Investigation of the effect of the bed slope on extreme waves using First Order Reliability Method

Amin Ghadirian*, Henrik Bredmose

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review


The effect of bed slope on the force statistics and shape of the force time history around the force peak is investigated numerically with a fully nonlinear wave model and FORM analysis. The numerical model setup is validated by comparison of averaged experimental force and free surface elevation time series and the FORM results for the given force levels. The fully nonlinear FORM solution shows a good improvement over first-order and second-order results by increasing the asymmetry of the force history through the peak. The average deviation between FORM and the experimental curve is found to be at the level of 10% of the maximum force value. Next, the order statistics for force peaks from experimental and numerical 3 h realization are compared. Bootstrapping is applied to estimate the expected mean value of the inline forces for a given exceedance probability and a good match between the numerical and experimental order statistics is found. FORM predictions of the force peak histories for a cylinder on flat or sloping bed are next compared. The diameter, depth at the structure and significant wave height are kept identical between the two cases. It is found that the force histories are not changed significantly by the presence of the slope in the sea states with lower Ursell number. For the larger Ursell number sea states, the corresponding time histories of the free surface elevation, however, show a larger skewness for the flat bed case. Further, from the FORM analysis, the exceedance probabilities for a given force level at sloped bed is found to be larger than for flat bed. The ratio of exceedance probability is found to increase with force level except for the sea states with largest Ursell number where the numerical results are affected by strong nonlinearity. The higher-harmonic content of the force histories is further analyzed by the four-phase separation method. The analysis confirms that the relative contribution from the higher-harmonic force components increases with the force peak level and further shows that the contribution from the first, second and third harmonics are very similar between flat bed and sloped bed. The analysis further highlights the presence of spurious second-harmonic waves from the linear wave generation method and shows that these are stronger for the flat bed case. The phase shift between the different harmonics of free surface elevation and inline force time series is observed to be constant for different sea states and target forces irrespective of the bed slope. Besides the direct findings of the study, the paper demonstrates the applicability of the FORM method for determination of design waves based on certain load effects. Although difficulties may occur for breaking waves or very strong nonlinearity, the combination of FORM and a fully nonlinear wave model enables average probability levels and time histories for extreme events to be determined and applied as design waves specific to a given load level or response level.
Original languageEnglish
Article number102627
JournalMarine Structures
Number of pages30
Publication statusPublished - 2019


  • Bed slope
  • Extreme waves
  • First order reliability method

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