ON THE DISPLACEMENTS OF CONTAINER SHIP HULL GIRDER UNDER TORSION

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1 Radolav VZZ Unvery of pl FEB Ruñera Boškovća 1 pl Croaa Bože LZBT Unvery of pl Deparmen of rofeonal ude Lvanjka 5 1 pl Croaa Marko VUKOVĆ Unvery of pl FEB Ruñera Boškovća 1 pl Croaa ON THE DLCEMENT OF CONTNER H HULL GRDER UNDER TORON ummary n analycal approach o oron of hn-walled beam of open econ wh one plane of ymmery condered. The heory of oron of hn-walled beam of open econ wh nfluence of hear baed on he clacal Vlaov heory of hn-walled beam of open econ a well a he Umanky heory for cloed-open econ appled. The general ranvere load ac n he beam wall reduced o he momen of oron wh repec o he prncpal pole (oron/hear cenre) only. The beam wll be ubjeced o oron wh nfluence of hear wh repec o he prncpal pole and n addon o bendng due o hear n he horonal plane rough he prncpal pole. The obaned analycal expreon for dplacemen are appled n he analy of dplacemen of he modern conaner hp hull grder ubjeced o oron a well a n he paramerc analy of mple U econ. Comparon wh he fne elemen mehod by applyng hell elemen are provded. Key word: Theory of hn-walled beam; Toron; nfluence of hear; Open econ and cloed-open econ; nalyc; FEM O pomacma rupa konejnerkh brodova operećenh na uvjanje ažeak Ramaran je analčk prup uvjanju šapova ovorenog ovoreno-avorenog ankojenog prejeka jednom ravnnom merje. rmjenjena je eorja uvjanja šapova ovorenog ankojenog prejeka ujecajem mcanja na emeljma klačne eorje Vlaova a šapove ovorenog ankojenog prejeka e Umankog a šapove avoreno-ovorenog prejeka. Opće poprečno operećenje djeluje u jenama šapa reducrano na momene uvjanja u odnou na glavn pol (redše uvjanja/redše mcanja). Šap će b operećen na uvjanje ujecajem mcanja u odnou na glavn pol e dodano na avjanje bog mcanja u horonalnoj ravnn kro glavn pol. Dobven analčk ra uporebljen u u anal pomaka rupa modernog konejnerkog broda operećenog na uvjanje kao u paramearkoj anal jednoavnh U profla. Dana je uporedba meodom konačnh elemenaa ljukam elemenma. Ključne rječ: Teorja ankojenh šapova; uvjanje; ujecaj mcanja; ovoren avoreno- ovoren prejec; analka; MKE

2 1. nroducon n clacal heore of oron of hn-walled beam wh open cro-econ warpng of he cro-econ due o hear negleced [1]. nalogou o he advanced heore of bendng by an engneerng approach [5678] he concep of hear facor expanded o he analye of oron [ ]. n h paper analycal expreon for dplacemen of hn-walled beam of open cro-econ ubjeced o oron wh nfluence of hear are appled n paramerc analye of mple U-econ [11]. n he cae of cloed-open econ [11516] he analy gven for modern conaner hp hull grder cro-econ [161718]. Comparon wh he fne elemen mehod are provded.. ran and dplacemen The dplacemen of an arbrary pon ( x ) n he cae of oron of hn-walled beam of open econ wh one ax of ymmery can be expreed a dα dv u = y + γ d (1) α he angle of oron.e. he roaon of he cro-econ mddle lne a a rgd lne wh repec o a cro-econ pole ; = ( ) he ecoral coordnae wh repec o he pole ; v = v( x) he dplacemen of he pole n he y-drecon y = y( ) orhogonal coordnae; γ = γ ( x ) he hear ran n he mddle urface; he curvlnear coordnae of he mddle lne; ξ he angenal ax on he curvlnear coordnae ; Oxy he orhogonal coordnae yem he -ax he ax of ymmery; ϕ = ϕ( ) he angle beween he angen ξ and he y-ax (Fg1); = h d d = h d () = ( ) he ecoral coordnae for he pole ; h = h ( ) he dance of he angen hrough he arbrary pon a mddle lne from he pole ; ( = ) =. y ϕ O Eq. (1) may be wren a ξ Fg.1 Cro-econ coordnae lka 1. Koordnae poprečnog prejeka u = ϑ γ y + γ d () dα dv ϑ = γ = ; () ϑ = ϑ( x) he relave angular dplacemen of he mddle lne a he rgd lne wh repec o he pole and γ = γ ( x) angular dplacemen of he mddle lne a he rgd lne wh repec o he -ax. Thu aumed ha he mddle lne dplace n he longudnal drecon due o warpng a n he cae of ordnary heory of oron by he fr member of Eq. () and n

3 addon due o hear by he econd and hrd member of Eq. (). The dplacemen may be eparaed a α α α = + v = v (5) α = α ( x) he angular dplacemen of he cro-econ a plane econ wh repec o he pole a n he cae of clacal heore of hn-walled beam of open croecon a α = α ( ) and v = v ( x) are he addonal dplacemen due o hear. Then ϑ ϑ ϑ x = + γ = γ (6) dα dα dv ϑ = ϑ = γ = γ =. (7) The ran n he beam longudnal drecon may hen be expreed a u d α d v γ x x d y ξ ε = = + x. (8) x. ree and dplacemen By gnorng he normal ree n he ranvere drecon he Hooke low may be mplfed a σ Eε τ = Gγ (9) x = x E he modulu of elacy and G he hear modulu. Thu d α d v E τ σ x = E E y + d G. (1) x From he equlbrum of a dfferenal poron of he beam wall f τ = con. (11) x he followng oluon for he hear re can be wren E d v d α u τ = ( ) ( ) + (1) ( ) y d ( ) d = ; (1) = = ( ) he wall hckne. Eq. (1) may be rewren a E d v d u x α τ ξ = + = d d y = d d = d d (1) = d = d ; he curvlnear coordnae of he cu-off poron of he beam wall he free edge τ =.

4 aumed ha he normal re gven by Eq. (1) and hear ree gven by Eq. (1) and (1) are conan acro he wall hckne. ccordng o he aumpon ha croecon manan her hape durng deformaon he. Venan pure oron may be V V ncluded by a lnearly drbued componen τ V = τ ( x ) τ M V = η M = M ( x) he momen of pure oron; dα M = G = Gϑ (15) 1 = d (16) L o Thu he oal hear re τ o = τ ( x ) o V τ = τ + τ. (17). Equlbrum equaon From he equlbrum of a fne poron of he beam wall he followng equaon can be wren ( τ ) Fy co ϕ d x d ( τ ) dm = = M = hd m x + + = (18) x L L M% dm d = M% d = L x x L m ( ) = m x he momen of oron per un lengh. fer negrang by par by ubung Eq. (1) Eq. (18) become d v d α d v d α E + E = E + E m = (19) = y d = = y d = d () dm d α dϑ m = m + = m + G = m G (1) For he prncpal coordnae y and when = = () Eq. (19) become d v d x = E 5. nernal force and hear ree d α = m. () negraon of he hear re componen τ over he cro-econ gve τ coϕ d = M = τ h d () M = M( x) he ecoral momen of oron wh repec o he pole. ubuon of Eq. (1) no Eq. () gve

5 wren a d v = d α M = E. (5) Referrng o Eq. () and (5) can be wren dm = m. (6) Thu by ubung Eq.(5) no (1) he hear ree componen τ can fnally be τ M =. (7) 6. nernal force and normal ree negraon of he normal ree over he cro-econ gve B = σ x d σ x y d = (8) B he bmomen. By ubung Eq.(1) he followng equaon by paral negrang can be wren a d α d v B = E B E M = (9) E M = m d G L E B = m d G. () Referrng o Eq. (11)(5) (7) (9) and (9) he followng equaon may hen be wren d α db db d v dm E = + = M E = = (1) and accordng o Eq.() d α d B d B dm d v E = + = = m =. () Then accordng o Eq. (17) and (1) may be wren dm M = M + M m =. () Eq. (9) may be expreed a d α B κ d v κ m = y = m () E G GW κ d W = κ y = d (5) are he hear facor wh repec o he α -dplacemen and o he v-dplacemen durng α -dplacemen repecvely; h = d W = ; (6) h are he polar econd momen of area and he polar modulu of area repecvely; h = h ( ) 5

6 he dance beween he angen hrough he arbrary pon a mddle lne from he prncpal pole and h he dance of he angen hrough he arbrary arng pon M ( he prncpal coordnae equal o ero) from he prncpal pole. 7. Dfferenal equaon wh eparaed dplacemen ccordng o Eq. (5) Eq. () can be eparaed a follow d α B d α = κ = m. (7) E G ccordng o Eq. () and (7) one ha dα Mκ dv κ y = ϑ = = γ = M. (8) G GW aumed ha he angular dplacemen θ and γ do no depend on he boundary condon. The fr equaon of Eq. (7) he well known equaon of he clacal heory of oron of hn-walled beam; The econd of Eq. (7) and Eq. (8) ake no accoun he dplacemen due o hear. negrang Eq. (8) gve κ κ y α = B + Cα v = B + Cv (9) G GW C α and C v are he negraon conan. Eq. (9) can alo be wren a B B α = + Cα v = + Cv () G GW κ y W = Wy = (1) κ y are he hear polar econd momen of area and he hear polar modulu of area repecvely. Boundary condon wh repec o he hear can be defned a follow for he arng econ α = v =. () Hence referrng o () B Cα G B Cv = GW () y B he bmomen a x = x. The oal dplacemen hen are B B B B α = α + v =. (5) G GW For he hnged econ may be wren α α = α = ( α = α = ) x= x x= x x x = α = x x ( α = αb = x= x ) = B = B B y d α x= x d α = ( B = ) ; x= xb = ( B B = ). (6) 6

7 For he free econ: d α x= x 8. llurave example = ( B = ) d α x= x = ( M = ). (7) The range of example ha been carred ou by FEM ung uodek lgor mulaon ro n order o compare he reul wh hoe obaned analycally. hell elemen wh 5 DOF are ued. Meh wa generaed wh recangle elemen of h/ wdh (n longudnal drecon) and h/ hegh (n ranvere drecon). Tranvere daphragm of /1 hckne are modeled wh membrane elemen every h/ o preven he doron of he cro-econ. Due o ymmery only one half of he beam modeled (Fg. ). Fg. Boundary condon Fg. Rao of he angle of oron due o hear and he angle of oron by clacal heory of oron lka. Rubn uvje lka. Omjer kua uvjanja bog mcanja kua uvjanja po klačnoj eorj uvjanja Relavely hor beam wh l / h 1 condered. Fg. Dplacemen of he cro-econ mddle lne: a) due o angle of oron due o hear b) due o angle of oron due o hear and horonal dplacemen due o hear lka. omac rednje lnje poprečnog prejeka: a) bog kua uvjanja bog mcanja b) bog kua uvjanja bog mcanja horonalnog pomaka bog mcanja Fg. how he dplacemen of he cro-econ mddle lne due o hear for U- econ beam wh l=5h b=h and 1 = =h/ for he beam loaded by momen of oron per un lengh m n comparon wh FEM. hown analycal reul are n well agreemen wh hoe obaned by FEM. 7

8 mlar reul are obaned for U-econ beam wh l=5h b=15h and b=h. The angle of oron due o hear acheve value of 185% of α VL for b=h 161% of α VL for b=15h and 18% of α VL for b=h α VL he angle of oron obaned by clacal heory [1]. an example of a beam wh open-cloed cro-econ he conaner hp wh he followng characerc condered [19]: L=15 m; H=16 m; B=6 m; E=1 Ma; G=8769 Ma; ν=. The propere of he cro-econ are deermned by he program ekort [1]: = = mm ; h d = mm ; = 6 1 mm ; = mm ; W y = mm mm Fg. 5 how he dplacemen of he mdhp cro-econ due o hear. hp loaded 5 wh unformly drbued momen of oron per un lengh m = 9 1 N. Maxmal horonal dplacemen due o hear obaned by preened heory 5% of v VLmax and he ame dplacemen obaned by FEM 5% of v VLmax v VLmax maxmal horonal dplacemen obaned by he clacal heory [1]. Becaue of he addonal dplacemen of he prncpal pole v due o bendng n he horonal plane due o hear he dplacemen due o hear of a pon on he cenerlne very near one half of he boom hegh become (Fg. 5b) ( ) a noced n [16]. α h + h / + v d d Fg. 5 Dplacemen of he mdhp cro-econ: a) due o angle of oron due o hear b) due o angle of oron due o hear and horonal dplacemen due o hear lka 5. omac rednje lnje poprečnog prejeka broda: a) bog kua uvjanja bog mcanja b) bog kua uvjanja bog mcanja horonalnog pomaka bog mcanja 9. Concluon The heory of oron of hn-walled beam wh nfluence of hear for open econ wh one ax of ymmery appled n he analy of dplacemen of beam wh mple U-econ a well a conaner hp grder rucure. hear facor wh repec o he oron are gven n he analycal paramerc form n he cae of mple U-econ whle for he real hp rucure hey are obaned numercally by ung he approprae compuer program. 8

9 hown ha he beam wh he ngle ymmercal econ uch a U-econ or cloed open econ a conaner hp econ loaded o oron by couple n he croecon plane are alo ubjeced o bendng n he plane orhogonal o plane of ymmery. For modern conaner hp rucure he hear nfluence on dplacemen mall for real load condon. For conaner hp wh ngle de rucure h nfluence could be gnfcan. Comparon of he numercal reul wh he fne elemen mehod have hown an accepable agreemen for engneerng purpoe for early degn age procee. ppendx ropere of he U-econ Fg. 6. U-econ propere: a) mddle lne b) drbuon of funcon c) drbuon of funcon lka 6. Značajke U-prejeka: a) rednja lnja b) rapodjela funkcje c) rapodjela funkcje Drbuon of he acal momen of he cu-off poron of area (Fg.6): b b h : = ( h ) ; = ( h h ) ( h ) ( h ) ; b hb 1 b y : b 1 b = + 1 ; = ( h h ) h + h1. (8) hear facor accordng o Eq. (5) are [1]: 18ψ + ρ ( 1+ 6ψ ) 1 ( 5 6ψ ) ρ + κ y = ρ + ψ 1+ 6ψ ( )( ) ( ) ( ) 18ψ + ρ 1+ 6ψ 8 + 1ψ + 18ψ + ψρ κ = (9) 1ρ ( 1+ 6ψ ) ( + ψ ) b 1 = b1; = h; ψ = ; ρ =. 1 h Cloed-open cro-econ ecoral coordnae for he pole q = ( h )d (5) q hear flow n -h cloed conour obaned by olvng he yem of equaon: q qk = ; = d k k d 1 n (51) 9

10 T q = T = τ Gϑ q k hear flow n he branch ha belong boh o he -h nad k-h cloed conour and he area encloed by -h cloed conour. Then = d q and = = q = q y are acal momen of he cro-econ wh fcou cu and q are unknown hear flow obaned from he yem of Eq. (5) repecvely: y q and q y d Reference y qk = k k d d; q d qk = d; = k k [1] VLOV V. Z.: Thn-Walled Beam rael rogram for cenfc Tranlaon Ld d 1 n. (5) [] KOLLBRUNNER C. F. BLER K.: Toron n rucure prnger Heldeberg New York [] GJELVK.: The heory of hn-walled bar John Whley and on New York [] TMOHENKO.. MCCULLOUGH G. H.: Elemen and rengh of Maeral van Norand New York 199. [5] COWER G. R.: The hear coeffcen n Tmohenko beam heory Journal of ppled Mechanc [6] LKEY W. D.: naly and Degn of Elac Beam. Compuaonal Mehod John Wley & on New York. [7] EL FTM R. ZENZR H.: On he rucural behavour and he an Venan oluon n he exac beam heory Compuer and rucure Vol [8] VZZ R. BLGOJEVĆ B.: On he re drbuon n hn-walled beam ubjeced bendng wh nfluence of hear h nernaonal Congre of Croaan ocey of Mechanc epember [9] VZZ R.: nfluence of hear on oron of a hn-walled beam of open cro-econ (n Croaan) rojarvo 5: [1] ROBERT T. M. L-UBD H.: nfluence of hear deformaon on reraned oronal warpng of pulrued FR bar of open cro-econ Thn-Walled rucure [11] VZZ R.: Toron wh of hn-walled beam of open econ wh nfluence of hear nernaonal Journal of Mechancal cence [1] VZZ R.: nroducon o he analy of hn-walled beam (n Croaan) Kgen Zagreb 7. [1] MTOKOVĆ.: Bendng and oron of hn-walled beam of open econ wh nfluence of hear (n Croaan) hd The Faculy of Elecrcal Engneerng Mechancal Engneerng and Naval rchecure Unvery of pl pl 1. [1] LZBT B. MTOKOVĆ.: compuer program for calculang geomercal propere of ymmercal hn-walled rucure Tranacon of Famena [15] UMNK..: Kručenje gb ankoennjh avokonrukc GO Mokva 199. [16] ENJNOVĆ. TOMŠEVĆ. VLDMR N.: n advanced heory of hn-walled grder wh applcaon o hp vbraon Marne rucure [17] LZBT B.: The nfluence of doron of cro-econ o oron of hn-walled beam wh open and cloed-open econ (n Croaan) hd The Faculy of Elecrcal Engneerng mechancal Engneerng and Naval rchecure Unvery of pl pl 11. [18] VLDMR N.: Hdroelacy and fague rengh of large conaner hp (n Croaan) hd The Faculy of Mechancal Engneerng and Naval rchecure Unvery of Zagreb Zagreb 11. [19] URŠĆ J.: hp rengh (n Croaan) FB Zagreb