O,*C”’%*

THE SOCIETY OF NAVAL ARCHITECTS AND MARINE ENGINEERSOne World Tmde Center, Suite 1369, New York, N.Y. 1004S

*.$

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‘% ?epem to be presented at Extreme Loads Response SYmPOSi urn~ ~

: Arli”@a”, VA, October 1%20, 1981

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O.,ti, . #o

Application of Loading Predictions to Ship StructureDesign: A Comparative Analysis of MethodsDonaki Liu, Hsao Chen and Faith Lee,American Bureau of Shipping, New York, NY

ABSTRACT

This paper reviews two statistical/probabilistic methods, the well-kno”nlong-term exceedance probability predic-tion and Ochi 1s extreme value apDroach,and compares results with the methodembodied in the current ABS Rules forpredicting a ship!s dynamic verticalbending moment. Numerical calculationsare performed for a sample tanke~ usingthe three different methods. Structuralresponses are also analyzed, applyingdifferent dynamic bending moments andthe still-water bending moment to afinite element structural model

Numerical results for the sampletanker indicate that Lewis t long-termexceedance prediction for 20-25 years ofshiu service time (Drobabilitv level10-t .‘) is quite Cotipa.rablet: Ochi, sprobable extreme value. The two theore-tical results are also shown to be inclose agreement with the nominal dynamicbenGlng moment specified in the ABSRules requirements . The stress valueobtained from the structural analysisbased on Lewis 7 dynamic bending momentat 10-’ .‘ is comparable to the ABS re-quired nominal permissible sti-ess.

Ochit s design extreme value of ver-tical bending moment, incorporating a!IriskDarameter,~ of 0 .01=10-2 for thesample ship closely agrees with COPreS_pending exceedance prediction at a prob-ability level of 10-7. 6X10-2.10-9 6.The stress value for this case is hiEherthan the nominal permissible stress ~e-quired by the ABS Rules. This would in-dicate that if the design extreme verti-cal bending moment is used as the basisfor the design of the ship girder struc-ture, a higher permissible stress levelshould be established.

INTRODUCTION

Ever since St. Denis and Piersonpublished their well-known paper [1]*almost thirty years ago for handling the

* Number in brackets designates refe-rence at end of paper,

motions of ships in a confused sea, va-rious theoretical procedures have beendeveloped foI predicting the ship,s res-ponses during its service time. Ingeneral, the theoretical methods fallinto two categories : One is a loruz-termprediction of-the probability “of e~c&ed-ing different response levels, where theRoot-Mean-Squa?e values of ship responseare employed. Works by Jasper [2], Nor-denstrom [3], Band [4] and Lewis [5] forexample, are in this category. The othercategory consists of the work by Ochi[6, 7], where the extreme value theory isused. The longitudinal strength require-ments of classification socie;ies a;egenerally developed on the basis of ser-vice experience, using a probabilisticaPPrOach to the determination of loads.POT example, the theoretical procedureused by ABS in the development of itsrequirements employs the long-term ex-ceedance approach developed by Band andLewis .

The approach employed by the clas-sification societies for establishingRules requirements of ship longitudinalstrength can be considered as semi-probabilistic since loads are predictedby a probabilistic method, but strength(permissible stress) is deterministic.Since va~ious methods are often used inDractice. a brief review of the 10nx-~erm exchedance prediction of Lewis;Ochi 7s ext~eme value approach, and themethod involved in the .40SRules isgiven in the following.

Lewis [5] examined the 30-minuterecords of midship bending st?ess ob-tained every 4 hours from the c4-s-B5cargo ships Wolverine State and Hoosier— —= in several years 1 service In theNo?th Atlantic He found that the in-dividual data samples fitted closely tothe Raylei Eh distribution. Furthermore.dividing ail the stress data into wea-ther groups , the mean square values ofavailable stress samples in each weathergroup we~e found to follow the normaldistribution. Consequently, Band [4]suggested a.probability model where thetwo aforementioned distributions wereassumed to apply to a much larger “po-pulation” (or quantity of data) Hence,

249

I

an ideal cumulative distribution ofstresses (or bending moments) could beconstructed for each weather group bysumming up all of the many Rayleighdistributions and integrating to diffe-rent stress (or bending moment ) level BThe total cununulative distribution couldthen be obtained by combining all of theindividual cumulative distributions onthe basis of the actual percentage oftime each weathe~ group would be-encoun–tered by ships in service, and on thebasis of average weather on some parti-cular route such as the NoFth AtlantlcLittle and Lewis [8] further evaluatedthe idealization of statistical dataagainst the stress data collected fmmnfive ships with about 3-1/2 years * ser-vice . These five ships consisted offour tankers and one bulk ca’?ie?, andare 754.6 to 1076 feet in length. Inaddition, Lewis introduced the so-calledH-family wave spectra which were selec-ted In a random fashion from those givenby Pierson, et .al [9]. The H-familywave data consists of fine weathergroups in accordance with significantwave heights 10, 20, 30, 40 and 48.zfeet . With the exception of the lastgroup which contains 12 spectra, allother groups contain 10 spectra.

In contrast to Lewis , approach,Ochi [7] first analyzed wave data col-lected at Stations P.,B, C, D, I, J andK in the North Atlantic, and then de-rived two families of wave spect~a oftwo- and six- parameter, Perspectively,which can be considered to represent the‘ImeanNorth Atlantict! sea conditionsBoth the two– and six- pa~ameter spec-tral family consists of 18 different seaseverities (significant wave heights) .But the numbers of wave spectra in eachsea severity of the two families aredifferent . In the two-parameter family,for each significant wave height the~eare nine spectra. The six-parameterfamily has eleven spectra for each .slg-nificant wave height . FOY dynamic loadprediction, Ochi introduced an apprmachbased on extreme value theory. Thispredicts the highest response expectedin a shipvs lifetime by calculating theshort-term extreme value for the mostsevere sea condition expected. Ochialso developed a long-term extreme valuemethod where the frequency of varioussea severities , as well as other facto~~including wane spectral shapes, shipspeeds, and heading angles, are weightedaccording to their occum.ence during theship!s service life time. This approachis similar but not identical with theLewis method.

Prior to 1975, the basic concept ofABS Rules requi~ements fop p~ima~y, lon-gitudinal strength of the hull girderwas to prescr-ibe for each individualship a b~ section modulus based onits main geometric characteristics. Therequired section modulus amidships was

250

then specified in terms of the basicsection modulus and the maximum still-water bending moment in the governingloaded condition. With this sectionmodulus the maximum primary bendingstress in the hull zirder. which can beproduced when the m;ximum’possible lon-gitudinal bending moment is Imposed onthe ship could not exceed a nominal per-missible stress level, which varied withship length. The requirements for lon-gitudinal strength In the above des-cribed form were successfully applied toships for many years . Since 1975, ABSRules strength requirements have beenpresented in a different format, inorder to accommodate the recent findingsobtained from the long–term dynamic orwave load studies . For the purpose ofship longitudinal strength evaluation,ABS now employs the long-term responseprediction method suggested by Band andLewis to generate a long-tepm response- which Is used as a basis for ex-tending the existing ABS Rule.? to largerships . The new requirements are speci-fied In terms of the nominal still-waterand wave-induced vertical bending mo-ments Stress limits are then sDecifiedin conjunction with the longitudinalstrength requirements .

In view of the different assump-tions and different approaches employedin the above two theoretical methods fordetermining extreme loads, it is of in-terest In this paper eo investigate theship structural response for the dynamicwave loads which are determined bv eachof the different methods, and to ;ompare;~lm~;ith those given in the latest ABS

Ship structural response subjectto different loads at different Drob-ability/confidence levels will aiso becalculated, using a simplified finite-element structural model of a sampleship No attempt is made to compare orcorrelate the theories pertaining to thedifferent approaches. Mathematical de-rivations and proofs are avoided, withthe exception of those necessary forpresentation of the related topicsEvaluation and m?rification of the as-sumptions in”olved in developing thedifferent methods can be found in otherpublished literature. This paper beginswith a summary of eq”atio”s and formulasassociated with the three different me-thods for determining loads under con-sideration. It is followed by a SUIWlaPyof the H-family and the six-parameterwave suectral data. Theoretically cal-culateil dynamic loads based on th;sewave spectra, and the mathematicalstructural model of the sample ship arethen presented. In the remainder ofthis paper, computed structural resultsof’the sample ship, and concluding re-marks are given. It is the authors 8 hopethat through the numerical example, somelight can be shed on the application ofthe different statistical/probabilisticmethods for the evaluation of ship

—.

girder longitudinal strength.

PROCEDURES OF STATISTICAL/PROBABILISTICMETHODS FOR LOADS

The basic assumption involved inthis approach is that the short-termresponse of a ship in statistically un-changing seas is a nar?ow band process,and is distributed according to the Ray-leigh disti-ibution. That is, the prob-ability density function, f(x), of arandom variable x takes the fo~m:

where Pr(Wk) is the probability of en-countering the k-th weather group,Pr{x~xo Iw~] is the conditional probabi-lity which, in turn, has similar expres-sions of the riEht-hand side of eouatior:

Long-term Exceedance Probability (based (3) Lewis fur~her assumed that, ‘for aon RMS values) specific condition of loading, the prob-

ability distribution of A was a func-tion of ship speed and heading, and was,a normal distribution in a particularweather group. Substituting this normaldistribution into equation (4) it canbe realized that the analytic ~ntegr,ationis not possible. For numerical calcula–tion. a value of five times the standard

r(x) = ~exp (-xZ/E) (1)

where E>O is the parameter of Raylei&hdistribution for O<x<co. The parameterE is equal to twic~ the mean squarevalue, m., if the peak value x is asingle amplitude. If x is a doubleamplitude, E equals eight times mo.

dist.?~;k!i’;;go? A$(fla;;~u%”i~;ili;yvariation of sea state (weather) , shipspeed, loading condition, and headingangle, then, by definition, a condition–al distribution of y, with respectiveto K is

f(x,m = f(x lfi)g(/F),

which leads to the probabilityexceeding X. as

(-

(2)

of x

Pr(x~xo I =

1exP(-x~/E) g({~)dfi. (3)

o

The probability given by the above equa-tion is equal to the reciprocal of thenumber of cycles, n(xo), in which therandom variable x is expected to ex-ceed the value xo once. The recipro-cal of the number of cycles is also re-fei.redto as the proba~ility level.

Most presently used long-term res-ponse prediction methods based on RMSvalues generally follow the format shownin equation (3) Differences, however,exist in the form of the probability

distribution g(fi) suggested bY diffe-rent investigato~s, based on differentdata being employed in their analyses .In Lewisi approach, the probability iSseparated into t“o parts; one being theconditional probability that x exceedsXo within each of a number of weathergroups, the other being the probabilitywith which the ship encounters each ofthese weather groups. Thus ,(3) iS exPressed as

equat ion

pr{X~Xo} = ~ pr(Wk)pr{X~XOIWk], (4)k

deviation has been adopted to replace theinfinite upper limit of the integral likethe one in equation (3) . The finiteuPPeP limit of d~ has been shown byBand as the minimum value to insure ~uf-ficient accuracy in the final result ASa consequence, in Lewis 7 approach, theconditional probability that the variablex exceeds x. in a particular weathergFOUp , ‘k, ~an be e;aluated numericallyusing the equation as fallows :

Vk+sak

pF{X~Xo\wk} = llXJE.o niro~

where

~k .

(m+)’

20;

mean value of d~ in k-th wea-ther group, Wk

&i: &i,

variance of ~ about Uk in k-th

‘eather group ~ ‘k

~ i: ~fii-Pk) 2

The total number of lifetime loadcycles, n(xo), to be used for dete~mlningthe load which is expected to exceed Xo,1S giVen by Ochl, as in equation (11) ofthe next section.

Ochi’s Extreme Value Approach

The extreme value is defined as thelargest value of a random variable ex-pected to gcc”r in n observations .Denoting Y. as the modal “al”c or the..most probable largest value of the ex-treme probability di?,tyibution, it ca”be shown (see Hoffmgn a“d Karst [11] ,for example ) thattic form for large yn has the asympto-n:

251

where ~=0.572z...... the Eulerqs COn-stant , and E is deffned in equation(l). If the random process is narrowband, Ochi and Bolton [6] showed thenumber n In equation (6) can be ex-pressed by

n= * (.2/.. )1/2 (7)

where

T = time in hours

MO = area under the spectrum of I-an–dom variable considered (meansquare value of ship responsein the present wopk)

m,= area under the second momentof random variable considered.

Considering the first term on theright hand side of equation (6), a“dusing equation (7), 0chi8s short-termprobable extreme value approach is ob-tained as follows :

[

3600T (’m/mo)‘1l/2

jin. Etn {y2

(8)

Ochi and Bolton further showed thatan extreme value higher than ~n givenby equation (8) would occur with prob-ability O .632 for large n. This prob-ability value is too hich for DTedi .kion,.. .—.—...purpos; in practice. ‘“”Thu S , a risk para-meter a (usually taken to be 0.01) isLntmduced into equation (8). DenotinEthe recently obtained “alue by “Yn, th~nOchi, s short-term deslm extreme v.l I,e.-..i.e. , the expected highest value in a ‘fleet of (l/a) similar ships, is

,.

[ 1

1/2~n .

~ q}~Ln{ 3600T . (9)

In equations (8) and (9), m. and m, aredefined in equation (7) In addition,

T=

a=

k=

E=

longest duration of specifiedseas fn hours

risk parameter

number of encounter with aspecified sea in ship 1s servicetime

2mo, if the peak value is asingle amplitude; 8mo for dou-ble amplitude.

Equations (8) and (9) are the basis forthe subsequent sample calculation of theprobable a“d design extreme values.

Ochi also developed his own veFsionof the long-term response predictionmethods for the most probable extreme

value and the design extreme value. ‘Theprobability density function of hls long-term response is shown as follows :

ZZzEn*pipJpkpfif*(x)~(x) = ijkk

EEZZn*P~Pj PkPE ‘(lo)

ijkk

where

f*(x) =

ns

(mo), =

(m,), =

Pi>Pj >Pk,PE=

probability density functionfor short-term response,

average numbeF of responseper unit time of short-termresponse,

1= ~ (m,)*/(mo)* ,

mean square “alue of resp-onse spectrum,

second moment of responsespectrum,

probabilities of sea condi–tion, wave spectrum, head-ing angle and ship speed,respectively.

The total number, n, of responses ex-pected in the lifetime of a ship is

n = (EZZZn*PiPJPkP1) X T x (Go)’, (11)ijkk

n, of response for a

1-1F(in)=n. (12)

where T is the total exposure time ofa ship to the sea.

The cumulative distribution of thepro bable extreme value, F(~n), can beobtained by integrating equation (10).The function F(Yn) , as shown by Ochi,has the following relationship with thetotal number,large n:

[1-

.For the design extreme ValUe, Yn, equa-tion (12) is also applicable, except thatthe term on the right-hand side should bereplaced by n/a, where a is the riskparameter.

Approach in ABS Rules

For the purpose of numerical compa-rison with other methods, only the “er-tical bending moment is considered. Therequired bending moment for ship girderlongitudinal strength evaluation in thecurrent ABS Rules [10] expresses the to-tal bending moment amidzhipa, Mt3 asfollows :

Mt=MSw+Mw, (13)

where MSW and Mw are Yespecti”ely the

252

r

still-water bendin~ moment and wave-induced bending moment . These bendingmoment components are given by

MSw = CStLZ.5B(Cb + 0.5] (14)

and

Mw = c2L2BHeKb

where

cst=~.,,, + ~],o,

‘k”” + %+-’:

‘P”’ + %%+-’>

=[0.544110-2,

‘b’ -‘b” +“F’”++’” +=[0.275]10-8 ,

‘E’”’”-%&l’”-’>He,Meter= 0.0172L+3 .653,

=0.0181 L+3. 516,

(15)

61~L<lI0 ~

11O<L.16O m

16o<L:21o m

210<L:250 m

250<L~427 m

2001L<360 ft—

360<L~525 ft

525<Ls690 ft

690.L>820 ft

820.Lj1400 ft

61<L~150 m

150.L<220 m

=(4.5L-0.0071L1 +103)X10-2,220<L>305 m

=8.151, 305.Lz427 m

He, Feet= 0.0172L+ll. g8, 200<L~490 ft

‘0.0181 L+11.535, 490<L~720 ft

=(4.5 L-0.00216 L2+335 )x102,720< L~1000 ft

,=26.75, 1000< L314OO ft

Kb=l. o for c#o.80=1.4 -.o.5cb foy o.64<cb<o .80.

cb=blOck coefficient at SUmme?load waterline. Cb is not tobe taken less than 0.64. See[1OI for the details.

c,=(2.34cb+o .2)xlo-2 for metric

units=(6. !i3cb+0.5?)x10-4 for inch/pound units

L,B=Ship length and bean! in metersor feet. (See [10] for the de-tails) .

The still-water and wave-induced bendingmoments given by equations (14) and (15)are in met?ic ton-meters or ton-feetIt should be noted that the term H~ inequation (15) does not represent a waveheight Rather, it is a parameter forcalculating the required wave-inducedbending moment by equation (15). Thecombination of the wave-induced bendingmoment and the still-water bending mo-ment from equations (14) and (15) , to-gether with nominal permissible stress,would give rise to the section modulusrequired by ABS Rules. On the otherhand , for Cb~O .8 and the waterplanecoefficient >0 .90, the value of thetern!He of equation (15) is approxima-tely equal to the effective wave heightused in the quasi-static calculation ofbending moment . A more detailed discus-sion on this point can be found in thework by Stiansen and Chen [12].

The nominal psrmlssible bendingstress, in principle, should be a con-stant However, to account for the dif-ferent tolerated coyi-osion margins fardifferent ships, and possible dynamicloading components , such as lateralbending and vibratory bending, might notbe in the same proportion as the verti-cal bending, the nominal permissiblestress suggested in the ABS Rules varieswith ship! s length, L, as follows:

fp= 1.663 - ~ tons/cm2,

L-24”= 1.663 + ~ tons/cm2;

= 10.56 .- ~ tons/in2,

= 10.56 + ~ tons/in2,

61.L<240 m

240<L~427 m

(16)

200<L~790 ft

790<L~1400ft

where fp is the nominal permissiblestress for mild steel.

WAVE SPECTRAL DATA

In the present work, two differentwave spectral families are used in thedynamic ?esponse (vertical bending mo-ment amidships ) calculation foi.thesample ship; one is the H-family spectraintroduced by Lewis, the othei- is Ochi ,ssix-paramete? wave spectra of the ,,meanNorth Atlantic. ‘r

The H-family of wave spectra wasrandomly selected from those preparedby Pierson [9]. The primary recordsused in Piers onTs analysis were take”by the National Institute of Oceanogra-phy (U.K.) on British weather ships at ‘~

Stations A, I, J and K in the NorthAtlantic. A total or 52 wave spect?a ir ‘p

the H-family are grouped into 5 groupsas shown in Table I. The characteris-tics of the average spectrum of each Ofthe five groups in the H-family are shownin Table II. In Figure 1, the spectra. ofone group are displayed.

m ‘Table I - Percentage of occurrence and

number of spectra of H–familywave data

“,.,,.,.”. , , , , ,

,s ,.3,2 ,.0,, 8.s39,,.8,0 ,4.,7,m- ,.0,, 3,,5, ,.,,, ,8,,,, ,,,830

=’. 0.70, 2.,03 ,.,,, ,,,18 ,2,,20. 0.,,, ,.57, ,.,,, ,.,,6 ,.5,4

.9 0.4s, 1.2,, 2.,,6 ,,,,, ,..,,

m, 0.44, ,.,,, ,.,,, ,,,,, ,,,3,

~, 0.507 ,.4,, ,.3,, ,,71, ,,443. 0.,50 0.5,, ,.4,, ,.+O, ,,40,

Table II .

Notes: Hs

‘n

“P

.

Characteristics of the a“er -age spectra of the H-familywave data

= significant wave height inmeters

= moments of wave spectrumin metric units, forn=-1,0,1,2,3 and 4

= peak frequency in rad/sec.

I,‘,

Fig. 1 - H–family wave spectra for sig-nificant wave height 3.o5meters

Ochi’s six-parameter wave spectralfamily of the “mean North Atlantic,, con-sists of 18 groups in accordance withthe significant wave height. The sug-

254

gested pe~centage of occurrence by Ochl[7], the significant wave heights andthe durations of the sea states used inthe present work are shown in Table III.The assumed duration of each particula~sea state as shown in this table is .aD-proximately the same as that in Ochi ~ndMotter [13] , with the exception of thewave group of one meter significant waveheight

Slmifica, t . . . .“.,,,, {.,,.,, )

,,0

,.s

2,,

,.5

4.5

s.,

6.,

,.5

8.,

9.5

,0.,

,1.5

,2.5

>3.5

,4.5

1s.,

,6. ,

17.0

,.,. t L..

sea S,.., (),..,s,

,0. ,

46.0

46.0

46.0

46,0

45.,

39.8

36.1

30.1

,4,,

17.0

1,,,

,.3

5.,

4.,

3.3

3.3 I

~

Table III - Wave group, percentage ofocc”mence, and duration ofsea state of six-parameterwave family

Within each significant wave groupof the six-parameter wave family, thei-eare 11 members, of which the most prob-able spectrum is weighted by a factor of0.50, and each of all othei- spectra isweighted by 0.05. For the significantwave height specified in the above table,spectra are generated using the followingformula

s(o) = >4

x

.J.

(17)

[ 11b+0.25) ‘mj

T

where r(),.) is a gamma function, andj=l,2 Stan~S fOr the lower and higherfrequency components, r’especti”ely. In .equation (17), the Parameters HSj , timj,and i are functions of significant “a”eheigh~, and are gi”en in Table IV of Ochi[71. An example of the six-parameterfamily is shown in Figure 2.

r-

Fig. 2 - Family of six-parameter wavespectra for significant waveheight 3.50 meters

NUMERICAL EXAMPLE OF RESPONSE CALCULATION

As a numerical examDle, the threedifferent approaches discussed in theprevious sections have been applied toan existing tanke~ for the dynamic ver-tical bending moment amidships calcula-tion in the fully loaded condition of105,700.00 met~lc tons The hull girderscantling of this vessel was designedbased on the wave-induced bending momentrequired by the ABS Rules, which is345,8oo meter-tons , and an allowablestill-water bendina moment of 207.800meter-tons. The d=signed section’modul”,sis 333,9oo cm2-meters for mild steel,leading to a bending stress at midshipsection of 1.658 tons/cm’ which is theDeOmissible stress sDecified in the CUI.-rent ABS Rules (see ilso equation (lb)of this paper) Other information of thesample ship i~ shown in Table IV. InFigure 3, a sketch is presented showingthe p~ofile and cargo tank arrangement ofthe sample ship, where indication of theextent of the structural model used inthe stress analysis is also given.

m,, lag,, k,.,,” .er,.ndimla.s

B, t.,,.,,, mO.ld6d

D, depth mu!...

,, ,,.,, fun, ,0.,4

.b, blockcoefficient for . ..ign draft“, .,1, s,eed ,,.1,..,

Fig. 3 - Profiles and tank arrangementof the sample ship

the head sea with an interval of 30 de–grees, where the probability of encoun-tering the dlffe~ent heading angle isassumed to be the same . The responsespectra are calculated based on the H-family wave spectra and the transferfunction of vertical bendlnE moment bythe program ABS/SHIPMOTION, -wherein theship speed is taken as 75 percent ofdesign speed. ABS/SHIPMOTION program incomputing the transfer function followsthe stFiD method and the linearized shiDmotion theory. The mathematical deriva:tion and the computational procedure ofthe program are discussed in the worksof Raff [14], Kaplan [15], Kaplan, Sar-

gent and Silbei-t [16], and Stiansen andChen [12].

In the calculation of mean squareresuonse value for detez-mininz the con-ditional probability by equat;on (5) ,the cosine-square spreading function isemployed to simulate the short-crested-ness of the seaway . The spreading func-tion used in the computation is given as

f(llu)=; cost(llu) (18)

where v is the heading angle of theship witi+respect to the wave Furthe~-more, the ship ,s response in a seaway isconsidered as a random process which isnot necessarily narrow band. For thiscase, the mean square value of the Fes-ponse spectrum is obtained by multiply-ing the mea” square value of a narrowband process by a factor of (1-s2/2) asOchi and Bolton [61 suf?!zested. Here Estands for the b~o’idne;; ok responsesgectrum.

Table IV - Principal particulars of the The conditional probability func-sample ship tions based on the H-family wave spectra

Results of Long-term Exceedance Prob-are substituted into equation (4) for

ability Predictioncalculating the total probability, wherethe Der’centage of ‘acc”I’renceof each

When Lewis 8 long-term response pre-diction is used for computing the dynamicvertical bendin!zmoment . the H-familvwave spectra di&ussed in the previo; ssection aye applied. The conditionalprobability for each wave group is eva-luated according to equation (5) at 12heading angles from the following sea to

wave height group given in Table I Isemployed. Since the total probabilityis equal to the Reciprocal of the numberof response cycles, a cu.ve can be ob-tained to represent the response of ex-ceedance versus the number of resnonsecycles (hence the probability lev~i) .Shown in Figure 4 is such a curve forthe dynamic vertical bending mome?.: ‘~-

<255

amidships of the sample ship in the fullyloaded condition.

6.W05

r“’’’’’””’ ““i..,,,.,.,, ,.,..,

,.,., ,,~ab, ,,, ., ,...=,...,

Fig. 4 - Long-term vertical dynamic

bending moment at midship sec-

tion of the sample ship in fullYloaded condition, based on theH-family wave spectra

Assuming two-thirds of 20-year ser-vice time that the ship is exposed to thesea. the total number of vertical bendinxmom~nt cycles of the sample ship, n, i;calculated by equation (11) . It shouldbe noted that, for a non-narPow band res-ponse spectrum, the average number ofresponse per unit time of short-term resp-onse, ‘3, ‘n ‘qUatiOn (11), should bemodified as suggested by Ochi and Bolton.The number of ~esponse cycles, n, cal-culated for the sample ship in this man-ner is 10-7.’. Consequently, as can beseen from Figure 4, the dynamic verticalbending moment of exceedance in 20-year

service time is 3.61x10S mete~-tons atthe probability level of 10-7. s .

The concept of applying a risk para-meter, a, in Ochits design extremevalue can also be incorporated in thelong-term exceedance probability predic-tion. Karst showed in Appendix A ofReference [17] that, with (1-a) percent

assurance, the expected response to beexceeded should be read from the long-term response curve as the one in Figure4 at a probability level of an-’ , prcl-vlded Cl<<l and *>>1. This Is equiva–lent to the exceedance to be expectedonce in the service time of (l/a) similarships. Taking a=O.01, the expected ver-tical bending moment of the sample shipto be exceeded in the sea way representedby the H-family wave spectra is given atthe probability level of 10-8. G, which is4 .6x10’ meter-tons ,

Similar to the H-family wave data,the six-parameter wave spectra generatedby equation (17) based on the character-istics in Table III, are also employed inLewis 1 long-term response prediction forthe sample ship. The number of verticalbending moment cycles in 20-year servicetime is 107.”. The expected “ertical

bending moment to be exceeded at prob-ability levels of 10-7 .“ and 10-’.’1are respectively 3.68x105 and 4.54x10’meter-tons Comparing these resultswith those based an the H-family, it is-”lnterestinE to notice that lon~-termprediction-values of the two differentfamilies are very close .

Results of Ochiv s Extreme Values

The short-term probable extremevalue and the short-term design extremevalue of vertical bending moment of thesample ship, are calculated by equations(8) and (9)”. In the calculation, onlythe six-parameter wave spectral familyis employed. Assuming two-thirds of the20-year service time that the ship isexposed to the sea, the total time ofthe most severe sea state of 17 meterssignificant wave height that the shipwill encounter is 10.37 hours. If theaverage duration of this sea state is7.3 hours . the number of encounters withibis sea ;Ondition is about 3 times .For the sample ship, the calculationsshow that the vessel in head sea condi-tions give rise to the maximum verticalbending moment , which is 3. 42x10’ meter-tons o? the probable extreme value, andis 4.65x10’ meter-tons of the design ex-treme value with a risk parameter a=O.01.Since Ochi !s long-term extreme valuesshould be the same as the short-term ex-treme values, aB discussed by Ochi [7],the long-term extreme vertical bendingmoments for the sample ship In the pre-sent work are not presented. In orderto compare the results of the two sta-tistical/probab ilistic approaches andthat required by ABS Rules, Table V isprepared, from which it can be seen (a)There is practically no difference In thelong-term exceedance predictions usingthe H-family wave data and the six-para-meter spectra, (b) Comparing probable anddesibzn extreme values with lonx-ter’m ex-ceed~nce predictions at probab~listic le-vels of 10- 7.’ and 10-9.’, ~espectively,calculation shows that the results ofthese two different statistical/Drobabi-listic methods are comparable, and (c)The nominal vertical wave-induced bendingmoment requi~ed in ABS Rules is about 4percent less than the exceedance predic-tion at a DrObabilit!J level of 10-7. ‘.and about ihe same w~th the short-terkprobable extreme value.

r,ong-,.rm .E,..ed..ce OChi . ,.,,..,

Am ml.. 10-7.6 ~o., ., ,robable .e. im

,,,58,,0, 3,6,,,0, 4.6.10> 3.,2.1,, 4.65.,0,

Table V - Comparison of vertical wavebending moments amidships ofthe sample ship

Note: Bending moment in meter-tons.

256

STRUCTURAL RESPONSE ANALYSIS

The structural response analysis isperformed for the sample ship by usingthe program SHIPOPT developed by Hughes,Mistree and Zanic [18]. This program wasdeveloped for analyzing the st~ucturalresponse (stress, deflection, etc ), pi-e-dicting the critical OF failure values ofthe structural response, and optimizingthe scantllngs of all girders, frames andstiffened panels in segments of the hullgirder. To achieve the functions justmentioned, the ship structure is repre-sented by a finite-element structuralmodel which consists of a number of com-partments with the transverse bulkheadsas the boundaries. The basic type ofelements to represent the hull structureare:

(a) Multiribbed plane stress ele-ment for modeling the panels of stif-fened plating. The element is rloncon-t’orming, and has constant shear stressand linearily varying direct stresses.

(b) Composite beam element for nm-deling of bracketed beams attached toplating.

(c) Strut element for modeling ofpillars, screen bulkheads, passageway s.,and other portions of structure whichaFe not primarily structural membersbut , because of their inplane rigidity,do contribute to the stiffness of theoverall structure.

(dJ Basic plane stress element(triangular and quadrilateral element) .

In the present work, the midshipportion containing three compartments asindicated in Figure 3 is modeled by fi-nite elements . The midship portion ofthe ship is chosen, as it usually issubjected to the largest bending moment.Figure 5 illustrates a typical structu-ral section where the st~ake number andnode point number assigned in the finite-element structural model are also shown.Figure 6 shows one-half of the finite-element model with the centerline planeas the plane of structural symmetry.Figures 7 and 8 show the transversebulkhead at model Frame O and the webframe at model Frame g. For tbe arlaly-sis, the bending moments and shear for-ces are applied at the two end bulkheadsof the structural model The bendingmoments and shear forces on the boun~a-ries of the structural model are computedby setting the ship on a simple wave witha length approximately the same as theship,s length. The wave height is chosensuch that the total “ertical bending mo-ment is equal to the sum of the dynamicbending moment and the still-water bend-ing moment of the ship in the loadingcondition under consideration. Bendingmoments and shear forces at the two endboundaries of the structural model aye

obtained from the qua5i-static condition.Pressure exerted an the side and bottomshell structure is directly calculatedbased on the local draft of the ship inthe simple wave.

In view of the long-term exceedanceresults, which are “cry close to thoseby Ochi’s extreme value approach, asshown in Table V, structural analysisis performed for two cases where the dy-namic vertical bending moment amidshipsare 3.61x105 and U.6xIo’ meter-tons .These cases correspond to Lewist ~esultsat ~o-7.6 probability level (correspond-ing to Ochi 1s probable extreme value)and 10–’ .‘ (corresponding to Ochi’s de–sign extreme value ), respectively . Thelongitudinal inplane plating stressesfor these two cases are shown in Figure9, which indicates that fop the case of10–7 ‘ probability level, the maximumstress is 2,10 tons/cm2 in the deck and2.16 tOns/cm2 in the bottom plating.For the case of 10–9 ‘ probability levelthe maximum stress is 2.43 tons/cm2 inthe deck and 2.32 tons/cm2 in the bottomplating. It should be noted that theinplane stress consists of the primaryst~ess and the secondary stress Itshould also be noted that, as indicatedin Figure 5, the bottom and deck plating,and other parts of the structure are ofhigh-tensile steel. For these structuralmembers the allowable stress value ishigher than in mild steel by the ratio of

Node ,.,,., Nu,rb,rT s,,.,. Numbsr>

Fig. 5 - Typical section of structure,to~ether with strake and nodepoint number used in finite-element structural model

I

I

Ii—257

1/0.78 or 1.28. Applying this factor tothe stresses obtained from the finlte-

Pequired nominal permissible stress of1.658 tons/cm2. In the case of a 1O–’.’

element analysis, it can be seen that the probability level, the maximum Inplanestress level for the case of a 10- 7.’ stress exceeds the nominal permissibleprobability level is close to the ABS re- stress by about 14 percent

a w ,, m ,0, ,2, ,“0 ,,0 ,80 *O

1,,

1,,

,,7

1,0

,,,Model Frame

Fig. 6 - Finite-element structural model

I

1

5000 500

!99 Is la

5000 5000=. Snnn.“..

1

5000.,.5000 55 17

50s

‘Fig. 7 - Finite-element structural modelof transverse hulkhead at modelFrame O

I

Fig. 8 - Finite-element structural modelof web frame at model FFame 9

I

j

1-t258

. .

Fig. 9 - Stress values of midship sectionstructure

CONCLUDING REMARKS

The clynamlc bending moment of thesample ship has been computed usingdifferent methods. Corresponding struc-tural analysis is performed by using astructural finlte-element program. Basedon the results of the present study onthe sample ship, the following conclu-sions can be drawn:

1. The H-family wave data and Ochi’ssix-parameter wave spectral family of themean North Atlantic are comparable inpredicting the long-term response predic-tion. This may not be totally surpris-ing, because of the fact that the twofamilies of wave spectra aPe generatedfrom similar wave data in the North At-lantic.

2. Numerical comparison shows thatusing Ochils pz’obable extreme “alue ap-proach is equivalent to using the long-term exceedance approach, predicted at aprobability level corresponding to 20-Z5years service time. In the present studyfoF the sample ship, the probabilitylevel is 10-, .s for the ve~tical bentiing

moment. The “designtr extreme value with

a risk parameter of 0.01 coIv’esponds tothe long-term exceedance p~edictlon at acomparable probability level of 10–9 .6.

3. The dynamic vertical wave bendingmoment required by ABS Rules agrees verywell with the probable extreme value byOchi’s method and is about 4 percent lessthan the exceedance prediction at 10-7.6probability level for the sample ship.This indicates that the nominal wave load

259

predicted theoretically is in good as:ee -ment with ABS Rules .

4. For the case where the long-t err.dynamic bending moment at 10-7. ‘ prob-ability level is considered, the maximumin-plane stress calculated for the sam-ple ship is comparable to the nominalpermissible stress specified by ABSRules This indicates that the long-termexceedance predictions for 20-25 yearsservice time (similarly Ochi’s probableextreme values) are in line with thosespecified by ciassific”ation societies.On the other hand, if the dynamic bendingmoment at 10-9. K probability level (also,Ochils “desiRn” extreme value) is used inevaluating siiipgirder strength, it isadvisable to cansidey using a higherstress limit which is based on extremeload considerations .

5. The H-family and the six-parame-ter wave spectral data which representthe general sea conditions of the NorthAtlantic may not be adequate for the dy-namic load calculation for shipe in otherareas OF specific ship Toutes, such ascoastal water regions , the North Pacific,etc. . For ships in a particular oceanarea other than the North Atlantic, re-presentative wave data of the particularlocation similar to the two spect?alfamilies are needed.

6. The dynamic load, in particularthe vertical bending moment of the sam-ple ship presented in this paper, is cal-culated based on linear theory of shipmotion. For ships where the effects ofnonlinear motion. such as slamminx. flareimpact, green waier, etc ., are si~iifi-cant, further study is needed in order to,aPPIY the Probabilistic lstatl$tical me-thods in the determination of hull Elrderstrength.

ACKNOWLEDGEMENT

gratitude toward the staff ;embers of theR & D Department of the American BuFeauof Shipping for their assistance and sug-gestions during the course of this study .In particular, the author’s appreciateMI-.S. G. Stiansen, Vice President of ABSfor his encoui.agement of this I’eseaFchwork, and Dr. H. Y. Jan, Dr. M. K. Ochiand ProfessoF E. V. Lewis for their re-view and invaluable comments during thepreparation of this paper.

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Nordenstrom, N. , “On Estimation of 14Long-Term Distributions of Wave-Induced Midship Bending Momentsin Ships ,11Chalmers Tekniska Hog-skola, 1963.

Band, E. G. U.,15

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Ochi, M. K. and Bolt On, W. E. , “Sta-tistics for Pvedictlon of ShipPerformance in a Seaway, 1,Interna-tional Shipbuilding Progress,February, April and September 1973.

17

Ochi, M. K., “Wave Statistics for theDesign of Ships and Ocean Struc-tures, ” -. SNAME, 1978.

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PierSon, W. J., Moskowitz, L., andMehr, E., ‘tWaveSpectra Estimatedfrom Wave Records obtained by theOWS Weather Explorer and OWS Wea-ther Report er, ,, New York Uni Ver–

sitv Research Di”ision Reoort .Pari I, November 1962, P~rt Ii,March 1963, Part III, June 196s.

American Bureau of Shipping Rules forBuilding and Classing Steel Vessels1981

Hoffman, D. , and Karst , 0. J. , “TheTheory of the Rayleigh Distributionand Some of Its Applications, rqJournal of Ship Research, Vol. 19,September 1975.

Stiansen, S. G., and Chen, H. H.,I!AppliCa~io~ Of Probabilistic De–

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Ochi, M. K. and Motter, L. I. , ‘TPr.e.diction of ExtFeme Ship Responsein Rough Seas of the North Atlan-tic ,“ International Symposium onthe Dynamics of Marine Vehiclesand Structures in Waves, Univer-sity College London, ApFil 1974.

Raff, P.., !fProgram SCORES -ShipStructural Response in Wanes ,1,Ship Structures Committee ReportSSC-230, 1972.

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submitted to the American Bureau ofShipping, April 1974.

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Hoffman. D. and Lewis . E. V. . ‘,Aria-lysis”and Interpretation o; Full-Scale Data on Midship BendingStresses of DFy Car&o Ships, rtReport SSC-196, The Ship Structurecommittee, June 1969.

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