Dr. College of Civil & Hydroelectric Engineering of China Three Gorges University, Yichang,, China 2

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1 h 14 th World Conrnc on Earthquak Engnrng Octobr 12-17, 28, Bjng, Chna Study on th Flud-Sold Intracton Mchanm o Caoh Aquduct n South-to-North Watr Dvron Mddl Ln Projct Undr Condton o Smc Load PENG Hu 1 LIU D-u 2 IAN Bn 3 1 Dr. Collg o Cvl & Hydrolctrc Engnrng o Chna hr Gorg Unvrty, Ychang,, Chna 2 Proor, Collg o Cvl & Hydrolctrc Engnrng o Chna hr Gorg Unvrty, Ychang,, Chna 3 Proor, Collg o Cvl & Hydrolctrc Engnrng o Chna hr Gorg Unvrty, Ychang,, Chna E-mal:hpng1976@163.com ABSRAC: Bad on condrng th lud-old couplng problm n aquduct, th dynamc FEM modl o lud-old ntracton wa tablhd by u o Galrkn Mthod. And th mc rpon o Caoh aquduct tructur n th South-to-North Watr Dvron Mddl Ln Projct wa calculatd. h rult ndcat that mc rpon o th aquduct tructur ncra wth th r o watr lvl, and o th lud-old ntracton wa o grat nlunc on th mc rpon o th aquduct tructur. h calculatd rult could provd good ntructon to orthcomng aquduct contructon and opratonal managmnt and aty montorng. KEYWORDS: aquduct; lud-old ntracton; mc rpon; dynamc charactrtc; South-to-North Watr Dvron Projct 1.INRODUCION In ngnrng communty, th lud-old ntracton btwn om tructur n lud and lud tl a common problm that w otn ncountr. hr an xtnv manng or projct contructon to tudy th problm o lud-old ntracton. h problm o lud-old ntracton wa rtly put orward by H. M. Wtrgaard n th 193[1]. In Chna, Pro. Zhng Zh-mn took om rarch on coupld vbraton btwn watr and plat and cantlvr n 195[2~3].In th prod o 196 to 197, th tudy on lud-old ntracton n ovra wa actv and th man rarch objct concntratd on hydraulc tructur and drnt hp whch occurrd coupld vbraton caud by watr wav. h thory o hydrolatcty wa tablhd by a Brth pcalt R. E. D.Bhop at that tm[5~7]. At prnt, gnrally thr ar two mthod whch can b ud to rarch th problm o lud-old ntracton. h rt on analytcal-numrcal mulaton, Gr doubl aymptotc approxmaton(daa) commonly ud o ar[8]; th cond on numrcal mthod, ncludng FEM, BEM, Lagrang-Eulran Mthod and o on. In lnar lud-old ntracton ytm(lnar latc tructur and dal ncomprbl lud), thr ar alo two mthod otn ud to tudy lud-old ntracton problm. h rt on through takng dplacmnt vctor o tructur and lud a ld varabl to rarch lud-old ntracton problm, uch a dplacmnt-dplacmnt chm[9~1]; th cond on to u hybrd chm whch com rom tructur dplacmnt vctor and lud ld varabl, uch dplacmnt-prur chm, dplacmnt-vlocty potntal chm and o on[11~12]. In FEM, th lud ar ncomprbl th mthod o Numrcal Computng Mthod o Addtonal Watr Ma otn appld [13~14]. Whn w condr th charactr o lud comprblty, th Galrkn Mthod appld to contruct lud-old ntracton FEM ormula btwn aquduct and watr n th papr. h rarch ndcat that numrcal mthod can mulat th couplng ntracton btwn watr and hug watr-rtanng tructur and mc rpon wll. 2.CALCULAION PRINCIPLE Flud-old ntracton problm vry complcatd and t nvolv n tructur dynamc and lud dynamc. Uually, om uppo can b mad accordng to practcal ngnrng problm. A a rult, th calculaton modl wll b mpld and cncy wll b mprovd ctvly. In th papr, w not only condr bg tructur coupld vbraton, but alo condr th ntracton btwn watr and bg tructur undr condton o mall luctuaton. In practcal calculaton, n ordr to mak calculaton convnnt w otn uppo that lud ar nvcd and rrotatonal low. Howvr, n mot condton, vn though w mak om uppo w can not obtan analytcal oluton. So w can not but tnd to numrcal mulaton mthod. Actually, om tructur wth rgular hap lk cylndr mayb u pcal uncton uch a Bl uncton to gt dal analytcal oluton. Accordng to rrnc 8, through comprhnvly condrng Nav-Stok quaton and contnuty quaton th FEM quaton wth dplacmnt-prur chm ( u, p) o unorm, nvcd and

2 h 14 th World Conrnc on Earthquak Engnrng Octobr 12-17, 28, Bjng, Chna comprbl lud undr condton o mall luctuaton ar contructd. 2.1 h dynamc modl o lud-old ntracton ytm and t bac quaton and boundary condton Hr w uppo that watr nvcd, comprbl and ur mall dturbanc. At th am tm, w can know that watr r urac appar mall luctuaton bcau aquduct locat n low mc ntnty ara. W tak aquduct tructur a lnar latc body bcau durng th cour o opraton th aquduct tructur don t allow crackng. Hr, VandV rpctvly rprnt old doman and lud doman. tand or ntrac btwn old and lud. S b lud rgd xd ntrac. S dnd a watr r urac. Su old dplacmnt boundar. S σ tand or orc boundar mpod on old. ln unt vctor o lud boundar. ntrac btwn old and lud 2.1.1Flud doman ( V ) h quaton or lud ld ar ltd a ollow: n xtrnal normal n xtrnal normal ln unt vctor o old boundar. Any pont at nand n ar oppot drcton. p 1 && p= (1), 2 c Hr p lud prur and ctand or ound vlocty n lud h boundary condton o lud p On rgd xd boundary condton ( S b) : = n (2) On watr r urac ( S ) : p 1 + && p = (3) z g 2.1.3Sold doman ( V ) h quaton or old ld ar xprd a ollow: σ + = ρ u&& (4) j, j Hr σj ar old tr componnt, u ar old dplacmnt componnt, ar old body orc componnt, ρtand or old dnty Sold body boundary condton h orc boundary condton ( S σ ) can b xprd a ollow: σ n = (5) j j h dplacmnt boundary condton ( Su) can b wrttn a ollow: u = u (6) Hr and u rpctvly rprnt th known urac orc componnt and dplacmnt componnt Condton on ntrac btwn old and lud Kntc condton on th ntrac ( ) ar that normal vlocty hould kp contnuou:

3 h 14 th World Conrnc on Earthquak Engnrng Octobr 12-17, 28, Bjng, Chna v = v n = v n = v n = v (7) n n By man o moton quaton o nvcd, wakly dturbd lud, th quaton 7 can b changd a ollow: p + ρ u&& n = (on ntrac S ) (8) n Hr utand or old dplacmnt vctor, ρ lud dnty. h contnuou condton o orc on ntrac ( ) ar xprd a ollow: σ n = τ n = τ n (9) j j j j j j h quaton 9 man that normal orc on ntrac ar contnuou. Hr componnt. I th lud ar nvcd thnτ j can b wrttn a ollow: τ j tand or lud tr tnor τ j = pδ (1) j By ubttuton quaton 1 nto quaton 9 w can gt: σ jnj = pn (on ntrac S ) (11) 2.2 Etablhmnt o lud-old ntracton FEM quaton bad on Galrkn Mthod 2.2.1Contructng ntrpolaton uncton For lud prur pattrn appld and prur dtrbuton n lud lmnt can b xprd a ollow: m pxyzt (,,,) N ( xyz,, ) p () t = Np (12) = 1 Hr mar nod numbr o lud lmnt, p ar nod prur vctor o lud lmnt, N ntrpolaton uncton corrpondng to nod No., N ntrpolaton uncton matrx. For old bod dplacmnt pattrn ud and dplacmnt dtrbuton n old lmnt can b ltd a ollow: u u m m u ( xyzt,,,) = v N( xyz,, ) v = N( xyz,, ) a () t = Na (13) = 1 = 1 w w Hr mar nod numbr o old lmnt, a ar nod dplacmnt vctor o old lmnt, N ntrpolaton uncton corrpondng to nod No., N ntrpolaton uncton matrx Drvd olvng quaton by man o Galrkn Mthod h wghtd rdual oluton o Galrkn Mthod or bac quaton and boundary condton o lud-old ntracton n lud doman can b wrttn a ollow: 1 p 1 p p δ p( p && pdv ) δ p( ) ds δ p( && p+ ) ds δ p( + ρ && ) ds = (14) u n, 2 c n V Sb g z n S For old doman uppod that dplacmnt boundary condton hav bn atd and thn th ormula n old doman can b xprd:

4 h 14 th World Conrnc on Earthquak Engnrng Octobr 12-17, 28, Bjng, Chna δu ( σ + ρ u&& ) dv δu ( σ n ) ds δu ( σ n pn ) ds = (15) j, j j j j j V Sσ hrough applcaton o ntgraton by part to rt tm δp( p, ) dv o ormula (14), w can obtan ormula (16):,, 2 c g V S V 1 1 [( δ p ) p + && pdv ] + δ p( && pds ) + δ p( ρ && ) ds = (16) u n Smlarly by u o ntgraton by part to rt tm δu ( σ, ) dv o ormula (15) and ubttutng t nto phycal quaton, thn w can gt ormula: j jkl kl V Sσ V j j [ δε D ε + δu ( ρ u&& )] dv δuds + δu ( pn ) ds = (17) By ubttuton ormula (12) and ormula (13) nto ormula (16) and ormula (17) rpctvly and at th am tm condrng th arbtrarn o ollow: δpandδ u, th FEM quaton o lud-old ntracton can b xprd a 1 M && a KS Q ρ a F + = Q M p&& p K (18) Hr par prur vctor on lud nod, aar dplacmnt vctor on old nod, Q lud-old couplng matrx, M and Kar global lud ma matrx and global lud tn matrx rpctvly, Mand Kar global old ma matrx and global old tn matrx rpctvly, Far xtrnal load vctor mpod on old. Each lmnt matrx corrpondng to global matrx can b xprd: M 1 1 = NN dv + NN ds ; = N N K dv ; Q ρ ds x x = NnN ; M = ρnn dv ; 2 c g V S = V K BDB dv ; F = N dv + N ds,hr B dplacmnt-tran rlaton matrx o old. From V V Sσ V abov calculaton proc w can that M cont o two part. h rt part MV and th cond part M.So w can know that M = M + M.Hr V MV lmnt ma matrx caud by comprbl lud, M lmnt ma matrx caud by r urac wav problm. Whn condrng th rpon o aquduct undr condton o arthquak, th FEM quaton o lud-old ntracton can b wrttn a ollow: 1 1 M && a M a& KS Q S ρ a& K Q α β ρ a F = Q M p&& Q M p& p & p K K (19)

5 h 14 th World Conrnc on Earthquak Engnrng Octobr 12-17, 28, Bjng, Chna 1 M a& KS Q Hr α ρ a& C= + β dampng matrx, αand βar proportonal ma and Q M p& p & K tn dampng cocnt rpctvly. 3.FEM MODEL OF AQUEDUC Caoh Aquduct locat n Shnxng own, Manchng County, Baodng Cty, Hb Provnc. h aquduct a larg pan cro tructur n South-to-North Watr Dvron Mddl Ln Projct. h dgnd dcharg 125 m 3 /,th maxmum dgnd dcharg 15 m 3 /.h pan o aquduct 3m and th tructur o ngl conncton o thr lum wth mult-d wall appld. Each lum cro cton z m and th thckn o d wall.6m. At th top o d wall 2m thck pdtran plat ar pavd. h thckn o mddl wall btwn ach lum.7m. Smlarly, 2.7m thck pdtran plat ar pavd on th top o mddl wall. Longtudnal prtrd tructur ar ud by d wall and mddl wall. Sd rb and bottom rb ar xd n aquduct body. On th top o th d wall and mddl wall, pull rod ar alo contructd. h pacng o d rb 2.5m, th wdn and hght o d rb ar.5m and.7m rpctvly. h pacng o bottom rb alo 2.5m, th wdn and hght o bottom rb ar.5m and 1.1m rpctvly. h wdn and hght o pull rod ar.3m and.4m rpctvly. h matral o aquduct C5 grad rnorcd concrt whch maxmum comprv trngth about 5MPa. Fgur on h cro cton o aquduct 3.1 h 3-D FEM modl h ngl-pan aquduct 3m long and 22m wd. In th papr ttrahdron lmnt ud to calculat aquduct tructur. h rod lmnt appld to mulat tl trand and om tl bar. A total numbr o lmnt ar ubdvdd. h X-coordnat n th am drcton wth watr low. h Y-coordnat paralll to aquduct wd drcton. h Z-coordnat n th am drcton wth lvaton o aquduct. h 3-D FEM modl obtand by ot ANSYS. ELEMENS NOV :4:3 Y Z X Fgur two h 3-D FEM modl o aquduct

6 h 14 th World Conrnc on Earthquak Engnrng Octobr 12-17, 28, Bjng, Chna 3.2 Calculaton ca In ordr to nur a opraton and obtan rlabl conomc bnt, t ncary to tak numrcal mulaton on hgh trngth and larg pan aquduct tructur. Accordng to ngnrng practc and projct contructon plannng, x calculaton ca ar chon n th papr. 1thr lum pang watr undr condton o dgnd watr dpth(4.15m) and condrng aquduct gravty; 2thr lum pang watr undr condton o ncrad watr dpth(4.792m) and condrng aquduct gravty; 3thr lum pang watr undr condton o bankull watr dpth(5.4m) and condrng aquduct gravty; 4thr lum pang watr undr condton o bankull watr dpth(5.4m) and condrng aquduct gravty and wnd load; 5only mddl lum pang watr undr condton o ncrad watr dpth(4.792m) and condrng aquduct gravty; 6only two d lum pang watr undr condton o ncrad watr dpth(4.792m) and condrng aquduct gravty. h calculaton otwar wa l-dvlopd and th otwar could ntrac wth ANSYS and Autocad. In th pat v yar, th otwar hav bn uccully appld to om hug hydropowr projct n Chna, uch a hr Gorg Projct, Jnpng arch dam and Xaowan arch dam. 3.3 h choong o mc wav Whn takng mc rpon calculaton on aquduct, t ncary to choo raonabl mc wav bcau wavorm hav proound ct on mc rpon o aquduct. Accordng to t condton o aquduct, or th ak o thoroughly rlctng mc rpon o aquduct undr condton o drnt mc wav, n th papr EL-Cntro wav, at wav, Paadna wav ar chon a ground acclraton. h mc wav nput mthod ltd a ollow: whn tranvr xctaton ourc happnng thn EW componnt o EL-Cntro wav, at wav, Paadna wav hould b nputtd. I vrtcal xctaton ourc takng plac, thn NS componnt o EL-Cntro wav, at wav hav to b nputtd. h concrt numrcal valu and maxmum ampltud ar conrmd accordng to rrnc [15]. 3.4 Calculaton rult Bor undrtakng dynamc analy watr can b takn a a knd o addtonal ma o aquduct tructur. Whn calculaton w can know that n th ntak and outlt o aquduct watr r and th boundary condton n ntak and outlt o aquduct ar p/ n=. Calculaton rult manly nclud thr part. h rt part tranvr dplacmnt n bottom plat n th mddl o thr lum rpctvly. h cond part longtudnal dplacmnt n bottom plat n th mddl o thr lum rpctvly. h thr part rotaton angl o mddl cro cton ntrwnng ax o thr lum rpctvly (bcau o mor calculaton ca, th calculatd tr and dplacmnt contour plot ar canclld). Man calculaton rult ar ltd a ollow: 3.4.1ranvr dplacmnt n bottom plat n th mddl o thr lum h maxmum tranvr dplacmnt n bottom plat n th mddl o thr lum rpctvly ar ltd n tabl on undr condton o EW componnt xctaton o EL-Cntro wav, at wav, Paadna wav. From tabl on w can know that th dtrbuton law and changng trnd o tranvr dplacmnt n bottom plat n th mddl o thr lum ar mlar. Howvr, tranvr dplacmnt n bottom plat n th mddl lum bggr than that o n two d lum. Wth watr dpth ncrang, th ma o aquduct ncra too. A a rult, tranvr dplacmnt n bottom plat n th mddl o thr lum gradually ncrad. Furthrmor, undr condton o cro-wnd load acton, tranvr dplacmnt n bottom plat n th mddl o thr lum a lttl largr than that o wthout condrng cro-wng load acton. h ncrang ampltud not obvou whch ndcat that watr can ctvly rduc tranvr mc rpon. Fgur 1 th maxmum tranvr dplacmnt n bottom plat n th mddl o thr lum Maxmum tranvr dplacmnt(mm) poton ca1 ca2 ca3 ca4 ca5 ca6 Bottom plat n th mddl o rght lum Bottom plat n mddl lum Bottom plat n th mddl o lt lum Rotaton angl o mddl cro cton ntrwnng ax o thr lum

7 h 14 th World Conrnc on Earthquak Engnrng Octobr 12-17, 28, Bjng, Chna h maxmum rotaton angl o mddl cro cton ntrwnng ax o thr lum rpctvly ar ltd n tabl two undr condton o EW componnt xctaton o EL-Cntro wav, at wav, Paadna wav. From calculaton rult, th dtrbuton law and changng trnd o rotaton angl o mddl cro cton ntrwnng ax o thr lum ar am. Actually, rotaton angl n th mddl lum bggr than that o two d lum. Smlarly, condrng cro-wnd load acton th maxmum rotaton angl o mddl cro cton ntrwnng ax o thr lum rpctvly ar largr that that o wthout condrng cro-wnd load. Wth watr dpth ncrang, th ma o aquduct ncra too. A a rult, th maxmum rotaton angl o mddl cro cton ntrwnng ax o thr lum rpctvly ncra too. In act, th dtrbuton law and changng trnd o rotaton angl o mddl cro cton ntrwnng ax o thr lum bor and atr watr lvl changng ar am. So w can know that lud-old ntracton obvouly act th valu o rotaton angl but l obvouly act dtrbuton law o rotaton angl. Flud-old ntracton calculaton alo ndcat that mc rpon aroud by longtudnal wav l than that o tranvr wav. abl 2 h maxmum rotaton angl o mddl cro cton ntrwnng ax o thr lum Maxmum rotaton angl(rad) poton ca1 ca2 ca3 ca4 ca5 ca6 Cro cton n th mddl o rght lum Cro cton n mddl lum Cro cton n th mddl o lt lum Longtudnal dplacmnt n bottom plat n th mddl o thr lum h maxmum longtudnal dplacmnt n bottom plat n th mddl o thr lum rpctvly ar ltd n tabl thr undr condton o NS componnt xctaton o EL-Cntro wav, at wav. From calculaton rult w can know that th maxmum longtudnal dplacmnt tak plac n artht pont on aquduct along mc tranmon drcton (rght lum ar rom that o mc ourc). Howvr, no mattr whthr condrng lud-old ntracton or not, wth watr dpth ncrang, th ma o aquduct ncra too. A a rult, longtudnal dplacmnt n bottom plat n th mddl o thr lum gradually ncrad. Furthrmor, undr condton o cro-wnd load acton, longtudnal dplacmnt n bottom plat n th mddl o thr lum a lttl largr than that o wthout condrng cro-wng load acton. h ncrang ampltud not obvou whch ndcat that watr can ctvly rduc longtudnal mc rpon. ranvr wav mpo lttl ct on longtudnal dplacmnt. abl 3 th maxmum longtudnal dplacmnt n bottom plat n th mddl o thr lum Maxmum longtudnal dplacmnt(mm) poton ca1 ca2 ca3 ca4 ca5 ca6 Bottom plat n th mddl o rght lum Bottom plat n mddl lum Bottom plat n th mddl o lt lum Vrtcal dplacmnt n bottom plat n th mddl o thr lum h maxmum vrtcal dplacmnt n bottom plat n th mddl o thr lum aroud by tranvr wav and longtudnal wav ar vry mall rpctvly. Calculaton ndcat that no mattr whthr condrng lud-old ntracton or not th vrtcal dplacmnt chang a lttl. It how that lud-old ntracton ha lttl nlunc on aquduct vrtcal dplacmnt. 4.CONCLUSIONS (1) Rult o drnt calculaton ca ndcat that condrng lud-old ntracton n aquduct and drnt watr dpth hav clo rlaton wth aquduct mc rpon. It ncary to tudy lud-old ntracton mchanm undr condton o mc wav acton[15]. (2) Drnt mc rpon can b obtan undr condton o drnt mc wav xctaton. Although calculaton rult changng trnd ar mlar th valu o rult ar drnt. So, n practcal projct w mut

8 h 14 th World Conrnc on Earthquak Engnrng Octobr 12-17, 28, Bjng, Chna condr raonabl xctaton drcton and mc wav. (3) Undr condton o cro-wnd load acton th maxmum longtudnal and tranvr dplacmnt ar a lttl largr than that o wthout cro-wnd acton. It ndcat that watr body can ctvly rduc mc rpon o aquduct. (4) h maxmum longtudnal dplacmnt aroud by tranvr wav vry mall. h mc rpon o aquduct brought by longtudnal wav acton l than that o tranvr wav acton. ACKNOWLEDGEMEN: h rarch wa upportd by Natonal Natural Scnc Foundaton o Chna (NSFC, wth grantd numbr: ). W thank Proor Huang Da-ha or commnt on th manucrpt. REFERENCES [1]Wtrgaard H M. Watr prur on dam durng arthquak[j].ran. o Amrcan Socty o Cvl Engnrng,1933,59 (3) : [2]Zhng Zh-mn. h vbraton o plat undr condton o lud acton[j].acta Mchanca Snca, 958,2 (1) : [3]Zhng Zh-mn, Ma Zong-ku. Fr vbraton o cantlvr undr condton o latral lud acton[j]. Acta Mchanca Snca,1959,3 (2) : [4]Gpanpan J E. Flud-old ntracton[a].h Wntr Annual Mtng o th ASME[C].Pttbrgh, Pnnylvana, [5]Bhop R E D, Prc W G. Hydrolatcty o Shp[M].Cambrdg :Cambrdg Unvrty Pr,1979. [6]Znkwcz O C, Btt P. Flud-tructur dynamc ntracton and wav orc: An Introducton to numrcal tratmnt [J]. Intrnatonal Journal or Numrcal Mthod n Engnrng, 1978,13 (1) :1-16. [7]FENG Zhn2xng, Hung C.h coupld BE/ FE algorthm or lud2tructur ntracton : 2-D and 3-D numrcal rult[j]. Acta Mchanca and Scnta, 1987,7(2). [8]Lu Xn-n.Advancd Structural Dynamc[M].Shangha: Shangha Jaotong Unvrty Pr, [9]2Alrdo B, Rodolo R. Fnt lmnt computaton o th vbraton mod o a lud-old ytm, [J]. Comput Mthod, Appl Mh Eugrg, 1994, [1]Alrdo B, Rodolo R, Rodrguz R. Fnt lmnt oluton o ncomp rbl lud-trucur vbraton p roblm [J]. Intrnatonal Journal or numrcalmthod n ngnrng, 1997:4-42. [11]Lu W K, Chang H G. Multdcp lnary and ntracton problm: A mthod o computaton or lud tructur ntracton,[j] Computr & Structur, 1985, 2 :1-3. [12]Da Da-nong, Wang Xu-chng, Du Qng-hua.A modl analy or th dynamc rpon o lud-tructur ytm[j].acta Mchnca Solda Snca, 199, 11 (4) : [13]Jn Zhan-l,Wang Zong-l,L Hong-yun. Numrcal Computng Mthod o Addtonal Watr Ma whn th Structur Vbrat n th Innt Lqud[J]. Journal o Shangha Jaotong Unvrty, 2, 34 (8). [14]Wang Xu-chng. Fnt Elmnt Mthod[M].Bjng: nghua Unvrty Pr,July,23 [15]Wang Bo, Chn Hua, Xu W. Nonlnar mc rpon analy o aquduct contructd wth lad rubbr barng or vbraton olaton [J].Watr Rourc and Hydropowr Engnrng, Jun,25,Vol.25 No.3:5-7.