Results from full scale measurements and predictions of wave bending moments acting on ships

RESULTS FROM FULL SCALE MEASUREMENTS

AND PREDICTIONS OF WAVE BENDING MOMENTS

ACTING ON SHIPS

(Project S-15)

by Rutger Bennet

Allan Ivarson and Nils Nordenström

1962

Report No. 32

STIFTELSEN FÖR

SKEPPSBYGGNADSTEKNISK FORSKNING

THE SWEDISH SHIPBUILDING RESEARCH FOUNDATION

Foreword

In l95 and 1959 the Swedish Shipbuilding Research Foundation

issued the reports No 13 resp0 15 about stress and motion measure-ments on ships at sea. The first report described the theoretica:l basis for a statistical analysis of measurements while the second report included test results from the fast-going cargo ships

N/S Canada and M/S Minnesota

Since then the stress irteasurements have been continued and extended to several ships of different types. Furthermore in co-operation between the Foundation and the Institution of Ship Design at Chalmers University of Technology, headed by professor Anders Svennerud, a research group has been established for

studies and analyses in connection with the measurements. The leader of the group has been civilingenjör Rutger Bennet of the Swedish Shipbuilding Research Foundation0

This research work has been possible through a generous grant from

Axel och Margaret Ax:son Johnsons stiftelse för allinän-nyttiga ändamAl

(Axel and Margaret Ax:son Johnsons Foundation)

The measurements on board ships have been carried out through the courtesy of the following owners:

Esso Petroleum Co. Ltd. Grand Cape Tankers, Inc. The Grängesberg Co.

The Johnson Line Shipping Co. Swedish Lloyd Steamship Co.

The Transatlantic Shipping Co. Ltd. Swedish American Line

Many people have participated in the measurements and given valuable help herewith. Particularly the officers on board the ships have shown great interest to the measurements.

The Stress Measurement Committee of the Reseach Foundation has discussed the results at different intervals.

Gothenburg, October, 1962 Lennart Swenson

In two previous reports (SSF 13 and 15) The Swedish Shipbuilding Research Foundation has described results of

statistical analysis of stress measurements on ships at sea. In this report results are presented of continued measurements on six other ships of various types.

Stresses have been measured in the strength deck or in the shell near the stringer corner, and as near amid-ships as practicable. In one ship stresses have also been measured at the forward and aft quarter lengths. The simple

beam theory has been assumed to be valid, and this has been confirmed by static tests on four of the ships.

For general conclusions measured wave stresses are of 1imnted value, as they depend on the actual section modulus at the gauge location of the test ship. In order to make a true comparison between the results from the various ships, the stresses have been converted to a dimensionless bending moment factor m, defined by

z

B

S umnmar y

where s = measured stress

Z = calculated section modulus at gauge location

For a given ship type the moment factor has been found to decrease with increasing ship length. The results show

an essential difference between fast cargo liners and ships of tanker type with large block coefficient. The latter type get the highest extreme values, while the bending moment which is exceeded with great probability (more than 1/100) is rather larger in the fast ships0 This may be even more pronounced if whipping and slamming stresses are taken into account.

A method for determining design values of bending moments using the tests results is proposed. A probability

of exceedance P = l0 is suggested for extreme design value. It is pointed out that the largest permitted wave stress generally must be smaller than the permitted total

-2-stress, produced by the sum of wave and still water bending moment. The total stress should be determined by the yield

point, with regard to unavoidable stress concentrations, while the maximum permitted waite stress is restricted by

considerations of fatigue and cumulative damage.

Regard±ng the longitudinal distribution of wave stresses,

tests on one of the ships have shown mean values of about O%

of the midship stress at the forward l/L length, and about

70% at the aft l/L. length.

The longitudinal stresses produced by horizontal bending are generally less than 2+0% of the vertical stresses. The only slightly affect the stresses at the deck edge, so the difference between the values from the port side gauge and the mean values of the port and starboard sides has been found to be less than 10%.

b1e of contents

Sid.

Ship Data, Measuring Equipment and Extent of

Measurements 1

Analysis and Results ₃

Introduction ₃

Statistical analysis of stress records 2+

Estimation of the r-value ₅

The wave system ₅

Influence of the wave state on the behding moment ₇

Long-term distribution of the root mean square value

Average expected long-term distribution of wave

stresses ₉

Design bending moment 10

Longitudinal distribution of wave stresses 12

Still water bending moments ₁₃

1.

Shin Data, Measuring Euiment and Extent of Measurements

All relevant data for the test ships are found in Table

I.

The measuring equipment consists of a Philips PR 2210

automatic measuring bridge as the measuring and recording

unit and a couple of specially designed extensometers as the

gauge unit,

The extensometer is described in ref. 1.

The design of the extensometer is such that there will

be full compensation for temperature changes.

In order to divide the bending moment into a vertical and

a horizontal part there are two extensometers, one on each

side of the ship.

By a special connection-box and coupler it

is possible to select between three different combinations of

strain signals, namely the sum or the difference of the

star-board arid port side signals or the port side signal only.

The fist two of these signals will give the vertical and the

horizontal stress respectively.

On one ship in the series extensometers are also installed

at the qu'ter1erigths on the port side.

The coupler is also

extended, and thus it is possible to measure the port side

stress at each quarterlength in addition to the three above

mentioned combinations amidships.

On the dry cargo ships the extensometers have been placed

in the cargo hold in front of the bridge.

They are mounted

on the inside of the side shell about 200 mm from the stringer

angle, except in ship C, when they are on the underside of the

deck.

_{On the tankers and the ore carrier they are mounted on}

the deck amidships.

They are protected against damage and

damp by strong covers.

The cables are placed in pipes mounted

on the deck.

It is very easy to calibrate the equipment in a

mechani-cal way, by inserting thin sheets between the extensometer arid

its connecting rod.

The extensometer has been de8igned by Mr, Sten Brämberg,

Götaverken, who has also been responsible for the lay out and

installation of the equipment.

the tanker type (D - G). The results are given in the tabl.e of ship data table I.

For the dry cargo ships (A - C) it has not been possible to obtain a sufficiently large change in bending moment during ordinary service conditions to permit an accurate calibration.

On all ships the gauges are placed in such position that there will be no risk of local deflections.

According to several previous tests also on dry cargo ships it is resonable to assume the beam theory to be valid on all the test ships.

Thus the calculated section moduli according to table I have been used to convert the measured stresses into bending

moment s,

2.

Analysis arid Results

Entroduction

The statistical methods of analysing measurements of wave

bending stresses in ships' hulls are now universally adopted,

and several papers have been written on results of such

ana-lyses (6.11). Different purposes may be set up for this

research:

To determine mean values of actual stresses, occurring

with various statistical frequencies,

To develop a rational method for determining design

bending moments,

To establish the influence of various factors on the

wave bending moment, and on the whole to explain the

true nature of the wave moment mechanism.

Depending on which of these purposes the researcher is

aiming at, the methods of statistical analysis must be

diffe-rent.

It is impossible to attain them all by one single

method.

Purpose No. 3 above can not be achieved solely through

full scale measurements, but will ultimately be a result of

hydrodynatnic research, model tests and full scale tests.

Purpose No, 2, a design method, cannot be finally attained

until more knowledge is gained about actual values of stresses

or bending moments.

_{It is not yet definitely clear whether}

hull scantlings should be decided by the ultimate extreme

stress which may occur once during the lifetime of a ship, or

if a large number of smaller stresses may be of greater

im-portance either for the safety or for the economy.

It follows

that the first thing that should be aimed at is to determine

average long term distributions of wave stresses which may be

expected with a high degree of confidence to occur in various

types of ships on various trade routes.

Many different methods have been suggested to establish

such long term distributions of wave stresses,

The most

di-rect way is naturally to measure stresses during

_a

sufficient-ly long time and then plot all the measured values

_{on various}

this method is not sufficient. The most important of all the factors which have influence on the bending stress is the

state of weather and waves, and it is practically impossible to ensure a true, average distribution of wave states during a limited period of tests. In order to obtain results of a general value in a shorter time, the statistical distribu-tion of stresses must be separated in the test analysis and

afterwards brought together through a weighting procedure applied to the measured stress distribution.

The present analysis of stress values, measured on seven ships of different types, has shown without any doubt that no simple distribution function of the usual type with only two parameters is generally valid during long periods. Such a distribution must necessarily have at least one more

parameter, which can take care of the different weather

distributions on different routes, and also of the different influences of each weather state on ships of different length and type.

Perhaps the most important of the different purposes that may be set up for the research on wave bending moments is to find rather rough values of the order of magnitudes of these moments. Before going further to establishing more exact values, we must get a general picture of the problem. It is a fact that very little is still known about which con-ditions really are the most dangerous for any specific ship. The question of whether extreme values or fatigue is the

cause of fractures is far from settled. It is quite possible that for some ship types ,a.nd on some routes the extreme values are critical, while in other cases fatigue is decisive. Thus it has been found valuable to calculate theoretical, avarage long term distributions of wave stresses, which don't give

exact measured values for any special ship, but rather show the average values to be expected for a large number of similar ships during a very long time. These distributions are believed to give a picture of some fundamental diff e-rences between long tankers and shorter, fast cargo liners. Statistica,l analysis of stress records

amplitudes are Rayleigh distributed during a fairly short

term0

This has been shown to be valid by so many independent

researches during the last years, that no more proof seems to

be necessary0

It has also been indirectly confirmed during the

present work0

The charac;eristic properties of the Rayleigh

function have thus been used for a comparatively rapid method

of estimating the root mean scuare value, which is the

para-meter of this distribution.

This paraneter will be designated

by the letter r.

The parameter r is decisive for the amplitude

distribu-tion during a period corresponding to some thousands of stress

variations0

All relevant external factors are rarely constant

for longer than that0

The variation during a long period may

be considered to consist of a large number of short term

distri-butions, each with its separate value of r.

This parameter

can consequently be regarded as a statistical variable with a

distribution of its own, which can be used for a calculation

of the long term distribution of the stress amplitudes

them-selves

Estirnationof the r--value

The method used for the estimation of the parameter r is

based on the theoretical extreme value distribution, derived

by CARTWRIGHT (2).

Each record was divided in groups of n

variations hog - sag, and in each group the largest amplitude

was measured.

If a record contained N such groups, a sample

of N values of

the largest of n

was obtained0

The mean of

this sample 1:ns computed, and with the aid of the theoretical

value of the largest of n, the roct mean square could he

calculated, The method is discussed in Appendix I.

For most of the ships hogging and sagging stresses have

been read off separately, using as mean level a visually

esti-mated line through all the snia.11 variations.

The wire system

During these tests the state of the wives has only been

determined through visual observations.

_{For several}

rea-sons the 3eaufort scale has been used to define the wave system.

in-experienced personnel) while on the other hand ship officers

ai'e generally accustomed to the Beaufort scale. Even if many

subjective and irrational factors contribute to the choice of this number at each special occasion, it is reasonable to

suppose that this estimate is in mean correct0 More or less

unconsciously the behaviour of the ship is also taken into account at this choice, which may be considered rather as an

van,

The Beaufort scale is originally made up according to the appearance of the sea, the relative amount of foam and

white caps etc. Afterwards the average wind velocity

corre-sponding to each Beaufort number has been determined by

mea-surements0 There are, however, large variations in these

wind speeds, especially at very low and very high Beauforts. This is evident from a comparison between statistics of wind velocities on the North Atlantic and similar statistics from

entries in the log books of ships, made in the saine area. In

fig. 2 the statistical distribution of Beauforts as measured on the North Atlantic weather ships () is compared to

corre-sponding distributions from ship logs. Wind speeds

correspon-ding to B. O - ₃ are measured only 21 % of the time, while

this wave state is observed from ships during a much larger

percentage of time. This is not necessarily depending on bad

judgement on the bridge, but may be explained by the time

nee-ded for the full development of the sea, During a large

pro-portion of the time the sea has been estimated as B. O - 3, the wind has actually been stronger but has not had the

neces-sary time to raise the sea, And wave bending moments are

caused by the sea, not by the wind.

During each separate measuring period at th present tests the sea has been estimated by the ships officers in

the same way as usual, In order to adjust the observed stress

distributions according to the average distributions of wave states on different trade routes, it has therefore been consi-dered necessary to make a special investigation of log books

from several ships. The resulting distributions are found

B,

O-3

!-

5

6-7

- 9

10 - 12

-7-The Beaufort scale has been divided in the following

groups:

calm, or slight sea

moderate sea

rough sea

very rough sea

extremely rough sea

It cannot be considered justified to make a finer

distinc--tion than this, because of' all uncertainties in the estirnabion0

The group limits make a rather natural subdivision, according

to the given designations.

Influence of the wave state on the bendgmment

If the wave state is only defined by the Beaufort sc1e,

the variation of the observed r-value within each group must

necessarily be large.

This variation actually contains the

distributions of all factors having influence on the bending

stress, viz0 course, speed and displacement0

It also

indu-des the dviatìcns of the estimates of both wave state and

r-value.

This distribution of r-values within each separate

weather group has been found to be very nearly normal,

An

example of this is given in fig. 3,

In Table III are given tJie mean values and standard

de-viations of the parameter r for all the test shipsmeasured in

different wave conditions.

These mean values are also given

in fig. La - !g as function of wave state,

Because of the

large number of measurements the means may be considered to

give a picture of the average influence of the waves on the

different ships.

It is interesting to note a principal difference between

the r-curves in fig.

2+ for ships A - C on one side, and for

ships D - G on the other.

For the first group a maximum

r-va--lue is obtained already in fairly moderate weather, while for

the others r is increasing all the way up to the most severe

weather,

As will be seen later on this fact has a very

im-portant effect on the resulting long term distribution of

wave stresses0

-Long-term distribution of the root mean scuare value

It has sometimes been suggested that the distribution of

the parameter r should be logarithmically normal. The

devia-tion which has been found, especially from the upper part of this distribution (6) has been attributed to the facts, that the tests have not been carried out for a sufficiently long time, and that the test ships have not met with really bad weather during this time.

This explanation does not seem to be adequate for all

ships. In several cases the log-normal distribution gives

much too high values for low probability of exceedance, in spite of the fact that the measurements contain a greater percentage of tests in hard weather than should be normal ac-cording to meteorological statistics.

If the r-distribution is to be used for estimating ex-treme values it must be valid up to at least four times the

standard deviation, and nearly all experimental distributions

have deviated upwards in this region. If a constant value

c is added and log (r + e) is used as variable, this will have the desired effect of straightening the experimental points. In this way a remarkably good fit has been obtained to the theoretical distribution for samples of several hundred

r-va-lues.

The constant c may be considered as a new parameter, which is determined by the weather distribution and by the

influence of various weather states on the ship. No general

method has yet been found for calculating c, other than as the value giving the best statistical result.

If a. long-term distribution of r could really be found, which is valid up to a sufficiently high probability level, this could be used to calculate an expected distribution of

the individual stress amplitudes. Every point on the

r-distri-bution corresponds to a Rayleigh distrir-distri-bution of stresses, and by integration the probability of exceeding any given stress

level may be computed. However, this needs a very accurate

knowledge of the upper part of the r-distribution, which is

not yet available, Work along these lines is going on, and

be possible to calculate the direct influence of the different weather intensities on various trade routes in this way,

Averae ex.ected lon-term distribution of wave stresses

The best way to obtain long-terni stress distributions is probably through integration of the r-distribution, as

des-cribed in the last paragraph0 It is, however, also possible

to calculate such distributions directly from the curves of

r as functions of Beaufort in fig. L1 This method is based

on the observed normal distribution of Beaufort numbers in

fig. 2. The theory is derived in Appendix III and the results

of the calculations are given in fig. 5a - 5g. To facilitate

a comparison between the different ships all these

distribu-tions are based on the same weather distribution, which is

the average of all trade routes according to fig. 2.

In fig. 5 is used a dimensionless bending moment factor in, defined by the equation

L3 _{B CWL}

The wave bending moment Mw is here calculated from measured stress with a section modulus, which for most of the ships has been confirmed by static calibrations of the hulls (Table 1).

The waterline coefficient is considered to be in better agree-ment with theory than the more usual block coefficient (10).

All wave moments and stresses in the figures are half

the peak-to-peak values, For the ships with block

coeffici-ents larger than .75 no significant difference has been found

between hogging and sagging, both amplitudes being equal to

half the total variation. For the dry cargo ships with lower

block coefficient the relation is roughly the same as calcu-lated statically in a sine wave.

It is important to emphasise that these distributions

are not actually observed, but calculated on certain

assump-tions from a limited number of measured values, They are to be looked upon as mean distributions, which are expected in

the average on a large number of ships during a very long

time. Any single ship, e.g. one of the test ships, may very

peri J_O peri

-od which is essentially shorter than the service life. It has

not yet been possible to find realistic confidence intervals, so nothing definite can be said about the amount of departure

from these lines that may be expected. It is hoped that

futu-re futu-research will provide such confidence intervals. _{For the}

purpose of this investigation the mean, expected distributions seem to be of greater value than the results from any specific ship, because it is possible to derive a general method of determining design bending moments for the average ship.

A better picture of the distributions is obtained from the cross curves with constant probability as parameber,

which are shown in fig. 6-e. The curves are drawn as

func-tians of ship length with respectively wave stress, moment

factor and effective wave height as ordinates, The effective

wave height is defined as the height of a sine wave, which by a conventional static calculation without correction for the Smith effect gives the same wave bending moment as the measu-red one.

A was mentioned before in connection with the influence of the waves, the distributions must be devided in two

separa-te groups. One group consists of dry cargo ships less than

150 m with blockcoefficient between .60 and .70, the other group is made up of tanker type ships from 150 to 250 m with

CB between .75 and .EO. This difference is evident from all

results. The reason must be a combination of length, speed

and natural pitching period in relation to the most frequent

wave lengths and periods. The longer ships have a higher

pitching period and generally a lower speed, which gives reso-nance and unfavourable phase conditions in quite different wave systems than is the case for the cargo liners, A special

investigation along these lines is planned, For the present

the two groups must be treated separately, Design bending moment

Several methods have been suggested for estimating the

extreme bending moment to be used for design (6,11). They

have all been based on distributions, either of initial valu-es or of extremvalu-es, which seem to overvalu-estimate the risk of

ex-involve statistical extrapolation, and it is then important that the basic distribution is valid up to very small

probabi-lities of exceedance. This condition has not been fulfilled

by the methods suggested before.

The wave bending moment which is expected to be exceeded with any given probability may be cetermined from curved of

the type shown in fig. 7. It is suggested here to use P = l0°

as a "design level. The total number of bending moment varia-tions during a service life of 20 - 30 years is of the order

of magnitude of 5.l0 Considering the rather small difference

in moment if this number is altered by a factor of 2 or 3, it

is not justified to make a more elaborate calculation0 It

is also believed .that no wave bending moments have ever been

recorded in ships of these types larger than what _corres:on

to P = l0 in fig. 5. This value is regarded as an average

upper limit of expected wave bending moment.

The maximum wave moment to be used for design is then

Mw m L3 _{B CWL}

(1)

where m = moment factor with P = l0 in fig. 7

To .obtain a design section modulus the still water

ben-ding moment must of course also be taken into account., The

basic figure should be the maximum permitted total stress ₍₂ The complete expression for the section modulus Z is then

z Msw

Mw

(2)

(o

Now a second condition must be imposed on this modulus, besides that the total stress is less thanff0 The maximum wave stress must be kept below another limit, which may be called(11. It

is probable, from considerations of fatigue, that

1 always

has to be smaller than

6

_{even if the still water stress is}

zero.

The maximum permitted stress should only be governed

by the yield stress, with proper regard to stress

concentra-tions. If , is the largest expected wave stress and

j'

6.

a

is the value exceeded 10 times during a service life of 5l0

-

12

-= O.2Ji

for tankers (3)

t7= 0.3((l

for Dry Cargo ships (1+)

In order to keep 6'a at the same value for all ships, the maximum permitted wave stress consequently may be 1.5 times

higher in a tanker than in a D.C. ship. For instance, if

= 2.0 kg/mm2, the design value of will be 10.0 kg/mm2

for tanker type ships and

6.7

kg/mm2 for D.C. ships.

The following expressions are finally obtained for the design section modulus:

Msw

Mw

Z.

_-

(6)

'

I

Mw is obtained from equation (i) and fig. 7. Equation (6) is to be used if

/

.6

Msw<U0,.

1 _Mw

(- _J.

It is naturally possible to derive equations for

fitting the points in fig. 7. The curves drawn in the figu-re have the following equations:

Tankers: m =

0.067

L°

D.C. ships: m = 0.0092

L05

Applied to equation (1) these expressions agree well with the

present practice of assuming Mw proportional to a power of L

between 2.3 and 205. More measurements are necessary, treated

in the same way as has been done here, before such formulas may be definitely established.

Longitudinal distribution of wave stresses

The ship D has been fitted with extensometers both

amid-ships and at the quarterlengths. At the latter points,

how-ever, only the stress of the port side has been measured.

This investigation is based on a smaller sample of measure-ments than the previous results amidships.

The comparison between the stresses at the three points has been made in terms of the parameter r, the valuesof

13

-which are given in the table below:

Rayleigh parameter of stress amplitudes, r kg/mm2 Weather group/ Forward 1/Lt. L Aft 1/4 L amidships

Beaufort mean stand dey, mean stand mean stand dev

dey

From these values the long term distributions of wave moments have been calculated in the same way as has been described

before. These distributions are found in fig. 9.

At the assumed design level of P = l0 the stresses at

the forward and aft quarterlengths are respectively about

0 % and 70 % of the stress amidships. At the pro.bability

level of P =

io2,

however, the corresponding percentages are

about 90 and 0,

These figures, show that the maximum bending moment in reality is situated at some distance forward of amidships,

which also conforms with some recent model investigations. (7) Still water bendinE moments

The measuring equipment is fitted with a recorder, which

has a wide scale, This will give the advantage, that although

the recorder is very sensitive, there is no need to balance the zero drift caused by the different loading conditions of

the ship. It is thus easy to use the equipment as a "load

indicator",

In order to make use of this quality the zero level is balanced according to the calculated still water stress of

the ship. In some cases this proceeding has functioned

ex-cellently, while in others it has failed because of

equip-ment error causing a zero drift. For three of the ships it

has been possible to state the variations in the still water

bending moment (SWEM) at sea. The results are given in fig.

10-12 for some typical voyages.

Generally the SWEM at sea seems to be rather constant during

the voyages, On ship B some measurements have been made

at noon, which show ari increased hogging stress, This must

1/ 0-3 B 0.55 0.39 0.47 0.17 0.60 0.30

2/ 4-5 B 0.59 0.22 0.55 0,25 0.65 0.23

3/ 6-7 B

0.1

0.32 0.7e 0.31 0.99 0.33

be due to thermal stresses. The gauges on this ship are

mounted on the side shell, and if the sun is shining straight

on the deck plating, there must be a tensile stress in the

aide shLl

at the gauge

positions.

The tankers show a remarkable hogging stress just before

loading. At these occasions the cargo tanks will be empty and

sometimes the freshwater and fuel oil tanks forward are filled

up in order to reduce the trim. In this condition there must also be great shearing forces acting on the ship.

During the ballast voyages there will be great variations

in the SWUM caused by the shifting of ballast during the

tank-cleaning. During these conditions the measuning equipment has

in some cases been used as load indicator by the officers.

By improved installation the above mentioned equipment error

seems to have been overcome, so that this quality of the

equipment could now be used more generally.

In order to examine the SWBM variations in more general way a number of cargo distributions have been calculated in

the following way.

The SWBM may be expressed thus:

SWflM=M

M

-M

w dw d

where

M = moment of light ship

MdW = moment of cargo, bunkers etc.

Md = hydrostatical moment

The moments may be expressed by dimensionless constants k, kdw and kd,

which

aro defined in the following way

The resuitsof the calculations are given in table IV.

=

kL'W

L = Length between p.p.

Mdw =

kdwLDW

W = Weight of light ship

Md = kdLD

DW = Deadweight

P(x) = 1 - e

A straight line in the coordinate system of fig. L. is the

usual exponential distribution with n = 1. The Rayleigh

dis-tribution has n = 2. For the dry cargo ships A-C the value

of n is greater than unity, for most of the tankers n is very nearly unity, and for the ore carrier D n is slightly less

than unity. In all cases the parameter ari _{is the mean of}

The value of n is obviously dependent on o factors, the

weather conditions and the response of the ship to each

sepa-rate weather state. The latter factor is naturally the most

important of all problems in this work.

If.it is assumed that log (r + c) is normally distribu-ted, the parameter c must be closely linked with the exponent

n in equation (10) above. Research is going on to find some

way of determining these constants, and if this can be done it will be possible to calculate curves like fig. 7 for any given ship on any route.

The influence of whipping stresses has not been properly

investigated. The records generally show a larger tendency

for whipping in the cargo liners than in the tankers.

Especi-ally the very fast ships with service speeds above 1 knots

may suffer severe whipping and slamming during such conditions

that full speed can only just be maintained, Most often the

speed has been reduced very soon after such conditions have been reached, and this has immediately reduced the vibrations, In such ships, where large vibratory stresses are frequent,

the number of amplitudes below about 2 kg/mm2 will be increased, thus making the curved form of the distributions in fig. 5 a

-5 c even more pronounced. This is a strong reason for

limi-ting the maximum permitted wave stress to a rather low

value in fast cargo liners, and may even be the real motive for the influence of speed on the design section modulus. Otherwise the speed seems to have a rather small influence on the low frequent wave bending moment.

(x/a)

15

-General conclusions

The distributions in figures 5 show the general trend of

the stress variation. It is probable that the distributions

for all ships may be represented by the function.

n

Books consulted

Bennet, R.

Stress and Motion Measurements on Ships at Sea.

The Swedish Shipbuilding Research Foundation

Report l3(i95) and 15(1959)

2

Cartwright, D,E,

On estimating the mean energy of sea waves from the

highest waves in a record.

Proc0 of Royal Society London, A 2L4.7,

(l95)

3

Chapman, J.C.

The Interaction between a Ship's Hull and a Long

Superstructure.

Preprint, Inst. of Naval Architects. 6 March 1957.

4.

Crawford, L., Ruby, W.

Model Tests on Hull-Dechhouse Interaction,

Ship Structure Committee,

SSC-67, January 1955.

5

Hamrin, C-J.

Spänningsmätningar i fartyg till sjöss.

(Stress Measurements on Ships at Sea).

Inst. f. Skeppsbyggnad, KTH.

Report E-e-3, 1961.

E

Jasper, N.H. et.al.

Report of the Committee on Response to Wave Loads.

International Ship Structure Committee, Glasgow 1961.

David Taylor Model Basin, Report 1537,

7

Ltveit, M., MUrer, Chr., Vedeler, B,, Christensen, Hj.

Wave Loads on a T-2 Tanker Model,

European Shipbuilding, 10(1961) 1.

16

-¿3

Roll, H,U,

Height, Length and Steepness of Sea

North Atlantic,

Soc. of Naval Architects and Marine

Technical and Research Bulleting No

New York l95.

Waves in the

Engineers.

l-19,

9

Vasta, J,

Lessons Learned from Full Scale Ship StructuralTests.

Trans. Society of Naval Architects arid Marine Engineers,

l95.

17

-10 Vossers, G.

Fundamentals of the Behaviour of Ships in Waves. Netherlands Ship Model Basin.

Publ, No. 151 A mt. Shipb. Progr.

6(1959) 13

and following.

11 Yuille, I.M,

Longitudinal Strength of Ships Royal Inst. of Naval Arch,

I

appendix I

Estimation of the ararneters of the hort-term distribution

When describing the bending moment variations due to wave loads in terms of estimates of statistic parameters, it is convenient to assume that the variations during a short time can be regarded as a stationary random process.

In practice however, the process is not stationary as the conditions such as sea state and heading of the ship is not constant, but investigations (an exhaustive literature survey is given in (6))indicate that the assumption generally leads to distributions to which the observed data fits good

for records of comparatively short time, such as our records of about a quarter of an hour.

As a check of the assumption of stationarity, an additio-nal series of measurements has been carried out on ship D.

Each measurement was started half an hour after the end of the

ordinary ones. The duration of the measurements was a

quar-ter of an hour, The analysis of these 2+ pairs of

measure-ments shows no significant deviation from the hypothesis that the differences between the estimated parameter of the first

and second measurement of the same occasion are due to the error introduced by the method of estimations

Thus the process can be said to be stationary at least during the time of measurement.

Concerning the randomness it can be stated that the

ben-ding moment variations are not random, There is a correlation

between succeeding crests which can be observed by eye on the records and also estimated by computing the autocorrelation function as made in (1) for a typical bending moment record. There it is shown that the only correlation of importance is between succeeding crests, and that the autocorrelation

func-tion for that case is numerically about 0.5. As the energy

spectrum is not infinitely narrow, the correlation function can not show the entire correlation, so 0.5 is an

underesti-mated value. The underestimation is however small, because

the spectrum is narrow (see e.g. energy spectra in (l))and

at any rate more narrow than that of sea waves. The reason

1:2

to pick out frequencies in the neighbourhood of its own natural piching frequency, unless the wave frequency is lower than

that of the ship0 In that case the wave frequency is dominant.

This tendency has been observed in most records and is also in

agreement with general theories for resonance fenomenons. The

comparison with sea waves is interesting because Cartwright (2) has shown that for a sea wave record extended over 224. hours and with autocorrelation between succeeding crests of about 0.5 the effect of correlation is negligible0

Another reason to neglect correlation - and this reason is in itself sufficient - is that the observed data generally fits very well to the distribution obtained by neglecting correlation.

It can thus be concluded that we do not introduce any con-siderable error by neglecting correlation.

Cartwright (2) has given a method to estimate the root mean square deviation Vn (notations from (2))and the relative

standard error in the estimate, from the largest of N

variations of a stationary random process knowing the spectrum

width ' and the autocorrelation of the record0 The spectrum

width can be estimated from the ratio: of the number of mean

value crossings and turning points of the record.

One example of the use of Cartwrights method on bending moment variations is given in (5)

In order to investigate the influence of and the error introduced by the simplifying assumption .= 0, corresponding to an infinitely narrow spectrum, all 155 measurements froni ship

B were analyzed both with the actual value on ¿. and with

= O. Table V gives some caracteristic values of ¿ and the corresponding values of r (the notation is explained

be-low). It can be observed that is far from zero but that

the influence on r is little. We will therefore use o.

Cartwright points out that the standard error in the esti-mate is reduced by a considerable amount by dividing a record

into subgroups and use the average of the largest values in

these groups. We have used about 10 subgroups of 12 periods

each, and have thereby reduced the relàtive standard error approximately from 0.11 to 0.05.

In order to facilitate the evaluation of our relatively large number of records, and get a high degree of accuracy, a program has been developed for the Facit EDB electronic computer, using the asympthotic expressions derived in (2).

This program computes ¿, M1, r, D and y, where r is

and y is the normalized extreme value according to equation 2. For ¿ = O the distribution function considered in (2) equals the Rayleigh distribution which is caracterized by the

single parameter r.

According to the Rayleigh distribution the probability to exceed the value x1 is

2

p(v1) = e'

where and V1 = xl r 2 - vi N

We will use equation L. in connection with the longterm

dis-tribution. Eauation L can also be written:

x = 2(elog(N/f))

₍₅₎

To understand the meaning of f we consider a great number of similar ships being subjected to weather conditions which

give all ships the same r-value. If f = 0.01 then on the

average one ship out of a hundred will get bending moments

larger than the value computed from equation 50

1:3

(i)

(2)

is the normalized variable. By introducing f as the risk, or

the expected number of samples of size N from a population with parameter r where at least one value of the variable is larger than x1, and assuming that the probability to exceed the largest of N variations is 11M, which holds good for large N, we get

p(v ) =

-N (3)

(L)

I:L.

distribution is to plot the observed values of v versus N in a diagram containing the expected value of y as a function of

N and the confidence limits for y. This has been made for

ship B. See diagrams 13 and lL.. The points in the diagrams

scatter at random around the expected value and within the

expected confidence limits. This shows that the Rayleigh

dis-tribution is well suited to describe the bending moment varia-tions, at least the extreme ones ir which we are especially

Appendix II

Sirnplified calculation of the RavleiEh parameter

In a truncated Rayleigh distribution with the point of truncation x1, the probability that a value will exceed x is:

2 2 X2

-X1

z

x)

= Ç p(x) d x = e R X. J-From observations

If these probabilities are eqalled

2 2

X.

-

X

1

R1

ln n1 - ln ni

n. = number of variations exceeding x.

1 1

n1 = number of variations exceeding x1 It can be shown (1) that if

n1 = all the stress variations (which means x1 = o)

the greatest efficiency of the estimation will be about 65 % and occurs if n. = 0.20 ri 1 1 Thus n. i n =

1n5

ln ._]. n. i = r1 = X. 1 ln 5

Appendix III

Calculation of probability of exceedance

The intention of the calculation is to get a long term

distribution of the wave-induced stresses of a ship,

When analyzing the stress measurements, the result will be a lot of r-values where r is the parameter in the Rayleigh distri-bution.

Before the calculation is made, these values have to be divided into five different groups concerning the weather and sea conditions of the actual measurements.

The weather groups are:

Group Sea state/Wind force Beaufor't

1

O-3

2

4-5

3

6-7

L.

- 9

5 10 - 12

The mean values and standard deviations of the r-values of each group are calculated.

For the calculation it is assumed that the r-values within each weather group are normally distributed. Examples of how this distribution fits are shown in fig0 3.

No t at ions

ni. = mean value of r-values of the group

= standard deviation

x = stress variations

= stress variation level (1, 2, 3, .... 10 kg/rrim2) For each weather group r is normally distributed. For each r-value x is Rayleigh distributed.

Put r - ru. i

u-r = u-ru. + i

The probability to exceed x: P(x) 2 2

X.

X.

trI

Px.)=e

=e

J

The probability that u will happen: P(u)du u2

1 2

ï(u)du ₌

The probability of exceeding x in the i:th group:

Ql

_j

P(x.) . u) du

The probability of exceeding x in all groups:

Q = 2 u2 Ç

-

+ur

1 = ,e e du -m/ i e -mAr u = +3 1 e Q. = P. J_.J i 5

o. => o.

_{- i3}

jr=i n. 111:2 u= -m/ú where Lu = 0.1

The different weather groups are weighted by a certain coefficient P. i 2 I X. 2i I. .1 u + u«1 2

For the numerical treatment we put

X.

'm. +

i u 2

dulus Meas. point Clcu-lated m' Calculated atatical stress vari-ations in sine L/20-2 wave kg/mm Hogging 4,1 6.3

7.2

.6

9.7

10.7

9.2

Sagging

4.9

.4

.7

10.5

11,5

12.7

11.0

Calibra-tions of ship bodies: Me asur ed stress varia-tion: - - -

4.6

3.0

6,7

4.2

Calcula-ted s.

y.

-

-

-

4.3

2.9

6.2

4.1

Table I Shin data Ship A B C D E F C

Type of Dry Dry Dry Ore Tanker Tanker Tanker

vessel cargo cargo cargo Carrier

¡Tanker

Deadweight

₄₆₀₀

tons

60O

9O5

₂₁₇₀₀

34200

45OO 65oo

Length bet- _97.. ween pp. n

l3.l

141,7

170.7 l9,l

214.9

23&4

Breath

_rnoul-14,5

ded m

19,2

19.5

22.7

26,

31.1

35..4

Depth moul-

9,22

ded

11,7

12,1

_13.5

_14.3

15.2

₁₇₁

Draught m

6,9

_7.

,1

9.4

_10,7

_11.5

_13,1

Machinery

₃₀₀₀

₉₀₀₀

_2x7000

_{30O 2x9150}

₁₆₅₀₀

₂₂₀₀₀

BHF 111F BHF IHP IHP SHP SHP Speed knots

₁₄

₁₇

_19,5

_14,5

l6,

16.

16

Blockcoeff,

O,6

0,66

0,65

0,79

0.77

O,0

O.0

Waterline

0.1

area coeff.

0.79

O.0 0.4

o.6

0.

0.7

Prism.

0.59

coeff, Section mo-

2.60

Q,6

5.67

0.67

5.2

O.O

O,1

11.6

l.7

O.1

25e0

0.1

439

Ship Measuring equipment being on board (months) Table II

Survey of measuring activities..

No. of ac-tive mease days

(days)

No., of mea- Most

se-surements vere we at h er (Beau-fort) Gr ea-t e sea-t measu-red stress range ₂ (kg/mm A B L.7 57

12

192

10 10 7o C

19

L.0 73 10 709 D

39

₁₃₉

_{3 ¿0} 11

17.0

E 63 73 'J-

12.0

F 1L. E3 3 1 9

15.0

G 30 ₂₅ 60 7

10.0

The measurements of this ship

are rather few and these figures are supposed.

o H) cf- (D Weather group 1 2 3 4 5 Ship m s m s m s m s m S A 0,25 0.14 0.37 0.22 0.69 0.15 0,59 0.17

o.o

0h20 B sag 0.52 0,22+ 0.71 0.25

O.1

0.20 1.3 0.25 1.16 0.21 B hog O.41f 0.19 0.60 O.i

0,63

0.12

0.95 0.1

0,96

0.17 C 0.53 0.10 0.74

0.19

1,12

0.22 1,07 0,25 0.94

0.16

D

o37

0.17 0.2+3 0.24 0.90 0.43 1.04 0.26 2,25 0035 E 0.3e 0,25 0.61 0.34 0.EE 0,32 1.5e 0.45 2.13 0.32 F 0.40 0,20 0.67 0.26 1.43 0,2e 1.76 0,14 2,57 0.30 x) x) x) G 0.40 0.20 0.45 0.25 0.92 0.30 1.40

0.20

2.00 0.20

Table IV

SWBM. Calculated stresses

and

coefficients.

Ship Voyage Caic. still

water2stress kd k_dw DW tons kg/mm B a 0.1771 0.2042

3975

b

+2.3

o.

l09

O226 6930

C +2.1

0.l3l

0.2246 293 a -0.1 0.2035

0,l7l

21030 b +0.5 0.203e

0.190e 2100

C -0.2 0.2036

o.162

21120 d

0.4

0.2040 0.1914 21900 a

-5.5

0.2014

01469 32900

b

-5.

0.2024 0.1474 33700 C

-5.3

0.2019 0,1530 33900 d

-5.3

0.2021 0.1672 34250 e

-5.4

0.2021

01551

34300 f

-5.5

0.2024 0,1559 34400 g

-5.5

0.2022 0,1532 34400 h 0.2020 0.1541 34400

i

-4.5

0.2024

0,1612 3L00

a

-1,4

0. 19 0,150 19370 b

-0.1

0.206 0.195

4630

C

-1.4

0.206 0,1 46610 d _{-2.4 0,207}

0.17

¿f9590 e

-24

0.211

0.19

4960

f

-4.5

0.207

0.14

49990

tri

r = arithmetic mean of

s = stndrd deviation of

max = m.xumum ok served min = minimum observed

mr = geometric mean of r. Corresponds to arithmetic mean

aI

J..og r,

Sr = standard deviation of log r.

Table V

Influence of spectrum width

Ship m ç s r

6max

C, min r (o) s r(0)

mr ()

S

H _OOL7

O.2

O.l

_0.977

_1.000

_________ ______________ ___________

wflr.ilin

H 'ri

0__09/

OZI

09/

Q.Ç/ 1 I I OfrI 0Cl

O"I

0/1

00/

¶06

09

OZ

09

02

0

OC

0

0/

I

il

ulIluuIllIll & p. F MOTOR ROOM tU ÚTC T.

FR(Si-i lulL OIL

SCALE O IO IO O 40 50 iSo LU' mw

..

-1' I I DY TAPIKS DEEP TANK tORE EiSTORE IOR(P(M PUI1P ROOM u o I io to 30 40 00 MITR!

\L_

-I I.-1 - I - ¡sJRJc!yu' - _-

jCr-

'

-á- % ¡t4jr' ;.i -r --

----

_-ip-

--r

N T

!

j. L L . i. ..--j -. 'L___-- , I . *.

-hj

l-J.

H

h-j s

ii.,....,iJ I'UY DISTiLUBUTI(..$ Oi JJFCRT UMBPS CN TL{3 ATLANTIC OCEAN.

General ro

Tanke r

ut e s

/

routes

North Atlentic Ocean routes,

Ï

, North accord ng

z

to Roll -I

o

z

4

6

a

Io

t

09...

P,,

'.1

i,

1

1P7 95 00 70 '0

4°

3°

to

'o

i, 3 2t C. i 0.0$ Coi

Fig. 2.

an, Atlantic Oc

IEATHER GROUP 2 t.Q l , t

r*

z. w t- 51 T-ec

tL

R tO V. I.E AMIflSH I to t t 1,6 r

OXM1ß

1,6rIk V.lbrV WE&THER GOUP3

i

x-c

g

6-8 L-9 $ t

-I dUiS v9 90K ¿I;OI 6-8 L-9 2-O I 2

9

O

61

L9

-r

2-O g g V dIHS 6-8 L-9 2-' I t

_

s c O z I

'¼J

Wj s 2 2W Wi I 2 2

2I;Ol

64

L-9 -4 t I

Zt-6-9 L-9 -v Q-O t I f

g.

I dUI I 2 Q 7Ut! w

j

9 I 2 Q Wi 6-9 L-9

94

Q-O 4 4 4 I a I Q

STRESSES EXPECTED TO BE E)CEEDED WiTH VAtOUS PßA1LITIES P

Fig. 6.

o. u, & u) Cu

9

G cE ECJ

o

eJ

E?

BENDING MOMENT RCTO m EXPECTED TO BE EXCEEDED WITH VARIOUS POBAILI1IES P.

Fig. 7.

'-I

/

i

I

1/

/

/ /

iÌE

EFFECTIVE WAVE HEIG1T EXPECTED TO BE

EXCEEDED WiTh VAIOJS POBAILI11ES P.

Fig.

. (u g W) .

z

uJ -J Q,

-iI

0. W 0.

I

J.0

s 0. 0. U IAJ

ii:

s kp/m lo 9 &

i

crq t, 'Q 5 ¡t z &

i

m G 5 ¡t

suip D

i. STRESS AT PORT SIDE

I -

AHIDSAIPS

ViL FORWAD

3

-'----

_l/i+LAFT

't. VEQT1CAL 5TES _{AHIDS1PS(FROH FI6.5D)}

3 z I o

rj H o

vit

_'r/mm

t-SHIP B HOG VOYAGEa HOG HOG T I I I I _{I I} I

t

_{3 5 6} 7 5DAYS g io SHIP B VOYAGE C WEST BOUND IB tO I I I 12 l 19 E1 BOUND i WEST BOUND z WEST BOUND ¼ 3 5 I - SAG y PCP/mr I I I i L. _{5 6 7} i 6 MEASURED AT NOON

i

t

1 J. _{5 DAYS}

HIPB

VOYWE d DAYS ¿ DM6

5-k

t-HOG I I Ib 1$ 28 I 5 bS '.1 I I t I I t I I I I I t t I I t t t I I I

1t,569IO

TI 19 O

323

1 'iO 'it

DAYS

LO PD

SfrG

BALLkST

VARIATIONS IN STILL WAIEQ

BENDING MOMENTS AT SEA.

I

17

Fig. 11.

jo .l. s DAYS LOPtO

1-

2-

k- 5- 6-6 11 1 I sçI 5 15 KPJm,, SHIP E HOG BM.LST

z-K' Z

_P,mm

t-

i-O HOG Zß I I I I I I. jtg _t9 31 VOVAE LOtD I I 35 3 3Ç1 SHIP F DP1YS Z l-J. 3 SAG H t) _I HOG

Bt11T

Z-I -Ib O I , I II

1-t

Fig. 13.

Scatter of measured extreme values. Ship

no 2. Sagging. q:: 2 i

t'C

L o

l'i

I p s I

.k\

.

.

,..a\41..

-..s . s s p

Pp

II

4 Pp

.

s I

I..

p t ... s. s p p

4 s.s

s p,sp t s I o o o o

Ñ

o

c%i

o

Fig. 12+. Scatter of measured extreme values. Ship no 2. Hogging.

r -1 f I j

i

I

t

s. s t I e fr

..

s

r

j

o o