Marine Structures 67 (2019) 102631

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Marine Structures
journal homepage: www.elsevier.com/locate/marstruc

Dynamic design considerations for offshore wind turbine jackets

T
supported on multiple foundations
Saleh Jalbia, Georgios Nikitasa, Subhamoy Bhattacharyaa,∗, Nicholas Alexanderb
a
University of Surrey, Guildford, United Kingdom
b
University of Bristol, Bristol, United Kingdom

A R T IC LE I N F O ABS TRA CT

Keywords: To support large wind turbines in deeper waters (30–60 m) jacket structures are currently being
Offshore wind turbines considered. As offshore wind turbines (OWT's) are effectively a slender tower carrying a heavy
Jacket structures rotating mass subjected to cyclic/dynamic loads, dynamic performance plays an important role in
Natural frequency the overall design of the system. Dynamic performance dictates at least two limit states: the
Rocking mode of vibration
Fatigue Limit State (FLS) and the overall deformation in the Serviceability Limit State (SLS). It
Sway-bending mode of vibration
Multiple foundations
has been observed through scaled model tests that the first eigen frequency of vibration for OWTs
supported on multiple shallow foundations (such as jackets on 3 or 4 suction caissons) corre-
sponds to low frequency rocking modes of vibration. In the absence of adequate damping, if the
forcing frequency of the rotor (so called 1P) is in close proximity to the natural frequency of the
system, resonance may occur affecting the fatigue design life. A similar phenomenon commonly
known as “ground resonance” is widely observed in helicopters (without dampers) where the rotor
frequency can be very close to the overall frequency causing the helicopter to a possible collapse.
This paper suggests that designers need to optimise the configuration of the jacket and choose the
vertical stiffness of the foundation such that rocking modes of vibration are prevented. It is
advisable to steer the jacket solution towards a sway-bending mode as the first mode of vibration.
Analytical solutions are developed to predict the eigen frequencies of jacket supported offshore
wind turbines and validated using the finite element method. Effectively, two parameters govern
the rocking frequency of a jacket: (a) the ratio of the super-structure stiffness (essentially the
lateral stiffness of the tower and the jacket) to the vertical stiffness of the foundation; (b) the
aspect ratio (the height to width ratio) of the jacket. A practical example considering a jacket
supporting a 5 MW turbine is considered to demonstrate the calculation procedure which can
allow designers to choose a foundation. It is anticipated that the results will have an impact in
choosing foundations for jackets.

1. Introduction

1.1. Wind turbines supported on monopiles and jackets on piles/suction caissons

Jackets or seabed frames supported on multiple foundations are currently being installed to support offshore wind turbines in
deep waters ranging between 23 m and 60 m, see for example Borkum Riffgrund 1 (Germany, water depth 23–29 m), Alpha Ventus
(Germany, water depth 28–30 m), Aberdeen Offshore wind farm (Scotland, water depth 20–30 m) (4C Offshore Limited [1,2]). The

∗
Corresponding author.
E-mail address: S.Bhattacharya@surrey.ac.uk (S. Bhattacharya).

https://doi.org/10.1016/j.marstruc.2019.05.009
Received 11 September 2018; Received in revised form 23 January 2019; Accepted 8 May 2019
0951-8339/ © 2019 Elsevier Ltd. All rights reserved.
S. Jalbi, et al. Marine Structures 67 (2019) 102631

Fig. 1. Schematic of a 3-legged jacket supported on suction caissons (Photo by Ørsted).

jackets are typically designed as three or four legged and are supported on either deep foundations (piles) or shallow foundations
(suction caissons). The height of the jackets currently in use is between 30 and 35 m and is governed by water depth above the
mudline and the wave height (50 year return period) following the guidance of DNV-OS-J101 (2014). However, it is expected that
future offshore developments will see jacket heights up to 65 m to support larger turbines (12 MW–20 MW) in deeper waters. Fig. 1
shows a schematic of a 3-legged jacket inspired by some recent offshore developments.
There are obvious differences between the behaviour of jacket supported wind turbines and monopile supported ones as illu-
strated through Figs. 2–4. The differences can be classified into two distinct types:

(a) For monopile supported wind turbines, the overturning moment resulting from wind and waves is transferred to the supporting
ground through moment and the monopile acts as a moment resisting foundation. On the other hand, for a jacket, the overturning
moment is transferred through axial push-pull (in combination with the lateral base shear to maintain lateral equilibrium), see
Fig. 2 for a schematic diagram.
(b) The modes of vibration for monopile supported wind turbines or for that matter any foundation supported on piles will be sway
bending as the foundation is very stiff compared to the tower, see Fig. 3. For the corresponding jacket supported wind turbines on
shallow foundations, the first mode of vibration is most likely to be rocking due to the relatively lower vertical stiffness of shallow
foundations as shown in Fig. 4. Further details on different types of modes of vibrations are discussed in Refs. [8–10].

Fig. 2. Schematic of a load transfer for two types of foundation systems. Note: The aim of figure is to show how the overturning moment is resisted. To
maintain equilibrium, both types of foundations will have a lateral resistance component at the mudline.

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S. Jalbi, et al. Marine Structures 67 (2019) 102631

Fig. 3. Sway-bending mode of vibration for pile supported wind turbines. It may be noted that the foundation is very stiff vertically.

Fig. 4. Rocking mode of vibration.

One of the aims of this paper is to highlight the importance of avoiding rocking type vibrations for wind turbine support structures
by learning lessons from an equivalent problem from aerospace industry – the “helicopter ground resonance”. OWT jackets supported
on shallow foundations are a new innovation which lack a track record of dynamic and long-term performance. For this reason, it is
important to learn lessons from dynamically similar types of engineering problems and of close similarity is ground resonance in
helicopters. It is therefore considered useful to study the problem.
The other aims and the scope of this paper are as follows:

(a) Develop and validate analytical solutions to study the vibration of offshore wind turbine jackets supported on shallow founda-
tions and piled foundations.

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S. Jalbi, et al. Marine Structures 67 (2019) 102631

Fig. 5. Ground Resonance of a helicopter.

(b) To find out the mechanics of the problem based on non-dimensional groups that can characterise the different vibration modes of
the system and identify the controlling parameters affecting the vibration modes.
(c) To provide insights for enhanced dynamic performance and develop simple design rules.
(d) To demonstrate the practical implications by taking an example and show the calculation procedure.

2. Ground resonance of helicopters & OWT structures supported on shallow foundations

Fig. 5 shows still photographs from the well-known helicopter resonance problem known as ground resonance, the video can be
accessed in Ref. [13]. Effectively, due to the imbalance in the helicopter rotor the RPM (Revolutions Per Minute) induced oscillations
get in phase with the rocking frequency of the helicopter on its landing gears. This leads to collapse and the experiment is sche-
matically shown in Fig. 6. The helicopter starts rocking about the two landing pads (skids) until the stresses induced through
resonance exceed the strength of the structural material and connections eventually causing failure. There are many similarities
between these two systems: both are essentially a structural beam carrying a heavy rotating mass resting on multiple supports, see Fig. 7.
Mathematically, the mass and the stiffness matrices in a dynamic formulation will be similar. The structures in both systems will
rock and there is considerable amount of energy in these modes of vibration. However, the difference is the plane of rotation of the
rotors and the rotor speed. It may be noted that the resonance phenomenon to be studied is irrespective of the planes of rotation. The
objective in this study is to learn from other engineering disciplines given that wind turbine jackets supported on suction caissons are
new structures with no track record. As the motion under consideration is rocking, the vertical stiffness of the supports is a governing

Fig. 6. Rocking motion of a helicopter getting tuned with the RPM of helicopter rotor.

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S. Jalbi, et al. Marine Structures 67 (2019) 102631

Fig. 7. Similarities between a helicopter and an offshore wind turbine.

parameter. For a jacket structure, at the onset the vertical stiffness may not be identical and therefore they are shown as K1 and K2. It
is clear that resonance must be avoided and this emphasizes the importance of understanding the subtle aspects of the dynamic
behaviour of jacket supported wind turbines not only in relation to FLS (Fatigue Limit State) and SLS (Serviceability Limit State) but
also from the point of view of monitoring and O & M (Operation and Maintenance).
Moreover, it is interesting to note how the target frequency in the soft-stiff frequency is shifting with turbine size. For instance, a
Vestas 8 MW OWT has a soft-stiff frequency band of 0.2–0.24 Hz which is very close to the predominant North sea frequency of
0.1 Hz. This is even more challenging for Chinese wind farms as the predominant wave frequency for Bohai and Yellow sea is 0.2 Hz.
Thus, even though the amplitude of the 1P and 3P excitations are relatively low, wave loads (which is also have a close forcing
frequency) have a considerably higher energy content. This higher energy content in combination with a low vertical stiffness will
induce a rocking type vibration, and though this rocking might not have ultimate failure effects as in the case of the helicopter, it may
have further implications on the fatigue performance of the structure and opens the door to further research needed in this area with
considerations of the correct energy content of the loads together with the incorporation of damping into the analysis.
The dynamic performance of jacket supported OWTs incorporating Soil Structure Interaction (SSI) is an area of active research
[14–16]. The dynamic response of jackets under the action of waves of different periods and energy is studied by Ref. [14] using
Finite Element analysis where the dynamic amplification factors (DAFs) are evaluated. The study shows that depending on the wave
amplitude and period, the DAF may reach values of 1.2–1.3 which is significant given the magnitude of wave loads. Studies by Ref.
[22] also modelled OWT jackets on a fixed base and assessed the fatigue damage on different types of welded joints. It was concluded
that the interaction of both wind and wave loads have to considered when assessing the fatigue damage with wind loads providing
the dominant contribution to the cumulative damage. Moreover, numerical studies by Ref. [15] show the importance of incorporating
the flexibility of the foundation in understanding the modes of vibration of the system when predicting the structural response. The
SSI effect was introduced through distributed springs along the depth of the foundation. Similarly [16], studied the effect of non-
linearity of the ground profiles in loose sands, medium sands and dense sands and concluded that the effect of SSI becomes pre-
dominant in looser sands. Other work by Ref. [19] showed, through numerical analysis, that incorporating SSI effects alters the
natural frequency and the dynamic response of the leg and bracing members. Moreover, the study also showed that incorporating pile
group effects has a noticeable effect on the fatigue analysis of the structure. The literature above builds upon previous work on SSI
effects on jackets supporting oil and gas decks/platforms where [20] also performed a numerical study on a jacket supported on piles
and showed that SSI reduces the natural period with an emphasis on the effect of the top soil layers on the frequency and [21]
performed scaled model tests showing the importance of SSI in predicting the response of offshore jackets to random loads.
Rocking type modes of vibration have been observed in small scale tests for jacket/seabed frame supported on shallow foun-
dations, see Refs. [8,10]. For offshore wind turbines, rocking modes can be quite complex where the vertical motion of the foundation
interacts with the flexible bending modes of the tower together with the 1P rotor frequency and 2P/3P blade passing frequency. In
some cases, depending on the stiffness and mass distribution of the superstructure (jacket and the tower with the huge RNA mass), the
superstructure may or may not be in phase with the rocking motion of the foundations [9]. Furthermore, rocking modes of vibration
will have a lower frequency which may be close to the wave frequency given the wave will have a higher energy of excitation. It is
therefore advisable to avoid rocking modes for jackets supported shallow foundations. Judging from the literature above, a better
understanding of the modes of vibration of the system is crucial for the dynamic analysis and assessing the fatigue life of the structure.
The next section of the paper derives an analytical expression for rocking modes of vibration for OWT jackets supported on shallow
foundations.

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Fig. 8. Mechanical representation of jacket supported offshore wind turbines.

3. Analytical solution for rocking modes of vibration for OWT jackets supported on shallow foundations

3.1. Simplified mechanical representation of the vibrating system

Fig. 8 shows an idealization of the vibration problem in hand i.e. eigen values of a jacket supported on multiple foundations
following the work of [10,18]. The vibration of such a complex system is a 3D problem where oscillations may occur over multiple
coupled planes depending on the locations of the centre of mass and centre of stiffness of the foundations. Under certain circum-
stances, a 3D problem can be simplified into 2D, where vibrations in orthogonal planes may be uncoupled and studied separately.
This is generally true if the centre of mass of the foundation coincides with centre of stiffness and can be applied for different
foundation arrangements. Examples of different foundation configurations in relation to the centre of mass and centre of stiffness are
shown in Appendix 1. The foundation will vibrate in two principle axes i.e. highest variance of moment of inertia. The foundation can
be modelled as two springs connected by a rigid base with a lumped mass m1, whilst the superstructure (the jacket and wind turbine
tower) can be modelled as an equivalent beam with a lumped mass at the tip. In the analysis, m2 represents the mass of the Rotor-
Nacelle Assembly (RNA) together with the total structural mass of the tower and the jacket and as shown is lumped at the tower tip.
In this paper, m2 has been computed using a FE package. However a detailed example on the methods of calculation for m2 (i.e. how
to lump the tower and jacket mass to the tower tip) and kt using simple spreadsheet programs is provided in Ref. [17] and sum-
marized in Appendix 2. Furthermore, guidance on the computation of the vertical stiffness of shallow caissons is provided in
Appendix 3. This two-dimensional (2D) mechanical model can be applied to both three legged or four legged jackets as shown in
Figs. 9–12. For four legged jackets, vibration can occur at X-X′ or Y-Y’ planes as shown in Figs. 9 and 10. It may be noted that a four-
legged jacket on shallow foundations may vibrate in a diagonal plane of the conventional orthogonal plane. Similarly, for three
legged jackets the rocking vibration modes will have three axes of symmetry as shown in Figs. 8 and 9. Further discussion and the
impact of three axes of symmetry on dynamic soil-structure interaction can be found in Ref. [9].
It may be noted that this method assumes the presence of translational restraints in the lateral direction at foundation level and
only the vertical stiffness is considered due to the load transfer mechanism. Typically, the inherent lateral stiffness of the foundation
will be sufficient, and the value of the vertical stiffness will govern the first natural frequency as shown in Ref. [17]. It was found in
Ref. [17] that the idealization of the foundations as lumped vertical springs (used in this paper) provides a close match with literature
that utilized distributed p-y and t-z springs to represent the foundations. In practice, one requires to carry out a refined analysis and
this can be modelled by adding lateral springs (KL) in addition to the vertical springs (k1 and k2). Thus, after the selection of a certain
foundation size using the proposed simplified method (which only includes vertical springs) designers are encouraged to further
refine structural models to include the lateral stiffness at the foundation level rather than a lateral restraint.

3.2. Mathematical derivation of the mass and stiffness matrices

The mass and stiffness matrices were assembled using Euler Lagrange's equations of potential and kinetic energy following the
work of [3–7]. The allowable degrees of freedom are the movement of the rigid base, the rotation of the rigid base, and the bending of
the tower as shown in Fig. 13 where u1 and u2 represent the vertical translations of the springs, ug represents the translation of the
centre of mass of the base m1. uT is the total translation of the tower and is composed of two components: (a) the translation of m2 due

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S. Jalbi, et al. Marine Structures 67 (2019) 102631

Fig. 9. Rocking modes for four legged jackets about X-X′ and Y-Y′ planes.

Fig. 10. Rocking modes about diagonal planes.

to rocking (where the tower is assumed to be rigid and does not bend) which can be simply computed as htanϴ; (b) u3 which is the
translation due to bending of the tower. Fig. 13 is a schematic showing the mass terms, the stiffness terms, and the degrees of freedom
for a square base jacket.
For the jacket vibrating about X-X′ and Y-Y’ axes, k1 and k2 can be computed using equations (1)–(4).

k1 = kA + kB (1)

k2 = kC + kD (2)

For vibrating about Y-Y’

k1 = kA + kD (3)

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Fig. 11. Rocking modes for three legged jackets.

Fig. 12. Planes of symmetry.

Fig. 13. (a): Plan view of the foundation (b) stiffness and mass idealization of the system (c) degrees of freedom of the system.

k2 = kB + kC (4)

Using kinematic equations (5) and (6), the end displacements of the base (u1 and u2) are related to the small angle of rotation ϴ.
Equation (7) links the displacement of the tip of the tower with the movement of the base. It may be noted that there are relative
movements between rocking and pure bending.

u1 + u2
uG =
2 (5)

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u2 − u1
θ=
L (6)

uT = u3 − h tan θ (7)

For small angle rotations tan θ ≈ θ

h
uT ≈ u3 − (u2 − u1)
L
As per equation Lagrange's equation is as follows
d ∂T ∂T ∂U
− + = (pi ). i = 1,2,3
dt ∂q˙ i ∂qi ∂qi (8)

where q1 → u1 and. q2 → u2
Since the objective is to find the natural frequency of the system under free vibration, no external forces are applied on all degrees
of freedom.
(p1 ) = (p2 ) = (p3 ) = 0

The Kinetic Energy of System T is given by three components as shown in equation (9): the kinetic energy due to the translational
acceleration of m1 (rigid base); the kinetic energy due to the angular acceleration of the rigid base; and the translation acceleration of
the lumped mass m2 in the lateral and vertical directions (uT and uG respectively)
1 1 2 1 1
T= m1 u˙ G 2 + IG θ˙ + m2 u˙ T 2 + m2 u˙ G 2
2 2 2 2 (9)

where IG is the moment of inertia of the rigid base.

Which can be further simplified to:
2
1 u˙ + u˙ 2 2 1 1 u˙ − u˙ 1 2 1 h
T= (m1 + m2) ⎛ 1 ⎞ + ⎛ m1 L2 ⎞ ⎛ 2 ⎞ + m2 ⎛u3 − (u˙ 2 − u˙ 1) ⎞
2 ⎝ 2 ⎠ 2 ⎝ 12 ⎠⎝ L ⎠ 2 ⎝ L ⎠
Further Algebraic simplification:

1 1 1 2h h2
T= (m1 + m2)(u˙ 12 + 2u˙ 1 u˙ 2 + u˙ 22) + m1 (u˙ 12 − 2u˙ 1 u˙ 2 + u˙ 22) + m2 ⎛u˙ 32 −
⎜ (u˙ 3 u˙ 2 − u˙ 3 u˙ 1) + 2 (u˙ 12 − 2u˙ 1 u˙ 2 + u˙ 22) ⎞
⎟

8 24 2 ⎝ L L ⎠
The potential Energy of System U is given by equation (10) and is also formed of 3 components: the extension in springs k1 and k2,
and the bending deformation of the tower with stiffness kt. It is important to note that u3 rather uT is used in the potential energy
evaluation as only the deformation due to kt is evaluated
1 1 1
U= k1 u12 + k2 u22 + kt u32
2 2 2 (10)

The partial derivatives for the kinetic Energy T in equation (9) are evaluated in equations (11)–(13):

∂T 1 1 hu˙ h2
= (m1 + m2)(u˙ 1 + u˙ 2) + m1 (u˙ 1 − u˙ 2) + m2 ⎛ 3 + 2 (u˙ 1 − u˙ 2) ⎞
⎜ ⎟

∂u˙ 1 4 12 ⎝ L L ⎠

d ∂T 1 1 hu¨ h2
= (m1 + m2)(u¨1 + u¨2) + m1 (u¨1 − u¨2) + m2 ⎛ 3 + 2 (u¨1 − u¨2) ⎞
⎜ ⎟

dt ∂u˙ 1 4 12 ⎝ L L ⎠ (11)

∂T 1 1 −hu˙ 3 h2
= (m1 + m2)(u˙ 1 + u˙ 2) + m1 (u˙ 2 − u˙ 1) + m2 ⎛ + 2 (u˙ 2 − u˙ 1) ⎞
⎜ ⎟

∂u˙ 2 4 12 ⎝ L L ⎠

d ∂T 1 1 −hu¨3 h2
= (m1 + m2)(u¨1 + u¨2) + m1 (u¨2 − u¨1) + m2 ⎛ + 2 (u¨2 − u¨1) ⎞
⎜ ⎟

dt ∂u˙ 2 4 12 ⎝ L L ⎠ (12)

∂T h
= m2 ⎛u˙ 3 + (u˙ 1 − u˙ 2) ⎞
∂u˙ 3 ⎝ L ⎠

d ∂T h
= m2 ⎛u¨3 + (u¨1 − u¨2) ⎞
dt ∂u˙ 3 ⎝ L ⎠ (13)

The equations (11)–(13) can be written in Matrix format as shown in equation (14), which is analogous to a mass matrix mul-
tiplied by an acceleration matrix

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S. Jalbi, et al. Marine Structures 67 (2019) 102631

2 h2
⎡ m1 + m2 + m2 h2 m1
+
m2
− m2
h
m2 L ⎤
L2
⎢3 4 L 6 4
⎥ ⎡ u¨1 ⎤
2 h2
= ⎢ m1 + m2 − m2 h2 m1
+
m2
+ m2 − m2 L ⎥ ⎢u¨2 ⎥ = [M ][u¨]
h
⎢6 4 L 3 4 L2 ⎥⎢ ⎥
⎢ ⎣u¨3 ⎦
m2 ⎥
h h
m2 L − m2 L (14)
⎣ ⎦
The derivatives of kinetic energy with respect to translation are zero as shown in equation 15
∂T ∂T ∂T
= = =0
∂u1 ∂u2 ∂u3 (15)
The partial derivatives for the potential energy U in the equations are evaluated in equations (16)–(18):
∂U
= k1 u1
∂u1 (16)

∂U
= k2 u2
∂u2 (17)

∂U
= kt u3
∂u3 (18)
Equations (16)–(18) can be written in Matrix format as shown in equation (19), which is analogous to a stiffness matrix multiplied
by a translation matrix

⎡ k1 0 0 ⎤ ⎡ u1 ⎤
= ⎢ 0 k2 0 ⎥ ⎢u2 ⎥ = [K ][u]
⎢ 0 0 k ⎥ ⎣u3 ⎦
⎣ t⎦ (19)
The equation of motion is then [M ][u¨] + [K ][u] = 0 where M and K are as per equations (14) and (19).
Hence the 3 natural frequencies are the eigen vector solutions which can be solved using any standard mathematics software..
([K ] − ω2 [M ])[u] = 0

eig [M−1K ] (20)

Similarly, if the vibration occurs along the diagonal axes, a third spring kG and a displacement uG is added to the system as shown
in Fig. 14. The potential energy of the system previously described in equation (10) such that it can be recalculated as equation (21).
1 1 1 1
U= k1 u12 + k2 u22 + kG uG2 kt u32
2 2 2 2 (21)
As per equation (1), equation (17) can be simplified as
1 1 1 u + u2 2 1
U= k1 u12 + k2 u22 + kG ⎛ 1 ⎞ + kt u32
2 2 2 ⎝ 2 ⎠ 2
The partial derivatives for the potential energy U in equation (21) are evaluated in equations (22)–(24)
∂U 1
= k1 u1 + kG (u1 + u2)
∂u1 4 (22)

∂U 1
= k2 u2 + kG (u1 + u2)
∂u2 4 (23)

Fig. 14. (a): Plan view of the foundation (b) stiffness and mass idealization of the system (c) degrees of freedom of the system.

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S. Jalbi, et al. Marine Structures 67 (2019) 102631

∂U
= kt u3
∂u3 (24)
Similarly, equation (19) can be adjusted to:
1 1
⎡ k1 + 4 kG k
4 G
0⎤ u
1
⎢ 1 1 ⎥ ⎡u ⎤
=⎢ k G k2 + k G 0 ⎥ ⎢ 2⎥
4 4
⎢ ⎥ ⎣u3 ⎦
⎣ 0 0 kt ⎦ (25)
where
k1 = kB (26)

k1 = kD (27)

kG = kA + kC (28)
Moreover as L1 = 2 L , equation (10) can be changed to
2 h2
⎡ m1 + m2 + m2 h 2 m1
+
m2
− m2 m2
h
⎤
2L2
⎢3 4 2L 6 4 2L
⎥ ⎡ u¨1 ⎤
⎢ h2 h2
− m2 2 L ⎥ ⎢u¨2 ⎥
m1 m2 m1 m2 h
= + − m2 + + m2
⎢6 4 2L2 3 4 2L2 ⎥⎢ ⎥
⎢ h h ⎥ ⎣u¨3 ⎦
⎢ m2 2 L − m2 m2 ⎥
⎣ 2L ⎦ (29)
The methodology above shows how the fundamental natural frequencies of the system can be computed analytically. As three
degrees of freedom are allowed, the modes of vibration will be computed. For the purpose of the problem in hand, only the first mode
of vibration which will be either rocking type vibration (Fig. 4) or sway-bending (Fig. 3) are of main interest.
It is important to highlight that the distribution of accelerating mass can be arbitrarily chosen, and respective Euler-Lagrange
equations must be formed. It is convenient and also typical in literature to lump the distributed mass at the tip of the cantilever tower.
It may be also noted that the provided formulations could be reconstructed in different ways such as splitting the mass of the base to
individual masses over the springs, which can be useful if designers have special mass requirements over individual caissons. From
the formulations presented, it is clear that different parameters such as the foundation stiffness (k1 and k2) and the geometrical aspect
ratio (h/L) are the main parameters affecting the first mode of vibration type of the system. The next section takes a practical example
to show the effect of influencing parameters.

4. Non-dimensional study of an example jacket on multiple foundations

For the purpose of this investigation and verifying the obtained mass and stiffness matrices, the jacket of of the Upwind project is
considered. Essentially, this is four-legged jacket structuresupporting a 5 MW wind turbine in deeper waters and the details can be
found in Ref. [11] and schematically shown in Fig. 15. The report also shows how different jacket arrangements and dimensions can
be optimized to obtain a satisfactory design. Other necessary information is shown in Table 1 and data pertaining to 5 MW reference
wind turbine can be found in Ref. [12].
The jacket supported system was analysed using the analytical expression derived in Section 3 as well as finite element package
SAP2000 for different values of kv. After obtaining kt and m2 from the fixed base finite element model, the mass and stiffness matrices
are constructed to obtain equations (14) and (19). Consequently, a parametric study is conducted to understand the variation of first
fundamental frequency (f0) with increasing vertical stiffness of the springs kv. Finite element analysis is carried out for the following
purposes:

(1) To obtain the fundamental natural frequency using modal analysis to compare with the analytical solution developed in Section
3.
(2) To obtain kt i.e. stiffness of the tower in the equivalent mechanical model by applying a unit load at the tower tip.
(3) To obtain the equivalent accelerating mass of the superstructure m2 (jacket, tower, and lumped mass of the RNA). After the fixed-
k
base natural frequency (ffb) is obtained for the full model shown, the accelerating mass m2 is obtained using m2 = 2.
⎛2πf ⎞
⎜ fb ⎟
⎝ ⎠
Alternatively, m2 can be calculated using the method provided in Appendix 2

It is important to note that the finite element results have been performed through a linear eigenvector analysis on SAP2000. The
jacket was constructed using beam elements with moment releases at the ends. The tower consisted of a non-prismatic section with a
linear variation of the moment of inertia. As for the accelerating masses, the mRNA was modelled through a lumped mass at the tower
top and the program automatically calculates the accelerating mass of the jacket and the tower (superstructure). The foundation
supports were modelled using linear springs, this however is an idealization that assumes equivalent axial stiffness of the foundations
in both the push-in and pull-out direction. In reality the stiffness is non-isotropic and slight differences in stiffness are expected.
Typical deflected mode shapes from the software output is shown in Fig. 16. Fig. 17 shows a comparison between the analytical

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S. Jalbi, et al. Marine Structures 67 (2019) 102631

Fig. 15. Schematic for example problem and details used for Finite Element Model.

Table 1
Jacket and Tower Properties of example problem.
Mass of Rotor-Nacelle Assembly 350 tons

Tower Height 70.4 m

Tower Bottom Diameter 6 m (27 mm thick)
Tower Top Diameter 3.87 m (20 mm thick)
Jacket Bottom Width 12 m
Jacket Top Width 8m
Jacket Height 70.15 m
Jacket External Legs 1.2 m (35 mm thick)
Jacket Braces 0.8 m (20 mm thick)

model and the finite element analysis. The closed form solution provided by Jalbi and Bhattacharya [17] is also used to verify the
solutions.
Few points may be noted from the graphs:

(1) From Fig. 17, it is clear that the analytical solution matches quite well with the finite element analysis which demonstrates that
the Euler-Lagrange mass and stiffness matrices obtained are valid. For low vertical stiffness of the foundation, rocking is the
dominant vibration mode, see Fig. 16(a). Also as the vertical stiffness of the foundation increases, the vibration mode moves to
sway-bending and the corresponding 1st natural frequency increases and approaches the fixed base natural frequency.
(2) The parameter dictating whether the system vibrates in a rocking or sway bending mode is the ratio of foundation vertical
stiffness (kV) to superstructure stiffness (kt). At low foundation stiffness, the structure is more susceptible to rocking, whilst at
higher foundation stiffness values sway-bending vibration governs. It is important to note that in the rocking vibration region any
change in vertical stiffness results in an abrupt changes in the frequency of the system. Therefore, to avoid rocking an optimi-
zation of the relative stiffness may be carried out.
(3) Rocking modes are low frequency and it may interfere with the 1P frequencies of the rotor. Using simple geometrical construction
as shown in Fig. 17, one can determine the threshold vertical stiffness of the foundation to find the theoretical boundary of two
types of vibration mode. Below the threshold vertical stiffness of the foundation, rocking mode of vibration is dominant. Based on
the analysis carried out by Ref. [9], it is shown that most monopile supported wind turbine are close to the fixed base frequency

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S. Jalbi, et al. Marine Structures 67 (2019) 102631

Fig. 16. Typical output from Finite Element model showing rocking and sway-bending modes of vibration; (a) Rocking mode of vibration for low kV
values; (b) Sway bending mode for high kV values. Note: In a 3D analysis, the natural frequencies in the two orthogonal directions will be almost
identical (See Fig. 9) if the spring stiffness below each foundation is the same i.e. rocking may occur in fore-aft and side to side vibrations of the
structure. It is important to remember that wave loads can change directions making the structure prone to rocking in both directions.

Fig. 17. Variation of the normalised 1st natural frequency of the system (f0/ffb) with the normalised vertical stiffness of the foundation (kv/kt).

i.e. value of f0/ffb close to 0.9. In the absence of monitoring data of jackets supported on shallow foundations, it is suggested to
having the vertical stiffness of the foundation such that sway bending mode of vibration governs, this is also discussed in Jalbi &
Bhattacharya [29].

Further analysis has been carried out to study the effect of aspect ratio h/L. For the simplified equivalent model, it is assumed that
the stiffness of the superstructure (kt) does not change with an increasing aspect ratio (by increasing L and keeping h constant). To
verify this assumption a study was performed on the model shown in Fig. 15 where the bottom width of the jacket was varied and the
top width was kept constant at 8 m. The fixed base natural frequency was then recorded for the different cases as shown in Fig. 18. It
may be noted that the fixed based frequency does not greatly change with increasing length, which means that the analytical method
could be used using a constant kt to study the effect of varying aspect ratio. Fig. 19 shows the similar results for different aspect ratios
of the jacket. It is clear from the figure that the transition between rocking and sway-bending mode is also affected by the aspect ratio.
As expected, higher aspect ratios (lower foundation width) makes the jacket system more susceptible to rocking. Higher h/L values
will lead to a lower foundation width and will require higher vertical stiffness of the foundation to engineer towards sway-bending
mode.
It is important to state that though the provided formulation results in 3 natural frequencies, special care should be taken when
assessing the 2nd and 3rd frequencies. This is because the value of m2 (which depends on the accelerating mass of the tower and the
jacket), is calculated using either substitution from the FEA or using the Appendix 2 is dependent on the function of the first mode of

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S. Jalbi, et al. Marine Structures 67 (2019) 102631

Fig. 18. Variation of the fixed base natural frequency with increasing jacket bottom width.

Fig. 19. Effect of increasing the aspect ratio on the modes of vibration of the system.

vibration. For preliminary designs, an accurate estimate of the first natural frequency would be sufficient. However, designers willing
to calculate subsequent frequencies must derive m2 using the second and third modes of vibration respectively either by finding it
using FEA software as shown above or by changing the mode shape equation in Appendix 2 to the 2nd and 3rd modes of vibration.
It may be also noted that the method presents the first estimate for preliminary design and providing design considerations. For
detailed nfa (natural frequency analysis), it is suggested that the mass matrix should also consist of the following.

• Mass of tower equipment such as the flanges.

• Mass of working platforms such as such as boat landings, access ladders, resting platforms, and external platforms.
• The mass of the transition piece, where a methodology to include this is also provided in Ref. [17].
• The mass of any heavy grouted connections (if present).
• Environmental conditions such as mass of marine growth and mass of corrosion allowance.
Other environmental factors influencing the stiffness of foundations such as scour should also be considered such as the study
shown in Ref. [18].
Finally, designers using the provided formulations need to keep in mind current design standards regarding the target frequency
for soft-stiff design (which is usually placed between 1P and 3P). For instance, the (DNVGL-ST-0126, 2016) recommends that the
natural frequency should have a safety factor margin of 10% on the maximum and minimum rotor speeds (soft-stiff design region).

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S. Jalbi, et al. Marine Structures 67 (2019) 102631

Similarly, the recommended values should also consider the ground material stiffness values when performing natural frequency
analysis (nfa). Typically, the characteristic soil conditions (material safety factor = 1) are used for natural frequency analysis.

5. Conclusions and recommendations

Jackets supported on shallow foundations are being considered as foundation solutions for deeper water offshore wind farms. As
offshore wind turbines are dynamically sensitive, modes of vibration are essential design considerations to satisfy the design limit
states. This paper shows that depending on the vertical stiffness of the shallow foundation, a jacket structure may exhibit either
rocking modes of vibration or sway-bending modes. As rocking modes of vibration are low frequency, these can get tuned with the
rotor frequency causing resonance type effects. Drawing an analogy from the well-known helicopter ground resonance problem, this
study suggests that rocking modes of vibration may be avoided to ensure intended performance in its full design life. Analytical
solutions are presented for eigen frequencies of a jacket system and are validated with finite element analysis. A jacket may be
engineered towards a no-rocking solution by optimising two parameters: (a) ratio of the vertical stiffness of the foundation stiffness to
the lateral superstructure stiffness; (b) the aspect ratio of the jacket-tower geometry. A low value of vertical foundation stiffness
values together with a high aspect ratio will promote a rocking mode of vibration. On the other hand, a high vertical stiffness of the
foundation with lower aspect ratios (broader base L of the jacket) will encourage a sway-bending mode. Furthermore, the study
shows that the transition from rocking to sway-bending is non-linear and depends not only on the aspect ratio but also on the ratio of
vertical stiffness of the foundation and lateral stiffness of jacket-tower configuration. A practical method is shown to choose the
vertical stiffness of the foundation to avoid rocking.

Acknowledgements

The authors would like to thank Hassan Moharam and Saeed Alqahtani for their help in the analysis.

Appendix 1. Example configuration of foundations

This section of the appendix shows the centre of mass and centre of stiffness for different foundation arrangements. The centre of
stiffness of the foundation “springs” can be defined as the arithmetic mean position of all the spring stiffness values or in other words
if the stiffness of all the foundations were to act at a single point. This is done to show that uncoupling between oscillations in
orthogonal planes may be permitted in certain situations.

Square foundation arrangement

Figure A1 shows the plan view of a square arrangement. The foundations are replaced with linear springs with identical stiffness
“k”. The base members are assumed to be homogenous with the same density and cross-section, and since they all have the same
length, all members have the same mass “m”.
Hence, the centre of mass:

⎡L ⎤ ⎡0⎤
L L
4m ⎡ X ⎤ = m ⎡ 2⎤ ⎡2⎤
⎢ ⎥ + m⎢ L ⎥ + m⎢ ⎥ + m⎢ L ⎥
⎢Y⎥
⎣ ⎦ ⎣0⎦ ⎣2⎦ ⎣L ⎦ ⎣2⎦ (A.1)
L
⎡X⎤ = ⎡ 2 ⎤
⎢Y⎥ ⎢ L ⎥
⎣ ⎦ ⎢ ⎣2⎦
⎥ (A.2)
Centre of Stiffness:

0 0
4k ⎡ X ⎤ = k ⎡ ⎤ + k ⎡ ⎤ + k ⎡ L ⎤ + k ⎡ L ⎤
⎢Y⎥ ⎢0⎥ ⎢0⎥ ⎢L ⎥ ⎢L ⎥ (A.3)
⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
L
⎡X⎤ = ⎡ 2 ⎤
⎢Y⎥ ⎢ L ⎥
⎣ ⎦ ⎢ ⎣2⎦
⎥ (A.4)
Judging from Eqns A.2 and A.4, the coordinated of the centre of mass and centre of stiffness coincide which means uncoupling of
orthogonal directions is permissible.

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S. Jalbi, et al. Marine Structures 67 (2019) 102631

Figure A.1. Plan view of a square foundation arrangement

Symmetric Triangle

Similarly, for a symmetric triangle as shown in schematic Figure A2, the centre of mass may be computed as:
L 3L
L ⎡ ⎤ ⎡ ⎤
3m ⎡ X ⎤ = m ⎡ 2⎤
4 4
⎢Y⎥ ⎢ ⎥ + m⎢
⎢ 3L
⎥ + m⎢
⎥ ⎢ 3L
⎥
⎥
⎣ ⎦ ⎣0⎦ ⎣ 4 ⎦ ⎣ 4 ⎦ (A.5)
L
⎡ ⎤
⎡X⎤ = ⎢ 2
⎥
⎢Y⎥ ⎢ 3L
⎥
⎣ ⎦ (A.6)
⎣ 6 ⎦

And the centre of stiffness as shown in equation A.7

L
0 ⎡ ⎤
3k ⎡ X ⎤ = k ⎡ ⎤ + k ⎡ L ⎤ + k ⎢
2
⎥
⎢Y⎥ ⎢0⎥ ⎢0⎥ ⎢ 3L
⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (A.7)
⎣ 2 ⎦

L
⎡ ⎤
⎡X⎤ = ⎢ 2
⎥
⎢Y⎥ ⎢ 3L
⎥
⎣ ⎦ (A.8)
⎣ 6 ⎦

In a similar manner to the square foundations, uncoupling may be performed on symmetric triangles with identical foundations.

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S. Jalbi, et al. Marine Structures 67 (2019) 102631

Figure A.2. Plan view of a symmetric triangle foundation arrangement2

Asymmetric Triangles

Consider the asymmetric triangle shown in Figure A3. Retaining the assumption that the members of the foundation have the
same density and cross-section, the mass of the horizontal member is 2 times the mass of other members due to its length.

Figure A.3. Plan view of an asymmetric triangle foundation arrangement3

Centre of mass:
2L
⎡ 2L ⎤ ⎡ ⎤ ⎡3 2L
⎤
(2 + 2 )m ⎡ X ⎤ = 2 m⎢ 2 ⎥ + m⎢ 4
⎥ + m⎢ 4
⎥
⎢Y⎥ ⎢ 2L ⎥ ⎢ 2L ⎥
⎣ ⎦ ⎣ 0 ⎦ ⎣ 4 ⎦ ⎣ 4 ⎦ (A.9)

2L
⎡ ⎤
⎡X⎤ = ⎢ 2
⎥
⎢ Y ⎥ ⎢ (−1 + 2 )L ⎥
⎣ ⎦ (A.10)
⎣ 2 ⎦
Similarly, the centre of stiffness is
2L
0 ⎡ ⎤
3k ⎡ X ⎤ = k ⎡ ⎤ + k ⎡ 2 L ⎤ + k ⎢ 2
⎥
⎢Y⎥ ⎢0⎥ ⎢ 0 ⎥ ⎢ 2L ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (A.11)
⎣ 2 ⎦

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S. Jalbi, et al. Marine Structures 67 (2019) 102631

2L
⎡ ⎤
⎡X⎤ = ⎢ 2
⎥
⎢Y⎥ ⎢ 2L ⎥
⎣ ⎦ (A.12)
⎣ 6 ⎦

Judging from Eqns A.10 and A.12, vibrations across orthogonal planes cannot be assessed independently and a 3D Lagrange
formulation is required. If however, the mass of the horizontal member is m rather than 2 m (Due to a smaller cross section for
instance) the centre of mass equation becomes as follows:
2L
⎡ 2L ⎤ ⎡ ⎤ ⎡3 2L
⎤
3m ⎡ X ⎤ = m ⎢ 2 ⎥ + m ⎢ 4
⎥ + m⎢ 4
⎥
⎢Y⎥ ⎢ 2L ⎥ ⎢ 2L ⎥
⎣ ⎦ ⎣ 0 ⎦ ⎣ 4 ⎦ ⎣ 4 ⎦ (Eq A.13)

2L
⎡ ⎤
⎡X⎤ = ⎢ 2
⎥
⎢Y⎥ ⎢ 2L ⎥
⎣ ⎦ (A.14)
⎣ 6 ⎦

Now judging from equation (A.14) with A.12, a match is observed and thus decoupling may occur even with an asymmetrical
arrangement.

Appendix 2. Calculation of lumped mass m2

At this stage, designers can estimate the distributed mass of the jacket and tower in kg/m as shown in Figure A4. The first step in
obtaining m2 is to obtain the equivalent distributed mass of the tower and jacket system meq. The Kinetic Energy of the system is
calculated as per Eq A.15

KE = ∫ m(z)φ2dz (A.15)
where m(z) and φ are the mass and eigen mode function of a continuous cantilever system. Equating the Kinetic energy of the tower-
jacket system with the equivalent beam

∫ m(z)φ2dz = ∫ meq φ2dz

Further simplification leads to
z(i) hJ hJ +hT
n
∑i= 1 mi ∫ φ12 dz mJ ∫ φ12 dz + mT ∫ φ12 dz
∫ m(z)φ12 dz z(i−1) 0 hJ
m eq = = =
∫φ12dz z(i) hT +hJ
∫ φ12 dz ∫ φ12 dz
0 0 (A.16)
The value of the integral of the square of the first mode function can be evaluated using Eq A.17
1 −β12 2λ1 β1 2λ1 1 +β12 2λ1 β12 2λ1 2β1 λ λ
∫ φ12 dz = z+ 4λ1
sin L
z− 2λ1
cos L
z+ 4λ1
sinh L
z+ 2λ1
cosh L
z− λ1
sin L1 z×sinh L1 z−
L L L L L
1 +β12 λ λ 1 −β12 λ λ
λ1
sin L1 z×cosh L1 z− λ1
cos L1 z×sinh L1 z
L L (A.17)

Figure A.4. Equivalent distributed mass of a jacket4

Such that
L= hJ + hT
And.Where λ1 and β1 dimensionless natural frequency parameters of an Euler-Bernoulli cantilever beam.
λ = 1.8751 and

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S. Jalbi, et al. Marine Structures 67 (2019) 102631

cosλ1 + coshλ1
β1 = −
sinλ1 + sinhλ1
The second step is to obtain the equivalent lumped mass at the tip of the tower in kg.
Which is calculated as
m2 = ε(m eq htotal)+ MRNA

where ε has been derived to be ε = 0.243.

It may be reminded that this method is applicable to the first mode of vibration for higher mode of vibrations the second mode φ2
should be used rather than φ1

Appendix 3. Computation of the vertical stiffness for shallow foundations kv

Table A.1 provides guidance on how to compute the vertical stiffness for shallow embedded foundations. It must be mentioned
that the method presented in this paper assumes a “linear” response of the foundations to obtain the natural frequencies.

Table A.1
Guidance on the selection of vertical

Shallow foundations

Reference Applicability Vertical stiffness

(Gazetas, 19- For rigid shallow embedded foundations in homogeneous ground 2

91) [23] profiles kv =
2.01Gs DC
(1 −υs) (1.02 + 0.1 ) ⎛⎜⎝1 + 0.51 ( ) ⎞⎟⎠
LC
DC
LC 3
DC

(Wolf & Dee- For rigid shallow embedded foundations in homogenous ground
ks, 2004) profiles
kv =
2Gs DC
(1 −υs) (1 + 1.08 )
LC
DC
[24]
(Doherty et a- For rigid shallow caissons in homogenous, parabolic, and linear Solutions for vertical stiffness of caissons provided in tabular format and is
l., 2005) ground profiles dependent on relative soil to pile stiffness, embedment ratio, and ground
[25] profile stiffness variation with depth
(Skau et al., For flexible shallow suction caissons. Dependent on finite element Adjusted the macro-element model provided in Ref. [27] (which assumes
2018) [2- soil model for the extraction of the macro-element model rigid behaviour) where the bending of the caisson lid in the vertical direction
6] inherently reduces the stiffness of the foundation in addition to changing the
volume of the soil plug, i.e. changing the stress state of the soil which also
reduces the stiffness. This has also been observed by site measurements shown
in Ref. [28]

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