On the determination of the statistical properties of the behaviour of moored tankers

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C60797

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WORKSHOP OE

ON. THE DETERMINATION OF THE

'

STATISTICAL PROPERTIES OF THE

' BEHAVIOUR OF MOORED TANKERS

By: J.A. PinkSter (MARIN)

ON THE DETERMINATION OF THE STATISTICAL PROPERTIES OF THE BEHAVIOUR OF MOORED TANKERS

J.A. Piaster

Maritime Research Institute Netherlands (MARIN)

SUMMARY

The motions and mooring loads of permanently moored storage/production tankers moored in harsh environment areas are dominated by large amplitude low frequency componentt which are strongly related to the low-frequency

second order wave drift forces. ASsessment

_of

these motion and mooring load

components in the early design stage of

a

mooring system is of paramount

importance. Such estimates can, depending on the complexity of the system

characteristics, be based on frequency-domain or on time-domain analyses.

Frequency,-domain analyses can be employed if the system properties such as

the static-,restoring characteristics and the motion damping are linear.

When this is not the case, time-domain analyses in the form of simulation computations are used. Time-domain computations, however, require

substan-tial computing power mainly due to the complexity of the time domain-formulation of the second order wave drift forces. In this paper a simple method to compute second order drift forces is described Which allows

time-domain analyses of the low frequency Options and mooring loads to be

carried out on desk-top micro-computers. The applicability of the method is

demonstrated by comparisons between results

_of

computations and of model

tests with a tanker moored in irregular head seas. The simulation method allows very long duration simulations to be carried out thus opening the way to systematic investigations of the statistical properties of the low-frequency behaviour of moored vessels. This can be of importance for the

selection of, for instance, the duration of model tests for the final

design of the system, or the method with which model test results are

extrapolated in order to arrive at the design loads for the system.

1. INTRODUCTION

Simulation methods to determine the

be-haviour of moored vessels in all six degrees of freedom taking into account, in the most

complete way, all relevant hydrodynamic

phenomena have been under development for

the last fifteen years See Ref. [1]

through Ref. [6]. These investigations hive revealed that, in order to be able to

simu-late the general case

of

a vessel moored by

a system with non-linear restoring charac-teristics, in shallow water Said in irregular directionally spread seas, with the effect

of current , Will require the computational

power

of the

largest Mainframe computers

available. It is expected that such methods

will find general application

in

due course

but there will always remain a need for

computational methods which allow quick

assessments of the major behavioural aspects

1

-of such moored vessels in the early design. stage when the system description is still very general and without detail.

For permanently moored storage/production Vessels a condition of prime importance for

the design is the case of high irregular

head seas. This is valid for the design sea condition Which is generally referred to as the 100 years storm. In this environmental Condition it is assumed that the directions

of waves, wind, and current co-incide. The

Vessel Will align with the environmental

effects thus leading to the assumption of

head waves for this condition.

Model tests carried out in the past have

shown that the head wave, condition often

leads to the highest mooring loads. It is

therefore useful to simplify simulation

com-putations to take into account only this

are not relevant. In its simplest form the simulation computations are limited to the

low-frequency surge motions i.e.,

_only

one

degree of freedom is considered.

Obviously this is a tremendous simplifi-cation but one, as will be shown, makes it possible to carry out meaningfull simulation

computations on ordinary micro-computers

thug bringing Such analyses within the reach of many and which also makes it possible to

investigate some of the

Major

characteris-tics of a mooring system at a very early

stage.

In the next sections the background to

this method will be given and the

assump-tions which make simplification of the more general method possible. As is usually the case, simplification of a more general meth-od Means that restrictions must be placed on the use of the simplified method, these will also be discussed. The time domain simula-tions of the low-frequency surge mosimula-tions of a 300 kDWT tanker moored in irregular head

seas by means of

a

linear and a non-linear

mooring system will be compared with results of model tests in order to Show the applica-bility of the method.

One of the problems with respect to simu-lation computations and model tests for the low-frequency motions and mooring loads is

related to the statistical reliability Of

the results when considering the restricted

duration of the simulations and tests in

relation to the number of oscillations of

the low-frequency components. The simulation method allows long duration simulation com-putations to be carried out which give more insight in the typical properties of these

motion and force components. Examples are

given of the effect Of simulation duration

of such quantities as the most probable

maximum values of motions and mooring

forces. Time domain simulations were also

carried out in order to investigate the

effect of changes in the mooring system

properties on the maximum loads in the

system.

2. LOW-FREQUENCY SURGE MOTIONS IN HEAD

WAVES

Based on potential theory, the surge

motions of a vessel in irregular head waves are described by the following equation of motion: (M+m)R+

f

K(t-T) X(t)dr 4I(x) = F(t) (1) -m in which: = surge motion F(t) = environmental forces

f(x) = restoring characteristics of the

mooring system

= mass of the vessel in air = retardation function = mass coefficient

The retardation function K and the mass co-efficient m are found from (See Ref. [3]):

K(t) =

f

b( w) cos(wt) dw ₍₂₎ 0 m = a(w1)

f

K(t) sin(w't) dt . w 0 (3) Where.:

a(w') = frequency dependent added mass in surge

b(w) = frequency dependent potential surge

damping

= an arbitrarily chosen frequency

At very low motion frequencies the effect

of the frequency dependency of the added

mass and the potential damping are small. From model tests and computations it has

been demonstrated that the low-frequency

motions of large vessels Wored in deeper water take place mainly about the natural

frequency of the motion as it is dictated by the mooring stiffness on the one hand and by

the surge virtual mass on the other hand.

The equation of motion can then be

simplified as follows:

(M+a)i + bX + f(x) = F(t) (A)

in which:

a = surge added mass b = damping coefficient

In this case the added nags and damping

coefficients are constant. It Should be

mentioned that, for instance for vessels

moored to jetties in shallower water the

above simplification cannot be made and the full expression given by equation (1) must be used. In such cases the mooring system is

generally stiff and also strongly

non-linear. See Ref. [1]. In this paper we

restrict ourselves to vessels moored by

relatively soft mooring systems in deeper

water. In the following section the methods used to determine the various quantities in

the simplified equation of motion Will be

discussed.

-Coefficients in the equation of mbtion: Added mass:

The added mass coefficient for

a

given determined from 3-4 potential

case Can be

Ref. [3], from motion decay

computations, testing. Damping:

The surge damping coefficient is

a

rather

complicated quantity in so far as it is made up of a number of contributions each of

which are of sufficient significance to

war-rant -evaluation. According to Wichers [7]

the damping may be expressed

_as

follows:

b =b+b+b+ b

_{SW u wa wi}

in which:

= damping in still water = damping due to current = damping due to waves = damping due to Wind

The still water damping of a given vessel

may be obtained from still water motion

decay tests. No reliable computational

procedure exists as yet for this quantity. The damping due to the presence of current

may be estimated from the slope of the

longitudinal current force as a function of the relative current speed:

b = F (V

+ X)

(6)

CU CU Cu

dx

Taking into account that the current force is a quadratic function of the relative

current speed, the following result is

found:

bcu _V

cu in which:

mean longitudinal current at current speed Vcu.

The surge damping due to the waves is dependent on the square of the wave

ampli-tudes and On the wave frequency and,for a

given wave frequency is found from the slope of the surge drift force in relation to the forward speed of the vessel:

bwa(w)

_{Fd. ('}

x w)

dx

In Ref. [1], WiChers has given results of

the surge damping coefficient for the case of a.fully loaded 200 kDWT tanker.

The mean value of the wave damping

coefficient for a given sea state is

subsequently found from:

b= .f

_wa S-(w) 11 dw wa( 0

in which:

S-(W) = wave spectral density

The damping due to wind is found in the

same way as the current damping, i.e. from

the slope of the longitudinal wind force as a function of the relative wind speed. The following result is found:

Fcu (Vcu ) =

2F

(V ) cu Cu (5) (8) (9)

3

2 Fwi(Vwi) bwi = V (10) wi in which:

Fwi(Vwi) = mean longitudinal wind force at wind speed Vw/

Restoring forces:

For most mooring systems the static re-storing force f(x) determined as a function of the _{static horizontal offset x of the}

vessel is used in the equation of motion.

This is allowed since in most cases the

frequencies of motion involved are very much

lower than the natural frequency of the

actual mooring system, in other words, at

the natural surge period of the vessel, the mooring system does not display any dynamic

magnification effects. This does not mean,

however, that all low-frequency forces

exerted by the mooring system are solely due to the static restoring characteristics of

the system. In Ref. [8] it is shown that in

the case of mooring systems based on chains,

'dynamic magnification effects may occur

which can result in low-frequency mooring

system forces which find their origin in

wave-frequency motion components.

In the aforegoing we have discussed the

coefficients on the left-hand-side of the simplified equation of motion. We will now

concentrate on the right-hand-side of the

equation which describes the environmental

forces acting on the vessel. Environmental forces:

In general the environmental forces

acting on the vessel can be described as

follows:

F(t) = Fwi(t) + Fwd(t) + Fcu(t) (11)

The wind and current forces are often

as-sumed to be constant, although it

is

known

that. dynamic components can

also

be present.

The low-frequency second order wave drift

forces in an arbitrary irregular head sea

are described by the following equation:

-140 440 F(t) =

f

f

h(TT2)C(t-T)c(t-T2)dt1dt2 (12) in which: C(t) h(2)(T1,r2)

= wave elevation record at the mean position of the vessel = second order impuis response

function of the drift force The second order impuls response function is found from the frequency-domain quadratic transfer function of the drift forces which can be determined based On .3e1 diffraction (7)

calculations.

See

Ref.

[4]. The quadratic

transfer function of the drift force

repre-sents

the

mean

and

low-frequency

forces

exerted on the vessel in regular waves of a

single frequency and in regular wave groups

consisting of two regular waves.

The above equation

(12)

for

the

time-domain representation of the drift force in

irregular waves

represents

a

considerable

task from the point of view of computational

effort. The subsequent solution to the

equa-tion of moequa-tion (4) requires only a fracequa-tion

of

that

effort.

The

simplified

method

alluded to

in

the introduction is mainly

based on an empirical simplification of the

model used to describe the excitation. It

can be

shown

that

the

second order wave

drift force excitation on

the vessel can

also be described in terms of a mean force

component,

and

a

varying force component

which, in turn can be described by a

spec-tral density and a distribrution. The mean

surge drift force follows from:

r b kW)

-kW)

. IF =

--- 0W

(13)

d C Ca

in which:

of this assumption lies in the fact that the

low-frequency surge motions are only lightly

damped and are very Eolith Concentrated in a

narrow band About the natural surge

frequen-cy of the vessel. In such cases the

calcu-lated motion

response based

on the white

noise assumption is very close to the true

value based on a frequency dependent

spec-tral density. Since the value of interest in

the

excitation

is the value Valid

at

the

natural frequency of the surge motion this

should be used.

However,

since the natural

frequency of

the surge motions are usually

very low, it is convenient to use the value

of

the

excitation spectrum valid

for the

frequency zero:

[St(w)770141244,

(15)

SF = SF(0) = 8 f

0

Ca

In Figure

1

an example is given Of the

spectral denSity of the low-frequency surge

force on 'a large tanker in head waves as

calculated based on the complete formulation

of

equation (14).

In the same figure the

approximation is also shown.

10 104

Spectral density. conplete expression Approximation

15 104

-in rad/s

Fig. 1. Spectral density of low-frequency

surge drift forces on a 200 kDOWT

tanker

As

can

be

teen,

the

approximation

stays

within 20% of the true value for values as

high as 0.05 rad/s. In general the natural

frequencies

of

interest

of

large

tankers

moored in deeper water Will not exceed about

0.03 xad/s.

(

_SF(0)

\

, . .

\

\\

\

\

1 Sf(u)

\

\

II

\

..-ii=---.../ ...6._

\

_\

\

i I Approximate range of Surge frequencies

\

\

_... N. I, . natural I 1

7(u0) = mean drift force in regular waves

with frequency w and amplitude Ca

The

spectral

density

of

the

drift

force

follows from:

SF(u) = 81 scoo

s(w4.10[T(w,w+u)]2dw

(14)

0

in which:

T(w,w+u) = quadratic transfer function of

the drift force in regular wave

groups with component frequencies

Ed and w4.11

respectively

The distribution function of the varying

part of the low-frequency drift forces is a

complicated quantity which depends both 04

the spectrum and distribution of the

irregu-lar waves

and

on

the

quadratic

transfer

function T(w,w+v). See Ref. [4].

It is,

however, related to the

distribu-tion funcdistribu-tion of the low-frequency part of

the

square

of

the wave elevation record,

which

is,

of Course,

not

surprising. The

low-frequency part of the square of the wave

elevation

is

exponentially distributed as

will be shown by modeltest results in this

paper. For the simplified model of the drift

force excitation use made. of this same

dis-tribution.

A second assumption Made for the

Simpli-fied model of

the excitation is

that

the

spectral density is constant over the

low-frequency range. This is what is known as a

!White noise'

assumption. The justification

0.010

p(F)

0.005

Fig. 2. Distribution of drift forces

In Figure. 2 an example is given of the

distribution of the low-frequency surge

force On a large tanker in irregular head

seas. In this case it concerns measurements carried out by MARIN. In the same figure a line is drawn showing the exponential dis-tribution assumed in the simplified excita-tion model. As can be seen, the agreement is

reasonable. So far, we have

only

described

the assumptions which underlie the simpli-fied model, which were as follows:.

White noise Spectral density

Exponential distribution

-The following expression for the time-domain representation of the force complies with these requirements:

F5-0F(B+1) Sign(d)

+ 1

(16)

in which:

B = ln(rnd(a)) 04tud(a)41

and P(rnd(a))=rnd(a) = mean surge force, sum of mean wave,

wind and current forces

Fd - mean surge wave drift force given by equation (13)

aF'represents the rms of the drift forces

as found from the product of the spectral

density and the frequency band of the exci-tation:

.

W 14

AS = [SF(°) * _1T-1

with:

SF(0) = spectral density of the wave drift force for zero frequency given by

(18)

-5-equation (15)

At = time interval used

in the

evaluation

of the force during the simulation

As

will

be apparent, equation (16)

repre-sents uncorrelated, exponentially

distrib-uted white noise with Mean value

correspond-ing to the mean environmental force and rms following from the selected time step of the force evaluation and the requited value of the spectral density using equation (18).

An example of the time trace of the force

record produced

in

this way is shown in

Figure 3.

in min.

Fig. 3. Drift force record

It can be seen that the force record

con-tains large and rapid-Variations in the

force which is also observed to be decidedly

Skew in distribution having very large

variations to one side and no

zero-cros-sings. It Should be kept in mind, however, that these variations take place at

frequen-cies dictated by the selected time

step.

This is normally chosen to be in the

order of 5 to 10 times Smaller than the

natural

period

of the moored vessel thus

ensuring that the motion response to the

highest frequencies in the rapid,

uncor-related force variations are negligibly

small.

In order to solve the simplified equation

of motion given by equation (4), standard

methods are available. For our particular

case we have Chosen to take advantage of the

fact that only one degree of freedom is

being considered. This allows easy

implemen-tation of the analytical solution of the

Motion

over

the time. step of the .process.

The procedure is as follows: At time t the excitation force is evaluated from equation

(16). This force Is assumed constant over

the time Step. The Motion and motion velo-city form the initial conditions at time t

and the analytical solution

for

the

lin-earized equation Of motion is Used to

pro-1 11

II

I' I' I'

I1

% 1 \ 1 \ 1 \ I k Complete expression Approximation I I I

\

_\

\

1 I I I I

I..,

., ... ., .. .... .100 0 -100 -200 F in t -300 -400 IC 200 100 F in tf

duce the values of the motion and the motion

velocity at time t + time step. Note that use is made of the linearized equation of motion, non-linear mooring systems are dealt

with applying local linearization Of the

restoring force relationship f(Z).

This method of solving the equation of

motion has turned out to be .quite usefull

for application on the micro-computer in

terms of computation time.. On a popular

brand

of

16-bit micro-computer,

without-floating-point co-processor, a total time

including disk I/0 to store the results

amounts to some 120 s for a 12 hour full

scale simulation.

In the following section, results of

time-domain simulation computations of

the

low-frequency surge motions will be compared

with results of model tests.

As will be

obvious from the aforegoing description of the simplified method for the simulations, a

deterministic comparison will not be pop-sible owing to the fact that the simulated force record is generated based on a random number procedure. Only a more sophisticated method of generating the forte such as given

by equation (12), Which can make use of a

measured wave record can be used for

deter-ministic comparisons.. See., for instance,

Ref. [4]. Our comparison will therefore be

based on statistical results obtained from

the simulated time record of the surge

motion and from the measured time record of the motions taken from model tests in

irreg-ular head waves. Before going into the

detail of the comparisons the model tests

will be treated briefly. A more detailed

account of the tests

is

given in Ref. [6].

3. MODEL TESTS WITH A 300 KDWT TANKER IN

IRREGULAR HEAD WAVES Test set-up:

Model tests were carried out in the Wave and

Current Basin of MARIN with a scale 1:100

model of a fully loaded 300 kDWT tanker

moored in irregular head seas by means of a linear and a non-linear mooring system. The main particulars of the vessel are given in Table 1 while a body plan is given in Figure (4). The basin measures 60x40 m. The water depth corresponded to 100 m full scale. Both mooting systems consisted of spring packages attached to the vessel at deck level.

The test set-up is shown

in

Figure 5. For

the linear mooring system the stiffness in the surge direction amounted to 15.5 tf/m.

Table 1

Main particulars 300 kDWT tanker

Fig. 4. Body plan of 300 kDWT tanker

+X

r

Designation Symbol Unit 300 kTDN

VLCC

Length between perpendiculars L m 347.6

Breadth B m 53.57

Draft fore TF m ' 21.19

Draft mean TM m 21.19

Draft aft TA in 21.19

Displacement v m3 335,415

Centre of gravity above base YU in 14.94

Metacentric height 'CR in 7.04

Centre of buoyancy forward

of station 10 LCB in 10.56

Longitudinal radius of gyration k

YY in 86.90

Natural roll period T

V s 15.1

A.P. F.P.

Fig. 5. Model test set-up

The restoring characteristics of the

non-linear mooring system are given in Figure 6. Wave conditional

The following sea conditions were tested:

Significant wave height : 13.0 m

Mean wave period : 12.0 s

PiersonMoskowitz--Non-linear mooring

Linear-mooring (15.5 tf/m)

Fig. 6. Restoring characteristics of mooring System

The test duration amounted to 12 hours

for

the full scale Vessel. In Figure 7, the

spectrum

of

the undisturbed measured wave

elevation record is given . The spectrum of

the low-frequency part of the square of the measured wave elevation which contains in-formation on the wave grouping phenomenon is

given in Figure 8 in which it is compared

with the theoretical spectrum based on the

random wave Model. The rando# wave model

predicts that, given the spectral density

_of

the wave elevation record and assUbing that

the record is normally distributed, the

spectral density of the low=frequeicy part of the square of the wave elevation follows from: CID SA2(w) ge 8

f

s

s

_;(wft) Aw 0 (19)

-7-40.0 20.0 0.0 5PCCT1611 tbasoresi IfIct- 12.6 is ; Ti - 14.0 Thoorsti...1. ( P.M.; Pict- 3.0.i ; Ti - 12.0.

0.0

Fig. 7. Spectrdifi of irregular waves

Note the similarity of this expression with equation (14) for the Spectral density

of the low-frequency drift

force!

The

dis-tribution function of this Signal is given in Figure 9 in which it is compared with the theoretical distribution which it also based on the random wave model. This model

pre,-Aicts that the low-frequency part fo the

square of the wave elevation it

exponential-].y distributed

as

was already mentioned in

a

previous section of this papaer. The distri-bution is given2by the following expression:

A

p(A)

1-

6 110

-0

0.5 1.0

NRVE_ntomicyINmon

1.5

(20)

in Which:

mo

f sc(odw

0

The results shown in both figures indi-cate that the irregular waves generated in

the basin correspond reasonably well With

the random wave model.

_Imo i I t 1 t i 1000 1 1 I t _ t 750 t I I 1 .

f(x)

in tf

no

/

I t I t 250 1 1 t

/

1

60 40 20

/

ie

ei

-250 0 -20

x in m

-40 -60

/

/

e

/

/

/

-/

e -750

4E00.0

3000.0

0.0

TEST C. 7499

ormlito rnOm Low encoutmcy PART OF SQUARED NAvE RECORD ---,- DERIVED TNEORETICALLT BASED ON SPECTRUm OF MEASURED NRvE

, -0.00

Fig. 8. Spectrum of wave groups

Results

of

model tests:

The Measured Surge motions were statisti-cally analysed to obtain the following data:

mean

root-mean-square

maximum positive value - maximun negative value

distribution of all measured data distribution of extremes (peaks and troughs)

The results of these analyses will be

discussed further it relation to the results of the simulation calculations.

4. SIMULATION CALCULATIONS

Input data:

The input data for the time-domain simu-lations consisted of data on the still Water damping, the wave damping, the virtual mass of the vessel, the restoring force and the

mean and low-frequency wave drift forces.

Since the tests Were carried out in Waves

only, wind and current effects in the damp-ing or the excitation were not included-.

025 050

CROUP FREOUENCT IN NM'S

075

TEST NO. 7492

DERIVED mon Lownotoubicy !ARTorSOURREO NRvE RECORD

----,OCRICDTPEORCiICALLYBASEDMSPIXTRUPIorMERSUREDwAvE

tk\

0.0 50.0

U' le

Fig. 9. Distribution

of

wave groups

The restoring force characteristics for the calculations for the non-linear mooring

system are given

in

Figure 6. The mooring

stiffness for the linearly moored vessel

amounted tO 15.5 tf/m.

The still water damping coefficient and the virtual mass were Obtained from motion

decay tests, the following results being

found:

- Still water damping coefficient:

bsw = 29.6 tfs/m

Vittual mass ,(M+a) = 38,940 tfs/m

calculated based on data given in Ref.

The wave damping coefficient

was

[7]

for a similar tanker and using equation

_(9),.

The following result was found: Wave damping coefficient bwa = 51.0 tft/m

The total surge dadping

follows

from:

b = bs + bwa_w - = 80.6 tft/m

The mean longitudinal _wave drift force and

the spectral density of the drift force at zero frequency were calculated based on a

3-d diffraction calculations data base_for

.similar tanker hull forms incorporated in

10

10

the MARIN WWCFORC program Which also incor-porates data on wind and current forces. The

following values were found: - Mean surge drift force

F = -175.6 tf - Spectral density

S = 206,073 tfs

Results of computations:

The results of computations were subjected to the same analyses as the model test data.

LINEAR MIX0II6STSTPI 45 100 1 60 hou T in rs

Fig. 10. Measured surge motions

in

irrgulsr head seas

5. COMPARISON OF RESULTS OF MODEL TESTS

AND SIMULATION COMPUTATIONS

In Figures 10 and 11 comparisons are

given of time traces of the measured and

computed time traces of the surge motions of -the tanker.

LINEAR 11082I1116 57srEPI

15 If E -J1

a

..mmEmme=mom=sm===mmimm. ilf101111170111.1=1:=COI

recnow wpm

1, Tr

I T

'17

1!.''!!

Lill

I

Le ! r ! I " V4.5 I LI 75 hairs

The results of the statistical analyses of

the surge motions from the tests and

simulation computations for the linear and the non-linear mooring systems for the full

12 hour duration are compared in Table 2.

In Figures 12 and 13 comparisons are

given of the distributions of all measured and simulated data and the distributions Of the extremes. A number of simulation

compu-tations were carried out

in

order to obtain

an indication of the variability of the dis-tributions.

-9-From the comparisons It can be concluded that the simulations give a good

represen-tation of the vessel motions both for the

linear and for the non-linear mooring

systems. The diStribution functions of _the

motions show the same characteristics with

respect to the influence of the mooring

system linearity. For instance, in Figure 13

it is seen that both measured data and

simulated data for the non-linear mooring system show the same deviation from the line denoting the normal distribution, i.e., the

sm-LINEARmom svsn.

20

3

a

positive surge motions have 4 larger proba-bility than the negative surge motions rela-tive to the normal distribution. This is in keeping with the increasing mooring system

stiffness for negative motions Which

pre-vents the development of larger negative

I I I ..17:w2.cmaTir. atm . T

..Y1

V I V7.' Tim to hours 1131411lEA2 CMOS 5751th IWIMIIMAW211 r =MUM

motions. The reverse is true for the posi-tive motion components.

The comparison of the statistical data given in Table 2 indicates that the mean and

root-mean square of the surge motions are

predicted reasonably accurately by the

simu-lations. Some differences are. seen in the

maximum values. As indicated in the section dealing with the background to the simula-tion method, this is attributed to the fact that the method of simulation is based on a random number process which restricts corn-Fig. 11. Simulated surge motion in irregular head seas

1 75

Tut in bars

05.0 92.0 55.0 50.0 70.0 130. 60.0 40.0 30. 55.0 10.0 5.0 2.0 1.0 0.5 0.2 0.1 LINEAR SYSTEM .

Fig. 12. Distribution of low-frequency Surge motions Table 2

Results of analyses of measured and computed Surge Motions

Duration of tests and simulations 12 hours

parisons to statistical data. A direct com-parison of data on maximum values can only be made if the simulation procedure is based on the sate wave trace as Was applied in the model test program.

6. APPLICATION OF SIMULATION COMPUTATIONS

In this Section we will treat three

examples of the application of the

simpli-fied simulation procedure. The first of

these is concerned with the statistical

variability of the low-frequency motions due to finite test/simulation duration effects. The Second is concerned with the reliability of short duration tests/Simulations with

NON-LINEAR SYSTEM

(a-i)/0.

respect to the

conclusions Which

are drawn

based on comparisons of results of sensitiv-ity studies. The third example concerns the

statistical variability of the most probable

maximum values of low-frequency mooring

loads and motions obtained from limited

duration tests/simulations.

Variability of low-frequency motions: Due

to the low. natural frequencies of moored

Vessel*, the number of oscillations in a

measured Motion record obtained from Short duration model tests may be is low as 5 to 10 for 4 test duration corresponding to 30

minutes full scale. It is often desired

in

such Cases to have an indication of the Test No. 7105 a Run No. 41 0 Run No. 42 A Run No. 43 V Run No. 44 D. Run No. 45 ---- Theory (Gauss) , Measured Simulations 0 4 -2 -1 0 2 7 uns.0 99.6 69.5 09.0 66.0 65.0 60.0 65.0 60.0 70.0 60.0 50.0 40.0 60.0-20.0 15:0 10.0 5.0 2.0 1.0 0.5 0.2 0.1 Test No. 7512 a Run No. 41 0 Run No: 42 A Run No. 43 V Run NO. 44 C Bun No. 45 ,,-Theory (Gauss) - Measured Siaulations -. . V -3 t -2

-Linear system Nom-linear system

mean r.m.s.. A MAX. 4- A MAX. - mean rxi.s. A MAX. + A MAX.

-Computed -10.97 16.82 ' 44.43 -74.10 -10.75 16.35 60.09 -54:48

0.1 0.1 ..._ 504 LINEAR SYSTEM TestNo. 7505 Sioulation ---- Theory (Rayleigh)

tI

INN

sample variance of such quantities, i.e. an

indication of the possible spread in the

outcome of, say

the mean and

root-mean-square values of the motions, if the test

had been carried out in the same sea condi-tions but for different realizacondi-tions of the wave record.

In order to show that the simple simula-tion procedure can be an aid in such cases

we will revert to the case treated in the

previous section in which results were given

for an uninterrupted duration of 12 hours

full scale. The sample variance can be

demonstrated by dividing the full 12 hour

record into 30 minute sections, and each

section individually analysed, in this case with respect to the mean, root-mean-square,

maximum and minimum surge motions. The

results of this exercise is shown in Figure 14 for the case of the linear mooring system

and in Figure 15 for the case of the

non-linear mooring system.

In Figures 16 and 17 the same data is

shown for the case that the 12 hour record

1 1 NON-LINEAR SYSTEM Test No. 7512 Siss11ation 'OA 00.0 0.1 .1034 _MA 0.0 -MA X11114CER IN

Fig. 13. Distribution of surge motion extremes

CRC575

-100.3

is divided into sections of 1.5 hours. In

each figure the corresponding data are also

given for the model test .data. The result

shown in the figure indicate that the compu-tations confirm the magnitude of the sample variance found in the model tests.

Application to sensitivity analyses: In

order to demonstrate the applicability -for sensitivity analyses we will again use the 300 kDWT tanker. We will take the case that it is required to determine the influence of a 20% increase in the stiffnets of a linear mooring system on the maximum mooring loads. We will assume that this can be investigated

by comparitive tests/simulations with a

duration of 0.5, 1. and 1.5 hours. Due to the

sample variance effect, there is a

possibil-ity that comparison of two tests , carried

out under identical environmental conditions but with slightly different mooring system characteristic6 will lead to incorrect con-clusions. We will investigate the probabil-ity of incorrect conclusions being drawn by carrying out 10 simulation computations for

_ 6

\

\ '

V

\

_\

\ \_\ GIINA rul .. -'MARS 100 es es Go 70 .eo so 40 30 20 10 5 0:1 0 -50.0

x-myric* (11)

so

-SO

Linear system Duraticm of each cycle 1 hour

MEASURED coPIPUTED XMAS.. eNsleivri, 0%.2/11,:sto., PEAR 1111.11111111II-I ... 24 1 24 Cycleneater

Fig. 14. Variance of surge motion data from 24 consecutive runs of 0.5 hours

Non-linear system Duration of each cycle hour

MEASURED COMPUTED

Fig. 15. Variance of sutge.motion data from 24 consecutive runs of 0.5 hours the original mooring system stiffness and 10 simulations for the altered mooring system stiffness:. The environmental forces will be the same for both stiffnesses, thus

simulat-ing

the ease of repeated model teSts Using

the same Wave train. Comparison of all 10

cases with respect to the maxim= mooring

force

will

reveal the probability level that

an arbitrary realization can lead to

incor-rect conclusions. This process will be

repeated for the three mentioned simulation durations.

For the input the same wave conditions

will be used as given in the previous sec-tion dealing with the cOmparidon between

Fig. 16. Variance of surge motion data from 8 consecutive runs of 1.5 hours

so

-so

KAN

Linear system Duration of each cycle 11 hours

MEASURED COMPUTED """.41".". 8.8.5. XPAX. of'd KAN

NODAinear system Duration of each cycle 11 hour

MEASURED COMPUTED

mu,.

o-.0,N,,e,./chNr.""as,ssuNb

Fig. 17. Variance of Surge motion data from 8 consecutive tuns of 1.5 hours model tests and simulations. The stiffness

of

the

linear mooring system amount to 15.5

tf/m and 18.6 tf/m.

-The results of the ten simulations in terms of the maximum mooring force are given in Table 3.

The results in the tables show that for a test duration of 30 minutes the simulations show that in six out of ten cases the maxi-mum mooring force will be reduced when the mooring stiffness is increased. However, for durations of 1 hour and 1.5 hours, seven out of ten simulations indicate that the maximum

.._{.1111111fill1111111-11-1-1} 1.1111.11.11111111111(1.12.1 1 / I .- .---. I 1

24 1 24 8 1 8

Cycle nuefter _{Cycle lumber}

S-1171011

(M)

mooring force will indrease due to the

increase

in

stiffness. The results of this

exercise show that care has to be exercised in drawing conclusions from simulations or

model tests which have a relatively short

duration from the point Of view of the

phenomena being investigated.

The Most probable maximum mooring forces:

The design loads in a mooring system are

often based on the most probable maximum of

the force which will occur in a selected

sea-condition for some assumed duration of the particular condition. The most probable

maximum of a quantity is the value of the

force for which the distribution function of the extremes of the force is at its maximum.

Table 3

Influence of mooring stiffness on maximum mooring forces

in which:

N = number of oscillations of the quantity in the assumed duration of the storm condition

In general the distribution of the

extremes of a quantity such as a mooring

force will not be in accordance with the

Rayleigh formulation In such cases the

probability level can be deduced directly from the distribution of the extremes by

-13-determining the force value at the peak of

the distribution. One of the problems

associated with this approach is the amount

Of data, in terms of the duration of the

record on which the distribution function is

based. In most cases only a limited amount

of data is available which means that the

results will always be influenced by finite sample effects.

For the case in hand we will assume that

we may use equation (21) to determine the probability level at Which the distribution of the extremes must be intersected in order to obtain the most probable maximum Value.

For the 300 kDWT tanker moored by means of the non-linear mooring system, simulation computations were carried out to determine the statistical variance of the most

proba-3000 -E. 2000 '4-S max Fmin

Run duration in hour

0.5 1.0 1.5 Run No . Mooring stiffness Mooring stiffness Mooring stiffness in tf/m it tf/m in tf/M 15.5 18.6 15.5 18.6 15.5 18.6 1 768 990 827 990 828 990 2 827 878 717 704 802 828 3 594 564 802 828 993 954 4 717 704 993 954 961 1124 5 802 828 728 736 735 760 6 572 570 967 1124 728 887 7 806 778 735 759 613 806 8 993 954 728 888 764 718 9 626 736 649 727 820 864 10 728 686 613 806 1129 971

For a quantity of Which the distribution of the extremes conforms with the Rayleigh

distribution, the most probable maximum

value is found by intersecting the

distri-bution of the extremes at the probability

level found from the following equation:

p(F)

= x 100% (21)

Fig. 18. Influence of simulation duration on variability of most probable

maximum mooring force in 3 hours ble maximum mooring force for the same

Sea-condition as used before. It was assumed

that the most probable maximum was to be

determined for a storm duration

of

3 hours.

In order to determine the distribution of

the extremes from which the Most probable

value was to be determined simulations were

10 20

carried out for durations of 6 hours, 12

hours and for 18 hours. In order to deter-mine the effect of sample variance, 10 sim-ulations Were carried out for each duration. For each simulation the most probable maxi-mum mooring force Was calculated according

to the procedure outlined above. Finally,

the Mean and root-Mean-square of the most

probable maximum mooring force values were determined from the 10 simulations carried out for each duration value. The results of the computations are shown in Figure 18.

This figure shows that as the test

dura-tion

is

increased, so the root-mean-square

value of the most probable maximum Mooring force decreases thus Making it more probable that the results based on a single simula-tion (test) will yield data Which is Close to the 'true' value.

7. CONCLUSIONS

In this paper a staple time-domain sim-ulation method for estimating the low-fre-quency surge motions of moored vessels has

been discuiled. The method is based on a

random number simulation model for the

slowly-varying second .order wave drift

forces applying simplifications which are

applicable to the case soft-moored vessels

in relatively deep mater. Statistical

comparisons of the computed and measured

motions of

a

300 kDWT tanker in irregular

head seas Show that the simulation method

leads to results Which can be useful in the early design stage of a moored vessel.

ExaMples of applications of the simulation

calculations were given which show the

influence of finite sample effects on the

lowfrequence Motions and mooring forces of the 300 kDWT tanker.

REFERENCES

Hsu, F.H. and Blerikarn, R.A.:

_"Analysis

of peak mooring forces caused by slow

vessel drift oscillations in random

seas", Paper No. 1159, OTC HoustOn, 1970.

Ramery, G.F.M. and Hermans, A.J.: "The

slow drift oscillation of a moored object

in random seas", Paper No. 1500, OTC

Houston, 1970.

0Ortmerssen, G. Van: "The Motions of a

moored ship in wave", NSMB Publication No. 510, 1976.

Pinkster, J.A.: "Low frequency second

order wave exciting forces on floating

structures", NS! B PublicatiOn No. 6504:

1980.

Wichera, J.E.W.W.: "Progress in computer simulations of SPM Moored vessels", Paper No. 5175, OTC Houston, 1986.

Pinkster, "Drift forces in

direc-tional. seas", Marintec China '85, Shanghai, 1985.

Wichars, J.E.W.: "On the low frequency

MotiOnt of a Vessel moored in high seas", Paper No. 4437, 1982.

Bo6m, H.J.J. van den: "Dynamic behaviour

of mooring lines", paper NO. B4, BOSS 1985, Delft, 1985.