Monte Carlo Simulation Approach for Assessment of Imprecise Reliability

A Dissertation

entitled

An Efficient Method to Assess Reliability under Dynamic Stochastic Loads

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

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

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

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

meet the target reliability based on Monte Carlo simulation and Tail-fitting method is presented.

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.

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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……………………………………………………………………………………………….. 18

3. Probabilistic Re-analysis (PRRA) for Efficient Calculation of Reliability 21

3.1. PRRA for Static Problems………………………………………...... 21

3.1.1. Monte Carlo Simulation……………...... 22

3.1.2. Importance Sampling……………...... 26

3.1.3. PRRA ……………...... 29

3.2. PRRA for Random Vibration Problems……………………………………….…………………………………………………..…. 32

3.2.1. Monte Carlo Simulation of Random Processes……………………………….………………… 32

3.2.2. Using PRRA to Estimate Response Attributes…………………………..……………………… 36

3.2.3. Practical Considerations……………...... 39

3.3. Discussion…………...……………………………………………………………………………………………………………………………………………………………. 42

4. Using PRRA to Estimate High-Cycle Fatigue Damage 45

4.1. Approach………………………………………………………………………………….………………………………………………………………………………………... 45

4.2. Application…………………………..………………………………….……………………………………………………………………………………….……....……… 50

4.3. Discussion…………………………………………………………………………………………………………………………………………………………..…………….. 83

5. Using PRRA to Estimate Probability of First Excursion Failure 86

5.1. Statistics of a Narrow-Band Process………………………………………..………………………………………………………….……. 86

5.2. Estimation of Average Up-crossing Rate………………………………………………………..……………………………..……. 90

5.3. Estimation of Probability of Failure………………………………………………………………………………………………..……….. 91

5.4. Application…………………………………….…………………………………..…………………….……………………………………………………………..………. 92

5.4.1. Estimation of Standard Deviation Using PRRA………………………..……………………. 92

5.4.2. Estimation of Average Up-crossing Using PRRA………………………..………….…… 94

5.4.3. Estimating Probability of Failure Using PRRA………………………………….…………….. 97

5.5. Discussion…………………………………………………………..………………………………………………………………………………………………………… 112

6. Integrating Subset Simulation with PRRA to Estimate

the Reliability of Dynamic Systems 114

6.1. Metropolis-Hastings Algorithm…………………………………………………………………………………………..……..……………… 115

6.2. Subset Simulation to Estimate Low Failure Probabilities…………………….……..………………….… 117

6.3. Performing Subset Simulation by Using Shinozuka’s Method……….……………………...... 120

6.4. Integrating PRRA with Subset Simulation ………………………………………………………………………….………. 121

6.5. Application……………………………………………………………………..………………………………………………………………………………………..….. 125

6.6. Discussion…………………………………………...……………………………………………………………………………………………………………………... 153

7. Probabilistic Design of an Offshore Wind Turbine 155

7.1. Model the Loading Environment………………………………………………………………………………………..…………………… 156

7.1.1. Copulas for Modeling Dependence…………………………………………………………………………… 157

7.1.2. Joint Distribution of the Wind and Wave Data

for the Lake Erie Site………………………………………………………………………………………………………………. 158

7.2. Find the Target Probability of Failure…………………………..…….……………………………………………….……………… 169

7.3. Probabilistic Design of the System………………………………………..…………………………………………...…………………. 169

7.3.1. Approach……………………………………………………………………………………………………………………………..…...………… 169

7.3.2. Tail Modeling Method………………………………………………………………………………..……………...………… 171

7.3.3. Monte Carlo Simulation of Extreme 1-year Condition……………………..…….. 174

7.3.4. Importance of Ice Impact in Lake Erie…………………………………………….…………………… 183

7.4. Discussion……………………………………………….……….…………………………………………………………………………………………………………… 187

8. Summary & Conclusion 188

References 195

Appendix 206

List of Tables

3.1 Pf and required number of replications to achieve accuracy……………..…... 26

4.1 Fatigue damage obtained by MCs (1-DOF beam)…………………………………………… 56

4.2 Fatigue damage using PRRA and its comparison

with MCs (1-DOF beam)………………………………………………………………………………………………………… 57

4.3 Pierson-Moskowitz spectra parameters for the

spectra shown in Figure 4-9 ………………………………………………………………………………………………… 59

4.4 Fatigue damage using PRRA and its comparison with

MCs for different spectra (1-DOF beam)……………………………………………..………………….. 60

4.5 Fatigue damage using PRRA and its comparison with

normalized PRRA for different spectra (1-DOF beam)……………………………….. 63

4.6 Properties of the suspension spring………………………………………………………………………………. 64

4.7 Fatigue damage by MCs for the sampling and the true spectra

(quarter car model)……………………………………………………………………………………………………….…………………. 69

4.8 Comparison of MCs with PRRA (quarter car model)………………………….……… 69

4.9 Pierson-Moskowitz spectra parameters for the

spectra shown in Figure 4-19…………………………………………………………………………………………….… 70

4.10 Fatigue damage calculated using PRRA and MCs for

different spectra (quarter car model)…………………………………………………………………………….. 71

4.11 Fatigue damage using PRRA and its comparison with

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normalized PRRA for different spectra (quarter car model)……………………. 73

4.12 Fatigue damage by MCs for the wind turbine example…………………………………. 81

4.13 Fatigue damage using PRRA and comparison with MCs

for wind turbine example………………………………………………………………………………………………………… 81

4.14 Fatigue damage using PRRA and comparison with

normalized PRRA (wind turbine)………………………………………………………………………………….. 82

5.1 Standard deviations of the process and that of its derivative

by MCs for the sampling and the true spectrum…………………………………………… .93

5.2 Standard deviation using PRRA and comparison with MCs………………………..93

5.3 Average up-crossing using PRRA and comparison with MCs……………….….95

5.4 Variances obtained by fitting Eq. 5.1 into results……………………………………………… 95

5.5 Probability of failure by MCs for the sampling and

the true spectrum (1-DOF beam)………………………………………………………………………………………… 98

5.6 Probability of failure using PRRA for the true

spectrum (1-DOF beam)………………………………………………………………………………………………………….…. 98

5.7 Probability of exceeding 650 MPa estimated using PRRA

and MCs for different spectra (1-DOF beam)……………………………………………………. 101

5.8 Probability of exceeding 650 MPa using PRRA and comparison

with normalized PRRA (1-DOF beam)…………………………………………………………….……….. 102

5.9 Probability of failure by MCs for the sampling and the

true spectra (quarter car model)…………………………………………………………………………..……………. 103

5.10 Probability of failure by MCs for the sampling and

the true spectrum (quarter car model)…………………………………………………………….…………. 104

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5.11 Probability of exceeding 650 MPa estimated using PRRA

and MCs for different spectra (quarter car model)………………………………..……………107

5.12 Probability of exceeding 650 MPa using PRRA and comparison

with normalized PRRA (quarter car model)………………………………………………..………… 108

5.13 Probability of failure estimated by MCs for the sampling

and true spectra (wind turbine)……………………………………………………………………….……………….. 110

5.14 Probability of failure estimated using PRRA and comparison

with MCs (wind turbine)……………………………………………………………………….………………………………. 111

6.1 Probability of failure by MCs for 1-DOF system

with 500,000 replications………………………………………………………………………………………………………. 128

6.2 Probability of failure using SS for 1-DOF system………………………………..……..… 130

6.3 Probability of failure using MCs for 1-DOF system for

the true spectrum……………………………………………………………………………………….…………………………………….. 134

6.4 Properties of the suspension spring………………………………………………………………………..………. 140

6.5 Probability of failure by MCs with 500,000 replications………………………..…. 141

6.6 Probability of failure by SS for quarter car system………………………….………………. 144

6.7 Probability of failure by MCs for quarter car system

for the true spectrum………………………………………………………………………………..…………………………………… 146

6.8 Pierson-Moskowitz spectra parameters for the spectra

shown in Figure 6-24……………………………………………………………………………………………………………….….. 150

7.1 Main buoy characteristics………………………………………………………………………………………….……………. 159

7.2 Extreme wind, wave and period (25 years)………………………………………………….…………. 161

7.3 Summary of results for the tower……………………………………………………………..……………………… 181

7.4 Summary of results for the blades……………………………………………………………………..……………. 182

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List of Figures

2-1 Single degree of freedom system……………………………………………………………………………..…………… 8

2-2 Major contributions in random vibrations………………………………………………………………… 12

3-1 Importance sampling……………………………………………………………………………..………………………………………. 28

3-2 Use of PRRA to estimate the expectation of an

attribute of a dynamic system…………………………………………………………………………………………….... 37

3-3 Flowchart of PRRA to estimate the expectation of an

attribute of a dynamic system …………………………………………………………………….………………………. 38

3-4 Support of sampling PDF covers the support of true PDF……………….……….. 39

3-5 Support of sampling PDF does not cover that of true PDF……………………….. 41

3-6 Choosing a proper sampling spectrum………………………………………………………………………… 43

3-7 Combining two spectra to extend the support of sampling spectra…….. 44

4-1 A typical S-N curve……………………………………………………………………………………………………………………….. 47

4-2 A beam structure under random load………………………………………………………………….………… 51

4-3 Sampling and true spectra. The damped natural

frequency of the system is 12.22 rad/sec…………………………………………………………..………. 53

4-4 Sample realization of the exciting force…………………………………………………………….……….. 54

4-5 Sample stress time histories…………………………………………………………………………………..………………. 54

4-6 Goodman equivalent stress histogram (1-DOF beam)………………………..………… 55

4-7 Likelihood ratios for the 1,000 replications…………………………………………………..………… 57

4-8 Fatigue damage as a function of the number of

replications (1-DOF beam)…………………………………………………………………………………………………….. 58

4-9 Sampling spectrum and different true spectra (1-DOF beam)………………. 59

4-10 Fatigue damage calculated using PRRA and MCs

for different spectra (1-DOF beam)……………………………………………………………………………….. 61

4-11 The COV fatigue damage calculated using PRRA and MCs

for different spectra (1-DOF beam)………………………………………………………….……………………. 62

4-12 Quarter car model…………………………………………………………………………………………………………………………….. 63

4-13 Sampling and true PSDs of the road elevation……………………………………………………... 66

4-14 Sample road elevations based on sampling spectrum…………………………………….. 66

4-15 Sample shear stress time history in the suspension spring…………………..…… 67

4-16 Sample shear stress time over entire replication………………………………………………….. 67

4-17 Goodman equivalent stress histogram

(nonlinear quarter car model)………………………………………………………………………………….……………. 68

4-18 Fatigue damage vs. the number of replications

(quarter car model)………………………………………………………………………………………………….……………………… 69

4-19 Sampling spectrum and different true spectra

(quarter car model)…………………………………………………………………………………………….…………………………… 70

4-20 Fatigue damage calculated using PRRA and MCs for

different spectra (quarter car model)……………………………………………………………………………… 71

4-21 The COV fatigue damage calculated using PRRA and MCs

for different spectra (quarter car model)……………………………………………………….……………. 72

4-22 An offshore wind turbine with monopile platform…………………………………….……….. 74

xvi

4-23 Convergence test of the wave elevation methods………………………………………………. 76

4-24 Wave elevation considered for validating

the custom wave kinematics code…………………………………………………………………………………….. 77

4-25 Fore-aft bending moment at the base of tower

by the two approaches………………………………………………………………………………………………………………… 77

4-26 The sampling and the true spectra in the wind turbine example……..…… 79

4-27 Sample wave elevations according to the sampling spectrum……………..… 80

4-28 Sample fore-aft bending moment at the base of tower………………………………… 80

4-29 Fatigue damage as number of simulation increases (wind turbine)…… 82

5-1 Distribution of the peaks for a narrow-band process

(Rayleigh distribution)……………………………………………………………………………………….……………………… 88

5-2 Probability of failure as threshold value……………………………………………………………..……… 89

5-3 Average up-crossing rate by MCs and using PRRA……………………………………….. 96

5-4 Probability of failure by MCs and using PRRA (1-DOF beam)………… 99

5-5 Probability of failure by MCs and using PRRA for 1-DOF beam

(Stress level 650 MPa)………………………………………………………………………………………….…………………. 100

5-6 Probability of failure by MCs and by PRRA (quarter car model)……. 105

5-7 Probability of failure by MCs and by PRRA (stress level 650 MPa)

(quarter car model)…………………………………………………………………………………………..………………………… 106

5-8 Coefficient of variation of the probability of first excursion failure

by MCs and by PRRA (stress level 650 MPa) (quarter car model)… 106

5-9 Sampling and true Pierson-Moskowitz wave spectra ………………………………… 109

5-10 Probability of excursion using PRRA and MCs

xvii

(wind turbine example)……………………………………………………………………………………………..……………. 111

5-11 The COV using PRRA and MCs (wind turbine example) …………………… 112

6-1 First two frequencies causing failure…………………………………………………….…………………… 120

6-2 First two frequencies causing failure (considering symmetry)………….. 121

6-3 Subset-PRRA simulation flowchart…………………………………………………………………………… 124

6-4 Input spectrum as combination of two Pierson-Moskowitz espectra…………….. 126

6-5 (a) Frequency response function (b) Phase angle function…………………… 127

6-6 Values of the first 2 frequencies for Markov chains

(Initial points: blue crosses, remaining points of each chain: red dots)……………………. 129

6-7 Probability of up-crossing a threshold during a 200 second period

(MCs with 500,000 replications vs. SS with 35,000 replications)……. 131

6-8 The COV of up-crossing probability for MCs and SS…………………………..….… 132

6-9 Sampling and true spectra for 1-DOF example………………………………………………… 133

6-10 Probability of first excursion failure…………………………………………………………………………. 135

6-11 The COV by different methods (35,000 rep.) ………………………………….……………… 135

6-12 Probability of first excursion failure by Subset-PRRA versus PRRA

(95% CI by 500,000 MCs)…………………………………………………………………………………………………. 136

6-13 Coefficient of variation by Subset-PRRA versus PRRA

(with 35,000 replications) …………………………………………………………………………………………………….. 137

6-14 Linear quarter car model………………………………………………………………….……………………………………. 138

6-15 a) Frequency response functions, b) Phase angle functions…………….……. 139

6-16 Values of the first 2 frequencies for Markov chains

(Initial points: blue crosses, remaining points of each chain: red dots)………. 143

xviii

6-17 Probability of up-crossing a threshold during a 20 second period…… 144

6-18 Coefficient of variation for SS and MCs………………………………………………………………… 145

6-19 Sampling and true spectra for the linear quarter car example……………… 145

6-20 Probability of first excursion failure……………………………………………………………………………. 147

6-21 The COV for the true and the sampling spectra

using different methods………………………………………………………………………………..………………………… 147

6-22 Probability of first excursion failure by Subset-PRRA versus PRRA

(95% CI by 500,000 MCs)…………………………………………………………………………………………………… 148

6-23 The COV by Subset-PRRA versus PRRA and MCs

(with 40,000 replications)………………………………………………………………………..…………………………… 149

6-24 The different true spectra with the same amount of energy …………...…… 150

6-25 Probability of first excursion for the thresholds of 950 and 1175 MPa

for the spectra shown in Figure 6-24 ……………………………………………………………………… 151

6-26 Coefficient of variation using PRRA, Subset-PRRA and MCs

with 40,000 rep. for the threshold of 950 MPa…………………………………………….…… 152

6-27 Coefficient of variation using PRRA, Subset-PRRA and MCs

with 40,000 rep. for the threshold of 1170 MPa……………………………………………… 152

7-1 Location of the reference buoy in Lake Erie

(Courtesy of Google Earth)………………………………………………………………………………………………… 159

7-2 Scatter diagram of wind speed versus significant wave

height for the data taken in 2002……………………………………………………………………………….…… 160

7-3 Goodness of fit test for wind data

(Red spots represent the data)…………………………………………………………………………………………… 162

xix

7-4 The CDF of a) wind speed b) significant wave height (Hs)………………..… 163

7-5 Copula of wind speed and significant wave height…………………………….………… 165

7-6 The PDF of wind speed and significant wave height………………………..………… 165

7-7 Simulated and observed values of the wind speed and

significant wave height (Hs) for the Lake Erie site…………………………….…………. 167

7-8 The CDF of peak wave period (Tp)…………………………………………………..………………………… 167

7-9 Simulated and observed values of the wave period (Tp) and

significant wave height (Hs) for the Lake Erie site…………………………..…………… 168

7-10 Generalized Pareto distributions for different shape parameters

(Ref. Kim et al., 2006)……………………………………………………………………………………………………………… 172

7-11 Tail modeling…………………………………………………………………………………….……………………………………………… 173

7-12 Simulated values of extreme 1-year triplets of wind speed,

significant wave height and period for the Lake Erie site ……………….……. 175

7-13 Combinations of the wave height, wind speed and wave

period used to perform MCs (40 seeds)………………………………………………..…………………. 176

7-14 The CDF of the fore-aft bending moment at the base of tower ………… 177

7-15 The CDF of the out-of-plane bending moment

at the root of the blades…………………………………………………………………………………………………………… 178

7-16 Pareto distribution fitted to the empirical CDF

(a) wind= 22.7 m/sec, wave= 2.41 m, period= 5.72 sec …………………………… 179 (b) wind= 14.4 m/sec, wave= 2.03 m, period= 4.12 sec

7-17 Ice coverage of the Great Lakes in winter time………………………………..………………. 183

7-18 Ice keel (ref. C-Core report, 2008)…………………………………………………………..…………………… 184

7-19 Schematic of the ice impact model………………………………………………………………………………. 185

7-20 Comparison of the fore-aft bending moment at

the base of tower after ice impact………………………………………………………………………………….. 186

7-21 Tower top displacement after ice impact………………………………………………………………… 186

xxi

List of Abbreviations

Ave……………………………………………………………………………………………………………………...... Average

CDF…………………………………………….……………….. Cumulative Distribution Function

CI ……………………………………………………………………………………………………….Confidence Interval

COV ……………………………………………………………………………………..Coefficient of Variation

FEA…………………………………………………………………………………..….. Finite Element Analysis

MC………………………………………………………………………………………………………………….….. Monte Carlo

MCs……………………………………………………………………………………..… Monte Carlo simulation

MCMC……………………………………………………………..………. Markov Chain Monte Carlo

PDF…………………………………………..……………………………… Probability Density Function

PRRA……………………………………………………………………………..…. .Probabilistic Re-analysis

PSD………………………………………………………………………………………….Power Spectral Density

SS……………………………………………………………………………………………………………Subset Simulation

StDev…………………………………………………………………………Standard Deviation of Mean

Subset-PRRA…………………..… Subset simulation integrated with PRRA

xxii

List of Symbols

……………………………………………………………………………………………………………………………………………………………………………………… correlation factor

…………………………………………………………………………………………………………………………………………………… coefficient of variation

, , ……………………………………………. scale, shape and location parameters in Weibull distribution th k… ……………………………………………………………………………………………………………………….likelihood ratio for k replication

………………………………………………………………………………………………………………………………………………………………………… frequency of maxima   a ……………………………………………………………………………………………………………………….. frequency of upcrossing level a

………………………………………………………………………………………………………………………………………………………………………….. damping ratio

a……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………… alternating stress

 e ……………………………………………………………………………………………………………………………………………………………… equivalent stress

m…… ……………………………………………………………………………………………………………………………………………………………………… .. mean stress

, ……………………………………………scale and shape parameters of generalized Pareto distribution

 ˆ ……………………………………………………………………………………… standard deviation of probability of failure Pf

………………………………………………………………………………………………………………………………………………………………………………phase angles

d …………………………………………………………………………………………………..damped natural frequency of the system

n………………………………………………………………………………………………………………………………………………………………………………………………………………. natural frequency if the system

p ……………………………………………………………………………………………………………………………………………………………. peak frequency

A, B……………………………………………………………………………………………. Pierson-Moskowitz spectrum parameters

a,b …………………………………………………………………………………………………………………………………………………………. Wohler parameters

C………………………………………………………..………………………………………………………………………………………………. damping coefficient

D ……………………………………………………………………….…………………………………………………………………………………..………………………. fatigue damage

D()……………………………………………………………………………………………………………………………………………………. dynamic stiffness matrix

E()……………………………………………………………………………………..……………………………………………. expectation of a random variable

E ………………………………………………………………………………………………………………………………………………………………………… module of elasticity

F(t)…………………………………………………………………………………………………………………..…………………………………………………………. load time history f X x)( ………………………………….………….……….. joint probability density function of random variables X

G…………………………………………………………………………………………………………………………………………………………………………………………… shear modulus G(X)………………………………………………….………………………………………………………………………………………………………… performance function Hs………………………………………………………………………………………………………………………………………………………………….significant wave height

I() …………………………………………………………………………..………………………………………………………………………………………………. indicator function

Kw ……… ……………………………………………………………………………………………………………………………………………………………………………………… …… Wahl factor

N ……………………………………………………………………………………………………………………………………………….. number of replications

Pf ………………………………………`………………………………………… …………………………………………….……probability of failure of a system

xxiii

Pp(a)………………………………………………………………………………………………………………….probability of peak exceeding level a P(Fi|Fi-1)……………………………………………………………………………………………………………………probability of event Fi given Fi-1 ˆ Pf ………………………………………………….………………………..………. estimator of the probability of failure of a system

R() …………………………………………………………….…………………………………………………………………………………………. autocorrelation function

S()………………………………………………………………………………………………………………………………….. power spectral density function

S ………………………………………………………………………………………..…………………………………………………………………………………………………………. stress cycle

Sut ……………………………………………………………………………………………………….………………………………………………………………………………. ultimate stress

Tp………… ……………………………………………………………………………………………………….………………………………………………………………………….. spectral period

W()……………………………………………………………………………………………………..……………………………………………………………. white noise process

(x1, x2, …, xn)……………………………………….sample of n random realizations of random variable X X……………………………………………………………………………………………………………………………………………………….. vector of random variables

xxiv

Chapter 1

Introduction

1.1 Problem Definition and Significance

Different types of mechanical failure may occur due to random vibration. The most common of these is fatigue failure, which is caused by the initiation and the gradual propagation of cracks in a critical region where the highest stresses are observed. In real- life applications, 50 to 90 percent of service failures in machines are due to fatigue

(Stephens et. al, 2000). Fatigue failure is commonly classified as high-cycle fatigue or low-cycle fatigue depending on the number of cycles to reach failure. High-cycle fatigue typically requires more than 105 cycles to failure, where the stress is low and the strain is primarily in the elastic region. In low-cycle fatigue the stress is high enough to have plastic deformation and requires typically less than 104 cycles to observe failure.

However, in most real life applications with variable amplitude loadings, there is not a clear distinction between high-cycle and low-cycle fatigue. In high-cycle fatigue, while the bulk of the material experiences elastic deformation, plastic deformation at the microscopic scale results in global failure. In low-cycle fatigue, plastic deformations occur in macroscopic scales at critical locations such as crack tips, notches or in the areas with high stress concentrations.

Another important mode of failure occurs when the response first crosses a threshold.

This failure mode is called first excursion failure. Other forms of failure due to random vibration may be defined. For instance, failure may occur when a response spends too much of its time out of a limit.

In many engineering designs, there are significant uncertainties in the load models.

For example, there is always uncertainty in the power spectral density (PSD) of the wave loads on an offshore platform or the wind spectrum of a wind farm. Therefore, it is important to evaluate the sensitivity of performance measures, such as the fatigue damage or the probability of first excursion failure, to changes in the PSD of the input. These analyses require Monte Carlo simulation (MCs) in the time domain with thousands of replications, which can be impractical for real-life complex systems.

Development of efficient Probabilistic Re-analysis (PRRA) methods can help engineers to bypass such problems by reducing the required number of simulations.

These methods have been applied successfully to static problems. This dissertation focuses on the configuration and application of PRRA to random vibration problems.

1.2 Dissertation Objective and Scope

Most reliability studies involve static problems in which the system performance is constant with time. Uncertainties are represented by random variables in these problems.

However, the applied loads on real-life systems and their properties vary in time. A car driven on a rough road, a ship sailing in stormy seas and an aircraft flying in bad weather are examples of systems under dynamic excitations. Assessment of the safety of these systems requires representing uncertainties by random (stochastic) processes.

The first objective of this dissertation is to extend the PRRA method from static problems involving random variables to dynamic problems involving stochastic processes. The proposed method enables predicting the response attributes of a dynamic system for different PSDs of the load by performing a single MCs. This work focuses on two types of failure, fatigue failure and first excursion failure. Nevertheless, the intention is not to propose a new model to quantify the high-cycle fatigue damage. The stress-life approach, which is commonly used in high-cycle fatigue, is used to demonstrate the application of the proposed method in the estimation of fatigue damage. However, the approach that is proposed in this dissertation could be applied to other fatigue damage models that exist in the literature.

The second objective of this dissertation is to improve the efficiency of Subset

Simulation (SS) for dynamic systems. Au & Beck (2001) introduced SS to estimate the low probabilities of failure more efficiently than the MC simulations. Subset Simulation is based on the idea that a small failure probability can be calculated as a product of larger conditional probabilities of intermediate events. This method is more efficient than standard MCs because it is much faster to calculate several large probabilities than a single low probability. However, the method is often impractical for random vibration problems because it requires generating the sample values of the excitation at hundreds or thousands of points in time. In this research, the efficiency of SS is improved by utilizing

Shinozuka’s method (1972) to calculate the time series of input loads. Moreover,

Shinozuka’s method enables integrating SS with PRRA to perform the sensitivity reliability analysis of highly reliable systems more efficiently.

As for the third objective of this dissertation, a methodology for the probabilistic design of wind turbine structure under the Great Lakes environment is presented. To perform this, copulas (Nelsen, 2006) are used to build the joint distribution of the wind and the wave data of the Lake Erie site.

Modeling the dependence between wind and wave in an offshore wind farm is very expensive, as it requires large amount of data. Many researchers, similar to the approach presented in the IEC standards (IEC 61400-3 Ed.1), assume that significant wave height follows standard distributions conditional on wind speed. This assumption can lead to considerable errors in the estimated reliability of an offshore wind turbine structure. The proposed approach is more complete because the joint distribution is obtained without making any assumption on the conditional distributions. Then an approach for the probabilistic design of the structure of an offshore wind turbine is presented.

1.3 Dissertation Contributions

The first contribution of this dissertation is to extend PRRA to dynamic stochastic problems. Most researchers generate the time series of the loads by passing a white noise through a filter representing the load spectrum. In order to apply PRRA, the excitation is calculated by using Shinozuka’s method (1972) in which random loads are represented by a series of cosines for which the frequencies are drawn from the spectrum of the load.

This enables converting a dynamic problem to a static problem with the frequencies as random variables.

The second contribution is the integration of Subset Simulation (Au & Beck, 2001) with PRRA. This goal is achieved by performing SS by utilizing Shinozuka’s method

(1972). The commutative property of Shinozuka’s equation assists in reducing the

dimensions of the space of random variables of a dynamic system, thereby reducing the computational cost of SS. Moreover, it enables estimating the probability of failure of highly reliable systems (failure probability less than or equal to 10-3) for different PSDs of the load by reusing the results of a single SS. This new approach will be more efficient than regular PRRA, which is based on MCs.

The third contribution of this dissertation is presenting a methodology for probabilistic design of the structure of an offshore wind turbine. This method is based on building the joint PDF of wind and wave condition using copula. The required load capacity is calculated in order to meet the target structural reliability.

1.4 Outline of the Dissertation

In Chapter 2, an overview and a brief literature survey is presented. In Chapter 3, application of PRRA for static problems is explained, the challenge in applying PRRA to dynamic problems is addressed and the method to resolve it is presented. In Chapter 4, the approach to estimate the high-cycle fatigue damage using PRRA is presented, and the method is illustrated through some examples. In Chapter 5, first the statistics of narrow- band processes is reviewed then the application of PRRA in estimation of the probability of the first excursion failure is presented. Subset Simulation can be used to estimate the probability of failure of highly reliable systems. In Chapter 6, first a new approach to perform SS by utilizing Shinozuka’s method (1972) is demonstrated. Then a methodology to integrate SS with PRRA to estimate the sensitivity reliability of dynamic systems is presented. In Chapter 7, the wind and wave data of a potential offshore wind farm in Lake Erie is reviewed and the loading environment is modeled. Then an approach to the probabilistic design of a structure of a monopile 5 MW offshore wind turbine is

presented. In the last chapter, the summary and the conclusions of the results are presented, and at the end, suggestions for future research are presented.

Chapter 2

Overview and Brief Literature Survey

This chapter is organized into four sections. First, an overview of random vibration is presented. The second section reviews MCs and the major efforts to decrease its computational cost. Then, a relatively new approach called Probabilistic Re-analysis is reviewed. This is the focus of this dissertation. Finally, the chapter discusses major challenges in the simulation of offshore wind turbines.

2.1 Random Vibration

Random vibration and probabilistic engineering mechanics play an important role in the assessment of safety and the reliability of systems. Powerful computers enable analysts to develop realistic models of stochastic systems, and analyze them using commercial software for FEA and multi body systems simulation.

Paez (2006, 2011 and 2012) reviewed the development of random vibration from a historical point of view. Here the history of random vibration is summarized.

The random vibration era began by Einstein (1905,1956), when he developed a mathematical theory to represent random processes and derived the governing equations of the distribution of the motion of a particle suspended in a fluid, which is known as

Brownian motion. Einstein’s work yielded the probabilistic description of the motion of a mass attached via a viscous damper to a fixed boundary excited by white noise. This development can be considered as the first solution to a random vibration problem (Paez,

2012).

In structural dynamics, the analysts are interested in the calculation of the oscillatory response of structures that are modeled with springs and dampers under random loads.

The simplest form of a vibrating structure is a one-degree-of-freedom (1-DOF) system as shown in Figure 2-1. Smoluchowski (1916) and Furth (1917) independently tried to analyze such a structure systematically.

Figure (2-1): Single degree of freedom system

Ornstein (1919) developed the idea of analyzing random vibrations, directly, considering the equation of motion of a single degree of freedom system,

  tFtXKtXCtXM )()()()( (2.1)

In the above equation, M, C, K and F(t) are mass, damping coefficient, the stiffness of the spring and excitation force, respectively. Ornstein’s study built the foundation for the current methods for random vibration analysis.

Uhlenbeck & Ornstein (1930) calculated the moments of the displacement and velocity for a free particle in Brownian motion. They showed that the motion of a free particle follows standard normal distribution. The application of moments in treating Eq.

2.1 was an important innovation as it built a basis for modern solution techniques that are used to analyze random vibration problems.

Wiener (1930) inspired by the work of Schuster (1899, 1900) made an important breakthrough in the mathematical theory of random vibration by introducing the concept of the power spectral density function. This function can only be used to describe stationary random processes whose statistical properties are constant in time. Wiener used the autocorrelation function to define PSD. Autocorrelation and PSD functions form a Fourier transform pair (Newland, 1993). Independently, Khintchine (1934) published a paper in which he defined the power spectral density function.

Crandall (1958) introduced the topic of random vibrations of mechanical systems to practicing engineers. Crandall popularized the fundamental relation of random vibration, in Eq. 2.2, among practitioners (Paez, 2012),

2 yy  SHS xx  )()()( (2.2)

In the above equation, H() is the frequency response function of a linear system, and

Sxx() and Syy() are PSDs of the excitation and the response, respectively.

Wang & Uhlenbeck (1945) extended the theory of Brownian motion that was already developed by Einstein (1905) for a single particle, to a system of coupled oscillating particles. This work appears to be the first to use the fundamental relation of random vibration (Eq. 2.2) in matrix form.

Miles (1954) wrote a paper in which the stress spectrum of a structure subjected to a random loading is determined by the approximation of a single-degree-of-freedom using a cumulative damage rule. This is probably the first study of fatigue due to random vibration (Paez, 2012).

Hurty (1965) described a method to define the deflection of a structure based on the mode shapes of its components. The method was restricted to a linear structure with force-deflection and force-velocity properties. This was significant because it enabled reducing the computational cost of the structural dynamics analysis.

Non-stationary random processes attracted attention in the 1940s when they were required in quantum mechanics investigations. Several approaches for modeling non- stationary random processes were proposed in the following years. Fung (1955) developed a means for characterizing a non-stationary random process in both the time and frequency domains. Priestley (1965, 1967) defined a framework that is considered the fundamental definition of non-stationary random processes. In his first paper,

Priestley (1965) developed the so called “evolutionary power spectra” theory to perform the spectral analysis of non-stationary processes by considering the variation of spectra in time. In the second paper, Priestley (1967) tried to demonstrate the practical application of his theory by explaining the underlying physical principles.

Nikolaidis et al. (1989) and Jan et al. (2003) studied a special type of nonstationary, random excitation, called cyclostationary. The main characteristic of this type of excitation is that its statistical properties (e.g., the RMS) vary periodically in time in contrast to a traditional, random stationary model, which assumes constant statistical properties. Many engineering structures, such as a submarine propeller, a turbine blade,

and an internal combustion engine, are subjected to this type of excitation. A cyclostationary model can yield considerably more accurate estimates of the RMS of the response of a system as compared to a traditional stationary model.

In real life, the physical structures are nonlinear and react randomly to the excitation.

Several researchers tried to develop a general theory for analysis of stochastic structures.

Ghanem & Spanos (1991) presented a technique called stochastic finite elements, which has a good potential for a wide range of applications. A key element is that randomness is represented by random processes or random fields (that is random functions of time and space) instead of random variables. This provides a more realistic representation of randomness in system properties and the applied excitations.

Early treatments of nonlinear random vibrations were not applicable to practical nonlinear structures. Then, it was recognized that nonlinear behavior in many, stiff structures does not actually come from the behavior of the overall structure, but from the localized nonlinear effects such as mechanical joints. In other words, although the main parts of structures behave approximately linearly, only a few components behave nonlinearly. This finding led to practical methods for analysis of nonlinear structures

(Simmermacher et. al, 2005 and Hasselman et al., 2010).

Figure 2-2 summarizes the above review of random vibration.

Figure (2-2): Major contributions in random vibrations (extracted from Paez, 2012)

The response of mechanical systems under random loads is usually studied using

MCs that is reviewd in the next section.

2.2 Monte Carlo simulation

The reliability study of mechanical systems under random loads usually yields the evaluation of an integral over the domain of variables. This integral usually can not be evaluated in closed form. Monte Carlo simulation (MCs) is a general technique to

evaluate such integrals numerically. Metropolis & Ulam (1949) published the first paper on MCs and named it after a famous casino in Monaco. Although the idea of MCs existed before Metropolis & Ulam published their paper (e.g. Hall 1873), they pioneered the application of MCs in treating a complex problem using ENIAC1, the first general- purpose electronic computer.

Monte Carlo simulation is an accurate and robust tool in estimating the reliability of both static and dynamic systems. In static problems, the input variables and the system response are random variables, which are time invariant. However, in dynamic systems, both the input loads and response attributes are time dependent. The additional variability due to the variation in time makes MCs for dynamic problems more expensive than for static problems. In spite of access to powerful computers today, the computational cost of

MCs is high, making it impractical for most real life complex problems with high dimensions. That is why MCs is mostly used for validating the new reliability assessment techniques, rather than for reliability analysis of alternative designs.

Many researches try to improve the efficiency of MCs by reducing the variance of the estimates from MCs. There are several variance reduction techniques in the literature

(Rubinstein & Kroese, 2008). Among them, Importance Sampling (IS) and Conditional

Monte Carlo methods are the most effective methods. Importance Sampling frequently reduces variance dramatically, sometimes in the order of millions (Rubinstein & Kroese,

2008). The first application of IS was in the estimation of probabilities of nuclear particles penetrating shields (Kahn 1950). Other applications of IS are available in the literature, for example, in the reliability assessment of digital communications (Davis

1987, Hahn & Jeruchim 1987), in simulation of flows in porous media (Lu & Zhang,

1 Electronic Numerical Integrator And Computer

2003) or in engineering analysis of stochastic processes (Moy 1965). Hesterberg (1988) reviewed other early applications of IS.

In many real life applications with small failure regions, standard MCs that samples directly from the true distributions is difficult. To generate samples from small failure regions, Markov Chain Monte Carlo (MCMC) is used alternatively. Metropolis et al.

(1953) first used this generic technique while working on the Manhattan project during

WWII. There are many modifications of Metropolis’ original algorithm, most importantly the Metropolis-Hastings algorithm (1970). Gibbs sampling is another prominent MCMC algorithm, which was proposed by Geman brothers (1984), and was named after the physicist J. W. Gibbs.

Au & Beck (1999) presented a MCMC sampling method for evaluating the multidimensional integrals in reliability analysis of structures. Later, Au & Beck (2001) extended this method for calculating very low failure probabilities (less than 10-3) by expressing them as products of conditional probabilities. They called this new method

Subset Simulation (SS). Although SS reduces the computational cost of MCs by several orders of magnitude, it requires a new simulation every time the load changes.

Many researchers have tried to reduce the computation cost of the reliability assessment of dynamic systems by introducing analytical techniques. Most efforts have been focused on the estimation of up-crossing rate and first excursion failure (e.g. Rice

1944, Shinozuka 1964, Veneziano et al., 1977, Schall et al., 1991 and Engelung et al.,

1995). These techniques are not as accurate as MCs because they are based upon simplifying assumptions that impose limitations on their application.

Singh & Mourelatos (2010) proposed an autoregressive, moving average methodology to calculate the cumulative probability of failure of a dynamic system with random properties, driven by an ergodic input random process. They calculate the time- dependent reliability, at a “long” time using an extrapolation procedure of the failure rate.

However, this approach requires hundreds or thousands of random variables to represent the process.

Frequency domain methods are efficient in fatigue reliability analysis. All these methods are based on the determination of rainflow ranges (Matsuishi & Endo, 1968) and fatigue life from the PSD of stress. Bendat (1964) took the first significant step in this area by estimating the fatigue life from the moments of the PSD of stress. For a narrow- band process, the density of peaks is the same as the density of ranges, and the distributions of the peaks follow the Rayleigh distribution. However, Bendat’s method yields conservative fatigue lives for broad-band processes.

Wirsching (1977) tried to improve Bendat’s solution for broad-band processes by introducing an empirical correction factor that was a function of irregularity factor and the S-N curve slope (Stephens et. al, 2000). Lutes (1984) compared different methods and concluded that, contrary to what Wirsching suggested, an analyst may need to use some bandwidth parameter other than the irregularity factor in order to obtain consistent results using the correction factor.

Dirlik (1985) claimed that the rainflow ranges also depend on the first spectral moment of the PSD. He did not propose a correction factor, but instead obtained expressions for the probability density functions of rainflow ranges, using time domain

MC simulations. Dirlik’s solution is a function of the first four moments of the PSD

function. His empirical formula proved to be far superior to the previously available methods that are based on correcting the narrow-band solution. Bishop (1988) used a theoretical approach that slightly improved the accuracy of Dirlik’s empirical approach.

Many comparative studies have been conducted to test the accuracy of different frequency domain methods for different applications. For example, Bishop et al. (1993) took data from an operating wind turbine and calculated the fatigue life using both frequency domain methods and using time domain simulations for different load cases, and compared the results. Their study showed that results from Dirlik’s method agree better with those from time domain simulation, when compared to other methods. For automotive applications, Quigley & Lee (2012) found that Dirlik's method underestimates the fatigue damage for automotive components by up to 30%. However, compared to other frequency domain methods, they found that Dirlik’s method is the most accurate.

Overall, fatigue frequency domain methods are efficient, but they are not as accurate as time domain MC simulations. The main challenge in applying these methods is the calculation of the spectral moments of the PSD of the stresses. For linear systems with known frequency response function, Eq. 2.2 can efficiently calculate the PSD of output, as a function of the PSD of the load. However, for nonlinear systems, it is quite challenging to find the spectral moments of the output with adequate accuracy. This is especially true for the higher degree moments. For this reason, in many cases where high accuracy is required, the time domain MCs is preferred.

2.3 Probabilistic Re-analysis

Probabilistic Re-analysis borrows the idea from IS. In this method, the expectation of a function is estimated for multiple probability distributions that could represent the excitation by re-weighting the results of MCs for the sampling distribution.

Beckman & Mckay (1987) presented two efficient methods that allow an analyst to calculate the mean value of a function of random variables when the analyst changes the probability distributions of the input variables without rerunning the computer codes for

MCs.

Fonseca et al. (2005) used PRRA for optimization problems in which design variables are the mean values of random variables, but they did not call their approach PRRA.

Fonseca et al. (2007) also presented efficient techniques to identify and quantify variability in the parameters from experimental data by maximizing the likelihood of the measurements, using Monte Carlo or perturbation methods for the likelihood computation.

Farizal & Nikolaidis (2007) used the PRRA approach for estimating lower and upper bounds of the probability of failure of a system when the parameters of the probability distributions of the random input variables are known up to their bounds. They demonstrated the approach on a single degree of freedom system with a dynamic vibration absorber with uncertain natural frequencies. They showed that PRRA can estimate the bounds of the probability of failure at one fiftieth of the cost of MCs.

Li & Nikolaidis (2010) used PRRA to evaluate and optimize discrete event systems, such as an assembly line or a call center. They demonstrated the approach on a drilling center and an electronic parts factory.

All above studies deal with static problems in which the random variables, and the response are invariant with time. This study deals with dynamic problems, where the excitation and response are random processes (functions of time).

2.4 Simulation of Offshore Wind Turbines

Offshore wind energy is a vast renewable energy resource with potential to produce a major portion of the global electricity need. In the United States, the potential of offshore wind energy is roughly estimated to be four times as large as the current total electricity production (Musial & Ram, 2010).

Offshore wind turbines have advantages over their onshore counterparts. Firstly, wind is stronger and steadier offshore than onshore. Secondly, large offshore wind farms can be constructed in vast areas with no obstructions. Thirdly, large cities with high population density are located near shore.

On the negative side of offshore wind farms, installation cost is higher and accessibility is more difficult, resulting in higher capital and maintenance costs.

Moreover, the environmental condition at sea is harsher, in general. For example high corrosion, marine growth, wave and ice loads, and the risk of hurricanes, are specific challenges for offshore wind farms. These new challenges make design and construction of offshore wind turbines more complicated, compared to land-based wind turbines.

Offshore wind turbines are operated in stochastic loading environment. Statistical extrapolation of extreme loads is being increasingly used in the design of offshore wind turbines. The International Electrotechnical Commission (IEC) design standard (IEC

61400 ed.1, 2005) has mandated the use of load extrapolation techniques to determine the

long-term loads. This requires performing MCs with thousands of replications (Moriarty,

2008), which can be computationally expensive.

The extrapolation methods are relatively better understood for land-based wind turbines (Moriarty et. al, 2004). However, for offshore turbines this is a more challenging task. The additional randomness due to wave loads adds new dimensions to the MCs integral and it can often become impractical to perform the MCs over the entire domain. Therefore, it is very important to develop efficient alternative extrapolation techniques for offshore wind turbine design.

One challenge in offshore wind energy is modeling the dependence of random processes representing the loading environment. Modeling the dependence between wind and wave is very expensive because it requires large amount of data. In the IEC-61400-3 standard (2009), two approaches are suggested to model the dependence between wind and wave. The first approach assumes that the significant wave height has a normal distribution conditional on the wind speed. The second approach assumes that this distribution is log-normal. Assuming a fixed conditional distribution can lead to considerable errors in the estimated reliability of an offshore wind turbine structure. An alternative method to the above approaches is based on the copula functions (Nelsen,

2006), which is more complete since the joint PDF of the wind and wave is built without making any assumption on the conditional distributions. Nikolaidis & Mourelatos (2010) investigated the effectiveness of copulas for modeling the dependence of random variables and compared it with a MCs using dispersive sampling.

Several studies in recent years have focused on the above issues in the offshore environment and have compared alternative methods to extract turbine extreme loads

(Cheng, 2002).

The environment contour approach is an alternative extrapolation method that is a simplified version of the inverse first-order reliability method (Winterstein et al., 1993).

The former method requires one to search for the design point by only considering environmental states defined on a loading contour associated with a target return period

(Saranyasoontorn & Manuel, 2006). This method is more efficient than a direct MCs method.

Subset Simulation is another method that can reduce the computational cost of a reliability study of wind turbines (e.g. Sichani & Nielsen, 2012). Norouzi & Nikolaidis

(2012) presented a new approach to perform SS more efficiently by utilizing Shinozuka’s method (1972), where the loads applied to the system are represented by PSD functions.

Since both wind and wave loads are represented by PSD functions, this new approach can be used to reduce the cost of the reliability study of offshore wind turbines.

Chapter 3

Probabilistic Re-analysis (PRRA) for Efficient Calculation of Reliability

Overview

In this chapter first the application of PRRA in static problems is reviewed. In section

3.1.1 standard MCs is described, and in section 3.1.2 the IS method is explained. Section

3.1.3 discusses the implementation of PRRA for static problems. Then, in section 3.2, the proposed methods to apply PRRA to dynamic stochastic problems are demonstrated. In section 3.2.1 MCs of random processes is explained, and in section 3.2.2 application of

PRRA to estimate response features in a stochastic problem is described. Section 3.2.3 addresses practical considerations associated with the use of PRRA, and the last section discusses the applicability of PRRA.

3.1 PRRA for Static Problems

This section describes standard MCs. Then it explains IS method, which is the most important technique to reduce the standard deviation of the estimates of MC simulations.

In the last part of this section the application of PRRA to estimate the expectation of a

function for true distribution by re-weighting the results of the MCs of sampling distribution is explained.

3.1.1 Monte Carlo simulation (MCs)

In many problems, the reliability of a system is very close to one. Therefore, it is more convenient to measure the reliability of a system by estimating its probability of failure. The probability of failure is calculated by performing a multi-dimensional integration of the joint PDF over failure region as shown below (Melchers, 1999):

f   X )( dfP xx (3.1) G x 0)(

where fX x)( is the joint probability distribution of vector X. The failure domain can be irregular with multiple disjoint regions. The above formula can be used when the failure region is known. In practice, most of the time, the failure region is unknown, and the probability of failure is calculated by the following equation (Melchers, 1999),

 )(( dfGIP . (3.2) f  X xx0x)

In the above equation GI (  0x)  is called indicator function, and is defined as follows:

 Gif x i  0)(1 I x)(   . (3.3)  0 otherwise

Evaluation of indicator function can be very expensive. Usually comprehensive analysis tools such as FEA software are required to figure out whether or not failure has occurred.

Monte Carlo simulation is a robust tool to calculate reliability for problems to which analytical methods (Eq. 3.2) are not applicable. Monte Carlo is suitable for problems with multiple failure regions or multiple failure modes. In this method, sample values of the random variables are drawn from the PDF of the inputs, and the attribute of interest is calculated for every sample. Then, the expected value of the results is calculated. Monte

Carlo simulation methods are gaining more popularity as computers are becoming computationally more powerful and models of systems are becoming more complex.

A Monte Carlo simulation consists of the following steps:

a) List all the random variables involved in the problem.

b) Generate input values from the PDF function of each variable.

c) Check if the system of interest fails for all the input values in step b.

d) Find relative frequency of failure.

Random variables are drawn until the expected value of accuracy is achieved. The accuracy is usually measured by the standard deviation of the probability of failure. The third step is computationally the most expensive part of a MCs.

To generate the observations of the inputs according to their PDFs, several methods exist (Rubinstein & Kroese, 2008). The most common algorithms are the Acceptance-

Rejection and Inverse-Transform methods. In the first method, sampling values for a PDF function f(x) are generated by using an auxiliary distribution g(x), and the samples are accepted provided that f(x) / cg(x) >U , where c > 1 is an appropriate bound on f(x) / g(x) and U is a random number drawn uniformly from [0,1]. This method is suitable to problems for which sampling from f(x) is difficult and its performance depends on a proper choice of c. The latter method, Inverse-Transform is straightforward and more

common. In this algorithm, a random number uniformly drawn from the range of U [0, 1] is used to extract an observation from a distribution. In order to do that, the inverse of

Cumulative Distribution Function (CDF) is solved for the drawn pseudo number, U, as follows:

 1 UFX )( . (3.4)

Popular programming tools like MATLAB and MATHCAD or compilers like

FORTRAN have built-in functions to generate pseudo random numbers, and can generate observations according to popular distributions such as Normal, Weibull and Gumbel

(1958). In order not to repeat the observations, one should make sure that different random seeds are used to generate observations.

In MCs the probability of failure is approximated by the following equation:

Gn (  0x) Pˆ  . (3.5) f K

The “hat” signifies an estimator of a quantity. In the above equation, K is the total number of trials and Gn (  0x)  denotes the number of trials for which the indicator function becomes unity. The accuracy of Pf increases with the number of trials.

Monte Carlo simulation using Eq. 3.5 is called Standard Monte Carlo simulation or

Crude Monte Carlo simulation since it calculates Pf using samples generated directly from f X x)( . In this research, standard Monte Carlo simulation is called as Monte Carlo simulation or it is abbreviated to MCs.

ˆ It can be proven that the standard deviation of the estimate P f from equation (3.5) is

(Shooman, 1968),

ˆ ˆ f 1(  PP f )  ˆ  . (3.6) Pf K

Usually sampling stops when the coefficient of variation (COV) gets smaller than 0.1.

ˆ The coefficient of variation is defined as the ratio of the standard deviation of the P f to

ˆ the mean of P f . Roughly, in order to halve the standard deviation, the number of replications should be quadrupled.

The following is a general algorithm for MCs to calculate reliability of a system:

a) Generate K trials of the random variables from f X x)(

b) Define the limit state function

c) Calculate the failure indicator function by Eq. 3.3 for every trial

d) Calculate the estimator of Pf using Eq. 3.5.

e) Calculate the standard deviation of Pf by Eq. 3.6.

f) Perform another trial until the required accuracy for Pf is achieved.

The following table shows the number of replications which is required to have the standard deviation of Pf smaller than 10 percent of Pf .

Table (3.1): Pf and required number of replications to achieve accuracy Required rep. Pf to have COV 0.1

1.E-01 1,000 1.E-02 10,000 1.E-03 100,000 1.E-04 1,000,000 1.E-05 10,000,000 1.E-06 100,000,000 1.E-07 1,000,000,000

In practice, most of the time a computationally expensive analysis tool, such as FEA software should be employed to evaluate the indicator function. This makes standard

-3 MCs impractical when the probability of failure is too small (Pf <10 ) as shown in Table

3.1.

3.1.2 Importance Sampling

The idea of IS comes from the fact that in a MCs certain values of the random variables affect the results more than others. If these important values appear more frequently in a sample, then the estimator of the variance could be reduced and consequently the accuracy could be improved. Importance sampling is the most fundamental variance reduction technique which often leads to dramatic variance reduction in particular when estimating low failure probabilities.

In order to estimate the probability of failure the following integral should be evaluated,

T T f  )(( dfGIP xx0x) (3.7)  X

where in the above equation f T x)( is the true PDF of interest. X

Two major steps are involved in IS. The first step is the choice of an appropriate auxiliary distribution which is called sampling distribution. In IS, instead of using the true PDF, the probability of failure for the sampling PDF is estimated by Eq. 3.7. The use of a sampling PDF, f S x)( results in a biased estimation of the variable of the interest. X

Therefore, in the second step, the results are weighted by the likelihood ratio of the true and sampling PDFs to correct the bias (Melchers, 1999) (Eq. 3.8). A good sampling distribution results in many failures with high likelihood.

f T x)( T X S f GIP ( 0x) )( df xx (3.8)  f S x)( X X

The above integral could be approximated by the following equation,

K T S 1 f xk )( ˆ T S X Pf  GI ( k  0)x . (3.9) K  f S xS )( k1 X k

In Eq. 3.9 the observations of random variables are drawn from the sampling distribution instead of the true one.

Then the standard deviation could be estimated by,

K T S 1 f xk )( S X 2 ˆ T 2  ˆT  GI ([ k  0)x  (] PK f ) . (3.10) Pf KK  )1(  f S xS )( k1 X k

In most engineering problems, a designer deals with stress-strength distributions and wants to evaluate the following integral.

 T ),( dSudSSuSfSSuIP (3.11) f  0

Figure 3-1 illustrates both stress and strength distributions. Note that all samples generated out of the interference range are wasted because they do not result in failure.

Figure (3-1): Importance sampling

A good sampling density is the one which covers the interference region as shown in

Figure 3-1. The samples generated by this auxiliary distribution, S SuSf ),( result in failure more frequently than the true distribution, T SuSf ),( .

T T SuSf ),( S f SSuIP  0 ),( dSudSSuSf . (3.12)  S SuSf ),(

The stress and strength distributions are usually independent and the above equation could be discretized as follows,

1 K T S T SufSf S )()( Pˆ T  S SSuI S  0 k k . (3.13) f  k k S S S S K k1 k SufSf k )()(

Then the standard deviation could be estimated by,

K T S T S 1 SS k SufSf k )()( 2 ˆ T 2  ˆT  [ SSuI kk  0  (] PK f ) . (3.14) Pf  S S S S KK  )1( k1 k SufSf k )()(

In practice, the distribution of load or stress is usually fixed therefore the PDF of the stress cancels from the Eqs. (3.13) and (3.14).

The choice of a sampling PDF can significantly affect the efficacy of IS. The optimum sampling PDF for estimating the probability of an event is the true PDF of the random variables truncated in the failure domain, and normalized by the probability of the event. Using this PDF, the unknown probability is calculated by performing a single replication. Similarly, for estimation of the mean value of a function, the optimum sampling PDF is the product of the true PDF times this function, normalized by the mean value of the function. However, both results have only theoretical value because determination of the optimum sampling PDF requires the probability of failure or the mean value of the function, which is to be estimated.

3.1.3 Probabilistic Re-analysis (PRRA)

The idea of PRRA is borrowed from IS and it is used to estimate the expectation of a function instead of the probability of failure. Assume that X is a random vector with r

T random variables, X  21 ,...,, XXX r . Function H is defined as H = g(X), where g can

have closed form equation or be calculated numerically. Note that H is a random variable itself. The expected value of H can be calculated using,

T   T )()())(()( dgfgEHE (3.15) X  X xxx

T where f X (x) is the joint probability distribution of vector X. Often, Eq. 3.15 must be evaluated numerically. In order to estimate accurately ET(H), many sample values of X should be drawn from the true PDF. The required sample size can be very large for complicated functions of many random variables. The efficiency of the above equation is

S improved by sampling from another distribution f X (x) . This idea is similar to IS method.

Importance sampling was originally used to reduce the variance of an estimate from

MCs, while here a similar idea is used to estimate the expectation of a function. Then,

ET(H) can be estimated by using Eq. 3.16,

f T x)( T  gEHE X))(()(  X S )()( dgf xxx . (3.16)  S X f X x)(

T Assume that K independent sample values of X,  21 ,...,, xxx r  are drawn from the

th sampling distribution. The likelihood ratio k for the k sample value is defined as follows,

f T x )( X k . (3.17) k  S f X xk )(

Then an unbiased estimator of the expected value of H, T HE )( for the true PDF of X is,

K  g )(  xkk ˆ T HE )(  k1 . (3.18) K

The “hat” signifies an estimator of a quantity. Then the standard deviation of this estimator is,

K 1 S 2 ˆ T 2  T   x  ()( HEKg ))( . (3.19) ˆ HE )( kk KK  )1( k1

The confidence interval (CI) is calculated as follows,

CI ˆ T )(  ZHE  , ˆ T )(  ZHE  (3.20)  )%1(   2/1 ˆT HE )(  2/1 ˆT HE )( 

where Z1-/2 is the 1-/2 percentile of a standard normal distribution and  is the error percentile. For instance, for a 95% CI, let  =0.05, then Z1-/2 equals 1.96.

Equation 3.20 defines the Wald CI, and it has the following interpretation: If an analyst repeats the simulation, each time calculating a different CI, then 95% of the time the above CI will contain the expectation of H, on average. There are alternative methods to calculate the CI (Brown et al., 2001). However, in this work Eq. 3.20 which is derived by the virtue of Central Limit Theorem (Vining & Kowalski, 2010) is used to calculate the CI.

Once function g(X) is calculated for a sample drawn from the sampling PDF, the expectation of this function can be calculated very efficiently for other PDFs. In order to

calculate the expected value of H for a new PDF, only the likelihood ratio of the new distribution with respect to the sampling PDF needs to be recalculated.

If the support of the sampling PDF contains the support of the true PDF then the average likelihood ratio converges to unity with the number of replications. If the support of the sampling PDF does not contain the support of the true PDF, then the average likelihood ratio will be smaller than one. This will be proven in section 3.2.3.

The PRRA algorithm consists of the following steps:

S a) Select a sampling distributionf X (X )

b) Generate sample values from the sampling distribution

c) Calculate the target function g(X) for the sample from step (b)

d) Find the likelihood ratio for the drawn sample from step (b)

e) Estimate the expectation of the target function for the true distribution by Eq. 3.18.

3.2 PRRA for Random Vibration Problems

In the previous section the application of PRRA to estimate the expectation of a function of random variables was presented. Here, a new approach to extend the application of PRRA to the estimation of some statistic of a stochastic process is proposed .

3.2.1 Monte Carlo simulation of random processes

The algorithm to perform MCs for a stochastic process is similar to the one, which was explained in the section 3.1.1 for static problems except that, for static problems, inputs are random variables whereas for a stochastic process inputs are time series. A random variable does not vary with time whereas a random process or stochastic process

does vary. To conduct a MCs for static problems random variables are drawn using

Acceptance-Rejection or Inverse-Transform methods as explained in section 3.1.1 to generate observations. For a stochastic process, time series of inputs should be generated instead of variables. After the response is calculated, the output of the system will be another time series with different statistics.

There are several methods to generate the time series of a Gaussian process, which are classified in three groups as below:

1. White noise filtration method: In this method, Gaussian white noise is passed

through a filter representing the input spectrum to calculate the time histories.

2. The random phase angle method: In this method, the time series are produced by

the sum of cosine series, where each cosine function has a random frequency and

a random phase angle with a fixed amplitude dependent on the input spectrum.

3. The random coefficient method: In this approach, the realizations are generated

by the summation of a number of sine and cosine functions, where each sine and

each cosine function has random, normally distributed amplitude dependent on

the input spectrum.

The first method is very common, and is widely used by researchers. In this method, first a realization of a white Gaussian noise process W() is generated. Then the time series are calculated by the following equation (Papoulis & Pillai, 2002):

 1 )(  )(2)(   tj deSWtL  . (3.21) 2  

However, in the next section it is shown that the PRRA method can be applied to stochastic processes if the time series are calculated by the second method.

The second and third methods resemble Fourier series. Shinozuka (1972) proposed a method which generates a time history by superimposing a number of cosines. The frequencies of these cosines are drawn from a PDF that is equal to the PSD of the random process normalized by the area under it,

S )( f )(  (3.22)  2 where S )( is the PSD of the random process, and  is the square root of the area under one-sided PSD function, S )( ,

    )( dS  . (3.23) 0

It can be shown that if a process L(t) is stationary and Gaussian with zero mean, then its sample time history can be approximated as follows,

N 2 tL )(  t  )(cos (3.24)  N ii i1

where  is a random phase angle which is uniformly drawn from [0, 2]. Frequencies,i

are distributed according to f )( in Eq. 3.22.

The required number of harmonics in Eq. 3.24 depends on the shape of the PSD. In calculation of low failure probabilities due to first excursion, the failure threshold is an

important consideration too. For narrowband processes, 10 terms are usually sufficient for an accurate representation of the random process. It is recommended to test the sensitivity of the estimates with respect to the number of frequencies in order to determine the required number of terms in Eq. 3.24.

The PSD is calculated from observed records by estimating the autocorrelation function first and then taking its Fourier transform, or by directly estimating the PSD from these observations. In the first approach, the autocorrelation function is estimated first by,

1 n Rˆ  )(   tLtL )()( . (3.25)  i n i1

In Eq. 3.25, ti are equally spaced values of time. Then the PSD is estimated by taking the Fourier transform of the autocorrelation function,

 1 Sˆ )(  Rˆ  )(cos)( dt . (3.26)   0

By calculating the above equation for many signals and finding the average value, this value converges to the PSD function.

In summary MCs for a stochastic process consists of the following steps:

a) Draw a sample of values of the random variables from the input PDFs.

b) Generate time histories of the input loads using Shinozuka’s method.

c) Using a simulation tool calculate the response of the system.

d) Using the response time histories estimate the desired attribute of the response.

Similar to static MCs the most expensive step is the third one, in which an analysis tool such as FEA software should be used to calculate the response of the system.

3.2.2 Using PRRA to estimate response attributes

In section 3.1.3 the application of PRRA to estimate the expectation of a function of random variables was explained. Here, the use of PRRA to estimate some statistic of a stochastic process during a period is demonstrated. The system excitations, and the resulting response, are stochastic processes. Frequencies and phase angles are drawn from the input PDFs, and they are used to generate time histories of the input loads by

Shinozuka’s method. Using Shinozuka’s method (1972) enables describing a stochastic process with random variables. The response of the system is another stochastic process that can be calculated by using an appropriate simulation tool such as commercial FEA software. Figure 3-2 summarizes the method for calculating a statistic of a random process using simulation.

To perform MCs all four steps shown in Figure 3-2 should be taken. The third step is the most expensive. As the figure shows, PRRA helps the analyst to bypass this step thereby reducing the computational effort.

Step 1: Draw a sample of values of the random variables from the input PDFs.

Step 2: Generate time histories of the input loads using Shinozuka’s method.

Step 3: Using a simulation tool calculate the PRRA response of the system.

Step 4: Using the response time histories estimate the desired attribute of the response.

Figure (3-2): Use of PRRA to estimate the expectation of an attribute of a dynamic system

The frequencies in Eq. 3.24 are the random variables used in the Eq. 3.17 to calculate the likelihood ratio for every replication of a simulation. In Eq. 3.17, the numerator and the denominator are the joint PDF of the true and the sampling distributions, respectively.

Since the frequencies are extracted independently from the PDF of the sampling distribution, Eq. 3.17 becomes,

N f T  )(   i . (3.27) k  S i1 f i )(

The PDF of the phase angles does not appear in the above equation because it remains unchanged for the sampling and the true simulations. Since in the definition of the likelihood ratio the PDF functions are used, in order to use PRRA the sampling and true spectra should contain the same amount of energy.

Load realizations are generated using Shinozuka’s method, and the corresponding time histories at a critical point of the structure are calculated by a simulation tool such as a FEA code. The next step is to estimate a response attribute of interest for every replication. It should be noted that not all types of response attributes can be estimated using PRRA. Only those response attributes, which can be averaged over different replications, can be estimated using PRRA. For instance up-crossing rate, autocorrelation function, standard deviation, cumulative fatigue damage or probability of failure can be estimated using PRRA for the true spectrum by using the results that have been already obtained for the sampling spectrum. Figure 3-3 demonstrates the method on a stochastic process.

Step: MCs1, PRRA1 Draw sample values of the frequencies and phase angles from the sampling PDFs.

Step: MCs2 Step: PRRA2 Generate time histories of the Calculate likelihood ratio input loads using Shinozuka’s method.

Step: MCs3 Monte Carlo simulation simulation Carlo Monte

Use a simulation tool to calculate PRRA the response of the system.

Step: PRRA3 Step: MCs4 Find the desired attribute for the Use the response time histories to true distribution by multiplying estimate the desired attribute of likelihood ratio by the response the response. of the sampling spectrum

Figure (3-3): Flowchart of PRRA to estimate the expectation of an attribute of a dynamic system

Simulation continues until the desired standard deviation or the desired CI is achieved.

3.2.3 Practical considerations

The mean value and standard deviation of the likelihood ratio can guide an analyst in selecting a suitable sampling PDF. These statistics can be calculated at minimal cost because this does not require computation of the response of the system. Here the mean value of the likelihood ratio in the special case where only one frequency is used in

Shinozuka’s method (1972) is calculated. It is easy to show that the same results for the mean value of the likelihood ratio are obtained for any number of frequencies.

Let f T )( be the PDF of the true distribution and f S )( be the sampling distribution of the frequencies as shown in Figure 3-4. In this case, the support of the true distribution is a subset of that of the sampling PDF.

Figure (3-4): Support of sampling PDF covers the support of true PDF.

For K replications, an unbiased estimator of the mean value of the likelihood ratio is,

1 K 1 K f T  )( ˆ  k (3.28) k  S K k1 K k1 f k )(

th th where  k is the frequency in the k replication. The mean value of the k term in the sum in the above equation is,

 T f )( S E k )(  )( df  . (3.29) 0 f S )(

Looking at Figure 3-4, the above equation is simplified to,

 3 T E k  df   1)()(   ,...,1[ Kk ] . (3.30)  2

The rightmost hand side of the above equation is the area under the true PDF, which is one. Therefore, the mean value of the likelihood ratio in Eq. 3.30 is one. In practical terms, this means that the average likelihood ratio should converge to unity with the number of replications, if the support of the sampling PDF contains the support of the true PDF.

A large value of the standard deviation of the likelihood ratio (normally higher than

10% of the average likelihood ratio) usually suggests an inaccurate estimator.

Now consider the case where the sampling PDF does not cover the true PDF as shown in Figure 3-5. The integration in Eq. 3.29 is still performed over the support of the sampling PDF, but Eq. 3.30 yields an area under a portion of the true PDF that is less than one. Therefore, even if an unlimited number of replications are performed, the average likelihood ratio would still be less than one.

Figure (3-5): Support of sampling PDF does not cover that of true PDF.

Normally a mean value of the likelihood ratio that differs significantly from one indicates a biased estimate. However, if the dependent function whose mean value is to be estimated is non-zero only over a narrow region of the sample space, then a sampling

PDF that covers only this region is used. Otherwise, many sample values for which the likelihood ratio is known to be zero are wasted. In this case, the average likelihood ratio differs from one, and the results of PRRA could still be accurate.

The deviation of the average likelihood ratio from unity indicates a biased estimation.

In order to improve the estimate, Fonseca et al. (2007), and Ridgeway and Madigan

(2003) suggest normalizing the estimator in Eq. 3.18 by the average likelihood ratio.

However, this could reduce accuracy if the sampling PDF does not cover the true PDF, as mentioned above. Moreover, if the estimator of the mean value in Eq. 3.18 is normalized by the average likelihood ratio a biased estimator is calculated. This will be demonstrated in the examples in the next chapters.

3.3 Discussion

A methodology to extend the application of PRRA from static problems to random vibration problems was proposed in this chapter. This method estimates mean values of attributes, such as the probability of first excursion failure, whose estimators are averages over multiple response time histories.

This approach can not be applied to deterministic loads for which time histories are defined with fixed frequencies. In order to use PRRA, the input loads should be

Gaussian random processes that are represented by their PSD functions.

It is important to note that in order to use PRRA, the PSD functions of the sampling and true excitations should contain the same amount of energy. That is, the areas under all PSD functions should be identical.

The required number of harmonics in Shinozuka equation (Eq. 3.24) depends on the shape of the PSD function. For narrow-band processes, considering 10 frequencies is usually sufficient; however, for broad-band processes more harmonics could be required.

In the applications that are considered, PRRA can yield accurate results for up to 20 frequencies. An analyst should perform a sensitivity study to find the required number of harmonics in Eq. 3.24 before performing any simulation.

Adding new terms to Sinozuka’s equation, causes excessive variability of the likelihood ratio, which leads to a deterioration in accuracy. Despite the lack of adequate accuracy, the result of PRRA would be still useful because it can qualitatively predict the sensitivity of response to the input loads. If the user of the method can perform a sufficiently large number of replications, then the user can still estimate the response attributes for the true PDF.

The efficacy of PRRA depends on the proper selection of the sampling spectrum. The selection depends upon three factors: first on the natural frequencies of the system, second on the range of the PSDs of the excitation, and third on the statistics of the likelihood ratio. A suitable spectrum for PRRA should cause the system to vibrate significantly. Therefore, among all load spectra, the analyst should select the one with the highest energy near to the natural frequencies of the system (Figure 3-6).

Before finalizing the choice of the sampling PSD, the designer should consider the statistics of the likelihood ratio. This can be calculated very efficiently from Eq. 3.28. A sampling spectrum that yields an average likelihood ratio close to unity and a COV close to 10% for most input spectra, should yield accurate results.

Figure (3-6): Choosing a proper sampling spectrum

As discussed in section 3.2, it is important that the support of the sampling spectrum contain that of the true spectrum. In order to increase the coverage of the support of the sampling spectrum, one can use a combination of two (or more) sampling spectra as shown in Figure 3.7. In this approach, 50% of the time the frequencies are drawn from the first sampling spectrum, and 50% of the time from the second. The resultant spectrum

is shown in bold in Figure 3.7. Note that the resultant spectrum is not necessarily from the same family of spectra as the original spectra.

Figure (3-7): Combining two spectra to extend the support of sampling spectra

The formulation of PRRA that is presented in this work considers a single excitation load. This method can be applied to the systems with two (or more) input loads. To do that, Eq. 3.27 should include the corresponding PDFs of all input loads. However, considering more loads increases the variability of likelihood ratio and results in the deterioration in accuracy.

Chapters 4 and 5 present the application of PRRA to the estimation of fatigue damage and the probability of first excursion failure, respectively.

Chapter 4

Using PRRA to Estimate High-Cycle Fatigue Damage

4.1 Approach

In Chapter 3 the use of PRRA to estimate some statistic of a stochastic process during a period was presented. This chapter focuses on the application of PRRA to estimate high-cycle fatigue damage. It is widely accepted by practicing design engineers that in high-cycle fatigue, stress is low and the deformation is primarily in the elastic region. In reality there is not a clear distinction between high-cycle and low-cycle fatigue, since in typical variable amplitude loadings occasionally overloads causing inelastic deformations are intermixed with many low amplitude loads causing elastic deformations. Plastic deformations under cyclic loads can occur in macroscopic scale such as at a notch or crack tip area, or at microscopic localized region, such as near metallurgical dislocations.

Metallurgical discontinuities can increase local stress or strain and cause inelastic deformation. The scale of inelastic deformation is very small and localized. Therefore, in so called high-cycle fatigue, while the bulk of the material experiences elastic deformation, plastic deformation at microscopic scale can lead to global failure. For additional readings on this issue, refer to Stephens et al., (2000).

The intention of this chapter is not to present a new approach to quantify high-cycle fatigue damage, but instead to demonstrate the use of PRRA in estimation of fatigue damage according to the stress-life method that is commonly used in high-cycle fatigue.

Nevertheless, the approach that is presented in this chapter is a mathematical method that can be applied to other fatigue damage models that exist in the literature (Fatemi & Yang,

1998) to quantify low-cycle or high-cycle fatigue damage.

Figure 3-2 summarized the application of PRRA for calculating a statistic of a random process. The interesting attribute in a high-cycle fatigue analysis can be fatigue damage. Load realizations are generated using Shinozuka’s method (1972), and the corresponding stress time histories at a critical point of the structure are calculated by a simulation tool such as a FEA code. The next step is to estimate the fatigue damage for every replication.

Numerous tests have established that under constant amplitude loading and in non- aggressive environment, ferrous materials have an endurance limit. The endurance limit of a material is defined as the highest level of alternating stress that can be withstood indefinitely without failure (Shigley et al., 2003). For the stresses higher than the endurance limit, the fatigue damage is quantified by using Basquin’s equation,

1 S  D  )( b (4.1) a where a and b are Wohler parameters, which depend on material properties, and S is the alternating stress for a given cycle. Wohler parameters are estimated from the stress-life curve known as S-N curve as shown in Figure 4-1. The endurance limit appears as a knee

in a log-log coordinate. The fatigue damage is neglected for the alternating stresses less than the endurance limit.

Figure (4-1): A typical S-N curve

For a non-zero mean stress Goodman’s relation, Eq. 4.2 can be used to find the equivalent fully reversed stress (Stephens et al., 2000) ,

   a . (4.2) e  1 m Sut

In the above equation,  a and  m are the alternating and mean stresses, respectively,

Sut is the ultimate stress, and  e is Goodman equivalent stress. According to Eq. 4.2, for a given level of alternating stress when the tensile mean stress increases the fatigue life decreases.

For a narrow-band process, the peaks and the valleys, and consequently, the corresponding stress cycles, are clearly distinguishable. However, for the broad-band

case, a cycle counting method should be employed in order to find the effective cycles, which are needed to evaluate Eq. 4.1.

Researchers originally developed cycle counting techniques to identify hysteresis loops in uniaxial loading. The most popular and probably the best method of cycle counting is rainflow cycle method that was first proposed by Matsuishi & Endo (1968).

The rainflow method properly accounts for the sequence of cycles by identifying closed hysteresis loops in a single channel stress-strain response. However, for out-of-phase multiaxial loading the definition of a hysteresis loop is not as clear as it is for uniaxial loading and many investigators try to find a proper cycle counting method for the multiaxial loading with multiple channels (e.g. Langlais et al., 2003). Despite the above fact, in this study the rainflow cycle counting algorithm developed by Downing and Socie

(1982) is used for counting the cycles.

After counting the effective cycles, using the linear damage accumulation model known as Palmgren-Miner rule, the total damage for a stress history is calculated as follows,

n 1 S  D  (i ) b . (4.3)  a i1

Failure occurs once the total damage becomes unity (Palmgren 1924, Miner 1945).

The linear damage rule has at least three deficiencies. First, it does not account for loading sequence effect. Second, it is independent from load-level, and third it lacks accounting for load-interaction effect. For the sake of the simplicity of the linear rule, it is

widely used by practicing engineers. There are other cumulative fatigue damage models in the literature (Fatemi & Yang, 1998), but in this study, the linear damage rule is used.

For a structure under random loads, multiple realizations of the loads are generated and the resulting system response is calculated. Then the total damage for every stress time history is calculated, and the expected damage is estimated as shown below,

1 K ˆ DE )(  D . (4.4)  k K k1

In the above equation, K is the total number of replications. Simulation stops when a desirable level of accuracy is achieved.

The standard deviation of the mean damage is calculated as follows,

1 K S  2  [ ˆ DEKD )]( 2 . (4.5) ˆ DE )(  k KK  )1( k1

Consider that MCs is performed for a sampling PDF of the frequencies f S )( , obtained from Eq. 3.22, and the total damage is estimated for every realization. Then, total damage for every replication of the true load PDF f T )( is calculated using,

T S k   DD kk  ..2,1 Kk (4.6)

where k is the likelihood ratio defined in Eq. 3.27. The frequencies in Eq. 3.24 are the random variables used in the Eq. 3.27 to calculate the likelihood ratio for every replication of

a simulation. In Eq. 3.27, the numerator and the denominator are the PDFs of the true and the sampling spectra, respectively.

The total damage for the true PSD is,

1 K ˆ T DE )(   D S (4.7)  kk K k1 where K is the number of replications. The standard deviation of the mean fatigue damage can be estimated by using Eq. 4.8 as follows,

K 1 2    S  [ ˆ T DEKD )]( 2 . (4.8) ˆT HE )( kk KK  )1( k1

The CI is calculated by Eq. 3.20.

4.2 Application

This section demonstrates the method on three examples. The first involves a beam structure under a stochastic, time varying load. In the second example, the fatigue damage in the suspension spring of a quarter car model is studied. Finally, the method is used to estimate the fatigue damage in an offshore wind turbine. In all three examples, the fatigue damage for several PSD functions that could represent the load is estimated.

First the fatigue damage for a sampling distribution is calculated using MCs, and then the same damage for each admissible PSD is estimated using PRRA.

Example 1: Fatigue damage in a linear, 1-DOF beam

A linear one degree of freedom system is studied here (Figure 4-2). Random load,

F(t), is applied to the concentrated mass located at mid-span of the beam made of 4340

quenched and tempered steel (E = 210 GPa ,  = 7850 Kg/m3). The mass of the beam is negligible compared to the concentrated mass (M = 100 Kg); therefore this system is modeled as a single degree of freedom system. The equation of motion for this system is,

  tFyKyCyM )( (4.9) where in the above equation spring rate is,

6 IE K  . (4.10) L 3

Figure (4-2): A beam structure under random load

Equation 4.9 is solved numerically in the time domain by using the state-space approach (Inman, 2001).

The un-damped and damped natural frequencies of the system are,

6 IE n  LM 3 . (4.11)

2 nd 1 

In the above equation, I is the moment of inertia of the beam, and  is the damping ratio, which is assumed equal to 0.3. The un-damped and damped natural frequencies of the above system are 12.81 rad/sec and 12.22 rad/sec, respectively.

The force, F(t), has a Pierson-Moskowitz spectrum. Samples of frequencies for the same family of spectra are drawn. This spectrum is described by two parameters A and B as follows,

B A 4 S )(  e  . (4.12)  5

The power spectral period or the peak frequency (p) of a Pierson-Moskowitz spectrum depends only on B,

4 25.0   ( B ) . (4.13) p 5

The energy of a PSD is measured by the area under its curve. For a Pierson-

Moskowitz spectrum, the area under the spectrum is calculated by,

A Area  . (4.14) 4 B

If the PSD of the excitation is specified instead of its PDF, then the area under the sampling and the true spectra should be the same in order to use PRRA.

Parameters A and B for the sampling spectrum are selected such that the peak frequency is close to the damped natural frequency of the system. For the sampling distribution, these parameters are considered to be 5109 N 2sec-4 and 2.7861104 sec-4,

respectively. This yields a peak frequency of about 12.22 rad/sec for the sampling PSD.

Assume that for the true spectrum A and B are 6.5109 N 2sec-4 and 3.6219104 sec-4 with a peak frequency of 13.05 rad/sec (Figure 4-3). Note that the sampling and true spectra contain the same energy. Since the sampling spectrum peak is closer to the damped natural frequency of the system, more damage is expected to be inflicted to the system for the sampling than the true spectrum.

Figure (4-3): Sampling and true spectra. The damped natural frequency of the system is 12.22 rad/sec.

Realizations of the force time history are calculated by Shinozuka’s method. Then the corresponding stress time histories at the critical element of the beam located at the extreme fiber of the clamped end of the beam are calculated. There is only a fluctuating uniaxial normal stress due to the shear force that is applied to the middle mass. Two sample time histories for the random load are shown in Figure 4-4.

Figure (4-4): Sample realization of the exciting force

The displacement of the middle mass does not exceed the 10% of the beam length; therefore the stress at the base of the beam can be calculated by using the mechanics of material formulas.

There is a mean stress resulting from the weight of the middle mass (M) ( m  235.4

MPa), which is added to the alternating stress due to the random load. Two sample stress histories corresponding to the loads shown in Figure 4.4 are shown in Figure 4.5.

Figure (4-5): Sample stress time histories

Effective fatigue cycles are calculated using the rain flow counting method, and then equivalent Goodman stress is calculated by Eq. 4.2. In Figure 4.6, the histogram of

Goodman equivalent stress for one sample replication from the sampling spectrum is shown.

Figure (4-6): Goodman equivalent stress histogram (1-DOF beam)

Fatigue damage is evaluated for every effective cycle by Eq. 4.1, where a is the ultimate stress of 4340 quenched and tempered steel (1,240 MPa) and b = -0.06. No endurance limit is considered for this example. In other words, it is assumed that the sloping line in the S-N curve continues until it intersects the horizontal axis. This might result in overestimating the fatigue damage, but does not have any effect on the applicability of PRRA.

The total fatigue damage is estimated by Palmgren-Miner rule (Eq. 4.3). In order to check the accuracy of PRRA, the beam structure was analyzed for both the true and sampling spectra for 250, 500 and 1,000 replications. Each simulation was performed for

3,600 seconds. The total fatigue damage for the sampling and the true PDFs using MCs

are shown in Table 4.1. The standard deviation of the mean fatigue damage decreases with the number of replications. Roughly, in order to halve the standard deviation, the number of replications should be quadrupled.

Table (4.1): Fatigue damage obtained by MCs (1-DOF beam) Sampling True spec.* spec.* Ave 2.01E-04 8.85E-05 1,000 rep. StDev 1.27E-05 6.83E-06 Ave 2.10E-04 8.40E-05 500 rep. StDev 1.98E-05 1.00E-05 Ave 2.14E-04 9.08E-05 250 rep. StDev 2.74E-05 1.72E-05 * 6 hours computational time for each MCs

In order to produce the time histories of the force, 10 frequencies were drawn from the sampling spectrum. Since the frequencies are independent, the likelihood ratio for every realization is calculated by,

10 S T  )( 10 f T  )(   i  i (4.15) k S S ii1 S i )( 1 f i )(

T S where S i )( and S i )( are the true and the sampling spectra , respectively and i are the frequencies drawn when performing MCs for the sampling distribution. In Figure

4-7, the likelihood ratios for 1,000 replications are shown.

Figure (4-7): Likelihood ratios for the 1,000 replications

For every time history generated from the sampling spectrum, the fatigue damage is calculated. Then it is scaled by the likelihood ratio in Eq. 4.6. Finally, the average fatigue damage with 250, 500 and 1,000 replications is calculated by Eq. 4.7. The results obtained using PRRA are compared with those from MCs in Table 4.2.

Table (4.2): Fatigue damage using PRRA and its comparison with MCs (1-DOF beam) Likelihood 95 % PRRA* COV ratio Confidence Interval Ave 0.9760 8.62E-05 1,000 rep. 0.05 [7.77E-05,9.47E-05] StDev 0.0269 4.35E-06 Ave 0.9822 8.44E-05 500 rep. 0.07 [7.22E-05,9.66E-05] StDev 0.0389 6.23E-06 Ave 0.9169 9.54E-05 250 rep. 0.11 [7.52E-05,1.16E-04] StDev 0.0580 1.03E-05 * Almost no additional computational cost for PRRA

The estimates of the mean damage from PRRA agree well with those from MCs. In

Figure 4-8, the solid horizontal line shows the estimate from standard MCs with 1,000

replications, which is considered as a reference. The dashed lines show the 95% CIs from

PRRA as a function of the number of replications. The accuracy of PRRA improves with the number of replications, as expected. This is because the standard deviation of the mean fatigue damage decreases, and so does the width of the 95% CI. Moreover, the average likelihood ratio converges to unity and its standard deviation decreases with the number of replications.

Figure (4-8): Fatigue damage as a function of the number of replications (1-DOF beam)

Performing MCs with 1,000 replications in series for the true spectrum by a desktop computer took six hours (real time), whereas the fatigue damage was calculated almost instantaneously using PRRA. The superior efficiency of PRRA enables a designer to examine the sensitivity of the fatigue damage to changes in the load spectrum, at practically no cost.

The accuracy of PRRA deteriorates when the true spectrum deviates significantly from the sampling distribution. To study the efficiency of the PRRA, simulations for other input spectra shown in Figure 4-9 are performed.

Figure (4-9): Sampling spectrum and different true spectra (1-DOF beam)

All the true spectra shown in Figure 4-9 are Pierson-Moskowitz according to Eq. 4.12 with different coefficients as listed in the Table 4.3. The first two spectra are shifted toward the left of the sampling spectrum, and the remaining five spectra are shifted to the right. All spectra contain the same amount of energy.

Table (4.3): Pierson-Moskowitz spectra parameters for the spectra shown in Figure 4-9 A B (KN) 2sec -4 sec -4 sampling spec. 5 2.7861104 true spec. 1 2.5 1.3931104 true spec. 2 3.5 1.9503104 true spec. 3 6.5 3.6219104 true spec. 4 7.5 4.1792104 true spec. 5 9 5.0150104 true spec. 6 12.5 6.9653104 true spec. 7 15 8.3583104

Table 4.4 shows the results obtained using PRRA, and compares them to those from

MCs with 1,000 replications. It also shows the COV of the estimates from PRRA and those from MCs.

Table (4.4): Fatigue damage using PRRA and its comparison with MCs for different spectra (1-DOF beam) Likelihood COV COV PRRA * MCs * ratio (MCs) (PRRA) Ave ------2.0126E-04 ---- sampling spec. 0.06 StDev ------1.2682E-05 ---- Ave 0.7396 4.2181E-04 5.8793E-04 true spec. 1 0.04 0.25 StDev 0.0934 1.0338E-04 2.3059E-05 Ave 0.9969 4.1825E-04 4.3059E-04 true spec. 2 0.05 0.11 StDev 0.0492 4.5247E-05 2.2375E-05 Ave 0.9760 8.6178E-05 8.8468E-05 true spec. 3 0.08 0.05 StDev 0.0269 4.3490E-06 6.8319E-06 Ave 0.9700 4.9269E-05 5.0683E-05 true spec. 4 0.09 0.05 StDev 0.0480 2.6287E-06 4.4431E-06 Ave 0.9779 2.1676E-05 2.0568E-05 true spec. 5 0.11 0.06 StDev 0.0956 1.2254E-06 2.2396E-06 Ave 1.0932 3.4764E-06 3.1552E-06 true spec. 6 0.11 0.08 StDev 0.3157 2.6803E-07 3.3567E-07 Ave 1.2036 1.0182E-06 7.3313E-07 true spec. 7 0.10 0.10 StDev 0.5205 1.0394E-07 7.4922E-08 * 6 hours computational time for each MCs and no additional cost for PRRA

The results of PRRA and those of MCs for the second to the sixth spectra agree well.

The sixth case is interesting because, although the standard deviation of the likelihood ratio exceeds 30% of the mean, the results of PRRA show the COV of 8%. This is probably due to luck. Usually, when the standard deviation of the average likelihood ratio exceeds 10% or when the average likelihood ratio deviates significantly from unity,

PRRA becomes unreliable. By moving farther from the sampling spectrum, the average likelihood ratio deviates significantly from unity and its standard deviation increases.

The results are plotted in Figure 4-10. By moving toward the left the CI becomes rapidly wider compared to the right. The reason is that by moving toward the left the support of the true spectrum departs from that of the sampling.

Figure (4-10): Fatigue damage calculated using PRRA and MCs for different spectra (1-DOF beam)

The COV of the results using PRRA is greater than those using MCs, but for the third to seventh spectra MCs has larger COV as shown in Figure 4-11. The sampling spectrum compared to the first and the second spectra is farther from the natural frequency of the system and that is why the accuracy of PRRA is lower than MCs. However, compared to the remaining spectra sampling spectrum is closer to the natural frequency and consequently it causes more damage to the structure. This improves the accuracy of

PRRA compared to MCs.

Figure (4-11): The COV fatigue damage calculated using PRRA and MCs for different spectra (1-DOF beam)

Fonseca et al. (2007), and Ridgeway & Madigan (2003) suggest normalizing an estimator by the average likelihood ratio. The results of PRRA in Table 4.4 after normalization are compared with those of original PRRA in Table 4.5.

In this example, with the exception of the fifth spectrum, normalizing the results of

PRRA improves the accuracy. For the fifth spectrum, normalization deteriorates the accuracy of the estimates from PRRA. The error of the result for the first spectrum reduces significantly after normalizing; however, the result is still unacceptable since the standard deviation is too large. The next two examples show and explain that normalizing the estimates of PRRA by the average likelihood ratio does not always improve accuracy.

Moreover, PRRA could yield accurate results even when the average likelihood ratio differs significantly from one.

Table (4.5): Fatigue damage using PRRA and comparison with normalized PRRA (1-DOF beam) normalized PRRA error error PRRA Ave 4.2181E-04 5.7035E-04 true spec. 1 28.3% 3.0% StDev 1.0338E-04 1.3978E-04 Ave 4.1825E-04 4.1955E-04 true spec. 2 2.9% 2.6% StDev 4.5247E-05 4.5388E-05 Ave 8.6178E-05 8.8295E-05 true spec. 3 2.6% 0.2% StDev 4.3490E-06 4.4558E-06 Ave 5.0683E-05 5.0793E-05 true spec. 4 2.8% 0.2% StDev 4.4431E-06 2.7100E-06 Ave 2.0568E-05 2.2166E-05 true spec. 5 5.4% 7.8% StDev 2.2396E-06 1.2531E-06 Ave 3.4764E-06 3.1801E-06 true spec. 6 10.2% 0.8% StDev 2.6803E-07 2.4518E-07 Ave 1.0182E-06 8.4600E-07 true spec. 7 38.9% 15.4% StDev 1.0394E-07 8.6357E-08

Example 2: Fatigue damage in a nonlinear quarter car model

In this example, fatigue damage to a suspension spring in a quarter car model is studied (Figure 4-12). The model is excited by road elevation, u(t), which is random.

In this model, masses m1 and m2 are 75 and 300 Kg, respectively. Damping coefficients are c1 = 4,000 and c2 = 7,000 Kg/sec. The spring rates are k1 = 400,000 and k2 = 40,000 N/m. A nonlinear hardening for the suspension spring is considered with cubic hardening coefficient equal to k3=200,000 N/m.

Figure (4-12): Quarter car model

The equations of motion for the above model are,

3   )(  )( 213221212212111 )(  ucukxxkxkxkkxcxccxm 1111 . (4.16)  3   xxkxxkxxcxm 21321221222  0)()()(

The natural frequencies of the system are 10.1 rad/sec and 76.1 rad/sec, respectively.

The above system of equations is solved numerically by the forth order Rung-Kutta method, using a state-space approach (Inman, 2001).

The suspension spring is subjected to a uniaxial shear stress consisting of a static component due to the weight of the car and an alternating component resulting from the road elevation.

The static component due to the weight of the vehicle, which acts as mean shear

stress, is  m  191.2 MPa.

The properties of the suspension spring are presented in Table 4.6.

Table (4.6): Properties of the suspension spring Quantity Value (units)

Wire diameter, d 1.91 cm Number of active turns, N 10 Coil diameter, D 15 cm

Ultimate stress, Sut 1240 MPa

Endurance limit, Sf 558 MPa Shear modulus, G 81 GPa Spring index, C=D/d 7.092

Spring rate, k2 40000 N/m

The spring rate k2, is calculated using the following equation (Shigley et al., 2003)

dG k  . (4.17) 8 CN 3

First, the dynamic response of the system is obtained, and the relative displacement of the first and the second masses is calculated. The dynamic component of the shear stress of the suspension spring is calculated using,

8 2 ykD   K w (4.18)  d 3

 xxy 21 (4.19) where in the above equation, y is the deflection of the suspension spring.

In Eq. 4.18 Kw is called Wahl factor that is to account for the effect of curvature, which is a function of the spring index, C, and is calculated using the following equation

(Shigley et al., 2003),

C 14 615.0 K   . (4.20) w C  44 C

The spring is considered to be made of 4340 quenched and tempered steel (E = 210

GPa , G = 81 GPa ,  = 7850 Kg/m3 , a = 1240 MPa , b = - 0.06, Sf = 558 MPa).

Contrary to the previous example endurance limit is considered, therefore all stress ranges below endurance limit are assumed to inflict no damage to the suspension spring.

Similar to the previous example, for the sampling and the true PSDs, the Pierson-

Moskowitz spectrum is used (Eq. 4.11). Parameters A and B of the sampling spectrum are

250 m2sec-4 and 4,000 sec-4. For the true spectrum, these parameters are 312.5 m2sec-4 and 5,000 sec-4, respectively. The sampling and the true spectra are shown in Figure 4-13.

Compared to the sampling spectrum, the centroid of the true spectrum is closer to the first

natural frequency of the system; therefore greater fatigue damage is expected for the true spectrum.

Figure (4-13): Sampling and true PSDs of the road elevation

One sample history of the road elevation, generated from the sampling spectrum, is shown in Figure 4-14.

Figure (4-14): Sample road elevations based on sampling spectrum

The shear stress history in the suspension spring, due to the road elevation in Figure

4-14, is shown in Figure 4-15.

Figure (4-15): Sample shear stress time history in the suspension spring

For this example, the simulation time is 1800 seconds. A sample shear stress history over the entire 1800 sec period is shown in Figure 4-16.

Figure (4-16): Sample shear stress over entire replication

The stress signal is not narrow-band. Therefore, the rainflow cycle counting method is used to find the effective cycles (Downing & Socie, 1982). Then using Goodman equation (Eq. 4.2) the equivalent fully reversed stress cycles are obtained.

In Figure 4-17, the histogram of Goodman equivalent stress for one sample replication from the sampling spectrum is shown.

Figure (4-17): Goodman equivalent stress histogram (nonlinear quarter car model)

In Figure 4-17, all stress ranges less than 558 MPa are neglected in the calculation of the fatigue damage.

Ten thousand replications for both the sampling and the true spectra are run, and the fatigue damage is calculated by using both MCs and PRRA. The results from MCs are summarized in the Table 4.7. Performing MCs took about 2 days, whereas the results of

PRRA were obtained almost instantaneously. As expected, higher fatigue damage is observed for the true spectrum.

In Table 4.8, the results of MCs are compared to those from PRRA.

The COV of PRRA decreases with the number of replications. The fatigue damage falls within the 95% CI for 1000, 2500, 5000 and 10,000 replications (Figure 4-18). By definition, if repeatedly the same interval is calculated, this interval will enclose the true mean value of the fatigue damage 95% of the time, on average.

Table (4.7): Fatigue damage by MCs for the sampling and the true spectra (quarter car model) sampling true spec.* spec.* Ave 2.18E-05 5.19E-05 10,000 rep. StDev 1.50E-06 3.07E-06 Ave 2.16E-05 4.92E-05 5000 rep. StDev 2.11E-06 3.45E-06 Ave 2.15E-05 5.10E-05 2500 rep. StDev 2.48E-06 4.83E-06 * ~2days computational time for each MCs

Table (4.8): Comparison of MCs with PRRA (quarter car model) Likelihood 95 % PRRA COV ratio Confidence Interval Ave 5.76E-5 10,000 rep. 0.12 [4.68E-05,6.84E-05] StDev 5.51E-6 Ave 5.43E-5 5,000 rep. 0.13 [4.68E-05,6.84E-05] StDev 6.71E-6 Ave 5.05E-5 2,500 rep. 0.20 [4.68E-05,6.84E-05] StDev 6.59E-6 * Almost no additional cost for PRRA

Figure (4-18): Fatigue damage vs. the number of replications (quarter car model)

Now six additional spectra are considered to investigate the efficacy of PRRA for larger deviations from the sampling spectrum. The additional spectra, which are listed in

Table 4.9, are Pierson-Moskowitz and contain the same amount of energy as the sampling spectrum.

Table (4.9): Pierson-Moskowitz spectra parameters for the spectra shown in Figure 4-19 A B m2sec-4 sec-4 sampling spec. 250 4,000 true spec. 1 62.5 1,000 true spec. 2 125 2,000 true spec. 3 187.5 3,000 true spec. 4 312.5 5,000 true spec. 5 375 6,000 true spec. 6 437.5 7,000 true spec. 7 500 8,000

The first three spectra are shifted toward the left of the sampling spectrum, and the remaining four are shifted to the right. Since the four spectra on the right of the sampling spectrum are closer to the natural frequency of the system, greater damage is expected for them. The spectra listed in Table 4.9 are shown in Figure 4.19.

Figure (4-19): Sampling spectrum and different true spectra (quarter car model)

Table 4.10 shows the corresponding fatigue damage for the spectra in Figure 4-19.

Table (4.10): Fatigue damage calculated using PRRA and MCs for different spectra (quarter car model) Likelihood COV COV PRRA MCs ratio (MCs) (PRRA) sampling Ave ------2.85E-4 0.07 ---- spec. StDev ------2.13E-5 Ave 0.177 1.01E-8 1.31E-8 true spec. 1 0.68 0.22 StDev 0.077 2.23E-9 8.87E-9 Ave 0.882 6.88E-7 9.06E-7 true spec. 2 0.28 0.09 StDev 0.096 6.19E-8 2.58E-7 Ave 0.993 5.56E-6 5.27E-6 true spec. 3 0.16 0.05 StDev 0.014 3.01E-7 8.51E-7 Ave 1.002 5.52E-5 4.92E-5 true spec. 4 0.07 0.13 StDev 0.007 6.91E-6 3.45E-6 Ave 1.004 1.20E-4 1.24E-4 true spec. 5 0.05 0.13 StDev 0.015 1.61E-5 6.23E-6 Ave 1.006 2.12E-4 2.38E-4 true spec. 6 0.05 0.19 StDev 0.026 3.95E-5 1.30E-5 Ave 1.005 3.35E-4 3.84E-4 true spec. 7 0.04 0.25 StDev 0.040 8.28E-5 1.45E-5 * ~2 days computational time for each MCs and no additional cost for PRRA

The fatigue damages along with their corresponding 95% CI are plotted in Figure 4-

20.

Figure (4-20): Fatigue damage calculated using PRRA and MCs for different spectra (quarter car model)

The COV of the fatigue damage is shown in Figure 4-21. PRRA yields accurate results for the third to the fifth spectra; the error is about 10 %, and the COV of the mean damage is about 10%. When the true spectrum deviates more from the sampling spectrum, the accuracy of PRRA deteriorates. The results for the first and the second spectra are surprising because, although the average likelihood ratio deviates considerably from unity, PRRA fairly yields accurate results. For the sixth and the seventh spectra, although the error is small, the standard deviation of the mean damage is about 20% of the mean, which results in an wide confidence interval. However, for all spectra, PRRA predicts the fatigue damage trend well. The mean damage predictions from PPRA correlate well with those from MCs; the Pierson’s correlation coefficient of the two sets of predictions in Table (4.10) is 0.999.

Figure (4-21): The COV of fatigue damage using PRRA and MCs for different spectra (quarter car model)

In Table (4.11), the original results of PRRA are compared with those normalized by the mean value of the likelihood ratio.

Table (4.11): Fatigue damage using PRRA and comparison with normalized PRRA (quarter car model) normalized PRRA error error PRRA Ave 1.01E-8 5.72E-8 true spec. 1 22.9% 337% StDev 2.23E-9 1.26E-8 Ave 6.88E-7 7.80E-7 true spec. 2 24.0% 13.9% StDev 6.19E-8 1.26E-8 Ave 5.56E-6 5.60E-6 true spec. 3 5.5% 6.2% StDev 3.01E-7 1.26E-8 Ave 5.52E-5 5.51E-5 true spec. 4 12.2% 12.0% StDev 6.91E-6 1.26E-8 Ave 1.20E-4 1.20E-4 true spec. 5 3.2% 3.6% StDev 1.61E-5 1.26E-8 Ave 2.12E-4 2.11E-4 true spec. 6 10.9% 11.5% StDev 3.95E-5 1.26E-8 Ave 3.35E-4 3.33E-4 true spec. 7 12.8% 13.2% StDev 8.28E-5 1.26E-8

For the first, third, fifth, sixth and seventh spectra normalizing does not increase the accuracy, specially for the first spectrum normalizing significantly deteriorates the accuracy.

Example 3: Fatigue damage inflicted by waves on an offshore wind turbine

In this section, fatigue damage to the tower of an offshore wind turbine with a monopile platform (Figure 4-22) is studied. To demonstrate the performance of PRRA for this example only wave loads exerted on the platform are considered.

Figure (4-22): An offshore wind turbine with monopile platform

The wind turbine model is based on the monopile concept of the 5 MW machine introduced by the National Renewable Energy Laboratory (NREL) (Jonkman et al.,

2007). The wind turbine is clamped to the seabed, and the water depth is 20 m. FAST that is developed by NREL is used to perform simulations (Jonkman et al., 2004). In FAST, wave elevations are generated by filtering a white noise process through a wave spectrum

(Jonkman, 2007).

FAST uses Box-Muller method to generate white Gaussian noise (Jonkman, 2007). In this method two randomly independent variables which are drawn uniformly from [0, 1] are turned into two standard normal numbers. Box-Muller transforming equations are as follows:

U  U U )2cos()ln(2 1 2 (4.21) V  U1 U 2 )2sin()ln(2

where U1 and U2 are two uniform random numbers within [0, 1] and U and V are standard normal random numbers. In FAST (Jonkman, 2007) the two random numbers are stored as real and imaginary component as below

  U1 2   UjU 2  ))(2sin())(2cos())((ln2   0  W )(   0   0 (4.22)   U U   Uj  ))(2sin())(2cos())((ln2  0  1 2 2 

In this approach, the wave elevations are generated by the following equation,

 1  )(  )(2)( tj deSWt  . (4.23) 2  

In the above equation, S() is two-sided PSD function of the wave elevation.

To use PRRA, wave elevations can not be calculated by the above approach.

Therefore, a custom computer code to generate the wave elevation using Shinozuka’s method (1972) is developed. The method implemented in FAST and the method that uses

Shinozuka’s equation (1972) are equally efficient in converging to the target PSD of the wave condition. In order to compare the performance of the two methods 100 wave elevations by FAST and by the developed code are calculated. In Figure 4-23, the results of the two methods are compared. Both methods are equally efficient and converge well to the target spectrum by 100 replications.

Figure (4-23): Convergence test of the wave elevation methods

In FAST, hydrodynamic loads are calculated by Morison equations (Jonkman, 2007).

In order to calculate the loads, the wave kinematics under the sea surface should be calculated. The linear wave theory along with Wheeler’s stretching method is used to calculate the wave kinematics (Wheeler, 1970) (see appendix A). The wave kinematics is calculated and the response of the structure is calculated by FAST. To validate the developed code, first FAST along with its built-in wave kinematics is used to calculate the fore-aft bending moment at the base of tower. Then FAST and the custom wave code are used to perform the same analysis. Figure 4-24 shows the wave elevation that is considered in this comparison study. This signal contains three harmonics.

Figure (4-24): wave elevation considered for validating the custom wave kinematics code

Figure 4-25 shows the fore-aft bending moment at the base of tower by the two approaches. The results agree well. Wheeler’s stretching method (Wheeler, 1970) is used in both simulations.

Figure (4-25): Fore-aft bending moment at the base of tower by the two approaches

To perform PRRA for this example the developed wave kinematics code is used. The largest load is the fore-aft bending moment at the mud-line. Pierson-Moskowitz spectrum

is used to model the wave elevation. The two-parameter Pierson-Moskowitz spectrum in the IEC-61400-3 (2009) standard is characterized by the power spectral period (Tp) and the significant wave height (Hs). Significant wave height is the mean height of the one- third highest waves. The location of the peak frequency is related to Tp, and the energy content of the waves depends on Hs. Equation 4.11 is used to describe Pierson-

Moskowitz spectrum. In order to do so the parameters A and B in terms of Hs and Tp are as follows:

5 2 2 4  HA s ( ) 16 Tp . (4.24) 2 B  (25.1 )4 Tp

The first natural frequency of the tower is 1.88 rad/sec (Jonkman et al., 2007). For the sampling spectrum, the spectral period of 3 sec and the significant wave height of 6 m are considered, and for the true spectrum, only the spectral period changes to 2.8 sec. In order to perform PRRA, the same energy under the true and sampling spectra should be preserved. This condition is satisfied by considering that the significant wave height for the sampling spectrum is identical to that of the true spectrum. For the sampling spectrum, the peak frequency is 2.07 rad/sec and for the true one is 2.26 rad/sec as shown in Figure 4-26. Since the peak frequency of the sampling spectrum is closer to the natural frequency of the tower, this spectrum inflicts more damage on the platform.

Figure (4-26): The sampling and the true spectra in the wind turbine example

The wave kinematics used in FAST and in the custom code is 2D, and it is assumed that the angle of incident waves remains zero. As a result of the waves’ impact with the tower, at the base of the tower, the side-to-side and the torsional moments are negligible compared to the fore-aft moment. The fore-aft and side-to-side shear forces are negligible compared to the axial load due to the weight of the wind turbine. Therefore, there are only two normal stresses, one alternating normal stress due to the fore-aft bending moment and other compressive stress due to the weight of the structure, which is added as a mean stress to the alternating stress due to the wave loads. Therefore the stress state in this example is assumed to be uniaxial.

Two sample paths of the wave elevation generated from the sampling spectrum are illustrated in Figure 4-27.

Figure (4-27): Sample wave elevations according to the sampling spectrum

In Figure 4-28, two sample fore-aft bending moment time histories at the base of the tower corresponding to the wave elevations in Figure 4-27 are shown.

Figure (4-28): Sample fore-aft bending moment at the base of tower

The compressive axial load due to the weight of the whole structure at the base of the tower is 8.584 MN. The cross section of the tower at the base is hollow and circular with outer diameter of 6 m and thickness of 27 mm. The tower is assumed to be made of

ASTM-A36 standard structural steel with Wohler parameters, a and b equal to 400 MPa and -0.11, respectively. No endurance limit is considered for this example.

Monte Carlo simulation with 1,000 replications was performed for both the sampling and the true spectra. Each replication contains 20 minutes of the system response. The results of the fatigue damage using MCs are shown in Table 4.12.

Table (4.12): Fatigue damage by MCs for the wind turbine example sampling spec.* true spec.* Ave 3.5881E-06 1.2464E-06 1,000 rep. StDev 3.6054E-07 1.4757E-07 Ave 3.6272E-06 1.1732E-06 500 rep. StDev 5.3142E-07 2.1623E-07 Ave 3.7241E-06 9.3626E-07 250 rep. StDev 7.2003E-07 1.8902E-07 * 15 hours computational time for each MCs with 1,000 replications

The results from PRRA and their comparison with MCs are shown in Table 4.13. As in the previous two examples, the accuracy of PRRA improves with number of replications.

Table (4.13): Fatigue damage using PRRA and comparison with MCs for wind turbine example

Likelihood PRRA* COV ratio (PRRA) Ave 0.9525 1.3540E-06 1,000 rep. 0.08 StDev 0.0274 1.0834E-07 Ave 0.9689 1.3885E-06 500 rep. 0.12 StDev 0.0390 1.6163E-07 Ave 1.0152 1.4253E-06 250 rep. 0.15 StDev 0.0567 2.1994E-07 * Almost no additional cost for PRRA

As shown in Figure 4-29, the width of the 95 % CI decreases with the number of replications.

Figure (4-29): Fatigue damage as number of simulation increases (wind turbine)

Table 4.14 shows that the accuracy of the results of PRRA does not increase after normalizing the estimates of the damage by the average likelihood ratio.

Table (4.14): Fatigue damage using PRRA and comparison with normalized PRRA (wind turbine) normalized PRRA error error PRRA Ave 1.3540E-06 1.4215E-06 1,000 rep. 8.6% 14.0% StDev 1.0834E-07 1.1374E-07 Ave 1.3885E-06 1.4332E-06 500 rep. 11.4% 15.0% StDev 1.6163E-07 1.6683E-07 Ave 1.4253E-06 1.4040E-06 250 rep. 14.4% 12.7% StDev 2.1994E-07 2.1665E-07

Performing MC simulation took about 15 hours on a desktop computer, whereas the results of PRRA were obtained instantaneously.

Note that the fatigue life is very sensitive to the value of exponent b. A designer should calculate the fatigue life for different values of this exponent. Alternatively, the designer could estimate a probability distribution of b and compute the resulting

distribution of the fatigue life. The results of PRRA for the wind turbine example deteriorate faster because of the short simulated periods. Overall, in the wind turbine similar trends to those in the first two examples are observed. The purpose of this last example was to demonstrate the application of PRRA to complex systems. In order to perform a thorough fatigue analysis, a more detailed model of the structure with considering all stress components should be used.

4.3 Discussion

In this chapter, the efficacy of PRRA was examined to estimate the fatigue damage in a structure.

Probabilistic Re-analysis can reduce the cost of fatigue sensitivity reliability analysis significantly provided that the input load is respresented by a PSD function. Another efficient way to reduce the cost of such analyses is the use of fatigue frequency domain methods. Among these methods, Dirlik’s (1985) empirical expression is widely used by practitioners. Dirlik’s expression calculates stress ranges and fatigue damage from a PSD function. This method requires calculation of the PSD of the output for the PSD of input load. For linear systems for which the transfer function is known Dirlik’s method is very efficient as the PSD of output can be easily calculated from the PSD of load. This enables performing fatigue sensitivity analysis very efficiently for linear systems. However, for nonlinear systems, the calculation of the PSD of output can be challenging and most researchers prefer a time domain approach to estimate the fatigue damage. To perform fatigue sensitivity analysis, the approach using PRRA is advantageous to Dirlik’s method as it can be applied to either linear or nonlinear systems. However, PRRA requires performing MCs in time domain for one PSD of the load.

The approach that was presented in this chapter has limitations that are highlighted below.

The first limitation is inherited from the rainflow cycle counting algorithm that was used in order to identify and count the effective fatigue cycles. This cycle counting technique is not valid for multiaxial loading with multiple channels.

Another limitation comes from the application of the linear damage rule. This rule is used in this study because of its popularity among practicing engineers.

The efficacy of the method was studied on three examples. For the 1-DOF beam and the wind turbine, no endurance limit was considered. This might result in overestimating the fatigue damage. Based on the results from the examples, the following observations are made.

– The standard deviation of the fatigue damage decreases with the number of

replications. Roughly, halving the standard deviation requires quadrupling the

number of replications.

– A mean value of the likelihood ratio that is significantly different than 1 does not

necessarily imply a biased estimator. This is the case when the fatigue damage is

zero for a range of frequencies of the sampling PSD function that are outside of

the support of the true spectrum.

– The accuracy of PRRA decreases as the true spectrum deviates significantly from

the sampling spectrum.

– Most of the time, the standard deviation of the results of PRRA is larger than that

of the sampling MCs. This is due to the variability of the likelihood ratio. In order

to reduce the standard deviation, the sampling PSD must be carefully chosen by

considering the frequency response of the system.

– Normalizing the results of PRRA by the average likelihood ratio, as some authors

have suggested, does not always improve the accuracy.

The difference between the computational cost of MCs and PRRA is significant for large real life structures. The required time to run PRRA is negligible, and it is insensitive to the size or the complexity of the structure, whereas performing MCs for large structures is far more expensive.

The presentation and examples suggest that PRRA is a promising tool for assessment of the sensitivity of the fatigue reliability of real-life complex structures to the PSD of the applied loads.

Chapter 5

Using PRRA to Estimate Probability of First Excursion Failure

Overview

In the design of a system under random vibration, first excursion failure is an important mode of failure in which failure occurs once the response of a system first crosses a threshold. In this chapter first the statistics of a narrow-band process is reviewed, then in section 5.2 the application of PRRA to estimate average up-crossing is presented. In section 5.3 the PRRA method is used to find the probability of first excursion failure and in section 5.4 the application of PRRA on three examples is presented. The first of which deals with a nonlinear beam; the second one with a nonlinear quarter car model; and the third with an offshore wind turbine.

5.1 Statistics of a Narrow-Band Process

Assume that process x(t) is narrow-band, ergodic, stationary and Gaussian, then the average frequency of up-crossing of a level a is calculated by (Newland, 1993):

a2  2  1 x 2 x a  e (5.1) 2  x

where in the above equation  2 and  2 are the variances of the process x(t) and its x x derivative, respectively. The variances are obtained as follows:

  2  )( dS   xx  (5.2)  2  2 dS  .)( x  x 

To estimate the zero up-crossing Eq. 5.1 is simplified to:

 1  x 0  . (5.3) 2  x

 Notice that 0 is obtained by averaging across the ensemble and it would not be equal to the average up-crossing along the time axis unless the process is ergodic.

The probability that any peak value chosen at random exceeds a threshold value a, is

(Bendat, 1964):

a2 a 2 aP )(  e 2 x . p 2 (5.4)  x

The above equation is called Rayleigh distribution which is plotted below:

Figure (5-1): Distribution of the peaks for a narrow-band process (Rayleigh distribution)

Therefore, variance is the only parameter that is required to have the distribution of the peaks for a narrow-band process.

Using Eq. 5.4, the probability that a peak exceeds a, is:

2  a 2 )(  edaaP 2 x  p (5.5) a

For a narrow-band process, the frequency of maxima of a process x(t) is equal to the zero up-crossing rate of its derivative. Therefore using Eq. 5.3 the frequency of maxima can be calculated by:

1    x . (5.6) 2  x

Let the process is ergodic and the peaks are randomly distributed along the time axis, then the probability of crossing a threshold, a, or the probability of failure within a given time T is:

  ePF a T .1 (5.7)

 In the above equation a is the up-crossing rate for the level of a, which according to

Eq. 5.1 is a function of  x and x . As shown in Figure 5-2 as the value of the threshold increases the probability of exceedance within a period decreases and the reliability of the system increases. In other words, as the capacity of the system increases the probability of failure decreases and the reliability of the system increases.

Figure (5-2): Probability of failure as threshold value

To summarize, Eq. 5.7 calculates the probability of first excursion failure within a period, T, for a narrow-band ergodic stationary Gaussian process. In the next section it is shown that how PRRA is used to find the variance of a process and that of its derivative that are required to evaluate the probability of first excursion failure according to Eq. 5.7.

5.2 Estimation of Average Up-crossing Rate

A designer is usually seeking to estimate the probability of failure for very rare events and the average up-crossing is not the parameter in which he is interested. However, here it is shown that using the equations that were presented in the previous section the average up-crossing rates of low thresholds can be used to estimate the probability of failure for higher thresholds. Then, PRRA is used to estimate the average up-crossing and the reliability of a system with narrow-band, Gaussian and ergodic response.

Equation 5.7 can be used to find the probability of failure within a period if the average up-crossing for the threshold a, is known. Average up-crossing is a function of

 2 and  2 according to Eq. 5.1, so if one can estimate them in some way, then the x x reliability of the system can be assessed according to Eq. 5.7. Finding average up- crossing is less expensive for low thresholds than for high thresholds. PRRA is used to estimate average up-crossing for low thresholds to calculate the corresponding variances and reliability for very high thresholds, which is far more expensive to evaluate. This approach is valid for narrow-band, ergodic, stationary and Gaussian processes. In the following paragraph the proposed method is explained then in section 5.4 the application of the above approach on some examples is presented.

The flowchart which was shown in Figure 3-3 is generic. In order to estimate average up-crossing one just needs to calculate that in step 4 for the sampling spectrum and then calculate its counterpart for the true spectrum. Load realizations should be generated using Shinozuka’s method (1972), and the corresponding time histories can be calculated by a simulation tool such as a FEA code. The time histories could be stress or displacement history at a critical point of the structure. Average up-crossings are

calculated and PRRA is run. The standard deviations and the corresponding CIs can be calculated by the Eqs. 3.19 and 3.20, respectively. An alternative approach is to estimate the standard deviation of the true spectrum directly through the flowchart shown in

Figure 3-3. These two approaches are compared in one example.

5.3 Estimation of Probability of Failure

As was mentioned in the previous section, PRRA can be used to obtain average up- crossing. Estimation of up-crossing rate or probability of failure for lower thresholds is much less expensive than those of high thresholds. This is because up-crossing rate for a high threshold is scarce and many replications should be run in order to capture high level up-crossings. PRRA can also be used to find the probability of failure. The attribute of the system in this case is the value of the indicator function. In step 4 of the flowchart

3.3 the value of the indicator function should be evaluated for the sampling spectrum.

According to the definition in Eq. 3.3, the indictor function becomes unity once failure occurs and it gets zero once system survives. Similar to Eq. 3.27 the likelihood ratio, k is calculated and the value of indication function for every replication is calculated by Eq.

3.3. Finally the probability of failure for the true spectrum is estimated by the following equation,

K T 1 S P  GI   .]0)([ (5.8) f  xk k K k1

In the above equation, K is the number of replications which is used to run MCs for the sampling spectrum. Notice that the value of the indicator function on the right hand side of the above equation is only evaluated for the sampling spectrum. The standard

deviation of the results for the sampling spectrum is calculated by Eq. 3.6 which is valid for standard MCs, but for the true spectrum the standard deviation is estimated by Eq.

5.9,

K 1 S 2 T 2  T  GI xk k  PK f )()0)(( (5.9) Pf  KK  )1( k1

The CI is calculated using Eq. 3.20.

The efficacy of PRRA to estimate low probabilities of failure is significantly superior when compared to MCs. In the next section the application of PRRA to estimate the probability of failure on some examples is shown.

5.4 Application

5.4.1 Estimation of standard deviation using PRRA

Example: Nonlinear quarter car model

In this example the same nonlinear quarter car model in section 4.2 is studied without considering the mean stress due to the weight of the vehicle. In fatigue analysis mean stress affects the fatigue life, whereas in an up-crossing problem most researchers do not consider mean values. This does not have any effect on the performance of PRRA.

Input spectra similar to those considered in section 4.2 are considered here. For the sampling spectrum Pierson-Moskowitz spectrum Eq. 4.11 is considered similar to the previous example with parameters A and B of 250 m2sec-4 and 4,000 sec-4, respectively.

For the true spectrum, these parameters are 312.5 m2sec-4 and 5,000 sec-4, respectively.

The sampling and the true spectra are shown in Figure 4-13.

One approach to calculate the standard deviation of the process and that of its derivative is to estimate them directly using PRRA. In order to do that MCs is run with

50,000 replications with 1800 sec of time duration for the sampling spectrum, the corresponding standard deviations are estimated, the variances for the true spectrum are estimated using PRRA, and then the results are compared with those obtained by running

MCs for the true spectrum. The results are summarized in the following table.

Table (5.1): Standard deviations of the process and that of its derivative by MCs for the sampling and the true distribution

sampling spec.* true spec.*     x x x x Ave 125.6 1686.6 133.4 1861.1 10,000 rep. StDev 0.18 6.62 0.18 6.83 Ave 125.5 1687.9 133.5 1865.1 20,000 rep. StDev 0.13 4.70 0.12 4.85 Ave 125.6 1687.2 133.5 1864.5 50,000 rep. StDev 0.08 2.95 0.08 3.05 * ~24 hours computational time for each MCs running 10 replications in parallel

In Table 5.2, the results of MCs are compared to those from PRRA.

Table (5.2): Standard deviation using PRRA and comparison with MCs

Likelihood PRRA PRRA ratio   x x Ave 0.9984 133.3 error 1866.5 error 10,000 rep. StDev 0.0071 1.11 0.2% 18.79 0.5% Ave 0.9986 133.4 error 1869.1 error 20,000 rep. StDev 0.0050 0.78 0.1% 13.27 0.3% Ave 1.0023 133.9 error 1870.9 error 50,000 rep. StDev 0.0032 0.49 0.3% 8.28 0.3% * Almost no additional cost for PRRA

Excellent agreement between the results of PRRA with those from MCs is observed.

Estimating the standard deviation of the derivative of a process is always more challenging than the process itself. That is why, in Table 5.1 and in Table 5.2 lower accuracies are observed for  , using MCs and using PRRA. In the next part the x

variances of both sampling and true spectra are estimated using average up-crossing rate based on the approach that was presented earlier in this chapter.

5.4.2 Estimation of average up-crossing using PRRA

Example: Nonlinear quarter car model

At first the performance of PRRA in the estimation of average up-crossing of the true spectrum is studied. The same nonlinear quarter car model that was described in the previous part is used here. First, MCs with 1000 replications is run for the sampling spectrum and the number of up-crossings during 1800 seconds of each simulation is calculated for different stress levels. Monte Carlo simulation for the true spectrum is used to validate the results obtained using PRRA.

Table 5.3 shows and compares the results of PRRA with those of MCs for the true and sampling spectra. The accuracy of PRRA deteriorates as the stress level becomes higher.

Let assume that the up-crossing rates follow a narrow-band process according to Eq.

5.1. The standard deviations of the process and that of its derivative are estimated by fitting Eq. 5.1 into the results shown in Table 5.3. The variances are calculated by least square method by using the results of low and high 95% CIs as shown in Table 5.4.

Table (5.3): Average up-crossing using PRRA and comparison with MCs Sampling True True spec. spec. spec. Stress level MCs with 1000 PRRA* COV error (MPa) rep.* AveUp 3857.1 3936.3 3983.17 0 0.03 1.2% StDev 36.7 34.7 107.2 50 AveUp 3582.6 3684.2 3730.70 0.03 1.3% StDev 35.6 33.9 102.5 AveUp 2868.1 3017.0 3062.78 100 0.03 1.5% StDev 32.6 31.6 89.64 AveUp 1972.3 2158.0 2198.53 150 0.03 1.9% StDev 27.80 27.67 71.51 AveUp 1160.6 1343.4 1376.99 200 0.04 2.5% StDev 21.40 22.13 52.08 AveUp 581.0 724.6 748.07 250 0.05 3.2% StDev 14.45 15.71 34.52 AveUp 48.9 78.7 84.43 375 0.10 7.2% StDev 2.91 3.69 8.29 AveUp 26.02 44.64 48.38 400 0.12 8.4% StDev 1.91 2.51 5.83 AveUp 13.32 24.04 26.71 425 0.15 11.1% StDev 1.21 1.64 4.02 AveUp 6.61 12.40 14.36 450 0.19 15.8% StDev 0.766 1.05 2.78 AveUp 3.05 6.09 7.22 475 0.25 18.6% StDev 0.463 0.66 1.83 AveUp 1.41 2.86 3.66 500 0.32 27.8% StDev 0.274 0.41 1.16 AveUp 0.59 1.32 1.68 525 0.42 27.7% StDev 0.158 0.25 0.71 AveUp 0.24 0.56 0.80 550 0.60 41.4% StDev 0.100 0.15 0.48 AveUp 0.11 0.23 0.40 575 0.73 73.3% StDev 0.059 0.09 0.29 AveUp 0.05 0.11 0.22 600 0.86 104.3% StDev 0.038 0.05 0.19 AveUp 0.03 0.05 0.13 625 0.92 156.1% StDev 0.023 0.03 0.12 AveUp 0.01 0.02 0.07 650 1.00 171.6% StDev 0.013 0.02 0.07 AveUp 0.01 0.01 0.03 675 1.00 200.9% StDev 0.006 0.01 0.03 AveUp 0.002 0.01 0.01 700 1.00 N/A StDev 0.002 0.01 0.01 * 6 hours computational time for each MCs and no additional cost for PRRA

Table (5.4): Variances obtained by fitting Eq. 5.1 into results

Using Using low 95 % CI high 95% CI  x 124.3 137.6

 x 1981.4 2810

The resulting curves according to Eq. 5.1 are plotted in Figure 5-3. The average up- crossings for this example especially for lower thresholds are covered well by narrow- band process assumption (Eq. 5.1). For higher thresholds the CI becomes wide and the narrow-band curve falls outside the CI, because the accuracy of the estimates deteriorate for higher thresholds.

Figure (5-3): Average up-crossing rate by MCs and using PRRA

In the second approach it is assumed that the process is narrow-band while in the first one no assumption is made to estimate the variances. Therefore the approach presented in the previous section also works for a broad-band process as long as it is Gaussian. For this example, the two approaches show fair agreements in estimating the variance of the process. Estimation of higher moments of a PSD function from time histories is always challenging. That is why the variance of the response derivative as shown in Table 5.4 shows wider CI. By increasing the number of replications, the estimate of the second moment would improve. However, the results indicate that the narrow-band assumption is not a valid for this example.

For a narrow-band process, using the formulation presented above either approaches

(preferably first one) can be used to estimate the variance of the process, and that of its derivative to estimate the probability of failure or the reliability of the system for a true spectrum using PRRA.

5.4.3 Estimating probability of failure using PRRA

Example 1: Probability of failure in a linear beam

In this section the same linear beam model as the one in section 4.2 is used without considering the mean stress due to the weight of the middle mass. Similar to the fatigue problem Pierson-Moskowitz (Eq. 4.11) spectrum is used as input and the simulation is repeated to calculate probability of failure for different thresholds. According to Eq. 4.11

A and B for the sampling spectrum are considered as 5109 N 2sec-4 and 2.7861104 sec-4, respectively. This results in a peak frequency of about 12.22 rad/sec for the sampling

PSD. Assume that for the true spectrum A and B are 6.5109 N 2sec-4 and 3.6219104 sec-

4 with a peak frequency of 13.05 rad/sec (Figure 4-3). Since the sampling spectrum peak is closer to the damped natural frequency of the system, higher probability of failure is expected for the sampling than the true spectrum. Monte Carlo simulation with 200,000 replications is performed to estimate the probability of failure for different levels for the sampling spectrum and then the results and PRAA are used to estimate the probability of failure for the true spectrum. Then MCs with 200,000 replications is performed with

3600 sec of time duration for the true spectrum. Then the results are compared with those obtained using PRRA. Table 5.5 shows that for higher thresholds, more replications are required in order to achieve desirable accuracy (measured by the coefficient of variation of the probability of failure). The results of PRRA for the true spectrum for different

numbers of replications are shown in Table 5.6. The results of PRRA for different thresholds are compared with those of the MCs in Figure 5-4. The 95% CIs calculated using PRRA (Eq. 3.20) are shown in the same figure. The 95% CI becomes wider as the stress threshold increases.

Table (5.5): Probability of failure by MCs for the sampling and the true distribution (1-DOF beam)

Sampling spec.* True spec.* Stress 50,000 100,000 200,000 50,000 100,000 200,000 levels rep. rep. rep. rep. rep. rep. (MPa) P 8.62E-02 8.60E-02 8.65E-02 2.79E-02 2.79E-02 2.85E-02 575 f StDev 1.26E-03 8.86E-04 6.29E-04 7.37E-04 5.20E-04 3.72E-04 P 1.02E-02 1.06E-02 1.07E-02 2.30E-03 2.36E-03 2.38E-03 625 f StDev 4.50E-04 3.24E-04 2.30E-04 2.14E-04 1.53E-04 1.09E-04 P 2.34E-03 2.50E-03 2.55E-03 5.20E-04 4.80E-04 5.60E-04 650 f StDev 2.16E-04 1.58E-04 1.13E-04 1.02E-04 6.93E-05 5.29E-05 P 3.20E-04 2.90E-04 3.45E-04 6.00E-05 4.00E-05 3.00E-05 680 f StDev 8.00E-05 5.38E-05 4.15E-05 3.46E-05 2.00E-05 1.22E-05

Pf 8.00E-05 6.00E-05 5.00E-05 2.00E-05 1.00E-05 5.00E-06 700 StDev 4.00E-05 2.45E-05 1.58E-05 2.00E-05 1.00E-05 5.00E-06 * ~ 1 day computational time for each MCs with 200,000 replications (20 rep in parallel)

To estimate the probability of failure using PRRA , likelihood ratios are calculated by

Eq. 4.15.

Table (5.6): Probability of failure using PRRA for the true distribution (1-DOF beam) True spec. using PRRA Stress levels 50,000 100,000 200,000 (MPa) rep. rep. rep.

Pf 2.87E-02 2.85E-02 2.88E-02 575 StDev 5.27E-04 3.69E-04 2.62E-04 P 2.48E-03 2.56E-03 2.52E-03 625 f StDev 1.36E-04 9.75E-05 6.73E-05 P 650 f 4.60E-04 4.74E-04 4.90E-04 StDev 5.59E-05 3.83E-05 2.71E-05 P 680 f 5.22E-05 4.53E-05 5.48E-05 StDev 1.85E-05 1.12E-05 8.30E-06

Pf 4.01E-06 4.44E-06 4.91E-06 700 StDev 2.50E-06 2.44E-06 2.07E-06 * Almost no additional cost for PRRA

The results of PRRA are obtained by multiplying the results of sampling spectrum by the likelihood ratio. Thus, the variability of likelihood ratio results in a larger standard deviation for the results obtained using PRRA compared to that of sampling spectrum.

However, an appropriate sampling spectrum can yield more accurate results for PRRA compared to those of true spectrum by MCs (similar to this example).

Figure (5-4): Probability of failure by MCs and using PRRA (1-DOF beam)

The results of PRRA and MCs agree well, and the predictions of MCs are within the

95% CI calculated using PRRA. Therefore, the deviations of the results of the two methods can be attributed to sampling variability. Note that the results of PRRA are demonstrating smaller coefficient of variation when compared to those of MCs for the true spectrum. Since the peak frequency of the sampling spectrum is closer to the natural frequency of the system, higher probability of failure for the sampling simulation is expected. Higher probabilities of failure always need smaller number of replications to

achieve desired accuracy. Therefore, performing MCs with 200,000 replications presents better accuracy for the sampling spectrum than for the true one. This justifies the contrary observation of smaller coefficient of variation for PRRA compared to MCs.

Performing MCs with 200,000 replications took one day to run on a desktop computer (running 20 replications in parallel) whereas the results of PRRA for the true spectrum are calculated almost instantaneously after performing MCs for the sampling spectrum.

Now let's look at the performance of PRRA in the estimation of the probability of failure for the 6 other spectra as shown in Figure 4-9. The stress level of 650 MPa is chosen to compare the results. The probability of failure and the 95% CI for all spectra are shown in Figure 5-5.

Figure (5-5): Probability of failure by MCs and using PRRA for 1-DOF beam (Stress level 650 MPa)

100

For all the spectra other than the sixth and the seventh ones, the estimates of the probability of exceeding 650 MPa by MCs with 200,000 replications fall within the 95%

CIs which are estimated using PRRA. However, by deviating more from the sampling spectrum the CI widens. The results are summarized in Table 5.7. No failure was observed for the seventh true spectrum by MCs even with 200,000 replications. For the sixth spectrum the COV from the MCs is too high while the results from PRRA present an acceptable CI. For the seventh spectrum PRRA gives an upper-bond for the probability of failure while no failure was observed by MCs.

Table (5.7): Probability of exceeding 650 MPa estimated using PRRA and MCs for different spectra (1-DOF beam) Likelihood COV COV PRRA MCs ratio (MCs) (PRRA) sampling P ------2.55E-03 f ------spec. StDev ------1.13E-04 P 0.9034 1.84E-02 1.90E-02 true spec. 1 f 0.02 0.16 StDev 0.0287 2.93E-03 3.06E-04 P 0.9896 1.06E-02 1.04E-02 true spec. 2 f 0.02 0.08 StDev 0.0048 7.98E-04 2.26E-04 P 1.0000 4.90E-04 5.60E-04 true spec. 3 f 0.09 0.06 StDev 0.0019 2.71E-05 5.29E-05 P 0.9990 1.54E-04 1.80E-04 true spec. 4 f 0.17 0.07 StDev 0.0034 1.10E-05 3.00E-05 P 0.9979 2.53E-05 2.50E-05 true spec. 5 f 0.45 0.11 StDev 0.0064 2.84E-06 1.12E-05 P 1.0070 3.33E-07 5.00E-06 true spec. 6 f 1.0 0.34 StDev 0.0214 1.15E-07 5.00E-06 P 1.0250 1.52E-08 No true spec. 7 f NA 0.59 StDev 0.0428 8.94E-09 failure * ~ 1 day computational time for each MCs and no additional cost for PRRA

PRRA yields an acceptable COV for the second to the fifth spectra. For the second and third, the coefficient of variation for both MCs and PRRA are about 0.1 hence the results of PRRA are validated by MCs. The coefficient of variation using PRRA increases as the true spectrum deviates more from the sampling spectrum. Note that the

COV grows faster when the true spectrum moves towards the left of the sampling

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spectrum rather than towards its right. This is because the support of the true spectrum departs from that of the sampling spectrum when it moves to the left. Nevertheless the results of PRRA for the other spectra are still useful because they show that it is more likely to observe failure for the true spectra on the left of the sampling spectrum. For the forth to the seventh spectra the COV by MCs is undesirably high and in order to improve the accuracy more replications should be used. However, running more replications may not be necessary since the results of PRRA shows that it is less likely to observe failure for the forth to the seventh spectra.

In Table 5.8, the original results of PRRA are compared with those normalized by the mean value of the likelihood ratio.

Table (5.8): Probability of exceeding 650 MPa using PRRA and comparison with normalized PRRA (1-DOF beam) Normalized PRRA error error PRRA P 1.84E-02 2.04E-02 true spec. 1 f 3.2% 7.4% StDev 2.93E-03 3.24E-03 P 1.06E-02 1.07E-02 true spec. 2 f 1.9% 2.9% StDev 7.98E-04 8.06E-04 P 4.90E-04 4.90E-04 true spec. 3 f 12.5% 12.5% StDev 2.71E-05 2.71E-05 P 1.54E-04 1.54E-04 true spec. 4 f 14.4% 14.4% StDev 1.10E-05 1.10E-05 P 2.53E-05 2.53E-05 true spec. 5 f 1.2% 1.2% StDev 2.84E-06 2.84E-06 P 3.33E-07 3.33E-07 true spec. 6 f 93.3% 93.3% StDev 1.15E-07 1.15E-07 P 1.52E-08 1.48E-08 true spec. 7 f NA NA StDev 8.94E-09 8.72E-09

For the first and the second spectra normalizing deteriorates the accuracy and for the other spectra it does not change the results considerably.

Example 2: Probability of failure in a nonlinear quarter car model

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In this example the probability of failure of a quarter car model is estimated using

PRRA. This model is the same as the one in section 5.4.2 that was used to estimate average up-crossing rate. Similarly, the mean stress due to the weight of the vehicle is not considered. For input spectra similar to those considered in section 4.2 Pierson-

Moskowitz spectrum (Eq. 4.11) for the sampling spectrum is considered with parameters

A and B of 250 m2sec-4 and 4,000 sec-4 , respectively. For the true spectrum, these parameters are assumed to be 312.5 m2sec-4 and 5,000 sec-4, respectively. The sampling and the true spectra were shown in Figure 4-13.

Monte Carlo simulation with 200,000 replications is run to estimate the probability of failure for different levels for the sampling spectrum, and then PRRA is used to estimate the probability of failure for the true spectrum. Then MCs with 200,000 replications is used for the true spectrum and the results are compared with those from PRRA. The replications are run for 1800 sec of time duration.

As shown in the Table 5.9 as threshold becomes higher greater number of replications is required in order to achieve desirable accuracy.

Table (5.9): Probability of failure by MCs for the sampling and the true spectra (quarter car model)

Sampling spec.* True spec.* Stress 50,000 100,000 200,000 50,000 100,000 200,000 levels rep. rep. rep. rep. rep. rep. (MPa) P 1.63E-01 1.61E-01 1.60E-01 3.01E-01 2.99E-01 2.98E-01 500 f StDev 1.65E-03 1.16E-03 8.20E-04 2.05E-03 1.45E-03 1.02E-03 P 4.57E-02 4.49E-02 4.46E-02 1.07E-01 1.06E-01 1.05E-01 550 f StDev 9.34E-04 6.55E-04 4.62E-04 1.38E-03 9.73E-04 6.85E-04 P 8.60E-03 9.00E-03 9.01E-03 2.65E-02 2.66E-02 2.62E-02 600 f StDev 4.13E-04 2.99E-04 2.11E-04 7.18E-04 5.09E-04 3.57E-04 P 3.40E-03 3.59E-03 3.54E-03 1.15E-02 1.17E-02 1.14E-02 625 f StDev 2.60E-04 1.89E-04 1.33E-04 4.77E-04 3.39E-04 2.37E-04 P 1.42E-03 1.46E-03 1.27E-03 4.44E-03 4.65E-03 4.55E-03 650 f StDev 1.68E-04 1.21E-04 7.96E-05 2.97E-04 2.15E-04 1.50E-04 * ~1 day computational cost for each MCs with 200,000 replications (20 rep in parallel)

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To estimate the probability of failure using PRRA, likelihood ratios are calculated by Eq. 4.15. The results of PRRA for the true spectrum with different number of replications are shown in the Table 5.10.

Table (5.10): Probability of failure by MCs for the sampling and the true distribution (quarter car model) True spec. using PRRA Stress levels 50,000 100,000 200,000 (MPa) rep. rep. rep. P 500 f 3.03E-01 2.99E-01 2.96E-01 StDev 3.43E-03 2.42E-03 1.70E-03 P 1.07E-01 1.05E-01 1.04E-01 550 f StDev 2.37E-03 1.67E-03 1.17E-03 P 2.55E-02 2.64E-02 2.63E-02 600 f StDev 1.30E-03 9.29E-04 6.55E-04 P 1.08E-02 1.13E-02 1.12E-02 625 f StDev 8.76E-04 6.29E-04 4.46E-04 P 650 f 4.81E-03 4.98E-03 4.44E-03 StDev 6.00E-04 4.34E-04 2.94E-04 * Almost no additional cost for PRRA

Note that the standard deviation of the results using PRRA is larger when compared to that of MCs and similar to MCs as number of replications increases the accuracy improves. The results of PRRA for different thresholds are compared with those of the

MCs in Figure 5-6. Low and high 95% CIs are also calculated using PRRA (Eq. 3.20) and are shown in Figure 5-6. Notice that the 95% CI widens as the stress level becomes larger.

The agreement between the results of PRRA and those of MCs is excellent and the results of MCs are within the 95% CI calculated using PRRA. Performing MCs with

200,000 replications took one day to run on a desktop computer whereas the results of

PRRA for the true spectrum are calculated almost instantaneously after performing MCs for the sampling spectrum.

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Figure (5-6): Probability of failure by MCs and by PRRA (quarter car model)

Now, the performance of PRRA in the estimation of the probability of failure for the six other spectra as shown in Figure 4.19 is studied. The stress level of 650 MPa is selected to compare the results. The probability of failure and the 95% CI for all spectra are shown in Figure 5-7.

For the all spectra with the exception of the first and the second, the estimates of the probability of exceeding 650 MPa by MCs with 200,000 replications fall within the 95%

CIs which are estimated using PRRA. For the second spectra the results of MCs are not reliable due to the probability of failure being low and performing MCs with 200,000 replications does not yield an accurate result. For the first spectrum, no failure occurs with performing MCs with 200,000 replications.

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Figure (5-7): Probability of failure by MCs and by PRRA (stress level 650 MPa) (quarter car model)

The results are summarized in Table 5.11, and the COV of the results using PRRA and MCs are illustrated in Figure 5-8.

Figure (5-8): Coefficient of variation of the probability of first excursion failure by MCs and by PRRA (stress level 650 MPa) (quarter car model)

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Probabilistic Re-analysis yields to acceptable COV for the second to the seventh spectra. For the third to the sixth, the COV for both MCs and PRRA are about 0.1 hence the results of PRRA are validated by MCs. The COV using PRRA increases as the true spectrum deviates more from the sampling spectrum.

Note that the COV grows faster when the true spectrum moves towards the left of the sampling spectrum rather than towards the right. This is because the support of the true spectrum departs from that of the sampling spectrum when it moves to the left.

Nevertheless, the results of PRRA for the first and second spectra are still useful because they show that it is less likely to observe failure for the true spectra on the left of the sampling spectrum. For the first spectrum no failure occurred by MCs. For the second spectrum the COV of the results by MCs is high, thus more replications are required to acquire desirable accuracy. Nevertheless, more replications might not be necessary since the results using PRRA indicates that it is less likely to observe failure for the first spectrum.

Table (5.11): Probability of exceeding 650 MPa estimated using PRRA and MCs for different spectra (quarter car model) Likelihood COV COV PRRA MCs ratio (MCs) (PRRA) P ------sampling spec. f ------StDev ------P 0.2424 8.39E-08 true spec. 1 f NA NA 0.23 StDev 0.0424 3.85E-08 P 0.9300 1.55E-05 3.00E-05 true spec. 2 f 0.41 0.12 StDev 0.0315 3.55E-06 1.22E-05 P 0.9988 2.24E-04 2.40E-04 true spec. 3 f 0.14 0.07 StDev 0.0032 3.11E-05 3.46E-05 P 0.9988 4.44E-03 4.55E-03 true spec. 4 f 0.03 0.07 StDev 0.0016 5.76E-04 1.50E-04 P 0.9969 1.15E-02 1.19E-02 true spec. 5 f 0.02 0.07 StDev 0.0033 1.66E-03 2.42E-04 P 0.9949 2.43E-02 2.19E-02 true spec. 6 f 0.01 0.08 StDev 0.0057 3.93E-03 3.27E-04 P 0.9933 4.40E-02 4.64E-02 true spec. 7 f 0.01 0.09 StDev 0.0091 8.03E-03 4.70E-04 * ~1 day computational time for each MCs versus no additional cost for PRRA

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The COV of the results of PRRA for the first to the third spectra are less than those of

MCs because compared to those spectra the sampling spectrum is closer to the natural frequency of the system. However for the forth to the seventh spectra the results of PRRA show higher COV for a similar reason.

In Table 5.12, the original results of PRRA are compared with those normalized by the mean value of the likelihood ratio.

Table (5.12): Probability of exceeding 650 MPa using PRRA and comparison with normalized PRRA normalized PRRA error error PRRA P 8.39E-08 3.46E-07 true spec. 1 f NA NA StDev 3.85E-08 1.59E-07 P 1.55E-05 1.67E-05 true spec. 2 f 48.3% 44.4% StDev 3.55E-06 3.82E-06 P 2.24E-04 2.24E-04 true spec. 3 f 6.7% 6.6% StDev 3.11E-05 3.11E-05 P 4.44E-03 4.45E-03 true spec. 4 f 2.4% 2.3% StDev 5.76E-04 5.77E-04 P 1.15E-02 1.15E-02 true spec. 5 f 3.4% 3.1% StDev 1.66E-03 1.67E-03 P 2.43E-02 2.44E-02 true spec. 6 f 11% 11.5% StDev 3.93E-03 3.95E-03 P 4.40E-02 4.43E-02 true spec. 7 f 5.2% 4.5% StDev 8.03E-03 8.08E-03

For the sixth spectrum, the normalizing deteriorates the accuracy. For the other spectra normalizing slightly improves the accuracy.

Example 3: Probability of failure in an offshore wind turbine

In this part the same 5 MW monopile offshore wind turbine model (Jonkman et al.,

2007) that was considered in Chapter 4 is used. In Chapter 4, the wave loads were only considered, but here both wind and wave loads are considered to assess the efficacy of

PRRA in the estimation of the probability of first excursion failure. KAIMAL (IEC-

61400-1 2005) spectrum is used to calculate wind loads and Pierson-Moskowitz spectrum

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(IEC-61400-3, 2009) is used to model wave elevations. The wind spectrum is kept unchanged and only the wave spectrum is changed. To perform simulations TurbSim

(Jonkman & Buhl, 2007) developed by NREL is used to generate wind fields. It is assumed that the average wind speed is 8 m/sec and according to the IEC 61400-1 (2005) standard the turbulence is characterized as “B”. Normal turbulence model (NTM) is considered according to the IEC standard. The wind loads for both the sampling and the true simulation are the same as described above. For the wave spectra Pierson-Moskowitz spectrum (Eq. 4.24) with significant wave height of 4.1 m and with power spectral periods of 10 sec and 9 sec are considered, respectively, for the sampling and the true simulations (Figure 5-9). The environmental conditions are selected according to a possible offshore wind farm in Lake Erie.

Figure (5-9): Sampling and true Pierson-Moskowitz wave spectra

The wave elevations and the wave kinematics are calculated by the code developed as described in Chapter 4 and the simulations are performed using FAST (Jonkman et al.,

2004) for 100 sec of time duration. The probability of exceeding a threshold for the fore- aft bending moment at the base of tower in 100 sec of simulation time is calculated. The

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first 30 sec of simulation results are not considered to avoid the influence of any transient effects.

Table 5.13 summarizes the results that are obtained for 5 different fore-aft bending moment levels by performing MCs with 10,000 replications. The average likelihood ratio and its standard deviation are as 1.009 and 0.016, respectively.

The probability of exceeding higher thresholds is always less than that of lower ones and this results in higher coefficients of variation for higher thresholds as shown in Table

5.13.

The results obtained using PRRA and their comparisons with MCs are summarized in

Table 5.14 and shown in Figures 5-10,11.

Table (5.13): Probability of failure estimated by MCs for the sampling and true spectra (wind turbine)

sampling spec.* true spec.* MCs MCs COV Moment level 10,000 10,000 (true spect.) (KN.m) rep. rep.

Pf 0.370 0.440 70,000 0.01 StDev 0.005 0.005 P 0.141 0.182 75,000 f 0.02 StDev 0.003 0.004 P 0.076 0.103 77,500 f 0.03 StDev 0.003 0.003 P 0.040 0.055 80,000 f 0.04 StDev 0.002 0.002 P 0.020 0.030 82,500 f 0.07 StDev 0.001 0.002 * ~ 5 days computational time for each MCs

Performing MCs with 10,000 replications took 5 days on a desktop computer, whereas the results of PRRA for the true spectrum were calculated almost instantaneously after performing MCs for the sampling spectrum.

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Table (5.14): Probability of failure estimated using PRRA and comparison with MCs (wind turbine)

true spec. error COV Moment level PRRA % (KN.m)

Pf 0.440 70,000 0.0% 0.03 StDev 0.013 P 0.181 75,000 f 0.5% 0.05 StDev 0.009 P 0.096 77,500 f 6.8% 0.07 StDev 0.007 P 0.054 80,000 f 1.8% 0.09 StDev 0.005

Pf 0.027 82,500 10% 0.11 StDev 0.003 * Almost no additional cost for PRRA

Figure (5-10): Probability of excursion using PRRA and MCs (wind turbine example)

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Figure (5-11): The COV using PRRA and MCs (wind turbine example)

Similar to the trends of MCs, as the threshold level increases the corresponding coefficient of variation and the corresponding error increases. Contrary to the expectation, smaller error for the threshold level of 80,000 compared to 77,500. This should be attributed to luck because the coefficient of variation is higher for the larger threshold. Overall, the results of PRRA agree well with those of MCs.

5.5 Discussion

This chapter demonstrated the application of PRRA to estimate the probability of failure for systems under random dynamic loads.

Probabilistic Re-analysis can estimate the expected probability of failure accurately.

In general, PRRA is less accurate than MCs. However, if the sampling spectrum is chosen so that it causes more failures than the true spectrum, and the likelihood ratios exhibit low variation, PRRA can be more accurate than standard MCs.

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The method was demonstrated on three examples. Observations similar to those in

Chapter 4 are made as follows.

– The standard deviation of the estimate of the probability of failure decreases with

the number of replications. Roughly, halving the standard deviation requires

quadrupling the number of replications.

– The accuracy of PRRA decreases as the true spectrum deviates significantly from

the sampling spectrum.

– Most of the time, due to the variability of the likelihood ratio, the standard

deviation of the results of PRRA is larger than that of the sampling MCs. In order

to reduce the standard deviation, the sampling PSD must be carefully chosen by

considering the natural frequencies of the system.

Probabilistic Re-analysis runs almost instantaneously, and it is insensitive to the size and complexity of the structure.

In the design of real life systems, the designer is interested in estimating the probabilities of failure lower than 10-3. Integrating PRRA with the tail fitting methods

(Kim & Ramu, 2006) can potentially improve the efficiency of the sensitivity reliability analysis of highly reliable systems.

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Chapter 6

Integrating Subset Simulation with PRRA to Estimate the

Reliability of Dynamic Systems

Overview

Monte Carlo simulation is a robust tool for reliability assessment, but it can be impractical for most real-life problems due to its high computational cost. This is particularly true for dynamic systems in which response is driven by a time varying excitation with low probability of failure (e.g. less than 10-3).

Reliability can be measured by the calculation of the probability of the response crossing a given threshold during a time period. For real-life complex systems failure region is usually small and is very hard to locate and it is very challenging to find its boundaries. Markov Chain Monte Carlo (MCMC) is a very efficient method for generating samples from the failure region of reliable systems. MCMC was first presented by Metropolis et al. (1953) and was later generalized by Hastings (1970).

Au & Beck (2001) introduced SS, which is based on the idea that a small failure probability can be calculated as a product of larger conditional probabilities of intermediate events. This method is more efficient than MCs as it is much faster to

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calculate several large probabilities rather than a single low probability. The conditional probabilities are estimated by means of Markov Chain Monte Carlo simulation (MCMC)

(Melcher, 1999).

In this chapter a method which combines SS with Shinozuka’s method (1972) is proposed in order to improve the efficiency of SS. In the first section, MCMC and the

Metropolis-Hastings algorithm (1953, 1970) are reviewed. Then, in section 6.2 SS method is explained. In section 6.3, the advantage of running SS by Shinozuka’s method

(1972) is discussed, and in section 6.4, a new approach to integrate PRRA with SS is proposed. In section 6.5 the application of the proposed method on two examples is demonstrated. In the last section the results are discussed.

6.1 Metropolis – Hastings Algorithm

Metropolis-Hastings (1953, 1970) algorithm is a sequential approach that generates a

S Markov Chain whose states follow a target PDF, fX (x) . A state of the chain is a vector containing the random variables in the reliability assessment problem. The algorithm starts from an initial state x0 , and generates a sequence of states , xx 21 ,... as follows.

Repeat k times:

1. Draw candidate state xi from transition PDF h / XX ii 1( xx ii 1)/ . This is the PDF

of state xi conditioned upon the previous state xi1 .

S S 2. Make xi the next state of the chain with probability of  X i ff X xx i1 ),(/)(min 1 .

S S Set  xx ii 1 with the remaining probability, 1max  f X i f X xx i1 0),(/)( . The

algorithm returns to the previous state in this case.

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The transition PDF is symmetric, h / XX ii 1 xx ii  )/(  h /1 XX ii 1 1 xx ii )/( . This algorithm produces a sequence of states with stationary PDF equal to the target one, in other words, if a particular state follows the target PDF, so does the next state.

Markov Chain Monte Carlo enables the user to explore the failure region more efficiently than standard MCs because it uses information from the current state to move to the next state. This is especially important when the failure region is small. On the other hand, it generates dependent states, which increases the variability of the estimates of the probability of failure. In addition, the analyst must well understand the problem and the shape of the failure region. If the failure region consists of multiple disjoint regions (islands) or loosely connected regions, the analyst must generate multiple sequences of points to cover each region with enough points.

The user has to select the transition PDF. In this study, the transition PDF is the probability density of random vector)xi1 1.(   , where  can be a Gaussian random variable, with zero mean value. The components of the vector are statistically independent. The choice of the standard deviation of  is challenging. A small value produces a sequence of highly depended states, with the attendant reduction in the accuracy of the estimates of the probability of failure. On the other hand, a large value of the standard deviation of  increases the frequency that the algorithm returns to the previous state. A standard deviation in the range from 0.05 to 0.3 produced acceptable results in this study.

It is challenging to apply MCMC to a problem with many random variables. In this case, the Metropolis algorithm tends to return to the same state very frequently. Au &

Beck (1999) suggested a modified algorithm that changes one random variable at a time

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in step 1 of MCMC. This algorithm is used in this study because it needs fewer steps than the original to estimate the probability of failure.

6.2 Subset Simulation to Estimate Low Failure Probabilities

Reliability is quantified by the probability of failure and it is assessed by performing

MCs. As mentioned in the section 3.1.1 when the probability of failure is low performing

MCs becomes impractical. Au & Beck (2001) introduced SS to address this drawback. In

SS an expensive problem of estimating a rare probability is broken into several sub- problems with higher probability of failures which are less expensive to solve. The determination of sub-problems is the key step in performing SS. Sub-problems are affected by setting the intermediate probabilities which are needed to approach the final target probability of failure. Let consider a series of thresholds as ,LL 21 to Lr which r is an unknown number of subset subdivisions. Then failure regions F1 to Fr are defined as follows,

i )(:  i  ..2,1 riLxGxF . (6.1)

Then the total probability of failure is the product of intermediate conditional probabilities, FFP ii 1)( and FP 1)( as following:

r f  1  FFPFPP ii 1)()( (6.2) i1

The accuracy of SS depends on the determination of the intermediate failure regions.

The analyst would need to run standard MCs to initiate SS and to calculate FP 1)( . After

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getting the result of the initial MCs it is more convenient to set subsets. The designer can consider subsets such that nearly equal partial failure probabilities are obtained for each subset.

In brief, SS would consist of the following steps:

(1). Run standard MCs to determine FP 1)( and failure region F1 to initiate SS. FP 1)( is

calculated by Eq. 6.3.

K 1 1 ˆ FP )(  )1(  LxGI ))(( (6.3) 1  k 1 K1 k1

(2). Obtain the respective probability functions for each subset by Eq. 6.4.

 i xfLxGI )())(( Fxf i )(  (6.4) FP i )(

(3). Generate samples with the application of MCMC in conjunction with the Metropolis-

Hastings (Hastings, 1970) algorithm to estimate conditional FFP ii 1)( by the

following equation

K 1 i ˆ FFP )(  i)(  LxGI ))(( (6.5) ii 1  k i Ki k1

(4). Calculate the total probability of failure by Eq. 6.2.

The accuracy of the results is usually measured by the COV. The COV of the initial

MCs is estimated using,

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 FP 1)(1 1  . (6.6) )( KFP 11

The COV for the estimates of the intermediate conditional probabilities can be estimated as follows (Au & Beck, 2001):

 FFP ii 1)(1 i    i )1(  ...2 ri . (6.7) 1 )( KFFP iii

Suppose that the number of Markov chains with distinct starting points is Nc then the correlation factor is calculated as follows,

NN 1 c  Nj    c  j)()1(2 (6.8) i  i j1 N

where i ( j) is the correlation coefficient of the Markov chain samples, and j is the separation between two values of the displacement in the chain. In practice, the correlation coefficient between two points in the chain vanishes beyond an upper limit that is less than N. This limit is used in Eq. 6.8 in order to reduce the computational effort of the method.

Assuming independence between the intermediate conditional probabilities in Eq. 6.2, it can easily be shown that the COV of the estimate of the total probability of failure from

SS can be approximated by the following equation (see appendix B):

r 2   i . (6.9) i1

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6.3 Performing Subset Simulation by Using Shinozuka’s Method

According to Shinozuka’s method (Eq. 3.24) every signal is generated by a summation of several cosine series. Addition operator has a commutative property which means that it does not matter how the cosine terms in Shinozuka’s equation are ordered.

This property results in a symmetric failure region in the space of random frequencies.

This property enables performing SS more efficiently by breaking the failure region into smaller regions. Then samples are drawn from these smaller regions instead of drawing samples from the whole design space. This improves the efficiency of SS when compared to the original SS. It is easier to visualize the idea by a considering only two terms in

Shinozuka’s equation. If only two terms are considered in Eq. 3.24, a failure region may look like Figure 6-1. The failure region is symmetric (Figure 6-1) since if the combination of (1, 2) yields failure, the combination of (2, 1) would result in failure, too.

Figure (6-1): First two frequencies causing failure

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Therefore sample frequencies can be drawn from the lower half of the failure region

(Figure 6-2) as shown in the following figure. This significantly improves the efficiency of the method.

Figure (6-2): First two frequencies causing failure (considering symmetry)

6.4 Integrating Subset Simulation with PRRA

In this section PRRA method is integrated with SS. This enables performing a reliability study more efficiently when compared to standard MCs or SS. This goal is achieved by estimating the probabilities that are required to estimate the total probability according to Eq. 6.2 for a new spectra using PRRA . This is similar to the approach that was presented for the estimation of fatigue damage and the probability of first excursion failure using PRRA in Chapters 4 and 5. Similar to PRRA method, the application of

Shinozuka’s method is the key idea because it enables calculating the likelihood ratios.

As first step SS is performed for the sampling spectrum as follows,

S SS S f r  121 FFPFFPFPFP rr 1)(...)()()( . (6.10)

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Then the probability of failure for the first level for the true spectrum is estimated as following,

T 1 N f xS )( ˆ T S X f FP 1)(  ( )x  LGI 1 . (6.11) N  f S xS )( i1 X

The above equation is identical to Eq. 5.8 that was presented in Chapter 5 for PRRA.

For higher thresholds the following equation to estimate the probability of first excursion failure is used,

ˆ T ˆ S ˆ T f )(  fi i1)( PRRA ii 1  ..3,2)( riFFPFPFP . (6.12)

Using PRRA an equation to calculate the conditional probability in the above equation for the true spectrum should be derived. In order to do that the following integral should be evaluated,

T  ()(  T )( dfLGIFFP . (6.13) ii 1  x) i xx DF 1-i

The above equation is rewritten as,

T T f x)( S ii 1  ()( x)  LGIFFP i )( df xx . (6.14)  f S x)( DF 1-i

Assume that Markov chain Metropolis-Hastings (1953, 1970) algorithm is used to generate samples with Q steps for a Markov chain with one initial point. Then the above equation is discretized as following,

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Q 1 f T xS )( ˆ T FFP )(  ( S )x  LGI . (6.15) ii 1  q i S S Q q1 f x )(

If multiple initial points are used then the above equation is converted to,

T 1 P Q f xS )( ˆ T FFP )(  ( S )x  LGI pq . (6.16) ii 1  pq i S S QP p11q f xpq )(

The above equation is rewritten as,

P Q T 1 FFP )(  ( S  LGI  (6.17) PRRA ii 1  pq )x pqi PQ p11q

where in the above equation, pq , is likelihood ratio.

Assuming independence between the selected P initial points the standard deviation of the estimate of above conditional probability can be calculated by,

2 P  Q   1 1 S T 2  T   pq )x  LGI  pqi   PRRA FFPP ii 1))(()(( PRRA FFP ii 1)( PP  )1( Q p1  q 1 

(6.18)

ˆ T The coefficient of variation of the estimate of Pf Fi )( is estimated by,

2 2 T   S  ˆ  ..3,2 ri (6.19) f FP i )( f FP i1)( FFP iiPRRA 1)(

The Subset-PRRA simulation presented above is summarized in Figure 6-3. To perform SS all SS1-7 steps shown in Figure 6-3 should be taken. The SS2 and SS5 steps

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are computationally most expensive part of the simulation. As the figure shows, PRRA helps bypassing these two steps thereby reducing the computational effort significantly.

Step: SS1, Subset-PRRA1

Draw sample values of the frequencies and phase angles from their sampling distributions.

Step: Subset-PRRA2

Calculate likelihood ratios for Step: SS2 each simulated time history

Perform initial MCs and calculate the system response Initial MCs Step: Subset-PRRA3 Step: SS3 Estimate the probability of failure of Calculate the probability of first threshold for the true distribution failure by scaling by the likelihood ratio. (Eq. 6.11)

Step: SS4, Subset-PRRA4

Setup Markov chains using the results of the initial MCs and generate samples by Metropolis- Hastings algorithm Step: Subset-PRRA5

Subset Simulation Calculate likelihood ratios for

each simulated time history Subset-PRRA Step: SS5

Calculate the system response Markov chain Markov chain Step: Subset-PRRA6

Estimate the conditional Step: SS6 probabilities of failure for the true distribution by scaling by the Calculate the conditional likelihood ratio. (Eq. 6.17) probabilities of failure

Step: SS7 Step: Subset-PRRA7 Calculate the total Calculate the total probability of probability of failure for failure for sampling distribution by true distribution by Eq. Eq. 6.2. 6.12.

Figure (6-3): Subset-PRRA simulation flowchart

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In section 6.5 the application of Subset-PRRA in estimating the probability of failure of a linear 1-DOF and quarter car models is demonstrated. The efficacy of Subset-PRRA is compared with that of the PRRA method in estimating reliability.

6.5 Application

Example 1: Probability of failure in a 1-DOF system

In part (a) of this example, the probability of first excursion failure by SS is estimated using the approach that was proposed in this chapter. In part (b), using the results of part

(a) and Subset-PRRA the probability of first excursion failure for different PSDs of excitation are estimated. Then in part (c) the probability of first excursion failure is estimated using PRRA and its efficiency is compared with that of Subset-PRRA. The results are validated by MCs with 500,000 replications.

Part (a): Subset Simulation

In this example a linear 1-DOF system is considered. It is assumed that the system has a mass of 1 Kg and natural frequency of n=0.4 rad/sec. A damping ratio of  = 0.05, is considered which result in damped natural frequency of d =0.399 rad/sec. The system is excited by a random load, which is a stationary, Gaussian random process, whose spectrum is the average of two Pierson-Moskowitz (Eq. 4.11) spectra. The resultant PSD function no longer belongs to Pierson-Moskowitz family.

For the first Pierson-Moskowitz spectrum (Eq. 4.11), the parameters A and B equal to

5 m2sec-4 and 0.0562 sec-4, and for the second spectrum, the parameters A and B equal to

20 m2sec-4 and 0.225 sec-4. To generate realizations by Shinozuka’s method, frequencies

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are drawn from each spectrum 50 percent of the time. The resultant spectrum is the average of the two Pierson-Moskowitz spectra as shown in Figure 6-4. The two spectra are chosen such that the peak frequency of the resultant spectrum is close to the natural frequency of the system.

Figure (6-4): Input spectrum as combination of two Pierson-Moskowitz espectra

The linearity of the system allows using the frequency response method to calculate the system response under random loads. The frequency response function and the phase angle of the response are calculated as follows:

1 H )(  k 2  2      2 1      )2( (6.20)       n   n

  2    Arc   2,)(1tan)(    n n 

The above functions are plotted in Figure 6-5.

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(a) (b)

Figure (6-5): (a) Frequency response function (b) Phase angle function

Since the system is linear, the total response of the system is the linear superposition of the responses for every harmonic in the input signal included in the Shinozuka’s Eq.

3.24. Monte Carlo simulation with 500,000 replications is performed in order to obtain the probability of failure for different thresholds as shown in Table 6.1. These results are used as reference to validate the results from the proposed approach.

The probability of up-crossing thresholds from 240 m to 420 m is estimated using SS.

First, the probability of up-crossing 240 m is estimated using MCs with 5,000 replications. This simulation generates around 450 up-crossings. The largest two frequencies in these replications are shown in Figure 6-6. Then the corresponding probabilities for thresholds in the range [240 m, 300 m], conditioned upon up-crossing

240 m, are calculated using MCMC simulation with 400 initial states and 25 steps for every state. This process is repeated for the ranges of [300 m, 360 m] with 80 states and

125 steps for each state, and [360 m, 420 m] with 25 states and 400 steps for each state.

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Thus, the total of 10,000 steps for every range is used. Finally, the up-crossing probability for each threshold is calculated from Eq. 6.2.

Table (6.1): Probability of failure using MCs for 1-DOF system with 500,000 replications

Threshold 500,000 COV m rep. P 240 f 0.0860 0.005 StDev 0.0004 P 0.0543 260 f 0.006 StDev 0.0003 P 0.0328 280 f StDev 0.0003 0.008 P 0.0195 300 f 0.010 StDev 0.0002 P 0.0115 320 f 0.013 StDev 0.0002 P 6.54E-03 340 f 0.017 StDev 1.14E-04 P 3.65E-03 360 f 0.023 StDev 8.53E-05 P 1.99E-03 380 f 0.032 StDev 6.31E-05 P 1.05E-03 400 f 0.044 StDev 4.58E-05 P 420 f 5.38E-04 0.061 StDev 3.28E-05 * ~5 days computational time for MCs with 500,000 rep.

Standard normal PDF with zero mean is used to generate Markov chains. In Figure

6-6, the two highest frequencies of the harmonics of the initial points for the three thresholds are shown with blue crosses and red dots represent the rest of the points for each chain.

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a) Threshod 240 m b) Threshold 300 m

c) Threshold 360 m

Figure (6-6): Values of the first 2 frequencies for Markov chains (Initial points: blue crosses, remaining points of each chain: red dots)

All points in the above figure are in the first quadrant because the second frequency is always less than or equal to the first. Note that most samples are populated close to the tip

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of the triangle which is close to the natural frequency of the system. The above figure demonstrates the fluctuation of the frequencies through the Markov chains.

The estimated probabilities of up-crossing from SS are compared to those from MCs with 500,000 time histories. Table 6.2 summarizes the results. Figure 6.7 shows up- crossing probabilities obtained from MCs and SS agree well. Note that SS requires calculation of only 35,000 time histories: 5,000 for the initial threshold of 240 m, plus

3×10,000 for the three thresholds.

Table (6.2): Probability of failure by SS for 1-DOF system

35,000 Conditional Threshold rep. probability P 240 f 8.88E-02 N/A COV 0.045 P 5.27E-02 P(260/240) 0.5931 260 f COV 0.075 COV 0.0596 P 3.16E-02 P(280/240) 0.3561 280 f COV 0.060 COV 0.0397 P 1.92E-02 P(300/240) 0.2158 300 f COV 0.045 COV 0.0001 P 1.13E-02 P(320/300) 0.5888 320 f COV 0.045 COV 0.0021 P 6.60E-03 P(340/300) 0.3446 340 f COV 0.045 COV 0.0037 P 3.93E-03 P(360/300) 0.205 360 f COV 0.045 COV 0.0001 P 2.35E-03 P(380/360) 0.5975 380 f COV 0.046 COV 0.0053 P 1.16E-03 P(400/360) 0.2955 400 f COV 0.052 COV 0.0251 P 6.59E-04 P(420/360) 0.1678 420 f COV 0.074 COV 0.0590

The COV is estimated differently from the approach presented in section 6.2. Since the initial states are randomly selected from the results of the MCs with 5,000 replications, the standard deviation of the probability of failure is calculated similar to

Eq. 3.19. The standard deviation of each estimate of the conditional probabilities in

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Table 6.2 and the COV of the up-crossing probability of each level are calculated by Eq.

3.19 and Eq. 6.9, respectively.

Figure 6-7 demonstrates the result of SS with 35,000 replications and those of MCs with 500,000 replications for different thresholds. The probabilities of excursion are estimated during 200 sec of the system response for the sampling spectrum.

Figure (6-7): Probability of up-crossing a threshold during a 200 second period (MCs with 500,000 replications vs. SS with 35,000 replications)

Figure 6-8 shows the COV for the SS with 35,000 replications and those of MCs with the same number of replications. Note that for low thresholds where the up-crossing probabilities are higher, the COV for SS is higher than that of MCs. However, for higher thresholds with lower probability of up-crossing, SS is more accurate than MCs as evidenced by the much lower values of the COV of the results of this method.

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Figure (6-8): The COV of up-crossing probability for MCs and SS

If a few states with long Markov chains are used, the approach presented in section

6.2 should be used to estimate the COV of the failure probability. A similar study with only one initial state with a very long chain (Norouzi & Nikolaidis, 2012) have found that it is more efficient to explore the failure region by using multiple initial states with shorter Markov chains, rather than a single or few states with very long chains. The main contribution to the total coefficient of variation of the failure probability is due to the variation of the probability of up-crossing the level of 240 m.

Part (b): Subset-PRRA simulation

In this part Subset-PRRA method is used to estimate the probability of first excursion for another input spectrum as shown in Figure 6-9 using the results of previous section.

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The new spectrum also belongs to the Pierson-Moskowitz family (Eq. 4.11) with parameters A and B as 12 m2sec-4 and 0.135 sec-4 , respectively.

Figure (6-9): Sampling and true spectra for 1-DOF example

The natural frequency of the system, which is 0.4 rad/sec, is closer to the main body of the sampling spectrum than the true one. Therefore, more failures are observed in a simulation for the sampling spectrum than the true one.

Monte Carlo simulation with 500,000 replications is performed to estimate the probability of failure for different thresholds for the true spectrum. The results are summarized in Table 6.3. The estimated failure probabilities for thresholds of 340 m and

360 m are not accurate, even with 500,000 replications.

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Table (6.3): Probability of failure by MCs for 1-DOF system for the true spec.

Threshold 500,000 COV (m) rep. P 240 f 4.86E-03 0.02 StDev 9.84E-05 P 2.35E-03 260 f 0.03 StDev 6.85E-05 P 1.03E-03 280 f 0.04 StDev 4.54E-05 P 4.46E-04 300 f 0.07 StDev 2.99E-05 P 1.90E-04 320 f 0.10 StDev 1.95E-05 P 8.80E-05 340 f 0.15 StDev 1.33E-05 P 3.60E-05 360 f 0.24 StDev 8.49E-06 P 2.20E-05 380 f 0.30 StDev 6.63E-06 P 8.00E-06 400 f 0.50 StDev 4.00E-06 P 420 f 2.00E-06 1.00 StDev 2.00E-06 * ~5 days computational time for MCs with 500,000 rep.

As expected, the probability of first excursion failure is lower for the true spectrum than for the sampling one. The result of SS from part (a) is used to perform Subset-PRRA to estimate the corresponding probabilities of failure for the true spectrum. The results are shown in Figure 6-10.

Figure 6-11 illustrates the variation of the COV for SS for the sampling spectrum that was obtained in previous section along with the corresponding results from Subset-

PRRA, MCs for the true spectrum with 35,000 replications.

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Figure (6-10): Probability of first excursion failure (No additional computational cost for Subset-PRRA)

Figure (6-11): The COV by different methods (35,000 rep.)

It is interesting that the accuracy of estimates from Subset-PRRA is higher than that from MCs with the same number of replications (35,000 rep.). The reason is that Subset-

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PRRA simulation uses the results of SS, which are far more accurate than those of MCs for high thresholds (e.g., thresholds higher than 300 m). After applying PRRA, the accuracy deteriorates slightly compared to the sampling spectrum but it is still superior to

MCs.

Part (c): Comparison of PRRA with Subset-PRRA

The probability of failure for the true spectrum is estimated using PRRA method and is compared with those from MCs and Subset-PRRA in Figure 6-12.

Figure (6-12): Probability of first excursion failure by Subset-PRRA versus PRRA (95% CI by 500,000 MCs) (No additional computational cost for Subset-PRRA & PRRA)

The CI in Figure 6-12 is estimated by MCs with 500,000 replications. The results of both PRRA and Subset-PRRA simulations fall within the CI. The COV of the results are compared in Figure 6-13.

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Figure (6-13): Coefficient of variation by Subset-PRRA versus PRRA (with 35,000 replications)

The coefficient of variation of the results of PRRA is smaller than that by Subset-

PRRA for low thresholds. However for high thresholds with small probability of failure

(<10-3) Subset-PRRA is more accurate. For thresholds 340 m and 360 m, the results of

Subset-PRRA are more accurate than those of MCs for sampling spectrum. The COV of the last threshold (420 m) for PRRA is smaller than that of Subset-PRRA, this should be attributed to luck. Overall, for very low probability of failure Subset-PRRA would yield more accurate results than PRRA or even MCs (not always).

Example 2: Probability of failure in a linear quarter car model

The probability of failure in a suspension spring of a linear quarter car model is studied (Figure 6-14) in this section. The model is excited by road elevation, u(t), which is random.

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Figure (6-14): Linear quarter car model

In the above model, masses m1 and m2 are 75 and 300 Kg. Damping coefficients are c1 = 4000 and c2 = 7000 Kg/sec and the spring rates are k1 = 400 and k2 = 40 kN/m.

The equations of motion are as follows,

  )(  )(  ucukxkxkkxcxccxm 1111221212212111  . (6.21)   xxkxxcxm 21221222  0)()(

The above system of equations is solved by frequency response approach (Inman 2001).

In order to do that the above equations are rewritten as

  tFtXKtXCtXM )()()()( . (6.22)

The dynamic stiffness matrix is defined as below:

() 2   jCMKD . (6.23)

Load vector is defined as follows

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  11  jck  F )(    . (6.24)  0 

The response of the system is simply calculated by:

 1 FDX  )()()( . (6.25)

From the above equation, the frequency response functions that relate the road elevation to the displacements of first and second masses are calculated. They are named as H1() and H2(). There is a phase difference between the response of each mass (1 )( , 2 )(

) and the excitation. These functions are plotted in Figure 6-15.

Figure (6-15): a) Frequency response functions, b) Phase angle functions

The excitation is calculated by Shinozuka Eq. 3.24. Since the system is linear the total solution of the system is the linear combination of the response for every individual

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harmonics in the Shinozuka Eq. 3.24. Thus, the response of the first and second masses is calculated using the following equations:

N  2   Htx )(1)( cos( t   ))(1 1  i iii  i1  N  (6.26) N  2   Htx )(2)( cos( t   ))(2 2  i ii i  i1  N 

Then the displacement of the suspension spring is simply calculated by subtraction of the above equations. Frequency response method is the most efficient method to calculate response of the system. However, it is only valid for a linear system.

The suspension spring is subjected to an alternating shear stress resulting from the road elevation. Properties of the suspension spring are presented in Table 6.4.

Table (6.4): Properties of the suspension spring Quantity Value (units) Wire diameter, d 1.91 cm Number of active turns, N 10 Coil diameter, D 15 cm

Ultimate stress, Sut 1240 MPa Shear modulus, G 81 GPa Spring index, C=D/d 7.85

Spring rate, k2 40 KN/m

The spring rate k2, is calculated using Eq. 4.17.

The dynamic response of the system is obtained, the deflection of the suspension spring (x1-x2) is calculated, and then using Eq. 4.18 the shear stress time history is calculated. The material properties of the spring are as follows (E = 210 GPa , G = 81

GPa ,  = 7850 Kg/m3)

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The PSD of the road elevation is considered to be Pierson-Moskowitz spectrum (Eq.

4.11) with A and B as 570 m2sec-4 and 3,654 sec-4. Monte Carlo simulation with 500,000 replications with 20 sec of response time is run. Then the probabilities of failure for different thresholds are calculated. The results which are summarized in Table 6.5 are later used for validation of the results obtained by the proposed approach.

Table (6.5): Probability of failure by MCs* with 500,000 replications Stress levels 500,000 COV (MPa) rep. P 770 f 1.27E-01 0.00 StDev 4.70E-04 P 7.19E-02 815 f 0.01 StDev 3.65E-04 P 3.84E-02 860 f 0.01 StDev 2.72E-04 P 1.94E-02 905 f 0.01 StDev 1.95E-04 P 9.33E-03 950 f 0.01 StDev 1.36E-04 P 4.27E-03 995 f 0.02 StDev 9.22E-05 P 1.82E-03 1040 f 0.03 StDev 6.03E-05 P 7.32E-04 1085 f 0.05 StDev 3.82E-05 P 2.80E-04 1130 f 0.08 StDev 2.37E-05 P 1.16E-04 1175 f 0.13 StDev 1.52E-05 * ~ 5 days computational cost for MCs

The probability of up-crossing of several thresholds is estimated from 770 MPa to

1175 MPa using SS. First, the probability of up-crossing the threshold of 770 MPa is calculated using MCs with 10,000 replications. This simulation generates several failure points. Five hundreds of these are selected to set up the MCMC. Then the corresponding probabilities are calculated for thresholds in the range from 770 MPa to 905 MPa, conditioned upon up-crossing the previous threshold, by using the Metropolis-Hastings algorithm (1953, 1970) with 20 states. This process is repeated for the ranges of [905

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MPa, 1040 MPa] with 200 initial states and 50 states. For the last range of [1040 MPa ,

1175 MPa] 20 initial states are used with 500 states. The up-crossing probability for each threshold is calculated by Eq. 6.2.

Figure 6-16 shows the variation of the first two frequencies of the harmonics of the excitation for three thresholds. The cross marks the initial state. All points in the figure are in the lower triangle because the second frequency is constrained to be less than or equal to the first.

In the Table 6.6 the corresponding conditional probabilities and the estimated probability of failure for different thresholds are summarized.

The estimated probabilities of up-crossing from SS are compared with the results from standard MCs with 500,000 time histories. Figure 6-17 shows that the results of the two methods agree well. Subset Simulation required only calculation of 40,000 time histories: 10,000 for the initial threshold of 770 MPa, plus 3×10,000 for the three thresholds.

The coefficient of variation is estimated similar to the approach explained in the previous example. Figure 6-18 shows that the COV for the SS with 40,000 replications and those of MCs with the same number of replications are comparable. For the low thresholds where the up-crossing probabilities are higher the COV for SS is higher than that of MCs. However, for higher thresholds with lower probability of up-crossing, SS is superior to MCs because it results in smaller COV.

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Figure (6-16): Values of the first 2 frequencies for Markov chains (Initial points: blue crosses, remaining points of each chain: red dots)

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Table (6.6): Probability of failure by SS for the quarter car model

Threshold 40,000 Conditional MPa rep. probability P 770 f 1.21E-01 N/A COV 0.027 P 7.00E-02 P(815|770) 0.580 815 f COV 0.036 COV 0.023 P 3.87E-02 P(860|770) 0.321 860 f COV 0.050 COV 0.042 P 2.02E-02 P(905|770) 0.167 905 f COV 0.071 COV 0.065 P 1.01E-02 P(950|905) 0.500 950 f COV 0.081 COV 0.030 P 4.54E-03 P(995|905) 0.225 995 f COV 0.092 COV 0.052 P 1.77E-03 P(1040|905) 0.088 1040 f COV 0.113 COV 0.083 P 7.26E-04 P(1085|1040) 0.410 1085 f COV 0.139 COV 0.031 P 2.78E-04 P(1130|1040) 0.157 1130 f COV 0.155 COV 0.075 P 1.13E-04 P(1175|1040) 0.064 1175 f COV 0.183 COV 0.123

Figure (6-17): Probability of up-crossing a threshold during a 20 second period

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Figure (6-18): Coefficient of variation for SS and MCs

Part (b): Subset-PRRA simulation

Here, the results obtained in previous part are used to estimate the probability of first excursion for another input spectrum that is shown in Figure 6.19. This spectrum belongs to the Pierson-Moskowitz family (Eq. 4.11) with parameters A and B as 468 m2sec-4 and

3,000 sec-4, respectively.

Figure (6-19): Sampling and true spectra for the linear quarter car example

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Monte Carlo simulation with 500,000 replications and estimated the probability of failure for different thresholds for the true spectrum. The results are summarized in Table

6.7. The estimated failure probabilities for thresholds of 770 MPa and 1175 MPa are not accurate, even with 500,000 replications.

Table (6.7): Probability of failure by MCs* for quarter car system for the true spectrum

Threshold 500,000 COV (MPa) rep. P 770 f 7.42E-02 0.00 StDev 3.71E-04 P 3.92E-02 815 f 0.01 StDev 2.75E-04 P 1.96E-02 860 f 0.01 StDev 1.96E-04 P 9.26E-03 905 f 0.01 StDev 1.35E-04 P 4.18E-03 950 f 0.02 StDev 9.13E-05 P 1.75E-03 995 f 0.03 StDev 5.91E-05 P 7.24E-04 1040 f 0.05 StDev 3.80E-05 P 2.84E-04 1085 f 0.08 StDev 2.38E-05 P 1.00E-04 1130 f 0.14 StDev 1.41E-05 P 1175 f 2.60E-05 0.28 StDev 7.21E-06 * ~5 days computational cost for MCs with 500,000 rep.

The results from part (a) are used to perform Subset-PRRA to estimate the corresponding probabilities of failure of the true spectrum. The results of MCs and those from Subset-PRRA for the true spectrum are shown in Figure 6.20. The CI shown in

Figure 6-20 was estimated using MCs with 500,000 replications.

Figure 6-21 illustrates the variation of the COV of the SS for the sampling spectrum from the previous section along with the corresponding results from Subset-PRRA and

MCs for the true spectrum with 40,000 replications.

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Figure (6-20): Probability of first excursion failure (No additional computational cost for Subset-PRRA)

Figure (6-21): The COV for the true and the sampling spectra using different methods

It is interesting that the accuracy of estimates by Subset-PRRA is higher than that from MCs with the same number of replications (40,000 rep.). The reason for this is due

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to the fact that Subset-PRRA simulation is built upon the results of the SS of sampling spectrum with high accuracy. The accuracy deteriorates slightly compared to the sampling spectrum but it is still superior to MCs.

Part (c): Comparison of PRRA with Subset-PRRA

The probability of failure for the true spectrum is estimated using PRRA and using

MCs of the sampling spectrum with 40,000 replications. The results by different methods are plotted in Figure 6-22.

Figure 6-23 illustrates the variation of the COV for the SS of the sampling spectrum from the previous section along with the corresponding results of PRRA-Subset, MCs and PRRA for the true spectrum with 40,000 replications.

Figure (6-22): Probability of first excursion failure by Subset-PRRA versus PRRA (95% CI by 500,000 MCs) (No additional computational cost for Subset-PRRA or PRRA)

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Figure (6-23): The COV by Subset-PRRA versus PRRA and MCs (with 40,000 replications)

The accuracy of estimates by Subset-PRRA is higher than those from MCs with the same number of replications (40,000 rep.). The reason for this is due to the fact that

Subset-PRRA simulation is built upon the results of the SS of the sampling spectrum with high accuracy. The accuracy deteriorates slightly compared to the sampling spectrum, yet is still superior to that of the MCs of the true spectrum. For higher thresholds with small probability of failure (<10-4) Subset-PRRA is more accurate than PRRA. The results of

Subset-PRRA are less accurate for the lower thresholds because 10,000 replications

(MCs) are used in order to initiate the SS.

In order to study the efficacy of PRRA and Subset-PRRA other spectra are considered according to Figure 6-24. All spectra are Pierson-Moskowitz with parameters shown in Table 6.8. The probability of first excursion is estimated for all spectra using

PRRA and Subset-PRRA and MCs is used to validate the results.

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Figure (6-24): The different true spectra with the same amount of energy

Monte Carlo simulation with half a million replications is used for all the spectra shown in Figure 6-24. The corresponding probability of first excursion for different thresholds of 950 and 1170 MPa are plotted along with their corresponding 95% CI in

Figure 6-25.

Table (6.8): Pierson-Moskowitz spectra parameters for the spectra shown in Figure 6.24 A B m2sec-4 sec-4 sampling spec. 570 3,654 true spec. 1 312 2,000 true spec. 2 390 2,500 true spec. 3 468 3,000 true spec. 4 663 4,250 true spec. 5 780 5,000 true spec. 6 850 5,450 true spec. 7 936 6,000

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Figure (6-25): Probability of first excursion for the thresholds of 950 and 1175 MPa for the spectra shown in Figure 6.24, by MCs with 500,000 replications (~5 days computational cost for each MCs with 500,000 rep.)

Monte Carlo simulation and SS with 40,000 replications are performed for the sampling spectrum and the results are used to perform PRRA and Subset-PRRA. The

COV by different methods for all the spectra shown in Figure 6-24 are calculated for different thresholds. The results corresponding to the first threshold (950 MPa) are shown in Figure 6-26 and the counterpart results for the second threshold (1170 MPa) are shown in Figure 6-27.

According to Figure 6-26 for the low threshold of 950 MPa, the COV of PRRA is less than that of Subset-PRRA. This is similar to the trends that were observed before.

Since PRRA is initiated with MCs with 40,000 replications it is more accurate for lower thresholds whereas for the higher threshold of 1170 MPa with low probabilities of failure, as shown in Figure 6-27, Subset-PRRA is more accurate. The results of both

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PRRA and Subset-PRRA deteriorate by increasing the deviation of the true spectra from the sampling one.

Figure (6-26): Coefficient of variation using PRRA, Subset-PRRA and MCs with 40,000 rep. for the threshold of 950 MPa

Figure (6-27): Coefficient of variation using PRRA, Subset-PRRA and MCs with 40,000 rep. for the threshold of 1170 MPa

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6.6 Discussion

In this chapter, a new approach was proposed to improve the efficiency of SS. This approach takes advantage of the symmetry of the response function in Sinozuka’s method to reduce the size of the space of random variables.

Moreover, an efficient method called Subset-PRRA was presented to estimate the probability of first excursion failure for systems under random dynamic loads. The efficacy of the proposed method was compared to PRRA. Subset-PRRA uses the results of the SS for the sampling spectrum to estimate the probability of first excursion for another spectrum. Both methods are applicable to both linear or nonlinear systems. The method was applied to two examples, and the following conclusions are reached.

– Probabilistic Re-analysis can estimate the expected probability of failure accurately.

In general, PRRA is less accurate than MCs. However, if the sampling spectrum is

chosen such that it causes more failures than the true one, PRRA can become more

accurate than standard MCs.

– Subset-PRRA can estimate the expected probability of failure for high thresholds

more accurately than MCs and PRRA. However, for lower thresholds with high

probability, PRRA is more accurate than Subset-PRRA. The choice of sampling

spectrum is also important to achieve higher accuracy using Subset-PRRA. For this

purpose, the user should choose a sampling spectrum that causes many failures.

However, for large-scale complex systems, the selection of a good sampling

spectrum might be challenging.

– The accuracy of both PRRA and Subset-PRRA deteriorate when the deviation of

the true spectrum from the sampling spectrum increases.

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Probabilistic Re-analysis and Subset-PRRA are performed almost instantaneously, and they are insensitive to the size and complexity of the structure. Both methods can reduce the computational cost of sensitivity reliability analysis of structures under random dynamic loads.

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Chapter 7

Probabilistic Design of an Offshore Wind Turbine

Overview

The objective of this chapter is to develop and validate a method and a computer program for reliability assessment and probabilistic design of an offshore wind turbine.

The scope is limited to the first excursion failure of the base of the monopile tower at mud line, and the root of the blades.

This part of the study focuses on the National Renewable Energy Laboratory (NREL) offshore 5-MW baseline wind turbine (Jonkman et al., 2007). This is a conventional three-bladed upwind variable-speed variable blade-pitch-to-feather-controlled turbine.

For offshore application, in this study it is assumed that this turbine is installed on a monopile platform in Lake Erie, 8 miles away off the coast of Cleveland (Marschall et al., 2009).

NREL has developed a FAST model of the 5-MW baseline wind turbine to support concept studies aimed at assessing offshore wind technology (Jonkman et al., 2004).

This FAST model is used to predict the response of this wind turbine to wind and waves.

This model was already used in Chapters 4 and 5 to demonstrate PRRA for the estimation of fatigue damage and first excursion failure.

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FAST uses linear wave theory (Jonkman, 2007) to calculate the hydrodynamics loads.

This might result in inaccurate estimation of the hydrodynamic loads, especially for steep waves in shallow waters, as shown in Appendix A.

The main tasks of the work in this chapter are as follows. In Section 7.1, the available metrological data of the tentative location of an offshore wind farm in Lake Erie is used to model the loading environment. In section 7.2, the target probability of failure that is required to have a desirable service life for an offshore wind turbine is calculated.

Section, 7.3 demonstrates a probabilistic design approach for a wind turbine structure under the Lake Erie loading environment. The focus is on the estimation of the required load capacity of the monopile at the mud line to meet a target annual probability of failure, corresponding to a desired service life. Since the wind turbine in this study is still at the concept stage, only two failure modes are used; first excursion failure of the monopile at the mud line and that of a blade at the root. A practical design of a wind turbine structure requires a more detailed analysis considering all failure modes. The approach in this chapter is applicable to other failure modes of offshore wind turbines in different loading environments.

7.1 Model the Loading Environment

Wave heights and wind speeds are dependent; large waves are associated with high wind speeds and wave periods. It is important to consider this dependence in order to avoid overestimating the reliability. A complete model of the wave elevation and the wind speed or the wave elevation and the spectral period consists of the joint PDFs of these quantities. However, it is impractical to develop such a model because a large amount of data is required to find a family of joint probability distributions that represent

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the relationship between the wind speed and wave height or the wave height and the spectral period. Many researchers assume that the wave height follows one of the standard distributions such as Gumbel (1958), Weibull (1951) or Generalized Extreme

Distribution conditioned on the wind speed according to the approach presented in the

IEC standards (IEC 61400-3 ed.1, Annex G). This assumption can lead to significant errors in the estimated reliability of an offshore wind turbine structure because the type of the probability distribution of the wave height could depend on the value of the wind speed. The following paragraph proposes an approach to circumvent this difficulty.

In order to model the joint distribution using the meteorological data at a proposed location of a wind farm in Lake Erie (Marschall et al., 2009) copulas are used (Nelsen,

2006). Then, MCs is used for generating sample values of the wind and wave, and the samples are compared with the data from Lake Erie site to validate the selected copula model. This approach is more complete because the joint distribution of the wind speed and the wave height and period is obtained without making any assumptions on the conditional distributions of these quantities.

Subsection 7.1.1 introduces the theory of copulas. Subsection 7.1.2 reviews available meteorological data and uses copulas to find the joint distribution of the wind and wave conditions at the Lake Erie site. In the same subsection, MCs is performed and validated by the constructed model using the data from the National Oceanic and Atmospheric

Administration (NOAA) for the Lake Erie environment.

7.1.1 Copulas for modeling dependence

Copulas are functions that couple a multivariate probability distribution function of a set of random variables to the one-dimensional marginal probability distribution

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functions of these variables. Sklar's theorem (Sklar 1959, Nelsen 2006) provides the theoretical foundation for the application of copulas. Let FUV be a joint CDF with margins FU and FV. Then, according to Sklar’s theorem, there exists a copula function, C, such that for all combinations of values of the cumulative probability distribution functions, u, v,

UV  VU vFuFCvuF ))(),((),( . (7.1)

For continuous FU and FV, C will be unique. Conversely, if C is a copula and FU and

FV are distribution functions, then function F defined by Eq. 7.1 is a joint distribution function with margins FU and FV

Copulas enable a designer to construct the joint CDF of random variables in two steps. The first step is to estimate the marginal probability distributions of the random variables using data. In the second step, the CDF functions are integrated by a copula function. For further reading on copulas refer to Nelson (2006). In the next section, the above steps are demonstrated by constructing the joint distribution of the wind speed, the significant wave height and the spectral period from data at a tentative location of a wind farm in Lake Erie.

7.1.2 Joint distribution of the wind and wave data for the Lake Erie site

In this section, the available meteorological data that are available through the

NOAA2 buoy (station 45005) are reviewed. This buoy is located at position 41°40'36" N

82°23'54" W (Figure 7-1). Table 7.1 lists the conditions at the location of the buoy and its main characteristics.

2 http://www.ndbc.noaa.gov/

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Figure (7-1): Location of the reference buoy in Lake Erie (Courtesy of Google Earth)

Although the buoy is 28 nautical miles from Cleveland and 40 km from the outmost corner of the potential offshore wind farm (Marshall et al., 2009), the wave data may be assumed representative of the coastal area of Lake Erie and the project area.

Table (7.1) Main buoy characteristics Site elevation 173.9 m above mean sea level Height at which air 4 m above site elevation temperature is measured Anemometer height 5 m above site elevation Barometer elevation 173.9 m above mean sea level Water depth 12.6 m

The first step in order to obtain a copula is to estimate the marginal PDFs of the significant wave height and the wind speed from the data. The Gumbel, Generalized

Extreme and Weibull probability distributions are good candidates for characterizing extreme loads/effects (Nikolaidis et al., 2010).

More than 30 years of historical data (since 1980) for wind speed, wave elevation, and dominant wave period are available in the NOAA website for the buoy number

45005. The measured wind speed at the buoys is calculated by averaging over an eight-

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minute period. The significant wave height is calculated during the 20 minute sampling period, and the dominant wave period is the period of the waves with maximum energy.

After removing the unreliable values, the triplets of extreme wind speed, significant wave height and wave period are extracted for each year. A scatter diagram of the wind speed and wave height are plotted to find the extreme condition for every year.

A sample scatter diagram of year 2002 is illustrated in Figure 7-2. Every dot represents a sample pair of wind speed and significant wave height taken during this year.

For each year, the extreme pair is selected by considering primarily the wind speed. The wind speed and wave height are strongly correlated according to the above figure. The point at the rightmost corner is selected as the extreme condition for each year and identified the corresponding period from the data. Table 7.2 summarizes the data.

Figure (7-2): Scatter diagram of wind speed versus significant wave height for the data taken in 2002

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Table (7.2): Extreme wind, wave and period (25 years) Year Wind speed Significant wave height Wave period m/sec m sec 1980 14.7 2.20 5.00 1981 17.4 2.50 5.00 1982 15.7 2.30 4.80 1983 13.8 2.30 5.00 1984 16.1 1.90 4.50 1985 12.9 2.10 5.90 1988 18.4 1.00 3.60 1989 13.4 1.10 3.80 1990 16.9 2.40 5.60 1991 15.3 2.50 2.90 1992 18.0 2.70 5.60 1993 17.3 2.50 5.00 1994 16.8 2.00 5.60 1995 17.1 3.40 8.30 1997 18.0 2.60 5.88 1998 19.0 2.85 6.25 1999 16.4 2.11 4.55 2000 16.6 2.26 5.56 2001 18.3 2.45 5.88 2002 18.5 2.98 5.24 2003 18.8 2.97 6.25 2004 17.6 2.54 5.56 2005 17.6 2.72 5.56 2006 17.4 2.29 5.00 2009 17.7 2.70 5.88

Two copulas are used to approximate a) the joint probability distribution of the wind

speed, and the significant wave height, and b) the significant wave height and the period using the data. First, a distribution is fitted to the wind and wave data in Table 7.2. Chi- square goodness-of-fit test is used (Vining & Kowalski, 2010) to measure the quality of fitness by MINITAB software. For a Weibull distribution, a p-value is estimated as 0.25, which exceeds the confidence level of 0.05. Another parameter to examine the quality of the fitness is the Anderson-Darling statistic (AD in Figure 7-3). Anderson-Darling parameter provides a measure of how far the data fall from the fitted line in a probability plot, especially closer to the tails. A smaller Anderson-Darling statistic indicates that there is no reason to suspect that the data do not follow Weibull distribution. A designer should consider both the AD statistic and the p-value to assess the quality of the fit. For

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this case, the combination of a high p-value and a low AD statistic means that the Weibull distribution fits well to the wind data (Figure 7-3).

The three-parameter Weibull PDF is,

  1  t  t  )(   tf )(   e    (7.2)  )(  where in the above equation  is the location parameter and ,  are the scale and shape parameters, respectively.

By considering the location parameter as zero, the shape and scale parameters are equal to 13.87 and 17.47, respectively. Similarly, Weibull distribution is fitted to the wave data and the shape parameter and scale parameters are equal to 5.63 and 2.57.

Figure 7-4 shows the CDFs of the wind and wave.

Figure (7-3): Goodness of fit test for wind data (Red spots represent the data)

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Figure (7-4): The CDF of a) wind speed b) significant wave height (Hs)

In the next step, the marginal CDF’s are integrated into a joint CDF by using a copula function. Many families of copulas have been proposed and studied including the

Clayton, Frank, Farlie-Gumbel-Morgenstern and normal families (Nelsen, 2006). The dependence between the wind and wave data is modeled by using Frank’s copula,

  vu  )1)(1(   LogvuC  1),(   (7.3)   1  where u and v are the values of the CDFs of the wind speed and significant wave height.

In Frank’s copula, a single parameter, , controls the strength of dependence. A zero value of  means perfect dependence, and unit value means independence.

The value of parameter  is estimated from the data or engineering judgment. There are two methods to estimate  from data; one is from Kendall’s coefficient (Nikolaidis et al., 2011) and the other is from the maximum pseudo-likelihood method. The latter

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method is used to find  for the wind and wave data. For this purpose, a pseudo- likelihood function is defined by,

n   vucL ),()( (7.4)   i1

where  ,( vuc ) is the joint PDF which is calculated by,

 2 vuC ),( vuc ),(   . (7.5)   vu

This function equals to 1 for unit , and

vu Ln   )1()(  vuc ),(  (7.6) (    vuvu )2 otherwise.

The log-likelihood function is the logarithm of Eq. 7.4. Then the value of  is found such that it maximizes the log-likelihood function ( = 0.0087).

Figure 7-5 illustrates the copula according to Eq. 7.3. The horizontal axes show the values of the CDFs of the wind speed and significant wave height. Figure 7.6 shows the corresponding joint PDF of the wind speed and the significant wave height.

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Figure (7-5): Copula of wind speed and significant wave height

Figure (7-6): The PDF of wind speed and significant wave height

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Before using any results, the accuracy of the model should be assessed. There are several methods in the literature to test the goodness-of-fit of copula to data (e.g. Genest et al., 2006, Wang & Wells, 2000). However, in order to evaluate this model, MCs is used (Nikolaidis et al., 2011). The simulated pairs of the wind speed and the wave heights drawn from the copula model are compared with the observed pairs from Table 7.2.

Fifty random pairs of values of the wind speed and significant wave heights are generated by using the constructed copula model. First, 50 random values of the CDF of one variable (u) from a uniform distribution between 0 and 1 are drawn, and another set of 50 values of a variable t also are drawn from a uniform distribution between 0 and 1.

Then the corresponding values of the CDF of variable, v, are calculated by the following equation,

1 tt   uu v  Ln( ) . (7.7) Ln  )( t   uu

The value of the v falls within the range of [0, 1].

Then the wind speed and the significant wave height pairs corresponding to each pair of values of variables u and v are calculated. The 50 simulated pairs along with the 25 pairs from the Lake Erie site are plotted in Figure 7.7. There is no reason to consider that the two sets of sample values in this figure were drawn from different probability distributions.

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Figure (7-7): Simulated and observed values of the wind speed and significant wave height (Hs) for the Lake Erie site

The above procedure is repeated to obtain joint distribution function of the significant wave height and wave period. A normal distribution with a mean of 5.28 and a standard deviation of 1.02 presents the best fit to the data. Figure 7-8 shows the CDF of the dominant wave period.

Figure (7-8): The CDF of peak wave period (Tp)

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Using the maximum pseudo-likelihood the dependence parameter  equals to

0.00297. Fifty samples pairs of values of the wave height and period are calculated and are illustrated and compared with the 25 observations in Figure 7-9.

Figure (7-9): Simulated and observed values of the wave period (Tp) and significant wave height (Hs) for the Lake Erie site

Again, the simulated pairs of the wave period and the significant wave height agree well with the observations. Therefore, the simulated results are representative of the measurements at the Lake Erie site.

In section 7.3, the distributions developed in this section are used to demonstrate a methodology to design an offshore wind turbine with a required reliability that corresponds to a desirable service life.

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7.2 Finding the Target Probability of Failure

The design life of a wind turbine should be at least 20 years. In probabilistic design, a designer calculates the required annual failure probability for specified probability of failure in N years as follows. Let PN be the maximum allowable failure probability of a turbine for a N-year period. Assume independence between the failures in different years. Then, the target probability of failure for one year (P1) is,

1 N P1  PN )1(1 . (7.8)

-4 -4 For example, if the required failure probability over 20 years is 10 (P20=10 ), then the

-6 corresponding probability of failure in one-year is P1=510 .

7.3 Probabilistic Design of the System

7.3.1 Approach

In order to calculate the probability of failure of an offshore wind turbine for a specific service life and failure threshold at a given site the following integral should be evaluated,

  ),,(),,( dTdVdHTHVfLTHVGIP , (7.9) f  ps ps ps

where V, Hs and Tp denote the wind speed, significant wave height and spectral period, respectively. The term in the bracket,  ( ps ),,  LTHVGI , represents the indicator

function that is one when the performance function, ,( HVG s ,Tp ) exceeds the load, L,

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and is zero otherwise. In order to evaluate the above equation the joint PDF f (V, Hs, Tp) and the load threshold, L needs to be known.

In the probabilistic design, a designer is interested in estimating the required load capacity of the system (L), to achieve a target probability of failure for a design service life. In this case, L is the unknown that should be calculated from Eq. 7.9. Then, the designer specifies the most economical combination of materials and dimensions so as to satisfy the loading capacity requirement.

Evaluating Eq. 7.9 in closed form is impractical and alternatively it is approximated by MCs as shown below (see section 3.1.1),

  LTHVGIn )),,(( Pˆ  ps (7.10) f N where in the above equation, n denotes the number of samples for which failure occurs.

This equation is equivalent to Eq. 3.5. Similarly, the accuracy of the estimate of the

ˆ probability of failure ( Pf ) can be quantified by calculating the standard deviation of this probability from Eq. 3.6.

The approach presented below is followed to estimate the probability of failure using

MCs. First, a triplet of the sample values of the wind speed, the significant wave height, and spectral period is drawn from their joint PDFs. The sampling approach that was presented in section 7.1 is used for this purpose.

The second step is to generate time histories of the wave elevation and the wind speed for each pair of values of the wind speed and the significant wave height with drawn spectral period. The wave elevation is characterized by the Pierson Moscowitz PSD according to Eqs. 4.11 and 4.24. Time histories of the wave elevation are generated

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using Shinozuka’s method (1972). In this step, the custom code that explained in Chapter

4 (see appendix A) is used to calculate the wave kinematics. To calculate the time histories of the wind speed at the hub height of the wind turbine, TurbSim (Jonkman &

Buhl, 2007) is used. TurbSim is a tool developed by the NREL to calculate wind fields according to IEC standards. The loads at the base of the monopile is calculated for each time history of the wave elevation and the wind speed by using FAST (Jonkman et al.,

2004).

The next step is the calculation of the probability of up-crossing several thresholds.

In this process, relatively high up-crossing probabilities in the range from 0.1 to 0.01 are estimated. The calculation of lower probabilities requires an inordinate amount of computational effort. The tail modeling method is effective for calculating such probabilities at a reasonable cost (Maes & Breitung, 1993, Kim & Ramu, 2006,

Mourelatos et al., 2009). Such methods are reviewed in the next subsections, and the simulation results are presented.

7.3.2 Tail modeling method

The tail modeling method is based on the property of tail equivalence. Tail equivalent distributions exhibit approximately the same behavior for extreme values of x.

Two distribution functions FX(x) and GX(x) are called tail equivalent (Maes & Breitung,

1993) if,

 xF )(1 Lim X 1. (7.11)  X xG )(1 x 

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For a random input variable x the performance function, y(x), will also be a random variable, but have a different distribution. Let a be a large threshold of y and define z=y- a. Then based on the extreme value theory (Castillo, 1988) for z>0 the distribution of FZ

(z) conditional on the threshold of a can be approximated by the generalized Pareto distribution as follows,

1       )1,0max(1 ifz   0 zF )(    . (7.12) Z z   )exp(1 if   0  

In the above equation,  and  are the scale and shape parameters that must be determined. The generalized Pareto distribution for different values of the shape parameters is shown in Figure 7-10.

Figure (7-10): Generalized Pareto distributions for different shape parameters (Ref. Kim et al., 2006)

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The tail modeling method consists of three steps. First N sample points of the random response y(x) are generated using MCs. Subsequently, a threshold value a is selected, and the Na sample points out of N, for which the performance function exceeds a (Figure

7-11), are identified. Proper selection of a is important and has been the subject of extensive research (Boos 1984, Hasofer 1996 and Caers et al., 1998). Hasofer (1996)

0.5 suggests that Na ≈ 1.5 N .

Figure (7-11): Tail modeling

Finally, the shape and scale parameters ξ and σ are estimated by approximating the tail–model with an empirical CDF. The maximum likelihood method (Prescott &

Walden, 1980) or the least square (Kim et al., 2006) methods is used for this purpose. To this end the following minimization problem should be solved,

N ˆ 2  yFFMin iYi , (7.13) ,   NNi a

ˆ where the empirical CDF, Fi can be estimated by (Kapur, 1977),

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i  3.0 Fˆ   2,1 Ni (7.14) i N  4.0 

and yF iY is the CDF by the generalized Pareto distribution that can be calculated by,

1    yF 1(1  Fˆ  ay ))(1,0max() (7.15) Y a 

ˆ In the above equation, Fa is the empirical CDF associated with threshold a. Only the tail part of the data is used in estimating the generalized Pareto’s distribution parameters.

The approach that is based on the safety index to solve the minimization problem in Eq.

7.13 is used here (Mourelatos et al., 2009). More details on tail modeling can be found in

(Caers et al., 1998 and Kim et al., 2006).

7.3.3 Monte Carlo Simulation of extreme 1-year condition

In order to perform the simulation, sample triplets of the wind speed, significant wave height, and period are drawn from their corresponding joint distributions. The wind data at the buoy was recorded at 5 m above sea level. Therefore the wind speed at the hub height (87.6 m) should be calculated. This is done by using the power shear law (IEC

61400-1 ed.3). Marschall et al., (2009) reported that the shear factor is equal to 0.09 for the potential wind farm in Lake Erie. Although this value is comparably low for an offshore wind farm, it is used to calculate the wind speed along the height.

According to Marschall et al., (2009), the turbulence intensity is highest (~11%) for wind speeds coming from east and southeast. This is to be expected, because in this direction, lower wind speeds are observed and turbulence is usually greater at lower wind speeds. For the other wind direction sectors the turbulence intensity is in the range of 5

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to 8%. These numbers are low compared to wind industry standards (IEC 61400-1 ed. 3).

In the simulations, the highest turbulence intensity that has been reported to estimate the extreme loads is used.

In Figure 7-12 some sample values of the wind speed, wave height and period that are extracted from their corresponding distributions are shown.

Figure (7-12): Simulated values of extreme 1-year triplets of wind speed, significant wave height and period for the Lake Erie site

Monte Carlo simulation continues for the triplets in the above figure until the desired confidence in the estimated up-crossing probability is achieved. To measure the accuracy of the results by the COV of this probability; the simulation is stopped once the COV of

10% is achieved. In order to use the tail fitting method the failure probabilities of up to

1% should be estimated accurately. To calculate the required load capacity of tower, the maximum fore-aft bending moment at the tower mud line is considered. This bending

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moment is the largest at the base of tower. Similarly, for the blades the maximum bending moment at the root of the blades is considered to estimate the load capacity of the blades. However, the methodology presented here is applicable to estimate the probability of upcrossing a threshold for other loads such as side-to-side bending moment at the base of the tower or out-of-plane bending moment at the root of the blades or the combination of two or three loads.

Figure 7-13, shows the 40 triplets from Figure 7-12 that are used to perform simulations. Every triplet is Figure 7-13 represents an annual extreme wind, wave height, and period. For each triplet, MCs with 10,000 replications is performed, for 10 minutes and the largest bending moment in every replication is calculated at the base of tower and at the root of the blades. Performing this simulation took almost one month on 10 Dell

Precision T7400 desktop computers.

Figure (7-13): Combinations of the wave height, wind speed and wave period used to perform MCs (40 seeds)

Using Eq. 7.14 the CDF of the fore-aft bending moments and the bending moment at the root of the blades is calculated and shown in Figures 7-14, 15.

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In Figure 7-14, the rightmost curve corresponds to the triplet with wind speed of 14.4 m/sec and wave height and wave period of 2.03 m and 4.12 sec, respectively. The wind speed of the triplet immediately to the left of this curve is 15.6 m/sec with a wave height and period 2 m and 4.07 sec, respectively. The CDF at the extreme left of Figure 7.14 corresponds to the triplet with wind speed of 22.5 m/sec, wave height of 2.84 m and period of 6.27 sec. The wind speed for the corresponding CDFs increases from right to left. Likewise, in Figure 7.15, the wind speed corresponding to the CDF of the bending moment at the root of the blades increases from right to left. The explanation for this apparent paradox is that the pitching control mechanism of the blades mitigates the average loads as the wind speed increases.

Figure (7-14): The CDF of the fore-aft bending moment at the base of tower

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Figure (7-15): The CDF of the out-of-plane bending moment at the root of the blades

The tail fitting method shows that although the average loads increase with decreasing the wind speed, these loads do not dominate the probabilistic design. The reason is that for higher thresholds the tail of the CDF of the bending moment for low wind speeds fades away faster than those for higher speeds (Figure 7-16).

The generalized Pareto distribution is fitted to every CDF in Figures 7-14 and 15 and the load capacity that is required to meet the target probability of failure of 510-6 during one year is calculated. Figure 7-16 shows the fitted CDFs of the bending moment the largest and lowest wind speeds.

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Figure (7-16): Pareto distribution fitted to the empirical CDF (a) wind= 22.7 m/sec, wave= 2.41 m, period= 5.72 sec (b) wind= 14.4 m/sec, wave= 2.03 m, period= 4.12 sec

In Figure 7-16b, the tail for the triplet with lower wind speed of 14.4 m/sec dies rapidly whereas for the higher wind speed Figure 7-16a the tail exhibits a plateau. Similar trends are observed for other wind speeds. Therefore, in the first excursion failure analysis of the tower base, higher wind speeds would dominate the design, as expected.

The required load capacities for the tower for all 40 triplets of Figure 7-13 are summarized in Table 7.3. The maximum load capacity is 155,918 KN.m. The results are sorted by the required load capacity calculated by the tail fitting method. The column on its left is the maximum bending moment that is obtained from MCs with 10,000 replications for every extreme 1-year load condition. The results for low wind speeds are closer to the bottom of table where bending moments are smaller.

The last column in Table 7.3 shows the required load capacity divided by the maximum load from MCs with 10,000 replications. The largest ratio is 1.26 that

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corresponds to the first triplet. This means that in order to meet the target reliability the loads from the MCs should be augmented by about 26%.

In order to estimate the 95% CI, Bootstrap method (Efron, 1979) with sample size of

150 is used. The standard deviation of the mean load capacity is estimated as 1,111.9

KN.m and the mean load capacity as 136,959.9 KN.m. Using these values the 95% CI is

(122,872.2, 152,328.6) KN.m. This means that 95% of the time, the required load capacity of the structure for the tower would fall within the above range, on average. The upper bound of the CI can be chosen as the design value. This means that if the structure of the tower is able to carry loads higher than 152,328.6 KN.m, the probability of failure during 1-year would be less than 510-6.

This procedure is repeated for the blades and the results are summarized in Table 7.4.

Similar to the tower, lower wind speeds are not critical. The maximum load capacity is

19,485 KN.m. Similarly, the last column in Table 7.4 shows the required load capacity divided by the maximum load from MCs with 10,000 replications. The largest ratio is

1.21 that corresponds to the first triplet.

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Table (7.3): Summary of results for the tower Annual extreme condition Probability of exceedance Maximum Required Wind Wave Period 87,500 KN.m 90,000 KN.m 105,000 KN.m load capacity m/sec m sec Pf COV Pf COV Pf COV KN.m KN.m 21.6 3.30 7.32 0.04 0.05 0.03 0.06 0.02 0.08 123300 155918 1.26 19.4 1.90 3.17 0.14 0.03 0.10 0.03 0.07 0.04 139900 151924 1.09 19.5 2.61 5.34 0.12 0.03 0.09 0.03 0.06 0.04 119900 150971 1.26 19.8 2.03 3.09 0.15 0.02 0.11 0.03 0.08 0.03 141800 147452 1.04 19.5 2.65 6.54 0.12 0.03 0.08 0.03 0.06 0.04 131500 146345 1.11 19.3 2.16 4.35 0.14 0.02 0.10 0.03 0.07 0.04 142100 146238 1.03 19.1 1.84 4.05 0.14 0.02 0.10 0.03 0.07 0.04 128000 145023 1.13 21.5 2.31 7.12 0.12 0.03 0.09 0.03 0.06 0.04 126800 143619 1.13 21.4 2.56 5.08 0.14 0.02 0.10 0.03 0.07 0.04 136200 143546 1.05 21.6 2.83 5.02 0.05 0.04 0.03 0.06 0.02 0.07 125800 142111 1.13 20.1 2.72 5.05 0.10 0.03 0.07 0.04 0.05 0.05 133200 141823 1.06 21.4 2.43 3.84 0.05 0.04 0.03 0.05 0.02 0.07 121000 140133 1.16 19.3 1.96 4.90 0.13 0.03 0.09 0.03 0.07 0.04 127800 139144 1.09 21.6 1.91 5.24 0.03 0.06 0.02 0.07 0.01 0.09 127400 138843 1.09 21.5 2.55 4.90 0.14 0.02 0.10 0.03 0.07 0.04 127200 136763 1.08 17.7 2.09 6.09 0.34 0.01 0.28 0.02 0.22 0.02 123800 136194 1.10 21.1 3.08 5.54 0.06 0.04 0.04 0.05 0.03 0.06 124900 136160 1.09 17.1 1.97 6.24 0.48 0.01 0.40 0.01 0.32 0.01 120000 136066 1.13 21.0 2.89 4.56 0.07 0.04 0.05 0.04 0.03 0.06 124700 135330 1.09 21.3 1.66 2.91 0.04 0.05 0.02 0.06 0.01 0.08 119000 134638 1.13 21.2 2.69 6.62 0.04 0.05 0.03 0.06 0.02 0.08 116700 134512 1.15 18.3 1.83 3.22 0.26 0.02 0.20 0.02 0.15 0.02 122000 134512 1.10 16.8 1.46 4.16 0.12 0.03 0.08 0.03 0.06 0.04 122700 134401 1.10 21.5 2.42 5.66 0.04 0.05 0.03 0.06 0.02 0.08 120900 134289 1.11 18.0 1.51 6.93 0.27 0.02 0.21 0.02 0.17 0.02 128300 133982 1.04 19.4 2.22 3.72 0.14 0.02 0.10 0.03 0.07 0.04 130200 133573 1.03 22.1 2.40 5.96 0.13 0.03 0.09 0.03 0.07 0.04 126100 133172 1.06 19.3 2.23 5.24 0.13 0.03 0.10 0.03 0.07 0.04 127100 132821 1.05 19.3 2.50 6.20 0.13 0.03 0.09 0.03 0.07 0.04 128400 131775 1.03 18.6 1.66 5.16 0.19 0.02 0.15 0.02 0.11 0.03 121400 130988 1.08 17.8 2.44 4.76 0.35 0.01 0.28 0.02 0.23 0.02 126700 130982 1.03 22.7 2.41 5.72 0.13 0.03 0.09 0.03 0.07 0.04 124800 130927 1.05 16.8 1.60 5.22 0.56 0.01 0.48 0.01 0.39 0.01 123500 130178 1.05 22.5 2.84 6.27 0.03 0.06 0.01 0.08 0.01 0.11 116900 130110 1.11 17.5 1.90 6.11 0.38 0.01 0.31 0.02 0.24 0.02 124600 130028 1.04 19.8 2.82 6.31 0.10 0.03 0.07 0.04 0.05 0.04 119800 129571 1.08 18.2 1.11 4.79 0.23 0.02 0.18 0.02 0.14 0.02 118900 129018 1.09 22.5 3.12 3.83 0.05 0.04 0.03 0.06 0.02 0.07 118700 128062 1.08 14.4 2.03 4.12 1.00 0.00 0.99 0.00 0.95 0.00 120400 127248 1.06 15.6 2.00 4.07 0.91 0.00 0.84 0.00 0.75 0.01 124400 125923 1.01

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Table (7.4): Summary of results for the blades Annual extreme condition Probability of exceedance Maximum Required Wind Wave Period 13,000 KN.m 13,500 KNm 14,000 KNm load capacity m/sec m sec Pf COV Pf COV Pf COV KN.m KN.m 21.5 2.31 7.12 0.03 0.06 0.01 0.09 0.01 0.13 16060 19485 1.21 21.4 2.56 5.08 0.03 0.06 0.01 0.09 0.01 0.13 16060 19054 1.19 19.8 2.82 6.31 0.02 0.07 0.01 0.11 0.00 0.15 16280 18850 1.16 19.3 2.23 5.24 0.03 0.06 0.02 0.08 0.01 0.12 16490 18772 1.14 21.5 2.42 5.66 0.00 0.16 0.00 0.23 0.00 0.30 16560 18727 1.13 21.6 1.91 5.24 0.00 0.17 0.00 0.24 0.00 0.30 15460 18699 1.21 19.4 1.90 3.17 0.03 0.06 0.01 0.08 0.01 0.12 16030 18695 1.17 21.2 2.69 6.62 0.00 0.16 0.00 0.22 0.00 0.33 16020 18656 1.16 19.1 1.84 4.05 0.03 0.06 0.02 0.08 0.01 0.12 16070 18656 1.16 20.1 2.72 5.05 0.01 0.08 0.01 0.13 0.00 0.16 16130 18625 1.15 19.5 2.65 6.54 0.03 0.06 0.01 0.09 0.01 0.13 16240 18545 1.14 21.4 2.43 3.84 0.00 0.15 0.00 0.21 0.00 0.29 16590 18403 1.11 21.6 3.30 7.32 0.00 0.17 0.00 0.24 0.00 0.30 15450 18386 1.19 19.3 2.16 4.35 0.03 0.06 0.02 0.08 0.01 0.13 16090 18352 1.14 21.6 2.83 5.02 0.00 0.16 0.00 0.23 0.00 0.28 16530 18339 1.11 21.3 1.66 2.91 0.00 0.15 0.00 0.20 0.00 0.26 16630 18331 1.10 21.5 2.55 4.90 0.03 0.06 0.01 0.08 0.01 0.13 16110 18305 1.14 22.1 2.40 5.96 0.03 0.06 0.01 0.08 0.01 0.13 16080 18220 1.13 16.8 1.60 5.22 0.21 0.02 0.11 0.03 0.04 0.05 15970 18190 1.14 16.8 1.46 4.16 0.03 0.06 0.01 0.08 0.01 0.13 16070 18060 1.12 19.5 2.61 5.34 0.02 0.06 0.01 0.10 0.01 0.14 16280 17998 1.11 22.5 3.12 3.83 0.00 0.23 0.00 0.35 0.00 0.45 15060 17993 1.19 22.7 2.41 5.72 0.03 0.06 0.01 0.09 0.01 0.13 16070 17778 1.11 19.4 2.22 3.72 0.03 0.06 0.01 0.09 0.01 0.14 16180 17606 1.09 19.3 2.50 6.20 0.03 0.06 0.02 0.08 0.01 0.12 16100 17540 1.09 19.3 1.96 4.90 0.03 0.06 0.02 0.08 0.01 0.12 16100 17526 1.09 18.0 1.51 6.93 0.08 0.03 0.04 0.05 0.02 0.07 16060 17455 1.09 21.1 3.08 5.54 0.01 0.13 0.00 0.17 0.00 0.23 16750 17432 1.04 15.6 2.00 4.07 0.45 0.01 0.23 0.02 0.10 0.03 16580 17368 1.05 22.5 2.84 6.27 0.00 0.21 0.00 0.32 0.00 0.41 15140 17263 1.14 21.0 2.89 4.56 0.01 0.12 0.00 0.16 0.00 0.23 16770 17166 1.02 14.4 2.03 4.12 0.74 0.01 0.42 0.01 0.17 0.02 16670 17092 1.03 19.8 2.03 3.09 0.03 0.06 0.02 0.08 0.01 0.12 16070 17074 1.06 17.1 1.97 6.24 0.17 0.02 0.08 0.03 0.04 0.05 16380 17021 1.04 17.5 1.90 6.11 0.12 0.03 0.06 0.04 0.03 0.06 16070 17010 1.06 17.7 2.09 6.09 0.10 0.03 0.06 0.04 0.02 0.06 16130 16956 1.05 18.2 1.11 4.79 0.07 0.04 0.04 0.05 0.02 0.07 16410 16891 1.03 17.8 2.44 4.76 0.09 0.03 0.05 0.04 0.02 0.06 16130 16646 1.03 18.6 1.66 5.16 0.05 0.04 0.03 0.06 0.01 0.09 15970 16576 1.04 18.3 1.83 3.22 0.06 0.04 0.03 0.05 0.02 0.08 16090 16213 1.01

In order to estimate the 95% CI of the load capacity for the blades, Bootstrap method

(Efron, 1979) with sample size of 150 is used. The standard deviation of the mean load capacity is estimated as 131.7 KN.m and the mean load capacity as 17,901.9 KN.m. Using these values the 95% CI is (16,537.3, 19,460) KN.m. This means that the required load

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capacity of the structure for the tower would fall within the above range 95% of the time, on average. The upper bound of the CI, 19,460 KNm can be chosen as a design value.

This means that if the structure of the blade is able to carry loads higher than 19,460

KN.m, the probability of failure during 1-year is less than 510-6.

7.3.4 Importance of ice impact in Lake Erie

The NOAA data shows that the lake Erie freezes over very frequently, making ice a significant concern (Figure 7-17). Due to the shallowness of Lake Erie compared to other lakes, its entire surface can freeze during winter. This makes ice loads a serious concern for deployment of an offshore wind turbine in Lake Erie.

In addition to ice sheets, ice ridges form in Lake Erie that can cause deep keels of rubble to form under the surface of the ice. These rubble keels can reach all the way down to the lakebed (C-Core report, 2008). It is not known how frequently these ice ridge keels occur. Figure 7-18 shows schematic of an ice keel.

Figure (7-17): Ice coverage of the Great Lakes in winter time

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Figure (7-18): Ice keel (ref to C-Core report, 2008)

The foundation must be strong enough to plow through the ice sheet (including any ice ridge keels). Using a cone shaped structure instead of a cylinder can cause the ice sheet to bend and snap instead of crushing. This significantly reduces the ice loads (IEC

61400-3 ed1).

Norouzi et al. (2011) demonstrated the importance of ice loads on the structural integrity of a monopile wind turbine. In this study, they consider the same 5 MW wind turbine with the monopile platform that was studied in this chapter. The platform is considered to be made of ASTM-A36 standard structural steel with yield stress of 250

MPa and the ultimate stress of 400 MPa. The bending moment that causes first yielding at the base is equal to 1.8105 KNm. Then overturning moment that results in yield in the whole cross section is 2.43105 KNm. Considering the ultimate stress as 400 MPa the ultimate bending moment is approximately 3105 KNm. These estimates are used in this preliminary study. A designer should conduct a thorough FEA to calculate the load capacities more accurately.

Two impact forces are considered, one at the sea level with a magnitude of 3.6 MN, and the other below sea level at one third of depth from sea level with magnitude of 7 MN

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along the direction shown in Figure 7-19. These loads are calculated from the extreme ice conditions on Lake Erie (C-Core report, 2008). The 3.6 MN is the force required to break the ice sheet, and the 7 MN is the force to fail the ice keel. The direction is chosen such that it increases the fore-aft bending moment at the base of the tower, where the bending moment is maximum.

Figure (7-19): Schematic of the ice impact model

The natural frequency of the tower is about 0.3 Hz . Therefore, for the dynamic case a load period of 3 sec is considered. Two scenarios are considered and compared. In the first, the wind turbine is assumed to be operating at a rated wind speed of 11.5 m/sec, and the significant wave height is at its extreme 50 year equal to 4.1 m. In the second case, the wind turbine is assumed to be operating at the same rated wind speed and that a dynamic triangular ice load with 6 peaks each with 3 sec period is applied (according to the IEC 61400-3 ed.1 standard). The fore-aft bending moment at the base of tower is

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calculated using MSC ADAMS and FAST as simulation tools Figure 7-20. The bending moment at the base of the tower significantly increases due to the ice impact, making it a serious concern.

Figure (7-20): Comparison of the fore-aft bending moment at the base of tower after ice impact

The displacement of the tower top under the ice impact is plotted in Figure 7-21.

Figure (7-21): Tower top displacement after ice impact

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7.4 Discussion

In this chapter, a methodology for probabilistic design of an offshore wind turbine structure was presented. The wind turbine model that was studied was based on the

NREL 5 MW offshore wind turbine on a monopile platform (Jonkman et. al, 2007).

The proposed approach was based on modeling the loading environment by means of copula functions (Nelsen, 2006). This approach is different from the method presented in the IEC standard (IEC 61400-3 ed 1), which uses load factors and assumes that wave height conditional on wind speed follows standard distribution. The author believes that the proposed method is more complete because it is based on obtaining the joint PDF of the wind and wave without making any assumptions on the distribution of the wave height conditioned on the wind speed.

As a conclusion, compared to wind and wave loads, the ice impact loads are a serious consideration in the design of offshore wind turbines in the Great Lakes. Therefore, future investigators must quantify and document the statistical properties of these loads and analyze their effect on the structural integrity of a wind turbine. A thorough study of the ice impact is beyond the scope of this dissertation, and the study presented above only highlights the significance of the ice impact on the structural integrity of any potential offshore wind turbine in Lake Erie. For further reading on the importance of ice loads, refer to Eric Wells (2012).

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Chapter 8

Summary & Conclusion

This dissertation focused on the applications of PRRA to estimate the statistics of random processes (functions of time). This method enables an analyst to estimate a design attribute for different power spectral densities of the excitation at approximately the cost of a single MCs. This goal was accomplished by scaling the results of the simulation by the likelihood ratios of the sampling and the true power spectra.

Probabilistic Re-analysis is applicable to linear or nonlinear systems provided that the input load is Guassian represented by a PSD function. In order to apply PRRA, the energy content of the PSDs of the excitation should be identical, and the support of the sampling spectrum should cover that of the true spectrum.

A designer can predict the accuracy of PRRA before performing structural analysis by examining the mean value and standard deviation of the likelihood ratios of the generated sample. A small coefficient of variation (COV<0.1) and a mean value that is close to 1 suggest an accurate estimate of the response attribute. A designer calculates these statistics at practically no cost by dividing PDFs.

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The first two chapters defined the problem of assessing the reliability of a dynamic system driven by stochastic, time varying loads, and explained its significance. They also reviewed previous work. Chapter 3 presented the PRRA method.

In Chapter 4, the efficacy of PRRA was examined to estimate the fatigue damage in a structure. The approach was applied to estimate the high-cycle fatigue damage with uniaxial stress condition using the linear damage accumulation rule.

In Chapter 5, PRRA was demonstrated to estimate of the probability of first excursion failure. The method was applied to examples, including an offshore wind turbine, considering changes in the wave spectrum. This method can be integrated with the tail fitting methods (Maes & Breitung, 1993, Kim & Ramu, 2006, Mourelatos et al., 2009) to estimate the probability of failure of highly reliable systems.

The observations and conclusions of Chapters 4 and 5 on the application of PRRA for the estimation of fatigue damage and the probability of first excursion failure, are summarized below.

o The average value and standard deviation of the likelihood ratio enable an analyst to predict the quality of a sampling distribution and the accuracy of

the results before performing the simulation. The closer the mean value to 1

and the smaller the standard deviation, the better the sampling distribution.

o The standard deviation of the estimated probabilities decreases with the number of replications. However, the law of diminishing returns applies:

roughly, halving the standard deviation requires quadrupling the number of

replications.

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o The accuracy of PRRA deteriorates when the true distribution deviates significantly from the sampling distribution.

o A mean value of the likelihood ratio that is significantly different than 1 does not necessarily imply a biased estimator. This is the case when the fatigue

damage is zero for a range of frequencies of the sampling PSD function.

o Most of the time, the standard deviation of the results using PRRA is larger than that of the sampling MCs. This occurs because the results of PRRA are

calculated by multiplication of sampling MCs by the likelihood ratio, and, in

most cases, the variation of the likelihood ratio increases the variability of the

results from PRRA. In order to achieve a small standard deviation from

PRRA, the sampling PSD must be carefully chosen by considering the

frequency response of the system of interest. A proper selection of the

sampling spectrum can lead to considerable improvement in accuracy.

o Normalizing the results of PRRA by the average likelihood ratio, as some authors have suggested, does not always improve the accuracy and produces a

biased estimator. In addition, it makes the estimator of the mean value biased.

o The difference between the computational cost using MCs or using PRRA is significant for large real-life structures. The required time to run PRRA is

very short, and it is not affected by the size or the complexity of the structure

while performing MCs for large structures is significantly more expensive.

Chapter 6 proposed an approach to estimate low probabilities of failure more efficiently than SS. This is important in safety assessment of highly reliable systems.

This approach integrates Shinozuka’s method for simulation of random processes with

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SS. Shinozuka’s method enables reducing the dimension of the problem by exploiting the symmetry of the space of the random variables (which are the frequencies and phase angles of the harmonics of the excitation). The choice of the intermediate probabilities in each simulation, and the step size are crucial to the effectiveness of the proposed method.

A small step size might slow down the exploration of the space of the random variables, whereas a large step size causes the Markov chain to return frequently to the same state.

The PRRA methodology was applied to SS, which enabled performing PRRA without running MCs. This new approach was named Subset-PRRA. Subset-PRRA is more efficient than PRRA as the latter relies on the results of a MCs, whereas Subset-

PRRA reuses the results of SS. In order to use the proposed method effectively, an analyst should understand the problem and explore all disjoint failure regions.

The observations and conclusions of Chapter 6 on integrating SS with PRRA to estimate the reliability of dynamic systems are summarized as below.

o Performing SS by utilizing Shinozuka’s equation is more efficient than performing original SS because it enables a user to reduce the dimension of

the problem by exploiting the symmetry of the space of the random variables.

o For low thresholds where the up-crossing probabilities are higher, the COV of SS is higher than that of MCs. However, for high thresholds with low

probability of failure, SS is significantly more efficient and accurate.

o In order to use SS effectively, an analyst should understand the problem and explore all disjoint failure regions. This could be very difficult for complex

systems.

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o Integration of SS with PRRA enables an analyst to evaluate reliability more efficiently than SS.

Chapter 7 presented a probabilistic approach for the probabilistic design of an offshore wind turbine structure. The objective was to calculate the required load capacities of the tower and the blades to achieve the target reliability. The method was demonstrated on a wind turbine in Lake Erie. Copulas were used to approximate the joint distributions of the wind speed, the wave height, and the period at a tentative location of a wind turbine farm.

Monte Carlo simulations with 10,000 replications were performed for every extreme

1-year load condition. Then the largest loads were calculated at the base of tower and the root of the blades.

An analyst can use the results of Chapter 7 to build a response surface. For this study, a response surface is a function that maps the input load parameters, wind speed, wave height and wave period to an output feature that can be maximum load at the base of the tower or that the root of the blades. The response surface enables predicting response attributes for new loading conditions without performing MCs.

The observations and conclusions of Chapter 7 on the probabilistic design of an offshore wind turbine are summarized as below.

o A copula is an effective tool to model the dependence of wind and wave loads and simulate these loads in order to assess the structural safety of a turbine.

o In reliability analysis of the baseline 5 MW monopile offshore wind turbine, the average loads corresponding to exceedance probabilities higher than 1%

are higher for average wind speeds close to the rated speed (e.g. 14 m/sec)

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than for high average speeds (e.g. 22 m/sec) because the control mechanism of

the blades mitigates these loads. However, the control system is not able to

mitigate the extreme loads corresponding to high wind speeds with annual

probabilities of exceedance less than 1% effectively.

o Only the reliability of the structure by considering the first excursion failure mode was studied. Assessment of the fatigue reliability of the structure

should be conducted.

o Ice impact is a major concern for any wind farm in Lake Erie. A thorough reliability analysis considering statistical ice loads should be performed prior

to deployment of any wind turbine in Lake Erie.

Overall, PRRA is a non-intrusive method. This means that PRRA does not

change the way that the indicator function is evaluated in a MCs. Therefore, it can

be applied to any structure under random dynamic loads that are represented by

PSD functions. However, the efficacy of PRRA can vary depending on the

complexity of the structure.

Future Research

The application of the PRRA method that was presented in this work is limited to the

PSDs with the same energy content. One should extend the application to the problems in which the energy of the input PSD functions changes.

In this work, the PRRA method was used to estimate the structural reliability of an offshore wind turbine. However, in the examples that were considered in Chapters 4, 5 only change in the wave spectrum was considered. In order to be able to apply PRRA to

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determine the effect of changes in the the wind spectrum, an analyst needs to implement

Shinozuka’s (1972) equation into Turbsim (Jonkman & Buhl, 2007).

Subset Simulation is more efficient than MCs. Using the approach that was presented in Chapter 6, the efficiency of SS can be improved. In addition, Subset-PRRA enables decreasing the computational cost of the reliability analysis considerably. The computational cost reduction is significant for large-scale structures. However, applying the proposed methods to large-scale structures such as an offshore wind turbine requires developing proper tools.

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Appendix A

Wave Kinematics

In aero-elastic codes (such as GH Bladed) that are used to analyze wind turbines, wave kinematics can be modeled using different methods. One method is based upon

Airy theory (Newman, 1997) with incorporating a stretching method such as Wheeler’s stretching (Wheeler, 1970) to extrapolate wave kinematics to instantaneous water surface.

The second method is a nonlinear solution such as stream function model (Dean, 1965), which includes the water surface elevation. Compared to Stokes’s nonlinear theory

(1847, 1880) applying stream function is more convenient and for this reason many commercial software use stream functions to model nonlinear waves.

Depending on the location of an offshore wind turbine and the environment condition, the appropriate wave theory should be employed to model the hydrodynamic forces. All theories try to find an approximate solution to the same differential equation with proper boundary conditions. The theories differ in the extent they satisfy the boundary conditions at the wave surface. International Electrotechnical Commission standard, IEC 61400–3 ed. 1 presents guidance on the selection of a suitable wave theory based on the environmental conditions at a given site (Figure A-1). An appropriate wave model depends on the relationship between the wave height, the water depth and the

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wave period. Wave with low steepness in deep waters can be modeled accurately with linear theory. However to model extreme events where the height to length ratio gets large value, the higher order stream functions should be used. In the study by Camp et al.

(2003) the effect of the wave model on the calculated overturning moment at the wind turbine tower base is studied. The difference in load level between the nonlinear and the linear wave theories reduces when the wave height decreases. However, the difference between the overturning moments from linear and nonlinear wave theory increases significantly by approaching the breaking wave limits especially for shallow waters

(Camp et al., 2003).

In Figure A-1, H, L, T and d denote wave height, wave length, wave period, and water depth, respectively and g denotes gravity acceleration.

For further reading about wave theories and their ranges of application, please refer to

ISO19901-1 standard.

In developing the custom wave kinematics code that was used in Chapters 4 and 5 to demonstrate the application of PRRA on an offshore wind turbine, linear wave theory along with Wheeler’s stretching method (1970) was used to calculate the wave kinematics. A short explanation of the linear wave theory and Wheeler’s stretching metod is presented in the following sections.

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Figure (A-1): Choosing a suitable wave kinematic model based on water depth and wave height (Ref. IEC 61400-3 ed.1)

Linear regular wave theory

Small height waves in deep water are linear in nature. According to Figure A-1, Airy theory can be used over a wide range of depths. As the wave height increases or the water depth reduces, the nonlinearity of wave increases and Airy theory does not yield an accurate result.

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The 2D linear wave theory by assuming small amplitude, inviscid, incompressible, irrotational flow with no currents, is the solution to the following equation, which is known as Laplace equation,

2 2   0 (A-1)  2 yx 2 where x and y are the spatial coordinates along the direction of the propagation of the wave and  is the velocity potential function. Then the kinematics of water particles will be functions of the potential function as follows,

 u  x (A-2)  . v  y

The solution to Eq. A-1 is determined by defining appropriate boundary conditions.

The boundary condition at the free surface and at the bottom of the fluid should be satisfied. The first boundary condition is that the pressure at free water surface is constant and is equal to the atmospheric pressure. The second one lies on the fact that there is no flow in the vertical direction at the seabed. Last boundary condition is that a parcel of water at the surface remains at the surface. In Figure A-2, the definitions of the linear wave theory are shown.

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Figure (A-2): Regular wave

A Cartesian coordinate system with y=0 at the still water level as shown in Figure A-

2 is used to describe the problem. The parameters involved in the solution are listed as below:

(x,t) : the free water surface elevation t : time u,v : velocity components in the x and y directions (x,y,t) : the two dimensional velocity potential function  : water density g : gravitational acceleration H : wave height L : wave length k : wave number which equals 2/L T : wave period  : wave frequency d : mean depth

The solution to Laplace equation (Eq. A-1) subjected to the boundary condition that is described above is:

 dykgH )(cosh  tyx ),,(  tkx )sin( . (A-3) 2  dk )(cosh

In which wave frequency,  can be calculated by,

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1   kdgk )tanh( 2 . (A-4)

Two alternative ways to write the above equation are,

2 g d L  (  )2tanh() , (A-5)  2 L

1  gL d  2 C  (  )2tanh() . (A-6)  2 L  where in the last equation C denotes wave speed which is called “celerity”. Then the free surface elevation is,

H  tx ),(   tkx )(cos . (A-7) 2

In deep water where d/L is less than 0.5 Eq. A-5 is simplified because tanh(2d/L) can be approximated by 1, therefore,

2 g L  . (A-8) d  2

The above equation in terms of wave period is simplified to,

2 d  56.1 TL . (A-9) where subscript “d” denotes deep water.

For deep water the velocity potential function is as following,

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gH  tyx ),,(  ky tkxe )sin( (A-10) 2 

Then the velocity components (u, v) are as follows,

H tyxu ),,(  ky tkxe )cos( 2 (A-11) H tyxv ),,(  ky tkxe )sin( 2

Acceleration components are simply obtained by taking the derivative of the above equations with respect to time as follows,

2 H ky x tyxa ),,(  tkxe )sin( 2 (A-12) H 2 tyxa ),,(  ky tkxe )sin( y 2

There is motion in the wave therefore the pressure distribution is no longer be hydrostatic along the vertical direction. The distribution of the gauge pressure is,

ky g    gytxegtyxp .),(),,( (A-13)

In shallow water d/L<0.05 and tanh(2d/L) can be approximated by 2d/L. Then Eqs

(A-5, 6) are simplified to,

 gdTL . (A-14)  gdC

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For shallow water the velocity potential function is similar to Eq. A-3. The velocity components are as follows,

H tyxu ),,(  tkx )cos( 2kd (A-15)   dyH tyxv ),,(  tkx )sin( 2 d

The acceleration components are simply calculated by taking derivative of above equations with respect to time as follows,

H 2 tyxa ),,(  tkx )sin( x 2kd (A-16)  2  dyH tyxa ),,(  tkx )cos( y 2 d and the gauge pressure distribution is as following,

g     gytxgtyxp .),(),,( (A-17)

For all depths the wave length, L can be calculated by iteration from the following equation,

11 d   )2tanh( . (A-18) d LL L

The water particle velocities in x and y direction could be obtained by taking derivation of the potential function as shown in Eq. A-2, therefore,

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 dykgkH )(cosh tyxu ),,(  tkx )cos( 2 dk )(cosh . (A-19)  dykgkH )(sinh tyxv ),,(  tkx )sin( 2 dk )(cosh

Since the velocities presented here are describing the water particle, to find the corresponding acceleration the derivative of the velocities with respect to time is taken as follows,

 dykgkH )(cosh tyxa ),,(  tkx )sin( x 2 dk )(cosh . (A-20)  dykgkH )(sinh tyxa ),,(  tkx )cos( y 2 dk )(cosh

The distribution of the gauge pressure is,

 dyk )(cosh ),,(  gtyxp   gytx .),( (A-21) g dk )(cosh

The first term in the above equation is known as the pressure response factor which tends to zero by moving toward the basin of the sea.

Wheeler’s stretching method

As described in the previous part, in the linear wave theory the boundary condition is applied to the still water level while the water free surface varies with time. There are several stretching methods such as linear, vertical and Wheeler’s stretching in the literature to extrapolate the wave kinematics to instantaneous water surface. Zhang et al.

(1991) compared these methods with the experimental results and indicated that there is not a superior method which can predict wave kinematics accurately for both narrow-

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band and broad-band waves. In the IEC-61400-3 Ed. 1 standard, Wheeler’s stretching

(1970) is considered as one of the suitable methods for offshore wind turbine application.

For demonstrating the application of PRRA, Wheeler’s method is chosen to calculate wave kinematics. However this does not limit the application of PRRA to Wheeler’s method. In Wheeler’s stretching (1970) the wave kinematics that is calculated using linear theory from sea bed to the still water level is shifted to new location proportional to their elevation above the seabed from sea bed to the instantaneous free water surface. The vertical coordinate y, which varies from sea bed to instantaneous water level (), is mapped onto a computational vertical coordinate, yc, by the following equation,

yd )( y  . (A-22) c d 

Utilizing wave kinematics by Shinozuka’s method

In order to utilize the linear wave theory with Shinozuka’s method (1972) for every harmonics that is considered in Eq. 3.24 its corresponding wave number, k, is calculated by Eq. A-4. Then the wave length by 2/k is calculated for every harmonics and using

Eqs. A-19, A-20 and A-21 the velocity, acceleration and pressure fields are calculated by superimposing the results of all harmonics. In Shinozuka’s equation the amplitudes of all harmonics are identical as a function of the area under the spectrum curve. Finally using

Wheeler’s stretching method by Eq. A-22 the wave kinematics are mapped to the instantaneous water surface and are input into FAST.

References

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Offshore Wind Turbines at Exposed Sites. Final Report of the OWTES Project, Garrad Hassan and Partners Ltd., Bristol, UK, November 2003.

Dean, R. G., Stream Function Representation of Nonlinear Ocean Waves, J. Geophysics. Res., 70 (1965) pp. 4561–4572.

IEC 61400–3 Ed.1, Wind Turbines – Part 3: Design Requirements for Offshore Wind Turbines, International Electro technical Commission (IEC), 2009

ISO 19901-1, Petroleum and natural gas industries – Specific requirements for offshore structures – Part 1: Metocean design and operating conditions, 2005.

Newman, J. N., Marine Hydrodynamics, The MIT Press, Cambridge, MA, USA, 1997.

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Stokes,G.G., On the Theory of Oscillatory Waves, Trans. Camb. Phil. Soc., 8 (1847) pp. 441-455.

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Wheeler, J.D., Method for Calculating Forces Produced by Irregular Waves, Journal of Petroleum Technology 22 (1970) pp. 359-367.

Zhang, J., Randall, R.E. and Spell, C.A. 1991. On Wave Kinematics Approximate Methods. 23rd Annual Offshore Technology Conference, OTC 6522, 6-9 May, Houston, TX.

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Appendix B

Derivation of the equation for the coefficient of variation of the total probability of an event in Markov Chain Monte Carlo

Simulation

This appendix presents the derivation of Eq. 6.9. Assume that the total probability of

P is estimated by the product of the probabilities P1 to Pn as follows:

P = P1 P2 . . . Pn (B-1)

Then the variance of P is:

n 2 n1 n 2  P)(  2  P   P)()(   P      2      PPPP (B-2)  P P i  P P jiji  i  PP  i  PP  j  i1 i11ij i   j PP

where in the above equation PiPj is the correlation coefficient.

Suppose independent probabilities P1 to Pn, then the second term is zero and the above equation is expanded as following:

2 2 2 2 2 2 2  P  32 ...PPP n   31 ...PPP n    ...... PPP n121   (B-3) P1 P2 Pn

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The mean value of P can be estimated by:

 21 ...PPPP n (B-4)

Dividing both sides of Eq. B-3 by the square of Eq. B-4 yields to:

2 2 2 ...PPP  2 ...PPP  2 ...PPP  2  2 32 n P 31 n P  n121  P P  1  2 ... n (B-5) 2 2 2 2 P 21 ...PPP n 21 ...PPP n 21 ...PPP n

Simplifying the above equation yields to:

2  2  2  2 PP P P 1 2 ... n (B-6) 2 2 2 2 P P1 P2 Pn which is equivalent to Eq. 6.9.

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