THE EFFECT OF GROUND SETTLEMENTS ON THE AXIAL RESPONSE OF PILES: SOME CLOSED FORM SOLUTIONS CUED/D-SOILS/TR 341 (Aug 2005) By A. Klar and K.

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1 THE EFFECT OF GROUND SETTEMENTS ON THE AXIA RESPONSE OF PIES: SOME COSED FORM SOUTIONS CUED/D-SOIS/TR 4 Aug 5 By A. Klr d K. Sog

2 Klr d Sog "Th Effct of Groud Displcmt o Axil Rspos of Pils: Som Closd Form Solutios, THE EFFECT OF GROUND SETTEMENTS ON THE AXIA RESPONSE OF PIES: SOME COSED FORM SOUTIONS A. Klr d K. Sog Cmbridg Gotchicl Rsrch Group, Uivrsity of Cmbridg UK ABSTRACT: Simpl closd form xprssios r prstd for th problm of groud sttlmt ffct o th xil rspos of vrticl pil. Th solutios r bsd o Wilr modl, d llow vlutio of soil-pil itrctio udr vrity of gr fild groud sttlmt profil. A xmpl o soil-pil itrctio subjct to tul iducd groud sttlmts is giv to dmostrt th us of th proposd solutios. INTRODUCTION This ot prsts svrl closd form solutios tht dscrib th xil bhviour of pil subjct to vrticl groud movmts such s by rby tul costructio. Th Wilr ssumptio is utilid to driv th closd form solutios d hc thy my oly b rgrdd s rough stimtio of th rl problm. Howvr, s th solutios c b computd usig hd hld clcultor or sprdsht progrm, thy r cosidrd to b vlubl for th iitil vlutio bfor coductig mor tim cosumig umricl lyss. Thy c lso b usd to s bchmr for vlidtio of umricl cod bsd o Wilr modl. Th solutios prstd i this ot r for pil i homogous soil. Vrticl groud movmts my rsult from umrous sourcs.g. tullig iducd groud dformtio, hv by xcvtio, cosolidtio sttlmt, tc.. Ech of ths problms is ssocitd with prticulr profil of groud sttlmts d spcific solutio my b foud. For xmpl, Poulos d Dvis 98 drivd cotiuum lstic solutio for th cs of soil sttlmt tht dcrss lirly with dpth. ogth t l. proposd solutio for pils bhviour du to tullig iducd groud dformtio usig boudry lmt lysis. Although ths solutios r vry vlubl, thy r ssocitd with spcific groud sttlmt pttr d cot b usd for othr css. Th closd form solutios prstd i this ot utilis grl shp fuctios for th gr fild groud sttlmts i.. th vrticl displcmts tht would occur i th bsc of th pil. Ths grl shp fuctios my b fittd to th ticiptd sttlmt profil, which my b obtid from fild msurmts, lyticl solutios, or mpiricl rltios. Mthods of fittig shp fuctios r ot covrd i this ot. FORMUATION Th formultio is bsd o th followig four ssumptios: [ th soil is homogous, [ th soil rspos t y dpth is fuctio of sttlmt solly t tht dpth bsd o grlid Wilr ssumptio, [ o rltiv slippg occurs btw th pil d th soil, d [4 th pil is lstic with costt xil stiffss. Th followig drivtio bgis by ssumig tht thr is o bs rsistc. Howvr, ltr i this ot, th solutios r xtdd to iclud this ffct. Tchicl Rport of th Uivrsity of Cmbridg CUED/D-SOIS/TR 4 Aug 5 Pg

3 Klr d Sog "Th Effct of Groud Displcmt o Axil Rspos of Pils: Som Closd Form Solutios, Th diffrtil qutio for th Wilr problm of th xil rspos of vrticl pil is s follows: y EA y Eq. whr EA is th xil rigidity of th pil, y is th vrticl sttlmt t dpth, is th subgrd modulus, [ i F/ Fforc, gth. If th pil is lodd loclly by forc P t crti dpth, ζ, s Fig., it will xpric th followig displcmts t dpth : ζ P Pf wh ζ EAsih[ y P f, ζ Eq. ζ P Pf wh ζ EAsih[ whr / EA. Th govrig Wilr qutio icludig th groud sttlmt ffct is: y EA y y Eq. whr y is th gr fild sttlmt with dpth. Th ffct of gr fild dformtio my b rltd to distributd lod o th pil, qul to dp y d. P y Fig. xtrl lodig o pil systm By usig th suprpositio pricipl of th Wilr ssumptio, th complt xil bhviour of th pil my b obtid s: dp y f, τ y τ dτ f, τ y τ dτ f, τ y τ dτ Eq. 4 whr fuctios f d f r dfid i Eq.. This itgrl c b solvd lyticlly for som gr fild shp fuctios giv i th followig sctio. Tchicl Rport of th Uivrsity of Cmbridg CUED/D-SOIS/TR 4 Aug 5 Pg

4 Klr d Sog "Th Effct of Groud Displcmt o Axil Rspos of Pils: Som Closd Form Solutios, Tchicl Rport of th Uivrsity of Cmbridg CUED/D-SOIS/TR 4 Aug 5 Pg Th bov formultio ssums tht th pil is ot rstrid t its hd. I rlity, ulss th suprstructur is stticlly dtrmid, this would ot b th cs. Howvr, if th suprstructur is rltivly flxibl, or tht ll pils r idticl d suffr th sm groud movmt, this ssumptio c b cosidrd vlid. COSED FORM SOUTIONS POYNOMIA GREEN FIED GROUND DISPACEMENT: Polyomil fits r vry populr; th tools for fittig polyomil to giv dt st r commo. A gr fild groud sttlmt profil with dpth c b pproximtd by polyomils up to powr q: q q y Eq. 5 A lyticl solutio of th itgrl i Eq. 4 usig Eq. 5 is possibl. Du to th lirity of th problm, th solutio my b obtid by suprpositio of solutio for ch powr i th polyomil sris. q y sih[ sih[ ; φ φ Eq. 6 Th bov qutio my ot b suitbl for us with hd hld clcultor, d it is prhps mor pproprit to us sprd sht. I my css, howvr, high ordr polyomils r ot rquird; thrfor simpl closd form solutio usig low ordr polyomil is prstd xt. RD ORDER POYNOMIA Cosidr tht gr fild groud sttlmt profil with dpth c b dscribd usig third ordr polyomil: y Eq. 7 Th solutio of Eq. 4 usig Eq. 7 my b xprssd s: sih[ sih[ sih[ sih[ sih[ y Eq. 8 Not tht solutios for polyomils of ordr smllr th thr r icludd i th bov xprssio; tht is, ths c b obtid dirctly by omittig trms ivolvig th ucssry polyomil cofficits.

5 Klr d Sog "Th Effct of Groud Displcmt o Axil Rspos of Pils: Som Closd Form Solutios, Th pil displcmt profil giv i Eq. 8 is of th shp: y y g Eq. 9 Thrfor, fuctio g my b lood upo s corrctio fuctio for th cs ssumig th pil follows th gr fild sttlmts. It is lso vidt from th trms ivolvd i g, tht th pil simply follows th groud displcmt s pprochs ifiit. Tht is, s EA dcrss, th pil must follow th gr fild soil sttlmt bcus th pil hs o rigidity to ltr it. Sic th stris i th pil r of itrst for xil forc vlutio, th drivtiv of th pil displcmts c b computd s: y sih[ sih[ sih[ sih[ sih[ Eq. sih[ sih[ sih[ sih[ sih[ COSINE FUNCTION I som css gr fild groud sttlmts my fittd with cosi fuctio. I dditio, if Fourir lysis is coductd o th groud sttlmt dt, th cosi my b usd s fudmtl solutio for pil rspos t giv wv lgth. Assumig tht th gr fild groud sttlmt profil with dpth is rprstd by th followig: Acos ω b Eq. y Th corrspodig solutio bcoms: si[ b si[ ω b cos[ ω b ω ω si[ bsih[ sih[ y A Eq. ω Ad th drivtiv is: si[ b si[ ω b si[ ω b sih[ si[ b y sih[ A ω ω Eq. EXPONENTIA FUNCTION If th gr fild groud sttlmt profil is rprstd by: y Axp[ b Eq. 4 Th corrspodig solutio bcoms: b b b sih[ y A Eq. 5 b Ad th drivtiv is: b b sih[ sih[ y sih[ A b Eq. 6 b Tchicl Rport of th Uivrsity of Cmbridg CUED/D-SOIS/TR 4 Aug 5 Pg 4

6 Klr d Sog "Th Effct of Groud Displcmt o Axil Rspos of Pils: Som Closd Form Solutios, INCUDING THE EFFECT OF TIP RESISTANCE By rcogiig tht th rsistc forc ctig o th bottom tip of th pil is qul to P K y y whr K B is th stiffss of th bs i [F/, o c us b B [ suprpositio to formult qutio which icluds th bs rsistc ffct. Th rsultt displcmt t th bottom of th pil is qul to: o bs EA y K B coth[ y y Eq. 7 EA K B coth[ whr y o bs is th solutio obtid ssumig o bs rsistc i.. Eqs. 6, 8, d 5 with. Oc th displcmt t th tip of th pil is dtrmid by Eq. 7, complt profil of displcmt d stri is fsibl. This is chivd by suprimposig th prviously giv solutio for ro bs rsistc with Eq. for P K y y, d ζ r substitutd. which [ B Although it ws sttd rlir tht th proposd solutios r for homogous soil, it is lso vlid for pils which thir tip is i diffrt lyr, sic K B my b rltd to this lyr. EXAMPE To illustrt th us of th proposd procdur, lt us cosidr problm of tullig r pil s show i Fig.. A 5 m dimtr tul is costructd t dpth of m. A m log pil of.5 m dimtr xists 6m wy from th tul ctrli. Th dditiol movmts tht occur i th pil du to th tullig r of itrst. W dopt Sgst 987 solutio for soil movmt du to tullig. Fig. shows th gr fild sttlmt profil with tul volum loss of.5% for th giv tul gomtry. A smll hv is obsrvd t dpths blow 4m. Abov this, groud sttls with mximum sttlmt of bout mm t dpth of 5m. Th shft bhviour of th pil is xprssd by th Rdolph d Wroth 978 modl, of which th lod trsfr fuctio is πg / log.5 ν / r, whr G is th shr modulus, ν is Poisso's rtio, d r r th lgth d rdius of th pil. I this xmpl, th shr modulus G is MP d th groud dformtio occurs is i udrid coditio ν.5. Th EA vlu of th pil is 5.9 x 9 N i.. cocrt pil. Th problm is solvd first by ssumig ro bs rsistc. Fig. shows th sttlmt profil obtid from Sgst s solutio solid li d th bst fit usig th rd ordr polyomil fuctio dottd li; th cofficits of th rd ordr polyomil r lso giv. Fig. b prsts th rsultt pil displcmts solid li from this fit dottd li usig Eq. 8. Filly, th dditiol stri/xil forc du to th tullig c b computd usig th stri profil drivd from Eq. d th forc profil is show s dottd li i Fig. c ot tht forc F i th figur is positiv i tsio. Ths forcs r dditiol to th os origitd from th suprstructur lod. Th forcs log th pil du to th suprstructur my b obtid dirctly from us of Eq. by positioig th locl forc t th top of th pil. I Fig. c, th "ccurt" vlus solid li r bsd o Sgst s groud displcmt d ot o th rd ordr polyomil fit. Thy wr obtid by high ccurcy umricl solutio of Eq.. For this prticulr problm, th diffrc btw th ccurt d rd ordr polyomil fit is % t th mximum stri vlu. With rgrd to pil displcmt, th diffrc ws much smllr, roud.%. Tchicl Rport of th Uivrsity of Cmbridg CUED/D-SOIS/TR 4 Aug 5 Pg 5

7 Klr d Sog "Th Effct of Groud Displcmt o Axil Rspos of Pils: Som Closd Form Solutios, Fig. 4 shows th xil forc distributio wh th bs rsistc is t ito ccout. Th bs stiffss ws vlutd s K B 4r G υ Rdolph d Wroth, 978. Wh th bs rsistc is icludd, similr diffrc btw th ccurt d th rd ordr polyomil fit is obsrvd s bfor. Nothlss th grmt is vry good cosidrig th mout of ffort rquird to stblish th vlus with th currt mthod. It should b otd tht i this spcific problm, th tullig procss iducs ot oly vrticl compots of soil dformtio, but lso horiotl. Ths will rsult i ltrl lodig of th pil, d will ffct it flxurl bhviour, rsultig i chgs i bdig momt vlus. Flxurl bhviour is ot covrd i th currt ot. P 5m Gr fild sttlmt for tul volum loss of.5%.5m 5m.5m GP,.5 5m m Fig. Schmtics of th xmpl problm Tchicl Rport of th Uivrsity of Cmbridg CUED/D-SOIS/TR 4 Aug 5 Pg 6

8 Klr d Sog "Th Effct of Groud Displcmt o Axil Rspos of Pils: Som Closd Form Solutios, Fig. Rsults for th xmpl problm 5 Dpth m Fig. 4 Rsults icludig bs rsistc CONCUSIONS Closd form mthmticl xprssios tht llow vlutio of th xil rspos of pil udr th ffct of vrticl groud movmts r giv. Thy r bsd o Wilr typ soil modl d thrfor c b usd i th iitil vlutio of th problm bfor prformig mor tim cosumig umricl lysis of soil-pil itrctio. Th solutios r drivd usig thr diffrt curv fittig fuctios for th gr fild groud sttlmt profil. Ech fuctio rsults i simpl closd form solutio tht c sily b computd by hd hld clcultor or b implmtd ito sprd sht progrm. Usig th suprpositio pricipl, th Tchicl Rport of th Uivrsity of Cmbridg CUED/D-SOIS/TR 4 Aug 5 Pg 7

9 Klr d Sog "Th Effct of Groud Displcmt o Axil Rspos of Pils: Som Closd Form Solutios, fuctios my lso b usd togthr. A xmpl problm of tullig ffcts o pil ws usd to dmostrt th us of th prst solutios. Th solutios prstd i this ot rfr oly to th ffct o th xil rspos du to vrticl groud movmts. I problms whr th gr fild groud dformtio hs substtil horiotl compot, bdig bhviour of th pil will supplmt to th xil o. Th currt solutios do ot covr this issu d furthr ivstigtio is dd to iclud this ffct. REFERENCES ogth, N., Poulos, H.G. d Xu. K.J. "Groud d Pil-Group Rsposs du to Tullig," Soils d Foudtios Vol. 4 pp Poulos, H.G. d Dvis, E.H. 98 "Pil Foudtios Alysis d Dsig," Joh Wily d Sos. Nw Yor. Rdolph, M.F. d Wroth C.P. 978 "A Alysis of th Dformtio of Vrticl odd Pils," J. Gotch. Egg. Div. ASCE. 4 GT pp Sgst C. 987 "Alysis of Udrid Soil Dformtio du to Groud loss," Gotchiqu 7, No. - Tchicl Rport of th Uivrsity of Cmbridg CUED/D-SOIS/TR 4 Aug 5 Pg 8

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