Elastic Analysis of Pavement Structure with Application of Vertical and Centripetal Surface Forces

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1 Elstic nlysis of Pvmnt Stct with ppliction of Vticl nd ntiptl Sfc Focs Min. W. SIR ilt Envionmnt Ptoi Soth fic Fjinmi K. & Mtsi K. ptmnt of ivil nd Envionmntl Engining Tokyo nki Univsity Sitm pn Tkmi Ino NIPPO opotion Tokyo pn STRT: ontct stsss t th ti/pvmnt intfc fo sttiony o olling whl compis not only of vticl stss componnt bt lso cntiptl stss s wll. In this sch whl lod in th fom of coniclly distibtd cntiptl lod nd nifomly distibtd vticl lod w considd nd nw dvlopmnt fo closd fom soltion ws intodcd. Th soltion ws tilid to nly th-lyd pvmnt stct in od to clify th inflnc of cntiptl lod on th pvmnt sponss. High tnsil stsss t th sfc of th pvmnt long th dg of th lod w fond. Fthmo tnsil stsss w fond t th bottom of th top ly blow th cnt of th lod. KEY WORS: ntiptl lod vticl lod lin lstic thoy Hnkl tnsfom.. INTROUTION Until cntly vticl lod only ws sd in th nlysis of pvmnts. Howv svl schs hv shown xistnc of complx contct stsss t th ti/pvmnt intfc. ccoding to R in ddition to vticl lod t th ti/pvmnt intfc th psnc of hoiontl tosionl s wll s cntiptl lods w lso potd. Whil tosionl lod occs nd ctin condition only vticl hoiontl nd cntiptl lods w fond to ct t ny tim th whl is sttiony o olling. Up ntil now mch ttntion hs bn focsd on th nlysis of vticl lod only with th ssmption tht ccking t th bottom of sphlt mixt slts fom th tnsil stin d to ction of vticl loding t th pvmnt sfc. Howv if pvmnt ccking is csd by tnsil stsss o stins only thn it wold b difficlt to xplin th cs of top-down ccking Mtsno nd Nishiw 99 Mys t l. 998 tht hs now sttd to civ lot of ttntion nd covg. This pp looks t th cntiptl lod which is nomlly ignod ding pvmnt nlysis bt foms pt of th totl whl lod cting on th pp sfc. Svl schs hv shown xistnc of focs cting fom th ti dg towds th cnt of th contct Tilking nd Robts 987 b 963 Wtnb. This typ of sh foc is wht is fd to s cntiptl lod in this pp nd modlling fo lod distibtion giving sltnt foc ql to o ws dvlopd. It hs lso bn potd tht contct stsss t th ti/pvmnt intfc vy

2 complx t l lb 999 dvlopd modl to xpss stss distibtion of th complx contct stsss fo s in finit lmnt nlysis. t pption of inpt dt nd vltion of th slts fom FEM pckg is tim consming. Fthmo mintining high cccy fo sponss sch s stsss t points of intst is ticky nd nds spcil considtion nd tchniq. In pn th widly sd softw fo pvmnt nlysis ISR ong t l. 979 hvon ELS GMES Mtsi t l. Min nd Mtsi 4. hvon nd ELS consid vticl lod only s xtnl lod whil ISR nd GMES consid not only vticl lod bt lso hoiontl lod s xtnl lods. lso thos hv psntd th closd fom soltions fo tosionl sfc lod Min t l. 5 nd sfc momnt lod Min t l. 6. Howv thos not w of ny pbliction tht hs dlt with closd fom soltion fo cntiptl lod cting t th sfc of mltilyd pvmnt systm. This pp psnts thoticl dvlopmnt of closd fom soltions of sponss d to th ction of cntiptl lod by dictly pplying Hnkl tnsfom to Nvi s qtions. This ppoch is on of th distinctiv fts of this pp.. OUNRY ONITIONS N GOVERNING EQUTIONS In xi-symmtic lstic poblms th displcmnts in hoiontl nd vticl dictions spctivly. Stsss will b nd in th hoiontl nd vticl nd cicl diction spctivly s shown on Fig. ccoding to Tilking nd Robts 987 th lod distibtion fo cicl cntiptl lod cting on th pvmnt sfc will b clos to coniclly distibtd shp s shown on Fig. Tking into considtion this distibtion cntiptl lod my b modlld sch tht it is o t th cnt of th contct nd mximm q t th xtm dg of th contct. Whn cicl vticl nd cntiptl lods ct t th sfc of th pvmnt th bondy condition my b xpssd s follow: p > q > b wh is th dis of th contct whs Eqtion psnts bondy conditions fo nifomly distibtd vticl lod whil Eqtion b psnts bondy conditions fo th cntiptl lod tht is dictd fom th dg to th cnt of th lodd. Th loding distibtion ws modlld s shown on Fig 3. Lod y x Fig : Stsss on n infinitsiml cb in smi-infinit systm.

3 If ppochs infinity ll sponss will b o: b Th stss-displcmnt ltionship in xi-symmtic poblm is s shown blow: 3 3b 3c 3d wh Lmé constnts. Sh Stss kp is-ply Rdil Stl Plis Ti Width mm -5 Fig : ntiptl ti lods t th sfc of smi-infinity systm Tilking nd Robts 987. q Fig 3: ntiptl lod modl t th pvmnt sfc.

4 Th qilibim qtions in cylindicl coodint systm my b wittn s follows: F 4 F 4b wh F F body focs in nd dictions spctivly. ssming body focs to b o nd sbstitt Eqtion 3 into Eqtion 4 to obtin Nvi qtions fo xi-symmtic poblms in tms of displcmnt s follows: 5 5b 3. SOLUTIONS FOR THE GOVERNING EQUTIONS 3. Smi-infinity systm Hnkl tnsfom of th bondy conditions psntd in Eqtion yilds: p d p d 6 q d q d 6b wh nd ssl fnctions of typ on nd nd ods spctivly. Pfoming Hnkl tnsfom on th Nvi qtions nd ng to obtin Eqtion 7 s follows: d d d d 7 d d d d 7b wh d 8 d 8b ncl fom Eqtion 7 to obtin: d d 9 This givs: { } { } Eqtion psnts th ltionship btwn Lmé constnts nd Poisson s tio: Sbstitting Eqtion into Eqtion 7b to obtin:

5 Thn on pfoming Hnkl tnsfom of Eqtion 3c nd sbstitt Eqtions nd ng yilds: { } { } d 3 Similly pfoming Hnkl tnsfom on Eqtion 3d nd sbstitt Eqtions nd ng to obtin: { } { } d 4 Rnging Eqtions 3 nd 4 nd xpss in Mtix fom s follows: [ ] P 5 wh [P ] is 4 4 mtix whos lmnts s shown in Tbl. Fthmo nd cofficints of intgtion tht my b dtmind by sing th bondy conditions. In od to dtmin th following Hnkl tnsfom is pfomd. d H 6 d H 6b Sbstitting Eqtions 3b into Eqtion 6 yilds: [ ] P H H 7 wh [P ] is 4 mtix whos lmnts s shown in Tbl. Whn ll sponss will b ql to o nd fo tht to b t thn. onsqntly pplying bondy conditions xpssd in Eqtion 6 into Eqtion 5 yilds: 8 Solving th bov qtion givs: 9 9b Rsponss fo smi-infinity systm my b obtind by sbstitting Eqtion 9 nd into Eqtion 5 nd pfom Hnkl invs tnsfom on th slting.

6 d d d d 3 Fthmo fom Eqtions 6 nd 7: [ H ] d 4 [ H ] d 5 Extnsion of th soltions to mltilyd systm is xplind in dtild in th pp pblishd by som of th thos Min nd Mtsi WORKE EXMPLE 4. Th-Ly systm Fig 4 shows th-ly systm with th fist ly cm scond ly 3cm nd thid ly is smi-infinity. Yong s modli fo th th lys E 4MP E 3MP nd E 3 8MP. Poisson s tio fo ll th lys is.35. 5cm dis lod is ssmd to ct t th sfc. Th lod typs 49kN nifomly distibtd vticl lod with pss of p.694mp nd coniclly distibtd cntiptl lod with dg pss of q.694mp. Th ppos is to invstigt th inflnc of cntiptl lod on th sponss. Th following nlysis ws pfomd. nlysis fo vticl lod only nlysis fo cntiptl lod only nd 3 nlysis fo simltnos ction of vticl nd hoiontl lods. Fig 5 shows compisons of th vition in th hoiontl diction of nd t th sfc of th fist ly fo th th typs of lod css tht w nlyd. Fig 6 shows compisons of noml stsss t th bottom of th fist ly. Fig 5 shows n xistnc of vy high tnsil stss n 5cm d to cntiptl loding bt tnsil sfc stsss dcss considbly whn vticl loding is lso considd. t cm is compssiv nd d to th ffct of cntiptl loding th is n incs in th compssiv stss by bot 6%. Evn thogh fom cntiptl lod is slightly tnsil t th sfc wh 5cm s Fig 5 b spposition with slts fom vticl lod givs compssiv stsss ll coss th sfc. Th compssiv stss is mximm t cm which is n incs 5cm h cm h 3cm h 3 E 4MP E 3MP E 3 8MP Fig 4: Th-lyd pvmnt modl.

7 of bot 6% compd to th slts fom vticl lod only. Fo th cs of cntiptl lod is o t th sfc whs mtchs wll with th xtnl pss fom th vticl lod s Fig 5 c. Fig 5d shows vition of t th sfc. Und vticl lod is o nd th slt fo cntiptl lod mtchs wll with th xtnl sh stss. oth vticl nd cntiptl lods slt in tnsil stsss nd t th bottom of th fist ly Figs 6 nd 6b nd th totl stss blow th cnt of th lod is.8mp which shows th inflnc of cntiptl loding is bot % of th vticl loding fo ths two stsss. t th bottom of th fist ly Fig 6c is compssiv with th vl of.36mp which is bot % high thn fom vticl lod only. Fom th slts psntd bov it ws fond tht th inflnc of cntiptl loding on pvmnt sponss is limitd to th nighbohood of th lodd.... [MP] Vticl lod ntiptl lod Vticl ntiptl lods [cm] sfc of fist ly [MP] Vticl lod ntiptl lod Vticl ntiptl lods [cm] b sfc of fist ly

8 c [MP] sfc of fist ly Vticl lod ntiptl lod Vticl ntiptl lods [cm] [MP] Vticl lod ntiptl lod Vticl ntiptl lods [cm] d sfc of fist ly Fig 5: Noml nd sh stsss t th sfc of fist ly...5. Vticl lod ntiptl lod Vticl ntiptl lods [MP] bottom of fist ly [cm]

9 ..5. Vticl lod ntiptl lod Vticl ntiptl lods [MP] b bottom of fist ly [cm].. c [MP] bottom of fist ly Vticl lod ntiptl lod Vticl ntiptl lods [cm] Fig 6: Noml stss t th bottom of fist ly. 5. OSERVTIONS N ONLUSIONS In this sch dvlopmnt of closd fom soltions fo cntiptl loding ws psntd sing dict Hnkl tnsfom of Nvi qtions instd of th clssic ppoch which ss Michll displcmnt fnctions. oth ppochs w fond to giv simil slts lthogh th shps of qtions diffnt. Mintining high lvl of compttionl cccy spcilly n th sfc is vy difficlt nd most softw lik ISR ong t l. 979 dvisd som modifictions to impov th cccy. This sch dos not mploy ny of thos sttgis bcs of th obstnss of th nmicl lgoithms sd. Rslts obtind hv mtchd wll th xtnl bondy conditions which is n indiction of th lvl of cccy of th softw GMES dvlopd. Th following conclsions w dwn fom th slts obtind. Tnsil stss slting fom th ction of cntiptl lod only is vy high long th dg of th lodd 5cm bt this ffct is cnclld ot by high compssiv stsss fom th vticl lod. t th sfc of th fist ly th compssiv stsss nd ttind thi mximm vls whn cm nd th inflnc of cntiptl lod ws bot 6%.

10 Fthmo t th bottom of th fist ly both nd w tnsil nd th contibtion of cntiptl lod ws fond to b bot %. 3 Whn > 3cm th inflnc of cntiptl lod bcoms ngligibl. This is n indiction tht th inflnc of cntiptl lod is highly concnttd in th nighbohood of th lodd. In light of th nlyticl dvlopmnt psntd in this pp ft sch pln wold b: to look t how nonlin tmpt distibtion in th sphlt mix inflncs th slts to consid ti/pvmnt contct stsss in th nlysis nd 3 to consid vltion of fctos oth thn mximm stss fo s to impov stndds fo ftig fil. REFERENES b E.S Sh Lods on Pvmnts. Pblic Rods onl of Highwy Rsch Vol.3 No.6 pp lb R Intodcing Impovd Loding ssmptions into nlyticl Pvmnt Modls sd on Msd ontct Stsss of Tis. Intntionl onfnc on ccltd Pvmnt Tsting Rno NV. M. Fish. nd oost F tmintion of Pnmtic Ty/Pvmnt Intfc ontct Stsss nd Moving Lods nd Som Effcts on Pvmnts with Thin sphlt Sfcing Lys. 8th Intntionl onfnc on sphlt Pvmnts Sttl Wshington Vol. pp ong.l. Pt M.G.F. nd Koswgn.R ompt pogm ISR lyd systms nd noml nd tngntil sfc lods. Extnl Rpot MSR.6.73 Koninklijk/Shll-Lbotoim mstdm. Min.W. Fjinmi K. Mtsi K. nd Ino T. Mlitilyd lstic nlysis fo Sfc Momnt Loding Tnspottion Rsch od 6 to b psntd. Min.W. Fjinmi K. Mtsi K. nd Ino T. Pvmnt Rsponss d to Tosionl Sfc Loding Pocdings of 7 th Intntionl onfnc on ing pcity of Rods Rilwys nd ifilds NTNU Tondhim Nowy n 5. Min. W. nd Mtsi K. 4. vloping Softw fo Elstic nlysis of Pvmnt Stct Rsponss to Vticl nd Hoiontl Sfc Lodings. Tnspottion Rsch Rcods No. 896 pp Mtsi K. Min.W. nd Ino T.. xi-symmtic nlysis of Elstic Mltily Systm onsiding Intfc Slips. Intntionl onl of Pvmnts Vol. No. pp Mtsno S nd Nishiw T.99. Mchnism of Longitdinl Sfc cking of sphlt Pvmnts Pocdings 7th Intntionl onfnc on sphlt Pvmnts Vol.. pp Mys L.. Roq R. nd Rth.E Mchnism of Sfc-Inititd Longitdinl Whl Pth cks in High-Typ itminos Pvmnts. onl of th ssocition of sphlt Pving Tchnologists Vol.67. Tilking.T. nd Robts F.L Ti ontct Pss nd Its Effct on Pvmnt Stin. onl of Tnspottion Engining SE Vol.3 No. pp Wtnb T.. Ti Tlks. Rvisd Vsionpns Stndds ssocition S pn. pns Woking Gop fo Pvmnt Tchnology Stndds. Rvision of Gidlin fo sign of Pvmnts nd th ction of Spcil Typs of Lods sign of Pvmnt nctions. onl of Rod onstction R pp pns