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Ultimate Longitudinal Strength Analysis of a DoubleSkin Tanker by Idealized Structural Unit MethodJ.K. Paik, PusanNationalUniversity,Pusan,Korea

ABSTRACT

In this study, midship section of adoubleskin tankerat 280,000 dwt isdesigoed.Inordertomake a systemsafetyassessmentof this hull girder in thelongitudinalstrength point of view,idealizedstructuralunitmethodis appliedsuchthattheprogressivecollapsebehaviourunder sagging or hogging bending moment isefficiently analyzed. For this purpose, anidealized plate elem~nt subjected to hiaxial.load is developed taking account of theinfluence of initial Mperfections as Well asthe interaction effect between local andglobal buckling in the StNCtUre; Aparametric. analysis with varying themagnitude of’ initial imperfections is carriedout. Based.on the results obtained here, asystem safety assessment is made using theconventional measure of safety such asreserve strength factor and safety index.Fromthe results of the present study, thefollowing conclusion is made:

i) the initial imperfections existing inthe local plate elempnt reduce the ultimatelongitudinal bending mnment of ship’s hullgirder,

ii) the double skin hull girder designedin this study is considered tn have asufficient safety in the longitudinalstrength point of view, and

iii) the present m&hnd is useful whenultimate collapse strength-based safetyassessmnt or optimization, is made in thestructural desigo stage for” a new type ofhull girder. ,“

INTRODUCTION

Recently, in nrder to minimizethe riskof ocean pollution due to cargo oil leakagein the event of accidents such as groundingor collision, the needs for construction of adouble skin type of hull, structure based on anewdesign concept are increasing in tankerindustry [l, 2]. AlSO, for achieving more

reliable and economicaldesign of structures,a system safety assessment which refers tothe redundancy of the overall structurebecomesof great interest [31. In general, theprocedure of system safety assessment ofstructures is split into two categories : oneis a cleterministic approach and another is

probabilistic one. The former measuresdeterministically the margin between theultimate capacity of tbe overall structureand the applied loads. In the latterapproach, reliability or probability offailure of the structure is evaluated, inwhich each design parameter is treated as arandom variable. For both approaches,however, the key issue is to obtain theultimate collapse strength of the overallstructure under external loading. Here, inorder to determine a more reasonable layoutof each structural member, structuraldesigyer is preferable to have understandingof detail progressive collapse history aswell as ultimate collpase capacity.

In usual, ship’s strength for the over-all structure is classified in threecategories such as longitudinal strength,transverse strength and torsional strength.Thus if designers are going to makea systemsafety assessment of ship’s hull girder inthe longitudinal strength point of view, thenthey should perform the ultimate longitudinalbending strength analysis of ship’s hullgirder subjected to sagging or hoggingbending moment.With the increasing of theexternal bending moment, hnwever, ship’s hullgirder eihibits hi@ly complicated nonlinearbehaviour associated with geometric andmaterial nonlinearity. Also since shipstructures are fabricated by heating processsuch as welding, flame cutting and sn on,initial imperfections in the form of initialdeflection and residual stress are alwayspresent in the structural members and thesemakestructural behaviour morecomplexone.

For the nonlinear analysis of structure,finite element method(FEM)is one of the mostpowerful approach, However, a directapplication of the conventional FEMto theultimate longitudinal strength analysis ofship’s hull girder which is very large andcomplexstructure is considered to be nearlyimpossible because a tremendous amount ofhuman labor and computational efforts isrequired. Therefore, many structuraldesigneri or analyzers have a great interestabout howto reduce the computingtime neededin the nonlinear analysis of structures whilegaining tbe reliablesolution. In thisrespect,Ueda and Raslied[41proposed an

H-1 .

efficient and accurate numerical method forthe nonlinear analysis of large sizestructures, namedidealized structural unitmethod(ISUM). The main idea behind thismethodologyis i~ modelling the objectivestructure to be analyzed by using very largesize structural element, named idealizedstructural element, comparing with theconventional finite element, which results inthe considerable reduction of thecomputational efforts because the number”ofunknownsin structural stiffness equation isgreatly decreased. In general, for theanalysis of a certain type of structure,various “kinds of the idealized structuralelement formulated in advanceare needed likein the conventional FEM.Usually each basicalstructural member composing the objectivestructure to be analyzed is chosen as anidealized structural element. By idealizingand formulating the nonlinear behaviour ofthe actual structural member, an idealizedstructural element is developed. Throughmanyapplication examplesof ISUMto various typesof steel structures. such as transverserings [4, 5], “upperdecks[6,71, double bottomstroctures[8-101, hull girder[ 11-131 andtubular off shore structures 14-211, it iswell knownthat ISUMis a very powerful anduseful metbodolo~ for the nonlinear analysisof large size steel structures.

In this study, midship section of adouble skin hull structure at 280,000 dwt isdesigned and in order to make an ultimatelongitudinal strength-based system safetyassessment of this hull girder, ISUMisapplied such that the progressive collapsebehaviour under sagging or hogging bendingmoment is efficiently analyzed. For thispurpose, an idealized plate element subjectedto biaxial load is formulated taking accountof geometric and material nonlinearity. Theinteraction effect betweenlocal and globalbuckling in the structure is taken intoconsideration. Also, tbe influence of initialimperfections in -the form of initialdeflection and residual stress on strengthand ‘stiffness of tbe structure is considered..A computer program, ALPS/ISUM[221 iscompletedbased on the present theory. Averification of the present theory is madecomptiring the present solution with theexisting numerical and experimental resultfor unit plate memberand welded box-girder.Thenthe present method is applied to theultimate longitudinal strength analysis ofthe example double skin hull girder. Aparametric analysis varying the magnitude ofinitial imperfections is carried out. Thereserve strength factor and”safety index areevalauted on the basis of the ultimatelongitudinal bending strength obtained here.

DESIGNOF MIDSHIPSECTIONOFADOUBLESKINHULLGIRDER

Fig. 1 Midshipsection of a 280Kdouble skinhull structure

the drauglt is 21 m. The ship has eight cargotanks having a hull form of double-skinnedbottom, side and deck. The longitudinalbulkhead located at center line and thetransverse bulkhead betweencargo tanks havealso a double-skinned structure. Thelongitudinal girder” is arranged in doubleskin deck and double bottom. Stringer isinstalled in double side structure and thelongitudinal bulkhead. In the present tanker,however, no floor members are arranged andall the plate membershave no stiffeners inthe longitudinal as well as transversedirection. The detail midship section of tbeship is shown in Fig. 1. The length of astandard cargo tarik is 24 m and transversespace is 5 m. The double bottom depth andside-skin width are 2.5 m and 3.5 m,respectively. For simplicity, all structuralmembers are considered to be built of 36HThigh-tensile steel whose yield stress is 36kghn’. The”section modulus of the presentscantling at midship section is checkedbeinggreater than the re,quiied one from theexisting role of classification societies. -The present double skin type of oil tauker isconsidered to have merits in prevention ofoil pollution, cargo oil handling,maintenance of cargo oil quality ~d so on.

MEASUREOFSYSTEM5ilFETY

Based on ‘the -’ultimate “longitudinalbending strength obtained in this study, ‘asystem safety assessnknt is made using twokinds of measureof safety such as reservestrength factor and safety index which aredefined as :

-.,....

.-

..

Midshipsection of a double skin tankerat 280,000 dwt is designed. Principalparticulars of the ship is 310 m in length,56.85.min breadth and 31 m in depth. Also

H-2

“liult7“-Gr (I)

.....

RJli -H.-% (2)/9= *1,* + S.s + S#

where, y :

d :

FL :

mmve tirengthfaotor●8fety indexmomn of theultinta longltudhallmmil~ strengthmoanof the totil externallmndiqmmnt (=ih+%)Mm of the ●till -tar kmndl~mmnt●m of the W8ve-hduoed Imndlngmonnt

&Ii ,S.,% : atm- devhthn of theulti-ute kmgltudlnd hmlhg etmngth,the sti11 inter Imdlm =oment aridthe wwe-induoed Lmnding mment,remotively

In the *kve equ8tl~., ii.lt la idoull-ted by ISUN formulatedIn thi~ study snd also% snd ~. me .mtimtecl by[23]

H. “= 213HiHw ,=2/,3Mu (3)

wham, )h “d HW H tlm d i 11 uater d thewme-lmhmd lmnding -nk estimated byuei~ .tbeaxlsting rule of ● olmoiflmthnsmiety, reuaotlvaly, in whloh AM rule[241iU employed@ the prment ●seament,

,“AIEomoh stanchrddevistlonim aaloulm-ted by using‘“omffioientof v8ri=tion(COV)as

Sul,t= Cavult ys, ❑,covp (4)*’ =CCIV. R ..

where, Covult, Cov, 8nd Cov” denotemef f loient8, of variation for’the u1tlutelongitudinal hndlng ntrength, the ati11water knding moment and tb” wwe-hduoedLmnding,~ent, rai~otively. In th prmentuulyula, ei.ohCOV 1s aseuwd to im 10S, Mand ~ fOr c~ult, Cov. Uld Cuvw,mgpotively.

THEORY OF ULTIMATS ~GITUDINAL BENDINGSTSQiG’fllMfiYSIS BYIsull

The ship hul1 girdar designedin thisrtudy mnsist~ of only plsta medmm in deok,Imt- snd -Me stmotu.m. Lan@tudlndglmier d ●l& strhgar am -1oo oomidemdto h plste mmlmr. Am lndiasted in Flg.2,eaoh plate memlw oompming the hul1 girderim trmted m unit elementwhioh is oalled mldasllzedplste elemnt.

In gerieral,tha plate alemnt in khipotmotureo 1s eubJeoted to in-plain adl=teral loads. But only lmplme Imdu meMen into wnmideration in the prmentetudy. The Iqltudiml oompman+on ortanslon by the soting of ~hg or h~ingkniding moment on the hull girder ia&vel~d in the elemnt. me M.ermtimmovement btween the ●dJuent plate elements

‘t

...Q,.,.~,,.

Fig. 2 Nodal forces and nodal displacementsof the idealized plate element

developsthe transversecompressionortension.Themagnitudeof theshearingforceis supposedtobe verysmallin the presentproblemthatthehull girderis under theactionofverticalbendingmoment such thatno occurence of shear buckling in plateelement is expected. Thus the dominantcomponentsof the external load acting on theplate element are considered to be biaxialloads, as shown in Fig.3. Tbe boundarycondition of the plate element is assumed tobe simply supported and the edges remainstraight even after the lateral deflection isoccured.

The initial imperfections in the form ofinitial deflection and welding residualstress are always present in real plateelement. The actual shape of the initialdeflection of the panel is very complex asshownin Fig. 3 but for simplicity two kindsof the simplified shape of the initialdeflection existing in the plate element areassumed as indicated in Fig. 4. Fig.4. aindicates the case that the shape of theinitial deflection matches the buckling modeof the plate, while a simplified form of“hungryhorse’s back” is given in Fig.4. b.The configuration of the initial deflectionis depicted by Fourier series function havingsufficient terms of the half-wave number.The distribution of the welding residualstress can be idealized as described inFig.3.

!

t

t B,—a,

Fig.3 Configuration of the rectangular plateelement

H-3

I J I>?? >,2

ti.. $,xws-. yn~n~

(b)

Fig.4 The “simplified shape of the initialdeflection existing in plate element

Tbera are several type of the 1iaitdste in stee1”etmatures but the pmmentstudy oonuldem only the ultieate 001 lapseetnngth mmrnlated with the large def leotionend yieldingof the plate, In addlthri, threebselo aeaumptlom era aede in thie utudy.Fimt, the dynmio IosdIng ie treated by theequivalent etatla one uuoh that the hullgirder 1s oonni&md to h mubjeatedto only

statia laadhg. %oond, the external Iosdeare pm@ional 1y applied. Third, theinfluenae of the lateral Ioade on thehhavlour of plste ●lenent itself 1s mtaooated for.

Fomdation d Id-l 1= d PlateElement Under~

fbiel Foros and Nmlal Dimlaoernentsof

Lba Elemn t. A: desorihd in the previousueotion,very large size platewemlmrwhiohis basioal structural.ooewnent in theetruoture iB ohosen m unit elenent in thepresent study, uee Fig.2. The prment etudystte~ptsto replaoe the defleotedplate withan equivalent fist pl-te whloh hoe zerodefleetion by evsluating the reduot ion ofin-planestiffneaudue to lnteml deflectionin -dvenoe.Thus rotationaldegree of freedomin the ele=ent hooeea to k removed. Thenthe hkviour of the prezsntplste element 1sexpressedby the ntil foroe veotor {R} end

the ridel displaomentthree degreesof freedompint of the ele~ent.

{R} = { Li RYi RmtRYS Rm Rx4

{u}= { U1 vi Wi UzU4 V4 w~ }T

veotor {U} havingSt eaoh oorner nods1

&z Ryz Rsz Rxs ,RY4 Rm4 }TV2 w% us Vs Wa

(5)

where, L, RY and R= lndioateaxialforaen inx, y and z dlmation and u, v end w denotediaplaoementu in x, y end z dimotion,mu~otively,

~e~ent re1stion, In order

to take ewount of the intermtion effepthtween lrnmlend overallbuoklingin theetruotum,.geometrianonlhear Mzavimr oftho plateelementshouldk ooneidemd,Thelnom~ental e~reeolon of the m 1stionahipbstwecnthe strainend displeoe~ent takingaaoountof the lme de”fleotion ef feot of theelement ie e=erged by the follwlng eatrlxfore.

{M’] = [BPl{AUi+ [Cl [G]{hu}+ i/2[ACl[GlMJ} (6)

where,{As}={hx AzY. AhY}T : imm~ent ofretrain oo~~nenta

$AU}={AuiAvt Awl Aue Ava AWSAus k Ays Au Av4 AW41T

[BPl{AU}={Au,s Av,y Au,Y+Av,X}r

[G] {AU}={AW,x Aw,~}To

[cl=c J

0’= V,y,yw,

Avereue Strnsa-Strain Relation fIkflmted Plate El-t in Elaatiow e. %the mal plmte ● lement, the initialdeflectionare alwaye pmuent. The in-planentiffness of the initiallY defIeotedpl-tedeom?ue?grsdusl’lyfreetheImginnlngas theoomp~ssive load lng-asee. In the pmeentatudy~ the .defleoted’p,lateiu mplaoed by anequivalent flit p,lste‘ whioh has “zerodeflootion.”For this purpse the mduot ion“ofthe in-pl~ Etiffnese due to the exietenoeof the laterul dafleation is agelytioallYevaIuated in““advanoe.

Fig.5 repmoent ❑ a typioa1 membranestmes dietributlon of defleotedplate underthe eotion of the axisl oonpmeaion. Sinoethe erMes of the eIoaent ie ooneidemd tore~in - atra”ight~displaoenente ,.intremveme dim”otion

& = / f%lwO o, b

u“/E.uyIw“O

6y = f IfYlxmo or .

v/E.ux Ix=o

the -verage Sxi*lthe lagitudinel andare oaloulatedby’

* = / (i/E.ffxl~O.. hm“rb)dx@ s f “(UE.Ul~r=Om. .or m)dy

whe’m lh ad “SY denote the average ax@ldi~plaaements in X“ and Y dimotlon,mnpotively. Also E 1s Young’smodulusand u10 Poisson’s ratio.

The lnteumt ion ~or the etmsm distri-bution in ~.17)”’”is”’fide ●lo~” the length orbmsdth of the&fined by

result is

(8)

where, u.=- and UY1* ire the nveregemembranestksaes of the efement in x tid Y dimation,mhfiot ivel y. Al go uxm=x and a~”ai denote theeeximi ■e~brene”~tmseea that em devolopedin the oorner of the deflkotedplate ● shawriii FiE.5, ., :

.. ......

.--. -,

H-4

I

Ytha plate Mae the aaae fora with that ofthe lniti81geoaetry.

~.,l--)’

E-——-- “-----------

E,L I~-,,!lr..-1

,- —-’ A-i--r. t

,—

Fig.5 Membranestressdistributionof tiledeflectedplate subjectedto aixalcompression

substituting Eq.(8) into Eq.(7) anddividing esoh tem in hth hand sidis ofM, (7) by length or breadth of the element,the average ■nbrme strainooa~nentB in xand y dlmotion - given a~ :

whom , Exav and EY.. denote the average

membrane atr.alnoompcmnt~ of the defleetedplate in x aid Y dimotlon, reupotively.

On the other hand, slnoe the aagnitudeof the ehearing foma aotlng on the plateelement’“i!ndaroonalderat Ion 1s very mal 1 andthe Ehazrhg defomat ion is mp~md to k1hear;’the’ average shearing strain of theelment madu

7XY.T = 2( l+v)/E‘tXy~V (lo)

where, )“XyfiT and mv~w am the average shearstrainand atrms, respsotlvely.

The aaxiaua membrane strmaea aim.. andU=-X In ~. (9) oan Im oaloulatedby solvingthe governi~ -squation of the pldo e leaenttakhg amount of the Mtial l~pmfeotiona,Aa ~entioned in the previom eeation, olnoethe doninant loadi~ mm~nent- aotlng on theplate ele”aent‘ am biaxial lad and themoumnoe of the ~heti buokl1~ is note-ted, the governing equation‘of theelemnt whioh lB e“ubJeotmd-to the blaxlsll~d IIJdlmot-ly solved in this study. Theuhapm of tha lriitial&f leotion.of the plateele~nt takns the follining fora .ualngFourier ●erleu funotlon.

“wo=~ Wom ain(mnx/a)ein(Wfi) (11)

where, the mbmript zero lndloatee the.hit ltilmnf l~t lon ind W.. mpreaenta theknoun ~ffioient for the‘initialdefleotlon.AI SO, the ‘Aa~ Of theadded def Ieotion of

-x V= ❑in(amda)aln(w/b) (12)

where, W. indioatea the unknown ooffiolentfor the added def1eotion.

It 1s known that the l-e deflectionlmhavlour of the plate element iu aalnlygoverned by the hal f+ve oqmnent for the

buokl i- ade[6, 251, Thus, the other terns inEqa.(11) and (12) exoept for the buoklingmde oompnent UY b removed, Themsf om, thepmaent study se leota only the buokl ing ~mieooa~nent as the defleot ion paraaeter whensolving the governing equation so that theauaaation syakl in Eqa.(11) and (12) 1Sdisappamd.

wo = WO ein(tia)ein(wy/b)w = W ~in(am/a)aln(ny/b) (13)

where, m 1s redefined and indioatea thehalfave nuatmr aatohing with the buoklingtie of the ele~ent,in whloh the subsoriptm●ttaohed in eaoh ooeffiolentwm oaltted.

The half+ave nulmr E in Eq.(13) la de-~ndent on the au~ot ratio am we11 as theratio of the biaxial loadingmn~nents auohthat it MY k altered if the loadingratioia ohangad in the subsequentloading❑tep andthen in deteralnedas an integerthat shouldb mtiofied the fol lowlngorlterlon [26-261.

(m*b*/a~+l)~/{m~b~ /azwy../r}.r}<{(n+l)~b*/a*+l}z/{(~+l)bz/a*+oy.~/u...}

(14)By applylng Galerkin methml for the

●olving of the equlIibriuaand oo~patlbi1ityequationof the plate elenent,the followingequatlon with tigard to the unknown-ffioient of the added deflmtlon iae~erged [6,27].

ciw3+csfl+caw+c4=o (15)

vhem, Cl = E(m%r/a4 + u=/b4)C* = 3woci

Cs = 2WOzCi + 16u2D/t(~z/a2+ l/bz)z- 16{n2(ox.=+ ur.x)/a*

+ (dya-~roy)/bz }C4= - lSWO{#(Ux~~ + up.x)/az

+ (uya-+ urwy)/b2}D = Ets/12(1-v~)

Uroy and U..V h Eq. (15) lndloatea theeffeotive ooapmmlve reEidual ❑tmaaea ofthe plate element in x and Y dlmotlon,ms~otively. If ths aotual oompmeaivemaidual stmmes in x emd Y direotlon that

~ pmaent in tba Elddle plate aa shown inFiE.3 are denoted by drx ~d Ury,mamot ive1Y, then the ef feetive ooapmus lvemnidual ntmaaeB are definedby [6,271

Ui-. z = Urx{l_O. tirx/(00 + Urx)l

U.. y = Ury{l-O. tiry/(UO + dry)}

(16)

H-5

where, uo indioatesthe yield st~ss of themsteria1.

,,

q. (15) 1s explicitly solvedby US@C~O’ a method and lf the unknownomffloiant W ia given, then ■ewbrene stremdistributionof the element 1S pmoiselyobtained. The isximm and dnime stremeaare ‘ given by the following explioitfom[6,27].

umax ■ o.5(u=.x*-.=I n) CCS(2m=t/b)+ o,5(uxwx*4uxmir4)

Upax ■ o. 5(uyw**mim) cm(2rqYt/u)+ .o.’5(uym Fwmlm) (17)

usmi m=VrmTWrX-e%2/8az <E.W. (W + 2WO)UmImWva.wrr rz/8b2,E.W.(W + 2VO)

(1s)

where, a%=,.mWX~W+m2Uz/8=Z.E ,V. (W+2WO)UnmX*SOYaT+ ua/8bz,E.W.(W+2WO)rlx = u,=/(ao + u,=) .b/2tllv = %d(uo ● U.Y) .s/2t

where, aubaorlpt mea end aln den&e. themsxlmm end dnhum ntmises, rea~otivsly,

Subat~kutl~ Eq. (17) into Eq. (9) endtaking SqB. (9) and (10) by the inormentalfore, the f,ol,lwi.~ relationship btweenwemge utrermlnorment and 8varege straininorment of the drifleated plate element iFJa~rged,

{k} = [D]s{A&} (19)

where, {k} ={AE} ❑

[D]s =

elaetio range

{b... My.. ATXY..}T{h?...A.%..4XY=.}Tthe strem-atraln eatrix in the

c:: :“7-’k O e4J

(ooz..dwxm.) /E,(6uxmx/wyav-v)/E(euysm./eox.v-u)/E(MYSnXleOyay)/E,2(l+u)/E

Thus, by using [Dl~ given in Eq.’(l9),the defleoted plate element is mplaaed bythe equivalent flat plste element whioh hmzero deflkotion,

~antisl El@io Stlffneaa HetrLZ. Byspplyhg the prinoiple of the virtual work,the .tengentialelmtio stiffness matrix ofthe pl~te elenent is derived. During theaoting of the virtual ,dieplaoement~thefollwing ●quilibrium equation ❑hould bsatisfied.

The left@nd ulde of theahva equationlndlo=te~thework@oneby theetiernelforoeend the righthand side1s tbe strainonorgytitomd in the element while the v~rtu-1

dlsplaoe~ent is applied, Also, J-(.)dvolrepresents the lnte~mtion for the entire

voluaa of the element end prefix 6 denotesthe mount due to the virtual diaplanenent.By aaklng Eq,(6) s dlff”erentiationwithrespeot to the dinplaoe~e”nkmn~nentu, thevirtual etmin S{AZ} down Ln Eq,(20) isgiven by

6{AEI ❑ [BP]6{AUI+[C+AC][G]6{AU}(21)

By mbntltuting Eqn,(19) end (21) intoEq.(20) end remving the unbzlenoe fomewhioh i. due to the dleom@noy bstween theexterm 1 fome end the internal fome, thetangentialalmtlo etlffnem equetlonof thedefleoted plate eleaent 1s given by thefollowingfern.

{AR} = [K]‘{AU} (22)

wherm, [K]~ io the tagenti-1 elmtio@tlffneee matrix of the element. Also, thetangential elmrklo etiffness eatrix [1]E inEq,(22) .i~ given “by spplylng u@eted~sJ@en appmaoh.

txlg = f~ [~]r[D]$[BP]dvol+ fF[G]7[uh][G]tkol (23)

atrese-average

.,Ftiot1g; In the present

study, the plate”hha$loitr is expmeeed byUHing three degreesof freedoa●t.eeoh omnerride1.mint. Moreoversinoe the equivalentflatplate ele~entia e-ployedinsteadOf the

defleotedplate, the dlspluoenentfunotionofin-planeSS we11 aa out-of-plsnedefo~t iontekea the following1inem fore.

u = al +aax+asy+a4xy+W2( b2-y2 ) (24.a)v = b’+bnx+bay+bdxy+sJ2(a~-xa) (24.b)w = o’ +onX+oaY+04W (24.0)

where, omfflolents a’, a2,..# are the”unknwn oonatenta whioh. are e~rassed interns of. the nmlal,,displmemente. AlEOIMS. (24a) end (24.b)lndiakte the in-plane.dieplaoe~entfumt ion, h whloh the lsat tamat eeoh right hand ulde .1s added auoh thstthe deering strain inside the.plate ele~entramine uniform %, (24.o) repmae;tu theout-f-plane dhpl ~oement fvnot ion. Aword-Ingly, ❑ubotltuting Eq..(24)-intoEq.(6) endperforatingthe integr+tion of Eq.(23) for the ,entire voluae of the plate ele~ent, .eaohooawnent of the ●t!ffnem eatrix wi11 @oblmined, 1,

,.Cri rbn of the Ele-

~. With the lnommi~of the erternalload, the region where the etreea leveI is

hi~ kooees to b yielded.Aa a,mault, thein-plane atiffneeaof ,~he elament deoreaaeafurther and than the. e lewent maoheu theultimate 1imit et,ate.,me membrane stress

H’6

dlstrlbut ion of the deflegted plate elementis mn-uniform end aime tho present pldeelement ie do~inantly mbJeoted to biaxialload, the mrmr or the middle mint of., the.

edges in tho longitudinal or transversedireotion is likely b h ylel&d. If theoorner where the aaximm atmm oon~nent isexiutadkaoees to h yielded,then the plateelement oan not oarry the further lnormeinsof the external load,Aleo elnoe the edges ofthe plsta element remain etmight, thetenaile membrane utmss MY develop in themid+dge of the plate depmdhg on thsloading ratio in the long~tudinalendtmnevereedireotion.As longse the regionwhere tenslle ❑tress.aote is under elast10mndltlon, the plste oen mslet againstabrupt inoreaee of the lsteral deflootioneven though tzone ~rtlon inside of theeleeent has almsady ylalded. HMver, if thetensile pert le yielded then the platee 1ement 001 lap~ss lmmadiate 1Y. Therefore,theultlnte limit state of the plate element isjudged whether or not the oorner or theaid-edgehaoms to k yielded in the presentetudy. Aooordlngly, mbst ltuting the membraneatreas ooapnente at emh oheoking ~intinto Mlsaa’a yielding - oondltion, thefol lowingfomul-aem e~erged,

where, dx=..,UWU : Eq.(17)

u==im, @mln :,%.(18)‘r(-xvmv) ~ : the average shearing

etrema

Also, index nuebr lndlo-taethe plaati-0lky oheoklng~int. Thus, lf any one vmlueof r ,givenin !lq.(25) ie equal to or greater.then zero, then it ia .oomldezwd that th~plete eleeent~has 001lapeed.

~st-U.ltieste Stiffneas MetrI& In tie~at-ulil~te .renge:of the ,6lement, theaverage internal stmmoea follw theunl-ding path even though the deforaationinorppsee oontlnuomly. In gen~ral, it ie●smmed that the m.gni tude of_ the etreeaooa~ents it the yielded ~int lU not

ohansed ao seriouslY. The, when anY one ofplaetloi,ty,,oheokhzgpint .10 yielded, that1s, the el,e~nt ml lapeed, the eaxlmuB■embrme etrese ooa~nentn -t the yieldedpdnt am aup~eed to- maain oonatant in thesubsequent inoreaeing of, the defomet ion.Just after the elenent reeohee tb ultiaete1imit sta$e, the =en.&ene ●treea oommnentaat the yieldedpint era nqerioal lY given by

where, eupmorlpt u indioateathe value Just●ftor the e1ement m 11apeed~

The average =edn-enestreseoompnentaof the a1●meritin the pst-ul ti~ate ra~ti

era given by ueing the oonoept of theeffeotivevidth[271i

UX=V = bdb .ox=~x”UYLV = ada .aY9~Xu (27)

where, b. end s. repmaent the offootivewidths in the 1ongitudina1 end tranevereedireotion,reepeotively,

With the imrmning of the deformation,the of foot ive widths oont inueously deorea~e

even in the ~t-ult hate range. Hem, thereduot ion tendamy of the effeet ive widths inthe Iwt-ult iaete range la auppaued to bename with the pm-ult hate range, The, theeffeotive wldthn mad

bdb = ux~.=la-..=aO/a = av~ rwlun~xg (2a)

where, uxaT*, ayav* : the virtual average■embrane etmeeee

~=mfi=*,~max, : t~ virt~l -i,~

MembraneatreBeeaAlso, the asterlakhdloatee that the streeslEI● virtualone in the Wut-ultlzate range.

Sinoe the large defomat ion hae henmoumd, the aagnltude of the effeotivewidth is oonaldemd to b very ma 11 auohthat.the lnfluenoeof the initial deflection16 amumd to h not ao aerloue in the~et-ultlmate range of the element[291. Foruiapl ioity, therefore, the virtual averagemembrene~tmos lzi●xpmmed in terne of thevirtua 1 aaxiaum neabrene et rem that dma notaooount for the lnfluenoe of the initialdefleotion[28],that is,

0. . . . =.,b. ~ .,,, ‘s’”O’””’” - “’”-”” “ ‘*ba - ‘=bsl

OZ.. * = .,5.: -,b, ‘-b’ “=””’” “ “1 ‘r”””” “ ‘y”’ - *’”]

b,=mbj=l -- ,

h, . 2m4bz/*4 . llbn +[+ -+)’

(2Q)

Shoe the edges of the elementreeainetralsht, the virtual aaxlmm membraneetmeeee are oaloulmtedby ua~ng th{retrainaa

Uxmnx* ❑ E/( l-u*)( CX.T+PCY.T)u=.=. ❑ E/(l+*)(lJcxaT+CY.T)

(30)

By eubgtltuting Eqa. (29) endEq. (2B), the effeotive widths

average

30) intoin the

~st-ult hate range em expreeeedin teme ofthe ave~age etmln oonponent. Aleo❑ulmtltuting the effeotive widthR intoM. (27). tk average etmee-average strainrelation in the Pet-u ltleate range le●xpli”oitlygiven,that ie,

H-7

lh the other hind, ●inoa the aoumnoa”of the dmmr huokllngis not qted in the

pmmnt elamnt, the s.heml~ 8tiffmm ofti element i- eumed to b fullyOffmtiveeven in the ~t*lti=t* ~. Thun, theeue relation btveen the slmaring stmm endetmln of Sq, ( 10) 1- employed, TakingEqa. (31) and ( 10) by tlm lmmMlrtml fore,the fol lwhg stm-~tmin mlatla in the~t-lt lute F-O 18 en-d.

{MI = [Dl”{M} (32)

v~m, [D] u in tb ~t-lt lmeta strem-etrainUtrsx.

“cli&o

■ &&oo&

Conmquent 1y, the ~st-ult lute St lff-mna metrix of the elnmnt iB obtained by

rnpltiing [DIE in ~. (29) with [D]” in

M. (32). tin,

[lIu=J-IBPI 7[DIU[BFIdvol●IwIGIWn] [G]dvol (33)

where, [I]u 10 the pmt-ultimeta wtlffnennmtrix of the element.

CHRACTERISHCS OP THE PSESENT IDEALIZEDPLATE~T

A oomputnr p~, ALW13UM [22] iaoomplated on the bis Of the theory&morihd in the prevloue ●ution. Sinm theMLbjeot 0011,i&Md brill@ Up a blgblynod in- proble~, ul inommntilete~by-etep ~dum Immd on the upietedhgrangien mppromh 1s applied.The MOWMYend effiolemy of the ~ lB verlf~ed~ing with the pmeent mlution end tkexletl~ e~rimtml end numrionl meultfor unit plde memhr ead ml&d kx-gl~rin thh ■mtlon.

Uede ●t.al. [301 mnduotid ultimte■tmngth mrie~ temtm for th Mlded qmplaten 9ubjmted to Uniexisl ampmmlon.Teble 1 mpmeents tlm ~td 1- of the-Ohll. Time klnde of pl~te thlokmm thstH 4. 5m(A~woinn), 9.~(B-m~lrnn) -d12. 7m(C-~aimn) uem teatd, The bmdmymnditiom of thp epolmn in the e~rimnt1- ●et to IM simply eu~ed. Tb megnitudeof the wlding ampmssln msiduel etresafor emh ~lmn on uhloh Ueds”~ mmumd

..2C -i /

●

,-

‘:$p.mt solutionFig.6‘ Comparison with the prese

and Ueda’s experimental result for

squareplatesunderuniaxial compre-

ssion1,! .,

. . ..

&tm 1- heed i- demribd in Teble1. Emoh~ohen 1- tilled by wing Jut oneldmllmd pl~te elemnt fomulated in tbepmwnt etudy d t~ total numlmm of therxxhl mint md tbe &greec of fmeda efterthe reetrtlnt oonditlon is inttiuoed m S11fmr. A ooaprleon btwen tbe ~lmntilmmlt md * pmment mlutiom i- ude inTeble.1 end Fig.S. It.1- oleer’thmt thepmmnt. ●olutlon 1s in :- egmemmt withIkdX’● ~lmntil “meults. Alm minm theoomuting the mqulmd h the Prewnt.mlyds wm mkt 20 moonde by uehg IBMFWAT atiputer, tlm..Pm*ant Mtbai i~mmldemd IxI km vem effkoient for thepmeihel uee. :..,

of seaQIKu14r Plmtee

In”omlm to lnveetlgetetlm lnflueme oftherepot ratio on the :ultimte ‘wtmngth ofthe plate ●lment, “)’!elmt~l-tlo lo&fleotion endyd~ of inlthl lY &fleatedmotmgwler plwte~ 1* ~rfomed by applyingthe nonl.lmm finite elamnt methd devel~by the euthor[3il:“Thee mnult- em tlmn~pamd with tb” mlution obt-imd by thepmeent methcul. T~ble 2 “ldioetos thedimmlom of the p14tes. The uxlmemmgnltudeof the lnitid defleotlonis eet toh 10 ~-nt of the plate tkiokmm. Thechew of the inltitl defleotlon of themntenguler plstea~!ms snwmed ae dmwa inFig.4.b. m tiflgumtion “of thl! initialdeflation is depioted by Fumier mrie~funot ton W’ing 16 tam-. k “rentiomdurlier, the pmtimt ‘tlnmryeeleote”only thehuokl1X ~de o~nt u the def leot ion

H-8

Table 1

--.,,,

kdmll

b.

A-1A-2A-3A-4A-5A-6A-7A-8A-9A -10A -11A -12A -13A -14A -15An-16A -17A -18B-1B-2B-3B-4B-5B-6B-7B-8B-9B -10B -11,B -12B -13B -14B -15B -16B -17B -18B -19B-B +1c-1c-2c-3c-4c-5C-6C-7C-8c-9c -loc -11c -12c -13c -14c -15

t

G4,504.w4.w4.B4.%4.w4.w4.504.w4.w4.m4,344.344*344.34.344*34Leael,m8.~9.m8.M8.M8.M8*M8.209.W9*m8.~8.M8.~Q.m8.858.658.838.ma,m8.m2.M2.m2.m2.M2.W3,=3.m

;:2,913*m2*m3.m2.m2. w

*

:Iq6#)28.49B. 56assam25.8726.372&0726.45S.se26.862a,m2s. 16a.olZLol=.4223.M2L05ZL8731.a31. m31.M23.7131.W31.273.91m, 2132.1128.6325.07m. smmm. 5225.3925.60%. Ill26.2a25,652s.5525.5525.m25.m25.W25.a3W. 1626.5226.51m.4626.34m.3226.47ma26.32m 153.26

Wt

111.11111.11111.11111-.11111.11111.11111.11111.11111.11111.11111.11111.11115.21115.21115.21115.21115.21115.21w.5256.6256.8255.=56.82M.82%.8256.5256.8256.W55.w%.62S.82%.8255.sSaw55.0756.6357.4756.8257.4736*m39a36.=36.s38.7a37.563?.5638.7637.5938.76n.5638.763?.6638*7a38.76

kwt

O.mO.mO.aO.aO.m0.240.270.310.s0,360,89l.m0.=0.521,071.2a1.311.74O.wO.mO.wO.mo.%0.310.310.380.61o*M0.720,730.750,820.64O.mO.a0.480.460.71o*nO.mO.wO.mO.mO.mO.mo.B0.250.260.s0.420,43O*430.520.53

*=

1#*)

2.a2.a2.m2.m2.m2.m2.m2.m2.m2*m2.m2.m5.W5.mS.m5.m5.m5.m2.m2.m2.m2.m2.m2.m2,m2.m2.m2.m2.W2*a2.m2*m2.m5.a5.m5.aJ5.WS.m5.m2.a2.W2.m2.m2.m2.W2.m2.m2.m2.m2.m2.m2.m2.W2.m

EXP.

Uu

11.1111.w10.m11.M11.W11,*11.5111.1111.2610.m10.m9.566,667.127.W7.817.2a7.8S27.m%.5228.89W.4021.73=.14=.lam.a19.56law16.~18,3la.1817.m14.6a18.~18.9113.5514.3414.0511.4624.1924.84a.5625.a32s,86n.el24.!m24.6724*aU 5aZ1.mB.7821.11Z2.31ama

-U-

0.360.450.410.460.450,440.410.420.420.410,30.380.320.32O,m0.340.35O.m0.870.850.a0.860.5d0,740,750.730.610.630.670.600.mo.5a0.560.mo.mo.Ho.Mo*550.450,w0.960.6al,mo.wo.al0.820.810.820.810.740.760.720.740.76

THE.

%

12.7612.1011.M11.671L6711,5112.1511.4211.5611.4611.189,647m&7.W6,846.465.756.0124.4a24.46iM.48u.12m.7419.8319.66la.5217,4415.3219.2115.6815.M15.3212.m19.3E19.25lz.1512.8511.m“i3.26=.21n.13m,13m.m28.1424.66.~.8124.25a.s22.25mm6521,25m.6119.6919.69

WO

0.450.460.450.450.450.440.430.430.430.420.400,360,340.320.260.2a0.270.Bo.m0.780.760.840.650.830.M0.610.Ho.a0.530.520.520.wo.m0.760.720.520.w0.440.520.620.620.920.920.a0.u0.810.mo.m0.730.710.mo.m0.650.65

....

C.35

w%%ah b t B m ‘(%) - % L *(* y

(d (d (la UP) m Am. .

1 m 10 w 27.46 m. m 2%21 5x5 1.= a.o2 m 10 s m.m m.m 32.S7 10X5 0.- 4s,33 m 10 s 26.M 31.6E 31.41 15%s 0.= 72.44 m 10 33 29.m =.34 32.61 WX5 0.= lm.55 m 10 3 2e.m 31.67 32.U 25x5 0.m14 141.1

Fig.7 comparisonwith thepresent”solutionandFEM resultforrectangularplatesunderuniaxialcompression

pma.meter. Table 2 givesthe buaklhg ■mleoo~pmant of the initlal deflqotion for eaohplate. The yield atmm of the mtarinl isamumed to Im 36 kg/u2. end Young’a ■mlulusla 21~ kg/u2. The .mlding raaiduel utreaaim not present, In FE analysis, due to thegeometriouyuetry,m quu-terof theplateis■del led,The meshsizenumk.rvaryings~psotratio lE given in Table 2, in whioh thereo~ 1ar p1●te e1ement hwing four oornernodsl pints ie employed.Slnoe the unlosdingedges of the a~olmen, have no reutraintgtothe in-plum ■ovement, it ia emoted thstthe U1tlaate utrength 10 undereatlasted byakut 10 pxuent leas than the ome that theedgee reasin ●traight. Thus, Table 2 ●laooomparm with the preeant ❑olution and theoorreotedultlmte strmgth by addhg UP 10~maent, hoauee the ideallzod plmteelsmentma formulatedunderthe wnsideration thatthe edgesreasinatrtight. Table2 givesthe ~pmiaon lmhwen the aolutlonaby F’Dlandp-sent ntbd. Fig.7 ●lmwatbeo-e oftheultimte ❑trengthwithvaryingthesawotratlo of tbe plate.Aoouraoyof the ‘pmnentmlution la auffloient mparing with ~

raault. ~ oo~putititimes required in FEand yaia by ual~ CYEEi 932/31 oomputer -given in Table 2, while the Prmmnt analyaiameded only 20 aeoondaby ISM PC/AT oomputer,It is obviousthst the presentmethmilavery*ffioient,

0,0 Y0.0 0,4 ‘o 8“ ,.2 1.6 2,0

Fig.8

!0

oe

u/uo

The load-displacement behaviour of asquare plate under uniaxial displace-ment obtained,, by the present methodwithvarying the magnitude of theinitial deflection

1 0xbx+=500x 500*6, 9[mh,

& . 36ta&,#, E . zloookg{~~/.

W+= 0.c5 &.:’o

i : “0’1’”::;/\4°/

Fig.9 The load-displacement behaviour of asquare plate under uniaxial displace-ment obtained by the present methodh’ith,vaving the magnitude of theresidual stress

‘.._..

.......

,,----

H-10

l?9Rt-fJltlute Edaviour of swam PIlteia1 coelmeissi~

In order to deaomztmte the .*PP1iosbi-llty of the present theom in the

peat-ulthzte range, ~t-ultlaate tmhaviourof simply sup~rted square platen eubjoatedto uniaxial 00~pm&lBiOIS is analyzed. Fig.8and 9 ~apreaentthe 1osd-~horteningourve foran iqmfeet square plate ❑ubJeqted to

uniaxlsldlsplaoezent.Hem, Flg.8 and 9 amrmulte obtsined with varying the azgoltudeof the lnitisl deflection and weldingresidual stress, rea~otlvely. It 1s olearthat the present ~ethui gives s rmaomble❑olution even in the pat-ult hate range ofthe ele~ent.

Go1lame Strer@h of Welded Box-colu.nq

Umai et,al. [32] oonduoted ultiaateetmngth series tests for w lded hx-ooluanm in order to inventigate theintermtlon effeot lmtmen lmal and ‘overal1buokllng of the struoture,Fig,10 ahwe theconfiguration of the s~i~en and theirdetai1 dimemions am given in Table3, Amongtheme,S-aerieeindioateuthatthe sham oflmx aaotionof tha spmlmen is nearly mquare,while R-serleempre~entu tbe s~oi~en hsvingthe motangulzr sham of the ukotign. Also,shorterwwoimen siuohm S-10 andR-10serieshas only end-dlaphrus whioh era inatalled inLmth ende of the ❑truotum but longerepeoimen has tm more inni&4isphrazu inaddition to and-chphrua au Awn in Fig.10.The lmx oolmm are mnpaad of thin plateelements and eaoh pltta eleaent is imlelledby am ideslized pllte eleaent.Hem, disp~uIs nluo adelled by an ldetilizedplateelement.Aa Sk in Fig.10, the end-diaphruis lmsted alightly imide fm~ ‘bth ende andfor the si~plioity of the mmlel 11- the platefragmanta outside of the end-diaphraz iBnagleated in the pmaent analyeis. Thelmmdary and loading oomllt’ion era sham inFig. 10. The losding is ino~mtally appliedby ● dimplaoeaent oonttil, A “oompariaonhtween the ultimate atmngth obtzined bye~iaent and preoent analyslu im aade in.Table.3~d -Fig.il, It in observedthat thepresent oolution ie in ~ agmezent withthe a~.riaentsi mault.. The oonputing .timarequired,in the preue.ntanalyaiu was about 30seoonda for ahotier 001umn and 5 minutes forlongerool.~ by using CYBER 932/31 ooaputer.

APPLICATIONOF THE ~T METHOD TO OSJIW-TIV2 lWBLE sKIN NULL GIRDER

Struotural Hmlel1lng

Fig.12 shown a stmotural edellingusing the ,idealizedplate element. A halflangth of ona oargo hold ImstA” at.~idehip..is ohoeen an the extent of the analysis,Also boauue of the ayzaetry with.regardtothe oenter 1he, a half breadth of the hul1girder lamielled. Fig, 13 mp~eants out1i.neof the kundary and loading oondition. Thelongltudlnsldieplaoe=nt along the left-end

Fig.10 Usami’stest set-upforweldedbox-column

1.20 -

1

o : 10 SariesA : 35 series

50 %ries

/

ox● 65 Seties o

x

0.0+CM K 1 20

&;,’Pp (E~P.)

Fig.11 Comparisonwith presentsolution andUsami’s experimental result for

weldedbox-column

of tha hold is metmined. The uyuetriooondition is intduoed -t the oenter 1lne.The bottm -t the loading position is fixedt.mdzrdthe dapth dizwotlon. Tha vertioalhnding no~ent is generated by thediaplam~ent oontrol with ma-t to thaneutra1 axle auoh that the orom-seot ion ofthe hul1 at the loadingpeitlon kaepsplanestate, The neutral axle in Waltionad ●t13,016 ~’ almve the ktto~ keel as shmm inFig.1. Seoauae of the notion of the vertloalhnding eoeent, if the up-r deok panel orhttoe plmtl~ that in nubJeated to axialooapmaaion 10 yielded in the earller etagethan the side -hell plating,the ~sition ofthe neutml axia ie emoted to h novedt- the op~sita part that ie under thetamile loading.Au a meult, the side ohel1plating contributes to eu~tain the furtherinomaeing of the extersul lasd instead ofthe fd led deok or Imttoe uubstruotureendthen. ● alight inomaae of the oaloulstad loadoarwing oapaoity lB a~oted. Aooording tothe maeamh’ result of Raf.[33],hver, theInfluanos of the ohange of neutral axle~aition on the ultiazte Iongitudlnslstrengthof the hul1 girder iu not no aariousand thuu for aimpl loity the present study~tulsteo thst the pmit ion of the neutralaxlu 1s not o-d evan after lmal failureszm wo~d, whioh maultn in the allghtundematiaztion of the ultiaate longitudinalotmn@h of the hul1 girder. Tha totalnuahre of ldesllzed plate eleeentsand ndalpints employed in the pmeent analyelm are

H-11

Y(m)

iii181217M263151

217263

1511622162472a31511812171s1181217247263151

181

217

m-(-)

246.5w. 5m. 53!0.O267.0M.o

%0.5263,0

215.5

%9.5265.5266.5163.5m.5246.0165.5197.0m.5240.0m. 5132.5

171.5

191.5

.

KEiiio. m0.7=0.6570,576l.m

o, 6Z?0.M6

0.M2

O.m0.6210.w0,7400.6720,6m0.7430.731O.m0.6360.5760.w

0.637

0.565

THL

W3n)

m5.5m. az61.7269.3=.5m. 1~. 6*m. 1“316.7ml,wm. 5m. 12M,0.=.8273.9ml.9a. 9264.s.165.4~.am,4255.1m. 2165.5167.8-211.5”

178,8=

s. 5

196. 1*

KIiiFD. ~0,763D.6760.6120.916l*OIW0.mO.ZC”0.6h0.816

0.w0.6150.v0.814,0.nl0.7120.6320.7720.7120.676,0.6Mo*mo.N*o.76s0.,=”O.mO.*

*iwb,. 2)*ls-sm arl”R”lne@m- l.rdl*wth+#”k

!.. W dim areP ad ~ar. _i@Y3)V41M aflnltlal&dUJ’tlam~ w *ial mti .- */us*.m7+.136(s~. 112)ml -.314129(XMH

4)x WIM oflnit@lXrar&tiam- ~ * a“+ls. ”m */m, 112,&w!K,29Jd@d Wda. 1“

5)Vslwz mrld 4 ~idc(*) wm mlmlatd - -lchmtlm @ m.

lwiml a’lmm ad m ~tlal *lmtim6)V-ha ~ m astaridc(~)m ~,- tlmmml~~ ~

*/@. 2,m. ad d dM.3,,.

116”-d 162, rnrqmotive-ly. Sarlmi analysis of the raeult of’ W “’and ~ ease am “obaemed.20 oases danotod in Treble4 was ~rfomed Fig.14, reptiaents “the Vartioal hndihgvarying the mgnltude of the initial mment-mtat ion ourve of the hul1 girder,deflebtioiand“kldirq ranldualetmuu. S and Aoodlng to tie 8aggirig or h~ingH a&lea denote the aagghg and h~igg oonditlon, the deok .p@el’or kttoz plstingoonditi~,,”res~otively. For alaplioltY,*I1 fdled fimt. k s result, the ‘overal1plate elenenta are aup~ued to have the same lmndhg etlffnem of the hul1 girder wmrat 10 of ,.theinitlal lB~rfeatlona ❑uoh ae” p~eiively “;d&~med. After the ~Wdt and OAU.. Tbn kinds of sha~ of the longitudinal girder in desk panel “or btto~initial deflection sti in Fig.4 are pl-ting failed, finallY the hull girderoomidemd. The ooapmsalve residual sttisain the tr&sveree dhotlon of the hul1

oollap-d m a whole.

girder is set to @ zam in the present Next, the influunoe of the initialanalyeis.For etih.oaloulatloniabout only 7 deflaotlon on the : ultlzate longitudinal~initues were mqulmd by USIX HIF%W120 -tmngth la lnvkutlgatad. Au deaorlbd insu~r’ Binl*omputer, Fig.4, tw ‘kinds of dmm of the initial

deflectionam oonsl~md. Fiu.16.* and 1$.bare ~BultB obt~ined for th;”ehqm of theinitial defhotion ~of Flg.4. a and 4.b,

-..,

....

First, in omhm to investlgate the mapotlvaly. AoooAi~” to them resu’lts,

nonlinearbahavlourof the hul1 girderwith when’”the initial’.deflection of the plate

the lnamashg the vertioal bending ■oment, elementhas the sha~ of Fig, 4.a thataatahaa.

H-12

\t-l” m.”+

m

J----- y

.

-i

17–1b H“’’’’’”lr

Fig.13 Outline of the loading and bounda~condition

Fig.12 Stmctural modelling by using theidealized plate element

.-.,,

LaMA

z........F&...........

Wdt

0.s0.20.51.0O.*0.20.51.00.0s

.O.M0.20.51.0O.&0.2.0.51.0O.*

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5rJuo-

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.

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9mpofInitialmfldiul

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the buokllng“’tie of the plste, the initi-1deflectionmduoea the ultiaete lonuitudlnal

longitudinaldimotlon of the hul 1 gimier.It 1s obeemed that bmauae of the exietenoe

strength of” the hul1 gitier. Ho&or, theEham of Fig.4,b gltis rhe to op~sito or noeffed on tha ultimate longitudinalstrength.This phenomenon io due thet if theahe~ of the initial deflection is not cameM buokling mxle, the hdtiml defleotloninoremeu the ultimate oo~pmsrlve ❑ttingthof the plate[34-36]. Fig. 16 lndloateo”thevertiosl hndlm aoaent-rotstion oume with

of the oowprenuive “meidual strase theulthate longitudinal utmn@h of the hul1girder io eericuely reduaed. The =saeas=entmrnult of the reserve etmn@h fmtor and themfety index h given in T~ble 5. It isobeerved ~hat the mment double akin hul1girder hae uuffioient safety in thelongitudinalBtmngth @nt of view lmoauBe

,... reeerve etmngth faotor and safety index emvarying the &nltude of tb ootipreee Ive oonsidemd to Im relatively l-e.reeldual etmaa whloh ie exlnted in the

H-13 .

Fig. 14 The vertical bending moment-rotationcurve for SAand HAcase

Fig. 15.a The vertical bending moment-rotation curve of the hull girdervarying the magnitude of initialdeflection with buckling modeshape

Fig.15. b

Fig.16

w -..”

1S+. of l.,,,-! D,,k, +,mn,

HOKINC

~–30

The vertical bending moment-rotation. . . ,,curve ot the hull glraer Varying tnemagoitudeof the residual stress

In order to aab ● 8YUte~88fetYaESl?BS-nent of ship’a hull gitiar in thelongitudinal ❑trengkh pint of vleu,

evaluation of ultimate longitudinal, Ixmdins❑trangth is one of key isauea. Alnostruotum1 designers are neoasuam to havedetd 1 underotandlng of prcgraauive 001 lapsehistory m wll *S ultlaate 001 lapae oapaoity

uuoh that they.o“m’”dctaraina the stmoturallsyout~mom reagombly and efficiently.Inthis study, an mtta~t’ is m-de suoh that thePmasive OD1lapoe strength of ship’s hul1girder under sagging or ~ing bndingapaant in effloientlyanalyzedby usingidealized atruotllfklUnit methui. Anideallzedplateeletinteubjeotad to biaxisll-d 1s foraul~ted”taklrg aooount of theinflueme of the lnltial l~~rfeotions aswe11 m the interwtion effeot lmtween theltmal tid overal1 buokl ins in the atmotura.A ooaputer ~aa, ALH/ISUM in oompletedbaaed on the pfi.sentthaow ~ the MOLWUOYand ●ffloienoy of the pmuent xethml amverified through oomparlng the praaent

zolutlonwith the existing emrimentsl andnuaarloalreoult. The pmaent aethod la thensppliad’to the ultiaate longitudinalatrnngthanalysitiof 2801 double wkin tankar aa aninple exple. A pammetrlo analysis withvarying thn initial. deflation and weldingNuldIM atrana 1~” ~rfomad. Smed on themmlt n obtminad ham, mseoment of thamaerve atranuth faotor and mfety index is

/110 0,0 .+...is

m,fi =0,5de. It 10- obearved that tbe lniti~l

Q<Q=....-.....-“3 ●’” ‘::

l~~rfeotions axlathg in the. lmal plate

5k?c41.,.!.l*{,,t+;Q.. elemnt mduoe system aafat~ of ●hlp’s hul1

-7::G;%Gj~”

gimlar and. the dou~ skin hul1 girder&aimed in this study .ks uuffloient aafety

.3GJ Iu,. in ~ lo@tudhal nt&r@h pint of view..”

The vertical bending moment- Alth&h the awple hull Btruotumrotation curve of the hull girdervarying the magnitude of initial designedin thin otu~: la + present ●oaewhat

deflection with the, shape of hungry unraaliatlot~ of the doub,leskin tanker,

horse’s back it is found fmrn the.results of this study

that the prauent,#ethmiom h appl led to the~nlinem ~lysis of large SIZO plated

,... ...

H-14

k 7 BCem m. (A (a (a

.& 1662.4 436,2 w. 9 1,566 2,817m Iw. 1 - -. 1,= 2.T?2s lm.4 - - 1.395 2,1Q5m 1436.5 - . 1.=1 1.WSZ 1817.9 “ - 1.652 3.247SF 16%.6 - - 1.= 2.674,W 17B.8 - - 1.625 3.U3H 1569.8 “ . 1.471 2.4mSI 1241.4 - - 1.148 0.871SJ lm6.2 - - 1.W7 O,ulH4 zm37.a - - 1.W 4,WHB m18.4 - - 1.i!47 3.653E 1622.5 - - i.n8 3.574m Im. 1 - - 1,626 3,175MI 2112.0 - . Lw 4. lWW 21m.o - - LM7 4.Wm ml.3 - “ Lm4 4.M1m m. 1 “ - 1.918 4.mHI lm.3 - - 1.S78 2.b6sNJ 1361.1 “ - I.m 1.*

d m, &EntiwelY

struotum euoh as the double#kin hul 1 girder 2.

Ixsamuae the oomputing tim required le verymall while givl~ a remnable result. Thue,it w1l1 be ueeful wlmn ultiaate oollapae❑tren@-baeed eafety aaseament or 3.

opthlzetlon Js eade in the ❑truotural deeignutege for a new t~“ of hul1 gl”kder,However,sinoe the ~hip’e hull .girdar 1S usual lYooa~aed of etiffened plate ele~ent, theideallzed uti”ffe~d pl~te ale~ent should k

develo-d in order to ozrry out the non1ineer 4.

enalysle for’ wre malletlo t~ of hulletruature. Alm in general, koauee theunstiffened end stiffened pl ●te ● 1eaent aresubjeotedto shemlng fome and lateral loadas well ●a .biexiallads, the influenae ofthe other loadingoonponentRon the Imhaviour 6.

of plate ● 1●merit itself- ehould hIncorporated-into eeoh Meal ized atruoturale1ement,

AC~~T6.

The present e@ly was oonduatedunderthe finenalzl eupprt of Korea Soienoe end~ineerhg Foundnt ion. (Projeot No.:’ KEKF -

69+206W) .

RSPEIWCH7.

i. T. Oboto et ,al, , “Strength Evalue-tion..of Novel Unidlmotlonal-GirderSyetem Prduot Oi1 C~ier by SE1ia-bilityAmelysie”,~, SNANE,1985.

RINA, “NewGenerat ion Supmtankero

Abe 100, ~ DkT”, The Naval Arahi-

MC *p., 1990.

E. Nikolaldim end K. KaPuIla, “Syetem

l?al i-hi 1 lty end Redundancy of Marine

Struotvree : A Review of the Stite

of the ~“s ~V01,34, No,l, 19S0,

Y,Uede end S. U. H. Raehed, “An Ultl-aate Trenuveme StrengthAnalysisofship Stmoturea”,H 81 Arohiteotv of1:4 (in Japeneee).

Y,llede~d S,H.H.Raehed,‘TheIdea-1ized Struotuml Unit Flethalend IteAppli oat lon to Deep Girder Struot-Umu”, Comwterm k Struature8,vOl.la, -1984.

Y. Ueda, S,H.H.Raehedand J.K.Paik,‘Plateand Stiffened Plate Unite afthe Ideal ized Struotuml Unit Method(let ileprt)”, J. of %miety of

Japan , VOI.156,

Y.Uede, S.M.H,Reshedend J,K.Paik,‘Plateand StiffenedPlate Units of

the Idesllzed Struotuml Unit Iiethui(2nd Remrt)”. J. f scalety of

~g VO1.159,

1986 (in Japaneae),

H-15

8, Y.1.leda,S.M.H.Rashedand M.Katayama,“Ultimate Strength Analysis ofDoubleBottomStructures by Idea-lized Structural Unit Method”,J-Societv of Naval Architects ofw, VO1.138, 1975 (in Japanese).

9. Y.L!eda et. al., “UltimateStrengthAnalysisof DoubleBottomStructuresin StrandingConditions”,_3rd International SymposiumonPractical Desire of ShiDs and Mobileu, Trondheim,Norway,1987.

10 J.K.Paiket.al.,“DamageandStreng-th AnalysisofIloubleBottomStructuresinStranding”,,Proc.oftheAutumnMeetinRof SoczetvofNavalArchitectsof Korea,1990.(inKorean).

11. J .K.Paik, “Ultimate Strength Analy-sis of Ship Structures by IdealizedStructural Unit Method”,Dr. Disser-tation, Os&kaUniversity, Japan,1987 (in Japanese).

12. J .K.Paik, “Ultimate LongitudinalStrength Analysis of the DoubleSkinTanker by Idealized Structural UnitMethod”,Proc. of the AutunmMeetingof Societv of Naval Architects ofK=, 1990 (in Korean).

13. J. K.Paik and D.H.Lee, “UltimateLongitudinal Strength-Based Safetyand Reliability Assessment of Ship’sHull Girder!’, J. of Society of NavalArchitects of Japan\ Vol.168, 1990.

.14. S;M.H.Rashed, “Ultimate Strength andPost-Ultlmate Strength BehaviourofDamagedTubular Structural Members”,Division of MarineStructures,NorwegianInstitute of Technology,Report No. SK/R52,1980.

15: Y.Ueda and ..S. H,H.Rashed, “BehaviourOf: DamagedTubular Structural

~‘ Members”, J. of EnerW ResourcesTechnolow , ASNE, VO1.1O7, 1985.

16. Y.Ueda, S.!l. H:Rashed and K.Nakacbo,“New Efficient and Accurate Methodof Nonlinear Analysis of Offshor~Frames(The Idealized Structural UnitMethod)”, J. of Energy ResourcesTechnolow , ASME,VO1.1O7,1985.

17. S.Pl.H.Rashed et. al., “ UltimateStrength of Jack-Up Rigs in Survivaland Puncli-ThroughConditions”, “~of 3rd International SymposiumonPractical DesiKOof Ships and MobileUnits Trondheim,Norway,1987.—7

18. T.Yao,.J .Taby and T’.Moan,“UltimateStrength and“Post-Ultimate StrengthBehaviour‘of DamagedTubular Membersin Offshore Structures”, J. ofOffshore !lecbanics”and Arm

Engineerin~,-ASHE,Vol.110,1988.

19. “J:K.Paik aridB.C.Shin, “TheoreticalandExperimentalStudyfortheProgressiveCollapseStrengthAnaly-sisofTubularOffshoreStructures”,Proc. of 1st Pacif it/Asia OffshoreMececanicsSvnmosium,Seoul, Korea,

20. J. K.Paik. and H.K.Lim,“AStudy onthe Ultiiate Strength Analysis ofFrameStructures by IdealizedStructural Unit Method”,Proc. ,ofthe AutumnMeetin~of ComputationalStructural Engineeri-m?Institute ofm, 1990.

21. J .K.Paik amdH.K.Lim,“ProgressiveCollapse Strength Analysis ofJacket TypeOffshore Structure”,Proc. of, the Autumnfleetin~ of KoreaCormnitteefor OceanResources &Engineering, 1990 (in Korean).

22. J.K.Paik, ALPS/ISUMUser’s Manual(Version 6.5), A Computerprogr~for Analysis of Large PlatedStructure by Idealized StructuralUnit Method,Departmentof NavalArchitecture, Pusan National Univer-sity, Report No. PNUNA-SE-04,1990.

23., D..K.Hart, S.E.Rutherford and A.H.S.Wiclhm, “Structural ReliabilityAnalysis of Stiffened Panels”,Trans. RINA,1985.

24. ABS;Rules for Eiuilding and ClassingSteel Vessels’, 1987.

25. J“.K.Paikand G.Kim,“Estimation ofthe Ultimate”’Compressive Strength ofActual Ship Panels with ComplexInitial’ DefIection”, J. of Societyof Naval Architects of Korea,VO1.25,No.1, 1988 (in Korean).

26. S.P.Timoshenkoand J. M.Gere, Theoryof Elastic Stability, SecondEdition, McGraw-Hill,1963.

-27. Y.Ueda, SJ4.H;Rashedadd J. K.Paik,“Effective Width of RectangularPlates Subjected to CombinedLoads”*J. of Society of Naval Architects!XG!IW!7 vol. ~sg! 1986 (inJapanese).

28. J. K.Paik, “AnAnalytical Solution ofNonlinear Behaviour for SimplySupported Rectangular PlatesSubjected to ‘Biaxial Compression”,Proc. of the AutumnSymposiumofKorea $bici Structure ConUess,Society of Naval Architects ofKorea. 1990 (in Korean1.

......

29. J-.”Rhodes; !’Effective Widths inPlate Buckling”, in DevelopmentsinThin-WalledStrictures, Edited by J.

....

H-16

Rhodesand A.C. Walker, AppliedScience Pubishers, 1982.

30. Y.Uedaet. al., “Ultimate Strength ofSquare Plates Subjected toCompression(1st Report) - Effectsof Initial Deflection and WeldingResidual Stress -“, J. of Society ofNaval Architects of Japan, Vol.137,1975 (in Japanese).

31. j. K.Paik and C.Y,Kim, “A SimplifiedFinite Element Method for theUltimate Strength Analysis of Plateswith Initial Imperfections”, J-Society of Naval Architects ofKorea VO1.26, No.1, 1989.—7

32. T.Usamiet. al., “Tests on theInteraction Strength BetweenLocaland Overall Buckling of WeldedBoxCOIUMIK”, J. of Societv of CivilEngineering of Japan, Vol.308, 1981[in Japanese).

33. J. K. Paik, “Utimate LongitudinalStrength-Based SystemSafetyAssessmentof a NewSuezm&SizedDoubleSkin Tanker”, J. of Societyof Naval Architects of Japan,Vol.169, 1991 (to be Appeared).

34. M. Kmiecik, “Factors AffectingLoad-Car~ing Gapacityof Plates”,ScientificProceedingsNo.115/4,ShipbuildingInstitute,SzczecinTechnicalUniversity,1981.

35. J. Czujko, “Collapse Analysis ofPlates Subjected to BiaxialCompressionand Lateral Load”,NorwegianInstitute of Technology,University of Trondheim,Norway,Nov.,1983.

36. T. Jastrzebski, “AllowableDistortions in Shipbuilding Plates-SomeTechnical Aspects-”, BulletinTechniaue du BureauVeritas, No.4,1988.