Ocean Engineering xxx (xxxx) xxx

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Ocean Engineering
journal homepage: www.elsevier.com/locate/oceaneng

Mooring system fatigue analysis of a floating offshore wind turbine

Carlos Barrera, Tommaso Battistella, Raúl Guanche *, In
~igo J. Losada
Environmental Hydraulics Institute, Universidad de Cantabria (IHCantabria), Avda. Isabel Torres, 15, Parque Científico y Tecnol�
ogico de Cantabria, 39011, Santander,
Spain

A R T I C L E I N F O A B S T R A C T

Keywords: Mooring systems are under a cyclic loading process caused by the randomness of metocean conditions, which
Fatigue could lead to a fatigue failure of the station keeping system. The present paper presents an innovative meth­
Mooring system odology for the assessment of floating offshore wind turbine mooring system fatigue considering the full lifetime
Floating offshore wind turbine
of the structure. The method integrates the impact of the life cycle metocean conditions over the dynamic
Non-linear interpolation techniques
performance of the platform thanks to coupled numerical models, selection and non-linear data interpolation
techniques and commonly accepted fatigue approaches. One of the benefits of using this methodology is that
there are no uncertainties due to the selection of a reduced set of sea states. The methodology is applied to a set of
moorings with different properties in the DeepCwind platform to evaluate the solution which offers the best
compromise between size and fatigue damage. Results show that the best long-term mooring behaviour is
achieved with a weight of approximately 300 kg/m. A comparison is conducted between the fatigue damage
obtained through the life-cycle method and conventional methods. The mean differences observed between the
standard and the new method proposed are between 13% and 49% depending on the use of the S–N or T-N
curves.

1. Introduction motions and natural periods. Hence, station keeping systems based on
mooring lines and anchors are crucial to guarantee structure surviv­
Wind industry has experienced a huge growth in recent years moti­ ability and its components (e.g. power cable) under different metocean
vated by the need for energy sources alternative to fossil fuels. In conditions. Traditionally, a successful mooring design considers several
particular, offshore wind energy technology evidences important po­ limit states (LS) (DNVGL-OS-E301, 2018) as follows: ultimate (ULS),
tential in the coming years. In fact, this trend is being led by the Euro­ accidental (ALS), fatigue (FLS) and service (SLS). These limit states
pean Union with a total installed offshore wind capacity of 15,780 MW contribute to properly ensuring the resistance of the mooring and its
in 2017 (Wind Europe, 2018a). The dominant substructures in offshore service criteria.
wind farms are fixed foundations including monopiles, jackets, and This article is focused on the analysis of fatigue loads (FLS) on the
gravity base foundations (Wind Europe, 2018a). However, new tech­ mooring lines of a FOWT. Moorings are under continuous cyclic meto­
nological advances point to offshore floating wind farms in intermediate cean loads, therefore fatigue damage is a potential failure mechanism.
and deep waters. These new solutions are an opportunity for countries Fatigue damage can be evaluated by means of either an S–N or a T-N
with important wind resources but with narrow continental shelves. curve. These curves relate a constant stress (S) or tension (T) range with
Hywind Scotland was the first floating offshore wind farm in the world the maximum number of cycles until component failure (N). Numerous
with a total of five floating spar buoys installed in 2017. After this investigations have been conducted since the 1980s to understand the
success and according to European policies, floating offshore wind farms fatigue failure mechanism of offshore mooring chains. Building on prior
could provide between 4 and 5 GW by 2030 (Wind Europe, 2018b). research on this topic, van Helvoirt (Van Helvoirt, 1982) described an
Floating offshore wind turbines (FOWTs), in comparison with fixed experimental test campaign related to the static and fatigue strength of
foundations, have a higher level of design complexity. Despite the sys­ stud-link chains and connecting links under high load cycles in the
tem stability is mainly driven by the floating platform characteristics, marine environment. Lereim (1985) presented a complete study of chain
the mooring system plays a critical role in the design of structure reliability including experimental and numerical assessments. He

* Corresponding author.
E-mail address: guancher@unican.es (R. Guanche).

https://doi.org/10.1016/j.oceaneng.2019.106670
Received 5 April 2019; Received in revised form 27 September 2019; Accepted 31 October 2019
0029-8018/© 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Carlos Barrera, Ocean Engineering, https://doi.org/10.1016/j.oceaneng.2019.106670
C. Barrera et al. Ocean Engineering xxx (xxxx) xxx

proposed an S–N curve for chain links based on a fatigue crack propa­ the FOWT lifetime. A maximum dissimilarity selection technique is then
gation analysis for a corrosive environment assuming different initial applied to select the most representative sea state subset from this
surface crack depths. The American Petroleum Institute (API) (API RP database. FOWT dynamics and mooring fatigue are evaluated by means
2FP1, 1993) proposed a standard based on a normalised tension range to of a numerical model for the full subset of sea states. Finally, the FOWT
define the fatigue lifetime for each mooring component. Different T-N dynamics and mooring fatigue results are rebuilt for the full lifetime
curves are defined according to floating offshore structure experiments. using a radial basis function (RBF) interpolation technique.
Later, Det Norske Veritas (DNV) (DNV-OS-E301, 2001) published
different design S–N curves to estimate the fatigue life. These two 2.1. Site assessment and metocean database
standards are widely used as references for fatigue design by industry
and researchers (Lassen and Syvertsen, 1997) (Xue et al., 2018) (Thies Long-term analysis requires the use of databases including time se­
et al., 2014). ries of the relevant environmental parameters to assess wind and wave
As has been shown, early investigations have been focused on conditions at a given offshore location. They are usually built upon
building fatigue damage curves to allow a safe mooring design. metocean reanalysis techniques and provide metocean parameters
Currently, the mooring integrity management is a current topic among extended over several decades. The metocean database used in this work
researchers and engineers. Issues such as residual stress, anomalous is based on the reanalysis developed by IHCantabria and BiMEP in the
loading modes or corrosion are receiving increased awareness by the framework of the TRLþ project (Metocean Analysis of BiMEP for
offshore industry. Martinez et al. (2017) estimated how residual stresses Offshore Design, 2017).
generated during the manufacturing process contribute to the fatigue Wind data are obtained from the Seawind database (Menendez et al.,
life of mooring chains depending on the loading mode. Rampi et al. 2014). Wind data are modelled with the Weather Research & Fore­
(2015) described a new fatigue mechanism based on a combination of casting model and the Advanced Research dynamical solver module,
high pretension levels and motions generating out-of-plane bending developed by the National Center for Atmospheric Research (Skamarock
fatigue loading and proposed a new S–N fatigue curve diagram. Gabri­ et al., 2008). Wind speed (W) and direction (β) for the 1985–2015 period
elsen et al. (2018) conducted mooring fatigue tests considering surface at 10 m above the sea surface are provided with a resolution of 1 h. Wind
roughness, corrosion pits and mean loads using chain segments recov­ speed at the nacelle height is obtained using the empirical expression for
ered from a floating structure in the North Sea. Their results concluded the wind power law (Jonkman and Kilcher, 2012) (Emeis, 2013). Wave
that the degradation of the chains reduces the fatigue capacity although data are taken from the Global Ocean Waves database (Reguero et al.,
the fatigue design capacity is still above the S–N design curve given by 2012). Significant wave height (Hs), wave peak period (Tp) and direc­
(DNVGL-OS-E301, 2018). tion (α), among other wave parameters, for the 1985–2015 period are
As seen in the previous literature review, a complete analysis of the provided with a 1-h resolution. The numerical simulation of the dynamic
different mechanisms that induce mooring fatigue already has been response of the FOWT and its mooring system over several decades using
carried out. However, there are significantly less investigations of long- hourly metocean time series would require a huge computational effort.
term fatigue performance evaluation. Traditionally, a set of environ­ As a consequence, the number of sea states to be simulated must be
mental states is chosen to discretise the long-term environmental con­ reduced by using a reliable selection technique.
ditions (occurrence matrix) (DNVGL-OS-E301, 2018) (API, 2008) but
this selection may affect the long-term fatigue life of mooring chains of
2.2. Maximum dissimilarity selection technique
FOWTs. The present paper proposes a new methodology to estimate the
long-term fatigue analysis involving long-term metocean databases,
In general, the objective of the selection techniques is to reduce the
advanced selection methods, FOWT numerical models and non-linear
large data amounts provided by metocean databases at given locations
interpolation techniques. All of these combined allow us to recreate
into a representative subset maintaining the variability of the original
the long-term damage to a mooring system with a low numerical cost.
The new methodology is applied to different mooring systems in order to
select the most appropriate mooring on a FOWT located in the BiMEP
test site (north of Spain) considering the influence of all metocean
conditions on the floating structure life-cycle. The reference platform
used in this work is the DeepCwind semisubmersible platform (Rob­
ertson, Jonkman, Wendt, Goupee, Dagher).
This paper is organised as follows. In section 2, the new methodol­
ogy, the metocean databases, the advanced selection techniques, the
FOWT numerical model, the methods to evaluate the fatigue damage
and the non-linear interpolation techniques are described. Section 3
describes the case study involving the site and the definition of FOWT,
the selected sea states and the long-term results. Finally, a discussion of
the obtained results and the main conclusions of this investigation are
presented in sections 4 and 5, respectively.

2. Methodology to predict mooring fatigue damage and FOWT

dynamics

The most accurate method to estimate the mooring fatigue response

is a dynamic analysis in the time domain (API, 2008) where all non­
linearities and dynamics are captured. The main disadvantage of this
method is the excessive computational cost associated with the evalu­
ation of all observed sea states at the target location. The proposed
methodology attempts to provide a more efficient approach by following
a different set of steps, as shown in Fig. 1. The first step is to collect from Fig. 1. Methodology to predict the lifetime mooring fatigue damage and
a metocean database all historical sea states for a time period equal to FOWT dynamics.

2
C. Barrera et al. Ocean Engineering xxx (xxxx) xxx

data. The maximum dissimilarity algorithm (MDA) (Camus et al., 2011) phase and Qj;α ðωl Þ ¼ ql;α ðωl Þ eiφlj;α , where ql;α is the amplitude of the first-
is used to select this representative subset. The samples included in this order force per wave amplitude unit and φlj;α is its phase. The symbol *
subset are hourly sea states of dimension 5 defined by W, β, Hs, Tp and α. denotes the complex conjugate. Qj;α ðωl ; ωm Þ represents the quadratic
From a database P including I sea states, pii ¼ {p1, p2, …, pI}, a
transfer function (QTF) associated with the frequency difference be­
representative data subset O with J sea states, ojj ¼ {o1, o2, …, oJ}, is
tween pairs of wave components.
selected, with J < I. The selection starts by choosing an initial sea state
The mooring system is simulated using a dynamic model that allows
from the full database P. In this case, the sea state containing the
the estimation of mooring loads with higher accuracy than a quasi-static
maximum significant wave height is chosen as the initial sea state. The
model (Barrera et al., 2019a) (Robertson et al., 2017) although it is
rest of the sea states are iteratively selected, calculating the dissimilarity
computationally more demanding. The dynamic formulation is based on
between of the remaining sea states in the database and the sea states
Newton’s second law (5) and is solved by a finite element method as
added to the subset by choosing those with the largest dissimilarity at
follows:
each iteration. During the application of the algorithm, the subset O is
� �
composed of R sea states, orr ¼ {o1, o2, …, oR}, with R < J. The selection ∂2 !r ∂ Ts ∂! r !
ρ0 2 ¼ þ f ð1 þ eÞ (5)
finishes when the number of required sea states, J, is reached. The ∂t ∂s 1 þ e ∂s
dissimilarity (di) is evaluated by means of the Euclidean-Circular norm
( k k) between the vectors p and o. The dissimilarity value is taken ac­ where ρ0 is the linear weight, ! r is the position vector, s is the longitu­
cording to Polinsky (Polinsky et al., 1996) as follows: !
dinal coordinate, e is the deformation, Ts is the tension and f is the sum
� !
di ¼ minimum
pii oR 1
(1) of external forces acting on the cable. External forces ð f Þ result from the
minimum ​ pii orr ; rr ¼ 1; …; R 2 ! ! !
sum of the buoyancy force ð f hg Þ, the normal ð f dn Þ and tangential, ð f dt Þ
!
components of the drag forces, the inertial force ð f i Þ and the seabed
2.3. Numerical model description ! !
contact force in the normal ð f sb;n Þ and horizontal ð f sb;t Þ directions.
The numerical model used this work was presented and validated in Their formulations are given by the following equations:
(Barrera et al., 2019a). However, a brief description is provided next ! ! ! ! ! ! ! ρ ρw !
f ¼ f hg þ f dn þ f dt þ f i þ f sb f hg ¼ ρ0 c g
because some differences with respect to the original model have been ð1 þ eÞ ρc
implemented. The numerical model is built by coupling a hydrody­
namic, an aerodynamic and a mooring model. The hydrodynamic ! 1 ! 1 !
f dn ¼ CDN dρw j!
v n j!
vn f dt ¼ CDT dρw j!
v t j!
vt f i
component models the behaviour of the FOWT in the frequency domain 2 2
using potential flow theory based on a boundary element method (BEM), 2
πd !
¼ CI ρ an
which is transformed to the time domain using the relationship proposed 4 w
by Ogilvie (1964). The BEM model used is ANSYS AQWA (ANSYS � xy �
AQWA, 2013). Hydrodynamic results are computed together with the ! ! r_ r_xy
f sb; ​ n ¼ ​ d½ðzbot ​ rz ÞKG r_z KB �rz f sb; ​ t ¼ ​ fhg Kμ min ;1
aerodynamic and mooring results by means of the Cummins equation vμ k r_xy k
(Cummins, 1962), a second-order ordinary differential equation with a (6)
convolution integral applied to solve the radiation problem (2) as
follows: where ! g is gravitational acceleration, ρc is the cable density and ρw is
Z t the density of water. CDN and CDT are the normal and tangential drag
ðM þ A∞ Þ€kðtÞ þ _ τÞdτ þ GkðtÞ ¼ Fe ðtÞ þ Fw ðtÞ þ Fm ðtÞ
Kðt τÞkð (2) coefficients, respectively; d is the mooring diameter; CI is a hydrody­
0 namic mass coefficient; ! v and !a are the velocity and acceleration
denoted by the subscripts n and t, which are the decompositions into the
where M is the inertia matrix of the floating structure, A∞ is the added
normal and tangential directions, respectively; zbot is the vertical coor­
mass matrix at infinite frequency, K is the retardation matrix, G is the
dinate of the seabed; and rz is the vertical projection of the position
hydrostatic stiffness matrix, t is time, τ is the integration variable of the
vector and r_z its velocity. Constants KG and KB represent the stiffness and
convolution integral and k;€ k;_ and ​ k are the floating platform acceler­
viscous coefficients, respectively. In the horizontal term, Kμ is the co­
ation, velocity and displacement, respectively. External forces are rep­
efficient of kinetic friction corresponding to a maximum velocity, vμ , and
resented by wave excitation forces (Fe ), wind forces (Fw ) and mooring
r_xy represents the velocity of the horizontal projection of the position
system forces (Fm ).
vector.
Wave excitation forces include both the first and second order
The main difference with respect to the numerical model used in
difference-frequency components, expressed in the time domain by the
(Barrera et al., 2019a) is the aerodynamic model. Normally, aero­
following equations:
dynamic forces are estimated through blade element momentum theory
�X
L � (BEMT) (Hansen, 2015). However, fatigue analysis requires a huge
Feð1Þ;j;α ¼ Re Al Qj;α ðωl Þeiðωl tÞ ; j ¼ 1; 2; 3; 4; 5; 6 (3) number of simulations to characterise properly the mooring damage. A
simplification of the aerodynamic model is implemented here in order to
l¼1

�X
L X
L X
L � reduce the computational cost but retaining a sufficient level of accuracy
Feð2 Þ;j;α ¼ Re Al A*l Qj;α ðωl ; ωl Þ þ 2 Al A*m Qj;α ðωl ; ωm Þeiðωl ωm Þt
in the estimation of wind forces, evidenced by the validations shown in
l¼1 l¼1 m¼lþ1 section 3.4. The aerodynamic model calculates the thrust force by means
(4) of a thrust coefficient defined for different relative wind speeds seen by
the rotor (Martini et al., 2016) (Karimirad and Moan, 2012). It is
where Feð1Þ is the first-order wave excitation force and Feð2 Þ the second- assumed that only the normal component of the rotor is generating a
order wave excitation force with j representing the degrees of freedom. L force and that the nacelle is always aligned with the wind direction. The
is the number of wave components, Al eiðωl tÞ the complex wave thrust and thrust coefficients are obtained from simulations made with
component, Al the complex wave amplitude, ωl the wave frequency and i FAST (Jonkman and Buhl, 2005) considering a rigid tower and constant
represents the imaginary number. Qj;α ðωl Þ stands for the first-order and turbulent winds defined across the rotor following the well-known
complex excitation transfer function associated with ωl , j and α. Al can power law with the exponent equal to 0.14. Ten iterations per wind
be written as Al ¼ al eiεl ; where al is the wave amplitude, εl is the wave

3
C. Barrera et al. Ocean Engineering xxx (xxxx) xxx

speed are simulated in the turbulent cases and the mean values of thrust sea state. The tension at the fairlead is mainly induced by the platform
and the thrust coefficient are adopted. Fig. 2 shows the thrust force for translational movements (surge, sway and heave) (Barrera et al., 2019b)
different wind speeds considering constant (static thrust) and turbulent and, therefore, their assessment has an important impact on the fatigue
(dynamic thrust) wind. Comparing both approximations, it should be evaluation. For this reason, floating platform rotations are not a primary
noted that an important discrepancy in the thrust estimation is found focus of investigation in this work. Additionally, the nacelle acceleration
between 9.5 m/s and 12.5 m/s that is higher for constant speeds. No is analysed as a potential parameter that influences the wind turbine
substantial discrepancies appear outside this range of simulated speeds. production as it is assumed that excessive tower top accelerations would
The differences are mainly related to the thrust variability resulting in trigger the turbine shut down.
turbulent wind simulations, which reaches the rated thrust value only Different theories can be considered to estimate the fatigue damage
when the incoming wind speed approaches the rated wind speed or in (Fatemi and Yang, 1998). However, two approaches have been estab­
case sharp gusts occur. Provided that wind fluctuations prevent the lished as the most trustworthy: the crack growth approach and the S–N
development of the aerodynamic loads generated under static condi­ approach.
tions, the thrust coefficient law calculated for turbulent wind will be The crack growth approach is based on fracture mechanics and as­
used. The thrust curves have been calculated using a conventional pitch sumes that the strength of a component fails when an initial crack grows
controller. These curves have a negative slope for wind speed above the to a critical crack size. In spite of the fact that this method considers the
rated value and may introduce negative damping at the pitch natural load sequence in the crack growth, different previous works have evi­
frequency (Larsen and Hanson, 2007). A filter is implemented to remove denced that most mooring chain fatigue lifetime is spent on crack
the contribution of the pitch natural frequency band from the calculated initiation (P�erez-Mora et al., 2015). Therefore, it seems appropriate to
relative wind speed to avoid this effect before evaluating the thrust force adopt a fatigue criterion based on crack initiation. The present work
(Martini et al., 2016) (Karimirad and Moan, 2012). A smooth transition evaluates the cumulative fatigue damage through S–N and T-N ap­
of the trust curve is added from the thrust curve value at 25 m/s to 0 at proaches. The S–N approach assumes that fatigue failure occurs when a
26 m/s, avoiding unrealistic thrust jumps when the wind speed fluctu­ number of cycles, N, is reached. N is a function of the constant cyclic
ates around 25 m/s. stress range, S, applied to the specimen. This approach provides
The relative wind speed seen by the rotor and the thrust force can be different S–N curves according to the type of material. These curves are
formulated following: modelled from a linear regression of normalised experimental test re­
sults. The number of fatigue cycles (N) for a particular range of constant
!
v rotor ¼ !
v ðv��! ��! �����!
SWL þ wSWL � rSWL rotor Þ (7)
cyclic stress (S) is formulated in (9) as follows:

where ! v rotor is the relative wind speed seen by the rotor, ! v is the un­ N ¼ aS m
(9)
disturbed wind speed, v��! SWL is the platform velocity at the sea water level,
log N ¼ log a m log S
���! is the platform angular velocity, and r�����
w SWL
�!
SWL rotor is the position vector
where a is the intercept parameter of the S–N curve and m is the slope of
between the sea water level and the rotor axis at the tower centreline.
the S–N curve.
Also:
It should be noted that there is no endurance limit in mooring S–N
1 curves. The fatigue damage accumulation during the mooring life-cycle
Trotor ¼ Arotor ρa CT v2rotor (8)
2 is evaluated according to Palmgren-Miner’s rule (Palmgrem, 1924)
(Miner, 1945). This rule assumes linear accumulation damage without
where Trotor is the thrust force, Arotor is the rotor area, ρa is the air density considering the load sequence in the life-cycle.
and CT is the thrust coefficient. The parameters (a and m) (Table 1) for a chain mooring with the
corrosive influence of seawater are found in DNVGL–OS–E301
2.4. Dynamics and fatigue evaluation (DNVGL-OS-E301, 2018). This standard provides the parameters as a
function of a stud chain or studless chain regardless of the steel grade.
The main objective of this work is to evaluate the mooring fatigue Recent studies have revealed the importance of the steel grade on the
damage considering different mooring properties. Fatigue analysis re­ S–N curve estimation (Arredondo et al., 2016) and the possible con­
quires the determination of the tension of each mooring line for every servative results of the curve proposed by the standard. It should be

Fig. 2. Thrust and thrust coefficient for different wind speeds.

4
C. Barrera et al. Ocean Engineering xxx (xxxx) xxx

Table 1 effective cycles is required to evaluate the fatigue damage through the
S–N fatigue curve parameters according to DNVGL–OS–E301. S–N and T-N curves. Different time-domain cycle counting methods are
S–N FATIGUE CURVE PARAMETERS used to obtain the equivalent response between regular and irregular
tension series. The most popular are the peak counting, the range
COMPONENT a m
counting, the level-crossing counting and the rainflow counting methods
STUD CHAIN 1.2 10 11
3 (ASTM E1049-1985, 1985). The rainflow cycle counting method is the
STUDLESS CHAIN 6.0 1010 3 most accurate and widely used method to estimate the fatigue damage
according to (API, 2008) (Dowling, 1972) (Watson and Dabell, 1975). It
noted that the measured nominal tension (Tn ) on moorings must be was proposed by (Matsuishi and Endo, 1968) and a new equivalent
transformed to a nominal stress (σ n ) considering the chain nominal cross definition was performed by (Rychlik, 1987).
sectional ðAc Þ as the representative area (10). The cross-sectional area to Fatigue damage in this work is determined making use of Palmgren-
be considered is twice the chain link (dc denotes the link diameter) area Miner’s rule (12) (Palmgrem, 1924) (Miner, 1945) in connection with
as follows: the rainflow counting method proposed by (Rychlik, 1987) and imple­
� � � mented by (The WAFO group, 2017). The fatigue damage of a particular
σn ½MPa� ¼ Tn ½N� Ac mm2 sea state is assessed as the sum of the individual tension/stress ranges
�� (10)
Ac ¼ πd2c 2 indicated by the rainflow algorithm during the sea state duration.
Finally, the total fatigue damage is obtained by adding all the sea states
Another standard widely used in the mooring fatigue design is API- during the life-cycle causing the failure if the damage is higher than 1 as
RP 2SK (API, 2008). This standard proposes the use of a T-N curve, follows:
similar to S–N but considering the tension range and not the stress range.
Xn ðSk Þ 1 X
The T-N curves define the number of cycles to failure, N, when mooring Damage ðsea stateÞ ¼ ¼ n ðSk Þ*ðSk Þm (12)
is repeatedly cycled by means of a given effective tension range (11). NðSk Þ a
The effective tension range is defined as a relation between the tension
range (T) and the reference breaking strength (RBS). The parameters a where n is the number of cycles in the sea state with the stress/tension
and m for a chain mooring considering a T-N curve are collected in range interval Sk , N is the number of cycles to failure at the normalised
Table 2. stress/tension range Sk , provided by the appropriate S–N or T-N curve.
� � m Sk is the succession of stress/tension ranges obtained by rainflow
N ¼ a T= (11) counting.
RBS
2.5. Reconstruction of fatigue and dynamics: radial basis function
The reference breaking strength is usually provided by mooring
interpolation technique
manufacturers (Vicinay Cadenas brochure., 2018). An R4S studless
chain mooring is considered for the purpose of this work. The R4S me­
Once the fatigue damage and dynamics have been evaluated for the
chanical properties are shown in Table 3.
selected subset of metocean data according to 2.2, it is possible to
Frequency domain or time domain approaches can be considered to
interpolate results in the original set of metocean conditions by means of
predict the fatigue damage due to low frequency and wave frequency
a non-linear interpolation technique called Radial Basis Function (RBF)
tensions (DNVGL-OS-E301, 2018) (API, 2008).
(Rippa, 1999). This method aims at finding an objective function (c)
Frequency domain methods have been discussed by several re­
through an approximation function (~c) built as a weighted sum of basic
searchers. A bimodal theoretical models for predicting fatigue damage
symmetric radial functions and a linear polynomial as follows:
under stationary and non-stationary Gaussian processes was presented
by (Jiao and Moan, 1990). Later, the previous theoretical models to X
J
�
estimate fatigue damage were imporved incorporating a trimodal cðpÞ ffi ~cðpÞ ¼ uðpÞ þ ajj ϕ p ojj (13)
spectral formulation to account for other processes such as jj¼1

vortex-induced vibrations (VIV) or wind loads at the low and wave

where pðhÞ is a linear polynomial equal to the multivariate data
frequencies (Gao and Moan, 2008). Previous studies have been incor­
dimension (mv) (W, β, Hs, Tp and α), ajj are the RBF adjustment co­
porated into design codes (DNVGL-OS-E301, 2018) (API, 2008). The
efficients, ∅ð Þ is the basic radial function and ojj are the approximation
standards admit three possible frequency domain methods to evaluate
centres.
fatigue damage: a simple summation of low frequency and wave fre­
The linear polynomial is defined on a monomial basis fu0 ; u1 ; …;
quency fatigue damage independently, the combined spectrum of low
umv g, including a number of monomials of degree 1 and a monomial of
and wave frequencies and the combined spectrum with a dual
degree 0, where b ¼ fb0 ; b1 ; …; bmv g are the coefficients of these mo­
narrow-banded correction factor.
nomials. Coefficients a and b are calculated by enforcing the interpola­
Despite the considerable computational cost, the time domain
tion constraints as follows:
approach is the most accurate methodology for predicting mooring fa­
� �
tigue response (API, 2008) because all nonlinearities related to mooring ~c ojj ¼ c ojj ; jj ¼ 1; …; J (14)
stiffness, seabed friction, drag and damping are taken into account. The
S–N and T-N curves are presented for regular stress/tension ranges. A Gaussian expression (15) is used as the radial basis function in this
However, the mooring response is irregular due to the randomness of work. The shape of the radial basis function is dominated by parameter
metocean loads. Hence, a conversion of tension/stress time histories to q0. The optimal q0 can be estimated by (Rippa, 1999) or by means of a
sensitivity analysis. Values of 0.1 and 0.175 are used to estimate the
response of fatigue damage and tension, respectively.
Table 2 � �2
T-N fatigue curve parameters according to API-RP 2SK
p ojj
� q0
ϕ p ojj ¼ e (15)
T-N FATIGUE CURVE PARAMETERS

COMPONENT a m

STUD CHAIN 1000 3

STUDLESS CHAIN 316 3

5
C. Barrera et al. Ocean Engineering xxx (xxxx) xxx

Table 3
R4S mechanical properties (Vicinay Cadenas brochure., 2018).
Ultimate strength (MPa Yield strength (MPa Reduction of area (% Elongation (% Design temperature Proof loada (kN min.) Break loada (kN
min.) min.) min) min) (� C) min.)
Stud Studless
chain chain

960 700 50 12 20 0.0240aZl 0.0213aZl 0.0304aZl

a
Zl ¼ d2l (44–0.08dl); dl: link diameter. Chain weight per metre: stud chain ¼ 0.0219 d2l ; studless chain ¼ 0.02 d2l .

3. Case study: description and results year long BiMEP metocean hourly time series (Metocean Analysis of
BiMEP for Offshore Design, 2017). Fig. 6 shows the full dataset and the
3.1. Site assessment selected subset. Each database sample contains five variables (wind
speed, wind direction, significant wave height, wave direction and peak
The location selected for this study is the BiMEP test site ( 2.894� , period). The number of database samples (sea states) is 271,728 and
43.563� ), an area offshore the town of Armintza on the Basque Coast they are represented by small black circles in this figure. Based on a
(north of Spain). The site water depth ranges between 50 and 90 m. The preliminary sensitivity analysis, the number of selected samples in the
selected period is between 1985 and 2015. Therefore, a period of 30 subset is 1,000, which are represented by means of large blue circles in
years is taken as the floating structure life-cycle which results in a total the figure. Additionally, 225 samples represented as red squares are
of 271,728 1-h sea states (DNV-OS-J101, 2014). Each sea state contains selected to validate the interpolation technique of multivariate data
data of significant wave height, wave period, wave direction, wind based on the radial basis function (RBF) approach presented in section
speed at 90 m above the sea water level and wind direction. Wind and 2.5. As a result of this selection, a data subset with high variability is
wave roses for this location are presented in Fig. 3. Waves mainly come chosen including operational and extreme sea states. The selected 1,225
from the north-west and wind has three predominant directions: east, sea states are used to generate wind and wave synthetic time series, and
south and west. Wind and wave directions, unless otherwise stated, by means of a wind turbine numerical model to predict the floating
considers the north direction at 0� with positive angles clockwise. structure dynamics. Finally, the RBF is used to transfer all dynamics in
terms of movements, tensions and fatigue damage to the original
database.
3.2. FOWT definition

The DeepCwind semisubmersible platform (Robertson, Jonkman,

3.4. Wind turbine numerical model validation
Wendt, Goupee, Dagher) with a 5 MW wind turbine (Jonkman et al.,
2009) is considered in this work. Station keeping is provided by three
A complete validation of the numerical model was performed in
equal chain mooring lines in catenary configuration separated by angles
(Barrera et al., 2019a) against laboratory tests published in the context
of 120� with a length of 835.5 m. The design weight of each mooring is
of the OC5 Project (Robertson et al., 2017) based on results pertaining to
125.6 kg/m in 200 m water depth. Despite the water depth at BiMEP is
static tension, decay tests and regular and irregular waves with and
lower than 200 m, it has been chosen to maintain the original configu­
without wind. However, a new validation of experimental tests of
ration of the DeepCwind mooring system. However, five other types of
irregular waves with wind is incorporated in this work because the
mooring systems (MS) with a different weight and pretension level are
aerodynamic module is now based on a look-up table of thrust co­
investigated. The most important features of these mooring systems are
efficients and not on the blade element momentum theory (BEMT) used
shown in Table 4. A schematic side view is presented in Fig. 4 for the six
in (Barrera et al., 2019a).
mooring systems evaluated. As shown in Fig. 5, the DeepCwind platform
Results are validated for all wind approaches comparing the move­
has been oriented with the main wind direction (west). The coordinate
ments and the tensions at the fairlead of the mooring lines. However, the
system used in this work is set such that the x-axis is directed from west
focus of this investigation is only on the tension validation. It should be
to east and the y-axis from south to north.
noted that the experimental tests were conducted in a single direction
with waves and wind oriented from the negative to the positive x-axis.
3.3. Metocean conditions The rotor torque was not considered in the experiments. Therefore, the
mooring line responses are equal in M1 and M3. The results of this new
The maximum dissimilarity selection technique is applied to the 30 validation for wind and irregular waves are presented in Fig. 7 through

Fig. 3. Wind and wave roses at the BiMEP test site.

6
C. Barrera et al. Ocean Engineering xxx (xxxx) xxx

Table 4
Mooring system parameters.
MOORING WEIGHT (kg/ PRETENSION LINK DIAMETER EQUIVALENT DIAMETER AXIAL STIFFNESS MINIMUM BREAKING
SYSTEM m) (kN) (mm) (m) (N) STRENGTH

R4S (N)

MS 1 92 811 68 0.1199 5.43Eþ08 5,420,364

MS 2 125.6 1112 79 0.1398 7.45Eþ08 7,200,284
MS 3 200 1770 100 0.1763 1.16Eþ09 10,944,000
MS 4 308 2735 124 0.2186 1.77Eþ09 15,930,028
MS 5 462 4117 152 0.2679 2.62Eþ09 22,363,193
MS 6 634 5675 178 0.3138 3.55Eþ09 28,664,642

Fig. 4. Mooring schematic side view.

theoretical distribution function.

The input of the validated numerical model are wind and wave time
series generated synthetically from the selected 1,000 sea states using
the maximum dissimilarity algorithm as explained in section 3.3. The
numerical model outputs are the time series of the movements (surge,
sway and heave), tensions (M1, M2 and M3) and the nacelle accelera­
tion. These time series are fitted to a generalised extreme value (GEV)
distribution function because the sea states include both operational and
extreme cases. The MPM of the distribution function is selected next as
representative of each time series. The correlation coefficients between
the empirical and GEV distribution for movements, tensions and nacelle
acceleration are presented in Table 5. The correlation coefficients are
obtained as the average of the 1,000 sea states for each variable and type
of mooring. A good correlation is found between the empirical and
theoretical data with values between 0.98 and 1.
The long-term values are obtained through a non-linear interpolation
technique based on the radial basis function (RBF) method previously
introduced in section 2.5. The non-linear interpolation technique re­
quires two development steps. First, the construction of the interpola­
Fig. 5. Coordinate system and location of the different mooring lines (M1, M2, tion function, based on the numerical simulations using the 1,000 sea
M3) in DeepCwind. states selected by the maximum dissimilarity algorithm. Second, the
accuracy verification provided by the non-linear interpolation. This
Fig. 9 considering three approaches: the BEMT, the thrust coefficients verification is conducted through the selection of 225 additional sea
for constant wind speed and the thrust coefficients for turbulent wind states shown in Fig. 6. The response to these additional sea states is
speed. In the coming validation figures, these approaches are called the predicted by means of the non-linear interpolation (RBF) and the nu­
control (dashed blue line), quasi-static (dotted red line) and quasi- merical simulation. Comparisons between the responses of the RBF and
dynamic (dashed-dotted green line), respectively. In general, the the simulation are presented in Fig. 10 through Fig. 13 for mooring
agreement between the different numerical approaches and the exper­ system 3. The graphs represent on the x-axis the response given by the
imental test results is very good. In this work, the quasi-dynamic RBF and on the y-axis the numerical model response (simulation). If the
approach is selected to estimate wind forces for the reasons explained RBF and simulation responses are coincident, they will be represented
in section 2.3(See. Fig. 8). by a point in the bisector of the graph. In contrast, if they are not
coincident, they will be far off the bisector. A linear fit line is built with
3.5. FOWT simulation for fatigue analysis the RBF and the simulation data to analyse its position with respect to
the bisector. In general, RBF presents a good agreement with the
The long-term FOWT analysed variables are platform movements, simulation due to the proximity between the fit line and the bisector
which are the most probable maximum (MPM) of surge, sway and (See. Figs. 11 and 12).
heave; moorings, which are the MPM of tension and fatigue damage for From the 1,000 simulated cases for each type of mooring, 30 years of
each mooring line; and nacelle acceleration, which is the MPM and data are rebuilt using the RBF technique for the 13 variables. Each
percentiles 90%, 95% and 99%. A total of 13 variables are considered. variable can be represented as a function of wind characteristics (speed
The MPM is obtained fitting the empirical data of each variable to a

7
C. Barrera et al. Ocean Engineering xxx (xxxx) xxx

Fig. 6. Selection of a representative metocean database by means of the maximum dissimilarity algorithm.

Fig. 7. Tensions: irregular waves and steady wind (Irregular waves: Hs ¼ 7.1 m, Tp ¼ 12.1 s, γJONSWAP ¼ 2.2 (γJONSWAP is defined as the peak enhancement factor in
the JONSWAP spectrum; Wind: RPM (revolutions per minute) ¼ 12.1, W ¼ 12.91 m/s).

and direction) or wave characteristics (significant height, direction and wind direction (Fig. 3) at the target location and the dependence of
peak period). Fig. 14 shows an example of this reconstruction consid­ thrust force on wind speed.
ering the fatigue damage on M2 of mooring system 3. Mooring line M2 is The influence of the different mooring systems proposed in Table 4
located in the west direction (270� ) and, therefore, the damage will be a on the tension and fatigue damage is displayed in Fig. 15 through
maximum at that direction. As the main direction of wave propagation is Fig. 18. Platform movements determine the tension at the mooring
at 315� , M2 is the mooring line most exposed to wave action. The higher system. Thus, higher platform movement produces lower tension while a
the significant wave height, the more fatigue-accumulated damage. In major restriction to the movement induces higher tension in the mooring
addition, according to Fig. 14, peak periods between 13 s and 21 s system. A wide range of platform movements can be found depending on
generate high fatigue damage. As far as the wind is concerned, there is a the mooring system from movements of up to 14 m in MS 1, with ten­
range of directions between 225� and 315� that generate high fatigue sions of 500 kN to restricted movements to 2.5 m in MS 6, with tensions
damage on M2. Unlike waves, higher wind speeds do not generate the of approximately 6,200 kN.
higher fatigue damage. The reason is that the turbine thrust force is According to the mooring system configuration, a predominant
maximal at 12 m/s according to Fig. 2. A significant fatigue damage is directional sector can be assigned to each mooring line. Consequently,
found starting at wind speed of 10 m/s. However, it should be noted that the higher fatigue damages come from the sector between northwest by
high wind speeds may be associated with high wave heights and north (NWbN) and east (E) in M1 (330� - 90� ), between southwest by
therefore the fatigue damage will be dominated by the significant wave south (SWbS) and northwest by north in M2 (210� - 330� ) and between
height rather than the wind speed. Further results will only be shown as east and southwest by south in M3 (90� - 210� ). As shown in Fig. 3,
a function of wind characteristics due to the important variability of directions coming from NWbN and E are less likely and for this reason

8
C. Barrera et al. Ocean Engineering xxx (xxxx) xxx

Fig. 8. Tensions: irregular waves and dynamic wind (Irregular waves: Hs ¼ 7.1 m, Tp ¼ 12.1 s, γJONSWAP ¼ 2.2; Wind (NPD spectrum): RPM ¼ 12.1, W ¼ 13.05 m/s).

Fig. 9. Tensions: white noise wave and steady wind (White noise: Hs ¼ 10.5 m, Tp ¼ 6 s–26 s, γJONSWAP ¼ 2.2; Wind: RPM ¼ 12.1, W ¼ 12.91 m/s).

Table 5
Correlation coefficients between the empirical and GEV distribution for movements, tensions and acceleration of the 1,000 sea states selected by the maximum
dissimilarity algorithm.
MOORING SYSTEM SURGE SWAY HEAVE TENSION M1 TENSION M2 TENSION M3 ACCELERATION

MS 1 0.9913 0.9912 0.9879 0.9944 0.9935 0.9952 0.9962

MS 2 0.9921 0.9922 0.9880 0.9950 0.9941 0.9958 0.9958
MS 3 0.9926 0.9932 0.9884 0.9956 0.9947 0.9964 0.9959
MS 4 0.9931 0.9942 0.9888 0.9960 0.9955 0.9969 0.9959
MS 5 0.9938 0.9944 0.9892 0.9964 0.9962 0.9970 0.9958
MS 6 0.9936 0.9945 0.9895 0.9967 0.9966 0.9972 0.9965

M1 will have minor fatigue damage. In general, fatigue damage de­ roses, this direction corresponds to the extreme events of this location,
creases with higher mooring weights. By comparing the different with significant wave heights between 7 m and 10 m and wind speeds
mooring systems, the high fatigue damage achieved in MS 1 (92 kg/m) is between 25 m/s and 40 m/s.
remarkable, as is how it decreases with the increase in mooring weight The developed method allows capturing the relevance and contri­
until reaching the lowest damage in MS 6 (634 kg/m). In addition, it bution of directionality to fatigue damage and selection of a suitable
should be noted that although the weight increment reduces the fatigue mooring system to the prevailing metocean conditions in the target
damage, all mooring lines show a high fatigue damage coinciding with location. In consonance with the obtained results, the most appropriate
the sector between 250� and 330� . According to the wind and wave mooring system for the proposed FOWT and location could be MS 4

9
C. Barrera et al. Ocean Engineering xxx (xxxx) xxx

Fig. 10. Comparison of movements (MPM) in mooring system 3: simulation & RBF.

Fig. 11. Comparison of tensions (MPM) in mooring system 3: simulation & RBF.

Fig. 12. Comparison of fatigue damage in mooring system 3: simulation & RBF.

(FLS) stated in the DNVGL–OS–E301 (DNVGL-OS-E301, 2018) is 5 for

mooring lines which are not regularly inspected, in tension-tension
processes and when the accumulated damage (dF) is lower than 0.8. If
dF is higher than 0.8, the safety factor is defined by the expression 5 þ 3
((dF-0.8)/0.2). Conversely, the API (API, 2008) recommends a value of 3
as safety factor. The proposed methodology may lead to a reduction in
the safety factors by considering all the climate variability in the target
location.
In addition to the uncertainty associated with the selection of design
Fig. 13. Comparison of accelerations (MPM and percentiles 95% and 99%) in sea states, there are other elements that may affect an accurate fatigue
mooring system 3: simulation & RBF. definition such as the design fatigue curve selection; the randomness of
wind and wave time series; the mooring line pretension and the
because it drastically reduces the fatigue damage found in MS 1, MS 2 corrosion.
and MS 3. No relevant improvement is provided by MS 5 and MS 6(See.
Figs. 16 and 17). 4.1. Selection of design sea states and fatigue curves

4. Discussion In this section, the proposed methodology, which is based on the

estimation of fatigue damage through the simulation of all sea states in
The reported methodology allows estimation of the dynamics and the FOWT life-cycle, is compared with the fatigue damage provided by
fatigue damage considering all metocean conditions throughout the assuming that the long-term behaviour can be represented by a limited
FOWT life-cycle. In general, the standards (DNVGL-OS-E301, 2018) number of sea states as proposed by standards (DNVGL-OS-E301, 2018)
(Section 6.3) (API, 2008) (Section 6.3) propose the selection of a limited and (API, 2008). In general, these standards recommend setting a
number of long-term representative sea states and perhaps this is the discrete number of sea states between 10 and 50.
reason for the extremely conservative safety factors recommended by To select the discrete sea states, the metocean database is divided
these standards. In particular, the safety factor for the fatigue limit state into four ranges of significant wave height: 0–1.5 m, 1.5–3 m, 3–4.5 m

10
C. Barrera et al. Ocean Engineering xxx (xxxx) xxx

Fig. 14. Reconstruction of fatigue damage (T-N approach) on M2 of mooring system 3 as function of wind (W and β) and waves (Hs, Tp and α).

Fig. 15. Reconstruction of tensions (MPM) on M2 for all mooring systems as a function of wind characteristics.

and higher than 4.5 m. The average wave height of each range is set as range. The percentage of occurrence of each sea state is obtained from
the representative height of each subset. In turn, for each range of sig­ the presentation probability in the metocean database. The selected sea
nificant wave height three ranges of wind speed are set: 0–7 m/s, 7–14 states are presented in Table 6. Fig. 19 shows the distribution of the sea
m/s and higher than 14 m/s. For each range, wind speed is estimated as states selected to reproduce the long-term environmental conditions and
the average of the speeds in each range. Finally, four ranges of peak estimate the mooring fatigue damage.
periods are proposed for each speed range: 0–6 s, 6–9 s, 9–12 s and The fatigue curves used in this work are the T-N curve and the S–N
higher than 12 s. As with the significant wave height and the wind speed, curve corresponding to studless chains proposed by API (API, 2008) and
the average is the representative value for each peak period range. In all, DNV (DNVGL-OS-E301, 2018), respectively. The fatigue damage results
48 sea states result from this discretisation. However, a total of 35 cases using both approaches for each of the six mooring systems (Table 4) with
are set as a consequence of the fact that the metocean database does not three mooring lines considering the matrix of 35 cases and the life-cycle
contain data in some of the previously set ranges. Wave and wind di­ simulation are presented in Fig. 20. According to the obtained results,
rections are obtained as the most likely value corresponding to each the following conclusions can be drawn.

11
C. Barrera et al. Ocean Engineering xxx (xxxx) xxx

Fig. 16. Reconstruction of fatigue damage on M1 for all mooring systems as a function of wind characteristics.

Fig. 17. Reconstruction of fatigue damage on M2 for all mooring systems as a function of wind characteristics.

First, the S–N curve provides estimations of more conservative the percentage difference between the life-cycle method and the discrete
damages than does the T-N. The reason for this is that the S–N curve sea states matrix according to both the S–N and T-N curves. Mean dif­
provides the same fatigue damage regardless of the grade of the chain ferences between 13% and 49% are obtained in the fatigue damage
steel, while the T-N curve takes into account the breaking strength of the estimation between both approaches.
steel and therefore a higher steel quality generates less fatigue damage. Finally, MS 1 and MS 2 present an important fatigue damage. In both
Different investigations have shown the importance of steel quality on mooring systems, M2 is the mooring line that suffers the most fatigue
the evaluation of fatigue damage (P� erez-Mora et al., 2015) (Arredondo damage. The reason is because these mooring systems have large ranges
et al., 2016). of movement and therefore large stress/tension ranges. In contrast, M2
Second, the fatigue damage obtained using the life-cycle method and M3 are the mooring lines with the most important damage in MS 3,
presented in this work is higher than that presented using the discrete MS 4, MS 5 and MS 6 due to the presence of wind in the region between
sea states proposed by the standards. Consequently, simulating the east and west according to Fig. 3. These results show that the mooring
FOWT life-cycle can reduce the uncertainty in the selection of sea states line characteristics can influence the distribution of fatigue damage for
and thus can estimate a more accurate fatigue damage. Table 7 shows all lines included in the mooring system.

12
C. Barrera et al. Ocean Engineering xxx (xxxx) xxx

Fig. 18. Reconstruction of fatigue damage on M3 for all mooring systems as a function of wind characteristics.

statistical uncertainty into the results due to the randomness of waves

Table 6
and wind. The generation of wind and wave time series by wind and
Selection of sea states.
wave spectra uses random phases for transforming spectra into the time
Cases Hs (m) Tp (s) α W (m/s) В % domain. Therefore, multiple time series meet the criteria characterising
1 1 5 0 4 0 3.22 in the spectra. This fact causes an uncertainty in the estimation of fatigue
2 1 8 320 4 320 7.67 damage due to the presence of random phases. To highlight the
3 1 10 310 4 90 17.15 importance of random phases in the fatigue damage assessment, twenty
4 1 13 310 4 90 6.20
5 1 5 40 9.5 90 3.23
wave and wind time series are generated for the same sea state. MS 2 and
6 1 8 310 9.5 90 4.71 the sea state defined by Hs ¼ 3.35 m, Tp ¼ 9.21 s, α ¼ 316.09� , W ¼
7 1 10 310 9.5 90 8.40 12.10 m/s, β ¼ 328.09� are considered in these simulations. Fig. 21
8 1 13 310 9.5 90 3.15 shows the total lifetime of each mooring line if the previous sea state is
9 1 5 290 16 270 0.39
repeated cyclically. As seen, lines M1 and M2 are those most exposed to
10 1 8 290 16 190 0.26
11 1 10.5 300 16 190 0.68 this sea state and, therefore, they have a shorter lifetime until failure. It
12 1 13 310 16 190 0.38 is noteworthy to highlight that with the same sea state, there are lifetime
13 2.1 5.6 10 4.2 30 0.03 differences of up to 4.16 and 3.42 years, with average values of 6.81 and
14 2.1 8 310 4.2 220 0.38 8.09 years considering the mooring lines M1 and M2, respectively.
15 2.1 10.5 10 4.2 30 4.08
16 2.1 14 340 4.2 330 8.06
These results show the importance of randomness on the estimation of
17 2.1 5.5 10 10.5 40 0.27 fatigue damage.
18 2.1 7.5 310 10.5 270 2.27
19 2.1 10.5 310 10.5 270 6.38
4.3. Effect of corrosion on fatigue life
20 2.1 13.5 310 10.5 190 8.03
21 2.1 5.5 290 16.5 270 0.24
22 2.1 7.5 310 16.5 270 1.16 Another important process in mooring fatigue life is corrosion. A
23 2.1 10.5 310 16.5 270 1.51 steel chain permanently in contact with sea water suffers degradation of
24 2.1 13.5 310 16.5 190 1.79
its physical and mechanical characteristics. The standards assess this
25 3.6 11 310 4.5 240 0.02
26 3.6 15 310 4.5 190 1.15
process as a section loss (DNVGL-OS-E301, 2018). The section loss ratio
27 3.6 11 310 10.8 250 0.35 depends on the type of water (polar, temperate or tropical), the type of
28 3.6 14.5 310 10.8 270 3.22 inspection and the part of the mooring involved (bottom, catenary,
29 3.6 8 0 17.5 300 0.16 splash zone). To assess the impact of corrosion on fatigue damage, a
30 3.6 10.5 310 17.5 270 1.33
fatigue analysis over M1 is performed considering MS 2, a corrosion rate
31 3.6 14 310 17.5 270 1.87
32 5.4 17 310 5 190 0.08 of 0.2 mm/year and the sea state proposed in 4.2. Fatigue damage
33 5.4 16 310 11.1 200 0.49 evolution for M1 is presented in Fig. 22. Fatigue damage is calculated for
34 5.4 11 310 21 290 0.15 the initial state and every five elapsed years. According to the obtained
35 5.4 15 310 21 270 1.54
results, the importance of considering corrosion in the fatigue design is
relevant because damage may double the initial values at the end of the
4.2. Influence of the randomness of waves and wind time series on fatigue lifetime of the structure.
damage evaluation
4.4. Influence of mooring line pretension on fatigue and power production
The most accurate method to estimate fatigue damage is by means of
a time domain method (API, 2008). However, this method introduces a The fatigue curves proposed by the standards do not consider the
mean stress effect when the fatigue cycle number is estimated for a

13
C. Barrera et al. Ocean Engineering xxx (xxxx) xxx

Fig. 19. Summary of selected sea states to reproduce the long-term environmental conditions.

Fig. 20. Fatigue damage for different mooring systems and fatigue curves.

Table 7
Percentage difference between the life-cycle method and the sea states matrix according to S–N and T-N curves.
MOORING SYSTEM MS1 MS2 MS3 MS4 MS5 MS6 MEAN VALUES

MOORING LINE M1 M2 M3 M1 M2 M3 M1 M2 M3 M1 M2 M3 M1 M2 M3 M1 M2 M3 M1 M2 M3

T-N (%) 51 11 5 59 12 12 50 13 24 37 14 18 41 8 26 35 23 30 46 13 19
S–N (%) 58 15 2 62 15 12 50 16 16 42 16 17 44 10 31 38 7 36 49 13 19

particular stress range. Goodman, Gerber or Soderberg corrections are are presented in Table 8. Two configurations are considered: the FOWT
commonly used to take into account the mean stress effect on the ma­ static position and a second one considering a sea state. MS 2 and the sea
terial fatigue behaviour (Rodríguez et al., 2005) (Bannantine et al., state proposed in 4.2 are chosen to estimate the correction factors. The
1990). These corrections relate the stress amplitude (Δσ) for a mean steel quality is set to be R4S with an ultimate strength of 960 MPa and a
stress (σm), with the stress that would provide the same fatigue life with yield strength of 700 MPa (Vicinay Cadenas brochure., 2018). The re­
a mean stress equal to zero (Δσ0), by means of the following expression: sults show that the mean stress reduces fatigue life. The Soderberg
� � �n � correction is the most conservative while the Gerber correction presents
σm
Δσ ¼ Δ σ 0 1 (16) values close to 1. However, the results clearly show the importance of
σR
considering the mean stress in the evaluation of fatigue damage.
where n ¼ 1 for the Goodman and Soderberg corrections, n ¼ 2 for the
5. Conclusions
Gerber correction, σR is the yield strength for the Soderberg correction
and σR is the ultimate strength for the Goodman and Gerber corrections.
This paper presents an innovative methodology to assess the fatigue
Mean stress correction factors through the use of the ratio Δσ= Δσ 0

14
C. Barrera et al. Ocean Engineering xxx (xxxx) xxx

Fig. 21. Elapsed time until fatigue failure. MS 2.

Sea state: Hs ¼ 3.35 m, Tp ¼ 9.21 s, α ¼ 316.09� , W ¼ 12.10 m/s, β ¼ 328.09�

Fig. 22. Evolution of fatigue damage due to corrosion. MS 2, M1

Sea state: Hs ¼ 3.35 m, Tp ¼ 9.21 s, α ¼ 316.09� , W ¼ 12.10 m/s, β ¼ 328.09�

life-cycle of FOWT moorings considering the time series of metocean the mooring fatigue damage. This numerical model is first validated
conditions during the full lifetime. It is shown that the present method with experimental tests to demonstrate the model reliability in the
reduces the uncertainty related to the discrete selection of representa­ prediction of dynamics. Finally, the results are propagated for all the sea
tive long-term sea states proposed by the standards. This approach in­ states contained in the metocean database, a total of 271,728 sea states,
tegrates multivariate metocean data selection techniques, FOWT making use of non-linear interpolation techniques based on radial basis
numerical models, fatigue modelling methods and non-linear interpo­ functions. This non-linear interpolation is validated with 225 additional
lation techniques. numerical cases for the following variables: platform movements (surge,
From a 30-year metocean reanalysis hourly dataset, a subset of 1,000 sway and heave), mooring tension, fatigue damage and nacelle accel­
sea states is selected using a maximum dissimilarity algorithm. These sea erations. A correlation coefficient between 0.96 and 0.99 is obtained for
states are simulated using hydrodynamic, aerodynamic and mooring all variables. This approach allows the evaluation of the FOWT dy­
coupled numerical models to obtain the FOWT dynamic response and namics throughout its life-cycle at a reduced computational cost and

15
C. Barrera et al. Ocean Engineering xxx (xxxx) xxx

Table 8 corrosion, mean tension effects or residual stress from chain

Mean stress correction factors. Ratio.Δσ =Δσ0 manufacturing. A sensitivity analysis to some key fatigue parameters is
FOWT STATIC POSITION: M1, M2, M3 also conducted in this work. In particular, the wave and wind random­
ness, corrosion and mean tension effects are evaluated in this research.
MOORING MEAN STRESS MEAN STRESS CORRECTIONS
SYSTEM (MPa) The wave and wind randomness is analysed for an operational sea state
GOODMAN SODERBERG GERBER with the turbine working close to the rated wind speed. Twenty cases are
MS 1 111.6563 0.8837 0.8405 0.9865 proposed to evaluate the time series statistical uncertainty. The lifetime
MS 2 112.5471 0.8828 0.8392 0.9863 until fatigue failure evidences a difference among all cases of around
MS 3 112.6817 0.8826 0.8390 0.9862
four years for the mooring lines with the greatest loading. Corrosion is a
MS 4 113.2385 0.8820 0.8382 0.9861
MS 5 113.4420 0.8818 0.8379 0.9860 complex degradation process of the physical and mechanical properties
MS 6 114.0265 0.8812 0.8371 0.9859 of moorings. Fatigue damage could increase up to a factor of two be­
SEA STATE: M1
tween the initial state and the elapsed life-cycle considering a degra­
dation ratio of 0.2 mm/year according to the results presented. Finally,
MOORING MEAN STRESS MEAN STRESS CORRECTIONS
mooring pretension is another important source of uncertainty in the
SYSTEM (MPa)
GOODMAN SODERBERG GERBER evaluation of fatigue damage. Its effect can be evaluated by means of
MS 1 146.5869 0.8473 0.7906 0.9767 different formulations proposed in the fatigue theory. The different
MS 2 137.5512 0.8567 0.8035 0.9795 approaches show an increase of up to 20% in the stress amplitude used
MS 3 128.5214 0.8661 0.8164 0.9821 to estimate the fatigue damage.
MS 4 123.7932 0.8710 0.8232 0.9834
MS 5 121.0556 0.8739 0.8271 0.9841
MS 6 120.1624 0.8748 0.8283 0.9843 Acknowledgements
SEA STATE: M2
The authors acknowledge financial support from the Spanish Min­
MOORING MEAN STRESS MEAN STRESS CORRECTIONS
SYSTEM (MPa)
istry of Science, Innovation and Universities to PhD candidate Carlos
GOODMAN SODERBERG GERBER Barrera S�anchez through his research training scholarship under Grant
MS 1 153.0608 0.8406 0.7813 0.9746 Agreement No. BES-2014-070381. Raúl Guanche also acknowledges
MS 2 142.6110 0.8514 0.7963 0.9779 financial support from the Ramon y Cajal Program (RYC-2017-23260) of
MS 3 131.8447 0.8627 0.8117 0.9811
the Spanish Ministry of Science, Innovation and Universities. This work
MS 4 125.8337 0.8689 0.8202 0.9828
MS 5 122.4716 0.8724 0.8250 0.9837
is part of the ACOPLE (Ana �lisis del COmportamiento dina �mico de
MS 6 121.2557 0.8737 0.8268 0.9840 PLataformas Eo �licas flotantes para la optimizacio
�n del disen
~ o de aguas
profundas) research project (Grant Agreement No. ENE2017-89716-R
SEA STATE: M3
within the National Programme for Research, Development and Inno­
MOORING MEAN STRESS MEAN STRESS CORRECTIONS
vation Aimed at the Challenges of Society, call 2017. Spanish Ministry of
SYSTEM (MPa)
GOODMAN SODERBERG GERBER Science, Innovation and Universities).
MS 1 71.9353 0.9251 0.8972 0.9944
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