Ocean Engineering 172 (2019) 629–640
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Ocean Engineering
journal homepage: www.elsevier.com/locate/oceaneng
Structural optimization based design of jacket type sub-structures for 10 MW
offshore wind turbines
T
Anand Natarajan∗, Mathias Stolpe, Wilfried Njomo Wandji
Technical University of Denmark, DTU Wind Energy, Frederiksborgvej 399, 4000, Roskilde, Denmark
ARTICLE INFO
ABSTRACT
Keywords:
Wind energy
Structural optimization
Sensitivity analysis
Offshore sub-structures
Fatigue
Welded joints
The design of offshore support structures for wind turbines of 10 MW capacity presents a challenge due to
potential resonance problems from the low rotor speeds during operation. The present work delineates the
optimization based design of jacket type sub-structures at 50 m water depths for a 10 MW turbine by exploring
the frequency constraint space of the sub-structure and confirming feasibility of a cost effective design away
from rotor excitation. The conceptual design is made using a two-level optimization framework. The outer
design problem, i.e. overall jacket dimensioning, is solved by a derivative free optimization method. The inner
problem which consists of member sizing is solved using a robust and efficient Sequential Quadratic
Programming method. The objective of the optimization is innovatively chosen as to minimize the fundamental
natural frequency of the structure, while subject to frequency constraints, tower top displacement constraints
and member ultimate stress constraints. The resulting design is modified during the process of verifying that all
ultimate and fatigue limit states are met using fully coupled aero-hydro-elastic simulations. The final design is a
low mass four-legged jacket that fully complies with offshore structural standard requirements for all design
limit states, while not being affected by rotor excitation, thus being best suited for long operating life.
1. Introduction
Towards reducing the Levelized Cost of Energy (LCOE), the size of
offshore wind turbines has significantly increased to 9.5 MW capacities
in recent years (de Vries, 2017). The design of sub-structures for such
large wind turbines is challenging since the support structure natural
frequencies can be within the ranges of rotor harmonics, which results
in resonances (Von Borstel, 2013). This problem is the result of two
critical outcomes of traditional wind turbine design. Firstly, larger rotors for bigger capacities result in lower rotational speeds (P), due to tip
speed constraints, implying that multiples of rotor speed for a 3-bladed
rotor such as 3P (3 times rotor speed) and 6P (6 times rotor speed) are
lowered and within the range of structural natural frequencies. Secondly, sub-structures such as jackets are intrinsically stiff structures
whose natural frequencies tend to lie within certain ranges and difficult
to alter in the traditional design process (Passon, 2015).
Of the commercially installed offshore wind turbines presently,
most are supported on monopile foundations whereas jackets constitute
only about 5% of the population (Kallehave et al., 2015). This is because a significant portion of the installations have been made at mean
water depths less than 30 m, for which monopiles are well suited. To
meet the European offshore wind targets of 2030 and beyond
∗
(European Environmental Agency, 2009), it is necessary to also install
large number of wind turbines in deeper waters (40 m–50 m), thus requiring more jacket type sub-structures as suited at those depths.
Usually such structures are based on design philosophies spin-off from
the oil and gas industry that has been using jackets for several decades.
However in the case of the wind turbine, a significant part of the mechanical loads on the jacket stem from the wind turbine rotor, which in
turn is wind driven, as differing from an oil rig, where the loading is
primarily wave driven. The wind driven loads are due to turbulence
primarily from the wakes within offshore wind farms and this can result
in significant fatigue damage to jacket welded joints. The structural
design of jackets is usually based on ISO 19902 and complimented by
regional standards and certification body guidelines (Branner et al.,
2013). Due to the large variability in the loads, the lifetime of the jacket
structure can vary greatly from welded joint to welded joint (Conti
et al., 2018) and this is based on a required annual reliability level
determined from an extensive processing of load simulation results
(Branner et al., 2013). The effects of wind turbulence are amplified in
the presence of support structure excitation through the rotor harmonics (Conti et al., 2018). The excitation range of the turbine can be
identified using a Campbell diagram, through which the rotor speed
values and its multiples thereof that contribute most to fatigue damage
Corresponding author.
E-mail addresses: anat@dtu.dk (A. Natarajan), matst@dtu.dk (M. Stolpe).
https://doi.org/10.1016/j.oceaneng.2018.12.023
Received 11 July 2018; Received in revised form 14 November 2018; Accepted 5 December 2018
0029-8018/ © 2018 Elsevier Ltd. All rights reserved.
Ocean Engineering 172 (2019) 629–640
A. Natarajan et al.
may be determined. Further, the fatigue damage is usually direction
dependent, that is based on the wind/wave direction, the fatigue damage incurred by different jacket members varies. The American Petroleum Institute (API) (API, 2005) states a minimum of eight directional load analyses are required for symmetrical platforms with
additional directions required for unsymmetrical structures to ensure
appropriate structural integrity.
In the present study, the design of a jacket at 50 m water depth is
assisted by structural optimization techniques. Optimization methods
have been used by others in literature for offshore structures (Toğan
et al., 2010) and integrated with wind turbine design (Fischer et al.,
2012). Recently various models and methods for structural optimization of jacket support structures have been proposed in a number of
articles, see e.g. (Chew et al., 2015, 2016; Oest et al., 2017, 2018;
Sandal et al., 2018a, 2018b), and (Stolpe and Sandal, 2018). The objective is usually to minimize weight or cost and the design variables
are generally dimensions of the structural members. While weight is a
relevant objective, since the natural frequency of the structure is inversely proportional to the square root of mass, minimization of the
mass can imply that the jacket frequency will not be sufficiently reduced. Since the jacket structure is inherently very stiff, minimization
of mass with constraints on displacements may also increase natural
frequencies. For 10 MW wind turbines, this implies excitation from
rotor harmonics and thereby also increase in the fatigue damage due to
resonance issues. Also the mass of the offshore support structure increases approximately quadratically with wind turbine power capacity
for fixed water depth (Chaviaropoulos et al., 2014), as the wind turbine
grows from conventional 5 MW capacities (Jonkman et al., 2009) to the
10 MW scale. This implies that jacket natural frequencies are constrained by the natural scale law as not being flexible in the vertical
direction. Thus a new approach to optimize jacket structures is needed
that does not seek to minimize mass, but instead seeks to avoid resonance of jacket members while satisfying all other constraints. This
work proposes to minimize the fundamental structural frequency of the
jacket-wind turbine structure and not the mass, while subject to constraints on tower top displacement, natural frequency ranges and
member ultimate and fatigue stress limits. It is proposed that the optimization will allow for reduction in jacket stiffness and not increase in
mass towards the minimization of the first fundamental frequency.
Structural optimization is a multi-disciplinary and active research
field covering many aspects of optimal design of load-carrying mechanical structures and it is today an indispensable tool both for academia and in industrial applications, see e.g. (Bendsøe and Sigmund,
2003). Structural optimization of frame structures has received considerable attention in the literature. Applications include stiffness optimization in (Fredricson et al., 2003) and (Fredricson, 2005), crashworthiness design of cars using transient analysis in (Pedersen, 2003,
2004), and reliability-based optimization in e.g. (Mogami et al., 2006).
Structural optimization of frames has, besides the mentioned literature
on jacket optimization, also received some attention in (offshore) wind
energy. Structural optimization of towers are considered in e.g. (Negm
and Maalawi, 2000), (Yoshida, 2006) and (Uys et al., 2007). Reliability
based design optimization of towers for mass minimization with constraint on yield stress, buckling stress and natural frequencies is presented in (Toğan et al., 2010). Optimization of an offshore support
structure using integrated design of controllers and dimensioning is
presented in (Fischer et al., 2012).
In this article we present an approach to perform simultaneous
overall dimensioning and preliminary structural optimization of jacket
sub-structures for offshore wind turbines. The structure is described by
two types of design variables allowing for both changes to the geometry
as well as the dimensions of the members in the jacket. The choice of
design parametrization thus allows for greater design freedom than
normally seen in structural optimization of jacket structures. The outer
variables model overall dimensions of the structure, such as the jacket
base and top widths and the transition piece height. The outer variables
also model the locations of the X- and K-joints in conventional jacket
structures, i.e. changes to the geometry of the structure are allowed.
The inner design variables represent the dimensions of the members in
the ground structure representing the jacket structure, i.e. diameters
and thickness distribution of the members that constitute the X-braces
and legs. These dimensions are not allowed to become zero since this
would change the topology of the jacket. Although this is potentially
desirable it also causes both theoretical and numerical issues and is left
for future research. The structural response is computed by the finite
element method for linear elasticity based on Timoshenko beam elements, see e.g. (Cook et al., 1989). The conceptual decisions on number
of legs or on number of levels of X-braces are not included in the optimization problem at this point but are instead taken care of by
parametric studies. The (realistic) number of combinations of legs and
X-braces is fairly modest and enumeration is possible. The proposed
problem formulation contains constraints on the member dimensions,
static constraints for example on displacements and local stresses, and
bounds on fundamental frequencies. Although the optimal design problems are far from containing all relevant constraints and dynamic
loads, the approach is sufficiently competent to find realistic structures
for the conceptual design phase in a relatively short period of time and
computational effort.
The outer problem is solved by a derivative free optimization
method, see e.g. (Conn et al., 2008), since analytical sensitivities for the
considered type of variables are difficult to derive and they are also
expensive to estimate using finite differences. Furthermore, the outer
problem has a rather small number of variables and only linear constraints; thus is a good candidate for derivative free methods. The inner
problem formulations are solved using robust and efficient modern
derivative based optimization methods such as primal-dual interior
point methods see e.g. (Nocedal and Wright, 1999) or by Sequential
Quadratic Programming (SQP) methods, see e.g. (Boggs and Tolle,
1995). These methods are known to provide good results and require a
modest number of function evaluations on many problems. These
methods require accurate and efficient sensitivity analysis of the objective and constraint functions. The following sections delineate the
optimization framework conducive for best in class large wind turbine
jacket design and its design verification.
2. Design methodology
The design of jacket type sub-structures for 10 MW wind turbines is
highly complex due to the significant excitation caused by the rotor
rotational harmonics (Von Borstel, 2013). While for other types of substructures such as monopiles in shallow water depths, this problem may
be avoided using active turbine control, these are not applicable for
jackets due to the intrinsically stiff nature of the structure. Offshore
wind turbine jacket structures are designed using a variety of standards
that are applied at different stages of the design process. The mechanical loads on the jacket are determined using a series of aeroelastic load
simulations as obtained from a model of the offshore turbine system
with wind/wave inputs (Natarajan, 2014). These load simulations
follow the IEC 61400-3 (International, 2009) design load cases and the
IEC 61400-1 (International, 2005) wind conditions. The loads on the
wind turbine structures including the jacket are computed as a time
series as resulting from blade aerodynamics and the Morison equation
(Morison et al., 1950) for wave loads. These load time series from
various wind/wave conditions are processed to determine the 50-year
extreme load at different jacket joints combined with the partial safety
factors for the loads and material as given in the IEC 61400-1. The
stress at each member and joint of the jacket are determined using
appropriate structural analysis, considering also the required stress
concentration factors. The joints are typically divided into Y-, K- and Xjoints. Y-joints are used for connections between the braces and legs of
the structure, but at the top or at the bottom, i.e. with no braces further
above or below, respectively. K-joints are similar to Y-joints but connect
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A. Natarajan et al.
the legs to in-between successive braces. The X-joints constitute the
center of braces. The lifetime fatigue damage at each location of the
jacket structure is calculated by aggregation of the individual fatigue
damage due to each stress cycle, computed from load simulations and
weighted with the probability of occurrence of the generating wind/
wave state. The aggregation is made using the Palmgren Miner's rule
(Miner, 1945).
The above design procedure is time consuming since several turbulent wind time series, each of 10 min duration have to be simulated
per mean wind speed bin to achieve statistically acceptable fatigue and
ultimate design load levels. Moreover, changes in the jacket design
made to satisfy stress or life criteria can result in further load simulations to re-evaluate the design loads, since the natural frequencies or
mode shapes of the sub-structure coupled to the wind turbine may have
changed. To be computationally efficient, the design process followed
herein is to assume that the tower top or rotor design loads do not
change, when the jacket design is altered in the optimization process
and the design of the jacket is made uncoupled with the wind turbine.
In literature, it has been shown that the uncoupled formulation results
in a more conservative jacket design (Haselbach et al., 2013). The
specific steps followed are as:
where ˜ 1 (w) is the first fundamental frequency of the structure defined
by the outer design variables w, A is a given matrix and b is a given
vector, both of appropriate dimensions. The outer problem is a linearly
constrained nonlinear problem. The objective function in (1) is obtained by attempting to solve an inner optimization problem which is a
structural sizing problem. Note that the inner optimization problem can
be in-feasible if the outer design variables are inappropriately chosen.
The design variables in the inner problem are also continuous and
model member dimensions, i.e. wall thickness distribution and member
ni
diameters. Let v
be a vector of inner design variables where ni is
the number of inner design variables. For each member in the jacket
ground structure we assign two design variables. One variable describes
the wall thickness of the member while the other describes the inner
radius of the cross-section. The members are assumed to have constant
cross-sections over the length. The inner problem is formulated as
˜ 12 (w)= minimize
n
v
subject to
•
j (ul (v),
2.1. Problem formulations
The traditional iterative methods to design jackets at 10 MW would
either yield very heavy structures to survive the fatigue damage for 25
years lifetime or would have fatigue failure prematurely (Von Borstel,
2013). Further fatigue damage constraints on structural members
during the optimization are challenging to implement or need simplified evaluations using frequency domain fatigue computations, which
do not capture all the load combinations shown in time domain simulations (Schafhirt and Muskulus, 2014). Unlike these common structural optimization approaches, which have mass minimization as the
objective, the objective for the present problem is to minimize the first
fundamental frequency of the structure, while keeping further few
natural frequencies within defined bounds as a means to maintain required fatigue life.
The outer optimization problem determines the overall dimensions
of a sub-structure where the connectivity is assumed to be given. The
no
continuous outer design variables are denoted by w
where no is
the number of outer variables. They model, among other things, the
base width, the top width, the positions of joints, such as X- and K-joints
in classical jacket structures. The feasible set of the outer problem is
assumed to be described only by linear constraints. With these kind of
constraints, it is possible to model box bounds on the variables. The
outer problem is modelled as
w
no
b
x k)
¯/
k = 2, …, kmax
l S
M
l
D, j
J, k
K
(2)
2.2. Structural analysis
The structural analysis is based on an in-house finite element program for linear elasticity, see e.g. (Cook et al., 1989). The code uses
tubular beam elements based on Timoshenko beam theory for the legs,
braces, and tower. The code is specifically designed with structural
optimization in mind, i.e. it is designed to efficiently compute both
relevant functions (displacements, stresses, eigenfrequencies, etc.) and
their derivatives with respect to the design variables. The stiffness
d × d and the mass matrix become non linear mappings
matrix Kw (v)
in the design variables. Here d is the number of (non fixed) degrees of
freedom. Throughout this article we do not allow that members vanish
from the original structure, i.e. the inner design variables are not allowed to become zero. Thus, both the stiffness and mass matrices are
positive definite for all inner design variables satisfying the box constraints. The notations Kw (v) and M w (v) are used to emphasize that the
stiffness and mass matrices are parametrized by the outer design
˜ 12 (w)
subject to Aw
W
¯ k2
The set W contains linear constraints, such as bounds, involving
only the design variables. The generalized displacements ul (v) are the
solution to the static equilibrium equations (cf. below). The given
parameters k2 < ¯ k2 are lower and upper bounds on the kth funda¯
mental eigenfrequency. The inner problem (2) contains the possibility
to include local displacement constraints, i.e. constraints specific to a
location or degree of freedom in the structure. Each row in the user
supplied matrix Pl can be used to define a local displacement constraint.
It is assumed that Pl is independent of the inner and outer design
variables. The bounds on the displacement constraints are described by
the (design independent) vectors p̄l .
The inner problem (2) also contains the possibility to include local
strength constraints based on the von Mises stress. The allowed limit on
the von Mises stress is denoted by ¯ . It is assumed that the same material is used for the entire structure and that the material partial safety
factor M 1 is prescribed. The index set J contains the indices of the
finite elements for which the von Mises stress constraints are included
in the formulation and the set K contains a list of locations within the
finite elements.
Note that the inner problem (2) is a non-convex optimization problem and hence it cannot be guaranteed that the method finds a
minimum. Furthermore, due to the non-convexity it cannot be assured
that the problem actually is infeasible in the situations that the method
deems it numerically infeasible. The overall structural optimization
process to design the jacket is summarized in the flow chart given in
Fig. 1.
ultimate stress constraints on all jacket members and a damage
equivalent stress constraint based on S-N curve at the extreme load
level.
Fully coupled aeroelastic simulations with normal turbulence and
normal waves is run on the optimized jacket with 10-min wind/
wave time series input in mean wind speed bins between 5 m/s 25 m/s and with changing rotor directions over 360° in steps of
22.5°.
Based on the results of the load simulations done, the fatigue lifetime and ultimate limit states are verified. The jacket members are
re-sized at locations where the limit state assessments indicate insufficient fatigue life or insufficient margins in ultimate limit state.
minimize
2
2
k (v)
¯k
Pl ul (v) p¯ l
v
• The tower top design driving 50-year extreme thrust and side force
are applied as static loads obtained from (Bak et al., 2013).
• The optimization is run with tower top displacement constraints,
•
2
1 (v)
i
(1)
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A. Natarajan et al.
a
I
Kw
j (v) = aj (v) K j + I j (v) K j +
Kaj ,
(6)
j (v) aj (v) K j .
KIj ,
and K j are independent of the inner design
where the matrices
variables. Here, aj (v) is the area, j (v) is the shear correction factor, and
I j (v) is the second moment of area of the j-th design element, respectively. The matrices Kaj , KIj , and K j are constructed by an assembly
process over all finite elements which are related to the j-th design
element. Similarly, the mass matrix M (v) is on the form
Mw
j (v)
Mw (v) = M0 +
(7)
j
where M0 is the mass matrix (in global coordinates) for the part of the
structure which is independent of the design variables. The element
mass matrices Mw
j (v) can be written as
a
I
Mw
j (v) = aj (v) M j + I j (v) M j .
(8)
where the matrices Maj and MIj are independent of the design variables.
It follows that analytical sensitivities of the mass and stiffness matrices
can be efficiently and accurately computed.
2.3. Design sensitivity analysis
This section contains a presentation of the required sensitivity
analysis for the considered objective and constraint functions used in
the inner problem (2). Design sensitivity analysis is extensively covered
in e.g. the monographs (Choi and Kim, 2005a, 2005b). Throughout this
section the sub index l is dropped for notational simplicity.
2.3.1. Static response
Let c (v) denote the static constraint function c (v) = pT u (v) where
d is a given (design independent) vector. The partial derivative
p
with respect to the i-the design variable of c (v) is given by
c (v)
vi
for k = 1, …, m,
c (v)
=
vi
0
2
2
(3)
Kw (v)
Kw
j (v)
= K0 +
j
+ pT K
K (v)
u (v) +
vi
T
f (v)
vi
N
Kw
j (v)
vi
j =1
with
Kw
j (v)
2
d.
Due to the assumptions stated above, the stiffness matrix
on the form
T
Kw (v)
=
vi
(4)
Kw (v)
K (v)
K 1(v) f (v)
vi
where u (v) is the unique solution to the static equilibrium equation (3).
Note that this sensitivity analysis can be extended to von Mises stress
constraints. The details are omitted.
The analysis outlined above requires computations of the sensitivities of the mass and stiffness matrices, respectively. Due to the partitioning of these (design) element stiffness and mass matrices outlined in
(5)–(6) and (7)–(8) this becomes a fairly uncomplicated operation, e.g.
where the eigenvectors are assumed to be M-orthonormal, i.e.,
zTi M w zk = ik . The frequencies are assumed to be ordered such that
2
1
f (v)
=
vi
1 (v) f (v) ·
vi
+ pT K 1(v)
where p is normally referred to as a pseudo-load. Then the sensitivities
are given by
where the index set S represents the static loads. The static loads are
modelled as design dependent, i.e., fl is a function of the design variables. This is used to model self-weight. The first m fundamental frequencies k2 are determined from the generalized eigenvalue problem
2
w
k M (v) zk ,
K 1(v)
f (v)
vi
K (v) = p,
variables w. For example, two different choices of outer design variables may lead to different number of degrees of freedom for the inner
optimal design problem. The stiffness and mass matrices are reduced in
size to deal with the Dirichlet boundary conditions. The equilibrium
equations for the static load case l are given by
Kw (v) zk =
= pT
Let η be the solution to the adjoint problem
Fig. 1. Overview of the structural optimization process to develop the jacket for
the 10 MW turbine to satisfy ultimate limit states (ULS) and fatigue limit states
(FLS).
S,
u
vi
pT K 1(v)
=
Kw (v) ul = fl (v) for all l
= pT
vi
is
=
aj (v)
vi
Kaj +
I j (v)
vi
KIj +
( j (v) aj (v))
vi
K j.
For circular cross-sections, areas, inertia, and shear correction factors are given analytically and computing sensitivities of element
stiffness matrices is possible. Similar observations also hold for the
element mass matrices.
(5)
where K 0 is the stiffness matrix (in global coordinates) for the part of
the structure which is independent of the design variables, e.g. the
tower and the model of the rotor-nacelle assembly. The stiffness matrices Kw
j (v) can be written as
2.3.2. Eigenfrequencies
Sensitivities of fundamental eigenfrequencies
if they are multiple. In this case the function
632
2
k (v ) are problematic
2
k (v) becomes non
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A. Natarajan et al.
differentiable, see e.g. (Seyranian et al., 1994) and the references
therein. It is therefore assumed, and the numerical experiments support
this assumption, that the first few fundamental frequencies are distinct.
The reason for this is the break of symmetry due to the inertias in the
model of the rotor nacelle assembly. The sensitivity of k2 with respect
to the design variable vi is in the case of a simple eigenfrequency given
by (see e.g. (Fox and Kapoor, 1968) and (Seyranian et al., 1994))
2
k
vi
= zTk
K (v)
vi
2
k
Table 1
DTU 10 MW reference turbine properties (data from (Bak et al., 2013)).
M (v)
zk
vi
where, again, the eigenvectors are assumed to be M-orthonormal, i.e.,
zTi Mzk = ik .
3. Implementation and parameters
The two-node finite elements are based on Timoshenko beam theory
and use standard shape functions and the integration of the finite element stiffness, mass matrices and loads are computed using Gauss
quadrature. The shear correction factors for the cylindrical Timoshenko
beams are computed using the results in (Hutchinson, 2001). Marine
growth and corrosion protection allowance have not been included in
the numerical experiments. Secondary structures such as boat landings,
J-tubes, sacrificial anodes, and ladders are not included in the models.
The jacket structures are assumed to be fully clamped at the sea-bed,
hence modelling of the actual soil-foundation coupling is not taken into
account.
The airgap between the mean water level and the tower bottom is
26 m. The model of the tower used in the numerical experiments closely
follows the model of the reference presented in (Bak et al., 2013). The
tower is partitioned into ten prismatic segments with circular crosssections each with constant wall thickness and diameter. The diameter
of a segment in the prismatic model is the mean of the end diameters in
the corresponding conical segment. The rotor-nacelle assembly is
modelled as a lumped mass of 676,723 kg at the tower top (at height
115.63 m) together with a moment of inertia about the x-axis of
1.66⋅108 kgm2, a moment of inertia about the y-axis of 1.27⋅108 kgm2,
and a moment of inertia about the z-axis of 1.27⋅108 kgm2. The highest
horizontal thrust force at tower top (located at 115.63 m) is set to
4800 kN with a simultaneous fore-aft bending moment of 18000 kNm
(Bak et al., 2013). Both include the partial safety factor for ultimate
loads. A concentrated vertical load from the tower top mass is also
included in the model. These loads are assumed to be independent of all
design variables. The Young's modulus and density for steel are
throughout assumed to be E = 210 GPa, and ρ = 7850.0 kg/m3 for the
jacket structures. For the tower the density is adjusted following (Bak
et al., 2013) to 8500.0 kg/m3 to account for the mass of secondary
structures. The Poisson's ratio is set to ν = 0.3.
The outer optimization problem Eqn. (1) is solved by the Generalized Pattern Search (GPS) method implemented in the MATLAB routine
patternsearch, see e.g. (Audet and Dennis, 2003) and (Abramson et al.,
2009). The options for the GPS method are largely set to default values.
We however restrict the number of GPS iterations to 1000 and the
number of function evaluations, i.e. the number of inner problems
solved, to 500. A time-limit of 24 h is also imposed.
The nonlinear inner optimization problems, Eqn. (2) are solved by
the robust and efficient SQP method implemented in SNOPT version 7.1
(Gill et al., 2002). The solver options are to a large extent set to default
values. The optimality and feasibility tolerances are increased compared to the default values to reduce the number of objective and
constraint function and gradient evaluations. They are increased from
10−6 to 10−3 and 10−5, respectively. Manufacturing or design catalog
requirements, corrosion allowance, effects of icing or add-ons such as
appurtenances are not considered in this process.
The structural analysis routines used in the optimization method
have been validated by exporting the jacket structure and tower,
Parameter
Value
Control
Cut-in, rated, cut-out wind speed
Rated power
Rotor, hub diameter
Hub height
Drivetrain
Minimum, maximum rotor speed
Maximum generator speed
Gearbox ratio
Maximum tip speed
Hub overhang
Shaft tilt, coning angle
Rotor mass including hub
Nacelle mass
Tower mass
Variable speed Collective pitch
5 m/s, 11.4 m/s, 25 m/s
10 MW
178.3 m, 5.6 m
119.0 m
Medium speed, Multiple-stage Gearbox
6.0 rpm, 9.6 rpm
480.0 rpm
50
90.0 m/s
7.1 m
5.0, −2.5
227,962 kg
446,036 kg
628,442 kg
modelled as a frame structure, to the commercial finite element software ABAQUS. ABAQUS was used to validate the implementation by
comparing global responses of the jacket and tower such as the lowest
eigen frequencies and tower top displacements for some of the applied
loads. Validation has also been done in a similar way for local stresses.
The analytical design sensitivity analyses implemented in the optimization framework have been compared to the results from finite differences and found to give accurate results.
4. Wind turbine setup
The DTU 10 MW reference wind turbine (Bak et al., 2013) is utilized
in the computation of the design loads on the jacket. The system level
properties of the DTU 10 MW turbine are given in Table 1.
The initial reference for the jacket of this 10 MW wind turbine is
taken from the INNWIND. EU project deliverable D4.31 (Von Borstel,
2013), which satisfies the ultimate load conditions, but does not fulfill
the fatigue life requirements of 25 years. This is due to the significant
3P excitation of the support structure from the rotor. The fully coupled
design load simulations on the turbine with jacket are done using the
aeroelastic software HAWC2 (Larsen and Hansen, 2012). The software
HAWC2 utilizes a multibody formulation, which couples different
elastic bodies together using Timoshenko beam finite elements. The
Mann wind turbulence model (Mann, 1994) is used in the load case
simulation to represent the input wind turbulence over the rotor.
Random Gaussian 10-min turbulent wind realizations are used as input
to simulate normal operation over 11 equally spaced mean wind speed
bins (5 m/s-25 m/s) in 16 equally spaced directions, aligned with the
rotor in each case and with 10 yaw misalignment. The input ocean
waves are assumed to be aligned with the wind. Random wave kinematics are utilized according to the linear Airy model with Wheeler
stretching and nonlinear wave components are not considered. The
hydrodynamic forces on the jacket are computed using the Morison
equation (Morison et al., 1950).
5. Conceptual design of the jacket
The description of the reference turbine (Bak et al., 2013) lists the
1P frequency range to be [0.1,0.16] Hz The 3P range is thus
[0.3,0.48] Hz and the 6P range is [0.6,0.96] Hz. Further the wave
excitation frequency is between the frequencies [0.1,0.2] Hz. The
natural frequencies of the support structure (including jacket) should
thereby be in the interval [0.2,0.28], or in the interval [0.54,0.58], or
above 1.0 Hz. For the present design case we have targeted to place the
two first fundamental frequencies in the first interval, the higher frequencies in the second interval or above 1.0 Hz.
The outer variables are bounded by the values listed in Table 2. The
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Ocean Engineering 172 (2019) 629–640
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the braces are also of significant cross-sectional area, thus allowing for
thick welded joints at the intersections. Since the stress hot spots are at
the welded joints, this optimized design enables the jacket to withstand
fatigue and ultimate design stress limit. While the optimizer had fixed
constraints on fatigue and ultimate stresses using fixed tower top loads
and assumed max load cycles in fatigue, there has been no coupled
iterations with an aeroelastic software to verify that the fatigue and
ultimate loads are indeed within the design margins. The next sections
verify the found jacket design using full coupled aeroelastic simulations.
Table 2
Bounds on the outer design variables.
Description
Lower bound [m]
Upper bound [m]
Base width
Top width
Transition jacket height
16
8
8
32
16
12
Table 3
Bounds on the inner design variables.
Description
Lower bound [mm]
Upper bound [mm]
Legs wall thickness
Braces etc. wall thickness
Legs inner radius
Braces etc. inner radius
40
20
250
250
120
100
1500
500
6. Verification of the design integrity of the optimized jacket
The optimized jacket needs to be assessed using fully coupled
aeroelastic simulations under normal wind turbulence and normal sea
state conditions to confirm that its fatigue lifetime is satisfactory and
that the ultimate loads during operation do not exceed the standstill
extreme loads used as constraints in the optimization process. The IEC
61400-3 (International, 2005) recommends that load assessment is
carried out considering variable wind and wave directions. The primary
verification concerns the ultimate limit state, where the extreme
stresses are checked against the allowable stress, the buckling limit
state, where the compressive stress in the slender members are seen not
to produce any instability and the fatigue limit state, where the accumulated damage during the structure lifetime should not lead to fatigue
failure. The ultimate and fatigue limit states are based on stresses whose
magnitudes are magnified at the welded joints compared to their
nominal values. Thus, it is sufficient to carry out the verification at the
welded joints, where failures generally occur for beam with constant
section and unique material. In this study, the verification will be
carried out at all the lowest members of the jacket and at the members
near the transition piece, as these are the most vulnerable locations,
that is at the fixed boundaries.
(relative) positions of the X- and K-joints are also included in the list of
outer design variables. The locations of the K-joint are allowed to move
over the length of the leg and are only constrained not to coincide. The
locations of the X-joints are allowed to move in the middle of the plane
described by the two incident legs. Again, the included constraints
prevent two joints to coincide, i.e. joints cannot vanish from the final
structure. The outer problem has in total 11 continuous variables.
The inner variables are bounded by the values listed in Table 3.
Furthermore, we include a number of constraints linking the variables
in the various members. The members of the X-braces all have the same
dimensions in every section and the same applied for the leg members.
The inner problem has, before reductions through variable linking, 226
design variables.
The tower top is not allowed to displace more than 3.2 m in any
direction for any load. We have also included local static stress constraints. The von Mises stresses are constrained not to exceed 355 MPa
anywhere in the jacket sub-structure.
6.1. Environmental conditions and load assessment
The site specific metocean conditions assumed in the load simulations during wind turbine operation are presented in Table 6.
Several loads simulations of Design Load Case (DLC) 1.2 are made
using HAWC2 with the input wind/wave conditions shown in Table 6.
It is particularly important that the load simulations include the rotor
facing different directions over a 360 polar, so as to account for a
variation in dynamic loading over the different faces of the jacket.
Random Gaussian 10-min turbulent wind realizations are used as input
to simulate normal operation over 11 mean wind speed bins spanning
from 5 m/s - 25 m/s in 16 equally spaced directions collinear with the
rotor in each case and with +/-10-degree yaw misalignment. Thus each
mean wind speed has 3 yaw mis-alignments and 16 wind directions to
simulate with one random wind turbulence seed. The input waves are
assumed to be aligned with the wind. Random wave kinematics are
computed according to the linear Airy model with Wheeler stretching.
The hydrodynamic forces are calculated based on the Morison equation.
The structure is assumed to be embedded (all degrees of freedom are
fully restrained) at the seabed, implying the soil is not considered in the
simulation. This set of conditions provides 11 × 1 × 3 × 16 = 528
load time series of 10-min duration. This provides a high fidelity set of
simulations to verify the limit states on all the members of the jacket.
The internal loads are observed over several members of the jacket,
which are typically the most vulnerable (Von Borstel, 2013). Each
jacket leg is numbered: Leg 1, Leg 2, Leg 3 or Leg 4. Similarly, each side
is named from the boundary-leg number: Side 1-2, Side 2-3, Side 3-4, or
Side 4-1. On a side, the connections of the lowest braces to the legs at
the floor just above the mud brace are also numbered 1 or 2 according
to their position left or right, respectively. Thus, the hotspot appellation
is formed from the side number and its position: 1–2.1, 1–2.2, or 2–3.1
for example. Based on the load time series computed from the hydro-
5.1. Optimization results
The results from the optimization process are shown in Table 4 and
Table 5, from which it can be seen that the resulting jacket has a low
mass of 654 tons, where most of the mass rests with the braces (see
Table 4).
As can be seen from Table 4, the optimization moves towards enabling similar diameters and wall thicknesses for the braces and legs.
Even though the legs are wider and thicker as in conventional jackets,
Table 4
Overview of the geometry and masses of the four legged jackets depicted in
Fig. 2.
Description
Unit
Dimension
Base width
Top width
Transition jacket height
Jacket legs inner radius
Jacket legs max wall thickness
Jacket legs min wall thickness
Level 1 X-brace inner radius
Level 1 X-brace wall thickness
Level 2 X-brace inner radius
Level 2 X-brace wall thickness
Level 3 X-brace inner radius
Level 3 X-brace wall thickness
Level 4 X-brace inner radius
Level 4 X-brace wall thickness
Total jacket mass
Total legs mass
Total X-braces mass
[m]
[m]
[m]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[t]
[t]
[t]
12.0
7.25
10
297
74
45
250
20
250
21
250
100
250
20
654
197
457
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Ocean Engineering 172 (2019) 629–640
A. Natarajan et al.
Table 5
The first five natural frequencies for the entire system for the four legged-jacket with four levels of X-braces shown in Table 4. The mode shapes are shown in Fig. 3.
Mode/
1st Bending
1st Bending
1st Torsion
2nd Bending
2nd Bending
Frequency [Hz]
Side-side
0.2381
Fore-aft
0.2391
0.5726
Side-side
1.24
Fore-aft
1.315
Fig. 2. Overview of the four-legged jacket obtained by structural optimization.
servo-aero-elastic simulation, nominal axial stresses are calculated at a
given location by taking the absolute maximum over eight angular
positions at a joint as depicted in Fig. 4 and given by:
x(
)=
Mz r
Fx
+
sin( )
A
I
My r
I
cos( )
Fig. 3. Overview of the bending modes of the optimized jacket with the 10 MW
turbine.
concentration factor for in plane moment (SCFMIP ), and the stress
concentration factor for out of plane moment (SCFMOP ) are estimated
from DNV RP C203 (Det Norske Veritas (DNV), 2013). The characteristic stresses around the weld circumference are computed using
(9)
Where Fx , Mz , and My are respectively the axial force in the member (x
direction) and the two bending moments in the member about the two
axes perpendicular to the axial direction. The geometry parameters, r,
A, and I are the outer radius, the cross section area, and the cross
section second moment respectively, while θ is the circumferential
coordinate measured from axis y, along the saddle direction at the joint.
At welded joints, the nominal axial stress x ( ) , that is the stress calculated at a typical section of the beam is magnified using stress concentration factors (SCFs) to obtain the characteristic axial stress k . For
each side (brace side and chord side) of the welds, the stress concentration factor at the saddle for axial load (SCFAS ), the stress concentration factor at the crown for axial load (SCFAC ), the stress
k(
) = SCFAS
2
SCFMOP
+ SCFAC 1
My
I /r
2
N
M
+ SCFMIP x sin
A
I /r
cos
(10)
where 0 < < /2 is the positive value of the angle between the considered point line and x-axis.
6.2. Verification of ultimate load limit states
The design stresses are checked against the steel material yield
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Table 6
Metocean conditions considered in the simulation process.
Table 8
Comparison of slenderness ratios of braces and legs.
Mean
wind
speed (m/
s)
Turbulence
intensity (%)
Significant
wave height
(m/s)
Peak spectral
frequency (s)
Expected
annual
frequency (hr/
yr)
5
7
9
11
13
15
17
19
21
23
25
18.95
16.75
15.60
14.90
14.40
14.05
13.75
13.50
13.35
13.20
13.00
1.140
1.245
1.395
1.590
1.805
2.050
2.330
2.615
2.925
3.255
3.600
5.820
5.715
5.705
5.810
5.975
6.220
6.540
6.850
7.195
7.600
7.950
933.75
1087.30
1129.05
1106.75
1006.40
820.15
633.00
418.65
312.70
209.90
148.96
a
fk
+
1
[(
fa
m1/(1
a / fE )
)2 + (
m2 /(1
a / fE )
1
)2 ] 2
1.0
(11)
Member
Threshold
Slenderness
Leg level 1
Leg level 2
Leg level 3
Leg level 4
Brace level
Brace level
Brace level
Brace level
Brace level
Brace level
Brace level
Brace level
Mud brace
1478.87
1478.87
1478.87
1478.87
1478.87
1478.87
1478.87
1478.87
1478.87
1478.87
1478.87
1478.87
1478.87
1646.30
936.32
1047.00
1030.10
1684.60
1459.20
1192.20
1039.60
792.57
686.62
930.67
804.18
4254.10
1
1
2
2
3
3
4
4
lower
upper
lower
upper
lower
upper
lower
upper
DNV-RP-C202 (Det Norske Veritas (DNV), 2011) recommends as stability requirement.
to column buckling. As the members are subjected to axial force combined with bending moments, Eulers formula for buckling is not sufficient to assess the stability. The buckling check is required (Det Norske
kL
E
Veritas (DNV), 2011) if (i )c2 > 2.5 f , where k is the effective length
c
y
factor, Lc is the member length, ic = I / A is the radius of gyration, and
E is the material elastic modulus. Table 8 presents the slenderness of all
members and compares with the threshold. It is seen that three members require further assessment: lowest part of leg 1 and X-brace portions and the mud-braces.where a is the design compression stress, m1
is the design bending stress about the given axis, fa is the design local
buckling strength, fk is the design column buckling strength, and fE is
the Euler buckling strength. Details of the procedure can be found in
(Det Norske Veritas (DNV), 2011). Table 9 presents the utilization ratios for each of the three members whose slenderness ratios are more
than the limit in Table 8. The buckling utilization ratios are shown in
Table 9 to be significantly lower than one, indicating the survival of the
members with respect to the column buckling limit state.
Based on Tables 7–9, it is concluded that the optimized jacket satisfies the necessary ultimate load and buckling constraints.
Fig. 4. Depiction of the eight circumferential locations at a welded joint for
stress computation (from (Det Norske Veritas (DNV), 2013)).
stress. At the ultimate limit state, the associated utilization ratio
uLS = m c l v / fy , where the partial safety factors, taken from
(International, 2005): m = 1.30 , c = 1.10 are associated with the material properties and to the components consequence of failure, respectively. IEC 61400-1 ed.3 (International, 2005) recommends that the
partial safety factor related to normal conditions is l = 1.35 corresponding to the mean of the extreme loads. However, since all extreme
load cases have not been run here, l is taken to be 1.50 and corresponding to the absolute maximum of the extreme loads. The constant
fy = 355 MPa is the steel yield limit. The ultimate stress over all jacket
joints is determined using the load simulation results with load extremes occurring in different directions and contemporaneous loads
corresponding to the extreme value. The peak ultimate stresses are seen
to occur at the lowest braces pf the jacket. The distribution of the utilization ratio for braces is shown in Table 7 wherein the utilization ratio
for braces are all much lower than one. A failure in ultimate strength
limit state would occur only if a magnification of extreme loads by a
factor of 3 occurs over what is presently seen. This would be unrealistic
even if all IEC load case situations had been simulated.
In Table 7, the node ID corresponds to leg number - brace connecting to leg number. node number on that brace. This 1–2.2 refers to
the second node on the brace connecting leg 1 with leg 2.
The optimized jacket members are furthermore verified with respect
6.3. Verification of fatigue limit state and predicted lifetime
In order to assess the fatigue limit state, the design axial stress, d , is
computed from the simulated stress amplitudes at all mean wind speed
bins across 16 equally distributed wind directions as explained in
Section 6.1. Life computation of the structural members are based on
the Palmgren Miner's rule using the design S-N curve, i.e. the T-curve in
sea water with cathodic protection as per DNV RP C203 (Det Norske
Veritas (DNV), 2013) and using the recommended Fatigue Design
Factor (FDF) (Det Norske Veritas (DNV), 2013) of 3 as recommended
for welded joints below the water level with moderate consequences of
failure.
The critical hotspots across the optimized jacket are examined,
which are the welded joints between the legs and braces or between
two braces. The damage equivalent stress can be computed using (9)
and (12), where nj is the number of design stress cycles accumulated
Table 7
ULS utilization factors at the lowest braces of the jacket.
Node ID
1–2.1
1–2.2
2–3.1
2–3.2
3–4.1
3–4.2
4–1.1
4–1.2
Utilization ratio
0.325
0.275
0.322
0.279
0.276
0.325
0.321
0.279
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If the four joints shown in Table 10 times four sides along with the
leg connection to the joints are modified to have sufficient fatigue
lifetime, then the jacket can be confirmed to satisfy the fatigue limit
state fully. The fully coupled load simulations described in Section 6.1
can be re-used to determine if the desired fatigue lifetime is met, provided the re-design does not alter the natural frequencies of the support
structure. The re-design of the four failing joints needs to be done based
on two aspects: 1) the influence of the stress concentration factors and
2) the stress at the welded joints.
With respect to the SCFs, it can be readily see from Eq. (13) that
increase in τ or γ would directly increase the SCF in the axial direction.
However the relationship of the SCF with β is less straight forward,
since it appears in interaction with other variables in those equations.
Thus it can be concluded that to lower the SCFs at a welded joint, τ or γ
would need to be as low as feasible and the influence of β needs to be
investigated. Fig. 5 shows the variation in the SCF at a tubular welded
joint in the different loading directions. It can be readily seen that β
should be high in order to reduce the SCF. Thus the diameter of the
brace should be as near to the diameter of the connecting leg as possible
at the welded joint.
Therefore increase in β is beneficial to reduce stresses, while Eq.
(13) shows that to lower SCF, τ or γ needs to be reduced, which implies
the thickness of the brace to be small in comparison to the thickness of
the leg and the thickness of the leg is small in comparison to its diameter. However a higher β values not only reduces the SCF, but also
allows for a low stress state, since it encourages a larger brace diameter.
Thus the diameter of the brace at the 4 joints with insufficient life in
Table 10 are first increased to a level that is just short of the diameter of
the leg. If the fatigue life of these joints is still less than required (due to
high stress cycles), then the thickness of the brace is moderately increased (to increase section area), until the desired fatigue life is obtained. Here it is assumed that S-N curves with cathodic protection are
to be used. The FDF above mean sea level for the X-braces was assumed
to be 2.0 as allowed by the (Det Norske Veritas (DNV), 2013), while the
FDF below mean sea level was taken as 3.0. No other partial safety
factors were considered in the evaluation of fatigue life. The required
changed geometry of the affected joints as compared with the output
from the optimization procedure is given in Table 11. It can be seen that
there is an increase in outer diameter of the top brace near the X-joint,
but a significant increase in its wall thickness. The corresponding Kjoint on leg-3 that connects to the brace below also is seen to require
significant increase in its wall thickness and diameter. With these
changes to the optimized jacket geometry, the minimum design fatigue
life was determined to be 26 years at the top X-joint. All other hot spots
in the jacket had a fatigue life greater than 26 years.
With the corrections described in Table 11, the overall jacket mass
was increased from 654 tons reported earlier to 861 tons. However the
first few frequencies of the support structure are reduced by less than
5%. This is due to the high stiffness of the jacket and the fact that all the
members changed are only localized to the four joints that had insufficient life without any change made to remaining structure or the
transition piece. This also shows that localize changes to the jacket
structure have little affect on the overall natural frequencies of the
support structure, which is the essential premise for not using minimization of mass as an objective in the structural optimization of the
jacket.
Since the natural frequencies were altered by less than 5%, it is
assumed that the load simulations need not be repeated on the modified
jacket as the underlying dynamics of the wind turbine and support
structure remain the same. However the sixteen wind/wave directions
that were used in the above verification can be further permuted to
identify the variation in the fatigue damage as a function of the discretization in the number of wind/wave directions. The variation in
fatigue life with varying wind directions is ascertained by bootstrapping, that is repeated sampling of a sub-set of directions from the
original sixteen, for example in twelve different directions. Using the
Table 9
Buckling utilization ratios for critical members.
Member type
Legs
X-brace
Mud brace
Utilization Ratio
0.057
0.049
0.167
over all the 10-min load simulation results at a particular mean wind
speed, i, Wni is the annual expected number of 10-minute samples for
that mean wind speed and Neq is the total number of stress cycles of the
year.
25
eq
=
Wni
i=5
1
m
N
n m
j = 1 j dj
Neq
(12)
The stresses themselves are computed using (10) where the impact
of the stress concentration factor on the total stress at a point is to be
quantified. The stress concentration factors at a welded joint are to be
determined empirically using the relations in DNV RP C203 (Det Norske
Veritas (DNV), 2013), which are typically a function of four sizing
parameters: 1) The ratio of wall thickness of the brace to the wall
thickness of the leg at the joint (τ), 2) The ratio of the outer diameter of
the leg to twice the thickness of the leg γ, 3) The ratio of the outer
diameter of the brace to the outer diameter of the leg β and 4) For Kand Y-joints, the ratio of the gap between the intersection of two connections to the diameter of the leg ζ. The stress concentration in the
axial direction on the chord side of a K-joint is typically expressed as
(Det Norske Veritas (DNV), 2013).
SCFAC =
0.9 0.5 (0.67
+ 0.29
2
+ 1.16 )sin( )
0.38tan 1 (8
sin(
sin(
max )
min )
0.3
0.3
max
(1.64
min
(13)
))
Similar empirical equations in the bending directions, as well as for
other types of joints (X, Y) can be found in (Det Norske Veritas (DNV),
2013). Table 10 provides the variation of the fatigue life over all joints
on one side of the jacket, where the bottom joint connecting the mudbrace to the leg is modelled as a K-joint and as a Y-joint. A similar (but
not identical) life distribution can be shown for the other jacket sides as
well. Since the predicted life shown in Table 10 does not include the
FDF, the required fatigue life of each joint should be at least 75 years.
As seen in Table 10, four of the welded joints fail this criteria. K-joints
and Y-joints are quite similar in classification, but in the case of Yjoints, the axial component of the brace connection is counter acted by
the shear force in the chord or leg section. Thus the end connections of
the jacket at the base should be modelled as Y-joint connections.
Table 10 shows the effect of the classification on the bottom welded
joint, where using the SCFs of a Y-joint results in significantly lower
fatigue life than for a K-joint.
Table 10
Computed fatigue life of welded joints in years on one face with variation between K-joint and Y-joint on the bottom joint.
Joint Type
Location
Life (bottom Y-joint)
Life (bottom K-joint)
K-joint/Y-joint
K-joint
K-joint
K-joint
K-joint
K-joint
K-joint
Y-joint
X-joint
X-joint
X-joint
X-joint
Mud-brace
Level1
Level1-Level2
Level2
Level2-Level3
Level3
Level3-Level4
Level4-transition
Level1
Level2
Level3
Level4
9
1498
1945
339
13780
22612
12
37546
241
22
152082
36
41
1498
1945
339
13780
22612
12
37546
241
22
152082
36
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Ocean Engineering 172 (2019) 629–640
A. Natarajan et al.
Fig. 5. Overview of the variation in stress concentration factors with variation in β.
same load time series at each mean wind speed, but now sampled from
each different subset of 12 wind directions over the 360° polar around
the jacket, the distribution of fatigue damage on the joints at the braces
and legs is determined. If the average damage over 26 years for such a
sampling is lower than unity at the weakest joint, then the jacket design
can be said to be robust to variation in wind/wave directional loading.
Fig. 6 depicts the damage distribution and the mean damage over eight
points on the chord side and brace side on the top K-joint on leg 3,
which was originally the weakest joint from the optimization result.
The results shown consider the Weibull parameters for each wind direction. Fig. 6 shows that the re-designed joint dimensions possess the
mean net damage level to be below 1 for a design life of 26 years regardless of the number of load directions and regardless of symmetry of
the loading directions.
As a comparison the jacket used in (Conti et al., 2018) designed in
the INNWIND. EU project for the 10 MW turbine was more than 1000
tons and requires specialized post welding grinding treatment to meet
25 years fatigue lifetime at all joints. Thus the present re-designed
jacket of 861 tons for the 10 MW turbine is a low mass jacket
configuration that has moved natural frequencies away from excitation
and satisfies all limits states so as to obtain 25 years lifetime without
post-weld treatments.
7. Conclusions and future research
It was shown that using the single objective of minimization of the
fundamental frequency in the structural optimization of offshore wind
turbine jackets enables the design of cost effective long life jackets. This
optimization objective was shown to exploit the full displacement
constraints available to significantly reduce the inherent stiffness of the
jacket structure to not only minimize its natural frequencies, but also
reduce the mass. The placing of the structural frequencies away from
the regions of harmonic excitation allows alleviation of the fatigue
damage. The resulting optimized jacket had a low fundamental frequency which was below the 3P excitation range and its first five frequencies were away from all excitations. The structure further had a
low mass for supporting the 10 MW wind turbine, and satisfied all ultimate stress and displacement constraints. Further it was shown by
Table 11
Revised Welded Joint dimensions.
Joint location
Outer diameter [mm]
Revised
Optimized
X-joint center level4
X-joint leg connect level4
K-joint level3 to level 4
X-Joint center level3
Y-Joint level1 mud brace
Mud brace leg connection
Wall thickness [mm]
Revised
Optimized
540.0
687.3
540.0
540.0
570.0
684.4
690.0
700.0
690.0
600.0
660.0
684.4
638
20.0
46.4
20.0
20.0
20.0
44.4
80.0
60.0
80.0
20.0
80.0
80.0
Ocean Engineering 172 (2019) 629–640
A. Natarajan et al.
Fig. 6. The fatigue damage distribution at the top K-joint showing maximum damage based on bootstrapping over all wind directions, the mean damage is shown as
the red line with the highest mean damage being 0.98. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of
this article.)
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A. Natarajan et al.
subsequent load analysis of the jacket, that there was also no buckling
problems. A detailed fatigue damage analysis showed that while the
optimized jacket produced the required fatigue life time of 25 years or
greater at most members which are typically highly stressed, it did not
possess satisfactory lifetime at four welded members, near the transition piece and at the base. These jacket members were subsequently redesigned, which increased the lifetime above 25 years without impacting the natural frequencies of the structure. The re-designed jacket
mass increased by 30% over the initial optimized jacket. The final solution still possessed natural frequencies away from excitation and the
first fundamental frequency was less than 5% away from the result of
the structural optimization, thus satisfying all stated requirements.
Future research is recommended in allowing the optimization to
consider a probability of fatigue damage exceeding a threshold versus a
deterministic approach, so as to determine the probability that the
lifetime of certain members of the jacket will be below 25 years.
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Acknowledgements
The research leading to these results has received funding from the
European Community’s Seventh Framework Programme under grant
agreement No. 308974 (INNWIND.EU). The support is gratefully acknowledged.
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