Frequency-domain fatigue analysis of wide-band stationary Gaussian processes using a trimodal spectral formulation

In this paper, a procedure for fatigue analysis of a general wide-band stationary Gaussian process is developed in the frequency domain using a trimodal spectral formulation, based on a generalization of the principle proposed by Jiao and Moan for predicting bimodal fatigue damage. The novel method approximates the rainflow cycles with large amplitudes as the sum of the envelopes of the process components. First, the method is derived and the corresponding accuracy is demonstrated for ideal trimodal Gaussian processes whose spectra exhibit peaks at three well separated modes, each assumed to be narrow-banded. Hermite numerical integration method is applied to evaluate the fatigue damage due to a Rayleigh sum distribution since there is no closed-form solution for a random variable which is the sum of more than two Rayleigh random variables. Then, the method is further developed and applied to general wide-band Gaussian processes by dividing the spectrum into three segments with the same variances and calculating the fatigue damage in the same way as for the ideal trimodal processes. Based on extensive time series simulated with spectra of wave- and wind-induced structural responses and with generally defined wide-band spectra, the rainflow cycle counting method has been applied to estimate the fatigue damage in the time domain in order to check the accuracy of the proposed method and other frequency-domain methods. It is found that the narrow-band assumption can be very well made for a process with a Vanmarcke’s bandwidth parameter which is less than 0.5. For most of the simulated processes with bandwidth parameters between 0.5 and 0.85, the proposed method slightly overestimates the fatigue damage on average. If the bandwidth is greater than 0.85, the fatigue damage obtained by the trimodal formulation may be significantly overestimated in some cases, however spectra with very high bandwidth parameters might be unrealistic. Anyway, it has been shown that the overestimation can be alleviated by using a formulation with more modes for these cases. Moreover, two empirical formulae for wide-band fatigue damage estimation, one derived by Dirlik and the other proposed by Benasciutti and Tovo, have also been verified to be very accurate but slightly underestimate the fatigue damage for a wide range of the bandwidth parameters considered in this paper.