Displacement and Stress Monitoring of a Chemical Tanker Based on Inverse Finite Element Method

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1 Dsplacmnt and Strss Montorng of a Chmcal Tanr Basd on Invrs Fnt Elmnt Mthod Adnan Kfal*, Eran Otrus** * Dpartmnt of Naval Archtctur, Ocan and Marn Engnrng Unvrsty of Strathclyd 100 Montros Strt Glasgow G4 0LZ, Untd Kngdom Phon: +44 (0) Fax: +44 (0) mal: adnan.fal@strath.ac.u ** Dpartmnt of Naval Archtctur, Ocan and Marn Engnrng Unvrsty of Strathclyd 100 Montros Strt Glasgow G4 0LZ, Untd Kngdom Phon: +44 (0) Fax: +44 (0) mal: ran.otrus@strath.ac.u Abstract: Ral-tm rconstructon of full fld structural dsplacmnts, strans, and strsss by usng surfac stran masurmnts obtand from on-board stran snsors s commonly rfrrd to as shap- and strss-snsng. For ths purpos, a computatonally accurat, robust, and rapd algorthm namd as nvrs Fnt Elmnt Mthod (FEM) was rcntly dvlopd. Th man goal of ths study s to prform dsplacmnt and strss montorng of a typcal chmcal tanr md-shp basd on FEM mthodology. Th numrcal mplmntaton of th FEM algorthm s don by consdrng four-nod nvrs quadrlatral shll (QS4) lmnt. In ordr to dmonstrat th capablty of th currnt approach, a long barg that has a cross-scton dntcal to a typcal chmcal tanr s modlld wth QS4 lmnts. Thn, hydrodynamc loads of th barg for a crtan frquncy of wavs ar calculatd by usng n-hous hydrodynamc softwar. Thn, ths forcs ar appld to a FEM modl of barg and structural rspons s computd by usng n-hous fnt lmnt softwar. Th rsults obtand from FEM analyss s utlzd as a sourc to smulat n-stu stran data usd n FEM analyss as nput. Fnally, FEM and FEM dsplacmnts ar compard and th ffcts of locatons and numbr of snsors on FEM soluton accuracy ar dscussd. 1

2 Kywords: Shap Snsng; Strss Snsng; Invrs Fnt Elmnt Mthod; Chmcal Tanr. 1. Introducton A catastrophc falur and snng of shps rsult n crucal fnancal losss, human lf loss, and polluton of marn nvronmnt. Harsh marn nvronmnt togthr wth strong wathr condtons ar vry lly to caus structural damags on shp structurs bcaus shps ar xposd to long-trm cyclc loadngs comng from contnuous sa wavs as wll as short-trm xtrm loadngs such as rogu wavs, ranstorms, strong gals, saquas tc. Morovr, contact btwn sa watr and matral of shps (hgh strngth stl n most cass) causs fast corroson, roson and hnc thcnss rducton. Ths phnomnon trggrs ntaton of damag growth and ncrass th sz of an xstng damag. Hnc, a catastrophc falur of shps mght b unavodabl, unlss an approprat on-st assssmnt of structural ntgrty s accomplshd. Structural Halth Montorng (SHM) s a multdscplnary tchnology ara that addrsss ths problms through provdng rlabl ral-tm nformaton from a structur about ts global or local structural condton by ntgraton of snsng systms on th structur. In othr words, prvnton of aformntond srous damag cass and dtald structural managmnt ncludng nspcton and mantnanc plans can b achvd by prformng an accurat SHM. Dynamcally tracng th thr dmnsonal dsplacmnt and strss flds of a structur by utlzng a ntwor of n-stu stran snsors and masurd strans, gnrally nown as shap- and strss-snsng, s a fundamntal tchnology for SHM procdur. Snc SHM systms potntally allow marn ndustry to ncras both human and nvronmntal safty n conjuncton wth rducton of mantnanc costs, dffrnt montorng schms hav bn usd for numrous yars on dffrnt typs of vssls. A typcal hull structur montorng systm confguraton for bul carrr or tanr s proposd by LR (2004). Th rqurmnts that hull montorng systms must mt ar rgulatd by class socts such as ABS (1995) and DNV (2011). Ths ruls spcfy th confguraton and typs of snsors to b usd, as wll as masurmnt accuracy, data storag and data analyss. Howvr, ths gudlns provd only a global outln on how data procssng should ta plac and thr has bn lttl chang n ths basc confguraton snc IMO orgnally ntroducd rqurmnts n

3 Apart from rgulatons dfnd by svral class socts and IMO rgardng hull structural montorng, SHM of shps s consdrd as an mportant ara of study by many rsarchrs. Phlps and Morrs (2013) provdd a gnral rvw of th tchncal aspcts of avalabl hull structural montorng systms wth consdraton of th dffrncs btwn nstallatons on navy shps as compard to commrcal shps. Van dr Cammn (2008) ndcatd hull structural montorng systms that may b usd for th assssmnt of motons and xtrm vnts such as slammng strss and fatgu. Torldsn (2005) statd an outln of a shp hull halth montorng nstalld on th Royal Norwgan. Thr xstng SHM systm conssts of a ntwor of fbroptc snsors to masur stran and tmpratur, X-band wav radar to stmat wav hght and drcton masurmnts, and a mcrowav altmtr mountd n th bow to masur th oncomng wav profl. Andrsson t al. (2011) proposd a hull condton montorng systm for damag montorng of shps whch ar mad of fbr rnforcd plastcs. Thr tchnqu uss acoustc vbratons that xst n th hull structur and on-board snsors data as nput. Whn a snsor s clos to a damagd part of th hull, spcfc ndcatons wll b dtctd by th damag ndcaton algorthm. Sls (2012) xprmntally montord fatgu crac ntaton and propagaton of alumnum shp structurs by usng a shp structural rlablty program ncludng a SHM tool as a part. Thr global mthod rqurs modal analyss and local mthod s dpndnt on th rlaton btwn lamb wav propagaton and snsor dagnostcs. Zhu and Frangopol (2013) usd SHM data obtand from snsors to mprov th accuracy and rdundancy of rlablty assssmnt of th shp cross-sctons. Pror load ffcts ar updatd accordng to SHM data rlatd to th wav-nducd load by usng Baysan updatng mthod. Ths study concludd that ntgraton of th SHM data can consdrably dcras th uncrtanty n a dstrbuton paramtr, and hnc updatd prformanc ndcators com closr to corrct valus. Hagman t al. (2013) dvlopd a hull fatgu montorng systm for Floatng Producton Storag and Offloadng (FPSO) vssls. Thr fatgu prdcton tchnqu was dvlopd basd on Baysan Statstcs. Also, thr systm asssss th fatgu consumpton from nvronmntal condtons and masurd strans sparatly, and compars wth dsgn fatgu consumpton n ordr to dfn conclusons on prformanc of hull structur. Majwsa t al. (2014) prsntd an xprmntal approach for SHM of salng shps by usng Fbr Bragg Gratng (FBG) snsors. Thy nstalld FBG snsors grd to th formast of a salng shp n ordr to dtrmn th stran/strss lvl of th formast durng dffrnt shp opratons. Nchols t al. 3

4 (2014) dscrbd th structurd dcson mang (SDM) procss for usng avalabl nformaton (loadng data, modl output, tc.) and producng a plan of acton for mantanng th structur. Thr xampl shows that SDM mnmzs both transt tm and probablty of falur through a usr-dfnd cost functon. Howvr, dvlopng ths approach rqurs modls that dscrb th loadng data, prdct th structural rspons to th load, and forcast accumulatd damag. Most of th aformntond SHM approachs don t ta nto account th advancd structural topologs and boundary condtons. Morovr, thy mostly rqur suffcntly accurat loadng nformaton vn though t s not asy to stmat dynamc loads of wavs and wnds du to th complxty and statstcal fatur of ocanographc phnomna. Furthrmor, som of thm ar not approprat for us n ral-tm du to th tm-consumng analyss. A novl shap- and strss-snsng algorthm namd as nvrs Fnt Elmnt Mthod (FEM) was dvlopd by Tsslr and Spanglr (2003, 2005) at NASA Langly Rsarch Cntr. FEM algorthm rconstructs th structural dformatons, strans and strsss by utlzng th stran data obtand from a ntwor of on-board stran snsors locatd at varous sts of a structur. FEM mthodology s basd upon th mnmzaton of a wghtd-last-squar functonal. In othr words, th formulaton nvolvs th ntr structural gomtry that s dscrtzd by usng sutabl nvrs fnt lmnts n whch th masurd stran data ar adaptd to th lmnt strans n a last-squar sns. Unl othr dvlopd SHM mthods, FEM mthodology posssss a gnral applcablty to complx structurs subjctd to complcatd boundary condtons n ral-tm (Tsslr and Spanglr 2005). Snc only on-board stran masurmnts ar usd as nput to th FEM formulaton, accurat structural dformd shaps of ngnrng structurs that ar undr any typ of statc and/or dynamc loadngs can b obtand wthout pror nowldg of matral proprts. Onc th structural dformd shap s rconstructd, th full fld strans can b calculatd by utlzng th dsplacmnts. Thn, th thr-dmnsonal strss stat of th structur can b valuatd from th full fld strans and th matral proprts of th structur. Fnally, thr-dmnsonal strsss can b adaptd to an quvalnt strss by mans of an approprat falur crtron whch can nabl ral-tm damag prdcton. FEM framwor has alrady shown to b prcs, powrful and suffcntly fast for ral-tm applcatons (Ghrlon t al. 2012, 2014). FEM formulaton s applcabl to montor thn and modratly thc bam, plat and shll structurs bcaus nvrs bam, fram, or plat and shll 4

5 fnt lmnts can b smply usd for dscrtzaton of th physcal doman. Crraccho t al. (2010) and Ghrlon t al. (2011, 2012, 2014) dvlopd a robust nvrs-fram fnt lmnt by adoptng nmatc assumptons of Tmoshno bam thory ncludng strtchng, bndng, transvrs shar and torson dformaton mods. Th capablty of thr nvrs-fram lmnt s numrcally and xprmntally vrfd for shap-snsng of thr-dmnsonal fram structurs undrgong statc and/or dampd harmonc xctatons. Furthrmor, Tsslr and Spanglr (2003, 2004) focusd on th nvrs problm of rconstructng th thr-dmnsonal dsplacmnts n plat and shll structurs from stran snsor masurmnts. Basd upon frst-ordr shar dformaton thory, Tsslr and Spanglr (2004) dvlopd a thr-nod nvrs shll lmnt (MIN3) by usng lowst-ordr ansoparamtrc C 0 contnuous functons,.., lnar n-plan dsplacmnts and bndng rotatons, and a constrand typ quadratc dflcton. Th prcson of MIN3 lmnt was also confrmd by usng xprmntally masurd-stran data (Quach t al. 2005, Vazquz t al. 2005). Morovr, Tsslr t al. (2012) rcntly mprovd MIN3 lmnt formulaton to rconstruct dformd shap of plat and shll structurs undrgong larg dsplacmnts. Furthrmor, Crraccho t al. (2013, 2015) nhancd th Tsslr and Spanglr (2003, 2005) FEM formulaton for shap- and strss-snsng of mult-layrd compost and sandwch structurs. Thy rformulatd th orgnal plat/shll FEM formulaton to nclud th nmatc assumptons of rcntly dvlopd Rfnd Zgzag Thory by Tsslr t al. (2009, 2010). Howvr, all proposd FEM formulatons as a numrcal or xprmntal applcaton on ngnrng structurs hav bn lmtd to th SHM of arospac vhcls (Tsslr t al. 2011, Ghrlon t al. 2013). Th man and novl am of ths wor s to apply FEM mthodology for shap- and strsssnsng of a chmcal tanr for th frst tm n th ltratur. Frst, a four-nod nvrs quadrlatral shll (QS4) lmnt formulaton s prsntd undr th assumptons of Mndln s frst-ordr plat thory. Thn, as a practcal applcaton to marn structurs, a long barg that has a cross-scton smlar to a typcal chmcal tanr s consdrd and analysd basd on th prsntd FEM algorthm. In ordr to ma th applcaton mor ralstc, a hydrodynamc modl of th barg s prformd by usng th hull form (panls) as wll as ts wttd surfac proprts and wav proprts such as hght, angl, and frquncy as nput. As an output, hydrodynamc shp loadng and rgd body moton of th barg ar obtand. Thn, fnt lmnt analyss of th modl s prformd by usng th loadng calculatd from hydrodynamc analyss. 5

6 Th stran output of fnt lmnt modl s usd to rprsnt th xprmntal stran masurmnts (.., n-stu stran data or smulatd snsor-stran). Fnally, th FEM modl s prformd for dsplacmnt and strss montorng of th chmcal tanr cross-scton by utlzng th smulatd stran data obtand from varous locatons of th structur. Fnally, th ffcts of snsor postons and numbr of snsors on FEM soluton accuracy ar dscussd. 2. Invrs fnt lmnt formulaton for shlls 2.1. Invrs quadrlatral shll lmnt A four-nod nvrs quadrlatral shll (QS4) lmnt s usd to dmonstrat th FEM formulaton. In ordr to dfn th local lmnt functons, a local coordnat systm xyz s locatd wth rfrnc to QS4 lmnt and global Cartsan coordnat systm XYZ as shown n Fgur 1a. Th QS4 lmnt has sx dgrs-of-frdom (DOF) pr nod, as shown n Fgur 1b, and thy consst of translatons u, v, and, w as wll as rotatons x, y, and z. Fgur 1 (a) QS4 lmnt showng global and local coordnat systms. (b) Nodal DOF n th local coordnat systm xyz. Th orgn of xyz systm s postond at th cntrod of lmnt s md-plan quadrlatral along unform shll thcnss z t, t. Whn assmblng th lmnts, transformaton of nodal DOF from th local to global coordnat systm can asly b don by utlzng global XYZ coordnats of nods. Th QS4 lmnt formulaton s drvd by frst wrtng mmbran and 6

7 bndng stffnss matrcs sparatly and fnally addng thm togthr. Ths procdur s a common and suffcntly accurat way to obtan th lmnt formulaton. Thr componnts of th dsplacmnt vctor of th lmnt ar dscrbd accordng to th assumptons of Mndln s frst-ordr plat thory. Th stran dsplacmnt rlatons of lnar lastcty thory ar drvd by calculatng th rlvant drvatvs of th thr componnts of th dsplacmnt vctor. For brvty, ths rlatons can b wrttn n compact vctor forms by utlzng nodal dsplacmnt vctor of QS4 lmnt u and drvatvs of shap functons m B, B, s B as xx m yy ( u ) z( u ) B u zb u (1a) xy and xz s g( u ) B u (1b) yz whr T u u 1 u2 u3 u 4 (1c) and T u u v w x y z wth ( 1,2,3,4) (1d) Th ntrpolaton functons (or shap functons) usd to dscrb mmbran, bndng curvaturs, and transvrs shar actons of QS4 lmnt ar analogous to four-nod flat shll lmnt dscrbd by Coo (1994) and MIN4 (Mndln-typ, four-nods) lmnt provdd by Tsslr and Hughs (1983). Hrn, shap functons of th dflcton varabls ar drvd from bquadratc srndpty ntrpolaton, whras th normal rotatons ar ntrpolatd by usng standard blnar functons. Usng ths ntrpolaton functons for th nmatc fld of th QS4 lmnt lads to C 0 contnuty, thus nsurng th lmnt ntror and dg ntrfac contnuts. Th mmbran strans u ( ) ar assocatd wth th strtchng of th mddl surfac. Thrfor, m B matrx stands for th drvatvs of th shap functons assocatd wth th 7

8 mmbran bhavour. Furthrmor, u ( ) and gu ( ) transvrs shar strans, rspctvly. Hnc, B and ar th bndng curvaturs and th s B matrcs ar th corrspondng drvatvs of shap functons usd to dfn bndng bhavour of th lmnt. Th xplct forms of th B m, B, and s B matrcs as wll as th ntrpolaton functons of QS4 lmnt ar gvn n Appndx A. It s mportant to mnton that zz has no rol n th ntrnal wor du to th plan strss assumpton zz Input data from n-stu stran snsors Dscrt n-stu stran masurs obtand from th snsors placd on-board structur ar fundamntal nput of th FEM mthodology. Convntonal stran rostts or mbddd fbroptc snsor ntwors ncludng Fbr Bragg Gratng (FBG) snsors and Snsng Fbr Optc Cabls ar promsng tchnology to collct larg amount of on-board stran data. Fbr-optc systms potntally offr a numbr of advantags for marn structur nstallatons as thy ar lghtwght, hgh spd, and not affctd by lctromagntc ntrfrnc and thy rqur no rcalbraton onc nstalld. Fgur 2 llustrats th postons of th n-stu stran rostts on an QS4 lmnt s surfacs for computng th rfrnc plan strans and curvaturs. Fgur 2 Dscrt surfac strans masurd at locaton x from stran rostts nstrumntd on top and bottom of QS4 lmnts. 8

9 Tsslr and Spanglr (2005) ndcatd that th rfrnc plan strans and curvaturs can b calculatd by usng th surfac strans masurd at n dscrt locatons ( x x, y, t) ( 1,..., n) as xx xx 1 yy yy wth ( 1, n) (2a) 2 xy xy xx xx 1 yy yy wth ( 1, n) (2b) 2t xy xy In Eqs. (2), th masurd surfac strans ar dnotd as (,, ) xx yy xy and (,, ) xx yy xy whr suprscrpts + and rprsnt th top and bottom surfac locatons, and th followng notaton s usd. In fact, th rfrnc plan strans and curvaturs x x ar assocatd wth th mmbran strans u ( ) and bndng curvaturs u ( ) dscrbd n Eq. (1a). In th dformaton of thn shlls, contrbutons of g ar much smallr compard to th bndng curvaturs. Snc most of th marn structurs ar sutabl to b modlld by usng thn shlls, th g contrbutons can b safly omttd Wghtd last-squars functonal of nvrs fnt lmnt mthod By tang nto account th mmbran, bndng and transvrs shar dformatons of th ndvdual lmnt, th nvrs fnt lmnt mthod rconstructs th dformd shap by mnmzng an lmnt functonal, namly a wghtd last-squars functonal ( u ) ( ) ( ) ( ) ( w w wg ) u u u g u g (3a) wth rspct to th unnown dsplacmnt DOF (Tsslr and Spanglr 2005). Th squard norms xprssd n Eq. (3a) can b wrttn n th form of th normalzd Eucldan norms as 9

10 2 n 1 ( u ) ( ) 2 dxdy n u (3b) A 1 2 n 2 2t 2 ( u ) ( ) dxdy n u (3c) A n 1 g( u ) g ( ) 2 dxdy n g u g (3d) whr A A rprsnts th ara of th lmnt. Durng th mplmntaton of th QS4 lmnt, all th ntgratons gvn n Eqs. (3b-3d) ar carrd out va four-pont Gaussan quadratur for a constant thcnss lmnt. Th wghtng constants w, w, and w g n Eq. (3a) ar postv valud and stand for ndvdual scton strans. Thy control th complt cohrnc btwn thortcal stran componnts and thr xprmntally masurd valus. If all n-stu strans,, and g ar nown by utlzng th masurd surfac strans collctd from on-board stran snsors, th wghtng constants ar w w w 1 for th g squard norms gvn n Eqs. (3b-3d). Howvr, for any cas of mssng n-stu stran componnt, th corrspondng wghtng constants can b adjustd to a small postv coffcnt such as 4 10 (Tsslr t al., 2011). Morovr, for thn shlls, th wghtng constant g w can b st as wg 4 10, whras th rmanng wghtng constants can b spcfd as w w 1 bcaus th dsplacmnts du to th transvrs shar strans bndng curvaturs g ar much smallr compard to. Ths adjustmnt of wghtng constants nsurs satsfactory rgularzaton of Eq. (3a) and provds accurat solutons for thn shlls. If non of th aformntond stran snsors s placd to an nvrs lmnt, th lmnt s mssng all of th n-stu stran componnts,, and g, so t s calld stranlss nvrs lmnt. In ordr to nsur th satsfactory rgularzaton of FEM modl, all of th wghtng coffcnts should b adjustd to a small postv coffcnt w w w 4 g 10 for a stranlss nvrs lmnt. Hnc, th ncssary ntrpolaton functonal connctvty can b stablshd btwn th lmnts that hav stran-snsor data, vn f th FEM modl has vry spars masurd stran data. Not that ncrasng th numbr of stranlss nvrs lmnts may dcras th accuracy of th soluton. 10

11 By vrtu of ths assumptons, all stran compatblty rlatons ar xplctly satsfd so that Eq. (3a) can b mnmzd wth rspct to nodal dsplacmnt vctor as ( u ) u f 0 u Aftr th mnmzaton, th rsultng lmnt matrx quaton s obtand as (4a) u f (4b) whr s th lmnt lft-hand-sd matrx, functon of th masurd stran valus, and f s th lmnt rght-hand-sd vctor that s a u s th nodal dsplacmnt vctor of th lmnt. Onc th local matrx quatons ar dfnd, th lmnt contrbutons to th global lnar quaton systm of th dscrtzd structur can b obtand as KU F (4c) whr K s ndpndnt of th masurd stran valus, U s th global nodal dsplacmnt vctor, and F s th global functon of th masurd stran valus. Although th global lft-handsd matrx K ncluds th rgd body moton mod of th dscrtzd structur, t can b rducd to a postv dfnt matrx by prscrbng problm-spcfc dsplacmnt constrants. Snc K rmans unchangd for a gvn dstrbuton of stran snsors, t nds to b nvrtd only onc durng montorng procss. As a rsult, soluton of ths quaton systm can b don vry rapdly. On th othr hand, th rght-hand-sd vctor F nds to b updatd durng th montorng procss bcaus t s dpndnt on th dscrt surfac stran data obtand from nstu stran snsors. Fnally, th matrx vctor multplcaton DOF vctor U, whch provds th dformd structural shap at any ral-tm. 1 K F gvs rs to th unnown Onc th prmary unnowns, namly nodal dsplacmnts, ar calculatd at any ral-tm, th strans at any pont n an QS4 lmnt can also b calculatd at any ral-tm by usng th drvatvs of shap functons. Aftr dtrmnng th ral-tm strans, ral-tm strsss can b obtand by usng consttutv quatons. Bhatt (2006) dscrbd drct fnt lmnt mthod strss calculatons and proposd svral tchnqus ncludng ntrpolaton-xtrapolaton of strsss, and avrag nodal strsss n ordr to calculat lmnt strsss mor prcsly. Snc 11

12 th prsntd FEM formulaton nhrts standard dsplacmnt-basd formulaton, th strss calculaton mthods proposd by Bhatt (2006) can b drctly followd. 3. FEM analyss of a chmcal tanr cross-scton A typcal chmcal tanr cross-scton as wll as ts structural componnts s prsntd n Fgur 3 (Doum, 2003). A long barg that has a vry smlar cross-scton to ths typcal chmcal tanr s dsgnd and modlld wth QS4 lmnts n ordr to dmonstrat th robustnss and practcalty aspcts of FEM mthodology for prformng an accurat shap- and strss-snsng opraton. Isomtrc vw of th barg cross-scton and dtald dmnsons of ts structural componnts ar ncludd n Appndx B. Fgur 3 Vw of a tan nsd a typcal chmcal tanr (Doum, 2003) For smplcty, t s dsrabl to dfn only on axs systm and th dfnd global Cartsan coordnat systm has ts orgn on th stll watr plan algnd vrtcally wth th shp s cntr of gravty. Th x-y plan s concdnt wth th calm watr lvl whr x-axs s along th man drcton of th shp, th y-axs ponts at th port sd of th shp, and z-axs s postv upwards 12

13 .. n th oppost drcton of gravty. Accordng to th dfnd Cartsan coordnat systm, man partculars of th barg ar lstd n Tabl 1. Tabl 1 Man partculars of barg Partcular Valu Unt Lngth (ovr all) 100 m Bradth (mouldd) 20 m Dpth (mouldd) 10.5 m Dsgnd load draft (mouldd) 8.25 m Bloc coffcnt (at dsgnd draft) m 3 /m 3 Dsplacmnt at dsgnd draft tonns Dadwght at dsgnd draft tonns Lght shp wght tonns Vrtcal cntr of buoyancy m Vrtcal cntr of gravty m Vrtcal cntr of dadwght m Vrtcal cntr of lght shp m Radus of gyraton around x axs m Radus of gyraton around y axs m Radus of gyraton around z axs m Radus of gyraton for roll-yaw product of nrta 0 m It has bn assumd that total numbr of cargo holds s ght, fram spacng btwn th transvrs frams s 1.25 m, and all th structural componnts ncludng plats, stffnrs, and bulhads hav th unform thcnss of 12 mm. All structural componnts of th barg ar mad of stl havng lastc modulus of 210 GPa and Posson s rato of 0.3. In ordr to rprsnt complxty of th structur mor clarly, svral somtrc vws of th barg structural modl ar ncludd n Appndx B. In ths study, th soluton s obtand by combnng thr dffrnt tools. Frst, a hydrodynamc analyss s prformd to calculat th hydrodynamc forcs. By usng ths forcs as a loadng condton, FEM analyss of th barg s prformd to calculat th numrcal stran data to b usd as an nput for th FEM analyss. Fnally, FEM analyss s prformd by smulatd stran data obtand from drct FEM analyss. 13

14 Th dsgn of marn structurs such as shps, offshor and coastal structurs s sgnfcantly affctd by wav-body dynamcs. Thrfor, hydrodynamc analyss of rgd bods that ar frly oscllatng undr th fr watr surfac s xtrmly mportant. An n-hous panl mthod cod, whch s a frquncy-doman hydrodynamc softwar, s utlzd to prdct th motons and wav loads of th barg. Th barg s assumd to mov wth zro forward spd n dp watr at angl β = 180 to rgular snusodal wavs whch dscrbs th wavs comng from ahad, namly had sas condton. Sx DOF motons of th barg ar calculatd for a wav ampltud of 1 m and wav frquncs startng from 0.2 rad/s to 1.5 rad/s. Full hydrodynamc modl of th barg s dscrtzd by usng 2508 flat quadrlatral panls as shown n Fgur 4. Fgur 4 Full hydrodynamc modl of th barg Th dscrtzd body, namly coordnats of nods usd to gnrat th panls and nodal connctvty of ths panls, s th man nput for th n-hous frquncy-doman hydrodynamc softwar. Th softwar frst calculats radaton and dffracton sourc strngths and thn vlocty potntals at th cntrods of th hydrodynamc panls for rqustd hadng and frquncy basd on thr dmnsonal potntal flow thory and zro spd Grn s functon (Whausn and Laton, 1960; Bc and Lon, 1989; and Papanolaou and Schlln, 1992). Snc th barg floats n had sas condton, th n-hous panl mthod cod tas nto account th advantag of th latral symmtry condton (x-z plan symmtry) and gnrats th sourc strngth rsults by usng only 1254 panls. By utlzng th vlocty potntals of radatd and dffractd wavs, th softwar solvs a complx form of lnar and harmonc oscllatory rgd 14

15 body moton quaton wth rspct to th barg cntr of gravty and gnrats th sx DOF moton rsults of th barg. Th numrcal sway, roll and yaw moton ampltuds wth rspct to barg cntr of gravty ar nglgbly small and n fact ths rsults ar thortcally zro bcaus th barg s affctd by had sa wavs only. Th moton ampltuds wth rspct to th barg s cntr of gravty for th rmanng drctons namly surg, hav, and ptch ar llustratd n Fgurs 5. Fgur 5 Barg s moton ampltuds wth rspct to wav frquncs [rad/s] Th chang of th moton ampltuds n surg, hav and ptch drctons s sgnfcant around th frquncy valu of 0.5 rad/s n trms of structural analyss bcaus th oscllatory dynamc prssur calculatd at ths frquncy causs th hghst vrtcal bndng momnt dstrbuton along th lngth of th barg. Th hydrodynamc forc varaton along th barg undrwatr panls du to th oscllaton at 0.5 rad/s s plottd n Fgur 6. 15

16 Fgur 6 Barg s total hydrodynamc forc [N] dstrbuton for th frquncy valu of 0.5 rad/s For a floatng structur, t s mportant to obtan qulbrum bfor prformng a drct FEM analyss bcaus an mbalancd modl causs an unralstc rsult. Thrfor, computd hydrodynamc forcs and thr corrspondng nrta loads ar appld to th drct FEM modl of th structur n ordr to fnd a ralstc global structural rspons of th barg. Applcaton of nrta loads s don by assocatng th acclraton vctor to ach fnt lmnt whl hydrodynamc forcs ar appld to th nods of ach fnt lmnt blow stll watr. Barg gomtry and ts constrant condtons as wll as loadng condtons ar symmtrc wth rspct to both x-z and y-z plan. So, thr s no nd to prform drct FEM analyss of th ntr structur. Hnc, only on quartr of th barg s dscrtsd by usng 6908 nods and lmnts as shown n Fgur 7. Fgur 7 On quartr of th barg msh 16

17 For th applcaton of constrant condtons, translaton along normal drcton of symmtry plan and rotatons around th symmtry plan axal drctons ar fxd for ach symmtry plan. Morovr, th structural modl s fxd by an artfcal support along z-drcton n ordr to dsabl th rgd body moton. Aftr prformng th FEM analyss, th racton forc calculatd at ths artfcal support s chcd to nsur that t dosn t affct th actual qulbrum of hydrodynamc prssurs and nrta loads. Th loadng condtons ar also appld by consdrng th symmtry plans. As mntond arlr, th rsultng total dflcton and rotaton obtand from drct FEM analyss ar usd to produc th smulatd snsor-stran data. 4. Cas studs for FEM analyss Thr dffrnt cas studs of th barg ar consdrd basd on FEM mthodology by usng dffrnt numbr of stran rostts and thr altrd locatons. An FEM modl, whch s composd of QS4 lmnts and gomtrcally dntcal to th quartr msh modl usd n drct FEM analyss (Fgur 7), s adoptd n th followng FEM analyss. Th stran rostts hav to b placd at th top and bottom surfacs of th QS4 lmnts whos rsultng dformatons xhbts both strtchng and bndng actons du to th complxty of th barg structur. Howvr, for th QS4 lmnts whos md-plans ar suprmposd on th x-z and y-z symmtry plans of th barg, th stran rostts ar only locatd at th top surfac of ths lmnts. Snc th matral proprts ar symmtrc wth rspct to md-plans of ths QS4 lmnts and thr rsultng dformatons ar du to strtchng only, th stran dstrbutons ar symmtrc wth rspct to thr md-plans. In th frst cas, all th QS4 lmnts usd n dscrtzaton of FEM modl ar assumd to b nstalld wth stran snsors n ordr to xamn th rsults whn on-to-on mappng of th rlvant stran data from drct FEM to FEM analyss s mad. Thrfor, th total numbr of QS4 lmnts that ar nstalld wth stran rostts s from whch 688 and 236 of thm blongs to th x-z and y-z symmtry plans of th barg, rspctvly. As mntond arlr, th stran rostts ar placd at th cntrods of th top surfacs of ths 924 QS4 lmnts, whras th stran rostts ar postond on th cntrods of both th top and bottom surfacs of th rmanng QS4 lmnts. Thus, th total numbr of th stran rostts s Snc th 17

18 barg structur s constructd wth thn shlls, th wghtng constants assocatd wth th transvrs shar strans ar st as w g 10 4, whras th rmanng wghtng constants for mmbran strans and bndng curvaturs ar rspctvly st as w w 1. Th total dsplacmnt rsults found n FEM analyss ar shown togthr wth th rfrnc FEM rsults n Fgur 8. Fgur 8 (a) FEM total dsplacmnt [m] dstrbuton for on-to-on stran data. (b) Drct fnt lmnt total dsplacmnt [m] dstrbuton Accordng to th dstrbutons, both maxmum dsplacmnts ar approxmatly 4.9 mm confrmng that FEM mthodology can captur vry promsngly xpctd rsults whn all th structural componnts n on quartr of th barg ar nstalld wth stran snsors. Onc a full fld dformd shap of th barg s obtand, full fld thr-dmnsonal strss stat of th barg body can b calculatd by usng dsplacmnt rsults. Thn, thr-dmnsonal strsss can b convrtd nto an quvalnt strss by utlzng an approprat falur crtron. Th von Mss falur crtron s th most commonly usd falur crtron for mtals, thus full fld von Mss strsss on th top surfacs of th shlls ar calculatd n FEM analyss and 18

19 compard wth thos found n drct FEM soluton. Both von Mss strss dstrbutons ar plottd togthr n Fgur 9 and contours of strss colours ar dntcally matchng ach othr. Morovr, th dffrnc btwn th maxmum von Mss rsults s 2.23% provng th supror accuracy of FEM algorthm. Fgur 9 (a) FEM von Mss strss [Pa] dstrbuton for on-to-on stran data. (b) Drct fnt lmnt vonmss strss [Pa] dstrbuton Scondly, FEM analyss of th barg s prformd whn all stran rostts usd n th frst cas study ar rmovd xcpt th ons glud on th cntral dc stffnr, th cntral longtudnal bulhad, and th cntral grdr. Ths cas study s don to assss th prcson of FEM mthodology whn thr ar mssng n-stu stran masurmnts. Aftr rmoval of th stran rostts, FEM analyss of th barg s conductd by usng th stran data collctd from 688 QS4 lmnts whch ar suprmposd on th x-z symmtry plan of th barg. Th stran rostts ar only postond on th top surfacs cntrods of ths 688 QS4 lmnts. Thrfor, 19

20 total numbr of stran rostts s 688 for ths cas study. Th xact locatons of th stran rostts ar clarly llustratd n Fgur 10. Fgur 10 Exact locatons of th stran rostts on x-z symmtry plan For QS4 lmnts on whch th stran data s avalabl, th mmbran stran s wghtng constants and bndng curvatur s wghtng constants ar both st as w w 1, whl thr transvrs shar stran s wghtng constants ar st as w g For QS4 lmnts that do not hav any snsors, namly stranlss QS4 lmnts, thr wghtng constants ar st as w w w 4 g 10. In both FEM and drct FEM analyss, th von Mss strsss ar valuatd accordng to th top surfacs of th shlls. Th total dsplacmnts as wll as th von Mss strsss obtand from FEM analyss ar rspctvly compard wth drct FEM rsults as dpctd n Fgur

21 Fgur 11 (a) FEM total dsplacmnt [m] dstrbuton for only stran data collctd from x-z symmtry plan. (b) Drct fnt lmnt total dsplacmnt [m] dstrbuton. Fgur 12 (a) FEM von Mss strss [Pa] dstrbuton for only stran data collctd from x-z symmtry plan. (b) Drct fnt lmnt vonmss strss [Pa] dstrbuton 21

22 Th FEM-rconstructd dsplacmnt and von Mss strss flds farly agr wth th rfrnc dsplacmnt and von Mss strss rsults. Th rror btwn th maxmum dsplacmnts s 1.41%, whras th rror s 3.74% for von Mss strss comparson. Hnc, ths accuracy confrms th robustnss of FEM framwor vn f thr ar mssng n-stu stran masurmnts. Although th rsults obtand n th frst two cas studs ar compltly satsfactory, th proposd numbr of stran rostts may b hgh for a practcal applcaton. Thrfor, n th thrd cas study, th FEM analyss of th barg s prformd basd on th sam msh, but only usng th stran data obtand from stran rostts locatd at th cntral dc stffnr, th cntral grdr and svral crtcal locatons on th cntral longtudnal bulhad ncludng th nutral axs of th barg, th dg nar to dc, and th dgs nar to th transvrs bulhad. Ths cas study not only asssss th practcal applcablty, but also xamns th prcson of FEM formulaton accordng to th ffct of snsor locatons and numbr of snsors on soluton accuracy. Aftr lmnatng majorty of th stran snsors, th stran data collctd from 196 QS4 lmnts ar usd to prform shap- and strss-snsng of th barg. Smlar to th scond cas study, ach of ths 196 QS4 lmnts ar nstalld wth only on stran snsor on th cntrod of th top surfac bcaus ths nvrs lmnts blong to x-z symmtry plan. Thrfor, th total numbr of stan rostts s dcrasd to 196 and th xact locatons of ths snsors ar clarly prsntd n Fgur 13. Fgur 13 Exact locatons of th stran rostts on th cntral dc stffnr, th cntral grdr, and th cntral longtudnal bulhad 22

23 As n th scond cas study, th wghtng coffcnts ar st to w w w 4 g 10 for ach stranlss QS4 lmnts, whras thy ar st to w w 1 and w g 4 10 for ach QS4 lmnt on whch th stran snsors ar locatd. Th total dsplacmnts dstrbutons obtand from FEM analyss ar consstnt wth drct FEM rsults as rprsntd n Fgur 14. Fgur 14 (a) FEM total dsplacmnt [m] dstrbuton for stran data collctd from th cntral dc stffnr, th cntral grdr, and th cntral longtudnal bulhad. (b) Drct fnt lmnt total dsplacmnt [m] dstrbuton. Th rror of th maxmum dsplacmnt producd by th FEM soluton s 12.85% wth rspct to th FEM maxmum dsplacmnt. Followng to th dsplacmnt calculaton, th von Mss strsss ar computd accordng to th top surfacs of th shlls n both FEM and drct FEM analyss. Although a larg amount of stran-snsor data s not avalabl for ths cas study, smlar to th dsplacmnt comparson, von Mss strss dstrbutons ar qut cohrnt whn thy ar compard as shown n Fgur

24 Fgur 15 (a) FEM von Mss strss [Pa] dstrbuton for stran data collctd from th cntral dc stffnr, th cntral grdr, and th cntral longtudnal bulhad. (b) Drct fnt lmnt vonmss strss [Pa] dstrbuton Th rror btwn th maxmum FEM-rconstructd and drct FEM von Mss strsss s 15.56%. Ths numrcal rsults hav vrfd that t s stll possbl to rconstruct suffcntly accurat dformatons and von Mss strsss, vn f a larg amount of stran-snsor data s not avalabl. Consquntly, t can b ndcatd that th FEM-rconstructd dsplacmnt and von Mss strss rsults ar qut prcs n trms of practcal ngnrng applcaton. Accordng to rsults of thr cas studs, t can b concludd that FEM mthodology s a promsng tchnology for accomplshng accurat dsplacmnt and strss montorng of marn structurs. 5. Concludng rmars Dsplacmnt and strss montorng of a chmcal tanr md-shp scton by usng FEM mthodology s accomplshd n ths wor. Th prsntd FEM formulaton s basd upon th 24

25 mnmzaton of wghtd-last-squars functonal and rqurs dscrt stran data obtand from on-board snsors n ordr to rconstruct th dsplacmnt, stran, and strss flds. In-hous hydrodynamc and fnt lmnt softwar ar utlzd for smulatng th on-board stran-snsor data n ordr to rprsnt a floatng structur n ral sa nvronmnt. Frst, hydrodynamc loads ar obtand and usd for th loadng condton of a drct FEM analyss of th chmcal tanr md-shp scton. Onc smulatd stran data s obtand from drct FEM analyss, FEM analyss of thr dffrnt cass ncludng dffrnt numbr of stran snsors for th sam modl ar xamnd. Th comparsons of th rsults obtand from both FEM and drct FEM analyss dmonstrat that dsplacmnt and strss montorng of th chmcal tanr can b accuratly prformd basd on FEM mthodology. Morovr, th numrcal rsults confrmd that rlatvly accurat dformd shaps and von Mss strsss can stll b rconstructd by xplotng th wghtng constants n th last-squars functonal of FEM, vn though a larg amount of n-stu stran data s absnt or only a rlatvly spars stran data s collctd. Furthrmor, t has bn obsrvd that usng th stran data obtand from stran rostts locatd at th cntral dc stffnr, th cntral longtudnal bulhad, and th cntral grdr can b suffcnt to rconstruct a prcs global dformd shap and von Mss strsss causd by vrtcal bndng momnt and shar forc du to oscllatory motons. Consquntly, t can b ndcatd that th cntral dc stffnr, th cntral longtudnal bulhad, and th cntral grdr ar th optmum locatons for placng on-board snsors for a longtudnally framd structur. As a rsult, t can b concludd that FEM algorthm s vry promsng systm for prformng a prcs shap- and strss-snsng of marn structurs. Acnowldgmnt Th authors would l to acnowldg Dr. Olgun Hzr for hs prcous gudanc durng th mplmntaton of th n-hous hydrodynamc softwar. 25

26 Appndx A Th drvatvs of shap functons m B, B, s B whch ar gvn n Eqs. (1a-b) ar dfnd as B B B B B m m m m m (A.1) B B 1 B2 B3 B 4 (A.2) s s s s s B B 1 B2 B3 B 4 (A.3) whr B B B , x, x m 0 N,y M, x s N N N L M,y, x,y, x N, 0 x N,y 0 0 ( 1, 2,3, 4) N N 0,x,y 0 0 N, x L, x M, x N N,y L,y N M,y 0 L (A.4) wth th shap functons ar gvn as N N N N N (1 s)(1 t) (A.5) 4 (1 s)(1 t) (A.6) 4 (1 s)(1 t) (A.7) 4 (1 s)(1 t) (A.8) 4 2 (1 s )(1 t) (A.9) 16 2 (1 s)(1 t ) N6 (A.10) 16 26

27 N N 7 8 and 2 (1 s )(1 t) (A.11) 16 2 (1 s)(1 t ) (A.12) 16 L1 y14 N8 y21n5 (A.13) L2 y21n5 y32 N6 (A.14) L3 y32n6 y43n7 (A.15) L4 y43n7 y14 N8 (A.16) M1 x41n8 x12 N5 (A.17) M2 x12 N5 x23n6 (A.18) M3 x23n6 x34 N7 (A.19) M4 x34 N7 x41n8 (A.20) Not that xj and yj can b xprssd n trms of local coordnats of QS4 lmnt as xj x x j ( 1,2,3,4; j 1,2,3,4) yj y y j and th parnt spac coordnats ar dfnd as st, 1, 1. (A.21) 27

28 Appndx B Cross-scton of th prsntd barg modl s smlar to a typcal chmcal tanr cross-scton. Dtald dmnsons of structural componnts n th barg modl s cross-scton and ts somtrc vw s prsntd n Fgur B1-B2, rspctvly. Fgur B1 Dtald dmnsons [mm] of structural componnts n half cross-scton of th barg 28

29 Fgur B2 Isomtrc vw of barg cross-scton 29

30 Appndx C Svral somtrc vws of th barg structural modl ar llustratd n Fgurs C1-C3 n ordr to show ts structural topology. Cargo tans n th full barg modl can b sn n Fgur C1. Structural dtals of th quartr barg modl s shown n Fgur C2. A dtald vw of doubl sd and doubl bottom framng s prsntd n Fgur C3. Fgur C1 Isomtrc vw of cargo tans Fgur C2 Structural dtals n th quartr barg modl 30

31 Fgur C3 Doubl sd and doubl bottom framng dtals 31

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