Application of an advanced beam theory to ship hydroelastic analysis

Transcription

1 Applcaon of an advancd ba hory o hp hydrolac analy I. njanovć,. Toašvć, N. Vladr, M. Toć Unvry of Zagrb, Faculy of Mchancal Engnrng and Naval Archcur, Ivana Lučća 5, Zagrb, CROATIA Š. Malnca Burau Vra, 67/71 Boulvard du Châu, 900 Nully-ur-n, FRANCE ABTRACT: Modrn a ranpor rqur buldng of Vry Larg Conanr hp (VLC), whch ar rlav flxbl rucur. Barng n nd h fac, and akng no accoun h pd of VLC, obvou ha hr naural frqunc could fall no h rang of h ncounr frqunc n an ordnary a pcru. Prn Clafcaon Rul for hp dgn and conrucon don covr uch condon coplly. Th ncourag cn and ngnr o dvlop or powrful and rlabl ool for h analy of hp bhavor n a and o prov h Rul. Hydrolac analy of VLC o b appropra oluon for h challngng probl. Mhodology of hydrolac nvgaon bad on ahacal odl whch nclud rucural, hydroac and hydrodynac ubodl whch ar abld no hydrolac on. Th hydrolac probl can b olvd a dffrn lvl of coplxy and accuracy. I obvou ha h b way o condr 3D FEM rucural odl and 3D hydrodynac odl, bu h approach would b oo xpnv, pcally n prlnary dgn ag. A h lvl would b or appropra o coupl 1D FEM odl of hp hull wh 3D hydrodynac odl. In h papr, h pha gvn on h advancd ba odl whch nclud har nflunc on oron a an xnon of har nflunc on bndng, and conrbuon of ranvr bulkhad o hull ffn. Bd rucural odl, hydroac and hydrodynac ubodl, a conuv par of hydrolac odl ar brfly dcrbd. Vrfcaon of propod nurcal procdur don by corrlaon analy of h ulaon rul and h aurd on for flxbl barg, for whch h rul ar avalabl n h lraur. Nurcal xapl, whch nclud copl hydrolac analy of 7800 TEU conanr hp, alo gvn. In h ca, valdaon of 1D FEM odl chckd by corrlaon analy wh h vbraon rpon of h fn 3D FEM odl. Th oband rul confr ha advancd hn-walld grdr hory a raonabl choc for drnng wav load ffc on VLC, n prlnary dgn ag. 1 INTRODUCTION Rapd ncra n hp ranpor nduc buldng of Vry Larg Conanr hp (VLC), whch ar rlav lndr, fa and qu flxbl hp. Bcau of h faur, rucural naural frqunc of VLC could fall no h rang of ncounr frqunc n an ordnary a pcru. I vry poran o hav rlabl and powrful dgn ool o avod ho ronan ag durng h navgaon. Th clacal hor for drnaon of hp oon and wav load, a for xapl (alvn al. 1970), ar bad on h aupon ha h hp hull a rgd body. Uually, h wav load oband accordng o h hor pod o h lac 3D FEM odl of hp rucur n ordr o analyz global rngh, a wll a local rngh wh r concnraon rlad o fagu analy. Alhough h abov approach good nough for hp wh clod cro-con and ordnary hach opnng uch a ankr, bulk carrr or gnral cargo hp, no rlabl a hould b for ulra larg conanr hp du o uual nflunc of h wav load and rucur rpon (njanovć al. 009a). Thrfor, a or rlabl oluon rqur analy of wav load and hp vbraon a a coupld hydrolac probl (Bhop & Prc, 1979). Th vry poran for pulv load uch a hp lang whch cau whppng. Nurcal procdur for hp hydrolac analy rqur dfnon of rucural odl, hp and cargo a drbuon, and gorcal odl of hp urfac (njanovć al. 007, 008a, 009b). In h papr, h pha gvn on advancd nurcal procdur bad on h ba and hnwalld grdr hor for calculaon of dry naural vbraon of conanr hp, a an poran p n hr hydrolac analy (njanovć al. 009c). Th hory nclud har nflunc on oron a an xnon of har nflunc on bndng, a wll a conrbuon of ranvr bulkhad o hull ff-

2 n. Bd advancd ba hory, hodology of hp hydrolac analy brfly dcrbd and llurad. Alo, hor dcrpon of hydroac and hydrodynac ubodl, and hydrolac odl gvn. Appld nurcal procdur a wll a dvlopd copur cod vrfd. Fnally, h rul of hydrolac analy of 7800 TEU conanr hp ar gvn and proprly nrprd. METHODOLOGY OF HIP HYDROELATIC ANALYI A nond bfor, rucural odl, hp and cargo a drbuon and gorcal odl of hp urfac hav o b dfnd o ak hydrolac analy of h hp. A h bgnnng of h analy, dry naural vbraon hav o b calculad, and afr ha odal hydroac ffn, odal addd a, dapng and odal wav load ar drnd. Fnally, w naural vbraon a wll a h ranfr funcon (RAO) for drnng hp rucural rpon o wav xcaon ar oband (njanovć al. 008a, 009b). vrcal vbraon, a wll a danc bwn h forr and har cnr for horzonal and oronal vbraon ar nglgbl for convnonal hp. Thrfor, n h abov ca hp hull vbraon can b analyzd paraly. Howvr, h har cnr n hp wh larg hach opnng locad oud h cro-con,.. blow h kl, and hrfor h couplng of horzonal and oronal vbraon xrly hgh. Th abov probl rahr coplcad du o gorcal dconnuy of h hull cro-con. Th accuracy of h oluon dpnd on h rlably of ffn parar drnaon,.. of bndng, har, oronal and warpng odul. Th fn ln hod a powrful ool o olv h abov probl n a uccful way. On of h fr oluon for coupld horzonal and oronal hull vbraon, dalng wh h fn ln chnqu, gvn n (Kawa, 1973, njanovć & Grubšć, 1991). Gnrald and provd oluon ar prnd n (Pdrn, 1985, Wu & Ho, 1987). In all h rfrnc, h drnaon of hull ffn bad on h clacal hn-walld grdr hory, whch do no gv a afacory valu for h warpng odulu of h opn cro-con (Halu & Tonnn, 197, Vlaov, 1961). Apar fro ha, h fxd valu of ffn odul ar drnd, o ha h applcaon of h ba hory for hull vbraon analy ld o a fw low naural od only. Ohrw, f h od dpndn ffn parar ar ud h applcaon of h ba hory can b xndd up o h nh naural od (njanovć & Fan, 1989, 199, 1997). 3. Ouln of an advancd ba hory Rfrrng o h flxural ba hory (njanovć & Grubšć, 1991), h oal ba dflcon, w, con of h bndng dflcon, w b, and h har dflcon, w,.., Fgur w= w + w. (1) b Fgur 1. Mhodology of h hydrolac analy. 3 TRUCTURAL MODEL BAED ON ADVANCED BEAM THEORY 3.1 Gnral rark A hp hull, a an lac non-prac hn-walld grdr, prfor longudnal, vrcal, horzonal and oronal vbraon. nc h cro-conal cnr of gravy and cnrod, a wll a h har cnr poon ar no dncal, coupld longudnal and vrcal, and horzonal and oronal vbraon occur, rpcvly. Th danc bwn h cnr of gravy and cnrod for longudnal and Th har dflcon a funcon of w b EIb wb w =, () GA whr E and G ar h Young' and har odulu, rpcvly, whl I b, and A ar h on of nra of cro-con and har ara, rpcvly. Th angl of cro-con roaon caud by h bndng dflcon wb ϕ =. (3) Th cro-conal forc ar h bndng on and h har forc

3 M = EI b w b w Q= GA = EI x, (4) b w x 3 b 3 Fgur. Ba bndng and oron.. (5) Concrnng oron, h oal w angl, ψ, con of h pur w angl, ψ, and h har conrbuon, ψ,.., Fgur ψ = ψ + ψ. (6) Rfrrng o h analogy of oron and bndng (Pavazza, 005), h har angl dpnd on h w angl, larly o Eq. () EIw ψ ψ =, (7) GI x whr I w h warpng odulu and I h har nra odulu. Th cond ba dplacn, whch cau warpng of cro-con (larly o h cro-con roaon du o bndng) a varaon of h pur w angl ψ ϑ =. (8) Th conal forc nclud h oal orqu, T, whch con of pur oronal orqu, T, and h warpng orqu T w.. T = T + T, (9) w whr T = GI ψ (10) 3 ψ ψ Tw = GI = EIw (11) 3 x x and h bon gvn by ψ Bw = EIw. (1) 1D FEM procdur for vrcal hp hull vbraon wll known n lraur. Coupld horzonal and oronal vbraon ar a or coplx probl. Du o analogy bwn bndng and oron h a hap funcon, rprnd by Hran polynoal, ar ud. Th arx fn ln quaon for coupld vbraon yld (njanovć, 1998) = + f k δ δ, (13) whr f nodal forc vcor, δ nodal dplacn vcor, k ffn arx, and a arx. Th quan con of flxural and oronal par f P U =, δ = R V (14) kb 0 =, b k = 0 kw. (15) w Vcor of nodal forc and dplacn ar Q(0) T (0) (0) (0) M Bw P=, R = (16) Ql () Tl () Ml () B () w l w(0) ψ (0) (0) (0) ϕ ϑ U=, V =. (17) wl () ψ () l ϕ() l ϑ() l In h abov forula ybol Q, M, T and B w dno har forc, bndng on, orqu and warpng bon, rpcvly. Alo, w, φ, ψ and ϑ ar dflcon, roaon of cro-con, w angl and varaon, rpcvly. Th ubarc, whch ar pcfd n (njanovć al. 009c), hav h followng anng: k b bndng - har ffn arx k w warpng - oron ffn arx b har - bndng a arx w oron - warpng a arx = T har - oron a arx.

4 I vdn ha couplng bwn horzonal and oronal vbraon ralzd hrough h a arx du o ccnrcy of h cnr of gravy and har cnr. Bfor ablng of fn ln ncary o ranfor Eq. (13) n uch a way ha all h nodal forc a wll a nodal dplacn, Eq. (16) and (17), ar rlad o h fr and hn o h cond nod. Furhror, Eq. (13) ha o b ranford fro local o global coordna y. Th orgn of h forr locad a h har cnr, and of h lar a h ba ln. 3.3 Conrbuon of ranvr bulkhad o hull ffn Th probl for conanr hp xnvly analyzd n (njanovć al. 008b), whr oronal odulu of hp cro-con ncrad proporonally o h bulkhad ran nrgy. Th bulkhad condrd a an orhoropc pla wh vry rong ool (zlard, 004). Th bulkhad ran nrgy drnd for h gvn warpng of cro-con a a boundary condon. Th warpng cau bulkhad crwng and bndng. Hr, only h rvw of h fnal rul prnd. Th bulkhad dflcon (axal dplacn) gvn by h followng forula, Fgur 3: y z z u( y, z) = y ( z d) + 1 b H H ψ, (18) whr H h hp hgh, b on half of bulkhad bradh, d h danc of warpng cnr fro doubl boo nural ln, y and z ar ranvr and vrcal coordna, rpcvly, and ψ h varaon of w angl. U 1 ν 35b 105H Hb + ( 1 ν) 75 Eψ H 3b 8Hb g = y z ν y z ( ) (19) whr y, z and ar h avrag on of nra of cro-con and oronal odulu pr un bradh, rpcvly. Th ool ran nrgy coprd of h bndng, har and oronal conrbuon U = hI ( ) b h Ib 9bI 7 1 ν 3 b b A 10 1 ( + ν ) Eψ (0) whr I b, A and I ar h on of nra of cro-con, har ara and oronal odulu, rpcvly. Quany h h ool danc fro h nnr boo, Fgur 4. Fgur 4. Longudnal con of conanr hp hold. Th quvaln oronal odulu yld, Fgur 4 ( + ν ) a 41 C I = I, (1) l1 Il0 whr a h wb hgh of bulkhad grdr (fra pacng), l 0 h bulkhad pacng, l 1 = l 0 a h n lngh, and C h nrgy coffcn Ug + U C =. () Eψ Th cond r n (1) h an conrbuon of h bulkhad a h clod cro-con gn of hp hull, and h hrd on copr h bulkhad ran nrgy. Fgur 3. hap of bulkhad dforaon. Th bulkhad grllag ran nrgy nclud vrcal and horzonal bndng wh conracon, and oron (njanovć al. 008b). 3.4 Naural vbraon analy If h FEM approach ud (1D or 3D odl), h govrnng quaon of dry naural vbraon yld (Bah, 1996) ( Ω ) K M δ = 0, (3)

5 whr K ffn arx, M a arx, Ω dry naural frquncy and δ dry naural od. A oluon of h gnvalu probl (3) Ω and δ ar oband for ach h -h dry od, whr = 1,...N, N oal nubr of dgr of frdo. Now naural od arx can b conud [, ] = 1 N δ δ δ δ δ (4) and h odal ffn and a can b drnd (njanovć, 1998) = T T k δ K δ, = δ Mδ. (5) nc h dry naural vcor ar uually orhogonal, arc k and ar dagonal. Tr k and Ω rprn ran and knc nrgy of h -h od rpcvly. No ha gnrally h fr x naural frqunc Ω ar zro wh corrpondng gnvcor rprnng h rgd body od. A a rul, h fr x dagonal ln of k ar alo zro, whl h fr hr ln n ar qual o rucur a, h a n all drcon x, y, z, and h nx hr ln rprn h a on of nra around h corrpondng coordna ax. If 1D analy appld, h ba od ar prad o h hp wd urfac ung h xpron for vrcal vbraon (njanovć al. 009a) d wv h = ( z z N) + w vk, (6) d x and for coupld horzonal and oronal vbraon dw d h ψ h = y + u dx dx + w + ( z z ) j ψ yk, [ ψ ] h (7) whr w hull dflcon, ψ w angl, y and z ar coordna of h pon on hp urfac, and z N and z ar coordna of cnrod and har cnr rpcvly, and u = u( x, y, z ) h cro-con warpng nny rducd o h wd urfac (njanovć al. 009d). 4 HYDRODYNAMIC MODEL Th couplng procdur do no dpnd on h ud hydrodynac odl, and hrfor dcrbd hr for h zro pd ca, a h pl on. Haronc hydrolac probl condrd n frquncy doan and hrfor w opra wh aplud of forc and dplacn. In ordr o prfor h couplng of rucural and hydrodynac odl, ncary o xpr h xrnal prur forc n a convnn annr (Malnca al. 003). Fr, h oal hydrodynac forc F h ha o b pl no wo par: h fr par F R dpndng on h rucural dforaon, and h cond on rprnng h pur xcaon. Furhror, h odal uprpoon hod can b ud. Vcor of h wd urfac dforaon H (x, y, z) can b prnd a a r of dry naural od h (x, y, z). Th ponal hory aupon ar adopd for h hydrodynac par of h probl. Whn h hory, h oal vlocy ponal ϕ, n h ca of no forward pd, dfnd wh h Laplac dffrnal quaon and h gvn boundary valu. Furhror, h lnar wav hory nabl h followng dcopoon of h oal ponal (njanovć al. 008a) ga ν ( z+ x) ϕ = ϕ + ϕ ω ξ ϕ, ϕ =, (8) ω F DI N I D j Rj I j= 1 whr ϕ I ncdn wav ponal, ϕ D dffracon ponal, ϕ Rj radaon ponal and A and ω rprn wav aplud and frquncy rpcvly. Onc h ponal ar drnd, h odal hydrodynac forc ar calculad by prur work ngraon ovr h wd urfac,. Th oal lnard prur can b found fro Brnoull' quaon p = ωρϕ ρ gz. (9) Fr, h r aocad wh h vlocy ponal ϕ condrd and ubdvdd no xcaon and radaon par DI F = ωρ ( ϕ + ϕ ) hn d, (30) F I D N R = j Rj j= 1 ρω ξ ϕ hn d. (31) DI Thu, F rprn h odal prur xcaon. Now on can dcopo (31)no h odal nra forc and dapng forc aocad wh acclraon and vlocy, rpcvly N a R = R( ) = j j, j = R Rj d j= 1 F F ω ξ A A ρ ϕ hn, (3) N v R = = j j j = Rj j= 1 F I( F ) ω ξ B, B ρωi ϕ hn d.(33) whr A j and B j ar ln of addd a and dapng arc, rpcvly. Drnaon of addd a and dapng for rgd body od a wll-known procdur n hp hydrodynac. Now h a procdur xndd o h calculaon of h quan for lac od. Th hydroac par of h oal prur, ρgz n (9), condrd whn h hydroac odl.

6 5 HYDROTATIC MODEL Hydrolacy a known u for any yar, and hr ar fw oluon for rorng ffn (Prc & Wu, 1985, Nwan, 1994, Huang & Rgg, 000, Malnca, 003). In h udy conn forulaon of rorng ffn ud (njanovć al. 009a, b), and condnd for gvn blow. Th rorng ffn con of hydroac and gravy par. Work of h hydroac prur, whch rprn h gnralzd forc, can b drvd n h followng for ( ) d h F = ρg Hz + Z H Hn, (34) whr Halon dffrnal opraor, H dplacn vcor, d dffrnal of wd urfac, Z dph and n un noral vcor. Accordng o dfnon, h ffn rlaon bwn ncrnal forc and dplacn, o drnd fro h varaonal quaon ( ) d h δf = ρg Hz + Z H δhn. (35) Furhror, h odal uprpoon hod ud, and h varaon rand o od,.. odal forc and dplacn N N N h h δ F δfj, = ξ j j, δ = jδξ j j= 1 j= 1 j= 1 = H h H h. (36) In ha way, Eq. (35) dcopod no h odal quaon ( ) h = N + p nh j j j j=1 δ F C C ξ δξ, (37) whr ( ) p j nh C = ρg h h nd, C = ρg Zh h n d, (38) j z j j ar ffn coffcn du o prur, and noral vcor and od conrbuon, rpcvly. larly o h prur par, h gnralzd gravy forc rad ( ) d F = g ρ H H V, (39) V z whr ρ and V ar rucur dny and volu, rpcvly. In ordr o oban conn varaonal quaon, ncary o rcly follow h dfnon of ffn and o vary dplacn vcor n (39) and no drvav ( ) d δf = g ρ δh H V. (40) V z Applcaon of h odal uprpoon hod lad o h odal varaonal quaon = N δ F C ξδξ, (41) j j j=1 whr j ( ) d C = g ρ h h V, (4) j z V ar h gravy ffn coffcn. Fnally, h copl rorng ffn coffcn ar oband by ung up conuv par C = C p + C nh + C. (43) j j j j 6 HYDROELATIC MODEL Afr h dfnon of h rucural, hydroac and hydrodynac odl, h hydrolac odl can b conud. Th govrnng arx dffrnal quaon for coupld hp oon and vbraon dducd k+ C ω( d+ B( ω) ) ω ( + A( ω) ) ξ = F, (44) whr k, d, and ar rucural ffn, dapng and a arc, rpcvly, C rorng ffn, B(ω) hydrodynac dapng, A(ω) addd a, ξ odal aplud, F wav xcaon and ω ncounr frquncy. All quan, xcp ω and ξ, ar rlad o h dry od. Th oluon of (44) gv h odal aplud ξ and dplacn of any pon of h rucur oband by rrackng o (36). 7 VERIFICATION OF PROPOED NUMERICAL PROCEDURE Th copur ofwar DYANA for hp hydrolac analy, bad on h prnd hory, ha bn dvlopd. Boh hory and cod ar chckd by corrlaon analy of h ulaon rul and h aurd on for a flxbl gnd barg conng of 1 ponoon, for whch rul ar avalabl (Malnca al. 003, Ry al. 006), Fgur 5. Good agrn bwn aurd and calculad ranfr funcon of horzonal bndng on and orqu, a funcon of wav prod, T, pl ha dvlopd procdur can b ud for hp hydrolac analy, Fgur 6, 7 (Malnca al. 003, 007). Alo, convrgnc of h appld odal uprpoon hod n hydrolac analy confrd n h ca of h a flxbl barg wh no forward pd (Toašvć, 007). Fgur 8 how abolu aplud valu ξ of noralzd od. Th fr hr od ar rlad o way, roll and yaw, whl h ranng od ar lac.

7 8 NUMERICAL EXAMPLE For h lluraon purpo, hydrolac analy of 7800 TEU VLC don, Fgur 9. Fgur 5. Barg n wav. Fgur 6. Tranfr funcon of barg horzonal bndng on, χ = 60. Fgur 7. Tranfr funcon of barg orqu, χ = 60. Fgur TEU Conanr hp. Fgur 8. Modal aplud of coupld horzonal and oronal vbraon of flxbl barg, V = 0 kn, χ = Parcular of h analyzd hp Th an vl parcular ar h followng: Lngh ovrall L oa = 334 Lngh bwn prpndcular L pp = 319 Bradh B = 4.8

8 Dph Draugh Dplacn, full load Dplacn, balla Engn powr hp pd H = 4.6 T = 14.5 f = b = P = 6960 kw v = 5.4 kn. Th dhp con, whch how a doubl kn rucur wh h wb fra and longudnal, prnd n Fgur 10. Row and r of conanr a h dhp con ar ndcad n Fgur 11. Vrcal poon of nural ln, dforaon (har, oronal) cnr, and cnr of gravy ar alo arkd n h fgur. Larg danc bwn gravy cnr and dforaon cnr cau hgh couplng of horzonal and oronal vbraon. Tabl 1 and, whl h ool parar ar gvn n Tabl 3. Tor. odulu I ( 4 ) I ( 4 ) c () ( 3 ) ( 3 ) Horz Vr Tabl 1. ffn parar of wargh bulkhad. Tor. Grd Mon pacng of nra Mon odulu r of nra Tabl. ffn parar of uppor bulkhad. Tor. Grd Mon pacng of nra Mon odulu r of nra Tor. odulu I ( 4 ) I ( 4 ) c () ( 3 ) ( 3 ) Horz Vr Tabl 3. ool ffn parar. har ara Mon of nra Tor. odulu A ( ) I ( 4 ) I ( 4 ) Th bulkhad ran nrgy, drnd accordng o forula prnd n Chapr 3 gvn n Tabl 4. Tabl 4. Bulkhad ran nrgy, / ( ) U Eψ. Wargh bulkhad uppor bulkhad Enrgy coffcn Grllag ool Grllag ool C, Eq. (C5) (1) () (3) (4) (5) = [(1)+ ()+ (3)+ (4)]/ Fgur 10. Mdhp con of h analyzd hp. Th hp hull ffn propr ar calculad by progra TIFF, bad on h hory of hn-walld grdr (njanovć & Fan, 199, 1993). Th gorcal propr rapdly chang valu n h ngn and uprrucur ara du o clod hp cro-con. Th pcally pronouncd n ca of oronal odulu, whch ak qu all valu for opn cro-con and rahr hgh for h clod on (Toašvć, 007). Influnc of h ranvr bulkhad akn no accoun by ung h quvaln oronal odulu for h opn cro-con nad of h acual valu,.. I =.4I. Th valu appld for all hp- * cro con a h fr approxaon. Th ffn parar of h bulkhad grdr ar ld n Fgur 11. Conanr drbuon a dhp con.

9 8. Valdaon of 1D FEM odl Th rlably of 1D FEM analy vrfd by 3D FEM analy of h condrd hp. For h purpo, h lgh wgh loadng condon of dry hp wh dplacn =3369 akn no accoun. Th laral and brd vw of h fr donanly oronal od of h wd urfac, drnd by 1D odl, hown n Fgur 1. Th fr 3D dry coupld naural od of h copl hp rucur hown n Fgur 13, whr Y and Z ar vrcal and ranvral ax, rpcvly. I lar o ha of 1D analy for h wd urfac. Warpng of h ranvr bulkhad, whch ncra h hull oronal ffn, vdn. Qu good agrn achvd. Valu of naural frqunc for hghr od ar or dffcul o corrla, nc rong couplng bwn global hull od and local ubrucur od of 3D analy occur. 8.3 Rul of h hp rpon calculaon Tranfr funcon of oronal on and horzonal bndng on a h dhp con ar hown n Fgur 14 and 15, rpcvly. Thy ar copard o h rgd body on drnd by progra HYDROTAR. Vry good agrn oband n h lowr frquncy doan, whr h hp bhav a a rgd body. Dcrpanc ar vry larg a h ronanc of h lac od, a xpcd. Fgur 1. Th fr donanly oronal od, laral and brd vw, lgh wgh, 1D odl. Fgur 14. Tranfr funcon of oronal on, χ=10, U=5 kn, x= fro AP. Fgur 13. Th fr donanly oronal od, laral and brd vw, lgh wgh, 3D odl. Th fr four corrpondng naural frqunc oband by 1D and 3D analy ar copard n Tabl 5. Tabl 5. Dry naural frqunc, lgh wgh, ω [rad/]. Mod Vr. Horz. + or. no. 1D 3D 1D 3D Mod no (H0 + T1) (H1 + T) (H + T3) (H3 + T4) Fgur 15. Tranfr funcon of horzonal bndng on, χ=10, U=5 kn, x= fro AP. Ncary condon for convrgnc of conal forc o zro valu a h wav frquncy approach o zro can b ud a a bnchark for valdaon of h rorng ffn. Fgur 16 how h zood ranfr funcon of oronal on drnd by h drc ngraon and hr forulaon of rorng ffn n h hydrolac approach: conn on fro h papr, yrc arx Cj = ( Cj + C j ) / oband by h nu nrgy hod, and hybrd arx n whch p p n n h h C = C, C = C, C = C, C = C, j j j j j j j j

10 (Malnca, 003). Only h conn rorng ffn af h abov condon a h rgd body oluon do. In h ca of yrc and hybrd arc h hp no qulbrad. Morovr, h conn rorng ffn phaz h roll ronanc a 0.3 rad/. Pur w angl ψ ralzd around h har cnr,.c., and owha allr hn h w angl drnd by 3D FEM odl. If h har w angl ψ addd o ψ around h doubl boo cnrod, valu of h oal w angl approach ha of 3D analy. A a rul, h w cnr drnd, T.C. 9 CONCLUDING REMARK Fgur 16. Zood ranfr funcon of oronal on, χ=10, U=5 kn, x= fro AP. har nflunc on oron nvgad n h ca of a ponoon wh h cro-con qual o h dhp con of h condrd 7800 conanr hp. On nd of h ponoon fxd and anohr loadd wh h concnrad orqu. Calculaon prford analycally by ployng h advancd ba hory and nurcally by 3D FEM odl. Roaon angl of h fr ponoon nd ar hown n Fgur 17. Fgur 17. Tw angl a h ponoon nd. Ulra larg conanr hp ar qu flxbl o hy rch h bound of prn clafcaon rul for rlabl rucural dgn. Thrfor, hydrolac analy ha o b prford (njanovć al. 008a, 009a) Th llurav nurcal xapl of h 7800 TEU conanr hp how ha h dvlopd hydrolacy hory, ulzng h provd 1D FEM rucural odl and 3D hydrodynac odl, an ffcn ool for applcaon n hp hydrolac analy. Th oband rul pon ou ha h ranfr funcon of hull conal forc n ca of ronan vbraon (prngng) ar uch hghr han n ronan hp oon. Vry good agrn bwn hp rpon drnd by hydrolac analy and rgd body analy n vcny of zro frquncy oband du o u of h conn rorng ffn. Th boh oluon convrg o zro, a frquncy approach zro valu. Th ud advancd ba odl of hp hull, bad on advancd hn-walld grdr hory wh ncludd har nflunc on oron and conrbuon of ranvr bulkhad o ffn, a raonabl choc for drnng wav load ffc. Howvr, r concnraon n hach cornr calculad drcly by h ba odl undrad. Th probl can b ovrco by applyng ubrucur approach,.. 3D FEM odl of ubrucur wh pod boundary condon fro ba rpon. In any ca, 3D FEM odl of copl hp prfrabl fro h vwpon of drnng r concnraon. In ordr o copl hydrolac analy of conanr hp and confr poranc for hp afy, ncary o procd furhr o hp oon calculaon n rrgular wav for dffrn a a, bad on h known ranfr funcon. Th nclud drnaon of global wav load,.. bndng and oronal on and hr convron no r, r concnraon n crcal ara of hp rucur, pcally n hach cornr du o rrand warpng, and fagu of rucural dal. A h nd of a copl nvgaon, whch alo ha o nclud odl and full-cal aurn, wll b pobl o dcd on h xn of

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