Stochastic model for joint wave and wind loads on offshore structures

The stochastic wave load environment of offshore structures is of such a complicated nature that any engineering analysis requires extensive simplifications. This concerns both the transformation of the wave field velocities and accelerations to forces on the structure and the probabilistic description of the wave field itself. In this paper the last issue is in focus. The modeling follows the traditional structure of subdividing the time development of the wind driven wave process into sea states within each of which the wave process is modeled as a stationary process. The wave process of each sea state is modeled as an affinity in height and time of a Gaussian process defined by a normalized dimensionless spectrum of Pierson-Moskowitz type. The affinity factors are the socalled significant wave height $H_s$ and the characteristic zero upcrossing time $T_z$. Based on measured data of $(H_s,T_z)$ from the North Sea a well fitting joint distribution of $(H_s,T_z)$ is obtained as a so-called Nataf model. Since the wave field is wind driven, there is a correlation between the time averaged wind velocity pressure $Q$ and the characteristic wave height in the stationary situation. Using the Poisson process model to concentrate on those load events that are of importance for the evaluation of the safety of the structure, that is, events with $Q$ larger than some threshold $q_0$, available information about the wind velocity pressure distributionin high wind situations can be used to formulate a Nataf model for the joint conditional distribution of $(H_s,T_z,Q)$ given that $Q>q_0$. The distribution of the largest wave height during a sea state is of interest for designing the free space between the sea level and the top side. An approximation to this distribution is well known for a Gaussian process and by integration over all sea states given $Q>q_0$, the distribution is obtained that is relevant for the free space design. However, for the forces on the members of the structure also the wave period is essential. Within the linear wave theory (Airy waves) the drag term in the Morison force formula increases by the square of the ratio between the wave height and the wave length, and the mass force term increases proportional to the ratio of the wave height and the square of the period. For a strongly narrow band Gaussian process Longuet-Higgins has derived a joint distribution of the height and the period. However, simulations show that the Pierson-Moskowitz spectrum (or any other standard spectrum for wind driven sea waves of similar bandwidth such as the JONSWAP spectrum) does not provide a sufficiently narrow banded process for the distribution of Longuet-Higgins to make a good fit. Surprisingly it turns out that the random time $L$ between two consecutive 0-upcrossings and the random wave height $H$ observed between the two 0-upcrossings behaves such that $L$ and the ratio $H/L$ are practically uncorrelated and both normally distributed except for clipping the negative tails. This result is of global nature and is therefore very difficult if not impossible to obtain by analytical mathematical reasoning. Keywords: Extreme wind driven sea waves, Local maxima and period properties of Gaussian process, Nataf model for wave and wind data, Offshore structure loads, Sea wave stochastics during wind storm, Wave and wind loads.