A Dissertation entitled An Efficient Method to Assess Reliability under Dynamic Stochastic Loads by Mahdi Norouzi Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Engineering _______________________________________________ Dr. Efstratios Nikolaidis, Committee Chair _______________________________________________ Dr. Abdollah Afjeh, Committee Member _______________________________________________ Dr. Sorin Cioc, Committee Member _______________________________________________ Dr. Ali Fatemi, Committee Member _______________________________________________ Dr. Mehdi Pourazadi, Committee Member _______________________________________________ Dr. Larry Viterna, Committee Member _______________________________________________ Dr. Patricia R. Komuniecki, Dean College of Graduate Studies The University of Toledo December 2012 Copyright 2012, Mahdi Norouzi This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author An Abstract of An Efficient Method to Assess Reliability under Dynamic Stochastic Loads by Mahdi Norouzi Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Engineering The University of Toledo December 2012 The objective of this research is to develop an efficient method to study the reliability of dynamic large complex engineering systems. In design of real-life dynamic systems, there are significant uncertainties in modeling the input. For instance, for an offshore wind turbine, there are considerable uncertainties in the power spectral density functions of the wave elevations or the wind speeds. Therefore, it is necessary to evaluate the reliability of a system for different power spectral density functions of the input loads. The reliability analysis of dynamic systems requires performing Monte Carlo simulations in time domain with thousands of replications. The computational cost of such analyses is prohibitive for most real-life complex systems. In this study, a new method is proposed to reduce the computational cost of the reliability study of dynamic systems. This method is applicable to the dynamic systems in which the loads are represented using power spectral density functions. This goal is achieved by estimating the reliability for several power spectral densities of a load by re- weighting the results of a single Monte Carlo simulation for one power spectral density function of the load. The proposed approach is based on Probabilistic Re-analysis method that is similar to the idea of Importance Sampling. That is the main variance reduction iii technique, which is used to lower the computational cost of Monte Carlo simulation. The proposed method extends the application of the Probabilistic Re-analysis, which has already been applied to static problems, to dynamic problems. Static problems are modeled using random variables that are invariant with time whereas in dynamic systems both the excitation and the response are stochastic processes varying with time. Utilizing Shinozuka’s method is the key idea because it enables representing a time varying random process in terms of random variables. This new approach can significantly lower the cost of the sensitivity reliability analysis of dynamic systems. This study also presents a new approach to apply Subset Simulation efficiently to dynamic problems. Subset Simulation is more efficient than Monte Carlo simulation in estimating the probability of first excursion failure of highly reliable systems. This method is based on the idea that a small failure probability can be calculated as a product of larger conditional probabilities of intermediate events. The method is more efficient because it is much faster to calculate several large probabilities than a single low probability. However, Subset Simulation is often impractical for random vibration problems because it requires considering numerous random variables that makes it very difficult to explore the space of the random variables due to its large dimension. A new approach is proposed in this research to perform Subset Simulation that utilizes Shinozuka’s equation to calculate the time series of the loads from a power spectral density function. The commutative property of Shinozuka’s equation enables taking advantage of its symmetry, thereby reducing the dimension of the space of the random variables in dynamic problems. Therefore, performing Subset Simulation using the new approach is more efficient than the original Subset Simulation. In addition, Shinozuka’s iv equation assists in integrating Subset Simulation with Probabilistic Re-analysis. This new method, which is called Subset-PRRA, is more efficient than regular Probabilistic Re- analysis as the latter is based on Monte Carlo simulation, whereas Subset-PRRA reuses the results of Subset Simulation. For an offshore wind turbine, the wind and waves are represented by power spectral density functions; Subset-PRRA seems to be a promising tool to cut the computational cost of the sensitivity analysis of first excursion reliability of an offshore wind turbine. The application of the Probabilistic Re-analysis in reliability analysis of an offshore wind turbine is demonstrated in this research through two examples in which only changes in the power spectral density function of the wave elevation are considered. The method is also applicable to the case that the wind spectrum changes, but requires calculation of wind field time histories using Shinozuka’s method. Finally, a probabilistic approach for the structural design of an offshore wind turbine under the Lake Erie environment is presented. To perform probabilistic design, the dependence between wind, wave and period should be modeled accurately. Modeling the dependence between wind and wave is expensive, as it requires a large amount of data. Many researchers, similar to the approach presented in the International Electrotechnical Commission standards, assume that wave height follows standard distributions conditional on wind speed. In this work, an alternative approach is used that is based on the application of copulas. This approach is more complete because the joint distribution is obtained without making any assumption on the conditional distributions. Using the joint distribution, a methodology to find the required load capacity of the structure to v meet the target reliability based on Monte Carlo simulation and Tail-fitting method is presented. vi To Three Women in My Life To my grandmother, Tooba, who died in 2008, for all the good memories we shared To my mother, Golzar, for all her relentless support and unconditional love throughout my whole life To my lovely wife, Bahareh, for all her support to make studying in a foreign country a comfortable experience Thanks to my father, Ali for all his hard work to support his family, and my siblings for their encouragement. I could not have completed my Ph.D. without their support. vii Acknowledgements I would like to express my sincere gratitude to my advisor, Dr. Efstratios Nikolaidis, for the patient guidance, encouragement and advice he has provided throughout my time as his student. I specially thank him for drawing my attention to the importance of probability and statistics in mechanical design. I would also like to thank Dr. Abdollah Afjeh, the chairman of the MIME department, for his support and valuable guidance, and also for providing the funding that allowed me to undertake this research. Thank you to Dr. Larry Viterna, Dr. Sorin Cioc and Mr. Robert Kozar for being in my Ph.D. committee and for their valuable suggestions during our weekly meetings over last 3 years. I would like to thank Dr. Ali Fatemi, and Dr. Mehdi Pourazady for evaluating this work and for their valuable suggestions that helped me to improve my dissertation. Many thanks to my friends and colleagues, Adrian Sescu, Eric Wells, Brett Andersen, Linhao Wu, Jihan Mussarat and Jin Wu Lee for their friendship, and for all their contributions to this project. Thank you to the MIME faculty and staff, especially Ms. Debbra Kraftchick and Ms. Emily Lewandowsky for their hard work and support. This research was conducted with the financial support of the U.S. Department of Energy; grant DE-EE0003540, under the direction of Michael Hahn, the project manager. This support is greatly appreciated. viii Table of Contents Abstract…………………………………………………………………………………… iii Acknowledgements……………………………………………………………………… …. viii Table of Contents ……………………………… ……………………………………… ix List of Tables ………………………………………… ……………………………… …xii List of Figures ………………………………………… ………………………… …… xv List of Abbreviations………………………………… …………………… ……………… xxii List of Symbols ………………………… ……………… ……………….…… xxiii 1. Introduction 1 1.1. Problem Definition and Significance…………………………..……………….....……………………………………………………….... 1 1.2. Dissertation Objective and Scope…………………………………………………………………………………………………….…..……… 2 1.3. Dissertation Contributions…………………………………………………………………………………………………………………....…..……….…. 4 1.4. Outline of the Dissertation…………………………………………………………………………………………………………………..………..…...… 5 2. Overview and Brief Literature Survey 7 2.1. Random Vibration……………………………………………………………………………………………………………………..………………………………… 7 2.2. Monte Carlo simulation…………………………………………………………..………..……………………………….…………………………………. 12 2.3. Probabilistic Re-analysis…………………………………………………………………………………………………………………..……………….. 17 2.4. Simulation of Offshore Wind Turbines……………………………………………………………………………………………….