Restoring Stiffness in the Hydroelastic Analysis of Marine Structures

Transcription

1 DC 6955 Ivo EJAOIĆ eve ADŽIĆ oa LADIMIR Restorg tffess te droeastc Aass of Mare tructures Reve aer e restorg stffess tat coues dsaceets ad deforatos as ver ortat roe droeastc aass of are structures e robe of ts foruato s qute coe ad s st dscussed te reevat terature I te aer te recet foruatos of restorg stffess are correated ad aaed Due to soe equvaet ters of te restorg ad geoetrc stffess as a resut of coo oad te ufed stffess s estabsed ad coared t te coete restorg stffess o te reevat terature It s foud out tat te e forua deas t ore ters ad tat uder soe assutos t s reduced to te o coete restorg stffess e ufed stffess costtuto s aaed troug derved aatca foruae for rsatc otoo Its cosstec s ceced for te rgd bod dsaceets Aso uerca resuts of te droeastc resose of segeted barge are correated t avaabe ode test resuts oe ssues tat are ortat for ractca eetato te droeastc code for febe structures are descrbed Keords: geoetrc stffess droeastct restorg stffess segeted barge Autors Address (Adresa autora): verst of Zagreb Facut of Mecaca Egeerg ad ava Arctecture Ivaa Lučća 5 Zagreb Croata E-a: voseaovc@fsbr Receved (Preo): -5-4 Acceted (Prvaćeo): -- Oe for dscusso (Otvoreo a rasravu): -- Povrata rutost u droeastčo aa oors ostruca Preged rad Povrata rutost oa sreže oae deforace gra važu uogu u droeastčo aa oors ostruca eo forurae redstava sože robe oš e uve redet rasrave u stručo teratur ovo čau atuae foruace ovrate rutost usoređee su aarae Zbog e evvaet čaova ovrate geoetrse rutost ao reutata aedčog oterećea ostavea e sedea rutost usoređea s uuo ovrato rutost oato u teratur staoveo e da ova forua a vše čaova a uvođee odgovarauć retostav reducraa e a oat ob oete ovrate rutost Kosttuca sedee rutost aaraa e ute vede aatč forua a ratča oto ea osteca roverea e a reru rutoga tea aođer uerč reutat droeastčog odva segete barže usoređe su s dostu reutata odesog stvaa avedee su ee uute a eetacu vedeog agorta u rogra a droeastču aau oors ostruca Kuče reč: geoetrsa rutost droeastčost ovrata rutost segeta barža Itroducto Recet treds creasg vesse se ad seed o oe ad ad otato of s structure o te oter resut a qute febe s u atura vbratos of suc ss ca eas fa to resoace t te ecouter frequec a ordar sea state Large cotaer ss are a seca subect of vestgato sce te are artcuar sestve to quarterg seas due to reduced torsoa stffess caused b arge atc oegs [] e s resose s usua cassfed as srgg or g deedg o stead-state or traset ature of vbratos reat effort s ut to vestgato of fud-structure teracto bot fro etree oadg ad fatgue ot of ve c are ortat for s structure desg ad safet [] e s droeastc aass cudes defto of structura ode ass dstrbuto restorg stffess added ass dag ad ave ectato [] I ste of te fact tat s droeastct as bee a o subect for a ears [4] soe questos rea oe Costtuto of restorg stffess c s ot as stragtforard as oe coud age s oe of te [5] Basca tere are to aroaces a ure droecaca oe ad aoter tat eteds to te structure Wt te forer aroac te e-o Prce ad Wu foruato o te basc drostatc ressure ter s cosdered [6] ea forua reresets a eteso gvg te ecessar drostatc ressure coeffcets [7] oever eter of tose to foruatos gves te coete restorg stffess coeffcets for te rgd bod otos because te gravt art s ssg e above sortcog s reoved b Rggs [8] secfg e oda ressure coeffcets ad addg te gravt ter te for so abe Eq () e de otato ad 65

2 suato coveto s used for coactess ere X Y ad Z are goba coordates; s te atura ode (rgd or eastc); s te etted surface; s te ut ora vector to te etted surface; ρ ρ are te fud ad structure ass destes ad g s gravt costat A furter roveet s doe b uag ad Rggs [9] offerg a cobed droeastc ad structura foruato of restorg stffess Eq () abe ere σ s stress tesor due to gravt ad drostatc oads e ast tegra Eq () actua reresets te e-o geoetrc stffess atr e et to detca eressos c are obtaed dfferet a are gve b Maeca ad Mo [ ] Eq () abe e frst oe s based o te varatoa rce e te atter s derved b eog vector dffereta ad tegra cacuus [] e ast cou abe Eq (4) reresets te foruato based o varatoa rce [] Frst te eerg of voved forces s estated ad te t s vared er dsaceet ad ode atude e sae eressos are obtaed drect b secag te geera forua Eq () for rgd bod oto [4] oever te source equato (4) [] bot rgd bod ad eastc odes are equa vauated s qute ortat fact s coeted ater o I [4] soe oter trasforatos of geera forua Eq () reated to Eqs () () ad (4) are aso derved ad dscussed at s ore terestg fro teoretca ta ractca ot of ve e ut ora vector to te etted surface [8 9] s drected to te fud I order to ae coarso of dfferet restorg stffess foruatos easer sg of Eqs () ad () abe s caged Based o eerece ad coarso of dfferet o foruatos so abe a ssteatc a te foog cocusos ca be dra: tffess s defed as a reato betee force ad dsaceet e geeraed force s ared t de ad terefore te restorg stffess defto te sae de ust be added to oda dsaceet ad ot to deforato e dervatves Eq (4) satsfes ts basc codto c esures te covergece of trasfer fucto of bedg oet to ero as ecouter frequec aroaces ero vaue [] Eq () as te sae tes as Eq (4) but t ooste ode dces ad o te corresodg quattes a tree ters tat accordg to Pot s ot correct ad cosequet eads to a error trasfer fuctos 4 ar stuato s t Eq () e ode varato ter C Eq (b) as e as C Eq (d) ave ooste ode dces ad ta tose Eq (4) tat aso resuts t soe dscreaces [] s s a cato of g of dsaceet varato ad ts gradet 5 Eq () cudes geoetrc stffess to te coete restorg stffess It s so [9] tat cotas gravt stffess C Eq (4d) so tat t does ot ect occur te foruato O te oter sde a addtoa ter O as a cotrbuto fro te boudar stress dstrbuto aears e sae quattes te coo ters of Eqs () ad (4) e C ad C are ared t dces ad resectve e obectve of te aer s to derve coete restorg stffess for seder are structures a trasaret a corsg geoetrc stffess Eq (e) ad cosstet restorg stffess for ss Eq (4) ad coare t t te o oe Eq () I addto te teto s to derve te coete restorg stffess for a otoo aatca order to ustrate te costtutg rocess ad ae sca eag of eac ter recogabe Moda restorg stffess drostatc stffess e restorg stffess cossts of drostatc ad gravt arts I order to secf te forer t s ecessar to detere te or of ressure gog fro te ta etted surface osto (r) to te stataeous osto ( r ) ere r s te osto vector r = r+ ad s te dsaceet vector e ressure or s dfferece betee to otetas W P r P r = c are scaar fuctos of te vector varabes ar to te aor seres easo vct of a gve ot oe ca rte b egectg sa ters of ger order (5) abe Actua foruatos of oda restorg stffess abca Atuae foruace odae ovrate rutost 66

3 ere D s drectoa dervatve te a of dsaceet vector us Eq (5) s reduced to ad sce ere Z Z s te vertca coordate of te etted surface ad s ts ut ora vector te coordate sste used droeastc aass Fgure eds = P ( r )= P( r)+ D P r ( ) W D P r = = P r ρg Zd = + + W ρ g D Z ZD Z D d ere D Z Z D D =( ) = ( )=( ) ( )=( ) (6) (7) (8) (9) () ac- ad s te ato oerator Deterato of D cordg to defto Eqs () s ot coveet sce t requres cacuato of ora vector dervatves erefore t s ore ratoa to use te verse verso obtaed fro te cotuu ecacs [5] c te dsaceet dervatves are voved Aed A us oe fds te foog dett equato D ( )= ( ) () ere te secod ter Eq () cotas te ode drectoa dervatve Eqs () us substtutg Eq () to Eq (9) eds W = g + Z () Accordg to defto te force s reato betee creeta or ad dsaceet so t s detered fro te varatoa equato assug costat force t a sa dsaceet δw = ρg + Z δd () Due to reasos of uerca reducto te oda suerosto etod s used ad te varato s trastted to odes e geeraed forces W ad ode atudes I tat a Eq () s decoosed to oda equatos ere (4) (5) (6) are te oda stffess coeffcets due to ressure ad ora vector ad ode cotrbutos resectve ravt stffess ar to te ressure art Eqs (8) (9) ad () te or of gravt force reads (7) ere ρ s ad are structure dest ad voue resectve I order to obta cosstet varatoa equato t s ecessar to strct foo te defto of geeraed force ad to var dsaceet vector Eq (7) ad ot ts gradet us (8) Acato of oda suerosto etod eads to te oda varatoa equato ere δw = δw = ξ δ = δξ = = = are te gravt stffess coeffcets = δw = C + C ξ δξ C = ρg d C = ρg Z d ( ) W = D g g s s ρ d = ρ ( ) d ~ d s δw = g ρ δ δw = = C ξ δξ C = g ρ d s (9) () Fgure Coordate sste droeastc aass a Koordat sustav u droeastčo aa 67

4 Restorg stffess Fa te coete restorg stffess coeffcets are obtaed b sug u ter costtutve arts ere te eaded for () () () (4) e above eressos ca be aso rtte de otato as stated abe Eq (4) Moda structura stffess overg foruae tructura stffess cossts of covetoa stffess ad geoetrc stffess Bot are detered fro te basc foruae of te teor of eastct ad cotuu ecacs Deste te fact tat structura stffess s e-o te structura aass ere t s derved te sae aer as restorg stffess due to reasos of ter reatos arato of stra eerg rtte te de otato reads δw = σ δε d (5) ere σ ad ε are te stress ad stra tesors resectve e forer s a fucto of te atter (6) ere λ ad μ are te Laé costats ad δ s te Kroecer sbo [5] tra tesor ca be eressed t dsaceets ere ε ad ε * are ear ad o-ear ters Covetoa stffess C = C + C + C C g = ρ + + d C = ρg Z + + (7) (8) (9) tra eerg Eq (5) ca aso be rtte te atr otato e ear tesor Eq (8) eads to te covetoa stffess e o-ear oe Eq (9) resuts t o-ear geoetrc stffess used arge dsaceet ad structura stabt aases [6] us ( ) ( ) + + d C g = s + + ρ d σ = λδ ε + με ε = ε + ε * ε = + * ε = ere σ ad ε are stress ad stra vectors t s ters () () D ad Λ are eastct atr ad atr dffereta oerator resectve ubsttutg Eq () to Eq () eds () ere ad ξ are oda atr ad ode atude vector resectve B substtutg Eq () to Eq () oe fds ere (4) (5) are covetoa stffess atr ad stra-dsaceet trasforato atr resectve [7] tructura robes are ordar soved b te fte eeet etod I tat case oda atr reresets atr of sae (teroato) fuctos e above foruato s gve for sods but t ca be eas adoted for t-aed structura eeets as costtutve arts of are structures eoetrc stffess For ear droeastc aass of are structures eared geoetrc stffess s suffcet erefore costat stress s tae to accout us after sertg Eq(9) for oear art of stra tesor to (5) te foog eresso s obtaed: (6) e oda suerosto ters Eq (4) ead to te oda equato ere δw = σ δε d ó = Då å = Λ = δw = ( Λ ) D ( Λ δ ) d () Furterore te oda suerosto etod Eqs (4) s used = { ξ }= ξ δw = ξ δξ = B DBd B = Λ δw = σ δ + δ d δw (7) (8) s geoetrc stffess atr Eresso (8) s derved for sods oever t ca be aso used for t-aed are structures us Eq (8) aed to te ate fte eeet te oca coordate sste taes te e-o for [8 9 ] = ξ δξ = σ d = 68

5 = σ σ (9) ere are defecto sae-fuctos e σ σ ad σ = σ are te ebrae stress cooets 4 fed geoetrc ad restorg stffess σ d σ I structura aass of are structures covetoa stffess K geoetrc stffess K ad restorg stffess C are used K s basc stffess of a structure e te acato of K ad C deeds o aass cocered as e as o te te of structure K s used te structura stabt aass ad te utate stregt aass of s structure Acato of C s ecessar te s droeastc aass For seder structures e foatg arorts LP etc bot K ad C ave to be used I tat case uo of K ad C as to be detered sce tese atrces ave soe ters of equvaet sese due to te sae etera oad e structure egt ad drostatc ressure ece oe ca rte Fgure Boudar codto for stresses eressed t drostatc ressure σ = a Rub uvet a areae raže drostats tao σ = Z = C = + C (4) ere Z = C (4) s te tersecto of cosdered to sets reated to vertca drecto If C s used aoe te K ters reated to aa ad trasverse drectos are otted e ecessar stffess Z ca be detered b trasforg Eq (e) abe va tegrato b arts I te stadard rocedure uv d = uv vu d dfferet cobatos of u ad v are ossbe oever o te coue u = σ ad dv = d ead to te trasforatos coatbe to Eq (4) c dsaceet of te ode occurs accordace t te cocuso ecto us accordg to [9] oe fds ere c = + + = σ d (4) (4) (44) (45) At te etted surface ad t te structure voue te foog boudar ad equbru codtos ave to be satsfed [4] Fgures ad resectve = σ d c d = σ Fgure Equbru codto of stresses ad gravt oad df = dq a vet ravoteže areaa gravtacsog oterećea df = dq σ = ρgz σ = ρ g (46a b) e σ = σ = ubsttutg Eqs (46a b) to Eqs (4) ad (44) resectve eds Z = ρ g Z d Z g = ρ d (47) (48) 69

6 drostatc ressure ad structure egt are equbrated etera oad ad te are cuded restorg stffess C ers Z ad Z are equvaet arts of geoetrc stffess detered va stresses as tera structura reacto to te osed oad ece te tersecto atr Eq (4) s (49) Furterore te ufed stffess Eq (4) b eog te costtutve arts of C Eq (4) ad Eq (49) for Z reads C Z C C C Z Z = + = (5) ers C Eq (4d) ad Z Eq (48) (5) ave te sae factors ad are codesed to oe eresso abe fed geoetrc ad restorg stffess abca Obedea geoetrsa ovrata rutost rasfer of Eq (4a) Eq (4b) Eq (48) Eq (5) Eq (e) (5) e ufed stffess Eq (5) s so te eaded for abe ubered as Eq (5) e artcuar ters are trasferred fro te o ad derved foruas as t s dcated te frst cou of te tabe Coarg Eq (5) t Eq () abe t s obvous tat te forer as oe ter ore ta Z te atter e C ad ters Z ad are dfferet I te case of rgd bod odes te rotato ages are utua deedet e = = ad te serca stra Z Z Z = + Eq (5d) s reduced to te Z = so tat ter C eastc odes Eq (5c) ca be rtte te foog for Z Z = ( + ) ere b eog Eq (c) for eds Z C = g + d otato C (5) (54) If te structure ass s codesed te etted se of te tcess roortoa to Z ad e t = ρ Z te voue tegra (5) ca be trasfored to te surface tegra ρ s Z C = ρg Z + d (55) ρ s Eq (5) e (5) Cotrbuto fro g (5a) Pressure C d C g Z d Z g Z d Z g s d d ( ) (5b) (5c) (5d) (5e) Z + = ρ g Z + d ora vector ad ode Boudar stress (eastc bod) ravt oad ad stress eoetrc stffess It s obvous tat te doat ter for = (54) s caceed t (55) ece oe ca eect tat te geera case su of ters (5) ad (54) reresets a sa dfferece of to arge quattes ad terefore ca be egected te ufed restorg ad geoetrc stffess Eq (5) I tat a te sfed ufed stffess s reduced to te so caed coete restorg stffess Eq () e ufed geoetrc ad restorg stffess Eq (5) s derved a trasaret a b te set teor aroac O te cotrar Eq () s obtaed b eog a advaced ad qute dfferet rocedure based o a cosstet earato of te etera drostatc ressure ad te tera structura stresses [9] Actua dervato of Eq () ad Eq (5) started fro te ooste sdes e reatos used trasforato of Eq () [9 4] are eoed dervato of Eq (5) ce te sfed ufed stffess s equa to te coete restorg stffess te crce s cosed Eq () Acato of Eq () s referabe due to ess uber of costtutve ters ad setr Based o te above facts trasforato ad secaato of Eq () for rgd bod oto s a reverse ad adequate ste fro te ufed stffess ot of ve [4] It dsutes eactess of Eq (4) for eastc odes oever Eq (4) s derved for s structures a drect ad trasaret a based o te or of etera forces tout ag te dstcto betee rgd bod ad eastc odes 5 uerca rocedure for droeastc aass Mateatca droeastc ode corses structura drostatc ad drodac oe [4] ere te vestgato s focused o restorg stffess so structura ode s ot treated deta e te drodac ode s ot cosdered at a It s so tat te gravt stffess s te art of te geoetrc stffess reated to vertca drecto I s droeastc aass t s suffcet to tae gravt stffess to accout e for seder structures as foatg arorts ad teso eg atfors te ufed geoetrc ad restorg stffess as to be used [5] ce te statc cofgurato s referet oe for dac resose ad vbratos te droeastc aass for seder structures soud be cossted of te foog stes: D FEM stregt aass st ater K δ = F ere oad cossts of te equbrated egt ad buoac Cacuato of geoetrc stffess K based o te o statc stress dstrbuto fro te revous ste Correcto of oda coordates D FEM ode due to statca dsaceets (f sgfcat) = + δ ad cosequet defto of e etted surface 4 Dr atura vbratos aass of te statca equbrated FEM ode as te referet state b oe of te egevaue equatos: = ( + ) = K Ω M δ K K Ω M δ (56a b) 5 Cacuato of oda atrces ad ; or = + ad C resectve 6 Cacuato of drodac coeffcets B(ω) ad A(ω) ad ave oad F 7 outo of te equato for fud-structure teracto: δ b B ω δ A ω δ F C + ( + ) + ( + ) = (57) 7

7 ere accordg to (56a b) * = + C resectve C = + ad C 6 Potoo ufed geoetrc ad restorg stffess 6 eera Acato of revous reseted teor s ustrated te case of a rsatc otoo e robe s soved aatca accordg to Eq () order to get better sgt to sca bacgroud troug costtuto of goverg equatos ad to rove cosstec of te teor ere F = ρ gb + ( ) Potoo eads eera data: = = for frot ead = = for aft ead Z = Z = d = Bd botto C = ρgb ( ) dz = F C = ρ gb ( ) dz = B d d d d ( ) d = ρ gb Z + dz = = B d d d d d + d d (6) (64) (65) (66) Fgure 4 Potoo a artcuars a 4 obae ačae otoa e a otoo artcuars are so Fgure 4 e oda dsaceet vector due to vertca defecto reads d = ( ) + (58) d Z ere s te vertca osto of eutra e (cetrod) of otoo cross-secto us = = = = d = = Z (59) d I te cosdered case = = d ad d = ad Z cosequet C = Eq (5) Aso at te botto ad Z eads = ad = tat gves + = Eq (54) Based o te above cocuso Eq () s used for deterg te ufed restorg ad geoetrc stffess 6 Restorg stffess Eqs (a) (b) (c) Potoo botto eera data: d = Bd Z = = = ere 6 eoetrc stffess Eq (e) eera data: d = dad = σ ( + )+ σ A = σ A F gb = ρ + gb = ρ B ( Z ) 4 = ρ gb + + d d d d + da d = d d d d + σ (67) (68) (69) da d e frst ter Eq (69) t curvatures s eggbe coared to te oters e secod ter s reated to te aa drecto ere σ s equa to te ead drostatc ressure us oe fds te coresso aa force Fgure 5 Fgure 5 tatc boudar oad a 5 tatčo rubo oterećee C = ρ gb Z d = F C = ρgb d d d d d d d F d (6) (6) = ρ gb Z d = F d d d d d (6) 7

8 (7) tat gves stffess of te aa drecto X d d = (7) d d d I order to detere stffess of te vertca drecto t s ecessar to a tegrato b arts for te trd ter Eq (69) ( u = σ dv = d ) Actua ts s te reverse rocedure of tat used foruato of ufed restorg stffess Eq (4): (7) e frst ter Eq (7) s ut order b tag σ ( )= ad σ ρg = to accout e secod ter t σ = gρ occurs te for c reresets statc oet of te ass dstrbuto er otoo cross-secto t resect to te cetrod ce ρ d A= = ρb for Eq (7) oe ca rte A ere ad s te cetre of gravt coordate Fgure 64 fed stffess (7) (74) After secfg te costtutve arts t s ossbe to assebe te ufed stffess I tat rocess soe ters are caceed: of te botto t te frst ter of Z ; C t te frst ter of bot for te eads us te ufed stffess s reduced to ere ϕ = d ad d A σ da= ρgb ZdZ = = (75) (76) If reato F = F s used Eq (75) ca be reseted te recogabe setrc for Z = B ( σ ) d σ = ( ) g ρ Z A Z = F + F ( g) A F = ρgb d d A d d d d d d d d d g ( ) ( ) = ρgbi + F F I F ϕ da d + ( ) F( ϕ ) g d d I = I d = d d d ( = ρgbi ) ( + F F I ) F ϕ ϕ g + ( ) (77) Cosstec of te above foruae ca be ceced te case of rgd bod odes e eave atura odes = gves = ρ glb e te stffess vaue equa to tat te s drostatcs [] e tc atura ode s = ϕ = ( L so tat I ) ( = ad I ) = L at gves glb g BL = ρ + + ρ (78) ce = LB s te dsaced voue ad BL I = s WLY te ogtuda oet of erta of te aterae area te ufed stffess taes te for (79) It s detca to te drostatc eresso because te coordate of te cetre of buoac taes vaue = B [] 7 Potoo vbratos e oda suerosto etod utg dr odes s coo used to sove droeastc robes I te cosdered case of oogeous otoo te dffereta equato of vertca dr atura vbratos reads 4 d (8) 4 d Ω EI = t te foog e-o odes for te case of free bea [] etrc odes At-setrc odes g I = ρ + WLY cβ cosβ = + = 4 cβ cosβ β = β = 65 β = sβ sβ = + = 5 sβ sβ β = β = 95 β = β = π 4 atura frequeces ar Ω β EI = = (8) (8) (8) B assug forced resose te seres of te atura odes = ξ = (84) 7

9 ad eog te varatoa etod te foog sste of te agebrac equatos s costtuted ( ) ( ) EI I + F F I ρ gb ω ( ) I + F ϕ F ϕ ξ = g t = qd = + ( ) (85) ere ω s te ave frequec ad ( ) I = (86) d d d d d ( ) ( ) It s ecessar to ot out tat I = I = f due to ortogoat of dr odes Eq (8) ca be reseted te atr otato ( ) = (87) ere = E/I () ad = I () are ordar dagoa bedg stffess ad ass atrces resectve ad s te ufed geoetrc ad restorg stffess atr Eq () 8 Bea fte eeet + ω ξ F I te fte eeet etod te sae fuctos are used te geera foruato of te goverg atrces stead of atura odes For te ordar to ode bea fte eeet for feura vbratos te erta ooas of te trd order are used = a { ξ }; = 4 ; = ere a are te coeffcets ξ = L (88) s te o-desoa coordate ad L s te eeet egt bos ad {} deote ro ad cou vectors resectve Matr of te sae fucto coeffcets reads a = L L L (89) L L us for te tegra atrces Eqs (76) ad (86) oe fds 56 L 54 L I ( L 4 ) = L L L 4 56 L (9) 4L I ( ) I ( ) 6 L 6 L 4 = L L L L 6 L 4L 6 L 6 L = L L L L 6 L L (9) (9) Boudar vaues of te sae fuctos tat dffer fro ero are te foog: ϕ L ϕ L = = = = 4 terefore te boudar vaue atr eds L ϕ (9) It s obvous tat te tegra atrces are setrc as e as te coue of te boudar atrces After te tegra ad boudar atrces are detered te fte eeet stffess ad ass roertes ca be coeted accordg to Eq (85) ere a araeters deed o te fte eeet cross-secto Cosstec of te derved fte eeet roertes ca be ceced case of te rgd bod odes of sef equbrated eeet e oda dsaceet vector for eave reads tat gves (94) (95) us accordg to Eq (85) oe fds te foog vaues for te stffess ad ass coeffcets: e tc oda vector reads tat eads to = L = L ϕ ϕ I I I = = L ( ) ( ) ( ) ( ϕ ) L = = = = = ρgbl = L I I I ( ) = L = = L ( ) ( ) ( ϕ ) L = = L (96) (97) (98) Fa te otoo stffess ad ass roertes Eq (85) tae te foog for: g I L = = ρ + WL t = (99) It s obvous tat te eave ad tc restorg stffess are equa to tose fro te s drostatcs [] t 7

10 9 Iustratve eae I order to aae restorg stffess et us cosder a qute febe segeted barge cosstg of equa rsatc otoos [ ] e otoos are coected b eas of to stee bars soeat above te dec eve as so Fgure 6 o cofguratos t bar rofes of 5 4 ad 5 6 are used e a barge artcuars are te foog: oever ts aa art I s egected ere sce aa force s ot used s ogtuda aass as a art of geoetrc stffess due to ver sa fuece o resose O te oter ad ter ( F + F g) I ( ) s te art of ordar restorg stffess C Eq (4) erefore te otoo vbratos are cacuated t C c s obtaed fro Eq (75) b gorg ( ) ( ) C = ρgbi + F F I F ϕ F ϕ g ( ) () abe Moda restorg stffess aatca C [ ] abca Modaa ovrata rutost aatč C [ ] g [ ] Fgure 6 Barge cross-secto a 6 Poreč rese barže Youg s oduus of stee bars E = / Moet of erta of cross-secto o bars 54 I = 5-4 o bars 56 I = 8-4 Potoo egt = 9 Barge egt (otoos + cearaces) L = 445 Barge breadt B = 6 Barge egt = 5 Barge draugt = ertca coordate of gravt cetre fro atere = 8 ertca coordate of eutra e fro atere = 5 Barge ass M = 76 g Dstrbuted ass = M = 7 L g Dstrbuted ass oet of erta J = 75 g I droeastc aass te barge s treated as a oou e basc quattes occurrg te stffess atrces tae te foog vaues: ρgb = / F = 97 F = 48 F 97 g = 48 e tegra ad boudar atrces are secfed Aed B e geoetrc stffess atr Eqs (7) ad (7) reads ( F F g ) = + + I ( ) () abe 4 Moda restorg stffess uerca C [ ] abca 4 Modaa ovrata rutost uerč C [ ] g [ ] e restorg stffess atr detered aatca b Eq () ad aatca odes Eqs (8) ad (8) s so abe for te to rgd odes ad to setrc ad at-setrc eastc odes It s aso cacuated uerca accordg to te geera foruae Eq (4) ad uerca atura odes abe 4 Dagoa ters abes ad 4 c are redoat agree qute e eseca te rgd bod oes a dffereces betee te ters of eastc odes ad dscreaces of off-dagoa ters are a te resut of sa dsagreeet betee uerca ad aatca detered atura odes e frst to eastc vertca atura odes of te etted surface obtaed b sreadg te bea dsaceets ad d/d Eq (58) are so Fgures 7 ad 8 For ustrato uroses te trasfer fuctos of vertca bedg oet ead sea for te bot barge cofguratos are so Fgure 9 e are cacuated uerca b rogra DYAA [4] e te drodac coeffcets (added ass ad dag) are detered b rogra YDROAR [5] e eastc atura odes are tae to accout as e as te ass oet of erta J va corresodg etc eerg E = ω J I () added to Eq (85) c cages te sae of ger atura odes ad eve of resose ad gves ore reastc resuts e basc codto for te cosstet restorg stffess s covergece of trasfer fucto to ero as ave frequec aroaces ero vaue s codto s satsfed for te eastc as e as for te rgd bod resose Large dffereces betee 74

11 Fgure 7 e frst dr atura ode of vertca vbratos a 7 Prv su rrod ob verta vbraca Fgure 8 e secod dr atura ode of vertca vbratos a 8 Drug su rrod ob verta vbraca Fgure 9 rasfer fucto of vertca bedg oet at ds a 9 Preosa fuca vertaog oeta savaa a sred barže tese resoses at te ver begg of te ave frequec ω are te resut of ver sa bedg stffess edec of eastc resose to foo te rgd bod oe for ger stffess s obvous e resoat ea for te softer ad stffer barge cofgurato occurs at ω = 6 ad 8 rad/s resectve A seres of ode tests o te segeted barge as bee coducted BO Frst (ouo Frace) order to vadate uerca rocedure Fgure [] uerca ad easured trasfer fuctos of te frst ode feura deforato for te bot barge cofguratos are so Fgures ad Feura deforato s defed as te dfferece betee tc ages of te frst ad ast otoo ese tc ages are qute sestve araeters to descrbe te barge feura deforato ad tat coud be oe of te reasos for dsagreeet betee te cacuated ad easured resoses c afests t dfferet ea vaues ad corresodg resoat frequeces Aoter 75

12 Fgure egeted barge a egeta barža Fgure rasfer fucto of atude of te st eastc ode EI = 7 restorg stffess C a Preosa fuca atude rvog eastčog oba EI = 7 ovrata rutost C Fgure rasfer fucto of atude of te st eastc ode EI = 6 restorg stffess C a Preosa fuca atude rvog eastčog oba EI = 6 ovrata rutost C reaso of dscreaces coud be te aout of te assued dag [] If Fgures ad are coared t s obvous tat te resoat ea te forer s st to to eas te atter due to creased feura stffess Correato of te oter trasfer fuctos s uc better ta tose so Fgures ad [ 4] abe 5 fed stffess atr aatca [ ] abca 5 Obedea atrca rutost aatč [ ] [ ] If te ufed geoetrc ad restorg stffess s tae to accout accordg to Eq (77) ts setrc for s otceabe abe 5 e forced resose s detered for te frst barge cofgurato e EI = 7 ad ecudg ass rotato Eq () Defecto s descrbed t 8 atura odes e trasfer fucto of vertca bedg oet s so Fgure ad coared t tat detered t ordar restorg stffess C abe 4 oe dffereces are otceabe at resoat frequec ω = 6 rad/s It s ecessar to ot out tat te trasfer fucto of bedg oet detered t te ufed stffess does ot coverge to ero as te ave frequec aroaces ero vaue Fgure 4 at eggbe resduu s ossb resut of aa force actg far fro te eutra as tat soud be furter vestgated Fg 6 76

13 Fgure rasfer fucto of vertca bedg oet a Preosa fuca vertaog oeta savaa Fgure 4 Zooed trasfer fucto of vertca bedg oet a 4 Deta reose fuce vertaog oeta savaa Cocuso e aer teds to sed ore gt o st oe ad caegg robe of restorg stffess costtuto droeastc aass of are structures sua restorg stffess as to be tae to accout togeter t geoetrc stffess Due to soe equvaet arts ts aer te are assebed a sca trasaret aer Its sfed foruato s detca to oe so [9] c s derved a drect a uder soe assutos Cosstec of te ufed geoetrc ad restorg stffess s ceced b aatca souto for rsatc otoo ad satsfg of te rgd bod equbru e-o te s drostatcs Aatca foruae offer te ossbt to foo sca bacgroud of eac costtutve art ad ter utua terferece Derved teor acato s ustrated case of a segeted barge for c te eereta data are avaabe It s so tat basc codto for eac restorg stffess foruato s covergece of eastc resose to ero as te ave frequec aroaces to ero vaue Bot te aatca souto of te otoo ufed geoetrc ad restorg stffess ad uerca eae of te segeted barge ca be used as becar droeastc aass of are structures as for stace [6] eoetrc stffess ca be decoosed to tree arts oe osed b gravt vertca drecto aoter b drostatc ressure ad te trd oe deedet o curvature e forer s cuded te ordar restorg stffess so tat case of sa aa ad trasverse cotrbuto t s ossbe to oerate t ordar restorg stffess tout geoetrc stffess uc stuato occurs ss t o st ater stress dstrbuto If s droeastc aass s aded t geoetrc stffess cuded te te addtoa aa stress caused b resstace ad trust ca aso be tae to accout e geoetrc stffess as to be accouted for droeastc aass of seder structures e foatg arorts ad teso eg atfors I te forer case geoetrc stffess reduces te tota structure stffess due to coresso e te atter case t s creased e te strgs of usca struets a equbru dsturbace at ero ave frequec c occurs f geoetrc stffess s cuded soud be furter vestgated A advatage of ag ufed geoetrc ad restorg stffess a case s a ossbt to geerate geoetrc stffess b stadard rocedure te ost coerca FEM acages for structura aass us o tegrato over etted surface reas to be addtoa doe t te oda suerosto etod Oterse te cacuato of gravt stffess soud be erfored b tegrato er structure voue ad tat coud reset a coe robe If geoetrc stffess s cuded droeastc aass te dr atura vbratos ca be cacuated t te coete structura stffess I tat case oe ste of oda suerosto cacuato s avoded Oe of te furter tass droeastc aass of are structures s ust surface tegrato tsef ad fdg te a o to overcoe te ga betee te structura FEM es ad ae et of etted surface Aso te acato of te reseted teor for a ratoa ad reabe droeastc aass of a arge cotaer s s a addtoa caegg tas e advaced restorg stffess foruato t geoetrc stffess cuded Eq () s ot et geera acceted aog drodacs eerts sce for te t sees strage tat structure artcates dro-ecaca resstace e structure s eected to be voved o te covetoa stffess B foruatg te ufed geoetrc ad restorg stffess ad ts sfcato detca to Eq () a ore trasaret a ts sef-cotaed aer t detaed aatca cosderato case of a rsatc otoo ad segeted barge ts avodace coud be overcoe Acoedget e vestgato as bee carred out t te atoa roect of te Croata Mstr of cece Educato ad orts etted: Load ad Resose of tructures ad te E FP7 Proect LC (oos for tra Large Cotaer s) e autors eress ter grattude to te oaševć PD FAME- A verst of Zagreb for uerca cacuato of te barge droeastc resose Refereces [] RIA 8 Desg ad Oeratos of Cotaer s Lodo [] PAYER : ecoogca ad Ecooc Icatos of Mega-Cotaer Carrers AME rasactos o 9-77

14 [] EJAOIĆ I MALEICA Š OMAŠEIĆ : Ivestgato of droeastct Ocea Eg [4] BIOP R E D PRICE W : droeastct of s Cabrdge verst Press 979 [5] EJAOIĆ I OMIĆ M OMAŠEIĆ : A Ect Foruato for Restorg tffess ad Its Perforace droeastct Ocea Eg [6] PRICE W W Y: droeastct of Mare tructures eoretca ad Aed Mecacs Esever cece Pubsers B [7] EWMA J : Wave Effects o Deforabe Bodes A Ocea Res [8] RI R: drostatc tffess of Febe Foatg tructures Proceedgs of te Iteratoa Worso o er Large Foatg tructures aaa Jaa [9] A L L RI R: e drostatc tffess of Febe Foatg tructures for Lear droeastct Mar truct 9-6 [] MALEICA Š: oe Asects of drostatc Cacuatos Lear eaeeg Proceedgs of te 4 t A Coferece Paero Ita [] MALEICA Š MOLI B REMY F EJAOIĆ I: droeastc Resose of a Barge to Iusve ad o- IusveWave Loads Proceedgs of droeastct Mare ecoog 7-5 [] KREYZI E: 99 Advaced Egeerg Mateatcs We [] EJAOIĆ I OMAŠEIĆ LADIMIR OMIĆ M MALEICA Š: droeastc Aass t ostcated Bea Mode ad Cosstet Restorg tffess droeastct Mare ecoog 9 verst of outato 69-8 [4] RI R: Coarso of Foruatos for te drostatc tffess of Febe tructures Joura of Offsore Mecacs ad Arctc Egeerg 9 o /45 [5] ALFIREIĆ I: Itroducto to Cotuu Mecacs ode Maretg Zagreb ( Croata) [6] ZIEKIEWICZ OC: e Fte Eeet Metod Egeerg cece Mcra- 97 Lodo [7] ORIĆ J: e Fte Eeet Metod ode aretg eča ga 4 Zagreb ( Coata) [8] OLAD J BELL K: Fte Eeet Metods tress Aass ar 97 [9] EJAOIĆ I: Fte Eeet Metod tructures verst of Zagreb 998 Zagreb ( Croata) [] ZILARD R: eores ad Acato of Pate Aass We 4 [] AME Prces of ava Arctecture Jerse Ct 988 [] EJAOIĆ I ĆAIPOIĆ I OMAŠEIĆ : Coued orota ad orsoa bratos of a Febe Barge Egeerg tructures [] REMY F MOLI B LEDOX A: Eereta ad uerca tud of te Wave Resose of a Febe Barge droeastct Mare ecoog 6 Wu Ca [4] OMAŠEIĆ : droeastc Mode of Dac Resose of Cotaer s Waves PD ess FAME- A 7 verst of Zagreb Zagreb ( Croata) [5] Bureau ertas YDROAR ser s Maua 6 Pars [6] RI R IIMI KM A LL: o becar robes for tree-desoa ear droeastct Joura of Offsore Mecacs ad Arctc Egeerg o 9 7 Aed A Foruato of ora vector drectoa dervatve Wt cotuu ecacs te foog reato betee te actua ad te referet dffereta surface ests [5]: d = F( F ) d (A) ere F s te deforato gradet atr c te case of te dsaceet vector taes te for + F = + + (A) us for te drectoa dervatve of ora vector oe ca rte D d d ( )= = F( F ) I d (A) ere I s te ut atr ad s te ut ora vector e verso of atr F ad ts trasose after egectg te sa ters of ger order ead to te foog forua D ( )= (A4) Eq (A4) ca aso be reseted te vector otato f te cotets t te bo bracets are eteded ad subtracted t te ters ad resectve s eads to D = ( ) + + (A5) 78

15 Aed B Itegra atrces I te droeastc aass of barge as a bea oe s faced t tca tegra atrces (76) ad (86) c ca be soved aatca for te aatca atura odes Eqs (8) ad (8) e frst atr s dagoa due to ortogoat of te atura odes ( ) I = d = + ( tgβ + tgβ 4 cos β β )+ = 4 cos β (B) ere te secod ter reresets te frequec equato equa to ero [] at eads to te vaue ( ) I = (B) c s aso vad for at-setrc eastc odes I a sar a te oter to tegra atrces ca be detered too uerca vaues of te tegra atrces ad te boudar vaue atr are sted abes B-B4 resectve Matr I () s aso dagoa due to ortogoat of te atura odes abe B Itegra atr I () abca B Matrca tegraa I () g abe B Itegra atr I () abca B Matrca tegraa I () g abe B Itegra atr I () abca B Matrca tegraa I () g abe B4 Boudar vaue atr ϕ abca B4 Matrca rub vredost ϕ ( ) B og te basc atrces te costtutve barge stffess ad ass atrces ca be detered It s obvous tat asetr of te restorg stffess C s caused b asetr of te boudar atr abe B4 Aso te ufed geoetrc ad restorg stffess atr ca be cacuated accordg to Eq (77) 79