Citation |

- Permanent Link:
- http://ufdc.ufl.edu/UFE0051616/00001
## Material Information- Title:
- Distribution Factors for Construction Loads and Girder Capacity Equations
- Creator:
- Honig, Jeffrey M
- Place of Publication:
- [Gainesville, Fla.]
Florida - Publisher:
- University of Florida
- Publication Date:
- 2017
- Language:
- english
- Physical Description:
- 1 online resource (165 p.)
## Thesis/Dissertation Information- Degree:
- Master's ( M.E.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Civil Engineering
Civil and Coastal Engineering - Committee Chair:
- CONSOLAZIO,GARY R
- Committee Co-Chair:
- HAMILTON,HOMER ROBERT,III
## Subjects- Subjects / Keywords:
- construction -- construction-load -- distribution -- distribution-factor -- factor -- girder -- girder-stability -- load -- stability -- sweep -- thermal -- thermal-sweep -- wind
Civil and Coastal Engineering -- Dissertations, Academic -- UF - Genre:
- bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) born-digital ( sobekcm ) Electronic Thesis or Dissertation Civil Engineering thesis, M.E.
## Notes- Abstract:
- During the process of constructing a highway bridge, there are several construction stages that warrant consideration from a structural safety and design perspective. The first objective of the present study was to use analytical models of prestressed concrete girders (Florida-I Beams) at multiple stages of construction to update previously developed capacity equations for wind load and gravity load. Updated analytical bridge models were developed that accounted for a revised definition of lateral girder sweep--one that accounted for both maximum allowable fabrication tolerance as well as transverse thermal gradients (i.e., thermally induced sweep). Subsequently, analytical parametric studies were conducted to update--using the revised definition of sweep--previously developed girder capacity equations. The updated capacity equations take into consideration different Florida-I Beam cross-sections, span lengths, wind loads, skew angles, and brace stiffnesses. A second objective in this study was to use finite element analyses of partially constructed bridge systems--consisting of multiple Florida-I Beam (FIBs) with construction loads--to quantify distribution factors for interior and exterior girder end shear forces and maximum girder moments. A large-scale parametric study was conducted with consideration of different Florida-I Beam cross-sections, span lengths, girder spacing, deck overhang widths, skew angles, number of girders, number of braces, and bracing configurations (K-brace and X-brace) to quantify shear and moment distribution factor data. These data were subsequently used to develop empirical construction stage distribution factor (DF) equations at multiple levels of design conservatism. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (M.E.)--University of Florida, 2017.
- Local:
- Adviser: CONSOLAZIO,GARY R.
- Local:
- Co-adviser: HAMILTON,HOMER ROBERT,III.
- Statement of Responsibility:
- by Jeffrey M Honig.
## Record Information- Source Institution:
- UFRGP
- Rights Management:
- Applicable rights reserved.
- Classification:
- LD1780 2017 ( lcc )
## UFDC Membership |

Downloads |

## This item has the following downloads: |

Full Text |

PAGE 1 DISTRIBUTION FACTORS FOR CONSTRUCTION LOADS AND GIRDER CAPACITY EQUATIONS By JEFFREY MARK HONIG A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ENGINEERING UNIVERSITY OF FLORIDA 201 7 PAGE 2 201 7 Jeffrey Mark Honig PAGE 3 3 ACKNOWLEDGMENTS First and foremost, I would like to express my sincere gratitude to my advisor, Dr. Gary Cons olazio, for his motivation and guidance throughout my graduate education. He is an excellent role model and has greatly influenced my professional and technical skills as an engineer I am greatly honored and humbled to have had the opportunity to work with him I would also like to thank Dr. H.R. (Trey) Hamilton for serving on my supervisory committee. Finally, I would like to thank my family and friends for providing support throughout this process PAGE 4 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ ............... 3 LIST OF TABLES ................................ ................................ ................................ ........................... 7 LIST OF FIGURES ................................ ................................ ................................ ......................... 8 ABSTRACT ................................ ................................ ................................ ................................ ... 13 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .................. 15 1.1 Background ................................ ................................ ................................ ....................... 15 1.2 Objectives ................................ ................................ ................................ ......................... 16 1.3 Scope of Work ................................ ................................ ................................ .................. 17 2 PHYSICAL DESCRIPTION DURING CONSTRUCTION ................................ ................. 19 2.1 Overview ................................ ................................ ................................ ........................... 19 2.2 Geometric Parameters ................................ ................................ ................................ ....... 19 2.3 Bearing Pads ................................ ................................ ................................ ..................... 20 2.4 Bracing ................................ ................................ ................................ .............................. 20 3 GIRDER SWEEP INCLUDING THERMAL GRADIENT EFFECTS ................................ 25 3.1 Overview ................................ ................................ ................................ ........................... 25 3.2 Literature Review: Thermal Sweep ................................ ................................ .................. 25 3.3 Thermal Sweep for Florida I Beams ................................ ................................ ................ 29 4 DEVELOPMENT OF UNANCHORED SINGLE GIRDER WIND CAPACITY EQUATION ................................ ................................ ................................ ............................ 35 4.1 Overview ................................ ................................ ................................ ........................... 35 4.2.1 Modeling of End Supports ................................ ................................ ...................... 38 4.2.2 Bearing Pad Selection ................................ ................................ ............................. 38 4.2.3 Axial Load Selection ................................ ................................ .............................. 39 4.2.4 Girder Slope Selection ................................ ................................ ............................ 39 4.2.5 Load Application to Individual Bridge Girders ................................ ...................... 40 4.3 Parametric Study of Unanchored Individual Bridge Girders ................................ ........... 41 4.3.1 Selection of Parameters ................................ ................................ .......................... 41 4.3.2 Updated Wind Capacity of a Single Unanchored Girder ................................ ....... 42 PAGE 5 5 5 DEVELOPMENT OF UNANCHORED STRUT BRACED TWO GIRDER BUCKLING CAPACITY EQUATION ................................ ................................ ................. 48 5.1 Overview ................................ ................................ ................................ ........................... 48 5.2 Review of Multi Girder System Related Information from BDK75 977 33 ................... 48 5.2.1 Preliminary Sensitivity Studies ................................ ................................ .............. 48 5.2.2 Strut Braces ................................ ................................ ................................ ............ 48 5.2.3 Moment Resisting Braces ................................ ................................ ...................... 49 5.2.4 Modeling of Braces ................................ ................................ ................................ 49 5.2. 5 Modeling of Bridge Skew and Wind Load ................................ ............................. 51 5.3 Parametric Study of System Capacity of Unanchored Two Girder System in Zero Wind ................................ ................................ ................................ ................................ .... 52 5.3.1 Parameters ................................ ................................ ................................ .............. 52 5.3.2 Updated System Capacity of Unanchored Two Girder System in Zero Wind ...... 53 5.4 Moment Resisting Brace: Limited Scope Parametric Study ................................ ............ 54 5.4.1 Selection of Parameters for Limited Scope Moment Resisting Brace Parametric Study ................................ ................................ ................................ .......... 55 5.4.2 Up dated System Capacity of Moment Resisting Brace ................................ ......... 57 6 PROCEDURES DEVELOPMENT FOR CONSTRUCTION LOAD DISTRIBUTION FACTOR EQUATIONS ................................ ................................ ................................ ......... 68 6.1 Overview ................................ ................................ ................................ ........................... 68 6.2 Modeling Multi Girder Bridge Systems during Construction ................................ .......... 69 6.3 Application of Construction Loads ................................ ................................ ................... 71 6.3.1 Construction Load Groups Considered ................................ ................................ .. 73 6.3.2 Application of Construction Loads ................................ ................................ ........ 73 7 DEVELOPMENT OF CONSTRUCTION LOAD DISTRIBUTION FACTOR EQUATIONS ................................ ................................ ................................ ......................... 85 7.1 Construction Load Distribution Factor Parame tric Study ................................ ................ 85 7.1.1 Scope ................................ ................................ ................................ ...................... 85 7.1.2 Special Cases ................................ ................................ ................................ .......... 8 6 7.2 Defi nition of Distribution Factors ................................ ................................ .................... 86 7.2.1 Distribution Factor Sensitivities ................................ ................................ ............. 88 7.2.2 Illustrative Examples ................................ ................................ .............................. 89 7.2.3 Selection of Culled Data ................................ ................................ ......................... 89 7.2.4 Key Parameters Exhibiting Sensitivity ................................ ................................ ... 90 7.3 Formation of Baseline Empirical Distribution Factor Equations ................................ ..... 91 7.4 Modifications to Achieve Desired Level of Prediction Error ................................ ........... 92 7.5 Final Distribution Factor Equations for Design ................................ ................................ 95 7.5.1 Application of Proposed Method ................................ ................................ ............ 95 7.5.2 Prediction Error for Full (Unculled) Parametric Data Set ................................ ...... 96 7.5.3 Proposed Method Compared to Traditional Tributary Area Method for Load Group 2 ................................ ................................ ................................ ......... 97 PAGE 6 6 8 SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS ................................ ....... 105 8.1 Summary and Conclusions ................................ ................................ ............................. 105 8.2 Recommendations ................................ ................................ ................................ ........... 106 APPENDIX A CROSS SECTIONAL PROPERTIES OF FLORIDA I BEAMS ................................ ........ 108 B ................................ .......... 112 C EXAMPLE CALCULATIONS: TEMPORARY BRACING ASSESSMENT FOR AN FIB BRIDGE ................................ ................................ ................................ ........................ 121 D DETAILED ILLUSTRATIONS: QUANTIFYING EXTERIOR AND INTERIOR GIRDER END SHEAR FORCES AND MAXIMUM MOMENTS WITH CONSTRUCTION LOADS APPLIED ................................ ................................ ................ 127 E EXAMPLE CALCUL ATIONS: QUANTIFYING EXTERIOR AND INTERIOR GIRDER END SHEAR FORCES AND MAXIMUM MOMENTS WITH CONSTRUCTION LOADS APPLIED ................................ ................................ ................ 136 F DETAIL ED ILLUSTRATIONS: DISTRIBUTION FACTORS FOR CASES WITH INTERIOR BRACING ................................ ................................ ................................ ......... 152 LIST OF REFERENCES ................................ ................................ ................................ ............. 163 BIOGRAPHICAL SKETCH ................................ ................................ ................................ ....... 165 PAGE 7 7 LIST OF TABLES Table page 4 1 Span length ranges for FIBs ................................ ................................ ............................... 44 5 1 Empirically determined values of for different numbers of interior braces .................. 59 5 2 Self weight ( w sw ) of each FIB cross sectional shape ................................ ........................ 59 5 3 Span length va lues used in moment resisting brace parametric study from BDK75 977 33 ................................ ................................ ................................ ................................ 59 5 4 Other parameter values used in moment resisting brace parametric study from BDK75 977 33 ................................ ................................ ................................ .................. 59 5 5 Selected span length values used in the present moment resisting brace parametri c study ................................ ................................ ................................ ................................ ... 60 5 6 Other selected parameter values used in the present moment resisting brace parametric study ................................ ................................ ................................ ................. 60 6 1 Varying finishing machine load ................................ ................................ ......................... 75 6 2 Summary of construction Load Group 1 loads in parametric studies ................................ 75 6 3 Summary of constructio n Load Group 2 loads in parametric studies ................................ 75 7 1 Span length values used in the distribution factor parametric study ................................ 98 7 2 Other parameter values used in the distribution factor parametric study .......................... 98 7 3 Constants for distribution factors ( DF ) calculation ................................ ........................... 98 7 4 Distribution factor ( DF ) exceedance values ................................ ................................ ...... 98 A 1 Definitions of cross sectional properties required for use of a warping beam element ... 110 A 2 Cross section al properties of Florida I Beams ................................ ............................... 110 PAGE 8 8 LIST OF FIGURES Figure page 1 1 Prestressed concrete girders braced together for stability ................................ .................. 18 1 2 Bridge construct ion loads ................................ ................................ ................................ .. 18 2 1 Girder system ................................ ................................ ................................ ..................... 22 2 2 Defini tion of grade (elevation view) ................................ ................................ .................. 22 2 3 Definition of cross slope (section view) ................................ ................................ ............ 22 2 4 Definition skew angle (plan view) ................................ ................................ ..................... 23 2 5 Definition of camber (elevation view) ................................ ................................ ............... 23 2 6 Definition of sweep (plan view) ................................ ................................ ........................ 23 2 7 Girder system with quarter point bracing ................................ ................................ .......... 23 2 8 Perpendicular brace placement on skewed bridge (plan view) ................................ .......... 24 2 9 Common brace types ................................ ................................ ................................ .......... 24 3 1 Transverse thermal gradients of prestressed concrete bridge girders proposed by Lee (2012) ................................ ................................ ................................ ................................ 31 3 2 Top flange transverse thermal gradient ................................ ................................ .............. 32 3 3 Transverse t hermal gradients for FIB bridge girders ................................ ......................... 33 3 4 Comparison of sweep ratios computed for BT 63 girder (at varying span lengths) as computed using different methods ................................ ................................ ..................... 33 3 5 Thermal sweep data for FIB sections ................................ ................................ ................. 34 3 6 Thermal sweep ratios for FIB sections ................................ ................................ .............. 34 4 1 Finite element model of a single FIB (isometric view) ................................ ..................... 44 4 2 Representation of sweep in FIB model (plan view) ................................ ........................... 44 4 3 Representation of camber in FIB model (elevation view) ................................ ................. 45 4 4 Bearing pad stiffness springs in FIB model (isometric view) ................................ ............ 45 4 5 Representation of wind load in structural models ................................ .............................. 46 PAGE 9 9 4 6 Wind capacities for unanchored FIB s ................................ ................................ ................ 46 4 7 Wind capacity of an unanchored FIB ................................ ................................ ................ 47 5 1 Examples of strut bracing ................................ ................................ ................................ .. 60 5 2 Collapse mechanism possible with strut bracing ................................ ............................... 60 5 3 Examples of moment resisting braces. ................................ ................................ .............. 61 5 4 Representation of brace configurations in FIB system models ................................ ......... 61 5 5 Proposed wind load shielding model for stability evaluation from BDK75 977 33 ......... 62 5 6 Effect of bridge skew on wind loading of braced 3 girder system (plan view) ................. 62 5 7 System capacities of unanchored two girder strut bra ced FIB systems in zero wind ....... 63 5 8 System capacity of an unanchored strut braced two girder FIB system in zero wind as predicted by C 0 Equation ................................ ................................ .............................. 64 5 9 System capacity of moment resisting two girder FIB system ................................ ........... 65 5 10 Absolute error of system capacity quantities predicted by Eqn. (5 4) from BDK75 977 33 with updated C 0 equation [Eqn. (5 3)] ................................ ................................ .. 66 5 11 Comparison of predicted capacities ................................ ................................ ................... 66 5 12 Comparison of selected parametric study data from BDK75 977 33 with updated parametric study data (thermal sweep included) ................................ ............................... 67 5 13 Absolute difference of current to previ ous parametric study system capacity quantities ................................ ................................ ................................ ............................ 67 6 1 Stay in place formwork (section view) ................................ ................................ ............. 75 6 2 Temporary support brackets used to support deck overhangs during construction ........... 76 6 3 Cantilever overhang support ed by overhang brackets ................................ ....................... 76 6 4 Details of overhang formwork support brackets and loads ................................ ............... 77 6 5 Typical bridge deck finishing machine in operation ................................ .......................... 77 6 6 Overhang bracket components and geom etry ................................ ................................ .... 78 6 7 Details of overhang bracket model ................................ ................................ .................... 78 6 8 Cross sectional view of overall braced girder system model ................................ ............ 79 PAGE 10 10 6 9 Isometric view of braced girder system model ................................ ................................ .. 79 6 10 Cross sectional summary of construction Load Group 1 ( LG1 ) loads .............................. 80 6 11 Construction Load Group 1 as a function of finishing machine location (Bridge with only end span braces; no interior braces) ................................ ................................ .......... 80 6 12 Construction Load Group 1 as a function of finishing machine location (Bridge with end span and midspan bracing) ................................ ................................ ......................... 80 6 13 Construction Load Group 1 as a function of finishing machine location (Bridges with third point bracing) ................................ ................................ ................................ ............ 81 6 14 Construction Load Group 1 as a function of finishing machine location (Bridges with quarter point bracing) ................................ ................................ ................................ ........ 81 6 15 Cross sectional summary of construction Load Group 2 ( LG2 ) loads .............................. 81 6 16 Construction Load Group 2 with incremental deck load (Bridge with only end span braces; no interior braces) ................................ ................................ ................................ .. 82 6 17 Construction Load Group 2 with incremental deck load (Bridge with end span and midspan bracing) ................................ ................................ ................................ ................ 82 6 18 Construction Load Group 2 with incremental deck load (Bridges with third point bracing) ................................ ................................ ................................ .............................. 82 6 19 Construction Load Gro up 2 with incremental deck load (Bridges with quarter point bracing) ................................ ................................ ................................ .............................. 83 6 20 Eccentric reaction forces from loads applied to SIP forms, and statically equivalent nodal force and moment applied to top of girder ................................ ............................... 83 6 21 All construction loads (LG1 and LG2) converted to equivalent nodal loads .................... 84 7 1 DF sensitivity to number of girders ................................ ................................ ................... 99 7 2 DF sensitivity to girder depth ................................ ................................ ............................ 99 7 3 Shear (V) prediction error for the culled data set using DF V EXT LG1 in conjunction with a static beam analysis, without introduction of ................................ ..................... 100 7 4 Shear (V) prediction error for the culled data set using DF V EXT LG1 in conjunction with a static beam analysis, shifted with to a 50% exceedance level ........................... 100 7 5 Shear (V) prediction error for the culled data set using DF V EXT LG1 in conjunction with a static beam analysis, shifted with to an 84% exceedance level ......................... 101 PAGE 11 11 7 6 Shear (V) prediction error for the culled data set using DF V EXT LG1 in conjunction with a static beam analysis, shifted with to a 95% exceedance level ........................... 101 7 7 Shear (V) prediction error for the culled data set using DF V EXT LG1 in conjunction with a static beam analysis, shifted with to a 98% exceedance level ........................... 102 7 8 Computation of exterior girder end shear force for construction load group LG1 .......... 102 7 9 Prediction error for V EXT LG1 using Eqn. ( 7 17 ) and a 50% exceedance level .................. 102 7 10 Prediction error for V EXT LG1 using Eqn. ( 7 17 ) and a 95% exceedance level ................. 103 7 11 Prediction error for V EXT LG2 ................................ ................................ ............................ 103 7 12 Prediction error for V INT LG2 ................................ ................................ ............................. 103 7 13 Prediction error for M EXT LG2 ................................ ................................ ........................... 104 7 14 Prediction error for M INT LG2 ................................ ................................ ............................ 104 A 1 Coordinate system used in the calculation of cross sectional properties ......................... 111 F 1 Load Group 1 loads with the finishing machine located at the midspan to produce maximum girder moments ................................ ................................ ............................... 156 F 2 Brace configurations considered in the parametric st udy for 5 girder bridge systems .... 156 F 3 Bridge cross section with only end span bracing ................................ ............................ 1 57 F 4 Bridge cross section with interior midspan timber X bracing ................................ ......... 157 F 5 Bridge cross section w ith interior midspan steel X bracing ................................ ............ 157 F 6 Bridge cross section with interior midspan steel K bracing ................................ ............ 157 F 7 Bridge cross section midspan deflection without interior bracing ................................ .. 158 F 8 Bridge cro ss section midspan deflection with interior timber X bracing ........................ 158 F 9 Bridge cross section midspan deflection with interior steel X bracing ........................... 158 F 10 Bridge cross section midspan deflection with interior steel K bracing ........................... 158 F 11 5 girder, FIB78, 180 ft span, 6 ft girder spacing, 25 in. deck overhang, 0 deg. skew bridge configuration ................................ ................................ ................................ ......... 159 F 12 Moment (M EXT LG1 ) prediction error for all bridge configurations (36,288 cases) using DF M EXT LG1 in conjunction with a static beam analysis, shifted with to a 95% exceedance ................................ ................................ ................................ ....................... 160 PAGE 12 12 F 13 than ...... 160 F 14 Load Group 1 loads applied at the midspan for a typical span length ............................. 160 F 15 Timber X bracing cases ................................ ................................ ................................ ... 161 F 16 Steel X bracing cases ................................ ................................ ................................ ....... 161 F 17 Steel K bracing cases ................................ ................................ ................................ ....... 162 PAGE 13 13 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 Engineering DISTRIBUTION FACTORS FOR CONSTRUCTION LOADS AND GIRDER CAPACITY EQUATIONS By Jeffrey Mark Honig August 2017 Chair: Gary Consolazio Major: Civil Engineering During the process of constructing a highway bridge, there are several construction stages that warrant consideration from a structural safety and design perspective. The first objective of the present study was to use analytical models of prestressed conc rete girders (Florida I Beams) at multiple stages of construction to update previously developed capacity equations for wind load and gravity load. Updated analytical bridge models were developed that accounted for a revised definition of lateral girder sw eep one that accounted for both maximum allowable fabrication tolerance as well as transverse thermal gradients (i.e., thermally induced sweep). Subsequently, analytical parametric studies were conducted to update using the revised definition of sweep prev iously developed girder capacity equations. The updated capacity equations take into consideration different Florida I Beam cross sections, span lengths, wind loads, skew angles, and brace stiffnesses. A second objective in this study was to use finite ele ment analyses of partially constructed bridge systems consisting of multiple Florida I Beam (FIBs) with construction loads to quantify distribution factors for interior and exterior girder end shear forces and maximum girder moments. A large scale parametr ic study was conducted with consideration of different Florida I Beam cross sections, span lengths, girder spacing, deck overhang widths, PAGE 14 14 skew angles, number of girders, number of braces, and bracing configurations (K brace and X brace) to quantify shear a nd moment distribution factor data. These data were subsequently used to develop empirical construction stage distribution factor ( DF ) equations at multiple levels of design conservatism. PAGE 15 15 CHAPTER 1 INTRODUCTION 1 1 Background During the process of constructing a highway bridge, there are several construction stages that warrant consideration f rom a structural safety and design perspective. Initially, individual girders are lifted by crane and placed into position atop flexible bearing pads located on the bridge supports (e.g., abutments or piers). The most critical phase of construction, with r egard to stability, is after girder placement (prior to the casting of the deck), when girders are supported only by bearing pads and can be subject to high lateral wind loads. The stage at which wind loading is often most critical occurs when the first gi rder is erected. At this stage of construction, other girders are not present to brace against, hence the initial girder can only be anchored to the pier at the ends. For bridge designs in which girder stability is a primary concern, girder erection can so metimes be scheduled to minimize the exposure period for the initial girder, so that statistically it is less likely that peak wind forces will occur. However, meeting such a schedule is not always feasible, and adverse weather conditions cannot necessaril y be anticipated. 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 t he effects of wind loads. Furthermore, placement of all girders into their final position constitutes another distinct structural stage that must be assessed for safety. In this structural configuration, it is typical for temporary braces (Figure 1 1 ) to be installed between the individual girders to form a more stable structural unit (Consolazio and Edwards, 2014). Addition ally, one or more girders may also be anchored to the bridge supports (Consolazio et al., 2013). Structurally, the system at this PAGE 16 16 stage consists of individual girders, bearing pads, braces, potentially anchors, and support structures (i.e., substructures). After continued construction progress, another key stage will be reached wherein stay in place (SIP) forms have been installed between the girders and overhang formwork (and associated overhang support brackets) have been eccentrically attached to the ext erior (fascia) structural) concrete deck (Figure 1 2 placement, most of the construction loads are applied eccentrically to over hang formwork and to stay in place forms. Consequently, both interior and exterior girder moments and end shear forces produced by construction loads must be considered in the bridge design process. Furthermore, several geometric parameters influence the m agnitude and distribution of maximum girder moments and girder end shears that are caused by construction loads. 1 2 Objectives One objective of the present study was to use analytical mode ls of prestressed concrete girders (Florida I Beams), at multiple stages of construction, to update previously developed capacity equations (Consolazio et al., 2013) for wind load and gravity load. Specifically, the analytical models were updated to accoun t for a revised definition of sweep one that included not only fabrication sweep, but also thermally induced sweep. An additional objective was to use finite element analysis models of partially constructed bridge systems consisting of multiple Florida I B eam (FIBs) with construction loads applied (Consolazio and Edwards, 2014) to quantify interior and exterior girder end shear forces and maximum girder moments. Computed end shear forces and maximum moments were subsequently used to develop empirical constr uction stage distribution factor ( DF ) equations. PAGE 17 17 1 3 Scope of Work In FDOT project BDK75 977 33 (Consolazio et al., 2013), results from multiple analytical parametric studies were used to de velop simplified girder capacity and bridge capacity equations. Girder sweep values were intended to account for fabrication tolerances, limited to in. for every 10 ft of girder length, and not to exceed 1.5 in. In the present study, a literature review was conducted to develop an updated (revised) definition of initial girder sweep one in which the effects of transverse thermal gradients (i.e., thermal sweep) were added and the previously imposed m aximum limit of 1.5 in. removed. Using the revised defini tion of sweep, analysis procedures and finite element models of unanchored FIB bridge girders with wind loads were used to revise the BDK75 977 33 equation for unanchored girder wind capacity [ P max ,0 reported as Eqn. (8.2) in Consolazio et al. (2013)]. Ad girder strut braced system buckling capacity in zero wind [ C 0 reported as Eqn. (9.2) in Consolazio et al. (2013)] was updated to reflect the revised definition of sweep. A limited scope parametric stu dy was subsequently conducted to ensure that the equation for multi girder moment resisting braced system capacity [ C reported as Eqn. (9.23) in Consolazio et al. (2013)], supplemented by the updated baseline equation ( C 0 ) remained conservative relative t o corresponding capacities computed using finite element analyses. Bridge modeling and analysis procedures were developed and used to conduct a large scale construction load parametric study covering typical ranges of possible bridge system configurations. Girder end shear forces and maximum moments due to superimposed construction loads were quantified and subsequently used to develop empirical distribution factor ( DF ) equations. PAGE 18 18 Figure 1 1 Prestressed concrete girders braced together for stability (Photo credit: FDOT) Figure 1 2 Bridge const ruction loads (Photo credit: Gomaco) PAGE 19 19 CHAPTER 2 PHYSICAL DESCRIPTION DURING CONSTRUCTION 2 1 Overview Girder types under investigation in this study were Florida I Beams (FIBs), a group of standard cross sectional shapes of varying depths that are commonly employed in Florida bridge designs. These beams are typically cast offsite, transported to the construction site, then lifted into position one at a time by crane, where they are placed on elastomeric bearing pads and braced toget structures under consideration will be provided along with definition of relevant terminology. 2 2 Geo metric P arameters The term girder system will be used to refer to a group of two or more FIBs braced together in an evenly spaced row (Figure 2 1 ). In addition to span length and lateral spacing, there are several geometric parameters that define the shape and placement of the girders within a system: Grade : Longitudinal incline of the girders, typically expressed as a percentage o f rise per unit of horizontal length (Figure 2 2 ). Cross slope : The transverse incline (slope) of the deck, expressed as a percentage, which resu lts 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 t he bottom flange; expressed as the maximum vertical deviation from a perfectly straight line connecting one end of the girder to the other. Sweep : Lateral bowing of the girder (Figure 2 6 ), expressed as the maximum horizontal deviation from a perfectly straight line connecting one end of the girder to the other. PAGE 20 20 2 3 Bearing P ads FIB bridge girders rest directly on steel reinforced elastomeric 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 m ovement of a girder relative to the substructure (with the exception of vertical uplift) must displace the top surface of the pad relative to the bottom surface. As a result, the girder support conditions in all six degrees of freedom (three translations, and three rotations) 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. Bearing pad sti ffnesses in this study were quantified using calculation procedures developed and experimentally validated in a previous study (BDK75 977 33, Consolazio et al., 2013) for typical Florida bridge bearing pads. 2 4 Bracing As adjacent girders are erected during the bridge construction process, girder to girder braces (henceforth referred to simply as braces ) are used to connect the girders together into a single structural unit. At a minimum, b races are installed near the ends of the girders (close to the supporting piers); such braces are referred to as end span braces In addition, intermediate span braces spaced at unit fractions (1/2, 1/3, 1/4) of the girder length may also be included. For example, quarter point (1/4 span) bracing divides the girder into four (4) equal unbraced lengths (Figure 2 7 ). When skew is present, brace point locations are longitudinally offset (Figure 2 8 ) between adjacent girders because FDOT Design Standard No. 20005: Prestressed I Beam Temporary Bracing (FDOT, 2014a) requires that all braces be placed perpendicular to the girders. PAGE 21 21 Braces are typically constructed from timber or steel members, but individual brace designs are left to the discretion o f the contractor, so a variety of different bracing configurations are possible. Common types of braces used in practice in Florida include X braces (Figure 2 9 a) and K braces (Figure 2 9 b). Braces are typically attached to the girders using bolted connections or welded to cast in steel plates. PAGE 22 22 Figure 2 1 Girder system Figure 2 2 Definition of grade (elevation view) Figure 2 3 Definition of cross slope (section view) PAGE 23 23 Figure 2 4 Definition skew angle (plan view) Figure 2 5 Definition of camber ( elevation view) Figure 2 6 Definition of sweep (plan view) Figure 2 7 Girder system with quarter point bracing PAGE 24 24 Figure 2 8 Perpendicular brace placement on skewed bridge (plan view) A B Figure 2 9 Common brace types. A) X brace. B) K br ace PAGE 25 25 CHAPTER 3 GIRDER SWEEP INCLUDING THERMAL GRADIENT EFFECTS 3 1 Overview A primary objective of the present study involved revising previously developed girder capacity equations (for wind load and gravity load) to account for a revised definition of girder sweep. Previous girder capacity equations were developed from finite element analysis (FEA) models used in BDK75 977 33 (Consolazi o et al., 2013), where girder capacity was reached when a girder (or girder system) became unstable. Lateral deflection, and ultimately instability of the analytical models was initiated by the introduction of girder imperfections (i.e., sweep). In the pre sent study, the definition of girder sweep has been revised by removing the previously imposed maximum fabrication limit of 1.5 in. and by including the effects of transverse thermal temperature gradients (i.e., thermal sweep). 3 2 Literature Review: Thermal Sweep In Lee (2010), equations for determining maximum vertical and lateral deflections caused by solar induced thermal gradients were presented for four (4) typical AASHTO PCI bridge g irder types. Lee used finite element heat transfer models, validated using experimental test data, to calculate non linear temperature gradients for prestressed concrete girders of varying cross sectional shapes. Each finite element model incorporated envi ronmental conditions that included the solar radiation level (based on geographic location), ambient air temperature, and wind data recorded over a 30 year period) o n each finite element model, Lee computed transverse non linear temperature gradients in the top flange, web, and bottom flange of four (4) typical AASHTO PCI bulb tee bridge girders. Primary environmental conditions employed in the study were established for the Atlanta, Georgia geographical area. However, to further assess PAGE 26 26 temperature gradients for different geographical locations, Lee also evaluated extreme environmental conditions for seven (7) additional cities, distributed across the continental Unite d States. Of the eight (8) cities considered, Atlanta, Georgia was closest in proximity to the state of Florida, therefore data for Atlanta were used in the present study (BDV31 977 46) to quantify thermally induced sweep values for Florida FIB girder sect ions. Using both experimentally measured temperature data as well gradients computed from finite element thermal analyses, Lee (2012) also proposed a set of simplified transverse temperature gradients (replicated in Figure 3 1 ) for the top flange, web, and bottom flange of prestressed concrete girders. Using the proposed simplified gradients, maximum transverse d eflection (i.e., sweep) induced by thermal gradients can be calculated for general bridge girder shapes and arbitrary span lengths by employing procedures such as moment curvature analysis. To assess the accuracy of both the maximum thermal sweep equations and the simplified thermal gradients, thermal sweep data were analytically computed using the equations proposed by Lee, and then compared to experimentally measured thermal sweep data published in the literature. Experimental data used in this evaluation process were obtained from an investigation of a bridge collapse in Arizona (CTL Group, 2007), and from research conducted for the Georgia DOT (Hurff, 2010). After computing tolerance limits on fabrication related sweep imperfection, as set forth in the P CI Bridge Design Manual (2011), Hurff (2010) demonstrated that maximum experimentally measured thermal sweep was as much as 48% of the maximum allowable fabrication sweep. In a separate experimental study, measured sweep data reported by the CTL Group indi cated a maximum thermal sweep of 0.65 in. for a 114 ft girder. At this magnitude, thermal sweep was approximately 46% of the allowable fabrication sweep for a girder of the same length. Analytical sweep values computed using the maximum sweep equations pro posed PAGE 27 27 by Lee (2010) were found to be in good agreement with the experimental data reported by Hurff (2010) and the CTL Group (2007). The simplified gradients proposed by Lee (2012) were derived from non linear temperature gradients computed from finite ele ment thermal analyses (FEA) of typical BT 63 and Type V girder cross sections. Each such simplified gradient (Figure 3 2 a) consis ted of a ensure that conservatively large predictions of maximum th ermal sweep were obtained for gradients was omitted from the simplified gradients proposed by Lee (2012, Figure 3 2 b). (i.e., more conservative) thermal sweep quantities than those computed using the more complex, two sided gradients. To avoid introducing excess conservatism into the thermal sweeps 2012). Using the two sided non linear FEA temperature gradients documented by Lee (2010), a temperature ratio ( ) was computed as: ( 3 1 ) 3 2 c) of the gradient was then defined as maximizing at temperature T R,SIMP (Figure 3 2 c) where: ( 3 2 ) and (2012). The simplified gradient proposed by Lee (2012) for the bottom flange also omitted an PAGE 28 28 ascending right side branch; therefore, a quantity, T R,SIMP was similarly approximated for the 3 1 produced final gradients for top flange, web, and bottom flange (Figure 3 3 ) that were used in the present study. Lee (2010) determined that winter and summer seasonal conditions produced the maximum and minimum transverse ther mal movements, respectively. Because winter conditions produced the largest thermal movements, for conservativism, Lee (2010) recommended that winter temperature gradient data (e.g., T 1,SIMP ) be used in computing prestressed concrete bridge girder thermal environmental (seasonal) conditions in Atlanta, Georgia. Florida, however, is located in a more temperate climate (environment) than Atlanta, Georgia. Consequently, to avoid i ntroducing excess conservatism into the calculation of thermal sweep for Florida I Beams, an average of the winter and summer thermal sweeps proposed by Lee was used in the present study. Before applying this approach to the calculation of thermal sweep va lues for FIB girders, it was first applied to the BT 63 girder cross section studied by Lee to ensure that suitable results was defined as the maximum thermal s weep divided by the maximum allowable fabrication sweep. Using the maximum thermal sweep equations provided in Lee (2010), calculations were performed for a BT 63 girder at varying span lengths for winter conditions, summer conditions, and the average of t hese two (Figure 3 4 lines). Next, thermal gradients were formed per Figure 3 3 applied to the BT 63 cross section for various span lengths, and thermal sweep ratios were computed (Figure 3 4 Figure 3 4 it is evident that there is good agreement between the BT 63 results derived from PAGE 29 29 approximate Figure 3 3 ) was deemed to be suitable for use in computing thermal sweep data for Florida I Beam sections at varying span lengths. 3 3 Thermal Sweep for Florida I Beams Total thermal sweep deflections were calculated for each Florida I Beam (FIB) cross sectional shape at multiple span lengths (minimum practical, intermediate, and maximum practical) using the gradient formation approach described above and using moment curvature analysis (see Appendix B ). For each pair of FIB section type and span length, upper and lower bounding values of thermal swee p (Figure 3 5 ) were calculated using winter temperature data (Lee, 2012) and summer temperature data (Lee, 2010). Subsequently, the tot al thermal sweep 3 6 ) by dividing by the allowable fabric ation sweep. Partially as a consequence of the fact that all FIB sections have the same top and bottom flange width and geometry, the computed sweep ratios were found to correlate to span length in an approximately linear manner. Based on this observation, individual linear best fit relationships between thermal sweep ratio and span length were constructed from the winter sweep data and summer sweep data. A seasonally averaged Florida I Beam (FIB) thermal sweep ratio relationship (Figure 3 6 ) was then constructed by averaging the best fit winter and summer curves. Functionally, this relationship has the form: ( 3 3 ) where sr FIB is the seasonally averaged FIB sweep ratio, and L is the span length in ft. To compute total lateral sweep imperfections for use in girder stabil ity analyses, thermal sweep was superimposed with the maximum allowable fabrication sweep as: PAGE 30 30 ( 3 4 ) Substituting Eqn. ( 3 3 ) into Eqn. ( 3 4 ), and defining maximum allowable fabrication sweep specified in the PCI Bridge Design Manual in. for every 10 ft of girder length, the total lateral sweep used in the pres ent study for FIB girders was: ( 3 5 ) which can be mathematically simplified to: ( 3 6 ) where is the total sweep (lateral imperfection) in inches, and L is the span length in ft. PAGE 31 31 A B C Figure 3 1 Transverse thermal gradients of pre stressed concrete bridge girders proposed by Lee (2012) A ) Top flange B ) Web C ) Bottom flange PAGE 32 32 A B C Figure 3 2 Top flange transverse thermal gradient. A) From finite element analysis of a typical BT 63 girder (Lee, 2010). B) Simpli fied gradient proposed by Lee (2012). C) Approximated right side ascending branch (BDV31 977 46). PAGE 33 33 A B C Figure 3 3 Transverse thermal gr adients for FIB bridge girders. A ) Top flange B ) Web C ) Bottom flange Figure 3 4 Comparison of sweep ratios computed for BT 63 gir der (at varying span lengths) as computed using different methods PAGE 34 34 Figure 3 5 Thermal sweep data for FIB sections Figure 3 6 Thermal sweep ratios for FIB sections PAGE 35 35 CHAPTER 4 DEVELOPMENT OF UNANCHORED SINGLE GIRDER WIND CAPACITY EQUATION 4 1 Overview To update previously developed girder capacity equations considering the revised definition of sweep systems of FIB girders were modeled (Figure 4 1 ) and structurally analyzed using the ADINA (2016) finite element analysis code. The models incorporated bearing pad support stiffnesses, and were capable of capturing buckling behavior of FIBs, while remaining computationally efficient enough that thousands of parametric analyses could be performed. Models analyzed in the present study were developed in a semi automated fashion by extending a modeling methodology developed in a previous study (BDK75 977 33, Consolazio et al., 2013) to include the r evised definition of sweep. In the global coordinate system employed in 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 corresp onding to the same directions, with the origin at one end of the girder at the centroid of the cross section. Girder buckling capacities were quantified using geometrically nonlinear large displacement analyses in which static loads were applied in increme ntal steps, taking into account the deformed state of the structure at each load step. Instability was initiated by introducing girder fabrication imperfections (i.e., sweep) into the models, so that each load step (load increment) caused the models to def orm to ever greater levels. By analyzing the load displacement results (using a modified Southwell analysis; see Consolazio et al., 2013), it was possible to determine when girder displacements began growing asymptotically, indicating a collapse. PAGE 36 36 4 2 Modeling of Bridge Girders Each bridge girder was modeled using warping beams an advanced beam element formulation provided by ADINA (2016) that incorporates degrees of freedom representi ng torsionally induced out of plane warping of the cross section. Warping beams are primarily intended for modeling thin walled sections for which warping effects can significantly affect structural response, but they also have several additional refinemen ts that make them superior to standard Hermitian beam elements for buckling analysis applications. For example, offsets between the shear center and the centroid of asymmetric cross sections are accounted for automatically, and the kinematic formulation of the element includes coupling between bending and torsional deformation modes. Warping beams require the calculation of a comprehensive set of cross sectional properties, several of which require knowledge of the warping function which cannot be calculat ed in closed form (for complex shapes) and must therefore instead be solved for numerically. Details relating to the section properties previously calculated in BDK75 977 33 for FIB cross sectional shapes are included in Appendix A of the present report. Construction sweep tolerances implemented in the present study (BDV31 977 46) were determined from Eqn. ( 3 6 ), which includes maximum allowable fabrication imperfections in. for every 10 ft of girder length and an additional sup erimposed thermal sweep. To ensure conservative calculations of buckling capacity, all FIBs were modeled with the maximum allowable sweep ( u max ), as determined from span length and Eqn. ( 3 6 ). Geometrically, sweep was implemented using a sinusoidal function (Figure 4 2 ) with the maximum allowable sweep at midspan, so that the lateral deviation, u at any position, v along the girder length was: PAGE 37 37 ( 4 1 ) During early phases of constructing a bridge, the deck is not present 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 act ive). It because the additional elevation of the girder center of gravity reduces buckling capacity by a small amount. In BDK75 977 33, to establish maximum probabl e 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 fi nding the span length at which camber was maximized. Long term creep effects were ignored and it was further assumed that cracking did not occur. From these calculations, it was determined that 3.25 in. was a reasonable upper bound camber for FIB girders d uring 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 weight 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 deflection 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 38 38 ( 4 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 strong axis moment of inertia. Because 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 4 3 ) so that the vertical deviation, w at any point, v along the girder length was: ( 4 3 ) 4 2 1 Modeling of End Supports Girder support stiffnesses were modeled with six (6) geometrically linear springs to represent the stiffness of the bearing pad in each degree of freedom, wi th each spring corresponding to one of the four (4) main deformation modes of the pad: shear, axial, torsion, and roll (Figure 4 4 ). Pad stiffnesses were computed using the methods discussed in Consolazio et al. (2013). 4 2 2 Bearing Pad Selection Seven (7) standard types of elastomeric bearing pad are provided in Design Standard No. 20510: Composite Elastomeric Bearing Pads Prestressed Florida I Beams (FDOT, 2012a) for use with FIBs. During design, selection of the type of pad that will be used i n 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 length). As such, it is not appropriate to assu me that for each FIB type, there is a specific corresponding type of bearing pad that would be utilized. Hence, in BDK75 977 33 and in the PAGE 39 39 present study, it was conservatively assumed that the pad type with the lowest roll stiffness (which will produce 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, using the grillage method discussed in Consolazio et al. (2013), the Type J bearing pad was selected for u se. 4 2 3 Axial Load Selection The amount of axial load applied to a bearing pad does not change the initial linear portion of the roll stiffness curve, b ut 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. Therefore, 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 Bea ms (IDS 20010; FDOT, 2012b) and the minimum expected axial pad load 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 t his 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. 4 2 4 Girder Slope Selection Overturning (roll) pad 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: Comp osite Elastomeric Bearing Pads Prestressed Florida I Beams (IDS 20510; FDOT, 2012c), 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 PAGE 40 40 eliminate slope. T herefore, 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 in BDK75 977 33, it was determined that a reasonable upper limit for the SDL negated slope was 0.01 rad. Additionally, AASHTO LRFD (2010) earing 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. 4 2 5 Load Application to Individual Bridge Girders Two types of structural loads were included in the models: wind and gravity. Lateral wind loads, calculated for each girder using the design drag coefficients proposed in Consolazio et al. (2013) for FIBs, were applied to the girder elements as tributary n odal loads (Figure 4 5 a). Small overturning moments were also 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 t he girder) (Figure 4 5 b). Wind loads were always applied in the direction of increasing girder sweep. Gravity load was applied as a vertic proportional body force) in units of g the acceleration due to gravity, so that a load of 1 g represented the self weight of the girder. In field conditions, girders are always subjected to a constant gravity load of 1 g. However, in the structural models analyzed in this study, gravity loading was used to initiate instability. After wind loads were applied, gravity load was linearly ramped up beyond 1 g if possible until girder instability occurred. Subsequently, the c apacity of the system was PAGE 41 41 expressed as a gravity load (in g), which can also be thought of as a capacity to demand ratio. For example, if a system became unstable at a gravity load of 1.5 g, then the ratio of capacity (1.5 g) to demand (1 g) was 1.5. 4 3 Pa rametric S tudy of Unanchored Individual Bridge Girders Using the updated sweep Eqn. ( 3 6 ), a parametric study was performed using ADINA to analyze finite element models of single unanchored Florida I Beams (FIBs) over a range of span lengths. Each stability analysis in the parametric study was performed for the purpose of quantifying the maximum lateral wind load that could be applied before girder instability occurred. For each model, the vertical load carrying capacity was evaluated several times at different lateral wind pressures, iterating until the vertical capacit y was within 1% of 1 g (i.e., girder self weight). The resulting wind pressure for each case was denoted as the lateral wind capacity of that girder. 4 3 1 Selection of Parameters Girder parameters that were varied in the parametric study were the FIB cross section depth (in.) and span length (ft). All eight (8) standard FIB cross sections were included in the study, with depths ranging from 36 in. to 96 in. In BDK75 977 33 (Consolazio et al., 2013), bridge grade was found to have negligible effect on wind capacity; therefore in the present study, all analyses were performed at a level (0%) grade. Material properties assumed for the prestressed concrete FIBs were f c = 8.5 ksi, unit weight = 150 pcf, and Poisson's ratio = 0.2. Using these values and equations provided in PCI (2010), the concrete elastic modulus was computed to be E = 5589 ksi. It should be noted that in BDK75 977 33 (Consolazio et al., 2013), the conc rete strength was assumed to be f c = 6.5 ksi (with corresponding E = 4887 ksi). In a related follow up study, BDK75 977 70 (Consolazio and PAGE 42 42 Edwards, 2014), the concrete strength was adjusted to f c = 8.5 ksi to better represent typical conditions in Florid a bridge construction practice. In the present study, the updated value of f c = 8.5 ksi was retained. Span length ranges for each FIB section used in the parametric study were in accordance with those determined previously in BDK75 977 33. Maximum and mi nimum span lengths were based on design aids included in Instructions for Design Standard No. 20010: Prestressed Florida I Beams (IDS 20010; FDOT, 2012b). This document provides estimated span lengths (Table 4 1 ) for FIBs with different lateral spacings, based on representative bridge design calculations. Maximum span lengths were based on a girder spacing of 6 ft and an environment cl ft girder spacing lengths 10 ft longer than the maximums considered in BDK75 977 33 were added t o the parametric study. Span lengths were incremented at 5 ft intervals over the final chosen ranges (Table 4 1 ). 4 3 2 Updated Wind Capacity of a Single Unanchored Girder The relationships between wind capacity and span length, as previously determined in BDK75 977 33 (without inclusi on of thermal sweep), and as determined in the present parametric study (with inclusion of thermal sweep) are compared in Figure 4 6 As expected, including thermal sweep had the effect of reducing wind load capacities, particularly at longer span lengths. Also, similar to the BDK75 977 33 results, capacity data from the present study indicated that span length wa s the strongest predictor of wind capacity. Girder depth was found to have a secondary influence in that increasing the depth decreased the wind capacity (due to the PAGE 43 43 In BDK75 977 33, the data shown in Figure 4 6 a were used to construct the following wind capacity equation [cited as Eqn. (8.2) in Consolazio et al. (2013)]: ( 4 4 ) where P max ,0 is the wind capacity in psf, L is the span length in ft, and D is the FIB cross section of Eqn. ( 4 4 ), 2) the updated capacity data shown in Figure 4 6 b, and 3) an improved curve fitting process, the following updated wind capacity equation for unanchored girders was constructed: ( 4 5 ) where P max ,0 is the wind capacity in psf, L is the span length in ft, and D is the FIB cross section depth in inches. When forming Eqn. ( 4 4 ) in BDK75 977 33, a multi stage curve fitting process was used in which individual curves were initially fit to the capacity vs. span length data for each FI B type, followed by a subsequent curve fitting process that was applied to the coefficients from the individual FIB fits. In contrast, when forming Eqn. ( 4 5 ) in the present study, error in the curve fitting process was simultaneously minimized across the entire data set (i.e., all span lengths and all girder types/depths shown in Figure 4 7 b) at one time. (Subsequent to this process, minor adjustments were made to the fitting coefficients to ensure conservative capacities were obtained for all data points in the data set). As Figure 4 7 process yielded a capac ity equation that fit the trends in the underlying data better than did the original equation developed in BDK75 977 33. PAGE 44 44 Table 4 1 Span length ranges for FIBs Values fr om FDOT IDS 20010 Section Min length (ft) Max length (ft) Final range 80 105 75 120 126 95 140 113 142 110 155 124 155 120 170 142 173 135 190 151 182 145 195 159 191 155 205 FIB 175 208 170 225 Spacing 12 ft 6 ft Environment Extremely aggressive Moderately Aggressive Figure 4 1 Finite element model of a single FIB (isometric view) Figure 4 2 Representation of sweep in FIB model (plan view) PAGE 45 45 Figure 4 3 Representation of camber in FIB model (elevation view) Figure 4 4 Bearing pad stiffness springs in FIB model (isometric view) PAGE 46 46 A B Figure 4 5 Representation of w ind load in structural models. A ) Lateral nodal loads (top view) B ) Overturning moments (section view) A B Figure 4 6 Wind capacities for unanchored FIBs. A ) Data from BDK75 977 33 (with out inclusion of thermal sweep). B ) Data from present study (with inclusion of thermal sweep) PAGE 47 47 A B Figure 4 7 Wind capacity of an unanchored FIB. A ) Data from Figure 4 6 a and Eqn. ( 4 4 ) [i.e., Eqn. (8.2) from BDK75 977 33 ]. B ) Data from Figure 4 6 b and Eqn. ( 4 5 ) developed in present study PAGE 48 48 CHAPTER 5 DEVELOPMENT OF UNANCHORED STRUT BRACED TWO GIRDER BUCKLING CAPACITY EQUATION 5 1 Overview To develop an updated unanchored strut braced two girder buckling capacity ( C 0 ) equation accounting for the revised definition of sweep multi girder systems of FIBs were mode led and analyzed using ADINA (2016). Structural modeling techniques used in the present study matched those previously developed in BDK75 977 33 (Consolazio et al., 2013). For the convenience of the reader, a summary is provided below of material in Consol azio et al. (2013) that is pertinent to the development of a revised baseline capacity ( C 0 ) equation. 5 2 Review of M ulti G irder System R elated I nformation from BDK75 977 33 5 2 1 Preliminary S ensitivity S tudies In BDK75 977 33, several system parameters were identified as having negligible influence on system capacity. Con sequently, 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. Preliminary studies revealed that braces were naturally divided into two categories that had very different effects on system behavior: strut braces and moment resisting braces As a result, separate parametric studies were performed for each brace category. 5 2 2 Strut Braces Top struts and parallel struts (Figure 5 1 ) are both examples of strut braces, which include (but are not limited to) all brace designs consisting solely of horizontal compression members. In BDK75 977 33, it was found that all strut brace designs are essentially interchangeable with regard to lat eral stability. That is, a girder system braced with top struts has PAGE 49 49 the same capacity as an otherwise identical system braced instead with parallel struts (or any other type of strut brace). As a result, the capacity of a strut braced system is also insens itive to girder spacing (which only affects the length and thus the axial stiffness of the strut members). Strut braces can be defined (or identified) by their lack of resistance to girder overturning. In a small displacement (geometrically linear) analysi s of a system with zero bearing pad rotational stiffness, a strut with ideal pin connections forms a collapse mechanism (Figure 5 2 ) that allows the connected girders to rotate freely in unison. Therefore, struts can only provide stability by coupling the girders 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. 5 2 3 Moment R esisting Braces X braces and K braces (Figure 5 3 ) are both examples of moment resisting braces, which are capable of resisting girder overturning. Unlike struts, the system capacity provided by different moment resisting bra ce 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 always the m ost unstable bridge cross section possible. In the presence of moment resisting bracing, the additional roll stiffness and stability provided by anchors is typically negligible. 5 2 4 Modeling of Braces Because the design of bracing has historically been left to the discretion of the contractor, a wide variety of bracing configurations are used in practice. Consequently, in BDK75 977 33, it was not possible for every potential brace configuration to be represented in the parametric PAGE 50 50 studies. After conducting a survey of bracing designs used in the construction of bridges throughout Florida, four (4) representative brace configurations were identified: Top str ut (Figure 5 4 a) : a horizontal timber compression strut situated between the edges of the top flanges. The top strut is typically nailed to the underside of a slightly Parallel strut (Figure 5 4 b) : Two (or more) horizontal timber compression struts wedged in place be tween the girder webs. X brace (Figure 5 4 c) : Two diagonal timber members wedged between the webs that pe. A steel bolt typically passes through both members at the crossing point to create a hinge. K brace (Figure 5 4 d) : Steel members (typica lly steel angles) welded together into a shaped frame and welded or bolted to 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 conservative with regard to girder stability. Pins and hinges were modeled with beam end releases and nodal constraints, respectively. In BDK75 977 33, during the survey of bracing d esigns, the vast majority of timber braces that were encountered were composed of 4x4 Southern Pine sawn lumber. According to the National Design Specification for Wood Construction (AF&PA, 2005), 4x4 Southern Pine x section an d an elastic modulus of E = 495 ksi (based on an E min of 550 ksi for 4 inch properties were used to model all 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, PAGE 51 51 K x x E = 29000 ksi. 5 2 5 Modeling of Bridge Skew and Wind Load In BDK75 977 33, the final proposed design wind loads (Figure 5 5 ) were based on a and et al., 2013). An initial pressure coefficient ( C P ) (i.e., C D,eff ) was a ssigned to G1 based on the type of girder section: 2.0 for FIBs. Girder G2 was assigned a C P of 0 (i.e., no load) while G3 and all subsequent girders were assigned a C P equal to half of the load on the windward girder. In a braced system of girders, the in troduction of bridge skew causes the girders to become staggered longitudinally. This affects system capacity in two ways (Figure 5 6 ): Brace placement : Because girders are installed perpendicular to the girder axes (per Design Standard No. 20005: Prestressed I Beam Temporary Bracing FDOT, 2014a), the region within which braces can be placed is smaller (shorter) than the span length of the girders. 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 completely shielded with respect to wind load. Rather, an end portion of each girder is exposed to full (unshielded) wind pressure. In BDK75 977 33, the aerodynamic properties of the may result from the presence of upwind girders are unknown. The magnitude of both of these effects is a function of the girder offset length ( L offset see Figure 5 6 ) which is depen dent on both skew angle and girder spacing. Conducting wind tunnel testing to experimentally quantify the effects of skew on girder end shielding was outside the scope of BDK75 977 33. Consequently, the non uniform wind pressure distribution shown for leew ard girders in Figure 5 6 b was an approximation based on engineering judgment. Lacking wind tunnel confirmation of this approximation, it was deemed unwarranted to model this distribution in detail. Inst ead, a simplified, but statically similar, PAGE 52 52 representation was used in which the wind load on each girder was modeled as a single, weighted average uniform pressure along the entire length of the girder. The uniform wind load applied to each partially shiel ded girder ( P ) [reported as Eqn. (9.1) in Consolazio et al., 2013] was computed as a weighted average of the shielded and unshielded wind loads, as follows: ( 5 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 produced by skew. 5 3 Pa rametric S tudy of S ystem C apacity of U nanchored T wo G irder S ystem in Z ero W ind Using the updated sweep Eqn. ( 3 6 ), a parametric study was performed using ADINA (2016) to analyze finite element models of strut braced systems, consisting of two (2) Florida I Beams (FIBs) without anchors and witho ut wind load, over a range of span lengths. Each stability analysis in the parametric study was performed for the purpose of quantifying the system capacity in units of g (the acceleration due to gravity), representing the total gravity load that can be ap plied before the system becomes unstable (collapses). For each model, gravity load was incrementally increased, iterating until the vertical capacity was reached (i.e., until a system instability occurred). 5 3 1 Parameters the FIB cross section depth (in.) and span length (ft). All eight (8) standard FIB cross sections were included, with depths ranging from 36 in. to 96 in. Material prope rties assumed for the prestressed concrete FIBs were f c = 8.5 ksi, unit weight = 150 pcf, and Poisson's ratio = 0.2. Using these values and equations provided in PCI (2010), the concrete elastic modulus was computed to be E = 5589 ksi. As noted in Section 4 3 in BDK75 977 33 (Consolazio et al., PAGE 53 53 2013), the concrete strength was assumed to be f c = 6.5 ksi (w ith corresponding E = 4887 ksi). In the present study, however, the concrete strength was adjusted to f c = 8.5 ksi to better represent typical conditions in Florida bridge construction practice. Span length ranges for each FIB section considered in the un anchored two girder zero wind parametric study were consistent with those previously discussed in Section 4 3 with maximum span lengths increased by 10 ft, per FDOT request. Additionally, span lengths were incremented at 5 ft intervals over the final chosen ranges (recall Tabl e 4 1 ), as opposed to the 10 ft increments used in BDK75 977 33 for the unanchored two girder zero wind study. 5 3 2 Updated System C apacity of U nanchored T wo G irder S ystem in Z ero W ind The relationships between system capacity (in terms of g) and span length, as previously determined in BDK75 977 33 (without inclusion of thermal sweep), and as determined in the present study (with inclusion of thermal sweep) for unanchored two girder strut braced systems in zero wind are compared in Figure 5 7 As expected, system capacity decreased due to the increase in lateral sweep. In both sets of data shown in Figure 5 7 system capacity is correlated with span length ( L ), but the FIB girder depth ( D ) has essentially negligible effect o n capacity. In BDK75 977 33, the data shown in Figure 5 7 a were used to construct the following baseline capacity ( C 0 ) equation of an unanchored strut braced system of two girders in zero wind, reported as Eqn. (9.2) in Con solazio et al. (2013): ( 5 2 ) where C 0 is in g and L is the span length in ft (see Figure 5 8 a). For the present study, using the updated baseline capacity data shown in Figure 5 7 b (with the inclusion of thermal sweep), the following updated baseline capacity ( C 0 ) equation of an unanchored two girder strut braced system in zero wind was constructed: PAGE 54 54 ( 5 3 ) where C 0 is in g and L is the span length in ft. Forming Eqn. ( 5 3 ) involved establishing the curve that fit the data in Figure 5 7 b with the minimum root mean square (RMS) error, then rounding and adjusting the fitting coefficients for 1) convenience of use and, 2) to minimize excess conservatism for span lengths exceeding approximately 140 ft (Figure 5 8 b). 5 4 Moment R esisting B race: L imited S cope Parametric S tudy In BDK75 977 33, the baseline capacity ( C 0 ) equation for an unanchored strut braced two girder system in zero wind was used to develop a set of final system capacity ( C ) equations, reported as Eqn. (9.22) and Eqn. (9.23) in Co nsolazio et al. (2013), for strut braced and moment braced systems, respectively. Development of the final capacity equation was achieved by applying correction factors (developed from parametric studies) to the baseline ( C 0 ) equation. The final system cap acity equation for moment resisting braced systems, reported as Eqn. (9.23) in Consolazio et al. (2013) is repeated here for convenience of reference: ( 5 4 ) wher e 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 unshielded wind load in psf, is the average wind load per girder in psf, k brace is the effective brace stiffness in k ip 5 1 w sw is the girder self weight in lbf/ft (Table 5 2 ), and C 0 [which has been updated in the present study as Eqn. ( 5 3 )] is in g. PAGE 55 55 To account for the effects of wind pressure on system capacity, an average wind pressure per girde r, [reported as Eqn. (9.7) in Consolazio et al., 2013] was defined: ( 5 5 ) where n is the number of girders in the bridge and is the sum of the individual wind pressures on all girders. Recalibration of the additional correction factors in Eqn. ( 5 4 ) to account for an updated definition of sweep (i.e., with inclusion of thermal sweep) was outside the scope of the present study. However, a limited scope parametric study was conducted to compare system capacities computed using full FEA bridge models to cap acities computed using Eqn. ( 5 4 ) supplemented with the updated baseline capacity ( C 0 ) equation [i.e., Eqn. ( 5 3 )]. 5 4 1 Selection of Parameters for Limited Scope Moment Resisting Brace Parametric Study System parameters varied in the moment resisting brace parametric studies that were conducted in BDK75 977 33, and in the present study, were as follows: FIB cross section d epth (in.) Span length (ft) Wind pressure (psf) Effective brace stiffness (kip ft/rad) Number of interior brace points Skew angle (deg.) In BDK75 977 33, seven (7) of the eight (8) standard FIB cross sections were considered. (The use the cross section is so shallow that use of moment resisting braces is unwarranted and unfeasible). For each FIB, capacity analyses were performed for every combination of span length, wind pressure, effective brace stiffnesses, and number of interior brace points sampled from the values listed in Table 5 3 and Table 5 4 Only two girder systems were considered because it was determined from the preliminary sensitivity studies that when PAGE 56 56 moment resis ting braces are used, the two girder system is always the least stable phase of construction. Wind pressure loads were applied to the girders using the shielding pattern proposed in BDK75 977 33 (shown in Figure 5 5 ), and using the wind pressures listed in Table 5 4 Wind pressures specified in the table refer to the unshielded pressure load applied to the windward girder (G1). Hence, in accordance with the wind load model propos ed in BDK75 977 33, the first shielded girder (G2) received no wind load and all subsequent girders (G3, G4, etc.), if any, received half of the listed pressure load. The maximum wind pressure of 160 psf was determined using the Structures Design Guideline s (FDOT, 2012d) by assuming a pressure coefficient of C P = 2.0, a basic wind speed of V = 150 mph, a bridge elevation 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). For the moment resisting braces, the number of interior brace points varied from 0 (end bracing only) to 3 (end bracing plus quarter point interior bracing), and girders in each system model were spaced at 6 ft on center. Furthermore, in moment resisting braces, changes in girder spacing produce changes in the geometric configuration of the brace members, thus changing the effective stiffness of the braces. Such changes can significantly affect system capacity and must be considered. In moment resisting brace para metric studies, the effects of changing girder spacing were accounted for by varying the effective brace stiffness parameter, even though the physical length of the reference brace remained a constant 6 ft. In the present project (BDV31 977 46), parameters for a limited scope parametric study were determined by selecting a reduced set of FIB cross section depths, span lengths, and skew parameters from those used in the previous moment resisting brace parametric study (BDK75 977 33). Final parameters selecte d for the limited scope parametric study are listed in Table 5 5 PAGE 57 57 and Table 5 6 (and can be compared to the complete list of parameter values used in BDK75 977 33, shown in Table 5 3 and Table 5 4 ). To cover a range of upper and lower bounding sections were selected. Similarly, for each FIB cross section de pth, minimum span length, intermediate span length, and a maximum span length were selected. Skew angles of 0 and 25 were selected to cover a wide range of common skew angles used in bridge design. Parametric values of the remaining parameters (i.e., uns hielded wind pressure, effective brace stiffness, and number of interior brace points) were the same as used in BDK75 977 33. In total, 1,440 models were analyzed. 5 4 2 Updated System Capacity of Moment Resisting Brace Results from the limited scope parametric study and corresponding predicted capacities ( C ) determined using Eqn. ( 5 3 ) and Eqn. ( 5 4 ) [i.e., Eqn. (9.23) from BDK75 977 33] are displayed in Figure 5 9 Results are sorted by descending capacity and separated by FIB girder type and skew angle to visualize the level of conservatism in predicted capacities. Results from BDK75 977 33 corresponding to the parameters selected for the present study (i.e., parameters listed in Table 5 5 and Table 5 6 ) are shown in Figure 5 9 a, and compared with predicted system capacity ( C ) quantities using Eqn. (9.23) and Eqn. (9.2) from BDK75 977 33. Results from the presen t study are compared with predicted system capacities ( C ) calculated using Eqn. ( 5 3 ) and Eqn. ( 5 4 ) in Figure 5 9 b. The level of conservatism in the final moment resisting braced system capacity ( C ) equation with an updated baseline ( C 0 ) equation is displayed in the form of a histogram in Figure 5 10 In BDK75 977 33, systems with capacities exceeding 10 g (i.e., extremely stable systems) were excluded from consideration in the development of the final moment resisting braced system capacity predict ion equation. Under the same criteria, predicted capacities for PAGE 58 58 essentially all systems (99%) considered in the present limited scope parametric study (with an updated definition of sweep) were conservative. The moderate increase in conservatism evident in Figure 5 9 b (i.e., a larger gap between parametric capacities and predicted capacities), compared with Figure 5 9 a, is the result of two influences. First, the reduced capacities predicted by the baseline capacity ( C 0 ) Eqn. ( 5 3 ) (accounting for thermal sweep) propagate through Eqn. ( 5 4 ) to yield lower system capacity ( C ) values. This downward influence of C 0 on computed C values is illustrated in Figure 5 11 Second, the parametric finite element data shown in Figure 5 9 b are obtained from models that employ a concrete strength of f c = 8.5 ksi, rather than the f c = 6.5 ksi that was assumed in BDK75 977 33. Due to the increase in concrete strength f c and corresponding increase in concrete modulus E c the capacities computed by finite element analysis increase (Figures 5 12 and 5 13 ). Thus the combined effects of reduced C 0 and increased E c moderately increase the gap betwee n empirically predicted capacity and finite element computed capacity. PAGE 59 59 Table 5 1 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 5 2 Self weight ( w sw ) of each FIB cross sectional shape (from FDOT, 2012b) Cross section w sw (lbf/ft) 1037 1103 1146 1190 1278 Table 5 3 Span length values used in moment r esisting brace parametric study from BDK75 977 33 Span length, L (ft) 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 Table 5 4 Other parameter values used in moment resisting brace parametric study from BDK75 977 33 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 PAGE 60 60 Table 5 5 Selected span length values used in the present moment resisting brace parametric study 110 145 170 120 155 180 130 165 190 140 175 200 150 185 205 220 Table 5 6 Other s elected parameter values used in the present moment resisting brace parametric study 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 25 80 400,000 2 120 600,000 3 160 A B Figure 5 1 Examples of strut bracing A ) Top strut B ) Parallel struts A B Figure 5 2 Collapse mechanism possible with strut bracing A ) Undeformed configuration B) Collapse mechanism PAGE 61 61 A B Figure 5 3 Examples of moment resisting braces A ) X brace B ) K brace A B C D Figure 5 4 Representation of brace configurations in FIB system models A ) Top strut brace B ) Parallel strut brace C ) X brace D ) K brace PAGE 62 62 Figure 5 5 Proposed wind load shielding model for stability evaluation from BDK75 977 33 A B Figure 5 6 Effect of bridge skew on wind loading of bra ced 3 girder system (plan view). A ) Unskewed system B ) Skewed system PAGE 63 63 A B Figure 5 7 System capacities of unanchored two girder strut braced FIB systems in zero wind. A ) Data from BDK75 977 33 (without inclusion of thermal sweep). B ) Data from present study (with inclusion of thermal sweep) PAGE 64 64 A B Figure 5 8 System capacity of an unanchored strut braced two girder FIB system in zero wind as predicted by C 0 Equation. A ) Data from Figure 5 7 a and Eqn. (9.2) from BDK75 977 33 B ) Data from Figure 5 7 b and Eqn. ( 5 3 ) developed in present study PAGE 65 65 A B Figure 5 9 System capacity of moment resisting two girder FIB system. A ) Partial data from BDK75 977 33 and predicted capacity from Eqn. (9.23) from BDK75 977 33 B ) Updated FEA data developed in present study, and predicted capacity from Eqn. ( 5 3 ) (present study) and Eqn. ( 5 4 ) PAGE 66 66 Figure 5 10 Absolute error of system capacity quantities predicted by Eqn. ( 5 4 ) from BDK75 977 33 with updated C 0 equation [Eqn. ( 5 3 )] (Note: negative absolute error indicates conservative prediction of capacity) Figure 5 11 Comparison of predicted capac ities PAGE 67 67 Figure 5 12 Comparison of selected parametric study data from BDK75 977 33 with updated parametric study data (thermal sweep included) Figure 5 13 Absolute difference of current to previous parametric study system capacity quantities (Note: positive absolute difference indicates increased system capacity from previous study) PAGE 68 68 CHAPTER 6 PROCEDURES DEVELOPMENT FOR CONSTRUCTION LOAD DISTRIBUTION FACTOR EQUATIONS 6 1 Overview In the development of distribution factors for girder end shears a nd moments that are induced by bridge construction loads, the deck placement (concrete application and finishing) was the construction stage and process considered. Components of the bridge construction loads considered were as follows: Concrete deck : Thro ughout the deck placement and finishing process, the wet (plastic) concrete has negligible stiffness Consequently, a non composite girder system must support these construction loads. However, in the final bridge condition, the bridge deck works together with the girders as a composite system to resist and distribute loads to the supporting girders. Since the wet (non structural) concrete load is incrementally applied to bridges in the longitudinal direction, this load is treated as a variable length load in the finite element analyses. Partial application of concrete deck loads to the girder system will be further explained later in this report. Stay in place formwork : Stay in place (SIP) formwork systems support intra girder loads (wet concrete) that span transversely between girder top flanges (Figure 6 1 ). Stay in place forms consist of corrugated metal panels that are attached to the tips of the top flange of adjacent girders. The connection between the SIP forms and the girder flange is considered to be incapable of transmitting moments, the refore the SIP forms are Overhang formwork : It is typical for the deck of a bridge to extend past the exterior (fascia) girders, thereby producing cantilevered overhangs (Figure 6 2 ). During construction, overhang brackets (Figure 6 3 ) are used to temporarily support the cantilever portion of the wet deck slab that extends beyond exterior girders. These temporary structural bracket systems support the overhang formwork, wet concrete, construction walkway, workers and concrete finishing machine. In BDK75 977 70, a survey of representative literature from overhang bracket manufacturers was conducted to quantify representative cross sectional properties and longitudinal spacing requirements. Most commercially available formwork systems con sist of timber joists and sheathing supported on steel bridge overhang brackets (Figure 6 4 ). It is important to no te that all of the gravity loads supported by the overhang brackets are eccentric relative to the exterior girders, and as such apply torque loads to the exterior girders in the overall cross sectional system. Finishing machine : Bridge deck finishing machi nes (Figure 6 5 ) spread, compact, and finish the freshly placed wet concrete deck surface. The finishing machine is an open stee l frame that is supported at the extremities of the bridge width on the overhang PAGE 69 69 brackets described above. Drive wheels (commonly referred to as bogies) move the paver longitudinally along the length of the bridge and are eccentrically supported by screed rails (Figure 6 4 ) on each side of the bridge. A suspended paving carriage with augers, drums, and floats finishes the concrete surface as it moves transversely from side to side across the width of the bridge (perpendicular to the longitudinal movement of the finishing machine along the length of the bridge). Concrete is typically placed just ahead of the travelling f inishing machine using separate equipment, such as a pump. Live loads : Live loads that are present during the deck finishing process consist of workers, temporary materials, and supplementary construction equipment. For modeling purposes, these loads are t reated as either uniform pressure loads, or as line loads, as will be discussed in greater detail later. 6 2 Modeling Multi Girder Bridge Systems during Construction Numerical modeling and a nalysis techniques developed in FDOT study BDK75 977 70 (Consolazio and Edwards, 2014), were extended in the present study for purposes of quantifying girder forces induced by construction loads. Each numerical finite element model was suitable for analyzi ng construction loads acting on systems of precast concrete girders (Florida I Beams) braced together (including the influence of brace configuration, bearing pad stiffness, etc.). The modeling techniques allow for consideration of different Florida I Beam cross sections, span lengths, girder spacings, deck overhang widths, skew angles, number of girders, number of braces, and bracing configurations (K brace and X brace). Additionally, partial coverage of wet (non structural) concrete load and variable posi tioning of deck finishing machine loads were considered. In all cases, structural element forces were determined using large displacement (i.e., geometrically nonlinear) analyses, in which static loads were applied to the models in incremental steps, takin g into account the deformed state of the structure at each step. Construction loads applied beyond the lateral extents of an exterior girder are structurally supported during construction by overhang brackets Specifically, the finishing machine, formwork, overhang wet concrete, and construction worker live loads are typical loads supported by overhangs. To define the lateral eccentricity of the overhang construction loads, two offset PAGE 70 70 parameters were established (in BDK75 977 70). To be consistent with the FDOT Instructions for Design Standard No. 20010: Prestressed Florida I Beams (FDOT, 2014b), the concrete finishing machine was offset 2.5 in. from the overhang edge (Figure 6 6 ). In the FDOT Concrete I girder Beam Stability Program in addition to providing calculations for determining bracing adequacy and girder stability, several recommended values for t he overhang geometry are specified, including a 2 ft worker platform width. Therefore, for all the numerical studies conducted herein, the worker platform was assumed to extend 2 ft beyond the finishing machine supports (Figure 6 6 ). In the finite element bridge models, components of the overhang brackets were modeled with beam elements, using representative cross sectional propertie s obtained from a survey of overhang bracket manufacturers. To represent the offset eccentricities between the girder centroid and bracket connection points, the deformable overhang bracket elements were connected to girder warping beam elements using rigi d links (Figure 6 7 ). To model interaction between the overhang bracket and the girder bottom flange, two co located but separate nodes were used: one at the bottom vertex of the metal overhang bracket, and a second at the end of the rigid link representing the surface of the girder bottom flange. At this location, the overhang bracket bears against (i.e., is in compressive contact with) the g irder bottom flange. To model this behavior structurally, a constraint condition was defined such that the lateral (X direction) translations of the two co located nodes were constrained to match, while permitting independent movements (relative slip) in t he vertical direction. Overhang bracket nodes were positioned (Figure 6 7 ) to define: the three corners of the triangular system; and all lo cations of load discontinuities (i.e., deck overhang edge) and load application points (i.e., finishing machine and worker line load application points). The worker PAGE 71 71 line load was conservatively applied to the center of the worker platform width. Thus, the load application of the worker line load was laterally offset in the X direction 12 in. from the assumed finishing machine application point and 14.5 in. from the deck overhang edge (Figure 6 7 ). Combining each of the previously mentioned modeling components, an overall illustration of a typical FIB system model is presented in Figure 6 8 Based on a review of literature obtained from typical overhang bracket manufacturers, brackets were commonly found to be spaced at between 4 ft and 6 ft on center longitudinally along the span length of a bridge. Therefore, an average longitudinal spacing of 5 ft was used for all brackets (Figure 6 9 ). Shown in Figure 6 8 and Figure 6 9 are rigid vertical elements (links) extending from girder centroid to girder top surface which were included in the model for application of construction loads on each girder. These rigid elements account for the verti cal eccentricity between the girder centroid and the girder top surface (where loads were applied). It was determined that brace forces induced by construction loads were not sensitive to changes in the longitudinal spacing of the rigid vertical elements, consequently the rigid links were given an arbitrary longitudinal spacing of 1 ft in the span direction. 6 3 Application of Construction Loads During bridge construction, self weight (i.e., gravity) loads from girders, braces, formwork, and overhang brackets act in combination with a variety of superimposed loads. A key load among the superimposed loads is the application of concrete finishing machine weight. Finishing machines are supported near the extremities of the bridge width by several wheels. The common Terex Bid Well 4800 machine has a total wheel base of approximately 8 ft in the longitudinal (bridge span) direction. Since this wheel base is small relative to the typical span lengths of prestressed girder bridges, the finishing machine wheel reaction forces were idealized as single concentrated loads (one load on each side of the bridge, equal to half the total machine PAGE 72 72 weight). Construction loading conditions specified in the Structur es Design Guidelines (FDOT, weight shall vary as a function of bridge width, as indicated in Table 6 1 For the present study, the FDOT minimum bridge width specification was reduced from 26 ft to 0 ft, to enable inclusion of bridge configurations narrower than 26 was defined to be the completed (finished) bridge deck width (i.e., from edge to edge of the deck, excluding temporary overhang formwork). As concrete placement and finishing of the bridge deck progresses, concentrated live loads (e.g., workers standing o n the overhang platforms) will be applied at the lateral extremities of a bridge. To account for such loads, the AASHTO Guide Design Specifications for Bridge Temporary Works factored) be applied along the outside edge of each deck overhang. The line load is stipulated to be applied as a moving load (i.e., co located with the finishing machine position) but with a fixed longitudinal length of 20 ft, so as to not introduce excessive conservatism. F or numerical modeling purposes, when the finishing machine was either at the start or the end of a bridge, the worker line load was applied over the first or last 20 ft of the structure, respectively. For all other cases, where the finishing machine was po sitioned at an interior brace point, the 20 ft worker line load was longitudinally extended 10 ft ahead of and behind the finishing machine. Additionally, in accordance with AASHTO design specifications, an un factored 20 psf construction live load was al so applied to each numerical bridge model analyzed. To be consistent with FDOT guidelines, the 20 psf live load was extended over the entire bridge width, and was extended 50 ft in longitudinal direction, but centered on the finishing machine location ( Str uctures Design Guidelines FDOT, 2016). For load cases where the finishing machine was PAGE 73 73 located at either the start or the end of a bridge (i.e., within 25 ft of either end), the construction live load was applied over the first or last 50 ft of the bridge length. In physical bridge construction, wet concrete pressure load is applied to a bridge (by way of stay in place forms) over incrementally increasing lengths, as deck placement progresses. The wet concrete is typically placed just ahead of the moving finishing machine using a pump, therefore, in a vast majority of paving situations, the location of the finishing machine and the end of the concrete coverage will coincide. For purposes of numerical modeling, the loading conditions considered in this study included placement of concrete deck loading over the full length of bridge as well as partial lengths of bridge. For partial coverage cases, the position of the finishing machine was taken to coincide with the location of the furthest placed concrete. 6 3 1 Construction Load Groups Considered Construction loads were separated into two groups (Table 6 2 and Table 6 3 ) so that dis tribution factors could be computed separately for each group. The key distinction between the two load groups was that loads included in Load Group 1 were considered to be live loads whereas loads in Load Group 2 were considered to be dead loads To ensu re that maximum (i.e., controlling) distribution factors were quantified in the construction load parametric study, multiple load cases for each load group were analyzed. Construction Load Group 1 load cases are shown in Figures 6 11 6 14 Similarly, Construction Load Group 2 loads were analyzed with multiple load cases where the wet concrete deck location was incrementally advanced in the longitudinal span direction (stopping at each brace point), as illustrated in Figures 6 16 6 19 6 3 2 Application of Construction Loads Construction loads that are ap plied between adjacent girders (e.g., on the stay in place forms) produce vertical reaction forces that act on the tips of the girder top flanges. Since all Florida I Beams have a top flange width of 48 in., the lateral eccentricity between the girder PAGE 74 74 cent roid and the formwork reaction force (Figure 6 20 ) is 24 inches (half of the girder top flange width). For numerical modeling and an alysis purposes, each eccentric reaction force of this type was converted into statically equivalent forces and moments (Figure 6 20 ) which were then applied along the centerlines of the girders. Consequently, all intra girder distributed pressure loads that were applied over the width of the stay in place formwork were converted into equivalent nodal forces and moments. Other types of construction loads, such as the overhang loads (overhang formwork, worker line load, etc.), were applied directly to nodes in the structural model based on the appropriate tributary areas (Figure 6 21 ). PAGE 75 75 Table 6 1 Varying finishing machine load (based on FDOT Structures Design Guidelines 2016) Bridge Width, W (ft) Total Weight of finishing machine (kips) 0 < W 7 32 < W 11 56 < W 13 80 < W 16 Table 6 2 Summary of construction Load Group 1 loads in parametric studies Construction Load Group 1 Load type Load Reference Live load Temporary 20 psf (for 50 ft, longitudinally) I Worker line load Temporary 75 lb/ft (for 20 ft, longitudinally) I Finishing machine Temporary Varies with bridge width (Table 6 1 ) I I: per FDOT Structures Design Guidelines (2016) Table 6 3 Summary of construction Load Group 2 loads in parametric studies Construction Load Group 2 Load type Load Reference Wet concrete deck Permanent I Wet concrete build up Permanent 50 lb/ft I Stay in place forms Permanent 20 psf I Overhang formwork Temporary 10 psf II Overhang brackets Temporary Self weight I: per FDOT Structures Design Guidelines (2016) II: per FDOT recommendations Figure 6 1 Stay in place formwork (section view) PAGE 76 76 Figure 6 2 Temporary support brackets used to support deck overhangs during construction Figure 6 3 Cantilever overhang supported by overhang brackets (Photo credit : Clifton and Bayrak, 2008) PAGE 77 77 Figure 6 4 Details of overhang formwork support brackets and loads Figure 6 5 Typical bridge deck finishing machine in operation (Photo credit: Gomaco) PAGE 78 78 Figure 6 6 Overhang bracket components and geometry Figure 6 7 Details of overhang bracket model PAGE 79 79 Figure 6 8 Cross sectional view of overall braced girder system model Figure 6 9 Isometric view of braced girder system model PAGE 80 80 Figure 6 10 Cross sectional summary of constr uction Load Group 1 ( LG1 ) loads Figure 6 11 Construction Load Group 1 as a function of finishing machine location (Bridge with only end span braces; no interior braces) Figure 6 12 Construction Load Group 1 as a functio n of finishing machine location (Bridge with end span and midspan bracing) PAGE 81 81 Figure 6 13 Construction Load Group 1 as a function of finishing machine location (Bridges with third point bracing) Figure 6 14 Construction Load Group 1 as a function of finishing machine location (Bridges with quarter point bracing) Figure 6 15 Cross sectional summary of construction Load Group 2 ( LG2 ) loads PAGE 82 82 Figure 6 16 Construction Load Group 2 with incremental deck load (Bridge with only end span bra ces; no interior braces) Figure 6 17 Construction Load Group 2 with incremental deck load (Bridge with end span and midspan bracing) Figure 6 18 Construction Load Group 2 with incremental deck load (Bridges with third point bracing) PAGE 83 83 Figure 6 19 Construction Load Group 2 with incremental deck load (Bridges with quarter point bracing) Figure 6 20 Eccentric reaction forces from loads applied to SIP forms, and statically equivalent nodal force and moment applied to top of girder PAGE 84 84 Figure 6 21 All construction loads (LG1 and LG2) converted to equivalent nodal loads PAGE 85 85 CHAPTER 7 DEVELOPMENT OF CONSTRUCTION LOAD DISTRIBUTION FACTOR EQUATIONS 7 1 Construction Load Distribution Factor Parametric Study 7 1 1 Scope To develop empirical construction load distribution factor ( DF ) equations, a large scale construction load distribution factor parametric study was conducted. Three dimensional (3 D) structural analyses were conducted (using automation scripts) to generate and analyze structural analysis models using the ADINA (2016) finite element code. The parametric study was conducted to quantify girder end shear forces and maximum girder moments. Since several geometric parameters influence the magnitude and distribution of maximum girder moments and girder end shear forces caused by construction loads, it was necessary to conduct a parametric study covering a wide range of possible parameters: Construction Load Group ( LG1 or LG2 ) FIB cross section depth (in.) Span length (ft) Skew angle (deg.) Number of bra ce points (end span only, 1/2 point, 1/3 point, 1/4 point) Brace type (K brace or X brace) X brace material type (steel or timber) Deck overhang width (in.) Girder spacing (ft) Number of girders Finishing machine location or non structural deck dead load location Specific parameter values that were included in the parametric study which involved 290,304 separate analyses are listed in Table 7 1 and Table 7 2 Span lengths were chosen to represent a shorter than typical length a practical minimum, an intermediate length, and a practical maximum length (per design FDOT aids cited in Consolazio and Edwards, 2014). Additional geometric parameters, such as skew angle, deck PAGE 86 86 overhang width, and girder spacing, were selected to cover a range of typical bridge configurations, as determined in Consolazio and Edwards (2014) based on a survey of typical Florida bridges and FDOT design recommendations. In regard to the choice of cross brace (X brace) construction material i.e., timber or s teel that was implemented in the numerical bridge models, the original intent was to identify whichever material produced the more conservative results, and then include only that material in the final parametric study. However, after conducting a prelimin ary sensitivity study, it was determined that timber X braces produced larger exterior girder end shear forces and moments in some cases, while steel X braces produced larger interior girder end shear forces and moments. Consequently, separate analyses of both timber X brace configurations and steel X brace configurations were modeled for each bridge included in the parametric study. 7 1 2 Special Cases As a consequence of including timber X braces in the parametric study, some Load Group 2 (LG2) models were found through analysis to be unstable. However, all such cases were found to correspond to long span bridges with end span only timber X bracing. Using timber X bracing only at the ends of a bridge is not typical practice and does not meet standard bracing design requirements. Consequently, each model that was determined to be unstable in this manner was removed from the distribution factor parametric stu dy. 7 2 Definition of Distribution Factors After analyzing all parametric models using ADINA (2016), girder end shear forces and maximum girder moments were quantified and processed into distribution factors, which were defined as: PAGE 87 87 ( 7 1 ) ( 7 2 ) where DF V is the girder end shear force distribution factor, V GIRDER is the girder end shear force, is the sum of the girder end shear forces at the same end of the bridge system, DF M is the maximum girder moment distribution factor, M GIRDER is the maximum girder moment (along the span length), and is the sum of the maximum girder moments in the entire system Note that based on this definition, two (2) girde r end shear forces (and subsequent end shear force distribution factors) and one (1) maximum girder moment (and subsequent maximum girder moment distribution factor) were quantified for every girder in every bridge model analyzed in the parametric study. Due to the eccentric nature of most construction loads where loads such as the finishing machine load are applied to overhang formwork and are therefore applied indirectly to exterior girders distribution factors were further separated into exterior girder factors and interior girder factors. In total, the following eight (8) distribution factors for Load Groups 1 and 2 were defined: ( 7 3 ) ( 7 4 ) ( 7 5 ) PAGE 88 88 ( 7 6 ) ( 7 7 ) ( 7 8 ) ( 7 9 ) ( 7 10 ) where, DF V EXT LG1 is the end shear force distribution factor for an exterior girder with construction Load Group 1 loads applied, V EXT LG1 is the end shear force for an exterior girder with LG1 loads applied, and is the sum of the girder end shear forces at the same end of the bridge system. Similarly, for the remaining distribution factors, the subscripts EXT and INT are used to distinguish between exterior and interior girders, respectively, while subscripts LG1 and LG2 are used to distinguish be tween the application of Load Group 1 and Load Group 2 loads, respectively. Distribution factors were computed only for superimposed construction loads, not self weight. To quantify distribution factors due only to superimposed construction loads, moments and end shear forces produced only by gravity (i.e., moments and end shear forces due to girder self weight and brace self weight alone) were subtracted from moments and end shear forces produced by the combined effects of gravity and superimposed construc tion loads. 7 2 1 Distribution Factor Sensitivities Due to the large number of parameters considered in the parametric study (11 in total), supplementary limited scope sensitivity studies were conducted prior to the development of PAGE 89 89 empirical distribution factor equations. These additional studies were used t o quantify the sensitivity of computed distribution factors to variations of a single parameter (while all remaining parameters were kept constant). Additionally, the sensitivity studies were used to determine if any of the 11 parameters could be omitted f rom the final empirical distribution factor development process. 7 2 2 Illustrative Examples After conducting several limited scope sensitivity studies, i t was determined that distribution factors were most sensitive to geometric parameters that influence the overall bridge geometry. For example, an increase in the number of girders was found to decrease certain distribution factor quantities, as shown in F igure 7 1 However, other parameters were found to have little or no influence on distribution factors. For example, girder depth w as determined to be minimally influential, as shown in Figure 7 2 7 2 3 Selection of Culled Data As previously noted, two (2) separate girder end shear forces (and subsequent end shear force distribution factors) and one (1) maximum girder moment (and subsequent maximum girde r moment distribution factor) were quantified for every girder in every model analyzed in the parametric study. However, considering the intended purpose of the parametric study to provide the designing engineer with a method for quantifying maximum interi or and exterior girder end shear forces and moments in a construction stage structural system for both loading conditions some distribution factors quantified in the parametric study were non critical. In conducting the parametric study, multiple models we re analyzed with all parameters being identical except for the finishing machine location or deck placement location (as multiple models were used to investigate girder end shear forces and moments for different stages of the entire construction phase of t he bridge). Therefore, it was not of interest to use distribution PAGE 90 90 factors associated with non critical construction stages in the development of empirical distribution factor equations. For example, for cases where LG1 loads (including the finishing machine) were applied near one end of the bridge system, the girder end shear forces at the other (far) end of the bridge were relatively low, compared to girder end shear forces near the applied load. Therefore, including d istribution factors associated with low (non critical) girder end shear forces in the development of empirical distribution factor equations would not produce desired distribution factor prediction equations (or subsequent maximum girder end shear force pr edictions). As a result, only distribution factor quantities associated with maximum interior and exterior girder end shear forces and moments were used to develop empirical DF equations, thus culling cases from the complete large scale parametric study da data set. 7 2 4 Key Parameters Exhibiting Sensitivity Based on results from the sensitivity studies, it was determined that empirical DF equations, for both LG1 and LG2 would be developed considering the following parameters: Number of girders Span length (ft) Deck overhang width (ft) Girder spacing (ft) Ske w angle (deg.) Furthermore, from the perspective of static equilibrium, deck overhang width and girder spacing have approximately inverse influences on distribution factors. For this reason, these two parameters were combined into a single dimensionless ra tio (ft/ft), reducing the total number of independent parameters considered in the empirical fitting process to four (4). PAGE 91 91 Based on the reduction of parameters considered in the empirical fitting process (from 11 to 4), distribution factor data were corresp ondingly culled. Specifically, distribution factor data were reduced in volume by grouping together all models that had matching values of: number of girders, span length, deck overhang width, girder spacing, and skew angle. For each such group of analyzed models, the values DF V EXT DF V INT DF M EXT and DF M INT were quantified. These four distribution factors were defined as the distribution factors associated with the largest (i.e., most critical) interior or exterior girder end shear force or maximum mo ment for that group of analyzed models This grouping process further culled the distribution factor data such that there were 1,836 data values for each distribution factor and for each Load Group (3 girder number configurations 1 7 span lengths 3 deck overhang widths 3 girder spacing dimensions 4 skew angles = 1836). 7 3 Formation of Baseline Empirical Distribution Factor Equations From the culled distribution factor data, empirical fits for LG1 and LG2 were developed, using a root mean square (RMS) error minimization scheme, where the generalized functional form of all eight (8) distribution f actor ( DF ) equations was defined as: ( 7 11 ) where DF is the distribution factor, N is the number of girders, L is the span length in ft, OH is the deck overhang w idth in ft, S is the girder center to center spacing in ft, is the skew angle in deg., and a 1 through a 9 are fitting constants. Eqn. ( 7 11 ) is the most general form used to generate empirical DF fits. However, based on sensitivity study results, the optional skew angle term was later removed from a majority (but not all) of the DF prediction equations. For PAGE 92 92 conditions where DF was found to be insensitive to skew, the number of critical distribution factor data points used in the fitting process was further culled to 459. In order to compute values of the best fit parameters a 1 through a 9 in an optimal manner, an error function minimization process was used. The root mean square (RMS) error function was defined as the square root of the sum of the squares of the distribution factor prediction errors differences between reduced distribution factor data and empirical distribution factor predictions accumulated across the entire reduced data set and divided by the number of cases considered. Minimizing the RMS error function with respect to the fitting parameters a 1 through a 9 produced prelimin ary empirical distribution factor prediction equations. 7 4 Modifications to Achieve Desired Level of Prediction Error As a consequence of developing fits with an RMS error minimization sche me, the empirically predicted distribution factors were conservative in 50% of cases when compared to distribution factors computed from corresponding finite element analyses. Moreover, the empirically predicted DF values themselves are less important than the values of shear (V) and moment (M) that are produced by application of the DF equations in design. The distribution factors developed here were intended to be used in conjunction with static analyses (i.e., a simple one dimensional beam models) to pre dict interior and exterior girder end shear forces and moments. Therefore, an automation script was developed to conduct a static (beam) analysis for every case (geometric configuration) analyzed in the parametric study. Using the empirically fit distribut ion factor equations, interior and exterior girder end shear force predictions and maximum interior and exterior girder moment predictions were compared to the corresponding finite element results to determine the level of conservatism produced by applicat ion of the predicted DF quantities. A normalized prediction error was defined as: PAGE 93 93 ( 7 12 ) ( 7 13 ) where E N is the normalized error, DF is the empirically predicted distribution factor, [V|M] indicates shear (V) or moment (M), [V|M] STATIC is the maximum end shear force or moment quantified using a static (beam) analysi s, and [V|M] FEA is the girder end shear force or maximum girder moment quantified from finite element analysis. Using the empirically developed DF equation in conjunction with a static analysis, a final interior or exterior girder end shear force or maximu m moment prediction, [V|M] PREDICTION was quantified and compared with the end shear force or moment quantified using a finite element analysis. Per this definition, normalized error values greater than or equal to 1.0 indicate a conservative prediction of [V|M] and values less than 1.0 indicate an unconservative prediction of [V|M] When girder end shear forces and moments computed using empirically predicted DF values and simplified beam analysis were compared to shears and moments computed from three di mensional FEA, it was found that final girder [V|M] PREDICTION values exhibited conservatism in excess of the 50% conservatism that was built into the empirical DF equations. An illustration of this additional conservatism in the prediction of exterior gird er end shear forces with LG1 loads applied is shown in Figure 7 3 The level of excess conservati sm varied between each of the remaining seven (7) DF equations, when used in conjunction with a static beam equation to permit desired levels of exceedance ( i.e., a desired level of conservatism) to be achieved. The exceedance parameter was introduced by applying a shift to the normalized error in the following manner: PAGE 94 94 ( 7 14 ) ( 7 15 ) ( 7 16 ) where C is a constant used to shift the normalized error to a desired exceedance level, and is the defined exceedance factor. As shown above, was used to relate the desired shift of the normalized error to the final form of the DF equation. Additionally, the exceedance factor provided the ability to achieve specific levels of conservatism, without necessitating adjustments to any of the empirically fit constants, a 1 through a 9 in the DF equations. To provide future flexibility in the implementation of the equ ations developed from this study, values were quantified at four (4) different levels of exceedance for each distribution factor DF expression. Each value was quantified in an iterative manner to achieve a desired exceedance level of predicted girder e nd shear force or maximum girder moment. Exceedance levels of 50% (mean [V|M] prediction error of zero), 84%, 95%, and 98% were identified as being of future value to the FDOT. In the 95% case, [V|M] predictions computed using the empirically predicted DF values and simplified beam analysis would be conservative (relative to FEA prediction) in 95% of cases. The levels of 84% and 98% corresponded to exceedance thresholds at 1 standard d eviations above zero mean error, respectively. To illustrate the influence of establishing at different levels of exceedance, normalized errors for exterior girder end shear force predictions with LG1 loads applied are shown in Figure 7 4 Figure 7 7 PAGE 95 95 7 5 Final Distribution Factor Equations for Design The final (general) form of the empirically developed distribution factor equations was defined as: ( 7 17 ) where DF is the predicted distribution factor, N is the number of girders in the bridge cross section, L is the span leng th in ft, OH is the deck overhang width in ft, S is the girder center to center spacing in ft, is the skew angle in deg., a 1 through a 9 are empirical fit constants listed in Table 7 3 and is exceedance factor selected to achieve a desired level of exceedance (i.e., conservatism) listed in Table 7 4 (Detailed summaries and illustrations of normalized error for each value are provided in Appendix D for all cases: interior and exterior girder end shear forces and moments). 7 5 1 Application of Proposed Method To illustrate the application of the proposed method for quantifying interior and exterior girder end shear forces and moments, an example involving the calculation of exterior girder end shear, fo r construction load group LG1 is presented. After collapsing all construction loads (point, line, and pressure) into an equivalent one dimensional beam loading diagram (Figure 7 8 ), a simple static beam analysis is performed to compute the maximum girder end shear. The exterior girder end shear force for construction load group LG1 is then determined as: ( 7 18 ) where V EXT LG1 is the empirically predicted exterior girder end shear force, V STATIC LG1 is the maximum end shear force quantified from the static beam analysis, and DF V EXT LG1 is the LG1 exterior girder end shear force distribution factor computed using Eqn. ( 7 17 ), Table 7 3 and PAGE 96 96 Table 7 4 (As noted previously, comprehensive illustrations of the approach, for interior and exterior girder end shear forces and moments, are shown in Appendix D ). 7 5 2 Prediction Error for Full (Unculled) Parametric Data Set In Figure 7 9 the level of conser vatism incorporated into the empirical prediction of exterior girder end shear force is compared for two different data sets. In Figure 7 9 a, normalized errors (50% exceedance; Table 7 4 ; = in the formation of the empirical DF equation. The 459 cases included in this data set include only critical loading conditions and critical structural configurations (i.e., those producing ma ximum girder forces). In Figure 7 9 b, normalized errors (again computed using = 0.01) are plotted f or critical loading conditions, but for all structural configurations (36,288 cases rather than 459). Evident in Figure 7 9 b is the fact that the use of the proposed empirical prediction method produces a 70% exceedance (rather than 50%) when applied to all structural configurations. Thus, application of the proposed method to a much larger and more gene ralized set of structures (than was used in the fitting process) is found to be conservative, but not overly so. In Figure 7 10 an analogous example is presented but for a different exceedance level. In Figure 7 10 a, normalized errors are plotted only for critical loading conditions and critical structural configurations (459 cases; 95% exceedance; Table 7 4 ; =+0.31). In Figure 7 10 b, normalized errors (again computed using =+0.31) are plotted for critical loading conditions, but for all structural configurations (36,288 cases). In Figure 7 10 b, use of the proposed empirical prediction method produces a 96% exceedance (rather than 95%) when applied to all structural configurations considered in the study. T hus, once again, application of the proposed method to a larger and more generalized set of structures is found to be conservative, but not overly so. PAGE 97 97 7 5 3 Proposed Method Compared to Traditional T ributary Area M ethod for Load Group 2 In structural design for construction Load Group 2 loads, it is conventional practice to compute girder end shear forces and moments by assuming that the distribution of load shall be based on using a tributary area for each girder in the bridge system. Therefore, it was desirable to compare interior and exterior girder end shear force and maximum moment predictions from the tributary area design method to girder end shear for ce and maximum moment values computed using the proposed DF equation approach of the present study. In Figure 7 11 Figure 7 14 values computed using the proposed DF method [using Eqn. ( 7 17 ), a 95% exceedance level, and a static analysis] are compared to values computed using the conventional tributary area method. In each case, the shear values h ave been normalized by FEA results from the culled data set (459 cases). The culled data set was selected as opposed to the full data set for comparison, due to redundant predictions that arise when using the tributary area method for specific bridge param eters. For example, two cases with different skew angles (and all remaining parameters the same) will produce equal predictions for both cases, using the tributary area method. Similarly, parameters such as the number of braces and girder depth will not in fluence quantities computed using the tributary area method. As a result, only the culled data set was considered. As shown in Figure 7 11 Figure 7 14 predictions using the traditional tributary area method relative to the corresponding FEA results were less conservative compared to the proposed DF equation approach. T his observation can be attributed to the assumed simplification in the transfer (distribution) of load using the tributary area method. Finite element analysis, from which the proposed DF expressions were derived, more accurately model the distribution of load, as compared to the simplified tributary area method. PAGE 98 98 Table 7 1 Span length values used in the distribution factor parametric study Span length, L (ft) 40 50 60 60 70 80 80 90 110 120 130 140 160 170 110 130 140 150 160 180 190 130 150 160 170 180 200 210 Table 7 2 Other parameter values used in the distribution factor parametric study Skew angle Intermediate span brace points, n i Deck overhang width, (in.) Girder spacing, (ft) Girders, n g 0 0 25 6 3 15 1 48 9 5 30 2 72 12 9 45 3 Table 7 3 Constants for distribution factors ( DF ) calculation DF equation a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 DF V EXT LG1 0.26 0.60 0.41 0.03 0.03 1.76 0.27 0.10 0.02 DF V INT LG1 0.90 0.13 0.11 0.01 0.02 4.80 0.02 0.03 0.01 DF M EXT LG1 0.23 0.47 0.33 2.51 0.09 27.00 0.09 0 0 DF M INT LG1 0.06 1.94 1.22 0.53 0.17 8.63 0.03 0 0 DF V EXT LG2 0.01 0.78 0.93 0.91 0.06 0.81 0.36 0 0 DF V INT LG2 0.03 0.89 1.08 1.04 0.04 10.16 0.16 0 0 DF M EXT LG2 0.06 1.66 0.77 2.29 0.01 24.58 0.17 0 0 DF M INT LG2 0.01 0.72 1.09 18.19 0.01 14.01 0.16 0 0 Table 7 4 Distribution factor ( DF ) exceedance values constant 50% exceedance 84% exceedance 95% exceedance 98% exceedance DF V EXT LG1 0.01 0.12 0.31 0.53 DF V INT LG1 0.04 0.14 0.29 0.42 DF M EXT LG1 0.04 0.01 0.08 0.14 DF M INT LG1 0.03 0.06 0.12 0.20 DF V EXT LG2 0.09 0.03 0.12 0.15 DF V INT LG2 0.05 0.05 0.17 0.27 DF M EXT LG2 0.01 0.06 0.15 0.17 DF M INT LG2 0.01 0.06 0.11 0.15 PAGE 99 99 Figure 7 1 DF sensitivity to number of girders Figure 7 2 DF sensitivity to girder depth PAGE 100 100 Figure 7 3 Shear (V) prediction error for the culled data set using DF V EXT LG1 in conjunction with a static beam analysis, without introduction of (Note: an exceedance of 57% condition, i.e., 50% exceedance) Figure 7 4 Shear (V) prediction error for the culled data set using DF V EXT LG1 in conjunction with a static beam analysis, shifted with to a 50% exceedance level PAGE 101 101 Figure 7 5 Shear (V) prediction error for the culled data set using DF V EXT LG1 in conjunction with a static beam analysis, shifted with to an 84% exceedance level Figure 7 6 Shear (V) prediction error for the culled data set using DF V EXT LG1 in conjunction with a static beam analysis, shifted with to a 95% exceedance level PAGE 102 102 Figure 7 7 Shear (V) predi ction error for the culled data set using DF V EXT LG1 in conjunction with a static beam analysis, shifted with to a 98% exceedance level Figure 7 8 Computatio n of exterior girder end shear force for construction load group LG1 A B Figure 7 9 Prediction error for V EXT LG1 using Eqn. ( 7 17 ) and a 50% exceedance level A ) For the reduced data set B ) For the complete large scale parametric study PAGE 103 103 A B Figure 7 10 Prediction error for V EXT LG1 using Eqn. ( 7 17 ) and a 95% exceedance level. A ) For the reduced data set B ) For the complete large scale parametric study A B Figure 7 11 Prediction error for V EXT LG2 using A ) Eqn. ( 7 17 ) and a 95% exceedance level. B ) Traditional tributary area method A B Figure 7 12 Prediction error for V INT LG2 using. A ) Eqn. ( 7 17 ) and a 95% exceedance level B ) Traditional tributary area method PAGE 104 104 A B Figure 7 13 Prediction error for M EXT LG2 using A ) Eqn. ( 7 17 ) and a 95% exceedance level B ) Traditional tributary area method A B Figure 7 14 Prediction error for M INT LG2 using. A ) Eqn. ( 7 17 ) and a 95 % exceedance level B ) Traditional tributary area method PAGE 105 105 CHAPTER 8 SU MMARY, CONCLUSIONS, AND RECOMMENDATIONS 8 1 Summary and Conclusions In the first phase of this study, finite element models of Florida I Beams were developed and analyzed for the purpose of updating previously developed capacity equations for wind load and gravity load. Using data available in published engineering literature, an appr oximate representation of thermally induced girder sweep was developed for bridges that are located in Florida. Importantly, thermal sweep values were found to be on the same order of magnitude as those corresponding to allowable fabrication (i.e., constru ction) sweep tolerances. Finite element analyses of numerical models that included the effects of thermal sweep, revealed as expected that girder capacities were diminished by inclusion of thermal sweep. Consequently, updated capacity equations Eqn. ( 4 5 ) and Eqn. ( 5 3 ) that acc ount for thermal sweep were developed to replace design capacity equations previously developed in FDOT study BDK75 977 33 (Consolazio et al., 2013). Additionally, a limited scope parametric study indicated that the updated baseline gravity load capacity E qn. ( 5 3 ) can be used in conjunction with Eqn. (9.23) from Consolazio et al. (2013) to account for a number of additional system characteristics (brac e stiffness, number of braces, etc.). In the second phase of this study, finite element analyses of partially constructed bridge systems consisting of multiple Florida I Beam (FIBs) with construction loads were used to quantify distribution factors ( DF ) fo r interior and exterior girder end shear forces and maximum girder moments. A large scale parametric study was conducted, with consideration of different Florida I Beam cross sections, span lengths, girder spacing, deck overhang widths, skew angles, number of girders, number of braces, and bracing configurations (K brace and X brace), to quantify shear and moment distribution factor data. These data were quantified separately for PAGE 106 106 two different construction load groups and subsequently used to develop empiri cal distribution factor equations for use in bridge design. The proposed DF equations incorporate the use of an exceedance factor that can be selected to achieve various desired levels of conservatism. The proposed empirical DF equations, when used in conj unction with simple static beam analyses, provide an efficient and accurate means of conservatively computing girder end shear forces and maximum girder moments during the construction phase, without the need for detailed finite element analysis. 8 2 Recommendations Based on the analytical data generated during this study, the following recommendations are suggested: The newly developed unanchored girder wind capacity ( P max ,0 ) Eqn. ( 4 5 ), which incorporates the influence of estimated thermal sweep (reproduced below), should be used as a replacement for the co rresponding equation previously developed in FDOT study BDK75 977 33 [see Eqn. (8.2) in Consolazio et al., (2013) which has been reproduced herein as Eqn. ( 4 4 )]: (4 5) where P max ,0 is the wind capacity in psf, L is the span length in ft, and D is the FIB cross section depth in inches. The newly developed baseline buckling capacity ( C 0 ) equation for an unanchored two girder strut braced system in zero wind, Eqn. ( 5 3 ), which incorporates the influence of estimated thermal sweep (and is reproduced belo w), should be used as a replacement for the corresponding equation previously developed in FDOT study BDK75 977 33 [see Eqn. (9.2) in Consolazio et al., (2013) which has been reproduced herein as Eqn. ( 5 2 )]. Additionally, the newly developed C 0 Eqn. ( 5 3 ) can be used in conjunction with Eqn. (9.23 ) from FDOT study BDK75 977 33 (Consolazio et al. 2013), which has been reproduced herein as Eqn. ( 5 4 ): (5 3) where C 0 is in g and L is the span length in ft. PAGE 107 107 It is recommended that the distribution factors ( DF ) expressions developed in this study Eqn. ( 7 17 ), Table 7 3 and Table 7 4 (reproduced below) be used in the calculation of girder end shear forces and moments that are caused by the application of construction loads. It has been demonstrated that the use of these distribu tion factor expressions coupled with simple beam analyses produces more accurate shear and moment data than does the traditional tributary width analysis approach: ( 7 17) where DF is the predicted distribution factor, N is the number of girders in the bridge cross section, L is the span length in ft, OH is the deck overhang width in ft, S is the girder center to center spacing in ft, is the skew angle in deg., a 1 through a 9 are empirical fit constants listed in Table 7 3 and is exceedance factor selected to achieve a desired level of exceedance (i.e., conservatism) listed in Table 7 4 Table 7 2 Constants for distribution factors ( DF ) calculation DF equation a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 DF V EXT LG1 0.26 0.60 0.41 0.03 0.03 1.76 0.27 0.10 0.02 DF V INT LG1 0.90 0.13 0.11 0.01 0.02 4.80 0.02 0.03 0.01 DF M EXT LG1 0.23 0.47 0.33 2.51 0.09 27.00 0.09 0 0 DF M INT LG1 0.06 1.94 1.22 0.53 0.17 8.63 0.03 0 0 DF V EXT LG2 0.01 0.78 0.93 0.91 0.06 0.81 0.36 0 0 DF V INT LG2 0.03 0.89 1.08 1.04 0.04 10.16 0.16 0 0 DF M EXT LG2 0.06 1.66 0.77 2.29 0.01 24.58 0.17 0 0 DF M INT LG2 0.01 0.72 1.09 18.19 0.01 14.01 0.16 0 0 Table 7 3 Distribution factor ( DF ) exceedance values constant 50% exceedance 84% exceedance 95% exceedance 98% exceedance DF V EXT LG1 0.01 0.12 0.31 0.53 DF V INT LG1 0.04 0.14 0.29 0.42 DF M EXT LG1 0.04 0.01 0.08 0.14 DF M INT LG1 0.03 0.06 0.12 0.20 DF V EXT LG2 0.09 0.03 0.12 0.15 DF V INT LG2 0.05 0.05 0.17 0.27 DF M EXT LG2 0.01 0.06 0.15 0.17 DF M INT LG2 0.01 0.06 0.11 0.15 PAGE 108 108 APPENDIX A CROSS SECTIONAL PROPERTIES OF FLORIDA I BEAMS In this study, finite element models were analyzed to evaluate girder shear forces and moments in Florida I Beams (FIBs). In each model, the FIBs were modeled 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 corresponding numeric values that were calculated for each FIB cross sectional shape. Mathematical definitions of cross sectional properties that are required to use the warping beam element in ADINA are listed in Table A 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 A 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 therefore 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 was 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 typically refers to a particular solution ( in Table A 1 ) corresponding to a state of pure 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 109 109 warping beam properties. However, it is possible to calculate the coordinates of the shear center, x s and y s (Table A 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 were 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 a s 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 (non zero) cross sectional properties calculated for each FIB shape are summarized in Table A 2 PAGE 110 110 Table A 1 Definitions of cross sectional properties required for use of a warping beam element Property Integral form Units Description A in 2 Cross sectional area I xx in 4 Strong axis moment of inertia I yy in 4 Weak axis moment of inertia I xy in 4 Product of inertia x s in X coordinate of shear 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/ weak axis bending coupling term I yr in 5 Twist/strong axis bending coupling term I in 6 Twist/warping coupling term I rr in 6 Wagner constant Table A 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 ) 81 283 3.00 30,864 81 540 3.46 31,885 81 798 3.81 32,939 82 055 4.07 33,973 1059 82 314 4.27 35,041 1101 82 484 4.38 35,693 1,314,600,000 1143 1,087,800 82 657 4.46 36,421 104,350,000 10,504,000 1,781,400,000 1227 1,516,200 83,002 4.56 37,859 142,280,000 15,336,000 3,107,900,000 PAGE 111 111 Figure A 1 Coordinate system used in the calculation of cross sectional properties PAGE 112 112 APPENDIX B Presented in this appendix is a calculation worksheet that was prepared to determine thermal sweep for a Florida I Beam, at an arbitrary span length. Furthermore, thermal sweep quantities (and subsequent thermal sweep ratios) were computed using transverse temperature gradients proposed in the present study, based on a literature review conducted in the present study (Lee, 2010). PAGE 113 113 PAGE 114 114 PAGE 115 115 PAGE 116 116 PAGE 117 117 PAGE 118 118 PAGE 119 119 PAGE 120 120 PAGE 121 121 APPENDIX C EXAMPLE CALCULATIONS: TEMPORARY BRACING ASSESSMENT FOR AN FIB BRIDGE Presented in this appendix is an updated version of a calculation worksheet that was previously presented in the final report for FDOT study BDK75 977 33 (Consolazio et al., 2013). Example calculations demon strating the assessment of temporary bracing requirements for a typical FIB bridge are illustrated. The updated calculation worksheet shown in this appendix utilizes equations from Consolazio et al. (2013) together with Eqn. ( 4 5 ) and Eqn. ( 5 3 ) from the present study. PAGE 122 122 PAGE 123 123 PAGE 124 124 PAGE 125 125 PAGE 126 126 PAGE 127 127 APPENDIX D DETAILED ILLUSTRATIONS: QUANTIFYING EXTERIOR AND INTERIOR GIRDER END SHEAR FORCES AND MAXIMUM MOMENTS WITH CONSTRUCTION LOADS APPLIED Presented in this appendix are detailed illustrations of the proposed method for quantifying in terior and exterior girder end shear forces and maximum moments. In each case, a simply supported static beam analysis is presented in conjunction with the proposed construction load distribution factor ( DF ) equations. PAGE 128 128 Load Group 1 Exterior girder end shear force prediction where: V EXT LG1 = Load Group 1 exterior girder end shear force prediction V STATIC LG1 = Maximum Load Group 1 end shear force using a static analysis (as shown above) DF V EXT LG1 = Load Group 1 exterior girder end shear force distribution factor = Exceedance factor (selected from figures below) N = Number of girders L = Span length (ft) OH = Deck overhang width (ft) S = Girder center to center spacing (ft) = Skew angle (deg.) Note: Normalized error defined as where normalized error indicates a conservative prediction PAGE 129 129 Load Group 1 Interior girder end shear force prediction where: V INT LG1 = Load Group 1 interior girder end shear force prediction V STATIC LG1 = Maximum Load Group 1 end shear force using a static analysis (as shown above) DF V INT LG1 = Load Group 1 interior girder end shear force distribution factor = Exceedance factor (selected from figures below) N = Number of girders L = Span length (ft) OH = Deck overhang width (ft) S = Girder center to center spacing (ft) = Skew angle (deg.) Note: Normalized error defined as where normalized error indicates a conservative prediction PAGE 130 130 Load Group 1 Exterior girder maximum moment prediction where: M EXT LG1 = Load Group 1 exterior girder maximum bending moment prediction M STATIC LG1 = Maximum Load Group 1 moment using a static analysis (as shown above) DF M EXT LG1 = Load Group 1 exterior girder maximum moment distribution factor = Exceedance factor (selected from figures below) N = Number of girders L = Span length (ft) OH = Deck overhang width (ft) S = Girder center to center spacing (ft) Note: Normalized error defined as where normalized error indicates a conservative prediction PAGE 131 131 Load Group 1 Interior girder maximum moment prediction where: M INT LG1 = Load Group 1 interior girder maximum bending moment prediction M STATIC LG1 = Maximum Load Group 1 moment using a static analysis (as shown above) DF M INT LG1 = Load Group 1 interior girder maximum moment distribution factor = Exceedance factor (selected from figures below) N = Number of girders L = Span length (ft) OH = Deck overhang width (ft) S = Girder center to center spacing (ft) Note: Normalized error defined as where normalized error indicates a conservative prediction PAGE 132 132 Load Group 2 Exterior girder end shear force prediction where: V EXT LG2 = Load Group 2 exterior girder end shear force prediction V STATIC LG2 = Maximum Load Group 2 end shear force using a static analysis (as shown above) DF V EXT LG2 = Load Group 2 exterior girder end shear force distribution factor = Exceedance factor (selected from figures below) N = Number of girders L = Span length (ft) OH = Deck overhang width (ft) S = Girder center to center spacing (ft) Note: Normalized error defined as where normalized error indicates a conservative prediction PAGE 133 133 Load Group 2 Interior girder end shear force prediction where: V INT LG2 = Load Group 2 interior girder end shear force prediction V STATIC LG2 = Maximum Load Group 2 end shear force using a static analysis (as shown above) DF V INT LG2 = Load Group 2 interior girder end shear force distribution factor = Exceedance factor (selected from figures below) N = Number of girders L = Span length (ft) OH = Deck overhang width (ft) S = Girder center to center spacing (ft) Note: Normalized error defined as where normalized error indicates a conservative prediction PAGE 134 134 Load Group 2 Exterior girder maximum moment prediction where: M EXT LG2 = Load Group 2 exterior girder maximum bending moment prediction M STATIC LG2 = Maximum Load Group 2 moment using a static analysis (as shown above) DF M EXT LG2 = Load Group 2 exterior girder maximum moment distribution factor = Exceedance factor (selected from figures below) N = Number of girders L = Span length (ft) OH = Deck overhang width (ft) S = Girder center to center spacing (ft) Note: Normalized error defined as where normalized error indicates a conservative prediction PAGE 135 135 Load Group 2 Interior girder maximum moment prediction where: M INT LG2 = Load Group 2 interior girder maximum bending moment prediction M STATIC LG2 = Maximum Load Group 2 moment using a static analysis (as shown above) DF M INT LG2 = Load Group 2 interior girder maximum moment distribution factor = Exceedance factor (selected from figures below) N = Number of girders L = Span length (ft) OH = Deck overhang width (ft) S = Girder center to center spacing (ft) Note: Normalized error defined as where normalized error indicates a conservative prediction PAGE 136 136 APPENDIX E EXAMPLE CALCULATIONS: QUANTIFYING EXTERIOR AND INTERIOR GIRDER END SHEAR FORCES AND MAXIMUM MOMENTS WITH CONSTRUCTION LOADS APPLIED Presented in this appendix is an example calculation worksheet that demonstrates (by example) how the proposed distribution factor equations are used in conjunction with a simply supported static beam analysis to predict girder end shear forces and maximum girder moments. PAGE 137 137 PAGE 138 138 PAGE 139 139 PAGE 140 140 PAGE 141 141 PAGE 142 142 PAGE 143 143 PAGE 144 144 PAGE 145 145 PAGE 146 146 PAGE 147 147 PAGE 148 148 PAGE 149 149 PAGE 150 150 PAGE 151 151 PAGE 152 152 APPENDIX F DETAILED ILLUSTRATIONS: DISTRI BUTION FACTORS FOR CASES WITH INTERIOR BRACING Presented in this appendix is an illustration of why the number of braces which may influence the distribution of load between interior and exterior girders was omitted from the final DF equation [i.e., Eqn. ( 7 17 )] as a parameter. For Load Group 1, finishing machine and worker line loads are applied to the lateral extremities of the bridge by means of the overhang brackets. The Load Group 1 loads applied (indirectly) to the exterior girders are noticeably different from the uniform live load that is applied (indirectly) to the interior girders. Due to the eccentric nature of the finishing ma chine and worker line loads, the presence of additional interior bracing can influence the amount of load that is distributed from exterior to interior girders. That is, adding additional interior braces increases the transfer (distribution) of the lateral extremity loads to interior girders. The loading condition where the number of braces is typically most influential occurs when the finishing machine load is applied at the midspan. In this condition, the finishing machine is at the furthest possible dist ance from the end span bracing, thus reducing the influence of the end span braces and increasing the influence of interior braces. To illustrate the influence of interior bracing on distribution factors, the calculation of maximum exterior girder moments for Load Group 1 (see Figure F 1 ) is given focus in this appendix. In the distribution factor parametric study carried out in this project, the number of brace points ranged from two (2) (i.e., only end span bracing) to five (5) (i.e., quarter point bracing). combinations w ere considered: Timber X bracing Steel X bracing PAGE 153 153 Steel K bracing As a result, for a single bridge configuration (i.e., a system defined by single parameter values for span length, girder depth, skew angle, overhang width, girder spacing, and number of girders), twelve (12) possible bracing arrangements were considered in the distribution factor parametric study (see Figure F 2 ). To illustrate how load distribution within a bridge system varies as the number of braces is increased, the following baseline bridge configuration was selected: 180 ft span lengths 6 ft girder spacings 25 in. deck overhang widths 0 deg. skew angle Load Group 1 loads applied with the finishing located at the midspan With these parameter choices fixed, four (4) different bracing conditions were selected for depiction: With only end span bracing With timber end sp an and midspan X bracing With steel end span and midspan X bracing With steel end span and midspan K bracing These cases are shown in Figures F 3 F 6 and the corresponding bridge system responses for each case depict ed as the midspan deflection for the girder system cross section are shown in Figures F 7 F 10 Midspan deflections for the end span bracing case were found to be essentially the same for all three of the bracing m aterial configuration combinations, and as a result, the material (steel or timber) and bracing configuration (X or K brace) was not included in the description. Theoretically, loads applied to a bridge girder system will be more evenly distributed among exterior and interior girders as interior bracing points are introduced. For the end span PAGE 154 154 only bracing case, bracing stiffness provided to the girder system is not sufficient to distribute loads which are located at the midspan and the lateral extremities to interior girders. Therefore, loads at the extremities are carried solely by the exterior girders, and midspan deflections of the exterior girders are larger, relative to the interior girders (Figure F 7 ). With the addition of an interior midspan bracing point, exterior and interior girders are integrated together as a system. However, as shown in Figures F 8 F 10 significant load sharing only occurs when there is sufficient brace stiffness. That is, addition of a timber X brace the lowest stiffness brace type considered in the par ametric study at midspan provides a minimal change in load distribution and reduction of displacement for the exterior girders. In contrast, addition of a steel K brace the highest stiffness brace type considered in the parametric study at midspan effectiv ely redistributes loads at the extremities to interior girders, reducing the difference in displacement between exterior and interior girders (as shown in Figure F 10 ). In Figure F 11 responses are shown for all twel ve (12) of the bracing arrangements considered in this study. As noted in Chapter 7 a culled dataset was used i n development of the recommended distribution factor equations. However, it is beneficial to illustrate the level of conservatism present in the empirical prediction of exterior girder maximum moment over the full data set as well. In Figure F 12 normalized errors of exterior girder moment (computed using the proposed DF M EXT LG1 equation with: 95% ex ceedance; Table 7 4 ; =0.08) are shown for the full data set, that is all structural configurations; finishing machine at midspan; 36,288 cases. These data were then separated into groups corresponding to the three different bracing material configuration combinations considere d (i.e., steel K bracing, steel X bracing, and timber X bracing (12,096 cases each). These data were further separated into sub groups consisting of: i ) cases with interior bracing at typical span lengths, and ii ) cases with end span bracing only, or PAGE 155 155 with shorter than than less than 90 ft), the uniform live load in Load Group 1 acts over nearly the entire span length (Figure F 13 ) resulting in a response that differs from that which occurs at more typical span lengths (see Figure F 14 ). For the typical bridge configuration selected for illustration, and superimposed Load Group 1 (with finishing machine located at midspan), the distri bution of maximum moment among exterior and interior girders is shown in Figures F 15 F 17 Normalized prediction errors in exterior girder moment, computed using the proposed DF M EXT LG1 equation (95% exceedance; Table 7 4 ; =0.08), are further separated into three groups based on brace stiffness. As illustrated in Figures F 15 F 17 a moderate increase in prediction conservatism is observed for cases with interior bracing, but only for the steel bracing cases. Additionally, the degree of prediction conser vatism is smaller for steel X bracing than steel K bracing, due to the lower stiffness of steel X braces as compared to steel K braces. Overall, differences in the conservatism of moment prediction for timber X braced bridges versus steel K braced bridges was not sufficient (~10% 20%) to warrant introducing additional terms (number of braces, brace stiffness) into the recommended distribution factor equations. PAGE 156 156 Figure F 1 Load Group 1 loads with the finishing machine located at the midspan to produce maximum girder moments A B C Figure F 2 Brace configurations considered in the parametric study for 5 girder bridge systems with. A) Timber X bracing. B) Steel X bracing. C) Steel K bracing. PAGE 157 157 A B Figure F 3 Bridge cross section with only end span bracing. A) Isometric view. B) Cross section at the midspan. A B Figure F 4 Bridge cross section with interior midspan timber X bracing. A) Isometric view. B) Cross section at the midspan. A B Figure F 5 Bridge cross sectio n with interior midspan steel X bracing. A) Isometric view. B ) Cross section at the midspan A B Figure F 6 Bridge cross section with interior midspan steel K bracing A ) Isometric view B ) Cross section at the midspan PAGE 158 158 Figure F 7 Bridge cross section midspan deflection without inter ior bracing Figure F 8 Bridge cross section midspan deflection with interior timber X bracing Figure F 9 Bridge cross section midspan deflection with interior steel X bracing Figure F 10 Bridge cross section midspan deflection with interior steel K bracing PAGE 159 159 A A B C C D E E F Figure F 11 5 girder, FIB78, 180 ft span, 6 ft girder spacing, 25 in. deck overhang, 0 deg. skew bridge configuration A ) deformed shapes for timber X bracing B ) midspan displacement quantities for timber X bracing C ) deformed shapes for steel X bracing D ) midspan displacement quantities for steel X bracing E ) deformed s hapes for steel K bracing. F ) midspan displacement quantities for steel K bracing PAGE 160 160 Figure F 12 Moment (M EXT LG1 ) prediction error for all bridge configurations (36,288 cases) using DF M EXT LG1 in conjunction with a static beam analysis, shifted with to a 95% exceedance ( Note: bridge configurations include different brace materials and configurations i.e., steel K bracing, steel X bracing, and timber X bracing are separate bridge configurations) A B Figure F 13 than A ) Isometric view B ) Elevation view A B Figure F 14 Load Group 1 loads applied at the midspan for a typical span length A ) Isometric view B ) Elevation view PAGE 161 161 A B Figure F 15 Timber X bracing cases A ) Moment for each girder at the midspan f or the typical bridge configuration B ) Moment (M EXT LG1 ) prediciton for the timber X brace data set using DF M EXT LG1 in conjunction with a static beam analysis, shifted with to a 95% exceedance A B Figure F 16 Steel X bracing cases A ) Moment for each girder at the midspan for the typical bridge configuration. B ) Moment (M EXT LG1 ) prediciton for the steel X brace data set using DF M EXT LG1 in conjunction with a static beam analysis, shifted with to a 95% exceedance PAGE 162 162 A B Figure F 17 Steel K bracing cases A ) Moment for each girder at the midspan for the typical bridge configuration. B ) Moment (M EXT LG1 ) prediciton for the steel K brace data set using DF M EXT LG1 in conjunction with a static beam analysis, shifted with to a 95% exceedance PAGE 163 163 LIST OF REFERENCES AASHTO ( American Association of State Highway and Transportation Officials) (2008). Guide Design Specification for Bridge Temporary Works AASHTO, Washington, D.C AASHTO ( American Association of State Highway and Transportation Officials) (2010). LRFD Bridge Design Specifications: 5 th Edition AASHTO, Washington, D.C ADINA (2016). Theory and Modeling Guide, Volume 1: ADINA Solids & Structures ADINA R&D, Inc, Watertown, MA. AF&PA (American Forest & Paper Association) (2005). National Design Specification for Wood Construction and Supplement AF&PA, Washington, D.C. Clifton, S. and Bayrak, O. ( 2008). Bridge Deck Overhang Construction, Technical Report IAC: 88 5DD1A003 2, University of Texas, Austin, TX. Consolazio, G., Gurley, K., and Harper, Z. (2013). Bridge Girder Drag Coefficients and Wind Related Bracing Recommendations Structures Research Report No. 2013/87322, University of Florida, Gainesville, FL. Consolazio, G. and Edwards, S. (2014). Determination of Brace Forces Caused by Construction Loads and Wind Loads During Bridge Construction Structures Research Report No. 2014/101350 102056, University of Florida, Gainesville, FL. CTL Group (2007). Investigation of Red Mountain Freeway Bridge Girder Collapse Arizona Department of Transportation Bridge Group, CTL Group Project No. 262291, Skokie, IL. FDOT (Florida Department of Transportation) (2012a). Design Standard No. 20510: Composite Elastomeric Bearing Pads Prestressed Florida I Beams FDOT, Tallahassee, FL. FDOT (Florida Department of Transportation) (2012b). Instructions for Design Standard No. 20010: Prestressed Florida I Beams FDOT Tallahassee, FL. FDOT (Florida Department of Transportation) (2012c). Instructions for Design Standard No. 20510: Composite Elastomeric Bearing Pads Prestressed Florida I Beams FDOT, Tallahassee, FL. FDOT (Florida Department of Transportation) (2012d) Structures Manual Volume I: Structures Design Guidelines FDOT, Tallahassee, FL. FDOT (Florida Department of Transportation) (2014a). Design Standard No. 20005: Prestressed I Beam Temporary Bracing FDOT, Tallahassee, FL. FDOT (Florida Department of Transportation) (2014b). Instructions for Design Standard No. 20010: Prestressed Florida I Beams FDOT, Tallahassee, FL. PAGE 164 164 FDOT (Florida Department of Transportation) (2016). Structures Manual Volume I: Structures Design Guideline s FDOT, Tallahassee, FL. Hurff, J (2010). Stability of Precast Prestressed Concrete Bridge Girders Considering Imperfections and Thermal Effects Doctoral dissertation, School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlan ta. Lee, J (2010). Experimental and Analytical Investigation of the Thermal Behavior of Prestressed Concrete Bridge Girders Including Imperfections Doctoral dissertation, School of Civil and Environmental Engineering, Georgia Institute of Technology, Atl anta. Lee, J ASCE Journal of Bridge Engineering 17 (3), 547 556. PCI (2010). PCI Design Handbook: 7th Edi tion, Precast/Prestressed Concrete Institute, Chicago, IL. PCI (2011). PCI Bridge Design Manual, 3rd Edition, Precast/Prestressed Concrete Institute, Chicago, IL. PAGE 165 165 BIOGRAPHICAL SKETCH Jeff Honig was born in Gaithersburg, Maryland, in 1991. In June 2010, he started his career at the University of Florida, where he received the degree of Bachelor of Science in civil engineering in May 2015. He then enrolled in graduate school at the University of Florida where he received a Master of Engineering in civil engineering in August 2017, with an emphasis in civil structures. |