HORIZONTAL IMPEDANCE FUNCTION OF SINGLE PILE IN SOIL LAYER WITH VARIABLE PROPERTIES

Transcription

1 13 th World Confrnc on Earthquak Engnrng Vancouvr, B.C., Canada August 1-6, 4 Papr No. 485 ORIZONTAL IMPEDANCE FUNCTION OF SINGLE PILE IN SOIL LAYER WIT VARIABLE PROPERTIES Mngln Lou 1 and Wnan Wang Abstract: An quvalnt mthod s suggstd to analyz th horzontal mpdanc functon of sngl pl n horzontal sol layr wth varabl proprts. In ths approach, thr man stps ar ncludd. Frst, a smplfd mthod, whch s stablshd on th on-dmnsonal shar wav quaton and modal prturbaton tchnqu, s dvlopd to dtrmn th mods of th complcatd sol layr wth varabl proprts. Thn an quvalnt homognous sol layr s suggstd to nstad of th varabl-proprty sol layr. Th quvalnt physcal paramtrs of th quvalnt homognous sol layr ar dtrmnd by wghtd ntgral. Fnally, th mpdanc functon of th pl n th quvalnt homognous sol layr can b consdrd as th on of th complcatd sol layr approxmatly. Numrcal xampl vrfs that rasonabl prcson can b achvd. Kywords: Impdanc functon; sngl pl; sol layr 1. Introducton Pl foundaton s a popular typ of th dp foundaton of larg-scal structur. Th namc ntracton btwn sol, pl foundaton and structur undr ssmc xctaton s an mportant problm n th ssmc stablty rsarch of hgh-rsng buldngs and long-span brdgs. In th substructur mthod for sol-structur ntracton (SSI) problm, th mpdanc functon or matrx of th pl foundaton s usually ntroducd n th quaton of moton of th structur as th consdraton of th SSI ffct upon th namc bhavor. Many usful rsarch works hav bn don for dtrmnng th mpdanc functon or matrx of sngl and group pl foundaton [1-4]. owvr, most of ths works ar only avalabl to homognous half plan (or spac) and layrd sol. Fnt lmnt mthod (FEM) and boundary lmnt mthod (BEM) or othr numrcal mthods can b usd to solv th mpdanc functon of th pl foundaton n mor complcatd sol layr, but numrous calculaton has to b carrd out. In th ngnrng practc, th proprts of th sol mdum vary wth th sol dpth for most of sol st. Thrfor, t s an ntrstng work to dvlop a smplfd mthod wth hgh prcson for dtrmnng th mpdanc functon of pl foundaton for structur dsgn and analyss. As known, tm-lag has an mportant nflunc on th ffctvnss of th structural control. It also rqurs th smplfd mthod wth hgh prcson mthod to achv th fast computaton for th onln analyss f th sol-pl-structur ntracton should b consdrd n th ssmc control [5,6]. As a basc work, a smplfd mthod for calculatng th horzontal mpdanc functon of pl n horzontal varabl-proprty sol layr s dscussd n ths papr.. Impdanc Functon of Pl n omognous Sol Layr Th approach suggstd by Novak [1] s adoptd n ths papr to calculat th horzontal mpdanc functon of sngl pl n homognous sol layr. But th sol s assumd as lnar vscous-lastc mda nstad of lnar lastc mda n th approach. It mans that th computng formula of th mpdanc functon of th pl wll b sam, only th complx vscous-lastc modulus of th sol s usd to rplac th modulus of lastcty n th quaton of moton. 3. Modal Prturbaton Tchnqu Th partal dffrntal quaton of moton for dtrmnng th namc bhavor of th horzontal sol.. layr wth varabl proprty undr th horzontal ssmc xctaton u g ( can b wrttn as followng: u( y, u( y, u( y,.. ρ( + c( + [ ] = ρ( u ( g (1) t t y y whr u ( y, s th rlatv horzontal dsplacmnt to th rock surfac moton u g ( ; ρ (, c ( 1 Tong Unvrsty, Stat Ky Laboratory for Dsastr Rducton n Cvl Engnrng, P. R. Chna, profssor Tong Unvrsty, P. R. Chna, graduat studnt

2 and G ( ar th mass dnsty, dampng coffcnt and shar lastc modulus at vrtcal locaton y from th rock surfac, rspctvly. Applyng sparaton of varabls, th shar vbraton modal charactrstcs of th varabl-proprty sol layr can obtand from th followng quaton: d dϕ ( [ ] + λ ρ( ϕ ( = () Obvously, t s dffcult to solv analytcally Eq. () to gt th gnvalu λ and vbraton functon ϕ ( of th shar mod. Lt = G +, ρ( = ρ + ρ( (3) whr G and ρ ar arthmtc avrag of th shar lastc modulus and mass dnsty ovr th dpth of th sol layr rspctvly, 1 1 G =, ρ = ρ( th As known, th shar vbraton modal paramtrs λ and ( of th homognous sol layr wth shar lastc modulus G, mass dnsty ρ and sam dpth can b asly dtrmnd from th constant coffcnt ordnary dffrntal quaton: d ( G + λ ( = ρ (4) Basd on th modal proprty λ and ( of th homognous sol layr, th gnvalu λ and shar vbraton functon ϕ ( of th th mod of th varabl-proprty sol layr can b xprssd approxmatly as λ = λ + λ ϕ ( = ( + ( = ( + a ( (6) Aftr substtutng Eqs. (5) and (6) nto Eq. () and prmultplyng ( on both sds of Eq. (), an algbrac quaton wll b obtand by ntgratng from to along th layr dpth, whr λ n k= 1 k ( m δ + m k + λ [ n k = 1 k k ) + k= 1 k ( m δ + m n ) a m k = 1 k [( λ λ ) m δ + m k k k ] + ( λ m k k k k ] a ) = ρ = s th gnralzd mass of ϕ ( ; = m k ρ ( ( k (, = ' ' kk ( k ( (8) h m In Eq. (7), thr ar n unknown varabls, that s { a } = a 1, a, L, a 1, λ, a + 1, La n. Whn crculatng from 1 to n, a nonlnar algbrac quaton can b formd to dtrmn th varabl a. Thn th th gnvalu λ and modal shap functon ϕ ( can b obtand mmdatly from Eq. (5) and (6). If th scond ordr small quantty n Eq.(7) s nglctd, a vry smpl xprsson can b gottn: k λ m k λ m λ =, a = (9) m λ λ ) m vctor { } ( k T (5) (7)

3 4. Equvalnt omognous Sol Layr It had bn vrfd that th namc flxblty coffcnt of th varabl-proprty sol layr could b computd approxmatly by man of th quvalnt homognous sol layr [7]. Ths tchnqu wll b appld to solv th problm of th mpdanc functon of sngl pl n th horzontal sol layr wth varabl proprts. Usng ths quvalnt tchnqu, th problm to obtan th mpdanc functon of sngl pl n varabl-proprty sol layr s dsplacd by th problm of solvng th mpdanc functon of th sngl pl n th homognous sol layr wth sam dpth. It bcoms asy to solv th problm, bcaus thr hav bn svral thortcal and analytcal mthod to solv th lattr problm. Two basc prncpls wll b usd n th quvalnt procdur. Frst, th statc quvalnt prncpl s appld to fnd th quvalnt lastc modulus for th varabl-proprty sol layr. And thn th namc quvalnt prncpl, whch rqurs to kp th sam fundamntal frquncy for th varabl-proprty sol layr and ts quvalnt homognous sol layr, s usd to dtrmn th quvalnt mass dnsty. Dtal of th quvalnt mthod s dscrbd as followng. 4.1 Equvalnt Elastc Modulus As known, th lastc modulus of sol mdum s th ky factor to nflunc th statc dformaton of th sol layr. Thrfor, th frst thng to do n th quvalnt procdur s to choos th rasonabl quvalnt formula for dtrmnng Young s lastc modulus E and shar lastc modulus G. Thr formulas ar lstd for tral analyss and comparson. 1 (1) G = f ( f ( (1) α whr α s a constant. It s obtand from th condton that whn s qual to G, two sds of Eq. (1) must b sam, that s, = ' α f ( f ( 1 () G = f ( (11) α whr α = f ( (3) G 1 α = f ( f ( whr α = [ f ( ] f ( In abov formulas, G s th quvalnt shar modulus; f ( s a wghtd functon and s assgnd as th frst modal functon φ 1( y ) of th varabl-proprty sol layr that can b dtrmnd by th prturbaton mthod n abov scton. If G ( s dsplacd by E ( n Eqs. (1)-(1), th quvalnt Young s lastc modulus E can b obtand. (1) 4. Equvalnt Posson s Rato Th quvalnt Posson s rato can b dtrmnd from quvalnt Young s lastc modulus and shar modulus undr th sotropy assumpton of th sol mdum: E µ = 1 (13) G 4.3 Equvalnt Dampng Rato If dampng form of sol mdum s assumd as a hystrtc dampng, th complx shar modulus may b xprssd as: G ( = [ 1+ ζ ( ] (14) n whch, ζ ( s dfnd as th quvalnt hystrtc dampng factor; = 1. For th homognous sol layr, th complx shar modulus can b xprssd corrspondngly as:

4 G = G (1 + ζ ) (15) n whch ζ s dfnd as th quvalnt hystrtc dampng factor. Du to both of th dampng forc and th lastc rsstanc ar proportonal to th dsplacmnt ampltud, ζ can b dtrmnd by usng smlar quatons shown n Eqs. (1) to (1), xcpt usng ζ ( n plac of. 4.4 Equvalnt Mass Dnsty Accordng to th scond quvalnt prncpl, th quvalnt mass dnsty of th quvalnt sol layr s dtrmnd by: ρ = π hω 1 G (16) n whch ω 1 s th fundamntal frquncy of th varabl-proprty sol layr and can b calculatd by modal prturbaton tchnqu mntond last scton n ths papr. 5. Numrcal Exampl A horzontal sol layr wth 9 groups of varabl proprts shown n tabl 1 s takn as th numrcal xampl. In th tabl, z, E, V, ζ and µ ar th thcknss, Young s lastc modulus, shar wav vlocty, hystrtc dampng factor and Posson s rato of ach sol layr. Th frst layr s locatd at th top of th varabl-proprty sol layr. Tabl1 Th paramtrs of varabl-proprty sol layr Layr numbr Z (m) E ( MPa ) V ( m/ s ) ζ µ Th proprts of th quvalnt homognous sol layr, dcdd by usng thr basc Eqs. (1) to (1) rspctvly, ar lstd n Tabl,. Tabl Proprts of th quvalnt homognous sol layr Equvalnt basc quaton E (MPa) ( m / s) ζ µ (MPa) V G Eq.(1) Eq.(11) Eq.(1) Th pl, 34 mtrs n lngth and 1.4 mtrs n damtr, s a stl pp flld wth concrt. Th horzontal mpdanc functon of th pl n ths quvalnt homognous sol layrs can b asly obtand by usng th mthod suggstd by M. Novak [1]. Ths solutons ar takn as th approxmat horzontal mpdanc functon of th pl n th varabl-proprty sol layr. Th ral and magnary part of th mpdanc functons gottn from usng dffrnt quvalnt basc quaton, that s Eq. (1), (11) or (1), ar shown n Fgur 1 and.

5 K C f ( z ) f ( z ) a. Equaton (1) a. Equaton (1) K f ( z ) C f ( z) b. Equaton (11) b. Equaton (11) K D f ( z) f ( z ) c. Equaton (1) c. Equaton (1) Fgur 1 Ral part of mpdanc functon Fgur Imagnary part of mpdanc functon Othrws, th horzontal mpdanc functon had bn calculatd bfor ths rsarch by fnt lmnt mthod (FEM) and th smplfd mthod. Ths rsults ar takn for th comparson. All of comparsons of th currnt rsults and prvous rsults ar shown n th Fgurs 1 and. Th sold ln and th dashd ln rprsnt th rsults obtand by th FEM and approxmat mthod suggstd by G.Gaztas and R. Dobry [4], rspctvly. Th dot lns show th rsults obtand by ths papr usng basc Eq. (1), (11) and (1) rspctvly. From th rsults, t can b shown that, th choc of quvalnt quaton for lastc modulus has a lttl nflunc on th magnary part of th mpdanc functon but mor mportant ffct on th ral part of th approxmat rsults. From ths comparsons, th bst prcson s achvd n th cas shown n Fgur 1a and Fgur a. Bcaus th quvalnt quaton (1) dscrbs th rasonabl rlaton that th works by th ntrnal shar strss n both sol layrs ar qual to. So, th quvalnt quaton (1) s rcommndd to us n th quvalnt procdur. Comparng wth th approxmat mthod dvlopd by G. Gaztas and R. Dobry, th approxmat mthod suggstd by ths papr s mor asly to gt th numrcal rsults. Bcaus, t s not ncssary to calculat th statc stffnss and dflcton ln of th pl by FEM to dtrmn th radaton and matral dampng of sol mdum.

6 6. Concluson A smplfd mthod for dtrmnng th horzontal mpdanc functon of sngl pl n th horzontal sol layr wth varabl proprts s suggstd n ths papr. In ths mthod, th statc and namc quvalnt prncpls and th modal prturbaton tchnqu ar appld. Th numrcal rsults show th mthod has good prcson. 7. Acknowldgmnts Ths rsarch s sponsord by Natonal Natural Scnc Foundaton of Chna through Grant No Th support s gratfully acknowldgd. 8. Rfrncs 1. Novak, M., Dynamc Stffnss and Dampng of Pls, Can. Gotch. J. 1974, 11(1): Nogam, T. and Novak, M., Rsstanc of Sol to A orzontal Vbraton Pl, Earthquak Engnrng and Structural Dynamc,, 1977, 5(3): Novak, M. and Aboul-Ella, Impdanc Functons of Pls n Layrd Mda, J. Engnrng Mchancs Dvson, ASCE, 1991, 14(3): Gaztas, G. and Dobry, R., orzontal rspons of ps n layrd sols, J. Gotchncal Engnrng Dvson, 1984, 11(1): Lou, M. and Wu, J., Effcts of Sol-Structur Intracton on Structur Vbraton Control, Proc. of Symposum on Dynamc Sol-Structur Intracton, Bng, Chna, August, 1997, pp Wu, J. and Lou, M., Effcts of Sol-Structur Intracton on Actv Varabl Stffnss Control, J.of Tong Unvrsty, 1996, 4(Suppl.): Lou, M., An Approach for Dtrmnng Dynamc Flxblty of Layrd Nonhomognous Sol Foundaton, Chns J. of Computatonal Mchancs, 1998, 15(1): 75-79