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PAGE 1 1 BRIDGE GIRDER DRAG COEFFICIENTS AND WIND RELATED BRACING RECOMMENDATIONS By ZACHARY HARPER A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE O F MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2013 PAGE 2 2 2013 Zachary Harper PAGE 3 3 ACKNOWLEDGMENTS I would not have been able to complete this thesis and the associated research without the continual guidance and support of my advisor, Dr. Gary Consolazio With h is deep engineering knowledge, attention to detail, and commitment to excellence he has played a far larger part than any other individual in making me the engineer and researcher I am today Dr. Kurt Gurley wind engineering exp ertise has been invaluable to this research I would also like to thank Dr. H.R. (T rey ) Hamilton and Dr. Ron Cook for serving on my supervisory committee. In addition to faculty, I would like to acknowledge the support, advice, and f riendship of the fellow engineering graduate students with whom I have shared an office including Dr. Michael Davidson, Daniel Getter, Me gan Beery, Natassia Brenkus, John Wilkes and Sam Edwards The past few years would have been much more difficult (and much less fun) without them. In particular, I would like to highlight the contribution of Danie l Getter to my personal and professional development. Daniel is much too generous with his time, and has always been willing to offer his vast technical knowled ge and sage advice Additionally, I would like to thank Megan Beery for her willingness to discuss the details of her research ( upon which my own is partly b ased ) and Sam Edwards for his assistance with the preparation of this manuscript. Finally, and mo st importantly, I want to thank my parents, Bill and Patricia Harper. None of what I have accomplished would have been possible without the ir 26 years of unwavering patience, love and support PAGE 4 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ ............... 3 LIST OF TABLES ................................ ................................ ................................ ........................... 8 LIST OF FIGURES ................................ ................................ ................................ ......................... 9 ABSTRACT ................................ ................................ ................................ ................................ ... 16 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .................. 18 1.1 Introduction ................................ ................................ ................................ ................... 18 1.2 Objectives ................................ ................................ ................................ ...................... 19 1.3 Scope of Work ................................ ................................ ................................ ............... 19 2 PHYSICAL DESCRIPTION OF BRIDGES DURING CONSTRUCTION ......................... 22 2.1 Introduction ................................ ................................ ................................ ................... 22 2.2 Geometric Parameters ................................ ................................ ................................ ... 22 2.3 Bearing Pads ................................ ................................ ................................ ................. 23 2.4 Sources of Lateral Instability ................................ ................................ ........................ 23 2.5 Lateral Wind Loads ................................ ................................ ................................ ....... 25 2.6 Temporary Bracing ................................ ................................ ................................ ....... 25 2.6.1 Anchor Bracing ................................ ................................ ................................ 25 2.6.2 Girder to Girder Bracing ................................ ................................ .................. 26 3 BACKGROUND ON DRAG COEFFICIENTS ................................ ................................ .... 34 3.1 Introduction ................................ ................................ ................................ ................... 34 3.2 Dimensionless Aerodynamic Coefficients ................................ ................................ .... 34 3.3 Terminology Related to Aerodynamic Coefficients ................................ ..................... 37 3.4 Current Wind Design Practice in Florida ................................ ................................ ...... 38 3.5 Literature Review: Drag Coefficients for Bridge Girders ................................ ............. 41 4 WIND TUNNEL TESTING ................................ ................................ ................................ ... 49 4.1 Introduction ................................ ................................ ................................ ................... 49 4.2 Testing Configurations ................................ ................................ ................................ .. 49 4.2.1 Number of Girders ................................ ................................ ............................ 50 4.2.2 Spacing ................................ ................................ ................................ .............. 50 4.2.3 Cross Slope ................................ ................................ ................................ ....... 51 4.2.4 Wind Angle ................................ ................................ ................................ ....... 51 4.3 Testing Procedure ................................ ................................ ................................ .......... 52 PAGE 5 5 5 WIND TUNNEL RESULTS AND ANALYSIS ................................ ................................ .... 57 5.1 Introduction ................................ ................................ ................................ ................... 57 5.2 Aerodynamic Coefficients for Individual Girders ................................ ........................ 57 5.3 Examination of Shielding Trends ................................ ................................ ................. 59 5.4 Effective Drag Coefficient ................................ ................................ ............................ 61 5.5 Proposed Wind Loads for Design ................................ ................................ ................. 64 5.6 Proposed Procedure for Calculation of Brace Forces ................................ ................... 66 6 BEARING PADS ................................ ................................ ................................ ................... 82 6.1 Intr oduction ................................ ................................ ................................ ................... 82 6.2 Behavior of Pads in Compression ................................ ................................ ................. 83 6.3 Behavior of Pads in Roll Rotation ................................ ................................ ................ 85 6.4 Calculation of Shear and Torsion Stiffness ................................ ................................ ... 86 6.5 Calculation of Axial Stiffness ................................ ................................ ....................... 86 6.5.1 Stiffn ess of Neoprene Layers ................................ ................................ ............ 87 6.5.2 Model Dimensions and Meshing ................................ ................................ ...... 87 6.5.3 Loading and Boundary Conditions ................................ ................................ ... 88 6.5.4 Material Model ................................ ................................ ................................ .. 88 6.5.5 Experimental Validation ................................ ................................ ................... 90 6.6 Calculation of Nonlinear R oll Stiffness Curves ................................ ............................ 90 6.6.1 Grillage Model ................................ ................................ ................................ .. 90 6.6.2 Spring Stiffness Distribution in Grillage Model ................................ ............... 91 6.6.3 Incorporating Girder Slope ................................ ................................ ............... 92 6.7 Simplified Method for Calculating Axial Stiffness and Instantaneous Roll Stiffnesses ................................ ................................ ................................ ..................... 93 6.7.1 Axial Stiffness ................................ ................................ ................................ ... 94 6.7.2 Basic Derivation of Instantaneous Roll Stiffness of a Continuous Grillage ..... 96 6.7.3 Incorporating Girder Slope ................................ ................................ ............... 97 7 MODEL DEVELOPMENT ................................ ................................ ................................ .. 114 7.1 Introduction ................................ ................................ ................................ ................. 114 7.2 Modeling of Bridge Girders ................................ ................................ ........................ 115 7.3 Modeling of End Supports ................................ ................................ .......................... 117 7.3.1 Pad Selection ................................ ................................ ................................ ... 117 7.3.2 Axial Load Selection ................................ ................................ ....................... 118 7.3.3 Girder Slope Selection ................................ ................................ .................... 118 7.4 Modeling of Braces and Anchors ................................ ................................ ................ 119 7.5 Loads ................................ ................................ ................................ ........................... 121 7.6 Modified Southwell Buckling Analysis ................................ ................................ ...... 122 8 PARAMETRIC STUDY OF INDIVIDUAL BRIDGE GIRDERS ................................ ..... 132 8.1 Introduction ................................ ................................ ................................ ................. 132 8.2 Selection of Para meters ................................ ................................ ............................... 132 PAGE 6 6 8.3 Results ................................ ................................ ................................ ......................... 134 8.3.1 Wind Capacity of a Single Unanchored Girder ................................ .............. 135 8.3.2 Wind Capacity of a Single Anchored Girder ................................ .................. 136 9 PARAMETRIC STUDY OF BRACED MULTI GIRDER SYSTEMS .............................. 145 9.1 Pr eliminary Sensitivity Studies ................................ ................................ ................... 145 9.1.1 Strut Braces ................................ ................................ ................................ ..... 145 9.1.2 Moment Resisting Braces ................................ ................................ ............... 146 9.2 Modeling of Bridge Skew and Wind Load ................................ ................................ 147 9.3 Selection of Parameters for Strut Brace Parametric Study ................................ ......... 148 9.4 Results of Strut Brace Parametric Study ................................ ................................ ..... 149 9.4.1 System Capacity of Unanchored Two Girder System in Zero Wind ............. 151 9.4.2 System Capacity Increase from Inclusion of Anchor ................................ ..... 151 9.4.3 System Capacity Reduction from Erection of Additional Girders .................. 152 9.4.4 System Capacity Reduction from Inclusion of Wind Load ............................ 153 9.4.5 Consideration of Skew ................................ ................................ .................... 155 9.5 Stiffness of Moment Res isting Braces ................................ ................................ ........ 156 9.6 Selection of Parameters for Moment Resisting Brace Parametric Study ................... 158 9.7 Results of Moment Resisting Br ace Parametric Study ................................ ............... 159 9.7.1 System Capacity Increase from Inclusion of Moment Resisting End Braces ................................ ................................ ................................ .............. 160 9.7.2 System Capacity Increase from Installation of Braces at Interior Points ........ 161 9.7.3 System Capacity Reduction from Inclusion of Wind Load ............................ 162 9.7.4 Consideration of Skew ................................ ................................ .................... 164 9.8 Incorporation of Aerodynamic Lift ................................ ................................ ............. 164 10 CONCLUSIONS AND RECOMMENDATIONS ................................ ............................... 186 10.2 Drag Coefficients ................................ ................................ ................................ ........ 186 10.3 Individual Unbraced Florida I Beams ................................ ................................ ......... 188 10.4 Braced Systems of Multiple Florida I Beams ................................ ............................. 189 10.5 Future Research ................................ ................................ ................................ ........... 190 APPENDIX A DIMENSIONED DRAWINGS OF WIND TUNNEL TEST CONFIGURATI ONS .......... 193 B TABULATED RESULTS FROM WIND TUNNEL TESTS ................................ .............. 202 C CROSS SECTIONAL PROPERTIES OF FLORIDA I BEAMS ................................ ........ 220 D PROPERTIES OF FLORIDA BEARING PADS ................................ ................................ 224 E PLOTS OF CAPACITY PREDICTION EQUATIONS ................................ ...................... 236 LIST OF REFERENCES ................................ ................................ ................................ ............. 246 PAGE 7 7 BIOGRAPHICAL SKETCH ................................ ................................ ................................ ....... 249 PAGE 8 8 LIST OF TABLES Table page 3 1 Summary of a erodynamic coefficients ................................ ................................ .............. 45 3 2 Pressure coefficients in FDOT Structures Design Guide (SDG) ................................ ....... 45 3 3 Drag coefficients ( C D ) of thin w alled I shapes ................................ ................................ 45 4 1 Testing configurations ................................ ................................ ................................ ....... 54 4 2 Wind tunnel test scaling ................................ ................................ ................................ ..... 54 5 1 Aerodynamic coefficients of bridge girder cross sectional shapes ................................ .... 69 5 2 Extreme combinations of tested wind angle and cross slope ................................ ............ 69 8 1 Parameter values used in parametric study for each FIB cross section ........................... 139 8 2 Range of allowable span lengths for FIBs ................................ ................................ ....... 140 9 1 Parameter values used in strut brace parametric study ................................ .................... 168 9 2 Girder offset lengths in model for each skew angle ................................ ........................ 168 9 3 Parameter values used in moment resisting brace parametric study ............................... 169 9 4 Empirically determined values of for different numbers of interior braces ................ 169 9 5 Selfweight ( w sw ) of each FIB cross sectional shape (from FDOT, 2012) ...................... 169 B 1 Meaning of letters in configuration IDs ................................ ................................ ........... 202 C 1 Definitions of cross sectional properties required for use of a warping beam element ... 222 C 2 Cross sectional properties of Florida I Beams ................................ ................................ 222 D 1 Bearing pad dimensions and computed stiffnesses ................................ .......................... 225 PAGE 9 9 LIST OF FIGURES Figure page 1 1 Prestressed c oncrete girders braced together for stability ................................ .................. 21 2 1 Girder system ................................ ................................ ................................ ..................... 27 2 2 Definition of grade (side view) ................................ ................................ .......................... 27 2 3 Definition of cross slope (section view) ................................ ................................ ............ 27 2 4 Definition of skew (top view) ................................ ................................ ............................ 28 2 5 Definition of camber (elevation view) ................................ ................................ ............... 28 2 6 Definition of sweep (plan view) ................................ ................................ ........................ 28 2 7 Rollover instability of girder ................................ ................................ .............................. 28 2 8 Lateral torsional instability of girder ................................ ................................ ................. 29 2 9 Increase in secondary effects due to higher application of vertical load ........................... 29 2 10 Effects of wind on stability of girder. ................................ ................................ ................ 30 2 11 Common anchor types. ................................ ................................ ................................ ...... 31 2 12 Chain braces on Florida Bulb Tee during transportation ................................ .................. 32 2 13 Perpendicular brace placement on skewed bridge ................................ ............................. 32 2 14 Common brac e types. ................................ ................................ ................................ ......... 33 3 1 Two dimensional bridge girder cross section with in plane line loads ............................. 45 3 2 Definition of C D C L C SD and C SL ................................ ................................ ................ 46 3 3 Center of pressure of a bridge girder ................................ ................................ ................. 46 3 4 Definition of C T and C PT ................................ ................................ ................................ .. 46 3 5 Velocity pressure exposure coefficient used by FDOT ................................ ..................... 47 3 6 Examples of I shaped girders (steel plate girders and Florida I Beams) for which C P = 2.2 (per FDOT, 2012e) ................................ ................................ ................................ ... 47 3 7 Example open top box girder cross section for which C P = 1.5 (per FDOT, 2012e) ....... 47 PAGE 10 10 3 8 Drag coefficients for rectangular sections with v arious width to depth ratios .................. 48 3 9 Drag coefficients for plate girder shapes and rectangles with various width to depth ratios ................................ ................................ ................................ ................................ ... 48 4 1 Girder cross sections used in study (drawn to scale) ................................ ......................... 54 4 2 Girder groupings investigated in study ................................ ................................ .............. 55 4 3 Parameters definiti ons for each testing configuration ................................ ....................... 55 4 4 Wind angle sign convention ................................ ................................ ............................... 55 4 5 Equivalence between wind angle and cross slope for box girders ................................ .... 56 5 1 Effect of wind angle on individual girder drag coefficients ( C D ) ................................ ..... 69 5 2 Effect of wind angle on individual girder lift coefficients ( C L ) ................................ ........ 70 5 3 Effect of wind angle on individual torque coefficients ( C T ) ................................ ............. 70 5 4 Drag coefficients of WF Plate gir ders in 5 girder configurations (0 Wind) .................... 71 5 5 Effect of adding additional girders ................................ ................................ .................... 71 5 6 Effect of wind angle on C D ................................ ................................ ............................... 72 5 7 Interaction between wind angle and cross slope. ................................ .............................. 72 5 8 Ten (10) girder models tested at wind angles producing maximum shielding .................. 73 5 9 Ten (10) girder models tested at wind angles producing minimum shielding ................... 73 5 10 Effect of wind angle on two (2) Box girder syste m drag coefficients ( C D ) ...................... 74 5 11 Lift coefficients on all I shaped girder test configurations (plate girders and FIBs) ......... 74 5 12 Tor que coefficients on all I shaped girder test configurations (plate girders and FIBs) ... 75 5 13 Transformation of C T to C PT ................................ ................................ ........................... 76 5 14 Moment load expressed as equivalent drag force. ................................ ............................. 76 5 15 Comparison between maximum C D and maximum C D,eff ................................ .............. 77 5 16 Proposed design l oads for plate girders ................................ ................................ ............. 78 5 17 Proposed design loads for FIBs ................................ ................................ ......................... 78 PAGE 11 11 5 18 Representation of positive and negative drag loads as a comb ined compression load ...... 79 5 19 Proposed brace force design loads for plate girders ................................ .......................... 79 5 20 Proposed brace force design loads for FI Bs ................................ ................................ ...... 80 5 21 Simplified brace force analysis. ................................ ................................ ......................... 81 6 1 Location and structure of neoprene bearing pads ................................ ............................ 101 6 2 Bulging of neoprene layers under compression. ................................ .............................. 102 6 3 Distribution of reaction force under bearing pad subjected to uniform axial load .......... 103 6 4 Behavior of bearing pads during girder rollover. ................................ ............................. 104 6 5 Dimensions of a bearing pad ................................ ................................ ............................ 104 6 6 Axial stiffness of pad as individual layer stiffnesses combined in series ........................ 105 6 7 Finite element model of elastomer layer ................................ ................................ .......... 105 6 8 Validation of neo Hookean material model. ................................ ................................ .... 106 6 9 Simplified grillage model of a bearing pad ................................ ................................ ...... 107 6 10 Standard FDO T bearing pads used for experimental verification ................................ ... 108 6 11 Distribution of stiffness to grillage springs. ................................ ................................ ..... 109 6 12 Comparison of e xperimentally measured bearing pad roll stiffnesses and roll stiffnesses predicted by the proposed computation method ................................ ............ 110 6 13 Bearing pad slope. ................................ ................................ ................................ ............ 111 6 14 Comparison of experimentally measured bearing pad roll stiffnesses and roll stiffnesses predicted by the proposed computation method with non zero slope. ........... 112 6 15 Coord inate system of continuous grillage (plan view). ................................ ................... 112 6 16 Continuous grillage with imposed differential angle ................................ ....................... 113 6 17 Comparison between Equation 6 28 and the square root approximation ........................ 113 7 1 Finite element model of a single FIB (isometric view) ................................ ................... 126 7 2 Repr esentation of sweep in FIB model (plan view) ................................ ......................... 126 7 3 Representation of camber in FIB model (elevation view) ................................ ............... 126 PAGE 12 1 2 7 4 Bearing pad st iffness springs in FIB model (isometric view) ................................ .......... 127 7 5 Representation of brace configurations in FIB system models. ................................ ...... 128 7 6 Calcula tion of rotational stiffness of anchor ................................ ................................ .... 129 7 7 Longitudinally inclined anchors on skewed bridges ................................ ....................... 129 7 8 Representation of wind lo ad in structural models. ................................ ........................... 129 7 9 Southwell method for determining critical buckling load ( ). ................................ ......... 130 7 10 Southwell analysis of non hyperbolic displacement load data obtained from a large displacement structural analysis ................................ ................................ ...................... 131 7 11 Determination of buckling capacity using modified Southwell approach ....................... 131 8 1 Summary of single girder wind load parametric study results ................................ ........ 140 8 2 Wind capacities of unanchored FIBs at various span lengths ................................ .......... 141 8 3 Wind capacity of an unanchored girder as predicted by Equation 8 2 ............................ 141 8 4 Wind capacity of an unanchored girder as predicted by simplified Equation 8 3 ........... 142 8 5 Comparison of basic and simplified unanchored girder wind capacity equat ions, Equations 8 2 and 8 3, respectively ................................ ................................ ................. 142 8 6 ................................ 143 8 7 Rejection of artificially ............................ 143 8 8 Anchor stiffness coefficient Equation 8 5 compared to parametric study results ........... 144 8 9 Comparison of wind capacity results computed using the combination of Equations 8 2 and 8 6 versus corresponding parametric study results ................................ ............. 144 9 1 Examples of strut bracing ................................ ................................ ................................ 170 9 2 Collapse mechanism possible with strut bracing ................................ ............................. 170 9 3 Examples of moment resisting braces. ................................ ................................ ............ 170 9 4 Effect of bridge skew on wind loading of braced 3 girder syste m (plan view). .............. 171 9 5 Summary of strut brace parametric study results ................................ ............................. 171 9 6 System capacities of unanchored two girder strut br aced systems in zero wind at various span lengths ................................ ................................ ................................ ......... 172 PAGE 13 13 9 7 System capacity of an unanchored strut braced two girder system in zero wind as predicted by Equation 9 2 ................................ ................................ ................................ 172 9 8 Effect of anchor ................................ ................ 173 9 10 Relative error of system capacity values predicted by Equation 9 4 ............................... 174 9 11 Rela tive error of system capacity values predicted by Equation 9 6 ............................... 174 9 12 FIB system with anchor stiffness of 1600 kip ft/rad ................................ ................................ ............................... 175 9 13 Approximate linear relationship between system capacity and square root of average wind pressure ................................ ................................ ................................ ................... 175 9 14 Quadratic surface (Equat ion 9 9) fitted to wind pressure coefficient values ................... 176 9 15 Absolute error of system capacity values predicted by Equation 9 10 ............................ 176 9 16 Absolute error of system capacity values predicted by Equation 9 10 for strut braced systems, including systems with non zero skew angles ................................ .................. 177 9 17 Brace designs in brace inventory (each implemented at three different spacings and three different FIB depths) ................................ ................................ ............................... 178 9 18 Model used to compute effective stiffness of brace configurations ................................ 178 9 19 Reference brace configuration used in parametric studies ................................ .............. 179 9 20 Cross ................................ ................................ ................................ .. 179 9 21 Sti ffness of every brace in brace inventory ................................ ................................ ...... 179 9 22 Stiffness of every X brace in brace inventory ................................ ................................ 180 9 23 Summary of moment resisti ng brace parametric study results ................................ ........ 180 9 24 Equation 9 ................................ ................. 181 9 25 Equation 9 ................................ ................. 181 9 26 Re lative error of system capacity values predicted by Equation 9 13 ............................. 182 9 27 Relative error of system capacity values predicted by Equation 9 14 ............................. 182 9 28 Approximate linear relationship between system capacity and square root of average wind pressure ................................ ................................ ................................ ................... 183 PAGE 14 14 9 29 Quadratic surface (Equation 9 16) fitted to wind pressure coeffic ient values and adjusted to produce conservative results in 95% of cases ................................ ............... 184 9 30 Absolute error of system capacity values predicted by Equation 9 17 for moment resisting braced systems ................................ ................................ ................................ ... 184 9 31 Absolute error of system capacity values predicted by Equation 9 17 for moment resisting braced systems, including systems with non zero skew angles ........................ 185 10 1 Girder cross sectional shapes tested in the wind tunnel ................................ .................. 191 10 2 Recommended design wind loads for systems of adjacent girders ................................ .. 191 10 3 Recommended structural analysis model for use in determining brace forces. ............... 192 C 1 Coordinate system used in the calculation of cross sectional properties ......................... 223 D 1 Bearing pad dimensions and variables ................................ ................................ ............. 225 E 1 Prediction of system capacity for 2 girder, unanchored strut braced systems in 0 psf wind (Equation 9 10) ................................ ................................ ................................ ....... 236 E 2 Prediction of system capacity for 2 girder, unanchored strut braced systems in 40 psf wind (Equation 9 10) ................................ ................................ ................................ ....... 237 E 3 Prediction of system capacity for 2 girder, unanchored strut braced systems in 80 psf wind (Equation 9 10) ................................ ................................ ................................ ....... 237 E 4 Prediction of system capacity for 2 girder, unanchored str ut braced systems in 120 psf wind (Equation 9 10) ................................ ................................ ................................ 238 E 5 Prediction of system capacity for 2 girder, unanchored strut braced systems in 160 psf wind (Equation 9 10) ................................ ................................ ................................ 238 E 6 Prediction of system capacity for end braced systems in 0 psf wind with moment resisting braces with k brace = 15,000 kip ft/rad (Equation 9 17) ................................ ... 239 E 7 Prediction of system capacity for end braced systems in 0 psf wind with moment resisting braces with k brace = 200,000 kip ft/rad (Equation 9 17) ................................ 239 E 8 Prediction of system capacity for end braced systems in 0 psf wind with moment resisting braces with k brace = 400,000 kip ft/rad (Equation 9 17) ................................ 240 E 9 Prediction of system capacity for end braced systems in 0 psf wind with momen t resisting braces with k brace = 600,000 kip ft/rad (Equation 9 17) ................................ 240 E 10 Prediction of system capacity for systems in 0 psf wind with moment resisting braces ( k brace = 200,000 kip ft/rad) w ith no interior brace points (Equation 9 17) ....... 241 PAGE 15 15 E 11 Prediction of system capacity for systems in 0 psf wind with moment resisting braces ( k brace = 200,000 kip ft/rad) with 1 interior brace po int (Equation 9 17) .......... 241 E 12 Prediction of system capacity for systems in 0 psf wind with moment resisting braces ( k brace = 200,000 kip ft/rad) with 2 interior brace points (Equation 9 17) ......... 242 E 13 Prediction of system capacity for systems in 0 psf wind with moment resisting braces ( k brace = 200,000 kip ft/rad) with 3 interior brace points (Equation 9 17) ......... 242 E 14 Prediction of system capacity for systems with moment resisting braces ( k brace = 200,000 kip ft/rad) with 1 interior brace point in 0 psf wind (Equation 9 17) ................ 243 E 15 Prediction of system capacity for systems with moment resisting braces ( k brace = 200,000 kip ft/rad) with 1 interior brace point in 40 psf wind (Equation 9 17) .............. 243 E 16 Predi ction of system capacity for systems with moment resisting braces ( k brace = 200,000 kip ft/rad) with 1 interior brace point in 80 psf wind (Equation 9 17) .............. 244 E 17 Prediction of system capacity for systems with moment resisting braces ( k brace = 200,000 kip ft/rad) with 1 interior brace point in 120 psf wind (Equation 9 17) ............ 244 E 18 Prediction of system capacity for systems with momen t resisting braces ( k brace = 200,000 kip ft/rad) with 1 interior brace point in 160 psf wind (Equation 9 17) ............ 245 PAGE 16 16 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science BRIDGE GIRDER DRAG COEFFICIENTS AND WIND RELATED BRACING RECOMMENDATIONS By Zachary Harper May 2013 Chair: Gary Consolazio Major: Civil Engineering A key objective of this study was t o experimentally quantify wind load coefficients (drag, torque, and lift) for common bridge girder shapes, and to quantify shielding effects arising from aerodynamic interference between adjacent girders. Wind tunnel tests were performed on reduced scale m odels of Florida I Beam (FIB), plate girder, and box girder cross sectional shapes to measure the aerodynamic properties of individual girders as well as systems of multiple girders. The focus of this study was on construction stage structural assessment u nder wind loading conditions, therefore the multiple girder systems that were considered did not have a bridge deck in place (and therefore air flow between adjacent girders was permitted). Results from the wind tunnel tests were synthesized into simplifi ed models of wind loading for single and multiple girder systems and conservative equations suitable for use in bridge design were developed. Separate wind load cases were developed for assessing overall system stability and required brace strength Also included in this study was the development of procedures for assessing temporary bracing requirements to resist wind load during bridge construction. Numerical finite element models and analysis techniques were developed for evaluating the stability of pre cast concrete girders (Florida I Beams), both individually and in systems of multiple girders braced together. PAGE 17 17 A sub component of this effort resulted in the development of a new calculation procedure for estimating bearing pad roll stiffness, which is kno wn to affect girder stability during construction. After integrating the improved estimates of wind loads and bearing pad stiffnesses into finite element models of individual and multiple girder braced systems, several large scale parametric studies were p erformed (in total, more than 50,000 separate stability analyses were conducted). The parametric studies included consideration of different Florida I Beam cross sections, span lengths, wind loads, skew angles, anchor stiffnesses, and brace stiffnesses. Re gression analyses were performed on the parametric study results to develop girder capacity prediction equations suitable for use in the design of temporary bracing for Florida I Beams during construction PAGE 18 18 CHAPTER 1 IN TRODUCTION 1 1 Introduction Prestressed concrete girders are commonly used in bridge construction because they are an economical choice for supporting very long spans. For example, the 96 in ch deep Florida I Beam (FIB), one of the standard girder designs employed by the Florida Department of Transportation (FDOT), is able to support spans of 200 ft or more. However, as such girders increase in span length, they become more susceptible t o issues of lateral instability The most critical phase of construction, with regard to stability, is after girder placement (prior to the casting of the deck), when girders are supported only by flexible bearing pads and can be subject to high lateral w ind loads. In many bridge designs, girders may be positioned (laterally spaced) near enough to one another that a single unstable girder can knock over adjacent girders, initiating a progressive collapse that can result in severe economic damage and risk t o human life. To prevent such a scenario, it is typical for girders to be temporarily braced together (Fig. 1 1 ) to form a more stable structural unit During the construction phase, wind loads tend to control the desig n of temporary bracing, so it is important that such loads be known as accurately as possible. Lateral wind loads are generally calculated using a drag coefficient a dimensionless quantity that relates the wind pressure on an object to its size and wind s peed. However, the drag coefficients of most common bridge girder cross sectional shapes have not been adequately addressed in the literature Furthermore, once multiple adjacent girders have been placed, the leading girder acts as a windbreak and disrupt s the airflow over subsequent girders, resulting in a phenomenon referred to as aerodynamic interference ( or shielding ) At common girder spacings, the alteration to the windstream will reduce or even reverse the direction of wind pressure on leeward girde rs. A PAGE 19 19 thorough understanding of this shielding effect is necessary to develop appropriately conservative bracing design forces. However this area has also received little attention in the literature 1 2 Objectives The primary objective of this research was to experimentally quantify drag coefficients for common bridge girder shapes as well as shielding effects arising from the aerodynamic interference between adjacent girders, and to synth esize the results into a set of conservative design parameters that can be used to compute lateral wind loads for design and construction calculations. A secondary objective was to use analytical models of braced girder systems to develop recommendations f or temporary bracing of prestressed concrete girders (FIBs) subjected to the new design wind loads 1 3 Scope of Work Experimental testing : Wind tunnel tests were performed to measure the ae rodynamic coefficients (drag, lift, and torque) of five (5) bridge girder cross sectional shapes [two (2) plate girder; two (2) FIB; and one open top box], chosen to be representative of a wide range modern Florida bridges. In addition to measuring the aer odynamic coefficients of the individual girders, tests were performed on groups of adjacent girders in a variety of common configurations in order to quantify the shielding effects caused by aerodynamic interference. Design wind loads : Measurements from th e wind tunnel tests were analyzed to identify common trends and to develop a conservative set of simplified wind load parameters that are suitable for use in design Analysis method for bearing pad stiffnesses : Experimental bearing pad stiffness measuremen ts from a previous FDOT research project (BDK75 977 03, Consolazio et al. 2012) were used to develop and validate a new analytical method for estimating the girder support stiffnesses provided by steel reinforced elastomeric bearing pads System level anal ytical models : Analytical models were developed that were capable of evaluating the lateral stability of Florida I Beams (FIBs). The models incorporated the estimated support stiffnesses provided by standard FDOT bearing pads and were capable of capturing system level behavior of multiple girders braced together with any of several common brace types PAGE 20 20 Wind load capacity of individual FIBs : An analytical parametric study was conducted to determine a simplified equation for estimating the maximum wind pressur e that an individual (unbraced) FIB can resist without becoming unstable Recommendations for temporary bracing : Analytical parametric studies were conducted using the system level models and the design wind loads to evaluate temporary bracing requirement s for FIB systems in a variety of configurations. In addition to general recommendations for temporary bracing design, the results of the parametric study were used to develop simplified equations for estimating the capacity of braced systems of FIBs PAGE 21 21 Figure 1 1 Prestressed concrete girders braced together for stability (photo courtesy of FDOT) PAGE 22 22 CHAPTER 2 PHYSICAL DESCRIPTION OF BRIDGES DUR ING CONSTRUCTION 2 1 Introduction This study is concerned with the stability of long span prestressed concrete girders during the construction process. Specifically, the girders under investigation are Florida I Beams (FIBs), a family of standard cross sectional shapes of varying depths that are commonly employed in bridge designs in Florida. These beams are typically cast offsite, transported to the construction site by truck, then lif ted into position one at a time by crane, where they are placed on elastomeric bearing pads and braced together for stability. It is this stage of construction, prior to the casting of the deck that is primarily of interest. In this chapter, a physical des cription of the construction stage bridge structures under consideration in this study will be provided along with the definition of relevant terminology 2 2 Geometric P arameters The term girder system will be used to refer to a group of one or more FIBs braced together in an evenly spaced row (Figure 2 1 ). In addition to span length and spacing, there are several geometric parameters that define the shape and placement of the girders within a system. They a re : Grade : Longitudinal incline of the girders, typically expressed as a percentage of rise per unit of horizontal length (Figure 2 2 ) Cross slope : The transverse incline (slope) of the deck, expressed as a percentage, which results in girders that are staggered vertically (Figure 2 3 ) Skew angle : Longitudinal staggering of girders, due to pier caps that are not perpendicular to the girder axes (Figure 2 4 ) Camber : Vertical bowing of the girder (Figure 2 5 ) due to prestressing in the bottom flange expressed as the maximum vertical deviation from a perfectly straight line connecting one end of the girder to th e other. Note that the total amount of vertical camber immediately following girder placement is larger than the camber in the completed bridge structure because the weight of the deck is not yet present PAGE 23 23 Sweep : Lateral bowing of the girder (Figure 2 6 ) due to manufacturing imperfections, expressed as the maximum horizontal deviation from a perfectly straight line connecting one end of the girder to the other. 2 3 Beari ng P ads Bridge girders rest directly on steel reinforced neoprene bearing pads which are the only points of contact between the girder and the substructure. There is generally sufficient friction between the pad and other structural components so that any movement of a girder relative to the substructure (with the exception of vertical uplift) must also move the top surface of the pad relative to the bottom surface. As a result, the girder support conditions in all six degrees of freedom can be represented as finite stiffnesses that correspond to the equivalent deformation modes of the pad. These deformation modes fall into four categories: shear, compression (axial), rotation (e.g. roll), and torsion. Calculation of these stiff nesses is addressed in Chapte r 6 2 4 Sources of Lateral Instability Girder instability arises when the structural deformations caused by application of a load act to increase the moment a rm of that load to such an extent that equilibrium cannot be achieved. The additional moment (often called the secondary effects ) causes the structure to deform further, which increases the moment arm even more. In a stable system, this process continues u ntil the structure converges on a deformed state in which static equilibrium is achieved. However, if the load exceeds some critical value (i.e., the buckling load), the system becomes unstable, in which case the process diverges and the structural deforma tions increase without bound (i.e., the structure collapses). Long span bridge girders are susceptible to two primary modes of instability: girder rollover and lateral torsional buckling Girder rollover refers to the rigid body rotation of a girder with s weep imperfections resting on end supports (i.e., bearing pads) that have a finite roll stiffness. Sweep imperfections cause the force resultant of the girder self weight ( F PAGE 24 24 centerline of the supports (Figure 2 7 ). The eccentric vertical load imparts an overturning moment that causes the rigid girder to rotate until an equivalent restoring moment is generated at the supports. However, during rotation, the eccentricity of the load is increased, creating the potential for instability. If the weight of the girder is high enough and/or the bearing pad roll stiffness is low enough, the process will continue until the girder rolls off the pad Lateral torsional buckling is a similar phenom enon that occurs in flexible girders, even if the supports are rotationally rigid. In this case, the eccentric load induces lateral torsional deformations in the girder that increase the load eccentricity (Figure 2 8 ). If the load is high enough to generate instability, the girder continues to deform until material failure (e.g., cracking) and, ultimately, structural collapse Both girder rollover and lateral torsional buckling have been studied thoroughly in isola tion. However, in real girders, the instability modes are coupled : any additional load eccentricities caused by girder rollover will induce additional lateral torsional buckling, and vice versa. It is not sufficient to perform separate analyses of each mod e and superpose the results Deviations from ideal straightness tend to increase the potential for girder instability. This is most intuitive in the case of sweep: a higher initial eccentricity induces more overturning moment. However, increasing vertical camber can also make a beam less stable by elevating the center of gravity of the girder. A higher load application point will displace farther laterally under the same amount of initial deformation (Figure 2 9 ) increasing the magnitude of the secondary effects. Effectively, two equal loads that are applied at different elevations will force a girder to roll/deform different amounts before reaching equilibrium. For a long span girder, this difference can me an the difference between stable equilibrium being achieved, or buckling instability occurring PAGE 25 25 2 5 Lateral Wind Loads In addition to gravity induced self weight, girder systems are also su bjected to intermittent lateral wind loads of varying intensity throughout the construction process. Wind loads are generally modeled as uniform pressure loads applied to girders in the lateral (transverse) direction. These types of loads can have a severe ly destabilizing effect on girder systems. Because the force resultant at the center of pressure ( W ) is offset from the bearing pad supports, large overturning moments can be generated that contribute directly to girder rollover. Furthermore, the wind forc e causes the girders to bend laterally (about their weak axes). This can increase the eccentricity of the self weight, increasing the potential for instability (Figure 2 10 ) 2 6 Temporary Bracing During construction, girders are often braced to prevent lateral instability from arising. Usually, these braces are temporary and are removed after the deck is cast. Bracing is divided into two basic types: anchor bracing and girder to girder bracing 2 6 1 Anchor B racing Because the first girder in the erection sequence has no adjacent girders to brace against anchors are used to brace the ends of the girder to the pier. Anchors can take the form of inclined structural members such as telescoping steel rods (Figure 2 11 a) or tension only members such as cables (Figure 2 11 b) or chains (Figure 2 11 c). In addition to their lateral incline, it is common for anchors to also be inclined inward (towards the center of the span) so that they can reuse the same precast connectio ns that are used to stabilize girders during transportation (Figure 2 12 ). Anchors are generally not as effective as girder to girder bracing; because they can only restrain the girders at the ends, they can prevent gi rder rollover but not lateral torsional buckling. PAGE 26 26 For this reason, anchors are generally only used on the first girder to be erected and are not used on subsequent girders. 2 6 2 Girder to G irder B racing As adjacent girders are erected, girder to girder braces (henceforth known simply as braces ) are used to connect the girders together into a single structural unit. Because the braces can be installed at in terior points (i.e., away from the girder ends), they are capable of providing resistance to both lateral torsional and rollover instabilities. Typically, interior braces are spaced at unit fractions of the girder length. For example, third point bracing d ivides the girder into three equal unbraced lengths. Brace point locations are offset somewhat in skewed bridges because Design Standard No. 20005 : Prestressed I Beam Temporary Bracing ( FDOT, 2012 a ) requires that all braces be placed perpendicular to the g irders (Fig. 2 13 ). Braces are typically constructed from timber or rolled steel members, but individual brace designs are left to the discretion of the contractor, so a wide variety of bracing configurations are used i n practice. Common types of brace include X braces (Figure 2 14 a), K braces (Figure 2 14 b), and simple compression struts (Fig 2 14 c). Braces are attached to the g irders via bolted connections, welded to cast in steel plates, or simply wedged tightly in place between the girders. In the latter case, an adjustable tension tie, such as a threaded bar (Figure 2 14 d), is normally included to prevent the girders from separating far enough for the braces to become dislodged PAGE 27 27 Figure 2 1 Girder system Figure 2 2 Definition of grade (side view) Figure 2 3 Definition of cross slope (sec tion view) PAGE 28 28 Figure 2 4 Definition of skew (top view) Figure 2 5 Definition of camber (elevation view) Figure 2 6 Definition of sweep (plan view) Figure 2 7 Rollover instability of girder PAGE 29 29 Figure 2 8 Lateral torsional instability of girder Figure 2 9 Increase in secondary effects due to higher application of vertical load PAGE 30 30 A B Figure 2 10 Effects of wind on stability of girder A) Girder without wind load B ) Girder with wind load PAGE 31 31 A B C Figure 2 11 Common anchor types A) Structural member. B ) Cable C ) Chain PAGE 32 32 Figure 2 12 Chain braces on Florida Bulb T ee during transportation (photo courtesy of FDOT) Figure 2 13 Perpendicular brace placement on skewed bridge PAGE 33 33 A B C D Figure 2 14 Common brace types A) X brace B ) K brace C ) Compression strut D ) Tension tie PAGE 34 34 CHAPTER 3 BACKGROUND ON DRAG COEFFICIENTS 3 1 Introduction In order to calculate the wind load on a bridge girder, it is necessary to know the drag coefficient for the girder cross s ectional shape. The drag coefficient is a type of aerodynamic coefficient : a dimensionless factor that relates the magnitude of the fluid force on a particular geometric shape to the approaching wind speed. Drag coefficients are typically a function of the relative orientation of the object with the direction of the impinging wind. 3 2 Dimensionless Aerodynamic C oefficients Fluid forces arise when a solid body is submerged in a moving fluid. As the fluid flow is diverted around the body, a combination of inertial and frictional effects generates a net force on the body. It is observed that this force called aerodynamic force ( F ) when the fluid under consideration is air is directly proportiona l the dynamic pressure ( q ) of the fluid: ( 3 1 ) where is the mass density of the fluid and V is the flow velocity (engel and Cimbala, 2006). Dynamic pressu re can be considered as the kinetic energy density of the fluid. This offers an intuitive explanation for its proportional relationship to aerodynamic force, which is, at the most fundamental level, the cumulative effect of innumerable microscopic collisio ns with individual fluid particles. Similarly, if the dimensions of the body are scaled up, it is observed that the aerodynamic force increases quadratically, reflecting the fact that the increased surface area results in a greater total number of collisio ns. These proportional relationships can be combined and expressed as : ( 3 2 ) PAGE 35 35 where L 0 and L 1 are arbitrary reference lengths and C F is a combined proportionality factor, called a force coefficient The selection of L 0 and L 1 does not affect the validity of Equation 3 2 as long as they both scale with the structure. However, it is important to be consistent; force coefficient s that use different reference lengths are not directly comparable, and a coefficient for which the reference lengths are not explicitly known is useless for predicting aerodynamic forces. In structural applications, it is common for the product L 0 L 1 to be expressed in the form of a reference area, A which is typically taken as the projected area of the structure in the direction of wind By an analogous process, it is possible to derive a moment coefficient ( C M ), which normalizes aerodynamic moment load i n the same way that the force coefficient normalizes aerodynamic force. The only difference is that aerodynamic moment grows cubically with body size rather than quadratically (because the moment arms of the individual collisions grow along with the surfac e area). Therefore, the moment proportionality expression is : ( 3 3 ) As with the force coefficient, the reference lengths must be known in order to properly interp ret the C M However, with moment coefficients, it is equally important to know the center of rotation about which the normalized moment acts. Together, C F and C M are called aerodynamic coefficients and they can be used to fully describe the three dimensio nal state of aerodynamic load on a structure (for a particular wind direction) When working with bridge girders, or other straight, slender members, it is often convenient to assume that the length of the girder is effectively infinite. This simplifies en gineering calculations by reducing the girder to a two dimensional cross section subjected to in plane aerodynamic line loads (Figure 3 1 ). Depending on the direction of wind, out of plane PAGE 36 36 forces and moments may ex ist, but they generally do not contribute to the load cases that control design and can therefore be considered negligible. In two dimensions, the proportionality expressions for the aerodynamic coefficients become : ( 3 4 ) ( 3 5 ) where is a distributed force (force per unit length) and is a distributed torque (moment p er unit length). Note that two dimensional aerodynamic coefficients can be used interchangeably in the three dimensional formulation if one reference length ( L 0 ) is taken to be the out of plane length of the girder. All further discussions of aerodynamic c oefficients in this report will use the two dimensional formulation unless stated otherwise. The remaining reference lengths ( L 1 and L 2 ) will always be taken as the girder depth, D so that the force and moment coefficients are defined as : ( 3 6 ) ( 3 7 ) Aerodynamic coefficients are sometimes called shape factors because they represent the contribution of the geometry of an object (i.e., the way airflow is diverted around it), independent of the scale of the object or the intensity of the flow. Because of the complexity of the differential equations governing fluid flow, the a erodynamic coefficients of a structure are not calculated from first principles but can, instead, be measured directly in a wind tunnel using reduced scale models PAGE 37 37 3 3 Terminology Related to Aerodynamic C oefficients Aerodynamic force on a body is typically resolved into two orthogonal components, drag and lift. These components have corresponding force coefficients: the drag coefficient ( C D ) and lift coefficient ( C L ). In this report, drag is defined as the lateral component of force and lift is defined as the vertical component of force, regardless of the angle of the applied wind In several subfields of fluid dynamics, it is more conventional to define drag as the component of force along th e direction of the wind stream and lift as the component perpendicular to the wind stream. However, this is inconvenient when evaluating wind loads on stationary structures (e.g. bridge girders) because the angle of the wind stream can change over time. W here necessary in this report, the names stream drag ( C SD ) and stream lift ( C SL ) (Figure 3 2 ) will be used to refer to the force components that are aligned with, and perpendicular to, the wind stream. Finally, the term pressure coefficient ( C P ), is an alternative name for C D, and is often used in design codes to indicate that it is to be used to calculate a wind pressure load ( P ) rather than a total force, as in : ( 3 8 ) This is advantageous because it obviates the need to explicitly specify the characteristic dimensions that were used to normalize the coefficient. Instead, denormalization occurs implicitly when the press ure load is applied over the projected surface area of the structure. Unfortunately, this approach breaks down when working with drag and lift coefficients together. If drag and lift are both represented as pressure loads, then the areas used to normalize the coefficients will differ (unless by chance the depth and width of the structure are equal). As a result, the magnitudes of the coefficients are not directly comparable that is, equal coefficients PAGE 38 38 will not produce loads of equal magnitude and they canno t be treated mathematically as components of a single force vector, which complicates coordinate transformations and other operations. For this reason, the term pressure coefficient is not used in this report, except when in reference to design codes that use the term. In this report, the term torque coefficient ( C T ) refers to the in plane moment that acts about the centroid of the cross section. This is a convenient choice of axis because it coincides with the axes of beam elements in most structural analy sis software. Loads calculated from C D C L and C T can be applied directly to beam nodes (located at the centroid of the cross section) to correctly model the two dimensional state of aerodynamic load. However, most design codes represent wind load as a un iform pressure load that produces a resultant force acting at a location called the center of pressure (Figure 3 3 ), which is typically assumed to correspond to the mid height of the cross section. For rea sons that are explained fully in Chapter 5 it is occasionally more convenient to work with a torque coefficient that acts about that center of pressure. In such circumstances, the term pressure torque coefficient ( C PT ) will be used to differentiate it from the C T which always acts about the centroid (Figure 3 4 ). A summary of the different types of aerodynamic coefficient used in this report is presented in Table 3 1 3 4 Current Wind D esig n P ractice in Florida Bridge structures in Florida are designed in accordance with the provisions of the Structures Design Guidelines (SDG ; FDOT, 2012 e ). As with most modern design codes, the wind load provisions in the SDG are based on Equation 3 8 with additional scale factors included to adjust the intensity of the wind load a ccording to the individual circumstances of the bridge. Specifically, Section 2.4 of the SDG gives the equation : PAGE 39 39 ( 3 9 ) where P Z is the design wind pressure (ksf), K Z is the velocity pressure exposure coefficient, V is the basic wind speed (mph), and G is the gust effect factor. The constant term, 2.56 10 represents the quantity 1 2 from equation 3 8 expr essed in derived units of (ksf)/(mph) 2 Each county in Florida is assigned a basic wind speed, V adapted from wind maps published by the American Society of Civil Engineers (ASCE 2006), which are based on statistical analyses of historical wind speed rec ords compiled by the National Weather Service. Statistically, V represents the peak 3 second gust wind speed for a 50 year recurrence interval. In other words, if the average wind speeds during every 3 second time interval were recorded over a period of 50 years, V is the expected value of the maximum speed that would be recorded. It is important to note that this does not mean that Florida bridges are only designed to resist 50 year wind loads. Different load combinations use load factors for wind that eff ectively adjust the recurrence interval up or down. For example, the Strength III limit state, as stipulated by the SDG, includes a wind load factor of 1.4, which increases the recurrence interval to approximately 850 years (FDOT 2009). Load combinations f or scenarios that do not include extreme wind speeds stipulate that the wind load be calculated using a basic wind speed of 70 mph, regardless of the location of the structure Basic wind speeds published by ASCE are based on measurements taken at an eleva tion of 33 ft and are not directly applicable to structures at other elevations. Wind that is closer to ground level is slowed by the effect of surface friction, resulting in a vertical wind gradient called the atmospheric boundary layer (Holmes, 2007). Th e purpose of the velocity pressure exposure coefficient, K Z is to modify the wind pressure load to account for differences in elevation. Because surface roughness of the terrain is known to reduce the steepness of the PAGE 40 40 gradient, ASCE divides terrains into three exposure categories, B, C, and D, and provides equations for each category. However, for simplicity, the SDG conservatively assumes that all Florida structures are in the Exposure C category. As a result, the equation for K Z in Florida is : ( 3 10 ) where z is the elevation above ground (ft). Note that K Z is equal to unity at an elevation of 33 ft (corresponding to the wind speed measurements) and that wind spee d is assumed to be constant for elevations of 15 ft or less ( Figure 3 5 ) Wind is characteristically gusty and turbulent, producing dynamic structural loads that can fluctuate significantly over short periods of time. However, it is simpler and more efficient to design structures to resist static loads. Furthermore, wind tunnel measurements of static force coefficients are typically performed in steady flow (with a major exception being site specific wind tunn el testing, which models a proposed structure along with its surrounding terrain for the express purpose of capturing turbulent loads). The gust effect factor, G modifies the static design wind pressure so as to envelope the effects of wind gustiness and dynamic structural response on peak structural demand. For aerodynamically rigid bridge structures, defined as those with spans less than 250 ft and elevations less than 75 ft, the SDG prescribes a gust effect factor of 0.85. By this definition, the vast m ajority of precast prestressed concrete girder bridges in Florida are aerodynamically rigid. It is noted that G actually reduces the design wind pressure on rigid bridges, reflecting the fact that peak gust pressures are unlikely to occur over the entire s urface area of such structures simultaneously (Solari and Kareem, 1998). The SDG further provides specific guidance on the calculation of wind loads during the bridge construction stage (as opposed to the calculation of wind loads on the completed bridge PAGE 41 41 s tructure). If the exposure period of the construction stage is less than one year, a reduction factor of 0.6 on the basic wind speed is allowed by the SDG. During active construction, the basic wind speed can be further reduced to a base level of 20 mph. T emporary bracing must be designed for three load cases: Girder Placement (construction active), Braced Girder (construction inactive), and Deck Placement (construction active). Calculation of wind pressure using Equation 3 9 requires that an appropriate pressure coefficient ( C P ) be determined for the structure under consideration. Pressure coefficients are provided by the SDG for several broad categories of bridge component as indicated in Table 3 2 In the Girder Placement and Braced Girder load cases noted above, pressure coefficients are needed for girders without deck forms or a completed deck in place. As Table 3 2 indic ates, the SDG provides two such values of C P depending on the shape of the girder cross section: C P = 2.2 for I shaped girders (Figure 3 6 ), and C P = 1.5 for box or U shaped girders (Figure 3 7 ) 3 5 Literature Review: Drag Coefficients for Bridge G irders The wind load provisions in the SDG are, for the most part, well supported by research. The main exception is the pressure coefficients (drag coefficients) prescribed for girders in partially erected bridges without deck forms or a completed deck in place. While experimentally measured drag coefficients have been published for simp le geometric shapes, truss members, buildings, and complete bridge superstructures, there has been little investigation of the aerodynamic properties of individual bridge girder shapes in the literature, and none specifically addressing the Florida FIB sha pes. In lieu of more specific information, the SDG pressure coefficients (noted in Table 3 2 ) are based on the assumption that the drag (or pressure) coefficient ( C D ) of a girder can be approximated by the C D of a rec tangle with the same width to depth ratio. Drag coefficients for rectangles with various width to depth ratios, taken from PAGE 42 42 Holmes (2007) and other sources, are shown in Figure 3 8 It is clear that there is sig nificant variation of C D as the width to depth ( W / D ) ratio changes. Also shown in the figure are W / D ranges for typical girder types common to the state of Florida. Finally, W / D values for the specific girder cross sectional shapes tested (in a wind tunnel ) in this study are also indicated (additional details regarding these shapes will be provided in Chapter 4 ). While drag coefficients for typical concrete bridge girder I shapes could not be located in the literature, t here have been some published studies of thin walled I shapes characteristic of rolled steel members (Table 3 3 ). Maher and Wittig (1980) measured C SD C SL and C T for a truss bridge member with a width t o depth ratio of 1.23. In head on wind (0 angle of attack), the C D was reported as 1.9. Similarly, Grant and Barnes (1981) performed wind tunnel tests on several structural members, including an I shape with a width to depth ratio of approximately 0.64 (e xact dimensions were not given) which had a C D of 2.2. In a general reference text, Simiu and Miyata (2006) provided several plots of drag coefficients for a wide range of shapes. These plots included two data points for I shapes with ratios of 0.5 0 ( C D = 1.87) and 1.0 0 ( C D = 1.78). Some of the most widely published coefficients for I shapes were originally produced by the Swiss Society of Engineers and Architects (SIA) for Assumptions, Acceptance and Supervision of Building s (1956, English translation reproduced in Davenport, 1960). Normen 160 contained pressure coefficient specifications for a wide variety of structures and structural components that, at the time, were considered the most refined and comprehensive treatment of the subject (Davenport, 1960). Tables of drag and lift coefficients from Normen 160 including I shapes with width to depth ratios of 0.48 ( C D = 2.05) and 1.0 0 ( C D = 1.6) have since been reproduced in multiple sources, including the Commentary of the PAGE 43 43 Na tional Building Code (NBC) of Canada (NRC, 2005; Sachs, 1978; Scruton and Newberry, 1963). The exact origins of the coefficients are unknown, but the NBC commentary states that Other jurisdictions provide varyi ng levels of guidance regarding drag coefficients for I shapes. In Japan, the de facto design code (AIJ, 2004) includes a C D of 2.1 for an I shape with a width to depth ratio of 0.5 0 The AIJ commentary cites an unobtainable Japanese language paper as the source of this value. Great Britain, like the FDOT, assumes that the girder cross sections are aerodynamically similar to rectangles, and provides a plot (reproduced in Figure 3 8 ) for selecting the coefficient based on the width to depth ratio of the cross section (BSI, 2006). edged structural ESDU, a non governmental organization that produces engineering reference materials, has performed its own literature review of drag coefficients for structural members and it has published a reference (ESDU, 1982) that synthesizes data from multiple sources, including several of those discussed above and several foreign langu age sources. Drag coefficients are provided for I shapes with width to depth ratios of 0.5 0 ( C D = 1.94) and 1.0 0 ( C D = 1.62), with an estimated uncertainty of approximately 15%. Interpolation between the two data points is encouraged. All of the I shapes investigated in the literature are for basic truss or building members and did not include any width to depth ratios less than approximately 1/2. However, most steel I shapes used in long span bridge girders have width to depth ratios that range roughly fr om 1/6 to 1/3. Because C D tends to vary with width to depth ratio, there is no reason to believe that the results of these studies are directly applicable to steel bridge girders. Furthermore, when the data PAGE 44 44 are plotted (Figure 3 9 ), it becomes clear that the equivalent rectangle is a poor (albeit conservative) predictor of aerodynamic properties. Regarding box girders, the SDG provides a value of 1.5, which is a common choice for box shaped bridge decks However, before the deck is cast, the top of the girder is open. A search of the literature found only one source that discusses the aerodynamic properties of open top box girders. Myers and Ghalib (n.d.) used a two dimensional computational fluid dynami cs analysis to calculate the drag on a pair of such girders. While coefficients for the individual girders were not provided, they concluded that drag coefficients can be significantly higher on a girder with an open top. PAGE 45 45 Table 3 1 Summary of aerodynamic coefficients Symbol Coefficient n ame Description C D Drag Component of force in horizontal (lateral) direction C L Lift Component of force in vertical direction C SD Stream Drag Component of force parallel to wind stream C SL Stream Lift Component of force perpendicular to wind stream C P Pressure Alternat ive name for C D C T Torque Torque measured about centroid C PT Pressure Torque Torque measured about center of pressure Table 3 2 Pressure coefficients in FDOT Structures Design Guide (SDG) Bridge c omponent C P Substructure 1.6 Girders with deck forms 1.1 Completed s uperstructure 1.1 I shaped b ridge g irders 2.2 Box and U sh aped g irders 1.5 Table 3 3 Drag coefficients ( C D ) of thin walled I shapes Width to depth ratio ( W / D ) Data s ource 0.48 0.5 0 0.64 1. 0 0 1.23 Maher and Wittig (1980) 1.90 Grant and B arnes (1981) 2.20 Simiu and Miyata (2006) 1.87 1.78 SIA Normen 160 (1956) 2.05 1.60 AIJ (2004) 2.10 ESDU (1982) 1.94 1.62 Figure 3 1 Two dimensional bridge girder cross section with in plane line loads PAGE 46 46 Figure 3 2 Definition of C D C L C SD and C SL (shown in positive dire ction except when noted) Figure 3 3 Center of pressure of a bridge girder Figure 3 4 Definition of C T and C PT (shown in positive direction) PAGE 47 47 Figure 3 5 Velocity pressure exposure coefficient used by FDOT Figure 3 6 Examples of I shaped girders (ste el plate girders and Florida I B eams) for which C P = 2.2 ( per FDOT 2012e) Figure 3 7 Example open top box girder cross section for which C P = 1.5 ( per FDOT 2012e) PAGE 48 48 Figure 3 8 Drag coefficients for rectangular sections w ith various width to depth ratios Figure 3 9 Drag coefficients for plate girder shapes and rectangles with various width to depth ratios PAGE 49 49 CHAPTER 4 WIND TUNNEL TESTING 4 1 Introduction A key goal of this research was the characterization of temporary bracing requirements for long span prestressed concrete girders, prior to the casting of the deck. Wind tunnel tests were an important component of this investigation because lateral wind loads tend to control bracing designs. However, the aerodynamic properties of most common bridge girder shapes have not been widely studied, so additiona l goals of this research were to use wind tunnel testing to quantify aerodynamic properties (e.g., drag coefficients) and to develop a simplified loading procedure that can be applied to wide variety of common girder shapes. Because nearly all bridge struc tures are made up of multiple girders positioned side by side, it was necessary to investigate the effect of shielding (i.e., aerodynamic interference), in which the windward girder acts as a wind break and reduces the total force on subsequent girders. Wi nd tunnel tests were therefore performed on groups of identical girders positioned in several different testing configurations 4 2 Testing C onfigurations Five different girder cross sectio nal shapes (Figure 4 1 ) were selected as being representative of a wide range of modern Florida bridges: 78 in ch deep Florida I Beam ( ) : Of the most commonly used FIB shapes, the 78 FIB is the deepest and is most susceptible to instability. 45 in ch deep Florida I Beam ( ) : All FIB shapes have identical flanges, with the differences in girder depth arising from differences in the height of the web. The 45 FIB was included in the study to quantify the effect of changing the FIB depth, and to ensure that the resulting design loads would be applicable to a range of FIB shapes. Wide flange plate g irder ( WF Plate ) : Drag coefficients of I shaped girders have been studied for width to depth ratios ranging from 1:1 to 2:1 (see Chapter 2 ). However, built up steel plate girders commonly used to support bridge decks tend to be much deeper than they are wide. The WF Plate girder considered in this study has an 8 ft deep web and PAGE 50 50 2 8 wide flanges, resulting in a width to depth ratio of 3:1, representing the approximate lower bound for bridge girders. Narrow flange plate g irder ( NF Plate ) : The NF Plate girder is identical to the WF Plate, but with flanges that are only one ha lf as wide. This gives it an upper bound width to depth ratio of 6:1. Open top box girder ( Box ) : The aerodynamic properties of box girder bridges have been studied, but experimental studies have not been performed on box girders with an open top (without t he deck in place). A survey of existing box girder bridges was used to develop a representative 6 ft deep cross section. These sections were tested individually, as well as in groups of 2, 5, and 10 (Fig. 4 2 ). Fully dimensioned drawings of these girder cross sections and each test configuration are included in Appendix A The full set of test configurations is available in Table 4 1 Each testing configuration can be described by a unique combination of spacing, cross slope, and number of girders (Fig 4 3 ): 4 2 1 Number o f G irders In addition to tests of individual girders, wind tunnel tests were performed on 2 girder, 5 girder, and 10 girder configurations. Each girder in a given test configuration was referred to by a sequential number starting with the windward girder, G1. In most configurations, individual force measurements were recorded for each girder. The only exceptions were the 5 girder configurations of the NF Plate, 78 FIB, and 45 FIB where measurements were only recorded for girders G1 G3. 4 2 2 Spacing Spacing refers to the horizontal center to center distance betwe en girders. Characteristic maximum and minimum spacings were determined for each girder type (Table 4 1 ) based on a survey of existing bridge designs and consultations with the FDOT. Each testing configur ation for a given girder type uses either the maximum or minimum spacing. PAGE 51 51 4 2 3 Cross S lope Most bridge decks are designed with a cross slope of 2% or gre ater, and the girders are usually aligned vertically along that slope so that they can evenly support the deck. Therefore, all of the FIBs and plate girders were tested at +2% cross slope. Steel bridges can have a greater amount of horizontal curvature, so higher cross slopes are often included to improve vehicle handling. To account for this, the WF Plate girders were also tested in configurations with +8% cross slope. Additionally, because the top and bottom flanges of the FIB cross sections differ in wid th and shape, the exposed portions of the shielded girders have a different shape if the cross slope is negative (or, equivalently, if the wind blows from the opposite direction). To account for this, the FIB sections were also tested in configurations wit slope. In contrast to I shaped girders, box girders are not aligned vertically when supporting a cross sloped deck. Instead, the girders are inclined to follow the cross slope (see Figure 4 5 ). As a result, the box girders were only tested in 0% (unsloped) configuration but the range of tested wind angles was increased, as described below. 4 2 4 Wind Angle In practical bridge construction situations, the direction of wind flow will not always be perfectly horizontal. To account for the natural variation in wind angle (and at the recommendation of a commercial wind tunnel test facility) each bridg e configuration was tested 4 4 ). In the case of the box girder, such a change in wind angle is geometrically equivalent to the way the girders are rotated to support a cross sloped deck (Figure 4 5 ). As a result, the box girder was of win d angle and 5 (8.7%) of cross slope. PAGE 52 52 4 3 Testing P rocedure The Boundary Layer Wind Tunnel Laboratory at the University of Western Ontario (UWO) was contracted to fabricate the test specime ns and to perform all wind tunnel measurements. Based on the size of the UWO wind tunnel, the girder models were constructed at reduced scale, with air flow properties similarly adjusted so that the resulting forces would be applicable at full scale. All t esting was performed in smooth flow, with turbulence intensities less than 0.5%. Because the tested cross sections were sharp edged, it was expected that the measured wind forces would not be sensitive to Reynolds number. The specific Reynolds numbers at w hich the tests were performed, and results reported, are shown in Table 4 2 Further, the assertion that wind forces would not be sensitive to Reynolds number was verified by UWO by additionally performing selected tests at lower Reynolds numbers (approximately 33% smaller than those shown in Table 4 2 ). Results from these additional tests did not reveal any obvious Reynolds number sensitivities. The scaled g irder models were all 7 ft long (equivalent to 175 ft and 196 ft girders at full scale) and were constructed to be fully rigid, without exhibiting any aeroelastic effects. An adjustable frame was used to keep the girders properly oriented relative to each other in each test configuration. To measure wind induced girder forces at varying wind angles of attack, the entire bridge cross sectional assembly was rotated in place relative to the wind stream. Wind forces on the girders in each test configuration wer e measured individually with a high precision load balance that recorded the time averaged horizontal load (drag), vertical load (lift), and torque (overturning moment). These loads were then normalized to produce the aerodynamic coefficients for drag ( C D ) lift ( C L ), and torque ( C T ). Finally, the torque coefficient was adjusted so that it represented the torque about the centroid of the section, rather than the PAGE 53 53 torque about the point of measurement (which was at mid height for the I shaped girders and at a n arbitrary point for the box girders). For additional details regarding the wind tunnel test procedures, please see Appendix E PAGE 54 54 Table 4 1 T esting configurations Cross section Min s pacing Max s pacing Tested w ind a ngles (deg) WF Plate 10 ft 14 ft NF Plate 10 ft 14 ft 78 FIB 10 ft 13 ft 45 FIB 10 ft 13 ft Box 20 ft 22 ft 0, +5, +10 2 g irder m odels 5 g irder m odels 10 girder m odels Cr oss section Min s p. Max s p. Inst. Min s p. Max s p. Inst. Max Sp. Inst. WF Plate 2%, 8% 2%, 8% All 2%, 8% 2%, 8% All 8% All NF Plate 2% 2% All 2% 2% G1 G3 78 FIB 2% 2% All 2% 2% G1 G3 All 45 FIB 2% 2% All 2% 2% G1 G3 All Box 0% 0% All Table 4 2 Wind tunnel test scaling Cross section Model s cale Reynolds n umber WF Plate 1:25 77000 NF Plate 1:25 77000 78 FIB 1:28 56000 45 FIB 1:28 33000 Bo x 1:25 59000 Figure 4 1 Girder cross sections used in study (drawn to scale) PAGE 55 55 Figure 4 2 Girder groupings investigated in study (girder to girder spacing not drawn to scale; cross slopes vary) Figure 4 3 Paramete rs definitions for each testing configuration Figure 4 4 Wind angle sign convention PAGE 56 56 Figure 4 5 Equivalen ce between wind angle and cross slope for box girders PAGE 57 57 CHAPTER 5 WIND TUNNEL RESULTS AND ANALYSIS 5 1 I ntroduction Wind tunnel tests were performed on the bridge girder test conf igurations described in Chapter 4 In addition to aerodynamic coefficients for individual girders, groups of laterally spaced girders were teste d to quantify shielding effects, identify trends, and assess the importance of girder spacing, girder cross slope and wind angle. From analysis of the results, a set of simplified design loads was developed for assessing the stability of a single girder or a system of braced girders. Additionally, a separate loading procedure was identified for conservatively predicting internal forces in brace members. The complete set of wind tunnel test data is available in Appendix B 5 2 Aerodynamic Coefficients for Individual G irders Of the wind tunnel tests that were performed, the most fundament and broadly applicable results were the aerodynamic coeffi cients measured for the individual cross sectional shapes (Table 5 1 ). In addition to being measured in level (0) wind, the aerodynamic coefficients were measured in a range of angles of attack in order to determine how the coefficients were affected by variation in wind angle. With regard to girder stability, the drag coefficient (Figure 5 1 ) is the most critical aerodynamic coefficient. Of the five (5) cross sectional shapes that were tested, the plate girder sections had the highest drag coefficients and were the least sensitive to wind angle, with both the wide flange and narrow flange varieties having coefficients that ranged from 2.12 to 2.13. The FIB sections had comparatively lowe section had by far the most sensitivity to wind angle, ranging from 1.68 to 1.93, with a median PAGE 58 58 value of 1.81 in level (0) wind. It is worth noting that the FDOT SDG currently overpredicts the C D of plate girders and FIBs, but under predicts the C D of box girders with an open top. Lift coefficients ( C L ) for the five cross sectional shapes are plotted in Figure 5 2 Because of their vertical symmetry, the plate girder sections generate negligible lift, with no coefficient exceeding a magnitude of 0.05. In contrast, the unsymmetrical FIB sections (with differently shaped flanges on top and bottom) generate significant uplift. For both FIBs, lift is fr om 0.24 to 0.48. Finally, because of its angled webs, the box girder generates far more uplift than the other sections, with C L nearly as much lift ( C L = 1.62) as it does drag ( C D = 1.68) To rque coefficients ( C T ) for the five cross sectional shapes are plotted in Figure 5 3 Qualitatively, the torque data are broadly similar to the lift data in that the symmetrical plate girder sections generate negligible torque the FIB sections generate a small amount, and the box Another differ ence in coefficient trends is that for the box girder, in terms of relative magnitude, C T (ranging from 0.69 to 0.85) is less sensitive to wind angle than C L (ranging from 0.71 to 1.62). (Recall from Chapter 3 that w hile the magnitudes of the force coefficients, C D and C L can be directly compared, C T a moment coefficient, is normalized differently, so absolute comparisons between the numeric values of C T and the values of C D and C L are meaningless.) PAGE 59 59 5 3 Examination of Shielding T rends Groups of multiple girders were tested in several differ ent configurations (see Chapter 4 ) to quantify shielding effects. Because th e largest variety of tests was performed using WF plate girders, data from those tests will be used for demonstration when discussing most shielding trends. In the case of I shaped girders (plate girders and FIBs), the most fundamental shielding trend that was observed was a basic down then up pattern (Figure 5 4 ). While the windward girder (G1) experiences the highest drag force in the system, there is a sharp and immediate reduction in drag so that the drag coef ficient of the first shielded girder (G2) becomes negative (indicating that the drag force acts in the opposite direction, against the wind stream). In some cases, such as the one shown in Figure 5 4 drag someti mes continues to decrease, so that the girder with the most negative drag force is either G2 or G3. Upon reaching the most negative value, drag then slowly increases for subsequent girders, with the drag coefficient gradually becoming less negative and the n increasingly positive. The first shielded girder with a positive drag coefficient is generally G3, G4 or G5. As can be seen in Figure 5 4 girder spacing and, to a lesser extent, cross slope can affect the shie lding pattern, but not enough to disrupt the overall trend. In general, a larger spacing decreases the total amount of shielding, but, as will be discussed, the effect of cross slope is dependent on the wind angle of attack. In addition to shielding effect s, which propagate down stream, the presence of shielded girders can modify the drag on girders that are farther up stream (Figure 5 5 ). In the case of a two girder system, the presence of the shielded girder (G 2) increases the total drag on the windward girder (G1). However, as additional shielded girders are added, they tend to reduce the drag on up stream girders. As a result, the largest drag force drag on the windward girder (G1) PAGE 60 60 and the most negative drag o n the first shielded girder (G2) usually both occur in the two girder configuration. All of the previously discussed examples have been in perfectly level wind (0). However, changing the wind angle can alter the shielding pattern. The effect of wind angle tends to be strongest on the more leeward (down stream) girders (Figure 5 6 ). Note that in this example (with 8% cross slope), a wind angle of +5 provides the most total shielding (i.e., the most overa Whether a change in wind angle will increase or decrease the total shielding in a particular testing configuration depends on the cross slope of the system (Figure 5 7 ). Maximum shielding is achieved when the wind angle is equal to the angle of the cross slope, so that as much of the shielded girder as possible is blocked along the direction of the wind stream. As the differ ence between the cross slope and wind angle increases, larger portions of the shielded girders are exposed and the total amount of shielding decreases. This also explains the previously noted trend that increasing girder spacing reduces shielding, as this exposes more of the girder for a given angle. However, while changing girder spacing can amplify or attenuate the shielding effect, it does not alter the sign of the C D values for shielded girders. From knowledge of the cross slope, it is possible to deter mine best and worst case wind angles (Table 5 2 ). When the shielding is close to maximum, the drag on leeward girders tends to plateau (Figure 5 8 ), even if the plateau In the tested cases where the wind angle was most different from the cross slope (Figure 5 9 ), the drag continued to increase on each subsequent girder, un til reaching either a plateau or a PAGE 61 61 change in direction at approximately G9. Note that the WF plate girder almost reaches the same amount of drag on G9 as it does on G1. Box girders were only tested in groups of two (2), with girders spaced at 20 ft and 22 ft (Figure 5 10 ). Girder spacing was found to have almost no effect on the drag force on either the windward girder (G1) or the shielded girder (G2). By comparison, the wind angle was a much stronger influence. In the most sen spaced group resulted in the C D 5 4 Effective Drag C oefficient As dis cussed in Chapter 3 the Structures Design Guidelines (SDG ; FDOT, 2012e ), along with most design codes, assumes that horizontal wind can be approximated as a uniform pressure load. This is convenient because a single coefficient ( C D ) is all that is necessary to characterize the aerodynamic properties of a structure. However, the results of the wind tunnel tests have shown that aerodynamic loads on bridge girders can include lift forces and torques that are too large to be considered negligible. Lift coefficients for I shaped girders (FIBs and plate girders) can be as large as 0.5 (Figure 5 11 (Figure 5 12 ). The additional structural demand contributed by lift and torque should therefore be included when evaluating girder stability. As will be presently shown, it is possible to define an effective drag coefficient ( C D,eff ) that conservatively combines the e ffects of both drag and torque into a single coefficient that can be used in design codes as if it were a standard drag coefficient. (It is not possible to incorporate lift in the same manner, but the effect of lift will be accounted for in the proposed gi rder capacity equations presented later in this report.) C T represents aerodynamic torque measured about the centroid of the section. However, in the SDG, wind load computed from C D is applied at the center of pressure which is assumed PAGE 62 62 to be at mid height of the girder. In the case of asymmetric girder shapes such as FIBs, the center of pressure is separated from the centroid by a vertical distance, (Figure 5 13 a). For consistency, before C T can be incorporated into C D,eff it must be transformed into the pressure torque coefficient ( C PT ), which represents the torque about the center of pressure (rather than about the cent roid). An equation for calculating C PT in terms of C D and C T can be derived in closed form. As discussed in Chapter 3 C D and C T represent a force and moment, and applied at the centroid (Figure 5 13 b). From principles of statics, this state of load is equivalent to a single 5 13 c), where: ( 5 1 ) When the same state of load is considered from the center of pressure (Figure 5 13 d), the moment that is generated ( M P ) is equal to : ( 5 2 ) which can be combined with Equation 5 1 to create the expression: ( 5 3 ) Based on concepts presented in Chapter 3 these forces are related to their corresponding aerodynamic coefficients as : ( 5 4 ) where D is the depth of the girder cross section. After substituting the expressions above into Equation 5 3 and solving for C PT the final equation for the transformation is: PAGE 63 63 ( 5 5 ) Once the value of C PT has been determined, it is necessary to represent it in the form of a drag coefficient so that it can be added to C D to form C D,eff In other words, the moment lo ad, M P (Figure 5 14 a) must be replaced by a drag force, F M that produces an equivalent amount of moment. After normalizing that force in the manner of a drag coefficient, it can be directly added to C D : ( 5 6 ) In the field, drag force generates an equal but opposite reaction force at the support (Figure 5 14 b), producing a force couple with an arm eq ual to half of the girder depth, D As a result, the force necessary to generate M P is: ( 5 7 ) Substituting the expression for M P from Equation 5 4 into the equation above yields: ( 5 8 ) which can be substituted into Equation 5 6 resulting in: ( 5 9 ) In this expression, the term 2 C PT can be thought of as a correction factor that ensures equivalence of moment by giving up equivalence of lateral force. In the majority of cases, the resulting value of C D,eff is greater than C D meaning that C D,eff conservatively overpredicts the amount of lateral force in the system. However, in some cases (e.g. when C D is positive and C PT is negative), the C D,eff expression given in Equation 5 9 underpredicts the amount o f lateral force PAGE 64 64 in order to achieve equivalence of moment. In these cases, the basic (rather than effective) C D coefficient is the conservative choice because it overpredicts moment but correctly predicts lateral force. Ensuring that neither lateral force nor moment are underpredicted can be achieved by redefining C D,eff as: ( 5 10 ) The definition of C D,eff given in Equation 5 10 was therefore used to develop the proposed design loads in this study. It is important to note that Equations 5 5 and 5 9 are only valid for aerodynamic coefficients that are normalized by girder depth. T his is true of all coefficients presented in this report, but these equations cannot be applied to coefficients from other sources that are normalized by different reference lengths. 5 5 Pr oposed Wind Loads for D esign After calculating effective drag coefficients for every wind tunnel test conducted in this study, the results were synthesized into simplified loads suitable for use in the design of girder bracing. Potential design loads were evaluat ed according to three criteria: Conservatism : Design loads must be conservative or they are useless. However, overly conservative design loads are also undesirable. Part of the motivation for studying shielding was to allow for reduced (i.e., less c onservative) wind loads on shielded girders. Generality : To maximize utility, design loads must be applicable to as wide a range of designs as possible, including cross sections, spacings, or cross slopes that were not directly tested. Consequently, attemp ting to develop design loads that recreated the tested load measurements as closely as possible was considered counterproductive. Simplicity : Simplicity in design codes is advantageous, but it must be balanced against the drawbacks of overconservatism. In general, the addition of significant mathematical or procedural complexity in exchange for a slight reduction in conservatism was considered undesirable. PAGE 65 65 When designing girder bracing for lateral stability, the worst case distribution of wind load is when the total drag force (i.e., the sum of the force on each girder in the system) is as large as possible. Negative drag on any girder is transferred through the braces to the rest of the system and acts to resist collapse. (A separate load case that maximize s the internal forces of individual brace members is described in the next section.) Therefore, design loads for stability should be based on configurations where girder drag is the largest (or least negative). Additionally, it is worth noting that because the highest drag cases tend not to coincide with high torque cases, the use of C D,eff instead of C D does not significantly increase total structural demand or conservatism (Figure 5 15 ). Girder spacing and cross slope were rej ected as possible input parameters meaning that, for example, two otherwise identical bridge designs with different girder spacings would have the same design loads because of the relatively small effect they have in isolation (recall Figure 5 4 ) and the complexity of their interactions with other parameters (such as wind angle). Additionally, because (for budgetary reasons) the 10 girder groups were only tested at the maximum spacing and cross slope, including sp acing and cross slope as parameters would have required extrapolation of their effect on the 5 most leeward girders (G6 G10). Instead, it was considered more conservative to envelope the C D,eff values for every combination of spacing and cross slope that w as tested in order to identify the worst case. Similarly, the width to depth ratio of the cross section was rejected as an input parameter because of the small differences observed between the WF Plate and NF Plate girders and IB, and because data was not collected for intermediate shapes. Instead, only the type of section plate girder or Florida I Beam was considered significant enough to modify the design loads. PAGE 66 66 Finally, the number of girders in the system was considered as an input parameter. While it was observed that adding additional downwind girders tended to reduce wind force on up stream girders (recall Figure 5 5 ), the fact that data is only available for 2, 5, and 10 girder groups makes the effect impossible to predict with any certainty. For example, because the drag on the windward girder (G1) increases in a 2 girder group and decreases in a 5 girder group, it is unclear whether it would increase or decrease for a 3 girder group. For this reason, it was decided to assign a single design load to each girder that would envelope all cases. Also evident from Figure 5 5 is the fact that the drag coefficients in a 5 girder model tend to rebound from the negative range at a higher rate than the 10 girder models, with the result that the drag force on the final girders is approximately equal. It is conceivable that this pattern holds true for 3 girder models as well. Indeed, in the case relative to the girder spacing, than the other sections), G3 sometimes sustains the maximum drag out of all the shielded girders (recall Figure 5 8 ). Therefore, it was assumed that gi rders G3 G10 are all potentially capable of being exposed to the maximum shielded drag. The final proposed desi gn loads are based on the basic down then up trend that was observed in all tested configurations. An initial pressure coefficient ( C P ) (i.e., C D ,eff ) is assigned to G1 based on the type of girder section: 2.5 for plate girders (Figure 5 16 ) and 2.0 for FIBs (Figure 5 17 ). Girder G2 is assigned a C P of 0 (i.e., no load) whil e G3 and all subsequent girders are assigned a C P equal to half of the load on the windward girder. 5 6 Proposed Procedure for Calculation of Brace F orces As previously discussed, system le vel stability is most critical when the total unidirectional wind load on the system is as high as possible. Brace designs must provide sufficient stiffness to keep the system stable under such loading conditions. However, to reach a PAGE 67 67 stable equilibrium pos ition, it is equally important that the strengths of the individual brace members not be exceeded, because an individual brace failure can initiate a progressive collapse. Individual brace forces (as opposed to overall system lateral loads) are maximized w hen differential wind forces on adjacent girders act to maximize compression of the braces that connect the girders together. Because the design loads proposed in the previous section are intended to maximize overall system loads, and not individual brace compression forces, a separate load case is required for evaluating brace strength in compression. Wind force differentials are always highest between the windward girder (G1) and the first shielded girder (G2), with the positive drag on G1 and the negativ e drag on G2 combining to produce a total compressive brace load (Figure 5 18 ) that consistently exceeds that of any other pair of adjacent girders. Recall from Figure 5 5 that the 2 girder configuration (with no additional down stream girders) has both the most positive drag on G1 and the most negative drag on G2. Because typical erection sequences always include a two girder phase (even if only briefly), the wind load on such systems was selected as the controlling load case for brace force determination. Upon checking every tested 2 girder configuration, the worst case compression load for plate girders was a system with a C D,eff 5 19 ). Similarly, for FIBs, the worst case had a C D,eff 5 20 ). The combined effect of C D,eff for G1 and G2 is then equivalent to using a single combined coefficient of 2.88 for plate girders and 2.60 for FIBs. To ensure that that brace force calculations remain conservative for untested girder configurations, it is recommended that these values be rounded up to 3.0 for plate girders and 2.75 for FIBs. Even with well defined wind loads, a pair of braced girders is a three dimensional structural system with several sources of variability. A simplified structural model (Figure 5 21 ) PAGE 68 68 is therefore proposed for conservative determination of brace forces using the loads described above. This approach is not restricted to any particular brace configuration: the core of the approach is a structural model of the brace, the configuration and implementation of which is left to the judgment of the Engineer of Record. Fixed boundary conditions are applied at every node where the brace connects to G2, while rigid links are used to link all of the G1 connec tion nodes to a pin located at the base of G1. An additional rigid link connects the base pin to the center of pressure where the total tributary wind load is applied as a single horizontal force. In the field, the connections between the brace elements an d the girders are likely to be neither perfectly fixed nor perfectly pinned, but rather achieve some intermediate level of moment transfer. If desired and appropriate, brace elements may include partial end releases at the girder connection points. This ca n lessen the resulting brace forces somewhat. However, it is important not to underpredict the amount of fixity in the connections or unconservative results may be obtained. Given the significant amount of uncertainty that is generally involved in such det erminations, it is recommended that full fixity be provided in the model unless reliable partially restrained connection information is available PAGE 69 69 Table 5 1 Aerodynamic coefficients of b ridge girder cross sectional shapes 0 wind Minimum value Maximum value SDG Cross section + C D + C L + C T + C D + C L + C T + C D + C L + C T C D WF Plate + 2.12 + 0.03 + 2.12 + 0.03 + 2.13 ++ 0.00 + 0.03 2.2 NF Plate + 2.12 + 0.00 + 2.12 + 0.00 + 2.13 + + 0.01 2.2 78" FIB + 1.89 + 0.15 + 0.11 + 1.89 + 0.13 + 0.10 + 1.91 ++ 0.18 + 0.12 2.2 45" FIB + 1.85 + 0.37 + 0.04 + 1.81 + 0.24 + 0.01 + 1.85 ++ 0.48 + 0.08 2.2 Box + 1.81 + 1.22 + 0.73 + 1.68 + 0.71 + 0.69 + 1.93 ++ 1.62 + 0.85 1.5 Table 5 2 Extreme combinations of tested wind angle and cross slope Cross slope Ideal shielding angle Best tested case Worst tested case +2% +1.15 + 0 + 0 +5 +8% +4.57 +5 Figur e 5 1 Effect of wind angle on individual girder drag coefficients ( C D ) PAGE 70 70 Figure 5 2 Effect of wind angle on individual gir der lift coefficients ( C L ) Figure 5 3 Effect of wind angle on individual torque coefficients ( C T ) PAGE 71 71 Figure 5 4 Drag coeffici ents of WF Plate girders in 5 gi r der configurations (0 Wind) Figure 5 5 Effect of adding additional girders (WF Plate, 14 ft spacing, 8% cross slope, 0 Wind) PAGE 72 72 Figure 5 6 Effect of wind angle on C D (WF Plate girder, 14 ft spacing, 8% cross slope) A B C Figure 5 7 Interaction between wind angle and cross slope A ) +2% cross slope B slope C ) 8% cross slope PAGE 73 73 Figure 5 8 Ten (10) girder models tested at wind angles producing maximum shielding Figure 5 9 Ten (10) girder models tested at wind angles producing minimum shielding PAGE 74 74 Figure 5 10 Effect of wind angle on two (2) Box girder system drag coefficients ( C D ) Figure 5 11 Lift coefficients on all I shaped girder test configurations (plate girders and FIBs) PAGE 75 75 Figure 5 12 Torque coefficients on all I shaped girder test configurations (plate girders and FIBs) PAGE 76 76 A B C D Figure 5 13 Transformation of C T to C PT A) Section dimensions. B) Force/moment pair at section centroid. C) Equivalent elevated f orce. D) Force/moment pair at center of pressure. A B Figure 5 14 Moment load expressed as equivalent drag force A) Moment ab out center of pressure. B) Equivalent drag force. PAGE 77 77 A B Figure 5 15 Comparison between maximum C D and maximum C D,eff for. A) All plate girder sections. B ) A ll FIB sections PAGE 78 78 Figure 5 16 Proposed design loads for plate girders Figure 5 17 Proposed design loads for FIBs PAGE 79 79 A B Figure 5 18 Representation of positive and negative drag loads as a combined compression load Figure 5 19 Proposed brace force design loads for plate girders PAGE 80 80 Figure 5 20 Proposed brace force design loads for FIBs PAGE 81 81 A B Figure 5 21 Simplified brace force analysis. A ) Example X brace B ) equivalent structural model for brace force determination PAGE 82 82 CHAPTER 6 BEARING PADS 6 1 Introduction When analyzing the stability of girder systems, it is important to consider the support conditions of the girders. In order to determine the support stiffness in each degree of freedom, it is necessary to calculate bearing pad stiffness in each of four pad deformation modes: shear, torsion, compression (axial) and rotation. While relatively simple methods are available for the calculation of s hear and torsional stiffnesses, the calculation of axial and rotational stiffnesses requires more advanced calculation methods. Empirical equations for calculating the compressive stiffness of a pad are available in the literature (Gent, 2001; Stanton et a l., 2008; Podolny and Muller, 1982), but they frequently produce results that differ significantly from each other One proven reliable method for quantifying compressive pad stiffness is the finite element method, but most bridge engineers have limited exp erience in modeling incompressible materials such as elastomer. In addition to axial deformations, bearing pads are susceptible to roll rotations about two orthogonal axes; roll about the transverse centerline ( bending roll ) is typically the result of the end rotations of the girder as it bends about its major axis, while rotation about the longitudinal centerline ( overturning roll ) corresponds to overturning rotations of the girder at the supports. Estimation of these stiffnesses is often required for cons truction and design calculations, but methods for calculating such stiffnesses are not comprehensively addressed in the literature. For example, overturning roll stiffness at the supports is of particular importance during the construction (prior to castin g of the deck) of long span prestressed concrete girder bridges, as it can have a significant influence on the lateral stability of an unbraced girder, and is sometimes the only source of structural resistance to overturning moments generated by lateral lo ads (e.g., PAGE 83 83 wind) or eccentric gravity loads. Recent experimental measurements of bearing pad roll stiffness (Consolazio et al., 2012), which extended into the nonlinear range, have provided an opportunity to develop an empirical calculation method capable of approximating roll stiffness while preserving nonlinear effects. The types of bearing pads under consideration in this study consist of rectangular steel plates separated by layers of neoprene rubber (Figure 6 1 ). An external neoprene cover layer, typically thinner than the internal layers, surrounds the pad on all sides, sealing the steel reinforcement against corrosive agents in the environment. During the vulcanization process, the neoprene becomes fully bon ded to the steel. Because the elastic modulus of the steel is so much greater than that of the neoprene, it can be considered to be effectively rigid. 6 2 Behavior of Pads in Compression El astomeric rubbers, such as neoprene, are almost completely incompressible (i.e., with a > 0.49), and when subjected to a uniaxial compressive force, they tend to expand laterally to preserve their volume. However, in a steel reinforced be aring pad, the steel plates are stiff enough to effectively restrain all movement of the neoprene at the steel neoprene interfaces. As a result, when a pad is compressed, the neoprene layers respond by bulging outward at the edges (Figure 6 2 ). Restraint of this expansion by the steel plates makes the pad much stiffer in compression than an unreinforced pad with equivalent thickness and volume of elastomer. At extreme levels of compression, the stiffness becomes n onlinear as the bulging displaces a significant portion of elastomer outside of the primary load path, reducing the effective layer thickness and stiffening the pad. However, if the pad has been properly sized according to the provisions of AASHTO (201 0 ) o r similar, then determining just the initial linear stiffness is sufficient for most relevant bridge engineering calculations. PAGE 84 84 One consequence of the characteristic bulge response is an uneven distribution of compressive stiffness throughout the pad such t hat local compressive stiffness at any point is a function of the distance from the center of the pad. Near the center, there is a large amount of surrounding rubber that must be displaced laterally in order for the neoprene layers to bulge at the edges. I n contrast, near the edges of the pad, the rubber is less confined and comparatively little force is required for the rubber to bulge. This confinement effect can be demonstrated by prescribing a uniform downward displacement on the top surface of a finite element model of a layer of neoprene and plotting the distribution of the reaction forces (Figure 6 3 ). Because every point on the pad is compressed (deformed) vertically the same amount, the react ion force distribution is proportional to the distribution of local stiffness. Gent (2001) and others have published correlations between the axial stiffness of a bearing pad and the shape factor S of the internal elastomer layers of the pad. The shape f actor is a dimensionless ratio between the load area (i.e. plan view area of the pad) and the bulge area (i.e. the perimeter area). Most shape factor based methods use the same basic functional form: ( 6 1 ) (where B a is an empirically determined constant) to calculate an effective elastic modulus for compression, E c that incorporates the additional restraint provided by the steel reinforcement. Stanton et a l. (2008) have refined this approach, providing a method for determining B a that takes into account the bulk compressibility of the elastomer. In contrast, Podolny and Muller (1982) have provided an empirical formula: ( 6 2 ) PAGE 85 85 that does not incorporate the shape factor, but does include a constant, C that changes depending on the aspect ratio of the pad. 6 3 Behavior of Pads in Roll Rotation As a girder rolls, the top surface of the pad becomes angled relative to the bottom surface, lifting upward from one side of the pad and depressing the opposite side. Because the pad is already compressed by the w eight of the girder it supports, the entire width of the pad initially remains in contact with the girder and the initial roll stiffness of the pad is linear. However, if the roll angle becomes large enough, the girder will begin to lift off the pad, and t he roll stiffness will become nonlinear (Figure 6 4 ). The critical roll angle at which this occurs is dependent on the initial compression load. As more of the pad becomes disengaged, the roll stiffness softens u ntil the moment vs. rotation (angle) curve becomes horizontal ( plateaus ) and the girder rolls off of the pad. In 2012 an experimental study (Consolazio et al., 2012) demonstrated that an increase in the initial compression load results in an approximatel y proportional increase in the plateau value (while the initial roll stiffness remains essentially unaffected). Also demonstrated in the study were the effects of the geometric orientation of the girder centerline relative to the longitudinal centerline of the pad. Specifically, the two types of orientation angle considered were slope (divergence of the centerlines in elevation view) and skew (divergence in plan view). It was found that overturning roll stiffness was significantly reduced when skew was pres ent, and that this effect was exacerbated by the simultaneous inclusion of a non zero slope angle. These experimental findings confirmed results from an earlier study (Consolazio et al., 2007) in which the interaction between skew and slope was examined an alytically. The effect of slope alone (with no skew) was less conclusive in the experimental study due to scatter in the experimental PAGE 86 86 data, but the majority of the test specimens exhibited at least some reduction in roll stiffness associated with an increa se in slope angle. From the experimental study, it was concluded that bearing pad skew has the potential to drastically reduce girder stability during construction and should be avoided. Data from the study were used to support implementation of a departme ntal (FDOT) design policy change requiring that bearing pads be aligned with bridge girders thus eliminating skew between girder and pad. For this reason, the effects of bearing pad skew have been ignored in the present study and report. 6 4 Calculation of Shear and Torsion Stiffness Because shear and torsional deformation modes involve shear, but not compression of the elastomer, the pad can be treated as a linear elastic shear deformable m aterial. Only the basic dimensions of the pad (Figure 6 5 ) and the shear modulus, G are then required to calculate the shear and torsional stiffnesses from basic principles of mechanics as follows: ( 6 3 ) ( 6 4 ) The torsional constant, J for a rectangular pad can be calculat ed from a formula provided by Roark (Young and Budynas, 200 2 ): ( 6 5 ) 6 5 Calculation of Axial S tiffness Finite element analysis can be used to determine the axial stiffness of a bearing pad but most bridge engineers have limited experience modeling rubber with three dimensional solid elements. The following describes a simple, accurate, numerically stable, and computationally PAGE 87 87 efficient modeling approach for quantifying axial stiffness, without the need for specialized material testing. 6 5 1 St iffness of Neoprene Layers As previously mentioned, steel reinforcing plates are stiff enough, relative to neoprene, that they can be treated as effectively rigid. As a result, the overall axial stiffness of a pad is equivalent to the individual stiffnesse s of the neoprene layers (including the top and bottom cover layers) combined in series (Figure 6 6 ). It is only necessary to model the individual neoprene layers and then combine the results to determine t he axial stiffness of the pad. Because all internal layers typically have the same thickness, generally it will only be necessary to create two finite element layer models an internal layer model and a cover layer model. 6 5 2 Model Dimensions and Meshing While it is important to include the cover layers at the top and bottom of the pad, the side layer that surrounds the perimeter of the pad does not contri bute significantly to pad stiffness and does not need to be included in the finite element models. Instead, both layer models (internal and cover) should have the same plan view dimensions as the steel reinforcing plates, differing only in thickness. The u se of tri quadratic solid elements (e.g. 20 node or 27 node) is highly recommended to avoid shear locking, as the higher order shape functions employed by such elements more naturally approximate the curvature of the elastomeric bulge. A much smaller quan tity of solid elements is then required, significantly reducing the computational burden. A mesh convergence study has demonstrated that if 27 node solid elements are used, a layer model need only be two elements thick, and the plan view dimensions of the elements can be as large as 1 in. on either side (as long as the model has a minimum subdivision of eight elements in both directions) (Figure 6 7 ). PAGE 88 88 6 5 3 Loading and Boundary Conditions All of the nodes on the bottom surface of each elastomer layer model should be fixed in place, while the nodes on the top surface are constrained together into a nodal rigid body, representing the restraint provided by the steel reinforcement (or by friction between the pad and the girder or substructure). Application of a uniform axial pressure load of 100 200 psi is sufficient to characterize the initial axial stiffness of typica l pads. 6 5 4 Material Model Rubbers and other incompressible materials are generally modeled as hyperelastic materials, meaning that the mechanical prope rties are defined by a strain energy density function, W ( 1 2 3 ) which relates the total strain energy per unit volume to the deformed state of the material. Each hyperelastic model uses a different form of W which is nearly always written in terms of the principal stretch ratios, 1 2 and 3 which rep resent the material deformation. Stretch is defined as the ratio of deformed length to undeformed length, so the principal stretches can be related to the principal strains as: ( 6 6 ) In most hyperelastic materials, the functional form of W is selected empirically, and requires two or more material parameters which must be determined from experimental testing of specially prepared material specimens. Material testi ng is often not feasible for bridge design and, in most circumstances, the only available material data for the neoprene in a bearing pad is the shear modulus, G In some cases, only a durometer hardness value may be available, which can be converted into an approximate shear modulus empirically (Podolny and Muller, 1982; AASHTO, 2010). PAGE 89 89 In contrast, a neo Hookean hyperelastic model (Haslach and Armstrong, 2004) employs a strain energy density function that is not empirical, but is based on a statistical mec hanical analysis of linked polymer chains. This results in a relatively simple strain energy density function: ( 6 7 ) with only one material parameter, C 1 It can be demonstrated (Treloar, 1975) that for consistency with linear elasticity, C 1 is equal to half of the shear modulus, so no material testing is required as long as the shear modulus is known. The neo Hookean model is only accurate for small strains, but this range is sufficient for capturing the initial axial stiffness of a neoprene layer. It is noted that many finite element software packages do not explicitly offer a neo Hookean material option because it is a degenerate form of the more general Mooney Rivlin model: ( 6 8 ) Standard neo Hookean behavior can be achieved by selecting a Mooney Rivlin material model and setting C 2 equal to zero (Bathe, 1996). Both th e neo Hookean and Mooney Rivlin material formulations assume fully incompressible behavior, which is a reasonable assumption for elastomeric layer models. However, it is also common for finite element packages to include compressible behavior by adding a v olumetric strain term to the strain energy density function. This requires that the user supply a finite value for the bulk modulus, K which can be used together with G to calculate K ). Layer models are no t highly sensitive to PAGE 90 90 changes in K as long as a reasonably high value of K is used. A default value of 200 ksi is recommended by Gent (2001) for situations where the actual value of K is unknown. Alternatively, the AASHTO LRFD (AASHTO, 2010) suggests a val ue of 450 ksi, and o thers, such as Bradley and Chang (1998), have reported measurements as high as 470 ksi for individual pad rubber specimens. 6 5 5 Expe rimental Validation In order to validate the finite element analysis approach, experimental axial compression tests were performed on two standard types of Florida bridge bearing pads to measure stiffness. Two pads of each type were tested (i.e., four pads total) with each pad loaded to a maximum pressure of 1 ksi over three complete load cycles. It was found that the average measured axial stiffness for each pad type was within 2% of the stiffness predicted by the corresponding finite element models when u 6 8 ). When K is doubled to 400 ksi, the calculated pad stiffnesses only increases by about 15%. These findings st situations, but higher accuracy can be achieved if the value of K is more precisely known. 6 6 Calculation of Nonlinear Roll Stiffness Curves In roll, different areas of the pad are comp ressed by different amounts, so when computing the equilibrium position of the girder, it is necessary to consider the non uniform distributions of deformation and axial stiffness across the pad. It is also necessary to include the softening effect of lift off. 6 6 1 Grillage Model Estimation of the roll stiffness of a bearing pad can be accomplished using a simplified grillage model which divides the pad into discrete rectangular regions and models each region with a spring representing the stiffness contribution of that region. Compression only springs are PAGE 91 91 used to allow the girder to liftoff from the pad. The upper nodes of the springs are linked toge ther into a rigid body which represents the top surface of the pad (i.e., the surface in contact with the girder), while the bottom nodes are fixed in place. The grillage model must first be compressed by a uniform axial load, representing the weight of th e girder. Then an overturning moment can be applied to the top surface and a structural analysis can be used to determine the resulting roll angle (Figure 6 9 ). If the overturning moment is increased increment ally, a complete stiffness curve can be generated. 6 6 2 Spring Stiffness Distribution in Grillage Model Several methods of distributing stiffness values to the springs in the grillage model were considered and the resulting stiffness curves were compared to experimental roll stiffness measurements (Consolazio et al., 2012) that were obtained for three standard FDOT pad types (Figure 6 10 ), designated A, B, and C. Empirically, the best approach was found to involve the use of a parabolic bubble function (Figure 6 11 ) to assign stiffnesses to the grillage springs. This function approximates the shape (but not the magnitude) of the true axial stiffness distribution within the pad (recall Figure 6 3 ). The bubble function must be scaled so that its max imum value (at the center of the pad) is equal to the full axial stiffness of the pad normalized by the pad area and multiplied by the tributary area of a single region. The full axial stiffness of the pad can be obtained using the finite element procedure outlined in the previous section. Note that while the value of the bubble function is zero at the pad edges, the outermost grillage springs are sampled at the center of their respective tributary regions, so they will have small non zero stiffnesses. The roll stiffness curves obtained from the grillage approach show close agreement to experimentally measured curves (Figure 6 12 ) that were obtained for a variety of pad dimensions and axial load level s. Also, the grillage approach correctly exhibits the proportional relationship PAGE 92 92 between initial axial load and roll stiffness plateau value (Consolazio et al., 2012). A mesh convergence study has demonstrated that a grillage of 1 in. x 1 in. square regions provides sufficient discretization to fully capture the nonlinear roll stiffness of pad sizes that are typical of bridge construction. Because the method is not highly sensitive to small changes in axial stiffness, the choice of bulk modulus ( K ) has minim al effect on the resulting roll stiffness curves. 6 6 3 Incorporating Girder Slope During bridge construction, the stage at which prestressed concrete gir ders are most susceptible to lateral instability is immediately after girder placement and before the casting of the deck. Girder stability at this stage is dependent on the overturning roll stiffness of the bearing pads, especially if the girders are unbr aced (Mast, 1993). However, because the weight of the deck is not yet present, and therefore is not available to counteract vertical camber of the girder, significant slopes (Figure 6 13 a) can be induced at th e girder ends. (These slopes will be reduced or eliminated later, after the deck has been cast.) If the weight of the girder does not compress the pad sufficiently, an edge region of the pad may not be in contact with the girder and therefore will be unabl e to contribute to the overall roll stiffness. Because sloped contact on a bearing pad has been shown to have a detrimental effect on girder stability (Consolazio et al., 2012), the effect of slope should be considered when estimating the roll stiffness of the supports for use in lateral stability calculations. Slope can be incorporated into the grillage model as an angular deformation that is imposed about the transverse centerline of the grillage (Figure 6 13 b). The angular deformation and initial axial load must be applied to the top surface prior to applying the overturning moment about the longitudinal centerline. If the slope angle is large enough (or the initial axial load is small enough), the grillage m corresponding approximately to the region of the pad not in contact with the girder in which the compression only springs PAGE 93 93 generate no force. As roll angle increases, the superposition of roll angle and slope angle wi ll cause the liftoff region to change shape. The resulting roll stiffnesses curves do not match experimental measurements (Figure 6 14 ) as closely as in the unsloped cases, however they are found to be conse rvatively low. In each case, the initial stiffness predicted by the grillage approach consistently remains within 40% of the lowest measured stiffness for that case. 6 7 Simplified Method f or Calculating Axial Stiffness and Instantaneous Roll S tiffnesses The calculation methods described in the previous sections produce accurate results, but require the use of finite element software ( to compute k axial ) and structural analysis software ( to c ompute k roll ). For bridge designers, the use of such software may be time consuming and impractical particularly if the analyses have to be repeated several times during an iterative design process. W hile the grillage method is capable of produc ing comple te nonlinear roll stiffness curves, some applications require only knowledge of the initial (instantaneous) roll stiffness. For such cases, it is possible to derive an expression for the initial roll stiffness of the grillage in closed form, obviating the need to construct and analyze a structural model. This is accomplished by considering a continuous grillage : a grillage discretized into an infinite number of springs, each representing an infinitesimal differential area of the pad, dA Such a grillage can be treated mathematically as a continuum, and properties (such as roll stiffness) arising from the aggregated actions of individual springs can be determined in closed form by integrating over the area of the pad. In the sections below, simplified methods for computing axial pad stiffness, k axial and instantaneous pad roll stiffness k roll are described PAGE 94 94 6 7 1 Axial S tiffness I n the finite element a pproach described earlier individual finite element models are constructed for each elastomer layer in a bearing pad, and the result ing stiffnesses ( k layer ) are combined in series to produce an overall axial stiffness for the pad ( k axial ). A simplified me thod proposed by Stanton et al. (2008) can be used in lieu of the finite element models to compute the stiffnesses of individual elastomer layers in closed form, in terms of the layer dimensions and ations produce k layer values that are consistently within 2% of the equivalent finite element model results. As noted in the discussion of the finite element approach, the plan view dimensions of the steel reinforcing plates ( L s and W s ) should be used in p lace of the nominal pad dimensions ( L and W ) because the side cover layer of rubber does not contribute significantly to the axial resistance of the pad. However, the stiffnesses of the top and bottom cover layers should be included in the final calculatio n of k axial Like many empirical expressions for layer axial stiffness available in the literature, such as that provided by Gent (2001), the method suggested by Stanton calculates an effective compression modulus, E c (Equation 6 1 ) in terms of the dimensionless shape factor ( S ) which can be calculated for a layer with thickness, t as follows: ( 6 9 ) The eff ective compression modulus E c can be interpreted as the hypothetical elastic modulus that would be required for an equivalent unreinforced elastomeric layer ( with the same dimensions as the reinforced layer) to exhibit the same axial stiffness as the reinf orced layer when loaded in pure compression. B y definition, the axial stiffness of the layer is: PAGE 95 95 ( 6 10 ) where B a is a fitting parameter. This can also be express ed in terms of the shear modulus ( G ) as: ( 6 11 ) due to the near incompressibility of the elastomer. d the developmen t of a method by which B a is selected. B a is shown to depend, in part, on the bulk compressibility of the elastomer layer, as measured by the compressibility index defined as: ( 6 12 ) and Stanton develops an empirical equation for B a follows: ( 6 13 ) Using Equations 6 11 6 12 and 6 13 k layer can be computed for every elastomer layer in th e pad, and the total axial stiffness can be computed as: ( 6 14 ) In most cases, only two unique values of k layer will need to be computed: one for the internal el S in addition to the elastomer material properties, so B a must be recalculated for each uniquely dimensioned layer, even if the elastomer properties remain constant. PAGE 96 96 6 7 2 Basic D erivation of I nstantaneous R oll Stiffness of a Continuous G rillage The following derivations employ a principal coordinate sys tem ( x y ) in which roll occurs about the y axis. In these coordinates, the pad (and, therefore, the grillage) extends 0.5W in the x direction and 0.5L in the y direction (Figure 6 15 ). Stiff ness of roll about the x axis can be obtained by transposing the values of W and L Instantaneous roll stiffness of a continuous grillage is determined by imposing a differential angular displacement in the direction of roll ( ) and computing the total re storing moment ( dM ) generated by the resulting spring forces. Recall from Figure 6 11 that the stiffness of each spring varies depending on its location within the grillage according to a scal ed bubble function expressed in normalized coordinates. In principal coordinates, the stiffness at every point ( x,y ), is: ( 6 15 ) When a differential roll angle ( ) is imposed about the y axis an axial displacement field is produced, so that every spring displaces a vertical distance of: ( 6 16 ) depending on its distance from the roll axis. At every point ( x,y ), the total axial restoring force is therefore the product of k spring ( x,y ) z ( x,y ) Because each spring has a moment arm of x (the distance from the y axis), the total restoring moment exerted by the deformed gr illage of springs can be computed with the following integral: ( 6 17 ) PAGE 97 97 Substituting Equations 6 15 a nd 6 16 into Equation 6 17 rewriting A pad as the product LW and dividing through by results in an integral expression for the instantaneous roll stiffness of the grillage: ( 6 18 ) in terms of L W k axial and the coordinate variables x and y To evaluate the surface integral in closed form, it is necessary to reformulate it as a double integral in x and y, evaluated over the plan view dimensions of the pad: ( 6 19 ) which reduces to a simple closed form expression: ( 6 20 ) in terms of only the total axial stiffness of the pad ( k axial ) and the width of the pad in the direction perpendicular to the roll axis ( W ). 6 7 3 Incorporating Girder S lope By integrating over the entire plan view area of the bearing pad ( i. e., the grillage), the preceding derivation assumes that the entire surface of the pad is in contact with the girder. This assumption may not hold if there is a non zero slope angle ( ), as the total axial load exerted on the pad by the girder may not be sufficient to compress the pad far enough to achieve full contact (recall Figure 6 13 ) Such a condition reduc es the effective area of th e pad that contributes to roll stiffness. This phenomenon can be accounted for in the calculation by altering the limits of integration to include only the region of the bearing pad grillage that is in initial contact with the girder as follows: PAGE 98 98 ( 6 21 ) where p p C losed form evaluation of the modified integral produces the expression: ( 6 22 ) which reduces to Equation 6 20 when p = 1. For a given configuration, the value of p depends on the total distance that the grillage is initially compressed, which is a function of both and the initial axial load res ulting from girder self weight ( F axial ). When F axial is applied, the sloped upper surface of the grillage deforms downward (increasing the contact area) until the total restoring force in the compressed springs achieves equilibrium with F axial From static s, the force equilibrium equation for the continuous grillage is: ( 6 23 ) z ( x,y ) is the displacement field imposed on the bearing pad grillage by the sloped surface of the girder Slope induced di splacement s, z ( x,y ) do not vary in the x direction, and can therefore be expressed as a line in the y z plane, with slope a nd y intercept p as follows : ( 6 24 ) L + pL ) is the y coordinate of p in principal coordinates. Substituting Equations 6 24 and 6 15 into Equation 6 23 and reformulating it as a double integral (which must also include p in the limits of integration), results in the following equation: ( 6 25 ) PAGE 99 99 ( 6 26 ) in which p is the only unknown. Rearranging the terms of Equation 6 26 yields : ( 6 27 ) revealing p to be the root of a quartic equation. For polynomials of degree < 5, general solutions for the roots can be expressed as closed form equations in terms of the polynomial coefficients. In the case of Equation 6 27 there are four roots and four corresponding equations (omitted here for brevity). Recall that the quantity p is only meaningful over the interval 0 p 1, and note that if p = 0, must also be 0; if p = 1, must also be 1; and within that interval, increas es monotonically with p Consequently, solutions for p need only be defined over the interval, 0 1. Upon substituting the polynomial coefficients from Equation 6 27 ( quartic in p : 1 ; cubic in p 2 ; quadra tic in p : 0 ; linear in p : 0 ; and constant: ) into the four root equations, the only one that results in a positive real root within the intended range for 0 1 reduces to: ( 6 28 ) Equation 6 28 which is exact but somewhat cumbersome, can be closely and conservatively approximated as the much simpler (Figure 6 17 ). In practice, given the empirical approximations introduced by the grillage representation of a bearing pad and the inherent variability in pad construction and behavior, the error introduced by us ing in plac e of Equation 6 28 is insignificant. Substituting in the definition of from Equation 6 27 the final expression for the approximate instantaneous roll stiffness of a rec tangular bearing pad is: PAGE 100 100 ( 6 29 ) where k axial is the total axial stiffness of the pad, F axial is the initial axial load (i.e., the reaction on the pad due to gird er weight), is the girder slope angle, and L and W are the plan view dimensions of the pad (perpendicular to and parallel to the roll axis, respectively). PAGE 101 101 Figure 6 1 Location and structure of neoprene bearing pads PAGE 102 102 A B C Figure 6 2 Bulging of ne oprene layers under compression. A ) Illustration of bulging B ) 2 in. x 2 in. pad in compression C ) 12 in. x 23 in. pad in compression (photo courtesy of FDOT) PAGE 103 103 Figure 6 3 Distribution of reaction force under bearing pad subjected to unifor m axial load (FEA results) PAGE 104 104 A B C Figure 6 4 Behavior of bear ing pads during girder rollover. A ) Girder liftoff from pad B ) Nonlinear roll stiffness curve C ) Equivalent conceptual model Figure 6 5 Dimensions of a bearing pad PAGE 105 105 Figure 6 6 Axial stiffness of pad as individual layer stiffnesses combined in series Figure 6 7 Finite element model of elastomer layer PAGE 106 106 A B Figure 6 8 Validatio n of neo Hookean material model. A ) P ad A. B ) Pad B PAGE 107 107 Figure 6 9 Simplified grillage model of a bearing pad PAGE 108 108 Bearing pad type A B C Bearing pad length, L (in.) 11 14 12 Bearing pad width, W (in.) 24 24 23 Bearing pad height, H (i n.) 1 29/32 2 9/16 2 9/16 Number of internal plates 3 4 4 Figure 6 10 Standard FDOT bearing pads used for experimental verification PAGE 109 109 A B Figure 6 11 Distribution of stiffness to grillage springs A ) Normalized coordinate s ystem B ) Scaled bubble function PAGE 110 110 Figure 6 12 Comparison of experimentally measured bearing pad roll stiffnesses and roll stiffnesses predicted by the proposed computation method PAGE 111 111 A B Figure 6 13 Bearing pad slope. A ) Sloped girder in partial contact with pad B ) Grillage model incorporating slope PAGE 112 112 Figure 6 14 Com parison of experimentally measured bearing pad roll stiffnesses and roll stiffnesses predicted by the proposed computation method with non zero slope Figure 6 15 Coordinate system of continuous grillage (plan view) PAGE 113 113 Figure 6 16 Continuous grillage with imposed differential angle (Example spring shown, all others o mitted for clarity) Figure 6 17 Comparison between Equation 6 28 and the square root approximation PAGE 114 114 CHAPTER 7 MO DEL DEVELOPMENT 7 1 Introduction In addition to proposing design wind loading drag coefficients, a secondary goal of this research was to investigate temporary bracing requirements for Florida I Beams (FIBs) subjected to wind loads. To that end, finite element models (Figure 7 1 ) were developed for evaluating the lateral stability of braced systems of FIBs, using the ADINA finite element code. The models incorporated bearing pad support stiffnesses (as discussed in the previous chapter), and were capable of capturing system level buckling behavior of braced FIBs, while remaining computationally efficient enough that thousands of parametric analy ses could be performed. In the global coordinate system of the models, X corresponded to the transverse direction, Y to the longitudinal direction, and Z to the vertical direction. A local girder coordinate system (u,v,w) was also used corresponding to the same directions, with the origin at one end of the girder at the centroid of the cross section. Buckling capacities were determined using large displacement analyses, in which static loads were applied to the models in incremental steps, taking into accou nt the deformed state of the structure at each step. Instability was initiated by the presence of girder fabrication imperfections (i.e., sweep) in the models, so that every load step caused the models to deform further in the direction of the final buckle d shape. By tracking the displacement history at each step, it was possible, using a modified version of a method originally proposed by Southwell (1932), to determine when the displacements began to grow asymptotically, indicating a collapse. PAGE 115 115 7 2 Modeling of Bridge Girders Bridge girders were modeled using warping beams an advanced beam element formulation provided by ADINA that possesses a 7 th degree of freedom in each end node, representi ng the torsionally induced out of plane warping of the cross section (ADINA, 2012). Warping beams are primarily intended for modeling thin walled sections for which warping effects can significantly affect structural response, but they also have several ad ditional refinements that make them superior to standard Hermitian beam elements in buckling applications. For example, offsets between the shear center and the centroid of asymmetric cross sections are accounted for automatically, and the kinematic formul ation of the element includes coupling between bending and torsional deformation modes. Warping beam cross sections require the calculation of a comprehensive set of cross sectional properties, several of which require knowledge of the warping function wh ich cannot be calculated in closed form and must be solved for numerically. Details relating to the section properties that were calculated in this study for the FIB cross sectional shapes are provided in Appendix C Material properties assumed for the prestressed concrete FIBs were f c = 6.5 ksi, unit weight = 150 pcf, and Poisson's ratio = 0.2. Using these values and the PCI Design Handbook ( PCI, 2010), the concrete elastic modulus was computed to be E = 48 87 ksi. Construction tolerances for FIBs are specified in the Standard Specifications for Road and Bridge Construction ( FDOT, in. for every 10 ft of girder length, but not to exceed 1.5 in. To ensure conservative buckl ing capacity results, all FIBs were modeled with the maximum allowable sweep ( u max ) for their length. Geometrically, sweep was implemented using a sinusoidal function (Figure 7 2 ) with the maximum allowable sweep at midspan, so that the lateral deviation, u at every point along the girder length, v was: PAGE 116 116 ( 7 1 ) During early phases of bridge construction, the bridge deck is not pres ent and hence the weight of the deck is not yet present. Consequently, the girders will have more camber at this stage than they will have in the completed configuration of the bridge (when deck self weight is active). In this study, it was important to re because the additional elevation of the girder center of gravity reduces buckling capacity by a small amount. (recall Figure 2 9 ). To establish maximum probable girder camber for use in model development, trial beam designs were produced for all eight (8) FIB cross sections with the goal of maximizing camber. This was accomplished by placing all prestressing tendons as low as possible in the bottom flange and finding the span length at which camber was maximized. For the purposes of these designs, long term creep effects were ignored and it was assumed that no cracking occurred. From these designs, it was determined that 3.25 in. was a reasonable upper bound for FIB cambe r during construction. It is important to recognize that the measured camber of a bridge girder in the field is a superposition of two independent deflections: an upward deflection caused by prestress forces and a downward deflection caused by the self wei ght of the girder. The initial (undeformed) geometry of a finite element model should represent its free body state, prior to the application of any external loads, including gravity loads. Therefore, it was necessary to add additional camber to the models to offset the expected self weight deflection. In other words, the geometric camber included in the finite elements models represented only the upward deflection caused by prestressing so that after self weight was applied to the model, the total deflecti on would match the camber that would be measured in the field. As a result, each girder model was assigned a maximum geometric camber ( w max ) of: PAGE 117 117 ( 7 2 ) where A is the girder cross sectional area, is the unit weight of the concrete, L is the span length, E is the elastic modulus, and I is the major axis moment of inertia. Because the geometric camber in the models represented upward deflections caused by straight prestressing tendons (which generate a constant internal moment throughout the length of the beam), the girder camber was implemented with a parabolic shape (Figure 7 3 ) so that the vertical deviation, w at eve ry point along the girder length, v was : ( 7 3 ) 7 3 Modeling of E nd S upports Girder support sti ffnesses were modeled with six (6) geometrically linear springs to represent the stiffness of the bearing pad in each degree of freedom, with each spring corresponding to one of the four (4) main deformation modes of the pad: shear, axial, torsion, and rol l (Figure 7 4 ). These stiffnesses were obtained using the calculatio n methods discussed in Chapter 6 The roll stiffness springs (in both the overturning and bending directions) were assigne d nonlinear moment rotation curves that captured the softening effects of partial girder liftoff from the pad. The remaining pad stif fnesses were treated as linear. 7 3 1 Pad Selection Seven (7) standard types of elastomeric bearing pad are provided in Design Standard No. 20510 : Composite Elastome ric Bearing Pads Prestressed Florida I Beams (FDOT, 2012 c ) for use with FIBs. During design, selection o f the type of pad that will be used in a particular bridge is based on thermal expansion and live load deflection limit states of the completed bridge, neither of which can be predicted based solely on girder dimensions (cross sectional and span PAGE 118 118 length). A s such, it is not appropriate to assume that for each FIB type, there is a specific corresponding type of bearing pad that would be utilized. Hence, in this study, it was conservatively assumed that the pad type with the lowest roll stiffness (which will p roduce the lowest buckling capacity) would be used in conjunction with all FIB types. After calculating the roll stiffness of every standard FDOT FIB pad type (see Appendix D for details), using the grillage meth od that was discussed in Chapter 6 the Type J bearing pad was selected for use in this study. 7 3 2 Axial L oad S election In Chapter 6 it was noted that the amount of axial load applied to a pad does not change the initial linear portion of the roll stiffness curve, but it does affect the moment required to initiate girder roll off from the pad. Reducing the compressive axial load on a pad reduces the moment that is required to cause girder roll off. Additionally, reducing girder span length reduces girder self weight which, in turn, reduces the axial loads on the bearing pads. Ther efore, to be conservative in this study, the minimum length ranges for each FIB shape were determined from design aids in Instructions for Design Standard No. 20010 : Prestressed Florida I Beams ( IDS 20010; FDOT, 2012 b ) and the minimum expected axial pad lo ad was calculated for each FIB shape. (These calculations assumed that the girders were simply supported. Additionally, the effects of wind uplift forces were conservatively ignored.) Using this process, a single worst case (minimized) roll stiffness curve was calculated for each type of FIB, resulting in a total of seven (7) bearing pad moment rotation curves. 7 3 3 Girder S lope S election I n Chapter 6 it was also noted that overturning roll stiffness is reduced by the presence of girder slope, which can arise from a combination of girder camber and bridge grade. According to Instructions for Design Standard No. 20510: Composite Elastomeric Bearing Pads PAGE 119 119 Prestressed Florida I Beams ( IDS 20510 ; FDOT, 2012 d ), the maximum expected slope angle in the completed bridge is 0.0125 rad, because if this angle is exceeded, beveled bearing plates must be installed to elimin ate slope. Therefore, the maximum expected camber prior to the casting of the deck is the sum of 0.0125 rad and any camber induced slope that is negated by the downward deflection under the weight of the deck and other superimposed dead loads (SDL). After a series of trial beam design calculations was performed, it was determined that a reasonable upper limit for the SDL negated slope was 0.01 rad. Additionally, AASHTO LRFD ng pad slope angle. The maximum completed slope of 0.0125 rad, the SDL negated slope of 0.01 rad, and the slope uncertainty of 0.005 rad combined for a total maximum slope angle of 0.0275 rad. This was conservatively rounded up to a slope angle of 0.03 rad which was used to compute the bearing pad overturning roll stiffness curves. 7 4 Modeling of B race s and A nchors Because the design of bracing has historically been left to the discretion o f the contractor, a wide variety of bracing configurations are used in practice. Consequently, in this study it was not possible for every potential brace configuration to be represented in the parametric studies. After conducting a survey of bracing desi gns used in the construction of bridges throughout Florida, four (4) representative brace configurations were identified: Top strut (Figure 7 5 a) : a horizontal timber compression strut situated between the ed ges of the top flanges. The top strut is typically nailed to the underside of a slightly longer timber member, creating lips that rest on the top of the flanges. Parallel strut (Figure 7 5 b) : Two (or more) horizontal timber compression struts wedged in place between the girder webs. X brace (Figure 7 5 c) : Two diagonal timber members wedged between the webs that bolt typically passes through both members at the crossing point to create a hinge. PAGE 120 120 K brace (Figure 7 5 d) : Steel members (typically steel angles) welded together into a shaped frame and welded or bolted t o steel plates cast into the webs. The majority of brace designs that were encountered were variations of one of these four basic configurations. For analysis purposes, braces were modeled primarily with beam elements, with each brace member represented by a single element. At the girder connection points, rigid links were used to connect the braces to the girder elements (i.e., warping beams located at the girder centroids). It was assumed that the brace girder connections were ideal pins, which was conser vative with regard to girder stability. Pins and hinges were modeled with beam end releases and nodal constraints, respectively. During the survey of bracing designs, the vast majority of timber braces that were encounter ed were comp o sed of 4x4 Southern Pi ne sawn lumber. According to the National Design Specification for Wood Construction (AF&PA, 2005), 4x4 Southern Pine has a x square cross section and an elastic modulus of E = 495 ksi (based on an E min of 550 ksi for 4 inch grade lumber and a Wet Service Factor of 0.9). These properties were used to model al l timber brace members including the top strut, parallel strut, and X brace. Based on a typical bridge bracing design that was acquired during the survey, K brace members were x x lastic modulus of E = 29000 ksi. In contrast to braces, girder anchors were not modeled with structural elements. Instead, the additional roll stiffness provided by the anchors ( k roll,anchor ) was quantified directly and added to the bearing pad support stiffness ( k roll,overturning ). It wa s assumed that only one FIB in each bridge cross section was anchored and that anchors at each end of the girder were of equal stiffness. PAGE 121 121 Anchor roll stiffness is a function of the axial stiffness of the anchor ( k anchor ), the radial distance from the cente r of rotation to the anchor connection point ( R ), and the angle between the anchor member and the tangential force exerted by the girder ( ) (Figure 7 6 ), and can be calculated as follows: ( 7 4 ) It is important to note that may become a three dimensional angle if the anchor is inclined longitudinally (away from the girder ends) with the result that the roll stiffness provided by the anchor may be reduced. In the presence of girder skew, this practice can also cause paired tension only anchors (i.e., chains or cables) to be of different lengths (Figure 7 7 ), in which case the average length is used to compute k anchor 7 5 Loads Two types of structural load were included in the models: wind loads and gravity load. Lateral wind loads were calculated for each girder in the system usi ng the design drag coefficients proposed earlier for FIBs (recall Figure 5 17 ) and were applied to the girder elements as tributary nodal loads (Figure 7 8 a). Small overturning moments were a lso applied at each node to compensate for the eccentricity between the centroid of the cross section (where the nodes and elements were located) and the center of pressure (where the lateral load was assumed to act on the girder) (Figure 7 8 b). Wind loads were always applied in the direction of increasing girder sweep. Gravity was applied as a vertical acceleration load (mass proportional body force) in units of g the acceleration due to gravity, so that a load of 1 g rep resented the self weight of the model. In field conditions, girders are always subjected to a constant gravity load of 1 g. In the structural models analyzed in this study, however, gravity loading was used to initiate instability. PAGE 122 122 After wind loads were ap plied, gravity load was linearly ramped up beyond 1 g if possible until girder instability occurred. Subsequently, the capacity of the system was expressed as a gravity load (in g), which can also be thought of as capacity to demand ratio. For example, if the system became unstable at a gravity load of 1.5 g, then the ratio of capacity (1.5 g) to demand (1 g) would 1.5. 7 6 Modified Southwell Buckling A nalysis To assess system stability from the results of the large displacement analyses, it was necessary to define the system capacity in terms of displacement load results data. In typical buckling problems, as the displacements increase, the applied load approaches an asymptote called the crit ical buckling load (where the displacements are considered to be infinite). In this study, the location of the asymptote was determined using a method originally proposed by Southwell (1932) for use with axially loaded columns. Southwell was able to demons trate mathematically (using the governing differential equation of an axially loaded column with a non zero sweep) that that expected shape of the displacement load curve (using the lateral displacement of the beam at midspan) is a rectangular hyperbola (F igure 7 9 a) of the form: ( 7 5 ) where is the horizontal asymptote (and therefore the critical buckling load). The value of can be determined using a Southwell Plot (Figure 7 9 b), in which the midspan displacement ( x) is plotted as a function of the ratio of d isplacement to load ( x/y ). By rearranging Equation 7 5 it becomes evident that the resulting relationship is linear: ( 7 6 ) PAGE 123 123 w ith the critical buckling load being equal to the inverse slope of the line (Figure 7 9 b). The critical buckling load can therefore be determined by applying linear least squares regression to the transformed analysis results (i.e., pairs of x and x / y values). Using this technique, reliable determination of the critical load can be obtained from displacement load data, even if only a portion of the overall displacement load plot is available. (This aspect of the method is part icularly useful for physical testing, as specimens do not need to be loaded all the way to failure in order for the buckling capacity to be quantified.) Theoretically, the mathematical foundation for the Southwell method is only valid for axial column buck ling. For this reason, several authors (Massey, 1963; Trahair, 1969; Meck, 1977) have published alternative methods mathematically formulated for lateral stability problems (based on the governing differential equation for lateral torsional buckling of a b eam). However, despite these developments, studies involving experimental test programs have frequently demonstrated that the Southwell method works well for lateral torsional buckling (Mandal and Calladine, 2002), and at least one survey of the different methods on the same set of experimental data (Kalkan, 2010), found the Southwell method to be superior to the supposedly more refined alternatives. Mandal and Calladine (2002) have published a discussion of this apparent contradiction which provides a math ematical explanation for why the Southwell method produces excellent results even in lateral torsional buckling applications. The large displacement structural analyses performed in this study did not exhibit pure lateral torsional buckling, but included s everal additional components (e.g., flexible bearing pad support conditions, lateral wind loads) that significantly complicate the governing differential equations. Consequently, there was no mathematical justification for using any particular method. Howe ver, when the methods (those of Southwell, Massey, Trahair, and Meck) were PAGE 124 124 evaluated using displacement load data from the structural analyses, it was found that the Southwell method produced excellent results, while the alternative methods produced unusab le results. This was attributed to the fact that the alternative methods were more dependent upon the underlying assumptions (e.g., pinned but torsionally rigid beam support conditions) and were less tolerant of small deviations from the ideal shape of the displacement load curve. It was concluded that the Southwell method for determining critical buckling load was an acceptable choice for any stability problem exhibiting asymptotic behavior because fitting a hyperbola to the data is a robust way of approxi mating the location of the asymptote, even if the data is not strictly hyperbolic (Figure 7 10 ). As a result, in this study, the Southwell method was used to determine the critical buckling loads from lateral girder displacement data computed at midspan. For models with multiple girders, a Southwell analysis was performed on displacement data for each girder, and the smallest resulting buckling load was used. In a physical bridge system, girders are not capable of su staining arbitrarily high levels of lateral displacement, as is implied by using the critical buckling load (the asymptote) as the definition of system capacity. Therefore, a modified version of the Southwell method, developed for use in a previous study ( BDK75 977 load versus midspan displacement curve (Figure 7 11 ) at which the tangent slope of the fitted hyperbola drops below 10% of the initial slope (at the origin). It can be demonstrated that this procedure is mathematically equivalent to multiplying the value of the asymptotically quantified critical buckling lo ad by a scale factor of 0.684. Hence, the complete procedure used in the present study for quantifying system capacity was as follows: PAGE 125 125 Wind loads (if any) were applied to the model. Gravity load was linearly and incrementally increased until the model fail ed to converge (i.e., until a system instability occurred). Displacement vs. load curves were produced for each girder in the system, using the lateral displacement of the girders at midspan. Southwell analyses were performed to locate the asymptotes (crit ical buckling loads) of the displacement load curves. The minimum critical buckling load from among all girders in the model was selected and multiplied by 0.684 (to apply the 10% rule) to calculate the system capacity. PAGE 126 126 F igure 7 1 Finite element model of a single FIB (isometric view) Figure 7 2 Representation of sweep in FIB model (plan view) Figure 7 3 Representation of camber in FIB model (elevation view) PAGE 127 127 Figure 7 4 Bearing pad stiffness springs in FIB model (isometric view) PAGE 128 128 A B C D Fi gure 7 5 Representation of brace conf igurations in FIB system models. A) Top strut brace. B) Parallel strut brace. C) X brace. D ) K brace PAGE 129 129 Figure 7 6 Calculation of rotational stiffness of anchor Figure 7 7 Longitudinally inclined anchors on skewed br idges A B Figure 7 8 Representation of wind load in structural models. A ) Lateral nodal loads (top view) B ) Overturning moments (section view) PAGE 130 130 A B Figure 7 9 Southwell method for determining critical buckling load ( ). A ) Displacement l oad curve (rectan gular hyperbola). B ) Southwell plot PAGE 131 131 Figure 7 10 Southwell analysis of non hyperbolic displacement load data obtained from a large displacement structural analysis Figure 7 11 Determination of buckling capacity using modified Southwell approach (Adapted from Consolazio et al. 2012) PAGE 132 132 CHAPTER 8 PARAMETRIC STUD Y OF INDIVIDUAL BRIDGE GIRDERS 8 1 Introduction During the bridge construction process, the stage at which wind loading is often most critical is when the first girder is erected. At th is stage there are no other girders to brace against, hence the initial girder cannot be braced at interior points, and can only be anchored to the pier at the ends. For bridge designs in which girder stability is a primary concern, girder erection can som etimes be scheduled to minimize the exposure period for the initial girder, so that it is statistically unlikely that peak wind forces will occur. However, meeting such a schedule is not always feasible. For example, strong afternoon thunderstorms can form rapidly in Florida during the summer months. In such situations, it is important to be able to assess, in advance, whether anchor bracing will be needed to prevent girder collapse under the effects of thunderstorm force winds. To investigate this scenario a parametric study was performed, using finite element models of single Florida I Beams (FIBs) over a range of span lengths, both with and without anchor bracing in place. For each model, the system capacity was evaluated several times at different wind pressures, iterating until the capacity was within 1% of 1 g (i.e., the capacity to demand ratio was approximately unity). For each such case, the resulting wind pressure was termed the wind capacity of that girder, representing the maximum wind load that can be sustained by the girder without collapsing. Using the results of the parametric study, equations were developed for predicting the wind capacity of a single FIB. 8 2 Selection of P ara meters The girder parameters that were varied in the parametric study were as follows: FIB cross section depth (in) Span length (ft) PAGE 133 133 Rotational stiffness of anchor bracing (kip ft/rad) All eight (8) standard FIB cross sections were included in the study, with depths ranging from 36 in. to 96 in. For each FIB, wind capacity analyses were performed for every combination of span length and anchor stiffness, sampled from the values listed in Table 8 1 Bridge grade was also considered as a potential parameter, but was rejected after preliminary analyses showed that it had a negligible effect on wind capacity. Therefore, all analyses were performed on girder models with a level (0%) grade. Maximum an d minimum span lengths used in the parametric study were based on design aids included in Instructions for Design Standard No. 20010: Prestressed Florida I Beams (IDS 20010; FDOT, 2012b), which provides estimated span lengths (Table 8 2 ) for FIBs with different lateral spacings, based on representative bridge design calculations. Maximum lengths minimum lengths assumed a 12 ft s ensure that the considered length ranges included all reasonable beam designs, the basic ranges taken from IDS 20010 were extended by 3 ft on each end, and then extended further so that range limits were even multiples of 5 ft. The parametric study included span lengths chosen at 5 ft intervals over the final ranges. A survey of bracing designs used in girder bridges constructed in Florida revealed anchors with equivalent rotational stiffnesses ranging fr om 500 to 50,000 kip ft/rad. However, it was found that when large anchor stiffnesses were included in girder models, particularly for shorter span, shallower girders less prone to instability, the models no longer exhibited a normal buckling response. In moderate cases, this caused the computed wind capacity to be artificially inflated, and, in the most extreme cases, the wind capacity could not be computed at all (i.e., the PAGE 134 134 wind pressures required to initiate girder instability under self weight were so l arge well beyond material strength limits that the displacement load results ceased to be characteristically asymptotic and the modified Southwell analysis method was no longer applicable). Therefore, a maximum practical anchor stiffness (at which a wind c apacity could be computed) was established for each FIB through trial and error, and the parametric study included ten (10) evenly distributed stiffness values up to and including that maximum (in addition to the unanchored case, with zero anchor stiffness ). In practice, the maximum practical anchor stiffness was smaller for FIBs with shorter span lengths, but it was not feasible to determine a different maximum stiffness for every FIB at every span length. Instead, a maximum anchor stiffness was establishe d based on the maximum span length for each FIB, and the parametric study was carried out with the understanding that not every combination of parameters would result in a computable wind capacity. At the minimum span length, it was common to obtain result s for only the lowest 4 or 5 stiffness values. There were also two cases (the 215 ft/rad) for which a meaningful wind capacity did not exist because the girder was inherently unstable, collapsing under less than 1 g in the absence of any wind. In total, 781 wind capacity analyses were attempted in the parametric study, of which 471 produced results. (The 471 computed wind capacities included the some of the artificially inflated values described above. I dentification and rejection of those data points is discussed in more detail later in the chapter.) 8 3 Results Wind capacities computed from the parametric study are summarized in Figure 8 1 Visually, the data are divided into eight (8) major groups, each of which corresponds to one of the tested FIB cross sections, ranging fro 471). Each group contains several subgroups visually identifiable as diagonal lines of PAGE 135 135 consecutive data points that represent the wind capacities computed for a single span length (at several different a subgroups, consists of the artificially inflated wind capacities that occurred at higher anchor stiffnesses. The results of the parametric study were used to develop an equation for estimating the wind capacity ( P max ) of any FIB. This was accomplished by first developing an equation for the wind capacity of an unanchored FIB ( P max ,0 ), in terms of the section depth ( D ) and span length ( L ) and then determining a linear correction factor that incorporated the effect of the anchor stiffness ( k ). 8 3 1 Wind C apacity of a S ingle U nanchored G irder The relationship between wind capacity a nd span length for unanchored FIBs is plotted in Figure 8 2 It is evident from the plot that span length is the strongest predictor of wind capacity, which declines sharply as span length increases. Wind capa city is also reduced when the girder depth increases, which can be attributed to the larger sail area (projected area) over which the wind pressure is applied. There is no data for the 215 capacity to demand ratio was less than 1 prior to the application of wind. In the final equation, this situation will be indicated by producing a negative wind capacity value. The functional form found to be the closest fit to each of the FIB curves was an exponential relationship : ( 8 1 ) where a b and c are fitting parameters. For each FIB, a separate exponential curve fit was performed to relate wind capacity to span length, L It was found that the value of b in these curve fits was approximately constant, while the variance in the a and c terms had a similar PAGE 136 136 exponential relationship with FIB depth, D Substituting curve fits for a and c into the original exponential fit resulted in a final equation for wind capacity of an unanchored girder: ( 8 2 ) where P max ,0 is in psf, L is the span length in ft, and D is the FIB cross section depth in inches. Equation 8 2 predicts conservative wind capacities for every case analyzed in the parametric study (Figure 8 3 ). Due to the complexity of Equation 8 2 an alternative, simplified equation was also produced that took the form of a basic exponential function that enveloped all of the data points (Figure 8 4 ). The simplified e quation is a function of span length only: ( 8 3 ) where P max ,0 is in psf, L is the span length in ft. The simplified Equation 8 3 is easier to use, but, as shown in Figure 8 5 produces more conservative results. In practice, either Equation 8 2 or 8 3 can be used to compute conservative estimates of unanchored girder wind load capacity. 8 3 2 Wind Capacity of a Single A nchored G irder The relationship between wind capacity and anchor rotational stiffness ( k roll,anchor ) for 8 6 with separate curves for each tested span length. (Data for the other FIB sections is qualitatively similar.) As expected, the stability provided by the anchor stiffness which adds to the roll stiffness of the bearing pad tends to increase wind capacity monotonically relative to the unanchored case (i.e., the case where k roll,anchor = 0). For each curve, the relationship between wind capacity and anchor stiffness follows the same basic pattern: a steady linear increase followed by a much sharper increase in the last 1 3 data points. PAGE 137 137 As previously described, the sud den increase is artificial a moderate form of the same numerical problems noted earlier and the inflated data points must be discarded. Based on an examination of system characteristics (span length, etc.) and the analysis results, it was determined that m eaningful results data were restricted to the linear portion of each curve and that points contained within the nonlinear portions should be rejected. Hence, starting with the first three (3) points in each curve, a linear least squares regression was perf ormed and the resulting line was extrapolated to predict the wind capacity of the next point. If the predicted wind capacity was within 5% of the computed value then the point was accepted, the regression line was recomputed (to include the new point). The process was then repeated on the next point in the curve. If a point failed the test, it was considered to be outside the linear range of the curve, and all remaining points were rejected (Figure 8 7 ). Because the y intercept of each curve in Figure 8 7 is equal to the unanchored wind capacity ( P max ,0 ), the slope ( m ) of each regression line can be thought of as an anchor stiffness coefficient such that the total wind capacity ( P max ) of the anchored girder is calculated as follows: ( 8 4 ) After computing m for every tested combination of girder cross section and span length, it was found to be primarily correlated with span length (Figure 8 8 ). An exponential least squares curve fit was performed, resulting in an equation for m as a function of span length: ( 8 5 ) where m has units of psf/(kip ft/rad), and L is the span length in ft. Equation 8 5 conservatively underpredicts nearly all compu ted values of m In the few cases where m is slightly over predicted, the amount of unconservatism is either negligibly small PAGE 138 138 or is compensated for by conservatism in the determination of P max ,0 As a result, the wind capacity of a single anchored girder c an be predicted as follows: ( 8 6 ) where k roll,anchor is in kip ft/rad, L is the span length in ft, and P max ,0 is in psf and is calculated using either Equation 8 2 or 8 3 When Equation 8 2 is used, the majority of wind capacities predicted by Equation 8 6 fall within 10% (see Figure 8 6 ) of the corresponding values computed in the parametric study. PAGE 139 139 Table 8 1 Parameter values use d in parametric study for each FIB cross section Span length, L (ft) 75 95 110 120 135 145 155 170 80 100 115 125 140 150 160 175 85 105 120 130 145 155 165 180 90 110 125 135 150 16 0 170 185 95 115 130 140 155 165 175 190 100 120 135 145 160 170 180 195 105 125 140 150 165 175 185 200 110 130 145 155 170 180 190 205 160 175 185 195 210 180 215 Anchor rotational stiffness, k roll,anchor (kip ft/rad) 0 0 0 0 0 0 0 0 15 30 50 75 125 160 210 410 30 60 100 150 250 320 420 820 45 90 150 225 375 480 630 1230 60 120 200 300 500 640 840 1640 75 150 250 375 625 800 1050 2050 90 180 300 450 750 960 1260 2460 105 210 350 525 875 1120 1470 2870 120 240 400 600 1000 1280 1680 3280 135 270 450 675 1125 1440 1890 3690 150 300 500 750 1250 1600 2100 4100 PAGE 140 140 Table 8 2 Range of allowable span lengths for FIBs Values from IDS 20010 Cross section Min l ength (ft) Max l ength (ft) Final tested range FIB 80 105 75 110 FIB 98 126 95 130 FIB 113 142 110 145 FIB 124 155 120 160 FIB 142 173 135 180 FIB 151 182 145 185 FIB 159 191 155 195 FIB 175 208 170 215 Spacing 12 ft 6 ft Environment Extremely a ggressive Moderately a ggressive Figure 8 1 Summary of single girder wind load parametric study results PAGE 141 141 Figure 8 2 Wind capacities of unanchored FIBs at various span lengths Figure 8 3 Wind capacity of an unanchored girder as predicted by Equation 8 2 PAGE 142 142 Figure 8 4 Wind capacity of an unanchored girder as predicted by simplified Equation 8 3 Figure 8 5 Comparison of basic and simplified unanchored girder wind capacity equations, Equations 8 2 and 8 3 respectively ( o nly data for FIBs with depths 7 2 in. or greater shown) PAGE 143 143 Figure 8 6 Figure 8 7 Rejection of artificially PAGE 144 144 Figure 8 8 Anchor stiffness coefficient Equation 8 5 compared to parametric study results Figure 8 9 Comparison of wind capacity results computed using the combination of Equations 8 2 and 8 6 versus corresponding parametric study results (n ote: negative relative error indicates that the combination of Equations 8 2 and 8 6 is conservative relative to the parametric study data) PAGE 145 145 CHAPTER 9 PARAMETRIC STUDY OF BRACED MULTI GIRDER SYSTEMS 9 1 Preliminary Sensitivity Studies Fully characterizing a braced multi girder system requires a large number of geometric parameters. Consequently, conducting parametric studies in which all possible combinations of these parameters are co nsidered (even if only a few discrete values are selected per parameter) would require hundreds of thousands of analyses to be performed. To avoid such a situation, several limited scope preliminary sensitivity studies were performed to help guide the desi gn of efficient final parametric studies. As a result of these preliminary investigations the details of which will be omitted here for brevity several system parameters were identified as having negligible influence on system capacity. Consequently, these parameters were not varied in the final parametric studies. The parameters were: Bridge grade : All analyses were performed on girder models with level (0%) grade. Cross slope : Multi slope. Location of anchored girder in bridge cross section : In cases where anchors were included, they were always attached to the most leeward (downwind) girder. The preliminary studies also revealed that braces were naturally divided into two categories that had very different effects on system behavior: strut braces and moment re s isting braces As a result, separate parametric studies were performed for each brace category. 9 1 1 Strut Braces Top struts and parallel struts (Figure 9 1 ) are both examples of strut braces, which include (but are not limited to) all brace designs consisting solely of horizontal compression members. Some what surprisingly, it was found that all strut brace designs are essentially interchangeable with regard to lateral stability. That is, a girder system braced with top struts has the same capacity as an otherwise identical system braced instead with parall el struts (or any PAGE 146 146 other type of strut brace). As a result, the capacity of a strut braced system is also insensitive to girder spacing (which only affects the length and thus the axial stiffness of the strut members). Additionally, there is no measurable g ain in system capacity that results from the installation of struts at interior brace points. End bracing alone is sufficient to mobilize all of the girder buckling capacity that can be provided by strut braces. It is also worth noting that the capacity of a strut braced system decreases as additional girders are added. Strut braces can be defined (or identified) by their lack of resistance to girder overturning. In a small displacement (geometrically linear) analysis of a system with zero bearing pad rotat ional stiffness, a strut with ideal pin connections forms a collapse mechanism (Figure 9 2 ) that allows the connected girders to rotate freely in unison. Therefore, struts can only provide stability by coupling the gird ers together, and resistance to collapse is primarily provided by the roll stiffness of the anchors and, to a lesser extent, the roll stiffness of the bearing pad supports. This behavior is also the reason that adding girders to a system reduces stability: each new girder adds additional wind load (and additional bearing pad stiffness) while the number of anchors remains constant. 9 1 2 Mome nt Resisting Braces X braces and K braces (Figure 9 3 ) are both examples of moment resisting braces, which are capable of resisting girder overturning. Unlike struts, the system capacity provided by different moment resist ing brace designs varies significantly, and capacity can be increased by the installation of braces at interior brace points. Systems with moment resisting braces become more stable as additional girders are added; hence a two girder system is nearly alway s the most unstable bridge cross section possible. In the presence of moment resisting bracing, the additional roll stiffness and stability provided by anchors is typically negligible. PAGE 147 147 9 2 M odeling of Bridge Skew and Wind Load In a braced system of girders, the introduction of bridge skew causes the girders to become staggered longitudinally. This affects system capacity in two ways (Figure 9 4 ): Brace placement : Because girders are installed perpendicular to the girder axes (per Design Standard No. 20005 : Prestressed I Beam Temporary Bracing FDOT, 2012a), the region within which braces can be placed is smaller (shorter) than the span length of the gird ers. As a result, girder stability can increase due to the reduced distance between brace points. Incomplete shielding : In a skewed system, none of the girders are completed shielded with respect to wind load. Rather, an end portion of each girder is expos ed to full (unshielded) wind pressure. The aerodynamic properties of the exposed end region have presence of upwind girders are unknown. The magnitude of both of these effects is a function of the girder offset length ( L offset ), (Figure 9 4 ) which is dependent on both skew angle and girder spacing. Conducting wind tunnel testing to experimentally quantify the effects of skew on gir der end shielding was outside the scope of this study. Consequently, the non uniform wind pressure distribution shown for leeward girders in Figure 9 4 b is an approximation based on engineering judgment. Lacking wind tunnel confirmation of this approximation, it was deemed unwarranted to model this distribution in detail in the parametric studies. Instead, a simplified, but statically similar, representation was used in which the wind load on each girder was modeled a s a single, weighted average uniform pressure along the entire length of the girder. The uniform wind load applied to each partially shielded girder ( P ) was computed as a weighted average of the shielded and unshielded wind loads, as follows: ( 9 1 ) where P U is the unshielded wind load (on the windward girder), P S is the shielded wind load, L is the girder length, and L offset is the length of girder offset pro duced by skew. PAGE 148 148 9 3 Selection of Parameters for Strut Brace Parametric Study System parameters that were varied in the strut brace parametric study were as follows: FIB cross section depth (i n) Span length (ft) Number of girders Rotational stiffness of anchor bracing (kip ft/rad) Wind pressure (psf) Skew angle (deg) All eight (8) standard FIB cross sections were included in the study, with depths ranging from 36 in. to 96 in. For each FIB, ca pacity analyses were performed for every combination of span length, anchor stiffness, wind pressure, skew angle, and number of girders, sampled from the values listed in Table 9 1 for a total of 37,8 00 analyses. Maximum and minimum span lengths were based on the same span length ranges used for the single girder study (see Chapter 8 ). However, to reduce the number of analyses to a feasible level, the st udy used lengths chosen at 10 ft intervals instead of 5 ft intervals. When necessary, the upper limit of the range was increased by 5 ft so that the total range was evenly divisible into 10 ft intervals. Similarly, the maximum anchor stiffnesses were the s ame that were used for the single girder study, but five (5) evenly distributed values were used instead of ten (10). As noted earlier, preliminary analyses demonstrated that the capacity of a strut braced system continues to decrease as more girders are a dded. A practical upper bound of nine (9) girders was selected to be representative of wide bridge cross sections while simultaneously limiting the finite element models to a manageable size (number of nodes, elements, and degrees of freedom). Wind pressur e loads were applied to the girders using the shielding pattern proposed in Chapter 5 and using the wind pressures listed in Table 9 1 Wind pressures specified in the table PAGE 149 149 refer to the unshielded pressure load applied to the windward girder (G1). Hence, in accordance with the model proposed in Chapter 5 the first shielded girder (G2) received no wind load and all subsequent girders (G3, G4, e tc.), if any, received half of the listed pressure load. The maximum wind pressure of 160 psf was determined using the Structures Design Guide lines (FDOT, 2012e) by assuming a pressure coefficient of C P = 2.0, a basic wind speed of V = 150 mph, a bridge el evation of z = 75 ft, a gust effect factor of G = 0.85 and a load multiplier of ws = 1.4 (corresponding to the Strength III limit state). During preliminary sensitivity studies, the effect of girder spacing (and therefore, axial strut stiffness) was found to be small enough so as to have no practical effect on the determination of system capacity. There was, however, a very slight (< 1%) reduction in capacity for a system spaced at 6 ft on center relative to an otherwise equivalent system spaced at 14 ft ( an 80% difference in strut length). Therefore, for conservatism, all systems in the parametric study were spaced at 6 ft on center. However, skew effects, which are a function of the girder offset length ( L offset ), are minimized when the smallest girder sp acing is selected. Therefore, to maintain conservatism in the system models, the girders were offset longitudinally as if they were spaced at 14 ft (Table 9 2 ). Essentially, the effective skew angle in each model was greater th an the nominal bridge skew, so that a conservative girder offset was produced. In this way, brace placement and wind loads were modeled conservatively while maintaining a 6 ft spacing. 9 4 R esults of Strut Brace Parametric Study The results of the strut brace parametric study are summarized in Figure 9 5 Recall from Chapter 7 that the system capacity computed for each cas e represents the total gravity load (in g) that can be resisted by the system without failing due to lateral instability (primarily a buckling phenomenon in systems where an anchor is present). Five (5) main groups are visible in the PAGE 150 150 summary of results, ea ch of which corresponds to a different wind pressure value (in increasing order from 0 to 160 psf). Each group contains eight (8) subgroups corresponding to the tested FIB cross sections each of which is composed of five (5) or six (6) adjacent groups of d ata points arranged in a steep diagonal line corresponding to the tested span lengths. Variation within the data groups reflects the relatively small effects of anchor stiffness and the number of girders in the system. The results were used to develop an e quation for capacity ( C ) of a strut braced system of FIBs. This was accomplished by first determining the baseline capacity of an unanchored system of two girders in zero wind conditions ( C 0 ), and then developing correction factors to adjust the system cap acity upward in response to anchor stiffness and downward in response to wind load and the erection of additional girders. A wide range of FIB system capacities (up to 10 g) were present in the parametric study results, but under heavy wind, a large number of systems also had a capacity of 0, which the capacity equations indicate by computing a negative capacity value. Two techniques were used to simplify interpretation of the parametric study results. The large number of interacting parameters and the wide range of the capacity values made it impractical to produce equations with a uniform level of conservatism throughout the data set. Therefore, capacity prediction equations were considered satisfactory if they conservatively enveloped 95% of the data poin ts, and as long as none of the unconservative cases were more than 5% in error with regard to computed capacity. Also, while it was important that the equations be accurate and not overly conservative for systems where stability was likely to control desig n, the same characteristics are less important for extremely stable systems. Therefore, when appropriate, to simplify the data interpretation process systems with capacities PAGE 151 151 greater than 3.0 g were excluded from curve fits or other trend determinations, so long as the resulting equations met the conservatism requirements over the full range of data. 9 4 1 System Capacity of Unanchored Two Girder System in Zero W ind The baseline system capacity results for strut braced systems of two (2) FIBs with no anchor and no wind are plotted in Figure 9 6 the FIB depth ( D ) has almost no effect on capacity, which is correlated primarily with span length ( L ). Therefore, an equation for C 0 in terms of L was produced using an exponential curve fit: ( 9 2 ) where C 0 is in g and L is the span length in ft. Equation 9 2 conservatively envelopes 95% of the available data points (Figure 9 7 ). 9 4 2 System Capacity Increase from Inclusion of A nchor The inclusion of an anchor tends to increase the capacity of the s ystem linearly with the roll stiffness of the anchor, as shown in Figure 9 8 sections are qualitatively similar, though the linearity is noisier for shallower FIB secti ons. The linear relationship between buckling capacity and anchor stiffness is similar to the effect of anchor stiffness on wind capacity (discussed in Chapter 8 ) and, as with wind capacity, anchor stiffness coefficients were determined by using linear regressions to calculate the slope ( m a ) of the anchor stiffness curves. Anchor stiffness coefficients ( m a ) were calculated for every curve, each corresponding to a unique combination of span length and FIB dept h. However, to facilitate interpretation of the data, the m a values for curves within which every capacity value was greater than 3.0 were rejected. This prevented the data obtained from such highly stable systems from obscuring PAGE 152 152 overall trends that were pe rtinent to systems of primary interest in this study. The remaining (i.e., non rejected) values of m a were found to be most strongly correlated to D therefore regression was used to construct a best fit linear relationship between m a and D This relations hip (Figure 9 9 dashed line) overpredicted m a in roughly half of the observed cases (which is consistent with the concept of a best fit line). Because overprediction of m a sometimes (but not always) result ed in unconservative capacity predictions, the fit parameters were adjusted (Figure 9 9 solid line) such that when the equation was later incorporated into the system capacity equation, all of the system c apacity values in the data set (except those that exceeded 3.0 g) were conservatively predicted (Figure 9 10 ). The resulting equation for m a was: ( 9 3 ) where m a has units of g/(kip ft/rad), and D is the FIB cross section depth in inches. Equation 9 3 was incorporated into the system capacity equation as follows: ( 9 4 ) where C is the buckling capacity in g, k roll,anchor is the anchor rotational stiffness in kip ft/rad, D is the FIB cross section depth in in., and C 0 is ca lculated using Equation 9 2 and is in g. 9 4 3 System C apacity R eduction from E rection of A dditional G i rders As noted earlier in Section 9 1 1 the erection of additional girders causes a reduction in system capacity because the resistance provided by the anchor is shared by multiple girders. Therefore, it was hypothe sized that the effect could be accounted for by restating Equation 9 4 in terms of the average anchor stiffness: PAGE 153 153 ( 9 5 ) where n is the number of girders in the system. Because Equation 9 4 was fitted to data where n = 2, the stiffness values that were used in that fit were equivalent to the quantity Substituting this into Equation 9 4 resulted in a new expression: ( 9 6 ) When the data set was expanded to includ e systems with between two (2) and nine (9) girders, Equation 9 6 was found to predict the system capacities with approximately the same degree of accuracy (Figure 9 11 ) that Equation 9 4 achieved for the data set restricted to two girder systems only, with conservative capacity predictions in 94.3% of cases. 9 4 4 System C apacity R eduction from I nclusion of W ind L oad The process of evaluating and predicting the effects of wind pressure on system capacity employed a concept similar to the average anchor stiffness concept introduce d in the previous section; the use of an average wind pressure per girder, : ( 9 7 ) where n is the number of girders in the bridge and is the sum of the individual wind pressures on all girders. Due to the nature of the shielding pattern that was employed, the average wind pressure ( ) for unskewed systems was always equal to one half of t he unshielded wind pressure ( P U listed previously in Tables 9 1 and listed later in Table 9 3 ), regardless of the number of girders in the system. I n contrast, for skewed systems is larger due to the fact that a portion of each girder remains unshielded. PAGE 154 154 As shown in the example cases presented in Figure 9 12 the decrease in system capacity produced by wind pressure is not linear. However, it was found to be approximately linear with respect to the square root of the wind pressure in most cases (Figure 9 13 ). This fact made it possible to define t he wind pressure coefficient, m w as the slope obtained from linear regression of system capacities expressed as a function of This linear regression process was carried out to obtain a value of m w for every unique combination of D, L, and The wind pressure coefficients, m w thus computed, were found to be nonlinearly correlated with both span length and average anchor stiffness. Therefore, an expression for the wind pressure coefficient was fitted to bo th variables simultaneously as a quadratic surface of the form : ( 9 8 ) where a b c d e and f are fitting parameters. When multivariate least squares regressi on was performed, the parameters b and e were found to be negligibly small and so those terms were omitted and the remaining terms were adjusted such that more than 95% of the results were conservatively predicted by the final surface fit (Figure 9 14 ), which was: ( 9 9 ) where m w is in g/( ), L is the span length in ft, and is the anchor roll stiffness in kip ft/rad. Incorporating Equations 9 9 and 9 7 into Equation 9 6 yields the system buckling capacity equation: ( 9 10 ) PAGE 155 155 where C is in g, L is the span length in ft, D is the FIB cross section depth in in., is the average wind load per girder in psf is the average anchor roll stiffness per girder in kip ft/rad and C 0 is calculated using Equation 9 2 and is in g. Once the parametric study data set was expanded to include cases with non zero wind, relative error ceased to be a useful metric for evaluating the accuracy of the system capacity prediction equation. This is because a large number of the cases had buckling capacity values at or near 0, where relative error is ill defined. However, the distribution of absolute error in predicted buckling capacity (Figure 9 15 ) shows that Equation 9 10 conservatively predicts the system capacity in more than 95% of cases. Plots illust rating the conservatism of Equation 9 10 relative to the parametric study results for selected cases are available in Appendix E 9 4 5 Consideration of S kew Inclusion of skew effects was found to reduce computed system capacities by as much as 30 40% for large skew angles. However, it was also determined th at, as long as the average wind pressure per girder, calculated using Equation 9 7 was based on skew modified wind loads calculated using Equation 9 1 the error distribution for Equation 9 10 was not significantly altered by inclusion of skew (Figure 9 16 ). Approximately 8% of all predictions, including cases with severe skew an gles (those approaching 50 ), were unconservative An unconservative prediction rate of 8%, which was larger than the previously targeted 5% criterion, was considered acceptable in this situation given that the distribution of bridge skew angles is biased toward smaller, rather than larger, skew angles. For this reason, and for the sake of simplicity no further modifications were made to Equation 9 10 to account for the effects of bridge skew. PAGE 156 156 9 5 Stiffness of Moment Resisting Braces In order for the results of the parametric studies to be as widely applicable as possible, it was necessary to define an effective brace stiffness ( k eff ): a single numerica l value describing the stability contribution of a moment resisting brace that can be computed for any brace configuration. Ideally, all braces configurations with the same k eff would be interchangeable (like strut braces). Unfortunately, in practice, the best that could be achieved was a strong correlation between k eff and system capacity. To evaluate potential brace stiffness definitions, an inventory of brace configurations was developed from fifteen (15) different X and K brace designs (Figure 9 17 ). Each design was implemented at three (3) different FIB depths (54, 78 and 96 in.) and at three (3) different girder spacings (6, 10, and 14 ft) for a total of 135 unique brace configurations in the inventory. The effectivenes s of each potential definition of k eff was tested by adjusting the elastic moduli of all brace configurations in the inventory such that the braces all had the same computed k eff A limited scope parametric study was then performed for each brace configura tion and the resulting capacity values were compared to determine how close to equal they were. After testing several potential definitions of k eff in this manner, the best correlation between k eff and system capacity was obtained from the use of a rotatio nal stiffness computed using a simplified brace model (Figure 9 18 ). The simplified model is similar to that which was proposed for evaluating brace forces (recall Chapter 5 ), b ut with ideal pins at the girder connection points and with a unit torque load applied at the girder center of rotation. In the parametric study for systems with moment resisting braces (discussed later in this chapter), k brace was the only parameter relat ed to the structural configuration of the braces that was varied. As a result, k brace was the only such parameter included in the proposed system PAGE 157 157 capacity prediction equation that was developed using the results of that study. It is intended that practicin g engineers will create a structural model of the type shown in Figure 9 18 to evaluate k brace for a potential brace design, and then use the resulting k brace in the capacity prediction equation to evaluate the sta bility of the system. However, because k brace is not a perfect predictor of system capacity, it is probable that different brace configurations having the same value of k brace might result in structural systems that have somewhat differing capacities, even though the proposed equation would predict identical capacities. These differences in capacity (termed ancillary effects ) are attributed to the involvement of brace deformation modes that are not perfectly represented by the simplified brace model (used t o compute k brace ), and to changes in brace geometry that occur as the system deforms. (Note that both of these phenomena were included in the large displacement numerical analyses of system capacity that were performed in the parametric studies). To separa te the ancillary effects from effects attributable to k brace a single structural configuration of brace was sought for use throughout the moment resisting brace parametric study, with different values of k brace achieved by altering the elastic modulus of the brace members. This reference brace configuration was chosen from the brace inventory using the results of the limited scope bracing parametric study. Every brace in the limited scope bracing study was adjusted (by calibration of material properties) t o have the same value of k brace so that differences in capacity between otherwise identical systems were a reflection only of the ancillary effects. Relative differences in capacity were then evaluated for every combination of brace design (Figure 9 17 ) and girder spacing. The combination that produced the lowest capacity on average, which turned out to be a K brace at 6 ft girder spacing (Figure 9 19 ), was then selected as the reference brace Consequently, an arbitrary brace configuration designed by PAGE 158 158 a practicing engineer is likely to have ancillary effects that only increase the capacity of the system relative to the reference brace used in this study, and the capacity of the system predicte d by the equations developed in this study is therefore likely to be conservatively low. In all subsequent parametric studies discussed in this chapter, different values of k brace were achieved by altering the elastic modulus of the brace members in the re ference brace shown in Figure 9 19 9 6 Selection of Parameters for Moment Resisting Brace Parametric Study System parameters that were varied in the moment r esisting brace parametric study were as follows: FIB cross section depth (in) Span length (ft) Wind pressure (psf) Effective brace stiffness (kip ft/rad) Number of interior brace points Skew angle (deg) Seven (7) of the eight (8) standard FIB cross sectio was excluded because the cross section (Figure 9 20 ) is so shallow that usage of moment resisting braces is unwarranted and unfeasible.) For each FIB, capacity analyses were p erformed for every combination of span length, wind pressure, effective brace stiffnesses, and number of interior brace points sampled from the values listed in Table 9 3 for a total of 17,760 analyses. This study only considered two girder systems because it was determined from sensitivity studies that when moment resisting braces are used, the two girder system is always the least stable phase of construction. Span lengths, skew angles, and wind pressure values were id entical to those used in the strut brace parametric study, while the number of interior brace points varied from 0 (end bracing only) to 3 (end bracing with quarter point interior bracing). As in the strut brace study, the PAGE 159 159 girders in each system model were spaced at 6 ft on center (because the reference brace configuration was 6 ft wide). For systems with moment resisting braces, changes in girder spacing produce changes in the geometric configuration of the brace members, thus changing the effective stiffn ess of the braces. Such changes can significantly affect system capacity and must be considered. In the moment resisting brace parametric study, the effects of changing girder spacing were accounted for by varying the effective brace stiffness parameter, e ven though the physical length of the reference brace remained a constant 6 ft. Effective brace stiffness values were selected using the brace inventory. Effective stiffness was computed for each brace (Figure 9 21 ) using the unadjusted material properties of timber and steel and stiffness values of 200,000, 400,000, and 600,000 kip ft/rad were selected to cover the range of representative values. However, the spread of values in Figure 9 21 corresponds primarily to the K braces, with all but one of the X brace stiffnesses confined to the leftmost column. When examining the distribution of k eff for X braces alone (Figure 9 22 ), it is clear that they are clustered at a much lower stiffness range. Therefore, to ensure adequate coverage for X braces, an additional representative value of 15,000 kip ft/rad was chosen as the fourth value. 9 7 Results of Moment Resisting Brace Parametric Study Results from the moment resisting brace parametric study are summarized in Figure 9 23 As with the earlier summary of strut brace para metric study results (Figure 9 5 ), the results shown in Figure 9 23 are divided into five (5) main groups representing the different wind pressures, then subgroups for the different cross sections and span lengths. The subgroups are less visually distinctive in Figure 9 23 than in Figure 9 5 because in moment resisting brace PAGE 160 160 cases, the effective brace stiffness has a much greater influence on the system capacity than does the anchor stiffness or the number of girders. In order to develop a capacity prediction equation for moment resisting braced systems, it was useful to establish a baseline capacity ( C 0 ) to which correcti on factors could be added to account for the effects of bracing stiffness and wind pressure. However, it would have been illogical to include systems with zero (0) effective brace stiffness in the study, because this would have been equivalent to simultane ously analyzing two individual (structurally independent) girders. Instead, a system with zero effective brace stiffness was defined as being strut braced, so that the girders were structurally connected but no moment resistance was provided. Therefore, da ta points from Figure 9 6 were included in the interpretation of the moment resisting brace study results, and Equation 9 2 was used as the baseline of the capacity prediction equation. All cases for which the cap acity exceeded 10 g were excluded from consideration because the behavior of such systems did not exhibit meaningful lateral instability. Aside from this change, the same criteria were used in developing the capacity prediction equation as were used for th e strut braced study: conservative capacity predictions in 95% of cases, with the unconservative cases not in error by more than 5%. 9 7 1 System C ap acity I ncrease from I nclusion of M oment R esisting E nd B races Inclusion of moment resisting braces at the girder ends increases the capacity of the system, however, as the effective stiffness of the end braces increases, the additional stability produced di minishes in magnitude, resulting in capacity vs. effective brace stiffness curves that tend to plateau (Figure 9 24 ). It was determined that the functional form that was the closest fit to this behavior was a rectangular hyp erbola: PAGE 161 161 ( 9 11 ) increase in capacity from the baseline value ( C 0 ) each effective brace stiffness curve (co rresponding to a unique combination of FIB depth and span length), was translated vertically so that it passed through the origin. A The distri bution of values in the data set exhibited no particular trends, so a constant value of 1,000,000 (the mean value of rounded off) was selected. However, the term showed a clear dependence on span length, so an exponential regression fit (Figure 9 25 ) was performed, resulting in the expression: ( 9 12 ) and Equations 9 12 and 9 11 were incorporated together to form the capacity prediction equation: ( 9 13 ) where C is in g, k brace is the effective moment resisting brace stiffness in kip ft/rad, C 0 is calculated using Equation 9 2 and is in g, and L is in ft. Equation 9 13 meets the 95 % criteri on for conservative capacity prediction. 9 7 2 System C apacity I ncrease from I nstallation of B races at I nterior P oints If braces are installed at interior points (in addition to braces at the girder ends), the additional brace stiffness increases the overall system capacity. However, the incremental increase in system capacity that is achieved by the addition of each new interior br ace PAGE 162 162 diminishes. For example, the additional stability provided by a midpoint brace is greater than the additional stability provided by replacing the midpoint brace with two (2) third point braces (all else being equal). However, there were no clearly obse rvable trends between the number of interior brace points (denoted n i ) and the system capacity that could be quantitatively related to the geometric parameters the system. Therefore the effect of interior braces was accounted for by introducing an empirica lly determined scale factor, into the brace stiffness term in the capacity prediction equation: ( 9 14 ) where C and C 0 are in g L is the span length in ft, k brace is the effective brace s tiffness in kip ft/rad, and is a dimensionless scale factor that is equal to 1 when n i is 0. To determine appropriate values of for n i > 0, three subsets of the parametric study results data were produced, corresponding to the non zero values of n i (1, 2, and 3). The subsets were restricted to cases where C < 3.0 g to ensure that each subset was representative of the (through trial and error), such that more than 95% the capacity values predicted by Equation 9 14 were conservative. Those values of were then reduced by approximately the same proportion until 95% conservatism was achieved over the full data set (Fig ure 9 27 ), including those cases g. Final values for use in the capacity prediction equation are listed in Table 9 4 9 7 3 System C apacity R eduction from I nclusion of Wind L oad The process by which the effect of average wind pressure, on system capacity, C was interpreted and predicted fo r moment resisting braced systems was very similar to that used for PAGE 163 163 the strut braced systems (recall Section 9 4 4 ). As in the case of strut braced systems, a linear relationship (Figure 9 28 ) was observed for moment resisting braced systems between the square root of average wind pressure ( ) and system capacity, and the corresponding slope was defined as the wind pressure coefficient ( m w ). For moment res isting braced systems in which the effective brace stiffness was small, the relationship between C and was not as linear as it was in strut braced systems, but for the sake of consistency, the assumption of linearity was consider ed an acceptable approximation. Linear regressions were performed to determine values of m w for every unique combination of FIB depth, span length, effective brace stiffness, and number of interior brace points ( n i ). The resulting values of m w were observe d to vary nonlinearly with both span length and effective brace stiffness, so a quadratic surface was fitted to both variables simultaneously, in the form: ( 9 15 ) where a b c d e and f are fitting parameters. In the resulting equation, the b parameter was found to be negligibly small, so it was discarded and the remaining parameters were adjusted such that system capacities were predicted conservatively in 95 % of cases. The final curve fit (Figure 9 29 ) was: ( 9 16 ) where m w is in g/( ), L is th e span length in ft, and k brace is the effective brace stiffness in kip ft/rad. Equation 9 16 was incorporated into Equation 9 14 to produce th e final capacity prediction equation for moment resisting braced systems: PAGE 164 164 ( 9 17 ) where C is the system capacity in g, L is the span length in ft, D is the FIB cr oss section depth in in., is the average wind load per girder (calculated using Equation 9 7 ) in psf, k brace is the is the effective brace stiffness in kip 9 4 and C 0 (calculated using Equation 9 2 ) is in g. As shown in the distribution of absolute prediction error (Figure 9 30 ), Equation 9 17 conservatively predicts the system capacity of a moment resisting braced system in over 95% of cases. 9 7 4 Consideration of S kew For moment resisting braced systems, it was found that Equation 9 17 predicted the capacity of skewed systems with approximately the same accuracy as was achieved for unskewed systems (Figure 9 31 ). Conservative capacity predictions were obtained in approximately 92% of cases (i.e., approximately 8% of cases were unconservative). Therefore, for the same reasons that were described in earlier in Section 9 4 5 (for strut braced systems), no further modifications were made to Equation 9 17 to account for the effects of bridge skew. 9 8 Incorporation of Ae rodynamic Lift In addition to horizontal wind pressure (drag), FIBs subjected to wind flow can also experience vertical lift forces and torques. As discussed in Chapter 5 the drag coefficient ( C D ) can be modified to in clude the structural demand associated with aerodynamic torque to form an effective drag coefficient ( C D,eff ) that represents both drag and torque. However, lift cannot be accommodated in the same manner and must be accounted for separately. Because lift a cts along the same vertical axis as gravity, it directly affects system capacity by either cancelling out PAGE 165 165 (offsetting) a portion of the girder self weight (positive lift; increased system capacity), or by adding to the total downward load on the girders (n egative lift; decreased system capacity). If lift force is expressed as an equivalent load acting on the girder (in units of g), in can be either directly added to or subtracted from the system capacity. Recall from Figure 5 11 that FIBs can have lift coefficients ( C L ) as large as 0.5. For conservatism, it was deemed necessary to assume that every girder in the system might have a C L Recalling concepts introduced in Chapter 3 the force coefficient Equation (3.6) can be rearranged and applied both to drag forc e ( D ) and lift force ( L ) as: ( 9 18 ) where [as was defined in Equation 3 1 ], and both F D and F L are proportional to their respective coefficients ( C L and C D ) by the same proportionality factor ( qD ). It follows therefore that: ( 9 19 ) The drag forc e, D, can be expressed in terms of the system parameters as: ( 9 20 ) where D is in lbf/ft (force per unit length of beam), D is in inches, P U is the unshielded wind pressure in psf, and 12 is a unit conversion factor. Substituting Equation 9 20 into Equation 9 19 employing a C L servatism, as noted above), and adopting a drag PAGE 166 166 coefficient of C D = 2.0 (based on the proposed design loads from Chapter 5 ), an expression for L in terms of the system parameters is obtained: ( 9 21 ) where L is in lbf/ft. Note that by expressing L in terms of the design wind load, any additional design factors (e.g., gust effect factor) are automatically inc orporated. L can be converted into units of g by dividing it by the girder self weight ( w sw ) in lbf/ft. Values of w sw are provided by the Structures Design Guidelines ( FDOT, 2012 e ) for each of the eight FIB cross sections (Table 9 5 ). Therefore, the final system capacity equation for strut braced systems is: ( 9 22 ) and the final system capacity equation for moment resisting braced systems is: ( 9 23 ) where C is the system capacity in g, L is the span length in ft, D is the FIB cross section depth in in., P U is the unshielde d wind load in psf, is the average wind load per girder (calculated using Equation 9 7 ) in psf, is the average anchor roll stiffness per girder (calculated using Equation 9 5 ) in kip ft/rad k brace is the is the effective brace stiffness in kip PAGE 167 167 determined from Table 9 4 w sw is the girder self weight in lbf/ft (from Table 9 5 ), and C 0 (calculated using Equation 9 2 ) is in g. PAGE 168 168 Table 9 1 Parameter values used in strut brace parametric study Span length, L (ft) 9 75 95 110 120 135 145 155 170 85 105 120 130 145 155 165 180 95 115 130 140 155 165 175 190 105 125 140 150 165 175 185 200 115 135 150 160 175 185 195 205 185 220 Anchor rotational stiffness, k anchor,roll (kip ft/rad ) 0 0 0 0 0 0 0 0 30 60 100 150 250 320 420 820 60 120 200 300 500 640 840 1640 90 180 300 450 750 960 1260 2460 120 240 400 600 1000 1280 1680 3280 150 300 500 750 1250 1600 2100 4100 Unshielded wind pressure, P U (psf) Number of girders, n Skew angle 0 2 0 40 3 2 80 5 5 120 7 10 160 9 25 50 Table 9 2 G irder offset lengths in model for each skew angle Nominal skew angle Offset length 2 5 10 25 50 PAGE 169 169 Table 9 3 Parameter values used in moment resisting brace parametric study Span length, L (ft) 95 110 120 135 145 155 170 105 120 130 145 155 165 180 115 130 140 155 165 175 190 125 140 150 165 175 185 200 135 150 160 175 185 195 205 185 220 Unshielded wind pressure, P U (psf) Eff. brace stiffness, k brace (kip ft/rad) Int. brace points, n i Skew angle 0 15,000 0 0 40 200,000 1 2 80 400,000 2 5 120 600,000 3 10 160 25 50 Table 9 4 Empirically determined values of for different numbers of interior braces n i Brace locations 0 End bracing 1.0 1 Midpoint bracing 1.4 2 Third point bracing 1.6 3 Quarter point bracing 1 .7 Table 9 5 Selfweight ( w sw ) of each FIB cross sectional shape (from FDOT, 2012) Cross section w sw (lbf/ft) 1037 1103 11 46 1190 1278 PAGE 170 170 A B Figure 9 1 Examples of strut bracing. A ) top strut B ) parallel struts A B Figure 9 2 Collapse mechanism possible with strut bracing A ) Undeformed configuration B ) Collapse mechanism A B Figure 9 3 Examples of moment resisting braces A ) X brace B ) K brace PAGE 171 171 A B Figure 9 4 Effect of bridge skew on wind loading of braced 3 girder system (plan view) A ) Unskewed system B ) Skewed system Figure 9 5 Summary of strut brace parametric study results PAGE 172 172 Figure 9 6 System capacities of unanchored two girder strut braced systems in zero wind at various span lengths Figure 9 7 System capacity of an unanchored strut braced two girder system in zero wind as predicted by Equation 9 2 PAGE 173 173 Figure 9 8 Figure 9 9 Anchor stiffness coefficient Equation 9 3 compared t o parametric study results PAGE 174 174 Figure 9 10 Relative error of system capacity values predicted by Equation 9 4 (n ote: negative relat ive error indicates conservative prediction of capacity) Figure 9 11 Relative error of system capacity values predicted by Equation 9 6 (n ot e: negative relative error indicates conservative prediction of capacity) PAGE 175 175 Figure 9 12 FIB system with anchor stiffness of 1600 ki p ft/rad Figure 9 13 Approximate linear relationship between system capacity and square root of FIB system with anchor stiffness of 1600 kip ft/rad) PAGE 176 176 Figure 9 14 Quadratic surface (Equation 9 9 ) fitted to wind pressure coefficient values Figure 9 15 Absolute error of system capacity values predicted by Equation 9 10 PAGE 177 177 Figure 9 16 Absolute error of s ystem capacity values predicted by Equation 9 10 for strut braced systems, including systems with non zero skew angles PAGE 178 178 Figure 9 17 Brace designs in brace inventory (each implemented at three different spacings and three different FIB depths) Figure 9 18 Model used to compute effective stiffness of brace configurations (X brace shown) PAGE 179 179 Figure 9 19 Reference brace configuration used in parametric studies Figure 9 20 Cross Figure 9 21 Stiffness of every brace in brace inventory PAGE 180 180 Figure 9 22 Stiffness of every X brace i n brace inventory Figure 9 23 Summary of moment resisting brace parametric study results PAGE 181 181 Figure 9 24 Equation 9 12 Figure 9 25 Equation 9 12 PAGE 182 182 Figure 9 26 Relative error of system capacity values predicted by Equation 9 13 ( n ote: negative relative error indicates conservative prediction of capacit y) Figure 9 27 Relative error of system capacity values predicted by Equation 9 14 PAGE 183 183 Figure 9 28 Approximate linear relationship between system capacity and square root of average wind pressure ( d ata shown are for 160 FIB systems with third point bracing) PAGE 184 184 Figure 9 29 Quadratic surface (Equation 9 16 ) fitted to wind pressure coefficient values and adjusted to produce conservative results in 95% of cases Figure 9 30 Absolute error of system capacity values predicted by Equation 9 17 for moment resisting braced systems PAGE 185 185 Figure 9 31 Absolute error of system capacity values predicted by Equation 9 17 for moment resisting braced systems, including systems with non zero skew angles PAGE 186 186 CHAPTER 10 CONCLUSIONS AND RECOMMENDATIONS 10 1 Introduction In this study, design issues related to wind loading and the stability of long span braced bridge girders were investiga ted. Wind tunnel tests were performed to measure the aerodynamic properties (drag coefficients, lift coefficients, torque coefficients, and shielding effects) of several common cross sectional shapes used for bridge girders. Additionally, numerical models were employed to evaluate the stability of braced systems of Florida I Beams (FIBs) subjected to wind loads. Based on the results of these investigations, conclusions and recommendations are discussed in the sections that follow. 10 2 Drag Coefficients Drag coefficients ( C D ) were measured using wind tunnel testing for five (5) different girder cross sectional shapes (Figure 10 1 ) that are typical of bridge construction in the state of Fl not exceed 1.95, and the drag coefficients of built up steel plate girders did not exceed 2.15. In both cases, the Structures Design Guidelines (SDG; FDOT, 2012e) currently conservatively overpredicts the drag coefficient by prescribing a value of 2.2. In contrast, for the open top box not exceed 1.95, but this value exce eds the value of 1.5 currently prescribed by the SDG. Wind tunnel test results also indicated significant shielding effects when multiple adjacent girders were subjected to lateral wind. In general, the windward girder (G1) acted as a windbreak, causing th e drag force on subsequent girders to be reduced sharply enough that the drag coefficient of the first shielded girder (G2) was typically negative (indicating that the drag force acted in the opposite direction, i.e., against the wind). Drag forces on shie lded girders (G2, PAGE 187 187 G3, etc.) tended to follow a down then up pattern : drag coefficients dropped to their most negative value at girder G2 or G3, then gradually grew more positive until a positive plateau value was reached farther down stream. Based on the w ind tunnel test results, shielding patterns for specific bridge cross sectional configurations could not be predicted with certainty, but were influenced by the interaction of cross slope, wind angle, section depth, and girder spacing. In addition to drag (horizontal) forces, it was observed that bridge girders subjected to lateral wind can also be subjected to both lift (vertical) forces and torques that are too large to be considered negligible. To address this issue, the concept of an effective drag coef ficient ( C D,eff ) was developed to envelope the combined effects of both drag and torque. The effective drag coefficient can be used in design calculations in the same manner that a standard drag coefficient is used. Unless project specific wind tunnel test results are available, the following pressure coefficients ( C P ) are recommended for systems of adjacent girders (Figure 10 2 ): Assign the windward girder (G1) an initial C P depending on the ty pe of section ( C P = 2.0 for FIBs, C P = 2.5 for plate girders). Assign the first shielded girder (G2) no wind load ( C P =0). Assign all subsequent shielded girders (G3 and greater) a C P equal to one half of the initial C P that was assigned to the windward gi rder (G1) These design loads are intended for use in system stability analyses, and incorporate the structural demand associated with both aerodynamic drag and aerodynamic torque. It is important to note that the wind loads that produce the greatest pote ntial for lateral instability in a braced girder system are not generally the loads that produce the largest individual brace forces. Therefore, two separate wind load cases are required when designing braces to withstand both limit states. Individual brac e forces are maximized when wind forces on adjacent girders act in opposite directions (thus compressing elements of the brace). In contrast, PAGE 188 188 lateral instability is maximized when girder wind loads act together in the same direction to maximize the total l ateral force exerted on the braced girder system. When evaluating brace forces (as opposed to system stability), it is recommended that a C P of 2.75 be used for FIBs and a C P of 3.0 be used for plate girders. These coefficients approximate the maximum tota l compressive brace load that arises from the combination of a large positive load on the windward girder (G1) and a smaller, negative load on the first shielded girder (G2). To ensure conservative determination of brace forces, it is recommended that a st ructural analysis be performed on a girder and brace sub assembly model (Figure 10 3 ). Required characteristics of the model include a pinned support at the base of G1, a fixed boundary condi tions in place of G2, and full moment transfer (i.e., no pins or end releases) at the brace girder connection points. 10 3 Individual Unbraced Florida I Beams An investigation into the stabi lity of individual unbraced girders supported by bearing pads, both with and without anchors in place, was a key component of this study. Numerical analysis techniques were developed to compute the wind capacity of such girders: i.e., the critical wind loa d at which a girder collapses under its own self weight. A parametric study was performed in which the wind capacity was computed for all eight (8) FIB cross sections at a variety of span lengths, both unanchored and with anchors of varying stiffnesses. Fr om the results of the parametric study, regression techniques were used to develop an empirical equation for computing the wind capacity of an unanchored girder (Chapter 8 Equation 8 2 ). Parametric study results were similarly used to develop a capacity modification (correction) factor to account the increase in wind capacity that is produced by the presence of an anchor of specified stiffness. Combining the una nchored girder capacity equation with the PAGE 189 189 correction factor produced a generalized wind capacity prediction equation (Chapter 8 Equation 8 6 ) that is recommended for us e in assessing the stability of individual (anchored or unanchored) Florida I Beams (FIBs) subjected to wind loading. 10 4 Braced Systems of Multiple Florida I Beams For a collection of gird ers braced together into a single structural system, numerical analysis techniques were developed to compute the system capacity in units of g (the acceleration due to gravity), representing the total gravity load that can be applied before the system beco mes unstable (collapses). Sensitivity studies were performed to evaluate the influence of a number of geometric parameters on the system capacity. From the results of the sensitivity studies, it was concluded that girder braces can be divided into two basi c categories: strut braces, which merely connect the girders together with axial stiffness but without providing any overturning resistance, and moment resisting braces, which resist girder overturning. The two categories of brace have very different effec ts on the capacity of a girder system. For strut braced systems, it was determined that varying the properties of the brace members had essentially no effect on system capacity. It was also found that no significant increase in system capacity was achieved by installing braces at interior brace points in addition to the girder end points. That is, a strut braced system with both end braces and interior braces has nearly the same capacity as a system with end braces only. In contrast, with moment resisting b races, the properties and geometric configuration of the brace members were found to have a very significant effect on system capacity, as was the presence of additional moment resisting braces at interior brace points. In order to predict the effect that a particular moment resisting brace design would have on system capacity, an effective brace stiffness was defined, the value of which is computed using a simplified structural model. PAGE 190 190 Two large scale parametric studies were performed: one for strut braced systems and one for systems with moment resisting braces. Structural parameters that were varied included the FIB cross section type, span length, skew angle, anchor stiffness (for strut braced systems), and both effective brace stiffness and number of int erior brace points (for systems with moment resisting braces). Wind loads associated with lateral drag force, torque, and vertical uplift were all taken into account. From the results of these parametric studies, system capacity prediction equations were d eveloped and recommended for use in assessing the stability of multi girder systems with strut braces (Chapter 9 Equation 9 22 ) and moment resisting braces (Chapter 9 Equation 9 23 ). 10 5 Future Research Wind tunnel testing performed in the present study foc used on individual girders and groups of unconnected girders without any additional components present. No consideration was given to the changes in aerodynamic properties that might occur when stay in place deck forms or overhangs are present. Additionall y, it was not within the scope of the wind tunnel testing conducted in this study to quantify changes in drag force that might occur when an otherwise shielded girder is partially exposed due to bridge skew. Hence, it may be appropriate to address these is sues with future wind tunnel testing. Alternatively, it may be possible to use a computational fluid dynamics (CFD) analysis approach to investigate one or both of these situations. Such an approach could potentially be validated using the wind tunnel meas urements obtained during the present study. PAGE 191 191 Figure 10 1 Girder cross sectional shapes tested in the wind tunnel Figur e 10 2 Recommended design wind loads for systems of adjacent girders PAGE 192 192 A B C Figure 10 3 Recommended structural analysis model for use in determining brace forces A) Strut brace. B ) X brace C ) K Brace PAGE 193 193 APPENDIX A DIMENSIONED DRAWINGS OF WIND TUNNEL TEST CONFIGURATIONS This appendix includes dimensioned drawings of every girder configuration that was subjected to wind tunnel testing. PAGE 194 194 PAGE 195 195 PAGE 196 196 PAGE 197 197 PAGE 198 198 PAGE 199 199 PAGE 200 200 PAGE 201 201 PAGE 202 202 APPENDIX B TABULATED RESULTS FROM WIND TUNNEL TESTS This appendix contains results from all o f the wind tunnel tests that were performed, including drag, lift, and torque coefficients, as well as effective drag coefficients (discussed in Chapter 5) that combine the structural demand of both drag and torque. Results for each test configuration are given an ID code consisting of a letter and 2 numbers. The letter describes the geometric arrangement of the girders (Table B 1 ), the first number is the number of girders and the second number is the girder being measured. For example, the designation B5 3 refers to the third (3) WF Plate girder in a group of five (5) with a spacing of 14 ft and a cross slope of 2%. Table B 1 Meaning of letters in configuration IDs Configuration ID l etter Section Cross slope Spacing (ft) A WF Plate + 2% 10 B WF Plate + 2% 14 C WF Plate + 8% 10 D WF Plate + 8% 14 E NF Plate + 2% 10 F NF Plate + 2% 14 G + 2% 10 H + 2% 13 I 13 J + 2% 10 K + 2% 13 L 13 M Box + 0% 20 N Box + 0% 22 PAGE 203 203 Testing Configuration A Cross section: WF Plate Spacing: 10 ft Cross slope: +2% Drag coefficient ( C D ) Indv. Wind Angle 2.13 2.23 1.99 + 0.03 0.34 2.13 2.25 2 .05 0.29 +0. 0 2.12 2.26 2.08 0.27 + 2.5 2.13 2.25 2.06 0.29 + 5.0 2.13 2.24 2.00 0.31 Lift coe fficient ( C L ) Indv. Wind Angle + 0.00 + 0.02 + 0.05 + 0.00 + 0.01 + 0.03 +0. 0 + 0.00 + 0.00 + 2.5 + 0.01 + 0.03 + 0.06 + 0.07 + 5.0 + 0.03 + 0.01 + 0.05 + 0.06 + 0.14 Torque coefficient ( C T ) Indv. Wind Angle 0. 03 0.03 + 0.00 0.03 0.00 + 0.00 0.00 0.03 0.03 + 0.00 0.03 0.00 + 0.00 0.00 +0. 0 0.03 0.03 0.03 0.00 0.00 + 2.5 0.03 0.03 0.03 0.00 + 0.01 0.01 + 5.0 0.03 0.03 0.03 0.00 + 0.02 0. 01 Effective drag coefficient ( C D,eff ) Indv. Wind Angle 2.19 2.30 2.04 + 0.03 0.34 2.19 2.31 2.10 0.29 +0. 0 2.18 2.32 2.13 0.28 + 2.5 2.18 2.31 2.12 0.31 + 5.0 2.18 2.30 2.06 0.34 PAGE 204 204 Testing Configuration B Cross section: WF Plate Spacing: 14 ft Cross slope: +2% Drag coefficient ( C D ) Indv. Wind Angle 2.13 2.19 1.89 0.26 0.64 2.13 2.20 1.97 0.15 0.64 +0. 0 2.12 2.22 2.01 0.10 0.57 + 2.5 2.13 2.21 1.98 0.14 0.65 + 5.0 2.13 2.19 1.90 0.23 0.71 Lift coefficient ( C L ) Indv. B2 Wind Angle + 0.00 + 0.02 + 0.04 + 0.03 + 0.00 + 0.01 + 0.03 + 0.02 +0. 0 + 0.00 + 2.5 + 0.03 + 0.01 + 0.05 + 0.07 + 5.0 + 0.08 + 0.02 + 0.07 + 0.08 Torque coefficient ( C T ) Indv. Wind Angle 0.03 0.03 0.03 + 0.00 1 + 0.00 0.03 0.03 0.03 + 0.00 + 0.00 +0. 0 0.03 0.03 0.03 + 0.00 + 0.00 + 0.01 + 2.5 0.03 0.03 0.03 + 0.00 + 0.00 + 0.01 + 0.02 + 5.0 0.03 0.03 0.02 + 0.00 + 0.01 + 0.03 Effective drag coefficient ( C D,eff ) Indv. Wind Angle 2.19 2.25 1.95 0.27 0.64 2.19 2.26 2.03 0.15 0.64 +0. 0 2.18 2.2 7 2.06 0.11 0.59 + 2.5 2.18 2.27 2.03 0.15 0.68 + 5.0 2.18 2.24 1.94 0.24 0.78 PAGE 205 205 Testing Configuration C Cross section: WF Plate Spacing: 10 ft Cross slope: +8% Drag coefficient ( C D ) Indv. Wind Angle 2.13 2.24 1.86 + 0.27 0.49 2.13 2.24 1.94 + 0.10 0.40 +0. 0 2.12 2.26 2.00 0.33 + 2.5 2.13 2.26 2.05 0.27 + 5.0 2.13 2.26 2.05 0.28 Lift coefficient ( C L ) Indv. C5 Wind Angle + 0.00 + 0.03 + 0.06 + 0.00 + 0.02 + 0.06 +0. 0 + 0.01 + 0.04 + 2.5 + 0.01 + 0.00 + 5.0 + 0.00 + 0.01 + 0.03 + 0.07 Torque coefficient ( C T ) Indv. Wind Angle 0.03 0.03 + 0.00 0.03 0.00 + 0.00 + 0.01 0.02 0.03 0.03 + 0.00 0.03 0.00 + 0.00 + 0.00 0.02 +0. 0 0.03 0.03 + 0.00 0.03 0.00 + 0.00 0.01 + 2.5 0.03 0.03 0.03 0.00 + 0.00 0.01 + 5.0 0.03 0.03 0.03 0.00 + 0.00 0.01 Effective drag coefficient ( C D,eff ) Indv. Wind Angle 2.19 2.30 1.91 + 0.30 0.52 2.19 2.30 1.99 + 0.11 0.45 +0. 0 2.18 2.31 2.07 15 0.36 + 2.5 2.18 2.31 2.11 0.29 + 5.0 2.18 2.31 2.10 0.29 PAGE 206 206 Testing Configuration D Cross section: WF Plate Spacing: 14 ft Cross s lope: +8% Drag coefficient ( C D ) Indv. Wind Angle 2.13 2.22 1.77 0.47 1.05 2.13 2.21 1.85 0.27 0.84 +0. 0 2.12 2.22 1.93 0.22 0.65 + 2.5 2.13 2.24 1.99 0.24 0.65 + 5.0 2.13 2.25 1.97 0.19 0.67 Lift coefficient ( C L ) Indv. Wind Angle + 0.0 0 + 0.04 + 0.04 + 0.00 + 0.03 + 0.04 +0. 0 + 0.01 + 0.03 + 2.5 + 0.00 + 5.0 + 0.02 + 0.00 + 0.03 + 0.06 + 0.07 Torque coefficient ( C T ) Indv. Wind Angle 0.03 0.03 0.03 0.00 + 0.00 0.01 0.04 0.03 0.03 0.03 0.00 0.00 0.04 +0. 0 0.03 0.03 0.03 0.00 0.00 0.03 + 2.5 0.03 0.03 0.03 0.00 0.00 0.02 + 5.0 0.03 0.03 0.03 0.00 0.02 0.01 Effective drag coefficien t ( C D,eff ) Indv. Wind Angle 2.19 2.28 1.83 0.48 1.12 2.19 2.27 1.91 0.27 0.91 +0. 0 2.18 2.27 1.99 0.23 0.70 + 2.5 2.18 2.29 2.04 0.24 0.69 + 5.0 2.18 2.30 2.02 0.23 0.69 PAGE 207 207 Testing Configuration D Cross section: WF Plate Spacing: 14 ft Cross slope: +8% Drag coefficient ( C D ) Wind Angle 1.54 0.15 0.51 0.75 0.99 1.15 1.31 1.23 1.65 0.04 0.35 0.49 0.63 0.80 0.96 1.00 +0. 0 1.77 0.05 0.34 0.38 0.42 0.53 0.60 0. 70 + 2.5 1.80 0.04 0.34 0.37 0.38 0.44 0.47 0.56 + 5.0 1.79 0.04 0.34 0.38 0.38 0.42 0.44 0.52 Lift coefficient ( C L ) D1 Wind Angle + 0.00 0.04 0.06 0.06 0.04 0.01 0.02 0.05 0.05 0.05 0.02 +0. 0 0.01 0.02 0.02 0.03 0.02 + 2.5 + 0.00 0.00 0.00 0.00 0.00 0.00 + 5.0 + 0.00 + 0.04 + 0.04 0.03 0.03 0.03 0.02 0.02 + 0.03 Torque coefficient ( C T ) Wind Angle 0.00 0.0 0 0.00 + 0.00 0.00 0.00 0.01 + 0.01 + 0.00 +0. 0 0.00 0.00 0.00 + 0.00 + 0.00 + 0.00 + 2.5 0.01 0.00 0.01 + 0.00 + 0.00 + 0.00 + 0.00 + 0.00 + 5.0 0.01 0.00 0.01 Effective drag coefficient ( C D,eff ) Wind Angle 1.54 0.16 0. 51 0.75 0.99 1.15 1.31 1.23 1.65 0.05 0.35 0.49 0.63 0.80 0.96 1.00 +0. 0 1.77 0.06 0.34 0.38 0.42 0.53 0.60 0.70 + 2.5 1.82 0.04 0.34 0.37 0.38 0.44 0.47 0.56 + 5.0 1.80 0.04 0.34 0.38 0.38 0.42 0.44 0.52 PAGE 208 208 Testing Configuration E Cross section: NF Plate Spacing: 10 ft Cross slope: +2% Drag coefficient ( C D ) Indv. Wind Angle 2.13 2.22 2.01 2.12 2.23 2.06 +0. 0 2.12 2.25 2.09 + 2.5 2.13 2.24 2.07 + 5.0 2.12 2.23 2.02 0 Lift coefficient ( C L ) Indv. Wind Angle + 0.00 + 0.00 +0. 0 + 0.00 + 0.00 0.01 + 2.5 + 0.00 + 0.00 + 0.01 + 5.0 + 0.00 + 0.00 + 0.02 Torque coefficient ( C T ) Indv. Wind Angle 0.01 0.01 + 0.00 0.01 + 0.00 0.01 0.01 + 0.00 0.01 + 0.00 + 0.00 +0. 0 0.00 0.01 0.01 + 2.5 0.00 0.00 0.00 + 5.0 0.00 0.00 0.00 M Effective drag coefficient ( C D,eff ) Indv. Wind Angle 2.14 2.23 2.03 2.13 2.24 2.07 +0. 0 2.12 2.26 2.11 + 2.5 2.13 2.24 2.07 + 5.0 2.12 2.23 2.02 PAGE 209 209 Testing Configuration F Cross section: NF Plate Spacing: 14 ft Cross slope: +2% Drag coefficient ( C D ) Indv. Wind Angle 2.13 2.22 1.90 2.12 2.20 1.96 +0. 0 2.12 2.20 1.99 + 2.5 2.13 2.21 1. 95 + 5.0 2.12 2.22 1.90 Lift coefficient ( C L ) Indv. Wind Angle .5 + 0.00 +0. 0 + 0.00 + 0.00 + 2.5 + 0.01 + 0.00 + 0.04 + 5.0 + 0.03 + 0.01 + 0.06 Torque coefficient ( C T ) Indv. Wind Angle 0.01 0.02 0.13 0.01 0.02 0.13 +0. 0 + 0.00 0.01 0.13 + 2.5 + 0.00 0.07 0.1 3 + 0.00 + 5.0 + 0.00 0.13 0.12 Effective drag coefficient ( C D,eff ) Indv. Wind Angle 2.14 2.26 2.16 36 2.13 2.23 2.22 +0. 0 2.12 2.22 2.25 + 2.5 2.13 2.35 2.20 + 5.0 2.12 2.48 2.14 PAGE 210 210 Testing Configuration G Cross section: Spacing: 10 ft Cross slope: +2% Drag coefficient ( C D ) Indv. Wind Angle 1.90 1.95 1.57 1.89 1.97 1.65 0.22 +0. 0 1.89 1.97 1.71 + 2.5 1.90 1.98 1.70 + 5.0 1.91 2.00 1.66 Lift coefficient ( C L ) Indv. Wind Angle 0.18 0.24 0.24 0.17 0.20 0.22 +0. 0 0.15 0.16 0.15 + 2.5 0.14 0.13 + 0.07 0.09 + 0.02 + 0.06 + 5.0 0.13 0.09 + 0.18 0.04 + 0.03 + 0.07 Torque coefficient ( C T ) Indv. Wind Angle 0.12 0.14 0.17 0.12 0.12 0.17 + 0. 0 0.11 0.12 0.16 + 2.5 0.10 0.10 0.16 + 5.0 0.10 0.12 0.15 + 0.00 Effective drag coefficient ( C D,eff ) Indv. G5 Wind Angle 1.93 2.01 1.74 1.92 1.99 1.80 +0. 0 1.89 1.98 1.85 + 2.5 1.90 1.98 1.82 + 5.0 1.91 2.02 1.78 PAGE 211 211 Testing Configuration H Cross section: Spacing: 13 ft Cross slope: +2% Drag coefficient ( C D ) Indv. Wind Angle 1.90 1.87 1.54 1.89 1.88 1.61 +0. 0 1.89 1.90 1.65 + 2.5 1.90 1.90 1.64 + 5.0 1.91 1.91 1.58 Lift coefficient ( C L ) Indv. Wind Angle 0.18 0.24 0.21 0.17 0.20 0.19 +0. 0 0.15 0.17 0.14 + 2.5 0.14 0.14 + 0.07 0.09 + 0.02 + 0.07 + 5.0 0.13 0.10 + 0.22 0.05 + 0.03 + 0.09 Torque coefficient ( C T ) Indv. Wind Angle 0.12 0.18 0.16 0.12 0.17 0.16 +0. 0 0.11 0.16 0.16 + 2.5 0.10 0.15 0.15 + 5.0 0.10 0.17 0.14 + 0.00 M0 Effective drag coefficient ( C D,eff ) Indv. Wind Angle 1.93 2.01 1.69 1.92 2.00 1.75 +0. 0 1.89 2.01 1.78 + 2. 5 1.90 1.98 1.75 + 5.0 1.91 2.04 1.69 PAGE 212 2 12 Testing Configuration H Cross section: Spacing: 13 ft Cross slope: +2% Drag coeffi cient ( C D ) Wind Angle 1.41 0.18 0.39 0.45 0.55 0.63 0.74 0.75 1.52 0.10 0.27 0.31 0.33 0.35 0.40 0.46 +0. 0 1.57 0.04 0.26 0.28 0.29 0.2 9 0.31 0.35 + 2.5 1.56 0.05 0.35 0.27 0.26 0.26 0.27 0.31 + 5.0 1.48 0.07 0.33 0.31 0.36 0.47 0.57 0.58 Lift coefficient ( C L ) Wind Angle 0.16 + 0.01 0.02 + 0.02 + 0.00 0.15 0.00 +0. 0 0.10 + 0.00 + 0.02 0.02 + 0.02 + 0.02 + 0.02 + 0.00 + 2.5 0.05 + 0.02 + 0.09 + 0.07 + 0.04 0.07 + 0.08 + 0.08 + 0.09 + 0.12 + 5.0 0.02 + 0.03 + 0.09 + 0.09 + 0.05 0.06 + 0.06 + 0.07 + 0.11 + 0.22 Torque coefficient ( C T ) Wi nd Angle 0.06 + 0.00 + 0.01 0.01 0.01 0.01 0.01 0.01 + 0.00 0.07 + 0.00 + 0.01 0.00 0.00 0.00 0.00 0.00 +0. 0 0.08 + 0.01 0.01 0.01 0.01 0.01 0.01 + 0.01 + 2.5 0.08 + 0.00 0.01 0.01 0.01 0.01 0.01 + 0. 02 + 5.0 0.08 + 0.00 0.01 0.01 0.01 0.02 0.03 + 0.04 Effective drag coefficient ( C D,eff ) Wind Angle 1.41 2 0.18 0.39 0.45 0.55 0.63 0.74 0.75 1.52 0.11 0.27 0.31 0.33 0.35 0.40 0.46 +0. 0 1.57 0.04 0.26 0.28 0.29 0.29 0.31 0.35 + 2.5 1.56 0.05 0.35 0.27 0.26 0.26 0.27 0.31 + 5.0 1.48 0.07 0.33 0.31 0.36 0.47 0.57 0.59 PAGE 213 213 Testing Configuration I Cross section: Spacing: 13 ft Cross slope: Drag coefficient ( C D ) 9 Wind Angle 1.52 0.13 0.28 0.32 0.29 0.32 0.36 0.42 1.57 0.05 0.27 0.31 0.27 0.27 0.29 0.32 +0. 0 1.55 0.05 0.30 0.33 0.28 0.28 0.30 0.33 + 2.5 1.46 0.07 0.36 0.37 0.38 0.47 0.5 7 0.60 + 5.0 1.37 0.13 0.41 0.53 0.68 0.89 1.02 0.94 Lift coefficient ( C L ) Wind Angle + 0.14 05 + 0.09 + 0.00 + 0.01 +0. 0 + 0.04 + 0.03 + 0.06 + 0.05 + 0.04 + 0.04 + 0.05 + 0.05 + 0.05 + 0.08 + 2.5 + 0.01 + 0.04 + 0.09 + 0.07 + 0.01 + 0.02 + 0.02 + 0.01 + 0.04 + 0.16 + 5.0 + 0.04 + 0.08 + 0.04 + 0.01 + 0.00 + 0.02 + 0.06 + 0.15 Torque coefficient ( C T ) Wind Angle 0.06 + 0.00 + 0.01 0.01 0.01 0.01 0. 01 0.01 0.01 0.07 + 0.00 0.01 0.01 0.01 0.01 0.01 0.01 +0. 0 0.08 + 0.00 + 0.00 0.01 0.01 0.01 0.01 0.02 0.02 + 2.5 0.08 + 0.00 0.01 0.02 0.02 0.03 0.04 0.04 + 5.0 0.07 + 0.00 + 0.00 0.02 0.03 0.04 0.06 0.07 0.07 Effective drag coefficient ( C D,eff ) Wind Angle 1.52 0.13 0.28 0.32 0.29 0.32 0.36 0.42 1.57 5 0.06 0.27 0.31 0.27 0.27 0.29 0.32 +0. 0 1.55 0.05 0.30 0.33 0.28 0.28 0.30 0.33 + 2.5 1.46 0.07 0.36 0.37 0.38 0.47 0.58 0.62 + 5.0 1.37 0.13 0.41 0.53 0.70 0.91 1.05 0.98 M0 PAGE 214 214 Testing Configuration J Cross section: Spacing: 10 ft Cross slope: +2% Drag coefficient ( C D ) Indv. Wind Angle 1.81 1.59 1.38 + 0.14 1.8 3 1.63 1.44 +0. 0 1.85 1.66 1.48 + 2.5 1.85 1.67 1.48 + 5.0 1.84 1.65 1.43 + 0.16 Lift coefficient ( C L ) Indv. Wind Angle 0.48 0.41 0.32 0.42 0.38 0.30 +0. 0 0.37 0.32 + 0.04 0.26 + 0.06 + 0.02 + 2.5 0.30 0.23 + 0.29 0.17 + 0.19 + 0.19 + 5.0 0.24 0.15 + 0.47 0.09 + 0.17 + 0.26 Torque coefficient ( C T ) Indv. Wind Angle 0.08 0.07 0.07 0.01 0.07 0.07 0.05 0.01 +0. 0 0.04 0.07 + 0.01 0.04 0.00 + 0.01 + 2.5 0.03 0.04 + 0.03 0.03 0.01 + 0.02 + 5.0 0.01 0.02 + 0.03 0.01 0.01 + 0.01 Effective drag coefficient ( C D,eff ) Indv Wind Angle 1.81 1.59 1.38 + 0.14 1.83 1.63 1.44 +0. 0 1.85 1.66 1.48 + 2.5 1.85 1.67 1.48 + 0.03 + 5.0 1.84 1.65 1.43 + 0.16 PAGE 215 215 Testing Configuration K Cross section: Spacing: 13 ft Cross slope: +2% Drag coefficient ( C D ) Indv. Wind Angle 1.81 1.54 1.38 0.51 1.83 1.58 + 0.04 1.44 0.48 +0. 0 1.85 1.60 1.47 0.56 + 2.5 1.85 1.60 1.45 0.53 + 5.0 1.84 1.60 1.40 0.66 0 Lift coefficient ( C L ) Indv. Wind Angle 0.48 0.33 0.28 0.42 0.30 0.27 +0. 0 0.37 0.27 0.24 + 0.05 + 0.09 + 2.5 0.30 0.21 + 0.22 0.17 + 0.22 + 0.11 + 5.0 0.24 0.16 + 0.47 0.12 + 0.27 + 0.11 Torque coefficient ( C T ) Indv. Wind Angle 0.08 0.07 0.06 + 0.04 0.07 0.07 0.05 + 0.03 +0. 0 0.04 0.04 0.04 + 0.01 + 0.02 + 2.5 0.03 0.02 + 0.03 0.01 + 0.02 + 5.0 0.01 0.02 + 0.03 0.01 + 0.01 M Effective drag coefficient ( C D,eff ) Indv. Wind Angle 1.81 1.54 1.38 0.54 1.83 1.58 + 0.04 1.44 0.49 +0. 0 1.85 1.60 1.47 0.56 + 2.5 1.85 1.60 1.45 0.53 + 5.0 1.84 1.60 1.40 0.66 PAGE 216 216 Testing Configuration K Cross section: Spacing: 13 ft Cross slope: +2% Drag coefficient ( C D ) Wind Angle 1.38 0.48 0.55 0.44 0.44 0.46 0.51 0.57 0.67 1.49 0.63 0.52 0.36 0.33 0.31 0.33 0.36 0.41 +0. 0 1.52 0.72 0.47 0.35 0.29 0.26 0.26 0.28 0.32 + 2.5 1.51 0.69 0.44 0.32 0.27 0.27 0.28 0.29 0.33 + 5.0 1.38 0.61 0.46 0.36 0.34 0.34 0.38 0.45 0.52 Lift coefficient ( C L ) 7 Wind Angle 0.26 0.04 0.23 + 0.15 0.11 +0. 0 0.18 + 0.18 0.09 + 0.04 + 0.02 + 0.01 + 0.02 + 0.02 + 0.01 + 2.5 0.12 + 0.11 + 0.0 8 0.09 + 0.10 + 0.11 + 0.11 + 0.12 + 0.14 + 0.16 + 5.0 0.08 + 0.22 + 0.06 0.13 + 0.16 + 0.17 + 0.17 + 0.18 + 0.22 + 0.28 Torque coefficient ( C T ) W ind Angle 0.20 0.07 0.13 0.11 0.12 0.12 0.14 0.16 0.18 0.20 0.15 0.11 0.08 0.07 0.08 0.08 0.09 0.10 +0. 0 0.19 0.16 0.08 0.07 0.05 0.05 0.05 0.04 0.06 + 2.5 0.19 0.11 0.07 0.06 0.05 0.05 0.05 0.05 0.05 + 5.0 0.16 0.09 0.09 0.07 0.06 0.05 0.06 0.05 0.05 Effective drag coefficient ( C D,eff ) Wind Angle 1.63 0.58 0.76 0.61 0.6 3 0.65 0.75 0.82 0.96 1.75 0.86 0.68 0.48 0.44 0.43 0.46 0.50 0.57 +0. 0 1.75 0.96 0.59 0.44 0.36 0.34 0.34 0.34 0.41 + 2.5 1.73 0.84 0.54 0.39 0.34 0.33 0.35 0.36 0.40 + 5.0 1.56 0.73 0.58 0.46 0.42 0.40 0.46 0.51 0.57 PAGE 217 217 Testing Configuration L Cross section: Spacing: 13 ft Cross slope: Drag coefficient ( C D ) Wind Ang le 1.47 0.65 0.52 0.38 0.34 0.33 0.35 0.39 0.43 1.53 0.69 0.48 0.35 0.28 0.27 0.27 0.28 0.31 +0. 0 1.52 0.62 0.42 0.32 0.27 0.27 0.27 0.28 0.30 + 2.5 1.42 0.58 0.44 0.34 0.32 0.33 0.37 0.39 0.49 + 5.0 1.29 0.24 0.27 0.51 0.50 0.48 0.58 0.72 0.74 0.89 Lift coefficient ( C L ) Wind Angle 0.29 0.08 0.01 0.24 0.13 0.05 +0. 0 0.18 + 0.00 0.05 0.05 + 0.06 + 0.06 + 0.07 + 0.08 + 0.08 + 0.09 + 2.5 0.12 + 0.23 0.04 0.09 + 0.13 + 0.13 + 0.13 + 0.14 + 0.15 + 0.22 + 5.0 0.08 + 0.14 0.21 0.12 + 0.13 + 0.13 + 0.13 + 0. 16 + 0.17 + 0.24 Torque coefficient ( C T ) Wind Angle 0.21 0.14 0.12 0.08 0.07 0.08 0.08 0.09 0.10 0.21 0.11 0. 08 0.05 0.04 0.05 0.04 0.05 0.05 +0. 0 0.19 0.08 0.05 0.05 0.03 0.04 0.04 0.03 0.04 + 2.5 0.17 0.06 0.05 0.04 0.04 0.04 0.04 0.05 0.04 + 5.0 0.15 0.04 0.05 0.04 0.04 0.05 0.06 0.06 0.06 Effective drag coefficient ( C D,eff ) Wind Angle 1.75 0.86 0.70 0.51 0.45 0.45 0.47 0.54 0.59 1.79 0.84 0.58 0.42 0.33 0.34 0.33 0.35 0.38 +0. 0 1.75 0.71 0.48 0.38 0.31 0.32 0.32 0.32 0.35 + 2.5 1.61 0.64 0.50 0.39 0.36 0.37 0.41 0.44 0.53 + 5.0 1.45 0.32 0.55 0.54 0.52 0.62 0.77 0.78 0.92 PAGE 218 218 Testing Configuration M Cross section: Box Spacing: 20 ft Cross slope: 0% Drag coefficient ( C D ) Indv. Wind Angle 1.74 1.42 + 0.65 1.68 1.51 + 0.34 1.81 1.69 1.93 1.75 10.0 1.87 1.74 + 0.17 Lift coefficient ( C L ) Indv. Wind Angle 1.16 1.32 1.62 1.63 1.22 0.87 1.03 0.59 + 0.67 10.0 0.71 0.26 + 0.94 Torque coefficient ( C T ) Indv. 1 Wind Angle 0.85 0.72 + 0.02 0.78 0.73 0.73 0.67 0.73 0.66 10.0 0.69 0.61 + 0.10 Effective drag coefficient ( C D,eff ) Indv. Wind Angle 3.20 2.6 7 + 0.65 3.03 2.78 + 0.34 +0. 0 3.04 2.81 + 2.5 3.15 2.84 + 5.0 3.01 2.74 + 0.35 PAGE 219 219 Testing Configuration N Cross section: Box Spacing: 22 ft Cross slope: 0% Drag coefficient ( C D ) Indv. Wind Angle 1.74 1.39 + 0.76 1.68 1.48 + 0.45 1.81 1.68 1.93 1.73 10.0 1.87 1.73 + 0.30 Lift coefficient ( C L ) Indv. Wind Angle 1.16 1.37 1.62 1.65 1.22 0.85 1.03 0.61 + 0.72 10.0 0.71 0.22 + 1.01 Torque coefficient ( C T ) Indv. Wind Angle 0.85 0.71 + 0.10 0.78 0.71 0.73 0. 65 0.73 0.64 + 0.03 10.0 0.69 0.61 + 0.14 Effective drag coefficient ( C D,eff ) Indv. Wind Angle 3.20 2.64 + 0.86 3.03 2.71 + 0.45 3.04 2.76 3.15 2.77 10.0 3.01 2.73 + 0.54 PAGE 220 220 APPENDIX C CROSS SECTIONAL PROPERTIES OF FLORIDA I BEAMS In this study, finite element models were analyzed to evaluate the lateral stability of Florida I Beams (FIBs). In each model, the FIBs were model ed using warping beams specialized beam elements available in the ADINA finite element code, which require the calculation of a comprehensive set of cross sectional properties. This appendix provides mathematical definitions of all such properties and cor responding numeric values that were calculated for each FIB cross sectional shape. Definitions of the cross sectional properties that are required to use the warping beam element in ADINA are listed in Table C 1 Each property requires the evaluation of an integral over the area of the cross section, in which the integrands are written in terms of coordinates x and y, referenced to the geometric centroid of the section (Figure C 1 ). Some properties also require knowledge of the warping function ( x,y ) which represents the torsionally induced out of plane warping displacements per rate of twist at every point on the cross section. (The units of are therefor e in/(rad/in ) or in 2 .) For general cross sectional shapes (e.g., an FIB), analytical (closed form) solutions for ( x,y ) do not exist; instead the warping field ( x,y ) must be solved numerically. In this study, the calculation of ( x,y ) for each FIB shape w as accomplished by discretizing the cross sectional shape into a high resolution mesh of thousands of two dimensional triangular elements, and then employing a finite element approach to solve the governing differential equation. In general, solutions for ( x,y ) change depending on the assumed location of the center of in Table C 1 ) corresponding to a state of p ure torsion i.e., torsion about the shear center. As a result, prior knowledge of the location of the shear center is required to compute several of the PAGE 221 221 warping beam properties. However, it is possible to calculate the coordinates of the shear center, x s and y s (Table C 1 ), using an alternative solution to the warping function ( c ) where the center of twist is assumed to be located at the centroid of the section. Therefore, two different warping functions w ere computed for each FIB section: first the section centroid was used to compute c and then the location of the shear center, obtained from c was used to compute as well as the remaining cross sectional properties. Because all FIB cross sections are symmetric about the y axis, I xy x s I xr and I have a value of zero (0) by definition. The remaining cross section al properties calculated for each FIB shape are summarized in Table C 2 PAGE 222 222 Table C 1 Definitions of cross sectional properties required for use of a warping beam element Property Integral form Units Description A in 2 Cr oss sectional area I yy in 4 Strong axis moment of inertia I xx in 4 Weak axis moment of inertia I xy in 4 Product of inertia x s in X coordinate of s hear center y s in Y coordinate of shear center J in 4 St. Venant torsional constant C in 6 Warping constant I xr in 5 Twist/strong axis bending co upling term I yr in 5 Twist/weak axis bending coupling term I in 6 Twist/warping coupling term I rr in 6 Wagner constant Table C 2 Cross sectional properties of Florida I Beams Section A (in 2 ) I yy (in 4 ) I xx (in 4 ) y s (in) J (in 4 ) C (in 6 ) I yr (in 5 ) I rr (in 6 ) 81283 3.00 30,864 250 81540 3.46 31,885 81798 3.81 32,939 82055 4.07 33,973 FIB 1059 82314 4.27 35,041 1101 82484 4.38 35,693 1,314,600,000 1143 1,087,800 82657 4.46 36,421 104,350,000 10,504,000 1,781,400,000 1227 1,516,200 8 3,002 4.56 37,859 142,280,000 15,336,000 3,107,900,000 PAGE 223 223 Figure C 1 Coordinate system used in the calculation of cross sec tional properties PAGE 224 224 APPENDIX D PROPERTIES OF FLORIDA BEARING PADS This appendix includes computed stiffnesses (Table D 1 ) for several standard types of FDOT bearing pad, produced using the calculation methods discus sed in Chapter 6 Pad types D, E, F, G, H, J, and K are current designs intended for use with Florida I Beams (FIBs), while pad types A, B, and C were formerly provided for use with Florida Bulb Tees (FBTs). While Florida Bulb Tee girders are no longer used in new bridge designs, they are still in service in existing bridges throughout Florida, thus stiffness data for pad types A, B, and C are included here for completeness. Each pad type is defined by four design parameters (F igure D 1 ): the plan view length and width ( L and W ), elastomer shear modulus ( G ), and the number of internal elastomer layers ( n ). Linear stiffnesses corresponding to bearing pad shear, torsion, axial compressio n, overturning roll (about the y axis) for the zero slope condition, and bending roll (about the x axis) are presented in Table D 1 Unlike bending roll, overturning roll stiffness varies depending on the total axia l load (i.e., girder weight) when the slope angle is non zero. Consequently, nonlinear overturning roll stiffness curves for several combinations slope angle and axial load are provided on the following pages. On the pages that follow, each curve correspon ds to a unique combination of pad type, slope angle, and axial load (the quantities noted in kips). On each curve, K r is the initial overturning roll stiffness in kip ft/rad. PAGE 225 225 Table D 1 Bearing pad dimensions and computed stiffnesses Pad Type L (in) W (in) G (psi) n K shear (kip/ft) K axial (kip/ft) k torsion (kip ft/rad) k rol l,overturning for zero slope (kip ft/rad) k roll,bending (kip ft/rad) FBTs A 11 24 110 2 232 71000 46.4 1330 B 14 24 110 3 222 85300 64.0 2590 C 12 23 150 3 248 72200 55.8 1610 FIBs D 32 110 2 225 45900 28.1 E 10 32 110 2 282 81400 52.4 12900 1260 F 10 32 110 3 211 57300 39.3 0 G 10 32 150 3 288 72700 53.6 11500 1130 H 10 32 150 4 230 56300 42.8 J 10 32 150 5 192 45900 35.7 K 12 32 150 5 230 70200 58.7 11100 1560 Figure D 1 Bearing pad dimensions and variables PAGE 226 226 Pad Type A PAGE 227 227 Pad Type B PAGE 228 228 Pad Type C PAGE 229 229 Pad Type D PAGE 230 230 Pad Type E PAGE 231 231 Pad Type F PAGE 232 232 Pad Type G PAGE 233 233 Pad Type H PAGE 234 234 Pad Type J PAGE 235 235 Pad Type K PAGE 236 236 APPENDIX E PLOTS OF CAPACITY PREDICTION EQUATIONS This appendix contains plots of subsets of the parametric study results along with the corresponding capacity prediction equations as reported in Chapter 9 The intent of the plots is to illustrate the level of conservatism of the capacity prediction equations relative to the data, as well as the sensitivity of select s ystem parameters. Figures E 1 E 5 illustrate the effect that changes in wind load have on the capacity of strut braced girder systems. For girder systems with moment resisting braces, Figures E 6 E 9 show the effect of changes in the effective brace stiffness ( k brace ), Figures E 10 E 13 show the effect of changes in the number of interior brace points, and Figures E 14 E 18 show the effect of changes in wind load. Figure E 1 Prediction of system capacity for 2 girder, unanchored strut braced systems in 0 psf wind (Equation 9 10 ) PAGE 237 237 Figure E 2 Prediction of system capacity for 2 girde r, unanchored strut braced systems in 40 psf wind (Equation 9 10 ) Figure E 3 Prediction of system capacity for 2 girder, unanchored strut braced systems in 80 psf wind (Equation 9 10 ) PAGE 238 238 Figure E 4 Prediction of system capacity for 2 girder, unanchored strut braced systems in 120 psf wind (Equation 9 10 ) Figure E 5 Prediction of system capacity for 2 girder, unanchored strut braced systems in 160 psf wind (Equation 9 10 ) PAGE 239 239 Figure E 6 Prediction of system capacity for end braced systems in 0 psf wind with moment resisting braces with k brace = 15,000 kip ft/rad (Equation 9 17 ) Figure E 7 Prediction of system capacity for end braced systems in 0 psf wind with moment resisting braces with k brace = 200,000 kip ft/r ad (Equation 9 17 ) PAGE 240 240 Figure E 8 Prediction of system capacity for end braced systems in 0 psf wi nd with moment resisting braces with k brace = 400,000 kip ft/rad (Equation 9 17 ) Figure E 9 Pr ediction of system capacity for end braced systems in 0 psf wind with moment resisting braces with k brace = 600,000 kip ft/rad (Equation 9 17 ) PAGE 241 241 Figure E 10 Prediction of system capacity for systems in 0 psf wind with moment resisting braces ( k brace = 200,000 kip ft/rad) with no interior brace points (Equation 9 17 ) Figure E 11 Prediction of system capacity for systems in 0 psf wind with moment resisting braces ( k brace = 200,000 kip ft/rad) with 1 interior brace point (Equation 9 17 ) PAGE 242 242 Figure E 12 Prediction of system capacity for syst ems in 0 psf wind with moment resisting braces ( k brace = 200,000 kip ft/rad) with 2 interior brace points (Equation 9 17 ) Figure E 13 Prediction of system capacity for systems in 0 psf wind with moment resisting braces ( k brace = 200,000 kip ft/rad) with 3 interior brace points (Equation 9 17 ) PAGE 243 243 Figure E 14 Prediction of system capacity for systems with moment resisting braces ( k brace = 200,000 kip ft/rad) with 1 interior brace point in 0 psf wind (Equation 9 17 ) Figure E 15 Prediction of system capacity for systems with moment resisti ng braces ( k brace = 200,000 kip ft/rad) with 1 interior brace point in 40 psf wind (Equation 9 17 ) PAGE 244 244 Figure E 16 Prediction of system capacity for systems with moment resisting braces ( k brace = 200,000 kip ft/rad) with 1 interior brace point in 80 psf wind (Equation 9 17 ) Figure E 17 Prediction of system capacity for systems with moment resisting braces ( k brace = 200,000 kip ft/rad) with 1 interior brace point in 120 psf wind (Equation 9 17 ) PAGE 245 245 Figure E 18 Prediction of system capacity for systems with moment resisting braces ( k brace = 200,000 kip ft/rad) with 1 interior brace point in 160 psf wind (Equation 9 17 ) PAGE 246 246 LIST OF REFERENCES AASHTO ( American Association of State Highway and Transportation Officials) (2010 ). 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The Physics of Rubber Elasticity: 3 rd Edition Clarendon Press, Oxford. Young, W. C. and Budynas, R. G. (2002 ). : 7 th Edition McGraw Hill, New York NY PAGE 249 249 BIOGRAPHICAL SKETCH The author was born in Tallahassee, Florida, in 1986. He began attending the University of Florida in August 2004, where he received the degree of Bachelor of Science in mechanical engineering in May 2009. He then enrolled in graduate school at the University of Florida where he received a Master of Science in civil engineering in May 2013, with an emphasis in civil structures. 