Parametic study of kinematic soil-pile interaction in two layer soil profile

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1 Scintific Cooprations Journal of Civil Enginring and Architctur, Vol., Issu., August Paramtic study of kinmatic soil-pil intraction in two layr soil profil Irshad Ahmad Univrsity of Enginring and Tchnology Pshawar, Pakistan Abstract In a sismic nvironmnt, soil displacmnt affcts th pil motion, and th pil motion affcts th wav fild surrounding it, which in turn affcts th pil displacmnt. This intr-dpndncy of soil and pil motion is calld kinmatic intraction. In crtain cass, this must b considrd in th strss calculations for pils. On such cas is whn a pil passs through soil layrs of sharply diffrnt shar moduli. Damag to pil is most imminnt at two soil layrs intrfac bcaus of high strains. In this papr diffrntial quation of kinmatic intraction is solvd and programmd in MatLab. Th solution is basd on Winklr formulation calld Bam on a Dynamic Winklr Formulation (BDWF). A dtaild paramtric study is carrid out to indicat th ffct of diffrnt paramtrs on pil bnding momnt. Kywords Pil, Kinmatic intraction; Soil-pil intraction, Sit rspons analysis I. INTRODUCTION During arthquak shaking, th suprstructur vibrats rsulting in th dvlopmnt of inrtia forcs in th suprstructur, which ar transmittd to th pil. Th pil must rsist ths inrtia forcs. Anothr forc xprincd by th pil during arthquak shaking is du to th vibration of ground itslf. Whn th ground vibrats, it imposs displacmnts on th pil mbddd in it. Th pil displacmnt rsulting from inrtial forcs and ground vibration modifis th wav fild around th pil, which in turn influncs th pil movmnt. This intraction btwn soil and pil is calld kinmatic intraction. Whil inrtia forcs hav rcivd gratr attntion and a bulk of knowldg is availabl in th litratur rgarding how to dal with inrtial loading, kinmatic intraction rmains rlativly lss xplord. Eurocod 8 is th only cod which considrs kinmatic intraction for pils. Many pil failurs causd by kinmatic intraction hav bn documntd in litratur, particularly whn pils pass through layrs of varying shar moduli []. Larg soil strains ar imposd at th layrs intrfac, rsulting in high pil bnding momnt [, 3, 4]. Thrfor it is vry important to study th paramtrs affcting kinmatic intraction. In this papr, th solution of diffrntial quation govrning kinmatic intraction is prsntd which yilds th pil rotation, displacmnt, bnding momnts, and shars dvlopd du to kinmatic intraction of a soil-pil. Soil with two layrs having diffrnt soil shar stiffnss is considrd. Th problm of kinmatic intraction of pil can b solvd by most rigorous mthods using thr dimnsional finit lmnts. Howvr, in this papr Winklr foundation is usd which is th simplst mthod for solving kinmatic soil-pil intraction problm. This mthod, mor spcifically calld Bam on Dynamic Winklr Formulation, is discussd in som dtail. First th diffrntial quation govrning th linar sit rspons is solvd for a two layr soil profil. Th rsulting solution is thn usd to solv th diffrntial quation for kinmatic intraction. Th solution is prsntd both for nd baring pils and fr standing pils by imposing rlvant boundary conditions on th solution. Th solution can b asily implmntd in any softwar. II. PILE SOIL SYSTEM Th pil soil systm considrd in this rsarch is shown in Fig.. Th pil passs through a two-layr soil dposit which has diffrnt shar moduli. Th cas of fr had and fr tip of pil ar considrd. Th bdrock that undrlis th soil layrs is subjctd to S-wav that propagats vrtically upward. Th S- wav imposs on bdrock a harmonic displacmnt of ug (t) =Ug xp(it) whr Ug is th amplitud of harmonic displacmnt and is th xcitation frquncy. Group ffct is not takn into considration as this is of lss importanc in kinmatic intraction [5, 6]. III. WINKLER FORMULATION Bam on Dynamic Winklr Formulation (BDWF) has bn xtnsivly usd to solv soil pil intraction bcaus of its simplicity. This mthod is usd in this papr to solv th pil soil systm subjctd to arthquak xcitation (Fig.). Th BDWF is schmatically shown in Fig., which consists of a bd of springs with linar stiffnss cofficint (kx) and a bd of dashpots with dashpot cofficint cx usd to connct pil to th soil all along its lngth. This spring and dashpot systm rsist th pil motion. Th fr fild displacmnt uff (,t) is transfrrd to th pil through th springs and dashpots systm. IV. DIFFERENTIAL EQUATION GOVERNING KINEMATIC RESPONSE OF PILES Th diffrntial quation for th stady-stat pil displacmnt to th harmonic xcitation is givn

2 Scintific Cooprations Journal of Civil Enginring and Architctur, Vol., Issu., August U '''' () 4 Upp() α U () pp ff whr 4 mpω Sx Sx, α, Sx K ic ω E I E x x p p p I p () mp is th pil mass pr unit lngth, Ip is th scond momnt of ara about cntroid of th pil sction, Ep is th pil lastic modulus, Sx is complx impdanc function, and Uff() and Upp() ar th amplituds of fr fild ground displacmnt and of pil at dpth, rspctivly. Th drivativs of Uff() and Upp() ar diffrntials with rspct to dpth. uppr soil bottom soil bdrock Figur Pil mbddd in a soil dposit composd of two layrs of soil Sismic fr fild motion - Uff() i G i, i i G, h h h3 ug =Ug it k x ( c x () Sismic pil motion Upp() Pil Soi Spring stiffnss (Kx) is takn from [7] and th distributd dashpot/lngth (cx) ar thos considrd by [8, 9]. V. SOLUTION OF DIFFERENTIAL EQUATION Equation is solvd using rlvant boundary and compatibility conditions of th pil soil systm. Upp Th gnral solution of Equation is [ i D i D ] su () D3 ff D4 whr s=/(q*4-4), q* = [/(Vs + is)]/ is complx wav numbr. In th wav numbr quation th shar wav vlocity (Vs) and damping ratio (s) of th corrsponding soil layr should b usd. D, D, D3, and D4 ar arbitrary constants to b valuatd through boundary conditions. Using subscript for th uppr soil layr and for th lowr soil layr and introducing local coordinats and (Fig. ), Equation can b r-writtn as Upp ( ) [ suff ( ) s Uff ( ) [ Upp ( ) i i ] ] Th Uff and Uff trms ar valuatd through fr fild sit rspons analysis. Equation 3 and quation 4 hav ight constants. Thir valus can b dtrmind using boundary and compatibility conditions of th pil soil systm. Th four boundary conditions ar th known pil momnt and shar at th pil had and tip. Th four compatibility conditions ar th pil displacmnt and rotation at th soil layrs intrfac. Ths conditions ar discussd blow. () D D D3 D4 (3) D5 D6 D7 D8 (4) A. Boundary conditions Vrtical Shar Wavs Figur Pil soil systm modlld as BDWF Th momnt, M and shar, V at th pil had and bas ar ro M( 0) E U ' pp ' p I p ( 0) 0 V( 0) E U ' pp '' p I p ( 0) 0 (5) (6)

3 Scintific Cooprations Journal of Civil Enginring and Architctur, Vol., Issu., August M( h) E U ' pp ' p I p ( h) 0 V( h) E U ' pp ' ' p I p ( h) B. Continuity conditions at intrfac of two soil layrs Th displacmnt and rotation at th soil layrs intrfac ar continuous: Upp( h) Upp ( 0) ' ' Upp ( h) Upp ( 0) Th continuity of momnt and shar yilds: M( V( h h ) ) M( 0) V( 0) (7) (8) (9) () () Using ths boundary and continuity conditions, Equations 3 and 4 can b solvd for th cofficints from D to D8. On th right sid of Equation 3 and 4, appar fr fild amplituds. Ths amplituds and thir drivativs at particular dpths can b obtaind from sit rspons analysis. Th diffrntial quation of wav propagation problm in two layr soil is solvd blow. C. Sit Rspons Analysis of Two layr soil systm To captur th rspons of fr fild ground motion, th soil is assumd to bhav as a Klvin-Voigt solid, th gnral wav quation is uff(,t)=a i(t+q*) + B -i(t+q*) (9) Applying local coordinats and uff(,t)=a i(t+q*) + B -i(t+q*) (0) uff(,t)=a i(t+q*) + B -i(t+q*) () whr uff and uff ar fr fild displacmnts at any dpth in th top and bottom soil layrs, rspctivly. Th constants A, B, A, and B ar found by using following boundary and continuity conditions: shar strss at ground surfac is ro, shar strsss in th two soil layrs at th intrfac ar qual, and displacmnt at th bottom of soil is qual to bdrock displacmnt. VI. PARAMETRIC STUDY Maximum pil bnding momnt which occurs at or closd to th intrfac of two soil layrs is th focus of this paramtric study. This momnt is normalid as Mn, max Mmax 4 ω d u Mmax = maximum pil bnding momnt (0) Mn,max = maximum normalid pil bnding momnt u bdrock acclrati on amplitud Th ffcts of th following paramtrs on th Mn,max ar valuatd. Ths paramtrs ar: ratio of xcitation frquncy to fundamntal frquncy of soil dposit (/ ), lngth to diamtr ratio of pil (L/d) ratio, ratio of th lastic modulus of pil to th lastic modulus of uppr soil layr (Ep/Es), ratio of th shar wav vlocitis of th two soil layrs (V/V), h/l, and h3/l. Following paramtrs ar considrd constant in th analysis Dnsity of top soil layr and bottom soil layr = 8 kn/m 3 Dnsity of pil =.6 dnsity of top soil layr Soil damping ratios = 0% Poisson s ratio of soil = 0.4 Amplitud of th bdrock acclration = m/s A. Effct of Frquncy of Excitation () To valuat th ffct of xcitation frquncy on th bnding momnt of pil at th two soil layrs intrfac, th following ight cass givn in Tabl ar considrd. Th h/l and h3/l ar considrd 0.5 and rspctivly for all th cass. Para- mtrs TABLE. PARAMETERS CONSIDERED FOR EVALUATION OF FREQUENCY EFFECTS ON BENDING MOMENT Cass Ep/Es V/V d/l Th normalid bnding momnts for all th cass ar shown as a function (/ ) in Fig. 3. Th bnding momnts along th pil lngth rsulting for diffrnt frquncis of xcitations ar also valuatd for a soil pil systm Ep/Es=000, L/d=0, Vs/Vs=/5, and h/l=0.5. Th soil pil systm is xcitd by,, and 0.5. Th rsults ar shown in Fig. 4. B. Effct of L/d, Ep/Es, V/V, and h/l Th variation of paramtrs considrd ar: h3/l=, Ep/Es= 000, 5000, 0000; V/V=, 0; H/L = 0.5, and 0.5., and a rang of L/d valus varying from 0 to 40. Th rsulting graphs ar shown in Fig. 5, and Fig. 6.

4 Scintific Cooprations Journal of Civil Enginring and Architctur, Vol., Issu., August Figur 3 Effct of frquncy ratio on Mn,max Figur 5 Effct of L/d on Mn,max for h/l=0.5 and h3/l= Figur 6 Effct of L/d on Mn,max for h/l=0.5 and h3/l= Figur 4 Effct of frquncy ratio on Mn along th pil lngth C. Effct of h3/l Th abov analysis ar rpatd for h3/l = 3. Th graphs ar shown in Fig. 7, and 8 for h/l=0.5 and h/l=0.5 rspctivly. Figur 7 Effct of L/d on Mn,max for h/l=0.5 and h3/l=3

5 Scintific Cooprations Journal of Civil Enginring and Architctur, Vol., Issu., August-05 4 th rat of incras in Mn,max with L/d is not vry pronouncd. 6. Effct of h/l: incras in th thicknss of th uppr softr soil layr compard to th lngth of pil significantly incrass th Mn,max. This conclusion is basd on comparing th Mn,max in Fig. 5 and Fig Rfrring to Fig. 7 and Fig. 8. Th ffct of all th paramtrs as dscribd in para-3 abov is studid for h3/l=3. It is sn that th Mn,max incrass with h3/l. This is xpctd as th fr fild motion incrass as th h3 incrass. REFERENCES Figur 8 Effct of L/d on Mn,max for h/l=0.5 and h3/l=3 VII. DISCUSSION. Rfrring to Fig. 3, th normalid maximum bnding momnt (Mn,max) of pil along its lngth is valuatd at diffrnt frquncy ratios (/). It is found that th Mn,max is gratst at frquncy ratio, /=. This signifis th gratr contribution of first natural mod of vibration to pil bnding momnt. This trnd is found for all th ight cass shown in Tabl. Scondly, bnding momnt is largst for cas-8 of tabl- for /=, and last for cas-. This show strong influnc of othr paramtrs on th bnding momnt. Influnc of xcitation frquncy on bnding momnt is also valuatd at diffrnt frquncis of xcitations (, 0.5, and ) and rsult shown in Fig.-4, which again shows th significant ffct of first natural mod of vibration (/=).. Rfrring to Fig.-5. Th ffct of various soil pil paramtrs on th Mn,max is shown in Fig.-5, and 6. Ths soil pil paramtrs ar Ep/Es, V/V, L/d, and h/l. h3/l= in all th cass. 3. As Ep/Es incrass for constant valu of V/V, th Mn, max incrass. This is bcaus th stiffnss of pil incrass with rspct to th top soil layr stiffnss. 4. Th ratio of th shar wav vlocity (V/V) of th soils considrd in th study, affcts Mn,max in similar fashion as discussd in para-3 abov. As th top soil layr bcoms lss stiff compard to bottom soil layr, th Mn,max incrass. 5. Effct of L/d: As th L/d ratio incrass, th Mn, max incrass. Howvr, for low valus of Ep/Es, or V/V, [] Miuno, H. (987). Pil damag during arthquaks in Japan. Dynamic Rspons of Pil Foundations Amrican Socity of Civil Enginrs, Nw York: (d. T. Nogami), pp [] Ahmad, M.H. El Naggar, and A.N. Khan, Artificial nural ntwork application to stimat kinmatic soil pil intraction rspons paramtrs, Journal of Soil dynamics and Earthquak Enginring, 7, 007, [3] Ahmad, M.H. El Naggar, and A.N. Khan, Maximum kinmatic pil bnding momnt in layrd soil profil: artificial nural ntwork approach, Go-Dnvr, Nw Paks in Gotchnics, Dnvr, Co, Fbruary 8-, 007. [4] Nikolauo A., Mylonakis G., Gatas G. (995). Kinmatic Bnding Momnts in Sismically Strssd Pils, NCEER Tchnical Rport, Buffalo, NY. [5] Gatas G, Fan K, Taoh T, Shimiu K, Kavvadas M, Makris N. Sismic pil-group structur intraction, Pils undr dynamic loads, Gotchnical spcial publication no. 34, 99. p [6] Kaynia, A. (997), Earthquak-inducd forcs in pils in layrd mdia, Gotchnical Spcial Publication, 70, ASCE, [7] Kavvadas M. and G. Gatas (993), Kinmatic sismic rspons and bnding of fr-had pils in layrd soil Gotchniqu, 43, No., 07-. [8] Gatas, G. & Dobry, R. (984a). Horiontal rspons of pils in layrd soils J. Gotch. Enggn Div. Am. Soc. Civ. Engrs. 0, No.,0-40. [9] Gatas, G. & Dobry, R. (984b). Simpl radiation damping modl for pils and footings. Horiontal rspons of pils in layrd soils. J. Gotch. Enggn Div. Am. Soc. Civ. Engrs., 0, No.6,