1 Introduction. M.P. Santisi d Avila 1 and F. Lopez Caballero 2. Abstract

Transcription

1 Analyss o Nonlnar Sol-Structur Intracton Ects on th rspons o hr-dmnsonal Fram Structurs usn a On-Drcton hr-componnt Wav Propaaton Modl M.P. Sants d Avla 1 and F. Lopz Caballro 2 1 Laborator J.A. Dudonné Unvrsté d Nc-Sopha Antpols, Franc 2 Laborator MSS-Mat, CntralSupélc, Pars, Franc Abstract In ths papr, a modl o on-drctonal propaaton o thr-componnt ssmc wavs n a nonlnar multlayrd sol prol s coupld wth a mult-story multspan ram modl to consdr, n a smpl way, th sol-structur ntracton modlld n a nt lmnt schm. Modln th thr-componnt wav propaaton nabls th cts o a sol multaxal strss stat to b takn nto account. hs rduc sol strnth and ncras nonlnar cts, compard wth th axal strss stat. h smultanous propaaton o thr componnts allows th prdcton o th ncdnt drcton o ssmc loadn at th round surac and th analyss o th bhavor o a ram structur shakn by a thr-componnt arthquak. A paramtrc study s carrd out to charactrz th chans n th round moton du to dynamc aturs o th structur, or drnt ncdnt wavld proprts and sol nonlnar cts. A ssmc rspons dpndn on paramtrs such as th rquncy contnt o sol and structur and th polarzaton o ssmc wavs s obsrvd. Kywords: sol-structur ntracton, ram, wav propaaton, ssmc load, nt lmnt. 1 Introducton h ssmc rspons o structurs dpnds on round moton aturs and on mchancal proprts o structur and sol. h stron round moton at th surac o a sol basn, shakn th bas o a structur, s also nluncd by th dynamc proprts o th structur t-sl. h stratraphy o th sol prol and th mchancal aturs o th sol mody th ssmc wavs, propaatn rom th bdrock to th surac, and consquntly th ssmc loadn at th bas o structurs. Furthrmor, structur oscllaton at th sol surac mods th round 1

2 moton. hs ct, known as sol-structur ntracton (SSI), s nluncd by th drnc n prncpal rquncs o th sol and structur, du by drncs n trms o mass and stnss. h wav propaaton o an arthquak alon a sol prol, whr a r surac s assumd at th top, dos not allow tak nto account th cts o SSI (Saz t al. [13]). h rsarch, dscrbd n ths papr, s drctd towards th analyss o th bhavor o a systm composd o a ram structur ovr surac sol layrs, undr ssmc loadn. h on-dmnsonal (1D) multlayrd sol prol s dscrtzd by thrnod ln nt lmnts and a thr-dmnsonal (3D) consttutv rlatonshp dscrbs th nonlnar sol bhavor undr cyclc loadn. h 3D ram dscrtzaton s prormd by usn two-nod bam nt lmnts, wth sx drs o rdom pr nod. Fram structurs wth shallow oundaton, assumd to b rd, can b modlld as rdly connctd to th sol at th sol surac, whr th thr componnts o th ssmc moton ar transmttd rom th sol to ram bas. h layrd sol systm s modlld as prmary substructur and th ramd structur as mult-connctd scondary substructur, jond at th round surac lvl. h thr-componnt (3C) ssmc wav s propaatd alon a horzontally layrd sol basn rom sol-bdrock ntrac to th round surac whr a multstory mult-span ram structur s connctd. h mutual nlunc o sol and structur on thr rspons to ssmc loadn s studd. Modln th thr-componnt wav propaaton, th mchancal coupln o multaxal strss n th sol, nducn rducd sol strnth and ncrasn nonlnar cts, can b analyzd. h ncdnt drcton o th ssmc loadn at th round surac, shakn th ram structur bas, can b takn nto account. Drnt sol proprts and sol stratraphs, wth consqunt varaton n nonlnar proprts and mpdanc contrast btwn sol layrs, mody ssmc wavs by amplcaton or damplcaton cts. h ssmc rspons o sol prols can b sncantly drnt n th cass o r surac and th prsnc o a structur. h proposd modl s a drct soluton mthod, smultanously modln structur and round moton by a lobal dynamc qulbrum quaton or th sol-structur systm. A on-drctonal thrcomponnt (1D-3C) propaaton modl (cod SWAP_3C by Sants d Avla t al. [14], [15]) s adoptd, whr th thr componnts o ssmc wavs ar smultanously propaatd n on drcton, rom th sol-bdrock ntrac. An absorbn condton s assumd at sol-bdrock ntrac. A nonlnar consttutv rlaton o th Masn-Prandtl-Ishlnsk-Iwan typ (MPII) s adoptd or sol undr multaxal cyclc loadn and a lnar bhavor s adoptd or ram bams. h modl s not dpndnt on th adoptd consttutv rlatonshps. h thr componnts o ssmc moton ar valuatd at th bas o th 3D ram structur, takn nto account sol nonlnarty, mpdanc contrast n a multlayrd sol and sol-structur ntracton. h round moton s dducd consdrn th mpdanc contrast btwn sol and structur, du to thr drnc n trms o mass and stnss. h proposd 1D-3C propaaton modl takn nto account SSI s mplmntd n a cod calld SFRIN_3C (Sol-FRam INracton_3Componnts). A paramtrc analyss s undrtakn to obsrv th cts o SSI or drnt combnatons o sol aturs, dynamcal proprts o structurs and arthquak 2

3 rquncy contnt. Ground moton tm hstors at th surac, prols o strss, stran and moton componnts wth dpth and strss-stran hystrss loops at a xd dpth ar stmatd or th sol stratcaton. Prncpal rquncs and modal shaps o th ram structur ar valuatd, as wll as dormaton durn th tm hstory. 2 3-Componnt arthquak propaaton n nonlnar sol h thr componnts o th ssmc moton ar propaatd nto a multlayrd column o nonlnar sol rom th top o th undrlyn lastc bdrock, by usn a nt lmnt schm. Alon th horzontal drcton, at a vn dpth, th sol s assumd to b a contnuous, homonous and nnt mdum. Sol stratcaton s dscrtzd nto a systm o horzontal layrs, paralll to th xy plan, usn quadratc ln lmnts wth thr nods (Fur 1). hr s not stran varaton n x- and y-drcton. Shar and prssur wavs propaat vrtcally n z-drcton. 2.1 Spatal dscrtzaton h sol prol s dscrtzd nto n quadratc ln lmnts and consquntly nto n 2 n 1 nods (Fur 1), havn as drs o rdom th thr dsplacmnts n drctons x, y and z. Fnt lmnt modln o th horzontally layrd sol systm rqurs spatal dscrtzaton, to prmt th problm soluton, and th nonlnar mchancal bhavor o sol dmands tm dscrtzaton o th procss and lnarzaton o th consttutv bhavor n th tm stp. Accordnly, th ncrmntal qulbrum quaton n dynamc analyss, ncludn compatblty condtons, thr-dmnsonal consttutv rlaton and boundary condtons, s xprssd n th matrx orm as & & (1) M D C D K D F whr D s th assmbld 3 n -dmnsonal nodal dsplacmnt vctor o round, D and D ar th vlocty and acclraton vctors, rspctvly,.. th rst and scond tm drvatvs o th dsplacmnt vctor. 3n 3n M and K ar th assmbld -dmnsonal mass and stnss matrx, rspctvly. C and F ar th assmbld 3n 3n -dmnsonal dampn matrx and th 3 n -dmnsonal load vctor, rspctvly, drvd rom th mposd absorbn boundary condton, as xpland n Scton 2.2. h Fnt Elmnt Mthod, as appld n th prsnt rsarch, s compltly dscrbd n th works o Batoz and Dhatt [3], Cook t al. [5] and Rddy [17]. h mass matrx dmnsonal matrcs as whch ar xprssd by M and stnss matrx M and K rsult rom th assmbla o K, rspctvly, corrspondn to th lmnt, 3

4 h h M A N N d z K A B E B d z (2) a,b 0 0 whr h s th nt lmnt lnth n vrtcal drcton, A s th lmnt ara n th horzontal plan and s th sol dnsty assumd constant n th lmnt. Fur 1: Spatal dscrtzaton o a horzontally layrd sol, orcd at ts bas by a 3-C arthquak and wth a ram structur at th top. h tannt consttutv 6 6 -dmnsonal matrx E, n quaton (2)b, s valuatd by an ncrmntal consttutv rlatonshp as σ E ε, whr th 6 - dmnsonal strss and stran vctors ar rspctvly dnd as ollows: σ ε x x y y x y zy zx zz zy zx zz (3) a,b accordn to th hypothss o nnt horzontal sol. h wav propaaton modl s not dpndnt on th adoptd consttutv rlatonshp or sols. In quaton (2)a, N z s th 3 9 -dmnsonal shap uncton matrx, such that u z, t N z d t (Cook t al. [5]), whr th thr trms o u ar th sol dsplacmnts n x -, y - and z -drcton and 4

5 d u u u u u u u u u (4) x y z x y z x y z s th vctor o dsplacmnts n drctons x, y and z o th thr nods o lmnt. Quadratc shap unctons, trms o th shap uncton matrx N, corrspondn to th thr-nod ln lmnt usd to dscrtz th sol column, ar dnd accordn to Cook t al. [5]. h trms o th 6 9 -dmnsonal matrx B z, n quaton (2)b, ar th spatal drvatvs o th shap unctons, accordn to compatblty condtons and to th hypothss o no stran varaton n th horzontal drctons x and y. Gvn that th stran vctor s rlatd to dsplacmnt vctor as ε u and s a matrx o drntal oprators dnd n such a way that compatblty quatons ar vrd, consquntly, t s ε B d and B N. z t h dampn matrx C and load vctor condtons and ar dnd n Scton Boundary and ntal condtons F n quaton (1) dpnd on boundary h systm o horzontal sol layrs s boundd at th bottom by a sm-nnt lastc mdum rprsntn th ssmc bdrock (Fur 2). h ollown condton, mplmntd by Joynr and Chn [11] n a nt drnc ormulaton and usd by Bardt and obta [2], s appld at th sol-bdrock ntrac to tak nto account th nt rdty o th bdrock: p σ c u u & 2 & (5) 1 0 h strsss normal to th sol column bas at th bdrock ntrac ar s a 3 3 h paramtrs b p σ and c -dmnsonal daonal matrx whos trms ar v, v and b sb b sb b p b, v and v ar dnsty, shar and prssur wav vlocts n sb pb th bdrock, rspctvly. Accordn to quaton (5), th dampn matrx 1 load vctor F, or th rst lmnt 1, ar dnd by v. 1 C and th 1 1 C A N c N F A N c 2 u& (6) a,b z0 z0 C and F ar a zro-matrx and vctor, rspctvly, or th othr lmnts all ovr th sol prol. h thr trms o vctor u& ar th vlocts at th sol-bdrock 1 ntrac (nod 1 n Fur 1) n x -, y - and z -drcton, rspctvly. rms o th 3 -dmnsonal vctor u& ar th nput ncdnt vlocts, n th undrlyn lastc 0 mdum, n drctons x, y and z, rspctvly. h 3-Componnt halvd outcroppn bdrock snals u& (Fur 2) ar propaatd. 0 5

6 h absorbn boundary condton (5), assumd at th sol-bdrock ntrac, allows nry to b radatd back nto th undrlyn mdum. hs condton can b asly modd to us downhol rcords, assumn an mposd moton at th bas o th sol prol (rst nod n Fur 1), accordn to Sants d Avla and Smblat [16]. h sol prol s boundd at th bottom by th lastc bdrock and t s connctd to th ram structur at th top. Global qulbrum quaton or th sol-ram systm s drctly solvd, by mposn boundary condtons only at th sol-bdrock ntrac. Gravty load s mposd as statc ntal condton n trms o stran and strss n ach nod. Fur 2: Rprsntaton o ssmc snals at drnt ponts o th modl schm: outcroppn bdrock, bas o th sol prol, r surac at th top o sol prol, bas and top o th ram structur. 2.3 Consttutv modl Modln th propaaton o a thr-componnt arthquak n stratd sols rqurs a thr-dmnsonal consttutv modl or sol. h so-calld Masn- Prandtl-Ishlnsk-Iwan (MPII) consttutv modl, sustd by Iwan [9] and appld by Joynr [10] and Joynr and Chn [11] n a nt drnc ormulaton, s usd n th prsnt work to proprly modl th nonlnar sol bhavor n a nt lmnt schm. h MPII modl s usd to rprsnt th bhavor o matrals satsyn Masn crtron [12] and not dpndn on th numbr o loadn cycls. h strss lvl dpnds on th stran ncrmnt and stran hstory but not on th stran rat. hs rholocal modl has no vscous dampn. h nry dsspaton procss s purly hystrtc and dos not dpnd on th rquncy. Iwan [9] proposs an xtnson o th standard ncrmntal thory o plastcty (Fun [6]), modyn th 1D approach by ntroducn a amly o yld suracs. H modls nonlnar strss-stran curvs usn a srs o mchancal lmnts, havn drnt stnss and ncrasn sldn rsstanc. h MPII modl taks nto 6

7 account th nonlnar hystrtc bhavor o sols n a thr-dmnsonal strss stat, usn an lasto-plastc approach wth hardnn, basd on th dnton o a srs o nstd yld suracs, accordn to von Mss crtron. h shar modulus s stran-dpndnt. h MPII hystrtc modl or dry sols, usd n th prsnt rsarch, s appld or strans n th ran o stabl nonlnarty. h man atur o th MPII rholocal modl s that th only ncssary nput data, to dnty sol proprts n th appld consttutv modl, s th shar modulus dcay curv G vrsus shar stran. h ntal lastc shar modulus G v 0 s mdum 2, dpnds on th mass dnsty and th shar wav vlocty n th v. h P-wav modulus s M v 2 p, dpndn on th prssur wav vlocty n th mdum v, charactrzs th lontudnal bhavor o sol. p In th prsnt study th sol bhavor s assumd adquatly dscrbd by a hyprbolc strss-stran curv (Hardn and Drnvch [7]). hs assumpton ylds a normalzd shar modulus dcay curv, usd as nput curv rprsntn sol charactrstcs, xprssd as G G 1 1 (7) 0 r whr s a rrnc shar stran provdd by tst data corrspondn to an actual r tannt shar modulus quvalnt to 50% o th ntal shar modulus. h appld consttutv modl (Iwan [9]; Joynr 1975 [10]; Joynr and Chn 1975 [11]) dos not dpnd on th hyprbolc backbon curv. It could ncorporat also shar modulus dcay curvs obtand rom laboratory dynamc tsts on sol sampls. Accordn to Joynr [10], th actual stran lvl and th stran and strss valus at th prvous tm stp allow to valuat th tannt consttutv matrx E n quaton (2) and th strss ncrmnt σ (Sants d Avla t al. [14]). 3 3D ram structur modln undr 3C ssmc loadn h rspons o a rular ram composd by horzontal and vrtcal bam lmnts, alon thr orthoonal drctons x, y and z, shakn by th thr componnts o a ssmc moton, s modlld. Bams ar assumd composd by a contnuous and homonous mdum wth constant cross-scton alon thr lontudnal axs. h hypothss o plan cross-scton, not ncssarly prpndcular to th bam axs, s assumd or bams durn dormaton. Bam cross-sctonal paramtrs ar th constant ara A, th momnts o nrta I and y I wth rspct to y and z axs, rspctvly, shap actors and or transvrs z y z shar and th scond momnt o ara J. Matral paramtrs ar th comprsson modulus E, shar modulus G E 2 1, whr s th Posson s rato, and mass dnsty. 7

8 3.1 Spatal dscrtzaton h 3D ram structur s modlld by a systm o on-dmnsonal 2-nod bam lmnts. Each nod has 6 drs o rdom n th xyz lobal coordnat systm, that ar th dsplacmnts n x-, y- and z-drcton and rotatons around th sam axs. h 6-dmnsonal dsplacmnt vctor o a nrc nod n a bam, paralll to x -axs, s dnd as u u u u (8) x y z x y z h corrspondn 6-dmnsonal vctor o nontrval strans or a 3D bam s ε (9) xx xz xy h ncrmntal orm o dynamc qulbrum quaton o th analyzd 3D ram n matrx orm s (10) M D C D K D 0 h dmnson o quaton (10) s 6 n b, whr n s th numbr o nods o th b ram ncludn th bas. D, D and D ar th 6 n b -dmnsonal assmbld vctor o nodal dsplacmnt, vlocty and acclraton, rspctvly. h consstnt mass matrx M, th dampn matrx C and stnss matrx (Rddy [17]) accordn to th omtry o th ram. h consstnt mass matrx dmnsonal matrcs as 6n 6n b K ar assmbld b -dmnsonal M and stnss matrx K rsult rom th assmbla o M and K, rspctvly, o ach bam lmnt, n lobal coordnats xyz, whch ar xprssd as (Batoz and Dhatt [3]) M Λ M Λ K Λ K Λ (11) a,b whr Λ s th dmnsonal rotaton matrx, assocatd to bam, that allow to transorm dsplacmnts and rotatons rom local coordnats xyz to lobal coordnats xyz. M and K, xprssd n local coordnats xyz, or a 2-nod bam lmnt o lnth L n x drcton, ar L L M A N N d x K B E B d x (12) a,b 0 0 whr A s th bam cross-sctonal ara and s th matral dnsty assumd 8

9 constant n th lmnt. A 2-nod ntrdpndnt ntrpolaton lmnt (Rddy [17]) s usd n a nt lmnt schm, basd on Hrmt cubc ntrpolaton o dsplacmnts and an ntrdpndnt quadratc ntrpolaton o rotatons, so that dsplacmnt rst drvatv s a polynomal wth th sam dr o rotatons. h dmnsonal shap uncton matrx N s dnd accordn to th transormaton u x, t N x d t, or ach bam lmnt havn nods j and l, whr j j j j j j l l l l l l u u u u u u j l x y z x y z x y z x y z d d d (13) s a 12-dmnsonal tm dpndnt nod dsplacmnt vctor. Shap unctons n matrx N ar dnd accordn to Rddy [17]. h dmnsonal matrx B, n quaton (12)b, s dnd n such a way that compatblty quatons ε u ar vrd, whr s a matrx o drntal oprators. h 6 6 -dmnsonal consttutv matrx E, n quaton (12)b, s dnd accordn wth an ncrmntal consttutv rlatonshp such as S E ε, whr N V V M M M S (14) y z x y z s th vctor o ntrnal orcs. h ram modl s ndpndnt rom th slctd consttutv law. In ths rsarch, a lnar bhavor or th ram structur matral s assumd, wth d a E A, G A, G A, G J, E I, E I y z y z Dampn matrx Scton 3.3. E. C n quaton (10) dpnd on modal analyss and t s dnd n 3.2 Boundary and ntal condtons h 3D ram s rdly connctd at th round surac. Consquntly, rotatons o th ram bas nods ar assumd null. h shallow oundaton s assumd to b rd. Accordnly, all nods o th ram bas ar supposd to b submttd to th sam round moton. Statc loadn conuraton rprsnts th ntal condton or th ram structur. h lnar lastc soluton o statc qulbrum quaton o th analyzd 3D ram s K D F R (15) whr F s a 6 n b -dmnsonal vctor o statc nodal loads, obtand by assmbln th 6-dmnsonal load vctors o ram nods 9

10 F F F M M M (16) x y z x y z whch trms ar th xtrnal orcs drctly appld n ach nod. h 6 dmnsonal vctor R s assmbld by usn th 12 -dmnsonal racton orc vctors that ar R Λ R, or ach bam, n lobal coordnat systm. h trms o vctor R ar racton orcs to loads n bam, xprssd n local coordnats xyz. 3.3 Modal analyss h systm o quatons (10) s composd by 6 n quatons rlatd to th moton o ram bas nods and 6 n quatons corrspondn to th moton o th othr ram nods. Accordnly, quaton (10) can b wrttn as b n - b M M D C C D K K D b b b b b b b b b b b b 0 t t t M M D C C D K K D b b b 0 (17) whr and b ndcat ach trm rlatd wth ram and boundary (sol-ram ntrac), rspctvly. Fundamntal xd-bas rquncs o th ram structur ar obtand solvn th 2 K M Φ 0, whr ar th natural 6 n -dmnsonal nproblm anular rquncs and Φ s th modal matrx. h mass-normalzaton o matrx Φ mpls a 6 n Φ M Φ I and -dmnsonal dntty matrx. Dampn matrx can b valuatd as [4]), whr 2 d a 0 2,..., n Φ K Φ 2 Ω, whr d a,... 1 n Ω and I s 1 1 C Φ Ξ Φ M Φ Ξ Φ M (Chopra Ξ Ω. h dampn rato s know 0 or typcal matrals mployd n rular 3D rams. It s assumd or 0 typcal rnorcd concrt buldns. Othrws, dampn matrx can b stmatd by th Raylh approach, as C a M a K, dpndn on mass and 0 1 stnss matrcs, wth a 2 and 2 and C ar null. b Sol-structur ntracton modln a C, C b h 3-Componnt moton at th bas o ram structur s assumd concdnt wth round moton at th top o sol prol (Fur 3). h sam loadn moton s appld to all column bass, rducn drs o rdom o th ram bas to only bb 10

11 thr dsplacmnts at th sol-ram ntrac lvl ( 6n 3). Rd rotatons o th b oundaton ar assumd null, supposn that surac wavs ar nlbl, accordn to th mployd 1D wav propaaton modl. hs analyss consdrs nrtal ntracton modyn th round moton, du to nrta orts nducd by structur mass at th sol-structur conncton lvl. Knmatc ntracton, nducd by stnss varaton btwn sol and structur oundaton at th most surac sol layrs, s assumd nlbl or shallow oundatons and vrtcal propaaton. Fur 3: Spatal dscrtzaton o sol (3-nod ln lmnts) and ram (2-nod bam lmnts) wth a conncton nod and lastc boundary condton. h qulbrum condton undr dynamc loadn or th sol-ram systm s dnd accordn to th concpt o coupln a prmary substructur wth a multconnctd scondary substructur, jond at th round surac lvl (Fur 1). Equatons (1) can b wrttn as M M D C C D K K D b b b M M b b b D C C b b b b D K K b b b b D b 0 (18) ndcatn wth and b ach trm rlatd wth round and boundary (ntrac btwn sol and ram), rspctvly. C, C and C ar null. h only non- b b zro trms n C and ar thos rlatd to th nod at sol-bdrock ntrac, accordn to th adoptd boundary condton (s Scton 2.2). h total dmnson bb 11

12 o quaton systm (18) s 3 n 6 n 3. It s dcomposd n two blocks o 6 n quatons rlatd to sol moton varabls, wth n th numbr o nt lmnts mployd n th sol dscrtzaton, and 3 quatons assocatd to round moton at th surac. Equaton (17) s rwrttn as M M D 0 0 D K K D b b b b b b b b b 0 t t t M M b D 0 C D K K b D 0 (19) consdrn that C, b bb C and C ar null (s Scton 3.3). b h total dmnson o quaton systm (19) s 6 n 6 n 6 n 3 6 n. It s b b composd o two blocks o 3 quatons rlatd to th moton o ram bas, accordn to th hypothss o qual ssmc loadn at th bas o all ram columns and nlbl bas rotaton, and 6 n quatons corrspondn to ram nod moton. A condnsaton o 6 n b drs o rdom o bas nods s appld, allown to consdr only 3 drs o rdom. Combnn quatons (18) and (19), w obtan th ollown qulbrum quaton: M D C D K D F (20) whr th (6 n 3 6 n ) (6 n 3 6 n ) -dmnsonal mass, dampn and stnss matrcs ar, rspctvly, M M 0 C 0 0 K K 0 b b M M M M M C K K K K K b b b b b b b b b b b b 0 M M 0 0 C 0 K K b b (21) h load and dsplacmnt ncrmnt vctors ar, rspctvly, F 0 0 D D D D b t (22)a,b h 6 n -dmnsonal rlatv dsplacmnt vctor o ram nods s valuatd as t s D D D, whr t D s th vctor o total dsplacmnt ncrmnt and s 1 D K K D s th dsplacmnt vctor du to th statc applcaton o b b bas nod dsplacmnt D (Chopra [4]). b 12

13 4.1 m dscrtzaton m ntraton s don accordn to Nwmark s procss. h ncrmntal dynamc qulbrum quaton (20) can b wrttn as (23) k k k k k k k M D C D K D F accordn to tm dscrtzaton. h subscrpt k ndcats th tm stp t and k th traton o th problm solvn procss. Equaton (23) bcoms % (24) k k k k K D F A h quvalnt stnss matrx and th quvalnt load vctor ar, rspctvly, 2 t t K% 1 M C K k k k k t & A 1 M C D 1 2 M 2 1 t C D& k k k k 1 k k k 1 (25)a,b Equaton (24) rqurs an tratv solvn, at ach tm stp k, to corrct th 1 tannt stnss matrx K. Startn rom th stnss matrx K K, k k k 1 valuatd at th prvous tm stp, th valu o matrx K s updatd at ach k traton. An lastc lnar bhavor s assumd or th rst traton at th rst tm stp. h nodal dsplacmnt D D D s obtand and stran k k 1 k ncrmnts ar dducd rom th dsplacmnt ncrmnts D D D. Strss k k k 1 ncrmnts and th tannt consttutv matrx consttutv rlatonshps or th sol. Matrcs K, k E ar obtand throuh th k C and th quvalnt stnss k matrx K % ar thn calculatd and th procss rstarts. h corrcton procss k contnus untl th drnc btwn two succssv approxmatons s rducd to 1 a xd tolranc, accordn to D D D, whr k k k otal nodal vlocty and acclraton ar valuatd by t 1 2 D& D& D D& t D& 2 D& D& 1 1 t D 1 t D& k k k k D& k 1 k k 1 k k 1 k 1 (26) Atrwards, th nxt tm stp s analyzd. h hypothss o lnar acclraton n th tm stp s assumd and th choc o th two paramtrs and 0.6 uarants uncondtonal stablty o th tm ntraton schm and numrcal dampn proprts to damp hhr mods (Huhs [8]). 13

14 5 Analyss o th local sol-structur ntracton h nlunc o 3-Componnt shakn vs 1C moton and local cts dus to mpdanc contrast n multlayrd sol ar xtnsvly dscrbd by Sants t al. [14], whr th sam 1D-3C wav propaaton modl s usd. In ths rsarch, coupln o sol and ram s nvstatd, to show th ntracton cts rproducd by a on-drctonal wav propaaton modl assmbld wth 3D multstory mult-span ram modl. Concrnn th ssmc loadn, th acclraton snal adoptd n ths analyss s a rcord rom th 2009 L Aqula arthquak n Cntral Italy. h halvd acclraton (Fur 4), havn pak round acclraton (PGA) qual to 1.14 m/s 2, s ntratd and orcd at th bas o sol prol. Fur 4: On-componnt acclraton rcord rom th 2009 L Aqula arthquak. h proposd modl s vrd by comparson wth GEFDyn cod (Aubry t al. [1]), n th cas o 1C-propaaton, lnar bhavor o sol and structur havn on dr o rdom. 5.1 SSI vs r ld condton A on-story on-span ram on th homonous sol prol S1 s analyzd. Stratraphy and sol proprts, as th dnsty and shar and prssur vlocty n th mdum, v and, rspctvly, ar rportd n abl 1. Bam lmnt s v p dmnsons, vrtcal load, dampn rato and th adoptd matral proprts 0 (comprsson modulus E, Posson rato and dnsty ) ar dscrbd n abl 2 (ram F). h nlunc o sol column cross-sctonal ara n SSI ct s nvstatd, by valuatn th sol-surac-bdrock transr uncton (F), th top-ram-sol-surac F and acclraton, vlocty, shar strss and stran prols wth dpth, or drnt valus o th sd a o sol squar cross-scton, assumd constant wth dpth. h prncpal rquncy o sol prol v 4 H H z s rproducd n all th cass, wth v and H th shar vlocty n th mdum and th sol column s hht, rspctvly (Fur 5a). Whn adoptn a 1 k m, th SSI s not obsrvd. h sol mass s prdomnant and a soluton clos to th r-ld condton s s 14

15 obtand (FF curv n Fur 5). On th contrary, whn adoptn a 3 0 m, th ct o th structur s ovrstmatd. Shar strans and strsss appar not null at th sol-ram ntrac (Fur 5c), that rprsnts a not ralstc soluton. h xd bas bhavor s wll rproducd or a 5 0 m. A sol column wth sd a 5 0 m s adoptd n th ollown computatons. Fur 5: (a) Sol-surac-bdrock ransr Functon. (b) op-ram-sol-surac ransr Functon. (c) Acclraton, vlocty, strss and stran prols wth dpth. z H v p v s z H v p v s m m k/m 3 m/s m/s m m k/m 3 m/s m/s > > abl 1: Proprts o th homonous sol prols S1 (lt) and R1 (rht). Usn a 1D sol modl, th sol prcvs th buldn as an lmntary oscllator (on dr o rdom) charactrzd by ts mass and stnss. h numbr o storys and spans mods th total mass and stnss and, consquntly, th SSI 15

16 ct, but th nlunc o an ncrasn loor ara, du to an ncrasn numbr o spans, s not capturd by a 1D sol modl. For ths rason, th sam sol ara s assumd n all cass n ths analyss. Fram Floors Spans Spans Scton L E 0 x y cm m N/mm 2 k/m 3 kn/m % F x R x S x x abl 2: Proprts o analyzd ram structurs. 5.2 Inlunc o rquncy contnt h snl-rquncy Mavrods-Papaorou wavlt s adoptd to study th ct o rquncy contnt o round moton, shakn rd and sot sol prols, coupld wth rd and sot ram structurs at th top. h outcrop moton s obtand by th ollown xprsson: u t u 2 1 c o s 2 n t t 2 c o s 2 t t 2 0 t 2 t (27) 0 0 m a x whr u, and t n 0 m a x 2 ar th acclraton pak, prncpal rquncy and 2 duraton, rspctvly. h numbr o paks s assumd n 5 and u 3.5 m s. 0 m a x Acclraton snals ar halvd to rmov th r surac ct and ntratd, to obtan ncdnt vlocts, bor bn orcd at th bas o th sol prol. h nlunc o arthquak rquncy contnt n structural rspons s analyzd studyn th bhavor o a thr-story on-span rd ram R (prncpal rquncy 5.8 H z ) shakd by an 1C ncdnt wav havn rquncy 6 H z 0, n both cass o rd sol R1 ( H z ) and sotr sol S1 ( H z ). hs stuaton s compard wth th bhavor o a sotr ram S (prncpal rquncy 3.5 H z ) shakd by an ncdnt wav havn rquncy 3.5 H z, n both 0 cass o rd sol R1 and sotr sol S1. Bam lmnt dmnsons, vrtcal load, dampn rato and th adoptd matral proprts ar dscrbd n abl 2 or 0 th analyzd rd and sotr ram. hs analyss conrms that a rd ram n a sot sol s lss strssd by a ssmc loadn havn a rquncy contnt clos to th ram prncpal rquncy, than n th cas o rd sol (havn prncpal rquncy clos to that o ram and quak). Smlarly, a sot ram n a rd sol s lss strssd by a quak havn a rquncy contnt clos to th ts prncpal rquncy, than n th cas o sotr sol (havn prncpal rquncy clos to that o ram and quak). Numrcal rsults n trms o horzontal acclraton at th top o th ram structur ar showd n abl 3. 16

17 (Fram) (Sol) 0 (Quak) ü Hz Hz Hz m/s 2 R-R R-S S-S S-R abl 3: Horzontal acclraton at th top o th ram structur. 5.3 Inlunc o sol nonlnarty Nonlnar aturs ar charactrzd, n th adoptd MPII consttutv modl or sols, by th shar modulus dcay curv. A hyprbolc rst loadn curv s assumd and th shar modulus dcay curv s dnd by Equaton (7). h shar stran r, rlatd wth a 50% dcay o shar modulus, s vard to ncras nonlnarty n th multlayrd sol prol (abl 4) and obsrv th assocatd cts. h appld ncdnt wav s th 1C snal shown n Fur 4. Fur 6: (a) Sol-surac-bdrock ransr Functon. (b) Shar strss-shar stran hystrss loop. (c) Acclraton, vlocty, strss and shar stran prols wth dpth. 17

18 Nonlnar cts lad to strnth rducton, stran ncrasn, and dcrmnt o prncpal rquncs (Fur 6). Nonlnar bhavor can rduc round moton paks at th round surac. z H v p v s m m k/m 3 m/s m/s > abl 4: Proprts o a multlayrd sol 5.4 Ssmc wav polarzaton vs ram plan rularty h cas o a thr-story on-span ram structur undr ssmc loadn wth drnt polarzaton s analyzd. Rctanular cross-scton o bam lmnts s 30x90cm and th othr proprts ar rportd n abl 2 (ram ). All columns ar n th sam poston. hs ram s str n x -drcton, snc columns hav mor nrta aanst horzontal actons n ths drcton. h sam ncdnt wav moton rprsntd n Fur 4 s appld n x -and y -drcton as 1-Componnt moton, and smultanously as a 2-Componnt ssmc loadn. h sol rrnc shar stran s r Fur 7: Sol-surac acclraton n th drctons whr th ram s str (a) and sotr (b), or 1- and 2-Componnt moton. A rducton o PGA s obsrvd rom on to two componnts (rom 2 2 u to u ). h PGA s ratr n y -drcton (rom x m a x m s 2 u m s to y m a x x m a x m s 2 u m s ), whr th ram s sotr, and lowr n x - y m a x 18

19 drcton, whr th ram s str. Acclratons at th sol surac and at th top o th ram structur ar lowr n th drcton whr th structur s str. 6 Conclusons A modl o on-drctonal thr-componnt ssmc wav propaaton n a nonlnar multlayrd sol prol s coupld wth a mult-story mult-span ram modl to consdr th sol-structur ntracton n a nt lmnt schm. Computaton tm s lowr, compard wth a 3D spatal dscrtzaton o sol and th boundary condton at th sol-bdrock ntrac s dnd n only on nod. Modln th smultanous thr-componnt wav propaaton nabls th analyss o th sol multaxal strss stat that rducs th sol strnth and ncrass nonlnar cts. h varaton o ncdnt drcton o ssmc loadn at th round surac can b takn nto account and th bhavor o a ram structur shakn by a thr-componnt arthquak can b obsrvd. A snstvty analyss s carrd out to dn th approprat sol column crosssctonal ara, allown to apprcat th Sol-Structur-Intracton cts, wthout ovrstmat th nlunc o structur. Usn a 1D sol modl, th sol prcvs th buldn as an lmntary oscllator (on dr o rdom) charactrzd by ts mass and stnss. h numbr o storys and spans mods th total mass and stnss and, consquntly, th SSI ct, but th nlunc o an ncrasn loor ara s not capturd by a 1D sol modl. Local cts n th sol, dus to nonlnar bhavor, mpdanc contrast btwn layrs and wavld polarzaton, ar rproducd by th proposd modl. h 3D modl o th ram structur allows to stmat th rspons to a ssmc acclraton at th bas and to valuat dsplacmnt componnts and ntrnal orcs. A lnar bhavor s assumd n ths analyss or bams, but th proposd modl s not dpndnt on th adoptd consttutv rlatonshp. hs 1D-3C-propaaton-3D-ram modl allows to conrm that a buldn s lss strssd by a ssmc wav havn a rquncy contnt clos to th buldn prncpal rquncy, t s placd on a sol wth a vry drnt prncpal rquncy, rathr than n th cas whr sol and structur rquncy contnt ar clos tothr. h acclraton at th sol surac and at th top o a buldn ar rducd n th drcton whr th ram structur s str aanst horzontal actons. Furthr work would rqur a thr-dmnsonal spatal dscrtzaton o sol, to tak nto account th nlunc o buldn loor ara and cts o spatal varablty n th ssmc loadn. Acknowldmnts hs rsarch has rcvd undn rom th Europan Unon's Svnth Framwork Proram (FP7/ ) undr rant armnt n (INEGER projct) as part o th PEPS Éalté Acton (MISS projct). For th scond author, a part o th rsarch rportd n ths papr has bn 19

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