Foundation Models for the Dynamic Response

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Foundation models for the dynamic response

of offshore wind turbines

.

M.B. Zaaijer, MSc Delft University of Technology, The Netherlands

SYNOPSIS

Due to the variety of excitation frequencies and the larger influence of the foundation on the wind turbine response,

modelling of the dynamic behaviour of the foundation becomes a more pronounced issue for offshore wind turbines.This paper investigates the sensitivity of the support structure’s natural frequency to variation in models for pile

foundations of a monotower, tripod and lattice tower. The use of a stiffness matrix at mudline provides a significantreduction of complexity and results in acceptable loss of accuracy. Uncoupled springs and an effective fixity depth

model are discouraged. For the tripod and lattice structure of this study lateral flexibility of the foundation is moreimportant than axial flexibility.

INTRODUCTION

An important aspect of extreme- and fatigue loading of the support structure of an offshore wind energy converter (OWEC) is its dynamic response. Its dynamic behaviour differs in some important aspects from that of platforms for theoffshore oil industry and of onshore wind energy converters. The natural frequency of an OWEC is wedged betweendifferent excitation frequencies, whereas the natural frequency of a platform for the offshore oil industry is usually

designed to be above all main excitation frequencies. The geometry and dimensions of offshore foundations differ fromtypical onshore solutions, resulting particularly in a larger influence of foundation stiffness.

Due to the variety of excitation frequencies and the larger influence of the foundation on the wind turbine response,modelling of the dynamic behaviour of the foundation becomes a more pronounced issue for offshore wind turbines.This paper investigates the sensitivity of the support structure’s natural frequency to variation in models for pile

foundations. The assessment is performed for a monotower, a tripod and a lattice tower. These support structures aredesigned for North Sea conditions with water depths between 15 and 25 m and for a 3 MW turbine. The monotower and

lattice tower are based on the design solutions of the Opti-OWECS project [4]. The tripod is designed by Heerema and was used in a study of the simulation of offshore wind turbines under stochastic loading [5]. The main dimensions of the support structures are given in Table 1.

Table 1 Main properties of analysed support structures

Monotower Tripod Lattice tower

Hub height (m) 60 69.2 61

Rotor-nacelle mass (kg) 130,000 130,000 130,000

Water depth (m) 21 25 25Piles

Cross section (m x m) ∅ 3.5 x 0.075 ∅ 1.2 x 0.02 ∅ 1.2 x 0.02

Penetration (m) 50 50 50Tower

Pile distance to centre (m) - 30.75 28.9

Braces (m x m) - ∅ 1.75 x 0.038 -

Diameter (m) 2.8 - 3.5 (one transition) 3 - 4.5 (in steps) 0.32 - 0.85 (different members)Wall thickness (m) 0.02 - 0.075 (in steps) 0.025 - 0.045 (in steps) 0.012 - 0.03 (different members)

1st Natural frequency (Hz) 0.291 0.455 0.725

2nd Natural frequency (Hz) 1.33 1.25 2.26

. Authors’ BiographiesMichiel Zaaijer is presently research scientist in offshore wind energy at Delft University of Technology. His research focuses ondynamic behaviour of offshore turbines and integrated wind farm design methods. After his M.Sc. in physics in 1993 he has worked

for several years as a researcher in the field of aircraft navigation.

Marine Renewable Energy Conference (MAREC), Newcastle, UK, September 2002.



The foundation piles of all structures are larger than strictly required to provide sufficient bearing. However, these pileshave the same stiffness as infinitely long piles and therefore the dynamic behaviour of the support structures isinsensitive to pile penetration depth. For the analysis of the natural frequencies of the support structures a finite element

model of the construction is made in ANSYS®. For the foundation, a finite element model of the pile with distributed

springs for pile-soil interaction is used as a reference model (see Section ‘Description of foundation models’).

PHYSICAL BACKGROUND FOR FREQUENCY ANALYSIS

A structure in free vibration does not exchange energy with the environment. Thus, during each vibration cycle kineticand strain energy in the structure are exchanged. The kinetic and strain energy can be calculated for an assumed

vibration, ( ) ( )st z Ψ⋅ , where z gives the time dependency and Ψ the vibration shape parameterised with s . For a

harmonic vibration, equation of the maximum kinetic and strain energy leads to the following expression for the

vibration frequency, ω :

( ) ( )( )

( ) ( )∫∫

⋅Ψ⋅

⋅Ψ⋅=

dsssm

dss f sk

2

2

2ω , (1)

with: ( )sk = distributed stiffness,

( )sm = distributed mass,

( )( )s f Ψ = local displacement or curvature, depending on the type of strain energy.

The lowest possible frequency that can be obtained in this way is the first natural frequency of the system and thecorresponding vibration shape is the first mode shape. This method is called after Rayleigh and is described for instancein [3].

To get more insight, Rayleigh’s method is used to obtain an expression for the sensitivity of the first natural frequency,

1ω , to changes in a physical parameter, x . Derivation of Equation 1 leads to:

( )( )

∫

∫∫

∫

∫∫⋅Ψ⋅

⋅∂

Ψ∂⋅⋅−⋅

∂

Ψ∂⋅

+⋅Ψ⋅

⋅Ψ⋅∂

∂−⋅Ψ⋅

∂

∂

=∂

∂

dsm

ds x

mds x

f k

dsm

ds x

mds f

x

k

x 2

22

2

2

2222 ω ω

ω . (2)

Because the first natural frequency is the minimum value obtained with Equation 1, by varying the vibration shape Ψ ,

the derivative of Equation 1 with fixed k and m must be zero around the first mode shape, 1Ψ , resulting in:

( )

0

1

1

2

22

2

2

=⋅Ψ⋅

⋅∂

Ψ∂⋅⋅−⋅

∂

Ψ∂⋅

=∂

∂

Ψ=Ψ

Ψ=Ψ ∫∫∫

dsm

ds x

mds x

f k

x

ω ω

. (3)

Combining Equations 2 and 3 for 1ω ω = and 1Ψ=Ψ leads to:

( )

∫∫∫

⋅Ψ⋅

⋅Ψ⋅∂

∂−⋅Ψ⋅

∂

∂

=∂

∂

dsm

ds x

mds f

x

k

x 21

21

2212

1ω

ω . (4)

Substituting k a x

k ⋅=

∂

∂1 and ma

x

m⋅=

∂

∂2 in Equation 4 shows that the distribution of strain energy, ( )2Ψ⋅ f k , and

kinetic energy, 22 Ψ⋅⋅mω , are useful parameters to analyse both the absolute natural frequency of Equation 1 and the

sensitivity of Equation 4. The latter substitution is based on the idea that relevant changes in stiffness and mass areseveral percent of the absolute stiffness and mass. For each of the support structures these parameters are plotted inFigure 1, scaled with the total energy of the vibration. Table 2 gives the integrated relative energy values for parts of the

structure.




0.00 0.02 0.04

-50

-25

0

25

50

75

100

H e i g h t a b o v e s e a b e d

( m )

0.00 0.02 0.04

Relative energy (per m)

-50

-25

0

25

50

75

100

0.00 0.02 0.04

-50

-25

0

25

50

75

100

Kinetic (structure)

Strain (structure)

Strain (soil - lateral)

Strain (soil - axial)

Kinetic rotor-nacelle: 0.78 Kinetic rotor-nacelle: 0.62 Kinetic rotor-nacelle: 0.74


Figure 1 Relative energy distribution of support structures

For all support structures the kinetic energy is concentrated in the rotor-nacelle assembly, with a small contribution of the tower near the top. The jagged character of the distributions is caused by the transitions in the structure and meshingof the finite element model.The monotower strain energy increases where the tower diameter decreases at approximately 12 m, but experiences adip at the very rigid grouted connection between 16 and 21 m. Strain energy decrease toward the top, where the bendingcurvature decreases and the largest bending in the pile occurs a few meter below the seabed. Strain energy in the tripod

largely occurs near the joint of the braces at 25 m. Below the joint the high stiffness of the tripod results in very smalldeformations. The large, but narrow, peak of strain energy near the seabed is caused by the accumulated strain energyof the base members. The strain energy distribution of the lattice tower is very jagged, due to changes in the geometryand in member dimensions. Apparently the section around 25 m is fairly rigid.For all structures the strain energy associated with the lateral compression of the soil is concentrated in the first 10 m below the seabed. The strain energy associated with the axial shear of the soil, which is only relevant for the tripod and

lattice tower, is very small.

Table 2 Relative energy of parts of the support structures


Rotor-nacelle 0.778 0.618 0.743Tower 0.222 0.382 0.257

Relative kinetic energy(-)

Pile 0.00032 0.000019 0.00025Tower 0.693 0.970 0.904Pile 0.236 0.016 0.040

Soil - lateral 0.071 0.0087 0.046

Relative strain energy(-)

Soil - axial - 0.0053 0.010

Table 2 shows that contribution of the kinetic energy of the pile is negligible, which is due to the small displacements.

Mass of the soil is not included in the finite element model, but it is expected that kinetic energy of the soil is similarlynegligible. For the monopile the strain energy in the pile exceeds the strain energy in the soil, which indicatesdominance of pile stiffness over soil stiffness. The difference is less pronounced for the tripod and the lattice tower. Theresults for the relative strain energy of the foundations lead to expect the largest sensitivity to foundation parametersand model for the monopile. The tripod and particularly the lattice tower have larger strain energy in lateralcompression of the soil than in axial shear. This indicates larger sensitivity to lateral behaviour, especially because pile

strain energy is expected to be contained mainly in lateral bending.




DESCRIPTION OF FOUNDATION MODELS

Introduction

The following models for the foundation are analysed in this paper:

• finite element model with distributed springs for lateral and axial soil stiffness,

• effective fixity length of the foundation pile,

• stiffness matrix of foundation behaviour at mudline,

• uncoupled lateral, axial and rotational springs at mudline.

The finite element model is used as reference. The models are illustrated in Figure 2, which also shows two differentapproaches to obtain the lateral and rotational spring stiffness.

Pile wall

⎥⎦

⎤⎢⎣

⎡⋅⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

θθθθ

θu

k k

k k

M

F

x

x xx

θF M

u

Ignore Ignore uθApplied force/moment

θF M

u

Ignore M Ignore F

Forced displacement/rotation

Reference:Distributed springs 1. Fixity length

2. Stiffness matrix

3. Uncoupled springs 3.a Force method 3.b Displacement method

Figure 2 Foundation models

Reference model

The reference model with distributed springs is commonly applied in the offshore oil and gas industry. The stiffness of the springs in the reference model is determined according to the recommendations in [1] for cyclic loading. Thesprings represent lateral resistance, internal and external shaft friction and pile tip and plug resistance. In accordancewith Winkler’s assumption the springs are considered to be independent. The springs have a non-linear stress-strainrelation and are linearised around unloaded conditions.

Effective fixity length

A simple model of the clamping effect of the soil is replacement of the soil by rigid clamping of the pile at an effectivedepth below the seabed. This model is sometimes used for (preliminary) dynamic analysis of structures for the offshoreoil and gas industry, using tabulated values for the effective fixity length. The values proposed by in [2] are given inTable 3 as a function of the soil type and pile diameter, D, along with values obtained from analysis of an offshore wind

turbine support structure [6].

Table 3 Effective fixity lengths, suggested in literature

Configuration Effective fixity length

Stiff clay 3.5 D - 4.5 D

Very soft silt 7 D - 8 D

General calculations 6 D

From measurement of an offshore turbine (500 kW) 3.3 D - 3.7 D

Stiffness matrix

For pile foundations a stiffness matrix can express the stiffness of the pile-soil system at the seabed. The stiffnessmatrix gives the forces, F , and moments, M , for displacements and rotations of the pile head. The relevant degrees of

freedom of a laterally loaded pile are the horizontal translation, u, in the plane of interest and the rotation, θ , about the

horizontal axis perpendicular to this plane. Two methods are used to obtain the values for the elements of the stiffnessmatrix, as described below:




Load displacement analysis with p-y curves

The unknown elements of the stiffness matrix can be solved from the set of algebraic equations that is obtained fromtwo load case analyses of the reference model. For the applied load cases the deflection of the foundation is in the linear

region of the stress-strain curves, which is representative for small vibrations. The second load case is sufficientlydifferent from the first one to give independent equations. For the tripod and lattice tower the stiffness matrix iscombined with a spring for the vertical degree of freedom. The spring stiffness is determined from an axial load case of the reference model.

Randolph elastic continuum model

Randolph performed dimension analysis and finite element analysis of piles in an elastic continuum to obtain anexpression for the pile head flexibility [7]. This resulted in parameterised flexibility matrices for piles in an elasticcontinuum with a constant soil shear modulus and with a linearly increasing soil shear modulus. The latter is the preferred model for sandy soil, where the soil shear modulus varies with depth and generally increases due to increasingeffective vertical soil pressure. The stiffness matrix for Randolph’s approach is the inverse of the flexibility matrix heobtained in his study. The rate of change of the soil shear modulus is obtained from a linear fit to the actual modulus.

For the tripod and lattice tower the stiffness matrix is not combined with a spring for the vertical degree of freedom.

Uncoupled springs

In this model the coupled stiffness of the pile head is simplified to independent springs for each relevant degree of freedom. In Figure 2 the springs are illustrated for lateral translation and rotation. For the tripod and lattice tower this is

combined with a spring for the vertical degree of freedom.To obtain the stiffness of the spring elements the two approaches of Figure 2 are applied, as described below:

Force method

In the ‘Force method’ the stiffness is obtained by application of a force or moment on the pile head in the referencemodel, without constraining the response. The response will consist of a rotation and a translation of the pile head, butonly the corresponding degree of freedom will be considered.

Displacement method

In the ‘Displacement method’ a deflection is applied to the pile head in the reference model and the corresponding

required force is used. The force corresponding to the other degree of freedom is ignored. The resulting stiffness isequivalent to the diagonal terms of the stiffness matrix for the same load conditions.

COMPARISON OF FOUNDATION MODELSFor each of the support structures and foundation models the first and second natural frequency are determined and normalised with the results for the reference model. The results are plotted in Figure 3.

W i n k l e r F E M ( r e f e r e n c e )

8 D

6 D

4 D

2 D

( S e a b e d ) 0 D

F E M - b a s e d

R a n d o l p h

F o r c e m e t h o d

D i s p l a c e m e n t m e t h o d

O n l y a x i a l s p r i n g s

O n l y l a t e r a l s t i f f n e s s m a t r i x

O n l y r o t a t i o n ( F o r c e m e t h o d )

0.8

1.0

1.2

1 s t n a t u

r a l f r e q u e n c y ( n o r m a l i s e d )

Effective fixity depth

Springs

Stiffnessmatrix

Monotower

Tripod

Lattice tower

W i n k l e r F E M ( r e f e r e n c e )

8 D

6 D

4 D

2 D

( S e a b e d ) 0 D

F E M - b a s e d

R a n d o l p h

F o r c e m e t h o d

D i s p l a c e m e n t m e t h o d

O n l y a x i a l s p r i n g s

O n l y l a t e r a l s t i f f n e s s m a t r i x

O n l y o t a t i o n ( F o r c e m e t h o d )

0.0

1.0

2.0

3.0

2 n d n a t u

r a l f r e q u e n c y ( n o r m a l i s e d )

Effective fixity depth Stiffnessmatrix

Springs

Figure 3 Predicted natural frequencies for several foundation models




Since the effective fixity depth cannot be determined a priori in a rigorous way, a variety of clamping depths isanalysed. Additionally, the effect of constraining some degrees of freedom is determined to assess the relativeimportance of lateral and rotation flexibility of the foundation. Note that only the Randolph stiffness matrix and the

effective fixity depth model are independent of the reference model, which is used for pre-analysis of the other foundation models. Several observations from Figure 3 are highlighted below:

Effective fixity depth

Both first and second natural frequency of the tripod and lattice tower correspond with the reference value for aneffective fixity depth of approximately 6 times the pile diameter. This is in agreement with the suggested value for

general calculations for offshore platforms. This can be expected, since the vibration shape and pile diameter are similar to that of fixed space frame type offshore structures. The effective fixity depth of the tubular tower gives better resultsfor lower values, in the order of 4 times the pile diameter. This is in agreement with earlier studies of monopile behaviour. The difference between the appropriate values may partly be contributed to the larger diameter of the pile, but the different mode shape of the vibration is probably also an important aspect.

The results of this model are very sensitive to the selected effective fixity depth, particularly for the tubular tower. Thetabulated values of the effective fixity depth as shown in Table 3, show a large variation as a function of soil conditions.Therefore, large inaccuracies of natural frequency must be anticipated for a priori assumed fixity depths.

Randolph’s linear elastic model

The first natural frequencies obtained with Randolph’s model correspond within 2.5% with the finite element model.Similar results were found for two other locations, with slightly different soil conditions. It must be noted thatRandolph’s model assumes that the piles are longer than a critical pile length and thus have the same behaviour asinfinitely long piles. The lengths of the piles used in this study are also selected to show the same behaviour as infinitelylong piles. When the pile length is below the critical pile length the increased flexibility of the pile will not be revealed by Randolph’s model. Furthermore, the natural frequency with the finite element foundation model is obtained for thelinear region of the stress-strain curves. When large deflections are expected the non-linear effect will not be revealed

with Randolph’s model, since this is a linear model. Note that part of the error made with Randolph’s model for thetripod and lattice tower is caused by the omission of axial pile flexibility.

Uncoupled lateral and rotational springs

For the tubular tower the springs determined with the force method give closer correspondence of the first naturalfrequency than those obtained with the displacement method. This can be expected, since the shapes of the pile

deflections of the first approach are close to the mode shape of the vibration, whereas those of the second approach arevery different. For a similar reason the displacement method gives better results for the first natural frequency of thetripod and lattice tower. For higher natural frequencies the mode shapes of the vibration tend to deviate from the staticdeflection shapes and give worse results.

Lateral behaviour and rotation

For the monotower use of a rotational spring only does not result in a large difference with the results with uncoupled

springs. However, part of the horizontal flexibility is implicitly present when the stiffness of the rotational spring isdetermined with the force method. For the tripod and lattice tower, the use of only a lateral stiffness matrix gives lessdeviation than the model with only axial springs. This indicates that the natural frequencies of the tripod and latticetower are dominated by the lateral behaviour of the piles. Due to the lateral flexibility of the piles, which is at least anorder of magnitude higher than the axial flexibility, the horizontal translation of the tower is a more importantcontribution to the vibration than its rotation.

SENSITIVITY FOR FOUNDATION PARAMETERS

A previous study focussed on the sensitivity of the natural frequencies to changes in physical parameters of the system,see [8]. The main results for foundation parameters are repeated here, to assess the differences found between different

foundation models against the background of expected variation and inaccuracy. A distinction is made between theuncertainty of a parameter, its variation during the lifetime of the wind turbine and its variation within the wind farm.For several parameters that characterise the foundation, an indication of its variation is obtained from a literature surveyand for this range the natural frequency has been determined with the finite element model. The results are summarised in Table 4.




Table 4 Sensitivity of first natural frequency to typical variation in physical parameter

Parameter Monotower Tripod Lattice

(Cohensionless soil) Uncertainty

%

Lifetime

%

Farm

%

Uncertainty

%

Lifetime

%

Farm

%

Uncertainty

%

Lifetime

%

Farm

%

Effective soil unit weight 0 0 - 0.03 0.03 - 0.05 0.05 -

Friction angle 4 4 - 0.9 0.9 - 3 3 -

Coefficient of lateral earth pressure - - - 0.05 0.05 - 0.09 0.09 -

Modulus of subgrade reaction 2 2 - 0.4 0.4 - 2 2 -

General scour - - 2 - - 0.7 - - 3Local scour - - 4 - - 0.3 - - 1

Postholing gap - - 6 - - 0.5 - - 3

Maximum 4 4 6 0.9 0.9 0.7 3 3 3

As expected, the monotower is more sensitive to variation of soil parameters than the tripod and the lattice tower.Considering the low relative strain energy in the soil for all support structures (see Table 2), the sensitivity is rather high. This is due to the very large variations in soil stiffness that occur. Scour results in a significant sensitivity, becausethe soil strain energy is concentrated in the upper soil layers, whose supporting function vanishes.

DISCUSSION AND CONCLUSIONS

The foundation models of this study are based on three different bases. The effective fixity depth model uses tabulated

values of the clamping depth. The linear elastic model developed by Randolph uses an analytical expression for the

elements of the stiffness matrix, based on pile and soil parameters. The reference model uses a finite elementrepresentation of the pile and non-linear pile-soil interaction. The other models share the finite element basis of thereference model, using a static pre-analysis to obtain model parameters.

Suitable values for the effective fixity depth for a monopile and for the tripod and lattice tower differ, probably due to

the different mode shapes of the vibration. Besides, the appropriate value of the fixity depth depends on both pilestiffness and soil properties and this dependency is not rigorously represented in tabulated values. Due to the largesensitivity of the predicted natural frequency to the effective fixity depth this model is strongly discouraged as an a priori model for analysis beyond an initial guess of support structure behaviour. An effective fixity depth could bedetermined from a reference model or from measurements to reduce complexity of the foundation model. However, asensitivity study is always recommended when this model is applied.

The results obtained with Randolph’s linear elastic model are similar to the results of the reference model. However, the

soil of this case study is nearly uniform, which makes it particularly suitable for the elastic model. Randolph’s method will not reveal the influence of pile length and loading conditions and is therefore less suitable for piles with a penetration less than a critical pile length and for deflections outside the linear region of the soil-structure interaction.

The first and second natural frequency obtained with a stiffness matrix with coupled lateral behaviour gives very good correspondence with the finite element foundation model for all assessed support structure types. The observed

difference is far less than expected uncertainties in foundation behaviour. This stiffness matrix has far less degrees-of-freedom than the comprehensive finite element model and will therefore reduce computations in dynamic analyses.Uncoupled springs for lateral displacement, rotation and axial displacement give larger errors, which can exceed theexpected errors due to uncertainties. Since no significant reduction of degrees-of-freedom is obtained, the use of

uncoupled springs is not recommended.

For the monotower the expected dominance of rotation flexibility is confirmed by the results. For the tripod and lattice

tower the lateral flexibility of the piles appear to be much more important than the axial flexibility. However, therelative importance of lateral and axial foundation behaviour depends mainly on lateral and axial foundation stiffness,spacing between the piles and the mass distribution of the tower and turbine. It requires further analysis to assess if the

conclusions for these cases are generally valid.

Note that the finite element model, which has been used as a reference throughout this study, will also differ fromreality. This model is known to generally underestimate soil stiffness, resulting in a low prediction of the naturalfrequency. Additionally, errors can be made by imperfections in or incompleteness of the information of the turbine and support structure and by other modelling aspects.

ACKNOWLEDGEMENT

The author would like to acknowledge the contributions made by the partners in the OWTES project to this paper.

The project Design methods for Offshore Wind Turbines at Exposed Sites is being funded by the European Commissionunder contract number JOR3-CT98-0284, with co-funding of NOVEM under contract 224.750-9854.




REFERENCES

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3. Clough, R. W., Penzien, J., Dynamics of Structures, McGraw-Hill, New York, 1993.4. Ferguson, M.C. (editor); Kühn, M.; Bussel, G.J.W. van; Bierbooms, W.A.A.M.; Cockerill, T.T.; Göransson, B.;

Harland, L.A.; Vugts, J.H.; Hes, R., Opti-OWECS Final Report Vol. 4: A Typical Design Solution for an Offshore

Wind Energy Conversion System, Institute for Wind Energy, Delft, 1998.5. Kühn, M., Simulation of Offshore Wind Turbines Under Stochastic Loading, Proceedings of the 5

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Wind Energy Association Conference and Exhibition, Greece, October 1994.

6. Kühn, M., Overall Dynamics of Offshore Wind Energy Converters, Opti-OWECS Final Report Vol. 2: Methods

Assisting the Design of Offshore Wind Energy Conversion Systems, Part D, Institute for Wind Energy, Delft, 1998.7. Randolph, M.F., The Response of Flexible Piles to Lateral Loading, Géotechnique, Vol. 31, no. 2, 19818. Zaaijer, M.B., Vugts, J.H., Sensitivity of Dynamics of Fixed Offshore Support Structures to Foundation and Soil

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Documents

Foundation Models for the Dynamic Response