncertainty due to large variabilities asso-ciated with their input parameters In order to incorporatethese variabilities a factor of safety or a reliability factor isused in the pavement design process For a given designperiod (119910119903) or given 119879 the reliability (119877) of pavement sectioncan be varied by varying the layer(s) thicknesses or 119873 Inother words for an in-service pavement or given 119873 thereliability varies with time (119905) or traffic repetitions Reliability(119877) as a function of time (119905) is the survivability (119878) of thestructure All the roads do not fail at the time equal to theirrespective design period That is how the time that it actuallyfails becomes a randomvariablewhich dealswithmortality or

failure of the system with timeThe randomness of the failuretime (119910119903) shall follow certain probability distribution

Probabilistically the failure time that is the design life(119910119903) of pavement structures is a considerable uncertainparameter Survival (119878(119905)) function is a property of therandom variable ldquo119910119903rdquo 119878(119905 = 119905

1) represents the possibility that

the structure would not fail at time 1199051for given ldquo119910119903rdquo Such

stochastic failure information would help while developingprobabilistic pavement management strategies and life cycleanalysis [1ndash4] Survival analysis is widely used in reliabilityengineering where the death or failure of the system dealswith time Various researchers [3ndash9] studied about fatiguesurvival of asphalt pavements with rehabilitation and overlayperspectives However the survivability of an in-servicepavement section would depend upon the survivability con-sidered at the time of initial design or construction aspectsThis paper focuses on the survival analysis based on fatigueand rutting design perspectives in asphalt pavements M-E pavement design principle is considered in the presentanalysis

Hindawi Publishing CorporationJournal of EngineeringVolume 2016 Article ID 8359103 7 pageshttpdxdoiorg10115520168359103

2 Journal of Engineering

This paper has six sections of which this is Section 1 Thescope and objectives of the study are presented in Section 2Section 3 discusses the brief analysis of hazard and survivalof pavement structures based on pavement reliability Thedeveloped survival and distribution functions of failure timehave been presented in Section 4 Section 5 explains thevalidation and discussions of survival function Finally theconclusions are placed as Section 6

2 Scope and Objectives

M-E pavement design method is popularly being adopted invarious guidelines [10ndash17] In reliability based M-E designprocess and to calculate the reliability of pavement sectionnormally a parameter named as damage factor (119863) is evalu-ated as given in

119863 =119879

119873 (1)

where 119879 is number of axles repetitions (eg in terms ofstandard axles load) at the end of design period (119910119903) and 119873is fatigue or rutting life (in terms of standard axles load) ofthe pavement section 119863 may represent the fatigue damageor rutting damage depending upon fatigue or rutting life Forgiven 119879 and119873 the reliability (119877) for fatigue or rutting failurecan be obtained as given in

119877 = prob (119879 lt 119873) = prob (119863 lt 1)

= prob (ln119863 lt 0)

(2)

119877 can be estimated for any known distributions of 119879 and119873 and 119877 = 05 when 119879 = 119873 Details about the reliabilityconsiderations in asphalt pavements can be seen elsewhere[18ndash31] Kalita and Rajbongshi [23] concluded that both 119879and119873 (fatigue or rutting case) follow lognormal distributionwith 95 confidence level irrespective of the distributions ofvarious input parameters of 119879 and119873 Thus the parameter119863also follows lognormal or ln119863 follows normal distributionMean (120583ln119863) and standard deviation (120590ln119863) of ln119863 can bedetermined as given in the following equations respectively

120583ln119863 = ln (119863) = ln (119879) minus ln (119873) (3)

120590ln119863 = radicln (1 + COV2119863)

= radicln (1 + COV2119879) + ln (1 + COV2

119873)

(4)

where 120583119883is mean of the random variable 119883 120590

119883is standard

deviation of the random variable 119883 and COV119883is coeffi-

cient of variation (COV) of random variable 119883 It may bementioned that the deterministic values of lognormal 119879 and119873 (fatigue or rutting case) represent the median of theirrespective distributions which divides the total probabilityinto two equal halves [32] Therefore the deterministic valueof 119863 also represents the median of random parameter 119863 orln(119863) value represents the mean of normal parameter ln119863(refer to (3)) The COV

119863may vary in the range of 316ndash

731 in case of fatigue and 395ndash942 in case of rutting[23]

From design perspective for any given 119879 the pavementreliability (119877) as a function of design layer(s) thicknesses (ℎ

119894)

can be expressed as given in

119877 (ℎ119894) = prob [ln119863(ℎ

119894) lt 0] = N(minus

120583ln119863 (ℎ119894)

120590ln119863)

= N(minusln (119879) minus ln [119873 (ℎ

119894)]

120590ln119863)

(5)

where N(119911) indicates normal probability corresponding tothe standard normal deviate 119911 and N(119911 = 0) = 05 It may bementioned that 120590ln119863 is constant for given COV

119879and given

COV119873(refer to (4)) In other words for given 119873 of an in-

service pavement the number of traffic repetitions (119879) varieswith time and thus 119877 also varies with time In this case 119877 asa function of time (119905) can be expressed as given in

119877 (119905) = prob [ln119863 (119905) lt 0] = N(minus120583ln119863 (119905)

120590ln119863)

= N(minusln [119879 (119905)] minus ln (119873)

120590ln119863)

(6)

Equation (6) indicates the survival of the pavementstructure either for fatigue or for rutting failure It describesthe ability of the structure to function under stated conditionsfor the pr-specified time of failure (119910119903) The fact remainsthat due to uncertain prediction in damage (119863) parametera pavement may fail early or later than that of 119910119903 That is thefailure time (119910119903) is a random parameter and is expected tofollow certain probability distribution Onemay be interestedto know the failure probability of the in-service pavementsection at any given time (119905) specifically for life cyclemanage-ment and rehabilitation purposes [1ndash7 9] Thus it needs thesurvival analysis of pavement structures based on the designperspectives To this effect it is necessary to establish thesurvival (119878(119905)) function and probability distribution (119865(119905)) ofrandom variable ldquo119910119903rdquo This forms the scope of the presentstudy The objectives of the present work are (a) to analyzereliability variations with time for different design life anddifferent variabilities of the damage parameter (fatigue orrutting) and (b) to derive survival function and probabilitydistribution function of failure time (119910119903) for fatigue andrutting failures

3 Survival and Hazard Analysis of Pavements

Reliability closely relates to safety or risks of failure in anyuncertain system or system component In pavement systemfatigue and rutting are two primary failure components thatassociated with large uncertainty The time when it wouldfail cannot not be ascertained deterministically and that ishow the deterministic failure time that is the design life (119910119903)parameter becomes a randomvariable Considering ldquo119910119903rdquo as arandomvariable the fatigue or rutting failure probability thatis 1 minus119877(119905) can be obtained as given in (6) The 119877(119905) functionrepresents the survival (119878(119905)) of fatigue or rutting in pavement

Journal of Engineering 3

0001020304050607080910

Relia

bilit

y R

(t)

N = 1711msayr = 26 yrsCOV of D = 50

5 10 15 20 25 30 35 40 45 500Time t (yrs)

(a) Reliability variation with time

000005010015020025030035040045050

Haz

ard

rate

h(t

)

COV of D = 50

5 10 15 20 25 30 35 40 45 500Time t (yrs)

N = 1711msayr = 26 yrs

(b) Hazard with time

Figure 1 Reliability and hazard as a function of time

sections and therefore at this juncture the 119878(119905) function maybe expressed as given in

119878 (119905) = 119877 (119905) = N(minusln [119879 (119905)] minus ln [119873]

120590ln119863) 119905 ge 0 (7)

The failure rate or hazards (ℎ(119905)) function may beexpressed as given in [33 34]

ℎ (119905) = minus119889

119889119905ln [119878 (119905)] = minus 119889

119889119905ln [119877 (119905)] 119905 ge 0

or 119878 (119905) = expminusint119905

0ℎ(1199051015840)1198891199051015840

119905 ge 0

(8)

The expression of 119878(119905) in (7) has no closed form solutionand therefore an attempt has been made to derive 119878(119905)

through numerical solution To find the time dependentvariations in 119878(119905) or ℎ(119905) (refer to (7)) the traffic (119879(119905))variation is expressed as given in

119879 (119905) = 365119860(1 + 119903)

119905minus 1

119903= 119875

(1 + 119903)119905minus 1

119903 (9)

where 119860 is annual average daily traffic at the time of openingpavement to the traffic in terms of a common axles load (egstandard axles) 119875 is traffic repetitions (in standard axles) atthe base year and 119903 is annual traffic growth rate Undermixedtraffic conditions the different axle loads may be convertedinto standard axles using load different equivalency factors[10 14 17 35] It may be mentioned that 119905 in (9) is in yearsand 119879(119905 = 119910119903) is equal to the life of pavement (119873) A genericform of M-E fatigue and rutting equations may be expressedas given in [11 14 35]

119873 = 1198961(1

120576119905

)

1198962

(1

1198641

)

1198963

for fatigue case

= 1198881(1

120576119911

)

1198882

for rutting case

(10)

where 120576119905is initial critical horizontal tensile strain at the bot-

tom of asphalt layer 120576119911is initial critical vertical compressive

strain at the top of subgrade layer 1198641is the initial stiffness

of asphalt material and 1198961 1198962 1198963 1198881 and 119888

2are regression

constants 120576119905and 120576

119911can be obtained using any pavement

analysis programThe following data are used in the presentLet for an asphalt pavement the base year traffic (119875) be

0231 million standard axles (msa) The design life (119910119903) is 26years and the traffic growth rate is 75 per annum COV ofthe damage (119863) parameter may be taken as 50 [23] Thatis from (4) 120590ln119863 can be obtained as 0472 Using (9) thedesign traffic repetitions (119879(119905 = 119910119903)) that is the pavementlife (119873) are obtained as 1711msa Thus from (7) and (8) thenumerical values of 119878(119905) (or 119877(119905)) and ℎ(119905) can be determinedfor different time ldquo119905rdquo This is shown in Figure 1

From Figure 1 it is observed that the hazard (ℎ(119905))function may be illustrated using the bathtub concept andcontaining two parts (i) the first part (119905 lt 1199101199032) is at constantfailure rate known as stable life and (ii) the second part(119905 ge 1199101199032) is at increasing failure rate known as wear-outlife [33 34] Similar results are also observed for different 119910119903(or 119873) and different 120590ln119863 values The ℎ(119905) variations for twomore cases as 119910119903 = 20 years with COV

119863= 40 (or 120590ln119863 =

0385) and 119910119903 = 30 years with COV119863= 60 (or 120590ln119863 =

0555) are presented in Figure 2 Similar bathtub curves ofℎ(119905) function could be seen for all cases where ℎ(119905) asymp 0

(constant) for 119905 lt 1199101199032 and there is increasing ℎ(119905) for 119905 ge1199101199032 Section 4 elaborates the ℎ(119905) and 119878(119905) functions usingthe bathtub analysis

4 Survival Function and Distribution

The bathtub curve is popularly being used in reliabilityengineering that involves significant uncertainty and risks offailure The failure rates (ℎ(119905)) of fatigue and rutting failurescan be represented using the bathtub concept as presented inFigures 1 and 2 ℎ(119905) 119878(119905) probability density function (pdf)


Fatigue

Rutting

0001020304050607080910

Prob

abili

ty R

(t) o

r S(t

)

5 10 15 20 25 30 35 40 45 50 55 600Time t (yrs)

Fatigue R(t)Fatigue S(t)

Rutting R(t)Rutting S(t)

yr = 32 yrs N = 4023msaCOV of D = 5914

COV of D = 6797yr = 35 yrs N = 5050msa

Figure 2 Hazard function with different design life

and probability distribution function (PDF) of failure time(119910119903) are interrelated and can be expressed as given in [33 34]

ℎ (119905) =119891 (119905)

119878 (119905)=

119891 (119905)

1 minus 119865 (119905) (11)

where 119891(119905) is pdf of random variable 119910119903 119865(119905) is PDF of 119910119903119891(119905) and 119865(119905) can also be expressed as given in

119891 (119905) = ℎ (119905) 119878 (119905) =119889

119889119905119865 (119905)

or 119865 (119905) = 1 minus expminusint119905

0ℎ(1199051015840)1198891199051015840

= 1 minus 119878 (119905)

(12)

For constant ℎ(119905) that is the first part of the bathtub(119905 lt 1199101199032) 119891(119905) and 119865(119905) can be represented by exponentialdistribution [33 34] For ℎ(119905) with increasing rate that is thesecond part of the bathtub (119905 ge 1199101199032) 119891(119905) and 119865(119905) canbe represented byWeibull distributionThus mathematicallythe 119891(119905) 119865(119905) ℎ(119905) and 119878(119905) functions can be expressed asgiven in

119891 (119905) =

1

120573expminus(119905120573) 119905 lt

119910119903

2 for exponential dist

119896

120582119896(119905 minus 120574)

119896minus1 expminus((119905minus120574)120582)119896

119905 ge 120574 (=119910119903

2) for Weibull dist

(13)

119865 (119905) =

1 minus expminus(119905120573) 119905 lt119910119903

2

1 minus expminus((119905minus120574)120582)119896

119905 ge 120574 (=119910119903

2)

(14)

ℎ (119905) =

1

120573 119905 lt

119910119903

2

119896

120582119896(119905 minus 120574)

119896minus1 119905 ge 120574 (=

119910119903

2)

(15)

119878 (119905) =

expminus(119905120573) 119905 lt119910119903

2

expminus((119905minus120574)120582)119896

119905 ge 120574 (=119910119903

2)

(16)

where 1120573 is mean failure rate (constant) for the exponentialdistribution 120582 is scale parameter 119896 is shape parameter and120574 is location parameter (=1199101199032) of theWeibull distribution Itmay be mentioned that a special case of Weibull distribution(ie 119896 = 1) is an exponential or shifted exponentialdistribution ldquo119910119903rdquo in (13)ndash(16) indicates the deterministicfailure time In other words using (9) the ldquo119910119903rdquo for givenpavement life (119873) can be calculated as given in

119910119903 =ln (119873119903119875 + 1)ln (1 + 119903)

(17)

Themean (120583119910119903) median (med

119910119903) and standard deviation

(120590119910119903) of the three parametric Weibull random variables (119910119903)

are given in [33 34]

120583119910119903= 120582Γ (1 +

1

119896) + 120574 (18)

med119910119903= 120582 (ln 2)1119896 + 120574 (19)

120590119910119903= 120582 [Γ (1 +

2

119896) minus Γ (1 +

1

119896)

2

]

05

(20)

where Γ(sdot) is gamma function and 120574 = 1199101199032 in the presentcase As seen in Figures 1 and 2 the constant failure rate ℎ(119905) asymp0 that is 1120573 asymp 0 or 119878(119905) asymp 1 for 119905 lt 120574 (=1199101199032) That isprobabilistically it is expected that there would not fail anypavement before half of the design period For the wear-outlife period (ie 119905 ge 120574) with increasing failure rate 119896 gt 1 [3334] 119896 gt 1 indicates that there is aging of the pavement systemwhich shows more likelihood to fail as time goes on

At this stage the parameters 119896 and 120582 are unknownMoreover comparing (7) and (16) one may conclude that 119896and 120582 shall depend upon 120590ln119863 (or COV119863) 119910119903 and119873 119896 and 120582


cannot be derived analytically since 119877(119905) or 119878(119905) function hasno closed form solution In order to find 119896 and 120582 parametersa simulation study has been performed with different valuesof COV

119863(=40 to 80) 119910119903 (=10 to 30 years) and 119873 (=10 to

40msa) Through simulation it is attempted to find the bestpossible value of 119896 so that 119878(119905) in (16) matches with 119877(119905) in (7)for 119905 ge 120574 for each set of data From the study it is observedthat the parameter 119896may be correlated as 119896 = 05119873028120590ln119863where 119873 is in msa From (19) and considering med

119910119903= 119910119903

corresponding to 50 reliability 120582 can be expressed as 120582 =

120574(ln 2)1119896 Thus (13)ndash(16) may be rewritten as

119891 (119905)

=

119896

120582119896(119905 minus 120574)


119905 ge 120574 (=119910119903

2)

0 elsewhere

(21)

119865 (119905) =


119905 ge 120574 (=119910119903

2)

0 elsewhere(22)

ℎ (119905) =

119896

120582119896(119905 minus 120574)

119896minus1 119905 ge 120574 (=

119910119903

2)

0 elsewhere(23)

119878 (119905) =

expminus((119905minus120574)120582)119896

119905 ge 120574 (=119910119903

2)

1 elsewhere(24)

where 119896 = 05119873028

120590ln119863 and 120582 = 120574(ln 2)1119896 and 120574 =

1199101199032 where 119873 may be either fatigue or rutting life andaccordingly 119863 may be either fatigue or rutting damagefactor Section 5 presents the validation and discussion on thesurvival function for fatigue and rutting failures

5 Validation and Discussion on Survival

To validate the survival (119878(119905)) function in (24) the followingdata are used

The annual average daily traffic (119860) of an asphalt pave-ment is 1000 vehiclesday that is base year traffic (119875) is 365 times1000 standard axles = 0365msa Traffic growth rate (119903) is 7per annum The COV

119879= 35 and COV

119873= 45 for fatigue

life and 55 for rutting life Fatigue and rutting lives of thepavement section are 4023msa and 505msa respectively

Using (17) the design period (119910119903) of the pavement sectionturns out to be 32 years and 35 years for fatigue and ruttingcases respectively That is 120574 = 16 years and 175 years forfatigue and rutting cases respectively Using (4) COV

119863can

be calculated as 5914 and 6797 for fatigue and ruttingcases respectively Thus the 120590ln119863 values for fatigue andrutting cases are obtained as 05477 and 0616 respectivelyThus the reliability (119877(119905)) values for different 119905 can beestimated using (6) This is shown in Figure 3

For known 119873 and 120590ln119863 the 119896 and 120582 parameters in (24)can be obtained as 119896 = 2568 and 120582 = 18454 for fatiguecase and 119896 = 2433 and 120582 = 20345 for rutting case Thusthe survival (119878(119905)) function for different 119905 can be evaluated

000

005

010

015

020

025

030

035

040

045

050

Haz

ard

rate

h(t

)

5 10 15 20 25 30 35 40 45 50 55 600Time t (yrs)

yr = 20 yrs N = 1000msa COV of D = 40yr = 30 yrs N = 2389msa COV of D = 60

Figure 3 Comparison of reliability and survival functions forfatigue and rutting distresses

using (24)This is depicted in Figure 3 It can be seen that119877(119905)and 119878(119905) represent the sameprobability variationwith time fordifferent 119910119903 119873 and COV

119863values (both fatigue and rutting

cases) Thus at any given time (119905) the survivability either forfatigue or for rutting failure may be predicted using (24) Inother words at any given reliability (119877) level the reliable life(119884119903119877) of pavement can be estimated as given in

119884119903119877= 120582 (minus ln119877)1119896 + 120574 (25)

where 119884119903119877is the probable life (fatigue or rutting) at reliability

equal to 119877To find the survivability of pavement structures basically

it needs three parameters namely 119896 120582 and 120574 These param-eters can be calculated for given COV

119879 COV

119873 119873 and 119910119903

For different values of COV119879 COV

119873119873 and 119910119903 (as example

cases) the 119896 120582 and 120574 values are shown in Table 1Further for these 119896 120582 and 120574 values the reliable life (119884119903

119877)

at different reliability (119877) levels is calculated using (25) Thisis given in Table 2 For three different cases of 119896 120582 and 120574 thepdf (119891(119905)) and PDF (119865(119905)) of failure time (119910119903) are shown inFigure 4 It may be mentioned that 119865(119905 = 119910119903) = 05 that is50 survivability or reliability at the age of design life

6 Conclusions

This paper presents the survival analysis of asphalt pavementsfor fatigue and rutting failures It may be concluded that thesurvival of pavement structures can well be represented bythe three parametric (119896 120582 and 120574) Weibull distributions 119896120582 and 120574 parameters are derived through simulation Theseparameters can be determined for any given design life(119910119903) and known COVs of traffic (COV

119879) and pavement life

(COV119873) parameters

Survival (119878(119905)) at any given time (119905) indicates the abilityto function and 1 minus 119878(119905) indicates the failure probability ThePDF 119865(119905) (=1minus 119878(119905)) of the failure time (119910119903) follows bimodal


Table 1 Parameters of survival (119878(119905)) function

COV119879COV119873

COV119863

120590ln119863119873 = 20msa and 2120574 = 119910119903 = 10 yrs 119873 = 40msa and 2120574 = 119910119903 = 20 yrs 119873 = 60msa and 2120574 = 119910119903 = 30 yrs

119896 120582 119896 120582 119896 120582

30 42 5313 0499 2320 5856 2817 11390 3156 1684737 45 6059 0559 2069 5969 2512 11571 2814 1708741 58 7490 0667 1733 6177 2105 11902 2358 17523

0

002

004

006

008

01

012

014

016

018

pdf f

(t)

5 10 15 20 25 30 35 40 45 500Time t (yrs)

k = 2320 120582 = 5856 120574 = 5

k = 2512 120582 = 11571 120574 = 10

k = 2358 120582 = 17523 120574 = 15

(a) Probability density

k = 2320 120582 = 5856 120574 = 5

k = 2512 120582 = 11571 120574 = 10

k = 2358 120582 = 17523 120574 = 15

0001020304050607080910

PDF F

(t)

5 10 15 20 25 30 35 40 45 500Time t (yrs)

(b) Probability distribution

Figure 4 Probability density and probability distribution of failure time (119910119903)

Table 2 Reliable life (119884119903119877) for different parameters of survival

function

119896 120582 2120574 = 119910119903 119877 = 85 119877 = 90 119877 = 95 119877 = 982320 5856 10 77 72 66 612069 5969 10 75 70 64 591733 6177 10 72 67 61 572817 11390 20 160 151 140 1292512 11571 20 156 147 135 1242105 11902 20 150 141 129 1193156 16847 30 245 233 216 1992814 17087 30 240 227 209 1932358 17523 30 231 217 200 183

distribution exponential distribution followed by Weibulldistribution It is seen that for 119905 lt 1199101199032 119865(119905) is exponentialwith zero failure rateThat is no failure is expectedwithin halfof the design life and thus there may not be any schedulingfor rehabilitation activity within this period This is a stablelife period which may be identified as a maintenance-freeperiod based on fatigue and rutting considerations For 119905 gt1199101199032119865(119905) followsWeibull distributionwith increasing failurerate This is the wear-out life where the number of failureevents increases at increasing rate The present study hasalso worked out the most reliable life (119884119903

119877) of the pavement

corresponding to any reliability level Such probabilistic

information would be useful for the engineerscontractorswhile preparing reliablemanagement strategies maintenancescheduling and life cycle analysis based on initial design ofpavement structures

Competing Interests

The authors declare that they have no competing interests

References

[1] P C Anastasopoulos F L Mannering and J E HaddockldquoEffectiveness and service livessurvival curves of various pave-ment rehabilitation treatmentsrdquo Final Report FHWAINJTRP-200912 Joint Transportation Research Program IndianaDepartment of Transportation and Purdue University WestLafayette Ind USA 2009

[2] J T Harvey A Rezaei and C Lee ldquoProbabilistic approach tolife-cycle cost analysis of preventive maintenance strategies onflexible pavementsrdquo Transportation Research Record no 2292pp 61ndash72 2012

[3] K Svenson ldquoEstimated lifetimes of road pavements in Swedenusing time-to-event analysisrdquo ASCE Journal of TransportationEngineering vol 140 no 11 2014

[4] Y Wang and D Allen ldquoStaged survival models for overlayperformance predictionrdquo International Journal of PavementEngineering vol 9 no 1 pp 33ndash44 2008


[5] P C Anastasopoulos and F L Mannering ldquoAnalysis of pave-ment overlay and replacement performance using randomparameters hazard-based duration modelsrdquo Journal of Infras-tructure Systems vol 21 no 1 Article ID 04014024 2014

[6] C Chen R CWilliamsMGMarasinghe et al ldquoAssessment ofcomposite pavement performance by survival analysisrdquo Jour-nal of Transportation Engineering vol 141 no 9 Article ID04015018 2015

[7] Q Dong and B Huang ldquoEvaluation of influence factors oncrack initiation of LTPP resurfaced-asphalt pavements usingparametric survival analysisrdquo ASCE Journal of Performance ofConstructed Facilities vol 28 no 2 pp 412ndash421 2014

[8] Y Wang K C Mahboub and D E Hancher ldquoSurvival analysisof fatigue cracking for flexible pavements based on long-term pavement performance datardquo Journal of TransportationEngineering vol 131 no 8 pp 608ndash616 2005

[9] J Yang and S-H Kim ldquoInvestigating the performance of as-built and overlaid pavements a competing risks approachrdquoInternational Journal of Pavement Engineering vol 16 no 3 pp191ndash197 2015

[10] American Association of State Highway and TransportationOfficials (AASHTO) Guide for Design of Pavement StructureAASHTO Washington DC USA 1993

[11] Asphalt Institute (AI)Thickness DesignmdashAsphalt Pavements forHighways and Streets Manual Series No 1 Asphalt Institute(AI) Lexington Ky USA 9th edition 1999

[12] Austroads Pavement Design Austroads Sydney Australia2004

[13] French French Design Manual for Pavement Structures GuideTechnique LCPC and SETRA Francaise 1997

[14] Indian Roads Congress (IRC) Guidelines for the Design of Flex-ible Pavements Second Revision IRC37-2012 Indian RoadsCongress (IRC) New Delhi India 2012

[15] National Cooperative Highway Research Program (NCHRP)ldquoMechanistic-empirical design of new and rehabilitatedpavement structuresrdquo NCHRP Project 1-37A TransportationResearch Board Washington DC USA 2004

[16] Shell Shell Pavement Design ManualmdashAsphalt Pavement andOverlays for Road Traffic International Petroleum CompanyLimited London UK 1978

[17] Transport Research Laboratory (TRL)AGuide to the StructuralDesign of Bitumen-Surfaced Roads in Tropical and Sub-TropicalCountries vol 31 ofOverseas Road Note Overseas Center TRLLondon UK 4th edition 1993

[18] J P Aguiar-Moya and J A Prozzi ldquoEffect of field variabilityof design inputs on the MEPDGrdquo in Proceedings of the Trans-portation Research Board 90th Annual Meeting WashingtonDC USA 2011

[19] D Bush Incorporation of Reliability into Mechanistic-EmpiricalPavement Design in Washington and California University ofWashington Seattle Wash USA 2004

[20] K H Chua A D Kiureghian and C L Monismith ldquoStochasticmodel for pavement designrdquo ASCE Journal of TransportationEngineering vol 118 no 6 pp 769ndash786 1992

[21] Central Road Research Institute (CRRI) ldquoDevelopment ofmethods such as benkelman beam deflection method for eval-uation of structural capacity of existing flexible pavements andalso for estimation and design of overlays for strengthening ofany weak pavementrdquo Final Report Research Scheme R-6 Min-istry of Surface Transport Government of India New DelhiIndia 1995

[22] D M Dilip P Ravi and G L S Babu ldquoSystem reliability analy-sis of flexible pavementsrdquo Journal of Transportation Engineeringvol 139 no 10 pp 1001ndash1009 2013

[23] K Kalita and P Rajbongshi ldquoVariability characterisation ofinput parameters in pavement performance evaluationrdquo RoadMaterials and Pavement Design vol 16 no 1 pp 172ndash185 2015

[24] W Kenis and W Wang Pavement Variability and ReliabilityUS Department of Transportation Federal Highway Admin-istration 2004

[25] S Kim H Ceylan and K Gopalakrishnan ldquoEffect of M-Edesign guide inputs on flexible pavement performance predic-tionsrdquo Road Materials and Pavement Design vol 8 no 3 pp375ndash397 2007

[26] A Maji and A Das ldquoReliability considerations of bituminouspavement design by mechanistic-empirical approachrdquo Interna-tional Journal of Pavement Engineering vol 9 no 1 pp 19ndash312008

[27] P Rajbongshi ldquoReliability based cost effective design of asphaltpavements considering fatigue and ruttingrdquo International Jour-nal of Pavement Research and Technology vol 7 no 2 pp 153ndash158 2014

[28] P Rajbongshi and A Das ldquoOptimal asphalt pavement designconsidering cost and reliabilityrdquo Journal of Transportation Engi-neering vol 134 no 6 pp 255ndash261 2008

[29] J Retherford and M McDonald ldquoReliability methods applica-ble to mechanistic-empirical pavement design methodrdquo Trans-portation Research Record vol 2154 pp 130ndash137 2010

[30] D H Timm B Briggison and D E Newcomb ldquoVariability ofmechanistic-empirical flexible pavement design parametersrdquo inProceedings of the 5th International Conference on the BearingCapacity of Roads and Airfields vol 1 pp 629ndash638 NorwegianUniversity of Science and Technology Trondheim Norway1998

[31] S FWojtkiewicz L Khazanovich G Gaurav and R VelasquezldquoProbabilistic numerical simulation of pavement performanceusing MEPDGrdquo Road Materials and Pavement Design vol 11no 2 pp 291ndash306 2010

[32] P Rajbongshi A comprehensive design approach for asphaltpavements using mechanisic-empirical framework [PhD thesis]Civil Engineering Department IIT Kanpur India 2008

[33] J P Klein andM LMoeschberger Survival Analysis Techniquesfor Censored andTruncatedData Springer NewYorkNYUSA1997

[34] J F Lawless Statistical Models and Methods for Lifetime DataJohn Wiley amp Sons Hoboken NJ USA 2nd edition 2002

[35] YHHuangPavement Analysis andDesign Pearson EducationNew Jersey NJ USA 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



This paper has six sections of which this is Section 1 Thescope and objectives of the study are presented in Section 2Section 3 discusses the brief analysis of hazard and survivalof pavement structures based on pavement reliability Thedeveloped survival and distribution functions of failure timehave been presented in Section 4 Section 5 explains thevalidation and discussions of survival function Finally theconclusions are placed as Section 6

2 Scope and Objectives

M-E pavement design method is popularly being adopted invarious guidelines [10ndash17] In reliability based M-E designprocess and to calculate the reliability of pavement sectionnormally a parameter named as damage factor (119863) is evalu-ated as given in

119863 =119879

119873 (1)

where 119879 is number of axles repetitions (eg in terms ofstandard axles load) at the end of design period (119910119903) and 119873is fatigue or rutting life (in terms of standard axles load) ofthe pavement section 119863 may represent the fatigue damageor rutting damage depending upon fatigue or rutting life Forgiven 119879 and119873 the reliability (119877) for fatigue or rutting failurecan be obtained as given in

119877 = prob (119879 lt 119873) = prob (119863 lt 1)

= prob (ln119863 lt 0)

(2)

119877 can be estimated for any known distributions of 119879 and119873 and 119877 = 05 when 119879 = 119873 Details about the reliabilityconsiderations in asphalt pavements can be seen elsewhere[18ndash31] Kalita and Rajbongshi [23] concluded that both 119879and119873 (fatigue or rutting case) follow lognormal distributionwith 95 confidence level irrespective of the distributions ofvarious input parameters of 119879 and119873 Thus the parameter119863also follows lognormal or ln119863 follows normal distributionMean (120583ln119863) and standard deviation (120590ln119863) of ln119863 can bedetermined as given in the following equations respectively

120583ln119863 = ln (119863) = ln (119879) minus ln (119873) (3)

120590ln119863 = radicln (1 + COV2119863)

= radicln (1 + COV2119879) + ln (1 + COV2

119873)

(4)

where 120583119883is mean of the random variable 119883 120590

119883is standard

deviation of the random variable 119883 and COV119883is coeffi-

cient of variation (COV) of random variable 119883 It may bementioned that the deterministic values of lognormal 119879 and119873 (fatigue or rutting case) represent the median of theirrespective distributions which divides the total probabilityinto two equal halves [32] Therefore the deterministic valueof 119863 also represents the median of random parameter 119863 orln(119863) value represents the mean of normal parameter ln119863(refer to (3)) The COV

119863may vary in the range of 316ndash

731 in case of fatigue and 395ndash942 in case of rutting[23]

From design perspective for any given 119879 the pavementreliability (119877) as a function of design layer(s) thicknesses (ℎ

119894)

can be expressed as given in

119877 (ℎ119894) = prob [ln119863(ℎ

119894) lt 0] = N(minus

120583ln119863 (ℎ119894)

120590ln119863)

= N(minusln (119879) minus ln [119873 (ℎ

119894)]

120590ln119863)

(5)

where N(119911) indicates normal probability corresponding tothe standard normal deviate 119911 and N(119911 = 0) = 05 It may bementioned that 120590ln119863 is constant for given COV

119879and given

COV119873(refer to (4)) In other words for given 119873 of an in-

service pavement the number of traffic repetitions (119879) varieswith time and thus 119877 also varies with time In this case 119877 asa function of time (119905) can be expressed as given in

119877 (119905) = prob [ln119863 (119905) lt 0] = N(minus120583ln119863 (119905)

120590ln119863)

= N(minusln [119879 (119905)] minus ln (119873)

120590ln119863)

(6)

Equation (6) indicates the survival of the pavementstructure either for fatigue or for rutting failure It describesthe ability of the structure to function under stated conditionsfor the pr-specified time of failure (119910119903) The fact remainsthat due to uncertain prediction in damage (119863) parametera pavement may fail early or later than that of 119910119903 That is thefailure time (119910119903) is a random parameter and is expected tofollow certain probability distribution Onemay be interestedto know the failure probability of the in-service pavementsection at any given time (119905) specifically for life cyclemanage-ment and rehabilitation purposes [1ndash7 9] Thus it needs thesurvival analysis of pavement structures based on the designperspectives To this effect it is necessary to establish thesurvival (119878(119905)) function and probability distribution (119865(119905)) ofrandom variable ldquo119910119903rdquo This forms the scope of the presentstudy The objectives of the present work are (a) to analyzereliability variations with time for different design life anddifferent variabilities of the damage parameter (fatigue orrutting) and (b) to derive survival function and probabilitydistribution function of failure time (119910119903) for fatigue andrutting failures

3 Survival and Hazard Analysis of Pavements

Reliability closely relates to safety or risks of failure in anyuncertain system or system component In pavement systemfatigue and rutting are two primary failure components thatassociated with large uncertainty The time when it wouldfail cannot not be ascertained deterministically and that ishow the deterministic failure time that is the design life (119910119903)parameter becomes a randomvariable Considering ldquo119910119903rdquo as arandomvariable the fatigue or rutting failure probability thatis 1 minus119877(119905) can be obtained as given in (6) The 119877(119905) functionrepresents the survival (119878(119905)) of fatigue or rutting in pavement


0001020304050607080910

Relia

bilit

y R

(t)


5 10 15 20 25 30 35 40 45 500Time t (yrs)


000005010015020025030035040045050

Haz

ard

rate

h(t

)

COV of D = 50

5 10 15 20 25 30 35 40 45 500Time t (yrs)






120590ln119863) 119905 ge 0 (7)


ℎ (119905) = minus119889

119889119905ln [119878 (119905)] = minus 119889

119889119905ln [119877 (119905)] 119905 ge 0

or 119878 (119905) = expminusint119905

0ℎ(1199051015840)1198891199051015840

119905 ge 0

(8)



119879 (119905) = 365119860(1 + 119903)

119905minus 1

119903= 119875

(1 + 119903)119905minus 1

119903 (9)


119873 = 1198961(1

120576119905

)

1198962

(1

1198641

)

1198963

for fatigue case

= 1198881(1

120576119911

)

1198882

for rutting case

(10)





2are regression

constants 120576119905and 120576





119863= 40 (or 120590ln119863 =







Fatigue

Rutting

0001020304050607080910

Prob

abili

ty R

(t) o

r S(t

)

5 10 15 20 25 30 35 40 45 50 55 600Time t (yrs)







ℎ (119905) =119891 (119905)

119878 (119905)=

119891 (119905)

1 minus 119865 (119905) (11)


119891 (119905) = ℎ (119905) 119878 (119905) =119889

119889119905119865 (119905)


0ℎ(1199051015840)1198891199051015840

= 1 minus 119878 (119905)

(12)


119891 (119905) =

1

120573expminus(119905120573) 119905 lt

119910119903


119896

120582119896(119905 minus 120574)


119905 ge 120574 (=119910119903

2) for Weibull dist

(13)

119865 (119905) =


2


119905 ge 120574 (=119910119903

2)

(14)

ℎ (119905) =

1

120573 119905 lt

119910119903

2

119896

120582119896(119905 minus 120574)

119896minus1 119905 ge 120574 (=

119910119903

2)

(15)

119878 (119905) =

expminus(119905120573) 119905 lt119910119903

2

expminus((119905minus120574)120582)119896

119905 ge 120574 (=119910119903

2)

(16)


119910119903 =ln (119873119903119875 + 1)ln (1 + 119903)

(17)





120583119910119903= 120582Γ (1 +

1

119896) + 120574 (18)

med119910119903= 120582 (ln 2)1119896 + 120574 (19)

120590119910119903= 120582 [Γ (1 +

2

119896) minus Γ (1 +

1

119896)

2

]

05

(20)





119863(=40 to 80) 119910119903 (=10 to 30 years) and 119873 (=10 to


119910119903= 119910119903



119891 (119905)

=

119896

120582119896(119905 minus 120574)


119905 ge 120574 (=119910119903

2)

0 elsewhere

(21)

119865 (119905) =


119905 ge 120574 (=119910119903

2)

0 elsewhere(22)

ℎ (119905) =

119896

120582119896(119905 minus 120574)

119896minus1 119905 ge 120574 (=

119910119903

2)

0 elsewhere(23)

119878 (119905) =

expminus((119905minus120574)120582)119896

119905 ge 120574 (=119910119903

2)

1 elsewhere(24)

where 119896 = 05119873028

120590ln119863 and 120582 = 120574(ln 2)1119896 and 120574 =





119879= 35 and COV




119863can



000

005

010

015

020

025

030

035

040

045

050

Haz

ard

rate

h(t

)

5 10 15 20 25 30 35 40 45 50 55 600Time t (yrs)






119884119903119877= 120582 (minus ln119877)1119896 + 120574 (25)




119879 COV

119873 119873 and 119910119903


119873119873 and 119910119903 (as example


119877)


6 Conclusions







COV119879COV119873

COV119863


119896 120582 119896 120582 119896 120582

30 42 5313 0499 2320 5856 2817 11390 3156 1684737 45 6059 0559 2069 5969 2512 11571 2814 1708741 58 7490 0667 1733 6177 2105 11902 2358 17523

0

002

004

006

008

01

012

014

016

018

pdf f

(t)

5 10 15 20 25 30 35 40 45 500Time t (yrs)

k = 2320 120582 = 5856 120574 = 5

k = 2512 120582 = 11571 120574 = 10

k = 2358 120582 = 17523 120574 = 15


k = 2320 120582 = 5856 120574 = 5

k = 2512 120582 = 11571 120574 = 10

k = 2358 120582 = 17523 120574 = 15

0001020304050607080910

PDF F

(t)

5 10 15 20 25 30 35 40 45 500Time t (yrs)




function

119896 120582 2120574 = 119910119903 119877 = 85 119877 = 90 119877 = 95 119877 = 982320 5856 10 77 72 66 612069 5969 10 75 70 64 591733 6177 10 72 67 61 572817 11390 20 160 151 140 1292512 11571 20 156 147 135 1242105 11902 20 150 141 129 1193156 16847 30 245 233 216 1992814 17087 30 240 227 209 1932358 17523 30 231 217 200 183





Competing Interests


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0001020304050607080910

Relia

bilit

y R

(t)


5 10 15 20 25 30 35 40 45 500Time t (yrs)


000005010015020025030035040045050

Haz

ard

rate

h(t

)

COV of D = 50

5 10 15 20 25 30 35 40 45 500Time t (yrs)






120590ln119863) 119905 ge 0 (7)


ℎ (119905) = minus119889

119889119905ln [119878 (119905)] = minus 119889

119889119905ln [119877 (119905)] 119905 ge 0

or 119878 (119905) = expminusint119905

0ℎ(1199051015840)1198891199051015840

119905 ge 0

(8)



119879 (119905) = 365119860(1 + 119903)

119905minus 1

119903= 119875

(1 + 119903)119905minus 1

119903 (9)


119873 = 1198961(1

120576119905

)

1198962

(1

1198641

)

1198963

for fatigue case

= 1198881(1

120576119911

)

1198882

for rutting case

(10)





2are regression

constants 120576119905and 120576





119863= 40 (or 120590ln119863 =







Fatigue

Rutting

0001020304050607080910

Prob

abili

ty R

(t) o

r S(t

)

5 10 15 20 25 30 35 40 45 50 55 600Time t (yrs)







ℎ (119905) =119891 (119905)

119878 (119905)=

119891 (119905)

1 minus 119865 (119905) (11)


119891 (119905) = ℎ (119905) 119878 (119905) =119889

119889119905119865 (119905)


0ℎ(1199051015840)1198891199051015840

= 1 minus 119878 (119905)

(12)


119891 (119905) =

1

120573expminus(119905120573) 119905 lt

119910119903


119896

120582119896(119905 minus 120574)


119905 ge 120574 (=119910119903

2) for Weibull dist

(13)

119865 (119905) =


2


119905 ge 120574 (=119910119903

2)

(14)

ℎ (119905) =

1

120573 119905 lt

119910119903

2

119896

120582119896(119905 minus 120574)

119896minus1 119905 ge 120574 (=

119910119903

2)

(15)

119878 (119905) =

expminus(119905120573) 119905 lt119910119903

2

expminus((119905minus120574)120582)119896

119905 ge 120574 (=119910119903

2)

(16)


119910119903 =ln (119873119903119875 + 1)ln (1 + 119903)

(17)





120583119910119903= 120582Γ (1 +

1

119896) + 120574 (18)

med119910119903= 120582 (ln 2)1119896 + 120574 (19)

120590119910119903= 120582 [Γ (1 +

2

119896) minus Γ (1 +

1

119896)

2

]

05

(20)





119863(=40 to 80) 119910119903 (=10 to 30 years) and 119873 (=10 to


119910119903= 119910119903



119891 (119905)

=

119896

120582119896(119905 minus 120574)


119905 ge 120574 (=119910119903

2)

0 elsewhere

(21)

119865 (119905) =


119905 ge 120574 (=119910119903

2)

0 elsewhere(22)

ℎ (119905) =

119896

120582119896(119905 minus 120574)

119896minus1 119905 ge 120574 (=

119910119903

2)

0 elsewhere(23)

119878 (119905) =

expminus((119905minus120574)120582)119896

119905 ge 120574 (=119910119903

2)

1 elsewhere(24)

where 119896 = 05119873028

120590ln119863 and 120582 = 120574(ln 2)1119896 and 120574 =





119879= 35 and COV




119863can



000

005

010

015

020

025

030

035

040

045

050

Haz

ard

rate

h(t

)

5 10 15 20 25 30 35 40 45 50 55 600Time t (yrs)






119884119903119877= 120582 (minus ln119877)1119896 + 120574 (25)




119879 COV

119873 119873 and 119910119903


119873119873 and 119910119903 (as example


119877)


6 Conclusions







COV119879COV119873

COV119863


119896 120582 119896 120582 119896 120582

30 42 5313 0499 2320 5856 2817 11390 3156 1684737 45 6059 0559 2069 5969 2512 11571 2814 1708741 58 7490 0667 1733 6177 2105 11902 2358 17523

0

002

004

006

008

01

012

014

016

018

pdf f

(t)

5 10 15 20 25 30 35 40 45 500Time t (yrs)

k = 2320 120582 = 5856 120574 = 5

k = 2512 120582 = 11571 120574 = 10

k = 2358 120582 = 17523 120574 = 15


k = 2320 120582 = 5856 120574 = 5

k = 2512 120582 = 11571 120574 = 10

k = 2358 120582 = 17523 120574 = 15

0001020304050607080910

PDF F

(t)

5 10 15 20 25 30 35 40 45 500Time t (yrs)




function

119896 120582 2120574 = 119910119903 119877 = 85 119877 = 90 119877 = 95 119877 = 982320 5856 10 77 72 66 612069 5969 10 75 70 64 591733 6177 10 72 67 61 572817 11390 20 160 151 140 1292512 11571 20 156 147 135 1242105 11902 20 150 141 129 1193156 16847 30 245 233 216 1992814 17087 30 240 227 209 1932358 17523 30 231 217 200 183





Competing Interests


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Fatigue

Rutting

0001020304050607080910

Prob

abili

ty R

(t) o

r S(t

)

5 10 15 20 25 30 35 40 45 50 55 600Time t (yrs)







ℎ (119905) =119891 (119905)

119878 (119905)=

119891 (119905)

1 minus 119865 (119905) (11)


119891 (119905) = ℎ (119905) 119878 (119905) =119889

119889119905119865 (119905)


0ℎ(1199051015840)1198891199051015840

= 1 minus 119878 (119905)

(12)


119891 (119905) =

1

120573expminus(119905120573) 119905 lt

119910119903


119896

120582119896(119905 minus 120574)


119905 ge 120574 (=119910119903

2) for Weibull dist

(13)

119865 (119905) =


2


119905 ge 120574 (=119910119903

2)

(14)

ℎ (119905) =

1

120573 119905 lt

119910119903

2

119896

120582119896(119905 minus 120574)

119896minus1 119905 ge 120574 (=

119910119903

2)

(15)

119878 (119905) =

expminus(119905120573) 119905 lt119910119903

2

expminus((119905minus120574)120582)119896

119905 ge 120574 (=119910119903

2)

(16)


119910119903 =ln (119873119903119875 + 1)ln (1 + 119903)

(17)





120583119910119903= 120582Γ (1 +

1

119896) + 120574 (18)

med119910119903= 120582 (ln 2)1119896 + 120574 (19)

120590119910119903= 120582 [Γ (1 +

2

119896) minus Γ (1 +

1

119896)

2

]

05

(20)





119863(=40 to 80) 119910119903 (=10 to 30 years) and 119873 (=10 to


119910119903= 119910119903



119891 (119905)

=

119896

120582119896(119905 minus 120574)


119905 ge 120574 (=119910119903

2)

0 elsewhere

(21)

119865 (119905) =


119905 ge 120574 (=119910119903

2)

0 elsewhere(22)

ℎ (119905) =

119896

120582119896(119905 minus 120574)

119896minus1 119905 ge 120574 (=

119910119903

2)

0 elsewhere(23)

119878 (119905) =

expminus((119905minus120574)120582)119896

119905 ge 120574 (=119910119903

2)

1 elsewhere(24)

where 119896 = 05119873028

120590ln119863 and 120582 = 120574(ln 2)1119896 and 120574 =





119879= 35 and COV




119863can



000

005

010

015

020

025

030

035

040

045

050

Haz

ard

rate

h(t

)

5 10 15 20 25 30 35 40 45 50 55 600Time t (yrs)






119884119903119877= 120582 (minus ln119877)1119896 + 120574 (25)




119879 COV

119873 119873 and 119910119903


119873119873 and 119910119903 (as example


119877)


6 Conclusions







COV119879COV119873

COV119863


119896 120582 119896 120582 119896 120582

30 42 5313 0499 2320 5856 2817 11390 3156 1684737 45 6059 0559 2069 5969 2512 11571 2814 1708741 58 7490 0667 1733 6177 2105 11902 2358 17523

0

002

004

006

008

01

012

014

016

018

pdf f

(t)

5 10 15 20 25 30 35 40 45 500Time t (yrs)

k = 2320 120582 = 5856 120574 = 5

k = 2512 120582 = 11571 120574 = 10

k = 2358 120582 = 17523 120574 = 15


k = 2320 120582 = 5856 120574 = 5

k = 2512 120582 = 11571 120574 = 10

k = 2358 120582 = 17523 120574 = 15

0001020304050607080910

PDF F

(t)

5 10 15 20 25 30 35 40 45 500Time t (yrs)




function

119896 120582 2120574 = 119910119903 119877 = 85 119877 = 90 119877 = 95 119877 = 982320 5856 10 77 72 66 612069 5969 10 75 70 64 591733 6177 10 72 67 61 572817 11390 20 160 151 140 1292512 11571 20 156 147 135 1242105 11902 20 150 141 129 1193156 16847 30 245 233 216 1992814 17087 30 240 227 209 1932358 17523 30 231 217 200 183





Competing Interests


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119863(=40 to 80) 119910119903 (=10 to 30 years) and 119873 (=10 to


119910119903= 119910119903



119891 (119905)

=

119896

120582119896(119905 minus 120574)


119905 ge 120574 (=119910119903

2)

0 elsewhere

(21)

119865 (119905) =


119905 ge 120574 (=119910119903

2)

0 elsewhere(22)

ℎ (119905) =

119896

120582119896(119905 minus 120574)

119896minus1 119905 ge 120574 (=

119910119903

2)

0 elsewhere(23)

119878 (119905) =

expminus((119905minus120574)120582)119896

119905 ge 120574 (=119910119903

2)

1 elsewhere(24)

where 119896 = 05119873028

120590ln119863 and 120582 = 120574(ln 2)1119896 and 120574 =





119879= 35 and COV




119863can



000

005

010

015

020

025

030

035

040

045

050

Haz

ard

rate

h(t

)

5 10 15 20 25 30 35 40 45 50 55 600Time t (yrs)






119884119903119877= 120582 (minus ln119877)1119896 + 120574 (25)




119879 COV

119873 119873 and 119910119903


119873119873 and 119910119903 (as example


119877)


6 Conclusions







COV119879COV119873

COV119863


119896 120582 119896 120582 119896 120582

30 42 5313 0499 2320 5856 2817 11390 3156 1684737 45 6059 0559 2069 5969 2512 11571 2814 1708741 58 7490 0667 1733 6177 2105 11902 2358 17523

0

002

004

006

008

01

012

014

016

018

pdf f

(t)

5 10 15 20 25 30 35 40 45 500Time t (yrs)

k = 2320 120582 = 5856 120574 = 5

k = 2512 120582 = 11571 120574 = 10

k = 2358 120582 = 17523 120574 = 15


k = 2320 120582 = 5856 120574 = 5

k = 2512 120582 = 11571 120574 = 10

k = 2358 120582 = 17523 120574 = 15

0001020304050607080910

PDF F

(t)

5 10 15 20 25 30 35 40 45 500Time t (yrs)




function

119896 120582 2120574 = 119910119903 119877 = 85 119877 = 90 119877 = 95 119877 = 982320 5856 10 77 72 66 612069 5969 10 75 70 64 591733 6177 10 72 67 61 572817 11390 20 160 151 140 1292512 11571 20 156 147 135 1242105 11902 20 150 141 129 1193156 16847 30 245 233 216 1992814 17087 30 240 227 209 1932358 17523 30 231 217 200 183





Competing Interests


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











COV119879COV119873

COV119863


119896 120582 119896 120582 119896 120582

30 42 5313 0499 2320 5856 2817 11390 3156 1684737 45 6059 0559 2069 5969 2512 11571 2814 1708741 58 7490 0667 1733 6177 2105 11902 2358 17523

0

002

004

006

008

01

012

014

016

018

pdf f

(t)

5 10 15 20 25 30 35 40 45 500Time t (yrs)

k = 2320 120582 = 5856 120574 = 5

k = 2512 120582 = 11571 120574 = 10

k = 2358 120582 = 17523 120574 = 15


k = 2320 120582 = 5856 120574 = 5

k = 2512 120582 = 11571 120574 = 10

k = 2358 120582 = 17523 120574 = 15

0001020304050607080910

PDF F

(t)

5 10 15 20 25 30 35 40 45 500Time t (yrs)




function

119896 120582 2120574 = 119910119903 119877 = 85 119877 = 90 119877 = 95 119877 = 982320 5856 10 77 72 66 612069 5969 10 75 70 64 591733 6177 10 72 67 61 572817 11390 20 160 151 140 1292512 11571 20 156 147 135 1242105 11902 20 150 141 129 1193156 16847 30 245 233 216 1992814 17087 30 240 227 209 1932358 17523 30 231 217 200 183





Competing Interests


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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