Observations versus Analyses: Lateral Earth Pressures on an Embedded Foundation during Earthquakes and Forced Vibration Tests

Transcription

1 Procdings Third UJNR Workshop on Soil-Structur Intraction, March 9-3, 4, Mnlo Park, California, USA. Obsrvations vrsus Analyss: Latral Earth Prssurs on an Embddd Foundation during Earthquaks and Forcd Vibration Tsts Michio Iguchi, a) and Chikahiro Minowa b) Th charactristics of arth prssurs during forcd vibration tsts and arthquaks ar studid on th basis of obsrvations rcordd on both sids of a larg-scal shaking tabl foundation. It is found that th arth prssurs on both sids of th foundation during arthquaks tnd to b inducd in phas for lowr frquncis containd in th surfac ground motions, and out-of phas for highr frquncis. It is strssd that th in-phas phnomnon of th arth prssurs can not b xplaind by a convntional assumption of vrtical incidnc of sismic wavs. It is also rvald that th arth prssurs ar inducd in rlation to th horizontal vlocitis of th foundation for rathr lowr frquncis, and to acclration rspons for highr frquncis. Ths obsrvations can wll b simulatd by a simplifid lumpd-mass modl connctd in sris quivalntly substitutd for th latral layrd soil of th foundation. INTRODUCTION Transmission of ground motions into a structur and rsistanc of th supporting soil during vibration of a suprstructur ar soil-structur intraction phnomna accompanying with gnration of dynamic arth prssurs on th foundation. Th obsrvation of arth prssurs during arthquaks, thrfor, would play a ky rol in undrstanding of th transmission mchanism of ground motions as wll as rsistanc mchanism of th surrounding soil for a suprstructur. Th invstigations of arth prssurs during arthquaks hav bn conductd so far on th basis of thortical, obsrvational and xprimntal points of viw. Abov all, obsrvations of arth prssurs for actual structurs or with us of modl structurs mbddd in an actual soil hav bn providing valuabl data in clarifying th gnration mchanism of arth prssurs on mbddd foundations. Basd on ths obsrvations, th frquncy charactristics of arth prssurs, th distribution of prssurs on latral sids of mbddd foundations, phas charactristics of arth prssurs inducd on two opposit sids of th mbddd foundation hav bn xtnsivly studid. A dtaild rviw about a) Scinc Univ. of Tokyo, Faculty of Scinc and Enginring, Yamazaki 64, Noda City 78-85, Japan. b) Indpndnt Administrativ Institut, National Rsarch Institut for Earth Scinc and Disastr Prvntion, Tsukuba City 35-8, Japan.

2 masurmnts of arth prssurs during arthquaks has bn prsntd by Ostadan and Whit (998). It is worthwhil noticing that following distinctiv faturs on th arth prssurs hav bn prsntd and discussd on th basis of arthquak obsrvations; () Th obsrvations includ such that arth prssur on th opposit sids of th mbddd foundation during arthquaks wr inducd in-phas, i.. pull-and-pull or push-and-push phnomna wr obsrvd (Sakai t al. 996, Uchiyama t al. 999, Minowa t al. ). It is obvious that th fact cannot b xplaind by a convntional assumption of a vrtical incidnc of sismic wavs. () As for th arth prssurs xamind in rlation to rsponss of foundation, thr hav bn prsntd som obsrvations showing diffrnt tndncis. On is that th arth prssurs on th sids of mbddd foundations ar strongly rlatd to th vlocity motions of th foundation (Matsumoto t al. 99, Sakai t al. 996), th othrs ar with acclration motions of th foundation (Onimaru t al. 994). Thr is also an obsrvation showing that th arth prssur is causd by th rlativ motion btwn th structur and th surrounding soil (Kazama t al. 988, Ostadan t al. 998). It should b notd that ths phnomna ar nithr wll documntd nor xplaind from thortical point of viw. Som simplifid modls to prdict th arth prssurs on th rigid wall during arthquaks hav bn prsntd (Scott 973, Vltsos t al. 994a, Vltsos t al. 994b). Unfortunatly, ths modls fail to xplain abov dscribd phnomna. Th obsrvations of arth prssurs inducd by arthquak ground motions hav bn conductd on both sids of a larg-scal shaking tabl foundation in Tsukuba. Simultanous obsrvations of fr-fild ground motions as wll as th rspons of th foundation hav bn mad (Iguchi t al., Iguchi t al. ). Th dns arthquak obsrvations around or on th shaking tabl foundation prmit us to study th charactristics of arth prssurs in rlation to th ground motions and to th rspons of foundation as wll. Forcd vibration tsts hav bn also prformd for th foundation to obtain th arth prssurs during th xcitations. Th objctiv of this papr is to analyz th rcords of th arth prssurs obsrvd on th latral sids of th foundation during arthquaks and th forcd vibration tsts as wll. Spcial mphasis is placd to xtract abov dscribd phnomna on th basis of th obsrvd data. A simplifid analysis modl of a lumpd-mass systm connctd in sris is also prsntd to simulat numrically th obsrvations. Th focus of this papr is to show and discuss how xtnt could w xplain th obsrvations with us of th simplifid modl. FOUNDATION AND EARTH PRESSURE MEASUREMENT Figur shows a larg-scal shaking tabl foundation in National Rsarch Institut for Earth Scinc and Disastr Prvntion in Tsukuba. Th siz of th foundation is 5m 39m in plan and th bas of th foundation is dirctly supportd on firm sand at a dpth of 8.m. Th sction of th foundation and soil profil ar shown in figurs (a) and. Th wight of th foundation and th shaking tabl is about,6tf (3.7MN) and 8tf (.76MN), rspctivly. Th total wight is approximatly amount to,tf including th wight of stl-mad supr-structur and corrsponds almost to th xcavatd soil of th foundation. Th fundamntal frquncy of th soil-foundation systm is about 4.Hz in th x-dirction

3 4m X X' (a) Sction and Dploymnt of Sismomtrs. Y X X' (b) Plan and Location of Earth Prssur Gaugs. Y' Figur. Larg shaking tabl foundation and location of sismomtrs and arth prssur gaugs. Figur. Soil profil and soil constants. that was obsrvd by th forcd vibration tst. It has bn confirmd that th foundation bhavs as a rigid body within frquncis lss than Hz. Th soil constants at th sit wr masurd to a dpth of about 4m as shown in figur, and mor dtails may b found lswhr (Minowa t al. 99). Figur (b) shows th location of arth prssur gaugs dployd on th sids of th foundation. On ach sid of th foundation, fiv arth prssur gaugs hav bn instrumntd at diffrnt dpths, but on installd on th wst sid was out of function and is omittd from th figur. Th simultanous obsrvations of arthquak ground motions hav bn conductd at dpths of m and 4m blow th soil surfac. W rfr th ground motions rcordd at th dpth m to as surfac ground motions. Rgarding th foundation, th arthquak rsponss of th foundation hav bn obsrvd at svral points with acclrographs and a vlocity sismograph as shown in figur (a). Th horizontal rspons of th foundation is rprsntd by sismograms rcordd at th point S3, which is locatd almost at th cntr of th foundation. Earth prssur during forcd vibration tsts Th arth prssurs hav bn masurd during th stp-swp forcd vibration tsts. A harmonic xcitation forc was gnratd by driving a shaking tabl in th frquncy rang from to Hz with a frquncy stp. Hz. Th horizontal xcitation was applid at m blow th foundation surfac. Two lvls of xcitation forcs wr prformd in th sris of xprimnts: on was th tst conductd by driving th shaking tabl so as to gnrat th acclration of about gals on th shaking tabl, which is quivalnt to about tf of xcitations, and th othr was th tst of about 5 gals throughout th frquncis. Thr was no vidnc of nonlinar phnomna such as sparation of th latral soil from th foundation as far as bing judgd by inspction of th rcordd wavforms of th arth prssurs. 3

4 E.P/Disp(kgf/cm /cm) Phas(dg) E.P/Disp(kgf/cm /cm) Phas(dg) EC Frquncy(Hz) EC Frquncy(Hz) ES-.5 WS-.5 wsgt 5 5 Frquncy(Hz) WS-.5 ES Frquncy(Hz) E.P/Disp(kgf/cm /cm) Phas(dg) E.P/Disp(kgf/cm /cm) Phas(dg) Frquncy(Hz) WC-4 EC-4 EC-4 WC Frquncy(Hz) EN-.5 WN Frquncy(Hz) WN-.5 EN Frquncy(Hz) E.P/Disp(kgf/cm /cm) Phas(dg) EC-7 WC Frquncy(Hz) WC-7 EC Frquncy(Hz) Figur 3. Amplitud and phas of arth prssurs on both sids of foundation normalizd by unit displacmnt of foundation. E.P/Vl(kgf/cm /cm/s) Phas(dg) EC Frquncy(Hz) EC Frquncy(Hz) E.P/Vl(kgf/cm /cm/s) Phas(dg) EC-4 WC Frquncy(Hz) WC-4 EC Frquncy(Hz) E.P/Vl(kgf/cm /cm/s) Phas(dg) EC-7 WC Frquncy(Hz) WC-7 EC Frquncy(Hz) Figur 4. Amplitud and phas of arth prssurs on both sids of foundation normalizd by unit vlocity of foundation. Figurs 3 and 4 show th frquncy charactristics of arth prssurs obsrvd on both sids of th foundation. Th rsults shown in figur 3 ar th amplitud and phas charactristics of arth prssurs normalizd by unit horizontal displacmnt of th foundation. Th rsults of arth prssurs shown in figur 3 indicat that for lowr frquncis lss than 3 4Hz th amplituds of th arth prssur ar almost constant and tnds to incras for highr frquncis. As for th phas charactristics, it will b noticd 4

5 that th arth prssurs ar inducd in-phas to th displacmnt motion of th foundation. This fact indicats that th sismic dformation mthod, which is widly usd in th sismic rspons analysis of undrground structurs, may not b valid for highr frquncis. Similarly, figur 4 shows th charactristics of arth prssurs normalizd by unit horizontal vlocity of th foundation. An inspction of ths rsults shown in figur 4 rvals that th magnituds of arth prssur varis with frquncis but th phass tnd to b almost constant in highr frquncis mor than 3 4Hz. This indicats that th arth prssurs ar inducd in phas to th vlocity rspons of th foundation in th frquncy rang mor than 3 4Hz. Th abov mntiond frquncis 3 4 Hz may b prsumd as th fundamntal frquncy of two layring surfac soil. Th fundamntal frquncy can b approximatd by mans of a quartr-wav-lngth mthod, and will b givn as follows with th soil constants shown in figur. H H = + = 4 Vs Vs f 3.5Hz whr H and H ar th thicknss of th first and th scond layrs of surfac soil, and V s and V S ar corrsponding shar wav vlocitis. Thus, 3.5Hz may b considrd to b th fundamntal frquncy of th two layrd surfac soil. Earth prssur obsrvation during arthquaks Th obsrvations of arth prssurs inducd by arthquak ground motions hav bn conductd for six yars from 99 to 996 and th rcords of about 3 arthquaks had bn obtaind. Th dtails of th arthquak rcords may b found lswhr (Iguchi t al., Iguchi t al. ). In ordr to discuss th charactristics of th arth prssurs in rlation to frquncy componnt containd in th arthquak ground motions on th soil surfac, th rcordd arthquak motions was catgorizd into thr groups (groups A, B and C). Th grouping was mad according to th prdominant frquncy componnts includd in th arthquak acclration motions (NS componnt) rcordd on th soil surfac; th arthquak ground motions including prdominantly th lowr frquncis lss than Hz wr catgorizd into group A; th arthquak motions with highr componnts (mor than about 3 to 4Hz) wr groupd into C, and group B was charactrizd by th motions having intrmdiat frquncy componnts btwn groups A and C. Tabl shows th arthquak paramtrs of th thr rcords slctd as th rprsntativ motions pickd up from ach group. Figur 5 shows th normalizd Fourir spctra of th rprsntativ motions chosn from th rspctiv groups. Th spctra ar smoothd by using th Parzn window with a bandwidth of.hz. Tabl. Earthquak paramtrs Eq. No Dat Epicntr Dpth (Km) Magnitud. PGA (gal) Group 3 99 Oct 9 N36.8, E C Oct N3., E B Sp N35.7, E A 5

6 Figur 5. Normalizd Fourir spctra of ground motions of Group A, B and C. Phas charactristics of arth prssurs inducd on both sids of th foundation can b xtractd by calculating th motion products of th arth prssurs rcordd on both sids. Th computd rsults of th motion products ar shown in figurs 6(a), (b) and (c) for th rprsntativ motions of ach group. Th positiv valu of th product can b intrprtd as th arth prssurs on both sids bing inducd in phas on both sids, and th ngativ valus of th product, on th othr hand, imply th gnration of arth prssurs with a phas lag out of 8 dgrs on two sids. Inspction of th rsults shown in figur 6(a) indicats that for th ground motions of group A th arth prssurs inducd on both sids ar almost in phas throughout th rcord. It should b noticd that th in-phas phnomnon of th arth prssurs cannot b xplaind by th convntional assumption of th vrtical incidnc of sismic wavs. As for group B shown in figur 6(b), th arth prssurs inducd on both sids of th foundation ar showing out-of phas in th primary portion of th tim history and tnd to shift to in-phas with progrss of tim. With rgard to th motions of group C, which contains highr frquncis, out-of-phas and in-phas phnomna ar rcognizd altrnatly throughout th whol tim history as shown in figur 6(c). Summarizing th rsults shown in figurs 6(a), (b) and (c), it may b notd that th phas charactristics of arth prssurs inducd on both sids of th foundation ar strongly affctd by frquncy componnts containd in th ground motions and tnd to b inducd in phas for th ground motions with lowr frquncis. Anothr point to b discussd is what componnts of foundation motions ar closly rlatd to th arth prssurs. Obsrvd tim historis of arth prssurs and vlocity rspons of foundation ar shown simultanously in figurs 7(a), (b) and (c) for rspctiv ground motions of groups A, B and C. It will b noticd that th arth prssurs ar closly corrlatd with vlocity motions of foundation for groups A and B. As for group C, on th othr hand, th clos corrspondnc btwn th two is not rcognizd as shown in figur 7(c). Figur 8 shows tim historis of arth prssurs and acclration motions of th foundation from to sc (top) and to sc (bottom). A strong corrlation btwn th arth prssurs and acclration motions of th foundation for group C is rcognizd. Finally, figurs 9(a) to () show Fourir amplitud spctrum ratios of th arth prssurs obsrvd at diffrnt points to th vlocity motions of foundation. Th rsults ar smoothd by us of Parzn window with a bandwidth of. Hz. It will b noticd that th spctrum ratios ar almost constant for frquncis lss than 4 5Hz and tnd to incras almost linarly for highr frquncy. This fact indicats that arth prssurs ar inducd in accordanc with 6

7 vlocity motions of th foundation for lowr frquncis and to acclration motions for highr frquncis. (a) Tim historis (top) and motion product (bottom). (Group A). (b) Tim historis (top) and motion product (bottom). (Group B). (c) Tim historis (top) and motion product (bottom). (Group C). Figur 6. Tim historis and motion products of arth prssurs inducd on opposit sids of th foundation during arthquaks. 7

8 (a) Group A (b) Group B (c) Group C Figur 7. Tim historis of arth prssurs and vlocity rsponss of foundation. Figur 8. Tim historis of arth prssur and acclration rspons of foundation (Group C). Figur 9. Spctrum ratios of arth prssurs to vlocity rspons of foundation. 8

9 ANALYSIS OF LATERAL SOIL RESISTANCE Dscription of problm and formulation Th objctiv of this chaptr is to prsnt formulation for analyss of th latral soil rsistanc to th foundation whn xcitd by a latral harmonic forc applid on th foundation. Th foundation-soil systm considrd in this study is shown in figur. Th latrally xtnding soil is composd of two horizontal layrs with hystrtic matrial damping, which is supposd to b supportd rigidly at th bas. Th soil is assumd to b composd of two-dimnsional mdium of plan strain with unit width. Th coordinat systms x and z ar chosn as shown in figur. On th assumption that th vrtical displacmnts in ach layr ar ngligibly small compard to th horizontal componnt, th quations of harmonic motion for ach layr can b writtn by, Layr ; ( ) u u + ih E + G ρω + u = x z Layr ; ( ) u u + ih E + G ρω + u = () x z whr ω =circular frquncy of harmonic xcitation, u l = horizontal displacmnt for l -th layr ( l =, ), ρ l = mass dnsity of mdium, and h l = matrial damping factor of hystrtic typ, and E l = Young s modulus of soil, which can b xprssd with Lam s constants Gl, λ l as follow. E = λ + G, l =, (3) l l l () Figur. Analysis modl of foundation-latral soil systm. Figur. Coordinat systm and arth prssur inducd on th intrfac btwn foundation and th latral soil. 9

10 Undr th condition of absnc of vrtical displacmnts, th normal strss in th soil σ ( x, z) may b xprssd with horizontal displacmnt as follows. x ul σ x( xz, ) = (+ ihl) El x, l =, (4) Th boundary conditions on th surfac, on th intrmdiat soil intrfac and at th bottom of soil may b xprssd by, du z = H + H = (5-) ; dz z= H + H z = H ; u ( H ) = u ( H ) (5-) du du (+ ih ) G = (+ ih ) G (5-3) dz z= H dz z= H z = ; u () = (5-4) whr H and H ar th thicknss of th first and th scond layr, rspctivly. In a similar mannr, th boundary conditions on th latral surfac of th foundation and th radiation condition at infinity may b xprssd by u x = ; (+ ih ) E = p( z), H z H + H x u (+ ih ) E = p( z), z H x (6-) (6-) x= ; u, u (6-3) whr p( z ) is th amplitud of latral normal strss that th foundation xrts on th latral surfac of th layrd soil. In analyzing of Eqs () and () undr th boundary conditions abov, w summariz ths two quations as follows for convninc. * u d * du E ( z) + G ( z) ρ( z) ω u + = x dz dz whr, u( x, z) ; H < z < H+ H uxz (, ) = u( x, z) ;< z < H (7) (8) and, * (+ ih ) G ; H < z < H + H G ( z) = ( + ih) G ; < z < H * (+ ih ) E ; H < z < H + H E ( z) = ( + ih) E ; < z < H (9-) (9-)

11 ρ( z) ; H < z < H+ H ρ( z) = (9-3) ρ( z) ;< z < H Employing th mthod of sparation of variabls for th displacmnt uxz (, ), w writ as u( x, z) uxz (, ) = = XxUz ( ) ( ) () u( x, z) As for th displacmnt componnt along th dpth of soil U( z ), w adopt th fundamntal mod shap of th two layring soil. Ltting U( z ) b as U( z), H z H+ H U( z) = U( z), z H th mod shaps for th first and scond layrs U( z), U( z ) may b givn by () ω U ( z) = Ccos ( H + H z), H z H+ H (-) Vs ω cos H Vs ω U( z) = C sin z, z H (-) ω sin H V V s s whr C = intgral constant, and Vs, V s ar shar wav vlocitis of soil layrs as givn by V = G G, V = (3) s s ρ ρ It may b asily shown that ths functions satisfy th boundary conditions of Eqs (5). In Eqs () ω is th fundamntal circular frquncy of th shar-column of two layring soil without damping, and is givn by th minimum root of th following charactristic quation. G ω ω G ω ω cos cos H + sin H sin H = (4) V V V V V V s s s s s s Th participation factor of th shar-column whn vibrating with th fundamntal mod may b givn by β = H+ H ρ( z) U( z) dz H+ H ρ( ){ ( )} z U z dz Nxt, substitution from Eq. () into Eq. (7) givs * d X d * du E ( z) U + X G ( z) ρ( z) ω U X + = dx dz dz Multiplying Eq. (6) by ( ) and multiplying β on both sids of th quation, thn w will arriv at (5) (6) U z and intgration with rspct to z in th rang [, H H ] +,

12 d X nk s k X + m ω X = (7) dx This quation of motion is quivalnt to that of a systm which is composd of an axial rod supportd on continuously distributd shar soil spring. Th paramtrs apparing in this quation m, k and k ar givn by s n { } H + H m β ρ( z) U( z) dz s = (8-) H + H d * du k = β G ( z) U( z) dz dz dz H H * du ( ) + = β G z dz dz (8-) { } k H H β + E * n ( z ) U ( z ) dz = (8-3) It should b notd that ths ar complx valud xcpt for m. Substituting from Eq. (5) into Eqs (8-) and (8-), w will obtain othr forms for m, k as s m H+ H { ρ( z) U( z) dz } = H+ H { } ρ( z) U( z) dz (9) s k = H+ H * du G ( z) dz z= * du G ( z) dz dz () Equivalnt multi-lumpd-mass modl Introducing a discrtization procdur for Eq. (7) with an qually spacd finit diffrnc approximation, it is possibl to rwrit as ( ω ) K X + K + K M X K X = () n j n s j n j+ whr X j is th horizontal displacmnt of th j-th mass, and othr paramtrs apparing in this quations ar dfind as nk = nk, sk = xsk, M = xm () x in which x is a mass intrval of spacing. W will notic that Eq. () is nothing but an quation of motion of a lumpd-mass systm connctd in sris as shown in figur, and

13 th constants of th systm ar thos givn in Eq. (). Th quivalnt mass hight systm may b dtrmind as th valu of z that satisfis th following quation. H of th β U( z) = (3) Th quivalnt rsultant forc that th foundation xrts on th first nd mass will b xprssd as follow. P H + H = β p z U z dz (4) ( ) ( ) If w assum that th latral normal strss on th intrfac btwn th foundation and th latral soil is uniformly distributd throughout th dpth z ( p( z) = p ), thn Eq. (4) rducs to (5) H + H P = p U z dz β ( ) whr p is th amplitud of th normal strss. A big advantag of th rduction into a lumpd-mass systm is that thus obtaind discrt systm prmits us to xtnd to nonlinar analyss of th latral soil. Figur. Multi-lumpd-mass systm. Analysis of finit diffrnc quation W try to solv th finit diffrnc quation shown in Eq. (). Th closd form solution of this quation has bn prsntd by Tajimi (99). Following th Tajimi s approach, w assum for th solution of Eq. () that satisfis th radiation condition of Eq. (6-3) as X j j = A α,,, j = (6) whr A is an arbitrary constant to b dtrmind by th boundary condition at j =. Substituting from Eq. (6) into Eq. () w obtain nk + K M ω M Ch α = = + ( ω ω ) (7) s nk nk It should b notd that α is a function of frquncy and a complx valu of R( α ). In this quation a rlation K M = k m = ω, in which ω indicats th fundamntal s s 3

14 circular frquncy of th shar-column of th layrd soil, is mad us of. Though th valu α may b dtrmind by solving Eq. (7), th analysis of th transcndntal quation is not ncssary as sn blow. Nxt stp is to dtrmin th arbitrary constant A in Eq. (6). Th constant A may b dtrmind from th boundary conditions for th first mass. Taking into account that th govrning rgion of th nd mass is a half of th othr masss, th quation of motion for th first mass ( j = ) may b givn as follows. nk + sk M ω X n K X = P Substituting from Eq. (6) into Eq. (8), it may b found that A is givn by P n K (8) α A= (9) Shα Substitution Eq. (9) into Eq. (6) lads to th solution of th finit diffrnc quation as shown by X P α ( j ) j = (3) n KShα By stting j = in this quation, w will obtain th rlationships btwn th quivalnt xcitation forc P and th displacmnt of th first mass as shown by P n K Shα X = (3) In this quation n K Sh α implis th quivalnt dynamic stiffnss of th latral layring soil of a unit width prpndicular to th plan, in which n K is th latral stiffnss of th lumpdmass systm whn th scond mass kpt immovabl and Sh α rprsnts th ffcts of th othr masss connctd latrally. Making us of th rlation Ch α Sh α = and from Eq. (7), w will obtain an xplicit form for Sh α as M M Sh α = ( ) ( ) K ω ω + ω ω (3) n 4 nk Finally, substituting from Eq. (3) into Eq. (5), th magnitud of arth prssur p gnratd on th sid of th foundation may b valuatd by p KShα = X β U( z) dz H + H It should b notd that X in Eq. (33) is quivalnt to th amplitud of a horizontal displacmnt of th foundation at th quivalnt hight H, which may b valuatd on th basis of th obsrvations during th forcd vibration tsts. (33) 4

15 ANALYSIS OF EARTH PRESSURE DURING EARTHQUAKES Analysis modl and assumptions Th modl adoptd in this study for th analysis of arth prssurs during arthquaks is illustratd in figur 3. It consists of a rigid foundation supportd by sway and rocking springs and dashpots which rprsnt radiation damping into subsoil blow foundation, and two sris of lumpd-mass systm rprsnting th dynamic rsistanc of th latrally xtnding soil. As dscribd arlir th obsrvd rsults of th arth prssurs inducd on th opposit sids of th foundation during arthquaks cannot b xplaind by an assumption of vrtical incidnc of sismic wavs. In th analysis of th soil-foundation systm subjctd to ground motions, thrfor, w assum in this papr an obliqu incidnc of sismic wavs that propagat in th x-dirction. In formulation of rspons analyss of th systm, th whol systm is dividd into thr substructurs: two of thm ar right and lft latral lumpd-mass systms as shown in figur 4, and th othr is th rigid foundation subjctd to th ground motions at th bas and horizontal forcs from th latral soil. Th input motions into th foundation from th latral soil ar includd in th formulation of th latral lumpd-mass systms. Th coordinat x is dfind as shown in figur 3 with th origin at th cntr of th foundation bas. In th sismic rspons analysis of th whol systm, it will bcom a ky stp to valuat th rspons of latral lumpd-mass systms whn subjctd to both th ground motions and horizontal forcs xrtd from th latral soil. Figur 3. Lumpd-mass modl for rspons analysis of foundation-latral soil systm. (a) (b) Figur 4. (a) Lumpd-mass systm for th latral soil of th right sid, and (b) th lft sid. 5

16 Earthquak rspons analyss of systm Th rspons of th lumpd-mass systm connctd in sris shown in figur 4 may b xprssd by th sum of two rsponss. On is th rspons to a harmonic xcitation applid at th nd mass, which has bn givn bfor, and th othr is on to th harmonic ground motions. Thus, th total rspons of th systm may b xprssd by th sum of two rsponss as X = X + X (34) () () j j j whr th first trm of th right sid rprsnts th rspons to th xcitation applid at th nd mass, and th scond to th ground motions. Making us of th rsult xprssd in Eq. (3) th rspons to th xcitation at th nd mass of th right sid shown in figur 4(b) may b xprssd by X P = (36) () R α ( j ) j n KShα whr P R indicats th quivalnt latral forc gnratd on th intrfac btwn th foundation and th latral soil xtnding to th right dirction. Nxt stp is to analyz th lumpd-mass systm whn subjctd to travling ground motions at th bas of ach mass. A harmonic sismic wavs travling to x-dirction with an apparnt wav vlocity c on th surfac of th supporting bd rock, may b xprssd as follows. u ( x, t) u ω g i ( t x/ c) = (37) whr u is th amplitud of th harmonic ground motions. Ltting th rspons of th j th mass of th systm b govrning th lumpd-mass systm may b xprssd by ( ω ) () () () n j n s j n j+ s g j () i t X j ω, th quation of motion K X + K + K M X K X = K U, j =,3, (38) whr U g j is th amplitud of ground motion at th j th mass, which is dfind as follows with wav numbr k = ω / c: Ugj ikx = u j (39) in which x j is horizontal distanc from th cntr of th foundation to th j th mass, and may b givn as, x = L/ + x( j ) (4) j with full lngth of th foundation L and th spacing of mass x. For th first mass, th quation of motion may b xprssd by ( ω ) K + K M X K X = K U (4) () () n s n s g Th gnral solution for Eq. (38) will b givn as follows. 6

17 { / x( j ) } () α j ik L + X j = HR( ω) A + u (4) whr A is an arbitrary constant to b dcidd so as to satisfy Eq. (4), and is givn as sin k x ( α ikl /) A= i (43) Shα In Eq. (4), H R H R is dfind as ω ( ω) = n K ω ω + M ( cosk x) whr ω indicats th fundamntal circular frquncy of th latral layring soil and dfind as s K ω = (45) M Th function H R is intrprtd as a frquncy rspons factor of th multi-lumpd-mass systm whn subjctd to non-vrtical incidnc of sismic wavs. Thus, th total harmonic rspons of th first mass in th right sid whn subjctd to both th horizontal xcitation from th foundation and th ground motions, will b obtaind by substitution of Eqs (36) and (4) with Eq. (43) into Eq. (34), and it lads to ikl / sin k x X = HR( ω) i u + P R (46) Shα n K Shα In a similar mannr, th harmonic rspons of th first mass in th lft sid of th foundation i t X ω will b finally xprssd by ikl / sin k x X = HL( ω) + i u P L (47) Shα n K Shα whr P L is an amplitud of th horizontal xcitation applid at th first mass in th lft sid. Th H L in Eq. (47) is th sam as H R, which is dfind in Eq. (44). It should b notd hr that th rsponss of th first masss in th right and lft sids of th foundation, X and X, ar qual to th foundation rspons of th foundation at th lvl of th quivalnt hight of th mass systm. Formulation for rspons analysis of foundation Th final stp of th analyss is th analysis of rspons of th rigid foundation subjctd to xcitations from th both sids of th foundation and arthquak ground motions at th bas. In valuation of th foundation input motion whn subjctd to obliqu incidnt sismic wavs, an avraging procdur proposd by Iguchi (973) is adoptd. Th horizontal * i t componnt of th foundation input motion U ω may b valuatd by (44) 7

18 U ω = ug ( xtda, ) A (48) A * i t whr A is bas ara of th foundation. Substitution from Eq. (37) into Eq. (48) and intgration ovr th ara lads to * sin( kl / ) U = u (49) kl / Assuming vrtical incidnc of incoming wavs ( c or k ) in Eq. (49), th amplitud of th foundation input motion will b qual to that of th incidnt wav and thus * U = u. Th quation of motion for th rigid foundation subjctd to both horizontal xcitations from th latral sids of th foundation and th ground motions may b xprssd as follows. KH M hg M/ H ω KR /( H ) hm G / H J/( H) H Φ * P R = ω M U B hg / H (5) PL whr and Φ ar horizontal and rocking motions at th bas of th foundation, h G = th hight of th gravity cntr of th foundation from th bas, B = width of th foundation, H = quivalnt hight of mass connctd in sris, M = mass of th foundation, and J = a mass momnt of inrtia of th foundation with rspct to its bottom, which may b valuatd by J = J + h M (5) G with a mass momnt of inrtia J with rspct to th gravity cntr of th foundation. In Eqs (46) and (47), it should b rmindd that X and X ar qual to th horizontal displacmnts of th foundation at th quivalnt hight of th mass systm. Thus w hold X X H U * = = + Φ + (5) Substituting from Eq. (5) into Eqs (46) and (47), w will obtain th xprssions for th rsultant arth prssurs on th latral sids of th foundation P and P. P = K Shα( + H Φ + U ) H( ω) K ( Shα isin k x) u (53) * ikl / R n n P = K Shα( + H Φ + U ) + H( ω) K ( Shα + isin k x) u (54) * ikl / L n n whr w st H( ω) = H ( ω) = H ( ω). R L Substituting from Eqs. (53) and (54) into Eq. (5), th quation of motion for th foundation may b finally xprssd by KH + KL KL M hg M/ H ω KL KR /( H ) K L hm G / H J/( H) + H Φ R L 8

19 = ω M U K U * * L hg / H sin kl / sin k x + H ( ω) KL cos kl/ u Shα (55) whr K H and K R ar th horizontal and rocking springs dfind at th bas of th foundation, which ar th function of frquncis and complx valud. Ths complx springs rprsnt th soil stiffnss valuatd xcluding th ffcts of latral soil. In addition, K in Eq. (55) is a complx spring for th latral sid of th foundation. L K L = (56) BnK Shα Onc th sway and rocking rsponss of th foundation ar solvd from Eq. (55) for givn ground motions, th rsultant arth prssurs on th latral sids of th foundation can b valuatd from Eqs (53) and (54). Th arth prssur in a unit ara undr th assumption of uniform distribution of th contact strss may b valuatd from Eq. (33). Thortical discussion of arth prssur during arthquaks Basd on Eqs (53) and (54) th xprssions of th latral arth prssurs on both sids of th foundation undr th action of arthquak ground motions may b rwrittn as follow. sin kl / sin k x P = K Shα( + H Φ + U ) H ( ω) K Shα cos kl / u Shα * R n n cos kl / sin k x + ih ( ω) n K Shα sin kl /+ u Shα (57) sin kl / sin k x P = K Shα( + H Φ + U ) + H ( ω) K Shα cos kl / u Shα * L n n cos kl / sin k x + ih ( ω) n K Shα sin kl /+ u (58) Shα It should b notd hr that comprssiv soil prssurs ar supposd to b positiv. Th first trms of th right hand sid of Eqs (57) and (58) corrsponds to arth prssurs inducd on th sids of th foundation by th horizontal motion of th foundation whn th latral soils ar fr from th ground motions. Ths trms will rsult in causing out-of-phas arth prssurs on th opposit sids of th foundation. Th scond and th third trms of ths quations corrspond to th arth prssurs inducd by ground shaking whn solly th latral soils ar subjctd to arthquak motions whil th foundation is kpt immovabl. By furthr inspction of ths trms, w will notic that th scond trms of Eqs (57) and (58) will caus out-of-phas arth prssurs on th opposit sids, and on th othr hand th third trm th in-phas prssurs. This fact indicats that th third trms of Eqs (57) and (58) will cancl ach othr, and will not b ffctiv as input motions into foundation from th latral sids of th foundation. For th cas c or k, which corrsponds to th vrtical incidnc of sismic wavs, th third trms of Eqs (57) and (58) will disappar and th phnomnon of in-phas arth prssurs will nvr occur. Th arth prssurs for th group C, which contains 9

20 prdominantly highr frquncis in th ground motions, hav shown a clos corrlation with th acclration motion of th foundation rspons. This tndncy may b xplaind by inspction of Eqs (57) and (58) as follows. If considring th casω, w will obtain as n K Shα M ω, H ( ω) K Shα const. n As a rsult, with incras of ω th first trm will b dominant comparing to th othr trms. This implis that th first trms of th quations tnd to hav significant ffct on th arth prssurs for larg valu ofω, and to caus th out-of-phas arth prssurs on th opposit sids of th foundation. It is obvious that th first trms of Eqs (57) and (58) ar proportional to th acclration rspons of th foundation. Finally w will xtnd th discussion to th cas of static arth prssurs by considring ω in Eq. (44). For ω, w will obtain HR ( ω) = H( ω), K Shα const n. and k. With considration of ths limits, th xprssions of th arth prssurs on both sids of th foundations will b givn as follows. P ( + H Φ + U u ), P ( + H Φ + U u ) * * R L * W will notic that ( + H Φ + U u) rprsnts th longitudinal strain at th intrfac btwn th foundation and th surrounding soil. On th othr hand, th longitudinal particl vlocity of soil at th intrfac is qual to th vlocity of th foundation. Thus, rminding th fact that for on dimnsional wav propagation of an lastic mdium th longitudinal strain is proportional to th particl vlocity of th mdium (Minowa t al. ), th abov quations may b rwrittn as, P ( + H Φ + U ), P ( + H Φ + U ) * * R L Ths xprssions indicat that for lowr frquncis th arth prssurs will b inducd in rspons to th vlocity motion of th foundation. This may b paraphras as th arth prssur will b inducd in proportional to th rlativ horizontal displacmnt motions btwn th foundation and th surrounding soil. It may b summarizd that th simplifid analysis modl prsntd in this papr can succssfully xplain th obsrvations of arth prssurs during arthquaks. COMPARISON OF ANALYTICAL RESULTS WITH OBSERVATIONS Paramtrs of lumpd mass systm - FORCED VIBRATION TEST- Th paramtrs of th multi-lumpd-mass systm shown in figurs 9 and 3 wr dtrmind basically on th basis of th soil constants shown in figur. As th damping constants of soil, howvr, wr not masurd w wr obligd to assum th damping factors of a hystrtic typ and w assumd for th first and th scond soil layrs as h = h = h and h =.. Additionally, th computd rsults of th first to th third frquncis of th layrd soil valuatd from Eq. (4) wr f = 4.4 Hz, f = 9.8 Hz, f3 = 8.4Hz. Thus obtaind fundamntal frquncy f = 4.4Hz is largr than f 3.5Hz that was xpctd by obsrvations. Th discrpancy btwn ths two may b attributd mainly

21 to th assumption of a rigid bas undrlying th surfac layrs. With rfrnc to th obsrvations, w assumd th fundamntal frquncy of th layrd soil as f = 3.5Hz. Th othr paramtrs of th lumpd mass systm ar shown in Tabl. Th spacing btwn masss was assumd to b constant all through th systm as x = m. Tabl Paramtrs of th lumpd mass systm Participation factor of th first mod β =.4 Equivalnt mass hight H = 5.7 [ m ] Equivalnt mass M =.9 tf sc / m Equivalnt shar spring constant K = s ( i ) tf / m 5 Equivalnt axial spring constant K = (.9 + n.8 i ) tf / m Simulation of arth prssur during forcd vibration tsts Figur 5 shows th comparison of th amplitud and phas charactristics of arth prssurs valuatd numrically on th basis of Eq. (33) with thos of obsrvations inducd on th latral sid of foundation during th forcd vibration tsts. Th compard rsults shown in figur 5 indicats that whras th rsult of amplitud at WC-7 is slightly largr than th obsrvations, th numrically valuatd rsults of th amplitud and phas charactristics ar wll corrsponding to th obsrvations as a whol. (a) WC-.5 (b) WC-4 Figur 5. Comparison of arth prssurs btwn analyss and obsrvations (Amplitud and phas normalizd by unit displacmnt of foundation). Th rsults shown in figur 5 ar th arth prssurs normalizd by a unit displacmnt of th foundation. Similarly, figur 6 shows th arth prssurs normalizd by a unit vlocity rspons of th foundation. It should b rmindd that th arth prssurs on th latral sids of th foundation ar inducd in accordanc with th vlocity rspons of th foundation for

22 highr frquncis mor than 3 4Hz. This tndncy is succssfully xplaind by a simplifid multi-lumpd-mass modl dvlopd in this study. (a) WC-.5 (b) WC-4 Figur 6. Comparison of arth prssurs btwn analyss and obsrvations (Amplitud and phas charactristics normalizd by unit vlocity of foundation). NUMERICAL ANALYSIS OF EARTH PRESSURE DURING EARTHQUAKES Evaluation of sway and rocking stiffnss In ordr to valuat numrically th arthquak rspons of th foundation on th basis of Eq. (55), it is rquird to obtain th sway and rocking stiffnss for th foundation motions. In th numrical analyss, ths valus wr dtrmind on th basis of th sway and rocking rsponss of th foundation obsrvd during th forcd vibration tsts. Lt th horizontal and rocking rsponss of th foundation b dnotd by i ω t iωt, Φ for harmonic xcitations i t applid on th foundation P ω as shown in figur 7(a). Th sway and rocking stiffnss of th soil undrlying th foundation, K H and K R, which ar shown in figur 7(b), may b valuatd by K = P+ M ω ( + h Φ ) ( + H Φ ) K H g L K = Ph + M ω h + Jω Φ ( + H Φ ) H K R p g L Φ whr h p is th hight of th applid forc ( h p =7.m), and K L is th latral soil stiffnss for th foundation, which is dfind in Eq. (56). It should b notd that, Φ in Eqs (59) and (6) ar complx valud amplituds of th foundation. Figurs 8(a) and (b) show th rsults of K H and K R valuatd on th basis of Eq. (59) and (6). Inspction of figur 8 rvals (59) (6)

23 that ths valus vary strongly with frquncis highr than 7 Hz, that is probably du to th ffct of dformation of th foundation bas. Ground motions at th foundation bas As mntiond in th arlir sction, th phas charactristics of arth prssurs bing inducd in-phas on both sids of th foundation cannot b xplaind by an assumption of th vrtical incidnc of sismic wavs. Th sam phnomnon was obsrvd in obsrvations using a small modl foundation (Uchiyama t al. 999). Uchiyama t al. (999) trid to xplain th phnomnon by using FEM and assuming a non-vrtical incidnc of sismic wavs. Following Uchiyama t al. (999) w assum hr obliqu incidnc of wavs. Consquntly, in valuation of th foundation rsponss and arth prssurs during arthquaks, it is rquird to dtrmin th apparnt wav vlocity c travling horizontally in th soil at th lvl of th foundation bas, as wll as th tim history of th ground motions at that dpth. As for th apparnt wav vlocity c, th valu would dpnd mainly on shar wav vlocity of th supporting soil and on th azimuth and incidnt angl of sismic wavs, among which th lattr two cannot b dcidd from th obsrvd data. A paramtric study, thrfor, was conductd with rspct to c by changing th valu in th rang c= 3,, m/ sc. By inspction of th rsults of th paramtric studis, c=, m/ sc was chosn as th appropriat valu rgardlss of th arthquak motions. (a) Foundation Rspons to Excitation. (b) Rsistant Modl of Foundation to Excitation. Figur 7. Forcd xcitation tst and quivalnt rsistanc modl of foundation. (a) Sway Spring. (b) Rocking Spring. Figur 8. Complx stiffnss at th bas of foundation. 3

24 On th othr hand, as for th tim historis of th ground motions at th lvl of th foundation bas, which is supportd at 8.m from th soil surfac, th ground motions at th dpth hav not bn obsrvd dirctly. Thrfor, th ground motions at th lvl wr valuatd numrically by mans of th Thomson-Haskl s mthod and with us of th ground motions rcordd in th fr fild at th dpths of m and 4m. Th dtails of th valuation of th ground motions at this sit may b found lswhr (Iguchi t al. ) and will not b rpatd hr. Comparison of vlocity rspons of foundation Th vlocity rsponss of th foundation valuatd numrically on th basis of Eq. (55) ar compard with obsrvations. Figurs 9(a), (b) and (c) show th compard rsults in trms of Fourir amplituds btwn th two. As sn from th figurs, fairly good agrmnt btwn two ar obtaind for groups A and B up to Hz, whras slight discrpancis may b rcognizd for group C. It was confirmd that th simplifid analysis modl proposd in this papr givs satisfactory rsults and may b applicabl to numrical simulation analyss of th mbddd foundation dirctly supportd on a firm soil. Comparison of tim historis of arth prssurs Figurs (a), (b) and (c) show th compard rsults of numrically valuatd tim historis of th arth prssurs inducd on both sids of th foundation with thos of obsrvations for groups A, B and C. In th calculations, th apparnt wav vlocity c was assumd to b,m/s. As sn from figur, whras somwhat discrpancy was rcognizd for group C, fairly good agrmnt btwn th numrically valuatd rsults and obsrvations wr dtctd for groups A and B. On of th main rasons of th discrpancy might b attributd to uncrtaintis inhrnt to ground motions with high frquncis. It was confirmd that th multi-lumpd-mass modl prsntd in this papr is applicabl not only for rspons analyss of a foundation but for th numrical simulation of th arth prssurs inducd by arthquak ground motions. (a) Group A. (b) Group B. (c) Group C. Figur 9. Comparison of vlocity rspons of foundation: Analyss vs. obsrvations. 4

25 . (a) Group A. (b) Group B. (c) Group C. Figur. Comparison of arth prssurs on th lft and right sids of foundation inducd by arthquak ground motions: Analyss vs. obsrvations. CONCLUSIONS Th charactristics of arth prssurs inducd on th latral sids of an mbddd foundation during th forcd vibration tsts and arthquaks hav bn studid on th basis of 5

26 obsrvations. In addition, a simplifid analysis modl, which is composd of a lumpd-mass systm connctd in sris, was prsntd to conduct simulation analyss of th arth prssurs. Th studis hav ld us following conclusions () Th arth prssurs during th forcd vibration tsts tnd to b inducd associatd with th displacmnt motion of th foundation for frquncis lowr than th fundamntal frquncy of th latral soil. For frquncis highr than th fundamntal frquncis of th soil, on th othr hand, th arth prssurs tnd to b causd in-phas to th vlocity motion of th foundation. () Th charactristics of arth prssurs inducd by arthquak motions ar strongly dpndnt on frquncy componnt includd in th ground motions. (3) Th arth prssurs during arthquaks ar strongly rlatd to th horizontal vlocitis of th foundation for rathr lowr frquncis and to acclration rspons for highr frquncis. (4) A phnomnon of arth prssurs bing inducd in-phas on opposit sids of th foundation was obsrvd for lowr frquncis includd in th ground motions. For highr frquncis, on th othr hand, th arth prssurs tnd to b inducd out-of-phas on th opposit sids of th foundation. (5) A simplifid analysis modl prsntd in this papr has bn provd to b ffctiv in xplanation of th obsrvations of arth prssurs inducd by forcd vibration tsts. (6) Th analysis modl could simulat satisfactorily th obsrvations of arth prssurs inducd by arthquak motions xcpt for ground motions including highr frquncis. (7) Th phnomnon of arth prssurs bing inducd in-phas during arthquaks could b xplaind with us of th analysis modl and by assuming obliqu incidnc of sismic wavs. (8) Th tndncis of arth prssurs on th sids of an mbddd foundation bing inducd in clos corrlation with acclration or displacmnt motions of a foundation could b xplaind thortically with us of th simplifid analysis modl prsntd in this papr. ACKNOWLEDGEMENT This papr is th rvision of prviously publishd paprs (Iguchi t al., Iguchi t al. ) and w xprss our gratitud to th coworkrs of th paprs, Ms. A. Sadohira and Mr. T. Mutoh. REFERENCES Iguchi, M., 973. Sismic rspons with considration of both phas diffrnc of ground motion and soil-structur intraction, Proc Japan Earthquak Enginring Symposium, -8. Iguchi, M., Unami, M., Yasui, Y., and Minowa, C.,. A not on ffctiv input motions basd on arthquak obsrvations rcordd on a larg shaking tabl foundation and th surrounding soil (in Japans), J. Struct. Constr. Eng., AIJ, No. 537,

27 Iguchi, M., Yasui, Y., and Minowa, C.,, On Effctiv Input Motions: Obsrvations and Simulation Analyss, Proc. of th Scond U.S.-Japan Workshop on Soil-Structur Intraction, Building Rsarch Institut, Ministry of Land, Infrastructur and Transport of Japan, pp Iguchi, M., Sadohira, A., and Minowa, C.,. Obsrvation and analysis of dynamic arth prssur on th sids of a larg-scal shaking-tabl foundation during xcitation tsts (in Japans), J. Struct. Constr. Eng., AIJ, No. 549, Iguchi, M., Mutoh, T., and Minowa, C.,. Obsrvation and analysis of sismic arth prssurs on th sids of a larg-shaking tabl foundation (in Japans), J. Struct. Constr. Eng., AIJ, No. 56, Kazama, M., and Inatomi, T., 988. A study on sismic stability of larg scal mbddd rigid structurs, Proc. 9 th World Conf. on Earthquak Enginring (Tokyo-Kyoto), Vol. Ⅲ, Ⅲ -69 to Ⅲ Matsumoto, H., Ariizumi, K., Kuniyoshi, H., Chiba, O., and Watakab, M., 99. Earthquak obsrvation of dply mbddd building structur, Proc. 8 th Japan Earthquak Enginring Symposium, Minowa, C., Ohyagi, N., Ogawa, N., and Ohtani, K., 99. Rspons data of larg shaking tabl foundation during improvmnt works (in Japans), Tch. Not of th Notional Rsarch Institut for Earth Scinc and Disastr Prvntion, Rport No. 5. Minowa, C., Iguchi, M., and Iiba, M.,. Masurmnt of latral arth prssurs on an mbddd foundation during arthquaks, Strong Motion Instrumntation for Civil Enginring Structurs, d. by M. Erdik t al., Kluwr Acadmic Publishrs, Onimaru, S., Sugawara, R., Utak, T., Sugimoto, M., and Ohmiya, Y., 994. Dynamic charactristics of arth prssur on sismic array obsrvation (in Japans), Proc. 9 th Japan Earthquak Enginring Symposium, Ostadan, F., and Whit, W. H., 998. Latral sismic soil prssur: An updatd approach, Proc. UJNR Workshop on Soil-Structur Intraction, Mnlo Park, California, - to -36. Sakai, S., Masuda, A., Imamura, A., Kishino, Y., Ishii, T., Yotsuda, H., and Inou, T., 996. Dynamic arth prssur of dply mbddd building structur, Part 3 Study on dynamic latral prssur on basmnt wall (in Japans), Summaris of Tchnical Paprs of Annual Mting, AIJ, Scott, R. F., 973. Earthquak inducd prssurs on rtaining walls, Proc. 5 th World Conf. Earthq. Engng, Rom, Italy, Vol. Ⅱ, 6-6. Tajimi, H., 99. Earth prssurs on basmnt walls inducd by soil amplification ffcts, Proc. 8 th Japan Earthquak Enginring Symposium,

28 Uchiyama, S., and Yamashita, T., 999. Exprimntal and analytical studis of latral sismic arth prssur on a dply mbddd building modl (in Japans), J. Struct. Constr. Eng., AIJ, No. 56, 5-. Vltsos, A. S., and Younan, A. H., 994a. Dynamic soil prssurs on rigid vrtical walls, Earthq. Engng Struct. Dyn., Vol. 3, Vltsos, A. S., and Younan, A. H., 994b. Dynamic modling and rspons of soil-wall systms, J. of Gotchnical Enginring, ASCE, Vol., No.,