RISK-BASED REPLACEMENT STRATEGIES FOR REDUNDANT DETERIORATING REINFORCED CONCRETE PIPE NETWORKS

RISK-BASED REPLACEMENT STRATEGIES FOR REDUNDANT DETERIORATING REINFORCED CONCRETE PIPE NETWORKS Bryan Adey, Olivier Bernard 2 and Bruno Gerard 2 Division of Mainenance and Safey, Faculy of Archiecure, Civil and Environmenal Engineering, Swiss Federal Insiue of Technology, Lausanne, Swizerland, h://mcswww.efl.ch 2 Oxand S.A. Avon / Fonainebleau, France, www.oxand.com ABSTRACT This aer gives an examle of how redicive models of he deerioraion of reinforced concree ies and he consequences of failure can be used o develo risk-based relacemen sraegies for redundan reinforced concree ie neworks. I also shows how an accurae deerioraion redicion can lead o a reducion of agency coss, and illusraes he limiaion of he incremenal inervenion se algorihm. The main conclusion is ha he use of redicive models, such as hose develoed by Oxand S.A., in he deerminaion of relacemen sraegies for redundan reinforced concree ie neworks can lead o a significan reducion in overall coss for he owner of he srucure. Keywords: Oimal managemen sraegies, redundan ie neworks, incremenal inervenion se algorihm, deerioraing concree ies INTRODUCTION Reinforced concree ie neworks used o ransor waer deeriorae wih ime due o environmenal condiions, such as chloride-induced corrosion of he seel reinforcemen, and if no mainained will evenually fail []. The risks associaed wih he failure of reinforced concree ie neworks, defined herein as he robabiliy of failure mulilied by he consequences of failure, and how hey change wih ime, lay a crucial role in deermining he oimal relacemen sraegies for he ies wihin hese neworks [2]. Consequences of failure include he inerruion o service and he damage ha resuls from he failure iself, such as he flooding of he buildings ha house he ies or nearby buildings or roads. Alhough risks can be diminished by eriodically relacing deerioraed ies [3], here are oenially high coss associaed wih relacing ies, including he cos of removing he exising ies, he cos of he new ies and he cos of service inerruion of a emorary closure. Oimal relacemen sraegies for he ies mus herefore be deermined by minimising boh he risks and he cos of relacemen for he ies in he nework, as well as how hese risks and coss change wih ime and he effeciveness of he relacemen in reducing fuure risks. Oimal relacemen sraegies are herein referred o wih he abbreviaion for oimal managemen sraegy, OMS. Relacemen sraegies are one ye of managemen sraegy. In his aer an examle of how OMS s can be deermined for redundan reinforced concree ie neworks using redicive models of deerioraion and considering boh he consequences of failure and he redundancy of he nework is given. RISK-BASED OBJECTIVE FUNCTION To deermine risk-based OMS s he objecive funcion is defined o minimise cumulaive overall coss (boh agency coss and risks (Eq.. By using Eq. as he objecive funcion boh he redundancy of he nework and he deerioraion of he ies, are aken ino consideraion in he esimaion of he robabiliy of failure of he sysem, P fsys. The coss of failure, C f are hen mulilied by he P fsys and added o he agency coss, C a. The minimisaion of hese overall coss hroughou he invesigaed ime eriod will give he OMS. /9

Minimise where: T ( C a = T ( P fsys C f = T T ( Ca + ( Pfsys C f λ = ( = = = cumulaive agency coss; and (2 = cumulaive risk (3 arallel The robabiliy of failure of a arallel sysem, P fsys, is given by Eq. 4, and he robabiliy of failure of he branches in a series arallel sysem, which are in series, P fsys, is given by Eq. 5. n branch ( fi arallel P fsys = P (4 where: i= branch P fi = he robabiliy of failure of he branches in he arallel sysem n series P fi = P where: i= secion fi (5 ion P sec fi = he robabiliy of failure of he secions in he branches The deerioraion is aken ino consideraion by assuming ha he ies are in a consan condiion sae for each ime inerval in he invesigaed ime eriod and evaluaing he robabiliy of he ies in each of hese condiion saes for each successive ime inerval. The robabiliy of assing beween condiion saes from one ime inerval o he nex is described using Markov models. In Markov models he condiion raings ake he form of discree saes in order o reduce he comlexiy associaed wih coninuous ranking sysems [4]. PREDICTIVE MODELS OF PIPE DETERIORATION Markov models and semi-markov models are used o model he deerioraion of he ies. Markov models are commonly used, in managemen sysems, o model he deerioraion of infrasrucure asses, such as ies [5] and road bridges [4]. Semi-Markov models, however, have been used o incororae changes in failure mode ha may occur as a funcion of boh ime and he number of revious breaks [6]. As an unchanging failure mode is assumed in his examle Markov models are considered adequae. A Markov model describes a sochasic rocess where he condiional robabiliy of any fuure even, such as being in condiion sae j, given any as even and he resen sae = i, is indeenden of he as even and deends only on he resen sae. The condiional robabiliies P{ + = j = i} are called ransiion robabiliies. The ransiion robabiliies in a Markov model can be deermined using Poisson and negaive binomial based regression echniques [7, 8] and can be correlaed o acual deerioraion models [9] albei no erfecly. The ossible ransiion robabiliies are ofen shown in marix form. The form of he Markov models used in his examle (Eq. 6 is based on he five-sae model shown in Fig. 2/9

2 3 4 5 ( + + + 2 3 4 5 ( + + 23 2 2 24 25 3 ( 34 3 23 + 35 4 24 34 4 45 5 5 25 35 45 (6 where ij = ransiion robabiliy from condiion sae i in year o condiion sae j in year +. For examle, column row (in bold, Eq. shows he robabiliy of being in condiion sae a + if he ie is in condiion sae a ime. Noe he ij = for i > j. This imoses he consrain ha ies canno imrove in condiion. Also 55 = because his is he wors ossible condiion sae and i is an absorbing sae, i.e. once a ie has enered his sae i canno leave wihou an inervenion. 4 3 35 2 23 34 45 2 3 4 5 22 33 44 55 24 25 5 Figure. Sae ransiion diagram for five-sae ie examle The robabiliy of being in condiion sae j, in year + can be deermined hrough he alicaion of he oal robabiliy heorem (Eq.7. + j where = j i= i P ij i = he robabiliy of being in condiion sae i, in year. EAMPLE NETWORKS To illusrae how redicive models of he deerioraion of redundan reinforced concree ie neworks and he consequences of failure can be used o develo risk-based relacemen sraegies a simle arallel nework is used (Fig. a. To illusrae he limiaions of an incremenal inervenion se algorihm boh he wo-secion arallel nework (Fig. a and he four-secion arallel nework (Fig. b are used. The difference beween he wo neworks is ha in nework, branch and branch 2 have only one secion each, whereas in nework 2 branch and branch 2 have wo secions each. Each secion in boh neworks consiss of 5 ies in series. The ies in he nework are classified ino 5 differen condiion saes (CS, which are defined in Table [3]. (7 3/9

Branch Branch Secion Branch 2 Secion Secion 2 Secion 3 Secion 3 Secion 4 (a Figure. (a Nework, (b Nework 2 Branch 2 (b Table. Condiion saes for underground reinforced concree ies [2] Condiion sae Physical descriion Near erfec condiion 2 Some suerficial deerioraion 3 Serious deerioraion, requiring subsanial mainenance 4 Level of deerioraion affecs he fabric of he asse, requiring major reconsrucion or refurbishmen 5 Level of deerioraion is such o render he asse unserviceable I is assumed ha insecion of he enire nework is erformed rior o he deerminaion of he OMS, and ha one quarer of he ies (375 in each secion is in each of he condiion saes, i.e. here are 375 ies iniially in CS, CS2, CS3 and CS4 for each of he secions. All ie deerioraion is assumed o be he same and is described by he medium deerioraion marix shown Fig. 2..995 Slow deerioraion: Medium deerioraion: Fas deerioraion:.5.99..98.2.995.5.99..98.2.995.5.99..98.2.995.5.99..98 (a (b (c.2 Figure 2. The deerioraion marices for (a slow deerioraion, (b medium deerioraion and (c fas deerioraion The four ossible inervenions on each secion of each nework are o relace all of he ies ha are iniially in each condiion sae, i.e. CS4, CS3, CS2 and CS. This means ha here are 8 ossible inervenions on nework and 6 ossible inervenions on nework 2 (Table 2. All inervenions are assumed o cos mu (mu = moneary unis. I is assumed ha if an inervenion is no done and failure (defined as CS5 occurs, ha he ie where he leak occurs is relaced and he res of he nework is no. This has he effec of leaving he nework in basically he same overall condiion sae, i.e. he robabiliy of failure of his secion of ie is basically no changed, and he execed failure coss in he ucoming year herefore remain unchanged. The cos of failure is mu, or imes he inervenion cos. 4/9

Table 2. Inervenions Nework Nework 2 Inervenion Pie secion Pie grous Inervenion Pie secion Pie grous Inervenion Pie secion Pie grous Pies iniially in CS4 Pies iniially in CS4 9 3 Pies iniially in CS4 2 Pies iniially in CS3 2 Pies iniially in CS3 3 Pies iniially in CS3 3 Pies iniially in CS2 3 Pies iniially in CS2 3 Pies iniially in CS2 4 Pies iniially in CS 4 Pies iniially in CS 2 3 Pies iniially in CS 5 3 Pies iniially in CS4 5 2 Pies iniially in CS4 3 4 Pies iniially in CS4 6 3 Pies iniially in CS3 6 2 Pies iniially in CS3 4 4 Pies iniially in CS3 7 3 Pies iniially in CS2 7 2 Pies iniially in CS2 5 4 Pies iniially in CS2 8 3 Pies iniially in CS 8 2 Pies iniially in CS 6 4 Pies iniially in CS OPTIMAL REPLACEMENT STRATEGIES The OMS s were deermined for a -year eriod wih no resricion on he number of inervenions er ime inerval (each ime inerval consiss of 5 years using an incremenal inervenion se algorihm. An incremenal inervenion se algorihm means ha he oimal inervenion and ime of he oimal inervenion were deermined one inervenion a a ime. For examle, o deermine he oimal inervenions for a wo-inervenion managemen sraegy, inervenion is firs deermined and hen i is assumed ha his inervenion (boh he ies o relace and he ime o relace hem would no be affeced by erforming inervenion 2. Inervenion 2 is hen deermined. This ye of algorihm is no always valid as can be seen when comaring he OMS s for he wo neworks (Table 3. Comlee enumeraion and dynamic rogramming were used o find for each successive inervenion. Number of inervenions Pie secion Pie grou Table 3. Ranking of inervenions Examle nework Examle nework 2 Incremenal Inervenion reducion in Pie Pie Inervenion ime overall coss secion grou ime Incremenal reducion in overall coss 4 642 4-77 2 3 2 79 2 4 4526 3 2 6 354 3 2 743 4-5 2 3 2 9486 5 4-5 2 5 8 6 3-62 2 2 6 324 7 2 3-9 -52 8 3 4 3-95 2-52 9 3 3 3-93 2 4-5 3 2 5-99 4-5 3 5-3 -63 2 3 3 5-2 3-62 3 3 4 4-2 2-86 4 3 2 5-2 2 3-9 5 2-2 5-98 6 3-5 -98 7-4 5-98 8 3 3-2 4 5-98 9 4 2-3 5-5/9

The OMS s, which can be read from Table 3 once he number of inervenions desired are known, are deermined for he boh neworks assuming discoun raes of %. For examle, if i is desired o have only 4 inervenions in he invesigaed -year eriod he OMS is o relace he ie grous 4, 3, 2 and a, 2, 6 and resecively for nework. The negaive values in Table 3 indicae where i is no longer beneficial o erform an inervenion, i.e. he agency coss are higher han he ossible reducion in execed failure coss. All values are rounded o he neares moneary uni. In Tabéle 3, his means ha i is only beneficial, given he C f, C a and deerioraion marices used, o relace ie grous 4, 3 and 2 in branch for boh neworks (secion for nework and secions and 2 for nework 2. These inervenions are shown in bold in Table 3. On nework he seven mos beneficial inervenions are o relace he ies on branch from bes o wors (ie grous 4, 3, 2, and, and hen 4, 3, 2 again a ime inervals, 2, 6,,,, and 3, resecively. I is no unil he robabiliy of sysem failure is sufficienly small hroughou he enire invesigaed ime eriod (Fig. 3a ha relacing ies in he second branch of he arallel nework becomes he mos beneficial. This occurs a he 8 h inervenion, shown in bold wih squares in Fig. 3a. Fig. 3a shows he cumulaive robabiliy of sysem failure of nework and nework 2. The decreasing robabiliy of failure of he neworks (in he direcion of he arrow occurs because each successive curve is generaed using a managemen sraegy wih an addiional inervenion. For examle, in Fig. 3a., he bold line wih squares is he cumulaive robabiliy of failure of nework when an OMS consising of 8 inervenions is used. The bold line wih diamonds in Fig. 3a is he cumulaive robabiliy of failure of nework when an OMS consising of 9 inervenions is used..... Pfsys.. Pfsys....... 5 5 2 Time inervals (a. 5 5 2 Time inervals (b Figure 3. The cumulaive robabiliy of sysem failure for all of he OMS s shown in Table 3 (a nework (b nework 2 On nework 2 all inervenions are found o be on branch. This is an error due o he use of he incremenal inervenion se algorihm, which only looks a one inervenion a a ime. I should sugges reairing nework 2 in he same order ha i suggess reairing nework. The inervenions ha should be he same, i.e. he ies and he locaion of he ies ha should be relaced, bu are no, are shown as shaded cells in Table 3. Insead of suggesing ha he ies on branch 2 (secions 3 and 4 in nework 2 are he mos beneficial o relace, as hey are on nework (secion 3 inervenions 8 and 9, he incremenal inervenion se algorihm suggess, ha he oimal inervenions are on branch (secions and 2 - inervenions 5, 6, 7, and 8. By no changing branches on nework 2 he cumulaive robabiliy of sysem failure hroughou he -year (2 ime inerval eriod remains much higher han for nework. The cumulaive robabiliy of sysem failure afer he inervenion 8, and afer inervenion 8 and 9, on branch 2, on nework, are in bold in Figure 3a. I can be seen ha here are order of magniude dros in he robabiliy of sysem failure wih successive inervenions. The cumulaive robabiliy of sysem failure afer each of he four successive inervenions ha should be he equivalen of he inervenions 8 and 9 on nework, on nework 2, are in bold in Figure 3b. I can be seen ha here is very lile furher reducion in he cumulaive robabiliy of sysem failure. This inabiliy o swich branches occurs because, by only looking a one inervenion a a ime he incremenal inervenion se algorihm never sees he fuure benefi of doing an inervenion on he more deerioraed branch in he arallel nework, i.e. if only he benefis of erforming one inervenion a a ime are comared he larges benefi will always come from erforming he inervenion on he branch ha has already had one inervenion if he robabiliy of failure of he oher branch is high (which in his examle i is. The incremenal inervenion se algorihm should herefore only be used in siuaions where i is no required o see he benefi of fuure inervenions. 6/9

ACCURATE DETERIORATION PREDICTION To invesigae he imorance of accurae deerminaion of deerioraion seeds on redundan reinforced concree ie neworks, he OMS s consising of u o 9 inervenions are deermined for nework using hree differen deerioraion seeds; slow, medium and fas (Table 4. Slow, medium and fas deerioraion seeds are defined in his examle by he deerioraion marices shown on Fig. 2. Slow deerioraion is defined as having a.5 % chance ha a ie will ass ou of is condiion sae in one ime inerval. Medium deerioraion is defined as having a % chance ha a ie will ass ou of is condiion sae in one ime inerval. Fas deerioraion is defined as having a 2 % chance ha a ie will ass ou of is condiion sae in one ime inerval. A discoun rae of % is used. For he slow deerioraion seed i can be seen ha no inervenions were seleced afer he 3 h inervenion. This is because he imrovemen in he robabiliy of sysem failure is small enough o be negligible. Any addiional exendiure afer 2 inervenions is simly a wase of agency money. Table 4. OMS for examle nework Number of inervenions in OMS Secion Grou Inervenion ime Deerioraion seed Fas Medium Slow Cumulaive overall coss Secion Grou Inervenion ime Cumulaive overall coss Secion Grou Inervenion ime Cumulaive overall coss - - - 8993 - - - 8953 - - - 8664 4 47 4 2533 4 698 2 3 452 3 83 3 422 3 2 5 26 2 5 459 2 4 3 4 9 98 9 5 3 4 2 47 5 4 9 78 4 9 56 3 3 2 5 6 3 74 3 623 3 2 7 6 7 2 2 755 2 2 72 3 2 7 8 3 4 2 844 3 4 2 87 3 3 8 9 3 3 2 95 3 3 2 9 3 4 2 9 3 2 4 2 3 2 4 3 2 5 3 3 3 4 2 3 3 4 2 3 3 4 2 4 4 2 3 3 4 4 3 3 4 3 3 3 4 3 2 4 4 3 2 4 4 - - - 4 5 2 5 2 5 - - - 5 6 3 9 6 3 9 6 - - - 6 7 2 7 7 - - - 7 8 3 2 8 3 2 8 - - - 8 9 4 2 9 4 9 - - - 9 The cumulaive overall coss for he OMS s for he hree deerioraion seeds for nework are shown in Figure 4. The number of inervenions in he OMS ha will maximize cumulaive overall savings is indicaed wih a large circle. I can be seen ha for slow deerioraion a relacemen sraegy wih, 2 and 3 inervenions will resul in savings of 745, 824 and 8353 mu, resecively (Fig. 4a. If a fourh inervenion is done here will acually be a decrease in overall savings o 8257 mu. This is because he agency coss for he inervenion are higher han he ossible reducion in execed failure coss. For medium deerioraion a relacemen sraegy wih, 2 or 3 inervenions will resul in 642, 739 and 8494 mu, resecively (Fig. 4b. A relacemen sraegy wih 4 inervenions will also decrease overall savings (8443 mu. If he deerioraion seed is fas, he 6 inervenion OMS will resul in he larges overall savings, 8288 mu, and erforming he sevenh inervenion will resul in a decrease in overall savings (Fig. 4c. 7/9

The imorance of accurae esimaion of deerioraion seed lies in he abiliy o deermine he aroriae number of inervenions o have in he OMS. For examle, in nework, if here is medium deerioraion and i is wrongly esimaed o be fas, he agency coss will be % higher han necessary. The cumulaive agency coss for he OMSs for he slow, medium and fas deerioraion seeds are shown in Fig. 4d. Of course his deends on he C f, C a and he exac deerioraion marices used o describe he deerioraion. If here is slow deerioraion seed and i is esimaed o be a medium deerioraion seed he same inervenions will be recommended and he agency coss (and overall coss will be no differen. Cumulaive overall savings 2 8 6 4 2 8 6 4 2 5 5 2 Number of inervenions in OMS (a Cumulaive overall savings 2 8 6 4 2 8 6 4 2 5 5 2 Number of inervenions in OMS (b Cumulaive overall savings 2 8 6 4 2 8 6 4 2 5 5 2 Number of inervenions in OMS (c (d Figure 4. Cumulaive overall coss vs. number of inervenions erformed from oimal sequences for nework (a slow deerioraion, (b medium deerioraion, (c fas deerioraion, and (d Cumulaive agency coss for each inervenion Cumulaive agency coss 6 5 4 3 2 Slow Medium Fas Deerioraion seed CONCLUSIONS This examle shows ha: Risk-based oimal managemen sraegies can be deermined for redundan reinforced concree ie neworks using redicive models of he deerioraion and considering boh he consequences of failure and he redundancy of he nework 2 The incremenal inervenion se algorihm can only be used on neworks where i is no required o see he benefi of fuure inervenions. 3 Predicive deerioraion models can be used o deermine risk-based relacemen sraegies for redundan reinforced concree ie neworks aking ino consideraion he funcioning of he nework as a whole. 4 The use of risk-based relacemen sraegies can deermine when addiional agency sending is unnecessary. 5 Accurae deerioraion redicion can resul in subsanial savings in agency coss. Fuure work needs o concenrae on he accuracy of he deerioraion models used in redicing fuure deerioraing of underground reinforced concree ies. 8/9

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