13 th World Conference on Earthquake Engineering Vancouver, B.C., Canada August 1-6, 2004 Paper No Susumu NAKAMURA 1 and Motoki KAZAMA 2

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1 3 th World Confrnc on Earthqua Enginring Vancouvr, B.C., Canada Augut -6, 4 Papr No. 85 PROPOSAL OF ONE-DIMENSIONAL DYNAMIC RESPONSE ANALYSIS METHOD OF EMBANKMENT AND SUPPORT GROUND SYSTEM CONSIDERING DYNAMIC INTERACTION WITH THE SIDE GROUND Suumu NAKAMURA and Motoi KAZAMA SUMMARY Thi papr prnt a on dimnional dynamic rpon analyi mthod to valuat th dynamic rpon of an mbanmnt and a upport ground ytm bad on th tiffn matrix mthod to conidr th dynamic intraction btwn an mbanmnt and a upport ground. Th motion of quation for a mbanmnt i formulatd undr th aumption that th hap of mbanmnt i a arbitrary trapzoid and th mbanmnt i dividd into om horizontally layrd thin lmnt. Th whol tiffn matrix for an mbanmnt and a upport ground ytm i obtaind by conidring not only th dynamic intraction btwn an mbanmnt and a upport ground but alo that btwn th mbanmnt-upport ground ytm and th id layrd ground. In ordr to vrify an accuracy of th propod mthod, imic rpon analyi for two-dimnional FE modl of mbanmnt and upport ground i carrid out. Furthrmor, thi mthod i applid to timat th dynamic rpon of th rivr di whr ha bn uffrd by th northrn Miyagi arthqua, July 6, 3 Japan. It i found that th dynamic intraction btwn mbanmnt-upport ground ytm and id layrd ground i vry important rol for valuating th dynamic rpon of mbanmnt and that th natural priod at t mod for th rivr di and upport ground ytm calculatd by th propod mthod i good agrmnt with that obtaind by th analyi of maurd micro-trmor on th top of th rivr di. INTRODUCTION A grat dal of arth tructur uch a a mbanmnt and a rivr di ha rpatdly uffrd by th arthqua. Howvr, it i difficult to dign th tructur which will not hav any damag for th trong arthqua ground motion. Bcau, if th tructur which atifid th rquirmnt mntiond abov could b dignd, a cal of th dignd tructur i too big to xit in thi ral pac and much xpn ar rquird to contruct th tructur. Thrfor, it i ncary for th dign concpt that th damag of a tructur i conidrd to b undr a allowabl limit. In ordr to tablih uch a dign procdur, th dformation caud by a failur of ground ha to b valuatd quantitativly. FEM [] and liding bloc mthod propod by Nwmar[] hav bn ud to valuat th dformation. Th formr ha an advantag to b abl to conidr th nonlinar bhavior Aociat Profor, Nihon Univrity, Koriyama, Japan. Emai:-na@civil.c.nihon-u.ac.jp Profor, Tohou Univrity, Sndai, Japan. m-azama@civil.tohou.ac.jp

2 ,EIF=?AAJ u(z,t) ->=AJ = B t t + dz z A(z) t dz >=OAH-AAJB->=AJ Figur. Schmatic illutration of Layrd mbanmnt lmnt of oil dirctly undr th complx oil and contructing condition. Howvr, an accuracy of th analyi i rmarably dpnd on a quality of information and a prformanc of a nginr for modling a matrial paramtr and a ground tructur. Nwmar' mthod ha bn commonly ud bcau th am lip circl with th ordinary dign i poibl to b ud for th tability analyi during arthqua in th xiting dign cod in Japan. Howvr, a dynamic rpon of a liding bloc in a arth tructur i not conidrd in th mthod. Thn, it i difficult to valuat a larg dformation accuratly for a trong arthqua motion. Rcntly, modifid Nwmar mthod which ta into account of th ffct by modling a mbanmnt a ingl dgr of frdom ytm ha bn propod [3],[4]. Furthrmor, it ha bn pointd out that many damag for th rivr di hav obrvd at th mbanmnt contructd on th oft ground [5.] Thi vidnc how it i important to conidr th dynamic intraction btwn th mbanmnt and th upport ground. Thi rport dcrib two ubjct. On i to propo on-dimnional dynamic rpon analyi mthod for a mbanmnt conidring th dynamic intraction with not only th upport ground jut blow a mbanmnt but alo th id layrd ground. Scond i th application of th mthod for valuating th dynamic bhavior of mbanmnt uffrd by th northrn Miyagi arthqua, July 6,3. Hnc, th mthod i formulatd by u of th tiffn matrix mthod propod by aul [6]. Th accuracy of th mthod i vrifid by th comparion with th analytical rult obtaind by two-dimnional FE mthod. ANALYTICAL METHOD FOR A EMBANKMENT AND A SUPPORT GROUND SYSTEM Fundamntal olution for layrd mbanmnt lmnt Th following gnral aumption for a on-dimnional analyi ar ud to modl an mbanmnt a a on-dimnional ytm: () vrtical incidnt wav i conidrd a an input motion. () a dformation at ach lmnt i only a horizontal dirctionfurthrmor, th hap of mbanmnt i aumd to b a trapzoid. A layrd mbanmnt lmnt a hown in Fig. i dfind to b horizontally licd layr who thicn and ara of a horizontal clo ction at an arbitrary dpth z i dx, A(z) rpctivly. Equilibrium quation () i introducd bad on th balanc btwn inrtia forc and har forc applid to boundary urfac at th lmnt. Hr, G, r rprnt har modulu and ma dnity rpctivly. Equation () xpr an ara of a clo ction at an arbitrary dpth z conidring th lop angl of mbanmnt. A variabl z i changd to Z by u of quation (3) aftr ubtituting A(z) hown in quation () to quation (). Thn, th partial diffrntial quation (4) i obtaind. Nxt, a horizontal diplacmnt u rprnt a th function of pac variabl and tim variabl bad on parating variabl tchniqu. Aftr ubtituting th nw diplacmnt into quation (4), a variabl z i changd to x uing quation (6). Thn, quation () i xprd a Bl quation (5) with rpct to a patial function f( x ).

3 ->=AJ ( u, P ) =OAHA@->=AJ -AAJ C m, D m B, m z m h m G m = ( u, P ) 5KHB=?A=OAH ( u, P ) B, m+ Figur. Charactritic of m th layrd mbanmnt lmnt! =IA=OAH ( u3, P3 ) Figur.3 Modl for a mbanmnt and upport ground ytm g Az () u = g t z u ( ( ) ) z Az () = Bz () = a z + B = a z + B Z = a z B u u u Z = avs( Z + ) d z f() df() z w w + + z f () = x = t Z Z dz Z dz a VS a V Z S d f( x) df( x) + + f( x) = dx x dx It i wll nown that gnral olution of Bl quation i xprd by u of Bl function and Numann function at th ordr. Thn, a diplacmnt and har tr in a layrd mbanmnt lmnt a a tationary olution with rpct to a circular frquncy w i xprd a th following quation (8) with two unnown cofficint C, D. Hr, rprnt a wav numbr w /V. B uz (, w) = [ C J{ z ( + )} a B + D Y{ ( z + )}]xp( iwt) a z G uz (, w) B t(, w) = = -G[ C J{ ( z + )} z a B + D Y { ( z + )}]xp( iwt) a Stiffn matrix of a layrd mbanmnt lmnt and a layrd upport ground Layrd mbanmnt lmnt Th diplacmnt and th har tr at uppr and lowr boundary of m th lmnt in a mbanmnt a hown in Fig. ar obtaind by u of quation (8). By combining th four quation at both boundari, quation (9) i obtaind by liminating th unnown cofficint. Th quation ha a important rlationhip to connct th diplacmnt and th har tr at btwn m th layr with tho at m+ th layrby u of th mthod propod by aul t al, th quation (9) i changd to th rlationhip btwn th diplacmnt at m and m+ th boundary and th har forc at tho boundary. Th obtaind quation () i th am with a tatic govrning quation by u of FE

4 [ ] rprnt tiffn matrix. m m mthod. K P, P m+ ar calculatd har forc by multiplying unit ara to har tr and tho ign i noticd to b dtrmind bad on a ign with rpct to a forc. Subcript for a diplacmnt and a har forc rprnt location numbr of boundary undr th top of mbanmnt and i corrpond to a nodal point which ha bn ud in FE mthod. Th componnt of th tiffn matrix ar hown in quation (). m m Ï um+ m um E E um Ì = [ E ] Ï Ì = m m Ó m+ Ó m È Î Í ÏÌ t t E E Ó t m Ï Pm m m um Ì = Ï t Ì = [ K ] Ó-Pm Ó m Ï Ì + t + Ó um+ m m um = È m m Î Í ÏÌ Ó u m+ m J - J Y =-Gmm J - J m J - J Y = Gm m J (h m) - J m J - J Y = Gm m J - J m J - J Y =-Gmm J - J Layrd upport ground In a layrd upport ground jut blow a mbanmnt mntiond abov, tiffn matrix propod for SH wav fild by Kaul t al i ud for xpring th rlationhip btwn diplacmnt and har forc. Th rlationhip and ach componnt of tiffn matrix ar hown in quation () and (3) rpctivly. G n, n and h n rprnt har modulu, wav numbr and thicn in th n th layr rpctivly. Ï Pn Ï n n un Ì = Ì = K Ó Pn+ - [ ] Ï n xp( i n hn) + xp( -i n hn) t =-ign n Ì xp( i n hn) -xp(-i n hn) Ó t n+ Ó un+ n ign n n n un = È = n n Î Í i n hn - -i n h xp( ) xp( n) n n ÏÌ Ó u = n+ n n = Ambling whol tiffn matrix and dynamic rpon analyi mthod Conidring forc balanc btwn th bottom of mbanmnt and th top of th upport ground jut blow th mbanmnt, a th whol tiffn matrix [ K ] for mbanmnt and upport ground ytm a hown in fig.3 i ambld by u of th tiffn matrix for a mbanmnt lmnt and layrd upport ground lmnt. Thn, th diplacmnt and har forc rlationhip i xprd a quation (3). Th modl a hown in fig.3 conit of a layrd mbanmnt lmnt and thr layrd lmnt in upport ground including ba layr. Hr, U,U S, P and P rprnt th diplacmnt vctor in th mbanmnt, that in th upport ground, th har forc vctor in th mbanmnt and that in th upport ground rpctivly. Th valu of ach vctor componnt a th rpon ar xprd by quation (4). Excpt th nodal forc on th ba layr, th all har forc ar zro. Each componnt of th whol tiffn matrix K [ ] i xprd a quation (5). Hr,

5 5E?K ->=AJ,EIF=?A=J EJAH=?JE S P - S P,EIF=?A=J@EIJHE>KJE BIE?K)MEJDKJ?IE@AHEC@O=E? EJAH=?JE 5E?K 5E@A=OAHA@/HK@,EIF=?A=J@EIJHE>KJEB?IE@AHEC@O=E? EJAH=?JE>AJMAAIE?K )=@IE?K Figur.4 Schmatic illutration for th dynamic intraction btwn th mbanmnt - upport Ground ytm and th id layrd ground ubcript, for an ach componnt rprnt mbanmnt and upport ground rpctivly and ubcript ij rprnt a raw and a column for a tiffn matrix of ach layr lmnt. Finally, uprcript i rprnt a numbr of lmnt. Th componnt 3 in quation (5) rprnt th tiffn for conidring th ba layr a th latic layr and i xprd in quation (6). So far, th formulation for th mbanmnt and upport ground ytm conidring th dynamic intraction btwn th mbanmnt and th upport ground jut blow th mbanmnt i dcribd. Th ondimnional ytm i namd a a oil column A a hown in fig.4. Nxt, Anothr dynamic intraction btwn th oil column A and th id layrd ground hown in fig.4 alo ha to b conidrd. Th id layrd ground which i namd a th oil column B a hown in fig.4 don t hav a mbanmnt. Thn, th diplacmnt and har forc rlationhip i ay to b xprd a quation (7). Th additional forc vctor P for th layr of both oil column undr th mbanmnt i applid to atify with th condition that ach componnt of th rpon diplacmnt vctor U in th oil column A caud by th imic forc P B on th ba layr and th additional forc i qual to ach componnt of that in th oil column B. Undr th condition mntiond abov, quilibrium matrix quation for ach oil column i xprd a quation (8) and (9). By adding both quation, th diplacmnt and har forc rlationhip for a whol ytm conidring two ind of th dynamic intraction i obtaind a th quation () A a imic rpon analyi for th mbanmnt and upport ground ytm, input motion u B on th latic ba layr i conidrd a an incidnt componnt (E). Th har forc P B on th ba layr i xprd a quation () by u of th input motion and rprnt th applid imic forc for th whol ytm. Thn, a rpon diplacmnt i calculatd by multiplying th invr whol tiffn matrix K [ ] - to th nodal forc vctor a hown in quation (). K U P [ ] Ï Ì = Ó US Ï Ì Ó P (3), { } = { } = { } = = B = U ( u ), U ( u, u, u ) P P P P P T ( ) ( ), (,, ) (,, P ) B T B T (4)

6 ! "# [ ] = K È Í Í Í Í Î Figur.5 FE modl for th mbanmnt and th upport ground ytm (5), = i B G B (6) Ï = Pf Ì P [ ]{ U }+ [ ]{ U } = P Ó PB Ì (9), [ f K]{ U } = Ï Ì () Ó PB Ó PB È U U Í ÏÌ Î U U Ó Ï = Ì Ó ÓP B () T P { u u u u B = 3} u3, (), - T K P (3) [ fk]{ U f } = = [ ] { B } VERIFICATION OF THE PROPOSED METHOD WITH THE COMPARISON OF THE RESULTS OBTAINED BY TWO-DIMENSIONAL FE ANALYSIS In ordr to vrify th propod on-dimnional imic rpon analyi mthod for th mbanmnt and th upport ground ytm, an analytical rult obtaind by th propod mthod i compard with an analytical rult calculatd by two-dimnional FE mthod. Stada [7] i ud for th analytical cod for two-dimnional FE analyi. Th analytical FE modl i hown in Fig.5. Th numbr of nodal point and that of lmnt ar 93 and 88 rpctivly. A th hap charactritic of mbanmnt, th hap i a trapzoid. Th lop angl i 45 dgr. Th hight and th width of th top ar m and 5m rpctivly. Furthrmor, th har wav vlocity i 5m/. Th thicn and th har wav vlocity of th upport ground ar 3m and m/ rpctivly. In ordr to calculat th vicou damping for conidring th ba layr a th latic layr, th har wav vlocity i 5m/. Th ma dnity and th poion ratio for all lmnt ar.8t/m 3 and.49 rpctivly. Th imic rcord obrvd at Fuiai du to th Hyogon Nanbu arthqua i ud a th input motion. Fig.6 how th comparion of th rpon acclration tim hitori at th top of th mbanmnt obtaind by D FE analyi to that obtaind by th propod D analyi. Furthrmor, Fig.7 how th comparion of th frquncy tranfr function obtaind by - dimnional FE analyi and that obtaind by th propod on-dimnional analyi mthod.

7 Figur.6 Comparion of th acclration tim hitory at th top of th mbanmnt obtaind by D FE analyi to that obtaind by th propod D analyi Th frquncy tranfr function btwn th top of th mbanmnt and th ba layr i calculatd a th pctral ratio. It i found that both th rpon charactritic for th propod D analyi with rpct to an acclration and a frquncy tranfr function ar good agrmnt with tho for D FE analyi. Thrfor, th propod D analyi ha a good accuracy to apply th dynamic rpon of th mbanmnt conidring th dynamic intraction with th upport ground. Figur.7 Comparion of th frquncy tranfr Function by D FE analyi to that by th propod D analyi DYNAMIC BEHAVIOR OF THE SUFFERED RIVER DIKE DUE TO THE NORTHERN MIYAGI EARTHQUAKE, JULY 6,3 Ovrviw of th arthqua and th damag of rivr di Th northrn Miyagi arthqua which had 3 vnt happnd within a day at July 6, 3 and caud a riou damag. Main hoc ha a dip-lip typ fault and th JMA magnitud i 6.. Epicntr of ach arthqua ar locatd around Aahi Mountain. Th focal dpth of main hoc i about m. Among th tructural damag, many damag of th rivr di wr caud by th main hoc. Th mot riou damag of rivr di at Kimazua along Naru rivr i hown in photo. Th location of th damagd rivr di around Kimazua along Naru rivr and th clo ction of th ground tructur along Naru rivr i hown in fig.8. It i found that th dpth from th bottom of mbanmnt to th ba roc vari thinnr from th lowr tram of th rivr to th uppr tram. Th varianc of th thicn i caud by th chang of th oft clayy oil which xit undr th mbanmnt. Th har wav vlocity th oil i about m/. Th damag ha bn uppod to b caud by th failur of th upport ground du to Photo. Damag of rivr di at Kimazua (Offic of lowr tram Kitaami Rivr contruction, Minitry of tranportation)

8 a) Comparion of th natural priod at t mod by microtrmor with that obtaind by th propod mthod Elvation(m).. -. Ac Ban A A Figur.9 Comparion of th natural priod at t mod obtaind by micro-trmor -. Ac Ac A A Ba Roc Sunayama Nigo Furudat Ditrict Ditrict Ditrict :Damagd Ara b) Th rlationhip btwn th damagd location of rivr di() and th ground tructur along Naru rivr Figur.8 Th comparion of natural priod for th rivr di and th upport ground ytm and th ground tructur along Naru rivr Figur. Comparion of th frquncy tranfr function obtaind by th propod mthod liqufaction and th failur of th mbanmnt du to th trong arthqua motion. Epcially, th damag at th uppr tram around Furuda ha uppod to b caud by th liqufaction. Dynamic bhavior of th uffrd rivr di In ordr to ma clar th vibrating charactritic of th di-upport ground ytm, micro-trmor wa maurd at 9 it on th top of rivr di. Th obrvd it wa locatd at almot vr.5m along th rivr. Th natural priod of th di-upport ground ytm wa obtaind a th prdominant priod of H/V pctral ratio a hown in Fig.9 and th natural priod at ach obrvd it i hown in Fig.8a). Hr, th componnt with rpct to th longitudinal dirction of th rivr di i ud a th horizontal componnt. It i found th natural priod chang from.3 cond to.65 cond with th chang of th dpth btwn th bottom of th mbanmnt and th ba layr. Furthrmor, th natural priod at t mod for th thr it whr ar locatd 3.5m, 5.5m and 6.5m from a front of th rivr wa valuatd bad on th frquncy tranfr function hown in Fig. calculatd by th propod D mthod and wr hown in Fig.8a) a olid circl. Th ground tructur modl on th ba roc hown in Fig. 8b) i conidrd a th analytical modl for calculating th frquncy tranfr function. It i found that th prdominant priod obtaind by th

9 a)comparion of Acclration tim hitori(ns) a)shar tr b)acclration in th mbanmnt Figur. Comparion of th maximum rpon at th two it b)comparion of fourir pctrum Figur. Charactritic of th obrvd rcord around th front of Naru rivr for th offhor Miyagi arthqua, May 6 and th northrn Miyagi Earthqua, July 6 micro-trmor i good agrmnt with th natural priod at t mod calculatd for th ground tructur modl ovr th ba roc by th propod D mthod. It i noticd that th dynamic intraction btwn th mbanmnt and th upport ground including th id layrd ground i important to valuat th vibrating charactritic of th rivr di. Finally, th imic rpon at two it whr wa locatd 3.5m and 6.5m from th front of rivr wr compard ach othr by u of th propod D mthod. Th imic rcord obrvd in th ground at Naaita around th front of Naru rivr wr ud a th input motion. Not only th NS componnt obrvd at th main hoc but alo that obrvd at th offhor Miyagi arthqua wr u for th analyi. Tho acclration tim hitory and fourir pctrum ar hown in Fig.. Th maximum rpon of har tr and acclration with dpth ar hown in Fig.. Hr, th convrgd har modulu and damping contant obtaind by a nonlinar dynamic rpon analyi in frquncy domain for th layrd ground undr th rivr di ar ud a th matrial proprti of oil for conidring th non-linarity of oil. It i found that ach dynamic rpon for th rcord obrvd at th northrn Miyagi arthqua ar thr or four tim to tho for th rcord obrvd at th arthqua, May 6. At th northrn Miyagi arthqua, th rpon undr th dpth of 3.m at th it locatd 3.5m i largr than that at th it locatd 6.5m. It i found that th diffrnc of th mbanmnt rpon i caud by th diffrnc of th thicn for th oft clayy layr and that th rult i on of th raon why th diffrnc of failur mod at ach it i occurrd. CONCLUDING REMARKS On of th ubjct on thi rport i to propo on-dimnional dynamic rpon analyi mthod for a mbanmnt conidring th dynamic intraction with th upport ground jut blow a mbanmnt and with th id layrd ground. Hr, a fw fatur of thi mthod i that th tiffn matrix mthod propod by aul [6] i ud to formulat th govrning quation of th

10 motion and that th tiffn matrix for th layrd mbanmnt lmnt i introducd. Twodimnional FE analyi i ud to ma ur th accuracy of th mthod. Scond ubjct i that thi propod mthod i applid to valuat th dynamic bhavior of mbanmnt uffrd by th northrn Miyagi arthqua, July 6,3. Major rult ar a follow: () Th propod D analyi i applicabl to valuat th dynamic rpon of mbanmnt accuratly bcau th analytical rult by D FE analyi i good agrmnt with tho by th propod D analyi. () Th prdominant priod of th vibration for th rivr di i corrpond to th natural priod at t mod for th rivr di and upport ground ytm. (3) Th thicn of oft clayy oil undr th mbanmnt and th charactritic of th arthqua motion i a important rol to valuat th damag dgr of th rivr di. REFERENCES. Rarch Committ of Simic Dign Mthod for Earth Structur againt Strong Earthqua Motion Rarch Rport on Simic Dign Mthod for Earth Structur againt Strong Earthqua Ground Motion. Committ of Earthqua Enginring: Japan Socity of Civil Enginr; (in Japan). Nwmar N. Effct of Earthqua on Dam and Embanmnt Gotchniqu, London: 965; 5() 3. Rail way Rarch Intitut Railway Structur Dign Standard and it xplanation Maruzn: 999 (in Japan) 4. Stvn L. Kramr, Matthw W. Smith Modifid Nwmar Modl for Simic Diplacmnt of Compliant Slop J. Gotch Eng, ASCE: 997; 3(7) 5. Hamid R. RAZAGHI, Eiji YANAGISAWA, Motoi KAZAMA An Approach to imic Prmannt Diplacmnt of Slop J. Gotch. Eng.JSCE: ; No.659/III-5 6. Saai Y Simic Damag of Rivr Di Tuchi-to-Kio: 98; Vol.8,No.8(7) (in Japan) 7. Yohida,N. STADAS, A Computr Program for Static and Dynamic Analyi of Ground and Soil-Structur intraction Program, Rport Soil Dynamic Group. Th Univrity of Britih Colombia, Vancouvr, Canada: 993