tim/PAPERS/AMM_2016.pdf · Applied Mathematical Modelling 40 (2016) 8217–8243 Contents lists available at ScienceDirect Applied Mathematical Modelling journal homepage:

Applied Mathematical Modelling 40 (2016) 8217–8243

Contents lists available at ScienceDirect

Applied Mathematical Modelling

journal homepage: www.elsevier.com/locate/apm

Undamped eigenperiods of a sea-based gravity monotower

O.M. Faltinsen

a , A.N. Timokha

a , b , ∗

a Centre for Autonomous Marine Operations and Structures, Norwegian University of Science and Technology, NO-7491 Trondheim,

Norway b Institute of Mathematics, National Academy of Sciences of Ukraine, Tereschenkivska 3, Str., 01601 Kiev, Ukraine

a r t i c l e i n f o

Article history:

Received 22 August 2015

Revised 26 February 2016

Accepted 13 April 2016

Available online 27 April 2016

Keywords:

Monotower

Coupled dynamics

Sloshing

Variational methods

a b s t r a c t

An analytically approximate method was proposed in [1] to estimate undamped eigenpe-

riods of a sea-based gravity monotower by accounting for sloshing inside the shaft. These

results are generalised to include the soil feedback and the inertia moment of the top

rigid body as well as to describe the external hydrodynamic loads using the linear three-

dimensional potential flow theory of an incompressible liquid. A mathematical model of

the multicomponent mechanical system is presented. The virtual work principle is used

to express the Euler–Bernoulli governing equation and all fluid-structure dynamic trans-

mission conditions. Numerical examples of the eigenperiods versus geometric and physi-

cal input parameters typical for the Draugen platform and some monopiles are given. The

highest eigenperiod of the horizontal vibrations belongs to experimentally-known ranges.

These eigenperiods increase with increasing mass, radius of gyration and mass centre of

the top body; they also increase with the shear modulus of the soil. Three classes of eigen-

modes are detected. They express dominant character of structural vibrations and sloshing,

respectively, or a mixed type. Sloshing is less important for existing monopiles.

© 2016 Elsevier Inc. All rights reserved.

1. Introduction

Sea-based gravity monotowers are used in the oil/gas production and the wind energy industry. The tower shaft is nor-

mally filled with water [1–6] . A rigid body (operational platform, rotor, nacelle, etc.) is installed at the tower top. Horizontal

and vertical structural vibrations are affected by the soil as well as the exterior and interior hydrodynamic loads. Excitations

could be related to incident (periodic or/and impact) wave loads and Earthquake. The two main differences of our mono-

tower problem relative to the elevated water tank problem [7,8] are that we have to consider hydrodynamic loads associated

with the outer water and that sloshing loads are not applied to the tower top but the shaft wall. The latter is like in the

fuel rocket problem [9,10] .

Pursuing a non-phenomenological theory of the eigenvalue problem of a sea-based gravity monotower, [1, Section 5.4.5] ,

proposed an approximate mechanical/mathematical model in which: (i) soil beneath the monotower is rigid, (ii) a mass

point ( not rigid body! ) is attached to the tower top, (iii) strip theory with frequency-independent added mass described

the external hydrodynamic loads in the undamped eigenvalue problem. The present paper derives a more sophisticated

mechanical/mathematical model assuming that (i ∗) the elastic soil feedback matters, (ii ∗) a rigid body ( not mass point! ) is

∗ Corresponding author at: Centre for Autonomous Marine Operations and Structures, Norwegian University of Science and Technology, NO-7491 Trond-

heim, Norway. Tel.: +47 73595524.

E-mail addresses: [email protected] (O.M. Faltinsen), [email protected] (A.N. Timokha).

http://dx.doi.org/10.1016/j.apm.2016.04.003

S0307-904X(16)30203-7/© 2016 Elsevier Inc. All rights reserved.


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8218 O.M. Faltinsen, A.N. Timokha / Applied Mathematical Modelling 40 (2016) 8217–8243

attached to the tower top, (iii ∗) the external hydrodynamic loads follow from considering the linear three-dimensional water

wave problem with body interactions. Excitations are not considered . Emphasis is on a theoretical estimate of undamped

eigenperiods . Numerical examples deal with the Draugen monotower and a sample gravity monopiles from the wind energy

industry.

Elastic, rigid, and hydrodynamic elements of a multicomponent mechanical system modelling a sea-gravity monotower

are introduced in Section 2 . Its central element is the so-called inverted elastic pendulum [7,8] , i.e. a vertical Euler–Bernoulli

beam, which is set upon a stiff/hard soil and has a rigid body attached to the top. The axial elastic beam oscillations are ne-

glected since they are characterised by much lower eigenperiods than periods of external wave excitations. The soil elasticity

introduces restoring forces and moments applied to the lower beam end. The beam is characterised by noncontinuous phys-

ical and geometrical input parameters and interacts with the inner (in shaft) and external (sea water) liquids. Soil, exterior

and interior water are coupled with the beam motions but not directly with each other. As a consequence, the soil-related

response and the hydrodynamic loads become only functions of the structural deflection and velocity. The virtual work

principle plays the role of a variational governing equation of the beam. By assuming the coordinate axes are parallel to the

principal axes of inertia of the top rigid body decouples the unknown variables so that one can analyse, independently, the

free unforced horizontal structural vibrations in the two vertical coordinate planes as well as the vertical unforced structural

motions (as a rigid body).

Section 3 evaluates the undamped eigenperiod of the vertical vibrations. For an almost cylindrical tower that is typical

for the sea-based gravity monopiles of the wind turbine industry, the external hydrodynamic loads due to the vertical

tower vibrations can be neglected. However, they must be included for the Draugen platform, which has a massive deeply

submerged caisson. Computations show that the corresponding nondimensional added mass is weakly frequency-dependent.

For the Draugen platform, including the added mass effect increases the undamped eigenperiod for vertical motions by about

30% (e.g., from 0.375 to 0.43 s).

In Section 4 , the aforementioned decoupling of the unknown variables makes it possible to concentrate on undamped

horizontal eigenoscillations in a vertical coordinate plane. The virtual work principle from Section 2 derives a variational

governing equation of the corresponding eigenvalue problem restricted by three functional constraints, which express the

hydrodynamic loads associated with the external flow, the Stokes–Joukowski potential (assuming frozen inner free surface)

and sloshing. The constraints appear as three linear operators defined on the trial structural eigenmodes. When the trial

eigenfrequencies belong to a subset of the natural sloshing frequencies, the sloshing-related functional constraint becomes

mathematically undefined. We select three classes of eigenperiods and modes. The first class ({ T k } and W

( k ) ( z )) neglects

sloshing, i.e. what we call the structural eigenperiods and modes. Describing them implies finding the structural eigenoscilla-

tions affected by the external frequency-dependent added mass and the frequency-independent internal added mass associ-

ated with the Stokes–Joukowski potential (see, more details on the Stokes–Joukowski potential in [1, chapter 5] ). The second

class of eigenperiods { T sk } is the natural sloshing periods . The third class corresponds to the coupled structure-and-sloshing

eigenoscillations with eigenperiods { T ck }, structural W

(k ) c and sloshing-related φ

W

(k ) c

eigenmode-components. Relationships

between these three classes versus geometric and physical parameters are described by using a projective scheme based on

the Trefftz method for the hydrodynamic problems and a special functional basis for the structural eigenmodes W

( k ) ( z ) (or

W

(k ) c ). The basis provides a natural classification of the structural motions distinguishing, in particular, the rigid-body beam

motions and the beam deflections considered in [1, Section 5.4.5] . The convergence is reported in Section 4 (regarding the

Trefftz method) and in Section 5 (for the entire projective scheme).

Section 5 collects and discusses a series of numerical examples associated with the Draugen platform and a sample

gravity monopiles whose geometric and physical parameters are provided in Appendices C and B, respectively. The section

starts with analysing assumption (iii). We show that there are clear three-dimensional flow effects at the free surface and

at the caisson of the Draugen platform. The frequency-dependency matters only near the mean sea surface.

In Section 5.2 , numerical studies of the highest eigenperiod for the Draugen monotower are conducted. A good agreement

with the full-scale observed period range is documented. Sloshing cannot be neglected. Furthermore, even though the model

assumes the caisson as a piece of the Euler–Bernoulli beam, computations confirm as expected that the caisson moves like

a rigid body and the speculatively-taken Young’s modulus for this beam piece weakly affects that.

Evaluating a sample gravity monopiles proposed in Appendix B to fit a future, much more massive wind-energy construc-

tion, one can expect T 1 ≈ T s 1 for certain soil characteristics and parameters of the top-attached rigid body (here, rotor and

nacelle). Numerical studies in Section 5.3 establish qualitatively the same behaviour of the monopiles as it was described in

Section 5.2 (the Draugen monotower). However, computations show that sloshing loads can, generally, be neglected. Another

matter would be if a Tuned Liquid Damper is installed close to the top of the monopiles.

2. Mechanical/mathematical model

2.1. Introductory remarks

A mathematical model of the considered multicomponent engineering system is presented in a dimensional form. The

model can be rewritten in a nondimensional form after choosing the characteristic size and time as the tower radius at the

mean free surface and the highest eigenperiod of the corresponding cantilever beam.

O.M. Faltinsen, A.N. Timokha / Applied Mathematical Modelling 40 (2016) 8217–8243 8219

a b

Fig. 1. A schematic view of the considered mechanical system in panel (a). The actual geometric proportions are not reflected. The long vertical axisym-

metric structure consists, formally, of a caisson and a tower. A rigid body is attached to the top. The caisson is set upon a circular fundament which

transmits the stiff/hard soil elastic feedback. There are inner (tower shaft) and external (sea water) liquids with free surfaces. The contained liquid has

no contact with the soil but only with the solid walls and bottom; the liquid mass and volume are constant values. Panel (b) illustrates three typical

monotower/monopiles shapes that can be associated with gravity monopiles of the wind energy industry (top and middle) and the Draugen platform

(bottom).

Fig. 1 (a) schematically depicts the fluid-structure-soil mechanical (mathematical) model. There are liquids with free sur-

faces inside and outside. The vertical liquid levels can theoretically differ. A tall and hollow axisymmetric vertical structure

(henceforth, monotower ) consists, formally, of a long tower keeping almost constant internal and external radii at the mean

free surface zone and a heavyweight cylindrical caisson (supporting base). The latter is set upon an underground fundament,

which is a rigid circular plate of a relatively small height. A rigid body is attached to the monotower top. The constructed

mechanical model has to describe the sea-based gravity monotowers associated with shapes in Fig. 1 (b).

The monotower. The vertical monotower length is sufficiently larger than its maximum diameter to adopt, for the lower

modes, the Euler–Bernoulli beam mathematical model with, generally speaking, noncontinuous geometrical and physical

characteristics. Since it facilitates the derivation to pursue a uniform analytical scheme, the caisson is considered as a piece

of the entire Euler–Bernoulli beam. This is consistent for some wind turbine monopiles but not for the Draugen platform

whose parameters are given in Appendix C . Physically, the caisson should then move as a rigid body. However, our extensive

numerical experiments showed that the beam caisson model reflects this physical expectation, i.e. it moves as a rigid body.

A rigid circular fundament transmits the soil-induced restoring forces and moments to the lower beam end. The inertia

forces and moments of the top rigid body are applied to the upper beam end. In other words, we deal with what is often

called an inverted beam pendulum on an elastic support [7,8] whose motions are, in addition, affected by hydrodynamic

loads. Soil and liquids interact with the monotower (beam) but not directly with each others. The axial elastic oscillations

have much lower eigenperiods than periods of external excitations and are therefore neglected [7,8] . Replacing the tower by

an equivalent uniform bar with the same Young’s modulus, density and length (from our numerical examples) results in a

highest eigenperiod about 0.1 s while the highest eigenperiod of the transverse oscillations is about 4 s.

Structural motions are considered in the Earth-fixed coordinate system Oxyz whose vertical axis coincides with the sym-

metry axis in its upright static state. The vertical axis Oz is counter-directed to the gravity acceleration vector g and the Oxy

plane is superposed with the horizontal seabed. The horizontal structural deflections in Oxz and Oyz are governed by func-

tions v 1 ( z, t ) and v 2 (z, t) , 0 < z < L = l 1 + l 2 , respectively. Vertical (heave) beam motions (as a rigid body) are possible due to

the soil-induced vertical restoring force. They are described by the generalised coordinate v 3 (t) = η3 (t) = ηp 3 (t) where η3 ( t )

and ηp 3 (t) describe small-amplitude heave motions in the Oxyz and O p x p y p z p coordinate systems introduced at the lower and

upper beam ends, respectively (see, Fig. 2 ). The velocity at a fixed point of the beam is the vector ( ̇ v 1 (z, t) , ˙ v 2 (z, t) , ˙ v 3 (t)) .

The z -variable geometric and physical characteristics of the beam are the inner radius R i ( z ), the external radius R e ( z ), the

Young modulus E ( z ), the second moment of inertia I(z) =

1 4 π(R 4 e (z) − R 4

i (z)) , the beam mass density ρb ( z ), and the cross-

sectional area of the beam S b (z) = π(R 2 e (z) − R 2 i (z)) . Fig. 2 illustrates that these functions can have jumps at z = h 1 and l 2


Fig. 2. A schematic (actual geometric proportions are not kept) meridional cross-section and a three-dimensional view of the monotower. The Oxyz coordi-

nate system is Earth-fixed. The coordinate system O p x p y p z p is rigidly fixed with the upper beam end in its static state; its axes are parallel to the Oxyz -axes.

Function R i ( z ) denotes the shaft radius (equal to zero as 0 < z < h 1 ). Function R e ( z ) governs the external radius: r 1 = R e (l 2 +) , r 2 = R e (l 2 −) ; h is the liquid

depth in the shaft, and d is the sea depth. We assume that 0 < h 1 < l 2 , which means that the shaft bottom is between the seabed and the caisson roof

levels. The circular fundament has radius r 4 ≥ r 2 ; L = l 1 + l 2 .

and defines

r 1 = R e (l 2 +) ; r 2 = R e (l 2 −) = R e (0) ; r 3 = R e (d) ; R 0 = R i (h 1 +) ; r 0 = R i (h + h 1 )

with r 1 < r 2 . We assume, for clarity, that h 1 < l 2 ( the shaft continues into the caisson ).

If necessary, the entire vertical beam will be treated as a beam–beam–beam mechanical system whose three elements

have continuous characteristics with respect to z (the beam–shell–beam and beam–beam–beam mechanical systems are

studied in [11] ). This divides v i ( z, t ), 0 < z < L , into the three sets: v (1) i

(z, t) on (0, h 1 ), v (2) i

(z, t) on ( h 1 , l 2 ), and v (3) i

(z, t)

on ( l 2 , L ) so that v (k ) i

(z, t) , i = 1 , 2 , k = 1 , 2 , 3 , possess up to fourth-order continuous derivatives by z on the corresponding

sub-intervals but the only first-order derivative is continuous at h 1 and l 2 . The latter fact implies the kinematic transmission

conditions

v (1) i

(h 1 , t) = v (2) i

(h 1 , t) , (v (1)

i

)′ (h 1 , t) =

(v (2)

i

)′ (h 1 , t) ,

v (2) i

(l 2 , t) = v (3) i

(l 2 , t) , (v (2)

i

)′ (l 2 , t) =

(v (3)

i

)′ (l 2 , t) , i = 1 , 2 ,

(1)

which should be a priori satisfied; the prime denotes the z -derivative. The dynamic transmission conditions express the

continuity of the shear force and the bending moment at z = h 1 and l 2 ; these conditions are derivable from our variational

formulation.

The forces and moments applied to the lower beam end. The monotower is set upon a rigid circular fundament whose small–

amplitude motions are determined by η1 ( t ) (surge), η2 ( t ) (sway), η3 ( t ) (heave), η4 ( t ) (roll), and η5 ( t ) (pitch) associated with

the Oxyz -coordinate system motions (see, notations in Fig. 2 ). Yaw motions are not considered, i.e. η6 (t) = 0 . We follow a

Winkel type approach for a single-layer soil, which has demonstrated applicability for the monotower and elevated water

tank problems (see, [12–14] and references therein). Multi-layer soil and scouring effects are not analysed. The approach

adopts the following empirical formulas expressing the soil-induced restoring forces ( F 1 ( t ), F 2 ( t ), F 3 ( t )) and moments ( F 4 ( t ),

F 5 ( t ), 0) (relative to O ) transmitted by the rigid circular fundament to the lower beam end

F 1 = −k 1 η1 = −k 1 v 1 (0 , t) , F 2 = −k 2 η2 = −k 2 v 2 (0 , t) , (2a)

F 3 = −k 3 η3 (t) = −k 3 v 3 (t) , (2b)

F 4 = −k 4 η4 (t) = −k 4 v ′ 2 (0 , t) , F 5 = −k 5 η5 (t) = k 5 v ′ 1 (0 , t) , (2c)

where the stiffness coefficients k i are

k 1 , 2 =

8 G s r 4 2 − ν

, k 3 =

4 G s r 4 1 − ν

, k 4 , 5 =

8 G s r 3 4

3(1 − ν ) , (3)

s s s


with the shear modulus, G s , Poisson’s ratio, νs , and the fundament radius, r 4 . Because we concentrate on undamped motions,

the soil-induced damping terms are not introduced in (2) . The empirical relations (3) are taken from [13,14] where the

Draugen platform problem with the soil effect was successfully analysed by a phenomenological model. The actual values of

G s and νs can, for instance, be found in [3,15] . The soil-induced restoring forces and moments (2) are kept decoupled in the

Oxz and Oyz planes as well as in the vertical direction, i.e., v 1 ( z, t ) only yields non-zero F 1 ( t ) and F 5 ( t ), v 2 ( z, t ) exclusively

influences F 2 ( t ) and F 4 ( t ), and v 3 ( t ) affects the vertical force F 3 ( t ) only.

The inertia forces and moments applied to the upper beam end. A rigid body with the mass M p is attached to the upper beam

end. The centre of gravity of the rigid body is assumed belonging to the symmetry axis, its coordinate is (0 , 0 , z C p ) in the

coordinate system O p x p y p z p ( Fig. 2 ). We assume that Oz is the principal axis of inertia of the rigid body and choose Ox and

Oy ( O p x p and O p y p are parallel to Ox and Oy , respectively) being parallel to the principal axes of inertia of the rigid body.

Under these assumptions, the diagonal inertia tensor

I p 1

= I p 11

=

∫ h p

0

∫ S p

ρp (y 2 p + z 2 p ) d Sd z p , I p 2

= I p 22

=

∫ h p

0

∫ S p

ρp (x 2 p + z 2 p ) d Sd z p ,

I p 3

= I p 33

=

∫ h p

0

∫ S p ρp (x 2 p + y 2 p ) d Sd z p , (4)

( I p i j

= 0 , i � = j, ρp = ρp (x p , y p , z p ) is the body mass density) emerges.

The six generalised coordinates ηp 1 (t) , ηp

2 (t) , ηp

3 (t) , ηp

4 (t) , ηp

5 (t) , and ηp

6 (t) = 0 associated with the small-magnitude

translatory and angular motion of the O p x p y p z p -system read as

ηp 1

= v 1 (L, t) , ηp 2

= v 2 (L, t) , ηp 3

= v 3 , ηp 4

= −v ′ 2 (L, t) , ηp 5

= v ′ 1 (L, t) . (5)

The forces ( P 1 ( t ), P 2 ( t ), P 3 ( t )) and moments ( P 4 ( t ), P 5 ( t ), 0) associated with the rigid body inertia can therefore be expressed

as

P 1 = M p

(−η̈p

1 − z C p ̈η

p 5

)= −M p

(v̈ 1 (L, t) + z C p ̈v ′ 1 (L, t)

)P 2 = M p

(−η̈p

2 + z C p ̈η

p 4

)= −M p

(v̈ 2 (L, t) + z C p ̈v ′ 2 (L, t)

)P 3 = −M p ̈v 3 ,

(6a)

P 4 = z C p M p

(η̈p

2 + gηp

4

)− I p

11 ̈ηp

4

= z C p M p

(v̈ 2 (L, t) − gv ′ 2 (L, t)

)+ I p

11 ̈v ′ 2 (L, t) ,

(6b)P 5 = z C p M p

(−η̈p

1 + gηp

5

)− I p

22 ̈ηp

5

= z C p M p

(−v̈ 1 (L, t) + gv ′ 1 (L, t)

)− I p

22 ̈v ′ 1 (L, t) , P 6 = 0 .

Henceforth, the dot denotes the time-derivative.

Because of the chosen axes direction, formulas (6) decouple the structural vibrations in the coordinate planes Oxz and

Oyz; v 1 ( z, t ) only yields non-zero P 1 ( t ) and P 5 ( t ) but v 2 ( z, t ) exclusively influences P 2 ( t ) and P 4 ( t ). Vertical motions do not

depend on horizontal vibrations ( v 3 ( t ) affects only the vertical force P 3 ( t )).

The hydrodynamic forces and moments. Inviscid incompressible liquids with irrotational flows are assumed. Adopting this

hydrodynamic model is consistent with our undamped linear analysis neglecting, by definition, dissipation and nonlinear-

ity. Linear sloshing in clean tanks [1] is well described within the model framework. Inviscid potential flow theory for

external linear surface wave loads is common for the so-called large-volume structures [16] , i.e. the type of structures con-

sidered in this paper. The theory is not applicable when viscous flow separation has a dominant role. If we consider a

circular stationary cross-section of diameter D , and ambient flow velocity amplitude U m

and period T , experiments show

that U m

T / D should be less than approximately 2 [16] to avoid flow separation. The latter guidance applies to our prob-

lem by replacing U m

by the relative horizontal velocity amplitude along the submerged structure. The fact that a viscous

boundary layer is present for unseparated flows has a small influence. The standard linear surface wave problems can

then be formulated in the inner (sloshing), Q 0 i , and exterior (sea waves), Q 0 e , domains as shown in Fig. 2 which intro-

duces the mean wetted structural surfaces W 0 i = { (R i (z) , θ, z) : h 1 < z < h 1 + h, 0 < θ ≤ 2 π} , W 0 e = { (R e (z) , θ, z) : 0 < z <

l 2 , l 2 < z < d, 0 < θ ≤ 2 π} , B 0 i = { (r, θ, h 1 ) : 0 ≤ r < R 0 , 0 < θ ≤ 2 π} , and B 0 e = { (r, θ, h 1 ) : r 1 < r < r 2 , 0 < θ ≤ 2 π} as well

as the mean free surfaces �0 i = { (r, θ, h 1 + h ) : 0 ≤ r < r 0 , 0 < θ ≤ 2 π} and �0 e = { (r, θ, d) : r 2 < r, 0 < θ ≤ 2 π} . The in-

ner domain Q 0 i is confined by W 0 i , B 0 i , and �0 i . The exterior domain Q 0 e is confined by W 0 e , B 0 e , �0 e , and the seabed

B 0 s = { (r, θ, 0) : r 2 < r, 0 < θ ≤ 2 π} . The cylindrical coordinate system Or θz (linked with Oxyz ) is naturally introduced.

Accounting for the linear surface wave theory, the velocity potentials ϕ i ( r, θ , z, t ) (sloshing) and ϕ e ( r, θ , z, t ) (sea waves)

are defined in Q 0 i and Q 0 e , respectively. They satisfy the following governing (Laplace) equation and boundary conditions in

the cylindrical coordinates ( r, θ , z ) (see, e.g., chapters 3 and 5 in [1] ):

∇

2 ϕ i,e = 0 in Q 0 i,e , (7a)

∂ϕ i,e / ∂n =

(cos θ ˙ v 1 + sin θ ˙ v 2 − R

′ i,e ̇ v 3

)/

√

1 + R

′ 2 i,e

on W 0 i,e , (7b)


∂ϕ i,e / ∂z =

˙ v 3 − r cos θ ˙ v ′ 1 − r sin θ ˙ v ′ 2 on B 0 i,e , (7c)

∂ϕ i,e / ∂z =

˙ ζ0 i,e , ˙ ϕ i,e + g ζ0 i,e = 0 on �0 i,e , (7d)

where z = h + h 1 + ζ0 i (r, θ, t) and z = d + ζ0 e (r, θ, t) determine the free-surface elevations of the interior and exterior liq-

uids, respectively, and n is the outer (from interior to exterior) normal vector. The function ζ 0 i should satisfy the volume

conservation condition in the Earth-fixed coordinate system, ∫ �0 i

˙ ζ0 i dS = v 3 (t) . The exterior boundary problem requires the

zero–Neumann boundary condition on the horizontal seabed, i.e.

∂ϕ e / ∂z = 0 on B 0 s , (8)

and, in the frequency domain, the corresponding radiation condition at the infinity (as r → ∞ ). The present paper does

not consider incident waves. Initial conditions (initial surface shape and velocity at t = t 0 ) are required in the time-domain

formulation (7), (8) . For the eigenvalue problem, the initial conditions are not needed but the Sommerfeld condition for

outgoing waves should be used.

The only inhomogeneous quantities in (7) are associated with

˙ v i , i = 1 , 2 , 3 , appearing in the Neumann boundary condi-

tions (7b) and (7c) so that one can find the θ–Fourier solution

ϕ i,e (r, θ, z, t) = ϕ

(0) i,e

(r, z, t | ̇ v 3 (t ) , init.cond. ) + ϕ

(1) i,e

(r, z, t| ̇ v 1 (z, t) , init.cond. ) cos θ

+ ϕ

(2) i,e

(r, z, t| ̇ v 2 (z, t) , init.cond. ) sin θ + [residual terms] , (9)

where the residual terms are associated with quantities proportional to sin m θ and cos m θ , m ≥ 2. For the time-domain

problem, the residual terms do not depend on v i but only on the initial conditions. The terms do not appear in the eigen-

value problem.

Hydrodynamic loads on the interior wetted shaft surface are caused by the linear dynamic pressure −ρl ˙ ϕ i where ρ l is

the contained liquid density and give rise to the horizontal forces (per unit length) along the Oz -axis in the Ox and Oy

directions, respectively,

f ki (z, t) = −χ[ h 1 ,h 1 + h ] (z) πρl R i (z) ˙ ϕ

(k ) i

(R i (z) , z, t| ̇ v k (z, t) , init.cond. ) , (10)

k = 1 , 2 , where χ [ a, b ] ( z ) is the Heaviside function χ[ a,b] (z) = 1 , if a ≤ z ≤ b , and = 0 otherwise. The vertical force reduces,

for the linear sloshing problem of a liquid mass M l in a closed basin (no contacts with the soil as well as there are in-

flows/outflows), to the liquid inertia (see, details in [1, Eq. (5.38)] where the mean liquid mass centre belongs to the sym-

metry axis)

f 3 i (t) = −M l v̈ 3 (t) . (11)

The hydrodynamic moments around the shaft bottom centre (0, 0, h 1 ) are

f ki (t) = πρl (−1) k ∫ R 0

0

r 2 ˙ ϕ

(6 −k ) i

(r, h 1 , t| ̇ v 6 −k (z, t) , init.cond. ) dr, k = 4 , 5 . (12)

Analogously, the hydrodynamic loads of the external sea water cause the hydrodynamic force (per unit length)

f ke (z, t) = χ[0 ,d] (z) ρw

πR e (z) ˙ ϕ

(k ) e (R e (z) , z, t| ̇ v k (z, t) , init.cond. ) , (13)

k = 1 , 2 ; the vertical hydrodynamic force

f 3 e (t) = 2 πρw

(−

∫ d

l 2

R e (z) R

′ e (z) ˙ ϕ

(0) e (R e (z) , z, t | ̇ v 3 (t ) , init.cond. ) dz +

∫ r 2

r 1

r ˙ ϕ

(0) e (r, l 2 , t | ̇ v 3 (t ) , init.cond. ) dr

)(14)

and the moments around (0, 0, l 2 ) due to the hydrodynamic pressure acting on the horizontal annular roof B 0 e of the caisson

are

f ke (t) = πρw

(−1) k ∫ r 2

r 1

r 2 ˙ ϕ

(6 −k ) e (r, l 2 , t| ̇ v 6 −k (z, t) , init.cond. ) dr, (15)

k = 4 , 5 , where ρw

is the sea water density. When we deal with the upper and middle monotower shapes in Fig. 1 (b), the

hydrodynamic moments vanish.

The residual terms in ( 9 ) do not affect the hydrodynamic forces and moments ( 10 )–( 15 ) and, therefore, can be ignored.

In addition to the above-introduced hydrodynamic forces and moments (10) –(15) , there exists the vertical quasi-static

restoring force due to external sea water,

f̄ 3 e (t) = −πρw

g(r 2 2 − r 2 3 ) v 3 (t) , (16)

where r 2 is the monotower radius at the seabed level and r 3 is the radius at the mean sea water free surface. Expres-

sion (16) follows from considering the change in the external hydrostatic load. The contained liquid volume is a constant

value; it does not yield a quasi-static vertical force. The quasi-static moment exists caused by a local (at each fixed z ) beam

inclination. The moment will further be handled for the vertical structure (beam) following the procedure from [17] and [7] .

We see that the liquid-related vertical forces depend only on v 3 ( t ), the horizontal forces and moments in the Oxz plane

are exclusively affected by ˙ v 1 (z, t) , but the horizontal forces and moments in the Oyz plane are only linear functions of

˙ v 2 (z, t) . In other words, the hydrodynamic loads in the Oxz - and Oyz -planes are decoupled within the framework of our

linear hydrodynamic model.


(

(

(

(

(

2.2. Free linear oscillations

External forces and moments are functions of v i and initial conditions of the linear surface-wave problem (7), (8) . This

section gives the governing dynamic equations with respect to v i (z, t) , i = 1 , 2 , in terms of the virtual work principle. Based

on the third Newton law, an ordinary differential equation with respect to v 3 ( t ) will be formulated.

2.2.1. Beam vibrations

Following [7,11] , the beam vibrations will be described using the virtual work principle [18] . According to this variational

principle, the full variation of the potential energy,

δW =

∫ L

0

EI

2 ∑

k =1

(v ′′ k δv ′′ k ) dz, (17)

should be equal to the sum of virtual works caused by different-type loads applied to the beam, i.e.

δW =

N f ∑

i =1

δA i , (18)

where admissible functions v k , and their variations δv k , k = 1 , 2 , are continuous together with the first-order derivatives by

z that automatically provides the kinematic transmission conditions (1) . The following loads ( N f = 6 ) should be accounted

for in the studied case:

a) the horizontal inertia force of the beam;

b) the body-and-liquids weight (quasi-static) moments due to a local beam inclination;

c) the forces and moments (6) associated with inertia of the top-installed rigid body;

d) the soil-induced restoring forces and moments (2) ;

e) the horizontal hydrodynamic forces (10) (contained liquid) and (13) (sea water);

(f) the hydrodynamic moments at z = h 1 (shaft bottom, (12) ) and l 2 (the ring-shaped caisson roof, (15) ).

(a) The virtual work due to inertial properties of the beam is

δA 1 = −∫ L

0

ρb S b

2 ∑

k =1

( ̈v k δv k ) dz, (19)

where ρb ( z ) is the mass density and S b ( z ) is the cross-sectional area.

(b) Following [17] and [7] , one can define the virtual work caused by the quasi-static (structural and liquid) moment due to

the beam inclination at each z as

δA 2 =

∫ L

0

N(z) 2 ∑

k =1

(v ′ k δv ′ k ) dz, N(z) = N b (z) + N i (z) + N e (z) , (20)

where N ( z ) is continuous on (0, L ), except at z = h 1 and l 2 , and includes three summands responsible for the structure,

internal and external liquids.

The structure-related summand,

N b (z) = M p g + ρb S b (z) g(L − z) , (21)

accounts for the top rigid body weight, M p g , and the beam weights; the quantity (21) was already included in analysis of

an elevated water tank [7] . The contained liquid summand reads as

N i (z) = χ[ h 1 ,h 1 + h ] (z) ρl S i (z) g(h 1 + h − z) , (22)

( S i (z) = πR 2 i (z) is the area of the shaft cross-section) but

N e (z) = −χ[0 ,d] (z) ρw

S e (z) g(d − z) , (23)

( S e (z) = πR 2 e (z) is the area of the external beam cross-section) expresses the same for the external sea water.

(c) Employing (6) derives the virtual work due to the top rigid body inertia

δA 3 =

2 ∑

k =1

[− M p

(v̈ k (L, t) + z C p ̈v ′ k (L, t)

)δv k (L, t) +

(z C p M p

(−v̈ k (L, t) + gv ′ k (L, t)

)− I p

3 −k ̈v ′ k (L, t)

)δv ′ k (L, t)

]. (24)

(d) The virtual work yielded by the soil-induced restoring forces and moments (2) reads as

δA 4 = F 1 δv (0 , t) + F 2 δw (0 , t) + F 4 δw

′ (0 , t) − F 5 δv ′ (0 , t) = −2 ∑

j=1

(k j v j (0 , t) δv j (0 , t) + k 6 − j v ′ j (0 , t) δv ′ j (0 , t)) . (25)


(e) Horizontal hydrodynamic forces (per unit length) due to internal and external liquids yield the virtual work

δA 5 = δA 5 i + δA 5 e , (26)

where, using (10) , the contained liquid leads to the virtual work

δA 5 i =

∫ L

0

2 ∑

j=1

( f ji δv j ) dz = −πρl

∫ h 1 + h

h 1

R i (z) 2 ∑

j=1

( ˙ ϕ

( j) i

(R i (z) , z, t| ̇ v j (z, t)) δv j ) dz. (27)

Using (13) , the sea water causes the virtual work

δA 5 e =

∫ L

0

2 ∑

j=1

( f je δv j ) dz = πρw

∫ d

0

R e (z) 2 ∑

j=1

( ˙ ϕ

( j) e (R e (z) , z, t| ̇ v j (z, t) δv j ) dz. (28)

(f) Similarly to δA 5 , the virtual work due to hydrodynamic moments at z = h 1 (the bottom of the shaft) and l 2 (the ring-

shaped caisson roof) reads as

δA 6 = f 4 i (−δv ′ 2 (h 1 , t)) + f 5 i δv ′ 1 (h 1 , t) ︸︷︷︸ δA 6 i

+ f 4 e (−δv ′ 2 (l 2 , t)) + f 5 e δv ′ 1 (l 2 , t) ︸︷︷︸ δA 6 e

, (29)

where

δA 6 i = −πρl

∫ R 0

0

r 2 2 ∑

j=1

[ ˙ ϕ

( j) i

(r, h 1 , t| v j (z, t)) δv j (h 1 , t)] dr, (30)

and

δA 6 e = −πρw

∫ r 2

r 1

r 2 2 ∑

j=1

[ ˙ ϕ

( j) e (r, l 2 , t| v j (z, t)) δv j (l 2 , t)] dr. (31)

The virtual work principle (18) with (19) –(31) derives the Euler–Bernoulli beam equation and the dynamic bound-

ary/transmission conditions provided that all kinematic relations are a priori satisfied and the three main terms of the

Fourier solution (9) are known. Even though this differential statement is not used in the present paper, it is formulated

in Appendix A . The boundary problem couples the beam deflections v j (z, t) , j = 1 , 2 , and needs initial conditions (initial

perturbations and velocities). The solution also depends on initial conditions for the linear surface (internal and external)

problems (7), (8) . Both variational Eq. (18) and its differential analogy in Appendix A clearly demonstrate decoupling of v 1 and v 2 . This means that one can independently consider the free beam vibrations in Oxz and Oyz , respectively.

2.2.2. Vertical beam motions

The small-magnitude vertical beam motion (as a rigid body) is described by the generalised coordinate v 3 ( t ), which does

not depend on v j (z, t) , j = 1 , 2 . The governing equation with respect to v 3 ( t ), follows from the third Newton law and reads,

accounting for (11), (14) and (16) , as

(M p + M b + M l ) ̈v 3 + (k 3 + πρw

g(r 2 2 − r 2 3 )) v 3 = f 3 e (t) , (32)

where M l is the contained liquid mass and M b is the total beam mass. The right-hand side of (32) is dependent on

˙ v 3 (t) .

Eq. (32) describes the free vertical structural oscillation subject to its initial vertical perturbations ( v 3 ( t 0 ) and

˙ v 3 (t 0 ) ) and

initial axisymmetric perturbations (surface patterns and velocities at t = t 0 ) of the sea water. The latter perturbations are

implicitly included in ϕ

(0) e and, in turn, f 3 e ( t ) defined by (14) .

3. Vertical eigenoscillations

Looking for a single-harmonic (eigen) solution of (32) postulates

v 3 = cos (σ t) , ϕ e = ϕ

(0) e = −σ

[sin (σ t) ψ 0 (r, z| σ 2 ) + cos (σ t) ψ̄ 0 (r, z| σ 2 )

], (33)

where the velocity potential components ψ 0 and ψ̄ 0 cause added-mass and wave-radiation damping components, respec-

tively. These components can be found from the corresponding boundary value problems in the meridional cross-section D e


Fig. 3. Meridional cross-section and geometric notations of the hydrodynamic problems.

of Q e (see, Fig. 3 ) following from substituting (33) into (7), (8) . The Sommerfeld radiation condition,

lim

r→ 0 r 1 / 2

(∂ψ 0 / ∂r − K ψ̄ 0

)= 0 , (34)

should be added where K is the unknown σ -dependent wave number.

Satisfying the governing equation, the Sommerfeld condition, and the boundary conditions on S 0 and S 4 gives, using

separation of spatial variables, r and z , that

ψ 0 = C 0 Z 0 (z) J ′ 0 (Kr 3 ) J 0 (Kr) + Y ′ 0 (Kr 3 ) Y 0 (Kr)

J ′ 0

2 (Kr 3 ) + Y ′

0

2 (Kr 3 )

+

q 3 ∑

j=1

C i Z j (z) K 0 (K j r)

K 0 (K j r 3 ) =

q 3 ∑

j=0

C i φ(0) j

(r, z| σ 2 ) , (35a)

ψ̄ 0 = C 0 Z 0 (z) Y ′ 0 (Kr 3 ) J 0 (Kr) − J ′ 0 (Kr 3 ) Y 0 (Kr)

J ′ 0

2 (Kr 3 ) + Y ′

0

2 (Kr 3 )

+

q 3 ∑

j=1

C̄ i φ(0) j

(r, z| σ 2 )

= C 0 φ̄(0) 0

(r, z;σ 2 ) +

q 3 ∑

j=1

C̄ i φ(0) j

(r, z| σ 2 ) , q 3 → ∞ , (35b)

where

Z 0 (z) =

cosh (Kz)

cosh (Kd) , K tanh (Kd) =

σ 2

g , (36a)

Z j (z) = cos (K j z) , −K j tan (K j d) =

σ 2

g , j ≥ 1 (36b)

compute K and K j as functions of σ 2 . The coefficients { C j , j = 0 , ..., q 3 } = c and { ̄C j , j = 1 , ..., q 3 } = c̄ can be found by satis-

fying the boundary conditions

∂ψ 0

∂n e = − R

′ e (z) √

1 + R

′ 2 e (z)

on S 1 , ∂ψ 0

∂n e = 1 on S 2 ,

∂ψ 0

∂n e = 0 on S 3 ; (37a)

∂ ψ̄ 0

∂n e = 0 on S 1 ∪ S 2 ∪ S 3 . (37b)


Using (35a) in the standard variational scheme leads to the system of linear algebraic equations M φ0 c = p in which the

right-hand side and the matrix elements are

p n = −∫ d

l 2

R e (z) R

′ e (z) φ(0)

n (R e (z) , z| σ 2 ) dz +

∫ r 2

r 1

rφ(0) n (r, l 2 | σ 2 ) dr, (38a)

m

φ0

nk =

∫ d

0

R e (z)

[

∂φ(0) k

∂r − R

′ e

∂φ(0) k

∂z

]

r= R e (z)

φ(0) n (R e (z) , z| σ 2 ) dz

+

∫ r 2

r 1

r ∂φ(0)

k

∂z (r, l 2 ;σ 2 ) φ(0)

n (r, z| σ 2 ) dr, n, k = 0 , . . . , q 3 . (38b)

Solving the matrix problem computes C 0 , which is the same for (35a) and (35b) . Employing similar variational scheme

to satisfy (37b) leads to the matrix problem M̄ φ0 ̄c = p̄ where M̄ φ0

= { m

φ0

nk , n, k = 1 , ..., q 3 } , p̄ = { ̄p n , n = 1 , ..., q 3 } :

p̄ n = −C 0

∫ d

0

R e (z)

[∂ φ̄(0)

0

∂r − R

′ e

∂ φ̄(0) 0

∂z

]r= R e (z)

φ(0) n (R e (z) , z| σ 2 ) dz − C 0

∫ r 2

r 1

r ∂ φ̄(0)

0

∂z (r, l 2 | σ 2 ) φ(0)

n (r, z| σ 2 ) dr. (39)

When neglecting the damping and substituting ϕ e = −σ sin (σ t) ψ 0 (r, z| σ 2 ) into (14) and (32) , we obtain

σ 2 0 =

k 3 + πρw

g(r 2 2 − r 2 3 )

M p + M b + M l

= σ 2

(1 +

A 33 (σ 2 )

M p + M b + M l

), (40)

where

A 33 (σ2 ) = −2 πρw

q 3 ∑

j=0

C 0 j (σ2 ) p j (σ

2 ) (41)

is the frequency-dependent heave added mass and σ 0 is the eigenfrequency of the ‘dry’ structure.

The added-mass and wave-radiation damping components associated with vertical motions are small for the monopiles

outlined in Appendix B . However, A 33 / (M p + M b + M l ) is non-negligible for the Draugen monotower described in

Appendix C . The reasons are that R e ( z ) is clearly far from a constant value and the non-negligible contribution from the

ring-shaped roof B 0 e . The approximate solution (35) provides a fast stabilisation of significant figures in (41) so that five

digits are stabilised as q 3 ≥ 30. The result is A 33 / (M p + M b + M l ) ≈ 0 . 31 as 0 . 1 < T = 2 π/σ < 0 . 6 s, i.e. the added-mass co-

efficient weakly depends on T . The eigenperiod for the ‘dry’ monotower is T 0 = 2 π/σ0 = 0 . 375 s. Accounting for the finite

added-mass contribution corrects T 0 to T c0 = 2 π/σc0 = 0 . 43 s where σ 2 c0

is the σ 2 -root of (40) . For T ≈ T 0 , the coefficient C 0 in (35) is small, its value is comparable with the computational error. As a consequence, one can neglect the contributions

to the outgoing wave component in ψ 0 and ψ̄ 0 . An explanation is the fact that the associated underwater flows are mainly

caused by the vertical caisson motions. Since the considered period is small and the related wave length, as a consequence,

is very small relative to the distance between caisson and the mean free surface, the water-wave generation is negligible.

4. Horizontal eigenvibrations—statement and solution method

4.1. Eigensolution—structural, sloshing, and coupled eigenoscillations

Because v 1 and v 2 are decoupled, we will focus, for brevity, on structural eigenoscillations in the Oyz -plane by introducing

the eigensolution

v 2 = cos (σ t) W (z) , (42a)

ϕ i = sin θ ϕ

(2) i

= −σ sin (σ t) sin θ[�W

(r, z) + φW

(r, z| σ 2 ) ], (42b)

ϕ e = sin θ ϕ

(2) e = −σ sin θ

[sin (σ t) ψ W

(r, z| σ 2 ) + cos (σ t) ψ̄ W

(r, z| σ 2 ) ], (42c)

where σ is the unknown circular eigenfrequency and W ( z ) is the corresponding structural eigenmode .

Substituting (42) into the virtual work principle (18) and excluding the wave-radiation damping component ψ̄ W

derives

the variational equation , ∫ L

0

(EI W

′′ δW

′′ + N W

′ δW

′ ) dz − z C p M p g W

′ (L ) δW

′ (L )

+ k 2 W (0) δW (0) + k 4 W

′ (0) δW

′ (0) = σ 2

[∫ L

0

ρb S b W δW dz + M p W (L ) δW (L )

+ M p z C p [ W

′ (L ) δW (L ) + W (L ) δW

′ (L )] + I p 11

W

′ (L ) δW

′ (L )


+ πρl

(∫ h 1 + h

h 1

R i (�W

(R i (z) , z) + φW

(R i (z) , z;σ 2 )) δW dz

+

∫ R 0

0

r 2 (�W

(r, h 1 ) + φW

(r, h 1 | σ 2 )) δW

′ (h 1 ) dr

)

+ πρw

(−∫ d

0

R e ψ W

(R e (z) , z| σ 2 ) δW dz +

∫ r 2

r 1

r 2 ψ W

(r, l 2 | σ 2 ) δW

′ (l 2 ) dr

)], (43)

with respect to σ 2 and W ( z ). The trial functions W and δW should be continuous together with the first-order derivative on

the whole interval (0, L ). Functions �W

, φW

, and ψ W

( ψ̄ W

) can be considered as linear functional constraints , i.e. these are

linear operators defined on admissible W .

Substituting (42b) into the internal hydrodynamic problem (7) and following the linear sloshing analysis [1, chapter 5] de-

rives the first and second functional constraints as the W -dependent solutions of the Neumann boundary problem,

r 2 (

∂ 2 �W

∂r 2 +

∂ 2 �W

∂z 2

)+ r

∂�W

∂r − �W

= 0 in D i , ∂�W

∂n i

=

∂�W

∂z = 0 on L 0 ,

∂�W

∂n i

= −∂�W

∂z = rW

′ | z= h 1 on L 2 , ∂�W

∂n i

=

W (z) √

1 + R

′ 2 i (z)

on L 1 , (44)

with respect to the generalised Stokes–Joukowski potential �W

(characterised by the frozen inner free surface). The sloshing-

related component φW

is governed by the Robin boundary value problem

r 2 (

∂ 2 φW

∂r 2 +

∂ 2 φW

∂z 2

)+ r

∂φW

∂r − φW

= 0 in D i ,

∂φW

∂n i

= 0 on L 1 ∪ L 2 , −σ 2 φW

+ g ∂φW

∂z = σ 2 �W

on L 0 (45)

(see, geometric notations in Fig. 3 ).

The Stokes–Joukowski potential �W

is uniquely determined by W ( z ). The sloshing-related component φW

depends on

W ( z ) through �W

, which must be known prior to solving the Robin problem (45) .

The third functional constraint is associated with the added-mass component ψ W

. Inserting (42c) into (7), (8) leads to

r 2 (

∂ 2 [ ψ, ψ̄ ] W

∂r 2 +

∂ 2 [ ψ, ψ̄ ] W

∂z 2

)+ r

∂[ ψ, ψ̄ ] W

∂r − [ ψ, ψ̄ ] W

= 0 in D e , (46a)

g ∂[ ψ, ψ̄ ] W

∂z = σ 2 [ ψ , ψ̄ ] W

on S 0 , ∂[ ψ , ψ̄ ] W

∂z = 0 on S 4 , (46b)

∂ψ W

∂n e =

∂ψ W

∂z = −rW

′ (l 2 ) on S 2 , ∂ψ W

∂n e =

W (z) √

1 + R

′ 2 e (z)

on S 1 ∪ S 3 , (46c)

∂ ψ̄ W

∂n e = 0 on S 1 ∪ S 2 ∪ S 3 (46d)

(see, Fig. 3 ) which is uniquely solvable when adding the Sommerfeld radiation condition

lim

r→ 0 r 1 / 2

(∂ψ W

/ ∂r − K ψ̄ W

)= 0 , (47)

where K is the wave number (the σ 2 -dependent root of (36a) ).

Whereas (44) and (46) –(47) are uniquely solvable for any admissible W , the Robin problem (45) has no solutions for

σ coinciding with a subset of the natural (eigen) sloshing frequencies σsk = 2 π/T sk , k ≥ 1 . When the trial frequency σ ap-

proaches σ sk , a resonant sloshing occurs leading to a dominant character of the sloshing-related quantity φW

with an infinite

sloshing-related added mass. Keeping away from σ sk allows, sometimes, for neglecting sloshing loads which are smaller than

other forces and moments applied to the structure (see, [1, chapter 5] ) and the inner liquid flows are mainly determined

by the Stokes–Joukowski potential �W

. Excluding φW

from the variational Eq. (43) leads to what we call the structural

eigenoscillations with eigenperiods T k = 2 π/σk and modes W

( k ) ( z ). Finally, when sloshing loads matter, we arrive at the cou-

pled structure-sloshing eigenoscillations with eigenperiods T ck = 2 π/σck , k ≥ 1 (that are, generally speaking, not equal to either

T k or T sk ), structural W

(k ) c (z) and sloshing φ

W

(k ) c

eigenmode components.


4.2. Projective scheme

Accounting for the linear functional constraints (44) and (46) –(47) , we introduce a continuous (together with the first-

order derivative) functional basis { ξ k ( z ), 0 < z < L } and pose the approximate eigensolution as

W (z) =

Q ∑

k =1

b k ξk (z) , (48a)

�W

(r, z) =

Q ∑

k =1

b k �ξk (r, z) , ψ W

(r, z| σ 2 ) =

Q ∑

k =1

b k ψ ξk (r, z| σ 2 ) , (48b)

where �ξk (r, z) and ψ ξk

(r, z| σ 2 ) are solutions of (44) and (46) –(47) , respectively, when W = ξk , k ≥ 1 and a trial σ 2 is

adopted. The weight coefficients b = { b k } and σ 2 are the unknowns.

The sums (48b) express an analytical approximation of the first and third functional constraints . When sloshing is ne-

glected ( φW

= 0 ), b = { b k } and σ 2 can be found from the variational Eq. (43) . The procedure computes the structural eigen-

frequencies σk = 2 π/T k and modes W

( k ) ( z ).

To include sloshing and compute the coupled eigenoscillations, we adopt the approximate solution

φW

(r, z) =

Q s ∑

k =1

βk �k (r, z) , β = { βk } (49)

which is based on �k ( r, z ) associated with the first antisymmetric natural sloshing modes [cos θ �k ] and [sin θ �k ] satisfy-

ing

r 2 (

∂ 2 �k

∂r 2 +

∂ 2 �k

∂z 2

)+ r

∂�k

∂r − �k = 0 in D 0 ,

∂�k

∂n i

= −∂�k

∂z = 0 on L 2 ,

∂�k

∂n i

=

(∂�k

∂r − R

′ i

∂�k

∂z

)/

√

1 + R

′ i

2 = 0 on L 1 , ∂�k

∂z = κk �k on L 0 . (50)

The spectral boundary problem (50) has positive eigenvalues κk and the traces f k (r) = �k (r, h 1 + h ) , k ≥ 1 on L 0 constitute

an orthogonal functional basis [1, chapter 4] providing ∫ r 0

0

r f k f n dr = δkn

∫ r 0

0

r f 2 k dr = δkn || f k || 2 , (51)

where δkn is the Kronecker delta. The eigenvalues κk determine the aforementioned natural (eigen) sloshing frequencies and

periods

σsk =

√

gκsk , T sk = 2 π/σsk . (52)

The approximate solution (49) satisfies all relations of (45) except the Robin condition on L 0 . Substituting (49) and

(48b) into this condition, multiplying the result by ρl πκk r f k (r) , k = 1 , . . . , Q s , and integrating over (0, r 0 ) lead to the alge-

braic equations

ρl π diag [κk || f k (r) || (σ 2

sk − σ 2 )]

β = σ 2 D b , (53)

where the Q s × Q -matrix D consists of the elements

{ D k, j = ρl πκk

∫ r 0

0

r f k (r ) �ξ j (r, h 1 + h ) dr, k = 1 , .., Q s ; j = 1 , .., Q} . (54)

Solving the linear Eqs. (53) with respect to β gives an approximation of the second functional constraint .

By adopting (48a) , the trial functions δW (z) = ξk (z) , k = 1 , ..., Q, the approximate constraints (48b) and the approximate

solution (49) , the variational Eq. (43) results in the linear algebraic equations,

(M

A − σ 2 M

B (σ 2 )) b = σ 2 D

T β, (55)

with respect to b . The matrix D is defined by (54) and the symmetric Q × Q matrices M A and M B ( σ2 ) consist of the elements

{ m

A ( ξm

, ξ n )} and { m

B ( ξm

, ξ n | σ 2 )} defined by the quadratic forms (the symmetry in the square brackets can be shown by

using the second Green identity)

m

A (ζ , ξ ) = m

A (ξ , ζ ) =

∫ L

0

(E(z) I(z) ξ ′′ (z) ζ ′′ (z) + N(z) ξ ′ (z) ζ ′ (z)) dz

−z C p M p gξ′ (L ) ζ ′ (L ) + k 2 ξ (0) ζ (0) + k 4 ξ

′ (0) ζ ′ (0) , (56a)


m

B (ζ , ξ | σ 2 ) = m

B (ξ , ζ ) =

∫ L

0

ρb (z) S b (z) ξ (z) ζ (z) dz + M p ξ (L ) ζ (L )

+ πρl

[∫ h + h 1

h 1

R i (z)�ξ (R i (z) , z) ζ (z) dz +

∫ R 0

0

r 2 �ξ (r, h 1 ) ζ′ (h 1 ) dr

]

+ πρw

[−

∫ d

0

R e (z) ψ ξ (R e (z) , z;σ 2 ) ζ (z) dz +

∫ r 2

r 1

r 2 ψ ξ (r, l 2 ;σ 2 ) ζ ′ (l 2 ) dr

]+ M p z

C p

[ξ ′ (L ) ζ (L ) + ξ (L ) ζ ′ (L )

]+ I p

11 ξ ′ (L ) ζ ′ (L ) . (56b)

The derivations use the Green identity

D k, j = ρl π

(∫ h + h 1

h 1

R i (z) �k (R i (z) , z) ξ j (z) dz +

∫ R 0

0

r 2 �k (r, h 1 ) ξ′ j (h 1 ) dr

)

= ρl π

∫ L 0 + L 1 + L 2

�k

∂�ξ j

∂n i

dS = ρl π

∫ L 0 + L 1 + L 2

�ξ j

∂�k

∂n i

dS

= ρl π

∫ r 0

0

r

[�ξ j

∂�k

∂z

]z= h + h 1

dr = ρl πκk

∫ r 0

0

r �ξ j (r, h 1 + h )�k (r, h 1 + h ) dr.

When neglecting the sloshing effect ( β = 0 ), the linear algebraic Eqs. (55) yield the generalised spectral matrix problem

(M

A − σ 2 M

B (σ 2 )) b = 0 , (57)

whose eigensolution, σ k and W

( k ) ( z ) by (48a) (based on the eigenvectors b k ), approximates the structural eigenoscillations .

The matrix M B ( σ2 ) depends on the spectral parameter σ 2 . A numerical solution of (57) can be obtained by an iterative

procedure.

When structural vibrations and sloshing are coupled, combining (53) and (55) leads to the generalised spectral matrix

problem ([M A 0

0 E A

]− σ 2

c

[M B (σ

2 ) D

T

D E B

])[b

β

]= 0 , (58)

where E A = ρl π g diag [ κ2 k || f k || ] and E B = ρl π diag [ κk || f k || ] . The eigenvalues σ 2

ck and vectors ( b , β) k following from (58) ap-

proximate the coupled eigenoscillations , eigenperiods T ck = 2 π/σck and modes by (48) and (49) .

4.3. Functional base { ξ k ( z )} and { �k }

Functions ξk (z) , k = 1 , . . . , Q, must be continuous, together with the first-order derivative, on (0, L ) but the second-order

derivatives of ξ k ( z ) must have jumps at h 1 and l 2 . Bearing in mind the admissible physical scenarios, one can construct such

a specific functional basis by following the procedure below.

First , the functional basis should contain ξ1 (z) = 1 and ξ2 (z) = z describing small–amplitude horizontal deviations due

to translatory and angular motions of the beam as a rigid body. The remaining functions should satisfy the clamped end

conditions, ξk (0) = ξ ′ k (0) = 0 , k ≥ 3 . These conditions disregard the soil elasticity and, therefore, only ξ 1 and ξ 2 account for

the soil feedback.

Secondly , considering the fully motionless caisson [1, Section 5.4.5] requires excluding ξ1 (z) = 1 and ξ2 (z) = z but may

adopt the functional basis

ξ (1) k

(z) =

⎧ ⎨

⎩

0 , z < l 2 ,

(z − l 2 ) 2 P k

(2 z − L − l 2

L − l 2

), l 2 < z,

(59)

where P k ( α) are the Legendre polynomials computed, e.g., by the recurrence formulas

P 0 = 1 , P 1 = α, kP k = (2 k − 1) αP k −1 − (k − 1) P k −2 ,

P ′ k

= αP ′ k −1

+ kP k −1 , P ′′ k

= αP ′′ k −1 + (k + 1) P ′ k −1

.

The completeness of (59) on ( l 2 , L ) follows from Müntz’ theorem (see, [19,20] , and references therein).

Thirdly , the following admissible (and complete on ( h 1 , l 2 )) functional set implies the motionless caisson along 0

< z < h , the beam type vibrations of the caisson along h < z < l , and the tower motions (interval ( l , L )) as a
1 1 2 2


rigid body:

ξ (2) k

(z) =

⎧ ⎪ ⎨

⎪ ⎩

0 , z < h 1 ,

(z − h 1 ) 2 P k

(2 z − l 2 − h 1

l 2 − h 1

), h 1 < z < l 2 ,

( l 2 − h 1 ) 2 P k ( 1) + 2( l 2 − h 1 )

[P k ( 1) + P ′

k ( 1)

]( z − l 2 ) , l 2 < z.

(60)

Finally , the flexibility of the caisson along 0 < z < h 1 can be accounted for by adding the functional set

ξ (3) k

(z) =

⎧ ⎨

⎩

z 2 P k

(2 z − h 1

h 1

), 0 < z < h 1 ,

h

2 1 P k (1) + 2 h 1

[P k (1) + P ′

k (1)

](z − z 1 ) , h 1 < z

(61)

which satisfies a clamped end condition at z = 0 and represents small-amplitude rigid-body motions along h 1 < z < L .

In summary, based on the introduced functional sets, we can describe different physical scenarios. The most general

scenario accounting for the structural flexibility on the interval 0 < z < L as well as for the elastic soil feedback requires

the functional basis

ξm

(z) =

{1 , z,

{ξ (1)

k (z)

}, {ξ (2)

k (z)

}, {ξ (3)

k (z)

}}. (62)

The Müntz theorem proves its completeness. If we want to ignore the elastic soil feedback, functions 1 and z must be

excluded. Neglecting the caisson flexibility (moves like a rigid body) on 0 < z < h 1 or/and h 1 < z < l 2 implies excluding

{ ξ (2) k

(z) } or/and { ξ (3) k

(z) } , respectively.

The eigenfunctions { �k } by (50) can be analytically introduced when remembering that we deal with a deep water

sloshing in an axisymmetric tank which keeps an almost upright cylindrical shape near the mean free surface. Under these

conditions, �k may be approximated by

�n (r, z) =

J 1 (K n r)

J 1 (K n r 0 ) exp (K n (z − h 1 − h )) , J ′ 1 (αn ) = 0 , αn = K n r 0 ,

κn = K n , σ 2 n = gκn = gK n , || f n || 2 = r 2 0 (α

2 n − 1) / 2 α2

n (63)

associated with the antisymmetric natural sloshing modes in an upright circular cylindrical tank [1, Section 4.3.2.2] ,. For the

considered shafts, we justified the approximate eigenfunctions (63) numerically by using the analytically-oriented method

from [21] . The discrepancy for the lowest eigenvalue ( k = 1 ) was less than 0.1% and it becomes almost negligible for k ≥ 2.

4.4. The Trefftz approximation of �ξ ( r, z ) and ψ ξ ( r, z | σ 2 )

Approximating �ξ ( r, z ). When the shaft has an upright circular cylindrical shape, R i (z) = r 0 = const, the analytical solution

is

�ξ (r, z) =

ξ ′ | z= h 1 2 h

[(z − h − h 1 )

2 r − 1 4

r 3 ]

+

q 1 ∑

k =0

b (ξ )

k

I 1 (s k r)

I ′ 1 (s k r 0 )

cos (s k (z − h 1 )) ︸︷︷︸ φ(1)

k (r,z)

, s k =

πk

h

, q 1 → ∞ , (64)

where φ0 0

= r/r 0 , I 1 ( · ) is the modified Bessel function of the first kind, and

b (ξ ) 0

=

1

h

∫ h 1 + h

h 1

f (z) dz, b (ξ )

k =

2

h

∫ h 1 + h

h 1

f (z) cos (s k (z − h 1 )) dz, k ≥ 1 ,

f (z) = ξ (z) − ξ ′ (h 1 )(2 h ) −1 ([ z − h − h 1 ]

2 − 3 4

r 2 0

).

When R i (z) � = const, expression (64) with a finite integer q 1 can be interpreted as an approximate solution of (44) , which

exactly satisfies the Neumann boundary conditions on L 2 and L 0 (see, Fig. 3 ). To approximate the Neumann boundary con-

dition on L 1 by finding appropriate coefficients b (ξ ) k

, the standard projective scheme can be employed which leads to the

system of linear algebraic equations M φ1 b (ξ ) = q (ξ ) , where b (ξ ) = (b

(ξ ) 0

, ..., b (ξ ) q 1

) , M φ1 = { m

φ1

nk } and q (ξ ) = { q (ξ )

0 , . . . , q

(ξ ) q 1

} :

m

φ1

nk =

∫ h 1 + h

h 1

R i (z)

[

∂φ(1) k

∂r − R

′ i (z)

∂φ(1) k

∂z

]

r= R i (z)

φ(1) n (R i (z) , z) dz, (65a)


q (ξ ) n =

{∫ h 1 + h

h 1

R i (z) [ ξ (z) − ξ ′ (h 1 )

2 h

((z − h − h 1 )

2 − 3 4

R

2 i (z)

− 2 R

′ i (z)(z − h − h 1 ) R i (z)

)] φ(1)

n (R i (z) , z) dz

}. (65b)

The approximate solution (64) provides a fast convergence in the corresponding integrals of (56b) so that q 1 ≥ 30 sta-

bilises from six to nine significant figures for the considered shaft shapes. The solution does not necessarily provide a

uniform convergence. Our numerical experiments showed that q 1 ≥ 50 enables stabilisation from three to four significant

figures of �ξk (R i (z) , z) , h 1 < z < h + h 1 in a uniform metrics, again, for the considered shafts.

Approximating ψ ξ ( r, z | σ 2 ). When W = ξ is a trial function, the two solution components of (46) can be approximated by

the Trefftz method suggesting

ψ ξ = c (ξ ) 0

Z 0 (z) J ′ 1 (Kr 3 ) J 1 (Kr) + Y ′ 1 (Kr 3 ) Y 1 (Kr)

J ′ 1

2 (Kr 3 ) + Y ′

1

2 (Kr 3 )

+

q 2 ∑

j=1

c (ξ ) i

Z j (z) K 1 (K j r)

K 1 (K j r 3 ) =

q 2 ∑

j=0

c (ξ ) i

φ(2) j

(r, z| σ 2 ) , (66a)

ψ̄ ξ = c (ξ ) 0

Z 0 (z) Y ′ 1 (Kr 3 ) J 1 (Kr) − J ′ 1 (Kr 3 ) Y 1 (Kr)

J ′ 1

2 (Kr 3 ) + Y ′

1

2 (Kr 3 )

+

q 2 ∑

j=1

c̄ (ξ ) i

φ(2) j

( r, z| σ 2 )

= c (ξ ) 0

˜ φ(2) 0

(r, z| σ 2 ) +

q 2 ∑

j=1

c̄ (ξ ) i

φ(2) j

(r, z| σ 2 ) (66b)

which satisfies, automatically, (46a) and (46b) .

The added mass terms are caused by (66a) , which is our main focus. To find c (ξ ) k

, k = 0 , ..., q 2 , we use the standard

projective scheme leading to the system of linear algebraic equations M φ2 c (ξ ) = p

(ξ ) within the elements

m

φ2

nk =

∫ d

0

R e (z)

[

∂φ(2) k

∂r − R

′ e

∂φ(2) k

∂z

]

r= R e (z)

φ(2) n (R e (z) , z| σ 2 ) dz

+

∫ r 2

r 1

r ∂φ(2)

k

∂z (r, l 2 ) φ

(2) n (r, l 2 | σ 2 ) dr, (67a)

p (ξ ) n =

∫ d

0

R e (z) ξ (z) φ(2) n (R e (z) , z| σ 2 ) dz −

∫ r 2

r 1

r 2 ξ ′ (l 2 ) φ(2) n (r, l 2 | σ 2 ) dr. (67b)

Using the approximate solution (66a) provides a fast convergence in the corresponding integrals of (56b) so that q 2 ≥40 stabilises from six to nine significant figures of the integrals for the considered monotowers and σ close to the low-

est structural eigenfrequencies. The numerical tests showed a weak uniform convergence of ψ ξk (R e (z) , z| σ 2 ) , 0 < z < L so

that the method only enables to stabilise from two to four significant figures as q 2 ≥ 60. Accounting for the corner-point

singularities at the caisson roof edge should improve the convergence [22,23] .

5. Numerical examples

The lower and middle cases in Fig. 1 (b) can be associated with the Draugen monotower ( Appendix C ), and a sample

monopiles ( Appendix B ), respectively. Our numerical examples adopt the input data from the appendices.

5.1. Hydrodynamic loads

When analysing the eigenperiods, the mathematical model [1, Section 5.4.5] approximates the external added-mass loads

by using strip theory. Adopting (42a) , the horizontal hydrodynamic force per unit length by (13) is then approximated as

f 2 e (z, t) ≈ χ[0 ,d] (z) ρw

πσ 2 R

2 e (z) W (z) cos (σ t) . (68)

Our analysis uses the ψ W

( r, z )-component in (42c) , which computes this force as

f 2 e (z, t) = −χ[0 ,d] (z) ρw

πσ 2 R e (z) ψ W

(R e (z) , z| σ 2 ) cos (σ t) . (69)


a b

Fig. 4. Comparison of the ψ W -based prediction of the horizontal hydrodynamic loads and the strip theory approximation (see Eq. (70) ) for W = ξ1 (z)

(panel a) and W = ξ2 (z) (panel b). The Draugen platform case. The computed values are scaled by max 0 < z < L | R e ( z ) W ( z )|) and presented along the hori-

zontal axis; z is the dimensional vertical coordinate. The strip theory results are T independent and marked by the solid lines. The dotted lines represent

the T -dependent results computed from −ψ ξm by the Trefftz method from Section 4.4 . The trial period T changes from 0.1 to 6 s, the dotted lines cannot

be visually distinguished when 0.1 � T � 3 s.

Comparing (68) and (69) shows that

−ψ W

(R e (z) , z| σ 2 ) = R e (z) W (z) , 0 < z < d (70)

according to the strip theory.

We can test the validity of the strip theory by examining the left- and right-hand sides of (70) on the trial functions, i.e.

for W = ξm

(z) with a fixed m , for admissible periods T , which typically belong to the range 0.1 � T � 6 s. The functional

base (62) allows for a clear physical treatment so that ξ 1 corresponds to the horizontal translatory vibrations, ξ 2 implies the

horizontal vibration due to an angular motions of the vertical structure around O (as a solid body) and ξ (1) k

, k ≥ 1 describe

the beam-like vibrations with a motionless rigid caisson. The comparison results are drawn in Figs. 4 and 5 for the Draugen

monotower. The numerical values are scaled by max 0 <z<L

| R e (z) W (z) | . Fig. 4 focuses on the results for ξ 1 ( z ) (panel a) and ξ 2 ( z ) (panel b). The major difference between the strip theory

(the solid lines) and the ψ W

-based computations (the dotted lines) is at the seabed and near the mean free surface. The

difference at the seabed is T -independent for the tested 0.1 s ≤ T ≤ 6 s. The difference near the free surface depends on T

for 3 s � T . Moreover, the dotted lines change their behaviour at the mean free surface as 6 s � T . There is a thin z -layer

near the free surface where the result corresponds to a negative sectional added mass when 0.1 s � T � 3 s. We did not

study this phenomenon, which could be due to a weak (non-uniform) convergence at z = d.

A significant difference at the seabed in a consequence of flows excited by the caisson, in general, and its roof motions,

in particular. For both horizontal (a) and angular (b) excitations of the monotower (moving as a rigid body), the strip theory

overpredicts the jump at the roof level, z = l 2 , which is due to the pressure difference between point ( r 1 , l 2 ) and ( r 2 , l 2 )

(see, Fig. 2 ). The difference is larger in panel (b) since the angular caisson motions excite a significant flow in the vertical

direction. This, probably, affects a sufficient difference for z > l 2 which has a local character only in the case (a).

As long as we assume that the caisson does not move (as in [1, Section 5.4.5] ), the difference at the seabed disappears.

This fact is illustrated in Fig. 5 , which compares the ψ W

-based and strip-theory predictions with the trial functions W =ξ (1)

m

, m = 1 , ..., 4 . Under certain circumstances, these functions can be associated with the lowest beam eigenmodes W

( m ) ( z )

as giving a dominant contribution into the modes when the caisson is motionless. Fig. 5 detects a difference only at the

mean free surface. The difference becomes smaller with increasing the base function number k (treated as increasing the

eigenmode number) and T .

A series of numerical experiments were conducted to evaluate a ‘strip theory’–like approximation of the internal hydro-

dynamic forces on the shaft walls following from (10) when neglecting the sloshing component. In this case, the hydrody-

namic loads are associated with the Stokes–Joukowski potential with the following analogy of (68) :

f 2 i (z, t) ≈ χ[ h 1 ,h 1 + h ] (z) ρl πσ 2 R

2 i (z) W (z) cos (σ t) . (71)


a b

c d

Fig. 5. The same as in Fig. 4 but for the unmovable caisson and ξ (1) x (z) , k = 1 , 2 , 3 , 4 in (70) . Cases (a) – ξ (1)

1 , (b) – ξ (1)

2 , (c) – ξ (1)

3 , and (d) — ξ (1)

4 .

These numerical experiments showed that the approximation is satisfactory when the caisson does not move angularly

(only horizontal motions are allowed). The angular motions yield non-negligible moment applied to the shaft bottom whose

prediction requires involving the Stokes–Joukowski potential.

5.2. Modelling the Draugen monotower

To implement the constructed mathematical model, Appendix C introduces and discusses the mean input parameters

scenario for the Draugen platform. A difficulty consists of an adequate modelling of the composite caisson, which we as-

sume can be replaced by an upright circular cylindrical beam with equivalent mass and volume. This replacement does

not guarantee a correct estimate of the corresponding added mass. In addition, the actual circular caisson radius r 2 is of

comparable size with its height l 2 . Thus, it can hardly be considered as an Euler–Bernoulli beam; a rigid body model may

look preferably. This means that testing our mathematical model for the Draugen monotower should justify that (a) this

model can correctly predict the highest eigenperiods which should be, according to full-scale observations, between 3.9

and 4.2 s; (b) even though the caisson is considered as a piece of the Euler–Bernoulli beam, it moves like a rigid body

and the speculatively-varying Young’s modulus (defined in Appendix C ) does not affect this motion type as well as weakly


Fig. 6. Structural eigenmodes W

(k ) (z) , k = 1 , ..., 4 computed with the mean input parameters scenario from Appendix C . Sloshing is neglected. Integers

denote the mode numbers k . Along with the entire eigenmode shapes, a 100-m zone is shown to demonstrate a linear behaviour of W

( k ) ( z ) along the

caisson, 0 < z < l 2 = 38 m. This means that the caisson of the Draugen monotower moves like a rigid body set upon an elastic soil.

changes the highest eigenperiods; (c) sloshing in the tower shaft may matter. The mathematical model will also be used for

a parameter study of the undamped eigenperiods.

Our projective scheme demonstrates a fast convergence to the four higher structural eigenperiods T m

= 2 π/σm

, m =1 , ..., 4 and the seven highest coupled structure-sloshing eigenperiods, T ck , k = 1 , ..., 7 . From ten to twenty base functions

{ ξ (1) k

(z) } as well as from two to five base functions { ξ (2) k

(z) } and { ξ (3) k

(z) } are needed to stabilise at least five significant

figures of these eigenperiods and from two to five significant figures of the corresponding eigenmodes.

The mean input parameters scenario—eigenperiods. By adopting this scenario from Appendix C returns the two highest struc-

tural eigenperiods (sloshing is neglected) T 1 = 4 . 12 and T 2 = 1 . 55 s. The natural sloshing periods are approximated by (63) .

These are computed as T s 1 = 4 . 05 , T s 2 = 2 . 38 , T s 3 = 1 . 88 , T s 4 = 1 . 61 , and T s 5 = 1 . 42 s. The coupled eigenperiods are esti-

mated as T c1 = 4 . 16 s, T c2 = 4 . 00 s, T c3 = 2 . 38 s ≈ T s 2 , T c4 = 1 . 88 s ≈ T s 3 , T c5 = 1 . 61 s ≈ T s 4 , and T c6 = 1 . 55 s ≈ T 2 . The highest

eigenperiods belong to the full-scale observed range, 3.9 < T c 2 < [ T 1 , T s 1 ] < T c 1 < 4.3 s.

The coupled eigenperiods T cm

are very close to either sloshing or structural eigenperiods except for T c 1 and T c 2 which are,

generally, different from T 1 and T s 1 . Alike in the elevated water tank case [7] , this fact enables to distinguish structural-type

(for T c6 = 1 . 55 s ≈ T 2 ), sloshing-type (for T c3 = 2 . 38 s ≈ T s 2 , T c4 = 1 . 88 s ≈ T s 3 , and T c5 = 1 . 61 s ≈ T s 4 ) and mixed eigenmotions

(expected for the two highest eigenmotions associated with T c1 = 4 . 16 s and T c2 = 4 . 00 s). The eigenmotions of the structural

type should identify structural vibrations for which the beam deflections are not accompanied by a considerable sloshing,

the sloshing-type eigenmotions have to imply a resonant sloshing with a negligible structural deflection, and the mixed

eigenoscillations should demonstrate the mutually-excited structural and sloshing motions.

The mean input parameters scenario—eigenmodes. Our first focus is on structural eigenmodes when sloshing is neglected.

Fig. 6 demonstrates the corresponding normalised ( ∫ L

0 (W

(k ) ) 2 dz = 1 ) eigenmodes within the framework of the mean input

parameters scenario. Their profiles differ from those for a cantilever beam (clamped at z = 0 ). The difference is mainly

explained by the noncontinuous character of S b ( z ), E ( z ), and ρb ( z ) and the soil elasticity. A visual emphasis is placed on

the interval 0 < z < l 2 corresponding to the caisson height (marked as ‘support’). Our mathematical model suggests that

caisson is a piece of the Euler–Bernoulli beam. However, calculations show that the caisson moves like a rigid body set upon

an elastic soil. These motions are a superposition of a horizontal drift at z = 0 and an inclination around O and, therefore,

W

( k ) ( z ) behave as linear functions along the interval 0 < z < l 2 = 38 m.

Accounting for sloshing suggests the eigenmode pairs (W

(m ) c (z) , φ

W

(m ) c

(r, z)) for each T cm

, m = 1 , . . . . The first element,

W

(m ) c (z) , is responsible for structural deflections but the second one, φ

W

(m ) c

(r, z) , characterises the sloshing amplifica-

tion. The latter becomes clearly visualised when introducing the free-surface profile F (m ) (y ) = φW

(m ) (y, h + h 1 ) , 0 ≤ y < r 0 , =
c


Fig. 7. The eigenmodes (W

(m ) c (z) , φ

W (m ) c

(r, z)) corresponding to the eigenperiods T cm , m = 1 , . . . , 6 . Here, F (m ) (y ) = φW (m )

c (y, h + h 1 ) , 0 ≤ y < r 0 , =

−φW (m )

c (y, h + h 1 ) , −r 0 < y < 0 specify the sloshing-related component by the free-surface profile in the Oyz cross-plane. The mean input parameters sce-

nario for the Draugen monotower is adopted. The dimensions are in meters. The eigenmodes are scaled so that the maximum wave elevation at the shaft

wall is equal to 1 m.

−φW

(m ) c

(y, h + h 1 ) , −r 0 < y < 0 associated with the Oyz meridional cross-section of the shaft. Adopting the mean input pa-

rameters scenario, Fig. 7 shows eigenmodes (W

(m ) c (z) , φ

W

(m ) c

(r, z)) for T cm

, m = 1 , . . . , 6 . The dimensions are in meters. The

eigenmodes are scaled requiring the maximum wave elevation at the shaft wall is equal to 1 m. The figure confirms our

expectations regarding structural-type (at T 2 ≈ T c 6 ), sloshing-type (at T c 3 ≈ T s 2 , T c 4 ≈ T s 3 , T c 5 ≈ T s 4 ), and mixed (at T c 1 and

T c 2 ) eigenmotions. For the sloshing-type eigenmodes, we see a small (but not zero) structural deflections accompanied by

a non-negligible wave elevation with the surface profile close to that defined by �m

(r, h + h 1 ) , m = 2 , . . . , 5 , (according to

(63) ). The sloshing-type eigenmodes are of less interest from a practical point of view. Occurrence of the mixed eigenmodes

at T c 1 and T c 2 is explained by the relatively-large sloshing loads when the lowest sloshing mode is exited due to the mutual

resonance T 1 ≈ T s 1 .

On the caisson elasticity. Apart from the mean input parameters scenario, Appendix C specifies a series of uncertain phys-

ical parameters, or, more precisely, parameters whose actual values are not constant but rather belong to wide admis-

sible ranges. We conducted several numerical experiments in evaluating how sensitive the eigenperiods are to changing

these parameters in the aforementioned ranges. The primary focus was on the Young modulus E ( z ), 0 < z < l 2 which is

speculatively chosen for the artificial circular cylindrical caisson. The mean input parameters scenario equals the modulus

to E b = 3 . 25 × 10 10 N/m

2 (the value is associated with the reinforced concrete). We conducted numerical experiments in

studying the eigenperiods versus E 0 = E(z) = const , 0 < z < l 2 . Our computations showed that T ck , k = 1 , . . . , 6 vary within

to 0.2% as E 0 changes between 2 . 0 × 10 10 N/m

2 (1.5 times less than the lower bound of E b for the reinforced concrete,

3 . 0 × 10 10 N/m

2 ) and 7 . 0 × 10 10 N/m

2 (twice larger than the upper bound of E b , 3 . 5 × 10 10 N/m

2 ). In all these computations,

the caisson moves like a rigid body.

The eigenperiods versus G s . A particular emphasis was placed on the soil shear modulus G s which was estimated in

Appendix C belonging to the range 10 7 � G s � 10 9 N/m

2 . The structural ( T k , the solid lines) and sloshing ( T sk , the dashed

lines) eigenperiods versus G s are shown in Fig. 8 . All other physical parameters are taken within the framework of the mean

input parameters scenario. The figure also shows the full-scale observations for the highest eigenperiods T 1 and T s 1 (the

range from 3.9 to 4.3 s). The test value G s = 2 . 456 × 10 8 N/m

2 was accepted in [13] . It returns T 1 = 4 . 22 s and T s 1 = 4 . 05 s

which belong to the observed range. The paper [3] states that the soil at the 6+ meters deep down of the seabed level is

the silky, sand, very stiff to very hard clay. The corresponding interval of G s is marked to show that the computed values

T 1 and T s 1 belong to the observation zone for this soil type. If the soil is soft enough, G s � 2 . 4 × 10 8 N/m

2 , the structural

eigenperiods T 1 and T 2 rapidly increase.

Accounting for sloshing introduces the eigenperiods T ck whose values versus G s are demonstrated in Fig. 9 by the solid

lines (within the framework of the mean input parameters scenario). The dashed lines are introduced to specify the struc-

tural and sloshing eigenperiods. The figure shows that T ck , k ≥ 3 are very close to either T k , k ≥ 2 or T sk , k ≥ 2. As a result,

the dashed lines are invisible, except, perhaps, in an extremely small vicinity of the intersection points P , k ≥ 2 where T
k 2


Fig. 8. Structural eigenperiods T 1 and T 2 (in seconds, the solid lines) as functions of the soil shear modulus G s (N/m

2 ) within the framework of the mean

input parameters scenario (except for G s ) for the Draugen platform. The dashed lines mark the G s -independent natural sloshing periods T sk . Experimental

range is indicated.

Fig. 9. The coupled T ck (the solid lines, in seconds), structural T k and sloshing T sk (dashed lines) eigenperiods as functions of the soil shear modulus G s (N/m

2 ) when all other input physical parameters are chosen according to the mean input parameters scenario for the Draugen platform. Points P k show

intersections of the T 2 -curve with the horizontal lines at T sk , k ≥ 2. The experimental range is indicated.

becomes equal to T sk , k ≥ 2. All these eigenperiods correspond to either structural- or sloshing-type eigenmodes. Depending

on G s , the structural-type eigenmode may be the third one in the left of P 2 , the fourth one between P 2 and P 3 ), the fifth one

between P 3 and P 4 and the sixth one in the right of P 4 . One can follow a fixed eigenperiod T ck , k ≥ 3 by the dotted arrow.

The mixed eigenmodes (see, the first two columns in Fig. 7 ) correspond to T c 1 and T c 2 . We distinguish them by the solid

lines in Fig. 9 which confirm that T c 2 < T s 1 < T 1 < T c 1 . The equality T 1 = T s 1 is not possible for the mean input parameters

scenario and the tested values of G s . However, decreasing the top body mass M p from M = 2 . 3 × 10 7 to M = 2 . 0 × 10 7 kg

makes this equality possible at a certain G s (see, Fig. 10 ). For this value of M p , T c 2 < min ( T s 1 , T 1 ) ≤ max ( T s 1 , T 1 ) < T c 1 . When

| T 1 − T s 1 | increases, T c 1 → max ( T 1 , T s 1 ) and T c 2 → min ( T 1 , T s 1 ). The closeness to T s 1 makes the corresponding eigenmode of

the sloshing type.

The first structural eigenperiod versus M p , z p C

and r g . Our extensive numerical studies with different input data, which are

close to the mean input parameters scenario, showed that, normally, the relative differences | T c1 − T 1 | /T 1 and | T c2 − T 1 | /T 1 do

not exceed 2.5%. This means that the input parameter analysis should concentrate on evaluating the structural eigenperiod

T 1 . This highest eigenperiod versus the top body characteristics, M p (mass), z p C

(mass centre) and r g (radius of gyration

computes I p 11

), are shown in Figs. 11 and 12 . We mark the full-scale expectation for T 1 (between 3.9 and 4.3 s) as well as


Fig. 10. The eigenperiod T 1 (no sloshing), T s 1 (the natural sloshing period), T c 1 and T c 2 (coupled eigenperiods) versus G s within the framework of the mean

input parameters scenario for the Draugen platform but with M p = 2 . 0 × 10 7 kg. Point P 1 specifies the case when T 1 = T s 1 .

Fig. 11. The highest eigenperiod T 1 versus r g : the graph 1 corresponds to z p C

= 2 . 0 m

2 , 2 – z p C

= 5 . 0 m

2 , 3 – z p C

= 10 . 0 m

2 , 4 – z p C

= 13 . 75 m

2 , 5 – z p C

=

18 . 0 m

2 , and 6 – z p C

= 23 . 0 m

2 . The mean input parameters scenario is adopted from Appendix C for the Draugen platform but radius of gyration, r g , and

the mass centre, z p C , of the top platform change. The point M corresponds to the mean input parameters scenario with r g = 28 . 0 m and z p

C = 13 . 75 m.

Fig. 12. The mean values input parameters scenario is taken from Appendix C for the Draugen platform but the mass, M p , and the mass centre, z p C , of the

top platform change. The highest eigenperiod T 1 versus z p C

: the graph 1 corresponds to M p = 1 . 7 × 10 7 kg, 2 – M p = 2 . 0 × 10 7 kg, 3 – M p = 2 . 3 × 10 7 kg, 4–

M p = 2 . 6 × 10 7 kg, and 5 – M p = 2 . 8 × 10 7 kg. The point M corresponds to the mean input parameters scenario with z p C

= 13 . 75 m and M p = 2 . 3 × 10 7 kg.


Fig. 13. The same as in Fig. 10 but for the sample gravity monopiles/monotower in Appendix B . The eigenperiods at G s = 5 . 0 × 10 8 N/m

2 are equal to

T 1 = 3 . 0203 s, T 2 = 0 . 6845 s, T s 1 = 3 . 0262 s, T c1 = 3 . 0498 s, and T c2 = 2 . 9968 s.

introduce M to indicate the mean input parameters scenario. These figures show that T 1 increases with increasing r g , z p C ,

and M p . The eigenperiod T 1 is less sensitive to r g but the mass centre position z C p and the mass M p are of importance.

5.3. Modelling a sample gravity monopiles of the wind energy industry

In future studies of gravity monopiles, the main purpose is to fill up the knowledge gaps about the external and internal

hydrodynamic effects following the established design trend but not necessarily to fit the monopiles built today. Perspective

monopiles gets larger due increased dimensions and forces of turbines now approaching 8 MW and may reach 10 MW—

rotor diameter 160–200 m. The base diameter will be up to 10 m and 7.6 m at the mean sea level but 10 m at the mean

free surface may also be assumed. The sea water depth will typically be from 25 to 45 m, accounting for the tidal effect.

The shaft is filled with water and, in contrast to existing gravity monopiles for which T s 1 is far from T 1 ≈ 3 s, sloshing may

theoretically matter due to increase of the shaft radius, the contained liquid mass, and the sea-water depth. The steel wall

thickness should be from 60 to 100 mm max.

Bearing in mind those future gravity monopiles, a sample monopiles is proposed in Appendix B for which both the

external sea water and the sloshing loads may have an effect on the undamped eigenperiods. The latter implies, in par-

ticular, adopting the maximum expected water depth (about 45 m) and the inner shaft radius at the mean free surface

level about 4 m for getting T 1 ≈ T s 1 close to 3 s (this is a typical highest eigenperiod for the monopiles). The gravity

monopiles/monotower are shaped as in the middle case of Fig. 1 (b).

Extensive numerical experiments with the sample gravity monopiles in Appendix B showed the same qualitative features

on the eigenoscillations of structural, sloshing and mixed type and the eigenperiods behaviour versus G s , M p , r g , and z C p . A

primary emphasis was on T c 1 and T c 2 whose values versus G s are compared with T 1 and T s 1 in Fig. 13 . The figure illustrates

that the relative differences, | T c1 − T 1 | /T 1 and | T c2 − T 1 | /T 1 (requiring T c 1 and T c 2 correspond to the mixed-type eigenmo-

tions), are rather small for this sample, it is less than 1% and reaches a local maximum only in a small neighbourhood of P 1 where T 1 = T s 1 . The mixed eigenoscillations are only detected in this neighbourhood. Physically, this means that the sloshing

loads cannot influence the dynamics of the monopiles.

6. Conclusions

The present paper generalises semi-analytical studies of the sea-based gravity monotowers presented in [1, Section 5.4.5] .

We account for an elastic soil feedback, attach a rigid body (not a mass point) to the tower top and compute the external

hydrodynamic loads as they follow from considering the linear three-dimensional water wave problem with the body in-

teraction. A focus is on linear free oscillations of the mechanical system and associated undamped eigenperiods and modes.

Structural, soil-induced and hydrodynamic damping sources and nonlinearity (may matter for sloshing [24,25] ) are men-

tioned but not analysed.

The new mechanical/mathematical model accounts for the soil restoring forces and moment, the inertia forces and mo-

ments of the top-installed body, and the hydrodynamic loads as they follow from the linear water wave theory. The vari-

ational governing equation of the structural motions expresses the virtual work principle. The principal axes of inertia of

the top-installed body are chosen being parallel to the horizontal coordinate axes. Based on the latter choose, the math-

ematical statement becomes decoupled with respect to the unknown variables describing the free oscillations in the two


perpendicular vertical coordinate planes. One can also independently consider the vertical free oscillations of the monotower

(as a rigid body) which become possible due to the soil elasticity.

The undamped vertical eigenoscillations are not affected by sloshing of the contained liquid, which can be treated as

frozen. The soil feedback matters and, depending on the monotower shape, external sea water can yield a non-negligible

heave added mass. For instance, for the Draugen platform, the heave added mass increases the eigenperiod from 0.375 s

(‘dry’ structure) to 0.43 s.

Because of the aforementioned decoupling, without loss of generality, the undamped eigenoscillations are considered in

the Oyz -plane. Mathematically, the problem appears as a variational equation with respect to the trial beam deflections and

eigenfrequencies, which has three linear functional constraints returning, for the trial deflections, the generalised Stokes–

Joukowski potential, the added-mass component of the external velocity potential, and a sloshing-related component. A

projective scheme is proposed to solve this equation in which the Trefftz method approximates the functional constraints.

Three classes of eigenperiods implying the natural sloshing periods, the structural eigenperiods (sloshing is neglected), and

the coupled eigenperiods (the sloshing loads are accounted for) are introduced. A special functional basis is proposed al-

lowing for a clear physical treatment and having non-continuous second-order derivatives at certain values of the vertical

coordinate.

Comparing our Trefftz approximation of the external hydrodynamic loads with a frequency-independent strip-theory

approximation from [1, Section 5.4.5] (see, assumption (iii) in Introduction) concludes that the latter approximation is, gen-

erally speaking, not accurate enough in the numerically-studied cases. This is, in particular, due to the soil-induced caisson

motions leading to non-negligible flows at the seabed but a difference near the free surface also matters.

Adopting the Draugen monotower for numerical experiments shows that our mathematical model computes the highest

eigenperiod belonging to the full-scale observation range, between 3.9 and 4.2 s. Computations also confirm that the caisson,

even though it is modelled by a piece of the Euler–Bernoulli beam, moves like a rigid body and the speculatively-chosen

Young’s modulus from Appendix C does not affect that fact. The numerical studies also detect three types of eigenmodes.

Two of these types imply structural-type (sloshing can be neglected) and sloshing-type (structural vibrations are small)

eigenmotions, respectively. The eigenmotions of the mixed type (sloshing loads and structural vibrations mutually interact)

are specified only for the two highest eigenperiods. Dependencies of these eigenperiods on the stiffness of the soil and the

features of the top rigid body are examined. The eigenperiods increase with increasing the shear modulus of the soil, G s , the

radius of gyration, r g , the vertical coordinate of the centre of gravity, z p C , and the mass, M p , of the top-attached body. They

are less sensitive to r g but the mass centre position z C p and the rigid body mass M p are of importance. Our numerical studies

have also dealt with a sample gravity monopiles from Appendix B . Features of the eigenperiods and modes are qualitatively

the same as in the previous section but sloshing can normally be neglected when analysing the undamped eigenoscillations

of the sample monopiles. In summary , our numerical studies showed that the eigenperiods may significantly vary with the

soil shear modulus as well as with the inertia moment of the top-installed rigid body. The soil-induced caisson mobility

and associated external and internal hydrodynamic loads must therefore be accounted for. The internal hydrodynamic loads

require the Stokes–Joukowski potential, especially, when the shaft bottom performs angular motions.

Future studies in modelling the sea-based gravity towers should focus on excited motions. A challenge is to include

ringing and springing [26] . The multimodal method can help modelling the sloshing-induced loads for the forced vibrations.

The Euler–Bernoulli beam model is commonly accepted for the gravity platforms, monopoles and elevated water tanks.

However, testing the Timoshenko beam model is of interest. The proposed analytical variational technique allows for that.

We assume a flat seabed near the vertical structure. Estimating the scouring effect suggesting a non-flat seabed could be an

independent task. The same is for estimating the multi-layer soil effect.

Acknowledgement

The authors acknowledge the financial support of the Centre of Autonomous Marine Operations and Systems (AMOS)

whose main sponsor is the Norwegian Research Council (project number 223254-AMOS ).

Appendix A. The differential formulation following from the virtual work principle

The calculus of variations in the virtual work principle (18) yields the fourth-order derivatives by z appearing after in-

tegration by part. Because the second- and higher-order derivatives are not necessarily continuous at z = h 1 and l 2 , the

beam is treated as the beam–beam–beam system. The kinematic transmission conditions (1) are postulated. We arrive at

the Euler–Bernoulli beam equations

(EI(v ( j) k

) ′′ ) ′′ + ρb S b ̈v ( j) k

+ (N(v ( j) k

) ′ ) ′ = −χ[ h 1 ,h 1 + h ] πρl R i ˙ ϕ

(k ) i

(r, z, t| ̇ v k (z, t))

+ χ[0 ,d] (z) πρw

R e (z) ˙ ϕ

(k ) e (r, z, t| ̇ v k (z, t)) , k = 1 , 2 , (A.1)

on the intervals (0 , h 1 ) , j = 1 , (h 1 , l 2 ) , j = 2 , and (l 2 , L ) , j = 3 , which describe the horizontal beam vibrations , and the fol-

lowing natural boundary conditions at the upper beam end

−(EI(v (3) k

) ′′ ) ′ + M p

(v̈ (3)

k + z C p ( ̈v

(3) k

) ′ )

− N(v (3) k

) ′ = 0 , (A.2a)

http://dx.doi.org/10.13039/501100005416


EI(v (3) k

) ′′ + z C p M p

(v̈ (3)

k − g(v (3)

k ) ′ )

+ I p 3 −k

( ̈v (3) k

) ′ = 0 , (A.2b)

at z = L, k = 1 , 2 , and at the lower end,

(EI (v (1) k

) ′′ ) ′ + N (v (1) k

) ′ + k k v (1) k

= 0 , −EI(v (1) k

) ′′ + k 6 −k (v (1) k

) ′ = 0 , (A.3)

at z = 0 , k = 1 , 2 . The boundary conditions (A .2) and (A .3) account for the top plate inertia and soil elasticity, respectively.

Adopting the kinematic transmission condition (1) leads to the following natural (dynamic) transmission conditions

(EI (v (2) k

) ′′ ) ∣∣

h 1 + − (EI (v (1)

k ) ′′ )

∣∣h 1 −

= ρl π

∫ R 0

0

r 2 ˙ ϕ

(k ) i

(r, h 1 , t| ̇ v k (z, t)) dr, (A.4a)

(EI (v (3) k

) ′′ ) ∣∣

l 2 + − (EI (v (2)

k ) ′′ )

∣∣l 2 −

= ρw

π

∫ r 2

r 1

r 2 ˙ ϕ

(k ) e (r, l 2 , t| ̇ v k (z, t)) dr, (A.4b)

implying a jump for the bending moment at z = h 1 and z = l 2 , respectively, caused by the extra hydrodynamic loads at the

shaft bottom and the caisson ring-type root ( k = 1 , 2 ). The transmission conditions for the shear forces

(EI (v (2) k

) ′′ ) ′ ∣∣

h 1 + = (EI (v (1)

k ) ′′ ) ′

∣∣h 1 −

, (EI (v (3) k

) ′′ ) ′ ∣∣

l 2 + = (EI (v (2)

k ) ′′ ) ′

∣∣l 2 −

, (A.5)

naturally appear as well ( k = 1 , 2 ).

The kinematic transmission conditions (1) should also be fulfilled. The problem requires initial conditions for v k and

˙ v k implying the initial beam shape and horizontal velocity as well as the initial conditions for the surface wave problems (7),

(8) . Because Ox and Oy are parallel to the principal axes of inertia of the top plate, problem (A .1) –(A .5) becomes decoupled

with respect to v 1 and v 2 . The decoupling is impossible when I p 12

� = 0 .

Appendix B. A sample gravity monopiles of the wind energy industry

The existing sea-based gravity monopiles are, normally characterised by a highest eigenperiod of about 3 s and a rel-

atively small internal shaft radius that never provide the closeness of the natural sloshing period to these 3 s. However,

as we explained in the beginning of Section 5.3 , a future gravity monopiles may be about 30–40% bigger that could lead

to the aforementioned closeness. Composing such a future gravity monopiles we took typical geometric sizes from [5,6] by

Fig. B.14. The dimensions of the sample sea-based gravity monopiles. The adopted values (in meters) are l 2 = 10 , h 1 = 1 , L = 130 , h 2 = 4 , h 3 = 6 , d = 45 , r 2 =

5 , r 1 = 4 . 4 , r 3 = 4 . 25 , r 5 = 3 . 8 , r 6 = 4 . 1 , t 1 = 0 . 06 , r 0 = r 3 − t 1 , R 0 = r 1 − t 1 .


increasing them approximately 1.35 times. In addition, we increased the thickness and assumed the mean sea depth equal

to 45 m. The result is shown in Fig. B.14 . It provides r 0 = 4 . 19 m at the inner mean free surface level and T s 1 = 3 . 0262 s.

The physical parameters are ρb = 7 . 85 × 10 3 kg/m

3 and E = 2 . 1 × 10 11 N/m

2 for the steel and ρl = ρw

= 1 . 025 ×10 3 kg/m

3 for the sea water. The soil properties are taken the same as for the Draugen platform. We assumed, in addi-

tion, that the underground circular fundament has the radius r 4 = 20 m that is twice larger than the caisson (base) radius.

The top rigid body characteristics are determined by those of the rotor and the nacelle. We followed the same procedure

as for the geometric sizes, namely, existing values from [5,6] were increased by approximately 35% and, thereafter, slightly

varied to reach T 1 ≈ T s 1 . The procedure led to the trial values M p = 4 . 75 × 10 5 kg, r g = 5 m, z p C

= 5 m.

Appendix C. The Draugen platform

The Draugen platform possesses specific geometric and physical input parameters which characterise the monotower and

its support (caisson). There are three media: the liquids (contained and sea water), the reinforced concrete, and the soil. One

should also know the mass M p , the mass centre z p , and the inertia tensor I

p for the top rigid body (operational platform).

C 11

Fig. C.15. The meridional cross-section of the Draugen monotower without internal structures (left) and the adopted idealisation suggesting a replacement

of the seven-cell supporting substructure by an upright circular cylindrical body. The vertical dimensions, radii, and thickness (t = ) are presented in meters.

The idealised structure dimensions are fully defined by the two functions, R i ( z ) (the shaft radius) and R e ( z ) (the external radius), 0 < z < L = l 1 + l 2 . The

functions have jumps at z = h 1 (for R i ( z )) and z = l 2 (for R e ( z )). The equivalent radius r 2 = R e (0) is chosen to save the supporting structure volume. There

are skirts confining the soil layer of the 9 m in the depth which corresponds to a circular rigid fundament. The cells are coupled by a special tri-cell joining

system.


Fig. C.16. The linear-spline approximation of R i ( z ) and R e ( z ) fits the idealised Draugen monotower shape. It provides jumps at z 2 and z 3 (stars denotes that

point) but a cubic-spline smoothening in an ε-neighbourhood of z k , k = 4 , 5 , 6 ( ε ≈ 3 m in calculations) is employed to keep continuous the first-order

derivative. The tower has constant radius r 0 for z 5 < z < z 6 as specified for the Draugen platform. The thickness values are t 3 = 1 . 9 , t 4 = 1 . 65 , t 5 = t 6 = 0 . 7 ,

and t 7 = 2 . 0 m. The control z -points are z 1 = 0 , z ∗2 = h 1 = 7 . 0 , z ∗3 = l 2 = 38 . 0 , z 4 = 70 . 0 , z 5 = 237 . 5 , z 6 = 258 . 3 , and z 7 = L = 280 . 1 m with the control radii

R 0 = 20 . 6 , r 1 = 22 . 5 , r 2 = 52 . 1 , r 0 = 7 . 5 , and r 5 = 13 . 0 m.

The contained liquid and the sea water have the same mass density, ρl = ρw

= 1 . 025 × 10 3 kg/m

3 . The reinforced con-

crete of the monotowers has the mass density in the relatively-narrow range 2 . 4 × 10 3 < ρb < 2 . 5 × 10 3 kg/m

3 . The fixed

mean value ρb = 2 . 45 × 10 3 kg/m

3 is adopted. This value is used in [13] . The Young modulus E b of the reinforced concrete

belongs to the range 3 . 0 × 10 10 < E b < 3 . 5 × 10 10 N/m

2 (private communication with Steinar Hetland, Kvaerner Concrete So-

lutions AS). The paper [13] uses E b = 3 . 1 × 10 10 N/m

2 , which is close to the mean value E b = 3 . 25 × 10 10 N/m

2 adopted in

our calculations. The stiffness coefficients in (3) include the shear modulus G s and Poisson’s ratio νs of the soil. The pa-

per [13] suggested G s = 2 . 456 × 10 8 N/m

2 and νs = 1 / 3 , which can be adopted for the sandy clay [15, Appendix C] . The

report in [3] on the soil conditions under the Draugen platform writes that the upper 6–7 m soil layer is silty, sandy, very

soft to medium clay, with gravel and occasionally cobbles for which G s varies from 1 . 5 × 10 7 N/m

2 (at the seabed level) to

3 . 0 × 10 7 N/m

2 (at the 6 m depth), but the deeper soil is the silty, sandy, very stiff to very hard clay with gravel and cobbles

for which 5 . 0 × 10 8 < G s < 10 9 N/m

2 . This means that G s may vary in a wide range around the mean value 5 . 0 × 10 8 N/m

2

which is the lower bound for the sandy silty, sandy, very stiff to very hard clay. Poisson’s ratio negligibly changes for the

aforementioned soil types.

The open literature contains a very limited information on the top operational platform. In a private communication,

Steinar Hetland (Kvaerner Concrete Solutions AS) informed us that the platform is characterised by M p = 2 . 3 × 10 7 kg, z p =13 . 75 m, and the radius of gyration r g (rotation about the short axis of the rectangular plate going through S p , see, Fig. 2 ) is

about 28 m which derives, according to Steiner’s (parallel axis) theorem, I p 11

= I S p 11

+ z 2 p M p = M p (r 2 g + z 2 p ) = 2 . 238 × 10 10 kgm

2 .

These values are chosen as part of our mean input parameters scenario. Our parameter analysis examines how M p , z p C , and

I p 11

affect the eigenperiods.

The geometric shape of the Draugen monotower is shown in Fig. C.15 by the meridional cross-sections of the structure

with the cell-type supporting base (left) and its idealisation (right). The vertical dimensions, the radii, and the thickness of

the walls are measured in meters. The thickness of the tower varies from 1.5 to 3.36 m around the upper tower edge (at

z = L ). Accounting for that as well as for the internal tower structures, e.g., vertical pipes (these are not shown in the figure)

breaks, generally speaking, the entire structure symmetry. However, we neglect that and, therefore, the mean thickness value

2 m at z = L is adopted. The supporting structure consists of seven cells with spheric roofs and bottoms linked alongside by

a specific joining system as shown in Fig. C.15 . Saving the original geometric dimensions of the tower, except at z = L, we

replace the cells by an “equivalent” cylinder of the same volume and mass. The volume conservation condition computes the

external cylinder radius r 2 = 52 . 1 m. These and other external/internal radii are shown in Fig. C.16 where we demonstrate

how functions R i ( z ) and R e (z) , 0 < z < L = l 1 + l 2 are approximated by a linear spline (see, also the function definitions in

Fig. C.15 ).

The mass conservation condition for the idealised circular cylindrical support shows, after tedious but straightforward

derivations, that ρb ( z ) is a piece-wise function with ρb (z) ≈ 1 . 25 × 10 3 kg/m

3 , 0 < z < l 2 and ρb (z) ≈ 2 . 45 × 10 3 kg/m

3 , l 2< z < L . Another piece-wise function is E ( z ) which is a constant and equal to E b for l 2 < z < L but the Young modulus

function possesses an uncertain value for 0 < z < l 2 and dramatically depends on the cell joining system as well as other

features of the complex multicomponent supporting structure.

We introduce a mean input parameters scenario implying the idealised structure shape in Fig. C.16 and the following

input parameters: ρl = ρw

= 1 . 025 × 10 3 kg/m

3 , ρb (z) = 1 . 25 × 10 3 kg/m

3 , 0 < z < l 2 and ρb (z) = 2 . 45 × 10 3 kg/m

3 , l 2 <

z < L , E(z) = E b = 3 . 25 × 10 10 N/m

2 , 0 < z < L , G s = 5 . 0 × 10 8 N/m

2 , and ν = 1 / 3 , M p = 2 . 3 × 10 7 kg, z p = 13 . 75 m, and

I p 11

= 2 . 238 × 10 10 kgm

2 . By dealing with uncertainties implies studying the effect of (i) the speculatively–defined E ( z ) on the


interval 0 < z < l 2 , (ii) the shear modulus G s changing in a wide range about the mean value G s = 5 . 0 × 10 8 N/m

2 , the

range should cover G s = 2 . 456 × 10 8 N/m

2 used in [13] , (iii) M p , z p C , and I

p 11

characterising the top operational platform;

these values change due to cargos, cranes, and other auxiliary structures installed on the top platform.

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