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Flexuralresponseoffootingonreinforcedgranularbedsofvariablesubgrademodulus

ARTICLEinINTERNATIONALJOURNALOFGEOTECHNICALENGINEERING·JANUARY2008

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ArindamDey

IndianInstituteofTechnologyGuwahati

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Availablefrom:ArindamDey

Retrievedon:05February2016

4 r y 1 - - * - - . , - . . . - . . - - . . , . - . ' - . .

tuifldam Dela and Pnbit K. Bosuahal

Flexurql response of fooling on reinforcedgronulor beds o{ voriqble subgrodemodulus

ABSTRACT: The papd p.tuiG to the sudy ofih. Aeru,L rtsponse ofd sh'Iow fooring ftsting on a comPrctd lGnuhr b.JoErllng d deposft ol poor Srduhr soil, {ith rein forene.i G g geogrid and Seocelt Pl'.ed * rM iniclfi.e oi th e $o mdiiThe foorins dnd Ei ordent d. onsidecd d tee ended 6*ms The granular.oil nedia rc nlIentcd bI r *ries ofwinHeisp;ng3, Tskng hb -6ut L're efied ofconfining pre$uro 'long the footing aDd ieinfo&n. dr1tbcansoilinteifre Erio$ distiburion of 3ubgnd. nodulus along the ldgth oa fooring dd Einbrcement (uniform l in.r xnd 'lu ld

ntic) de co$ideEd. Governing equitio$ for rh. above Pnblm re dsi€d md subsequenr\' nom ilnld, thnh is $.n

solv.d qusins Finitc Diltemce Metiod (FDM) in @niundionuid cans_seidelltedtiveTecnniqde (cslr) compirnon or

rh. obhhed sobtions with those.vdllible in litenhre for (he.rre ofunilomsubgndenodulus *hibitcd a d.vi ionoa219i

40% in rie setlem.nr chanderi*i.. of rie einfo(ed bund.tion bed Prrmetic studies s€c p{lirmcd ro $ldv rll eraed

olviiidus p'lmeb*, !!.1 ic EIatiE nexlFl rigi&ty ofbeans (nl ftldriE lenglh ofbams (li), relativc sifins. otuoik (r:r)

Elarive deDrh ofDlaement dfr.inforement (H'), effa.i ofintcrfac. friction and efa.d ofmfttion jn $bgndc nodulus Thcd.rloD€d mdhod has be.n foind ro be tdnd. h pftdicti.g the beh.vioi of €infor.d found.tion be&

KEYwonDs: Gmst nthdi.i R.infored Found.tlon Be4 varying suh8dde Modnlus Nunni..l modcl Finite DilTcrtn..

Mdhod (mM). Go$ SsideL lteation Tfthnique (GSIT)

INTRODUCTION

If the rinforcing .Lm.nt of r found'rion b.d is nade up orgeor/nridics such G.ogrid, Comar or Geocel Con.Frsor Mddlic drip., it n .sentisl b .onsider their bendingHishce in the mabti3 ro Pfdid thnd.s ofReinao(ed Foudarion Be& (RlB)

EDgiEd4lid|6hldfot

orT<hdosrqftPhoN No, +eL !&ea (M), otr-e 7s6 (o),Fn6D,593s,Enil|dq@&di

,PrcfesifuPifud!..odE4iqcil&hdb'ddfutd

Ph{c Nq 6r ae me (o), ot2je v4 (R),

ln d*eloping narhendicd ndd.ls for pirdi.tiDg IhcBponse of such qstns $Enl appne.hmics aPpor.!.Bing rheory of elNi.iy solDtions,Iinit ll.hents 6a Iumpcd panneter noddr hly b.ldopted Ingenuiry lles in rhc applicafion of tha. p.ov.nmdhods b nere re6 6r rdt{is The rcsulting dilGledilequarions .an .iih.r be solved rnrhically or nurd.rlly (by6nite Elment or nnit diffcnn.e mthodt. In llnit elemeitndhod, insbd ol going for adnliond aornn dion ..dcd.*in appba.h, dned fornnlations cin iho b. rdolhdA lmped panneer nodel t tr vcry .oDnnon ind siDph.pproi.h io simiat €ngineering beh ioi These mdhodsd. stil being u.ed in n$ech in gcotchnical .ngineeringmd in pnctic.. A choice of illc m.rhod Min\.depe.& onthe hdi[anry md $se wih vhich it ai b. $cd to solv. rhepmblem. Any nathod rhar cin reasonably prudnt drbdravior ofa toundntion stu.torc h g.od so lons tho ooddprmetN cm be ddemined $itAour nu.h ofa problem. Iiendq ktlment beiry infredids its not inrpprcprii€ ro

hroadqml d,ri ofcot.hnkrl EBiddic cmo ,: o9rr4)

Fenw rqqa rrsidry o, foorhs d ahrorc€nmt (9Fe d W bqs o,6oh9 aid dibrc€md q,

F e d w { d ^ * s d s i n € d a a ,Fc d w deph d ds6M or Edm6m{t l4J

Fe n w ui r {e !n or cmpdsd ru ry)Vddbiisercefr.hibo'dGdMiolsb.danodu!s

I

se lfinkler splhgs to ide.lize L\e loudadon. In dis papertft lacr aplrord h$ b..n Bed

Frkn{ md ,o!es (200t) presmtd r tuit. rlenmt sinlliton io nodcl a lq* ol ssd overlyilg . ldyer ol gp6yn-theti. r.infocnenr bd dry in 1sup€i soft sbe b shdy lneinfluence of bcndins rtlffne$ (ndual rigidity) of the rinfo(eneDt on the barins @pr.ity o, tie foedation. fteb.'hg .4aciry of the foundltion M obsrved to incre6.NiL\ d. ndr3l rigidiry of the Einbrcemerr Ho*fti inthe$oveiudyn\e. ecrolnexunlrigidityofcitrfomentdr dE irnprwned in rlc stlment chmcdisriB oflherci.lorcetl foundrtion bed us noi studi.d. Considding ih*dr nrdulG ofsubgntle cactjon ft uitom .long the lengthoftne iooring nd tne rehforcemen! Maheh{di et il (200a)mlde Gs orHctryis Do,lel tHetuii (lea5)1 b dalze theprcblen bd obhin.d a closedjom sobtion. But, thedsunrption or udiaorn distibutid ofihe sukrade noddBmy be f lrom the cllitt $e ssc of slrss developedbd.rl, dr looriDg wis along the lmgln of the bem ddtheFforc thc subgnde nodd6 n ds lkely ro var] Assuch, $. $ortNorkis ext dedmdrgeHalpnftdu.hdb*n dev.lop.d rld liamtd h rhis p3pd klhg inba(o!ft lne vfjxtion oa $bglxdc nod'nus 3long d)c ldgrhofdf lbotirs and rtinro(eftcnt in th. u.lysis.

2. PROBABLE VARIATION OF SUBGRADEMODULUS'Lhe vdr onnod'nus ofs$gnde ft&tion is gmerdly deb-Dtud bI Fh@ Ioad esr S.v.iJ nertrods hrv. b.en d.veLoped 1o r \zc sonjhcture inhadionPloblm bded on

IhH{t model or i$ eiecion. Hovw*, t1E s.Iysis b6edor dr .orcpt ol flodds of sub8hd. r€.cdon dne.dy

trnired lLon tlaE load 6b dos not indude uy fu&-nenlal soil propcins ud L\acfor. mkB dE apprach I€srtracdve. To Gmove thi! dFvbac&. sftEl r$dcha3 [8lot(rr37), Grrh (1e43), v6ic (le6l), Bddd (1e62, re63),vh$v ard rmnricv 0e66)l haE povided corElatiotrbeslten lnc lrodlls ofsubgsd. rescrio. (l) ud tle nD{b

nental soil!Io!*tis lik d. modula ofeloticity (r) mdde Poissont ntio (vl A siv ol ldatore in Lhis respdEEals th* dre nagnihd. of , is not condmt for a $iln.dim. n rdis with the depL\ depmdiry or L\e confiningF$uq ddburden press; ndtuil deposidon.l prcGsad the otrolidation ?rc6s [Madock md tues (re60)].Mado& ud Rde (1960) 6ued so bdic forns ofrdr-tion of nod'nu of.hsricity witf, deptb a powr fib.tionEiarioa (xt = lt), 6d r poLynonid 6D.tion Eidion(4 = to + trz + l:l), {heF, . is tne &ptn fion rlE gomdsuf3a, ud t, h l! k, re the coffhnb of wiadon.t nrd,ln md Hehdi (196r) obsffd dEr th. d6rcebomd noddN (t )of.ohaioir.rs ioils incr$ed si$ !5etnft4 in confining pr6s@. Mitliouled setu oe6s)ob3er€d that tie elsdc ioduls of.o6edonle$ soils n notd fundion ofndiaun pdn.ipd ste$, but a fucdon oldEcodning pHsurq ud a pnboli. cwe ffted very wdl siih

Wnd a footi4 fitI .oncenhed lo.d dt ia nid{pu nplaed or the soil nediB, disribltior of6\e contuingpi.!sre bmerth the footing is such rh* it i5 lrHind tolardr ilEnid spm of the fooring md pmSrcsively de.na* towd,lr

/ . ,Yn € equtio l, - {&l- , pmpa?d hy lubu

\P ' l0t63) 6 be sed b debmine tne ralue ofmo<lutu orelasticity ofsoil6 . tundion ofthe .onfini4 presuE! wheB 4is dE nodds ol eldticity, d3 is lne .fiedive confningstie$, (r is lhe noduru nMber,tr is tle ibosphri. prssm, iltl , is a pE numbs ruSing Eon 0 io 1 FrcD riestudis ndde by Lade md Nelen (1937) ei!1 ftsrrd to thenodel prnmeEn (lor smd)' soil6 l(i Eried fron 50 - 100,ad r r&ied f,an 0.4s io 0.55), it a be dsB€d da lenod,nB of.hsdcityis n.dlyapmbolic tunction oftne con-fini4 presuc Cocldedrg thlr vdtation of nodulB ofeldticity m'l refmhg the relstiotr pnpo*d by liot 0946)or v6ic (1961) b.Meen modurus of eldticity ud rle sub-

- - - J - < -

gnde nodulns, it is logi..l to.ondrl. rhd ihe dlgmd. frodulus b.ncdh th. fooiti.q ph(d or soilfi.diufrnnotx.onnrni'burl'ontdu,y.onljdrlahl) d.ng th.hngrh ol|II looli,,g r i n,rdior ol

ro .he& the aborc Folo5irior a {N\' rsJund.rtaken tu drin,tc th. mltrN olrhe moilrl[ ofsubgrade ediondlfuidioD oaoinniigprsnr..The ft$lt m p(snied bftnr.s tolldr Iri3trc Lshoss ihc plor of dsrn nodnlLsofsol sr tuodiorof.oninirg prc$ui d.t.niful k.d tnxiil t-c

otuknbuf and st(rr 1965). l n oLsr\.d thir rhcpinbolic .ufles fitred io ihe rl.t, f.r L.rh loos. r'dd.nsc sxid, {ifi .odn.i.Dh ol .otrdition L,.irg0 ees and 0ee7 rurpzdrrl)1 (hnL iMnitd i s.od

K- ll:!fiia 11,.r. rrr-ro.tL -, lr rr

^d . r I . , . \ o '( === ; - \ :

1 \ ! L . r ed r .t1 v't \t,

' tErl

it is obsrved lhat |lovided olltrr p..i,.dtrs rcNin.oNant (andthelft .onn.nt for. FrtnuhrFrob-kfr), kq(qriB add rr'tl=(r = ..r,, s ind..rji L.lepedirL Bi.Li qmhiD hnLm Nd roddcni.e modulN of $bsnde c(n.D ao! ti,otrr.onfining presnres .nd ihe c$lt {c ar.'D hFjsuie 2 Ir n obsercd i.on dr isurc $n pmbolic.uds .onld be usd to RrErnt Llr rrrixrion ofnodtrlus of b8rd. rci.rion rlilh.oinrmll rrer.ure, rhe.o.fi .i.nt oa.otrddjotr b.i rg 0.99s lor thcboth the 1r"$ or iDi, iidndidg rSood Jl rx..lorthe aboye 5n,drin thtFipri ftrtiods jn rl[ n'bgride noddus neE consid{ed .lons tl,.l.n31l' ofihe aooting od thp reinlorc.ndt rs rrll

3, SIATEI\,IENT OFTHE PROBLEM

Fis.r I shorLs !.u!h.. rooridg rni,ig or x d.Nll.omprded grannhrnll unddiinbli r.o.lnD,Llrdeposit.Ageogrid, g€ontor g.o-1l1trfdnphadsEinforceneni at ihe inElface or ths \.il n.di.. Borht|[ looting md Rinforenent aE id.il;ed n elnftb6ns of n.xunl.igidin rrrr rnd Erlr rcspc.rirL:The hngrhs oa r|[ fooring d (iofor.cNnr rc rll,aDd 24 r?.divdy. ]! .or-ntird lo.d, Q, i.s tthe nid{pan of rhr li.ridg. Tli. rrit rrisht of thc

e3*! r !n @di4 0€$! € lMriolt

70so

z

i i $

Fhure5'coNgghGdldydioi.dfs

top d L,otum soil Dedil !t JL rd 1? rcsPdivelt Ssd,liinpli4i',s wirqr of subgnde nodrnus rlolg the lenglhof tlr looling md rlt drlorament htrve 6.cn bleD htocollidcr 'ior 'l l,!,\ xc {il PerbolicFIur r ift DijJ q,D d Progr$ntly d{Rdi4 ro I nln'iDil vrle. r rM aig$, (ii) Lntr rL!L. rt rhe DiJ{pD rnd p,osrt$ilel/ d{ntrsing to r nini.rd vrluc rlr !d$!, (ii) PrLaboli.yilu. * tlrt .,id 5p rnd pogr$i\tn slu! d rL! !dg*, (i) Liftai vvJu. at tL! rinl rNn md Foglc$ntly nrrdin8 io a nqiNl Rlu! n [Lr !dg*, ind {v) Unilonn subgmde noduls|ioDgho* dr! l{,sLr oltlt fooulgtor i,d LoroD soil h|{. htrv. subgmde modulur l(i) itrdlr(a) E.!ed,r!,1 yhde a is the dntrft non L\e nid sprnoa dr b{Ds. Dri,g ddl..Ed by dtfo,-n,.nr eiFr$.r! i rsdh Ensile force, T(a) risirgdue to tlr indoD 6 oo rl,t sunoudnrg gnnuhr Feda Thcef!donrlighi of rhc top BnNltrDcdiumon tne Fintodngh)x I d i,.D4,orcd b,\ codidcri,rg utiloLnr surhar8eord rh! lu,!!h or rh dnlorcing la}tr' r dn oldt $udIrs! b,lddn N rhe erild of tlre ruirrion of.ubg d. nrdul$.n11,!sd uat F:poi$ of tinfored foundriion sJr'

4,ANALYSIS

4, I covernlng Dilielentidl EquoiionItlldi,rg ro fir(rc I, rlr govdnl,g difi{.ntirl equLioN forthenduiilFrror!!otdr loodng is $rith d follo{sl

t l - = l t ) = r r , - v , r t r , rY r ! < r< I ' r ' r

: :

It:

:

P .",",d:--G:7j -T

6rcedc6^!ldr beds or wdoble 5!

ln the psiN eqution, /r,/, * $. dllMwrd defledio$of ihe lbohng md EiniorceEe.l and ?ft) n ln. conhd

P6sue benedti ihe fooring,Similsr\. ior th. dnfored.nl tne gmning equitio.s

, , L ' , , , ,4 ' 4l ! 41"rH {arrr- ' r , )r (2.)41' dx a' '

=hH (*(r) + 4(:l)yr+*(rr, o<r<l

",,.trrtff-fffr.," o.L't- tF -h@r, \3 '<11

r (x )=2 [1E,03r< ! (3 )

ln rh. abore eqnarions ?:(i) is the cd.rad PHsuiebeneiil rhe reinforen.n! and E is th. thicln.s of the

nr senenl polFoni,l rui.tion of L\e subgradehoddus of the onpa.ed grsdd fll dd th. ude+ing pool/lo6e gdubr nedlr rlo.g the l.ng& of foo!ins Md Eintoremenl is sssed s folft:

t rc) =t lo- inr * ,1 0<x<4 (a)

k2G)-kr-4:F- ktt o<a<t2 \5)

ln tle abde equdions trc, in dd fttr de the .olfubof Bdario. of 3ubgtrd. noddu ror comladed gtuul& fill, md, 40, *r md ftr, aE th. con$inb ofvariationof subg6d. nod'n$ 6r Poornoo$ soil ProPs choieof the nagnibd. ol the consEnb l.ad. b difaercnt w _

,iions ofrA. subar.ds nodulu6 Jong the ld8!h orrh.tooiins .nd relnrorcenot, ftB. c !! fouous (i) rr =

t, = tr = *x= o, flie subghdt modul4 G mifomthtuushotri dr.leigth of lh.boiing md reinrom.nt(ijl *n.lD,,!x. [, > 0 The!ubgFd.modul6!,ristionR quaddik with mffd qlu. d ihe midipr rd

nininal varu. ar 6\e eds., (iii) lrr, kn' ka, k 2 < a.the suhgnde noddB Ention is guadFri. uih nini_msl €l@ ar rh. nid'spu ud ndinal hlue ii rh.edgA. ( v r r x 4 r ' o& l i l [ r=0 :T f i e iubg radenodul8 wnarion r rier sth ndinal rnl!. t r]]e mid_

spo ,nd niniml €lue 6t the .dges, dd (v) ku, kr < 0

& ftD, tz = 0: $. subcnde mod'nus wiabon is linar'iih ninin'l vdue at th. Did{tm dd ndin.l hluar bc dg4,

4.2. Non-Dhensionot tom of Governing

R.dangl4 xqurion I ro obhin an crpF$ion rorl:, sbscquendy difecntaring &e er?ft$ion four 1imc5, and subdioting in Eq$tion 2, the Sov{ning ditreientirl equdion forthe nexui.l rspoB. of th. fooiing Ms o[bined, Nhich isexpEsed in t non-dift.nsion.l forn s lolo\.:

(rb) ',#'",#.4#.".#,+ff'* *tt'*,r'

d H*4ib dldv Fe doEed

"-"::;;6--^*'**"\l'l"'

: : : - : . . . ' . " - ' * " " * , ,

. . . . ' i , .*,""--""",. . . ,

! i

: -

iF

9aq;

J,

q d ' t r d c d W , 8 e | a d g d L l y dll

d ':ff ".,tt't' r.r^",fi., +",:,r,,'t *a.^-liri"4'-ii,r'l:, (,,7o: i,, r

ln tlre prelioA equdion, i! - N7 aE rhe non-,limnNhosc .xlicsions de lroyided in

&pa Lr r, I is dr mr dine$ioml disbnc tom rieDrid sp.n of ihelooLing,/i nthenon.diBsionddendionot rhN looliig, li is 'l,c non dimaBion,l bjt sei*t of rhe.omprd.d gFnubr fl1, H'is the non.dinsional thi.kn.soldr!.oDprdcd gnn,ir6ll,Misinon dinensionrllensrtrotllr Ri,aordn.ll ah!{r to rhe chanddisric lmgrh ofr looi,3, l,r is ,. lchdt l.nglh ol lhe fooring and the

,tiilordfuDr, ult i is rhe rclariv. ndui,l i8idiry ol rhelooti,rg ud thc rci,rforceftni

ln ! sinil{ l.!:hion, lne non.dineNional equationsgomri,rs Ihe den(tion belx\ior of dnformen! n .ied a3

I! lnd $ove equ*ions, ,: is !\e nor diDsiond denediorof the ftinforcenBi. li n tn€ non din.nsionil unit weighrofdtgru'n!,lri,iri!ft rhenon-dinensioDtsubgladetuoduls of the comprctd gn'ir 6ll md Md*lling pooisoil, ir is dr drddrisri.leng$ offooring and ir is rle rtI.dir .hd&E!isri. lengrh of footing sd thc dnfo(emer.

For the $ke of brevjty dd spra only 6e linal fonn olthe goydning equnjon is pissenred hae s}jlping the nr*.ncdirte ddivdtions. Er"E$ions for aI thc mn.dinensionalformare pmeNd tnc'Nohtioi edion.!t ,,,r, r r ",

!-. .,u",t",^., r.!r, r : a:+ ru'{i(rL.+r,)ri= 1is{; +i/{*'}j' 0<r" r t,

::+...""i_---

--i-a

9 o ' F o . - r oa l

4.3. Bo!ndory CondilionsTh. bouodlry.ondition5forrh.pmblcnne(i) Atthefrid span of the aooling, sloPe n z.o and ihe derforc is halfihat ol rhe .onantded l.d 'pPlicd d themid{prn o{fooiing, (ii)Ar the edse orrootii& bd'ding mom.it d shrr foc is z.rc, (iii) At the md.span ol rtinforceDhl slopc .nd shar fot. is nqrnd (ir) Ar rh. edge of Einfor.neni, the bendingnomot ind shear fo(e is zerc. The boundrry condirions cn b. evE$.d in non dimenrioill form $

i :

; !

l*')*(f

(?i".. tr-,)*(+rrh{e, ? (:r) * (he lon din.nsionil te'.ih lotc gcn

4.4, Melhod otsolution: Finile Difieence

I

IrI

I

I

I

,tr.')"".(*-j',,",#.,J

'c.14=ol

(#-')-

The difi.rnril .quations sotrning ihe nexunldspons of th! looiing xnd the tinfo(€nent aEexprs$cd in a nnil. difcrnce forn. CentJ difi.tnft(henre hs b€en usd ro r.nrtt the diffeEntirl equlriodsro difGrer a . qudions. Due io loxding and geonetn,l synmerry orly one half of the aood4 l[sbeen inalFed H ofthefooiing{trrnunbd ofnod* Thc difcrena forn orihe goveniiie equation for flexudl rcsPonse of footing n

I , . r ,+ . , + , r . . .c ' . . r ' ] - " r ,v1n+ !L . r , . . 1+ . . , i +c . r ' . , , ,

1,, the abw. equaiion, CJ r - C asioml.odfi.ien6 used in tbe firih diltcrn.e fornu

.h.i.o Ensnee'ino

hlioD rh! !:tprsrior {e snq r ApFondn ll.flrlf ol r|! rqofttrtqldt Ms discrizd into ,rnddr. The n.ru,n Hponr! ol dr reinfor.enmt isgorrrd br cqdions 7 and s Lrhnh cli b. .xpt$edr diiticur! lonls s lolo!5:

\li|t jDt \t\,,iD) \jDir 1tt\+Diti tt2

: .

c, .i);.+q l1t \+GLtt,+c itr'\tr\+Gh2y''.,,2= t )H1,1R1, , (D)

l,! rtrborq,DroorD-:n.n illdr;oDilco.ffi.iq'b Bcdb tne tuiit,l ifG.are io,d,lhrion rJhose eve$ioN eN givdr in

Fqullion (10) r$ appltd d.rh node ofdtIooLjirs utL, propr i,rorponrion olbounddy conJirior i rl,! bdddrt nod$. A se( oflinn.qurions\robh rd, \hnh sr roLrd by caurlseid.lItdlion L.driqu! (GslT) to obrain the ,letl*tionpirlr ol llt foodng. Ontu it \rs dncrnined, prolils

otuL!Ln,liigDomni,shelrro(e,hdconradP6

5.RESULTs AND DISCUSSIONS

A.oi,pucr.ode\!! $ritcr using ATrAB rndthcfollorlDg $udi.s rrre .arjcd 0tr1 s.quDti.ny: (ilcor$3or! r4\. (ii)coftch$s of the derdoP.d.oJ. ,u ri,! ioluriur obblred, nnd (jii) P:nmdrnsbJI fh*4e dii.usdh ddril

5,1. convergence siudyconriJrn nuJIr$.ided oul to deErmme de mrSe oiDsh slgNDt i/e or& vhnh dc dcncdion, bedingDonkd nrtri inr- .nd .onhd !lur i)J uur{31 bI the choice ol the size of Nsh $s.Drdn. rl! botds lrs diid.d iNo r fiiit nunb{ ofrr5noai lL,c siuc of llndr y!. .ominloldy d{nNd iretrch iLri LiqiJ *noD diBaxion.ldefledionnndbend.irgn,o rtrftheD .sFrof ,.aood{x rci orcemntrrcF En)dld }d irdrtion. For i q.picrl combindion ori)iput praD.{! thcinnl.n.eof mD &rfldonalsgmenti?. (rIgiris tom 0.05.o 0 000625 .odsspondne to 25 to1000 rcJlt on ft mn dinensionrl mid{pln foolingdenatur l,s Lrd shour ii Figuc 5. It was obsNed rhatdr NN,td$l ,on dt.!vd), coa3,rnr tt ir drorn snil{ ihrn 0.00125 Gorsponding

:g

to r 000 nod* or norc). Tl)e optim]l element size n d)us cl,o$as 0.00D5 (1000nod*) md lne$me G ued ror furthe

[,

I

5,2.Coirectness ofthe developed code ond

5 2 | Canpatisan \|ilh HetTyiS nadel

To .lre.k the cor(hr$ of the denloped .odc a degaEaEd solu.ion ton $s !ru*nr $trdy vr conprrd\rith rhsolurion prcvided br Haenyi (19a6) for a finit bean resti,,lon ar undnfored ssbgnde and subje.kd to a .onantdc,loxd n rrr nid sptrn. ror srh ot conparno4 dr pekn$odI ss degenentd bI loN{ins the masnitude oftr:4 nr rcry sml klue so l]]d it is nedy rnoved froD dr .otulubrional nodul. dd lne problen G nde ro ilpilienr

t|6Xurc€spons6o'loo]hgoneinJorc€d9rcn!]orbedsofW[ab|es!b

uneirfo@d foudation bad fie cohlanson orfts'l]bis povided in Figue 6. It b obserud thd the solniionsm in vcry 8!od rgEenenr to a.h oth{, d d.ndionl.s dEn 296 being obsned onlt at d v.ry snau Esion

HeEnyi (leao hld pmvided lolution for an infiniEbad ph.ed on ! Einlored foun&rion and a.ed upon

6r ! concentaied load ar the nid{pin of de footin!riowe€n the pEvd tudy pmphsizes o! rie flmnlresponse of linit. bems lh!s, I sbdy \€. lndshks$i.g lnal'rd eror &.hnique in.mdng the lengd ofthe 6niie beams duins the nention ?roce$ sud thdih.y aft sft.hdly long mough b b. beh.ving a, infi-

nie beam3 amlosdls ro rhat oi Hdcnyis nodel(Herenri, l9a6). Ior dt !&r ofPritf fd conparnonth. lengih ofrhe loori4 and ninforcro be .quali the str(naqe on A. Einfo(efr.ni md the

ensio. gen{*ed due to L\e ni.tion on the rtinfo@'m.nr wE neslered. The Fsulrs 3E Provided in FEtrt7. ft lmdical solntion tu obsftd to ba in dcdl.nt isEenent niti Hetenrit .olutivdi,tion b.ing s% for ih. foding den.dion and t0%for Eintu(sned den dion. It M! obsred rhd ibr r

nn e bem, if rhe din€ndonle$ Pa'metn 1,1 (l -

chaad.rftc l.ngtl ora bean) eeded 5 7 (i.c >

L,3r), the belm b€hdv.d rs a long h.m, vhi.h i3

sE!&r rhan the linit (7.r > r)$sge$ed in litntuE(Hden]1 lea6i vedq 1e61 dd s.lnduFi reT9)

5.2.2 Canqdhsan wlh gevi.us t$erch

Figuc 3 sho* the conPd3dn ofdel{tion pbl'ls ofrh. fooring nd einfoftm.nt obt,lned fmn the pres

enr tudI {ith tie solution! rPorted by Mdhemrwi etal. (2004). llre resdk sho{ an dedwjation b.tueen ri. so being lss th.n 296. For iheske of.onpdison, loothg d.fle.iion for rhe undnrorced .ondiiion ha. Jso ben in.ba.d I6m obsftdrhit rh. indlsion of the rinforenent rh. intrticeof the soil media r.du.ed Lhe iooti'g snleneni tv rndinum of 1696. Tl1. 6gw slso shds L\e defledionprcfils of hoiing and teinforement sing vriablesunsnde noduls (Qu.d*tic Eiaiion vith mdim,rr.alue at tI. nid{pr ud progrelikly d.ue.in3 to i

ni.inal vrue { d€ edge!) in ihe analFb Signiticmtdeviarionq 7.596 rd 1s96, wie obs.ned in trms ofsidendt i. foding rnd Einfot.m.ni EsPectiftlvwirh defledion prof'16 obkincd by sing u.ilorn sub'(ide nodulls. nNs considedion of variaLl. sub

;Ede rcebon r ois rhe le"grh or the rootns ddE'nfoftement hd signihent iflu.n.e on rh€ ar e

FFE15(aI rtrdvaidtrdrbsademodusprciielF:n6d.ur+

fie!€ i5 F). barbi d milrm4s oi

nent 6pons. oa L\e Einlbred iobd io. bed oYd lnd'bm the sine ohri'ed rvith uDiform nonnhB of $bsndrflction. A. .!c6, it .ffed shonld not be ieglcdcd

5,3. Porcmetric studiesThe mnge oflaftod.B consil*ed in r nudl is high_lightd in Tabb r. The flffrl rigrditt of i g.oslnihc(ic(S@grid, g.omt or seoel) n .omideFblr .hall (in rhcordn ol l0r) witt esped ro rhe flerurdl rigidiq or r conqerefdoting (in L\e ordd of r0?). Thus $c thtiE fleruin rigidiiy is .hosn to bc in rhe odd oa t01 Th. litcnt a! ErtdthrrbeFndad!3!itod.of2 4tim*lh.footiDgl.ngih,$clen3& ofL\e Einfortenenr dos not alled th. bodng spxcityafdr RIB. Hen.e n Fnge of1 4rns.hdtn fdr rddntlengd. Mag.ibde of I indi.itcd .guxl hnslh tor todtingbean and silaoftcmeni. Teuasii (rti5) mnied out phie

i".i

?E1,

I

i

:ii

It

nico Engin*,.9

load tess for i tfgs nmbd of soils to dekrDine iheD.glit e of inodulus of subgftdobstrved that the moduls otslbg!'de re&tion for looesands vtriied ftoDr 20 - 60 bns/cu, fi (3 - 23 Mlb/n),while it vrid anD 300 1000bN/crft.(ll0-370MPtn) lor den.e smds, ThB, a ruge of5 - 100 is cho-setr for reldile nifin*s oa soG. Tlle mge ld r.lstikdclrh oa phcDdt of Einfocftent ud ietauw Ditseighr ol lill is .ho$r on the b*is of edier dudi6rcvealed in lir*{uF (Mrh$h\Ei eh]., ?004).

5 i.l . Ehe.l ct Relotive F|*El Rigllw af balhg

F€ur e(x) shovs the elT&t orrado ofndual rieidrty ofrootins and lchaorcenent (i) on ln. defledion at ln.lid spm d * rhe edge o,footing. it w,s obserkd th*d I Ns inrcred, the nrid.pr dsn .iion oalh. fao!ing dc.nNcd d the e4e dene.tion lJeso4 ddbodl lns vdus bc.dN nsrt oNht .ftd a c*tij,rcla.iye nciu.l rigidity (r > 30000)in ]l nrid{pM dea<ion d(Eased by 796 mtl dE edgedefledion increaed by 1696, Figue 9(b) shos L\edefledion Plof'ls of fooring io! Eios sagniiud.s ofL T[e ndi'DNr denection o.cdFd at the nid{pu ofthe rooti.g, d gndurllr dKered dd be.me niri-nr.l d dr cdge ft va obsrved lnrt sher ine rl.tivefldunl ligid,ty vs loN (R = 100001 i\ere us a signif-i.ut difieH.e (lrei) bawen ll. deflc.lon n t}le nid-span ! c4c of rl]e footinB But as ihe nagnitude ofRvas in(ssd, the dife(n.s in rhe defle.non tetw.6the Did spm xnd fooring beme le$er md le$ei lor I= 30000, tl)e dif€ren.e beme ody 596. thu, within&rs. h n br 700%, the diiaren.e ftduced bI 1396.I hn obs.d ior Na probibl) dE to L\e &d dEt as theflqural rigidity of d]. footing incase4 the footingb(ame noF rjgid, the contr.t sk$s {ee dDrcng,{ydislibtrted rlong ihe logth ofihe footi4i ud th6 rithmoc rigidity, dr loothg denedior tuded to be noreunirornr (Mudhr 1974). Ii ua obsded tfid mndnnorioDrl dkhne of0.29 ftoD ihe did4pu offoo!ing N4 . rtiri.:l point $ at this poirt, dr dell.ction oftft fooiing did not depend upon R. Froh non-dinen-sional dinorie 0 ro 0.2t, dr mr dinensimsl deflecdond<csed virh in.cxe in l dd beyond dtn poinfihedeflc.don pattr sot re€6ed.

o-'-{d"t*{ l ! ! ! 3 ! d 0 . { M . ]

FrguE 16 (al. liapff dddm orr6t

i i '

I: '

I

t,

I

l i

: .

i '

Fsuo 15 lb). Notrnn8iiod dderd

5.3.2. Etecl ol Ealoti@ Lehglh ol t@ting ahd

F4re r0 slDws lle fridiid otdE non-dioe*ioral deflectior ar 1he nid spm of footing with the rehtiR len8i{ ofhotiry md ftinaortehed rt * obsftd Lhd uftn A.hft$e in dE ulue ofli, rh. rd dln.nsion.l d.tu.tion offootins it Eid{pu ri..rG.d dd bem. redly odutaftd lr' = 3.5. For M ind.de in ln. 'due of r, by 30096, rhenon-dinensional denedion offootirc ar mid spm d€desedby 3496. T[us, it M obs4d rlat b@sirg rhe lensrh oflne EinloEm{t betlnd 3.5 rins rhe tength offootiig didmt inprov. ih. d.tuction ch@ct rinj6 of the einbkdiounaauon 6ed, wlicn @ in pdi.y riti dre 6db obtdnod

frexuo €epon.e ol loolino on €ir oaed grctu or beds ol voiable sLbg€de m'dllus 209

bv hrioN re*dched (Fn3!szl md hfron 1934lidido et .1., Lest, 1e36, le37i saktl md Dat, le37i Dd,19s9i (ning ?t d.. 1993, 1994 Ond .t €r'' l99a)

Fs@ 17, ry*d dbdMhi or rci drm

*ll s r 6ion nembe. the ellect oihoih bcing tspoNible

for the fd&rior in sttlcn€nt ofttte rinforc.d fomdatidn

bed. It ws obsewed rhat fo. 100% incnxse in !h' biction

coefhddi, the non-din.nsional dcncction oflooting d nid'

.pm dedesed by 7616. This u* dnc to thc lird thrt a lars{i;t rase tu.tion coefficient EsulGd in x l.rger tnsne 6rce

in &e rin6cmenr, $hich in tum lutrhd redu'ed dt ser

denflr of the rti.Jored foundation bcd

5.3,3. Ef,@l af Ileldh@ Sliltn*s oJ Soi!5 (k)

FioE I I show ihe nnation of non_dime'sional mid

sp;n md edg. defletion of fooring witl the r&tiE

iiffn.$ or 3oils (t,). Il m obsded tnar the noF

din€nslon:l d.flecrions inct*.d with rhe inc@'e in

ft- md b.c'me nearlv a!)'6Proric after a Padicularta]ug(i.= 60r B€rondrhi5 qlue the reliM6nllnesor!oil5hd negligibh €fted on dre dei.tion (hsrEn'trG d

$! rtinforc.d ion'dation bed l.da$ ln *' cpre_

sded the ercsnce of water gduld nedi!6 b'lo$

dienroise;upp-50' lqs. rnd s su.n dir defl€dioB

inrt.a€d wirh rhe rnqos. in k.Ifor 190096 increase i! *. the 'on_dim'nsiontl nid{pan

defl€ctio! inc@kd by 2ls96. rd that thg edge aeflec

5.3.4. El@t af etallw deplh af pl@ehent al

etnfttben't betfu laoting (Ha)

Fidr. 12 .hows tne kriation of llle non_diFcnsional

Did rD:n d€fledionorfooting w'rh therhtivedepfhor

daft;ent orftinioGneot b.lo{ botin8 (Hi) lt \6t

;b$d.d dEr * rh. dep& ofthe Placenent of ftinroft-

n.nt below footirg {d ln.Eae4 rh;enEtion also i.desed, Bqond a dePth orplaenem

of 1.2 hns rhe foor ns mdL\, rhe !

AmDrok Thirvsindk,iiE oi dr' f'd thd ihe eifed

oi Eintorins continuoudy dsF6.d uftn indeding

d.dh Gor 900% inde*. in th. relarire d'pth of plaa

d;d, tr. non-dinsion.l defl&tion at nid_span or

foorins ind.6ed by scn6) od qoid a ceniin depth or

Dl@;ent, ihe dnforcinS.ff.d {d n'glieible on dt

;.fl{tion .hdddisiic of th. Einlocd foundation

t.d. Wiih very l,rge d.pds ofpl'ament, ihe Pbbt2mapprca.hed one rhrr ofuieinfor.ed foundation b'd

5.3 5. Etect ol wtialian ol c@ftciehl ol

Fielr 13 sho{. tie uriation ol non_dimenlional

dei edlon of boiins d nid span widr ihe .o'6cient of

irreface fri.rion (!). ft Ms ob$fled that the tiiction

slso had a ocidsable efr€d on theterini.s of fti'ford lomdation bea .ve' vhen lne

ddudl risid'ry of fooi ig 3nd EiitoiideFd in the inalFis Thur tr.oLrld be qid Lhn 6e

rcbfo(cD{r behatd bolh at a fldal smb{ as

!

I

!iIt

c,t '

!,E'!'

5.3,6. Elfect al Rebtive unit @ight ai @tupdcled lill

Fisur 14 sho$ rhc rtistion ol4on dinension l nid{pmdenedior of food,s Nith Eldire unii weBht ol liU (Yl 1tN6 olisrvld ih* * dt Elative uii veislt of rhe 6ninaeled Ly 67L the nor dinensionil didjpan defle.non.lso inqss.d by 2796 uril it becde nedy synpb6c for

r'>0.17.T hc$lueoly,lep€nd.dondreuit*ightorcon_pr.E,l tJ L l, hns,horrh. boMg 1lL) bd tle.oncenkaiedlord a. rillpL q o,htr h.@A con*rnc frhe mit \e4ht or(h. coBprced fitl r$ incresed, the such4! on dre rein-focelrrt Nn inc(sed, md subsequendy it rsulted in dtnddal *tlemhr of fodi$, whi.i n obsffid h Figue

5.3.7.Va .tian ar subga.te madutus

Srudies NcF uddbken io dedDrine the effect orvdiarionof subg?dc urdL N on lnc setdemed cha6deristis ofEinfo(ed founddion bed coNid{:bl'" chinge Hsob$^rd h (he stle ut chan.t ristid uiL\ the .hdge in$bsnd. odulus. Fisue t5G) sho* d tlli.d plot ofEria'dor nr subgrde Do,lului Qur&dic vadatjon of subgradenodulus Nid, nuimu mlue rt dt nid*pd md ni,iiBalat llt eds6 n co$idtrcd hea subgr.de moddB at edge({r.),\r. \n.d b) co6dtrig ddeEnt lrelod oflub

r"a;m,J,ls j nLd.pr (1,,,d,*) 'nG R$i,Ds ii djJ'fe* $bg.',j| Nod'nus profJes ; slmM in Figue t3(r)t!. =s!, r/,,,r-4, d dkd r tug{r non-sLform dnu6F'io; orth! !L\;Lde muddu5 rloog L\e hngln otiF footingdd rnrfuandt. ldre = 95% l'/qq indiEkd ! nedlyuniloim distibution of subsm,le ftoduls :long L\e lengthof rhc lootirg rnd reirfdrccnen( Figuie r5(b) 3ho\es wia-tion of non dnn.nrioDd d.nection of footing ror wious.ulgn,le Drodultrs distibutio8 lt ua obseded thd $ thedi$iLurion of ruLgiide noduls b&Me nore md noienon.loiaonn, ihe non'din.nsional defleciion of the footing\rnt on ituprovbs For 95% ftdoction h *r4r tlE r€dudion

de Don.di nsioDrl denedion at nid{PB oflioting slsobsnrd 1o be 269i. Thuc it nly he $&d dd otuid*ingunilorm 'tistibulion of subgiade nodulus .rong lne lengihof dr fooltg mly lerd to ov*esin'ftd predidion ofstucDar drmcGriltns of r.blorc€d fouddion !ed, ud myl.rd b uncconolri.:r dsjgn pddnq

Srddis N* udertlrcn to check L\e dfect of vdioNdistiburion olsubgnde nod'nu on ihe stdenedt ch*rcterfui.s ol th. rcinlorced foudrtor bed 61 1 Pddcdd 3etoriryui panudca. risue L6(d snows if,e non aimmsionilden{don of foolidg r nndjlan for vtios dntibudon ofsubs dc modulN along de Lergth of fooring. It rnsobsryed lh1r r1E uiform dn.ibtrtion of srbsade nodds

rarnied in tle hjgher nor.dihensio!61 den.dior of footingrhm the oins dlslihutios. It m fouowed by lhed udquadnric distibuiion uidr mimum v.le of nod'nus airne nidipr, m<l rhe nininm non-dimension l defl.ctions v.ru obtded tor quidnric ud liner distnbdo.swiL\ nininu nlue oi tlE nod'nus at the nid{pln or tne

IigG l6(b) shous tne non.,linensional denecrion rlo'nles ol footing for }dio6 distibutioN oisubgnde noddsslong i\e len8lh of dr footi4, It us obsryed tn wilhftspcd b lne uifoin distibution ot subg4d. noddu({hkh is smerdly 6ed ii wiou ua}6isl t!. orier fornof distdbdim shfled. s,grincst ddiation in rhe !on'dine$ion l def,edion cnlddistic ofln. fooii4, Th.deviation! 6r q$dntic and thed disrributions viL\ mdi_hs r.alu. of subsdde nod'nus at nid-spd or roothg m29x rd z% BpectiEly, thile iIEt for quadnri. and lineddiiriburioN uitl ninimum v.l@ di nidjPan offooting iie2696 and 409!, lmn 6se obsmtions, it nsy b. $bd dEr.osiddltion ol uniloin distibution of slbgrade noddspsulbd h nu.n high{ del]{don thm the otier forn ofdisaibltions, ud hene thii @sidention nay lead to uneco'mni.rl ilesigns in lalge nagnitu&s Thus, proPs judgmentis fta$n.r Egdding the choi@ oftle a.hDl distibution of$bgrrtl. noduls dd vdnetl agais. popd oTdiNnhtion to gain a jstiied idight inro rhe ptublen ofbehavior ofrei orced fouddm bed, 6Pe.iaIY

5.3.3 fypicol distibutian af nan4ihqslanal

bending honent alohg the laofng

Figuc l7 sho{s the ttpi.d distibudon ofnon-diftensionalhending norenr elong $e footing for rdou distibution ofsubodo noddus It {d observed drt though the non-dinensional beiding nonent ar Lbe nid{pm Ms neeLy L\esm. for aI lhe dndbutions, 6ur it wied lignificddy .iongrhe len8rl of the footing iar dlffeftnt wildon ol subgRdenodulus. Tlle lruinum b{djng nomst wied signiiicmdy {ith the .hoic. of di$ribltion oi subgiide noddBTft pmbolic dd lincd distibutios ofsubgftde nodulsvitn mdi6u ralue * nid+Pd of fooring udtrestimtdlho n*inum non-dinesiond beding nonent 4obrahed with biforh djstrjbutior orsubgt.d. noduls hy dBlsni[de of 596 ud ]% 6p<tivdy! sh€as tne panbolcmd liner djstihutios ol subgad. nod'nu wirh nininmnlue d the nid-spr offooring ovei$ti@ted L\c sm. bynaSnituds of 10396 od 479{ iespetiv+ ThB, it G eviden.tid drft is a Dnked efi6t otdE wioB distibldon ofsub-gr.de nodns o. lle bendiry non.nt gen ntd ihroughout the le4ti of dr footirg, whicn in .s6 shodd not bs

rd

lorc€d sbtuorbeds orvoioble.ubs6de moduLs 2ll

negl.cted to amid 3. undere*ind$ign depending on rhe choice dd er$l distiburion ofsubgr e modllG ten.ath the footilg simibr obsmtion,*eE nade sith t}le non dimen.ion,l nininm bendinsnoment Shemted in the fooring.

5 3,9,l@ical distibttian ol n64lhensi.nol

@hbct pte$ue oktlg lhe ftbting

Figure l3 sio* the ,listibutlon ofnon dimen.iontl co.hcrpEsutr .lo$ rhe footing for hnols distibutioB of 3ub_g$de nod!l$. lt N* obsned rhd the dbtibudon of @nract pft$ue foUo{ed ! sinilar Eriarion d Art asmed fortie subgrad. nod'nus. A[ th$e ftious dntributions ofcon'rad prcsure de v€U dodnentd i. rious lit*b6 byTshebohriotr (Lett) hd Bdd{ 0962).

6.CONCLUSION

The fouo{in8 con.lNions are nlda fmn dr. studl a pre'

. Co.vergenc study 6ri.d out ro ddenine th.optinal ron dine$ional size ofrhe melh legdentbqlnd whi.h the solurion of diffeftla equ*ion3a&ined dability sioftd dlar the non-dinensionarsiz. shodd be 0.00125 korFspondins to r di.'cErization of tie f,alf of the rooring into 1000

, conpanson of th. degenddted sotutior for sses{i&ot any clnlorcenent {irh H{enl! soltrtion(Hdeni, leao sho*d m drtned agl.dent, lled.viation being 8ry snau (le$ ilhn 29{,) and con_nied b a smal rqlon ovd the l,ooting

. Soltriion lion ti. Present lbdt rhen coDPdedsith lhe solurion for Einforc€d foundation BingHee.yi nodel (HerdI, 1946) shorcd ! lery goodagEemeit, the ndinun friaiion b.h{een rhe twosolulions being les L\d t96 - t09i It r% alsoob$rRdthdNhenthenlg bde of the pirmetri,rdeeded5.7(i.e. > rir), the b.am behared s

, conparison sith paviou. 6.d.h (M.nsh{:ri da1., 2004) sho{ed a €ry Sood agftmenl, th. Dui_nm ddidlon being od/ 2%. ft rm ,lso ob$iEdth indusion of einfoment.t tie inierG.e of rh.soil nedir Edu&d the fooling sdleheni by 1696coNidenng u.ifon dfttibuto. of subgnde nodL us. Conddding quaddiic whdon of subSradanodul$ 'ld] ndinun d tie nid sPd of lbotingmd mininal at ih. edgs, a tuih* Rduciion in 3e!

deDentbyT.596- l5jis!.ot&rv.d Thus irmiyb..onchd.d rhd rhe eff<t ofrirlir! sutsdd. modu'1us should not be nqleded.

. As rhe ielative fdui.l rigidity (n) $as in(asd by

denedio n of aooring dedcased b Tei aD d the edsedefl.dion indeded 6y 16%, aid boih thc nlmsb{ffie nesill conshnt rfkr I > 30000, indndingn.glj8i6le eftd ofli beFnd ihis viltrc. It N6 ilsoobseRed that * tia nignitudc of n incraed, the.etdenent ol the footing tnded b b. m.rc ,ndmore unilorm due to incrcls. in th. rigiditv oldrfooting. Al a non dinension.l dnhn.c of 0.29 fromrhe nidnpan of fooiing, m nifl(iion point in thedcn.dior pit*n of the lootig ns obrmd. Tldpati.ul,r point Ms in,lepeDdeDr.tuh. .hxigc in n.

. The non-dinensionil Rtlencni ol thc lootingd.d.s.d with inftIins rclililc lcng'h (l), buibe.ine n€ y oshnt bqond i vrlrc of l" > r.5,afts {hi.h lnert ws nqligible jmprcrment in thesefrlenent chrilcErnks of rhc tooring.

. As the relatiR $ilhcs orihe soik (rJ idond ty190096, tienon dincnsionilnidaP.n defl.dionorfooting inde6ed by 2ls96 nqondimlueoft.=60,th. (htiE stifire$ oi 3oik hd i.gligiblc ctrcd onrhe 3ettlnem chamctrhhc ofthc reinfoted foh-

. '$nth the lndescd d.p(h of Pli.eDent of th. rin'

forcment belos ris footing, drc rtlrD€nt .r rh.footing Ms lbo iicftsed ht bcime ne r con-srr beyond a deprh of Lz tim* $c hoting ridthindic*i.g thd hcpnd whi.h (hc bed cslon$

4pnsched that of an unr.info(.d lonndation bcd.. It ws ob$ryed tid nntion xlso h.d i significint

.ff.d on the flmBl resPons of thc dnfo(edtdundriion bed. ]!ith l00ri ind.is. in tbc rriction.oefli.i.nl rh. non dimesional frid sPin d.nction oi footing d.cftrsed bI769'0. LaIg{ lndionEsdted in Lhe ddelopnent olhighft tnsih forcc inthe ielnao(6ent, thus inproving th. $tl.ncnt of

lron rh. abde sbdies, it {s obsenrd thd €rio$ dn'hibutior ofs$gnde nodul8 along the l.ngth ofthe footingdd dnfocndt hd a sEnincnt df.ct on the enlefreie.ponse of a footing pla<d on dic ninforccd found{ionbed. ObsefttioN of the 3bNe r.!l1s iidi.d.d that consideriry hriable srbgFde nodulus, th*e Ln! x signifi.anrtdudion of2496 - 40% in ine non ddre foorins oer dnd 1hN. rh. rne obhil*l ronsid.ringuniforn dntiburidn of .ubgnde nodflrc. Thur it mat

sded thi .onsideLnrg nnifom distibutior of subgnlehodd$ resulcd in ovdestnnded prednlion of dcflecriorrMtr ulnrs vrrins dis!$ution or su4lde noddus, ddthus 'n!/ lsd to Me.ononn:l dsign nr hge masnitudsThrs pFpe iudgNnt should 6. Brde uI e .noosins thctlistibuLion ol nLbglrde modurds ro $dy de setdenent.h .Fristn5 or lM rchforced fouddion bcd

APPENDIX - I

_,="..r.._*,,(-"

+=furr

l , "k ;

('r.,')

''d;.i',.d)##.# 'r0r rorv. 6M 2N -.

""(-J {"J.('J (a-f '*

" #i #'#.#i di.#,

APPENDIX - II

".-(^",I-(^".f^ 3N' r!, N' N|*-(^,I '(-J't^'.t ' -a&:I

. ?3N' 7N. 6\ ]\' '=(-J (^"1'-o,J-n"j-(",J

6N 2N. N-' (a"")' (a'.I (&"f (&"t (^,")'

. 3N rN ,N ,N .'"-('J '-J'(^.J-4"J

*-(';j'-r^'"t

- ( i " " I

*--(^,J--GJ-

, , .a.e,i!r!4.M.{(Ra.-4&.)te , ) L&.)

*,-,A,J'-la,.l-

""' - (q^,"I

-'--i&^'J-

J,"

o4)

(3r).fJr'-i*"':*' r'"

,. -["\ (i' "r)4'4'a(4i'];,)'*4'{i':(4j'q,l^

, -[]L{i a;r",{l)-Do,r,,,;,'.(d'r,)"", rli'; ({!+ ri ). loqd'l d,41,;

n{{ . rii {iti'i ', nq. t)

V,

. _ _,*ilr1tdr.q ".

rislxrive lensfi oI the fooling ind dt

ApuE nunbdnngingtom 0 to l

Nunb{ornodes iD hia3Panof foot

Ahosphed. p{sna (,aDbu, Le63i

conrdprcsur i the biseollootingmd rtinlorcemenf esP{tivellDahnft fron mid sp?n ol looting

Non dinensional dishn.c hon ihefridipan orihe rooring (/L)DorrHid defledions of footi.g nxd

Non-din.nsioDil.o.fli!i!nr lrd itrFinite Dift r.m lonnf hrion

Eldii. modqlus olsoit {rhrlo.li *

nhnn nbonndmodtrl$ of iilqrntrdanin & Hirri!'li, re6r)Flquol rigidiq ol tlt lod;g.i

Thi.kns of .onrr(bd !dnf hr liLl /De h ofph.onor of r;r.nndt

Non.din.nsionxl ihi.kn*5 of ..

Moddus nuDb{ (li'bf, re63iL.d.

No. dinensionil lh3rh otrhe ninlor.Pncnt rhiit io {lr dDn.t|nti.ler$hoflhc li,.ti,ig, (= rrlrr)Non dindrionil.oca[ri.nt

conftnttnl !.rd.di,,g.t ur mnl

Itelxrivc flmml rig ilofrhc footingard th. ritrforc.m.nt (t1rr,I:rtcherctris(k l.i-qrh ot tooinrg

NOTATION'nic fouo,iing tmbols are us,l in ift nocl&, ri,h. il - con darb o r v,iiiton (Mdlo.k &

t,,

l#D.pin beh{ ihe ground surrart

= SlbSddemodrlusofrhe chidft filland md&lFng pootloos soiLi

= connanb ofwixrion ofsubgrademodllus for .onprcted gnnuhr nI

= conshib ol nriation of sL$Cmdemodulus ror Mderhng Pootloos

= Non'din.nsional $bCnde nodulusotrhe .onPktd sonuhr 6ll ind

= Rd{i'€ siilTnes ofsoils { = kr/kr)= LcngLh of aoodns an,l ninforement

tuliivc .hindd5rn lcxrh offoorins an,l Ih. ninfor.ncnl(= rl,R,Teisilc fo(e.rnhg dtr. to lrntionfroi surotrndinJ !ndtrls'ncdiiNon-dim.nsionil tniilr lo(. g. rlncd dE b nntioi {rirr. 11,Unit reighG of the.oDptrrd gritrn|In andunddlriigroonlo.rs.il

Non diDenri.drlnlil rtighr of.on

chreernri. lc'srh of b.1d'

rl,!l !,rCdel6.i.nl of intrta- Iri.rionlfedn€ .oic ng rc*(r UxDbf,r96:]t lad. ddNdson l9s')

(4r,

)

REFERENCES

Bedcn, 1- (1e62) "Distilution or.onhci piesure udsiouldrtiod Gtrtdut{/4 Vol.12, pp 131 19s.

Brd , L. {196r) "Th! l{ilt]d model ,nd ib ippljerion tosoif st!d!,rl!,{t.aiVol.4r,pP279 230.

Biot,M.A.(1937) !.rding ofm inffnie hem on o ela3ticfoudation'/orr,.r oJ the A?Pnzd MthanktTd,sdi,,tr

"lirrs /rsMr, voL 4, At - 47.

D!s, D. M. (1ese) "roundatios o sod uderlain ry softclay Nid, gcodrile ar smd day ilerrjt{ C.6titheti6co,/dsr., sr Di.go, u.s.a, pp 20r - 2lr.

Irldc. A. rd lond, c. t. L P (2001) "wlen bendLrs s1iff'nss of g.os:/rdrtic dnforcenent is impo(mf'Cnstnth.ti6 h11eflnLional,\al. 8,rto. 5, pp. 445 - $a.

Inssry, tr I. and rxlfron, E. (1$4 "Be{ing opeity olreinfo(ed sd .ubgrad$" /ornaL aJ Gcakxhninl!)A;,?d,rg Dreh,"x, ASCE, Vol. t r0, No, 10, Pp. i500

Gdn! L. A. (l9.lr) On the $,jrUd zilmdnm il?othanlat b.tna Ptigaana|. Mqtandikai M?*hanikajr'ol. 7,No. a, pp. 2t3 r00. (In R$im) (& in SelBduri,

Glido, v ,\., liesiad<li, G. L. dd s'n!vu, M. t. O9as)'Bcring (prtity of SeoExriL i.inforcd foudado."Plamtitss oJ the Etcr.tttt Lkt,4tia,at catede anSdiL Me.hr"n: nnd Fatntdtio, E ginei'C, s^rFFn.i\.q U.s A. , pp.1777- 1730.

Cuido, v A., ch g, D. K and Sweenqr M, A. (1936)'companron ofBco8rid .rd geon:xde reinlbrc.d erL{lchatliat c.ok.lni:al lohlal,vol. zr, pp. a15 - 44a.

cuido, V A., Krueprd, I, D ud sw.net M. A. O9l7)''Pl.b load t$s on geogiid reinfoi..a edtf, slabdcea:! ttxs Canfztd.a Nd an dii US,A' pp 216

Iid$r! M. (re46) ,rd'a or riGfic Fauddtiaht rheoryvith Ar?tiaLkhs ir the fda aJ cirit Ensinuing,Uniusity of Mi.hiFn Prac An'-{Jboi

taDardinxn, o. U 3|d HdLrdi, A. N. (196r) "Rebou'l chF.ctdstts ol.ohsio.less soill Pmedj"Es aJthe SaondtnteflatiaBt conleftnd on soiL Metha,i.s tndl:a r"ddtiar EryiEting, lrpu, pp. I - 4.

ldbq N. (D63) "soil omprsiibility d ddenined byoldoncrr md rifi:r bb'?me'diryr o/.r' tmPanCa"feftn.c o, Soil MdaniB and Founaationt g',ptll, rvie.brdon,Gtrlldy,Vol. Lpp, t9 2s.

Khinsj K, H., DN, B. M., Pui v K, coolc L L md Yen, S.C.{199:r) lh€ bedns €p.city of r sti! foadatid ongeaEtid4n\taftd sba: Galzttilz 4d GMeflb M,

Khins, K. H., D6, B. M., Pui v L, Yd, s. C. md coob E.E. (r99a) Tdundation on sblg s.ad udslian by se.ldly vith seosrid it the interfact 6cdE til. andGnnmbtu"a,'ld.I3,pp. r99 - 2a6.

hdc P V ad Nel$n, R, B. (1937)'Modeling rie e16ti.behavior of gFnular soilt tr.datlonal lorthdl JalAiaqtuoL and Nunei'L Methad. ir c@hkhdni6vol.

M:Mshrtri, ?, Bsudh , P K. and ChMdF, S. (2004)nnalFis ol bs@s on reinlo(ed grmdr b€ds"Gaslnttcti: IntundtianaL\ol !t,No, 6 pp 470 - 430.

Mab\loui H, M. dd sbld, l. l, 0e6s) "ractm iduftcingL're noduls oielctj.lty ofdry sod ?ft.al,rjr o/rr,6th Inkrnatidnat cadetend aJ san M..hdhi6 chdFaunditjon Engin*ting. Manaeal,\al L pp. ?93 - 302.

Msdoda H. md Reek, L. C. (1960) {G€ndalizd solurionslot I ei{y lorded pild .l,uflal o/ the Soil Me.haniBard Faunad M Divinon, Ptut .nirgs oJ thz ASCE, vol,36, No. SM5, pp. 6r - 91.

Muin, v N s, (D7a) s'rl M&r,ni6 ann Founddtkht Atedirl, Dhmpd R3j, Ldh.

Oma M.T., Ds, B. M., Puri v K, Yfl, S. C. ud Cook, !. E.(leea) "B.rins 6pacit/ or lomdation on gosridiein'tot@d snt, Peedires ol tt rhituth InbnationalconJ.rchd on sdiL M.d'd'ks Md Fatndathr!'grnr'irg, New Delhl, !odl4 pp 1279 - t232,

$lii, J. P. ud DA, n. M. (1e37) "ModeL 6b ror stip foudation on .liy reinfo(ed widt g.ot.xtile lay(d'rrMspotutian Pz'ca4l1 Reat4 rr51w 4a - 45.

sermdmi ,a. P s. OeTe) ,rari.,,l,dtt\is aJ soit-stttdute

T.mghi, K. (19ss) "Azluarion of coefiicienb of subgtsdcrcKtjDn G{kxhniquq]/oL s,pp.297 - 126.

Tschebobrioff, G. 9 (r97r) Fllrddriat R.kking dd Eaihsiru.tu, Mccnw Hq l$g.hsh4 ,apm.

vsic, A. B. (1961) "Bendiry ofbam, a$i$ on isdopicdastk satit lot al oJ Enqkeenry Meerni: DirkkaPra.. aJth. ASCE"VoL A7, EM2, N t5 - 5t.

\4sor V z md rzonti.i N. N (19.:6) Be6 Pt ts akdSrdl& o, grJrtr &odatlor, Is6el Ptoglm of S.ieftin.Trusldiotr, NTIS Na N67-r4233. (As in Selvadud,