MODELING TIME VARYING AND MULTIVARIATE ENVIRONMENTAL CONDITIONS FOR EXTREME LOAD PREDICTION ON OFFSHORE STRUCTURES IN A RELIABILITY PERSPECTIVE ZHANG YI NATIONAL UNIVERSITY OF SINGAPORE 2014 MODELING TIME VARYING AND MULTIVARIATE ENVIRONMENTAL CONDITIONS FOR EXTREME LOAD PREDICTION ON OFFSHORE STRUCTURES IN A RELIABILITY PERSPECTIVE ZHANG YI (B. Eng. Nanyang Technological University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2014 To my mother ACKNOWLEDGEMENT I would like to express my sincere gratitude to my supervisors, Professor Quek Ser Tong and Professor Michael Beer, for their excellent guidance and warm encouragement throughout my Ph.D study. Their precious judgments and critical comments in the revision of my writing have helped me to refine my research tremendously. I am grateful to them for their contribution, not only the research works, but also to teaching me how to become a better researcher. I would like to thank Dr Zhang Mingqiang for his generously sharing with me his knowledge on the uncertainty modeling and experience of Ph.D studies. I could never adequately express all the helps and supports that he has given to me. I like to share my joy of completing the thesis with my friends, especially Dr Zhang Zhen, Miss Liu Mi, Miss Ge Yao, Mr Dai Jian, Dr Wang Yanbo, Dr Wang Li, Dr Ye Feijian, Mr Lu Yitan and Mr Luo Min. I also want to extend my sincere thanks to the other colleagues from the structural lab of National University of Singapore. Their helpful discussion and persistent friendship has made my Ph.D study quite enjoyable and fruitful. My deepest gratitude goes to my family for their help, encouragement and support through all these years. Most importantly, I owe my loving thanks to my mother for her unlimited love and warm care. Without her continuous support and encouragement, I never would have been able to achieve my goals. i   Finally, I wish to acknowledge the heartwarming support provided by my dear wife, who has always been the person most understand me and give me the unflagging love. ii   TABLE OF CONTENTS TITLE PAGE DECLARATION PAGE ACKNOWLEDGEMENT . i TABLE OF CONTENTS iii SUMMARY vii LIST OF TABLES x LIST OF FIGURES . xiii LIST OF SYMBOLS xvii Chapter 1.1 Introduction . Background 1.1.1 Robust Extreme Models 1.1.2 Time Varying Environment 1.1.3 Multivariate Environment . 1.1.4 Efficient Methods for Multivariate Analysis 1.2 Objectives and Scope of Thesis . 1.3 Limitations . 1.4 Organization of Thesis . 10 Chapter Literature Review . 14 2.1 Environment Modeling in Analysis of Offshore Structures . 14 2.2 Framework of Reliability Analysis 19 2.2.1 Measures of Reliability . 21 2.2.2 Simulation Methods 22 2.2.3 Transformation Techniques 26 2.3 Long Term Assessment Criteria . 28 2.4 Extreme Value Theory . 32 2.4.1 Asymptotic Model 33 iii   2.4.2 2.5 Inference for the Extreme Value Distribution 35 Concluding Remarks 39 Chapter Establishing Robust Extreme Value Model .41 3.1 Introduction 42 3.2 Peak-Over-Threshold (POT) Method . 44 3.2.1 Pareto Family 44 3.2.2 Poisson-GPD Model . 45 3.2.3 Declustering 48 3.2.4 Parameter Estimate Method 53 3.3 Uncertainty Assessment of POT Method . 57 3.3.1 Effects of Tail Behavior 59 3.3.2 Effects of Noise . 64 3.3.3 Effects of Range of Dependency 70 3.4 Effects of Nonstationarity through Random Set Approach . 76 3.4.1 Review of Random Set and Dempster-Shafer Structure . 77 3.4.2 Selection of Threshold and Time Span . 80 3.4.3 Uncertainty Quantification 85 3.5 Concluding Remarks 89 Chapter Modeling the Time Varying Environmental Condition for Offshore Structural Analysis 92 4.1 Introduction 93 4.2 Field Data at Ocean Site . 96 4.2.1 Seasonal Characteristics 96 4.2.2 Directional Characteristics 98 4.3 Test for Stationarity of Poisson-GPD model 103 4.3.1 Segmentation Algorithm for Seasonality 104 4.3.2 Segmentation Algorithm for Directionality 113 4.4 Time Varying Modeling . 118 4.4.1 2D Fourier Series Characterization . 118 iv   4.4.2 4.5 Model Validation 123 Static Push-Over Analysis 129 4.5.1 Structural Model Description 129 4.5.2 Reliability Analysis with Importance Sampling . 132 4.6 Concluding Remarks 139 Chapter Modeling the Multivariate Environmental Condition for the Offshore Structural Analysis . 142 5.1 Introduction 142 5.2 Bivariate Models for Sea State Parameters 145 5.2.1 Conditional Joint Distribution Model . 145 5.2.2 Nataf Model 148 5.3 Copula Theory 150 5.3.1 Definition and Basic Properties 150 5.3.2 Examples of Copula 153 5.3.3 Dependence Concepts . 158 5.4 Comparative Study in Multivariate Modeling . 160 5.4.1 Data Pre-treatment 160 5.4.2 Application of Bivariate Models . 167 5.4.3 Results and Discussions 175 5.5 Time Domain Structural Analysis 180 5.5.1 Proposed Discretized Copula Approach . 182 5.5.2 Structural Analysis of a Fixed Offshore Platform 187 5.5.3 Results and Discussions 196 5.6 Concluding Remarks 200 Chapter Conclusions 202 6.1 Summary of Thesis . 202 6.2 Recommendation of Future Works 204   v   REFERENCES 209 Appendix A. Detailed Information of Four Discretization Steps 223 Appendix B. Uncertainty Assessment in POT Method . 239 Appendix C. Selection of Threshold and Time Span in POT Method 254 Appendix D. Testing Values of Threshold and Time Span in POT Approach for Each Identified Time Sectors . 259 Appendix E. USFOS Program Input for the Example Structure . 264 Appendix F. Example of Constructing a Random Set Model . 285 Appendix G. Information of Selected Wave Data and Basic Linear Wave Theory 288   vi   PIPE PIPE PIPE PIPE PIPE PIPE PIPE PIPE PIPE PIPE PIPE PIPE PIPE PIPE PIPE ' ' ' 58 59 60 61 62 63 64 68 69 70 71 83 119 120 121 .800 .800 .800 .800 .700 .650 .600 3.000 3.000 3.250 2.000 .55000 .700 .500 .700 .040 .035 .030 .025 .020 .030 .020 .069 .060 .125 .070 .00100 .050 .055 .045 Mat ID E-mod Poiss Yield Density Thermal MISOIEP 2.100E+11 3.000E-01 3.550E+08 7.850E+03 MISOIEP 2.100E+11 3.000E-01 3.400E+08 7.850E+03 MISOIEP 2.100E+11 3.000E-01 3.200E+08 7.850E+03 MISOIEP 2.100E+11 3.000E-01 3.100E+08 7.850E+03 MISOIEP 2.100E+11 3.000E-01 3.100E+08 4.830E+03 MISOIEP 2.100E+11 3.000E-01 3.400E+08 1.480E+04 MISOIEP 11 2.100E+11 3.000E-01 3.400E+08 1.470E+04 MISOIEP 12 2.100E+11 3.000E-01 3.400E+08 1.260E+04 MISOIEP 13 2.100E+11 3.000E-01 3.400E+08 1.170E+04 MISOIEP 14 2.100E+09 3.000E-01 1.000E+20 .000E+00 MISOIEP 16 2.100E+11 3.000E-01 1.000E+15 .000E+00 MISOIEP 17 2.100E+11 3.000E-01 3.100E+08 .000E+00 Loc-Coo UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC dx 10 11 12 14 21 23 24 25 27 28 29 30 dy .000 .000 -.698 .000 .000 .000 1.000 .812 .754 .707 .707 -.707 -.707 dz .000 1.000 .000 .822 -.822 -1.000 .000 .000 .000 -.707 .707 -.707 .707 279 1.000 .000 .716 .569 .569 .000 .000 .583 .657 .000 .000 .000 .000 .000E+00 .000E+00 .000E+00 .000E+00 .000E+00 .000E+00 .000E+00 .000E+00 .000E+00 .000E+00 .000E+00 .000E+00 UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC 38 39 40 43 44 47 48 49 50 53 55 66 67 74 76 93 94 97 98 101 103 119 120 123 124 125 126 127 129 136 140 143 153 154 159 161 162 163 164 165 166 167 168 169 170 173 -1.000 .664 -.664 .638 -.638 .665 .664 -.665 .664 -.664 -.664 .699 -.699 .699 -.699 .726 -.726 .764 -.764 .726 -.726 .746 -.746 .806 -.806 .990 -.990 .746 -.746 .698 -.754 .890 .000 .000 -.812 -.819 -.825 -.819 -.819 -.825 -.819 -.031 .031 .000 .000 .031 .000 .747 .747 .770 .770 -.747 .748 -.747 -.748 -.748 .748 .715 .715 -.715 -.715 .688 .688 .645 .645 -.688 -.688 .666 .666 .592 .592 .142 .142 -.666 -.666 .000 .000 .000 .694 -.694 .000 .068 .067 -.068 -.068 -.067 .068 .869 .869 .868 .868 .869 280 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .716 .657 .456 .720 .720 .583 .570 .561 -.570 .570 .561 -.570 -.495 .495 -.496 .496 -.495 UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC 174 175 176 177 178 179 180 181 182 183 184 187 188 191 192 193 194 195 196 197 198 199 200 202 203 205 209 210 211 214 217 219 223 226 229 232 233 251 252 253 254 256 257 258 260 261 -.031 -.812 -.842 -.812 -.812 -.842 -.812 -.034 .034 .000 .000 .034 -.034 -.803 -.779 -.834 -.803 -.803 -.779 -.834 -.779 -.803 .042 -.042 .000 .000 .039 -.042 -.039 -.998 -.998 -.998 -.998 -.998 -.998 -.998 .674 .051 .063 .063 .051 -.999 -.999 -.998 -1.000 -1.000 .869 .069 .064 -.069 -.069 -.064 .069 .841 .841 .840 .840 .841 .841 -.070 .074 .065 .070 .070 -.074 -.065 .074 -.070 .741 .741 .740 .740 .785 .741 .785 .007 .007 -.007 .007 .007 -.007 -.007 -.599 -.737 .781 -.781 .737 -.002 .002 .000 -.002 .002 281 .495 .580 .536 -.580 .580 .536 -.580 -.541 .541 -.543 .543 -.541 .541 -.592 .623 .548 .592 -.592 .623 .548 -.623 .592 .670 -.670 .673 -.673 -.618 .670 .618 .062 .061 -.062 -.062 -.061 .062 .061 .433 .673 .621 .621 .673 -.032 -.032 -.063 -.031 -.031 UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC UNITVEC ' ' ' ' ' ' ' Ecc-ID 263 264 265 266 267 278 279 281 364 365 366 367 368 369 370 371 372 412 414 416 418 420 422 424 426 428 430 432 434 440 442 446 448 450 452 -.992 -.905 -.926 -.905 -.926 -.779 .042 -.998 -.765 .787 -.787 .765 -.765 .787 -.787 .765 -.674 -.468 .820 .000 -.780 -.820 .820 -.780 -.820 -.108 -.108 .000 .000 .108 .108 .468 -.468 .468 .000 Ex .000 -.156 .138 .156 -.138 -.074 .741 -.007 .112 .125 .125 .112 -.112 -.125 -.125 -.112 .599 .874 .148 .993 .120 .148 -.148 -.120 -.148 .880 -.880 .887 -.887 .880 -.880 .874 -.874 -.874 -.993 Ey .129 .395 -.350 .395 -.350 -.623 -.670 -.061 .634 .604 .604 .634 .634 .604 .604 .634 .433 .133 .553 .118 .614 .553 .553 .614 .553 .464 .464 .461 .461 .464 .464 .133 .133 .133 .118 Ez Mat ID S P R I N G C H A R. SPRIDIAG 19 6.72000E+09 6.72000E+09 2.07600E+10 5.42000E+11 5.42000E+11 2.02000E+11 Load Case Node ID NODELOAD 101 NODELOAD 102 LOAD INTENSITY .00000E+00 .00000E+00 -1.94880E+07 .00000E+00 .00000E+00 -1.59120E+07 282 NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 103 104 107 108 109 110 111 112 113 114 115 116 117 118 119 120 130 132 134 135 201 202 203 204 205 206 301 302 303 304 305 306 307 308 311 312 401 402 403 404 405 406 407 408 411 412 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 283 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 -1.59120E+07 -1.94880E+07 -1.50000E+07 -1.50000E+07 -1.50000E+07 -1.50000E+07 -7.50000E+06 -7.50000E+06 -7.50000E+06 -7.50000E+06 -7.50000E+06 -1.29000E+07 -7.50000E+06 -7.50000E+06 -1.29000E+07 -7.50000E+06 -1.70000E+06 -1.70000E+06 -1.70000E+06 -1.70000E+06 -2.76000E+05 -1.48000E+05 -1.48000E+05 -4.17000E+05 -8.07000E+05 -2.93000E+05 -2.10000E+05 -2.10000E+05 -2.10000E+05 -2.10000E+05 -7.04000E+05 -1.85000E+05 -3.46000E+05 -3.70000E+05 -8.79000E+05 -8.72000E+05 -1.27500E+06 -1.27500E+06 -1.27500E+06 -1.27500E+06 -6.84000E+05 -1.15000E+05 -4.37000E+05 -3.76000E+05 -4.82000E+05 -7.84000E+05 NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD NODELOAD ' ' ' ' ' ' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 501 502 503 504 505 506 507 508 511 512 513 516 601 602 603 604 605 606 607 608 611 612 613 614 701 702 703 704 Load Case Elem ID Load Case Elem ID .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 -1.35600E+06 -1.35600E+06 -1.35600E+06 -1.35600E+06 -7.10000E+05 -1.15000E+05 -4.27000E+05 -3.75000E+05 -1.15000E+05 -8.74000E+05 -1.15000E+05 -7.35000E+05 -5.29200E+06 -5.29200E+06 -5.29200E+06 -5.29200E+06 -1.15000E+05 -1.15000E+05 -2.61000E+05 -1.98000E+05 -1.15000E+05 -4.68000E+05 -1.15000E+05 -3.92000E+05 -1.17620E+07 -1.17620E+07 -1.17620E+07 -1.17620E+07 LOAD INTENSITY Press1 Press2 Press3 Load Case Acc_X Acc_Y Acc_Z GRAVITY .0000E+00 .0000E+00 -9.8100E+00 284 Press4 Appendix F. Example of Constructing a Random Set Model As described in Chapter 3, the random set which is also sometimes referred to as a Dempster-Shafer structure consists of a finite number of focal sets. For example, for discrete random variable xi in the space X. Random sets are a collection of many imprecise observations    Ai : i  1, , n of the given fundamental set X and its probability weight mapping function: m :    0,1 (F.1) where mi = m(Ai) ∑m(Ai) = 1. This gives a measure of the degree of confidence for the observations of X. For the occurrence event E in the space of X, the plausibility measure of is defined by: Pls  E    Ai E  m  Ai  (F.2) and the belief function is given by: Bel  E    m A  Ai  E i (F.3) These plausibility and the belief functions can be interpreted as a prescription of a set of probability function. Or in other words, it gives the upper and lower probabilities of a certain set of probability distributions. For demonstration purpose, the following shows an example of how to construct the random set model. 285 Random Set Example The uncertainty of X is represented by three random focal sets as: A1   2, 4 , A2  3,8 , A3  1,5 and associated weights are given by: m  A1   0.2 , m  A2   0.3 , m  A3   0.5 These are presented in Fig. F.1 (a) and the probability weights which are represented by the three bins are also illustrated in this figure, see Fig. F.1 (b). CDF CDF  A3 , m3  1.0 1.0  A2 , m2  0.8 0.6 0.6  A1 , m1  0.4 0.2  1 , m1  (a) 0.8 0.4 0.2 10 X (b) 10 X Figure F.1 Random set (a) and its assigned probability weight (b). The imprecise probability (probability box) used to represent the random set is illustrated in Fig. F.2 (a) where the focal sets are plotted as a stack and the height of the lines is determined by the cumulative sum of weights (illustrated by the dotted line). The random set can also be visualized by its contour function which assigns each singleton x its plausibility Pls(x). The value is obtained by 286 adding all the probability m(Ai) of those focal sets Ai to which x belongs. This is illustrated in Fig. F.2 (b). CDF CDF 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 (a) 10 X Figure F.2 Probability box (a) and contour function (b). 287 (b) 10 X Appendix G. Information of Selected Wave Data and Basic Linear Wave Theory G.1 Wave Information Studies The wave data used in this thesis are taken from the Wave Information Studies (WIS) which is a US Army Corps of Engineers (USACE) sponsored project that generates consistent, hourly, long-term (20+ years) wave climatologies along all US coastlines, including the Great Lakes and US island territories. The WIS program originated in the Great Lakes in the mid 1970s and migrated to the Atlantic, Gulf of Mexico and Pacific Oceans. The currently available domains are depicted in Fig. G.1. The official website of WIS is at http://wis.usace.army.mil/. Figure G.1 WIS data domain. 288 This site provides access to the database of wave information for a densely-spaced series of wave gauges in water depths of 15-20 m and for a lessdense series in deeper water (100 m or more). Data available from each site include hourly wind speed, wind direction, and bulk wave parameters (significant wave height, period, and direction). Discrete directional wave spectra at to 3hour intervals are also available. A suite of tabular and graphic products for each location is also provided in the website. Figure G.2 Geological location of the selected buoy. In this thesis, the set of data is taken from a buoy (No. 82283) locating in the south coast of Alaska (56.5oN 203.25oE), see Fig. G.2. The water depth at this location is 124m. The selected data set containing 25 years of hourly wave records (1985/1/1 01:00 to 2010/1/1 01:00) is filtered. In Chapter 3, the significant wave height HS time series data are utilized. In Chapter 4, the record of significant wave height HS, wave observation time t and wave directions θS are been used. In Chapter 5, a set of ocean parameters are been used, these include significant wave height HS, peak wave period TP, wave direction θS, wind speed VW and wind direction θW. 289 G.2 Linear Wave Theory The basic wave theory for analyzing an offshore structure relevant to the current study is presented. Detailed information could be found from the design codes (DNV 2007) and books (Sarpkaya and Isaacson 1981). Direction of wave Mean sea level Surface elevation η(x,t) Z X Water depth d Wave particle velocity and acceleration , z+d Seabed Figure G.3 Regular wave propagation properties (Sarpkaya and Isaacson 1981). The linear wave theory is the most common way of representing the properties of a wave particle. The theory generally states that when the wave height H is much smaller compared to the wave length λ and still water depth d, the Airy wave theory may be adopted to give a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The surface elevation of a wave, denoted as η(x, t), can be expressed in terms of the 290 amplitude a=H/2, time t and spatial location x in the direction of wave propagation by:   x, t   a cos  x  t  (G.1) where κ=2π/λ is the wave number and ω=2π/T is the angular frequency of the wave, in which T is the wave period. Based on the solution of the Laplace equation in terms of the velocity potential and the use of linearized boundary condition, the horizontal water particle velocity and acceleration at depth z (for z≤0) measured from the mean sea level can be obtained by the derivative to the potential function which are expressed as: u  x, z , t   a cosh   z  d   u  x, z, t   a sinh  d  cos  x  t  cosh   z  d   sinh  d  sin  x  t  (G.2) (G.3) In deep water, when κd>π, the above formulae can be approximated by: u  x, z, t   ae z cos  x  t  (G.4) u  x, z, t   a 2e z sin  x  t  (G.5) Usually, the ocean data are collected based on measurements at regular intervals (e.g. hourly record). The random behavior of ocean waves can be described by a statistical model. The variation of the physical variable within the time interval is assumed to be stationary and can be described by a stochastic 291 process. Based on the property of a zero-mean stationary process, the property of a sea state within such a short period can be conveniently represented in the frequency domain by the wave spectrum S(ω). Several well-established wave spectra are available in the literature. The most commonly applied wave spectrums are the Pierson-Moskowitz (P-M) spectrum and the JONSWAP spectrum (DNV 2007). The Pierson-Moskowitz (P-M) spectrum is given by:    4  5 S PM     H S P   exp      16   P   (G.6) where HS is the significant wave height, TP is the peak period, ωP=2π/T is the angular spectral peak frequency. JONSWAP spectrum SJ(ω) is formulated as a modification to P-M spectrum with the consideration of a developing sea state in a fetch limited situation: S J    A  S PM        P   exp  0.5     P    (G.7) where ζ is the spectral width parameter evaluated on the values of ω and ωP: 0.07 for   P  0.09 for   P   (G.8) Aγ=1-0.287ln(γ) is the normalizing factor, γ is the non-dimensional peak shape parameter and the following value can be applied: 292  5    T    exp  5.75  1.15 P  HS      for     for 3.6  TP  3.6 HS TP 5 HS for  (G.9) TP HS Once the wave spectrum is established, the irregular random waves in the short term stationary sea state can be easily represented by a summation of sinusoidal wave components. The simulation of the surface elevation in time domain could be obtained by a superposition of large number of independent linear waves corresponding to different amplitudes, frequencies and arbitrary phase angles. The series representation of the elevation corresponding to a wave spectrum is given by: N   x, t    cos  i x  i t  i  (G.10) i 1 where κi and ωi are the wave number and discrete frequency which have the same meaning as in Eq. (5.26). θi are random phases, uniformly distributed between and 2π, mutually independent of each other and of the amplitude ai. The amplitude of the ith component, which is Rayleigh distributed, is given by  S i  i (G.11) where S(ωi) is the value of the ith component in the established wave spectrum and is the difference between successive frequencies. To simulate the random waves accurately, the simulation must contain enough 293 number of wave components. Or in other words, the value of ∆ωi needs to be small. The linear wave representation also allows the superposition of the water particle velocity and acceleration components. The water horizontal velocity and acceleration for this simulated sea state is given by: N u  x, z, t    i 2S i  i i 1 N u  x, z, t    i2 2S i  i i 1 cosh  i  z  d   sinh  i d  cosh  i  z  d   sinh  i d  cos  i x  it  i  (G.12) sin  i x  i t  i  (G.13) which is employed as the basic simulation technique in the calculation of random wave loading to be used to compute the hydrodynamic response of offshore structures. 294 [...]... to characterize the variations of extreme values with time A collected group of data is selected to demonstrate that such a discretized model can provide a more reasonable and accurate characterization of each parameter of interest This approach of incorporating the time varying effect is examined through the reliability analysis of an existing offshore platform The results show that incorporating the... Fourier characterizations of shape and scale parameter changes along the time and direction axes 121 Figure 4.12 CDF of original data and simulated data for θ (-10o~62o] 124 Figure 4.13 CDF of original data and simulated data for θ (62o~134o] 125 Figure 4.14 CDF of original data and simulated data for θ (134o~206o] 125 Figure 4.15 CDF of original data and simulated data for θ (206o~278o]... 5.12 Tail fittings of marginal parametric model for (a) HS, (b) TP, and (c) VW 174  Figure 5.13 Comparison of contour plot between original data and (a) copula approach, (b) Nataf model, (c) conditional joint model for HS and TP 178 Figure 5.14 Comparison of contour plot between the original data and (a) copula approach (b) Nataf model (c) conditional joint model for HS and. ..SUMMARY With the changing environmental conditions experienced over the last decade, design against failures of offshore structures has becomes even more challenging Complexities exist in various steps, from translating and modeling of the environmental data, appropriate structural analysis, reliability assessment, installation, operations and maintenance This is compounded by the presence of climatic... established and used in the characterization of environmental factors in the offshore industry (Muir & El-Shaarawi 1986) The parameters governing the models are estimated statistically using data collected from monitoring stations Often only the extreme values govern the design loads and the design life span becomes a primary determining variable A commonly adopted approach is to use extreme value statistical... has yet to be completely answered It is therefore of interest to perform a comparative study highlighting the characteristics in each approach in modeling multivariate data 1.1.4 Efficient Methods for Multivariate Analysis To assess the long term performance of a structure within a multivariate environment associated with many environment conditions requires many numerical simulations and is computationally... variables associated with the environmental loads, the dependencies between these variables, and investigating the impact of the uncertainties and dependencies on the long term assessment of a typical marine structure The wave parameters which directly influence the loadings on offshore structures are studied in this work Considering the long time life-span and the limited data normally available, extreme. .. presented 2 To manage the non-stationarity inherent in long term data by developing a discrete statistical model to represent the time varying effects in the ocean parameters and show the importance of this in a reliability analysis A segmentation approach is employed and 2-D Fourier transforms is used in conjunction with a probabilistic model to reflect the smooth transition in the parameters amongst the... (Menendez et al 2009), in particular accounting for all the uncertainties associated with such an approach (Vanem and Bitner-Gregersen 2012) 1.1.3 Multivariate Environment It is natural that the environment parameters, such as wave period, significant wave height and wind speed, are correlated To assume that they are independent may lead to unconservative results and hence is a potential cause for under-design... storm and tsunami Under such environment, many unfavorable phenomena like marine corrosion, marine growth, foundation scouring, material deterioration and fatigue damage will cause a weakening of the overall strength of the structure and thus lead to an unexpected accident Historical records depicted our inadequate understanding of the ocean environment leading to accidents with drastic consequences and . MODELING TIME VARYING AND MULTIVARIATE ENVIRONMENTAL CONDITIONS FOR EXTREME LOAD PREDICTION ON OFFSHORE STRUCTURES IN A RELIABILITY PERSPECTIVE ZHANG YI NATIONAL UNIVERSITY. SINGAPORE 2014 MODELING TIME VARYING AND MULTIVARIATE ENVIRONMENTAL CONDITIONS FOR EXTREME LOAD PREDICTION ON OFFSHORE STRUCTURES IN A RELIABILITY PERSPECTIVE ZHANG YI (B. Eng. Nanyang. provide a more reasonable and accurate characterization of each parameter of interest. This approach of incorporating the time varying effect is examined through the reliability analysis of an existing