A method to determine the response of ships in irregular waves

DIVISION OF SHIP DESIGN

A METHOD TO DETERMINE

THE RESPONSE OF SHIPS

IN IRREGULAR WAVES

by

RUTGER BENN ET

Sponsored by the Swedish Technical Research Council Gothenburg, September, 19b6

The mathematical f oi-mula for average wave spectra proposed by

Pierson & Moskowitz is used as a basis for a simple practical method to determine the expected response of ships in any

desired sea state.

The spectrum is applied to regular wave response amplitude

operators, obtained either from model tests or from calcul-ations according to the strip theory. It is shown how general

dimensionless irregular response operators can be derived for a given hull form, which cover all possible combinations of

significant wave height and average apparent period. The

expected response in an arbitrary sea state can be directly determined from these functions, without the need for further

calculations.

An attempt is made to construct a table of six typical sea

states, with corresponding wave and wind conditions. It is

shown by numerical examples that good agreement may be ob-tained between the expected response determined with this

Page

IJÍTRODIJCTION

...

. . .

...

1 WAVE STATISTICS ... 2

THE WAVE SPECTRUM

_...4

THE RESPONSE SPECTRUM ₉

THE SPECTRUM RESPONSE OPERATOR 10

RESULTS OF SPECTRWI RESPONSE OPERATOR CALCULATIONS 11

SELECTING AVERAGE llAVE SPECTRA 12

ADAPTING SPECTRUM PARATERS TO

OBSERVED VALUES 14

PRACTICAL APPLICATIONS ₁₆

The statistical concept of treating both the irregular surface of the sea and the equally irregular variation of the _response

of ships in such a sea as a normal stochastic process is now

universally adopted. A great proportion of the research, which

has led -to development 0±' practical methods for statistical

analysis of full scale data, has been within the structural

design field and based on stress measurements _{on ships at sea.}

Relatively less full scale work has been devoted to ship

motions than to the bending stresses in the hull. However,

the same methods that have been found useful to evaluate design stresses can also be used for finding the probability of slam-ming a-t a certain speed, or the expected value of vertical acceleration due to the combined pitching and heaving motion.

The essential feature of -the mathematical model for treating irregular waves is that the wave surface is assumed to consist

of a great number of regular components superposed _{on one}

another in random phase. _{If these components are Imown,} certain statistical proper-ties of the resulting irregular surface can be determined. _{Most important from the naval}

architect's point of view is that the statistical _distribution

of -the resulting wave heights can be calculated. Inversely,

the amplitude and frequency of the components _{can be computed} from a sample of the irregular surface, such _{as for instance}

a record from a wave meter.

A great deal of the previous work on statistical prediction _of

ship behavior in waves hi.s been aimed at finding extreme values of various responses, which are expected to occur with a

certain probability during the entire service life oÍ' the ship.

Several methods have been suggested for evaluating long term distributions of, for instance, longitudinal bending moment

/1, 2, 3, 4, 5, 6, 7/. In many cases, however, a more specific

knowledge is desired of the expected behavior of a ship in any sea state, and not necessarily the worst one. The long term distribution, including the largest expected value, must be the natural basis for design of the hull structure, and

there-fore of special interest for the classification societies,

while "average" values, both of stresses and of accelerations

operator for judging the service economy of the ship.

The most important responses, from the operator's point of view, are the vertical accelerations due to the longitudinal motions, and the relative motion between the bow and the water

surface. The violence of the motion and the frequency of

occurrence of slamming are the decisive factors for the speed that can be maintained in a rough sea, and a great deal of

research is at present going on, in order to find slamming

criteria in terms of maximum permitted values of relative motion and velocity /8, 9/. To make full use of weather routing, the speed that can be maintained in the conditions predicted for the voyage must be accurately Iaaown, in order to find a least-time track. One must then be able to determine

quickly the probability of slamming at various speeds in sea

states, defined by the predicted values of apparent period and significant wave height.

The oceanographic research on wave spectra is at present very

active. _{Two slightly different basic types of spectra have}

recently been proposed, which are considered to agree with a great number of actually recorded spectra on the North Atlantic. SCOTT /10/ has suggested a modified Darbyshire-spectrum, which has been adopted by the British Towing Tank Panel, and

PIERSON-LIOSKOWITZ /ii/ have proposed a mathematical formulation, which

was recommended by the ISSC Committee 1, "Environmental

Conditions", 1964 /12/. In this paper the PIERSON-MOSKO\IITZ-ISSC spectrum has been used for calculations of general

irregular responso operators.

1[A1JE STATISTICS.

So much has been published during the last ten years about the statistical consequences of the wave spectrum concept, that

only a very short recapitulation of some essential facts should

be necessary here.

Mathematically, the definition of the energy spectrum _{can be} arrived at along two different paths, both of course leading

to the same result. The wave surface can be treated as a

normal, stationary, stochastic process, and the spectrum

func-N

The parameter R is directly proportional to the spectrum area, with a factor of proportionality which is depending on whether single or double amplitudes are considered.

Strictly speaking, the Rayleigh distribution is only a special

case of a more general distribution, and it is only exactly

valid for a so called narrow spectrum. The exact expression

for a spectrum of arbitrary width has been treated by

CARTWRIGHT, but experience has shovo. that it iâ hardly ever

necessary to take spectrum width into account for practical

calculations.

However, the Rayleigh distribution can be considered a special case also from another point of view. The parameter R, which is the spectrum area, is only remaining constant as long as

wind conditions are constant, which usually only is the case

during short periods of time. During a voyage, or a year, or a ship's service life, a great many different wave spectra

2 X

R

tion. One can also start with the assumption, that the irregular waves are made up of a large number of harmonic components, superposed in random phase. It is then easy to show, by elementary analysis, that the ordinate of the result-ing function will be normally distributed with zero mean and variance equal to half the sum of the squared amplitud.es of all the components. The total area under the spectrum is equal

to the sum of the squared amplitudos of the components, which

consequently is twice the variance of the distribution of the irregular surface ordinates.

Por practical use one is not so much interested in the entire wave surface, but rather in the height of the wave crests over

the adjacent troughs. These points can loe seen as the envelope

of the surface, and their statistical distribution has been

shovm to follow the Rayleigh function:

f(x) = e

(and hence R-values) will be encountered. The parameter R can therefore itself be considered as a statistical variable with

a certain distribution. For this reason the Rayleigh distribu-tian has been called a "short term distribution", and a sum

of such distributions is called a "long term distribution".

For long tarin conditions good agreement has been obtained with the Weibull distribution, which has the following general

equation: k-1 f(x) _kXak e k a N (3a)

]ior k = 2, the Rayleigh distribution is obtained as a special

case.

The exponent k can be considered as an "irregularity

para-meter". Â single regular wave is exactly represented by k = (see fig. i), as the amplitude is constant at all probability levels. The greatest degree of regularity normally found in a wave system corresponds to k = 2 (the Rayleigh distribution), but very likely somewhat larger values of k may be found. in a

swell, may be up to 3 or 3.5. During a long period with very varying wind conditions k will in general be less than 2.

(For the long term variation of the wave bending moment, k has bean estimated to between 1.0 and 1.25).

This interpretation of the exponent k in the Weibull

distribu-tion was originally suggested by ROOF in 1940.

TKB WAVE SPECTRUM.

For calculation of the behavior of a ship in a certain sea

state, a mathematical formulation of the wave spectriun is required in terms of some observable properties of the sea. When waves are observed, for instance from the weather ships,

they are usually defined by two quantities, an average ob-served height, and an average obob-served period. It is

gene-k (X

"a

rally believed that the observed height has a definite

re-lation to the root mean square height (which is the parameter

in the Rayleigh distribution and the square root of the

spectru.m area), and that the observed period is a function of the peak frequency of the spectrum, or of one of its moments.

The previously mentioned PIERSON-MOSKOWITZ-ISSC spectrum

con-tains two parameters, a height H1 and a period T1, which can

be considered -to represent the observed values. The form of the spectrum equation is such, that it is not necessary to

define the parameters until after the response spectrum is

calculated. The question of adapting the observed values of height and period to the corresponding parameters of the

spectrum is therefore deferred to a later chapter.

The spectrum has the following equation (fig. 2):

-BT

s(f) = A H T4 f5

e

where s(f) = spectrum ordinate

f = frequency of component

T1 = a characteristic period in the spectrum H1 = a characteristic height of the resulting

irregular wave system

A and B = constants

In the form suggested by the ISSC Committee 1,

as the significant crest-to-trough height, and

was chosen so that the spectrum area was equal is correct from the point of view of' a strict definition of the energy spectrum. For practi it is much more convenient to let H equal the (which is also Darbyshire's definition), which by putting (4a) H1 is defined the constant A

to H/16, which

mathematical

cal use, however, spectrum area is accomplished

A=4B

(5)

The numerical value of B depends on the definition of T1.

With A = 4 B the following general expression is obtained for

H2 m = B4 r (i -n and hence: = H (spectrum area) H2 ni1 = _{1.2254 B1/4} T1 in2 = . 1.7725 B1/2 1114 = J

(That ni4 = means that the spectrum has infinite width, according to Cartwright's definition.)

According to the ISSC formulation

T1 =

and B = 1.2254k = 0.44

II' T1 is related to the second moment:

T1 = 1m0/m2

B = _{1.7725_2 = 0.32}

It is not necessary at this stage to take any definite posi-tien to the question of how the parameter T1 shall be related

to the visually observed average apparent period Tobs. For

all calculations in the present paper T1 has been defined as the first moment according to equation (a), and consequently

B = 0.44. As will be shown later, the relation between T1

and Tobs can be given any desired valuo after the final

response calculation is performed.

(6)

The wave spectrum can now be written as:

5(f)

_-0.44(T1f4

2

= 1.76 T1(T1f)

H1

where H = area of spectrum

and T1 = m0/m1 = 1.09

Calculations of response spectra are generally based either on

model tests or on computed values in a series of regular waves.

The relative wave lengths are then a more natural parameter

than the wave frequencies. To avoid having to convert the amplitude operators to a frequency scalo, the wave spectrum

can be rewritten in terms of component wave lengths, X. For the harmonic components the theoretical relation naturally

holds:

g = 2 ri f2

(12)

One must, however, be very careful in using wave lengths in coimoction with irregular sea, because no wave lengths in the

usual senso actually exist. Only the imaginary components have a fixed length. By defining a characteristic "vave

length" as

7

---T

'1 - 2u 1

it is to be remembered that this is only a definition, and X1 cannot be visually estimated in the irregular system.

If the nondimcnsional relative frequency (T1f) is used for

abscissa, the spectrum is written:

s(T1f) - s(f)

d(Tf)

=

1.76

(T1f)5

-0.44(T1f) e

(4b)

(13)

(4c)

i

Since T1f

= V

_{T '}

X

and d(T1f) _{= (_)}

(1)2

d

one will have

X

X2

x

-d-

= 0.88 (-r--) 1

111

It has previously been shown by LEWIS /13/ how responso

cal-culations are greatly simplified if both spectrum and

ampli-tude operator are expressed as functions of the logarithm of

the wave frequency. Mathematically it is insignificant if the wave length is substituted for the frequency, especially

if it is made relativo to the ship length, L. A wave with length L has frequency

- g

- V 2uL

and consequently x/L

=

The final form of the spectrum in terms of the component slope

can therefore be written:

s(log x/x1)/x2 A -0.44

=0.88e

(14) (4e)

This spectrum is entirely non-dimensional and will express the

slope (ratio wave height/wave length) of the spectrum com-ponents as function of (log x/x1) = log 1/(T1f)2, It is once moro pointed out that X. ought to be considered as a period T1, according to equ. (13),

Gr1 =

p g L3 B

for bending moment,

G9 = 9 radians, for pitch angle,

Gz = for heave,

GA = for vertical acceleration,

G5 for relativo motion,

G = g.- for relative velocity.

This response factor as a function of (log X/L) is the ampli-tude operator to be used with the wave spectrum (4e). The area of the response spectrum will then be obtained through

the following operation:

While T1 will later be assumed to have a definite relation to an observed average period of thc wave system, X1 has no lmown

relation to any observed "average wave length".

THE RESPONSE SPECTRUM.

The response of a ship in a regular wave can in general be

expressed by the response factor

G

(15)

whore G = a non-dimensional response h = height of the regular wave X = length of the regular wave

Examples of G are

M

ç-i

G(log X/L)

s(log x/x1)/x2 d(log x/x1) = j

h/X

H/X

(1 6a)

and the wave spectrum can be given the form

This operation is illustrated in fig. _{3, right hand} side of equation (16a)is the area of

_curve(3'

_{in that figuro.} iince the wave spectrum was constructed so that its area

equals H, G1 in (16will corresond to the

_{same fractile in} the G-distribution as H.1 in the H-distribution. _{Both G and H} are as usual assumed to follow the Rayleigh distribution. _If, therefore, H1 is taken to represent the significant height, then G1 will be the significant response (= mean of the upper third), and if the root mean square height is used for H1, G1 will be the root mean square response.

SPECTRUN RESPONSE OPERATOR.

Each calculation of G1 according to equation (16a) refers to given values of X1 and of L, and consequently to a given X1/L. As a result of the logarithmic scale, _{a change in this ratio}

is effected through a simple shifting of the relative positions

of the wave spectrum and the amplitude operator. The distance in the logarithmic scale between the zero points of the two curves is always equal tc log X1/L. A series of response

spectrum areas can then be calculated for various values of

X1/L. The results plotted versus X1/L

= g T/2n L may be

called a ectrum response operator, fig. 4a. Prom this graph the response of geometrically similar ships of any desired

length can be found for wave conditions, characterized by _any

desired values of H1 and T1.

If both the amplitude operator and the wave spectrum are expressed in terms of wave slope, the result of the spectrum calculation will also be in teims of H1/X1, which might be

called an "effective slope". _{It is, of course, not really a}

wave slope, because the wave with height H1 can not he

expected to have the period T1 or the length X1. The

ampli-tude operator can equally well be expressed in terms of only

the wave height,

G L_ G

X2

s(log x/x1) X 2

-0.44(ç)

0.88 e

An operation similar to equation (16a) will then yield the result:

jG2clo X/L) s(logx/x1) G L2

h/L2

d(log X/X1)

=

The same result can of course also be achieved by multiplying

the right hand side 0±' equation (16a)by L/X1. The spectrum

response operator will then be obtained in the form shown in

fig. 4b.

The wave spectrum was defined so that T1 (or X1) corresponded to the first moment of the frequency spectrum. A different

definition of T1 will only mean a parallel shift along the

abscissa of the spectrum, and consequently also a simple parallel shift of the resulting spectrum response operator, as long as the logarithmic scale is used. If, therefore, the

average observed period in the wave system is assumed to he

Tobs = C

T1

thon it will follow that

G1 G1

HIÌT

= (HlTh)T - _T

ob

1c obi

If, for instance, Tobs is considered to correspond to the second moment of the spectrum, then o

=

1b9 and a period of

1 .09 Tobs ought to be used as T1 to obtain the response

distribution in a certain spectrum.

RESULTS OP SPECTRUM RESPONSE OPERATOR CALCULATIONS.

A series of response spectra have been calculated for two

ships, a cargo liner with block coefficient 0.65, and a bulk

(16h) (18)

carrier with block 0.78.

The amplitude operators in regular

waves were obtained from strip theory calculations, published

in ref. /14/, where they also viere shovni to agree rather well

with model tests.

The spectrum operators arc given in fig.

5 - 12.

G1 L

The spec±nlm operator

_H -

is very similar to the regular

amplitude operator

and a direct comparison can ho

made between the response in o regular wave with height H and

length X, and in an irregular system with significant height

H1 and average apparent period T1

(or average "length" x1).

Some examples of such a comparison are shown in fig. 13 - 14.

It is evident from these curves that it is practically

im-possible to draw any conclusions regarding the seakeoping

abilities of a ship from tests in regular waves without any

consideration for the actual wave spectra.

Normal values of

T1 in those wave spectra that arc usually encountered at sea

are between 6 and 10 seconds, corresponding to X1 between 50

and 150 meters.

By far the most frequent value of Ti is 7

-- 8 sec. (75 -- 100

m).

The relation between the response in

regular waves with length approximately equal to ship length,

and the actual response in average conditions at sea, is

therefore dependent on the length of the ship in question,

and no general such relation can be determined.

It is also

different for different responses.

SELECTING AVERAGE WAVE SPECTRA.

The oceanographic research on wind waves has mostly been

directed towards finding good estimates of fully developed

spectra during various wind conditions.

The PIERSON-LIOSKOWITZ

spectrum, which is used in the present paper, is also primarily

intended to represent such fully developed seas.

Normally

such ideal conditions are very rare, because the wind is seldom

steady enough from the seme direction during sufficiently long

timo.

In the general case the wave system will consist of a

mixture of old and new systems superposed on one another, very

often with several different directions.

This may not affect

the shape of the spectrum as much as the area (which is

propor-tional to the significant hoight) and the frequency range.

The normal effect of the actual stato of development is that

the spectra have a larger area in the low-frequency rango at

low wind speeds, and a smaller area at high wind speeds than

is expected during fully developed conditions. The reason is

obviously that there is usually a swell left over from previous stronger winds, which will be more noticeable at light wind, while on the other hand strong winds will rarely blow long enough to build up the low-frequency part of the spectrum. The formulae which have been suggested, for instance in ref. /15/, for the relation between wind speed and

signi-ficant height therefore show a poor agreement with observ-ations, except at the most common wind speeds of about 20 knots. Below this speed actual waves are higher, and above this speed they are very much lower than predicted by the

formulae.

In order to estimate ship behavior in average conditions, such that a ship must be expected to encounter during any normal voyage, a reliable approximation of the shape, area, and

frequency range of average spectra is needed. One may assume

that the previously mentioned equation for the spectrum shape

still can be used, if only the area and average frequency are adjusted to agree with normal conditions.

The most extensive published statistics on wave observations in various parts of the oceans is found in the rcport of

Committee i to the ISSO in 1964 /12/. These observations are not related to wind speeds. By comparing the relative fre-quency of observed heights and periods with the corresponding figures in ROLL, ref. /16/, a tentativo table of six typical sea stats has been produced (Table I). The significant

heights and average apparent periods given in this table are considered to be close to the values, which may be expected

to occur during normal conditions Sea states II- III are encountered during practically every voyage, their combined probability is nearly 50 %. As will be shova later with numerical examples the wave bending moments and ship motions

predicted in these sea states agree well with full scale tests during similar conditions. State IV is rather common on the

North Atlantic during winter, but is very rare in most other

sea areas. State V and VI are extreme conditions, that are encountered only exceptionally, and then only for a few hours.

will, when it is used, have the effect that these sea states may never be met at all. However, it is recognized that

ships must be designed to be able to withstand them.

KDAPTING- THE SPECTRUM PARAJ1ETERS H1 AI'T]) T1 TO OBSERV]flJ VALUES. The significant height and the average apparent period given

in Table I are observed values. The observed height has gene-rally been assumed to be the mean of the largest third of all

heights, usually denoted H1113. Various investigations have found different relations between H1113 and

H,

the visual height. NORDENSTRÖM at the Norske Ventas has suggested that Hv is smaller than H1113 when the height is less than 8 m, and

greater when it is more than 8 m. In another case a comparison with a limited number of recorded spectra indicated that _Hv might be more close to the root mean square height, which is

0.71

. H1113,

As

wave heights above 8 n are rarely reported

and belong to the extreme part of the distribution, Hv can in general be considered to be somewhat smaller than _H1,,,3, say about 85 %, at least for deteinining values of relatively great frequency of occurrence, which is the chief purpose in

this paper. Por the determination of long-term distributions a much more detailed investigation is needed of the true

probability of

H.

As a reasonable maximum in a short-tern distribution the

greatest expected value out of 10,000 is often used. Another common value is the mean of the greatest 10 %. All such

values are directly proportional to the root mean square and

to the mean of the upper third. With the approximate relation

between

H1,3

and H that was mentioned earlier, and assuming an average number of 500 - 600 waves per hour, Table II. is

proposed as a rather rough but fairly reliable estimate of

wave heights.

The result of the spectrum response calculation is obtained

as the ratio between the response amplitude and the wave

height, which have the same probability according to the

Rayleigh distribution. The response expected to be exceeded

GL H

G-Q H1

L

The response having the expected frequencies of Table II is consequently directly obtained by multiplying the spectrum operator by the wave heights in that table. The operator must then be taken at the appropriate value of T1.

Many investigations of recorded wave spectra have indicated that the average observed apparent period is somewhat smaller

than the period T1 that corresponds to the first moment of the frequency spectrum. The second moment has generally been

assumed to give a better agreement, which for the mathematical model used here means that T = 0.92 T1, (see equ. (7)). If a fully integrated calculation is performed in order to produce a complete long term distribution, this question becomes im-portant. For determining the approximate order of maaitude of average responses in the most frequently occurring sea

states, a difference of 10 % in T1 has only a small influence on the result. The inaccuracy in the estimation of T is

certainly much greater than 10 , and it has therefore not

been considered necessary to depart from the recommendation

of the ISSC Committee 1, which was to take T = T1. In the

chapter on numerical examples it is shoi how to estimate the

range of response, caused by a possible error in the average apparent period. If nothing else is stated, calculations aro made on the assumption that _T = T1 = m0/m1.

PRACTICAL APPLICATIONS.

The response of a ship in a given wave spectrum is very much

dependent on the length. If the irregular response operators are plotted versus ship length L, as in fig. 15 - 17, it is evident that the maximum response per unit wave height occurs at different values of T1 for different values of L. Por the important responses the following relationship generally holds approximately:

(T1)

= Vo.5 L

(22)

where (T1

max = T1 in the spectrum causing maximum

response per wave height,

L = ship length in metres.

The most frequent value of T1 in average weather conditions is 7 - 8 seconds, which means that ships up to about 130 m spend a comparatively greater part of their time in unfavorable sea

conditions than the longer ships.

The bending moment coefficient for the 0.65 block coeff. ship has been calculated in the sea states of Table I and plotted

with length as parameter in fig. 18. The moments are given

as percent of the value in a spectrum with H1,,,3 = 10 m and T1 = 10 s, which is assumed to be close to the greatest moment ever expected to occur. One can see here that ships up to

120 m get 50 % of their maximum already in 4.5 m signif. height

while i-t takes 6 . for a 225 m ship to get such a high value.

The moment coefficient can be written in terms of stress and

section modulus:

z

a

p g L3 B = p g L

In the rules of the classification societies the section

modulus has been calculated from formulae o± the type:

M

(23)

where _{has varied between 0.5 and 0.3, which is the value now} used by L.R. and D.N.V.

Equations (23) and (24) give an expression for the expected

wave stress in ships built to these rules:

iur

a.

- (

gL3

B

L1 ₍₂₅₎

In fig. 19 this stress is shown as percent of the stress in a 120 m ship.

It can thus be shown that the method of determining the scantlings of the midship section in itself leads to very different stresses in ships of different length. The gradual decrease of the factor in equ. (24) has been a step in the right direction, but it has not been enough to give equal relative strength to all ships. This would mean an entirely

new approach to the influence of length in the design formula.

It is interesting to compare fig. 19 with the fracture statistics published by Murray in ref. /17/. It is shown

there that ships of about 120 m in length have had a greater number of cracks per ship year than longer ships, especially

those built during the period when section modulus was

pro-portional to L25.

Corn arison with full scale results.

M/S Canada.

The sea states in Table I have been used for a comparison with

results of full scale measurements of wave bending stresses, previously published in ref. /18/. The range of calculated

values shown in fig. 20 arc obtained from the range of

signi-ficant height and average apparent period given in the table.

In the sea states corresponding to Beaufort 4 - 7 the agreement is quite good. These conditions are very common, and expected to he encountered on the North Atlantic about 45 % of the time0 In 2 - 3 Bft measured values arc usually greater than expected,

probably because there is a great deal of swell present which

assuming a T1 of 7 - 8 sec. even at this low wind speed. In

more severe weather, above 8 B±'t, measured values fall below

theory. Generally very few tests are carried out in such weather, but the tendency is clear, and it also agrees with

other similar investigations. It is probable that this is a result of several factors, the most important perhaps being a wider spreading of the energy in severe spectra, and that

ships are rarely heading directly into the waves in such con-ditions but rather handled in a way to ease the wave loads. It is also possible that the normal efforts of most captains

to avoid excessive weather conditions (quite apart from any systematic weather routing) will have the effect, that only the milder forms of the sea states 7 - 9 are in fact normally

encountered.

Regarding the fairly good agreement in the most frequent sea

conditions, it is to be remembered that no reduction has been

made for energy spreading or different course angles to the

dominant wave direction. The average combined effect of these factors is estimated to between 20 and 30 %.

M/S Jordaens and. M/S Mineral Serain.

Professor Aertssen has made very extensive sea trials on two ships, /19/. The instmmentation included a shipborne wave recorder, and both bending moment and wave spectra were computed. A response amplitude operator could then be cal-culated and applied to a very severe spectrum which had been

recorded at another occasion, thus extrapolating the measured

values to those severe conditions. The two wave spectra had

the following characteristics:

111/3 9.0 m, 10.5 sec., (corresponding T1 = 11.5 sec) 111/3 = 16.7 m, = 14.0 sec., (corresponding T1 = 15.25 sec).

(The periods 10.5 and 14 sec. are assumed to be the theoretical

average apparent periods, i.e. the second moment of the

frequency spectrum. As T1 is defined as the first moment of the spectrum, the correction factor 0.92_1 = 1.09 has been

JORDAENS. Fig. 6 is calculated for a ship with a hull form

very similar to the JORDAENS.

2

g T1

Sea state A: = 1.4 for T1 = 11.5 and L = 146 in.

2TrL ¡

M,

Corresponding

2 - 11.5 10 (from fig. 6)

pgL BH1/3

Ship dimensions: p g L2 B = 4.4 1O ton

M.1/3 = 11.5 10 4.4 10 9.0 = 45 500 tm

for H1/3 = 9.0 in.

45 500

16 200 tin - 2.83

-According to the Rayleigh distribution, the following

peak-to-peak bending moments are expected:

The maximum recorded B.M. in this sea state was 72 300 tin.

g Sea state B: M1/3 = 8.0 . p g L2 B 111/3 M1/3 = 58 500 tin \/ = 20 800 tm 2îrL = 2.52 Maximum of 100 = 4.35 \/ = 70 500 tin t' t' 1 10 300 000 000 = = = 4.80 5.25 6.06

\/T

= V'È = \/ = 77 600 -tin 85 000 -tin 98 000 tm

Maximum of loo =

90 500 tin

U

300 = 100 000 tin

"

1 000 = 109

000 tin

10 000 = 126 000 tin

From the spectrum extrapolation 107 000 tin was predicted as the expected maximum of 10 000.

MINERAL SEBAING-. This ore carrier has a hull form similar to the ship in fig. 10. Using the same procedure as for the

JORDAENS, the following prediction is made:

2 g T1 Sea state A:

_{- - 0.94}

2 TIL M1/3 2 =

15.5

10 (from fig. 10)

pgL BH1/3

p g L2 B = 15.6 10 ton M11,13 = 217 000 tin

\/TT=

77 000 tin Maximum of 100 = 335 000 tin 300 = 370 000 tm i 000 = 405 000 tm 10 000 = 467 000 tin

The maximum recorded B,M. in this sea state was 341 000 tin.

2 g T1 Sea state B: = 1.66 2 uL M1/3 = 13.0 p g L2 B

M1/3 = 338 000 tm

/= 120 000 tm

Maximum of loo = 520 000 tin 300 = 575 000 tin

1 000 = 630 000 tm

t! _{10 000 725 000 tin}

From the spectrum extrapolation 532 000 tin was predicted as the expected maximum of 10 000.

Calculation of expected_behavior of a 140 in bulk carrier in an average rough oea.

A wave system with significant height in and average apparent

period about 8 sec. seems to be a fair estimate of conditions

that are typical on the North Atlantic. Such waves may occur

in anything between 5 and 7 Beaufort, and must be expected on

almost every voyage, at least during the winter period. They correspond to Sea State Group III in Table I, but are just within the possible range of Group II.

Spectrum operators for a typical 14 000 tdw bulk carrier may

be determined from fig. 9 - 12. A ship is chosen with the

following main dimensions:

= 140 m B

_=19.5m

d =

8.8m

CB = 0.78 2 g T1

= 0.71 for T1 = 8 s, and L = 140 in. 2 nL

The response is sought for a speed of 14.5 knots, which is a

Froude number F = 0.21.

From fig. 9 - 12 the following response operators are obtained, after having inserted the appropriate ship dimensions:

These operators shall be multiplied by significant height to

give significant double amplitude of response. Significant single amplitudes will then be:

Using the multipliers in Table II, the average expected maxima

during various time intervals can be determined:

Bend. stress amidsh. kg/nm2 2.1 2.5 2.8 3.5

x) Assuming a section modulus Z = 6.4 rn3.

Tith a draft forward of more than 8 ni, there seems to be no risk of the stem leaving the water, and slamming should be no problem in these waves at a speed of 14.5 imots. As the

motions are very little affected by the draft, one may draw the conclusion that 5 m is the minimum ballast draft forward

Average maximum every

minute 10 min 30 min 20 hours

Vert. acc. at F.P., g 0.26 0.32 0.35 0.44

Rel. motion at F..P., rn 4.3 5.2 5.7 7.3

Rel. velocity at F.P., rn/s 3.9 4.7 5.2 6.6

Bend. mom. amidah., t ni 13 500 16 250 18 000 22 700 Vertical acceleration: A1,3 = 0.205 g

Relative motion: 1/3 = 3.38 rn Relative velocity: S.1/3 = 3.06 rn/s Bending moment: _1.11/3 10550 t Vertical acceleration at P.P.: Relative motion at P.P.: Relative velocity at P.P.:

Bending moment amidships:

-= = 0.138

225

ni 2.04

-7040

that can be periitted if this speed is desired to be main-tained. With a draft less than 5 na, stem emergence may be so

frequent, that the resulting slamming will make a speed

reduction necessary. The probability of slamming, or the

expected numbers of slam per unit time can be calculated from these values of relative motion and relative velocity, using

REFERENCES.

/i/ Bennet, Ivarson & Nordenström: Results from full scale

measurements and predictions of wave bending moments

acting on ships. Rep. No. 32 of the Swedish Ship-ouilding Research Foundation, 1962.

/2/ Ivarson, A.: Ore carrier model tests. Rep. No. 35 of the Swedish Shipb. Ros. Pound., 1963.

/3/ Nordenström, N.: On estimation of long-term

distri-butions of wave induced midship bending moments in ships. Chalmers University of Technology, 1963.

/4/ Nordenström, N.: Statistics and wave loads. Chalmers

Univ. of Teclin., 1964.

/5/ Yuille, I.M.: Longitudinal strength of ships. Trans.

R.I.N.A. 1963.

/6/ Masuda, Y.: Statistical amidship bending moment for

ships. Journ. of Zosen Kiokai. Vol. 111 (1962), June.

/7/ ISSC 1964. Report of Committee 2b-II.

/a/ Ochi, M.K.: Prediction of occurrence and severity of

ship slamming at sea. 5th Symposium on Naval Hydro-dynamics, Bergen, 1964.

/9/ Ochi, M.K.: ]Ixtreme behavior of a ship in rough seas.

SNATì _1964.

/io/ Scott, J.R.: A sea spectrum for model tests and

long-term ship prediction. Journ. of Ship Research Vol. 9

(1965):3 p. 145.

/11/ Pierson, W.J. & Moskowitz, L.: A proposed spectral form for fully developed wind seas based on the similarity

theory of S.A. Kitaigorodskii. N.Y. Univ., Department

of Meteorology and Oceanography, Geophysical Sciences

Lab. Rep. 63 - 12.

/12/ ISSC 1964. Report of Committee 1.

/13/ Lewis, E.V. & Bonnet, R.: Lecture notes on ship motions in irregular seas. Webb Inst. of Naval Arch., 1963.

/14/ Ivarson, A. & Thomsson, O.: Comparison between model

tests and calculated values of ship behavior in regular

waves (in Swedish). Chalmers Univ. of Tochn., 1965.

/15/ Moskowitz, L.: Estimates of the power spectra for fully

developed seas for wind speeds of 20 to 40 knots. N.Y.

Univ., Department of Meteorology and Oceanography, Geophysical Sciences Lab. Rep. 63 - 11.

/16/ Roll, H.U.: Height, length, and steepness of sea waves

in the North Atlantic. SNA Techn. and Research Bull. 1 _{- 19, 1958,}

/17/ Murray, J.M.: Development of' basis of longitudinal strength standards for merchant ships. R.I.N.A.,

May 1966.

/18/ Bonnet, R.: Stress and motion measurements on ships at

sea. European Shipbuilding 1959, Nos.

5 & 6.

/19/

Aortssen, G.: Service-performance and soakeeping trials on M.V. JOPDAENS. R.I.N.A., LIay

1966.

Sea state Approxiriate wind Significant height Apparent period Expected frequency Group Number Speed

Imots Beaufort Range m Average m Range Average 2 slight sea

4 - 10

2 - 3

0.75 -

1.5 0.75 5 - 8 6

20 %

II

slight-

3-4

moderate 11 - 21

4 - 5

1.0 - 3.0

2.0 6 - 7

6.5

31 %

5-6

rough

22 - 33

6

- 7

2.5 -

4.5

3.5

7 -

9

7.5

14 %

IV

_{very rough}7

34-47 8-9

4.0-6.5

5.0

7-9

8

3.5%

V 8 Extremely rough

48 - 55

10

6.0 - 8.5

7.5

8 - 10

9

0.5 %

VI

_niountainous9 > 56

11 - 12

> 8

10

9 -

12 10

0.05 %

TABLE II. Average highest waves.

Mean highest every minute .. . . . 1.28 H1/3

1.50

One wave every ten minutes higher than -..

1.54

H11,,3 1.75 J:I

H 11 thirty minutes higher than _..

. . 1.70 H11i3 2.00 H

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