Analytical Approach to Predicting Temperature Fields in Multi-Layered Pavement Systems

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1 Analycal Appoach o Pdcng Tmpau lds n ul-layd Pavmn Sysms ong Wang, Jffy R. Rosl, P.E., mb ASCE and a-zh Guo 3 Absac: An accua and apd smaon of h pavmn mpau fld s dsd o b pdc pavmn sponss and fo pavmn sysm dsgn. In hs pap, an nnovav mhod o dv h hocal soluon of an axsymmc mpau fld n a mul-layd pavmn sysm s psnd. Th mul-layd pavmn sysm was modld as a wodmnsonal ha ansf poblm. Th mpau a any locaon, and any m n an N- lay pavmn sysm can b calculad by usng h dvd analycal soluon. Hankl ngal ansfom wh spc o h adal coodna s uld n h dvaon of h soluon. Th npolaoy gonomc polynomals basd on dsc ou ansfom a usd o f h masud a mpaus and sola adaon nnss dung a day, whch a ssnal componns n h bounday condon fo h undlyng ha ansf poblm. A ORTRAN pogam was codd o mplmn hs analycal soluon. asud fld mpau suls fom a gd pavmn sysm dmonsa ha h dvd analycal soluon gnas asonabl mpau pofls n h conc slab. CE aabas subc hadngs: Tmpau dsbuon; Layd sysms; lxbl pavmns; Rgd pavmns; Ha ansf; Tmpau ffcs. Gadua Rsach Asssan Cospondng auho, p. of Cvl and Envonmnal Engnng, Unvsy of Illnos a Ubana-Champagn. B3 Nwmak Cvl Engnng Laboaoy, C-5, 5 Noh ahws Avnu, Ubana, IL 68. Tl: E-mal: dongwang@uuc.du Assoca Pofsso, p. of Cvl and Envonmnal Engnng, Unvsy of Illnos a Ubana-Champagn. Nwmak Cvl Engnng Laboaoy, C-5, 5 Noh ahws Avnu, Ubana, IL 68. Tl: E-mal: osl@uuc.du 3 Emus Pofsso, School of Tanspoaon Scnc and Engnng, Habn Insu of Tchnology. No., Hah Avnu, Nangang sc, Habn, Chna, 59. Tl:

2 Wang, Rosl and Guo INTROUCTION I s wll known ha mpau vaaon n pavmn lays play an mpoan ol n h pfomanc of boh flxbl and gd pavmn sysms. In flxbl pavmn sysms, h sufac lay s usually mad of ho-mx asphal HA, whch s a vsco-lasc maal and s bhavo s hghly lad o s mpau,.., HA sponds lk an lasc sold und low mpau and san condons; on h oh hand, also acs as a vscous maal a hgh mpau n h sns ha h dfomaon du o affc loadng canno fully covd whn a fn m pod und h unloadng condon. Thfo, an accua pdcon of h mpau pofl n h HA lay s dsd whn slcng h asphal bnd and pdcng pfomanc. o gd pavmn dsgn, h hmal culng sss n h conc slab canno b gnod 3 and by som mann mus b addd o h affc loadng ssss 4,5. In od o accualy capu h ccal hmal ssss n h PCC slab, h mpau pofl houghou h day mus b known. any sach ffos hav bn akn on dvlopng dffn mahmacal modls o pdc mpau pofl whn a pavmn sysm. os of publshd suls on hs opc can b fd no sascs-basd modls o ha ansf modls. Sascs-Basd Pavmn Tmpau Pdcon odls Th sascs-basd gsson fomula a usually dvlopd basd on lag daabass of clmac, moologcal and gogaphcal facos, such as a mpau, wnd spd, sola adaon and laud c, as wll as h masud fld pavmn mpaus. Rumny and Jmn 7 appoxmad mpau a h sufac and a a -nch dph basd on a mpau and houly sola adaon. Lukann al. 8 pdcd h svn-day avag hgh pavmn mpau usng svn-day avag hgh a mpau. o cnly, fndf al. 9 calculad h maxmum and mnmum mpau a any dph by usng a mpau, daly sola adaon and dph whn h pavmn. Empcal fomula a usually appld o apdly pdc can xm mpaus whn a pavmn sysm o a spcfc mpau a a gvn pavmn dph. Howv, h dsadvanag of hs yps of fomula s ha hy gv asonabl pdcon fo h npu daa ncludd whn h ognal sampl daabas, bu do no guaan h accuacy of pdcon fo h npu daa ousd h ognal sampl daabas. Ha Tansf odls Th xsng ha ansf modls ha pdc pavmn mpau pofl a usually solvd usng a numcal mhod, whch ypcally conss of fou sps. sly, h govnng quaon o accoun fo h ha conducon whn a pavmn mus b s up, whch s usually a ondmnsonal - o wo-dmnsonal - ha ansf modl psnd by a mdpndn paal dffnal quaon PE. Scondly, an appopa bounday condon mus b sablshd lnkng h clmac paams wh h pavmn sufac mpau. Ths lnk s accomplshd by analyng h ngy balanc a h pavmn sufac. Thdly, h spaal doman nds o b dscd usng a numcal mhod, such as fn-dffnc mhod,

3 Wang, Rosl and Guo 3 fn-lmn mhod, c., whch suls n a lag sysm of odnay dffnal quaons OEs n m. ouhly, an appopa m-ngao s qud o solv hs OEs. o xampl, hs m-ngao can b h a lna mul-sp mhod o a Rung-ua mhod. mpsy and Thompson w on of h fs sachs o dvlop a numcal smulaon appoach by usng - ha ansf modl and an xplc fn-dffnc mhod. Hsh al. poposd a h-dmnsonal numcal modl o calcula h mpau dsbuon whn conc pavmn. Rcnly, Rasmussn al. and Schndl al. 3 poposd modls o pdc h mpau dsbuon n h aly-ag PCC pavmn by ncopoang boh h clmac facos and h ha of hydaon of cmnous maals no h modls usng a fn lmn o a - fn-dffnc mhod, spcvly. Yavuuk al. 4 smulad mpau flucuaons n asphal pavmns du o hmal nvonmnal condons by usng a - fn-dffnc mhod. Analycal Appoachs As fa as analycal soluon of mpau pofls hough a mul-lay pavmn sysm s concnd, vy fw suls a avalabl du o h complxy ncound n dvng h closd-fom analycal soluon. Bab 6 calculad h maxmum pavmn mpau fom wah pos fo a -lay sysm. Solamanan and nndy 5 poposd a smpl analycal quaon o pdc h maxmum pavmn sufac mpau basd on maxmum a mpau and houly sola adaon. Lang and Nu 6 dvd a closd-fom analycal soluon of mpau dsbuon n a 3-lay sysm usng a smplfd bounday condon, whch only nvolvd h convcon of ha bwn h amosph and pavmn sufac bu no any sola adaon ffc. Th man hudl assocad wh h numcal mhods fo pdcng pavmn mpau fld s ha h nal pavmn mpau dsbuon calld nal condon mus b avalabl n od o calcula mpau fld fo h la m, snc a m-dpndn PE poblm ssnally nds o b solvd. Howv, h nal pavmn mpau pofl s ypcally no avalabl. To mov hs hudl, a - axsymmc appoach fo analycally pdcng h mpau fld n an N-layd pavmn sysm s poposd n hs pap whn h nal pavmn mpau pofl s no known, whch xnds h analycal soluon fo a 3-lay pavmn sysm by Lang and Nu 6. oov, h poposd modl handls masud sola adaon and consds adaon. Th poposd mpau soluon s vald fo pavmn sysms wh N-lays, a any dph of pavmn,, adal dcon, and m, dung h day usng masud clmac daa and pavmn maal pops. Ths modl can pdc mpau pofls n boh mul-layd flxbl and gd pavmn sysms assumng h hmal pops of h maals a known. ATHEATICAL TEPERATURE OEL Th mpau dsbuon n a mul-layd pavmn sysm, shown n g., can b modld as a ha ansf poblm, wh h lay hcknss m, λ hmal conducvy cal/m-h-ºc, α hmal dffusvy m /h, and,, mpau ºC fo lay. Th T

4 Wang, Rosl and Guo 4 hcknss of h las lay subgad, hn, s assumd o b nfn along h posv dcon. All maals n h mul-layd sysms a assumd o b connuous, homognous, and soopc. Th mpau pofl n h lay, T,, s assumd o b axsymmc. On advanag of hs assumpon s ha h hmal ssss du o mpau chang can b asly ncopoad wh h affc loadng ssss by usng h layd lasc hoy, snc h la s also consdd o b axsymmc 7. Lay h, λ, α, T,, Lay h, λ, α, T,, Lay n,,,, Z λ n α n gu: ul-layd Pavmn Sysm In cylndcal coodna sysm, h - axsymmc ha ansf poblm can b modld wh h followng govnng m-dpndn PE: T n T α T fo H H wh H h k ; k axsymmc poblm., Laplac opao n cylndcal coodna fo h I s assumd ha h mpau and ha flow a connuous along h nfac of wo conscuv lays,.. T, H, T, H, a

5 Wang, Rosl and Guo 5 T T λ, H, λ, H, b Th boundd mpau a an nfn dph s gvn by T n,, as 3 wh s a consan. Th bounday condons BCs play an mpoan ol n compung h mpau pofl n mul-layd pavmn sysms. Th mpau a h pavmn sufac s galy nfluncd by moologcal and gogaphcal facos, such as a mpau, sola adaon, wnd spd, laud and lvaon of h pavmn, 5, 8. Th BCs can b s up by analyng h ngy balanc a h pavmn sufac. ahmacally, s xpssd as 9 q P R 4 wh q ha flux no pavmn cal/m -h; P ha flux causd by h convcon du o mpau dffnc bwn amosph and pavmn sufac cal/m -h; R n sola adaon flux cal/m -h. By ou s law of conducon, ha flow no pavmn, q, s pond o h dcon of mpau gadn T and calculad as follows T q λ,, 5 Th convcon ngy, P, s calculad as P B Ta T,, 6 Wh T a a mpau C; T,, pavmn sufac mpau C; B pavmn sufac convcon coffcn cal/m -h- C, whch pmaly dpnds on h sufac oughnss, wnd spd, a mpau and pavmn sufac mpau. Banco al. appoxmad as: B v Wh v wnd spd m/s; and.8598 s h un convson consan whn h un s convd fom W/m - C o cal/m -h- C. Th n adaon ngy, R, s gvn by

6 Wang, Rosl and Guo 6 R asq 7 Wh a s sufac absopvy o h oal sola adaon dmnsonlss; Q sola adaon flux cal/m h; adaon flux md by pavmn sufac cal/m h. Subsung quaons 5-7 no quaon 4 ylds h followng BC T λ,, asq B Ta T,, 8 Eqs. -3 and 8 consu h mahmacal modlng of mpau fld n mul-layd pavmn sysms. In od o analycally dv h soluon of mpau fld, connuous funcons fo psnng Q, and T a a dsd. In hs pap, Q and T a a assumd o b masud a a half-hou nval dung a day. Inpolaoy gonomc polynomals, basd on dsc ou ansfom, can hn b usd o f h daly masud daa of Q and T a as dscbd blow. Sola Radaon, Q m a a m mπ kπ kπ Q cos a cos sn k bk k wh 9 a b k k m k π q cos m m m m q m kπ sn m m fo ach k,, Lm fo ach k,, Lm a b wh m 4 f 48 qually spacd sampl pons a usd; a : a.m. and s h masud sola adaon nnsy valu a m.5* fo,, L m. Ambn Tmpau, T a q Th appoxmaon of ambn mpau, T a can b analogously ad by c Ta cm mπ cos m k c k kπ cos d k kπ sn Wh

7 Wang, Rosl and Guo 7 c d k k m k π T cos m m m m T m kπ sn m m fo ach k,, Lm fo ach k,, Lm a b Wh m 4 and T s h masud a mpau a m as dfnd abov. Iadaon Engy md by pavmn sufac A fouh od quaon o accoun fo h adaon md by pavmn sufac was usd n fnc, whch was gvn by 4 4 ρ p NW σ[ εt,, Ta G J ] 3 wh N cloud-bas faco; W pcnag of cloud cov a ngh; σ Sfan-Bolmann consan; ε mssvy of adaon by pavmn sufac; T,, Rankn mpau of pavmn sufac; Ta Rankn a mpau; G, J, ρ and p a modl paams. Bab 6 ook adaon no consdaon by dscounng h daly sola adaon nnsy. o smplcy, h nonlnay n h BC mposd by s avodd n hs pap by modfyng h convcon coffcn, B and pavmn sufac absopvy a s 9. Tmpau Vaaon Along h Radal con nally, o accoun fo h mpau vaaon along h adal dcon, a non-dmnsonal calbaon m f s noducd n h BC as follows μ f 4 wh μ can b fd by usng h masud pavmn mpau n h adal dcon. In hs pap, μ. o. /m s assumd whn h mpau dffnc bwn h cn of pavmn coss scon and should s 5 o C, spcvly 9. Also h un of s n ms n q. 4. Thus, h BCs may b modfd as T asq λ,, B f Ta T,, 5 B Subsung qs. 9,, and 4 no q. 5 and akng modfyng B and yld h followng BCs a s no consdaon by

8 Wang, Rosl and Guo 8 T λ,, B{ m k μ [ c π wh. c as a [ B k cm as am B π sn m as π as ak sn k d k bk sn k ]] T,, } B B 6 THEORETICAL SOLUTION O PAVEENT TEPERATURE PROILE Th snusodal ms n h gh hand sd of q. 6 only dff n magnuds, fquncs, and phas angls. In od o facla psnaon of h analycal soluon of,,, h followng sampl BC s noducd fom whch h soluon of T T,, wll b dvlopd: T μ λ,, B{ Asn ω T,, } 7 wh A consan C, psnng h magnud of ach sn funcon n q. 6. I s cla ha h fnal soluon of T,, basd on h lna BC n q. 6 can b oband by usng h pncpl of supposon,.., quals h summaon of mpau valus basd on a asc ach snusodal m n q. 6 plus h consan ms,.. and n q. 6. uhmo, B s obsvd ha sn ω on h gh hand sd of q. 7 can b lad o h complx valu by usng h Eul fomula: ω ω cos ω sn ω 8 wh s h magnay un numb wh. Snc lang sn ω wh ω can galy facla h dvaon of h analycal soluon, h mpau dsbuon s fs dvd n h complx plan, Y,,, hn h dsd soluon of mpau pofl fo h - ha ansf poblm, T,, s smply h magnay pa of Y,,. To asly psn h dvaon, h govnng quaon and h consan condons fo hs - ha ansf poblm a psnd n ms of h complx vaabls, Y,, as follows: Two-mnsonal Ha Tansf Equaon Y α Y fo H 9 H

9 Wang, Rosl and Guo 9 wh H h k k Inlay Conac Condon Y, H, Y, H, a Y Y λ, H, λ, H, b Boundd Tmpau Valu a Infn ph n,, Y as wh consan. Bounday Condon Y μ ω λ,, B{ A Y,, } Th followng oulns h man sps nvolvd n dvng h analycal soluon of,, : By usng h appoach of spaaon of vaabls, Y, Y w, u, Y, can b xpssd as, 3 I follows ha Y ωy 4 Subsung of qs. 3, 4 no qn. 9 ylds ω u, α u, 5 3 Hankl ngal ansfom s mployd o solv q. 5. L φ b h Hankl ansfom of od o of an abay funcon φ and hn fom fnc 3 φ φ J d 6

10 Wang, Rosl and Guo Wh J h fs knd of Bssl funcon of od o. Th nvs Hankl ansfom of od o of φ s φ φ d J 7 Also, h Hankl ansfom of od o of d d d d φ φ s gvn by fomula n fnc 3 φ φ d J d d d d 8 4 Applyng Hankl ansfom of od o o h boh sds of q. 5 wh spc o n conuncon wh q. 8 ylds,, u u α ω 9 5 Solvng q. 9 n h Hankl doman gvs, α ω α ω C u 3 wh a consans fo lay, whch a dmnd by usng h consan condons. C, 6 Takng nvs Hankl ansfom of od o of q. 3 and consdng q. 3 yld h mpau of lay n h complx plan, Y,, as follows: ω d J C Y N N ] [,, 3 Wh and a dfnd n h Appndx. N 7 mnng coffcns of and : C a Th laonshp bwn and can b xplod n h followng maxvco fom by usng q. 3 and h nlay conac condons sad n q. :, C C,

11 Wang, Rosl and Guo H H H H C P P P P C 3 Wh a dfnd n h Appndx.,,, P P P P b Th cusv fomula lnkng and can b fuh dducd fom q. 3 as follows: C, C, C R R R R C H H H H 33 wh ; a dfnd n h Appndx, and q. 33 can b asly povd by usng h mhod of mahmacal nducon. Ln,3,,, l k R kl c Boundd soluon fo s Y n,, a ndcas ha n fom q. 3, and h laonshp bwn and can b fuh dvd by sng n q. 33 as follows: C n C R R n n H 34 d and can b oband by usng q. 34 n conuncon wh h B.C. n q.. uhmo, and fo h lay can b solvd by usng q. 33. C C h Onc and a dmnd, h dsd soluon C T,, fo h lay s smply h magnay pa of n q. 3. Th xpsson fo h Y,, T,, s gvn n q. 35 wh all h symbols dfnd n h Appndx. Δ ] [ sn [,, H N T δ ω 35 δ ω d J N H ] sn ] [ Δ Numcal Implmnaon of Analycal Soluon As shown n qn. 35, an ngal wh spc o angng fom o mus b solvd n od o calcula h mpau dsbuon T,, fo h lay, and hs can b solvd numcally. A compu canno handl an h upp lm of ngaon, hus, an appopa uncad upp lm,, has o b chosn, whch can b dmnd by a convgnc s of. On such convgnc s xampl s shown n Tabl n h nx scon. In hs x u T,,

12 Wang, Rosl and Guo pap, h compos 6-pon Gaussan Quadau fomula s mployd o solv h ngal n q. 35,.., nval [, x u ] s fsly dvdd no som small subnvals, hn h 6-pon Gaussan Quadau fomula s appld n ach subnval o oban h soluon fo T,,. OEL VERIICATION WITH IEL ATA A ORTRAN compu pogam was dvlopd o pdc h mpau pofl n a mullayd pavmn sysm by usng h abov analycal soluon of h mpau fld. o h modl valdaon, h compud mpau pofl n a connuously nfocd conc pavmn CRCP s scon s compad wh masud fld daa fom h Advancd Tanspoaon Rsach and Engnng Laboaoy ATREL n Illnos, USA 4. Th CRCP s scons conss of conc slab.54m wh connuous nfocng sl, asphal conc bas.m and aggga subbas.5m all suppod on a sly-clay sol lay. Tmpaus a fv dffn slab dph locaons,..,.54m,.76m,.7m,.778m and.86m fom pavmn sufac, along wh sval clmac paams, such as wnd spd, wnd dcon and sola adaon w masud a a half-hou nval. In hs pap, mpaus n CRCP s scon a h afomnond fv slab dph locaons w connuously pdcd usng h abov analycal soluon fo 7.5 hous a a half-hou nval n boh wn and summ condons,.. sang fom : a.m. on Januay, 3 unl :3 p.m. on Januay 4, 3 and fom : a.m. on Jun 8, 3 unl :3 p.m. on Jun 3, 3. Half-hou masud a mpau and sola adaon fo ach h-day pod w mployd o gna h fng npolaoy gonomc polynomals as shown n gus. Th oh npu paams ncludng h ypcal hmal pops fo hs pavmn maals 5, a lsd n Tabl. To dmn h appopa upp lm, x u, of h ngal n q. 35, numcal convgnc ss fo h nvs Hankl ngal ansfom s cad ou usng h abov npu paams. Th numcal mplmnaon ndcad ha h mpop ngal n q. 35 usually convgd fas as valu of ncasd, hus h dmnsonlss quany x u s slcd as H n xu n 3I 36 3 wh n an ng funcon convng s agumn no an lags ng lss han o qual o slf; I s numb; sum of hcknss of pavmn lays xcp subgad lay m. H n Tabl shows dffn upp ngal lms xu whn I n q. 36 fo fv slab dph locaons. Tabl 3 llusas h convgnc of h nvs Hankl ngal ansfom n pdcng mpaus a hs fv dffn slab dphs a 4: a.m. on Jun 8, 3. Basd on h convgnc s, h mnmum valu fo h upp ngal lm x u n Tabl 3 Ts No. sll gav 3 sgnfcan fgu accuacy n h pdcd mpau a all dphs. Tabl : Inpu paams n modl vfcaon

13 Wang, Rosl and Guo 3 Paams Valu Thmal conducvy cal/h-m- C PCC slab.85 Asphal conc bas.38 Aggga subbas.58 Subgad. Thmal dffusvy m /h PCC slab.35 Asphal conc bas. Aggga subbas.3 Subgad.3 Calbad Absopvy, a.5 s Paam o accoun fo mpau vaaon. along h adal dcon /m, μ Radal coodna m,. odfd wnd spd m/s, v 3.5 o 5. Tabl : Valus of upp ngal lm, x whn I fo dffn dphs u m x u dmnsonlss Tabl 3: Numcal convgnc s suls fo h nvs Hankl ngal ansfom pdcd mpaus a fv dffn slab dphs, C Ts No. Slab ph Z m I

14 Wang, Rosl and Guo 4 a b c d gu. asud and fd a mpau a,c and sola adaon nnsy b,d fo h days n Januay and Jun 3, spcvly. Th pdcd and masud pavmn mpaus a plod n gus 3 and 4 fo Januay and Jun 3, spcvly, a.54 m,.76 m,.7 m, and.86 m. 4

15 Wang, Rosl and Guo 5 a b gu 3. Pdcd T p and masud T m mpau fo.54 m,.76 m a and.7 m,.86 m b n Januay 3 a gu 4. Pdcd T p and masud T m mpau fo.54 m,.76 m a and.7 m,.86 m b n Jun 3 om gus 3 and 4, s cla ha h lags mpau flucuaon wh m occus na h pavmn sufac.54 m among h plod fou mpau dphs. Th fah fom h pavmn sufac, h mpau vaaon wh m s lss. Th hghs pavmn mpau of 4 C occud a.54 m dung h hghs masud a mpau of 7.5 C on Jun 8, 3. I s obsvd ha h dvd hocal soluon pdcs asonably good pavmn mpau pofl compad o h masud daa. Th maxmum o bwn h pdcd and masud mpau s aound 3 C fo hs wo, h-day sng suls xcp fo on pacula cas,.. n pdcng h mpau a.54 m fom h pavmn sufac a 8: a.m. on Jun 8, 3, wh h o bwn h pdcd and masud mpau s aound 5 C. Tabl 4 psns h man os and sandad dvaons bwn h pdcd T p and masud T m mpau fo ach of fv dffn slab dph locaons. Th man o s h gas na h boom of h slab bu h sandad dvaon s h lags na h op of h slab wh mpau flucuaons a h gas. Th mpau dscpancy bwn h pdcd and masud valus com fom many facos, such as h os nvolvd n slcng h appopa maal hmal paams,.g. hmal conducvy and hmal dffusvy; os nvolvd n h connuous nfacal ha flux assumpons, snc dffn lvls of ha flow ssanc may xs n h nfac of wo conscuv pavmn lays; dp sol b 5

16 Wang, Rosl and Guo 6 mpau ffcs; adaon occung a ngh and mpau masumn o. Th adaon a ngh and h ffc of h dp sol mpau a lkly h mao asons fo h mpau dscpancy bwn h pdcd and masud valus. Th adaon n hs pap s consdd only by adusng h absopvy of h conc and convcon coffcn. uhmo, h dp sol mpau canno b cunly consdd wh h poposd analycal appoach. Tabl 4: an os bwn pdcd and masud mpau a dffn dphs C,.. T p T m valus n back dno sandad dvaons of mpau o a Slab ph Locaon m Jan. o Jan. 4, Jun 8 o Jun 3, Commns on h Poposd Analycal Soluon Th poposd analycal soluon can apdly, y asonably, pdc h mpau fld n h N-layd pavmn sysm wh lmd npu daa, whch s spcally mpoan fo chaacng fld mpau pofl whn usng h fallng wgh dflcom W s dvc. In pacula, h nal pavmn mpau pofl s no qud n od o mplmn hs soluon. A smplfcaon of h poposd modl o accommoda lmd wah daa could b mad o us us h maxmum and mnmum a mpaus fo h day along wh h pak sola adaon nnsy fo h day. uhmo, fo W sng conducd dung h daym hous, only h sufac lay s mpau pofl a qud hus allowng fo usag of a mo alsc sufac absopvy valu. Th poposd analycal soluon can also sv as a dvng ngn o gna h nal pavmn mpau pofl fo oh numcal amns of mpau fld, such as n smulang mpau voluon of hydang conc pavmn wh h ha of hydaon plays an mpoan ol and s valu s lad o h nal pavmn mpau pofl. Also, hs - modl sul can b asly mplmnd fo - cas by smply sng and Bssl funcon J n q. 35. CONCLUSIONS Th hocal soluon of - axsymmc mpau fld n a mul-layd pavmn sysm s succssfully dvd. Th mpau a any pavmn locaon, and m n an N-layd pavmn sysm can b calculad by usng hs soluon und h cylndcal coodna sysm. Hankl ngal ansfomaon wh spc o h adal coodna s uld n h dvaon of soluon. Th npolaoy gonomc polynomals ha a basd on dsc ou ansfom a usd o f h masud a mpaus and sola adaon nnss dung a day, whch a ssnal componns n h bounday condon fo h

17 Wang, Rosl and Guo 7 undlyng ha ansf poblm. ld mpau sng suls dmonsa ha h dvd analycal soluon gnas alsc mpau pofls n a conc slab fo a 4-layd gd pavmn sysm. Th advanag of hs fomulaon s ha can apdly pdc h pavmn mpau fld fo sho m duaons wh lmd npu daa and dos no qu h nal mpau fld o b known. ACNOWLEGEENT nancal suppo fo hs sudy was basd on h suls of ICT-R57, Evaluaon And Implmnaon of Impovd CRCP and JPCP sgn hods o Illnos. ICT-R57 was conducd n coopaon wh h Illnos Cn fo Tanspoaon; h Illnos pamn of Tanspoaon, vson of Hghways; and h U.S. pamn of Tanspoaon, dal Hghway Admnsaon. APPENIX Ths appndx summas h man vaabls and symbols usd n h dvaon and fomulaon of analycal soluon of mpau fld n N-layd pavmn sysm 5. In h followng, h subscp uns fom o n xcp sad xplcly ohws, h supscp k,l uns fom o wh undsandng ha hy a no akn as pows; symbol lk CA n CA s ad as a sngl vaabl. n oal lays of pavmn sysm ncludng subgad adal coodna n h cylndcal coodna sysm m vcal coodna n h cylndcal coodna sysm m T,, funcon of h lay mpau fld ºC h hcknss of h pavmn lay m,,, L n λ hmal conducvy of h pavmn lay cal/m-h-ºc α hmal dffusvy of h pavmn lay m /h H h,,, Ln A- q ha flux no pavmn cal/m -h P ha flux causd by h convcon du o mpau dffnc bwn amosph and pavmn sufac cal/m -h R n sola adaon flux cal/m -h T a a mpau C A magnud of wav mod usd n h modl bounday condon C B pavmn sufac convcon coffcn cal/m -h- C C, ngaon consans fo lay a sufac absopvy o h oal sola adaon dmnsonlss s

18 Wang, Rosl and Guo 8 Q sola adaon flux cal/m h adaon flux md by pavmn sufac cal/m h μ paam usd n h adal mpau vaaon funcon f Y,, funcon of h lay mpau fld n h complx plan ºC ω fquncy of wav mod usd n h modl bounday condon /h ansfomaon vaabl usd n h Hankl ngal ansfom magnay un numb wh ω v A- 4 α v A-3 v N A-4 v γ acan A-5 v CA H N N,,, L n A-6a CB H N N CC γ C,,, L n A-6b γ,,, Ln A-6c λ λ v v,,, Ln A-6d CH H H A-6 C H A-6f H N N H N N γ γ P [ C P H N N H N N γ γ [ C P H N N H N N γ γ [ C H N N H N N γ γ P [ C wh,, Ln n q. A-7a o A-7d. kl ] ] ] ] A-7a A-7b A-7c A-7d kl kl β,,, Ln A-8 P kl kl kl ψ R A-9 kl kl R P H H H R P R P R A- H H H R P R P R A-a A-b

19 Wang, Rosl and Guo 9 R H H H P R P R H H H P R P R R wh,3ln n q. A-a-A-d. A-c A-d [cos CA C cos CA CC ] [sn CA C sn CA CC ] A-a [cos CB C cos CB CC ] [sn CB C sn CB CC ] A-b [cos CB C cos CB CC ] [sn CB C sn CB CC ] A-c [cos CA C cos CA CC ] [sn CA C sn CA CC ] A-d wh,l n n q. A-a-A-d. Agumn assocad wh,, L n can b dmnd as follows: β L Φ cos CA C cos CA CC, Ψ sn CA C sn CA CC Ψ If Φ > & Ψ o Φ > & Ψ, hn β acan Φ A-3a Ψ If Φ < & Ψ o Φ < & Ψ, hn β acan π Φ A-3b 3 If Φ & Ψ >, hn β A-3c 3π 4 If Φ & Ψ <, hn β A-3d 5 If Φ & Ψ, hn A-3 β π β, β, β kl kl kl kl β CH {[ can b oband analogously. A-4a ψ A-4b [ CH cos β ψ sn β ψ C C cos β sn β ψ ψ ] } ] A-5a {[ [ CH CH cos β ψ sn β ψ C C cos β sn β ψ ψ ] } ] A-5b

20 Wang, Rosl and Guo {[ [ [ CH CH CH {[ CH cos β sn β sn β ψ ψ β ψ cos ψ C C C C cos β sn β sn β ψ cos β ψ ψ ] ψ ] } ] ] } A-5c A-5d wh,3l n n q. A-5a-A-5d. kl Agumn ψ assocad wh, Ln can b oband analogously n dmnng β abov. B λ kl G v G v B λ n n H n n H H H n n n n cos ψ cos ψ sn ψ sn ψ n n n ψ n ψ ψ ψ n n n n γ v cosγ B λ γ v snγ A-6a A-6b Agumn τ assocad wh G and G can b oband analogously n dmnng β abov by sng G Φ and Ψ. G μa B μ G G Γ λ μ G G A-7a n H Γ Γ A-7b n H ψ ψ n ψ n ψ H ψ ψ n ψ n ψ SC Γ A-8a S Γ A-8b SE τ A-8c S τ A-8d SG Γ A-8 SH Γ A-8f S τ A-8g SJ τ A-8h wh,3ln n q. A-8a-A-8h. Δ Γ A-9a δ A-9b τ

21 Wang, Rosl and Guo Δ A-9c δ Γ ψ ψ n τ n A-9d Δ A-a [ SC cos S S cos SE ] [ SC sn S S sn SE ] Δ A-b [ SG cos SJ SH cos S ] [ SG sn SJ SH sn S ] Δ n A-c wh,3ln n q. A-a and,3l n n q. A-b. C τ Γ A-a ψ n ψ n τ Γ δ H Δ δ H Δ A-b C,,3Ln A-c,,3L n A-d A- n wh,3ln n q. A-c and,3l n n q. A-d, and agumns δ, δ can b oband analogously n dmnng β, fo xampl, o dmn δ, placng Φ, Ψ n A- 3 by Φ SC cos S S cos SE Ψ SC sn S S sn SE REERENCES. Huang, Y. H. Pavmn Analyss and sgn, nd don, Pason Educaon, Inc., 4.. Tschogl, N. W. Th phnomnologcal Thoy of Lna Vscolasc Bhavo: An Inoducon, Spng-Vlag, Wsgaad, H.. Analyss of Ssss n Conc Pavmn u o Vaaons of Tmpau. Pocdngs, Hghway Rsach Boad, Vol. 6, 96, pp a,. I., and E. J. Banbg. sgn of Zo-annanc Plan Jond Conc Pavmn. Publcaon HWA-R-77-, Vol.. HWA, U.S. pamn of Tanspoaon, ARA, Inc. Gud fo chansc-empcal sgn of Nw and Rhablad Pavmn Sucus. nal Rpo. Naonal Coopav Hghway Rsach Pogam Poc -37A, Bab, E. S. Calculaon of axmum Pavmn Tmpaus fom Wah Rpos. Hghway Rsach Boad Bulln 68, Naonal Rsach Councl, Washngon,.C., 957, pp Rumny, T. N., and R. A. Jmn. Pavmn Tmpaus n h Souhws. Hghway Rsach Rcod No. 36, Naonal Rsach Councl, Washngon,.C., 969, pp Lukann, E. O., C. Han, and E. L. J. Skok. Pobablsc hod of Asphal Bnd Slcon Basd on Pavmn Tmpau. In Tanspoaon Rsach Rcod 69, TRB, Naonal Rsach Councl, Washngon,.C., 998, pp. -.

22 Wang, Rosl and Guo 9. fndf, B.., I. L. Al-Qad, and S.. fndf. odl o Pdc Pavmn Tmpau Pofl: vlopmn and Valdaon. ASCE Jounal of Tanspoaon Engnng, Vol. 3, No., 6, pp mpsy, B. J., and. R. Thompson. A Ha Tansf odl fo Evaluang os Acon and Tmpau Rlad Effcs n ullayd Pavmn Sysms. Hghway Rsach Rcod No. 34, Naonal Rsach Councl, Washngon,.C., 97, pp Hsh, C.., C. Qn, and E. E. Ryd. vlopmn of Compu odlng fo Pdcon of Tmpau sbuon Insd Conc Pavmns. Rpo Numb L/OT/SO/9-374, nal Rpo o loda pamn of Tanspoaon, Rasmussn, R. O., J.. Ru,.. Royck, and B.. ccullough. Consucng Hgh- Pfomanc Conc Pavmns wh HWA HIPERPAV Sysms Analyss Sofwa. In Tanspoaon Rsach Rcod 83, Tanspoaon Rsach Boad, Washngon,.C.,, pp Schndl, A.., J.. Ru, R. O. Rasmussn, G.. Chang, and L. G. Wahn. Conc Pavmn Tmpau Pdcon and Cas Suds wh h HWA HIPERPAV modls. Cmn and Conc Composs 6 4, pp Yavuuk, C.,. saba, and A.. Chasson. Assssmn of Tmpau lucuaons n Asphal Pavmns u o Thmal Envonmnal Condons Usng a Two-mnsonal, Tansn n-ffnc Appoach. Jounal of aals n Cvl Engnng, Vol. 7, No. 4, 5, pp Solamanan,., and T. W. nndy. Pdcng axmum Pavmn Sufac Tmpau Usng maxmum A Tmpau and Houly Sola Radaon. In Tanspoaon Rsach Rcod 47, Tanspoaon Rsach Boad, Washngon,.C., 993, pp Lang, R. Y., and Y-Z. Nu. Tmpau and Culng Sss n Conc Pavmns: Analycal Soluon. ASCE Jounal of Tanspoaon Engnng, Vol. 4, No., 998, pp Bums,.. Th Gnal Thoy of Ssss and splacmns n Layd Sol Sysms. Jounal of Appld Physcs, Vol. 6, 945, pp , 6-7, Saub, A. L., H. N. J. Schnck, and. E. Pybycn Bumnous Pavmn Tmpau Rlad o clma. Hghway Rsach Rcod 56, 968, pp Wu, Ganchang. Th Analyc Thoy of h Tmpau lds of Bumnous Pavmn Ov Sm-gd Roadbas. Appld ahmacs and chancs Englsh vson, Vol. 8, No., b Banco,.A., R. A. nds, and E. H. abll. Ha of Hydaon Effcs n Conc Sucus. ACI a. J., 89, 99, pp Budn, R. L., and J.. as. Numcal Analyss. 7 h don, Books/Col,.. Pows, avd L. Bounday Valu Poblms, 4 h don, Hacou Acadmc Pss, Snddon, I. N. Th Us of Ingal Tansfoms, cgaw-hll Book Company, ohl, E., G. Long, and J. Rosl. Consucon of Exndd Lf Connuously Rnfocd Conc Pavmn a ATREL. Tanspoaon Engnng Ss No. 6, Illnos Coopav Hghway and Tanspoaon Ss No. 8, UILU-ENG--9, Unvsy of Illnos, Ubana, IL,, 54 pp. 5. Wang,. Analycal Soluon of Tmpau ld and Thmal Sss n ul-layd Asphal Pavmn Sysms.. Sc. Thss, School of Tanspoaon Scnc and Engnng, Habn Insu of Tchnology, 996 n Chns.