Prediction of extreme ship responses in rough seas of the North Atlantic

Paper 20. Prediction of Extreme Ship Respo...3es in

Rough Sea8 of the North Atian

ISSUE UN

Era

boratori um vow Stheepettydromechanica

Archief

.Mekelweg 2,2828 CD Dellft of occurrence of. very seVeraitiePfkr lt?FrAtafi% tt@eliqe`35 hence it is ...cireMaure to estimate how many times the

see., .., : e

s.ap w,li staen severe seas ea. iife.jme of 20 years. Hov.,eve.14;:seve.rity of the sea in .the Atlantic Ocean .. has .been "identified ' through Oceanographic. research. For' .example. the severest.seas ever observed (measured)

-in the

.entire' Atlantic Ocean have a signifieant -wave height of approximately 45 ft (14 cm). It occurred four

times in 1.3-years in a-particular area in the North

Atlantic Ocean It is certain,. therefore. that a sea of this . severity will occur again .sometime in the future

some-where in that area; hence for a ship whose route is

expected to pass -this area, consideration must be given in the design so that she will be safe in the event she

enconiiiers-this severe sea.

-The procedure for ptedietion is as follows:

(i). .Fifst, estimate the 'severest sea condition expected in the .serVice'area of the 'ship.- Evaluate the extreme values .in thiS and various 'other seas below this severest condi-tion: Estimate the .duration of each storm from the available 'data. since the e,xtreme-valtie is a -funetion of ship operation time in the storm.'

EStii-nate.the.ship-speed in each sea assuming that the ship is headed into the sea-whiCh is -the usual situation .v.,:hen:.a'Ship encounters a severe storm. In estimating the 'speed, tWo. factors' should be considered': the natural eed 'reduction, due to the added resistance of the ship. in seaWay; and the voluntary speed reduction by the ship's eaplain.in 'order to ease severe motions.

(iii). Then. -estimate by applying order- statistics- the ex-treme:Value. of ship responses for a given sea state. dura--.don, and ship speed.

(iv) Estimate two 'extreme :Value's: that which is 'most likely to occur Under the situation _considered; and that which is unlikely to, occur with a preassigned. assurance.. The latter isethe -yalne' for Which the ship will be safe while she is operating in theservice areas, and it maybe used for design consideration.

As. an -example of the 'apPlication of this method, numerical' 'calculations. are made for the light and full

draught conditionSof the.- Mariner (526 ft; 160.m).

-Extreme Values of pitch' and heave motions, vertical

acceleration of the bow.., and wave-induced bending -

-moment (for- full draught only) are obtained. Long-crested seas' .having the Bretschneider two-parameter wave spectium are used, sincLt.ship responses in long-

-crested head seas are greater than in short--crested seas. However, for the wave-induced bending moment, a

comparison between long- and short-crested seas is made.

M. K. Ochi*.and E.yotter*

SYNOPSIS

This paper presents a. method to predict extreme ship responses id-,severe S-c:Os The severity and duration of rough seas are estiMated from analysis of available data observed., in the North. Atlantic Ocean: Ship speed in each transient stage of the storm is estimated.by taking into account two faCtors: the natural speed reduction due ' to the adeneel.'resistanee Of a ship in a seaway; and the voluntary-specd reduction by the ship's captain in order to ease severe motions. Extreme responses most likely to occur, as well as thoSe to be used for design considera-tion, are estimated. applying order statistics As an example, numerical calculation is made for light and

full draught ,:. (Itiszlitions of the Mariner (526 ft; '160m), and extreme values of pitch and 'heave motions, vertical acceleration at the bow; and wave-induced hull-bending moment (fOr full draught only) are obtained. Also, .a comparison of he extreme values for long and shorte

creasted seas:is:made for the bending moment.

INTRODUCTIONS

RECENT .P0.Grti--.1.Ss in application of statistics to naval engineering, pertnitS the estimation of the magnitudes of.

ship responses 'such as motions and wave-induced bend-. ing moment,at an early design stage. Perhaps one of the incist useful pieces of information to be estimated at the desigi stage is the..extrerne value of ship responses. Here, the extreme Value is .defined as the largest value which the ship n3.ayeexperience in her lifetime. . .

TwO..different approaches may be applied in predicting extreme ;Values: one involves the long-term prediction 'method, the other. short-term. The Ione:term prediction considers ail variations. of the response at every cycle of . waVe, 'encounter. :regardless of their Magnitnde..The

disadvantage of applying this method in estimating eitrerne.ivalue is that a .cdrisiderable percentage of small

magnitudes of response in relatively mild seas (whimh ii.appiiittitly do not contribute to the extreme value) are. "Ir _{included :in .the estimation. As will be seen later in a}

numerical..example the magnitude of ship response sAll t. not reach.a-level critical for safe navigation irrespective .orhow long she operates in mild seas, while the magnii.ude will reach the critical level within a few hours in.sel.iere seas Therefore, as far as estimation. of extreme vaiue.s. is -concerned, it appears to be appropriate to consider only severe seas and apply the short-term prediction method. This paper discusses a method to 'predict exiTeme Motions and wave-induced bending moments of a ship. .operating in rough seas in the North Atlantic Ocean. The underlying concept of the present method is that the ship has to be. designed to be safe in the event she encounters the severest storm expected on the .service 'route. It is rioted, however, that insufficient information is .available at present to statistically estimate the frequency

The An of this paper 'was accepted for publication on 4th January 1974

" _{Ship Performance Department, Naval Ship Research and Development}

Center; Bethseda, Maryland 2004, USA.

1:1' Notation

g _{Gravitational acceleration}

coo Encounter

frequency- of ship and waves

[= V(ni2ini0)].

.-Area under the speetral density ftinction of bow

motion . . . .

, Area under 2nd moment of spectral density

function of bow motion _

f'

Effective freeboard

f

Actual freeboard

Ship length Ship breadth

LE Length of the entrance on the waterplane Froude number

Ar0 Amplitude of ship vertical motion relative to

waves at f.p.

/1,3 Amplitude of ship vertical motion relative to waves at Station 3

/1,3 Amplitude of ship vertical velocity relative to waves at Station 3

Aso Amplitude of absolute ship motion at f.p. d3 Ship draught at Station 3

r0(t) Relative motion at f.p. so(t) Absolute motion at f.p.

Positive peak value of a steady-state random process (0 < co)

Bandwidth parameter (24) Extreme value in n observations

2

ESTIMATION OF SEVEREST SEA

STATES IN THE NORTH ATLANTIC

OCEAN

In order to evaluate the extreme values of ship motions and wave-induced bending moments for design con,-sideration, it is first necessary to estimate severe sea _states and their duration in the service areaof the ship. It is known from available literature regarding sea states. in the North Atlantic Ocean (1)*--(11) that very severe seas (State 8) are expected in winter near Station I (19W and 59 N). and J (20 W and 52 N). Although. sea states in the middle Atlantic Ocean, in theneighbourhood of 30 N, are much less severe than Stations I. and J, 'a significant wave height of 30 ft (9 m) is expected in the winter season. It is also -generally known that the'waves

in the East Atlantic are higher and longer than in the

western region of the ocean.

Roll (9) and Walden (10) have presented the regional anomalies'. of wave. heights from the data accuitulated by weather ships. For example. the contour-lines labelled with numbers in Fig. 1 are those obtained by Walden (10). The numbers. indicate the sea severity- at the individual locations as a percentage of the eombined average of all

-':0

:: w Al , CD j 1 . I ;i5.;.' -''' , _m ,

i

r.-' . \-',3 k

r'

I. L...,4c.,, ,f \ i --.

\

....

/I

--,.-?---;,ii...,. :),.-,1

(f-

(I `-'7'...1,41

6

, . , 8 I //I r '''. iI), I f ,r ,,II ,i ("' \ ( , " I ' 1 -r ' ' -

' i

' " I,

1 - ç)I

^- . 1 I . -

c,"(r

,j,./i.ip-r-S).)1 ) ) , 1 r;.."-.1;;7 ";://iiijc,.. ?...--\rr r77 N k'-'. !, L."--z"7,----.

r

---1,,,,---;'Y .--/ I , . ?-;---, (....,'...' -.1.) is.\. .,,, ; ..--- , ').v:_-! -\ -,., ,,. ,....-,-:. ,.

,,i..

,.._-_,...,,-- _..,..,,, ,..--.--, (---.77. ..\\,..,

'

Nrff- -i14-- ---: ..- ' / 1 ' C.NT,.42. I.:N,., I,2E D

...-1

!..?!'?! JJ.f.

Hog-1--,f,I YrPtly. ii,_FCS'S D.\ l'A

I lilatEST SF..:.11::").:;1 OVE :".7;T-'1'. .. - - f 1 '

7

'.11 7( 1.2 W . W 2C 73

Fig. 1. Zone of equal highest significant wave height

Reference.i: are given in Appenthly, 2.. 200

observations. Based on Walden's information shown in

Fig. 1, the zone in which the highest significant wave

heights are nearly equal will be determined. For this purpose Station K will be considered as a reference point.. The highest significant wave height measured at Station K is 35.1 ft (10.7) (2). Then; using Walden's numbers indicating wave severity, the highest significant wave height at Stations I and J are 47.2 ft (14.4 m) and 44.5 ft (13.6 m), respectively. On the other hand, the actually measured highest significant wave height was 44.8 ft (13.66 m) at Station 1(4) and 47.0 ft (14.34m) at Station I .(3). If the significant wave heights are classified into zones

of every 5-ft interval, then Stations I and J are both categorized in a 45-ft zone (42.5-47.5 ft) as compared with a 35-ft zone (32.5-37.5 ft) for Station K. Four zones ranging 30 ft to 45 ft drawn in Fig. I are thus deterinined in which the expected highest significant wave heights are the indiCated values.

Since the extreme value of ship response is a function of ship operation time in a given sea, the duration of each

sea state has to be considered. For this purpose, 15.

minute samples of data obtained every: 3 hours during storms (1). (2) and (3) are analyzed and the duration of each sea state obtained. Unfortunately, the analysis is limited to significant wave heights higher than 204

(6.1 m). since the duration of seas less than this could not be determined explicitly due to the lack of sufficient information. However, this does not cause a problem for

the prediction of extreme values, since, such values increaSe significantly for the first tea-hours (approx. imately) and then increase very slowly thereafter, as will be shown later in connection with Fig. 16. Thus, the time duration necessary for prediction in seas of significant WaVe height of 15 ft (4.6m) will be estimated by extra-polating the available information.

Figs 2 and 3 show examples of variation of significant -wave height observed at Stations Jand K, respectively.

0 6 12 18 0 6 12 08 0 6

TIME

I--OEC. 10, 1958-1 DEC. 11 DEC.

Fig. 2. Variation of significant wave height observed at

Station J in December 1958 (2)

As can be seen in Fig. 2. rough Seas of significant wave height of 20 ft (6.1 m) or higher continued for nearly 2-2'7 days in December 1958 at Station J. In this severe seJ condition, significant wave heights of 30 ft (9.15 ni) or higher lasted for 17 h in which the sea reached. a levelei 45 ft (13.7 m) for 3 h. On the other hand, in March 1956.,

the severest storm ever observed at Station K lastedfor.

several days in which seas of significant wave height 20 It (6.1 m) or higher continued for 38 h 'including4111

for which significant wave. height was at the 35 ft (10.7 level (within 32.5-37.5 ft) as can be seen in Fig. 3. Intin=

r . 1 r_____ 1 . 1 _,.. 1

pr.

1_ ..../A 1 EITNTto I

r

i i 1 I I__ 4...---_I__ 1:__ 1

-in ave his int. ion ers ave 5 ft .8 ft lly n J nes oth red nes ned hts lion ach 15-ed a

storm, for a relatively large percentage of the time (22 h), the significant wave height was at the 30 ft (9.15 m) level ring _{as compared with only 6 h at the 25 ft (7.6 m) level. A} n of _{imilar analysis was made on available data of severe}

IS _{conditions. and the results are summarized in Fig. 4}

20 ft _{in which the duration is plotted for every 5 ft-interval of} not significant wave height.

ient for lues rox-1 40 Will g time cant z 3° tral 50 o 40 20 10

PREDICTION OF EXTREME SHIP RESPONSE

SEA STATE

4. Level of significant wave height and its persistence in the North Atlantic Ocean

Since the duration of each sea state varies randomly. the envelope of the time duration may be drawn in Fig. 4. and it can be considered as the longest duration of each 5-ft interval of significant wave height. For example, a sea of significant wave height in the 30 ft (9.15m) level Would not last more than 25h during one storm; hence

it _{is sufficient to evaluate the extreme values of ship}

way responses for 25 h of ship operation in thissea. For seas

1

earl of significant wave height of 15 ft (4.6 ni), a 45-h duration e se will be used by extending the envelope in Fig. 4 aswas

) o stated earlier.

j

vet0 _{Next, it is necessary to determine the wave period for}

19.5 _{which the wave spectrum peaks (modal period) for each} d fot significant

wave height in order to establish a

two-ht _{parameter wave spectrum. For this, the modal periods} 4-14 of the_{wave spectra observed at Station 1(4) are averaged}

7 ilsi for the same significant wave height.-and the results are

n thiishoi,vn in Fig. 5. Included also in the figure are the

...

IN ROUGH SEAS OF tHE NORTH ATLANTIC

theoretical modal periods for the Pierson- Moskowitz wave spectra. As can be seen in the figure. the observed peak periods are consistently lower than those given by

PiersonMoskowitz for severe seas; for this reason

Brctschneider's two-parameter wave spectrum will be used in the present study with the relationship between modal periods and significant wave height given by the Station I data in Fig. 5.

15 20 25

SIGNIFICANT

Table I summarizes the modal period and duration of each significant wave height used for evaluating the extreme value of ship responses in the present study. As can be seen in the table. a significant wave height of the 45 ft (13.7 m) level is the severest in the entire North Atlantic Ocean; seas of this severity were observed four times in 13 years (1954-66), twice at Stations I and J each, and the perseverance of these storms was not more than 5 h in each occurrence. As was mentioned in the Intro-duction. it

is premature from the available data to

surmise how frequently this level of severity may be

repeated. and hence how many times the ship will

encounter this sea in her lifetime of 20 years. However. it is certain that seas of this severity will occur again sometime in

the future somewhere in the domain

labelled as the 45-ft zone in Fig. 1. Hence, for a ship whose route is expected to cross the Atlantic Ocean just south of Iceland, consideration must be given in the design so that the ship may navigate safely in the event she encounters this severe storm. For a ship expected_to take the southern route passing near Station K, the severity of the sea for design consideration may be

relaxed to the level of 35 ft: however, the duration of this sea has to be extended to 15h.

Fig. _{5. Relationship between significant wave .height and}

modal period in the North Atlantic Ocean

Table 1. Significant wave height, modal period, and

. duration used in the computation

Level of significant Range. Modal period, Duration,

wave height. ft It s h 45 42.5-473 15.8 5 40 37.5-42.5 15.3 10 35 323-37.5 14.5 15 ! P I ERSON- MOSKOWITZ-. 41110...r.11011

loorei.

111 STATION I L_eir

III.

ENVELOPE 30 27.5-32.5 13.6 25 25 22.5-27.5 12.6 35 20 17.5-223 11.5 42 15 12.5-17.5 103 45 30 35 40 45

`NAVE HEIGHT IN. FEET,

20 25 30 35 40 45

SiGNIFIGANT WAVE HEIGHT IN FT.

18 12 18 0 12 18

TIME

MAR. 21, 1956 22

Fig. 3- Variation of significant wave height observed at

Station K in March 1956 (2)

3

_{ESTIMATION OF SHIP SPEED IN A}

SEAWAY

Another prerequisite for predicting the extreme value of a ship responses in a seaway is the estimation of ship speed in each sea state. This is particularly important for evaluating ship motion which is extremely sensitive L\. tp speed in contrast to the wave-induced bending moment.

Two types of speed reduction in a seaway must be considered: one due to the added resistance and reduced propulsive efficiency associated with waves and ship motions; the other due to a voluntary slow-down by the ship's captain in order to ease severe ship motions. The former may be called the natural speed reduction which is experienced in all sea states, and it depends on the hull form and the seakeeping characteristics of a ship. On the other hand, the latter is the speed loss unavoidable in relatively severe seas, and it depends on the personal judgment of a ship's operator. The procedure for estimat-ing these two types of speed reduction will be discussed separately.

3.1

Speed reduction due to added

resistance in a seaway

The speed reduction due to the added resistance and

reduced propulsive efficiency in a seway may be estimated

at the design stage from available theories on added resistance in waves (12)(15). Recently, Strom-Tejsen etal. (16) have made a comparison between experimentally obtained added resistances and predicted values com-puted by various methods, and found that the available theories are sufficiently accurate for practical application. In particular, the method developed by Gerritsma and Btfekelman (12) appears to provide a prediction technique

equally accurate for several ship forms used in their comparison. Hence, Gerritsma and Bdtkelman's method will be used in evaluating the added resistance of the Mariner, used in the present study. The body plan and the characteristics of the Mariner are givenin Fig. 6 and

W '6 2 6 67...

t

202

Fig. 6. Body plan of the Mariner

Table 2 respectively. Included also in Table 2 are the conditions for which the computation was carried out.

Fin. 7 shows the e.h.p. in various sea states for the full draught condition, Which is evaluated as a summation of the calm water e.h.p. and the computed added e.h.p. in

2

Table Z Dimensions and general characteristics of

Mariner 563.6 ft 528.0 ft 76.0 ft 44.5 ft 29.8 ft 20 knots 0.624 0.635 0.745 0.983 29.8 ft 29.8 ft 16.4 ft 23.6 ft 20 30 40

SIGNIFICANT WAVE HEIGHT IN FEET

00

SEA STATE

Fig. 8. Ship speed in various sea states (Mariner, full

draught)

waves. Since the e.h.p. in calm water is 10 370hp for the

design speed of 20 knots, the speeds in various sea

states for a given constant 10 370 hp are read off from the figure. The ship speed thus obtained is shown in Fig. 8 identified as the 'natural speed reduction from

e.h.p. in waves'.

Another method of estimating the natural speed

reduction is based on s.h.p. using the efficiency e.h.p./s.h.p.

1.

A

4 0. 4 N-. k C) (t,4-) 0. k :'2 i%.

/A

4

,

..,,

.

C3 _*.

Agrri

,

4 _,,4

vt--10,4111

DESIGN SPEED

i

IASI V:. SPEED I I I

r--

_I VOLUNTARY I SPEED REDUCTION 1 ---,,,..

inal-i i I I NATURAL SPEED ,RECUSTI.ON

(FROM EHP IN WAVES)

---.(FROM SHP IN WAVES) -- _SHIP IN A SEAWAY SPEED 11113 r 1 ...

Ina=

., . A "La ...MA ..". .,, 1 , ' ie_.-14.P.I49.i.lb;r3M I i , 1 ..., CEV-Clilla"kI\ . I

i I1:"

.

Wank

I * ' I .,f / ,I /. ' ./I ii

/

--.1 I

s ..,\ 1 i I .'"- r7 I . %IV 1/ Imilim. /II

I 111111

Mk \ `'

i ,,,_:: ` 4 I .E., I , , -, ...-. cec.cr FE '-1 I1,i . 1 l'.;.:t.a e.s:1....xLIE pa 0:0iAlf _{' ' 1 r-cr '} I -111K. 1 I I _.i \ ... ''. ' I i! I: . 7 1--₁

Eft N.

1....ii.' I .1 i

..,\

/ 1II

' 7' ..__\LLse.k4A.A.)-__11221, _rtrittie.p.ka ,..,.. ....),_,...._..,_,1 '_{...i.);_....6aboi:...f6L.L._} 2 14 16 16 20

SHIP SPEED IN KNOTS

Fig. 7. E. h. p in various sea states as a function of ship speed (Mariner, full draught)

Length overall

Length between perpendiculars Breadth, maximum. moulded Depth, main deck at side

Draught, maximum

Speed, sustained at 29.8 ft draught Block coefficient

Prismatic coefficient

Midship coefficient

Water-plane coefficient Computations

(Full draught) Draught fore

Draught aft

(Light draught) Draught fore

Draught aft 111, 666 rm. a 1 10 8

a

6 4 2

e. in various sea states. Unfortunately, no data regarding

the efficiency in waves is available for the Mariner except value 0.73 obtained in calm water. The estimation, ,iiowever. may be made by using the data obtained on the Jordaens by Aertssen (17). Since the stern profile of the Jordaens is very similar to that of the Mariner, the added s.h.p. of the Mariner in waves can be estimated from the information on the propulsive efficiency e.h.p./d.h.p. in various sea states and the shafting efficiency d.h.p./s.h.p. obtained on the Jordaens. Then, by the same procedure as for e.h.p. in waves, the speeds in various sea states can be estimated for? constant s.h.p. of 14 200 hp, which is required for the design speed of 20 knots in calm water.

The speed in waves thus obtained is also included in

Fig. 8 labelled as the 'natural speed reduction from s.h.p. in waves'.

3.2

Voluntary speed reduction in a

seaway

.

The voluntary speed reduction in a sesway may be

predicted at the design stage if we estimate the attainable maximum speed below which the severity of ship motions and behaviour are acceptable to the ship's captain. One I.hod to determine the criteria for this acceptable level of severity is to analyse results of data obtained from full-scale trials s'o that the limiting severity is expressed in terms of probability. Among others, results obtained from a series of full-scale trials carried out by Aertssen will be used to establish the criteria, since his results provide comprehensive information concerning the relationship between speed reduction, power, and ship performance in various sea states (17H19).

From the Aertssen series of full-scale trials on service performance, it is understood that voluntary reduction in speed is expected in rough seas when at least one of the following conditions occurs. These are:

Number of occurrences of appreciable slamming reaches a certain limit. It is estimated that three times in every 100 pitch oscillations appears to be reasonable.

Significant value of the bow acceleration reaches a certain limiting level. 0.4 g in amplitude.

Number of occurrences of deck wetness reaches a certain limit, seven times in every 100 pitch oscillations.

(ri, \ Number of occurrences of propeller emergence :hes a certain limit which appears to be a function of loading condition and sea state.

In particular. Aertssen's analyses have indicated that Items (b) and (c) are the dominant parameters which

contribute to speed reduction for the full draught

condition, while Items (a) and (b) are those for the light draught condition. Hence, the criteria for the acceptable limit below which no voluntary speed reduction is expected may be written in terms of the probabilities (abbreviated Pr). These are:

[For full draught]

Pr {Occurrence of deck wetness}:.<. 0.07 and/or

Pr{Significant ampl.of>.-.. 0.4g} < Preassigned

bow acceleration specified value 41) [For light draught]

PREDICTION OF EXTREME SHIP RESPONSES IN ROUGH SEAS OF THE NORTH ATLANTIC

Pr

I

Occurrence of

slam impact < 0.03 and/or

pr

Significant amp' of

0.4 g Preassigned

bow acceleration

'

specified value ' 42) 1

In order to express the above probabilities in mathe-matical formulae for computation, the following con-siderations will be given:

In evaluating the probability of occurrence of one and/or the other event, the joint probability of the two events will be considered. This is necessary since two events, deck wetness and high acceleration at the bow for example. are not statistically independent: hence the effect of correlation between them has to be included in the evaluation.

The probability concerning the significant amplitude of acceleration in the criteria will be converted to the probability of amplitude of vertical motion at the bow for the convenience of evaluating the joint probability. For full draught condition, the preassigned specified value involved in the criteria on bow acceleration is

chosen as the same probability as that specified for deck wetness, i.e. 0.07, so that the two conditions given in equation (1) can be expressed in a single joint probability function. Similarly, for the light draught condition, the preassigned probability is chosen as 0.03, the same as for slamming. This yields the following formulae which can be replaced for the 2nd condition given in equations (1) and (2), respectively:

[For fiill draught] Significant arnpl. of

Pr. -

0.4g} bow acceleration Vertical bow 0.459g

= Pr

;-'. < 0.07 ...(3) motion _(Do

[For light draught] Significant ampl. of

0A4

bow acceleration Vertical bow 0.527g

= Pr

2 < 0.03 ...(4) motion

The derivation of the above two formulae is given in Appendix 1 of this paper.

The effective freeboard is considered in evaluating

the probability of deck wetness. For this, Tasaki's

empirical formula established through model experiments will be used (20):

f' = f

(0.75) L

(I)

.9.2 LE

In Aertssen's criteria, slamming is not defined as every impact delivered on the 'bottom but as a severe impact which may subsequently cause the captain to reduce speed in storm weather. For example, severe impacts which induced a whipping stress of 5.9 MN/al' (850 lbf/in2) at the midship was considered as slamming. Hence it may be appropriate to evaluate the probability of occurrence of impact at Station 3. This is because

results of model experiments have shown that an impact

Pr

{

and/or { ro

= Pr A > f' l_J Aso >

0.459g}2

c)o

= 1 - Pr

_{{Ao < f, A50r} ' < 0.459g) .. .. 0.07 [For light draught]

Pr Occurrence of slam impact

and/or Significant amplitude of bow

-acceleration will exceed 0.4g 0.527g1

= Pr

I/4,3 > d3, A13 A50> (02

0 J

= 1 - Pr {Ai.3 <r*} Pr Ar3 < d3, A

I

0.527g}2

0o

....5. 0.03 ...(7)

It is noted in equation (7) that relative motion and velocity are both normal random processes and hence they are treated as statistically independent.

The ship speeds which satisfy the equality in the above

eqnations are the maximum attainable speeds below which no voluntary speed reduction is expected for full and light draught conditions. respectively.

In evaluating eqUations (6) and (7), however, the joint probabilities (Aro and Aso) and (Ar3 and Aso) have to be obtained. In general, application of the multiple joint Rayleigh probability function to the problem of speed loss in a seaway pertinent to all draught conditions is rather complicated. This will be discussed in detail in a separate paper. For the present problem, however, only two variables are considered for each full and light draught condition. Hence, the two-dimensional joint Rayleigh probability function will be applied assuming that the relative motion as well as the absolute vertical motion are narrow-banded normal random processes.

Consider the case of full draught condition,and let the

relative and asbsolute motions at f.p., ro(r) and so(t),

respectively, be written as follows:

1.ro(c) = A0(t) cos { wet + cro(t)} i

1 = rop) cos eV r(t) sin co,t

isc(?)

= .4(r) cos

{Wet_{t + cso(01}

= sop) coseact

-

so(t)_{sin Jet}

Then. tile two-dimensiona! joint Rayleigh probability /04

near Station 2 and forward does not cause appreciable hull response: but whenever an impact is applied near Station 3. impacts are also delivered to locations further forward and this produces an appreciable hull response. Thus. taking these four considerations into account, the criteria for voluntary speed reduction can be

ex-pressed as follows: [For full draught]

Pr.

{Occurrence of

deck wetness

1 Significant amplitude of bow acceleration will exceed 0.4g

cot) J

VEHICLES

density function, .4,0 and Aso. applicable to the _present case can be written as follows (21)-(23):

f(A A Ar0 AsO

(

p

AroA so) ,o, so) -0 1 p2 0.030.4

P =

E p2)20."00.1 ,./(Cr25 + Qr2s) 6r06s0

at% = var [ro]

var [roj

= var [sos] = var [sos]

Crs = area under the co-spectrum of r 0(t) and s0(t)

= area under the quadrature spectrum ofro(t1

andso(t)

I_1_ I_II%0_2 0.2_1.0O cr.°. s20 no. s20)

s

1

d33 (0.2 .2 ors0.0)

1E1, r00 50

10( )= modified Bessel function of the zero-th order. From equation (9), the joint probability involved in equation (6) can be evaluated by

0A591. 6t0

{

Pr A,0

_{.0 < f', Aso <}

-P0.459gIco3 ./.'

=

jf (A

ro, Aso) dAro dAso ...(10)

0 o

The joint probability involved in equation (7) can also be solved in a similar fashion.

A numerical calculation of the voluntary speed

reduction was carried out on the Mariner in various sea states for the original freeboard f = 36.7 ft (11.2m). and the result is shown in Fig. 8. As can be seen in the figure, ship speed in a seaway is expressed by a combina-tion of two speed-reduccombina-tion curves (the natural and the voluntary). The voluntary speed reduction does not come

into effect for the Mariner in seas below a high S ca State 6, but the reduction is significant inseas of State 7

or higher. _.

A similar calculation of the speed in a seway was made for the light draught condition of the Mariner, and the result is shown in Fig. 9.

4

_{EVALUATION OF EXTREME VALUES}

In the previous two sections, severity and duration of sea states in the North Atlantic Ocean and sustained ship speed in each sea have been obtained. Extreme values ol ship responses can now be evaluated by applying order statistics to the probability function which represents the statistical properties of ship responses for a giVCD

speed in a given sea. The Rayleigh probability function is

x exp (-.(d1 1A0 + d334)}

a 0 Crs

_Qrs

0

Q

Crs Crs Qrs Cr2s0 0

QC,5 0

_tsO

I

where

Z=

ANIMA NEGATIVE)

PREDICTION OF EXTREME SHIP RESPONSES

10 20 s0 40

SIGNIFICANT WAVE HEIGHT IN FEET

1-0-4®

I ®

-SEA STATE

Fig. 9. Ship speed in various sea states (Mariner, light

draught)

often used for evaluating the statistical characteristics of

the maxima (peak values) of a steady-state random process. The Rayleigh probability function, however,

.1:.4;-!d

been developed under the assumption that the

iLponse spectrum

is narrow-bandedwhich is not

always the case in practice.

The probability function Of the maxima of a random process having an arbitrary bandwidth spectrum. a, is given in references (24), (25). _{while the probability} function of the positive maxima, defined as the peak value which will occur throughout the range of 0 to so, is discussed in reference (26). That is, the probability density function of the positive peak values (see Fig. 10) is given by f() =--1 ± ,j2ir 2s2 inoi

1/ m

0 a eXp). \12 )2

vi(1 e2) exp

v"7o.

x11

(

-

4

e2)

e )1]

...(11) a NI mol

where

= positive peak value (0 < cc) E = bandwidth parameter (24) 1 cD(u) =

v2n

Lc.

exp du 2 2 MAMMA (POSITIVE) MAXIMA (NEGATIVE)

Fig. 10. Explanatory sketch of a random process

IN ROUGH SEAS OF THE NORTH ATLANTIC

By applying order statistics, the probability .density function of the extreme 'value (the largest peak value) denoted by y. in a observations becomes

g(y) = _

(12) where

F() =

f'j()dc:

From the above equation, the extreme value which is most likely to occur in n observations (the most probable extreme value) can be obtained as the modal value of .g(7). Fig. 11 is an explanatory sketch showing the

CC C6' EXTREME VALUE A Y 'n YIIa)

Most Probable Extreme Value for

Extreme Value Design Consideration

Fig. 11. Explanatory sketch of probability density function

of extreme value

probability density function of the extreme value. The modal value denoted by peaks the probability density function. and it is obtained by letting the derivative of equation (12) with respect to ;/. be zero. That is

g(y)=O

= 0 ...(13)

dy. "

On the other hand, the expected number Of positive maxima in T hours can be expressed by (26):.

TI + ,/(1

a2) im2

n (60)2 ...(14)

47r .,j(1 E2) V m.

Then, from equations (11) through (14) the most probable extreme value. can be written as a function of time T by

(m)27. m

5;.

2

Ni2

In

\/P)1,

.17770 for amplitude . -(15)

7r _mo

Since this extreme value is Most likely to occur, it may be reasonable to compare with the largest vaiue obtained in model experiments or full-scale trials. However, the probability that an extreme value larger than will occur is rather high. It can be proved that. if the number of observations (wave encounters) is large. then the probability that the extreme value will exceed

.7). is theoretically 1 = 0.632, regardless of the

spectrum band.width. It appears, therefore, that the most probable extreme value is too low to be

con-sidered for engineering design consideration. In order to

-1-INATU-RAL

SPEED REDUCTION

.

.

-''..e.1.--,(FOM EHP IN WAVES) l'-(FROM S.HP IN WAVES) I 1 ! ! I 1 k___ I -'SHIP SPEED IN A SEAWAY i I I I I i i r-bESISN I SPEED, I VOLUNTARY . SPEED i'EDUCTION 16 2!' 12 La 8 LJ 0.

provide a certain amount of margin above Z, for safe design, we may evaluate another extreme value, denoted by 9.(cc) shown in Fig. 11, for which the probability of being exceeded is a preassigned small value a. That is, 7(2) may be obtained by finding the solution of the

following equation:

g(y0) dy..= a ...(16)

Under the assumption that n is large and s 0.9, the solution is given by (26):

ni2

fl(c()= \112111[(6

)2T

22ct

NIGT31

°

(,IM) for amplitude ...(17) cc in the above formula is specified by the designer; hence it may be called the extreme value for design considera-tion. If a is chosen to be 0.01, then it is possible to estimate

the extreme value which is unlikely to occur with 99 per cent assurance. This implies that one ship in 100

sister ships operating in the same statistical environment may experience a value greater than the predicted value 2). Another interpretation may be made: that a ship

may experience a greater value than the predicted

extreme value once in 100 encounters in the same

statistical environment. If greater assurance for safety is required, one may choose a smaller a depending on the

particular situation. However, from a comparison of the predicted and observed extreme values carried out in the model experiments, a = 0.01 appears to be reasonably safe (26).

Figure 12 shows the most probable extreme values-and the extreme values for the design consideration of

pitching motions (amplitudes) of Mariner in various sea

16

206

SIGNIFICANT WAVE HEIGHT IN FEET

SEA STATE

Fig. 12. Significant and extreme values of pitching motion

in various sea states (Mariner)

states for the full and light draught conditions. Included also in the figure arc the .significant pitch.ainplitudes for reference. It is noted that ship speeds in each sea state and the. duration of the sea used in the computation are different: namely ship speed is significantly low and ship operation time is short in severe seas. Nevertheless, the

extreme values of pitching motion in severe seas are much greater than those in mild seas. A similar trend can be observed for the heaving motion as is shown in Fig. 13. However, vertical acceleration at the bow which is the 2nd derivative of the combined pitch and heave motions, including the phases between them, shows an

40 30 Ui 2 t: 20 0. L., 10 Ui 0 10 20 30 40 SIGNIFICANT WAVE HEIGHT IN FEET

SEA STATE

Fig. 13. Significant and extreme values of heaving motion

in various sea states (Mariner)

FULL DRAFT LIGHT DRAFT

:1.01

= 0.8 0 Ui 0.2 MOST PROBABLE EXTREME YALU:. '

EXTREME VALUE FOR DESIGN CONSIDERATION (cc =0.01.) I I 1 I SIGNIFICANT i ' ! 1---! i 1 _j______ 1 [ I I. 10 20 30 40

SIGNIFICANT WAVE HEIGHT IN FEET

.SEA STATE

Fig. 14, Significant and extreme values of vertical accelera-tion at the bow in various sea .states (Mariner)

entirely different trend as can be seen in Fig. 14. That is, the extreme values of the acceleration in severe seas are somewhat less than those in mild seas. This is because one of the conditions in determining ship speed in a

seaway is the limiting of the significant bow acceleration to a certain level. 0.4g. This implies that (vinio) equations (15) and (17) is not significantly different in all se.a states. On the other hand, the. number of en-counters with waves, n, involved in the equations is. larger in mild seas; this results in somewhat larger

extreme values in mild seas than in severe seas.

FULL DRAFT DRAFT

---LIGHT

1

(it.

EXTREME DESIGN 0 VALUE 01) CONSIDERATION FOR ' PROBABLE MOST EXTREME VALUE . ...---"--SIGNIFICANT .1 I I I EXTREME VALUE 1 I FOR F---1 ;---. DESIGN ZONSIDERATION (i:t =.0.01)1

\ !

I ! 1 JrLOST I PROBABLE 1 .

/

EXTREME ; VALUE I

\

! 1 DRAFT SIGNIFICANT FULL i --- LIGHT DRAFT I I

In order to check the validity of the computed results,

3comparison is made between predicted and measured

-",..tes. Figure 15 shows a comparison of significant aMplitudes ofpitching motion in various sea states for light draught condition. Open circles in the figure are those obtained from model experiments carried out in irregular seas generated in the towing tank (27). Although the shape of the sea spectra generated in the towing tank

issomewhatdifferent from that Used in thc computation,

rood agreement can be seen between computed and

measured significant pitching motions. Extreme values were measured during experiments in irregular seas of

significant Wave height 31.2 ft (9.5m) for 30-minute operation (converted to full-scale) at 10-knot speed

see Table 2 in reference (27). Hence, the most probable extreme values are computed for the same significant wave height, ship speed and operation time but using

9

6

o

PREDICTION OF EXTREME SHIP RESPONSES

10 10 30 40

SIGNIFICANT WAVE HEIGHT IN FEET

SEA STATE

F;g. 15. Comparison between computed and observed

significant pitching motion in various sea states (Mariner, light draught)

Table 3. Comparison of predicted and observed extreme values (amplitudes) (Mariner, significant wave height

31.2 ft, ship speed 10 knots, 30-minute observation)

Bretschneider's wave spectra. Table 3 shows the results of comparison. As can be seen in the table, the predicted

most-probable extreme values agree reasonably well %nil those measured. The maximum discrepancy is of the order of 13 per cent in this example.

As an example of the rate of increase of the extreme values with time, the extreme values ofpitchingmotion a a Sea State 7 (significant wave height 25 ft; 7.6 m) are thown in Fig. 16. Ascan be seen in the figure, the extreme

IN ROUGH SEAS OF THE NORTH ATLANTIC

values increase significantly during the first ten hours (approximately) and thereafter increase very slowly with time for this size of ship.

As mentioned earlier, the computations are made in seas -categorized by Bretschneider's two-parameter wave spectrum. Hence, it may be of considerable interest to compare the extreme values evaluated by using these sea spectra with those using the spectra actually measured

TIME IN HOURS

Fig. 16. Extreme values of pitching motion as a function of

time (Mariner, light draught, significant ways, height 25 ft)

at Station I. In particular, it is of interest to compare the results in long and short-crested seas, since the computa-tions so far have been limited to long-crested seas.

Figures 17 and 18 show comparisons of the extreme values of the wave-induced bending moments in head long and short-crested seas using 230 wave spectra

MOST PR-03A3LE VALUE

10 20 30 40 50

L006-MSTD).SEAS. STATION I

.0 .SNORT-CRESTED:WAS. SiATICI I

FOR DESIGN CONSIDERATION

(cc 0.01)

60 70

SEA STATE

Fig. 17. Comparison between the most probable extreme bending moment using two-parameter long-crested sea

spectra and measured spectra at Station I (Mariner , full

draught)

measured at Station I by Weather Reporter & Weather Adviser (4). The cosine-squared spreading function is used to obtain the short-crested seas, and the results of computation in seas of significant wave height greater than 15 ft (4.6 m) are shown in the figures. It is noted that the ship may move somewhat faster in head short-crested seas than in long-crested seas. However, speed in the

La

Vertical

_Pitch, Heave, acceleration degrees ft at Station 2,

Observed extreme Lieu

draught

value

Predicted most probable

9.60 19.3 0.92 extrdme value .- 9.84 17.1 0.80 Observed extreme

draught

value

Predicted most probable

8.85 173 0.80 extreme.value 9.94 18.2 0.77

10 20 30 40

mune

NAVE MISFIT IN FEET

I.% 12 La E10 8 x 6 o-4 w ce LI

short-crested seas is assumed the same as that in long-crested seas for the same significantwave height, since the magnitude of wave-induced bending moment does not change significantly with ship speed.

As can be seen in Figs. 17 and 18, the computed bending moments in lone-crested seas using the Bret-schneider two-parameter wave spectrum give the average

of those in long-crested seas using the wave spectra

measured at Station I for the significant wave height less than 30 ft (9.15 m). For significant wave height above 30 ft, however, the two-parameter spectra give somewhat

691 STATE

Fig. 18. Comparison between the extreme value of bending

moment for design consideration using two-parameter

long-crested sea spectra and measured spectra at Station I (Mariner, full draught)

(about 5 to 15 per cent) higher values than the_measured spectra. This is because the actually measured _spectra are more narrow-banded than the two-parameter spectra, in general. and the former have less wave energy than the latter near the frequency where bending moment peaks: In short-crested seas. the extreme bending moments are reduced by 10 to 20 per cent in comparison with those in long-crested seas. The general trend is for the percentage difference to increase with sea severity.

It is estimated from Fig. 18 that the extreme bending

moment for design consideration with a

= 0.01 is 315 x 103 ft-tonf for full draught for the Mariner to be operated throughout the entire Atlantic Ocean. It is of interest to note here that reference (28) gives a bending moment of 381 x 103 ft-tonf calculated at the design stage

of the Mariner following the standard procedure of

assuming the ship is in static equilibrium on a trochoidal

wave of length equal to the ship length and having _a

height equal to 1/20 of the length.

5

_CONCLUSlONS

A method to predict extreme ship responses such as the largest value of motions and wave-induced bending moments which the ship may experience in the North Atlantic Ocean is discussed. The underlying concept of the present method is that the ship has to be designed to be safe in the event she encounters the severest storm

expected on the service route. For this purpose, the

following procedures are taken:

(i) The severity and duration of rough seas in the

208

Atlantic Ocean are estimated from the available data. and the extreme values of ship responses are estimated in the severest and various other seas below theseverest conditions expected in the service area of the ship.

Ship speed in each sea is estimated by taking into account the natural speed reduction due to the added resistance of a ship in a seaway and the voluntary speed reduction due to the added resistance of a ship in aseaway and the voluntary speed reduction by the ship's captain in order to ease severe motion.

The extreme values of ship responses are estimated by applying order statistics for a given sea state, duration. and ship speed.

The extreme value which is most likely to occur and that which is unlikely to occur with a preassigned small probability are estimated. The latter may be called the extreme value for design consideration.

As an example of the application of the present method, numerical calculations are made on the Mariner (526 ft; 160m). Extreme values of pitching, heaving motions, and vertical acceleration at the bow are obtained in seas of various significant wave heights up to 45 ft (13.7m). Long-crested seas having Bretschneider's two-parameter wave spectrum are used in the computation. where the

modal period for each significant _{wave height is the}

average value of those obtained from the spectra observed at Station I. Furthermore, a comparison of the_extreme values is made of the wave-induced bending _moment using the long-crested two-parameter_{wave spectra and}

the wave spectra actually measured

_{at Station I. In}

particular, the computations are made for the latter

spectra for long-crested as well as short-crested _seas.

Results of the computations show that the bending

moment in the short-crested seas are 10 to 20 per cent less than those in long-crested seas. The general trend

is _{for the percentage difference to increase with sea}

severity. It is estimated that the extreme bending moment (amplitude) for design consideration is 315 x 103 ft-tonf

for the Mariner to be operated throughout _{the entire}

Atlantic Ocean as compared with 381 _{x 103 ft-tonf by} the standard calculation in regularwaves of length equal to ship length and height of 1/20 of wavelength.

APPENDIX 1

Conversion of significant acceleration

to-probability of bow motion

A method to convert the probability of the significant amplitude of vertical acceleration at the ship bow to the probability of amplitude of vertical bow motion is as

follows:

Assume that the amplitude of acceleration, x, follows

the Rayleigh probability law with the parameter, R.

Then, one of the criteria for the acceptable limit below which no voluntary speed reduction is _{expected for the} full draught condition is given by:

Significant amplitude = 1.42 (.\/R) = 0.4g ...(A-1)

Choose a value, xo. such that the

probability of -exceeding xo is the same as that imposed for deck_wetness. That is:

Pr Ix > x0} f(x)dx = 0.07 u

280 TWO-PARAMETER I.LcS-C STED SEAS

240 (m 0.13.) I I. ED I ...iiiiill1111

MIIIMMill

40 i 1

13115-CHEM SEAS. STATION

o SHORT-CFES1E2 SEAS. STATION

I 1 I I II , I 10 70 30 40

;

where

I

PREDICTION OF EXTREME SHIP RESPONSES IN ROUGH SEAS OF THE NORTH ATLANTIC

= Pr

IT-From equations (A.1) and (A.2) we have

=

-

(o4)

xo In(0.07)] = 0.459g ...(A.3)

1.42

Then, the 2nd condition given in equation (1) of the text can be written by

Pr ?...- 0.491 bow acceleration Significant ampl. of amplittide 0.459g} {Bow acceleration {Bow motion 0.459 g

= Pr

amplitude _(°o2 0.07 ...(A.4)

For the light draught condition, the probabjlity in

equation (A.1) is replaced by 0.03, and hence 0.459 g in equation (A.3) becomes 0.527 g in this case. Numbers volved in equation (A.4) should be revised accordingly.

APPENDIX 2

REFERENCES

MOSKOWITZ, L.. PIERSON, W. J. and MEHR. E. 'Wave spectra estimated from wave records obtained by the OVv'S Weather explorer and the OWS Weather Reporter: I', New York Unir. Coll. of Eng., Res. Div. 1962.

1MOSKOWTZ, L., PIERSON, W. J. and MEHR. E. 'Wave spectra estimated from wave records obtained by the OWS Weather Explorer and the OWS Weather Reporter: New York Univ. Coll. of Eng., Res. Division, 1963.

Moskowrrz, L., P/ERSON. W. J. and MEHR. E. 'Wave spectra estimated from wave records obtained by the OWS Weather Explorer and OV.'S Weather Reporter: III', New. .York Univ. Coll. of Eng., .Res. Div., 1965.

MILES, M. 'Wave spectra estimated from a. stratified sample of 323 North Atlantic wave records', Nat. Res. Coun. Div. Mech. Eng. rep. LTR-S1-1-118, 1971.

HOGREN, N. and LOMB, F. E. Ocean wave statistics 1967 (HM Stationary Office, London).

BRErscHN.-Eioat, C. L. 'Maximum sea state for the North Atlantic hurricfre belt', Fundamentals of ocean engineering, Part 3

Ocean industry 1967.

'1 DRAPER, L. and WHrrAKER, N. 'Waves at ocean weather ship station Julien', Deutsche Hydro. Zeit 1965, 18 (1).

-11

DRAPER. 1.. and SQUIRE. E. M. -Waves at Ocean weather ship station India', Transaction. Royal Inst. Nay. Arch. 1967. 109. Roi.i., H. U. 'Height. length and steepness of seawaves in the

North Atlantic and dimensions of seawaves as function of wind force' (English transn). Soc. Nav. Arch. Ma,. Enur. Tech. &

Res. Bull. 1-19, 1958.

WALDEN, H. 'Die eigenschaften der meeresweilen in Nord-atlan tischen Ozean', Deutscher Wetterdienst. Ein:eircrliffent-lichungen, Nr. 41, 1964.

BRETSCHNEIDER, C. L. et al. 'Data for hish wave conditions observed by the OWS WEATHER REPORTER in December

1959'. Deutsch.? Hydro. Zeit. 1962, 15(6).

GERRITSNIA, J. and BEUKELNIAN, W. 'Analysis of the resistance

- increase in waves of a fast cargo ship', Inter. Shipbuild. Frog. 1972, 19 (237).

MARUO, H. 'The excess resistance of a ship in rough seas', Inter. Shipbuild. Frog. 1957, 4 (35).

JoosEN, W. P. A. 'Added resistance of ships in waves'. Proc. 6th Symp. Naval Hydrodynamics, ACR 136, Office of Navai Research, 1966.

BOESE. P. 'Eine einfache methode zur berixhnung der v.ieder-sandserhiihuna eines schiffes im seegang'. Schi..Tsrechnik 1970, 17 (86).

STRONI-TEMEN, J., YEE, H. Y. H. and MoRAN. D. D. 'Add resist-ance in waves, Ann. Mtg Soc. Nay, Arch. Mar. Eng. 1973. AERTSSEN, G. 'Service performance and seakeepina trials on

MN' Jordaens. Trans. Roy. Inst. Nor. Arch. 1966. 10S (4). AERTSSEN, G. 'Labouring of ships in rough seas with special

emphasis on the fast ship', Soc. Nov. Arch. Mar. Engr. Diamond Jubilee Int. Mtg 1968.

AERTSSEN, G. and VAN SUNS. M. F. 'Service performance and seakeepine trials on a large container ship', Trans. Roy. Inst. Nat'. Arch. 1972.

-TASAKI, R. 'On shipping water in head waves', J. Soc. Nor. Arch. Japco: 1960, 107 (in Japanese).

DAVENPORT, W. B. and Roor. W. L. An introduction to the theory of random' signals and noise 1958 (McGraw-Hill, New York). MIDDLFTON. D. An introduction to statistical communication

theory 1960 (McGraw-Hill, New York).

OcHt, M. K. and BOLTON. W. E. 'Statistics for prediction of ship performance in a seaway',. Inter. Shipbuild. Prog. 1973, 20 (222), (224) and (229).

CAR-rwRioa-r, D. E. and LoNourr-HrooiNs, M. S. 'The statistical distribution of the maxima of a random function'. Proc. Roy. Soc. London ser. A 1956 237.

HusToN, W. B. and SKOPINSKI. T. H. 'Probability and frequency characteristics of some flight buffet loads', NAC.-I tech. note

3733. 1956.

-OcHT, M. K. 'On prediction of extreme values', J. ship Res. 1973 17(1).

OcHt, M. K. 'Extreme behaviour of' a ship in rough seas', Trans. Soc. Nat'. Arch. Mar. Eng. 1964 72.

Russo, V. L. and SuiLivAN, E. K. 'Desien of the Mariner-type ship', Trans. Soc. Nay. Arch.. Mar. Eng. 1953 61.