Some aspects of model tests in designing mooring systems in the open sea

Presented at the second WEGENT Graduate School

14 March, 1979, Wageningen. TECHNISCHE UNIVERSITEIT Laboratorium voor Scheepshydromechanlca Archief Mekelweg 2, 2628 CD Deift Tel.: 015-786873 Fax: 015- 781838

SOME ASPECTS OF MODEL TESTS IN DESIGNING MOORING SYSTEMS IN THE OPEN SEA

SOME ASPECTS OF MODEL TESTS IN DESIGNING MOORING SYSTEMS IN OPEN SEA

J.E.W. Wichers Netherlands Ship Model Basin The Netherlands

INTRODUCTION

Some aspects of model testing with mooring systems of floating

structures in the open sea will be dealt with. Up to now the role of model tests in determining certain design values are still

indis--pensable. For a general and broad outline of model tests in the

field of mooring, see ref. (1) . Ref. (1) includes the purpose of the

model tests, the information necessary to set _{up test programs, the}

scope of tests, the characteristics simulated, measurements carried

out, possible sources of errors and the analysis of results.

In this paper more specified aspects of model tests concerning en-vironmental conditions and mooring systems will be treated.

When dealing with mooring systems, the determination of the

environ-mental input for the model tests must be considered carefully.

Several investigations have established wave spectrum formulations. The wave spectrum formulations are based on either:

- a known significant wave height and a wave period

or:

- the known wind velocity and fetch.

For the different wave spectrum formulations the spectral shapes have been computed and presented. Besides the fact that _{the chosen}

wave spectrum will have consequence for the high frequency behaviour

of the moored structure it will also effect the low frequency mo-tians of the structure. From the wave spectrum the _{spectrum of the} square of the wave envelope can be derived, see ref. (9) . The square

of the wave envelope can be related to the low _{frequency wave drift} forces. The wave drift forces can induce large amplitude low

fre-quency horizontal motions of the moored structure. For different

wave spectrum formulations the spectrum of the square of the _wave envelope has been computed and presented.

After this introduction regarding the environmental _{input some} as-pects of the experimentally determined low and high _frequency mo-tions of a semi-submersible crane vessel anchored in a linear moor-ing system will be discussed.

out in order to determine the forces in and the motions of the moor-ing structure. A method to predict short term extreme values is

shown.

Some remarks will be given on deep water riser systems.

Model test experience of bow hawser type mooring systems and the theoretical approach for computer simulation will be discussed briefly.

1. ENVIRONMENTAL CONDITIONS - WAVE SPECTRA

a. Wave spectrum formulation based on a wave period and the signifi-cant wave height

Darbyshire (1957)

From the analysis of wave records taken in the North Atlantic Ocean including the Irish Sea, Darbyshire derived for the unlimited fetch the following (open sea) spectrum, see ref. (2)

0.271 w1/ e S (w) = 0.2377 w p in which x =

0.05320.2.r(-1)±0.264}

w p

w1/3 = 4.,/ = significant wave height in m

m0 ₌

_f

S (w) dw = area under spectrum in m2 o

271

w = = frequency, at which the maximum wave energy

occurs.

By means of numerical integration the following relationship _{can be} derived:

2it 4.8

-w

=7:;-=

P

IP pl

in which = 2 is called the first period of the spectrum and

1

is physically related to the average wave period.

271 4.3

- wp

= ₌

in which

'2 271 is called the second period of the spectrum

and is physically related to the average zero

{0. 2.(-1)

}2

w p

uperossing wave period.

The r- and s-exponent wave spectra

The r- and s-exponent wave spectra are of the following type:

B s A w S

(w) =-. e

r 2

in which w =

-These spectrum formulations are valid for fully developed seas. The exponents in this spectrum formulation, the author, the year of publishing and the references are given in the following table:

short fetch

The full description for the spectral formulation of the Neumann,

the Fisher and Roll _{and the Pierson-Moskowitz spectrum based on the} average wave period or the average zero uperossing period

2 and the significant wave height

w1/3 are given on the next page.

Author r s year ref.

Neumann

Fisher and Roll Burling Ni taigorodskii Pierson Moskowitz Phillips 6 5 5.5 5 5 2 2 2 4 3 1954 1956 1955 1964 1966 (3) (4)

in which w1/3 =

=2r

m1 m

=2

-2 co m_{n =}

_f

w'.S (w) dw (n = O, i or 2)

Of these spectra the Pierson-Moskowitz spectrum is the most frequent-ly applied one for fulfrequent-ly developed seas. Pierson and Moskowitz have derived the wave spectral formulation for fully developed seas from analysis of wave spectra measured on the North Atlantic Ocean.

By means of numerical integration the following relationships be-tween various wave periods valid for the Pierson-Moskowitz spectrum can be derived, see table on the next page.

4 Author

S(w)/(Ç1/3

)2

= - s 69. 76 59. 22

(1.w)2

(2.w)2

Nemann

3623 .e 2538 _6.e

(.w)6

50.26 _39.47

(1.w)2

22

Fisherand Roll 315.8 -.e 194.85.e

(1.w) 691 496 Kitaigorodskii 172.8 124 5.e .e Pierson

(1.w)5

(2.w) Moskowitz

According to ref. (5) for fully developed seas, the following rela-tionship between various wave heights will exist:

in which is the mean of the wave height of the nth _{highest part}

of the wave height distribution.

The Jonswap spectrum (1973)

An extensive wave measurement program known as the Joint North Sea Wave Project (Jonswap) was carried out in 1968 and 1969 along a line extending over loo miles (160 km) into the North Sea from Silt

Island. From the analysis of the measured spectra, a Jonswap wave spectral formulation was derived which is representative of wind-generated seas with a fetch limitation, see ref. (6)

In this section in which the spectral formulation is based on a pe-riod and the significant wave height the Jonswap spectrum has the

following form: -1) t1

-

1

25 ()

_{- 2} - w e 2e 2 -5 ctg w e Example: T 0.71 T 2 _p T 1 T p 2

Average wave period = 1 0.77 1.086

Period of the spectral component associated with maximum wave

energy = 1.296 1 1.408

Average zero uperossing

period T2 = 0.92 0.71 1 Example: 1.27 H1/3 H1/3 H1/10 Average wave height 1 0.63 0.49 Significant wave height H1/3 1.59 1 0.78 2.03 1.27 1

witha =c forw < w

a p

o for w > w

b p

in which a, y,

0a and = spectral shape parameters

g gravity constant 9.81 rn/sec.2 2îr

w

p T0 = frequency associated with the spectral component

with maximum wave energy

2r

w = --- = wave frequency

A definition sketch of the parameters for the description of the Jonswap spectrum is given in Figure 1.

/

Db

(Jon swap) Ç max.

in rad.sec.

Fig. 1. Definition sketch of Jonswap

For the mean Jonswap spectrum the authors of ref. (6) found the following spectral shape coefficients:

y = 3.3

o = 0.07

a

0.09

In the case that only the significant wave height

w1/3 and the average period are known, the following procedure can be applied:

- for the standard deviations

0a and the mean values respectively

0 .07 and 0.09 will be taken to be constant;

- by means of numerical integration the following relationship

be-tween Tf

. for the Jonswap spectra (ca = 0.07 and °b = 0.09)

as a function of the y-value can be established; see tab1e next page.

6 S; max. (Jonswap) Gamma - S (P.M.) Ç max. (Pie rson-Moskowitz) Ç max.

m

in which T = 2ir

-1

=

2rr\/--Using the results of the table at a chosen Y-value and the given average wave period the period of the spectral component associ-ated with the maximum energy T can be calculated.

- the significant wave height

w1/3 is known: (

_j2

w1/3)2

= 16f S(w)dw

1.25

()-4

2

= 16 .

nf

g w e . . dw

::rom this computation the value of the spectral shape coefficient a can be determined.

By means of this procedure the wave spectrum can be described for a given

w1/3 and

For reason of comparison the spectral shapes of the above-mentioned

wave spectra have been computed for the same significant wave height

'w1/3 = 5.5 m,and average period 11.5 sec.

The result is shown in the figure on the next page.

I

T/T1 T/2 T1/T2

1.0 (P.M.) 1.296 1.408 1.086 2.0 1.240 1.338 1.079 3.0 1.206 1.295 1.073 3.3 (mean) 1.198 1.285 1.072 4.0 1.183 1.264 1.069 5.0 1.165 1.240 1.065 6.0 1.151 1.221 1.061

20 12 8 4 o Irregular sea: = 5.50 ni T = 11.50 sec. (w)

In ref. (7) a study has been carried out using the P.M. spectrum and the Jonswap spectrum consisting of the same P.M. spectrum multiplied by the "peak enhancement" factor. The P.M. spectrum formulation is:

691

SpM

=

w1/32

T1 172.8 e (i.w) = 0.77 T in which T1 w1/3 = 8 Neumann and Roll (Gamma = (Gamma = Pierson-rioskowitz 3.3) 5.3) - --Darbys hire Fisher Jonswsp Jonswap ¡I\

\\

I ¡: 7. ¡

//I

\

/

!/

,/

_1/'

o 0.4 0.8 12 j in rad.sec

Fig. 2. Spectral shapes for constant and

Note on the P.M. spectrum and the Jonswap spectrum containing

The Jonswap spectrum formulation based on the same P.M. spectrum will be: ,w - 1 2 2 e 2o S (w) = S (w) . y Jonswap P.M. For o = 0.07 for w < w p = 0.09 for w > w p

and the same w for both spectra in ref. (7) the following relation-ship between the spectra was found:

From this table it can be found that for the mean Jonswap spectrum

(y = 3.3) the ratio between the area under Jonswap spectrum and the P.M. spectrum amounts to 1.52 and the ratio between the average period of the P.M. spectrum and the Jonswap spectrum will be

0.92.

To compare the spectrum of Pierson-Moskowitz and Jonswap for the same peak frequency w and significant wave height

w1/3 in ref. (7), the following formulations were proposed;

691 4 172.8 (P1.w)

SpM (w)

₌

_{w1/3)r ()5}

e (1.296.1.w-1)2 2iî e

Sj

(w) = S (w). . y onswap P M r 27r I in which 0.07 for w 1.296 1 2 2o y m0 JONSWAP F -(m1/m0) JONSWAP m0 P.M. (m1/m0) P.M. 1 1.0 _1.0 2 1.24 0.95 3 1.46 0.93 3.3 1.52 0.92 4 1.66 0.91 5 1.86 0.90 6 2.04 0.89

and the factor F can be read off from the table.

In Figure 3 the spectral shapes for the different wave spectrum for-mulations have been computed for a constant significant wave height

w1/3 and the constant peak period T.

Irregular sea: w1/3 = 5.50 m ; Tp = 14.90 sec. lb 12 8 4 'o Ne umano and Roll (Gamma = 3.3) (Gamma = 5.0) Pierson-Moskowitz ._Darbyahire Jonswap Jonswap ___Fisher --/)! \t :i '\

(4

,

Ii

1i

j)

o 0 4 0.8 1.2 W in rad.sec

b. Wave spectrum formulation based on wind velocity and fetch.

Normally in the field of Ocean Engineering the wind strength is in-dicated by the Beaufort number. Beaufort indicates the wind velocity

10 metres above the sea surface. In the following table the Beaufort number and the corresponding wind velocity in knots and rn/sec. are

shown:

The above table is valid for 10 m above the sea surface. In case the

wind velocity is wanted on another altitude above the _{sea surface,}

the following formule according ref. (8) can be used:

U(z) = U(lO)

(z)l/7

in which U(lO) = wind velocity 10 rn above the sea surface (= Beaufort velocity)

U(z) _{= wind velocity z m above the sea surface}

Darbyshire (ref. (2))

From the analysis of wave records taken in the North Atlantic Ocean including the Irish Sea,Darbyshire derived for the unlimited fetch for fully developed open sea the following spectra formulation:

-

2

-(m-w p 0.053(w-w +0.264)

S(w) = 0.238

1/3 e p in which: = in ft. T = = 1.94 U0.5 + 2.5 ± l0 U4 in sec. p

Beaufort knots rn/sec.

1 2 _1.03 2 5 2.57 3 9 4.63 4 _{13 6.68} 5 18 9.25 6 24 12.30 7 30 15.42 8 37 19.02 9 44 22.62 10 52 26.73 11 60 30.84

U is the wind velocity in knots and is assumed to be at 10 m

above sea surface.

Pierson-Moskowitz (ref. (4))

Pierson and Noskowitz analyzed wave signals, which were recorded on floating weather stations in the North Atlantic Ocean. Wave spectra were determined for fully developed seas and can be described by the

following formulation: -4 (WU 2 -5 g

S(w)=ag w

e in which g = 9.81 m.sec.2 a 8.1 . 10 0.74

U is the wind velocity in m.sec. at 19.5 m above the sea surface,

while 2 = 4

/E

0.21 .- in m wl/3 O g

5-

U U T = = 2îr (---) - 7.16 - in sec. p 4 g g Neumann (ref. (3))

The empirically determined spectra for fully developed _{sea in deep} water based on wind velocity according Neumann _{can be written as:}

-2 - 2 (-) 2 -6 g

S(w) = c g

w e in which c

0.025 sec.1

g = 9.81

m.sec.2

U is the wind velocity in m.sec. _{at 7.5 m above the} sea surface,

whi le

Ç1/3

4 = 7.0 . 10 U5 in m

T = = 2Tv\/ij . 7.7 in sec.

Jonswap (ref. (6))

The Jonswap spectrum is a function of two parameters, _{fetch length X} and wind speed U. Because of the inclusion of the fetch _the wind-generated seas must be considered as not fully developed _seas.

The from the North Sea measurements derived wave spectrum formula-tion based on the parameters X and U reads:

w 2 -1) 2 -5 S (w) = .g .w .e p .y in which c = c for w < w a p o for w > w b p = 0.076 (-i- .X)022 = 2î .

3.5 () (X)033

g = gravity constant = 9.81

m.sec.2

U = wind velocity in m.sec. at 10 metres above the sea surface

X

=fetchinm

The from the Jonswap measurements derived relations in spectral shape parameters, non-dimensional fetch and non-dimensional peak frequency have been summarized in Figure 4.

In order to compare the spectral shapes of the mentioned wave spectra for the same Beaufort, calculations have been carried out. For all the wind-generated seas spectra Beaufort 7 has been assumed, while in addition for the Jonswap spectrum a fetch of 100 km is

taken. The calculated input data for the spectrum formulations are

given in the table below.

Beaufort = 7: 0)30) = 30 knots = 15.42 rn.sec.1

Darbyshire Ptersorp-Moskowitz Neumann JONSWAP 0(10) = 30 knots IS.2 0(19.5) = 0(10) = 16.96 = (19.5)1/7 m.sec. U(7.5) = 0(10). 0(10) 15.42 m.sec.

() =

= 14.80 m.sec.1 -Y = 3.3 o = 0.07 = 0.09 X = 10 m T w1/3 = = = 10.83 0.58 10.32 sec. rad.sec. ft. =3.14 m 12.38 0.508 6.16 sec. rad.sec. m 11.62 sec. 0.54 rad.sec.3 5.19 m 7.012 sec. 0.896 rad.sec.1 o 0.0122

T,.

Ca.fIary-n.e. data ma sod'rJed where po'bIn)

'n 'O," \ \4\ 'n' n» * o :

Icki aLO) _{14'n7r.e' 40,1cr et 1(1953)}

' '''7 OaII,.rIolrC t'l$7) "''2c"., '.;..';.:' o lsayao (lObe) 0n 'A.' Lro 11770)

o fat,alO7t) p'.4 P..,nOrt fan04n,,,l0 (1064) 'i. r,*l. f .11

''171) V.,nont, Stony 114711 4 ibiS Eap,rlrr,.nf tO' 10 IO' IO' IO' tO' PrnIl.ps Conflict

fetch scafed vrordrng Io Y.rlagoro.iskr'. SmjJI-feIi-h d,ìfa aro obta'nnd horn wind-wave tank,.

(.pili.ry - *,,','c dale waS Ci,,,I,'d *Iin,o poosblo.) f.iaSoroeotnls by Srihc'vlioJ (1907) aod Tobe (1071) wo, .4,1 from W J Pro, non arid ISA SLc1 (f973)

c'a o

Fig. 4. Parameters derived from the

Jonswap measurements (ref. 6)

4. .4 4 + # 4 * * +

t..

e. + + . * 4+ 4+ 44. 4 0% * " * : ,4 4 4 >,

Pock nitape parameter, y, non-dimensional fetch a - gal U!0

4)

-o

n'I o o

o

r,

k

k

-o

te

-o

o IO -p.,'' L. 'f h 'I' ''r_n ° f a 'f'' f' I. t, btIlIrfOt ff962) a' L t) a.fnenti fof.4j a, L Vra'. nf.ff _rEO) 4. T. E 'cr''' r") P", lacho. 4+ ₊ + IO O IO' O' IO' IO' Peak f.oqanny

Ir Inh .ca'ed e'aco'd,'9 r, Krlai1o'odskil. Dala al snaefi Ie:ches ero cblained loro ,vrrtd-wano

o

-o c'i o o o o o b .2 a 170 07 DI .000 IO, lo, A I0' to' to' IO' 0° "n "n 'n' -n' "'n b b r i o (0 r,') od

O

6

4

Irregular sea: Beaufort 7

Neumann : = 5.90 T = 11.62 sac. : = 6.16 o _{12.38 sec.} : = 3.14 m = 10.83 sec. (Gamma = 3.3): = 2.92 o 7.01 sec. ____Pierson-?loskowjtz Darbyshire .Jonswap

I'

I \ I I I I

\\\\

/

/\\

j

_/

II

\

\;\lOQkm

/\\ Fetch -

___i

J//

-0 0.4 0.8 1.2 16 w in rad.sec. Fis. 5. Spectral shapes at Beaufort 7

c. Wave group spectra related to the wave spectra

In the foregoing the different wave spectrum formulations are re-viewed.

The wave spectral components of the spectrum will cause, according to the linear theory, the wave exciting forces and moments on the moored vessel and mooring system inducing the high frequency motions of the vessel and the high frequency motions of and forces in the moor-ing system.

The spectral shape involved, however, will not only effect the high frequency behaviour of the system. In ref. (9) it is shown that

from the normal wave spectrum S.(w) the spectrum of the square of the wave envelope can be derived theoretically. This feature is important because of the analogy which exists between the low frequency wave

heights of the incident waves, see ref. (10) , (11) and (9) . A short

review on the wave group spectra of ref. (9) is given followed by the computations of the wave group spectra derived from the in section a. mentioned wave spectra.

We assume that the waves behave as a random Gaussian process. The wave height in a fixed point, in irregular waves, may be written as follows:

N

(t) = sin (w.t

+ E.)

i=1

where:

E. = random phase angle.

In amplitude modulated form this becomes: (t) = A(e) . sin (w0t + s(t))

where:

The behaviour of A(t) contains information with respect to the grouping of waves.

Squaring the wave height (t) and taking only the low frequency part

gives:

= ½ IZ cos {(w. -

w.)t +

(E1

-= ½ A2(t)

This shows that the low frequency part of the square of the wave height also contains information on wave grouping. It follows that:

A2(t) = 2 12(t)

Based on the assumption that the wave elevations are normally

distriu-ted, it can be shown that the spectral density of A (t), Sq(P) is

related to the normal wave spectrum S(w) of the waves as follows:

S (ii) = 8 ₅ S (w) . S (w + p) . dw

g

The distribution function of A2(t) is, for a narrow band spectrum:

16

A(t) =

envelope = I

1J

_{cos {(w. - EJt +} (E.

-i

J

-i

-i

E(t) =

slowly varying phase angle

p (A2) - . e

2m0

where:

m0 =

JS(w)

. dw

O

From the above, it follows that knowledge of S(w) and the assumption

that the wave height is normally distributed is sufficient to calcu-late the spectral density and distribution function A2(t)

From the foregoing it follows that the spectral density and distri-bution may also be calculated from the low frequency part of the

square of the record of the wave height measured in the basin. If the waves are completely random then the spectral density and distribution

obtained from and based on the normal (first) spectrum of the waves should correspond with the spectral density and distribution of

2.12

calculated directly from the wave record.

The distribution function p(A2) and p(2.12) and the spectral den-sity of the waves and of A2 and 2.12 are shown in Figure 6 and 7,

respectively for an irregular wave train generated in a basin of the N.S.M.B.

As mentioned, the wave envelope is related to the low-pass filtered square of the wave elevation:

A(t) =\/2.12(t)

In order to show that this operation does indeed represent the wave envelope, results of an analyzed irregular wave train are plotted in Figure 8, showing the wave elevation (t) and the corresponding

wave envelope A(t)

4_ 2 S g ( )I) (m4sec) 2_ 0-S (W) (m?sec) o. i 0.01 ao01 o

THEORETICAL SPECTRUM DERIVED FROM _{S ( w}

SPECTRUM OF LOW FREQUENCY PART OF SQUARE OF MEASURED WAVE RECORD SgI.1)

WAVE REGISTRATION LASTED 210 min.

0.5 1.0

Ui ).L (rad/sec.)

Fig 6. SPECTRA OF WAVES AND THE LOW FREQUENCY

PART OF THE SQUARE OF A WAVE RECORD

SPECTRUM OF MEASURED WAVES S(W)

1.5

FROM WAVE RECORD THEORETICAL

(WAVE REGIS TRATION LASTED 210 m n.

2.5

A22 (m2)

J-o

5,0 7.5

Fig. 7. DISTRIBUTION FUNCTION 0F THE LOW FREQUENCY PART OF THE SQUARE OF A WAVE RECORD

2-(m)O

-2

2

(m)O

-2

2

(m)O

-2

f

'J +

t

t

t

Fig8.

IRREGULAR WAVES, WAVE ENVELOPE

WAVE

(t)

ENVELOPE A(t)

Ç (t)

A(t)

WAVE AND

ENVELOPE

Using the spectral density formulation of A2 (t) , which means the

spectrum of the square of the wave envelope or the wave group spectrum according to:

S (bi) = 8 J S (u) .

S (w+p)

. thAi

g

computations have been carried out on the different afore-mentioned wave spectrum formulations.

In Figure 9 the wave group spectra have been computed for 6 differ-ent wave spectrum formulations each having the same significant wave height

w1/3 = v = 5.5 m and the same average wave period = 11.50 sec.

Irregular sea: L1/3 5.50 o = 11.50 sec.

200 160 120 40 o 20 16 Neumann Pierson-Moskowitz Darbyshire

Fisher and Roll Jonewap (Canina = 3.3) .Jonswap (Gamina = 5.0) -12 ci 68 .J, -J .. \:

\.\

_'.4:

II

'

/ ji 1/ ji1' -o 0 4 0.8 1.2 w, p in rad.sect

In Figure lO the wave group spectra have been computed for 6 differ-cnt wave spectrum formulations each having the same significant wave height

w1/3 = 5.5 m and the same peak period T = 14.9 sec.

Irregular sea: w1/3 5.50 ru ; T = 14.90 sec. p u a ti 80 o 200 160 120 40 O -¿L) 16 12 Neumann and Roll (Gamma = (Gamina = Pierson-Moskowitz 3.3) 5.0) -Darbyshire Fisher Jonswap Jonswap --\ k

¡

1' --, _-4 o 04 08 1.2 a, u in rad.sec

Fig. 10. Spectra of waves and the square of the wave envelope

From the results it can be concluded that the spectral density of the square of the wave groups will be substantially influenced by the shape of the chosen wave spectrum.

If a floating structure is moored in a soft mooring system with

natural frequencies < OO5 rad.sec. it appears that the low

frequency motions will be more effected by the peaked Jonswap spectrum than by the flattened Fisher-Roll spectrum.

It must be mentioned that for mooring systems with natural fre-quencies of ± 0.12 rad.sec. the low frequency motions will not

significantly be influenced by the different wave spectra. The spectral components of the square of the wave envelopes for the frequencies in this region are approximately the same.

2. SEMI-SUBMERSIBLE VESSEL MOORED IN A LINEAR SPREAD MOORING

In Figure 11 the layout of a semi-submersible crane vessel1 anchored

in a spread mooring system consisting of steel wires with a linear load-elongation characteristic, is shown.

Fig. ii. Semi-submersible crane vessel in a linear spread

mooring system

In order to determine the workability (or the down-time) of the vessel one may for instance require predictions of the motions of

the tip of the crane boom for several sea conditions.

Knowing the response amplitude operators of the tip of the crane

boom the high frequency motions for all kinds of narrow band wave spectra can be determined by means of the well known spectral anal-ysis technique. A review of the procedure on the spectral technique to determine the high frequency heave motion of the vessel is shown

below:

S (w) = {H (w)}2.

S(w)

z z ç

where: S(w) = energy spectrum of the heave motion z S(w) = energy spectrum of irregular sea

From the heave spectrum S(w) it follows that:

m0

₌

_f

S()dw = area

of the spectrum

Assuming that the wave elevation is a random variable with a normal

distribution and a narrow band energy spectrum, one may also assume that the same holds true for the heave motions. Consequently the

normal distribution is valid for the elevations and the Rayleigh

distribution for the amplitudes.

Normal distribution for the elevation:

-2

(z-z) i

2m

f(z) = . e zO .rn zO

Rayleigh distribution for the amplitudes:

2 (z a z a 2 ra

z) =- e

zU a zu

in which: _{Za = amplitude of heave motion} = mean value of heave motion

The probability of exceedance of a certain value of the heave amplitude can be calculated as:

2 (z ) 2 a _q Za

_{2 m0}

_{2 m} ?r(za > q) ₌

_f

-- .

e dz = e zU ra a q zU

The assumption as to the narrowness of the spectrum enables us to determine some interesting properties of the heave motion:

o =1ra z Oz 2/rn Oz m zU =2ir iz ra zi

while further in irregular waves of which the _{wave heights conform} with the Rayleigh distribution the expected maximum heave amplitude z _{can be calculated according ref. (12)}

a max. Z = 2 {(lnN)½ _{+ 0.29 (lnN)}} a max. 24

in which N = number of oscillations

The motions of the tip of the crane boom, however, consist of high and low frequency motions. By applying the already mentioned high frequency theory only a part of the necessary information on the motions will be obtained.

In the Figures 12 and 13 the measured heave and surge spectra of the

crane tip are shown for the semi-submersible crane vessel moored in

the linear spring system under operational condition in head waves.

These figures clearly indicate that the energy of the low frequency

motions and the high frequency motions are of the same magnitude, which means that the low frequency motions are as important as the high frequency motions.

ol SURGE SPECTRUM Head waves condition T. T ,17f\OP,,ationa1 HEAVE SPECTRUM Head wave s Operational condition O 0 4 0.8 1.2

16

20

W in rad.sec1

Fig. 12. Surge spectrum of tip of crane boom

O o 4 0.8 1.2 1.6

20

W in rad.sec' c-i ai (fc c'4 ai o 3

The duration of the test has to be lasted long enough to achieve a sufficient number of low frequency oscillations for reliable

statistical and spectral treatment. The duration time depends on the natural periods of the systems.

From Figures 12 and 13 it can be seen that the total energy of the

spectrum consists of two separate spectra. Of these spectra the high frequency motions are induced by the wave exciting forces andmoments and the low frequency motions are caused by the wave drift forces and moments and are concentrated about the resonance frequencies of

the vessel.

According to Figures 12 and 13 the total energy of the motion u can

be written as:

m

=m

+m

uot uoh uol

in which: mO

= total energy of motion u

mUOh = high frequency part of the spectrum

= low frequency part of the spectrum

The high and low frequency spectra can be considered as narrow band spectra. Of these spectra the standard deviation can be deter-mined:

G =v'm

uh uoh

a

=/m

uf

ud

or m = sum of the variances = o 2 + o 2

uot uh uf

For the determination of the low frequency variance Gui2 in differ-ent wave conditions anumber of model tests have been carried out. These model tests were performed in wave conditions corresponding to Pierson-Moskowitz spectra having different significant wave heights

w1/3 and average wave periods .

The results have been presented in the Figures 14 and 15.

N N b o 4 2 2 w1/3 in m

Fig. 14. The mean and standard deviation of the low-frequency _surge motion of the crane boom tip

4 6 2 2 w1/3 in m T = 11.9 i sec. (P.M. ) .-= 12.2 i sec. (P.M.

-01

-J_1

-

0.-J-sec. i (P.M.) L = 12.2 sec. = A i. O sec.

-3 10 o 2 o 2 _{8 10} +

--From the Figures 14 and 15 it can be seen that the mean u and the standard deviation of the low frequency motions are proportion-al to the square of the significant wave height. Using these re-suits it is now possible to predict the variance of the low fre-quency part of the motion spectrum

012

as function of the signifi-cant wave height of a P.M. spectrum.

The prediction of the high-frequency variance 0uh2 can be made by

means of the spectral technique using the P.M. spectrum with the

same significant wave height:

0 2 =

J{H

()}2

_S(w)dw

uh u

o

Up to now using anarbitrar P.M. spectrum the variances _011h2 and

012

can be determined. However, in order to predict the total mo-tion, a combination of the variances or standard deviations have to be made. This problem is still in study.

3. NON-LINEAR MOORING SYSTEMS

In the foregoing section a linear mooring system has been dealt

with. In this section attention will be given to a non-linear

mooring system. A non-linear mooring system means that the

re-storing force versus the horizontal displacement of the vessel does not give a linear relation. In Figure 16 yoke mooring systems in relative shallow water are shown.

Tri-axial swivel universal joint Tri-axial swivel universal joint Buoyancy chamber Riser column

Base universal joint Piled base

MOORING YOKE

Riser column

Base universal joint Piled base

////\\\\\\\//////\\\\\\\/////

Fig. 16. Yoke mooring systems

Of these systems the restoring force versus the surge displacement (of the top of the riser) is almost linear at first, however, at larger displacements the system becomes non--linear. See the Ligure on the next page (Fig. ₁₇₎

MOORING YOKE

V

Buoyancy chamber

BARGE

Surge displacement of top mooring riser

Fig. 17. Static-load displacement curve

Because of the rather soft spring system of the mooring and the relatively small damping of the moored barge for low frequency horizontal motions, the wave drift force induced horizontal

mo-tionscanbe large, thus causing high forces in the mooring system.

An example of the surge motion record of yoke mooring systems in irregular waves is shown in Figure 18.

IRREGULAR WAVES

SURGE OF TOP MOORING RISER

29

w-NON-LINEAR STATIC-LOAD DISPLACEMENT CURVE

/

/

/

z

Fig. 18. Example of the surge motion registration of yoke mooring systems in irregular waves

For non-linear mooring systems model tests are indispensable.

The model tests are often carried out in storm wave spectra in

combination with current and wind. The purpose of the tests is to

optimize the system and to obtain design values for the strength

of the mooring system. In more moderate weather conditions model tests are often carried out to obtain data for fatigue analysis of the structure and to study the operational behaviour. In Figure 19 a model set-up of a mooring system is shown.

The designer, having selected a design sea condition, desires infor-mation about the probability that a certain (e.g. the measured) maximum motion or force will occur in that particular sea state.

Using the theory of extreme values the distribution function of

these values can be determined.

For records of quantities which have a linear response with respect

to the wave motions it may be assumed that the distribution

func-tions of the peak values are of the same type as that of the wave motion. For this type of initial distribution functions

Longuet-Higgins and Cartwright (ref.12 and 13) compute the distribution functions of the extreme values. It appears that they are of the

type:

-y

P(x )

=l_ee

max.

For E = O (narrow spectrum) the quantity y satisfied:

-2

2

(x

-x)

-x

max. O

y =

_2m0

in which: = mean value

2 m0 inN

= number of oscillations = root-mean square value

Of the considered mooring systems, however, the quantities have no

linear response to wave motion. Then the theoretical distribution functions of the extreme values are not valid. However, when many tests have been conducted, a large number of records of such a non-linear quantity is available.

Due to the random nature of the tests the measured maximum values do not need to be equal to those measured during a similar test

Fig. 19. Model set-up of a mooring system Yoke-structure Universal j oint Base

r

T h r e e - comp one n t strain gauge force trans-ducer Tri-axial swi ve i universal joint Two-directional strain gauge bending moment transducer in-stalled in the mooring riser Ele c t r on i c -magnetic motion transducer mea-suring roll Electronic-magnetic motion transducer mea-suring pitch Three-component strain gauge force trans-ducer

height and mean period.

The other values as mean value, significant value and standard de-viation, however, would be the same in both cases since the tests lasted long enough to be stationary ergodic.

Therefore the maximum values may be related to the mean and signif-icant values measured during the various tests.

From each of the record the value

u

-u

a max.

- u can be obtained.

u = mean value

u = maximum peak value

a max.

Ual/3 = significant value = mean value of 1/3 highest top

values

u

-u

the distribution of a max. - is a distribution of extreme values,

u

-u

al/3

which can be expected during a time period corresponding with the

duration of the test.

By means of the experimentally obtained data the distribution func-tian of the extreme values can be established. The result is given in Figure 20.

100 -J n o 50 n -H rd C) 10 Q) 4 o >1 2 -J -r-1 -H o o 0.1

R__

_____

i.

-R'

-

_R

Forces u a max. -32

-u

u

Fig. 20. Distribution function of extreme values determined by means of experimental data

As an example for 1/250 probability of occurrence, which corresponds to a Q.49 probability of exceedance in 30 minutes the following values were read from the horizontal axis:

motions f = 3.11 forces f = 4.63

A maximum value of a motion with a probability of 0.4% can now be

calculated:

u (1/250 occurrence) = 3.11 (Ü1/3 u) +

a max.

In order to extrapolate to longer periods of time than 30 minutes the following method can be used:

From the distribution function of maximum values the probability

P1, that a certain maximum value of f will be exceeded during a period of 30 minutes, can be determined. Consequently the

probabili-ty that the maximum value will be less than f equals (1 -P1)

From this it follows that during a period of n times 30 minutes the

probability that the maximum value will be less than f amounts to (1

pjfl

Thus the probability P2 that the maximum value is larger than f

during a period of n times 30 minutes statisfies P2 = 1-(1

Example: What is probability P1 in case P2 is 0.4% for a one hour period?

60

n =

P2 = 0.004

Leads toP1 = 0.002 0.20%

The maximum value of the motion mentioned before would then become

3.20

'a1/3

+ U

It must be taken in mind that the reliability of the calculations is dependent on the number of experimental extreme values.

4. A REMARK ON DEEP WATER RISER SYSTEMS

By shifting more into deeper water, designs are made of deep water production riser systems. Designs are under study for water depth up to 5000 ft. see (ref. 14) . An example of a deep water production riser system is given in Figure 21. (See next page)

Tri-axial swivel Rigid arm universal joint Hinges

I

Universal joint Base

Fig. -21. A deep water production riser system

In this example the deep water production riser _{system consists of} an articulated riser, which is connected to the production/storage tanker by means of a rigid yoke construction. Besides _{the function} of the riser mooring the mooring system also leads _{the crude} trans-portation tubes and hoses from the base up to the tanker as the before mentioned shallow water mooring systems.

In this section only attention is given to high _{frequency forces in} the risr system. Due to the presence of the waves the buoy and riser column will be excited by the wave forces (and _{the high frequency} tanker motions) . For one-dimensional waves the wave exciting forces

can be calculated by the Morison equation. This force will be the right hand term of tne equation of motions of the pendulum. The whole equation can be based on the relative motion principle. From this calculation a one-dimensional bending moment and shear _force dis-tribution in the cross-section over the length of the riser system will be obtained from experiments, however, it was found that the bending moment and shear force distributions in a cross-section of the buoy or the riser are not one-dimensional but strongly two-dimensional as is shown in Figure 22. (See next page)

PRODUCTION AND STORAGE TANKER Buoy

Inertia section Universal joint

Buoyant riser

o o +1500 3 +1000 3 33 +500

/

3_ 7 -1000 -1500 Mr

= \/(MX)

2 ± (My) 2

in which: Mx and My are the th sample in t1e sector. In each sector the following quantities were determined: - the mean value:

N1 +M

t3

22 ₃

23

2 3 2

33

3 3 2 '11111_Ii 3 22 -500 3, _{2 22333}

333

2 3 3 2 3 3 3 3 -1500 -1000 -500 0 +500 +1000 +1500 in ton.m

Fig. 22. Distribution of bending moment components in the buoyant riser of a deep water riser system measured in a storm wave spectrum

The symbols in Figure 22 and the procedure of analysis of the (two-dimensional) signals of the bending moments are described below.

Both components of a signal Ux(t) and My(t) were plottedin a

plane with respect to the system of co-ordinates Mx and My. The centre cf gravity of all the digitized values of Mx(t) and My(t) corresponds to the intersection point of Mx and My . With

mean mean

the centre of gravity as the origin of a new system of co-ordinates Mx' and My' the plane will be divided in 72 sectors of 5 degrees. In each sector all the digitized points will be considered. The length of the radius of the nth point in a sector will be:

33

233

2 2 3 3 3 2 2 1111111, 2, 1 2 3 3 3 +MX 3 3 3 3 3

in which N1 = number of points in the sector. - the root-mean square value:

(Mr1

-1)2

n r

1 n=1

1

- the maximum lenqth of the radius: Mr

max.

The results of the calculations in each sector are presented as a

plot, in which on the centre line of each sector the following fig-ures are shown:

- the mean value Ir represented by the figure "1"

- the sum of the mean value and two times the r.m.s. value (Er1 +

2e) represented by the figure "2"

1

- the maximum length of the radius Mrmax represented by the figure

3H

In the distribution plot only the original system of co-ordinates Mx and My (= the absolute zero axis) is presented. The figures "1",

"2" and "3" in a sector should be in line. However, a small devia-tion can occur due to the size of the plotting grid. If a combina-tion of "1", "2" or/and "3" coincides, a cross mark (x) _{is plotted.} From the result in Figure 22 it can be concluded that the magnitude of the transverse bending moment My is of the same magnitude as

the in-line bending moment Mx. Moreover, both maximum components appear to occur at approximately the same time-point. The wave forces on a cylinder, however, are dependent on the KeLilegan-Car-penter Number.

For different KC-Numbers on and the physical background of the transverse forces and bending moments on cylinders an extensive treatment is given byref. (15) and (16).

5. BOW HAWSER TYPE MOORING SYSTEMS

Over the world bow hawser type mooring systems exist. In most of the cases the mooring terminal consists of:

cantenary anchor leg mooring (CALM) single anchor leg mooring (SALM) fixed mooring tower

tensioned leg mooring

exposed location single buoy mooring

f. articulated mooring tower: - space frame tower

- monotube tower

- more articulated tower

All these mooring systems are called single point moorings (SPM's) The concept of a SPM is to moor the tanker by its bow to one mooring point, so that it may swing freely around this point, thus mini-rnizing forces due to wind, waves and current. Means must be pro-vided to transfer oil through the mooring point to (or from) the

tanker. A common feature of SPM's for tankers is the co-axial ar-rangement of the mooring system and the oil transfer system.

To predict the design values and to study the behaviour of the SPM

systems in operational and storm condition the aid of model tests is indispensable of the bow hawser type mooring systems, in operational condition the tanker is moored by means of a hawser. The spring characteristics of hawsers are non-linear as is indicated in Fig-ure 23, which is derived from ref. (17) . Obviously, the non-linear elasticity has to be simulated in model tests.

loo 80 c.c . .4_J cc so 60 Q4J os ow wo 20 o 0 10 20 30

Percent elongation (from O load) based on broken-in length

Fig. 23. Broken-in nylon load elongation curves based on broken-in length

As a result of waves, wind and current the bow hawser force shows a dominant slowly varying force due to the low frequency horizontal ship motions with a high frequency force superimposed due to the

I) / /' i. Three-strand '--i' / I Eight-strand I, /1 I/i / / J. r.

7/

,/

'

Double-braid

Slow oscillating horizontal motions may ever occur due to steady wind and/or current. For some combinations of wind and current condi-tions, bow hawser length and loading condition, the total system may be unstable. A mathematical model to judge SPM stability is developed in ref. (18) . In Figure 24 an example is given which

shows the stability as a function of bow hawser length and the angle between wind and current for fixed combinations of wind and current

ve oca. 220 ÇDWT TANKER-BALLASTED 200 E z = w 150 o z o w 0 TOO-NC o -J COMPUTED

O i (NOT CURRENT-4SIINOTS WIND

A OBI<NOT CURRENT-5KÑOTS WIND

o-o

0 45

D IN DEGREES

Fig. 24. STABILITY CRITERION FOR A 220- KDWT TANKER N BALLASTED CONDITION AS FUNCTION OF

BOW- HAWSER

In case the tanker is unstable the tanker starts low frequency mo-Ntions behind the SPM. In Figure 25 only the results of a model test

on a tanker moored to a SPI-1 in wind and current are shown. The fishtailing and galoping of the tanker is shown by the _{surge and} sway at the fore perpendicular and the yaw angle. In this case an exceptionally long length of the bow hawser is used to demonstrate

clearly the effect of this parameter on the stability _{of the}

system.

38

UNSTABLE

ASTERN PROPULSION PN O TON ASTERN PROPULSION PN 15 TONS

BUOY - HAWSER ELASTICITY ASTI IN

STABLE

Sway at F.P. in in Yaw in rad. Bow hawser force in tons 80 -80L 0.2 -0.2 200 100 O Ballasted tanker

Buoy-hawser length unloaded: 105 in

Buoy-hawser elasticity : Non-linear

Wind velocity : 25.7 xn/sec.

k-Unidirectional Current velocity: 1.03 rn/sec.j

Fig. 25. Time history of the tanker behaviour moored to an SPM in wind and current

A mathematical model to describe the behaviour of a tanker moored to a SPM in wind and current only is derived in ref. _{(19) .} In this model the non-linear equations of motion are resolved in the time domain by means of simulation techniques. In considering the low

frequency motions of a tanker moored to a SPM in wind, current and irregular waves, the mean and oscillating wave drift forces have to be considered in addition to the afore-rnentioned effects. Moreover, a hydrodynamic reaction force called wave damping force has to be taken into account, see ref. (20) . This subject is still

in study.

L

REFERENCES

Pinkster, J.A.. and Remery, G.F.M.: "The role of model tests in the design of single point mooring terminals". Paper 2212, off-shore Technology Conference, Houston, 1975.

Remery, G.F.M.: "Model testing for the design of offshore

structures". Symposium on Offshore Hydrodynamics, August _1971, Wageningen.

Darbyshire, J.: "Ocean wave spectra". Proceedings of _a confer-ence, Prentice-Hall, 1963.

Neumann, G.: "On ocean wave spectra and a new method of fore-casting wind-generated sea". Beach Erosion Board, U.S. _Army Corps of Engineers, Tech. Mem. 43, 1953.

Pierson, W.J. and Moskowitz, L.: "A proposed spectral _{form for} fully developed wind seas based on the similarity theory of S.A. Kitagorodskii". Journal of Geoph. Res., 69, No. 2, Dec.

1964.

Pierson, W.J., Neumann, G. and James, R.W.: "Practical methods _for observing and forecasting ocean waves by _{means of wave spectra} and statistics". H.O. Pub. No. 603, Hydrographic Office, U.S. Navy, 1955.

Hasselmann, K. _{et al.: "Measurements of wind and wave growth} and swell decay during the Joint Nortii Sea Wave Project _(JONSWAP)".

Ergnzungsheft Nr. 12, Reihe A, Deutsche Hydrographischen

Zeit-schrift 1973.

Report of Committee I. One of the 6th International Ship Struc-tures Congress, Boston, 1976 on "Environmental Conditions".

Brettschneider, C.L.: "Wave and wind loads". Section 12 of handbook of ocean and underwater engineering, McGraw-Hill Book Company, New York (1969)

Pinkster, J.A.: "Wave drifting forces". _{WEGENT Second Graduate} School, Aachen March 1979.

Remery, G.F.M. and Hermans, A.J.: "The slow drift oscillation of a moored object in random seas". Soc. of Petroleum Eng. Journal, 1972.

Pinkster, J.A.: "Low frequency phenomena associated with vessels moored at sea". Paper SPE 4837. European Spring Meeting of SPE-AIME, Amsterdam 1974.

Longuet-Higgins, M.S.: "On the statistical distribution of the heights of sea waves". J. Mar. Res. XI, 3, 1952.

Cartwright, D.E. and Longuet-Higgins, M.S.: "The statistical distribution of the maxima of a random function". Proc. Royal Soc. of London, Ser. A 237, No. 1203 (1956).

Gunderson, R.H. and Lunde, P.A.: "Exxon's new deep water produc-tion riser". Ocean Industry, November 1978.

Chakrabati, S.K., Wolbert, A.L. and Tam, W.A.: "Wave forces on vertical circular cylinders". Journal of the Waterways, harbours and Coastal Engineering division, May 1976.

Sarpkaya,T.: "Forces on cylinders and spheres in a sinusoidally oscillating fluid". Journal of Applied Mechanics, March 1975.

Woehieke, S.P.,Flory, J.F. and Sherrard, J.R.: "Hawser system

design for single point moorings". Paper 3156, Offshore Tech-nology Conference, Houston, 1978.

Wichers, J.E.W.: "On the slow motions of tankers moored to single point mooring systems". Journal of Petroleum Technology, June

1978.

Wichers, J.E.W.: "Slowly oscillating mooring forces in single point mooring systems", to be presented at the BOSS'79 - London. Wichers, J.E.W. and van Sluijs, M.F.: "On the influence of waves on the low frequency hydrodynamic coefficients of moored ves-sels", Paper 3625, OffshoreTechnology Conference, Houston, 1979.