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Cross Correlation Functions 332 D STATISTICAL ANALYSIS OF WIND LOADINGS AND RESPONSES OF A TRANSMISSION TOWER STRUCTURE by SIEW HOCK LIEW, B.S. A DISSERTATION IN CIVIL ENGINEERING Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Approved Accepted May, 1988 ©1989 SIEW HOCK LIEW All Rights Reserved ACKNOWLEDGMENTS I wish to express my deepest appreciation to my committee chairman. Dr. H. Scott Norville, for his confi­ dence in me during the entire course of this researcr. project. His encouragement and valuable advice as a committee chairman, as a major advisor, and as a friend, has instilled a value of perseverance in me. I also ^-'ish to express my sincere appreciation to Dr. K. C. M^?nt^^, for his expertise and advice in wind engineering, to Dr. T. 0. Lewis for his advice in statistics and in time series, to Dr. W. P. Vann and Dr. E. Blair for t-heir valuable suggestions and comments on this dissertation manuscript, and to Mr. M. L. Levitan for his permisf5ion to use some of the time histories plots. I am most grateful to my wife, Lily, for her patience and understanding throughout the course of my graduate work, and to my parents for their moral support -^nd instilling in me the value of education from the very beginning of my life. 11 TABLE OF CONTENTS PAGE ACKNOWLEDGMENTS i i LIST OF TABLES vi LIST OF FIGURES vii 1. INTRODUCTION 1 2. LITERATURE REVIEW 7 3. SITE DESCRIPTION, INSTRUMENTATION, AND DATA COLLECTION SYSTEM 20 3.1 Introduction 20 3.2 Location 20 3.3 Instrumentation 27 4. STATISTICAL ANALYSIS OF TIME SERIES RECORD 35 4.1 Introduction 35 4.2 Data Handling 36 4.3 Time Series Analysis 38 4.4 Stationarity of a Time Series 43 4.5 Selected Time Histories 48 4.6 Autocorrelation of Selected Time Histories 61 4.7 Cross Correlation of Selected Time Histories 79 4.S Discussion of Results 119 5. FREQUENCY DOMAIN ANALYSIS OF SINGLE LOAD AND SINGLE RESPONSE SYSTEM 127 5.1 Introduction 127 i i i PAGE 5.2 Transfer Function Relating Loads With Responses 128 5.3 Power Spectra of Time Series 131 5.4 Cross Spectra of Bivariate Time Series 142 5.5 Frequency Response Function of a Linear System 146 5.6 Illustration and Applications of Frequency Response Function 149 5.6.1 Power Spectra Plots for Selected Time Histories 149 5.6.2 Frequency Response Function of Transmission Tower 163 5.6.3 Coherence Function of Transmission Tower 191 5.7 Conclusions and Discussion of Results 212 6. FREQUENCY RESPONSE FUNCTION OF MULTIPLE INPUTS AND A SINGLE OUTPUT 218 6.1 Introduction 218 6.2 General Requirements 220 6.3 Frequency Response Function Relating Two Loadings and One Response 223 6.4 Numerical Examples of Transmission Tower Subjected to Two Loadings and One Response 231 6.5 Discussion of Results 246 7. TIME SERIES ANALYSIS AND MODELING 252 7.1 Introduction 252 7.2 Definition of a Time Series 252 IV PAGE 7.3 Concept in Time Series Modeling 255 7.4 Models for Stationary and Nonstationary Time Series 258 7.5 Power Spectrum of a Linear Process 273 7.6 Numerical Examples of Time Series Modeling 277 7.6.1 Time Series Models 277 7.6.2 Power Spectra of Modeled Time Series 280 7.7 Discussion of Results 281 8. CONCLUSION 283 LIST OF REFERENCES 290 APPENDICES 297 A. TIME HISTORIES PLOTS 298 B. AUTOCORRELATION FUNCTIONS 324 C. CROSS CORRELATION FUNCTIONS 332 D. POWER SPECTRA 355 E. FREQUENCY RESPONSE FUNCTION 376 F. COHERENCE FUNCTION 399 G. MOVING AVERAGE PARAMETERS 411 LIST OF TABLES TABLE PAGE 3.1 Twenty Three Segments of Data 21 3.2 File Description in Mode 22 28 4.1 Selected Records for Analysis 49 4.2 Mean Load Values From The Conductors (kN) 56 4.3 Mean Swing Angles (Degrees) 56 4.4 Mean Transverse Load Values From the Conductors (kN) 57 4.5 Mean Longitudinal Load Values From the Conductors (kN) 57 4.6 Mean Leg Stresses of the Transmission Tower Due to Wind and Total Conductor Loads (MPa) 58 5.1 Lag aind Spectral Windows (Jenkins and Watts, 1968) 137 5.2 Properties of Spectral Windows (Jenkins and Watts, 1968) 140 7.1 General Models 271 7.2 Identification of Time Series Models 279 7.3 Chi-Square Statistics of the Autocorrelation of the Residuals 279 VI LIST OF FIGURES FIGURE PAGE 3.1 Topography of Site and Orientation of Power Lines 22 3.2 Elevation Along the Test Line (Vertical Scale Exaggerated) 25 3.3 Schematic of Tower 16/4 26 3.4 Schematic Locations of Strain Gages Designated by the Corners of the Tower 32 4.1 Twenty Three Available Wind Records 37 4.2 Methodology for Time Series Analysis 39 4.3a Time History for Wind Speed at 34.7 m for Record N05 51 4.3b Time History for Wind Speed at 34.7 m for Record N07 51 4.4a Time History for Wind Speed at 34.7 m for Record NOS 52 4.4b Time History for Wind Speed at 34.7 m for Record N16 52 4.5 Locations of Selected Strain Gages for Analysis 55 4.6a Autocorrelation Functions of Original and First Differenced Series of WS05 for Record N05 63 4.6b Autocorrelation Functions of Original and First Differenced Series of WS05 for Record NOS 63 4.7a Autocorrelation Functions of LC03 for Record N05 65 4.7b Autocorrelation Functions of First Differenced Series of LC03 for Record N05 65 vii FIGURE PAGE 4.8a Autocorrelation Functions of LC04 for Record N05 67 4.8b Autocorrelation Functions of First Differenced Series of LC04 for Record N05 67 4.9a Autocorrelation Functions of LC05 for Record NOS 68 4.9b Autocorrelation Functions of First Differenced Series of LC05 for Record N05 68 4.lOa Autocorrelation Functions of Original and First Differenced Series of LC03 for Record NOS 69 4.lOb Autocorrelation Functions of Original and First Differenced Series of LC04 for Record NOS 69 4.11 Autocorrelation Functions of Original and First Differenced Series of LC05 for Record NOS 71 4.12a Autocorrelation Functions of Original and First Differenced Series of SG05 for Record N05 73 4.12b Autocorrelation Functions of Original and First Differenced Series of SGll for Record N05 73 4.13a Autocorrelation Functions of Original and First Differenced Series of SG06 for Record NOS 75 4.13b Autocorrelation Functions of Original and First Differenced Series of SG12 for Record NOS 75 4.14a Autocorrelation Functions of Original and First Differenced Series of SG05 for Record NOS 76 VI 1 1 FIGURE PAGE 4.14b Autocorrelation Functions of Original and First Differenced Series of SGll for Record NOS 76 4.ISa Autocorrelation Functions of Original and First Differenced Series of SG06 for Record NOS 77 4.15b Autocorrelation Functions of Original and First Differenced Series of SG12 for Record NOS 77 4.16a Cross Correlation Functions of WS05 and SG05 for Record NOS 83 4.16b Cross Correlation Functions of WS05 and SG06 for Record NOS S3 4.17a Cross Correlation Functions of WS05 and SG05 for Record NOS 85 4.17b Cross Correlation Functions of WS05 and SGll for Record NOS 85 4.ISa Cross Correlation Functions of WSOS and SG06 for Record NOS 86 4.18b Cross Correlation Functions of WSOS and SG12 for Record NOS 86 4.19a Cross Correlation Functions of Total Load Values From West Conductor and SGOS for Record NOS 90 4.19b Cross Correlation Functions of Total Load Values From West Conductor and SG06 for Record NOS 90 4.20a Cross Correlation Functions of Total Load Values From East Conductor and SGOS for Record NOS 91 4.20b Cross Correlation Functions of Total Load Values From East Conductor and SG06 for Record N05 91 IX FIGURE PAGE 4.21a Cross Correlation Functions of Total Load Values From Center Conductor and SGOS for Record NOS 92 4.21b Cross Correlation Functions of Total Load Values From Center Conductor and SG06 for Record NOS 92 4.22a Cross Correlation Functions of Total Load Values From West Conductor and SGOS for Record NOS 94 4.22b Cross Correlation Functions of Total Load Values From West Conductor and SG06 for Record NOS 94 4.23a Cross Correlation Functions of Total Load Values From West Conductor and SGll for Record NOS 96 4.23b Cross Correlation Functions of Total Load Values From West Conductor and SG12 for Record NOS 96 4.24a Cross Correlation Functions of Total Load Values From East Conductor auid SGOS for Record NOS 97 4.24b Cross Correlation Functions of Total Load Values From East Conductor and SG06 for Record NOS 97 4.2Sa Cross Correlation Functions of Total Load Values From East Conductor and SGll for Record NOS 98 4.25b Cross Correlation Functions of Total Load Values From East Conductor and SG12 for Record NOS 98 4.26a Cross Correlation Functions of Total Load Values From Center Conductor and SGOS for Record NOS 100 FIGURE PAGE 4.26b Cross Correlation Functions of Total Load Values From Center Conductor and SG06 for Record NOS 100 4.27a Cross Correlation Functions of Longitudinal Load Values From West Conductor and SGll for Record NOS 103 4.27b Cross Correlation Functions of Longitudinal Load Values From West Conductor and SG12 for Record NOS 103 4.2Sa Cross Correlation Functions of Longitudinal Load Values From East Conductor and SGll for Record NOS 104 4.2Sb Cross Correlation Functions of Longitudinal Load Values From East Conductor and SG12 for Record NOS 104 4.29a Cross Correlation Functions of Longitudinal Load Values From West Conductor and SGOS for Record NOS 106 4.29b Cross Correlation Functions of Longitudinal Load Values From West Conductor and SG12 for Record NOS 106 4.30a Cross Correlation Functions of Longitudinal Load Values From East Conductor and SGOS for Record NOS 107 4.30b Cross Correlation Functions of Longitudinal Load Values From East Conductor and SGll for Record NOS 107 4.31a Cross Correlation Functions of Longitudinal Load Values From Center Conductor
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