Journal of Engineering Science and Technology Review 9 (6) (2016s) Research Article. Bai Shi-ye* and Zhu Jie-jiang

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1 Jestr Joural of Egieerig Siee ad Tehology eview 9 (6) (06s) 69-5 esearh Artile Coetivity eliability Aalysis of oad etwork of Multiple OD Pairs based o the Strutural eliability of Joit Failure Modes Bai Shi-ye* ad Zhu Jie-jiag Departmet of Civil Egieerig, Shaghai Uiversity, Shaghai, 000, Chia eeived 8 Marh 06; Aepted 5 Deember 06 Abstrat JOUAL OF Egieerig Siee ad Tehology eview I order to improve operatioal effiiey of traffi road, it is eessary to arry out the exploratio o the reliability of road etwork. The oetivity reliability is the ritial idex to alulate the reliability of the road etwork. Based o the joit failure modes of fatigue rakig ad permaet deformatio, the oetivity reliability model of road etwork was established, whih was further itrodued ito the reliability aalysis of road etwork system. By osiderig the ifluees of differet OD (Origi-Destiatio) pairs ad their weightig oeffiiets o the system reliability, the reliability model of omplex road etwork of multiple OD pairs was fially ostruted. Furthermore, oetivity reliability of omplex road etwork was alulated o the basis of the Mote Carlo method ad the breadth-first searh method. The ifluees of the differet road elemets ad odes o the system reliability were aalyzed by the umerial examples. I additio, the ifluees of reiforemet ad expasio of the road etwork o the system reliability were also evaluated. esults show that, although the preseted models based o multiple OD pairs ad its differet weighig oeffiiets are loser to the atual road etwork system, it is safer to defie the road etwork reliability by sigle OD pair. Moreover, the failure of the odes, espeially those adjaet to the startig ode, auses more damages to the road etwork system tha the road setio does, whih derease the reliability of the road etwork system. The reovery of the road setios adjaet to the startig odes a ot oly effetively improve the reliability of the system but also redue eoomi osts. Keywords: Joit Failure Modes; Weightig Coeffiiet of Multiple OD Pairs; Coetivity eliability of oad etwork; eiforemet ad Expasio of oad etwork. Itrodutio With the rapid developmet of ostrutio of urba road, the ogestio problem of urba road traffi has attrated more attetios. Thus, the aalysis o oetivity reliability model of road etwork plays a key role i dealig with this problem. Presetly, the sope of studies o the traffi etwork reliability maily iludes three aspets, amely oetivity reliability [-], travel time reliability [-6] ad apaity reliability [-9]. The oetivity reliability was the earliest researh field amog the researh o the road etwork reliability. It bega to attrat more oers after the earthquake i Kobe, Japa. After multiple 8-magitude earthquakes took plae i Chia, the researh o oetivity reliability beame oe of the most popular researh field [0-5]. Oly whe the road etwork remais oetive, the ormal travel a be guarateed. Therefore, the oetivity reliability is the ritial idex to alulate the reliability of the road etwork.. State of the Art The oetivity reliability was firstly proposed by Mie ad Kawai i 98 [6]. They defied it as the probability of the oetivity betwee two poits i the road etwork. I address: bai88060@sia.om ISS: 9-06 Easter Maedoia ad Thrae Istitute of Tehology. All rights reserved. 989, Iida ad Wakabayashi studied the oetivity reliability of the speial OD pairs based o the miimal path set algorithm ad ut set algorithm [],whih exteded the oetivity reliability of two poits i road etwork to the whole etwork. Bell et al. adopted the graph theory to alulate the oetivity reliability [8]. I their study, oly two ruig states (omplete oetio ad omplete iterruptio) were osidered, ad the importae of the road setio was igored. I 996, Hu preseted a ew method to ompute reliability of disjoited etwork [9]. I 005, Stephe ad David proposed a way to estimate the probability distributio of total etwork travel time [0].I 008, Zhag et al. iitially preseted the etwork apaity reliability idex based o topologial struture []. I 009, Erik ivestigated the way i whih the geographial disparities depeded o the road etwork struture ad travel patters. He proposed aggregate supply-side as well as demad-side idiators, ad ombied them i statistial regressio models []. I 0, Keetsu developed two models whih simultaeously estimated the value of travel time ad its reliability based o the route hoie behavior of risk-averse driver []. I 05, Chog et al. preseted a bilevel programmig model based o the ubloked reliability of urba etwork []. Amog the previous studies o oetivity reliability of the whole etwork, the road strutural reliability ad etwork reliability were treated as two mai researh subjets that were idepedet from eah other from the

2 Bai Shi-ye ad Zhu Jie-jiag/Joural of Egieerig Siee ad Tehology eview 9 (6) (06) 69-5 perspetive of maro maagemet. However, there is a ievitable lik betwee these two subjets. The road strutural reliability is the foudatio of researhig the road etwork reliability. The hage of the road strutural reliability must affet the pavemet bearig apaity, whih a determie the road bearig traffi flow. The hage of the traffi flow of road setio will ifluee the traffi flow of the adjaet road. At the same time, the whole road etwork reliability will also udergo hages. Therefore, it is essetial to explore the reliability of road etwork o the basis of the road struture reliability. I additio, graph approahes are usually used i the traditioal oetivity reliability alulatio. This kid of method is with high auray ad a be applied ito the oetivity reliability alulatio of low desity ad small-sale road etwork. evertheless, it is ot appliable to the large-sale ad ompliated traffi etwork beause the alulatio amout is too huge ad it is diffiult to reflet the real situatio of the road etwork. The traditioal alulatio of oetivity reliability of road etwork is mostly based o sigle OD pairs. Eve if multiple OD pairs are itrodued ito the reliability alulatio of road etwork, the model a t distiguish the differet effets betwee differet OD pairs. I fat, differet OD pairs have a differet ifluee o the road etwork performae ad damage probability. Thus, by osiderig the importae of differet OD pairs, the road etwork reliability evaluatio a be better omplied with the atual situatio. The remaider of this study was orgaized as follows. Setio established the reliability model based o the joit failure modes of fatigue rakig ad permaet deformatio, ad alulated the reliability of asphalt pavemet struture. By itroduig the strutural reliability ito the road etwork oetivity aalysis, the oetivity reliability model of road etwork was ostruted based o the omplex multiple OD pairs ad weightig oeffiiets. Furthermore, the Mote Carlo method ad the bread-first searh method were utilized to obtai the oetivity reliability of the omplex road etwork system. Setio aalyzed the ifluee of the road setio ad odes failure o the oetivity reliability of road etwork system through umerial examples. Moreover, this setio disussed the effet of reiforemet ad expasio of the road etwork o the reliability reovery of road etwork. Fially, Setio 5 summarized the olusios.. Methodology. eliability of Asphalt Pavemet Struture Uder Multiple Failure Modes.. Fatigue rakig failure mode The trasfer futios of fatigue rakig failure mode are geerally utilized to alulate the umber of load repetitios that lead to the failure of the pavemet. I this study, the trasfer futio of fatigue rakig is take as the followig form: f t = ε E () where is the umulative umber of stadard axle f load repetitio that produes 0% rakig area; ε is the tesile strai at the bottom of asphalt surfae layer; E is the t 0 elasti modulus of asphalt surfae layer;,, ad are regressio oeffiiets of the fatigue rakig equatio... uttig failure mode The relatioship betwee the umulative umber of stadard axle load repetitio that leads to permaet ruttig deformatio ad the vertial ompressive strai at the top of the subgrade is desribed as the followig trasfer futio: r 5 = ε () Where is the umulative umber of stadard axle r load repetitio that produes.mm ruttig; ε is the vertial ompressive strai at the top of the subgrade; ad 5 are regressio oeffiiets of the ruttig equatio... eliability of asphalt pavemet struture As for most of the pratial egieerig strutures, there are atually multiple failure modes. Thus, the reliability of the asphalt pavemet should be defied as the system reliability. For the asphalt pavemet struture, the system will fail if either fatigue rakig or ruttig ours. The failure probability of two failure modes of the pavemet struture a be expressed as follows: p Z = = f f x dx P 0 X ( ) U i Zi 0 i = U where Z i is the futioal equatio (, represets fatigue rakig ad ruttig evet respetively). Its futio is give as follows: Z ( x) = f d ; Z( x) = r d () where d is the limitig umber of traffi ylig load repetitio. The Mote Carlo Method is widely applied i reliability aalysis of egieerig strutures. Futio M(x) is utilized to deote the failure state (M(x)=) or the safe state (M(x)=0), ad x is radom variable that depeds o parameters suh as material, load, ad struture size. Based o geeratio algorithm, a sequee of iput variable x is obtaied aordig to the joit statistial distributio. The the failure probability of the asphalt pavemet struture a be estimated by the followig formulatio: p f s M ( xi ) = s where s is the total simulatio umber of Mote Carlo Method. Firstly, fatigue rakig ad ruttig are determied respetively by the tesile strai ε t at the bottom of the asphalt surfae layer ad the vertial ompressive strai i () (5) ε at the top of the subgrade. The pavemet desig geeral software KEPAVE is utilized to predit the ritial strai

3 Bai Shi-ye ad Zhu Jie-jiag/Joural of Egieerig Siee ad Tehology eview 9 (6) (06) 69-5 ε t ad ε. Seodly, aordig to the equatios of () ad (), the umber of load repetitios that produe fatigue rakig ad ruttig respetively is obtaied. Fially, the failure probability futio () is solved by Mote Carlo Method. Furthermore, the joit failure probability based o two failure modes ad the reliability of asphalt pavemet struture are obtaied.. oad Coetivity eliability of Complex oad etwork with Multiple OD Pairs.. eliability of road setio Based o the above failure modes, the reliability of the road setio is desribed as follows: = P <, < ) (6) ijk ( f d r d where ijk is the reliability of the k-th road setio of the j-th path of the i-th OD pair... eliability of path The path is a route from the startig poit of ay OD pair to the edig poit. It is omposed of multiple road setios i series. Thus, the path reliability a be defied as: m ij = ijk k = where ij is the reliability of the j-th path of the i-th OD pair... OD pair reliability The OD pair is regarded as multiple paths i parallel, show as follows: i = ( ij ) (8) j= where i is the reliability of the i-th OD pair... The traditioally defied reliability of road etwork The reliability of road etwork is expressed as follows: i = (9) where is the traditioally defied reliability of the road etwork...5 The improved reliability model with multiple OD pairs ad its weighig oeffiiets Previous alulatio method of the road etwork reliability is based o the average reliability of all the OD pairs. However, the relative importae of differet road setios ad odes have t bee osidered. Therefore, the road etwork reliability model should be improved as follows: P et = [ wi () POD () i] wi () () (0) where P et is the reliability of road etworks based o the preseted model, P OD (i) is the reliability of the i-th OD pair, wi () is weightig oeffiiets of the i-th OD pair, ad is the umber of all OD pairs. Whe the umber of odes is m, m( m ) =. The weightig oeffiiet of the i-th OD pair is expressed as follows: P G + () P G ( ) α α wi = i i where α ad α are weightig oeffiiets of populatio umbers ad GDP respetively, α+ α =, P i is the populatio of the i-th OD pair, G i is the GDP of the i-th OD pair, P is the populatio of all OD pairs, ad G is the GPD of all OD pairs...6 The alulatio of the oetivity reliability of road etwork Based o the Mote Carlo method, the oetivity reliability of road etwork is alulated through MATLAB programmig. The implemetatio proedure is show i Figure. Firstly, some parameters are predefied iludig road setio umbers r, reliability matrix of road setio a(i,j) (i ad j are the odes of the road setio), oetivity oeffiiet matrix A(i,j) (A(i,j)= whe the road setio with odes i ad j is oetive, or A(i,j)=0), ad failure iformatio matrix of road setio (S, ) (S deotes simulatio umber, ad sigifies ode umber.) Seodly, the demad oetivity reliability of road setios is postulated to obey the uiform distributio i the iterval of [0, ]. Whe the demad oetivity reliability of road setio is lower tha the oetivity reliability, the the road setios are liked, whih meas A(i,j)=, or A(i,j)=0. Besides, the failure iformatio matrix of road setio is assiged. Thirdly, the road setio of the etwork is searhed by breadth-first-searh method, i order to judge the oetivity of the OD pair. If oetive, the umulative time F ireases by oe time, or it will ot irease. Loop omputatio is performed to aquire satisfyig simulatio umbers. Fially, the oetivity reliability of road etwork a be obtaied.. esult Aalysis ad Disussio. Example As show i Figure, a ertai area of the road etwork i Shaghai is seleted as the omputatioal example. The improved reliability model is hose to alulate the reliability of the road etwork. Aordig to the above road etwork system, some mai road setios are hose to be aalyzed ad marked i red, as show i Figure. Fially, the simplified road etwork system is obtaied ad preseted i Figure. There are 5 odes (~5) i the road etwork system, iludig road setios (~), ad 9 OD pairs (,, 5,,, 5,,,ad 5).

4 Bai Shi-ye ad Zhu Jie-jiag/Joural of Egieerig Siee ad Tehology eview 9 (6) (06) 69-5 Fig.. The sketh graph of road etwork Fig.. The alulatio proedure of the road etwork oetivity reliability Fig.. The simplified road etwork system The road setio reliability is based o the joit failure modes of struture. The weightig oeffiiets of populatio ad GPD are both 0.5. All the road setios reliability ad path parameters are show i Table ad Table respetively. Fig.. A ertai area of the road etwork i Shaghai Table. The road setio reliabilities i the etwork system oad setio umber eliability Path umber reliability Pi / Table. Parameters of all road setios i the etwork system oad setio umber Gi / Path umber Pi / Gi / I this study, the breadth-first-searh method is ombied with the Mote Carlo method, whih is used to alulate the reliability of all OD pairs i the road etwork system. The simulatio umber is oe hudred thousad. The reliability results of all OD pairs ad the weightig oeffiiets are listed i Table.

5 Bai Shi-ye ad Zhu Jie-jiag/Joural of Egieerig Siee ad Tehology eview 9 (6) (06) 69-5 Table. The reliabilities of all OD pairs i road etwork system OD pair Start Ed 5 5 eliability Weightig oeffiiet As show i Table, the miimum reliability of all OD pair is 0.86 ad the maximum is The weightig oeffiiets of differet OD pairs are differet, so that the relative otributios of differet OD pairs are also diverse. Based o Equatio (0), the reliability of road etwork system is alulated as Whe this value is ompared with the differet sigle OD pair results show i Table, it a be foud that the aurate system reliability aot be evaluated by usig a sigle OD pair. For example, the system reliability based o the OD pair is 0.99, whereas the reliability based o multiple OD pairs model is The maximum error betwee them reahes.6%, ad it is ot safe. Therefore, the improved evaluatio model i this study is loser to the real egieerig situatio, whih a lay a good foudatio for the omputatio ad aalysis of the reiforemet ad expasio of more omplex road etwork systems.. The Ifluee of Failure ad eiforemet of oad o System eliability.. oad setio failure O the basis of the above example results, the ifluee of the failure of road setio ad odes o system reliability is further aalyzed. I this study, oly oe OD pair (~5) is seleted, while all the other parameters remai uhaged. The simulatio time is oe hudred thousad uder the Mote Carlo method. There are three forms of road setio failures, amely the adjaet road setio failure, the seod adjaet road setio failure ad o-adjaet road setio failure. Three ases are hose to be aalyzed, ad their system reliabilities are respetively alulated ad show i Table - 6. with the previous omplete road etwork system, whe the road setios adjaet to the startig ad edig poits fail. It dereases by as muh as 6% whe the seodary adjaet road setios fail, ad oly % whe the o-adjaet road setios fail. By omparig the above three ases with eah other, it a be foud that the road setios adjaet to the startig ad edig poits ifluee the system reliability most sigifiatly, followed by the seodary adjaet road setios. That is to say, the o-adjaet road setios ifluee o is the least promiet, whih suggests that the road etwork system is stable... oad ode failure There are two forms of road ode failure, amely the adjaet ad o-adjaet road ode failures. Two ases are hose to be aalyzed, ad their system reliabilities are alulated ad show i Table ad Table 8. Tab.. The road etwork reliability of adjaet road odes to ad 5 Failure road ode eliability Table 8. The road etwork reliability of o-adjaet road odes to ad 5 Failure road ode eliability Failure road 9 0 ode Failure road ode Table. The road etwork reliability of adjaet road setios to ad 5 Failure road setio 9 eliability Tab.5. The road etwork reliability of seodly adjaet road setios to ad 5 Failure road setio 6 9 eliability Failure road setio 5 8 eliability Table 6. The road etwork reliability of o-adjaet road setios to ad 5 Failure road setio eliability Failure road setio 6 0 eliability From Table -6, it a be see that the road etwork system reliability a delie by as muh as % ompared From Table ad Table 8, it a be see that the road etwork system reliability delies by as muh as % ompared with the previous omplete road etwork system whe the road odes adjaet to the startig ad edig poits fail. It dereases by % whe the o-adjaet road odes fail. I aordae to the result of ompariso betwee the above two ases, the road odes adjaet to the startig ad edig poits more ifluee the system reliability (as show i Tables -6). I additio, the oadjaet ode failure ifluees the system reliability more tha the o-adjaet road setio failure does. Thus, it is more importat to esure the safety ad reliability of road odes rather tha road setios for the same road etwork system... oad reiforemet ad expasio I order to further disuss the ifluee of the reiforemet ad expasio of road o the road etwork system reliability, the ase of failure is aalyzed. The other parameters are the same as those i the previous example. Whe the odes fail, there are two approahes to reover the reliability of the road etwork system, amely () eiforemet of the other road setios, () Expasio of

6 Bai Shi-ye ad Zhu Jie-jiag/Joural of Egieerig Siee ad Tehology eview 9 (6) (06) 69-5 the road etwork. I our subsequet studies, the ifluee of the two approahes o the reliability of road system will be disussed. As fails, the other road setios adopt the same form of reiforemet, ad the the reliability of reifored road setios will be ehaed. The reiforemet forms are give i Table 9. Table 9. eliability after reiforemet eiforemet level of other road 0% 5% 0% 5% 0% setios eliability eiforemet level of other road 5% 0% 5% 0% setios eliability Obviously, the system reliability ireases oliearly with the improvemet of the road reiforemet level. Whe the reiforemet degree ireases from 0 to 0%, the reliability grows from to 0. (risig by %). Whe the reiforemet degree ireases from 0% to 0%, the reliability grows from 0. to 0.86 (risig by 9%). As the reiforemet degree otiues to grow to 0%, the reliability reahes 00%. Cosequetly, it is a effetive way to improve the system reliability by reiforig the ofailure road setios. Fig. 6. The expasio of etwork--approah As show i Figure 6, is added, ad the iitial oditio P () =0.8 is set, after the failure of. The, the reliability of the road etwork system is alulated as based o Approah. Compared with the previous system, the reliability has a otable irease. Fig.5. The expasio of etwork--approah Whe the ode fails, the expasio of the etwork a be used to reover the reliability of the road system. Figures 5- display three forms of expasio. As depited i Figure 5, the failure of will lead to the failure of road setios, ad. At this time, the reliability of the road etwork system is By addig the ode 6 ad road setio ad, the reliability a be reovered. The iitial oditio (P() =0.8, P() =0.88) is set. The, the road etwork system reliability is alulated as based o Approah. Compared with the previous road etwork system, it is ot effetive to improve the reliability of road etwork system with this approah. Fig.. The expasio of etwork--approah As show i Figure, is added, ad the iitial oditio P() =0.8 is set after the failure of. The, the reliability of the road etwork system is alulated as based o Approah. Compared with the previous system, the reliability also has a remarkable irease. I the light of the ompariso results amog the three approahes of expadig the road etwork, although oe ode ad two road setios are added ito the system, Approah has a less ifluee o the ehaemet of reliability, whih suggests that differet forms of expasio will otribute differetly to the system reliability. By otrast, Approah ad oly add oe road setio, ad the reliability ireases remarkably. As a result, the expasio of the etwork should osider the atual travelig demad so as to improve the system reliability.

7 Bai Shi-ye ad Zhu Jie-jiag/Joural of Egieerig Siee ad Tehology eview 9 (6) (06) Colusio I order to ease the traffi ogestio, the exploratio o reliability of road etwork were arried out. The oetio reliability problem seems more promiet. Firstly, the reliability model of a omplex road etwork with multiple OD pairs was established. Seodly, based o the Mote Carlo method ad the breast-first searh method, the oetivity reliability of the omplex road etwork was alulated. Fially, the followig olusios were obtaied: () It is safer to use a sigle OD pair to defie the road etwork reliability, whereas the preseted model based o multiple OD pairs with differet weighig oeffiiets is more pratial to desribe the atual road etwork system. () The failure of odes have a greater ifluees o the system reliability ompared with that of the road setios. For this reaso, the ode desig is more importat i the road etwork desig ad plaig. () The reiforemet of the damaged road setios ad the expasio of the etwork will irease the oetivity reliability of the etwork. However, three approahes of expadig the etwork are ompared, ad the results show that the expasio of the road setios adjaet to the travelig poits is a more effetive ad eoomial way to reover the reliability of road etwork system. I this study, the aalysis o road etwork oetivity reliability model is oly appliable to the situatio where road etwork system is damaged immediately by atural disasters or ma-made deed.this is just a kid of the routie road etwork failure.thus, more omplex fators,suh as the dyami of road etwork, should be take ito osideratio to modify the preset road etwork model of oetivity reliability. eferees. Qia YS, Wag M, Kag HX, Zeg JW, Liu YF. Study o the road etwork oetivity reliability of valley ity based o omplex etwork, Mathematial Problems i Egieerig, 0, 0, pp. -. Qia Y S, Wag M, Kag H X, et al. 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