Sway-Rocking Spring System Applicable to Short-Rigid Monopile Foundations


Sway-Rocking (SR) spring system is a foundation modelling method in which a 2 × 2 spring stiffness matrix represents the foundation stiffness at seabed level. Since each individual element of the spring stiffness matrix is a function of a few physical properties of the soil and pile, it is the simplest and computationally cheapest method to estimate the deformation at the seabed level and the natural frequency of an offshore wind turbine structure. Formulae currently available are only applicable to conventional long-flexible piles or perfectly rigid foundations such as footings or caissons. This study aims to extend the application of the SR spring system to a so-called short-rigid pile range, which lays between the two ranges: long-flexible and perfectly rigid. Monopiles, the most commonly selected type of offshore wind foundations, are categorized in the short-rigid pile range. By conducting three-dimensional finite element analyses, the short-rigid pile behaviour is investigated. Based on the results, a new set of formulae of the SR spring stiffness, which is applicable to a wide range (long-flexible, short-rigid and perfectly rigid) of piles, is proposed. The new formulae also improve the accuracy of frequency estimation for practical use cases.

This is a preview of subscription content, access via your institution.

Fig. 1

source: Wind-Europe 2017, 4C Offshore 2018 and IRENA 2019)

Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14


H :

Lateral force at the seabed (N)

M :

Moment at the seabed (Nm)

u :

Lateral displacement (m)

\(\theta\) :

Rotational displacement (rad)

\(K_{L}\) :

Lateral spring stiffness (N/m)

\(K_{LR}\) :

Coupling spring 1 stiffness (N/rad)

\(K_{RL}\) :

Coupling spring 2 stiffness (Nm/m)

\(K_{R}\) :

Rotational spring stiffness (Nm/rad)

\(\eta_{L}\) :

\(K_{L}\) Normalized by \(E_{s}\) and \(D_{p}\) (–)

\(\eta_{LR}\) :

\(K_{LR}\) Normalized by \(E_{s}\) and \(D_{p}\) (–)

\(\eta_{RL}\) :

\(K_{RL}\) Normalized by \(E_{s}\) and \(D_{p}\) (–)

\(\eta_{R}\) :

\(K_{R}\) Normalized by \(E_{s}\) and \(D_{p}\) (–)

\(E_{s}\) :

Young’s modulus of soil (Pa)

\(E_{p}\) :

Effective pile stiffness (Pa)

\(\nu\) :

Poisson’s ratio (–)

\(L_{p}\) :

Embedded length of pile (m)

\(D_{p}\) :

Diameter of pile (m)

\(t_{p}\) :

Wall thickness of pile (m)

\(EI_{p}\) :

Bending stiffness of pile (Nm2)

\(L_{t}\) :

Tower height (m)

\(D_{t}\) :

Diameter of tower (m)

\(t_{t}\) :

Wall thickness of tower (m)

\(EI_{t}\) :

Bending stiffness of tower (Nm2)

\(K_{t}\) :

Tower stiffness in lateral way (N/m)

\(M_{eff}\) :

Effevtive mass at the top of tower (kg)

\(m_{t}\) :

Total mass of tower (kg)

\(m_{RNA}\) :

Mass of RNA (kg)

\(\upomega _{1}\) :

Angular velocity (1st mode) (rad/s)

\(\upomega _{FB}\) :

Fixed-base angular velocity (rad/s)

\(f_{1}\) :

Natural frequency (1st mode) (Hz)

\(f_{FB}\) :

Fixed-base natural frequency (Hz)

\(\zeta_{L}\) :

\(K_{L}\) normalized by \(K_{t}\) and \(L_{t}\) (–)

\(\zeta_{LR}\) :

\(K_{LR}\) normalized by \(K_{t}\) and \(L_{t}\) (–)

\(\zeta_{R}\) :

\(K_{R}\) normalized by \(K_{t}\) and \(L_{t}\) (–)

\(\chi\) :

Soil-pile-tower interaction factor (–)

\(\upomega _{D}\) :

Angular velocity with damping (rad/s)

\(\xi\) :

Damping ratio (–)

\(E_{p_{dynamic}}\) :

Dynamic pile effective stiffness (Pa)


  1. Arany L, Bhattacharya S, Macdonald J, Hogan SJ (2014) Simplified critical mudline bending moment spectra of offshore wind turbine support structures. Wind Energy 18(2015):2171–2197. https://doi.org/10.1002/we.1812

    Article  Google Scholar 

  2. Arnay L, Bhattacharya S, Adhiakari S, Hogan SJ, Macdonald J (2015) An analytical model to predict the natural frequency of offshore wind turbines on three-spring flexible foundations using two different beam models. Soil Dyn Earthq Eng 74:40–45. https://doi.org/10.1016/j.soildyn.2015.03.007

    Article  Google Scholar 

  3. Arany L, Bhattacharya S, Macdonald J, Hogan SJ (2016) Closed form solution of Eigen frequency of monopile supported offshore wind turbines in deeper waters incorporating stiffness of substructure and SSI. Soil Dyn Earthq Eng 83:18–32. https://doi.org/10.1016/j.soildyn.2015.12.011

    Article  Google Scholar 

  4. Bacaër N (2011) Verhulst and the logistic equation (1838). In: A short history of mathematical population dynamics. Springer-Verlag, London, pp 35–39. https://doi.org/10.1007/978-0-85729-115-8_6

  5. Bhatacharya S (2019) Design of foundation for foundations for offshore wind turbines. Wiley, New Jersey

    Google Scholar 

  6. Byrne BW et al (2020) Editorial: geotechnical design for offshore wind turbines monopiles. Geotéchnique 70(11):943–944. https://doi.org/10.1680/jgeot.2020.70.11.943

    Article  Google Scholar 

  7. Carter BJP, Kulhawy FH (1992) Analysis of laterally loaded shaft in rock. J Geotech Eng 118(6):839–855. https://doi.org/10.1061/(ASCE)0733-9410(1992)118:6(839)

    Article  Google Scholar 

  8. Chowdhury I, Tarafdar R, Ghosh A, Dasgupta SP (2016) Dynamic response of cylindrical structure considering coupled soil-structure interaction under seismic loading. Bull Earthq Eng 14:2329–2360. https://doi.org/10.1007/s10518-016-9909-4

    Article  Google Scholar 

  9. DNV (2002) Guidelines for deign of wind turbines. DNV/Risø, London

    Google Scholar 

  10. DNVGL-RP-C212 (2017) Recommended practice. DNVGL-RP-C212 Offshore soil mechanics and geotechnical engineering. DNV GL AS, Edition August 2017

  11. DNVGL-ST-0126 (2018) Standard. DNVGL-ST-0126 Support structures for wind turbines. DNV GL AS, Edition July 2018

  12. EN1998–5 (2004) Eurocode 8: Design of structures for earthquake resistance—part 5: foundations, retaining structures and geotechnical aspects. The European Union, 36 (Annex C)

  13. Fleming WGK, Weltman AJ, Randolph MF et al (1992) Pilling engineering. Blackie, Glasgow

    Google Scholar 

  14. Gazetas G (1984) (1984) Seismic response of end-bearing single piles. Int J Soil Dyn Earthq Eng 3(2):82–93. https://doi.org/10.1016/0261-7277(84)90003-2

    Article  Google Scholar 

  15. Gürgöze M (2005) On the representation of a cantilevered beam carrying a tip mass by an equivalent spring-mass system. J Sound Vib 282:538–542. https://doi.org/10.1016/j.jsv.2004.04.006

    Article  Google Scholar 

  16. Higgins W, Vasquez C, Basu D, Griffiths DVG (2013) Elastic solutions for laterally loaded piles. J Geotech Geoenviron Eng ASCE 139(7):1096–1103. https://doi.org/10.1061/(ASCE)GT.1943-5606.0000828

    Article  Google Scholar 

  17. IRENA (2019) A global energy transformation paper—Future of wind. https://www.irena.org/-/media/Files/IRENA/Agency/Publication/2019/Oct/IRENA_Future_of_wind_2019.pdf. Accessed 15 Nov 2020

  18. Novak M, Nogami T (1977) Soil-pile interaction in horizontal vibration. Earthq Eng Struct Dyn 5:263–281. https://doi.org/10.1002/eqe.4290050305

    Article  Google Scholar 

  19. Okada T, Unjoh S (2007) Analytical study on the damping factor of bridge foundations. JSCE J Earthq Eng 29:381–388. https://doi.org/10.11532/proee2005a.29.381

    Article  Google Scholar 

  20. Randolph MF (1981) The response of flexible piles to lateral loading. Geotéchnique 31(2):247–259. https://doi.org/10.1680/geot.1981.31.2.247

    Article  Google Scholar 

  21. Richards FJ (1959) A flexible growth function for empirical use. J Exp Bot 10(2):290–301

    Article  Google Scholar 

  22. Shadlou M, Bhattacharya S (2016) Dynamic stiffness of monopiles supporting offshore wind turbine generators. Soil Dyn Earthq Eng 88:15–32. https://doi.org/10.1016/j.soildyn.2016.04.002

    Article  Google Scholar 

  23. Wind-Europe (2017) The European offshore wind industry Key trends and statistics 2016. https://windeurope.org/wp-content/uploads/files/about-wind/statistics/WindEurope-Annual-Offshore-Statistics-2016.pdf. Accessed 12 April 2020

  24. Zaaijer MB (2006) Foundation modelling to assess dynamic behaviour of offshore wind turbines. Appl Ocean Res 28:45–57. https://doi.org/10.1016/j.apor.2006.03.004

    Article  Google Scholar 

  25. Zhang F, Kimura M, Nakai T, Hoshikawa T (2000) Mechanical behavior of pile foundations subjected to cyclic lateral loading up to the ultimate state. Soils Found 40(5):1–17. https://doi.org/10.3208/sandf.40.5_1

    Article  Google Scholar 

  26. 4C Offshore (2018), Offshore wind farms database. https://www.4coffshore.com/windfarms/. Accessed 12 April 2020

Download references


The first author studied at ENPC to obtain the engineering degree besides the master degree of Tokyo Tech, financially supported by BGF and Tobitate. This research topic has been selected by the author who was inspired by her ENPC supervisors Jean-Michel Pereirra and Aphrodite Michael as well as her colleagues during an internship at Mott MacDonald.

Author information



Corresponding author

Correspondence to Akihiro Takahashi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Yukiho Kamata: Formerly Department of Civil and Environmental Engineering, Tokyo Institute of Technology, Tokyo, Japan.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kamata, Y., Takahashi, A. Sway-Rocking Spring System Applicable to Short-Rigid Monopile Foundations. Geotech Geol Eng (2021). https://doi.org/10.1007/s10706-021-01678-2

Download citation


  • Offshore wind turbine structure
  • Monopile
  • Short-rigid pile
  • Sway-rocking spring system
  • Natural frequency