master in engineering of structures, foundations and...

MASTER IN ENGINEERING OF STRUCTURES, FOUNDATIONS AND MATERIALS

MASTER'S THESIS

INFLUENCE OF SOIL STIFFNESS ON THE

DYNAMIC RESPONSE OF AN OFFSHORE WIND

TURBINE

Rafael Luque Suárez

September 2015

ABSTRACT

Offshore wind energy industry has experienced a significant growth over the past 15 years,

and it is expected to continue its growth in the coming years. The expansion to increasingly

deep waters and the rise in power and size of the turbines have led to a need for more reliable

and optimized support designs, which requires an extensive knowledge of the behaviour of

these structures. This work focuses on the dynamic response of an offshore wind turbine

founded on a monopile and subjected to wind loading. Different soil properties have been

considered in order to cover the range of stiffness from a very loose to a very dense sand. In

this way, the influence of stiffness on the structure behaviour has been assessed. Static and

dynamic analyses have been carried out by means of a finite element model implemented in

Abaqus. Head displacement and stress at the tower base have been obtained as functions of

soil stiffness, and they have been used to calculate the dynamic amplification that is produced

when the natural frequency of the system soil‐foundation‐tower approaches the load

frequency. Two different approaches of soil modelling have been compared: soil modelled as a

continuum and soil simulated with non linear elastic springs. Finally, a reliability analysis to

assess the probability of resonance has been performed with an analytical model, in which soil

stiffness properties are considered as stochastic variables.

RESUMEN

La industria de la energía eólica marina ha crecido de forma significativa durante los últimos 15

años, y se espera que siga creciendo durante los siguientes. La construcción de torres en aguas

cada vez más profundas y el aumento en potencia y tamaño de las turbinas han creado la

necesidad de diseñar estructuras de soporte cada vez más fiables y optimizadas, lo que

requiere un profundo conocimiento de su comportamiento. Este trabajo se centra en la

respuesta dinámica de una turbina marina con cimentación tipo monopilote y sobre la que

actúa la fuerza del viento. Se han realizado cálculos con distintas propiedades del suelo para

cubrir un rango de rigideces que va desde una arena muy suelta a una muy densa. De este

modo se ha analizado la influencia que tiene la rigidez del suelo en el comportamiento de la

estructura. Se han llevado a cabo análisis estáticos y dinámicos en un modelo de elementos

finitos implementado en Abaqus. El desplazamiento en la cabeza de la torre y la tensión en su

base se han obtenido en función de la rigidez del suelo, y con ellos se ha calculado la

amplificación dinámica producida cuando la frecuencia natural del sistema suelo‐cimentación‐

torre se aproxima a la frecuencia de la carga. Dos diferentes enfoques a la hora de modelizar el

suelo se han comparado: uno utilizando elementos continuos y otro utilizando muelles

elásticos no lineales. Por último, un análisis de fiabilidad se ha llevado a cabo con un modelo

analítico para calcular la probabilidad de resonancia del sistema, en el que se han considerado

las propiedades de rigidez del suelo como variables aleatorias.

TABLE OF CONTENTS

1 INTRODUCTION AND PURPOSE .................................................................................................. 1

2 OFFSHORE WIND TURBINES. OVERVIEW .................................................................................... 3

2.1 TYPOLOGIES ....................................................................................................................................... 3

2.2 LOADS ............................................................................................................................................. 13

2.2.1 Loads on the rotor .............................................................................................................. 13

2.2.2 Loads on the tower ............................................................................................................. 15

2.3 NATURAL FREQUENCY AND MODAL ANALYSIS .......................................................................................... 18

2.4 FAILURE MODES ................................................................................................................................ 21

2.4.1 Limit States ......................................................................................................................... 21

2.4.2 Bearing capacity ................................................................................................................. 23

2.4.3 Examples of failure ............................................................................................................. 25

2.5 ADVANCED DESIGN ASPECTS ................................................................................................................ 28

2.5.1 Soil ‐ structure interaction .................................................................................................. 28

2.5.2 Long term deformations ..................................................................................................... 31

2.5.3 Probabilistic approach ........................................................................................................ 34

2.6 SOLUTIONS AND IMPROVEMENTS ......................................................................................................... 35

2.7 CODES ............................................................................................................................................ 36

3 CASE OF STUDY ........................................................................................................................ 39

3.1 INTRODUCTION ................................................................................................................................. 39

3.2 STRUCTURE AND LOAD PROPERTIES ....................................................................................................... 39

3.3 SOIL PROPERTIES ............................................................................................................................... 40

3.4 MODEL DESCRIPTION ......................................................................................................................... 41

3.5 ANALYSES PERFORMED ....................................................................................................................... 44

4 RESULTS AND DISCUSSION ....................................................................................................... 53

4.1 STATIC LOAD .................................................................................................................................... 53

4.2 FREE OSCILLATION ............................................................................................................................. 54

4.3 FORCED OSCILLATION ......................................................................................................................... 57

4.4 P‐Y AND M‐ CURVES ........................................................................................................................ 66 4.5 MODAL ANALYSIS .............................................................................................................................. 72

4.6 RELIABILITY ANALYSIS ......................................................................................................................... 75

4.6.1 Montecarlo ......................................................................................................................... 76

4.6.2 FORM .................................................................................................................................. 81

5 CONCLUSIONS .......................................................................................................................... 85

6 REFERENCES ............................................................................................................................. 87

A. APPENDIX ................................................................................................................................ 91

I. FREE OSCILLATION. DISPLACEMENTS ..................................................................................................... 91

II. FORCED OSCILLATION. DISPLACEMENTS ................................................................................................. 97

III. FORCED OSCILLATION. STRESSES ......................................................................................................... 103

IV. STATIC EQUIVALENT PLASTIC STRAIN. CASE E=20 MPA ........................................................................... 109

LIST OF TABLES AND FIGURES

Table 3.1: Soil Properties

Table 4.1: Eigen frequencies

Table 4.2: Comparison of both model results

Table 4.3: Adjusted distribution functions to Montecarlo sample

Table 4.4: FORM results

Figure 2.1: Foundation typologies

Figure 2.2: Gravity foundation

Figure 2.3: Installation stages of a suction anchor

Figure 2.4: Tripod foundation

Figure 2.5: Jacket foundation

Figure 2.6: Helical pile

Figure 2.7: Spudcan foundation

Figure 2.8: Floating turbines

Figure 2.9: Vertical axis turbine developed by Vertax

Figure 2.10: Gust factor

Figure 2.11: Effect of wind turbulence

Figure 2.12: Design approaches

Figure 2.13: Cambell diagram

Figure 2.14: Grouted connection

Figure 2.15: Scour hole at Scroby Sands

Figure 2.16: Soil idealization with springs

Figure 2.17: P‐y curves

Figure 2.18: Global foundation stiffness model

Figure 3.1: Global view of the model

Figure 3.2: Dynamic load

Figure 3.3: Equilibrium of forces

Figure 4.1: Horizontal displacement at the head. Static analysis

Figure 4.2: Horizontal displacement at the head. Free oscillation Case E=20 MPa

Figure 4.3: Natural frequency

Figure 4.4: Horizontal displacement at the head. Forced oscillation Case E=10 MPa (f0<)

Figure 4.5: Horizontal displacement at the head. Forced oscillation Case E=20 MPa (f0=)

Figure 4.6: Horizontal displacement at the head. Forced oscillation Case E=40 MPa (f0>)

Figure 4.7: Maximum horizontal displacement at the head. Forced oscillation

Figure 4.8: Dynamic amplification factor

Figure 4.9: Stress contour. Case E=20 MPa

Figure 4.10: Stress history at seabed level. Case E=20 MPa

Figure 4.11: Stress range at seabed level

Figure 4.12: P‐y curve at the base of the tower. Case E=20 MPa

Figure 4.13: M‐ curve at the base of the tower. Case E=20 MPa

Figure 4.14: Plastic equivalent strain. Case E=20 MPa. Increment 9, F=4000 kN

Figure 4.15: Normal forces on the pile. Case E20. Increment 3

Figure 4.16: P‐y curve. Case E=20 MPa. Depth 4.80m

Figure 4.17: Modes of vibration

Figure 4.18: Natural frequency distribution

Figure 4.19: Tested distribution functions

Influence of soil stiffness on the dynamic response of an offshore wind turbine

1

1 Introductionandpurpose

Offshore wind energy industry has grown significantly over the past 15 years, with

Europe leading the way in the development of offshore wind farms, and it will

continue expanding in the coming years. Offshore wind power capacity is expected to

reach a total of 75 GW worldwide by 2020.

The need to build wind turbines in increasingly deep waters and an increment in the

height of the tower to support more powerful turbines are two immediate

consequences of this growth. In this situation, horizontal loads on the structure caused

by wind and waves are dominant factors in its support design. Due to the variable

nature of these loads, a dynamic analysis is required to predict the response of the

structure, both in the short and the long term.

Fatigue damages are critical in these structures and can reduce significantly its life and,

therefore, its profitability. The reason is that offshore structures are subjected to

millions of loading cycles, whose stress amplitude determines the possible occurrence

of damage on the material. This stress amplitude will be higher, due to dynamic

amplification effects, if the excitation frequencies are close to the natural frequency of

the structure. Therefore, the dynamic behaviour of the structure, and particularly its

natural frequency, is a crucial factor in the design of a wind turbine support. This factor

is highly dependent on the stiffness of the foundation and on the characteristics of the

soil.

The influence of the foundation on the dynamic behaviour of an offshore wind turbine

will be analyzed in this document. A total of 9 soil cases with different degree of

stiffness have been covered. Static and dynamic analyses have been performed to

assess the influence of soil stiffness on displacement, natural frequency and stress on

the tower. Two different models have been tested in which the soil has been

implemented in two different ways: as continuous elements in a finite element

simulation, and as springs at the base of the tower. Finally, a reliability analysis has

been performed with an analytical model to obtain the probability of resonance,

considering soil stiffness properties as stochastic variables.

The main objectives to be attained by the present study are as follows:

To review the state of knowledge and summarize the main relevant aspects.

To define the values that geotechnical parameters can take in order to cover

several degrees of stiffness in a granular soil.

1. Introduction and purpose

2

Development in Abaqus of a numerical model of an offshore wind turbine

founded on monopile. Such model is subsequently employed to perform static

and dynamic analyses that illustrate the behaviour of interest.

To assess the influence of soil stiffness on the natural frequency of the

structure.

To obtain the dynamic amplification of the structure in displacements and

stresses as a function of soil stiffness.

To compare the results of two different soil modelling approaches: continuous

elements and springs

To estimate the probability of resonance when soil parameters are considered

as stochastic variables.

A better understanding of the dynamic response of the structure will lead to optimized

designs and more economical foundations; therefore, they could have a significant

economic relevance, as current foundation cost is usually about 35% of the total cost

of the structure. Accurate estimations of the deformability of the foundation will allow

to prevent large displacements and to reduce maintenance operations or reparations

of damage caused by excessive tilt, which will also imply a reduction in the shutdowns

of power generation. Eventually, this will result in an increment of the design life and

profitability of the wind turbine.


3

2 Offshorewindturbines.Overview

2.1 Typologies

There are different typologies available for offshore wind turbines support. The

foundation has great influence both on the behaviour and cost of the structure,

therefore the choice of its typology is a relevant aspect in the design process,

especially with the increase in size and water depth of the latest turbines. The designer

must take into account, among others, the following factors when choosing a typology:

size of the turbine, water depth, geotechnical characterization of soil, dominant loads,

stiffness, dynamic response, cost and building process.

The main types of foundations for offshore wind turbines are the following:

Gravity

Monopile and guyed monopile with tensioned wires

Suction caisson

Multipod

Jacket

Helical pile

Spudcan

Floating

Vertical axis

Figure 2.1: Foundation typologies: a) gravity, b) monopile, c) suction caisson, d) multipile, e) multipod caisson structure. (Byrne & Houlsby, 2006)

2. Offshore Wind turbines. Overview

4

A brief description of these types is made in the following paragraphs.

Gravity

Gravity foundations are characterized by a great plant surface to reduce the unit load

that receives. Its geometry is usually circular or polygonal in plant and the most

common material is reinforced concrete. These foundations are constructed onshore

and transported by ship. In some cases they have hollow modules to facilitate the

transport, and they are finally filled with ballast to reduce costs. It is necessary to

adapt the sea bed before its installation. They are often used with depths less than 20

meters and begin to be expensive from 10 meters.

Figure 2.2: Gravity foundation

Monopile

This is the most common type of foundation at intermediate water depth, 25 ‐ 30m. It

consists of a steel tubular pile, typically 4m or more in diameter, which is installed


5

either by drilling and grouting, or by driving, sometimes by a combination of these two

methods. It reduces the amount of material that a shallow foundation would need at

intermediate depths but the construction process is more complex. Its cost is highly

dependent on the necessary equipment for the installation, as well as the materials.

This typology has been used at Horns Rev, Denmark, and in several wind farms in UK.

The size of monopiles is continuously increasing and giant monopiles designs, in up to

60m water depth, are currently under study. Monopiles of 73.5m length have been

recently fabricated for Siemens' turbines for the German Baltic 2 project, in water

depths of 23 to 44m.

Suction caisson

They are also called suction anchors, suction buckets or skirted foundations. They are

designed to reduce the cost increment of gravity foundations at depths greater than

10m. This foundation type is like a large upturned bucket, cylindrical in shape with

larger diameters than monopiles, 10 ‐ 15m, and shallower penetration depths. They

are installed by sinking them down to the seabed and pumping water out of the cavity

by a pump. When water is pumped, a pressure difference is created and a downward

hydrostatic force on the top of the caisson pushes the foundation to the design depth.

The horizontality of the caisson must be controlled during the installation.

Suction caisson are expected to be suitable for foundations in soft cohesive sediments,

where they can be easily installed and the drainage conditions allow a suction pressure

to be developed in the cavity when waves are soliciting the tower. This suction and the

friction between caisson and soil counteract the uplift force.

This typology has been used in 2008 at Horns Rev II, Denmark, and in a trial foundation

of suction caissons at Frederikshavn, Denmark.


6

Figure 2.3: Installation stages of a suction anchor (Malhotra, 2011)

Multipod

As water becomes deeper than 30m and as turbines become larger, monopile designs

tend to be too expensive to be economically viable, and equally a single caisson would

be uneconomical. Multiple footing is a more attractive solution, either a tripod or a

tetrapod. This typology has a longer mechanical arm and resists the bending moment

caused by horizontal loads with less material. The feet of the structure can be founded

either with piles or suction caissons.

The foundation supports a simple steel structure, which supports the turbine tower.

This steel structure allows to reduce the free length of the tower, which results in a

stiffer structure. This fact can make it easier to meet the dynamic requirements.


7

Figure 2.4: Tripod foundation

Jacket

This type of foundation, which is used to depths of 60 meters, comes from the offshore

oil industry. It consists of a lattice structure anchored to the bottom by piles. They are

transported on barge and lowered onto previously executed piles. Beatrice offshore

wind plant, Scotland, provides an example of such facilities.

These structures have little sensitivity to large waves due to the truss geometry. In

addition, its stiffness minimizes the dynamic amplification of loads. The tubular joints

are stress concentration points and have to be designed carefully to support fatigue.


8

Figure 2.5: Jacket foundation

Helical pile

The helical pile has been used in onshore installations when large tension capacities

are required, but its use in offshore installations has not been tested yet. Recent

studies suggest the suitability of this typology for offshore wind turbines (Byrne &

Houlsby, 2015).

In a multipod foundation the upwind footing is probably under a significant tension

load for the ultimate limit state. The geometry of an helical pile can contribute to

support this load with less length than a conventional pile due to the contribution of

the plates.


9

Figure 2.6: Helical pile (Byrne & Houlsby, 2015)

Spudcan

Spudcan foundations are usually used in jack‐up rigs rather than in permanent towers.

These structures are used for oil and gas exploration, temporary production and

maintenance work, or in construction processes of offshore structures, such as

lowering a jacket foundation for a wind turbine.

The platform of a jack‐up unit is supported by independent legs, usually three in

triangular platforms, each resting on a large inverted conical footing, which is known

as a spudcan. Supdcans are usually circular or polygonal in plant with a shallow conical

underside and a central spigot that contributes to sliding resistance. Diameters of

more than 20m are usual in the latest designs.

Spudcans are preloaded before the beginning of the jack‐up operation by pumping

water into compartments in the hull. This preload causes the spudcans to penetrate

into the sea bed until the load is equilibrated by the resistance of the underlying soil. A

spudcan can penetrate in a soft soil between up to 2 or 3 diameters prior to reaching

equilibrium. The purpose of preloading is to penetrate the foundation sufficiently so

that its bearing capacity exceeds that required during extreme storm loading.


10

Figure 2.7: Spudcan foundation (Hossain, Hu, Randolph, & White, 2005)

Floating

As water depth continues to increase, fixed structures on the seabed become very

expensive or even technically unfeasible. Floating structures are a novel solution which

is being investigated for depths greater than 100m. It consists of floating the wind

turbine and fixing its position by cables which are anchored to the seabed.

Floating platforms can be divided into three main categories based on their strategy

used to achieve static stability:

1. Ballast stabiliser ‐ SPAR: Stability is achieved by using ballast weights positioned

in the lower part of a buoyancy tank, which creates a righting moment and

inertial resistance to pitch and roll motions.

2. Mooring lines stabiliser – Tension Leg Platform (TLP): Stability is achieved

through the use of mooring line tension.

3. Buoyancy stabiliser ‐ FLOAT: Stability is achieved through the use of distributed

buoyancy, taking advantage of weighted water plane area for righting moment.


11

Figure 2.8: Floating turbines (Breton & Moe, 2009)

Vertical axis

This is not a foundation itself but a kind of turbine with a different configuration and

therefore with different solicitations in its foundation. In a vertical axis wind turbine

the axis of rotation is perpendicular to the ground. They are always aligned with the

wind, so they do not need and adjustment when the wind direction changes. In

onshore installations, the mechanical power generation equipment can be located at

ground level, which makes easier its maintenance. They are not self‐starting currently,

so they require an outside power source to start the turbine.

Offshore floating vertical axis wind turbines have been studied in the Inflow project,

which started in 2009 financed by the European Union.


12

Figure 2.9: Vertical axis turbine developed by Vertax


13

2.2 Loads

Wind turbines are exposed to very specific actions. Due to the stochastic nature of

wind and waves, loads are variable and difficult to predict. In addition, structural

components and ground are subjected to fatigue, which is a critical state in these

structures. The effect is more critical in large wind turbines: by increasing its size,

complex aeroelastic interactions are created, vibrations and resonances are induced

and they may produce dynamic load amplification. A description of the main loads

which act in the rotor and tower is made in the next paragraphs.

2.2.1 Loadsontherotor

The forces acting on the rotor are attributable to the wind and the weight of the

structure. They can be classified according to their temporary effect in relation to

rotation of the rotor:

Aerodynamic loads of constant winds and centrifugal forces that generate

stationary loads independent of time as long as the rotor turns at constant

speed.

Stationary fields but spatially uneven in the path of the blades, which create

cyclical loads when the rotor is turning:

o The wind flow produces variable loads according to the revolution of

the rotor, since the wind strikes the blades asymmetrically. An

inevitable asymmetry is due to the increase in wind speed with height:

During each revolution, the rotor blades are subjected to higher wind

speeds in the upper part, and therefore to higher loads than the sector

closets to the ground.

o A similar asymmetry is caused by crossed winds which occur with rapid

changes in wind direction.

The inertia forces due to the dead weight of the rotor blades also cause

periodic loads. Furthermore, the gyroscopic loads that occur with rotor

orientation also vary depending on the number of revolutions of the rotor. As

a result of gravity, the blade has a variable bending which changes according to

the angle, thus being an important source of fatigue.

In addition to the fixed and cyclic loads, the rotor is exposed to non periodic

loads, highly variable in magnitude and space, caused by wind turbulence. This

turbulence contributes to fatigue, particularly of the rotor blades. In order to

consider different wind speeds, gust factors can be specified depending on the

duration of the gust. The frequency of occurrence can be related to the

average speed and the gust factor.

The following figures show the gust factor and the influence of wind turbulence on

dynamic loading in a specific wind turbine. The bending of the blades is calculated in


14

two different analysis: ignoring turbulence (dashed line) and including the effect of

turbulence (solid line). The values of deflection are almost double due to wind

turbulence.

Figure 2.10: Gust factor

Figure 2.11: Effect of wind turbulence


15

2.2.2 Loadsonthetower

The forces on the shaft of the turbine are due to environmental actions. Two

conditions can be distinguished depending of their probability of occurrence: service

conditions, which are expected to occur often, and extreme conditions, which occur

rarely during the life of the structure. The design loads of the structure will be different

in each condition. A description of the main loads acting on the tower of an offshore

wind turbine is made in the following paragraphs.

Swell

The swell is usually considered together with the flow and it affects the part of the

structure that is located below the height of the incident waves.

Waves are irregular in shape, varying in height, length, speed and frequency, and they

can act in different directions simultaneously. 'Sea States' can be used to characterized

the swell with stochastic models, that is, as a superposition of components, each of

which is a periodic wave with amplitude, frequency and direction of propagation and

there are random phase relationships between them. A design sea state must be

described by a wave spectrum, a significant wave height, a spectral frequency related

to a peak period and a mean wave direction.

It can be used either real or periodic waves as an abstraction of a real sea for design. A

deterministic design must specify wave height, period and direction.

Wind

Wind exerts considerable efforts in every part of the structure above the sea level.

Wind speed is classified into gusts under a minute duration and in wind velocities

which are extended along a minute or more, normally 10 minutes. Wind data are

fitted into a reference height, typically 10 meters above the mean sea level, by using

wind profiles. Mean wind velocity increase with height and instant speed varies around

the average value due to turbulences. Wind shear is lower in offshore turbines than in

onshore ones, due to the lower roughness of the sea.

Tide

Tides can be classified in astronomical, wind tides and tides caused by pressure

gradients. The latter two are usually considered together and are called storm surge.

The sum of these three is called storm tide.

Tides are considered indirectly when designing a marine structure because they affect

both ocean currents and sea level.

Currents


16

Although currents are usually variable in space and time, they are generally considered

as a flow with constant speed and direction, varying only with depth. There are four

types of currents:

Wind generated currents

Tidal currents

Subsurface currents

Near shore currents

Ice

When wind turbines are projected in an area where water can freeze or where ice can

come adrift, this factor must be considered and it must be taken into account aspects

such as geometry and nature of the ice, its concentration and distribution, type of ice,

mechanical properties, velocity, direction of drift and the probability of encountering

icebergs.

Other conditions

It is essential to collect any additional environmental information available, as it would

be the salinity of water, seismicity of the area, temperature and so on. These are

factors that may be involved in some aspects of the design.

2.2.2.1 Morrisonequation

The dynamic wave analysis is indicated when the foundation of the wind turbine

allows large movements of the structure with respect to the sea, as it is the case of a

floating structure, or when loads are close to the natural frequency of the structure.

Moreover, the natural frequency of the structure has to be fitted between the

frequency of the rotor, 1P, and the frequency of the blades, 2P or 3P, and sometimes

wave frequencies are in this interval and therefore could be near the natural frequency

of the structure. This is a determining factor in the fatigue analysis of the structure

because the stress amplitude of each cycle is larger if the natural and excitation

frequencies are similar.

The force caused by waves on the shaft of the turbine is time dependent and can be

obtained with the Morrison equation:

Where:


17

, ′: Components of the velocities of tower and water due to currents and waves

normal to the cylinder axis.

, ′: Components of the accelerations of tower and water normal to the cylinder axis.

: Water density

D: Effective diameter, including marine growth.

Cd: Drag coefficient

Cm: Inertia coefficient

d, (t): Limit values of water and wave depth integration

2.2.2.2 Characterizationofwindload

Wind actions on the structure are time dependent and cause pressure acting on the

surface and producing normal forces. The overall response of the structure to wind can

be considered as a superposition of a quasi‐static 'background' component and a

'resonant' component due to the excitation of the natural frequencies. In the turbines,

the resonant effect of wind is given by the rotation of the blades. The symmetry of the

support structure allows not to take into account directionality, however, according to

UNE ENV1991‐2‐4, the following instability dynamic phenomena must be considered:

vortex shedding, galloping, flutter, divergence, interference galloping.

The basic wind pressure is defined by the following equation:

Where:

a: Air density

Cs: Shape coefficient

Vt,z: Mean wind velocity during a period T and z meters over the mean sea level

Vertical angle between wind direction and cylindrical axis.


18

2.3 Naturalfrequencyandmodalanalysis

The response of an offshore wind turbine to wind and wave loads is highly dependent

on the natural frequency of the system hub‐tower‐foundation, due to the dynamic

nature of these loads and the slenderness of the system. The natural frequency will

determine the stress and strain amplitudes produced by loading cycles, which in turn

determine the fatigue failure of the structure. Therefore, an accurate estimation of this

parameter is essential to assess the working life of a wind turbine.

In order to avoid resonance, the natural frequency of the system and the main load

frequency must be far enough from each other. The most repetitive load of a wind

turbine is that generated by mass imbalances in the rotor, whose frequency is usually

called '1P'. The shadowing effect of wind is also related to this frequency: each time a

blade passes the tower, the wind force stop pushing directly on it. The frequency of

this effect is n*P, 'n' being the number of blades in the turbine, which is 3 in most

cases. Therefore, the natural frequency of the tower must be far from 1P and 3P.

The rotor of modern wind turbines does not operate at constant velocity but in a range

of different velocities, therefore, there are two ranges of operating frequencies around

1P and 3P. The natural frequency of the tower cannot be in any of these two ranges.

There are three classical approaches to classify the design of a wind turbine structure

according to its natural frequency, the frequency 'P' of the rotor and frequency '3P' of

the blades:

Soft‐Soft: The natural frequency is less than '1P'. This implies a high flexibility of the structure. Furthermore, the frequency of waves is usually within this range, which can lead to resonance.

Soft‐Stiff: The tower frequency lies between 1P and 3P. This is the most common design.

Stiff‐Stiff: The tower frequency is higher than the passing blade frequency 3P. This leads to very stiff and therefore expensive foundations.

The next figure illustrates the possible design approaches depending on the value of

the natural frequency (Bhattacharya, 2014). The figure also refers to possible long

term changes in the natural frequency caused by soil hardening or softening under

cyclic loading. Although these considerations are outside the scope of this study, it

basically means that if the soil has a hardening trend, the natural frequency of the

system will increase with time, therefore, its design value should be lower. In the case

of softening, the design value of the natural frequency should be higher and it will

decrease with time.


19

Figure 2.12: Design approaches

In the last figure, the dash line shows the 10% increment of the ranges 1P and 3P that

suggests the DNV code (2014). It can be deduced from the figure that the natural

frequency of an offshore wind turbine has to be fitted into a narrow range of values

and, therefore, has to be determined accurately. For a soft‐stiff configuration, the

Cambell diagram can be used to illustrate the safe region o a wind turbine tower (L. V.

Andersen, Vahdatirad, Sichani, & Sørensen, 2012):


20

Figure 2.13: Cambell diagram (L. V. Andersen et al., 2012)

The blue and green lines represent the frequencies 1P and 3P respectively, which

depend on the velocity of the rotor. The red line is the natural frequency of a given

tower and the two vertical black lines represents the minimum and maximum

operation velocities of the rotor. The design is safe if the red line does not cross the

green or blue line between the two black lines, that is, if the natural frequency is

always different from 1P or 3P in the operation range.

The way to calculate the natural frequency of a system is to solve the eigenvalue

problem associated with the motion equation of the system. In the case of a linear

system, the displacement is defined by the following differential equation (Clough &

Penzien, 1975):

∙ ∙ ∙

Where:

M: Mass matrix

U: Displacement vector


21

C: Damping matrix

K: Stiffness matrix

F: External forces vector

In the particular case of zero damping, the free vibration problem considering the

common hypothesis of an harmonic solution, U=A*sin(t+), can be expressed as follows:

∙ ∙ ∙

∙ ∙1∙

The equation above is the eigenvalue problem for the matrix K‐1*M. For a n‐degree of

freedoms system, there are n number of modes, each one with a frequency of

vibration. The lowest frequency is called the natural frequency of the structure. If the

system is linear, which implies that all elements of the stiffness matrix are constant,

and the stiffness matrix is symmetrical, the solution of the forced vibration problem

can be expressed as a superposition of the vibration modes:

∑ ∅ ∗ for i =1,...n

Where:

∅ : Mode shape coordinate representing the position of the i‐mass in the j‐node.

: Generalized coordinate representing the variation of the response in mode j

with time.

When the system is very big, the solution obtained by modal superposition stops to be

reliable. In most common structures, the solution can be expressed accurately enough

with a few modes of vibration.

2.4 Failuremodes

2.4.1 LimitStates

The main limit states which have to be analyzed in an offshore wind turbine are the

following:

Ultimate Limit State (ULS).

Fatigue Limit State (FLS).

Accidental Limit State (ALS).

Serviceability Limit State (SLS).


22

The Ultimate Limit State refers to the capacity of the foundation to resist the

maximum load. It is necessary to analyze the critical combination of moment, lateral

and axial load, which gives the failure envelope of the foundation. Although it is

necessary to study each case, the maximum load is usually caused by waves during a

storm. In this situation the power production is stopped in order to reduce the

maximum load caused by wind. Therefore the wind forces on blades are of minor

importance in this state. The DNV gives the following examples of ULS failures:

Loss of structural resistance (excessive yielding and buckling).

Failure of components due to brittle fracture.

Loss of static equilibrium of the structure, or of a part of the structure,

considered as a rigid body, e.g. overturning or capsizing.

Failure of critical components of the structure caused by exceeding the

ultimate resistance (which in some cases is reduced due to repetitive loading)

or the ultimate deformation of the components.

Transformation of the structure into a mechanism (collapse or excessive

deformation).

The Fatigue Limit State is the failure due to the effect of cyclic loading. Most loads that

excite wind tower turbines are cyclic, such as rotation of blades, wind or waves; so the

structure is affected by repetitive cycles of load. The repetition of these cycles, which

can be about N=108 throughout the life of a wind turbine, reduces the resistance of the

structure. The damage caused by fatigue is highly dependent on stress amplitude,

which is affected by the natural frequency of the system tower‐foundation. If the

excitation frequency is close to the natural frequency of the structure, the stress

amplitude of the load cycles is larger, therefore it is necessary to take into account the

dynamics effects. The stiffness of the foundation plays a fundamental role in the

mitigation or amplification of the dynamics effects.

The Accidental Limit State corresponds to maximum load‐carrying capacity for

accidental loads or post‐accidental integrity for damaged structures. Some examples of

ALS are the following:

Structural damage caused by accidental loads.

Ultimate resistance of damaged structures.

Loss of structural integrity after local damage.

The Serviceability Limit State gives the tolerance criteria applicable to normal use. A

critical aspect in this state is the tilt at the hub level over the life of the structure. To

obtain it, it is necessary to calculate with a certain degree of reliability the settlement

and inclination of the foundation. Usually, the tilt criterion is very exigent and leads to

a design with a very stiff foundation, which increases its cost, this being about 35% of


23

the cost of the structure. A full understanding of the behaviour of the foundation is

necessary in order to optimize the design of the system. The DNV code gives the

following examples of SLS:

Deflections that may alter the effect of the acting forces.

Deformations that may change the distribution of loads between supported

rigid objects and the supporting structure.

Excessive vibrations producing discomfort or affecting non‐structural

components.

Motions that exceed the limitation of equipment.

Differential settlements of foundations soils causing intolerable tilt of the wind

turbine.

Temperature‐induced deformations.

Taking into account the previous Limit States, the following design aspects have to be

considered in the design of a wind tower foundation:

Capacity. Critical in ULS and ALS.

Stiffness and deformation. Critical in SLS and FLS.

Long term repetition of cycles, up to N=108. Critical in FSL.

2.4.2 Bearingcapacity

The most critical aspect in the Ultimate and Accidental Limit States is the bearing

capacity of the foundation, especially in shallow foundations. They have some specific

particularities that must be taken into account in the design. In the following

paragraphs, some of these aspects will be described briefly.

In general, there are two situations that have to be studied depending on the soil

conditions:

Undrained clays

Fully or partially drained conditions in sands (depends on permeability)

The bearing capacity and failure envelopes for combined loading of a skirted

foundation, considering that shear strength varies with depth, have been studied by

Gourvenec and Randolph (2003). They concluded that in the plane V:H, with M=0, the

shape of the failure envelope is independent of the foundation geometry or the

degree of non homogeneity. In the plane V:M, with H=0, they found a simple power

law relationship with the non homogeneity ratio. However, in the plane H:M, where

the failure envelope is asymmetric, they did not find a simple relationship.

An undrained failure envelope was studied lately by some of these authors,

considering general 3D loading and shear strength heterogeneity (Gourvenec &


24

Barnett, 2011). The authors proposed closed form expressions to predict the ultimate

limit states V‐H‐M that provides the apex points of the failure envelope and the shape

of the normalized envelope. The size and shape of the envelope depends on load

combination, embedment ratio and degree of soil strength heterogeneity.

A six degree of freedom model based on work hardening plasticity models has been

developed to study the response of shallow foundations to general loading (Bienen,

Byrne, Houlsby, & Cassidy, 2006). This model provides the yield surface, plastic

potential expressions, a hardening law and the elastic stiffness derived from unload‐

reload loops of load.

Andersen (2009) studied the behaviour of soils under cyclic loading in both offshore

and onshore structures. He observed that the cyclic shear strength and the failure

mode under cyclic loading depend strongly on the strength path and the combination

of average and cyclic shear stresses. Andersen concluded that the foundation capacity

under cyclic loading can be determined on the basis of cyclic shear strength

determined in laboratory tests.

For a suction caisson there are two states which must be analyzed, each one with

different load configuration:

Installation

Service

In the installation stage it is necessary to estimate the self weight penetration of the

caisson and the suction required. These two parameters can be calculated from CPT

data and caisson geometry by using a model based on measured data (Houlsby, Ibsen,

& Byrne, 2005). The horizontality of the caisson during the installation can be

controlled by dividing the caisson in two section and measuring the pressure in each

one.

The cavity depth in spudcan foundations during the penetration stage is a critical

parameter and can trigger a soil flow failure. The cavity depth is limited then by this

failure, which can be more restrictive than the wall failure incorporated in the

guidelines, that is, collapse of the vertical sides of the soil (Hossain et al., 2005).

Houlsby et al. (2005) carried out laboratory model testing, centrifuge model testing,

field trials at reduced scale and a full scale installation of an offshore wind tower. From

these experiments they developed plasticity based models to represent the behaviour

of monopod and tetrapod caisson foundations under cyclic loading. They concluded

that stiffness and fatigue are as important for turbine design as ultimate capacity. They

also observed a stiffness reduction and an increase in hysteresis when the load

amplitude, either vertical or moment, increases.


25

In multipod caisson foundations the uplift capacity is a relevant parameter. A reliable

understanding of the tensile capacity can reduce the separation between foots and

therefore the size of the foundation. Centrifuge tests conducted under undrained

conditions have been carried out to study the relationship between peak uplift

resistance, embedment ratio, state of the skirt‐soil interface, displacement and loss of

suction (Mana, Gourvenec, & Randolph, 2013). They concluded that peak undrained

uplift resistance is mobilized at displacements between 2% and 5% of the foundation

diameter, increasing with the embedment ratio.

2.4.3 Examplesoffailure

A failure detected in several offshore wind turbines with monopile foundation is the

displacement of the transition piece. This piece is usually located at sea water level,

between the turbine steel tower and the monopile. The link between the transition

piece and the monopile is called the grouted connection, which is showed in the next

figure:

Figure 2.14: Grouted connection (Löhning, Voßbeck, & Kelm, 2013)

A failure in the grouted connection has been detected in hundreds of offshore wind

turbines all over Europe, several years after construction. The transition piece slid

several centimetres and this settlement had to be stopped by temporary brackets


26

(Löhning et al., 2013). The repetition of cycles of bending moment is believed to be

one of the main reasons of this failure. The stress amplitude of these cycles is a critical

parameter to analyze the fatigue failure of the grouted connection.

A very restrictive parameter in a wind turbine is the tilt of the tower. Predictions show

that cyclic displacements could be two to five times the static single load values,

depending on the pile size, stiffness and depth. Therefore, to estimate the tilt of the

tower along its whole life it is necessary to analyze the long term displacement under

cyclic loading. The usual limit criteria for the tilt is 0.25°, although there is little to no

monitoring of lateral movement in existing turbines to justify this criterion. A rational

analysis of the operational limits of the turbine could lead to a less restrictive tilt limit

and therefore to a reduction in the foundation size (Golightly, 2014).

The erosion of the sea bead around an offshore structure caused by waves or currents

is called scour. A sinking in the scour protection was observed at Horns Rev 1 Offshore

Wind Farm, Denmark. This farm was installed in 2002 and a survey in 2005 showed a

sinking of the scour protection adjacent to the piles up to 1.5m (Nielsen, Sumer, &

Petersen, 2014).

The next figure shows a scour hole measured at Scroby Sands Offshore Wind Farm,

east coast England:

Figure 2.15: Scour hole at Scroby Sands (Whitehouse, Harris, Sutherland, & Rees, 2011)

The scour can affect the following three main areas of an offshore wind turbine (J Van

der Tempel, Zaaijer, & Subroto, 2004):


27

Foundation length

Natural frequency

J‐tube

When scour occurs, the effective length of the foundation decreases. The resistance of

the upper layers of soil is removed and the overburden pressure around the lower part

of the pile is reduced. Therefore, scour can affect the bearing capacity of the pile.

Scour has also impact on the natural frequency of the structure, which decreases with

an increasing scour depth. This reduction in the natural frequency implies a reduction

in the stiffness, which can affect the stress range and the fatigue damage.

The J‐tube is used to support the power cable from the turbine to the seabed. When

scour occurs, the J‐tube could be free spanning over the scour hole, which could

damage the power cable.

In deep waters the wall thickness of a monopile is conditioned by the buckling failure.

De Vries and Krolis (2007) have studied the buckling of a section at the mud line in

several offshore wind turbines in water depths ranging from 20 to 50m. They

concluded that the mass of the support structure increases dramatically with

increasing water depth. A ratio of the wall thickness and the diameter of the monopile

of 1:80 is a good initial estimate.

A possible failure in a multipod foundation is the uplift of one of the legs due to a

lateral load. In a monopod foundation, the uplift capacity is also mobilized in part of

the section to resist a strong bending moment. In the particular case of a skirted

foundation, a negative excess pore pressure can be generated between the foundation

top plate and the confined soil plug during undrained uplift, allowing reverse end

bearing capacity to be mobilized. A possible failure in this situation is the loss of

suction under the top plate. In order to avoid this, it is necessary to determine the

minimum skirt depth to foundation diameter embedment ratio required to generate

negative excess pore pressure under the top cap. It is also important to determine

over what duration the negative excess pore pressure can be sustained (Mana,

Gourvenec, Randolph, & Hossain, 2012).

In the case of spudcan foundations, most codes consider the instability of the cavity

walls during the penetration stage, which limits the depth of the foundation. However,

a more critical failure during the penetration stage is the soil flow failure (Hossain et

al., 2005). The soil back flow into the cavity can occur due to:

Plastic flow around the spudcan edge

Collapse of the vertical cavity walls into the hole

A combination of these


28

Liquefaction is a phenomenon that produces a drastic reduction in the effective

pressure of the soil due to an increase in the pore pressure. De Groot, Kudella, Meijers

and Oumeraci (2006) studied the phenomenon of liquefaction in structures subjected

to wave loads. They considered four types of failures:

Liquefaction flow failure

Stepwise liquefaction failure

Stepwise failure

Wobble failure

The most spectacular failure type is the liquefaction flow failure. This is only possible in

the case of a subsoil of very loose sand or silt combined with a low drainage potential,

for example by the presence of a clay layer or large structure dimensions. The other

three failures are more likely to occur in other conditions. Their relevance increases

with decreasing relative density and decreasing drainage potential.

Other particular situations that must be considered are the following:

Backward rotation or self healing. Recovering of rotation by cyclic loading after

a huge load.

Lateral spread potential. Horizontal displacement of soil at seabed due to

liquefaction.

Pile driveability and hammer performance. Behaviour of the pile during

installation.

Corrosion.

Transportation conditions.

2.5 Advanceddesignaspects

2.5.1 Soil‐structureinteraction

When the deformations that are produced at foundation level are relevant and can

change the behaviour of the system, it is necessary to take into account the interaction

between soil and foundation, that is, how the deformability of soil affects the response

of the structure. The traditional approach is to model the soil as springs whose forces

are applied at discrete points. The original concept of a beam on a elastic foundation

was proposed by Winkler in 1867 and has been has been adapted and applied to

specific problems by several authors (Broms, 1964; Davisson, 1970; Matlock & Reese,

1960).


29

The following figure shows the scheme of these kinds of models in a laterally loaded

pile:

Figure 2.16: Soil idealization with springs

In structure engineering, springs are usually characterized by a constant stiffness,

which represents the relation between force and displacement: k=F/x. In the case of

soils, the relation between force and displacement is not always constant, therefore,

the stiffness of springs modelling soil depends on the displacement of the spring. The

relation between the soil lateral resistance force and the displacement is called the P‐y

curve. The following figure shows the P‐y curves of a laterally loaded pile:


30

Figure 2.17: P‐y curves (DNV, 1992)

The P‐y curves are different at each depth, given that soil deformability changes with

depth and confinement. Therefore, each spring has associated a different curve P‐y. As

the depth increases, the stiffness of the curve is higher. The concept of P‐y curve is

associated with lateral displacement, but can be also extended to the others degrees

of freedom. Thereby, t‐z curve relates the tangential force at the pile shaft and the

vertical displacement; and Q‐z curve relates the vertical force at pile tip and the

vertical displacement.

Instead of modelling the surrounding soil of the pile with a distribution of several

springs, the whole system foundation soil can be substituted by springs at the base of

the tower for each degree of freedom. The curves that represent the relation between

force or moment and displacement or rotation are called global stiffness curves. The

following figure shows an example of this kind of model:


31

Figure 2.18: Global foundation stiffness model (Adhikari & Bhattacharya, 2012)

Modelling the soil as a three dimensional continuous element is an alternative

approach to the springs. This concept was originally developed by Poulos (1971), and

was extended later to introduce soil non linearity and yield criterion (Budhu & Davies,

1988; Poulos, 1973). These methods are not so extended as the P‐y curves, because

they involve more calculations and do not provide single and practical steps to obtain

foundation displacements. Nowadays, the most versatile method based on soil

continuous elements is the finite element method, which allows to analyze a wide

range of soil behaviour.

2.5.2 Longtermdeformations

The guidelines published by Germanischer Lloyd demand an analysis of both short and

long term soil‐structure interaction under cyclic loading, although the methods and

extent for analysis are not specified. They recommend to use experience from past

projects, however, this information is missing in the case of new offshore wind

turbines, whose diameter and depth are steadily increasing.

Byrne and Houlsby (2000) proposed a model based on continuous hyperplasticity to

describe the cyclic loading behaviour of a suction caisson on sand. In essence the

theory replaces the plastic strain in conventional plasticity theory with a continuous

field of an infinite number of plastic strain components, each one associated with a

separate yield surface.


32

The High‐Cycle Accumulation (HCA) model for sand (Niemunis, Wichtmann, &

Triantafyllidis, 2005) quantify the phenomenon of accumulation of stress and strain for

a large number of cycles of small amplitude. It is an empirical model based on

laboratory triaxial compression and extension tests. The main parameters of the model

are the following:

Accumulation rate of strain

Strain amplitude

Number of cycles

Average value of mean pressure during a cycle

Average stress ratio

Void ratio

Change of the polarization of the strain loop

The model follows an explicit calculation strategy, which can be summarized in the

following points:

Calculation of the initial stress field. The initial density can be obtained from

CPT / SPT values.

Implicit calculation of at least two first load cycles to obtain the spatial field of

strain amplitude. The authors use the hypoplasticity model with intergranular

strain for this purpose.

Recording of the strain path during the second cycle at each integration point.

Evaluate the tensorial strain amplitude from the recorded strain path. The

amplitude is assumed constant over all subsequent cycles, until it is

recalculated in a control cycle. The load cycles are grouped in packages of

constant amplitude and average value of bending moment and shear.

Find the accumulation rate of strain as the product of the parameters of the

model.

Find the stress increment caused by a package of N cycles.

The HCA model's authors studied the application of their model to offshore wind

foundations. Based on laboratory tests, they calibrated the model for a typical North

Sea fine sand and the parameters were used for FE calculations of an offshore

monopile foundation (Wichtmann, Niemunis, & Triantafyllidis, 2008). They concluded

that the application of the HCA model to offshore wind foundations has the following

particularities:

The HCA model is based on test with N<106 cycles, whereas an OWT is

subjected about N=108 cycles throughout its life.

It is necessary to consider changes in the polarization of the cycles due to

variations of the direction of wind and wave loading.


33

In case of scour protection it is necessary to determine the constants of the

HCA model for this material.

Galindo, Illueca, and Jimenez (2014) employed the HCA model to account for

accumulated deformations in a gas turbine submitted to a high number of cycles. The

cases of fine, poorly graded and medium density sand were studied. They observed

that transient situations at equipment's start‐up could be more restrictive than

stationary situations corresponding to normal operation. During these situations, a

wide range of frequencies is traversed, including frequencies that could be similar to

the natural frequencies of the ground.

LeBlanc, Houlsby and Byrne (2009) have studied the response of a monopile

foundation in sand to long term cyclic lateral loading. They conducted laboratory tests

where a small scale pile was subjected to between 8000 and 60000 cycles of combined

moment and horizontal loading. The authors concluded that accumulated rotation is

largely affected by the cyclic characteristics of the load. The typical tolerances for

accumulated rotation are breached if the foundation is designed considering the ULS

load, which suggests that accumulated rotation is the primary design driver. They also

concluded that cyclic loading increases the pile stiffness, independently of relative

density, which contrasts with the current methodology of degrading static p‐y curves.

Cuellar (2011) studied both long and short term effects of cyclic loading in an offshore

monopile foundation. In the short term, the foundation can be affected by transient

episodes of softening as a consequence of pore pressure accumulation, whereas in the

long term, the hardening and soil densification can affect the serviceability of the

structure. The author used the Finite Element Method to analyze the short term

effects and studied the long term effects by means of model tests in a reduced scale.

His main conclusions were the following:

In the short term, cyclic lateral displacements caused by extreme loads

produce a net accumulation of pore pressure in the soil due to a progressive

reduction of pore volume and the inability of the soil to dissipate the

overpressure between consecutive cycles.

The accumulation of pore pressure in the short term produces a decrease of

effective stress in the soil and can lead to considerable plastic deformations.

An increase of pile diameter can have a beneficial effect on the accumulation

rate of pore pressure due to the lower levels of pile displacement and soil

compression.

In the long term, a progressive reduction of bending moments was observed

due to an increasing rigidization of the upper layers of soil.

The general trend in the long term can be described as an attenuating

incremental deformation.


34

The densification of soil around the pile causes a general subsidence,

independently of scour phenomena, and affects the dynamic behaviour of the

foundation. The increase in stiffness of the soil and the reduction in

embedment change the natural frequency of the structure.

Experimental model tests on shallow foundations on dense sand have shown that

cyclic loads following a single huge load lead to a backward rotation of the foundation

(Wienbroer, Zachert, & Triantafyllidis, 2011). The authors explain this rotation by

different compaction rates due to uneven void ratios on the leeward and the

windward side. They introduce the Backward Rotation Index, BRI, to quantify the

rotation. 1000 cycles are sufficient to revert about 70% of the rotation due to the

maximum force.

2.5.3 Probabilisticapproach

Uncertainties in values of model parameters are especially relevant in geotechnics.

Unlike concrete or steel, soil is not a manufactured product submitted to a quality‐

controlled process, therefore, it is difficult to guarantee a certain value of its

properties. A proper characterization of soil must take into account the grade of

uncertainty of its parameters. This characterization of uncertainties in the input

parameters of a given model allows designers to evaluate their impact on the results.

The aim of a reliability analysis is to quantify the probability of failure of a given design,

with the concept of "probability of failure" referring to situations in which the design is

not fulfilling its purpose, considering it in a general sense so that it can be applied to

any limit state.

There are several methods to assess the reliability of a design, each one with a

different degree of complexity. They can be divided into the following four levels

(Mínguez, 2003):

Level 1: These methods do not calculate probability of failure. The grade of

uncertainty is measured by partial safety factors, which are selected for each

variable, such as load or strength. It is the traditional way and the most used

in several codes.

Level 2: The probability of failure, Pf, is calculated from the integral of the

joint probability density function of all variables of the problem. The main

disadvantage of these methods is that the integral is difficult to calculate

because of the complexity of both the density function and the limit state

surface that defines the integration domain. It is therefore necessary to

approximate the density function, which at this level is done by taking the

two first moments of the joint probability density function.


35

Level 3: The probability of failure is calculated by means of the global joint

probability function. These methods calculate the better approximations to

the exact probability of failure, and require specific integration techniques

and methodologies. The most common methods in this level are FORM,

SORM and Montecarlo.

A fourth level is mentioned in some texts in which an economic factor is taken into

account. The aim of these methods is to minimize the cost or to maximize the profits.

They involve principles of engineering economic analysis under uncertainty and the

problem variables are usually cost and benefits of construction, maintenance, repair,

consequences of failure, interest on capital and so on.

Methods in level 2 work with independent variables which follow a normal

distribution. In the case of dependent variables, they have to be transformed. A linear

approximation (First Order) of the limit state surface is used to estimate the probability

of failure. The probability density function is characterized by its two first moments

(Second Moment). These methods take their name from the two mentioned

assumptions: FOSM (First Order Second Moment). The main limitation of these

methods is that they are only exact with normal distributions and linear limit state

surfaces.

The First Order Reliability Methods (FORM) also uses a linear approximation of the

limit state surface, but it works with the exact density functions of the variables. The

Second Order Reliability Methods approximate the limit state surface by means of a

second order polynomial surface. They are very precise methods and much more

efficient than Montecarlo simulations.

The Montecarlo method is based on computing a large number of simulations for

different realizations of the stochastic variables. This values are taken randomly, so the

number of simulations has to be large enough to cover a representative range of

events (Metropolis & Ulam, 1949).

2.6 Solutionsandimprovements

In this section, some solutions that have been either proposed or adopted previously

to mitigate the problems that affect these structures are described. Given that this is a

complex and very changing field, they will be only listed and described briefly. It is

outside the scope of this study to explain the details of each solution:

Tuned liquid column damper (TLCD). It is a U‐shaped tube which is partially

filled with liquid and attached to the structure. The liquid oscillations when the

tower is excited help to restore the equilibrium. They can increase the fatigue

life of the structure (Colwell & Basu, 2009).


36

Roughness in skirted foundations. An increase in roughness in the wall of this

foundation generates more friction between the steel and soil and improves

the uplift resistance.

Reduction of gap. A mitigation of gap initiation and propagation in skirted

foundations would increase the uplift strength.

Electrokinetic strengthening of soil. Laboratory experiments carried out on

clays surrounding skirted foundations have shown an increase in the undrained

shear strength of soil, which enhances the uplift resistance (Micic, Shang, & Lo,

2002).

Scour protection. Several studies have been carried out in order to design

protection measures against scour. The main solutions are blankets or a layer

of ballast around the foundation (De Vos, De Rouck, Troch, & Frigaard, 2011).

Reduction of cavity depth. The flow failure in spudcan foundations during the

penetration stage can be avoided by limiting the cavity depth (Hossain et al.,

2005).

In the case of grouted connections, there are three main measures that can be

taken to avoid fatigue failure:

Shear keys. In the form of welded‐on profiles, they increase the axial bearing

capacity. They contribute to transfer the bending moment by vertical

circumferential forces. They reduce the interface opening at the top and the

bottom.

Conical connections. They transfer the axial load without relying on the bond

capacity.

Additional supports.

2.7 Codes

The main codes, recommendations or guidelines that are used in the design and

construction of an offshore wind turbine are listed as follows. Some of them are

specific of wind turbines and others were developed for general offshore structures,

mainly for the petroleum industry. The main organizations that have developed these

codes are the International Organization for Standardization (ISO), Det Norske Veritas

(DNV) and Germanischer Lloyd (GL). Norsok standards (N) are developed by the

Norwegian petroleum industry.

ISO 19900 (2013). Petroleum and natural gas industries. General requirements for

offshore structures.


37

ISO 19902 (2007). Petroleum and natural gas industries. Fixed steel offshore

structures.

ISO 19903 (2006). Petroleum and natural gas industries. Fixed concrete offshore

structures.

DNV‐OS‐J101 (2014). Design of offshore wind turbine structures.

DNV‐RP‐C203 (2011). Fatigue design of offshore steel structures.

DNV‐RP‐C204 (2010). Design against accidental loads.

DNV‐RP‐C205 (2010). Environmental conditions and environmental loads.

DNV‐CN‐30.4 (1992). Foundations.

GL‐IV‐2 (2012). Guideline for the Certification of Offshore Wind Turbines.

GL‐IV‐6‐4 (2007). Offshore Technology. Structural design.

GL‐IV‐6‐7 (2005). Offshore installations. Guideline for the construction of fixed

offshore installations in ice infested waters.

N‐003 (2007). Actions and actions effects.

N‐004 (2013). Design of steel structures.

API‐RP‐2A‐LRFD (1993). Recommended practice for planning, designing and

constructing fixed offshore platforms. Load and resistance factor design.

API‐RP‐2A‐WSD (2014). Recommended practice for planning, designing and

constructing fixed offshore platforms. Working stress design


39

3 Caseofstudy

3.1 Introduction

The behaviour of an offshore wind turbine under both monotonic and cyclic loading

has been studied by means of a finite element model implemented in the software

Abaqus. The tower is founded on a steel monopile in sand. In order to study the

influence of the soil properties on the response of the structure, several cases have

been analyzed with different soil stiffness.

3.2 Structureandloadproperties

The geometry of the structure is typical of an offshore wind turbine installed in water

depths around 30m. Its main properties are as follows:

Length of the tower over the seabed: 100m

Foundation depth: 30m

External diameter of tower and monopile: 6m

Thickness: 0.075m

Steel density: 8450 kg/m3

Friction coefficient steel‐sand: 0.3

Young modulus of steel: 205 GPa

Poisson coefficient of steel: 0.3

Mass of nacelle, hub and rotor blades: 336 tn

The load acting on the structure is the wind force, which is applied as a horizontal

concentrated force in the rotor. Two different loads have been considered:

Monotonic load:

o Magnitude: 700 kN

Sinusoidal load:

o Mean: 600 kN

o Amplitude: 100 kN

o Angular velocity: 1.65 rad/s

3. Case of study

40

The load has been chosen from a study of a similar tower so that its magnitude is an

intermediate value between the normal operative case and the extreme sea state

(Garcés García, 2012). The maximum value of the cyclic force is equal to the static

force, so the structure is under the same load magnitude in both cases.

The angular velocity of the cyclic load has been chosen to be within the range of

natural frequencies obtained in the dynamical analysis for different soils. Therefore,

the dynamic amplification of the response of the structure can be studied for different

ratios of natural frequency to load frequency. The load frequency considered is high

for the most usual wind spectrums but is close to 3P, therefore, it is representative of

the shadowing effect produced by the blades.

3.3 Soilproperties

A total of ten different cases have been analyzed in order to study the influence of soil

stiffness in the response of the structure. Geometry, structure properties and load are

the same in each case; soil properties vary gradually from a very loose sand to a very

dense or cemented sand. A Mohr Coulomb failure criterion has been considered in the

constitutive model of the soil. The soil properties are summarized in the following

table:

Case E (Mpa)

(°)

(°)

c (kPa)

k0 (Kg/m3)

Very loose 1 5 30 8 0 0.33 0.50 1800

2 10 32 8 0 0.32 0.47 1810

Loose

3 15 33 8 0 0.31 0.45 1820

4 20 34 9 0 0.31 0.44 1830

5 25 35 9 0 0.3 0.43 1840

6 30 36 9 0 0.29 0.41 1855

Dense 7 40 37 10 0 0.28 0.40 1875

8 50 38 10 0 0.28 0.38 1900 Very dense 9 100 42 12 10 0.25 0.33 2000

Table 3.1: Soil Properties

Where:

E: Young Modulus

: Angle of internal friction

: Angle of dilatancy

c: Cohesion


41

: Poisson coefficient

K0: Lateral earth pressure coefficient at rest

: Mass density

As it is shown in the table, soil properties represents a granular soil with a gradually

increasing stiffness from Case 1 to 9.

3.4 Modeldescription

A 3D finite element model of the system tower‐foundation‐soil has been implemented

in the software Abaqus v6.12‐1 to obtain both the static and dynamic response of the

system in each case.

The tower and monopile have been modelled with shell elements, given that the steel

thickness is small compared with the other two dimensions. The top section of the

tower, where the load is applied, is modelled with a disc whose Yong Modulus is 100

times the steel modulus. This represents a rigid union between the tower and the

nacelle and local problems derived from the application point of the load are avoided.

The mass of the disc is the sum of the masses of the nacelle, hub and rotor blades. This

mass at the top of the tower, and specially its inertia, has a big effect on the dynamic

analysis of the system.

The soil surrounding the monopile has been modelled with continuous elements,

specifically with 8‐nodes bricks. The soil part consists of a concentric cylinder with the

monopile. The cylinder has a depth of 60m, which means that there are 30m of soil, 2

times the monopile length, under the tip of the foundation. The diameter of the

cylinder is 120m, so that distance from the external side of the soil part to the centre

of the monopile is 10 times its diameter.

Soil stresses on the external side of the model have been checked to ensure that

during the load state they remain the same as those in the gravitational state.

Therefore, the dimensions of the soil model are large enough to avoid that its

boundary conditions affect the behaviour of the tower.

Monopiles are usually installed by driving, which results in a hardening of the soil

around the tip of the monopile during the construction process. In order to take this

into account, the region around the tip of the pile has been modelled with an increase

of 20% in the Young modulus. The dimensions of this area are 0.5 monopile diameters

from the lateral side of the monopile and 1 diameter under its tip.

Given that the problem is symmetrical with respect to the plane formed by the tower

edge and the direction of the load, only half part of the system tower‐foundation‐soil

has been modelled, in order to reduce the computational cost. Therefore, the load

3. Case of study

42

applied to the head of the model has to be half the actual load. The displacements in

perpendicular direction to the plane of symmetry have to be zero, which is introduced

as a boundary condition. The rest of boundary conditions are applied to the external

sides of the soil part: zero vertical displacement at the base of the cylinder and zero

horizontal displacement, along any of the two edges, on the external vertical side of

the cylinder.

In order to simulate the contact between the soil and the monopile, master and slave

surfaces have been implemented. The external side of the monopile is the master

surface and the adjacent soil is the slave. The normal behaviour is defined as a hard

contact, which means that the master surface cannot cross the slave, and separation

of both surfaces is allowed after contact without any tensile strength. The tangential

behaviour is defined by a friction coefficient between both surfaces. A tangential force

is generated when there is relative displacement between both surfaces, its magnitude

being the normal force multiplied by the friction coefficient.

The soil column inside the monopile has not been modelled because the state of this

soil after the construction process and its contribution to friction resistance is not

clear. However, its weight has been taken into account as a pressure under the

monopile tip.

The following figure shows a global view of the mesh, main dimensions, loads and

boundary conditions, Ui being the displacement along the i‐axis:


43

Figure 3.1: Global view of the model

Some results from this model has been implemented in a reduced model to perform

other group of calculations. The reduced model consists on a beam element

representing the tower, with a point mass at the head. The system soil‐foundation has

been modelled as two spring connectors at the base of the tower. They affect two

degrees of freedom at the base: horizontal translation and rotation. The behaviour of

these connectors is defined by the non linear elastic curves of stiffness obtained from

the continuous model.

3. Case of study

44

3.5 Analysesperformed

Two main groups of analyses have been performed in this study. The first one has been

applied to the 9 cases of different soils detailed in the paragraph "Soil properties". Its

aim is to obtain the displacement at the head of the tower, the natural frequency of

the system and the range of cyclic stresses in the steel.

The second group of analyses has been applied to the case nº 4 and its aim is to obtain

the relationship between displacements and forces in the foundation. These results

are implement in a simplified model in which the soil is substituted by its reaction

forces and moments on the tower, which are obtained from the curves force‐

displacement (P‐y) and moment‐angle (M‐).

The first group of analyses, which are applied to the 9 cases, is formed by the following

states:

Static load

Free oscillation

Forced oscillation

The first state consists of a static analysis in which a constant horizontal load of 700 kN

is applied to the head of the tower. This state is divided into two consecutive load

steps: gravity and static load. The main result obtained is the horizontal displacement

at the head of the tower.

The free oscillation state consists of the application of a static load of 700 kN to the

head of the tower and then, from this equilibrium state, the load is removed and a

dynamic implicit analysis is performed in which the tower is oscillating freely during 20

seconds. There are three consecutive load steps in this state: gravity, static load and

free oscillation. The objective of this analysis is to obtain the oscillation frequency of

the tower, which is measured from the displacement at the head of the tower over

time.

The state of free oscillation is a non standard way to obtain the natural frequency of

the system. The usual way to calculate this in a structure is to perform a modal

analysis, however, in this case there are some particularities which difficult the

application of this method. The main one is the high number of degrees of freedom

that the soil part introduces in the model. The soil nodes result in a great amount of

modes of vibration that are not relevant or hardly contribute to the response of the

structure. For example, a vertical oscillation of the soil nodes located in the external

area of the model do not mobilize any mass of the tower. Another problem is that a

modal analysis implies a linear response and a symmetrical stiffness matrix. The


45

contact surface between pile and soil can introduce some asymmetrical elements in

the stiffness matrix, which invalidates or distorts the modal analysis.

The last of the analyses performed for each case is the forced oscillation state. It

consists of the application of a sinusoidal load to the head of the tower. The peak value

of this load is 700 kN, which is the magnitude of the load applied in the static state.

This forced oscillation state is dived into three load steps: gravity, transition and

sinusoidal load. The load has a been modelled with the following analytical expression:

600 100 ∙ ∙

Where:

: Angular velocity. =1.65 rad/s

t: Time, starting in the sinusoidal load state.

The transition step is an implicit dynamic analysis of 5 seconds in which the load

increases gradually from 0 to 700 kN in 5s. Then, the sinusoidal load is applied and

maintained during 40 seconds. The response of the structure is obtained during this

period from an implicit dynamic analysis. The following graph shows the load that is

introduced in the model over time, where the magnitude is half the actual load due to

the symmetry of the model:

Figure 3.2: Dynamic load

3. Case of study

46

The main results that are obtained from the forced oscillation analysis are the

horizontal displacement at the head of the tower and the stresses at the base of the

tower. The difference between the maximum and minimum stresses is the range of

cyclic stress at the base of the tower, which is a critical parameter in the fatigue failure

of the steel.

The second group of analyses has been performed in case nº 4 and is formed by the

following calculations:

P‐y and M‐ curves Modal analysis in a reduced numerical model

Stochastic analysis in an analytical model

The global P‐y curve represents the relationship between the horizontal reaction force

that the soil produces on the pile and the horizontal displacement at the base of the

tower. The global M‐ curve is the relationship between the resultant reaction moment that the soil produces respect to the base of the tower and the angle of

rotation at this point. Both curves are used for calculations in a reduced model. The

following figure shows a scheme of the equilibrium of forces in the simplified model:


47

Figure 3.3: Equilibrium of forces

Where:

F: Wind force

P1: Weight of the nacelle, mass and rotor blades

P2: Weight of the tower

Rh: Horizontal reaction force

Rv: Vertical reaction force

M: Reaction moment

3. Case of study

48

The soil is substituted by two degrees of freedom at the base of the tower: horizontal

displacement and rotation respect to an edge which is perpendicular to the scheme. In

order to simplify the analysis, the vertical displacement at the base of the tower is

considered to be zero. The two degrees of freedom of the foundation are contained in

the plane formed by the tower and the wind force F, therefore, it is a 2D model. Both

degrees of freedom are considered to be uncoupled and they are characterized by two

elastic springs. The horizontal reaction force Rh and the horizontal displacement at the

base of the tower lead to the P‐y curve. The reaction moment M and the angle of

rotation at the base of the tower lead to the M‐ curve.

Each point of the foundation behaviour curves is obtained from an equilibrium state

for a given value of the wind force, F. A total of ten points have been calculated to

define each curve, increasing gradually the value of F from 0 to 4000 kN and

performing a static analysis in each load increment. The reaction forces can be

obtained from the equilibrium equations:

1 ∗ 1 2 ∗ 2 ∗

Where:

d1: Horizontal displacement at the head of the tower

d2: Horizontal displacement at the middle point of the tower

Ht: Height of the tower

In addition to the global P‐y and M‐ curves, which refer to the base of the tower, the P‐y curve at a given depth has also been obtained, that is, the lateral reaction of the

soil per unit length at a given depth versus the lateral deflection. These kind of curves

are used in Winkler models, where the soil is substituted by several springs.

In the model analyzed, the P‐y curve for one of these springs have been obtained from

the reactions on the contact surface at a given depth. This curve can be compared with

that given in the DNV code, which is obtained from the following expression, valid for a

laterally loaded pile in a cohesionless soil:

∙ ∙∙∙

∙

Where:

A: Factor to account for cyclic or static loading. For static loading:


49

3 0.8 0.9

H: Length (m)

D: Pile diameter (m)

Pu: Lateral resistance of soil per unit length (kN/m)

k: Initial modulus of subgrade reaction (MN/m3). Obtained as a function of

x: Depth (m)

y: Lateral deflection (mm)

The global P‐y and M‐ curves are implemented in a simplified model with the scheme

given in figure 3.3. The soil part has been changed for springs and, therefore, a

considerable reduction in the degrees of freedom in the foundation is produced. This

reduction allows to perform a modal analysis to obtain the natural frequency of the

reduced model. This natural frequency has been calculated and compared with that

obtained by means of the free oscillation of the whole model. The reduced model has

also been used to obtain the dynamic response under a cyclic load. In this way, the

goodness of the soil behaviour curves obtained from a static analysis can be assessed

for their use in a dynamic analysis.

Finally, a stochastic analysis of the natural frequency is carried out by following the

analytical methodology proposed by Adhikari & Bhattacharya (2012). In this model, the

foundation behaviour is represented by two degrees of freedom at the base of the

tower: horizontal translation and rotation. The global P‐y and M‐ curves are used to obtain the mean values of the translation stiffness, Kt, and rotation stiffness, Kr. These

parameters are considered as independent stochastic variables characterized by a

lognormal distribution.

The natural frequency is obtained from the free vibration problem by considering no

force on the system. The details of the analysis process to reach the solution are given

by the authors. The problem is reduced to solve the following equation:

| | 0

Where R is the following 4x4 matrix:

3. Case of study

50

Being:

This equation in can be solved numerically by using the Newton‐Raphson method in

Matlab. The solution of the equation which is closest to zero gives the first natural

frequency of the system. All variables detailed above are constant values except for Kr

and Kt, which are stochastic variables. The equation will be solved for different values

of these two variables depending on the chosen method. Two different methods have

been used to obtain the probability of failure: Montecarlo and FORM (First Order

Reliability Method). They are detailed in the section "Reliability analysis".

In order to compare the efficiency of each method, the computational cost has been

defined as the number of times that the equation |R()|=0 has to be solved, which is


51

where the most number of operations are computed and, therefore, what consume

most part of the time.


53

4 Resultsanddiscussion

4.1 Staticload

A horizontal constant load of 700 kN has been applied to the head of the tower for the

9 different soils cases. A static analysis has been performed to obtain the horizontal

displacement at the head of the tower under this load state. The results are shown in

the following graph:

Figure 4.1: Horizontal displacement at the head. Static analysis

The horizontal axis of the graph represents the Yong modulus of each case, however,

this parameter is not the only one which varies, as it is detailed in the section "Soil

properties". This and the next graphs will be in general functions of the Young

modulus, as it is the most representative parameter of soil stiffness, but it has to be

taken into account that there are other soil properties, such as angle of friction,

dilatancy, Poisson coefficient, density and so on, that also change in each case.

0.000

10.000

20.000

30.000

40.000

50.000

60.000

70.000

0 10 20 30 40 50 60 70 80 90 100 110

U1 (cm

)

E (Mpa)

Soil

Rigid

0.25° limit

4. Results and discussion

54

The blue line represents the horizontal displacement at the head of the tower, U1,

obtained from the model for each case of soil. The dashed red line represents the

commonly used limit criterion for the tilt of the tower of 0.25°, which for a tower of

100m length results in a head displacement of 44 cm. The green continuous line

represents the head displacement for the case of a completely rigid foundation, that is,

all displacements and rotations at the base of the tower are zero. For the completely

rigid case the displacement is around 19 cm.

The displacement obtained from the model varies from 64 cm in the more flexible soil,

E=5 MPa, to 29 cm in the more rigid soil, E=100 MPa. As it is shown in the graph, the

displacement trends asymptotically towards the rigid value as the Young modulus

increases, but there is still a difference of 34% in the case E=100 MPa. The

displacement decreases sharply when the Young modulus increases from 5 to 20 MPa.

The reduction is slighter between 20 and 40 MPa, where the displacement varies from

37 to 33 cm. Over 40 MPa the displacement reduction is very small, even for large

increments in the Young Modulus.

In relation to the tilt criterion of 0.25°, the displacement is acceptable if the Young

modulus is equal to or greater than 15 MPa. That means that cases 1 and 2 would lead

to excessive displacements at the head of the tower, whereas the rest of the soils

would be stiff enough to maintain the displacement below the limit.

The influence of soil stiffness on the static displacement of the structure is more

relevant in the case of loose sands, which usually have a Young modulus below 20MPa.

In these cases, the difference between a very loose sand and a loose sand can be very

relevant, given that small increments in the Young modulus lead to large reductions in

the static displacement. An increment of 10 MPa in the Young modulus, from 5 to 15

MPa, implies a displacement reduction of 24 cm, which is around 38% of the

displacement. Dense sands have normally Young moduli over 50 MPa. In these cases

the difference between a dense sand and very dense sand does not have a relevant

influence on the static displacement. In this range, an increment of 50 MPa in the

Young modulus, from 50 to 100 MPa, leads to a displacement reduction lower than 3

cm, which is around 8% of the displacement.

4.2 Freeoscillation

An implicit dynamic analysis has been carried out in which the model oscillates freely

from a static equilibrium state. In the static equilibrium state a constant horizontal

load of 700 kN is applied to the head of the tower. Then, this load is removed and the

oscillations of the tower allows to measure the natural frequency. The displacement at

the head of the tower has been measured over time during 20s and the natural

frequency is obtained from this curve.


55

The next figure shows the displacement obtained for the case E=20 MPa:

Figure 4.2: Horizontal displacement at the head. Free oscillation Case E=20 MPa

The curve starts in t=2s with a displacement of 37cm at the head. The period from t=0

to t=2s corresponds to the static states of gravity and constant load. From this state

the system oscillates freely during 20s, until t=22s. Along this time almost five cycles

are repeated and the oscillation period can be measured. The period is very similar in

all cycles and the obtained value is the mean of them.

Although the period is maintained over time, the amplitude decreases in each cycle,

mainly due to dissipation of energy by friction. These losses of energy are largest when

the soil is more flexible, because the relative displacement between contact surfaces is

greater.

The next figure shows the natural frequencies calculated for each case:


56

Figure 4.3: Natural frequency

The blue line represents the natural frequency obtained for each soil and the green

line represents the natural frequency for the case of a completely rigid foundation,

that is, if the displacements and rotations at the base of the tower are zero. The

frequency for the rigid case can be obtained with the following expression (Jan van der

Tempel & Molenaar, 2002):

03.04 ∙ ∙

0.227 ∙ ∙ 4 ∙0.399

Where:

E: Young modulus of steel

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0 10 20 30 40 50 60 70 80 90 100 110

f0 (Hz)

E (MPa)

Soil

Rigid


57

I: Inertia of the tower section

M: Mass of the nacelle, hub and rotor blades

m: Mass of the tower per unit length

L: Height of the tower from seabed

As can be observed in the graph, the natural frequency increases with the Young

modulus, which means that, as it was expected, rigid soils lead to rigid soil‐structure

systems. The range of frequencies varies from 0.19 Hz in the case E=5 MPa to 0.30 Hz

in the case E=100 MPa. The highest value obtained, 0.30 Hz is still far from the rigid

case, 0.399 Hz, therefore the influence of a non rigid foundation is relevant on the

natural frequency.

In a similar way to the results of static displacement, the variation of natural frequency

is very sharp for values of E under 20 MPa. When the Young modulus varies from 5

MPa to 20 MPa the natural frequency increases 0.07 Hz, or 38%. However, when the

Young modulus varies from 50 to 100 MPa, the natural frequency increases only 0.01

Hz, or 3%. Therefore, the difference between a very loose sand and a loose sand is

much more relevant than between a dense sand and a very dense sand. Increases in

soil stiffness over values of Young modulus of 40 MPa produce very little variation in

the natural frequency.

It has to be mentioned that the frequency obtained for the case E=20 MPa has a value

of 0.263 Hz, which is equivalent to 1.65 rad/s. This is the angular velocity of the load

described in the section "Structure and loads properties". Therefore, resonance is

expected to occur around the case E=20 MPa in the forced oscillation of the system,

which is described in the following section.

4.3 Forcedoscillation

A horizontal sinusoidal load has been applied to the head of the tower and an implicit

dynamic analysis has been carried out to obtain the displacement at the head over

time and the maximum stresses in the steel. The response of the structure has been

obtained over time for 40s of cyclic loading. Before this 40s period there is a transition

state of 5s in which the load increases from zero to 700 kN. The load is defined by the

following equation:

600 100 ∙ ∙

Where:

: Angular velocity. =1.65 rad/s


58

t: Time, starting in the sinusoidal load state.

The next three graphs show the horizontal displacement of the head of the tower for

the cases E=10, 20 and 40 MPa. As it was showed in the section "Free oscillation", in

the case E=10 MPa the natural frequency is lower than the load frequency, both

frequencies are equal in the case E=20 MPa, and the natural frequency is higher than

the load frequency in the case E=40 MPa.

Figure 4.4: Horizontal displacement at the head. Forced oscillation Case E=10 MPa (f0<)


59

Figure 4.5: Horizontal displacement at the head. Forced oscillation Case E=20 MPa (f0=)

Figure 4.6: Horizontal displacement at the head. Forced oscillation Case E=40 MPa (f0>)

As can be observed in the three graphs, the amplitude of each cycle increases in all

cases over time during the first part of the loading period. In the cases E=10 and 40

MPa, the amplitude starts to decrease after 23 and 33s respectively. The 40s duration


60

of the cyclic loading has been chosen in order to analyze the response of the system a

few seconds after the instant that the amplitude starts to decrease, which is very close

to 40s in the cases whose natural frequency is around the load frequency.

In the case E=20 MPa the amplitude of the cycles is continuously increasing until the

end of the loading period. The natural frequency and the load frequency are equal and

resonance is produced due to the coupling of these two frequencies, therefore, the

amplitude increases steadily. Damping has not been considered specifically, but some

losses of energy by friction occur on the contact surfaces. This means that the cycle

amplitude would probably stop growing and stabilize after a longer period that the one

considered in this study. Due to the great computational cost that implies a longer

dynamic analysis, this testing has not been carried out and the maximum displacement

for the case E=20 has been taken within the period of 40s of cyclic loading.

The following graph shows the maximum displacements that have been obtained at

the head of the tower for each case:

Figure 4.7: Maximum horizontal displacement at the head. Forced oscillation

0

20

40

60

80

100

120

140

160

0 10 20 30 40 50 60 70 80 90 100 110

U1 (cm

)

E (MPa)

Dinamic

Static


61

The blue line represents the displacements under dynamic load and, in order to

compare them, the dashed red line represents the displacements obtained in the

section "Static load". The maximum tilt criterion of 0.25°, as it was specified in the

section "Static load", leads to a maximum displacement at the head of 44 cm for a

100m tower. This means that all cases are above this limit for the dynamic load state,

whereas in the static load state, only the cases E=5 and 10 MPa presented

unacceptable head displacements. Therefore, consideration of inertial effects is a

crucial issue in this kind of structures, given that the dynamic response of the system is

likely to be above the static. In this particular study, the dynamic amplification is high

enough to exceed the limit criterion and switch the response of the system from

acceptable to unacceptable within the whole range of soil stiffness.

The dynamic displacement curve is roughly parallel to the static curve when the Young

modulus is under 10 MPa or over 40 MPa, which means that variations in soil stiffness

outside this range produce small variations in displacements. However, between these

two values the dynamic amplification is higher and reach a maximum in E=20 MPa, due

to resonance effects. In this case the natural frequency, which was obtained in the

section "Free oscillation", has a value of 0.26 Hz, equal to the load frequency. There is

a range around this value where the dynamic amplification is higher than in the rest of

the domain. The following graphs shows the amplification factor versus the ratio of

frequencies:

Figure 4.8: Dynamic amplification factor

0

50

100

150

200

250

300

350

400

450

70 75 80 85 90 95 100 105 110 115 120

Udin / U

sta(%

)

f0/ (%)


62

Where:

Udin: Maximum displacement under cyclic load

Usta: Displacement under static load

f0: Natural frequency

: Load frequency. =0.263 Hz

The ratio f0/ ranges from 72% in the case E=5 MPa to 115% in the case E=100 MPa,

and there is dynamic amplification in all cases analyzed. The dynamic amplification

factor Udin/Usta reaches a maximum of 400% when the natural frequency is equal to the

load frequency. This factor decreases to 200% when the ratio f0/ is lower than 90% or higher than 110%. Increments of the ratio f0/ outside the range 90‐110% produce smaller variations of the amplification factor than those produced within this range.

Therefore, resonance effects are mainly produced when the ratio between natural

frequency and load frequency is 100% with an influence margin of 10%. This margin

of 10% around the load frequency is recommended by the code DNV (2014), which

suggests that the ranges of frequencies 1P and 3P to be avoided by the natural

frequency should be extended 10% at each end, P being the rotor frequency.

The displacement at the head of the tower, which has been analyzed above, is an

important factor in the Serviceability Limit State (SLS), but there are other limit states

that must be analyzed, as it was described in the section "Failure modes". Fatigue Limit

State (FLS) is critical in this kind of structures, due to the cyclic nature of loads, and the

stress range in the steel is the factor that determines this model of failure. The

following figure shows a stress contour of the tower for the case E=20 MPa:


63

Figure 4.9: Stress contour. Case E=20 MPa

The contour represents the values of S22, which is the normal stress along the local 2‐

axis, parallel to the global z‐axis represented in the figure. The values correspond to

the instant of maximum displacement, when tensile stresses are also maximum. Red

elements correspond to tensile stresses, which are positive, and blue elements to

compression stresses, which are negative. The point of highest stress is represented in

the figure, which is reached in the element 300 located at seabed level, on the left

side of the tower. Due to oscillation, the stress value in this element changes over

time, and the difference between the maximum and minimum value will determine

the fatigue failure. The following graph shows the value of S22 at the base of the tower

over time for the case E=20 MPa.


64

Figure 4.10: Stress history at seabed level. Case E=20 MPa

After the gravity step, t=1s, the base of the tower is compressed, so the graph starts

with a negative value of S22. Then, the amplitude is increasing steadily over time, in a

similar way that the displacements. In the rest of the cases the behaviour is also similar

to the displacements: the stress amplitude increases until certain instant and then

starts to decrease. The maximum and minimum values of stress have been obtained

for each case and its difference, the stress range, is represented in the following graph:


65

Figure 4.11: Stress range at seabed level

The tendency of the stress range showed in this graph is similar to that showed in the

maximum displacements at the head, except for the left side. As it was showed in the

figure 4.7, the displacements decrease from case E=5 MPa to 10 MPa due to an

increase in soil stiffness, and start to increase when the Young modulus is within the

resonance range. However, the stress range increases between E=5 and 10 MPa. The

soil flexibility in the case E=5 MPa produces more displacement at the head than in

case E=10 MPa, but less stress at the base. The monopile rotation in the first case is

higher than in the second, and so the head displacement, but the steel flexion is lower,

which results in a lower stress range.

From the stress range calculated above, the Fatigue Limit State can be analyzed by

means of the S‐N curves. The stress range is related to the number of cycles that the

steel can support before a fatigue failure is produced. The number of cycles decreases

when the stress range increases, therefore, structures within the range of resonance

will have a shorter lifespan than those outside this range.

0

20

40

60

80

100

120

140

160

180

200

0 10 20 30 40 50 60 70 80 90 100 110

S22 (MPa)

E (MPa)


66

4.4 P‐yandM‐curves

A static analysis has been performed in the model in order to obtain the P‐y and M‐ curves. These curves represent the stiffness of the system soil‐foundation and are

formed by the following parameters:

P: Horizontal reaction force at the base of the tower

y: Horizontal displacement at the base of the towe

M: Reaction moment at the base of the tower

: Angle of rotation at the base of the tower

A horizontal force F is applied at the head of the tower to obtain these four

parameters. Each curve has been defined with 9 points, in addition to the (0,0) point,

which have been calculated from 9 equilibrium states with different values of F. The

magnitude of F ranges from 0 to 4000 kN, leading to head displacements from 0 to

2.72m. Values of F over 4000 kN have been tested but not included, because they

produce excessive plastic strain around the pile and some convergence problems arise.

Moreover, head displacements higher than 2.72m are well above the range that have

been analyzed in this study. The maximum head displacement obtained in previous

analyses is 1.46m, as it is shown in the figure 4.7. Therefore, a maximum force of

4000kN is enough to define the stiffness curves of the foundation in this case.

The analysis of the stiffness curves has been focused only on the case E=20 MPa, due

to the high computational cost that it requires. There are two curves to obtain for each

case, each of them needs to be defined by 9 points. Each equilibrium state gives a

point for each curve, so it would be necessary to perform 81 equilibrium states to

obtain the curves of the 9 cases. The case E=20 MPa is considered to be representative

to analyze the modelling issues associated with stiffness curves.

The two following graphs represent the P‐y and M‐ curves which have been obtained according to the methodology detailed above:


67

Figure 4.12: P‐y curve at the base of the tower. Case E=20 MPa

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 5 10 15 20 25

P (kN

)

y (cm)


68

Figure 4.13: M‐ curve at the base of the tower. Case E=20 MPa

The derivative of the curves P‐y and M‐ represents the translational and rotational stiffness respectively:

These parameters are used in the next sections to perform an analysis with a reduced

numerical model and with an analytical model. Given that the curves are non linear, Kt

and Kr are not constant values, and they depend on the value of displacement or

rotation respectively. Kt ranges from 4.18107 N/m in y=0, to 1.49107 N/m in y=20cm;

and Kr ranges from 4.771010 N*m/rad in =0, to 2.391010 Nm/rad in =0.82°. The non linearity of these curves can be taken into account in the reduced numerical model of

the section "Modal Analysis", but not in the analytical model which has been used in

the section "Reliability Analysis".

0

50

100

150

200

250

300

350

400

450

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

M (MN*m

)

(°)


69

It has to be mentioned that these stiffness parameters, or modulus of subgrade

reaction as they are commonly known, are not intrinsic soil properties. They depend

on soil properties but also on geometric factors, such as pile length and diameter.

Therefore, stiffness curves have to be calculated for each problem. The advantage of

these curves is that they allow to reduce the computational cost by simplifying the

model. The problem is that they are geometry dependent, although there are models

to obtain them for some common cases.

The non linearity of the curves calculated above can be explained by the existence of

soil elements around the pile whose stresses have reached the failure criterion. That

means that these elements change their constitutive law from a linear elastic

behaviour to a perfectly plastic one, which leads to a non linear relationship between

forces and displacements at a global level. Plastic strains start to arise in increment 4,

when the force at the head has a value of 460 kN. From this point, stiffness curves start

to lose linearity. The first soil elements that present plastic strain are those located

close to the seabed level on the left side of the tower, where soil and monopile start to

separate. As the head force increases, plastic strain arise also at the base of the

monopile, due to separation between soil and steel, and at seabed level on the right

side, due to the compression of the monopile against soil. The following figure shows a

contour of the plastic strain around the monopile for increment 9, when the head

force value is 4000 kN:

Figure 4.14: Plastic equivalent strain. Case E=20 MPa. Increment 9, F=4000 kN


70

The contour represents the plastic equivalent strain, which is a scalar measure of the

plastic strain defined by the following expression:

1

:

In the last figure, dark blue colour represents elements that do not suffer any plastic

strain because they remain in elastic range. Elements from light blue to red presents

plastic strain, red being the colour which represents the maximum value of equivalent

plastic strain.

The curves obtained above are called global, and they represent the reactions that the

soil and foundation apply at the base of the tower. They can be used in a model where

the whole foundation is substituted by two springs. When the foundation is

substituted by several springs at different depths, such as in Winkler model, it is

necessary to use a P‐y curve for each spring. In the model analyzed, the P‐y curve at a

depth of 4.80m has been calculated from the reaction of soil on the contact surface.

The following figure shows the forces that the soil exerts on the pile in each node:

Figure 4.15: Normal forces on the pile. Case E20. Increment 3

The figure represents the normal forces on the pile for depths between 0 and 9.6m.

The values correspond to increment 3, when the head force has a value of 280 kN. The

resultant of these forces at each depth is the soil reaction at that depth. As can be

inferred from the figure, the resultant force at a given depth will be a negative vector


71

parallel to the x‐axis, or opposite to the movement. The y‐component of the resultant

is annulled by the reactions on the edge of the pile, which are the forces that the other

half exerts on the one modelled. The forces in the figure have to be multiplied by 2 in

order to take into account the other half of the pile. Then, they are divided by the

height of the element to obtain the normal reaction force per unit length of pile at a

given depth. Given that there is friction between soil and steel, there are also

tangential forces acting on the pile. The process is very similar to normal forces and

the resultant is also taken into account. The total resultant reaction with the average

lateral displacement of the nodes of a given section give a point of the curve P‐y. The

process is repeated increasing the value of the head force and solving the static

equilibrium for each one of them. The P‐y curve at a depth of 4.80m has been obtained

with 9 points. This curve is compared with the P‐y curve of the DNV code (2014) in the

following graph:

Figure 4.16: P‐y curve. Case E=20 MPa. Depth 4.80m

The red line represents the P‐y curve recommended in DNV code, as it is detailed in

the section "Analyses performed", and the blue line represents the P‐y curve obtained

from the model. Both curves start with a linear range and then the stiffness, which is

0

1000

2000

3000

4000

5000

6000

7000

0 20 40 60 80 100 120 140

P (kN

/m)

y (mm)

DNV

FEM


72

the derivative, decreases gradually. Both curves are practically coincident for small

displacements, but they start to separate as the displacement increases. It can be

observed in the figure that both curves start to separate from a value of 'y' under

10mm, which correspond with interval 4 in the FEM curve. As it was explained above,

plastic strains arise in this increment and they imply loss of linearity in these curves.

The DNV curve remains linear until values of 'y' around 40mm, then its stiffness

decrease and the difference between both curves remains roughly constant, even

seems to decrease at the end of the curve.

In view of the results discussed in the previous paragraph, it is necessary to further

study this methodology in order to reach a better understanding of the problem. Local

problems due to plastic areas can affect the soil reaction in a different way depending

on the depth. These local differences may lead to different tendencies in the P‐y

curves for each depth. The shape of the curve is highly affected by the failure criterion,

which determines the linear range of the curve. The estimation of the failure

parameters must be made carefully in order to avoid distorted results.

4.5 Modalanalysis

Natural frequency was obtained from the model by means of a dynamic analysis in

which the tower oscillates freely after imposing a displacement at the head. A modal

analysis cannot be performed in the whole due to the great number of degrees of

freedom that the soil part introduces in the model. This number of degrees of freedom

can be reduced by substituting the system soil‐monopile by two degrees of freedom at

the base of the tower: horizontal displacement and rotation around a perpendicular

axis. The behaviour of these two degrees of freedom is defined by the global P‐y and

M‐ curves obtained in the previous section.

The first 5 frequencies and modes of vibration have been calculated in the reduced

model by performing a modal analysis. The eigen‐frequencies obtained are detailed in

the following table:

Mode Frequency (Hz)

1 0.266

2 0.387

3 1.400

4 2.800

5 3.836

Table 4.1: Eigen frequencies


73

The modes of vibration associated with these frequencies are shown in the next figure:

Figure 4.17: Modes of vibration

The deformed tower is represented in green and the initial state in grey. The square at

the head of the tower represents the point mass of nacelle, hub and rotor blades. The

translational degree of freedom at the base of the tower has been introduced in the

simplified model along the x‐axis, and the rotational one around the z‐axis. The rest of

degrees of freedom at the foundation are impeded. Therefore, deformations in the x‐y

plane correspond to flexible foundation, whereas when they occur in the y‐z plane, the

response corresponds to a perfectly rigid foundation. In this way, the behaviour of

both cases be analyzed from the same modal analysis.


74

Mode 1 is related to deformations in the plane x‐y, where the foundation stiffness is

implemented. An horizontal load at the head along the x‐axis, as the one considered in

this study, will produce a deformed tower very similar to mode 1. Therefore, mode 1

has high contribution factor in the response of the structure under this kind of load.

The frequency associated with mode 1 is very similar to the frequency obtained in the

section "Free oscillation", which was 0.263. Therefore, the procedure of that section to

obtain the natural frequency is valid, as long as the initial state imposed is similar to

the first mode.

Mode 2 is very similar to mode 1, but it is contained in the plane y‐z. The only

difference between them is the displacement and rotation of the foundation. They are

both zero in mode 2 because the foundation degrees of freedom are impeded in the

plane of the deformation, unlike in mode 1 where they are nonzero. The frequency

associated with mode 2 is very close to that obtained in the section "Free oscillation"

with the analytical expression for a perfectly rigid foundation, which was 0.399 Hz.

Mode 3 is unlikely to occur with a load at the head, but it could be similar to the

deformation produced by waves, as they act in a lower part of the tower than the

wind. However, the value of the frequency associated with this mode is higher than

the usual frequencies of wave loads, so it is unlikely that both frequencies become to

couple.

Mode 4 would be similar to mode 3 but for a perfectly rigid foundation, and mode 5

could contribute together with mode 1 when the wind is acting as a linear load along

the tower. The frequencies of these modes are also too high to be in the range of wind

or waves frequencies.

In addition to the frequencies calculation, a dynamic analysis has been carried out in

the reduced model to obtain the response of the structure under a cyclic load. The

cyclic load is the same as that used in the rest of the study, which is described in the

section "Structure and load properties". In order to check the simplified model, a static

state has also been implemented. The results obtained are compared with those of the

whole model in the following table:

Simplified model

Complete model

Static displacement (cm) 37.7 37.3

Natural frequency (Hz) 0.266 0.263

Dynamic displacement (cm) 138.908 146.278

Table 4.2: Comparison of both model results


75

As it was expected, there is good agreement between the static displacements in both

models, given that the P‐y and M‐ curves implemented in the simplified model were

obtained from a static analysis. However, the dynamic displacements do not agree so

well, the difference between them being of more than 7 cm. When inertial effects are

relevant, parameters such as stiffness of the foundation or damping must be chosen

carefully. In this case, the static stiffness curves do not approximate well the response

of the system soil‐foundation when cyclic loading is involved. On the other hand, the

natural frequencies calculated with both methodologies are practically the same. As

the modal analysis of eigenvalues does not involve large displacements, inertial effects

do not affect so much the results and the static curves are valid to simulate the system

soil‐foundation.

4.6 Reliabilityanalysis

In this section, the natural frequency has been calculated from an analytical model

where translational and rotational stiffness, Kt and Kr respectively, have been

considered as stochastic variables. The mean values of these variables have been

obtained from the global stiffness curves calculated in the section "P‐y and M‐ curves". As the curves are non linear, increment 7 has been chosen as representative,

because the displacement at the head in this increment is within the range of that

caused by the load of the problem. Then, the stochastic variables are characterized as

follows:

KtlogN(mean=2.37*107 N/m, COV=0.3)

KrlogN(mean=3.30*1010 N*m/rad, COV=0.2)

Before starting the analysis it is necessary to obtain the parameters that define the

probability density functions of Kr and Kt, that is, and . These are the following for a lognormal distribution:

√

1

Where:

: Location parameter

: Scale parameter

m: Mean of the non logarithmic sample values

s: Variance of the non logarithmic sample values. ∙


76

The values obtained from the equations above are the following:

r=24.200 , r=0.198 , t=16.938 , t=0.294

The result of the analysis will be the probability that the natural frequency is within the

range of resonance of a load frequency. As it was discussed in the section "Forced

oscillation", there is a range of 10% around the load frequency that has to be avoided

by the natural frequency in order to prevent resonance. Therefore, the probability that

the natural frequency of the system is in the range of 10% around a given load frequency is going to be calculated.

Given that the mean values of Kt and Kr are calculated from the case E=20, whose

natural frequency is equal to the load frequency considered in the previous analyses,

the probability that they are in the range of resonance would be almost 100%. In this

section, a different load frequency will be considered in order to evaluate the

probability of resonance when the mean values of natural and load frequencies are

slightly far. Then, the load frequency considered in this section is the following:

Load frequency: =0.22 Hz

Range of resonance (10%): 0.20 ‐ 0.24 Hz

4.6.1 Montecarlo

The Montecarlo method is based on computing a large number of simulations for

different realizations of the stochastic variables. This values are taken randomly, so the

number of simulations has to be large enough to cover a representative range of

events (Metropolis & Ulam, 1949). The steps of the method are the following:

Fix a number of simulations, n.

Compute two series of n uniformly distributed values ranging between 0 and 1: Nr and Nt.

Each pair of values Nr and Nt is related to a pair of values Kr and Kt. These can be obtained from the inverse cumulative distribution function of each one. In this case we obtain two series of values for Kr and Kt with a log normal distribution.

Compute the non dimensional parameters associated with each pair of values Kr and Kt:


77

Solve the equation of the natural frequency. It has to be done numerically for each pair of values Kr and Kt using the Newton‐Raphson method. The root of the equation associated with the natural frequency is that which is closest to

zero and positive. This root is the non dimensional frequency 1:

| | 0 1

Compute the natural frequency:

12

∙

Compute the probability of failure, which in this case is the probability of resonance, by counting the number of events:

1

Where:

n1: Number of events in which (0.9*) < f1 < (1.1*)

The results presented are obtained after 7000 simulations, from which the following

results have been obtained:

Total simulations: n=7000

Failure events: n1=714

Probability of resonance: Pf=10.20%

Computational cost: 7000

From these results we can see that there is a probability of 10.20% that the natural

frequency of the structure will be within the resonance range of the studied load,

which will cause dynamic amplification problems. As we can see in the table 4.3 below,

the mean value of the sample is 0.262 Hz, which is higher than the frequency load 0.22

Hz. The probability of failure can be reduced by designing a stiffer foundation, which

will result in an increase in the mean value of the natural frequency and a reduction in

the number of events within the range of resonance. The analysed design could be

dangerous if the soil had a strain softening behaviour, which would cause a decrease in

the natural frequency and, therefore, an increase in the probability of failure.


78

Given that the result of the Montecarlo method is obtained from a series of different

values for the natural frequency, it is also possible not only to obtain the probability of

failure but also to analyse the distribution of these values and characterise the natural

frequency as a stochastic variable.

The distribution of natural frequencies obtained is the following:

Figure 4.18: Natural frequency distribution

The graph shown in the last figure is not actually an histogram but a discrete

probability distribution function. To obtain it, the original histogram has been

normalized, that is, the height of each bar has been divided by the total area. In this

way, the area of this normalized histogram is 1 and can be compared with other

probability distribution functions.

The range of resonance is also shown in the last figure in red line, from 0.20Hz to 0.24

Hz. The mean of the sample is 0.262 Hz, which is out of the range of resonance.

However, as it was detailed in the numerical results, there are 714 events within this

range. This mean of the natural frequency of 0.262 Hz obtained from the analytical

model is very close to the natural frequencies obtained in the sections "Free


79

oscillation" and "Modal analysis" with the whole and simplified numerical models,

which were 0.263 and 0.266 Hz respectively.

Once the histogram of natural frequencies is obtained, a probability density function

can be then adjusted to this histogram. The parameters of the theoretical distribution

have been obtained from the sample by means of a maximum likelihood approach

(Ledesma, Gens, & Alonso, 1996). To check the goodness of this adjustment, a

Kolmogorov‐Smirnov test (Massey Jr, 1951) has been carried out for each distribution

function. The methodology is the following:

Run the Montecarlo simulations and obtain the vector of natural frequencies.

A hypothesis about the probability distribution function has to be made. It has been tested four different PDF's:

o Lognormal o Weibull o Gamma o Beta

Compute the parameters of each distribution using the method of maximum likelihood. This method calculates the value of the two parameters of a given distribution, looking for the best adjustment to the sample.

Once the probability function is defined by its parameters, a Kolmogorov‐Smirnov test is performed to check the goodness of each hypothesis. They all have been performed at the 5% significance level.

The results of the Kolmogorov‐Smirnov tests at the 5% significance level performed for

each hypothesis are summarized in the following table. The parameters of each

probability distribution function obtained with the maximum likelihood method are

also detailed in the table.

Lognormal Weibull Gamma Beta Sample(MC)

KS Test, 5% Rejected Rejected Rejected No rejected

p‐value 0.11E‐02 5E‐15 1.13E‐02 4.47E‐02

or a ‐1.34 0.269 292.4 216.7

or b 5.87E‐02 18.4942 8.94E‐04 612.115

Mean 0.262 0.261 0.262 0.262 0.262

Variance 2.36E‐04 3.04E‐04 2.34E‐04 2.330E‐04 2.31E‐04

Table 4.3: Adjusted distribution functions to Montecarlo sample

These functions have been drawn in the next graph:


80

Figure 4.19: Tested distribution functions

The graph shows that the Lognormal, Gamma and Beta distributions are very close and

practically overlapped, whereas the Weibull distribution is far from the rest. Attending

to the numerical results of the table, the only hypothesis which has not been rejected

by the Kolmogorov‐Smirnov test is that the sample follows a Beta distribution. The p‐

value shown in the table is related to the probability that the theoretical function

adjust to the sample, and it allows to compare the different hypothesis. High values of

this parameter indicate a better fit than lower values. The p‐value is close in the

Gamma and Beta distributions, slightly higher in this last. The little difference between

these two values and the fact that the test rejects only one of them indicates that the

adjustment of the Beta distribution, although it has not been rejected, is in the limit of

rejection. In the case of the Weibull distribution, the p‐value is far from the other

three, and it can be seen in the graph that the function is too right skewed.

Attending to the statistics values, mean and variance, obtained by the maximum

likelihood method, it can be seen that they are very similar and close to the values

obtained from the sample, that is, from the results of the simulations. However, the


81

results of the Kolmogorov‐Smirnov test show that a good fit of these parameters is not

enough to ensure a good adjustment of the distribution function.

4.6.2 FORM

The First Order Reliability Method, FORM (Nikolaidis, Ghiocel, & Singhal, 2014), has

been used to obtain the probability of failure defined in section 2. The computational

cost of this method is considerably lower than that of the Montecarlo method, given

that it does not give a whole distribution of the natural frequency. The Hasofer Lind

and Rackwitz Fiessler method has been used to optimize the limit surface and find the

design point. The steps of the method are described in the following paragraphs.

The stochastic variables of the problem are Kr and Kt. The parameters of these

distributions were obtained in section 3:

,

,

The variables of R() are non dimensional and are obtained from the following

expressions:

From the parameters of the distribution of Kr and Kt can be obtained the parameters

of the distributions of r and t, taking into account the rules of a lognormal

distribution:

,

,

The variables r and t has to be normalized in order to apply the FORM method in a

normal standard space. It is necessary the following change of variable to express

them as normal standard distributions:

,

0,1

Where:


82

,

,

,

,

To compute the probability of failure it is necessary to define a limit state surface. In

this case the LSF is expressed in terms of f1, which will be obtained by solving the

equation R()=0, and the extreme values of the ranges 1P and 3P. It is considered

that a failure occurs when:

0.9 , 1.1

Therefore, there are four functions that define the limit state surfaces, taking into

account that the fail occurs when gk0:

, 1 ,

, , 1

The probability of failure is then:

2 0 1 0

The function g of the limit state surface must be in terms of the normalized variables

ur and ut. This can be done applying the following change of variable in each function

gk:

, , , k=1,...4

Finally, the HLRF method is applied to each function gk to obtain the reliability index . The steps of the HLRF method are the following:

Fix a starting point. U1=[0 , 0]

Obtain the reliability index for this point. 1=norm(U1)

Fix a tolerance level. tol=0.001

Obtain points Ui (i=1,...n) by iterations until tol<0.001 from these expression:

1‖ ‖

∙


83

| |

After a few iterations, the HLRF converges and the reliability index is obtained. The

probability that gk0 is then:

0

Where (‐) is the normal standard probability distribution function evaluated in ‐. These process is computed for each gk, k=1,2, to obtain the probability of failure as it

was detailed above.

The gradient of the function g(ur,ut) has been used in the HLRF method. Given that g

depends on f1 and this is obtained numerically by solving the equation |R()|=0, there is not an analytical expression for g(ur,ut). The gradient of g has to be approximated

numerically by the following expression:

, ≅, ,

2,

, ,2

The values of hr and ht has been defined in function of the length of each coordinate,

unless they are zero:

0| |10000

01

10000

0| |10000

01

10000

The computational cost of this method, defined as the number of times the equation

|R()|=0 has to be solved, is 5 per iteration and gk: 1 to evaluate g(U) and 4 to evaluate g(U).

The results are the following:

Iterat. HLRF Comp. cost

P(gk0)

g1 4 20 4.032 2.8*10‐5

g2 3 15 1.236 0.1082 Table 4.4: FORM results


84

The total computational cost of this method is the sum of the obtained for each gk:

Total computational cost: 35

This computational cost is much lower than the obtained with Montecarlo method,

7000. If the distribution of the natural frequency is not a necessary result in the

analysis, the FORM is a more efficient option to calculate probability.

The probability of failure is the difference of the obtained in g2 and g1:

. %

This result is very close to the 10.20% obtained in the previous section with

Montecarlo method.


85

5 Conclusions

Several analyses have been carried out in this study to analyze the influence of soil

properties on the response of an offshore wind turbine on a monopile foundation. A

finite element model has been implemented in which the soil is represented as a

continuous element with a Mohr‐Coulomb failure criterion. This model has been used

as a tool to obtain the response of the tower under several load states: static, free

oscillation and forced oscillation. P‐y and M‐ curves at the base of the tower have also been calculated from the model. These curves have been implemented in a

reduced model in which the system soil‐foundation is substituted by two degrees of

freedom at the base of the tower: horizontal translation and rotation. The behaviour

of these two degrees of freedom is determined by the P‐y and M‐ curves. A modal

analysis has been performed in the reduced model to obtain the vibration modes of

the system and its frequencies. Finally, a reliability analysis has been performed to

obtain the probability of resonance. In this analysis, translational and rotational

stiffness are considered as stochastic variables. The probability of resonance has been

calculated with two different methods: Montecarlo and FORM.

The results from the static analysis show that the head displacement decreases when

the soil stiffness increases. The curve of displacement tends to the value of a perfectly

rigid foundation, although the difference in the stiffest case is still considerable,

around 34%. The influence of soil stiffness is higher in the case of loose sands, where

small increments of stiffness lead to large increments in displacement.

Soil stiffness also affects the natural frequency of the system, as it is shown in the

results of the free oscillation analysis. As the soil stiffness increases, the natural

frequency is higher. There is a trend to the frequency of the perfectly rigid foundation,

but the difference between them is around 33% in the stiffest case. Again, the

influence of soil stiffness on natural frequency is higher in loose sands. In the case of

dense sands, the increments in natural frequency are very small when soil stiffness

increases.

The forced oscillation results show that dynamic amplification has been produced in all

cases, although it is much more relevant around the resonance range. The natural

frequency of the system has ranged from 72 to 115% of the load frequency. Within this

range, the dynamic amplification has been always higher than 175%, and has reached

a maximum of almost 400% when the natural frequency equals the load frequency.

There is a range of 10% around the load frequency where the amplification factor

starts to increase sharply. The steel stress range at the base of the tower, which

determines the fatigue life of the structure, has shown a similar behaviour to the

displacements. This stress range increases sharply when the natural frequency of the

system is within the interval of 10% around the load frequency.

5. Conclusions

86

The global P‐y and M‐ curves obtained from the incremental static analysis show a

range of linear behaviour when the displacements and rotations are small. The point

from which the slope of the curves starts to decrease coincides with the beginning of

plastic strain in elements around the pile. These plastic strains start to arise close to

the seabed level, on the side where the monopile separates from the soil. As the load

increases, they extend to the opposite side, due to compression in soil produced by

the monopile, and they also arise at the monopile tip, where the soil separates from

steel due to rotation.

The first five modes of vibration have been calculated from a modal analysis in the

reduced model. The first mode is the most influential in the response of the structure

under a horizontal load. Therefore, the first frequency of the system, or natural

frequency, can get coupled with the frequency of a horizontal load and produce

resonance. Attending to the shape of the third mode, it could be excited by wave

loads, as they are applied at a lower point. However, it is unlikely that frequency of the

third vibration mode could get coupled with waves frequencies, given that their

frequency is much lower. Comparing with the free oscillation analysis, the natural

frequencies calculated with both methods are practically the same. The static

displacements of the reduced model are also the same as those of the whole model,

but there is not good agreement in the dynamic displacements, due to fact that the

global stiffness curves were obtained from a static analysis.

The probability of resonance obtained from the reliability analysis has been very

similar in both methods, around 10%. The computational cost has been significantly

lower in the case of FORM method, 35, compared to Montecarlo, 7000. If the

information needed from the analysis is just the probability of failure, FORM is much

more efficient, however, more information about the distribution function of the

results can be obtained with Montecarlo. The distribution of natural frequency

obtained with Montecarlo is similar to a Beta distribution, although the adjustment is

not clear enough. The statistic parameters obtained by the maximum likelihood

method are very similar in the four distribution tested, and also similar to those of the

sample.

The variability of soil properties in the long term after the accumulation of a high

number of cycles, which can affect the natural frequency of the system, is proposed as

a future line of research. In addition, the analysis of possible solutions to minimize

damage would lead to more optimized designs.


87

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A. Appendix

I. Freeoscillation.Displacements

Head displacement. E=5MPa


AI. Free oscillations. Displacements

92




93



AI. Free oscillations. Displacements

94




95



97

II. Forcedoscillation.Displacements



AII. Forced oscillations. Displacements

98




99



AII. Forced oscillations. Displacements

100




101



103

III. Forcedoscillation.Stresses

Base S22 stress. E=5MPa


AIII. Forced oscillations. Stresses

104




105



AIII. Forced oscillations. Stresses

106




107



109

IV. Staticequivalentplasticstrain.CaseE=20MPa

Equivalent plastic strain. Case E=20 MPa. F=460 kN


AIV. Static equivalent plastic strain. Case E=20 MPa

110




111



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