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

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

SIEW HOCK LIEW

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 and SGOS for Record NOS 108 4.31b Cross Correlation Functions of Longitudinal Load Values From Center Conductor and SGI2 for Record NOS 108

XI FIGURE PAGE 4.32a Cross Correlation Functions of Transverse Load Values From West Conductor and SGOS for Record NOS 110 4.32b Cross Correlation Functions of Transverse Load Values From West Conductor and SGll for Record NOS 110 4.33a Cross Correlation Functions of Transverse Load Values From East Conductor and SGOS for Record NOS 112 4.33b Cross Correlation Functions of Transverse Load Values From East Conductor and SGll for Record NOS 112 4.34a Cross Correlation Functions of Transverse Load Values From Center Conductor and SGOS for Record NOS 113 4.34b Cross Correlation Functions of Transverse Load Values From Center Conductor and SGI2 for Record NOS 113 4.3Sa Cross Correlation Functions of Transverse Load Values From West Conductor and SGOS for Record NOS 115 4.3Sb Cross Correlation Functions of Transverse Load Values From West Conductor and SGll for Record NOS 115 4.36a Cross Correlation Functions of Transverse Load Values From East Conductor and SGOS for Record NOS 116 4.36b Cross Correlation Functions of Transverse Load Values From Center Conductor and SGOS for Record NOS 116 4.37a Cross Correlation Functions of Transverse Load Values From Center Conductor and SGOS for Record NOS US

Xll FIGURE PAGE 4.37b Cross Correlation Functions of Transverse Load Values From Center Conductor and SGI2 for Record NOS 118

5.1 Schematic Representation of a Linear System Subjected to a Single Load and a Single Response 129

5.2a Plot of Lag Windows From Table 5.1 (Jenkins and Watts, 1968) 138 5.2b Plot of Spectral Windows From Table 5.1 (Jenkins and Watts, 1968) 138 S.3a Power Spectrum of Wind Speed at 34.7 m for Record NOS 151 S.3b Power Spectrum of Wind Speed at 34.7 m for Record NOS 151 S.4a Power Spectrum of Transverse Load From West Conductor for Record NOS 153 5.4b Power Spectrum of Transverse Load From East Conductor for Record NOS 153 S.Sa Power Spectrum of Transverse Load From West Conductor for Record NOS 154 S.Sb Power Spectrum of Transverse Load From East Conductor for Record NOS 154 S.Sa Power Spectrum of Transverse Load From Center Conductor for Record NOS 156 5.6b Power Spectrum of Transverse Load From Center Conductor for Record NOS 156 5.7a Power Spectrum of Main Strain Gage Located at SE Corner for Record NOS 159 5.7b Power Spectrum of Main Strain Gage Located at NW Corner for Record NOS 159 S.Sa Power Spectrum of Main Strain Gage Located at SE Corner for Record NOS 161 xi i i FIGURE PAGE

S.Sb Power Spectrum of Main Strain Gage Located at NW Corner for Record NOS 161 5.9a Frequency Response Function Relating Wind Speed and SE Diagonal Leg Strain Record for Record NOS 165 5.9b Frequency Response Function Relating Wind Speed and NW Diaigonal Leg Strain Record for Record NOS 165

5.10a Frequency Response Function Relating Wind Speed and SE Main Leg Strain Record for Record NOS 167

5.10b Frequency Response Function Relating Wind Speed and SE Main Leg Strain Record for Record NOS 167 5.11a Frequency Response Function Relating Wind Speed and SE Diagonal Leg Strain Record for Record NOS 168 5.lib Frequency Response Function Relating Wind Speed and NW Diagonal Leg Strain Record for Record NOS 168

5. 12a Frequency Response Function Relating Total Load From East Conductor and SE Main Leg Strain Record for Record NOS 170 5.12b Frequency Response Function Relating Total Load From East Conductor and NW Main Leg Strain Record for Record NOS 170

5. 13a Frequency Response Function Relating Total Load From East Conductor and SE Diagonal Leg Strain Record for Record NOS 172

S. 13b Frequency Response Function Relating Total Load From East Conductor and NW Diagonal Leg Strain Record for Record NOS 172

5. 14a Frequency Response Function Relating Total Load From Center Conductor and SE Main Leg Strain Record for Record NOS 173

xiv FIGURE PAGE 5. 14b Frequency Response Function Relating Total Load From Center Conductor and NW Main Leg Strain Record for Record NOS 173

5.15a Frequency Response Function Relating Total Load From West Conductor and SE Main Leg Strain Record for Record NOS 174 5.15b Frequency Response Function Relating Total Load From West Conductor and NW Main Leg Strain Record for Record NOS 174

5.16a Frequency Response Function Relating Total Load From East Conductor and SE Main Leg Strain Record for Record NOS 176 5.16b Frequency Response Function Relating Total Load From East Conductor and NW Main Leg Strain Record for Record NOS 176

5. 17a Frequency Response Function Relating Transverse Load From West Conductor and SE Main Leg Strain Record for Record NOS 178

5.17b Frequency Response Function Relating Transverse Load From West Conductor and NW Main Leg Strain Record for Record NOS 178

5. 18a Frequency Response Function Relating Transverse Load From East Conductor and SE Diagonal Leg Strain Record for Record N05 179

5.18b Frequency Response Function Relating Transverse Load From East Conductor and NW Diagonal Leg Strain Record for Record NOS 179

5.19a Frequency Response Function Relating Transverse Load From Center Conductor and SE Main Leg Strain Record for Record NOS 180

5. 19b Frequency Response Function Relating Transverse Load From Center Conductor and NW Diagonal Leg Strain Record for Record NOS 180

XV FIGURE PAGE 5.20a Frequency Response Function Relating Transverse Load From West Conductor and SE Main Leg Strain Record for Record NOS 182 S.20b Frequency Response Function Relating Transverse Load From West Conductor and NW Main Leg Strain Record for Record NOS 182 5.21a Frequency Response Function Relating Transverse Load From West Conductor and SE Diagonal Leg Strain Record for Record NOS 183 5.21b Frequency Response Function Relating Transverse Load From West Conductor and NW Diagonal Leg Strain Record for Record NOS 183 5.22a Frequency Response Function Relating Transverse Load From East Conductor and SE Main Leg Strain Record for Record N08 184 5.22b Frequency Response Function Relating Transverse Load From East Conductor and NW Main Leg Strain Record for Record NOS 184 5.23a Frequency Response Function Relating Transverse Load From West Conductor and SE Main Leg Strain Record for Record NOS 186 5.23b Frequency Response Function Relating Transverse Load From West Conductor and NW Main Leg Strain Record for Record NOS 186 5. 24a Frequency Response Function Relating Longitudinal Load From East Conductor and NW Main Leg Strain Record for Record NOS 187 5.24b Frequency Response Function Relating Longitudinal Load From Center Conductor and NW Main Leg Strain Record for Record NOS 187 5.2Sa Frequency Response Function Relating Transverse Load From West Conductor and SE Main Leg Strain Record for Record N08 189

XVI FIGURE PAGE

S.2Sb Frequency Response Function Relating Transverse Load From West Conductor and NW Main Leg Strain Record for Record NOS 189

5.26a Frequency Response Function Relating Transverse Load From East Conductor and SE Main Leg Strain Record for Record NOS 190 5.26b Frequency Response Function Relating Trainsverse Load From Center Conductor and SE Main Leg Strain Record for Record NOS 190

5.27a Coherence Function Relating Wind Speed and SE Diagonal Leg Strain Record for Record NOS 192

5.27b Coherence Function Relating Wind Speed auid NW Diaigonal Leg Strain Record for Record NOS 192 S.2Sa Coherence Function Relating Wind Speed and SE Main Leg Strain Record for Record NOS 194 5.28b Coherence Function Relating Wind Speed and SE Main Leg Strain Record for Record NOS 194 5.29a Coherence Function Relating Total Load From West Conductor and SE Main Leg Strain Record for Record NOS 196

S.29b Coherence Function Relating Total Load From West Conductor and SE Diagonal Leg Strain Record for Record N08 196

5.30a Coherence Function Relating Total Load From East Conductor and SE Main Leg Strain Record for Record NOS 198

5.30b Coherence Function Relating Total Load From East Conductor and NW Main Leg Strain Record for Record NOS 198

XVI 1 FIGURE PAGE 5.31a Coherence Function Relating Transverse Load From West Conductor and SE Main Leg Strain Record for Record NOS 199

5.31b Coherence Function Relating Transverse Load From West Conductor and SE Diagonal Leg Strain Record for Record NOS 199 5.32a Coherence Function Relating Transverse Load From West Conductor and NW Main Leg Strain Record for Record NOS 201

5.32b Coherence Function Relating Transverse Load From West Conductor and NW Diagonal Leg Strain Record for Record NOS 201 5.33a Coherence Function Relating Transverse Load From West Conductor and SE Main Leg Strain Record for Record NOS 202 5.33b Coherence Function Relating Transverse Load From West Conductor and SE Diagonal Leg Strain Record for Record NOS 202 5.34a Coherence Function Relating Transverse Load From West Conductor and NW Main Leg Strain Record for Record N08 204 5.34b Coherence Function Relating Transverse Load From West Conductor and NW Diagonal Leg Strain Record for Record NOS 204 S.3Sa Coherence Function Relating Transverse Load From East Conductor and SE Main Leg Strain Record for Record NOS 205

5.35b Coherence Function Relating Transverse Load From East Conductor and SE Diagonal Leg Strain Record for Record NOS 205

5.36a Coherence Function Relating Longitudinal Load From West Conductor and SE Main Leg Strain Record for Record NOS 207

XVI 11 FIGURE PAGE 5.36b Coherence Function Relating Longitudinal Load From West Conductor and SE Diagonal Leg Strain Record for Record NOS 207

5.37a Coherence Function Relating Longitudinal Load From East Conductor and SE Main Leg Strain Record for Record NOS 208 5.37b Coherence Function Relating Longitudinal Load From East Conductor and SE Diaigonal Leg Strain Record for Record NOS 208 5.38a Coherence Function Relating Longitudinal Load From East Conductor and NW Main Leg Strain Record for Record NOS 210

5.38b Coherence Function Relating Longitudinal Load From East Conductor and NW Diagonal Leg Strain Record for Record NOS 210

6.1 Schematic Diagram for Multiple Inputs and Single Output System 221 6.2 Schematic Representation of Two Loadings and One Response 224 6.3a Frequency Response Function Relating WSOS and SGOS With WSOS and LC03 as Loadings and SGOS as Response for Record NOS 233

6.3b Frequency Response Function Relating LC03 and SGOS With WSOS and LC03 as Loadings and SGOS as Response for Record NOS 233

6.4a Frequency Response Function Relating WSOS and SG06 With WSOS and LC03 as Loadings and SG06 as Response for Record NOS 235

6.4b Frequency Response Function Relating LC03 and SG06 With WSOS and LC03 as Loadings and SG06 as Response for Record NOS 235

S.Sa Frequency Response Function Relating LC03 and SGOS With LC03 and LC04 as Loadings and SGOS as Response for Record NOS 236

XIX FIGURE PAGE 6.5b Frequency Response Function Relating LC04 and SGOS With LC03 and LC04 as Loadings and SGOS as Response for Record NOS 236 6.6a Frequency Response Function Relating LC03 and SG06 With LC03 and LC04 as Loadings and SG06 as Response for Record NOS 238 6.6b Frequency Response Function Relating LC04 and SG06 With LC03 and LC04 as Loadings and SG06 as Response for Record NOS 238 6.7a Frequency Response Function Relating LC03 and SGll With LC03 and LC04 as Loadings and SGll as Response for Record NOS 240 6.7b Frequency Response Function Relating LC04 and SGll With LC03 and LC04 as Loadings and SGll as Response for Record NOS 240 S.Sa Frequency Response Function Relating LC03 and SGOS With LC03 and LCOS as Loadings and SGOS as Response for Record NOS 241 S.Sb Frequency Response Function Relating LCOS and SGOS With LC03 and LCOS as Loadings and SGOS as Response for Record NOS 241 6.9a Partial Coherence Function Relating LC03 and SG06 Given LC04 for Record NOS 243 6.9b Partial Coherence Function Relating LC04 and SG06 Given LC03 for Record NOS 243 6.10a Partial Coherence Function Relating LC03 and SGOS Given LC04 for Record NOS 245 6.10b Partial Coherence Function Relating LC04 and SGOS Given LCOS for Record NOS 245 6.11a Partial Coherence Function Relating LC03 and SGll Given LC04 for Record NOS 247 6.11b Partial Coherence Function Relating LC04 and SGll Given LCOS for Record NOS 247

XX FIGURE PAGE 7.1 Stages in Time Series Modeling Using Box-Jenkins Methodology 257 7.2 Generation of Time Series From White Noise Through a Linear Filter 260 7.3 Estimated euid Modeled Power Spectra of WSOS for Record NOS 282 A.la Time History of Load From West Conductor for Record NOS 298 A.lb Time History of Load From West Conductor for Record NO? 298 A.2a Time History of Load From West Conductor for Record NOS 299 A.2b Time History of Load From West Conductor for Record NIB 299 A.3a Time History of Load From East Conductor for Record NOS 300 A.3b Time History of Load From East Conductor for Record N07 300 A.4a Time History of Load From East Conductor for Record NOS 301 A. 4b Time History of Load From East Conductor for Record N16 301 A. 5a Time History of Load From Center Conductor for Record NOS 302 A.5b Time History of Load From Center Conductor for Record N07 302 A. 6a Time History of Load From Center Conductor for Record NOS 303 A. 6b Time History of Load From Center Conductor for Record N16 303

XXI FIGURE PAGE A.7a Time History for West Conductor Longitudinal Swing for Record NOS 304 A. 7b Time History for West Conductor Longitudinal Swing for Record N07 304 A. 8a Time History for West Conductor Longitudinal Swing for Record NOB 305 A. 8b Time History for West Conductor Longitudinal Swing for Record NIB 305 A. 9a Time History for West Conductor Transverse Swing for Record NOS 306 A. 9b Time History for West Conductor Transverse Swing for Record N07 30B A. 10a Time History for West Conductor Transverse Swing for Record NOS 307 A. lOb Time History for West Conductor Transverse Swing for Record NIB 307 A.11a Time History for East Conductor Longitudinal Swing for Record NOS 308 A.lib Time History for East Conductor Longitudinal Swing for Record N07 308 A.12a Time History for East Conductor Longitudinal Swing for Record NOB 309 A. 12b Time History for East Conductor Longitudinal Swing for Record NIB 309 A. 13a Time History for East Conductor Transverse Swing for Record NOS 310 A. 13b Time History for East Conductor Transverse Swing for Record N07 310 A.14a Time History for East Conductor Transverse Swing for Record NOS 311

XXll FIGURE PAGE A.14b Time History for East Conductor Transverse Swing for Record NIB 311 A.15a Time History for Center Conductor Longitudinal Swing for Record NOS 312 A.15b Time History for Center Conductor Longitudinal Swing for Record N07 312 A.16a Time History for Center Conductor Longitudinal Swing for Record NOB 313 A.IBb Time History for Center Conductor Longitudinal Swing for Record NIB 313 A.17a Time History for Center Conductor Transverse Swing for Record NOS 314 A.17b Time History for Center Conductor Transverse Swing for Record N07 314 A.18a Time History for Center Conductor Trsuisverse Swing for Record NOB 315 A.18b Time History for Center Conductor Transverse Swing for Record N16 315 A.19a Time History for Stress in Southeast Main Leg for Record NOS 316 A.19b Time History for Stress in Southeast Main Leg for Record N07 316 A.20a Time History for Stress in Southeast Main Leg for Record N08 317 A.20b Time History for Stress in Southeast Main Leg for Record NIB 317 A.21a Time History for Stress in Southeast Diagonal Leg for Record NOS 318 A. 21b Time History for Stress in Southeast Diagonal Leg for Record N07 318

XXI11 FIGURE PAGE A.22a Time History for Stress in Southeast Diagonal Leg for Record NOS 319 A.22b Time History for Stress in Southeast Diagonal Leg for Record N07 319 A.23a Time History for Stress in Northwest Main Leg for Record NOS 320 A.23b Time History for Stress in Northwest Main Leg for Record N07 320 A.24a Time History for Stress in Northwest Main Leg for Record NOS 321 A.24b Time History for Stress in Northwest Main Leg for Record NIB 321 A.2Sa Time History for Stress in Northwest Diagonal Leg for Record NOS 322 A.25b Time History for Stress in Northwest Diagonal Leg for Record N07 322 A.2Ba Time History for Stress in Northwest Diagonal Leg for Record NOB 323 A.2Bb Time History for Stress in Northwest Diagonal Leg for Record N16 323 B.la Autocorrelation Functions of Original and First Differenced Series of WSOS for Record N07 324 B.lb Autocorrelation Functions of Original and First Differenced Series of WSOS for Record NIB 324 B.2a Autocorrelation Functions of Original and First Differenced Series of LC03 for Record N07 325 B.2b Autocorrelation Functions of Original and First Differenced Series of LC04 for Record N07 325

XXIV FIGURE PAGE B.3a Autocorrelation Functions of Original and First Differenced Series of LCOS for Record N07 326 B.3b Autocorrelation Functions of Original and First Differenced Series of LC03 for Record NIB 326 B.4a Autocorrelation Functions of Original and First Differenced Series of LC04 for Record NIB 327 B.4b Autocorrelation Functions of Original and First Differenced Series of LCOS for Record NIB 327 B.5a Autocorrelation Functions of Original and First Differenced Series of SGOS for Record N07 328 B.Sb Autocorrelation Functions of Original and First Differenced Series of SGll for Record N07 328 B.Ba Autocorrelation Functions of Original and First Differenced Series of SGOB for Record N07 329 B.Bb Autocorrelation Functions of Original and First Differenced Series of SG12 for Record N07 329 B.7a Autocorrelation Functions of Original and First Differenced Series of SGOS for Record NIB 330 B.7b Autocorrelation Functions of Original and First Differenced Series of SGll for Record NIB 330 B.Ba Autocorrelation Functions of Original and First Differenced Series of SGOB for Record NIB 331

XXV FIGURE PAGE B.Bb Autocorrelation Functions of Original and First Differenced Series of SG12 for Record NIB 331 C.la Cross Correlation Function of WSOS and SGll for Record NOS 332 C.lb Cross Correlation Function of WSOS and SGI2 for Record NOS 332 C.2a Cross Correlation Function of WSOS and SGOS for Record N07 333 C.2b Cross Correlation Function of WSOS and SGll for Record N07 333 C.3a Cross Correlation Function of WSOS and SGOB for Record N07 334 C.3b Cross Correlation Function of WSOS and SGI2 for Record N07 334 C.4a Cross Correlation Function of WSOS and SGOS for Record NIB 335 C.4b Cross Correlation Function of WSOS and SGOB for Record NIB 335 C.Sa Cross Correlation Function of Total Load Values From West Conductor and SGll for Record NOS 336 C.Sb Cross Correlation Function of Total Load Values From West Conductor and SGI2 for Record NOS 336 C.6a Cross Correlation Function of Total Load Values From East Conductor and SGll for Record NOS 337 C.Bb Cross Correlation Function of Total Load Values From East Conductor and SGI2 for Record NOS 337

XXVI FIGURE PAGE C.7a Cross Correlation Function of Total Load Values From Center Conductor and SGll for Record NOS 338 C.7b Cross Correlation Function of Total Load Values From Center Conductor and SGI2 for Record NOS 338 C.Ba Cross Correlation Function of Total Load Values From West Conductor aind SGOB for Record N07 339 C.Bb Cross Correlation Function of Total Load Values From West Conductor and SGI2 for Record N07 339 C.9a Cross Correlation Function of Total Load Values From East Conductor and SGOB for Record N07 340 C.9b Cross Correlation Function of Total Load Values From Center Conductor and SGOB for Record N07 340 C.lOa Cross Correlation Function of Total Load Values From West Conductor and SGOS for Record NIB 341 C.lOb Cross Correlation Function of Total Load Values From West Conductor and SGll for Record NIB 341 C.11a Cross Correlation Function of Total Load Values From Center Conductor and SGOS for Record NIB 342 C.lib Cross Correlation Function of Total Load Values From East Conductor and SGOS for Record NIB 342 C.12a Cross Correlation Function of Longitudinal Load Values From West Conductor and SGOS for Record NOS 343

xxvii FIGURE PAGE C.12b Cross Correlation Function of Longitudinal Load Values From East Conductor and SGOS for Record NOS 343 C.13a Cross Correlation Function of Longitudinal Load Values From East Conductor and SGOS for Record N07 344 C.13b Cross Correlation Function of Longitudinal Load Values From West Conductor and SGOS for Record N07 344 C.14a Cross Correlation Function of Longitudinal Load Values From West Conductor and SG06 for Record N07 345 C.14b Cross Correlation Function of Longitudinal Load Values From East Conductor and SG06 for Record N07 345 C.ISa Cross Correlation Function of Longitudinal Load Values From West Conductor and SGI2 for Record N07 346 C.15b Cross Correlation Function of Longitudinal Load Values From East Conductor and SGI2 for Record N07 346 C.16a Cross Correlation Function of Transverse Load Values From Center Conductor and SGOS for Record NOS 347 C.16b Cross Correlation Function of Transverse Load Values From Center Conductor aind SGll for Record NOS 347 C.17a Cross Correlation Function of Transverse Load Values From East Conductor and SGll for Record NOB 348 C.17b Cross Correlation Function of Transverse Load Values From Center Conductor and SGll for Record NOB 348

XXVlll FIGURE PAGE C.18a Cross Correlation Function of Transverse Load Values From West Conductor and SGOS for Record NO? 349 C.18b Cross Correlation Function of Transverse Load Values From West Conductor and SGll for Record NO? 349 C.19a Cross Correlation Function of Transverse Load Values From West Conductor and SGOB for Record NO? 350 C.19b Cross Correlation Function of Transverse Load Values From West Conductor and SGI2 for Record NO? 350 C.20a Cross Correlation Function of Transverse Load Values From Center Conductor and SGOS for Record NO? 351 C.20b Cross Correlation Function of Transverse Load Values From East Conductor and SGOS for Record NO? 351 C.21a Cross Correlation Function of Transverse Load Values From Center Conductor and SG06 for Record NO? 352 C.21b Cross Correlation Function of Transverse Load Values From Center Conductor and SGI2 for Record NO? 352 C.22a Cross Correlation Function of Transverse Load Values From Center Conductor and SGOS for Record N16 353 C.22b Cross Correlation Function of Transverse Load Values From Center Conductor and SG06 for Record N16 353 C.23a Cross Correlation Function of Transverse Load Values From Center Conductor and SGll for Record N16 354

XXIX FIGURE PAGE C.23b Cross Correlation Function of Transverse Load Values From Center Conductor and SGI2 for Record NIB 354 D.la Power Spectrum of Wind Speed at 34.7 m for Record NO? 355 D.lb Power Spectrum of Wind Speed at 34.7 m for Record NIB 355 D.2a Power Spectrum of Total Load Values From West Conductor for Record NOS 356 D.2b Power Spectrum of Longitudinal Load Values From West Conductor for Record NOS 356 D.3a Power Spectrum of Total Load Values From East Conductor for Record NOS 357 D.3b Power Spectrum of Longitudinal Load Values From East Conductor for Record NOS 357 D.4a Power Spectrum of Total Load Values From Center Conductor for Record NOS 358 D.4b Power Spectrum of Longitudinal Load Values From Center Conductor for Record NOS 358 D.Sa Power Spectrum of Total Load Values From West Conductor for Record NO? 359 D.Sb Power Spectrum of Transverse Load Values From West Conductor for Record NO? 359 D.Ba Power Spectrum of Longitudinal Load Values From West Conductor for Record NO? 360 D.6b Power Spectrum of Total Load Values From East Conductor for Record NO? 360 D.?a Power Spectrum of Transverse Load Values From East Conductor for Record NO? 361 D.?b Power Spectrum of Longitudinal Load Values From East Conductor for Record NO? 361

XXX FIGURE PAGE D.Ba Power Spectrum of Total Load Values From Center Conductor for Record NO? 362 D.Bb Power Spectrum of Transverse Load Values From Center Conductor for Record NO? 362 D.9a Power Spectrum of Longitudinal Load Values From Center Conductor for Record NO? 363 D.9b Power Spectrum of Total Load Values From West Conductor for Record NOB 363 D.lOa Power Spectrum of Longitudinal Load Values From West Conductor for Record NOS 364 D.10b Power Spectrum of Total Load Values From East Conductor for Record NOB 364 D.11a Power Spectrum of Total Load Values From West Conductor for Record NIB 365 D.lib Power Spectrum of Transverse Load Values From West Conductor for Record N16 365 D.12a Power Spectrum of Longitudinal Load Values From West Conductor for Record N16 366 D.12b Power Spectrum of Total Load Values From East Conductor for Record N16 366 D.13a Power Spectrum of Transverse Load Values From East Conductor for Record N16 367 D.13b Power Spectrum of Longitudinal Load Values From East Conductor for Record N16 36? D.14a Power Spectrum of Total Load Values From Center Conductor for Record N16 368 D.14b Power Spectrum of Transverse Load Values From Center Conductor for Record N16 368 D.15a Power Spectrum of Longitudinal Load Values From Center Conductor for Record N16 369

XXXI FIGURE PAGE D.IBa Power Spectrum of Diagonal Strain Gage Located at SE Corner for Record NOS 370 D.IBb Power Spectrum of Diagonal Strain Gage Located at NW Corner for Record NOS 370 D.17a Power Spectrum of Main Strain Gage Located at SE Corner for Record NO? 371 D.l?b Power Spectrum of Main Strain Gage Located at NW Corner for Record NO? 371 D.IBa Power Spectrum of Diagonal Strain Gage Located at SE Corner for Record NO? 372 D.18b Power Spectrum of Diagonal Strain Gage Located at NW Corner for Record NO? 372 D.19a Power Spectrum of Diagonal Strain Gage Located at SE Corner for Record NOS 373 D.19b Power Spectrum of Diagonal Strain Gage Located at NW Corner for Record NOS 373 D.20a Power Spectrum of Main Strain Gage Located at SE Corner for Record NIB 374 D.20b Power Spectrum of Main Strain Gage Located at NW Corner for Record NIB 374 D.21a Power Spectrum of Diagonal Strain Gage Located at SE Corner for Record N16 375 D.21b Power Spectrum of Diagonal Strain Gage Located at NW Corner for Record N16 375 E.la Frequency Response Function Relating Wind Speed and NW Main Leg Strain Record for Record NOS 376 E.lb Frequency Response Function Relating Wind Speed and NW Main Leg Strain Record for Record NOB 376

XXXll FIGURE PAGE E.2a Frequency Response Function Relating Wind Speed and SE Main Leg Strain Record for Record NO? 377 E.2b Frequency Response Function Relating Wind Speed and NW Main Leg Strain Record for Record NO? 377 E.3a Frequency Response Function Relating Wind Speed and SE Diagonal Leg Strain Record for Record NO? 378 E.3b Frequency Response Function Relating Wind Speed and NW Diagonal Leg Strain Record for Record NO? 378 E.4a Frequency Response Function Relating Total Load From West Conductor and SE Main Leg Strain Record for Record N05 379 E.4b Frequency Response Function Relating Total Load From West Conductor and NW Main Leg Strain Record for Record NOS 379 E.Sa Frequency Response Function Relating Total Load From Center Conductor and SE Diagonal Leg Strain Record for Record NOS 380 E.Sb Frequency Response Function Relating Total Load From Center Conductor and NW Diagonal Leg Strain Record for Record NOS 380 E.Ba Frequency Response Function Relating Total Load From West Conductor and SE Main Leg Strain Record for Record NO? 381 E.Bb Frequency Response Function Relating Total Load From West Conductor auid NW Main Leg Strain Record for Record NO? 381 E.7a Frequency Response Function Relating Total Load From West Conductor suid SE Diagonal Leg Strain Record for Record NO? 382

XXXI11 FIGURE PAGE E.?b Frequency Response Function Relating Total Load From West Conductor and NW Diagonal Leg Strain Record for Record NO? 382 E.Ba Frequency Response Function Relating Total Load From East Conductor and SE Main Leg Strain Record for Record NO? 383 E.Bb Frequency Response Function Relating Total Load From East Conductor euid SE Diagonal Leg Strain Record for Record NO? 383 E.9a Frequency Response Function Relating Total Load From West Conductor and SE Diagonal Leg Strain Record for Record NOS 384 E.9b Frequency Response Function Relating Total Load From West Conductor and NW Diagonal Leg Strain Record for Record NOS 384 E.10a Frequency Response Function Relating Total Load From Center Conductor and SE Diagonal Leg Strain Record for Record NOB 385 E.10b Frequency Response Function Relating Total Load From Center Conductor and NW Diagonal Leg Strain Record for Record NOS 385 E.11a Frequency Response Function Relating Transverse Load From West Conductor and SE Diagonal Leg Strain Record for Record NOS 386 E.lib Frequency Response Function Relating Transverse Load From West Conductor and NW Diagonal Leg Strain Record for Record NOS 386 E.12a Frequency Response Function Relating Transverse Load From East Conductor and SE Main Leg Strain Record for Record NOS 387 E.12b Frequency Response Function Relating Transverse Load From East Conductor and NW Main Leg Strain Record for Record NOS 387

xxxiv FIGURE PAGE E.13a Frequency Response Function Relating Transverse Load From Center Conductor and SE Main Leg Strain Record for Record NOB 388 E.13b Frequency Response Function Relating Transverse Load From Center Conductor and NW Main Leg Strain Record for Record NOS 388 E.14a Frequency Response Function Relating Transverse Load From West Conductor and SE Main Leg Strain Record for Record NO? 389 E.14b Frequency Response Function Relating Transverse Load From West Conductor and NW Main Leg Strain Record for Record NO? 389 E.ISa Frequency Response Function Relating Transverse Load From West Conductor and SE Diagonal Leg Strain Record for Record NO? 390 E.15b Frequency Response Function Relating Transverse Load From West Conductor and NW Diagonal Leg Strain Record for Record NO? 390 E.IBa Frequency Response Function Relating Transverse Load From East Conductor and SE Main Leg Strain Record for Record NO? 391 E.IBb Frequency Response Function Relating Transverse Load From East Conductor and SE Diagonal Strain Record for Record NO? 391 E.17a Frequency Response Function Relating Transverse Load From Center Conductor and SE Main Leg Strain Record for Record NO? 392 E.17b Frequency Response Function Relating Transverse Load From Center Conductor eind NW Main Leg Strain Record for Record NO? 392 E. 18a Frequency Response Function Relating Longitudinal Load From West Conductor and SE Diagonal Leg Strain Record for Record NOS 393

XXXV FIGURE PAGE E.18b Frequency Response Function Relating Longitudinal Load From West Conductor and NW Diagonal Leg Strain Record for Record NOS 393 E.19a Frequency Response Function Relating Longitudinal Load From Center Conductor and SE Main Leg Strain Record for Record NOS 394 E.19b Frequency Response Function Relating Longitudinal Load From Center Conductor and SE Diagonal Leg Strain Record for Record NOS 394 E.20a Frequency Response Function Relating Longitudinal Load From West Conductor and SE Diagonal Leg Strain Record for Record NO? 395 E.20b Frequency Response Function Relating Longitudinal Load From West Conductor and NW Diagonal Leg Strain Record for Record NO? 395 E.21a Frequency Response Function Relating Longitudinal Load From East Conductor and SE Main Leg Strain Record for Record NO? 396 E.21b Frequency Response Function Relating Longitudinal Load From East Conductor and NW Main Leg Strain Record for Record NO? 396 E.22a Frequency Response Function Relating Longitudinal Load From Center Conductor and SE Main Leg Strain Record for Record NO? 397 E.22b Frequency Response Function Relating Longitudinal Load From Center Conductor and SE Diagonal Leg Strain Record for Record NO? 397

XXXVl FIGURE PAGE E.23a Frequency Response Function Relating Longitudinal Load From Center Conductor and SE Diagonal Leg Strain Record for Record NOB 398 E.23b Frequency Response Function Relating Longitudinal Load From Center Conductor and NW Main Leg Strain Record for Record NOS 398 F.la Coherence Function Relating Wind Speed and NW Main Leg Strain Record for Record NOS 399 F.lb Coherence Function Relating Wind Speed and NW Main Leg Strain Record for Record NOB 399 F.2a Coherence Function Relating Wind Speed and SE Main Leg Strain Record for Record NO? 400 F.2b Coherence Function Relating Wind Speed and NW Main Leg Strain Record for Record NO? 400 F.3a Coherence Function Relating Wind Speed and SE Diagonal Leg Strain Record for Record NO? 401 F.3b Coherence Function Relating Wind Speed and NW Diagonal Leg Strain Record for Record NO? 401 F.4a Coherence Function Relating Total Load From West Conductor and SE Main Leg Strain Record for Record NO? 402 F.4b Coherence Function Relating Total Load From West Conductor and NW Main Leg Strain Record for Record NO? 402 F.Sa Coherence Function Relating Total Load From West Conductor and SE Diagonal Leg Strain Record for Record NO? 403

XXXVll FIGURE PAGE F.Sb Coherence Function Relating Total Load From West Conductor and NW Diagonal Leg Strain Record for Record NO? 403 F.Ba Coherence Function Relating Total Load From East Conductor and SE Main Leg Strain Record for Record NO? 404 F.Bb Coherence Function Relating Total Load From East Conductor and SE Diagonal Leg Strain Record for Record NO? 404 F.?a Coherence Function Relating Total Load From Center Conductor and SE Diagonal Leg Strain Record for Record NOB 405 F.7b Coherence Function Relating Total Load From Center Conductor and NW Diagonal Leg Strain Record for Record NOS 405 F.Sa Coherence Function Relating Transverse Load From Center Conductor euid SE Main Leg Strain Record for Record NO? 406 F.8b Coherence Function Relating Tranverse Load From Center Conductor and NW Main Leg Strain Record for Record NO? 406 F.9a Coherence Function Relating Transverse Load From East Conductor and SE Main Leg Strain Record for Record NO? 40? F.9b Coherence Function Relating Tranverse Load From East Conductor and SE Diagonal Leg Strain Record for Record NO? 40? F.lOa Coherence Function Relating Transverse Load From West Conductor and NW Main Leg Strain Record for Record NOS 408 F.10b Coherence Function Relating Tranverse Load From West Conductor and NW Diagonal Leg Strain Record for Record NOS 408

XXXVlll FIGURE PAGE F.11a Coherence Function Relating Longitudinal Load From West Conductor euid SE Diagonal Leg Strain Record for Record NO? 409 F.lib Coherence Function Relating Longitudinal Load From West Conductor and NW Diagonal Leg Strain Record for Record NO? 409 F.12a Coherence Function Relating Longitudinal Load From East Conductor and SE Main Leg Strain Record for Record NO? 410 F.12b Coherence Function Relating Longitudinal Load From East Conductor and NW Main Leg Strain Record for Record NO? 410

xxxix CHAPTER 1 INTRODUCTION

Transmission towers and conductors form an integral part of the national grid voltage networks. The primary responsibility of the transmission towers and conductors within the national grid voltaige networks is to transmit electricity from electric power plants to consumers. Therefore, continuous, uninterrupted, and proper func tioning of transmission towers and conductors are essen tial in order to meet the power demand in the nation. The structural reliability and the integrity of trans V mission towers and conductors are important facets of the national grid voltage networks. Therefore, a high stan dard for design must be enforced to preclude any struc tural failure even when the transmission towers and con ductors are subject to severe loading conditions. The locations of the transmission towers and conduc tors are diverse in nature. There are two extreme cases in which the locations of the transmission towers and conductors may vary. In the first extreme case, the transmission towers and conductors may be located in the isolated country-side far from the nearest civilization. In the second extreme case, the transmission towers and conductors may be located near densely populated metropolitan areas. These two extreme cases result in different geographical terrains which the transmission towers and conductors may encounter. The design of transmission tower structural members is dictated by the locations and the geographical terrains in which the transmission towers are to be erected. The design of the trainsmission towers is highly sen sitive to geographical terrain and geographical location of the transmission towers. In an effort to curtail the the perils faced by transmission tower structures under actual field conditions, two methods may be employed to \y perform the tasks. The first method involves a conserva tive design using standard design guidelines. The second method involves a proper in-depth study and research into the behavior of the transmission tower structures and conductors under various loading conditions. The former approach facilitates a design which may be overly con servative. The latter allows the designer to analyze the behavior of the transmission tower under static and dynamic loads. The results obtained from the analysis are then incorporated into the design of the transmission tower. Transmission towers and conductors are more sensitive to dynamic loads than to static loads. On the other V hand, dynamic loads, such as those produced by wind, tend to be stochastic in nature. The effect of dynamic load ings on trainsmission towers suid conductors must be thoroughly studied and fully understood before an effi cient design may be obtained. Dynamic loads are any loads for which the magnitude, direction, position, or any combination of these varies with time. Dynamic loads can be characterized as either deterministic or non-deterministic. If the time varia tion of the loads is fully known, then they are classi fied as deterministic loads. When the time variation of the loads is not fully known, then they are classified as non-deterministic or stochastic loads. Non-deterministic loads can be treated as random processes because at any point in time, these loadings are random variables. Wind loads on structures are characterized as dynamic loads because their magnitude, direction, and position varies with time. Wind loadings are stochastic because no two records of wind speed or wind direction resemble one another. Since they are stochastic, wind loads are treated as random processes. Because wind loadings are stochastic, the responses of transmission towers and conductors acted upon by wind loadings are stochastic, also. This is true for the responses of any flexible structure subjected to stochastic loads. The design of important structures to resist wind loads is an important topic in structural engineering. In most building codes, wind loadings are treated as quasi-static loads. For example, the ANSI ASS.1-1982 provides methodology to analyze wind loads acting on a structure. However, if serious doubt exists in designing structures highly sensitive to wind loads, the designer can resort to wind tunnel tests or perform a full-scale wind-structure interaction test. The wind tunnel tests or a full-scale wind structure interaction test permits the designer to simulate the actual field conditions. Ideal ly, the results obtained from such studies, represent the actual responses of a structure due to wind-structure interaction. The results obtained from such studies are expected to enhance the actual design of the transmission towers. Therefore, a more efficient and a more reliable design are obtained. Wind loads act on a transmission tower in two ways. First, the wind loads act directly on the transmission v/ tower itself. Second, the wind loads act on the con ductor which in turn transmits these loads to the trans mission tower. The latter case is thought to be more severe than the former. However, both cases are im- port2u:it in ascertaining the overall effect of wind loads on the transmission tower. The nature of this research is to study the statis tical properties of the wind and response data on a transmission line tower. These data were collected by the Bonneville Power Administration (BPA) in Oregon. These data consist of wind speed and direction records as well as associated response records of full-scale trans mission tower structure and conductors. The primary objectives are: (1) to study wind speed and associated load and response records for the transmission line system, (2) to determine the stationarity of the afore mentioned records, (3) to study the frequency content of these records, (4) to relate a single loading to a measure of response in the frequency domain by estimating the frequency response function, (5) to relate two loadings to a measure of response in the frequency domain and compare the quality of this relationship to that obtained previously for a single loading, and (6) to model wind speed time histories using time series method ology.

The outline of the material presented is as follows. A literature review of statistical and time series work, simulation techniques, and stochastic modeling of sta tionary and non-stationary processes performed in the past in engineering and non-engineering fields is pre sented in Chapter 2. The data collection and instru mentation system is described in Chapter 3. In Chapter 4, statistical analysis of selected time series records are performed. The frequency responses of single input- single output structural system are described in Chapter 5. The frequency responses of multiple inputs-single output structural system are described in Chapter 6. Time series modeling and simulation are performed in Chapter 7 Finally, the discussion of results obtained auid the directions of future research are described in Chapter 8. CHAPTER 2 LITERATURE REVIEW

A large volume of literature exists concerning wind- structure interaction and responses, random responses of structures, statistical analysis of random physical data, stochastic modeling and simulation of time series, and the application of spectra in engineering design and analysis. The data describing any random physical process are usually functions of time and are termed "time histo ries. " In order to study the frequency contents of a time history, a mathematical technique capable of trans forming information obtained in the time domain to the frequency domain must be utilized. The mathematical development of the Fast Fourier Transform (FFT) has enabled researchers to study the frequency contents of any realizable time history in the frequency domain. The FFT has the advantage over the classical Fourier Trans form in that the former has a smaller number of computa tions involved. Cooley, Lewis, and Welch (1967) provided an overview of the development of the FFT in data analysis and described the contributions of other inves tigators in the development of the FFT. Helms (1967) described two methods for using the FFT to reduce the 8 number of arithmetic operations. The first method is capable of formulating a filtering operation so that a simulation of independent variables can be handled by the second method known as the "overlap-adding" version of the FFT. Cooley, Lewis, and Welch (1969) defined the elementary properties of the finite Fourier Transform such as the convolution and term-by-term product opera tions. Their results were then used to give a simple derivation of the FFT algorithm. Holmes (1977) concluded that the frequency domain approach was the easiest to use and was the most efficient method when coupled with an Inverse FFT algorithm in simulating random wind data. The cornerstone of time series analysis is the con cept that the sequence of observations making up a time series constitutes a realization of jointly distributed random variables. This concept enables the analyst to model a time series based on its statistical properties. Box and Jenkins (1976) comprehensively described the techniques involved in the modeling of time series in the time domain. They considered three models which may be used to describe time series. These three models are the autoregressive model (AR), the moving-average model (MA), and the mixed model (ARMA, ARIMA). In the AR model, forecasting of a current observation in a time series is dependent on past observations plus a random shock or a white noise. In the MA model, fore casting of a current observation in a time series is dependent on a past observation plus several random shocks or white noises. In the mixed model, ARMA or ARIMA, forecasting of a current observation is dependent on past observations plus the random shocks or white noises. Box and Jenkins (1976) also presented the mathe matical techniques to obtain the proper transfer function in the time domain and the frequency response function in the frequency domain based on modeled time series of the input imposed upon a physical system and its associated output.

The advent of modern computers enabled the analyst to apply the theoretical mathematical development of the Fourier Transform and time series modeling to practical problems. Nau, Oliver, and Pister (1980) described the use of ARMA models to simulate and analyze real and artificial earthquake accelerograms. The results of the analysis indicated that a discrete-time ground accelera tion model consisting of a linear filter whose parameters were piecewise-constant on five second intervals was useful in structural response studies. This simulation, 10 coupled with white noise input modulated by a non- parametric variance envelope, provided a useful and prac tical characterization of the non-stationary features of earthquake accelerograms. Reed and Scanlan (1983) considered full-scale wind velocity and wind pressure time series data collected on two hyperbolic cooling towers. They utilized time series analysis and modeling to simulate wind loadings and to predict the transfer function models in the time domain with wind velocity as the input and wind pressure- differences as the output. Hamilton and Watts (1978) mathematically studied the behavior of the partial autocorrelation function of sea sonal time series based on a partial autocorrelation pattern. This pattern acted as a signature of the regu lar component of the model which is a simple composite of the autocorrelation and partial autocorrelation functions of the regular component. Newbold (1980) showed the adequacy of a fitted ARMA model based on a Lagrange multiplier test and on analysis of residual autocorrela tions. Ljung and Box (1978) considered the overall test for lack of fit in ARMA models using models proposed by Box and Pierce (1970). They considered the power of the tests and the test capabilities when non-normal series 11 were encountered. Similar modifications in the overall tests used for transfer function noise models were proposed. Godfrey (1979) considered the problem involved in testing the adequacy of an ARMA time series model. He proposed the use of Monte Carlo results on samples of finite size. Rao and Tong (1973) provided a method using the frequency domain approach to furnish tests to deter mine the time dependence of a tranfer function of a linear system. Hannan (1979) discussed the identification and parameterization of ARMA systems in relation to the use of certain canonical forms. Stanton (1965) estimated the frequency responses for transfer functions occurring in a multivariate block diagram representation of a turboalternator. He modeled the turboalternator as a linear system. The availability of Fourier Transforms enables most researchers to study the frequency content of a physi cally realizable time series. The transformation of a physically realizable time series into the frequency domain yields the power spectrum of the time series. Newbold (1981) gave an overview of some research in time series analysis with an emphasis on the methodological problems in linear model building and spectral analysis. 12

The analysis of both single and multiple time series was considered. Parzen (1961) gave a full account of the mathematical considerations involved in the estimation of spectra. Jenkins (1961) provided a set of general guide lines in the analysis of spectra. Hashish and Abu-Sitta (1974) described a procedure which considered the fre quency distribution of the pressure spectra and response spectra to predict the dynamic response of a cooling tower under the action of turbulent wind. Steinmetz, Billington, and Abel (1978) studied the dynamic behavior of a hyperbolic cooling tower subjected to wind using specific wind records and stochastic analysis of wind pressure spectra. They concluded that the results ob tained supported the quasi-static assumption normally used for cooling tower wind loading. Loh (1985) obtained the spectra of wind data collected from a typhoon. He estimated the turbulence spectra and cross spectra of horizontal typhoon wind fluctuations. Norville, Mehta, and Farwagi (1985) studied the wind load and response characteristics of several wind records acting on a traunsmission line tower. They also obtained the power spectra and the cross spectra of the records. Bhansali (1980) described the mathematics underlying the analysis of the inverse correlation function of a 13 stationary process with emphasis on window methods for estimating this function. Cleveland (1972) estimated the inverse autocorrelations of a time series using the auto correlations associated with the inverse of a spectral density estimate. He utilized autoregressive and spectral smoothing methods to obtain the estimates. The estimation of parameters used in modeling of time series is often tedious and time consuming. Most modern computing techniques for modeling time series use numerical solutions to increase computational efficiency. Reilly (1980) presented a computer-based procedure to identify the form and the characteristics of outliers by combining univariate methods proposed by Box and Jenkins (1976) with outlier detection to enheuice the analysis and the forecasting of time series data. Reilly (1981) des cribed a general algorithm for the Box-Jenkins modeling process. He related Box-Jenkins procedures to regression models which consisted of the identification, the estima tion and verification, and the forecasting phases. Next, literature involving practical engineering applications of the response of structures to random loadings, the response of structures due to wind-structure interaction, and the response of structures in full-scale testing are discussed. 14

Vibration of mechanical and structural systems has received attention in engineering requirements and prac tices. The engineering profession has recognized the random characteristics of physical vibrations which in troduce complexity in actual design of important struc tures. Crandall and Mark (1963) presented a very thor ough treatment of the raindom vibration of mechanical systems. They discuss extensively the characterization of random vibration, the transmission of random vibra tion, and the problem of failures resulting from random vibration of mechanical systems. Lin (1967) emphasized the analyses of the responses of practical structures of importance to civil engineers to random excitations. Bycroft (1960) investigated the suitability of white noise as a representation of earthquake excitation. All the literature reported herein discusses the random and stochastic nature of structural loadings and structural responses. The effects of wind forces on structures are an important design consideration for most engineered struc tures. Researchers in the area of wind engineering have recognized the fact that wind speed is random in nature and have attributed to the difficulty in studying the effect of wind-structure interaction to the randomness of 15 wind speed. Therefore, researchers in wind engineering have used simulation and stochastic modeling techniques to study the responses of structures to wind loads. Smith (1985) studied the dynamic response of tall stacks subjected to wind forces using a stochastic model. He identified critical and transcritical flow regimes using wind-force spectra. Matheson and Holmes (1980) simulated wind velocities using a Monte Carlo technique based on an Inverse Fast Fourier Transform and the equations of motion of the line were then solved numerically using a finite difference method. They concluded that at high velocities, dynaunic response is suppressed by high aero- dynaunic damping. Using wind records from several weather stations in the United States, Grigoriu and Longo (1985) described a statistical method to evaluate wind records and to estimate the wind speeds of various return periods using simplified non-stationary models. The design of structures to resists wind loads has been incorporated in most building codes at local and national levels for structures which are sensitive to wind loads or for structures in which the design of the structural members is highly dependent on the wind loads. ANSI ASS.1-1982 described quite comprehensively the design aspect of wind-structure interaction. ANSI 16

ASS.1-1982 is the most common reference for the designer to determine the magnitude of wind load which a struc ture will experience. The Guidelines for Transmission Line Structural Loading were prepared by the American Society of Civil Engineers' Committee on Electrical Transmission Structures in 1984. The purpose of the guidelines is to provide transmission line designers with procedures for the selection of design loads and load factors based on the latest research and state-of-the-art design techniques. The concepts of probability based design, and the nature and variability of loads used to design transmission line structures, were reviewed and discussed. In particular, the guidelines deal comprehen sively with wind loads on transmission line structures. Other loadings such as the conductor galloping effect, conductor longitudinal loads, and vibrations were also described.

The physical behavior of latticed structures such as transmission towers subjected to raindom loadings can be obtained by proper instrumentation of the structures to measure external loadings acting on the structures and the associated structural responses. The Electric Power Research Institute (EPRI) has, through the years, pub lished literature in the area of on-site instrumentation 17 of latticed structures. On-site instrumentation is still by far the primary means of evaluating both analysis methods and failure criteria. Arnold (1985) described the proper procedures in using strain gages as a major source for collecting important structural data on full- scale electric trauismission structures for the EPRI Transmission Line Mechanical Research Facility. These structural data were the member loads, the bending mo ments in latticed towers, and the stress distributions in steel poles. He also described proper gaiging procedures, calibration techniques, and error ainalysis of the strain gage data. Tuan, Potter, and Jackman (1985) installed load transducers and monitoring equipment used for data acquisition on a 161 kV steel pole. The data obtained were used to establish load statistics in reliability design practices. Kempner and Laursen (1979) investigated the effect of wind loadings on a 500 kV AC single circuit latticed transmission tower. They performed data analyses in an attempt to characterize the wind forces acting on the tower. Other related statistical properties calculated were power spectral densities, turbulence intensities, and wind gust factors. Stationarity tests were also performed on wind records. Kempner (1980) conducted full 18 scale structural tests on a non-energized mechanical test line transmission tower structure to characterize the static behavior of the structure and of the 1200 kV eight-conductor bundle. He obtained the torsional stiff ness values of the structure from the field data to be approximately twice as large as the theoretical values. Kempner (1980) performed structural dynamic tests on a non-energized mechanical test line traunsmission tower and 1200 kV conductor bundle system. He obtained the dynamic mode shapes, the tower structural daunping coefficients, the conductor bundle modal frequencies, and the tower longitudinal and transverse frequencies.

Chiu and Taoka (1971) computed the dynamic response of a 150 foot high free-standing, three-legged, latticed steel tower using actual wind velocity records, and digi tally simulated wind records. They found that for free standing structures the fundaimental mode of vibration predominates. In the fundamental mode of vibration, the structure has a fairly low daunping ratio. Peyrot (1985) derived a Load Resistance Factor Design format for high voltage electric transmission lines using a reliability approach. He assigned relative levels of reliability to different lines, to different structures within one line, and to different components within one structure. 19

Results obtained from the literature review indicate that: (a) modeling of physical time series is helpful in forecasting and in predicting future characteristics and responses of a system, (b) spectral analysis gives a very useful representation of the power distribution of random physical data in the frequency domain, and (c) frequency domain analysis for transfer functions is more meaningful in the context of interpretation because most dynamic analyses of structures use calculations and results in the frequency domain. An investigation of the behavior of transmission tower responses to wind loads will expand our knowledge concerning the behavior of transmission towers. In the next chapter, the on-site data collection system and instrumentation of the full-scale transmission tower structure are described in detail. CHAPTER 3 SITE DESCRIPTION, INSTRUMENTATION, AND DATA COLLECTION SYSTEM

3. 1 Introduction The Bonneville Power Administration (BPA) has con ducted studies on wind load response of transmission line systems by collecting and ainalyzing wind and response related data on a test line in the field. The location of the transmission line system and the structural configu ration of the transmission tower are described in section 3.2. The instruments used in collecting wind and related response data are described in section 3.3. The wind and the types of response data collected are also described in section 3.3. BPA has given special emphasis to data handling, data processing and filing because each record contains over 300,000 data points. Table 3.1 provides general information concerning the twenty-three records collected at BPA test site.

3^2 Location The BPA test site is located in Northern Oregon, between the Dalles and More, east of the Cascade Moun tains, and approximately 1.5 km east of the Deschutes River as shown in Figure 3.1. The terrain to the east of

20 21

Table 3.1 Twenty Three Segments of Data

MEAN WIND MEAN WIND RECORD DATE TIME DIRECTION SPEED (DEGREES) (MPS)

NOl 12/02/81 01.31.57 212 18.33 N02 12/05/81 06.42.45 179 17.43 NOS 12/15/81 16.11.48 224 15. 20 N04 12/16/81 08.30.32 — 1.34 NOS 12/16/81 16.10.28 81 15.65 NOS 01/14/82 10.57.52 300 15. 20 N07 01/16/82 19.04.51 258 23.25 NOS 01/31/82 07.36.32 300 20.56 N09 02/03/82 14.11.35 77 3. 13 NIO 02/14/82 13.05.39 268 14. 31 Nil 02/15/82 23.26.40 270 20. 12 N12 02/15/82 10.29.05 258 14.31 N13 02/16/82 00.38.56 277 20. 12 N14 03/08/82 16.03.42 — 0.00 NIS 03/11/82 14.52.13 267 17.88 N16 03/12/82 16.13.50 229 S.OS N17 04/12/82 01.10.26 243 15.65 NIS 04/13/82 15.29.13 253 18.32 N19 04/17/82 17.57.11 289 12.96 N20 04/20/82 22.03.35 100 14.31 N21 04/23/82 11.58.38 280 15.65 N22 04/28/82 12.28.13 286 16.54 N23 05/07/82 14.19.51 282 12.96 22

Figure 3.1 Topography of Site and Orientation of Power Lines 23 the measurement site consists mostly of smooth, rolling Wheatland. That portion of the terrain to the east which is uncultivated is covered with grass and shrubs. The terrain to the west of the measurement site consists of a deep wide canyon. The test site is located in this area for two major reasons. First, strong winds are frequent at the chosen site. Second, winds from the southwest direction are funneled up the side of the canyon which tends to provide wind patterns which are different from normal. These funneling effects were investigated by Kempner aind Laursen in 1979. The general topology in the vicinity of test lines and their orientation are shown in Figure 3.1. Three lines, orientated at an angle of about 8 degrees west of true north, form part of the Pacific Northwest-Southwest Intertie System. The first two lines east of the Des chutes River are 500 Kv energized lines. These two lines are referred to as John Dao^-Grizzly 1 (JD-G 1) and John Day-Grizzly 2 (JD-G 2). The third line is a non- energized mechanical test line. It was constructed in 1976 to study the behavior of 1100 Kv conductor bundles. This line is parallel to the other two lines, and is referred to as the MORO test line. 24

The perpendicular distances between JD-G 1 and JD-G 2, and between JD-G 2 and the mechanical test line are 150 feet and 125 feet, respectively. The distance between JD-G 1 and the Deschutes River is approximately 1.5 km. There are 5 towers located on line JD-G 2. These towers are numbered from 1 to 5, tower 1 being the tower at the northern end of the site. The distance between towers 4 and 5 is 1475 feet and this represents the longest span (Kempner and Laursen, 1977). Figure 3.2 presents a sche matic plaui of conductor spans between towers 3 and 4, and between towers 4 and 5. Tower 4 is an instrumented, delta configured tower. It is referred to as tower JD-G 16/4 because it is lo cated on mile 16 of JD-G 2. Tower 4 is similar to that shown in Figure 3.3. Tower 4 supports central, east, and west conductors. Overhead ground wires (OHGW) which are smaller in diauneter than conductors are located at a height of 111 feet. All the conductors are paired up in twin bundles, whereas overhead ground wires are single cables. The west conductor group bundle is 9 inches lower than the east conductor group bundle. This ar rangement reduces subconductor oscillations which occur during west winds. Tower 3 is located at a distance of 825 feet north of tower 4 of JD-G 2. Tower 3 also has a 25

N- -^

* effective half span lengths of conductors (Kadaba, 1988)

Figure 3.2 Elevation Along the Test Line (Vertical Scale Exaggerated) Anemometer (34.7 m) 9.1 m

Load ceil and Swing angle indicators

41 mm cfiameter

West conductor

Figure 3.3 Schematic of Tower 16/4 27 delta configuration, wheareas tower 5 has a non-delta configuration. The change in configurations between towers 4 and 5 results in the insulator assemblies of the outer phases at the test tower to be suspended in a non- vertical position (Kempner and Laursen, 1977).

3i_3 Instrumentation The Data Acquisition System is comprised of a mini computer, a soft magnetic disk and requisite signal con ditioning equipment to activate the entire system. The activated system can record data from 256 channels of instrumentation. All devices used in measuring wind and response related data are located on tower JD-G 16/4 and on towers 3,4, and 5 of JD-G 2. A total of 256 channels are available to collect wind and tower response data. Rather than recording all 256 channels simultaneously, various recording modes, comprised of several selected channels were established in the modes selected to record static or dynamic phenomena of interest. The data used in the research reported herein are recorded in Mode 22. Mode 22 is comprised of 38 channels for recording wind and associated structural response data. Table 3.2 gives an overall description of Mode 22. The 38 channels are used to record the following data: 3 conductor loads and 28

Table 3.2 File Description in Mode 22 FILE CHANNEL INSTRUMENT, LOCATION, HEIGHT LCOl 78 LOAD CELL 1 JD-G 16/4 EAST OHGW LC02 79 LOAD CELL 2 JD-G 16/4 WEST OHGW LCOS 80 LOAD CELL 3 JD-G 16/4 WEST CONDUCTOR LC04 81 LOAD CELL 4 JD-G 16/4 EAST CONDUCTOR LCOS 82 LOAD CELL 5 JD-G 16/4 CENTER CONDUCTOR SAOl 83 SWING ANGLE 1 JD-G 16/4 EAST OHGW +X DIR. SA02 84 SWING ANGLE 2 JD-G 16/4 EAST OHGW +Y DIR. SA03 85 SWING ANGLE 3 JD-G 16/4 WEST OHGW +X DIR. SA04 86 SWING ANGLE 4 JD-G 16/4 WEST OHGW +Y DIR. SA05 87 SWING ANGLE 5 JD-G 16/4 WEST COND +X DIR. SA06 88 SWING ANGLE 6 JD-G 16/4 WEST COND +Y DIR. SAO? 89 SWING ANGLE 7 JD-G 16/4 EAST COND +X DIR. SA08 90 SWING ANGLE 8 JD-G 16/4 EAST COND +Y DIR. SA09 91 SWING ANGLE 9 JD-G 16/4 CENT COND +X DIR. SAIO 92 SWING ANGLE 10 JD-G 16/4 CENT COND +Y DIR. SGOl 66 STRAIN GAGE 1 JD-G 16/4 NW 1 DIA SG02 67 STRAIN GAGE 2 JD-G 16/4 NE 2 MAIN SGOS 68 STRAIN GAGE 3 JD-G 16/4 NW 2 MAIN SG04 69 STRAIN GAGE 4 JD-G 16/4 SE 1 DIA SGOS 70 STRAIN GAGE 5 JD-G 16/4 SE 2 MAIN SGOS 71 STRAIN GAGE 6 JD-G 16/4 SE 3 DIA SGO? 72 STRAIN GAGE 7 JD-G 16/4 NW 2 MAIN SGOS 73 STRAIN GAGE 8 JD-G 16/4 SW 2 MAIN SG09 74 STRAIN GAGE 9 JD-G 16/4 NW 2 MAIN SGIO 75 STRAIN GAGE 10 JD-G 16/4 NW 1 DIA SGll 76 STRAIN GAGE 11 JD-G 16/4 NW 2 MAIN SG12 77 STRAIN GAGE 12 JD-G 16/4 NW 3 MAIN WDOl 159 WIND DIRECTION ANEM TWR 3, 47.4 M WD02 163 WIND DIRECTION ANEM TWR 4, 41.4 M WD03 168 WIND DIRECTION ANEM TWR 5, 39.3 M WD04 179 WIND DIRECTION ANEM JD-G 16/4, 34.7 M WDOS 181 WIND DIRECTION ANEM JD-G 16/4, 10.9 M WSOl 156 WIND SPEED IiOT WIRE ANEM, ATOP LIGHT POLE WS02 158 WIND SPEED PROP ANEM, MORO TWR 3, 47 .4 M WSOS 161 WIND SPEED PROP ANEM, MORO TWR 4, 41 .5 M WS04 16? WIND SPEED PROP ANEM, MORO TWR 5, 39 .3 M WSOS 178 WIND SPEED ]PRO P ANEM, JD-G 16/4, 34.'7 M WSOS 180 WIND SPEED PROP ANEM, JD-G 16/4, 10.<0 M 29

2 OHGW loads from load cells, 10 swing angles, 12 sets of strains, 5 wind directions, and 6 wind speeds. The following two requirements must be met simulta neously to trigger the data acquisition system to collect » and store data: 1. the wind speed must be equal to or greater than a triggering value of 18 mps (40.5 mph) for one minute, and 2. the ambient temperature must be equal to or greater than 39.2 degrees Fahrenheit (4 degrees Celsius). Once the system is activated, the recording mode samples the data for 10 or 12 minutes, depending upon the sam pling rate. At the end of the sampling time, the system cannot be automatically triggered again for 60 minutes. The system can be manually triggered within this 60 minute time period from terminals either on the site or at BPA Headquarters in Portland. Two sampling rates are used in data collection: 10 samples per second (sps) or 20 sps. In Table 3.2, LCOl, LC02, and SG01-SG12 are sampled at 20 sps. All other channels are sampled at 10 sps. In monitoring wind speeds, hot wire and propeller vane anemometers are used. The hot wire anemometer is 30 mounted on top a light pole at a height of 10 meters from the ground. The light pole is located at a distance of 200 feet ENE of tower 2. The exposed resistor of the hot wire anemometer acts as a voltage sensor. Changes in voltage associated with changes in windspeed are trans mitted to the Data Acquisition System. The main advan tage of using the hot wire anemometer is because of its high sensitivity; stalling speed is low since no friction is encountered by interacting mechanical parts. The propeller-vane anemometers measure both wind speed and direction. One of these anemometers is mounted at a height of 33.5 feet (10.21 m) from the ground on the northwest leg of tower JD-G 16/4. The other anemometer is located at an elevation of 110 feet (33.53 m) on the west side of tower JD-G 16/4. In the measurement of wind speed, the three blade propeller has a threshold speed of 19.12 mph (8.5 mps) and a distance constant of 10.35 feet (4.6 m). The distance constant of an anemometer is the length of air column necessary to pass a wind sensor after a stepwise change in speed for the sensor to regis ter 63 percent of the change in speed (Kempner and Laursen, 1981). The directional sensitivity of the ane mometers is 4.95 mph (2.2 mps) at 8 degrees with a 31 distance constant of 34.12 feet (10.4 m). The orientation of wind direction is such that:

1. a zero degree reading represents true north, and

2. a clockwise rotation corresponds to an increase in the wind direction reading (Kempner and Laursen, 1979).

Internal heaters enable the propeller anemometers to operate under cold weather conditions.

Strain gages are welded to main and diaigonal members of the base of the tower. The orientation auid locations of the strain gages are such that only axial strains in the members were measured. Strain gaiged members are designated by the corner of the tower in which they are located. The numbering of the strain gaged members is from left to right by looking at a main leg in the center of the tower. Member 2 is the main leg, whereas members 1 and 3 are the diagonal legs. Figure 3.4 represents schematic locations of strain gaiges described in Table 3.2. The accuracy of the strain readings is controlled by the installation of the strain gages. Signal condi tioning units located on the tower are used transmit strain gage signals to the Data Acquisition System. Norville, Mehta, and Farwagi (1985) described the mounting of strain gages on the members. 32

N e SGOl (NW 1 Dia) SGOS (NW 2 Main) SG07 (NW 2 Main) SG09 (NW 2 Main) SGIO (NW 1 Dia) NE SG02 (NE 2 Main) SGll (NW 2 Main) SG12 (NW 3 Dia)

SG04 (SE 1 Dia) SGOS (SW 2 Main) SGOS (SE 2 Main) SGOS (SE 3 Dia)

Figure 3.4 Schematic Locations of Strain Gages Designated by the Corner of the Tower 33

Load cells measure the maignitudes of the loads trans mitted to the tower attachment points by the conductors and the overhead ground wires. Axial loads can be meas ured in the insulator assemblies connected to the east and west groundwires, aind to the east, center and west conductors. The load cells in the overhead ground wires (OHGW) have a sampling rate of 20 sps, whereas the load cells in the 3 conductors have a sampling rate of 10 sps. Baldwin-Lima-Hamilton (BLH) strain gage load cells are used. BLH Type T3P1, rated at 5000 pounds, were used on the OHGW and BLH Type T2P1, rated at 20000 pounds, were used on the conductors (Kempner and Laursen, 1981).

Swing-angle indicators are used to measure the di rection of swing in the conductors and OHGW. The swing angle indicators measure longitudinal (parallel to line) and transverse (perpendicular to line) swings. Humphrey CP17-0601-1 pendulum swing-angle indicators are used. The maximum swing angle in either direction which the indicators can measure is 45 degrees. The zero position is vertical (Kempner and Laursen, 1979). Load cells and swing-angle indicators were installed in the linkage between the insulator string and the tower.

The twenty three available records are now considered for analysis. Four of the twenty three records were 34 selected for analysis purposes. The statistical analyses of these four records are described in detail in Chapter 4. The criteria used in selecting these four records from those available are also described in Chapter 4. CHAPTER 4 STATISTICAL ANALYSIS OF TIME SERIES RECORDS

4.1 Introduction The statistical analyses of selected time series records are discussed in this chapter. Table 3.2 summa rized the types of wind and response data collected by BPA for any one particular record. These wind and re sponse data were in the form of time series. In particu lar, these time series were the time series of conductor loads, swing angles of conductors, strain gages, wind directions, aind wind speeds. All the time series de scribed herein are stochastic in nature. Since these time series are stochastic, they can only be assessed in probabilistic terms. Therefore, statistical analyses were used to define the descriptive properties of the selected time series. The descriptive statistical properties of a selected time series are the mean, variaince, autocovariance func tion, and autocorrelation function. If two separate and causally related time series are analyzed, cross covariance functions and cross correlation functions are used as additional descriptive statistical properties.

35 36

4.2 Data Handling All the time series records used in subsequent auialy- sis were recorded on 9-track unlabeled, 1600 bytes per inch (BPI) magnetic tapes. There are twenty-three avail able wind auid trainsmission tower response records which were collected between December 1981 and Mao^ 1982 at the test site. Twenty-one of these records were collected under strong wind conditions. The other two records were obtained during relatively calm conditions to establish zero values as a reference in subsequent analysis. The average wind speeds amd directions of the twenty-three available records are shown in Figure 4.1. As discussed in section 3.3, each tape is divided into 38 channels. Each channel performs a different task and is labeled accordingly. Each recording number constitutes an index including 38 files. For example, N02.WS06 refers to the sixth wind speed file in record number 2. Files sampled at 20 sps contained 12000 observations and have a 10- minute record length. Files sampled at 10 sps contained 7200 observations and have a 12-minute record length. Section 3.3 provides information concerning sampling rates of the files. 37

300* • 60'

270' 90'

• 120' 240

JD-G 2

Figure 4.1 Twenty Three Available Wind Records 38

4i.3 Time Series Analysis

A time series is a collection of observations made sequentially in time. The sequence of N observations, X, X, , X making up a time series as a realiza- 1 2 N tion of jointly distributed random variables are usually taken at discrete time points equally spaced with time interval, Zjkt. In time series analysis, information per taining to the time series may be obtained in the time and in the frequency domain. Both domains provide useful engineering and statistical properties to the analyst. Figure 4.2 summarizes a general methodology used in time series analysis. The literature review discussed in Chapter 2 indicated that both time and frequency domains analyses are widely used in engineering analyses of time series. This chapter discusses only the time domain anal ysis of time series. In time domain analysis, the first step is to gener ate a time series plot. A time series plot is a plot of observations versus time. Visual inspection of the time series plot will often indicate statistical consistency of a time series. Features such as trends, periodicity, and discontinuities in the time series should be appar ent. Such features often assist in preliminary assess ment and validation of a time series. Validation of a 39 DOMRI N FREQUENC Y

od C < m £ 01 CK (D Ui O r 1 »-t u. 01 en I > Z2 Z (D O I ^ *" Uo . QC

0) TIM E TIM E CM HISTOR Y DOMRI N

cc oz CD •- O cr •-• 03 CO cn o z QC UI 40 time series is important if the results obtained from basic descriptive statistical analyses are to be meaning ful.

A statistical phenomenon that evolves in time ac cording to probabilistic laws is called a stochastic process. In analyzing a time series, the time series is regarded as a realization of a stochastic process.

The time series to be analyzed can then be thought of as one particular realization produced by the underlying stochastic process. After plotting the time series, basic descriptive statistical properties of the time series maiy be obtained. The mean and the variance are the basic statistical properties of the time series. The mean X of a time series can be estimated by Equation 4.1 as:

I ^ X = E[X ] = lT-2x ('^•l^ t '^ t=l t where N denotes the total number of observations in the time series. X defines the level about which the time series fluctuates. The variance of a time series meas ures the dispersion about the mean and is estimated by

Equation 4. 2: 41 2 2 , N (J = E[(X -X) ] =7rT-S(x -X) . (4.2) t N-'t=l t

The covariance between x and x separated by k inter- t t+k vals in time may be estimated by:

C = E[(X -X)(X -X)] k t t+k , N-K

^ t=l t t+k

Equation 4.3 is also referred to as the autocovariance function. Box and Jenkins (1969) suggested that the autocovariance function should be calculated for k=0,1,2, ,K, where K is not larger than N/4. Simi larly, the autocorrelation function can also be calcu lated from Equation 4.3. In most cases, the autocorrela tion function is used to describe the correlation in a time series because it is the normalized form of the autocovariance function. The autocorrelation function of a time series may be estimated by:

r = C /rr^ (4.4) k k' where -1 ^ r i 1 k 42

Since r = r , the autocorrelation function is neces- k -k sarily symmetric about zero, hence it is necessary only to consider that portion of the function for k > 0. In many instances, two separate time series are anal yzed jointly and the statistical properties are calcu lated. If X , X , , X comprise the observations 12 N from the first time series, and y ,y , ,y comprise 12 N the observations from the second time series, then the cross covariance functions between the two time series of the bivariate process at laig +k can be written as:

C (k) = EC(X -X)(Y -Y)] xy t t+k , N-K _ = Tr S (^ -^>(y -^^ (4.5a) •^ t»l t t+k and

C (k) = E[(Y -Y)(X -X)] yx t t+k N-K = J- y (y -Y)(x -X) (4.Sb) N ^^ t t+k where k=0, 1,2, . N Y = f^?, "t C (k) = C (-k) . xy yx 43

Since C (k)= C (-k), we need only define one function xy yx C (k) for k=0,+l,+2,+ The cross covariance xy function can be expressed in a normalized form. The cross correlation function is a normalized form of cross covariance function which is estimated by Equation 4.6

r (k) = C (k)/!^ (4.6) xy xy /^X^Y

where

N 2 CT - V. ^X N- 1 t = l t

N 2 (T> - V,i - V^fv -Y^ ^r N-l J. . +-

The cross correlation function is not symmetric about k=0 since it is not an even function. In general, the statistical analysis of a time series is not complete without examining the stationarity of the time series. In the following section, a brief dis cussion of stationarity of a time series is given.

^-•-i Stationarity of a Time Series A time series can either be stationary or non- stationary. In general, if a time series is stationary, its statistical properties -are independent of the time 44 origin. If a time series is non-stationary, its statis tical properties are dependent on the time origin. Nel son (1972) indicated that stationarity is a very strong condition to impose on a time series since stationary time series rarely exist in practice. A stationary time series may be categorized as either strictly stationary or weakly stationary. Most computations of descriptive statistical properties of a time series assume stationa rity. In the case of a non-stationary time series, the method advanced by Box and Jenkins (1976) may be used. This method is discussed in Chapter 7.

A stochastic process x is strictly stationary or t stationary in the strict sense if the higher-order mo ments and joint moments of the stochastic process x are t time invariant. For a stochastic process to be strict ly stationary, two conditions must be met: 1. the statistics of a process x must be the same t for any time lag k, that is, the statistical prop erties of X and x are the same, t • t+k 2. the joint distribution of a process x must be t invariant with regard to a displacement in time. If k and m represent any integers in time lags,

then P (x , , X ) = P (x , , X ). t t+k t+m t+k+m 45

These two criteria ma^^ be used to test whether or not two stochastic processes are strictly stationary. If x and t y represent two stochastic processes, then stationarity t in the strict sense requires that the joint statistics of X and y to be the same as the joint statistics of x t t t+k and y t+k A stochastic process is weadtly stationary if the moments through some specified order are invariant over time. Generally, if the first and second moments of a stochastic process are time invariant, then the sto chastic process is weaikly stationary. The general re quirements for a stochastic process to be weakly sta tionary are: 1. the expected values and the variances are the same for any lag k, that is

E(x ) = E(x ), and t t+k Var(x ) = Var(x ) t t+k 2. the autocorrelation depends only on time lag m, i.e.,

E[(x -X)(x -X)] = C . t t+m m The above criteria can be extended to two causally re lated stochastic processes. For two causally related 46 processes to be weakly stationarity requires that: 1. the cross correlation between two processes de pends only on time lag m, i.e.,

E[(x -X)(y -Y) = C (m). t t+m xy It is impossible to obtain a physical time series which is strictly stationary. Most physical time series are nonstationary in nature or at the very most, weakly stationary. In order to apply Equations 4. 1 through 4.6, it is necessary to assume or to obtain a time series which is weakly stationary. There are other available statistical methods of checking for stationarity of a stochastic process. The method proposed by Box and Jen kins (1976) is discussed below. Box and Jenkins (1976) suggesLcd using autocorrela tion plots as a means to check for stationarity. In general, a stationary time series tends to have an auto correlation function which dies off rapidly with in creasing time lags. The autocorrelation values are con sidered to be zero at laig greater than q if

^ 2 1/2 |r 1^ 2/fr: [l+2y!r] j>q- (4.7) .i "^/N it^ i 47

In many instances, if a time series is not stationary, the autocorrelation function tends to have high numerical values at large time lags. In other words, the auto correlation function dies off with increasing time lags very slowly, if at all, for a non-stationary time series. High values of the autocorrelation function at high time lags indicate that some form of non-stationarity exists. In many cases, taking the first difference of a non- stationary time series will often produce a stationary time series. This technique is used widely by statisti cians to model stationary time series from actual time histories which are non-stationary because the mathe matics involved in taiking the first difference of a non- stationary time series are less complicated. The auto correlation of the time series produced by taking the first difference of the original time series tends to die off rapidly if this results in a stationary time series. In general, a set of new observations is generated by first differencing of the original time series. The first difference of the original time series may be written as

z = X - X (4.8) t t t-1 48 where z is the new observation at time step t. A t complete discussion on time series modeling is given in Chapter 7.

4.5 Selected Time Histories Table 3.1 provided information regarding record num bers, dates, and times when the instrumentation system was activated, mean wind directions, and mean wind speeds during the recording process. Among these twenty-three available records, N04 and N14 represent zero wind speed recordings. The selected records used for analyses in the re search reported herein are shown in Table 4.1. Records N07, NOS, and N16 were recorded during west winds. Record NOS was recorded during an east wind. Four crite ria were used in selecting records NOS, N07, NOS, and N16. These criteria are: 1. One record with the highest mean wind speed from all west wind records was selected, i.e., record N07. 2. One record with the second highest mean wind speed from all west wind records was selected, i.e., record NOS. 49

Table 4.1 Selected Records for Analysis

Mean Wind Mean Wind Record Date Time Direction Speed (Degrees) (Meters/Second)

NOS 12/16/81 16.10.28 93.0 15.79 NO? 01/16/82 19.04.51 215.0 22.31 NOS 01/31/82 01.36.32 282.0 21.42 N16 03/12/82 15.13.50 220.0 9.99

Note: Wind speeds are monitored at 34.7 m height so

3. One record with the lowest non-zero mean wind speed from all west wind records was selected, i.e., record N16. 4. One record with the highest mean wind speed from all east wind records was selected, i.e., record NOS. Records NO? and NOS were obtained through the automatic triggering of the data acquisition system. On the con trary, records N16 and NOS were obtained by manually triggering the data acquisition system. In general, record NOS is from the WNW direction, record NO? is from the WSW direction, record N16 is from the SSW direction, and record NOS is from the east direction. The time history plots for one second interval averaiges of wind speeds at 34.7 m height are shown in Figures 4.3a, 4.3b, 4.4a, and 4.4b. Associated with these wind speeds, the time history plots of the conductor load values (LCOS, LC04, and LCOS) for west, east, and center conductors are given in Appen dix A (Figures A.1 through A.6). Similarly, the time history plots of the associated swing angles of the conductors (SAOS through SAIO) are shown in Appendix A 51 NOS. WSOS la/ia/Bi «.J0.j* nor BTTc 00/07/W INST. MI - la.u ira Tt>« - l.2flT UN 15.7« »n IHT. im s I6.7S ITS TOt = I.2S] KIN leT ST1I EV o.a »n Iter. MiN - it.zz m TtJt - S.UW RIN iwr TTOcev iKT. MIN - m.37 m T[>€ - S.SQO HIN M« »»TV1IIT 10 TIfC HISTIKT POR 1.000 XCtfO IKTBrfPL RVEKflGES Rn

(D 0. N

2 d-i UJ a. a

10 12 TIME (MINUTES!

Figure 4.3a Time History for Wind Speed at 34.7 m for Record NOS

N07.WS05 oi/ifl/aa ia.o«.si Nim attu rtar r««Mtcim JiMr IH<^ Hr=9i.Hr=9l.7 Nn fTDT UHlt OO/IIT/TJU «.51 tri wax. KW - 32.B i« TT>« - i.flsa NtN »€«» 3.70 r« IKT. HTK = 31.81 ITS TDC = 2.967 KIN IBT STU OEY = INST. MIN - 8.99 m ^Mr. - i.ryn NIN Iin 3TTJ DfV • 3.aj m INT. NiN - ta.aa m TI>€ - 7.OBI HIN Mfi PTVm • to TIW HI3TBRT R» 1.000 SECtJK] IHTERIPl RVtHCES

q a s TIME (MINUTES)

Figure 4.3b Time History for Wind Speed at 374 m for Record NO? 52 NOS. WSOS 01/31/82 01.38.32. Mi»a 3Tcn rior FNUWCIEK JO-* IB/H m=»i.maw. 77 N nor ORTC 0B/TJ7/'W IN3T. mx - 29.33 ITS TTM: - 0.733 KIN «(3R im? w-3 IMT. ¥F9. = 2B.73 HP3 TI>C = 0.750 NIN INSr STTJ DEV = 3.13 fTS INST. WN - B.os rr3 TDt - B.Rn NIN INT 5TD OCV - 3.0D ft-3 INT. NIN - fl.aa MPS TOt. - 1.883 NIN NW PT3/IirT • 10 TIW KISTORT RK 1.000 SECOO (MTEm. flVERnCES

TIME (MINUTESl

Figure 4.4a Time History for Wind Speed at 34.7 m for Record NOS

N16.WS05 03/13/82 1S.13.SQ. MiMi JLEU rtar muuciiK JD-* IB/^ Mr=w.7 N nor nmc 0B/TJ7/TJO 0.99 wa IM3T. MW - 13.3t Tt>e - l.isa NIN »CPM IMT. HPK = (3.21 m TOC = B.167 NtN tier 3Tn IEV 1.23 trs 1.33 m INST. MIN > B.BB m TTfC • S.02S UN INT STO otv INT. NIN - NPS TI>€ - 1.033 NIN fW PTS/JNT 10 Tl« HISTOTT MR 1.000 SECOO IMTEmn. RVCKflGES

== I*-

liJ a(n "^kAr^/^i^

-T 10 12 TIME (MINUTES)

Figure 4.4b Time History for Wind Speed at 34.7 m for Record N16 S3

(Figures A.7 through A.IS) and the time history plots of selected strain gages are shown in Figures A.19 through A.26. All the time histories plotted in Appendix A are for selected channels only. These channels are used for analysis purposes in subsequent sections. The selected channels are WSOS for wind speeds, LCOS-LCOS for conduc tor loads, SAOS-SAIO for swing angles, and SGOS, SG06, SGll, and SG12 for strain gages. Except for the selected strain gaiges, all other selected channels are used as inputs to the transmission tower structural system. The selected strain gages are used as outputs or responses from the transmission tower structural system. Strain gaiges 5 and 6 were mounted on the main member and the diaigonal member on the southeast corner of the transmission tower, respectively. Strain gaiges 11 and 12 were mounted on the main member and the diagonal member on the northwest corner of the transmission tower, res pectively. Earlier research by Norville, Mehta, and Farwagi (1984) in validating wind and response related data indicated that strain readings recorded by strain gages 3, 4, 7, 9, and 10 were erroneous. The most viable choice is to select strain gaiges 5 and 6 because both of these strain gages are located on the southeast corner. 54

Further, strain gage 5 is mounted on the main leg member whereas strain gage 6 is mounted on the diaigonal leg member. Similarly, strain gaiges 11 and 12 are selected because they are located on the northwest corner of the transmission tower. Additionally, strain gage 11 is mounted on the main leg member whereas strain gage 12 is mounted on the diagonal leg member. Figure 4.5 shows the locations of the selected strain gaiges.

Table 4.2 gives the mean values of the selected load records from the conductor. Table 4.3 gives the mean of the swing angles associated with the conductor motions in which the conductor is capable of swinging in the longi tudinal and transverse directions. Since the conductors can swing freely in the longitudinal and in the trans verse directions, the mean transverse and longitudinal load values are summarized in Table 4.4 and in Table 4.5, respectively. Table 4.6 gives the mean of the selected stresses from the strain gages mounted on the trans mission tower structural members. In Table 4.2, differences in load values for the same load cell can be attributed to different wind speeds and wind directions which affected the swing angles. Table 4.3 presents the results of mean swing angles. In Table 4.3, the mean swing angles associated with the 55

SGll (NW 2 Main) SG12 (NW 3 Dia) NW- -NE

<> SW- •SE SGOS (SE 2 Main) SG06 (SE 3 Dia)

Figure 4.5 Locations of Selected Strain Gages for Analysis 56

Table 4.2 Mean Load Values From The Conductors (kN)

Load Cell Records Number NOS NO? NOS N16

LCOS 29.00 27.07 28.41 28.85 LC04 26.25 28.09 28.26 28. 58 LCOS 28.25 27.28 28. 15 28. 16

Table 4.3 Mean Swing Angles (Degrees)

Swing Angle Records : Number NOS NO? NOS N16

SAOS 0.88 -0.49 -0.28 0. 19 SA06 -8.30 9.02 7.67 -1.25 : SAO? 1.59 1.08 1.58 1.96 : SAOS -2.02 9.99 19.98 5.97 : SA09 2.85 1.99 2.95 2.94 : SAIO -5.34 8.50 11.49 1.99 : 57

Table 4.4 Mean Transverse Load Values From the Conductors (kN)

Load Cell Records Number NOS NO? NOS N16

LC03 -2.84 2.82 5. 15 0.72 LC04 -2.62 2.80 4.82 0.80 LCOS -2.77 2.94 5.47 0.81

Table 4.5 Mean Longitudinal Load Values From the Conductors (kN)

Load Cell Records Number NOS NO? NOS N16

LC03 0. 30 0.70 0.60 0.40

LC04 0.25 0.50 0. 26 0. 35

LCOS 0. 12 0.60 0.39 0.37 58

Table 4.6 Mean Leg Stresses of the Transmission Tower Due to Wind and Total Conductor Loads (MPa)

Strain Gage Records ; Number NOS NO? NOS N16

SGOS -7.02 6.04 14.01 1.49 : SG06 0.59 -0.49 0.58 1.97 : SGll 8.64 -3.46 -14.69 0. 41 : SG12 0.59 -1. 16 -0.66 0.23 : 59 the transverse direction for swing angle indicators SA06, SAOS, and SAIO are generally larger than the mean swing angles associated with the longitudinal direction for swing angles indicators SAOS, SA07, and SA09 for the selected records. Since the transverse swing angles are larger than the longitudinal swing angles, it is expected that the transverse loads from the conductors are gener ally larger than the longitudinal loads from the same conductors. Negative values of the swing angles indicate that the conductors are s\faying to the west. Table 4.4 presents the mean transverse loads from the conductor. The transverse load at any instant is calcu lated by the following equation:

F = P(coscb)sin9 (4.9) T where P is the total conductor load recorded by the load cell, (b is the longitudinal angle recorded by the longi tudinal swing angle indicator, and Q is the transverse angle recorded by the transverse swing angle indicator. The highest transverse loads amongst the selected records occurred in record NOS. Table 4.5 presents the mean longitudinal loads from the conductor. The horizontal load at any instant is 60 calculated by the following equation:

F = P(sincb)cos9 (4. 10) H where P,

4. 6 Autocorrelation of Selected Time Histories In this section, only autocorrelations of selected time histories are described. In particular, one west wind and one east wind record are used as examples of the autocorrelation functions for all time histories. The autocorrelations of records NOS and NOS are described in this section, whereas the autocorrelations of records NO? and N16 are shown in Appendix B. As described in section 4.2, the autocorrelation function is a normalized version of the autocovariance function. The autocovariance function given in Equation 4.3 states that the covariance between observations X t and X is the expected product of their deviations from t+k the mean of the time histories. Qualitatively, if a higher than averaige observation tends to be followed by another higher than averaige observation k periods later, and likewise for lower than averaige observations, then the autocovariance between X and X is positive. The t t+k autocovariance is negative if a higher than average ob servation tends to be followed by a lower than average observation k periods later and vice versa. Such an interpretation of the autocorrelation function plots of a time series aids in understanding the nature of the time series. 62

If the original time series are stationary, further differencing of the original time series will produce another stationary time series. In other words, the autocorrelations of the original and first differenced series cut off and die off rapidly. The general auto correlation shapes for the original and first difference series would then be approximately identical to each other if the original series is stationary. In all the autocorrelation functions plots presented herein, two horizontal lines representing two standard errors are plotted. These two lines which are slightly above and below zero correlation horizontal line represent Equation 4.7. In most cases, it was found that the autocorrela tion values are considered zero beyond 150 to 300 lags.

The autocorrelation functions of the wind speed (WSOS) for records NOS and NOS are shown in Figures 4.6a and 4.6b, respectively. In both figures, the autocorre lation functions are plotted for the original and first differenced series of WSOS. For both records, all obser vations for the wind speed were used to calculate the autocorrelation values. In both plots, the autocorrela tion functions of the original wind speed time series die off very slowly. On the contrary, the autocorrelation functions of the first differenced series die off 63 AUTOCORRilATION VALUES OP N05.WS06 OMioouL Aim rasT oarxiatnx SMRTSS FOM ISO LACS SAMPUHeJUTM m tOSPS

Original

First Differenced

y^\/\,f*j\^y^'-'-i^^\^ry

-ca •p^T^TT^^^yTTTT w IB If w^»tPWf>yT^wryy^>YT^^py^N^yy^TWtft^ ^^^T'^rvw^rv

Lua.x Figure 4.6a Autocorrelation Functions of Original and First Differenced Series of WSOS for Record NOS

iUTOCORBELATION VALUES OF N08.WS05 CMOHAL Aim FVfST DOTSKMMC* SKWSS rOM 160 LAGS SAUPWfC XAJM " to SPS

tL* Original

•.I

•L2 First Differenced

na y^v\,j^-^Vy^\/\,^

-t.a ^^^^^^^^yy^i m i» n'w ••! •••^^y w^w^r^^yv^^^^^tn y> »m»• >• |• ^^^r^rn w *i ••yi ry tOB UB lAOS. r

Figure 4.6b Autocorrelation Functions of Original and First Differenced Series of WSOS for Record NOS 64 rapidly. The autocorrelation function of the original wind speed time series indicate that the wind speeds for records NOS and NOS are non-stationary. Similarly, Fig ures B. la and B. lb in Appendix B indicate that the wind speeds for records N07 and N16 are also non-stationary. The first differenced series of WSOS for records NOS, N07, NOS, and N16 intersect zero between 5 to 10 lags for the first time.

Next, the autocorrelation functions of conductor loads are calculated using all the observations available in the load cell time histories. The plots of the auto correlation functions of conductor loads for records N07 and N16 are given in Appendix B. The autocorrelation functions of the original and first differenced series of load cell (LCOS) for record NOS are shown in Figures 4.7a and 4.7b, respectively. Periodic fluctuation is noticed in the autocorrelation function of the original time history. The autocorrela tion function of the original time history dies off very slowly. On the contrary, the autocorrelation function of the first differenced series dies off with increasing lag in a periodical manner. The term "die off" in the con text presented herein refers to autocorrelation functions which exhibit gradual decrease in magnitude of the 65 AUTOCORRELATION VALXJBS OF NOft.LCOS ouaiNAL sours FOR tso LAGS SAMPlUfC JUIM m (OSPS

LB IB UBxr

Figure 4.7a Autocorrelation Functions of LC03 for Record NOS

AUTOCORRHAnON VALTJBS OF H0ft.LC03 FIRST DlFTKUFHCg SBUTS rOH 100 LAGS SAUPLUfC JIAJM ^ tOSPB

i.t

•.t

ft.a

LiLJLtilll. It .t. tj. tf jk !ii ,,iMtm «.• iiiiii'U ifrii.tiriiiijii nil If Ji.'iriii-iwiiiiii']fli';rii,ii'ij livi [|M[ii(fpr !'* MI '^ ^

-*.» 1 1 1' m tm iM Laos, K

Figure 4.7b Autocorrelation Functions of First Differenced Series of LCOS for Record NOS 66 correlation with increasing time lag. This decrease in magnitude of the autocorrelation functions can be rapid or slow with increasing time laig. Further, the auto correlation functions can decrease in magnitude either positively, negatively, or even in a daanped sinusoidal fashion.

The autocorrelation functions of the original and first differenced series of load cell (LC04) for record NOS are shown in Figures 4.Sa and 4.Sb, respectively. The autocorrelation function of the original time history exhibits periodical fluctuation and dies off very slowly. On the contrary, the autocorrelation function of the first differenced series dies off in a periodical manner. The autocorrelation functions of the original and first differenced series of load cell (LCOS) for record NOS are shown in Figures 4.9a and 4.9b, respectively. Again, the autocorrelation function of the original time history dies off slowly and exhibits a periodic fluctua tion. Also, the autocorrelation function of the first differenced series dies off in a periodic manner. The autocorrelation functions of the original and first differenced series of LCOS and LC04 are shown in Figures 4.10a and 4.10b, respectively. In both plots, the autocorrelation functions of the original time 67 AUTOCOBSELATION VALUES OF N0ft.LCC4 oaiaitAL ssFiss rott mo LACS SAMPLUteJUTM' tOSFS

\r Figure 4.Sa Autocorrelation Functions of LC04 for Record NOS

AUTOCORBEUnON VALUES OF N05.LC04 pmsTDifrKxgfrcK sgRas ron tso LAGS SAuruKCJuram tosps i.a

a.t

prFF"

-•.a rw999^wwjww9% m9mw^m^r^^wwfw^^wwww9y^^9^^9t9 p ninmifi^iBw >»•! i m i» iAes.r

Figure 4.8b Autocorrelation Functions of First Differenced Series of LC04 for Record NOS 68 AUTOCORRELATION VALVES OF NOft.LCOft oaiGif/AL ^iias roR iso LAGS SAUPLUfCJUTM - tOSPS

1.8 4

«k«

a.a

^pwv^r^i^^ww^wqr* F^PW*Y»^fT.W*.*T^rw»».ww^ \r Figure 4.9a Autocorrelation Functions of LCOS for Record NOS

AUTOCORREIATION VALUES OF N0&.LC09 FasTDOrxxgircs sgnas /CM tao LAGS SCAUrUMCJUTM' tOSPB

-«.a

LtBa,X

Figure 4.9b Autocorrelation Functions of First Differenced Series of LCOS for Record NOS 69 AUTOCORRELATION VALUES OF }}OS.I C03 SNCS^AL SBSIS^ rr>K fSO LACS FmaTDirrsxrNcg stjuws ran iso LACS SAUPUSe JUTM m iOSPS

a.a Original

a.a

a.t

OJ First Differenced

ii7

Ltas.r

Figure 4.lOa Autocorrelation Functions of Original and First Differenced Series of LCOS for Record NOS

AUTOCORRELATION VALUES OF N06.LC04 OUGDIAL SBHaS rOK tao LACS /jRSTlurrKRgfrcs SgRas ran tao LAGS SAUPUHC JUn ~ to SPS La

na Original

a.a

a.1

First Difference a.2

j(iiiiiiiMBiriBiirt«'.jt;tii>'kiiu'tvii*-:i M »* J» « ' t-J* • H \* \

-».» ^•^.^rvT^r^^^* I |l T I" ua *• uea.r

Figure 4.lOb Autocorrelation Functions of Original and First Differenced Series of LC04 for Record NOS 70 histories die off slowly. Also, the autocorrelation functions of the first differenced series oscillate about zero and die off slowly in a periodic manner in both plots. The autocorrelation function of the original time history shown in Figure 4.10a is similar to the auto correlation function of the original time history shown in Figure 4.10b. Similarity between the first differ enced series of LCOS and LC04 is also noticed.

Figure 4.11 shows the autocorrelation functions of the original and first differenced series of LCOS for record NOS. Again, the autocorrelation function of the original time history dies off slowly in a periodic manner. However, the autocorrelation function of the first differenced series oscillates about zero and dies off slowly in a periodic manner. In Figures 4.7 (a and b) through 4.11, the autocorre lation functions of the original time histories die off slowly in a periodic manner. Such periodic variations in the autocorrelation functions may be attributed to the fact that the conductors are oscillating at their own natural frequencies. Since the autocorrelation functions tend to exhibit a periodic cycle or seasonal variation, all the load cell time histories are non-stationary. In most cases, the autocorrelation functions of original 71

AUTOCORREIATION VALUES OF N08.LCC9 oaientdL sgRsts FOR lao LACS nUSTDirTJSXBJf eg SERBS FOR tao LAGS SAurtaejun - to SPS t-»4

Original

•J First Differenced

.«^ rJ.k *2»i II .t.iiiitiii*iiiiit.iic.«,*.«iiiiiiii:tt llJf^lfWlUflUlMW •*ntt

-«.« jvwv^F9^^f^'99m^^^yfw^wwwW9 9wjwwww9wmwwfm a>»aa^»a | w w 9WI a^a ^mww^

lABB. K

Figure 4.11 Autocorrelation Functions of Original and First Differenced Series of LCOS for Record NOS 72 time histories of LCOS, LC04, and LCOS for records NOS, N07, NOS, and N16 indicated two distinct lags due to periodic variations of the autocorrelation functions. These periods are approximately 24 and SO lags ( 2.4 and 8.0 seconds, respectively). The response of the transmission tower is represented in the form of stresses experienced by the structural members. First, the strain readings from the selected strain gaiges mounted on the main leg members (SGOS and SGll) and on the diagonal leg members (SGOS and SG12) are obtained. These strain readings are then converted into stresses using prescribed conversion factors for each strain gage under consideration. All the observations in the time histories of the strain gages were used to calculate the autocorrelation functions of SGOS, SG06, SGll, and SG12. The autocorrelation functions of the original and first differenced series of SGOS and SGll for records NOS are shown in Figures 4.12a and 4.12b. In both figures, the autocorrelation functions of the original time histo ries are similar to each other and die off slowly in a periodic manner. On the contrary, the autocorrelation functions of the first differenced series of SGOS and 73 AUTOCORBELATICN VALUES OP N0O.SGO6 QJUOOtAL Am FIRST DOTTRgttar SgRTSS rOM ISO LACS SAUPtOfC RATS - 20 SPS

iifst Differenced

-9.2

-•.«

yvff^F^^rwY'^^^^^^n^^^^^^^^^^^^^^^y^^^^^n^^^

Figure 4.12a Autocorrelation Functions of Original and First Differenced Series of SGOS for Record NOS

AUTOCORREIATION VALUES OP N09.SG11 OMOOULJim FIRST DIFFKRgtlCg SgRTSS FOR ISO LACS SAUPLoreRATg'iasps

i.a4

a.a' Original

a.a

oa First Differenced

m^t *^\^WY * n r»p •A.2

-•.«

-•.a .^ii.i w II H I j |i I 1 1 " *

LU9. K

Figure 4.12b Autocorrelation Functions of Original and First Differenced Series of SGll for Record NOS 74

SGll for record NOS oscillate about zero correlation and die off quickly. Figures 4.13a and 4.13b present the autocorrelation functions of the original and first differenced series of SG06 and SG12 for record NOS, respectively. The auto correlation functions of the original time histories for both plots are similar to one another, and are periodic in nature. The autocorrelation functions of the first differenced series for both plots die off in a periodic manner. Figures 4.14a and 4.14b present the autocorrelation functions of the original and first differenced series of SGOS and SGll for record NOS, respectively. Similarities between the autocorrelation functions of SGOS and SGll are noticed. The autocorrelation functions of the origi nal time histories die off slowly, wheareas the auto correlation functions of the first differenced series die off extremely slowly in a periodic manner. The autocorrelation functions of the original and first differenced series of SGOS and SG12 for record NOS are shown in Figures 4.ISa and 4.ISb. Similarity is again observed for the autocorrelation functions of the original time histories. The autocorrelation functions of the first differenced series in both figures die off 75 AUTOCORRELATION VALUES OP prOfl.S^»« QJUGDUL Jim FIRST DOTgRgtrOf SgRIgS FOR ISO lACS SJUtPUMG RAJg ~ 20 SPS

-*.i ^First Differenced -a.t

-•.a

-a.a • iBBin iaw^aaway • s lAoa.g Figure 4.13a Autocorrelation Functions of Original and First Differenced Series of SGOS for Record NOS

AUTOCORRELATION VALUES OP N00.SG12 OXOOUL Aim FIRST DOTgRMhnS SgRfSS FOR ISO [ACS SAMFlMCRAlg - Sn SPS

I.a4

f»Uly*u^Y »i Y * i? » *

First Differenced

•1 |i..ii..in Iiiii "I ' 0 ••«a«iDOLaDiaM UBS. X

Figure 4.13b Autocorrelation Functions of Original and First Differenced Series of SG12 for Record NOS 76 AUTOCORRELATION VALUE:J OP N08.SGO0 OJUaOiAL Aim FIRST DOTgRgtfCX SgRfSS FOR ISO UCS SAUKJ)ieRAIg^2aSPS

t.a

A.a riginal

a.t First Differenced

tua

a.a

-a-s

-•.t

-a.a

-•.a fV^W^MW^vwwv^if «*^F^^wr*vvvw«^^wvwwv«|«*v««f«««f f ««•» •••^ HiVaMill

uaa,g

Figure 4.14a Autocorrelation Functions of Original and First Differenced Series of SGOS for Record NOS

AUTOCORRELATION VALUES OP N06.SG1 \ OJUODiiLAim FIRSTDOTgRgtia SgRfSS FOR 160 UCS SAHPLUfCRATg - 2A £RS

Original

Ltaa.x

Figure 4.14b Autocorrelation Functions of Original and First Differenced Series of SGll for Record NOS 77 AUTOCORRELATION VALUES OP NOd.SGOd QIUGIHAL Aim FIRST DIFFSRgtfCg SgRTSS FOR ISO LACS SAia>tJJ/eRATg'20SPS

i.a

La t 1 ^^ Original

•.a'

a.t

OLS t I ' 1 JN IT XA fl i aj>. m I

-a.2

-%A First Differenced -a.a

-a.a • ai ai • » ua ua M> M Ltaa.g Figure 4.ISa Autocorrelation Functions of Original and First Differenced Series of SGOS for Record NOS

AUTOCORRELATION VALUES OP N08.SG12 OIUOOUL Aim FIRST DlFFgRgMCg SgRTSS FOR ISO UCS SAUPLU/C RAIg - J0 SPS

La 4 •Original a.a'

ika

a.t

iLa

a.a

-a.i

-a.t First Differenced -a.a

-a.a ...... iH..y»^y^^.i| .11 fi ittwmwwf^w |wi.ii.ii| a a m « toa Ui •> » LtBS, g

Figure 4.ISb Autocorrelation Functions of Original and First Differenced Series of SG12 for Record NOS 78 in a periodic manner. Also, except for the magnitude, the autocorrelation functions of the first differenced series are similar to the autocorrelation functions of the original time histories in both figures. In Figures 4. 12a through 4.15b, similarity between the autocorrelation functions of SGOS and SGll are observed. Also, similarities between the autocorrelation functions of SGOS and SG12 are noticed. This similarity is attributed to the fact that SGOS and SGll are the strain gaiges located on the main members which are oppo sitely configured to one another of the transmission tower. Similarly, SG06 and SG12 are located on the diaigonal members which are also oppositely configured to one another of the transmission tower (see Figure 4.5). A periodic trend of approximately 48 lags (2.4 second) was observed for SGOS and SGll, and a periodic trend of 20 lags (1.0 second) was observed for SGOS and SG12 for record NOS. For record NOS, no distinct lag was detected for SGOS and SGOS. However, a periodic trend of approxi mately 22 lags (1.1 second) was detected for SG06 and SG12. In all the selected strain gages, the autocorrela tion functions of the first differenced series indicate that the values of autocorrelation functions fluctuate about zero. This indicates that apart from differencing 79 the original time histories once, seasonal variations or trend must be considered in modeling the time histories. Finally, the autocorrelation functions of the original and first differenced series of SGOS, SG06, SGll,and SG12 for records NO? and N16 are shown in Figures B.Sa through B.Sb in Appendix B. The last step in time domain analysis involves studying the input and output statistical relationships of the time histories. The studies of these statistical relationships may be achieved through cross correlation functions of the causally related inputs and outputs.

4^7 Cross Correlation of Selected Time Histories In this section, cross correlation functions of the selected time histories are discussed. In particular, only the cross correlation functions between the selected inputs and the selected outputs are considered. The cross correlation functions are described by Equation 4.6. In calculating the cross correlation functions between any two selected time series, the sampling rate for the first time series should be equal to the sampling rate for the second time series. This enables a one to one statistical comparison between the first time series and the second time series. As discussed earlier, only so the cross correlation functions between the time series of the selected inputs, i.e., wind speed and load cells, and the time series of the selected outputs, i.e., strain gages need to be calculated. All the cross correlation functions are plotted for 300 lags (30 seconds) in this section. In calculating the number of cross correlation sets, all possible cross correlation sets between input and output must be considered. In the research reported here in, one common notation is used to describe any cross correlation functions between two causally related time histories. For example, the notation WSOS-SGOS is used to represent the cross correlation relating the input time history, WSOS, with the output time history, SGOS. For each record considered, only cross correlation sets between the selected input time histories and the select ed output time histories are calculated. Hence, there are 40 possible cross correlation sets. These cross correlation sets are as follow: 1. 4 sets relating wind speed (WSOS) with strain gages (SGOS, SGOS, SGll, and SG12), i.e., WS05-SG06, , WS0S-SG12. 2. 4 sets each relating total load cells values (LCOS, LC04, and LCOS) with strain gages (SGOS, 81

SGOS, SGll, and SG12), i.e., LC03-SG0S, , LC03-SG12, LC04-SG0S, , LC04-SG12, LCOS- SG03, , LC0S-SG12. 3. 4 sets each relating transverse load cell values (LCOS, LC04, and LCOS) with strain gages (SGOS, SG06, SGll, and SG12). The possible combinations in this case are similar to the combinations de scribed in 2 above. 4. 4 sets each relating longitudinal load cell values (LC03, LC04, and LCOS) with strain gages (SGOS, SG06, SGll, and SG12). The possible combi nations in this case are similar to the combina tions described in 2 above. The . transverse and longitudinal load values may be obtained by resolving the total load cell value into its transverse and its longitudinal components vjtilizing the transverse and longitudinal swing angles as described in the previous section. This enables finer details to be seen in calculating the cross correlation functions between load cells (total, transverse, and longitudinal) and strain gages. Since there are 160 possible cross correlation functions for all 4 selected records, only cross correlation functions of significant importance are presented. Most of the plots of the cross correlation 82 functions of the total load values from the conductors and the strain gaiges for the selected records which are not given in this section are given in Appendix C. Figures 4. 16a and 4.16b present the cross correlation functions relating WSOS with SGOS and WSOS with SG06, respectively, for record NOS. The cross correlation function relating WSOS with SGOS is negatively corre lated, dies off with increasing lag, and peaks at zero lag. The cross correlation function relating WSOS with SG06 for the same record does not peak at zero lag and the magnitudes of the cross correlation function are relatively smaller compared to the cross correlation function relating WSOS with SGOS for any time lag. This indicates that the wind speed has an immediate effect on the main members of the transmission tower. There is a time laig involved before the diaigonal members experience the effect of the wind speed since the cross correlation function relating WSOS with SG06 does not peak at zero laig. In the context presented herein, "time lag" refers to the time taken for the cross correlation functions to achieve an absolute maximum numerical value. Similar ly, the effect of the wind speed on the main member is more significant than the effect of the wind speed on the CBOSS CORSSIATION BBTWESN M09.WS0gMm M4.SC0S 83 SdMPUirajurs • t4 SPS

Figure 4. 16a Cross Correlation Functions of WSOS and SGOS for Record NOS

CROSS CORSBIATION BffTYEEN iR>e.gse6Afm Mog.soot aAMPUNoiun 3 f SPS

Figure 4.16b Cross Correlation Functions of WSOS and SG06 for Record NOS 83 CROSS CORRSUTION BKTWESN SAMFfJltaiUTg m to SPS

Figure 4.16a Cross Correlation Functions of WSOS and SGOS for Record NOS

CROSS CORRELATION BETWEEN ARM. WSOS AND MOO.SCOO aAMPUHaiLATB = to SPS

Figure 4. 16b Cr^^ss Correlation Functiuas uf WSOS and SG06 for Record NOS 84 diagonal member since the cross correlation function relating WSOS with SGOS has a larger maignitude. Figures 4. 17a and 4.17b present the cross correlation functions relating WSOS with SGOS and WSOS with SGll, respectively, for record NOS. The cross correlation function relating WSOS with SGOS is positively corre lated, and it does not die off quickly with increasing lag. On the contrary, the cross correlation function relating WSOS with SGll is negatively correlated. Fig ures 4.17a and 4.17b are similar to one another but in an opposite sense. The cross correlation functions for both cases have peak values at approximately zero laig. The cross correlation functions relating WSOS with SGOS and WSOS with SG12 for record NOS are shown respec tively in Figures 4.ISa and 4.18b. In both cases, the cross correlation functions are negatively correlated, and they are similar in shape. Also, the cross correla tion functions for Figures 4.18a and 4.ISb do not peak at zero lag. Figures C.la through C.4b in Appendix C give the plots of the cross correlation functions relating wind speed with strain gaiges for the selected records not shown in this section. Higher magnitudes of the cross correlation functions are observed relating wind speed with strain gages 85 CROSS CORRELATION BPTVEEN ifOO, WS09 Aim lt0t.SC06 SAMPUNC KATE • tO SPS

4.7

a.a I • a laa iia ua Ma JM SB aa

Figure 4.17a Cross Correlation Functions of WSOS and SGOS for Record NOS

CROSS COREtEUTION BETWEEN MM. WSOS AND itOa.SCI t SAMPfJNClUTg - to SPS

ILO

-aa

-i 0 I I } I I " I I M • • LSi la lao tia

Figure 4.17b Cross Correlation Functions of WSOS and SGll for Record NOS 86 CROSS CORRELATION BTTWESN »IM.WS08 AND JfOt.SCOt SAMMJNOIUTt = to SPS -aku

-a IB •*.»

-aa -«a ^a

A m tt M IB lii IM ail M

Figure 4.ISa Cross Correlation Functions of WSOS and SGOS for Record NOS

CROSS CORREUTION BETWEEN )foo.wsosANDint.seit aAldFUHClUTB » to SPS

Figure 4. 18b Cross Correlation Function.s of '-JoOo and SG12 for Record NOS 87 mounted on main leg members of the transmission tower than with strain gaiges mounted on diagonal leg members of the transmission tower for records NOS and NOS. On the contrary, higher magnitudes of the cross correlation functions are observed relating wind speed with strain gages mounted on diagonal leg members of the transmission tower than with strain gages mounted on main leg members for records NO? and N16. The cross correlation functions relating wind speed with strain gages mounted on main leg members of the transmission tower peak almost immediately for records NOS and NOS. However, the cross correlation functions relating wind speed with strain gages mounted on diaigonal leg members of the transmission tower do not peak imme diately but at a later time. Lag times of approximately 3.0 and 21.0 seconds are observed for record NOS and NOS, respectively. On the contrary, the cross correlation functions relating wind speed with strain gages mounted on both main and diagonal leg members of the transmission tower peak almost immediately for records NO? and N16. This means that the effect of wind speed is felt by both main and diagonal leg members immediately for records N07 and N16. If the cross correlation functions relating WSOS with SGOS are positively correlated, then the cross 88 correlation functions relating WSOS with SGll are nega tively correlated and vice versa. This relationship is expected since SGOS is located on the southeast corner of the transmission tower and SGll is located on the north west corner of the transmission tower (see Figure 4.5) and they are located diagonally opposite to one another. However, if the cross correlation functions relating WSOS with SGOS are positively correlated, then the cross cor relation functions relating WSOS with SGI2 are also posi tively correlated and vice versa. The next set of inputs to consider are the load cell values. The cross correlations relating load cells with strain gaiges may be performed for three cases. First, the cross correlations relating total load values with selected strain gaiges are performed. Second, the cross correlations relating longitudinal load values with se lected strain gaiges are performed by resolving the total load values into their longitudinal components utilizing the respective swing angles. Finally, the cross correla tions relating transverse load values with selected strain gaiges are performed by resolving the total load values into their transverse components utilizing the respective swing angles (see Equation 4.8). 89

The cross correlation functions relating load cells with strain gages may be obtained by first considering the total load values. The total load values are the load values obtained from the instrument without con sidering the effect of swing angles. The cross corre lation functions relating LC03 with SGOS and LCOS with SGOS are shown respectively in Figures 4.19a and 4. 19b for record NOS. The cross correlation function relating LCOS with SGOS indicate a definite periodic correlation. This periodic correlation tends to oscillate back and forth from negative correlation to positive correlation about the maignitude of zero correlation. Similarly, the cross correlation function relating LCOS with SGOS also indicates a definite periodic correlation. For both cases, the time taken from peak to peak of the cross correlation functions is about 25 laigs (2.5 seconds). Figures 4.20a and 4.20b present the cross correlation functions relating LC04 with SGOS and LC04 with SG06, respectively, for record NOS. Both cases indicate defi nite periodic correlation. Figures 4.21a and 4.21b present the cross correlation functions relating LCOS with SGOS and LCOS with SG06, respectively, for record NOS. Again, periodic correla tion occurs in both cases. Comparison is made between CROSS CORRBIATION BETWEEN RO^UOafTOtAL) Am ilO0.SG06 90 SAUPUHOHJOB - fO SPS

Figure 4.19a Cross Correlation Functions of Total Load Values From West Conductor and SGOS for Record NOS

CROSS CORRELATION BETWEEN MOgJJCOa(TtJtAL)AilD HOO-SOOO aAMPUWRATE m tO SPS

Figure 4.19b Cross Correlation Functions of Total Load Values From West Conductor and SG06 for Record NOS CROSS CORRELATION BETWEEN 91 JMiCX£0

Figure 4.20a Cross Correlation Functions of Total Load Values From East Conductor and SGOS for Record NOS

CROSS CORRELATION BETWEEN M00.£C04(T0tJiL) AJID MOO.SCOO SAMPintaiufB - to SPS

Fi.^ure 4. 20b Cross Correlation F'lnctions of To^.^.l Load Values From East Conductor ana SGC6 for Record NOS 92 CROSS CORRELATION BETWEEN H00.LCWS(TOtXL) AilD NOH.SGOS SAJUPLDfaiUTB » ^« SPS

-«L00

i ^la l-^iM

-aia

w^^^^^^^^r^ • a tt tt la liA III 311

Figure 4.21a Cross Correlation Functions of Total Load Values From Center Conductor and SGOS for Record NOS

CROSS CORRELATION BETWEEN g00.lC09(TOtAL) Aim MOO.SGOO SJUUPUJtaiUTB - *0 SPS

Figure 4.21b Cross Correlation Functions of TJ-.3I1 Load Values From Center Conductor and SG06 for Record NOS 93

Figures 4.19a, 4.20a, and 4.21a. No similarity in the shape of the cross correlation functions is noticed. Similar observation is obtained when comparison is made between Figures 4.19b, 4.20b, and 4.21b.

Discussion is now made pertaining to the cross corre lation functions relating total load values from the conductors with the strain gaiges. In Figures 4.19a through 4.21b, periodical cross correlation is expected because the conductor tends to oscillate at a certain frequency about its point of attachment when subjected to wind forces. Also, the cross correlation functions rela ting total load values from the conductors with the strain gages peaJk almost instantly. In general, the time taken from pealt of the cross correlation functions relating total load values from the conductors with strain gages is shorter for the strain gages mounted on the diagonal leg members than for those on the main leg members of the transmission tower. The cross correlation functions relating LCOS with SGOS and LCOS with SG06 for record NOS are shown respec tively in Figures 4.22a and 4.22b. In both cases, the time taken from peak to peak of the cross correlation functions is approximately 10 lags (1.0 second), and the cross correlation functions are periodic in nature. The 94 CROSS CORREUTION BETWEEN g0g.LC03fTOnL) AM) mO^SGOS SAidnorasAis - to SPS

Figure 4.22a Cross Correlation Functions of Total Load Values From West Conductor and SGOS for Record NOS

CROSS CORREUTION BETWEEN )Ut,LC0»(TOtiL) AMD MOO.SCOt SAidPinta JUTS - to SPS

Figure 4.22b Cross Correlation Functions o: To'il Load Values From West Conductor and SGOS for Record NOS 95 cross correlation function shown in Figure 4.22b oscil lates back and forth from positive correlation to nega tive correlation about zero value. Figures 4.23a and 4.23b present the cross correlation functions relating LCOS with SGll and LC03 with SG12, respectively, for record N08. In both cases, the cross correlation functions are periodic in nature. The cross correlation function shown in Figure 4.23a is mainly negatively correlated. Figures 4.24a and 4.24b present the cross correlation functions relating LC04 with SGOS and LC04 with SGOS, respectively, for record NOS. The cross correlation functions in both cases are periodic in nature. Figure 4.24a shows that the cross correlation function is mainly positively correlated. Also, Figure 4.24b indicates that the cross correlation function oscillates back and forth about zero value. The cross correlation functions relating LC04 with SGll and LC04 with SG12 for record NOS are shown res pectively in Figures 4.2Sa and 4.25b. The cross correla tion functions are periodic in nature in both cases. Similarities are noted when comparison is made for Fig ures 4-24b and 4.2Sb. 96 CROSS CORREUTION BETWEEN M0m.LC0a(TOTXL) AMD »O0.SGi 1 SAMPUHOKATB » tO SPS

Figure 4.23a Cross Correlation Functions of Total Load Values From West Conductor and SGll for Record NOS

CROSS CORREUTION BETWEEN )(0a.LC0a(TOtAL) AMD HOO.SGtZ SAUPiaaRATE m to SPS

'l 11 • ••I I" • na

Figure 4.23b Cross Correlation Functions of Total Load Values From West Conductor and SG12 for Record NOS 97 CROSS CORREUTION BETWEEN Ma.I£04fTOtLL) AMD MO0,SO06 SAidPUNaiUTE 'to SPS

-t.fl' I t I I I ta BO la IB aa

Figure 4.24a Cross Correlation Functions of Total Load Values From East Conductor and SGOS for Record NOS

CROSS CORREUTION BETWEEN MOa.LC04(TOTJLL) AMD MOO.SCOt SAUPLOfOJUTS 'to SPS

Figure 4 24b Cross Correlation Functions of To^al Load Values From East Conauoior and SGOS for Record NOS 98 CROSS CORRSUTION BETWEEN M0a.l£04^ntAL) AMD MOO.SGt 1 SJLiOfJMaiUTS • to SPS

ta la aa na

Figure 4.2Sa Cross Correlation Functions of Total Load Values From East Conductor and SGll for Record NOS

CROSS CORREUTION BETWEEN M0t.LC04(TOtiL) AMD nOO.SCtt SAMPLOfOlUTB - tO SPS

Figure 4.25b Cross Correlation Functions of Total Load Values From East Conductor and SG12 for Record NOS 99

Figures 4.26a and 4.26b present the cross correlation functions relating LCOS with SGOS and LC05 with SG06 respectively for record NOS. Again, the cross correla tion functions are periodic in nature in both cases. Comparison of Figures 4.22b, 4.24b, and 4.26b indicate that the cross correlation functions are similar to one another. Figures C.Sa through C.lib in Appendix C give the cross correlation functions of the total load values from the conductors and the strain gages for the selected records which are not shown in this section. The general discussions of the cross correlation functions of the total load values and the strain gaiges are now given for west wind records. The cross correlation functions relating total load values from the conductors with strain gaiges mounted on main leg members of the transmission tower peak almost immediately. The time taken from peak to peak of the cross correlation functions is consistently 10 lags or 1.0 second. Little variation is noticed in the shape of the cross correlation functions when the same strain gage is cross correlated with the total load values from different conductors. The cross correlation functions relating a particular total conductor load with the main strain gaiges are alwao's similar but in an opposite sense. CROSS CORREUTION BETWEEN 100 M0t.iJC0a(TOtlL) AMD MO0.SG06 aAMPUHOKATE - tO SPS

^ w *T !'• »*»T

Figure 4.26a Cross Correlation Functions of Total Load Values From Center Conductor and SGOS for Record NOS

CROSS CORREUTION BETWEEN MOt.Leoa(TOtlL) AMD MO0.SO00 aAUPLBtaiUTE 'to SPS

Figure 4.26b Cross Correlation Functions of To^al Load Values From Center Conductor and SG06 for Record NOS 101

Also, the values of the cross correlation functions rela ting total conductor loads with strain gages mounted on main leg members of the transmission tower are generally higher than the cross correlation functions relating total conductor loads with strain gaiges mounted on diago nal leg members. Finally, the explanation of the period ical nature of the cross correlation functions cannot be well accomplished without performing a frequency domain analysis. Any attempt to explain the periodic cycles will be addressed in the chapter dealing with frequency domain analysis. However, if the values of the cross correlation functions are positive, then the input is ahead of the response or output. On the other hand, if the values of the cross correlation functions are nega tive, the input is lagging behind the output. Upon obtaining the cross correlation relating total load cell values with strain gaiges, the swing angles of the load cells must be considered. The swing angles enable the load values to be resolved into the longitudi nal and transverse components. Next, the longitudinal load values are cross correlated with the strain gages and results obtained can be compared to the results obtained using only total load cell values. This gives 102 an added in-depth analysis into the behavior of the transmission tower system. Figures 4.27a and 4.27b present the cross correlation functions relating LC03 with SGll and LCOS with SG12 respectively for record NOS. In Figure 4.27a, the cross correlation function indicates a definite periodic corre lation. The time taken from peak to peak of the cross correlation function is approximately 20-25 lags (2.0-2.5 seconds). Although the cross correlation function is predominantly negatively correlated, small magnitudes of positive correlation occur. The cross correlation func tion continue to decrease periodically with increasing lags indicating that correlation eventually dies off with increasing lags. In Figure 4.27b, no definite period can be detected. The cross correlation function tend to oscillates about zero correlation and eventually dies off with increasing lag. The cross correlation functions relating LC04 with SGll and LC04 with SG12 for record NOS are shown respec tively in Figures 4.2Sa and 4.28b. In Figure 4.28a, the cross correlation function is negatively correlated, and it eventually dies off with increasing lags. No definite period can be detected. In Figure 4.2Sb, the cross corre lation function is negatively correlated. Also, no 103 CROSS CORREUTION BETWEEN Moesa>a(UiNG.)jLjmMoe.sGtt SAMPUNORATS = tO SPS

Figure 4.27a Cross Correlation Functions of Longitudinal Load Values From West Conductor and SGll for Record NOS

CROSS CORREUTION BETWEEN i»eaj£oafi4UfG.) AMB Moe.sGiz SAMPUNOIUTB - tO SPS

Figure 4.27b Cross Correlation Functions of Longitudinal Load Values From West Conductor and SG12 for Record NOS CROSS CORREUTION BETWEEN 104 M0aM»4(L0NG,) AMD M06.SG11 3AMPUK0RATE • tO SPS

i Aa

I^a a,a

'"I T»*'"! |inn f f faia^aya im fnp^y « fl M a la 10 m M M 9ft M

Figure 4.2Sa Cross Correlation Functions of Longitudinal Load Values From East Conductor and SGll for Record NOS

CROSS CORREUTION BETWEEN M0gM»4fL0Ma)AM» M0e.SG1t SAidKBiOlUTt 'to SPS

Figure 4.2Sb Cross Correlation Functions of Longitudinal Load Values From East Conductor ,ind SGl'C for Record NOS 105 definite period can be detected, and the cross corre lation function eventually dies off with increasing lag. In the case of west wind record NOS, Figures 4.29a and 4.29b present the cross correlation functions rela ting LC03 with SGOS and LCOS with SG12, respectively. The cross correlation functions shown in both figures osci llate about zero correlation and they eventually die off with increasing lag. Figures 4.30a and 4.30b present the cross correla tion functions relating LC04 with SGOS and LC04 with SGll respectively for record NOS. The cross correlation func tion shown in Figure 4.30a is positively correlated, and it eventually dies off with increasing lag. The cross correlation function shown in Figure 4.30b is negatively correlated, and it eventually dies off with increasing lag. Also, the cross correlation function shown in Figure 4.30a is similar, but in an opposite sense, to the cross correlation function shown in Figure 4.30b. Figures 4.31a and 4.31b present the cross correlation functions relating LCOS with SGOS and LCOS with SG12, respectively, for record NOS. In both figures, the cross correlation functions oscillate about zero correlation. Also, the cross correlation function shown in Figure 4.31a is similar to the cross correlation function shown 106 CROSS CORREUTION BETWEEN MOOMafSfLOIfC.) AMD M09.SCOW SAMPLING RATW « 10 SPS

Figure 4.29a Cross Correlation Functions of Longitudinal Load Values From West Conductor and SGOS for Record NOS

CROSS CORREUTION BETWEEN tfOOJJCOS^LONC J AND JVO«.SCtX SAMPLING RATg '10 SPS xa (La xa. a.n ».m xm X9 t,m ba a.a

•«,m

•*m

^^»^'^r^*'F"^^B^^^^^^^^^^^a^^^ '" •!

.,1 Figure 4.29b Cross Correlation Functions of Lotigi •A - Load Values From West Conductor and SG12 for Record NCS 107 CROSS CORRSUTION BETWEEN N90MX4(L0NG.} AND N00.SCOS SAMPtntG RATX ' 10 SPS

CM 1.17 xm %m cm cia

a.aa a.ai ^v^r*T^'^^^T'**T^'**T''^*'^.f^ ^FVX*.*v*VT*v^ Ma

Figure 4.30a Cross Correlation Functions of Longitudinal Load Values From East Conductor and SGOS for Record NOS

CROSS CORREUTION BETWEEN NVJJCOAiLONC) AND UOB^SCII SAMPtntG RATX '10 SPS

-«a

-ft 17

^f^w^^^ \ I rw^^mT^^-^^ I I I I'

Figure 4.30b Cross Correlation Functions of Longitudinal Load Values From East Conductor and SGll for Record NOS 108 CROSS CORREUTION BETWEEN NOOUJCOefLONG.) AND N00.SG09 SAMPLING RATg '10 SPS

Figure 4.31a Cross Correlation Functions of Longitudinal Load Values From Center Conductor and SGOS for Record NOS

CROSS CORREUTION BETWEEN NMLCCOOfUWC.; AND N00.SG12 SAMPLIMGRATg ' 10 SPS

ii 11 I • • I

Figure 4.31b Cross Correlation Functions of Longitudinal Load Values From Center Cond'^ctor and SGI2 for Record NOS 109 in Figure 4.31b. Finally, Figures C.12a through C.15b in Appendix C present the cross correlation functions rela ting longitudinal load values from the conductors with strain gages which are not shown in this section.

For all the selected records, the cross correlation functions indicate no definite period. Unless stated, all inferences are valid for the selected records. The cross correlation functions relating the longitudinal load values with the strain gages mounted on main leg members of the transmission tower die off with increasing lag. However, the cross correlation functions relating the longitudinal load values with the strain gages mounted on diagonal leg members of the transmission tower oscil late about zero correlation. For record N07, the cross correlation functions relating longitudinal load values with the strain gages mounted on diaigonal leg members aie similar to one another. Finally, the cross correlation functions relating transverse load values with strain gages are performed to complete the time domain input- output analysis. Figures 4.32a and 4.32b present the cross correlation functions relating LCOS with SGOS and LC03 with SGll, respectively, for record NOS. In both figures, the cross correlation functions are periodic in nature and they 110 CROSS CORRSUTION BETWEEN M06.LCOa(TRAMaT.)JND M06.SGOa SAMPUNCIUTS ' to SPS

Figure 4.32a Cross Correlation Functions of Transverse Load Values From West Conductor and SGOS for Record NOS

CROSS CORREUTION BETWEEN MOt.Leoa(TRAM3r.)JND X0e.SGI1 SAMPLOraiUTS ' to SPS

Figure 4.32b Cross Correlation Functions of 'r 11. s verse Load Values From West Conductor and SGll for Record NOS Ill eventually die off. The cross correlation function rela ting LC03 with SGOS shown in Figure 4.32a is mainly positively correlated. However, the cross correlation function relating LC03 with SGll shown in Figure 4.32b is mainly negatively correlated. Also, in both figures, the cross correlation functions are similar, but in an opposite sense, to one another.

The cross correlation functions relating LC04 with SGOS and LC04 with SGll for record NOS are shown respec tively in Figures 4.33a and 4.33b. The cross correlation function relating LC04 with SGOS shown in Figure 4.33a is positively correlated. However, the cross correlation function relating LC04 with SGll shown in Figure 4.33b is negatively correlated. The cross correlation function shown in Figure 4.33a is similar, but in an opposite sense, to the cross correlation function shown in Figure 4.33b. Figures 4.34a and 4.34b present the cross correlation functions relating LCOS with SGOS and LCOS with SG12, respectively, for record NOS. The cross correlation functions shown in Figures 4.34a and 4.34b are not simi lar. Also, the magnitudes of the cross correlation func tions shown in Figures 4.34a and 4.34b are relatively 112 CROSS CORREUTION BETWEEN X04.LC04(TRAMar.)AND Jf0(LSCO6 SAMPLBtaRAIS ' tO SPS

Figure 4.33a Cross Correlation Functions of Transverse Load Values From East Conductor and SGOS for Record NOS

CROSS CORREUTION BETWEEN M06.LeoMrRAirsr.)AND MOO^SGI 1 SIAJUPLWaJUTB ' to SPS

-ftOS

I:: i Am

I-fta J.»17 ^a

• •I I la II

Figure 4.33b Cross Correlation Functions of Transverse Load Values From East Conductor and SGll for Record NOS 113 CROSS CORRSUTION BETWEEN Jf06.LC0a(TRAMSr. )JND M06.SCOO SAitPLBiOIUTE - tO SPS

0.0

C4B

a.a

•*.m

-B.ai

a a M a <•

~~Figure 4.34a Cross Correlation Functions of Transverse Load Values From Center Conductor and SGOS for Record NOS~~

~~CROSS CORREUTION BETWEEN JfOALeoa(TRAMSr.)AND M06.SCI2 SAMPLBtaRATE ' tO SPS~~

~~-au~~

~~-«a~~

~~y^^^^^^^^^^^^^^^^^^^^ i»^yT»aa>y»aaii| |iiii 19 la la aa~~

Figure 4.34b Cross Correlation Functions of Transverse^ Load Values From Center Conductor and SGl^ for Record NOS 114 small in comparison with the maignitudes of the cross correlation functions shown in Figures 4.33a and 4.33b. Figures 4.3Sa and 4.35b present the cross correlation functions relating LCOS with SGOS and LCOS with SGll, respectively, for record NOS. In both figures, the mag nitudes of the cross correlation functions are high. Also, the cross correlation functions die off with increasing laig in both figures. The cross correlation function relating LC03 with SGOS shown in Figure 4.3Sa is positively correlated. However, the cross correlation function relating LC03 with SGll shown in Figure 4.3Sb is negatively correlated. The cross correlation function shown in Figure 4.35a is similar, but in an opposite sense, to the cross correlation function shown in Figure 4.35b. The cross correlation functions relating LC04 with SGOS and LCOS with SGOS for record NOS are shown in Figures 4.36a and 4.36b, respectively. In both figures, the maignitudes of the cross correlation functions are high and die off with increasing lag. The cross correla tion function relating LC04 with SGOS shown in Figure 4.36a is positively correlated. Similarly, the cross correlation function relating LCOS with SGOS shown in Figure 4.36b is also positively correlated. Th^ cross CROSS CORRSUTION BErWEBN lis Rotj^oacnuNs.) AND Noa.sGoa SAUPLOfajtATE ' to SPS (La 4

~~0.7~~

~~I ^'^^'^^T^^^^^T^^'^'^T^^^^^T^'^*^^ I I i f a ua iM la aia~~

~~Figure 4.SSa Cross Correlation Functions of Transverse Load Values From West Conductor and SGOS for Record NOS~~

~~CROSS CORREUTION BETWEEN NOgJjCOafTRANS.) AMD N00.SG11 BAMPUHCRATE ' tO SPS~~

~~11'~~

~~e I~~

~~\- a a 17~~

-1.0' ^««v>»f^w^^^^^wvyv^ivvv>f« rwj^mwwmjnv aaa y T*

~~Figure 4.3Sb Cross Correlation Functions of Transverse Load Values From West Conductor and SGll for Record NOS CROSS CORRSUTION BETWEEN 116 mgJJC04CFRANS.) AMD N0O.SCO6 SAMPUMO RATS ' tO SPS~~

~~Figure 4.36a Cross Correlation Functions of Transverse Load Values From East Conductor and SGOS for Record NOS~~

~~CROSS CORREUTION BETWEEN N0gJ£O0(TRANS.) AMD N0a.SGOe SAiOUMa RATE ' tO SPS~~

~~0.9~~

~~T^^^^^l.i • i i ^^^^^wf^^^'r^T^'w^-^^r^ IIIII I ) I I .i i 11 I I I I a a BiaaiMi^iMMSi~~

Figure 4.36b Cross Correlation Functions of Trrnsverse Load Values From Center Conductor and -fC^OS for Record NOS 117 correlation functions shown in Figures 4,SSa, 4.36a, and 4.36b are similar to one another. Figures 4.37a and 4.37b present the cross correlation functions relating LCOS with SGOS and LCOS with SG12 for record NOS, respectively. In both figures, the cross correlation functions are periodic in nature, and they are similar to one another. Also, the magnitudes of the cross correlation functions shown in Figures 4.37a and 4.37b are relatively small in comparison with the magni tudes of the cross correlation functions shown in Figures 4.35a through 4.36b. The general discussions regarding the cross correlation functions relating transverse load values and strains measured from gages are now given. Unless stated, inferences made are valid for all of the selected records. The maignitudes of the cross correlation functions of transverse load values and strains measured using gages mounted on main leg members of the transmission tower are higher than the magnitudes of the cross correlation func tions relating transverse load values and strains meas ured using gaiges mounted on diagonal leg members of the transmission tower except for record N07. Record NO? has •significantly higher magnitudes of cross correlation functions relating transverse load values and strains CROSS CORRSUTION BETWEEN lis N0gJ£06(7RANS.) AMD NOa,SGOO aAMPUNORATB ' tO SPS

~~-OCB .a.~~

~~-«.a -ftU -fta e AIM |-o»~~

~~|-ai7 j-oa~~

~~a a -a,~~

~~-aa~~

~~1 11 a a SB~~

~~Figure 4.37a Cross Correlation Functions of Transverse Load Values From Center Conductor and SGOS for Record NOS~~

~~CROSS CORREUTION BETWEEN MOgJ£0«(TRANS.)AMD N0O.SG12 aAltPlBfa RATE ' to SPS~~

Figure 4.37b Cross Correlation Functions of Transverse Load Values From Center Conductor and SG12 for Record NOS 119 measured using gages mounted on diagonal leg members of the transmission tower when compared with other records. Also, the cross correlation functions relating transverse load values with strains measured using gaiges mounted on main leg members are similar, but in an opposite sense, to one another for the same record. No similarity or trend is observed for the cross correlation functions relating transverse load values and strains measured using gaiges mounted on diagonal leg members for the same record.

The time taken from peak to peaLk of the cross corre lation functions relating transverse load values with strains measured for west wind records is approximately 10 lags (1.0 second). On the contrary, no definite lag is noticed for east wind record. Next, discussion of results obtained in this section is given in the follow ing section.

4^8 Discussion of Results In this section, the results obtained from sections 4.5 and 4.6 are discussed. Unless stated, the discussions given in this section are valid for all the selected records. The discussions based on the results of the autocorrelation functions presented in section 4.5 are considered first. 120

WSOS with SG06 and WSOS with SG12 are larger than the magnitudes of the cross correlation functions relating WSOS with SGOS and WSOS with SGll. The former indicates that wind speed is more significant on the response of main leg members of the transmission tower for records NOS, NOS, and N16. The latter indicates that wind speed is also significant in the response of diagonal leg members of the transmission tower for record N07.