Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: 2311-9020; ISBN: 978-972-752-165-4

3611

ABSTRACT: One of the important design drivers for offshore wind turbine (OWT) structures is the fatigue life. In order for such structures to make worthwhile investments, they need to be in operation for 20-30 years after installation. The wind turbine and the foundation are subject to fatigue damage from environmental loading (wind, waves) as well as from cyclic loading imposed through the rotational frequency (1P) through mass and aerodynamic imbalances and from the blade passing frequency (3P) of the wind turbine. Through dynamic amplification and resonance, the fatigue damage suffered by the structure can severely increase if the natural frequency of the wind turbine gets close to the frequency of excitation, thereby reducing the service lifetime of the OWT. Therefore, predicting the first natural frequency is of paramount importance. In this paper a mechanical and mathematical model is presented, which provides a good initial estimate of the natural frequency of OWTs for conceptual design. The soil-structure interaction (SSI) is modelled through a set of springs, which also includes the cross-coupling between the lateral and rotational stiffness of the foundation. Approximate analytical formulae are given to approximate the natural frequency. The results are compared to measured data as well as results from similar software. The sensitivity of the natural frequency of the structure to the stiffness parameters of the foundation are analysed and discussed.

KEY WORDS: Offshore wind turbine; Euler-Bernoulli beam theory; Soil-structure interaction; Natural frequency; Cross stiffness; Sensitivity analysis.

1 INTRODUCTION Offshore wind farms (OWF) are expected to become significant contributors to electricity production in the future in Europe and worldwide. To make them a cost-effective alternative to fossil fuel power plants, offshore wind turbines (OWTs) are usually designed to be operational for at least 20-30 years. OWTs are subjected to intensive dynamic loading in a wide frequency band during their lifetime. The main dynamic loads are the environmental loading from wind turbulence and wave loading, and mechanical loading from aerodynamic- and mass imbalance of the rotating rotor (1P frequency band) and blade passage (3P frequency band) in front of the tower. The structures need to survive a large number of load cycles and therefore fatigue damage is an important design driver in OWT technology.

Offshore wind turbines are slender columns with a heavy mass on top: they are dynamically sensitive structures [1]–[3]. Therefore, it is essential that the structure is designed such that its natural frequency is reasonably far from the frequency bands of the excitations in order to minimise fatigue damage and achieve a long service lifetime. Further details on the loading and the frequency bands associated with the loadings can be found in [1]. Designing the support structure and foundation to fit these criteria is a challenging task. It requires the estimation of the stiffness of the foundation, which involves soil-structure interaction (SSI), a source of uncertainty. Furthermore, there are also dynamic issues related to the soil stiffness properties, which may change over time due to cyclic/dynamic excitation, as was demonstrated in [4]–[6]

Change in the natural frequency of an OWT over time was reported in [7]. Measured natural frequencies at the Walney site were reported to be 6-7% higher than the design value [8]. Depending on the natural frequency of the wind turbine structure, three forms of design are adopted: soft-soft, soft-stiff and stiff-stiff. Among these, soft-stiff is the current preferred design option whereby the natural frequency is designed to be within 1P (rotational frequency) and 3P (blade passing frequency). It is to be noted here that neither underestimation nor overestimation of the natural frequency of the OWT is conservative, as the fatigue damage may increase due to dynamic amplification with frequency change in any direction. Some cases of fatigue type failure of OWTs (specifically failure of the grouted connection between the tower and the transition piece) have also been reported [9]. A posteriori changes in an offshore environment are very expensive, however, and are to be avoided.

In this paper an attempt is made to provide a simple and quick method to estimate the first natural frequency of an OWT for the conceptual design phase in order to provide a means for incorporating fatigue in the early stages of design. In this formulation only basic information about the particular wind turbine and site is required. Furthermore, analytical formulae are provided to analyse sensitivity of the natural frequency to changing soil parameters.

2 MODEL OF THE OWT CONSIDERING SSI INCLUDING CROSS-COUPLING TERM

A typical offshore wind turbine supported on a monopile foundation is shown in Figure 1. The main structural elements of an OWT are the rotor, nacelle, tower, substructure and

Dynamic soil-structure interaction issues of offshore wind turbines

Laszlo Arany1, Subhamoy Bhattacharya2, S. J. Hogan3, John Macdonald4

1Department of Engineering Mathematics, University of Bristol, Office 1.80 Queens Building, University Walk Clifton BS8 1TR, PH (+44) 7423 690 220, e-mail: laszlo.arany@bristol.ac.uk

2Department of Civil and Environmental Engineering, University of Surrey, Guildford GU27 XH, UK 3Department of Engineering Mathematics, University of Bristol, Office 2.26, Merchant Venturers’ Building, University Walk

Clifton BS8 1TR, PH (+44) (0) 117 331 5606, e-mail: s.j.hogan@bristol.ac.uk 4Department of Civil Engineering, University of Bristol, Office 2.36 Queens Building, University Walk Clifton BS8 1TR,

(+44) (0) 117 331 5735

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foundation. The slender columns are typically connected to the substructure via a transition piece (TP). The most common foundations are monopiles, gravity base and jacket structures [10], although floating turbines are also being tested. In this paper a simplified mechanical model is used, whereby the rotor-nacelle assembly is modelled as a top head mass with rotational inertia, the tower is modelled as an Euler-Bernoulli beam, and the foundation stiffness is modelled by three springs (lateral-, rotational- and cross springs).

Figure 1. Model of the offshore wind turbine.

For simplicity, several parameters are introduced in Table 1 (for derivation see see [1]). The three springs model of the foundation stiffness is described by Equation 1.

(1)

where , and are the lateral, rotational and cross stiffness, respectively, and are the displacement and slope at the foundation, respectively, and and are the reactions (see Figure 1 for coordinates). In absence of more detailed information and formulae, we used Eurocode 8 Part 5 [11] Some methods for obtaining the soil stiffness parameters are given in [1].

The tower of length is modelled as an Euler-Bernoulli beam, using the mass per length , equivalent bending stiffness . Equivalent bending stiffness needs to be calculated because the tower is tapered. The calculations are

omitted here and only the final results are given; details can be found in detail in [1], [2], [12]. The equivalent bending stiffness for the calculation of the non-dimensional axial force is given in Equation 2-4. · (2) (3)

1 1 (4)

where and are the bottom and top diameters of the tower, respectively, is the area moment of inertia of the tower cross section at the bottom.

The equivalent stiffness for the non-dimensional stiffness parameters is given in Equation 5-6.

· (5)

(6)

where is the bending stiffness at the top. The rotor-nacelle assembly is modelled as a lumped mass on

the top of the tower , with mass moment of inertia (or rotational inertia) . Due to gravity, this mass also exerts an axial force along the tower .

Table 1. Parameters of natural frequency calculation.

Dimensionless group Formula

Non-dimensional lateral stiffness

Non-dimensional rotational stiffness

Non-dimensional cross stiffness

Non-dimensional axial force

Mass ratio (top head mass / tower mass)

Non-dimensional rotary inertia

Frequency scaling parameter

Non-dimensional rotational frequency ( is the rotational frequency)

Ω

The Euler-Bernoulli beam equation with compressive axial

force and without excitation is written in Equation 7.

, ,

, 0 (7)

where is the displacement in the direction, is time, is the equivalent bending stiffness of the tower, is the

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

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mass per length of the tower, is the axial force (see Figure 1).

Assuming constant , and , using the parameters introduced in Table 1, separating variables with ,

and using the non-dimensional coordinate / , the Euler-Bernoulli beam equation can be simplified

to the form given in Equation 8

Ω 0 (8)

Looking for a solution in the form the following characteristic equation can be written:

Ω 0 (9)

Replacing with we get the following:

Ω 0 (10)

,Ω

Ω , Ω (11)

The solution can be written in the following form:

cos sin cosh

sinh (12)

with

| | and | | (13)

Using vector notation: cos sin cosh sinh

PP

The boundary conditions are also formulated with the non-dimensional parameters introduced in Table 1. The derivations are given in [1] in detail.

I. The sum of shear forces at the bottom is zero: 0 0 0 0 (14)

II. The sum of bending moments at the bottom is zero: 0, 0, 0, 0 (15)

III. The sum of shear forces at the top is zero: 1 1 Ω 1 0 (16)

IV. The sum of bending moments at the top is zero: 1 Ω 1 0 (17)

Substituting the solution for given in Equation 12 into the boundary conditions and looking for non-trivial solutions, one obtains the matrix shown in Equation 18.

sinh αΩ cosh cosh αΩ sinhcosh Ω sinh sinh Ω cosh

(18)

The determinant of this matrix is set to zero to find the natural frequency.

det 0 (19)

This determinant produces a non-linear transcendental equation, which has to be solved numerically.

3 APPROXIMATE FORMULAE In order to study the dependency of the natural frequency on the foundation stiffness parameters, analytical approximations are formulated to fit the solutions calculated by numerically solving the transcendental equation given in Equation 19. The natural frequency is expressed in terms of the six main parameters , , , , , as defined in Table 1 (see Equation 20.)

, , , , , (20)

First the fixed base value of the natural frequency is calculated, which is the natural frequency on a perfectly stiff foundation with ∞, ∞, 0. For this calculation some initial values of the axial force, mass ratio and rotational inertia parameters are selected: , , . Using these values the fixed base natural frequency is expressed as:

∞, ∞, 0, , , (21)

This fixed base frequency is practically calculated for a vertical cantilever beam carrying an end mass. This frequency can be calculated by standard formulae for uniform beams [13]:

and (22)

where is the mass of the rotor-nacelle assembly, is the mass of the tower, is the stiffness of the 1 degree of freedom system, is the length of the tower and is the equivalent bending stiffness of the tower. More accurate estimate of the natural frequency may be obtained by using the equivalent stiffness and equivalent mass formulae given in [3].

The dependency of the natural frequency on the parameters are determined by separation of variables. The flexible foundation is taken into account by the coefficients and as given in Equation 23.

· (23)

Once the fixed base natural frequency is available, these coefficients are calculated to incorporate the effects of a flexible foundation, using the three springs model shown in Equation 1. The two coefficients and represent the rotational and lateral stiffness dependency, respectively. Both coefficients are, however, dependent on all three variables, as shown in Equation 24.

, , , , . (24)

The expressions given in Equation 25 and 26 were found to approximate the numerically calculated curves well.

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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

3617

increases by 0.00257Hz. Similarly, the row corresponding to for example the mass ratio contains the negative value of -0.285, which corresponds to decreasing natural frequency with increasing mass ratio. Clearly, the slopes are meaningful only for small variations in the parameters, for significant changes the graphs in Figures 2-6 are to be used. According to Kühn [7], the soil stiffness parameters had an uncertainty of -20% to +40% in the case of the Lely A2 wind farm.

Table 4. Sensitivity of the natural frequency for parameters.

Sensitivity of OWT

Parameter A B C D

8.32E-06 5.08E-06 6.79E-08 3.49E-06

2.57E-03 2.97E-03 2.37E-03 1.14E-03

1.23E-04 1.03E-04 1.01E-05 5.21E-05

-1.59E-01 -9.88E-02 -7.52E-02 -7.17E-02

-2.85E-01 -1.60E-01 -1.70E-01 -1.45E-01

-7.95E-01 -4.91E-01 -3.79E-01 -3.66E-01

It is important to note it is more meaningful to incorporate

the orders of magnitude of the parameters in order to compare the sensitivity. For example, one can look at the change in the natural frequency if the parameters change by 1%. Table 5 shows the result of this analysis for the soil stiffness parameters.

Table 5. Frequency change with 1% parameter change.

Sensitivity of OWT

Parameter A B C D

0.00022 0.00030 0.00001 0.00027

0.00100 0.00118 0.00066 0.00089

-0.00021 -0.00029 -0.00001 -0.00027

It is clear from Table 5 that the natural frequency is most sensitive to the rotational stiffness, and therefore overturning moment resistance is the most important task. The lateral and cross stiffness parameters show equal sensitivity, but both are generally an order of magnitude smaller than the rotational stiffness.

7 CONCLUSION In this paper a methodology was presented for calculating the natural frequency of an offshore wind turbine structure on flexible foundation using the Euler-Bernoulli beam theory and a three spring model to take into account the flexible foundation and soil-structure interaction. The analysis yielded a non-linear transcendental equation that needs to be solved numerically.

For preliminary design it is useful to formulate a simplified expression for the natural frequency, therefore analytical

formulae were derived to approximate the numerically calculated natural frequencies. The approximation curves incorporate the dependence of the natural frequency on the lateral, rotational and cross stiffness parameters, as well as the axial force in the column, the mass ratio of the rotor-nacelle assembly and the tower, and the rotational inertia of the rotor-nacelle assembly. The analytical formulae were found to approximate the numerically obtained results reasonably well.

The approximate formulae were also useful to analyse the sensitivity of the natural frequency to each parameter. It was shown that the parameters with likely changes and high uncertainty are the foundation stiffness parameters. Among these soil stiffness parameters the rotational stiffness ranks as the most important variable with the highest sensitivity.

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towers on flexible foundations,” Shock Vib., vol. 19, no. 1, pp. 37–56, 2012.

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[3] S. Adhikari and S. Bhattacharya, “Vibrations of wind-turbines considering soil-structure interaction,” Wind Struct., vol. 14, no. 2, pp. 85–112, 2011.

[4] D. Lombardi, S. Bhattacharya, and D. Muir Wood, “Dynamic soil–structure interaction of monopile supported wind turbines in cohesive soil,” Soil Dyn. Earthq. Eng., vol. 49, pp. 165–180, Jun. 2013.

[5] S. Bhattacharya, N. Nikitas, J. Garnsey, N. A. Alexander, J. Cox, D. Lombardi, D. Muir Wood, and D. F. T. Nash, “Observed dynamic soil–structure interaction in scale testing of offshore wind turbine foundations,” Soil Dyn. Earthq. Eng., 2013.

[6] S. Bhattacharya and S. Adhikari, “Experimental validation of soil–structure interaction of offshore wind turbines,” Soil Dyn. Earthq. Eng., vol. 31, no. 5–6, pp. 805–816, May 2011.

[7] M. Kühn, “Soft or stiff: A fundamental question for designers of offshore wind energy converters,” in Proc. European Wind Energy Conference EWEC ’97, 1997.

[8] D. Kallehave and C. L. Thilsted, “Modification of the API p-y Formulation of Initial Stiffness of Sand,” in Offshore Site Investigation and Geotechnics: Integrated Geotechnologies - Present and Future, 2012.

[9] I. Lotsberg, “Structural mechanics for design of grouted connections in monopile wind turbine structures,” Mar. Struct., vol. 32, pp. 113–135, Jul. 2013.

[10] European Wind Energy Association, “The European offshore wind industry - key trends and statistics 2012,” 2013.

[11] European Committee for Standardization, “Eurocode 8: Design of Structures for earthquake resistance - Part 5: Foundations, retaining structures and geotechnical aspects,” 2003.

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[13] R. D. Blevins, Formulas for Natural Frequency and Mode Shape. Krieger Publishing Company, 1984.

[14] DONG Energy, “Walney Offshore Wind Farm - Facts of the project,” 2013. [Online]. Available: http://www.dongenergy.com/walney/about_walney/about_the_project/pages/facts.aspx. [Accessed: 06-May-2013].