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Title page
Contents
FOREWORD 2
EXECUTIVE SUMMARY 15
CHAPTER 1. INTRODUCTION 18
BACKGROUND 18
PROJECT OBJECTIVES AND TASKS 20
CHAPTER 2. COMPILATION OF EXISTING INFORMATION 21
REFERENCE MATERIALS 21
INVENTORY OF U.S. CABLE-STAYED BRIDGES 21
CHAPTER 3. ANALYSIS, EVALUATION, AND TESTING 22
MECHANICS OF WIND-INDUCED VIBRATIONS 22
Reynolds Number 22
Strouhal Number 22
Scruton Number 23
Vortex Excitation of an Isolated Cable and Groups of Cables 23
Rain/Wind-Induced Vibrations 24
Wake Galloping for Groups of Cables 25
Galloping of Dry Inclined Cables 26
WIND TUNNEL TESTING OF DRY INCLINED CABLES 27
Introduction 27
Testing 28
Results Summary 29
OTHER EXCITATION MECHANISMS 31
Effects Due to Live Load 31
Deck-Stay Interaction Because of Wind 32
STUDY OF MITIGATION METHODSThe development of recommended design 34
Linear and Nonlinear Dampers 34
Linear Dampers 35
Nonlinear Dampers 36
Field Performance of Dampers 37
Crosstie Systems 39
Analysis 41
Field Performance 44
Considerations for Crosstie Systems 46
Cable Surface Treatment 47
FIELD MEASUREMENTS OF STAY CABLE DAMPING 48
Leonard P. Zakim Bunker Hill Bridge (over Charles River in Boston, MA) 48
Sunshine Skyway Bridge (St. Petersburg, FL) 51
BRIDGE USER TOLERANCE LIMITS ON STAY CABLE VIBRATION 53
CHAPTER 4. DESIGN GUIDELINES 56
NEW CABLE-STAYED BRIDGES 56
General 56
Mitigation of Rain/Wind Mechanism 56
Additional Mitigation 56
Minimum Scruton Number 56
External Dampers 57
Cable Crossties 57
User Tolerance Limits 58
RETROFIT OF EXISTING BRIDGES 58
WORKED EXAMPLES 59
Example 1 59
Example 2 63
CHAPTER 5. RECOMMENDATIONS FOR FUTURE RESEARCH AND DEVELOPMENT 66
WIND TUNNEL TESTING OF DRY INCLINED CABLES 66
DECK-INDUCED VIBRATION OF STAY CABLES 66
MECHANICS OF RAIN/WIND-INDUCED VIBRATIONS 66
DEVELOP A MECHANICS-BASED MODEL FOR STAY CABLE VIBRATION ENABLING THE PREDICTION OF ANTICIPATED VIBRATION CHARACTERISTICS 67
PREDICT THE PERFORMANCE OF STAY CABLES AFTER MITIGATION USING THE MODEL 68
PERFORM A DETAILED QUANTITATIVE ASSESSMENT OF VARIOUS ALTERNATIVE MITIGATION STRATEGIES 69
IMPROVE UNDERSTANDING OF INHERENT DAMPING IN STAYS AND THAT PROVIDED BY EXTERNAL DEVICES 69
IMPROVE UNDERSTANDING OF CROSSTIE SOLUTIONS 70
REFINE RECOMMENDATIONS FOR EFFECTIVE AND ECONOMICAL DESIGN OF STAY CABLE VIBRATION MITIGATION STRATEGIES FOR FUTURE BRIDGES 70
APPENDIX A. DATABASE OF REFERENCE MATERIALS 72
APPENDIX B. INVENTORY OF U.S. CABLE-STAYED BRIDGES 91
APPENDIX C. WIND-INDUCED CABLE VIBRATIONS 96
APPENDIX D. WIND TUNNEL TESTING OF STAY CABLES 110
APPENDIX E. LIST OF TECHNICAL PAPERS 161
APPENDIX F. ANALYTICAL AND FIELD INVESTIGATIONS 163
APPENDIX G. INTRODUCTION TO MECHANICS OF INCLINED CABLES 220
APPENDIX H. LIVE-LOAD VIBRATION SUBSTUDY 231
APPENDIX I. STUDY OF USER COMFORT 263
REFERENCES AND OTHER SOURCES 266
Figure 1. Graph. Comparison of wind velocity-damping relation of inclined dry cable 30
Figure 2. Graph. Cable M26, tension versus time (transit train speed = 80 km/h (50 mi/h)) 31
Figure 3. Graph. Time history and power spectral density (PSD) of the first 2 Hz for deck at midspan (vertical direction) 32
Figure 4. Graph. Time history and power spectral density (PSD) of the first 2 Hz for cable at AS24 (in-plane direction) deck level wind speed 33
Figure 5. Deck level wind speed 33
Figure 6. Photo. Damper at cable anchorage 34
Figure 7. Drawing. Taut cable with linear damper 35
Figure 8. Graph. Normalized damping ratio versus normalized damper coefficient: Linear damper 36
Figure 9. Graph. Normalized damping ratio versus normalized damper coefficient (β = 0.5) 37
Figure 10. Photo. Fred Hartman Bridge 38
Figure 11. Photo. Cable crosstie system 40
Figure 12. Photo. Dames Point Bridge 41
Figure 13. Chart. General problem formulation 42
Figure 14. Chart. General problem formulation (original configuration) 42
Figure 15. Graph. Eigenfunctions of the network equivalent to Fred Hartman Bridge: Mode 1 43
Figure 16. Graph. Eigenfunctions of the network equivalent to Fred Hartman Bridge: Mode 5 43
Figure 17. Graph. Comparative analysis of network vibration characteristics and individual cable behavior: Fred Hartman Bridge 44
Figure 18. Chart. Fred Hartman Bridge, field performance testing arrangement 45
Figure 19. Drawing. Types of cable surface treatments 47
Figure 20. Graph. Example of test data for spiral bead cable surface treatment 48
Figure 21. Photo. Leonard P. Zakim Bunker Hill Bridge 48
Figure 22. Graph. Sample decay: No damping and no crossties 50
Figure 23. Graph. Sample decay: With damping and no crossties 50
Figure 24. Graph. Sample decay: With damping and crossties 51
Figure 25. Photo. Sunshine Skyway Bridge 51
Figure 26. Photo. Stay and damper brace configuration 52
Figure 27. Photo. Reference database search page 72
Figure 28. Photo. Reference database search results page 73
Figure 29. Photo. U.S. cable-stayed bridge database: Switchboard 92
Figure 30. Photo. U.S. cable-stayed bridge database: General bridge information 93
Figure 31. Photo. U.S. cable-stayed bridge database: Cable data 94
Figure 32. Photo. U.S. cable-stayed bridge database: Wind data 95
Figure 33. Graph. Galloping of inclined cables 101
Figure 34. Drawing. Aerodynamic devices 103
Figure 35. Drawing. Cable crossties 107
Figure 36. Drawing. Viscous damping 107
Figure 37. Drawing. Material damping 108
Figure 38. Drawing. Angle relationships between stay cables and natural wind (after Irwin et al.) 112
Figure 39. Photo. Cable supporting rig: Top 114
Figure 40. Photo. Cable supporting rig: Bottom 114
Figure 41. Drawing. Longitudinal section of the propulsion wind tunnel 116
Figure 42. Drawing. Cross section of the working section of propulsion wind tunnel 117
Figure 43. Photo. Data acquisition system 118
Figure 44. Photo. Airpot damper 120
Figure 45. Drawing. Cross section of airpot damper 121
Figure 46. Photo. Elastic bands on the spring coils 122
Figure 47. Drawing. Side view of setups 1B and 1C 124
Figure 48. Drawing. Side view of setups 2A and 2C 125
Figure 49. Drawing. Side view of setups 3A and 3C 126
Figure 50. Photo. Cable setup in wind tunnel for testing 127
Figure 51. Graph. Amplitude-dependent damping (A, sway; B, vertical) with setup 2C (smooth surface, low damping) 134
Figure 52. Graph. Divergent response of inclined dry cable (setup 2C; smooth surface, low damping) 135
Figure 53. Graph. Lower end X-motion, time history of setup 2C at U = 32 m/s (105 ft/s) 135
Figure 54. Graph. Top end X-motion, time history of setup 2C at U = 32 m/s (105 ft/s) 136
Figure 55. Graph. Lower end Y-motion, time history of setup 2C at U = 32 m/s (105 ft/s) 136
Figure 56. Graph. Top end Y-motion, time history of setup 2C at U = 32 m/s (105 ft/s) 137
Figure 57. Graph. Trajectory of setup 2C at U = 32 m/s (105 ft/s) 137
Figure 58. Graph. Lower end X-motion, time history of setup 2A at U = 18 m/s (59 ft/s) in the first 5 minutes 138
Figure 59. Graph. Top end X-motion, time history of setup 2A at U = 18 m/s (59 ft/s) in the first 5 minutes 138
Figure 60. Graph. Lower end Y-motion, time history of setup 2A at U = 18 m/s (59 ft/s) in the first 5 minutes 139
Figure 61. Graph. Top end Y-motion, time history of setup 2A at U = 18 m/s (59 ft/s) in the first 5 minutes 139
Figure 62. Graph. Lower end X-motion, time history of setup 2A at U = 18 m/s (59 ft/s) in second 5 minutes 140
Figure 63. Graph. Top end X-motion, time history of setup 2A at U = 18 m/s (59 ft/s) in second 5 minutes 140
Figure 64. Graph. Lower end Y-motion, time history of setup 2A at U = 18 m/s (59 ft/s) in second 5 minutes 141
Figure 65. Graph. Top end Y-motion, time history of setup 2A at U = 18 m/s (59 ft/s) in second 5 minutes 141
Figure 66. Graph. Lower end X-motion, time history of setup 2A at U = 19 m/s (62 ft/s) 142
Figure 67. Graph. Top end X-motion, time history of setup 2A at U = 19 m/s (62 ft/s) 142
Figure 68. Graph. Lower end Y-motion, time history of setup 2A at U = 19 m/s (62 ft/s) 143
Figure 69. Graph. Top end Y-motion, time history of setup 2A at U = 19 m/s (62 ft/s) 143
Figure 70. Graph. Lower end X-motion, time history of setup 1B at U = 24 m/s (79 ft/s) 144
Figure 71. Graph. Top end X-motion, time history of setup 1B at U = 24 m/s (79 ft/s) 144
Figure 72. Graph. Lower end Y-motion, time history of setup 1B at U = 24 m/s (79 ft/s) 145
Figure 73. Graph. Top end Y-motion, time history of setup 1B at U = 24 m/s (79 ft/s) 145
Figure 74. Graphic. Lower end X-motion, time history of setup 1C at U = 36 m/s (118 ft/s) 146
Figure 75. Graph. Top end X-motion, time history of setup 1C at U = 36 m/s (118 ft/s) 146
Figure 76. Graph. Lower end Y-motion, time history of setup 1C at U = 36 m/s (118 ft/s) 147
Figure 77. Graph. Top end Y-motion, time history of setup 1C at U = 36 m/s (118 ft/s) 147
Figure 78. Graph. Lower end X-motion, time history of setup 3A at U = 22 m/s (72 ft/s) 148
Figure 79. Graph. Top end X-motion, time history of setup 3A at U = 22 m/s (72 ft/s) 148
Figure 80. Graph. Lower end Y-motion, time history of setup 3A at U = 22 m/s (72 ft/s) 149
Figure 81. Graph. Top end Y-motion, time history of setup 3A at U = 22 m/s (72 ft/s) 149
Figure 82. Graph. Trajectory of setup 2A at U = 18 m/s (59 ft/s), first 5 minutes 150
Figure 83. Graph. Trajectory of setup 2A at U = 18 m/s (59 ft/s), second 5 minutes 150
Figure 84. Graphic. Trajectory of setup 2A at U = 19 m/s (62 ft/s) 151
Figure 85. Graphic. Trajectory of setup 1B at U = 24 m/s (79 ft/s) 151
Figure 86. Graphic. Trajectory of setup 1C at U = 36 m/s (119 ft/s) 152
Figure 87. Graph. Trajectory of setup 3A at U = 22 m/s (72 ft/s) 152
Figure 88. Graph. Wind-induced response of inclined dry cable (setup 2A; smooth surface, low damping) 153
Figure 89. Graph. Wind-induced response of inclined dry cable (setup 1B; smooth surface, low damping) 153
Figure 90. Graph. Wind-induced response of inclined dry cable (setup 1C; smooth surface, low damping) 154
Figure 91. Graph. Wind-induced response of inclined dry cable (setup 3A; smooth surface, low damping) 154
Figure 92. Graph. Wind-induced response of inclined dry cable (setup 3B; smooth surface, low damp 155
Figure 93. Graph. Critical Reynolds number of circular cylinder (from Scruton) 155
Figure 94. Graph. Damping trace of four different levels of damping (setup 1B; smooth surface) 156
Figure 95. Graph. Effect of structural damping on the wind response of inclined cable (setup 1B; smooth surface) 156
Figure 96. Graph. Surface roughness effect on wind-induced response of dry inclined cable (setup 3A; low damping) 157
Figure 97. Graph. Surface roughness effect on wind-induced response of dry inclined cable (setup 1B; low damping) 157
Figure 98. Graph. Surface roughness effect on wind-induced response of dry inclined cable (setup 2A; low damping) 158
Figure 99. Graph. Amplitude-dependent damping in the X-direction with setup 2A (frequency ratio effect) 158
Figure 100. Graph. Amplitude-dependent damping in the Y-direction with setup 2A 159
Figure 101. Graph. Wind-induced response of inclined cable in the X-direction with setup 2A (frequency ratio effect) 159
Figure 102. Graph. Wind-induced response of inclined cable in the Y-direction with setup 2A (frequency ratio effect) 160
Figure 103. Graph. Comparison of wind velocity-damping relation of inclined dry cable 160
Figure 104. Chart. Taut cable with a linear damper 165
Figure 105. Graph. Normalized damping ratio versus normalized damper coefficient 167
Figure 106. Chart. Cable with attached friction/viscous damper 169
Figure 107. Chart. Force-velocity curve for friction/viscous damper 169
Figure 108. Graph. Normalized damping ratio versus clamping ratio 171
Figure 109. Graph. Normalized viscous damper coefficient versus clamping ratio 171
Figure 110. Graph. Relationship between nondimensional parameters μ and κ with different values of the clamping ratio Θci for a friction/viscous damper 173
Figure 111. Graphic. Normalized damping ratio versus κ with varying μ 174
Figure 112. Graph. Normalized damping ratio versus normalized damper coefficient (β = 0.5) 176
Figure 113. Graph. Normalized damping ratio versus mode ratio (β = 1) 178
Figure 114. Graph. Normalized damping ratio versus amplitude ratio (β = 0.5) 178
Figure 115. Graph. Normalized damping ratio versus mode-amplitude ratio (β = 0) 178
Figure 116. Chart. General problem formulation 181
Figure 117. Chart. General problem formulation (original configuration) 184
Figure 118. Graph. Eigenfunctions of the network equivalent to Fred Hartman Bridge (1st-8th modes) 186
Figure 119. Graph. Comparative analysis of network vibration characteristics and individual cable behavior (Fred Hartman Bridge; NET_3C, original configuration; NET_3RC, infinitely rigid restrainers; NET_3CG, spring connectors extended to ground (restrainers 2,3)) 187
Figure 120. Chart. Generalized cable network configuration 190
Figure 121. Chart. Twin cable with variable position connector 191
Figure 122. Graph. Twin cable system, with connector location ξ = 0.35, example of frequency solution for linear spring model 193
Figure 123. Graph. Typical solution curves of the complex frequency for the dashpot 193
Figure 124. Chart. Intermediate segments of specific cables only 193
Figure 125. Chart. Fred Hartman Bridge (A-line) 3D network 194
Figure 126. Chart. Equivalent model 194
Figure 127. Graph. Frequency solutions (1st mode) for the damped cable network (A-line) 196
Figure 128. Graph. Complex modal form (1st mode) for the optimized system M1(uo) 196
Figure 129. Graphic. Damping versus mode number for Hartman stays A16 and A23 198
Figure 130. Graph. Stay vibration and damper force characteristics; stay A16 201
Figure 131. Graph. Stay vibration and damper force characteristics; stay A23 202
Figure 132. Chart. In-plane versus lateral RMS displacement for (A) AS16 and (B) AS23 206
Figure 133. Chart. Sample Lissajous plots of displacement for two records from AS16 207
Figure 134. Chart. Power spectral density of displacement of two records from AS16 208
Figure 135. Graph. Sample Lissajous plots of displacement for two records from AS23 209
Figure 136. Graph. Power spectral density of displacement of two records from AS23 209
Figure 137. Graph. In-plane versus lateral RMS displacement for (A) AS16 and (B) AS23 after damper installation 210
Figure 138. Graph. Lissajous and power spectral density plots of displacement for record A 211
Figure 139. Graph. Modal frequencies of stays (A) AS16 and (B) AS23 212
Figure 140. Graph. Second-mode frequency versus RMS displacement for stay AS16 213
Figure 141. Graph. Estimated modal damping of stay AS16 showing effect of damper 214
Figure 142. Graphic. Histogram of estimated damping for (A) mode 2 of AS16 and (B) mode 3 of AS23 214
Figure 143. Graphic. Dependence of modal damping on damper force 215
Figure 144. Graph. RMS damper force versus RMS displacements for (A) AS16 and (B) AS23 216
Figure 145. Chart. Damper force versus displacement and velocity for a segment of a sample record 217
Figure 146. Chart. Displacement and damper force time histories of a sample record 218
Figure 147. Drawing. Incline stay cable properties 220
Figure 148. Drawing. Definition diagram for a horizontal cable (taut string), compared to the definition diagram for an inclined cable 225
Figure 149. Graph. Cable √T/m versus cable unstressed length: Summary of Alex Fraser, Maysville, and Owensboro bridges 229
Figure 150. Graph. Cable frequency versus cable unstressed length: Summary of Alex Fraser, Maysville, and Owensboro bridges 230
Figure 151. Photo. RAMA 8 Bridge (artistic rendering) 231
Figure 152. Drawing. RAMA 8 Bridge computer model: XY, YZ, and ZX views 232
Figure 153. Chart. Independent cable M26 discretization 10-segment model: XZ view 234
Figure 154. Chart. Cable catenary 235
Figure 155. Chart. Cable modes: XZ, YZ, and XY views (as defined in figure 152) 236
Figure 156. Chart. Inextensible cable mode 1, in-plane: XY, YX, and XZ views 238
Figure 157. Drawing. Cable M26 discretization: 10-segment model, isometric view. Only cables M26 are shown. Other cables not shown for clarity 239
Figure 158. Drawing. Cable M26 discretization: 10-segment model, XZ view. Other cables not shown for clarity 239
Figure 159. Chart. Fundamental bridge modes 241
Figure 160. Chart. Additional bridge modes 242
Figure 161. Chart. Four first modes of the cables; XY, YZ, and XZ views 243
Figure 162. Chart. Four second modes of the cables; XY, YZ, and XZ views 243
Figure 163. Chart. Four third modes of the cables; XY, YZ, and XZ views 244
Figure 164. Chart. Nodes, members, and cables for comparison of results 245
Figure 165. Graph. RAMA 8 Bridge model damping versus frequency 250
Figure 166. Graph. Vertical displacements, velocities, and accelerations of node 427 versus time (train speed = 80 km/h (50 mi/h) 251
Figure 167. Graph. Member 1211: Bending moment versus time (train speed = 80 km/h (50 mi/h)) 252
Figure 168. Graph. Cable M26: Tension versus time (train speed = 80 km/h (50 mi/h)) 252
Figure 169. Graph. Difference in cable tension for cable M26 between the dynamic train load case and static train load case versus time (train speed = 80 km/h (50 mi/h)) 253
Figure 170. Graph. Cable M26 tension spectra (train speed = 80 km/h (50 mi/h)) 254
Figure 171. Graph. Global coordinate displacements (A, B, C) of cable M26 nodes (mm) versus time (train speed = 80 km/h (50 mi/h)) 256
Figure 172. Chart. Transformation from global coordinates to coordinates along the cable 257
Figure 173. Chart. Local coordinate displacements of nodes of cable M26 (mm). Displacements are shown for three nodes of the cable: At 1/4 span (closer to the tower), 1/2 span, and 3/4 span (closer to the deck; train speed = 80 km/h (50 mi/h) 258
Figure 174. Graph. Spectra for movements of cable M26 nodes: At 1/4 span (closer to the tower), 1/2 span, and 3/4 span (closer to the deck; frequency range = 0-2 Hz; train speed = 80 km/h (50 mi/h)) 259
Figure 175. Graph. Deck rotations and cable end rotations for cable M26: Dynamic (train speed = 80 km/h (50 mi/h)) and static 261
Figure 176. Graph. Deck rotations and cable end rotations for cable M21: dynamic (train speed = 80 km/h (50 mi/h)) and static 261
Figure 177. Graph. Effect of mode (constant amplitude and velocity) 264
Figure 178. Graph. Effect of velocity (constant amplitude) 264
Figure 179. Graph. Effect of amplitude (constant velocity) 265
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