A coupled model for an offshore pile, seabed and seawater interactions

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1 A coupled model or a oshore ple seabed ad seawaer eracos Auhor Lu Ja-Fe Jeg Dog-Sheg ublshed 007 Joural Tle Joural o Coasal Research Coprgh Saeme 007 CERF. The aached le s reproduced here accordace wh he coprgh polc o he publsher. lease reer o he oural's webse or access o he deve publshed verso. Dowloaded rom hp://hdl.hadle.e/007/69 Lk o publshed verso hp://cer-cr.org/de.php/eraoal-coasal-smposum/cs-007ausrala/-a-coupledmodel-or-a-oshore-ple-seabed-ad-seawaer-eraco---lu-ad-d-s-eg-pg-89-9 rh Research Ole hps://research-reposor.grh.edu.au

Joural o Coasal Research SI 50 89-9 ICS007 (roceedgs Ausrala ISSN 079.

2 Joural o Coasal Research SI ICS007 (roceedgs Ausrala ISSN A Coupled Model or a Oshore le Seabed ad Seawaer Ieraco Ja-Fe Lu ad Dog-Sheg Jeg Deparme o Cvl Egeerg School o Cvl Egeerg Jagsu Uvers Zheag Uvers o Sde Sde Jagsu.R.Cha 0. NSW 006 Ausrala. ldocor@ahoo.com d.eg@cvl.usd.edu.au ABSTRACT LU J.-F. AND JEN D.-S A coupled model or a oshore ple seabed ad seawaer eraco. Joural o Coasal Research SI 50 (roceedgs o he 9h Ieraoal Coasal Smposum old Coas Ausrala ISSN A coupled model s developed o vesgae he damc eraco bewee a oshore ple poroelasc seabed ad sea waer. The ple ad he seabed are reaed as sauraed porous meda descrbed b Bo s damc heor. The sea waer s regarded as a acousc medum whch s characersed b he Helmhol equao. Three boudar eleme ormulaos are cosruced or he ple he seabed ad he sea waer respecvel. The hree boudar eleme mehod ormulaos ad he cou codos (sress ad dsplaceme bewee he ple he seabed ad he sea waer are used o ormulae he coupled model or he ssem. Ar lear wave heor s used o evaluae he eeral orce appled o he ple ad he seabed due o he cde ad he scaered waer waves. The damc respose o he ssem s calculaed b he coupled boudar eleme mehod ormulao. A umercal eample s used o demosrae he capac o he coupled model. ADDITIONAL INDEX WORDS: lear waer waves ple boudar eleme mehod porous meda INTRODUCTION The ple oudao s a commo ad mpora srucure or oshore egeerg. For eample ma oshore wd arms are suppored b ple oudaos. Moreover varous oshore plaorms use ples o raser loads o seabed. eerall oshore ples ca be dvded o hree pars: he lowes par whch s embedded seabed he mddle par whch eracs wh seawaer ad he upper par whch s above seawaer. The possble loads assocaed wh he hree pars o he ple are as ollows: sesmc wave loads rom he seabed wave loads ad ce loads rom he waer rego ad wd loads rom he upper rego sa rom a wd urbe. Whe perormg a oshore ple oudao desg wo acors mus be ake o accou. The rs acor s he desg o he ple srucure sel whch makes he ple oudao ulll sregh sess ad sabl requreme. The secod oe s he desg o he seabed whch ams o guaraee he seabeds have eough sregh o ress lqueaco ad shear alure. Numerous researches have bee carred ou cocerg he aalss o oshore ples seabed ad waer waves. For eample he eecs o varous waves o oshore ples have bee vesgaed b researchers such as (MORISON e al. 950; MACCAMY ad FUCHS 95; CHAKARABARTI ad TAM 975. The oshore ple capac ad sresses ad sra o a ple due o wave loads were addressed b TAN (989 ad EICHER e al. (00 respecvel. MITWALLY ad NOVAK (987 used a lear aalss o aalse damc eraco bewee ple sol ple whe he ssem s subeced o a radom wave loadg. Moreover usg he cocep o damc p curves he respose o ed oshore plaorms o wave ad curre loadg whe akg o accou sol ple eraco was addressed b MOSTAFA ad NAAR (00. I s worh sressg ha curre researches abou oshore ples seabed ad seawaer mal ocussed o separae aspecs o he problem raher ha reaed hem as a coupled ssem. However whe subeced o damc loads resposes o he ple he seabed ad he seawaer are coupled ogeher. Thus a coupled model or he oshore ple he seabed ad he seawaer s crucal or a successul oshore ple oudao desg. Aoher lmao o esg researches cocerg he ple-seabed eraco s ha ol sgle phase medum model s used o descrbe he seabed. However s wellkow ha he seabed s a porous medum sauraed b seawaer. Furhermore he evaluao o he pore pressure o he seabed aroud oshore ples s crucal or he esmao o he rsk o lqueaco ad shear alure or he seabed. Cosequel s desrable o rea seabed as a sauraed porous medum ad esablsh a coupled model or he ple he seabed ad he seawaer whe he ssem s subeced o varous damc loads. I hs sud a coupled model s developed o vesgae he eraco bewee he ple he seabed ad he seawaer whe he ssem s subeced o lear waer waves. The ple ad he seabed are reaed as sauraed porous meda descrbed b Bo s damc heor (BIOT 96. The damc respose o he sea waer s descrbed b he Helmhol equao. Three drec boudar eleme ormulaos are esablshed or he ple he seabed ad he seawaer. The cou codos (sress ad dsplaceme bewee he ple he seabed ad he seawaer are used o couple he hree BEM ormulaos. The damc respose o he whole ssem due o waer waves s calculaed b he proposed coupled BEM model. To demosrae he ew model a umercal eample or he coupled ssem wll be preseed. BEM FORMULATIONS BEM Formulaos or he le ad he Seabed The cosuve equaos or a homogeeous porous medum have he orm (BIOT 96 Joural o Coasal Research Specal Issue

3 90 Lu ad Jeg σ = με + λδ αδ p ( e p = α Me + M ζ ( e = u ζ = w (a b where u ad w deoe he average sold dsplaceme ad he lrao dsplaceme o he pore lud; ε e are he sra esor ad he dlaao o he sold skeleo; ζ s he volume o lud eco o a u volume o he bulk maeral; σ s he sress o he bulk porous medum; p s he ecess pore pressure ad δ s he Kroecker dela. Moreover λ μ are Lame cosas o he sold skeleo; α M are BIOT S parameers. The equaos o moo or he bulk porous medum ad a u volume pore lud are epressed erms o he dsplacemes u ad w as ollows (BIOT 96 μ u ( + F = ρ u& + ρ w& (a + λ + α M + μ u + αmw η α Mu + Mw + = ρ u&& + m w&& + w& (b k where ρ b ρ deoe he bulk des o he porous medum ad he des o he pore lud ( ρb= ( φ ρs+ φρ ρ s s he des o he sold skeleo ad φ s he poros o he porous medum; F ad are bod orces o he porous medum ad he pore lud; m = a ρ / φ ad a s oruos; η ad k accou or he vscos o he pore lud ad he permeabl o he porous medum respecvel. Accordg o he aalss o BONNET (987 alhough wo dsplaceme vecors are used BIOT s heor here are ol our depede varables he wo-phase porous medum.e. hree dsplaceme compoes or he sold skeleo ad oe pore pressure or he pore lud. A our-varable-based complee boudar egral equao or sauraed porous meda was ormulaed b DOMINUEZ (99 ad ZIMMERMAN ad STERN (99. Thus or a sauraed porous medum he ollowg boudar egral equao ca be derved [ cu = U ξ ξ T ξ u ( ξ] d( ξ b = (5 where he care over a varable deoes he Fourer rasormed varable u ( ad ( are he geeralsed dsplaceme ad raco compoes (ZIMMERMAN ad STERN 99 wh { u } = { u ~ u u p} { = } = { w} ; c depeds ol = ~ upo he local geomer a he ad reduces o a geeralsed dela uco δ or sde ad o δ / or o a smooh poro o he boudar surace; he requec doma ree s uco U ad T ca be derved as Norrs (985 ad ZIMMERMAN ad STERN (99. Moreover w s he lrao dsplaceme o he pore lud alog he ouward ormal o he surace ad { } = { σ } wh = ~ = ~ beg he dreco cose o. Also he de = ~correspod o coordae ad respecvel. Implemeao o he boudar egral equao (5 o a specc doma volves he dscresao o he boudar surace b suable umber o dscree elemes he erpolao o he eld varable wh a eleme b he ode values o he eleme ad he egrao o shape uco kerel produc over each dscree eleme. As saed above he oshore ple s also reaed as a porous medum. Thus applg boudar egral equao (5 o he ple boudar: ad (Fgure ad ollowg he procedure ouled above he ollowg equao s obaed = H u (6 where he superscrp deoes he ple; ad H (6 are he coece mar obaed b egrag shape uco kerel producs over all he boudar elemes o he ple; u ad are he geeralsed dsplaceme ad raco vecors or all he ple odes ad have he epressos ( ( ( ( ( Np Np Np u = { u u... u ; u... u ; u... u ; ( Np ( Np ( Np ( ( Np T... ;... ;... } ( (... Np T ( ( p p p } { =... ; w w w (7 where N s he oal ode umber or he ple boudar he subscrp N or deog he ode umber. Mulplg boh sde o equao (6 b ( ad leg E = H leads o ( ( ( = E u (8 Dvdg u ad o hree pars correspodg o ad (Fgure respecvel ad rearragg he coece mar E ( (8 he oe has he ollowg paroed mar equao ( ( E E E u ( ( = E E E u (9 ( ( ( ( ( E E E u where he superscrp = or deoe he boudar surace = u = deoe he geeralsed ple dsplaceme ad ple raco vecors or he boudar ad respecvel ad have he ollowg epressos ( ( ( ( Np Np ( Np u = u u u u u u u ( ( Np T p p... p ( ( } Np = ( ( Np Np ( ( Np T... ;... ; w w... w } {... ;... ;... ; {... ; = (0 where N s he oal ode umber or he surace. I s worh pog ou ha o roud o he corers o he seabed ad he seawaer he suraces ad o he ple are separaed b a shor segme o ple surace respecvel. Lkewse usg he same egral equao (5 o he seabed ad ollowg he smlar procedure a paroed mar equao ca also be esablshed or he seabed ( S ( S ( S ( S E E u = ( S S S ( ( S E E u ( S where he superscrp S deoes he seabed ad u ( S = deoe he geeralsed seabed dsplaceme ad seabed raco vecors a he boudar ad (Fgure whch have smlar epressos as (0. Joural o Coasal Research Specal Issue

4 A Coupled Model or Oshore ple Seabed ad Seawaer Ieraco 9 COULED BEM MODEL FOR THE ILE THE SEABED AND THE SEAWATER I above seco hree dscree boudar ormulaos are esablshed or he ple he seabed ad he seawaer. I hs seco he hree dscree boudar ormulaos ad he erace codos bewee he ple he seabed ad he seawaer are used o cosruc he coupled boudar eleme ormulao or he ssem. Fgure A oshore ple embedded a poroelasc seabed ad subeced o sea waves BEM Formulao or he Seawaer As meoed prevousl he ple he seabed ad he seawaer here are reaed as a coupled orced-vbrao ssem subeced o eeral orce due o waer waves. I vew o hs he seawaer here ca be cosdered as a acousc medum characersed b he ollowg Helmhol equao p + k p = 0 ( where p s he pressure o he seawaer ad k s he wave-umber ( k = ω / v ω s he requec o he wave moo ad v s he acousc veloc o he seawaer. The dsplaceme o he seawaer has he orm ( W u = p ( ρω w where ρw s he des o he seawaer. For he seawaer above he seabed due o he presece o he e ree surace o he waer laer hus o save compuaoal me a boudar egral equao whch ca avod he dscresao o he ree surace s preerable. Cosequel he boudar egral equao ad ree s uco due TO SEYBERT ad WU (989 are used hs research cp( = [ p ( ξ( q ξ q ( ξ p(] ξ d( ξ ( where p ad q are he ree s uco or he hal space c depeds upo he local geomer a he po ad q( ξ = p( ξ/ ( ξ ( ξ s he ouward ormal or he boudar. Usg he boudar egral equao ( o he seawaer he ollowg paroed mar equao s obaed ( W ( W ( W ( W u E E p = ( W W W (5 ( W u E E p where he superscrp W deoes he seawaer ad u W ( W p = deoe he ormal seawaer dsplaceme ad he seawaer pressure vecors a he boudar ad ad have he ollowg orm ( W ( W ( W W Np { u u... u ( W ( W ( W W Np u = } p = { p p... p } (6 where = ad u ( W ( W p deoe he seawaer dsplaceme alog he ormal dreco a -h po o he surace ad he seawaer pressure a he same po. Cou codos bewee he hree domas The cou codos a he commo boudar bewee he ple ad he seabed (Fgure have he ollowg orm (DERESIEWICZ & SKALAK 96 ( u ( S = u ( ( S = (7 The cou codos bewee he ple ad he seawaer a her commo boudar (Fgure have he ollowg orm (STOLL ad KAN 98 ( p ( W = p ( u ( + w ( W = u ( ( W = ( W p ( + ( W ( W ( u s ad w = p + = 0 (8 where p are he ple pore pressure he ormal ple dsplaceme o he sold rame ad he ormal lrao dsplaceme o he ple pore lud a ; ( are he raco or he ad dreco a he ple ( W ( W surace ; are he dreco cose mar alog he ad dreco or he seawaer surace ; are he racos alog dreco appled o he ple surace χ ( due o he cde wave ad he scaered wave. Usg o represe he vecors p u w ad he he above vecors have he ollowg uorm epresso ( ( Np χ = { χ χ... χ } (9 Noe ha he vecor u ca be calculaed b he ple surace dsplaceme alog ad he correspodg dreco cose. ( W ( W The dreco cose vecors or he o he seawaer have he epresso ( W ( W ( W W Np = {... } = (0 The cou codos bewee he seabed ad he seawaer a he commo boudar (Fgure have he ollowg orm (STOLL ad KAN 98 ( S ( W ( S ( S ( W p = p u + w = u S = 0 ( S = 0 S W S = p + ( ( S where are he dreco raco o o he seabed due o he cde ad he scaered waer wave. Coupled BEM Formulao or he Ssem Usg he cou codos (7 bewee he ple ad he seabed o ad equaos (9 ad ( he ollowg equao or s obaed Joural o Coasal Research Specal Issue

5 9 Lu ad Jeg ( E + E u + E u + E u ( ( S ( ( ( S + E u = 0 ( Usg he cou codos (8 ad equaos (9 ad (5 he ollowg equao s obaed or he boudar ( W ( W p u ( W ( W ( ( ( [ p E E E ] u + 0 u ( W ( W u u w w S T T [ 000 E p + E p ] = [ 00 ] ( where u u have he epresso ( ( Np u = { u u... u } = ( Besdes or he boudar s assumed ha s kow a pror ad hus usg (9 oe has he ollowg equao ( ( ( T [ E E E ] [ u u u ] = (5 For he boudar usg he cou codo ( ad equaos ( ad (5 he ollowg mar equao s obaed ( S ( S u ( S ( S T [ E E ] [ ] p u ( S u W w S T ( S T [ 000 E p + E p ] = [ 00 ] 0 (6 Combao o equaos ( ( ad (5 (6 elds all he equaos or he ukows u u u ad u S. Oce ( u u u ad u S are deermed ukow racos alog he surace o he boudar ca be evaluaed b (9 ad (. I s worh emphassg ha whe calculag he eeral loads due o he cde wave ad he scaered wave he ple ad he seabed are assumed o be ed hus ad ( S ca be evaluaed uquel b he pressure o he cde ad he scaered waer wave. Also compared wh ad he luece o S s eglgble so s egleced hs sud. NUMERICAL EXAMLES I hs eample he embedded deph o he ple sde he seabed s h = 5 m ad he deph o he waer s h = 5 m. The legh o he ple whch s above he waer s h = m ad he dameer o he ple s d = 0. 6 m (Fgure. The alog he o he ple s supposed o vash. The waer wave s a lear harmoc wave wh agular requec beg /s ad /s respecvel ad he wave hegh s m. The parameers or he seabed ad he correspodg lud assume he ollowg values: 7 7 μ =.0 0 a λ =.0 0 a ρ s =.0 0 kg/m ρ =.0 0 kg/m φ = 0. η =.0 0 a.s 7 k =.0 0 m α = 0.85 M =.0 0 a a =. For he 8 8 ple μ =.0 0 a λ =.0 0 a ρ s =. 0 kg/m φ = 0.5 k =.0 0 m. The oher parameers or he ple ad s saurag lud assume he same values as hose or he seabed. The des ad he acousc veloc o he seawaer are equal o.0 0 kg/m ad 00 m/s respecvel. The shear orce ad he horoal dsplaceme o he ple or he wo cases wh ω = ad /s are ploed Fgures ad respecvel. The pore pressure o he seabed alog he ple sde wh he coordaes = 0 = 0. m ad = 5.0 ~ 0.0 m s show Fgure. Fgure shows ha he shear orce reaches s mamum a he erace bewee he seawaer ad he seabed. Also or he upper par o he ple he shear orce or he case o ω = /s s larger ha ha or he case o ω = /s whle or he lower par o he ple oppose edec occurs. For he horoal dsplaceme he dsplaceme or he case o ω = /s s larger ha ha o he case wh ω = /s. Moreover he ω = /s case acheves s mamum dsplaceme a he upper par o he ple whle or he ω = /s case he mamum dsplaceme occurs aroud h /( + h = 0.7. Fgure dcaes ha he pore pressure or he case o ω = /s s slghl larger ha ha or he case o ω = /s. Moreover he pore pressure decreases sharpl wh creasg deph o he seabed. Thus he upper par o he seabed has a hgh rsk o lqueaco ad shear alure due o he larger value o he pore pressure. Fgure. The shear orce o he ple subeced o lear waer waves wh wave hegh beg m ad ω = s -. Fgure. The horoal dsplaceme o he ple subeced o lear waer waves wh wave hegh beg m ad ω = s -. Joural o Coasal Research Specal Issue

6 A Coupled Model or Oshore ple Seabed ad Seawaer Ieraco 9 Fgure The pore pressure o he seabed alog he ple sde wh he coordaes = 0 = 0. m ad = 5.0 ~ 0.0 m whe he ple s subeced o lear waer waves wh wave hegh beg m ad ω = s -. CONCLUSION A requec doma coupled model accoug or he damc eraco bewee he ple he seabed ad he seawaer has bee developed hs paper. The coupled model s based o he boudar eleme mehod or he porous medum ad he acousc medum as well as he cou codos a he erace bewee he hree domas. Alhough ol he lear harmoc wave he requec doma s cosdered he paper usg he Fourer rasorm mehod radom waves he me doma ca also be ake o accou b our model. Moreover he ew model ca also be used o vesgae he damc respose o he ssem o olear waer waves sesmc waves wd loads or ce loads. Also he curre model ca be eeded o solve he problem o he eraco bewee seabed ple groups ad seawaves. A umercal eample demosrag he respose o he coupled ssem o lear sea waves s gve he paper whch dcaes ha wave loads ca geerae cosderable pore pressure a some pars o he seabed. Clearl he obaed pore pressure wll aclae he lqueaco ad he shear alure aalss or he seabed. EICHER J.A.; UAN H. ad JEN D.S. 00. Sress ad deormao o oshore ples uder srucural ad wave loadg. Ocea Egeerg 0 ( MACCAMY R.C. ad FUCHS R.A. 95. Waves orces o ples: a draco heor. U.S. Arm Corps o Egeerg Beach Eroso Board Washgo D.C. echcal Memoradum No. 69. MITWALLY H. ad NOVAK M Respose o oshore owers wh ple eraco. Joural o Egeerg Mechacs ASCE ( MORISON J.R.; O BRIEN M..; JOHNSON J.W. ad SCHAAF S.A The orces eered b surace waves o ples. Joural o eroleum Techolog eroleum Trasacos AIME MOSTAFA Y. E. ad NAAR M. H. E. 00. Respose o ed oshore plaorms o wave ad curre loadg cludg sol srucure eraco. Sol Damcs ad Earhquake Egeerg NORRIS A.N Radao rom a po source ad scaerg heor a lud-sauraed porous medum. Joural o he Acouscal Soce o Amerca SEYBERT A.F. ad WU T.W Moded Helmhol egral equao or bodes sg o a e plae. Joural o he Acouscal Soce o Amerca STOLL R. D. ad KAN T.K. 98. Releco o acousc waves a a waer-sedme erace. Joural o he Acouscal Soce o Amerca TAN W.H Uceraes oshore aal ple capac. I: (F.H. Kulhaw Ed.. Foudao Egeerg: Curre rcples ad racces. ASCE ress New York pp ZIMMERMAN C. ad STERN M. 99. Boudar eleme soluo o -D wave scaer problems a poroelasc medum. Egeerg Aalss wh Boudar Elemes -0. LITERATURE CITED BIOT M.A. 96. Mechacs o deormao ad acousc propagao porous meda. Joural o Appled hscs BONNET Basc sgular soluos or a poroelasc medum he damc rage. Joural o he Acouscal Soce o Amerca CHAKARABARTI S.K. ad TAM A Ieraco o waves wh large vercal clder. Joural o Shp Research 9. DERESIEWICZ H. ad SKALAK R. 96. O uqueess damc poroelasc. Bulle o he Sesmologcal Soce o Amerca DOMINUEZ J. 99. Boudar eleme approach or damc poroelasc problems. Ieraoal Joural or Numercal Mehods Egeerg Joural o Coasal Research Specal Issue

Quantum Mechanics II Lecture 11 Time-dependent perturbation theory. Time-dependent perturbation theory (degenerate or non-degenerate starting state)

Quantum Mechanics II Lecture 11 Time-dependent perturbation theory. Time-dependent perturbation theory (degenerate or non-degenerate starting state) Pro. O. B. Wrgh, Auum Quaum Mechacs II Lecure Tme-depede perurbao heory Tme-depede perurbao heory (degeerae or o-degeerae sarg sae) Cosder a sgle parcle whch, s uperurbed codo wh Hamloa H, ca exs a superposo