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1 Provded by he auhor(s) and Unversy College Dubln Lbrary n accordance wh publsher polces, Please ce he publshed verson when avalable Tle Deernaon of brdge lfee dynac aplfcaon facor usng fne eleen analyss of crcal loadng scenaros Auhors(s) Gonále, Aruro; Ragan, Parac; O'Bren, Eugene J; Capran, Coln C Publcaon dae Publcaon nforaon Engneerng Srucures, 30 (9): Seres Crcal Infrasrucure Group Publsher Elsever Lnk o onlne verson hp://dxdoorg/006/engsruc Ie record/ore nforaon hp://hdlhandlene/097/35 Publsher's saeen All rghs reserved Publsher's verson (DOI) 006/engsruc Downloaded T0::Z The UCD couny has ade hs arcle openly avalable Please share how hs access benefs you Your sory aers! (@ucd_oa) Soe rghs reserved For ore nforaon, please see he e record lnk above

2 Deernaon of brdge lfee dynac aplfcaon facor usng fne eleen analyss of crcal loadng scenaros Aruro Gonále, Parac Ragan, Eugene J OBren and Coln Capran Absrac The developen of accurae codes for he desgn of brdges and he evaluaon of exsng srucures requres adequae assessen of heavy raffc loadng and also he dynac neracon ha ay occur as hs raffc raverses he srucure Curren approaches generally frs calculae characersc sac load effec and hen apply an aplfcaon facor o allow for dynacs Ths neglecs he sgnfcanly-reduced probably of boh hgh sac loadng and hgh dynac aplfcaon occurrng sulaneously Ths paper presens an assessen procedure whereby only crcal loadng evens are consdered o allow for an effcen and accurae deernaon of ndependen values for characersc (lfee-axu) sac and oal (ncludng dynac neracon) load effecs Inally he crcal sac loadng scenaros for a chosen brdge are deerned fro Mone Carlo sulaon usng wegh-n-oon daa The developen of a daabase of 3-densonal fne eleen brdge and ruck odels allows for he analyss of hese varous cobnaons of vehcular loadng paerns The denfed crcal loadng scenaros are odelled and analysed ndvdually o oban he crcal oal load effec I s hen possble o oban a correlaon beween crcal sac load effec and correspondng oal load effec and o exrapolae o fnd a se-specfc dynac aplfcaon facor

3 Inroducon Correc evaluaon of he behavour of hghway brdges under heavy loadng s exreely poran boh for he enhanceen of desgn echnques, and for he assessen of exsng nfrasrucure I s wdely acceped ha codes for he desgn of new brdges are hghly conservave n her allowance for dynacs whch s approprae gven he sall argnal cos of ncreasng capacy n os cases, bu such conservas s napproprae for he assessen of hghway brdges [,] In he case of shor/edu span brdges (0-30 ), he crcal raffc loadng even ypcally consss of wo heavy rucks eeng or passng on he brdge These crcal evens are coonly obaned usng Mone Carlo sulaon n ande wh easured Wegh n Moon (WIM) daa [3,4,5] Once he wors sac case s known, he fnal raffc load s coonly calculaed hrough he applcaon of a Dynac Aplfcaon Facor (DAF), ha accouns for he dynac coponen conaned n he brdge response [6,7,8] DAF s defned as he rao of axu oal load effec o axu sac load effec for a gven loadng even (cobnaon of rucks crossng) I s known ha dynac neracon s nfluenced by nuerous brdge- and vehcle-dependen dynac paraeers, such as vehcle velocy, road profle, suspenson and yre sffness [9,0,] The dynac aplfcaon facors prescrbed n desgn/assessen codes are soees based on dynac load ess of exsng brdges and end o be conservave There s consderably dscrepancy aong he values recoended by dfferen codes due o he coplexy of he vehcle brdge neracon proble They ypcally sugges a dynac aplfcaon whch s funcon only of a few general paraeers (e, brdge lengh or naural frequency, nuber of lanes and

4 load effec) ha gnore any of hese sgnfcan brdge and ruck dynac characerscs As par of he SAMARIS proec [], experens ook place on a edu-span brdge showng evdence ha he dynac aplfcaon decreases as he gross vehcle wegh ncreases and also ha he dynac load facors for wo ruck loadng evens are less han for one ruck loadng evens Coplex dynac brdge ruck neracon odels have been developed, usng fne eleen packages, o ad undersandng of he neracon ha can be expeced n vehcle crossng evens [,3-6] Ths paper uses a fne eleen analyss approach o assess he levels of dynac neracon occurrng for he sascally obaned crcal loadng scenaros of a bea-and-slab brdge A procedure s descrbed by whch a sespecfc allowance for dynacs can be found Ths dynac allowance can be exrapolaed o he 00-year brdge lfee usng ulvarae exree value analyss [4] Descrpon of vehcle brdge neracon fne eleen odels An elaborae vehcle brdge neracon odel s necessary o allow for 3- densonal aspecs, such as he ransverse effec of ulple vehcles, ha have a sgnfcan nfluence on he oal response and spler odels can no capure Vehcle brdge neracon odellng The auhors use MSc/ASTRA [7] and a Lagrange echnque o odel he load posed by a ruck crossng a brdge The Lagrange ulpler forulaon allows 3

5 for he represenaon of he copably condon a he brdge/vehcle nerface hrough a se of auxlary funcons Cfuenes [8] uses hs forulaon o solve for he oon of a sngle crcular ass ovng a consan speed on a one-densonal brdge odel Ths approach can be exended o allow for he presence of ulple asses ravellng n gven pahs a dfferen speeds [5] The brdge srucure along any ravel pah s dvded no (-) fne eleens, wh coordnaes x,x,x adoped for he nodes The varables defnng he behavour of he brdge are: (x,) : vercal deflecon of he brdge n poson x a e, = () : deflecon of node a e, θ =θ () : roaon of node a e, ż = () : acceleraon of vercal dsplaceen n node a e, θ = ) θ ( : acceleraon of roaon n node a e The varables defnng he behavour of a seres of ovng asses (one per wheel or one per axle n a 3D or D proble respecvely) are: v = v () : e-dependen velocy of he vehcle If here are dfferen : ass of wheel, vehcles, v gh be dfferen for each vehcle w = w ( ) = ( ζ, ) : vercal dsplaceen of wheel easured wh respec o he horonal axs, R = R () : Ineracon force a conac pon of wheel, ζ = ζ () : Dsance x ravelled on he brdge by ovng wheel a e 4

6 5 The varable ζ denong poson of ass on he brdge a e can be defned as a funcon of he velocy npu v () as: = = = v d v ) ( ) ( ζ 0 ζ L () where e s he nsan a whch ass eners he brdge and L s he brdge lengh For an undaped odel, he equaon of oon of he brdge fne eleen odel can be wren as: = M f M f k k k k k k k k,,,,,,,, θ θ θ θ () where [ ] x s he ass arx of he fne eleen odel and [k ] x s he sffness arx, represenng he dynac characerscs of he brdge odel { } x s a vecor conanng dsplaceens ( ) and roaon (θ ) of he nodes, and { ż } x her acceleraon a e In he rgh-hand sde of he equaon, vecor {f} x represens he forces f () and oens M () acng on each node a a e due o he ovng loads A copably condon beween he vercal dsplaceen w () of each ass and he brdge a he conac pon us be esablshed a any e For hs purpose, a se of auxlary funcons A () and B () are defned for every ass, and he copably condon a he conac pon of ass s forulaed as [8]:

7 w ( ) = A ( ) ( ) B ( ) θ ( ) = = ; =,, p (3) where () and θ () are he dsplaceen and roaon a each node, s he oal nuber of brdge nodes, p s oal nuber of ovng loads, and A () and B () are he auxlary funcons for load A () and B () can adop dfferen values n each node a each nsan The shape of hese auxlary funcons s shown n Fg The funcons have ero value ou of he nerval beween adacen nodes A () and B () are copleely defned once v (), approach lengh, axle spacngs and he coordnaes of he brdge nodes are known Each axle akes a dfferen e o reach he sae node and each wheel of he sae axle follows a dfferen pah on he brdge Thus, he auxlary funcons are dfferen for each ass, and for each e Accordng o Cfuenes [8], he neracon force, R (), beween a ovng crcular ass and he brdge srucure, s coposed of he neral force due o vercal oon of he ass ( ( x, ) ( ξ, ) oon beween he brdge and he load ( v ), he Corols force due o relave ( x, ) x ( ξ, ) ), he cenrpeal force due o crcular oon followng he defored shape of he brdge acng owards he ( x, ) cenre of he ass ( v x ), and he wegh due o gravy force ( g ) ( ξ, ) Hence he neracon force can be defned as: 6

8 7 ) ( } { x g v v R ξ δ = (4) where ), ( ), ( x ξ =, ), ( ), ( x x ξ =, ), ( ), ( x x ξ = and δ s he Drac funcon If 0, = R x ζ, where ζ s he dsance ravelled by he ass as defned n Eq () By cobnng Eqs (3) and (4), he neracon force, R, beween he brdge and he h ass can also be expressed as: ] [ v v g w R = (5) Usng he Lagrange ulpler funcons and re-orderng ers: = v v A g R w ] [ (6) where v = v ( ), ha s, velocy of ass when reaches node The roughness of he paveen surface r(x) can be pored no Equaon (6) by akng no accoun ha vercal dsplaceen of he ass wll be equal o he vercal deforaon of he bea nus he deph of he rregulares a he sae locaon Ths gves: = r v v v A g R w ] [ (7)

9 8 An nal defleced shape of he brdge can be nroduced n he sae way Therefore, he oal force f () and oen M () acng on a brdge node a e due o p dfferen asses can be expressed usng he auxlary funcons as: p p p p p p p R B A B A B A R B A B A B A M f M f M f = (8) Fnally, he equaons of oon of he coplee ovng load plus fne eleen odel are gven by: [ ] [ ] [ ] r r r r p p p p p p A R B R A R S B R w R g c A B w R g c A B θ θ θ θ = = = = = = = (9)

10 where k k k k , k,, k, S =, k,, k, p [ ] (0) The las (p) rows n Eq (9) represen he equaons of oon of each ovng ass and he copably condon beween deflecons of he ovng asses and he brdge In hese las rows of he global load vecor, he paraeer c s gven by: v v v v c = A A θ A θ = x x = x x = x x = x x A =,,p () When he dynac sulaon of a ruck ravellng over a brdge s carred ou n he followng secon, hs er c s gnored Also, he cenrpeal and Corols forces are no aken no accoun as os of he vehcle ass (excep wheel ass) s no under crcular oon and he vehcle speeds are relavely sall Brdge and ruck fne eleen odels 9

11 The brdge chosen for hs sudy s he Mura Rver Brdge n Slovena The brdge s 3 long and has wo lanes of b-dreconal raffc flow The brdge, of bea and slab consrucon, s sply suppored and fors par of a larger srucure Fve concree longudnal beas suppor a concree slab, wh a layer of asphal acng as he road surface Fve concree daphrag beas are also presen n he ransverse drecon The brdge has been prevously odelled, nsruened and valdaed [9] and s odelled usng bea and plae eleens as shown n Fg The vbraonal ode shapes of he odel are conssen wh experenal resuls (frs naural longudnal frequency of 35 H, frs orsonal frequency of 46 H, and dapng 3%) The fne eleen ruck odels are odelled usng rgd bodes suppored by suspenson and yre syses The body ass n he rucks s dsrbued unforly hroughou he frae n addon o ass eleens represenng he cabn and cargo asses rgdly conneced o he frae The 5-axle and 4-axle vehcle odels allow for arculaon beween he racor and raler The 3-axle and -axle vehcle odels are rgd boded The suspensons and yres are odelled as sandard sprng dashpo syses, akng ypcal sffness and dapng values fro leraure [4,5] Fg 3 shows wo ypcal odes of vbraon for he ASTRA odel of he 5-axle ruck rucks Body and axle hop frequences vary wh laden wegh bu hey generally fall n he ranges [5-45 H] and [8-6 H] respecvely A sooh road profle (Class A accordng o he ISO sandard [0]) s used o exce dynacs, and a secon of hs profle s presened n Fg 4 3 Calculaon of crcal sac loadng evens 0

12 3 Sulaon se-up In order o conduc Mone Carlo sulaon of a grea nuber of sac load cases for he brdge s necessary oban fro he ASTRA brdge odel he sac response o an axle of un wegh a any pon on he ravel pah The load effec of neres s aken o be dspan sress Therefore, he nfluence lnes of dspan sress, for each of he 5 longudnal beas, for boh of he lane loadng possbles are obaned fro he FE odel I s noed ha he longudnal beas are no syercal abou he brdge cenre lne, and as a resul he nfluence lnes for boh allowable lane loadngs us be obaned Connenal European drvng laws are applcable (drvng on rgh) Consequenly he wheel pahs for rucks n eher lane are a 840 and 660 fro brdge cenre-lne for drver and passenger sdes respecvely A scheac layou of he brdge s shown n Fg 5 The nfluence lnes for each lane loadng are obaned by placng a sac load of 5 k on each of he wheel pahs for he respecve lane Ths s equvalen o a sac axle load of 0 k beng appled by a parcular ruck axle The loads are oved a ncreens n he drecon of raffc flow o oban he nfluence lnes of bea sress a dspan as shown n Fgs 6(a) and (b) for load drecons D and D respecvely The obaned nfluence lnes are hen noralsed o gve he response due o a un axle load Pece-wse polynoal equaons are used o algebracally odel he nfluence lnes obaned These can be used o rapdly calculae brdge sac response due o rando raffc flow Eq () gves he superposon equaon ε M ( ) = W I( x ( )) () sac = =

13 where ε sac () s he sac load effec a e, M s he nuber of rucks, s he nuber of axles n ruck, x () s he poson on he brdge of axle n ruck a nsan, I(x ) s he nfluence lne ordnae a he specfed locaon x a nsan, and W s he sac wegh of axle n ruck 3 Sascal calculaon of crcal loadng evens One week of WIM daa was aken fro he A6 oorway near Auxerre, France The se has 4 lanes of raffc ( n each drecon) bu only he raffc recorded n he slow lanes was used and s acknowledged ha hs resuls n conservave loadng In oal 7756 and 867 rucks were easured n he norh and souh slow lanes respecvely, wh an average daly ruck flow of 6744 rucks [4] Sascal dsrbuons were f o he raffc characerscs of he se for each lane The load effec chosen for analyss s he axu dspan sress n longudnal bea (Fg 5) Mone Carlo sulaon s appled, usng he fed dsrbuons o generae 0 years of b-dreconal, free-flowng raffc daa Ths raffc s passed over he nfluence lne for bea o deerne he sac load effecs ha resul In he case of edu span brdges (< 40 ), he crcal raffc even ypcally consss of wo heavy rucks crossng he brdge a he sae e However, all possble confguraons of vehcles and vehcle eeng evens were consdered, and soe of he wors cases conssed of 3-ruck even crossngs nvolvng rucks n he sae lane over bea Each year of sulaed raffc consss of 0 represenave perods, loosely ered onhs, of 5 workng days each (allowng for 5 weekends and publc

14 holdays per annu) The evens correspondng o onhly-axu sac load effec are reaned, for each onh n each of he 0 years of flow Ths s done o ne he nuber of evens ha are o be dynacally analysed, as well as provdng a shorer exrapolaon dsance Thus here are 00 crcal evens correspondng o he 00 onhly axa, reaned for dynac analyss The daa reaned for a ypcal crcal even s presened n Table (hs even consss of he eeng of a 5-axle vehcle n lane D wh a 4-axle vehcle n Lane D) Of he 00 onhly-axu evens, 0 are found o be one-ruck evens, 77 o be wo-ruck evens and 3 are hree-ruck evens The nfluence surface for bea s asyercal (Fg 6); herefore rucks n lane D donae, reducng he effec of rucks n lane D Hence he onhly-axu evens are derved fro he occurrence of heavy rucks n lane D, and rucks wh less exree gross wegh n lane D, as can be seen n Table 33 Dynac analyss of wors onhly loadng cases Each of he 00 cases s ndvdually odelled and sulaed usng MSc/ASTRA and he FE odels descrbed n Secon An enry s generaed n he assebled sffness arx of he vehcle-brdge ASTRA fne eleen syse as shown n Eq (9), and he neracon forces F a he conac pon of each wheel on he brdge are defned as n Eq (8) hrough a bulk daa fle Fro he generaed responses s possble o oban he axu oal load effec n bea, for each ndvdual loadng even The lengh of he approach s odfed o ensure ha vehcles ee a he prescrbed locaon n he onhly-axu sac load case A nu approach 3

15 lengh of 00 s specfed o ensure he dynac odels have acheved a suable level of sably before crossng he brdge Fgs 7(a) and (b) llusrae soe exaples of he onhly-axu evens and he correspondng oal sress a dspan; he prevalence of heavy rucks n lane D (op lane) s agan evden 4 Deernaon of se-specfc dynac aplfcaon facor The sac and oal sress resuls fro he FE sulaons of he 00 evens are presened n Fg 8, whch llusraes he scaer n he resuls A bes f lne hrough he daa corresponds o a rao of oal o sac of abou 05 Ths bes f s approxaely parallel o he bes f lne hrough he sac axa In Fg 9, he resuls are ranked by onhly-axu sac sress The sac response s approxaely lnear whle he oal response s ore varable Brdge DAF s also ploed n he fgure The Eurocode workng group recoends a DAF value of 7 for a brdge lengh of 3 and lanes [6], bu even hough s apparen ha DAF values of agnude o 5 ay be obaned for he chosen brdge and lgh rucks [6,9], hese hgh order DAF values are no evden n he analyss of he brdge s crcal loadng evens For exaple, whn hese 00 crcal loadng evens, he ean DAF s 035 wh a sandard devaon of 004 The rao of he axu oal response o he axu sac response over hs 0-year saple perod s 06, whch n hs case corresponds o he sae raffc even Therefore, he 00h wors sac load effec s % less han he load effec due o he heaves sacally weghed ruck Ths eans ha a sac loadng case below he 00h would requre a DAF well n excess of 3 (=*06) o cause hgher sran han he derved axu sress 4

16 5 Conclusons In hs paper, he curren eans of allowng for dynac neracon of brdge and ruck(s) have been revewed for a parcular brdge specfc and specfc raffc condons Mone Carlo sulaon of sac load effec has been used o oban onhly axu loadng evens, whch are hen odelled and analysed dynacally usng ASTRA o oban he oal load effec I has been shown ha by ndvdually assessng he dynac response due o crcal loadng a ore brdge specfc value of DAF can be obaned The procedure has been deonsraed for an exsng bea-andslab brdge wh a very good road profle subec o raffc fro a ypcal European roue Sulaon resuls have shown ha he dynac aplfcaon facor ay resul lower han he value proposed by desgn codes and ha accurae dynac fne eleen odellng can lead o sgnfcan savngs n srucural assessen Acknowledgeens The auhors wsh o acknowledge he connued fnancal asssance provded by he Irsh Research Councl for Scence, Engneerng and Technology (Ebark Inave) References 5

17 [] Gonale A, O Connor AJ, OBren EJ An assessen of he nfluence of dynac neracon odelng on predced characersc load effecs n brdges Proceedngs of he 3 rd Inernaonal Conference on Curren and Fuure Trends n Brdge Desgn, Consrucon and Manenance, Shangha, Chna; 003, 4-49 [] DIVIE Prograe, OECD Dynac neracon of heavy vehcles wh roads and brdges DIVIE Concludng Conference, Oawa, Canada, 997 [3] O Connor AJ, OBren EJ Traffc load odelng and facors nfluencng he accuracy of predced exrees Canadan Journal of Cvl Engneerng 005;3:70-78 [4] Capran C Probablsc analyss of hghway brdge raffc loadng PhD Thess, UCD School of Archecure, Landscape and Cvl Engneerng, Unversy College Dubln, Dubln, Ireland, 005 [5] owak AS, Hong YK Brdge lve load odels ASCE Journal of Srucural Engneerng 99;7(9): [6] Dawe P Traffc loadng on hghway brdges Thoas Telford, London, 003 [7] Chaeree S The desgn of odern seel brdges Oxford BSP Professonal Books, 99 [8] Zhang Q-L, Vrouwenvelder A, Wardener J Dynac aplfcaon facors and EUDL of brdges under rando raffc flows Journal of Engneerng Srucures 000; 3: [9] Green MF, Cebon D Dynac neracon beween heavy vehcles and hghway brdges Copuers and Srucures 995; 6():

18 [0] Brady SP, OBren EJ The effec of vehcle velocy on he dynac aplfcaon of wo vehcles crossng a sply suppored brdge ASCE Journal of Brdge Engneerng 006; () :50-56 [] OBren EJ, L Y, Gonále A Brdge roughness ndex as an ndcaor of brdge dynac aplfcaon Copuers and Srucures 006; 84: [] SAMARIS Prograe Gudance for he opal assessen of hghway srucures Susanable and Advanced Maerals for Road Infrasrucure, Delverable SAM-GE-D30, EU 6 h fraework, 006 [3] Kwasnewsk L L H, Wekeer J, Malachowsk J Fne eleen analyss of vehcle-brdge neracon Fne Eleens n Analyss and Desgn 006; 4(): [4] Krkegaard PH, elson SRK, Enevoldson I Heavy vehcles on nor hghway brdges dynac odelng of vehcles and brdges, Deparen of Buldng Technology and Srucural Engneerng, Aalborg Unversy, ISS R97, 997 [5] Baugärner W Brdge-vehcle neracon usng exended FE analyss Heavy Vehcle Syses, In Journal of Vehcle Desgn 999; 6(-4): [6] Ragan P, Gonále A, OBren E, Brady SP Transverse varaon of dynac effecs on bea-and-slab edu span brdges Proceedngs of he 6 h Inernaonal Conference on Srucural Dynacs, EURODY 005, CSoe & GI Schuëller (eds), Pars, France, 4-7 Sepeber; 005, [7] MSC/ASTRA for Wndows Advanced dynac analyss user s gude verson 70 The Maceal-Schwendler Corporaon, USA, 997 7

19 [8] Cfuenes AO Dynac response of a bea exced by a ovng ass Fne Eleens n Analyss and Desgn 5, Elsever Scence Publshers BV, Aserda, The eherlands; 989, [9] Brady SP, OBren EJ, Žndarč A The effec of vehcle velocy on he dynac aplfcaon of a vehcle crossng a sply suppored brdge ASCE Journal of Brdge Engneerng 006; ():4-49 [0] ISO 8608:995 Mechancal vbraon-road surface profles-reporng of easure daa, 995 8

20 Table Typcal wo-ruck eeng even daa fle Lane Axle Weghs (kg/00) Axle Spacngs () Velocy (/s) W W W3 W4 W5 AS AS AS3 AS4 Approach () D D

21 Fgure Auxlary funcons 0

22 Fgure Brdge odel

23 Fgure 3 Truck odel

24 Fgure 4 Road surface profle 3

25 Fgure 5 Scheac of brdge showng bea layou 4

26 Fgure 6(a) Mdspan nfluence lnes due o load ovng n lane D Fgure 6(b) Mdspan nfluence lnes due o load ovng n lane D 5

27 Fgure 7(a) Exaple of one-ruck crossng onhly axu even Fgure 7(b) Exaple of wo-ruck crossng onhly axu even 6

28 Fgure 8 Correlaon beween axu sac and oal sress 7

29 Fgure 9 Rankng of evens by axu sac response 8