Estimation of the Long Term Correlation Coefficients by Simulation

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This paper proposes a simulation method for obtaining the estimate of the long term correlation coefficients between different low-frequency wave-induced loads acting on a ship hull. They are essential part of the load combination procedures in design and strength evaluations. Existing theory is limited to linear time-invariant systems with weakly stationary stochastic inputs such as waves during a single sea state (short-term). The simulation treats the non-stationary wave elevations during the ship’s entire life (long-term) as a sequence of different stationary Gaussian stochastic processes. Different sea states (HS, T0, Wave Direction) are sampled, using rejection sampling, from the joint probability density functions fitted to every Marsden zone on the ship’s route. The time series of the loads are simulated from the load spectra for each sea state, including the effects of loading condition, heading, speed, seasonality and voluntary as well as involuntary speed reduction. The estimates of the correlation coefficients are then calculated from these time series. The simulation time can be significantly reduced (to the order of seconds rather than hours and days) by introducing the seasonal variations into a single voyage. It is proven that the estimate of the correlation coefficient, obtained by simulating only a single voyage, approaches the true correlation coefficient in probability as the number of simulated load values increases. The simulation method can also be used for finding the long-term exceedance probabilities of the peak values of individual loads as well as for analyzing various load combinations (linear and nonlinear). Related concepts and limitations of this method are demonstrated by an example of a containership operating between Boston, MA and Southampton, UK.