Response-based method for determining the extreme behaviour of floating offshore platforms

Response-based method for determining

the extreme behaviour of floating

offshore platforms

Said Mazaheri*, Martine J. Downie

School of Marine Science and Technology, University of Newcastle upon Tyne, NE1 7RU, UK

Received 23 March 2004; accepted 4 August 2004

Available online 28 October 2004

Abstract

The accurate prediction of extreme excursion and mooring force of floating offshore

structures due to multi-variete environmental conditions which requires the joint probability

analysis of environmental conditions for the worst case situation is still impractical as the

processing of large amount of met-ocean data is required. On the other hand, the simplified

multiple design criteria (e.g. the N-year wave with associated winds and currents) recommended

by API known as traditional method does lead neither to the N-year platform response nor to

the N-year mooring force. Therefore, in order to reduce the level of conservatism as well as

uncertainties involved in the traditional method the response-based method can be used as a

reliable alternative approach. In this paper this method is described. In order to perform the

calculations faster using large databases of sea states, Artificial Neural Networks (ANN) is

designed and employed. In the paper the response-based method is applied to a 200,000 tdw

FPSO and the results are discussed.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Floating offshore platforms; Artificial neural networks; Mooring system; Response-based method;

Hydrodynamics

Ocean Engineering 32 (2005) 363–393

www.elsevier.com/locate/oceaneng

0029-8018/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.oceaneng.2004.08.004

* Corresponding author. Tel.: C44 191 222 5863; fax: C44 191 222 5491.

E-mail address: [email protected] (S. Mazaheri).

http://www.elsevier.com/locate/oceaneng

Nomenclature

Ajk added-mass matrix

AP projected area normal to the force (m2)

Awp vessel’s water plane area (m2)

B beam of the hull (m)

BA beam of the aftermost section (m)

BS beam of the right end section (bow) (m)

Bjk the matrix of damping coefficients

Ccu current coefficient

CD drag coefficient

CT finite draft correction factor

Cii wave drift damping

Cjk the matrix of the hydrostatic restoring coefficients

D draft of vessel (m)

FA1 acceleration force in surge mode due to wave (kN)

FCurrent current force (kN)

FWave (freq) first-order oscillatory wave force (kN)

FWave (steady) steady wave force (kN)

FWave (slowdrift) slow drift wave force (kN)

FWave (steady) steady wind force (kN)

FWave (slowdrift) slow drift wind force (kN)

Fp1 dynamic pressure force in surge mode due to wave (kN)

Fi force in the ith mode (kN)

Fj the amplitude of the 1st order wave induced loads (kN)

FN Froude number�GM distance between the vessel’s metacentre and the vessel’s gravity points (m)

GT distance between the turret mooring point and the vessel’s centre of gravity

(m)

Hs significant wave height (m)

I(z) turbulence intensity

Iwp water plane moment of inertia (m4)

Kq the rotational mooring stiffness (kN m)

L length between perpendiculars (m)

L1 the integration domain over the no N-shadow part of the water plane curve

Lm the distance between the turret mooring point and the vessel’s centre of

gravity (m)�Mc current induced yaw moment (kN m)

Mi moment in the ith mode

Mjk mass matrix

P wave’s dynamic pressure (kN/m2)

Q dynamic magnification factor

S wetted surface of the hull (m)

S. Mazaheri, M.J. Downie / Ocean Engineering 32 (2005) 363–393364

S(f) wind spectrum

S(u) wave spectrum

SF(m) the spectral density of the slow drift excitation forces

Sjk mooring stiffness matrix (kN/m)

T mooring force (kN)

T1 mean wave period (s)

TH Horizontal mooring force (kN)

UC current velocity (m/s)

V undisturbed mean hourly wind speed (m/s)

VWind mean hourly wind speed (m/s)�Vwi mean hourly wind speed (m/s)

V 0(t) fluctuating wind gust velocity (m/s)

ZG centre of the gravity (m)

a wave’s particle acceleration (m2/s)

aij sectional added-mass component

bij sectional damping component

d damping factor

g gravity acceleration (m2/s)

h water depth (m)

k wave number

kx the mooring stiffness in x mode (kN/m)

ky the mooring stiffness in y mode (kN/m)

s cable length (m)

xy coordinate system based on the FPSO (Platform)

x 0y 0 coordinate system coincides with wave direction and wave’s crest

w submerged weight of mooring cable per unit length (N/m)

a wind direction in respect to x (deg.)

b current angle in respect to x (deg.)

g wave angle in respect to x (deg.)

x wave surface elevation (m)

xa wave amplitude (m)

hk vessel’s displacement (m)

_hk vessel’s velocity (m/s)

€hk vessel’s acceleration (m2/sec)

q 0i vessel’s rotational angle at time i

k surface drag coefficient

l wave length (m)

r water density (t/m3)

rair air density (t/m3)

f velocity potential

f cable angle in respect to x-axis (deg.)

fw cable angle at water line surface (deg.)

f0 cable angle at sea floor (deg.)

S. Mazaheri, M.J. Downie / Ocean Engineering 32 (2005) 363–393 365

u angular wave frequency (rad/s)

ue frequency of encounter (rad/s)

un natural frequency (rad/s)


1. Introduction

For the design and evaluation of mooring systems of floating offshore structures two

different environmental conditions are usually considered. These conditions are named as

the ‘maximum design condition’ and the ‘maximum operating condition’. According to

API (American Petroleum Institute., 1995) mooring systems should be designed for the

combination of the wind, wave and current conditions causing the extreme load, in the

design environment. In the practice this is often approximated by the use of multiple

design criteria, e.g. the N-year wave with associated winds and currents or the N-year wind

with associated waves and winds. It is obvious that the most sever directional combination

of wind wave and current forces should be specified according to the site’s environmental

condition. Moreover, API indicates that if the floating offshore structure is designed to

continue production during a severe storm, the maximum operating condition should be

the same as the maximum design condition. In practice, severe met-ocean parameters, i.e.

the N-year values of wind speed, wave height and current velocity are often assumed to

occur simultaneously (Standing et al., 1997). This leads neither to the N-year vessel’s

response nor to the N-year mooring force as the N-year values of met-ocean parameters are

not inline with the N-year values of vessel’s responses for a number of reasons.

In the first place the probability of occurrence of the N-year values of met-ocean

parameters with coincident propagation directions is extremely low. This means that the

N-year environmental condition is not the combination of the N-year values of individual

met-ocean parameters. To overcome this problem joint probability analysis of the

environmental conditions is necessarily required. The approach of determining the worst

case possibilities when more than one parameter is involved, for example for wave height

and period, wind speed and wave height and wave height and current, is commonly

adopted. However, when more than two environmental parameters affect the response,

then a multi-dimensional joint probability diagram should be used, which is very difficult

because a large amount of data is needed (Barltrop et al., 1998).

Secondly, the distribution of met-ocean parameters does not coincide with the

distribution of the vessel’s responses and mooring forces. This means that even

determining the N-year severe environmental condition accurately does not lead to the

N-year values of the vessel’s responses and mooring forces. For example Standing et al.

(1997) showed that using 100-year values of wind speed, wave height and current velocity

at the same time to calculate the vessel’s responses will end up with results with return

periods much longer than 100 years. In addition, the results presented by Incecik et al.

(1998) showed that the most severe mooring loads may not occur when wind, wave and

current are collinear and are at their maximum values.

In this paper a different approach to the response-based method is developed. It is based

on the response-based hydrodynamic model (SAMRES) developed by the first author


and makes use of the following theories. Artificial Neural Networks (ANN) is then

employed to simplify the response calculation of the vessel as the met-ocean parameters

vary. This simplification allows the response-based method to perform quickly whilst

processing long-term met-ocean data.

2. Response-based approach

A reliable method to predict the maximum motion responses of a turret-moored FPSO

subjected to arbitrary wind, current and wave loads for an N-year life period, which is

essential for the design of the mooring system, is still under development. Over the last

few years the response-based approach has been developed to reduce the level of

conservatism as well as the uncertainties involved in traditional methods for determining

vessel’s extreme responses due to arbitrary environmental loads over her service life. The

response-based approach was first applied to assess the extreme loading design

combinations on a fixed jacket platform by Marshall and Rezvan (1995). Later, Standing

et al. (1997) showed that by using the response-based method the 100-year maximum

resultant excursion of an FPSO reduces to about 75–80% of the maximum excursion

predicted using a traditional collinear combination of 100-year wind, 100-year current and

100-year waves. It is believed that the response-based calculation procedure will yield

more accurate results than those obtained by traditional methods. In the response-based

method one determines vessel responses using environmental data associated with the

place where the platform is going to be installed. For this purpose a model to predict the

responses of the vessel is required. In order to perform the calculations based on large

databases of sea states fast enough to be practical, some simplifications are needed. For

example, the time taken to process 6 months of data used in the spectral analysis with the

simple response model presented by Khor and Barltrop (1999) was about 4 days. Standing

et al. (1997) employed response surface modelling to simplify the response of the vessel as

a function of wind speed, wave height, spectral peak wave period, current speed, wind and

current heading angles. Using another approach, Incecik et al. (1998) developed the

structure variable method to transform the multi-variate environmental record into a

univariate time series of mooring forces. In this paper a different approach to the response-

based method is developed. This approach is based on the first author’s response-based

model (SAMRES) by employing Artificial Neural Networks (ANN) as a simplification

tool to predict the platform responses due to met-ocean parameters. This approach is

briefly illustrated by the flowchart in Fig. 1. As can be seen on the flowchart,

environmental data, the vessel’s particulars, a mathematical response model and artificial

neural networks (ANN) are the main components of the proposed response-based

approach. The procedures involved in the response-based approach entail the following

tasks:

(1)
Building up a mathematical model in order to determine the loads and motions of a
turret-moored FPSO due to wind, current and waves.

(2)
Obtaining the vessel’s responses by running the mathematical model over a
reasonable period of environmental data (e.g. 5 years).

Fig. 1. The flowchart of response-based approach.


(3)
Setting up an artificial neural network (ANN) that has been trained and cross validated
using sufficient data obtained from the mathematical model to ensure an accurate

representation of vessel responses as a function of environmental variables.

(4)
Using the ANN model to generate long-term vessel responses (e.g. using 25-year met-
ocean data).

(5)
Analysing the resulting vessel responses statistically to predict its maximum excursion
over an N-year life period.


3. Mathematical response model

A typical form of an FPSO is shown in Fig. 2. A right handed coordinate system at the

conjunction of the water surface and the axis passing through the platform’s centre of

gravity is considered so that the centre of gravity can be defined as (0,0,KZG). The general

form of the response equation for this platform can be written as

ðM CAÞdh2

dt2CB

dh

dtC ðC CSÞh

Z FwaveðfreqÞ CFWaveðsteadyÞ CFWaveðslowdriftÞ CFWindðsteadyÞ CFWindðslowdriftÞ

CFCurrent (1)

where M, A, B, C and S are matrices of mass, added-mass, damping, hydrostatic restoring

and mooring stiffness coefficients, respectively, which can be written for an FPSO with

lateral symmetry as follows:

Mjk Z

M 0 0 0 MZG 0

0 M 0 KMZG 0 0

0 0 M 0 0 0

0 KMZG 0 I4 0 0

MZG 0 0 0 I5 0

0 0 0 0 0 I6

266666666664

377777777775

Ajk Z

A11 0 0 0 0 0

0 A22 0 A24 0 A26

0 0 A33 0 A35 0

0 A42 0 A44 0 A46

0 0 A53 0 A55 0

0 A62 0 A64 0 A66

266666666664

377777777775

Fig. 2. Definition of the coordinate system and motions of a turret-moored FPSO.


Bjk Z

B11 0 0 0 0 0

0 B22 0 B24 0 B26

0 0 B33 0 B35 0

0 B42 0 B44 0 B46

0 0 B53 0 B55 0

0 B62 0 B64 0 B66

266666666664

377777777775

Cjk Z

0 0 0 0 0 0

0 0 0 0 0 0

0 0 C33 0 C35 0

0 0 0 C44 0 0

0 0 C53 0 C55 0

0 0 0 0 0 0

266666666664

377777777775

Sjk Z

S11 0 0 0 0 0

0 S22 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 S66

266666666664

377777777775

The details of obtaining the components of above matrices can be found in Faltinsen

(1990) and Salvesen et al. (1970).

3.1. Forces due to first-order waves

The velocity potential for an oblique wave with propagation angle of g in respect to x

axis can be defined as:

f Z0:5 gHs

u

cosh kðz ChÞ

cosh ðkhÞsinðkx0 KutÞ x0 Z x cos g Cy sin g (2)

Other wave components such as surface elevation, particle velocity, particle

acceleration and dynamic pressure can be derived from Eq. (2).

The wave induced surge force was calculated by considering the influence of the

platform’s lateral curvature. Strip theory is employed to calculate the sway and heave

forces by assuming that the length to beam ratio of the platform (FPSO) is not lower than

2.5 (ITTC, 1987). This means that the forces and moments are calculated by integrating

the inertia forces, which are the combination of the dynamic pressure and acceleration

Table 1

Main particulars of a 200,000 tdw FPSO

Length between perpendiculars (m) 310

Breadth (m) 47.20 m

Draft (m) 18.90

Volume of displacement (m3) 235,000

Block coefficient 0.85

Mid-ship section coefficient 0.995

Prismatic coefficient 0.855

Distance of centre of gravity to mid-ship section (m) 6.61

Height of centre of gravity (m) 13.32

Meta-centric height (m) 5.78

Longitudinal radius of gyration (m) 77.5

Transverse radius of gyration (m) 17.00


forces obtained at each cross section of the platform. The effects of wave direction and

shallow water conditions are also taken into account in deriving force equations.

Based on the method described above, a program was written in MATLAB and

executed for a 200,000 tdw FPSO whose particulars are given in Table 1. The results

obtained from the mathematical model showed a good correlation with those obtained by

Oortmerssen (1976) who used the 3D source technique (Figs. 3–12). This means that the

strip theory is consistent with the 3D source technique, particularly when lateral loads are

the aim of the calculation.

Fig. 3. Transfer function of wave induced surge force, gZ45.

Fig. 4. Transfer function of wave induced sway force, gZ45.

Fig. 5. Transfer function of wave induced sway force, gZ90.


Fig. 6. Transfer function of wave induced heave force, gZ0.

Fig. 7. Transfer function of wave induced roll moment, gZ45.


Fig. 8. Transfer function of wave induced roll moment, gZ90.

Fig. 9. Transfer function of wave induced pitch moment, gZ0.


Fig. 10. Transfer function of wave induced pitch moment, gZ45.


3.2. Forces due to steady waves

These forces are second-order forces which can be derived from the quadratic term of

the velocity potential. Havelock (1940) presented an empirical formula to calculate the

wave drift force acting on a fixed cylinder with vertical walls. This has been extended later

on by Besho (1958) to cover arbitrary bodies. Then Kwon (1982) has added three

coefficients to the Besho’s formula to take into account the effects of object’s speed, finite

draft and scattering. After that, Faltinsen (1990) presented a general frequency

independent formula to calculate the mean drift forces due to arbitrary waterline shaped

Fig. 11. Transfer function of wave induced yaw moment, gZ45.

Fig. 12. Transfer function of wave induced yaw moment, gZ90.


bodies. In order to include the effects of wave direction, wave frequency, current and

arbitrary waterline shaped bodies on the calculation of the mean drift forces Mazaheri

(2002) has redeveloped Faltinsen’s formula by adding the finite draft coefficient and

current coefficient as follows:

Fi Z CT Ccu;i

rgz2a

2

ðL1

sin2ðq CgÞni dl (3)

The current coefficient, Ccu, can be derived from the ship added resistance formula

proposed by Faltinsen et al. (1980) as:

Ccu;i Z 1 C2uUC;i

g

The above formula is applicable to bluff bodies when Froude number is equal or less

than 0.2.

CT coefficient is given as:

CT Z 1 KexpðK2kDÞ

By applying Eq. (3), the surge drift force for a stationary barge shape vessel with a

beam of B subjected to head sea waves can be simplified as:

F1 Z CT

rgz2a

2

ðB

sin2ðq C0Þdy Z CT

rgz2a

2B (4)


Now by defining the surge drift force coefficient R1(u) as

R1ðuÞ ZF1

12

rgz2aB

and substituting F1 from Eq. (4), the surge drift force coefficient becomes:

R1ðuÞ Z CT

This means that the surge drift force coefficient for a stationary rectangular barge-shape

vessel subjected to head sea waves is equal to the finite draft coefficient. This surge drift

force coefficient is compared with those obtained by Kwon (1982), Fujii and Takahashi

(1975), and Helvacioglu’s experiments (1990) favourably in high wave frequencies

(Fig. 13).

For an FPSO with a general water plane area shown in Fig. 14 by assuming that the

most left and right curvature parts can be replaced by a half-circle with a diameter equal to

the beam of the middle rectangular section, the mean drift loads in regular waves in surge,

sway and yaw modes can be written as:

F1 Z CT 1 C2uUCcos b

g

� rgz2

a

2

ðL1

sin2ðq CgÞsinq dl ðsurgeÞ (5)

Fig. 13. Comparisons between different surge drift force coefficients.

Fig. 14. The water plane area of a 200,000 tdw FPSO.


F2 Z CT 1 C2uUCsin b

g

� rgz2

a

2

ðL1

sin2ðq CgÞcos q dl ðswayÞ (6)

F6 Z CT 1 C2uUCsin b

g

� rgz2

a

2

ðL1

sin2ðq CgÞðx cos q Ky sin qÞ dl ðyawÞ (7)

In order to obtain the mean drift loads in irregular waves, the following equation can be

employed

�Fsi Z 2

ðN

0sðuÞ

Fiðu;b;gÞ

z2a

� du i Z 1;.; 6 (8)

where S(u) is the sea spectrum which can be defined according to 15th ITTC as

SðuÞ

H2s T1

Z0:11

2p

uT1

2p

� K5

exp K0:44uT1

2p

� K4 �where:ðN

0SðuÞdu Z m0

ðN

0uSðuÞdu Z m1

m1

m0

Z2p

T1

Hs Z 4ffiffiffiffiffiffim0

p


Fi(u,b,g) is the ith mean force component in regular waves with the circular frequency

of u and arbitrary direction of g in the presence of arbitrary current with angle of b.

Applying Eq. (8) to a general FPSO leads to the following mean drift loads in irregular

waves in surge, sway and yaw modes:

�Fs1 Z rg

H2s

16CT 1 CFn1

4p

T1

ffiffiffiffiL

g

s" #4

3r cos g

� ðSurgeÞ (9)

�Fs2 Z rg

H2s

16CT 1 CFn2

4p

T1

ffiffiffiffiB

g

s" #4

3r sin g CL2 sin gjsin gj

� ðSwayÞ (10)

�Fs6 Z rg

H2s

16CT 1 CFn2

4p

T1

ffiffiffiffiB

g

s" #2

3r singðL0

1 KL02Þ

�

C ðL022 KL0

12Þsin gjsin gjC1

3rðL0

1 CL02Þsin 2g

ðYawÞ (11)

3.3. Forces due to slow drift waves

The slow drift excitation loads can be written either in time series or in spectral form. It

is obvious that the spectral form can be much simpler to use for design calculations rather

than the time series form. The spectral density of the slow drift excitation loads can be

calculated according to Pinkster (1975) as:

SFðmÞ Z 8

ðN

0SðuÞSðu CmÞ

Fi u C m2

� �z2

a

� 2

du (12)

where Fi(uC(m/2)) is the mean wave load component at the frequency of uC(m/2) or in

other words we can say that ðFiðuC ðm=2ÞÞ=z2aÞ is the mean wave load coefficient at the

frequency of uC(m/2)

Fig. 15 shows the spectral density of slow drift excitation load for a 200,000 tdw FPSO

subjected to a head sea which can be represented by a P–M spectrum with significant wave

height of 5 m and the peak frequency of 9.61 s. The figure explains that the presence of the

current can only alter the lower frequency part of the spectrum by increasing the area

under that part of the spectrum.

3.4. Forces due to steady winds

The steady mean wind loads on an FPSO can be determined by using the following drag

formula

FW ;i Z 1=2rairCDAPV2 (13)

Fig. 15. Second order surge force spectrum.


where

rair Z 1:21!10K3ðt=m3Þ at 20 8C

CD is the drag coefficient, which varies for different type of structures. According to the

API the drag coefficient varies between 0.5 and 1.5.

V is undisturbed mean hourly wind speed at the location of the force centre and it can be

obtained from the following formula

�Vz Z �VzR

z

zR

� 0:125

where z is the height of the force centre and zR is the reference height.

3.5. Forces due to wind gusts

Wind gusts with significant energy at periods of the order of magnitude of minute can

produce slowly varying oscillations of marine structures. According to ISSC no appreciable

difference between various spectral formulations for dimensionless frequencies greater than

approximately 0.02 is existed. The Harris wind spectrum can be defined as


fSðf Þ

V210

Z4k ~f

ð2 C ~f2Þ5=6

(14)

~f Z Lf =V10

L is the length scale and k is the surface drag coefficient.

The fluctuating drag force due to gust can be obtained by defining the gust velocity as a

combination of mean wind velocity and the fluctuating gust velocity in the main drag force

equation. By neglecting the squared term of the fluctuating gust velocity, the fluctuating

drag force due to wind gust becomes:

F 0DðtÞ Z rairCDAPV 0ðtÞ (15)

Therefore, the power spectrum of the gust force in Hz can be determined as:

SWF ðf Þ Z ðCDAPrair

�VÞ2Sðf Þ (16)

The power spectrum can also be calculated in circular frequency by employing the

following equation:

SWF ðuÞ Z

1

2pSW

F ðf Þ (17)

3.6. Forces due to current

The longitudinal drag force due to an arbitrary current on an FPSO is mainly due to

frictional forces and can be calculated by the following approximate formula given by

Faltinsen (1990)

F1 Z0:075

ðlog10 Rn K2Þ21

2rSU2

C cos bjcos bj (18)

where

Rn ZUCLjcos bj

n

vZ1.19!10K6 m2/s for 15 8C water temperature. S is the wetted surface of the vessel.

The transverse current force can be calculated by writing the drag force formula

for the vessel’s cross section and integrating it over the vessel’s length. So, it can be

written as

FC2 Z

1

2rU2

Csin bjsin bj

ðL

CDðxÞDðxÞdx (19)

where CD(x) is the cross-sectional drag coefficient and D(x) is the sectional draught.

Current can also produce a yaw moment. This moment is the sum of the Munk moment

and the viscous moment due to cross-flow (Faltinsen, 1990) which can be calculated by


the following equation:

F6 Z1

2rUCsin bjsin bj

ðL

CDðxÞDðxÞx dx C1

2U2

CðA22 KA11Þsin 2b|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Munk moment

K kL2m|{z}

Due to mooring

(20)

where A11 and A22 are the vessel’s added masses in surge and sway, respectively.

3.7. Equilibrium angle

A turret moored FPSO subjected to met-ocean loads, i.e. wind, waves and current

weathervanes until the platform reaches an equilibrium position (Fig. 16). The first-order

wind and current forces are static forces. Meanwhile, the mean drift wave loads, which is

the result of the second-order wave effects is also a static force. These forces rotate

the vessel to its equilibrium position. In addition to these static forces, the first-order wave

forces which is time varying forces should be considered for determining the vessel’s

equilibrium angle. The reason is that the first-order wave loads are varying like a sine

function as shown in Fig. 17. Between t0 and t1 the vessel will rotate from its initial

position ‘P0’ to position ‘P1’ (Fig. 18). Therefore, the angle between the vessel and the

propagation direction of the incoming waves will reduce which cause a reduction in

rotational moment. As from this stage ‘t1’ towards ‘t2’ the direction of wave loads will

change then the vessel will rotate backward to position ‘P2’ where is somewhere between

‘P0’ and ‘P1’. At this stage the direction of wave loads will change again and the vessel

will rotate forward again to position ‘P3’. This forward and backward fluctuation will

continue until the vessel reaches its equilibrium position ‘Pn’. So, it can be concluded that,

however, the first-order wave forces are dynamic forces and fluctuate the vessel but these

forces will rotate the vessel to its equilibrium angle. A vessel subjected to environmental

loads would stand at an equilibrium heading angle where all rotational moments cancel

each other and there are no any resultant rotational moment acting on the vessel. The total

Fig. 16. Comparison between wave drift damping derived from different methods.

Fig. 17. The equilibrium angle of a turret-moored FPSO subjected to arbitrary wind, wave and current loads.


rotational moment can be calculated by adding up the following equations:

�Mwind Z1

2rairj �Vwisin aj �Vwisin a

XAo;n1CD;onxo C

ðLTB

LTA

CDiSixidx

�(21)

�MWave Z rg0:5Hw

1

k

sinh ðkhÞKsinh kðh KDÞ

cosh ðkhÞ

� ðLTB

LTA

2sinðkx cos gÞ�

!sin kBi

2sin g

� cos ut K2cosðkx cos gÞsin k

Bi

2sin g

� sin ut

�x dx

C0:5Hsu2 cosh k KD

2Ch

� �sinhðkhÞ

ðLTB

LTA

½a22;isinðkx cos gÞcos ut

Ka22;icosðkx cos gÞsin ut�x dx

(22)

�Mc Z1

2rU2

Csin bjsin bj

ðLTB

LTA

CDðxÞDðxÞx dx C1

2U2

CðA22 KA11Þsin 2b|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Munk moment

(23)

The equilibrium heading for a 200,000 tdw FPSO (Table 1) due to different set of

environmental conditions are given in Table 2.

Fig. 18. Typical time varying first-order wave loads.


The second order wind weathervaning effect can be analysed by calculating the

second order rotational moment about the turret-mooring point due to gusty wind.

Practically speaking a floating offshore platform can only be able to respond

dynamically to high speed short term gust (Barltrop et al., 1998, pages 2-33 and

2-34). In other words, there is not sufficient time for the low speed short term gust to

excite a floating offshore structure. For example, the 3-s fluctuating wind gust

velocity, V 0(t), for 30 m/s mean hourly wind speed would be around 14 m/s while this

figure for a 30-s gust velocity would be around 7 m/s. Applying the wind gust effect

on the equilibrium heading angle calculation of a turret-moored FPSO will lead to a

few degree fluctuation (Mazaheri, 2003).

The second-order wave effects on vessel’s equilibrium angle can be determined by

applying the slowly varying yaw drift excitation moment. It is expected that the second-

order wave effects fluctuate the floating platform just with a few degree amplitude around

its equilibrium angle. Analysing the second-order effects of wind and wave loads on a

200,000 tdw tanker showed that a few degree fluctuations can be expected. Also, a vessel

can be moved from one of her equilibrium angle to the other one if the first equilibrium

Fig. 19. Vessel fluctuations due to first-order wave loads.

Table 2

Equilibrium heading angle of a 200,000 tdw FPSO subjected to arbitrary wind, wave and current loads

Wind Wave Current Vessel’s

equilibrium

heading

angle (q)

a Vwi (m/s) g Hs (m) b Uc (m/s)

0 30 45 5 0 1 223

0 30 90 5 0 1 265

0 30 135 5 0 1 313

0 30 225 5 0 1 48

45 30 0 5 0 1 3

45 30 45 5 0 1 225

45 30 90 5 0 1 268

45 30 135 5 0 1 310

45 30 225 5 0 1 46


heading angle was happened to be in a position that wind and wave forces were acting

from the aft of the vessel (Mazaheri, 2003).

3.8. SAMRES model

SAMRES is a hydrodynamic response model for calculating responses of a floating

offshore platform subjected to arbitrary wind, wave and current loads. This hydrodynamic

response model is written in MATLAB and it is based on the theories described earlier. It

should be noted that the forces are calculated on the vessel at its equilibrium heading

angle. In general SAMRES solves Eq. (1). The total platform’s response in each mode of

motion is calculated as

�Xi Z XWaveðsteadyÞ;i C �XWaveðfreqÞ;i C �XWaveðslowdriftÞ;i CXWindðsteadyÞ;i

C �XWindðslowdriftÞ;i CXCurrent;i (24)

where

�Xi
is the mean square value of the vessel’s response due to wind, wave and
current loads in mode i,

XWaveðsteadyÞ;i
is the vessel’s response due to steady wave load in mode i, �XWaveðferqÞ;i is the mean square value of the vessel’s response due to frequency wave
load in i mode,
�XWaveðslowdriftÞ;i is the mean square value of the vessel’s response due to slow drift wave
load in i mode,

XWind(stready),i
is the vessel’s response due to steady wind load in i mode, �XWindðslowdriftÞ;i is the mean square value of the vessel’s response due to wind gust load in i
mode,

XCurrent
is the vessel’s response due to steady current load in i mode.


4. The ANN model

The ANN model can be most adequately characterised as a computational model with

additional abilities such as adaptation and learning. The interpolation, and in some cases

extrapolation, capability is very powerful particularly when mapping a multi-dimensional

input data space to a multi-dimensional output data space (Roskilly and Mesbahi, 1996).

Therefore, ANN as an powerful regression tool can be implemented together with

SAMRES hydrodynamic model to predict the platform’s excursions. In general SAMRES

model calculates platform responses as a function of met-ocean data and platform’s

particulars. So, for a specific platform it can be expressed as

Y Z f ðXÞ (25)

where Y is platform responses (output data), f is the hydrodynamic response function, X is

the met-ocean parameters (input data).

The values of input and output matrices (X and Y) should be normalised. The following

formula can be used for normalising input and output values:

Table 3

Some of the input and output series of the data used in ANN

Vwind(m/s) awi Hs (m) gwa current

(m/s)

bac Surge

(m)

Sway

(m)

Excur.

(m)

32 136 2 73 1 233 1.09 4.04 4.18

10 206 3 148 1 115 0.13 1.04 1.05

14 31 4 321 1 40 0.48 3.87 3.90

25 53 8 68 2 242 9.52 1.06 9.58

23 124 5 88 2 265 2.06 5.61 5.98

25 103 3 228 2 321 0.84 0.01 0.84

12 224 4 280 1 343 0.46 1.13 1.22

21 302 2 229 1 8 0.41 0.00 0.41

28 139 5 309 1 59 1.25 41.18 41.20

20 206 7 248 2 76 6.07 0.52 6.09

33 329 3 298 2 159 0.27 68.15 68.15

33 177 2 61 1 232 1.49 0.62 1.61

24 228 8 307 2 262 4.57 24.80 25.22

26 179 8 341 1 157 7.56 21.70 22.97

13 180 6 4 1 65 2.05 0.58 2.13

20 179 4 315 1 210 0.56 13.68 13.69

25 150 5 125 2 331 1.77 26.30 26.36

11 204 6 326 1 43 2.16 1.53 2.64

34 166 4 237 2 197 1.96 0.40 2.00

25 254 3 211 1 120 0.77 0.15 0.79

16 243 2 162 2 40 0.05 4.67 4.67

21 160 5 302 2 251 0.86 15.37 15.39

35 282 7 240 2 40 7.42 1.01 7.49

16 302 5 97 2 138 0.81 2.75 2.86

25 76 3 298 1 56 0.19 26.93 26.94


Normalised value ZReal value KMin: value

Max: value KMin: value

The ANN model tries to utilise f(X) according to the existing input data, X, and the

relevant output data, Y, obtained from the hydrodynamic mathematical model.

It should be emphasised that it is impractical to calculate the platform’s responses

using the SAMRES model with long-term met-ocean parameters as it takes a

considerable time. Therefore, in order to make the model more accessible and

practical, the use of ANN model has been adopted. This model has been designed,

tested and validated using the platform’s excursion data predicted by the SAMRES

model. Taking into account the nature of input and output data displayed as an

example in Table 3, a (6-10-3) multi-layer feed forward network, which has one

hidden layer containing 10 neurons, has been selected (Fig. 19). The number of

neurons in input and output layers has been chosen according to the number of input

and output parameters. But, the number of neurons in hidden layer has been

calculated (decided) on a trial and basis in which the minimum means square error

has been achieved in both training and cross-validation procedure. Eq. (1) for the

deigned ANN model can be written as:

Y Z a sig1

fsig½AX�g

� B Cb (26)

where YZ[Surge, Sway, Total], Surge is vessel’s excursion in surge mode, Sway is

vessel’s excursion in sway mode, and Total is total vessel’s excursion, a and b are

normalisation factors which are:

a Z

0:0360 K0:3100

0:0025 0:0450

0:1500 K0:2500

0:0025 0:0475

0:9000 K0:8500

0:0025 0:0500

b Z

0:0947 0:0486

0:0109 0:0500

0:0109 0:0492

A and B are coefficient matrices which are


A Z

4:3899 23:4794 9:8039 2:9633 6:8804 11:9349

1:8872 2:6946 7:9440 6:2699 0:2288 4:4961

0:5779 0:8103 7:3273 0:9930 0:3290 1:7961

5:3140 16:7633 12:1113 4:6270 6:6249 17:3256

3:4854 0:1798 0:3999 31:2892 0:4736 0:4161

3:3931 1:2402 0:1614 24:0280 0:2692 2:2755

27:5614 0:8846 2:0947 19:8817 1:7678 10:8667

0:71087 0:99665 9:01257 1:22143 0:40466 2:20914

6:53625 20:6188 14:89688 5:69117 8:148615 21:3104

4:28709 0:22114 0:491884 38:48569 0:58247 0:5118

BZ

0:2357 1:2071 2:6545 0:3262 0:6652 0:4590 0:0291 0:489236 0:638572 0:440601

1:3489 0:7285 1:3145 1:3087 3:4134 3:3863 1:6033 1:96311 3:27685 3:25081

1:2387 0:5467 1:2972 1:2180 3:1552 3:0949 1:5018 1:82698 3:02898 2:97106

sigðxÞZ1

1CeKx

,

X is the input matrix which can be defined as:

X Z ½HS;g;Vwind;a;UCurrent;b�

5. Implementing the approach to an FPSO

In order to implement the approach a 200,000 tdw FPSO whose particulars are

described in Table 1 is considered. The platform is assumed to be installed in a place

where the long term met-ocean data is assumed to occur as specified in Table 4. SAMRES

hydrodynamic model is used to produce enough input and output data in order to design

the ANN model as described earlier. Fig. 20 shows how the ANN model can precisely

predict the platform excursions in surface modes of motion. Then the designed ANN

model, Eq. (26), is applied to predict data series of platform’s excursion due to long term

met-ocean data. The result of this simulation is illustrated in the form of the annual

frequency of platform’s excursion in Fig. 21. Fisher Tippett (FT) distribution method is

then applied to predict the maximum platform’s excursion in various return periods.

Fig. 22 shows the platform’s excursion against KLN(KLN(P)) in which the return period

can be obtained. The relation between return period (N) and KLN(KLN(P)) is expressed

in Table 5.

Table 4

The range of environmental parameters

Statistics Parameters

V (m/s) a (deg.) Hs (m) g (deg.) Vc (m/s) b (deg.)

Minimum 10 0 2 0 1 0

Maximum 35 360 8 360 2 360

Fig. 20. Comparisons between Desired excursions and ANN’s predictions (cross-validation).

Fig. 21. A multi-layer feedforward network (6-10-3). Note: Hs is wave height; g, wave angle; Vwind, mean hourly

wind velocity; a, wind direction; Ucurrent, current velocity; b, current angle and W1/W2 is weighting factor.


Fig. 22. Annual frequency of vessel’s excursion.


6. Conclusions

The strip theory was used in developing mathematical response model (SAMRES)

instead of 3D analysis for reason of simplification and computational efficiency. The

features in the approach adopted include a method for accounting lateral curvature effects of

the hull on surge using Gauss’ theorem to handle the pressure integration. Another aspect

was the inclusion of the Munk moment (Faltinsen, 1990) in the calculation of the current

induced yaw. The surge mode was also included in the coupled equations of motions.

Table 5

The relation between return period ‘N’ and ‘KLN(KLN(P))’

Return Period (N) (Year) P KLN(KLN(P))

1.1 0.09 K0.87

1.2 0.17 K0.58

1.3 0.23 K0.38

1.4 0.29 K0.23

1.5 0.33 K0.09

1.6 0.38 0.02

1.7 0.41 0.12

1.8 0.44 0.21

1.9 0.47 0.29

2 0.50 0.37

5 0.80 1.50

10 0.90 2.25

20 0.95 2.97

25 0.96 3.20

50 0.98 3.90

100 0.99 4.60

Fig. 23. Vessel’s excursion prediction using FT method.


The results of this approach were shown to agree closely with those of Oortmerssen (1976)

who used the 3D source technique. Faltinsen’s wave drift formula (1990) was redeveloped

by adding finite draft coefficient. The results obtained from this formula compared

favourably with results obtained in Helvacioglu’s experiments (1990). In addition, the

influence of the current on the wave mean drift force was taken into account by considering

the current coefficient derived from the ship added resistance formula. The formula for the

calculation of the wave drift damping was extended to cover high wave frequencies as well

as low wave frequencies. The results compared with asymptotic formula (Fig. 23) showed

good agreement in high frequency band. The weathervaning effects were also included in

calculations of forces and excursions of the platform due to environmental loads. The

analysis of the second-order effects of wind and wave forces on the case study platform

showed that only a few degree fluctuations occurs around the platform’s equilibrium angle.

The designed ANN model in conjunction with the hydrodynamic SAMRES model was

used to carry out response based predictions to determine excursions of a series of

platforms due to long-term of met-ocean data. Fig. 24 shows how the ANN model can

considerably reduce the simulation time.

Fig. 24. Training and simulating tasks of the ANN model along with the required time for doing those tasks.


The response-based method applied to the case study platform subjected to long-

term met-ocean data showed that the traditional method based on the combination of

N-year wind, N-year wave and N-year current cannot lead to N-year platform’s

excursion. In the case study considered, applying the response-based method showed

that the 50-year return period of the platform’s excursion due to particular met-ocean

data is around 60 m (Fig. 22) which has occurred due to a combination of maximum

12-year wave height and 8-year wind speed. Using the traditional method which

means applying 50-year wave height and wind speed makes the platform to move

around 80 m. Therefore, it can be concluded that the traditional method based on the

combination of N-year wind, wave and current loads for predicting the N-year

platform excursion provides a very conservative estimate.

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Response-based method for determining the extreme behaviour of floating offshore platforms

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