Probability of Parametric Roll in Random Seaways

  • Jørgen Juncher JensenEmail author


The aim of the present chapter is to advocate for a very effective stochastic procedure based on the First-Order Reliability method (FORM) and Monte Carlo simulations (MCS), for prediction of parametric rolling in stationary stochastic seaways. Due to efficient optimization procedures implemented in standard FORM codes and the short duration of the time domain simulations needed, the calculation of the mean out-crossing rates of any given roll response is very fast. Thus complicated nonlinear effects can be included. Furthermore, the FORM analysis also identifies the most probable wave episodes leading to given roll responses. Because of the linearization in the FORM procedure the results are, however, only asymptotically exact and thus MCS often also needs to be performed. Here a scaling property inherent in the FORM procedure is investigated for use in MCS in order to reduce the necessary simulation time. More specifically, the MCS results for the reliability index β for parametric rolling of a ship suggest that the relation \(\beta = a + b/{H}_{\mathrm{s}}\) can provide an accurate scaling of the reliability index with significant wave height Hs. MCS can therefore be performed for sea states higher than the design sea state and thereafter be scaled down. From the reliability index the mean out-crossing rate and the probability of exceedance are then readily obtained.


Wave Height Monte Carlo Simulation Reliability Index Significant Wave Height Roll Angle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Adegeest, L. J. M, Braathen, A. Løseth, R. M.: Use of non-linear sea-loads nimulations in design of ships. Proc. PRADS’1998, Delft, pp. 53–58 (1998)Google Scholar
  2. 2.
    Alford, K. A., Troesch, A. W., McCue, L.: LS design wave elevations leading to extreme roll motions. Proc. STAB’2005, Istanbul, Turkey (2005)Google Scholar
  3. 3.
    Borge, J., Gonzáles, R., Hessner, K., Reichert, K., Soares, C. G.: Estimation of sea state directional spectra by using marine radar imaging of sea surface. Proc. ETCE/OMAE2000 Joint Conference, New Orleans, Louisiana, USA (2000)Google Scholar
  4. 4.
    Bulian, G., Francescutto, A. A.: Simplified modular approach for the prediction of the roll motion due to the combined action of wind and waves. Proc Instn Enggrs, Part M: J. Engineering for the Maritime Environment 218:189–212 (2004)CrossRefGoogle Scholar
  5. 5.
    Bulian, G.: Nonlinear parametric rolling in regular waves – a general procedure for the analytical approximation of the GZ curve and its use in time domain simulations. Ocean Engineering 32:309–330 (2005)CrossRefGoogle Scholar
  6. 6.
    Daalen, E. F. G., Boonstra, H., Blok, J. J.: Capsize probability analysis of a small container vessel. Proc- 8th Int. Workshop on Stability and Operational Safety of Ships, Istanbul, October 6–7 (2005)Google Scholar
  7. 7.
    Der Kiureghian, A.: The geometry of random vibrations and solutions by FORM and SORM. Probabilistic Engineering Mechanics 15:81–90 (2000)CrossRefGoogle Scholar
  8. 8.
    Det Norske Veritas: General purpose probabilistic analysis program, Version 4.4 (2003)Google Scholar
  9. 9.
    Dietz, J. S., Friis-Hansen, P., Jensen, J. J.: Most likely response waves for estimation of extreme value ship response statistics. Proc PRADS’2004, Travemünde, September, Germany (2004)Google Scholar
  10. 10.
    Ditlevsen, O., Arnbjerg-Nielsen, T.: Model-correction-factor method in structural reliability. Journal of Engineering 120:1–10 (1994)Google Scholar
  11. 11.
    France, W. N., Levadou, M., Treakle, T. W., Paulling, J. R., Michel, R. K., Moore, C.: An investigation of head-sea parametric rolling and its influence on container lashing systems. Marine Technology 40:1–19 (2003)Google Scholar
  12. 12.
    Fujimura, K., Der Kiureghian, A.: Tail-equivalent linearization method for nonlinear random vibration. Probabilistic Engineering Mechanics 22: 63–76 (2007)CrossRefGoogle Scholar
  13. 13.
    Hsieh, S.-R., Troesch. A. W., Shaw, S. W.: A nonlinear probabilistic method for predicting vessel capsize in random beam seas. Proc Royal Society London, Part A, 446:195–211 (1994)Google Scholar
  14. 14.
    IMO submission (SLF 50) by Germany: Proposal for an additional intact stability regulations. Subcommittee on Stability, Loadlines and on Fishing Vessels (2007)Google Scholar
  15. 15.
    de Kat, J. O.: ITTC specialist committee on stability in waves (2005)Google Scholar
  16. 16.
    Jensen, J. J., Mansour, A. E., Olsen, A. S.: Estimation of ship motions using closed-form expressions. Ocean Engineering 31:61–85 (2004)CrossRefGoogle Scholar
  17. 17.
    Jensen, J. J., Capul, J.: Extreme response predictions for jack-up units in second-order stochastic waves by FORM. Probabilistic Engineering Mechanics 21:330–337 (2006)CrossRefGoogle Scholar
  18. 18.
    Jensen, J. J., Olsen, A. S.: On the assessment of parametric roll in random sea. Proc. World Maritime Technology Conference, London, 6–10 March (2006)Google Scholar
  19. 19.
    Jensen, J. J., Pedersen, P. T.: Critical wave episodes for assessment of parametric roll Proc. IMDC’06, Ann Arbor, May, pp. 399–411 (2006)Google Scholar
  20. 20.
    Jensen, J. J.: Efficient estimation of extreme non-linear roll motions using the first-order reliability method (FORM). J. Marine Science and Technology 12:191–202 (2007)CrossRefGoogle Scholar
  21. 21.
    Jensen, J. J.: Extreme value predictions using Monte Carlo simulations with artificially increased load spectrum. Probabilistic Engineering Mechanics, 26:399–404 (2010)CrossRefGoogle Scholar
  22. 22.
    Koo, H., Der Kiureghian, A., Fujimura, K.: Design point excitation for nonlinear random vibrations. Probabilistic Engineering Mechanics. 20:136–147 (2005)CrossRefGoogle Scholar
  23. 23.
    Krüger, S., Hinrichs, R., Cramer, H.: Performance based approaches for the evaluation of intact stability problems. Proc. PRADS’2004, Travemünde, September, Germany (2004)Google Scholar
  24. 24.
    Kroeger, H.-P.: Roll Simulation von Schiffen im seegang. Schiffstechnik 33:187–216 (in German) (1986)Google Scholar
  25. 25.
    Lindgren, G.: Some properties of a normal process near a local maximum. Ann. Math. Stat. 41:1870–1883 (1970)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ness, O. B., McHenry, G., Mathisen, J., Winterstein, S. R.: Nonlinear analysis of ship rolling in random beam waves. Proc. STAR Symposium on 21st Century Ship and Offshore Vessel Design, Production and Operation, April 12–15, pp. 49–66 (1989)Google Scholar
  27. 27.
    Neves, M. A. S., Rodriquez, C. A.: A coupled third-order model of roll parametric resonance. Proc Maritime Transportation and Exploitation of Ocean and Coastal Resources, pp. 243–253, Taylor and Francis, London, UK (2005)Google Scholar
  28. 28.
    Nielsen, J. K., Hald, N. H., Michelsen, J., Nielsen, U. D., Baatrup, J., Jensen, J. J., Petersen, E. S.: Seasense – real-time onboard decision Support. Proc. World Maritime Technology Conference, London, 6-10 March (2006)Google Scholar
  29. 29.
    Nielsen, U. D.: Estimations of on-site directional wave spectra from measured ship responses. Marine Structures 19:33–69 (2006)CrossRefGoogle Scholar
  30. 30.
    Nielsen, U. D., Jensen, J. J.: Numerical simulations of the rolling of a ship in a stochastic sea – Evaluations by Use of MCS and FORM. Proc of 28th International Conference on Offshore Mechanics and Arctic Engineering (OMAE’09), Honolulu, USA, June, Paper No. 79765 (2009)Google Scholar
  31. 31.
    Pereira, R.: Simulation nichtlinearer Seegangslasten. Schiffstechnik 35:173-193 (in German) (1988)Google Scholar
  32. 32.
    Rathje, H.: Impact of Extreme Waves on Ship Design and Ship Operation. Proc Design & Operation for Abnormal Conditions, Royal Institution of Naval Architects, London, 26–27 January (2005)Google Scholar
  33. 33.
    Shin, Y. S., Belenky, V. L., Paulling, J. R., Weems, K. M., Lin, W. M.: Criteria for parametric roll of large container ships in head seas. Transactions of SNAME 112:14–47 (2004)Google Scholar
  34. 34.
    Spanos, D., Papanikolaou, A. A numerical simulation of a fishing vessel in parametric roll in head sea. Proc. 8th Int. Workshop on Stability and Operational Safety of Ships, Istanbul, October 6–7 (2005)Google Scholar
  35. 35.
    Spanos, D., Papanikolaou, A., Papatzanakis, G.: Risk-based onboard guidance to the master for avoiding dangerous seaways. 6th OSAKA Colloquium on Seakeeping and Stability of Ships, Osaka University, Japan, March 26–28 (2008)Google Scholar
  36. 36.
    Spyrou, K. J.: Designing against parametric instability in following seas. Ocean Engineering 26:625–653 (2000)CrossRefGoogle Scholar
  37. 37.
    Spyrou, K., Themelis, N.: Development of probabilistic procedures and validation – Alternative 2: Capsize mode analysis. Deliverables D235, SAFEDOR-D-235-2006-11-9-NTUA–rev-1 (2006)Google Scholar
  38. 38.
    Themelis, N., Spyrou, K., Niotis, S.: Implementation and application of probabilistic Procedures. Deliverables D236, SAFEDOR-D-236-2007-06-15-NTUA–rev-1 (2007)Google Scholar
  39. 39.
    Tonguc, E., Söding, H.: Computing capsizing frequencies of ships in a seaway. Proc. 3rd Int. Conf. on Stability of Ships and Ocean Vehicles STAB’86, Gdansk (1986)Google Scholar
  40. 40.
    Tromans, P. S., Anaturk, A. R., Hagemeijer, P.: A New Model for the Kinematics of Large Ocean Waves - Application as a Design Wave. Proc 1st Offshore and Polar Engineering Conference (ISOPE), Vol. 3, pp. 64–71 (1991)Google Scholar
  41. 41.
    Vidic-Perunovic, J., Jensen, J. J.: Parametric roll due to instantaneous volumetric changes and speed variations. Ocean Engineering 36:891–899 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.DTU Mechanical Engineering, Department of Mechanical EngineeringTechnical University of DenmarkKgs. LyngbyDenmark

Personalised recommendations