###

# OTRC Project Summary

### Project Title: |
Effects of Morison Equation Nonlinearity on Stochastic Dynamics and Fatigue of Offshore Structures |

### Prinicipal Investigators: |
Loren Lutes |

### Sponsor: |
National Science Foundation |

### Completion Date: |
July, 1992 |

### Final Report ID# |
A33(Click to view final report abstract) |

The primary purpose of this study is to investigate the dynamic response and fatigue damage accumulation of offshore structures subjected to nonlinear non-Gaussian wave loading. Various factors affecting the structural dynamic response and fatigue damage will be investigated. In particular, the factors include those related to the power spectral density function (PSD), such as the variance, average frequency and spectral bandwidth of the response stress time history, and those related to the probability distribution function (PDF), such as the kurtosis or coefficient of excess (COE) and skewness. With the focus of the study on the non-Gaussian effects, the objectives are to determine the extent to which non-Gaussianity (nonnormality) has a significant effect on the dynamic response and fatigue life prediction for fixed offshore platforms and deepwater compliant offshore structures, and to provide improve methods for predicting fatigue life of such structures in nonlinear and non-Gaussian situations.

In order to investigate the non-Gaussian effects, response moments higher than the second will be considered since the first to moments, mean and variance, do not contain any information regarding the non-Gaussianity of the process. In particular, the third and fourth moments are important for characterizing the non-Gaussianity. Partial measures of these two moments are given by the skewness and the kurtosis or the coefficient of excess of the random process. The fourth moment is especially important if the random process is symmetric about its mean value so that the third moment gives no new information. Thus, the kurtosis or the coefficient of excess will serve as the main index to measure the degree of non-Gaussianity of a symmetric random process, while skewness may also be important for an unsymmetric process.

In this study, the hydrodynamic wave loading will be modeled by the Morison equation (Morison et al. 1950). Five versions of the Morison equation commonly found in the literature will be included. These five formulations have different degrees of complexity, varying from the nonlinear fluid-structure interaction model to the simple linearized model. The structure will be idealized as a linear single degree of freedom (SDOF) system. Note that the equation of motion will become nonlinear if the excitation is modeled by the nonlinear fluid-structure interaction Morison equation model. It should be mentioned here that adequate description of the statistical characteristics of the system response requires knowledge of more than the one-dimensional probability distribution of the excitation. For example, calculation of the root-mean-squared (rms) value of the structural response requires prior knowledge of the autocorrelation function or power spectral density function of the excitation. Analogously, the calculation of the kurtosis (fourth moment) of the response requires prior knowledge for the excitation, of the fourth cumulant function in the time domain or the trispectral density function in the frequency domain. Therefore, the problem of determining the degree of non-Gaussianity (the and fourth order moment properties) of a non-Gaussian structural response is fare more complicated than that of finding the second order moment of a Gaussian process even when the system is linear and the kurtosis of the excitation is known.

The approaches used in this study will include both numerical simulations and analytical procedures. Numerical simulation can often help one to gain useful insight and information from which better understanding of the behavior of a very complicated structural system may be achieved. Numerical simulation is generally considered to be one of the most versatile techniques for solving stochastic dynamic response problems. On the other hand, it is always desirable to have analytical solutions since numerical simulation may be quite computational intensive and time-consuming. Due to the complex nature of the problem, it is not difficult to se that the study of the non-Gaussian effects poses new challenges for both numerical simulation and analytical approaches.

It is expected that this study will lead to better understanding of the non-Gaussian effects on the dynamic response and fatigue damage of offshore structures, and that it will ultimately lead to improved fatigue life prediction in the design and analysis of offshore structures.

**Related Publications:** Wang, J. and Lutes, L.D. “Nonlinear and Non-Gaussian Aspects of Morison Equation Induced Fatigue Damage,” Proceedings of the 23rd Offshore Technology Conference, Houston, Texas, May 1991, pp.451-460.

Wang, J. and Lutes, L.D. “A Non-Gaussian Fatigue Model for Offshore Structures,” Probablistic Mechanics and Structural and Geotechnical Reliability, Proceedings of the Sixth Conference, ASCE, Denver, Colorado, July, 1992, pp.463-466.

Lutes, L.D. and Wang, J., Discussion Closure, “Simulating of an Improved Gaussian Time History,” Journal of Engineering Mechanics, ASCE, Vol.118, No.6, pp.1276-1277, 1992.