Closed form solution of Eigen frequency of monopile supported offshore wind turbines in deeper waters incorporating stiffness of substructure and SSI

Offshore wind turbines (OWTs) are dynamically loaded structures and therefore the estimation of the natural frequency is an important design calculation to avoid resonance and resonance related effects (such as fatigue). Monopiles are currently the most used foundation type and are also being considered in deeper waters (>30 m) where a stiff transition piece will join the monopile and the tapered tall tower. While rather computationally expensive, high fidelity finite element analysis can be carried to find the Eigen solutions of the whole system considering soil–structure interaction; a quick hand calculation method is often convenient during the design optimisation stage or conceptual design stage. This paper proposes a simplified methodology to obtain the first natural frequency of the whole system using only limited data on the WTG (Wind Turbine Generator), tower dimensions, monopile dimensions and the ground. The most uncertain component is the ground and is characterised by two parameters: type of ground profile (i.e. soil stiffness variation with depth) and the soil stiffness at one monopile depth below mudline. In this framework, the fixed base natural frequency of the wind turbine is first calculated and is then multiplied by two non-dimensional factors to account for the foundation flexibility (i.e. the effect of soil–structure interaction). The theoretical background behind the model is the Euler–Bernoulli and Timoshenko beam theories where the foundation is idealised by three coupled springs (lateral, rocking and cross-coupling). 10 wind turbines founded in different ground conditions from 10 different wind farms in Europe (e.g. Walney, Gunfleet sand, Burbo Bank, Belwind, Barrow, Kentish flat, Blyth, Lely, Thanet Sand, Irene Vorrink) have been analysed and the results compared with the measured natural frequencies. The results show good accuracy (errors below 3.5%). A step by step sample calculation is also shown for practical use of the proposed methodology.

[1]  Lars Vabbersgaard Andersen,et al.  Effects of soil–structure interaction on real time dynamic response of offshore wind turbines on monopiles , 2014 .

[2]  Cen.  Foundations, retaining structures and geotechnical aspects , 2003 .

[3]  John P. Carter,et al.  Analysis of Laterally Loaded Shafts in Rock , 1992 .

[4]  Subhamoy Bhattacharya,et al.  Experimental validation of soil–structure interaction of offshore wind turbines , 2011 .

[5]  Patrick Guillaume,et al.  Experimental and computational damping estimation of an offshore wind turbine on a monopile foundation , 2013 .

[6]  Subhamoy Bhattacharya,et al.  Dynamic stiffness of pile in a layered elastic continuum , 2014 .

[7]  S. John Hogan,et al.  Simplified critical mudline bending moment spectra of offshore wind turbine support structures , 2015 .

[8]  Lars Bo Ibsen,et al.  Natural Frequency and Damping Estimation of an Offshore Wind Turbine Structure , 2012 .

[9]  B B Broms,et al.  Lateral Resistance of Piles in Cohesive Soils , 1964 .

[10]  Subhamoy Bhattacharya,et al.  Dynamic soil–structure interaction of monopile supported wind turbines in cohesive soil , 2013 .

[11]  Eduardo Kausel,et al.  The spring method for embedded foundations , 1978 .

[12]  Subhamoy Bhattacharya,et al.  Dynamics of offshore wind turbines supported on two foundations , 2013 .

[13]  J. Tempel,et al.  Design Methods for Offshore Wind Turbines at Exposed Sites , 2004 .

[14]  W. G. Versteijlen.  Estimation of the Vibration Decrement of an Offshore Wind Turbine Support Structure Caused by its Interaction with Soil , 2011 .

[15]  Emil Winkler,et al.  Die lehre von der elasticitaet und festigkeit mit besonderer rücksicht auf ihre anwendung in der technik, für polytechnische schulen, bauakademien, ingenieure, maschinenbauer, architecten etc. , 1867 .

[16]  H. Matlock.  Correlation for Design of Laterally Loaded Piles in Soft Clay , 1970 .

[17]  Bengt B. Broms,et al.  LATERAL RESISTANCE OF PILE IN COHESIONLESS SOILS , 1964 .

[18]  Lars Vabbersgaard Andersen,et al.  Cross-wind modal properties of offshore wind turbines identified by full scale testing , 2013 .

[19]  Milos Novak,et al.  Soil-pile interaction in horizontal vibration , 1977 .

[20]  Eduardo Kausel,et al.  An explicit solution for the Green functions for dynamic loads in layered media , 1981 .

[21]  M. B. Zaaijer,et al.  Foundation models for the dynamic response of offshore wind turbines , 2002 .

[22]  Subhamoy Bhattacharya,et al.  Observed increases in offshore pile driving resistance , 2009 .

[23]  John H G Macdonald,et al.  Dynamic soil-structure interaction issues of offshore wind turbines , 2014 .

[24]  Jessica Daecher,et al.  Pile Foundation Analysis And Design , 2016 .

[25]  Lymon C. Reese,et al.  Field Testing and Analysis of Laterally Loaded Piles om Stiff Clay , 1975 .

[26]  Amir M. Kaynia,et al.  Dynamic stiffness and seismic response of pile groups , 1982 .

[27]  A. B. Vesic,et al.  Bending of Beams Resting on Isotropic Elastic Solid , 1961 .

[28]  M. Randolph,et al.  The response of flexible piles to lateral loading , 1981 .

[29]  Varvara Zania,et al.  Natural vibration frequency and damping of slender structures founded on monopiles , 2014 .

[30]  Lars Vabbersgaard Andersen,et al.  Comparing Sources of Damping of Cross-Wind Motion , 2009 .

[31]  K. Terzaghi,et al.  EVALUATION OF COEFFICIENTS OF SUBGRADE REACTION , 1955 .

[32]  Milos Novak,et al.  Resistance of soil to a horizontally vibrating pile , 1977 .

[33]  Jose M. Roesset,et al.  Dynamic stiffness of piles , 1976 .

[34]  Farrel Zwerneman,et al.  Insights into Using the 22nd Edition of API RP 2A Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms - Working Stress Design , 2012 .

[35]  Ervin Bossanyi,et al.  Wind Energy Handbook , 2001 .

[36]  R B Peck,et al.  THE BEARING CAPACITY OF CLAYS , 1952 .

[37]  David-Pieter Molenaar,et al.  Wind Turbine Structural Dynamics – A Review of the Principles for Modern Power Generation, Onshore and Offshore , 2002 .

[38]  Subhamoy Bhattacharya,et al.  Dynamic Analysis of Wind Turbine Towers on Flexible Foundations , 2012 .

[39]  M T Davisson,et al.  LATERAL LOAD CAPACITY OF PILES , 1970 .

[40]  George Gazetas,et al.  Seismic response of end-bearing single piles , 1984 .

[41]  D. Muir Wood,et al.  Observed dynamic soil–structure interaction in scale testing of offshore wind turbine foundations , 2013 .

[42]  Dan Kallehave,et al.  Modification of the API P-y Formulation of Initial Stiffness of Sand , 2012 .

[43]  N. Jenkins,et al.  Wind Energy Handbook: Burton/Wind Energy Handbook , 2011 .

[44]  Lymon C. Reese,et al.  Field Testing and Analysis of Laterally Loaded Piles in Stiff Clay , 2006 .

[45]  John H G Macdonald,et al.  An analytical model to predict the natural frequency of offshore wind turbines on three-spring flexible foundations using two different beam models , 2015 .

[46]  Milos Novak,et al.  Dynamic Stiffness and Damping of Piles , 1974 .

[47]  Subhamoy Bhattacharya,et al.  Vibrations of wind-turbines considering soil-structure interaction , 2011 .